LMFDB - Embedded newform 756.4.x.a.125.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,4,Mod(125,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 3]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("756.125");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.x (of order $6$, degree $2$, 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:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			125.3
Character	$\chi$	$=$	756.125
Dual form			756.4.x.a.629.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.29874 + 14.3738i) q^{5} +(18.2297 + 3.26749i) q^{7} +O(q^{10})$	$q+(-8.29874 + 14.3738i) q^{5} +(18.2297 + 3.26749i) q^{7} +(-46.2764 + 26.7177i) q^{11} +(11.0474 + 6.37824i) q^{13} -96.7463 q^{17} +54.6996i q^{19} +(55.6727 + 32.1426i) q^{23} +(-75.2381 - 130.316i) q^{25} +(112.711 - 65.0740i) q^{29} +(-190.618 - 110.053i) q^{31} +(-198.250 + 234.915i) q^{35} +279.735 q^{37} +(-185.308 + 320.963i) q^{41} +(-153.876 - 266.520i) q^{43} +(163.683 + 283.506i) q^{47} +(321.647 + 119.131i) q^{49} -451.749i q^{53} -886.892i q^{55} +(258.739 - 448.148i) q^{59} +(-234.511 + 135.395i) q^{61} +(-183.360 + 105.863i) q^{65} +(-370.881 + 642.385i) q^{67} -914.198i q^{71} -337.210i q^{73} +(-930.906 + 335.849i) q^{77} +(-498.583 - 863.570i) q^{79} +(17.0271 + 29.4917i) q^{83} +(802.872 - 1390.61i) q^{85} -208.953 q^{89} +(180.551 + 152.371i) q^{91} +(-786.243 - 453.937i) q^{95} +(-1100.94 + 635.630i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 6 q^{7}+O(q^{10}) $ 48 * q + 6 * q^7	$ 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) $ 48 * q + 6 * q^7 + 12 * q^11 + 408 * q^23 - 600 * q^25 + 84 * q^29 + 336 * q^37 + 84 * q^43 + 318 * q^49 - 2964 * q^65 - 588 * q^67 - 2400 * q^77 + 204 * q^79 - 360 * q^85 - 1080 * q^91 - 300 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−8.29874	+	14.3738i	−0.742262	+	1.28564i	0.209202	+	0.977873i	$0.432914\pi$
$5$	−8.29874	+	14.3738i	−0.742262	+	1.28564i	−0.951463	+	0.307762i	$0.900420\pi$
$6$		0			0
$7$	18.2297	+	3.26749i	0.984314	+	0.176428i
$7$	18.2297	+	3.26749i	0.984314	+	0.176428i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−46.2764	+	26.7177i	−1.26844	+	0.732334i	−0.974693	−	0.223548i	$-0.928236\pi$
$11$	−46.2764	+	26.7177i	−1.26844	+	0.732334i	−0.293748	+	0.955883i	$0.594903\pi$
$12$		0			0
$13$	11.0474	+	6.37824i	0.235693	+	0.136077i	0.613196	−	0.789931i	$-0.289884\pi$
$13$	11.0474	+	6.37824i	0.235693	+	0.136077i	−0.377503	+	0.926009i	$0.623217\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−96.7463			−1.38026			−0.690130	−	0.723686i	$-0.742447\pi$
$17$	−96.7463			−1.38026			−0.690130	+	0.723686i	$0.742447\pi$
$18$		0			0
$19$			54.6996i			0.660471i	0.943899	+	0.330235i	$0.107128\pi$
$19$			54.6996i			0.660471i	−0.943899	+	0.330235i	$0.892872\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	55.6727	+	32.1426i	0.504720	+	0.291400i	0.730660	−	0.682741i	$-0.239212\pi$
$23$	55.6727	+	32.1426i	0.504720	+	0.291400i	−0.225941	+	0.974141i	$0.572546\pi$
$24$		0			0
$25$	−75.2381	−	130.316i	−0.601905	−	1.04253i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	112.711	−	65.0740i	0.721724	−	0.416688i	−0.0936628	−	0.995604i	$-0.529858\pi$
$29$	112.711	−	65.0740i	0.721724	−	0.416688i	0.815387	+	0.578916i	$0.196524\pi$
$30$		0			0
$31$	−190.618	−	110.053i	−1.10439	−	0.637618i	−0.167017	−	0.985954i	$-0.553413\pi$
$31$	−190.618	−	110.053i	−1.10439	−	0.637618i	−0.937370	+	0.348336i	$0.886747\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−198.250	+	234.915i	−0.957440	+	1.13451i
$36$		0			0
$37$	279.735			1.24292			0.621462	−	0.783445i	$-0.286539\pi$
$37$	279.735			1.24292			0.621462	+	0.783445i	$0.286539\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−185.308	+	320.963i	−0.705861	+	1.22259i	0.260519	+	0.965469i	$0.416106\pi$
$41$	−185.308	+	320.963i	−0.705861	+	1.22259i	−0.966380	+	0.257118i	$0.917227\pi$
$42$		0			0
$43$	−153.876	−	266.520i	−0.545717	−	0.945209i	−0.998561	−	0.0536194i	$-0.982924\pi$
$43$	−153.876	−	266.520i	−0.545717	−	0.945209i	0.452845	−	0.891589i	$-0.350409\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	163.683	+	283.506i	0.507990	+	0.879865i	0.999957	+	0.00925130i	$0.00294482\pi$
$47$	163.683	+	283.506i	0.507990	+	0.879865i	−0.491967	+	0.870614i	$0.663722\pi$
$48$		0			0
$49$	321.647	+	119.131i	0.937747	+	0.347320i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	451.749i		−	1.17080i	−0.810744	−	0.585401i	$-0.800937\pi$
$53$		−	451.749i		−	1.17080i	0.810744	−	0.585401i	$-0.199063\pi$
$54$		0			0
$55$		−	886.892i		−	2.17434i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	258.739	−	448.148i	0.570930	−	0.988880i	−0.425540	−	0.904939i	$-0.639916\pi$
$59$	258.739	−	448.148i	0.570930	−	0.988880i	0.996471	−	0.0839410i	$-0.0267507\pi$
$60$		0			0
$61$	−234.511	+	135.395i	−0.492230	+	0.284189i	−0.725499	−	0.688223i	$-0.758391\pi$
$61$	−234.511	+	135.395i	−0.492230	+	0.284189i	0.233269	+	0.972412i	$0.425058\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−183.360	+	105.863i	−0.349892	+	0.202010i
$66$		0			0
$67$	−370.881	+	642.385i	−0.676274	+	1.17134i	0.299821	+	0.953996i	$0.403073\pi$
$67$	−370.881	+	642.385i	−0.676274	+	1.17134i	−0.976095	+	0.217345i	$0.930260\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	914.198i		−	1.52810i	−0.645155	−	0.764051i	$-0.723207\pi$
$71$		−	914.198i		−	1.52810i	0.645155	−	0.764051i	$-0.276793\pi$
$72$		0			0
$73$		−	337.210i		−	0.540650i	−0.962769	−	0.270325i	$-0.912869\pi$
$73$		−	337.210i		−	0.540650i	0.962769	−	0.270325i	$-0.0871311\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−930.906	+	335.849i	−1.37775	+	0.497059i
$78$		0			0
$79$	−498.583	−	863.570i	−0.710062	−	1.22986i	−0.964833	−	0.262863i	$-0.915333\pi$
$79$	−498.583	−	863.570i	−0.710062	−	1.22986i	0.254771	−	0.967001i	$-0.418000\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	17.0271	+	29.4917i	0.0225176	+	0.0390016i	0.877065	−	0.480372i	$-0.159498\pi$
$83$	17.0271	+	29.4917i	0.0225176	+	0.0390016i	−0.854547	+	0.519374i	$0.826165\pi$
$84$		0			0
$85$	802.872	−	1390.61i	1.02451	−	1.77451i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−208.953			−0.248865			−0.124433	−	0.992228i	$-0.539711\pi$
$89$	−208.953			−0.248865			−0.124433	+	0.992228i	$0.539711\pi$
$90$		0			0
$91$	180.551	+	152.371i	0.207988	+	0.175526i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−786.243	−	453.937i	−0.849124	−	0.490242i
$96$		0			0
$97$	−1100.94	+	635.630i	−1.15241	+	0.665345i	−0.949474	−	0.313847i	$-0.898382\pi$
$97$	−1100.94	+	635.630i	−1.15241	+	0.665345i	−0.202937	+	0.979192i	$0.565049\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−307.124	−	531.954i	−0.302574	−	0.524073i	0.674144	−	0.738600i	$-0.264513\pi$
$101$	−307.124	−	531.954i	−0.302574	−	0.524073i	−0.976718	+	0.214526i	$0.931179\pi$
$102$		0			0
$103$	−903.410	−	521.584i	−0.864229	−	0.498963i	0.00119716	−	0.999999i	$-0.499619\pi$
$103$	−903.410	−	521.584i	−0.864229	−	0.498963i	−0.865426	+	0.501036i	$0.832952\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1128.83i		−	1.01989i	−0.860207	−	0.509945i	$-0.829666\pi$
$107$		−	1128.83i		−	1.01989i	0.860207	−	0.509945i	$-0.170334\pi$
$108$		0			0
$109$	1555.72			1.36707			0.683535	−	0.729918i	$-0.260442\pi$
$109$	1555.72			1.36707			0.683535	+	0.729918i	$0.260442\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−223.834	−	129.231i	−0.186341	−	0.107584i	0.403927	−	0.914791i	$-0.367645\pi$
$113$	−223.834	−	129.231i	−0.186341	−	0.107584i	−0.590269	+	0.807207i	$0.700978\pi$
$114$		0			0
$115$	−924.026	+	533.487i	−0.749268	+	0.432590i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1763.66	−	316.117i	−1.35861	−	0.243516i
$120$		0			0
$121$	762.167	−	1320.11i	0.572628	−	0.991820i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	422.841			0.302560
$126$		0			0
$127$	−586.759			−0.409972			−0.204986	−	0.978765i	$-0.565715\pi$
$127$	−586.759			−0.409972			−0.204986	+	0.978765i	$0.565715\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1132.43	+	1961.43i	−0.755275	+	1.30817i	0.189962	+	0.981791i	$0.439163\pi$
$131$	−1132.43	+	1961.43i	−0.755275	+	1.30817i	−0.945237	+	0.326384i	$0.894170\pi$
$132$		0			0
$133$	−178.730	+	997.159i	−0.116525	+	0.650110i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1654.71	+	955.347i	−1.03191	+	0.595772i	−0.917531	−	0.397665i	$-0.869821\pi$
$137$	−1654.71	+	955.347i	−1.03191	+	0.595772i	−0.114377	+	0.993437i	$0.536487\pi$
$138$		0			0
$139$	−1469.79	−	848.585i	−0.896879	−	0.517813i	−0.0206927	−	0.999786i	$-0.506587\pi$
$139$	−1469.79	−	848.585i	−0.896879	−	0.517813i	−0.876186	+	0.481973i	$0.839920\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−681.647			−0.398617
$144$		0			0
$145$			2160.13i			1.23717i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	140.294	+	80.9987i	0.0771364	+	0.0445347i	0.538072	−	0.842899i	$-0.319153\pi$
$149$	140.294	+	80.9987i	0.0771364	+	0.0445347i	−0.460936	+	0.887434i	$0.652486\pi$
$150$		0			0
$151$	1536.40	+	2661.13i	0.828017	+	1.43417i	0.899591	+	0.436732i	$0.143864\pi$
$151$	1536.40	+	2661.13i	0.828017	+	1.43417i	−0.0715744	+	0.997435i	$0.522802\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3163.77	−	1826.61i	1.63949	−	0.946558i
$156$		0			0
$157$	877.841	+	506.822i	0.446238	+	0.257636i	0.706240	−	0.707972i	$-0.250390\pi$
$157$	877.841	+	506.822i	0.446238	+	0.257636i	−0.260002	+	0.965608i	$0.583723\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	909.873	+	767.862i	0.445391	+	0.375876i
$162$		0			0
$163$	1297.23			0.623356			0.311678	−	0.950188i	$-0.399109\pi$
$163$	1297.23			0.623356			0.311678	+	0.950188i	$0.399109\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−716.233	+	1240.55i	−0.331879	+	0.574831i	−0.982880	−	0.184246i	$-0.941016\pi$
$167$	−716.233	+	1240.55i	−0.331879	+	0.574831i	0.651001	+	0.759077i	$0.274349\pi$
$168$		0			0
$169$	−1017.14	−	1761.73i	−0.462966	−	0.801880i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−987.659	−	1710.68i	−0.434048	−	0.751793i	0.563169	−	0.826341i	$-0.309582\pi$
$173$	−987.659	−	1710.68i	−0.434048	−	0.751793i	−0.997217	+	0.0745483i	$0.976249\pi$
$174$		0			0
$175$	−945.765	−	2621.47i	−0.408532	−	1.13237i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	2310.07i		−	0.964595i	−0.876007	−	0.482298i	$-0.839802\pi$
$179$		−	2310.07i		−	0.964595i	0.876007	−	0.482298i	$-0.160198\pi$
$180$		0			0
$181$		−	423.545i		−	0.173933i	−0.996211	−	0.0869665i	$-0.972283\pi$
$181$		−	423.545i		−	0.173933i	0.996211	−	0.0869665i	$-0.0277173\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2321.45	+	4020.87i	−0.922574	+	1.59795i
$186$		0			0
$187$	4477.06	−	2584.83i	1.75078	−	1.01081i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2030.43	+	1172.27i	−0.769197	+	0.444096i	−0.832588	−	0.553893i	$-0.813142\pi$
$191$	−2030.43	+	1172.27i	−0.769197	+	0.444096i	0.0633911	+	0.997989i	$0.479808\pi$
$192$		0			0
$193$	−2025.36	+	3508.02i	−0.755380	+	1.30836i	0.189805	+	0.981822i	$0.439214\pi$
$193$	−2025.36	+	3508.02i	−0.755380	+	1.30836i	−0.945185	+	0.326535i	$0.894119\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3659.21i		−	1.32339i	−0.749773	−	0.661695i	$-0.769837\pi$
$197$		−	3659.21i		−	1.32339i	0.749773	−	0.661695i	$-0.230163\pi$
$198$		0			0
$199$		−	557.050i		−	0.198433i	−0.995066	−	0.0992166i	$-0.968366\pi$
$199$		−	557.050i		−	0.198433i	0.995066	−	0.0992166i	$-0.0316337\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2267.33	−	817.999i	0.783918	−	0.282819i
$204$		0			0
$205$	−3075.65	−	5327.18i	−1.04787	−	1.81496i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1461.45	−	2531.30i	−0.483685	−	0.837768i
$210$		0			0
$211$	2195.03	−	3801.91i	0.716173	−	1.24045i	−0.246333	−	0.969185i	$-0.579226\pi$
$211$	2195.03	−	3801.91i	0.716173	−	1.24045i	0.962505	−	0.271262i	$-0.0874410\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5107.89			1.62026
$216$		0			0
$217$	−3115.32	−	2629.08i	−0.974569	−	0.822460i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1068.80	−	617.071i	−0.325318	−	0.187822i
$222$		0			0
$223$	2171.75	−	1253.86i	0.652157	−	0.376523i	−0.137125	−	0.990554i	$-0.543786\pi$
$223$	2171.75	−	1253.86i	0.652157	−	0.376523i	0.789282	+	0.614030i	$0.210453\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1344.84	+	2329.34i	0.393218	+	0.681073i	0.992872	−	0.119186i	$-0.0380286\pi$
$227$	1344.84	+	2329.34i	0.393218	+	0.681073i	−0.599654	+	0.800259i	$0.704695\pi$
$228$		0			0
$229$	−3562.03	−	2056.54i	−1.02788	−	0.593449i	−0.111507	−	0.993764i	$-0.535568\pi$
$229$	−3562.03	−	2056.54i	−1.02788	−	0.593449i	−0.916378	+	0.400314i	$0.868901\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			2621.07i			0.736960i	0.929636	+	0.368480i	$0.120122\pi$
$233$			2621.07i			0.736960i	−0.929636	+	0.368480i	$0.879878\pi$
$234$		0			0
$235$	−5433.43			−1.50825
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4446.09	+	2566.95i	1.20332	+	0.694738i	0.961292	−	0.275531i	$-0.0888538\pi$
$239$	4446.09	+	2566.95i	1.20332	+	0.694738i	0.242029	+	0.970269i	$0.422187\pi$
$240$		0			0
$241$	−568.896	+	328.452i	−0.152057	+	0.0877903i	−0.574098	−	0.818786i	$-0.694647\pi$
$241$	−568.896	+	328.452i	−0.152057	+	0.0877903i	0.422041	+	0.906577i	$0.361314\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4381.63	+	3634.67i	−1.14258	+	0.947797i
$246$		0			0
$247$	−348.887	+	604.290i	−0.0898752	+	0.155668i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1807.27			−0.454479			−0.227239	−	0.973839i	$-0.572970\pi$
$251$	−1807.27			−0.454479			−0.227239	+	0.973839i	$0.572970\pi$
$252$		0			0
$253$	−3435.10			−0.853609
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1044.71	−	1809.50i	0.253570	−	0.439196i	−0.710936	−	0.703256i	$-0.751729\pi$
$257$	1044.71	−	1809.50i	0.253570	−	0.439196i	0.964506	+	0.264061i	$0.0850619\pi$
$258$		0			0
$259$	5099.50	+	914.030i	1.22343	+	0.219286i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4219.35	−	2436.05i	0.989265	−	0.571152i	0.0842101	−	0.996448i	$-0.473163\pi$
$263$	4219.35	−	2436.05i	0.989265	−	0.571152i	0.905054	+	0.425296i	$0.139830\pi$
$264$		0			0
$265$	6493.37	+	3748.95i	1.50522	+	0.869042i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4640.31			1.05176			0.525882	−	0.850557i	$-0.323735\pi$
$269$	4640.31			1.05176			0.525882	+	0.850557i	$0.323735\pi$
$270$		0			0
$271$			5685.82i			1.27450i	0.770658	+	0.637249i	$0.219928\pi$
$271$			5685.82i			1.27450i	−0.770658	+	0.637249i	$0.780072\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6963.49	+	4020.37i	1.52696	+	0.881591i
$276$		0			0
$277$	2507.59	+	4343.28i	0.543923	+	0.942103i	0.998674	+	0.0514839i	$0.0163951\pi$
$277$	2507.59	+	4343.28i	0.543923	+	0.942103i	−0.454751	+	0.890619i	$0.650272\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4421.84	+	2552.95i	−0.938736	+	0.541979i	−0.889564	−	0.456811i	$-0.848992\pi$
$281$	−4421.84	+	2552.95i	−0.938736	+	0.541979i	−0.0491721	+	0.998790i	$0.515658\pi$
$282$		0			0
$283$	5499.39	+	3175.07i	1.15514	+	0.666921i	0.950135	−	0.311840i	$-0.100945\pi$
$283$	5499.39	+	3175.07i	1.15514	+	0.666921i	0.205006	+	0.978761i	$0.434279\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4426.87	+	5245.59i	−0.910486	+	1.07888i
$288$		0			0
$289$	4446.84			0.905117
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−762.297	+	1320.34i	−0.151993	+	0.263259i	−0.931960	−	0.362561i	$-0.881902\pi$
$293$	−762.297	+	1320.34i	−0.151993	+	0.263259i	0.779967	+	0.625820i	$0.215236\pi$
$294$		0			0
$295$	4294.41	+	7438.13i	0.847560	+	1.46802i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	410.027	+	710.188i	0.0793059	+	0.137362i
$300$		0			0
$301$	−1934.26	−	5361.38i	−0.370395	−	1.02666i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	4494.43i		−	0.843771i
$306$		0			0
$307$		−	633.080i		−	0.117693i	−0.998267	−	0.0588466i	$-0.981258\pi$
$307$		−	633.080i		−	0.117693i	0.998267	−	0.0588466i	$-0.0187423\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2222.36	−	3849.24i	0.405204	−	0.701833i	−0.589142	−	0.808030i	$-0.700534\pi$
$311$	2222.36	−	3849.24i	0.405204	−	0.701833i	0.994345	+	0.106197i	$0.0338673\pi$
$312$		0			0
$313$	−8892.94	+	5134.34i	−1.60594	+	0.927189i	−0.615673	+	0.788002i	$0.711116\pi$
$313$	−8892.94	+	5134.34i	−1.60594	+	0.927189i	−0.990266	+	0.139187i	$0.955551\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1625.88	+	938.700i	−0.288070	+	0.166318i	−0.637071	−	0.770805i	$-0.719854\pi$
$317$	−1625.88	+	938.700i	−0.288070	+	0.166318i	0.349001	+	0.937122i	$0.386521\pi$
$318$		0			0
$319$	−3477.25	+	6022.78i	−0.610309	+	1.05709i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	5291.98i		−	0.911621i
$324$		0			0
$325$		−	1919.55i		−	0.327623i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2057.54	+	5703.08i	0.344789	+	0.955687i
$330$		0			0
$331$	1549.03	+	2683.00i	0.257228	+	0.445531i	0.965498	−	0.260410i	$-0.0838576\pi$
$331$	1549.03	+	2683.00i	0.257228	+	0.445531i	−0.708271	+	0.705941i	$0.750524\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6155.69	−	10662.0i	−1.00394	−	1.73888i
$336$		0			0
$337$	−5445.10	+	9431.18i	−0.880158	+	1.52448i	−0.0289934	+	0.999580i	$0.509230\pi$
$337$	−5445.10	+	9431.18i	−0.880158	+	1.52448i	−0.851165	+	0.524899i	$0.824103\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	11761.5			1.86780
$342$		0			0
$343$	5474.29	+	3222.70i	0.861760	+	0.507317i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3226.80	+	1862.99i	0.499203	+	0.288215i	0.728385	−	0.685169i	$-0.240271\pi$
$347$	3226.80	+	1862.99i	0.499203	+	0.288215i	−0.229181	+	0.973384i	$0.573605\pi$
$348$		0			0
$349$	−529.889	+	305.931i	−0.0812730	+	0.0469230i	−0.540086	−	0.841610i	$-0.681608\pi$
$349$	−529.889	+	305.931i	−0.0812730	+	0.0469230i	0.458813	+	0.888533i	$0.348275\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2592.39	−	4490.16i	−0.390876	−	0.677017i	0.601689	−	0.798730i	$-0.294495\pi$
$353$	−2592.39	−	4490.16i	−0.390876	−	0.677017i	−0.992565	+	0.121713i	$0.961161\pi$
$354$		0			0
$355$	13140.5	+	7586.69i	1.96458	+	1.13425i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			10228.9i			1.50379i	0.659280	+	0.751897i	$0.270861\pi$
$359$			10228.9i			1.50379i	−0.659280	+	0.751897i	$0.729139\pi$
$360$		0			0
$361$	3866.96			0.563778
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4847.00	+	2798.42i	0.695078	+	0.401304i
$366$		0			0
$367$	−8636.46	+	4986.26i	−1.22839	+	0.709212i	−0.966693	−	0.255938i	$-0.917616\pi$
$367$	−8636.46	+	4986.26i	−1.22839	+	0.709212i	−0.261698	+	0.965150i	$0.584282\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1476.08	−	8235.27i	0.206562	−	1.15244i
$372$		0			0
$373$	−4230.45	+	7327.36i	−0.587251	+	1.01715i	0.407340	+	0.913277i	$0.366457\pi$
$373$	−4230.45	+	7327.36i	−0.587251	+	1.01715i	−0.994591	+	0.103872i	$0.966877\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1660.23			0.226807
$378$		0			0
$379$	−1.71044			−0.000231819			−0.000115910	−	1.00000i	$-0.500037\pi$
$379$	−1.71044			−0.000231819			−0.000115910		1.00000i	$0.500037\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1278.28	−	2214.04i	0.170540	−	0.295384i	−0.768069	−	0.640367i	$-0.778782\pi$
$383$	1278.28	−	2214.04i	0.170540	−	0.295384i	0.938609	+	0.344983i	$0.112115\pi$
$384$		0			0
$385$	2897.91	−	16167.8i	0.383613	−	2.14023i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2061.39	+	1190.14i	−0.268680	+	0.155123i	−0.628288	−	0.777981i	$-0.716244\pi$
$389$	−2061.39	+	1190.14i	−0.268680	+	0.155123i	0.359607	+	0.933104i	$0.382911\pi$
$390$		0			0
$391$	−5386.12	−	3109.68i	−0.696644	−	0.402208i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16550.4			2.10821
$396$		0			0
$397$			12885.5i			1.62898i	0.580177	+	0.814491i	$0.302983\pi$
$397$			12885.5i			1.62898i	−0.580177	+	0.814491i	$0.697017\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6388.40	−	3688.35i	−0.795565	−	0.459320i	0.0463529	−	0.998925i	$-0.485240\pi$
$401$	−6388.40	−	3688.35i	−0.795565	−	0.459320i	−0.841918	+	0.539605i	$0.818573\pi$
$402$		0			0
$403$	−1403.89	−	2431.61i	−0.173531	−	0.300564i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12945.1	+	7473.87i	−1.57657	+	0.910236i
$408$		0			0
$409$	5869.30	+	3388.64i	0.709580	+	0.409676i	0.810906	−	0.585177i	$-0.198975\pi$
$409$	5869.30	+	3388.64i	0.709580	+	0.409676i	−0.101325	+	0.994853i	$0.532308\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6181.06	−	7324.20i	0.736440	−	0.872641i
$414$		0			0
$415$	−565.212			−0.0668558
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2707.65	−	4689.79i	0.315698	−	0.546804i	−0.663888	−	0.747832i	$-0.731095\pi$
$419$	2707.65	−	4689.79i	0.315698	−	0.546804i	0.979586	+	0.201028i	$0.0644281\pi$
$420$		0			0
$421$	−409.136	−	708.644i	−0.0473636	−	0.0820361i	0.841372	−	0.540457i	$-0.181749\pi$
$421$	−409.136	−	708.644i	−0.0473636	−	0.0820361i	−0.888735	+	0.458421i	$0.848415\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7279.01	+	12607.6i	0.830785	+	1.43896i
$426$		0			0
$427$	−4717.47	+	1701.95i	−0.534647	+	0.192888i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	10300.0i		−	1.15113i	−0.817757	−	0.575564i	$-0.804783\pi$
$431$		−	10300.0i		−	1.15113i	0.817757	−	0.575564i	$-0.195217\pi$
$432$		0			0
$433$		−	16119.3i		−	1.78902i	−0.447047	−	0.894511i	$-0.647524\pi$
$433$		−	16119.3i		−	1.78902i	0.447047	−	0.894511i	$-0.352476\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1758.19	+	3045.27i	−0.192461	+	0.333353i
$438$		0			0
$439$	8192.95	−	4730.20i	0.890725	−	0.514260i	0.0165456	−	0.999863i	$-0.494733\pi$
$439$	8192.95	−	4730.20i	0.890725	−	0.514260i	0.874180	+	0.485603i	$0.161400\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3327.36	+	1921.05i	−0.356857	+	0.206032i	−0.667701	−	0.744429i	$-0.732722\pi$
$443$	−3327.36	+	1921.05i	−0.356857	+	0.206032i	0.310844	+	0.950461i	$0.399388\pi$
$444$		0			0
$445$	1734.05	−	3003.46i	0.184723	−	0.319950i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			15663.0i			1.64628i	0.567837	+	0.823141i	$0.307780\pi$
$449$			15663.0i			1.64628i	−0.567837	+	0.823141i	$0.692220\pi$
$450$		0			0
$451$		−	19804.0i		−	2.06770i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3688.50	+	1330.73i	−0.380043	+	0.137111i
$456$		0			0
$457$	−2476.59	−	4289.58i	−0.253501	−	0.439077i	0.710986	−	0.703206i	$-0.248249\pi$
$457$	−2476.59	−	4289.58i	−0.253501	−	0.439077i	−0.964487	+	0.264129i	$0.914915\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1755.21	+	3040.12i	0.177329	+	0.307142i	0.940965	−	0.338505i	$-0.109921\pi$
$461$	1755.21	+	3040.12i	0.177329	+	0.307142i	−0.763636	+	0.645647i	$0.776588\pi$
$462$		0			0
$463$	−7086.75	+	12274.6i	−0.711338	+	1.23207i	0.253017	+	0.967462i	$0.418577\pi$
$463$	−7086.75	+	12274.6i	−0.711338	+	1.23207i	−0.964355	+	0.264611i	$0.914756\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1922.47			0.190495			0.0952474	−	0.995454i	$-0.469636\pi$
$467$	1922.47			0.190495			0.0952474	+	0.995454i	$0.469636\pi$
$468$		0			0
$469$	−8860.05	+	10498.7i	−0.872323	+	1.03365i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	14241.6	+	8222.39i	1.38442	+	0.799294i
$474$		0			0
$475$	7128.24	−	4115.49i	0.688560	−	0.397541i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8556.48	−	14820.3i	−0.816192	−	1.41369i	−0.908469	−	0.417952i	$-0.862748\pi$
$479$	−8556.48	−	14820.3i	−0.816192	−	1.41369i	0.0922775	−	0.995733i	$-0.470585\pi$
$480$		0			0
$481$	3090.36	+	1784.22i	0.292948	+	0.169134i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	21099.7i		−	1.97544i
$486$		0			0
$487$	3067.21			0.285397			0.142699	−	0.989766i	$-0.454422\pi$
$487$	3067.21			0.285397			0.142699	+	0.989766i	$0.454422\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7470.97	−	4313.37i	−0.686681	−	0.396455i	0.115687	−	0.993286i	$-0.463093\pi$
$491$	−7470.97	−	4313.37i	−0.686681	−	0.396455i	−0.802367	+	0.596831i	$0.796426\pi$
$492$		0			0
$493$	−10904.4	+	6295.67i	−0.996167	+	0.575137i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2987.13	−	16665.6i	0.269600	−	1.50413i
$498$		0			0
$499$	6142.19	−	10638.6i	0.551026	−	0.954406i	−0.447174	−	0.894447i	$-0.647570\pi$
$499$	6142.19	−	10638.6i	0.551026	−	0.954406i	0.998201	−	0.0599589i	$-0.0190970\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3977.87			0.352613			0.176307	−	0.984335i	$-0.443585\pi$
$503$	3977.87			0.352613			0.176307	+	0.984335i	$0.443585\pi$
$504$		0			0
$505$	10195.0			0.898356
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8362.32	+	14484.0i	−0.728199	+	1.26128i	0.229444	+	0.973322i	$0.426309\pi$
$509$	−8362.32	+	14484.0i	−0.728199	+	1.26128i	−0.957644	+	0.287956i	$0.907024\pi$
$510$		0			0
$511$	1101.83	−	6147.25i	0.0953856	−	0.532169i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	14994.3	−	8656.97i	1.28297	−	0.740722i
$516$		0			0
$517$	−15149.3	−	8746.43i	−1.28871	−	0.744038i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9088.77			−0.764273			−0.382137	−	0.924106i	$-0.624812\pi$
$521$	−9088.77			−0.764273			−0.382137	+	0.924106i	$0.624812\pi$
$522$		0			0
$523$			13218.7i			1.10519i	0.833450	+	0.552596i	$0.186363\pi$
$523$			13218.7i			1.10519i	−0.833450	+	0.552596i	$0.813637\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18441.6	+	10647.2i	1.52434	+	0.880078i
$528$		0			0
$529$	−4017.20	−	6958.00i	−0.330172	−	0.571875i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4094.36	+	2363.88i	−0.332733	+	0.192103i
$534$		0			0
$535$	16225.6	+	9367.87i	1.31121	+	0.757025i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−18067.6	+	3080.72i	−1.44383	+	0.246189i
$540$		0			0
$541$	−20011.9			−1.59035			−0.795174	−	0.606382i	$-0.792620\pi$
$541$	−20011.9			−1.59035			−0.795174	+	0.606382i	$0.792620\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12910.5	+	22361.6i	−1.01472	+	1.75755i
$546$		0			0
$547$	8725.59	+	15113.2i	0.682046	+	1.18134i	0.974355	+	0.225015i	$0.0722430\pi$
$547$	8725.59	+	15113.2i	0.682046	+	1.18134i	−0.292309	+	0.956324i	$0.594424\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3559.52	+	6165.27i	0.275210	+	0.476678i
$552$		0			0
$553$	−6267.33	−	17371.8i	−0.481942	−	1.33585i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			15169.7i			1.15397i	0.816754	+	0.576986i	$0.195771\pi$
$557$			15169.7i			1.15397i	−0.816754	+	0.576986i	$0.804229\pi$
$558$		0			0
$559$		−	3925.82i		−	0.297039i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3488.62	+	6042.47i	−0.261151	+	0.452326i	−0.966548	−	0.256486i	$-0.917435\pi$
$563$	−3488.62	+	6042.47i	−0.261151	+	0.452326i	0.705397	+	0.708812i	$0.250769\pi$
$564$		0			0
$565$	3715.08	−	2144.90i	0.276628	−	0.159711i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−17354.8	+	10019.8i	−1.27865	+	0.738227i	−0.976600	−	0.215066i	$-0.931003\pi$
$569$	−17354.8	+	10019.8i	−1.27865	+	0.738227i	−0.302047	+	0.953293i	$0.597670\pi$
$570$		0			0
$571$	8327.71	−	14424.0i	0.610339	−	1.05714i	−0.380844	−	0.924639i	$-0.624366\pi$
$571$	8327.71	−	14424.0i	0.610339	−	1.05714i	0.991183	−	0.132500i	$-0.0423003\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	9673.40i		−	0.701580i
$576$		0			0
$577$			8678.08i			0.626123i	0.949733	+	0.313062i	$0.101355\pi$
$577$			8678.08i			0.626123i	−0.949733	+	0.313062i	$0.898645\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	214.035	+	593.262i	0.0152834	+	0.0423626i
$582$		0			0
$583$	12069.7	+	20905.3i	0.857419	+	1.48509i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5941.85	−	10291.6i	−0.417796	−	0.723645i	0.577921	−	0.816093i	$-0.303864\pi$
$587$	−5941.85	−	10291.6i	−0.417796	−	0.723645i	−0.995718	+	0.0924479i	$0.970531\pi$
$588$		0			0
$589$	6019.87	−	10426.7i	0.421128	−	0.729415i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	12702.9			0.879675			0.439838	−	0.898077i	$-0.355036\pi$
$593$	12702.9			0.879675			0.439838	+	0.898077i	$0.355036\pi$
$594$		0			0
$595$	19180.0	−	22727.2i	1.32152	−	1.56592i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−20674.6	−	11936.5i	−1.41025	−	0.814209i	−0.414840	−	0.909894i	$-0.636162\pi$
$599$	−20674.6	−	11936.5i	−1.41025	−	0.814209i	−0.995412	+	0.0956853i	$0.969496\pi$
$600$		0			0
$601$	−9016.55	+	5205.71i	−0.611968	+	0.353320i	−0.773735	−	0.633509i	$-0.781614\pi$
$601$	−9016.55	+	5205.71i	−0.611968	+	0.353320i	0.161767	+	0.986829i	$0.448281\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	12650.1	+	21910.5i	0.850079	+	1.47238i
$606$		0			0
$607$	−13446.8	−	7763.51i	−0.899158	−	0.519129i	−0.0222310	−	0.999753i	$-0.507077\pi$
$607$	−13446.8	−	7763.51i	−0.899158	−	0.519129i	−0.876927	+	0.480624i	$0.840410\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			4176.03i			0.276504i
$612$		0			0
$613$	−9275.88			−0.611173			−0.305587	−	0.952164i	$-0.598853\pi$
$613$	−9275.88			−0.611173			−0.305587	+	0.952164i	$0.598853\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2687.23	−	1551.47i	−0.175338	−	0.101232i	0.409762	−	0.912192i	$-0.365612\pi$
$617$	−2687.23	−	1551.47i	−0.175338	−	0.101232i	−0.585101	+	0.810961i	$0.698945\pi$
$618$		0			0
$619$	−5859.30	+	3382.87i	−0.380460	+	0.219659i	−0.678019	−	0.735045i	$-0.737161\pi$
$619$	−5859.30	+	3382.87i	−0.380460	+	0.219659i	0.297558	+	0.954704i	$0.403828\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3809.17	−	682.752i	−0.244962	−	0.0439067i
$624$		0			0
$625$	5895.72	−	10211.7i	0.377326	−	0.653548i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−27063.3			−1.71556
$630$		0			0
$631$	−4096.60			−0.258452			−0.129226	−	0.991615i	$-0.541249\pi$
$631$	−4096.60			−0.258452			−0.129226	+	0.991615i	$0.541249\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4869.36	−	8433.98i	0.304307	−	0.527075i
$636$		0			0
$637$	2793.53	+	3367.63i	0.173758	+	0.209467i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15136.8	+	8739.26i	−0.932713	+	0.538502i	−0.887669	−	0.460483i	$-0.847676\pi$
$641$	−15136.8	+	8739.26i	−0.932713	+	0.538502i	−0.0450444	+	0.998985i	$0.514343\pi$
$642$		0			0
$643$	−8965.15	−	5176.03i	−0.549846	−	0.317454i	0.199214	−	0.979956i	$-0.436161\pi$
$643$	−8965.15	−	5176.03i	−0.549846	−	0.317454i	−0.749060	+	0.662502i	$0.769495\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−13536.7			−0.822539			−0.411269	−	0.911514i	$-0.634914\pi$
$647$	−13536.7			−0.822539			−0.411269	+	0.911514i	$0.634914\pi$
$648$		0			0
$649$			27651.6i			1.67245i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−20174.8	−	11647.9i	−1.20904	−	0.698039i	−0.246490	−	0.969145i	$-0.579277\pi$
$653$	−20174.8	−	11647.9i	−1.20904	−	0.698039i	−0.962549	+	0.271106i	$0.912610\pi$
$654$		0			0
$655$	−18795.5	−	32554.8i	−1.12122	−	1.94202i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1032.74	−	596.254i	0.0610469	−	0.0352454i	−0.469166	−	0.883110i	$-0.655445\pi$
$659$	1032.74	−	596.254i	0.0610469	−	0.0352454i	0.530213	+	0.847865i	$0.322112\pi$
$660$		0			0
$661$	10922.7	+	6306.24i	0.642731	+	0.371081i	0.785666	−	0.618651i	$-0.212321\pi$
$661$	10922.7	+	6306.24i	0.642731	+	0.371081i	−0.142935	+	0.989732i	$0.545654\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−12849.8	−	10844.2i	−0.749312	−	0.632361i
$666$		0			0
$667$	8366.60			0.485691
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7234.87	−	12531.2i	0.416243	−	0.720954i
$672$		0			0
$673$	−12824.0	−	22211.8i	−0.734515	−	1.27222i	−0.954936	−	0.296813i	$-0.904076\pi$
$673$	−12824.0	−	22211.8i	−0.734515	−	1.27222i	0.220421	−	0.975405i	$-0.429257\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7049.87	+	12210.7i	0.400220	+	0.693201i	0.993752	−	0.111609i	$-0.0356005\pi$
$677$	7049.87	+	12210.7i	0.400220	+	0.693201i	−0.593533	+	0.804810i	$0.702267\pi$
$678$		0			0
$679$	−22146.8	+	7990.05i	−1.25172	+	0.451591i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	6066.68i		−	0.339875i	−0.985455	−	0.169938i	$-0.945643\pi$
$683$		−	6066.68i		−	0.339875i	0.985455	−	0.169938i	$-0.0543567\pi$
$684$		0			0
$685$		−	31712.7i		−	1.76888i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2881.37	−	4990.67i	0.159320	−	0.275950i
$690$		0			0
$691$	−27362.7	+	15797.9i	−1.50641	+	0.869725i	−0.506435	+	0.862278i	$0.669037\pi$
$691$	−27362.7	+	15797.9i	−1.50641	+	0.869725i	−0.999972	+	0.00744663i	$0.997630\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	24394.8	−	14084.4i	1.33144	−	0.768706i
$696$		0			0
$697$	17927.9	−	31052.0i	0.974271	−	1.68749i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	14828.2i		−	0.798936i	−0.916747	−	0.399468i	$-0.869195\pi$
$701$		−	14828.2i		−	0.798936i	0.916747	−	0.399468i	$-0.130805\pi$
$702$		0			0
$703$			15301.4i			0.820914i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3860.63	−	10700.9i	−0.205366	−	0.569235i
$708$		0			0
$709$	10513.5	+	18210.0i	0.556903	+	0.964584i	0.997753	+	0.0670034i	$0.0213438\pi$
$709$	10513.5	+	18210.0i	0.556903	+	0.964584i	−0.440850	+	0.897581i	$0.645323\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7074.80	−	12253.9i	−0.371604	−	0.643636i
$714$		0			0
$715$	5656.81	−	9797.88i	0.295878	−	0.512476i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	36268.4			1.88120			0.940601	−	0.339514i	$-0.110262\pi$
$719$	36268.4			1.88120			0.940601	+	0.339514i	$0.110262\pi$
$720$		0			0
$721$	−14764.7	−	12460.2i	−0.762642	−	0.643610i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−16960.4	−	9792.09i	−0.868819	−	0.501613i
$726$		0			0
$727$	20411.1	−	11784.3i	1.04127	−	0.601178i	0.121078	−	0.992643i	$-0.461365\pi$
$727$	20411.1	−	11784.3i	1.04127	−	0.601178i	0.920193	+	0.391465i	$0.128032\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	14886.9	+	25784.8i	0.753231	+	1.30463i
$732$		0			0
$733$	−15961.7	−	9215.47i	−0.804307	−	0.464367i	0.0406678	−	0.999173i	$-0.487051\pi$
$733$	−15961.7	−	9215.47i	−0.804307	−	0.464367i	−0.844975	+	0.534806i	$0.820385\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	39636.3i		−	1.98104i
$738$		0			0
$739$	−2029.42			−0.101020			−0.0505098	−	0.998724i	$-0.516085\pi$
$739$	−2029.42			−0.101020			−0.0505098	+	0.998724i	$0.516085\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1592.75	−	919.575i	−0.0786438	−	0.0454050i	0.460162	−	0.887835i	$-0.347791\pi$
$743$	−1592.75	−	919.575i	−0.0786438	−	0.0454050i	−0.538806	+	0.842430i	$0.681125\pi$
$744$		0			0
$745$	−2328.52	+	1344.37i	−0.114511	+	0.0661128i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3688.44	−	20578.3i	0.179937	−	1.00389i
$750$		0			0
$751$	1091.30	−	1890.19i	0.0530254	−	0.0918427i	−0.838294	−	0.545218i	$-0.816447\pi$
$751$	1091.30	−	1890.19i	0.0530254	−	0.0918427i	0.891320	+	0.453375i	$0.149780\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−51000.8			−2.45842
$756$		0			0
$757$	14605.5			0.701250			0.350625	−	0.936516i	$-0.385969\pi$
$757$	14605.5			0.701250			0.350625	+	0.936516i	$0.385969\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8971.11	−	15538.4i	0.427336	−	0.740167i	−0.569300	−	0.822130i	$-0.692786\pi$
$761$	8971.11	−	15538.4i	0.427336	−	0.740167i	0.996635	+	0.0819631i	$0.0261190\pi$
$762$		0			0
$763$	28360.3	+	5083.28i	1.34563	+	0.241189i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5716.80	−	3300.60i	0.269129	−	0.155381i
$768$		0			0
$769$	4600.21	+	2655.93i	0.215719	+	0.124545i	0.603966	−	0.797010i	$-0.293586\pi$
$769$	4600.21	+	2655.93i	0.215719	+	0.124545i	−0.388248	+	0.921555i	$0.626919\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	21593.6			1.00475			0.502373	−	0.864651i	$-0.332460\pi$
$773$	21593.6			1.00475			0.502373	+	0.864651i	$0.332460\pi$
$774$		0			0
$775$			33120.8i			1.53514i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−17556.6	−	10136.3i	−0.807483	−	0.466200i
$780$		0			0
$781$	24425.2	+	42305.7i	1.11908	+	1.93831i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−14569.9	+	8411.96i	−0.662451	+	0.382466i
$786$		0			0
$787$	−30052.6	−	17350.9i	−1.36120	−	0.785887i	−0.371413	−	0.928468i	$-0.621126\pi$
$787$	−30052.6	−	17350.9i	−1.36120	−	0.785887i	−0.989783	+	0.142581i	$0.954460\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3658.18	−	3087.22i	−0.164437	−	0.138772i
$792$		0			0
$793$	−3454.33			−0.154687
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5560.35	−	9630.81i	0.247124	−	0.428031i	−0.715603	−	0.698508i	$-0.753848\pi$
$797$	5560.35	−	9630.81i	0.247124	−	0.428031i	0.962727	+	0.270476i	$0.0871812\pi$
$798$		0			0
$799$	−15835.7	−	27428.2i	−0.701159	−	1.21444i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	9009.46	+	15604.8i	0.395936	+	0.685782i
$804$		0			0
$805$	−18587.9	+	6706.08i	−0.813836	+	0.293613i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			39619.1i			1.72180i	0.508777	+	0.860898i	$0.330098\pi$
$809$			39619.1i			1.72180i	−0.508777	+	0.860898i	$0.669902\pi$
$810$		0			0
$811$		−	2483.62i		−	0.107536i	−0.998553	−	0.0537679i	$-0.982877\pi$
$811$		−	2483.62i		−	0.107536i	0.998553	−	0.0537679i	$-0.0171231\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10765.4	+	18646.2i	−0.462693	+	0.801408i
$816$		0			0
$817$	14578.6	−	8416.93i	0.624283	−	0.360430i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	4241.97	−	2449.10i	0.180324	−	0.104110i	−0.407121	−	0.913374i	$-0.633467\pi$
$821$	4241.97	−	2449.10i	0.180324	−	0.104110i	0.587445	+	0.809264i	$0.300134\pi$
$822$		0			0
$823$	−7968.20	+	13801.3i	−0.337490	+	0.584549i	−0.983960	−	0.178390i	$-0.942911\pi$
$823$	−7968.20	+	13801.3i	−0.337490	+	0.584549i	0.646470	+	0.762939i	$0.276245\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	11183.9i		−	0.470256i	−0.971964	−	0.235128i	$-0.924449\pi$
$827$		−	11183.9i		−	0.470256i	0.971964	−	0.235128i	$-0.0755509\pi$
$828$		0			0
$829$			6180.21i			0.258924i	0.991584	+	0.129462i	$0.0413250\pi$
$829$			6180.21i			0.258924i	−0.991584	+	0.129462i	$0.958675\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−31118.2	−	11525.5i	−1.29433	−	0.479392i
$834$		0			0
$835$	−11887.7	−	20590.0i	−0.492682	−	0.853350i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10159.7	+	17597.1i	0.418059	+	0.724100i	0.995744	−	0.0921599i	$-0.0293771\pi$
$839$	10159.7	+	17597.1i	0.418059	+	0.724100i	−0.577685	+	0.816260i	$0.696044\pi$
$840$		0			0
$841$	−3725.25	+	6452.32i	−0.152743	+	0.264558i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	33763.8			1.37457
$846$		0			0
$847$	18207.6	−	21574.9i	0.738630	−	0.875235i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	15573.6	+	8991.42i	0.627328	+	0.362188i
$852$		0			0
$853$	−21731.2	+	12546.5i	−0.872289	+	0.503616i	−0.868108	−	0.496375i	$-0.834664\pi$
$853$	−21731.2	+	12546.5i	−0.872289	+	0.503616i	−0.00418087	+	0.999991i	$0.501331\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8181.75	−	14171.2i	−0.326118	−	0.564853i	0.655620	−	0.755091i	$-0.272407\pi$
$857$	−8181.75	−	14171.2i	−0.326118	−	0.564853i	−0.981738	+	0.190238i	$0.939074\pi$
$858$		0			0
$859$	30710.8	+	17730.9i	1.21984	+	0.704273i	0.964883	−	0.262680i	$-0.0846064\pi$
$859$	30710.8	+	17730.9i	1.21984	+	0.704273i	0.254954	+	0.966953i	$0.417940\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			33166.4i			1.30822i	0.756398	+	0.654111i	$0.226957\pi$
$863$			33166.4i			1.30822i	−0.756398	+	0.654111i	$0.773043\pi$
$864$		0			0
$865$	32785.3			1.28871
$866$		0			0
$867$		0			0
$868$		0			0
$869$	46145.2	+	26641.9i	1.80134	+	1.04001i
$870$		0			0
$871$	−8194.58	+	4731.14i	−0.318786	+	0.184051i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7708.28	+	1381.63i	0.297814	+	0.0533800i
$876$		0			0
$877$	−5617.94	+	9730.56i	−0.216311	+	0.374661i	−0.953677	−	0.300832i	$-0.902736\pi$
$877$	−5617.94	+	9730.56i	−0.216311	+	0.374661i	0.737367	+	0.675493i	$0.236069\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−25537.5			−0.976597			−0.488298	−	0.872677i	$-0.662382\pi$
$881$	−25537.5			−0.976597			−0.488298	+	0.872677i	$0.662382\pi$
$882$		0			0
$883$	17936.9			0.683606			0.341803	−	0.939772i	$-0.388962\pi$
$883$	17936.9			0.683606			0.341803	+	0.939772i	$0.388962\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2646.98	−	4584.71i	0.100200	−	0.173551i	−0.811567	−	0.584259i	$-0.801385\pi$
$887$	2646.98	−	4584.71i	0.100200	−	0.173551i	0.911767	+	0.410708i	$0.134719\pi$
$888$		0			0
$889$	−10696.5	−	1917.23i	−0.403541	−	0.0723304i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15507.7	+	8953.37i	−0.581125	+	0.335513i
$894$		0			0
$895$	33204.5	+	19170.6i	1.24012	+	0.715982i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−28646.4			−1.06275
$900$		0			0
$901$			43705.0i			1.61601i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6087.97	+	3514.89i	0.223614	+	0.129104i
$906$		0			0
$907$	−6770.38	−	11726.6i	−0.247857	−	0.429302i	0.715074	−	0.699049i	$-0.246393\pi$
$907$	−6770.38	−	11726.6i	−0.247857	−	0.429302i	−0.962931	+	0.269747i	$0.913060\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	23505.1	−	13570.7i	0.854840	−	0.493542i	−0.00744116	−	0.999972i	$-0.502369\pi$
$911$	23505.1	−	13570.7i	0.854840	−	0.493542i	0.862281	+	0.506430i	$0.169035\pi$
$912$		0			0
$913$	−1575.90	−	909.846i	−0.0571245	−	0.0329808i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−27052.9	+	32056.2i	−0.974226	+	1.15440i
$918$		0			0
$919$	−32872.5			−1.17994			−0.589969	−	0.807426i	$-0.700860\pi$
$919$	−32872.5			−1.17994			−0.589969	+	0.807426i	$0.700860\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5830.98	−	10099.5i	0.207940	−	0.360163i
$924$		0			0
$925$	−21046.7	−	36454.0i	−0.748122	−	1.29578i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	11828.2	+	20487.0i	0.417729	+	0.723528i	0.995711	−	0.0925216i	$-0.0294927\pi$
$929$	11828.2	+	20487.0i	0.417729	+	0.723528i	−0.577981	+	0.816050i	$0.696159\pi$
$930$		0			0
$931$	−6516.41	+	17594.0i	−0.229395	+	0.619354i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			85803.5i			3.00115i
$936$		0			0
$937$		−	26716.0i		−	0.931456i	−0.884928	−	0.465728i	$-0.845793\pi$
$937$		−	26716.0i		−	0.931456i	0.884928	−	0.465728i	$-0.154207\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19766.0	−	34235.6i	0.684752	−	1.18603i	−0.288762	−	0.957401i	$-0.593244\pi$
$941$	19766.0	−	34235.6i	0.684752	−	1.18603i	0.973515	−	0.228625i	$-0.0734229\pi$
$942$		0			0
$943$	−20633.2	+	11912.6i	−0.712524	+	0.411376i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	38984.8	−	22507.9i	1.33774	−	0.772343i	0.351265	−	0.936276i	$-0.385752\pi$
$947$	38984.8	−	22507.9i	1.33774	−	0.772343i	0.986471	+	0.163933i	$0.0524182\pi$
$948$		0			0
$949$	2150.81	−	3725.31i	0.0735702	−	0.127427i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	9240.19i		−	0.314081i	−0.987592	−	0.157040i	$-0.949805\pi$
$953$		−	9240.19i		−	0.314081i	0.987592	−	0.157040i	$-0.0501953\pi$
$954$		0			0
$955$		−	38913.4i		−	1.31854i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−33286.5	+	12009.0i	−1.12083	+	0.404370i
$960$		0			0
$961$	9327.94	+	16156.5i	0.313113	+	0.542327i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−33615.8	−	58224.3i	−1.12138	−	1.94229i
$966$		0			0
$967$	12391.8	−	21463.2i	0.412093	−	0.713766i	−0.583026	−	0.812454i	$-0.698131\pi$
$967$	12391.8	−	21463.2i	0.412093	−	0.713766i	0.995118	+	0.0986881i	$0.0314646\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−37516.1			−1.23991			−0.619953	−	0.784639i	$-0.712848\pi$
$971$	−37516.1			−1.23991			−0.619953	+	0.784639i	$0.712848\pi$
$972$		0			0
$973$	−24021.2	−	20272.0i	−0.791454	−	0.667925i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−14175.0	−	8183.92i	−0.464173	−	0.267991i	0.249624	−	0.968343i	$-0.419693\pi$
$977$	−14175.0	−	8183.92i	−0.464173	−	0.267991i	−0.713797	+	0.700352i	$0.753026\pi$
$978$		0			0
$979$	9669.60	−	5582.75i	0.315671	−	0.182253i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	14758.1	+	25561.7i	0.478850	+	0.829393i	0.999706	−	0.0242520i	$-0.00772041\pi$
$983$	14758.1	+	25561.7i	0.478850	+	0.829393i	−0.520856	+	0.853645i	$0.674387\pi$
$984$		0			0
$985$	52596.9	+	30366.8i	1.70140	+	0.982302i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	19783.9i		−	0.636087i
$990$		0			0
$991$	−13893.1			−0.445336			−0.222668	−	0.974894i	$-0.571477\pi$
$991$	−13893.1			−0.445336			−0.222668	+	0.974894i	$0.571477\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8006.95	+	4622.81i	0.255113	+	0.147289i
$996$		0			0
$997$	17139.3	−	9895.40i	0.544441	−	0.314333i	−0.202436	−	0.979296i	$-0.564886\pi$
$997$	17139.3	−	9895.40i	0.544441	−	0.314333i	0.746877	+	0.664962i	$0.231552\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.x.a.125.3		48
3.2	odd	2		252.4.x.a.41.8	✓	48
7.6	odd	2	inner	756.4.x.a.125.22		48
9.2	odd	6	inner	756.4.x.a.629.22		48
9.4	even	3		2268.4.f.a.1133.43		48
9.5	odd	6		2268.4.f.a.1133.6		48
9.7	even	3		252.4.x.a.209.17	yes	48
21.20	even	2		252.4.x.a.41.17	yes	48
63.13	odd	6		2268.4.f.a.1133.5		48
63.20	even	6	inner	756.4.x.a.629.3		48
63.34	odd	6		252.4.x.a.209.8	yes	48
63.41	even	6		2268.4.f.a.1133.44		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.x.a.41.8	✓	48	3.2	odd	2
252.4.x.a.41.17	yes	48	21.20	even	2
252.4.x.a.209.8	yes	48	63.34	odd	6
252.4.x.a.209.17	yes	48	9.7	even	3
756.4.x.a.125.3		48	1.1	even	1	trivial
756.4.x.a.125.22		48	7.6	odd	2	inner
756.4.x.a.629.3		48	63.20	even	6	inner
756.4.x.a.629.22		48	9.2	odd	6	inner
2268.4.f.a.1133.5		48	63.13	odd	6
2268.4.f.a.1133.6		48	9.5	odd	6
2268.4.f.a.1133.43		48	9.4	even	3
2268.4.f.a.1133.44		48	63.41	even	6

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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.x (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-8.29874 + 14.3738i) q^{5} +(18.2297 + 3.26749i) q^{7} +O(q^{10})\)	\(q+(-8.29874 + 14.3738i) q^{5} +(18.2297 + 3.26749i) q^{7} +(-46.2764 + 26.7177i) q^{11} +(11.0474 + 6.37824i) q^{13} -96.7463 q^{17} +54.6996i q^{19} +(55.6727 + 32.1426i) q^{23} +(-75.2381 - 130.316i) q^{25} +(112.711 - 65.0740i) q^{29} +(-190.618 - 110.053i) q^{31} +(-198.250 + 234.915i) q^{35} +279.735 q^{37} +(-185.308 + 320.963i) q^{41} +(-153.876 - 266.520i) q^{43} +(163.683 + 283.506i) q^{47} +(321.647 + 119.131i) q^{49} -451.749i q^{53} -886.892i q^{55} +(258.739 - 448.148i) q^{59} +(-234.511 + 135.395i) q^{61} +(-183.360 + 105.863i) q^{65} +(-370.881 + 642.385i) q^{67} -914.198i q^{71} -337.210i q^{73} +(-930.906 + 335.849i) q^{77} +(-498.583 - 863.570i) q^{79} +(17.0271 + 29.4917i) q^{83} +(802.872 - 1390.61i) q^{85} -208.953 q^{89} +(180.551 + 152.371i) q^{91} +(-786.243 - 453.937i) q^{95} +(-1100.94 + 635.630i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 6 q^{7}+O(q^{10}) \) 48 * q + 6 * q^7	\( 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) \) 48 * q + 6 * q^7 + 12 * q^11 + 408 * q^23 - 600 * q^25 + 84 * q^29 + 336 * q^37 + 84 * q^43 + 318 * q^49 - 2964 * q^65 - 588 * q^67 - 2400 * q^77 + 204 * q^79 - 360 * q^85 - 1080 * q^91 - 300 * q^95

Introduction

L-functions

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