LMFDB - Embedded newform 1682.2.b.k.1681.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1682,2,Mod(1681,1682)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1682, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1682.1681");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1682 = 2 \cdot 29^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1682.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:	$13.4308376200$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 37x^{14} + 548x^{12} + 4119x^{10} + 16415x^{8} + 33099x^{6} + 30128x^{4} + 10537x^{2} + 961$ x^16 + 37x^14 + 548x^12 + 4119x^10 + 16415x^8 + 33099x^6 + 30128x^4 + 10537*x^2 + 961
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1681.6
Root			$1.06263i$ of defining polynomial
Character	$\chi$	$=$	1682.1681
Dual form			1682.2.b.k.1681.11

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.06263i q^{3} -1.00000 q^{4} -4.09274 q^{5} +1.06263 q^{6} +2.34135 q^{7} +1.00000i q^{8} +1.87081 q^{9} +4.09274i q^{10} +4.82606i q^{11} -1.06263i q^{12} -4.41399 q^{13} -2.34135i q^{14} -4.34908i q^{15} +1.00000 q^{16} -2.51636i q^{17} -1.87081i q^{18} -2.07073i q^{19} +4.09274 q^{20} +2.48799i q^{21} +4.82606 q^{22} -2.02806 q^{23} -1.06263 q^{24} +11.7505 q^{25} +4.41399i q^{26} +5.17588i q^{27} -2.34135 q^{28} -4.34908 q^{30} -0.814848i q^{31} -1.00000i q^{32} -5.12833 q^{33} -2.51636 q^{34} -9.58253 q^{35} -1.87081 q^{36} +2.59308i q^{37} -2.07073 q^{38} -4.69045i q^{39} -4.09274i q^{40} +0.104627i q^{41} +2.48799 q^{42} -3.15502i q^{43} -4.82606i q^{44} -7.65675 q^{45} +2.02806i q^{46} -9.52807i q^{47} +1.06263i q^{48} -1.51809 q^{49} -11.7505i q^{50} +2.67397 q^{51} +4.41399 q^{52} -10.5005 q^{53} +5.17588 q^{54} -19.7518i q^{55} +2.34135i q^{56} +2.20043 q^{57} -8.95633 q^{59} +4.34908i q^{60} -5.91917i q^{61} -0.814848 q^{62} +4.38022 q^{63} -1.00000 q^{64} +18.0653 q^{65} +5.12833i q^{66} -5.20120 q^{67} +2.51636i q^{68} -2.15508i q^{69} +9.58253i q^{70} -5.54194 q^{71} +1.87081i q^{72} +2.52861i q^{73} +2.59308 q^{74} +12.4865i q^{75} +2.07073i q^{76} +11.2995i q^{77} -4.69045 q^{78} -13.5999i q^{79} -4.09274 q^{80} +0.112368 q^{81} +0.104627 q^{82} -13.9937 q^{83} -2.48799i q^{84} +10.2988i q^{85} -3.15502 q^{86} -4.82606 q^{88} -8.95536i q^{89} +7.65675i q^{90} -10.3347 q^{91} +2.02806 q^{92} +0.865884 q^{93} -9.52807 q^{94} +8.47497i q^{95} +1.06263 q^{96} -2.97302i q^{97} +1.51809i q^{98} +9.02865i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 16 q^{4} - 10 q^{5} - 2 q^{6} + 14 q^{7} - 26 q^{9} - 26 q^{13} + 16 q^{16} + 10 q^{20} + 14 q^{22} + 24 q^{23} + 2 q^{24} + 90 q^{25} - 14 q^{28} - 40 q^{30} + 8 q^{33} - 18 q^{34} + 26 q^{36} + 26 q^{38}+ \cdots - 2 q^{96}+O(q^{100})$ 16 * q - 16 * q^4 - 10 * q^5 - 2 * q^6 + 14 * q^7 - 26 * q^9 - 26 * q^13 + 16 * q^16 + 10 * q^20 + 14 * q^22 + 24 * q^23 + 2 * q^24 + 90 * q^25 - 14 * q^28 - 40 * q^30 + 8 * q^33 - 18 * q^34 + 26 * q^36 + 26 * q^38 - 68 * q^42 + 60 * q^45 + 54 * q^49 + 84 * q^51 + 26 * q^52 + 8 * q^53 - 4 * q^54 + 22 * q^57 + 16 * q^59 - 30 * q^62 - 24 * q^63 - 16 * q^64 - 30 * q^65 - 68 * q^67 + 22 * q^71 + 8 * q^74 - 18 * q^78 - 10 * q^80 + 24 * q^81 + 16 * q^82 + 20 * q^83 - 24 * q^86 - 14 * q^88 - 24 * q^91 - 24 * q^92 - 30 * q^93 - 26 * q^94 - 2 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times$ .

$n$	$843$
$\chi(n)$	$-1$

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

)

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

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.06263i	0.613511i	0.951788	+	0.306756i	$0.0992435\pi$
$3$	1.06263i	0.613511i	−0.951788	+	0.306756i	$0.900757\pi$
$4$	−1.00000	−0.500000
$5$	−4.09274	−1.83033	−0.915165	−	0.403080i	$-0.867940\pi$
$5$	−4.09274	−1.83033	−0.915165	+	0.403080i	$0.867940\pi$
$6$	1.06263	0.433818
$7$	2.34135	0.884947	0.442473	−	0.896782i	$-0.354101\pi$
$7$	2.34135	0.884947	0.442473	+	0.896782i	$0.354101\pi$
$8$	1.00000i	0.353553i
$9$	1.87081	0.623604
$10$	4.09274i	1.29424i
$11$	4.82606i	1.45511i	0.686048	+	0.727556i	$0.259344\pi$
$11$	4.82606i	1.45511i	−0.686048	+	0.727556i	$0.740656\pi$
$12$	− 1.06263i	− 0.306756i
$13$	−4.41399	−1.22422	−0.612110	−	0.790773i	$-0.709679\pi$
$13$	−4.41399	−1.22422	−0.612110	+	0.790773i	$0.709679\pi$
$14$	− 2.34135i	− 0.625752i
$15$	− 4.34908i	− 1.12293i
$16$	1.00000	0.250000
$17$	− 2.51636i	− 0.610307i	−0.952303	−	0.305153i	$-0.901292\pi$
$17$	− 2.51636i	− 0.610307i	0.952303	−	0.305153i	$-0.0987077\pi$
$18$	− 1.87081i	− 0.440954i
$19$	− 2.07073i	− 0.475058i	−0.971380	−	0.237529i	$-0.923662\pi$
$19$	− 2.07073i	− 0.475058i	0.971380	−	0.237529i	$-0.0763375\pi$
$20$	4.09274	0.915165
$21$	2.48799i	0.542925i
$22$	4.82606	1.02892
$23$	−2.02806	−0.422879	−0.211439	−	0.977391i	$-0.567815\pi$
$23$	−2.02806	−0.422879	−0.211439	+	0.977391i	$0.567815\pi$
$24$	−1.06263	−0.216909
$25$	11.7505	2.35011
$26$	4.41399i	0.865654i
$27$	5.17588i	0.996099i
$28$	−2.34135	−0.442473
$29$	0	0
$29$	0	0
$30$	−4.34908	−0.794030
$31$	− 0.814848i	− 0.146351i	−0.997319	−	0.0731755i	$-0.976687\pi$
$31$	− 0.814848i	− 0.146351i	0.997319	−	0.0731755i	$-0.0233133\pi$
$32$	− 1.00000i	− 0.176777i
$33$	−5.12833	−0.892728
$34$	−2.51636	−0.431552
$35$	−9.58253	−1.61974
$36$	−1.87081	−0.311802
$37$	2.59308i	0.426300i	0.977019	+	0.213150i	$0.0683723\pi$
$37$	2.59308i	0.426300i	−0.977019	+	0.213150i	$0.931628\pi$
$38$	−2.07073	−0.335917
$39$	− 4.69045i	− 0.751073i
$40$	− 4.09274i	− 0.647119i
$41$	0.104627i	0.0163400i	0.999967	+	0.00817002i	$0.00260063\pi$
$41$	0.104627i	0.0163400i	−0.999967	+	0.00817002i	$0.997399\pi$
$42$	2.48799	0.383906
$43$	− 3.15502i	− 0.481137i	−0.970632	−	0.240568i	$-0.922666\pi$
$43$	− 3.15502i	− 0.481137i	0.970632	−	0.240568i	$-0.0773338\pi$
$44$	− 4.82606i	− 0.727556i
$45$	−7.65675	−1.14140
$46$	2.02806i	0.299020i
$47$	− 9.52807i	− 1.38981i	−0.719100	−	0.694906i	$-0.755446\pi$
$47$	− 9.52807i	− 1.38981i	0.719100	−	0.694906i	$-0.244554\pi$
$48$	1.06263i	0.153378i
$49$	−1.51809	−0.216870
$50$	− 11.7505i	− 1.66178i
$51$	2.67397	0.374430
$52$	4.41399	0.612110
$53$	−10.5005	−1.44236	−0.721181	−	0.692747i	$-0.756400\pi$
$53$	−10.5005	−1.44236	−0.721181	+	0.692747i	$0.756400\pi$
$54$	5.17588	0.704349
$55$	− 19.7518i	− 2.66334i
$56$	2.34135i	0.312876i
$57$	2.20043	0.291454
$58$	0	0
$59$	−8.95633	−1.16601	−0.583007	−	0.812467i	$-0.698124\pi$
$59$	−8.95633	−1.16601	−0.583007	+	0.812467i	$0.698124\pi$
$60$	4.34908i	0.561464i
$61$	− 5.91917i	− 0.757872i	−0.925423	−	0.378936i	$-0.876290\pi$
$61$	− 5.91917i	− 0.757872i	0.925423	−	0.378936i	$-0.123710\pi$
$62$	−0.814848	−0.103486
$63$	4.38022	0.551856
$64$	−1.00000	−0.125000
$65$	18.0653	2.24073
$66$	5.12833i	0.631254i
$67$	−5.20120	−0.635428	−0.317714	−	0.948187i	$-0.602915\pi$
$67$	−5.20120	−0.635428	−0.317714	+	0.948187i	$0.602915\pi$
$68$	2.51636i	0.305153i
$69$	− 2.15508i	− 0.259441i
$70$	9.58253i	1.14533i
$71$	−5.54194	−0.657707	−0.328854	−	0.944381i	$-0.606662\pi$
$71$	−5.54194	−0.657707	−0.328854	+	0.944381i	$0.606662\pi$
$72$	1.87081i	0.220477i
$73$	2.52861i	0.295952i	0.988991	+	0.147976i	$0.0472758\pi$
$73$	2.52861i	0.295952i	−0.988991	+	0.147976i	$0.952724\pi$
$74$	2.59308	0.301440
$75$	12.4865i	1.44182i
$76$	2.07073i	0.237529i
$77$	11.2995i	1.28770i
$78$	−4.69045	−0.531089
$79$	− 13.5999i	− 1.53011i	−0.643967	−	0.765053i	$-0.722713\pi$
$79$	− 13.5999i	− 1.53011i	0.643967	−	0.765053i	$-0.277287\pi$
$80$	−4.09274	−0.457582
$81$	0.112368	0.0124853
$82$	0.104627	0.0115541
$83$	−13.9937	−1.53601	−0.768003	−	0.640446i	$-0.778750\pi$
$83$	−13.9937	−1.53601	−0.768003	+	0.640446i	$0.778750\pi$
$84$	− 2.48799i	− 0.271462i
$85$	10.2988i	1.11706i
$86$	−3.15502	−0.340215
$87$	0	0
$88$	−4.82606	−0.514460
$89$	− 8.95536i	− 0.949267i	−0.880184	−	0.474633i	$-0.842581\pi$
$89$	− 8.95536i	− 0.949267i	0.880184	−	0.474633i	$-0.157419\pi$
$90$	7.65675i	0.807092i
$91$	−10.3347	−1.08337
$92$	2.02806	0.211439
$93$	0.865884	0.0897880
$94$	−9.52807	−0.982746
$95$	8.47497i	0.869513i
$96$	1.06263	0.108455
$97$	− 2.97302i	− 0.301865i	−0.988544	−	0.150932i	$-0.951772\pi$
$97$	− 2.97302i	− 0.301865i	0.988544	−	0.150932i	$-0.0482276\pi$
$98$	1.51809i	0.153350i
$99$	9.02865i	0.907414i
$100$	−11.7505	−1.17505
$101$	− 1.34253i	− 0.133586i	−0.997767	−	0.0667932i	$-0.978723\pi$
$101$	− 1.34253i	− 0.133586i	0.997767	−	0.0667932i	$-0.0212768\pi$
$102$	− 2.67397i	− 0.264762i
$103$	−12.1572	−1.19788	−0.598941	−	0.800793i	$-0.704412\pi$
$103$	−12.1572	−1.19788	−0.598941	+	0.800793i	$0.704412\pi$
$104$	− 4.41399i	− 0.432827i
$105$	− 10.1827i	− 0.993731i
$106$	10.5005i	1.01990i
$107$	−0.164950	−0.0159463	−0.00797316	−	0.999968i	$-0.502538\pi$
$107$	−0.164950	−0.0159463	−0.00797316	+	0.999968i	$0.502538\pi$
$108$	− 5.17588i	− 0.498050i
$109$	−9.09810	−0.871440	−0.435720	−	0.900082i	$-0.643506\pi$
$109$	−9.09810	−0.871440	−0.435720	+	0.900082i	$0.643506\pi$
$110$	−19.7518	−1.88326
$111$	−2.75549	−0.261540
$112$	2.34135	0.221237
$113$	17.8409i	1.67833i	0.543873	+	0.839167i	$0.316957\pi$
$113$	17.8409i	1.67833i	−0.543873	+	0.839167i	$0.683043\pi$
$114$	− 2.20043i	− 0.206089i
$115$	8.30031	0.774007
$116$	0	0
$117$	−8.25774	−0.763428
$118$	8.95633i	0.824497i
$119$	− 5.89167i	− 0.540089i
$120$	4.34908	0.397015
$121$	−12.2909	−1.11735
$122$	−5.91917	−0.535896
$123$	−0.111180	−0.0100248
$124$	0.814848i	0.0731755i
$125$	−27.6282	−2.47114
$126$	− 4.38022i	− 0.390221i
$127$	4.78987i	0.425033i	0.977158	+	0.212516i	$0.0681658\pi$
$127$	4.78987i	0.425033i	−0.977158	+	0.212516i	$0.931834\pi$
$128$	1.00000i	0.0883883i
$129$	3.35263	0.295183
$130$	− 18.0653i	− 1.58443i
$131$	− 17.1054i	− 1.49450i	−0.664541	−	0.747252i	$-0.731373\pi$
$131$	− 17.1054i	− 1.49450i	0.664541	−	0.747252i	$-0.268627\pi$
$132$	5.12833	0.446364
$133$	− 4.84830i	− 0.420401i
$134$	5.20120i	0.449315i
$135$	− 21.1836i	− 1.82319i
$136$	2.51636	0.215776
$137$	17.8870i	1.52819i	0.645102	+	0.764097i	$0.276815\pi$
$137$	17.8870i	1.52819i	−0.645102	+	0.764097i	$0.723185\pi$
$138$	−2.15508	−0.183452
$139$	19.7082	1.67163	0.835814	−	0.549012i	$-0.184996\pi$
$139$	19.7082	1.67163	0.835814	+	0.549012i	$0.184996\pi$
$140$	9.58253	0.809872
$141$	10.1248	0.852666
$142$	5.54194i	0.465069i
$143$	− 21.3022i	− 1.78138i
$144$	1.87081	0.155901
$145$	0	0
$146$	2.52861	0.209269
$147$	− 1.61317i	− 0.133052i
$148$	− 2.59308i	− 0.213150i
$149$	8.42016	0.689807	0.344903	−	0.938638i	$-0.387912\pi$
$149$	8.42016	0.689807	0.344903	+	0.938638i	$0.387912\pi$
$150$	12.4865	1.01952
$151$	17.5963	1.43196	0.715981	−	0.698119i	$-0.245980\pi$
$151$	17.5963	1.43196	0.715981	+	0.698119i	$0.245980\pi$
$152$	2.07073	0.167958
$153$	− 4.70763i	− 0.380590i
$154$	11.2995	0.910539
$155$	3.33496i	0.267871i
$156$	4.69045i	0.375536i
$157$	− 14.1234i	− 1.12717i	−0.826058	−	0.563585i	$-0.809422\pi$
$157$	− 14.1234i	− 1.12717i	0.826058	−	0.563585i	$-0.190578\pi$
$158$	−13.5999	−1.08195
$159$	− 11.1582i	− 0.884905i
$160$	4.09274i	0.323560i
$161$	−4.74838	−0.374225
$162$	− 0.112368i	− 0.00882847i
$163$	− 11.5253i	− 0.902729i	−0.892340	−	0.451365i	$-0.850937\pi$
$163$	− 11.5253i	− 0.902729i	0.892340	−	0.451365i	$-0.149063\pi$
$164$	− 0.104627i	− 0.00817002i
$165$	20.9889	1.63399
$166$	13.9937i	1.08612i
$167$	−11.0161	−0.852450	−0.426225	−	0.904617i	$-0.640157\pi$
$167$	−11.0161	−0.852450	−0.426225	+	0.904617i	$0.640157\pi$
$168$	−2.48799	−0.191953
$169$	6.48329	0.498715
$170$	10.2988	0.789882
$171$	− 3.87395i	− 0.296248i
$172$	3.15502i	0.240568i
$173$	11.3228	0.860854	0.430427	−	0.902625i	$-0.358363\pi$
$173$	11.3228	0.860854	0.430427	+	0.902625i	$0.358363\pi$
$174$	0	0
$175$	27.5121	2.07972
$176$	4.82606i	0.363778i
$177$	− 9.51729i	− 0.715363i
$178$	−8.95536	−0.671233
$179$	−22.9508	−1.71542	−0.857711	−	0.514132i	$-0.828114\pi$
$179$	−22.9508	−1.71542	−0.857711	+	0.514132i	$0.828114\pi$
$180$	7.65675	0.570700
$181$	0.584454	0.0434421	0.0217211	−	0.999764i	$-0.493085\pi$
$181$	0.584454	0.0434421	0.0217211	+	0.999764i	$0.493085\pi$
$182$	10.3347i	0.766058i
$183$	6.28990	0.464963
$184$	− 2.02806i	− 0.149510i
$185$	− 10.6128i	− 0.780269i
$186$	− 0.865884i	− 0.0634897i
$187$	12.1441	0.888065
$188$	9.52807i	0.694906i
$189$	12.1185i	0.881495i
$190$	8.47497	0.614839
$191$	2.17573i	0.157430i	0.996897	+	0.0787150i	$0.0250817\pi$
$191$	2.17573i	0.157430i	−0.996897	+	0.0787150i	$0.974918\pi$
$192$	− 1.06263i	− 0.0766889i
$193$	10.9969i	0.791574i	0.918342	+	0.395787i	$0.129528\pi$
$193$	10.9969i	0.791574i	−0.918342	+	0.395787i	$0.870472\pi$
$194$	−2.97302	−0.213451
$195$	19.1968i	1.37471i
$196$	1.51809	0.108435
$197$	−2.58734	−0.184341	−0.0921703	−	0.995743i	$-0.529380\pi$
$197$	−2.58734	−0.184341	−0.0921703	+	0.995743i	$0.529380\pi$
$198$	9.02865	0.641638
$199$	−5.42095	−0.384281	−0.192141	−	0.981367i	$-0.561543\pi$
$199$	−5.42095	−0.384281	−0.192141	+	0.981367i	$0.561543\pi$
$200$	11.7505i	0.830888i
$201$	− 5.52696i	− 0.389842i
$202$	−1.34253	−0.0944599
$203$	0	0
$204$	−2.67397	−0.187215
$205$	− 0.428212i	− 0.0299076i
$206$	12.1572i	0.847030i
$207$	−3.79411	−0.263709
$208$	−4.41399	−0.306055
$209$	9.99348	0.691263
$210$	−10.1827	−0.702674
$211$	− 9.80716i	− 0.675152i	−0.941298	−	0.337576i	$-0.890393\pi$
$211$	− 9.80716i	− 0.675152i	0.941298	−	0.337576i	$-0.109607\pi$
$212$	10.5005	0.721181
$213$	− 5.88905i	− 0.403511i
$214$	0.164950i	0.0112758i
$215$	12.9127i	0.880638i
$216$	−5.17588	−0.352174
$217$	− 1.90784i	− 0.129513i
$218$	9.09810i	0.616201i
$219$	−2.68699	−0.181570
$220$	19.7518i	1.33167i
$221$	11.1072i	0.747150i
$222$	2.75549i	0.184937i
$223$	7.81817	0.523543	0.261772	−	0.965130i	$-0.415693\pi$
$223$	7.81817	0.523543	0.261772	+	0.965130i	$0.415693\pi$
$224$	− 2.34135i	− 0.156438i
$225$	21.9830	1.46553
$226$	17.8409	1.18676
$227$	6.97891	0.463206	0.231603	−	0.972810i	$-0.425603\pi$
$227$	6.97891	0.463206	0.231603	+	0.972810i	$0.425603\pi$
$228$	−2.20043	−0.145727
$229$	− 23.7679i	− 1.57063i	−0.619098	−	0.785314i	$-0.712502\pi$
$229$	− 23.7679i	− 1.57063i	0.619098	−	0.785314i	$-0.287498\pi$
$230$	− 8.30031i	− 0.547306i
$231$	−12.0072	−0.790017
$232$	0	0
$233$	4.73738	0.310356	0.155178	−	0.987887i	$-0.450405\pi$
$233$	4.73738	0.310356	0.155178	+	0.987887i	$0.450405\pi$
$234$	8.25774i	0.539825i
$235$	38.9959i	2.54381i
$236$	8.95633	0.583007
$237$	14.4517	0.938738
$238$	−5.89167	−0.381901
$239$	−6.83571	−0.442165	−0.221083	−	0.975255i	$-0.570959\pi$
$239$	−6.83571	−0.442165	−0.221083	+	0.975255i	$0.570959\pi$
$240$	− 4.34908i	− 0.280732i
$241$	4.12413	0.265659	0.132829	−	0.991139i	$-0.457594\pi$
$241$	4.12413	0.265659	0.132829	+	0.991139i	$0.457594\pi$
$242$	12.2909i	0.790087i
$243$	15.6471i	1.00376i
$244$	5.91917i	0.378936i
$245$	6.21314	0.396943
$246$	0.111180i	0.00708860i
$247$	9.14018i	0.581576i
$248$	0.814848	0.0517429
$249$	− 14.8702i	− 0.942358i
$250$	27.6282i	1.74736i
$251$	7.01765i	0.442950i	0.975166	+	0.221475i	$0.0710871\pi$
$251$	7.01765i	0.442950i	−0.975166	+	0.221475i	$0.928913\pi$
$252$	−4.38022	−0.275928
$253$	− 9.78752i	− 0.615336i
$254$	4.78987	0.300543
$255$	−10.9439	−0.685331
$256$	1.00000	0.0625000
$257$	−8.92910	−0.556982	−0.278491	−	0.960439i	$-0.589834\pi$
$257$	−8.92910	−0.556982	−0.278491	+	0.960439i	$0.589834\pi$
$258$	− 3.35263i	− 0.208726i
$259$	6.07131i	0.377253i
$260$	−18.0653	−1.12036
$261$	0	0
$262$	−17.1054	−1.05677
$263$	− 10.6660i	− 0.657691i	−0.944384	−	0.328845i	$-0.893340\pi$
$263$	− 10.6660i	− 0.657691i	0.944384	−	0.328845i	$-0.106660\pi$
$264$	− 5.12833i	− 0.315627i
$265$	42.9760	2.64000
$266$	−4.84830	−0.297269
$267$	9.51626	0.582386
$268$	5.20120	0.317714
$269$	− 1.83883i	− 0.112115i	−0.998428	−	0.0560577i	$-0.982147\pi$
$269$	− 1.83883i	− 0.112115i	0.998428	−	0.0560577i	$-0.0178531\pi$
$270$	−21.1836	−1.28919
$271$	10.4981i	0.637713i	0.947803	+	0.318857i	$0.103299\pi$
$271$	10.4981i	0.637713i	−0.947803	+	0.318857i	$0.896701\pi$
$272$	− 2.51636i	− 0.152577i
$273$	− 10.9820i	− 0.664659i
$274$	17.8870	1.08060
$275$	56.7088i	3.41967i
$276$	2.15508i	0.129720i
$277$	12.2091	0.733573	0.366787	−	0.930305i	$-0.380458\pi$
$277$	12.2091	0.733573	0.366787	+	0.930305i	$0.380458\pi$
$278$	− 19.7082i	− 1.18202i
$279$	− 1.52443i	− 0.0912650i
$280$	− 9.58253i	− 0.572666i
$281$	−32.3410	−1.92931	−0.964653	−	0.263525i	$-0.915115\pi$
$281$	−32.3410	−1.92931	−0.964653	+	0.263525i	$0.915115\pi$
$282$	− 10.1248i	− 0.602926i
$283$	−7.69360	−0.457337	−0.228669	−	0.973504i	$-0.573437\pi$
$283$	−7.69360	−0.457337	−0.228669	+	0.973504i	$0.573437\pi$
$284$	5.54194	0.328854
$285$	−9.00578	−0.533456
$286$	−21.3022	−1.25962
$287$	0.244969i	0.0144601i
$288$	− 1.87081i	− 0.110239i
$289$	10.6679	0.627526
$290$	0	0
$291$	3.15923	0.185197
$292$	− 2.52861i	− 0.147976i
$293$	− 28.4855i	− 1.66414i	−0.554671	−	0.832069i	$-0.687156\pi$
$293$	− 28.4855i	− 1.66414i	0.554671	−	0.832069i	$-0.312844\pi$
$294$	−1.61317	−0.0940819
$295$	36.6559	2.13419
$296$	−2.59308	−0.150720
$297$	−24.9791	−1.44944
$298$	− 8.42016i	− 0.487767i
$299$	8.95181	0.517697
$300$	− 12.4865i	− 0.720908i
$301$	− 7.38701i	− 0.425780i
$302$	− 17.5963i	− 1.01255i
$303$	1.42661	0.0819568
$304$	− 2.07073i	− 0.118765i
$305$	24.2256i	1.38716i
$306$	−4.70763	−0.269117
$307$	15.3241i	0.874592i	0.899318	+	0.437296i	$0.144064\pi$
$307$	15.3241i	0.874592i	−0.899318	+	0.437296i	$0.855936\pi$
$308$	− 11.2995i	− 0.643848i
$309$	− 12.9186i	− 0.734914i
$310$	3.33496	0.189413
$311$	14.2888i	0.810246i	0.914262	+	0.405123i	$0.132771\pi$
$311$	14.2888i	0.810246i	−0.914262	+	0.405123i	$0.867229\pi$
$312$	4.69045	0.265544
$313$	3.74014	0.211405	0.105703	−	0.994398i	$-0.466291\pi$
$313$	3.74014	0.211405	0.105703	+	0.994398i	$0.466291\pi$
$314$	−14.1234	−0.797030
$315$	−17.9271	−1.01008
$316$	13.5999i	0.765053i
$317$	33.7075i	1.89320i	0.322405	+	0.946602i	$0.395509\pi$
$317$	33.7075i	1.89320i	−0.322405	+	0.946602i	$0.604491\pi$
$318$	−11.1582	−0.625722
$319$	0	0
$320$	4.09274	0.228791
$321$	− 0.175281i	− 0.00978325i
$322$	4.74838i	0.264617i
$323$	−5.21070	−0.289931
$324$	−0.112368	−0.00624267
$325$	−51.8667	−2.87705
$326$	−11.5253	−0.638326
$327$	− 9.66794i	− 0.534638i
$328$	−0.104627	−0.00577707
$329$	− 22.3085i	− 1.22991i
$330$	− 20.9889i	− 1.15540i
$331$	− 18.1622i	− 0.998283i	−0.866520	−	0.499142i	$-0.833649\pi$
$331$	− 18.1622i	− 0.998283i	0.866520	−	0.499142i	$-0.166351\pi$
$332$	13.9937	0.768003
$333$	4.85117i	0.265842i
$334$	11.0161i	0.602773i
$335$	21.2872	1.16304
$336$	2.48799i	0.135731i
$337$	− 0.304315i	− 0.0165771i	−0.999966	−	0.00828855i	$-0.997362\pi$
$337$	− 0.304315i	− 0.0165771i	0.999966	−	0.00828855i	$-0.00263836\pi$
$338$	− 6.48329i	− 0.352645i
$339$	−18.9584	−1.02968
$340$	− 10.2988i	− 0.558531i
$341$	3.93251	0.212957
$342$	−3.87395	−0.209479
$343$	−19.9438	−1.07686
$344$	3.15502	0.170107
$345$	8.82018i	0.474862i
$346$	− 11.3228i	− 0.608716i
$347$	3.49887	0.187829	0.0939146	−	0.995580i	$-0.470062\pi$
$347$	3.49887	0.187829	0.0939146	+	0.995580i	$0.470062\pi$
$348$	0	0
$349$	26.3670	1.41139	0.705696	−	0.708515i	$-0.250634\pi$
$349$	26.3670	1.41139	0.705696	+	0.708515i	$0.250634\pi$
$350$	− 27.5121i	− 1.47058i
$351$	− 22.8463i	− 1.21944i
$352$	4.82606	0.257230
$353$	17.8689	0.951066	0.475533	−	0.879698i	$-0.342255\pi$
$353$	17.8689	0.951066	0.475533	+	0.879698i	$0.342255\pi$
$354$	−9.51729	−0.505838
$355$	22.6817	1.20382
$356$	8.95536i	0.474633i
$357$	6.26069	0.331351
$358$	22.9508i	1.21299i
$359$	− 2.89578i	− 0.152834i	−0.997076	−	0.0764168i	$-0.975652\pi$
$359$	− 2.89578i	− 0.152834i	0.997076	−	0.0764168i	$-0.0243480\pi$
$360$	− 7.65675i	− 0.403546i
$361$	14.7121	0.774320
$362$	− 0.584454i	− 0.0307182i
$363$	− 13.0607i	− 0.685508i
$364$	10.3347	0.541685
$365$	− 10.3490i	− 0.541689i
$366$	− 6.28990i	− 0.328779i
$367$	16.6465i	0.868942i	0.900686	+	0.434471i	$0.143065\pi$
$367$	16.6465i	0.868942i	−0.900686	+	0.434471i	$0.856935\pi$
$368$	−2.02806	−0.105720
$369$	0.195738i	0.0101897i
$370$	−10.6128	−0.551734
$371$	−24.5854	−1.27641
$372$	−0.865884	−0.0448940
$373$	−9.06657	−0.469449	−0.234725	−	0.972062i	$-0.575419\pi$
$373$	−9.06657	−0.469449	−0.234725	+	0.972062i	$0.575419\pi$
$374$	− 12.1441i	− 0.627957i
$375$	− 29.3586i	− 1.51607i
$376$	9.52807	0.491373
$377$	0	0
$378$	12.1185	0.623311
$379$	36.9042i	1.89564i	0.318801	+	0.947822i	$0.396720\pi$
$379$	36.9042i	1.89564i	−0.318801	+	0.947822i	$0.603280\pi$
$380$	− 8.47497i	− 0.434757i
$381$	−5.08988	−0.260762
$382$	2.17573	0.111320
$383$	5.74597	0.293606	0.146803	−	0.989166i	$-0.453102\pi$
$383$	5.74597	0.293606	0.146803	+	0.989166i	$0.453102\pi$
$384$	−1.06263	−0.0542273
$385$	− 46.2459i	− 2.35691i
$386$	10.9969	0.559728
$387$	− 5.90245i	− 0.300039i
$388$	2.97302i	0.150932i
$389$	1.82846i	0.0927066i	0.998925	+	0.0463533i	$0.0147600\pi$
$389$	1.82846i	0.0927066i	−0.998925	+	0.0463533i	$0.985240\pi$
$390$	19.1968	0.972067
$391$	5.10332i	0.258086i
$392$	− 1.51809i	− 0.0766750i
$393$	18.1767	0.916895
$394$	2.58734i	0.130348i
$395$	55.6608i	2.80060i
$396$	− 9.02865i	− 0.453707i
$397$	−19.5808	−0.982732	−0.491366	−	0.870953i	$-0.663502\pi$
$397$	−19.5808	−0.982732	−0.491366	+	0.870953i	$0.663502\pi$
$398$	5.42095i	0.271728i
$399$	5.15197	0.257921
$400$	11.7505	0.587526
$401$	12.0020	0.599349	0.299674	−	0.954042i	$-0.403122\pi$
$401$	12.0020	0.599349	0.299674	+	0.954042i	$0.403122\pi$
$402$	−5.52696	−0.275660
$403$	3.59673i	0.179166i
$404$	1.34253i	0.0667932i
$405$	−0.459893	−0.0228523
$406$	0	0
$407$	−12.5144	−0.620314
$408$	2.67397i	0.132381i
$409$	5.79580i	0.286584i	0.989680	+	0.143292i	$0.0457688\pi$
$409$	5.79580i	0.286584i	−0.989680	+	0.143292i	$0.954231\pi$
$410$	−0.428212	−0.0211479
$411$	−19.0074	−0.937564
$412$	12.1572	0.598941
$413$	−20.9699	−1.03186
$414$	3.79411i	0.186470i
$415$	57.2725	2.81140
$416$	4.41399i	0.216414i
$417$	20.9426i	1.02556i
$418$	− 9.99348i	− 0.488797i
$419$	18.9577	0.926145	0.463073	−	0.886320i	$-0.346747\pi$
$419$	18.9577	0.926145	0.463073	+	0.886320i	$0.346747\pi$
$420$	10.1827i	0.496866i
$421$	− 23.3207i	− 1.13658i	−0.822827	−	0.568292i	$-0.807604\pi$
$421$	− 23.3207i	− 1.13658i	0.822827	−	0.568292i	$-0.192396\pi$
$422$	−9.80716	−0.477405
$423$	− 17.8252i	− 0.866692i
$424$	− 10.5005i	− 0.509952i
$425$	− 29.5686i	− 1.43429i
$426$	−5.88905	−0.285325
$427$	− 13.8588i	− 0.670676i
$428$	0.164950	0.00797316
$429$	22.6364	1.09290
$430$	12.9127	0.622705
$431$	−5.71450	−0.275258	−0.137629	−	0.990484i	$-0.543948\pi$
$431$	−5.71450	−0.275258	−0.137629	+	0.990484i	$0.543948\pi$
$432$	5.17588i	0.249025i
$433$	− 16.0811i	− 0.772807i	−0.922330	−	0.386403i	$-0.873717\pi$
$433$	− 16.0811i	− 0.772807i	0.922330	−	0.386403i	$-0.126283\pi$
$434$	−1.90784	−0.0915794
$435$	0	0
$436$	9.09810	0.435720
$437$	4.19956i	0.200892i
$438$	2.68699i	0.128389i
$439$	−31.7509	−1.51539	−0.757695	−	0.652609i	$-0.773674\pi$
$439$	−31.7509	−1.51539	−0.757695	+	0.652609i	$0.773674\pi$
$440$	19.7518	0.941631
$441$	−2.84005	−0.135241
$442$	11.1072	0.528315
$443$	17.9434i	0.852518i	0.904601	+	0.426259i	$0.140169\pi$
$443$	17.9434i	0.852518i	−0.904601	+	0.426259i	$0.859831\pi$
$444$	2.75549	0.130770
$445$	36.6520i	1.73747i
$446$	− 7.81817i	− 0.370201i
$447$	8.94754i	0.423204i
$448$	−2.34135	−0.110618
$449$	− 3.20804i	− 0.151397i	−0.997131	−	0.0756984i	$-0.975881\pi$
$449$	− 3.20804i	− 0.151397i	0.997131	−	0.0756984i	$-0.0241186\pi$
$450$	− 21.9830i	− 1.03629i
$451$	−0.504938	−0.0237766
$452$	− 17.8409i	− 0.839167i
$453$	18.6984i	0.878525i
$454$	− 6.97891i	− 0.327536i
$455$	42.2972	1.98292
$456$	2.20043i	0.103044i
$457$	−34.3685	−1.60769	−0.803845	−	0.594839i	$-0.797216\pi$
$457$	−34.3685	−1.60769	−0.803845	+	0.594839i	$0.797216\pi$
$458$	−23.7679	−1.11060
$459$	13.0244	0.607926
$460$	−8.30031	−0.387004
$461$	29.4802i	1.37303i	0.727115	+	0.686515i	$0.240860\pi$
$461$	29.4802i	1.37303i	−0.727115	+	0.686515i	$0.759140\pi$
$462$	12.0072i	0.558626i
$463$	−32.1086	−1.49222	−0.746108	−	0.665825i	$-0.768080\pi$
$463$	−32.1086	−1.49222	−0.746108	+	0.665825i	$0.768080\pi$
$464$	0	0
$465$	−3.54384	−0.164342
$466$	− 4.73738i	− 0.219455i
$467$	34.6768i	1.60465i	0.596888	+	0.802324i	$0.296404\pi$
$467$	34.6768i	1.60465i	−0.596888	+	0.802324i	$0.703596\pi$
$468$	8.25774	0.381714
$469$	−12.1778	−0.562319
$470$	38.9959	1.79875
$471$	15.0080	0.691532
$472$	− 8.95633i	− 0.412248i
$473$	15.2263	0.700108
$474$	− 14.4517i	− 0.663788i
$475$	− 24.3322i	− 1.11644i
$476$	5.89167i	0.270044i
$477$	−19.6445	−0.899462
$478$	6.83571i	0.312658i
$479$	41.7958i	1.90970i	0.297089	+	0.954850i	$0.403984\pi$
$479$	41.7958i	1.90970i	−0.297089	+	0.954850i	$0.596016\pi$
$480$	−4.34908	−0.198508
$481$	− 11.4458i	− 0.521885i
$482$	− 4.12413i	− 0.187849i
$483$	− 5.04579i	− 0.229591i
$484$	12.2909	0.558676
$485$	12.1678i	0.552512i
$486$	15.6471	0.709765
$487$	−30.9514	−1.40254	−0.701271	−	0.712895i	$-0.747384\pi$
$487$	−30.9514	−1.40254	−0.701271	+	0.712895i	$0.747384\pi$
$488$	5.91917	0.267948
$489$	12.2471	0.553835
$490$	− 6.21314i	− 0.280681i
$491$	19.6356i	0.886140i	0.896487	+	0.443070i	$0.146111\pi$
$491$	19.6356i	0.886140i	−0.896487	+	0.443070i	$0.853889\pi$
$492$	0.111180	0.00501240
$493$	0	0
$494$	9.14018	0.411236
$495$	− 36.9519i	− 1.66087i
$496$	− 0.814848i	− 0.0365878i
$497$	−12.9756	−0.582036
$498$	−14.8702	−0.666347
$499$	40.0525	1.79300	0.896498	−	0.443048i	$-0.146103\pi$
$499$	40.0525	1.79300	0.896498	+	0.443048i	$0.146103\pi$
$500$	27.6282	1.23557
$501$	− 11.7061i	− 0.522988i
$502$	7.01765	0.313213
$503$	− 16.3539i	− 0.729185i	−0.931167	−	0.364592i	$-0.881208\pi$
$503$	− 16.3539i	− 0.729185i	0.931167	−	0.364592i	$-0.118792\pi$
$504$	4.38022i	0.195111i
$505$	5.49462i	0.244507i
$506$	−9.78752	−0.435108
$507$	6.88936i	0.305967i
$508$	− 4.78987i	− 0.212516i
$509$	−39.2493	−1.73969	−0.869847	−	0.493322i	$-0.835782\pi$
$509$	−39.2493	−1.73969	−0.869847	+	0.493322i	$0.835782\pi$
$510$	10.9439i	0.484602i
$511$	5.92036i	0.261901i
$512$	− 1.00000i	− 0.0441942i
$513$	10.7179	0.473205
$514$	8.92910i	0.393846i
$515$	49.7562	2.19252
$516$	−3.35263	−0.147591
$517$	45.9831	2.02233
$518$	6.07131	0.266758
$519$	12.0320i	0.528144i
$520$	18.0653i	0.792216i
$521$	10.9022	0.477633	0.238817	−	0.971065i	$-0.423241\pi$
$521$	10.9022	0.477633	0.238817	+	0.971065i	$0.423241\pi$
$522$	0	0
$523$	16.6229	0.726871	0.363435	−	0.931619i	$-0.381604\pi$
$523$	16.6229	0.726871	0.363435	+	0.931619i	$0.381604\pi$
$524$	17.1054i	0.747252i
$525$	29.2352i	1.27593i
$526$	−10.6660	−0.465058
$527$	−2.05045	−0.0893190
$528$	−5.12833	−0.223182
$529$	−18.8870	−0.821174
$530$	− 42.9760i	− 1.86676i
$531$	−16.7556	−0.727131
$532$	4.84830i	0.210201i
$533$	− 0.461824i	− 0.0200038i
$534$	− 9.51626i	− 0.411809i
$535$	0.675098	0.0291870
$536$	− 5.20120i	− 0.224658i
$537$	− 24.3883i	− 1.05243i
$538$	−1.83883	−0.0792775
$539$	− 7.32638i	− 0.315570i
$540$	21.1836i	0.911595i
$541$	− 42.7235i	− 1.83683i	−0.395620	−	0.918414i	$-0.629470\pi$
$541$	− 42.7235i	− 1.83683i	0.395620	−	0.918414i	$-0.370530\pi$
$542$	10.4981	0.450931
$543$	0.621060i	0.0266522i
$544$	−2.51636	−0.107888
$545$	37.2362	1.59502
$546$	−10.9820	−0.469985
$547$	−5.64174	−0.241223	−0.120612	−	0.992700i	$-0.538486\pi$
$547$	−5.64174	−0.241223	−0.120612	+	0.992700i	$0.538486\pi$
$548$	− 17.8870i	− 0.764097i
$549$	− 11.0736i	− 0.472612i
$550$	56.7088	2.41807
$551$	0	0
$552$	2.15508	0.0917262
$553$	− 31.8421i	− 1.35406i
$554$	− 12.2091i	− 0.518714i
$555$	11.2775	0.478704
$556$	−19.7082	−0.835814
$557$	13.7312	0.581809	0.290904	−	0.956752i	$-0.406044\pi$
$557$	13.7312	0.581809	0.290904	+	0.956752i	$0.406044\pi$
$558$	−1.52443	−0.0645341
$559$	13.9262i	0.589017i
$560$	−9.58253	−0.404936
$561$	12.9047i	0.544838i
$562$	32.3410i	1.36422i
$563$	− 8.73098i	− 0.367967i	−0.982929	−	0.183983i	$-0.941101\pi$
$563$	− 8.73098i	− 0.367967i	0.982929	−	0.183983i	$-0.0588993\pi$
$564$	−10.1248	−0.426333
$565$	− 73.0183i	− 3.07190i
$566$	7.69360i	0.323386i
$567$	0.263093	0.0110489
$568$	− 5.54194i	− 0.232535i
$569$	4.48796i	0.188145i	0.995565	+	0.0940726i	$0.0299886\pi$
$569$	4.48796i	0.188145i	−0.995565	+	0.0940726i	$0.970011\pi$
$570$	9.00578i	0.377211i
$571$	−24.5254	−1.02636	−0.513179	−	0.858282i	$-0.671532\pi$
$571$	−24.5254	−1.02636	−0.513179	+	0.858282i	$0.671532\pi$
$572$	21.3022i	0.890689i
$573$	−2.31200	−0.0965851
$574$	0.244969	0.0102248
$575$	−23.8307	−0.993810
$576$	−1.87081	−0.0779505
$577$	17.0299i	0.708963i	0.935063	+	0.354482i	$0.115343\pi$
$577$	17.0299i	0.708963i	−0.935063	+	0.354482i	$0.884657\pi$
$578$	− 10.6679i	− 0.443728i
$579$	−11.6857	−0.485640
$580$	0	0
$581$	−32.7641	−1.35928
$582$	− 3.15923i	− 0.130954i
$583$	− 50.6763i	− 2.09880i
$584$	−2.52861	−0.104635
$585$	33.7968	1.39733
$586$	−28.4855	−1.17672
$587$	20.5873	0.849730	0.424865	−	0.905257i	$-0.360322\pi$
$587$	20.5873	0.849730	0.424865	+	0.905257i	$0.360322\pi$
$588$	1.61317i	0.0665260i
$589$	−1.68733	−0.0695253
$590$	− 36.6559i	− 1.50910i
$591$	− 2.74940i	− 0.113095i
$592$	2.59308i	0.106575i
$593$	−3.53718	−0.145255	−0.0726273	−	0.997359i	$-0.523138\pi$
$593$	−3.53718	−0.145255	−0.0726273	+	0.997359i	$0.523138\pi$
$594$	24.9791i	1.02491i
$595$	24.1131i	0.988541i
$596$	−8.42016	−0.344903
$597$	− 5.76048i	− 0.235761i
$598$	− 8.95181i	− 0.366067i
$599$	17.0646i	0.697239i	0.937264	+	0.348620i	$0.113349\pi$
$599$	17.0646i	0.697239i	−0.937264	+	0.348620i	$0.886651\pi$
$600$	−12.4865	−0.509759
$601$	− 29.3558i	− 1.19745i	−0.800955	−	0.598725i	$-0.795674\pi$
$601$	− 29.3558i	− 1.19745i	0.800955	−	0.598725i	$-0.204326\pi$
$602$	−7.38701	−0.301072
$603$	−9.73046	−0.396255
$604$	−17.5963	−0.715981
$605$	50.3034	2.04512
$606$	− 1.42661i	− 0.0579522i
$607$	− 14.2568i	− 0.578664i	−0.957229	−	0.289332i	$-0.906567\pi$
$607$	− 14.2568i	− 0.578664i	0.957229	−	0.289332i	$-0.0934332\pi$
$608$	−2.07073	−0.0839792
$609$	0	0
$610$	24.2256	0.980867
$611$	42.0568i	1.70144i
$612$	4.70763i	0.190295i
$613$	18.1200	0.731859	0.365929	−	0.930643i	$-0.380751\pi$
$613$	18.1200	0.731859	0.365929	+	0.930643i	$0.380751\pi$
$614$	15.3241	0.618430
$615$	0.455033	0.0183487
$616$	−11.2995	−0.455270
$617$	− 10.5215i	− 0.423581i	−0.977315	−	0.211791i	$-0.932071\pi$
$617$	− 10.5215i	− 0.423581i	0.977315	−	0.211791i	$-0.0679295\pi$
$618$	−12.9186	−0.519663
$619$	18.1506i	0.729533i	0.931099	+	0.364766i	$0.118851\pi$
$619$	18.1506i	0.729533i	−0.931099	+	0.364766i	$0.881149\pi$
$620$	− 3.33496i	− 0.133935i
$621$	− 10.4970i	− 0.421229i
$622$	14.2888	0.572930
$623$	− 20.9676i	− 0.840050i
$624$	− 4.69045i	− 0.187768i
$625$	54.3223	2.17289
$626$	− 3.74014i	− 0.149486i
$627$	10.6194i	0.424098i
$628$	14.1234i	0.563585i
$629$	6.52512	0.260174
$630$	17.9271i	0.714233i
$631$	−41.9413	−1.66966	−0.834829	−	0.550509i	$-0.814434\pi$
$631$	−41.9413	−1.66966	−0.834829	+	0.550509i	$0.814434\pi$
$632$	13.5999	0.540974
$633$	10.4214	0.414214
$634$	33.7075	1.33870
$635$	− 19.6037i	− 0.777950i
$636$	11.1582i	0.442453i
$637$	6.70082	0.265496
$638$	0	0
$639$	−10.3679	−0.410149
$640$	− 4.09274i	− 0.161780i
$641$	10.5733i	0.417621i	0.977956	+	0.208810i	$0.0669591\pi$
$641$	10.5733i	0.417621i	−0.977956	+	0.208810i	$0.933041\pi$
$642$	−0.175281	−0.00691780
$643$	−26.1513	−1.03131	−0.515654	−	0.856797i	$-0.672451\pi$
$643$	−26.1513	−1.03131	−0.515654	+	0.856797i	$0.672451\pi$
$644$	4.74838	0.187113
$645$	−13.7215	−0.540282
$646$	5.21070i	0.205012i
$647$	−31.3433	−1.23223	−0.616116	−	0.787655i	$-0.711295\pi$
$647$	−31.3433	−1.23223	−0.616116	+	0.787655i	$0.711295\pi$
$648$	0.112368i	0.00441423i
$649$	− 43.2238i	− 1.69668i
$650$	51.8667i	2.03438i
$651$	2.02734	0.0794576
$652$	11.5253i	0.451365i
$653$	− 18.8481i	− 0.737582i	−0.929512	−	0.368791i	$-0.879772\pi$
$653$	− 18.8481i	− 0.737582i	0.929512	−	0.368791i	$-0.120228\pi$
$654$	−9.66794	−0.378046
$655$	70.0079i	2.73543i
$656$	0.104627i	0.00408501i
$657$	4.73055i	0.184557i
$658$	−22.3085	−0.869677
$659$	− 39.6720i	− 1.54540i	−0.634772	−	0.772700i	$-0.718906\pi$
$659$	− 39.6720i	− 1.54540i	0.634772	−	0.772700i	$-0.281094\pi$
$660$	−20.9889	−0.816993
$661$	10.9995	0.427831	0.213915	−	0.976852i	$-0.431378\pi$
$661$	10.9995	0.427831	0.213915	+	0.976852i	$0.431378\pi$
$662$	−18.1622	−0.705893
$663$	−11.8029	−0.458385
$664$	− 13.9937i	− 0.543060i
$665$	19.8429i	0.769473i
$666$	4.85117	0.187979
$667$	0	0
$668$	11.0161	0.426225
$669$	8.30784i	0.321200i
$670$	− 21.2872i	− 0.822395i
$671$	28.5663	1.10279
$672$	2.48799	0.0959765
$673$	−33.1651	−1.27842	−0.639209	−	0.769033i	$-0.720738\pi$
$673$	−33.1651	−1.27842	−0.639209	+	0.769033i	$0.720738\pi$
$674$	−0.304315	−0.0117218
$675$	60.8194i	2.34094i
$676$	−6.48329	−0.249357
$677$	− 22.4611i	− 0.863252i	−0.902053	−	0.431626i	$-0.857940\pi$
$677$	− 22.4611i	− 0.863252i	0.902053	−	0.431626i	$-0.142060\pi$
$678$	18.9584i	0.728092i
$679$	− 6.96088i	− 0.267134i
$680$	−10.2988	−0.394941
$681$	7.41602i	0.284182i
$682$	− 3.93251i	− 0.150583i
$683$	16.7244	0.639943	0.319971	−	0.947427i	$-0.396327\pi$
$683$	16.7244	0.639943	0.319971	+	0.947427i	$0.396327\pi$
$684$	3.87395i	0.148124i
$685$	− 73.2070i	− 2.79710i
$686$	19.9438i	0.761458i
$687$	25.2566	0.963598
$688$	− 3.15502i	− 0.120284i
$689$	46.3493	1.76577
$690$	8.82018	0.335778
$691$	1.93289	0.0735304	0.0367652	−	0.999324i	$-0.488295\pi$
$691$	1.93289	0.0735304	0.0367652	+	0.999324i	$0.488295\pi$
$692$	−11.3228	−0.430427
$693$	21.1392i	0.803013i
$694$	− 3.49887i	− 0.132815i
$695$	−80.6606	−3.05963
$696$	0	0
$697$	0.263280	0.00997243
$698$	− 26.3670i	− 0.998005i
$699$	5.03410i	0.190407i
$700$	−27.5121	−1.03986
$701$	15.0476	0.568340	0.284170	−	0.958774i	$-0.408282\pi$
$701$	15.0476	0.568340	0.284170	+	0.958774i	$0.408282\pi$
$702$	−22.8463	−0.862278
$703$	5.36957	0.202517
$704$	− 4.82606i	− 0.181889i
$705$	−41.4384	−1.56066
$706$	− 17.8689i	− 0.672505i
$707$	− 3.14332i	− 0.118217i
$708$	9.51729i	0.357682i
$709$	−36.7239	−1.37919	−0.689597	−	0.724193i	$-0.742212\pi$
$709$	−36.7239	−1.37919	−0.689597	+	0.724193i	$0.742212\pi$
$710$	− 22.6817i	− 0.851230i
$711$	− 25.4428i	− 0.954180i
$712$	8.95536	0.335616
$713$	1.65256i	0.0618887i
$714$	− 6.26069i	− 0.234300i
$715$	87.1843i	3.26051i
$716$	22.9508	0.857711
$717$	− 7.26385i	− 0.271273i
$718$	−2.89578	−0.108070
$719$	−21.6653	−0.807980	−0.403990	−	0.914763i	$-0.632377\pi$
$719$	−21.6653	−0.807980	−0.403990	+	0.914763i	$0.632377\pi$
$720$	−7.65675	−0.285350
$721$	−28.4642	−1.06006
$722$	− 14.7121i	− 0.547527i
$723$	4.38244i	0.162985i
$724$	−0.584454	−0.0217211
$725$	0	0
$726$	−13.0607	−0.484728
$727$	1.87449i	0.0695210i	0.999396	+	0.0347605i	$0.0110668\pi$
$727$	1.87449i	0.0695210i	−0.999396	+	0.0347605i	$0.988933\pi$
$728$	− 10.3347i	− 0.383029i
$729$	−16.2900	−0.603332
$730$	−10.3490	−0.383032
$731$	−7.93917	−0.293641
$732$	−6.28990	−0.232482
$733$	53.2656i	1.96741i	0.179791	+	0.983705i	$0.442458\pi$
$733$	53.2656i	1.96741i	−0.179791	+	0.983705i	$0.557542\pi$
$734$	16.6465	0.614435
$735$	6.60228i	0.243529i
$736$	2.02806i	0.0747551i
$737$	− 25.1013i	− 0.924618i
$738$	0.195738	0.00720521
$739$	20.1011i	0.739431i	0.929145	+	0.369716i	$0.120545\pi$
$739$	20.1011i	0.739431i	−0.929145	+	0.369716i	$0.879455\pi$
$740$	10.6128i	0.390135i
$741$	−9.71266	−0.356803
$742$	24.5854i	0.902560i
$743$	− 10.7246i	− 0.393449i	−0.980459	−	0.196725i	$-0.936970\pi$
$743$	− 10.7246i	− 0.393449i	0.980459	−	0.196725i	$-0.0630305\pi$
$744$	0.865884i	0.0317449i
$745$	−34.4615	−1.26257
$746$	9.06657i	0.331951i
$747$	−26.1795	−0.957859
$748$	−12.1441	−0.444032
$749$	−0.386206	−0.0141116
$750$	−29.3586	−1.07202
$751$	− 19.7336i	− 0.720090i	−0.932935	−	0.360045i	$-0.882761\pi$
$751$	− 19.7336i	− 0.720090i	0.932935	−	0.360045i	$-0.117239\pi$
$752$	− 9.52807i	− 0.347453i
$753$	−7.45719	−0.271755
$754$	0	0
$755$	−72.0169	−2.62096
$756$	− 12.1185i	− 0.440747i
$757$	− 1.39712i	− 0.0507793i	−0.999678	−	0.0253896i	$-0.991917\pi$
$757$	− 1.39712i	− 0.0507793i	0.999678	−	0.0253896i	$-0.00808264\pi$
$758$	36.9042	1.34042
$759$	10.4005	0.377516
$760$	−8.47497	−0.307419
$761$	−29.3599	−1.06430	−0.532148	−	0.846651i	$-0.678615\pi$
$761$	−29.3599	−1.06430	−0.532148	+	0.846651i	$0.678615\pi$
$762$	5.08988i	0.184387i
$763$	−21.3018	−0.771178
$764$	− 2.17573i	− 0.0787150i
$765$	19.2671i	0.696604i
$766$	− 5.74597i	− 0.207610i
$767$	39.5331	1.42746
$768$	1.06263i	0.0383445i
$769$	24.6926i	0.890439i	0.895421	+	0.445220i	$0.146874\pi$
$769$	24.6926i	0.890439i	−0.895421	+	0.445220i	$0.853126\pi$
$770$	−46.2459	−1.66659
$771$	− 9.48835i	− 0.341715i
$772$	− 10.9969i	− 0.395787i
$773$	9.23177i	0.332044i	0.986122	+	0.166022i	$0.0530922\pi$
$773$	9.23177i	0.332044i	−0.986122	+	0.166022i	$0.946908\pi$
$774$	−5.90245	−0.212159
$775$	− 9.57490i	− 0.343940i
$776$	2.97302	0.106725
$777$	−6.45157	−0.231449
$778$	1.82846	0.0655535
$779$	0.216655	0.00776247
$780$	− 19.1968i	− 0.687355i
$781$	− 26.7457i	− 0.957038i
$782$	5.10332	0.182494
$783$	0	0
$784$	−1.51809	−0.0542174
$785$	57.8034i	2.06309i
$786$	− 18.1767i	− 0.648343i
$787$	−6.56731	−0.234099	−0.117050	−	0.993126i	$-0.537344\pi$
$787$	−6.56731	−0.234099	−0.117050	+	0.993126i	$0.537344\pi$
$788$	2.58734	0.0921703
$789$	11.3340	0.403501
$790$	55.6608	1.98032
$791$	41.7719i	1.48524i
$792$	−9.02865	−0.320819
$793$	26.1271i	0.927802i
$794$	19.5808i	0.694896i
$795$	45.6677i	1.61967i
$796$	5.42095	0.192141
$797$	10.6456i	0.377088i	0.982065	+	0.188544i	$0.0603768\pi$
$797$	10.6456i	0.377088i	−0.982065	+	0.188544i	$0.939623\pi$
$798$	− 5.15197i	− 0.182378i
$799$	−23.9761	−0.848212
$800$	− 11.7505i	− 0.415444i
$801$	− 16.7538i	− 0.591966i
$802$	− 12.0020i	− 0.423804i
$803$	−12.2032	−0.430643
$804$	5.52696i	0.194921i
$805$	19.4339	0.684955
$806$	3.59673	0.126689
$807$	1.95400	0.0687840
$808$	1.34253	0.0472299
$809$	− 28.7914i	− 1.01225i	−0.862460	−	0.506126i	$-0.831077\pi$
$809$	− 28.7914i	− 1.01225i	0.862460	−	0.506126i	$-0.168923\pi$
$810$	0.459893i	0.0161590i
$811$	34.2025	1.20101	0.600505	−	0.799621i	$-0.294966\pi$
$811$	34.2025	1.20101	0.600505	+	0.799621i	$0.294966\pi$
$812$	0	0
$813$	−11.1556	−0.391244
$814$	12.5144i	0.438629i
$815$	47.1700i	1.65229i
$816$	2.67397	0.0936075
$817$	−6.53321	−0.228568
$818$	5.79580	0.202645
$819$	−19.3342	−0.675593
$820$	0.428212i	0.0149538i
$821$	−27.5825	−0.962637	−0.481319	−	0.876546i	$-0.659842\pi$
$821$	−27.5825	−0.962637	−0.481319	+	0.876546i	$0.659842\pi$
$822$	19.0074i	0.662958i
$823$	31.5932i	1.10127i	0.834746	+	0.550635i	$0.185614\pi$
$823$	31.5932i	1.10127i	−0.834746	+	0.550635i	$0.814386\pi$
$824$	− 12.1572i	− 0.423515i
$825$	−60.2606	−2.09801
$826$	20.9699i	0.729636i
$827$	− 33.6269i	− 1.16932i	−0.811278	−	0.584660i	$-0.801228\pi$
$827$	− 33.6269i	− 1.16932i	0.811278	−	0.584660i	$-0.198772\pi$
$828$	3.79411	0.131854
$829$	22.4142i	0.778478i	0.921137	+	0.389239i	$0.127262\pi$
$829$	22.4142i	0.778478i	−0.921137	+	0.389239i	$0.872738\pi$
$830$	− 57.2725i	− 1.98796i
$831$	12.9738i	0.450055i
$832$	4.41399	0.153028
$833$	3.82005i	0.132357i
$834$	20.9426	0.725183
$835$	45.0860	1.56026
$836$	−9.99348	−0.345632
$837$	4.21756	0.145780
$838$	− 18.9577i	− 0.654884i
$839$	18.7664i	0.647888i	0.946076	+	0.323944i	$0.105009\pi$
$839$	18.7664i	0.647888i	−0.946076	+	0.323944i	$0.894991\pi$
$840$	10.1827	0.351337
$841$	0	0
$842$	−23.3207	−0.803686
$843$	− 34.3667i	− 1.18365i
$844$	9.80716i	0.337576i
$845$	−26.5344	−0.912812
$846$	−17.8252	−0.612844
$847$	−28.7772	−0.988797
$848$	−10.5005	−0.360590
$849$	− 8.17547i	− 0.280581i
$850$	−29.5686	−1.01419
$851$	− 5.25891i	− 0.180273i
$852$	5.88905i	0.201755i
$853$	42.4529i	1.45356i	0.686870	+	0.726781i	$0.258984\pi$
$853$	42.4529i	1.45356i	−0.686870	+	0.726781i	$0.741016\pi$
$854$	−13.8588	−0.474240
$855$	15.8551i	0.542232i
$856$	− 0.164950i	− 0.00563788i
$857$	−42.4913	−1.45147	−0.725737	−	0.687972i	$-0.758501\pi$
$857$	−42.4913	−1.45147	−0.725737	+	0.687972i	$0.758501\pi$
$858$	− 22.6364i	− 0.772794i
$859$	33.4671i	1.14188i	0.820991	+	0.570941i	$0.193421\pi$
$859$	33.4671i	1.14188i	−0.820991	+	0.570941i	$0.806579\pi$
$860$	− 12.9127i	− 0.440319i
$861$	−0.260312	−0.00887141
$862$	5.71450i	0.194637i
$863$	−21.4668	−0.730739	−0.365369	−	0.930863i	$-0.619057\pi$
$863$	−21.4668	−0.730739	−0.365369	+	0.930863i	$0.619057\pi$
$864$	5.17588	0.176087
$865$	−46.3412	−1.57565
$866$	−16.0811	−0.546457
$867$	11.3361i	0.384994i
$868$	1.90784i	0.0647564i
$869$	65.6339	2.22648
$870$	0	0
$871$	22.9580	0.777903
$872$	− 9.09810i	− 0.308101i
$873$	− 5.56196i	− 0.188244i
$874$	4.19956	0.142052
$875$	−64.6872	−2.18683
$876$	2.68699	0.0907848
$877$	9.19139	0.310371	0.155186	−	0.987885i	$-0.450402\pi$
$877$	9.19139	0.310371	0.155186	+	0.987885i	$0.450402\pi$
$878$	31.7509i	1.07154i
$879$	30.2696	1.02097
$880$	− 19.7518i	− 0.665834i
$881$	− 15.6573i	− 0.527509i	−0.964590	−	0.263755i	$-0.915039\pi$
$881$	− 15.6573i	− 0.527509i	0.964590	−	0.263755i	$-0.0849609\pi$
$882$	2.84005i	0.0956296i
$883$	−54.7458	−1.84234	−0.921171	−	0.389159i	$-0.872766\pi$
$883$	−54.7458	−1.84234	−0.921171	+	0.389159i	$0.872766\pi$
$884$	− 11.1072i	− 0.373575i
$885$	38.9518i	1.30935i
$886$	17.9434	0.602821
$887$	6.00643i	0.201676i	0.994903	+	0.100838i	$0.0321524\pi$
$887$	6.00643i	0.201676i	−0.994903	+	0.100838i	$0.967848\pi$
$888$	− 2.75549i	− 0.0924683i
$889$	11.2148i	0.376131i
$890$	36.6520	1.22858
$891$	0.542295i	0.0181676i
$892$	−7.81817	−0.261772
$893$	−19.7301	−0.660242
$894$	8.94754	0.299251
$895$	93.9316	3.13979
$896$	2.34135i	0.0782190i
$897$	9.51249i	0.317613i
$898$	−3.20804	−0.107054
$899$	0	0
$900$	−21.9830	−0.732767
$901$	26.4232i	0.880283i
$902$	0.504938i	0.0168126i
$903$	7.84968	0.261221
$904$	−17.8409	−0.593381
$905$	−2.39202	−0.0795134
$906$	18.6984	0.621211
$907$	− 5.84164i	− 0.193968i	−0.995286	−	0.0969842i	$-0.969080\pi$
$907$	− 5.84164i	− 0.193968i	0.995286	−	0.0969842i	$-0.0309196\pi$
$908$	−6.97891	−0.231603
$909$	− 2.51161i	− 0.0833050i
$910$	− 42.2972i	− 1.40214i
$911$	20.9647i	0.694593i	0.937755	+	0.347296i	$0.112900\pi$
$911$	20.9647i	0.694593i	−0.937755	+	0.347296i	$0.887100\pi$
$912$	2.20043	0.0728634
$913$	− 67.5344i	− 2.23506i
$914$	34.3685i	1.13681i
$915$	−25.7429	−0.851036
$916$	23.7679i	0.785314i
$917$	− 40.0496i	− 1.32256i
$918$	− 13.0244i	− 0.429869i
$919$	47.8325	1.57785	0.788924	−	0.614491i	$-0.210638\pi$
$919$	47.8325	1.57785	0.788924	+	0.614491i	$0.210638\pi$
$920$	8.30031i	0.273653i
$921$	−16.2839	−0.536572
$922$	29.4802	0.970879
$923$	24.4621	0.805178
$924$	12.0072	0.395008
$925$	30.4701i	1.00185i
$926$	32.1086i	1.05516i
$927$	−22.7438	−0.747004
$928$	0	0
$929$	−25.6956	−0.843046	−0.421523	−	0.906818i	$-0.638504\pi$
$929$	−25.6956	−0.843046	−0.421523	+	0.906818i	$0.638504\pi$
$930$	3.54384i	0.116207i
$931$	3.14355i	0.103026i
$932$	−4.73738	−0.155178
$933$	−15.1838	−0.497095
$934$	34.6768	1.13466
$935$	−49.7027	−1.62545
$936$	− 8.25774i	− 0.269913i
$937$	39.4622	1.28917	0.644587	−	0.764531i	$-0.277029\pi$
$937$	39.4622	1.28917	0.644587	+	0.764531i	$0.277029\pi$
$938$	12.1778i	0.397620i
$939$	3.97440i	0.129700i
$940$	− 38.9959i	− 1.27191i
$941$	−30.6390	−0.998802	−0.499401	−	0.866371i	$-0.666447\pi$
$941$	−30.6390	−0.998802	−0.499401	+	0.866371i	$0.666447\pi$
$942$	− 15.0080i	− 0.488987i
$943$	− 0.212190i	− 0.00690985i
$944$	−8.95633	−0.291504
$945$	− 49.5981i	− 1.61343i
$946$	− 15.2263i	− 0.495051i
$947$	− 41.6241i	− 1.35260i	−0.736625	−	0.676301i	$-0.763582\pi$
$947$	− 41.6241i	− 1.35260i	0.736625	−	0.676301i	$-0.236418\pi$
$948$	−14.4517	−0.469369
$949$	− 11.1613i	− 0.362310i
$950$	−24.3322	−0.789440
$951$	−35.8187	−1.16150
$952$	5.89167	0.190950
$953$	21.2035	0.686848	0.343424	−	0.939180i	$-0.388413\pi$
$953$	21.2035	0.686848	0.343424	+	0.939180i	$0.388413\pi$
$954$	19.6445i	0.636016i
$955$	− 8.90469i	− 0.288149i
$956$	6.83571	0.221083
$957$	0	0
$958$	41.7958	1.35036
$959$	41.8798i	1.35237i
$960$	4.34908i	0.140366i
$961$	30.3360	0.978581
$962$	−11.4458	−0.369028
$963$	−0.308590	−0.00994419
$964$	−4.12413	−0.132829
$965$	− 45.0075i	− 1.44884i
$966$	−5.04579	−0.162346
$967$	2.94436i	0.0946844i	0.998879	+	0.0473422i	$0.0150751\pi$
$967$	2.94436i	0.0946844i	−0.998879	+	0.0473422i	$0.984925\pi$
$968$	− 12.2909i	− 0.395044i
$969$	− 5.53707i	− 0.177876i
$970$	12.1678	0.390685
$971$	− 40.0139i	− 1.28411i	−0.766660	−	0.642054i	$-0.778083\pi$
$971$	− 40.0139i	− 1.28411i	0.766660	−	0.642054i	$-0.221917\pi$
$972$	− 15.6471i	− 0.501880i
$973$	46.1438	1.47930
$974$	30.9514i	0.991747i
$975$	− 55.1153i	− 1.76510i
$976$	− 5.91917i	− 0.189468i
$977$	28.3655	0.907491	0.453746	−	0.891131i	$-0.350087\pi$
$977$	28.3655	0.907491	0.453746	+	0.891131i	$0.350087\pi$
$978$	− 12.2471i	− 0.391620i
$979$	43.2191	1.38129
$980$	−6.21314	−0.198471
$981$	−17.0208	−0.543433
$982$	19.6356	0.626596
$983$	8.30979i	0.265041i	0.991180	+	0.132521i	$0.0423071\pi$
$983$	8.30979i	0.265041i	−0.991180	+	0.132521i	$0.957693\pi$
$984$	− 0.111180i	− 0.00354430i
$985$	10.5893	0.337404
$986$	0	0
$987$	23.7058	0.754564
$988$	− 9.14018i	− 0.290788i
$989$	6.39856i	0.203462i
$990$	−36.9519	−1.17441
$991$	3.00560	0.0954762	0.0477381	−	0.998860i	$-0.484799\pi$
$991$	3.00560	0.0954762	0.0477381	+	0.998860i	$0.484799\pi$
$992$	−0.814848	−0.0258715
$993$	19.2997	0.612458
$994$	12.9756i	0.411561i
$995$	22.1866	0.703361
$996$	14.8702i	0.471179i
$997$	32.3634i	1.02496i	0.858700	+	0.512479i	$0.171273\pi$
$997$	32.3634i	1.02496i	−0.858700	+	0.512479i	$0.828727\pi$
$998$	− 40.0525i	− 1.26784i
$999$	−13.4215	−0.424637

Display

a_p

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p

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a_n

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a_n

instead) (See

a_n

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a_n

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up to: 50 250 1000 (See only

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a_p

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.2.b.k.1681.6		16
29.12	odd	4		1682.2.a.v.1.3	yes	8
29.17	odd	4		1682.2.a.u.1.6	✓	8
29.28	even	2	inner	1682.2.b.k.1681.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1682.2.a.u.1.6	✓	8	29.17	odd	4
1682.2.a.v.1.3	yes	8	29.12	odd	4
1682.2.b.k.1681.6		16	1.1	even	1	trivial
1682.2.b.k.1681.11		16	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1682.2.b.k.1681.6

Level	$1682$
Weight	$2$
Character	1682.1681
Analytic conductor	$13.431$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$