LMFDB - Embedded newform 4100.2.d.g.1149.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4100,2,Mod(1149,4100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4100.1149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4100 = 2^{2} \cdot 5^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4100.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:	$32.7386648287$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 27x^{12} + 261x^{10} + 1141x^{8} + 2289x^{6} + 1896x^{4} + 549x^{2} + 9 $ x^14 + 27x^12 + 261x^10 + 1141x^8 + 2289x^6 + 1896x^4 + 549x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.11
Root			$-1.68828i$ of defining polynomial
Character	$\chi$	$=$	4100.1149
Dual form			4100.2.d.g.1149.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.68828i q^{3} -4.03576i q^{7} +0.149714 q^{9} +O(q^{10})$	$q+1.68828i q^{3} -4.03576i q^{7} +0.149714 q^{9} +0.693145 q^{11} +2.78311i q^{13} -0.285400i q^{17} -2.88605 q^{19} +6.81349 q^{21} -6.88986i q^{23} +5.31760i q^{27} -3.46601 q^{29} +8.71226 q^{31} +1.17022i q^{33} +1.40908i q^{37} -4.69866 q^{39} +1.00000 q^{41} +0.924607i q^{43} -8.52760i q^{47} -9.28736 q^{49} +0.481836 q^{51} +3.50505i q^{53} -4.87245i q^{57} +4.69625 q^{59} -0.199732 q^{61} -0.604209i q^{63} -10.2442i q^{67} +11.6320 q^{69} +1.95448 q^{71} -5.03982i q^{73} -2.79737i q^{77} +5.73152 q^{79} -8.52844 q^{81} -5.03089i q^{83} -5.85159i q^{87} +10.7537 q^{89} +11.2320 q^{91} +14.7087i q^{93} -13.6022i q^{97} +0.103773 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 12 q^{9}+O(q^{10}) $ 14 * q - 12 * q^9	$ 14 q - 12 q^{9} - 2 q^{11} + 2 q^{19} - 2 q^{21} + 10 q^{29} + 8 q^{31} + 22 q^{39} + 14 q^{41} + 6 q^{49} + 54 q^{51} + 32 q^{59} + 20 q^{61} + 38 q^{69} + 54 q^{71} + 2 q^{79} + 54 q^{81} + 32 q^{89} + 18 q^{91} + 80 q^{99}+O(q^{100}) $ 14 * q - 12 * q^9 - 2 * q^11 + 2 * q^19 - 2 * q^21 + 10 * q^29 + 8 * q^31 + 22 * q^39 + 14 * q^41 + 6 * q^49 + 54 * q^51 + 32 * q^59 + 20 * q^61 + 38 * q^69 + 54 * q^71 + 2 * q^79 + 54 * q^81 + 32 * q^89 + 18 * q^91 + 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times$.

$n$	$1477$	$2051$	$3901$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.68828i	0.974728i	0.873199	+	0.487364i	$0.162042\pi$
$3$	1.68828i	0.974728i	−0.873199	+	0.487364i	$0.837958\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.03576i	− 1.52537i	−0.646768	−	0.762687i	$-0.723880\pi$
$7$	− 4.03576i	− 1.52537i	0.646768	−	0.762687i	$-0.276120\pi$
$8$	0	0
$9$	0.149714	0.0499046
$10$	0	0
$11$	0.693145	0.208991	0.104496	−	0.994525i	$-0.466677\pi$
$11$	0.693145	0.208991	0.104496	+	0.994525i	$0.466677\pi$
$12$	0	0
$13$	2.78311i	0.771895i	0.922521	+	0.385948i	$0.126125\pi$
$13$	2.78311i	0.771895i	−0.922521	+	0.385948i	$0.873875\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.285400i	− 0.0692198i	−0.999401	−	0.0346099i	$-0.988981\pi$
$17$	− 0.285400i	− 0.0692198i	0.999401	−	0.0346099i	$-0.0110189\pi$
$18$	0	0
$19$	−2.88605	−0.662104	−0.331052	−	0.943612i	$-0.607404\pi$
$19$	−2.88605	−0.662104	−0.331052	+	0.943612i	$0.607404\pi$
$20$	0	0
$21$	6.81349	1.48683
$22$	0	0
$23$	− 6.88986i	− 1.43663i	−0.695716	−	0.718317i	$-0.744913\pi$
$23$	− 6.88986i	− 1.43663i	0.695716	−	0.718317i	$-0.255087\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.31760i	1.02337i
$28$	0	0
$29$	−3.46601	−0.643621	−0.321811	−	0.946804i	$-0.604291\pi$
$29$	−3.46601	−0.643621	−0.321811	+	0.946804i	$0.604291\pi$
$30$	0	0
$31$	8.71226	1.56477	0.782384	−	0.622796i	$-0.214003\pi$
$31$	8.71226	1.56477	0.782384	+	0.622796i	$0.214003\pi$
$32$	0	0
$33$	1.17022i	0.203710i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.40908i	0.231651i	0.993270	+	0.115825i	$0.0369513\pi$
$37$	1.40908i	0.231651i	−0.993270	+	0.115825i	$0.963049\pi$
$38$	0	0
$39$	−4.69866	−0.752388
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	0.924607i	0.141001i	0.997512	+	0.0705006i	$0.0224597\pi$
$43$	0.924607i	0.141001i	−0.997512	+	0.0705006i	$0.977540\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.52760i	− 1.24388i	−0.783065	−	0.621939i	$-0.786345\pi$
$47$	− 8.52760i	− 1.24388i	0.783065	−	0.621939i	$-0.213655\pi$
$48$	0	0
$49$	−9.28736	−1.32677
$50$	0	0
$51$	0.481836	0.0674705
$52$	0	0
$53$	3.50505i	0.481456i	0.970593	+	0.240728i	$0.0773862\pi$
$53$	3.50505i	0.481456i	−0.970593	+	0.240728i	$0.922614\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 4.87245i	− 0.645372i
$58$	0	0
$59$	4.69625	0.611399	0.305700	−	0.952128i	$-0.401110\pi$
$59$	4.69625	0.611399	0.305700	+	0.952128i	$0.401110\pi$
$60$	0	0
$61$	−0.199732	−0.0255731	−0.0127865	−	0.999918i	$-0.504070\pi$
$61$	−0.199732	−0.0255731	−0.0127865	+	0.999918i	$0.504070\pi$
$62$	0	0
$63$	− 0.604209i	− 0.0761232i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.2442i	− 1.25153i	−0.780011	−	0.625765i	$-0.784787\pi$
$67$	− 10.2442i	− 1.25153i	0.780011	−	0.625765i	$-0.215213\pi$
$68$	0	0
$69$	11.6320	1.40033
$70$	0	0
$71$	1.95448	0.231954	0.115977	−	0.993252i	$-0.463000\pi$
$71$	1.95448	0.231954	0.115977	+	0.993252i	$0.463000\pi$
$72$	0	0
$73$	− 5.03982i	− 0.589866i	−0.955518	−	0.294933i	$-0.904703\pi$
$73$	− 5.03982i	− 0.589866i	0.955518	−	0.294933i	$-0.0952974\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.79737i	− 0.318790i
$78$	0	0
$79$	5.73152	0.644847	0.322423	−	0.946596i	$-0.395503\pi$
$79$	5.73152	0.644847	0.322423	+	0.946596i	$0.395503\pi$
$80$	0	0
$81$	−8.52844	−0.947605
$82$	0	0
$83$	− 5.03089i	− 0.552212i	−0.961127	−	0.276106i	$-0.910956\pi$
$83$	− 5.03089i	− 0.552212i	0.961127	−	0.276106i	$-0.0890441\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 5.85159i	− 0.627356i
$88$	0	0
$89$	10.7537	1.13989	0.569945	−	0.821683i	$-0.306964\pi$
$89$	10.7537	1.13989	0.569945	+	0.821683i	$0.306964\pi$
$90$	0	0
$91$	11.2320	1.17743
$92$	0	0
$93$	14.7087i	1.52522i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 13.6022i	− 1.38110i	−0.723286	−	0.690548i	$-0.757369\pi$
$97$	− 13.6022i	− 1.38110i	0.723286	−	0.690548i	$-0.242631\pi$
$98$	0	0
$99$	0.103773	0.0104296
$100$	0	0
$101$	−3.27533	−0.325907	−0.162954	−	0.986634i	$-0.552102\pi$
$101$	−3.27533	−0.325907	−0.162954	+	0.986634i	$0.552102\pi$
$102$	0	0
$103$	1.60047i	0.157699i	0.996887	+	0.0788495i	$0.0251247\pi$
$103$	1.60047i	0.157699i	−0.996887	+	0.0788495i	$0.974875\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.01175i	− 0.291157i	−0.989347	−	0.145578i	$-0.953496\pi$
$107$	− 3.01175i	− 0.291157i	0.989347	−	0.145578i	$-0.0465043\pi$
$108$	0	0
$109$	3.13904	0.300665	0.150333	−	0.988635i	$-0.451966\pi$
$109$	3.13904	0.300665	0.150333	+	0.988635i	$0.451966\pi$
$110$	0	0
$111$	−2.37891	−0.225797
$112$	0	0
$113$	− 13.9590i	− 1.31315i	−0.754260	−	0.656576i	$-0.772004\pi$
$113$	− 13.9590i	− 1.31315i	0.754260	−	0.656576i	$-0.227996\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.416670i	0.0385212i
$118$	0	0
$119$	−1.15181	−0.105586
$120$	0	0
$121$	−10.5195	−0.956323
$122$	0	0
$123$	1.68828i	0.152227i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.04514i	− 0.181476i	−0.995875	−	0.0907382i	$-0.971077\pi$
$127$	− 2.04514i	− 0.181476i	0.995875	−	0.0907382i	$-0.0289227\pi$
$128$	0	0
$129$	−1.56099	−0.137438
$130$	0	0
$131$	5.43798	0.475118	0.237559	−	0.971373i	$-0.423653\pi$
$131$	5.43798	0.475118	0.237559	+	0.971373i	$0.423653\pi$
$132$	0	0
$133$	11.6474i	1.00996i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 18.8530i	− 1.61072i	−0.592789	−	0.805358i	$-0.701973\pi$
$137$	− 18.8530i	− 1.61072i	0.592789	−	0.805358i	$-0.298027\pi$
$138$	0	0
$139$	−17.7958	−1.50942	−0.754711	−	0.656057i	$-0.772223\pi$
$139$	−17.7958	−1.50942	−0.754711	+	0.656057i	$0.772223\pi$
$140$	0	0
$141$	14.3970	1.21244
$142$	0	0
$143$	1.92910i	0.161319i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 15.6796i	− 1.29324i
$148$	0	0
$149$	14.6750	1.20223	0.601113	−	0.799164i	$-0.294724\pi$
$149$	14.6750	1.20223	0.601113	+	0.799164i	$0.294724\pi$
$150$	0	0
$151$	17.9622	1.46174	0.730871	−	0.682515i	$-0.239114\pi$
$151$	17.9622	1.46174	0.730871	+	0.682515i	$0.239114\pi$
$152$	0	0
$153$	− 0.0427284i	− 0.00345439i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.373333i	0.0297952i	0.999889	+	0.0148976i	$0.00474223\pi$
$157$	0.373333i	0.0297952i	−0.999889	+	0.0148976i	$0.995258\pi$
$158$	0	0
$159$	−5.91751	−0.469289
$160$	0	0
$161$	−27.8058	−2.19140
$162$	0	0
$163$	− 20.9452i	− 1.64056i	−0.571966	−	0.820278i	$-0.693819\pi$
$163$	− 20.9452i	− 1.64056i	0.571966	−	0.820278i	$-0.306181\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.33894i	0.180993i	0.995897	+	0.0904964i	$0.0288454\pi$
$167$	2.33894i	0.180993i	−0.995897	+	0.0904964i	$0.971155\pi$
$168$	0	0
$169$	5.25431	0.404177
$170$	0	0
$171$	−0.432081	−0.0330421
$172$	0	0
$173$	0.309212i	0.0235090i	0.999931	+	0.0117545i	$0.00374166\pi$
$173$	0.309212i	0.0235090i	−0.999931	+	0.0117545i	$0.996258\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.92857i	0.595948i
$178$	0	0
$179$	12.5487	0.937931	0.468966	−	0.883216i	$-0.344627\pi$
$179$	12.5487	0.937931	0.468966	+	0.883216i	$0.344627\pi$
$180$	0	0
$181$	−7.83563	−0.582418	−0.291209	−	0.956659i	$-0.594057\pi$
$181$	−7.83563	−0.582418	−0.291209	+	0.956659i	$0.594057\pi$
$182$	0	0
$183$	− 0.337204i	− 0.0249268i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.197824i	− 0.0144663i
$188$	0	0
$189$	21.4605	1.56102
$190$	0	0
$191$	8.44248	0.610877	0.305438	−	0.952212i	$-0.401197\pi$
$191$	8.44248	0.610877	0.305438	+	0.952212i	$0.401197\pi$
$192$	0	0
$193$	− 7.02942i	− 0.505989i	−0.967468	−	0.252994i	$-0.918585\pi$
$193$	− 7.02942i	− 0.505989i	0.967468	−	0.252994i	$-0.0814154\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 21.6298i	− 1.54106i	−0.637405	−	0.770529i	$-0.719992\pi$
$197$	− 21.6298i	− 1.54106i	0.637405	−	0.770529i	$-0.280008\pi$
$198$	0	0
$199$	−11.0545	−0.783633	−0.391816	−	0.920043i	$-0.628153\pi$
$199$	−11.0545	−0.783633	−0.391816	+	0.920043i	$0.628153\pi$
$200$	0	0
$201$	17.2951	1.21990
$202$	0	0
$203$	13.9880i	0.981763i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.03151i	− 0.0716947i
$208$	0	0
$209$	−2.00045	−0.138374
$210$	0	0
$211$	18.8420	1.29714	0.648568	−	0.761157i	$-0.275368\pi$
$211$	18.8420	1.29714	0.648568	+	0.761157i	$0.275368\pi$
$212$	0	0
$213$	3.29971i	0.226092i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 35.1606i	− 2.38686i
$218$	0	0
$219$	8.50862	0.574959
$220$	0	0
$221$	0.794300	0.0534304
$222$	0	0
$223$	9.39906i	0.629407i	0.949190	+	0.314704i	$0.101905\pi$
$223$	9.39906i	0.629407i	−0.949190	+	0.314704i	$0.898095\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	27.7210i	1.83991i	0.392028	+	0.919953i	$0.371774\pi$
$227$	27.7210i	1.83991i	−0.392028	+	0.919953i	$0.628226\pi$
$228$	0	0
$229$	−4.57037	−0.302019	−0.151009	−	0.988532i	$-0.548252\pi$
$229$	−4.57037	−0.302019	−0.151009	+	0.988532i	$0.548252\pi$
$230$	0	0
$231$	4.72274	0.310733
$232$	0	0
$233$	11.8057i	0.773414i	0.922203	+	0.386707i	$0.126388\pi$
$233$	11.8057i	0.773414i	−0.922203	+	0.386707i	$0.873612\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.67641i	0.628550i
$238$	0	0
$239$	27.4335	1.77452	0.887262	−	0.461265i	$-0.152604\pi$
$239$	27.4335	1.77452	0.887262	+	0.461265i	$0.152604\pi$
$240$	0	0
$241$	−13.0370	−0.839787	−0.419893	−	0.907573i	$-0.637933\pi$
$241$	−13.0370	−0.839787	−0.419893	+	0.907573i	$0.637933\pi$
$242$	0	0
$243$	1.55439i	0.0997144i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 8.03218i	− 0.511075i
$248$	0	0
$249$	8.49355	0.538257
$250$	0	0
$251$	−11.9854	−0.756512	−0.378256	−	0.925701i	$-0.623476\pi$
$251$	−11.9854	−0.756512	−0.378256	+	0.925701i	$0.623476\pi$
$252$	0	0
$253$	− 4.77567i	− 0.300244i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.89821i	0.555055i	0.960718	+	0.277528i	$0.0895150\pi$
$257$	8.89821i	0.555055i	−0.960718	+	0.277528i	$0.910485\pi$
$258$	0	0
$259$	5.68669	0.353354
$260$	0	0
$261$	−0.518910	−0.0321197
$262$	0	0
$263$	20.9916i	1.29440i	0.762321	+	0.647199i	$0.224060\pi$
$263$	20.9916i	1.29440i	−0.762321	+	0.647199i	$0.775940\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	18.1553i	1.11108i
$268$	0	0
$269$	12.1862	0.743006	0.371503	−	0.928432i	$-0.378843\pi$
$269$	12.1862	0.743006	0.371503	+	0.928432i	$0.378843\pi$
$270$	0	0
$271$	−3.71331	−0.225567	−0.112784	−	0.993620i	$-0.535977\pi$
$271$	−3.71331	−0.225567	−0.112784	+	0.993620i	$0.535977\pi$
$272$	0	0
$273$	18.9627i	1.14767i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.7985i	1.12949i	0.825264	+	0.564747i	$0.191026\pi$
$277$	18.7985i	1.12949i	−0.825264	+	0.564747i	$0.808974\pi$
$278$	0	0
$279$	1.30435	0.0780892
$280$	0	0
$281$	12.1757	0.726339	0.363170	−	0.931723i	$-0.381695\pi$
$281$	12.1757	0.726339	0.363170	+	0.931723i	$0.381695\pi$
$282$	0	0
$283$	27.0538i	1.60818i	0.594507	+	0.804090i	$0.297347\pi$
$283$	27.0538i	1.60818i	−0.594507	+	0.804090i	$0.702653\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.03576i	− 0.238223i
$288$	0	0
$289$	16.9185	0.995209
$290$	0	0
$291$	22.9644	1.34619
$292$	0	0
$293$	20.2066i	1.18048i	0.807227	+	0.590241i	$0.200967\pi$
$293$	20.2066i	1.18048i	−0.807227	+	0.590241i	$0.799033\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.68587i	0.213876i
$298$	0	0
$299$	19.1752	1.10893
$300$	0	0
$301$	3.73149	0.215080
$302$	0	0
$303$	− 5.52967i	− 0.317671i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 13.7021i	− 0.782018i	−0.920387	−	0.391009i	$-0.872126\pi$
$307$	− 13.7021i	− 0.782018i	0.920387	−	0.391009i	$-0.127874\pi$
$308$	0	0
$309$	−2.70204	−0.153714
$310$	0	0
$311$	26.7478	1.51673	0.758365	−	0.651830i	$-0.225998\pi$
$311$	26.7478	1.51673	0.758365	+	0.651830i	$0.225998\pi$
$312$	0	0
$313$	− 11.7722i	− 0.665407i	−0.943032	−	0.332703i	$-0.892039\pi$
$313$	− 11.7722i	− 0.665407i	0.943032	−	0.332703i	$-0.107961\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.6009i	0.707735i	0.935296	+	0.353867i	$0.115134\pi$
$317$	12.6009i	0.707735i	−0.935296	+	0.353867i	$0.884866\pi$
$318$	0	0
$319$	−2.40245	−0.134511
$320$	0	0
$321$	5.08467	0.283799
$322$	0	0
$323$	0.823679i	0.0458307i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.29957i	0.293067i
$328$	0	0
$329$	−34.4154	−1.89738
$330$	0	0
$331$	−6.14860	−0.337957	−0.168979	−	0.985620i	$-0.554047\pi$
$331$	−6.14860	−0.337957	−0.168979	+	0.985620i	$0.554047\pi$
$332$	0	0
$333$	0.210958i	0.0115604i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 30.8745i	− 1.68184i	−0.541160	−	0.840920i	$-0.682015\pi$
$337$	− 30.8745i	− 1.68184i	0.541160	−	0.840920i	$-0.317985\pi$
$338$	0	0
$339$	23.5667	1.27997
$340$	0	0
$341$	6.03886	0.327023
$342$	0	0
$343$	9.23122i	0.498439i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.0751i	1.13137i	0.824620	+	0.565687i	$0.191389\pi$
$347$	21.0751i	1.13137i	−0.824620	+	0.565687i	$0.808611\pi$
$348$	0	0
$349$	−7.71832	−0.413152	−0.206576	−	0.978431i	$-0.566232\pi$
$349$	−7.71832	−0.413152	−0.206576	+	0.978431i	$0.566232\pi$
$350$	0	0
$351$	−14.7994	−0.789936
$352$	0	0
$353$	24.3915i	1.29823i	0.760691	+	0.649114i	$0.224860\pi$
$353$	24.3915i	1.29823i	−0.760691	+	0.649114i	$0.775140\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.94457i	− 0.102918i
$358$	0	0
$359$	4.76516	0.251495	0.125748	−	0.992062i	$-0.459867\pi$
$359$	4.76516	0.251495	0.125748	+	0.992062i	$0.459867\pi$
$360$	0	0
$361$	−10.6707	−0.561618
$362$	0	0
$363$	− 17.7599i	− 0.932155i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	23.7591i	1.24022i	0.784516	+	0.620109i	$0.212911\pi$
$367$	23.7591i	1.24022i	−0.784516	+	0.620109i	$0.787089\pi$
$368$	0	0
$369$	0.149714	0.00779379
$370$	0	0
$371$	14.1456	0.734401
$372$	0	0
$373$	3.81407i	0.197485i	0.995113	+	0.0987427i	$0.0314821\pi$
$373$	3.81407i	0.197485i	−0.995113	+	0.0987427i	$0.968518\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 9.64628i	− 0.496808i
$378$	0	0
$379$	31.6225	1.62434	0.812170	−	0.583421i	$-0.198286\pi$
$379$	31.6225	1.62434	0.812170	+	0.583421i	$0.198286\pi$
$380$	0	0
$381$	3.45276	0.176890
$382$	0	0
$383$	− 23.5627i	− 1.20400i	−0.798497	−	0.601999i	$-0.794371\pi$
$383$	− 23.5627i	− 1.20400i	0.798497	−	0.601999i	$-0.205629\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.138427i	0.00703662i
$388$	0	0
$389$	−22.7152	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$389$	−22.7152	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$390$	0	0
$391$	−1.96637	−0.0994435
$392$	0	0
$393$	9.18083i	0.463111i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 8.07622i	− 0.405334i	−0.979248	−	0.202667i	$-0.935039\pi$
$397$	− 8.07622i	− 0.405334i	0.979248	−	0.202667i	$-0.0649609\pi$
$398$	0	0
$399$	−19.6640	−0.984433
$400$	0	0
$401$	−23.1682	−1.15696	−0.578482	−	0.815695i	$-0.696355\pi$
$401$	−23.1682	−1.15696	−0.578482	+	0.815695i	$0.696355\pi$
$402$	0	0
$403$	24.2472i	1.20784i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.976694i	0.0484129i
$408$	0	0
$409$	−15.7537	−0.778971	−0.389486	−	0.921033i	$-0.627347\pi$
$409$	−15.7537	−0.778971	−0.389486	+	0.921033i	$0.627347\pi$
$410$	0	0
$411$	31.8290	1.57001
$412$	0	0
$413$	− 18.9529i	− 0.932612i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 30.0443i	− 1.47128i
$418$	0	0
$419$	−1.42647	−0.0696875	−0.0348437	−	0.999393i	$-0.511093\pi$
$419$	−1.42647	−0.0696875	−0.0348437	+	0.999393i	$0.511093\pi$
$420$	0	0
$421$	−23.9743	−1.16844	−0.584218	−	0.811597i	$-0.698599\pi$
$421$	−23.9743	−1.16844	−0.584218	+	0.811597i	$0.698599\pi$
$422$	0	0
$423$	− 1.27670i	− 0.0620753i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.806071i	0.0390085i
$428$	0	0
$429$	−3.25686	−0.157242
$430$	0	0
$431$	−3.04590	−0.146716	−0.0733579	−	0.997306i	$-0.523372\pi$
$431$	−3.04590	−0.146716	−0.0733579	+	0.997306i	$0.523372\pi$
$432$	0	0
$433$	− 3.00742i	− 0.144527i	−0.997386	−	0.0722636i	$-0.976978\pi$
$433$	− 3.00742i	− 0.144527i	0.997386	−	0.0722636i	$-0.0230223\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.8844i	0.951202i
$438$	0	0
$439$	2.38614	0.113884	0.0569421	−	0.998377i	$-0.481865\pi$
$439$	2.38614	0.113884	0.0569421	+	0.998377i	$0.481865\pi$
$440$	0	0
$441$	−1.39045	−0.0662117
$442$	0	0
$443$	− 17.3305i	− 0.823397i	−0.911320	−	0.411698i	$-0.864936\pi$
$443$	− 17.3305i	− 0.823397i	0.911320	−	0.411698i	$-0.135064\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	24.7755i	1.17184i
$448$	0	0
$449$	−8.26959	−0.390266	−0.195133	−	0.980777i	$-0.562514\pi$
$449$	−8.26959	−0.390266	−0.195133	+	0.980777i	$0.562514\pi$
$450$	0	0
$451$	0.693145	0.0326389
$452$	0	0
$453$	30.3252i	1.42480i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 11.2319i	− 0.525408i	−0.964877	−	0.262704i	$-0.915386\pi$
$457$	− 11.2319i	− 0.525408i	0.964877	−	0.262704i	$-0.0846142\pi$
$458$	0	0
$459$	1.51764	0.0708376
$460$	0	0
$461$	0.898708	0.0418570	0.0209285	−	0.999781i	$-0.493338\pi$
$461$	0.898708	0.0418570	0.0209285	+	0.999781i	$0.493338\pi$
$462$	0	0
$463$	− 37.8663i	− 1.75980i	−0.475163	−	0.879898i	$-0.657611\pi$
$463$	− 37.8663i	− 1.75980i	0.475163	−	0.879898i	$-0.342389\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 22.4397i	− 1.03838i	−0.854657	−	0.519192i	$-0.826233\pi$
$467$	− 22.4397i	− 1.03838i	0.854657	−	0.519192i	$-0.173767\pi$
$468$	0	0
$469$	−41.3432	−1.90905
$470$	0	0
$471$	−0.630291	−0.0290423
$472$	0	0
$473$	0.640887i	0.0294680i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.524755i	0.0240269i
$478$	0	0
$479$	29.9212	1.36713	0.683567	−	0.729888i	$-0.260428\pi$
$479$	29.9212	1.36713	0.683567	+	0.729888i	$0.260428\pi$
$480$	0	0
$481$	−3.92161	−0.178810
$482$	0	0
$483$	− 46.9440i	− 2.13602i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.3207i	1.01145i	0.862696	+	0.505724i	$0.168774\pi$
$487$	22.3207i	1.01145i	−0.862696	+	0.505724i	$0.831226\pi$
$488$	0	0
$489$	35.3614	1.59910
$490$	0	0
$491$	−39.6418	−1.78901	−0.894505	−	0.447058i	$-0.852472\pi$
$491$	−39.6418	−1.78901	−0.894505	+	0.447058i	$0.852472\pi$
$492$	0	0
$493$	0.989200i	0.0445513i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.88781i	− 0.353816i
$498$	0	0
$499$	11.5616	0.517570	0.258785	−	0.965935i	$-0.416678\pi$
$499$	11.5616	0.517570	0.258785	+	0.965935i	$0.416678\pi$
$500$	0	0
$501$	−3.94879	−0.176419
$502$	0	0
$503$	− 15.1456i	− 0.675308i	−0.941270	−	0.337654i	$-0.890367\pi$
$503$	− 15.1456i	− 0.675308i	0.941270	−	0.337654i	$-0.109633\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.87074i	0.393963i
$508$	0	0
$509$	6.71503	0.297639	0.148819	−	0.988864i	$-0.452453\pi$
$509$	6.71503	0.297639	0.148819	+	0.988864i	$0.452453\pi$
$510$	0	0
$511$	−20.3395	−0.899766
$512$	0	0
$513$	− 15.3468i	− 0.677579i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.91087i	− 0.259960i
$518$	0	0
$519$	−0.522037	−0.0229149
$520$	0	0
$521$	13.6035	0.595979	0.297990	−	0.954569i	$-0.403684\pi$
$521$	13.6035	0.595979	0.297990	+	0.954569i	$0.403684\pi$
$522$	0	0
$523$	34.4680i	1.50718i	0.657343	+	0.753591i	$0.271680\pi$
$523$	34.4680i	1.50718i	−0.657343	+	0.753591i	$0.728320\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.48648i	− 0.108313i
$528$	0	0
$529$	−24.4701	−1.06392
$530$	0	0
$531$	0.703093	0.0305116
$532$	0	0
$533$	2.78311i	0.120550i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	21.1857i	0.914228i
$538$	0	0
$539$	−6.43749	−0.277282
$540$	0	0
$541$	−38.7637	−1.66658	−0.833292	−	0.552833i	$-0.813547\pi$
$541$	−38.7637	−1.66658	−0.833292	+	0.552833i	$0.813547\pi$
$542$	0	0
$543$	− 13.2287i	− 0.567699i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 29.5652i	− 1.26412i	−0.774921	−	0.632058i	$-0.782210\pi$
$547$	− 29.5652i	− 1.26412i	0.774921	−	0.632058i	$-0.217790\pi$
$548$	0	0
$549$	−0.0299027	−0.00127621
$550$	0	0
$551$	10.0031	0.426145
$552$	0	0
$553$	− 23.1310i	− 0.983632i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3.36776i	− 0.142696i	−0.997451	−	0.0713482i	$-0.977270\pi$
$557$	− 3.36776i	− 0.142696i	0.997451	−	0.0713482i	$-0.0227302\pi$
$558$	0	0
$559$	−2.57328	−0.108838
$560$	0	0
$561$	0.333982	0.0141007
$562$	0	0
$563$	− 6.45413i	− 0.272009i	−0.990708	−	0.136005i	$-0.956574\pi$
$563$	− 6.45413i	− 0.272009i	0.990708	−	0.136005i	$-0.0434262\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	34.4187i	1.44545i
$568$	0	0
$569$	−5.61628	−0.235447	−0.117723	−	0.993046i	$-0.537560\pi$
$569$	−5.61628	−0.235447	−0.117723	+	0.993046i	$0.537560\pi$
$570$	0	0
$571$	−33.8755	−1.41765	−0.708824	−	0.705386i	$-0.750774\pi$
$571$	−33.8755	−1.41765	−0.708824	+	0.705386i	$0.750774\pi$
$572$	0	0
$573$	14.2533i	0.595439i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 6.71150i	− 0.279404i	−0.990194	−	0.139702i	$-0.955386\pi$
$577$	− 6.71150i	− 0.279404i	0.990194	−	0.139702i	$-0.0446144\pi$
$578$	0	0
$579$	11.8676	0.493202
$580$	0	0
$581$	−20.3035	−0.842330
$582$	0	0
$583$	2.42951i	0.100620i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 48.1988i	− 1.98938i	−0.102928	−	0.994689i	$-0.532821\pi$
$587$	− 48.1988i	− 1.98938i	0.102928	−	0.994689i	$-0.467179\pi$
$588$	0	0
$589$	−25.1440	−1.03604
$590$	0	0
$591$	36.5171	1.50211
$592$	0	0
$593$	− 10.8452i	− 0.445360i	−0.974892	−	0.222680i	$-0.928519\pi$
$593$	− 10.8452i	− 0.445360i	0.974892	−	0.222680i	$-0.0714805\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 18.6631i	− 0.763829i
$598$	0	0
$599$	18.1359	0.741013	0.370506	−	0.928830i	$-0.379184\pi$
$599$	18.1359	0.741013	0.370506	+	0.928830i	$0.379184\pi$
$600$	0	0
$601$	−2.41959	−0.0986972	−0.0493486	−	0.998782i	$-0.515715\pi$
$601$	−2.41959	−0.0986972	−0.0493486	+	0.998782i	$0.515715\pi$
$602$	0	0
$603$	− 1.53370i	− 0.0624572i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1.61933i	− 0.0657266i	−0.999460	−	0.0328633i	$-0.989537\pi$
$607$	− 1.61933i	− 0.0657266i	0.999460	−	0.0328633i	$-0.0104626\pi$
$608$	0	0
$609$	−23.6156	−0.956953
$610$	0	0
$611$	23.7332	0.960144
$612$	0	0
$613$	− 0.323576i	− 0.0130691i	−0.999979	−	0.00653456i	$-0.997920\pi$
$613$	− 0.323576i	− 0.0130691i	0.999979	−	0.00653456i	$-0.00208003\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.6820i	1.23521i	0.786488	+	0.617605i	$0.211897\pi$
$617$	30.6820i	1.23521i	−0.786488	+	0.617605i	$0.788103\pi$
$618$	0	0
$619$	−8.94344	−0.359467	−0.179734	−	0.983715i	$-0.557524\pi$
$619$	−8.94344	−0.359467	−0.179734	+	0.983715i	$0.557524\pi$
$620$	0	0
$621$	36.6375	1.47021
$622$	0	0
$623$	− 43.3994i	− 1.73876i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.37732i	− 0.134877i
$628$	0	0
$629$	0.402151	0.0160348
$630$	0	0
$631$	−27.7892	−1.10627	−0.553135	−	0.833092i	$-0.686569\pi$
$631$	−27.7892	−1.10627	−0.553135	+	0.833092i	$0.686569\pi$
$632$	0	0
$633$	31.8105i	1.26436i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 25.8477i	− 1.02412i
$638$	0	0
$639$	0.292613	0.0115756
$640$	0	0
$641$	−4.41368	−0.174330	−0.0871650	−	0.996194i	$-0.527781\pi$
$641$	−4.41368	−0.174330	−0.0871650	+	0.996194i	$0.527781\pi$
$642$	0	0
$643$	36.4867i	1.43889i	0.694547	+	0.719447i	$0.255605\pi$
$643$	36.4867i	1.43889i	−0.694547	+	0.719447i	$0.744395\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.43532i	− 0.135056i	−0.997717	−	0.0675282i	$-0.978489\pi$
$647$	− 3.43532i	− 0.135056i	0.997717	−	0.0675282i	$-0.0215113\pi$
$648$	0	0
$649$	3.25518	0.127777
$650$	0	0
$651$	59.3609	2.32654
$652$	0	0
$653$	− 15.9796i	− 0.625329i	−0.949864	−	0.312665i	$-0.898778\pi$
$653$	− 15.9796i	− 0.625329i	0.949864	−	0.312665i	$-0.101222\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 0.754531i	− 0.0294371i
$658$	0	0
$659$	1.84762	0.0719732	0.0359866	−	0.999352i	$-0.488543\pi$
$659$	1.84762	0.0719732	0.0359866	+	0.999352i	$0.488543\pi$
$660$	0	0
$661$	−22.4375	−0.872716	−0.436358	−	0.899773i	$-0.643732\pi$
$661$	−22.4375	−0.872716	−0.436358	+	0.899773i	$0.643732\pi$
$662$	0	0
$663$	1.34100i	0.0520802i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	23.8803i	0.924649i
$668$	0	0
$669$	−15.8682	−0.613501
$670$	0	0
$671$	−0.138443	−0.00534455
$672$	0	0
$673$	1.22416i	0.0471879i	0.999722	+	0.0235939i	$0.00751088\pi$
$673$	1.22416i	0.0471879i	−0.999722	+	0.0235939i	$0.992489\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.90951i	0.0733884i	0.999327	+	0.0366942i	$0.0116827\pi$
$677$	1.90951i	0.0733884i	−0.999327	+	0.0366942i	$0.988317\pi$
$678$	0	0
$679$	−54.8953	−2.10669
$680$	0	0
$681$	−46.8008	−1.79341
$682$	0	0
$683$	0.272216i	0.0104160i	0.999986	+	0.00520802i	$0.00165777\pi$
$683$	0.272216i	0.0104160i	−0.999986	+	0.00520802i	$0.998342\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 7.71607i	− 0.294386i
$688$	0	0
$689$	−9.75494	−0.371634
$690$	0	0
$691$	−44.3159	−1.68586	−0.842928	−	0.538026i	$-0.819170\pi$
$691$	−44.3159	−1.68586	−0.842928	+	0.538026i	$0.819170\pi$
$692$	0	0
$693$	− 0.418805i	− 0.0159091i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 0.285400i	− 0.0108103i
$698$	0	0
$699$	−19.9312	−0.753869
$700$	0	0
$701$	1.38888	0.0524573	0.0262287	−	0.999656i	$-0.491650\pi$
$701$	1.38888	0.0524573	0.0262287	+	0.999656i	$0.491650\pi$
$702$	0	0
$703$	− 4.06666i	− 0.153377i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.2184i	0.497131i
$708$	0	0
$709$	27.9220	1.04863	0.524317	−	0.851523i	$-0.324321\pi$
$709$	27.9220	1.04863	0.524317	+	0.851523i	$0.324321\pi$
$710$	0	0
$711$	0.858088	0.0321808
$712$	0	0
$713$	− 60.0262i	− 2.24800i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	46.3154i	1.72968i
$718$	0	0
$719$	−4.11256	−0.153373	−0.0766864	−	0.997055i	$-0.524434\pi$
$719$	−4.11256	−0.153373	−0.0766864	+	0.997055i	$0.524434\pi$
$720$	0	0
$721$	6.45911	0.240550
$722$	0	0
$723$	− 22.0101i	− 0.818564i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	43.8918i	1.62786i	0.580965	+	0.813929i	$0.302675\pi$
$727$	43.8918i	1.62786i	−0.580965	+	0.813929i	$0.697325\pi$
$728$	0	0
$729$	−28.2096	−1.04480
$730$	0	0
$731$	0.263883	0.00976007
$732$	0	0
$733$	16.6831i	0.616204i	0.951353	+	0.308102i	$0.0996938\pi$
$733$	16.6831i	0.616204i	−0.951353	+	0.308102i	$0.900306\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 7.10073i	− 0.261559i
$738$	0	0
$739$	−42.0031	−1.54511	−0.772554	−	0.634949i	$-0.781021\pi$
$739$	−42.0031	−1.54511	−0.772554	+	0.634949i	$0.781021\pi$
$740$	0	0
$741$	13.5606	0.498160
$742$	0	0
$743$	3.55105i	0.130275i	0.997876	+	0.0651377i	$0.0207487\pi$
$743$	3.55105i	0.130275i	−0.997876	+	0.0651377i	$0.979251\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 0.753195i	− 0.0275580i
$748$	0	0
$749$	−12.1547	−0.444123
$750$	0	0
$751$	−19.8818	−0.725498	−0.362749	−	0.931887i	$-0.618162\pi$
$751$	−19.8818	−0.725498	−0.362749	+	0.931887i	$0.618162\pi$
$752$	0	0
$753$	− 20.2347i	− 0.737394i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 10.5110i	− 0.382030i	−0.981587	−	0.191015i	$-0.938822\pi$
$757$	− 10.5110i	− 0.382030i	0.981587	−	0.191015i	$-0.0611779\pi$
$758$	0	0
$759$	8.06266	0.292656
$760$	0	0
$761$	16.0828	0.583002	0.291501	−	0.956571i	$-0.405845\pi$
$761$	16.0828	0.583002	0.291501	+	0.956571i	$0.405845\pi$
$762$	0	0
$763$	− 12.6684i	− 0.458627i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.0702i	0.471936i
$768$	0	0
$769$	−7.79476	−0.281086	−0.140543	−	0.990075i	$-0.544885\pi$
$769$	−7.79476	−0.281086	−0.140543	+	0.990075i	$0.544885\pi$
$770$	0	0
$771$	−15.0227	−0.541028
$772$	0	0
$773$	− 2.07091i	− 0.0744854i	−0.999306	−	0.0372427i	$-0.988143\pi$
$773$	− 2.07091i	− 0.0744854i	0.999306	−	0.0372427i	$-0.0118575\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.60072i	0.344424i
$778$	0	0
$779$	−2.88605	−0.103403
$780$	0	0
$781$	1.35474	0.0484763
$782$	0	0
$783$	− 18.4308i	− 0.658664i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 31.0086i	− 1.10534i	−0.833401	−	0.552668i	$-0.813610\pi$
$787$	− 31.0086i	− 1.10534i	0.833401	−	0.552668i	$-0.186390\pi$
$788$	0	0
$789$	−35.4397	−1.26169
$790$	0	0
$791$	−56.3351	−2.00305
$792$	0	0
$793$	− 0.555876i	− 0.0197397i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	47.1477i	1.67006i	0.550207	+	0.835028i	$0.314549\pi$
$797$	47.1477i	1.67006i	−0.550207	+	0.835028i	$0.685451\pi$
$798$	0	0
$799$	−2.43378	−0.0861010
$800$	0	0
$801$	1.60998	0.0568858
$802$	0	0
$803$	− 3.49333i	− 0.123277i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20.5737i	0.724229i
$808$	0	0
$809$	4.74525	0.166834	0.0834171	−	0.996515i	$-0.473417\pi$
$809$	4.74525	0.166834	0.0834171	+	0.996515i	$0.473417\pi$
$810$	0	0
$811$	47.7650	1.67725	0.838627	−	0.544706i	$-0.183359\pi$
$811$	47.7650	1.67725	0.838627	+	0.544706i	$0.183359\pi$
$812$	0	0
$813$	− 6.26910i	− 0.219867i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2.66846i	− 0.0933575i
$818$	0	0
$819$	1.68158	0.0587592
$820$	0	0
$821$	31.2699	1.09133	0.545663	−	0.838005i	$-0.316278\pi$
$821$	31.2699	1.09133	0.545663	+	0.838005i	$0.316278\pi$
$822$	0	0
$823$	17.4865i	0.609541i	0.952426	+	0.304770i	$0.0985797\pi$
$823$	17.4865i	0.609541i	−0.952426	+	0.304770i	$0.901420\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.2936i	0.705678i	0.935684	+	0.352839i	$0.114784\pi$
$827$	20.2936i	0.705678i	−0.935684	+	0.352839i	$0.885216\pi$
$828$	0	0
$829$	−27.4912	−0.954808	−0.477404	−	0.878684i	$-0.658422\pi$
$829$	−27.4912	−0.954808	−0.477404	+	0.878684i	$0.658422\pi$
$830$	0	0
$831$	−31.7372	−1.10095
$832$	0	0
$833$	2.65062i	0.0918384i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	46.3283i	1.60134i
$838$	0	0
$839$	15.4470	0.533289	0.266644	−	0.963795i	$-0.414085\pi$
$839$	15.4470	0.533289	0.266644	+	0.963795i	$0.414085\pi$
$840$	0	0
$841$	−16.9868	−0.585751
$842$	0	0
$843$	20.5559i	0.707983i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	42.4544i	1.45875i
$848$	0	0
$849$	−45.6743	−1.56754
$850$	0	0
$851$	9.70833	0.332797
$852$	0	0
$853$	16.8982i	0.578582i	0.957241	+	0.289291i	$0.0934195\pi$
$853$	16.8982i	0.578582i	−0.957241	+	0.289291i	$0.906580\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 23.1431i	− 0.790553i	−0.918562	−	0.395277i	$-0.870649\pi$
$857$	− 23.1431i	− 0.790553i	0.918562	−	0.395277i	$-0.129351\pi$
$858$	0	0
$859$	41.6584	1.42137	0.710683	−	0.703512i	$-0.248386\pi$
$859$	41.6584	1.42137	0.710683	+	0.703512i	$0.248386\pi$
$860$	0	0
$861$	6.81349	0.232203
$862$	0	0
$863$	− 33.2548i	− 1.13201i	−0.824403	−	0.566003i	$-0.808489\pi$
$863$	− 33.2548i	− 1.13201i	0.824403	−	0.566003i	$-0.191511\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	28.5632i	0.970058i
$868$	0	0
$869$	3.97278	0.134767
$870$	0	0
$871$	28.5108	0.966051
$872$	0	0
$873$	− 2.03644i	− 0.0689231i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.46007i	0.251909i	0.992036	+	0.125954i	$0.0401993\pi$
$877$	7.46007i	0.251909i	−0.992036	+	0.125954i	$0.959801\pi$
$878$	0	0
$879$	−34.1144	−1.15065
$880$	0	0
$881$	14.5285	0.489479	0.244739	−	0.969589i	$-0.421298\pi$
$881$	14.5285	0.489479	0.244739	+	0.969589i	$0.421298\pi$
$882$	0	0
$883$	− 35.8195i	− 1.20542i	−0.797960	−	0.602710i	$-0.794087\pi$
$883$	− 35.8195i	− 1.20542i	0.797960	−	0.602710i	$-0.205913\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.44065i	0.249833i	0.992167	+	0.124916i	$0.0398662\pi$
$887$	7.44065i	0.249833i	−0.992167	+	0.124916i	$0.960134\pi$
$888$	0	0
$889$	−8.25368	−0.276819
$890$	0	0
$891$	−5.91145	−0.198041
$892$	0	0
$893$	24.6111i	0.823577i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	32.3731i	1.08091i
$898$	0	0
$899$	−30.1968	−1.00712
$900$	0	0
$901$	1.00034	0.0333263
$902$	0	0
$903$	6.29980i	0.209644i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0.436634i	0.0144982i	0.999974	+	0.00724909i	$0.00230748\pi$
$907$	0.436634i	0.0144982i	−0.999974	+	0.00724909i	$0.997693\pi$
$908$	0	0
$909$	−0.490362	−0.0162643
$910$	0	0
$911$	−20.2003	−0.669266	−0.334633	−	0.942348i	$-0.608612\pi$
$911$	−20.2003	−0.669266	−0.334633	+	0.942348i	$0.608612\pi$
$912$	0	0
$913$	− 3.48714i	− 0.115407i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 21.9464i	− 0.724733i
$918$	0	0
$919$	14.2264	0.469284	0.234642	−	0.972082i	$-0.424608\pi$
$919$	14.2264	0.469284	0.234642	+	0.972082i	$0.424608\pi$
$920$	0	0
$921$	23.1329	0.762255
$922$	0	0
$923$	5.43953i	0.179044i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.239613i	0.00786991i
$928$	0	0
$929$	8.08932	0.265402	0.132701	−	0.991156i	$-0.457635\pi$
$929$	8.08932	0.265402	0.132701	+	0.991156i	$0.457635\pi$
$930$	0	0
$931$	26.8037	0.878457
$932$	0	0
$933$	45.1578i	1.47840i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	13.3336i	0.435591i	0.975994	+	0.217796i	$0.0698866\pi$
$937$	13.3336i	0.435591i	−0.975994	+	0.217796i	$0.930113\pi$
$938$	0	0
$939$	19.8748	0.648591
$940$	0	0
$941$	37.4861	1.22201	0.611006	−	0.791626i	$-0.290765\pi$
$941$	37.4861	1.22201	0.611006	+	0.791626i	$0.290765\pi$
$942$	0	0
$943$	− 6.88986i	− 0.224365i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 16.1211i	− 0.523865i	−0.965086	−	0.261933i	$-0.915640\pi$
$947$	− 16.1211i	− 0.523865i	0.965086	−	0.261933i	$-0.0843598\pi$
$948$	0	0
$949$	14.0264	0.455315
$950$	0	0
$951$	−21.2738	−0.689849
$952$	0	0
$953$	6.88884i	0.223151i	0.993756	+	0.111576i	$0.0355898\pi$
$953$	6.88884i	0.223151i	−0.993756	+	0.111576i	$0.964410\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 4.05600i	− 0.131112i
$958$	0	0
$959$	−76.0860	−2.45694
$960$	0	0
$961$	44.9035	1.44850
$962$	0	0
$963$	− 0.450901i	− 0.0145301i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	57.0827i	1.83566i	0.396978	+	0.917828i	$0.370059\pi$
$967$	57.0827i	1.83566i	−0.396978	+	0.917828i	$0.629941\pi$
$968$	0	0
$969$	−1.39060	−0.0446725
$970$	0	0
$971$	−35.4428	−1.13741	−0.568707	−	0.822540i	$-0.692556\pi$
$971$	−35.4428	−1.13741	−0.568707	+	0.822540i	$0.692556\pi$
$972$	0	0
$973$	71.8197i	2.30243i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0.304691i	0.00974792i	0.999988	+	0.00487396i	$0.00155144\pi$
$977$	0.304691i	0.00974792i	−0.999988	+	0.00487396i	$0.998449\pi$
$978$	0	0
$979$	7.45388	0.238227
$980$	0	0
$981$	0.469957	0.0150046
$982$	0	0
$983$	− 53.8626i	− 1.71795i	−0.512016	−	0.858976i	$-0.671101\pi$
$983$	− 53.8626i	− 1.71795i	0.512016	−	0.858976i	$-0.328899\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 58.1027i	− 1.84943i
$988$	0	0
$989$	6.37041	0.202567
$990$	0	0
$991$	−46.9376	−1.49102	−0.745511	−	0.666493i	$-0.767795\pi$
$991$	−46.9376	−1.49102	−0.745511	+	0.666493i	$0.767795\pi$
$992$	0	0
$993$	− 10.3805i	− 0.329417i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.79233i	0.151775i	0.997116	+	0.0758873i	$0.0241789\pi$
$997$	4.79233i	0.151775i	−0.997116	+	0.0758873i	$0.975821\pi$
$998$	0	0
$999$	−7.49290	−0.237065

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4100.2.d.g.1149.11		14
5.2	odd	4		4100.2.a.j.1.5	yes	7
5.3	odd	4		4100.2.a.g.1.3	✓	7
5.4	even	2	inner	4100.2.d.g.1149.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4100.2.a.g.1.3	✓	7	5.3	odd	4
4100.2.a.j.1.5	yes	7	5.2	odd	4
4100.2.d.g.1149.4		14	5.4	even	2	inner
4100.2.d.g.1149.11		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.68828i q^{3} -4.03576i q^{7} +0.149714 q^{9} +O(q^{10})\)	\(q+1.68828i q^{3} -4.03576i q^{7} +0.149714 q^{9} +0.693145 q^{11} +2.78311i q^{13} -0.285400i q^{17} -2.88605 q^{19} +6.81349 q^{21} -6.88986i q^{23} +5.31760i q^{27} -3.46601 q^{29} +8.71226 q^{31} +1.17022i q^{33} +1.40908i q^{37} -4.69866 q^{39} +1.00000 q^{41} +0.924607i q^{43} -8.52760i q^{47} -9.28736 q^{49} +0.481836 q^{51} +3.50505i q^{53} -4.87245i q^{57} +4.69625 q^{59} -0.199732 q^{61} -0.604209i q^{63} -10.2442i q^{67} +11.6320 q^{69} +1.95448 q^{71} -5.03982i q^{73} -2.79737i q^{77} +5.73152 q^{79} -8.52844 q^{81} -5.03089i q^{83} -5.85159i q^{87} +10.7537 q^{89} +11.2320 q^{91} +14.7087i q^{93} -13.6022i q^{97} +0.103773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 12 q^{9}+O(q^{10}) \) 14 * q - 12 * q^9	\( 14 q - 12 q^{9} - 2 q^{11} + 2 q^{19} - 2 q^{21} + 10 q^{29} + 8 q^{31} + 22 q^{39} + 14 q^{41} + 6 q^{49} + 54 q^{51} + 32 q^{59} + 20 q^{61} + 38 q^{69} + 54 q^{71} + 2 q^{79} + 54 q^{81} + 32 q^{89} + 18 q^{91} + 80 q^{99}+O(q^{100}) \) 14 * q - 12 * q^9 - 2 * q^11 + 2 * q^19 - 2 * q^21 + 10 * q^29 + 8 * q^31 + 22 * q^39 + 14 * q^41 + 6 * q^49 + 54 * q^51 + 32 * q^59 + 20 * q^61 + 38 * q^69 + 54 * q^71 + 2 * q^79 + 54 * q^81 + 32 * q^89 + 18 * q^91 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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