LMFDB - Embedded newform 6000.2.f.p.1249.1

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:	$8$
Coefficient field:	8.0.566105760000.21
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 45x^{6} + 728x^{4} + 4965x^{2} + 11881 $ x^8 + 45x^6 + 728x^4 + 4965*x^2 + 11881
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3000)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$-3.97706i$ of defining polynomial
Character	$\chi$	$=$	6000.1249
Dual form			6000.2.f.p.1249.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.97706i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -2.97706i q^{7} -1.00000 q^{9} -5.59509 q^{11} -6.43501i q^{13} -1.61803i q^{17} -3.07599 q^{19} -2.97706 q^{21} +3.59509i q^{23} +1.00000i q^{27} -7.19894 q^{29} -5.69402 q^{31} +5.59509i q^{33} -9.27493i q^{37} -6.43501 q^{39} +11.4492 q^{41} -0.145898i q^{43} +12.9071i q^{47} -1.86287 q^{49} -1.61803 q^{51} -0.419089i q^{53} +3.07599i q^{57} +5.54379 q^{59} -11.8471 q^{61} +2.97706i q^{63} +6.91525i q^{67} +3.59509 q^{69} +6.81698 q^{71} +3.33500i q^{73} +16.6569i q^{77} -6.30329 q^{79} +1.00000 q^{81} -8.45903i q^{83} +7.19894i q^{87} +2.31206 q^{89} -19.1574 q^{91} +5.69402i q^{93} +9.44919i q^{97} +5.59509 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 6 q^{11} + 6 q^{21} - 30 q^{29} - 12 q^{31} - 6 q^{39} + 26 q^{41} - 38 q^{49} - 4 q^{51} + 16 q^{59} + 26 q^{61} - 10 q^{69} + 18 q^{71} + 42 q^{79} + 8 q^{81} - 24 q^{89} - 34 q^{91} + 6 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 - 6 * q^11 + 6 * q^21 - 30 * q^29 - 12 * q^31 - 6 * q^39 + 26 * q^41 - 38 * q^49 - 4 * q^51 + 16 * q^59 + 26 * q^61 - 10 * q^69 + 18 * q^71 + 42 * q^79 + 8 * q^81 - 24 * q^89 - 34 * q^91 + 6 * 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
$7$	− 2.97706i	− 1.12522i	−0.826722	−	0.562611i	$-0.809797\pi$
$7$	− 2.97706i	− 1.12522i	0.826722	−	0.562611i	$-0.190203\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−5.59509	−1.68698	−0.843492	−	0.537142i	$-0.819504\pi$
$11$	−5.59509	−1.68698	−0.843492	+	0.537142i	$0.819504\pi$
$12$	0	0
$13$	− 6.43501i	− 1.78475i	−0.451293	−	0.892376i	$-0.649037\pi$
$13$	− 6.43501i	− 1.78475i	0.451293	−	0.892376i	$-0.350963\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.61803i	− 0.392431i	−0.980561	−	0.196215i	$-0.937135\pi$
$17$	− 1.61803i	− 0.392431i	0.980561	−	0.196215i	$-0.0628652\pi$
$18$	0	0
$19$	−3.07599	−0.705681	−0.352840	−	0.935684i	$-0.614784\pi$
$19$	−3.07599	−0.705681	−0.352840	+	0.935684i	$0.614784\pi$
$20$	0	0
$21$	−2.97706	−0.649647
$22$	0	0
$23$	3.59509i	0.749628i	0.927100	+	0.374814i	$0.122293\pi$
$23$	3.59509i	0.749628i	−0.927100	+	0.374814i	$0.877707\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−7.19894	−1.33681	−0.668405	−	0.743797i	$-0.733023\pi$
$29$	−7.19894	−1.33681	−0.668405	+	0.743797i	$0.733023\pi$
$30$	0	0
$31$	−5.69402	−1.02268	−0.511338	−	0.859379i	$-0.670850\pi$
$31$	−5.69402	−1.02268	−0.511338	+	0.859379i	$0.670850\pi$
$32$	0	0
$33$	5.59509i	0.973980i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9.27493i	− 1.52479i	−0.647112	−	0.762395i	$-0.724023\pi$
$37$	− 9.27493i	− 1.52479i	0.647112	−	0.762395i	$-0.275977\pi$
$38$	0	0
$39$	−6.43501	−1.03043
$40$	0	0
$41$	11.4492	1.78806	0.894032	−	0.448004i	$-0.147865\pi$
$41$	11.4492	1.78806	0.894032	+	0.448004i	$0.147865\pi$
$42$	0	0
$43$	− 0.145898i	− 0.0222492i	−0.999938	−	0.0111246i	$-0.996459\pi$
$43$	− 0.145898i	− 0.0222492i	0.999938	−	0.0111246i	$-0.00354115\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.9071i	1.88270i	0.337430	+	0.941351i	$0.390442\pi$
$47$	12.9071i	1.88270i	−0.337430	+	0.941351i	$0.609558\pi$
$48$	0	0
$49$	−1.86287	−0.266124
$50$	0	0
$51$	−1.61803	−0.226570
$52$	0	0
$53$	− 0.419089i	− 0.0575663i	−0.999586	−	0.0287832i	$-0.990837\pi$
$53$	− 0.419089i	− 0.0575663i	0.999586	−	0.0287832i	$-0.00916323\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.07599i	0.407425i
$58$	0	0
$59$	5.54379	0.721740	0.360870	−	0.932616i	$-0.382480\pi$
$59$	5.54379	0.721740	0.360870	+	0.932616i	$0.382480\pi$
$60$	0	0
$61$	−11.8471	−1.51686	−0.758432	−	0.651753i	$-0.774034\pi$
$61$	−11.8471	−1.51686	−0.758432	+	0.651753i	$0.774034\pi$
$62$	0	0
$63$	2.97706i	0.375074i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.91525i	0.844832i	0.906402	+	0.422416i	$0.138818\pi$
$67$	6.91525i	0.844832i	−0.906402	+	0.422416i	$0.861182\pi$
$68$	0	0
$69$	3.59509	0.432798
$70$	0	0
$71$	6.81698	0.809027	0.404513	−	0.914532i	$-0.367441\pi$
$71$	6.81698	0.809027	0.404513	+	0.914532i	$0.367441\pi$
$72$	0	0
$73$	3.33500i	0.390332i	0.980770	+	0.195166i	$0.0625246\pi$
$73$	3.33500i	0.390332i	−0.980770	+	0.195166i	$0.937475\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.6569i	1.89823i
$78$	0	0
$79$	−6.30329	−0.709176	−0.354588	−	0.935023i	$-0.615379\pi$
$79$	−6.30329	−0.709176	−0.354588	+	0.935023i	$0.615379\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 8.45903i	− 0.928500i	−0.885704	−	0.464250i	$-0.846324\pi$
$83$	− 8.45903i	− 0.928500i	0.885704	−	0.464250i	$-0.153676\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.19894i	0.771808i
$88$	0	0
$89$	2.31206	0.245078	0.122539	−	0.992464i	$-0.460896\pi$
$89$	2.31206	0.245078	0.122539	+	0.992464i	$0.460896\pi$
$90$	0	0
$91$	−19.1574	−2.00824
$92$	0	0
$93$	5.69402i	0.590443i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.44919i	0.959420i	0.877427	+	0.479710i	$0.159258\pi$
$97$	9.44919i	0.959420i	−0.877427	+	0.479710i	$0.840742\pi$
$98$	0	0
$99$	5.59509	0.562328
$100$	0	0
$101$	16.2273	1.61468	0.807339	−	0.590089i	$-0.200907\pi$
$101$	16.2273	1.61468	0.807339	+	0.590089i	$0.200907\pi$
$102$	0	0
$103$	− 10.2990i	− 1.01479i	−0.861715	−	0.507393i	$-0.830609\pi$
$103$	− 10.2990i	− 1.01479i	0.861715	−	0.507393i	$-0.169391\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.10877i	− 0.107189i	−0.998563	−	0.0535946i	$-0.982932\pi$
$107$	− 1.10877i	− 0.107189i	0.998563	−	0.0535946i	$-0.0170679\pi$
$108$	0	0
$109$	15.1821	1.45418	0.727090	−	0.686542i	$-0.240872\pi$
$109$	15.1821	1.45418	0.727090	+	0.686542i	$0.240872\pi$
$110$	0	0
$111$	−9.27493	−0.880338
$112$	0	0
$113$	− 5.48090i	− 0.515600i	−0.966198	−	0.257800i	$-0.917002\pi$
$113$	− 5.48090i	− 0.515600i	0.966198	−	0.257800i	$-0.0829975\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.43501i	0.594917i
$118$	0	0
$119$	−4.81698	−0.441572
$120$	0	0
$121$	20.3050	1.84591
$122$	0	0
$123$	− 11.4492i	− 1.03234i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.39615i	0.301359i	0.988583	+	0.150680i	$0.0481462\pi$
$127$	3.39615i	0.301359i	−0.988583	+	0.150680i	$0.951854\pi$
$128$	0	0
$129$	−0.145898	−0.0128456
$130$	0	0
$131$	−1.05130	−0.0918528	−0.0459264	−	0.998945i	$-0.514624\pi$
$131$	−1.05130	−0.0918528	−0.0459264	+	0.998945i	$0.514624\pi$
$132$	0	0
$133$	9.15740i	0.794047i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.11311i	0.351407i	0.984443	+	0.175703i	$0.0562200\pi$
$137$	4.11311i	0.351407i	−0.984443	+	0.175703i	$0.943780\pi$
$138$	0	0
$139$	20.9559	1.77745	0.888726	−	0.458438i	$-0.151591\pi$
$139$	20.9559	1.77745	0.888726	+	0.458438i	$0.151591\pi$
$140$	0	0
$141$	12.9071	1.08698
$142$	0	0
$143$	36.0045i	3.01085i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.86287i	0.153647i
$148$	0	0
$149$	−10.8781	−0.891171	−0.445585	−	0.895239i	$-0.647004\pi$
$149$	−10.8781	−0.891171	−0.445585	+	0.895239i	$0.647004\pi$
$150$	0	0
$151$	−15.9530	−1.29824	−0.649120	−	0.760686i	$-0.724863\pi$
$151$	−15.9530	−1.29824	−0.649120	+	0.760686i	$0.724863\pi$
$152$	0	0
$153$	1.61803i	0.130810i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.8744i	1.74576i	0.487931	+	0.872882i	$0.337752\pi$
$157$	21.8744i	1.74576i	−0.487931	+	0.872882i	$0.662248\pi$
$158$	0	0
$159$	−0.419089	−0.0332359
$160$	0	0
$161$	10.7028	0.843498
$162$	0	0
$163$	− 18.1432i	− 1.42109i	−0.703654	−	0.710543i	$-0.748449\pi$
$163$	− 18.1432i	− 1.42109i	0.703654	−	0.710543i	$-0.251551\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.18910i	0.401545i	0.979638	+	0.200772i	$0.0643452\pi$
$167$	5.18910i	0.401545i	−0.979638	+	0.200772i	$0.935655\pi$
$168$	0	0
$169$	−28.4094	−2.18534
$170$	0	0
$171$	3.07599	0.235227
$172$	0	0
$173$	− 18.9771i	− 1.44280i	−0.692519	−	0.721399i	$-0.743499\pi$
$173$	− 18.9771i	− 1.44280i	0.692519	−	0.721399i	$-0.256501\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 5.54379i	− 0.416697i
$178$	0	0
$179$	−11.9744	−0.895007	−0.447503	−	0.894282i	$-0.647687\pi$
$179$	−11.9744	−0.895007	−0.447503	+	0.894282i	$0.647687\pi$
$180$	0	0
$181$	11.5881	0.861334	0.430667	−	0.902511i	$-0.358278\pi$
$181$	11.5881	0.861334	0.430667	+	0.902511i	$0.358278\pi$
$182$	0	0
$183$	11.8471i	0.875761i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.05305i	0.662024i
$188$	0	0
$189$	2.97706	0.216549
$190$	0	0
$191$	12.8322	0.928508	0.464254	−	0.885702i	$-0.346322\pi$
$191$	12.8322	0.928508	0.464254	+	0.885702i	$0.346322\pi$
$192$	0	0
$193$	9.15024i	0.658648i	0.944217	+	0.329324i	$0.106821\pi$
$193$	9.15024i	0.658648i	−0.944217	+	0.329324i	$0.893179\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.45187i	0.602171i	0.953597	+	0.301086i	$0.0973490\pi$
$197$	8.45187i	0.602171i	−0.953597	+	0.301086i	$0.902651\pi$
$198$	0	0
$199$	−18.0231	−1.27762	−0.638811	−	0.769364i	$-0.720574\pi$
$199$	−18.0231	−1.27762	−0.638811	+	0.769364i	$0.720574\pi$
$200$	0	0
$201$	6.91525	0.487764
$202$	0	0
$203$	21.4317i	1.50421i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 3.59509i	− 0.249876i
$208$	0	0
$209$	17.2104	1.19047
$210$	0	0
$211$	−7.11245	−0.489641	−0.244821	−	0.969568i	$-0.578729\pi$
$211$	−7.11245	−0.489641	−0.244821	+	0.969568i	$0.578729\pi$
$212$	0	0
$213$	− 6.81698i	− 0.467092i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.9514i	1.15074i
$218$	0	0
$219$	3.33500	0.225359
$220$	0	0
$221$	−10.4121	−0.700392
$222$	0	0
$223$	− 24.7383i	− 1.65660i	−0.560285	−	0.828300i	$-0.689308\pi$
$223$	− 24.7383i	− 1.65660i	0.560285	−	0.828300i	$-0.310692\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.2581i	1.34458i	0.740290	+	0.672288i	$0.234688\pi$
$227$	20.2581i	1.34458i	−0.740290	+	0.672288i	$0.765312\pi$
$228$	0	0
$229$	−2.20002	−0.145382	−0.0726908	−	0.997355i	$-0.523159\pi$
$229$	−2.20002	−0.145382	−0.0726908	+	0.997355i	$0.523159\pi$
$230$	0	0
$231$	16.6569	1.09594
$232$	0	0
$233$	− 1.55189i	− 0.101667i	−0.998707	−	0.0508337i	$-0.983812\pi$
$233$	− 1.55189i	− 0.101667i	0.998707	−	0.0508337i	$-0.0161878\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.30329i	0.409443i
$238$	0	0
$239$	−5.98690	−0.387260	−0.193630	−	0.981075i	$-0.562026\pi$
$239$	−5.98690	−0.387260	−0.193630	+	0.981075i	$0.562026\pi$
$240$	0	0
$241$	−17.4368	−1.12320	−0.561600	−	0.827409i	$-0.689814\pi$
$241$	−17.4368	−1.12320	−0.561600	+	0.827409i	$0.689814\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	19.7940i	1.25946i
$248$	0	0
$249$	−8.45903	−0.536069
$250$	0	0
$251$	−15.4580	−0.975698	−0.487849	−	0.872928i	$-0.662218\pi$
$251$	−15.4580	−0.975698	−0.487849	+	0.872928i	$0.662218\pi$
$252$	0	0
$253$	− 20.1149i	− 1.26461i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.74881i	− 0.608114i	−0.952654	−	0.304057i	$-0.901659\pi$
$257$	− 9.74881i	− 0.608114i	0.952654	−	0.304057i	$-0.0983414\pi$
$258$	0	0
$259$	−27.6120	−1.71573
$260$	0	0
$261$	7.19894	0.445603
$262$	0	0
$263$	7.81590i	0.481949i	0.970531	+	0.240975i	$0.0774670\pi$
$263$	7.81590i	0.481949i	−0.970531	+	0.240975i	$0.922533\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.31206i	− 0.141496i
$268$	0	0
$269$	−14.3640	−0.875790	−0.437895	−	0.899026i	$-0.644276\pi$
$269$	−14.3640	−0.875790	−0.437895	+	0.899026i	$0.644276\pi$
$270$	0	0
$271$	7.68418	0.466781	0.233390	−	0.972383i	$-0.425018\pi$
$271$	7.68418	0.466781	0.233390	+	0.972383i	$0.425018\pi$
$272$	0	0
$273$	19.1574i	1.15946i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 4.92307i	− 0.295799i	−0.989002	−	0.147899i	$-0.952749\pi$
$277$	− 4.92307i	− 0.295799i	0.989002	−	0.147899i	$-0.0472512\pi$
$278$	0	0
$279$	5.69402	0.340892
$280$	0	0
$281$	−15.1371	−0.903006	−0.451503	−	0.892270i	$-0.649112\pi$
$281$	−15.1371	−0.903006	−0.451503	+	0.892270i	$0.649112\pi$
$282$	0	0
$283$	20.8914i	1.24186i	0.783865	+	0.620931i	$0.213245\pi$
$283$	20.8914i	1.24186i	−0.783865	+	0.620931i	$0.786755\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 34.0849i	− 2.01197i
$288$	0	0
$289$	14.3820	0.845998
$290$	0	0
$291$	9.44919	0.553921
$292$	0	0
$293$	− 13.2820i	− 0.775940i	−0.921672	−	0.387970i	$-0.873176\pi$
$293$	− 13.2820i	− 0.775940i	0.921672	−	0.387970i	$-0.126824\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.59509i	− 0.324660i
$298$	0	0
$299$	23.1345	1.33790
$300$	0	0
$301$	−0.434347	−0.0250353
$302$	0	0
$303$	− 16.2273i	− 0.932234i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.650819i	0.0371442i	0.999828	+	0.0185721i	$0.00591202\pi$
$307$	0.650819i	0.0371442i	−0.999828	+	0.0185721i	$0.994088\pi$
$308$	0	0
$309$	−10.2990	−0.585887
$310$	0	0
$311$	−25.6128	−1.45237	−0.726183	−	0.687501i	$-0.758708\pi$
$311$	−25.6128	−1.45237	−0.726183	+	0.687501i	$0.758708\pi$
$312$	0	0
$313$	18.8372i	1.06474i	0.846511	+	0.532372i	$0.178699\pi$
$313$	18.8372i	1.06474i	−0.846511	+	0.532372i	$0.821301\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 5.57649i	− 0.313207i	−0.987662	−	0.156603i	$-0.949946\pi$
$317$	− 5.57649i	− 0.313207i	0.987662	−	0.156603i	$-0.0500544\pi$
$318$	0	0
$319$	40.2787	2.25518
$320$	0	0
$321$	−1.10877	−0.0618858
$322$	0	0
$323$	4.97706i	0.276931i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 15.1821i	− 0.839571i
$328$	0	0
$329$	38.4253	2.11846
$330$	0	0
$331$	13.3088	0.731518	0.365759	−	0.930710i	$-0.380810\pi$
$331$	13.3088	0.731518	0.365759	+	0.930710i	$0.380810\pi$
$332$	0	0
$333$	9.27493i	0.508263i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.2122i	0.883134i	0.897228	+	0.441567i	$0.145577\pi$
$337$	16.2122i	0.883134i	−0.897228	+	0.441567i	$0.854423\pi$
$338$	0	0
$339$	−5.48090	−0.297682
$340$	0	0
$341$	31.8586	1.72524
$342$	0	0
$343$	− 15.2935i	− 0.825774i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7.18651i	− 0.385792i	−0.981219	−	0.192896i	$-0.938212\pi$
$347$	− 7.18651i	− 0.385792i	0.981219	−	0.192896i	$-0.0617880\pi$
$348$	0	0
$349$	−19.6390	−1.05125	−0.525625	−	0.850717i	$-0.676168\pi$
$349$	−19.6390	−1.05125	−0.525625	+	0.850717i	$0.676168\pi$
$350$	0	0
$351$	6.43501	0.343476
$352$	0	0
$353$	− 10.6099i	− 0.564710i	−0.959310	−	0.282355i	$-0.908884\pi$
$353$	− 10.6099i	− 0.564710i	0.959310	−	0.282355i	$-0.0911156\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.81698i	0.254942i
$358$	0	0
$359$	−30.3040	−1.59938	−0.799691	−	0.600412i	$-0.795003\pi$
$359$	−30.3040	−1.59938	−0.799691	+	0.600412i	$0.795003\pi$
$360$	0	0
$361$	−9.53828	−0.502015
$362$	0	0
$363$	− 20.3050i	− 1.06574i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.16549i	− 0.0608383i	−0.999537	−	0.0304191i	$-0.990316\pi$
$367$	− 1.16549i	− 0.0608383i	0.999537	−	0.0304191i	$-0.00968421\pi$
$368$	0	0
$369$	−11.4492	−0.596021
$370$	0	0
$371$	−1.24765	−0.0647749
$372$	0	0
$373$	− 2.49400i	− 0.129135i	−0.997913	−	0.0645673i	$-0.979433\pi$
$373$	− 2.49400i	− 0.129135i	0.997913	−	0.0645673i	$-0.0205667\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	46.3253i	2.38587i
$378$	0	0
$379$	1.83492	0.0942534	0.0471267	−	0.998889i	$-0.484994\pi$
$379$	1.83492	0.0942534	0.0471267	+	0.998889i	$0.484994\pi$
$380$	0	0
$381$	3.39615	0.173990
$382$	0	0
$383$	− 16.0646i	− 0.820864i	−0.911891	−	0.410432i	$-0.865378\pi$
$383$	− 16.0646i	− 0.820864i	0.911891	−	0.410432i	$-0.134622\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.145898i	0.00741641i
$388$	0	0
$389$	25.1203	1.27365	0.636824	−	0.771009i	$-0.280248\pi$
$389$	25.1203	1.27365	0.636824	+	0.771009i	$0.280248\pi$
$390$	0	0
$391$	5.81698	0.294177
$392$	0	0
$393$	1.05130i	0.0530312i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	17.6798i	0.887326i	0.896194	+	0.443663i	$0.146321\pi$
$397$	17.6798i	0.887326i	−0.896194	+	0.443663i	$0.853679\pi$
$398$	0	0
$399$	9.15740	0.458443
$400$	0	0
$401$	2.43219	0.121458	0.0607289	−	0.998154i	$-0.480657\pi$
$401$	2.43219	0.121458	0.0607289	+	0.998154i	$0.480657\pi$
$402$	0	0
$403$	36.6411i	1.82522i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	51.8941i	2.57230i
$408$	0	0
$409$	20.3509	1.00629	0.503144	−	0.864202i	$-0.332176\pi$
$409$	20.3509	1.00629	0.503144	+	0.864202i	$0.332176\pi$
$410$	0	0
$411$	4.11311	0.202885
$412$	0	0
$413$	− 16.5042i	− 0.812117i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 20.9559i	− 1.02621i
$418$	0	0
$419$	24.0158	1.17325	0.586625	−	0.809859i	$-0.300456\pi$
$419$	24.0158	1.17325	0.586625	+	0.809859i	$0.300456\pi$
$420$	0	0
$421$	−32.6241	−1.59000	−0.795001	−	0.606608i	$-0.792530\pi$
$421$	−32.6241	−1.59000	−0.795001	+	0.606608i	$0.792530\pi$
$422$	0	0
$423$	− 12.9071i	− 0.627567i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	35.2694i	1.70681i
$428$	0	0
$429$	36.0045	1.73831
$430$	0	0
$431$	8.72399	0.420220	0.210110	−	0.977678i	$-0.432618\pi$
$431$	8.72399	0.420220	0.210110	+	0.977678i	$0.432618\pi$
$432$	0	0
$433$	33.2307i	1.59697i	0.602018	+	0.798483i	$0.294364\pi$
$433$	33.2307i	1.59697i	−0.602018	+	0.798483i	$0.705636\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 11.0585i	− 0.528998i
$438$	0	0
$439$	22.1581	1.05755	0.528773	−	0.848763i	$-0.322652\pi$
$439$	22.1581	1.05755	0.528773	+	0.848763i	$0.322652\pi$
$440$	0	0
$441$	1.86287	0.0887079
$442$	0	0
$443$	− 2.95085i	− 0.140199i	−0.997540	−	0.0700996i	$-0.977668\pi$
$443$	− 2.95085i	− 0.140199i	0.997540	−	0.0700996i	$-0.0223317\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.8781i	0.514518i
$448$	0	0
$449$	−7.48698	−0.353333	−0.176666	−	0.984271i	$-0.556531\pi$
$449$	−7.48698	−0.353333	−0.176666	+	0.984271i	$0.556531\pi$
$450$	0	0
$451$	−64.0593	−3.01643
$452$	0	0
$453$	15.9530i	0.749539i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.7258i	1.34374i	0.740671	+	0.671868i	$0.234508\pi$
$457$	28.7258i	1.34374i	−0.740671	+	0.671868i	$0.765492\pi$
$458$	0	0
$459$	1.61803	0.0755234
$460$	0	0
$461$	−16.0034	−0.745353	−0.372676	−	0.927961i	$-0.621560\pi$
$461$	−16.0034	−0.745353	−0.372676	+	0.927961i	$0.621560\pi$
$462$	0	0
$463$	− 16.4852i	− 0.766134i	−0.923721	−	0.383067i	$-0.874868\pi$
$463$	− 16.4852i	− 0.766134i	0.923721	−	0.383067i	$-0.125132\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.68700i	0.355712i	0.984057	+	0.177856i	$0.0569161\pi$
$467$	7.68700i	0.355712i	−0.984057	+	0.177856i	$0.943084\pi$
$468$	0	0
$469$	20.5871	0.950623
$470$	0	0
$471$	21.8744	1.00792
$472$	0	0
$473$	0.816313i	0.0375341i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.419089i	0.0191888i
$478$	0	0
$479$	16.0318	0.732514	0.366257	−	0.930514i	$-0.380639\pi$
$479$	16.0318	0.732514	0.366257	+	0.930514i	$0.380639\pi$
$480$	0	0
$481$	−59.6843	−2.72137
$482$	0	0
$483$	− 10.7028i	− 0.486994i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	18.6066i	0.843145i	0.906795	+	0.421573i	$0.138522\pi$
$487$	18.6066i	0.843145i	−0.906795	+	0.421573i	$0.861478\pi$
$488$	0	0
$489$	−18.1432	−0.820465
$490$	0	0
$491$	−3.62788	−0.163724	−0.0818619	−	0.996644i	$-0.526087\pi$
$491$	−3.62788	−0.163724	−0.0818619	+	0.996644i	$0.526087\pi$
$492$	0	0
$493$	11.6481i	0.524606i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 20.2945i	− 0.910334i
$498$	0	0
$499$	9.10435	0.407567	0.203783	−	0.979016i	$-0.434676\pi$
$499$	9.10435	0.407567	0.203783	+	0.979016i	$0.434676\pi$
$500$	0	0
$501$	5.18910	0.231832
$502$	0	0
$503$	1.80714i	0.0805763i	0.999188	+	0.0402881i	$0.0128276\pi$
$503$	1.80714i	0.0805763i	−0.999188	+	0.0402881i	$0.987172\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	28.4094i	1.26171i
$508$	0	0
$509$	−25.9576	−1.15055	−0.575275	−	0.817960i	$-0.695105\pi$
$509$	−25.9576	−1.15055	−0.575275	+	0.817960i	$0.695105\pi$
$510$	0	0
$511$	9.92849	0.439210
$512$	0	0
$513$	− 3.07599i	− 0.135808i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 72.2167i	− 3.17609i
$518$	0	0
$519$	−18.9771	−0.833000
$520$	0	0
$521$	−35.4236	−1.55193	−0.775967	−	0.630773i	$-0.782738\pi$
$521$	−35.4236	−1.55193	−0.775967	+	0.630773i	$0.782738\pi$
$522$	0	0
$523$	23.7691i	1.03935i	0.854364	+	0.519675i	$0.173947\pi$
$523$	23.7691i	1.03935i	−0.854364	+	0.519675i	$0.826053\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.21312i	0.401330i
$528$	0	0
$529$	10.0753	0.438058
$530$	0	0
$531$	−5.54379	−0.240580
$532$	0	0
$533$	− 73.6757i	− 3.19125i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11.9744i	0.516732i
$538$	0	0
$539$	10.4229	0.448946
$540$	0	0
$541$	0.931170	0.0400341	0.0200171	−	0.999800i	$-0.493628\pi$
$541$	0.931170	0.0400341	0.0200171	+	0.999800i	$0.493628\pi$
$542$	0	0
$543$	− 11.5881i	− 0.497292i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 28.6968i	− 1.22698i	−0.789701	−	0.613492i	$-0.789764\pi$
$547$	− 28.6968i	− 1.22698i	0.789701	−	0.613492i	$-0.210236\pi$
$548$	0	0
$549$	11.8471	0.505621
$550$	0	0
$551$	22.1439	0.943361
$552$	0	0
$553$	18.7653i	0.797980i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.5273i	1.20874i	0.796703	+	0.604371i	$0.206576\pi$
$557$	28.5273i	1.20874i	−0.796703	+	0.604371i	$0.793424\pi$
$558$	0	0
$559$	−0.938856	−0.0397094
$560$	0	0
$561$	9.05305	0.382220
$562$	0	0
$563$	35.6411i	1.50209i	0.660249	+	0.751047i	$0.270451\pi$
$563$	35.6411i	1.50209i	−0.660249	+	0.751047i	$0.729549\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.97706i	− 0.125025i
$568$	0	0
$569$	−28.8728	−1.21041	−0.605205	−	0.796070i	$-0.706909\pi$
$569$	−28.8728	−1.21041	−0.605205	+	0.796070i	$0.706909\pi$
$570$	0	0
$571$	16.7217	0.699782	0.349891	−	0.936790i	$-0.386219\pi$
$571$	16.7217	0.699782	0.349891	+	0.936790i	$0.386219\pi$
$572$	0	0
$573$	− 12.8322i	− 0.536074i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.5022i	0.478844i	0.970916	+	0.239422i	$0.0769580\pi$
$577$	11.5022i	0.478844i	−0.970916	+	0.239422i	$0.923042\pi$
$578$	0	0
$579$	9.15024	0.380271
$580$	0	0
$581$	−25.1830	−1.04477
$582$	0	0
$583$	2.34484i	0.0971135i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.5701i	0.931569i	0.884898	+	0.465785i	$0.154228\pi$
$587$	22.5701i	0.931569i	−0.884898	+	0.465785i	$0.845772\pi$
$588$	0	0
$589$	17.5148	0.721683
$590$	0	0
$591$	8.45187	0.347664
$592$	0	0
$593$	47.2237i	1.93924i	0.244608	+	0.969622i	$0.421341\pi$
$593$	47.2237i	1.93924i	−0.244608	+	0.969622i	$0.578659\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	18.0231i	0.737636i
$598$	0	0
$599$	−2.79672	−0.114271	−0.0571354	−	0.998366i	$-0.518197\pi$
$599$	−2.79672	−0.114271	−0.0571354	+	0.998366i	$0.518197\pi$
$600$	0	0
$601$	−14.2935	−0.583042	−0.291521	−	0.956564i	$-0.594161\pi$
$601$	−14.2935	−0.583042	−0.291521	+	0.956564i	$0.594161\pi$
$602$	0	0
$603$	− 6.91525i	− 0.281611i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	12.8876i	0.523090i	0.965191	+	0.261545i	$0.0842319\pi$
$607$	12.8876i	0.523090i	−0.965191	+	0.261545i	$0.915768\pi$
$608$	0	0
$609$	21.4317	0.868455
$610$	0	0
$611$	83.0577	3.36015
$612$	0	0
$613$	− 14.1722i	− 0.572411i	−0.958168	−	0.286206i	$-0.907606\pi$
$613$	− 14.1722i	− 0.572411i	0.958168	−	0.286206i	$-0.0923941\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 10.7558i	− 0.433014i	−0.976281	−	0.216507i	$-0.930534\pi$
$617$	− 10.7558i	− 0.433014i	0.976281	−	0.216507i	$-0.0694663\pi$
$618$	0	0
$619$	−40.2143	−1.61635	−0.808175	−	0.588942i	$-0.799545\pi$
$619$	−40.2143	−1.61635	−0.808175	+	0.588942i	$0.799545\pi$
$620$	0	0
$621$	−3.59509	−0.144266
$622$	0	0
$623$	− 6.88313i	− 0.275767i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 17.2104i	− 0.687319i
$628$	0	0
$629$	−15.0072	−0.598375
$630$	0	0
$631$	3.16710	0.126080	0.0630401	−	0.998011i	$-0.479920\pi$
$631$	3.16710	0.126080	0.0630401	+	0.998011i	$0.479920\pi$
$632$	0	0
$633$	7.11245i	0.282694i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	11.9876i	0.474965i
$638$	0	0
$639$	−6.81698	−0.269676
$640$	0	0
$641$	−8.93559	−0.352935	−0.176467	−	0.984306i	$-0.556467\pi$
$641$	−8.93559	−0.352935	−0.176467	+	0.984306i	$0.556467\pi$
$642$	0	0
$643$	− 17.5289i	− 0.691274i	−0.938368	−	0.345637i	$-0.887663\pi$
$643$	− 17.5289i	− 0.691274i	0.938368	−	0.345637i	$-0.112337\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.0600668i	0.00236147i	0.999999	+	0.00118073i	$0.000375840\pi$
$647$	0.0600668i	0.00236147i	−0.999999	+	0.00118073i	$0.999624\pi$
$648$	0	0
$649$	−31.0180	−1.21756
$650$	0	0
$651$	16.9514	0.664379
$652$	0	0
$653$	− 5.63396i	− 0.220474i	−0.993905	−	0.110237i	$-0.964839\pi$
$653$	− 5.63396i	− 0.220474i	0.993905	−	0.110237i	$-0.0351610\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3.33500i	− 0.130111i
$658$	0	0
$659$	−39.1355	−1.52450	−0.762252	−	0.647280i	$-0.775906\pi$
$659$	−39.1355	−1.52450	−0.762252	+	0.647280i	$0.775906\pi$
$660$	0	0
$661$	−13.8720	−0.539556	−0.269778	−	0.962922i	$-0.586950\pi$
$661$	−13.8720	−0.539556	−0.269778	+	0.962922i	$0.586950\pi$
$662$	0	0
$663$	10.4121i	0.404371i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 25.8809i	− 1.00211i
$668$	0	0
$669$	−24.7383	−0.956438
$670$	0	0
$671$	66.2855	2.55892
$672$	0	0
$673$	− 18.5323i	− 0.714369i	−0.934034	−	0.357185i	$-0.883737\pi$
$673$	− 18.5323i	− 0.714369i	0.934034	−	0.357185i	$-0.116263\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.6361i	1.36961i	0.728728	+	0.684804i	$0.240112\pi$
$677$	35.6361i	1.36961i	−0.728728	+	0.684804i	$0.759888\pi$
$678$	0	0
$679$	28.1308	1.07956
$680$	0	0
$681$	20.2581	0.776291
$682$	0	0
$683$	30.4749i	1.16609i	0.812440	+	0.583044i	$0.198139\pi$
$683$	30.4749i	1.16609i	−0.812440	+	0.583044i	$0.801861\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.20002i	0.0839361i
$688$	0	0
$689$	−2.69685	−0.102742
$690$	0	0
$691$	−16.5208	−0.628479	−0.314240	−	0.949344i	$-0.601750\pi$
$691$	−16.5208	−0.628479	−0.314240	+	0.949344i	$0.601750\pi$
$692$	0	0
$693$	− 16.6569i	− 0.632743i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 18.5252i	− 0.701691i
$698$	0	0
$699$	−1.55189	−0.0586977
$700$	0	0
$701$	6.68728	0.252575	0.126287	−	0.991994i	$-0.459694\pi$
$701$	6.68728	0.252575	0.126287	+	0.991994i	$0.459694\pi$
$702$	0	0
$703$	28.5296i	1.07601i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 48.3096i	− 1.81687i
$708$	0	0
$709$	4.84628	0.182006	0.0910029	−	0.995851i	$-0.470993\pi$
$709$	4.84628	0.182006	0.0910029	+	0.995851i	$0.470993\pi$
$710$	0	0
$711$	6.30329	0.236392
$712$	0	0
$713$	− 20.4705i	− 0.766627i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.98690i	0.223585i
$718$	0	0
$719$	36.6303	1.36608	0.683041	−	0.730380i	$-0.260657\pi$
$719$	36.6303	1.36608	0.683041	+	0.730380i	$0.260657\pi$
$720$	0	0
$721$	−30.6606	−1.14186
$722$	0	0
$723$	17.4368i	0.648480i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 2.36143i	− 0.0875807i	−0.999041	−	0.0437903i	$-0.986057\pi$
$727$	− 2.36143i	− 0.0875807i	0.999041	−	0.0437903i	$-0.0139434\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−0.236068	−0.00873129
$732$	0	0
$733$	3.19126i	0.117872i	0.998262	+	0.0589359i	$0.0187708\pi$
$733$	3.19126i	0.117872i	−0.998262	+	0.0589359i	$0.981229\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 38.6914i	− 1.42522i
$738$	0	0
$739$	−53.3412	−1.96219	−0.981094	−	0.193530i	$-0.938006\pi$
$739$	−53.3412	−1.96219	−0.981094	+	0.193530i	$0.938006\pi$
$740$	0	0
$741$	19.7940	0.727152
$742$	0	0
$743$	37.2497i	1.36656i	0.730157	+	0.683280i	$0.239447\pi$
$743$	37.2497i	1.36656i	−0.730157	+	0.683280i	$0.760553\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.45903i	0.309500i
$748$	0	0
$749$	−3.30089	−0.120612
$750$	0	0
$751$	−39.5540	−1.44334	−0.721672	−	0.692235i	$-0.756626\pi$
$751$	−39.5540	−1.44334	−0.721672	+	0.692235i	$0.756626\pi$
$752$	0	0
$753$	15.4580i	0.563319i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 17.0039i	− 0.618017i	−0.951059	−	0.309009i	$-0.900003\pi$
$757$	− 17.0039i	− 0.618017i	0.951059	−	0.309009i	$-0.0999972\pi$
$758$	0	0
$759$	−20.1149	−0.730123
$760$	0	0
$761$	40.6662	1.47415	0.737075	−	0.675811i	$-0.236207\pi$
$761$	40.6662	1.47415	0.737075	+	0.675811i	$0.236207\pi$
$762$	0	0
$763$	− 45.1979i	− 1.63627i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 35.6743i	− 1.28813i
$768$	0	0
$769$	−4.31197	−0.155494	−0.0777468	−	0.996973i	$-0.524773\pi$
$769$	−4.31197	−0.155494	−0.0777468	+	0.996973i	$0.524773\pi$
$770$	0	0
$771$	−9.74881	−0.351095
$772$	0	0
$773$	− 21.1022i	− 0.758993i	−0.925193	−	0.379497i	$-0.876097\pi$
$773$	− 21.1022i	− 0.758993i	0.925193	−	0.379497i	$-0.123903\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	27.6120i	0.990575i
$778$	0	0
$779$	−35.2176	−1.26180
$780$	0	0
$781$	−38.1416	−1.36481
$782$	0	0
$783$	− 7.19894i	− 0.257269i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 35.6296i	− 1.27006i	−0.772488	−	0.635029i	$-0.780988\pi$
$787$	− 35.6296i	− 1.27006i	0.772488	−	0.635029i	$-0.219012\pi$
$788$	0	0
$789$	7.81590	0.278253
$790$	0	0
$791$	−16.3169	−0.580164
$792$	0	0
$793$	76.2361i	2.70722i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 35.1353i	− 1.24455i	−0.782797	−	0.622277i	$-0.786208\pi$
$797$	− 35.1353i	− 1.24455i	0.782797	−	0.622277i	$-0.213792\pi$
$798$	0	0
$799$	20.8842	0.738830
$800$	0	0
$801$	−2.31206	−0.0816926
$802$	0	0
$803$	− 18.6596i	− 0.658484i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.3640i	0.505638i
$808$	0	0
$809$	0.0908359	0.00319362	0.00159681	−	0.999999i	$-0.499492\pi$
$809$	0.0908359	0.00319362	0.00159681	+	0.999999i	$0.499492\pi$
$810$	0	0
$811$	3.09125	0.108548	0.0542742	−	0.998526i	$-0.482715\pi$
$811$	3.09125	0.108548	0.0542742	+	0.998526i	$0.482715\pi$
$812$	0	0
$813$	− 7.68418i	− 0.269496i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0.448781i	0.0157009i
$818$	0	0
$819$	19.1574	0.669414
$820$	0	0
$821$	26.8171	0.935924	0.467962	−	0.883749i	$-0.344988\pi$
$821$	26.8171	0.935924	0.467962	+	0.883749i	$0.344988\pi$
$822$	0	0
$823$	− 8.66133i	− 0.301915i	−0.988540	−	0.150957i	$-0.951764\pi$
$823$	− 8.66133i	− 0.301915i	0.988540	−	0.150957i	$-0.0482356\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.9171i	1.90966i	0.297160	+	0.954828i	$0.403961\pi$
$827$	54.9171i	1.90966i	−0.297160	+	0.954828i	$0.596039\pi$
$828$	0	0
$829$	−27.2963	−0.948039	−0.474019	−	0.880514i	$-0.657197\pi$
$829$	−27.2963	−0.948039	−0.474019	+	0.880514i	$0.657197\pi$
$830$	0	0
$831$	−4.92307	−0.170779
$832$	0	0
$833$	3.01418i	0.104435i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.69402i	− 0.196814i
$838$	0	0
$839$	26.1547	0.902961	0.451481	−	0.892281i	$-0.350896\pi$
$839$	26.1547	0.902961	0.451481	+	0.892281i	$0.350896\pi$
$840$	0	0
$841$	22.8248	0.787062
$842$	0	0
$843$	15.1371i	0.521351i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 60.4492i	− 2.07706i
$848$	0	0
$849$	20.8914	0.716990
$850$	0	0
$851$	33.3442	1.14303
$852$	0	0
$853$	15.7614i	0.539660i	0.962908	+	0.269830i	$0.0869675\pi$
$853$	15.7614i	0.539660i	−0.962908	+	0.269830i	$0.913033\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 8.76868i	− 0.299532i	−0.988721	−	0.149766i	$-0.952148\pi$
$857$	− 8.76868i	− 0.299532i	0.988721	−	0.149766i	$-0.0478521\pi$
$858$	0	0
$859$	−39.6170	−1.35171	−0.675857	−	0.737032i	$-0.736227\pi$
$859$	−39.6170	−1.35171	−0.675857	+	0.737032i	$0.736227\pi$
$860$	0	0
$861$	−34.0849	−1.16161
$862$	0	0
$863$	− 15.7727i	− 0.536911i	−0.963292	−	0.268455i	$-0.913487\pi$
$863$	− 15.7727i	− 0.536911i	0.963292	−	0.268455i	$-0.0865132\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 14.3820i	− 0.488437i
$868$	0	0
$869$	35.2675	1.19637
$870$	0	0
$871$	44.4997	1.50781
$872$	0	0
$873$	− 9.44919i	− 0.319807i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 52.2422i	− 1.76409i	−0.471162	−	0.882047i	$-0.656165\pi$
$877$	− 52.2422i	− 1.76409i	0.471162	−	0.882047i	$-0.343835\pi$
$878$	0	0
$879$	−13.2820	−0.447989
$880$	0	0
$881$	13.3902	0.451127	0.225564	−	0.974228i	$-0.427578\pi$
$881$	13.3902	0.451127	0.225564	+	0.974228i	$0.427578\pi$
$882$	0	0
$883$	− 57.4075i	− 1.93192i	−0.258698	−	0.965958i	$-0.583293\pi$
$883$	− 57.4075i	− 1.93192i	0.258698	−	0.965958i	$-0.416707\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 36.0967i	− 1.21201i	−0.795462	−	0.606004i	$-0.792772\pi$
$887$	− 36.0967i	− 1.21201i	0.795462	−	0.606004i	$-0.207228\pi$
$888$	0	0
$889$	10.1105	0.339096
$890$	0	0
$891$	−5.59509	−0.187443
$892$	0	0
$893$	− 39.7023i	− 1.32859i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 23.1345i	− 0.772437i
$898$	0	0
$899$	40.9910	1.36713
$900$	0	0
$901$	−0.678101	−0.0225908
$902$	0	0
$903$	0.434347i	0.0144542i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 7.07317i	− 0.234861i	−0.993081	−	0.117430i	$-0.962534\pi$
$907$	− 7.07317i	− 0.234861i	0.993081	−	0.117430i	$-0.0374657\pi$
$908$	0	0
$909$	−16.2273	−0.538226
$910$	0	0
$911$	15.6951	0.520002	0.260001	−	0.965608i	$-0.416277\pi$
$911$	15.6951	0.520002	0.260001	+	0.965608i	$0.416277\pi$
$912$	0	0
$913$	47.3291i	1.56636i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.12979i	0.103355i
$918$	0	0
$919$	20.4509	0.674614	0.337307	−	0.941395i	$-0.390484\pi$
$919$	20.4509	0.674614	0.337307	+	0.941395i	$0.390484\pi$
$920$	0	0
$921$	0.650819	0.0214452
$922$	0	0
$923$	− 43.8673i	− 1.44391i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.2990i	0.338262i
$928$	0	0
$929$	−20.3367	−0.667227	−0.333613	−	0.942710i	$-0.608268\pi$
$929$	−20.3367	−0.667227	−0.333613	+	0.942710i	$0.608268\pi$
$930$	0	0
$931$	5.73016	0.187798
$932$	0	0
$933$	25.6128i	0.838524i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 60.3183i	− 1.97051i	−0.171086	−	0.985256i	$-0.554727\pi$
$937$	− 60.3183i	− 1.97051i	0.171086	−	0.985256i	$-0.445273\pi$
$938$	0	0
$939$	18.8372	0.614730
$940$	0	0
$941$	−6.18625	−0.201666	−0.100833	−	0.994903i	$-0.532151\pi$
$941$	−6.18625	−0.201666	−0.100833	+	0.994903i	$0.532151\pi$
$942$	0	0
$943$	41.1609i	1.34038i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 32.3809i	− 1.05224i	−0.850411	−	0.526120i	$-0.823646\pi$
$947$	− 32.3809i	− 1.05224i	0.850411	−	0.526120i	$-0.176354\pi$
$948$	0	0
$949$	21.4608	0.696646
$950$	0	0
$951$	−5.57649	−0.180830
$952$	0	0
$953$	38.2037i	1.23754i	0.785573	+	0.618770i	$0.212369\pi$
$953$	38.2037i	1.23754i	−0.785573	+	0.618770i	$0.787631\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 40.2787i	− 1.30203i
$958$	0	0
$959$	12.2450	0.395411
$960$	0	0
$961$	1.42191	0.0458681
$962$	0	0
$963$	1.10877i	0.0357298i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 31.7778i	− 1.02191i	−0.859609	−	0.510953i	$-0.829293\pi$
$967$	− 31.7778i	− 1.02191i	0.859609	−	0.510953i	$-0.170707\pi$
$968$	0	0
$969$	4.97706	0.159886
$970$	0	0
$971$	−0.870691	−0.0279418	−0.0139709	−	0.999902i	$-0.504447\pi$
$971$	−0.870691	−0.0279418	−0.0139709	+	0.999902i	$0.504447\pi$
$972$	0	0
$973$	− 62.3868i	− 2.00003i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 14.3197i	− 0.458128i	−0.973411	−	0.229064i	$-0.926433\pi$
$977$	− 14.3197i	− 0.458128i	0.973411	−	0.229064i	$-0.0735666\pi$
$978$	0	0
$979$	−12.9362	−0.413442
$980$	0	0
$981$	−15.1821	−0.484727
$982$	0	0
$983$	12.4302i	0.396461i	0.980155	+	0.198231i	$0.0635195\pi$
$983$	12.4302i	0.396461i	−0.980155	+	0.198231i	$0.936481\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 38.4253i	− 1.22309i
$988$	0	0
$989$	0.524517	0.0166787
$990$	0	0
$991$	−27.1963	−0.863918	−0.431959	−	0.901893i	$-0.642177\pi$
$991$	−27.1963	−0.863918	−0.431959	+	0.901893i	$0.642177\pi$
$992$	0	0
$993$	− 13.3088i	− 0.422342i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	47.1334i	1.49273i	0.665537	+	0.746364i	$0.268202\pi$
$997$	47.1334i	1.49273i	−0.665537	+	0.746364i	$0.731798\pi$
$998$	0	0
$999$	9.27493	0.293446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6000.2.f.p.1249.1		8
4.3	odd	2		3000.2.f.h.1249.8		8
5.2	odd	4		6000.2.a.bc.1.4		4
5.3	odd	4		6000.2.a.bl.1.1		4
5.4	even	2	inner	6000.2.f.p.1249.8		8
20.3	even	4		3000.2.a.j.1.4	✓	4
20.7	even	4		3000.2.a.o.1.1	yes	4
20.19	odd	2		3000.2.f.h.1249.1		8
60.23	odd	4		9000.2.a.t.1.4		4
60.47	odd	4		9000.2.a.y.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3000.2.a.j.1.4	✓	4	20.3	even	4
3000.2.a.o.1.1	yes	4	20.7	even	4
3000.2.f.h.1249.1		8	20.19	odd	2
3000.2.f.h.1249.8		8	4.3	odd	2
6000.2.a.bc.1.4		4	5.2	odd	4
6000.2.a.bl.1.1		4	5.3	odd	4
6000.2.f.p.1249.1		8	1.1	even	1	trivial
6000.2.f.p.1249.8		8	5.4	even	2	inner
9000.2.a.t.1.4		4	60.23	odd	4
9000.2.a.y.1.1		4	60.47	odd	4

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.97706i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -2.97706i q^{7} -1.00000 q^{9} -5.59509 q^{11} -6.43501i q^{13} -1.61803i q^{17} -3.07599 q^{19} -2.97706 q^{21} +3.59509i q^{23} +1.00000i q^{27} -7.19894 q^{29} -5.69402 q^{31} +5.59509i q^{33} -9.27493i q^{37} -6.43501 q^{39} +11.4492 q^{41} -0.145898i q^{43} +12.9071i q^{47} -1.86287 q^{49} -1.61803 q^{51} -0.419089i q^{53} +3.07599i q^{57} +5.54379 q^{59} -11.8471 q^{61} +2.97706i q^{63} +6.91525i q^{67} +3.59509 q^{69} +6.81698 q^{71} +3.33500i q^{73} +16.6569i q^{77} -6.30329 q^{79} +1.00000 q^{81} -8.45903i q^{83} +7.19894i q^{87} +2.31206 q^{89} -19.1574 q^{91} +5.69402i q^{93} +9.44919i q^{97} +5.59509 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 6 q^{11} + 6 q^{21} - 30 q^{29} - 12 q^{31} - 6 q^{39} + 26 q^{41} - 38 q^{49} - 4 q^{51} + 16 q^{59} + 26 q^{61} - 10 q^{69} + 18 q^{71} + 42 q^{79} + 8 q^{81} - 24 q^{89} - 34 q^{91} + 6 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 - 6 * q^11 + 6 * q^21 - 30 * q^29 - 12 * q^31 - 6 * q^39 + 26 * q^41 - 38 * q^49 - 4 * q^51 + 16 * q^59 + 26 * q^61 - 10 * q^69 + 18 * q^71 + 42 * q^79 + 8 * q^81 - 24 * q^89 - 34 * q^91 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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