LMFDB - Embedded newform 75.3.d.c.74.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,3,Mod(74,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.74");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	75.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.04360198270$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			74.3
Root			$1.65831 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	75.74
Dual form			75.3.d.c.74.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.31662 q^{2} +(-1.65831 - 2.50000i) q^{3} +7.00000 q^{4} +(-5.50000 - 8.29156i) q^{6} +9.94987 q^{8} +(-3.50000 + 8.29156i) q^{9} +O(q^{10})$	\(q+3.31662 q^{2} +(-1.65831 - 2.50000i) q^{3} +7.00000 q^{4} +(-5.50000 - 8.29156i) q^{6} +9.94987 q^{8} +(-3.50000 + 8.29156i) q^{9} +16.5831i q^{11} +(-11.6082 - 17.5000i) q^{12} -10.0000i q^{13} +5.00000 q^{16} +3.31662 q^{17} +(-11.6082 + 27.5000i) q^{18} -7.00000 q^{19} +55.0000i q^{22} -19.8997 q^{23} +(-16.5000 - 24.8747i) q^{24} -33.1662i q^{26} +(26.5330 - 5.00000i) q^{27} -33.1662i q^{29} +42.0000 q^{31} -23.2164 q^{32} +(41.4578 - 27.5000i) q^{33} +11.0000 q^{34} +(-24.5000 + 58.0409i) q^{36} -40.0000i q^{37} -23.2164 q^{38} +(-25.0000 + 16.5831i) q^{39} -16.5831i q^{41} +50.0000i q^{43} +116.082i q^{44} -66.0000 q^{46} -46.4327 q^{47} +(-8.29156 - 12.5000i) q^{48} +49.0000 q^{49} +(-5.50000 - 8.29156i) q^{51} -70.0000i q^{52} +46.4327 q^{53} +(88.0000 - 16.5831i) q^{54} +(11.6082 + 17.5000i) q^{57} -110.000i q^{58} -66.3325i q^{59} -8.00000 q^{61} +139.298 q^{62} -97.0000 q^{64} +(137.500 - 91.2072i) q^{66} +45.0000i q^{67} +23.2164 q^{68} +(33.0000 + 49.7494i) q^{69} +33.1662i q^{71} +(-34.8246 + 82.5000i) q^{72} +35.0000i q^{73} -132.665i q^{74} -49.0000 q^{76} +(-82.9156 + 55.0000i) q^{78} -12.0000 q^{79} +(-56.5000 - 58.0409i) q^{81} -55.0000i q^{82} -69.6491 q^{83} +165.831i q^{86} +(-82.9156 + 55.0000i) q^{87} +165.000i q^{88} +149.248i q^{89} -139.298 q^{92} +(-69.6491 - 105.000i) q^{93} -154.000 q^{94} +(38.5000 + 58.0409i) q^{96} -70.0000i q^{97} +162.515 q^{98} +(-137.500 - 58.0409i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 28 q^{4} - 22 q^{6} - 14 q^{9}+O(q^{10}) $ 4 * q + 28 * q^4 - 22 * q^6 - 14 * q^9	$ 4 q + 28 q^{4} - 22 q^{6} - 14 q^{9} + 20 q^{16} - 28 q^{19} - 66 q^{24} + 168 q^{31} + 44 q^{34} - 98 q^{36} - 100 q^{39} - 264 q^{46} + 196 q^{49} - 22 q^{51} + 352 q^{54} - 32 q^{61} - 388 q^{64} + 550 q^{66} + 132 q^{69} - 196 q^{76} - 48 q^{79} - 226 q^{81} - 616 q^{94} + 154 q^{96} - 550 q^{99}+O(q^{100}) $ 4 * q + 28 * q^4 - 22 * q^6 - 14 * q^9 + 20 * q^16 - 28 * q^19 - 66 * q^24 + 168 * q^31 + 44 * q^34 - 98 * q^36 - 100 * q^39 - 264 * q^46 + 196 * q^49 - 22 * q^51 + 352 * q^54 - 32 * q^61 - 388 * q^64 + 550 * q^66 + 132 * q^69 - 196 * q^76 - 48 * q^79 - 226 * q^81 - 616 * q^94 + 154 * q^96 - 550 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	3.31662			1.65831			0.829156	−	0.559017i	$-0.188821\pi$
$2$	3.31662			1.65831			0.829156	+	0.559017i	$0.188821\pi$
$3$	−1.65831	−	2.50000i	−0.552771	−	0.833333i
$3$	−1.65831	−	2.50000i	−0.552771	−	0.833333i
$4$	7.00000			1.75000
$5$		0			0
$5$		0			0
$6$	−5.50000	−	8.29156i	−0.916667	−	1.38193i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	9.94987			1.24373
$9$	−3.50000	+	8.29156i	−0.388889	+	0.921285i
$10$		0			0
$11$			16.5831i			1.50756i	0.657129	+	0.753778i	$0.271771\pi$
$11$			16.5831i			1.50756i	−0.657129	+	0.753778i	$0.728229\pi$
$12$	−11.6082	−	17.5000i	−0.967349	−	1.45833i
$13$		−	10.0000i		−	0.769231i	−0.923077	−	0.384615i	$-0.874334\pi$
$13$		−	10.0000i		−	0.769231i	0.923077	−	0.384615i	$-0.125666\pi$
$14$		0			0
$15$		0			0
$16$	5.00000			0.312500
$17$	3.31662			0.195096			0.0975478	−	0.995231i	$-0.468900\pi$
$17$	3.31662			0.195096			0.0975478	+	0.995231i	$0.468900\pi$
$18$	−11.6082	+	27.5000i	−0.644899	+	1.52778i
$19$	−7.00000			−0.368421			−0.184211	−	0.982887i	$-0.558973\pi$
$19$	−7.00000			−0.368421			−0.184211	+	0.982887i	$0.558973\pi$
$20$		0			0
$21$		0			0
$22$			55.0000i			2.50000i
$23$	−19.8997			−0.865206			−0.432603	−	0.901584i	$-0.642405\pi$
$23$	−19.8997			−0.865206			−0.432603	+	0.901584i	$0.642405\pi$
$24$	−16.5000	−	24.8747i	−0.687500	−	1.03645i
$25$		0			0
$26$		−	33.1662i		−	1.27562i
$27$	26.5330	−	5.00000i	0.982704	−	0.185185i
$28$		0			0
$29$		−	33.1662i		−	1.14366i	−0.820371	−	0.571832i	$-0.806233\pi$
$29$		−	33.1662i		−	1.14366i	0.820371	−	0.571832i	$-0.193767\pi$
$30$		0			0
$31$	42.0000			1.35484			0.677419	−	0.735597i	$-0.263098\pi$
$31$	42.0000			1.35484			0.677419	+	0.735597i	$0.263098\pi$
$32$	−23.2164			−0.725512
$33$	41.4578	−	27.5000i	1.25630	−	0.833333i
$34$	11.0000			0.323529
$35$		0			0
$36$	−24.5000	+	58.0409i	−0.680556	+	1.61225i
$37$		−	40.0000i		−	1.08108i	−0.841318	−	0.540541i	$-0.818220\pi$
$37$		−	40.0000i		−	1.08108i	0.841318	−	0.540541i	$-0.181780\pi$
$38$	−23.2164			−0.610957
$39$	−25.0000	+	16.5831i	−0.641026	+	0.425208i
$40$		0			0
$41$		−	16.5831i		−	0.404466i	−0.979337	−	0.202233i	$-0.935180\pi$
$41$		−	16.5831i		−	0.404466i	0.979337	−	0.202233i	$-0.0648199\pi$
$42$		0			0
$43$			50.0000i			1.16279i	0.813621	+	0.581395i	$0.197493\pi$
$43$			50.0000i			1.16279i	−0.813621	+	0.581395i	$0.802507\pi$
$44$			116.082i			2.63822i
$45$		0			0
$46$	−66.0000			−1.43478
$47$	−46.4327			−0.987931			−0.493965	−	0.869482i	$-0.664453\pi$
$47$	−46.4327			−0.987931			−0.493965	+	0.869482i	$0.664453\pi$
$48$	−8.29156	−	12.5000i	−0.172741	−	0.260417i
$49$	49.0000			1.00000
$50$		0			0
$51$	−5.50000	−	8.29156i	−0.107843	−	0.162580i
$52$		−	70.0000i		−	1.34615i
$53$	46.4327			0.876090			0.438045	−	0.898953i	$-0.355671\pi$
$53$	46.4327			0.876090			0.438045	+	0.898953i	$0.355671\pi$
$54$	88.0000	−	16.5831i	1.62963	−	0.307095i
$55$		0			0
$56$		0			0
$57$	11.6082	+	17.5000i	0.203652	+	0.307018i
$58$		−	110.000i		−	1.89655i
$59$		−	66.3325i		−	1.12428i	−0.827042	−	0.562140i	$-0.809978\pi$
$59$		−	66.3325i		−	1.12428i	0.827042	−	0.562140i	$-0.190022\pi$
$60$		0			0
$61$	−8.00000			−0.131148			−0.0655738	−	0.997848i	$-0.520888\pi$
$61$	−8.00000			−0.131148			−0.0655738	+	0.997848i	$0.520888\pi$
$62$	139.298			2.24675
$63$		0			0
$64$	−97.0000			−1.51562
$65$		0			0
$66$	137.500	−	91.2072i	2.08333	−	1.38193i
$67$			45.0000i			0.671642i	0.941926	+	0.335821i	$0.109014\pi$
$67$			45.0000i			0.671642i	−0.941926	+	0.335821i	$0.890986\pi$
$68$	23.2164			0.341417
$69$	33.0000	+	49.7494i	0.478261	+	0.721005i
$70$		0			0
$71$			33.1662i			0.467130i	0.972341	+	0.233565i	$0.0750392\pi$
$71$			33.1662i			0.467130i	−0.972341	+	0.233565i	$0.924961\pi$
$72$	−34.8246	+	82.5000i	−0.483674	+	1.14583i
$73$			35.0000i			0.479452i	0.970841	+	0.239726i	$0.0770576\pi$
$73$			35.0000i			0.479452i	−0.970841	+	0.239726i	$0.922942\pi$
$74$		−	132.665i		−	1.79277i
$75$		0			0
$76$	−49.0000			−0.644737
$77$		0			0
$78$	−82.9156	+	55.0000i	−1.06302	+	0.705128i
$79$	−12.0000			−0.151899			−0.0759494	−	0.997112i	$-0.524199\pi$
$79$	−12.0000			−0.151899			−0.0759494	+	0.997112i	$0.524199\pi$
$80$		0			0
$81$	−56.5000	−	58.0409i	−0.697531	−	0.716555i
$82$		−	55.0000i		−	0.670732i
$83$	−69.6491			−0.839146			−0.419573	−	0.907722i	$-0.637820\pi$
$83$	−69.6491			−0.839146			−0.419573	+	0.907722i	$0.637820\pi$
$84$		0			0
$85$		0			0
$86$			165.831i			1.92827i
$87$	−82.9156	+	55.0000i	−0.953053	+	0.632184i
$88$			165.000i			1.87500i
$89$			149.248i			1.67695i	0.544944	+	0.838473i	$0.316551\pi$
$89$			149.248i			1.67695i	−0.544944	+	0.838473i	$0.683449\pi$
$90$		0			0
$91$		0			0
$92$	−139.298			−1.51411
$93$	−69.6491	−	105.000i	−0.748915	−	1.12903i
$94$	−154.000			−1.63830
$95$		0			0
$96$	38.5000	+	58.0409i	0.401042	+	0.604593i
$97$		−	70.0000i		−	0.721649i	−0.932634	−	0.360825i	$-0.882495\pi$
$97$		−	70.0000i		−	0.721649i	0.932634	−	0.360825i	$-0.117505\pi$
$98$	162.515			1.65831
$99$	−137.500	−	58.0409i	−1.38889	−	0.586272i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	−18.2414	−	27.5000i	−0.178838	−	0.269608i
$103$			70.0000i			0.679612i	0.940496	+	0.339806i	$0.110361\pi$
$103$			70.0000i			0.679612i	−0.940496	+	0.339806i	$0.889639\pi$
$104$		−	99.4987i		−	0.956719i
$105$		0			0
$106$	154.000			1.45283
$107$	69.6491			0.650926			0.325463	−	0.945555i	$-0.394480\pi$
$107$	69.6491			0.650926			0.325463	+	0.945555i	$0.394480\pi$
$108$	185.731	−	35.0000i	1.71973	−	0.324074i
$109$	88.0000			0.807339			0.403670	−	0.914905i	$-0.367734\pi$
$109$	88.0000			0.807339			0.403670	+	0.914905i	$0.367734\pi$
$110$		0			0
$111$	−100.000	+	66.3325i	−0.900901	+	0.597590i
$112$		0			0
$113$	−102.815			−0.909871			−0.454935	−	0.890525i	$-0.650338\pi$
$113$	−102.815			−0.909871			−0.454935	+	0.890525i	$0.650338\pi$
$114$	38.5000	+	58.0409i	0.337719	+	0.509131i
$115$		0			0
$116$		−	232.164i		−	2.00141i
$117$	82.9156	+	35.0000i	0.708681	+	0.299145i
$118$		−	220.000i		−	1.86441i
$119$		0			0
$120$		0			0
$121$	−154.000			−1.27273
$122$	−26.5330			−0.217484
$123$	−41.4578	+	27.5000i	−0.337055	+	0.223577i
$124$	294.000			2.37097
$125$		0			0
$126$		0			0
$127$			190.000i			1.49606i	0.663663	+	0.748031i	$0.269001\pi$
$127$			190.000i			1.49606i	−0.663663	+	0.748031i	$0.730999\pi$
$128$	−228.847			−1.78787
$129$	125.000	−	82.9156i	0.968992	−	0.642757i
$130$		0			0
$131$		−	198.997i		−	1.51906i	−0.650469	−	0.759532i	$-0.725428\pi$
$131$		−	198.997i		−	1.51906i	0.650469	−	0.759532i	$-0.274572\pi$
$132$	290.205	−	192.500i	2.19852	−	1.45833i
$133$		0			0
$134$			149.248i			1.11379i
$135$		0			0
$136$	33.0000			0.242647
$137$	69.6491			0.508388			0.254194	−	0.967153i	$-0.418190\pi$
$137$	69.6491			0.508388			0.254194	+	0.967153i	$0.418190\pi$
$138$	109.449	+	165.000i	0.793106	+	1.19565i
$139$	−77.0000			−0.553957			−0.276978	−	0.960876i	$-0.589333\pi$
$139$	−77.0000			−0.553957			−0.276978	+	0.960876i	$0.589333\pi$
$140$		0			0
$141$	77.0000	+	116.082i	0.546099	+	0.823276i
$142$			110.000i			0.774648i
$143$	165.831			1.15966
$144$	−17.5000	+	41.4578i	−0.121528	+	0.287901i
$145$		0			0
$146$			116.082i			0.795081i
$147$	−81.2573	−	122.500i	−0.552771	−	0.833333i
$148$		−	280.000i		−	1.89189i
$149$			165.831i			1.11296i	0.830861	+	0.556481i	$0.187849\pi$
$149$			165.831i			1.11296i	−0.830861	+	0.556481i	$0.812151\pi$
$150$		0			0
$151$	172.000			1.13907			0.569536	−	0.821966i	$-0.307123\pi$
$151$	172.000			1.13907			0.569536	+	0.821966i	$0.307123\pi$
$152$	−69.6491			−0.458218
$153$	−11.6082	+	27.5000i	−0.0758705	+	0.179739i
$154$		0			0
$155$		0			0
$156$	−175.000	+	116.082i	−1.12179	+	0.744115i
$157$		−	250.000i		−	1.59236i	−0.605062	−	0.796178i	$-0.706852\pi$
$157$		−	250.000i		−	1.59236i	0.605062	−	0.796178i	$-0.293148\pi$
$158$	−39.7995			−0.251896
$159$	−77.0000	−	116.082i	−0.484277	−	0.730075i
$160$		0			0
$161$		0			0
$162$	−187.389	−	192.500i	−1.15672	−	1.18827i
$163$		−	35.0000i		−	0.214724i	−0.994220	−	0.107362i	$-0.965760\pi$
$163$		−	35.0000i		−	0.214724i	0.994220	−	0.107362i	$-0.0342404\pi$
$164$		−	116.082i		−	0.707816i
$165$		0			0
$166$	−231.000			−1.39157
$167$	−179.098			−1.07244			−0.536221	−	0.844078i	$-0.680149\pi$
$167$	−179.098			−1.07244			−0.536221	+	0.844078i	$0.680149\pi$
$168$		0			0
$169$	69.0000			0.408284
$170$		0			0
$171$	24.5000	−	58.0409i	0.143275	−	0.339421i
$172$			350.000i			2.03488i
$173$	278.596			1.61038			0.805192	−	0.593014i	$-0.202062\pi$
$173$	278.596			1.61038			0.805192	+	0.593014i	$0.202062\pi$
$174$	−275.000	+	182.414i	−1.58046	+	1.04836i
$175$		0			0
$176$			82.9156i			0.471111i
$177$	−165.831	+	110.000i	−0.936900	+	0.621469i
$178$			495.000i			2.78090i
$179$		−	116.082i		−	0.648502i	−0.945971	−	0.324251i	$-0.894888\pi$
$179$		−	116.082i		−	0.648502i	0.945971	−	0.324251i	$-0.105112\pi$
$180$		0			0
$181$	182.000			1.00552			0.502762	−	0.864425i	$-0.332317\pi$
$181$	182.000			1.00552			0.502762	+	0.864425i	$0.332317\pi$
$182$		0			0
$183$	13.2665	+	20.0000i	0.0724945	+	0.109290i
$184$	−198.000			−1.07609
$185$		0			0
$186$	−231.000	−	348.246i	−1.24194	−	1.87229i
$187$			55.0000i			0.294118i
$188$	−325.029			−1.72888
$189$		0			0
$190$		0			0
$191$			232.164i			1.21552i	0.794122	+	0.607758i	$0.207931\pi$
$191$			232.164i			1.21552i	−0.794122	+	0.607758i	$0.792069\pi$
$192$	160.856	+	242.500i	0.837793	+	1.26302i
$193$			25.0000i			0.129534i	0.997900	+	0.0647668i	$0.0206304\pi$
$193$			25.0000i			0.129534i	−0.997900	+	0.0647668i	$0.979370\pi$
$194$		−	232.164i		−	1.19672i
$195$		0			0
$196$	343.000			1.75000
$197$	218.897			1.11115			0.555577	−	0.831465i	$-0.312498\pi$
$197$	218.897			1.11115			0.555577	+	0.831465i	$0.312498\pi$
$198$	−456.036	−	192.500i	−2.30321	−	0.972222i
$199$	68.0000			0.341709			0.170854	−	0.985296i	$-0.445347\pi$
$199$	68.0000			0.341709			0.170854	+	0.985296i	$0.445347\pi$
$200$		0			0
$201$	112.500	−	74.6241i	0.559701	−	0.371264i
$202$		0			0
$203$		0			0
$204$	−38.5000	−	58.0409i	−0.188725	−	0.284514i
$205$		0			0
$206$			232.164i			1.12701i
$207$	69.6491	−	165.000i	0.336469	−	0.797101i
$208$		−	50.0000i		−	0.240385i
$209$		−	116.082i		−	0.555416i
$210$		0			0
$211$	77.0000			0.364929			0.182464	−	0.983212i	$-0.441593\pi$
$211$	77.0000			0.364929			0.182464	+	0.983212i	$0.441593\pi$
$212$	325.029			1.53316
$213$	82.9156	−	55.0000i	0.389275	−	0.258216i
$214$	231.000			1.07944
$215$		0			0
$216$	264.000	−	49.7494i	1.22222	−	0.230321i
$217$		0			0
$218$	291.863			1.33882
$219$	87.5000	−	58.0409i	0.399543	−	0.265027i
$220$		0			0
$221$		−	33.1662i		−	0.150074i
$222$	−331.662	+	220.000i	−1.49398	+	0.990991i
$223$		−	140.000i		−	0.627803i	−0.949456	−	0.313901i	$-0.898364\pi$
$223$		−	140.000i		−	0.627803i	0.949456	−	0.313901i	$-0.101636\pi$
$224$		0			0
$225$		0			0
$226$	−341.000			−1.50885
$227$	185.731			0.818198			0.409099	−	0.912490i	$-0.365843\pi$
$227$	185.731			0.818198			0.409099	+	0.912490i	$0.365843\pi$
$228$	81.2573	+	122.500i	0.356392	+	0.537281i
$229$	−372.000			−1.62445			−0.812227	−	0.583341i	$-0.801745\pi$
$229$	−372.000			−1.62445			−0.812227	+	0.583341i	$0.801745\pi$
$230$		0			0
$231$		0			0
$232$		−	330.000i		−	1.42241i
$233$	−119.398			−0.512440			−0.256220	−	0.966619i	$-0.582477\pi$
$233$	−119.398			−0.512440			−0.256220	+	0.966619i	$0.582477\pi$
$234$	275.000	+	116.082i	1.17521	+	0.496076i
$235$		0			0
$236$		−	464.327i		−	1.96749i
$237$	19.8997	+	30.0000i	0.0839652	+	0.126582i
$238$		0			0
$239$			232.164i			0.971396i	0.874127	+	0.485698i	$0.161435\pi$
$239$			232.164i			0.971396i	−0.874127	+	0.485698i	$0.838565\pi$
$240$		0			0
$241$	−413.000			−1.71369			−0.856846	−	0.515572i	$-0.827580\pi$
$241$	−413.000			−1.71369			−0.856846	+	0.515572i	$0.827580\pi$
$242$	−510.760			−2.11058
$243$	−51.4077	+	237.500i	−0.211554	+	0.977366i
$244$	−56.0000			−0.229508
$245$		0			0
$246$	−137.500	+	91.2072i	−0.558943	+	0.370761i
$247$			70.0000i			0.283401i
$248$	417.895			1.68506
$249$	115.500	+	174.123i	0.463855	+	0.699288i
$250$		0			0
$251$			248.747i			0.991023i	0.868601	+	0.495512i	$0.165019\pi$
$251$			248.747i			0.991023i	−0.868601	+	0.495512i	$0.834981\pi$
$252$		0			0
$253$		−	330.000i		−	1.30435i
$254$			630.159i			2.48094i
$255$		0			0
$256$	−371.000			−1.44922
$257$	−278.596			−1.08403			−0.542017	−	0.840368i	$-0.682339\pi$
$257$	−278.596			−1.08403			−0.542017	+	0.840368i	$0.682339\pi$
$258$	414.578	−	275.000i	1.60689	−	1.06589i
$259$		0			0
$260$		0			0
$261$	275.000	+	116.082i	1.05364	+	0.444758i
$262$		−	660.000i		−	2.51908i
$263$	−285.230			−1.08452			−0.542262	−	0.840210i	$-0.682432\pi$
$263$	−285.230			−1.08452			−0.542262	+	0.840210i	$0.682432\pi$
$264$	412.500	−	273.622i	1.56250	−	1.03645i
$265$		0			0
$266$		0			0
$267$	373.120	−	247.500i	1.39745	−	0.926966i
$268$			315.000i			1.17537i
$269$		−	464.327i		−	1.72612i	−0.505098	−	0.863062i	$-0.668544\pi$
$269$		−	464.327i		−	1.72612i	0.505098	−	0.863062i	$-0.331456\pi$
$270$		0			0
$271$	22.0000			0.0811808			0.0405904	−	0.999176i	$-0.487076\pi$
$271$	22.0000			0.0811808			0.0405904	+	0.999176i	$0.487076\pi$
$272$	16.5831			0.0609674
$273$		0			0
$274$	231.000			0.843066
$275$		0			0
$276$	231.000	+	348.246i	0.836957	+	1.26176i
$277$		−	210.000i		−	0.758123i	−0.925372	−	0.379061i	$-0.876247\pi$
$277$		−	210.000i		−	0.758123i	0.925372	−	0.379061i	$-0.123753\pi$
$278$	−255.380			−0.918633
$279$	−147.000	+	348.246i	−0.526882	+	1.24819i
$280$		0			0
$281$		−	198.997i		−	0.708176i	−0.935212	−	0.354088i	$-0.884791\pi$
$281$		−	198.997i		−	0.708176i	0.935212	−	0.354088i	$-0.115209\pi$
$282$	255.380	+	385.000i	0.905603	+	1.36525i
$283$		−	345.000i		−	1.21908i	−0.792755	−	0.609541i	$-0.791354\pi$
$283$		−	345.000i		−	1.21908i	0.792755	−	0.609541i	$-0.208646\pi$
$284$			232.164i			0.817478i
$285$		0			0
$286$	550.000			1.92308
$287$		0			0
$288$	81.2573	−	192.500i	0.282143	−	0.668403i
$289$	−278.000			−0.961938
$290$		0			0
$291$	−175.000	+	116.082i	−0.601375	+	0.398907i
$292$			245.000i			0.839041i
$293$	−318.396			−1.08668			−0.543338	−	0.839514i	$-0.682840\pi$
$293$	−318.396			−1.08668			−0.543338	+	0.839514i	$0.682840\pi$
$294$	−269.500	−	406.287i	−0.916667	−	1.38193i
$295$		0			0
$296$		−	397.995i		−	1.34458i
$297$	82.9156	+	440.000i	0.279177	+	1.48148i
$298$			550.000i			1.84564i
$299$			198.997i			0.665543i
$300$		0			0
$301$		0			0
$302$	570.459			1.88894
$303$		0			0
$304$	−35.0000			−0.115132
$305$		0			0
$306$	−38.5000	+	91.2072i	−0.125817	+	0.298063i
$307$			325.000i			1.05863i	0.848425	+	0.529316i	$0.177551\pi$
$307$			325.000i			1.05863i	−0.848425	+	0.529316i	$0.822449\pi$
$308$		0			0
$309$	175.000	−	116.082i	0.566343	−	0.375669i
$310$		0			0
$311$		−	397.995i		−	1.27973i	−0.768489	−	0.639863i	$-0.778991\pi$
$311$		−	397.995i		−	1.27973i	0.768489	−	0.639863i	$-0.221009\pi$
$312$	−248.747	+	165.000i	−0.797266	+	0.528846i
$313$			490.000i			1.56550i	0.622339	+	0.782748i	$0.286182\pi$
$313$			490.000i			1.56550i	−0.622339	+	0.782748i	$0.713818\pi$
$314$		−	829.156i		−	2.64062i
$315$		0			0
$316$	−84.0000			−0.265823
$317$	−212.264			−0.669602			−0.334801	−	0.942289i	$-0.608669\pi$
$317$	−212.264			−0.669602			−0.334801	+	0.942289i	$0.608669\pi$
$318$	−255.380	−	385.000i	−0.803082	−	1.21069i
$319$	550.000			1.72414
$320$		0			0
$321$	−115.500	−	174.123i	−0.359813	−	0.542439i
$322$		0			0
$323$	−23.2164			−0.0718773
$324$	−395.500	−	406.287i	−1.22068	−	1.25397i
$325$		0			0
$326$		−	116.082i		−	0.356079i
$327$	−145.931	−	220.000i	−0.446274	−	0.672783i
$328$		−	165.000i		−	0.503049i
$329$		0			0
$330$		0			0
$331$	−243.000			−0.734139			−0.367069	−	0.930194i	$-0.619639\pi$
$331$	−243.000			−0.734139			−0.367069	+	0.930194i	$0.619639\pi$
$332$	−487.544			−1.46851
$333$	331.662	+	140.000i	0.995983	+	0.420420i
$334$	−594.000			−1.77844
$335$		0			0
$336$		0			0
$337$			385.000i			1.14243i	0.820799	+	0.571217i	$0.193528\pi$
$337$			385.000i			1.14243i	−0.820799	+	0.571217i	$0.806472\pi$
$338$	228.847			0.677062
$339$	170.500	+	257.038i	0.502950	+	0.758225i
$340$		0			0
$341$			696.491i			2.04250i
$342$	81.2573	−	192.500i	0.237594	−	0.562865i
$343$		0			0
$344$			497.494i			1.44620i
$345$		0			0
$346$	924.000			2.67052
$347$	−295.180			−0.850662			−0.425331	−	0.905038i	$-0.639842\pi$
$347$	−295.180			−0.850662			−0.425331	+	0.905038i	$0.639842\pi$
$348$	−580.409	+	385.000i	−1.66784	+	1.10632i
$349$	−532.000			−1.52436			−0.762178	−	0.647368i	$-0.775870\pi$
$349$	−532.000			−1.52436			−0.762178	+	0.647368i	$0.775870\pi$
$350$		0			0
$351$	−50.0000	−	265.330i	−0.142450	−	0.755926i
$352$		−	385.000i		−	1.09375i
$353$	278.596			0.789225			0.394613	−	0.918848i	$-0.370879\pi$
$353$	278.596			0.789225			0.394613	+	0.918848i	$0.370879\pi$
$354$	−550.000	+	364.829i	−1.55367	+	1.03059i
$355$		0			0
$356$			1044.74i			2.93465i
$357$		0			0
$358$		−	385.000i		−	1.07542i
$359$		−	397.995i		−	1.10862i	−0.832310	−	0.554311i	$-0.812982\pi$
$359$		−	397.995i		−	1.10862i	0.832310	−	0.554311i	$-0.187018\pi$
$360$		0			0
$361$	−312.000			−0.864266
$362$	603.626			1.66747
$363$	255.380	+	385.000i	0.703526	+	1.06061i
$364$		0			0
$365$		0			0
$366$	44.0000	+	66.3325i	0.120219	+	0.181236i
$367$		−	180.000i		−	0.490463i	−0.969465	−	0.245232i	$-0.921136\pi$
$367$		−	180.000i		−	0.490463i	0.969465	−	0.245232i	$-0.0788640\pi$
$368$	−99.4987			−0.270377
$369$	137.500	+	58.0409i	0.372629	+	0.157293i
$370$		0			0
$371$		0			0
$372$	−487.544	−	735.000i	−1.31060	−	1.97581i
$373$			110.000i			0.294906i	0.989069	+	0.147453i	$0.0471075\pi$
$373$			110.000i			0.294906i	−0.989069	+	0.147453i	$0.952892\pi$
$374$			182.414i			0.487739i
$375$		0			0
$376$	−462.000			−1.22872
$377$	−331.662			−0.879741
$378$		0			0
$379$	533.000			1.40633			0.703166	−	0.711025i	$-0.251769\pi$
$379$	533.000			1.40633			0.703166	+	0.711025i	$0.251769\pi$
$380$		0			0
$381$	475.000	−	315.079i	1.24672	−	0.826980i
$382$			770.000i			2.01571i
$383$	79.5990			0.207830			0.103915	−	0.994586i	$-0.466863\pi$
$383$	79.5990			0.207830			0.103915	+	0.994586i	$0.466863\pi$
$384$	379.500	+	572.118i	0.988281	+	1.48989i
$385$		0			0
$386$			82.9156i			0.214807i
$387$	−414.578	−	175.000i	−1.07126	−	0.452196i
$388$		−	490.000i		−	1.26289i
$389$		−	99.4987i		−	0.255781i	−0.991788	−	0.127890i	$-0.959179\pi$
$389$		−	99.4987i		−	0.255781i	0.991788	−	0.127890i	$-0.0408206\pi$
$390$		0			0
$391$	−66.0000			−0.168798
$392$	487.544			1.24373
$393$	−497.494	+	330.000i	−1.26589	+	0.839695i
$394$	726.000			1.84264
$395$		0			0
$396$	−962.500	−	406.287i	−2.43056	−	1.02598i
$397$		−	20.0000i		−	0.0503778i	−0.999683	−	0.0251889i	$-0.991981\pi$
$397$		−	20.0000i		−	0.0503778i	0.999683	−	0.0251889i	$-0.00801873\pi$
$398$	225.530			0.566660
$399$		0			0
$400$		0			0
$401$		−	746.241i		−	1.86095i	−0.366357	−	0.930475i	$-0.619395\pi$
$401$		−	746.241i		−	1.86095i	0.366357	−	0.930475i	$-0.380605\pi$
$402$	373.120	−	247.500i	0.928160	−	0.615672i
$403$		−	420.000i		−	1.04218i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	663.325			1.62979
$408$	−54.7243	−	82.5000i	−0.134128	−	0.202206i
$409$	−77.0000			−0.188264			−0.0941320	−	0.995560i	$-0.530008\pi$
$409$	−77.0000			−0.188264			−0.0941320	+	0.995560i	$0.530008\pi$
$410$		0			0
$411$	−115.500	−	174.123i	−0.281022	−	0.423656i
$412$			490.000i			1.18932i
$413$		0			0
$414$	231.000	−	547.243i	0.557971	−	1.32184i
$415$		0			0
$416$			232.164i			0.558086i
$417$	127.690	+	192.500i	0.306211	+	0.461631i
$418$		−	385.000i		−	0.921053i
$419$			116.082i			0.277045i	0.990359	+	0.138523i	$0.0442353\pi$
$419$			116.082i			0.277045i	−0.990359	+	0.138523i	$0.955765\pi$
$420$		0			0
$421$	412.000			0.978622			0.489311	−	0.872109i	$-0.337248\pi$
$421$	412.000			0.978622			0.489311	+	0.872109i	$0.337248\pi$
$422$	255.380			0.605166
$423$	162.515	−	385.000i	0.384195	−	0.910165i
$424$	462.000			1.08962
$425$		0			0
$426$	275.000	−	182.414i	0.645540	−	0.428203i
$427$		0			0
$428$	487.544			1.13912
$429$	−275.000	−	414.578i	−0.641026	−	0.966383i
$430$		0			0
$431$		−	198.997i		−	0.461711i	−0.972988	−	0.230856i	$-0.925848\pi$
$431$		−	198.997i		−	0.461711i	0.972988	−	0.230856i	$-0.0741525\pi$
$432$	132.665	−	25.0000i	0.307095	−	0.0578704i
$433$			455.000i			1.05081i	0.850853	+	0.525404i	$0.176086\pi$
$433$			455.000i			1.05081i	−0.850853	+	0.525404i	$0.823914\pi$
$434$		0			0
$435$		0			0
$436$	616.000			1.41284
$437$	139.298			0.318760
$438$	290.205	−	192.500i	0.662568	−	0.439498i
$439$	−22.0000			−0.0501139			−0.0250569	−	0.999686i	$-0.507977\pi$
$439$	−22.0000			−0.0501139			−0.0250569	+	0.999686i	$0.507977\pi$
$440$		0			0
$441$	−171.500	+	406.287i	−0.388889	+	0.921285i
$442$		−	110.000i		−	0.248869i
$443$	527.343			1.19039			0.595196	−	0.803581i	$-0.297075\pi$
$443$	527.343			1.19039			0.595196	+	0.803581i	$0.297075\pi$
$444$	−700.000	+	464.327i	−1.57658	+	1.04578i
$445$		0			0
$446$		−	464.327i		−	1.04109i
$447$	414.578	−	275.000i	0.927468	−	0.615213i
$448$		0			0
$449$			82.9156i			0.184667i	0.995728	+	0.0923337i	$0.0294326\pi$
$449$			82.9156i			0.184667i	−0.995728	+	0.0923337i	$0.970567\pi$
$450$		0			0
$451$	275.000			0.609756
$452$	−719.708			−1.59227
$453$	−285.230	−	430.000i	−0.629646	−	0.949227i
$454$	616.000			1.35683
$455$		0			0
$456$	115.500	+	174.123i	0.253289	+	0.381848i
$457$		−	275.000i		−	0.601751i	−0.953663	−	0.300875i	$-0.902721\pi$
$457$		−	275.000i		−	0.601751i	0.953663	−	0.300875i	$-0.0972788\pi$
$458$	−1233.78			−2.69385
$459$	88.0000	−	16.5831i	0.191721	−	0.0361288i
$460$		0			0
$461$			596.992i			1.29499i	0.762068	+	0.647497i	$0.224184\pi$
$461$			596.992i			1.29499i	−0.762068	+	0.647497i	$0.775816\pi$
$462$		0			0
$463$			240.000i			0.518359i	0.965829	+	0.259179i	$0.0834520\pi$
$463$			240.000i			0.518359i	−0.965829	+	0.259179i	$0.916548\pi$
$464$		−	165.831i		−	0.357395i
$465$		0			0
$466$	−396.000			−0.849785
$467$	−13.2665			−0.0284079			−0.0142040	−	0.999899i	$-0.504521\pi$
$467$	−13.2665			−0.0284079			−0.0142040	+	0.999899i	$0.504521\pi$
$468$	580.409	+	245.000i	1.24019	+	0.523504i
$469$		0			0
$470$		0			0
$471$	−625.000	+	414.578i	−1.32696	+	0.880208i
$472$		−	660.000i		−	1.39831i
$473$	−829.156			−1.75297
$474$	66.0000	+	99.4987i	0.139241	+	0.209913i
$475$		0			0
$476$		0			0
$477$	−162.515	+	385.000i	−0.340701	+	0.807128i
$478$			770.000i			1.61088i
$479$			298.496i			0.623165i	0.950219	+	0.311583i	$0.100859\pi$
$479$			298.496i			0.623165i	−0.950219	+	0.311583i	$0.899141\pi$
$480$		0			0
$481$	−400.000			−0.831601
$482$	−1369.77			−2.84184
$483$		0			0
$484$	−1078.00			−2.22727
$485$		0			0
$486$	−170.500	+	787.698i	−0.350823	+	1.62078i
$487$			410.000i			0.841889i	0.907086	+	0.420945i	$0.138301\pi$
$487$			410.000i			0.841889i	−0.907086	+	0.420945i	$0.861699\pi$
$488$	−79.5990			−0.163113
$489$	−87.5000	+	58.0409i	−0.178937	+	0.118693i
$490$		0			0
$491$		−	265.330i		−	0.540387i	−0.962806	−	0.270193i	$-0.912912\pi$
$491$		−	265.330i		−	0.540387i	0.962806	−	0.270193i	$-0.0870877\pi$
$492$	−290.205	+	192.500i	−0.589847	+	0.391260i
$493$		−	110.000i		−	0.223124i
$494$			232.164i			0.469967i
$495$		0			0
$496$	210.000			0.423387
$497$		0			0
$498$	383.070	+	577.500i	0.769217	+	1.15964i
$499$	−322.000			−0.645291			−0.322645	−	0.946520i	$-0.604572\pi$
$499$	−322.000			−0.645291			−0.322645	+	0.946520i	$0.604572\pi$
$500$		0			0
$501$	297.000	+	447.744i	0.592814	+	0.893701i
$502$			825.000i			1.64343i
$503$	411.261			0.817617			0.408809	−	0.912620i	$-0.365944\pi$
$503$	411.261			0.817617			0.408809	+	0.912620i	$0.365944\pi$
$504$		0			0
$505$		0			0
$506$		−	1094.49i		−	2.16302i
$507$	−114.424	−	172.500i	−0.225687	−	0.340237i
$508$			1330.00i			2.61811i
$509$			431.161i			0.847075i	0.905879	+	0.423538i	$0.139212\pi$
$509$			431.161i			0.847075i	−0.905879	+	0.423538i	$0.860788\pi$
$510$		0			0
$511$		0			0
$512$	−315.079			−0.615389
$513$	−185.731	+	35.0000i	−0.362049	+	0.0682261i
$514$	−924.000			−1.79767
$515$		0			0
$516$	875.000	−	580.409i	1.69574	−	1.12482i
$517$		−	770.000i		−	1.48936i
$518$		0			0
$519$	−462.000	−	696.491i	−0.890173	−	1.34199i
$520$		0			0
$521$			281.913i			0.541100i	0.962706	+	0.270550i	$0.0872055\pi$
$521$			281.913i			0.541100i	−0.962706	+	0.270550i	$0.912794\pi$
$522$	912.072	+	385.000i	1.74726	+	0.737548i
$523$		−	1015.00i		−	1.94073i	−0.241651	−	0.970363i	$-0.577689\pi$
$523$		−	1015.00i		−	1.94073i	0.241651	−	0.970363i	$-0.422311\pi$
$524$		−	1392.98i		−	2.65836i
$525$		0			0
$526$	−946.000			−1.79848
$527$	139.298			0.264323
$528$	207.289	−	137.500i	0.392593	−	0.260417i
$529$	−133.000			−0.251418
$530$		0			0
$531$	550.000	+	232.164i	1.03578	+	0.437220i
$532$		0			0
$533$	−165.831			−0.311128
$534$	1237.50	−	820.865i	2.31742	−	1.53720i
$535$		0			0
$536$			447.744i			0.835344i
$537$	−290.205	+	192.500i	−0.540418	+	0.358473i
$538$		−	1540.00i		−	2.86245i
$539$			812.573i			1.50756i
$540$		0			0
$541$	912.000			1.68577			0.842884	−	0.538096i	$-0.180856\pi$
$541$	912.000			1.68577			0.842884	+	0.538096i	$0.180856\pi$
$542$	72.9657			0.134623
$543$	−301.813	−	455.000i	−0.555825	−	0.837937i
$544$	−77.0000			−0.141544
$545$		0			0
$546$		0			0
$547$			55.0000i			0.100548i	0.998735	+	0.0502742i	$0.0160095\pi$
$547$			55.0000i			0.100548i	−0.998735	+	0.0502742i	$0.983990\pi$
$548$	487.544			0.889679
$549$	28.0000	−	66.3325i	0.0510018	−	0.120824i
$550$		0			0
$551$			232.164i			0.421350i
$552$	328.346	+	495.000i	0.594829	+	0.896739i
$553$		0			0
$554$		−	696.491i		−	1.25720i
$555$		0			0
$556$	−539.000			−0.969424
$557$	1014.89			1.82206			0.911030	−	0.412341i	$-0.135289\pi$
$557$	1014.89			1.82206			0.911030	+	0.412341i	$0.135289\pi$
$558$	−487.544	+	1155.00i	−0.873734	+	2.06989i
$559$	500.000			0.894454
$560$		0			0
$561$	137.500	−	91.2072i	0.245098	−	0.162580i
$562$		−	660.000i		−	1.17438i
$563$	−119.398			−0.212075			−0.106038	−	0.994362i	$-0.533816\pi$
$563$	−119.398			−0.212075			−0.106038	+	0.994362i	$0.533816\pi$
$564$	539.000	+	812.573i	0.955674	+	1.44073i
$565$		0			0
$566$		−	1144.24i		−	2.02162i
$567$		0			0
$568$			330.000i			0.580986i
$569$		−	49.7494i		−	0.0874330i	−0.999044	−	0.0437165i	$-0.986080\pi$
$569$		−	49.7494i		−	0.0874330i	0.999044	−	0.0437165i	$-0.0139198\pi$
$570$		0			0
$571$	242.000			0.423818			0.211909	−	0.977289i	$-0.432032\pi$
$571$	242.000			0.423818			0.211909	+	0.977289i	$0.432032\pi$
$572$	1160.82			2.02940
$573$	580.409	−	385.000i	1.01293	−	0.671902i
$574$		0			0
$575$		0			0
$576$	339.500	−	804.282i	0.589410	−	1.39632i
$577$			665.000i			1.15251i	0.817269	+	0.576256i	$0.195487\pi$
$577$			665.000i			1.15251i	−0.817269	+	0.576256i	$0.804513\pi$
$578$	−922.022			−1.59519
$579$	62.5000	−	41.4578i	0.107945	−	0.0716024i
$580$		0			0
$581$		0			0
$582$	−580.409	+	385.000i	−0.997267	+	0.661512i
$583$			770.000i			1.32075i
$584$			348.246i			0.596311i
$585$		0			0
$586$	−1056.00			−1.80205
$587$	−626.842			−1.06787			−0.533937	−	0.845524i	$-0.679288\pi$
$587$	−626.842			−1.06787			−0.533937	+	0.845524i	$0.679288\pi$
$588$	−568.801	−	857.500i	−0.967349	−	1.45833i
$589$	−294.000			−0.499151
$590$		0			0
$591$	−363.000	−	547.243i	−0.614213	−	0.925961i
$592$		−	200.000i		−	0.337838i
$593$	859.006			1.44858			0.724288	−	0.689497i	$-0.242169\pi$
$593$	859.006			1.44858			0.724288	+	0.689497i	$0.242169\pi$
$594$	275.000	+	1459.31i	0.462963	+	2.45676i
$595$		0			0
$596$			1160.82i			1.94768i
$597$	−112.765	−	170.000i	−0.188887	−	0.284757i
$598$			660.000i			1.10368i
$599$		−	331.662i		−	0.553694i	−0.960914	−	0.276847i	$-0.910711\pi$
$599$		−	331.662i		−	0.553694i	0.960914	−	0.276847i	$-0.0892895\pi$
$600$		0			0
$601$	−343.000			−0.570715			−0.285358	−	0.958421i	$-0.592112\pi$
$601$	−343.000			−0.570715			−0.285358	+	0.958421i	$0.592112\pi$
$602$		0			0
$603$	−373.120	−	157.500i	−0.618773	−	0.261194i
$604$	1204.00			1.99338
$605$		0			0
$606$		0			0
$607$		−	1100.00i		−	1.81219i	−0.423073	−	0.906096i	$-0.639049\pi$
$607$		−	1100.00i		−	1.81219i	0.423073	−	0.906096i	$-0.360951\pi$
$608$	162.515			0.267294
$609$		0			0
$610$		0			0
$611$			464.327i			0.759947i
$612$	−81.2573	+	192.500i	−0.132773	+	0.314542i
$613$			290.000i			0.473083i	0.971621	+	0.236542i	$0.0760140\pi$
$613$			290.000i			0.473083i	−0.971621	+	0.236542i	$0.923986\pi$
$614$			1077.90i			1.75554i
$615$		0			0
$616$		0			0
$617$	−79.5990			−0.129010			−0.0645049	−	0.997917i	$-0.520547\pi$
$617$	−79.5990			−0.129010			−0.0645049	+	0.997917i	$0.520547\pi$
$618$	580.409	−	385.000i	0.939174	−	0.622977i
$619$	58.0000			0.0936995			0.0468498	−	0.998902i	$-0.485082\pi$
$619$	58.0000			0.0936995			0.0468498	+	0.998902i	$0.485082\pi$
$620$		0			0
$621$	−528.000	+	99.4987i	−0.850242	+	0.160223i
$622$		−	1320.00i		−	2.12219i
$623$		0			0
$624$	−125.000	+	82.9156i	−0.200321	+	0.132878i
$625$		0			0
$626$			1625.15i			2.59608i
$627$	−290.205	+	192.500i	−0.462846	+	0.307018i
$628$		−	1750.00i		−	2.78662i
$629$		−	132.665i		−	0.210914i
$630$		0			0
$631$	862.000			1.36609			0.683043	−	0.730378i	$-0.260656\pi$
$631$	862.000			1.36609			0.683043	+	0.730378i	$0.260656\pi$
$632$	−119.398			−0.188922
$633$	−127.690	−	192.500i	−0.201722	−	0.304107i
$634$	−704.000			−1.11041
$635$		0			0
$636$	−539.000	−	812.573i	−0.847484	−	1.27763i
$637$		−	490.000i		−	0.769231i
$638$	1824.14			2.85916
$639$	−275.000	−	116.082i	−0.430360	−	0.181662i
$640$		0			0
$641$		−	596.992i		−	0.931345i	−0.884957	−	0.465673i	$-0.845812\pi$
$641$		−	596.992i		−	0.931345i	0.884957	−	0.465673i	$-0.154188\pi$
$642$	−383.070	−	577.500i	−0.596682	−	0.899533i
$643$			1050.00i			1.63297i	0.577366	+	0.816485i	$0.304080\pi$
$643$			1050.00i			1.63297i	−0.577366	+	0.816485i	$0.695920\pi$
$644$		0			0
$645$		0			0
$646$	−77.0000			−0.119195
$647$	252.063			0.389588			0.194794	−	0.980844i	$-0.437596\pi$
$647$	252.063			0.389588			0.194794	+	0.980844i	$0.437596\pi$
$648$	−562.168	−	577.500i	−0.867543	−	0.891204i
$649$	1100.00			1.69492
$650$		0			0
$651$		0			0
$652$		−	245.000i		−	0.375767i
$653$	1207.25			1.84878			0.924389	−	0.381452i	$-0.124576\pi$
$653$	1207.25			1.84878			0.924389	+	0.381452i	$0.124576\pi$
$654$	−484.000	−	729.657i	−0.740061	−	1.11568i
$655$		0			0
$656$		−	82.9156i		−	0.126396i
$657$	−290.205	−	122.500i	−0.441712	−	0.186454i
$658$		0			0
$659$		−	812.573i		−	1.23304i	−0.787339	−	0.616520i	$-0.788542\pi$
$659$		−	812.573i		−	1.23304i	0.787339	−	0.616520i	$-0.211458\pi$
$660$		0			0
$661$	−98.0000			−0.148260			−0.0741301	−	0.997249i	$-0.523618\pi$
$661$	−98.0000			−0.148260			−0.0741301	+	0.997249i	$0.523618\pi$
$662$	−805.940			−1.21743
$663$	−82.9156	+	55.0000i	−0.125061	+	0.0829563i
$664$	−693.000			−1.04367
$665$		0			0
$666$	1100.00	+	464.327i	1.65165	+	0.697188i
$667$			660.000i			0.989505i
$668$	−1253.68			−1.87677
$669$	−350.000	+	232.164i	−0.523169	+	0.347031i
$670$		0			0
$671$		−	132.665i		−	0.197712i
$672$		0			0
$673$			210.000i			0.312036i	0.987754	+	0.156018i	$0.0498657\pi$
$673$			210.000i			0.312036i	−0.987754	+	0.156018i	$0.950134\pi$
$674$			1276.90i			1.89451i
$675$		0			0
$676$	483.000			0.714497
$677$	−79.5990			−0.117576			−0.0587880	−	0.998270i	$-0.518724\pi$
$677$	−79.5990			−0.117576			−0.0587880	+	0.998270i	$0.518724\pi$
$678$	565.485	+	852.500i	0.834048	+	1.25737i
$679$		0			0
$680$		0			0
$681$	−308.000	−	464.327i	−0.452276	−	0.681832i
$682$			2310.00i			3.38710i
$683$	−169.148			−0.247654			−0.123827	−	0.992304i	$-0.539517\pi$
$683$	−169.148			−0.247654			−0.123827	+	0.992304i	$0.539517\pi$
$684$	171.500	−	406.287i	0.250731	−	0.593986i
$685$		0			0
$686$		0			0
$687$	616.892	+	930.000i	0.897951	+	1.35371i
$688$			250.000i			0.363372i
$689$		−	464.327i		−	0.673915i
$690$		0			0
$691$	−713.000			−1.03184			−0.515919	−	0.856637i	$-0.672549\pi$
$691$	−713.000			−1.03184			−0.515919	+	0.856637i	$0.672549\pi$
$692$	1950.18			2.81817
$693$		0			0
$694$	−979.000			−1.41066
$695$		0			0
$696$	−825.000	+	547.243i	−1.18534	+	0.786269i
$697$		−	55.0000i		−	0.0789096i
$698$	−1764.44			−2.52786
$699$	198.000	+	298.496i	0.283262	+	0.427033i
$700$		0			0
$701$		−	1160.82i		−	1.65595i	−0.560767	−	0.827973i	$-0.689494\pi$
$701$		−	1160.82i		−	1.65595i	0.560767	−	0.827973i	$-0.310506\pi$
$702$	−165.831	−	880.000i	−0.236227	−	1.25356i
$703$			280.000i			0.398293i
$704$		−	1608.56i		−	2.28489i
$705$		0			0
$706$	924.000			1.30878
$707$		0			0
$708$	−1160.82	+	770.000i	−1.63957	+	1.08757i
$709$	248.000			0.349788			0.174894	−	0.984587i	$-0.444042\pi$
$709$	248.000			0.349788			0.174894	+	0.984587i	$0.444042\pi$
$710$		0			0
$711$	42.0000	−	99.4987i	0.0590717	−	0.139942i
$712$			1485.00i			2.08567i
$713$	−835.789			−1.17222
$714$		0			0
$715$		0			0
$716$		−	812.573i		−	1.13488i
$717$	580.409	−	385.000i	0.809497	−	0.536960i
$718$		−	1320.00i		−	1.83844i
$719$			198.997i			0.276770i	0.990379	+	0.138385i	$0.0441911\pi$
$719$			198.997i			0.276770i	−0.990379	+	0.138385i	$0.955809\pi$
$720$		0			0
$721$		0			0
$722$	−1034.79			−1.43322
$723$	684.883	+	1032.50i	0.947279	+	1.42808i
$724$	1274.00			1.75967
$725$		0			0
$726$	847.000	+	1276.90i	1.16667	+	1.75882i
$727$		−	10.0000i		−	0.0137552i	−0.999976	−	0.00687758i	$-0.997811\pi$
$727$		−	10.0000i		−	0.0137552i	0.999976	−	0.00687758i	$-0.00218922\pi$
$728$		0			0
$729$	679.000	−	265.330i	0.931413	−	0.363964i
$730$		0			0
$731$			165.831i			0.226855i
$732$	92.8655	+	140.000i	0.126865	+	0.191257i
$733$		−	770.000i		−	1.05048i	−0.850955	−	0.525239i	$-0.823976\pi$
$733$		−	770.000i		−	1.05048i	0.850955	−	0.525239i	$-0.176024\pi$
$734$		−	596.992i		−	0.813341i
$735$		0			0
$736$	462.000			0.627717
$737$	−746.241			−1.01254
$738$	456.036	+	192.500i	0.617935	+	0.260840i
$739$	−802.000			−1.08525			−0.542625	−	0.839975i	$-0.682570\pi$
$739$	−802.000			−1.08525			−0.542625	+	0.839975i	$0.682570\pi$
$740$		0			0
$741$	175.000	−	116.082i	0.236167	−	0.156656i
$742$		0			0
$743$	−1346.55			−1.81231			−0.906157	−	0.422941i	$-0.860998\pi$
$743$	−1346.55			−1.81231			−0.906157	+	0.422941i	$0.860998\pi$
$744$	−693.000	−	1044.74i	−0.931452	−	1.40422i
$745$		0			0
$746$			364.829i			0.489047i
$747$	243.772	−	577.500i	0.326335	−	0.773092i
$748$			385.000i			0.514706i
$749$		0			0
$750$		0			0
$751$	322.000			0.428762			0.214381	−	0.976750i	$-0.431227\pi$
$751$	322.000			0.428762			0.214381	+	0.976750i	$0.431227\pi$
$752$	−232.164			−0.308728
$753$	621.867	−	412.500i	0.825853	−	0.547809i
$754$	−1100.00			−1.45889
$755$		0			0
$756$		0			0
$757$			400.000i			0.528402i	0.964468	+	0.264201i	$0.0851082\pi$
$757$			400.000i			0.528402i	−0.964468	+	0.264201i	$0.914892\pi$
$758$	1767.76			2.33214
$759$	−825.000	+	547.243i	−1.08696	+	0.721005i
$760$		0			0
$761$			348.246i			0.457616i	0.973472	+	0.228808i	$0.0734828\pi$
$761$			348.246i			0.457616i	−0.973472	+	0.228808i	$0.926517\pi$
$762$	1575.40	−	1045.00i	2.06745	−	1.37139i
$763$		0			0
$764$			1625.15i			2.12715i
$765$		0			0
$766$	264.000			0.344648
$767$	−663.325			−0.864830
$768$	615.234	+	927.500i	0.801086	+	1.20768i
$769$	193.000			0.250975			0.125488	−	0.992095i	$-0.459950\pi$
$769$	193.000			0.250975			0.125488	+	0.992095i	$0.459950\pi$
$770$		0			0
$771$	462.000	+	696.491i	0.599222	+	0.903361i
$772$			175.000i			0.226684i
$773$	−417.895			−0.540614			−0.270307	−	0.962774i	$-0.587125\pi$
$773$	−417.895			−0.540614			−0.270307	+	0.962774i	$0.587125\pi$
$774$	−1375.00	−	580.409i	−1.77649	−	0.749883i
$775$		0			0
$776$		−	696.491i		−	0.897540i
$777$		0			0
$778$		−	330.000i		−	0.424165i
$779$			116.082i			0.149014i
$780$		0			0
$781$	−550.000			−0.704225
$782$	−218.897			−0.279920
$783$	−165.831	−	880.000i	−0.211790	−	1.12388i
$784$	245.000			0.312500
$785$		0			0
$786$	−1650.00	+	1094.49i	−2.09924	+	1.39248i
$787$			910.000i			1.15629i	0.815934	+	0.578145i	$0.196223\pi$
$787$			910.000i			1.15629i	−0.815934	+	0.578145i	$0.803777\pi$
$788$	1532.28			1.94452
$789$	473.000	+	713.074i	0.599493	+	0.903770i
$790$		0			0
$791$		0			0
$792$	−1368.11	−	577.500i	−1.72741	−	0.729167i
$793$			80.0000i			0.100883i
$794$		−	66.3325i		−	0.0835422i
$795$		0			0
$796$	476.000			0.597990
$797$	−1107.75			−1.38990			−0.694951	−	0.719057i	$-0.744574\pi$
$797$	−1107.75			−1.38990			−0.694951	+	0.719057i	$0.744574\pi$
$798$		0			0
$799$	−154.000			−0.192741
$800$		0			0
$801$	−1237.50	−	522.368i	−1.54494	−	0.652145i
$802$		−	2475.00i		−	3.08603i
$803$	−580.409			−0.722801
$804$	787.500	−	522.368i	0.979478	−	0.649712i
$805$		0			0
$806$		−	1392.98i		−	1.72827i
$807$	−1160.82	+	770.000i	−1.43844	+	0.954151i
$808$		0			0
$809$			1260.32i			1.55787i	0.627104	+	0.778935i	$0.284240\pi$
$809$			1260.32i			1.55787i	−0.627104	+	0.778935i	$0.715760\pi$
$810$		0			0
$811$	−858.000			−1.05795			−0.528977	−	0.848636i	$-0.677424\pi$
$811$	−858.000			−1.05795			−0.528977	+	0.848636i	$0.677424\pi$
$812$		0			0
$813$	−36.4829	−	55.0000i	−0.0448744	−	0.0676507i
$814$	2200.00			2.70270
$815$		0			0
$816$	−27.5000	−	41.4578i	−0.0337010	−	0.0508061i
$817$		−	350.000i		−	0.428397i
$818$	−255.380			−0.312201
$819$		0			0
$820$		0			0
$821$			696.491i			0.848345i	0.905581	+	0.424172i	$0.139435\pi$
$821$			696.491i			0.848345i	−0.905581	+	0.424172i	$0.860565\pi$
$822$	−383.070	−	577.500i	−0.466022	−	0.702555i
$823$			1060.00i			1.28797i	0.765038	+	0.643985i	$0.222720\pi$
$823$			1060.00i			1.28797i	−0.765038	+	0.643985i	$0.777280\pi$
$824$			696.491i			0.845256i
$825$		0			0
$826$		0			0
$827$	500.810			0.605575			0.302787	−	0.953058i	$-0.402083\pi$
$827$	500.810			0.605575			0.302787	+	0.953058i	$0.402083\pi$
$828$	487.544	−	1155.00i	0.588821	−	1.39493i
$829$	1038.00			1.25211			0.626055	−	0.779779i	$-0.284668\pi$
$829$	1038.00			1.25211			0.626055	+	0.779779i	$0.284668\pi$
$830$		0			0
$831$	−525.000	+	348.246i	−0.631769	+	0.419068i
$832$			970.000i			1.16587i
$833$	162.515			0.195096
$834$	423.500	+	638.450i	0.507794	+	0.765528i
$835$		0			0
$836$		−	812.573i		−	0.971977i
$837$	1114.39	−	210.000i	1.33140	−	0.250896i
$838$			385.000i			0.459427i
$839$		−	928.655i		−	1.10686i	−0.832896	−	0.553430i	$-0.813319\pi$
$839$		−	928.655i		−	1.10686i	0.832896	−	0.553430i	$-0.186681\pi$
$840$		0			0
$841$	−259.000			−0.307967
$842$	1366.45			1.62286
$843$	−497.494	+	330.000i	−0.590147	+	0.391459i
$844$	539.000			0.638626
$845$		0			0
$846$	539.000	−	1276.90i	0.637116	−	1.50934i
$847$		0			0
$848$	232.164			0.273778
$849$	−862.500	+	572.118i	−1.01590	+	0.673873i
$850$		0			0
$851$			795.990i			0.935358i
$852$	580.409	−	385.000i	0.681232	−	0.451878i
$853$		−	630.000i		−	0.738570i	−0.929316	−	0.369285i	$-0.879603\pi$
$853$		−	630.000i		−	0.738570i	0.929316	−	0.369285i	$-0.120397\pi$
$854$		0			0
$855$		0			0
$856$	693.000			0.809579
$857$	1296.80			1.51319			0.756593	−	0.653886i	$-0.226862\pi$
$857$	1296.80			1.51319			0.756593	+	0.653886i	$0.226862\pi$
$858$	−912.072	−	1375.00i	−1.06302	−	1.60256i
$859$	−307.000			−0.357392			−0.178696	−	0.983904i	$-0.557188\pi$
$859$	−307.000			−0.357392			−0.178696	+	0.983904i	$0.557188\pi$
$860$		0			0
$861$		0			0
$862$		−	660.000i		−	0.765661i
$863$	−484.227			−0.561098			−0.280549	−	0.959840i	$-0.590516\pi$
$863$	−484.227			−0.561098			−0.280549	+	0.959840i	$0.590516\pi$
$864$	−616.000	+	116.082i	−0.712963	+	0.134354i
$865$		0			0
$866$			1509.06i			1.74257i
$867$	461.011	+	695.000i	0.531731	+	0.801615i
$868$		0			0
$869$		−	198.997i		−	0.228996i
$870$		0			0
$871$	450.000			0.516648
$872$	875.589			1.00412
$873$	580.409	+	245.000i	0.664845	+	0.280641i
$874$	462.000			0.528604
$875$		0			0
$876$	612.500	−	406.287i	0.699201	−	0.463797i
$877$			840.000i			0.957811i	0.877867	+	0.478905i	$0.158966\pi$
$877$			840.000i			0.957811i	−0.877867	+	0.478905i	$0.841034\pi$
$878$	−72.9657			−0.0831045
$879$	528.000	+	795.990i	0.600683	+	0.905563i
$880$		0			0
$881$			464.327i			0.527046i	0.964653	+	0.263523i	$0.0848845\pi$
$881$			464.327i			0.527046i	−0.964653	+	0.263523i	$0.915116\pi$
$882$	−568.801	+	1347.50i	−0.644899	+	1.52778i
$883$		−	995.000i		−	1.12684i	−0.826171	−	0.563420i	$-0.809485\pi$
$883$		−	995.000i		−	1.12684i	0.826171	−	0.563420i	$-0.190515\pi$
$884$		−	232.164i		−	0.262629i
$885$		0			0
$886$	1749.00			1.97404
$887$	−477.594			−0.538437			−0.269219	−	0.963079i	$-0.586765\pi$
$887$	−477.594			−0.538437			−0.269219	+	0.963079i	$0.586765\pi$
$888$	−994.987	+	660.000i	−1.12048	+	0.743243i
$889$		0			0
$890$		0			0
$891$	962.500	−	936.947i	1.08025	−	1.05157i
$892$		−	980.000i		−	1.09865i
$893$	325.029			0.363975
$894$	1375.00	−	912.072i	1.53803	−	1.02021i
$895$		0			0
$896$		0			0
$897$	497.494	−	330.000i	0.554620	−	0.367893i
$898$			275.000i			0.306236i
$899$		−	1392.98i		−	1.54948i
$900$		0			0
$901$	154.000			0.170921
$902$	912.072			1.01117
$903$		0			0
$904$	−1023.00			−1.13164
$905$		0			0
$906$	−946.000	−	1426.15i	−1.04415	−	1.57412i
$907$		−	1050.00i		−	1.15766i	−0.815447	−	0.578831i	$-0.803509\pi$
$907$		−	1050.00i		−	1.15766i	0.815447	−	0.578831i	$-0.196491\pi$
$908$	1300.12			1.43185
$909$		0			0
$910$		0			0
$911$		−	729.657i		−	0.800941i	−0.916310	−	0.400471i	$-0.868847\pi$
$911$		−	729.657i		−	0.800941i	0.916310	−	0.400471i	$-0.131153\pi$
$912$	58.0409	+	87.5000i	0.0636414	+	0.0959430i
$913$		−	1155.00i		−	1.26506i
$914$		−	912.072i		−	0.997890i
$915$		0			0
$916$	−2604.00			−2.84279
$917$		0			0
$918$	291.863	−	55.0000i	0.317934	−	0.0599129i
$919$	−1342.00			−1.46028			−0.730141	−	0.683296i	$-0.760546\pi$
$919$	−1342.00			−1.46028			−0.730141	+	0.683296i	$0.760546\pi$
$920$		0			0
$921$	812.500	−	538.952i	0.882193	−	0.585181i
$922$			1980.00i			2.14751i
$923$	331.662			0.359331
$924$		0			0
$925$		0			0
$926$			795.990i			0.859600i
$927$	−580.409	−	245.000i	−0.626116	−	0.264293i
$928$			770.000i			0.829741i
$929$			464.327i			0.499814i	0.968270	+	0.249907i	$0.0804001\pi$
$929$			464.327i			0.499814i	−0.968270	+	0.249907i	$0.919600\pi$
$930$		0			0
$931$	−343.000			−0.368421
$932$	−835.789			−0.896770
$933$	−994.987	+	660.000i	−1.06644	+	0.707395i
$934$	−44.0000			−0.0471092
$935$		0			0
$936$	825.000	+	348.246i	0.881410	+	0.372057i
$937$		−	165.000i		−	0.176094i	−0.996116	−	0.0880470i	$-0.971937\pi$
$937$		−	165.000i		−	0.176094i	0.996116	−	0.0880470i	$-0.0280626\pi$
$938$		0			0
$939$	1225.00	−	812.573i	1.30458	−	0.865360i
$940$		0			0
$941$			232.164i			0.246720i	0.992362	+	0.123360i	$0.0393670\pi$
$941$			232.164i			0.246720i	−0.992362	+	0.123360i	$0.960633\pi$
$942$	−2072.89	+	1375.00i	−2.20052	+	1.45966i
$943$			330.000i			0.349947i
$944$		−	331.662i		−	0.351337i
$945$		0			0
$946$	−2750.00			−2.90698
$947$	−278.596			−0.294188			−0.147094	−	0.989122i	$-0.546992\pi$
$947$	−278.596			−0.294188			−0.147094	+	0.989122i	$0.546992\pi$
$948$	139.298	+	210.000i	0.146939	+	0.221519i
$949$	350.000			0.368809
$950$		0			0
$951$	352.000	+	530.660i	0.370137	+	0.558002i
$952$		0			0
$953$	195.681			0.205331			0.102666	−	0.994716i	$-0.467263\pi$
$953$	195.681			0.205331			0.102666	+	0.994716i	$0.467263\pi$
$954$	−539.000	+	1276.90i	−0.564990	+	1.33847i
$955$		0			0
$956$			1625.15i			1.69994i
$957$	−912.072	−	1375.00i	−0.953053	−	1.43678i
$958$			990.000i			1.03340i
$959$		0			0
$960$		0			0
$961$	803.000			0.835588
$962$	−1326.65			−1.37905
$963$	−243.772	+	577.500i	−0.253138	+	0.599688i
$964$	−2891.00			−2.99896
$965$		0			0
$966$		0			0
$967$		−	280.000i		−	0.289555i	−0.989464	−	0.144778i	$-0.953753\pi$
$967$		−	280.000i		−	0.289555i	0.989464	−	0.144778i	$-0.0462467\pi$
$968$	−1532.28			−1.58293
$969$	38.5000	+	58.0409i	0.0397317	+	0.0598978i
$970$		0			0
$971$		−	1044.74i		−	1.07594i	−0.842964	−	0.537970i	$-0.819192\pi$
$971$		−	1044.74i		−	1.07594i	0.842964	−	0.537970i	$-0.180808\pi$
$972$	−359.854	+	1662.50i	−0.370220	+	1.71039i
$973$		0			0
$974$			1359.82i			1.39612i
$975$		0			0
$976$	−40.0000			−0.0409836
$977$	1097.80			1.12365			0.561823	−	0.827257i	$-0.310100\pi$
$977$	1097.80			1.12365			0.561823	+	0.827257i	$0.310100\pi$
$978$	−290.205	+	192.500i	−0.296733	+	0.196830i
$979$	−2475.00			−2.52809
$980$		0			0
$981$	−308.000	+	729.657i	−0.313965	+	0.743789i
$982$		−	880.000i		−	0.896130i
$983$	1671.58			1.70049			0.850244	−	0.526389i	$-0.176455\pi$
$983$	1671.58			1.70049			0.850244	+	0.526389i	$0.176455\pi$
$984$	−412.500	+	273.622i	−0.419207	+	0.278071i
$985$		0			0
$986$		−	364.829i		−	0.370009i
$987$		0			0
$988$			490.000i			0.495951i
$989$		−	994.987i		−	1.00605i
$990$		0			0
$991$	452.000			0.456105			0.228052	−	0.973649i	$-0.426764\pi$
$991$	452.000			0.456105			0.228052	+	0.973649i	$0.426764\pi$
$992$	−975.088			−0.982951
$993$	402.970	+	607.500i	0.405811	+	0.611782i
$994$		0			0
$995$		0			0
$996$	808.500	+	1218.86i	0.811747	+	1.22375i
$997$		−	420.000i		−	0.421264i	−0.977565	−	0.210632i	$-0.932448\pi$
$997$		−	420.000i		−	0.421264i	0.977565	−	0.210632i	$-0.0675521\pi$
$998$	−1067.95			−1.07009
$999$	−200.000	−	1061.32i	−0.200200	−	1.06238i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.3.d.c.74.3		4
3.2	odd	2	inner	75.3.d.c.74.1		4
4.3	odd	2		1200.3.c.d.449.4		4
5.2	odd	4		75.3.c.c.26.2	yes	2
5.3	odd	4		75.3.c.f.26.1	yes	2
5.4	even	2	inner	75.3.d.c.74.2		4
12.11	even	2		1200.3.c.d.449.2		4
15.2	even	4		75.3.c.c.26.1	✓	2
15.8	even	4		75.3.c.f.26.2	yes	2
15.14	odd	2	inner	75.3.d.c.74.4		4
20.3	even	4		1200.3.l.f.401.2		2
20.7	even	4		1200.3.l.s.401.1		2
20.19	odd	2		1200.3.c.d.449.1		4
60.23	odd	4		1200.3.l.f.401.1		2
60.47	odd	4		1200.3.l.s.401.2		2
60.59	even	2		1200.3.c.d.449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.3.c.c.26.1	✓	2	15.2	even	4
75.3.c.c.26.2	yes	2	5.2	odd	4
75.3.c.f.26.1	yes	2	5.3	odd	4
75.3.c.f.26.2	yes	2	15.8	even	4
75.3.d.c.74.1		4	3.2	odd	2	inner
75.3.d.c.74.2		4	5.4	even	2	inner
75.3.d.c.74.3		4	1.1	even	1	trivial
75.3.d.c.74.4		4	15.14	odd	2	inner
1200.3.c.d.449.1		4	20.19	odd	2
1200.3.c.d.449.2		4	12.11	even	2
1200.3.c.d.449.3		4	60.59	even	2
1200.3.c.d.449.4		4	4.3	odd	2
1200.3.l.f.401.1		2	60.23	odd	4
1200.3.l.f.401.2		2	20.3	even	4
1200.3.l.s.401.1		2	20.7	even	4
1200.3.l.s.401.2		2	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	75.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.31662 q^{2} +(-1.65831 - 2.50000i) q^{3} +7.00000 q^{4} +(-5.50000 - 8.29156i) q^{6} +9.94987 q^{8} +(-3.50000 + 8.29156i) q^{9} +O(q^{10})\)	\(q+3.31662 q^{2} +(-1.65831 - 2.50000i) q^{3} +7.00000 q^{4} +(-5.50000 - 8.29156i) q^{6} +9.94987 q^{8} +(-3.50000 + 8.29156i) q^{9} +16.5831i q^{11} +(-11.6082 - 17.5000i) q^{12} -10.0000i q^{13} +5.00000 q^{16} +3.31662 q^{17} +(-11.6082 + 27.5000i) q^{18} -7.00000 q^{19} +55.0000i q^{22} -19.8997 q^{23} +(-16.5000 - 24.8747i) q^{24} -33.1662i q^{26} +(26.5330 - 5.00000i) q^{27} -33.1662i q^{29} +42.0000 q^{31} -23.2164 q^{32} +(41.4578 - 27.5000i) q^{33} +11.0000 q^{34} +(-24.5000 + 58.0409i) q^{36} -40.0000i q^{37} -23.2164 q^{38} +(-25.0000 + 16.5831i) q^{39} -16.5831i q^{41} +50.0000i q^{43} +116.082i q^{44} -66.0000 q^{46} -46.4327 q^{47} +(-8.29156 - 12.5000i) q^{48} +49.0000 q^{49} +(-5.50000 - 8.29156i) q^{51} -70.0000i q^{52} +46.4327 q^{53} +(88.0000 - 16.5831i) q^{54} +(11.6082 + 17.5000i) q^{57} -110.000i q^{58} -66.3325i q^{59} -8.00000 q^{61} +139.298 q^{62} -97.0000 q^{64} +(137.500 - 91.2072i) q^{66} +45.0000i q^{67} +23.2164 q^{68} +(33.0000 + 49.7494i) q^{69} +33.1662i q^{71} +(-34.8246 + 82.5000i) q^{72} +35.0000i q^{73} -132.665i q^{74} -49.0000 q^{76} +(-82.9156 + 55.0000i) q^{78} -12.0000 q^{79} +(-56.5000 - 58.0409i) q^{81} -55.0000i q^{82} -69.6491 q^{83} +165.831i q^{86} +(-82.9156 + 55.0000i) q^{87} +165.000i q^{88} +149.248i q^{89} -139.298 q^{92} +(-69.6491 - 105.000i) q^{93} -154.000 q^{94} +(38.5000 + 58.0409i) q^{96} -70.0000i q^{97} +162.515 q^{98} +(-137.500 - 58.0409i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{4} - 22 q^{6} - 14 q^{9}+O(q^{10}) \) 4 * q + 28 * q^4 - 22 * q^6 - 14 * q^9	\( 4 q + 28 q^{4} - 22 q^{6} - 14 q^{9} + 20 q^{16} - 28 q^{19} - 66 q^{24} + 168 q^{31} + 44 q^{34} - 98 q^{36} - 100 q^{39} - 264 q^{46} + 196 q^{49} - 22 q^{51} + 352 q^{54} - 32 q^{61} - 388 q^{64} + 550 q^{66} + 132 q^{69} - 196 q^{76} - 48 q^{79} - 226 q^{81} - 616 q^{94} + 154 q^{96} - 550 q^{99}+O(q^{100}) \) 4 * q + 28 * q^4 - 22 * q^6 - 14 * q^9 + 20 * q^16 - 28 * q^19 - 66 * q^24 + 168 * q^31 + 44 * q^34 - 98 * q^36 - 100 * q^39 - 264 * q^46 + 196 * q^49 - 22 * q^51 + 352 * q^54 - 32 * q^61 - 388 * q^64 + 550 * q^66 + 132 * q^69 - 196 * q^76 - 48 * q^79 - 226 * q^81 - 616 * q^94 + 154 * q^96 - 550 * q^99

Introduction

L-functions

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