LMFDB - Embedded newform 3150.3.c.d.449.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,3,Mod(449,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3150.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3150.c (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:	$85.8312832735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.9671731157401600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256 $ x^16 + 7x^12 + 753x^8 + 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.6
Root			$-0.941471 - 2.08559i$ of defining polynomial
Character	$\chi$	$=$	3150.449
Dual form			3150.3.c.d.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +2.32744i q^{11} +18.6912i q^{13} -3.74166i q^{14} +4.00000 q^{16} +12.0149 q^{17} -30.0698 q^{19} -3.29150i q^{22} -21.5664 q^{23} -26.4333i q^{26} +5.29150i q^{28} +23.6176i q^{29} -55.0615 q^{31} -5.65685 q^{32} -16.9917 q^{34} -38.8482i q^{37} +42.5252 q^{38} -39.3269i q^{41} -71.8398i q^{43} +4.65489i q^{44} +30.4995 q^{46} +80.6726 q^{47} -7.00000 q^{49} +37.3823i q^{52} -71.1923 q^{53} -7.48331i q^{56} -33.4003i q^{58} +51.3418i q^{59} -1.95806 q^{61} +77.8687 q^{62} +8.00000 q^{64} +67.6295i q^{67} +24.0298 q^{68} +56.1112i q^{71} -126.371i q^{73} +54.9396i q^{74} -60.1397 q^{76} -6.15784 q^{77} +35.7550 q^{79} +55.6166i q^{82} +39.6722 q^{83} +101.597i q^{86} -6.58301i q^{88} -56.3868i q^{89} -49.4521 q^{91} -43.1327 q^{92} -114.088 q^{94} -129.518i q^{97} +9.89949 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} + 64 q^{16} - 160 q^{19} - 224 q^{31} + 64 q^{34} - 64 q^{46} - 112 q^{49} - 160 q^{61} + 128 q^{64} - 320 q^{76} + 128 q^{94}+O(q^{100}) $ 16 * q + 32 * q^4 + 64 * q^16 - 160 * q^19 - 224 * q^31 + 64 * q^34 - 64 * q^46 - 112 * q^49 - 160 * q^61 + 128 * q^64 - 320 * q^76 + 128 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$	−1.41421	−0.707107
$2$	−1.41421	−0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	−2.82843	−0.353553
$9$	0	0
$10$	0	0
$11$	2.32744i	0.211586i	0.994388	+	0.105793i	$0.0337381\pi$
$11$	2.32744i	0.211586i	−0.994388	+	0.105793i	$0.966262\pi$
$12$	0	0
$13$	18.6912i	1.43778i	0.695123	+	0.718891i	$0.255350\pi$
$13$	18.6912i	1.43778i	−0.695123	+	0.718891i	$0.744650\pi$
$14$	− 3.74166i	− 0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	12.0149	0.706759	0.353380	−	0.935480i	$-0.385032\pi$
$17$	12.0149	0.706759	0.353380	+	0.935480i	$0.385032\pi$
$18$	0	0
$19$	−30.0698	−1.58262	−0.791312	−	0.611413i	$-0.790601\pi$
$19$	−30.0698	−1.58262	−0.791312	+	0.611413i	$0.790601\pi$
$20$	0	0
$21$	0	0
$22$	− 3.29150i	− 0.149614i
$23$	−21.5664	−0.937668	−0.468834	−	0.883286i	$-0.655326\pi$
$23$	−21.5664	−0.937668	−0.468834	+	0.883286i	$0.655326\pi$
$24$	0	0
$25$	0	0
$26$	− 26.4333i	− 1.01666i
$27$	0	0
$28$	5.29150i	0.188982i
$29$	23.6176i	0.814399i	0.913339	+	0.407200i	$0.133495\pi$
$29$	23.6176i	0.814399i	−0.913339	+	0.407200i	$0.866505\pi$
$30$	0	0
$31$	−55.0615	−1.77618	−0.888089	−	0.459672i	$-0.847967\pi$
$31$	−55.0615	−1.77618	−0.888089	+	0.459672i	$0.847967\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	−16.9917	−0.499754
$35$	0	0
$36$	0	0
$37$	− 38.8482i	− 1.04995i	−0.851117	−	0.524975i	$-0.824075\pi$
$37$	− 38.8482i	− 1.04995i	0.851117	−	0.524975i	$-0.175925\pi$
$38$	42.5252	1.11908
$39$	0	0
$40$	0	0
$41$	− 39.3269i	− 0.959192i	−0.877489	−	0.479596i	$-0.840783\pi$
$41$	− 39.3269i	− 0.959192i	0.877489	−	0.479596i	$-0.159217\pi$
$42$	0	0
$43$	− 71.8398i	− 1.67069i	−0.549723	−	0.835347i	$-0.685267\pi$
$43$	− 71.8398i	− 1.67069i	0.549723	−	0.835347i	$-0.314733\pi$
$44$	4.65489i	0.105793i
$45$	0	0
$46$	30.4995	0.663032
$47$	80.6726	1.71644	0.858219	−	0.513283i	$-0.171571\pi$
$47$	80.6726	1.71644	0.858219	+	0.513283i	$0.171571\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	37.3823i	0.718891i
$53$	−71.1923	−1.34325	−0.671625	−	0.740891i	$-0.734403\pi$
$53$	−71.1923	−1.34325	−0.671625	+	0.740891i	$0.734403\pi$
$54$	0	0
$55$	0	0
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	− 33.4003i	− 0.575867i
$59$	51.3418i	0.870200i	0.900382	+	0.435100i	$0.143287\pi$
$59$	51.3418i	0.870200i	−0.900382	+	0.435100i	$0.856713\pi$
$60$	0	0
$61$	−1.95806	−0.0320993	−0.0160497	−	0.999871i	$-0.505109\pi$
$61$	−1.95806	−0.0320993	−0.0160497	+	0.999871i	$0.505109\pi$
$62$	77.8687	1.25595
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	67.6295i	1.00940i	0.863296	+	0.504698i	$0.168396\pi$
$67$	67.6295i	1.00940i	−0.863296	+	0.504698i	$0.831604\pi$
$68$	24.0298	0.353380
$69$	0	0
$70$	0	0
$71$	56.1112i	0.790299i	0.918617	+	0.395150i	$0.129307\pi$
$71$	56.1112i	0.790299i	−0.918617	+	0.395150i	$0.870693\pi$
$72$	0	0
$73$	− 126.371i	− 1.73111i	−0.500817	−	0.865553i	$-0.666967\pi$
$73$	− 126.371i	− 1.73111i	0.500817	−	0.865553i	$-0.333033\pi$
$74$	54.9396i	0.742427i
$75$	0	0
$76$	−60.1397	−0.791312
$77$	−6.15784	−0.0799719
$78$	0	0
$79$	35.7550	0.452595	0.226297	−	0.974058i	$-0.427338\pi$
$79$	35.7550	0.452595	0.226297	+	0.974058i	$0.427338\pi$
$80$	0	0
$81$	0	0
$82$	55.6166i	0.678251i
$83$	39.6722	0.477979	0.238989	−	0.971022i	$-0.423184\pi$
$83$	39.6722	0.477979	0.238989	+	0.971022i	$0.423184\pi$
$84$	0	0
$85$	0	0
$86$	101.597i	1.18136i
$87$	0	0
$88$	− 6.58301i	− 0.0748069i
$89$	− 56.3868i	− 0.633560i	−0.948499	−	0.316780i	$-0.897398\pi$
$89$	− 56.3868i	− 0.633560i	0.948499	−	0.316780i	$-0.102602\pi$
$90$	0	0
$91$	−49.4521	−0.543430
$92$	−43.1327	−0.468834
$93$	0	0
$94$	−114.088	−1.21371
$95$	0	0
$96$	0	0
$97$	− 129.518i	− 1.33523i	−0.744506	−	0.667616i	$-0.767315\pi$
$97$	− 129.518i	− 1.33523i	0.744506	−	0.667616i	$-0.232685\pi$
$98$	9.89949	0.101015
$99$	0	0
$100$	0	0
$101$	− 19.7981i	− 0.196021i	−0.995185	−	0.0980103i	$-0.968752\pi$
$101$	− 19.7981i	− 0.196021i	0.995185	−	0.0980103i	$-0.0312478\pi$
$102$	0	0
$103$	110.632i	1.07410i	0.843551	+	0.537050i	$0.180461\pi$
$103$	110.632i	1.07410i	−0.843551	+	0.537050i	$0.819539\pi$
$104$	− 52.8666i	− 0.508332i
$105$	0	0
$106$	100.681	0.949821
$107$	174.264	1.62864	0.814318	−	0.580419i	$-0.197111\pi$
$107$	174.264	1.62864	0.814318	+	0.580419i	$0.197111\pi$
$108$	0	0
$109$	−104.227	−0.956208	−0.478104	−	0.878303i	$-0.658676\pi$
$109$	−104.227	−0.956208	−0.478104	+	0.878303i	$0.658676\pi$
$110$	0	0
$111$	0	0
$112$	10.5830i	0.0944911i
$113$	113.968	1.00857	0.504284	−	0.863538i	$-0.331756\pi$
$113$	113.968	1.00857	0.504284	+	0.863538i	$0.331756\pi$
$114$	0	0
$115$	0	0
$116$	47.2352i	0.407200i
$117$	0	0
$118$	− 72.6082i	− 0.615324i
$119$	31.7885i	0.267130i
$120$	0	0
$121$	115.583	0.955231
$122$	2.76911	0.0226977
$123$	0	0
$124$	−110.123	−0.888089
$125$	0	0
$126$	0	0
$127$	− 91.3596i	− 0.719367i	−0.933074	−	0.359684i	$-0.882885\pi$
$127$	− 91.3596i	− 0.719367i	0.933074	−	0.359684i	$-0.117115\pi$
$128$	−11.3137	−0.0883883
$129$	0	0
$130$	0	0
$131$	− 18.9042i	− 0.144307i	−0.997394	−	0.0721536i	$-0.977013\pi$
$131$	− 18.9042i	− 0.144307i	0.997394	−	0.0721536i	$-0.0229872\pi$
$132$	0	0
$133$	− 79.5573i	− 0.598175i
$134$	− 95.6425i	− 0.713750i
$135$	0	0
$136$	−33.9833	−0.249877
$137$	49.0678	0.358159	0.179080	−	0.983835i	$-0.442688\pi$
$137$	49.0678	0.358159	0.179080	+	0.983835i	$0.442688\pi$
$138$	0	0
$139$	127.291	0.915763	0.457881	−	0.889013i	$-0.348608\pi$
$139$	127.291	0.915763	0.457881	+	0.889013i	$0.348608\pi$
$140$	0	0
$141$	0	0
$142$	− 79.3533i	− 0.558826i
$143$	−43.5026	−0.304214
$144$	0	0
$145$	0	0
$146$	178.715i	1.22408i
$147$	0	0
$148$	− 77.6963i	− 0.524975i
$149$	− 222.551i	− 1.49363i	−0.665030	−	0.746817i	$-0.731581\pi$
$149$	− 222.551i	− 1.49363i	0.665030	−	0.746817i	$-0.268419\pi$
$150$	0	0
$151$	9.84451	0.0651954	0.0325977	−	0.999469i	$-0.489622\pi$
$151$	9.84451	0.0651954	0.0325977	+	0.999469i	$0.489622\pi$
$152$	85.0503	0.559542
$153$	0	0
$154$	8.70850	0.0565487
$155$	0	0
$156$	0	0
$157$	− 114.833i	− 0.731423i	−0.930728	−	0.365711i	$-0.880826\pi$
$157$	− 114.833i	− 0.731423i	0.930728	−	0.365711i	$-0.119174\pi$
$158$	−50.5652	−0.320033
$159$	0	0
$160$	0	0
$161$	− 57.0593i	− 0.354405i
$162$	0	0
$163$	226.318i	1.38845i	0.719757	+	0.694226i	$0.244253\pi$
$163$	226.318i	1.38845i	−0.719757	+	0.694226i	$0.755747\pi$
$164$	− 78.6537i	− 0.479596i
$165$	0	0
$166$	−56.1050	−0.337982
$167$	13.1518	0.0787532	0.0393766	−	0.999224i	$-0.487463\pi$
$167$	13.1518	0.0787532	0.0393766	+	0.999224i	$0.487463\pi$
$168$	0	0
$169$	−180.359	−1.06721
$170$	0	0
$171$	0	0
$172$	− 143.680i	− 0.835347i
$173$	−165.445	−0.956327	−0.478164	−	0.878271i	$-0.658697\pi$
$173$	−165.445	−0.956327	−0.478164	+	0.878271i	$0.658697\pi$
$174$	0	0
$175$	0	0
$176$	9.30978i	0.0528965i
$177$	0	0
$178$	79.7430i	0.447995i
$179$	− 32.5764i	− 0.181991i	−0.995851	−	0.0909955i	$-0.970995\pi$
$179$	− 32.5764i	− 0.181991i	0.995851	−	0.0909955i	$-0.0290049\pi$
$180$	0	0
$181$	−294.398	−1.62651	−0.813253	−	0.581910i	$-0.802305\pi$
$181$	−294.398	−1.62651	−0.813253	+	0.581910i	$0.802305\pi$
$182$	69.9359	0.384263
$183$	0	0
$184$	60.9989	0.331516
$185$	0	0
$186$	0	0
$187$	27.9640i	0.149540i
$188$	161.345	0.858219
$189$	0	0
$190$	0	0
$191$	256.682i	1.34389i	0.740602	+	0.671944i	$0.234540\pi$
$191$	256.682i	1.34389i	−0.740602	+	0.671944i	$0.765460\pi$
$192$	0	0
$193$	− 57.0950i	− 0.295829i	−0.989000	−	0.147915i	$-0.952744\pi$
$193$	− 57.0950i	− 0.295829i	0.989000	−	0.147915i	$-0.0472561\pi$
$194$	183.165i	0.944152i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	191.143	0.970271	0.485135	−	0.874439i	$-0.338770\pi$
$197$	191.143	0.970271	0.485135	+	0.874439i	$0.338770\pi$
$198$	0	0
$199$	−269.301	−1.35327	−0.676635	−	0.736319i	$-0.736562\pi$
$199$	−269.301	−1.35327	−0.676635	+	0.736319i	$0.736562\pi$
$200$	0	0
$201$	0	0
$202$	27.9987i	0.138607i
$203$	−62.4862	−0.307814
$204$	0	0
$205$	0	0
$206$	− 156.458i	− 0.759503i
$207$	0	0
$208$	74.7646i	0.359445i
$209$	− 69.9859i	− 0.334861i
$210$	0	0
$211$	−209.158	−0.991271	−0.495636	−	0.868531i	$-0.665065\pi$
$211$	−209.158	−0.991271	−0.495636	+	0.868531i	$0.665065\pi$
$212$	−142.385	−0.671625
$213$	0	0
$214$	−246.447	−1.15162
$215$	0	0
$216$	0	0
$217$	− 145.679i	− 0.671332i
$218$	147.399	0.676141
$219$	0	0
$220$	0	0
$221$	224.573i	1.01617i
$222$	0	0
$223$	− 90.4720i	− 0.405704i	−0.979209	−	0.202852i	$-0.934979\pi$
$223$	− 90.4720i	− 0.405704i	0.979209	−	0.202852i	$-0.0650211\pi$
$224$	− 14.9666i	− 0.0668153i
$225$	0	0
$226$	−161.175	−0.713166
$227$	209.496	0.922888	0.461444	−	0.887169i	$-0.347332\pi$
$227$	209.496	0.922888	0.461444	+	0.887169i	$0.347332\pi$
$228$	0	0
$229$	246.581	1.07677	0.538386	−	0.842698i	$-0.319034\pi$
$229$	246.581	1.07677	0.538386	+	0.842698i	$0.319034\pi$
$230$	0	0
$231$	0	0
$232$	− 66.8006i	− 0.287934i
$233$	330.414	1.41808	0.709042	−	0.705166i	$-0.249128\pi$
$233$	330.414	1.41808	0.709042	+	0.705166i	$0.249128\pi$
$234$	0	0
$235$	0	0
$236$	102.684i	0.435100i
$237$	0	0
$238$	− 44.9557i	− 0.188889i
$239$	− 311.038i	− 1.30141i	−0.759330	−	0.650706i	$-0.774473\pi$
$239$	− 311.038i	− 1.30141i	0.759330	−	0.650706i	$-0.225527\pi$
$240$	0	0
$241$	199.209	0.826593	0.413296	−	0.910597i	$-0.364377\pi$
$241$	199.209	0.826593	0.413296	+	0.910597i	$0.364377\pi$
$242$	−163.459	−0.675451
$243$	0	0
$244$	−3.91612	−0.0160497
$245$	0	0
$246$	0	0
$247$	− 562.040i	− 2.27547i
$248$	155.737	0.627973
$249$	0	0
$250$	0	0
$251$	− 42.0895i	− 0.167687i	−0.996479	−	0.0838436i	$-0.973280\pi$
$251$	− 42.0895i	− 0.167687i	0.996479	−	0.0838436i	$-0.0267196\pi$
$252$	0	0
$253$	− 50.1945i	− 0.198397i
$254$	129.202i	0.508669i
$255$	0	0
$256$	16.0000	0.0625000
$257$	273.951	1.06596	0.532978	−	0.846129i	$-0.321073\pi$
$257$	273.951	1.06596	0.532978	+	0.846129i	$0.321073\pi$
$258$	0	0
$259$	102.783	0.396844
$260$	0	0
$261$	0	0
$262$	26.7346i	0.102041i
$263$	22.2237	0.0845006	0.0422503	−	0.999107i	$-0.486547\pi$
$263$	22.2237	0.0845006	0.0422503	+	0.999107i	$0.486547\pi$
$264$	0	0
$265$	0	0
$266$	112.511i	0.422974i
$267$	0	0
$268$	135.259i	0.504698i
$269$	− 131.494i	− 0.488826i	−0.969671	−	0.244413i	$-0.921405\pi$
$269$	− 131.494i	− 0.488826i	0.969671	−	0.244413i	$-0.0785952\pi$
$270$	0	0
$271$	327.858	1.20981	0.604903	−	0.796299i	$-0.293212\pi$
$271$	327.858	1.20981	0.604903	+	0.796299i	$0.293212\pi$
$272$	48.0596	0.176690
$273$	0	0
$274$	−69.3924	−0.253257
$275$	0	0
$276$	0	0
$277$	245.825i	0.887454i	0.896162	+	0.443727i	$0.146344\pi$
$277$	245.825i	0.887454i	−0.896162	+	0.443727i	$0.853656\pi$
$278$	−180.017	−0.647542
$279$	0	0
$280$	0	0
$281$	460.199i	1.63772i	0.573994	+	0.818860i	$0.305393\pi$
$281$	460.199i	1.63772i	−0.573994	+	0.818860i	$0.694607\pi$
$282$	0	0
$283$	− 25.2621i	− 0.0892654i	−0.999003	−	0.0446327i	$-0.985788\pi$
$283$	− 25.2621i	− 0.0892654i	0.999003	−	0.0446327i	$-0.0142118\pi$
$284$	112.222i	0.395150i
$285$	0	0
$286$	61.5220	0.215112
$287$	104.049	0.362541
$288$	0	0
$289$	−144.642	−0.500491
$290$	0	0
$291$	0	0
$292$	− 252.742i	− 0.865553i
$293$	91.6688	0.312863	0.156431	−	0.987689i	$-0.450001\pi$
$293$	91.6688	0.312863	0.156431	+	0.987689i	$0.450001\pi$
$294$	0	0
$295$	0	0
$296$	109.879i	0.371214i
$297$	0	0
$298$	314.735i	1.05616i
$299$	− 403.100i	− 1.34816i
$300$	0	0
$301$	190.070	0.631463
$302$	−13.9222	−0.0461001
$303$	0	0
$304$	−120.279	−0.395656
$305$	0	0
$306$	0	0
$307$	− 211.387i	− 0.688558i	−0.938867	−	0.344279i	$-0.888123\pi$
$307$	− 211.387i	− 0.688558i	0.938867	−	0.344279i	$-0.111877\pi$
$308$	−12.3157	−0.0399860
$309$	0	0
$310$	0	0
$311$	− 271.854i	− 0.874128i	−0.899430	−	0.437064i	$-0.856018\pi$
$311$	− 271.854i	− 0.874128i	0.899430	−	0.437064i	$-0.143982\pi$
$312$	0	0
$313$	− 496.385i	− 1.58590i	−0.609289	−	0.792948i	$-0.708545\pi$
$313$	− 496.385i	− 1.58590i	0.609289	−	0.792948i	$-0.291455\pi$
$314$	162.399i	0.517194i
$315$	0	0
$316$	71.5100	0.226297
$317$	−598.335	−1.88749	−0.943746	−	0.330673i	$-0.892724\pi$
$317$	−598.335	−1.88749	−0.943746	+	0.330673i	$0.892724\pi$
$318$	0	0
$319$	−54.9686	−0.172315
$320$	0	0
$321$	0	0
$322$	80.6940i	0.250602i
$323$	−361.286	−1.11853
$324$	0	0
$325$	0	0
$326$	− 320.062i	− 0.981784i
$327$	0	0
$328$	111.233i	0.339126i
$329$	213.440i	0.648753i
$330$	0	0
$331$	−310.533	−0.938165	−0.469083	−	0.883154i	$-0.655415\pi$
$331$	−310.533	−0.938165	−0.469083	+	0.883154i	$0.655415\pi$
$332$	79.3444	0.238989
$333$	0	0
$334$	−18.5994	−0.0556869
$335$	0	0
$336$	0	0
$337$	160.169i	0.475280i	0.971353	+	0.237640i	$0.0763739\pi$
$337$	160.169i	0.475280i	−0.971353	+	0.237640i	$0.923626\pi$
$338$	255.067	0.754635
$339$	0	0
$340$	0	0
$341$	− 128.153i	− 0.375814i
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	203.194i	0.590679i
$345$	0	0
$346$	233.974	0.676226
$347$	312.663	0.901045	0.450522	−	0.892765i	$-0.351238\pi$
$347$	312.663	0.901045	0.450522	+	0.892765i	$0.351238\pi$
$348$	0	0
$349$	−498.867	−1.42942	−0.714709	−	0.699422i	$-0.753441\pi$
$349$	−498.867	−1.42942	−0.714709	+	0.699422i	$0.753441\pi$
$350$	0	0
$351$	0	0
$352$	− 13.1660i	− 0.0374034i
$353$	−322.080	−0.912409	−0.456205	−	0.889875i	$-0.650791\pi$
$353$	−322.080	−0.912409	−0.456205	+	0.889875i	$0.650791\pi$
$354$	0	0
$355$	0	0
$356$	− 112.774i	− 0.316780i
$357$	0	0
$358$	46.0700i	0.128687i
$359$	− 365.485i	− 1.01806i	−0.860748	−	0.509032i	$-0.830004\pi$
$359$	− 365.485i	− 1.01806i	0.860748	−	0.509032i	$-0.169996\pi$
$360$	0	0
$361$	543.195	1.50470
$362$	416.341	1.15011
$363$	0	0
$364$	−98.9043	−0.271715
$365$	0	0
$366$	0	0
$367$	49.5646i	0.135053i	0.997717	+	0.0675267i	$0.0215108\pi$
$367$	49.5646i	0.135053i	−0.997717	+	0.0675267i	$0.978489\pi$
$368$	−86.2655	−0.234417
$369$	0	0
$370$	0	0
$371$	− 188.357i	− 0.507701i
$372$	0	0
$373$	− 37.9572i	− 0.101762i	−0.998705	−	0.0508810i	$-0.983797\pi$
$373$	− 37.9572i	− 0.101762i	0.998705	−	0.0508810i	$-0.0162029\pi$
$374$	− 39.5471i	− 0.105741i
$375$	0	0
$376$	−228.177	−0.606853
$377$	−441.440	−1.17093
$378$	0	0
$379$	274.865	0.725237	0.362619	−	0.931938i	$-0.381883\pi$
$379$	274.865	0.725237	0.362619	+	0.931938i	$0.381883\pi$
$380$	0	0
$381$	0	0
$382$	− 363.004i	− 0.950272i
$383$	211.809	0.553025	0.276513	−	0.961010i	$-0.410821\pi$
$383$	211.809	0.553025	0.276513	+	0.961010i	$0.410821\pi$
$384$	0	0
$385$	0	0
$386$	80.7446i	0.209183i
$387$	0	0
$388$	− 259.035i	− 0.667616i
$389$	− 500.529i	− 1.28671i	−0.765569	−	0.643354i	$-0.777542\pi$
$389$	− 500.529i	− 1.28671i	0.765569	−	0.643354i	$-0.222458\pi$
$390$	0	0
$391$	−259.118	−0.662706
$392$	19.7990	0.0505076
$393$	0	0
$394$	−270.318	−0.686085
$395$	0	0
$396$	0	0
$397$	− 201.976i	− 0.508755i	−0.967105	−	0.254378i	$-0.918129\pi$
$397$	− 201.976i	− 0.508755i	0.967105	−	0.254378i	$-0.0818706\pi$
$398$	380.849	0.956906
$399$	0	0
$400$	0	0
$401$	− 685.665i	− 1.70989i	−0.518721	−	0.854944i	$-0.673592\pi$
$401$	− 685.665i	− 1.70989i	0.518721	−	0.854944i	$-0.326408\pi$
$402$	0	0
$403$	− 1029.16i	− 2.55375i
$404$	− 39.5962i	− 0.0980103i
$405$	0	0
$406$	88.3689	0.217657
$407$	90.4169	0.222155
$408$	0	0
$409$	−451.082	−1.10289	−0.551445	−	0.834211i	$-0.685923\pi$
$409$	−451.082	−1.10289	−0.551445	+	0.834211i	$0.685923\pi$
$410$	0	0
$411$	0	0
$412$	221.264i	0.537050i
$413$	−135.838	−0.328905
$414$	0	0
$415$	0	0
$416$	− 105.733i	− 0.254166i
$417$	0	0
$418$	98.9750i	0.236782i
$419$	− 652.743i	− 1.55786i	−0.627111	−	0.778930i	$-0.715763\pi$
$419$	− 652.743i	− 1.55786i	0.627111	−	0.778930i	$-0.284237\pi$
$420$	0	0
$421$	566.058	1.34456	0.672278	−	0.740299i	$-0.265316\pi$
$421$	566.058	1.34456	0.672278	+	0.740299i	$0.265316\pi$
$422$	295.794	0.700934
$423$	0	0
$424$	201.362	0.474911
$425$	0	0
$426$	0	0
$427$	− 5.18054i	− 0.0121324i
$428$	348.528	0.814318
$429$	0	0
$430$	0	0
$431$	− 174.960i	− 0.405939i	−0.979185	−	0.202969i	$-0.934941\pi$
$431$	− 174.960i	− 0.405939i	0.979185	−	0.202969i	$-0.0650592\pi$
$432$	0	0
$433$	3.68329i	0.00850645i	0.999991	+	0.00425322i	$0.00135385\pi$
$433$	3.68329i	0.00850645i	−0.999991	+	0.00425322i	$0.998646\pi$
$434$	206.021i	0.474703i
$435$	0	0
$436$	−208.453	−0.478104
$437$	648.497	1.48398
$438$	0	0
$439$	−459.976	−1.04778	−0.523891	−	0.851785i	$-0.675520\pi$
$439$	−459.976	−1.04778	−0.523891	+	0.851785i	$0.675520\pi$
$440$	0	0
$441$	0	0
$442$	− 317.594i	− 0.718538i
$443$	−132.661	−0.299461	−0.149731	−	0.988727i	$-0.547841\pi$
$443$	−132.661	−0.299461	−0.149731	+	0.988727i	$0.547841\pi$
$444$	0	0
$445$	0	0
$446$	127.947i	0.286876i
$447$	0	0
$448$	21.1660i	0.0472456i
$449$	− 181.615i	− 0.404488i	−0.979335	−	0.202244i	$-0.935177\pi$
$449$	− 181.615i	− 0.404488i	0.979335	−	0.202244i	$-0.0648233\pi$
$450$	0	0
$451$	91.5311	0.202951
$452$	227.936	0.504284
$453$	0	0
$454$	−296.271	−0.652580
$455$	0	0
$456$	0	0
$457$	− 31.5784i	− 0.0690992i	−0.999403	−	0.0345496i	$-0.989000\pi$
$457$	− 31.5784i	− 0.0690992i	0.999403	−	0.0345496i	$-0.0109997\pi$
$458$	−348.718	−0.761393
$459$	0	0
$460$	0	0
$461$	− 184.100i	− 0.399348i	−0.979862	−	0.199674i	$-0.936012\pi$
$461$	− 184.100i	− 0.399348i	0.979862	−	0.199674i	$-0.0639883\pi$
$462$	0	0
$463$	− 382.078i	− 0.825223i	−0.910907	−	0.412612i	$-0.864617\pi$
$463$	− 382.078i	− 0.825223i	0.910907	−	0.412612i	$-0.135383\pi$
$464$	94.4703i	0.203600i
$465$	0	0
$466$	−467.275	−1.00274
$467$	682.737	1.46196	0.730982	−	0.682396i	$-0.239062\pi$
$467$	682.737	1.46196	0.730982	+	0.682396i	$0.239062\pi$
$468$	0	0
$469$	−178.931	−0.381516
$470$	0	0
$471$	0	0
$472$	− 145.216i	− 0.307662i
$473$	167.203	0.353495
$474$	0	0
$475$	0	0
$476$	63.5769i	0.133565i
$477$	0	0
$478$	439.874i	0.920238i
$479$	478.628i	0.999223i	0.866250	+	0.499611i	$0.166524\pi$
$479$	478.628i	0.999223i	−0.866250	+	0.499611i	$0.833476\pi$
$480$	0	0
$481$	726.117	1.50960
$482$	−281.724	−0.584489
$483$	0	0
$484$	231.166	0.477616
$485$	0	0
$486$	0	0
$487$	− 412.636i	− 0.847303i	−0.905825	−	0.423651i	$-0.860748\pi$
$487$	− 412.636i	− 0.847303i	0.905825	−	0.423651i	$-0.139252\pi$
$488$	5.53823	0.0113488
$489$	0	0
$490$	0	0
$491$	− 166.144i	− 0.338380i	−0.985583	−	0.169190i	$-0.945885\pi$
$491$	− 166.144i	− 0.338380i	0.985583	−	0.169190i	$-0.0541151\pi$
$492$	0	0
$493$	283.763i	0.575584i
$494$	794.845i	1.60900i
$495$	0	0
$496$	−220.246	−0.444044
$497$	−148.456	−0.298705
$498$	0	0
$499$	152.599	0.305810	0.152905	−	0.988241i	$-0.451137\pi$
$499$	152.599	0.305810	0.152905	+	0.988241i	$0.451137\pi$
$500$	0	0
$501$	0	0
$502$	59.5235i	0.118573i
$503$	−886.442	−1.76231	−0.881155	−	0.472827i	$-0.843234\pi$
$503$	−886.442	−1.76231	−0.881155	+	0.472827i	$0.843234\pi$
$504$	0	0
$505$	0	0
$506$	70.9858i	0.140288i
$507$	0	0
$508$	− 182.719i	− 0.359684i
$509$	71.9303i	0.141317i	0.997501	+	0.0706584i	$0.0225100\pi$
$509$	71.9303i	0.141317i	−0.997501	+	0.0706584i	$0.977490\pi$
$510$	0	0
$511$	334.346	0.654297
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	−387.425	−0.753745
$515$	0	0
$516$	0	0
$517$	187.761i	0.363174i
$518$	−145.357	−0.280611
$519$	0	0
$520$	0	0
$521$	− 315.499i	− 0.605564i	−0.953060	−	0.302782i	$-0.902085\pi$
$521$	− 315.499i	− 0.605564i	0.953060	−	0.302782i	$-0.0979155\pi$
$522$	0	0
$523$	338.473i	0.647176i	0.946198	+	0.323588i	$0.104889\pi$
$523$	338.473i	0.647176i	−0.946198	+	0.323588i	$0.895111\pi$
$524$	− 37.8085i	− 0.0721536i
$525$	0	0
$526$	−31.4290	−0.0597509
$527$	−661.559	−1.25533
$528$	0	0
$529$	−63.8917	−0.120778
$530$	0	0
$531$	0	0
$532$	− 159.115i	− 0.299088i
$533$	735.065	1.37911
$534$	0	0
$535$	0	0
$536$	− 191.285i	− 0.356875i
$537$	0	0
$538$	185.961i	0.345652i
$539$	− 16.2921i	− 0.0302265i
$540$	0	0
$541$	−23.7715	−0.0439400	−0.0219700	−	0.999759i	$-0.506994\pi$
$541$	−23.7715	−0.0439400	−0.0219700	+	0.999759i	$0.506994\pi$
$542$	−463.661	−0.855462
$543$	0	0
$544$	−67.9666	−0.124939
$545$	0	0
$546$	0	0
$547$	− 277.586i	− 0.507469i	−0.967274	−	0.253734i	$-0.918341\pi$
$547$	− 277.586i	− 0.507469i	0.967274	−	0.253734i	$-0.0816589\pi$
$548$	98.1356	0.179080
$549$	0	0
$550$	0	0
$551$	− 710.177i	− 1.28889i
$552$	0	0
$553$	94.5988i	0.171065i
$554$	− 347.649i	− 0.627525i
$555$	0	0
$556$	254.582	0.457881
$557$	−1030.67	−1.85040	−0.925201	−	0.379479i	$-0.876103\pi$
$557$	−1030.67	−1.85040	−0.925201	+	0.379479i	$0.876103\pi$
$558$	0	0
$559$	1342.77	2.40209
$560$	0	0
$561$	0	0
$562$	− 650.820i	− 1.15804i
$563$	−730.338	−1.29722	−0.648612	−	0.761119i	$-0.724650\pi$
$563$	−730.338	−1.29722	−0.648612	+	0.761119i	$0.724650\pi$
$564$	0	0
$565$	0	0
$566$	35.7260i	0.0631202i
$567$	0	0
$568$	− 158.707i	− 0.279413i
$569$	− 1113.46i	− 1.95687i	−0.206555	−	0.978435i	$-0.566225\pi$
$569$	− 1113.46i	− 1.95687i	0.206555	−	0.978435i	$-0.433775\pi$
$570$	0	0
$571$	−400.319	−0.701083	−0.350542	−	0.936547i	$-0.614002\pi$
$571$	−400.319	−0.701083	−0.350542	+	0.936547i	$0.614002\pi$
$572$	−87.0052	−0.152107
$573$	0	0
$574$	−147.148	−0.256355
$575$	0	0
$576$	0	0
$577$	− 157.271i	− 0.272566i	−0.990670	−	0.136283i	$-0.956484\pi$
$577$	− 157.271i	− 0.272566i	0.990670	−	0.136283i	$-0.0435156\pi$
$578$	204.555	0.353901
$579$	0	0
$580$	0	0
$581$	104.963i	0.180659i
$582$	0	0
$583$	− 165.696i	− 0.284213i
$584$	357.431i	0.612039i
$585$	0	0
$586$	−129.639	−0.221227
$587$	330.729	0.563422	0.281711	−	0.959499i	$-0.409098\pi$
$587$	330.729	0.563422	0.281711	+	0.959499i	$0.409098\pi$
$588$	0	0
$589$	1655.69	2.81102
$590$	0	0
$591$	0	0
$592$	− 155.393i	− 0.262488i
$593$	53.0569	0.0894719	0.0447360	−	0.998999i	$-0.485755\pi$
$593$	53.0569	0.0894719	0.0447360	+	0.998999i	$0.485755\pi$
$594$	0	0
$595$	0	0
$596$	− 445.103i	− 0.746817i
$597$	0	0
$598$	570.070i	0.953294i
$599$	1107.12i	1.84828i	0.382059	+	0.924138i	$0.375215\pi$
$599$	1107.12i	1.84828i	−0.382059	+	0.924138i	$0.624785\pi$
$600$	0	0
$601$	484.792	0.806642	0.403321	−	0.915058i	$-0.367856\pi$
$601$	484.792	0.806642	0.403321	+	0.915058i	$0.367856\pi$
$602$	−268.800	−0.446512
$603$	0	0
$604$	19.6890	0.0325977
$605$	0	0
$606$	0	0
$607$	631.961i	1.04112i	0.853824	+	0.520561i	$0.174277\pi$
$607$	631.961i	1.04112i	−0.853824	+	0.520561i	$0.825723\pi$
$608$	170.101	0.279771
$609$	0	0
$610$	0	0
$611$	1507.86i	2.46786i
$612$	0	0
$613$	− 1098.94i	− 1.79272i	−0.443324	−	0.896361i	$-0.646201\pi$
$613$	− 1098.94i	− 1.79272i	0.443324	−	0.896361i	$-0.353799\pi$
$614$	298.947i	0.486884i
$615$	0	0
$616$	17.4170	0.0282743
$617$	1135.21	1.83989	0.919943	−	0.392053i	$-0.128235\pi$
$617$	1135.21	1.83989	0.919943	+	0.392053i	$0.128235\pi$
$618$	0	0
$619$	−250.735	−0.405064	−0.202532	−	0.979276i	$-0.564917\pi$
$619$	−250.735	−0.405064	−0.202532	+	0.979276i	$0.564917\pi$
$620$	0	0
$621$	0	0
$622$	384.459i	0.618102i
$623$	149.186	0.239463
$624$	0	0
$625$	0	0
$626$	701.995i	1.12140i
$627$	0	0
$628$	− 229.667i	− 0.365711i
$629$	− 466.757i	− 0.742063i
$630$	0	0
$631$	−0.958851	−0.00151957	−0.000759787	−	1.00000i	$-0.500242\pi$
$631$	−0.958851	−0.00151957	−0.000759787		1.00000i	$0.500242\pi$
$632$	−101.130	−0.160016
$633$	0	0
$634$	846.173	1.33466
$635$	0	0
$636$	0	0
$637$	− 130.838i	− 0.205397i
$638$	77.7373	0.121845
$639$	0	0
$640$	0	0
$641$	− 1061.83i	− 1.65652i	−0.560343	−	0.828260i	$-0.689331\pi$
$641$	− 1061.83i	− 1.65652i	0.560343	−	0.828260i	$-0.310669\pi$
$642$	0	0
$643$	− 303.612i	− 0.472180i	−0.971731	−	0.236090i	$-0.924134\pi$
$643$	− 303.612i	− 0.472180i	0.971731	−	0.236090i	$-0.0758660\pi$
$644$	− 114.119i	− 0.177203i
$645$	0	0
$646$	510.936	0.790923
$647$	−657.623	−1.01642	−0.508210	−	0.861233i	$-0.669693\pi$
$647$	−657.623	−1.01642	−0.508210	+	0.861233i	$0.669693\pi$
$648$	0	0
$649$	−119.495	−0.184122
$650$	0	0
$651$	0	0
$652$	452.636i	0.694226i
$653$	−31.5315	−0.0482871	−0.0241436	−	0.999709i	$-0.507686\pi$
$653$	−31.5315	−0.0482871	−0.0241436	+	0.999709i	$0.507686\pi$
$654$	0	0
$655$	0	0
$656$	− 157.307i	− 0.239798i
$657$	0	0
$658$	− 301.849i	− 0.458737i
$659$	671.484i	1.01894i	0.860487	+	0.509472i	$0.170159\pi$
$659$	671.484i	1.01894i	−0.860487	+	0.509472i	$0.829841\pi$
$660$	0	0
$661$	−843.588	−1.27623	−0.638115	−	0.769941i	$-0.720286\pi$
$661$	−843.588	−1.27623	−0.638115	+	0.769941i	$0.720286\pi$
$662$	439.160	0.663383
$663$	0	0
$664$	−112.210	−0.168991
$665$	0	0
$666$	0	0
$667$	− 509.345i	− 0.763636i
$668$	26.3036	0.0393766
$669$	0	0
$670$	0	0
$671$	− 4.55727i	− 0.00679176i
$672$	0	0
$673$	810.581i	1.20443i	0.798334	+	0.602214i	$0.205715\pi$
$673$	810.581i	1.20443i	−0.798334	+	0.602214i	$0.794285\pi$
$674$	− 226.514i	− 0.336074i
$675$	0	0
$676$	−360.719	−0.533607
$677$	−230.728	−0.340810	−0.170405	−	0.985374i	$-0.554508\pi$
$677$	−230.728	−0.340810	−0.170405	+	0.985374i	$0.554508\pi$
$678$	0	0
$679$	342.671	0.504670
$680$	0	0
$681$	0	0
$682$	181.235i	0.265741i
$683$	608.618	0.891095	0.445547	−	0.895258i	$-0.353009\pi$
$683$	608.618	0.891095	0.445547	+	0.895258i	$0.353009\pi$
$684$	0	0
$685$	0	0
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	− 287.359i	− 0.417673i
$689$	− 1330.67i	− 1.93130i
$690$	0	0
$691$	−228.711	−0.330985	−0.165493	−	0.986211i	$-0.552921\pi$
$691$	−228.711	−0.330985	−0.165493	+	0.986211i	$0.552921\pi$
$692$	−330.889	−0.478164
$693$	0	0
$694$	−442.172	−0.637135
$695$	0	0
$696$	0	0
$697$	− 472.509i	− 0.677918i
$698$	705.504	1.01075
$699$	0	0
$700$	0	0
$701$	− 51.0999i	− 0.0728958i	−0.999336	−	0.0364479i	$-0.988396\pi$
$701$	− 51.0999i	− 0.0728958i	0.999336	−	0.0364479i	$-0.0116043\pi$
$702$	0	0
$703$	1168.16i	1.66168i
$704$	18.6196i	0.0264482i
$705$	0	0
$706$	455.491	0.645171
$707$	52.3808	0.0740888
$708$	0	0
$709$	−481.591	−0.679254	−0.339627	−	0.940560i	$-0.610301\pi$
$709$	−481.591	−0.679254	−0.339627	+	0.940560i	$0.610301\pi$
$710$	0	0
$711$	0	0
$712$	159.486i	0.223997i
$713$	1187.48	1.66546
$714$	0	0
$715$	0	0
$716$	− 65.1528i	− 0.0909955i
$717$	0	0
$718$	516.874i	0.719880i
$719$	− 886.683i	− 1.23322i	−0.787270	−	0.616608i	$-0.788506\pi$
$719$	− 886.683i	− 1.23322i	0.787270	−	0.616608i	$-0.211494\pi$
$720$	0	0
$721$	−292.705	−0.405971
$722$	−768.194	−1.06398
$723$	0	0
$724$	−588.795	−0.813253
$725$	0	0
$726$	0	0
$727$	− 113.429i	− 0.156024i	−0.996952	−	0.0780119i	$-0.975143\pi$
$727$	− 113.429i	− 0.156024i	0.996952	−	0.0780119i	$-0.0248572\pi$
$728$	139.872	0.192132
$729$	0	0
$730$	0	0
$731$	− 863.149i	− 1.18078i
$732$	0	0
$733$	− 477.858i	− 0.651921i	−0.945383	−	0.325960i	$-0.894312\pi$
$733$	− 477.858i	− 0.651921i	0.945383	−	0.325960i	$-0.105688\pi$
$734$	− 70.0949i	− 0.0954972i
$735$	0	0
$736$	121.998	0.165758
$737$	−157.404	−0.213574
$738$	0	0
$739$	−1179.50	−1.59608	−0.798038	−	0.602607i	$-0.794129\pi$
$739$	−1179.50	−1.59608	−0.798038	+	0.602607i	$0.794129\pi$
$740$	0	0
$741$	0	0
$742$	266.377i	0.358999i
$743$	−126.573	−0.170354	−0.0851770	−	0.996366i	$-0.527146\pi$
$743$	−126.573	−0.170354	−0.0851770	+	0.996366i	$0.527146\pi$
$744$	0	0
$745$	0	0
$746$	53.6796i	0.0719566i
$747$	0	0
$748$	55.9281i	0.0747701i
$749$	461.059i	0.615566i
$750$	0	0
$751$	181.769	0.242035	0.121018	−	0.992650i	$-0.461384\pi$
$751$	181.769	0.242035	0.121018	+	0.992650i	$0.461384\pi$
$752$	322.690	0.429110
$753$	0	0
$754$	624.290	0.827971
$755$	0	0
$756$	0	0
$757$	42.6185i	0.0562992i	0.999604	+	0.0281496i	$0.00896148\pi$
$757$	42.6185i	0.0562992i	−0.999604	+	0.0281496i	$0.991039\pi$
$758$	−388.718	−0.512820
$759$	0	0
$760$	0	0
$761$	350.054i	0.459993i	0.973191	+	0.229996i	$0.0738714\pi$
$761$	350.054i	0.459993i	−0.973191	+	0.229996i	$0.926129\pi$
$762$	0	0
$763$	− 275.758i	− 0.361413i
$764$	513.365i	0.671944i
$765$	0	0
$766$	−299.543	−0.391048
$767$	−959.637	−1.25116
$768$	0	0
$769$	1261.14	1.63998	0.819988	−	0.572381i	$-0.193980\pi$
$769$	1261.14	1.63998	0.819988	+	0.572381i	$0.193980\pi$
$770$	0	0
$771$	0	0
$772$	− 114.190i	− 0.147915i
$773$	−1312.02	−1.69731	−0.848657	−	0.528943i	$-0.822588\pi$
$773$	−1312.02	−1.69731	−0.848657	+	0.528943i	$0.822588\pi$
$774$	0	0
$775$	0	0
$776$	366.331i	0.472076i
$777$	0	0
$778$	707.855i	0.909840i
$779$	1182.55i	1.51804i
$780$	0	0
$781$	−130.596	−0.167216
$782$	366.448	0.468604
$783$	0	0
$784$	−28.0000	−0.0357143
$785$	0	0
$786$	0	0
$787$	129.898i	0.165055i	0.996589	+	0.0825273i	$0.0262992\pi$
$787$	129.898i	0.165055i	−0.996589	+	0.0825273i	$0.973701\pi$
$788$	382.287	0.485135
$789$	0	0
$790$	0	0
$791$	301.532i	0.381203i
$792$	0	0
$793$	− 36.5984i	− 0.0461518i
$794$	285.637i	0.359744i
$795$	0	0
$796$	−538.601	−0.676635
$797$	−698.272	−0.876125	−0.438063	−	0.898944i	$-0.644335\pi$
$797$	−698.272	−0.876125	−0.438063	+	0.898944i	$0.644335\pi$
$798$	0	0
$799$	969.274	1.21311
$800$	0	0
$801$	0	0
$802$	969.677i	1.20907i
$803$	294.121	0.366278
$804$	0	0
$805$	0	0
$806$	1455.46i	1.80578i
$807$	0	0
$808$	55.9974i	0.0693037i
$809$	− 153.076i	− 0.189216i	−0.995515	−	0.0946079i	$-0.969840\pi$
$809$	− 153.076i	− 0.189216i	0.995515	−	0.0946079i	$-0.0301597\pi$
$810$	0	0
$811$	−431.260	−0.531764	−0.265882	−	0.964006i	$-0.585663\pi$
$811$	−431.260	−0.531764	−0.265882	+	0.964006i	$0.585663\pi$
$812$	−124.972	−0.153907
$813$	0	0
$814$	−127.869	−0.157087
$815$	0	0
$816$	0	0
$817$	2160.21i	2.64408i
$818$	637.927	0.779861
$819$	0	0
$820$	0	0
$821$	− 147.878i	− 0.180119i	−0.995936	−	0.0900595i	$-0.971294\pi$
$821$	− 147.878i	− 0.180119i	0.995936	−	0.0900595i	$-0.0287057\pi$
$822$	0	0
$823$	243.546i	0.295925i	0.988993	+	0.147963i	$0.0472715\pi$
$823$	243.546i	0.295925i	−0.988993	+	0.147963i	$0.952728\pi$
$824$	− 312.915i	− 0.379751i
$825$	0	0
$826$	192.103	0.232571
$827$	−551.557	−0.666937	−0.333469	−	0.942761i	$-0.608219\pi$
$827$	−551.557	−0.666937	−0.333469	+	0.942761i	$0.608219\pi$
$828$	0	0
$829$	68.0040	0.0820313	0.0410157	−	0.999159i	$-0.486941\pi$
$829$	68.0040	0.0820313	0.0410157	+	0.999159i	$0.486941\pi$
$830$	0	0
$831$	0	0
$832$	149.529i	0.179723i
$833$	−84.1044	−0.100966
$834$	0	0
$835$	0	0
$836$	− 139.972i	− 0.167430i
$837$	0	0
$838$	923.118i	1.10157i
$839$	527.646i	0.628899i	0.949274	+	0.314449i	$0.101820\pi$
$839$	527.646i	0.628899i	−0.949274	+	0.314449i	$0.898180\pi$
$840$	0	0
$841$	283.210	0.336754
$842$	−800.527	−0.950745
$843$	0	0
$844$	−418.316	−0.495636
$845$	0	0
$846$	0	0
$847$	305.804i	0.361044i
$848$	−284.769	−0.335813
$849$	0	0
$850$	0	0
$851$	837.814i	0.984505i
$852$	0	0
$853$	569.082i	0.667154i	0.942723	+	0.333577i	$0.108256\pi$
$853$	569.082i	0.667154i	−0.942723	+	0.333577i	$0.891744\pi$
$854$	7.32639i	0.00857891i
$855$	0	0
$856$	−492.893	−0.575810
$857$	−1078.60	−1.25858	−0.629288	−	0.777172i	$-0.716653\pi$
$857$	−1078.60	−1.25858	−0.629288	+	0.777172i	$0.716653\pi$
$858$	0	0
$859$	766.027	0.891766	0.445883	−	0.895091i	$-0.352890\pi$
$859$	766.027	0.891766	0.445883	+	0.895091i	$0.352890\pi$
$860$	0	0
$861$	0	0
$862$	247.430i	0.287042i
$863$	−753.851	−0.873524	−0.436762	−	0.899577i	$-0.643875\pi$
$863$	−753.851	−0.873524	−0.436762	+	0.899577i	$0.643875\pi$
$864$	0	0
$865$	0	0
$866$	− 5.20896i	− 0.00601497i
$867$	0	0
$868$	− 291.358i	− 0.335666i
$869$	83.2177i	0.0957626i
$870$	0	0
$871$	−1264.07	−1.45129
$872$	294.798	0.338071
$873$	0	0
$874$	−917.114	−1.04933
$875$	0	0
$876$	0	0
$877$	− 253.776i	− 0.289369i	−0.989478	−	0.144684i	$-0.953783\pi$
$877$	− 253.776i	− 0.289369i	0.989478	−	0.144684i	$-0.0462167\pi$
$878$	650.505	0.740894
$879$	0	0
$880$	0	0
$881$	841.004i	0.954602i	0.878740	+	0.477301i	$0.158385\pi$
$881$	841.004i	0.954602i	−0.878740	+	0.477301i	$0.841615\pi$
$882$	0	0
$883$	1084.98i	1.22874i	0.789017	+	0.614371i	$0.210590\pi$
$883$	1084.98i	1.22874i	−0.789017	+	0.614371i	$0.789410\pi$
$884$	449.145i	0.508083i
$885$	0	0
$886$	187.611	0.211751
$887$	598.029	0.674216	0.337108	−	0.941466i	$-0.390551\pi$
$887$	598.029	0.674216	0.337108	+	0.941466i	$0.390551\pi$
$888$	0	0
$889$	241.715	0.271895
$890$	0	0
$891$	0	0
$892$	− 180.944i	− 0.202852i
$893$	−2425.81	−2.71648
$894$	0	0
$895$	0	0
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	256.842i	0.286016i
$899$	− 1300.42i	− 1.44652i
$900$	0	0
$901$	−855.369	−0.949355
$902$	−129.445	−0.143508
$903$	0	0
$904$	−322.351	−0.356583
$905$	0	0
$906$	0	0
$907$	1465.14i	1.61537i	0.589618	+	0.807683i	$0.299279\pi$
$907$	1465.14i	1.61537i	−0.589618	+	0.807683i	$0.700721\pi$
$908$	418.991	0.461444
$909$	0	0
$910$	0	0
$911$	− 532.912i	− 0.584975i	−0.956269	−	0.292487i	$-0.905517\pi$
$911$	− 532.912i	− 0.584975i	0.956269	−	0.292487i	$-0.0944829\pi$
$912$	0	0
$913$	92.3349i	0.101133i
$914$	44.6585i	0.0488605i
$915$	0	0
$916$	493.162	0.538386
$917$	50.0159	0.0545430
$918$	0	0
$919$	491.253	0.534552	0.267276	−	0.963620i	$-0.413876\pi$
$919$	491.253	0.534552	0.267276	+	0.963620i	$0.413876\pi$
$920$	0	0
$921$	0	0
$922$	260.356i	0.282382i
$923$	−1048.78	−1.13628
$924$	0	0
$925$	0	0
$926$	540.340i	0.583521i
$927$	0	0
$928$	− 133.601i	− 0.143967i
$929$	− 1821.16i	− 1.96034i	−0.198149	−	0.980172i	$-0.563493\pi$
$929$	− 1821.16i	− 1.96034i	0.198149	−	0.980172i	$-0.436507\pi$
$930$	0	0
$931$	210.489	0.226089
$932$	660.827	0.709042
$933$	0	0
$934$	−965.537	−1.03376
$935$	0	0
$936$	0	0
$937$	1089.05i	1.16227i	0.813807	+	0.581135i	$0.197391\pi$
$937$	1089.05i	1.16227i	−0.813807	+	0.581135i	$0.802609\pi$
$938$	253.046	0.269772
$939$	0	0
$940$	0	0
$941$	− 768.831i	− 0.817036i	−0.912750	−	0.408518i	$-0.866046\pi$
$941$	− 768.831i	− 0.817036i	0.912750	−	0.408518i	$-0.133954\pi$
$942$	0	0
$943$	848.138i	0.899404i
$944$	205.367i	0.217550i
$945$	0	0
$946$	−236.461	−0.249959
$947$	553.026	0.583977	0.291988	−	0.956422i	$-0.405683\pi$
$947$	553.026	0.583977	0.291988	+	0.956422i	$0.405683\pi$
$948$	0	0
$949$	2362.02	2.48895
$950$	0	0
$951$	0	0
$952$	− 89.9114i	− 0.0944447i
$953$	−217.307	−0.228024	−0.114012	−	0.993479i	$-0.536370\pi$
$953$	−217.307	−0.228024	−0.114012	+	0.993479i	$0.536370\pi$
$954$	0	0
$955$	0	0
$956$	− 622.075i	− 0.650706i
$957$	0	0
$958$	− 676.882i	− 0.706557i
$959$	129.821i	0.135371i
$960$	0	0
$961$	2070.77	2.15480
$962$	−1026.88	−1.06745
$963$	0	0
$964$	398.418	0.413296
$965$	0	0
$966$	0	0
$967$	− 34.3234i	− 0.0354947i	−0.999843	−	0.0177474i	$-0.994351\pi$
$967$	− 34.3234i	− 0.0354947i	0.999843	−	0.0177474i	$-0.00564946\pi$
$968$	−326.918	−0.337725
$969$	0	0
$970$	0	0
$971$	1388.95i	1.43043i	0.698905	+	0.715214i	$0.253671\pi$
$971$	1388.95i	1.43043i	−0.698905	+	0.715214i	$0.746329\pi$
$972$	0	0
$973$	336.780i	0.346126i
$974$	583.556i	0.599134i
$975$	0	0
$976$	−7.83224	−0.00802483
$977$	−1088.47	−1.11410	−0.557048	−	0.830481i	$-0.688066\pi$
$977$	−1088.47	−1.11410	−0.557048	+	0.830481i	$0.688066\pi$
$978$	0	0
$979$	131.237	0.134052
$980$	0	0
$981$	0	0
$982$	234.964i	0.239271i
$983$	−1276.47	−1.29854	−0.649272	−	0.760557i	$-0.724926\pi$
$983$	−1276.47	−1.29854	−0.649272	+	0.760557i	$0.724926\pi$
$984$	0	0
$985$	0	0
$986$	− 401.302i	− 0.407000i
$987$	0	0
$988$	− 1124.08i	− 1.13773i
$989$	1549.32i	1.56656i
$990$	0	0
$991$	1461.19	1.47446	0.737229	−	0.675643i	$-0.236134\pi$
$991$	1461.19	1.47446	0.737229	+	0.675643i	$0.236134\pi$
$992$	311.475	0.313987
$993$	0	0
$994$	209.949	0.211216
$995$	0	0
$996$	0	0
$997$	− 1193.26i	− 1.19685i	−0.801177	−	0.598427i	$-0.795793\pi$
$997$	− 1193.26i	− 1.19685i	0.801177	−	0.598427i	$-0.204207\pi$
$998$	−215.808	−0.216240
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.3.c.d.449.6		16
3.2	odd	2	inner	3150.3.c.d.449.16		16
5.2	odd	4		3150.3.e.i.701.2		8
5.3	odd	4		630.3.e.a.71.6	yes	8
5.4	even	2	inner	3150.3.c.d.449.9		16
15.2	even	4		3150.3.e.i.701.6		8
15.8	even	4		630.3.e.a.71.4	✓	8
15.14	odd	2	inner	3150.3.c.d.449.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.3.e.a.71.4	✓	8	15.8	even	4
630.3.e.a.71.6	yes	8	5.3	odd	4
3150.3.c.d.449.3		16	15.14	odd	2	inner
3150.3.c.d.449.6		16	1.1	even	1	trivial
3150.3.c.d.449.9		16	5.4	even	2	inner
3150.3.c.d.449.16		16	3.2	odd	2	inner
3150.3.e.i.701.2		8	5.2	odd	4
3150.3.e.i.701.6		8	15.2	even	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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +2.32744i q^{11} +18.6912i q^{13} -3.74166i q^{14} +4.00000 q^{16} +12.0149 q^{17} -30.0698 q^{19} -3.29150i q^{22} -21.5664 q^{23} -26.4333i q^{26} +5.29150i q^{28} +23.6176i q^{29} -55.0615 q^{31} -5.65685 q^{32} -16.9917 q^{34} -38.8482i q^{37} +42.5252 q^{38} -39.3269i q^{41} -71.8398i q^{43} +4.65489i q^{44} +30.4995 q^{46} +80.6726 q^{47} -7.00000 q^{49} +37.3823i q^{52} -71.1923 q^{53} -7.48331i q^{56} -33.4003i q^{58} +51.3418i q^{59} -1.95806 q^{61} +77.8687 q^{62} +8.00000 q^{64} +67.6295i q^{67} +24.0298 q^{68} +56.1112i q^{71} -126.371i q^{73} +54.9396i q^{74} -60.1397 q^{76} -6.15784 q^{77} +35.7550 q^{79} +55.6166i q^{82} +39.6722 q^{83} +101.597i q^{86} -6.58301i q^{88} -56.3868i q^{89} -49.4521 q^{91} -43.1327 q^{92} -114.088 q^{94} -129.518i q^{97} +9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} + 64 q^{16} - 160 q^{19} - 224 q^{31} + 64 q^{34} - 64 q^{46} - 112 q^{49} - 160 q^{61} + 128 q^{64} - 320 q^{76} + 128 q^{94}+O(q^{100}) \) 16 * q + 32 * q^4 + 64 * q^16 - 160 * q^19 - 224 * q^31 + 64 * q^34 - 64 * q^46 - 112 * q^49 - 160 * q^61 + 128 * q^64 - 320 * q^76 + 128 * q^94

Introduction

L-functions

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Database

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