LMFDB - Embedded newform 8100.2.d.k.649.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(649,8100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8100.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.d (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:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	8100.649
Dual form			8100.2.d.k.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.85410i q^{7} +O(q^{10})$	$q+3.85410i q^{7} -4.85410 q^{11} +5.85410i q^{13} +7.85410i q^{17} -2.00000 q^{19} -1.85410i q^{23} +9.70820 q^{29} -10.7082 q^{31} +0.854102i q^{37} +8.56231 q^{41} +5.85410i q^{43} +6.70820i q^{47} -7.85410 q^{49} +1.85410i q^{53} -7.85410 q^{59} -5.85410 q^{61} -7.00000i q^{67} +9.00000 q^{71} +10.7082i q^{73} -18.7082i q^{77} +1.70820 q^{79} +6.70820i q^{83} +12.0000 q^{89} -22.5623 q^{91} -10.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 6 q^{11} - 8 q^{19} + 12 q^{29} - 16 q^{31} - 6 q^{41} - 18 q^{49} - 18 q^{59} - 10 q^{61} + 36 q^{71} - 20 q^{79} + 48 q^{89} - 50 q^{91}+O(q^{100}) $ 4 * q - 6 * q^11 - 8 * q^19 + 12 * q^29 - 16 * q^31 - 6 * q^41 - 18 * q^49 - 18 * q^59 - 10 * q^61 + 36 * q^71 - 20 * q^79 + 48 * q^89 - 50 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$.

$n$	$4051$	$6401$	$7777$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.85410i	1.45671i	0.685198	+	0.728357i	$0.259716\pi$
$7$	3.85410i	1.45671i	−0.685198	+	0.728357i	$0.740284\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.85410	−1.46357	−0.731783	−	0.681537i	$-0.761312\pi$
$11$	−4.85410	−1.46357	−0.731783	+	0.681537i	$0.761312\pi$
$12$	0	0
$13$	5.85410i	1.62364i	0.583911	+	0.811818i	$0.301522\pi$
$13$	5.85410i	1.62364i	−0.583911	+	0.811818i	$0.698478\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.85410i	1.90490i	0.304696	+	0.952450i	$0.401445\pi$
$17$	7.85410i	1.90490i	−0.304696	+	0.952450i	$0.598555\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1.85410i	− 0.386607i	−0.981139	−	0.193303i	$-0.938080\pi$
$23$	− 1.85410i	− 0.386607i	0.981139	−	0.193303i	$-0.0619202\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.70820	1.80277	0.901384	−	0.433020i	$-0.142552\pi$
$29$	9.70820	1.80277	0.901384	+	0.433020i	$0.142552\pi$
$30$	0	0
$31$	−10.7082	−1.92325	−0.961625	−	0.274367i	$-0.911532\pi$
$31$	−10.7082	−1.92325	−0.961625	+	0.274367i	$0.911532\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.854102i	0.140413i	0.997532	+	0.0702067i	$0.0223659\pi$
$37$	0.854102i	0.140413i	−0.997532	+	0.0702067i	$0.977634\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.56231	1.33721	0.668604	−	0.743619i	$-0.266892\pi$
$41$	8.56231	1.33721	0.668604	+	0.743619i	$0.266892\pi$
$42$	0	0
$43$	5.85410i	0.892742i	0.894848	+	0.446371i	$0.147284\pi$
$43$	5.85410i	0.892742i	−0.894848	+	0.446371i	$0.852716\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.70820i	0.978492i	0.872146	+	0.489246i	$0.162728\pi$
$47$	6.70820i	0.978492i	−0.872146	+	0.489246i	$0.837272\pi$
$48$	0	0
$49$	−7.85410	−1.12201
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.85410i	0.254680i	0.991859	+	0.127340i	$0.0406440\pi$
$53$	1.85410i	0.254680i	−0.991859	+	0.127340i	$0.959356\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.85410	−1.02252	−0.511258	−	0.859427i	$-0.670821\pi$
$59$	−7.85410	−1.02252	−0.511258	+	0.859427i	$0.670821\pi$
$60$	0	0
$61$	−5.85410	−0.749541	−0.374770	−	0.927118i	$-0.622278\pi$
$61$	−5.85410	−0.749541	−0.374770	+	0.927118i	$0.622278\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	10.7082i	1.25330i	0.779301	+	0.626650i	$0.215575\pi$
$73$	10.7082i	1.25330i	−0.779301	+	0.626650i	$0.784425\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 18.7082i	− 2.13200i
$78$	0	0
$79$	1.70820	0.192188	0.0960940	−	0.995372i	$-0.469365\pi$
$79$	1.70820	0.192188	0.0960940	+	0.995372i	$0.469365\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.70820i	0.736321i	0.929762	+	0.368161i	$0.120012\pi$
$83$	6.70820i	0.736321i	−0.929762	+	0.368161i	$0.879988\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	−22.5623	−2.36517
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.85410	−0.184490	−0.0922450	−	0.995736i	$-0.529404\pi$
$101$	−1.85410	−0.184490	−0.0922450	+	0.995736i	$0.529404\pi$
$102$	0	0
$103$	− 6.85410i	− 0.675355i	−0.941262	−	0.337677i	$-0.890359\pi$
$103$	− 6.85410i	− 0.675355i	0.941262	−	0.337677i	$-0.109641\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.29180i	0.511577i	0.966733	+	0.255789i	$0.0823351\pi$
$107$	5.29180i	0.511577i	−0.966733	+	0.255789i	$0.917665\pi$
$108$	0	0
$109$	2.85410	0.273373	0.136687	−	0.990614i	$-0.456355\pi$
$109$	2.85410	0.273373	0.136687	+	0.990614i	$0.456355\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4.85410i	− 0.456636i	−0.973587	−	0.228318i	$-0.926677\pi$
$113$	− 4.85410i	− 0.456636i	0.973587	−	0.228318i	$-0.0733225\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−30.2705	−2.77489
$120$	0	0
$121$	12.5623	1.14203
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 7.00000i	− 0.621150i	−0.950549	−	0.310575i	$-0.899478\pi$
$127$	− 7.00000i	− 0.621150i	0.950549	−	0.310575i	$-0.100522\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.7082	−1.37243	−0.686216	−	0.727398i	$-0.740730\pi$
$131$	−15.7082	−1.37243	−0.686216	+	0.727398i	$0.740730\pi$
$132$	0	0
$133$	− 7.70820i	− 0.668386i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 17.5623i	− 1.50045i	−0.661183	−	0.750225i	$-0.729945\pi$
$137$	− 17.5623i	− 1.50045i	0.661183	−	0.750225i	$-0.270055\pi$
$138$	0	0
$139$	16.2705	1.38005	0.690023	−	0.723787i	$-0.257600\pi$
$139$	16.2705	1.38005	0.690023	+	0.723787i	$0.257600\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 28.4164i	− 2.37630i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−14.8541	−1.20881	−0.604405	−	0.796677i	$-0.706589\pi$
$151$	−14.8541	−1.20881	−0.604405	+	0.796677i	$0.706589\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3.29180i	− 0.262714i	−0.991335	−	0.131357i	$-0.958067\pi$
$157$	− 3.29180i	− 0.262714i	0.991335	−	0.131357i	$-0.0419334\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.14590	0.563176
$162$	0	0
$163$	− 12.4164i	− 0.972528i	−0.873812	−	0.486264i	$-0.838359\pi$
$163$	− 12.4164i	− 0.972528i	0.873812	−	0.486264i	$-0.161641\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 16.1459i	− 1.24941i	−0.780862	−	0.624704i	$-0.785220\pi$
$167$	− 16.1459i	− 1.24941i	0.780862	−	0.624704i	$-0.214780\pi$
$168$	0	0
$169$	−21.2705	−1.63619
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.29180i	0.174242i	0.996198	+	0.0871210i	$0.0277667\pi$
$173$	2.29180i	0.174242i	−0.996198	+	0.0871210i	$0.972233\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.7082	1.39832	0.699158	−	0.714967i	$-0.253558\pi$
$179$	18.7082	1.39832	0.699158	+	0.714967i	$0.253558\pi$
$180$	0	0
$181$	12.4164	0.922904	0.461452	−	0.887165i	$-0.347329\pi$
$181$	12.4164	0.922904	0.461452	+	0.887165i	$0.347329\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 38.1246i	− 2.78795i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.41641	0.536632	0.268316	−	0.963331i	$-0.413533\pi$
$191$	7.41641	0.536632	0.268316	+	0.963331i	$0.413533\pi$
$192$	0	0
$193$	3.29180i	0.236949i	0.992957	+	0.118474i	$0.0378003\pi$
$193$	3.29180i	0.236949i	−0.992957	+	0.118474i	$0.962200\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.1459i	1.36409i	0.731311	+	0.682044i	$0.238909\pi$
$197$	19.1459i	1.36409i	−0.731311	+	0.682044i	$0.761091\pi$
$198$	0	0
$199$	−9.14590	−0.648336	−0.324168	−	0.946000i	$-0.605084\pi$
$199$	−9.14590	−0.648336	−0.324168	+	0.946000i	$0.605084\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	37.4164i	2.62612i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.70820	0.671531
$210$	0	0
$211$	−9.29180	−0.639674	−0.319837	−	0.947473i	$-0.603628\pi$
$211$	−9.29180	−0.639674	−0.319837	+	0.947473i	$0.603628\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 41.2705i	− 2.80162i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−45.9787	−3.09286
$222$	0	0
$223$	16.7082i	1.11886i	0.828876	+	0.559432i	$0.188981\pi$
$223$	16.7082i	1.11886i	−0.828876	+	0.559432i	$0.811019\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.00000i	0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$	3.00000i	0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$	0	0
$229$	−18.4164	−1.21699	−0.608495	−	0.793558i	$-0.708227\pi$
$229$	−18.4164	−1.21699	−0.608495	+	0.793558i	$0.708227\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.1246i	1.31841i	0.751965	+	0.659204i	$0.229106\pi$
$233$	20.1246i	1.31841i	−0.751965	+	0.659204i	$0.770894\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	13.4164	0.867835	0.433918	−	0.900953i	$-0.357131\pi$
$239$	13.4164	0.867835	0.433918	+	0.900953i	$0.357131\pi$
$240$	0	0
$241$	9.41641	0.606564	0.303282	−	0.952901i	$-0.401918\pi$
$241$	9.41641	0.606564	0.303282	+	0.952901i	$0.401918\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 11.7082i	− 0.744975i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.14590	0.261687	0.130843	−	0.991403i	$-0.458231\pi$
$251$	4.14590	0.261687	0.130843	+	0.991403i	$0.458231\pi$
$252$	0	0
$253$	9.00000i	0.565825i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.41641i	0.0883531i	0.999024	+	0.0441765i	$0.0140664\pi$
$257$	1.41641i	0.0883531i	−0.999024	+	0.0441765i	$0.985934\pi$
$258$	0	0
$259$	−3.29180	−0.204542
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 15.7082i	− 0.968609i	−0.874899	−	0.484305i	$-0.839073\pi$
$263$	− 15.7082i	− 0.968609i	0.874899	−	0.484305i	$-0.160927\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.27051	−0.382320	−0.191160	−	0.981559i	$-0.561225\pi$
$269$	−6.27051	−0.382320	−0.191160	+	0.981559i	$0.561225\pi$
$270$	0	0
$271$	18.4164	1.11872	0.559359	−	0.828926i	$-0.311048\pi$
$271$	18.4164	1.11872	0.559359	+	0.828926i	$0.311048\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.41641i	0.205272i	0.994719	+	0.102636i	$0.0327277\pi$
$277$	3.41641i	0.205272i	−0.994719	+	0.102636i	$0.967272\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.70820	0.400178	0.200089	−	0.979778i	$-0.435877\pi$
$281$	6.70820	0.400178	0.200089	+	0.979778i	$0.435877\pi$
$282$	0	0
$283$	5.14590i	0.305892i	0.988235	+	0.152946i	$0.0488760\pi$
$283$	5.14590i	0.305892i	−0.988235	+	0.152946i	$0.951124\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	33.0000i	1.94793i
$288$	0	0
$289$	−44.6869	−2.62864
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.1246i	1.17569i	0.808973	+	0.587846i	$0.200024\pi$
$293$	20.1246i	1.17569i	−0.808973	+	0.587846i	$0.799976\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.8541	0.627709
$300$	0	0
$301$	−22.5623	−1.30047
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 10.0000i	− 0.570730i	−0.958419	−	0.285365i	$-0.907885\pi$
$307$	− 10.0000i	− 0.570730i	0.958419	−	0.285365i	$-0.0921148\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.7082	−0.720616	−0.360308	−	0.932833i	$-0.617328\pi$
$311$	−12.7082	−0.720616	−0.360308	+	0.932833i	$0.617328\pi$
$312$	0	0
$313$	14.1459i	0.799573i	0.916608	+	0.399787i	$0.130916\pi$
$313$	14.1459i	0.799573i	−0.916608	+	0.399787i	$0.869084\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 28.4164i	− 1.59602i	−0.602641	−	0.798012i	$-0.705885\pi$
$317$	− 28.4164i	− 1.59602i	0.602641	−	0.798012i	$-0.294115\pi$
$318$	0	0
$319$	−47.1246	−2.63847
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 15.7082i	− 0.874028i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−25.8541	−1.42538
$330$	0	0
$331$	9.41641	0.517573	0.258786	−	0.965935i	$-0.416677\pi$
$331$	9.41641	0.517573	0.258786	+	0.965935i	$0.416677\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 31.7082i	− 1.72726i	−0.504130	−	0.863628i	$-0.668187\pi$
$337$	− 31.7082i	− 1.72726i	0.504130	−	0.863628i	$-0.331813\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	51.9787	2.81481
$342$	0	0
$343$	− 3.29180i	− 0.177740i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 15.4377i	− 0.828739i	−0.910109	−	0.414369i	$-0.864002\pi$
$347$	− 15.4377i	− 0.828739i	0.910109	−	0.414369i	$-0.135998\pi$
$348$	0	0
$349$	−10.2918	−0.550907	−0.275454	−	0.961314i	$-0.588828\pi$
$349$	−10.2918	−0.550907	−0.275454	+	0.961314i	$0.588828\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2.56231i	− 0.136378i	−0.997672	−	0.0681889i	$-0.978278\pi$
$353$	− 2.56231i	− 0.136378i	0.997672	−	0.0681889i	$-0.0217221\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.8541	1.20619	0.603097	−	0.797668i	$-0.293933\pi$
$359$	22.8541	1.20619	0.603097	+	0.797668i	$0.293933\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.5623i	0.551348i	0.961251	+	0.275674i	$0.0889010\pi$
$367$	10.5623i	0.551348i	−0.961251	+	0.275674i	$0.911099\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7.14590	−0.370997
$372$	0	0
$373$	24.1246i	1.24913i	0.780975	+	0.624563i	$0.214723\pi$
$373$	24.1246i	1.24913i	−0.780975	+	0.624563i	$0.785277\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	56.8328i	2.92704i
$378$	0	0
$379$	13.2705	0.681660	0.340830	−	0.940125i	$-0.389292\pi$
$379$	13.2705	0.681660	0.340830	+	0.940125i	$0.389292\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 15.9787i	− 0.816474i	−0.912876	−	0.408237i	$-0.866144\pi$
$383$	− 15.9787i	− 0.816474i	0.912876	−	0.408237i	$-0.133856\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.70820	0.340119	0.170060	−	0.985434i	$-0.445604\pi$
$389$	6.70820	0.340119	0.170060	+	0.985434i	$0.445604\pi$
$390$	0	0
$391$	14.5623	0.736447
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	35.5410i	1.78375i	0.452279	+	0.891876i	$0.350611\pi$
$397$	35.5410i	1.78375i	−0.452279	+	0.891876i	$0.649389\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.12461	0.255911	0.127955	−	0.991780i	$-0.459159\pi$
$401$	5.12461	0.255911	0.127955	+	0.991780i	$0.459159\pi$
$402$	0	0
$403$	− 62.6869i	− 3.12266i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4.14590i	− 0.205505i
$408$	0	0
$409$	19.2705	0.952865	0.476433	−	0.879211i	$-0.341930\pi$
$409$	19.2705	0.952865	0.476433	+	0.879211i	$0.341930\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 30.2705i	− 1.48951i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10.8541	−0.530258	−0.265129	−	0.964213i	$-0.585414\pi$
$419$	−10.8541	−0.530258	−0.265129	+	0.964213i	$0.585414\pi$
$420$	0	0
$421$	−7.70820	−0.375675	−0.187837	−	0.982200i	$-0.560148\pi$
$421$	−7.70820	−0.375675	−0.187837	+	0.982200i	$0.560148\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 22.5623i	− 1.09187i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.0000	1.01153	0.505767	−	0.862670i	$-0.331209\pi$
$431$	21.0000	1.01153	0.505767	+	0.862670i	$0.331209\pi$
$432$	0	0
$433$	24.2918i	1.16739i	0.811973	+	0.583695i	$0.198393\pi$
$433$	24.2918i	1.16739i	−0.811973	+	0.583695i	$0.801607\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.70820i	0.177387i
$438$	0	0
$439$	−10.5623	−0.504111	−0.252056	−	0.967713i	$-0.581107\pi$
$439$	−10.5623	−0.504111	−0.252056	+	0.967713i	$0.581107\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 25.8541i	− 1.22837i	−0.789164	−	0.614183i	$-0.789486\pi$
$443$	− 25.8541i	− 1.22837i	0.789164	−	0.614183i	$-0.210514\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13.8541	0.653815	0.326908	−	0.945056i	$-0.393993\pi$
$449$	13.8541	0.653815	0.326908	+	0.945056i	$0.393993\pi$
$450$	0	0
$451$	−41.5623	−1.95709
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.43769	0.160109	0.0800547	−	0.996790i	$-0.474491\pi$
$461$	3.43769	0.160109	0.0800547	+	0.996790i	$0.474491\pi$
$462$	0	0
$463$	− 9.85410i	− 0.457959i	−0.973431	−	0.228979i	$-0.926461\pi$
$463$	− 9.85410i	− 0.457959i	0.973431	−	0.228979i	$-0.0735389\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	26.9787	1.24576
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 28.4164i	− 1.30659i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−22.8541	−1.04423	−0.522115	−	0.852875i	$-0.674857\pi$
$479$	−22.8541	−1.04423	−0.522115	+	0.852875i	$0.674857\pi$
$480$	0	0
$481$	−5.00000	−0.227980
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 23.4164i	− 1.06110i	−0.847654	−	0.530549i	$-0.821986\pi$
$487$	− 23.4164i	− 1.06110i	0.847654	−	0.530549i	$-0.178014\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	35.8328	1.61711	0.808556	−	0.588419i	$-0.200249\pi$
$491$	35.8328	1.61711	0.808556	+	0.588419i	$0.200249\pi$
$492$	0	0
$493$	76.2492i	3.43409i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	34.6869i	1.55592i
$498$	0	0
$499$	−25.5623	−1.14433	−0.572163	−	0.820140i	$-0.693896\pi$
$499$	−25.5623	−1.14433	−0.572163	+	0.820140i	$0.693896\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.27051i	0.413352i	0.978409	+	0.206676i	$0.0662645\pi$
$503$	9.27051i	0.413352i	−0.978409	+	0.206676i	$0.933735\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	−41.2705	−1.82570
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 32.5623i	− 1.43209i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5.12461	−0.224513	−0.112257	−	0.993679i	$-0.535808\pi$
$521$	−5.12461	−0.224513	−0.112257	+	0.993679i	$0.535808\pi$
$522$	0	0
$523$	8.41641i	0.368024i	0.982924	+	0.184012i	$0.0589085\pi$
$523$	8.41641i	0.368024i	−0.982924	+	0.184012i	$0.941091\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 84.1033i	− 3.66360i
$528$	0	0
$529$	19.5623	0.850535
$530$	0	0
$531$	0	0
$532$	0	0
$533$	50.1246i	2.17114i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	38.1246	1.64214
$540$	0	0
$541$	−33.3951	−1.43577	−0.717884	−	0.696163i	$-0.754889\pi$
$541$	−33.3951	−1.43577	−0.717884	+	0.696163i	$0.754889\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 11.5836i	− 0.495279i	−0.968852	−	0.247639i	$-0.920345\pi$
$547$	− 11.5836i	− 0.495279i	0.968852	−	0.247639i	$-0.0796548\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−19.4164	−0.827167
$552$	0	0
$553$	6.58359i	0.279963i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 40.2492i	− 1.70541i	−0.522389	−	0.852707i	$-0.674959\pi$
$557$	− 40.2492i	− 1.70541i	0.522389	−	0.852707i	$-0.325041\pi$
$558$	0	0
$559$	−34.2705	−1.44949
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.2918i	0.981632i	0.871263	+	0.490816i	$0.163301\pi$
$563$	23.2918i	0.981632i	−0.871263	+	0.490816i	$0.836699\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9.70820	−0.406989	−0.203495	−	0.979076i	$-0.565230\pi$
$569$	−9.70820	−0.406989	−0.203495	+	0.979076i	$0.565230\pi$
$570$	0	0
$571$	−42.8328	−1.79250	−0.896249	−	0.443552i	$-0.853718\pi$
$571$	−42.8328	−1.79250	−0.896249	+	0.443552i	$0.853718\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 26.4164i	− 1.09973i	−0.835254	−	0.549865i	$-0.814679\pi$
$577$	− 26.4164i	− 1.09973i	0.835254	−	0.549865i	$-0.185321\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−25.8541	−1.07261
$582$	0	0
$583$	− 9.00000i	− 0.372742i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.0000i	0.619116i	0.950881	+	0.309558i	$0.100181\pi$
$587$	15.0000i	0.619116i	−0.950881	+	0.309558i	$0.899819\pi$
$588$	0	0
$589$	21.4164	0.882448
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 30.2705i	− 1.24306i	−0.783390	−	0.621530i	$-0.786511\pi$
$593$	− 30.2705i	− 1.24306i	0.783390	−	0.621530i	$-0.213489\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0.437694	0.0178837	0.00894185	−	0.999960i	$-0.497154\pi$
$599$	0.437694	0.0178837	0.00894185	+	0.999960i	$0.497154\pi$
$600$	0	0
$601$	3.14590	0.128324	0.0641619	−	0.997940i	$-0.479563\pi$
$601$	3.14590	0.128324	0.0641619	+	0.997940i	$0.479563\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	38.5410i	1.56433i	0.623070	+	0.782166i	$0.285885\pi$
$607$	38.5410i	1.56433i	−0.623070	+	0.782166i	$0.714115\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−39.2705	−1.58871
$612$	0	0
$613$	1.70820i	0.0689937i	0.999405	+	0.0344969i	$0.0109829\pi$
$613$	1.70820i	0.0689937i	−0.999405	+	0.0344969i	$0.989017\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 20.1246i	− 0.810186i	−0.914275	−	0.405093i	$-0.867239\pi$
$617$	− 20.1246i	− 0.810186i	0.914275	−	0.405093i	$-0.132761\pi$
$618$	0	0
$619$	−6.58359	−0.264617	−0.132308	−	0.991209i	$-0.542239\pi$
$619$	−6.58359	−0.264617	−0.132308	+	0.991209i	$0.542239\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	46.2492i	1.85294i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.70820	−0.267474
$630$	0	0
$631$	6.85410	0.272857	0.136429	−	0.990650i	$-0.456438\pi$
$631$	6.85410	0.272857	0.136429	+	0.990650i	$0.456438\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 45.9787i	− 1.82174i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.56231	−0.338191	−0.169095	−	0.985600i	$-0.554085\pi$
$641$	−8.56231	−0.338191	−0.169095	+	0.985600i	$0.554085\pi$
$642$	0	0
$643$	− 17.4377i	− 0.687676i	−0.939029	−	0.343838i	$-0.888273\pi$
$643$	− 17.4377i	− 0.687676i	0.939029	−	0.343838i	$-0.111727\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.2918i	0.443926i	0.975055	+	0.221963i	$0.0712465\pi$
$647$	11.2918i	0.443926i	−0.975055	+	0.221963i	$0.928754\pi$
$648$	0	0
$649$	38.1246	1.49652
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 10.8541i	− 0.424754i	−0.977188	−	0.212377i	$-0.931880\pi$
$653$	− 10.8541i	− 0.424754i	0.977188	−	0.212377i	$-0.0681205\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.2705	0.594855	0.297427	−	0.954744i	$-0.403871\pi$
$659$	15.2705	0.594855	0.297427	+	0.954744i	$0.403871\pi$
$660$	0	0
$661$	−27.8328	−1.08257	−0.541286	−	0.840839i	$-0.682062\pi$
$661$	−27.8328	−1.08257	−0.541286	+	0.840839i	$0.682062\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 18.0000i	− 0.696963i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	28.4164	1.09700
$672$	0	0
$673$	10.7082i	0.412771i	0.978471	+	0.206385i	$0.0661701\pi$
$673$	10.7082i	0.412771i	−0.978471	+	0.206385i	$0.933830\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.4164i	0.400335i	0.979762	+	0.200168i	$0.0641486\pi$
$677$	10.4164i	0.400335i	−0.979762	+	0.200168i	$0.935851\pi$
$678$	0	0
$679$	38.5410	1.47907
$680$	0	0
$681$	0	0
$682$	0	0
$683$	44.3951i	1.69873i	0.527804	+	0.849366i	$0.323015\pi$
$683$	44.3951i	1.69873i	−0.527804	+	0.849366i	$0.676985\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−10.8541	−0.413508
$690$	0	0
$691$	−30.1246	−1.14599	−0.572997	−	0.819557i	$-0.694219\pi$
$691$	−30.1246	−1.14599	−0.572997	+	0.819557i	$0.694219\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	67.2492i	2.54725i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.29180	−0.0865599	−0.0432800	−	0.999063i	$-0.513781\pi$
$701$	−2.29180	−0.0865599	−0.0432800	+	0.999063i	$0.513781\pi$
$702$	0	0
$703$	− 1.70820i	− 0.0644261i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 7.14590i	− 0.268749i
$708$	0	0
$709$	19.0000	0.713560	0.356780	−	0.934188i	$-0.383875\pi$
$709$	19.0000	0.713560	0.356780	+	0.934188i	$0.383875\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19.8541i	0.743542i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0.437694	0.0163232	0.00816162	−	0.999967i	$-0.497402\pi$
$719$	0.437694	0.0163232	0.00816162	+	0.999967i	$0.497402\pi$
$720$	0	0
$721$	26.4164	0.983798
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	17.4377i	0.646728i	0.946275	+	0.323364i	$0.104814\pi$
$727$	17.4377i	0.646728i	−0.946275	+	0.323364i	$0.895186\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−45.9787	−1.70058
$732$	0	0
$733$	16.4377i	0.607140i	0.952809	+	0.303570i	$0.0981786\pi$
$733$	16.4377i	0.607140i	−0.952809	+	0.303570i	$0.901821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33.9787i	1.25162i
$738$	0	0
$739$	−25.5623	−0.940325	−0.470162	−	0.882580i	$-0.655805\pi$
$739$	−25.5623	−0.940325	−0.470162	+	0.882580i	$0.655805\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	41.1246i	1.50872i	0.656463	+	0.754358i	$0.272052\pi$
$743$	41.1246i	1.50872i	−0.656463	+	0.754358i	$0.727948\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−20.3951	−0.745222
$750$	0	0
$751$	5.97871	0.218166	0.109083	−	0.994033i	$-0.465209\pi$
$751$	5.97871	0.218166	0.109083	+	0.994033i	$0.465209\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 0.729490i	− 0.0265138i	−0.999912	−	0.0132569i	$-0.995780\pi$
$757$	− 0.729490i	− 0.0265138i	0.999912	−	0.0132569i	$-0.00421992\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.1459	−1.02029	−0.510144	−	0.860089i	$-0.670408\pi$
$761$	−28.1459	−1.02029	−0.510144	+	0.860089i	$0.670408\pi$
$762$	0	0
$763$	11.0000i	0.398227i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 45.9787i	− 1.66020i
$768$	0	0
$769$	32.8541	1.18475	0.592375	−	0.805663i	$-0.298191\pi$
$769$	32.8541	1.18475	0.592375	+	0.805663i	$0.298191\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 33.5410i	− 1.20639i	−0.797595	−	0.603193i	$-0.793895\pi$
$773$	− 33.5410i	− 1.20639i	0.797595	−	0.603193i	$-0.206105\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−17.1246	−0.613553
$780$	0	0
$781$	−43.6869	−1.56324
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 8.85410i	− 0.315615i	−0.987470	−	0.157807i	$-0.949558\pi$
$787$	− 8.85410i	− 0.315615i	0.987470	−	0.157807i	$-0.0504425\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	18.7082	0.665187
$792$	0	0
$793$	− 34.2705i	− 1.21698i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.2705i	0.859706i	0.902899	+	0.429853i	$0.141435\pi$
$797$	24.2705i	0.859706i	−0.902899	+	0.429853i	$0.858565\pi$
$798$	0	0
$799$	−52.6869	−1.86393
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 51.9787i	− 1.83429i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32.1246	1.12944	0.564721	−	0.825282i	$-0.308984\pi$
$809$	32.1246	1.12944	0.564721	+	0.825282i	$0.308984\pi$
$810$	0	0
$811$	29.9787	1.05270	0.526348	−	0.850270i	$-0.323561\pi$
$811$	29.9787	1.05270	0.526348	+	0.850270i	$0.323561\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 11.7082i	− 0.409618i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22.4164	−0.782338	−0.391169	−	0.920319i	$-0.627929\pi$
$821$	−22.4164	−0.782338	−0.391169	+	0.920319i	$0.627929\pi$
$822$	0	0
$823$	49.5410i	1.72689i	0.504442	+	0.863446i	$0.331698\pi$
$823$	49.5410i	1.72689i	−0.504442	+	0.863446i	$0.668302\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.3951i	1.02217i	0.859531	+	0.511084i	$0.170756\pi$
$827$	29.3951i	1.02217i	−0.859531	+	0.511084i	$0.829244\pi$
$828$	0	0
$829$	−21.4164	−0.743823	−0.371911	−	0.928268i	$-0.621297\pi$
$829$	−21.4164	−0.743823	−0.371911	+	0.928268i	$0.621297\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 61.6869i	− 2.13733i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.8754	−0.444508	−0.222254	−	0.974989i	$-0.571341\pi$
$839$	−12.8754	−0.444508	−0.222254	+	0.974989i	$0.571341\pi$
$840$	0	0
$841$	65.2492	2.24997
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	48.4164i	1.66361i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.58359	0.0542848
$852$	0	0
$853$	− 16.1246i	− 0.552096i	−0.961144	−	0.276048i	$-0.910975\pi$
$853$	− 16.1246i	− 0.552096i	0.961144	−	0.276048i	$-0.0890249\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 10.8541i	− 0.370769i	−0.982666	−	0.185385i	$-0.940647\pi$
$857$	− 10.8541i	− 0.370769i	0.982666	−	0.185385i	$-0.0593531\pi$
$858$	0	0
$859$	−36.4164	−1.24251	−0.621256	−	0.783608i	$-0.713377\pi$
$859$	−36.4164	−1.24251	−0.621256	+	0.783608i	$0.713377\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.2705i	0.621935i	0.950420	+	0.310968i	$0.100653\pi$
$863$	18.2705i	0.621935i	−0.950420	+	0.310968i	$0.899347\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.29180	−0.281280
$870$	0	0
$871$	40.9787	1.38851
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 19.2705i	− 0.650719i	−0.945590	−	0.325359i	$-0.894515\pi$
$877$	− 19.2705i	− 0.650719i	0.945590	−	0.325359i	$-0.105485\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14.1246	−0.475870	−0.237935	−	0.971281i	$-0.576471\pi$
$881$	−14.1246	−0.475870	−0.237935	+	0.971281i	$0.576471\pi$
$882$	0	0
$883$	9.12461i	0.307068i	0.988143	+	0.153534i	$0.0490654\pi$
$883$	9.12461i	0.307068i	−0.988143	+	0.153534i	$0.950935\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.9787i	1.54381i	0.635735	+	0.771907i	$0.280697\pi$
$887$	45.9787i	1.54381i	−0.635735	+	0.771907i	$0.719303\pi$
$888$	0	0
$889$	26.9787	0.904837
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 13.4164i	− 0.448963i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−103.957	−3.46717
$900$	0	0
$901$	−14.5623	−0.485141
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.9787i	1.09504i	0.836793	+	0.547520i	$0.184428\pi$
$907$	32.9787i	1.09504i	−0.836793	+	0.547520i	$0.815572\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	14.1246	0.467969	0.233985	−	0.972240i	$-0.424823\pi$
$911$	14.1246	0.467969	0.233985	+	0.972240i	$0.424823\pi$
$912$	0	0
$913$	− 32.5623i	− 1.07766i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 60.5410i	− 1.99924i
$918$	0	0
$919$	4.43769	0.146386	0.0731930	−	0.997318i	$-0.476681\pi$
$919$	4.43769	0.146386	0.0731930	+	0.997318i	$0.476681\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	52.6869i	1.73421i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	54.7082	1.79492	0.897459	−	0.441098i	$-0.145411\pi$
$929$	54.7082	1.79492	0.897459	+	0.441098i	$0.145411\pi$
$930$	0	0
$931$	15.7082	0.514816
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 9.83282i	− 0.321224i	−0.987018	−	0.160612i	$-0.948653\pi$
$937$	− 9.83282i	− 0.321224i	0.987018	−	0.160612i	$-0.0513468\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−44.3951	−1.44724	−0.723620	−	0.690199i	$-0.757523\pi$
$941$	−44.3951	−1.44724	−0.723620	+	0.690199i	$0.757523\pi$
$942$	0	0
$943$	− 15.8754i	− 0.516974i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.5836i	0.636381i	0.948027	+	0.318191i	$0.103075\pi$
$947$	19.5836i	0.636381i	−0.948027	+	0.318191i	$0.896925\pi$
$948$	0	0
$949$	−62.6869	−2.03490
$950$	0	0
$951$	0	0
$952$	0	0
$953$	56.8328i	1.84100i	0.390748	+	0.920498i	$0.372216\pi$
$953$	56.8328i	1.84100i	−0.390748	+	0.920498i	$0.627784\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	67.6869	2.18572
$960$	0	0
$961$	83.6656	2.69889
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 9.83282i	− 0.316202i	−0.987423	−	0.158101i	$-0.949463\pi$
$967$	− 9.83282i	− 0.316202i	0.987423	−	0.158101i	$-0.0505372\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24.9787	0.801605	0.400803	−	0.916164i	$-0.368731\pi$
$971$	24.9787	0.801605	0.400803	+	0.916164i	$0.368731\pi$
$972$	0	0
$973$	62.7082i	2.01033i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 22.6869i	− 0.725819i	−0.931824	−	0.362909i	$-0.881783\pi$
$977$	− 22.6869i	− 0.725819i	0.931824	−	0.362909i	$-0.118217\pi$
$978$	0	0
$979$	−58.2492	−1.86165
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 57.5410i	− 1.83527i	−0.397420	−	0.917637i	$-0.630094\pi$
$983$	− 57.5410i	− 1.83527i	0.397420	−	0.917637i	$-0.369906\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	10.8541	0.345140
$990$	0	0
$991$	−29.8541	−0.948347	−0.474173	−	0.880431i	$-0.657253\pi$
$991$	−29.8541	−0.948347	−0.474173	+	0.880431i	$0.657253\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	37.1246i	1.17575i	0.808952	+	0.587874i	$0.200035\pi$
$997$	37.1246i	1.17575i	−0.808952	+	0.587874i	$0.799965\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.d.k.649.4		4
3.2	odd	2		8100.2.d.n.649.4		4
5.2	odd	4		8100.2.a.o.1.1	✓	2
5.3	odd	4		8100.2.a.q.1.2	yes	2
5.4	even	2	inner	8100.2.d.k.649.1		4
15.2	even	4		8100.2.a.p.1.1	yes	2
15.8	even	4		8100.2.a.r.1.2	yes	2
15.14	odd	2		8100.2.d.n.649.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8100.2.a.o.1.1	✓	2	5.2	odd	4
8100.2.a.p.1.1	yes	2	15.2	even	4
8100.2.a.q.1.2	yes	2	5.3	odd	4
8100.2.a.r.1.2	yes	2	15.8	even	4
8100.2.d.k.649.1		4	5.4	even	2	inner
8100.2.d.k.649.4		4	1.1	even	1	trivial
8100.2.d.n.649.1		4	15.14	odd	2
8100.2.d.n.649.4		4	3.2	odd	2

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:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.85410i q^{7} +O(q^{10})\)	\(q+3.85410i q^{7} -4.85410 q^{11} +5.85410i q^{13} +7.85410i q^{17} -2.00000 q^{19} -1.85410i q^{23} +9.70820 q^{29} -10.7082 q^{31} +0.854102i q^{37} +8.56231 q^{41} +5.85410i q^{43} +6.70820i q^{47} -7.85410 q^{49} +1.85410i q^{53} -7.85410 q^{59} -5.85410 q^{61} -7.00000i q^{67} +9.00000 q^{71} +10.7082i q^{73} -18.7082i q^{77} +1.70820 q^{79} +6.70820i q^{83} +12.0000 q^{89} -22.5623 q^{91} -10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 6 q^{11} - 8 q^{19} + 12 q^{29} - 16 q^{31} - 6 q^{41} - 18 q^{49} - 18 q^{59} - 10 q^{61} + 36 q^{71} - 20 q^{79} + 48 q^{89} - 50 q^{91}+O(q^{100}) \) 4 * q - 6 * q^11 - 8 * q^19 + 12 * q^29 - 16 * q^31 - 6 * q^41 - 18 * q^49 - 18 * q^59 - 10 * q^61 + 36 * q^71 - 20 * q^79 + 48 * q^89 - 50 * q^91

Introduction

L-functions

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