LMFDB - Embedded newform 1224.4.a.q.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,4,Mod(1,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1224.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1224 = 2^{3} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1224.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:	$72.2183378470$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 425x^{5} - 474x^{4} + 46988x^{3} + 195600x^{2} - 182556x - 1029768 $ x^7 - 2x^6 - 425x^5 - 474x^4 + 46988x^3 + 195600x^2 - 182556x - 1029768
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 13 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.70652$ of defining polynomial
Character	$\chi$	$=$	1224.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.6859 q^{5} -6.29875 q^{7} +54.0846 q^{11} -74.4441 q^{13} +17.0000 q^{17} -57.4880 q^{19} +89.3410 q^{23} +121.047 q^{25} +186.208 q^{29} -31.6646 q^{31} -98.8014 q^{35} +318.267 q^{37} +338.338 q^{41} +177.552 q^{43} -227.242 q^{47} -303.326 q^{49} -90.8549 q^{53} +848.365 q^{55} +704.687 q^{59} -619.461 q^{61} -1167.72 q^{65} +261.990 q^{67} +70.2650 q^{71} +244.089 q^{73} -340.665 q^{77} -187.386 q^{79} +581.247 q^{83} +266.660 q^{85} +966.375 q^{89} +468.904 q^{91} -901.750 q^{95} -227.941 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{5} + 26 q^{7} + 65 q^{11} + 17 q^{13} + 119 q^{17} + 125 q^{19} - 107 q^{23} + 176 q^{25} + 6 q^{29} + 44 q^{31} + 58 q^{35} + 254 q^{37} - 219 q^{41} + 419 q^{43} - 300 q^{47} + 1171 q^{49}+ \cdots + 2536 q^{97}+O(q^{100}) $ 7 * q + 3 * q^5 + 26 * q^7 + 65 * q^11 + 17 * q^13 + 119 * q^17 + 125 * q^19 - 107 * q^23 + 176 * q^25 + 6 * q^29 + 44 * q^31 + 58 * q^35 + 254 * q^37 - 219 * q^41 + 419 * q^43 - 300 * q^47 + 1171 * q^49 - 284 * q^53 + 633 * q^55 + 46 * q^59 + 638 * q^61 + 185 * q^65 + 796 * q^67 + 1472 * q^71 + 1712 * q^73 - 586 * q^77 + 1134 * q^79 + 596 * q^83 + 51 * q^85 - 694 * q^89 + 3822 * q^91 + 229 * q^95 + 2536 * 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
$5$	15.6859	1.40299	0.701494	−	0.712676i	$-0.252517\pi$
$5$	15.6859	1.40299	0.701494	+	0.712676i	$0.252517\pi$
$6$	0	0
$7$	−6.29875	−0.340100	−0.170050	−	0.985435i	$-0.554393\pi$
$7$	−6.29875	−0.340100	−0.170050	+	0.985435i	$0.554393\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	54.0846	1.48247	0.741233	−	0.671248i	$-0.234241\pi$
$11$	54.0846	1.48247	0.741233	+	0.671248i	$0.234241\pi$
$12$	0	0
$13$	−74.4441	−1.58824	−0.794118	−	0.607764i	$-0.792067\pi$
$13$	−74.4441	−1.58824	−0.794118	+	0.607764i	$0.792067\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	−57.4880	−0.694140	−0.347070	−	0.937839i	$-0.612823\pi$
$19$	−57.4880	−0.694140	−0.347070	+	0.937839i	$0.612823\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	89.3410	0.809951	0.404976	−	0.914327i	$-0.367280\pi$
$23$	89.3410	0.809951	0.404976	+	0.914327i	$0.367280\pi$
$24$	0	0
$25$	121.047	0.968373
$26$	0	0
$27$	0	0
$28$	0	0
$29$	186.208	1.19234	0.596171	−	0.802857i	$-0.296688\pi$
$29$	186.208	1.19234	0.596171	+	0.802857i	$0.296688\pi$
$30$	0	0
$31$	−31.6646	−0.183456	−0.0917279	−	0.995784i	$-0.529239\pi$
$31$	−31.6646	−0.183456	−0.0917279	+	0.995784i	$0.529239\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−98.8014	−0.477156
$36$	0	0
$37$	318.267	1.41413	0.707064	−	0.707150i	$-0.250019\pi$
$37$	318.267	1.41413	0.707064	+	0.707150i	$0.250019\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	338.338	1.28877	0.644385	−	0.764701i	$-0.277113\pi$
$41$	338.338	1.28877	0.644385	+	0.764701i	$0.277113\pi$
$42$	0	0
$43$	177.552	0.629683	0.314842	−	0.949144i	$-0.398049\pi$
$43$	177.552	0.629683	0.314842	+	0.949144i	$0.398049\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−227.242	−0.705246	−0.352623	−	0.935765i	$-0.614710\pi$
$47$	−227.242	−0.705246	−0.352623	+	0.935765i	$0.614710\pi$
$48$	0	0
$49$	−303.326	−0.884332
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−90.8549	−0.235470	−0.117735	−	0.993045i	$-0.537563\pi$
$53$	−90.8549	−0.235470	−0.117735	+	0.993045i	$0.537563\pi$
$54$	0	0
$55$	848.365	2.07988
$56$	0	0
$57$	0	0
$58$	0	0
$59$	704.687	1.55496	0.777478	−	0.628910i	$-0.216499\pi$
$59$	704.687	1.55496	0.777478	+	0.628910i	$0.216499\pi$
$60$	0	0
$61$	−619.461	−1.30023	−0.650113	−	0.759837i	$-0.725279\pi$
$61$	−619.461	−1.30023	−0.650113	+	0.759837i	$0.725279\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1167.72	−2.22827
$66$	0	0
$67$	261.990	0.477718	0.238859	−	0.971054i	$-0.423227\pi$
$67$	261.990	0.477718	0.238859	+	0.971054i	$0.423227\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	70.2650	0.117450	0.0587248	−	0.998274i	$-0.481297\pi$
$71$	70.2650	0.117450	0.0587248	+	0.998274i	$0.481297\pi$
$72$	0	0
$73$	244.089	0.391349	0.195674	−	0.980669i	$-0.437310\pi$
$73$	244.089	0.391349	0.195674	+	0.980669i	$0.437310\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−340.665	−0.504187
$78$	0	0
$79$	−187.386	−0.266868	−0.133434	−	0.991058i	$-0.542600\pi$
$79$	−187.386	−0.266868	−0.133434	+	0.991058i	$0.542600\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	581.247	0.768677	0.384339	−	0.923192i	$-0.374430\pi$
$83$	581.247	0.768677	0.384339	+	0.923192i	$0.374430\pi$
$84$	0	0
$85$	266.660	0.340274
$86$	0	0
$87$	0	0
$88$	0	0
$89$	966.375	1.15096	0.575481	−	0.817815i	$-0.304815\pi$
$89$	966.375	1.15096	0.575481	+	0.817815i	$0.304815\pi$
$90$	0	0
$91$	468.904	0.540160
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−901.750	−0.973870
$96$	0	0
$97$	−227.941	−0.238597	−0.119299	−	0.992858i	$-0.538065\pi$
$97$	−227.941	−0.238597	−0.119299	+	0.992858i	$0.538065\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1036.42	−1.02106	−0.510532	−	0.859859i	$-0.670551\pi$
$101$	−1036.42	−1.02106	−0.510532	+	0.859859i	$0.670551\pi$
$102$	0	0
$103$	1928.83	1.84518	0.922588	−	0.385786i	$-0.126070\pi$
$103$	1928.83	1.84518	0.922588	+	0.385786i	$0.126070\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1055.55	0.953686	0.476843	−	0.878989i	$-0.341781\pi$
$107$	1055.55	0.953686	0.476843	+	0.878989i	$0.341781\pi$
$108$	0	0
$109$	−759.560	−0.667456	−0.333728	−	0.942669i	$-0.608307\pi$
$109$	−759.560	−0.667456	−0.333728	+	0.942669i	$0.608307\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1248.02	−1.03898	−0.519488	−	0.854478i	$-0.673877\pi$
$113$	−1248.02	−1.03898	−0.519488	+	0.854478i	$0.673877\pi$
$114$	0	0
$115$	1401.39	1.13635
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−107.079	−0.0824865
$120$	0	0
$121$	1594.15	1.19771
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−62.0127	−0.0443727
$126$	0	0
$127$	1853.29	1.29490	0.647452	−	0.762106i	$-0.275835\pi$
$127$	1853.29	1.29490	0.647452	+	0.762106i	$0.275835\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1119.13	0.746402	0.373201	−	0.927751i	$-0.378260\pi$
$131$	1119.13	0.746402	0.373201	+	0.927751i	$0.378260\pi$
$132$	0	0
$133$	362.103	0.236077
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1420.65	−0.885943	−0.442972	−	0.896536i	$-0.646076\pi$
$137$	−1420.65	−0.885943	−0.442972	+	0.896536i	$0.646076\pi$
$138$	0	0
$139$	−1538.66	−0.938904	−0.469452	−	0.882958i	$-0.655548\pi$
$139$	−1538.66	−0.938904	−0.469452	+	0.882958i	$0.655548\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4026.28	−2.35451
$144$	0	0
$145$	2920.83	1.67284
$146$	0	0
$147$	0	0
$148$	0	0
$149$	131.493	0.0722973	0.0361487	−	0.999346i	$-0.488491\pi$
$149$	131.493	0.0722973	0.0361487	+	0.999346i	$0.488491\pi$
$150$	0	0
$151$	1521.08	0.819758	0.409879	−	0.912140i	$-0.365571\pi$
$151$	1521.08	0.819758	0.409879	+	0.912140i	$0.365571\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−496.687	−0.257386
$156$	0	0
$157$	−2501.05	−1.27137	−0.635686	−	0.771948i	$-0.719283\pi$
$157$	−2501.05	−1.27137	−0.635686	+	0.771948i	$0.719283\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−562.736	−0.275465
$162$	0	0
$163$	−1030.23	−0.495054	−0.247527	−	0.968881i	$-0.579618\pi$
$163$	−1030.23	−0.495054	−0.247527	+	0.968881i	$0.579618\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1051.50	0.487230	0.243615	−	0.969872i	$-0.421667\pi$
$167$	1051.50	0.487230	0.243615	+	0.969872i	$0.421667\pi$
$168$	0	0
$169$	3344.92	1.52249
$170$	0	0
$171$	0	0
$172$	0	0
$173$	989.481	0.434849	0.217424	−	0.976077i	$-0.430234\pi$
$173$	989.481	0.434849	0.217424	+	0.976077i	$0.430234\pi$
$174$	0	0
$175$	−762.442	−0.329344
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−715.209	−0.298644	−0.149322	−	0.988789i	$-0.547709\pi$
$179$	−715.209	−0.298644	−0.149322	+	0.988789i	$0.547709\pi$
$180$	0	0
$181$	2259.54	0.927901	0.463951	−	0.885861i	$-0.346432\pi$
$181$	2259.54	0.927901	0.463951	+	0.885861i	$0.346432\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4992.29	1.98400
$186$	0	0
$187$	919.439	0.359551
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4718.85	1.78766	0.893832	−	0.448401i	$-0.148007\pi$
$191$	4718.85	1.78766	0.893832	+	0.448401i	$0.148007\pi$
$192$	0	0
$193$	2633.97	0.982369	0.491184	−	0.871056i	$-0.336564\pi$
$193$	2633.97	0.982369	0.491184	+	0.871056i	$0.336564\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1073.63	−0.388287	−0.194144	−	0.980973i	$-0.562193\pi$
$197$	−1073.63	−0.388287	−0.194144	+	0.980973i	$0.562193\pi$
$198$	0	0
$199$	672.051	0.239399	0.119700	−	0.992810i	$-0.461807\pi$
$199$	672.051	0.239399	0.119700	+	0.992810i	$0.461807\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1172.88	−0.405516
$204$	0	0
$205$	5307.13	1.80813
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3109.22	−1.02904
$210$	0	0
$211$	457.238	0.149183	0.0745913	−	0.997214i	$-0.476235\pi$
$211$	457.238	0.149183	0.0745913	+	0.997214i	$0.476235\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2785.05	0.883438
$216$	0	0
$217$	199.447	0.0623934
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1265.55	−0.385204
$222$	0	0
$223$	−6005.02	−1.80326	−0.901628	−	0.432512i	$-0.857627\pi$
$223$	−6005.02	−1.80326	−0.901628	+	0.432512i	$0.857627\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4144.92	1.21193	0.605965	−	0.795492i	$-0.292787\pi$
$227$	4144.92	1.21193	0.605965	+	0.795492i	$0.292787\pi$
$228$	0	0
$229$	−2493.77	−0.719621	−0.359810	−	0.933025i	$-0.617159\pi$
$229$	−2493.77	−0.719621	−0.359810	+	0.933025i	$0.617159\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3122.25	0.877877	0.438939	−	0.898517i	$-0.355355\pi$
$233$	3122.25	0.877877	0.438939	+	0.898517i	$0.355355\pi$
$234$	0	0
$235$	−3564.48	−0.989452
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6578.13	1.78035	0.890175	−	0.455618i	$-0.150582\pi$
$239$	6578.13	1.78035	0.890175	+	0.455618i	$0.150582\pi$
$240$	0	0
$241$	5839.79	1.56089	0.780444	−	0.625226i	$-0.214993\pi$
$241$	5839.79	1.56089	0.780444	+	0.625226i	$0.214993\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4757.93	−1.24071
$246$	0	0
$247$	4279.64	1.10246
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2474.62	0.622298	0.311149	−	0.950361i	$-0.399286\pi$
$251$	2474.62	0.622298	0.311149	+	0.950361i	$0.399286\pi$
$252$	0	0
$253$	4831.97	1.20073
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2765.34	0.671195	0.335598	−	0.942005i	$-0.391062\pi$
$257$	2765.34	0.671195	0.335598	+	0.942005i	$0.391062\pi$
$258$	0	0
$259$	−2004.68	−0.480945
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4940.35	1.15831	0.579154	−	0.815218i	$-0.303383\pi$
$263$	4940.35	1.15831	0.579154	+	0.815218i	$0.303383\pi$
$264$	0	0
$265$	−1425.14	−0.330361
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2790.02	−0.632381	−0.316190	−	0.948696i	$-0.602404\pi$
$269$	−2790.02	−0.632381	−0.316190	+	0.948696i	$0.602404\pi$
$270$	0	0
$271$	8725.38	1.95583	0.977914	−	0.209009i	$-0.0670237\pi$
$271$	8725.38	1.95583	0.977914	+	0.209009i	$0.0670237\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6546.76	1.43558
$276$	0	0
$277$	7629.92	1.65501	0.827505	−	0.561458i	$-0.189760\pi$
$277$	7629.92	1.65501	0.827505	+	0.561458i	$0.189760\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4244.28	−0.901041	−0.450521	−	0.892766i	$-0.648762\pi$
$281$	−4244.28	−0.901041	−0.450521	+	0.892766i	$0.648762\pi$
$282$	0	0
$283$	553.651	0.116294	0.0581469	−	0.998308i	$-0.481481\pi$
$283$	553.651	0.116294	0.0581469	+	0.998308i	$0.481481\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2131.11	−0.438311
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7361.10	−1.46771	−0.733857	−	0.679304i	$-0.762282\pi$
$293$	−7361.10	−1.46771	−0.733857	+	0.679304i	$0.762282\pi$
$294$	0	0
$295$	11053.6	2.18158
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6650.91	−1.28639
$300$	0	0
$301$	−1118.35	−0.214156
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9716.78	−1.82420
$306$	0	0
$307$	−8081.25	−1.50235	−0.751175	−	0.660103i	$-0.770513\pi$
$307$	−8081.25	−1.50235	−0.751175	+	0.660103i	$0.770513\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5463.27	−0.996120	−0.498060	−	0.867142i	$-0.665954\pi$
$311$	−5463.27	−0.996120	−0.498060	+	0.867142i	$0.665954\pi$
$312$	0	0
$313$	−9631.37	−1.73929	−0.869644	−	0.493680i	$-0.835651\pi$
$313$	−9631.37	−1.73929	−0.869644	+	0.493680i	$0.835651\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3960.05	−0.701636	−0.350818	−	0.936444i	$-0.614096\pi$
$317$	−3960.05	−0.701636	−0.350818	+	0.936444i	$0.614096\pi$
$318$	0	0
$319$	10071.0	1.76761
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−977.297	−0.168354
$324$	0	0
$325$	−9011.20	−1.53800
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1431.34	0.239855
$330$	0	0
$331$	−2928.79	−0.486347	−0.243174	−	0.969983i	$-0.578188\pi$
$331$	−2928.79	−0.486347	−0.243174	+	0.969983i	$0.578188\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4109.53	0.670232
$336$	0	0
$337$	−8128.46	−1.31390	−0.656952	−	0.753933i	$-0.728154\pi$
$337$	−8128.46	−1.31390	−0.656952	+	0.753933i	$0.728154\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1712.57	−0.271967
$342$	0	0
$343$	4071.04	0.640862
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9798.14	1.51583	0.757913	−	0.652356i	$-0.226219\pi$
$347$	9798.14	1.51583	0.757913	+	0.652356i	$0.226219\pi$
$348$	0	0
$349$	9776.71	1.49953	0.749764	−	0.661706i	$-0.230167\pi$
$349$	9776.71	1.49953	0.749764	+	0.661706i	$0.230167\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5073.34	−0.764947	−0.382474	−	0.923966i	$-0.624928\pi$
$353$	−5073.34	−0.764947	−0.382474	+	0.923966i	$0.624928\pi$
$354$	0	0
$355$	1102.17	0.164780
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10473.6	−1.53976	−0.769881	−	0.638187i	$-0.779684\pi$
$359$	−10473.6	−1.53976	−0.769881	+	0.638187i	$0.779684\pi$
$360$	0	0
$361$	−3554.12	−0.518170
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3828.75	0.549057
$366$	0	0
$367$	−12038.7	−1.71230	−0.856151	−	0.516726i	$-0.827150\pi$
$367$	−12038.7	−1.71230	−0.856151	+	0.516726i	$0.827150\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	572.272	0.0800833
$372$	0	0
$373$	−5671.52	−0.787293	−0.393647	−	0.919262i	$-0.628787\pi$
$373$	−5671.52	−0.787293	−0.393647	+	0.919262i	$0.628787\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13862.1	−1.89372
$378$	0	0
$379$	−13320.7	−1.80538	−0.902691	−	0.430289i	$-0.858412\pi$
$379$	−13320.7	−1.80538	−0.902691	+	0.430289i	$0.858412\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7981.52	−1.06485	−0.532423	−	0.846478i	$-0.678719\pi$
$383$	−7981.52	−1.06485	−0.532423	+	0.846478i	$0.678719\pi$
$384$	0	0
$385$	−5343.63	−0.707368
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8497.74	1.10759	0.553795	−	0.832653i	$-0.313179\pi$
$389$	8497.74	1.10759	0.553795	+	0.832653i	$0.313179\pi$
$390$	0	0
$391$	1518.80	0.196442
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2939.31	−0.374412
$396$	0	0
$397$	−15542.3	−1.96485	−0.982425	−	0.186658i	$-0.940234\pi$
$397$	−15542.3	−1.96485	−0.982425	+	0.186658i	$0.940234\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−77.1854	−0.00961210	−0.00480605	−	0.999988i	$-0.501530\pi$
$401$	−77.1854	−0.00961210	−0.00480605	+	0.999988i	$0.501530\pi$
$402$	0	0
$403$	2357.24	0.291371
$404$	0	0
$405$	0	0
$406$	0	0
$407$	17213.3	2.09640
$408$	0	0
$409$	3722.48	0.450037	0.225018	−	0.974355i	$-0.427756\pi$
$409$	3722.48	0.450037	0.225018	+	0.974355i	$0.427756\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4438.64	−0.528841
$414$	0	0
$415$	9117.37	1.07844
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5305.35	0.618576	0.309288	−	0.950968i	$-0.399909\pi$
$419$	5305.35	0.618576	0.309288	+	0.950968i	$0.399909\pi$
$420$	0	0
$421$	6897.43	0.798480	0.399240	−	0.916846i	$-0.369274\pi$
$421$	6897.43	0.798480	0.399240	+	0.916846i	$0.369274\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2057.79	0.234865
$426$	0	0
$427$	3901.83	0.442208
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7033.81	0.786094	0.393047	−	0.919518i	$-0.371421\pi$
$431$	7033.81	0.786094	0.393047	+	0.919518i	$0.371421\pi$
$432$	0	0
$433$	−5805.81	−0.644364	−0.322182	−	0.946678i	$-0.604416\pi$
$433$	−5805.81	−0.644364	−0.322182	+	0.946678i	$0.604416\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5136.04	−0.562220
$438$	0	0
$439$	−15444.5	−1.67911	−0.839553	−	0.543278i	$-0.817183\pi$
$439$	−15444.5	−1.67911	−0.839553	+	0.543278i	$0.817183\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5028.01	−0.539251	−0.269625	−	0.962965i	$-0.586900\pi$
$443$	−5028.01	−0.539251	−0.269625	+	0.962965i	$0.586900\pi$
$444$	0	0
$445$	15158.4	1.61478
$446$	0	0
$447$	0	0
$448$	0	0
$449$	260.448	0.0273749	0.0136874	−	0.999906i	$-0.495643\pi$
$449$	260.448	0.0273749	0.0136874	+	0.999906i	$0.495643\pi$
$450$	0	0
$451$	18298.9	1.91056
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7355.17	0.757837
$456$	0	0
$457$	11910.0	1.21910	0.609550	−	0.792748i	$-0.291350\pi$
$457$	11910.0	1.21910	0.609550	+	0.792748i	$0.291350\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14793.3	1.49456	0.747281	−	0.664509i	$-0.231359\pi$
$461$	14793.3	1.49456	0.747281	+	0.664509i	$0.231359\pi$
$462$	0	0
$463$	−17198.0	−1.72626	−0.863132	−	0.504978i	$-0.831501\pi$
$463$	−17198.0	−1.72626	−0.863132	+	0.504978i	$0.831501\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4539.95	−0.449858	−0.224929	−	0.974375i	$-0.572215\pi$
$467$	−4539.95	−0.449858	−0.224929	+	0.974375i	$0.572215\pi$
$468$	0	0
$469$	−1650.21	−0.162472
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9602.82	0.933485
$474$	0	0
$475$	−6958.73	−0.672186
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2627.93	0.250674	0.125337	−	0.992114i	$-0.459999\pi$
$479$	2627.93	0.250674	0.125337	+	0.992114i	$0.459999\pi$
$480$	0	0
$481$	−23693.1	−2.24597
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3575.46	−0.334749
$486$	0	0
$487$	2561.77	0.238367	0.119184	−	0.992872i	$-0.461972\pi$
$487$	2561.77	0.238367	0.119184	+	0.992872i	$0.461972\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17310.7	1.59108	0.795542	−	0.605898i	$-0.207186\pi$
$491$	17310.7	1.59108	0.795542	+	0.605898i	$0.207186\pi$
$492$	0	0
$493$	3165.53	0.289185
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−442.581	−0.0399447
$498$	0	0
$499$	−1072.95	−0.0962558	−0.0481279	−	0.998841i	$-0.515326\pi$
$499$	−1072.95	−0.0962558	−0.0481279	+	0.998841i	$0.515326\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3933.18	−0.348651	−0.174326	−	0.984688i	$-0.555775\pi$
$503$	−3933.18	−0.348651	−0.174326	+	0.984688i	$0.555775\pi$
$504$	0	0
$505$	−16257.1	−1.43254
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2794.72	−0.243367	−0.121684	−	0.992569i	$-0.538829\pi$
$509$	−2794.72	−0.243367	−0.121684	+	0.992569i	$0.538829\pi$
$510$	0	0
$511$	−1537.45	−0.133098
$512$	0	0
$513$	0	0
$514$	0	0
$515$	30255.4	2.58876
$516$	0	0
$517$	−12290.3	−1.04550
$518$	0	0
$519$	0	0
$520$	0	0
$521$	6193.76	0.520832	0.260416	−	0.965496i	$-0.416140\pi$
$521$	6193.76	0.520832	0.260416	+	0.965496i	$0.416140\pi$
$522$	0	0
$523$	1789.39	0.149607	0.0748034	−	0.997198i	$-0.476167\pi$
$523$	1789.39	0.149607	0.0748034	+	0.997198i	$0.476167\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−538.298	−0.0444946
$528$	0	0
$529$	−4185.19	−0.343979
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−25187.3	−2.04687
$534$	0	0
$535$	16557.3	1.33801
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−16405.3	−1.31099
$540$	0	0
$541$	−253.559	−0.0201504	−0.0100752	−	0.999949i	$-0.503207\pi$
$541$	−253.559	−0.0201504	−0.0100752	+	0.999949i	$0.503207\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11914.4	−0.936432
$546$	0	0
$547$	−15242.0	−1.19141	−0.595705	−	0.803203i	$-0.703127\pi$
$547$	−15242.0	−1.19141	−0.595705	+	0.803203i	$0.703127\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10704.7	−0.827652
$552$	0	0
$553$	1180.30	0.0907619
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2150.93	−0.163623	−0.0818114	−	0.996648i	$-0.526071\pi$
$557$	−2150.93	−0.163623	−0.0818114	+	0.996648i	$0.526071\pi$
$558$	0	0
$559$	−13217.7	−1.00009
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6861.49	−0.513636	−0.256818	−	0.966460i	$-0.582674\pi$
$563$	−6861.49	−0.513636	−0.256818	+	0.966460i	$0.582674\pi$
$564$	0	0
$565$	−19576.3	−1.45767
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2803.64	−0.206564	−0.103282	−	0.994652i	$-0.532934\pi$
$569$	−2803.64	−0.206564	−0.103282	+	0.994652i	$0.532934\pi$
$570$	0	0
$571$	−8201.92	−0.601121	−0.300560	−	0.953763i	$-0.597174\pi$
$571$	−8201.92	−0.601121	−0.300560	+	0.953763i	$0.597174\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	10814.4	0.784335
$576$	0	0
$577$	4295.12	0.309893	0.154946	−	0.987923i	$-0.450480\pi$
$577$	4295.12	0.309893	0.154946	+	0.987923i	$0.450480\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3661.13	−0.261427
$582$	0	0
$583$	−4913.86	−0.349076
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22867.5	1.60791	0.803953	−	0.594693i	$-0.202726\pi$
$587$	22867.5	1.60791	0.803953	+	0.594693i	$0.202726\pi$
$588$	0	0
$589$	1820.34	0.127344
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−19302.1	−1.33666	−0.668332	−	0.743863i	$-0.732991\pi$
$593$	−19302.1	−1.33666	−0.668332	+	0.743863i	$0.732991\pi$
$594$	0	0
$595$	−1679.62	−0.115727
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2992.91	−0.204152	−0.102076	−	0.994777i	$-0.532548\pi$
$599$	−2992.91	−0.204152	−0.102076	+	0.994777i	$0.532548\pi$
$600$	0	0
$601$	−1387.51	−0.0941723	−0.0470862	−	0.998891i	$-0.514994\pi$
$601$	−1387.51	−0.0941723	−0.0470862	+	0.998891i	$0.514994\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	25005.6	1.68037
$606$	0	0
$607$	−28444.9	−1.90205	−0.951024	−	0.309117i	$-0.899966\pi$
$607$	−28444.9	−1.90205	−0.951024	+	0.309117i	$0.899966\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16916.8	1.12010
$612$	0	0
$613$	−26059.3	−1.71700	−0.858502	−	0.512811i	$-0.828604\pi$
$613$	−26059.3	−1.71700	−0.858502	+	0.512811i	$0.828604\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17412.1	−1.13612	−0.568058	−	0.822989i	$-0.692305\pi$
$617$	−17412.1	−1.13612	−0.568058	+	0.822989i	$0.692305\pi$
$618$	0	0
$619$	−20574.9	−1.33599	−0.667994	−	0.744166i	$-0.732847\pi$
$619$	−20574.9	−1.33599	−0.667994	+	0.744166i	$0.732847\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6086.95	−0.391442
$624$	0	0
$625$	−16103.5	−1.03063
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5410.53	0.342976
$630$	0	0
$631$	−7927.70	−0.500153	−0.250077	−	0.968226i	$-0.580456\pi$
$631$	−7927.70	−0.500153	−0.250077	+	0.968226i	$0.580456\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	29070.4	1.81673
$636$	0	0
$637$	22580.8	1.40453
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20190.2	1.24409	0.622047	−	0.782980i	$-0.286301\pi$
$641$	20190.2	1.24409	0.622047	+	0.782980i	$0.286301\pi$
$642$	0	0
$643$	1871.36	0.114773	0.0573865	−	0.998352i	$-0.481723\pi$
$643$	1871.36	0.114773	0.0573865	+	0.998352i	$0.481723\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9629.86	−0.585145	−0.292573	−	0.956243i	$-0.594511\pi$
$647$	−9629.86	−0.585145	−0.292573	+	0.956243i	$0.594511\pi$
$648$	0	0
$649$	38112.7	2.30517
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7956.02	0.476789	0.238395	−	0.971168i	$-0.423379\pi$
$653$	7956.02	0.476789	0.238395	+	0.971168i	$0.423379\pi$
$654$	0	0
$655$	17554.5	1.04719
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−31802.1	−1.87987	−0.939936	−	0.341352i	$-0.889115\pi$
$659$	−31802.1	−1.87987	−0.939936	+	0.341352i	$0.889115\pi$
$660$	0	0
$661$	−10768.1	−0.633632	−0.316816	−	0.948487i	$-0.602614\pi$
$661$	−10768.1	−0.633632	−0.316816	+	0.948487i	$0.602614\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5679.90	0.331213
$666$	0	0
$667$	16636.0	0.965739
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−33503.3	−1.92754
$672$	0	0
$673$	15170.9	0.868938	0.434469	−	0.900687i	$-0.356936\pi$
$673$	15170.9	0.868938	0.434469	+	0.900687i	$0.356936\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1102.92	0.0626127	0.0313064	−	0.999510i	$-0.490033\pi$
$677$	1102.92	0.0626127	0.0313064	+	0.999510i	$0.490033\pi$
$678$	0	0
$679$	1435.74	0.0811470
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2140.82	0.119936	0.0599678	−	0.998200i	$-0.480900\pi$
$683$	2140.82	0.119936	0.0599678	+	0.998200i	$0.480900\pi$
$684$	0	0
$685$	−22284.1	−1.24297
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6763.61	0.373981
$690$	0	0
$691$	28162.7	1.55045	0.775223	−	0.631687i	$-0.217637\pi$
$691$	28162.7	1.55045	0.775223	+	0.631687i	$0.217637\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24135.3	−1.31727
$696$	0	0
$697$	5751.75	0.312573
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−17802.0	−0.959160	−0.479580	−	0.877498i	$-0.659211\pi$
$701$	−17802.0	−0.959160	−0.479580	+	0.877498i	$0.659211\pi$
$702$	0	0
$703$	−18296.5	−0.981602
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6528.13	0.347264
$708$	0	0
$709$	7646.66	0.405044	0.202522	−	0.979278i	$-0.435086\pi$
$709$	7646.66	0.405044	0.202522	+	0.979278i	$0.435086\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2828.95	−0.148590
$714$	0	0
$715$	−63155.7	−3.30334
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11522.7	0.597671	0.298835	−	0.954305i	$-0.403402\pi$
$719$	11522.7	0.597671	0.298835	+	0.954305i	$0.403402\pi$
$720$	0	0
$721$	−12149.2	−0.627545
$722$	0	0
$723$	0	0
$724$	0	0
$725$	22539.8	1.15463
$726$	0	0
$727$	1027.74	0.0524301	0.0262151	−	0.999656i	$-0.491655\pi$
$727$	1027.74	0.0524301	0.0262151	+	0.999656i	$0.491655\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3018.38	0.152721
$732$	0	0
$733$	−8097.01	−0.408008	−0.204004	−	0.978970i	$-0.565396\pi$
$733$	−8097.01	−0.408008	−0.204004	+	0.978970i	$0.565396\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14169.6	0.708201
$738$	0	0
$739$	31129.4	1.54955	0.774773	−	0.632240i	$-0.217864\pi$
$739$	31129.4	1.54955	0.774773	+	0.632240i	$0.217864\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−31420.7	−1.55143	−0.775716	−	0.631083i	$-0.782611\pi$
$743$	−31420.7	−1.55143	−0.775716	+	0.631083i	$0.782611\pi$
$744$	0	0
$745$	2062.58	0.101432
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6648.67	−0.324349
$750$	0	0
$751$	−9157.73	−0.444967	−0.222484	−	0.974936i	$-0.571416\pi$
$751$	−9157.73	−0.444967	−0.222484	+	0.974936i	$0.571416\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	23859.4	1.15011
$756$	0	0
$757$	14631.0	0.702474	0.351237	−	0.936287i	$-0.385761\pi$
$757$	14631.0	0.702474	0.351237	+	0.936287i	$0.385761\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−17408.4	−0.829244	−0.414622	−	0.909994i	$-0.636086\pi$
$761$	−17408.4	−0.829244	−0.414622	+	0.909994i	$0.636086\pi$
$762$	0	0
$763$	4784.28	0.227002
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−52459.7	−2.46964
$768$	0	0
$769$	−9361.21	−0.438978	−0.219489	−	0.975615i	$-0.570439\pi$
$769$	−9361.21	−0.438978	−0.219489	+	0.975615i	$0.570439\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−13639.0	−0.634622	−0.317311	−	0.948322i	$-0.602780\pi$
$773$	−13639.0	−0.634622	−0.317311	+	0.948322i	$0.602780\pi$
$774$	0	0
$775$	−3832.89	−0.177654
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−19450.4	−0.894587
$780$	0	0
$781$	3800.26	0.174115
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−39231.1	−1.78372
$786$	0	0
$787$	−26004.1	−1.17782	−0.588912	−	0.808197i	$-0.700444\pi$
$787$	−26004.1	−1.17782	−0.588912	+	0.808197i	$0.700444\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	7860.99	0.353356
$792$	0	0
$793$	46115.2	2.06507
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13799.1	−0.613285	−0.306643	−	0.951825i	$-0.599206\pi$
$797$	−13799.1	−0.613285	−0.306643	+	0.951825i	$0.599206\pi$
$798$	0	0
$799$	−3863.11	−0.171047
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13201.5	0.580161
$804$	0	0
$805$	−8827.01	−0.386474
$806$	0	0
$807$	0	0
$808$	0	0
$809$	43491.0	1.89006	0.945031	−	0.326980i	$-0.106031\pi$
$809$	43491.0	1.89006	0.945031	+	0.326980i	$0.106031\pi$
$810$	0	0
$811$	36562.9	1.58311	0.791553	−	0.611101i	$-0.209273\pi$
$811$	36562.9	1.58311	0.791553	+	0.611101i	$0.209273\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16160.0	−0.694554
$816$	0	0
$817$	−10207.1	−0.437089
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−31722.8	−1.34852	−0.674258	−	0.738496i	$-0.735537\pi$
$821$	−31722.8	−1.34852	−0.674258	+	0.738496i	$0.735537\pi$
$822$	0	0
$823$	−27119.8	−1.14865	−0.574324	−	0.818628i	$-0.694735\pi$
$823$	−27119.8	−1.14865	−0.574324	+	0.818628i	$0.694735\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22931.8	0.964227	0.482113	−	0.876109i	$-0.339869\pi$
$827$	22931.8	0.964227	0.482113	+	0.876109i	$0.339869\pi$
$828$	0	0
$829$	−5879.04	−0.246306	−0.123153	−	0.992388i	$-0.539301\pi$
$829$	−5879.04	−0.246306	−0.123153	+	0.992388i	$0.539301\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5156.54	−0.214482
$834$	0	0
$835$	16493.7	0.683577
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2880.69	−0.118537	−0.0592684	−	0.998242i	$-0.518877\pi$
$839$	−2880.69	−0.118537	−0.0592684	+	0.998242i	$0.518877\pi$
$840$	0	0
$841$	10284.3	0.421680
$842$	0	0
$843$	0	0
$844$	0	0
$845$	52468.0	2.13604
$846$	0	0
$847$	−10041.1	−0.407341
$848$	0	0
$849$	0	0
$850$	0	0
$851$	28434.2	1.14537
$852$	0	0
$853$	−11673.4	−0.468569	−0.234284	−	0.972168i	$-0.575275\pi$
$853$	−11673.4	−0.468569	−0.234284	+	0.972168i	$0.575275\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1369.67	−0.0545940	−0.0272970	−	0.999627i	$-0.508690\pi$
$857$	−1369.67	−0.0545940	−0.0272970	+	0.999627i	$0.508690\pi$
$858$	0	0
$859$	7559.18	0.300251	0.150126	−	0.988667i	$-0.452032\pi$
$859$	7559.18	0.300251	0.150126	+	0.988667i	$0.452032\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28575.6	1.12714	0.563572	−	0.826067i	$-0.309427\pi$
$863$	28575.6	1.12714	0.563572	+	0.826067i	$0.309427\pi$
$864$	0	0
$865$	15520.9	0.610087
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−10134.7	−0.395623
$870$	0	0
$871$	−19503.6	−0.758729
$872$	0	0
$873$	0	0
$874$	0	0
$875$	390.602	0.0150912
$876$	0	0
$877$	26907.7	1.03604	0.518021	−	0.855368i	$-0.326669\pi$
$877$	26907.7	1.03604	0.518021	+	0.855368i	$0.326669\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	31659.4	1.21071	0.605353	−	0.795957i	$-0.293032\pi$
$881$	31659.4	1.21071	0.605353	+	0.795957i	$0.293032\pi$
$882$	0	0
$883$	−2895.58	−0.110355	−0.0551777	−	0.998477i	$-0.517573\pi$
$883$	−2895.58	−0.110355	−0.0551777	+	0.998477i	$0.517573\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17577.7	−0.665392	−0.332696	−	0.943034i	$-0.607958\pi$
$887$	−17577.7	−0.665392	−0.332696	+	0.943034i	$0.607958\pi$
$888$	0	0
$889$	−11673.4	−0.440397
$890$	0	0
$891$	0	0
$892$	0	0
$893$	13063.7	0.489540
$894$	0	0
$895$	−11218.7	−0.418993
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5896.20	−0.218742
$900$	0	0
$901$	−1544.53	−0.0571098
$902$	0	0
$903$	0	0
$904$	0	0
$905$	35442.8	1.30183
$906$	0	0
$907$	−9827.12	−0.359762	−0.179881	−	0.983688i	$-0.557571\pi$
$907$	−9827.12	−0.359762	−0.179881	+	0.983688i	$0.557571\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31873.7	−1.15919	−0.579595	−	0.814904i	$-0.696789\pi$
$911$	−31873.7	−1.15919	−0.579595	+	0.814904i	$0.696789\pi$
$912$	0	0
$913$	31436.6	1.13954
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7049.10	−0.253852
$918$	0	0
$919$	27402.8	0.983609	0.491804	−	0.870706i	$-0.336338\pi$
$919$	27402.8	0.983609	0.491804	+	0.870706i	$0.336338\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5230.81	−0.186538
$924$	0	0
$925$	38525.1	1.36940
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19977.0	0.705514	0.352757	−	0.935715i	$-0.385244\pi$
$929$	19977.0	0.705514	0.352757	+	0.935715i	$0.385244\pi$
$930$	0	0
$931$	17437.6	0.613850
$932$	0	0
$933$	0	0
$934$	0	0
$935$	14422.2	0.504445
$936$	0	0
$937$	27416.5	0.955878	0.477939	−	0.878393i	$-0.341384\pi$
$937$	27416.5	0.955878	0.477939	+	0.878393i	$0.341384\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3729.87	−0.129214	−0.0646070	−	0.997911i	$-0.520579\pi$
$941$	−3729.87	−0.129214	−0.0646070	+	0.997911i	$0.520579\pi$
$942$	0	0
$943$	30227.5	1.04384
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5509.23	0.189045	0.0945226	−	0.995523i	$-0.469868\pi$
$947$	5509.23	0.189045	0.0945226	+	0.995523i	$0.469868\pi$
$948$	0	0
$949$	−18171.0	−0.621554
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−26354.9	−0.895822	−0.447911	−	0.894078i	$-0.647832\pi$
$953$	−26354.9	−0.895822	−0.447911	+	0.894078i	$0.647832\pi$
$954$	0	0
$955$	74019.3	2.50807
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8948.31	0.301310
$960$	0	0
$961$	−28788.4	−0.966344
$962$	0	0
$963$	0	0
$964$	0	0
$965$	41316.1	1.37825
$966$	0	0
$967$	26496.4	0.881146	0.440573	−	0.897717i	$-0.354775\pi$
$967$	26496.4	0.881146	0.440573	+	0.897717i	$0.354775\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−8592.79	−0.283992	−0.141996	−	0.989867i	$-0.545352\pi$
$971$	−8592.79	−0.283992	−0.141996	+	0.989867i	$0.545352\pi$
$972$	0	0
$973$	9691.65	0.319322
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18182.1	0.595389	0.297695	−	0.954661i	$-0.403782\pi$
$977$	18182.1	0.595389	0.297695	+	0.954661i	$0.403782\pi$
$978$	0	0
$979$	52266.0	1.70626
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3369.08	−0.109315	−0.0546577	−	0.998505i	$-0.517407\pi$
$983$	−3369.08	−0.109315	−0.0546577	+	0.998505i	$0.517407\pi$
$984$	0	0
$985$	−16840.7	−0.544762
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15862.6	0.510013
$990$	0	0
$991$	−30421.4	−0.975144	−0.487572	−	0.873083i	$-0.662117\pi$
$991$	−30421.4	−0.975144	−0.487572	+	0.873083i	$0.662117\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	10541.7	0.335874
$996$	0	0
$997$	40464.6	1.28538	0.642691	−	0.766125i	$-0.277818\pi$
$997$	40464.6	1.28538	0.642691	+	0.766125i	$0.277818\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1224.4.a.q.1.7	yes	7
3.2	odd	2		1224.4.a.p.1.1	✓	7
4.3	odd	2		2448.4.a.bw.1.7		7
12.11	even	2		2448.4.a.bv.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1224.4.a.p.1.1	✓	7	3.2	odd	2
1224.4.a.q.1.7	yes	7	1.1	even	1	trivial
2448.4.a.bv.1.1		7	12.11	even	2
2448.4.a.bw.1.7		7	4.3	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:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1224.a (trivial)

\(f(q)\)	\(=\)	\(q+15.6859 q^{5} -6.29875 q^{7} +54.0846 q^{11} -74.4441 q^{13} +17.0000 q^{17} -57.4880 q^{19} +89.3410 q^{23} +121.047 q^{25} +186.208 q^{29} -31.6646 q^{31} -98.8014 q^{35} +318.267 q^{37} +338.338 q^{41} +177.552 q^{43} -227.242 q^{47} -303.326 q^{49} -90.8549 q^{53} +848.365 q^{55} +704.687 q^{59} -619.461 q^{61} -1167.72 q^{65} +261.990 q^{67} +70.2650 q^{71} +244.089 q^{73} -340.665 q^{77} -187.386 q^{79} +581.247 q^{83} +266.660 q^{85} +966.375 q^{89} +468.904 q^{91} -901.750 q^{95} -227.941 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{5} + 26 q^{7} + 65 q^{11} + 17 q^{13} + 119 q^{17} + 125 q^{19} - 107 q^{23} + 176 q^{25} + 6 q^{29} + 44 q^{31} + 58 q^{35} + 254 q^{37} - 219 q^{41} + 419 q^{43} - 300 q^{47} + 1171 q^{49}+ \cdots + 2536 q^{97}+O(q^{100}) \) 7 * q + 3 * q^5 + 26 * q^7 + 65 * q^11 + 17 * q^13 + 119 * q^17 + 125 * q^19 - 107 * q^23 + 176 * q^25 + 6 * q^29 + 44 * q^31 + 58 * q^35 + 254 * q^37 - 219 * q^41 + 419 * q^43 - 300 * q^47 + 1171 * q^49 - 284 * q^53 + 633 * q^55 + 46 * q^59 + 638 * q^61 + 185 * q^65 + 796 * q^67 + 1472 * q^71 + 1712 * q^73 - 586 * q^77 + 1134 * q^79 + 596 * q^83 + 51 * q^85 - 694 * q^89 + 3822 * q^91 + 229 * q^95 + 2536 * q^97

Introduction

L-functions

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