LMFDB - Embedded newform 6000.2.f.f.1249.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6000,2,Mod(1249,6000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6000.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6000 = 2^{4} \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6000.f (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:	$47.9102412128$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1500)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	6000.1249
Dual form			6000.2.f.f.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +3.23607i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +3.23607i q^{7} -1.00000 q^{9} +1.23607 q^{11} +0.381966i q^{17} +5.85410 q^{19} -3.23607 q^{21} +4.85410i q^{23} -1.00000i q^{27} -8.47214 q^{29} -7.61803 q^{31} +1.23607i q^{33} +2.00000i q^{37} -4.47214 q^{41} +5.70820i q^{43} -9.09017i q^{47} -3.47214 q^{49} -0.381966 q^{51} +5.09017i q^{53} +5.85410i q^{57} +11.2361 q^{59} -5.56231 q^{61} -3.23607i q^{63} +15.7082i q^{67} -4.85410 q^{69} -5.52786 q^{71} +3.52786i q^{73} +4.00000i q^{77} +13.8541 q^{79} +1.00000 q^{81} -6.32624i q^{83} -8.47214i q^{87} +2.76393 q^{89} -7.61803i q^{93} -12.0000i q^{97} -1.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 4 q^{11} + 10 q^{19} - 4 q^{21} - 16 q^{29} - 26 q^{31} + 4 q^{49} - 6 q^{51} + 36 q^{59} + 18 q^{61} - 6 q^{69} - 40 q^{71} + 42 q^{79} + 4 q^{81} + 20 q^{89} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 - 4 * q^11 + 10 * q^19 - 4 * q^21 - 16 * q^29 - 26 * q^31 + 4 * q^49 - 6 * q^51 + 36 * q^59 + 18 * q^61 - 6 * q^69 - 40 * q^71 + 42 * q^79 + 4 * q^81 + 20 * q^89 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6000\mathbb{Z}\right)^\times$.

$n$	$751$	$4001$	$4501$	$5377$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.23607i	1.22312i	0.791199	+	0.611559i	$0.209457\pi$
$7$	3.23607i	1.22312i	−0.791199	+	0.611559i	$0.790543\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.23607	0.372689	0.186344	−	0.982485i	$-0.440336\pi$
$11$	1.23607	0.372689	0.186344	+	0.982485i	$0.440336\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.381966i	0.0926404i	0.998927	+	0.0463202i	$0.0147494\pi$
$17$	0.381966i	0.0926404i	−0.998927	+	0.0463202i	$0.985251\pi$
$18$	0	0
$19$	5.85410	1.34302	0.671512	−	0.740994i	$-0.265645\pi$
$19$	5.85410	1.34302	0.671512	+	0.740994i	$0.265645\pi$
$20$	0	0
$21$	−3.23607	−0.706168
$22$	0	0
$23$	4.85410i	1.01215i	0.862489	+	0.506075i	$0.168904\pi$
$23$	4.85410i	1.01215i	−0.862489	+	0.506075i	$0.831096\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−8.47214	−1.57324	−0.786618	−	0.617440i	$-0.788170\pi$
$29$	−8.47214	−1.57324	−0.786618	+	0.617440i	$0.788170\pi$
$30$	0	0
$31$	−7.61803	−1.36824	−0.684120	−	0.729370i	$-0.739813\pi$
$31$	−7.61803	−1.36824	−0.684120	+	0.729370i	$0.739813\pi$
$32$	0	0
$33$	1.23607i	0.215172i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.47214	−0.698430	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214	−0.698430	−0.349215	+	0.937043i	$0.613552\pi$
$42$	0	0
$43$	5.70820i	0.870493i	0.900311	+	0.435246i	$0.143339\pi$
$43$	5.70820i	0.870493i	−0.900311	+	0.435246i	$0.856661\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.09017i	− 1.32594i	−0.748647	−	0.662969i	$-0.769296\pi$
$47$	− 9.09017i	− 1.32594i	0.748647	−	0.662969i	$-0.230704\pi$
$48$	0	0
$49$	−3.47214	−0.496019
$50$	0	0
$51$	−0.381966	−0.0534859
$52$	0	0
$53$	5.09017i	0.699189i	0.936901	+	0.349594i	$0.113681\pi$
$53$	5.09017i	0.699189i	−0.936901	+	0.349594i	$0.886319\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.85410i	0.775395i
$58$	0	0
$59$	11.2361	1.46281	0.731406	−	0.681943i	$-0.238865\pi$
$59$	11.2361	1.46281	0.731406	+	0.681943i	$0.238865\pi$
$60$	0	0
$61$	−5.56231	−0.712180	−0.356090	−	0.934452i	$-0.615890\pi$
$61$	−5.56231	−0.712180	−0.356090	+	0.934452i	$0.615890\pi$
$62$	0	0
$63$	− 3.23607i	− 0.407706i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.7082i	1.91906i	0.281602	+	0.959531i	$0.409134\pi$
$67$	15.7082i	1.91906i	−0.281602	+	0.959531i	$0.590866\pi$
$68$	0	0
$69$	−4.85410	−0.584365
$70$	0	0
$71$	−5.52786	−0.656037	−0.328018	−	0.944671i	$-0.606381\pi$
$71$	−5.52786	−0.656037	−0.328018	+	0.944671i	$0.606381\pi$
$72$	0	0
$73$	3.52786i	0.412905i	0.978457	+	0.206453i	$0.0661919\pi$
$73$	3.52786i	0.412905i	−0.978457	+	0.206453i	$0.933808\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.00000i	0.455842i
$78$	0	0
$79$	13.8541	1.55871	0.779354	−	0.626584i	$-0.215547\pi$
$79$	13.8541	1.55871	0.779354	+	0.626584i	$0.215547\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.32624i	− 0.694395i	−0.937792	−	0.347197i	$-0.887133\pi$
$83$	− 6.32624i	− 0.694395i	0.937792	−	0.347197i	$-0.112867\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 8.47214i	− 0.908308i
$88$	0	0
$89$	2.76393	0.292976	0.146488	−	0.989212i	$-0.453203\pi$
$89$	2.76393	0.292976	0.146488	+	0.989212i	$0.453203\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 7.61803i	− 0.789953i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 12.0000i	− 1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$	− 12.0000i	− 1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$	0	0
$99$	−1.23607	−0.124230
$100$	0	0
$101$	0.763932	0.0760141	0.0380070	−	0.999277i	$-0.487899\pi$
$101$	0.763932	0.0760141	0.0380070	+	0.999277i	$0.487899\pi$
$102$	0	0
$103$	− 2.29180i	− 0.225817i	−0.993605	−	0.112909i	$-0.963983\pi$
$103$	− 2.29180i	− 0.225817i	0.993605	−	0.112909i	$-0.0360168\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.3820i	1.39036i	0.718837	+	0.695179i	$0.244675\pi$
$107$	14.3820i	1.39036i	−0.718837	+	0.695179i	$0.755325\pi$
$108$	0	0
$109$	−1.61803	−0.154980	−0.0774898	−	0.996993i	$-0.524691\pi$
$109$	−1.61803	−0.154980	−0.0774898	+	0.996993i	$0.524691\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	− 1.67376i	− 0.157454i	−0.996896	−	0.0787271i	$-0.974914\pi$
$113$	− 1.67376i	− 0.157454i	0.996896	−	0.0787271i	$-0.0250856\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.23607	−0.113310
$120$	0	0
$121$	−9.47214	−0.861103
$122$	0	0
$123$	− 4.47214i	− 0.403239i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.41641i	0.835571i	0.908546	+	0.417786i	$0.137194\pi$
$127$	9.41641i	0.835571i	−0.908546	+	0.417786i	$0.862806\pi$
$128$	0	0
$129$	−5.70820	−0.502579
$130$	0	0
$131$	−10.9443	−0.956205	−0.478103	−	0.878304i	$-0.658675\pi$
$131$	−10.9443	−0.956205	−0.478103	+	0.878304i	$0.658675\pi$
$132$	0	0
$133$	18.9443i	1.64268i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.4721i	1.40731i	0.710542	+	0.703655i	$0.248450\pi$
$137$	16.4721i	1.40731i	−0.710542	+	0.703655i	$0.751550\pi$
$138$	0	0
$139$	5.14590	0.436469	0.218235	−	0.975896i	$-0.429970\pi$
$139$	5.14590	0.436469	0.218235	+	0.975896i	$0.429970\pi$
$140$	0	0
$141$	9.09017	0.765530
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.47214i	− 0.286377i
$148$	0	0
$149$	−23.1246	−1.89444	−0.947221	−	0.320581i	$-0.896122\pi$
$149$	−23.1246	−1.89444	−0.947221	+	0.320581i	$0.896122\pi$
$150$	0	0
$151$	5.85410	0.476400	0.238200	−	0.971216i	$-0.423443\pi$
$151$	5.85410	0.476400	0.238200	+	0.971216i	$0.423443\pi$
$152$	0	0
$153$	− 0.381966i	− 0.0308801i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 14.7639i	− 1.17829i	−0.808027	−	0.589145i	$-0.799465\pi$
$157$	− 14.7639i	− 1.17829i	0.808027	−	0.589145i	$-0.200535\pi$
$158$	0	0
$159$	−5.09017	−0.403677
$160$	0	0
$161$	−15.7082	−1.23798
$162$	0	0
$163$	− 2.76393i	− 0.216488i	−0.994124	−	0.108244i	$-0.965477\pi$
$163$	− 2.76393i	− 0.216488i	0.994124	−	0.108244i	$-0.0345228\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 7.85410i	− 0.607769i	−0.952709	−	0.303884i	$-0.901716\pi$
$167$	− 7.85410i	− 0.607769i	0.952709	−	0.303884i	$-0.0982836\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	−5.85410	−0.447674
$172$	0	0
$173$	− 23.8885i	− 1.81621i	−0.418740	−	0.908106i	$-0.637528\pi$
$173$	− 23.8885i	− 1.81621i	0.418740	−	0.908106i	$-0.362472\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	11.2361i	0.844555i
$178$	0	0
$179$	−23.8885	−1.78551	−0.892757	−	0.450539i	$-0.851232\pi$
$179$	−23.8885	−1.78551	−0.892757	+	0.450539i	$0.851232\pi$
$180$	0	0
$181$	−17.5623	−1.30540	−0.652698	−	0.757618i	$-0.726363\pi$
$181$	−17.5623	−1.30540	−0.652698	+	0.757618i	$0.726363\pi$
$182$	0	0
$183$	− 5.56231i	− 0.411177i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.472136i	0.0345260i
$188$	0	0
$189$	3.23607	0.235389
$190$	0	0
$191$	−12.9443	−0.936615	−0.468307	−	0.883566i	$-0.655136\pi$
$191$	−12.9443	−0.936615	−0.468307	+	0.883566i	$0.655136\pi$
$192$	0	0
$193$	− 21.1246i	− 1.52058i	−0.649582	−	0.760291i	$-0.725056\pi$
$193$	− 21.1246i	− 1.52058i	0.649582	−	0.760291i	$-0.274944\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.0344i	1.64114i	0.571549	+	0.820568i	$0.306343\pi$
$197$	23.0344i	1.64114i	−0.571549	+	0.820568i	$0.693657\pi$
$198$	0	0
$199$	−11.7984	−0.836365	−0.418182	−	0.908363i	$-0.637333\pi$
$199$	−11.7984	−0.836365	−0.418182	+	0.908363i	$0.637333\pi$
$200$	0	0
$201$	−15.7082	−1.10797
$202$	0	0
$203$	− 27.4164i	− 1.92425i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 4.85410i	− 0.337383i
$208$	0	0
$209$	7.23607	0.500529
$210$	0	0
$211$	−12.5623	−0.864825	−0.432412	−	0.901676i	$-0.642338\pi$
$211$	−12.5623	−0.864825	−0.432412	+	0.901676i	$0.642338\pi$
$212$	0	0
$213$	− 5.52786i	− 0.378763i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 24.6525i	− 1.67352i
$218$	0	0
$219$	−3.52786	−0.238391
$220$	0	0
$221$	0	0
$222$	0	0
$223$	26.8328i	1.79686i	0.439119	+	0.898429i	$0.355291\pi$
$223$	26.8328i	1.79686i	−0.439119	+	0.898429i	$0.644709\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 11.9098i	− 0.790483i	−0.918577	−	0.395242i	$-0.870661\pi$
$227$	− 11.9098i	− 0.790483i	0.918577	−	0.395242i	$-0.129339\pi$
$228$	0	0
$229$	3.14590	0.207887	0.103943	−	0.994583i	$-0.466854\pi$
$229$	3.14590	0.207887	0.103943	+	0.994583i	$0.466854\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	13.4164i	0.878938i	0.898258	+	0.439469i	$0.144833\pi$
$233$	13.4164i	0.878938i	−0.898258	+	0.439469i	$0.855167\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.8541i	0.899921i
$238$	0	0
$239$	0.763932	0.0494147	0.0247073	−	0.999695i	$-0.492135\pi$
$239$	0.763932	0.0494147	0.0247073	+	0.999695i	$0.492135\pi$
$240$	0	0
$241$	12.2705	0.790413	0.395207	−	0.918592i	$-0.370673\pi$
$241$	12.2705	0.790413	0.395207	+	0.918592i	$0.370673\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.32624	0.400909
$250$	0	0
$251$	−14.1803	−0.895055	−0.447528	−	0.894270i	$-0.647695\pi$
$251$	−14.1803	−0.895055	−0.447528	+	0.894270i	$0.647695\pi$
$252$	0	0
$253$	6.00000i	0.377217i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.38197i	− 0.585231i	−0.956230	−	0.292615i	$-0.905474\pi$
$257$	− 9.38197i	− 0.585231i	0.956230	−	0.292615i	$-0.0945256\pi$
$258$	0	0
$259$	−6.47214	−0.402159
$260$	0	0
$261$	8.47214	0.524412
$262$	0	0
$263$	17.6180i	1.08637i	0.839612	+	0.543187i	$0.182783\pi$
$263$	17.6180i	1.08637i	−0.839612	+	0.543187i	$0.817217\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.76393i	0.169150i
$268$	0	0
$269$	29.1246	1.77576	0.887879	−	0.460076i	$-0.152178\pi$
$269$	29.1246	1.77576	0.887879	+	0.460076i	$0.152178\pi$
$270$	0	0
$271$	−18.2705	−1.10985	−0.554927	−	0.831899i	$-0.687254\pi$
$271$	−18.2705	−1.10985	−0.554927	+	0.831899i	$0.687254\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 15.4164i	− 0.926282i	−0.886285	−	0.463141i	$-0.846722\pi$
$277$	− 15.4164i	− 0.926282i	0.886285	−	0.463141i	$-0.153278\pi$
$278$	0	0
$279$	7.61803	0.456080
$280$	0	0
$281$	−26.7639	−1.59660	−0.798301	−	0.602258i	$-0.794268\pi$
$281$	−26.7639	−1.59660	−0.798301	+	0.602258i	$0.794268\pi$
$282$	0	0
$283$	− 4.18034i	− 0.248495i	−0.992251	−	0.124248i	$-0.960348\pi$
$283$	− 4.18034i	− 0.248495i	0.992251	−	0.124248i	$-0.0396517\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 14.4721i	− 0.854263i
$288$	0	0
$289$	16.8541	0.991418
$290$	0	0
$291$	12.0000	0.703452
$292$	0	0
$293$	− 19.0344i	− 1.11200i	−0.831181	−	0.556002i	$-0.812335\pi$
$293$	− 19.0344i	− 1.11200i	0.831181	−	0.556002i	$-0.187665\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.23607i	− 0.0717239i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−18.4721	−1.06472
$302$	0	0
$303$	0.763932i	0.0438867i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.1803i	1.38004i	0.723788	+	0.690022i	$0.242399\pi$
$307$	24.1803i	1.38004i	−0.723788	+	0.690022i	$0.757601\pi$
$308$	0	0
$309$	2.29180	0.130376
$310$	0	0
$311$	14.2918	0.810413	0.405207	−	0.914225i	$-0.367200\pi$
$311$	14.2918	0.810413	0.405207	+	0.914225i	$0.367200\pi$
$312$	0	0
$313$	15.7082i	0.887880i	0.896056	+	0.443940i	$0.146420\pi$
$313$	15.7082i	0.887880i	−0.896056	+	0.443940i	$0.853580\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 32.8328i	− 1.84407i	−0.387101	−	0.922037i	$-0.626524\pi$
$317$	− 32.8328i	− 1.84407i	0.387101	−	0.922037i	$-0.373476\pi$
$318$	0	0
$319$	−10.4721	−0.586327
$320$	0	0
$321$	−14.3820	−0.802723
$322$	0	0
$323$	2.23607i	0.124418i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.61803i	− 0.0894775i
$328$	0	0
$329$	29.4164	1.62178
$330$	0	0
$331$	−24.0344	−1.32105	−0.660526	−	0.750803i	$-0.729667\pi$
$331$	−24.0344	−1.32105	−0.660526	+	0.750803i	$0.729667\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.1803i	0.990346i	0.868794	+	0.495173i	$0.164895\pi$
$337$	18.1803i	0.990346i	−0.868794	+	0.495173i	$0.835105\pi$
$338$	0	0
$339$	1.67376	0.0909063
$340$	0	0
$341$	−9.41641	−0.509927
$342$	0	0
$343$	11.4164i	0.616428i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.50658i	− 0.456657i	−0.973584	−	0.228329i	$-0.926674\pi$
$347$	− 8.50658i	− 0.456657i	0.973584	−	0.228329i	$-0.0733260\pi$
$348$	0	0
$349$	4.85410	0.259834	0.129917	−	0.991525i	$-0.458529\pi$
$349$	4.85410	0.259834	0.129917	+	0.991525i	$0.458529\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 22.8541i	− 1.21640i	−0.793783	−	0.608201i	$-0.791892\pi$
$353$	− 22.8541i	− 1.21640i	0.793783	−	0.608201i	$-0.208108\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.23607i	− 0.0654197i
$358$	0	0
$359$	−10.4721	−0.552698	−0.276349	−	0.961057i	$-0.589125\pi$
$359$	−10.4721	−0.552698	−0.276349	+	0.961057i	$0.589125\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	0	0
$363$	− 9.47214i	− 0.497158i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	19.4164i	1.01353i	0.862085	+	0.506764i	$0.169158\pi$
$367$	19.4164i	1.01353i	−0.862085	+	0.506764i	$0.830842\pi$
$368$	0	0
$369$	4.47214	0.232810
$370$	0	0
$371$	−16.4721	−0.855191
$372$	0	0
$373$	21.5279i	1.11467i	0.830288	+	0.557335i	$0.188176\pi$
$373$	21.5279i	1.11467i	−0.830288	+	0.557335i	$0.811824\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	−9.41641	−0.482417
$382$	0	0
$383$	− 21.2148i	− 1.08402i	−0.840371	−	0.542012i	$-0.817663\pi$
$383$	− 21.2148i	− 1.08402i	0.840371	−	0.542012i	$-0.182337\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 5.70820i	− 0.290164i
$388$	0	0
$389$	−6.65248	−0.337294	−0.168647	−	0.985677i	$-0.553940\pi$
$389$	−6.65248	−0.337294	−0.168647	+	0.985677i	$0.553940\pi$
$390$	0	0
$391$	−1.85410	−0.0937660
$392$	0	0
$393$	− 10.9443i	− 0.552065i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 6.29180i	− 0.315776i	−0.987457	−	0.157888i	$-0.949531\pi$
$397$	− 6.29180i	− 0.315776i	0.987457	−	0.157888i	$-0.0504685\pi$
$398$	0	0
$399$	−18.9443	−0.948400
$400$	0	0
$401$	18.3607	0.916889	0.458444	−	0.888723i	$-0.348407\pi$
$401$	18.3607	0.916889	0.458444	+	0.888723i	$0.348407\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.47214i	0.122539i
$408$	0	0
$409$	21.5623	1.06619	0.533094	−	0.846056i	$-0.321029\pi$
$409$	21.5623	1.06619	0.533094	+	0.846056i	$0.321029\pi$
$410$	0	0
$411$	−16.4721	−0.812511
$412$	0	0
$413$	36.3607i	1.78919i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.14590i	0.251996i
$418$	0	0
$419$	5.81966	0.284309	0.142155	−	0.989844i	$-0.454597\pi$
$419$	5.81966	0.284309	0.142155	+	0.989844i	$0.454597\pi$
$420$	0	0
$421$	−20.3262	−0.990640	−0.495320	−	0.868711i	$-0.664949\pi$
$421$	−20.3262	−0.990640	−0.495320	+	0.868711i	$0.664949\pi$
$422$	0	0
$423$	9.09017i	0.441979i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 18.0000i	− 0.871081i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.52786	0.362604	0.181302	−	0.983427i	$-0.441969\pi$
$431$	7.52786	0.362604	0.181302	+	0.983427i	$0.441969\pi$
$432$	0	0
$433$	27.4164i	1.31755i	0.752341	+	0.658774i	$0.228925\pi$
$433$	27.4164i	1.31755i	−0.752341	+	0.658774i	$0.771075\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	28.4164i	1.35934i
$438$	0	0
$439$	−15.4164	−0.735785	−0.367893	−	0.929868i	$-0.619921\pi$
$439$	−15.4164	−0.735785	−0.367893	+	0.929868i	$0.619921\pi$
$440$	0	0
$441$	3.47214	0.165340
$442$	0	0
$443$	20.7984i	0.988161i	0.869416	+	0.494080i	$0.164495\pi$
$443$	20.7984i	0.988161i	−0.869416	+	0.494080i	$0.835505\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 23.1246i	− 1.09376i
$448$	0	0
$449$	5.81966	0.274647	0.137323	−	0.990526i	$-0.456150\pi$
$449$	5.81966	0.274647	0.137323	+	0.990526i	$0.456150\pi$
$450$	0	0
$451$	−5.52786	−0.260297
$452$	0	0
$453$	5.85410i	0.275050i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.12461i	0.239719i	0.992791	+	0.119860i	$0.0382444\pi$
$457$	5.12461i	0.239719i	−0.992791	+	0.119860i	$0.961756\pi$
$458$	0	0
$459$	0.381966	0.0178286
$460$	0	0
$461$	−20.1803	−0.939892	−0.469946	−	0.882695i	$-0.655727\pi$
$461$	−20.1803	−0.939892	−0.469946	+	0.882695i	$0.655727\pi$
$462$	0	0
$463$	30.9443i	1.43810i	0.694957	+	0.719051i	$0.255423\pi$
$463$	30.9443i	1.43810i	−0.694957	+	0.719051i	$0.744577\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5.74265i	− 0.265738i	−0.991134	−	0.132869i	$-0.957581\pi$
$467$	− 5.74265i	− 0.265738i	0.991134	−	0.132869i	$-0.0424190\pi$
$468$	0	0
$469$	−50.8328	−2.34724
$470$	0	0
$471$	14.7639	0.680286
$472$	0	0
$473$	7.05573i	0.324423i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5.09017i	− 0.233063i
$478$	0	0
$479$	−11.1246	−0.508296	−0.254148	−	0.967165i	$-0.581795\pi$
$479$	−11.1246	−0.508296	−0.254148	+	0.967165i	$0.581795\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 15.7082i	− 0.714748i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 10.7639i	− 0.487760i	−0.969805	−	0.243880i	$-0.921580\pi$
$487$	− 10.7639i	− 0.487760i	0.969805	−	0.243880i	$-0.0784204\pi$
$488$	0	0
$489$	2.76393	0.124989
$490$	0	0
$491$	32.8328	1.48172	0.740862	−	0.671657i	$-0.234417\pi$
$491$	32.8328	1.48172	0.740862	+	0.671657i	$0.234417\pi$
$492$	0	0
$493$	− 3.23607i	− 0.145745i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 17.8885i	− 0.802411i
$498$	0	0
$499$	−28.6869	−1.28420	−0.642101	−	0.766620i	$-0.721937\pi$
$499$	−28.6869	−1.28420	−0.642101	+	0.766620i	$0.721937\pi$
$500$	0	0
$501$	7.85410	0.350895
$502$	0	0
$503$	− 32.9443i	− 1.46891i	−0.678656	−	0.734456i	$-0.737437\pi$
$503$	− 32.9443i	− 1.46891i	0.678656	−	0.734456i	$-0.262563\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	13.0000i	0.577350i
$508$	0	0
$509$	−29.8885	−1.32479	−0.662393	−	0.749156i	$-0.730459\pi$
$509$	−29.8885	−1.32479	−0.662393	+	0.749156i	$0.730459\pi$
$510$	0	0
$511$	−11.4164	−0.505032
$512$	0	0
$513$	− 5.85410i	− 0.258465i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 11.2361i	− 0.494162i
$518$	0	0
$519$	23.8885	1.04859
$520$	0	0
$521$	29.8885	1.30944	0.654720	−	0.755871i	$-0.272786\pi$
$521$	29.8885	1.30944	0.654720	+	0.755871i	$0.272786\pi$
$522$	0	0
$523$	39.8885i	1.74420i	0.489324	+	0.872102i	$0.337244\pi$
$523$	39.8885i	1.74420i	−0.489324	+	0.872102i	$0.662756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.90983i	− 0.126754i
$528$	0	0
$529$	−0.562306	−0.0244481
$530$	0	0
$531$	−11.2361	−0.487604
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 23.8885i	− 1.03087i
$538$	0	0
$539$	−4.29180	−0.184861
$540$	0	0
$541$	−16.2705	−0.699524	−0.349762	−	0.936839i	$-0.613738\pi$
$541$	−16.2705	−0.699524	−0.349762	+	0.936839i	$0.613738\pi$
$542$	0	0
$543$	− 17.5623i	− 0.753671i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 44.8328i	− 1.91691i	−0.285239	−	0.958456i	$-0.592073\pi$
$547$	− 44.8328i	− 1.91691i	0.285239	−	0.958456i	$-0.407927\pi$
$548$	0	0
$549$	5.56231	0.237393
$550$	0	0
$551$	−49.5967	−2.11289
$552$	0	0
$553$	44.8328i	1.90649i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.3607i	0.438996i	0.975613	+	0.219498i	$0.0704420\pi$
$557$	10.3607i	0.438996i	−0.975613	+	0.219498i	$0.929558\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−0.472136	−0.0199336
$562$	0	0
$563$	− 6.38197i	− 0.268968i	−0.990916	−	0.134484i	$-0.957062\pi$
$563$	− 6.38197i	− 0.268968i	0.990916	−	0.134484i	$-0.0429377\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.23607i	0.135902i
$568$	0	0
$569$	25.3050	1.06084	0.530419	−	0.847735i	$-0.322034\pi$
$569$	25.3050	1.06084	0.530419	+	0.847735i	$0.322034\pi$
$570$	0	0
$571$	37.8885	1.58559	0.792793	−	0.609491i	$-0.208626\pi$
$571$	37.8885	1.58559	0.792793	+	0.609491i	$0.208626\pi$
$572$	0	0
$573$	− 12.9443i	− 0.540755i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.1246i	0.962690i	0.876531	+	0.481345i	$0.159852\pi$
$577$	23.1246i	0.962690i	−0.876531	+	0.481345i	$0.840148\pi$
$578$	0	0
$579$	21.1246	0.877909
$580$	0	0
$581$	20.4721	0.849327
$582$	0	0
$583$	6.29180i	0.260580i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.0344i	0.537989i	0.963142	+	0.268994i	$0.0866913\pi$
$587$	13.0344i	0.537989i	−0.963142	+	0.268994i	$0.913309\pi$
$588$	0	0
$589$	−44.5967	−1.83758
$590$	0	0
$591$	−23.0344	−0.947510
$592$	0	0
$593$	− 20.5066i	− 0.842104i	−0.907036	−	0.421052i	$-0.861661\pi$
$593$	− 20.5066i	− 0.842104i	0.907036	−	0.421052i	$-0.138339\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 11.7984i	− 0.482875i
$598$	0	0
$599$	32.8328	1.34151	0.670756	−	0.741678i	$-0.265970\pi$
$599$	32.8328	1.34151	0.670756	+	0.741678i	$0.265970\pi$
$600$	0	0
$601$	−12.8541	−0.524330	−0.262165	−	0.965023i	$-0.584436\pi$
$601$	−12.8541	−0.524330	−0.262165	+	0.965023i	$0.584436\pi$
$602$	0	0
$603$	− 15.7082i	− 0.639688i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	48.2492i	1.95838i	0.202956	+	0.979188i	$0.434945\pi$
$607$	48.2492i	1.95838i	−0.202956	+	0.979188i	$0.565055\pi$
$608$	0	0
$609$	27.4164	1.11097
$610$	0	0
$611$	0	0
$612$	0	0
$613$	7.59675i	0.306830i	0.988162	+	0.153415i	$0.0490271\pi$
$613$	7.59675i	0.306830i	−0.988162	+	0.153415i	$0.950973\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 34.4508i	− 1.38694i	−0.720486	−	0.693469i	$-0.756081\pi$
$617$	− 34.4508i	− 1.38694i	0.720486	−	0.693469i	$-0.243919\pi$
$618$	0	0
$619$	23.7984	0.956537	0.478269	−	0.878214i	$-0.341265\pi$
$619$	23.7984	0.956537	0.478269	+	0.878214i	$0.341265\pi$
$620$	0	0
$621$	4.85410	0.194788
$622$	0	0
$623$	8.94427i	0.358345i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.23607i	0.288981i
$628$	0	0
$629$	−0.763932	−0.0304600
$630$	0	0
$631$	29.3050	1.16661	0.583306	−	0.812253i	$-0.301759\pi$
$631$	29.3050	1.16661	0.583306	+	0.812253i	$0.301759\pi$
$632$	0	0
$633$	− 12.5623i	− 0.499307i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.52786	0.218679
$640$	0	0
$641$	−41.7771	−1.65010	−0.825048	−	0.565063i	$-0.808852\pi$
$641$	−41.7771	−1.65010	−0.825048	+	0.565063i	$0.808852\pi$
$642$	0	0
$643$	− 4.65248i	− 0.183476i	−0.995783	−	0.0917379i	$-0.970758\pi$
$643$	− 4.65248i	− 0.183476i	0.995783	−	0.0917379i	$-0.0292422\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.0557i	0.749158i	0.927195	+	0.374579i	$0.122213\pi$
$647$	19.0557i	0.749158i	−0.927195	+	0.374579i	$0.877787\pi$
$648$	0	0
$649$	13.8885	0.545173
$650$	0	0
$651$	24.6525	0.966207
$652$	0	0
$653$	− 46.6869i	− 1.82700i	−0.406838	−	0.913500i	$-0.633369\pi$
$653$	− 46.6869i	− 1.82700i	0.406838	−	0.913500i	$-0.366631\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3.52786i	− 0.137635i
$658$	0	0
$659$	32.0689	1.24923	0.624613	−	0.780934i	$-0.285257\pi$
$659$	32.0689	1.24923	0.624613	+	0.780934i	$0.285257\pi$
$660$	0	0
$661$	18.2016	0.707961	0.353981	−	0.935253i	$-0.384828\pi$
$661$	18.2016	0.707961	0.353981	+	0.935253i	$0.384828\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 41.1246i	− 1.59235i
$668$	0	0
$669$	−26.8328	−1.03742
$670$	0	0
$671$	−6.87539	−0.265421
$672$	0	0
$673$	7.88854i	0.304081i	0.988374	+	0.152041i	$0.0485844\pi$
$673$	7.88854i	0.304081i	−0.988374	+	0.152041i	$0.951416\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 6.97871i	− 0.268214i	−0.990967	−	0.134107i	$-0.957183\pi$
$677$	− 6.97871i	− 0.268214i	0.990967	−	0.134107i	$-0.0428165\pi$
$678$	0	0
$679$	38.8328	1.49027
$680$	0	0
$681$	11.9098	0.456386
$682$	0	0
$683$	− 19.3262i	− 0.739498i	−0.929132	−	0.369749i	$-0.879444\pi$
$683$	− 19.3262i	− 0.739498i	0.929132	−	0.369749i	$-0.120556\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3.14590i	0.120023i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	4.56231	0.173558	0.0867791	−	0.996228i	$-0.472343\pi$
$691$	4.56231	0.173558	0.0867791	+	0.996228i	$0.472343\pi$
$692$	0	0
$693$	− 4.00000i	− 0.151947i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.70820i	− 0.0647028i
$698$	0	0
$699$	−13.4164	−0.507455
$700$	0	0
$701$	−1.88854	−0.0713293	−0.0356647	−	0.999364i	$-0.511355\pi$
$701$	−1.88854	−0.0713293	−0.0356647	+	0.999364i	$0.511355\pi$
$702$	0	0
$703$	11.7082i	0.441583i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.47214i	0.0929742i
$708$	0	0
$709$	−12.6180	−0.473880	−0.236940	−	0.971524i	$-0.576145\pi$
$709$	−12.6180	−0.473880	−0.236940	+	0.971524i	$0.576145\pi$
$710$	0	0
$711$	−13.8541	−0.519569
$712$	0	0
$713$	− 36.9787i	− 1.38486i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.763932i	0.0285296i
$718$	0	0
$719$	20.7639	0.774364	0.387182	−	0.922003i	$-0.373449\pi$
$719$	20.7639	0.774364	0.387182	+	0.922003i	$0.373449\pi$
$720$	0	0
$721$	7.41641	0.276201
$722$	0	0
$723$	12.2705i	0.456345i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.8328i	0.698470i	0.937035	+	0.349235i	$0.113559\pi$
$727$	18.8328i	0.698470i	−0.937035	+	0.349235i	$0.886441\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−2.18034	−0.0806428
$732$	0	0
$733$	− 10.1803i	− 0.376019i	−0.982167	−	0.188010i	$-0.939796\pi$
$733$	− 10.1803i	− 0.376019i	0.982167	−	0.188010i	$-0.0602036\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.4164i	0.715213i
$738$	0	0
$739$	−12.5623	−0.462112	−0.231056	−	0.972940i	$-0.574218\pi$
$739$	−12.5623	−0.462112	−0.231056	+	0.972940i	$0.574218\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 7.41641i	− 0.272082i	−0.990703	−	0.136041i	$-0.956562\pi$
$743$	− 7.41641i	− 0.272082i	0.990703	−	0.136041i	$-0.0434378\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.32624i	0.231465i
$748$	0	0
$749$	−46.5410	−1.70057
$750$	0	0
$751$	−8.90983	−0.325124	−0.162562	−	0.986698i	$-0.551976\pi$
$751$	−8.90983	−0.325124	−0.162562	+	0.986698i	$0.551976\pi$
$752$	0	0
$753$	− 14.1803i	− 0.516760i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	43.4853i	1.58050i	0.612785	+	0.790250i	$0.290049\pi$
$757$	43.4853i	1.58050i	−0.612785	+	0.790250i	$0.709951\pi$
$758$	0	0
$759$	−6.00000	−0.217786
$760$	0	0
$761$	−13.5279	−0.490385	−0.245192	−	0.969474i	$-0.578851\pi$
$761$	−13.5279	−0.490385	−0.245192	+	0.969474i	$0.578851\pi$
$762$	0	0
$763$	− 5.23607i	− 0.189558i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	2.14590	0.0773831	0.0386915	−	0.999251i	$-0.487681\pi$
$769$	2.14590	0.0773831	0.0386915	+	0.999251i	$0.487681\pi$
$770$	0	0
$771$	9.38197	0.337883
$772$	0	0
$773$	51.9230i	1.86754i	0.357874	+	0.933770i	$0.383502\pi$
$773$	51.9230i	1.86754i	−0.357874	+	0.933770i	$0.616498\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 6.47214i	− 0.232187i
$778$	0	0
$779$	−26.1803	−0.938008
$780$	0	0
$781$	−6.83282	−0.244497
$782$	0	0
$783$	8.47214i	0.302769i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 39.7082i	− 1.41544i	−0.706491	−	0.707722i	$-0.749723\pi$
$787$	− 39.7082i	− 1.41544i	0.706491	−	0.707722i	$-0.250277\pi$
$788$	0	0
$789$	−17.6180	−0.627219
$790$	0	0
$791$	5.41641	0.192585
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 20.7426i	− 0.734742i	−0.930074	−	0.367371i	$-0.880258\pi$
$797$	− 20.7426i	− 0.734742i	0.930074	−	0.367371i	$-0.119742\pi$
$798$	0	0
$799$	3.47214	0.122835
$800$	0	0
$801$	−2.76393	−0.0976587
$802$	0	0
$803$	4.36068i	0.153885i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	29.1246i	1.02523i
$808$	0	0
$809$	−49.4164	−1.73739	−0.868694	−	0.495349i	$-0.835040\pi$
$809$	−49.4164	−1.73739	−0.868694	+	0.495349i	$0.835040\pi$
$810$	0	0
$811$	−2.43769	−0.0855990	−0.0427995	−	0.999084i	$-0.513628\pi$
$811$	−2.43769	−0.0855990	−0.0427995	+	0.999084i	$0.513628\pi$
$812$	0	0
$813$	− 18.2705i	− 0.640775i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	33.4164i	1.16909i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	41.5279	1.44933	0.724666	−	0.689100i	$-0.241994\pi$
$821$	41.5279	1.44933	0.724666	+	0.689100i	$0.241994\pi$
$822$	0	0
$823$	14.7639i	0.514638i	0.966326	+	0.257319i	$0.0828392\pi$
$823$	14.7639i	0.514638i	−0.966326	+	0.257319i	$0.917161\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.1459i	0.561448i	0.959789	+	0.280724i	$0.0905745\pi$
$827$	16.1459i	0.561448i	−0.959789	+	0.280724i	$0.909425\pi$
$828$	0	0
$829$	−37.8541	−1.31473	−0.657364	−	0.753574i	$-0.728328\pi$
$829$	−37.8541	−1.31473	−0.657364	+	0.753574i	$0.728328\pi$
$830$	0	0
$831$	15.4164	0.534789
$832$	0	0
$833$	− 1.32624i	− 0.0459514i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.61803i	0.263318i
$838$	0	0
$839$	27.0132	0.932598	0.466299	−	0.884627i	$-0.345587\pi$
$839$	27.0132	0.932598	0.466299	+	0.884627i	$0.345587\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	− 26.7639i	− 0.921799i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 30.6525i	− 1.05323i
$848$	0	0
$849$	4.18034	0.143469
$850$	0	0
$851$	−9.70820	−0.332793
$852$	0	0
$853$	− 26.2918i	− 0.900214i	−0.892975	−	0.450107i	$-0.851386\pi$
$853$	− 26.2918i	− 0.900214i	0.892975	−	0.450107i	$-0.148614\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.9098i	1.19250i	0.802800	+	0.596249i	$0.203343\pi$
$857$	34.9098i	1.19250i	−0.802800	+	0.596249i	$0.796657\pi$
$858$	0	0
$859$	26.8328	0.915524	0.457762	−	0.889075i	$-0.348651\pi$
$859$	26.8328	0.915524	0.457762	+	0.889075i	$0.348651\pi$
$860$	0	0
$861$	14.4721	0.493209
$862$	0	0
$863$	− 17.5279i	− 0.596655i	−0.954464	−	0.298328i	$-0.903571\pi$
$863$	− 17.5279i	− 0.596655i	0.954464	−	0.298328i	$-0.0964288\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.8541i	0.572395i
$868$	0	0
$869$	17.1246	0.580913
$870$	0	0
$871$	0	0
$872$	0	0
$873$	12.0000i	0.406138i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34.5410i	1.16637i	0.812340	+	0.583184i	$0.198193\pi$
$877$	34.5410i	1.16637i	−0.812340	+	0.583184i	$0.801807\pi$
$878$	0	0
$879$	19.0344	0.642016
$880$	0	0
$881$	−16.5836	−0.558715	−0.279358	−	0.960187i	$-0.590122\pi$
$881$	−16.5836	−0.558715	−0.279358	+	0.960187i	$0.590122\pi$
$882$	0	0
$883$	10.1115i	0.340278i	0.985420	+	0.170139i	$0.0544216\pi$
$883$	10.1115i	0.340278i	−0.985420	+	0.170139i	$0.945578\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.9098i	0.769237i	0.923076	+	0.384618i	$0.125667\pi$
$887$	22.9098i	0.769237i	−0.923076	+	0.384618i	$0.874333\pi$
$888$	0	0
$889$	−30.4721	−1.02200
$890$	0	0
$891$	1.23607	0.0414098
$892$	0	0
$893$	− 53.2148i	− 1.78076i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	64.5410	2.15256
$900$	0	0
$901$	−1.94427	−0.0647731
$902$	0	0
$903$	− 18.4721i	− 0.614714i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 53.7082i	− 1.78335i	−0.452675	−	0.891676i	$-0.649530\pi$
$907$	− 53.7082i	− 1.78335i	0.452675	−	0.891676i	$-0.350470\pi$
$908$	0	0
$909$	−0.763932	−0.0253380
$910$	0	0
$911$	56.0689	1.85764	0.928822	−	0.370525i	$-0.120822\pi$
$911$	56.0689	1.85764	0.928822	+	0.370525i	$0.120822\pi$
$912$	0	0
$913$	− 7.81966i	− 0.258793i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 35.4164i	− 1.16955i
$918$	0	0
$919$	−3.25735	−0.107450	−0.0537251	−	0.998556i	$-0.517109\pi$
$919$	−3.25735	−0.107450	−0.0537251	+	0.998556i	$0.517109\pi$
$920$	0	0
$921$	−24.1803	−0.796769
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.29180i	0.0752725i
$928$	0	0
$929$	49.4853	1.62356	0.811780	−	0.583964i	$-0.198499\pi$
$929$	49.4853	1.62356	0.811780	+	0.583964i	$0.198499\pi$
$930$	0	0
$931$	−20.3262	−0.666166
$932$	0	0
$933$	14.2918i	0.467892i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 14.5836i	− 0.476425i	−0.971213	−	0.238213i	$-0.923438\pi$
$937$	− 14.5836i	− 0.476425i	0.971213	−	0.238213i	$-0.0765615\pi$
$938$	0	0
$939$	−15.7082	−0.512618
$940$	0	0
$941$	38.2918	1.24828	0.624138	−	0.781314i	$-0.285450\pi$
$941$	38.2918	1.24828	0.624138	+	0.781314i	$0.285450\pi$
$942$	0	0
$943$	− 21.7082i	− 0.706916i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.2148i	0.689388i	0.938715	+	0.344694i	$0.112017\pi$
$947$	21.2148i	0.689388i	−0.938715	+	0.344694i	$0.887983\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	32.8328	1.06468
$952$	0	0
$953$	12.7426i	0.412775i	0.978470	+	0.206387i	$0.0661707\pi$
$953$	12.7426i	0.412775i	−0.978470	+	0.206387i	$0.933829\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 10.4721i	− 0.338516i
$958$	0	0
$959$	−53.3050	−1.72131
$960$	0	0
$961$	27.0344	0.872079
$962$	0	0
$963$	− 14.3820i	− 0.463452i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 18.9443i	− 0.609207i	−0.952479	−	0.304603i	$-0.901476\pi$
$967$	− 18.9443i	− 0.609207i	0.952479	−	0.304603i	$-0.0985239\pi$
$968$	0	0
$969$	−2.23607	−0.0718329
$970$	0	0
$971$	30.5410	0.980108	0.490054	−	0.871692i	$-0.336977\pi$
$971$	30.5410	0.980108	0.490054	+	0.871692i	$0.336977\pi$
$972$	0	0
$973$	16.6525i	0.533854i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 17.7295i	− 0.567217i	−0.958940	−	0.283608i	$-0.908468\pi$
$977$	− 17.7295i	− 0.567217i	0.958940	−	0.283608i	$-0.0915316\pi$
$978$	0	0
$979$	3.41641	0.109189
$980$	0	0
$981$	1.61803	0.0516598
$982$	0	0
$983$	48.5066i	1.54712i	0.633723	+	0.773560i	$0.281526\pi$
$983$	48.5066i	1.54712i	−0.633723	+	0.773560i	$0.718474\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	29.4164i	0.936335i
$988$	0	0
$989$	−27.7082	−0.881070
$990$	0	0
$991$	−5.56231	−0.176692	−0.0883462	−	0.996090i	$-0.528158\pi$
$991$	−5.56231	−0.176692	−0.0883462	+	0.996090i	$0.528158\pi$
$992$	0	0
$993$	− 24.0344i	− 0.762710i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	6.00000i	0.190022i	0.995476	+	0.0950110i	$0.0302886\pi$
$997$	6.00000i	0.190022i	−0.995476	+	0.0950110i	$0.969711\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6000.2.f.f.1249.4		4
4.3	odd	2		1500.2.d.b.1249.1		4
5.2	odd	4		6000.2.a.s.1.1		2
5.3	odd	4		6000.2.a.i.1.2		2
5.4	even	2	inner	6000.2.f.f.1249.1		4
12.11	even	2		4500.2.d.a.4249.1		4
20.3	even	4		1500.2.a.g.1.1	yes	2
20.7	even	4		1500.2.a.c.1.2	✓	2
20.19	odd	2		1500.2.d.b.1249.4		4
60.23	odd	4		4500.2.a.d.1.1		2
60.47	odd	4		4500.2.a.h.1.2		2
60.59	even	2		4500.2.d.a.4249.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1500.2.a.c.1.2	✓	2	20.7	even	4
1500.2.a.g.1.1	yes	2	20.3	even	4
1500.2.d.b.1249.1		4	4.3	odd	2
1500.2.d.b.1249.4		4	20.19	odd	2
4500.2.a.d.1.1		2	60.23	odd	4
4500.2.a.h.1.2		2	60.47	odd	4
4500.2.d.a.4249.1		4	12.11	even	2
4500.2.d.a.4249.4		4	60.59	even	2
6000.2.a.i.1.2		2	5.3	odd	4
6000.2.a.s.1.1		2	5.2	odd	4
6000.2.f.f.1249.1		4	5.4	even	2	inner
6000.2.f.f.1249.4		4	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:	\( N \)	\(=\)	\( 6000 = 2^{4} \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6000.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +3.23607i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +3.23607i q^{7} -1.00000 q^{9} +1.23607 q^{11} +0.381966i q^{17} +5.85410 q^{19} -3.23607 q^{21} +4.85410i q^{23} -1.00000i q^{27} -8.47214 q^{29} -7.61803 q^{31} +1.23607i q^{33} +2.00000i q^{37} -4.47214 q^{41} +5.70820i q^{43} -9.09017i q^{47} -3.47214 q^{49} -0.381966 q^{51} +5.09017i q^{53} +5.85410i q^{57} +11.2361 q^{59} -5.56231 q^{61} -3.23607i q^{63} +15.7082i q^{67} -4.85410 q^{69} -5.52786 q^{71} +3.52786i q^{73} +4.00000i q^{77} +13.8541 q^{79} +1.00000 q^{81} -6.32624i q^{83} -8.47214i q^{87} +2.76393 q^{89} -7.61803i q^{93} -12.0000i q^{97} -1.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 4 q^{11} + 10 q^{19} - 4 q^{21} - 16 q^{29} - 26 q^{31} + 4 q^{49} - 6 q^{51} + 36 q^{59} + 18 q^{61} - 6 q^{69} - 40 q^{71} + 42 q^{79} + 4 q^{81} + 20 q^{89} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 - 4 * q^11 + 10 * q^19 - 4 * q^21 - 16 * q^29 - 26 * q^31 + 4 * q^49 - 6 * q^51 + 36 * q^59 + 18 * q^61 - 6 * q^69 - 40 * q^71 + 42 * q^79 + 4 * q^81 + 20 * q^89 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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