LMFDB - Embedded newform 3150.3.c.f.449.3

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} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$-0.796626 - 0.359610i$ of defining polynomial
Character	$\chi$	$=$	3150.449
Dual form			3150.3.c.f.449.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} -2.82843 q^{8} +3.78840i q^{11} -14.9496i q^{13} +3.74166i q^{14} +4.00000 q^{16} -0.335699 q^{17} +29.8955 q^{19} -5.35761i q^{22} +18.5255 q^{23} +21.1419i q^{26} -5.29150i q^{28} +11.1480i q^{29} +0.603437 q^{31} -5.65685 q^{32} +0.474751 q^{34} +41.9826i q^{37} -42.2786 q^{38} +35.7336i q^{41} -24.2590i q^{43} +7.57680i q^{44} -26.1991 q^{46} +50.3738 q^{47} -7.00000 q^{49} -29.8992i q^{52} -44.9810 q^{53} +7.48331i q^{56} -15.7657i q^{58} -16.1852i q^{59} -102.189 q^{61} -0.853389 q^{62} +8.00000 q^{64} -17.8987i q^{67} -0.671399 q^{68} +76.4296i q^{71} -63.2996i q^{73} -59.3723i q^{74} +59.7910 q^{76} +10.0232 q^{77} +60.6925 q^{79} -50.5349i q^{82} +129.480 q^{83} +34.3074i q^{86} -10.7152i q^{88} +15.0312i q^{89} -39.5529 q^{91} +37.0511 q^{92} -71.2394 q^{94} -59.4802i q^{97} +9.89949 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) $ 16 * q + 32 * q^4 + 64 * q^16 + 160 * q^19 + 224 * q^31 - 192 * q^34 - 64 * q^46 - 112 * q^49 - 288 * q^61 + 128 * q^64 + 320 * q^76 + 128 * q^79 - 448 * q^91

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
$11$	3.78840i	0.344400i	0.985062	+	0.172200i	$0.0550875\pi$
$11$	3.78840i	0.344400i	−0.985062	+	0.172200i	$0.944912\pi$
$12$	0	0
$13$	− 14.9496i	− 1.14997i	−0.818164	−	0.574985i	$-0.805008\pi$
$13$	− 14.9496i	− 1.14997i	0.818164	−	0.574985i	$-0.194992\pi$
$14$	3.74166i	0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	−0.335699	−0.0197470	−0.00987351	−	0.999951i	$-0.503143\pi$
$17$	−0.335699	−0.0197470	−0.00987351	+	0.999951i	$0.503143\pi$
$18$	0	0
$19$	29.8955	1.57345	0.786723	−	0.617306i	$-0.211776\pi$
$19$	29.8955	1.57345	0.786723	+	0.617306i	$0.211776\pi$
$20$	0	0
$21$	0	0
$22$	− 5.35761i	− 0.243528i
$23$	18.5255	0.805458	0.402729	−	0.915319i	$-0.368062\pi$
$23$	18.5255	0.805458	0.402729	+	0.915319i	$0.368062\pi$
$24$	0	0
$25$	0	0
$26$	21.1419i	0.813151i
$27$	0	0
$28$	− 5.29150i	− 0.188982i
$29$	11.1480i	0.384415i	0.981354	+	0.192208i	$0.0615647\pi$
$29$	11.1480i	0.384415i	−0.981354	+	0.192208i	$0.938435\pi$
$30$	0	0
$31$	0.603437	0.0194657	0.00973286	−	0.999953i	$-0.496902\pi$
$31$	0.603437	0.0194657	0.00973286	+	0.999953i	$0.496902\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	0.474751	0.0139633
$35$	0	0
$36$	0	0
$37$	41.9826i	1.13466i	0.823489	+	0.567332i	$0.192024\pi$
$37$	41.9826i	1.13466i	−0.823489	+	0.567332i	$0.807976\pi$
$38$	−42.2786	−1.11259
$39$	0	0
$40$	0	0
$41$	35.7336i	0.871550i	0.900056	+	0.435775i	$0.143526\pi$
$41$	35.7336i	0.871550i	−0.900056	+	0.435775i	$0.856474\pi$
$42$	0	0
$43$	− 24.2590i	− 0.564163i	−0.959391	−	0.282081i	$-0.908975\pi$
$43$	− 24.2590i	− 0.564163i	0.959391	−	0.282081i	$-0.0910248\pi$
$44$	7.57680i	0.172200i
$45$	0	0
$46$	−26.1991	−0.569545
$47$	50.3738	1.07178	0.535892	−	0.844287i	$-0.319975\pi$
$47$	50.3738	1.07178	0.535892	+	0.844287i	$0.319975\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 29.8992i	− 0.574985i
$53$	−44.9810	−0.848699	−0.424349	−	0.905499i	$-0.639497\pi$
$53$	−44.9810	−0.848699	−0.424349	+	0.905499i	$0.639497\pi$
$54$	0	0
$55$	0	0
$56$	7.48331i	0.133631i
$57$	0	0
$58$	− 15.7657i	− 0.271823i
$59$	− 16.1852i	− 0.274326i	−0.990548	−	0.137163i	$-0.956202\pi$
$59$	− 16.1852i	− 0.274326i	0.990548	−	0.137163i	$-0.0437984\pi$
$60$	0	0
$61$	−102.189	−1.67523	−0.837615	−	0.546260i	$-0.816051\pi$
$61$	−102.189	−1.67523	−0.837615	+	0.546260i	$0.816051\pi$
$62$	−0.853389	−0.0137643
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	− 17.8987i	− 0.267144i	−0.991039	−	0.133572i	$-0.957355\pi$
$67$	− 17.8987i	− 0.267144i	0.991039	−	0.133572i	$-0.0426448\pi$
$68$	−0.671399	−0.00987351
$69$	0	0
$70$	0	0
$71$	76.4296i	1.07647i	0.842794	+	0.538237i	$0.180909\pi$
$71$	76.4296i	1.07647i	−0.842794	+	0.538237i	$0.819091\pi$
$72$	0	0
$73$	− 63.2996i	− 0.867118i	−0.901125	−	0.433559i	$-0.857258\pi$
$73$	− 63.2996i	− 0.867118i	0.901125	−	0.433559i	$-0.142742\pi$
$74$	− 59.3723i	− 0.802328i
$75$	0	0
$76$	59.7910	0.786723
$77$	10.0232	0.130171
$78$	0	0
$79$	60.6925	0.768260	0.384130	−	0.923279i	$-0.374501\pi$
$79$	60.6925	0.768260	0.384130	+	0.923279i	$0.374501\pi$
$80$	0	0
$81$	0	0
$82$	− 50.5349i	− 0.616279i
$83$	129.480	1.56001	0.780003	−	0.625776i	$-0.215218\pi$
$83$	129.480	1.56001	0.780003	+	0.625776i	$0.215218\pi$
$84$	0	0
$85$	0	0
$86$	34.3074i	0.398923i
$87$	0	0
$88$	− 10.7152i	− 0.121764i
$89$	15.0312i	0.168890i	0.996428	+	0.0844451i	$0.0269117\pi$
$89$	15.0312i	0.168890i	−0.996428	+	0.0844451i	$0.973088\pi$
$90$	0	0
$91$	−39.5529	−0.434648
$92$	37.0511	0.402729
$93$	0	0
$94$	−71.2394	−0.757866
$95$	0	0
$96$	0	0
$97$	− 59.4802i	− 0.613197i	−0.951839	−	0.306599i	$-0.900809\pi$
$97$	− 59.4802i	− 0.613197i	0.951839	−	0.306599i	$-0.0991910\pi$
$98$	9.89949	0.101015
$99$	0	0
$100$	0	0
$101$	34.9641i	0.346179i	0.984906	+	0.173089i	$0.0553750\pi$
$101$	34.9641i	0.346179i	−0.984906	+	0.173089i	$0.944625\pi$
$102$	0	0
$103$	− 16.5291i	− 0.160476i	−0.996776	−	0.0802381i	$-0.974432\pi$
$103$	− 16.5291i	− 0.160476i	0.996776	−	0.0802381i	$-0.0255681\pi$
$104$	42.2839i	0.406576i
$105$	0	0
$106$	63.6128	0.600121
$107$	−117.574	−1.09883	−0.549413	−	0.835551i	$-0.685149\pi$
$107$	−117.574	−1.09883	−0.549413	+	0.835551i	$0.685149\pi$
$108$	0	0
$109$	−2.23755	−0.0205280	−0.0102640	−	0.999947i	$-0.503267\pi$
$109$	−2.23755	−0.0205280	−0.0102640	+	0.999947i	$0.503267\pi$
$110$	0	0
$111$	0	0
$112$	− 10.5830i	− 0.0944911i
$113$	89.0131	0.787726	0.393863	−	0.919169i	$-0.371138\pi$
$113$	89.0131	0.787726	0.393863	+	0.919169i	$0.371138\pi$
$114$	0	0
$115$	0	0
$116$	22.2961i	0.192208i
$117$	0	0
$118$	22.8894i	0.193978i
$119$	0.888177i	0.00746367i
$120$	0	0
$121$	106.648	0.881389
$122$	144.517	1.18457
$123$	0	0
$124$	1.20687	0.00973286
$125$	0	0
$126$	0	0
$127$	− 234.130i	− 1.84354i	−0.387732	−	0.921772i	$-0.626741\pi$
$127$	− 234.130i	− 1.84354i	0.387732	−	0.921772i	$-0.373259\pi$
$128$	−11.3137	−0.0883883
$129$	0	0
$130$	0	0
$131$	− 17.0511i	− 0.130161i	−0.997880	−	0.0650805i	$-0.979270\pi$
$131$	− 17.0511i	− 0.130161i	0.997880	−	0.0650805i	$-0.0207304\pi$
$132$	0	0
$133$	− 79.0960i	− 0.594707i
$134$	25.3125i	0.188900i
$135$	0	0
$136$	0.949501	0.00698163
$137$	177.901	1.29855	0.649274	−	0.760554i	$-0.275073\pi$
$137$	177.901	1.29855	0.649274	+	0.760554i	$0.275073\pi$
$138$	0	0
$139$	192.255	1.38313	0.691563	−	0.722316i	$-0.256922\pi$
$139$	192.255	1.38313	0.691563	+	0.722316i	$0.256922\pi$
$140$	0	0
$141$	0	0
$142$	− 108.088i	− 0.761181i
$143$	56.6351	0.396050
$144$	0	0
$145$	0	0
$146$	89.5192i	0.613145i
$147$	0	0
$148$	83.9651i	0.567332i
$149$	231.696i	1.55501i	0.628880	+	0.777503i	$0.283514\pi$
$149$	231.696i	1.55501i	−0.628880	+	0.777503i	$0.716486\pi$
$150$	0	0
$151$	−238.478	−1.57932	−0.789661	−	0.613543i	$-0.789744\pi$
$151$	−238.478	−1.57932	−0.789661	+	0.613543i	$0.789744\pi$
$152$	−84.5572	−0.556297
$153$	0	0
$154$	−14.1749	−0.0920448
$155$	0	0
$156$	0	0
$157$	− 226.506i	− 1.44271i	−0.692564	−	0.721356i	$-0.743519\pi$
$157$	− 226.506i	− 1.44271i	0.692564	−	0.721356i	$-0.256481\pi$
$158$	−85.8322	−0.543242
$159$	0	0
$160$	0	0
$161$	− 49.0140i	− 0.304435i
$162$	0	0
$163$	38.9379i	0.238883i	0.992841	+	0.119441i	$0.0381104\pi$
$163$	38.9379i	0.238883i	−0.992841	+	0.119441i	$0.961890\pi$
$164$	71.4671i	0.435775i
$165$	0	0
$166$	−183.113	−1.10309
$167$	−195.038	−1.16789	−0.583945	−	0.811793i	$-0.698492\pi$
$167$	−195.038	−1.16789	−0.583945	+	0.811793i	$0.698492\pi$
$168$	0	0
$169$	−54.4907	−0.322430
$170$	0	0
$171$	0	0
$172$	− 48.5180i	− 0.282081i
$173$	−98.7868	−0.571022	−0.285511	−	0.958375i	$-0.592163\pi$
$173$	−98.7868	−0.571022	−0.285511	+	0.958375i	$0.592163\pi$
$174$	0	0
$175$	0	0
$176$	15.1536i	0.0861000i
$177$	0	0
$178$	− 21.2574i	− 0.119423i
$179$	164.993i	0.921751i	0.887465	+	0.460875i	$0.152464\pi$
$179$	164.993i	0.921751i	−0.887465	+	0.460875i	$0.847536\pi$
$180$	0	0
$181$	112.556	0.621854	0.310927	−	0.950434i	$-0.399361\pi$
$181$	112.556	0.621854	0.310927	+	0.950434i	$0.399361\pi$
$182$	55.9363	0.307342
$183$	0	0
$184$	−52.3981	−0.284772
$185$	0	0
$186$	0	0
$187$	− 1.27176i	− 0.00680088i
$188$	100.748	0.535892
$189$	0	0
$190$	0	0
$191$	− 261.319i	− 1.36816i	−0.729406	−	0.684081i	$-0.760204\pi$
$191$	− 261.319i	− 1.36816i	0.729406	−	0.684081i	$-0.239796\pi$
$192$	0	0
$193$	268.372i	1.39053i	0.718754	+	0.695265i	$0.244713\pi$
$193$	268.372i	1.39053i	−0.718754	+	0.695265i	$0.755287\pi$
$194$	84.1176i	0.433596i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	−51.9378	−0.263644	−0.131822	−	0.991273i	$-0.542083\pi$
$197$	−51.9378	−0.263644	−0.131822	+	0.991273i	$0.542083\pi$
$198$	0	0
$199$	307.244	1.54394	0.771970	−	0.635659i	$-0.219272\pi$
$199$	307.244	1.54394	0.771970	+	0.635659i	$0.219272\pi$
$200$	0	0
$201$	0	0
$202$	− 49.4466i	− 0.244785i
$203$	29.4950	0.145295
$204$	0	0
$205$	0	0
$206$	23.3756i	0.113474i
$207$	0	0
$208$	− 59.7984i	− 0.287492i
$209$	113.256i	0.541895i
$210$	0	0
$211$	−112.097	−0.531265	−0.265632	−	0.964074i	$-0.585581\pi$
$211$	−112.097	−0.531265	−0.265632	+	0.964074i	$0.585581\pi$
$212$	−89.9621	−0.424349
$213$	0	0
$214$	166.275	0.776988
$215$	0	0
$216$	0	0
$217$	− 1.59654i	− 0.00735735i
$218$	3.16437	0.0145155
$219$	0	0
$220$	0	0
$221$	5.01857i	0.0227085i
$222$	0	0
$223$	− 192.031i	− 0.861124i	−0.902561	−	0.430562i	$-0.858315\pi$
$223$	− 192.031i	− 0.861124i	0.902561	−	0.430562i	$-0.141685\pi$
$224$	14.9666i	0.0668153i
$225$	0	0
$226$	−125.883	−0.557006
$227$	−7.21336	−0.0317769	−0.0158885	−	0.999874i	$-0.505058\pi$
$227$	−7.21336	−0.0317769	−0.0158885	+	0.999874i	$0.505058\pi$
$228$	0	0
$229$	336.703	1.47032	0.735159	−	0.677894i	$-0.237107\pi$
$229$	336.703	1.47032	0.735159	+	0.677894i	$0.237107\pi$
$230$	0	0
$231$	0	0
$232$	− 31.5314i	− 0.135911i
$233$	255.652	1.09722	0.548609	−	0.836079i	$-0.315158\pi$
$233$	255.652	1.09722	0.548609	+	0.836079i	$0.315158\pi$
$234$	0	0
$235$	0	0
$236$	− 32.3705i	− 0.137163i
$237$	0	0
$238$	− 1.25607i	− 0.00527761i
$239$	− 59.2293i	− 0.247822i	−0.992293	−	0.123911i	$-0.960456\pi$
$239$	− 59.2293i	− 0.247822i	0.992293	−	0.123911i	$-0.0395437\pi$
$240$	0	0
$241$	67.0919	0.278390	0.139195	−	0.990265i	$-0.455549\pi$
$241$	67.0919	0.278390	0.139195	+	0.990265i	$0.455549\pi$
$242$	−150.823	−0.623236
$243$	0	0
$244$	−204.378	−0.837615
$245$	0	0
$246$	0	0
$247$	− 446.926i	− 1.80942i
$248$	−1.70678	−0.00688217
$249$	0	0
$250$	0	0
$251$	− 303.866i	− 1.21062i	−0.795990	−	0.605310i	$-0.793049\pi$
$251$	− 303.866i	− 1.21062i	0.795990	−	0.605310i	$-0.206951\pi$
$252$	0	0
$253$	70.1822i	0.277400i
$254$	331.110i	1.30358i
$255$	0	0
$256$	16.0000	0.0625000
$257$	44.2141	0.172039	0.0860197	−	0.996293i	$-0.472585\pi$
$257$	44.2141	0.172039	0.0860197	+	0.996293i	$0.472585\pi$
$258$	0	0
$259$	111.075	0.428863
$260$	0	0
$261$	0	0
$262$	24.1139i	0.0920377i
$263$	296.519	1.12745	0.563724	−	0.825963i	$-0.309368\pi$
$263$	296.519	1.12745	0.563724	+	0.825963i	$0.309368\pi$
$264$	0	0
$265$	0	0
$266$	111.859i	0.420521i
$267$	0	0
$268$	− 35.7973i	− 0.133572i
$269$	6.16168i	0.0229059i	0.999934	+	0.0114529i	$0.00364566\pi$
$269$	6.16168i	0.0229059i	−0.999934	+	0.0114529i	$0.996354\pi$
$270$	0	0
$271$	71.3691	0.263355	0.131677	−	0.991293i	$-0.457964\pi$
$271$	71.3691	0.263355	0.131677	+	0.991293i	$0.457964\pi$
$272$	−1.34280	−0.00493676
$273$	0	0
$274$	−251.590	−0.918213
$275$	0	0
$276$	0	0
$277$	36.9936i	0.133551i	0.997768	+	0.0667754i	$0.0212711\pi$
$277$	36.9936i	0.133551i	−0.997768	+	0.0667754i	$0.978729\pi$
$278$	−271.889	−0.978018
$279$	0	0
$280$	0	0
$281$	74.7131i	0.265883i	0.991124	+	0.132942i	$0.0424423\pi$
$281$	74.7131i	0.265883i	−0.991124	+	0.132942i	$0.957558\pi$
$282$	0	0
$283$	37.7485i	0.133387i	0.997774	+	0.0666934i	$0.0212449\pi$
$283$	37.7485i	0.133387i	−0.997774	+	0.0666934i	$0.978755\pi$
$284$	152.859i	0.538237i
$285$	0	0
$286$	−80.0941	−0.280049
$287$	94.5421	0.329415
$288$	0	0
$289$	−288.887	−0.999610
$290$	0	0
$291$	0	0
$292$	− 126.599i	− 0.433559i
$293$	343.619	1.17276	0.586381	−	0.810035i	$-0.300552\pi$
$293$	343.619	1.17276	0.586381	+	0.810035i	$0.300552\pi$
$294$	0	0
$295$	0	0
$296$	− 118.745i	− 0.401164i
$297$	0	0
$298$	− 327.667i	− 1.09955i
$299$	− 276.949i	− 0.926252i
$300$	0	0
$301$	−64.1833	−0.213233
$302$	337.258	1.11675
$303$	0	0
$304$	119.582	0.393362
$305$	0	0
$306$	0	0
$307$	− 55.4240i	− 0.180534i	−0.995918	−	0.0902671i	$-0.971228\pi$
$307$	− 55.4240i	− 0.180534i	0.995918	−	0.0902671i	$-0.0287721\pi$
$308$	20.0463	0.0650855
$309$	0	0
$310$	0	0
$311$	452.240i	1.45415i	0.686560	+	0.727074i	$0.259120\pi$
$311$	452.240i	1.45415i	−0.686560	+	0.727074i	$0.740880\pi$
$312$	0	0
$313$	− 232.943i	− 0.744228i	−0.928187	−	0.372114i	$-0.878633\pi$
$313$	− 232.943i	− 0.744228i	0.928187	−	0.372114i	$-0.121367\pi$
$314$	320.328i	1.02015i
$315$	0	0
$316$	121.385	0.384130
$317$	389.212	1.22780	0.613899	−	0.789384i	$-0.289600\pi$
$317$	389.212	1.22780	0.613899	+	0.789384i	$0.289600\pi$
$318$	0	0
$319$	−42.2333	−0.132393
$320$	0	0
$321$	0	0
$322$	69.3162i	0.215268i
$323$	−10.0359	−0.0310709
$324$	0	0
$325$	0	0
$326$	− 55.0665i	− 0.168916i
$327$	0	0
$328$	− 101.070i	− 0.308140i
$329$	− 133.277i	− 0.405096i
$330$	0	0
$331$	−201.361	−0.608341	−0.304170	−	0.952618i	$-0.598379\pi$
$331$	−201.361	−0.608341	−0.304170	+	0.952618i	$0.598379\pi$
$332$	258.961	0.780003
$333$	0	0
$334$	275.825	0.825823
$335$	0	0
$336$	0	0
$337$	− 576.780i	− 1.71151i	−0.517379	−	0.855756i	$-0.673092\pi$
$337$	− 576.780i	− 1.71151i	0.517379	−	0.855756i	$-0.326908\pi$
$338$	77.0615	0.227993
$339$	0	0
$340$	0	0
$341$	2.28606i	0.00670399i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	68.6148i	0.199462i
$345$	0	0
$346$	139.706	0.403773
$347$	395.628	1.14014	0.570069	−	0.821597i	$-0.306917\pi$
$347$	395.628	1.14014	0.570069	+	0.821597i	$0.306917\pi$
$348$	0	0
$349$	378.937	1.08578	0.542890	−	0.839804i	$-0.317330\pi$
$349$	378.937	1.08578	0.542890	+	0.839804i	$0.317330\pi$
$350$	0	0
$351$	0	0
$352$	− 21.4304i	− 0.0608819i
$353$	−610.300	−1.72889	−0.864447	−	0.502724i	$-0.832331\pi$
$353$	−610.300	−1.72889	−0.864447	+	0.502724i	$0.832331\pi$
$354$	0	0
$355$	0	0
$356$	30.0624i	0.0844451i
$357$	0	0
$358$	− 233.336i	− 0.651776i
$359$	493.501i	1.37466i	0.726347	+	0.687328i	$0.241216\pi$
$359$	493.501i	1.37466i	−0.726347	+	0.687328i	$0.758784\pi$
$360$	0	0
$361$	532.740	1.47573
$362$	−159.178	−0.439717
$363$	0	0
$364$	−79.1059	−0.217324
$365$	0	0
$366$	0	0
$367$	384.894i	1.04876i	0.851485	+	0.524379i	$0.175703\pi$
$367$	384.894i	1.04876i	−0.851485	+	0.524379i	$0.824297\pi$
$368$	74.1021	0.201365
$369$	0	0
$370$	0	0
$371$	119.009i	0.320778i
$372$	0	0
$373$	659.925i	1.76924i	0.466316	+	0.884618i	$0.345581\pi$
$373$	659.925i	1.76924i	−0.466316	+	0.884618i	$0.654419\pi$
$374$	1.79855i	0.00480895i
$375$	0	0
$376$	−142.479	−0.378933
$377$	166.659	0.442066
$378$	0	0
$379$	157.587	0.415796	0.207898	−	0.978150i	$-0.433338\pi$
$379$	157.587	0.415796	0.207898	+	0.978150i	$0.433338\pi$
$380$	0	0
$381$	0	0
$382$	369.561i	0.967436i
$383$	6.80377	0.0177644	0.00888220	−	0.999961i	$-0.497173\pi$
$383$	6.80377	0.0177644	0.00888220	+	0.999961i	$0.497173\pi$
$384$	0	0
$385$	0	0
$386$	− 379.536i	− 0.983253i
$387$	0	0
$388$	− 118.960i	− 0.306599i
$389$	− 769.376i	− 1.97783i	−0.148484	−	0.988915i	$-0.547439\pi$
$389$	− 769.376i	− 1.97783i	0.148484	−	0.988915i	$-0.452561\pi$
$390$	0	0
$391$	−6.21901	−0.0159054
$392$	19.7990	0.0505076
$393$	0	0
$394$	73.4511	0.186424
$395$	0	0
$396$	0	0
$397$	− 533.930i	− 1.34491i	−0.740137	−	0.672456i	$-0.765240\pi$
$397$	− 533.930i	− 1.34491i	0.740137	−	0.672456i	$-0.234760\pi$
$398$	−434.509	−1.09173
$399$	0	0
$400$	0	0
$401$	− 622.370i	− 1.55204i	−0.630705	−	0.776022i	$-0.717234\pi$
$401$	− 622.370i	− 1.55204i	0.630705	−	0.776022i	$-0.282766\pi$
$402$	0	0
$403$	− 9.02115i	− 0.0223850i
$404$	69.9281i	0.173089i
$405$	0	0
$406$	−41.7122	−0.102739
$407$	−159.047	−0.390778
$408$	0	0
$409$	754.068	1.84369	0.921843	−	0.387564i	$-0.126683\pi$
$409$	754.068	1.84369	0.921843	+	0.387564i	$0.126683\pi$
$410$	0	0
$411$	0	0
$412$	− 33.0581i	− 0.0802381i
$413$	−42.8221	−0.103685
$414$	0	0
$415$	0	0
$416$	84.5677i	0.203288i
$417$	0	0
$418$	− 160.168i	− 0.383178i
$419$	− 312.486i	− 0.745789i	−0.927874	−	0.372895i	$-0.878365\pi$
$419$	− 312.486i	− 0.745789i	0.927874	−	0.372895i	$-0.121635\pi$
$420$	0	0
$421$	−159.259	−0.378287	−0.189144	−	0.981949i	$-0.560571\pi$
$421$	−159.259	−0.378287	−0.189144	+	0.981949i	$0.560571\pi$
$422$	158.529	0.375661
$423$	0	0
$424$	127.226	0.300060
$425$	0	0
$426$	0	0
$427$	270.367i	0.633178i
$428$	−235.149	−0.549413
$429$	0	0
$430$	0	0
$431$	610.194i	1.41576i	0.706331	+	0.707882i	$0.250349\pi$
$431$	610.194i	1.41576i	−0.706331	+	0.707882i	$0.749651\pi$
$432$	0	0
$433$	179.507i	0.414565i	0.978281	+	0.207283i	$0.0664620\pi$
$433$	179.507i	0.414565i	−0.978281	+	0.207283i	$0.933538\pi$
$434$	2.25786i	0.00520243i
$435$	0	0
$436$	−4.47509	−0.0102640
$437$	553.830	1.26734
$438$	0	0
$439$	385.194	0.877434	0.438717	−	0.898625i	$-0.355433\pi$
$439$	385.194	0.877434	0.438717	+	0.898625i	$0.355433\pi$
$440$	0	0
$441$	0	0
$442$	− 7.09733i	− 0.0160573i
$443$	98.2782	0.221847	0.110923	−	0.993829i	$-0.464619\pi$
$443$	98.2782	0.221847	0.110923	+	0.993829i	$0.464619\pi$
$444$	0	0
$445$	0	0
$446$	271.572i	0.608907i
$447$	0	0
$448$	− 21.1660i	− 0.0472456i
$449$	− 345.416i	− 0.769302i	−0.923062	−	0.384651i	$-0.874322\pi$
$449$	− 345.416i	− 0.769302i	0.923062	−	0.384651i	$-0.125678\pi$
$450$	0	0
$451$	−135.373	−0.300162
$452$	178.026	0.393863
$453$	0	0
$454$	10.2012	0.0224697
$455$	0	0
$456$	0	0
$457$	− 520.425i	− 1.13879i	−0.822066	−	0.569393i	$-0.807178\pi$
$457$	− 520.425i	− 1.13879i	0.822066	−	0.569393i	$-0.192822\pi$
$458$	−476.170	−1.03967
$459$	0	0
$460$	0	0
$461$	131.936i	0.286196i	0.989709	+	0.143098i	$0.0457064\pi$
$461$	131.936i	0.286196i	−0.989709	+	0.143098i	$0.954294\pi$
$462$	0	0
$463$	− 240.485i	− 0.519406i	−0.965688	−	0.259703i	$-0.916375\pi$
$463$	− 240.485i	− 0.519406i	0.965688	−	0.259703i	$-0.0836247\pi$
$464$	44.5922i	0.0961038i
$465$	0	0
$466$	−361.546	−0.775850
$467$	524.985	1.12416	0.562082	−	0.827081i	$-0.310000\pi$
$467$	524.985	1.12416	0.562082	+	0.827081i	$0.310000\pi$
$468$	0	0
$469$	−47.3554	−0.100971
$470$	0	0
$471$	0	0
$472$	45.7787i	0.0969888i
$473$	91.9028	0.194298
$474$	0	0
$475$	0	0
$476$	1.77635i	0.00373184i
$477$	0	0
$478$	83.7629i	0.175236i
$479$	259.839i	0.542461i	0.962514	+	0.271230i	$0.0874305\pi$
$479$	259.839i	0.542461i	−0.962514	+	0.271230i	$0.912569\pi$
$480$	0	0
$481$	627.623	1.30483
$482$	−94.8823	−0.196851
$483$	0	0
$484$	213.296	0.440694
$485$	0	0
$486$	0	0
$487$	− 16.7084i	− 0.0343089i	−0.999853	−	0.0171544i	$-0.994539\pi$
$487$	− 16.7084i	− 0.0343089i	0.999853	−	0.0171544i	$-0.00546070\pi$
$488$	289.034	0.592284
$489$	0	0
$490$	0	0
$491$	− 750.622i	− 1.52876i	−0.644765	−	0.764381i	$-0.723045\pi$
$491$	− 750.622i	− 1.52876i	0.644765	−	0.764381i	$-0.276955\pi$
$492$	0	0
$493$	− 3.74239i	− 0.00759106i
$494$	632.048i	1.27945i
$495$	0	0
$496$	2.41375	0.00486643
$497$	202.214	0.406869
$498$	0	0
$499$	263.681	0.528419	0.264210	−	0.964465i	$-0.414889\pi$
$499$	263.681	0.528419	0.264210	+	0.964465i	$0.414889\pi$
$500$	0	0
$501$	0	0
$502$	429.731i	0.856038i
$503$	383.069	0.761569	0.380784	−	0.924664i	$-0.375654\pi$
$503$	383.069	0.761569	0.380784	+	0.924664i	$0.375654\pi$
$504$	0	0
$505$	0	0
$506$	− 99.2526i	− 0.196151i
$507$	0	0
$508$	− 468.260i	− 0.921772i
$509$	− 850.177i	− 1.67029i	−0.550031	−	0.835144i	$-0.685384\pi$
$509$	− 850.177i	− 1.67029i	0.550031	−	0.835144i	$-0.314616\pi$
$510$	0	0
$511$	−167.475	−0.327740
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	−62.5282	−0.121650
$515$	0	0
$516$	0	0
$517$	190.836i	0.369122i
$518$	−157.084	−0.303252
$519$	0	0
$520$	0	0
$521$	− 435.203i	− 0.835323i	−0.908603	−	0.417661i	$-0.862850\pi$
$521$	− 435.203i	− 0.835323i	0.908603	−	0.417661i	$-0.137150\pi$
$522$	0	0
$523$	− 350.756i	− 0.670662i	−0.942100	−	0.335331i	$-0.891152\pi$
$523$	− 350.756i	− 0.670662i	0.942100	−	0.335331i	$-0.108848\pi$
$524$	− 34.1022i	− 0.0650805i
$525$	0	0
$526$	−419.341	−0.797226
$527$	−0.202573	−0.000384390 0
$528$	0	0
$529$	−185.805	−0.351237
$530$	0	0
$531$	0	0
$532$	− 158.192i	− 0.297353i
$533$	534.203	1.00226
$534$	0	0
$535$	0	0
$536$	50.6251i	0.0944498i
$537$	0	0
$538$	− 8.71393i	− 0.0161969i
$539$	− 26.5188i	− 0.0492000i
$540$	0	0
$541$	814.349	1.50527	0.752633	−	0.658440i	$-0.228783\pi$
$541$	814.349	1.50527	0.752633	+	0.658440i	$0.228783\pi$
$542$	−100.931	−0.186220
$543$	0	0
$544$	1.89900	0.00349081
$545$	0	0
$546$	0	0
$547$	187.880i	0.343474i	0.985143	+	0.171737i	$0.0549379\pi$
$547$	187.880i	0.343474i	−0.985143	+	0.171737i	$0.945062\pi$
$548$	355.802	0.649274
$549$	0	0
$550$	0	0
$551$	333.276i	0.604857i
$552$	0	0
$553$	− 160.577i	− 0.290375i
$554$	− 52.3168i	− 0.0944346i
$555$	0	0
$556$	384.509	0.691563
$557$	490.508	0.880624	0.440312	−	0.897845i	$-0.354868\pi$
$557$	490.508	0.880624	0.440312	+	0.897845i	$0.354868\pi$
$558$	0	0
$559$	−362.662	−0.648770
$560$	0	0
$561$	0	0
$562$	− 105.660i	− 0.188008i
$563$	479.230	0.851207	0.425604	−	0.904910i	$-0.360062\pi$
$563$	479.230	0.851207	0.425604	+	0.904910i	$0.360062\pi$
$564$	0	0
$565$	0	0
$566$	− 53.3844i	− 0.0943187i
$567$	0	0
$568$	− 216.176i	− 0.380591i
$569$	− 681.193i	− 1.19718i	−0.801057	−	0.598588i	$-0.795729\pi$
$569$	− 681.193i	− 1.19718i	0.801057	−	0.598588i	$-0.204271\pi$
$570$	0	0
$571$	727.119	1.27341	0.636707	−	0.771106i	$-0.280296\pi$
$571$	727.119	1.27341	0.636707	+	0.771106i	$0.280296\pi$
$572$	113.270	0.198025
$573$	0	0
$574$	−133.703	−0.232932
$575$	0	0
$576$	0	0
$577$	653.309i	1.13225i	0.824319	+	0.566125i	$0.191558\pi$
$577$	653.309i	1.13225i	−0.824319	+	0.566125i	$0.808442\pi$
$578$	408.548	0.706831
$579$	0	0
$580$	0	0
$581$	− 342.573i	− 0.589627i
$582$	0	0
$583$	− 170.406i	− 0.292292i
$584$	179.038i	0.306572i
$585$	0	0
$586$	−485.951	−0.829268
$587$	−615.483	−1.04852	−0.524262	−	0.851557i	$-0.675659\pi$
$587$	−615.483	−1.04852	−0.524262	+	0.851557i	$0.675659\pi$
$588$	0	0
$589$	18.0400	0.0306283
$590$	0	0
$591$	0	0
$592$	167.930i	0.283666i
$593$	−692.270	−1.16740	−0.583701	−	0.811968i	$-0.698396\pi$
$593$	−692.270	−1.16740	−0.583701	+	0.811968i	$0.698396\pi$
$594$	0	0
$595$	0	0
$596$	463.392i	0.777503i
$597$	0	0
$598$	391.666i	0.654959i
$599$	449.015i	0.749608i	0.927104	+	0.374804i	$0.122290\pi$
$599$	449.015i	0.749608i	−0.927104	+	0.374804i	$0.877710\pi$
$600$	0	0
$601$	402.229	0.669266	0.334633	−	0.942348i	$-0.391388\pi$
$601$	402.229	0.669266	0.334633	+	0.942348i	$0.391388\pi$
$602$	90.7688	0.150779
$603$	0	0
$604$	−476.955	−0.789661
$605$	0	0
$606$	0	0
$607$	− 473.443i	− 0.779971i	−0.920821	−	0.389986i	$-0.872480\pi$
$607$	− 473.443i	− 0.779971i	0.920821	−	0.389986i	$-0.127520\pi$
$608$	−169.114	−0.278149
$609$	0	0
$610$	0	0
$611$	− 753.069i	− 1.23252i
$612$	0	0
$613$	− 830.537i	− 1.35487i	−0.735581	−	0.677437i	$-0.763091\pi$
$613$	− 830.537i	− 1.35487i	0.735581	−	0.677437i	$-0.236909\pi$
$614$	78.3813i	0.127657i
$615$	0	0
$616$	−28.3498	−0.0460224
$617$	−193.710	−0.313954	−0.156977	−	0.987602i	$-0.550175\pi$
$617$	−193.710	−0.313954	−0.156977	+	0.987602i	$0.550175\pi$
$618$	0	0
$619$	558.554	0.902349	0.451174	−	0.892436i	$-0.351005\pi$
$619$	558.554	0.902349	0.451174	+	0.892436i	$0.351005\pi$
$620$	0	0
$621$	0	0
$622$	− 639.564i	− 1.02824i
$623$	39.7689	0.0638345
$624$	0	0
$625$	0	0
$626$	329.431i	0.526248i
$627$	0	0
$628$	− 453.012i	− 0.721356i
$629$	− 14.0935i	− 0.0224062i
$630$	0	0
$631$	−653.265	−1.03529	−0.517643	−	0.855597i	$-0.673190\pi$
$631$	−653.265	−1.03529	−0.517643	+	0.855597i	$0.673190\pi$
$632$	−171.664	−0.271621
$633$	0	0
$634$	−550.429	−0.868185
$635$	0	0
$636$	0	0
$637$	104.647i	0.164281i
$638$	59.7268	0.0936158
$639$	0	0
$640$	0	0
$641$	− 148.157i	− 0.231134i	−0.993300	−	0.115567i	$-0.963132\pi$
$641$	− 148.157i	− 0.231134i	0.993300	−	0.115567i	$-0.0368685\pi$
$642$	0	0
$643$	449.684i	0.699353i	0.936870	+	0.349677i	$0.113709\pi$
$643$	449.684i	0.699353i	−0.936870	+	0.349677i	$0.886291\pi$
$644$	− 98.0279i	− 0.152217i
$645$	0	0
$646$	14.1929	0.0219704
$647$	272.475	0.421137	0.210568	−	0.977579i	$-0.432469\pi$
$647$	272.475	0.421137	0.210568	+	0.977579i	$0.432469\pi$
$648$	0	0
$649$	61.3161	0.0944779
$650$	0	0
$651$	0	0
$652$	77.8758i	0.119441i
$653$	594.454	0.910343	0.455172	−	0.890404i	$-0.349578\pi$
$653$	594.454	0.910343	0.455172	+	0.890404i	$0.349578\pi$
$654$	0	0
$655$	0	0
$656$	142.934i	0.217888i
$657$	0	0
$658$	188.482i	0.286446i
$659$	− 423.092i	− 0.642022i	−0.947076	−	0.321011i	$-0.895977\pi$
$659$	− 423.092i	− 0.642022i	0.947076	−	0.321011i	$-0.104023\pi$
$660$	0	0
$661$	−252.437	−0.381901	−0.190951	−	0.981600i	$-0.561157\pi$
$661$	−252.437	−0.381901	−0.190951	+	0.981600i	$0.561157\pi$
$662$	284.767	0.430162
$663$	0	0
$664$	−366.226	−0.551545
$665$	0	0
$666$	0	0
$667$	206.523i	0.309630i
$668$	−390.075	−0.583945
$669$	0	0
$670$	0	0
$671$	− 387.133i	− 0.576950i
$672$	0	0
$673$	− 216.267i	− 0.321348i	−0.987008	−	0.160674i	$-0.948633\pi$
$673$	− 216.267i	− 0.321348i	0.987008	−	0.160674i	$-0.0513667\pi$
$674$	815.690i	1.21022i
$675$	0	0
$676$	−108.981	−0.161215
$677$	−828.658	−1.22402	−0.612008	−	0.790852i	$-0.709638\pi$
$677$	−828.658	−1.22402	−0.612008	+	0.790852i	$0.709638\pi$
$678$	0	0
$679$	−157.370	−0.231767
$680$	0	0
$681$	0	0
$682$	− 3.23298i	− 0.00474044i
$683$	−534.258	−0.782223	−0.391111	−	0.920343i	$-0.627909\pi$
$683$	−534.258	−0.782223	−0.391111	+	0.920343i	$0.627909\pi$
$684$	0	0
$685$	0	0
$686$	− 26.1916i	− 0.0381802i
$687$	0	0
$688$	− 97.0360i	− 0.141041i
$689$	672.449i	0.975978i
$690$	0	0
$691$	−506.311	−0.732723	−0.366361	−	0.930473i	$-0.619397\pi$
$691$	−506.311	−0.732723	−0.366361	+	0.930473i	$0.619397\pi$
$692$	−197.574	−0.285511
$693$	0	0
$694$	−559.502	−0.806199
$695$	0	0
$696$	0	0
$697$	− 11.9957i	− 0.0172105i
$698$	−535.898	−0.767762
$699$	0	0
$700$	0	0
$701$	− 588.963i	− 0.840176i	−0.907483	−	0.420088i	$-0.861999\pi$
$701$	− 588.963i	− 0.840176i	0.907483	−	0.420088i	$-0.138001\pi$
$702$	0	0
$703$	1255.09i	1.78533i
$704$	30.3072i	0.0430500i
$705$	0	0
$706$	863.094	1.22251
$707$	92.5062	0.130843
$708$	0	0
$709$	−18.2232	−0.0257027	−0.0128513	−	0.999917i	$-0.504091\pi$
$709$	−18.2232	−0.0257027	−0.0128513	+	0.999917i	$0.504091\pi$
$710$	0	0
$711$	0	0
$712$	− 42.5147i	− 0.0597117i
$713$	11.1790	0.0156788
$714$	0	0
$715$	0	0
$716$	329.987i	0.460875i
$717$	0	0
$718$	− 697.916i	− 0.972028i
$719$	− 813.874i	− 1.13195i	−0.824422	−	0.565976i	$-0.808499\pi$
$719$	− 813.874i	− 1.13195i	0.824422	−	0.565976i	$-0.191501\pi$
$720$	0	0
$721$	−43.7318	−0.0606543
$722$	−753.408	−1.04350
$723$	0	0
$724$	225.111	0.310927
$725$	0	0
$726$	0	0
$727$	1003.37i	1.38016i	0.723734	+	0.690079i	$0.242424\pi$
$727$	1003.37i	1.38016i	−0.723734	+	0.690079i	$0.757576\pi$
$728$	111.873	0.153671
$729$	0	0
$730$	0	0
$731$	8.14373i	0.0111405i
$732$	0	0
$733$	1223.13i	1.66867i	0.551259	+	0.834334i	$0.314148\pi$
$733$	1223.13i	1.66867i	−0.551259	+	0.834334i	$0.685852\pi$
$734$	− 544.323i	− 0.741584i
$735$	0	0
$736$	−104.796	−0.142386
$737$	67.8073	0.0920045
$738$	0	0
$739$	−639.556	−0.865435	−0.432717	−	0.901530i	$-0.642445\pi$
$739$	−639.556	−0.865435	−0.432717	+	0.901530i	$0.642445\pi$
$740$	0	0
$741$	0	0
$742$	− 168.304i	− 0.226824i
$743$	1325.46	1.78393	0.891964	−	0.452106i	$-0.149327\pi$
$743$	1325.46	1.78393	0.891964	+	0.452106i	$0.149327\pi$
$744$	0	0
$745$	0	0
$746$	− 933.275i	− 1.25104i
$747$	0	0
$748$	− 2.54353i	− 0.00340044i
$749$	311.073i	0.415317i
$750$	0	0
$751$	1055.16	1.40501	0.702503	−	0.711681i	$-0.252066\pi$
$751$	1055.16	1.40501	0.702503	+	0.711681i	$0.252066\pi$
$752$	201.495	0.267946
$753$	0	0
$754$	−235.691	−0.312588
$755$	0	0
$756$	0	0
$757$	908.785i	1.20051i	0.799809	+	0.600254i	$0.204934\pi$
$757$	908.785i	1.20051i	−0.799809	+	0.600254i	$0.795066\pi$
$758$	−222.861	−0.294012
$759$	0	0
$760$	0	0
$761$	− 218.340i	− 0.286912i	−0.989657	−	0.143456i	$-0.954178\pi$
$761$	− 218.340i	− 0.286912i	0.989657	−	0.143456i	$-0.0458216\pi$
$762$	0	0
$763$	5.91999i	0.00775884i
$764$	− 522.638i	− 0.684081i
$765$	0	0
$766$	−9.62198	−0.0125613
$767$	−241.963	−0.315466
$768$	0	0
$769$	−297.681	−0.387102	−0.193551	−	0.981090i	$-0.562000\pi$
$769$	−297.681	−0.387102	−0.193551	+	0.981090i	$0.562000\pi$
$770$	0	0
$771$	0	0
$772$	536.744i	0.695265i
$773$	−272.647	−0.352713	−0.176356	−	0.984326i	$-0.556431\pi$
$773$	−272.647	−0.352713	−0.176356	+	0.984326i	$0.556431\pi$
$774$	0	0
$775$	0	0
$776$	168.235i	0.216798i
$777$	0	0
$778$	1088.06i	1.39854i
$779$	1068.27i	1.37134i
$780$	0	0
$781$	−289.546	−0.370737
$782$	8.79501	0.0112468
$783$	0	0
$784$	−28.0000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 164.051i	− 0.208451i	−0.994554	−	0.104226i	$-0.966764\pi$
$787$	− 164.051i	− 0.208451i	0.994554	−	0.104226i	$-0.0332364\pi$
$788$	−103.876	−0.131822
$789$	0	0
$790$	0	0
$791$	− 235.506i	− 0.297732i
$792$	0	0
$793$	1527.69i	1.92646i
$794$	755.091i	0.950996i
$795$	0	0
$796$	614.488	0.771970
$797$	−416.976	−0.523182	−0.261591	−	0.965179i	$-0.584247\pi$
$797$	−416.976	−0.523182	−0.261591	+	0.965179i	$0.584247\pi$
$798$	0	0
$799$	−16.9105	−0.0211645
$800$	0	0
$801$	0	0
$802$	880.164i	1.09746i
$803$	239.804	0.298636
$804$	0	0
$805$	0	0
$806$	12.7578i	0.0158286i
$807$	0	0
$808$	− 98.8933i	− 0.122393i
$809$	− 1237.84i	− 1.53009i	−0.643975	−	0.765046i	$-0.722716\pi$
$809$	− 1237.84i	− 1.53009i	0.643975	−	0.765046i	$-0.277284\pi$
$810$	0	0
$811$	87.6470	0.108073	0.0540363	−	0.998539i	$-0.482791\pi$
$811$	87.6470	0.108073	0.0540363	+	0.998539i	$0.482791\pi$
$812$	58.9899	0.0726477
$813$	0	0
$814$	224.926	0.276322
$815$	0	0
$816$	0	0
$817$	− 725.234i	− 0.887679i
$818$	−1066.41	−1.30368
$819$	0	0
$820$	0	0
$821$	1432.46i	1.74477i	0.488818	+	0.872386i	$0.337428\pi$
$821$	1432.46i	1.74477i	−0.488818	+	0.872386i	$0.662572\pi$
$822$	0	0
$823$	699.888i	0.850410i	0.905097	+	0.425205i	$0.139798\pi$
$823$	699.888i	0.850410i	−0.905097	+	0.425205i	$0.860202\pi$
$824$	46.7512i	0.0567369i
$825$	0	0
$826$	60.5596	0.0733167
$827$	−1538.24	−1.86003	−0.930015	−	0.367522i	$-0.880206\pi$
$827$	−1538.24	−1.86003	−0.930015	+	0.367522i	$0.880206\pi$
$828$	0	0
$829$	−42.3166	−0.0510454	−0.0255227	−	0.999674i	$-0.508125\pi$
$829$	−42.3166	−0.0510454	−0.0255227	+	0.999674i	$0.508125\pi$
$830$	0	0
$831$	0	0
$832$	− 119.597i	− 0.143746i
$833$	2.34990	0.00282100
$834$	0	0
$835$	0	0
$836$	226.512i	0.270948i
$837$	0	0
$838$	441.922i	0.527353i
$839$	652.708i	0.777959i	0.921246	+	0.388980i	$0.127172\pi$
$839$	652.708i	0.777959i	−0.921246	+	0.388980i	$0.872828\pi$
$840$	0	0
$841$	716.721	0.852225
$842$	225.226	0.267490
$843$	0	0
$844$	−224.194	−0.265632
$845$	0	0
$846$	0	0
$847$	− 282.164i	− 0.333134i
$848$	−179.924	−0.212175
$849$	0	0
$850$	0	0
$851$	777.749i	0.913924i
$852$	0	0
$853$	− 348.369i	− 0.408404i	−0.978929	−	0.204202i	$-0.934540\pi$
$853$	− 348.369i	− 0.408404i	0.978929	−	0.204202i	$-0.0654600\pi$
$854$	− 382.357i	− 0.447724i
$855$	0	0
$856$	332.551	0.388494
$857$	−444.441	−0.518601	−0.259301	−	0.965797i	$-0.583492\pi$
$857$	−444.441	−0.518601	−0.259301	+	0.965797i	$0.583492\pi$
$858$	0	0
$859$	93.8325	0.109235	0.0546173	−	0.998507i	$-0.482606\pi$
$859$	93.8325	0.109235	0.0546173	+	0.998507i	$0.482606\pi$
$860$	0	0
$861$	0	0
$862$	− 862.945i	− 1.00110i
$863$	−1138.10	−1.31877	−0.659384	−	0.751806i	$-0.729183\pi$
$863$	−1138.10	−1.31877	−0.659384	+	0.751806i	$0.729183\pi$
$864$	0	0
$865$	0	0
$866$	− 253.861i	− 0.293142i
$867$	0	0
$868$	− 3.19309i	− 0.00367867i
$869$	229.928i	0.264589i
$870$	0	0
$871$	−267.578	−0.307208
$872$	6.32874	0.00725773
$873$	0	0
$874$	−783.234	−0.896148
$875$	0	0
$876$	0	0
$877$	965.193i	1.10056i	0.834980	+	0.550281i	$0.185479\pi$
$877$	965.193i	1.10056i	−0.834980	+	0.550281i	$0.814521\pi$
$878$	−544.746	−0.620440
$879$	0	0
$880$	0	0
$881$	518.314i	0.588324i	0.955756	+	0.294162i	$0.0950406\pi$
$881$	518.314i	0.588324i	−0.955756	+	0.294162i	$0.904959\pi$
$882$	0	0
$883$	− 1525.69i	− 1.72785i	−0.503621	−	0.863925i	$-0.667999\pi$
$883$	− 1525.69i	− 1.72785i	0.503621	−	0.863925i	$-0.332001\pi$
$884$	10.0371i	0.0113542i
$885$	0	0
$886$	−138.986	−0.156869
$887$	−1667.27	−1.87967	−0.939835	−	0.341629i	$-0.889021\pi$
$887$	−1667.27	−1.87967	−0.939835	+	0.341629i	$0.889021\pi$
$888$	0	0
$889$	−619.450	−0.696794
$890$	0	0
$891$	0	0
$892$	− 384.061i	− 0.430562i
$893$	1505.95	1.68639
$894$	0	0
$895$	0	0
$896$	29.9333i	0.0334077i
$897$	0	0
$898$	488.493i	0.543978i
$899$	6.72714i	0.00748292i
$900$	0	0
$901$	15.1001	0.0167593
$902$	191.446	0.212247
$903$	0	0
$904$	−251.767	−0.278503
$905$	0	0
$906$	0	0
$907$	214.983i	0.237026i	0.992952	+	0.118513i	$0.0378127\pi$
$907$	214.983i	0.237026i	−0.992952	+	0.118513i	$0.962187\pi$
$908$	−14.4267	−0.0158885
$909$	0	0
$910$	0	0
$911$	− 515.038i	− 0.565355i	−0.959215	−	0.282677i	$-0.908777\pi$
$911$	− 515.038i	− 0.565355i	0.959215	−	0.282677i	$-0.0912226\pi$
$912$	0	0
$913$	490.524i	0.537266i
$914$	735.992i	0.805243i
$915$	0	0
$916$	673.406	0.735159
$917$	−45.1129	−0.0491962
$918$	0	0
$919$	157.043	0.170884	0.0854421	−	0.996343i	$-0.472770\pi$
$919$	157.043	0.170884	0.0854421	+	0.996343i	$0.472770\pi$
$920$	0	0
$921$	0	0
$922$	− 186.586i	− 0.202371i
$923$	1142.59	1.23791
$924$	0	0
$925$	0	0
$926$	340.097i	0.367276i
$927$	0	0
$928$	− 63.0629i	− 0.0679557i
$929$	− 1002.16i	− 1.07875i	−0.842065	−	0.539376i	$-0.818660\pi$
$929$	− 1002.16i	− 1.07875i	0.842065	−	0.539376i	$-0.181340\pi$
$930$	0	0
$931$	−209.268	−0.224778
$932$	511.303	0.548609
$933$	0	0
$934$	−742.441	−0.794905
$935$	0	0
$936$	0	0
$937$	386.668i	0.412666i	0.978482	+	0.206333i	$0.0661531\pi$
$937$	386.668i	0.412666i	−0.978482	+	0.206333i	$0.933847\pi$
$938$	66.9707	0.0713973
$939$	0	0
$940$	0	0
$941$	− 440.452i	− 0.468068i	−0.972228	−	0.234034i	$-0.924807\pi$
$941$	− 440.452i	− 0.468068i	0.972228	−	0.234034i	$-0.0751927\pi$
$942$	0	0
$943$	661.983i	0.701997i
$944$	− 64.7409i	− 0.0685815i
$945$	0	0
$946$	−129.970	−0.137389
$947$	−255.582	−0.269886	−0.134943	−	0.990853i	$-0.543085\pi$
$947$	−255.582	−0.269886	−0.134943	+	0.990853i	$0.543085\pi$
$948$	0	0
$949$	−946.304	−0.997159
$950$	0	0
$951$	0	0
$952$	− 2.51214i	− 0.00263881i
$953$	−1609.14	−1.68849	−0.844247	−	0.535954i	$-0.819952\pi$
$953$	−1609.14	−1.68849	−0.844247	+	0.535954i	$0.819952\pi$
$954$	0	0
$955$	0	0
$956$	− 118.459i	− 0.123911i
$957$	0	0
$958$	− 367.467i	− 0.383578i
$959$	− 470.682i	− 0.490805i
$960$	0	0
$961$	−960.636	−0.999621
$962$	−887.592	−0.922653
$963$	0	0
$964$	134.184	0.139195
$965$	0	0
$966$	0	0
$967$	872.053i	0.901812i	0.892571	+	0.450906i	$0.148899\pi$
$967$	872.053i	0.901812i	−0.892571	+	0.450906i	$0.851101\pi$
$968$	−301.646	−0.311618
$969$	0	0
$970$	0	0
$971$	829.279i	0.854046i	0.904241	+	0.427023i	$0.140438\pi$
$971$	829.279i	0.854046i	−0.904241	+	0.427023i	$0.859562\pi$
$972$	0	0
$973$	− 508.658i	− 0.522773i
$974$	23.6293i	0.0242601i
$975$	0	0
$976$	−408.756	−0.418808
$977$	115.891	0.118620	0.0593098	−	0.998240i	$-0.481110\pi$
$977$	115.891	0.118620	0.0593098	+	0.998240i	$0.481110\pi$
$978$	0	0
$979$	−56.9443	−0.0581658
$980$	0	0
$981$	0	0
$982$	1061.54i	1.08100i
$983$	854.970	0.869756	0.434878	−	0.900489i	$-0.356791\pi$
$983$	854.970	0.869756	0.434878	+	0.900489i	$0.356791\pi$
$984$	0	0
$985$	0	0
$986$	5.29254i	0.00536769i
$987$	0	0
$988$	− 893.851i	− 0.904708i
$989$	− 449.411i	− 0.454409i
$990$	0	0
$991$	−1389.62	−1.40224	−0.701119	−	0.713045i	$-0.747316\pi$
$991$	−1389.62	−1.40224	−0.701119	+	0.713045i	$0.747316\pi$
$992$	−3.41356	−0.00344108
$993$	0	0
$994$	−285.973	−0.287700
$995$	0	0
$996$	0	0
$997$	1331.86i	1.33586i	0.744222	+	0.667932i	$0.232820\pi$
$997$	1331.86i	1.33586i	−0.744222	+	0.667932i	$0.767180\pi$
$998$	−372.902	−0.373649
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.3.c.f.449.3		16
3.2	odd	2	inner	3150.3.c.f.449.10		16
5.2	odd	4		630.3.e.b.71.2	✓	8
5.3	odd	4		3150.3.e.f.701.6		8
5.4	even	2	inner	3150.3.c.f.449.15		16
15.2	even	4		630.3.e.b.71.8	yes	8
15.8	even	4		3150.3.e.f.701.1		8
15.14	odd	2	inner	3150.3.c.f.449.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.3.e.b.71.2	✓	8	5.2	odd	4
630.3.e.b.71.8	yes	8	15.2	even	4
3150.3.c.f.449.3		16	1.1	even	1	trivial
3150.3.c.f.449.6		16	15.14	odd	2	inner
3150.3.c.f.449.10		16	3.2	odd	2	inner
3150.3.c.f.449.15		16	5.4	even	2	inner
3150.3.e.f.701.1		8	15.8	even	4
3150.3.e.f.701.6		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.64575i q^{7} -2.82843 q^{8} +3.78840i q^{11} -14.9496i q^{13} +3.74166i q^{14} +4.00000 q^{16} -0.335699 q^{17} +29.8955 q^{19} -5.35761i q^{22} +18.5255 q^{23} +21.1419i q^{26} -5.29150i q^{28} +11.1480i q^{29} +0.603437 q^{31} -5.65685 q^{32} +0.474751 q^{34} +41.9826i q^{37} -42.2786 q^{38} +35.7336i q^{41} -24.2590i q^{43} +7.57680i q^{44} -26.1991 q^{46} +50.3738 q^{47} -7.00000 q^{49} -29.8992i q^{52} -44.9810 q^{53} +7.48331i q^{56} -15.7657i q^{58} -16.1852i q^{59} -102.189 q^{61} -0.853389 q^{62} +8.00000 q^{64} -17.8987i q^{67} -0.671399 q^{68} +76.4296i q^{71} -63.2996i q^{73} -59.3723i q^{74} +59.7910 q^{76} +10.0232 q^{77} +60.6925 q^{79} -50.5349i q^{82} +129.480 q^{83} +34.3074i q^{86} -10.7152i q^{88} +15.0312i q^{89} -39.5529 q^{91} +37.0511 q^{92} -71.2394 q^{94} -59.4802i q^{97} +9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) \) 16 * q + 32 * q^4 + 64 * q^16 + 160 * q^19 + 224 * q^31 - 192 * q^34 - 64 * q^46 - 112 * q^49 - 288 * q^61 + 128 * q^64 + 320 * q^76 + 128 * q^79 - 448 * q^91

Introduction

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Modular forms

Varieties

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