LMFDB - Embedded newform 1350.4.c.h.649.2

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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} +19.0000i q^{7} -8.00000i q^{8} +O(q^{10})$	$q+2.00000i q^{2} -4.00000 q^{4} +19.0000i q^{7} -8.00000i q^{8} -12.0000 q^{11} +50.0000i q^{13} -38.0000 q^{14} +16.0000 q^{16} -126.000i q^{17} -29.0000 q^{19} -24.0000i q^{22} -18.0000i q^{23} -100.000 q^{26} -76.0000i q^{28} +102.000 q^{29} -265.000 q^{31} +32.0000i q^{32} +252.000 q^{34} -65.0000i q^{37} -58.0000i q^{38} -240.000 q^{41} -367.000i q^{43} +48.0000 q^{44} +36.0000 q^{46} -72.0000i q^{47} -18.0000 q^{49} -200.000i q^{52} +636.000i q^{53} +152.000 q^{56} +204.000i q^{58} +102.000 q^{59} -103.000 q^{61} -530.000i q^{62} -64.0000 q^{64} +52.0000i q^{67} +504.000i q^{68} +582.000 q^{71} +65.0000i q^{73} +130.000 q^{74} +116.000 q^{76} -228.000i q^{77} -173.000 q^{79} -480.000i q^{82} -498.000i q^{83} +734.000 q^{86} +96.0000i q^{88} -822.000 q^{89} -950.000 q^{91} +72.0000i q^{92} +144.000 q^{94} -821.000i q^{97} -36.0000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} - 24 q^{11} - 76 q^{14} + 32 q^{16} - 58 q^{19} - 200 q^{26} + 204 q^{29} - 530 q^{31} + 504 q^{34} - 480 q^{41} + 96 q^{44} + 72 q^{46} - 36 q^{49} + 304 q^{56} + 204 q^{59} - 206 q^{61} - 128 q^{64} + 1164 q^{71} + 260 q^{74} + 232 q^{76} - 346 q^{79} + 1468 q^{86} - 1644 q^{89} - 1900 q^{91} + 288 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 - 24 * q^11 - 76 * q^14 + 32 * q^16 - 58 * q^19 - 200 * q^26 + 204 * q^29 - 530 * q^31 + 504 * q^34 - 480 * q^41 + 96 * q^44 + 72 * q^46 - 36 * q^49 + 304 * q^56 + 204 * q^59 - 206 * q^61 - 128 * q^64 + 1164 * q^71 + 260 * q^74 + 232 * q^76 - 346 * q^79 + 1468 * q^86 - 1644 * q^89 - 1900 * q^91 + 288 * 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
$7$	19.0000i	1.02590i	0.858417	+	0.512952i	$0.171448\pi$
$7$	19.0000i	1.02590i	−0.858417	+	0.512952i	$0.828552\pi$
$8$	− 8.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	50.0000i	1.06673i	0.845885	+	0.533366i	$0.179073\pi$
$13$	50.0000i	1.06673i	−0.845885	+	0.533366i	$0.820927\pi$
$14$	−38.0000	−0.725423
$15$	0	0
$16$	16.0000	0.250000
$17$	− 126.000i	− 1.79762i	−0.438342	−	0.898808i	$-0.644434\pi$
$17$	− 126.000i	− 1.79762i	0.438342	−	0.898808i	$-0.355566\pi$
$18$	0	0
$19$	−29.0000	−0.350161	−0.175080	−	0.984554i	$-0.556019\pi$
$19$	−29.0000	−0.350161	−0.175080	+	0.984554i	$0.556019\pi$
$20$	0	0
$21$	0	0
$22$	− 24.0000i	− 0.232583i
$23$	− 18.0000i	− 0.163185i	−0.996666	−	0.0815926i	$-0.973999\pi$
$23$	− 18.0000i	− 0.163185i	0.996666	−	0.0815926i	$-0.0260006\pi$
$24$	0	0
$25$	0	0
$26$	−100.000	−0.754293
$27$	0	0
$28$	− 76.0000i	− 0.512952i
$29$	102.000	0.653135	0.326568	−	0.945174i	$-0.394108\pi$
$29$	102.000	0.653135	0.326568	+	0.945174i	$0.394108\pi$
$30$	0	0
$31$	−265.000	−1.53534	−0.767668	−	0.640848i	$-0.778583\pi$
$31$	−265.000	−1.53534	−0.767668	+	0.640848i	$0.778583\pi$
$32$	32.0000i	0.176777i
$33$	0	0
$34$	252.000	1.27111
$35$	0	0
$36$	0	0
$37$	− 65.0000i	− 0.288809i	−0.989519	−	0.144405i	$-0.953873\pi$
$37$	− 65.0000i	− 0.288809i	0.989519	−	0.144405i	$-0.0461267\pi$
$38$	− 58.0000i	− 0.247601i
$39$	0	0
$40$	0	0
$41$	−240.000	−0.914188	−0.457094	−	0.889418i	$-0.651110\pi$
$41$	−240.000	−0.914188	−0.457094	+	0.889418i	$0.651110\pi$
$42$	0	0
$43$	− 367.000i	− 1.30156i	−0.759267	−	0.650779i	$-0.774443\pi$
$43$	− 367.000i	− 1.30156i	0.759267	−	0.650779i	$-0.225557\pi$
$44$	48.0000	0.164461
$45$	0	0
$46$	36.0000	0.115389
$47$	− 72.0000i	− 0.223453i	−0.993739	−	0.111726i	$-0.964362\pi$
$47$	− 72.0000i	− 0.223453i	0.993739	−	0.111726i	$-0.0356380\pi$
$48$	0	0
$49$	−18.0000	−0.0524781
$50$	0	0
$51$	0	0
$52$	− 200.000i	− 0.533366i
$53$	636.000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	636.000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	0	0
$56$	152.000	0.362712
$57$	0	0
$58$	204.000i	0.461836i
$59$	102.000	0.225072	0.112536	−	0.993648i	$-0.464103\pi$
$59$	102.000	0.225072	0.112536	+	0.993648i	$0.464103\pi$
$60$	0	0
$61$	−103.000	−0.216193	−0.108097	−	0.994140i	$-0.534476\pi$
$61$	−103.000	−0.216193	−0.108097	+	0.994140i	$0.534476\pi$
$62$	− 530.000i	− 1.08565i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	52.0000i	0.0948181i	0.998876	+	0.0474090i	$0.0150964\pi$
$67$	52.0000i	0.0948181i	−0.998876	+	0.0474090i	$0.984904\pi$
$68$	504.000i	0.898808i
$69$	0	0
$70$	0	0
$71$	582.000	0.972827	0.486413	−	0.873729i	$-0.338305\pi$
$71$	582.000	0.972827	0.486413	+	0.873729i	$0.338305\pi$
$72$	0	0
$73$	65.0000i	0.104215i	0.998641	+	0.0521074i	$0.0165938\pi$
$73$	65.0000i	0.104215i	−0.998641	+	0.0521074i	$0.983406\pi$
$74$	130.000	0.204219
$75$	0	0
$76$	116.000	0.175080
$77$	− 228.000i	− 0.337442i
$78$	0	0
$79$	−173.000	−0.246380	−0.123190	−	0.992383i	$-0.539312\pi$
$79$	−173.000	−0.246380	−0.123190	+	0.992383i	$0.539312\pi$
$80$	0	0
$81$	0	0
$82$	− 480.000i	− 0.646428i
$83$	− 498.000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 498.000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	734.000	0.920340
$87$	0	0
$88$	96.0000i	0.116291i
$89$	−822.000	−0.979009	−0.489505	−	0.872001i	$-0.662822\pi$
$89$	−822.000	−0.979009	−0.489505	+	0.872001i	$0.662822\pi$
$90$	0	0
$91$	−950.000	−1.09436
$92$	72.0000i	0.0815926i
$93$	0	0
$94$	144.000	0.158005
$95$	0	0
$96$	0	0
$97$	− 821.000i	− 0.859381i	−0.902976	−	0.429690i	$-0.858623\pi$
$97$	− 821.000i	− 0.859381i	0.902976	−	0.429690i	$-0.141377\pi$
$98$	− 36.0000i	− 0.0371076i
$99$	0	0
$100$	0	0
$101$	1200.00	1.18222	0.591111	−	0.806590i	$-0.298689\pi$
$101$	1200.00	1.18222	0.591111	+	0.806590i	$0.298689\pi$
$102$	0	0
$103$	− 2041.00i	− 1.95248i	−0.216686	−	0.976241i	$-0.569525\pi$
$103$	− 2041.00i	− 1.95248i	0.216686	−	0.976241i	$-0.430475\pi$
$104$	400.000	0.377146
$105$	0	0
$106$	−1272.00	−1.16554
$107$	− 1278.00i	− 1.15466i	−0.816510	−	0.577331i	$-0.804094\pi$
$107$	− 1278.00i	− 1.15466i	0.816510	−	0.577331i	$-0.195906\pi$
$108$	0	0
$109$	205.000	0.180142	0.0900708	−	0.995935i	$-0.471291\pi$
$109$	205.000	0.180142	0.0900708	+	0.995935i	$0.471291\pi$
$110$	0	0
$111$	0	0
$112$	304.000i	0.256476i
$113$	1500.00i	1.24874i	0.781127	+	0.624372i	$0.214645\pi$
$113$	1500.00i	1.24874i	−0.781127	+	0.624372i	$0.785355\pi$
$114$	0	0
$115$	0	0
$116$	−408.000	−0.326568
$117$	0	0
$118$	204.000i	0.159150i
$119$	2394.00	1.84418
$120$	0	0
$121$	−1187.00	−0.891811
$122$	− 206.000i	− 0.152872i
$123$	0	0
$124$	1060.00	0.767668
$125$	0	0
$126$	0	0
$127$	− 416.000i	− 0.290662i	−0.989383	−	0.145331i	$-0.953575\pi$
$127$	− 416.000i	− 0.290662i	0.989383	−	0.145331i	$-0.0464247\pi$
$128$	− 128.000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	1902.00	1.26854	0.634269	−	0.773112i	$-0.281301\pi$
$131$	1902.00	1.26854	0.634269	+	0.773112i	$0.281301\pi$
$132$	0	0
$133$	− 551.000i	− 0.359231i
$134$	−104.000	−0.0670465
$135$	0	0
$136$	−1008.00	−0.635554
$137$	− 2712.00i	− 1.69125i	−0.533774	−	0.845627i	$-0.679227\pi$
$137$	− 2712.00i	− 1.69125i	0.533774	−	0.845627i	$-0.320773\pi$
$138$	0	0
$139$	367.000	0.223946	0.111973	−	0.993711i	$-0.464283\pi$
$139$	367.000	0.223946	0.111973	+	0.993711i	$0.464283\pi$
$140$	0	0
$141$	0	0
$142$	1164.00i	0.687892i
$143$	− 600.000i	− 0.350871i
$144$	0	0
$145$	0	0
$146$	−130.000	−0.0736909
$147$	0	0
$148$	260.000i	0.144405i
$149$	3006.00	1.65276	0.826380	−	0.563113i	$-0.190397\pi$
$149$	3006.00	1.65276	0.826380	+	0.563113i	$0.190397\pi$
$150$	0	0
$151$	2651.00	1.42871	0.714355	−	0.699783i	$-0.246720\pi$
$151$	2651.00	1.42871	0.714355	+	0.699783i	$0.246720\pi$
$152$	232.000i	0.123801i
$153$	0	0
$154$	456.000	0.238607
$155$	0	0
$156$	0	0
$157$	− 2801.00i	− 1.42385i	−0.702257	−	0.711924i	$-0.747824\pi$
$157$	− 2801.00i	− 1.42385i	0.702257	−	0.711924i	$-0.252176\pi$
$158$	− 346.000i	− 0.174217i
$159$	0	0
$160$	0	0
$161$	342.000	0.167412
$162$	0	0
$163$	1412.00i	0.678505i	0.940695	+	0.339253i	$0.110174\pi$
$163$	1412.00i	0.678505i	−0.940695	+	0.339253i	$0.889826\pi$
$164$	960.000	0.457094
$165$	0	0
$166$	996.000	0.465690
$167$	− 2154.00i	− 0.998093i	−0.866575	−	0.499046i	$-0.833684\pi$
$167$	− 2154.00i	− 0.998093i	0.866575	−	0.499046i	$-0.166316\pi$
$168$	0	0
$169$	−303.000	−0.137915
$170$	0	0
$171$	0	0
$172$	1468.00i	0.650779i
$173$	− 234.000i	− 0.102836i	−0.998677	−	0.0514182i	$-0.983626\pi$
$173$	− 234.000i	− 0.102836i	0.998677	−	0.0514182i	$-0.0163741\pi$
$174$	0	0
$175$	0	0
$176$	−192.000	−0.0822304
$177$	0	0
$178$	− 1644.00i	− 0.692264i
$179$	2472.00	1.03221	0.516106	−	0.856525i	$-0.327381\pi$
$179$	2472.00	1.03221	0.516106	+	0.856525i	$0.327381\pi$
$180$	0	0
$181$	−190.000	−0.0780254	−0.0390127	−	0.999239i	$-0.512421\pi$
$181$	−190.000	−0.0780254	−0.0390127	+	0.999239i	$0.512421\pi$
$182$	− 1900.00i	− 0.773832i
$183$	0	0
$184$	−144.000	−0.0576947
$185$	0	0
$186$	0	0
$187$	1512.00i	0.591275i
$188$	288.000i	0.111726i
$189$	0	0
$190$	0	0
$191$	−1524.00	−0.577344	−0.288672	−	0.957428i	$-0.593214\pi$
$191$	−1524.00	−0.577344	−0.288672	+	0.957428i	$0.593214\pi$
$192$	0	0
$193$	3335.00i	1.24383i	0.783086	+	0.621913i	$0.213644\pi$
$193$	3335.00i	1.24383i	−0.783086	+	0.621913i	$0.786356\pi$
$194$	1642.00	0.607674
$195$	0	0
$196$	72.0000	0.0262391
$197$	− 4440.00i	− 1.60577i	−0.596133	−	0.802886i	$-0.703297\pi$
$197$	− 4440.00i	− 1.60577i	0.596133	−	0.802886i	$-0.296703\pi$
$198$	0	0
$199$	−2192.00	−0.780838	−0.390419	−	0.920637i	$-0.627670\pi$
$199$	−2192.00	−0.780838	−0.390419	+	0.920637i	$0.627670\pi$
$200$	0	0
$201$	0	0
$202$	2400.00i	0.835957i
$203$	1938.00i	0.670054i
$204$	0	0
$205$	0	0
$206$	4082.00	1.38061
$207$	0	0
$208$	800.000i	0.266683i
$209$	348.000	0.115175
$210$	0	0
$211$	−1732.00	−0.565099	−0.282549	−	0.959253i	$-0.591180\pi$
$211$	−1732.00	−0.565099	−0.282549	+	0.959253i	$0.591180\pi$
$212$	− 2544.00i	− 0.824163i
$213$	0	0
$214$	2556.00	0.816470
$215$	0	0
$216$	0	0
$217$	− 5035.00i	− 1.57511i
$218$	410.000i	0.127379i
$219$	0	0
$220$	0	0
$221$	6300.00	1.91757
$222$	0	0
$223$	− 5989.00i	− 1.79844i	−0.437492	−	0.899222i	$-0.644133\pi$
$223$	− 5989.00i	− 1.79844i	0.437492	−	0.899222i	$-0.355867\pi$
$224$	−608.000	−0.181356
$225$	0	0
$226$	−3000.00	−0.882996
$227$	− 3300.00i	− 0.964884i	−0.875928	−	0.482442i	$-0.839750\pi$
$227$	− 3300.00i	− 0.964884i	0.875928	−	0.482442i	$-0.160250\pi$
$228$	0	0
$229$	−4379.00	−1.26364	−0.631818	−	0.775117i	$-0.717691\pi$
$229$	−4379.00	−1.26364	−0.631818	+	0.775117i	$0.717691\pi$
$230$	0	0
$231$	0	0
$232$	− 816.000i	− 0.230918i
$233$	− 2958.00i	− 0.831695i	−0.909434	−	0.415848i	$-0.863485\pi$
$233$	− 2958.00i	− 0.831695i	0.909434	−	0.415848i	$-0.136515\pi$
$234$	0	0
$235$	0	0
$236$	−408.000	−0.112536
$237$	0	0
$238$	4788.00i	1.30403i
$239$	−2676.00	−0.724251	−0.362126	−	0.932129i	$-0.617949\pi$
$239$	−2676.00	−0.724251	−0.362126	+	0.932129i	$0.617949\pi$
$240$	0	0
$241$	3626.00	0.969175	0.484588	−	0.874743i	$-0.338970\pi$
$241$	3626.00	0.969175	0.484588	+	0.874743i	$0.338970\pi$
$242$	− 2374.00i	− 0.630605i
$243$	0	0
$244$	412.000	0.108097
$245$	0	0
$246$	0	0
$247$	− 1450.00i	− 0.373527i
$248$	2120.00i	0.542823i
$249$	0	0
$250$	0	0
$251$	2046.00	0.514511	0.257256	−	0.966343i	$-0.417182\pi$
$251$	2046.00	0.514511	0.257256	+	0.966343i	$0.417182\pi$
$252$	0	0
$253$	216.000i	0.0536751i
$254$	832.000	0.205529
$255$	0	0
$256$	256.000	0.0625000
$257$	5700.00i	1.38349i	0.722143	+	0.691744i	$0.243157\pi$
$257$	5700.00i	1.38349i	−0.722143	+	0.691744i	$0.756843\pi$
$258$	0	0
$259$	1235.00	0.296290
$260$	0	0
$261$	0	0
$262$	3804.00i	0.896992i
$263$	7674.00i	1.79924i	0.436678	+	0.899618i	$0.356155\pi$
$263$	7674.00i	1.79924i	−0.436678	+	0.899618i	$0.643845\pi$
$264$	0	0
$265$	0	0
$266$	1102.00	0.254015
$267$	0	0
$268$	− 208.000i	− 0.0474090i
$269$	4326.00	0.980524	0.490262	−	0.871575i	$-0.336901\pi$
$269$	4326.00	0.980524	0.490262	+	0.871575i	$0.336901\pi$
$270$	0	0
$271$	6473.00	1.45095	0.725474	−	0.688250i	$-0.241621\pi$
$271$	6473.00	1.45095	0.725474	+	0.688250i	$0.241621\pi$
$272$	− 2016.00i	− 0.449404i
$273$	0	0
$274$	5424.00	1.19590
$275$	0	0
$276$	0	0
$277$	7825.00i	1.69732i	0.528936	+	0.848662i	$0.322591\pi$
$277$	7825.00i	1.69732i	−0.528936	+	0.848662i	$0.677409\pi$
$278$	734.000i	0.158354i
$279$	0	0
$280$	0	0
$281$	−2760.00	−0.585935	−0.292968	−	0.956122i	$-0.594643\pi$
$281$	−2760.00	−0.585935	−0.292968	+	0.956122i	$0.594643\pi$
$282$	0	0
$283$	− 4801.00i	− 1.00844i	−0.863574	−	0.504222i	$-0.831779\pi$
$283$	− 4801.00i	− 1.00844i	0.863574	−	0.504222i	$-0.168221\pi$
$284$	−2328.00	−0.486413
$285$	0	0
$286$	1200.00	0.248103
$287$	− 4560.00i	− 0.937869i
$288$	0	0
$289$	−10963.0	−2.23143
$290$	0	0
$291$	0	0
$292$	− 260.000i	− 0.0521074i
$293$	8166.00i	1.62820i	0.580724	+	0.814100i	$0.302769\pi$
$293$	8166.00i	1.62820i	−0.580724	+	0.814100i	$0.697231\pi$
$294$	0	0
$295$	0	0
$296$	−520.000	−0.102109
$297$	0	0
$298$	6012.00i	1.16868i
$299$	900.000	0.174075
$300$	0	0
$301$	6973.00	1.33527
$302$	5302.00i	1.01025i
$303$	0	0
$304$	−464.000	−0.0875402
$305$	0	0
$306$	0	0
$307$	577.000i	0.107268i	0.998561	+	0.0536338i	$0.0170804\pi$
$307$	577.000i	0.107268i	−0.998561	+	0.0536338i	$0.982920\pi$
$308$	912.000i	0.168721i
$309$	0	0
$310$	0	0
$311$	7656.00	1.39592	0.697961	−	0.716135i	$-0.254091\pi$
$311$	7656.00	1.39592	0.697961	+	0.716135i	$0.254091\pi$
$312$	0	0
$313$	2726.00i	0.492277i	0.969235	+	0.246138i	$0.0791618\pi$
$313$	2726.00i	0.492277i	−0.969235	+	0.246138i	$0.920838\pi$
$314$	5602.00	1.00681
$315$	0	0
$316$	692.000	0.123190
$317$	− 9450.00i	− 1.67434i	−0.546945	−	0.837169i	$-0.684209\pi$
$317$	− 9450.00i	− 1.67434i	0.546945	−	0.837169i	$-0.315791\pi$
$318$	0	0
$319$	−1224.00	−0.214830
$320$	0	0
$321$	0	0
$322$	684.000i	0.118378i
$323$	3654.00i	0.629455i
$324$	0	0
$325$	0	0
$326$	−2824.00	−0.479776
$327$	0	0
$328$	1920.00i	0.323214i
$329$	1368.00	0.229241
$330$	0	0
$331$	3089.00	0.512951	0.256476	−	0.966551i	$-0.417439\pi$
$331$	3089.00	0.512951	0.256476	+	0.966551i	$0.417439\pi$
$332$	1992.00i	0.329293i
$333$	0	0
$334$	4308.00	0.705758
$335$	0	0
$336$	0	0
$337$	7894.00i	1.27600i	0.770034	+	0.638002i	$0.220239\pi$
$337$	7894.00i	1.27600i	−0.770034	+	0.638002i	$0.779761\pi$
$338$	− 606.000i	− 0.0975209i
$339$	0	0
$340$	0	0
$341$	3180.00	0.505005
$342$	0	0
$343$	6175.00i	0.972066i
$344$	−2936.00	−0.460170
$345$	0	0
$346$	468.000	0.0727163
$347$	− 3450.00i	− 0.533734i	−0.963733	−	0.266867i	$-0.914012\pi$
$347$	− 3450.00i	− 0.533734i	0.963733	−	0.266867i	$-0.0859884\pi$
$348$	0	0
$349$	−2261.00	−0.346787	−0.173393	−	0.984853i	$-0.555473\pi$
$349$	−2261.00	−0.346787	−0.173393	+	0.984853i	$0.555473\pi$
$350$	0	0
$351$	0	0
$352$	− 384.000i	− 0.0581456i
$353$	− 11544.0i	− 1.74058i	−0.492539	−	0.870291i	$-0.663931\pi$
$353$	− 11544.0i	− 1.74058i	0.492539	−	0.870291i	$-0.336069\pi$
$354$	0	0
$355$	0	0
$356$	3288.00	0.489505
$357$	0	0
$358$	4944.00i	0.729884i
$359$	−198.000	−0.0291087	−0.0145544	−	0.999894i	$-0.504633\pi$
$359$	−198.000	−0.0291087	−0.0145544	+	0.999894i	$0.504633\pi$
$360$	0	0
$361$	−6018.00	−0.877387
$362$	− 380.000i	− 0.0551723i
$363$	0	0
$364$	3800.00	0.547182
$365$	0	0
$366$	0	0
$367$	− 10388.0i	− 1.47752i	−0.673970	−	0.738759i	$-0.735412\pi$
$367$	− 10388.0i	− 1.47752i	0.673970	−	0.738759i	$-0.264588\pi$
$368$	− 288.000i	− 0.0407963i
$369$	0	0
$370$	0	0
$371$	−12084.0	−1.69102
$372$	0	0
$373$	− 4867.00i	− 0.675613i	−0.941216	−	0.337807i	$-0.890315\pi$
$373$	− 4867.00i	− 0.675613i	0.941216	−	0.337807i	$-0.109685\pi$
$374$	−3024.00	−0.418094
$375$	0	0
$376$	−576.000	−0.0790025
$377$	5100.00i	0.696720i
$378$	0	0
$379$	−656.000	−0.0889089	−0.0444544	−	0.999011i	$-0.514155\pi$
$379$	−656.000	−0.0889089	−0.0444544	+	0.999011i	$0.514155\pi$
$380$	0	0
$381$	0	0
$382$	− 3048.00i	− 0.408244i
$383$	− 3078.00i	− 0.410649i	−0.978694	−	0.205324i	$-0.934175\pi$
$383$	− 3078.00i	− 0.410649i	0.978694	−	0.205324i	$-0.0658249\pi$
$384$	0	0
$385$	0	0
$386$	−6670.00	−0.879518
$387$	0	0
$388$	3284.00i	0.429690i
$389$	1644.00	0.214278	0.107139	−	0.994244i	$-0.465831\pi$
$389$	1644.00	0.214278	0.107139	+	0.994244i	$0.465831\pi$
$390$	0	0
$391$	−2268.00	−0.293344
$392$	144.000i	0.0185538i
$393$	0	0
$394$	8880.00	1.13545
$395$	0	0
$396$	0	0
$397$	− 5729.00i	− 0.724258i	−0.932128	−	0.362129i	$-0.882050\pi$
$397$	− 5729.00i	− 0.724258i	0.932128	−	0.362129i	$-0.117950\pi$
$398$	− 4384.00i	− 0.552136i
$399$	0	0
$400$	0	0
$401$	1776.00	0.221170	0.110585	−	0.993867i	$-0.464728\pi$
$401$	1776.00	0.221170	0.110585	+	0.993867i	$0.464728\pi$
$402$	0	0
$403$	− 13250.0i	− 1.63779i
$404$	−4800.00	−0.591111
$405$	0	0
$406$	−3876.00	−0.473800
$407$	780.000i	0.0949955i
$408$	0	0
$409$	−12842.0	−1.55256	−0.776279	−	0.630390i	$-0.782895\pi$
$409$	−12842.0	−1.55256	−0.776279	+	0.630390i	$0.782895\pi$
$410$	0	0
$411$	0	0
$412$	8164.00i	0.976241i
$413$	1938.00i	0.230903i
$414$	0	0
$415$	0	0
$416$	−1600.00	−0.188573
$417$	0	0
$418$	696.000i	0.0814413i
$419$	−11298.0	−1.31729	−0.658644	−	0.752455i	$-0.728870\pi$
$419$	−11298.0	−1.31729	−0.658644	+	0.752455i	$0.728870\pi$
$420$	0	0
$421$	3431.00	0.397189	0.198595	−	0.980082i	$-0.436362\pi$
$421$	3431.00	0.397189	0.198595	+	0.980082i	$0.436362\pi$
$422$	− 3464.00i	− 0.399585i
$423$	0	0
$424$	5088.00	0.582772
$425$	0	0
$426$	0	0
$427$	− 1957.00i	− 0.221794i
$428$	5112.00i	0.577331i
$429$	0	0
$430$	0	0
$431$	−10530.0	−1.17683	−0.588413	−	0.808560i	$-0.700247\pi$
$431$	−10530.0	−1.17683	−0.588413	+	0.808560i	$0.700247\pi$
$432$	0	0
$433$	3179.00i	0.352824i	0.984316	+	0.176412i	$0.0564492\pi$
$433$	3179.00i	0.352824i	−0.984316	+	0.176412i	$0.943551\pi$
$434$	10070.0	1.11377
$435$	0	0
$436$	−820.000	−0.0900708
$437$	522.000i	0.0571411i
$438$	0	0
$439$	−6047.00	−0.657420	−0.328710	−	0.944431i	$-0.606614\pi$
$439$	−6047.00	−0.657420	−0.328710	+	0.944431i	$0.606614\pi$
$440$	0	0
$441$	0	0
$442$	12600.0i	1.35593i
$443$	− 16554.0i	− 1.77540i	−0.460418	−	0.887702i	$-0.652300\pi$
$443$	− 16554.0i	− 1.77540i	0.460418	−	0.887702i	$-0.347700\pi$
$444$	0	0
$445$	0	0
$446$	11978.0	1.27169
$447$	0	0
$448$	− 1216.00i	− 0.128238i
$449$	−4446.00	−0.467304	−0.233652	−	0.972320i	$-0.575068\pi$
$449$	−4446.00	−0.467304	−0.233652	+	0.972320i	$0.575068\pi$
$450$	0	0
$451$	2880.00	0.300696
$452$	− 6000.00i	− 0.624372i
$453$	0	0
$454$	6600.00	0.682276
$455$	0	0
$456$	0	0
$457$	− 146.000i	− 0.0149444i	−0.999972	−	0.00747220i	$-0.997622\pi$
$457$	− 146.000i	− 0.0149444i	0.999972	−	0.00747220i	$-0.00237850\pi$
$458$	− 8758.00i	− 0.893525i
$459$	0	0
$460$	0	0
$461$	−9822.00	−0.992313	−0.496156	−	0.868233i	$-0.665256\pi$
$461$	−9822.00	−0.992313	−0.496156	+	0.868233i	$0.665256\pi$
$462$	0	0
$463$	7931.00i	0.796080i	0.917368	+	0.398040i	$0.130309\pi$
$463$	7931.00i	0.796080i	−0.917368	+	0.398040i	$0.869691\pi$
$464$	1632.00	0.163284
$465$	0	0
$466$	5916.00	0.588097
$467$	18714.0i	1.85435i	0.374631	+	0.927174i	$0.377770\pi$
$467$	18714.0i	1.85435i	−0.374631	+	0.927174i	$0.622230\pi$
$468$	0	0
$469$	−988.000	−0.0972742
$470$	0	0
$471$	0	0
$472$	− 816.000i	− 0.0795751i
$473$	4404.00i	0.428110i
$474$	0	0
$475$	0	0
$476$	−9576.00	−0.922091
$477$	0	0
$478$	− 5352.00i	− 0.512123i
$479$	12786.0	1.21964	0.609820	−	0.792540i	$-0.291242\pi$
$479$	12786.0	1.21964	0.609820	+	0.792540i	$0.291242\pi$
$480$	0	0
$481$	3250.00	0.308082
$482$	7252.00i	0.685310i
$483$	0	0
$484$	4748.00	0.445905
$485$	0	0
$486$	0	0
$487$	− 10400.0i	− 0.967698i	−0.875151	−	0.483849i	$-0.839238\pi$
$487$	− 10400.0i	− 0.967698i	0.875151	−	0.483849i	$-0.160762\pi$
$488$	824.000i	0.0764359i
$489$	0	0
$490$	0	0
$491$	19548.0	1.79672	0.898359	−	0.439261i	$-0.144760\pi$
$491$	19548.0	1.79672	0.898359	+	0.439261i	$0.144760\pi$
$492$	0	0
$493$	− 12852.0i	− 1.17409i
$494$	2900.00	0.264124
$495$	0	0
$496$	−4240.00	−0.383834
$497$	11058.0i	0.998026i
$498$	0	0
$499$	−14939.0	−1.34020	−0.670102	−	0.742269i	$-0.733750\pi$
$499$	−14939.0	−1.34020	−0.670102	+	0.742269i	$0.733750\pi$
$500$	0	0
$501$	0	0
$502$	4092.00i	0.363815i
$503$	− 4098.00i	− 0.363262i	−0.983367	−	0.181631i	$-0.941862\pi$
$503$	− 4098.00i	− 0.363262i	0.983367	−	0.181631i	$-0.0581376\pi$
$504$	0	0
$505$	0	0
$506$	−432.000	−0.0379540
$507$	0	0
$508$	1664.00i	0.145331i
$509$	−12216.0	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−12216.0	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	0	0
$511$	−1235.00	−0.106914
$512$	512.000i	0.0441942i
$513$	0	0
$514$	−11400.0	−0.978273
$515$	0	0
$516$	0	0
$517$	864.000i	0.0734984i
$518$	2470.00i	0.209509i
$519$	0	0
$520$	0	0
$521$	−18492.0	−1.55499	−0.777494	−	0.628890i	$-0.783510\pi$
$521$	−18492.0	−1.55499	−0.777494	+	0.628890i	$0.783510\pi$
$522$	0	0
$523$	− 5467.00i	− 0.457085i	−0.973534	−	0.228542i	$-0.926604\pi$
$523$	− 5467.00i	− 0.457085i	0.973534	−	0.228542i	$-0.0733959\pi$
$524$	−7608.00	−0.634269
$525$	0	0
$526$	−15348.0	−1.27225
$527$	33390.0i	2.75995i
$528$	0	0
$529$	11843.0	0.973371
$530$	0	0
$531$	0	0
$532$	2204.00i	0.179616i
$533$	− 12000.0i	− 0.975193i
$534$	0	0
$535$	0	0
$536$	416.000	0.0335233
$537$	0	0
$538$	8652.00i	0.693335i
$539$	216.000	0.0172612
$540$	0	0
$541$	6983.00	0.554940	0.277470	−	0.960734i	$-0.410504\pi$
$541$	6983.00	0.554940	0.277470	+	0.960734i	$0.410504\pi$
$542$	12946.0i	1.02597i
$543$	0	0
$544$	4032.00	0.317777
$545$	0	0
$546$	0	0
$547$	− 9281.00i	− 0.725461i	−0.931894	−	0.362730i	$-0.881845\pi$
$547$	− 9281.00i	− 0.725461i	0.931894	−	0.362730i	$-0.118155\pi$
$548$	10848.0i	0.845627i
$549$	0	0
$550$	0	0
$551$	−2958.00	−0.228702
$552$	0	0
$553$	− 3287.00i	− 0.252762i
$554$	−15650.0	−1.20019
$555$	0	0
$556$	−1468.00	−0.111973
$557$	− 1986.00i	− 0.151076i	−0.997143	−	0.0755382i	$-0.975933\pi$
$557$	− 1986.00i	− 0.151076i	0.997143	−	0.0755382i	$-0.0240675\pi$
$558$	0	0
$559$	18350.0	1.38841
$560$	0	0
$561$	0	0
$562$	− 5520.00i	− 0.414319i
$563$	12672.0i	0.948599i	0.880364	+	0.474299i	$0.157299\pi$
$563$	12672.0i	0.948599i	−0.880364	+	0.474299i	$0.842701\pi$
$564$	0	0
$565$	0	0
$566$	9602.00	0.713078
$567$	0	0
$568$	− 4656.00i	− 0.343946i
$569$	−8370.00	−0.616676	−0.308338	−	0.951277i	$-0.599773\pi$
$569$	−8370.00	−0.616676	−0.308338	+	0.951277i	$0.599773\pi$
$570$	0	0
$571$	−19675.0	−1.44198	−0.720992	−	0.692943i	$-0.756314\pi$
$571$	−19675.0	−1.44198	−0.720992	+	0.692943i	$0.756314\pi$
$572$	2400.00i	0.175435i
$573$	0	0
$574$	9120.00	0.663173
$575$	0	0
$576$	0	0
$577$	− 26951.0i	− 1.94451i	−0.233914	−	0.972257i	$-0.575154\pi$
$577$	− 26951.0i	− 1.94451i	0.233914	−	0.972257i	$-0.424846\pi$
$578$	− 21926.0i	− 1.57786i
$579$	0	0
$580$	0	0
$581$	9462.00	0.675645
$582$	0	0
$583$	− 7632.00i	− 0.542170i
$584$	520.000	0.0368455
$585$	0	0
$586$	−16332.0	−1.15131
$587$	− 10986.0i	− 0.772471i	−0.922400	−	0.386236i	$-0.873775\pi$
$587$	− 10986.0i	− 0.772471i	0.922400	−	0.386236i	$-0.126225\pi$
$588$	0	0
$589$	7685.00	0.537614
$590$	0	0
$591$	0	0
$592$	− 1040.00i	− 0.0722023i
$593$	11226.0i	0.777397i	0.921365	+	0.388699i	$0.127075\pi$
$593$	11226.0i	0.777397i	−0.921365	+	0.388699i	$0.872925\pi$
$594$	0	0
$595$	0	0
$596$	−12024.0	−0.826380
$597$	0	0
$598$	1800.00i	0.123089i
$599$	−11400.0	−0.777615	−0.388807	−	0.921319i	$-0.627113\pi$
$599$	−11400.0	−0.777615	−0.388807	+	0.921319i	$0.627113\pi$
$600$	0	0
$601$	−6469.00	−0.439062	−0.219531	−	0.975606i	$-0.570453\pi$
$601$	−6469.00	−0.439062	−0.219531	+	0.975606i	$0.570453\pi$
$602$	13946.0i	0.944180i
$603$	0	0
$604$	−10604.0	−0.714355
$605$	0	0
$606$	0	0
$607$	− 6383.00i	− 0.426817i	−0.976963	−	0.213409i	$-0.931543\pi$
$607$	− 6383.00i	− 0.426817i	0.976963	−	0.213409i	$-0.0684565\pi$
$608$	− 928.000i	− 0.0619003i
$609$	0	0
$610$	0	0
$611$	3600.00	0.238364
$612$	0	0
$613$	− 10843.0i	− 0.714428i	−0.934022	−	0.357214i	$-0.883727\pi$
$613$	− 10843.0i	− 0.714428i	0.934022	−	0.357214i	$-0.116273\pi$
$614$	−1154.00	−0.0758496
$615$	0	0
$616$	−1824.00	−0.119304
$617$	25236.0i	1.64662i	0.567594	+	0.823309i	$0.307874\pi$
$617$	25236.0i	1.64662i	−0.567594	+	0.823309i	$0.692126\pi$
$618$	0	0
$619$	−24809.0	−1.61092	−0.805459	−	0.592652i	$-0.798081\pi$
$619$	−24809.0	−1.61092	−0.805459	+	0.592652i	$0.798081\pi$
$620$	0	0
$621$	0	0
$622$	15312.0i	0.987066i
$623$	− 15618.0i	− 1.00437i
$624$	0	0
$625$	0	0
$626$	−5452.00	−0.348092
$627$	0	0
$628$	11204.0i	0.711924i
$629$	−8190.00	−0.519168
$630$	0	0
$631$	−11476.0	−0.724013	−0.362007	−	0.932176i	$-0.617908\pi$
$631$	−11476.0	−0.724013	−0.362007	+	0.932176i	$0.617908\pi$
$632$	1384.00i	0.0871085i
$633$	0	0
$634$	18900.0	1.18394
$635$	0	0
$636$	0	0
$637$	− 900.000i	− 0.0559801i
$638$	− 2448.00i	− 0.151908i
$639$	0	0
$640$	0	0
$641$	−25350.0	−1.56204	−0.781018	−	0.624509i	$-0.785299\pi$
$641$	−25350.0	−1.56204	−0.781018	+	0.624509i	$0.785299\pi$
$642$	0	0
$643$	− 14164.0i	− 0.868699i	−0.900744	−	0.434350i	$-0.856978\pi$
$643$	− 14164.0i	− 0.868699i	0.900744	−	0.434350i	$-0.143022\pi$
$644$	−1368.00	−0.0837061
$645$	0	0
$646$	−7308.00	−0.445092
$647$	− 7536.00i	− 0.457915i	−0.973436	−	0.228957i	$-0.926468\pi$
$647$	− 7536.00i	− 0.457915i	0.973436	−	0.228957i	$-0.0735316\pi$
$648$	0	0
$649$	−1224.00	−0.0740311
$650$	0	0
$651$	0	0
$652$	− 5648.00i	− 0.339253i
$653$	432.000i	0.0258889i	0.999916	+	0.0129445i	$0.00412047\pi$
$653$	432.000i	0.0258889i	−0.999916	+	0.0129445i	$0.995880\pi$
$654$	0	0
$655$	0	0
$656$	−3840.00	−0.228547
$657$	0	0
$658$	2736.00i	0.162098i
$659$	−10782.0	−0.637340	−0.318670	−	0.947866i	$-0.603236\pi$
$659$	−10782.0	−0.637340	−0.318670	+	0.947866i	$0.603236\pi$
$660$	0	0
$661$	26057.0	1.53328	0.766641	−	0.642076i	$-0.221926\pi$
$661$	26057.0	1.53328	0.766641	+	0.642076i	$0.221926\pi$
$662$	6178.00i	0.362711i
$663$	0	0
$664$	−3984.00	−0.232845
$665$	0	0
$666$	0	0
$667$	− 1836.00i	− 0.106582i
$668$	8616.00i	0.499046i
$669$	0	0
$670$	0	0
$671$	1236.00	0.0711107
$672$	0	0
$673$	19091.0i	1.09347i	0.837306	+	0.546734i	$0.184129\pi$
$673$	19091.0i	1.09347i	−0.837306	+	0.546734i	$0.815871\pi$
$674$	−15788.0	−0.902272
$675$	0	0
$676$	1212.00	0.0689577
$677$	1044.00i	0.0592676i	0.999561	+	0.0296338i	$0.00943412\pi$
$677$	1044.00i	0.0592676i	−0.999561	+	0.0296338i	$0.990566\pi$
$678$	0	0
$679$	15599.0	0.881642
$680$	0	0
$681$	0	0
$682$	6360.00i	0.357092i
$683$	− 26046.0i	− 1.45918i	−0.683883	−	0.729592i	$-0.739710\pi$
$683$	− 26046.0i	− 1.45918i	0.683883	−	0.729592i	$-0.260290\pi$
$684$	0	0
$685$	0	0
$686$	−12350.0	−0.687355
$687$	0	0
$688$	− 5872.00i	− 0.325389i
$689$	−31800.0	−1.75832
$690$	0	0
$691$	−28060.0	−1.54479	−0.772397	−	0.635140i	$-0.780942\pi$
$691$	−28060.0	−1.54479	−0.772397	+	0.635140i	$0.780942\pi$
$692$	936.000i	0.0514182i
$693$	0	0
$694$	6900.00	0.377407
$695$	0	0
$696$	0	0
$697$	30240.0i	1.64336i
$698$	− 4522.00i	− 0.245215i
$699$	0	0
$700$	0	0
$701$	−16794.0	−0.904851	−0.452426	−	0.891802i	$-0.649441\pi$
$701$	−16794.0	−0.904851	−0.452426	+	0.891802i	$0.649441\pi$
$702$	0	0
$703$	1885.00i	0.101130i
$704$	768.000	0.0411152
$705$	0	0
$706$	23088.0	1.23078
$707$	22800.0i	1.21285i
$708$	0	0
$709$	−2495.00	−0.132160	−0.0660802	−	0.997814i	$-0.521049\pi$
$709$	−2495.00	−0.132160	−0.0660802	+	0.997814i	$0.521049\pi$
$710$	0	0
$711$	0	0
$712$	6576.00i	0.346132i
$713$	4770.00i	0.250544i
$714$	0	0
$715$	0	0
$716$	−9888.00	−0.516106
$717$	0	0
$718$	− 396.000i	− 0.0205830i
$719$	1950.00	0.101144	0.0505721	−	0.998720i	$-0.483896\pi$
$719$	1950.00	0.101144	0.0505721	+	0.998720i	$0.483896\pi$
$720$	0	0
$721$	38779.0	2.00306
$722$	− 12036.0i	− 0.620407i
$723$	0	0
$724$	760.000	0.0390127
$725$	0	0
$726$	0	0
$727$	− 16127.0i	− 0.822720i	−0.911473	−	0.411360i	$-0.865054\pi$
$727$	− 16127.0i	− 0.822720i	0.911473	−	0.411360i	$-0.134946\pi$
$728$	7600.00i	0.386916i
$729$	0	0
$730$	0	0
$731$	−46242.0	−2.33970
$732$	0	0
$733$	− 12634.0i	− 0.636627i	−0.947986	−	0.318313i	$-0.896884\pi$
$733$	− 12634.0i	− 0.636627i	0.947986	−	0.318313i	$-0.103116\pi$
$734$	20776.0	1.04476
$735$	0	0
$736$	576.000	0.0288473
$737$	− 624.000i	− 0.0311877i
$738$	0	0
$739$	10420.0	0.518682	0.259341	−	0.965786i	$-0.416495\pi$
$739$	10420.0	0.518682	0.259341	+	0.965786i	$0.416495\pi$
$740$	0	0
$741$	0	0
$742$	− 24168.0i	− 1.19573i
$743$	21936.0i	1.08311i	0.840664	+	0.541557i	$0.182165\pi$
$743$	21936.0i	1.08311i	−0.840664	+	0.541557i	$0.817835\pi$
$744$	0	0
$745$	0	0
$746$	9734.00	0.477731
$747$	0	0
$748$	− 6048.00i	− 0.295637i
$749$	24282.0	1.18457
$750$	0	0
$751$	−21715.0	−1.05512	−0.527558	−	0.849519i	$-0.676892\pi$
$751$	−21715.0	−1.05512	−0.527558	+	0.849519i	$0.676892\pi$
$752$	− 1152.00i	− 0.0558632i
$753$	0	0
$754$	−10200.0	−0.492655
$755$	0	0
$756$	0	0
$757$	19849.0i	0.953004i	0.879173	+	0.476502i	$0.158096\pi$
$757$	19849.0i	0.953004i	−0.879173	+	0.476502i	$0.841904\pi$
$758$	− 1312.00i	− 0.0628681i
$759$	0	0
$760$	0	0
$761$	29952.0	1.42675	0.713377	−	0.700781i	$-0.247165\pi$
$761$	29952.0	1.42675	0.713377	+	0.700781i	$0.247165\pi$
$762$	0	0
$763$	3895.00i	0.184808i
$764$	6096.00	0.288672
$765$	0	0
$766$	6156.00	0.290372
$767$	5100.00i	0.240092i
$768$	0	0
$769$	−17582.0	−0.824477	−0.412239	−	0.911076i	$-0.635253\pi$
$769$	−17582.0	−0.824477	−0.412239	+	0.911076i	$0.635253\pi$
$770$	0	0
$771$	0	0
$772$	− 13340.0i	− 0.621913i
$773$	− 19194.0i	− 0.893092i	−0.894761	−	0.446546i	$-0.852654\pi$
$773$	− 19194.0i	− 0.893092i	0.894761	−	0.446546i	$-0.147346\pi$
$774$	0	0
$775$	0	0
$776$	−6568.00	−0.303837
$777$	0	0
$778$	3288.00i	0.151517i
$779$	6960.00	0.320113
$780$	0	0
$781$	−6984.00	−0.319984
$782$	− 4536.00i	− 0.207426i
$783$	0	0
$784$	−288.000	−0.0131195
$785$	0	0
$786$	0	0
$787$	9823.00i	0.444920i	0.974942	+	0.222460i	$0.0714087\pi$
$787$	9823.00i	0.444920i	−0.974942	+	0.222460i	$0.928591\pi$
$788$	17760.0i	0.802886i
$789$	0	0
$790$	0	0
$791$	−28500.0	−1.28109
$792$	0	0
$793$	− 5150.00i	− 0.230620i
$794$	11458.0	0.512127
$795$	0	0
$796$	8768.00	0.390419
$797$	22188.0i	0.986122i	0.869995	+	0.493061i	$0.164122\pi$
$797$	22188.0i	0.986122i	−0.869995	+	0.493061i	$0.835878\pi$
$798$	0	0
$799$	−9072.00	−0.401682
$800$	0	0
$801$	0	0
$802$	3552.00i	0.156391i
$803$	− 780.000i	− 0.0342785i
$804$	0	0
$805$	0	0
$806$	26500.0	1.15809
$807$	0	0
$808$	− 9600.00i	− 0.417979i
$809$	9654.00	0.419551	0.209775	−	0.977750i	$-0.432727\pi$
$809$	9654.00	0.419551	0.209775	+	0.977750i	$0.432727\pi$
$810$	0	0
$811$	−32377.0	−1.40186	−0.700931	−	0.713229i	$-0.747232\pi$
$811$	−32377.0	−1.40186	−0.700931	+	0.713229i	$0.747232\pi$
$812$	− 7752.00i	− 0.335027i
$813$	0	0
$814$	−1560.00	−0.0671720
$815$	0	0
$816$	0	0
$817$	10643.0i	0.455755i
$818$	− 25684.0i	− 1.09782i
$819$	0	0
$820$	0	0
$821$	−23952.0	−1.01819	−0.509093	−	0.860712i	$-0.670019\pi$
$821$	−23952.0	−1.01819	−0.509093	+	0.860712i	$0.670019\pi$
$822$	0	0
$823$	5072.00i	0.214822i	0.994215	+	0.107411i	$0.0342561\pi$
$823$	5072.00i	0.214822i	−0.994215	+	0.107411i	$0.965744\pi$
$824$	−16328.0	−0.690307
$825$	0	0
$826$	−3876.00	−0.163273
$827$	− 3024.00i	− 0.127152i	−0.997977	−	0.0635760i	$-0.979749\pi$
$827$	− 3024.00i	− 0.127152i	0.997977	−	0.0635760i	$-0.0202505\pi$
$828$	0	0
$829$	3049.00	0.127740	0.0638698	−	0.997958i	$-0.479656\pi$
$829$	3049.00	0.127740	0.0638698	+	0.997958i	$0.479656\pi$
$830$	0	0
$831$	0	0
$832$	− 3200.00i	− 0.133341i
$833$	2268.00i	0.0943356i
$834$	0	0
$835$	0	0
$836$	−1392.00	−0.0575877
$837$	0	0
$838$	− 22596.0i	− 0.931463i
$839$	−40374.0	−1.66134	−0.830671	−	0.556764i	$-0.812043\pi$
$839$	−40374.0	−1.66134	−0.830671	+	0.556764i	$0.812043\pi$
$840$	0	0
$841$	−13985.0	−0.573414
$842$	6862.00i	0.280855i
$843$	0	0
$844$	6928.00	0.282549
$845$	0	0
$846$	0	0
$847$	− 22553.0i	− 0.914912i
$848$	10176.0i	0.412082i
$849$	0	0
$850$	0	0
$851$	−1170.00	−0.0471294
$852$	0	0
$853$	40322.0i	1.61852i	0.587449	+	0.809261i	$0.300132\pi$
$853$	40322.0i	1.61852i	−0.587449	+	0.809261i	$0.699868\pi$
$854$	3914.00	0.156832
$855$	0	0
$856$	−10224.0	−0.408235
$857$	− 19932.0i	− 0.794474i	−0.917716	−	0.397237i	$-0.869969\pi$
$857$	− 19932.0i	− 0.794474i	0.917716	−	0.397237i	$-0.130031\pi$
$858$	0	0
$859$	−34553.0	−1.37245	−0.686224	−	0.727390i	$-0.740733\pi$
$859$	−34553.0	−1.37245	−0.686224	+	0.727390i	$0.740733\pi$
$860$	0	0
$861$	0	0
$862$	− 21060.0i	− 0.832142i
$863$	− 6918.00i	− 0.272875i	−0.990649	−	0.136438i	$-0.956435\pi$
$863$	− 6918.00i	− 0.272875i	0.990649	−	0.136438i	$-0.0435653\pi$
$864$	0	0
$865$	0	0
$866$	−6358.00	−0.249485
$867$	0	0
$868$	20140.0i	0.787553i
$869$	2076.00	0.0810397
$870$	0	0
$871$	−2600.00	−0.101145
$872$	− 1640.00i	− 0.0636897i
$873$	0	0
$874$	−1044.00	−0.0404048
$875$	0	0
$876$	0	0
$877$	− 6497.00i	− 0.250157i	−0.992147	−	0.125079i	$-0.960082\pi$
$877$	− 6497.00i	− 0.250157i	0.992147	−	0.125079i	$-0.0399183\pi$
$878$	− 12094.0i	− 0.464866i
$879$	0	0
$880$	0	0
$881$	44082.0	1.68577	0.842883	−	0.538096i	$-0.180856\pi$
$881$	44082.0	1.68577	0.842883	+	0.538096i	$0.180856\pi$
$882$	0	0
$883$	10127.0i	0.385958i	0.981203	+	0.192979i	$0.0618149\pi$
$883$	10127.0i	0.385958i	−0.981203	+	0.192979i	$0.938185\pi$
$884$	−25200.0	−0.958787
$885$	0	0
$886$	33108.0	1.25540
$887$	− 12936.0i	− 0.489682i	−0.969563	−	0.244841i	$-0.921264\pi$
$887$	− 12936.0i	− 0.489682i	0.969563	−	0.244841i	$-0.0787358\pi$
$888$	0	0
$889$	7904.00	0.298191
$890$	0	0
$891$	0	0
$892$	23956.0i	0.899222i
$893$	2088.00i	0.0782444i
$894$	0	0
$895$	0	0
$896$	2432.00	0.0906779
$897$	0	0
$898$	− 8892.00i	− 0.330434i
$899$	−27030.0	−1.00278
$900$	0	0
$901$	80136.0	2.96306
$902$	5760.00i	0.212624i
$903$	0	0
$904$	12000.0	0.441498
$905$	0	0
$906$	0	0
$907$	45853.0i	1.67864i	0.543640	+	0.839319i	$0.317046\pi$
$907$	45853.0i	1.67864i	−0.543640	+	0.839319i	$0.682954\pi$
$908$	13200.0i	0.482442i
$909$	0	0
$910$	0	0
$911$	34530.0	1.25580	0.627898	−	0.778296i	$-0.283916\pi$
$911$	34530.0	1.25580	0.627898	+	0.778296i	$0.283916\pi$
$912$	0	0
$913$	5976.00i	0.216623i
$914$	292.000	0.0105673
$915$	0	0
$916$	17516.0	0.631818
$917$	36138.0i	1.30140i
$918$	0	0
$919$	31015.0	1.11326	0.556632	−	0.830759i	$-0.312093\pi$
$919$	31015.0	1.11326	0.556632	+	0.830759i	$0.312093\pi$
$920$	0	0
$921$	0	0
$922$	− 19644.0i	− 0.701671i
$923$	29100.0i	1.03774i
$924$	0	0
$925$	0	0
$926$	−15862.0	−0.562913
$927$	0	0
$928$	3264.00i	0.115459i
$929$	36186.0	1.27796	0.638980	−	0.769224i	$-0.279357\pi$
$929$	36186.0	1.27796	0.638980	+	0.769224i	$0.279357\pi$
$930$	0	0
$931$	522.000	0.0183758
$932$	11832.0i	0.415848i
$933$	0	0
$934$	−37428.0	−1.31122
$935$	0	0
$936$	0	0
$937$	1441.00i	0.0502406i	0.999684	+	0.0251203i	$0.00799688\pi$
$937$	1441.00i	0.0502406i	−0.999684	+	0.0251203i	$0.992003\pi$
$938$	− 1976.00i	− 0.0687832i
$939$	0	0
$940$	0	0
$941$	33168.0	1.14904	0.574520	−	0.818491i	$-0.305189\pi$
$941$	33168.0	1.14904	0.574520	+	0.818491i	$0.305189\pi$
$942$	0	0
$943$	4320.00i	0.149182i
$944$	1632.00	0.0562681
$945$	0	0
$946$	−8808.00	−0.302720
$947$	− 42054.0i	− 1.44305i	−0.692387	−	0.721527i	$-0.743441\pi$
$947$	− 42054.0i	− 1.44305i	0.692387	−	0.721527i	$-0.256559\pi$
$948$	0	0
$949$	−3250.00	−0.111169
$950$	0	0
$951$	0	0
$952$	− 19152.0i	− 0.652017i
$953$	− 26460.0i	− 0.899395i	−0.893181	−	0.449698i	$-0.851532\pi$
$953$	− 26460.0i	− 0.899395i	0.893181	−	0.449698i	$-0.148468\pi$
$954$	0	0
$955$	0	0
$956$	10704.0	0.362126
$957$	0	0
$958$	25572.0i	0.862415i
$959$	51528.0	1.73506
$960$	0	0
$961$	40434.0	1.35726
$962$	6500.00i	0.217847i
$963$	0	0
$964$	−14504.0	−0.484588
$965$	0	0
$966$	0	0
$967$	29239.0i	0.972350i	0.873861	+	0.486175i	$0.161608\pi$
$967$	29239.0i	0.972350i	−0.873861	+	0.486175i	$0.838392\pi$
$968$	9496.00i	0.315303i
$969$	0	0
$970$	0	0
$971$	54288.0	1.79422	0.897109	−	0.441810i	$-0.145664\pi$
$971$	54288.0	1.79422	0.897109	+	0.441810i	$0.145664\pi$
$972$	0	0
$973$	6973.00i	0.229747i
$974$	20800.0	0.684266
$975$	0	0
$976$	−1648.00	−0.0540484
$977$	− 38634.0i	− 1.26511i	−0.774516	−	0.632554i	$-0.782007\pi$
$977$	− 38634.0i	− 1.26511i	0.774516	−	0.632554i	$-0.217993\pi$
$978$	0	0
$979$	9864.00	0.322017
$980$	0	0
$981$	0	0
$982$	39096.0i	1.27047i
$983$	19548.0i	0.634267i	0.948381	+	0.317133i	$0.102720\pi$
$983$	19548.0i	0.634267i	−0.948381	+	0.317133i	$0.897280\pi$
$984$	0	0
$985$	0	0
$986$	25704.0	0.830205
$987$	0	0
$988$	5800.00i	0.186764i
$989$	−6606.00	−0.212395
$990$	0	0
$991$	28445.0	0.911791	0.455896	−	0.890033i	$-0.349319\pi$
$991$	28445.0	0.911791	0.455896	+	0.890033i	$0.349319\pi$
$992$	− 8480.00i	− 0.271412i
$993$	0	0
$994$	−22116.0	−0.705711
$995$	0	0
$996$	0	0
$997$	48706.0i	1.54718i	0.633689	+	0.773588i	$0.281540\pi$
$997$	48706.0i	1.54718i	−0.633689	+	0.773588i	$0.718460\pi$
$998$	− 29878.0i	− 0.947667i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.c.h.649.2		2
3.2	odd	2		1350.4.c.m.649.1		2
5.2	odd	4		1350.4.a.c.1.1	✓	1
5.3	odd	4		1350.4.a.y.1.1	yes	1
5.4	even	2	inner	1350.4.c.h.649.1		2
15.2	even	4		1350.4.a.q.1.1	yes	1
15.8	even	4		1350.4.a.k.1.1	yes	1
15.14	odd	2		1350.4.c.m.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.4.a.c.1.1	✓	1	5.2	odd	4
1350.4.a.k.1.1	yes	1	15.8	even	4
1350.4.a.q.1.1	yes	1	15.2	even	4
1350.4.a.y.1.1	yes	1	5.3	odd	4
1350.4.c.h.649.1		2	5.4	even	2	inner
1350.4.c.h.649.2		2	1.1	even	1	trivial
1350.4.c.m.649.1		2	3.2	odd	2
1350.4.c.m.649.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} +19.0000i q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} +19.0000i q^{7} -8.00000i q^{8} -12.0000 q^{11} +50.0000i q^{13} -38.0000 q^{14} +16.0000 q^{16} -126.000i q^{17} -29.0000 q^{19} -24.0000i q^{22} -18.0000i q^{23} -100.000 q^{26} -76.0000i q^{28} +102.000 q^{29} -265.000 q^{31} +32.0000i q^{32} +252.000 q^{34} -65.0000i q^{37} -58.0000i q^{38} -240.000 q^{41} -367.000i q^{43} +48.0000 q^{44} +36.0000 q^{46} -72.0000i q^{47} -18.0000 q^{49} -200.000i q^{52} +636.000i q^{53} +152.000 q^{56} +204.000i q^{58} +102.000 q^{59} -103.000 q^{61} -530.000i q^{62} -64.0000 q^{64} +52.0000i q^{67} +504.000i q^{68} +582.000 q^{71} +65.0000i q^{73} +130.000 q^{74} +116.000 q^{76} -228.000i q^{77} -173.000 q^{79} -480.000i q^{82} -498.000i q^{83} +734.000 q^{86} +96.0000i q^{88} -822.000 q^{89} -950.000 q^{91} +72.0000i q^{92} +144.000 q^{94} -821.000i q^{97} -36.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} - 24 q^{11} - 76 q^{14} + 32 q^{16} - 58 q^{19} - 200 q^{26} + 204 q^{29} - 530 q^{31} + 504 q^{34} - 480 q^{41} + 96 q^{44} + 72 q^{46} - 36 q^{49} + 304 q^{56} + 204 q^{59} - 206 q^{61} - 128 q^{64} + 1164 q^{71} + 260 q^{74} + 232 q^{76} - 346 q^{79} + 1468 q^{86} - 1644 q^{89} - 1900 q^{91} + 288 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 - 24 * q^11 - 76 * q^14 + 32 * q^16 - 58 * q^19 - 200 * q^26 + 204 * q^29 - 530 * q^31 + 504 * q^34 - 480 * q^41 + 96 * q^44 + 72 * q^46 - 36 * q^49 + 304 * q^56 + 204 * q^59 - 206 * q^61 - 128 * q^64 + 1164 * q^71 + 260 * q^74 + 232 * q^76 - 346 * q^79 + 1468 * q^86 - 1644 * q^89 - 1900 * q^91 + 288 * q^94

Introduction

L-functions

Modular forms

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