LMFDB - Embedded newform 1350.4.c.c.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(649,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1350.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.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:	$79.6525785077$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.4.c.c.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} +34.0000i q^{7} +8.00000i q^{8} +O(q^{10})$	$q-2.00000i q^{2} -4.00000 q^{4} +34.0000i q^{7} +8.00000i q^{8} -48.0000 q^{11} -70.0000i q^{13} +68.0000 q^{14} +16.0000 q^{16} +27.0000i q^{17} -119.000 q^{19} +96.0000i q^{22} -51.0000i q^{23} -140.000 q^{26} -136.000i q^{28} +30.0000 q^{29} -133.000 q^{31} -32.0000i q^{32} +54.0000 q^{34} -218.000i q^{37} +238.000i q^{38} +156.000 q^{41} -88.0000i q^{43} +192.000 q^{44} -102.000 q^{46} +516.000i q^{47} -813.000 q^{49} +280.000i q^{52} -639.000i q^{53} -272.000 q^{56} -60.0000i q^{58} +654.000 q^{59} +461.000 q^{61} +266.000i q^{62} -64.0000 q^{64} -182.000i q^{67} -108.000i q^{68} +900.000 q^{71} +704.000i q^{73} -436.000 q^{74} +476.000 q^{76} -1632.00i q^{77} +1375.00 q^{79} -312.000i q^{82} +915.000i q^{83} -176.000 q^{86} -384.000i q^{88} +1116.00 q^{89} +2380.00 q^{91} +204.000i q^{92} +1032.00 q^{94} +16.0000i q^{97} +1626.00i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} - 96 q^{11} + 136 q^{14} + 32 q^{16} - 238 q^{19} - 280 q^{26} + 60 q^{29} - 266 q^{31} + 108 q^{34} + 312 q^{41} + 384 q^{44} - 204 q^{46} - 1626 q^{49} - 544 q^{56} + 1308 q^{59} + 922 q^{61} - 128 q^{64} + 1800 q^{71} - 872 q^{74} + 952 q^{76} + 2750 q^{79} - 352 q^{86} + 2232 q^{89} + 4760 q^{91} + 2064 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 - 96 * q^11 + 136 * q^14 + 32 * q^16 - 238 * q^19 - 280 * q^26 + 60 * q^29 - 266 * q^31 + 108 * q^34 + 312 * q^41 + 384 * q^44 - 204 * q^46 - 1626 * q^49 - 544 * q^56 + 1308 * q^59 + 922 * q^61 - 128 * q^64 + 1800 * q^71 - 872 * q^74 + 952 * q^76 + 2750 * q^79 - 352 * q^86 + 2232 * q^89 + 4760 * q^91 + 2064 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	34.0000i	1.83583i	0.396780	+	0.917914i	$0.370128\pi$
$7$	34.0000i	1.83583i	−0.396780	+	0.917914i	$0.629872\pi$
$8$	8.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−48.0000	−1.31569	−0.657843	−	0.753155i	$-0.728531\pi$
$11$	−48.0000	−1.31569	−0.657843	+	0.753155i	$0.728531\pi$
$12$	0	0
$13$	− 70.0000i	− 1.49342i	−0.665148	−	0.746712i	$-0.731631\pi$
$13$	− 70.0000i	− 1.49342i	0.665148	−	0.746712i	$-0.268369\pi$
$14$	68.0000	1.29813
$15$	0	0
$16$	16.0000	0.250000
$17$	27.0000i	0.385204i	0.981277	+	0.192602i	$0.0616926\pi$
$17$	27.0000i	0.385204i	−0.981277	+	0.192602i	$0.938307\pi$
$18$	0	0
$19$	−119.000	−1.43687	−0.718433	−	0.695596i	$-0.755141\pi$
$19$	−119.000	−1.43687	−0.718433	+	0.695596i	$0.755141\pi$
$20$	0	0
$21$	0	0
$22$	96.0000i	0.930330i
$23$	− 51.0000i	− 0.462358i	−0.972911	−	0.231179i	$-0.925742\pi$
$23$	− 51.0000i	− 0.462358i	0.972911	−	0.231179i	$-0.0742583\pi$
$24$	0	0
$25$	0	0
$26$	−140.000	−1.05601
$27$	0	0
$28$	− 136.000i	− 0.917914i
$29$	30.0000	0.192099	0.0960493	−	0.995377i	$-0.469379\pi$
$29$	30.0000	0.192099	0.0960493	+	0.995377i	$0.469379\pi$
$30$	0	0
$31$	−133.000	−0.770565	−0.385282	−	0.922799i	$-0.625896\pi$
$31$	−133.000	−0.770565	−0.385282	+	0.922799i	$0.625896\pi$
$32$	− 32.0000i	− 0.176777i
$33$	0	0
$34$	54.0000	0.272380
$35$	0	0
$36$	0	0
$37$	− 218.000i	− 0.968621i	−0.874896	−	0.484311i	$-0.839070\pi$
$37$	− 218.000i	− 0.968621i	0.874896	−	0.484311i	$-0.160930\pi$
$38$	238.000i	1.01602i
$39$	0	0
$40$	0	0
$41$	156.000	0.594222	0.297111	−	0.954843i	$-0.403977\pi$
$41$	156.000	0.594222	0.297111	+	0.954843i	$0.403977\pi$
$42$	0	0
$43$	− 88.0000i	− 0.312090i	−0.987750	−	0.156045i	$-0.950125\pi$
$43$	− 88.0000i	− 0.312090i	0.987750	−	0.156045i	$-0.0498745\pi$
$44$	192.000	0.657843
$45$	0	0
$46$	−102.000	−0.326937
$47$	516.000i	1.60141i	0.599058	+	0.800706i	$0.295542\pi$
$47$	516.000i	1.60141i	−0.599058	+	0.800706i	$0.704458\pi$
$48$	0	0
$49$	−813.000	−2.37026
$50$	0	0
$51$	0	0
$52$	280.000i	0.746712i
$53$	− 639.000i	− 1.65610i	−0.560653	−	0.828051i	$-0.689450\pi$
$53$	− 639.000i	− 1.65610i	0.560653	−	0.828051i	$-0.310550\pi$
$54$	0	0
$55$	0	0
$56$	−272.000	−0.649063
$57$	0	0
$58$	− 60.0000i	− 0.135834i
$59$	654.000	1.44311	0.721555	−	0.692357i	$-0.243427\pi$
$59$	654.000	1.44311	0.721555	+	0.692357i	$0.243427\pi$
$60$	0	0
$61$	461.000	0.967623	0.483811	−	0.875172i	$-0.339252\pi$
$61$	461.000	0.967623	0.483811	+	0.875172i	$0.339252\pi$
$62$	266.000i	0.544872i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 182.000i	− 0.331863i	−0.986137	−	0.165932i	$-0.946937\pi$
$67$	− 182.000i	− 0.331863i	0.986137	−	0.165932i	$-0.0530631\pi$
$68$	− 108.000i	− 0.192602i
$69$	0	0
$70$	0	0
$71$	900.000	1.50437	0.752186	−	0.658951i	$-0.229000\pi$
$71$	900.000	1.50437	0.752186	+	0.658951i	$0.229000\pi$
$72$	0	0
$73$	704.000i	1.12873i	0.825527	+	0.564363i	$0.190878\pi$
$73$	704.000i	1.12873i	−0.825527	+	0.564363i	$0.809122\pi$
$74$	−436.000	−0.684919
$75$	0	0
$76$	476.000	0.718433
$77$	− 1632.00i	− 2.41537i
$78$	0	0
$79$	1375.00	1.95822	0.979111	−	0.203325i	$-0.0651748\pi$
$79$	1375.00	1.95822	0.979111	+	0.203325i	$0.0651748\pi$
$80$	0	0
$81$	0	0
$82$	− 312.000i	− 0.420178i
$83$	915.000i	1.21005i	0.796206	+	0.605026i	$0.206837\pi$
$83$	915.000i	1.21005i	−0.796206	+	0.605026i	$0.793163\pi$
$84$	0	0
$85$	0	0
$86$	−176.000	−0.220681
$87$	0	0
$88$	− 384.000i	− 0.465165i
$89$	1116.00	1.32917	0.664583	−	0.747215i	$-0.268609\pi$
$89$	1116.00	1.32917	0.664583	+	0.747215i	$0.268609\pi$
$90$	0	0
$91$	2380.00	2.74167
$92$	204.000i	0.231179i
$93$	0	0
$94$	1032.00	1.13237
$95$	0	0
$96$	0	0
$97$	16.0000i	0.0167480i	0.999965	+	0.00837399i	$0.00266555\pi$
$97$	16.0000i	0.0167480i	−0.999965	+	0.00837399i	$0.997334\pi$
$98$	1626.00i	1.67603i
$99$	0	0
$100$	0	0
$101$	348.000	0.342844	0.171422	−	0.985198i	$-0.445164\pi$
$101$	348.000	0.342844	0.171422	+	0.985198i	$0.445164\pi$
$102$	0	0
$103$	− 412.000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 412.000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	560.000	0.528005
$105$	0	0
$106$	−1278.00	−1.17104
$107$	900.000i	0.813143i	0.913619	+	0.406571i	$0.133276\pi$
$107$	900.000i	0.813143i	−0.913619	+	0.406571i	$0.866724\pi$
$108$	0	0
$109$	115.000	0.101055	0.0505275	−	0.998723i	$-0.483910\pi$
$109$	115.000	0.101055	0.0505275	+	0.998723i	$0.483910\pi$
$110$	0	0
$111$	0	0
$112$	544.000i	0.458957i
$113$	966.000i	0.804191i	0.915598	+	0.402096i	$0.131718\pi$
$113$	966.000i	0.804191i	−0.915598	+	0.402096i	$0.868282\pi$
$114$	0	0
$115$	0	0
$116$	−120.000	−0.0960493
$117$	0	0
$118$	− 1308.00i	− 1.02043i
$119$	−918.000	−0.707167
$120$	0	0
$121$	973.000	0.731029
$122$	− 922.000i	− 0.684213i
$123$	0	0
$124$	532.000	0.385282
$125$	0	0
$126$	0	0
$127$	− 1406.00i	− 0.982381i	−0.871052	−	0.491190i	$-0.836562\pi$
$127$	− 1406.00i	− 0.982381i	0.871052	−	0.491190i	$-0.163438\pi$
$128$	128.000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	246.000	0.164070	0.0820348	−	0.996629i	$-0.473858\pi$
$131$	246.000	0.164070	0.0820348	+	0.996629i	$0.473858\pi$
$132$	0	0
$133$	− 4046.00i	− 2.63784i
$134$	−364.000	−0.234663
$135$	0	0
$136$	−216.000	−0.136190
$137$	− 519.000i	− 0.323658i	−0.986819	−	0.161829i	$-0.948261\pi$
$137$	− 519.000i	− 0.323658i	0.986819	−	0.161829i	$-0.0517393\pi$
$138$	0	0
$139$	−1316.00	−0.803034	−0.401517	−	0.915852i	$-0.631517\pi$
$139$	−1316.00	−0.803034	−0.401517	+	0.915852i	$0.631517\pi$
$140$	0	0
$141$	0	0
$142$	− 1800.00i	− 1.06375i
$143$	3360.00i	1.96488i
$144$	0	0
$145$	0	0
$146$	1408.00	0.798130
$147$	0	0
$148$	872.000i	0.484311i
$149$	−372.000	−0.204533	−0.102267	−	0.994757i	$-0.532609\pi$
$149$	−372.000	−0.204533	−0.102267	+	0.994757i	$0.532609\pi$
$150$	0	0
$151$	−1456.00	−0.784686	−0.392343	−	0.919819i	$-0.628335\pi$
$151$	−1456.00	−0.784686	−0.392343	+	0.919819i	$0.628335\pi$
$152$	− 952.000i	− 0.508009i
$153$	0	0
$154$	−3264.00	−1.70793
$155$	0	0
$156$	0	0
$157$	− 956.000i	− 0.485969i	−0.970030	−	0.242984i	$-0.921874\pi$
$157$	− 956.000i	− 0.485969i	0.970030	−	0.242984i	$-0.0781264\pi$
$158$	− 2750.00i	− 1.38467i
$159$	0	0
$160$	0	0
$161$	1734.00	0.848810
$162$	0	0
$163$	− 2446.00i	− 1.17537i	−0.809089	−	0.587686i	$-0.800039\pi$
$163$	− 2446.00i	− 1.17537i	0.809089	−	0.587686i	$-0.199961\pi$
$164$	−624.000	−0.297111
$165$	0	0
$166$	1830.00	0.855636
$167$	3111.00i	1.44154i	0.693177	+	0.720768i	$0.256211\pi$
$167$	3111.00i	1.44154i	−0.693177	+	0.720768i	$0.743789\pi$
$168$	0	0
$169$	−2703.00	−1.23031
$170$	0	0
$171$	0	0
$172$	352.000i	0.156045i
$173$	− 2397.00i	− 1.05341i	−0.850047	−	0.526707i	$-0.823427\pi$
$173$	− 2397.00i	− 1.05341i	0.850047	−	0.526707i	$-0.176573\pi$
$174$	0	0
$175$	0	0
$176$	−768.000	−0.328921
$177$	0	0
$178$	− 2232.00i	− 0.939862i
$179$	540.000	0.225483	0.112742	−	0.993624i	$-0.464037\pi$
$179$	540.000	0.225483	0.112742	+	0.993624i	$0.464037\pi$
$180$	0	0
$181$	2333.00	0.958069	0.479035	−	0.877796i	$-0.340987\pi$
$181$	2333.00	0.958069	0.479035	+	0.877796i	$0.340987\pi$
$182$	− 4760.00i	− 1.93865i
$183$	0	0
$184$	408.000	0.163468
$185$	0	0
$186$	0	0
$187$	− 1296.00i	− 0.506807i
$188$	− 2064.00i	− 0.800706i
$189$	0	0
$190$	0	0
$191$	−2730.00	−1.03422	−0.517110	−	0.855919i	$-0.672992\pi$
$191$	−2730.00	−1.03422	−0.517110	+	0.855919i	$0.672992\pi$
$192$	0	0
$193$	− 4570.00i	− 1.70443i	−0.523188	−	0.852217i	$-0.675258\pi$
$193$	− 4570.00i	− 1.70443i	0.523188	−	0.852217i	$-0.324742\pi$
$194$	32.0000	0.0118426
$195$	0	0
$196$	3252.00	1.18513
$197$	− 675.000i	− 0.244121i	−0.992523	−	0.122060i	$-0.961050\pi$
$197$	− 675.000i	− 0.244121i	0.992523	−	0.122060i	$-0.0389501\pi$
$198$	0	0
$199$	3112.00	1.10856	0.554281	−	0.832330i	$-0.312993\pi$
$199$	3112.00	1.10856	0.554281	+	0.832330i	$0.312993\pi$
$200$	0	0
$201$	0	0
$202$	− 696.000i	− 0.242428i
$203$	1020.00i	0.352660i
$204$	0	0
$205$	0	0
$206$	−824.000	−0.278693
$207$	0	0
$208$	− 1120.00i	− 0.373356i
$209$	5712.00	1.89047
$210$	0	0
$211$	2441.00	0.796424	0.398212	−	0.917294i	$-0.369631\pi$
$211$	2441.00	0.796424	0.398212	+	0.917294i	$0.369631\pi$
$212$	2556.00i	0.828051i
$213$	0	0
$214$	1800.00	0.574979
$215$	0	0
$216$	0	0
$217$	− 4522.00i	− 1.41462i
$218$	− 230.000i	− 0.0714567i
$219$	0	0
$220$	0	0
$221$	1890.00	0.575272
$222$	0	0
$223$	− 3418.00i	− 1.02640i	−0.858270	−	0.513198i	$-0.828461\pi$
$223$	− 3418.00i	− 1.02640i	0.858270	−	0.513198i	$-0.171539\pi$
$224$	1088.00	0.324532
$225$	0	0
$226$	1932.00	0.568649
$227$	4377.00i	1.27979i	0.768464	+	0.639894i	$0.221022\pi$
$227$	4377.00i	1.27979i	−0.768464	+	0.639894i	$0.778978\pi$
$228$	0	0
$229$	−4187.00	−1.20823	−0.604115	−	0.796897i	$-0.706473\pi$
$229$	−4187.00	−1.20823	−0.604115	+	0.796897i	$0.706473\pi$
$230$	0	0
$231$	0	0
$232$	240.000i	0.0679171i
$233$	− 1098.00i	− 0.308723i	−0.988014	−	0.154361i	$-0.950668\pi$
$233$	− 1098.00i	− 0.308723i	0.988014	−	0.154361i	$-0.0493320\pi$
$234$	0	0
$235$	0	0
$236$	−2616.00	−0.721555
$237$	0	0
$238$	1836.00i	0.500043i
$239$	−6474.00	−1.75217	−0.876084	−	0.482158i	$-0.839853\pi$
$239$	−6474.00	−1.75217	−0.876084	+	0.482158i	$0.839853\pi$
$240$	0	0
$241$	3251.00	0.868943	0.434472	−	0.900686i	$-0.356935\pi$
$241$	3251.00	0.868943	0.434472	+	0.900686i	$0.356935\pi$
$242$	− 1946.00i	− 0.516916i
$243$	0	0
$244$	−1844.00	−0.483811
$245$	0	0
$246$	0	0
$247$	8330.00i	2.14585i
$248$	− 1064.00i	− 0.272436i
$249$	0	0
$250$	0	0
$251$	1728.00	0.434543	0.217272	−	0.976111i	$-0.430284\pi$
$251$	1728.00	0.434543	0.217272	+	0.976111i	$0.430284\pi$
$252$	0	0
$253$	2448.00i	0.608318i
$254$	−2812.00	−0.694648
$255$	0	0
$256$	256.000	0.0625000
$257$	5469.00i	1.32742i	0.747990	+	0.663710i	$0.231019\pi$
$257$	5469.00i	1.32742i	−0.747990	+	0.663710i	$0.768981\pi$
$258$	0	0
$259$	7412.00	1.77822
$260$	0	0
$261$	0	0
$262$	− 492.000i	− 0.116015i
$263$	− 3216.00i	− 0.754019i	−0.926209	−	0.377010i	$-0.876952\pi$
$263$	− 3216.00i	− 0.754019i	0.926209	−	0.377010i	$-0.123048\pi$
$264$	0	0
$265$	0	0
$266$	−8092.00	−1.86523
$267$	0	0
$268$	728.000i	0.165932i
$269$	8010.00	1.81553	0.907766	−	0.419476i	$-0.137786\pi$
$269$	8010.00	1.81553	0.907766	+	0.419476i	$0.137786\pi$
$270$	0	0
$271$	−3805.00	−0.852905	−0.426453	−	0.904510i	$-0.640237\pi$
$271$	−3805.00	−0.852905	−0.426453	+	0.904510i	$0.640237\pi$
$272$	432.000i	0.0963009i
$273$	0	0
$274$	−1038.00	−0.228861
$275$	0	0
$276$	0	0
$277$	− 3224.00i	− 0.699319i	−0.936877	−	0.349660i	$-0.886297\pi$
$277$	− 3224.00i	− 0.699319i	0.936877	−	0.349660i	$-0.113703\pi$
$278$	2632.00i	0.567830i
$279$	0	0
$280$	0	0
$281$	−4530.00	−0.961698	−0.480849	−	0.876803i	$-0.659671\pi$
$281$	−4530.00	−0.961698	−0.480849	+	0.876803i	$0.659671\pi$
$282$	0	0
$283$	− 3292.00i	− 0.691481i	−0.938330	−	0.345740i	$-0.887628\pi$
$283$	− 3292.00i	− 0.691481i	0.938330	−	0.345740i	$-0.112372\pi$
$284$	−3600.00	−0.752186
$285$	0	0
$286$	6720.00	1.38938
$287$	5304.00i	1.09089i
$288$	0	0
$289$	4184.00	0.851618
$290$	0	0
$291$	0	0
$292$	− 2816.00i	− 0.564363i
$293$	− 7953.00i	− 1.58573i	−0.609397	−	0.792866i	$-0.708588\pi$
$293$	− 7953.00i	− 1.58573i	0.609397	−	0.792866i	$-0.291412\pi$
$294$	0	0
$295$	0	0
$296$	1744.00	0.342459
$297$	0	0
$298$	744.000i	0.144627i
$299$	−3570.00	−0.690496
$300$	0	0
$301$	2992.00	0.572944
$302$	2912.00i	0.554857i
$303$	0	0
$304$	−1904.00	−0.359217
$305$	0	0
$306$	0	0
$307$	5290.00i	0.983441i	0.870753	+	0.491720i	$0.163632\pi$
$307$	5290.00i	0.983441i	−0.870753	+	0.491720i	$0.836368\pi$
$308$	6528.00i	1.20769i
$309$	0	0
$310$	0	0
$311$	5358.00	0.976927	0.488464	−	0.872584i	$-0.337558\pi$
$311$	5358.00	0.976927	0.488464	+	0.872584i	$0.337558\pi$
$312$	0	0
$313$	5600.00i	1.01128i	0.862744	+	0.505640i	$0.168744\pi$
$313$	5600.00i	1.01128i	−0.862744	+	0.505640i	$0.831256\pi$
$314$	−1912.00	−0.343632
$315$	0	0
$316$	−5500.00	−0.979111
$317$	− 7341.00i	− 1.30067i	−0.759649	−	0.650334i	$-0.774629\pi$
$317$	− 7341.00i	− 1.30067i	0.759649	−	0.650334i	$-0.225371\pi$
$318$	0	0
$319$	−1440.00	−0.252741
$320$	0	0
$321$	0	0
$322$	− 3468.00i	− 0.600199i
$323$	− 3213.00i	− 0.553486i
$324$	0	0
$325$	0	0
$326$	−4892.00	−0.831113
$327$	0	0
$328$	1248.00i	0.210089i
$329$	−17544.0	−2.93991
$330$	0	0
$331$	380.000	0.0631018	0.0315509	−	0.999502i	$-0.489955\pi$
$331$	380.000	0.0631018	0.0315509	+	0.999502i	$0.489955\pi$
$332$	− 3660.00i	− 0.605026i
$333$	0	0
$334$	6222.00	1.01932
$335$	0	0
$336$	0	0
$337$	− 434.000i	− 0.0701528i	−0.999385	−	0.0350764i	$-0.988833\pi$
$337$	− 434.000i	− 0.0701528i	0.999385	−	0.0350764i	$-0.0111675\pi$
$338$	5406.00i	0.869963i
$339$	0	0
$340$	0	0
$341$	6384.00	1.01382
$342$	0	0
$343$	− 15980.0i	− 2.51557i
$344$	704.000	0.110341
$345$	0	0
$346$	−4794.00	−0.744876
$347$	− 8004.00i	− 1.23826i	−0.785287	−	0.619131i	$-0.787485\pi$
$347$	− 8004.00i	− 1.23826i	0.785287	−	0.619131i	$-0.212515\pi$
$348$	0	0
$349$	−1109.00	−0.170096	−0.0850479	−	0.996377i	$-0.527104\pi$
$349$	−1109.00	−0.170096	−0.0850479	+	0.996377i	$0.527104\pi$
$350$	0	0
$351$	0	0
$352$	1536.00i	0.232583i
$353$	− 7662.00i	− 1.15526i	−0.816298	−	0.577630i	$-0.803977\pi$
$353$	− 7662.00i	− 1.15526i	0.816298	−	0.577630i	$-0.196023\pi$
$354$	0	0
$355$	0	0
$356$	−4464.00	−0.664583
$357$	0	0
$358$	− 1080.00i	− 0.159441i
$359$	8478.00	1.24638	0.623192	−	0.782069i	$-0.285836\pi$
$359$	8478.00	1.24638	0.623192	+	0.782069i	$0.285836\pi$
$360$	0	0
$361$	7302.00	1.06459
$362$	− 4666.00i	− 0.677457i
$363$	0	0
$364$	−9520.00	−1.37083
$365$	0	0
$366$	0	0
$367$	− 13286.0i	− 1.88971i	−0.327489	−	0.944855i	$-0.606202\pi$
$367$	− 13286.0i	− 1.88971i	0.327489	−	0.944855i	$-0.393798\pi$
$368$	− 816.000i	− 0.115590i
$369$	0	0
$370$	0	0
$371$	21726.0	3.04032
$372$	0	0
$373$	3080.00i	0.427551i	0.976883	+	0.213775i	$0.0685760\pi$
$373$	3080.00i	0.427551i	−0.976883	+	0.213775i	$0.931424\pi$
$374$	−2592.00	−0.358367
$375$	0	0
$376$	−4128.00	−0.566184
$377$	− 2100.00i	− 0.286885i
$378$	0	0
$379$	−10109.0	−1.37009	−0.685045	−	0.728500i	$-0.740218\pi$
$379$	−10109.0	−1.37009	−0.685045	+	0.728500i	$0.740218\pi$
$380$	0	0
$381$	0	0
$382$	5460.00i	0.731303i
$383$	− 8727.00i	− 1.16431i	−0.813080	−	0.582153i	$-0.802211\pi$
$383$	− 8727.00i	− 1.16431i	0.813080	−	0.582153i	$-0.197789\pi$
$384$	0	0
$385$	0	0
$386$	−9140.00	−1.20522
$387$	0	0
$388$	− 64.0000i	− 0.00837399i
$389$	2712.00	0.353480	0.176740	−	0.984258i	$-0.443445\pi$
$389$	2712.00	0.353480	0.176740	+	0.984258i	$0.443445\pi$
$390$	0	0
$391$	1377.00	0.178102
$392$	− 6504.00i	− 0.838014i
$393$	0	0
$394$	−1350.00	−0.172619
$395$	0	0
$396$	0	0
$397$	8818.00i	1.11477i	0.830255	+	0.557384i	$0.188195\pi$
$397$	8818.00i	1.11477i	−0.830255	+	0.557384i	$0.811805\pi$
$398$	− 6224.00i	− 0.783872i
$399$	0	0
$400$	0	0
$401$	3306.00	0.411705	0.205853	−	0.978583i	$-0.434003\pi$
$401$	3306.00	0.411705	0.205853	+	0.978583i	$0.434003\pi$
$402$	0	0
$403$	9310.00i	1.15078i
$404$	−1392.00	−0.171422
$405$	0	0
$406$	2040.00	0.249368
$407$	10464.0i	1.27440i
$408$	0	0
$409$	−6401.00	−0.773861	−0.386930	−	0.922109i	$-0.626465\pi$
$409$	−6401.00	−0.773861	−0.386930	+	0.922109i	$0.626465\pi$
$410$	0	0
$411$	0	0
$412$	1648.00i	0.197066i
$413$	22236.0i	2.64930i
$414$	0	0
$415$	0	0
$416$	−2240.00	−0.264002
$417$	0	0
$418$	− 11424.0i	− 1.33676i
$419$	2256.00	0.263038	0.131519	−	0.991314i	$-0.458015\pi$
$419$	2256.00	0.263038	0.131519	+	0.991314i	$0.458015\pi$
$420$	0	0
$421$	1811.00	0.209650	0.104825	−	0.994491i	$-0.466572\pi$
$421$	1811.00	0.209650	0.104825	+	0.994491i	$0.466572\pi$
$422$	− 4882.00i	− 0.563156i
$423$	0	0
$424$	5112.00	0.585520
$425$	0	0
$426$	0	0
$427$	15674.0i	1.77639i
$428$	− 3600.00i	− 0.406571i
$429$	0	0
$430$	0	0
$431$	−5454.00	−0.609536	−0.304768	−	0.952427i	$-0.598579\pi$
$431$	−5454.00	−0.609536	−0.304768	+	0.952427i	$0.598579\pi$
$432$	0	0
$433$	2990.00i	0.331848i	0.986139	+	0.165924i	$0.0530607\pi$
$433$	2990.00i	0.331848i	−0.986139	+	0.165924i	$0.946939\pi$
$434$	−9044.00	−1.00029
$435$	0	0
$436$	−460.000	−0.0505275
$437$	6069.00i	0.664347i
$438$	0	0
$439$	−9371.00	−1.01880	−0.509400	−	0.860530i	$-0.670133\pi$
$439$	−9371.00	−1.01880	−0.509400	+	0.860530i	$0.670133\pi$
$440$	0	0
$441$	0	0
$442$	− 3780.00i	− 0.406779i
$443$	6171.00i	0.661835i	0.943660	+	0.330918i	$0.107358\pi$
$443$	6171.00i	0.661835i	−0.943660	+	0.330918i	$0.892642\pi$
$444$	0	0
$445$	0	0
$446$	−6836.00	−0.725771
$447$	0	0
$448$	− 2176.00i	− 0.229478i
$449$	−4122.00	−0.433250	−0.216625	−	0.976255i	$-0.569505\pi$
$449$	−4122.00	−0.433250	−0.216625	+	0.976255i	$0.569505\pi$
$450$	0	0
$451$	−7488.00	−0.781810
$452$	− 3864.00i	− 0.402096i
$453$	0	0
$454$	8754.00	0.904946
$455$	0	0
$456$	0	0
$457$	− 7076.00i	− 0.724292i	−0.932121	−	0.362146i	$-0.882044\pi$
$457$	− 7076.00i	− 0.724292i	0.932121	−	0.362146i	$-0.117956\pi$
$458$	8374.00i	0.854348i
$459$	0	0
$460$	0	0
$461$	762.000	0.0769846	0.0384923	−	0.999259i	$-0.487745\pi$
$461$	762.000	0.0769846	0.0384923	+	0.999259i	$0.487745\pi$
$462$	0	0
$463$	8822.00i	0.885514i	0.896642	+	0.442757i	$0.146000\pi$
$463$	8822.00i	0.885514i	−0.896642	+	0.442757i	$0.854000\pi$
$464$	480.000	0.0480247
$465$	0	0
$466$	−2196.00	−0.218300
$467$	4977.00i	0.493165i	0.969122	+	0.246583i	$0.0793076\pi$
$467$	4977.00i	0.493165i	−0.969122	+	0.246583i	$0.920692\pi$
$468$	0	0
$469$	6188.00	0.609244
$470$	0	0
$471$	0	0
$472$	5232.00i	0.510217i
$473$	4224.00i	0.410613i
$474$	0	0
$475$	0	0
$476$	3672.00	0.353584
$477$	0	0
$478$	12948.0i	1.23897i
$479$	−10104.0	−0.963807	−0.481903	−	0.876224i	$-0.660054\pi$
$479$	−10104.0	−0.963807	−0.481903	+	0.876224i	$0.660054\pi$
$480$	0	0
$481$	−15260.0	−1.44656
$482$	− 6502.00i	− 0.614436i
$483$	0	0
$484$	−3892.00	−0.365515
$485$	0	0
$486$	0	0
$487$	− 14924.0i	− 1.38865i	−0.719663	−	0.694323i	$-0.755704\pi$
$487$	− 14924.0i	− 1.38865i	0.719663	−	0.694323i	$-0.244296\pi$
$488$	3688.00i	0.342106i
$489$	0	0
$490$	0	0
$491$	1146.00	0.105332	0.0526662	−	0.998612i	$-0.483228\pi$
$491$	1146.00	0.105332	0.0526662	+	0.998612i	$0.483228\pi$
$492$	0	0
$493$	810.000i	0.0739971i
$494$	16660.0	1.51735
$495$	0	0
$496$	−2128.00	−0.192641
$497$	30600.0i	2.76177i
$498$	0	0
$499$	14965.0	1.34254	0.671268	−	0.741215i	$-0.265750\pi$
$499$	14965.0	1.34254	0.671268	+	0.741215i	$0.265750\pi$
$500$	0	0
$501$	0	0
$502$	− 3456.00i	− 0.307269i
$503$	15525.0i	1.37619i	0.725619	+	0.688097i	$0.241554\pi$
$503$	15525.0i	1.37619i	−0.725619	+	0.688097i	$0.758446\pi$
$504$	0	0
$505$	0	0
$506$	4896.00	0.430146
$507$	0	0
$508$	5624.00i	0.491190i
$509$	8196.00	0.713716	0.356858	−	0.934159i	$-0.383848\pi$
$509$	8196.00	0.713716	0.356858	+	0.934159i	$0.383848\pi$
$510$	0	0
$511$	−23936.0	−2.07215
$512$	− 512.000i	− 0.0441942i
$513$	0	0
$514$	10938.0	0.938627
$515$	0	0
$516$	0	0
$517$	− 24768.0i	− 2.10695i
$518$	− 14824.0i	− 1.25739i
$519$	0	0
$520$	0	0
$521$	4932.00	0.414731	0.207365	−	0.978264i	$-0.433511\pi$
$521$	4932.00	0.414731	0.207365	+	0.978264i	$0.433511\pi$
$522$	0	0
$523$	− 5938.00i	− 0.496464i	−0.968701	−	0.248232i	$-0.920150\pi$
$523$	− 5938.00i	− 0.496464i	0.968701	−	0.248232i	$-0.0798495\pi$
$524$	−984.000	−0.0820348
$525$	0	0
$526$	−6432.00	−0.533172
$527$	− 3591.00i	− 0.296824i
$528$	0	0
$529$	9566.00	0.786225
$530$	0	0
$531$	0	0
$532$	16184.0i	1.31892i
$533$	− 10920.0i	− 0.887425i
$534$	0	0
$535$	0	0
$536$	1456.00	0.117331
$537$	0	0
$538$	− 16020.0i	− 1.28378i
$539$	39024.0	3.11852
$540$	0	0
$541$	−6730.00	−0.534834	−0.267417	−	0.963581i	$-0.586170\pi$
$541$	−6730.00	−0.534834	−0.267417	+	0.963581i	$0.586170\pi$
$542$	7610.00i	0.603095i
$543$	0	0
$544$	864.000	0.0680950
$545$	0	0
$546$	0	0
$547$	17656.0i	1.38010i	0.723761	+	0.690051i	$0.242412\pi$
$547$	17656.0i	1.38010i	−0.723761	+	0.690051i	$0.757588\pi$
$548$	2076.00i	0.161829i
$549$	0	0
$550$	0	0
$551$	−3570.00	−0.276020
$552$	0	0
$553$	46750.0i	3.59496i
$554$	−6448.00	−0.494493
$555$	0	0
$556$	5264.00	0.401517
$557$	− 7974.00i	− 0.606587i	−0.952897	−	0.303294i	$-0.901914\pi$
$557$	− 7974.00i	− 0.606587i	0.952897	−	0.303294i	$-0.0980863\pi$
$558$	0	0
$559$	−6160.00	−0.466083
$560$	0	0
$561$	0	0
$562$	9060.00i	0.680023i
$563$	− 25332.0i	− 1.89630i	−0.317824	−	0.948150i	$-0.602952\pi$
$563$	− 25332.0i	− 1.89630i	0.317824	−	0.948150i	$-0.397048\pi$
$564$	0	0
$565$	0	0
$566$	−6584.00	−0.488951
$567$	0	0
$568$	7200.00i	0.531876i
$569$	1038.00	0.0764767	0.0382383	−	0.999269i	$-0.487825\pi$
$569$	1038.00	0.0764767	0.0382383	+	0.999269i	$0.487825\pi$
$570$	0	0
$571$	15671.0	1.14853	0.574265	−	0.818669i	$-0.305288\pi$
$571$	15671.0	1.14853	0.574265	+	0.818669i	$0.305288\pi$
$572$	− 13440.0i	− 0.982438i
$573$	0	0
$574$	10608.0	0.771375
$575$	0	0
$576$	0	0
$577$	916.000i	0.0660894i	0.999454	+	0.0330447i	$0.0105204\pi$
$577$	916.000i	0.0660894i	−0.999454	+	0.0330447i	$0.989480\pi$
$578$	− 8368.00i	− 0.602185i
$579$	0	0
$580$	0	0
$581$	−31110.0	−2.22145
$582$	0	0
$583$	30672.0i	2.17891i
$584$	−5632.00	−0.399065
$585$	0	0
$586$	−15906.0	−1.12128
$587$	− 9141.00i	− 0.642742i	−0.946953	−	0.321371i	$-0.895856\pi$
$587$	− 9141.00i	− 0.642742i	0.946953	−	0.321371i	$-0.104144\pi$
$588$	0	0
$589$	15827.0	1.10720
$590$	0	0
$591$	0	0
$592$	− 3488.00i	− 0.242155i
$593$	5247.00i	0.363353i	0.983358	+	0.181677i	$0.0581524\pi$
$593$	5247.00i	0.363353i	−0.983358	+	0.181677i	$0.941848\pi$
$594$	0	0
$595$	0	0
$596$	1488.00	0.102267
$597$	0	0
$598$	7140.00i	0.488255i
$599$	24162.0	1.64813	0.824067	−	0.566492i	$-0.191700\pi$
$599$	24162.0	1.64813	0.824067	+	0.566492i	$0.191700\pi$
$600$	0	0
$601$	14357.0	0.974433	0.487217	−	0.873281i	$-0.338012\pi$
$601$	14357.0	0.974433	0.487217	+	0.873281i	$0.338012\pi$
$602$	− 5984.00i	− 0.405132i
$603$	0	0
$604$	5824.00	0.392343
$605$	0	0
$606$	0	0
$607$	− 3152.00i	− 0.210767i	−0.994432	−	0.105384i	$-0.966393\pi$
$607$	− 3152.00i	− 0.210767i	0.994432	−	0.105384i	$-0.0336071\pi$
$608$	3808.00i	0.254005i
$609$	0	0
$610$	0	0
$611$	36120.0	2.39159
$612$	0	0
$613$	4592.00i	0.302560i	0.988491	+	0.151280i	$0.0483395\pi$
$613$	4592.00i	0.302560i	−0.988491	+	0.151280i	$0.951661\pi$
$614$	10580.0	0.695397
$615$	0	0
$616$	13056.0	0.853963
$617$	7359.00i	0.480166i	0.970752	+	0.240083i	$0.0771746\pi$
$617$	7359.00i	0.480166i	−0.970752	+	0.240083i	$0.922825\pi$
$618$	0	0
$619$	15712.0	1.02022	0.510112	−	0.860108i	$-0.329604\pi$
$619$	15712.0	1.02022	0.510112	+	0.860108i	$0.329604\pi$
$620$	0	0
$621$	0	0
$622$	− 10716.0i	− 0.690792i
$623$	37944.0i	2.44012i
$624$	0	0
$625$	0	0
$626$	11200.0	0.715083
$627$	0	0
$628$	3824.00i	0.242984i
$629$	5886.00	0.373116
$630$	0	0
$631$	−3175.00	−0.200309	−0.100154	−	0.994972i	$-0.531934\pi$
$631$	−3175.00	−0.200309	−0.100154	+	0.994972i	$0.531934\pi$
$632$	11000.0i	0.692336i
$633$	0	0
$634$	−14682.0	−0.919711
$635$	0	0
$636$	0	0
$637$	56910.0i	3.53981i
$638$	2880.00i	0.178715i
$639$	0	0
$640$	0	0
$641$	96.0000	0.00591540	0.00295770	−	0.999996i	$-0.499059\pi$
$641$	96.0000	0.00591540	0.00295770	+	0.999996i	$0.499059\pi$
$642$	0	0
$643$	− 18070.0i	− 1.10826i	−0.832430	−	0.554130i	$-0.813051\pi$
$643$	− 18070.0i	− 1.10826i	0.832430	−	0.554130i	$-0.186949\pi$
$644$	−6936.00	−0.424405
$645$	0	0
$646$	−6426.00	−0.391374
$647$	− 1341.00i	− 0.0814840i	−0.999170	−	0.0407420i	$-0.987028\pi$
$647$	− 1341.00i	− 0.0814840i	0.999170	−	0.0407420i	$-0.0129722\pi$
$648$	0	0
$649$	−31392.0	−1.89868
$650$	0	0
$651$	0	0
$652$	9784.00i	0.587686i
$653$	− 24495.0i	− 1.46794i	−0.679183	−	0.733969i	$-0.737666\pi$
$653$	− 24495.0i	− 1.46794i	0.679183	−	0.733969i	$-0.262334\pi$
$654$	0	0
$655$	0	0
$656$	2496.00	0.148556
$657$	0	0
$658$	35088.0i	2.07883i
$659$	12378.0	0.731682	0.365841	−	0.930677i	$-0.380781\pi$
$659$	12378.0	0.731682	0.365841	+	0.930677i	$0.380781\pi$
$660$	0	0
$661$	−24442.0	−1.43825	−0.719125	−	0.694880i	$-0.755457\pi$
$661$	−24442.0	−1.43825	−0.719125	+	0.694880i	$0.755457\pi$
$662$	− 760.000i	− 0.0446197i
$663$	0	0
$664$	−7320.00	−0.427818
$665$	0	0
$666$	0	0
$667$	− 1530.00i	− 0.0888183i
$668$	− 12444.0i	− 0.720768i
$669$	0	0
$670$	0	0
$671$	−22128.0	−1.27309
$672$	0	0
$673$	2378.00i	0.136204i	0.997678	+	0.0681019i	$0.0216943\pi$
$673$	2378.00i	0.136204i	−0.997678	+	0.0681019i	$0.978306\pi$
$674$	−868.000	−0.0496055
$675$	0	0
$676$	10812.0	0.615157
$677$	− 5478.00i	− 0.310985i	−0.987837	−	0.155492i	$-0.950304\pi$
$677$	− 5478.00i	− 0.310985i	0.987837	−	0.155492i	$-0.0496964\pi$
$678$	0	0
$679$	−544.000	−0.0307464
$680$	0	0
$681$	0	0
$682$	− 12768.0i	− 0.716880i
$683$	8595.00i	0.481521i	0.970585	+	0.240760i	$0.0773968\pi$
$683$	8595.00i	0.481521i	−0.970585	+	0.240760i	$0.922603\pi$
$684$	0	0
$685$	0	0
$686$	−31960.0	−1.77877
$687$	0	0
$688$	− 1408.00i	− 0.0780225i
$689$	−44730.0	−2.47326
$690$	0	0
$691$	−31615.0	−1.74051	−0.870254	−	0.492603i	$-0.836046\pi$
$691$	−31615.0	−1.74051	−0.870254	+	0.492603i	$0.836046\pi$
$692$	9588.00i	0.526707i
$693$	0	0
$694$	−16008.0	−0.875584
$695$	0	0
$696$	0	0
$697$	4212.00i	0.228897i
$698$	2218.00i	0.120276i
$699$	0	0
$700$	0	0
$701$	29790.0	1.60507	0.802534	−	0.596606i	$-0.203485\pi$
$701$	29790.0	1.60507	0.802534	+	0.596606i	$0.203485\pi$
$702$	0	0
$703$	25942.0i	1.39178i
$704$	3072.00	0.164461
$705$	0	0
$706$	−15324.0	−0.816893
$707$	11832.0i	0.629403i
$708$	0	0
$709$	−3818.00	−0.202240	−0.101120	−	0.994874i	$-0.532243\pi$
$709$	−3818.00	−0.202240	−0.101120	+	0.994874i	$0.532243\pi$
$710$	0	0
$711$	0	0
$712$	8928.00i	0.469931i
$713$	6783.00i	0.356277i
$714$	0	0
$715$	0	0
$716$	−2160.00	−0.112742
$717$	0	0
$718$	− 16956.0i	− 0.881326i
$719$	28314.0	1.46861	0.734307	−	0.678817i	$-0.237507\pi$
$719$	28314.0	1.46861	0.734307	+	0.678817i	$0.237507\pi$
$720$	0	0
$721$	14008.0	0.723558
$722$	− 14604.0i	− 0.752776i
$723$	0	0
$724$	−9332.00	−0.479035
$725$	0	0
$726$	0	0
$727$	− 56.0000i	− 0.00285684i	−0.999999	−	0.00142842i	$-0.999545\pi$
$727$	− 56.0000i	− 0.00285684i	0.999999	−	0.00142842i	$-0.000454681\pi$
$728$	19040.0i	0.969326i
$729$	0	0
$730$	0	0
$731$	2376.00	0.120218
$732$	0	0
$733$	− 34432.0i	− 1.73503i	−0.497413	−	0.867514i	$-0.665717\pi$
$733$	− 34432.0i	− 1.73503i	0.497413	−	0.867514i	$-0.334283\pi$
$734$	−26572.0	−1.33623
$735$	0	0
$736$	−1632.00	−0.0817341
$737$	8736.00i	0.436628i
$738$	0	0
$739$	1051.00	0.0523162	0.0261581	−	0.999658i	$-0.491673\pi$
$739$	1051.00	0.0523162	0.0261581	+	0.999658i	$0.491673\pi$
$740$	0	0
$741$	0	0
$742$	− 43452.0i	− 2.14983i
$743$	39144.0i	1.93278i	0.257084	+	0.966389i	$0.417238\pi$
$743$	39144.0i	1.93278i	−0.257084	+	0.966389i	$0.582762\pi$
$744$	0	0
$745$	0	0
$746$	6160.00	0.302324
$747$	0	0
$748$	5184.00i	0.253403i
$749$	−30600.0	−1.49279
$750$	0	0
$751$	−1735.00	−0.0843023	−0.0421512	−	0.999111i	$-0.513421\pi$
$751$	−1735.00	−0.0843023	−0.0421512	+	0.999111i	$0.513421\pi$
$752$	8256.00i	0.400353i
$753$	0	0
$754$	−4200.00	−0.202858
$755$	0	0
$756$	0	0
$757$	− 6698.00i	− 0.321589i	−0.986988	−	0.160795i	$-0.948594\pi$
$757$	− 6698.00i	− 0.321589i	0.986988	−	0.160795i	$-0.0514057\pi$
$758$	20218.0i	0.968801i
$759$	0	0
$760$	0	0
$761$	38766.0	1.84660	0.923302	−	0.384074i	$-0.125479\pi$
$761$	38766.0	1.84660	0.923302	+	0.384074i	$0.125479\pi$
$762$	0	0
$763$	3910.00i	0.185520i
$764$	10920.0	0.517110
$765$	0	0
$766$	−17454.0	−0.823288
$767$	− 45780.0i	− 2.15518i
$768$	0	0
$769$	−23501.0	−1.10204	−0.551019	−	0.834492i	$-0.685761\pi$
$769$	−23501.0	−1.10204	−0.551019	+	0.834492i	$0.685761\pi$
$770$	0	0
$771$	0	0
$772$	18280.0i	0.852217i
$773$	− 3591.00i	− 0.167088i	−0.996504	−	0.0835442i	$-0.973376\pi$
$773$	− 3591.00i	− 0.167088i	0.996504	−	0.0835442i	$-0.0266240\pi$
$774$	0	0
$775$	0	0
$776$	−128.000	−0.00592130
$777$	0	0
$778$	− 5424.00i	− 0.249948i
$779$	−18564.0	−0.853818
$780$	0	0
$781$	−43200.0	−1.97928
$782$	− 2754.00i	− 0.125937i
$783$	0	0
$784$	−13008.0	−0.592566
$785$	0	0
$786$	0	0
$787$	20716.0i	0.938305i	0.883117	+	0.469152i	$0.155440\pi$
$787$	20716.0i	0.938305i	−0.883117	+	0.469152i	$0.844560\pi$
$788$	2700.00i	0.122060i
$789$	0	0
$790$	0	0
$791$	−32844.0	−1.47636
$792$	0	0
$793$	− 32270.0i	− 1.44507i
$794$	17636.0	0.788260
$795$	0	0
$796$	−12448.0	−0.554281
$797$	42981.0i	1.91024i	0.296211	+	0.955122i	$0.404277\pi$
$797$	42981.0i	1.91024i	−0.296211	+	0.955122i	$0.595723\pi$
$798$	0	0
$799$	−13932.0	−0.616869
$800$	0	0
$801$	0	0
$802$	− 6612.00i	− 0.291119i
$803$	− 33792.0i	− 1.48505i
$804$	0	0
$805$	0	0
$806$	18620.0	0.813724
$807$	0	0
$808$	2784.00i	0.121214i
$809$	2268.00	0.0985644	0.0492822	−	0.998785i	$-0.484307\pi$
$809$	2268.00	0.0985644	0.0492822	+	0.998785i	$0.484307\pi$
$810$	0	0
$811$	11756.0	0.509012	0.254506	−	0.967071i	$-0.418087\pi$
$811$	11756.0	0.509012	0.254506	+	0.967071i	$0.418087\pi$
$812$	− 4080.00i	− 0.176330i
$813$	0	0
$814$	20928.0	0.901138
$815$	0	0
$816$	0	0
$817$	10472.0i	0.448432i
$818$	12802.0i	0.547202i
$819$	0	0
$820$	0	0
$821$	8646.00	0.367537	0.183768	−	0.982970i	$-0.441170\pi$
$821$	8646.00	0.367537	0.183768	+	0.982970i	$0.441170\pi$
$822$	0	0
$823$	10784.0i	0.456752i	0.973573	+	0.228376i	$0.0733415\pi$
$823$	10784.0i	0.456752i	−0.973573	+	0.228376i	$0.926659\pi$
$824$	3296.00	0.139347
$825$	0	0
$826$	44472.0	1.87334
$827$	− 42597.0i	− 1.79110i	−0.444957	−	0.895552i	$-0.646781\pi$
$827$	− 42597.0i	− 1.79110i	0.444957	−	0.895552i	$-0.353219\pi$
$828$	0	0
$829$	26458.0	1.10847	0.554237	−	0.832359i	$-0.313010\pi$
$829$	26458.0	1.10847	0.554237	+	0.832359i	$0.313010\pi$
$830$	0	0
$831$	0	0
$832$	4480.00i	0.186678i
$833$	− 21951.0i	− 0.913034i
$834$	0	0
$835$	0	0
$836$	−22848.0	−0.945233
$837$	0	0
$838$	− 4512.00i	− 0.185996i
$839$	11496.0	0.473046	0.236523	−	0.971626i	$-0.423992\pi$
$839$	11496.0	0.473046	0.236523	+	0.971626i	$0.423992\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	− 3622.00i	− 0.148245i
$843$	0	0
$844$	−9764.00	−0.398212
$845$	0	0
$846$	0	0
$847$	33082.0i	1.34204i
$848$	− 10224.0i	− 0.414025i
$849$	0	0
$850$	0	0
$851$	−11118.0	−0.447850
$852$	0	0
$853$	21548.0i	0.864935i	0.901650	+	0.432467i	$0.142357\pi$
$853$	21548.0i	0.864935i	−0.901650	+	0.432467i	$0.857643\pi$
$854$	31348.0	1.25610
$855$	0	0
$856$	−7200.00	−0.287489
$857$	− 6261.00i	− 0.249559i	−0.992185	−	0.124779i	$-0.960178\pi$
$857$	− 6261.00i	− 0.249559i	0.992185	−	0.124779i	$-0.0398223\pi$
$858$	0	0
$859$	3355.00	0.133261	0.0666305	−	0.997778i	$-0.478775\pi$
$859$	3355.00	0.133261	0.0666305	+	0.997778i	$0.478775\pi$
$860$	0	0
$861$	0	0
$862$	10908.0i	0.431007i
$863$	19701.0i	0.777091i	0.921429	+	0.388546i	$0.127022\pi$
$863$	19701.0i	0.777091i	−0.921429	+	0.388546i	$0.872978\pi$
$864$	0	0
$865$	0	0
$866$	5980.00	0.234652
$867$	0	0
$868$	18088.0i	0.707312i
$869$	−66000.0	−2.57641
$870$	0	0
$871$	−12740.0	−0.495612
$872$	920.000i	0.0357284i
$873$	0	0
$874$	12138.0	0.469764
$875$	0	0
$876$	0	0
$877$	− 16292.0i	− 0.627300i	−0.949539	−	0.313650i	$-0.898448\pi$
$877$	− 16292.0i	− 0.627300i	0.949539	−	0.313650i	$-0.101552\pi$
$878$	18742.0i	0.720401i
$879$	0	0
$880$	0	0
$881$	9270.00	0.354500	0.177250	−	0.984166i	$-0.443280\pi$
$881$	9270.00	0.354500	0.177250	+	0.984166i	$0.443280\pi$
$882$	0	0
$883$	38486.0i	1.46677i	0.679814	+	0.733384i	$0.262060\pi$
$883$	38486.0i	1.46677i	−0.679814	+	0.733384i	$0.737940\pi$
$884$	−7560.00	−0.287636
$885$	0	0
$886$	12342.0	0.467988
$887$	− 1893.00i	− 0.0716581i	−0.999358	−	0.0358290i	$-0.988593\pi$
$887$	− 1893.00i	− 0.0716581i	0.999358	−	0.0358290i	$-0.0114072\pi$
$888$	0	0
$889$	47804.0	1.80348
$890$	0	0
$891$	0	0
$892$	13672.0i	0.513198i
$893$	− 61404.0i	− 2.30102i
$894$	0	0
$895$	0	0
$896$	−4352.00	−0.162266
$897$	0	0
$898$	8244.00i	0.306354i
$899$	−3990.00	−0.148024
$900$	0	0
$901$	17253.0	0.637936
$902$	14976.0i	0.552823i
$903$	0	0
$904$	−7728.00	−0.284325
$905$	0	0
$906$	0	0
$907$	− 44876.0i	− 1.64287i	−0.570302	−	0.821435i	$-0.693174\pi$
$907$	− 44876.0i	− 1.64287i	0.570302	−	0.821435i	$-0.306826\pi$
$908$	− 17508.0i	− 0.639894i
$909$	0	0
$910$	0	0
$911$	23802.0	0.865637	0.432819	−	0.901481i	$-0.357519\pi$
$911$	23802.0	0.865637	0.432819	+	0.901481i	$0.357519\pi$
$912$	0	0
$913$	− 43920.0i	− 1.59205i
$914$	−14152.0	−0.512152
$915$	0	0
$916$	16748.0	0.604115
$917$	8364.00i	0.301204i
$918$	0	0
$919$	24784.0	0.889607	0.444803	−	0.895628i	$-0.353274\pi$
$919$	24784.0	0.889607	0.444803	+	0.895628i	$0.353274\pi$
$920$	0	0
$921$	0	0
$922$	− 1524.00i	− 0.0544363i
$923$	− 63000.0i	− 2.24666i
$924$	0	0
$925$	0	0
$926$	17644.0	0.626153
$927$	0	0
$928$	− 960.000i	− 0.0339586i
$929$	−9060.00	−0.319967	−0.159983	−	0.987120i	$-0.551144\pi$
$929$	−9060.00	−0.319967	−0.159983	+	0.987120i	$0.551144\pi$
$930$	0	0
$931$	96747.0	3.40575
$932$	4392.00i	0.154361i
$933$	0	0
$934$	9954.00	0.348720
$935$	0	0
$936$	0	0
$937$	− 6176.00i	− 0.215327i	−0.994187	−	0.107663i	$-0.965663\pi$
$937$	− 6176.00i	− 0.215327i	0.994187	−	0.107663i	$-0.0343369\pi$
$938$	− 12376.0i	− 0.430800i
$939$	0	0
$940$	0	0
$941$	−4182.00	−0.144877	−0.0724385	−	0.997373i	$-0.523078\pi$
$941$	−4182.00	−0.144877	−0.0724385	+	0.997373i	$0.523078\pi$
$942$	0	0
$943$	− 7956.00i	− 0.274743i
$944$	10464.0	0.360778
$945$	0	0
$946$	8448.00	0.290347
$947$	− 44079.0i	− 1.51254i	−0.654260	−	0.756270i	$-0.727020\pi$
$947$	− 44079.0i	− 1.51254i	0.654260	−	0.756270i	$-0.272980\pi$
$948$	0	0
$949$	49280.0	1.68567
$950$	0	0
$951$	0	0
$952$	− 7344.00i	− 0.250021i
$953$	12726.0i	0.432566i	0.976331	+	0.216283i	$0.0693934\pi$
$953$	12726.0i	0.432566i	−0.976331	+	0.216283i	$0.930607\pi$
$954$	0	0
$955$	0	0
$956$	25896.0	0.876084
$957$	0	0
$958$	20208.0i	0.681514i
$959$	17646.0	0.594180
$960$	0	0
$961$	−12102.0	−0.406230
$962$	30520.0i	1.02287i
$963$	0	0
$964$	−13004.0	−0.434472
$965$	0	0
$966$	0	0
$967$	− 45218.0i	− 1.50374i	−0.659314	−	0.751868i	$-0.729153\pi$
$967$	− 45218.0i	− 1.50374i	0.659314	−	0.751868i	$-0.270847\pi$
$968$	7784.00i	0.258458i
$969$	0	0
$970$	0	0
$971$	−3978.00	−0.131473	−0.0657364	−	0.997837i	$-0.520940\pi$
$971$	−3978.00	−0.131473	−0.0657364	+	0.997837i	$0.520940\pi$
$972$	0	0
$973$	− 44744.0i	− 1.47423i
$974$	−29848.0	−0.981922
$975$	0	0
$976$	7376.00	0.241906
$977$	− 23466.0i	− 0.768417i	−0.923246	−	0.384209i	$-0.874474\pi$
$977$	− 23466.0i	− 0.768417i	0.923246	−	0.384209i	$-0.125526\pi$
$978$	0	0
$979$	−53568.0	−1.74876
$980$	0	0
$981$	0	0
$982$	− 2292.00i	− 0.0744813i
$983$	47913.0i	1.55462i	0.629120	+	0.777308i	$0.283415\pi$
$983$	47913.0i	1.55462i	−0.629120	+	0.777308i	$0.716585\pi$
$984$	0	0
$985$	0	0
$986$	1620.00	0.0523238
$987$	0	0
$988$	− 33320.0i	− 1.07293i
$989$	−4488.00	−0.144297
$990$	0	0
$991$	31997.0	1.02565	0.512825	−	0.858493i	$-0.328599\pi$
$991$	31997.0	1.02565	0.512825	+	0.858493i	$0.328599\pi$
$992$	4256.00i	0.136218i
$993$	0	0
$994$	61200.0	1.95286
$995$	0	0
$996$	0	0
$997$	45628.0i	1.44940i	0.689064	+	0.724701i	$0.258022\pi$
$997$	45628.0i	1.44940i	−0.689064	+	0.724701i	$0.741978\pi$
$998$	− 29930.0i	− 0.949316i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.c.c.649.1		2
3.2	odd	2		1350.4.c.r.649.2		2
5.2	odd	4		270.4.a.k.1.1	yes	1
5.3	odd	4		1350.4.a.n.1.1		1
5.4	even	2	inner	1350.4.c.c.649.2		2
15.2	even	4		270.4.a.a.1.1	✓	1
15.8	even	4		1350.4.a.bb.1.1		1
15.14	odd	2		1350.4.c.r.649.1		2
20.7	even	4		2160.4.a.t.1.1		1
45.2	even	12		810.4.e.x.271.1		2
45.7	odd	12		810.4.e.d.271.1		2
45.22	odd	12		810.4.e.d.541.1		2
45.32	even	12		810.4.e.x.541.1		2
60.47	odd	4		2160.4.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.a.1.1	✓	1	15.2	even	4
270.4.a.k.1.1	yes	1	5.2	odd	4
810.4.e.d.271.1		2	45.7	odd	12
810.4.e.d.541.1		2	45.22	odd	12
810.4.e.x.271.1		2	45.2	even	12
810.4.e.x.541.1		2	45.32	even	12
1350.4.a.n.1.1		1	5.3	odd	4
1350.4.a.bb.1.1		1	15.8	even	4
1350.4.c.c.649.1		2	1.1	even	1	trivial
1350.4.c.c.649.2		2	5.4	even	2	inner
1350.4.c.r.649.1		2	15.14	odd	2
1350.4.c.r.649.2		2	3.2	odd	2
2160.4.a.j.1.1		1	60.47	odd	4
2160.4.a.t.1.1		1	20.7	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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.00000 q^{4} +34.0000i q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q-2.00000i q^{2} -4.00000 q^{4} +34.0000i q^{7} +8.00000i q^{8} -48.0000 q^{11} -70.0000i q^{13} +68.0000 q^{14} +16.0000 q^{16} +27.0000i q^{17} -119.000 q^{19} +96.0000i q^{22} -51.0000i q^{23} -140.000 q^{26} -136.000i q^{28} +30.0000 q^{29} -133.000 q^{31} -32.0000i q^{32} +54.0000 q^{34} -218.000i q^{37} +238.000i q^{38} +156.000 q^{41} -88.0000i q^{43} +192.000 q^{44} -102.000 q^{46} +516.000i q^{47} -813.000 q^{49} +280.000i q^{52} -639.000i q^{53} -272.000 q^{56} -60.0000i q^{58} +654.000 q^{59} +461.000 q^{61} +266.000i q^{62} -64.0000 q^{64} -182.000i q^{67} -108.000i q^{68} +900.000 q^{71} +704.000i q^{73} -436.000 q^{74} +476.000 q^{76} -1632.00i q^{77} +1375.00 q^{79} -312.000i q^{82} +915.000i q^{83} -176.000 q^{86} -384.000i q^{88} +1116.00 q^{89} +2380.00 q^{91} +204.000i q^{92} +1032.00 q^{94} +16.0000i q^{97} +1626.00i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} - 96 q^{11} + 136 q^{14} + 32 q^{16} - 238 q^{19} - 280 q^{26} + 60 q^{29} - 266 q^{31} + 108 q^{34} + 312 q^{41} + 384 q^{44} - 204 q^{46} - 1626 q^{49} - 544 q^{56} + 1308 q^{59} + 922 q^{61} - 128 q^{64} + 1800 q^{71} - 872 q^{74} + 952 q^{76} + 2750 q^{79} - 352 q^{86} + 2232 q^{89} + 4760 q^{91} + 2064 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 - 96 * q^11 + 136 * q^14 + 32 * q^16 - 238 * q^19 - 280 * q^26 + 60 * q^29 - 266 * q^31 + 108 * q^34 + 312 * q^41 + 384 * q^44 - 204 * q^46 - 1626 * q^49 - 544 * q^56 + 1308 * q^59 + 922 * q^61 - 128 * q^64 + 1800 * q^71 - 872 * q^74 + 952 * q^76 + 2750 * q^79 - 352 * q^86 + 2232 * q^89 + 4760 * q^91 + 2064 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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