LMFDB - Embedded newform 1620.3.g.b.161.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(161,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1620.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$44.1418028264$
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} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} + \cdots + 531441 $ x^12 - 2x^11 - 11x^10 + 60x^9 - 120x^8 - 342x^7 + 2709x^6 - 3078x^5 - 9720x^4 + 43740x^3 - 72171x^2 - 118098*x + 531441
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.9
Root			$-2.85525 + 0.920635i$ of defining polynomial
Character	$\chi$	$=$	1620.161
Dual form			1620.3.g.b.161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{5} -1.18917 q^{7} +O(q^{10})$	$q+2.23607i q^{5} -1.18917 q^{7} +4.19702i q^{11} -14.3123 q^{13} -18.7074i q^{17} +33.8986 q^{19} +13.4998i q^{23} -5.00000 q^{25} +35.3702i q^{29} -9.95632 q^{31} -2.65907i q^{35} -19.3047 q^{37} +64.6225i q^{41} -41.5544 q^{43} -67.2114i q^{47} -47.5859 q^{49} -30.0712i q^{53} -9.38482 q^{55} -4.23260i q^{59} -87.6420 q^{61} -32.0032i q^{65} -35.8080 q^{67} -24.5173i q^{71} +11.9305 q^{73} -4.99099i q^{77} -38.1715 q^{79} +24.0977i q^{83} +41.8311 q^{85} -44.4494i q^{89} +17.0198 q^{91} +75.7996i q^{95} +63.9343 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7}+O(q^{10}) $ 12 * q + 12 * q^7	$ 12 q + 12 q^{7} + 60 q^{13} + 72 q^{19} - 60 q^{25} + 24 q^{31} - 12 q^{37} - 228 q^{43} - 96 q^{49} - 120 q^{55} + 156 q^{61} + 336 q^{67} - 24 q^{73} - 240 q^{79} + 60 q^{85} + 120 q^{91} - 192 q^{97}+O(q^{100}) $ 12 * q + 12 * q^7 + 60 * q^13 + 72 * q^19 - 60 * q^25 + 24 * q^31 - 12 * q^37 - 228 * q^43 - 96 * q^49 - 120 * q^55 + 156 * q^61 + 336 * q^67 - 24 * q^73 - 240 * q^79 + 60 * q^85 + 120 * q^91 - 192 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	−1.18917	−0.169882	−0.0849410	−	0.996386i	$-0.527070\pi$
$7$	−1.18917	−0.169882	−0.0849410	+	0.996386i	$0.527070\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.19702i	0.381547i	0.981634	+	0.190774i	$0.0610996\pi$
$11$	4.19702i	0.381547i	−0.981634	+	0.190774i	$0.938900\pi$
$12$	0	0
$13$	−14.3123	−1.10095	−0.550473	−	0.834853i	$-0.685552\pi$
$13$	−14.3123	−1.10095	−0.550473	+	0.834853i	$0.685552\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 18.7074i	− 1.10044i	−0.835021	−	0.550218i	$-0.814545\pi$
$17$	− 18.7074i	− 1.10044i	0.835021	−	0.550218i	$-0.185455\pi$
$18$	0	0
$19$	33.8986	1.78414	0.892069	−	0.451899i	$-0.149253\pi$
$19$	33.8986	1.78414	0.892069	+	0.451899i	$0.149253\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	13.4998i	0.586947i	0.955967	+	0.293474i	$0.0948113\pi$
$23$	13.4998i	0.586947i	−0.955967	+	0.293474i	$0.905189\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	35.3702i	1.21966i	0.792532	+	0.609831i	$0.208763\pi$
$29$	35.3702i	1.21966i	−0.792532	+	0.609831i	$0.791237\pi$
$30$	0	0
$31$	−9.95632	−0.321171	−0.160586	−	0.987022i	$-0.551338\pi$
$31$	−9.95632	−0.321171	−0.160586	+	0.987022i	$0.551338\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.65907i	− 0.0759735i
$36$	0	0
$37$	−19.3047	−0.521750	−0.260875	−	0.965373i	$-0.584011\pi$
$37$	−19.3047	−0.521750	−0.260875	+	0.965373i	$0.584011\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	64.6225i	1.57616i	0.615573	+	0.788080i	$0.288925\pi$
$41$	64.6225i	1.57616i	−0.615573	+	0.788080i	$0.711075\pi$
$42$	0	0
$43$	−41.5544	−0.966382	−0.483191	−	0.875515i	$-0.660522\pi$
$43$	−41.5544	−0.966382	−0.483191	+	0.875515i	$0.660522\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 67.2114i	− 1.43003i	−0.699109	−	0.715015i	$-0.746420\pi$
$47$	− 67.2114i	− 1.43003i	0.699109	−	0.715015i	$-0.253580\pi$
$48$	0	0
$49$	−47.5859	−0.971140
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 30.0712i	− 0.567382i	−0.958916	−	0.283691i	$-0.908441\pi$
$53$	− 30.0712i	− 0.567382i	0.958916	−	0.283691i	$-0.0915589\pi$
$54$	0	0
$55$	−9.38482	−0.170633
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4.23260i	− 0.0717390i	−0.999356	−	0.0358695i	$-0.988580\pi$
$59$	− 4.23260i	− 0.0717390i	0.999356	−	0.0358695i	$-0.0114201\pi$
$60$	0	0
$61$	−87.6420	−1.43675	−0.718377	−	0.695654i	$-0.755115\pi$
$61$	−87.6420	−1.43675	−0.718377	+	0.695654i	$0.755115\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 32.0032i	− 0.492358i
$66$	0	0
$67$	−35.8080	−0.534448	−0.267224	−	0.963634i	$-0.586106\pi$
$67$	−35.8080	−0.534448	−0.267224	+	0.963634i	$0.586106\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 24.5173i	− 0.345314i	−0.984982	−	0.172657i	$-0.944765\pi$
$71$	− 24.5173i	− 0.345314i	0.984982	−	0.172657i	$-0.0552352\pi$
$72$	0	0
$73$	11.9305	0.163432	0.0817159	−	0.996656i	$-0.473960\pi$
$73$	11.9305	0.163432	0.0817159	+	0.996656i	$0.473960\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.99099i	− 0.0648180i
$78$	0	0
$79$	−38.1715	−0.483184	−0.241592	−	0.970378i	$-0.577670\pi$
$79$	−38.1715	−0.483184	−0.241592	+	0.970378i	$0.577670\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	24.0977i	0.290334i	0.989407	+	0.145167i	$0.0463719\pi$
$83$	24.0977i	0.290334i	−0.989407	+	0.145167i	$0.953628\pi$
$84$	0	0
$85$	41.8311	0.492130
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 44.4494i	− 0.499431i	−0.968319	−	0.249716i	$-0.919663\pi$
$89$	− 44.4494i	− 0.499431i	0.968319	−	0.249716i	$-0.0803371\pi$
$90$	0	0
$91$	17.0198	0.187031
$92$	0	0
$93$	0	0
$94$	0	0
$95$	75.7996i	0.797891i
$96$	0	0
$97$	63.9343	0.659117	0.329558	−	0.944135i	$-0.393100\pi$
$97$	63.9343	0.659117	0.329558	+	0.944135i	$0.393100\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.9617i	0.118433i	0.998245	+	0.0592163i	$0.0188602\pi$
$101$	11.9617i	0.118433i	−0.998245	+	0.0592163i	$0.981140\pi$
$102$	0	0
$103$	−163.968	−1.59193	−0.795963	−	0.605346i	$-0.793035\pi$
$103$	−163.968	−1.59193	−0.795963	+	0.605346i	$0.793035\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	102.401i	0.957019i	0.878082	+	0.478509i	$0.158823\pi$
$107$	102.401i	0.957019i	−0.878082	+	0.478509i	$0.841177\pi$
$108$	0	0
$109$	−56.4485	−0.517876	−0.258938	−	0.965894i	$-0.583373\pi$
$109$	−56.4485	−0.517876	−0.258938	+	0.965894i	$0.583373\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 172.404i	− 1.52570i	−0.646573	−	0.762852i	$-0.723799\pi$
$113$	− 172.404i	− 1.52570i	0.646573	−	0.762852i	$-0.276201\pi$
$114$	0	0
$115$	−30.1864	−0.262491
$116$	0	0
$117$	0	0
$118$	0	0
$119$	22.2464i	0.186944i
$120$	0	0
$121$	103.385	0.854422
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	−195.853	−1.54215	−0.771073	−	0.636746i	$-0.780280\pi$
$127$	−195.853	−1.54215	−0.771073	+	0.636746i	$0.780280\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	113.767i	0.868449i	0.900805	+	0.434224i	$0.142978\pi$
$131$	113.767i	0.868449i	−0.900805	+	0.434224i	$0.857022\pi$
$132$	0	0
$133$	−40.3114	−0.303093
$134$	0	0
$135$	0	0
$136$	0	0
$137$	181.026i	1.32136i	0.750669	+	0.660679i	$0.229732\pi$
$137$	181.026i	1.32136i	−0.750669	+	0.660679i	$0.770268\pi$
$138$	0	0
$139$	−229.109	−1.64827	−0.824134	−	0.566395i	$-0.808338\pi$
$139$	−229.109	−1.64827	−0.824134	+	0.566395i	$0.808338\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 60.0689i	− 0.420063i
$144$	0	0
$145$	−79.0901	−0.545449
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 270.020i	− 1.81221i	−0.423049	−	0.906107i	$-0.639040\pi$
$149$	− 270.020i	− 1.81221i	0.423049	−	0.906107i	$-0.360960\pi$
$150$	0	0
$151$	−163.141	−1.08040	−0.540202	−	0.841535i	$-0.681652\pi$
$151$	−163.141	−1.08040	−0.540202	+	0.841535i	$0.681652\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 22.2630i	− 0.143632i
$156$	0	0
$157$	−134.600	−0.857322	−0.428661	−	0.903465i	$-0.641015\pi$
$157$	−134.600	−0.857322	−0.428661	+	0.903465i	$0.641015\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 16.0536i	− 0.0997118i
$162$	0	0
$163$	−159.658	−0.979500	−0.489750	−	0.871863i	$-0.662912\pi$
$163$	−159.658	−0.979500	−0.489750	+	0.871863i	$0.662912\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 111.154i	− 0.665593i	−0.942999	−	0.332796i	$-0.892008\pi$
$167$	− 111.154i	− 0.665593i	0.942999	−	0.332796i	$-0.107992\pi$
$168$	0	0
$169$	35.8416	0.212080
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 147.351i	− 0.851740i	−0.904784	−	0.425870i	$-0.859968\pi$
$173$	− 147.351i	− 0.851740i	0.904784	−	0.425870i	$-0.140032\pi$
$174$	0	0
$175$	5.94587	0.0339764
$176$	0	0
$177$	0	0
$178$	0	0
$179$	253.022i	1.41353i	0.707449	+	0.706765i	$0.249846\pi$
$179$	253.022i	1.41353i	−0.707449	+	0.706765i	$0.750154\pi$
$180$	0	0
$181$	−212.017	−1.17136	−0.585682	−	0.810541i	$-0.699173\pi$
$181$	−212.017	−1.17136	−0.585682	+	0.810541i	$0.699173\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 43.1667i	− 0.233334i
$186$	0	0
$187$	78.5154	0.419868
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 331.430i	− 1.73524i	−0.497230	−	0.867619i	$-0.665650\pi$
$191$	− 331.430i	− 1.73524i	0.497230	−	0.867619i	$-0.334350\pi$
$192$	0	0
$193$	332.558	1.72310	0.861549	−	0.507674i	$-0.169494\pi$
$193$	332.558	1.72310	0.861549	+	0.507674i	$0.169494\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	321.219i	1.63056i	0.579070	+	0.815278i	$0.303416\pi$
$197$	321.219i	1.63056i	−0.579070	+	0.815278i	$0.696584\pi$
$198$	0	0
$199$	−208.913	−1.04981	−0.524907	−	0.851160i	$-0.675900\pi$
$199$	−208.913	−1.04981	−0.524907	+	0.851160i	$0.675900\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 42.0613i	− 0.207199i
$204$	0	0
$205$	−144.500	−0.704880
$206$	0	0
$207$	0	0
$208$	0	0
$209$	142.273i	0.680733i
$210$	0	0
$211$	220.609	1.04554	0.522769	−	0.852474i	$-0.324899\pi$
$211$	220.609	1.04554	0.522769	+	0.852474i	$0.324899\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 92.9185i	− 0.432179i
$216$	0	0
$217$	11.8398	0.0545613
$218$	0	0
$219$	0	0
$220$	0	0
$221$	267.746i	1.21152i
$222$	0	0
$223$	287.252	1.28812	0.644062	−	0.764973i	$-0.277248\pi$
$223$	287.252	1.28812	0.644062	+	0.764973i	$0.277248\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	261.515i	1.15205i	0.817432	+	0.576024i	$0.195397\pi$
$227$	261.515i	1.15205i	−0.817432	+	0.576024i	$0.804603\pi$
$228$	0	0
$229$	198.449	0.866589	0.433294	−	0.901252i	$-0.357351\pi$
$229$	198.449	0.866589	0.433294	+	0.901252i	$0.357351\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	182.063i	0.781387i	0.920521	+	0.390693i	$0.127765\pi$
$233$	182.063i	0.781387i	−0.920521	+	0.390693i	$0.872235\pi$
$234$	0	0
$235$	150.289	0.639529
$236$	0	0
$237$	0	0
$238$	0	0
$239$	242.641i	1.01523i	0.861583	+	0.507617i	$0.169473\pi$
$239$	242.641i	1.01523i	−0.861583	+	0.507617i	$0.830527\pi$
$240$	0	0
$241$	−317.765	−1.31853	−0.659263	−	0.751912i	$-0.729132\pi$
$241$	−317.765	−1.31853	−0.659263	+	0.751912i	$0.729132\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 106.405i	− 0.434307i
$246$	0	0
$247$	−485.167	−1.96424
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 64.0280i	− 0.255092i	−0.991833	−	0.127546i	$-0.959290\pi$
$251$	− 64.0280i	− 0.255092i	0.991833	−	0.127546i	$-0.0407100\pi$
$252$	0	0
$253$	−56.6589	−0.223948
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 464.202i	− 1.80623i	−0.429397	−	0.903116i	$-0.641274\pi$
$257$	− 464.202i	− 1.80623i	0.429397	−	0.903116i	$-0.358726\pi$
$258$	0	0
$259$	22.9567	0.0886359
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 56.0964i	− 0.213294i	−0.994297	−	0.106647i	$-0.965988\pi$
$263$	− 56.0964i	− 0.213294i	0.994297	−	0.106647i	$-0.0340115\pi$
$264$	0	0
$265$	67.2413	0.253741
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 29.8910i	− 0.111119i	−0.998455	−	0.0555594i	$-0.982306\pi$
$269$	− 29.8910i	− 0.111119i	0.998455	−	0.0555594i	$-0.0176942\pi$
$270$	0	0
$271$	−166.826	−0.615594	−0.307797	−	0.951452i	$-0.599592\pi$
$271$	−166.826	−0.615594	−0.307797	+	0.951452i	$0.599592\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 20.9851i	− 0.0763094i
$276$	0	0
$277$	196.811	0.710509	0.355254	−	0.934770i	$-0.384394\pi$
$277$	196.811	0.710509	0.355254	+	0.934770i	$0.384394\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	332.781i	1.18427i	0.805838	+	0.592136i	$0.201715\pi$
$281$	332.781i	1.18427i	−0.805838	+	0.592136i	$0.798285\pi$
$282$	0	0
$283$	−466.933	−1.64994	−0.824971	−	0.565176i	$-0.808808\pi$
$283$	−466.933	−1.64994	−0.824971	+	0.565176i	$0.808808\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 76.8474i	− 0.267761i
$288$	0	0
$289$	−60.9675	−0.210960
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.5233i	0.0734582i	0.999325	+	0.0367291i	$0.0116939\pi$
$293$	21.5233i	0.0734582i	−0.999325	+	0.0367291i	$0.988306\pi$
$294$	0	0
$295$	9.46438	0.0320827
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 193.213i	− 0.646197i
$300$	0	0
$301$	49.4154	0.164171
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 195.973i	− 0.642536i
$306$	0	0
$307$	452.833	1.47503	0.737514	−	0.675332i	$-0.236000\pi$
$307$	452.833	1.47503	0.737514	+	0.675332i	$0.236000\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	564.296i	1.81446i	0.420638	+	0.907229i	$0.361806\pi$
$311$	564.296i	1.81446i	−0.420638	+	0.907229i	$0.638194\pi$
$312$	0	0
$313$	−461.320	−1.47387	−0.736933	−	0.675966i	$-0.763727\pi$
$313$	−461.320	−1.47387	−0.736933	+	0.675966i	$0.763727\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	336.760i	1.06233i	0.847267	+	0.531167i	$0.178246\pi$
$317$	336.760i	1.06233i	−0.847267	+	0.531167i	$0.821754\pi$
$318$	0	0
$319$	−148.449	−0.465358
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 634.156i	− 1.96333i
$324$	0	0
$325$	71.5614	0.220189
$326$	0	0
$327$	0	0
$328$	0	0
$329$	79.9261i	0.242936i
$330$	0	0
$331$	−421.793	−1.27430	−0.637149	−	0.770741i	$-0.719886\pi$
$331$	−421.793	−1.27430	−0.637149	+	0.770741i	$0.719886\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 80.0691i	− 0.239012i
$336$	0	0
$337$	−118.423	−0.351404	−0.175702	−	0.984443i	$-0.556219\pi$
$337$	−118.423	−0.351404	−0.175702	+	0.984443i	$0.556219\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 41.7869i	− 0.122542i
$342$	0	0
$343$	114.857	0.334861
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 170.000i	− 0.489914i	−0.969534	−	0.244957i	$-0.921226\pi$
$347$	− 170.000i	− 0.489914i	0.969534	−	0.244957i	$-0.0787739\pi$
$348$	0	0
$349$	620.255	1.77723	0.888617	−	0.458650i	$-0.151667\pi$
$349$	620.255	1.77723	0.888617	+	0.458650i	$0.151667\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 409.408i	− 1.15979i	−0.814690	−	0.579897i	$-0.803093\pi$
$353$	− 409.408i	− 1.15979i	0.814690	−	0.579897i	$-0.196907\pi$
$354$	0	0
$355$	54.8223	0.154429
$356$	0	0
$357$	0	0
$358$	0	0
$359$	138.441i	0.385630i	0.981235	+	0.192815i	$0.0617618\pi$
$359$	138.441i	0.385630i	−0.981235	+	0.192815i	$0.938238\pi$
$360$	0	0
$361$	788.116	2.18315
$362$	0	0
$363$	0	0
$364$	0	0
$365$	26.6775i	0.0730890i
$366$	0	0
$367$	79.0717	0.215454	0.107727	−	0.994181i	$-0.465643\pi$
$367$	79.0717	0.215454	0.107727	+	0.994181i	$0.465643\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	35.7599i	0.0963879i
$372$	0	0
$373$	103.770	0.278203	0.139102	−	0.990278i	$-0.455579\pi$
$373$	103.770	0.278203	0.139102	+	0.990278i	$0.455579\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 506.228i	− 1.34278i
$378$	0	0
$379$	315.487	0.832419	0.416210	−	0.909269i	$-0.363358\pi$
$379$	315.487	0.832419	0.416210	+	0.909269i	$0.363358\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 414.185i	− 1.08142i	−0.841208	−	0.540711i	$-0.818155\pi$
$383$	− 414.185i	− 1.08142i	0.841208	−	0.540711i	$-0.181845\pi$
$384$	0	0
$385$	11.1602	0.0289875
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 69.1732i	− 0.177823i	−0.996040	−	0.0889115i	$-0.971661\pi$
$389$	− 69.1732i	− 0.177823i	0.996040	−	0.0889115i	$-0.0283388\pi$
$390$	0	0
$391$	252.546	0.645898
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 85.3541i	− 0.216086i
$396$	0	0
$397$	347.995	0.876562	0.438281	−	0.898838i	$-0.355587\pi$
$397$	347.995	0.876562	0.438281	+	0.898838i	$0.355587\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	733.596i	1.82942i	0.404114	+	0.914709i	$0.367580\pi$
$401$	733.596i	1.82942i	−0.404114	+	0.914709i	$0.632420\pi$
$402$	0	0
$403$	142.498	0.353592
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 81.0224i	− 0.199072i
$408$	0	0
$409$	94.1455	0.230185	0.115092	−	0.993355i	$-0.463284\pi$
$409$	94.1455	0.230185	0.115092	+	0.993355i	$0.463284\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.03330i	0.0121872i
$414$	0	0
$415$	−53.8841	−0.129841
$416$	0	0
$417$	0	0
$418$	0	0
$419$	104.621i	0.249692i	0.992176	+	0.124846i	$0.0398436\pi$
$419$	104.621i	0.249692i	−0.992176	+	0.124846i	$0.960156\pi$
$420$	0	0
$421$	553.083	1.31374	0.656868	−	0.754005i	$-0.271881\pi$
$421$	553.083	1.31374	0.656868	+	0.754005i	$0.271881\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	93.5371i	0.220087i
$426$	0	0
$427$	104.222	0.244079
$428$	0	0
$429$	0	0
$430$	0	0
$431$	469.242i	1.08873i	0.838849	+	0.544364i	$0.183229\pi$
$431$	469.242i	1.08873i	−0.838849	+	0.544364i	$0.816771\pi$
$432$	0	0
$433$	426.644	0.985321	0.492660	−	0.870222i	$-0.336024\pi$
$433$	426.644	0.985321	0.492660	+	0.870222i	$0.336024\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	457.624i	1.04720i
$438$	0	0
$439$	−573.146	−1.30557	−0.652786	−	0.757543i	$-0.726400\pi$
$439$	−573.146	−1.30557	−0.652786	+	0.757543i	$0.726400\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	104.597i	0.236111i	0.993007	+	0.118055i	$0.0376661\pi$
$443$	104.597i	0.236111i	−0.993007	+	0.118055i	$0.962334\pi$
$444$	0	0
$445$	99.3918	0.223352
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 401.878i	− 0.895051i	−0.894271	−	0.447526i	$-0.852305\pi$
$449$	− 401.878i	− 0.895051i	0.894271	−	0.447526i	$-0.147695\pi$
$450$	0	0
$451$	−271.222	−0.601379
$452$	0	0
$453$	0	0
$454$	0	0
$455$	38.0574i	0.0836427i
$456$	0	0
$457$	96.3184	0.210762	0.105381	−	0.994432i	$-0.466394\pi$
$457$	96.3184	0.210762	0.105381	+	0.994432i	$0.466394\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 74.4306i	− 0.161455i	−0.996736	−	0.0807274i	$-0.974276\pi$
$461$	− 74.4306i	− 0.161455i	0.996736	−	0.0807274i	$-0.0257243\pi$
$462$	0	0
$463$	397.861	0.859310	0.429655	−	0.902993i	$-0.358635\pi$
$463$	397.861	0.859310	0.429655	+	0.902993i	$0.358635\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	643.763i	1.37851i	0.724520	+	0.689254i	$0.242062\pi$
$467$	643.763i	1.37851i	−0.724520	+	0.689254i	$0.757938\pi$
$468$	0	0
$469$	42.5819	0.0907931
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 174.405i	− 0.368720i
$474$	0	0
$475$	−169.493	−0.356828
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 176.404i	− 0.368275i	−0.982900	−	0.184138i	$-0.941051\pi$
$479$	− 176.404i	− 0.368275i	0.982900	−	0.184138i	$-0.0589492\pi$
$480$	0	0
$481$	276.295	0.574418
$482$	0	0
$483$	0	0
$484$	0	0
$485$	142.962i	0.294766i
$486$	0	0
$487$	−494.859	−1.01614	−0.508069	−	0.861317i	$-0.669640\pi$
$487$	−494.859	−1.01614	−0.508069	+	0.861317i	$0.669640\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 184.494i	− 0.375752i	−0.982193	−	0.187876i	$-0.939840\pi$
$491$	− 184.494i	− 0.375752i	0.982193	−	0.187876i	$-0.0601603\pi$
$492$	0	0
$493$	661.685	1.34216
$494$	0	0
$495$	0	0
$496$	0	0
$497$	29.1553i	0.0586626i
$498$	0	0
$499$	324.553	0.650407	0.325203	−	0.945644i	$-0.394567\pi$
$499$	324.553	0.650407	0.325203	+	0.945644i	$0.394567\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 626.131i	− 1.24479i	−0.782702	−	0.622397i	$-0.786159\pi$
$503$	− 626.131i	− 1.24479i	0.782702	−	0.622397i	$-0.213841\pi$
$504$	0	0
$505$	−26.7471	−0.0529646
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 447.601i	− 0.879373i	−0.898151	−	0.439686i	$-0.855090\pi$
$509$	− 447.601i	− 0.879373i	0.898151	−	0.439686i	$-0.144910\pi$
$510$	0	0
$511$	−14.1875	−0.0277641
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 366.644i	− 0.711931i
$516$	0	0
$517$	282.088	0.545624
$518$	0	0
$519$	0	0
$520$	0	0
$521$	181.821i	0.348984i	0.984659	+	0.174492i	$0.0558284\pi$
$521$	181.821i	0.348984i	−0.984659	+	0.174492i	$0.944172\pi$
$522$	0	0
$523$	−293.719	−0.561604	−0.280802	−	0.959766i	$-0.590600\pi$
$523$	−293.719	−0.561604	−0.280802	+	0.959766i	$0.590600\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	186.257i	0.353429i
$528$	0	0
$529$	346.756	0.655493
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 924.896i	− 1.73527i
$534$	0	0
$535$	−228.976	−0.427992
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 199.719i	− 0.370536i
$540$	0	0
$541$	343.593	0.635107	0.317553	−	0.948240i	$-0.397139\pi$
$541$	343.593	0.635107	0.317553	+	0.948240i	$0.397139\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 126.223i	− 0.231601i
$546$	0	0
$547$	−41.1575	−0.0752422	−0.0376211	−	0.999292i	$-0.511978\pi$
$547$	−41.1575	−0.0752422	−0.0376211	+	0.999292i	$0.511978\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1199.00i	2.17604i
$552$	0	0
$553$	45.3926	0.0820842
$554$	0	0
$555$	0	0
$556$	0	0
$557$	976.675i	1.75346i	0.480986	+	0.876728i	$0.340279\pi$
$557$	976.675i	1.75346i	−0.480986	+	0.876728i	$0.659721\pi$
$558$	0	0
$559$	594.739	1.06393
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 442.626i	− 0.786191i	−0.919498	−	0.393096i	$-0.871404\pi$
$563$	− 442.626i	− 0.786191i	0.919498	−	0.393096i	$-0.128596\pi$
$564$	0	0
$565$	385.508	0.682315
$566$	0	0
$567$	0	0
$568$	0	0
$569$	787.034i	1.38319i	0.722286	+	0.691594i	$0.243091\pi$
$569$	787.034i	1.38319i	−0.722286	+	0.691594i	$0.756909\pi$
$570$	0	0
$571$	−6.82746	−0.0119570	−0.00597851	−	0.999982i	$-0.501903\pi$
$571$	−6.82746	−0.0119570	−0.00597851	+	0.999982i	$0.501903\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 67.4989i	− 0.117389i
$576$	0	0
$577$	−45.4328	−0.0787396	−0.0393698	−	0.999225i	$-0.512535\pi$
$577$	−45.4328	−0.0787396	−0.0393698	+	0.999225i	$0.512535\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 28.6564i	− 0.0493225i
$582$	0	0
$583$	126.210	0.216483
$584$	0	0
$585$	0	0
$586$	0	0
$587$	99.2488i	0.169078i	0.996420	+	0.0845391i	$0.0269418\pi$
$587$	99.2488i	0.169078i	−0.996420	+	0.0845391i	$0.973058\pi$
$588$	0	0
$589$	−337.505	−0.573014
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 226.897i	− 0.382626i	−0.981529	−	0.191313i	$-0.938726\pi$
$593$	− 226.897i	− 0.382626i	0.981529	−	0.191313i	$-0.0612745\pi$
$594$	0	0
$595$	−49.7444	−0.0836040
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 753.014i	− 1.25712i	−0.777762	−	0.628559i	$-0.783645\pi$
$599$	− 753.014i	− 1.25712i	0.777762	−	0.628559i	$-0.216355\pi$
$600$	0	0
$601$	685.049	1.13985	0.569924	−	0.821697i	$-0.306972\pi$
$601$	685.049	1.13985	0.569924	+	0.821697i	$0.306972\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	231.176i	0.382109i
$606$	0	0
$607$	908.951	1.49745	0.748724	−	0.662882i	$-0.230667\pi$
$607$	908.951	1.49745	0.748724	+	0.662882i	$0.230667\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	961.949i	1.57438i
$612$	0	0
$613$	514.234	0.838881	0.419441	−	0.907783i	$-0.362226\pi$
$613$	514.234	0.838881	0.419441	+	0.907783i	$0.362226\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1150.27i	1.86429i	0.362089	+	0.932144i	$0.382064\pi$
$617$	1150.27i	1.86429i	−0.362089	+	0.932144i	$0.617936\pi$
$618$	0	0
$619$	−139.987	−0.226151	−0.113076	−	0.993586i	$-0.536070\pi$
$619$	−139.987	−0.226151	−0.113076	+	0.993586i	$0.536070\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	52.8580i	0.0848443i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	361.142i	0.574153i
$630$	0	0
$631$	−369.971	−0.586325	−0.293163	−	0.956063i	$-0.594708\pi$
$631$	−369.971	−0.586325	−0.293163	+	0.956063i	$0.594708\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 437.940i	− 0.689669i
$636$	0	0
$637$	681.063	1.06917
$638$	0	0
$639$	0	0
$640$	0	0
$641$	77.3829i	0.120722i	0.998177	+	0.0603611i	$0.0192252\pi$
$641$	77.3829i	0.120722i	−0.998177	+	0.0603611i	$0.980775\pi$
$642$	0	0
$643$	261.808	0.407167	0.203583	−	0.979058i	$-0.434741\pi$
$643$	261.808	0.407167	0.203583	+	0.979058i	$0.434741\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 617.428i	− 0.954293i	−0.878824	−	0.477147i	$-0.841671\pi$
$647$	− 617.428i	− 0.954293i	0.878824	−	0.477147i	$-0.158329\pi$
$648$	0	0
$649$	17.7643	0.0273718
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 674.478i	− 1.03289i	−0.856320	−	0.516446i	$-0.827255\pi$
$653$	− 674.478i	− 1.03289i	0.856320	−	0.516446i	$-0.172745\pi$
$654$	0	0
$655$	−254.390	−0.388382
$656$	0	0
$657$	0	0
$658$	0	0
$659$	384.485i	0.583436i	0.956504	+	0.291718i	$0.0942269\pi$
$659$	384.485i	0.583436i	−0.956504	+	0.291718i	$0.905773\pi$
$660$	0	0
$661$	−303.426	−0.459041	−0.229520	−	0.973304i	$-0.573716\pi$
$661$	−303.426	−0.459041	−0.229520	+	0.973304i	$0.573716\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 90.1389i	− 0.135547i
$666$	0	0
$667$	−477.490	−0.715877
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 367.835i	− 0.548189i
$672$	0	0
$673$	−560.124	−0.832279	−0.416139	−	0.909301i	$-0.636617\pi$
$673$	−560.124	−0.832279	−0.416139	+	0.909301i	$0.636617\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1016.56i	− 1.50157i	−0.660546	−	0.750786i	$-0.729675\pi$
$677$	− 1016.56i	− 1.50157i	0.660546	−	0.750786i	$-0.270325\pi$
$678$	0	0
$679$	−76.0290	−0.111972
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 88.7713i	− 0.129973i	−0.997886	−	0.0649863i	$-0.979300\pi$
$683$	− 88.7713i	− 0.129973i	0.997886	−	0.0649863i	$-0.0207004\pi$
$684$	0	0
$685$	−404.786	−0.590929
$686$	0	0
$687$	0	0
$688$	0	0
$689$	430.388i	0.624656i
$690$	0	0
$691$	807.252	1.16824	0.584119	−	0.811668i	$-0.301440\pi$
$691$	807.252	1.16824	0.584119	+	0.811668i	$0.301440\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 512.304i	− 0.737128i
$696$	0	0
$697$	1208.92	1.73446
$698$	0	0
$699$	0	0
$700$	0	0
$701$	201.018i	0.286758i	0.989668	+	0.143379i	$0.0457968\pi$
$701$	201.018i	0.286758i	−0.989668	+	0.143379i	$0.954203\pi$
$702$	0	0
$703$	−654.404	−0.930874
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 14.2245i	− 0.0201196i
$708$	0	0
$709$	391.321	0.551934	0.275967	−	0.961167i	$-0.411002\pi$
$709$	391.321	0.551934	0.275967	+	0.961167i	$0.411002\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 134.408i	− 0.188511i
$714$	0	0
$715$	134.318	0.187858
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1241.75i	1.72705i	0.504308	+	0.863524i	$0.331747\pi$
$719$	1241.75i	1.72705i	−0.504308	+	0.863524i	$0.668253\pi$
$720$	0	0
$721$	194.987	0.270439
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 176.851i	− 0.243932i
$726$	0	0
$727$	−200.423	−0.275685	−0.137843	−	0.990454i	$-0.544017\pi$
$727$	−200.423	−0.275685	−0.137843	+	0.990454i	$0.544017\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	777.376i	1.06344i
$732$	0	0
$733$	−120.721	−0.164695	−0.0823475	−	0.996604i	$-0.526242\pi$
$733$	−120.721	−0.164695	−0.0823475	+	0.996604i	$0.526242\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 150.287i	− 0.203917i
$738$	0	0
$739$	−471.461	−0.637972	−0.318986	−	0.947759i	$-0.603342\pi$
$739$	−471.461	−0.637972	−0.318986	+	0.947759i	$0.603342\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1302.68i	− 1.75327i	−0.481158	−	0.876634i	$-0.659784\pi$
$743$	− 1302.68i	− 1.75327i	0.481158	−	0.876634i	$-0.340216\pi$
$744$	0	0
$745$	603.783	0.810446
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 121.773i	− 0.162580i
$750$	0	0
$751$	−879.453	−1.17104	−0.585521	−	0.810657i	$-0.699110\pi$
$751$	−879.453	−1.17104	−0.585521	+	0.810657i	$0.699110\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 364.795i	− 0.483172i
$756$	0	0
$757$	−35.1814	−0.0464748	−0.0232374	−	0.999730i	$-0.507397\pi$
$757$	−35.1814	−0.0464748	−0.0232374	+	0.999730i	$0.507397\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1273.57i	− 1.67355i	−0.547549	−	0.836774i	$-0.684439\pi$
$761$	− 1273.57i	− 1.67355i	0.547549	−	0.836774i	$-0.315561\pi$
$762$	0	0
$763$	67.1270	0.0879778
$764$	0	0
$765$	0	0
$766$	0	0
$767$	60.5782i	0.0789807i
$768$	0	0
$769$	759.485	0.987626	0.493813	−	0.869568i	$-0.335603\pi$
$769$	759.485	0.987626	0.493813	+	0.869568i	$0.335603\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 165.240i	− 0.213765i	−0.994272	−	0.106883i	$-0.965913\pi$
$773$	− 165.240i	− 0.213765i	0.994272	−	0.106883i	$-0.0340869\pi$
$774$	0	0
$775$	49.7816	0.0642343
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2190.61i	2.81209i
$780$	0	0
$781$	102.899	0.131753
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 300.974i	− 0.383406i
$786$	0	0
$787$	575.504	0.731263	0.365632	−	0.930760i	$-0.380853\pi$
$787$	575.504	0.731263	0.365632	+	0.930760i	$0.380853\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	205.019i	0.259190i
$792$	0	0
$793$	1254.36	1.58179
$794$	0	0
$795$	0	0
$796$	0	0
$797$	607.615i	0.762378i	0.924497	+	0.381189i	$0.124485\pi$
$797$	607.615i	0.762378i	−0.924497	+	0.381189i	$0.875515\pi$
$798$	0	0
$799$	−1257.35	−1.57366
$800$	0	0
$801$	0	0
$802$	0	0
$803$	50.0726i	0.0623570i
$804$	0	0
$805$	35.8969	0.0445925
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 962.401i	− 1.18962i	−0.803867	−	0.594809i	$-0.797228\pi$
$809$	− 962.401i	− 1.18962i	0.803867	−	0.594809i	$-0.202772\pi$
$810$	0	0
$811$	−140.096	−0.172745	−0.0863723	−	0.996263i	$-0.527527\pi$
$811$	−140.096	−0.172745	−0.0863723	+	0.996263i	$0.527527\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 357.007i	− 0.438046i
$816$	0	0
$817$	−1408.64	−1.72416
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 323.656i	− 0.394222i	−0.980381	−	0.197111i	$-0.936844\pi$
$821$	− 323.656i	− 0.394222i	0.980381	−	0.197111i	$-0.0631559\pi$
$822$	0	0
$823$	−193.212	−0.234766	−0.117383	−	0.993087i	$-0.537450\pi$
$823$	−193.212	−0.234766	−0.117383	+	0.993087i	$0.537450\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 15.7916i	− 0.0190951i	−0.999954	−	0.00954754i	$-0.996961\pi$
$827$	− 15.7916i	− 0.0190951i	0.999954	−	0.00954754i	$-0.00303912\pi$
$828$	0	0
$829$	−265.576	−0.320357	−0.160178	−	0.987088i	$-0.551207\pi$
$829$	−265.576	−0.320357	−0.160178	+	0.987088i	$0.551207\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	890.209i	1.06868i
$834$	0	0
$835$	248.548	0.297662
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 879.816i	− 1.04865i	−0.851519	−	0.524324i	$-0.824318\pi$
$839$	− 879.816i	− 1.04865i	0.851519	−	0.524324i	$-0.175682\pi$
$840$	0	0
$841$	−410.050	−0.487574
$842$	0	0
$843$	0	0
$844$	0	0
$845$	80.1442i	0.0948452i
$846$	0	0
$847$	−122.943	−0.145151
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 260.610i	− 0.306240i
$852$	0	0
$853$	204.146	0.239327	0.119663	−	0.992815i	$-0.461818\pi$
$853$	204.146	0.239327	0.119663	+	0.992815i	$0.461818\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	213.457i	0.249075i	0.992215	+	0.124537i	$0.0397447\pi$
$857$	213.457i	0.249075i	−0.992215	+	0.124537i	$0.960255\pi$
$858$	0	0
$859$	647.353	0.753613	0.376806	−	0.926292i	$-0.377022\pi$
$859$	647.353	0.753613	0.376806	+	0.926292i	$0.377022\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 699.772i	− 0.810859i	−0.914126	−	0.405430i	$-0.867122\pi$
$863$	− 699.772i	− 0.810859i	0.914126	−	0.405430i	$-0.132878\pi$
$864$	0	0
$865$	329.487	0.380910
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 160.207i	− 0.184357i
$870$	0	0
$871$	512.494	0.588398
$872$	0	0
$873$	0	0
$874$	0	0
$875$	13.2954i	0.0151947i
$876$	0	0
$877$	−1424.96	−1.62481	−0.812406	−	0.583092i	$-0.801843\pi$
$877$	−1424.96	−1.62481	−0.812406	+	0.583092i	$0.801843\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1598.97i	1.81495i	0.420102	+	0.907477i	$0.361994\pi$
$881$	1598.97i	1.81495i	−0.420102	+	0.907477i	$0.638006\pi$
$882$	0	0
$883$	−1117.03	−1.26504	−0.632522	−	0.774542i	$-0.717980\pi$
$883$	−1117.03	−1.26504	−0.632522	+	0.774542i	$0.717980\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1166.01i	1.31455i	0.753651	+	0.657275i	$0.228291\pi$
$887$	1166.01i	1.31455i	−0.753651	+	0.657275i	$0.771709\pi$
$888$	0	0
$889$	232.903	0.261983
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2278.37i	− 2.55137i
$894$	0	0
$895$	−565.774	−0.632149
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 352.157i	− 0.391721i
$900$	0	0
$901$	−562.555	−0.624367
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 474.084i	− 0.523850i
$906$	0	0
$907$	−1762.49	−1.94320	−0.971602	−	0.236620i	$-0.923960\pi$
$907$	−1762.49	−1.94320	−0.971602	+	0.236620i	$0.923960\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1674.20i	1.83776i	0.394542	+	0.918878i	$0.370903\pi$
$911$	1674.20i	1.83776i	−0.394542	+	0.918878i	$0.629097\pi$
$912$	0	0
$913$	−101.139	−0.110776
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 135.289i	− 0.147534i
$918$	0	0
$919$	1136.29	1.23644	0.618221	−	0.786004i	$-0.287854\pi$
$919$	1136.29	1.23644	0.618221	+	0.786004i	$0.287854\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	350.898i	0.380171i
$924$	0	0
$925$	96.5237	0.104350
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1077.26i	1.15959i	0.814761	+	0.579796i	$0.196868\pi$
$929$	1077.26i	1.15959i	−0.814761	+	0.579796i	$0.803132\pi$
$930$	0	0
$931$	−1613.10	−1.73265
$932$	0	0
$933$	0	0
$934$	0	0
$935$	175.566i	0.187771i
$936$	0	0
$937$	1063.87	1.13540	0.567702	−	0.823234i	$-0.307833\pi$
$937$	1063.87	1.13540	0.567702	+	0.823234i	$0.307833\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1148.68i	1.22070i	0.792133	+	0.610349i	$0.208971\pi$
$941$	1148.68i	1.22070i	−0.792133	+	0.610349i	$0.791029\pi$
$942$	0	0
$943$	−872.391	−0.925123
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 115.892i	− 0.122378i	−0.998126	−	0.0611891i	$-0.980511\pi$
$947$	− 115.892i	− 0.122378i	0.998126	−	0.0611891i	$-0.0194893\pi$
$948$	0	0
$949$	−170.753	−0.179930
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1854.05i	− 1.94549i	−0.231881	−	0.972744i	$-0.574488\pi$
$953$	− 1854.05i	− 1.94549i	0.231881	−	0.972744i	$-0.425512\pi$
$954$	0	0
$955$	741.101	0.776022
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 215.271i	− 0.224475i
$960$	0	0
$961$	−861.872	−0.896849
$962$	0	0
$963$	0	0
$964$	0	0
$965$	743.622i	0.770593i
$966$	0	0
$967$	−912.894	−0.944048	−0.472024	−	0.881586i	$-0.656476\pi$
$967$	−912.894	−0.944048	−0.472024	+	0.881586i	$0.656476\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1732.32i	− 1.78405i	−0.451982	−	0.892027i	$-0.649283\pi$
$971$	− 1732.32i	− 1.78405i	0.451982	−	0.892027i	$-0.350717\pi$
$972$	0	0
$973$	272.451	0.280011
$974$	0	0
$975$	0	0
$976$	0	0
$977$	231.347i	0.236793i	0.992966	+	0.118396i	$0.0377754\pi$
$977$	231.347i	0.236793i	−0.992966	+	0.118396i	$0.962225\pi$
$978$	0	0
$979$	186.555	0.190557
$980$	0	0
$981$	0	0
$982$	0	0
$983$	779.201i	0.792677i	0.918104	+	0.396338i	$0.129719\pi$
$983$	779.201i	0.792677i	−0.918104	+	0.396338i	$0.870281\pi$
$984$	0	0
$985$	−718.268	−0.729207
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 560.976i	− 0.567215i
$990$	0	0
$991$	222.470	0.224490	0.112245	−	0.993681i	$-0.464196\pi$
$991$	222.470	0.224490	0.112245	+	0.993681i	$0.464196\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 467.144i	− 0.469491i
$996$	0	0
$997$	−453.424	−0.454788	−0.227394	−	0.973803i	$-0.573021\pi$
$997$	−453.424	−0.454788	−0.227394	+	0.973803i	$0.573021\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.g.b.161.9		12
3.2	odd	2	inner	1620.3.g.b.161.3		12
9.2	odd	6		540.3.o.b.341.5		12
9.4	even	3		540.3.o.b.521.5		12
9.5	odd	6		180.3.o.b.101.2	yes	12
9.7	even	3		180.3.o.b.41.2	✓	12
36.7	odd	6		720.3.bs.b.401.5		12
36.11	even	6		2160.3.bs.b.881.5		12
36.23	even	6		720.3.bs.b.641.5		12
36.31	odd	6		2160.3.bs.b.1601.5		12
45.2	even	12		2700.3.u.c.449.6		24
45.4	even	6		2700.3.p.c.1601.3		12
45.7	odd	12		900.3.u.c.149.9		24
45.13	odd	12		2700.3.u.c.2249.6		24
45.14	odd	6		900.3.p.c.101.5		12
45.22	odd	12		2700.3.u.c.2249.7		24
45.23	even	12		900.3.u.c.749.9		24
45.29	odd	6		2700.3.p.c.2501.3		12
45.32	even	12		900.3.u.c.749.4		24
45.34	even	6		900.3.p.c.401.5		12
45.38	even	12		2700.3.u.c.449.7		24
45.43	odd	12		900.3.u.c.149.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.o.b.41.2	✓	12	9.7	even	3
180.3.o.b.101.2	yes	12	9.5	odd	6
540.3.o.b.341.5		12	9.2	odd	6
540.3.o.b.521.5		12	9.4	even	3
720.3.bs.b.401.5		12	36.7	odd	6
720.3.bs.b.641.5		12	36.23	even	6
900.3.p.c.101.5		12	45.14	odd	6
900.3.p.c.401.5		12	45.34	even	6
900.3.u.c.149.4		24	45.43	odd	12
900.3.u.c.149.9		24	45.7	odd	12
900.3.u.c.749.4		24	45.32	even	12
900.3.u.c.749.9		24	45.23	even	12
1620.3.g.b.161.3		12	3.2	odd	2	inner
1620.3.g.b.161.9		12	1.1	even	1	trivial
2160.3.bs.b.881.5		12	36.11	even	6
2160.3.bs.b.1601.5		12	36.31	odd	6
2700.3.p.c.1601.3		12	45.4	even	6
2700.3.p.c.2501.3		12	45.29	odd	6
2700.3.u.c.449.6		24	45.2	even	12
2700.3.u.c.449.7		24	45.38	even	12
2700.3.u.c.2249.6		24	45.13	odd	12
2700.3.u.c.2249.7		24	45.22	odd	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.23607i q^{5} -1.18917 q^{7} +O(q^{10})\)	\(q+2.23607i q^{5} -1.18917 q^{7} +4.19702i q^{11} -14.3123 q^{13} -18.7074i q^{17} +33.8986 q^{19} +13.4998i q^{23} -5.00000 q^{25} +35.3702i q^{29} -9.95632 q^{31} -2.65907i q^{35} -19.3047 q^{37} +64.6225i q^{41} -41.5544 q^{43} -67.2114i q^{47} -47.5859 q^{49} -30.0712i q^{53} -9.38482 q^{55} -4.23260i q^{59} -87.6420 q^{61} -32.0032i q^{65} -35.8080 q^{67} -24.5173i q^{71} +11.9305 q^{73} -4.99099i q^{77} -38.1715 q^{79} +24.0977i q^{83} +41.8311 q^{85} -44.4494i q^{89} +17.0198 q^{91} +75.7996i q^{95} +63.9343 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7}+O(q^{10}) \) 12 * q + 12 * q^7	\( 12 q + 12 q^{7} + 60 q^{13} + 72 q^{19} - 60 q^{25} + 24 q^{31} - 12 q^{37} - 228 q^{43} - 96 q^{49} - 120 q^{55} + 156 q^{61} + 336 q^{67} - 24 q^{73} - 240 q^{79} + 60 q^{85} + 120 q^{91} - 192 q^{97}+O(q^{100}) \) 12 * q + 12 * q^7 + 60 * q^13 + 72 * q^19 - 60 * q^25 + 24 * q^31 - 12 * q^37 - 228 * q^43 - 96 * q^49 - 120 * q^55 + 156 * q^61 + 336 * q^67 - 24 * q^73 - 240 * q^79 + 60 * q^85 + 120 * q^91 - 192 * q^97

Introduction

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