LMFDB - Embedded newform 4368.2.h.o.337.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 273)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.4
Root			$-0.854638 - 0.854638i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.o.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.630898i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.630898i q^{5} +1.00000i q^{7} +1.00000 q^{9} +5.70928i q^{11} +(-2.17009 - 2.87936i) q^{13} -0.630898i q^{15} +1.07838 q^{17} -7.41855i q^{19} -1.00000i q^{21} -5.41855 q^{23} +4.60197 q^{25} -1.00000 q^{27} +6.68035 q^{29} -6.15676i q^{31} -5.70928i q^{33} -0.630898 q^{35} +3.41855i q^{37} +(2.17009 + 2.87936i) q^{39} +1.21235i q^{41} +12.6803 q^{43} +0.630898i q^{45} -6.04945i q^{47} -1.00000 q^{49} -1.07838 q^{51} -6.00000 q^{53} -3.60197 q^{55} +7.41855i q^{57} +1.36910i q^{59} +12.6537 q^{61} +1.00000i q^{63} +(1.81658 - 1.36910i) q^{65} -0.581449i q^{67} +5.41855 q^{69} +4.81432i q^{71} -3.81658i q^{73} -4.60197 q^{75} -5.70928 q^{77} +1.65983 q^{79} +1.00000 q^{81} +12.8865i q^{83} +0.680346i q^{85} -6.68035 q^{87} +6.20620i q^{89} +(2.87936 - 2.17009i) q^{91} +6.15676i q^{93} +4.68035 q^{95} -1.07838i q^{97} +5.70928i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{9} - 2 q^{13} - 4 q^{23} - 10 q^{25} - 6 q^{27} - 4 q^{29} + 4 q^{35} + 2 q^{39} + 32 q^{43} - 6 q^{49} - 36 q^{53} + 16 q^{55} + 28 q^{61} + 20 q^{65} + 4 q^{69} + 10 q^{75} - 20 q^{77} + 32 q^{79} + 6 q^{81} + 4 q^{87} - 8 q^{91} - 16 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^9 - 2 * q^13 - 4 * q^23 - 10 * q^25 - 6 * q^27 - 4 * q^29 + 4 * q^35 + 2 * q^39 + 32 * q^43 - 6 * q^49 - 36 * q^53 + 16 * q^55 + 28 * q^61 + 20 * q^65 + 4 * q^69 + 10 * q^75 - 20 * q^77 + 32 * q^79 + 6 * q^81 + 4 * q^87 - 8 * q^91 - 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			0.630898i			0.282146i	0.989999	+	0.141073i	$0.0450552\pi$
$5$			0.630898i			0.282146i	−0.989999	+	0.141073i	$0.954945\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			5.70928i			1.72141i	0.509103	+	0.860706i	$0.329977\pi$
$11$			5.70928i			1.72141i	−0.509103	+	0.860706i	$0.670023\pi$
$12$		0			0
$13$	−2.17009	−	2.87936i	−0.601874	−	0.798591i
$13$	−2.17009	−	2.87936i	−0.601874	−	0.798591i
$14$		0			0
$15$		−	0.630898i		−	0.162897i
$16$		0			0
$17$	1.07838			0.261545			0.130773	−	0.991412i	$-0.458254\pi$
$17$	1.07838			0.261545			0.130773	+	0.991412i	$0.458254\pi$
$18$		0			0
$19$		−	7.41855i		−	1.70193i	−0.525221	−	0.850966i	$-0.676017\pi$
$19$		−	7.41855i		−	1.70193i	0.525221	−	0.850966i	$-0.323983\pi$
$20$		0			0
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$	−5.41855			−1.12985			−0.564923	−	0.825144i	$-0.691094\pi$
$23$	−5.41855			−1.12985			−0.564923	+	0.825144i	$0.691094\pi$
$24$		0			0
$25$	4.60197			0.920394
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	6.68035			1.24051			0.620255	−	0.784401i	$-0.287029\pi$
$29$	6.68035			1.24051			0.620255	+	0.784401i	$0.287029\pi$
$30$		0			0
$31$		−	6.15676i		−	1.10579i	−0.833252	−	0.552893i	$-0.813524\pi$
$31$		−	6.15676i		−	1.10579i	0.833252	−	0.552893i	$-0.186476\pi$
$32$		0			0
$33$		−	5.70928i		−	0.993857i
$34$		0			0
$35$	−0.630898			−0.106641
$36$		0			0
$37$			3.41855i			0.562006i	0.959707	+	0.281003i	$0.0906671\pi$
$37$			3.41855i			0.562006i	−0.959707	+	0.281003i	$0.909333\pi$
$38$		0			0
$39$	2.17009	+	2.87936i	0.347492	+	0.461067i
$40$		0			0
$41$			1.21235i			0.189337i	0.995509	+	0.0946684i	$0.0301791\pi$
$41$			1.21235i			0.189337i	−0.995509	+	0.0946684i	$0.969821\pi$
$42$		0			0
$43$	12.6803			1.93373			0.966867	−	0.255279i	$-0.0821675\pi$
$43$	12.6803			1.93373			0.966867	+	0.255279i	$0.0821675\pi$
$44$		0			0
$45$			0.630898i			0.0940487i
$46$		0			0
$47$		−	6.04945i		−	0.882403i	−0.897408	−	0.441201i	$-0.854552\pi$
$47$		−	6.04945i		−	0.882403i	0.897408	−	0.441201i	$-0.145448\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−1.07838			−0.151003
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	−3.60197			−0.485689
$56$		0			0
$57$			7.41855i			0.982611i
$58$		0			0
$59$			1.36910i			0.178242i	0.996021	+	0.0891210i	$0.0284058\pi$
$59$			1.36910i			0.178242i	−0.996021	+	0.0891210i	$0.971594\pi$
$60$		0			0
$61$	12.6537			1.62014			0.810069	−	0.586334i	$-0.199430\pi$
$61$	12.6537			1.62014			0.810069	+	0.586334i	$0.199430\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	1.81658	−	1.36910i	0.225319	−	0.169816i
$66$		0			0
$67$		−	0.581449i		−	0.0710353i	−0.999369	−	0.0355177i	$-0.988692\pi$
$67$		−	0.581449i		−	0.0710353i	0.999369	−	0.0355177i	$-0.0113080\pi$
$68$		0			0
$69$	5.41855			0.652317
$70$		0			0
$71$			4.81432i			0.571354i	0.958326	+	0.285677i	$0.0922185\pi$
$71$			4.81432i			0.571354i	−0.958326	+	0.285677i	$0.907782\pi$
$72$		0			0
$73$		−	3.81658i		−	0.446697i	−0.974739	−	0.223349i	$-0.928301\pi$
$73$		−	3.81658i		−	0.446697i	0.974739	−	0.223349i	$-0.0716988\pi$
$74$		0			0
$75$	−4.60197			−0.531390
$76$		0			0
$77$	−5.70928			−0.650632
$78$		0			0
$79$	1.65983			0.186745			0.0933726	−	0.995631i	$-0.470235\pi$
$79$	1.65983			0.186745			0.0933726	+	0.995631i	$0.470235\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			12.8865i			1.41448i	0.706972	+	0.707241i	$0.250061\pi$
$83$			12.8865i			1.41448i	−0.706972	+	0.707241i	$0.749939\pi$
$84$		0			0
$85$			0.680346i			0.0737939i
$86$		0			0
$87$	−6.68035			−0.716208
$88$		0			0
$89$			6.20620i			0.657856i	0.944355	+	0.328928i	$0.106687\pi$
$89$			6.20620i			0.657856i	−0.944355	+	0.328928i	$0.893313\pi$
$90$		0			0
$91$	2.87936	−	2.17009i	0.301839	−	0.227487i
$92$		0			0
$93$			6.15676i			0.638426i
$94$		0			0
$95$	4.68035			0.480193
$96$		0			0
$97$		−	1.07838i		−	0.109493i	−0.998500	−	0.0547463i	$-0.982565\pi$
$97$		−	1.07838i		−	0.109493i	0.998500	−	0.0547463i	$-0.0174350\pi$
$98$		0			0
$99$			5.70928i			0.573804i
$100$		0			0
$101$	−7.23513			−0.719923			−0.359961	−	0.932967i	$-0.617210\pi$
$101$	−7.23513			−0.719923			−0.359961	+	0.932967i	$0.617210\pi$
$102$		0			0
$103$	−13.7587			−1.35569			−0.677844	−	0.735206i	$-0.737085\pi$
$103$	−13.7587			−1.35569			−0.677844	+	0.735206i	$0.737085\pi$
$104$		0			0
$105$	0.630898			0.0615693
$106$		0			0
$107$	10.0989			0.976297			0.488149	−	0.872761i	$-0.337672\pi$
$107$	10.0989			0.976297			0.488149	+	0.872761i	$0.337672\pi$
$108$		0			0
$109$			17.9421i			1.71855i	0.511518	+	0.859273i	$0.329083\pi$
$109$			17.9421i			1.71855i	−0.511518	+	0.859273i	$0.670917\pi$
$110$		0			0
$111$		−	3.41855i		−	0.324474i
$112$		0			0
$113$	10.6803			1.00472			0.502361	−	0.864658i	$-0.332465\pi$
$113$	10.6803			1.00472			0.502361	+	0.864658i	$0.332465\pi$
$114$		0			0
$115$		−	3.41855i		−	0.318781i
$116$		0			0
$117$	−2.17009	−	2.87936i	−0.200625	−	0.266197i
$118$		0			0
$119$			1.07838i			0.0988547i
$120$		0			0
$121$	−21.5958			−1.96326
$122$		0			0
$123$		−	1.21235i		−	0.109314i
$124$		0			0
$125$			6.05786i			0.541831i
$126$		0			0
$127$	−6.34017			−0.562599			−0.281300	−	0.959620i	$-0.590765\pi$
$127$	−6.34017			−0.562599			−0.281300	+	0.959620i	$0.590765\pi$
$128$		0			0
$129$	−12.6803			−1.11644
$130$		0			0
$131$	7.51745			0.656802			0.328401	−	0.944538i	$-0.393490\pi$
$131$	7.51745			0.656802			0.328401	+	0.944538i	$0.393490\pi$
$132$		0			0
$133$	7.41855			0.643270
$134$		0			0
$135$		−	0.630898i		−	0.0542990i
$136$		0			0
$137$		−	4.47414i		−	0.382252i	−0.981566	−	0.191126i	$-0.938786\pi$
$137$		−	4.47414i		−	0.382252i	0.981566	−	0.191126i	$-0.0612139\pi$
$138$		0			0
$139$	9.75872			0.827724			0.413862	−	0.910340i	$-0.364180\pi$
$139$	9.75872			0.827724			0.413862	+	0.910340i	$0.364180\pi$
$140$		0			0
$141$			6.04945i			0.509455i
$142$		0			0
$143$	16.4391	−	12.3896i	1.37470	−	1.03607i
$144$		0			0
$145$			4.21461i			0.350005i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$			1.46800i			0.120263i	0.998190	+	0.0601316i	$0.0191520\pi$
$149$			1.46800i			0.120263i	−0.998190	+	0.0601316i	$0.980848\pi$
$150$		0			0
$151$			13.3607i			1.08728i	0.839319	+	0.543639i	$0.182954\pi$
$151$			13.3607i			1.08728i	−0.839319	+	0.543639i	$0.817046\pi$
$152$		0			0
$153$	1.07838			0.0871817
$154$		0			0
$155$	3.88428			0.311993
$156$		0			0
$157$	−8.15676			−0.650980			−0.325490	−	0.945545i	$-0.605529\pi$
$157$	−8.15676			−0.650980			−0.325490	+	0.945545i	$0.605529\pi$
$158$		0			0
$159$	6.00000			0.475831
$160$		0			0
$161$		−	5.41855i		−	0.427042i
$162$		0			0
$163$			4.58145i			0.358847i	0.983772	+	0.179423i	$0.0574232\pi$
$163$			4.58145i			0.358847i	−0.983772	+	0.179423i	$0.942577\pi$
$164$		0			0
$165$	3.60197			0.280413
$166$		0			0
$167$			7.31124i			0.565761i	0.959155	+	0.282881i	$0.0912900\pi$
$167$			7.31124i			0.565761i	−0.959155	+	0.282881i	$0.908710\pi$
$168$		0			0
$169$	−3.58145	+	12.4969i	−0.275496	+	0.961302i
$170$		0			0
$171$		−	7.41855i		−	0.567311i
$172$		0			0
$173$	7.60197			0.577967			0.288983	−	0.957334i	$-0.406683\pi$
$173$	7.60197			0.577967			0.288983	+	0.957334i	$0.406683\pi$
$174$		0			0
$175$			4.60197i			0.347876i
$176$		0			0
$177$		−	1.36910i		−	0.102908i
$178$		0			0
$179$	−9.10504			−0.680543			−0.340271	−	0.940327i	$-0.610519\pi$
$179$	−9.10504			−0.680543			−0.340271	+	0.940327i	$0.610519\pi$
$180$		0			0
$181$	15.3607			1.14175			0.570876	−	0.821037i	$-0.306604\pi$
$181$	15.3607			1.14175			0.570876	+	0.821037i	$0.306604\pi$
$182$		0			0
$183$	−12.6537			−0.935387
$184$		0			0
$185$	−2.15676			−0.158568
$186$		0			0
$187$			6.15676i			0.450227i
$188$		0			0
$189$		−	1.00000i		−	0.0727393i
$190$		0			0
$191$	24.5692			1.77776			0.888881	−	0.458138i	$-0.151483\pi$
$191$	24.5692			1.77776			0.888881	+	0.458138i	$0.151483\pi$
$192$		0			0
$193$		−	13.5753i		−	0.977172i	−0.872516	−	0.488586i	$-0.837513\pi$
$193$		−	13.5753i		−	0.977172i	0.872516	−	0.488586i	$-0.162487\pi$
$194$		0			0
$195$	−1.81658	+	1.36910i	−0.130088	+	0.0980435i
$196$		0			0
$197$		−	2.94441i		−	0.209780i	−0.994484	−	0.104890i	$-0.966551\pi$
$197$		−	2.94441i		−	0.209780i	0.994484	−	0.104890i	$-0.0334491\pi$
$198$		0			0
$199$	10.5236			0.745998			0.372999	−	0.927832i	$-0.378330\pi$
$199$	10.5236			0.745998			0.372999	+	0.927832i	$0.378330\pi$
$200$		0			0
$201$			0.581449i			0.0410123i
$202$		0			0
$203$			6.68035i			0.468868i
$204$		0			0
$205$	−0.764867			−0.0534206
$206$		0			0
$207$	−5.41855			−0.376615
$208$		0			0
$209$	42.3545			2.92973
$210$		0			0
$211$	3.02052			0.207941			0.103971	−	0.994580i	$-0.466845\pi$
$211$	3.02052			0.207941			0.103971	+	0.994580i	$0.466845\pi$
$212$		0			0
$213$		−	4.81432i		−	0.329871i
$214$		0			0
$215$			8.00000i			0.545595i
$216$		0			0
$217$	6.15676			0.417948
$218$		0			0
$219$			3.81658i			0.257901i
$220$		0			0
$221$	−2.34017	−	3.10504i	−0.157417	−	0.208868i
$222$		0			0
$223$		−	22.6225i		−	1.51491i	−0.652885	−	0.757457i	$-0.726442\pi$
$223$		−	22.6225i		−	1.51491i	0.652885	−	0.757457i	$-0.273558\pi$
$224$		0			0
$225$	4.60197			0.306798
$226$		0			0
$227$			22.9444i			1.52287i	0.648239	+	0.761437i	$0.275506\pi$
$227$			22.9444i			1.52287i	−0.648239	+	0.761437i	$0.724494\pi$
$228$		0			0
$229$			21.5441i			1.42367i	0.702344	+	0.711837i	$0.252137\pi$
$229$			21.5441i			1.42367i	−0.702344	+	0.711837i	$0.747863\pi$
$230$		0			0
$231$	5.70928			0.375643
$232$		0			0
$233$	7.36069			0.482215			0.241107	−	0.970498i	$-0.422489\pi$
$233$	7.36069			0.482215			0.241107	+	0.970498i	$0.422489\pi$
$234$		0			0
$235$	3.81658			0.248966
$236$		0			0
$237$	−1.65983			−0.107817
$238$		0			0
$239$		−	15.5525i		−	1.00601i	−0.864284	−	0.503004i	$-0.832228\pi$
$239$		−	15.5525i		−	1.00601i	0.864284	−	0.503004i	$-0.167772\pi$
$240$		0			0
$241$			7.33403i			0.472426i	0.971701	+	0.236213i	$0.0759064\pi$
$241$			7.33403i			0.472426i	−0.971701	+	0.236213i	$0.924094\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	0.630898i		−	0.0403066i
$246$		0			0
$247$	−21.3607	+	16.0989i	−1.35915	+	1.02435i
$248$		0			0
$249$		−	12.8865i		−	0.816652i
$250$		0			0
$251$	13.6742			0.863108			0.431554	−	0.902087i	$-0.357965\pi$
$251$	13.6742			0.863108			0.431554	+	0.902087i	$0.357965\pi$
$252$		0			0
$253$		−	30.9360i		−	1.94493i
$254$		0			0
$255$		−	0.680346i		−	0.0426049i
$256$		0			0
$257$	−26.1256			−1.62967			−0.814834	−	0.579695i	$-0.803172\pi$
$257$	−26.1256			−1.62967			−0.814834	+	0.579695i	$0.803172\pi$
$258$		0			0
$259$	−3.41855			−0.212418
$260$		0			0
$261$	6.68035			0.413503
$262$		0			0
$263$	30.7792			1.89793			0.948965	−	0.315382i	$-0.102133\pi$
$263$	30.7792			1.89793			0.948965	+	0.315382i	$0.102133\pi$
$264$		0			0
$265$		−	3.78539i		−	0.232534i
$266$		0			0
$267$		−	6.20620i		−	0.379814i
$268$		0			0
$269$	5.39189			0.328749			0.164375	−	0.986398i	$-0.447439\pi$
$269$	5.39189			0.328749			0.164375	+	0.986398i	$0.447439\pi$
$270$		0			0
$271$		−	11.4186i		−	0.693628i	−0.937934	−	0.346814i	$-0.887264\pi$
$271$		−	11.4186i		−	0.693628i	0.937934	−	0.346814i	$-0.112736\pi$
$272$		0			0
$273$	−2.87936	+	2.17009i	−0.174267	+	0.131340i
$274$		0			0
$275$			26.2739i			1.58438i
$276$		0			0
$277$	13.1506			0.790144			0.395072	−	0.918650i	$-0.370720\pi$
$277$	13.1506			0.790144			0.395072	+	0.918650i	$0.370720\pi$
$278$		0			0
$279$		−	6.15676i		−	0.368595i
$280$		0			0
$281$		−	2.14834i		−	0.128160i	−0.997945	−	0.0640798i	$-0.979589\pi$
$281$		−	2.14834i		−	0.128160i	0.997945	−	0.0640798i	$-0.0204112\pi$
$282$		0			0
$283$	12.1978			0.725084			0.362542	−	0.931968i	$-0.381909\pi$
$283$	12.1978			0.725084			0.362542	+	0.931968i	$0.381909\pi$
$284$		0			0
$285$	−4.68035			−0.277240
$286$		0			0
$287$	−1.21235			−0.0715626
$288$		0			0
$289$	−15.8371			−0.931594
$290$		0			0
$291$			1.07838i			0.0632156i
$292$		0			0
$293$			7.15449i			0.417970i	0.977919	+	0.208985i	$0.0670159\pi$
$293$			7.15449i			0.417970i	−0.977919	+	0.208985i	$0.932984\pi$
$294$		0			0
$295$	−0.863763			−0.0502903
$296$		0			0
$297$		−	5.70928i		−	0.331286i
$298$		0			0
$299$	11.7587	+	15.6020i	0.680025	+	0.902285i
$300$		0			0
$301$			12.6803i			0.730883i
$302$		0			0
$303$	7.23513			0.415648
$304$		0			0
$305$			7.98318i			0.457116i
$306$		0			0
$307$			21.8432i			1.24666i	0.781959	+	0.623330i	$0.214221\pi$
$307$			21.8432i			1.24666i	−0.781959	+	0.623330i	$0.785779\pi$
$308$		0			0
$309$	13.7587			0.782706
$310$		0			0
$311$	7.51745			0.426275			0.213138	−	0.977022i	$-0.431632\pi$
$311$	7.51745			0.426275			0.213138	+	0.977022i	$0.431632\pi$
$312$		0			0
$313$	25.7009			1.45270			0.726349	−	0.687326i	$-0.241215\pi$
$313$	25.7009			1.45270			0.726349	+	0.687326i	$0.241215\pi$
$314$		0			0
$315$	−0.630898			−0.0355471
$316$		0			0
$317$			9.36910i			0.526221i	0.964766	+	0.263111i	$0.0847484\pi$
$317$			9.36910i			0.526221i	−0.964766	+	0.263111i	$0.915252\pi$
$318$		0			0
$319$			38.1399i			2.13543i
$320$		0			0
$321$	−10.0989			−0.563665
$322$		0			0
$323$		−	8.00000i		−	0.445132i
$324$		0			0
$325$	−9.98667	−	13.2507i	−0.553961	−	0.735018i
$326$		0			0
$327$		−	17.9421i		−	0.992203i
$328$		0			0
$329$	6.04945			0.333517
$330$		0			0
$331$		−	21.0472i		−	1.15686i	−0.815733	−	0.578429i	$-0.803666\pi$
$331$		−	21.0472i		−	1.15686i	0.815733	−	0.578429i	$-0.196334\pi$
$332$		0			0
$333$			3.41855i			0.187335i
$334$		0			0
$335$	0.366835			0.0200423
$336$		0			0
$337$	15.4452			0.841354			0.420677	−	0.907210i	$-0.361793\pi$
$337$	15.4452			0.841354			0.420677	+	0.907210i	$0.361793\pi$
$338$		0			0
$339$	−10.6803			−0.580077
$340$		0			0
$341$	35.1506			1.90351
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			3.41855i			0.184049i
$346$		0			0
$347$	−21.7321			−1.16664			−0.583319	−	0.812243i	$-0.698246\pi$
$347$	−21.7321			−1.16664			−0.583319	+	0.812243i	$0.698246\pi$
$348$		0			0
$349$		−	5.65983i		−	0.302964i	−0.988460	−	0.151482i	$-0.951596\pi$
$349$		−	5.65983i		−	0.302964i	0.988460	−	0.151482i	$-0.0484045\pi$
$350$		0			0
$351$	2.17009	+	2.87936i	0.115831	+	0.153689i
$352$		0			0
$353$			33.7237i			1.79493i	0.441087	+	0.897464i	$0.354593\pi$
$353$			33.7237i			1.79493i	−0.441087	+	0.897464i	$0.645407\pi$
$354$		0			0
$355$	−3.03734			−0.161205
$356$		0			0
$357$		−	1.07838i		−	0.0570738i
$358$		0			0
$359$			18.7031i			0.987114i	0.869714	+	0.493557i	$0.164303\pi$
$359$			18.7031i			0.987114i	−0.869714	+	0.493557i	$0.835697\pi$
$360$		0			0
$361$	−36.0349			−1.89657
$362$		0			0
$363$	21.5958			1.13349
$364$		0			0
$365$	2.40787			0.126034
$366$		0			0
$367$	−10.0722			−0.525766			−0.262883	−	0.964828i	$-0.584673\pi$
$367$	−10.0722			−0.525766			−0.262883	+	0.964828i	$0.584673\pi$
$368$		0			0
$369$			1.21235i			0.0631123i
$370$		0			0
$371$		−	6.00000i		−	0.311504i
$372$		0			0
$373$	4.24128			0.219605			0.109802	−	0.993953i	$-0.464978\pi$
$373$	4.24128			0.219605			0.109802	+	0.993953i	$0.464978\pi$
$374$		0			0
$375$		−	6.05786i		−	0.312826i
$376$		0			0
$377$	−14.4969	−	19.2351i	−0.746630	−	0.990660i
$378$		0			0
$379$			20.1978i			1.03749i	0.854929	+	0.518745i	$0.173601\pi$
$379$			20.1978i			1.03749i	−0.854929	+	0.518745i	$0.826399\pi$
$380$		0			0
$381$	6.34017			0.324817
$382$		0			0
$383$		−	7.04331i		−	0.359896i	−0.983676	−	0.179948i	$-0.942407\pi$
$383$		−	7.04331i		−	0.359896i	0.983676	−	0.179948i	$-0.0575930\pi$
$384$		0			0
$385$		−	3.60197i		−	0.183573i
$386$		0			0
$387$	12.6803			0.644578
$388$		0			0
$389$	−3.84324			−0.194860			−0.0974301	−	0.995242i	$-0.531062\pi$
$389$	−3.84324			−0.194860			−0.0974301	+	0.995242i	$0.531062\pi$
$390$		0			0
$391$	−5.84324			−0.295506
$392$		0			0
$393$	−7.51745			−0.379205
$394$		0			0
$395$			1.04718i			0.0526894i
$396$		0			0
$397$		−	14.4391i		−	0.724676i	−0.932047	−	0.362338i	$-0.881979\pi$
$397$		−	14.4391i		−	0.724676i	0.932047	−	0.362338i	$-0.118021\pi$
$398$		0			0
$399$	−7.41855			−0.371392
$400$		0			0
$401$			9.83483i			0.491128i	0.969380	+	0.245564i	$0.0789732\pi$
$401$			9.83483i			0.491128i	−0.969380	+	0.245564i	$0.921027\pi$
$402$		0			0
$403$	−17.7275	+	13.3607i	−0.883071	+	0.665543i
$404$		0			0
$405$			0.630898i			0.0313496i
$406$		0			0
$407$	−19.5174			−0.967444
$408$		0			0
$409$			7.54864i			0.373256i	0.982431	+	0.186628i	$0.0597560\pi$
$409$			7.54864i			0.373256i	−0.982431	+	0.186628i	$0.940244\pi$
$410$		0			0
$411$			4.47414i			0.220693i
$412$		0			0
$413$	−1.36910			−0.0673691
$414$		0			0
$415$	−8.13009			−0.399091
$416$		0			0
$417$	−9.75872			−0.477887
$418$		0			0
$419$	35.1506			1.71722			0.858610	−	0.512630i	$-0.171329\pi$
$419$	35.1506			1.71722			0.858610	+	0.512630i	$0.171329\pi$
$420$		0			0
$421$		−	25.6742i		−	1.25128i	−0.780110	−	0.625642i	$-0.784837\pi$
$421$		−	25.6742i		−	1.25128i	0.780110	−	0.625642i	$-0.215163\pi$
$422$		0			0
$423$		−	6.04945i		−	0.294134i
$424$		0			0
$425$	4.96266			0.240724
$426$		0			0
$427$			12.6537i			0.612355i
$428$		0			0
$429$	−16.4391	+	12.3896i	−0.793686	+	0.598177i
$430$		0			0
$431$			16.2329i			0.781910i	0.920410	+	0.390955i	$0.127855\pi$
$431$			16.2329i			0.781910i	−0.920410	+	0.390955i	$0.872145\pi$
$432$		0			0
$433$	−5.38735			−0.258900			−0.129450	−	0.991586i	$-0.541321\pi$
$433$	−5.38735			−0.258900			−0.129450	+	0.991586i	$0.541321\pi$
$434$		0			0
$435$		−	4.21461i		−	0.202075i
$436$		0			0
$437$			40.1978i			1.92292i
$438$		0			0
$439$	10.2413			0.488789			0.244395	−	0.969676i	$-0.421411\pi$
$439$	10.2413			0.488789			0.244395	+	0.969676i	$0.421411\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	−15.9421			−0.757434			−0.378717	−	0.925513i	$-0.623635\pi$
$443$	−15.9421			−0.757434			−0.378717	+	0.925513i	$0.623635\pi$
$444$		0			0
$445$	−3.91548			−0.185612
$446$		0			0
$447$		−	1.46800i		−	0.0694340i
$448$		0			0
$449$		−	13.2001i		−	0.622949i	−0.950254	−	0.311475i	$-0.899177\pi$
$449$		−	13.2001i		−	0.622949i	0.950254	−	0.311475i	$-0.100823\pi$
$450$		0			0
$451$	−6.92162			−0.325926
$452$		0			0
$453$		−	13.3607i		−	0.627740i
$454$		0			0
$455$	1.36910	+	1.81658i	0.0641845	+	0.0851627i
$456$		0			0
$457$			18.3090i			0.856458i	0.903670	+	0.428229i	$0.140862\pi$
$457$			18.3090i			0.856458i	−0.903670	+	0.428229i	$0.859138\pi$
$458$		0			0
$459$	−1.07838			−0.0503344
$460$		0			0
$461$		−	18.9399i		−	0.882118i	−0.897478	−	0.441059i	$-0.854603\pi$
$461$		−	18.9399i		−	0.882118i	0.897478	−	0.441059i	$-0.145397\pi$
$462$		0			0
$463$			1.74435i			0.0810667i	0.999178	+	0.0405334i	$0.0129057\pi$
$463$			1.74435i			0.0810667i	−0.999178	+	0.0405334i	$0.987094\pi$
$464$		0			0
$465$	−3.88428			−0.180129
$466$		0			0
$467$	18.1568			0.840194			0.420097	−	0.907479i	$-0.361996\pi$
$467$	18.1568			0.840194			0.420097	+	0.907479i	$0.361996\pi$
$468$		0			0
$469$	0.581449			0.0268488
$470$		0			0
$471$	8.15676			0.375843
$472$		0			0
$473$			72.3956i			3.32875i
$474$		0			0
$475$		−	34.1399i		−	1.56645i
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$		−	30.4619i		−	1.39184i	−0.718120	−	0.695919i	$-0.754997\pi$
$479$		−	30.4619i		−	1.39184i	0.718120	−	0.695919i	$-0.245003\pi$
$480$		0			0
$481$	9.84324	−	7.41855i	0.448813	−	0.338257i
$482$		0			0
$483$			5.41855i			0.246553i
$484$		0			0
$485$	0.680346			0.0308929
$486$		0			0
$487$			0.313511i			0.0142065i	0.999975	+	0.00710327i	$0.00226106\pi$
$487$			0.313511i			0.0142065i	−0.999975	+	0.00710327i	$0.997739\pi$
$488$		0			0
$489$		−	4.58145i		−	0.207180i
$490$		0			0
$491$	8.93600			0.403276			0.201638	−	0.979460i	$-0.435374\pi$
$491$	8.93600			0.403276			0.201638	+	0.979460i	$0.435374\pi$
$492$		0			0
$493$	7.20394			0.324449
$494$		0			0
$495$	−3.60197			−0.161896
$496$		0			0
$497$	−4.81432			−0.215952
$498$		0			0
$499$		−	33.9877i		−	1.52150i	−0.649046	−	0.760750i	$-0.724832\pi$
$499$		−	33.9877i		−	1.52150i	0.649046	−	0.760750i	$-0.275168\pi$
$500$		0			0
$501$		−	7.31124i		−	0.326642i
$502$		0			0
$503$	−33.5585			−1.49630			−0.748149	−	0.663530i	$-0.769057\pi$
$503$	−33.5585			−1.49630			−0.748149	+	0.663530i	$0.769057\pi$
$504$		0			0
$505$		−	4.56463i		−	0.203123i
$506$		0			0
$507$	3.58145	−	12.4969i	0.159058	−	0.555008i
$508$		0			0
$509$			24.8287i			1.10051i	0.834996	+	0.550256i	$0.185470\pi$
$509$			24.8287i			1.10051i	−0.834996	+	0.550256i	$0.814530\pi$
$510$		0			0
$511$	3.81658			0.168836
$512$		0			0
$513$			7.41855i			0.327537i
$514$		0			0
$515$		−	8.68035i		−	0.382502i
$516$		0			0
$517$	34.5380			1.51898
$518$		0			0
$519$	−7.60197			−0.333689
$520$		0			0
$521$	20.2290			0.886248			0.443124	−	0.896460i	$-0.353870\pi$
$521$	20.2290			0.886248			0.443124	+	0.896460i	$0.353870\pi$
$522$		0			0
$523$	40.1666			1.75636			0.878181	−	0.478328i	$-0.158757\pi$
$523$	40.1666			1.75636			0.878181	+	0.478328i	$0.158757\pi$
$524$		0			0
$525$		−	4.60197i		−	0.200846i
$526$		0			0
$527$		−	6.63931i		−	0.289213i
$528$		0			0
$529$	6.36069			0.276552
$530$		0			0
$531$			1.36910i			0.0594140i
$532$		0			0
$533$	3.49079	−	2.63090i	0.151203	−	0.113957i
$534$		0			0
$535$			6.37137i			0.275458i
$536$		0			0
$537$	9.10504			0.392911
$538$		0			0
$539$		−	5.70928i		−	0.245916i
$540$		0			0
$541$		−	18.2101i		−	0.782912i	−0.920197	−	0.391456i	$-0.871971\pi$
$541$		−	18.2101i		−	0.782912i	0.920197	−	0.391456i	$-0.128029\pi$
$542$		0			0
$543$	−15.3607			−0.659190
$544$		0			0
$545$	−11.3197			−0.484881
$546$		0			0
$547$	6.32580			0.270472			0.135236	−	0.990813i	$-0.456821\pi$
$547$	6.32580			0.270472			0.135236	+	0.990813i	$0.456821\pi$
$548$		0			0
$549$	12.6537			0.540046
$550$		0			0
$551$		−	49.5585i		−	2.11126i
$552$		0			0
$553$			1.65983i			0.0705830i
$554$		0			0
$555$	2.15676			0.0915492
$556$		0			0
$557$		−	10.6309i		−	0.450446i	−0.974307	−	0.225223i	$-0.927689\pi$
$557$		−	10.6309i		−	0.450446i	0.974307	−	0.225223i	$-0.0723110\pi$
$558$		0			0
$559$	−27.5174	−	36.5113i	−1.16386	−	1.54426i
$560$		0			0
$561$		−	6.15676i		−	0.259938i
$562$		0			0
$563$	37.0472			1.56135			0.780676	−	0.624936i	$-0.214875\pi$
$563$	37.0472			1.56135			0.780676	+	0.624936i	$0.214875\pi$
$564$		0			0
$565$			6.73820i			0.283478i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	25.7152			1.07804			0.539019	−	0.842293i	$-0.318795\pi$
$569$	25.7152			1.07804			0.539019	+	0.842293i	$0.318795\pi$
$570$		0			0
$571$	12.4969			0.522980			0.261490	−	0.965206i	$-0.415786\pi$
$571$	12.4969			0.522980			0.261490	+	0.965206i	$0.415786\pi$
$572$		0			0
$573$	−24.5692			−1.02639
$574$		0			0
$575$	−24.9360			−1.03990
$576$		0			0
$577$		−	3.91548i		−	0.163004i	−0.996673	−	0.0815018i	$-0.974028\pi$
$577$		−	3.91548i		−	0.163004i	0.996673	−	0.0815018i	$-0.0259716\pi$
$578$		0			0
$579$			13.5753i			0.564170i
$580$		0			0
$581$	−12.8865			−0.534624
$582$		0			0
$583$		−	34.2557i		−	1.41872i
$584$		0			0
$585$	1.81658	−	1.36910i	0.0751064	−	0.0566054i
$586$		0			0
$587$		−	6.94441i		−	0.286626i	−0.989677	−	0.143313i	$-0.954224\pi$
$587$		−	6.94441i		−	0.286626i	0.989677	−	0.143313i	$-0.0457756\pi$
$588$		0			0
$589$	−45.6742			−1.88197
$590$		0			0
$591$			2.94441i			0.121117i
$592$		0			0
$593$		−	29.9383i		−	1.22942i	−0.788754	−	0.614709i	$-0.789274\pi$
$593$		−	29.9383i		−	1.22942i	0.788754	−	0.614709i	$-0.210726\pi$
$594$		0			0
$595$	−0.680346			−0.0278915
$596$		0			0
$597$	−10.5236			−0.430702
$598$		0			0
$599$	−43.2905			−1.76880			−0.884402	−	0.466726i	$-0.845433\pi$
$599$	−43.2905			−1.76880			−0.884402	+	0.466726i	$0.845433\pi$
$600$		0			0
$601$	−31.2306			−1.27392			−0.636961	−	0.770896i	$-0.719809\pi$
$601$	−31.2306			−1.27392			−0.636961	+	0.770896i	$0.719809\pi$
$602$		0			0
$603$		−	0.581449i		−	0.0236784i
$604$		0			0
$605$		−	13.6248i		−	0.553925i
$606$		0			0
$607$	−4.96266			−0.201428			−0.100714	−	0.994915i	$-0.532113\pi$
$607$	−4.96266			−0.201428			−0.100714	+	0.994915i	$0.532113\pi$
$608$		0			0
$609$		−	6.68035i		−	0.270701i
$610$		0			0
$611$	−17.4186	+	13.1278i	−0.704679	+	0.531095i
$612$		0			0
$613$		−	0.366835i		−	0.0148163i	−0.999973	−	0.00740816i	$-0.997642\pi$
$613$		−	0.366835i		−	0.0148163i	0.999973	−	0.00740816i	$-0.00235811\pi$
$614$		0			0
$615$	0.764867			0.0308424
$616$		0			0
$617$		−	47.7237i		−	1.92128i	−0.277793	−	0.960641i	$-0.589603\pi$
$617$		−	47.7237i		−	1.92128i	0.277793	−	0.960641i	$-0.410397\pi$
$618$		0			0
$619$			19.7854i			0.795242i	0.917550	+	0.397621i	$0.130164\pi$
$619$			19.7854i			0.795242i	−0.917550	+	0.397621i	$0.869836\pi$
$620$		0			0
$621$	5.41855			0.217439
$622$		0			0
$623$	−6.20620			−0.248646
$624$		0			0
$625$	19.1880			0.767518
$626$		0			0
$627$	−42.3545			−1.69148
$628$		0			0
$629$			3.68649i			0.146990i
$630$		0			0
$631$		−	20.1522i		−	0.802247i	−0.916024	−	0.401124i	$-0.868620\pi$
$631$		−	20.1522i		−	0.802247i	0.916024	−	0.401124i	$-0.131380\pi$
$632$		0			0
$633$	−3.02052			−0.120055
$634$		0			0
$635$		−	4.00000i		−	0.158735i
$636$		0			0
$637$	2.17009	+	2.87936i	0.0859820	+	0.114084i
$638$		0			0
$639$			4.81432i			0.190451i
$640$		0			0
$641$	−12.1034			−0.478057			−0.239028	−	0.971013i	$-0.576829\pi$
$641$	−12.1034			−0.478057			−0.239028	+	0.971013i	$0.576829\pi$
$642$		0			0
$643$		−	20.6803i		−	0.815553i	−0.913082	−	0.407777i	$-0.866304\pi$
$643$		−	20.6803i		−	0.815553i	0.913082	−	0.407777i	$-0.133696\pi$
$644$		0			0
$645$		−	8.00000i		−	0.315000i
$646$		0			0
$647$	−40.8781			−1.60709			−0.803543	−	0.595247i	$-0.797054\pi$
$647$	−40.8781			−1.60709			−0.803543	+	0.595247i	$0.797054\pi$
$648$		0			0
$649$	−7.81658			−0.306828
$650$		0			0
$651$	−6.15676			−0.241302
$652$		0			0
$653$	6.31351			0.247067			0.123533	−	0.992340i	$-0.460577\pi$
$653$	6.31351			0.247067			0.123533	+	0.992340i	$0.460577\pi$
$654$		0			0
$655$			4.74274i			0.185314i
$656$		0			0
$657$		−	3.81658i		−	0.148899i
$658$		0			0
$659$	20.9360			0.815551			0.407775	−	0.913082i	$-0.366305\pi$
$659$	20.9360			0.815551			0.407775	+	0.913082i	$0.366305\pi$
$660$		0			0
$661$			22.2245i			0.864431i	0.901770	+	0.432216i	$0.142268\pi$
$661$			22.2245i			0.864431i	−0.901770	+	0.432216i	$0.857732\pi$
$662$		0			0
$663$	2.34017	+	3.10504i	0.0908848	+	0.120590i
$664$		0			0
$665$			4.68035i			0.181496i
$666$		0			0
$667$	−36.1978			−1.40158
$668$		0			0
$669$			22.6225i			0.874636i
$670$		0			0
$671$			72.2434i			2.78892i
$672$		0			0
$673$	−1.15061			−0.0443528			−0.0221764	−	0.999754i	$-0.507060\pi$
$673$	−1.15061			−0.0443528			−0.0221764	+	0.999754i	$0.507060\pi$
$674$		0			0
$675$	−4.60197			−0.177130
$676$		0			0
$677$	−33.5897			−1.29096			−0.645478	−	0.763779i	$-0.723342\pi$
$677$	−33.5897			−1.29096			−0.645478	+	0.763779i	$0.723342\pi$
$678$		0			0
$679$	1.07838			0.0413843
$680$		0			0
$681$		−	22.9444i		−	0.879232i
$682$		0			0
$683$			25.1110i			0.960846i	0.877037	+	0.480423i	$0.159517\pi$
$683$			25.1110i			0.960846i	−0.877037	+	0.480423i	$0.840483\pi$
$684$		0			0
$685$	2.82273			0.107851
$686$		0			0
$687$		−	21.5441i		−	0.821959i
$688$		0			0
$689$	13.0205	+	17.2762i	0.496042	+	0.658170i
$690$		0			0
$691$			6.30898i			0.240005i	0.992774	+	0.120002i	$0.0382902\pi$
$691$			6.30898i			0.240005i	−0.992774	+	0.120002i	$0.961710\pi$
$692$		0			0
$693$	−5.70928			−0.216877
$694$		0			0
$695$			6.15676i			0.233539i
$696$		0			0
$697$			1.30737i			0.0495201i
$698$		0			0
$699$	−7.36069			−0.278407
$700$		0			0
$701$	45.8843			1.73303			0.866513	−	0.499155i	$-0.166356\pi$
$701$	45.8843			1.73303			0.866513	+	0.499155i	$0.166356\pi$
$702$		0			0
$703$	25.3607			0.956497
$704$		0			0
$705$	−3.81658			−0.143741
$706$		0			0
$707$		−	7.23513i		−	0.272105i
$708$		0			0
$709$			46.4534i			1.74460i	0.488975	+	0.872298i	$0.337371\pi$
$709$			46.4534i			1.74460i	−0.488975	+	0.872298i	$0.662629\pi$
$710$		0			0
$711$	1.65983			0.0622484
$712$		0			0
$713$			33.3607i			1.24937i
$714$		0			0
$715$	7.81658	+	10.3714i	0.292324	+	0.387867i
$716$		0			0
$717$			15.5525i			0.580819i
$718$		0			0
$719$	7.20394			0.268661			0.134331	−	0.990937i	$-0.457112\pi$
$719$	7.20394			0.268661			0.134331	+	0.990937i	$0.457112\pi$
$720$		0			0
$721$		−	13.7587i		−	0.512402i
$722$		0			0
$723$		−	7.33403i		−	0.272756i
$724$		0			0
$725$	30.7427			1.14176
$726$		0			0
$727$	−13.6742			−0.507148			−0.253574	−	0.967316i	$-0.581606\pi$
$727$	−13.6742			−0.507148			−0.253574	+	0.967316i	$0.581606\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	13.6742			0.505759
$732$		0			0
$733$		−	7.70086i		−	0.284438i	−0.989835	−	0.142219i	$-0.954576\pi$
$733$		−	7.70086i		−	0.284438i	0.989835	−	0.142219i	$-0.0454237\pi$
$734$		0			0
$735$			0.630898i			0.0232710i
$736$		0			0
$737$	3.31965			0.122281
$738$		0			0
$739$		−	24.5814i		−	0.904243i	−0.891956	−	0.452122i	$-0.850667\pi$
$739$		−	24.5814i		−	0.904243i	0.891956	−	0.452122i	$-0.149333\pi$
$740$		0			0
$741$	21.3607	−	16.0989i	0.784705	−	0.591408i
$742$		0			0
$743$		−	24.3773i		−	0.894318i	−0.894455	−	0.447159i	$-0.852436\pi$
$743$		−	24.3773i		−	0.894318i	0.894455	−	0.447159i	$-0.147564\pi$
$744$		0			0
$745$	−0.926157			−0.0339318
$746$		0			0
$747$			12.8865i			0.471494i
$748$		0			0
$749$			10.0989i			0.369006i
$750$		0			0
$751$	24.6803			0.900599			0.450299	−	0.892878i	$-0.351317\pi$
$751$	24.6803			0.900599			0.450299	+	0.892878i	$0.351317\pi$
$752$		0			0
$753$	−13.6742			−0.498316
$754$		0			0
$755$	−8.42923			−0.306771
$756$		0			0
$757$	−41.3172			−1.50170			−0.750850	−	0.660473i	$-0.770356\pi$
$757$	−41.3172			−1.50170			−0.750850	+	0.660473i	$0.770356\pi$
$758$		0			0
$759$			30.9360i			1.12291i
$760$		0			0
$761$			15.9506i			0.578207i	0.957298	+	0.289104i	$0.0933572\pi$
$761$			15.9506i			0.578207i	−0.957298	+	0.289104i	$0.906643\pi$
$762$		0			0
$763$	−17.9421			−0.649549
$764$		0			0
$765$			0.680346i			0.0245980i
$766$		0			0
$767$	3.94214	−	2.97107i	0.142342	−	0.107279i
$768$		0			0
$769$		−	6.22446i		−	0.224460i	−0.993682	−	0.112230i	$-0.964201\pi$
$769$		−	6.22446i		−	0.224460i	0.993682	−	0.112230i	$-0.0357993\pi$
$770$		0			0
$771$	26.1256			0.940889
$772$		0			0
$773$		−	39.1545i		−	1.40829i	−0.710057	−	0.704145i	$-0.751331\pi$
$773$		−	39.1545i		−	1.40829i	0.710057	−	0.704145i	$-0.248669\pi$
$774$		0			0
$775$		−	28.3332i		−	1.01776i
$776$		0			0
$777$	3.41855			0.122640
$778$		0			0
$779$	8.99386			0.322238
$780$		0			0
$781$	−27.4863			−0.983535
$782$		0			0
$783$	−6.68035			−0.238736
$784$		0			0
$785$		−	5.14608i		−	0.183671i
$786$		0			0
$787$			32.2967i			1.15125i	0.817713	+	0.575626i	$0.195242\pi$
$787$			32.2967i			1.15125i	−0.817713	+	0.575626i	$0.804758\pi$
$788$		0			0
$789$	−30.7792			−1.09577
$790$		0			0
$791$			10.6803i			0.379749i
$792$		0			0
$793$	−27.4596	−	36.4345i	−0.975119	−	1.29383i
$794$		0			0
$795$			3.78539i			0.134254i
$796$		0			0
$797$	−30.2700			−1.07222			−0.536110	−	0.844148i	$-0.680107\pi$
$797$	−30.2700			−1.07222			−0.536110	+	0.844148i	$0.680107\pi$
$798$		0			0
$799$		−	6.52359i		−	0.230788i
$800$		0			0
$801$			6.20620i			0.219285i
$802$		0			0
$803$	21.7899			0.768950
$804$		0			0
$805$	3.41855			0.120488
$806$		0			0
$807$	−5.39189			−0.189803
$808$		0			0
$809$	39.5585			1.39080			0.695401	−	0.718622i	$-0.255227\pi$
$809$	39.5585			1.39080			0.695401	+	0.718622i	$0.255227\pi$
$810$		0			0
$811$			34.4079i			1.20822i	0.796899	+	0.604112i	$0.206472\pi$
$811$			34.4079i			1.20822i	−0.796899	+	0.604112i	$0.793528\pi$
$812$		0			0
$813$			11.4186i			0.400466i
$814$		0			0
$815$	−2.89043			−0.101247
$816$		0			0
$817$		−	94.0698i		−	3.29109i
$818$		0			0
$819$	2.87936	−	2.17009i	0.100613	−	0.0758290i
$820$		0			0
$821$		−	7.89269i		−	0.275457i	−0.990470	−	0.137728i	$-0.956020\pi$
$821$		−	7.89269i		−	0.275457i	0.990470	−	0.137728i	$-0.0439801\pi$
$822$		0			0
$823$	−29.6742			−1.03438			−0.517189	−	0.855871i	$-0.673022\pi$
$823$	−29.6742			−1.03438			−0.517189	+	0.855871i	$0.673022\pi$
$824$		0			0
$825$		−	26.2739i		−	0.914740i
$826$		0			0
$827$			14.1918i			0.493498i	0.969079	+	0.246749i	$0.0793624\pi$
$827$			14.1918i			0.493498i	−0.969079	+	0.246749i	$0.920638\pi$
$828$		0			0
$829$	3.71315			0.128963			0.0644815	−	0.997919i	$-0.479461\pi$
$829$	3.71315			0.128963			0.0644815	+	0.997919i	$0.479461\pi$
$830$		0			0
$831$	−13.1506			−0.456190
$832$		0			0
$833$	−1.07838			−0.0373636
$834$		0			0
$835$	−4.61265			−0.159627
$836$		0			0
$837$			6.15676i			0.212809i
$838$		0			0
$839$		−	48.6719i		−	1.68034i	−0.542322	−	0.840171i	$-0.682455\pi$
$839$		−	48.6719i		−	1.68034i	0.542322	−	0.840171i	$-0.317545\pi$
$840$		0			0
$841$	15.6270			0.538863
$842$		0			0
$843$			2.14834i			0.0739929i
$844$		0			0
$845$	−7.88428	−	2.25953i	−0.271228	−	0.0777301i
$846$		0			0
$847$		−	21.5958i		−	0.742041i
$848$		0			0
$849$	−12.1978			−0.418627
$850$		0			0
$851$		−	18.5236i		−	0.634981i
$852$		0			0
$853$			1.92777i			0.0660054i	0.999455	+	0.0330027i	$0.0105070\pi$
$853$			1.92777i			0.0660054i	−0.999455	+	0.0330027i	$0.989493\pi$
$854$		0			0
$855$	4.68035			0.160064
$856$		0			0
$857$	24.3980			0.833421			0.416710	−	0.909039i	$-0.363183\pi$
$857$	24.3980			0.833421			0.416710	+	0.909039i	$0.363183\pi$
$858$		0			0
$859$	15.1506			0.516932			0.258466	−	0.966020i	$-0.416783\pi$
$859$	15.1506			0.516932			0.258466	+	0.966020i	$0.416783\pi$
$860$		0			0
$861$	1.21235			0.0413167
$862$		0			0
$863$			49.7380i			1.69310i	0.532308	+	0.846551i	$0.321325\pi$
$863$			49.7380i			1.69310i	−0.532308	+	0.846551i	$0.678675\pi$
$864$		0			0
$865$			4.79606i			0.163071i
$866$		0			0
$867$	15.8371			0.537856
$868$		0			0
$869$			9.47641i			0.321465i
$870$		0			0
$871$	−1.67420	+	1.26180i	−0.0567282	+	0.0427543i
$872$		0			0
$873$		−	1.07838i		−	0.0364976i
$874$		0			0
$875$	−6.05786			−0.204793
$876$		0			0
$877$			9.72753i			0.328475i	0.986421	+	0.164238i	$0.0525164\pi$
$877$			9.72753i			0.328475i	−0.986421	+	0.164238i	$0.947484\pi$
$878$		0			0
$879$		−	7.15449i		−	0.241315i
$880$		0			0
$881$	6.12556			0.206375			0.103188	−	0.994662i	$-0.467096\pi$
$881$	6.12556			0.206375			0.103188	+	0.994662i	$0.467096\pi$
$882$		0			0
$883$	−1.03281			−0.0347567			−0.0173783	−	0.999849i	$-0.505532\pi$
$883$	−1.03281			−0.0347567			−0.0173783	+	0.999849i	$0.505532\pi$
$884$		0			0
$885$	0.863763			0.0290351
$886$		0			0
$887$	−39.0882			−1.31245			−0.656227	−	0.754564i	$-0.727849\pi$
$887$	−39.0882			−1.31245			−0.656227	+	0.754564i	$0.727849\pi$
$888$		0			0
$889$		−	6.34017i		−	0.212643i
$890$		0			0
$891$			5.70928i			0.191268i
$892$		0			0
$893$	−44.8781			−1.50179
$894$		0			0
$895$		−	5.74435i		−	0.192012i
$896$		0			0
$897$	−11.7587	−	15.6020i	−0.392612	−	0.520935i
$898$		0			0
$899$		−	41.1293i		−	1.37174i
$900$		0			0
$901$	−6.47027			−0.215556
$902$		0			0
$903$		−	12.6803i		−	0.421975i
$904$		0			0
$905$			9.69102i			0.322141i
$906$		0			0
$907$	−45.3751			−1.50665			−0.753327	−	0.657646i	$-0.771552\pi$
$907$	−45.3751			−1.50665			−0.753327	+	0.657646i	$0.771552\pi$
$908$		0			0
$909$	−7.23513			−0.239974
$910$		0			0
$911$	24.3090			0.805392			0.402696	−	0.915334i	$-0.368073\pi$
$911$	24.3090			0.805392			0.402696	+	0.915334i	$0.368073\pi$
$912$		0			0
$913$	−73.5729			−2.43491
$914$		0			0
$915$		−	7.98318i		−	0.263916i
$916$		0			0
$917$			7.51745i			0.248248i
$918$		0			0
$919$	−55.8453			−1.84217			−0.921084	−	0.389364i	$-0.872695\pi$
$919$	−55.8453			−1.84217			−0.921084	+	0.389364i	$0.872695\pi$
$920$		0			0
$921$		−	21.8432i		−	0.719759i
$922$		0			0
$923$	13.8622	−	10.4475i	0.456278	−	0.343883i
$924$		0			0
$925$			15.7321i			0.517267i
$926$		0			0
$927$	−13.7587			−0.451896
$928$		0			0
$929$		−	39.4680i		−	1.29490i	−0.762107	−	0.647452i	$-0.775835\pi$
$929$		−	39.4680i		−	1.29490i	0.762107	−	0.647452i	$-0.224165\pi$
$930$		0			0
$931$			7.41855i			0.243133i
$932$		0			0
$933$	−7.51745			−0.246110
$934$		0			0
$935$	−3.88428			−0.127030
$936$		0			0
$937$	−22.2122			−0.725640			−0.362820	−	0.931859i	$-0.618186\pi$
$937$	−22.2122			−0.725640			−0.362820	+	0.931859i	$0.618186\pi$
$938$		0			0
$939$	−25.7009			−0.838716
$940$		0			0
$941$		−	32.1483i		−	1.04801i	−0.851716	−	0.524003i	$-0.824438\pi$
$941$		−	32.1483i		−	1.04801i	0.851716	−	0.524003i	$-0.175562\pi$
$942$		0			0
$943$		−	6.56916i		−	0.213921i
$944$		0			0
$945$	0.630898			0.0205231
$946$		0			0
$947$		−	19.2390i		−	0.625184i	−0.949888	−	0.312592i	$-0.898803\pi$
$947$		−	19.2390i		−	0.625184i	0.949888	−	0.312592i	$-0.101197\pi$
$948$		0			0
$949$	−10.9893	+	8.28231i	−0.356728	+	0.268855i
$950$		0			0
$951$		−	9.36910i		−	0.303814i
$952$		0			0
$953$	54.3423			1.76032			0.880159	−	0.474678i	$-0.157436\pi$
$953$	54.3423			1.76032			0.880159	+	0.474678i	$0.157436\pi$
$954$		0			0
$955$			15.5006i			0.501588i
$956$		0			0
$957$		−	38.1399i		−	1.23289i
$958$		0			0
$959$	4.47414			0.144478
$960$		0			0
$961$	−6.90564			−0.222763
$962$		0			0
$963$	10.0989			0.325432
$964$		0			0
$965$	8.56463			0.275705
$966$		0			0
$967$			1.43084i			0.0460126i	0.999735	+	0.0230063i	$0.00732378\pi$
$967$			1.43084i			0.0460126i	−0.999735	+	0.0230063i	$0.992676\pi$
$968$		0			0
$969$			8.00000i			0.256997i
$970$		0			0
$971$	25.2450			0.810150			0.405075	−	0.914284i	$-0.367245\pi$
$971$	25.2450			0.810150			0.405075	+	0.914284i	$0.367245\pi$
$972$		0			0
$973$			9.75872i			0.312850i
$974$		0			0
$975$	9.98667	+	13.2507i	0.319829	+	0.424363i
$976$		0			0
$977$		−	7.31124i		−	0.233907i	−0.993137	−	0.116954i	$-0.962687\pi$
$977$		−	7.31124i		−	0.233907i	0.993137	−	0.116954i	$-0.0373129\pi$
$978$		0			0
$979$	−35.4329			−1.13244
$980$		0			0
$981$			17.9421i			0.572848i
$982$		0			0
$983$			21.0556i			0.671569i	0.941939	+	0.335785i	$0.109001\pi$
$983$			21.0556i			0.671569i	−0.941939	+	0.335785i	$0.890999\pi$
$984$		0			0
$985$	1.85762			0.0591887
$986$		0			0
$987$	−6.04945			−0.192556
$988$		0			0
$989$	−68.7091			−2.18482
$990$		0			0
$991$	−42.0410			−1.33548			−0.667739	−	0.744396i	$-0.732738\pi$
$991$	−42.0410			−1.33548			−0.667739	+	0.744396i	$0.732738\pi$
$992$		0			0
$993$			21.0472i			0.667912i
$994$		0			0
$995$			6.63931i			0.210480i
$996$		0			0
$997$	−9.20394			−0.291492			−0.145746	−	0.989322i	$-0.546558\pi$
$997$	−9.20394			−0.291492			−0.145746	+	0.989322i	$0.546558\pi$
$998$		0			0
$999$		−	3.41855i		−	0.108158i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.o.337.4		6
4.3	odd	2		273.2.c.b.64.1	✓	6
12.11	even	2		819.2.c.c.64.6		6
13.12	even	2	inner	4368.2.h.o.337.3		6
28.27	even	2		1911.2.c.h.883.1		6
52.31	even	4		3549.2.a.k.1.1		3
52.47	even	4		3549.2.a.q.1.3		3
52.51	odd	2		273.2.c.b.64.6	yes	6
156.155	even	2		819.2.c.c.64.1		6
364.363	even	2		1911.2.c.h.883.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.c.b.64.1	✓	6	4.3	odd	2
273.2.c.b.64.6	yes	6	52.51	odd	2
819.2.c.c.64.1		6	156.155	even	2
819.2.c.c.64.6		6	12.11	even	2
1911.2.c.h.883.1		6	28.27	even	2
1911.2.c.h.883.6		6	364.363	even	2
3549.2.a.k.1.1		3	52.31	even	4
3549.2.a.q.1.3		3	52.47	even	4
4368.2.h.o.337.3		6	13.12	even	2	inner
4368.2.h.o.337.4		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.630898i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.630898i q^{5} +1.00000i q^{7} +1.00000 q^{9} +5.70928i q^{11} +(-2.17009 - 2.87936i) q^{13} -0.630898i q^{15} +1.07838 q^{17} -7.41855i q^{19} -1.00000i q^{21} -5.41855 q^{23} +4.60197 q^{25} -1.00000 q^{27} +6.68035 q^{29} -6.15676i q^{31} -5.70928i q^{33} -0.630898 q^{35} +3.41855i q^{37} +(2.17009 + 2.87936i) q^{39} +1.21235i q^{41} +12.6803 q^{43} +0.630898i q^{45} -6.04945i q^{47} -1.00000 q^{49} -1.07838 q^{51} -6.00000 q^{53} -3.60197 q^{55} +7.41855i q^{57} +1.36910i q^{59} +12.6537 q^{61} +1.00000i q^{63} +(1.81658 - 1.36910i) q^{65} -0.581449i q^{67} +5.41855 q^{69} +4.81432i q^{71} -3.81658i q^{73} -4.60197 q^{75} -5.70928 q^{77} +1.65983 q^{79} +1.00000 q^{81} +12.8865i q^{83} +0.680346i q^{85} -6.68035 q^{87} +6.20620i q^{89} +(2.87936 - 2.17009i) q^{91} +6.15676i q^{93} +4.68035 q^{95} -1.07838i q^{97} +5.70928i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{9} - 2 q^{13} - 4 q^{23} - 10 q^{25} - 6 q^{27} - 4 q^{29} + 4 q^{35} + 2 q^{39} + 32 q^{43} - 6 q^{49} - 36 q^{53} + 16 q^{55} + 28 q^{61} + 20 q^{65} + 4 q^{69} + 10 q^{75} - 20 q^{77} + 32 q^{79} + 6 q^{81} + 4 q^{87} - 8 q^{91} - 16 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^9 - 2 * q^13 - 4 * q^23 - 10 * q^25 - 6 * q^27 - 4 * q^29 + 4 * q^35 + 2 * q^39 + 32 * q^43 - 6 * q^49 - 36 * q^53 + 16 * q^55 + 28 * q^61 + 20 * q^65 + 4 * q^69 + 10 * q^75 - 20 * q^77 + 32 * q^79 + 6 * q^81 + 4 * q^87 - 8 * q^91 - 16 * q^95

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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