LMFDB - Embedded newform 675.4.b.k.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,4,Mod(649,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, 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("675.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.b (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:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.2033649216.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 47x^{4} + 541x^{2} + 36 $ x^6 + 47x^4 + 541x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-5.20067i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.k.649.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.20067i q^{2} -19.0470 q^{4} -24.4013i q^{7} +57.4517i q^{8} +O(q^{10})$	$q-5.20067i q^{2} -19.0470 q^{4} -24.4013i q^{7} +57.4517i q^{8} -28.9839 q^{11} -65.3919i q^{13} -126.903 q^{14} +146.411 q^{16} -68.1718i q^{17} -104.424 q^{19} +150.736i q^{22} -154.807i q^{23} -340.082 q^{26} +464.772i q^{28} +205.658 q^{29} -18.2497 q^{31} -301.824i q^{32} -354.539 q^{34} +337.613i q^{37} +543.076i q^{38} -195.969 q^{41} +334.882i q^{43} +552.055 q^{44} -805.098 q^{46} +5.00398i q^{47} -252.425 q^{49} +1245.52i q^{52} -319.965i q^{53} +1401.90 q^{56} -1069.56i q^{58} -430.611 q^{59} +594.581 q^{61} +94.9106i q^{62} -398.396 q^{64} -195.876i q^{67} +1298.47i q^{68} +425.955 q^{71} +929.193i q^{73} +1755.82 q^{74} +1988.96 q^{76} +707.245i q^{77} -24.4296 q^{79} +1019.17i q^{82} -545.859i q^{83} +1741.61 q^{86} -1665.17i q^{88} +84.1332 q^{89} -1595.65 q^{91} +2948.60i q^{92} +26.0241 q^{94} -827.613i q^{97} +1312.78i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 46 q^{4}+O(q^{10}) $ 6 * q - 46 * q^4	$ 6 q - 46 q^{4} - 76 q^{11} - 216 q^{14} + 382 q^{16} - 374 q^{19} - 832 q^{26} + 320 q^{29} + 454 q^{31} - 34 q^{34} + 676 q^{41} + 3272 q^{44} - 2850 q^{46} + 394 q^{49} + 2508 q^{56} + 280 q^{59} + 1190 q^{61} + 1836 q^{64} + 1204 q^{71} + 5756 q^{74} + 1050 q^{76} - 1258 q^{79} + 7460 q^{86} + 4308 q^{89} - 880 q^{91} + 2216 q^{94}+O(q^{100}) $ 6 * q - 46 * q^4 - 76 * q^11 - 216 * q^14 + 382 * q^16 - 374 * q^19 - 832 * q^26 + 320 * q^29 + 454 * q^31 - 34 * q^34 + 676 * q^41 + 3272 * q^44 - 2850 * q^46 + 394 * q^49 + 2508 * q^56 + 280 * q^59 + 1190 * q^61 + 1836 * q^64 + 1204 * q^71 + 5756 * q^74 + 1050 * q^76 - 1258 * q^79 + 7460 * q^86 + 4308 * q^89 - 880 * q^91 + 2216 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 5.20067i	− 1.83871i	−0.393424	−	0.919357i	$-0.628709\pi$
$2$	− 5.20067i	− 1.83871i	0.393424	−	0.919357i	$-0.371291\pi$
$3$	0	0
$3$	0	0
$4$	−19.0470	−2.38087
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 24.4013i	− 1.31755i	−0.752341	−	0.658774i	$-0.771075\pi$
$7$	− 24.4013i	− 1.31755i	0.752341	−	0.658774i	$-0.228925\pi$
$8$	57.4517i	2.53903i
$9$	0	0
$10$	0	0
$11$	−28.9839	−0.794451	−0.397226	−	0.917721i	$-0.630027\pi$
$11$	−28.9839	−0.794451	−0.397226	+	0.917721i	$0.630027\pi$
$12$	0	0
$13$	− 65.3919i	− 1.39511i	−0.716530	−	0.697556i	$-0.754271\pi$
$13$	− 65.3919i	− 1.39511i	0.716530	−	0.697556i	$-0.245729\pi$
$14$	−126.903	−2.42260
$15$	0	0
$16$	146.411	2.28768
$17$	− 68.1718i	− 0.972593i	−0.873794	−	0.486296i	$-0.838347\pi$
$17$	− 68.1718i	− 0.972593i	0.873794	−	0.486296i	$-0.161653\pi$
$18$	0	0
$19$	−104.424	−1.26087	−0.630435	−	0.776242i	$-0.717124\pi$
$19$	−104.424	−1.26087	−0.630435	+	0.776242i	$0.717124\pi$
$20$	0	0
$21$	0	0
$22$	150.736i	1.46077i
$23$	− 154.807i	− 1.40345i	−0.712446	−	0.701727i	$-0.752413\pi$
$23$	− 154.807i	− 1.40345i	0.712446	−	0.701727i	$-0.247587\pi$
$24$	0	0
$25$	0	0
$26$	−340.082	−2.56521
$27$	0	0
$28$	464.772i	3.13691i
$29$	205.658	1.31689	0.658443	−	0.752631i	$-0.271215\pi$
$29$	205.658	1.31689	0.658443	+	0.752631i	$0.271215\pi$
$30$	0	0
$31$	−18.2497	−0.105734	−0.0528668	−	0.998602i	$-0.516836\pi$
$31$	−18.2497	−0.105734	−0.0528668	+	0.998602i	$0.516836\pi$
$32$	− 301.824i	− 1.66736i
$33$	0	0
$34$	−354.539	−1.78832
$35$	0	0
$36$	0	0
$37$	337.613i	1.50009i	0.661387	+	0.750044i	$0.269968\pi$
$37$	337.613i	1.50009i	−0.661387	+	0.750044i	$0.730032\pi$
$38$	543.076i	2.31838i
$39$	0	0
$40$	0	0
$41$	−195.969	−0.746469	−0.373234	−	0.927737i	$-0.621751\pi$
$41$	−195.969	−0.746469	−0.373234	+	0.927737i	$0.621751\pi$
$42$	0	0
$43$	334.882i	1.18765i	0.804594	+	0.593826i	$0.202383\pi$
$43$	334.882i	1.18765i	−0.804594	+	0.593826i	$0.797617\pi$
$44$	552.055	1.89149
$45$	0	0
$46$	−805.098	−2.58055
$47$	5.00398i	0.0155299i	0.999970	+	0.00776496i	$0.00247169\pi$
$47$	5.00398i	0.0155299i	−0.999970	+	0.00776496i	$0.997528\pi$
$48$	0	0
$49$	−252.425	−0.735934
$50$	0	0
$51$	0	0
$52$	1245.52i	3.32158i
$53$	− 319.965i	− 0.829256i	−0.909991	−	0.414628i	$-0.863912\pi$
$53$	− 319.965i	− 0.829256i	0.909991	−	0.414628i	$-0.136088\pi$
$54$	0	0
$55$	0	0
$56$	1401.90	3.34529
$57$	0	0
$58$	− 1069.56i	− 2.42138i
$59$	−430.611	−0.950182	−0.475091	−	0.879937i	$-0.657585\pi$
$59$	−430.611	−0.950182	−0.475091	+	0.879937i	$0.657585\pi$
$60$	0	0
$61$	594.581	1.24800	0.624002	−	0.781422i	$-0.285505\pi$
$61$	594.581	1.24800	0.624002	+	0.781422i	$0.285505\pi$
$62$	94.9106i	0.194414i
$63$	0	0
$64$	−398.396	−0.778118
$65$	0	0
$66$	0	0
$67$	− 195.876i	− 0.357166i	−0.983925	−	0.178583i	$-0.942849\pi$
$67$	− 195.876i	− 0.357166i	0.983925	−	0.178583i	$-0.0571513\pi$
$68$	1298.47i	2.31562i
$69$	0	0
$70$	0	0
$71$	425.955	0.711994	0.355997	−	0.934487i	$-0.384141\pi$
$71$	425.955	0.711994	0.355997	+	0.934487i	$0.384141\pi$
$72$	0	0
$73$	929.193i	1.48978i	0.667188	+	0.744889i	$0.267498\pi$
$73$	929.193i	1.48978i	−0.667188	+	0.744889i	$0.732502\pi$
$74$	1755.82	2.75824
$75$	0	0
$76$	1988.96	3.00197
$77$	707.245i	1.04673i
$78$	0	0
$79$	−24.4296	−0.0347917	−0.0173959	−	0.999849i	$-0.505538\pi$
$79$	−24.4296	−0.0347917	−0.0173959	+	0.999849i	$0.505538\pi$
$80$	0	0
$81$	0	0
$82$	1019.17i	1.37254i
$83$	− 545.859i	− 0.721877i	−0.932590	−	0.360938i	$-0.882456\pi$
$83$	− 545.859i	− 0.721877i	0.932590	−	0.360938i	$-0.117544\pi$
$84$	0	0
$85$	0	0
$86$	1741.61	2.18375
$87$	0	0
$88$	− 1665.17i	− 2.01714i
$89$	84.1332	0.100203	0.0501017	−	0.998744i	$-0.484045\pi$
$89$	84.1332	0.100203	0.0501017	+	0.998744i	$0.484045\pi$
$90$	0	0
$91$	−1595.65	−1.83813
$92$	2948.60i	3.34144i
$93$	0	0
$94$	26.0241	0.0285551
$95$	0	0
$96$	0	0
$97$	− 827.613i	− 0.866303i	−0.901321	−	0.433152i	$-0.857401\pi$
$97$	− 827.613i	− 0.866303i	0.901321	−	0.433152i	$-0.142599\pi$
$98$	1312.78i	1.35317i
$99$	0	0
$100$	0	0
$101$	823.576	0.811375	0.405688	−	0.914012i	$-0.367032\pi$
$101$	823.576	0.811375	0.405688	+	0.914012i	$0.367032\pi$
$102$	0	0
$103$	1171.19i	1.12040i	0.828359	+	0.560198i	$0.189275\pi$
$103$	1171.19i	1.12040i	−0.828359	+	0.560198i	$0.810725\pi$
$104$	3756.87	3.54223
$105$	0	0
$106$	−1664.03	−1.52477
$107$	− 1023.21i	− 0.924460i	−0.886760	−	0.462230i	$-0.847049\pi$
$107$	− 1023.21i	− 0.924460i	0.886760	−	0.462230i	$-0.152951\pi$
$108$	0	0
$109$	403.647	0.354700	0.177350	−	0.984148i	$-0.443247\pi$
$109$	403.647	0.354700	0.177350	+	0.984148i	$0.443247\pi$
$110$	0	0
$111$	0	0
$112$	− 3572.63i	− 3.01413i
$113$	1082.20i	0.900931i	0.892794	+	0.450465i	$0.148742\pi$
$113$	1082.20i	0.900931i	−0.892794	+	0.450465i	$0.851258\pi$
$114$	0	0
$115$	0	0
$116$	−3917.16	−3.13534
$117$	0	0
$118$	2239.46i	1.74711i
$119$	−1663.48	−1.28144
$120$	0	0
$121$	−490.935	−0.368847
$122$	− 3092.22i	− 2.29473i
$123$	0	0
$124$	347.601	0.251738
$125$	0	0
$126$	0	0
$127$	774.132i	0.540890i	0.962735	+	0.270445i	$0.0871709\pi$
$127$	774.132i	0.540890i	−0.962735	+	0.270445i	$0.912829\pi$
$128$	− 342.664i	− 0.236621i
$129$	0	0
$130$	0	0
$131$	−1214.04	−0.809702	−0.404851	−	0.914383i	$-0.632677\pi$
$131$	−1214.04	−0.809702	−0.404851	+	0.914383i	$0.632677\pi$
$132$	0	0
$133$	2548.09i	1.66126i
$134$	−1018.69	−0.656726
$135$	0	0
$136$	3916.58	2.46944
$137$	− 2300.15i	− 1.43441i	−0.696860	−	0.717207i	$-0.745420\pi$
$137$	− 2300.15i	− 1.43441i	0.696860	−	0.717207i	$-0.254580\pi$
$138$	0	0
$139$	1355.93	0.827396	0.413698	−	0.910414i	$-0.364237\pi$
$139$	1355.93	0.827396	0.413698	+	0.910414i	$0.364237\pi$
$140$	0	0
$141$	0	0
$142$	− 2215.25i	− 1.30915i
$143$	1895.31i	1.10835i
$144$	0	0
$145$	0	0
$146$	4832.43	2.73928
$147$	0	0
$148$	− 6430.51i	− 3.57152i
$149$	−259.845	−0.142868	−0.0714340	−	0.997445i	$-0.522758\pi$
$149$	−259.845	−0.142868	−0.0714340	+	0.997445i	$0.522758\pi$
$150$	0	0
$151$	−508.304	−0.273941	−0.136971	−	0.990575i	$-0.543737\pi$
$151$	−508.304	−0.273941	−0.136971	+	0.990575i	$0.543737\pi$
$152$	− 5999.34i	− 3.20139i
$153$	0	0
$154$	3678.15	1.92463
$155$	0	0
$156$	0	0
$157$	23.3052i	0.0118468i	0.999982	+	0.00592342i	$0.00188549\pi$
$157$	23.3052i	0.0118468i	−0.999982	+	0.00592342i	$0.998115\pi$
$158$	127.050i	0.0639721i
$159$	0	0
$160$	0	0
$161$	−3777.49	−1.84912
$162$	0	0
$163$	− 4032.10i	− 1.93754i	−0.247964	−	0.968769i	$-0.579762\pi$
$163$	− 4032.10i	− 1.93754i	0.247964	−	0.968769i	$-0.420238\pi$
$164$	3732.62	1.77725
$165$	0	0
$166$	−2838.83	−1.32733
$167$	671.911i	0.311341i	0.987809	+	0.155671i	$0.0497539\pi$
$167$	671.911i	0.311341i	−0.987809	+	0.155671i	$0.950246\pi$
$168$	0	0
$169$	−2079.10	−0.946337
$170$	0	0
$171$	0	0
$172$	− 6378.49i	− 2.82765i
$173$	1633.53i	0.717889i	0.933359	+	0.358944i	$0.116863\pi$
$173$	1633.53i	0.717889i	−0.933359	+	0.358944i	$0.883137\pi$
$174$	0	0
$175$	0	0
$176$	−4243.57	−1.81745
$177$	0	0
$178$	− 437.549i	− 0.184245i
$179$	−341.260	−0.142497	−0.0712485	−	0.997459i	$-0.522698\pi$
$179$	−341.260	−0.142497	−0.0712485	+	0.997459i	$0.522698\pi$
$180$	0	0
$181$	−1695.92	−0.696447	−0.348223	−	0.937412i	$-0.613215\pi$
$181$	−1695.92	−0.696447	−0.348223	+	0.937412i	$0.613215\pi$
$182$	8298.45i	3.37979i
$183$	0	0
$184$	8893.90	3.56341
$185$	0	0
$186$	0	0
$187$	1975.88i	0.772678i
$188$	− 95.3107i	− 0.0369747i
$189$	0	0
$190$	0	0
$191$	−726.451	−0.275205	−0.137603	−	0.990488i	$-0.543940\pi$
$191$	−726.451	−0.275205	−0.137603	+	0.990488i	$0.543940\pi$
$192$	0	0
$193$	4247.26i	1.58406i	0.610479	+	0.792032i	$0.290977\pi$
$193$	4247.26i	1.58406i	−0.610479	+	0.792032i	$0.709023\pi$
$194$	−4304.14	−1.59288
$195$	0	0
$196$	4807.94	1.75216
$197$	− 2678.52i	− 0.968713i	−0.874871	−	0.484357i	$-0.839054\pi$
$197$	− 2678.52i	− 0.968713i	0.874871	−	0.484357i	$-0.160946\pi$
$198$	0	0
$199$	−1486.48	−0.529517	−0.264759	−	0.964315i	$-0.585292\pi$
$199$	−1486.48	−0.529517	−0.264759	+	0.964315i	$0.585292\pi$
$200$	0	0
$201$	0	0
$202$	− 4283.15i	− 1.49189i
$203$	− 5018.32i	− 1.73506i
$204$	0	0
$205$	0	0
$206$	6090.97	2.06009
$207$	0	0
$208$	− 9574.12i	− 3.19157i
$209$	3026.62	1.00170
$210$	0	0
$211$	−4827.41	−1.57504	−0.787519	−	0.616291i	$-0.788635\pi$
$211$	−4827.41	−1.57504	−0.787519	+	0.616291i	$0.788635\pi$
$212$	6094.37i	1.97435i
$213$	0	0
$214$	−5321.37	−1.69982
$215$	0	0
$216$	0	0
$217$	445.317i	0.139309i
$218$	− 2099.23i	− 0.652193i
$219$	0	0
$220$	0	0
$221$	−4457.88	−1.35688
$222$	0	0
$223$	2774.48i	0.833153i	0.909101	+	0.416576i	$0.136770\pi$
$223$	2774.48i	0.833153i	−0.909101	+	0.416576i	$0.863230\pi$
$224$	−7364.91	−2.19683
$225$	0	0
$226$	5628.19	1.65655
$227$	5101.34i	1.49158i	0.666184	+	0.745788i	$0.267927\pi$
$227$	5101.34i	1.49158i	−0.666184	+	0.745788i	$0.732073\pi$
$228$	0	0
$229$	4097.83	1.18250	0.591249	−	0.806489i	$-0.298635\pi$
$229$	4097.83	1.18250	0.591249	+	0.806489i	$0.298635\pi$
$230$	0	0
$231$	0	0
$232$	11815.4i	3.34361i
$233$	357.613i	0.100549i	0.998735	+	0.0502747i	$0.0160097\pi$
$233$	357.613i	0.100549i	−0.998735	+	0.0502747i	$0.983990\pi$
$234$	0	0
$235$	0	0
$236$	8201.83	2.26226
$237$	0	0
$238$	8651.22i	2.35620i
$239$	351.682	0.0951818	0.0475909	−	0.998867i	$-0.484846\pi$
$239$	351.682	0.0951818	0.0475909	+	0.998867i	$0.484846\pi$
$240$	0	0
$241$	−6165.53	−1.64795	−0.823976	−	0.566624i	$-0.808249\pi$
$241$	−6165.53	−1.64795	−0.823976	+	0.566624i	$0.808249\pi$
$242$	2553.19i	0.678204i
$243$	0	0
$244$	−11325.0	−2.97134
$245$	0	0
$246$	0	0
$247$	6828.50i	1.75906i
$248$	− 1048.48i	− 0.268461i
$249$	0	0
$250$	0	0
$251$	3245.53	0.816160	0.408080	−	0.912946i	$-0.366198\pi$
$251$	3245.53	0.816160	0.408080	+	0.912946i	$0.366198\pi$
$252$	0	0
$253$	4486.90i	1.11498i
$254$	4026.00	0.994543
$255$	0	0
$256$	−4969.25	−1.21320
$257$	− 3552.19i	− 0.862178i	−0.902309	−	0.431089i	$-0.858129\pi$
$257$	− 3552.19i	− 0.862178i	0.902309	−	0.431089i	$-0.141871\pi$
$258$	0	0
$259$	8238.22	1.97644
$260$	0	0
$261$	0	0
$262$	6313.81i	1.48881i
$263$	4416.59i	1.03551i	0.855530	+	0.517754i	$0.173232\pi$
$263$	4416.59i	1.03551i	−0.855530	+	0.517754i	$0.826768\pi$
$264$	0	0
$265$	0	0
$266$	13251.8	3.05458
$267$	0	0
$268$	3730.85i	0.850366i
$269$	3419.93	0.775155	0.387578	−	0.921837i	$-0.373312\pi$
$269$	3419.93	0.775155	0.387578	+	0.921837i	$0.373312\pi$
$270$	0	0
$271$	716.407	0.160585	0.0802927	−	0.996771i	$-0.474415\pi$
$271$	716.407	0.160585	0.0802927	+	0.996771i	$0.474415\pi$
$272$	− 9981.12i	− 2.22498i
$273$	0	0
$274$	−11962.3	−2.63748
$275$	0	0
$276$	0	0
$277$	− 657.529i	− 0.142625i	−0.997454	−	0.0713124i	$-0.977281\pi$
$277$	− 657.529i	− 0.142625i	0.997454	−	0.0713124i	$-0.0227187\pi$
$278$	− 7051.72i	− 1.52135i
$279$	0	0
$280$	0	0
$281$	1513.91	0.321397	0.160698	−	0.987004i	$-0.448625\pi$
$281$	1513.91	0.321397	0.160698	+	0.987004i	$0.448625\pi$
$282$	0	0
$283$	3906.38i	0.820532i	0.911966	+	0.410266i	$0.134564\pi$
$283$	3906.38i	0.820532i	−0.911966	+	0.410266i	$0.865436\pi$
$284$	−8113.16	−1.69517
$285$	0	0
$286$	9856.89	2.03794
$287$	4781.91i	0.983509i
$288$	0	0
$289$	265.611	0.0540629
$290$	0	0
$291$	0	0
$292$	− 17698.3i	− 3.54697i
$293$	− 8048.76i	− 1.60483i	−0.596770	−	0.802413i	$-0.703549\pi$
$293$	− 8048.76i	− 1.60483i	0.596770	−	0.802413i	$-0.296451\pi$
$294$	0	0
$295$	0	0
$296$	−19396.4	−3.80877
$297$	0	0
$298$	1351.37i	0.262693i
$299$	−10123.1	−1.95797
$300$	0	0
$301$	8171.57	1.56479
$302$	2643.52i	0.503700i
$303$	0	0
$304$	−15288.9	−2.88447
$305$	0	0
$306$	0	0
$307$	− 101.564i	− 0.0188814i	−0.999955	−	0.00944068i	$-0.996995\pi$
$307$	− 101.564i	− 0.0188814i	0.999955	−	0.00944068i	$-0.00300511\pi$
$308$	− 13470.9i	− 2.49213i
$309$	0	0
$310$	0	0
$311$	−7684.59	−1.40113	−0.700567	−	0.713586i	$-0.747070\pi$
$311$	−7684.59	−1.40113	−0.700567	+	0.713586i	$0.747070\pi$
$312$	0	0
$313$	− 1345.15i	− 0.242915i	−0.992597	−	0.121457i	$-0.961243\pi$
$313$	− 1345.15i	− 0.242915i	0.992597	−	0.121457i	$-0.0387568\pi$
$314$	121.202	0.0217830
$315$	0	0
$316$	465.310	0.0828346
$317$	− 7622.33i	− 1.35051i	−0.737583	−	0.675257i	$-0.764033\pi$
$317$	− 7622.33i	− 1.35051i	0.737583	−	0.675257i	$-0.235967\pi$
$318$	0	0
$319$	−5960.76	−1.04620
$320$	0	0
$321$	0	0
$322$	19645.5i	3.40000i
$323$	7118.78i	1.22631i
$324$	0	0
$325$	0	0
$326$	−20969.6	−3.56258
$327$	0	0
$328$	− 11258.7i	− 1.89531i
$329$	122.104	0.0204614
$330$	0	0
$331$	−6585.09	−1.09350	−0.546751	−	0.837295i	$-0.684136\pi$
$331$	−6585.09	−1.09350	−0.546751	+	0.837295i	$0.684136\pi$
$332$	10397.0i	1.71870i
$333$	0	0
$334$	3494.39	0.572468
$335$	0	0
$336$	0	0
$337$	2946.94i	0.476351i	0.971222	+	0.238175i	$0.0765493\pi$
$337$	2946.94i	0.476351i	−0.971222	+	0.238175i	$0.923451\pi$
$338$	10812.7i	1.74004i
$339$	0	0
$340$	0	0
$341$	528.947	0.0840002
$342$	0	0
$343$	− 2210.14i	− 0.347919i
$344$	−19239.5	−3.01548
$345$	0	0
$346$	8495.44	1.31999
$347$	− 8493.48i	− 1.31399i	−0.753896	−	0.656994i	$-0.771828\pi$
$347$	− 8493.48i	− 1.31399i	0.753896	−	0.656994i	$-0.228172\pi$
$348$	0	0
$349$	−5646.54	−0.866053	−0.433027	−	0.901381i	$-0.642554\pi$
$349$	−5646.54	−0.866053	−0.433027	+	0.901381i	$0.642554\pi$
$350$	0	0
$351$	0	0
$352$	8748.03i	1.32464i
$353$	− 1221.93i	− 0.184240i	−0.995748	−	0.0921202i	$-0.970636\pi$
$353$	− 1221.93i	− 0.184240i	0.995748	−	0.0921202i	$-0.0293644\pi$
$354$	0	0
$355$	0	0
$356$	−1602.48	−0.238571
$357$	0	0
$358$	1774.78i	0.262011i
$359$	4151.44	0.610319	0.305160	−	0.952301i	$-0.401290\pi$
$359$	4151.44	0.610319	0.305160	+	0.952301i	$0.401290\pi$
$360$	0	0
$361$	4045.41	0.589796
$362$	8819.93i	1.28057i
$363$	0	0
$364$	30392.3	4.37635
$365$	0	0
$366$	0	0
$367$	7038.71i	1.00114i	0.865696	+	0.500569i	$0.166876\pi$
$367$	7038.71i	1.00114i	−0.865696	+	0.500569i	$0.833124\pi$
$368$	− 22665.5i	− 3.21065i
$369$	0	0
$370$	0	0
$371$	−7807.58	−1.09259
$372$	0	0
$373$	− 7119.57i	− 0.988303i	−0.869376	−	0.494152i	$-0.835479\pi$
$373$	− 7119.57i	− 0.988303i	0.869376	−	0.494152i	$-0.164521\pi$
$374$	10275.9	1.42073
$375$	0	0
$376$	−287.487	−0.0394309
$377$	− 13448.4i	− 1.83720i
$378$	0	0
$379$	−3372.29	−0.457053	−0.228526	−	0.973538i	$-0.573391\pi$
$379$	−3372.29	−0.457053	−0.228526	+	0.973538i	$0.573391\pi$
$380$	0	0
$381$	0	0
$382$	3778.03i	0.506024i
$383$	3958.63i	0.528138i	0.964504	+	0.264069i	$0.0850646\pi$
$383$	3958.63i	0.528138i	−0.964504	+	0.264069i	$0.914935\pi$
$384$	0	0
$385$	0	0
$386$	22088.6	2.91264
$387$	0	0
$388$	15763.5i	2.06256i
$389$	−9654.01	−1.25830	−0.629148	−	0.777285i	$-0.716596\pi$
$389$	−9654.01	−1.25830	−0.629148	+	0.777285i	$0.716596\pi$
$390$	0	0
$391$	−10553.4	−1.36499
$392$	− 14502.3i	− 1.86856i
$393$	0	0
$394$	−13930.1	−1.78119
$395$	0	0
$396$	0	0
$397$	− 10928.3i	− 1.38155i	−0.723068	−	0.690776i	$-0.757269\pi$
$397$	− 10928.3i	− 1.38155i	0.723068	−	0.690776i	$-0.242731\pi$
$398$	7730.70i	0.973631i
$399$	0	0
$400$	0	0
$401$	4085.57	0.508787	0.254393	−	0.967101i	$-0.418124\pi$
$401$	4085.57	0.508787	0.254393	+	0.967101i	$0.418124\pi$
$402$	0	0
$403$	1193.38i	0.147510i
$404$	−15686.6	−1.93178
$405$	0	0
$406$	−26098.7	−3.19028
$407$	− 9785.34i	− 1.19175i
$408$	0	0
$409$	−10156.3	−1.22786	−0.613930	−	0.789361i	$-0.710412\pi$
$409$	−10156.3	−1.22786	−0.613930	+	0.789361i	$0.710412\pi$
$410$	0	0
$411$	0	0
$412$	− 22307.6i	− 2.66752i
$413$	10507.5i	1.25191i
$414$	0	0
$415$	0	0
$416$	−19736.9	−2.32615
$417$	0	0
$418$	− 15740.4i	− 1.84184i
$419$	−15878.8	−1.85139	−0.925693	−	0.378275i	$-0.876517\pi$
$419$	−15878.8	−1.85139	−0.925693	+	0.378275i	$0.876517\pi$
$420$	0	0
$421$	−2279.85	−0.263926	−0.131963	−	0.991255i	$-0.542128\pi$
$421$	−2279.85	−0.263926	−0.131963	+	0.991255i	$0.542128\pi$
$422$	25105.8i	2.89604i
$423$	0	0
$424$	18382.5	2.10551
$425$	0	0
$426$	0	0
$427$	− 14508.6i	− 1.64431i
$428$	19489.0i	2.20102i
$429$	0	0
$430$	0	0
$431$	−1947.38	−0.217638	−0.108819	−	0.994062i	$-0.534707\pi$
$431$	−1947.38	−0.217638	−0.108819	+	0.994062i	$0.534707\pi$
$432$	0	0
$433$	12636.2i	1.40244i	0.712946	+	0.701219i	$0.247361\pi$
$433$	12636.2i	1.40244i	−0.712946	+	0.701219i	$0.752639\pi$
$434$	2315.95	0.256150
$435$	0	0
$436$	−7688.25	−0.844496
$437$	16165.6i	1.76957i
$438$	0	0
$439$	15849.8	1.72317	0.861585	−	0.507614i	$-0.169472\pi$
$439$	15849.8	1.72317	0.861585	+	0.507614i	$0.169472\pi$
$440$	0	0
$441$	0	0
$442$	23184.0i	2.49491i
$443$	− 17455.6i	− 1.87210i	−0.351872	−	0.936048i	$-0.614455\pi$
$443$	− 17455.6i	− 1.87210i	0.351872	−	0.936048i	$-0.385545\pi$
$444$	0	0
$445$	0	0
$446$	14429.2	1.53193
$447$	0	0
$448$	9721.41i	1.02521i
$449$	16068.1	1.68887	0.844435	−	0.535658i	$-0.179936\pi$
$449$	16068.1	1.68887	0.844435	+	0.535658i	$0.179936\pi$
$450$	0	0
$451$	5679.94	0.593033
$452$	− 20612.7i	− 2.14500i
$453$	0	0
$454$	26530.4	2.74258
$455$	0	0
$456$	0	0
$457$	− 11891.7i	− 1.21722i	−0.793468	−	0.608612i	$-0.791727\pi$
$457$	− 11891.7i	− 1.21722i	0.793468	−	0.608612i	$-0.208273\pi$
$458$	− 21311.4i	− 2.17428i
$459$	0	0
$460$	0	0
$461$	−2802.23	−0.283108	−0.141554	−	0.989931i	$-0.545210\pi$
$461$	−2802.23	−0.283108	−0.141554	+	0.989931i	$0.545210\pi$
$462$	0	0
$463$	− 12933.3i	− 1.29819i	−0.760707	−	0.649096i	$-0.775147\pi$
$463$	− 12933.3i	− 1.29819i	0.760707	−	0.649096i	$-0.224853\pi$
$464$	30110.6	3.01261
$465$	0	0
$466$	1859.83	0.184882
$467$	− 5748.11i	− 0.569573i	−0.958591	−	0.284787i	$-0.908077\pi$
$467$	− 5748.11i	− 0.569573i	0.958591	−	0.284787i	$-0.0919228\pi$
$468$	0	0
$469$	−4779.65	−0.470583
$470$	0	0
$471$	0	0
$472$	− 24739.3i	− 2.41254i
$473$	− 9706.17i	− 0.943531i
$474$	0	0
$475$	0	0
$476$	31684.3	3.05094
$477$	0	0
$478$	− 1828.98i	− 0.175012i
$479$	11217.3	1.07000	0.535002	−	0.844851i	$-0.320311\pi$
$479$	11217.3	1.07000	0.535002	+	0.844851i	$0.320311\pi$
$480$	0	0
$481$	22077.2	2.09279
$482$	32064.9i	3.03012i
$483$	0	0
$484$	9350.83	0.878177
$485$	0	0
$486$	0	0
$487$	8905.12i	0.828603i	0.910140	+	0.414301i	$0.135974\pi$
$487$	8905.12i	0.828603i	−0.910140	+	0.414301i	$0.864026\pi$
$488$	34159.7i	3.16872i
$489$	0	0
$490$	0	0
$491$	−6553.10	−0.602316	−0.301158	−	0.953574i	$-0.597373\pi$
$491$	−6553.10	−0.602316	−0.301158	+	0.953574i	$0.597373\pi$
$492$	0	0
$493$	− 14020.1i	− 1.28079i
$494$	35512.8	3.23440
$495$	0	0
$496$	−2671.96	−0.241884
$497$	− 10393.9i	− 0.938087i
$498$	0	0
$499$	4610.09	0.413579	0.206789	−	0.978385i	$-0.433698\pi$
$499$	4610.09	0.413579	0.206789	+	0.978385i	$0.433698\pi$
$500$	0	0
$501$	0	0
$502$	− 16878.9i	− 1.50069i
$503$	13069.1i	1.15850i	0.815151	+	0.579249i	$0.196654\pi$
$503$	13069.1i	1.15850i	−0.815151	+	0.579249i	$0.803346\pi$
$504$	0	0
$505$	0	0
$506$	23334.9	2.05012
$507$	0	0
$508$	− 14744.9i	− 1.28779i
$509$	−15930.8	−1.38727	−0.693635	−	0.720327i	$-0.743992\pi$
$509$	−15930.8	−1.38727	−0.693635	+	0.720327i	$0.743992\pi$
$510$	0	0
$511$	22673.6	1.96286
$512$	23102.1i	1.99410i
$513$	0	0
$514$	−18473.8	−1.58530
$515$	0	0
$516$	0	0
$517$	− 145.035i	− 0.0123378i
$518$	− 42844.3i	− 3.63411i
$519$	0	0
$520$	0	0
$521$	3654.38	0.307296	0.153648	−	0.988126i	$-0.450898\pi$
$521$	3654.38	0.307296	0.153648	+	0.988126i	$0.450898\pi$
$522$	0	0
$523$	− 5138.66i	− 0.429633i	−0.976654	−	0.214816i	$-0.931085\pi$
$523$	− 5138.66i	− 0.429633i	0.976654	−	0.214816i	$-0.0689153\pi$
$524$	23123.7	1.92780
$525$	0	0
$526$	22969.2	1.90400
$527$	1244.11i	0.102836i
$528$	0	0
$529$	−11798.1	−0.969681
$530$	0	0
$531$	0	0
$532$	− 48533.4i	− 3.95524i
$533$	12814.8i	1.04141i
$534$	0	0
$535$	0	0
$536$	11253.4	0.906854
$537$	0	0
$538$	− 17785.9i	− 1.42529i
$539$	7316.27	0.584664
$540$	0	0
$541$	6932.06	0.550892	0.275446	−	0.961317i	$-0.411175\pi$
$541$	6932.06	0.550892	0.275446	+	0.961317i	$0.411175\pi$
$542$	− 3725.80i	− 0.295271i
$543$	0	0
$544$	−20575.9	−1.62166
$545$	0	0
$546$	0	0
$547$	3423.11i	0.267572i	0.991010	+	0.133786i	$0.0427135\pi$
$547$	3423.11i	0.267572i	−0.991010	+	0.133786i	$0.957287\pi$
$548$	43810.8i	3.41516i
$549$	0	0
$550$	0	0
$551$	−21475.6	−1.66042
$552$	0	0
$553$	596.115i	0.0458398i
$554$	−3419.59	−0.262246
$555$	0	0
$556$	−25826.3	−1.96992
$557$	24489.2i	1.86291i	0.363856	+	0.931455i	$0.381460\pi$
$557$	24489.2i	1.86291i	−0.363856	+	0.931455i	$0.618540\pi$
$558$	0	0
$559$	21898.6	1.65691
$560$	0	0
$561$	0	0
$562$	− 7873.37i	− 0.590957i
$563$	− 10053.1i	− 0.752552i	−0.926508	−	0.376276i	$-0.877204\pi$
$563$	− 10053.1i	− 0.752552i	0.926508	−	0.376276i	$-0.122796\pi$
$564$	0	0
$565$	0	0
$566$	20315.8	1.50872
$567$	0	0
$568$	24471.8i	1.80777i
$569$	6670.45	0.491459	0.245729	−	0.969338i	$-0.420973\pi$
$569$	6670.45	0.491459	0.245729	+	0.969338i	$0.420973\pi$
$570$	0	0
$571$	4633.55	0.339594	0.169797	−	0.985479i	$-0.445689\pi$
$571$	4633.55	0.339594	0.169797	+	0.985479i	$0.445689\pi$
$572$	− 36099.9i	− 2.63884i
$573$	0	0
$574$	24869.1	1.80839
$575$	0	0
$576$	0	0
$577$	− 7045.15i	− 0.508307i	−0.967164	−	0.254154i	$-0.918203\pi$
$577$	− 7045.15i	− 0.508307i	0.967164	−	0.254154i	$-0.0817969\pi$
$578$	− 1381.35i	− 0.0994062i
$579$	0	0
$580$	0	0
$581$	−13319.7	−0.951108
$582$	0	0
$583$	9273.82i	0.658804i
$584$	−53383.7	−3.78259
$585$	0	0
$586$	−41859.0	−2.95082
$587$	8001.06i	0.562588i	0.959622	+	0.281294i	$0.0907636\pi$
$587$	8001.06i	0.562588i	−0.959622	+	0.281294i	$0.909236\pi$
$588$	0	0
$589$	1905.71	0.133316
$590$	0	0
$591$	0	0
$592$	49430.4i	3.43172i
$593$	6747.53i	0.467265i	0.972325	+	0.233632i	$0.0750612\pi$
$593$	6747.53i	0.467265i	−0.972325	+	0.233632i	$0.924939\pi$
$594$	0	0
$595$	0	0
$596$	4949.26	0.340150
$597$	0	0
$598$	52646.9i	3.60016i
$599$	−21547.3	−1.46978	−0.734890	−	0.678186i	$-0.762766\pi$
$599$	−21547.3	−1.46978	−0.734890	+	0.678186i	$0.762766\pi$
$600$	0	0
$601$	−12155.1	−0.824983	−0.412492	−	0.910961i	$-0.635341\pi$
$601$	−12155.1	−0.824983	−0.412492	+	0.910961i	$0.635341\pi$
$602$	− 42497.6i	− 2.87720i
$603$	0	0
$604$	9681.64	0.652219
$605$	0	0
$606$	0	0
$607$	− 16348.9i	− 1.09322i	−0.837388	−	0.546608i	$-0.815919\pi$
$607$	− 16348.9i	− 1.09322i	0.837388	−	0.546608i	$-0.184081\pi$
$608$	31517.7i	2.10232i
$609$	0	0
$610$	0	0
$611$	327.220	0.0216660
$612$	0	0
$613$	− 29955.5i	− 1.97372i	−0.161581	−	0.986859i	$-0.551659\pi$
$613$	− 29955.5i	− 1.97372i	0.161581	−	0.986859i	$-0.448341\pi$
$614$	−528.202	−0.0347174
$615$	0	0
$616$	−40632.4	−2.65767
$617$	− 2159.74i	− 0.140921i	−0.997515	−	0.0704603i	$-0.977553\pi$
$617$	− 2159.74i	− 0.140921i	0.997515	−	0.0704603i	$-0.0224468\pi$
$618$	0	0
$619$	−22100.8	−1.43507	−0.717535	−	0.696523i	$-0.754730\pi$
$619$	−22100.8	−1.43507	−0.717535	+	0.696523i	$0.754730\pi$
$620$	0	0
$621$	0	0
$622$	39965.0i	2.57629i
$623$	− 2052.96i	− 0.132023i
$624$	0	0
$625$	0	0
$626$	−6995.68	−0.446651
$627$	0	0
$628$	− 443.893i	− 0.0282058i
$629$	23015.7	1.45898
$630$	0	0
$631$	18360.1	1.15833	0.579164	−	0.815211i	$-0.303379\pi$
$631$	18360.1	1.15833	0.579164	+	0.815211i	$0.303379\pi$
$632$	− 1403.52i	− 0.0883372i
$633$	0	0
$634$	−39641.2	−2.48321
$635$	0	0
$636$	0	0
$637$	16506.6i	1.02671i
$638$	30999.9i	1.92367i
$639$	0	0
$640$	0	0
$641$	−21064.5	−1.29797	−0.648985	−	0.760802i	$-0.724806\pi$
$641$	−21064.5	−1.29797	−0.648985	+	0.760802i	$0.724806\pi$
$642$	0	0
$643$	− 10539.1i	− 0.646381i	−0.946334	−	0.323190i	$-0.895245\pi$
$643$	− 10539.1i	− 0.646381i	0.946334	−	0.323190i	$-0.104755\pi$
$644$	71949.8	4.40251
$645$	0	0
$646$	37022.4	2.25484
$647$	− 22553.4i	− 1.37043i	−0.728343	−	0.685213i	$-0.759709\pi$
$647$	− 22553.4i	− 1.37043i	0.728343	−	0.685213i	$-0.240291\pi$
$648$	0	0
$649$	12480.8	0.754873
$650$	0	0
$651$	0	0
$652$	76799.4i	4.61303i
$653$	22624.0i	1.35582i	0.735147	+	0.677908i	$0.237113\pi$
$653$	22624.0i	1.35582i	−0.735147	+	0.677908i	$0.762887\pi$
$654$	0	0
$655$	0	0
$656$	−28692.1	−1.70768
$657$	0	0
$658$	− 635.022i	− 0.0376227i
$659$	−6376.60	−0.376930	−0.188465	−	0.982080i	$-0.560351\pi$
$659$	−6376.60	−0.376930	−0.188465	+	0.982080i	$0.560351\pi$
$660$	0	0
$661$	−22097.5	−1.30029	−0.650146	−	0.759809i	$-0.725292\pi$
$661$	−22097.5	−1.30029	−0.650146	+	0.759809i	$0.725292\pi$
$662$	34246.9i	2.01064i
$663$	0	0
$664$	31360.5	1.83287
$665$	0	0
$666$	0	0
$667$	− 31837.2i	− 1.84819i
$668$	− 12797.9i	− 0.741264i
$669$	0	0
$670$	0	0
$671$	−17233.3	−0.991479
$672$	0	0
$673$	24033.0i	1.37653i	0.725461	+	0.688264i	$0.241627\pi$
$673$	24033.0i	1.37653i	−0.725461	+	0.688264i	$0.758373\pi$
$674$	15326.1	0.875873
$675$	0	0
$676$	39600.6	2.25311
$677$	− 179.638i	− 0.0101980i	−0.999987	−	0.00509901i	$-0.998377\pi$
$677$	− 179.638i	− 0.0101980i	0.999987	−	0.00509901i	$-0.00162307\pi$
$678$	0	0
$679$	−20194.9	−1.14140
$680$	0	0
$681$	0	0
$682$	− 2750.88i	− 0.154452i
$683$	− 30434.2i	− 1.70502i	−0.522709	−	0.852511i	$-0.675079\pi$
$683$	− 30434.2i	− 1.70502i	0.522709	−	0.852511i	$-0.324921\pi$
$684$	0	0
$685$	0	0
$686$	−11494.2	−0.639725
$687$	0	0
$688$	49030.5i	2.71696i
$689$	−20923.1	−1.15691
$690$	0	0
$691$	9792.73	0.539122	0.269561	−	0.962983i	$-0.413121\pi$
$691$	9792.73	0.539122	0.269561	+	0.962983i	$0.413121\pi$
$692$	− 31113.7i	− 1.70920i
$693$	0	0
$694$	−44171.8	−2.41605
$695$	0	0
$696$	0	0
$697$	13359.6i	0.726010i
$698$	29365.8i	1.59242i
$699$	0	0
$700$	0	0
$701$	8130.47	0.438065	0.219032	−	0.975718i	$-0.429710\pi$
$701$	8130.47	0.438065	0.219032	+	0.975718i	$0.429710\pi$
$702$	0	0
$703$	− 35255.0i	− 1.89142i
$704$	11547.1	0.618177
$705$	0	0
$706$	−6354.86	−0.338765
$707$	− 20096.4i	− 1.06903i
$708$	0	0
$709$	4859.95	0.257432	0.128716	−	0.991681i	$-0.458914\pi$
$709$	4859.95	0.257432	0.128716	+	0.991681i	$0.458914\pi$
$710$	0	0
$711$	0	0
$712$	4833.59i	0.254419i
$713$	2825.17i	0.148392i
$714$	0	0
$715$	0	0
$716$	6499.96	0.339267
$717$	0	0
$718$	− 21590.3i	− 1.12220i
$719$	−19463.0	−1.00952	−0.504762	−	0.863259i	$-0.668420\pi$
$719$	−19463.0	−1.00952	−0.504762	+	0.863259i	$0.668420\pi$
$720$	0	0
$721$	28578.6	1.47618
$722$	− 21038.8i	− 1.08447i
$723$	0	0
$724$	32302.2	1.65815
$725$	0	0
$726$	0	0
$727$	− 2432.66i	− 0.124102i	−0.998073	−	0.0620512i	$-0.980236\pi$
$727$	− 2432.66i	− 0.124102i	0.998073	−	0.0620512i	$-0.0197642\pi$
$728$	− 91672.8i	− 4.66706i
$729$	0	0
$730$	0	0
$731$	22829.5	1.15510
$732$	0	0
$733$	− 17967.6i	− 0.905386i	−0.891666	−	0.452693i	$-0.850463\pi$
$733$	− 17967.6i	− 0.905386i	0.891666	−	0.452693i	$-0.149537\pi$
$734$	36606.0	1.84081
$735$	0	0
$736$	−46724.4	−2.34006
$737$	5677.26i	0.283751i
$738$	0	0
$739$	−23473.0	−1.16843	−0.584214	−	0.811599i	$-0.698597\pi$
$739$	−23473.0	−1.16843	−0.584214	+	0.811599i	$0.698597\pi$
$740$	0	0
$741$	0	0
$742$	40604.6i	2.00895i
$743$	− 33559.2i	− 1.65702i	−0.559971	−	0.828512i	$-0.689188\pi$
$743$	− 33559.2i	− 1.65702i	0.559971	−	0.828512i	$-0.310812\pi$
$744$	0	0
$745$	0	0
$746$	−37026.5	−1.81721
$747$	0	0
$748$	− 37634.6i	− 1.83965i
$749$	−24967.6	−1.21802
$750$	0	0
$751$	−7782.75	−0.378158	−0.189079	−	0.981962i	$-0.560550\pi$
$751$	−7782.75	−0.378158	−0.189079	+	0.981962i	$0.560550\pi$
$752$	732.640i	0.0355274i
$753$	0	0
$754$	−69940.5	−3.37809
$755$	0	0
$756$	0	0
$757$	38154.3i	1.83189i	0.401300	+	0.915946i	$0.368558\pi$
$757$	38154.3i	1.83189i	−0.401300	+	0.915946i	$0.631442\pi$
$758$	17538.2i	0.840390i
$759$	0	0
$760$	0	0
$761$	19867.1	0.946363	0.473182	−	0.880965i	$-0.343105\pi$
$761$	19867.1	0.946363	0.473182	+	0.880965i	$0.343105\pi$
$762$	0	0
$763$	− 9849.52i	− 0.467335i
$764$	13836.7	0.655228
$765$	0	0
$766$	20587.5	0.971094
$767$	28158.5i	1.32561i
$768$	0	0
$769$	15710.8	0.736730	0.368365	−	0.929681i	$-0.379918\pi$
$769$	15710.8	0.736730	0.368365	+	0.929681i	$0.379918\pi$
$770$	0	0
$771$	0	0
$772$	− 80897.5i	− 3.77146i
$773$	25811.9i	1.20102i	0.799617	+	0.600510i	$0.205036\pi$
$773$	25811.9i	1.20102i	−0.799617	+	0.600510i	$0.794964\pi$
$774$	0	0
$775$	0	0
$776$	47547.8	2.19957
$777$	0	0
$778$	50207.3i	2.31365i
$779$	20463.9	0.941201
$780$	0	0
$781$	−12345.8	−0.565645
$782$	54885.0i	2.50982i
$783$	0	0
$784$	−36958.0	−1.68358
$785$	0	0
$786$	0	0
$787$	29242.6i	1.32450i	0.749281	+	0.662252i	$0.230399\pi$
$787$	29242.6i	1.32450i	−0.749281	+	0.662252i	$0.769601\pi$
$788$	51017.7i	2.30638i
$789$	0	0
$790$	0	0
$791$	26407.2	1.18702
$792$	0	0
$793$	− 38880.8i	− 1.74111i
$794$	−56834.6	−2.54028
$795$	0	0
$796$	28313.0	1.26071
$797$	− 32573.7i	− 1.44771i	−0.689955	−	0.723853i	$-0.742370\pi$
$797$	− 32573.7i	− 1.44771i	0.689955	−	0.723853i	$-0.257630\pi$
$798$	0	0
$799$	341.130	0.0151043
$800$	0	0
$801$	0	0
$802$	− 21247.7i	− 0.935514i
$803$	− 26931.6i	− 1.18356i
$804$	0	0
$805$	0	0
$806$	6206.39	0.271229
$807$	0	0
$808$	47315.8i	2.06010i
$809$	34644.9	1.50562	0.752812	−	0.658236i	$-0.228697\pi$
$809$	34644.9	1.50562	0.752812	+	0.658236i	$0.228697\pi$
$810$	0	0
$811$	−29057.9	−1.25815	−0.629077	−	0.777343i	$-0.716567\pi$
$811$	−29057.9	−1.25815	−0.629077	+	0.777343i	$0.716567\pi$
$812$	95583.9i	4.13096i
$813$	0	0
$814$	−50890.3	−2.19128
$815$	0	0
$816$	0	0
$817$	− 34969.8i	− 1.49748i
$818$	52819.3i	2.25768i
$819$	0	0
$820$	0	0
$821$	46709.5	1.98560	0.992798	−	0.119802i	$-0.0382259\pi$
$821$	46709.5	1.98560	0.992798	+	0.119802i	$0.0382259\pi$
$822$	0	0
$823$	− 3468.10i	− 0.146890i	−0.997299	−	0.0734450i	$-0.976601\pi$
$823$	− 3468.10i	− 0.146890i	0.997299	−	0.0734450i	$-0.0233993\pi$
$824$	−67286.8	−2.84472
$825$	0	0
$826$	54645.9	2.30191
$827$	− 42454.9i	− 1.78513i	−0.450920	−	0.892564i	$-0.648904\pi$
$827$	− 42454.9i	− 1.78513i	0.450920	−	0.892564i	$-0.351096\pi$
$828$	0	0
$829$	3933.47	0.164795	0.0823975	−	0.996600i	$-0.473742\pi$
$829$	3933.47	0.164795	0.0823975	+	0.996600i	$0.473742\pi$
$830$	0	0
$831$	0	0
$832$	26051.9i	1.08556i
$833$	17208.3i	0.715764i
$834$	0	0
$835$	0	0
$836$	−57647.9	−2.38492
$837$	0	0
$838$	82580.5i	3.40417i
$839$	32959.6	1.35625	0.678123	−	0.734948i	$-0.262794\pi$
$839$	32959.6	1.35625	0.678123	+	0.734948i	$0.262794\pi$
$840$	0	0
$841$	17906.1	0.734188
$842$	11856.7i	0.485285i
$843$	0	0
$844$	91947.6	3.74996
$845$	0	0
$846$	0	0
$847$	11979.5i	0.485974i
$848$	− 46846.5i	− 1.89707i
$849$	0	0
$850$	0	0
$851$	52264.8	2.10530
$852$	0	0
$853$	38845.8i	1.55927i	0.626235	+	0.779634i	$0.284595\pi$
$853$	38845.8i	1.55927i	−0.626235	+	0.779634i	$0.715405\pi$
$854$	−75454.3	−3.02341
$855$	0	0
$856$	58785.0	2.34723
$857$	29305.3i	1.16809i	0.811723	+	0.584043i	$0.198530\pi$
$857$	29305.3i	1.16809i	−0.811723	+	0.584043i	$0.801470\pi$
$858$	0	0
$859$	909.659	0.0361318	0.0180659	−	0.999837i	$-0.494249\pi$
$859$	909.659	0.0361318	0.0180659	+	0.999837i	$0.494249\pi$
$860$	0	0
$861$	0	0
$862$	10127.7i	0.400173i
$863$	47998.0i	1.89324i	0.322346	+	0.946622i	$0.395529\pi$
$863$	47998.0i	1.89324i	−0.322346	+	0.946622i	$0.604471\pi$
$864$	0	0
$865$	0	0
$866$	65716.6	2.57868
$867$	0	0
$868$	− 8481.94i	− 0.331677i
$869$	708.065	0.0276403
$870$	0	0
$871$	−12808.7	−0.498286
$872$	23190.2i	0.900594i
$873$	0	0
$874$	84071.7	3.25374
$875$	0	0
$876$	0	0
$877$	− 3258.01i	− 0.125445i	−0.998031	−	0.0627225i	$-0.980022\pi$
$877$	− 3258.01i	− 0.125445i	0.998031	−	0.0627225i	$-0.0199783\pi$
$878$	− 82429.8i	− 3.16842i
$879$	0	0
$880$	0	0
$881$	−33380.2	−1.27651	−0.638256	−	0.769824i	$-0.720344\pi$
$881$	−33380.2	−1.27651	−0.638256	+	0.769824i	$0.720344\pi$
$882$	0	0
$883$	33714.6i	1.28492i	0.766319	+	0.642460i	$0.222086\pi$
$883$	33714.6i	1.28492i	−0.766319	+	0.642460i	$0.777914\pi$
$884$	84909.2	3.23055
$885$	0	0
$886$	−90780.6	−3.44225
$887$	6218.80i	0.235408i	0.993049	+	0.117704i	$0.0375534\pi$
$887$	6218.80i	0.235408i	−0.993049	+	0.117704i	$0.962447\pi$
$888$	0	0
$889$	18889.8	0.712649
$890$	0	0
$891$	0	0
$892$	− 52845.5i	− 1.98363i
$893$	− 522.537i	− 0.0195812i
$894$	0	0
$895$	0	0
$896$	−8361.46	−0.311760
$897$	0	0
$898$	− 83565.1i	− 3.10535i
$899$	−3753.19	−0.139239
$900$	0	0
$901$	−21812.6	−0.806529
$902$	− 29539.5i	− 1.09042i
$903$	0	0
$904$	−62174.4	−2.28749
$905$	0	0
$906$	0	0
$907$	22878.6i	0.837566i	0.908086	+	0.418783i	$0.137543\pi$
$907$	22878.6i	0.837566i	−0.908086	+	0.418783i	$0.862457\pi$
$908$	− 97165.0i	− 3.55125i
$909$	0	0
$910$	0	0
$911$	−30144.8	−1.09631	−0.548157	−	0.836376i	$-0.684670\pi$
$911$	−30144.8	−1.09631	−0.548157	+	0.836376i	$0.684670\pi$
$912$	0	0
$913$	15821.1i	0.573496i
$914$	−61844.9	−2.23813
$915$	0	0
$916$	−78051.2	−2.81538
$917$	29624.1i	1.06682i
$918$	0	0
$919$	−3803.52	−0.136525	−0.0682625	−	0.997667i	$-0.521746\pi$
$919$	−3803.52	−0.136525	−0.0682625	+	0.997667i	$0.521746\pi$
$920$	0	0
$921$	0	0
$922$	14573.5i	0.520555i
$923$	− 27854.0i	− 0.993312i
$924$	0	0
$925$	0	0
$926$	−67262.0	−2.38700
$927$	0	0
$928$	− 62072.5i	− 2.19572i
$929$	−125.985	−0.00444934	−0.00222467	−	0.999998i	$-0.500708\pi$
$929$	−125.985	−0.00444934	−0.00222467	+	0.999998i	$0.500708\pi$
$930$	0	0
$931$	26359.3	0.927918
$932$	− 6811.45i	− 0.239395i
$933$	0	0
$934$	−29894.0	−1.04728
$935$	0	0
$936$	0	0
$937$	− 28107.9i	− 0.979984i	−0.871727	−	0.489992i	$-0.837000\pi$
$937$	− 28107.9i	− 0.979984i	0.871727	−	0.489992i	$-0.163000\pi$
$938$	24857.4i	0.865268i
$939$	0	0
$940$	0	0
$941$	−49194.9	−1.70426	−0.852130	−	0.523330i	$-0.824689\pi$
$941$	−49194.9	−1.70426	−0.852130	+	0.523330i	$0.824689\pi$
$942$	0	0
$943$	30337.3i	1.04763i
$944$	−63046.3	−2.17371
$945$	0	0
$946$	−50478.6	−1.73488
$947$	14498.0i	0.497490i	0.968569	+	0.248745i	$0.0800181\pi$
$947$	14498.0i	0.497490i	−0.968569	+	0.248745i	$0.919982\pi$
$948$	0	0
$949$	60761.7	2.07841
$950$	0	0
$951$	0	0
$952$	− 95569.8i	− 3.25361i
$953$	3201.79i	0.108831i	0.998518	+	0.0544155i	$0.0173296\pi$
$953$	3201.79i	0.108831i	−0.998518	+	0.0544155i	$0.982670\pi$
$954$	0	0
$955$	0	0
$956$	−6698.48	−0.226616
$957$	0	0
$958$	− 58337.6i	− 1.96743i
$959$	−56126.7	−1.88991
$960$	0	0
$961$	−29457.9	−0.988820
$962$	− 114816.i	− 3.84805i
$963$	0	0
$964$	117435.	3.92356
$965$	0	0
$966$	0	0
$967$	− 57781.9i	− 1.92155i	−0.277326	−	0.960776i	$-0.589448\pi$
$967$	− 57781.9i	− 1.92155i	0.277326	−	0.960776i	$-0.410552\pi$
$968$	− 28205.1i	− 0.936513i
$969$	0	0
$970$	0	0
$971$	2611.60	0.0863133	0.0431566	−	0.999068i	$-0.486259\pi$
$971$	2611.60	0.0863133	0.0431566	+	0.999068i	$0.486259\pi$
$972$	0	0
$973$	− 33086.4i	− 1.09013i
$974$	46312.6	1.52356
$975$	0	0
$976$	87053.4	2.85503
$977$	33598.3i	1.10021i	0.835096	+	0.550104i	$0.185412\pi$
$977$	33598.3i	1.10021i	−0.835096	+	0.550104i	$0.814588\pi$
$978$	0	0
$979$	−2438.51	−0.0796067
$980$	0	0
$981$	0	0
$982$	34080.5i	1.10749i
$983$	− 39484.8i	− 1.28115i	−0.767895	−	0.640575i	$-0.778696\pi$
$983$	− 39484.8i	− 1.28115i	0.767895	−	0.640575i	$-0.221304\pi$
$984$	0	0
$985$	0	0
$986$	−72913.7	−2.35501
$987$	0	0
$988$	− 130062.i	− 4.18809i
$989$	51841.9	1.66681
$990$	0	0
$991$	39918.6	1.27957	0.639786	−	0.768553i	$-0.279023\pi$
$991$	39918.6	1.27957	0.639786	+	0.768553i	$0.279023\pi$
$992$	5508.20i	0.176296i
$993$	0	0
$994$	−54055.2	−1.72487
$995$	0	0
$996$	0	0
$997$	25670.3i	0.815432i	0.913109	+	0.407716i	$0.133675\pi$
$997$	25670.3i	0.815432i	−0.913109	+	0.407716i	$0.866325\pi$
$998$	− 23975.6i	− 0.760454i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.b.k.649.1		6
3.2	odd	2		675.4.b.l.649.6		6
5.2	odd	4		135.4.a.g.1.3	yes	3
5.3	odd	4		675.4.a.q.1.1		3
5.4	even	2	inner	675.4.b.k.649.6		6
15.2	even	4		135.4.a.f.1.1	✓	3
15.8	even	4		675.4.a.r.1.3		3
15.14	odd	2		675.4.b.l.649.1		6
20.7	even	4		2160.4.a.be.1.1		3
45.2	even	12		405.4.e.t.271.3		6
45.7	odd	12		405.4.e.r.271.1		6
45.22	odd	12		405.4.e.r.136.1		6
45.32	even	12		405.4.e.t.136.3		6
60.47	odd	4		2160.4.a.bm.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.f.1.1	✓	3	15.2	even	4
135.4.a.g.1.3	yes	3	5.2	odd	4
405.4.e.r.136.1		6	45.22	odd	12
405.4.e.r.271.1		6	45.7	odd	12
405.4.e.t.136.3		6	45.32	even	12
405.4.e.t.271.3		6	45.2	even	12
675.4.a.q.1.1		3	5.3	odd	4
675.4.a.r.1.3		3	15.8	even	4
675.4.b.k.649.1		6	1.1	even	1	trivial
675.4.b.k.649.6		6	5.4	even	2	inner
675.4.b.l.649.1		6	15.14	odd	2
675.4.b.l.649.6		6	3.2	odd	2
2160.4.a.be.1.1		3	20.7	even	4
2160.4.a.bm.1.1		3	60.47	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.20067i q^{2} -19.0470 q^{4} -24.4013i q^{7} +57.4517i q^{8} +O(q^{10})\)	\(q-5.20067i q^{2} -19.0470 q^{4} -24.4013i q^{7} +57.4517i q^{8} -28.9839 q^{11} -65.3919i q^{13} -126.903 q^{14} +146.411 q^{16} -68.1718i q^{17} -104.424 q^{19} +150.736i q^{22} -154.807i q^{23} -340.082 q^{26} +464.772i q^{28} +205.658 q^{29} -18.2497 q^{31} -301.824i q^{32} -354.539 q^{34} +337.613i q^{37} +543.076i q^{38} -195.969 q^{41} +334.882i q^{43} +552.055 q^{44} -805.098 q^{46} +5.00398i q^{47} -252.425 q^{49} +1245.52i q^{52} -319.965i q^{53} +1401.90 q^{56} -1069.56i q^{58} -430.611 q^{59} +594.581 q^{61} +94.9106i q^{62} -398.396 q^{64} -195.876i q^{67} +1298.47i q^{68} +425.955 q^{71} +929.193i q^{73} +1755.82 q^{74} +1988.96 q^{76} +707.245i q^{77} -24.4296 q^{79} +1019.17i q^{82} -545.859i q^{83} +1741.61 q^{86} -1665.17i q^{88} +84.1332 q^{89} -1595.65 q^{91} +2948.60i q^{92} +26.0241 q^{94} -827.613i q^{97} +1312.78i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 46 q^{4}+O(q^{10}) \) 6 * q - 46 * q^4	\( 6 q - 46 q^{4} - 76 q^{11} - 216 q^{14} + 382 q^{16} - 374 q^{19} - 832 q^{26} + 320 q^{29} + 454 q^{31} - 34 q^{34} + 676 q^{41} + 3272 q^{44} - 2850 q^{46} + 394 q^{49} + 2508 q^{56} + 280 q^{59} + 1190 q^{61} + 1836 q^{64} + 1204 q^{71} + 5756 q^{74} + 1050 q^{76} - 1258 q^{79} + 7460 q^{86} + 4308 q^{89} - 880 q^{91} + 2216 q^{94}+O(q^{100}) \) 6 * q - 46 * q^4 - 76 * q^11 - 216 * q^14 + 382 * q^16 - 374 * q^19 - 832 * q^26 + 320 * q^29 + 454 * q^31 - 34 * q^34 + 676 * q^41 + 3272 * q^44 - 2850 * q^46 + 394 * q^49 + 2508 * q^56 + 280 * q^59 + 1190 * q^61 + 1836 * q^64 + 1204 * q^71 + 5756 * q^74 + 1050 * q^76 - 1258 * q^79 + 7460 * q^86 + 4308 * q^89 - 880 * q^91 + 2216 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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