LMFDB - Embedded newform 1152.4.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,4,Mod(1,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1152.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1152.a (trivial)

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:	yes
Analytic conductor:	$67.9702003266$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.4764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x - 6 $ x^3 - x^2 - 12*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.66160$ of defining polynomial
Character	$\chi$	$=$	1152.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.64640 q^{5} -4.98287 q^{7} +31.2585 q^{11} -46.5513 q^{13} +29.2243 q^{17} -87.1370 q^{19} -38.6541 q^{23} -50.2397 q^{25} -131.095 q^{29} +137.291 q^{31} -43.0839 q^{35} -51.9281 q^{37} -226.395 q^{41} +16.7260 q^{43} +110.031 q^{47} -318.171 q^{49} -142.198 q^{53} +270.274 q^{55} +547.476 q^{59} -9.17456 q^{61} -402.502 q^{65} +22.1369 q^{67} -1160.82 q^{71} +317.171 q^{73} -155.757 q^{77} -958.078 q^{79} +207.022 q^{83} +252.685 q^{85} -97.3529 q^{89} +231.959 q^{91} -753.421 q^{95} +1213.71 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 10 q^{5} + 6 q^{7} - 20 q^{11} + 46 q^{13} - 68 q^{17} + 68 q^{19} - 56 q^{23} + 53 q^{25} - 46 q^{29} - 226 q^{31} + 332 q^{35} + 66 q^{37} - 236 q^{41} + 212 q^{43} - 760 q^{47} + 327 q^{49} - 702 q^{53}+ \cdots - 1222 q^{97}+O(q^{100}) $ 3 * q - 10 * q^5 + 6 * q^7 - 20 * q^11 + 46 * q^13 - 68 * q^17 + 68 * q^19 - 56 * q^23 + 53 * q^25 - 46 * q^29 - 226 * q^31 + 332 * q^35 + 66 * q^37 - 236 * q^41 + 212 * q^43 - 760 * q^47 + 327 * q^49 - 702 * q^53 + 152 * q^55 + 600 * q^59 + 122 * q^61 - 1028 * q^65 - 760 * q^67 - 512 * q^71 - 330 * q^73 - 2024 * q^77 - 218 * q^79 + 2340 * q^83 - 392 * q^85 - 2616 * q^89 + 1372 * q^91 - 2680 * q^95 - 1222 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	8.64640	0.773358	0.386679	−	0.922214i	$-0.373622\pi$
$5$	8.64640	0.773358	0.386679	+	0.922214i	$0.373622\pi$
$6$	0	0
$7$	−4.98287	−0.269050	−0.134525	−	0.990910i	$-0.542951\pi$
$7$	−4.98287	−0.269050	−0.134525	+	0.990910i	$0.542951\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	31.2585	0.856800	0.428400	−	0.903589i	$-0.359077\pi$
$11$	31.2585	0.856800	0.428400	+	0.903589i	$0.359077\pi$
$12$	0	0
$13$	−46.5513	−0.993155	−0.496578	−	0.867992i	$-0.665410\pi$
$13$	−46.5513	−0.993155	−0.496578	+	0.867992i	$0.665410\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	29.2243	0.416937	0.208468	−	0.978029i	$-0.433152\pi$
$17$	29.2243	0.416937	0.208468	+	0.978029i	$0.433152\pi$
$18$	0	0
$19$	−87.1370	−1.05214	−0.526068	−	0.850442i	$-0.676334\pi$
$19$	−87.1370	−1.05214	−0.526068	+	0.850442i	$0.676334\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−38.6541	−0.350432	−0.175216	−	0.984530i	$-0.556062\pi$
$23$	−38.6541	−0.350432	−0.175216	+	0.984530i	$0.556062\pi$
$24$	0	0
$25$	−50.2397	−0.401918
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−131.095	−0.839439	−0.419719	−	0.907654i	$-0.637872\pi$
$29$	−131.095	−0.839439	−0.419719	+	0.907654i	$0.637872\pi$
$30$	0	0
$31$	137.291	0.795426	0.397713	−	0.917510i	$-0.369804\pi$
$31$	137.291	0.795426	0.397713	+	0.917510i	$0.369804\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−43.0839	−0.208072
$36$	0	0
$37$	−51.9281	−0.230728	−0.115364	−	0.993323i	$-0.536803\pi$
$37$	−51.9281	−0.230728	−0.115364	+	0.993323i	$0.536803\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−226.395	−0.862367	−0.431183	−	0.902264i	$-0.641904\pi$
$41$	−226.395	−0.862367	−0.431183	+	0.902264i	$0.641904\pi$
$42$	0	0
$43$	16.7260	0.0593184	0.0296592	−	0.999560i	$-0.490558\pi$
$43$	16.7260	0.0593184	0.0296592	+	0.999560i	$0.490558\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	110.031	0.341481	0.170741	−	0.985316i	$-0.445384\pi$
$47$	110.031	0.341481	0.170741	+	0.985316i	$0.445384\pi$
$48$	0	0
$49$	−318.171	−0.927612
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−142.198	−0.368535	−0.184268	−	0.982876i	$-0.558991\pi$
$53$	−142.198	−0.368535	−0.184268	+	0.982876i	$0.558991\pi$
$54$	0	0
$55$	270.274	0.662613
$56$	0	0
$57$	0	0
$58$	0	0
$59$	547.476	1.20806	0.604028	−	0.796963i	$-0.293561\pi$
$59$	547.476	1.20806	0.604028	+	0.796963i	$0.293561\pi$
$60$	0	0
$61$	−9.17456	−0.0192571	−0.00962854	−	0.999954i	$-0.503065\pi$
$61$	−9.17456	−0.0192571	−0.00962854	+	0.999954i	$0.503065\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−402.502	−0.768064
$66$	0	0
$67$	22.1369	0.0403649	0.0201824	−	0.999796i	$-0.493575\pi$
$67$	22.1369	0.0403649	0.0201824	+	0.999796i	$0.493575\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1160.82	−1.94034	−0.970170	−	0.242425i	$-0.922057\pi$
$71$	−1160.82	−1.94034	−0.970170	+	0.242425i	$0.922057\pi$
$72$	0	0
$73$	317.171	0.508521	0.254261	−	0.967136i	$-0.418168\pi$
$73$	317.171	0.508521	0.254261	+	0.967136i	$0.418168\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−155.757	−0.230522
$78$	0	0
$79$	−958.078	−1.36446	−0.682229	−	0.731138i	$-0.738989\pi$
$79$	−958.078	−1.36446	−0.682229	+	0.731138i	$0.738989\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	207.022	0.273779	0.136890	−	0.990586i	$-0.456289\pi$
$83$	207.022	0.273779	0.136890	+	0.990586i	$0.456289\pi$
$84$	0	0
$85$	252.685	0.322441
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−97.3529	−0.115948	−0.0579741	−	0.998318i	$-0.518464\pi$
$89$	−97.3529	−0.115948	−0.0579741	+	0.998318i	$0.518464\pi$
$90$	0	0
$91$	231.959	0.267208
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−753.421	−0.813678
$96$	0	0
$97$	1213.71	1.27045	0.635226	−	0.772327i	$-0.280907\pi$
$97$	1213.71	1.27045	0.635226	+	0.772327i	$0.280907\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1370.20	−1.34990	−0.674949	−	0.737864i	$-0.735834\pi$
$101$	−1370.20	−1.34990	−0.674949	+	0.737864i	$0.735834\pi$
$102$	0	0
$103$	149.956	0.143452	0.0717260	−	0.997424i	$-0.477149\pi$
$103$	149.956	0.143452	0.0717260	+	0.997424i	$0.477149\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1285.89	1.16179	0.580897	−	0.813977i	$-0.302702\pi$
$107$	1285.89	1.16179	0.580897	+	0.813977i	$0.302702\pi$
$108$	0	0
$109$	165.859	0.145747	0.0728736	−	0.997341i	$-0.476783\pi$
$109$	165.859	0.145747	0.0728736	+	0.997341i	$0.476783\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1870.37	−1.55708	−0.778539	−	0.627596i	$-0.784039\pi$
$113$	−1870.37	−1.55708	−0.778539	+	0.627596i	$0.784039\pi$
$114$	0	0
$115$	−334.219	−0.271010
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−145.621	−0.112177
$120$	0	0
$121$	−353.904	−0.265893
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1515.19	−1.08418
$126$	0	0
$127$	−688.949	−0.481373	−0.240686	−	0.970603i	$-0.577372\pi$
$127$	−688.949	−0.481373	−0.240686	+	0.970603i	$0.577372\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1536.00	1.02443	0.512217	−	0.858856i	$-0.328824\pi$
$131$	1536.00	1.02443	0.512217	+	0.858856i	$0.328824\pi$
$132$	0	0
$133$	434.192	0.283077
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2656.41	−1.65658	−0.828292	−	0.560296i	$-0.810687\pi$
$137$	−2656.41	−1.65658	−0.828292	+	0.560296i	$0.810687\pi$
$138$	0	0
$139$	2581.45	1.57522	0.787611	−	0.616173i	$-0.211318\pi$
$139$	2581.45	1.57522	0.787611	+	0.616173i	$0.211318\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1455.13	−0.850936
$144$	0	0
$145$	−1133.50	−0.649186
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1679.60	−0.923476	−0.461738	−	0.887016i	$-0.652774\pi$
$149$	−1679.60	−0.923476	−0.461738	+	0.887016i	$0.652774\pi$
$150$	0	0
$151$	976.195	0.526104	0.263052	−	0.964782i	$-0.415271\pi$
$151$	976.195	0.526104	0.263052	+	0.964782i	$0.415271\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1187.07	0.615149
$156$	0	0
$157$	988.907	0.502697	0.251348	−	0.967897i	$-0.419126\pi$
$157$	988.907	0.502697	0.251348	+	0.967897i	$0.419126\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	192.608	0.0942837
$162$	0	0
$163$	−2698.42	−1.29667	−0.648334	−	0.761356i	$-0.724534\pi$
$163$	−2698.42	−1.29667	−0.648334	+	0.761356i	$0.724534\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2629.01	−1.21820	−0.609098	−	0.793095i	$-0.708469\pi$
$167$	−2629.01	−1.21820	−0.609098	+	0.793095i	$0.708469\pi$
$168$	0	0
$169$	−29.9723	−0.0136424
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2874.29	−1.26317	−0.631584	−	0.775307i	$-0.717595\pi$
$173$	−2874.29	−1.26317	−0.631584	+	0.775307i	$0.717595\pi$
$174$	0	0
$175$	250.338	0.108136
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3631.19	1.51624	0.758122	−	0.652113i	$-0.226117\pi$
$179$	3631.19	1.51624	0.758122	+	0.652113i	$0.226117\pi$
$180$	0	0
$181$	−2284.59	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$181$	−2284.59	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−448.991	−0.178435
$186$	0	0
$187$	913.508	0.357232
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4074.14	−1.54343	−0.771713	−	0.635971i	$-0.780600\pi$
$191$	−4074.14	−1.54343	−0.771713	+	0.635971i	$0.780600\pi$
$192$	0	0
$193$	−764.596	−0.285165	−0.142582	−	0.989783i	$-0.545541\pi$
$193$	−764.596	−0.285165	−0.142582	+	0.989783i	$0.545541\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	146.623	0.0530277	0.0265138	−	0.999648i	$-0.491559\pi$
$197$	146.623	0.0530277	0.0265138	+	0.999648i	$0.491559\pi$
$198$	0	0
$199$	4183.78	1.49035	0.745176	−	0.666868i	$-0.232365\pi$
$199$	4183.78	1.49035	0.745176	+	0.666868i	$0.232365\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	653.229	0.225851
$204$	0	0
$205$	−1957.51	−0.666918
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2723.77	−0.901471
$210$	0	0
$211$	−4844.19	−1.58051	−0.790256	−	0.612777i	$-0.790052\pi$
$211$	−4844.19	−1.58051	−0.790256	+	0.612777i	$0.790052\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	144.620	0.0458743
$216$	0	0
$217$	−684.103	−0.214009
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1360.43	−0.414083
$222$	0	0
$223$	−6057.62	−1.81905	−0.909525	−	0.415649i	$-0.863554\pi$
$223$	−6057.62	−1.81905	−0.909525	+	0.415649i	$0.863554\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3417.50	0.999239	0.499620	−	0.866245i	$-0.333473\pi$
$227$	3417.50	0.999239	0.499620	+	0.866245i	$0.333473\pi$
$228$	0	0
$229$	−170.401	−0.0491720	−0.0245860	−	0.999698i	$-0.507827\pi$
$229$	−170.401	−0.0491720	−0.0245860	+	0.999698i	$0.507827\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6047.01	−1.70023	−0.850114	−	0.526599i	$-0.823467\pi$
$233$	−6047.01	−1.70023	−0.850114	+	0.526599i	$0.823467\pi$
$234$	0	0
$235$	951.370	0.264087
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2623.08	−0.709928	−0.354964	−	0.934880i	$-0.615507\pi$
$239$	−2623.08	−0.709928	−0.354964	+	0.934880i	$0.615507\pi$
$240$	0	0
$241$	2398.24	0.641013	0.320507	−	0.947246i	$-0.396147\pi$
$241$	2398.24	0.641013	0.320507	+	0.947246i	$0.396147\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2751.03	−0.717376
$246$	0	0
$247$	4056.34	1.04493
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1028.10	−0.258539	−0.129270	−	0.991609i	$-0.541263\pi$
$251$	−1028.10	−0.258539	−0.129270	+	0.991609i	$0.541263\pi$
$252$	0	0
$253$	−1208.27	−0.300251
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2966.66	−0.720058	−0.360029	−	0.932941i	$-0.617233\pi$
$257$	−2966.66	−0.720058	−0.360029	+	0.932941i	$0.617233\pi$
$258$	0	0
$259$	258.751	0.0620773
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1571.29	0.368402	0.184201	−	0.982889i	$-0.441030\pi$
$263$	1571.29	0.368402	0.184201	+	0.982889i	$0.441030\pi$
$264$	0	0
$265$	−1229.50	−0.285010
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−824.889	−0.186968	−0.0934839	−	0.995621i	$-0.529800\pi$
$269$	−824.889	−0.186968	−0.0934839	+	0.995621i	$0.529800\pi$
$270$	0	0
$271$	−4037.24	−0.904962	−0.452481	−	0.891774i	$-0.649461\pi$
$271$	−4037.24	−0.904962	−0.452481	+	0.891774i	$0.649461\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1570.42	−0.344363
$276$	0	0
$277$	4162.23	0.902830	0.451415	−	0.892314i	$-0.350919\pi$
$277$	4162.23	0.902830	0.451415	+	0.892314i	$0.350919\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5749.45	1.22058	0.610291	−	0.792178i	$-0.291053\pi$
$281$	5749.45	1.22058	0.610291	+	0.792178i	$0.291053\pi$
$282$	0	0
$283$	5401.34	1.13455	0.567273	−	0.823530i	$-0.307998\pi$
$283$	5401.34	1.13455	0.567273	+	0.823530i	$0.307998\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1128.10	0.232019
$288$	0	0
$289$	−4058.94	−0.826164
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4166.93	−0.830835	−0.415417	−	0.909631i	$-0.636364\pi$
$293$	−4166.93	−0.830835	−0.415417	+	0.909631i	$0.636364\pi$
$294$	0	0
$295$	4733.70	0.934259
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1799.40	0.348034
$300$	0	0
$301$	−83.3435	−0.0159596
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−79.3269	−0.0148926
$306$	0	0
$307$	−3302.85	−0.614018	−0.307009	−	0.951707i	$-0.599328\pi$
$307$	−3302.85	−0.614018	−0.307009	+	0.951707i	$0.599328\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5910.07	−1.07759	−0.538793	−	0.842438i	$-0.681120\pi$
$311$	−5910.07	−1.07759	−0.538793	+	0.842438i	$0.681120\pi$
$312$	0	0
$313$	5164.96	0.932719	0.466359	−	0.884595i	$-0.345565\pi$
$313$	5164.96	0.932719	0.466359	+	0.884595i	$0.345565\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3892.83	0.689726	0.344863	−	0.938653i	$-0.387925\pi$
$317$	3892.83	0.689726	0.344863	+	0.938653i	$0.387925\pi$
$318$	0	0
$319$	−4097.84	−0.719231
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2546.51	−0.438674
$324$	0	0
$325$	2338.73	0.399167
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−548.269	−0.0918755
$330$	0	0
$331$	11839.0	1.96595	0.982973	−	0.183752i	$-0.0588243\pi$
$331$	11839.0	1.96595	0.982973	+	0.183752i	$0.0588243\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	191.404	0.0312165
$336$	0	0
$337$	4362.10	0.705100	0.352550	−	0.935793i	$-0.385315\pi$
$337$	4362.10	0.705100	0.352550	+	0.935793i	$0.385315\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4291.52	0.681521
$342$	0	0
$343$	3294.53	0.518623
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5615.61	0.868766	0.434383	−	0.900728i	$-0.356966\pi$
$347$	5615.61	0.868766	0.434383	+	0.900728i	$0.356966\pi$
$348$	0	0
$349$	11024.7	1.69094	0.845468	−	0.534027i	$-0.179322\pi$
$349$	11024.7	1.69094	0.845468	+	0.534027i	$0.179322\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5004.03	0.754497	0.377249	−	0.926112i	$-0.376870\pi$
$353$	5004.03	0.754497	0.377249	+	0.926112i	$0.376870\pi$
$354$	0	0
$355$	−10036.9	−1.50058
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6283.22	0.923720	0.461860	−	0.886953i	$-0.347182\pi$
$359$	6283.22	0.923720	0.461860	+	0.886953i	$0.347182\pi$
$360$	0	0
$361$	733.848	0.106991
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2742.39	0.393269
$366$	0	0
$367$	−10006.5	−1.42325	−0.711627	−	0.702557i	$-0.752042\pi$
$367$	−10006.5	−1.42325	−0.711627	+	0.702557i	$0.752042\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	708.553	0.0991543
$372$	0	0
$373$	−6825.46	−0.947477	−0.473738	−	0.880666i	$-0.657096\pi$
$373$	−6825.46	−0.947477	−0.473738	+	0.880666i	$0.657096\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6102.65	0.833693
$378$	0	0
$379$	5566.81	0.754479	0.377240	−	0.926116i	$-0.376873\pi$
$379$	5566.81	0.754479	0.377240	+	0.926116i	$0.376873\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11103.5	1.48137	0.740683	−	0.671855i	$-0.234502\pi$
$383$	11103.5	1.48137	0.740683	+	0.671855i	$0.234502\pi$
$384$	0	0
$385$	−1346.74	−0.178276
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6661.90	0.868308	0.434154	−	0.900839i	$-0.357047\pi$
$389$	6661.90	0.868308	0.434154	+	0.900839i	$0.357047\pi$
$390$	0	0
$391$	−1129.64	−0.146108
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8283.93	−1.05521
$396$	0	0
$397$	−10836.2	−1.36990	−0.684951	−	0.728589i	$-0.740176\pi$
$397$	−10836.2	−1.36990	−0.684951	+	0.728589i	$0.740176\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	636.112	0.0792167	0.0396084	−	0.999215i	$-0.487389\pi$
$401$	636.112	0.0792167	0.0396084	+	0.999215i	$0.487389\pi$
$402$	0	0
$403$	−6391.08	−0.789981
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1623.20	−0.197688
$408$	0	0
$409$	−600.642	−0.0726157	−0.0363079	−	0.999341i	$-0.511560\pi$
$409$	−600.642	−0.0726157	−0.0363079	+	0.999341i	$0.511560\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2728.00	−0.325027
$414$	0	0
$415$	1790.00	0.211729
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11467.2	−1.33701	−0.668506	−	0.743707i	$-0.733066\pi$
$419$	−11467.2	−1.33701	−0.668506	+	0.743707i	$0.733066\pi$
$420$	0	0
$421$	1389.82	0.160892	0.0804462	−	0.996759i	$-0.474365\pi$
$421$	1389.82	0.160892	0.0804462	+	0.996759i	$0.474365\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1468.22	−0.167574
$426$	0	0
$427$	45.7156	0.00518111
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8972.88	1.00280	0.501402	−	0.865214i	$-0.332818\pi$
$431$	8972.88	1.00280	0.501402	+	0.865214i	$0.332818\pi$
$432$	0	0
$433$	766.999	0.0851262	0.0425631	−	0.999094i	$-0.486448\pi$
$433$	766.999	0.0851262	0.0425631	+	0.999094i	$0.486448\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3368.20	0.368703
$438$	0	0
$439$	9856.76	1.07161	0.535806	−	0.844341i	$-0.320008\pi$
$439$	9856.76	1.07161	0.535806	+	0.844341i	$0.320008\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9206.36	−0.987375	−0.493688	−	0.869639i	$-0.664351\pi$
$443$	−9206.36	−0.987375	−0.493688	+	0.869639i	$0.664351\pi$
$444$	0	0
$445$	−841.753	−0.0896694
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13424.9	1.41105	0.705524	−	0.708686i	$-0.250712\pi$
$449$	13424.9	1.41105	0.705524	+	0.708686i	$0.250712\pi$
$450$	0	0
$451$	−7076.79	−0.738876
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2005.61	0.206647
$456$	0	0
$457$	−5114.83	−0.523549	−0.261774	−	0.965129i	$-0.584308\pi$
$457$	−5114.83	−0.523549	−0.261774	+	0.965129i	$0.584308\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5927.65	0.598868	0.299434	−	0.954117i	$-0.403202\pi$
$461$	5927.65	0.598868	0.299434	+	0.954117i	$0.403202\pi$
$462$	0	0
$463$	−7364.01	−0.739168	−0.369584	−	0.929197i	$-0.620500\pi$
$463$	−7364.01	−0.739168	−0.369584	+	0.929197i	$0.620500\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17521.4	−1.73618	−0.868089	−	0.496408i	$-0.834652\pi$
$467$	−17521.4	−1.73618	−0.868089	+	0.496408i	$0.834652\pi$
$468$	0	0
$469$	−110.305	−0.0108602
$470$	0	0
$471$	0	0
$472$	0	0
$473$	522.830	0.0508240
$474$	0	0
$475$	4377.74	0.422872
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3411.89	0.325456	0.162728	−	0.986671i	$-0.447971\pi$
$479$	3411.89	0.325456	0.162728	+	0.986671i	$0.447971\pi$
$480$	0	0
$481$	2417.32	0.229149
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10494.2	0.982513
$486$	0	0
$487$	13943.4	1.29741	0.648704	−	0.761041i	$-0.275311\pi$
$487$	13943.4	1.29741	0.648704	+	0.761041i	$0.275311\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7307.60	0.671665	0.335832	−	0.941922i	$-0.390982\pi$
$491$	7307.60	0.671665	0.335832	+	0.941922i	$0.390982\pi$
$492$	0	0
$493$	−3831.16	−0.349993
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5784.22	0.522048
$498$	0	0
$499$	702.653	0.0630362	0.0315181	−	0.999503i	$-0.489966\pi$
$499$	702.653	0.0630362	0.0315181	+	0.999503i	$0.489966\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9917.71	0.879142	0.439571	−	0.898208i	$-0.355130\pi$
$503$	9917.71	0.879142	0.439571	+	0.898208i	$0.355130\pi$
$504$	0	0
$505$	−11847.3	−1.04395
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10111.7	0.880538	0.440269	−	0.897866i	$-0.354883\pi$
$509$	10111.7	0.880538	0.440269	+	0.897866i	$0.354883\pi$
$510$	0	0
$511$	−1580.42	−0.136817
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1296.58	0.110940
$516$	0	0
$517$	3439.40	0.292581
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16432.4	1.38180	0.690898	−	0.722952i	$-0.257215\pi$
$521$	16432.4	1.38180	0.690898	+	0.722952i	$0.257215\pi$
$522$	0	0
$523$	−5373.49	−0.449266	−0.224633	−	0.974443i	$-0.572118\pi$
$523$	−5373.49	−0.449266	−0.224633	+	0.974443i	$0.572118\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4012.23	0.331642
$528$	0	0
$529$	−10672.9	−0.877197
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10539.0	0.856464
$534$	0	0
$535$	11118.4	0.898483
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−9945.56	−0.794779
$540$	0	0
$541$	14302.1	1.13659	0.568295	−	0.822825i	$-0.307603\pi$
$541$	14302.1	1.13659	0.568295	+	0.822825i	$0.307603\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1434.09	0.112715
$546$	0	0
$547$	−1109.60	−0.0867335	−0.0433668	−	0.999059i	$-0.513808\pi$
$547$	−1109.60	−0.0867335	−0.0433668	+	0.999059i	$0.513808\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11423.2	0.883204
$552$	0	0
$553$	4773.98	0.367107
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20183.7	1.53539	0.767695	−	0.640816i	$-0.221404\pi$
$557$	20183.7	1.53539	0.767695	+	0.640816i	$0.221404\pi$
$558$	0	0
$559$	−778.618	−0.0589124
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3804.02	0.284761	0.142380	−	0.989812i	$-0.454524\pi$
$563$	3804.02	0.284761	0.142380	+	0.989812i	$0.454524\pi$
$564$	0	0
$565$	−16172.0	−1.20418
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6866.43	0.505898	0.252949	−	0.967480i	$-0.418600\pi$
$569$	6866.43	0.505898	0.252949	+	0.967480i	$0.418600\pi$
$570$	0	0
$571$	3856.38	0.282635	0.141317	−	0.989964i	$-0.454866\pi$
$571$	3856.38	0.282635	0.141317	+	0.989964i	$0.454866\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1941.97	0.140845
$576$	0	0
$577$	7348.06	0.530162	0.265081	−	0.964226i	$-0.414601\pi$
$577$	7348.06	0.530162	0.265081	+	0.964226i	$0.414601\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1031.57	−0.0736602
$582$	0	0
$583$	−4444.90	−0.315761
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21465.6	−1.50934	−0.754670	−	0.656105i	$-0.772203\pi$
$587$	−21465.6	−1.50934	−0.754670	+	0.656105i	$0.772203\pi$
$588$	0	0
$589$	−11963.1	−0.836896
$590$	0	0
$591$	0	0
$592$	0	0
$593$	11309.2	0.783156	0.391578	−	0.920145i	$-0.371929\pi$
$593$	11309.2	0.783156	0.391578	+	0.920145i	$0.371929\pi$
$594$	0	0
$595$	−1259.10	−0.0867527
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7950.33	0.542307	0.271153	−	0.962536i	$-0.412595\pi$
$599$	7950.33	0.542307	0.271153	+	0.962536i	$0.412595\pi$
$600$	0	0
$601$	10642.7	0.722338	0.361169	−	0.932500i	$-0.382378\pi$
$601$	10642.7	0.722338	0.361169	+	0.932500i	$0.382378\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3059.99	−0.205630
$606$	0	0
$607$	−13655.1	−0.913085	−0.456542	−	0.889702i	$-0.650912\pi$
$607$	−13655.1	−0.913085	−0.456542	+	0.889702i	$0.650912\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5122.08	−0.339144
$612$	0	0
$613$	8987.49	0.592172	0.296086	−	0.955161i	$-0.404318\pi$
$613$	8987.49	0.592172	0.296086	+	0.955161i	$0.404318\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12086.4	0.788624	0.394312	−	0.918977i	$-0.370983\pi$
$617$	12086.4	0.788624	0.394312	+	0.918977i	$0.370983\pi$
$618$	0	0
$619$	6815.67	0.442561	0.221280	−	0.975210i	$-0.428976\pi$
$619$	6815.67	0.442561	0.221280	+	0.975210i	$0.428976\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	485.097	0.0311958
$624$	0	0
$625$	−6821.00	−0.436544
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1517.56	−0.0961990
$630$	0	0
$631$	−24313.6	−1.53393	−0.766966	−	0.641688i	$-0.778234\pi$
$631$	−24313.6	−1.53393	−0.766966	+	0.641688i	$0.778234\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5956.93	−0.372273
$636$	0	0
$637$	14811.3	0.921263
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12727.5	−0.784251	−0.392125	−	0.919912i	$-0.628260\pi$
$641$	−12727.5	−0.784251	−0.392125	+	0.919912i	$0.628260\pi$
$642$	0	0
$643$	5323.52	0.326500	0.163250	−	0.986585i	$-0.447802\pi$
$643$	5323.52	0.326500	0.163250	+	0.986585i	$0.447802\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8083.11	0.491159	0.245579	−	0.969376i	$-0.421022\pi$
$647$	8083.11	0.491159	0.245579	+	0.969376i	$0.421022\pi$
$648$	0	0
$649$	17113.3	1.03506
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16001.4	−0.958934	−0.479467	−	0.877560i	$-0.659170\pi$
$653$	−16001.4	−0.958934	−0.479467	+	0.877560i	$0.659170\pi$
$654$	0	0
$655$	13280.9	0.792254
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.23917	0.000487030 0	0.000243515	−	1.00000i	$-0.499922\pi$
$659$	8.23917	0.000487030 0	0.000243515		1.00000i	$0.499922\pi$
$660$	0	0
$661$	5401.14	0.317821	0.158911	−	0.987293i	$-0.449202\pi$
$661$	5401.14	0.317821	0.158911	+	0.987293i	$0.449202\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3754.20	0.218920
$666$	0	0
$667$	5067.36	0.294166
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−286.783	−0.0164995
$672$	0	0
$673$	15105.6	0.865196	0.432598	−	0.901587i	$-0.357597\pi$
$673$	15105.6	0.865196	0.432598	+	0.901587i	$0.357597\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10985.6	−0.623648	−0.311824	−	0.950140i	$-0.600940\pi$
$677$	−10985.6	−0.623648	−0.311824	+	0.950140i	$0.600940\pi$
$678$	0	0
$679$	−6047.77	−0.341814
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5609.36	−0.314255	−0.157128	−	0.987578i	$-0.550223\pi$
$683$	−5609.36	−0.314255	−0.157128	+	0.987578i	$0.550223\pi$
$684$	0	0
$685$	−22968.4	−1.28113
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6619.50	0.366013
$690$	0	0
$691$	−9919.62	−0.546107	−0.273054	−	0.961999i	$-0.588034\pi$
$691$	−9919.62	−0.546107	−0.273054	+	0.961999i	$0.588034\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	22320.3	1.21821
$696$	0	0
$697$	−6616.24	−0.359553
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30526.1	1.64473	0.822365	−	0.568960i	$-0.192654\pi$
$701$	30526.1	1.64473	0.822365	+	0.568960i	$0.192654\pi$
$702$	0	0
$703$	4524.86	0.242757
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6827.51	0.363190
$708$	0	0
$709$	16212.0	0.858750	0.429375	−	0.903126i	$-0.358734\pi$
$709$	16212.0	0.858750	0.429375	+	0.903126i	$0.358734\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5306.87	−0.278743
$714$	0	0
$715$	−12581.6	−0.658078
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10848.8	0.562715	0.281357	−	0.959603i	$-0.409215\pi$
$719$	10848.8	0.562715	0.281357	+	0.959603i	$0.409215\pi$
$720$	0	0
$721$	−747.209	−0.0385957
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6586.18	0.337385
$726$	0	0
$727$	−9476.80	−0.483460	−0.241730	−	0.970344i	$-0.577715\pi$
$727$	−9476.80	−0.483460	−0.241730	+	0.970344i	$0.577715\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	488.805	0.0247320
$732$	0	0
$733$	30420.7	1.53290	0.766449	−	0.642305i	$-0.222022\pi$
$733$	30420.7	1.53290	0.766449	+	0.642305i	$0.222022\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	691.966	0.0345846
$738$	0	0
$739$	21172.7	1.05392	0.526961	−	0.849889i	$-0.323331\pi$
$739$	21172.7	1.05392	0.526961	+	0.849889i	$0.323331\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−35683.8	−1.76193	−0.880964	−	0.473183i	$-0.843105\pi$
$743$	−35683.8	−1.76193	−0.880964	+	0.473183i	$0.843105\pi$
$744$	0	0
$745$	−14522.5	−0.714177
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6407.44	−0.312580
$750$	0	0
$751$	22059.5	1.07186	0.535928	−	0.844264i	$-0.319962\pi$
$751$	22059.5	1.07186	0.535928	+	0.844264i	$0.319962\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8440.58	0.406866
$756$	0	0
$757$	184.774	0.00887149	0.00443575	−	0.999990i	$-0.498588\pi$
$757$	184.774	0.00887149	0.00443575	+	0.999990i	$0.498588\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−33873.0	−1.61353	−0.806764	−	0.590873i	$-0.798783\pi$
$761$	−33873.0	−1.61353	−0.806764	+	0.590873i	$0.798783\pi$
$762$	0	0
$763$	−826.456	−0.0392132
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−25485.7	−1.19979
$768$	0	0
$769$	−2115.36	−0.0991961	−0.0495980	−	0.998769i	$-0.515794\pi$
$769$	−2115.36	−0.0991961	−0.0495980	+	0.998769i	$0.515794\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	29141.3	1.35594	0.677969	−	0.735090i	$-0.262860\pi$
$773$	29141.3	1.35594	0.677969	+	0.735090i	$0.262860\pi$
$774$	0	0
$775$	−6897.47	−0.319696
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19727.4	0.907327
$780$	0	0
$781$	−36285.6	−1.66248
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8550.49	0.388764
$786$	0	0
$787$	36766.6	1.66530	0.832648	−	0.553803i	$-0.186824\pi$
$787$	36766.6	1.66530	0.832648	+	0.553803i	$0.186824\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9319.82	0.418931
$792$	0	0
$793$	427.088	0.0191253
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−37670.1	−1.67421	−0.837104	−	0.547044i	$-0.815753\pi$
$797$	−37670.1	−1.67421	−0.837104	+	0.547044i	$0.815753\pi$
$798$	0	0
$799$	3215.57	0.142376
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9914.30	0.435701
$804$	0	0
$805$	1665.37	0.0729150
$806$	0	0
$807$	0	0
$808$	0	0
$809$	28007.6	1.21718	0.608588	−	0.793487i	$-0.291736\pi$
$809$	28007.6	1.21718	0.608588	+	0.793487i	$0.291736\pi$
$810$	0	0
$811$	−9076.18	−0.392981	−0.196491	−	0.980506i	$-0.562955\pi$
$811$	−9076.18	−0.392981	−0.196491	+	0.980506i	$0.562955\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−23331.7	−1.00279
$816$	0	0
$817$	−1457.45	−0.0624110
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−20371.6	−0.865985	−0.432992	−	0.901398i	$-0.642542\pi$
$821$	−20371.6	−0.865985	−0.432992	+	0.901398i	$0.642542\pi$
$822$	0	0
$823$	10244.3	0.433892	0.216946	−	0.976184i	$-0.430390\pi$
$823$	10244.3	0.433892	0.216946	+	0.976184i	$0.430390\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−402.047	−0.0169051	−0.00845257	−	0.999964i	$-0.502691\pi$
$827$	−402.047	−0.0169051	−0.00845257	+	0.999964i	$0.502691\pi$
$828$	0	0
$829$	34019.0	1.42525	0.712623	−	0.701548i	$-0.247507\pi$
$829$	34019.0	1.42525	0.712623	+	0.701548i	$0.247507\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9298.32	−0.386756
$834$	0	0
$835$	−22731.5	−0.942102
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−13859.3	−0.570293	−0.285146	−	0.958484i	$-0.592042\pi$
$839$	−13859.3	−0.570293	−0.285146	+	0.958484i	$0.592042\pi$
$840$	0	0
$841$	−7203.11	−0.295343
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−259.153	−0.0105504
$846$	0	0
$847$	1763.46	0.0715384
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2007.24	0.0808545
$852$	0	0
$853$	−32174.9	−1.29150	−0.645748	−	0.763550i	$-0.723454\pi$
$853$	−32174.9	−1.29150	−0.645748	+	0.763550i	$0.723454\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−37549.3	−1.49669	−0.748344	−	0.663311i	$-0.769151\pi$
$857$	−37549.3	−1.49669	−0.748344	+	0.663311i	$0.769151\pi$
$858$	0	0
$859$	−42689.8	−1.69564	−0.847821	−	0.530283i	$-0.822086\pi$
$859$	−42689.8	−1.69564	−0.847821	+	0.530283i	$0.822086\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2022.74	0.0797856	0.0398928	−	0.999204i	$-0.487298\pi$
$863$	2022.74	0.0797856	0.0398928	+	0.999204i	$0.487298\pi$
$864$	0	0
$865$	−24852.2	−0.976881
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−29948.1	−1.16907
$870$	0	0
$871$	−1030.50	−0.0400886
$872$	0	0
$873$	0	0
$874$	0	0
$875$	7550.01	0.291699
$876$	0	0
$877$	−27663.2	−1.06513	−0.532565	−	0.846389i	$-0.678772\pi$
$877$	−27663.2	−1.06513	−0.532565	+	0.846389i	$0.678772\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−17663.5	−0.675482	−0.337741	−	0.941239i	$-0.609663\pi$
$881$	−17663.5	−0.675482	−0.337741	+	0.941239i	$0.609663\pi$
$882$	0	0
$883$	23255.2	0.886298	0.443149	−	0.896448i	$-0.353861\pi$
$883$	23255.2	0.886298	0.443149	+	0.896448i	$0.353861\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32189.4	−1.21851	−0.609253	−	0.792976i	$-0.708530\pi$
$887$	−32189.4	−1.21851	−0.609253	+	0.792976i	$0.708530\pi$
$888$	0	0
$889$	3432.94	0.129513
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9587.74	−0.359285
$894$	0	0
$895$	31396.7	1.17260
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−17998.2	−0.667711
$900$	0	0
$901$	−4155.63	−0.153656
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19753.5	−0.725556
$906$	0	0
$907$	−39926.1	−1.46166	−0.730829	−	0.682561i	$-0.760866\pi$
$907$	−39926.1	−1.46166	−0.730829	+	0.682561i	$0.760866\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2444.86	0.0889152	0.0444576	−	0.999011i	$-0.485844\pi$
$911$	2444.86	0.0889152	0.0444576	+	0.999011i	$0.485844\pi$
$912$	0	0
$913$	6471.22	0.234574
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7653.69	−0.275624
$918$	0	0
$919$	−33333.1	−1.19647	−0.598236	−	0.801320i	$-0.704132\pi$
$919$	−33333.1	−1.19647	−0.598236	+	0.801320i	$0.704132\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	54037.8	1.92706
$924$	0	0
$925$	2608.86	0.0927337
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−28429.6	−1.00403	−0.502016	−	0.864858i	$-0.667408\pi$
$929$	−28429.6	−1.00403	−0.502016	+	0.864858i	$0.667408\pi$
$930$	0	0
$931$	27724.5	0.975974
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7898.56	0.276268
$936$	0	0
$937$	17367.5	0.605518	0.302759	−	0.953067i	$-0.402092\pi$
$937$	17367.5	0.605518	0.302759	+	0.953067i	$0.402092\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−29272.3	−1.01408	−0.507040	−	0.861922i	$-0.669260\pi$
$941$	−29272.3	−1.01408	−0.507040	+	0.861922i	$0.669260\pi$
$942$	0	0
$943$	8751.12	0.302201
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12340.7	0.423461	0.211731	−	0.977328i	$-0.432090\pi$
$947$	12340.7	0.423461	0.211731	+	0.977328i	$0.432090\pi$
$948$	0	0
$949$	−14764.7	−0.505041
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5581.81	−0.189730	−0.0948650	−	0.995490i	$-0.530242\pi$
$953$	−5581.81	−0.189730	−0.0948650	+	0.995490i	$0.530242\pi$
$954$	0	0
$955$	−35226.6	−1.19362
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13236.5	0.445703
$960$	0	0
$961$	−10942.2	−0.367298
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6611.00	−0.220534
$966$	0	0
$967$	−34140.3	−1.13534	−0.567672	−	0.823255i	$-0.692156\pi$
$967$	−34140.3	−1.13534	−0.567672	+	0.823255i	$0.692156\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3902.45	0.128976	0.0644880	−	0.997918i	$-0.479459\pi$
$971$	3902.45	0.128976	0.0644880	+	0.997918i	$0.479459\pi$
$972$	0	0
$973$	−12863.0	−0.423813
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−39016.9	−1.27765	−0.638824	−	0.769353i	$-0.720579\pi$
$977$	−39016.9	−1.27765	−0.638824	+	0.769353i	$0.720579\pi$
$978$	0	0
$979$	−3043.11	−0.0993445
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−48792.0	−1.58314	−0.791569	−	0.611080i	$-0.790735\pi$
$983$	−48792.0	−1.58314	−0.791569	+	0.611080i	$0.790735\pi$
$984$	0	0
$985$	1267.76	0.0410093
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−646.529	−0.0207871
$990$	0	0
$991$	44042.7	1.41177	0.705885	−	0.708327i	$-0.250550\pi$
$991$	44042.7	1.41177	0.705885	+	0.708327i	$0.250550\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	36174.6	1.15258
$996$	0	0
$997$	25930.8	0.823706	0.411853	−	0.911250i	$-0.364882\pi$
$997$	25930.8	0.823706	0.411853	+	0.911250i	$0.364882\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.4.a.bb.1.3	yes	3
3.2	odd	2		1152.4.a.bf.1.1	yes	3
4.3	odd	2		1152.4.a.z.1.3	yes	3
8.3	odd	2		1152.4.a.bc.1.1	yes	3
8.5	even	2		1152.4.a.be.1.1	yes	3
12.11	even	2		1152.4.a.bd.1.1	yes	3
24.5	odd	2		1152.4.a.ba.1.3	yes	3
24.11	even	2		1152.4.a.y.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.4.a.y.1.3	✓	3	24.11	even	2
1152.4.a.z.1.3	yes	3	4.3	odd	2
1152.4.a.ba.1.3	yes	3	24.5	odd	2
1152.4.a.bb.1.3	yes	3	1.1	even	1	trivial
1152.4.a.bc.1.1	yes	3	8.3	odd	2
1152.4.a.bd.1.1	yes	3	12.11	even	2
1152.4.a.be.1.1	yes	3	8.5	even	2
1152.4.a.bf.1.1	yes	3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1152.a (trivial)

\(f(q)\)	\(=\)	\(q+8.64640 q^{5} -4.98287 q^{7} +31.2585 q^{11} -46.5513 q^{13} +29.2243 q^{17} -87.1370 q^{19} -38.6541 q^{23} -50.2397 q^{25} -131.095 q^{29} +137.291 q^{31} -43.0839 q^{35} -51.9281 q^{37} -226.395 q^{41} +16.7260 q^{43} +110.031 q^{47} -318.171 q^{49} -142.198 q^{53} +270.274 q^{55} +547.476 q^{59} -9.17456 q^{61} -402.502 q^{65} +22.1369 q^{67} -1160.82 q^{71} +317.171 q^{73} -155.757 q^{77} -958.078 q^{79} +207.022 q^{83} +252.685 q^{85} -97.3529 q^{89} +231.959 q^{91} -753.421 q^{95} +1213.71 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 10 q^{5} + 6 q^{7} - 20 q^{11} + 46 q^{13} - 68 q^{17} + 68 q^{19} - 56 q^{23} + 53 q^{25} - 46 q^{29} - 226 q^{31} + 332 q^{35} + 66 q^{37} - 236 q^{41} + 212 q^{43} - 760 q^{47} + 327 q^{49} - 702 q^{53}+ \cdots - 1222 q^{97}+O(q^{100}) \) 3 * q - 10 * q^5 + 6 * q^7 - 20 * q^11 + 46 * q^13 - 68 * q^17 + 68 * q^19 - 56 * q^23 + 53 * q^25 - 46 * q^29 - 226 * q^31 + 332 * q^35 + 66 * q^37 - 236 * q^41 + 212 * q^43 - 760 * q^47 + 327 * q^49 - 702 * q^53 + 152 * q^55 + 600 * q^59 + 122 * q^61 - 1028 * q^65 - 760 * q^67 - 512 * q^71 - 330 * q^73 - 2024 * q^77 - 218 * q^79 + 2340 * q^83 - 392 * q^85 - 2616 * q^89 + 1372 * q^91 - 2680 * q^95 - 1222 * q^97

Introduction

L-functions

Modular forms

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