LMFDB - Embedded newform 928.3.b.c.463.24

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [928,3,Mod(463,928)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(928, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("928.463");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 928 = 2^{5} \cdot 29 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	928.b (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:	$25.2861685326$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	no (minimal twist has level 232)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			463.24
Character	$\chi$	$=$	928.463
Dual form			928.3.b.c.463.33

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23647i q^{3} +6.52788i q^{5} +8.84035i q^{7} +7.47113 q^{9} +O(q^{10})$	$q-1.23647i q^{3} +6.52788i q^{5} +8.84035i q^{7} +7.47113 q^{9} +8.03032i q^{11} +23.9693i q^{13} +8.07155 q^{15} +3.61156i q^{17} -25.4722i q^{19} +10.9309 q^{21} -12.7195i q^{23} -17.6132 q^{25} -20.3661i q^{27} +(-28.6905 - 4.22540i) q^{29} -31.7948 q^{31} +9.92928 q^{33} -57.7087 q^{35} -28.0952 q^{37} +29.6374 q^{39} +58.3896i q^{41} -20.5039i q^{43} +48.7706i q^{45} +53.0390 q^{47} -29.1517 q^{49} +4.46560 q^{51} -77.4712i q^{53} -52.4209 q^{55} -31.4958 q^{57} -36.8773 q^{59} +114.321 q^{61} +66.0474i q^{63} -156.469 q^{65} -21.5732 q^{67} -15.7274 q^{69} +18.8089i q^{71} +105.400i q^{73} +21.7783i q^{75} -70.9908 q^{77} -17.8244 q^{79} +42.0580 q^{81} +0.565354 q^{83} -23.5758 q^{85} +(-5.22460 + 35.4751i) q^{87} -91.6468i q^{89} -211.897 q^{91} +39.3135i q^{93} +166.280 q^{95} -86.0400i q^{97} +59.9955i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 184 q^{9}+O(q^{10}) $ 56 * q - 184 * q^9	$ 56 q - 184 q^{9} - 280 q^{25} + 32 q^{33} - 96 q^{35} - 424 q^{49} - 32 q^{51} + 32 q^{57} - 512 q^{59} + 192 q^{65} - 352 q^{67} + 280 q^{81} - 128 q^{83} - 96 q^{91}+O(q^{100}) $ 56 * q - 184 * q^9 - 280 * q^25 + 32 * q^33 - 96 * q^35 - 424 * q^49 - 32 * q^51 + 32 * q^57 - 512 * q^59 + 192 * q^65 - 352 * q^67 + 280 * q^81 - 128 * q^83 - 96 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/928\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$639$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$		−	1.23647i		−	0.412158i	−0.978535	−	0.206079i	$-0.933930\pi$
$3$		−	1.23647i		−	0.412158i	0.978535	−	0.206079i	$-0.0660704\pi$
$4$		0			0
$5$			6.52788i			1.30558i	0.757541	+	0.652788i	$0.226401\pi$
$5$			6.52788i			1.30558i	−0.757541	+	0.652788i	$0.773599\pi$
$6$		0			0
$7$			8.84035i			1.26291i	0.775414	+	0.631453i	$0.217541\pi$
$7$			8.84035i			1.26291i	−0.775414	+	0.631453i	$0.782459\pi$
$8$		0			0
$9$	7.47113			0.830126
$10$		0			0
$11$			8.03032i			0.730029i	0.931002	+	0.365014i	$0.118936\pi$
$11$			8.03032i			0.730029i	−0.931002	+	0.365014i	$0.881064\pi$
$12$		0			0
$13$			23.9693i			1.84379i	0.387436	+	0.921897i	$0.373361\pi$
$13$			23.9693i			1.84379i	−0.387436	+	0.921897i	$0.626639\pi$
$14$		0			0
$15$	8.07155			0.538104
$16$		0			0
$17$			3.61156i			0.212445i	0.994342	+	0.106222i	$0.0338755\pi$
$17$			3.61156i			0.212445i	−0.994342	+	0.106222i	$0.966124\pi$
$18$		0			0
$19$		−	25.4722i		−	1.34064i	−0.742070	−	0.670322i	$-0.766156\pi$
$19$		−	25.4722i		−	1.34064i	0.742070	−	0.670322i	$-0.233844\pi$
$20$		0			0
$21$	10.9309			0.520517
$22$		0			0
$23$		−	12.7195i		−	0.553023i	−0.961011	−	0.276511i	$-0.910822\pi$
$23$		−	12.7195i		−	0.553023i	0.961011	−	0.276511i	$-0.0891784\pi$
$24$		0			0
$25$	−17.6132			−0.704527
$26$		0			0
$27$		−	20.3661i		−	0.754301i
$28$		0			0
$29$	−28.6905	−	4.22540i	−0.989328	−	0.145703i
$29$	−28.6905	−	4.22540i	−0.989328	−	0.145703i
$30$		0			0
$31$	−31.7948			−1.02564			−0.512820	−	0.858496i	$-0.671399\pi$
$31$	−31.7948			−1.02564			−0.512820	+	0.858496i	$0.671399\pi$
$32$		0			0
$33$	9.92928			0.300887
$34$		0			0
$35$	−57.7087			−1.64882
$36$		0			0
$37$	−28.0952			−0.759330			−0.379665	−	0.925124i	$-0.623961\pi$
$37$	−28.0952			−0.759330			−0.379665	+	0.925124i	$0.623961\pi$
$38$		0			0
$39$	29.6374			0.759935
$40$		0			0
$41$			58.3896i			1.42414i	0.702110	+	0.712068i	$0.252241\pi$
$41$			58.3896i			1.42414i	−0.702110	+	0.712068i	$0.747759\pi$
$42$		0			0
$43$		−	20.5039i		−	0.476836i	−0.971163	−	0.238418i	$-0.923371\pi$
$43$		−	20.5039i		−	0.476836i	0.971163	−	0.238418i	$-0.0766288\pi$
$44$		0			0
$45$			48.7706i			1.08379i
$46$		0			0
$47$	53.0390			1.12849			0.564245	−	0.825607i	$-0.309167\pi$
$47$	53.0390			1.12849			0.564245	+	0.825607i	$0.309167\pi$
$48$		0			0
$49$	−29.1517			−0.594933
$50$		0			0
$51$	4.46560			0.0875608
$52$		0			0
$53$		−	77.4712i		−	1.46172i	−0.682527	−	0.730861i	$-0.739119\pi$
$53$		−	77.4712i		−	1.46172i	0.682527	−	0.730861i	$-0.260881\pi$
$54$		0			0
$55$	−52.4209			−0.953108
$56$		0			0
$57$	−31.4958			−0.552557
$58$		0			0
$59$	−36.8773			−0.625040			−0.312520	−	0.949911i	$-0.601173\pi$
$59$	−36.8773			−0.625040			−0.312520	+	0.949911i	$0.601173\pi$
$60$		0			0
$61$	114.321			1.87411			0.937057	−	0.349177i	$-0.113539\pi$
$61$	114.321			1.87411			0.937057	+	0.349177i	$0.113539\pi$
$62$		0			0
$63$			66.0474i			1.04837i
$64$		0			0
$65$	−156.469			−2.40721
$66$		0			0
$67$	−21.5732			−0.321987			−0.160994	−	0.986955i	$-0.551470\pi$
$67$	−21.5732			−0.321987			−0.160994	+	0.986955i	$0.551470\pi$
$68$		0			0
$69$	−15.7274			−0.227933
$70$		0			0
$71$			18.8089i			0.264915i	0.991189	+	0.132457i	$0.0422867\pi$
$71$			18.8089i			0.264915i	−0.991189	+	0.132457i	$0.957713\pi$
$72$		0			0
$73$			105.400i			1.44383i	0.691981	+	0.721915i	$0.256738\pi$
$73$			105.400i			1.44383i	−0.691981	+	0.721915i	$0.743262\pi$
$74$		0			0
$75$			21.7783i			0.290377i
$76$		0			0
$77$	−70.9908			−0.921958
$78$		0			0
$79$	−17.8244			−0.225625			−0.112813	−	0.993616i	$-0.535986\pi$
$79$	−17.8244			−0.225625			−0.112813	+	0.993616i	$0.535986\pi$
$80$		0			0
$81$	42.0580			0.519234
$82$		0			0
$83$	0.565354			0.00681149			0.00340575	−	0.999994i	$-0.498916\pi$
$83$	0.565354			0.00681149			0.00340575	+	0.999994i	$0.498916\pi$
$84$		0			0
$85$	−23.5758			−0.277362
$86$		0			0
$87$	−5.22460	+	35.4751i	−0.0600529	+	0.407760i
$88$		0			0
$89$		−	91.6468i		−	1.02974i	−0.857268	−	0.514870i	$-0.827840\pi$
$89$		−	91.6468i		−	1.02974i	0.857268	−	0.514870i	$-0.172160\pi$
$90$		0			0
$91$	−211.897			−2.32854
$92$		0			0
$93$			39.3135i			0.422726i
$94$		0			0
$95$	166.280			1.75031
$96$		0			0
$97$		−	86.0400i		−	0.887010i	−0.896272	−	0.443505i	$-0.853735\pi$
$97$		−	86.0400i		−	0.887010i	0.896272	−	0.443505i	$-0.146265\pi$
$98$		0			0
$99$			59.9955i			0.606016i
$100$		0			0
$101$	−13.3964			−0.132637			−0.0663187	−	0.997798i	$-0.521125\pi$
$101$	−13.3964			−0.132637			−0.0663187	+	0.997798i	$0.521125\pi$
$102$		0			0
$103$			123.281i			1.19690i	0.801160	+	0.598450i	$0.204217\pi$
$103$			123.281i			1.19690i	−0.801160	+	0.598450i	$0.795783\pi$
$104$		0			0
$105$			71.3553i			0.679575i
$106$		0			0
$107$	121.653			1.13695			0.568473	−	0.822702i	$-0.307534\pi$
$107$	121.653			1.13695			0.568473	+	0.822702i	$0.307534\pi$
$108$		0			0
$109$			60.4520i			0.554606i	0.960783	+	0.277303i	$0.0894405\pi$
$109$			60.4520i			0.554606i	−0.960783	+	0.277303i	$0.910559\pi$
$110$		0			0
$111$			34.7390i			0.312964i
$112$		0			0
$113$		−	52.0761i		−	0.460851i	−0.973090	−	0.230425i	$-0.925988\pi$
$113$		−	52.0761i		−	0.460851i	0.973090	−	0.230425i	$-0.0740118\pi$
$114$		0			0
$115$	83.0315			0.722013
$116$		0			0
$117$			179.078i			1.53058i
$118$		0			0
$119$	−31.9274			−0.268298
$120$		0			0
$121$	56.5140			0.467058
$122$		0			0
$123$	72.1973			0.586970
$124$		0			0
$125$			48.2202i			0.385762i
$126$		0			0
$127$	−97.8761			−0.770678			−0.385339	−	0.922775i	$-0.625915\pi$
$127$	−97.8761			−0.770678			−0.385339	+	0.922775i	$0.625915\pi$
$128$		0			0
$129$	−25.3526			−0.196532
$130$		0			0
$131$			229.929i			1.75518i	0.479408	+	0.877592i	$0.340851\pi$
$131$			229.929i			1.75518i	−0.479408	+	0.877592i	$0.659149\pi$
$132$		0			0
$133$	225.183			1.69311
$134$		0			0
$135$	132.948			0.984797
$136$		0			0
$137$			81.0099i			0.591313i	0.955294	+	0.295656i	$0.0955384\pi$
$137$			81.0099i			0.591313i	−0.955294	+	0.295656i	$0.904462\pi$
$138$		0			0
$139$	−122.442			−0.880878			−0.440439	−	0.897782i	$-0.645177\pi$
$139$	−122.442			−0.880878			−0.440439	+	0.897782i	$0.645177\pi$
$140$		0			0
$141$		−	65.5814i		−	0.465116i
$142$		0			0
$143$	−192.481			−1.34602
$144$		0			0
$145$	27.5829	−	187.288i	0.190227	−	1.29164i
$146$		0			0
$147$			36.0454i			0.245207i
$148$		0			0
$149$		−	1.76709i		−	0.0118597i	−0.999982	−	0.00592983i	$-0.998112\pi$
$149$		−	1.76709i		−	0.0118597i	0.999982	−	0.00592983i	$-0.00188753\pi$
$150$		0			0
$151$		−	180.821i		−	1.19749i	−0.800939	−	0.598747i	$-0.795666\pi$
$151$		−	180.821i		−	1.19749i	0.800939	−	0.598747i	$-0.204334\pi$
$152$		0			0
$153$			26.9824i			0.176356i
$154$		0			0
$155$		−	207.553i		−	1.33905i
$156$		0			0
$157$	53.7550			0.342388			0.171194	−	0.985237i	$-0.445237\pi$
$157$	53.7550			0.342388			0.171194	+	0.985237i	$0.445237\pi$
$158$		0			0
$159$	−95.7912			−0.602460
$160$		0			0
$161$	112.445			0.698416
$162$		0			0
$163$			234.231i			1.43700i	0.695528	+	0.718499i	$0.255171\pi$
$163$			234.231i			1.43700i	−0.695528	+	0.718499i	$0.744829\pi$
$164$		0			0
$165$			64.8171i			0.392831i
$166$		0			0
$167$			223.946i			1.34099i	0.741913	+	0.670496i	$0.233919\pi$
$167$			223.946i			1.34099i	−0.741913	+	0.670496i	$0.766081\pi$
$168$		0			0
$169$	−405.528			−2.39957
$170$		0			0
$171$		−	190.306i		−	1.11290i
$172$		0			0
$173$		−	79.6404i		−	0.460349i	−0.973149	−	0.230175i	$-0.926070\pi$
$173$		−	79.6404i		−	0.460349i	0.973149	−	0.230175i	$-0.0739298\pi$
$174$		0			0
$175$		−	155.707i		−	0.889752i
$176$		0			0
$177$			45.5979i			0.257615i
$178$		0			0
$179$	−218.155			−1.21874			−0.609371	−	0.792885i	$-0.708578\pi$
$179$	−218.155			−1.21874			−0.609371	+	0.792885i	$0.708578\pi$
$180$		0			0
$181$		−	86.7152i		−	0.479090i	−0.970885	−	0.239545i	$-0.923002\pi$
$181$		−	86.7152i		−	0.479090i	0.970885	−	0.239545i	$-0.0769982\pi$
$182$		0			0
$183$		−	141.355i		−	0.772431i
$184$		0			0
$185$		−	183.402i		−	0.991362i
$186$		0			0
$187$	−29.0020			−0.155091
$188$		0			0
$189$	180.044			0.952612
$190$		0			0
$191$	−119.023			−0.623156			−0.311578	−	0.950221i	$-0.600858\pi$
$191$	−119.023			−0.623156			−0.311578	+	0.950221i	$0.600858\pi$
$192$		0			0
$193$			28.0803i			0.145494i	0.997350	+	0.0727468i	$0.0231765\pi$
$193$			28.0803i			0.145494i	−0.997350	+	0.0727468i	$0.976823\pi$
$194$		0			0
$195$			193.470i			0.992152i
$196$		0			0
$197$		−	70.7646i		−	0.359211i	−0.983739	−	0.179606i	$-0.942518\pi$
$197$		−	70.7646i		−	0.359211i	0.983739	−	0.179606i	$-0.0574821\pi$
$198$		0			0
$199$		−	281.244i		−	1.41329i	−0.707569	−	0.706644i	$-0.750208\pi$
$199$		−	281.244i		−	1.41329i	0.707569	−	0.706644i	$-0.249792\pi$
$200$		0			0
$201$			26.6747i			0.132710i
$202$		0			0
$203$	37.3540	−	253.634i	0.184010	−	1.24943i
$204$		0			0
$205$	−381.160			−1.85932
$206$		0			0
$207$		−	95.0292i		−	0.459078i
$208$		0			0
$209$	204.550			0.978709
$210$		0			0
$211$			251.923i			1.19395i	0.802260	+	0.596975i	$0.203631\pi$
$211$			251.923i			1.19395i	−0.802260	+	0.596975i	$0.796369\pi$
$212$		0			0
$213$	23.2568			0.109187
$214$		0			0
$215$	133.847			0.622545
$216$		0			0
$217$		−	281.077i		−	1.29529i
$218$		0			0
$219$	130.324			0.595087
$220$		0			0
$221$	−86.5665			−0.391704
$222$		0			0
$223$			219.801i			0.985653i	0.870128	+	0.492826i	$0.164036\pi$
$223$			219.801i			0.985653i	−0.870128	+	0.492826i	$0.835964\pi$
$224$		0			0
$225$	−131.590			−0.584846
$226$		0			0
$227$	450.034			1.98253			0.991264	−	0.131892i	$-0.0421052\pi$
$227$	450.034			1.98253			0.991264	+	0.131892i	$0.0421052\pi$
$228$		0			0
$229$	−354.060			−1.54611			−0.773057	−	0.634336i	$-0.781274\pi$
$229$	−354.060			−1.54611			−0.773057	+	0.634336i	$0.781274\pi$
$230$		0			0
$231$			87.7783i			0.379993i
$232$		0			0
$233$	382.640			1.64223			0.821116	−	0.570761i	$-0.193352\pi$
$233$	382.640			1.64223			0.821116	+	0.570761i	$0.193352\pi$
$234$		0			0
$235$			346.232i			1.47333i
$236$		0			0
$237$			22.0394i			0.0929933i
$238$		0			0
$239$			83.7394i			0.350374i	0.984535	+	0.175187i	$0.0560530\pi$
$239$			83.7394i			0.350374i	−0.984535	+	0.175187i	$0.943947\pi$
$240$		0			0
$241$	−243.346			−1.00973			−0.504867	−	0.863197i	$-0.668459\pi$
$241$	−243.346			−1.00973			−0.504867	+	0.863197i	$0.668459\pi$
$242$		0			0
$243$		−	235.299i		−	0.968308i
$244$		0			0
$245$		−	190.299i		−	0.776730i
$246$		0			0
$247$	610.552			2.47187
$248$		0			0
$249$		−	0.699046i		−	0.00280741i
$250$		0			0
$251$		−	313.357i		−	1.24844i	−0.781250	−	0.624218i	$-0.785418\pi$
$251$		−	313.357i		−	1.24844i	0.781250	−	0.624218i	$-0.214582\pi$
$252$		0			0
$253$	102.142			0.403723
$254$		0			0
$255$			29.1509i			0.114317i
$256$		0			0
$257$	102.651			0.399419			0.199710	−	0.979855i	$-0.436000\pi$
$257$	102.651			0.399419			0.199710	+	0.979855i	$0.436000\pi$
$258$		0			0
$259$		−	248.371i		−	0.958963i
$260$		0			0
$261$	−214.351	−	31.5685i	−0.821267	−	0.120952i
$262$		0			0
$263$	−46.1457			−0.175459			−0.0877294	−	0.996144i	$-0.527961\pi$
$263$	−46.1457			−0.175459			−0.0877294	+	0.996144i	$0.527961\pi$
$264$		0			0
$265$	505.723			1.90839
$266$		0			0
$267$	−113.319			−0.424416
$268$		0			0
$269$	404.551			1.50391			0.751953	−	0.659217i	$-0.229112\pi$
$269$	404.551			1.50391			0.751953	+	0.659217i	$0.229112\pi$
$270$		0			0
$271$	−277.241			−1.02303			−0.511516	−	0.859274i	$-0.670916\pi$
$271$	−277.241			−1.02303			−0.511516	+	0.859274i	$0.670916\pi$
$272$		0			0
$273$			262.005i			0.959726i
$274$		0			0
$275$		−	141.439i		−	0.514325i
$276$		0			0
$277$		−	225.726i		−	0.814894i	−0.913229	−	0.407447i	$-0.866419\pi$
$277$		−	225.726i		−	0.814894i	0.913229	−	0.407447i	$-0.133581\pi$
$278$		0			0
$279$	−237.543			−0.851409
$280$		0			0
$281$	−65.4123			−0.232784			−0.116392	−	0.993203i	$-0.537133\pi$
$281$	−65.4123			−0.232784			−0.116392	+	0.993203i	$0.537133\pi$
$282$		0			0
$283$	515.470			1.82145			0.910724	−	0.413015i	$-0.135524\pi$
$283$	515.470			1.82145			0.910724	+	0.413015i	$0.135524\pi$
$284$		0			0
$285$		−	205.601i		−	0.721405i
$286$		0			0
$287$	−516.184			−1.79855
$288$		0			0
$289$	275.957			0.954867
$290$		0			0
$291$	−106.386			−0.365589
$292$		0			0
$293$	170.234			0.581005			0.290502	−	0.956874i	$-0.406178\pi$
$293$	170.234			0.581005			0.290502	+	0.956874i	$0.406178\pi$
$294$		0			0
$295$		−	240.731i		−	0.816037i
$296$		0			0
$297$	163.547			0.550662
$298$		0			0
$299$	304.878			1.01966
$300$		0			0
$301$	181.262			0.602199
$302$		0			0
$303$			16.5643i			0.0546676i
$304$		0			0
$305$			746.273i			2.44680i
$306$		0			0
$307$		−	506.327i		−	1.64927i	−0.565663	−	0.824637i	$-0.691380\pi$
$307$		−	506.327i		−	1.64927i	0.565663	−	0.824637i	$-0.308620\pi$
$308$		0			0
$309$	152.433			0.493312
$310$		0			0
$311$	391.438			1.25864			0.629321	−	0.777145i	$-0.283333\pi$
$311$	391.438			1.25864			0.629321	+	0.777145i	$0.283333\pi$
$312$		0			0
$313$	−156.982			−0.501540			−0.250770	−	0.968047i	$-0.580684\pi$
$313$	−156.982			−0.501540			−0.250770	+	0.968047i	$0.580684\pi$
$314$		0			0
$315$	−431.149			−1.36873
$316$		0			0
$317$	−145.018			−0.457471			−0.228736	−	0.973489i	$-0.573459\pi$
$317$	−145.018			−0.457471			−0.228736	+	0.973489i	$0.573459\pi$
$318$		0			0
$319$	33.9313	−	230.394i	0.106368	−	0.722238i
$320$		0			0
$321$		−	150.421i		−	0.468602i
$322$		0			0
$323$	91.9944			0.284812
$324$		0			0
$325$		−	422.176i		−	1.29900i
$326$		0			0
$327$	74.7474			0.228585
$328$		0			0
$329$			468.883i			1.42518i
$330$		0			0
$331$			386.070i			1.16637i	0.812338	+	0.583187i	$0.198195\pi$
$331$			386.070i			1.16637i	−0.812338	+	0.583187i	$0.801805\pi$
$332$		0			0
$333$	−209.903			−0.630339
$334$		0			0
$335$		−	140.827i		−	0.420379i
$336$		0			0
$337$		−	150.855i		−	0.447641i	−0.974630	−	0.223820i	$-0.928147\pi$
$337$		−	150.855i		−	0.447641i	0.974630	−	0.223820i	$-0.0718529\pi$
$338$		0			0
$339$	−64.3908			−0.189943
$340$		0			0
$341$		−	255.322i		−	0.748746i
$342$		0			0
$343$			175.466i			0.511561i
$344$		0			0
$345$		−	102.666i		−	0.297584i
$346$		0			0
$347$	32.6034			0.0939579			0.0469790	−	0.998896i	$-0.485041\pi$
$347$	32.6034			0.0939579			0.0469790	+	0.998896i	$0.485041\pi$
$348$		0			0
$349$			470.346i			1.34770i	0.738869	+	0.673849i	$0.235360\pi$
$349$			470.346i			1.34770i	−0.738869	+	0.673849i	$0.764640\pi$
$350$		0			0
$351$	488.162			1.39078
$352$		0			0
$353$	300.585			0.851515			0.425758	−	0.904837i	$-0.360008\pi$
$353$	300.585			0.851515			0.425758	+	0.904837i	$0.360008\pi$
$354$		0			0
$355$	−122.782			−0.345866
$356$		0			0
$357$			39.4774i			0.110581i
$358$		0			0
$359$	304.011			0.846827			0.423413	−	0.905937i	$-0.360832\pi$
$359$	304.011			0.846827			0.423413	+	0.905937i	$0.360832\pi$
$360$		0			0
$361$	−287.835			−0.797326
$362$		0			0
$363$		−	69.8781i		−	0.192502i
$364$		0			0
$365$	−688.036			−1.88503
$366$		0			0
$367$	−577.287			−1.57299			−0.786495	−	0.617597i	$-0.788106\pi$
$367$	−577.287			−1.57299			−0.786495	+	0.617597i	$0.788106\pi$
$368$		0			0
$369$			436.236i			1.18221i
$370$		0			0
$371$	684.872			1.84602
$372$		0			0
$373$			443.642i			1.18939i	0.803952	+	0.594694i	$0.202727\pi$
$373$			443.642i			1.18939i	−0.803952	+	0.594694i	$0.797273\pi$
$374$		0			0
$375$	59.6231			0.158995
$376$		0			0
$377$	101.280	−	687.692i	0.268647	−	1.82412i
$378$		0			0
$379$			293.481i			0.774357i	0.922005	+	0.387179i	$0.126550\pi$
$379$			293.481i			0.774357i	−0.922005	+	0.387179i	$0.873450\pi$
$380$		0			0
$381$			121.021i			0.317641i
$382$		0			0
$383$			202.204i			0.527949i	0.964530	+	0.263974i	$0.0850334\pi$
$383$			202.204i			0.527949i	−0.964530	+	0.263974i	$0.914967\pi$
$384$		0			0
$385$		−	463.419i		−	1.20369i
$386$		0			0
$387$		−	153.188i		−	0.395833i
$388$		0			0
$389$	127.641			0.328126			0.164063	−	0.986450i	$-0.447540\pi$
$389$	127.641			0.328126			0.164063	+	0.986450i	$0.447540\pi$
$390$		0			0
$391$	45.9373			0.117487
$392$		0			0
$393$	284.302			0.723414
$394$		0			0
$395$		−	116.356i		−	0.294571i
$396$		0			0
$397$			29.5557i			0.0744477i	0.999307	+	0.0372239i	$0.0118515\pi$
$397$			29.5557i			0.0744477i	−0.999307	+	0.0372239i	$0.988149\pi$
$398$		0			0
$399$		−	278.434i		−	0.697828i
$400$		0			0
$401$	273.221			0.681350			0.340675	−	0.940181i	$-0.389344\pi$
$401$	273.221			0.681350			0.340675	+	0.940181i	$0.389344\pi$
$402$		0			0
$403$		−	762.100i		−	1.89107i
$404$		0			0
$405$			274.549i			0.677899i
$406$		0			0
$407$		−	225.613i		−	0.554333i
$408$		0			0
$409$			116.481i			0.284795i	0.989810	+	0.142398i	$0.0454812\pi$
$409$			116.481i			0.284795i	−0.989810	+	0.142398i	$0.954519\pi$
$410$		0			0
$411$	100.167			0.243714
$412$		0			0
$413$		−	326.009i		−	0.789367i
$414$		0			0
$415$			3.69056i			0.00889292i
$416$		0			0
$417$			151.397i			0.363061i
$418$		0			0
$419$	341.609			0.815296			0.407648	−	0.913139i	$-0.366349\pi$
$419$	341.609			0.815296			0.407648	+	0.913139i	$0.366349\pi$
$420$		0			0
$421$	98.8121			0.234708			0.117354	−	0.993090i	$-0.462559\pi$
$421$	98.8121			0.234708			0.117354	+	0.993090i	$0.462559\pi$
$422$		0			0
$423$	396.262			0.936789
$424$		0			0
$425$		−	63.6110i		−	0.149673i
$426$		0			0
$427$			1010.64i			2.36683i
$428$		0			0
$429$			237.998i			0.554774i
$430$		0			0
$431$		−	256.201i		−	0.594434i	−0.954810	−	0.297217i	$-0.903942\pi$
$431$		−	256.201i		−	0.594434i	0.954810	−	0.297217i	$-0.0960584\pi$
$432$		0			0
$433$			258.832i			0.597764i	0.954290	+	0.298882i	$0.0966138\pi$
$433$			258.832i			0.597764i	−0.954290	+	0.298882i	$0.903386\pi$
$434$		0			0
$435$	−231.577	−	34.1056i	−0.532361	−	0.0784036i
$436$		0			0
$437$	−323.995			−0.741407
$438$		0			0
$439$			435.711i			0.992508i	0.868177	+	0.496254i	$0.165292\pi$
$439$			435.711i			0.992508i	−0.868177	+	0.496254i	$0.834708\pi$
$440$		0			0
$441$	−217.796			−0.493869
$442$		0			0
$443$			108.453i			0.244815i	0.992480	+	0.122408i	$0.0390615\pi$
$443$			108.453i			0.244815i	−0.992480	+	0.122408i	$0.960938\pi$
$444$		0			0
$445$	598.259			1.34440
$446$		0			0
$447$	−2.18496			−0.00488805
$448$		0			0
$449$			233.260i			0.519509i	0.965675	+	0.259755i	$0.0836417\pi$
$449$			233.260i			0.519509i	−0.965675	+	0.259755i	$0.916358\pi$
$450$		0			0
$451$	−468.887			−1.03966
$452$		0			0
$453$	−223.581			−0.493557
$454$		0			0
$455$		−	1383.24i		−	3.04008i
$456$		0			0
$457$	−248.835			−0.544497			−0.272249	−	0.962227i	$-0.587767\pi$
$457$	−248.835			−0.544497			−0.272249	+	0.962227i	$0.587767\pi$
$458$		0			0
$459$	73.5535			0.160247
$460$		0			0
$461$	378.704			0.821484			0.410742	−	0.911752i	$-0.365270\pi$
$461$	378.704			0.821484			0.410742	+	0.911752i	$0.365270\pi$
$462$		0			0
$463$			768.303i			1.65940i	0.558208	+	0.829701i	$0.311489\pi$
$463$			768.303i			1.65940i	−0.558208	+	0.829701i	$0.688511\pi$
$464$		0			0
$465$	−256.634			−0.551900
$466$		0			0
$467$		−	38.3026i		−	0.0820184i	−0.999159	−	0.0410092i	$-0.986943\pi$
$467$		−	38.3026i		−	0.0820184i	0.999159	−	0.0410092i	$-0.0130573\pi$
$468$		0			0
$469$		−	190.714i		−	0.406640i
$470$		0			0
$471$		−	66.4667i		−	0.141118i
$472$		0			0
$473$	164.653			0.348104
$474$		0			0
$475$			448.647i			0.944520i
$476$		0			0
$477$		−	578.798i		−	1.21341i
$478$		0			0
$479$	−660.995			−1.37995			−0.689973	−	0.723835i	$-0.742378\pi$
$479$	−660.995			−1.37995			−0.689973	+	0.723835i	$0.742378\pi$
$480$		0			0
$481$		−	673.423i		−	1.40005i
$482$		0			0
$483$		−	139.035i		−	0.287858i
$484$		0			0
$485$	561.659			1.15806
$486$		0			0
$487$		−	137.898i		−	0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$487$		−	137.898i		−	0.283158i	0.989927	−	0.141579i	$-0.0452180\pi$
$488$		0			0
$489$	289.620			0.592271
$490$		0			0
$491$		−	186.714i		−	0.380274i	−0.981758	−	0.190137i	$-0.939107\pi$
$491$		−	186.714i		−	0.380274i	0.981758	−	0.190137i	$-0.0608931\pi$
$492$		0			0
$493$	15.2603	−	103.617i	0.0309539	−	0.210177i
$494$		0			0
$495$	−391.644			−0.791199
$496$		0			0
$497$	−166.278			−0.334562
$498$		0			0
$499$	−342.947			−0.687268			−0.343634	−	0.939104i	$-0.611658\pi$
$499$	−342.947			−0.687268			−0.343634	+	0.939104i	$0.611658\pi$
$500$		0			0
$501$	276.903			0.552701
$502$		0			0
$503$	363.632			0.722926			0.361463	−	0.932386i	$-0.382277\pi$
$503$	363.632			0.722926			0.361463	+	0.932386i	$0.382277\pi$
$504$		0			0
$505$		−	87.4499i		−	0.173168i
$506$		0			0
$507$			501.425i			0.989004i
$508$		0			0
$509$			37.9726i			0.0746023i	0.999304	+	0.0373012i	$0.0118761\pi$
$509$			37.9726i			0.0746023i	−0.999304	+	0.0373012i	$0.988124\pi$
$510$		0			0
$511$	−931.769			−1.82342
$512$		0			0
$513$	−518.771			−1.01125
$514$		0			0
$515$	−804.761			−1.56264
$516$		0			0
$517$			425.920i			0.823830i
$518$		0			0
$519$	−98.4734			−0.189737
$520$		0			0
$521$	−406.563			−0.780351			−0.390175	−	0.920741i	$-0.627586\pi$
$521$	−406.563			−0.780351			−0.390175	+	0.920741i	$0.627586\pi$
$522$		0			0
$523$	153.421			0.293348			0.146674	−	0.989185i	$-0.453143\pi$
$523$	153.421			0.293348			0.146674	+	0.989185i	$0.453143\pi$
$524$		0			0
$525$	−192.527			−0.366719
$526$		0			0
$527$		−	114.829i		−	0.217891i
$528$		0			0
$529$	367.214			0.694166
$530$		0			0
$531$	−275.515			−0.518862
$532$		0			0
$533$	−1399.56			−2.62581
$534$		0			0
$535$			794.137i			1.48437i
$536$		0			0
$537$			269.743i			0.502315i
$538$		0			0
$539$		−	234.098i		−	0.434319i
$540$		0			0
$541$	−58.5232			−0.108176			−0.0540880	−	0.998536i	$-0.517225\pi$
$541$	−58.5232			−0.108176			−0.0540880	+	0.998536i	$0.517225\pi$
$542$		0			0
$543$	−107.221			−0.197461
$544$		0			0
$545$	−394.623			−0.724079
$546$		0			0
$547$	97.3827			0.178031			0.0890153	−	0.996030i	$-0.471628\pi$
$547$	97.3827			0.178031			0.0890153	+	0.996030i	$0.471628\pi$
$548$		0			0
$549$	854.107			1.55575
$550$		0			0
$551$	−107.630	+	730.812i	−0.195336	+	1.32634i
$552$		0			0
$553$		−	157.574i		−	0.284944i
$554$		0			0
$555$	−226.772			−0.408598
$556$		0			0
$557$		−	363.577i		−	0.652741i	−0.945242	−	0.326371i	$-0.894174\pi$
$557$		−	363.577i		−	0.652741i	0.945242	−	0.326371i	$-0.105826\pi$
$558$		0			0
$559$	491.465			0.879186
$560$		0			0
$561$			35.8602i			0.0639219i
$562$		0			0
$563$		−	247.139i		−	0.438968i	−0.975616	−	0.219484i	$-0.929563\pi$
$563$		−	247.139i		−	0.438968i	0.975616	−	0.219484i	$-0.0704374\pi$
$564$		0			0
$565$	339.947			0.601676
$566$		0			0
$567$			371.807i			0.655744i
$568$		0			0
$569$		−	1066.94i		−	1.87511i	−0.347834	−	0.937556i	$-0.613083\pi$
$569$		−	1066.94i		−	1.87511i	0.347834	−	0.937556i	$-0.386917\pi$
$570$		0			0
$571$	−11.3272			−0.0198375			−0.00991874	−	0.999951i	$-0.503157\pi$
$571$	−11.3272			−0.0198375			−0.00991874	+	0.999951i	$0.503157\pi$
$572$		0			0
$573$			147.169i			0.256839i
$574$		0			0
$575$			224.031i			0.389620i
$576$		0			0
$577$		−	754.505i		−	1.30763i	−0.756652	−	0.653817i	$-0.773166\pi$
$577$		−	754.505i		−	1.30763i	0.756652	−	0.653817i	$-0.226834\pi$
$578$		0			0
$579$	34.7205			0.0599664
$580$		0			0
$581$			4.99792i			0.00860228i
$582$		0			0
$583$	622.118			1.06710
$584$		0			0
$585$	−1169.00			−1.99829
$586$		0			0
$587$	−56.5736			−0.0963775			−0.0481887	−	0.998838i	$-0.515345\pi$
$587$	−56.5736			−0.0963775			−0.0481887	+	0.998838i	$0.515345\pi$
$588$		0			0
$589$			809.885i			1.37502i
$590$		0			0
$591$	−87.4986			−0.148052
$592$		0			0
$593$	218.323			0.368167			0.184083	−	0.982911i	$-0.441068\pi$
$593$	218.323			0.368167			0.184083	+	0.982911i	$0.441068\pi$
$594$		0			0
$595$		−	208.418i		−	0.350283i
$596$		0			0
$597$	−347.752			−0.582498
$598$		0			0
$599$	−44.2277			−0.0738358			−0.0369179	−	0.999318i	$-0.511754\pi$
$599$	−44.2277			−0.0738358			−0.0369179	+	0.999318i	$0.511754\pi$
$600$		0			0
$601$			717.250i			1.19343i	0.802454	+	0.596714i	$0.203527\pi$
$601$			717.250i			1.19343i	−0.802454	+	0.596714i	$0.796473\pi$
$602$		0			0
$603$	−161.176			−0.267290
$604$		0			0
$605$			368.916i			0.609779i
$606$		0			0
$607$	233.407			0.384526			0.192263	−	0.981343i	$-0.438417\pi$
$607$	233.407			0.384526			0.192263	+	0.981343i	$0.438417\pi$
$608$		0			0
$609$	−313.612	−	46.1873i	−0.514963	−	0.0758412i
$610$		0			0
$611$			1271.31i			2.08070i
$612$		0			0
$613$			186.431i			0.304130i	0.988371	+	0.152065i	$0.0485922\pi$
$613$			186.431i			0.304130i	−0.988371	+	0.152065i	$0.951408\pi$
$614$		0			0
$615$			471.295i			0.766333i
$616$		0			0
$617$		−	51.9925i		−	0.0842665i	−0.999112	−	0.0421333i	$-0.986585\pi$
$617$		−	51.9925i		−	0.0842665i	0.999112	−	0.0421333i	$-0.0134154\pi$
$618$		0			0
$619$			582.738i			0.941419i	0.882288	+	0.470709i	$0.156002\pi$
$619$			582.738i			0.941419i	−0.882288	+	0.470709i	$0.843998\pi$
$620$		0			0
$621$	−259.048			−0.417146
$622$		0			0
$623$	810.190			1.30047
$624$		0			0
$625$	−755.105			−1.20817
$626$		0			0
$627$		−	252.921i		−	0.403383i
$628$		0			0
$629$		−	101.467i		−	0.161315i
$630$		0			0
$631$			925.171i			1.46620i	0.680122	+	0.733099i	$0.261927\pi$
$631$			925.171i			1.46620i	−0.680122	+	0.733099i	$0.738073\pi$
$632$		0			0
$633$	311.497			0.492096
$634$		0			0
$635$		−	638.923i		−	1.00618i
$636$		0			0
$637$		−	698.747i		−	1.09693i
$638$		0			0
$639$			140.524i			0.219912i
$640$		0			0
$641$		−	1030.85i		−	1.60819i	−0.594498	−	0.804097i	$-0.702649\pi$
$641$		−	1030.85i		−	1.60819i	0.594498	−	0.804097i	$-0.297351\pi$
$642$		0			0
$643$	1173.19			1.82456			0.912278	−	0.409572i	$-0.134322\pi$
$643$	1173.19			1.82456			0.912278	+	0.409572i	$0.134322\pi$
$644$		0			0
$645$		−	165.499i		−	0.256587i
$646$		0			0
$647$		−	976.029i		−	1.50855i	−0.656561	−	0.754273i	$-0.727990\pi$
$647$		−	976.029i		−	1.50855i	0.656561	−	0.754273i	$-0.272010\pi$
$648$		0			0
$649$		−	296.137i		−	0.456297i
$650$		0			0
$651$	−347.545			−0.533863
$652$		0			0
$653$	528.389			0.809171			0.404586	−	0.914500i	$-0.367416\pi$
$653$	528.389			0.809171			0.404586	+	0.914500i	$0.367416\pi$
$654$		0			0
$655$	−1500.95			−2.29153
$656$		0			0
$657$			787.454i			1.19856i
$658$		0			0
$659$			835.795i			1.26828i	0.773219	+	0.634139i	$0.218645\pi$
$659$			835.795i			1.26828i	−0.773219	+	0.634139i	$0.781355\pi$
$660$		0			0
$661$			691.575i			1.04626i	0.852254	+	0.523128i	$0.175235\pi$
$661$			691.575i			1.04626i	−0.852254	+	0.523128i	$0.824765\pi$
$662$		0			0
$663$			107.037i			0.161444i
$664$		0			0
$665$			1469.97i			2.21048i
$666$		0			0
$667$	−53.7451	+	364.930i	−0.0805773	+	0.547121i
$668$		0			0
$669$	271.778			0.406245
$670$		0			0
$671$			918.033i			1.36816i
$672$		0			0
$673$	−556.526			−0.826933			−0.413466	−	0.910519i	$-0.635682\pi$
$673$	−556.526			−0.826933			−0.413466	+	0.910519i	$0.635682\pi$
$674$		0			0
$675$			358.712i			0.531426i
$676$		0			0
$677$	157.217			0.232225			0.116113	−	0.993236i	$-0.462957\pi$
$677$	157.217			0.232225			0.116113	+	0.993236i	$0.462957\pi$
$678$		0			0
$679$	760.624			1.12021
$680$		0			0
$681$		−	556.456i		−	0.817115i
$682$		0			0
$683$	−1109.74			−1.62480			−0.812401	−	0.583099i	$-0.801840\pi$
$683$	−1109.74			−1.62480			−0.812401	+	0.583099i	$0.801840\pi$
$684$		0			0
$685$	−528.823			−0.772004
$686$		0			0
$687$			437.787i			0.637244i
$688$		0			0
$689$	1856.93			2.69511
$690$		0			0
$691$	−749.369			−1.08447			−0.542235	−	0.840227i	$-0.682422\pi$
$691$	−749.369			−1.08447			−0.542235	+	0.840227i	$0.682422\pi$
$692$		0			0
$693$	−530.381			−0.765341
$694$		0			0
$695$		−	799.287i		−	1.15005i
$696$		0			0
$697$	−210.877			−0.302550
$698$		0			0
$699$		−	473.125i		−	0.676859i
$700$		0			0
$701$			684.233i			0.976081i	0.872821	+	0.488041i	$0.162288\pi$
$701$			684.233i			0.976081i	−0.872821	+	0.488041i	$0.837712\pi$
$702$		0			0
$703$			715.648i			1.01799i
$704$		0			0
$705$	428.107			0.607245
$706$		0			0
$707$		−	118.429i		−	0.167509i
$708$		0			0
$709$			173.047i			0.244073i	0.992526	+	0.122036i	$0.0389424\pi$
$709$			173.047i			0.244073i	−0.992526	+	0.122036i	$0.961058\pi$
$710$		0			0
$711$	−133.168			−0.187297
$712$		0			0
$713$			404.415i			0.567202i
$714$		0			0
$715$		−	1256.49i		−	1.75733i
$716$		0			0
$717$	103.542			0.144410
$718$		0			0
$719$		−	947.226i		−	1.31742i	−0.752396	−	0.658711i	$-0.771102\pi$
$719$		−	947.226i		−	1.31742i	0.752396	−	0.658711i	$-0.228898\pi$
$720$		0			0
$721$	−1089.84			−1.51157
$722$		0			0
$723$			300.891i			0.416170i
$724$		0			0
$725$	505.331	+	74.4228i	0.697009	+	0.102652i
$726$		0			0
$727$	754.844			1.03830			0.519150	−	0.854683i	$-0.326249\pi$
$727$	754.844			1.03830			0.519150	+	0.854683i	$0.326249\pi$
$728$		0			0
$729$	87.5806			0.120138
$730$		0			0
$731$	74.0511			0.101301
$732$		0			0
$733$	−295.912			−0.403700			−0.201850	−	0.979416i	$-0.564695\pi$
$733$	−295.912			−0.403700			−0.201850	+	0.979416i	$0.564695\pi$
$734$		0			0
$735$	−235.300			−0.320136
$736$		0			0
$737$		−	173.239i		−	0.235060i
$738$		0			0
$739$		−	568.516i		−	0.769304i	−0.923062	−	0.384652i	$-0.874321\pi$
$739$		−	568.516i		−	0.769304i	0.923062	−	0.384652i	$-0.125679\pi$
$740$		0			0
$741$		−	754.932i		−	1.01880i
$742$		0			0
$743$	210.064			0.282724			0.141362	−	0.989958i	$-0.454852\pi$
$743$	210.064			0.282724			0.141362	+	0.989958i	$0.454852\pi$
$744$		0			0
$745$	11.5353			0.0154837
$746$		0			0
$747$	4.22383			0.00565439
$748$		0			0
$749$			1075.46i			1.43586i
$750$		0			0
$751$	−319.921			−0.425993			−0.212997	−	0.977053i	$-0.568322\pi$
$751$	−319.921			−0.425993			−0.212997	+	0.977053i	$0.568322\pi$
$752$		0			0
$753$	−387.459			−0.514553
$754$		0			0
$755$	1180.38			1.56342
$756$		0			0
$757$	1121.45			1.48144			0.740718	−	0.671816i	$-0.234485\pi$
$757$	1121.45			1.48144			0.740718	+	0.671816i	$0.234485\pi$
$758$		0			0
$759$		−	126.296i		−	0.166398i
$760$		0			0
$761$	−239.760			−0.315059			−0.157529	−	0.987514i	$-0.550353\pi$
$761$	−239.760			−0.315059			−0.157529	+	0.987514i	$0.550353\pi$
$762$		0			0
$763$	−534.417			−0.700415
$764$		0			0
$765$	−176.138			−0.230246
$766$		0			0
$767$		−	883.925i		−	1.15244i
$768$		0			0
$769$			337.494i			0.438874i	0.975627	+	0.219437i	$0.0704221\pi$
$769$			337.494i			0.438874i	−0.975627	+	0.219437i	$0.929578\pi$
$770$		0			0
$771$		−	126.925i		−	0.164624i
$772$		0			0
$773$	−411.788			−0.532714			−0.266357	−	0.963874i	$-0.585820\pi$
$773$	−411.788			−0.532714			−0.266357	+	0.963874i	$0.585820\pi$
$774$		0			0
$775$	560.008			0.722591
$776$		0			0
$777$	−307.105			−0.395244
$778$		0			0
$779$	1487.31			1.90926
$780$		0			0
$781$	−151.042			−0.193395
$782$		0			0
$783$	−86.0551	+	584.315i	−0.109904	+	0.746252i
$784$		0			0
$785$			350.906i			0.447014i
$786$		0			0
$787$	33.0333			0.0419737			0.0209869	−	0.999780i	$-0.493319\pi$
$787$	33.0333			0.0419737			0.0209869	+	0.999780i	$0.493319\pi$
$788$		0			0
$789$			57.0580i			0.0723168i
$790$		0			0
$791$	460.371			0.582012
$792$		0			0
$793$			2740.19i			3.45548i
$794$		0			0
$795$		−	625.313i		−	0.786557i
$796$		0			0
$797$	358.483			0.449791			0.224895	−	0.974383i	$-0.427796\pi$
$797$	358.483			0.449791			0.224895	+	0.974383i	$0.427796\pi$
$798$		0			0
$799$			191.554i			0.239742i
$800$		0			0
$801$		−	684.706i		−	0.854813i
$802$		0			0
$803$	−846.393			−1.05404
$804$		0			0
$805$			734.027i			0.911835i
$806$		0			0
$807$		−	500.217i		−	0.619847i
$808$		0			0
$809$			556.938i			0.688428i	0.938891	+	0.344214i	$0.111855\pi$
$809$			556.938i			0.688428i	−0.938891	+	0.344214i	$0.888145\pi$
$810$		0			0
$811$	−295.938			−0.364905			−0.182453	−	0.983215i	$-0.558404\pi$
$811$	−295.938			−0.364905			−0.182453	+	0.983215i	$0.558404\pi$
$812$		0			0
$813$			342.802i			0.421651i
$814$		0			0
$815$	−1529.03			−1.87611
$816$		0			0
$817$	−522.281			−0.639267
$818$		0			0
$819$	−1583.11			−1.93298
$820$		0			0
$821$			739.559i			0.900803i	0.892826	+	0.450402i	$0.148719\pi$
$821$			739.559i			0.900803i	−0.892826	+	0.450402i	$0.851281\pi$
$822$		0			0
$823$	−653.161			−0.793634			−0.396817	−	0.917898i	$-0.629885\pi$
$823$	−653.161			−0.793634			−0.396817	+	0.917898i	$0.629885\pi$
$824$		0			0
$825$	−174.886			−0.211983
$826$		0			0
$827$		−	897.006i		−	1.08465i	−0.840169	−	0.542325i	$-0.817544\pi$
$827$		−	897.006i		−	1.08465i	0.840169	−	0.542325i	$-0.182456\pi$
$828$		0			0
$829$	−349.807			−0.421962			−0.210981	−	0.977490i	$-0.567666\pi$
$829$	−349.807			−0.421962			−0.210981	+	0.977490i	$0.567666\pi$
$830$		0			0
$831$	−279.104			−0.335865
$832$		0			0
$833$		−	105.283i		−	0.126390i
$834$		0			0
$835$	−1461.89			−1.75077
$836$		0			0
$837$			647.538i			0.773641i
$838$		0			0
$839$	426.644			0.508515			0.254258	−	0.967137i	$-0.418169\pi$
$839$	426.644			0.508515			0.254258	+	0.967137i	$0.418169\pi$
$840$		0			0
$841$	805.292	+	242.458i	0.957541	+	0.288297i
$842$		0			0
$843$			80.8806i			0.0959438i
$844$		0			0
$845$		−	2647.24i		−	3.13282i
$846$		0			0
$847$			499.603i			0.589851i
$848$		0			0
$849$		−	637.365i		−	0.750725i
$850$		0			0
$851$			357.358i			0.419927i
$852$		0			0
$853$	785.349			0.920690			0.460345	−	0.887740i	$-0.347726\pi$
$853$	785.349			0.920690			0.460345	+	0.887740i	$0.347726\pi$
$854$		0			0
$855$	1242.30			1.45298
$856$		0			0
$857$	1087.79			1.26930			0.634652	−	0.772798i	$-0.281144\pi$
$857$	1087.79			1.26930			0.634652	+	0.772798i	$0.281144\pi$
$858$		0			0
$859$			559.776i			0.651660i	0.945428	+	0.325830i	$0.105644\pi$
$859$			559.776i			0.651660i	−0.945428	+	0.325830i	$0.894356\pi$
$860$		0			0
$861$			638.249i			0.741288i
$862$		0			0
$863$		−	1011.37i		−	1.17193i	−0.810337	−	0.585964i	$-0.800716\pi$
$863$		−	1011.37i		−	1.17193i	0.810337	−	0.585964i	$-0.199284\pi$
$864$		0			0
$865$	519.883			0.601021
$866$		0			0
$867$		−	341.213i		−	0.393556i
$868$		0			0
$869$		−	143.136i		−	0.164713i
$870$		0			0
$871$		−	517.094i		−	0.593678i
$872$		0			0
$873$		−	642.816i		−	0.736330i
$874$		0			0
$875$	−426.284			−0.487181
$876$		0			0
$877$		−	816.846i		−	0.931409i	−0.884940	−	0.465704i	$-0.845801\pi$
$877$		−	816.846i		−	0.931409i	0.884940	−	0.465704i	$-0.154199\pi$
$878$		0			0
$879$		−	210.490i		−	0.239466i
$880$		0			0
$881$			916.922i			1.04077i	0.853931	+	0.520387i	$0.174212\pi$
$881$			916.922i			1.04077i	−0.853931	+	0.520387i	$0.825788\pi$
$882$		0			0
$883$	−166.543			−0.188611			−0.0943054	−	0.995543i	$-0.530063\pi$
$883$	−166.543			−0.188611			−0.0943054	+	0.995543i	$0.530063\pi$
$884$		0			0
$885$	−297.658			−0.336336
$886$		0			0
$887$	−557.936			−0.629014			−0.314507	−	0.949255i	$-0.601839\pi$
$887$	−557.936			−0.629014			−0.314507	+	0.949255i	$0.601839\pi$
$888$		0			0
$889$		−	865.259i		−	0.973294i
$890$		0			0
$891$			337.739i			0.379056i
$892$		0			0
$893$		−	1351.02i		−	1.51290i
$894$		0			0
$895$		−	1424.09i		−	1.59116i
$896$		0			0
$897$		−	376.974i		−	0.420261i
$898$		0			0
$899$	912.210	+	134.346i	1.01469	+	0.149439i
$900$		0			0
$901$	279.792			0.310535
$902$		0			0
$903$		−	224.126i		−	0.248201i
$904$		0			0
$905$	566.066			0.625488
$906$		0			0
$907$		−	136.822i		−	0.150851i	−0.997151	−	0.0754257i	$-0.975968\pi$
$907$		−	136.822i		−	0.150851i	0.997151	−	0.0754257i	$-0.0240316\pi$
$908$		0			0
$909$	−100.086			−0.110106
$910$		0			0
$911$	1231.76			1.35210			0.676048	−	0.736858i	$-0.263691\pi$
$911$	1231.76			1.35210			0.676048	+	0.736858i	$0.263691\pi$
$912$		0			0
$913$			4.53997i			0.00497259i
$914$		0			0
$915$	922.748			1.00847
$916$		0			0
$917$	−2032.65			−2.21663
$918$		0			0
$919$		−	977.438i		−	1.06359i	−0.846873	−	0.531794i	$-0.821518\pi$
$919$		−	977.438i		−	1.06359i	0.846873	−	0.531794i	$-0.178482\pi$
$920$		0			0
$921$	−626.060			−0.679761
$922$		0			0
$923$	−450.837			−0.488448
$924$		0			0
$925$	494.846			0.534969
$926$		0			0
$927$			921.046i			0.993578i
$928$		0			0
$929$	−1547.17			−1.66542			−0.832708	−	0.553713i	$-0.813211\pi$
$929$	−1547.17			−1.66542			−0.832708	+	0.553713i	$0.813211\pi$
$930$		0			0
$931$			742.560i			0.797594i
$932$		0			0
$933$		−	484.003i		−	0.518760i
$934$		0			0
$935$		−	189.321i		−	0.202483i
$936$		0			0
$937$	73.0644			0.0779770			0.0389885	−	0.999240i	$-0.487586\pi$
$937$	73.0644			0.0779770			0.0389885	+	0.999240i	$0.487586\pi$
$938$		0			0
$939$			194.104i			0.206714i
$940$		0			0
$941$			189.365i			0.201238i	0.994925	+	0.100619i	$0.0320823\pi$
$941$			189.365i			0.201238i	−0.994925	+	0.100619i	$0.967918\pi$
$942$		0			0
$943$	742.688			0.787580
$944$		0			0
$945$			1175.30i			1.24371i
$946$		0			0
$947$		−	548.657i		−	0.579364i	−0.957123	−	0.289682i	$-0.906451\pi$
$947$		−	548.657i		−	0.579364i	0.957123	−	0.289682i	$-0.0935495\pi$
$948$		0			0
$949$	−2526.36			−2.66213
$950$		0			0
$951$			179.312i			0.188551i
$952$		0			0
$953$	−661.587			−0.694215			−0.347107	−	0.937825i	$-0.612836\pi$
$953$	−661.587			−0.694215			−0.347107	+	0.937825i	$0.612836\pi$
$954$		0			0
$955$		−	776.966i		−	0.813577i
$956$		0			0
$957$	−284.876	−	41.9552i	−0.297676	−	0.0438403i
$958$		0			0
$959$	−716.155			−0.746773
$960$		0			0
$961$	49.9103			0.0519358
$962$		0			0
$963$	908.887			0.943808
$964$		0			0
$965$	−183.305			−0.189953
$966$		0			0
$967$	45.5659			0.0471209			0.0235605	−	0.999722i	$-0.492500\pi$
$967$	45.5659			0.0471209			0.0235605	+	0.999722i	$0.492500\pi$
$968$		0			0
$969$		−	113.749i		−	0.117388i
$970$		0			0
$971$		−	279.275i		−	0.287616i	−0.989606	−	0.143808i	$-0.954065\pi$
$971$		−	279.275i		−	0.287616i	0.989606	−	0.143808i	$-0.0459348\pi$
$972$		0			0
$973$		−	1082.43i		−	1.11247i
$974$		0			0
$975$	−522.010			−0.535395
$976$		0			0
$977$	1194.22			1.22234			0.611168	−	0.791501i	$-0.290700\pi$
$977$	1194.22			1.22234			0.611168	+	0.791501i	$0.290700\pi$
$978$		0			0
$979$	735.953			0.751740
$980$		0			0
$981$			451.645i			0.460392i
$982$		0			0
$983$	1876.04			1.90849			0.954244	−	0.299030i	$-0.0966632\pi$
$983$	1876.04			1.90849			0.954244	+	0.299030i	$0.0966632\pi$
$984$		0			0
$985$	461.943			0.468977
$986$		0			0
$987$	579.763			0.587399
$988$		0			0
$989$	−260.800			−0.263701
$990$		0			0
$991$			1206.19i			1.21715i	0.793497	+	0.608574i	$0.208258\pi$
$991$			1206.19i			1.21715i	−0.793497	+	0.608574i	$0.791742\pi$
$992$		0			0
$993$	477.366			0.480731
$994$		0			0
$995$	1835.93			1.84515
$996$		0			0
$997$	1603.13			1.60795			0.803977	−	0.594660i	$-0.202713\pi$
$997$	1603.13			1.60795			0.803977	+	0.594660i	$0.202713\pi$
$998$		0			0
$999$			572.191i			0.572763i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	928.3.b.c.463.24		56
4.3	odd	2		232.3.b.c.115.11	✓	56
8.3	odd	2	inner	928.3.b.c.463.23		56
8.5	even	2		232.3.b.c.115.45	yes	56
29.28	even	2	inner	928.3.b.c.463.34		56
116.115	odd	2		232.3.b.c.115.46	yes	56
232.115	odd	2	inner	928.3.b.c.463.33		56
232.173	even	2		232.3.b.c.115.12	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.3.b.c.115.11	✓	56	4.3	odd	2
232.3.b.c.115.12	yes	56	232.173	even	2
232.3.b.c.115.45	yes	56	8.5	even	2
232.3.b.c.115.46	yes	56	116.115	odd	2
928.3.b.c.463.23		56	8.3	odd	2	inner
928.3.b.c.463.24		56	1.1	even	1	trivial
928.3.b.c.463.33		56	232.115	odd	2	inner
928.3.b.c.463.34		56	29.28	even	2	inner

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:	\( N \)	\(=\)	\( 928 = 2^{5} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	928.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.23647i q^{3} +6.52788i q^{5} +8.84035i q^{7} +7.47113 q^{9} +O(q^{10})\)	\(q-1.23647i q^{3} +6.52788i q^{5} +8.84035i q^{7} +7.47113 q^{9} +8.03032i q^{11} +23.9693i q^{13} +8.07155 q^{15} +3.61156i q^{17} -25.4722i q^{19} +10.9309 q^{21} -12.7195i q^{23} -17.6132 q^{25} -20.3661i q^{27} +(-28.6905 - 4.22540i) q^{29} -31.7948 q^{31} +9.92928 q^{33} -57.7087 q^{35} -28.0952 q^{37} +29.6374 q^{39} +58.3896i q^{41} -20.5039i q^{43} +48.7706i q^{45} +53.0390 q^{47} -29.1517 q^{49} +4.46560 q^{51} -77.4712i q^{53} -52.4209 q^{55} -31.4958 q^{57} -36.8773 q^{59} +114.321 q^{61} +66.0474i q^{63} -156.469 q^{65} -21.5732 q^{67} -15.7274 q^{69} +18.8089i q^{71} +105.400i q^{73} +21.7783i q^{75} -70.9908 q^{77} -17.8244 q^{79} +42.0580 q^{81} +0.565354 q^{83} -23.5758 q^{85} +(-5.22460 + 35.4751i) q^{87} -91.6468i q^{89} -211.897 q^{91} +39.3135i q^{93} +166.280 q^{95} -86.0400i q^{97} +59.9955i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 184 q^{9}+O(q^{10}) \) 56 * q - 184 * q^9	\( 56 q - 184 q^{9} - 280 q^{25} + 32 q^{33} - 96 q^{35} - 424 q^{49} - 32 q^{51} + 32 q^{57} - 512 q^{59} + 192 q^{65} - 352 q^{67} + 280 q^{81} - 128 q^{83} - 96 q^{91}+O(q^{100}) \) 56 * q - 184 * q^9 - 280 * q^25 + 32 * q^33 - 96 * q^35 - 424 * q^49 - 32 * q^51 + 32 * q^57 - 512 * q^59 + 192 * q^65 - 352 * q^67 + 280 * q^81 - 128 * q^83 - 96 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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