LMFDB - Embedded newform 1200.3.c.d.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(449,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1200.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.c (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:	$32.6976317232$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$-1.65831 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.d.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.65831 + 2.50000i) q^{3} +(-3.50000 + 8.29156i) q^{9} +O(q^{10})$	$q+(1.65831 + 2.50000i) q^{3} +(-3.50000 + 8.29156i) q^{9} -16.5831i q^{11} -10.0000i q^{13} +3.31662 q^{17} +7.00000 q^{19} +19.8997 q^{23} +(-26.5330 + 5.00000i) q^{27} -33.1662i q^{29} -42.0000 q^{31} +(41.4578 - 27.5000i) q^{33} -40.0000i q^{37} +(25.0000 - 16.5831i) q^{39} -16.5831i q^{41} -50.0000i q^{43} +46.4327 q^{47} +49.0000 q^{49} +(5.50000 + 8.29156i) q^{51} +46.4327 q^{53} +(11.6082 + 17.5000i) q^{57} +66.3325i q^{59} -8.00000 q^{61} -45.0000i q^{67} +(33.0000 + 49.7494i) q^{69} -33.1662i q^{71} +35.0000i q^{73} +12.0000 q^{79} +(-56.5000 - 58.0409i) q^{81} +69.6491 q^{83} +(82.9156 - 55.0000i) q^{87} +149.248i q^{89} +(-69.6491 - 105.000i) q^{93} -70.0000i q^{97} +(137.500 + 58.0409i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{9}+O(q^{10}) $ 4 * q - 14 * q^9	$ 4 q - 14 q^{9} + 28 q^{19} - 168 q^{31} + 100 q^{39} + 196 q^{49} + 22 q^{51} - 32 q^{61} + 132 q^{69} + 48 q^{79} - 226 q^{81} + 550 q^{99}+O(q^{100}) $ 4 * q - 14 * q^9 + 28 * q^19 - 168 * q^31 + 100 * q^39 + 196 * q^49 + 22 * q^51 - 32 * q^61 + 132 * q^69 + 48 * q^79 - 226 * q^81 + 550 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	1.65831	+	2.50000i	0.552771	+	0.833333i
$3$	1.65831	+	2.50000i	0.552771	+	0.833333i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−3.50000	+	8.29156i	−0.388889	+	0.921285i
$10$		0			0
$11$		−	16.5831i		−	1.50756i	−0.657129	−	0.753778i	$-0.728229\pi$
$11$		−	16.5831i		−	1.50756i	0.657129	−	0.753778i	$-0.271771\pi$
$12$		0			0
$13$		−	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
$16$		0			0
$17$	3.31662			0.195096			0.0975478	−	0.995231i	$-0.468900\pi$
$17$	3.31662			0.195096			0.0975478	+	0.995231i	$0.468900\pi$
$18$		0			0
$19$	7.00000			0.368421			0.184211	−	0.982887i	$-0.441027\pi$
$19$	7.00000			0.368421			0.184211	+	0.982887i	$0.441027\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	19.8997			0.865206			0.432603	−	0.901584i	$-0.357595\pi$
$23$	19.8997			0.865206			0.432603	+	0.901584i	$0.357595\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−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	$-0.736902\pi$
$31$	−42.0000			−1.35484			−0.677419	+	0.735597i	$0.736902\pi$
$32$		0			0
$33$	41.4578	−	27.5000i	1.25630	−	0.833333i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	40.0000i		−	1.08108i	−0.841318	−	0.540541i	$-0.818220\pi$
$37$		−	40.0000i		−	1.08108i	0.841318	−	0.540541i	$-0.181780\pi$
$38$		0			0
$39$	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	$-0.802507\pi$
$43$		−	50.0000i		−	1.16279i	0.813621	−	0.581395i	$-0.197493\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	46.4327			0.987931			0.493965	−	0.869482i	$-0.335547\pi$
$47$	46.4327			0.987931			0.493965	+	0.869482i	$0.335547\pi$
$48$		0			0
$49$	49.0000			1.00000
$50$		0			0
$51$	5.50000	+	8.29156i	0.107843	+	0.162580i
$52$		0			0
$53$	46.4327			0.876090			0.438045	−	0.898953i	$-0.355671\pi$
$53$	46.4327			0.876090			0.438045	+	0.898953i	$0.355671\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	11.6082	+	17.5000i	0.203652	+	0.307018i
$58$		0			0
$59$			66.3325i			1.12428i	0.827042	+	0.562140i	$0.190022\pi$
$59$			66.3325i			1.12428i	−0.827042	+	0.562140i	$0.809978\pi$
$60$		0			0
$61$	−8.00000			−0.131148			−0.0655738	−	0.997848i	$-0.520888\pi$
$61$	−8.00000			−0.131148			−0.0655738	+	0.997848i	$0.520888\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	45.0000i		−	0.671642i	−0.941926	−	0.335821i	$-0.890986\pi$
$67$		−	45.0000i		−	0.671642i	0.941926	−	0.335821i	$-0.109014\pi$
$68$		0			0
$69$	33.0000	+	49.7494i	0.478261	+	0.721005i
$70$		0			0
$71$		−	33.1662i		−	0.467130i	−0.972341	−	0.233565i	$-0.924961\pi$
$71$		−	33.1662i		−	0.467130i	0.972341	−	0.233565i	$-0.0750392\pi$
$72$		0			0
$73$			35.0000i			0.479452i	0.970841	+	0.239726i	$0.0770576\pi$
$73$			35.0000i			0.479452i	−0.970841	+	0.239726i	$0.922942\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	12.0000			0.151899			0.0759494	−	0.997112i	$-0.475801\pi$
$79$	12.0000			0.151899			0.0759494	+	0.997112i	$0.475801\pi$
$80$		0			0
$81$	−56.5000	−	58.0409i	−0.697531	−	0.716555i
$82$		0			0
$83$	69.6491			0.839146			0.419573	−	0.907722i	$-0.362180\pi$
$83$	69.6491			0.839146			0.419573	+	0.907722i	$0.362180\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	82.9156	−	55.0000i	0.953053	−	0.632184i
$88$		0			0
$89$			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
$92$		0			0
$93$	−69.6491	−	105.000i	−0.748915	−	1.12903i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	70.0000i		−	0.721649i	−0.932634	−	0.360825i	$-0.882495\pi$
$97$		−	70.0000i		−	0.721649i	0.932634	−	0.360825i	$-0.117505\pi$
$98$		0			0
$99$	137.500	+	58.0409i	1.38889	+	0.586272i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		−	70.0000i		−	0.679612i	−0.940496	−	0.339806i	$-0.889639\pi$
$103$		−	70.0000i		−	0.679612i	0.940496	−	0.339806i	$-0.110361\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−69.6491			−0.650926			−0.325463	−	0.945555i	$-0.605520\pi$
$107$	−69.6491			−0.650926			−0.325463	+	0.945555i	$0.605520\pi$
$108$		0			0
$109$	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	$0.650338\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	82.9156	+	35.0000i	0.708681	+	0.299145i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−154.000			−1.27273
$122$		0			0
$123$	41.4578	−	27.5000i	0.337055	−	0.223577i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	190.000i		−	1.49606i	−0.663663	−	0.748031i	$-0.730999\pi$
$127$		−	190.000i		−	1.49606i	0.663663	−	0.748031i	$-0.269001\pi$
$128$		0			0
$129$	125.000	−	82.9156i	0.968992	−	0.642757i
$130$		0			0
$131$			198.997i			1.51906i	0.650469	+	0.759532i	$0.274572\pi$
$131$			198.997i			1.51906i	−0.650469	+	0.759532i	$0.725428\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	69.6491			0.508388			0.254194	−	0.967153i	$-0.418190\pi$
$137$	69.6491			0.508388			0.254194	+	0.967153i	$0.418190\pi$
$138$		0			0
$139$	77.0000			0.553957			0.276978	−	0.960876i	$-0.410667\pi$
$139$	77.0000			0.553957			0.276978	+	0.960876i	$0.410667\pi$
$140$		0			0
$141$	77.0000	+	116.082i	0.546099	+	0.823276i
$142$		0			0
$143$	−165.831			−1.15966
$144$		0			0
$145$		0			0
$146$		0			0
$147$	81.2573	+	122.500i	0.552771	+	0.833333i
$148$		0			0
$149$			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	$-0.692877\pi$
$151$	−172.000			−1.13907			−0.569536	+	0.821966i	$0.692877\pi$
$152$		0			0
$153$	−11.6082	+	27.5000i	−0.0758705	+	0.179739i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	250.000i		−	1.59236i	−0.605062	−	0.796178i	$-0.706852\pi$
$157$		−	250.000i		−	1.59236i	0.605062	−	0.796178i	$-0.293148\pi$
$158$		0			0
$159$	77.0000	+	116.082i	0.484277	+	0.730075i
$160$		0			0
$161$		0			0
$162$		0			0
$163$			35.0000i			0.214724i	0.994220	+	0.107362i	$0.0342404\pi$
$163$			35.0000i			0.214724i	−0.994220	+	0.107362i	$0.965760\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	179.098			1.07244			0.536221	−	0.844078i	$-0.319851\pi$
$167$	179.098			1.07244			0.536221	+	0.844078i	$0.319851\pi$
$168$		0			0
$169$	69.0000			0.408284
$170$		0			0
$171$	−24.5000	+	58.0409i	−0.143275	+	0.339421i
$172$		0			0
$173$	278.596			1.61038			0.805192	−	0.593014i	$-0.202062\pi$
$173$	278.596			1.61038			0.805192	+	0.593014i	$0.202062\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−165.831	+	110.000i	−0.936900	+	0.621469i
$178$		0			0
$179$			116.082i			0.648502i	0.945971	+	0.324251i	$0.105112\pi$
$179$			116.082i			0.648502i	−0.945971	+	0.324251i	$0.894888\pi$
$180$		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
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	55.0000i		−	0.294118i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	232.164i		−	1.21552i	−0.794122	−	0.607758i	$-0.792069\pi$
$191$		−	232.164i		−	1.21552i	0.794122	−	0.607758i	$-0.207931\pi$
$192$		0			0
$193$			25.0000i			0.129534i	0.997900	+	0.0647668i	$0.0206304\pi$
$193$			25.0000i			0.129534i	−0.997900	+	0.0647668i	$0.979370\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	218.897			1.11115			0.555577	−	0.831465i	$-0.312498\pi$
$197$	218.897			1.11115			0.555577	+	0.831465i	$0.312498\pi$
$198$		0			0
$199$	−68.0000			−0.341709			−0.170854	−	0.985296i	$-0.554653\pi$
$199$	−68.0000			−0.341709			−0.170854	+	0.985296i	$0.554653\pi$
$200$		0			0
$201$	112.500	−	74.6241i	0.559701	−	0.371264i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−69.6491	+	165.000i	−0.336469	+	0.797101i
$208$		0			0
$209$		−	116.082i		−	0.555416i
$210$		0			0
$211$	−77.0000			−0.364929			−0.182464	−	0.983212i	$-0.558407\pi$
$211$	−77.0000			−0.364929			−0.182464	+	0.983212i	$0.558407\pi$
$212$		0			0
$213$	82.9156	−	55.0000i	0.389275	−	0.258216i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−87.5000	+	58.0409i	−0.399543	+	0.265027i
$220$		0			0
$221$		−	33.1662i		−	0.150074i
$222$		0			0
$223$			140.000i			0.627803i	0.949456	+	0.313901i	$0.101636\pi$
$223$			140.000i			0.627803i	−0.949456	+	0.313901i	$0.898364\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−185.731			−0.818198			−0.409099	−	0.912490i	$-0.634157\pi$
$227$	−185.731			−0.818198			−0.409099	+	0.912490i	$0.634157\pi$
$228$		0			0
$229$	−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
$232$		0			0
$233$	−119.398			−0.512440			−0.256220	−	0.966619i	$-0.582477\pi$
$233$	−119.398			−0.512440			−0.256220	+	0.966619i	$0.582477\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	19.8997	+	30.0000i	0.0839652	+	0.126582i
$238$		0			0
$239$		−	232.164i		−	0.971396i	−0.874127	−	0.485698i	$-0.838565\pi$
$239$		−	232.164i		−	0.971396i	0.874127	−	0.485698i	$-0.161435\pi$
$240$		0			0
$241$	−413.000			−1.71369			−0.856846	−	0.515572i	$-0.827580\pi$
$241$	−413.000			−1.71369			−0.856846	+	0.515572i	$0.827580\pi$
$242$		0			0
$243$	51.4077	−	237.500i	0.211554	−	0.977366i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	70.0000i		−	0.283401i
$248$		0			0
$249$	115.500	+	174.123i	0.463855	+	0.699288i
$250$		0			0
$251$		−	248.747i		−	0.991023i	−0.868601	−	0.495512i	$-0.834981\pi$
$251$		−	248.747i		−	0.991023i	0.868601	−	0.495512i	$-0.165019\pi$
$252$		0			0
$253$		−	330.000i		−	1.30435i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−278.596			−1.08403			−0.542017	−	0.840368i	$-0.682339\pi$
$257$	−278.596			−1.08403			−0.542017	+	0.840368i	$0.682339\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	275.000	+	116.082i	1.05364	+	0.444758i
$262$		0			0
$263$	285.230			1.08452			0.542262	−	0.840210i	$-0.317568\pi$
$263$	285.230			1.08452			0.542262	+	0.840210i	$0.317568\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−373.120	+	247.500i	−1.39745	+	0.926966i
$268$		0			0
$269$		−	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	$-0.512924\pi$
$271$	−22.0000			−0.0811808			−0.0405904	+	0.999176i	$0.512924\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	210.000i		−	0.758123i	−0.925372	−	0.379061i	$-0.876247\pi$
$277$		−	210.000i		−	0.758123i	0.925372	−	0.379061i	$-0.123753\pi$
$278$		0			0
$279$	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	$-0.115209\pi$
$282$		0			0
$283$			345.000i			1.21908i	0.792755	+	0.609541i	$0.208646\pi$
$283$			345.000i			1.21908i	−0.792755	+	0.609541i	$0.791354\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−278.000			−0.961938
$290$		0			0
$291$	175.000	−	116.082i	0.601375	−	0.398907i
$292$		0			0
$293$	−318.396			−1.08668			−0.543338	−	0.839514i	$-0.682840\pi$
$293$	−318.396			−1.08668			−0.543338	+	0.839514i	$0.682840\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	82.9156	+	440.000i	0.279177	+	1.48148i
$298$		0			0
$299$		−	198.997i		−	0.665543i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	325.000i		−	1.05863i	−0.848425	−	0.529316i	$-0.822449\pi$
$307$		−	325.000i		−	1.05863i	0.848425	−	0.529316i	$-0.177551\pi$
$308$		0			0
$309$	175.000	−	116.082i	0.566343	−	0.375669i
$310$		0			0
$311$			397.995i			1.27973i	0.768489	+	0.639863i	$0.221009\pi$
$311$			397.995i			1.27973i	−0.768489	+	0.639863i	$0.778991\pi$
$312$		0			0
$313$			490.000i			1.56550i	0.622339	+	0.782748i	$0.286182\pi$
$313$			490.000i			1.56550i	−0.622339	+	0.782748i	$0.713818\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−212.264			−0.669602			−0.334801	−	0.942289i	$-0.608669\pi$
$317$	−212.264			−0.669602			−0.334801	+	0.942289i	$0.608669\pi$
$318$		0			0
$319$	−550.000			−1.72414
$320$		0			0
$321$	−115.500	−	174.123i	−0.359813	−	0.542439i
$322$		0			0
$323$	23.2164			0.0718773
$324$		0			0
$325$		0			0
$326$		0			0
$327$	145.931	+	220.000i	0.446274	+	0.672783i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	243.000			0.734139			0.367069	−	0.930194i	$-0.380361\pi$
$331$	243.000			0.734139			0.367069	+	0.930194i	$0.380361\pi$
$332$		0			0
$333$	331.662	+	140.000i	0.995983	+	0.420420i
$334$		0			0
$335$		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	$0.806472\pi$
$338$		0			0
$339$	−170.500	−	257.038i	−0.502950	−	0.758225i
$340$		0			0
$341$			696.491i			2.04250i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	295.180			0.850662			0.425331	−	0.905038i	$-0.360158\pi$
$347$	295.180			0.850662			0.425331	+	0.905038i	$0.360158\pi$
$348$		0			0
$349$	−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
$352$		0			0
$353$	278.596			0.789225			0.394613	−	0.918848i	$-0.370879\pi$
$353$	278.596			0.789225			0.394613	+	0.918848i	$0.370879\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			397.995i			1.10862i	0.832310	+	0.554311i	$0.187018\pi$
$359$			397.995i			1.10862i	−0.832310	+	0.554311i	$0.812982\pi$
$360$		0			0
$361$	−312.000			−0.864266
$362$		0			0
$363$	−255.380	−	385.000i	−0.703526	−	1.06061i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			180.000i			0.490463i	0.969465	+	0.245232i	$0.0788640\pi$
$367$			180.000i			0.490463i	−0.969465	+	0.245232i	$0.921136\pi$
$368$		0			0
$369$	137.500	+	58.0409i	0.372629	+	0.157293i
$370$		0			0
$371$		0			0
$372$		0			0
$373$			110.000i			0.294906i	0.989069	+	0.147453i	$0.0471075\pi$
$373$			110.000i			0.294906i	−0.989069	+	0.147453i	$0.952892\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−331.662			−0.879741
$378$		0			0
$379$	−533.000			−1.40633			−0.703166	−	0.711025i	$-0.748231\pi$
$379$	−533.000			−1.40633			−0.703166	+	0.711025i	$0.748231\pi$
$380$		0			0
$381$	475.000	−	315.079i	1.24672	−	0.826980i
$382$		0			0
$383$	−79.5990			−0.207830			−0.103915	−	0.994586i	$-0.533137\pi$
$383$	−79.5990			−0.207830			−0.103915	+	0.994586i	$0.533137\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	414.578	+	175.000i	1.07126	+	0.452196i
$388$		0			0
$389$		−	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
$392$		0			0
$393$	−497.494	+	330.000i	−1.26589	+	0.839695i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	20.0000i		−	0.0503778i	−0.999683	−	0.0251889i	$-0.991981\pi$
$397$		−	20.0000i		−	0.0503778i	0.999683	−	0.0251889i	$-0.00801873\pi$
$398$		0			0
$399$		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	$-0.380605\pi$
$402$		0			0
$403$			420.000i			1.04218i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−663.325			−1.62979
$408$		0			0
$409$	−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
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	127.690	+	192.500i	0.306211	+	0.461631i
$418$		0			0
$419$		−	116.082i		−	0.277045i	−0.990359	−	0.138523i	$-0.955765\pi$
$419$		−	116.082i		−	0.277045i	0.990359	−	0.138523i	$-0.0442353\pi$
$420$		0			0
$421$	412.000			0.978622			0.489311	−	0.872109i	$-0.337248\pi$
$421$	412.000			0.978622			0.489311	+	0.872109i	$0.337248\pi$
$422$		0			0
$423$	−162.515	+	385.000i	−0.384195	+	0.910165i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−275.000	−	414.578i	−0.641026	−	0.966383i
$430$		0			0
$431$			198.997i			0.461711i	0.972988	+	0.230856i	$0.0741525\pi$
$431$			198.997i			0.461711i	−0.972988	+	0.230856i	$0.925848\pi$
$432$		0			0
$433$			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
$436$		0			0
$437$	139.298			0.318760
$438$		0			0
$439$	22.0000			0.0501139			0.0250569	−	0.999686i	$-0.492023\pi$
$439$	22.0000			0.0501139			0.0250569	+	0.999686i	$0.492023\pi$
$440$		0			0
$441$	−171.500	+	406.287i	−0.388889	+	0.921285i
$442$		0			0
$443$	−527.343			−1.19039			−0.595196	−	0.803581i	$-0.702925\pi$
$443$	−527.343			−1.19039			−0.595196	+	0.803581i	$0.702925\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−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
$452$		0			0
$453$	−285.230	−	430.000i	−0.629646	−	0.949227i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	275.000i		−	0.601751i	−0.953663	−	0.300875i	$-0.902721\pi$
$457$		−	275.000i		−	0.601751i	0.953663	−	0.300875i	$-0.0972788\pi$
$458$		0			0
$459$	−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	$-0.916548\pi$
$463$		−	240.000i		−	0.518359i	0.965829	−	0.259179i	$-0.0834520\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	13.2665			0.0284079			0.0142040	−	0.999899i	$-0.495479\pi$
$467$	13.2665			0.0284079			0.0142040	+	0.999899i	$0.495479\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	625.000	−	414.578i	1.32696	−	0.880208i
$472$		0			0
$473$	−829.156			−1.75297
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−162.515	+	385.000i	−0.340701	+	0.807128i
$478$		0			0
$479$		−	298.496i		−	0.623165i	−0.950219	−	0.311583i	$-0.899141\pi$
$479$		−	298.496i		−	0.623165i	0.950219	−	0.311583i	$-0.100859\pi$
$480$		0			0
$481$	−400.000			−0.831601
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	410.000i		−	0.841889i	−0.907086	−	0.420945i	$-0.861699\pi$
$487$		−	410.000i		−	0.841889i	0.907086	−	0.420945i	$-0.138301\pi$
$488$		0			0
$489$	−87.5000	+	58.0409i	−0.178937	+	0.118693i
$490$		0			0
$491$			265.330i			0.540387i	0.962806	+	0.270193i	$0.0870877\pi$
$491$			265.330i			0.540387i	−0.962806	+	0.270193i	$0.912912\pi$
$492$		0			0
$493$		−	110.000i		−	0.223124i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	322.000			0.645291			0.322645	−	0.946520i	$-0.395428\pi$
$499$	322.000			0.645291			0.322645	+	0.946520i	$0.395428\pi$
$500$		0			0
$501$	297.000	+	447.744i	0.592814	+	0.893701i
$502$		0			0
$503$	−411.261			−0.817617			−0.408809	−	0.912620i	$-0.634056\pi$
$503$	−411.261			−0.817617			−0.408809	+	0.912620i	$0.634056\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	114.424	+	172.500i	0.225687	+	0.340237i
$508$		0			0
$509$			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
$512$		0			0
$513$	−185.731	+	35.0000i	−0.362049	+	0.0682261i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	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	$0.912794\pi$
$522$		0			0
$523$			1015.00i			1.94073i	0.241651	+	0.970363i	$0.422311\pi$
$523$			1015.00i			1.94073i	−0.241651	+	0.970363i	$0.577689\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−139.298			−0.264323
$528$		0			0
$529$	−133.000			−0.251418
$530$		0			0
$531$	−550.000	−	232.164i	−1.03578	−	0.437220i
$532$		0			0
$533$	−165.831			−0.311128
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−290.205	+	192.500i	−0.540418	+	0.358473i
$538$		0			0
$539$		−	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	$0.180856\pi$
$542$		0			0
$543$	301.813	+	455.000i	0.555825	+	0.837937i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	55.0000i		−	0.100548i	−0.998735	−	0.0502742i	$-0.983990\pi$
$547$		−	55.0000i		−	0.100548i	0.998735	−	0.0502742i	$-0.0160095\pi$
$548$		0			0
$549$	28.0000	−	66.3325i	0.0510018	−	0.120824i
$550$		0			0
$551$		−	232.164i		−	0.421350i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1014.89			1.82206			0.911030	−	0.412341i	$-0.135289\pi$
$557$	1014.89			1.82206			0.911030	+	0.412341i	$0.135289\pi$
$558$		0			0
$559$	−500.000			−0.894454
$560$		0			0
$561$	137.500	−	91.2072i	0.245098	−	0.162580i
$562$		0			0
$563$	119.398			0.212075			0.106038	−	0.994362i	$-0.466184\pi$
$563$	119.398			0.212075			0.106038	+	0.994362i	$0.466184\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	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	$-0.567968\pi$
$571$	−242.000			−0.423818			−0.211909	+	0.977289i	$0.567968\pi$
$572$		0			0
$573$	580.409	−	385.000i	1.01293	−	0.671902i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			665.000i			1.15251i	0.817269	+	0.576256i	$0.195487\pi$
$577$			665.000i			1.15251i	−0.817269	+	0.576256i	$0.804513\pi$
$578$		0			0
$579$	−62.5000	+	41.4578i	−0.107945	+	0.0716024i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	770.000i		−	1.32075i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	626.842			1.06787			0.533937	−	0.845524i	$-0.320712\pi$
$587$	626.842			1.06787			0.533937	+	0.845524i	$0.320712\pi$
$588$		0			0
$589$	−294.000			−0.499151
$590$		0			0
$591$	363.000	+	547.243i	0.614213	+	0.925961i
$592$		0			0
$593$	859.006			1.44858			0.724288	−	0.689497i	$-0.242169\pi$
$593$	859.006			1.44858			0.724288	+	0.689497i	$0.242169\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−112.765	−	170.000i	−0.188887	−	0.284757i
$598$		0			0
$599$			331.662i			0.553694i	0.960914	+	0.276847i	$0.0892895\pi$
$599$			331.662i			0.553694i	−0.960914	+	0.276847i	$0.910711\pi$
$600$		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
$604$		0			0
$605$		0			0
$606$		0			0
$607$			1100.00i			1.81219i	0.423073	+	0.906096i	$0.360951\pi$
$607$			1100.00i			1.81219i	−0.423073	+	0.906096i	$0.639049\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	464.327i		−	0.759947i
$612$		0			0
$613$			290.000i			0.473083i	0.971621	+	0.236542i	$0.0760140\pi$
$613$			290.000i			0.473083i	−0.971621	+	0.236542i	$0.923986\pi$
$614$		0			0
$615$		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	$0.520547\pi$
$618$		0			0
$619$	−58.0000			−0.0936995			−0.0468498	−	0.998902i	$-0.514918\pi$
$619$	−58.0000			−0.0936995			−0.0468498	+	0.998902i	$0.514918\pi$
$620$		0			0
$621$	−528.000	+	99.4987i	−0.850242	+	0.160223i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$	290.205	−	192.500i	0.462846	−	0.307018i
$628$		0			0
$629$		−	132.665i		−	0.210914i
$630$		0			0
$631$	−862.000			−1.36609			−0.683043	−	0.730378i	$-0.739344\pi$
$631$	−862.000			−1.36609			−0.683043	+	0.730378i	$0.739344\pi$
$632$		0			0
$633$	−127.690	−	192.500i	−0.201722	−	0.304107i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	490.000i		−	0.769231i
$638$		0			0
$639$	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	$-0.154188\pi$
$642$		0			0
$643$		−	1050.00i		−	1.63297i	−0.577366	−	0.816485i	$-0.695920\pi$
$643$		−	1050.00i		−	1.63297i	0.577366	−	0.816485i	$-0.304080\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−252.063			−0.389588			−0.194794	−	0.980844i	$-0.562404\pi$
$647$	−252.063			−0.389588			−0.194794	+	0.980844i	$0.562404\pi$
$648$		0			0
$649$	1100.00			1.69492
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1207.25			1.84878			0.924389	−	0.381452i	$-0.124576\pi$
$653$	1207.25			1.84878			0.924389	+	0.381452i	$0.124576\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−290.205	−	122.500i	−0.441712	−	0.186454i
$658$		0			0
$659$			812.573i			1.23304i	0.787339	+	0.616520i	$0.211458\pi$
$659$			812.573i			1.23304i	−0.787339	+	0.616520i	$0.788542\pi$
$660$		0			0
$661$	−98.0000			−0.148260			−0.0741301	−	0.997249i	$-0.523618\pi$
$661$	−98.0000			−0.148260			−0.0741301	+	0.997249i	$0.523618\pi$
$662$		0			0
$663$	82.9156	−	55.0000i	0.125061	−	0.0829563i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	660.000i		−	0.989505i
$668$		0			0
$669$	−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	$0.950134\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−79.5990			−0.117576			−0.0587880	−	0.998270i	$-0.518724\pi$
$677$	−79.5990			−0.117576			−0.0587880	+	0.998270i	$0.518724\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−308.000	−	464.327i	−0.452276	−	0.681832i
$682$		0			0
$683$	169.148			0.247654			0.123827	−	0.992304i	$-0.460483\pi$
$683$	169.148			0.247654			0.123827	+	0.992304i	$0.460483\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−616.892	−	930.000i	−0.897951	−	1.35371i
$688$		0			0
$689$		−	464.327i		−	0.673915i
$690$		0			0
$691$	713.000			1.03184			0.515919	−	0.856637i	$-0.327451\pi$
$691$	713.000			1.03184			0.515919	+	0.856637i	$0.327451\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	55.0000i		−	0.0789096i
$698$		0			0
$699$	−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	$-0.310506\pi$
$702$		0			0
$703$		−	280.000i		−	0.398293i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	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
$712$		0			0
$713$	−835.789			−1.17222
$714$		0			0
$715$		0			0
$716$		0			0
$717$	580.409	−	385.000i	0.809497	−	0.536960i
$718$		0			0
$719$		−	198.997i		−	0.276770i	−0.990379	−	0.138385i	$-0.955809\pi$
$719$		−	198.997i		−	0.276770i	0.990379	−	0.138385i	$-0.0441911\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−684.883	−	1032.50i	−0.947279	−	1.42808i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			10.0000i			0.0137552i	0.999976	+	0.00687758i	$0.00218922\pi$
$727$			10.0000i			0.0137552i	−0.999976	+	0.00687758i	$0.997811\pi$
$728$		0			0
$729$	679.000	−	265.330i	0.931413	−	0.363964i
$730$		0			0
$731$		−	165.831i		−	0.226855i
$732$		0			0
$733$		−	770.000i		−	1.05048i	−0.850955	−	0.525239i	$-0.823976\pi$
$733$		−	770.000i		−	1.05048i	0.850955	−	0.525239i	$-0.176024\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−746.241			−1.01254
$738$		0			0
$739$	802.000			1.08525			0.542625	−	0.839975i	$-0.317430\pi$
$739$	802.000			1.08525			0.542625	+	0.839975i	$0.317430\pi$
$740$		0			0
$741$	175.000	−	116.082i	0.236167	−	0.156656i
$742$		0			0
$743$	1346.55			1.81231			0.906157	−	0.422941i	$-0.139002\pi$
$743$	1346.55			1.81231			0.906157	+	0.422941i	$0.139002\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−243.772	+	577.500i	−0.326335	+	0.773092i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−322.000			−0.428762			−0.214381	−	0.976750i	$-0.568773\pi$
$751$	−322.000			−0.428762			−0.214381	+	0.976750i	$0.568773\pi$
$752$		0			0
$753$	621.867	−	412.500i	0.825853	−	0.547809i
$754$		0			0
$755$		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	$0.914892\pi$
$758$		0			0
$759$	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	$0.926517\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	663.325			0.864830
$768$		0			0
$769$	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
$772$		0			0
$773$	−417.895			−0.540614			−0.270307	−	0.962774i	$-0.587125\pi$
$773$	−417.895			−0.540614			−0.270307	+	0.962774i	$0.587125\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	116.082i		−	0.149014i
$780$		0			0
$781$	−550.000			−0.704225
$782$		0			0
$783$	165.831	+	880.000i	0.211790	+	1.12388i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	910.000i		−	1.15629i	−0.815934	−	0.578145i	$-0.803777\pi$
$787$		−	910.000i		−	1.15629i	0.815934	−	0.578145i	$-0.196223\pi$
$788$		0			0
$789$	473.000	+	713.074i	0.599493	+	0.903770i
$790$		0			0
$791$		0			0
$792$		0			0
$793$			80.0000i			0.100883i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−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
$802$		0			0
$803$	580.409			0.722801
$804$		0			0
$805$		0			0
$806$		0			0
$807$	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	$-0.322576\pi$
$811$	858.000			1.05795			0.528977	+	0.848636i	$0.322576\pi$
$812$		0			0
$813$	−36.4829	−	55.0000i	−0.0448744	−	0.0676507i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	350.000i		−	0.428397i
$818$		0			0
$819$		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	$0.860565\pi$
$822$		0			0
$823$		−	1060.00i		−	1.28797i	−0.765038	−	0.643985i	$-0.777280\pi$
$823$		−	1060.00i		−	1.28797i	0.765038	−	0.643985i	$-0.222720\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−500.810			−0.605575			−0.302787	−	0.953058i	$-0.597917\pi$
$827$	−500.810			−0.605575			−0.302787	+	0.953058i	$0.597917\pi$
$828$		0			0
$829$	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
$832$		0			0
$833$	162.515			0.195096
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1114.39	−	210.000i	1.33140	−	0.250896i
$838$		0			0
$839$			928.655i			1.10686i	0.832896	+	0.553430i	$0.186681\pi$
$839$			928.655i			1.10686i	−0.832896	+	0.553430i	$0.813319\pi$
$840$		0			0
$841$	−259.000			−0.307967
$842$		0			0
$843$	497.494	−	330.000i	0.590147	−	0.391459i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−862.500	+	572.118i	−1.01590	+	0.673873i
$850$		0			0
$851$		−	795.990i		−	0.935358i
$852$		0			0
$853$		−	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
$856$		0			0
$857$	1296.80			1.51319			0.756593	−	0.653886i	$-0.226862\pi$
$857$	1296.80			1.51319			0.756593	+	0.653886i	$0.226862\pi$
$858$		0			0
$859$	307.000			0.357392			0.178696	−	0.983904i	$-0.442812\pi$
$859$	307.000			0.357392			0.178696	+	0.983904i	$0.442812\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	484.227			0.561098			0.280549	−	0.959840i	$-0.409484\pi$
$863$	484.227			0.561098			0.280549	+	0.959840i	$0.409484\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−461.011	−	695.000i	−0.531731	−	0.801615i
$868$		0			0
$869$		−	198.997i		−	0.228996i
$870$		0			0
$871$	−450.000			−0.516648
$872$		0			0
$873$	580.409	+	245.000i	0.664845	+	0.280641i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			840.000i			0.957811i	0.877867	+	0.478905i	$0.158966\pi$
$877$			840.000i			0.957811i	−0.877867	+	0.478905i	$0.841034\pi$
$878$		0			0
$879$	−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	$0.915116\pi$
$882$		0			0
$883$			995.000i			1.12684i	0.826171	+	0.563420i	$0.190515\pi$
$883$			995.000i			1.12684i	−0.826171	+	0.563420i	$0.809485\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	477.594			0.538437			0.269219	−	0.963079i	$-0.413235\pi$
$887$	477.594			0.538437			0.269219	+	0.963079i	$0.413235\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−962.500	+	936.947i	−1.08025	+	1.05157i
$892$		0			0
$893$	325.029			0.363975
$894$		0			0
$895$		0			0
$896$		0			0
$897$	497.494	−	330.000i	0.554620	−	0.367893i
$898$		0			0
$899$			1392.98i			1.54948i
$900$		0			0
$901$	154.000			0.170921
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$			1050.00i			1.15766i	0.815447	+	0.578831i	$0.196491\pi$
$907$			1050.00i			1.15766i	−0.815447	+	0.578831i	$0.803509\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			729.657i			0.800941i	0.916310	+	0.400471i	$0.131153\pi$
$911$			729.657i			0.800941i	−0.916310	+	0.400471i	$0.868847\pi$
$912$		0			0
$913$		−	1155.00i		−	1.26506i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1342.00			1.46028			0.730141	−	0.683296i	$-0.239454\pi$
$919$	1342.00			1.46028			0.730141	+	0.683296i	$0.239454\pi$
$920$		0			0
$921$	812.500	−	538.952i	0.882193	−	0.585181i
$922$		0			0
$923$	−331.662			−0.359331
$924$		0			0
$925$		0			0
$926$		0			0
$927$	580.409	+	245.000i	0.626116	+	0.264293i
$928$		0			0
$929$			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
$932$		0			0
$933$	−994.987	+	660.000i	−1.06644	+	0.707395i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	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	$0.960633\pi$
$942$		0			0
$943$		−	330.000i		−	0.349947i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	278.596			0.294188			0.147094	−	0.989122i	$-0.453008\pi$
$947$	278.596			0.294188			0.147094	+	0.989122i	$0.453008\pi$
$948$		0			0
$949$	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	$0.467263\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−912.072	−	1375.00i	−0.953053	−	1.43678i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	803.000			0.835588
$962$		0			0
$963$	243.772	−	577.500i	0.253138	−	0.599688i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			280.000i			0.289555i	0.989464	+	0.144778i	$0.0462467\pi$
$967$			280.000i			0.289555i	−0.989464	+	0.144778i	$0.953753\pi$
$968$		0			0
$969$	38.5000	+	58.0409i	0.0397317	+	0.0598978i
$970$		0			0
$971$			1044.74i			1.07594i	0.842964	+	0.537970i	$0.180808\pi$
$971$			1044.74i			1.07594i	−0.842964	+	0.537970i	$0.819192\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1097.80			1.12365			0.561823	−	0.827257i	$-0.310100\pi$
$977$	1097.80			1.12365			0.561823	+	0.827257i	$0.310100\pi$
$978$		0			0
$979$	2475.00			2.52809
$980$		0			0
$981$	−308.000	+	729.657i	−0.313965	+	0.743789i
$982$		0			0
$983$	−1671.58			−1.70049			−0.850244	−	0.526389i	$-0.823545\pi$
$983$	−1671.58			−1.70049			−0.850244	+	0.526389i	$0.823545\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	994.987i		−	1.00605i
$990$		0			0
$991$	−452.000			−0.456105			−0.228052	−	0.973649i	$-0.573236\pi$
$991$	−452.000			−0.456105			−0.228052	+	0.973649i	$0.573236\pi$
$992$		0			0
$993$	402.970	+	607.500i	0.405811	+	0.611782i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	420.000i		−	0.421264i	−0.977565	−	0.210632i	$-0.932448\pi$
$997$		−	420.000i		−	0.421264i	0.977565	−	0.210632i	$-0.0675521\pi$
$998$		0			0
$999$	200.000	+	1061.32i	0.200200	+	1.06238i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.3.c.c.26.1	✓	2	60.47	odd	4
75.3.c.c.26.2	yes	2	20.7	even	4
75.3.c.f.26.1	yes	2	20.3	even	4
75.3.c.f.26.2	yes	2	60.23	odd	4
75.3.d.c.74.1		4	12.11	even	2
75.3.d.c.74.2		4	20.19	odd	2
75.3.d.c.74.3		4	4.3	odd	2
75.3.d.c.74.4		4	60.59	even	2
1200.3.c.d.449.1		4	5.4	even	2	inner
1200.3.c.d.449.2		4	3.2	odd	2	inner
1200.3.c.d.449.3		4	15.14	odd	2	inner
1200.3.c.d.449.4		4	1.1	even	1	trivial
1200.3.l.f.401.1		2	15.8	even	4
1200.3.l.f.401.2		2	5.3	odd	4
1200.3.l.s.401.1		2	5.2	odd	4
1200.3.l.s.401.2		2	15.2	even	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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.65831 + 2.50000i) q^{3} +(-3.50000 + 8.29156i) q^{9} +O(q^{10})\)	\(q+(1.65831 + 2.50000i) q^{3} +(-3.50000 + 8.29156i) q^{9} -16.5831i q^{11} -10.0000i q^{13} +3.31662 q^{17} +7.00000 q^{19} +19.8997 q^{23} +(-26.5330 + 5.00000i) q^{27} -33.1662i q^{29} -42.0000 q^{31} +(41.4578 - 27.5000i) q^{33} -40.0000i q^{37} +(25.0000 - 16.5831i) q^{39} -16.5831i q^{41} -50.0000i q^{43} +46.4327 q^{47} +49.0000 q^{49} +(5.50000 + 8.29156i) q^{51} +46.4327 q^{53} +(11.6082 + 17.5000i) q^{57} +66.3325i q^{59} -8.00000 q^{61} -45.0000i q^{67} +(33.0000 + 49.7494i) q^{69} -33.1662i q^{71} +35.0000i q^{73} +12.0000 q^{79} +(-56.5000 - 58.0409i) q^{81} +69.6491 q^{83} +(82.9156 - 55.0000i) q^{87} +149.248i q^{89} +(-69.6491 - 105.000i) q^{93} -70.0000i q^{97} +(137.500 + 58.0409i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{9}+O(q^{10}) \) 4 * q - 14 * q^9	\( 4 q - 14 q^{9} + 28 q^{19} - 168 q^{31} + 100 q^{39} + 196 q^{49} + 22 q^{51} - 32 q^{61} + 132 q^{69} + 48 q^{79} - 226 q^{81} + 550 q^{99}+O(q^{100}) \) 4 * q - 14 * q^9 + 28 * q^19 - 168 * q^31 + 100 * q^39 + 196 * q^49 + 22 * q^51 - 32 * q^61 + 132 * q^69 + 48 * q^79 - 226 * q^81 + 550 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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