LMFDB - Embedded newform 1682.2.b.i.1681.8

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} + 30x^{10} + 341x^{8} + 1897x^{6} + 5456x^{4} + 7680x^{2} + 4096$ x^12 + 30x^10 + 341x^8 + 1897x^6 + 5456x^4 + 7680*x^2 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 58)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1681.8
Root			$-1.74168i$ of defining polynomial
Character	$\chi$	$=$	1682.1681
Dual form			1682.2.b.i.1681.5

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.74168i q^{3} -1.00000 q^{4} +0.494698 q^{5} +1.74168 q^{6} +3.13840 q^{7} -1.00000i q^{8} -0.0334417 q^{9} +0.494698i q^{10} +1.39672i q^{11} +1.74168i q^{12} +5.71545 q^{13} +3.13840i q^{14} -0.861605i q^{15} +1.00000 q^{16} +5.31800i q^{17} -0.0334417i q^{18} +4.96656i q^{19} -0.494698 q^{20} -5.46607i q^{21} -1.39672 q^{22} -0.813609 q^{23} -1.74168 q^{24} -4.75527 q^{25} +5.71545i q^{26} -5.16679i q^{27} -3.13840 q^{28} +0.861605 q^{30} +6.36328i q^{31} +1.00000i q^{32} +2.43263 q^{33} -5.31800 q^{34} +1.55256 q^{35} +0.0334417 q^{36} -8.92766i q^{37} -4.96656 q^{38} -9.95448i q^{39} -0.494698i q^{40} +4.01226i q^{41} +5.46607 q^{42} -1.39672i q^{43} -1.39672i q^{44} -0.0165436 q^{45} -0.813609i q^{46} +10.2982i q^{47} -1.74168i q^{48} +2.84952 q^{49} -4.75527i q^{50} +9.26224 q^{51} -5.71545 q^{52} +4.24359 q^{53} +5.16679 q^{54} +0.690954i q^{55} -3.13840i q^{56} +8.65014 q^{57} +11.1598 q^{59} +0.861605i q^{60} +4.86648i q^{61} -6.36328 q^{62} -0.104953 q^{63} -1.00000 q^{64} +2.82742 q^{65} +2.43263i q^{66} -7.13840 q^{67} -5.31800i q^{68} +1.41705i q^{69} +1.55256i q^{70} -6.03736 q^{71} +0.0334417i q^{72} -12.2792i q^{73} +8.92766 q^{74} +8.28215i q^{75} -4.96656i q^{76} +4.38345i q^{77} +9.95448 q^{78} -9.40223i q^{79} +0.494698 q^{80} -9.09921 q^{81} -4.01226 q^{82} +12.7094 q^{83} +5.46607i q^{84} +2.63080i q^{85} +1.39672 q^{86} +1.39672 q^{88} +1.79217i q^{89} -0.0165436i q^{90} +17.9373 q^{91} +0.813609 q^{92} +11.0828 q^{93} -10.2982 q^{94} +2.45695i q^{95} +1.74168 q^{96} -3.79630i q^{97} +2.84952i q^{98} -0.0467086i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 12 q^{4} - 4 q^{6} - 6 q^{7} - 24 q^{9} + 6 q^{13} + 12 q^{16} + 2 q^{22} - 28 q^{23} + 4 q^{24} + 8 q^{25} + 6 q^{28} + 54 q^{30} - 40 q^{33} + 12 q^{34} + 18 q^{35} + 24 q^{36} - 36 q^{38} + 20 q^{42}+ \cdots - 4 q^{96}+O(q^{100})$ 12 * q - 12 * q^4 - 4 * q^6 - 6 * q^7 - 24 * q^9 + 6 * q^13 + 12 * q^16 + 2 * q^22 - 28 * q^23 + 4 * q^24 + 8 * q^25 + 6 * q^28 + 54 * q^30 - 40 * q^33 + 12 * q^34 + 18 * q^35 + 24 * q^36 - 36 * q^38 + 20 * q^42 + 36 * q^45 + 2 * q^49 + 22 * q^51 - 6 * q^52 + 6 * q^53 + 46 * q^54 + 38 * q^57 + 38 * q^59 - 34 * q^62 + 66 * q^63 - 12 * q^64 + 22 * q^65 - 42 * q^67 + 18 * q^71 + 24 * q^74 + 20 * q^78 + 4 * q^81 - 30 * q^82 - 4 * q^83 - 2 * q^86 - 2 * q^88 + 44 * q^91 + 28 * q^92 - 2 * q^93 + 16 * q^94 - 4 * 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
$3$	− 1.74168i	− 1.00556i	−0.864415	−	0.502779i	$-0.832311\pi$
$3$	− 1.74168i	− 1.00556i	0.864415	−	0.502779i	$-0.167689\pi$
$4$	−1.00000	−0.500000
$5$	0.494698	0.221236	0.110618	−	0.993863i	$-0.464717\pi$
$5$	0.494698	0.221236	0.110618	+	0.993863i	$0.464717\pi$
$6$	1.74168	0.711037
$7$	3.13840	1.18620	0.593101	−	0.805128i	$-0.297904\pi$
$7$	3.13840	1.18620	0.593101	+	0.805128i	$0.297904\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−0.0334417	−0.0111472
$10$	0.494698i	0.156437i
$11$	1.39672i	0.421126i	0.977580	+	0.210563i	$0.0675297\pi$
$11$	1.39672i	0.421126i	−0.977580	+	0.210563i	$0.932470\pi$
$12$	1.74168i	0.502779i
$13$	5.71545	1.58518	0.792591	−	0.609754i	$-0.208732\pi$
$13$	5.71545	1.58518	0.792591	+	0.609754i	$0.208732\pi$
$14$	3.13840i	0.838771i
$15$	− 0.861605i	− 0.222465i
$16$	1.00000	0.250000
$17$	5.31800i	1.28980i	0.764265	+	0.644902i	$0.223102\pi$
$17$	5.31800i	1.28980i	−0.764265	+	0.644902i	$0.776898\pi$
$18$	− 0.0334417i	− 0.00788229i
$19$	4.96656i	1.13941i	0.821850	+	0.569703i	$0.192942\pi$
$19$	4.96656i	1.13941i	−0.821850	+	0.569703i	$0.807058\pi$
$20$	−0.494698	−0.110618
$21$	− 5.46607i	− 1.19279i
$22$	−1.39672	−0.297781
$23$	−0.813609	−0.169649	−0.0848246	−	0.996396i	$-0.527033\pi$
$23$	−0.813609	−0.169649	−0.0848246	+	0.996396i	$0.527033\pi$
$24$	−1.74168	−0.355519
$25$	−4.75527	−0.951055
$26$	5.71545i	1.12089i
$27$	− 5.16679i	− 0.994349i
$28$	−3.13840	−0.593101
$29$	0	0
$29$	0	0
$30$	0.861605	0.157307
$31$	6.36328i	1.14288i	0.820645	+	0.571439i	$0.193615\pi$
$31$	6.36328i	1.14288i	−0.820645	+	0.571439i	$0.806385\pi$
$32$	1.00000i	0.176777i
$33$	2.43263	0.423467
$34$	−5.31800	−0.912029
$35$	1.55256	0.262430
$36$	0.0334417	0.00557362
$37$	− 8.92766i	− 1.46770i	−0.679313	−	0.733849i	$-0.737722\pi$
$37$	− 8.92766i	− 1.46770i	0.679313	−	0.733849i	$-0.262278\pi$
$38$	−4.96656	−0.805682
$39$	− 9.95448i	− 1.59399i
$40$	− 0.494698i	− 0.0782187i
$41$	4.01226i	0.626610i	0.949652	+	0.313305i	$0.101436\pi$
$41$	4.01226i	0.626610i	−0.949652	+	0.313305i	$0.898564\pi$
$42$	5.46607	0.843433
$43$	− 1.39672i	− 0.212997i	−0.994313	−	0.106499i	$-0.966036\pi$
$43$	− 1.39672i	− 0.212997i	0.994313	−	0.106499i	$-0.0339640\pi$
$44$	− 1.39672i	− 0.210563i
$45$	−0.0165436	−0.00246617
$46$	− 0.813609i	− 0.119960i
$47$	10.2982i	1.50214i	0.660223	+	0.751070i	$0.270462\pi$
$47$	10.2982i	1.50214i	−0.660223	+	0.751070i	$0.729538\pi$
$48$	− 1.74168i	− 0.251390i
$49$	2.84952	0.407075
$50$	− 4.75527i	− 0.672497i
$51$	9.26224	1.29697
$52$	−5.71545	−0.792591
$53$	4.24359	0.582902	0.291451	−	0.956586i	$-0.405862\pi$
$53$	4.24359	0.582902	0.291451	+	0.956586i	$0.405862\pi$
$54$	5.16679	0.703111
$55$	0.690954i	0.0931682i
$56$	− 3.13840i	− 0.419386i
$57$	8.65014	1.14574
$58$	0	0
$59$	11.1598	1.45288	0.726438	−	0.687232i	$-0.241174\pi$
$59$	11.1598	1.45288	0.726438	+	0.687232i	$0.241174\pi$
$60$	0.861605i	0.111233i
$61$	4.86648i	0.623089i	0.950232	+	0.311544i	$0.100846\pi$
$61$	4.86648i	0.623089i	−0.950232	+	0.311544i	$0.899154\pi$
$62$	−6.36328	−0.808137
$63$	−0.104953	−0.0132229
$64$	−1.00000	−0.125000
$65$	2.82742	0.350699
$66$	2.43263i	0.299436i
$67$	−7.13840	−0.872094	−0.436047	−	0.899924i	$-0.643622\pi$
$67$	−7.13840	−0.872094	−0.436047	+	0.899924i	$0.643622\pi$
$68$	− 5.31800i	− 0.644902i
$69$	1.41705i	0.170592i
$70$	1.55256i	0.185566i
$71$	−6.03736	−0.716502	−0.358251	−	0.933625i	$-0.616627\pi$
$71$	−6.03736	−0.716502	−0.358251	+	0.933625i	$0.616627\pi$
$72$	0.0334417i	0.00394114i
$73$	− 12.2792i	− 1.43717i	−0.695439	−	0.718585i	$-0.744790\pi$
$73$	− 12.2792i	− 1.43717i	0.695439	−	0.718585i	$-0.255210\pi$
$74$	8.92766	1.03782
$75$	8.28215i	0.956341i
$76$	− 4.96656i	− 0.569703i
$77$	4.38345i	0.499541i
$78$	9.95448	1.12712
$79$	− 9.40223i	− 1.05783i	−0.848674	−	0.528917i	$-0.822598\pi$
$79$	− 9.40223i	− 1.05783i	0.848674	−	0.528917i	$-0.177402\pi$
$80$	0.494698	0.0553089
$81$	−9.09921	−1.01102
$82$	−4.01226	−0.443080
$83$	12.7094	1.39504	0.697520	−	0.716565i	$-0.254287\pi$
$83$	12.7094	1.39504	0.697520	+	0.716565i	$0.254287\pi$
$84$	5.46607i	0.596397i
$85$	2.63080i	0.285351i
$86$	1.39672	0.150612
$87$	0	0
$88$	1.39672	0.148891
$89$	1.79217i	0.189970i	0.995479	+	0.0949849i	$0.0302803\pi$
$89$	1.79217i	0.189970i	−0.995479	+	0.0949849i	$0.969720\pi$
$90$	− 0.0165436i	− 0.00174384i
$91$	17.9373	1.88034
$92$	0.813609	0.0848246
$93$	11.0828	1.14923
$94$	−10.2982	−1.06217
$95$	2.45695i	0.252078i
$96$	1.74168	0.177759
$97$	− 3.79630i	− 0.385456i	−0.981252	−	0.192728i	$-0.938267\pi$
$97$	− 3.79630i	− 0.385456i	0.981252	−	0.192728i	$-0.0617334\pi$
$98$	2.84952i	0.287845i
$99$	− 0.0467086i	− 0.00469439i
$100$	4.75527	0.475527
$101$	− 1.22182i	− 0.121576i	−0.998151	−	0.0607879i	$-0.980639\pi$
$101$	− 1.22182i	− 0.121576i	0.998151	−	0.0607879i	$-0.0193613\pi$
$102$	9.26224i	0.917098i
$103$	1.80814	0.178161	0.0890807	−	0.996024i	$-0.471607\pi$
$103$	1.80814	0.178161	0.0890807	+	0.996024i	$0.471607\pi$
$104$	− 5.71545i	− 0.560446i
$105$	− 2.70406i	− 0.263889i
$106$	4.24359i	0.412174i
$107$	4.15063	0.401257	0.200628	−	0.979667i	$-0.435702\pi$
$107$	4.15063	0.401257	0.200628	+	0.979667i	$0.435702\pi$
$108$	5.16679i	0.497174i
$109$	−14.5046	−1.38929	−0.694644	−	0.719354i	$-0.744438\pi$
$109$	−14.5046	−1.38929	−0.694644	+	0.719354i	$0.744438\pi$
$110$	−0.690954	−0.0658798
$111$	−15.5491	−1.47586
$112$	3.13840	0.296550
$113$	− 16.6834i	− 1.56944i	−0.619851	−	0.784720i	$-0.712807\pi$
$113$	− 16.6834i	− 1.56944i	0.619851	−	0.784720i	$-0.287193\pi$
$114$	8.65014i	0.810160i
$115$	−0.402491	−0.0375325
$116$	0	0
$117$	−0.191135	−0.0176704
$118$	11.1598i	1.02734i
$119$	16.6900i	1.52997i
$120$	−0.861605	−0.0786534
$121$	9.04918	0.822653
$122$	−4.86648	−0.440590
$123$	6.98807	0.630093
$124$	− 6.36328i	− 0.571439i
$125$	−4.82592	−0.431643
$126$	− 0.104953i	− 0.00934998i
$127$	− 12.6929i	− 1.12631i	−0.826351	−	0.563155i	$-0.809587\pi$
$127$	− 12.6929i	− 1.12631i	0.826351	−	0.563155i	$-0.190413\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−2.43263	−0.214181
$130$	2.82742i	0.247982i
$131$	14.7961i	1.29274i	0.763025	+	0.646369i	$0.223713\pi$
$131$	14.7961i	1.29274i	−0.763025	+	0.646369i	$0.776287\pi$
$132$	−2.43263	−0.211733
$133$	15.5870i	1.35157i
$134$	− 7.13840i	− 0.616663i
$135$	− 2.55600i	− 0.219986i
$136$	5.31800	0.456014
$137$	− 1.50925i	− 0.128944i	−0.997920	−	0.0644721i	$-0.979464\pi$
$137$	− 1.50925i	− 0.128944i	0.997920	−	0.0644721i	$-0.0205363\pi$
$138$	−1.41705	−0.120627
$139$	8.75639	0.742707	0.371353	−	0.928492i	$-0.378894\pi$
$139$	8.75639	0.742707	0.371353	+	0.928492i	$0.378894\pi$
$140$	−1.55256	−0.131215
$141$	17.9361	1.51049
$142$	− 6.03736i	− 0.506644i
$143$	7.98287i	0.667561i
$144$	−0.0334417	−0.00278681
$145$	0	0
$146$	12.2792	1.01623
$147$	− 4.96295i	− 0.409337i
$148$	8.92766i	0.733849i
$149$	4.74072	0.388375	0.194187	−	0.980964i	$-0.437793\pi$
$149$	4.74072	0.388375	0.194187	+	0.980964i	$0.437793\pi$
$150$	−8.28215	−0.676235
$151$	6.49415	0.528486	0.264243	−	0.964456i	$-0.414878\pi$
$151$	6.49415	0.528486	0.264243	+	0.964456i	$0.414878\pi$
$152$	4.96656	0.402841
$153$	− 0.177843i	− 0.0143777i
$154$	−4.38345	−0.353228
$155$	3.14790i	0.252846i
$156$	9.95448i	0.796996i
$157$	− 6.56139i	− 0.523656i	−0.965115	−	0.261828i	$-0.915675\pi$
$157$	− 6.56139i	− 0.523656i	0.965115	−	0.261828i	$-0.0843253\pi$
$158$	9.40223	0.748002
$159$	− 7.39097i	− 0.586142i
$160$	0.494698i	0.0391093i
$161$	−2.55343	−0.201238
$162$	− 9.09921i	− 0.714901i
$163$	− 5.65808i	− 0.443175i	−0.975140	−	0.221588i	$-0.928876\pi$
$163$	− 5.65808i	− 0.443175i	0.975140	−	0.221588i	$-0.0711239\pi$
$164$	− 4.01226i	− 0.313305i
$165$	1.20342	0.0936860
$166$	12.7094i	0.986442i
$167$	12.0891	0.935483	0.467742	−	0.883865i	$-0.345068\pi$
$167$	12.0891	0.935483	0.467742	+	0.883865i	$0.345068\pi$
$168$	−5.46607	−0.421717
$169$	19.6664	1.51280
$170$	−2.63080	−0.201773
$171$	− 0.166090i	− 0.0127012i
$172$	1.39672i	0.106499i
$173$	−11.6405	−0.885008	−0.442504	−	0.896767i	$-0.645910\pi$
$173$	−11.6405	−0.885008	−0.442504	+	0.896767i	$0.645910\pi$
$174$	0	0
$175$	−14.9239	−1.12814
$176$	1.39672i	0.105282i
$177$	− 19.4367i	− 1.46095i
$178$	−1.79217	−0.134329
$179$	18.3551	1.37193	0.685963	−	0.727637i	$-0.259381\pi$
$179$	18.3551	1.37193	0.685963	+	0.727637i	$0.259381\pi$
$180$	0.0165436	0.00123308
$181$	−9.89404	−0.735418	−0.367709	−	0.929941i	$-0.619858\pi$
$181$	−9.89404	−0.735418	−0.367709	+	0.929941i	$0.619858\pi$
$182$	17.9373i	1.32960i
$183$	8.47584	0.626552
$184$	0.813609i	0.0599801i
$185$	− 4.41650i	− 0.324707i
$186$	11.0828i	0.812629i
$187$	−7.42774	−0.543170
$188$	− 10.2982i	− 0.751070i
$189$	− 16.2154i	− 1.17950i
$190$	−2.45695	−0.178246
$191$	0.424259i	0.0306983i	0.999882	+	0.0153492i	$0.00488598\pi$
$191$	0.424259i	0.0306983i	−0.999882	+	0.0153492i	$0.995114\pi$
$192$	1.74168i	0.125695i
$193$	− 10.3248i	− 0.743193i	−0.928394	−	0.371597i	$-0.878810\pi$
$193$	− 10.3248i	− 0.743193i	0.928394	−	0.371597i	$-0.121190\pi$
$194$	3.79630	0.272558
$195$	− 4.92446i	− 0.352648i
$196$	−2.84952	−0.203537
$197$	−26.8428	−1.91247	−0.956234	−	0.292604i	$-0.905478\pi$
$197$	−26.8428	−1.91247	−0.956234	+	0.292604i	$0.905478\pi$
$198$	0.0467086	0.00331944
$199$	−6.80294	−0.482248	−0.241124	−	0.970494i	$-0.577516\pi$
$199$	−6.80294	−0.482248	−0.241124	+	0.970494i	$0.577516\pi$
$200$	4.75527i	0.336249i
$201$	12.4328i	0.876941i
$202$	1.22182	0.0859671
$203$	0	0
$204$	−9.26224	−0.648486
$205$	1.98486i	0.138629i
$206$	1.80814i	0.125979i
$207$	0.0272085	0.00189112
$208$	5.71545	0.396295
$209$	−6.93688	−0.479834
$210$	2.70406	0.186598
$211$	− 5.84061i	− 0.402084i	−0.979583	−	0.201042i	$-0.935567\pi$
$211$	− 5.84061i	− 0.402084i	0.979583	−	0.201042i	$-0.0644328\pi$
$212$	−4.24359	−0.291451
$213$	10.5151i	0.720485i
$214$	4.15063i	0.283731i
$215$	− 0.690954i	− 0.0471226i
$216$	−5.16679	−0.351555
$217$	19.9705i	1.35568i
$218$	− 14.5046i	− 0.982374i
$219$	−21.3864	−1.44516
$220$	− 0.690954i	− 0.0465841i
$221$	30.3948i	2.04457i
$222$	− 15.5491i	− 1.04359i
$223$	−21.9361	−1.46895	−0.734474	−	0.678637i	$-0.762571\pi$
$223$	−21.9361	−1.46895	−0.734474	+	0.678637i	$0.762571\pi$
$224$	3.13840i	0.209693i
$225$	0.159025	0.0106016
$226$	16.6834	1.10976
$227$	−2.15542	−0.143060	−0.0715302	−	0.997438i	$-0.522788\pi$
$227$	−2.15542	−0.143060	−0.0715302	+	0.997438i	$0.522788\pi$
$228$	−8.65014	−0.572870
$229$	− 9.29190i	− 0.614026i	−0.951705	−	0.307013i	$-0.900670\pi$
$229$	− 9.29190i	− 0.614026i	0.951705	−	0.307013i	$-0.0993296\pi$
$230$	− 0.402491i	− 0.0265395i
$231$	7.63456	0.502317
$232$	0	0
$233$	−11.7848	−0.772045	−0.386022	−	0.922489i	$-0.626151\pi$
$233$	−11.7848	−0.772045	−0.386022	+	0.922489i	$0.626151\pi$
$234$	− 0.191135i	− 0.0124949i
$235$	5.09448i	0.332327i
$236$	−11.1598	−0.726438
$237$	−16.3757	−1.06371
$238$	−16.6900	−1.08185
$239$	−14.3692	−0.929465	−0.464732	−	0.885451i	$-0.653849\pi$
$239$	−14.3692	−0.929465	−0.464732	+	0.885451i	$0.653849\pi$
$240$	− 0.861605i	− 0.0556164i
$241$	9.20432	0.592902	0.296451	−	0.955048i	$-0.404197\pi$
$241$	9.20432	0.592902	0.296451	+	0.955048i	$0.404197\pi$
$242$	9.04918i	0.581703i
$243$	0.347520i	0.0222934i
$244$	− 4.86648i	− 0.311544i
$245$	1.40965	0.0900595
$246$	6.98807i	0.445543i
$247$	28.3861i	1.80617i
$248$	6.36328	0.404068
$249$	− 22.1357i	− 1.40279i
$250$	− 4.82592i	− 0.305218i
$251$	5.91599i	0.373414i	0.982416	+	0.186707i	$0.0597815\pi$
$251$	5.91599i	0.373414i	−0.982416	+	0.186707i	$0.940219\pi$
$252$	0.104953	0.00661144
$253$	− 1.13638i	− 0.0714437i
$254$	12.6929	0.796421
$255$	4.58201	0.286937
$256$	1.00000	0.0625000
$257$	−26.7213	−1.66683	−0.833413	−	0.552650i	$-0.813617\pi$
$257$	−26.7213	−1.66683	−0.833413	+	0.552650i	$0.813617\pi$
$258$	− 2.43263i	− 0.151449i
$259$	− 28.0185i	− 1.74099i
$260$	−2.82742	−0.175349
$261$	0	0
$262$	−14.7961	−0.914104
$263$	− 23.3544i	− 1.44009i	−0.693926	−	0.720047i	$-0.744120\pi$
$263$	− 23.3544i	− 1.44009i	0.693926	−	0.720047i	$-0.255880\pi$
$264$	− 2.43263i	− 0.149718i
$265$	2.09930	0.128959
$266$	−15.5870	−0.955702
$267$	3.12138	0.191026
$268$	7.13840	0.436047
$269$	− 15.6093i	− 0.951714i	−0.879523	−	0.475857i	$-0.842138\pi$
$269$	− 15.6093i	− 0.951714i	0.879523	−	0.475857i	$-0.157862\pi$
$270$	2.55600	0.155553
$271$	− 0.219679i	− 0.0133446i	−0.999978	−	0.00667229i	$-0.997876\pi$
$271$	− 0.219679i	− 0.0133446i	0.999978	−	0.00667229i	$-0.00212387\pi$
$272$	5.31800i	0.322451i
$273$	− 31.2411i	− 1.89080i
$274$	1.50925	0.0911773
$275$	− 6.64177i	− 0.400514i
$276$	− 1.41705i	− 0.0852961i
$277$	3.75191	0.225431	0.112715	−	0.993627i	$-0.464045\pi$
$277$	3.75191	0.225431	0.112715	+	0.993627i	$0.464045\pi$
$278$	8.75639i	0.525173i
$279$	− 0.212799i	− 0.0127399i
$280$	− 1.55256i	− 0.0927831i
$281$	−24.4511	−1.45863	−0.729316	−	0.684177i	$-0.760161\pi$
$281$	−24.4511	−1.45863	−0.729316	+	0.684177i	$0.760161\pi$
$282$	17.9361i	1.06808i
$283$	−7.59751	−0.451625	−0.225813	−	0.974171i	$-0.572504\pi$
$283$	−7.59751	−0.451625	−0.225813	+	0.974171i	$0.572504\pi$
$284$	6.03736	0.358251
$285$	4.27921	0.253479
$286$	−7.98287	−0.472037
$287$	12.5921i	0.743286i
$288$	− 0.0334417i	− 0.00197057i
$289$	−11.2811	−0.663593
$290$	0	0
$291$	−6.61193	−0.387598
$292$	12.2792i	0.718585i
$293$	− 25.7223i	− 1.50271i	−0.659896	−	0.751357i	$-0.729400\pi$
$293$	− 25.7223i	− 1.50271i	0.659896	−	0.751357i	$-0.270600\pi$
$294$	4.96295	0.289445
$295$	5.52071	0.321428
$296$	−8.92766	−0.518910
$297$	7.21654	0.418746
$298$	4.74072i	0.274623i
$299$	−4.65014	−0.268925
$300$	− 8.28215i	− 0.478170i
$301$	− 4.38345i	− 0.252658i
$302$	6.49415i	0.373696i
$303$	−2.12802	−0.122252
$304$	4.96656i	0.284852i
$305$	2.40744i	0.137849i
$306$	0.177843	0.0101666
$307$	1.81399i	0.103530i	0.998659	+	0.0517649i	$0.0164847\pi$
$307$	1.81399i	0.103530i	−0.998659	+	0.0517649i	$0.983515\pi$
$308$	− 4.38345i	− 0.249770i
$309$	− 3.14920i	− 0.179152i
$310$	−3.14790	−0.178789
$311$	19.4100i	1.10064i	0.834953	+	0.550321i	$0.185495\pi$
$311$	19.4100i	1.10064i	−0.834953	+	0.550321i	$0.814505\pi$
$312$	−9.95448	−0.563561
$313$	33.8847	1.91528	0.957639	−	0.287972i	$-0.0929811\pi$
$313$	33.8847	1.91528	0.957639	+	0.287972i	$0.0929811\pi$
$314$	6.56139	0.370281
$315$	−0.0519202	−0.00292537
$316$	9.40223i	0.528917i
$317$	− 5.49761i	− 0.308777i	−0.988010	−	0.154388i	$-0.950659\pi$
$317$	− 5.49761i	− 0.308777i	0.988010	−	0.154388i	$-0.0493407\pi$
$318$	7.39097	0.414465
$319$	0	0
$320$	−0.494698	−0.0276545
$321$	− 7.22906i	− 0.403487i
$322$	− 2.55343i	− 0.142297i
$323$	−26.4121	−1.46961
$324$	9.09921	0.505511
$325$	−27.1785	−1.50759
$326$	5.65808	0.313372
$327$	25.2623i	1.39701i
$328$	4.01226	0.221540
$329$	32.3197i	1.78184i
$330$	1.20342i	0.0662460i
$331$	13.6463i	0.750067i	0.927011	+	0.375034i	$0.122369\pi$
$331$	13.6463i	0.750067i	−0.927011	+	0.375034i	$0.877631\pi$
$332$	−12.7094	−0.697520
$333$	0.298556i	0.0163608i
$334$	12.0891i	0.661487i
$335$	−3.53135	−0.192938
$336$	− 5.46607i	− 0.298199i
$337$	− 22.5673i	− 1.22932i	−0.788792	−	0.614660i	$-0.789293\pi$
$337$	− 22.5673i	− 1.22932i	0.788792	−	0.614660i	$-0.210707\pi$
$338$	19.6664i	1.06971i
$339$	−29.0571	−1.57816
$340$	− 2.63080i	− 0.142675i
$341$	−8.88770	−0.481296
$342$	0.166090	0.00898113
$343$	−13.0258	−0.703329
$344$	−1.39672	−0.0753059
$345$	0.701010i	0.0377411i
$346$	− 11.6405i	− 0.625795i
$347$	2.97305	0.159602	0.0798008	−	0.996811i	$-0.474572\pi$
$347$	2.97305	0.159602	0.0798008	+	0.996811i	$0.474572\pi$
$348$	0	0
$349$	−5.73913	−0.307208	−0.153604	−	0.988132i	$-0.549088\pi$
$349$	−5.73913	−0.307208	−0.153604	+	0.988132i	$0.549088\pi$
$350$	− 14.9239i	− 0.797717i
$351$	− 29.5305i	− 1.57622i
$352$	−1.39672	−0.0744453
$353$	6.47216	0.344478	0.172239	−	0.985055i	$-0.444900\pi$
$353$	6.47216	0.344478	0.172239	+	0.985055i	$0.444900\pi$
$354$	19.4367	1.03305
$355$	−2.98667	−0.158516
$356$	− 1.79217i	− 0.0949849i
$357$	29.0686	1.53847
$358$	18.3551i	0.970098i
$359$	10.7414i	0.566911i	0.958985	+	0.283455i	$0.0914808\pi$
$359$	10.7414i	0.566911i	−0.958985	+	0.283455i	$0.908519\pi$
$360$	0.0165436i	0 0.000871922i
$361$	−5.66670	−0.298247
$362$	− 9.89404i	− 0.520019i
$363$	− 15.7608i	− 0.827225i
$364$	−17.9373	−0.940172
$365$	− 6.07449i	− 0.317954i
$366$	8.47584i	0.443039i
$367$	18.9393i	0.988625i	0.869284	+	0.494313i	$0.164580\pi$
$367$	18.9393i	0.988625i	−0.869284	+	0.494313i	$0.835420\pi$
$368$	−0.813609	−0.0424123
$369$	− 0.134177i	− 0.00698498i
$370$	4.41650	0.229603
$371$	13.3181	0.691440
$372$	−11.0828	−0.574615
$373$	2.43402	0.126029	0.0630145	−	0.998013i	$-0.479929\pi$
$373$	2.43402	0.126029	0.0630145	+	0.998013i	$0.479929\pi$
$374$	− 7.42774i	− 0.384079i
$375$	8.40519i	0.434042i
$376$	10.2982	0.531087
$377$	0	0
$378$	16.2154	0.834031
$379$	− 17.1415i	− 0.880498i	−0.897876	−	0.440249i	$-0.854890\pi$
$379$	− 17.1415i	− 0.880498i	0.897876	−	0.440249i	$-0.145110\pi$
$380$	− 2.45695i	− 0.126039i
$381$	−22.1069	−1.13257
$382$	−0.424259	−0.0217070
$383$	23.7774	1.21497	0.607483	−	0.794332i	$-0.292179\pi$
$383$	23.7774	1.21497	0.607483	+	0.794332i	$0.292179\pi$
$384$	−1.74168	−0.0888796
$385$	2.16849i	0.110516i
$386$	10.3248	0.525517
$387$	0.0467086i	0.00237433i
$388$	3.79630i	0.192728i
$389$	11.5272i	0.584453i	0.956349	+	0.292227i	$0.0943962\pi$
$389$	11.5272i	0.584453i	−0.956349	+	0.292227i	$0.905604\pi$
$390$	4.92446	0.249360
$391$	− 4.32677i	− 0.218814i
$392$	− 2.84952i	− 0.143923i
$393$	25.7700	1.29992
$394$	− 26.8428i	− 1.35232i
$395$	− 4.65127i	− 0.234031i
$396$	0.0467086i	0.00234720i
$397$	28.9833	1.45463	0.727316	−	0.686303i	$-0.240767\pi$
$397$	28.9833	1.45463	0.727316	+	0.686303i	$0.240767\pi$
$398$	− 6.80294i	− 0.341001i
$399$	27.1476	1.35908
$400$	−4.75527	−0.237764
$401$	10.3297	0.515838	0.257919	−	0.966167i	$-0.416963\pi$
$401$	10.3297	0.515838	0.257919	+	0.966167i	$0.416963\pi$
$402$	−12.4328	−0.620091
$403$	36.3690i	1.81167i
$404$	1.22182i	0.0607879i
$405$	−4.50136	−0.223674
$406$	0	0
$407$	12.4694	0.618086
$408$	− 9.26224i	− 0.458549i
$409$	9.96171i	0.492575i	0.969197	+	0.246287i	$0.0792107\pi$
$409$	9.96171i	0.492575i	−0.969197	+	0.246287i	$0.920789\pi$
$410$	−1.98486	−0.0980252
$411$	−2.62863	−0.129661
$412$	−1.80814	−0.0890807
$413$	35.0237	1.72340
$414$	0.0272085i	0.00133722i
$415$	6.28733	0.308633
$416$	5.71545i	0.280223i
$417$	− 15.2508i	− 0.746835i
$418$	− 6.93688i	− 0.339294i
$419$	−28.3523	−1.38510	−0.692549	−	0.721371i	$-0.743512\pi$
$419$	−28.3523	−1.38510	−0.692549	+	0.721371i	$0.743512\pi$
$420$	2.70406i	0.131944i
$421$	9.51999i	0.463976i	0.972719	+	0.231988i	$0.0745230\pi$
$421$	9.51999i	0.463976i	−0.972719	+	0.231988i	$0.925477\pi$
$422$	5.84061	0.284317
$423$	− 0.344388i	− 0.0167447i
$424$	− 4.24359i	− 0.206087i
$425$	− 25.2885i	− 1.22667i
$426$	−10.5151	−0.509460
$427$	15.2729i	0.739109i
$428$	−4.15063	−0.200628
$429$	13.9036	0.671272
$430$	0.690954	0.0333207
$431$	−18.9749	−0.913990	−0.456995	−	0.889469i	$-0.651074\pi$
$431$	−18.9749	−0.913990	−0.456995	+	0.889469i	$0.651074\pi$
$432$	− 5.16679i	− 0.248587i
$433$	− 28.3148i	− 1.36072i	−0.732877	−	0.680361i	$-0.761823\pi$
$433$	− 28.3148i	− 1.36072i	0.732877	−	0.680361i	$-0.238177\pi$
$434$	−19.9705	−0.958613
$435$	0	0
$436$	14.5046	0.694644
$437$	− 4.04084i	− 0.193299i
$438$	− 21.3864i	− 1.02188i
$439$	19.8958	0.949577	0.474788	−	0.880100i	$-0.342525\pi$
$439$	19.8958	0.949577	0.474788	+	0.880100i	$0.342525\pi$
$440$	0.690954	0.0329399
$441$	−0.0952930	−0.00453776
$442$	−30.3948	−1.44573
$443$	35.5600i	1.68951i	0.535157	+	0.844753i	$0.320253\pi$
$443$	35.5600i	1.68951i	−0.535157	+	0.844753i	$0.679747\pi$
$444$	15.5491	0.737928
$445$	0.886584i	0.0420281i
$446$	− 21.9361i	− 1.03870i
$447$	− 8.25681i	− 0.390534i
$448$	−3.13840	−0.148275
$449$	10.8511i	0.512095i	0.966664	+	0.256048i	$0.0824204\pi$
$449$	10.8511i	0.512095i	−0.966664	+	0.256048i	$0.917580\pi$
$450$	0.159025i	0.00749649i
$451$	−5.60400	−0.263882
$452$	16.6834i	0.784720i
$453$	− 11.3107i	− 0.531424i
$454$	− 2.15542i	− 0.101159i
$455$	8.87357	0.416000
$456$	− 8.65014i	− 0.405080i
$457$	−32.3358	−1.51261	−0.756304	−	0.654221i	$-0.772997\pi$
$457$	−32.3358	−1.51261	−0.756304	+	0.654221i	$0.772997\pi$
$458$	9.29190	0.434182
$459$	27.4770	1.28251
$460$	0.402491	0.0187662
$461$	− 25.4047i	− 1.18321i	−0.806227	−	0.591607i	$-0.798494\pi$
$461$	− 25.4047i	− 1.18321i	0.806227	−	0.591607i	$-0.201506\pi$
$462$	7.63456i	0.355192i
$463$	−3.22072	−0.149680	−0.0748398	−	0.997196i	$-0.523845\pi$
$463$	−3.22072	−0.149680	−0.0748398	+	0.997196i	$0.523845\pi$
$464$	0	0
$465$	5.48263	0.254251
$466$	− 11.7848i	− 0.545918i
$467$	22.8565i	1.05767i	0.848724	+	0.528836i	$0.177371\pi$
$467$	22.8565i	1.05767i	−0.848724	+	0.528836i	$0.822629\pi$
$468$	0.191135	0.00883520
$469$	−22.4031	−1.03448
$470$	−5.09448	−0.234991
$471$	−11.4278	−0.526567
$472$	− 11.1598i	− 0.513669i
$473$	1.95082	0.0896987
$474$	− 16.3757i	− 0.752159i
$475$	− 23.6173i	− 1.08364i
$476$	− 16.6900i	− 0.764984i
$477$	−0.141913	−0.00649775
$478$	− 14.3692i	− 0.657231i
$479$	− 2.14546i	− 0.0980284i	−0.998798	−	0.0490142i	$-0.984392\pi$
$479$	− 2.14546i	− 0.0980284i	0.998798	−	0.0490142i	$-0.0156080\pi$
$480$	0.861605	0.0393267
$481$	− 51.0256i	− 2.32657i
$482$	9.20432i	0.419245i
$483$	4.44725i	0.202357i
$484$	−9.04918	−0.411326
$485$	− 1.87802i	− 0.0852766i
$486$	−0.347520	−0.0157638
$487$	21.7759	0.986761	0.493381	−	0.869814i	$-0.335761\pi$
$487$	21.7759	0.986761	0.493381	+	0.869814i	$0.335761\pi$
$488$	4.86648	0.220295
$489$	−9.85456	−0.445639
$490$	1.40965i	0.0636817i
$491$	38.8631i	1.75387i	0.480610	+	0.876935i	$0.340415\pi$
$491$	38.8631i	1.75387i	−0.480610	+	0.876935i	$0.659585\pi$
$492$	−6.98807	−0.315047
$493$	0	0
$494$	−28.3861	−1.27715
$495$	− 0.0231067i	− 0.00103857i
$496$	6.36328i	0.285720i
$497$	−18.9476	−0.849916
$498$	22.1357	0.991925
$499$	−39.2992	−1.75928	−0.879638	−	0.475644i	$-0.842215\pi$
$499$	−39.2992	−1.75928	−0.879638	+	0.475644i	$0.842215\pi$
$500$	4.82592	0.215822
$501$	− 21.0553i	− 0.940683i
$502$	−5.91599	−0.264043
$503$	− 28.0514i	− 1.25075i	−0.780324	−	0.625375i	$-0.784946\pi$
$503$	− 28.0514i	− 1.25075i	0.780324	−	0.625375i	$-0.215054\pi$
$504$	0.104953i	0.00467499i
$505$	− 0.604433i	− 0.0268969i
$506$	1.13638	0.0505183
$507$	− 34.2525i	− 1.52121i
$508$	12.6929i	0.563155i
$509$	1.70457	0.0755535	0.0377768	−	0.999286i	$-0.487972\pi$
$509$	1.70457	0.0755535	0.0377768	+	0.999286i	$0.487972\pi$
$510$	4.58201i	0.202895i
$511$	− 38.5369i	− 1.70477i
$512$	1.00000i	0.0441942i
$513$	25.6612	1.13297
$514$	− 26.7213i	− 1.17862i
$515$	0.894484	0.0394157
$516$	2.43263	0.107091
$517$	−14.3836	−0.632590
$518$	28.0185	1.23106
$519$	20.2739i	0.889927i
$520$	− 2.82742i	− 0.123991i
$521$	21.0834	0.923678	0.461839	−	0.886964i	$-0.347190\pi$
$521$	21.0834	0.923678	0.461839	+	0.886964i	$0.347190\pi$
$522$	0	0
$523$	−8.93606	−0.390747	−0.195373	−	0.980729i	$-0.562592\pi$
$523$	−8.93606	−0.390747	−0.195373	+	0.980729i	$0.562592\pi$
$524$	− 14.7961i	− 0.646369i
$525$	25.9927i	1.13441i
$526$	23.3544	1.01830
$527$	−33.8399	−1.47409
$528$	2.43263	0.105867
$529$	−22.3380	−0.971219
$530$	2.09930i	0.0911877i
$531$	−0.373201	−0.0161956
$532$	− 15.5870i	− 0.675783i
$533$	22.9319i	0.993291i
$534$	3.12138i	0.135076i
$535$	2.05331	0.0887723
$536$	7.13840i	0.308332i
$537$	− 31.9687i	− 1.37955i
$538$	15.6093	0.672964
$539$	3.97998i	0.171430i
$540$	2.55600i	0.109993i
$541$	− 11.9679i	− 0.514541i	−0.966339	−	0.257271i	$-0.917177\pi$
$541$	− 11.9679i	− 0.514541i	0.966339	−	0.257271i	$-0.0828232\pi$
$542$	0.219679	0.00943604
$543$	17.2322i	0.739506i
$544$	−5.31800	−0.228007
$545$	−7.17539	−0.307360
$546$	31.2411	1.33699
$547$	−24.1219	−1.03138	−0.515689	−	0.856776i	$-0.672464\pi$
$547$	−24.1219	−1.03138	−0.515689	+	0.856776i	$0.672464\pi$
$548$	1.50925i	0.0644721i
$549$	− 0.162743i	− 0.00694572i
$550$	6.64177	0.283206
$551$	0	0
$552$	1.41705	0.0603134
$553$	− 29.5079i	− 1.25480i
$554$	3.75191i	0.159404i
$555$	−7.69211	−0.326512
$556$	−8.75639	−0.371353
$557$	18.2020	0.771243	0.385622	−	0.922657i	$-0.373987\pi$
$557$	18.2020	0.771243	0.385622	+	0.922657i	$0.373987\pi$
$558$	0.212799	0.00900849
$559$	− 7.98287i	− 0.337639i
$560$	1.55256	0.0656076
$561$	12.9367i	0.546189i
$562$	− 24.4511i	− 1.03141i
$563$	31.5149i	1.32820i	0.747646	+	0.664098i	$0.231184\pi$
$563$	31.5149i	1.32820i	−0.747646	+	0.664098i	$0.768816\pi$
$564$	−17.9361	−0.755244
$565$	− 8.25323i	− 0.347216i
$566$	− 7.59751i	− 0.319347i
$567$	−28.5569	−1.19928
$568$	6.03736i	0.253322i
$569$	28.4651i	1.19332i	0.802494	+	0.596660i	$0.203506\pi$
$569$	28.4651i	1.19332i	−0.802494	+	0.596660i	$0.796494\pi$
$570$	4.27921i	0.179236i
$571$	−4.26294	−0.178399	−0.0891993	−	0.996014i	$-0.528431\pi$
$571$	−4.26294	−0.178399	−0.0891993	+	0.996014i	$0.528431\pi$
$572$	− 7.98287i	− 0.333781i
$573$	0.738923	0.0308690
$574$	−12.5921	−0.525583
$575$	3.86893	0.161346
$576$	0.0334417	0.00139340
$577$	− 22.5839i	− 0.940179i	−0.882619	−	0.470090i	$-0.844222\pi$
$577$	− 22.5839i	− 0.940179i	0.882619	−	0.470090i	$-0.155778\pi$
$578$	− 11.2811i	− 0.469231i
$579$	−17.9824	−0.747324
$580$	0	0
$581$	39.8872	1.65480
$582$	− 6.61193i	− 0.274073i
$583$	5.92710i	0.245475i
$584$	−12.2792	−0.508116
$585$	−0.0945539	−0.00390932
$586$	25.7223	1.06258
$587$	−33.5049	−1.38289	−0.691447	−	0.722427i	$-0.743026\pi$
$587$	−33.5049	−1.38289	−0.691447	+	0.722427i	$0.743026\pi$
$588$	4.96295i	0.204669i
$589$	−31.6036	−1.30220
$590$	5.52071i	0.227284i
$591$	46.7514i	1.92310i
$592$	− 8.92766i	− 0.366925i
$593$	−13.3565	−0.548488	−0.274244	−	0.961660i	$-0.588428\pi$
$593$	−13.3565	−0.548488	−0.274244	+	0.961660i	$0.588428\pi$
$594$	7.21654i	0.296098i
$595$	8.25650i	0.338484i
$596$	−4.74072	−0.194187
$597$	11.8485i	0.484928i
$598$	− 4.65014i	− 0.190159i
$599$	0.915670i	0.0374133i	0.999825	+	0.0187066i	$0.00595485\pi$
$599$	0.915670i	0.0374133i	−0.999825	+	0.0187066i	$0.994045\pi$
$600$	8.28215	0.338118
$601$	5.77455i	0.235549i	0.993040	+	0.117774i	$0.0375760\pi$
$601$	5.77455i	0.235549i	−0.993040	+	0.117774i	$0.962424\pi$
$602$	4.38345	0.178656
$603$	0.238720	0.00972144
$604$	−6.49415	−0.264243
$605$	4.47661	0.182000
$606$	− 2.12802i	− 0.0864449i
$607$	14.9822i	0.608108i	0.952655	+	0.304054i	$0.0983403\pi$
$607$	14.9822i	0.608108i	−0.952655	+	0.304054i	$0.901660\pi$
$608$	−4.96656	−0.201421
$609$	0	0
$610$	−2.40744	−0.0974743
$611$	58.8586i	2.38116i
$612$	0.177843i	0.00718887i
$613$	23.9141	0.965881	0.482940	−	0.875653i	$-0.339569\pi$
$613$	23.9141	0.965881	0.482940	+	0.875653i	$0.339569\pi$
$614$	−1.81399	−0.0732067
$615$	3.45699	0.139399
$616$	4.38345	0.176614
$617$	14.7070i	0.592080i	0.955176	+	0.296040i	$0.0956661\pi$
$617$	14.7070i	0.592080i	−0.955176	+	0.296040i	$0.904334\pi$
$618$	3.14920	0.126679
$619$	− 22.9142i	− 0.920998i	−0.887660	−	0.460499i	$-0.847671\pi$
$619$	− 22.9142i	− 0.920998i	0.887660	−	0.460499i	$-0.152329\pi$
$620$	− 3.14790i	− 0.126423i
$621$	4.20375i	0.168691i
$622$	−19.4100	−0.778271
$623$	5.62454i	0.225342i
$624$	− 9.95448i	− 0.398498i
$625$	21.3890	0.855560
$626$	33.8847i	1.35431i
$627$	12.0818i	0.482501i
$628$	6.56139i	0.261828i
$629$	47.4773	1.89304
$630$	− 0.0519202i	− 0.00206855i
$631$	−13.1112	−0.521947	−0.260974	−	0.965346i	$-0.584044\pi$
$631$	−13.1112	−0.521947	−0.260974	+	0.965346i	$0.584044\pi$
$632$	−9.40223	−0.374001
$633$	−10.1725	−0.404319
$634$	5.49761	0.218338
$635$	− 6.27914i	− 0.249180i
$636$	7.39097i	0.293071i
$637$	16.2863	0.645287
$638$	0	0
$639$	0.201900	0.00798702
$640$	− 0.494698i	− 0.0195547i
$641$	− 43.0960i	− 1.70219i	−0.525011	−	0.851096i	$-0.675939\pi$
$641$	− 43.0960i	− 1.70219i	0.525011	−	0.851096i	$-0.324061\pi$
$642$	7.22906	0.285308
$643$	8.64296	0.340845	0.170422	−	0.985371i	$-0.445487\pi$
$643$	8.64296	0.340845	0.170422	+	0.985371i	$0.445487\pi$
$644$	2.55343	0.100619
$645$	−1.20342	−0.0473846
$646$	− 26.4121i	− 1.03917i
$647$	4.79078	0.188345	0.0941725	−	0.995556i	$-0.469979\pi$
$647$	4.79078	0.188345	0.0941725	+	0.995556i	$0.469979\pi$
$648$	9.09921i	0.357451i
$649$	15.5870i	0.611844i
$650$	− 27.1785i	− 1.06603i
$651$	34.7821	1.36322
$652$	5.65808i	0.221588i
$653$	− 5.51910i	− 0.215979i	−0.994152	−	0.107990i	$-0.965559\pi$
$653$	− 5.51910i	− 0.215979i	0.994152	−	0.107990i	$-0.0344413\pi$
$654$	−25.2623	−0.987835
$655$	7.31958i	0.286000i
$656$	4.01226i	0.156653i
$657$	0.410637i	0.0160205i
$658$	−32.3197	−1.25995
$659$	− 1.84337i	− 0.0718075i	−0.999355	−	0.0359037i	$-0.988569\pi$
$659$	− 1.84337i	− 0.0718075i	0.999355	−	0.0359037i	$-0.0114310\pi$
$660$	−1.20342	−0.0468430
$661$	−26.0980	−1.01509	−0.507547	−	0.861624i	$-0.669448\pi$
$661$	−26.0980	−1.01509	−0.507547	+	0.861624i	$0.669448\pi$
$662$	−13.6463	−0.530378
$663$	52.9379	2.05594
$664$	− 12.7094i	− 0.493221i
$665$	7.71087i	0.299015i
$666$	−0.298556	−0.0115688
$667$	0	0
$668$	−12.0891	−0.467742
$669$	38.2055i	1.47711i
$670$	− 3.53135i	− 0.136428i
$671$	−6.79709	−0.262399
$672$	5.46607	0.210858
$673$	−21.9315	−0.845396	−0.422698	−	0.906271i	$-0.638917\pi$
$673$	−21.9315	−0.845396	−0.422698	+	0.906271i	$0.638917\pi$
$674$	22.5673	0.869261
$675$	24.5695i	0.945680i
$676$	−19.6664	−0.756400
$677$	− 10.1655i	− 0.390694i	−0.980734	−	0.195347i	$-0.937417\pi$
$677$	− 10.1655i	− 0.390694i	0.980734	−	0.195347i	$-0.0625832\pi$
$678$	− 29.0571i	− 1.11593i
$679$	− 11.9143i	− 0.457228i
$680$	2.63080	0.100887
$681$	3.75405i	0.143856i
$682$	− 8.88770i	− 0.340327i
$683$	20.4869	0.783910	0.391955	−	0.919984i	$-0.371799\pi$
$683$	20.4869	0.783910	0.391955	+	0.919984i	$0.371799\pi$
$684$	0.166090i	0.00635062i
$685$	− 0.746624i	− 0.0285271i
$686$	− 13.0258i	− 0.497329i
$687$	−16.1835	−0.617439
$688$	− 1.39672i	− 0.0532493i
$689$	24.2540	0.924006
$690$	−0.701010	−0.0266870
$691$	−30.8462	−1.17345	−0.586723	−	0.809788i	$-0.699582\pi$
$691$	−30.8462	−1.17345	−0.586723	+	0.809788i	$0.699582\pi$
$692$	11.6405	0.442504
$693$	− 0.146590i	− 0.00556850i
$694$	2.97305i	0.112855i
$695$	4.33177	0.164313
$696$	0	0
$697$	−21.3372	−0.808204
$698$	− 5.73913i	− 0.217229i
$699$	20.5252i	0.776336i
$700$	14.9239	0.564071
$701$	20.6761	0.780927	0.390463	−	0.920619i	$-0.372315\pi$
$701$	20.6761	0.780927	0.390463	+	0.920619i	$0.372315\pi$
$702$	29.5305	1.11456
$703$	44.3397	1.67230
$704$	− 1.39672i	− 0.0526408i
$705$	8.87294	0.334174
$706$	6.47216i	0.243583i
$707$	− 3.83456i	− 0.144213i
$708$	19.4367i	0.730476i
$709$	45.0200	1.69076	0.845382	−	0.534163i	$-0.179373\pi$
$709$	45.0200	1.69076	0.845382	+	0.534163i	$0.179373\pi$
$710$	− 2.98667i	− 0.112088i
$711$	0.314427i	0.0117919i
$712$	1.79217	0.0671644
$713$	− 5.17722i	− 0.193888i
$714$	29.0686i	1.08786i
$715$	3.94911i	0.147688i
$716$	−18.3551	−0.685963
$717$	25.0265i	0.934631i
$718$	−10.7414	−0.400866
$719$	−10.0777	−0.375834	−0.187917	−	0.982185i	$-0.560174\pi$
$719$	−10.0777	−0.375834	−0.187917	+	0.982185i	$0.560174\pi$
$720$	−0.0165436	−0.000616542 0
$721$	5.67466	0.211335
$722$	− 5.66670i	− 0.210893i
$723$	− 16.0310i	− 0.596198i
$724$	9.89404	0.367709
$725$	0	0
$726$	15.7608	0.584937
$727$	22.4140i	0.831289i	0.909527	+	0.415644i	$0.136444\pi$
$727$	22.4140i	0.831289i	−0.909527	+	0.415644i	$0.863556\pi$
$728$	− 17.9373i	− 0.664802i
$729$	−26.6924	−0.988606
$730$	6.07449	0.224827
$731$	7.42774	0.274725
$732$	−8.47584	−0.313276
$733$	− 3.94609i	− 0.145752i	−0.997341	−	0.0728760i	$-0.976782\pi$
$733$	− 3.94609i	− 0.145752i	0.997341	−	0.0728760i	$-0.0232177\pi$
$734$	−18.9393	−0.699064
$735$	− 2.45516i	− 0.0905601i
$736$	− 0.813609i	− 0.0299900i
$737$	− 9.97032i	− 0.367261i
$738$	0.134177	0.00493912
$739$	6.20181i	0.228137i	0.993473	+	0.114069i	$0.0363884\pi$
$739$	6.20181i	0.228137i	−0.993473	+	0.114069i	$0.963612\pi$
$740$	4.41650i	0.162354i
$741$	49.4395	1.81620
$742$	13.3181i	0.488922i
$743$	− 48.6919i	− 1.78633i	−0.449725	−	0.893167i	$-0.648478\pi$
$743$	− 48.6919i	− 1.78633i	0.449725	−	0.893167i	$-0.351522\pi$
$744$	− 11.0828i	− 0.406314i
$745$	2.34523	0.0859224
$746$	2.43402i	0.0891160i
$747$	−0.425025	−0.0155508
$748$	7.42774	0.271585
$749$	13.0263	0.475971
$750$	−8.40519	−0.306914
$751$	− 26.3802i	− 0.962626i	−0.876549	−	0.481313i	$-0.840160\pi$
$751$	− 26.3802i	− 0.962626i	0.876549	−	0.481313i	$-0.159840\pi$
$752$	10.2982i	0.375535i
$753$	10.3037	0.375489
$754$	0	0
$755$	3.21264	0.116920
$756$	16.2154i	0.589749i
$757$	38.1907i	1.38806i	0.719944	+	0.694032i	$0.244168\pi$
$757$	38.1907i	1.38806i	−0.719944	+	0.694032i	$0.755832\pi$
$758$	17.1415	0.622606
$759$	−1.97921	−0.0718408
$760$	2.45695	0.0891229
$761$	6.23834	0.226140	0.113070	−	0.993587i	$-0.463932\pi$
$761$	6.23834	0.226140	0.113070	+	0.993587i	$0.463932\pi$
$762$	− 22.1069i	− 0.800848i
$763$	−45.5211	−1.64798
$764$	− 0.424259i	− 0.0153492i
$765$	− 0.0879786i	− 0.00318087i
$766$	23.7774i	0.859111i
$767$	63.7830	2.30307
$768$	− 1.74168i	− 0.0628474i
$769$	28.1561i	1.01534i	0.861553	+	0.507668i	$0.169492\pi$
$769$	28.1561i	1.01534i	−0.861553	+	0.507668i	$0.830508\pi$
$770$	−2.16849	−0.0781468
$771$	46.5398i	1.67609i
$772$	10.3248i	0.371597i
$773$	35.0724i	1.26146i	0.776000	+	0.630732i	$0.217245\pi$
$773$	35.0724i	1.26146i	−0.776000	+	0.630732i	$0.782755\pi$
$774$	−0.0467086	−0.00167891
$775$	− 30.2591i	− 1.08694i
$776$	−3.79630	−0.136279
$777$	−48.7992	−1.75066
$778$	−11.5272	−0.413271
$779$	−19.9271	−0.713964
$780$	4.92446i	0.176324i
$781$	− 8.43248i	− 0.301738i
$782$	4.32677	0.154725
$783$	0	0
$784$	2.84952	0.101769
$785$	− 3.24591i	− 0.115851i
$786$	25.7700i	0.919184i
$787$	27.3021	0.973216	0.486608	−	0.873621i	$-0.338234\pi$
$787$	27.3021	0.973216	0.486608	+	0.873621i	$0.338234\pi$
$788$	26.8428	0.956234
$789$	−40.6758	−1.44810
$790$	4.65127	0.165485
$791$	− 52.3590i	− 1.86167i
$792$	−0.0467086	−0.00165972
$793$	27.8141i	0.987708i
$794$	28.9833i	1.02858i
$795$	− 3.65630i	− 0.129676i
$796$	6.80294	0.241124
$797$	40.9620i	1.45095i	0.688249	+	0.725475i	$0.258380\pi$
$797$	40.9620i	1.45095i	−0.688249	+	0.725475i	$0.741620\pi$
$798$	27.1476i	0.961014i
$799$	−54.7655	−1.93746
$800$	− 4.75527i	− 0.168124i
$801$	− 0.0599333i	− 0.00211764i
$802$	10.3297i	0.364753i
$803$	17.1506	0.605230
$804$	− 12.4328i	− 0.438471i
$805$	−1.26318	−0.0445211
$806$	−36.3690	−1.28104
$807$	−27.1863	−0.957004
$808$	−1.22182	−0.0429835
$809$	43.1355i	1.51656i	0.651927	+	0.758282i	$0.273961\pi$
$809$	43.1355i	1.51656i	−0.651927	+	0.758282i	$0.726039\pi$
$810$	− 4.50136i	− 0.158162i
$811$	−38.0001	−1.33436	−0.667182	−	0.744894i	$-0.732500\pi$
$811$	−38.0001	−1.33436	−0.667182	+	0.744894i	$0.732500\pi$
$812$	0	0
$813$	−0.382611	−0.0134187
$814$	12.4694i	0.437053i
$815$	− 2.79904i	− 0.0980463i
$816$	9.26224	0.324243
$817$	6.93688	0.242691
$818$	−9.96171	−0.348303
$819$	−0.599856	−0.0209607
$820$	− 1.98486i	− 0.0693143i
$821$	12.2176	0.426396	0.213198	−	0.977009i	$-0.431612\pi$
$821$	12.2176	0.426396	0.213198	+	0.977009i	$0.431612\pi$
$822$	− 2.62863i	− 0.0916840i
$823$	36.6269i	1.27673i	0.769733	+	0.638367i	$0.220390\pi$
$823$	36.6269i	1.27673i	−0.769733	+	0.638367i	$0.779610\pi$
$824$	− 1.80814i	− 0.0629896i
$825$	−11.5678	−0.402740
$826$	35.0237i	1.21863i
$827$	25.8144i	0.897655i	0.893618	+	0.448828i	$0.148158\pi$
$827$	25.8144i	0.897655i	−0.893618	+	0.448828i	$0.851842\pi$
$828$	−0.0272085	−0.000945560 0
$829$	− 15.7732i	− 0.547825i	−0.961755	−	0.273913i	$-0.911682\pi$
$829$	− 15.7732i	− 0.547825i	0.961755	−	0.273913i	$-0.0883179\pi$
$830$	6.28733i	0.218236i
$831$	− 6.53463i	− 0.226684i
$832$	−5.71545	−0.198148
$833$	15.1538i	0.525046i
$834$	15.2508	0.528092
$835$	5.98046	0.206962
$836$	6.93688	0.239917
$837$	32.8777	1.13642
$838$	− 28.3523i	− 0.979412i
$839$	− 2.15662i	− 0.0744547i	−0.999307	−	0.0372274i	$-0.988147\pi$
$839$	− 2.15662i	− 0.0744547i	0.999307	−	0.0372274i	$-0.0118526\pi$
$840$	−2.70406	−0.0932988
$841$	0	0
$842$	−9.51999	−0.328080
$843$	42.5860i	1.46674i
$844$	5.84061i	0.201042i
$845$	9.72893	0.334685
$846$	0.344388	0.0118403
$847$	28.3999	0.975832
$848$	4.24359	0.145726
$849$	13.2324i	0.454135i
$850$	25.2885	0.867389
$851$	7.26363i	0.248994i
$852$	− 10.5151i	− 0.360242i
$853$	21.6578i	0.741548i	0.928723	+	0.370774i	$0.120908\pi$
$853$	21.6578i	0.741548i	−0.928723	+	0.370774i	$0.879092\pi$
$854$	−15.2729	−0.522629
$855$	− 0.0821645i	− 0.00280997i
$856$	− 4.15063i	− 0.141866i
$857$	−12.6208	−0.431117	−0.215559	−	0.976491i	$-0.569157\pi$
$857$	−12.6208	−0.431117	−0.215559	+	0.976491i	$0.569157\pi$
$858$	13.9036i	0.474661i
$859$	19.4272i	0.662849i	0.943482	+	0.331424i	$0.107529\pi$
$859$	19.4272i	0.662849i	−0.943482	+	0.331424i	$0.892471\pi$
$860$	0.690954i	0.0235613i
$861$	21.9313	0.747418
$862$	− 18.9749i	− 0.646289i
$863$	45.9285	1.56342	0.781712	−	0.623640i	$-0.214347\pi$
$863$	45.9285	1.56342	0.781712	+	0.623640i	$0.214347\pi$
$864$	5.16679	0.175778
$865$	−5.75852	−0.195795
$866$	28.3148	0.962175
$867$	19.6480i	0.667281i
$868$	− 19.9705i	− 0.677842i
$869$	13.1323	0.445482
$870$	0	0
$871$	−40.7992	−1.38243
$872$	14.5046i	0.491187i
$873$	0.126955i	0.00429677i
$874$	4.04084	0.136683
$875$	−15.1456	−0.512016
$876$	21.3864	0.722579
$877$	8.09527	0.273358	0.136679	−	0.990615i	$-0.456357\pi$
$877$	8.09527	0.273358	0.136679	+	0.990615i	$0.456357\pi$
$878$	19.8958i	0.671452i
$879$	−44.8000	−1.51107
$880$	0.690954i	0.0232920i
$881$	3.00187i	0.101135i	0.998721	+	0.0505677i	$0.0161031\pi$
$881$	3.00187i	0.101135i	−0.998721	+	0.0505677i	$0.983897\pi$
$882$	− 0.0952930i	− 0.00320868i
$883$	38.6155	1.29952	0.649758	−	0.760141i	$-0.274870\pi$
$883$	38.6155	1.29952	0.649758	+	0.760141i	$0.274870\pi$
$884$	− 30.3948i	− 1.02229i
$885$	− 9.61530i	− 0.323215i
$886$	−35.5600	−1.19466
$887$	− 29.4281i	− 0.988100i	−0.869433	−	0.494050i	$-0.835516\pi$
$887$	− 29.4281i	− 0.988100i	0.869433	−	0.494050i	$-0.164484\pi$
$888$	15.5491i	0.521794i
$889$	− 39.8352i	− 1.33603i
$890$	−0.886584	−0.0297184
$891$	− 12.7090i	− 0.425768i
$892$	21.9361	0.734474
$893$	−51.1464	−1.71155
$894$	8.25681	0.276149
$895$	9.08024	0.303519
$896$	− 3.13840i	− 0.104846i
$897$	8.09905i	0.270420i
$898$	−10.8511	−0.362106
$899$	0	0
$900$	−0.159025	−0.00530082
$901$	22.5674i	0.751829i
$902$	− 5.60400i	− 0.186593i
$903$	−7.63456	−0.254062
$904$	−16.6834	−0.554881
$905$	−4.89456	−0.162701
$906$	11.3107	0.375773
$907$	− 4.66629i	− 0.154942i	−0.996995	−	0.0774709i	$-0.975316\pi$
$907$	− 4.66629i	− 0.154942i	0.996995	−	0.0774709i	$-0.0246845\pi$
$908$	2.15542	0.0715302
$909$	0.0408598i	0.00135523i
$910$	8.87357i	0.294156i
$911$	37.9571i	1.25758i	0.777577	+	0.628788i	$0.216449\pi$
$911$	37.9571i	1.25758i	−0.777577	+	0.628788i	$0.783551\pi$
$912$	8.65014	0.286435
$913$	17.7515i	0.587488i
$914$	− 32.3358i	− 1.06957i
$915$	4.19298	0.138616
$916$	9.29190i	0.307013i
$917$	46.4359i	1.53345i
$918$	27.4770i	0.906875i
$919$	25.3594	0.836530	0.418265	−	0.908325i	$-0.362638\pi$
$919$	25.3594	0.836530	0.418265	+	0.908325i	$0.362638\pi$
$920$	0.402491i	0.0132697i
$921$	3.15939	0.104105
$922$	25.4047	0.836659
$923$	−34.5062	−1.13579
$924$	−7.63456	−0.251159
$925$	42.4535i	1.39586i
$926$	− 3.22072i	− 0.105839i
$927$	−0.0604674	−0.00198601
$928$	0	0
$929$	36.8642	1.20947	0.604737	−	0.796425i	$-0.293278\pi$
$929$	36.8642	1.20947	0.604737	+	0.796425i	$0.293278\pi$
$930$	5.48263i	0.179783i
$931$	14.1523i	0.463824i
$932$	11.7848	0.386022
$933$	33.8060	1.10676
$934$	−22.8565	−0.747887
$935$	−3.67449	−0.120169
$936$	0.191135i	0.00624743i
$937$	2.33445	0.0762631	0.0381316	−	0.999273i	$-0.487859\pi$
$937$	2.33445	0.0762631	0.0381316	+	0.999273i	$0.487859\pi$
$938$	− 22.4031i	− 0.731487i
$939$	− 59.0163i	− 1.92592i
$940$	− 5.09448i	− 0.166164i
$941$	−8.78880	−0.286507	−0.143253	−	0.989686i	$-0.545756\pi$
$941$	−8.78880	−0.286507	−0.143253	+	0.989686i	$0.545756\pi$
$942$	− 11.4278i	− 0.372339i
$943$	− 3.26441i	− 0.106304i
$944$	11.1598	0.363219
$945$	− 8.02174i	− 0.260947i
$946$	1.95082i	0.0634266i
$947$	− 12.6976i	− 0.412618i	−0.978487	−	0.206309i	$-0.933855\pi$
$947$	− 12.6976i	− 0.412618i	0.978487	−	0.206309i	$-0.0661452\pi$
$948$	16.3757	0.531857
$949$	− 70.1811i	− 2.27818i
$950$	23.6173	0.766248
$951$	−9.57506	−0.310493
$952$	16.6900	0.540925
$953$	54.4562	1.76401	0.882005	−	0.471240i	$-0.156194\pi$
$953$	54.4562	1.76401	0.882005	+	0.471240i	$0.156194\pi$
$954$	− 0.141913i	− 0.00459460i
$955$	0.209880i	0.00679157i
$956$	14.3692	0.464732
$957$	0	0
$958$	2.14546	0.0693165
$959$	− 4.73663i	− 0.152954i
$960$	0.861605i	0.0278082i
$961$	−9.49127	−0.306170
$962$	51.0256	1.64513
$963$	−0.138804	−0.00447290
$964$	−9.20432	−0.296451
$965$	− 5.10765i	− 0.164421i
$966$	−4.44725	−0.143088
$967$	− 49.5652i	− 1.59391i	−0.604038	−	0.796955i	$-0.706443\pi$
$967$	− 49.5652i	− 1.59391i	0.604038	−	0.796955i	$-0.293557\pi$
$968$	− 9.04918i	− 0.290852i
$969$	46.0014i	1.47778i
$970$	1.87802	0.0602996
$971$	− 27.5824i	− 0.885162i	−0.896729	−	0.442581i	$-0.854063\pi$
$971$	− 27.5824i	− 0.885162i	0.896729	−	0.442581i	$-0.145937\pi$
$972$	− 0.347520i	− 0.0111467i
$973$	27.4810	0.881000
$974$	21.7759i	0.697746i
$975$	47.3363i	1.51597i
$976$	4.86648i	0.155772i
$977$	−44.5323	−1.42471	−0.712357	−	0.701818i	$-0.752372\pi$
$977$	−44.5323	−1.42471	−0.712357	+	0.701818i	$0.752372\pi$
$978$	− 9.85456i	− 0.315114i
$979$	−2.50316	−0.0800012
$980$	−1.40965	−0.0450298
$981$	0.485058	0.0154867
$982$	−38.8631	−1.24017
$983$	1.19667i	0.0381680i	0.999818	+	0.0190840i	$0.00607499\pi$
$983$	1.19667i	0.0381680i	−0.999818	+	0.0190840i	$0.993925\pi$
$984$	− 6.98807i	− 0.222772i
$985$	−13.2791	−0.423106
$986$	0	0
$987$	56.2904	1.79174
$988$	− 28.3861i	− 0.903083i
$989$	1.13638i	0.0361348i
$990$	0.0231067	0.000734378 0
$991$	−35.0275	−1.11269	−0.556343	−	0.830953i	$-0.687796\pi$
$991$	−35.0275	−1.11269	−0.556343	+	0.830953i	$0.687796\pi$
$992$	−6.36328	−0.202034
$993$	23.7674	0.754236
$994$	− 18.9476i	− 0.600982i
$995$	−3.36540	−0.106690
$996$	22.1357i	0.701397i
$997$	31.1728i	0.987254i	0.869674	+	0.493627i	$0.164329\pi$
$997$	31.1728i	0.987254i	−0.869674	+	0.493627i	$0.835671\pi$
$998$	− 39.2992i	− 1.24400i
$999$	−46.1273	−1.45940

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.2.b.i.1681.8		12
29.2	odd	28		58.2.d.b.25.2	yes	12
29.12	odd	4		1682.2.a.q.1.5		6
29.14	odd	28		58.2.d.b.7.2	✓	12
29.17	odd	4		1682.2.a.t.1.2		6
29.28	even	2	inner	1682.2.b.i.1681.5		12
87.2	even	28		522.2.k.h.199.1		12
87.14	even	28		522.2.k.h.181.1		12
116.31	even	28		464.2.u.h.257.1		12
116.43	even	28		464.2.u.h.65.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.d.b.7.2	✓	12	29.14	odd	28
58.2.d.b.25.2	yes	12	29.2	odd	28
464.2.u.h.65.1		12	116.43	even	28
464.2.u.h.257.1		12	116.31	even	28
522.2.k.h.181.1		12	87.14	even	28
522.2.k.h.199.1		12	87.2	even	28
1682.2.a.q.1.5		6	29.12	odd	4
1682.2.a.t.1.2		6	29.17	odd	4
1682.2.b.i.1681.5		12	29.28	even	2	inner
1682.2.b.i.1681.8		12	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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.i.1681.8

Level	$1682$
Weight	$2$
Character	1682.1681
Analytic conductor	$13.431$
Analytic rank	$0$
Dimension	$12$
Inner twists	$2$