LMFDB - Embedded newform 378.5.b.b.323.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,5,Mod(323,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("378.323");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	378.b (of order $2$, degree $1$, 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:	$39.0738460457$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 92x^{6} + 2949x^{4} + 37548x^{2} + 142884 $ x^8 + 92x^6 + 2949x^4 + 37548*x^2 + 142884
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.3
Root			$-2.54824i$ of defining polynomial
Character	$\chi$	$=$	378.323
Dual form			378.5.b.b.323.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} +17.8674i q^{5} -18.5203 q^{7} +22.6274i q^{8} +O(q^{10})$	\(q-2.82843i q^{2} -8.00000 q^{4} +17.8674i q^{5} -18.5203 q^{7} +22.6274i q^{8} +50.5366 q^{10} -45.4134i q^{11} -155.432 q^{13} +52.3832i q^{14} +64.0000 q^{16} +114.905i q^{17} -17.2296 q^{19} -142.939i q^{20} -128.449 q^{22} +55.0943i q^{23} +305.756 q^{25} +439.629i q^{26} +148.162 q^{28} -578.935i q^{29} +514.064 q^{31} -181.019i q^{32} +325.001 q^{34} -330.909i q^{35} +1287.01 q^{37} +48.7326i q^{38} -404.293 q^{40} -2151.69i q^{41} +2349.59 q^{43} +363.307i q^{44} +155.830 q^{46} -1965.66i q^{47} +343.000 q^{49} -864.809i q^{50} +1243.46 q^{52} -3259.56i q^{53} +811.419 q^{55} -419.066i q^{56} -1637.48 q^{58} +276.141i q^{59} -438.948 q^{61} -1453.99i q^{62} -512.000 q^{64} -2777.17i q^{65} -1761.95 q^{67} -919.241i q^{68} -935.951 q^{70} +1378.26i q^{71} +1873.02 q^{73} -3640.20i q^{74} +137.837 q^{76} +841.068i q^{77} +4585.41 q^{79} +1143.51i q^{80} -6085.89 q^{82} -7058.33i q^{83} -2053.05 q^{85} -6645.64i q^{86} +1027.59 q^{88} -7138.01i q^{89} +2878.64 q^{91} -440.755i q^{92} -5559.72 q^{94} -307.848i q^{95} -1109.01 q^{97} -970.151i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{4}+O(q^{10}) $ 8 * q - 64 * q^4	$ 8 q - 64 q^{4} - 32 q^{10} + 296 q^{13} + 512 q^{16} - 80 q^{19} + 800 q^{22} - 1464 q^{25} - 1024 q^{31} + 896 q^{34} + 904 q^{37} + 256 q^{40} - 3224 q^{43} - 4384 q^{46} + 2744 q^{49} - 2368 q^{52} + 8312 q^{55} + 2944 q^{58} + 128 q^{61} - 4096 q^{64} - 10152 q^{67} + 1568 q^{70} + 10632 q^{73} + 640 q^{76} - 4072 q^{79} - 3552 q^{82} + 13448 q^{85} - 6400 q^{88} - 6664 q^{91} - 384 q^{94} + 35616 q^{97}+O(q^{100}) $ 8 * q - 64 * q^4 - 32 * q^10 + 296 * q^13 + 512 * q^16 - 80 * q^19 + 800 * q^22 - 1464 * q^25 - 1024 * q^31 + 896 * q^34 + 904 * q^37 + 256 * q^40 - 3224 * q^43 - 4384 * q^46 + 2744 * q^49 - 2368 * q^52 + 8312 * q^55 + 2944 * q^58 + 128 * q^61 - 4096 * q^64 - 10152 * q^67 + 1568 * q^70 + 10632 * q^73 + 640 * q^76 - 4072 * q^79 - 3552 * q^82 + 13448 * q^85 - 6400 * q^88 - 6664 * q^91 - 384 * q^94 + 35616 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.82843i	− 0.707107i
$2$	− 2.82843i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	17.8674i	0.714696i	0.933971	+	0.357348i	$0.116319\pi$
$5$	17.8674i	0.714696i	−0.933971	+	0.357348i	$0.883681\pi$
$6$	0	0
$7$	−18.5203	−0.377964
$7$	−18.5203	−0.377964
$8$	22.6274i	0.353553i
$9$	0	0
$10$	50.5366	0.505366
$11$	− 45.4134i	− 0.375317i	−0.982234	−	0.187659i	$-0.939910\pi$
$11$	− 45.4134i	− 0.375317i	0.982234	−	0.187659i	$-0.0600899\pi$
$12$	0	0
$13$	−155.432	−0.919717	−0.459858	−	0.887992i	$-0.652100\pi$
$13$	−155.432	−0.919717	−0.459858	+	0.887992i	$0.652100\pi$
$14$	52.3832i	0.267261i
$15$	0	0
$16$	64.0000	0.250000
$17$	114.905i	0.397596i	0.980041	+	0.198798i	$0.0637037\pi$
$17$	114.905i	0.397596i	−0.980041	+	0.198798i	$0.936296\pi$
$18$	0	0
$19$	−17.2296	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$19$	−17.2296	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$20$	− 142.939i	− 0.357348i
$21$	0	0
$22$	−128.449	−0.265390
$23$	55.0943i	0.104148i	0.998643	+	0.0520740i	$0.0165832\pi$
$23$	55.0943i	0.104148i	−0.998643	+	0.0520740i	$0.983417\pi$
$24$	0	0
$25$	305.756	0.489210
$26$	439.629i	0.650338i
$27$	0	0
$28$	148.162	0.188982
$29$	− 578.935i	− 0.688389i	−0.938898	−	0.344195i	$-0.888152\pi$
$29$	− 578.935i	− 0.688389i	0.938898	−	0.344195i	$-0.111848\pi$
$30$	0	0
$31$	514.064	0.534926	0.267463	−	0.963568i	$-0.413815\pi$
$31$	514.064	0.534926	0.267463	+	0.963568i	$0.413815\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	325.001	0.281143
$35$	− 330.909i	− 0.270130i
$36$	0	0
$37$	1287.01	0.940107	0.470053	−	0.882638i	$-0.344235\pi$
$37$	1287.01	0.940107	0.470053	+	0.882638i	$0.344235\pi$
$38$	48.7326i	0.0337483i
$39$	0	0
$40$	−404.293	−0.252683
$41$	− 2151.69i	− 1.28001i	−0.768373	−	0.640003i	$-0.778933\pi$
$41$	− 2151.69i	− 1.28001i	0.768373	−	0.640003i	$-0.221067\pi$
$42$	0	0
$43$	2349.59	1.27073	0.635367	−	0.772210i	$-0.280849\pi$
$43$	2349.59	1.27073	0.635367	+	0.772210i	$0.280849\pi$
$44$	363.307i	0.187659i
$45$	0	0
$46$	155.830	0.0736438
$47$	− 1965.66i	− 0.889841i	−0.895570	−	0.444920i	$-0.853232\pi$
$47$	− 1965.66i	− 0.889841i	0.895570	−	0.444920i	$-0.146768\pi$
$48$	0	0
$49$	343.000	0.142857
$50$	− 864.809i	− 0.345924i
$51$	0	0
$52$	1243.46	0.459858
$53$	− 3259.56i	− 1.16040i	−0.814474	−	0.580200i	$-0.802974\pi$
$53$	− 3259.56i	− 1.16040i	0.814474	−	0.580200i	$-0.197026\pi$
$54$	0	0
$55$	811.419	0.268238
$56$	− 419.066i	− 0.133631i
$57$	0	0
$58$	−1637.48	−0.486765
$59$	276.141i	0.0793281i	0.999213	+	0.0396641i	$0.0126288\pi$
$59$	276.141i	0.0793281i	−0.999213	+	0.0396641i	$0.987371\pi$
$60$	0	0
$61$	−438.948	−0.117965	−0.0589826	−	0.998259i	$-0.518786\pi$
$61$	−438.948	−0.117965	−0.0589826	+	0.998259i	$0.518786\pi$
$62$	− 1453.99i	− 0.378250i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 2777.17i	− 0.657318i
$66$	0	0
$67$	−1761.95	−0.392503	−0.196251	−	0.980554i	$-0.562877\pi$
$67$	−1761.95	−0.392503	−0.196251	+	0.980554i	$0.562877\pi$
$68$	− 919.241i	− 0.198798i
$69$	0	0
$70$	−935.951	−0.191010
$71$	1378.26i	0.273409i	0.990612	+	0.136705i	$0.0436511\pi$
$71$	1378.26i	0.273409i	−0.990612	+	0.136705i	$0.956349\pi$
$72$	0	0
$73$	1873.02	0.351477	0.175738	−	0.984437i	$-0.443769\pi$
$73$	1873.02	0.351477	0.175738	+	0.984437i	$0.443769\pi$
$74$	− 3640.20i	− 0.664756i
$75$	0	0
$76$	137.837	0.0238637
$77$	841.068i	0.141857i
$78$	0	0
$79$	4585.41	0.734724	0.367362	−	0.930078i	$-0.380261\pi$
$79$	4585.41	0.734724	0.367362	+	0.930078i	$0.380261\pi$
$80$	1143.51i	0.178674i
$81$	0	0
$82$	−6085.89	−0.905100
$83$	− 7058.33i	− 1.02458i	−0.858813	−	0.512290i	$-0.828797\pi$
$83$	− 7058.33i	− 1.02458i	0.858813	−	0.512290i	$-0.171203\pi$
$84$	0	0
$85$	−2053.05	−0.284160
$86$	− 6645.64i	− 0.898545i
$87$	0	0
$88$	1027.59	0.132695
$89$	− 7138.01i	− 0.901150i	−0.892739	−	0.450575i	$-0.851219\pi$
$89$	− 7138.01i	− 0.901150i	0.892739	−	0.450575i	$-0.148781\pi$
$90$	0	0
$91$	2878.64	0.347620
$92$	− 440.755i	− 0.0520740i
$93$	0	0
$94$	−5559.72	−0.629213
$95$	− 307.848i	− 0.0341105i
$96$	0	0
$97$	−1109.01	−0.117867	−0.0589336	−	0.998262i	$-0.518770\pi$
$97$	−1109.01	−0.117867	−0.0589336	+	0.998262i	$0.518770\pi$
$98$	− 970.151i	− 0.101015i
$99$	0	0
$100$	−2446.05	−0.244605
$101$	− 15526.5i	− 1.52205i	−0.648721	−	0.761027i	$-0.724696\pi$
$101$	− 15526.5i	− 1.52205i	0.648721	−	0.761027i	$-0.275304\pi$
$102$	0	0
$103$	−17118.9	−1.61362	−0.806809	−	0.590813i	$-0.798807\pi$
$103$	−17118.9	−1.61362	−0.806809	+	0.590813i	$0.798807\pi$
$104$	− 3517.03i	− 0.325169i
$105$	0	0
$106$	−9219.44	−0.820526
$107$	4086.42i	0.356924i	0.983947	+	0.178462i	$0.0571121\pi$
$107$	4086.42i	0.356924i	−0.983947	+	0.178462i	$0.942888\pi$
$108$	0	0
$109$	9630.93	0.810616	0.405308	−	0.914180i	$-0.367164\pi$
$109$	9630.93	0.810616	0.405308	+	0.914180i	$0.367164\pi$
$110$	− 2295.04i	− 0.189673i
$111$	0	0
$112$	−1185.30	−0.0944911
$113$	5446.96i	0.426577i	0.976989	+	0.213288i	$0.0684174\pi$
$113$	5446.96i	0.426577i	−0.976989	+	0.213288i	$0.931583\pi$
$114$	0	0
$115$	−984.392	−0.0744342
$116$	4631.48i	0.344195i
$117$	0	0
$118$	781.045	0.0560935
$119$	− 2128.07i	− 0.150277i
$120$	0	0
$121$	12578.6	0.859137
$122$	1241.53i	0.0834140i
$123$	0	0
$124$	−4112.51	−0.267463
$125$	16630.2i	1.06433i
$126$	0	0
$127$	3615.47	0.224160	0.112080	−	0.993699i	$-0.464249\pi$
$127$	3615.47	0.224160	0.112080	+	0.993699i	$0.464249\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	−7855.02	−0.464794
$131$	− 13601.5i	− 0.792584i	−0.918125	−	0.396292i	$-0.870297\pi$
$131$	− 13601.5i	− 0.792584i	0.918125	−	0.396292i	$-0.129703\pi$
$132$	0	0
$133$	319.096	0.0180392
$134$	4983.53i	0.277541i
$135$	0	0
$136$	−2600.01	−0.140571
$137$	− 1456.98i	− 0.0776268i	−0.999246	−	0.0388134i	$-0.987642\pi$
$137$	− 1456.98i	− 0.0776268i	0.999246	−	0.0388134i	$-0.0123578\pi$
$138$	0	0
$139$	−2883.53	−0.149243	−0.0746217	−	0.997212i	$-0.523775\pi$
$139$	−2883.53	−0.149243	−0.0746217	+	0.997212i	$0.523775\pi$
$140$	2647.27i	0.135065i
$141$	0	0
$142$	3898.29	0.193329
$143$	7058.71i	0.345186i
$144$	0	0
$145$	10344.1	0.491989
$146$	− 5297.70i	− 0.248531i
$147$	0	0
$148$	−10296.0	−0.470053
$149$	− 40158.5i	− 1.80886i	−0.426623	−	0.904430i	$-0.640297\pi$
$149$	− 40158.5i	− 1.80886i	0.426623	−	0.904430i	$-0.359703\pi$
$150$	0	0
$151$	−3953.92	−0.173410	−0.0867051	−	0.996234i	$-0.527634\pi$
$151$	−3953.92	−0.173410	−0.0867051	+	0.996234i	$0.527634\pi$
$152$	− 389.861i	− 0.0168742i
$153$	0	0
$154$	2378.90	0.100308
$155$	9184.98i	0.382309i
$156$	0	0
$157$	15472.7	0.627722	0.313861	−	0.949469i	$-0.398377\pi$
$157$	15472.7	0.627722	0.313861	+	0.949469i	$0.398377\pi$
$158$	− 12969.5i	− 0.519528i
$159$	0	0
$160$	3234.34	0.126342
$161$	− 1020.36i	− 0.0393643i
$162$	0	0
$163$	37395.2	1.40748	0.703738	−	0.710460i	$-0.251513\pi$
$163$	37395.2	1.40748	0.703738	+	0.710460i	$0.251513\pi$
$164$	17213.5i	0.640003i
$165$	0	0
$166$	−19964.0	−0.724487
$167$	5416.39i	0.194212i	0.995274	+	0.0971061i	$0.0309586\pi$
$167$	5416.39i	0.194212i	−0.995274	+	0.0971061i	$0.969041\pi$
$168$	0	0
$169$	−4401.84	−0.154121
$170$	5806.92i	0.200931i
$171$	0	0
$172$	−18796.7	−0.635367
$173$	− 14162.2i	− 0.473194i	−0.971608	−	0.236597i	$-0.923968\pi$
$173$	− 14162.2i	− 0.473194i	0.971608	−	0.236597i	$-0.0760321\pi$
$174$	0	0
$175$	−5662.69	−0.184904
$176$	− 2906.46i	− 0.0938294i
$177$	0	0
$178$	−20189.3	−0.637209
$179$	16147.2i	0.503955i	0.967733	+	0.251977i	$0.0810809\pi$
$179$	16147.2i	0.503955i	−0.967733	+	0.251977i	$0.918919\pi$
$180$	0	0
$181$	26540.9	0.810138	0.405069	−	0.914286i	$-0.367248\pi$
$181$	26540.9	0.810138	0.405069	+	0.914286i	$0.367248\pi$
$182$	− 8142.03i	− 0.245805i
$183$	0	0
$184$	−1246.64	−0.0368219
$185$	22995.4i	0.671890i
$186$	0	0
$187$	5218.23	0.149225
$188$	15725.3i	0.444920i
$189$	0	0
$190$	−870.725	−0.0241198
$191$	2149.99i	0.0589344i	0.999566	+	0.0294672i	$0.00938106\pi$
$191$	2149.99i	0.0589344i	−0.999566	+	0.0294672i	$0.990619\pi$
$192$	0	0
$193$	26344.9	0.707265	0.353633	−	0.935384i	$-0.384946\pi$
$193$	26344.9	0.707265	0.353633	+	0.935384i	$0.384946\pi$
$194$	3136.76i	0.0833448i
$195$	0	0
$196$	−2744.00	−0.0714286
$197$	62745.0i	1.61676i	0.588658	+	0.808382i	$0.299656\pi$
$197$	62745.0i	1.61676i	−0.588658	+	0.808382i	$0.700344\pi$
$198$	0	0
$199$	−41364.4	−1.04453	−0.522264	−	0.852784i	$-0.674912\pi$
$199$	−41364.4	−1.04453	−0.522264	+	0.852784i	$0.674912\pi$
$200$	6918.48i	0.172962i
$201$	0	0
$202$	−43915.5	−1.07625
$203$	10722.0i	0.260187i
$204$	0	0
$205$	38445.1	0.914814
$206$	48419.5i	1.14100i
$207$	0	0
$208$	−9947.66	−0.229929
$209$	782.454i	0.0179129i
$210$	0	0
$211$	60192.2	1.35200	0.675998	−	0.736904i	$-0.263713\pi$
$211$	60192.2	1.35200	0.675998	+	0.736904i	$0.263713\pi$
$212$	26076.5i	0.580200i
$213$	0	0
$214$	11558.1	0.252383
$215$	41981.0i	0.908188i
$216$	0	0
$217$	−9520.60	−0.202183
$218$	− 27240.4i	− 0.573192i
$219$	0	0
$220$	−6491.35	−0.134119
$221$	− 17860.0i	− 0.365675i
$222$	0	0
$223$	54766.3	1.10129	0.550647	−	0.834738i	$-0.314381\pi$
$223$	54766.3	1.10129	0.550647	+	0.834738i	$0.314381\pi$
$224$	3352.53i	0.0668153i
$225$	0	0
$226$	15406.3	0.301635
$227$	23659.5i	0.459149i	0.973291	+	0.229575i	$0.0737335\pi$
$227$	23659.5i	0.459149i	−0.973291	+	0.229575i	$0.926267\pi$
$228$	0	0
$229$	−18364.7	−0.350198	−0.175099	−	0.984551i	$-0.556025\pi$
$229$	−18364.7	−0.350198	−0.175099	+	0.984551i	$0.556025\pi$
$230$	2784.28i	0.0526329i
$231$	0	0
$232$	13099.8	0.243382
$233$	71340.7i	1.31409i	0.753851	+	0.657046i	$0.228194\pi$
$233$	71340.7i	1.31409i	−0.753851	+	0.657046i	$0.771806\pi$
$234$	0	0
$235$	35121.2	0.635965
$236$	− 2209.13i	− 0.0396641i
$237$	0	0
$238$	−6019.10	−0.106262
$239$	43626.2i	0.763751i	0.924214	+	0.381876i	$0.124722\pi$
$239$	43626.2i	0.763751i	−0.924214	+	0.381876i	$0.875278\pi$
$240$	0	0
$241$	12447.2	0.214308	0.107154	−	0.994242i	$-0.465826\pi$
$241$	12447.2	0.214308	0.107154	+	0.994242i	$0.465826\pi$
$242$	− 35577.7i	− 0.607501i
$243$	0	0
$244$	3511.59	0.0589826
$245$	6128.52i	0.102099i
$246$	0	0
$247$	2678.03	0.0438957
$248$	11631.9i	0.189125i
$249$	0	0
$250$	47037.3	0.752596
$251$	78794.7i	1.25069i	0.780348	+	0.625345i	$0.215042\pi$
$251$	78794.7i	1.25069i	−0.780348	+	0.625345i	$0.784958\pi$
$252$	0	0
$253$	2502.02	0.0390886
$254$	− 10226.1i	− 0.158505i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 28911.2i	− 0.437723i	−0.975756	−	0.218861i	$-0.929766\pi$
$257$	− 28911.2i	− 0.437723i	0.975756	−	0.218861i	$-0.0702343\pi$
$258$	0	0
$259$	−23835.7	−0.355327
$260$	22217.3i	0.328659i
$261$	0	0
$262$	−38470.9	−0.560442
$263$	− 65442.0i	− 0.946118i	−0.881031	−	0.473059i	$-0.843150\pi$
$263$	− 65442.0i	− 0.946118i	0.881031	−	0.473059i	$-0.156850\pi$
$264$	0	0
$265$	58239.9	0.829333
$266$	− 902.540i	− 0.0127557i
$267$	0	0
$268$	14095.6	0.196251
$269$	104321.i	1.44167i	0.693107	+	0.720835i	$0.256241\pi$
$269$	104321.i	1.44167i	−0.693107	+	0.720835i	$0.743759\pi$
$270$	0	0
$271$	−31635.6	−0.430762	−0.215381	−	0.976530i	$-0.569099\pi$
$271$	−31635.6	−0.430762	−0.215381	+	0.976530i	$0.569099\pi$
$272$	7353.93i	0.0993989i
$273$	0	0
$274$	−4120.96	−0.0548905
$275$	− 13885.4i	− 0.183609i
$276$	0	0
$277$	−116481.	−1.51809	−0.759045	−	0.651039i	$-0.774334\pi$
$277$	−116481.	−1.51809	−0.759045	+	0.651039i	$0.774334\pi$
$278$	8155.86i	0.105531i
$279$	0	0
$280$	7487.61	0.0955052
$281$	− 17406.9i	− 0.220449i	−0.993907	−	0.110224i	$-0.964843\pi$
$281$	− 17406.9i	− 0.220449i	0.993907	−	0.110224i	$-0.0351570\pi$
$282$	0	0
$283$	−10431.0	−0.130242	−0.0651211	−	0.997877i	$-0.520743\pi$
$283$	−10431.0	−0.130242	−0.0651211	+	0.997877i	$0.520743\pi$
$284$	− 11026.0i	− 0.136705i
$285$	0	0
$286$	19965.0	0.244083
$287$	39849.8i	0.483796i
$288$	0	0
$289$	70317.8	0.841918
$290$	− 29257.4i	− 0.347888i
$291$	0	0
$292$	−14984.1	−0.175738
$293$	64199.7i	0.747822i	0.927465	+	0.373911i	$0.121983\pi$
$293$	64199.7i	0.747822i	−0.927465	+	0.373911i	$0.878017\pi$
$294$	0	0
$295$	−4933.92	−0.0566955
$296$	29121.6i	0.332378i
$297$	0	0
$298$	−113585.	−1.27906
$299$	− 8563.43i	− 0.0957867i
$300$	0	0
$301$	−43515.0	−0.480292
$302$	11183.4i	0.122619i
$303$	0	0
$304$	−1102.69	−0.0119318
$305$	− 7842.86i	− 0.0843092i
$306$	0	0
$307$	−65196.6	−0.691749	−0.345874	−	0.938281i	$-0.612418\pi$
$307$	−65196.6	−0.691749	−0.345874	+	0.938281i	$0.612418\pi$
$308$	− 6728.55i	− 0.0709283i
$309$	0	0
$310$	25979.1	0.270334
$311$	− 151896.i	− 1.57046i	−0.619206	−	0.785229i	$-0.712545\pi$
$311$	− 151896.i	− 1.57046i	0.619206	−	0.785229i	$-0.287455\pi$
$312$	0	0
$313$	−54154.6	−0.552773	−0.276387	−	0.961047i	$-0.589137\pi$
$313$	−54154.6	−0.552773	−0.276387	+	0.961047i	$0.589137\pi$
$314$	− 43763.5i	− 0.443867i
$315$	0	0
$316$	−36683.3	−0.367362
$317$	− 137384.i	− 1.36715i	−0.729878	−	0.683577i	$-0.760423\pi$
$317$	− 137384.i	− 1.36715i	0.729878	−	0.683577i	$-0.239577\pi$
$318$	0	0
$319$	−26291.4	−0.258364
$320$	− 9148.10i	− 0.0893370i
$321$	0	0
$322$	−2886.02	−0.0278347
$323$	− 1979.77i	− 0.0189762i
$324$	0	0
$325$	−47524.4	−0.449935
$326$	− 105770.i	− 0.995236i
$327$	0	0
$328$	48687.2	0.452550
$329$	36404.5i	0.336328i
$330$	0	0
$331$	−163805.	−1.49510	−0.747550	−	0.664205i	$-0.768770\pi$
$331$	−163805.	−1.49510	−0.747550	+	0.664205i	$0.768770\pi$
$332$	56466.6i	0.512290i
$333$	0	0
$334$	15319.9	0.137329
$335$	− 31481.4i	− 0.280520i
$336$	0	0
$337$	164077.	1.44473	0.722367	−	0.691510i	$-0.243054\pi$
$337$	164077.	1.44473	0.722367	+	0.691510i	$0.243054\pi$
$338$	12450.3i	0.108980i
$339$	0	0
$340$	16424.4	0.142080
$341$	− 23345.4i	− 0.200767i
$342$	0	0
$343$	−6352.45	−0.0539949
$344$	53165.1i	0.449272i
$345$	0	0
$346$	−40056.8	−0.334599
$347$	− 72809.2i	− 0.604683i	−0.953200	−	0.302341i	$-0.902232\pi$
$347$	− 72809.2i	− 0.604683i	0.953200	−	0.302341i	$-0.0977682\pi$
$348$	0	0
$349$	15848.6	0.130118	0.0650592	−	0.997881i	$-0.479276\pi$
$349$	15848.6	0.130118	0.0650592	+	0.997881i	$0.479276\pi$
$350$	16016.5i	0.130747i
$351$	0	0
$352$	−8220.71	−0.0663474
$353$	− 40491.9i	− 0.324951i	−0.986713	−	0.162476i	$-0.948052\pi$
$353$	− 40491.9i	− 0.324951i	0.986713	−	0.162476i	$-0.0519479\pi$
$354$	0	0
$355$	−24625.8	−0.195404
$356$	57104.1i	0.450575i
$357$	0	0
$358$	45671.2	0.356350
$359$	− 30479.4i	− 0.236492i	−0.992984	−	0.118246i	$-0.962273\pi$
$359$	− 30479.4i	− 0.236492i	0.992984	−	0.118246i	$-0.0377272\pi$
$360$	0	0
$361$	−130024.	−0.997722
$362$	− 75069.1i	− 0.572854i
$363$	0	0
$364$	−23029.2	−0.173810
$365$	33466.0i	0.251199i
$366$	0	0
$367$	−252948.	−1.87801	−0.939007	−	0.343898i	$-0.888253\pi$
$367$	−252948.	−1.87801	−0.939007	+	0.343898i	$0.888253\pi$
$368$	3526.04i	0.0260370i
$369$	0	0
$370$	65040.9	0.475098
$371$	60367.9i	0.438590i
$372$	0	0
$373$	131647.	0.946221	0.473110	−	0.881003i	$-0.343131\pi$
$373$	131647.	0.946221	0.473110	+	0.881003i	$0.343131\pi$
$374$	− 14759.4i	− 0.105518i
$375$	0	0
$376$	44477.8	0.314606
$377$	89985.1i	0.633123i
$378$	0	0
$379$	−64033.9	−0.445791	−0.222896	−	0.974842i	$-0.571551\pi$
$379$	−64033.9	−0.445791	−0.222896	+	0.974842i	$0.571551\pi$
$380$	2462.78i	0.0170553i
$381$	0	0
$382$	6081.08	0.0416729
$383$	− 272341.i	− 1.85659i	−0.371849	−	0.928293i	$-0.621276\pi$
$383$	− 272341.i	− 1.85659i	0.371849	−	0.928293i	$-0.378724\pi$
$384$	0	0
$385$	−15027.7	−0.101384
$386$	− 74514.7i	− 0.500112i
$387$	0	0
$388$	8872.11	0.0589336
$389$	− 285818.i	− 1.88882i	−0.328775	−	0.944408i	$-0.606636\pi$
$389$	− 285818.i	− 1.88882i	0.328775	−	0.944408i	$-0.393364\pi$
$390$	0	0
$391$	−6330.62	−0.0414088
$392$	7761.20i	0.0505076i
$393$	0	0
$394$	177470.	1.14323
$395$	81929.4i	0.525104i
$396$	0	0
$397$	201945.	1.28130	0.640652	−	0.767831i	$-0.278664\pi$
$397$	201945.	1.28130	0.640652	+	0.767831i	$0.278664\pi$
$398$	116996.i	0.738593i
$399$	0	0
$400$	19568.4	0.122303
$401$	− 58597.3i	− 0.364409i	−0.983261	−	0.182204i	$-0.941677\pi$
$401$	− 58597.3i	− 0.364409i	0.983261	−	0.182204i	$-0.0583232\pi$
$402$	0	0
$403$	−79902.1	−0.491981
$404$	124212.i	0.761027i
$405$	0	0
$406$	30326.5	0.183980
$407$	− 58447.3i	− 0.352838i
$408$	0	0
$409$	179325.	1.07200	0.535999	−	0.844219i	$-0.319935\pi$
$409$	179325.	1.07200	0.535999	+	0.844219i	$0.319935\pi$
$410$	− 108739.i	− 0.646871i
$411$	0	0
$412$	136951.	0.806809
$413$	− 5114.21i	− 0.0299832i
$414$	0	0
$415$	126114.	0.732263
$416$	28136.2i	0.162585i
$417$	0	0
$418$	2213.11	0.0126663
$419$	153806.i	0.876085i	0.898954	+	0.438042i	$0.144328\pi$
$419$	153806.i	0.876085i	−0.898954	+	0.438042i	$0.855672\pi$
$420$	0	0
$421$	−91254.8	−0.514863	−0.257432	−	0.966297i	$-0.582876\pi$
$421$	−91254.8	−0.514863	−0.257432	+	0.966297i	$0.582876\pi$
$422$	− 170249.i	− 0.956005i
$423$	0	0
$424$	73755.5	0.410263
$425$	35133.0i	0.194508i
$426$	0	0
$427$	8129.44	0.0445866
$428$	− 32691.4i	− 0.178462i
$429$	0	0
$430$	118740.	0.642186
$431$	− 348521.i	− 1.87618i	−0.346394	−	0.938089i	$-0.612594\pi$
$431$	− 348521.i	− 1.87618i	0.346394	−	0.938089i	$-0.387406\pi$
$432$	0	0
$433$	−86162.8	−0.459562	−0.229781	−	0.973242i	$-0.573801\pi$
$433$	−86162.8	−0.459562	−0.229781	+	0.973242i	$0.573801\pi$
$434$	26928.3i	0.142965i
$435$	0	0
$436$	−77047.4	−0.405308
$437$	− 949.252i	− 0.00497071i
$438$	0	0
$439$	−284089.	−1.47409	−0.737047	−	0.675842i	$-0.763780\pi$
$439$	−284089.	−1.47409	−0.737047	+	0.675842i	$0.763780\pi$
$440$	18360.3i	0.0948364i
$441$	0	0
$442$	−50515.6	−0.258572
$443$	− 77877.1i	− 0.396828i	−0.980118	−	0.198414i	$-0.936421\pi$
$443$	− 77877.1i	− 0.396828i	0.980118	−	0.198414i	$-0.0635790\pi$
$444$	0	0
$445$	127538.	0.644048
$446$	− 154902.i	− 0.778733i
$447$	0	0
$448$	9482.37	0.0472456
$449$	− 87226.0i	− 0.432667i	−0.976320	−	0.216333i	$-0.930590\pi$
$449$	− 87226.0i	− 0.432667i	0.976320	−	0.216333i	$-0.0694098\pi$
$450$	0	0
$451$	−97715.5	−0.480408
$452$	− 43575.7i	− 0.213288i
$453$	0	0
$454$	66919.2	0.324667
$455$	51433.9i	0.248443i
$456$	0	0
$457$	60354.0	0.288984	0.144492	−	0.989506i	$-0.453845\pi$
$457$	60354.0	0.288984	0.144492	+	0.989506i	$0.453845\pi$
$458$	51943.3i	0.247627i
$459$	0	0
$460$	7875.14	0.0372171
$461$	136850.i	0.643938i	0.946750	+	0.321969i	$0.104345\pi$
$461$	136850.i	0.643938i	−0.946750	+	0.321969i	$0.895655\pi$
$462$	0	0
$463$	167651.	0.782068	0.391034	−	0.920376i	$-0.372118\pi$
$463$	167651.	0.782068	0.391034	+	0.920376i	$0.372118\pi$
$464$	− 37051.8i	− 0.172097i
$465$	0	0
$466$	201782.	0.929203
$467$	− 188881.i	− 0.866074i	−0.901376	−	0.433037i	$-0.857442\pi$
$467$	− 188881.i	− 0.866074i	0.901376	−	0.433037i	$-0.142558\pi$
$468$	0	0
$469$	32631.7	0.148352
$470$	− 99337.7i	− 0.449696i
$471$	0	0
$472$	−6248.36	−0.0280467
$473$	− 106703.i	− 0.476929i
$474$	0	0
$475$	−5268.05	−0.0233487
$476$	17024.6i	0.0751385i
$477$	0	0
$478$	123394.	0.540054
$479$	396256.i	1.72705i	0.504305	+	0.863526i	$0.331749\pi$
$479$	396256.i	1.72705i	−0.504305	+	0.863526i	$0.668251\pi$
$480$	0	0
$481$	−200042.	−0.864632
$482$	− 35206.1i	− 0.151539i
$483$	0	0
$484$	−100629.	−0.429568
$485$	− 19815.2i	− 0.0842392i
$486$	0	0
$487$	−205617.	−0.866962	−0.433481	−	0.901163i	$-0.642715\pi$
$487$	−205617.	−0.866962	−0.433481	+	0.901163i	$0.642715\pi$
$488$	− 9932.27i	− 0.0417070i
$489$	0	0
$490$	17334.1	0.0721952
$491$	− 146443.i	− 0.607445i	−0.952761	−	0.303723i	$-0.901770\pi$
$491$	− 146443.i	− 0.607445i	0.952761	−	0.303723i	$-0.0982296\pi$
$492$	0	0
$493$	66522.6	0.273700
$494$	− 7574.61i	− 0.0310389i
$495$	0	0
$496$	32900.1	0.133732
$497$	− 25525.6i	− 0.103339i
$498$	0	0
$499$	−266336.	−1.06962	−0.534810	−	0.844973i	$-0.679617\pi$
$499$	−266336.	−1.06962	−0.534810	+	0.844973i	$0.679617\pi$
$500$	− 133042.i	− 0.532166i
$501$	0	0
$502$	222865.	0.884371
$503$	8405.60i	0.0332225i	0.999862	+	0.0166113i	$0.00528777\pi$
$503$	8405.60i	0.0332225i	−0.999862	+	0.0166113i	$0.994712\pi$
$504$	0	0
$505$	277417.	1.08780
$506$	− 7076.79i	− 0.0276398i
$507$	0	0
$508$	−28923.8	−0.112080
$509$	106126.i	0.409624i	0.978801	+	0.204812i	$0.0656584\pi$
$509$	106126.i	0.409624i	−0.978801	+	0.204812i	$0.934342\pi$
$510$	0	0
$511$	−34688.8	−0.132846
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	−81773.1	−0.309517
$515$	− 305870.i	− 1.15325i
$516$	0	0
$517$	−89267.3	−0.333973
$518$	67417.5i	0.251254i
$519$	0	0
$520$	62840.1	0.232397
$521$	68150.9i	0.251071i	0.992089	+	0.125535i	$0.0400648\pi$
$521$	68150.9i	0.251071i	−0.992089	+	0.125535i	$0.959935\pi$
$522$	0	0
$523$	−117480.	−0.429496	−0.214748	−	0.976669i	$-0.568893\pi$
$523$	−117480.	−0.429496	−0.214748	+	0.976669i	$0.568893\pi$
$524$	108812.i	0.396292i
$525$	0	0
$526$	−185098.	−0.669006
$527$	59068.6i	0.212684i
$528$	0	0
$529$	276806.	0.989153
$530$	− 164727.i	− 0.586427i
$531$	0	0
$532$	−2552.77	−0.00901962
$533$	334442.i	1.17724i
$534$	0	0
$535$	−73013.7	−0.255092
$536$	− 39868.3i	− 0.138771i
$537$	0	0
$538$	295063.	1.01941
$539$	− 15576.8i	− 0.0536168i
$540$	0	0
$541$	78291.9	0.267499	0.133750	−	0.991015i	$-0.457298\pi$
$541$	78291.9	0.267499	0.133750	+	0.991015i	$0.457298\pi$
$542$	89478.9i	0.304594i
$543$	0	0
$544$	20800.0	0.0702856
$545$	172080.i	0.579344i
$546$	0	0
$547$	−294769.	−0.985160	−0.492580	−	0.870267i	$-0.663946\pi$
$547$	−294769.	−0.985160	−0.492580	+	0.870267i	$0.663946\pi$
$548$	11655.8i	0.0388134i
$549$	0	0
$550$	−39273.9	−0.129831
$551$	9974.81i	0.0328550i
$552$	0	0
$553$	−84923.0	−0.277700
$554$	329459.i	1.07345i
$555$	0	0
$556$	23068.3	0.0746217
$557$	− 477782.i	− 1.54000i	−0.638047	−	0.769998i	$-0.720257\pi$
$557$	− 477782.i	− 1.54000i	0.638047	−	0.769998i	$-0.279743\pi$
$558$	0	0
$559$	−365202.	−1.16872
$560$	− 21178.2i	− 0.0675324i
$561$	0	0
$562$	−49234.1	−0.155881
$563$	450077.i	1.41994i	0.704232	+	0.709970i	$0.251292\pi$
$563$	450077.i	1.41994i	−0.704232	+	0.709970i	$0.748708\pi$
$564$	0	0
$565$	−97323.0	−0.304873
$566$	29503.2i	0.0920951i
$567$	0	0
$568$	−31186.4	−0.0966647
$569$	2913.20i	0.00899800i	0.999990	+	0.00449900i	$0.00143208\pi$
$569$	2913.20i	0.00899800i	−0.999990	+	0.00449900i	$0.998568\pi$
$570$	0	0
$571$	108683.	0.333342	0.166671	−	0.986013i	$-0.446698\pi$
$571$	108683.	0.333342	0.166671	+	0.986013i	$0.446698\pi$
$572$	− 56469.6i	− 0.172593i
$573$	0	0
$574$	112712.	0.342096
$575$	16845.4i	0.0509503i
$576$	0	0
$577$	388049.	1.16556	0.582780	−	0.812630i	$-0.301965\pi$
$577$	388049.	1.16556	0.582780	+	0.812630i	$0.301965\pi$
$578$	− 198889.i	− 0.595326i
$579$	0	0
$580$	−82752.5	−0.245994
$581$	130722.i	0.387255i
$582$	0	0
$583$	−148028.	−0.435518
$584$	42381.6i	0.124266i
$585$	0	0
$586$	181584.	0.528790
$587$	541195.i	1.57064i	0.619087	+	0.785322i	$0.287503\pi$
$587$	541195.i	1.57064i	−0.619087	+	0.785322i	$0.712497\pi$
$588$	0	0
$589$	−8857.11	−0.0255306
$590$	13955.2i	0.0400897i
$591$	0	0
$592$	82368.4	0.235027
$593$	− 545815.i	− 1.55216i	−0.630636	−	0.776079i	$-0.717206\pi$
$593$	− 545815.i	− 1.55216i	0.630636	−	0.776079i	$-0.282794\pi$
$594$	0	0
$595$	38023.1	0.107402
$596$	321268.i	0.904430i
$597$	0	0
$598$	−24221.0	−0.0677315
$599$	− 30038.9i	− 0.0837202i	−0.999123	−	0.0418601i	$-0.986672\pi$
$599$	− 30038.9i	− 0.0837202i	0.999123	−	0.0418601i	$-0.0133284\pi$
$600$	0	0
$601$	−250122.	−0.692473	−0.346237	−	0.938147i	$-0.612541\pi$
$601$	−250122.	−0.692473	−0.346237	+	0.938147i	$0.612541\pi$
$602$	123079.i	0.339618i
$603$	0	0
$604$	31631.4	0.0867051
$605$	224747.i	0.614021i
$606$	0	0
$607$	−270171.	−0.733265	−0.366632	−	0.930366i	$-0.619489\pi$
$607$	−270171.	−0.733265	−0.366632	+	0.930366i	$0.619489\pi$
$608$	3118.89i	0.00843709i
$609$	0	0
$610$	−22183.0	−0.0596156
$611$	305527.i	0.818402i
$612$	0	0
$613$	378425.	1.00707	0.503535	−	0.863975i	$-0.332033\pi$
$613$	378425.	1.00707	0.503535	+	0.863975i	$0.332033\pi$
$614$	184404.i	0.489140i
$615$	0	0
$616$	−19031.2	−0.0501539
$617$	580490.i	1.52484i	0.647083	+	0.762420i	$0.275989\pi$
$617$	580490.i	1.52484i	−0.647083	+	0.762420i	$0.724011\pi$
$618$	0	0
$619$	196149.	0.511922	0.255961	−	0.966687i	$-0.417608\pi$
$619$	196149.	0.511922	0.255961	+	0.966687i	$0.417608\pi$
$620$	− 73479.9i	− 0.191155i
$621$	0	0
$622$	−429627.	−1.11048
$623$	132198.i	0.340603i
$624$	0	0
$625$	−106040.	−0.271463
$626$	153172.i	0.390870i
$627$	0	0
$628$	−123782.	−0.313861
$629$	147884.i	0.373782i
$630$	0	0
$631$	69136.2	0.173639	0.0868194	−	0.996224i	$-0.472330\pi$
$631$	69136.2	0.173639	0.0868194	+	0.996224i	$0.472330\pi$
$632$	103756.i	0.259764i
$633$	0	0
$634$	−388581.	−0.966724
$635$	64599.1i	0.160206i
$636$	0	0
$637$	−53313.2	−0.131388
$638$	74363.4i	0.182691i
$639$	0	0
$640$	−25874.7	−0.0631708
$641$	− 426696.i	− 1.03849i	−0.854625	−	0.519245i	$-0.826213\pi$
$641$	− 426696.i	− 1.03849i	0.854625	−	0.519245i	$-0.173787\pi$
$642$	0	0
$643$	−129308.	−0.312755	−0.156377	−	0.987697i	$-0.549982\pi$
$643$	−129308.	−0.312755	−0.156377	+	0.987697i	$0.549982\pi$
$644$	8162.89i	0.0196821i
$645$	0	0
$646$	−5599.63	−0.0134182
$647$	71367.5i	0.170487i	0.996360	+	0.0852437i	$0.0271669\pi$
$647$	71367.5i	0.170487i	−0.996360	+	0.0852437i	$0.972833\pi$
$648$	0	0
$649$	12540.5	0.0297732
$650$	134419.i	0.318152i
$651$	0	0
$652$	−299162.	−0.703738
$653$	− 308085.i	− 0.722511i	−0.932467	−	0.361256i	$-0.882348\pi$
$653$	− 308085.i	− 0.722511i	0.932467	−	0.361256i	$-0.117652\pi$
$654$	0	0
$655$	243024.	0.566456
$656$	− 137708.i	− 0.320001i
$657$	0	0
$658$	102967.	0.237820
$659$	576984.i	1.32860i	0.747468	+	0.664298i	$0.231269\pi$
$659$	576984.i	1.32860i	−0.747468	+	0.664298i	$0.768731\pi$
$660$	0	0
$661$	13885.9	0.0317813	0.0158907	−	0.999874i	$-0.494942\pi$
$661$	13885.9	0.0317813	0.0158907	+	0.999874i	$0.494942\pi$
$662$	463310.i	1.05720i
$663$	0	0
$664$	159712.	0.362244
$665$	5701.42i	0.0128926i
$666$	0	0
$667$	31896.0	0.0716944
$668$	− 43331.1i	− 0.0971061i
$669$	0	0
$670$	−89042.8	−0.198358
$671$	19934.1i	0.0442744i
$672$	0	0
$673$	61470.0	0.135717	0.0678583	−	0.997695i	$-0.478383\pi$
$673$	61470.0	0.135717	0.0678583	+	0.997695i	$0.478383\pi$
$674$	− 464080.i	− 1.02158i
$675$	0	0
$676$	35214.7	0.0770603
$677$	553915.i	1.20855i	0.796774	+	0.604277i	$0.206538\pi$
$677$	553915.i	1.20855i	−0.796774	+	0.604277i	$0.793462\pi$
$678$	0	0
$679$	20539.2	0.0445497
$680$	− 46455.3i	− 0.100466i
$681$	0	0
$682$	−66030.8	−0.141964
$683$	− 111142.i	− 0.238253i	−0.992879	−	0.119127i	$-0.961991\pi$
$683$	− 111142.i	− 0.238253i	0.992879	−	0.119127i	$-0.0380094\pi$
$684$	0	0
$685$	26032.4	0.0554796
$686$	17967.4i	0.0381802i
$687$	0	0
$688$	150374.	0.317684
$689$	506641.i	1.06724i
$690$	0	0
$691$	−284092.	−0.594981	−0.297491	−	0.954725i	$-0.596150\pi$
$691$	−284092.	−0.594981	−0.297491	+	0.954725i	$0.596150\pi$
$692$	113298.i	0.236597i
$693$	0	0
$694$	−205936.	−0.427575
$695$	− 51521.2i	− 0.106664i
$696$	0	0
$697$	247240.	0.508924
$698$	− 44826.5i	− 0.0920076i
$699$	0	0
$700$	45301.5	0.0924520
$701$	125819.i	0.256041i	0.991772	+	0.128020i	$0.0408623\pi$
$701$	125819.i	0.256041i	−0.991772	+	0.128020i	$0.959138\pi$
$702$	0	0
$703$	−22174.6	−0.0448688
$704$	23251.7i	0.0469147i
$705$	0	0
$706$	−114528.	−0.229775
$707$	287554.i	0.575282i
$708$	0	0
$709$	−864378.	−1.71954	−0.859768	−	0.510686i	$-0.829392\pi$
$709$	−864378.	−1.71954	−0.859768	+	0.510686i	$0.829392\pi$
$710$	69652.3i	0.138172i
$711$	0	0
$712$	161515.	0.318605
$713$	28322.0i	0.0557115i
$714$	0	0
$715$	−126121.	−0.246703
$716$	− 129178.i	− 0.251977i
$717$	0	0
$718$	−86208.6	−0.167225
$719$	− 537857.i	− 1.04042i	−0.854038	−	0.520210i	$-0.825854\pi$
$719$	− 537857.i	− 1.04042i	0.854038	−	0.520210i	$-0.174146\pi$
$720$	0	0
$721$	317046.	0.609890
$722$	367764.i	0.705496i
$723$	0	0
$724$	−212327.	−0.405069
$725$	− 177013.i	− 0.336767i
$726$	0	0
$727$	−182656.	−0.345592	−0.172796	−	0.984958i	$-0.555280\pi$
$727$	−182656.	−0.345592	−0.172796	+	0.984958i	$0.555280\pi$
$728$	65136.3i	0.122902i
$729$	0	0
$730$	94656.0	0.177624
$731$	269980.i	0.505238i
$732$	0	0
$733$	690415.	1.28500	0.642499	−	0.766287i	$-0.277898\pi$
$733$	690415.	1.28500	0.642499	+	0.766287i	$0.277898\pi$
$734$	715445.i	1.32796i
$735$	0	0
$736$	9973.14	0.0184110
$737$	80015.9i	0.147313i
$738$	0	0
$739$	−447899.	−0.820146	−0.410073	−	0.912053i	$-0.634497\pi$
$739$	−447899.	−0.820146	−0.410073	+	0.912053i	$0.634497\pi$
$740$	− 183964.i	− 0.335945i
$741$	0	0
$742$	170746.	0.310130
$743$	− 156899.i	− 0.284212i	−0.989851	−	0.142106i	$-0.954613\pi$
$743$	− 156899.i	− 0.284212i	0.989851	−	0.142106i	$-0.0453873\pi$
$744$	0	0
$745$	717527.	1.29278
$746$	− 372353.i	− 0.669079i
$747$	0	0
$748$	−41745.9	−0.0746123
$749$	− 75681.6i	− 0.134905i
$750$	0	0
$751$	−733687.	−1.30086	−0.650431	−	0.759565i	$-0.725412\pi$
$751$	−733687.	−1.30086	−0.650431	+	0.759565i	$0.725412\pi$
$752$	− 125802.i	− 0.222460i
$753$	0	0
$754$	254516.	0.447686
$755$	− 70646.3i	− 0.123935i
$756$	0	0
$757$	1.11424e6	1.94440	0.972202	−	0.234142i	$-0.0752281\pi$
$757$	1.11424e6	1.94440	0.972202	+	0.234142i	$0.0752281\pi$
$758$	181115.i	0.315222i
$759$	0	0
$760$	6965.80	0.0120599
$761$	− 747308.i	− 1.29042i	−0.764006	−	0.645209i	$-0.776770\pi$
$761$	− 747308.i	− 1.29042i	0.764006	−	0.645209i	$-0.223230\pi$
$762$	0	0
$763$	−178367.	−0.306384
$764$	− 17199.9i	− 0.0294672i
$765$	0	0
$766$	−770296.	−1.31280
$767$	− 42921.2i	− 0.0729594i
$768$	0	0
$769$	965130.	1.63205	0.816024	−	0.578017i	$-0.196173\pi$
$769$	965130.	1.63205	0.816024	+	0.578017i	$0.196173\pi$
$770$	42504.7i	0.0716896i
$771$	0	0
$772$	−210759.	−0.353633
$773$	195019.i	0.326376i	0.986595	+	0.163188i	$0.0521776\pi$
$773$	195019.i	0.326376i	−0.986595	+	0.163188i	$0.947822\pi$
$774$	0	0
$775$	157178.	0.261691
$776$	− 25094.1i	− 0.0416724i
$777$	0	0
$778$	−808414.	−1.33560
$779$	37072.7i	0.0610913i
$780$	0	0
$781$	62591.3	0.102615
$782$	17905.7i	0.0292804i
$783$	0	0
$784$	21952.0	0.0357143
$785$	276457.i	0.448630i
$786$	0	0
$787$	700518.	1.13102	0.565510	−	0.824742i	$-0.308679\pi$
$787$	700518.	1.13102	0.565510	+	0.824742i	$0.308679\pi$
$788$	− 501960.i	− 0.808382i
$789$	0	0
$790$	231731.	0.371305
$791$	− 100879.i	− 0.161231i
$792$	0	0
$793$	68226.7	0.108495
$794$	− 571187.i	− 0.906019i
$795$	0	0
$796$	330915.	0.522264
$797$	− 6535.45i	− 0.0102887i	−0.999987	−	0.00514433i	$-0.998363\pi$
$797$	− 6535.45i	− 0.0102887i	0.999987	−	0.00514433i	$-0.00163750\pi$
$798$	0	0
$799$	225864.	0.353797
$800$	− 55347.8i	− 0.0864809i
$801$	0	0
$802$	−165738.	−0.257676
$803$	− 85060.2i	− 0.131915i
$804$	0	0
$805$	18231.2	0.0281335
$806$	225997.i	0.347883i
$807$	0	0
$808$	351324.	0.538127
$809$	− 1.10789e6i	− 1.69278i	−0.532566	−	0.846388i	$-0.678772\pi$
$809$	− 1.10789e6i	− 1.69278i	0.532566	−	0.846388i	$-0.321228\pi$
$810$	0	0
$811$	−961969.	−1.46258	−0.731289	−	0.682067i	$-0.761081\pi$
$811$	−961969.	−1.46258	−0.731289	+	0.682067i	$0.761081\pi$
$812$	− 85776.2i	− 0.130093i
$813$	0	0
$814$	−165314.	−0.249494
$815$	668155.i	1.00592i
$816$	0	0
$817$	−40482.4	−0.0606488
$818$	− 507208.i	− 0.758017i
$819$	0	0
$820$	−307560.	−0.457407
$821$	− 926435.i	− 1.37445i	−0.726445	−	0.687224i	$-0.758829\pi$
$821$	− 926435.i	− 1.37445i	0.726445	−	0.687224i	$-0.241171\pi$
$822$	0	0
$823$	608329.	0.898129	0.449064	−	0.893499i	$-0.351757\pi$
$823$	608329.	0.898129	0.449064	+	0.893499i	$0.351757\pi$
$824$	− 387356.i	− 0.570500i
$825$	0	0
$826$	−14465.2	−0.0212013
$827$	26569.8i	0.0388487i	0.999811	+	0.0194244i	$0.00618335\pi$
$827$	26569.8i	0.0388487i	−0.999811	+	0.0194244i	$0.993817\pi$
$828$	0	0
$829$	785642.	1.14318	0.571591	−	0.820538i	$-0.306326\pi$
$829$	785642.	1.14318	0.571591	+	0.820538i	$0.306326\pi$
$830$	− 356704.i	− 0.517788i
$831$	0	0
$832$	79581.3	0.114965
$833$	39412.5i	0.0567994i
$834$	0	0
$835$	−96776.7	−0.138803
$836$	− 6259.63i	− 0.00895646i
$837$	0	0
$838$	435030.	0.619485
$839$	338072.i	0.480270i	0.970739	+	0.240135i	$0.0771918\pi$
$839$	338072.i	0.480270i	−0.970739	+	0.240135i	$0.922808\pi$
$840$	0	0
$841$	372115.	0.526121
$842$	258108.i	0.364063i
$843$	0	0
$844$	−481538.	−0.675998
$845$	− 78649.4i	− 0.110149i
$846$	0	0
$847$	−232959.	−0.324723
$848$	− 208612.i	− 0.290100i
$849$	0	0
$850$	99371.0	0.137538
$851$	70906.7i	0.0979103i
$852$	0	0
$853$	742542.	1.02052	0.510262	−	0.860019i	$-0.329548\pi$
$853$	742542.	1.02052	0.510262	+	0.860019i	$0.329548\pi$
$854$	− 22993.5i	− 0.0315275i
$855$	0	0
$856$	−92465.2	−0.126192
$857$	419964.i	0.571808i	0.958258	+	0.285904i	$0.0922938\pi$
$857$	419964.i	0.571808i	−0.958258	+	0.285904i	$0.907706\pi$
$858$	0	0
$859$	843636.	1.14332	0.571662	−	0.820490i	$-0.306299\pi$
$859$	843636.	1.14332	0.571662	+	0.820490i	$0.306299\pi$
$860$	− 335848.i	− 0.454094i
$861$	0	0
$862$	−985766.	−1.32666
$863$	− 346514.i	− 0.465264i	−0.972565	−	0.232632i	$-0.925266\pi$
$863$	− 346514.i	− 0.465264i	0.972565	−	0.232632i	$-0.0747338\pi$
$864$	0	0
$865$	253042.	0.338190
$866$	243705.i	0.324959i
$867$	0	0
$868$	76164.8	0.101092
$869$	− 208239.i	− 0.275755i
$870$	0	0
$871$	273863.	0.360992
$872$	217923.i	0.286596i
$873$	0	0
$874$	−2684.89	−0.00351482
$875$	− 307995.i	− 0.402280i
$876$	0	0
$877$	316397.	0.411370	0.205685	−	0.978618i	$-0.434058\pi$
$877$	316397.	0.411370	0.205685	+	0.978618i	$0.434058\pi$
$878$	803524.i	1.04234i
$879$	0	0
$880$	51930.8	0.0670594
$881$	67739.5i	0.0872751i	0.999047	+	0.0436375i	$0.0138947\pi$
$881$	67739.5i	0.0872751i	−0.999047	+	0.0436375i	$0.986105\pi$
$882$	0	0
$883$	1.13315e6	1.45333	0.726666	−	0.686991i	$-0.241069\pi$
$883$	1.13315e6	1.45333	0.726666	+	0.686991i	$0.241069\pi$
$884$	142880.i	0.182838i
$885$	0	0
$886$	−220270.	−0.280600
$887$	− 925863.i	− 1.17679i	−0.808573	−	0.588396i	$-0.799760\pi$
$887$	− 925863.i	− 1.17679i	0.808573	−	0.588396i	$-0.200240\pi$
$888$	0	0
$889$	−66959.5	−0.0847244
$890$	− 360731.i	− 0.455411i
$891$	0	0
$892$	−438130.	−0.550647
$893$	33867.5i	0.0424698i
$894$	0	0
$895$	−288509.	−0.360174
$896$	− 26820.2i	− 0.0334077i
$897$	0	0
$898$	−246712.	−0.305942
$899$	− 297610.i	− 0.368237i
$900$	0	0
$901$	374540.	0.461370
$902$	276381.i	0.339700i
$903$	0	0
$904$	−123251.	−0.150818
$905$	474217.i	0.579002i
$906$	0	0
$907$	478626.	0.581811	0.290905	−	0.956752i	$-0.406043\pi$
$907$	478626.	0.581811	0.290905	+	0.956752i	$0.406043\pi$
$908$	− 189276.i	− 0.229575i
$909$	0	0
$910$	145477.	0.175676
$911$	− 388418.i	− 0.468018i	−0.972234	−	0.234009i	$-0.924815\pi$
$911$	− 388418.i	− 0.468018i	0.972234	−	0.234009i	$-0.0751845\pi$
$912$	0	0
$913$	−320543.	−0.384543
$914$	− 170707.i	− 0.204342i
$915$	0	0
$916$	146918.	0.175099
$917$	251904.i	0.299569i
$918$	0	0
$919$	−807823.	−0.956501	−0.478250	−	0.878223i	$-0.658729\pi$
$919$	−807823.	−0.956501	−0.478250	+	0.878223i	$0.658729\pi$
$920$	− 22274.2i	− 0.0263165i
$921$	0	0
$922$	387071.	0.455333
$923$	− 214225.i	− 0.251459i
$924$	0	0
$925$	393510.	0.459910
$926$	− 474189.i	− 0.553005i
$927$	0	0
$928$	−104798.	−0.121691
$929$	1.62506e6i	1.88294i	0.337094	+	0.941471i	$0.390556\pi$
$929$	1.62506e6i	1.88294i	−0.337094	+	0.941471i	$0.609444\pi$
$930$	0	0
$931$	−5909.75	−0.00681819
$932$	− 570726.i	− 0.657046i
$933$	0	0
$934$	−534237.	−0.612407
$935$	93236.2i	0.106650i
$936$	0	0
$937$	−1.17872e6	−1.34255	−0.671275	−	0.741208i	$-0.734253\pi$
$937$	−1.17872e6	−1.34255	−0.671275	+	0.741208i	$0.734253\pi$
$938$	− 92296.3i	− 0.104901i
$939$	0	0
$940$	−280970.	−0.317983
$941$	− 418331.i	− 0.472433i	−0.971700	−	0.236217i	$-0.924093\pi$
$941$	− 418331.i	− 0.472433i	0.971700	−	0.236217i	$-0.0759075\pi$
$942$	0	0
$943$	118546.	0.133310
$944$	17673.0i	0.0198320i
$945$	0	0
$946$	−301801.	−0.337240
$947$	− 860451.i	− 0.959458i	−0.877417	−	0.479729i	$-0.840735\pi$
$947$	− 860451.i	− 0.959458i	0.877417	−	0.479729i	$-0.159265\pi$
$948$	0	0
$949$	−291127.	−0.323259
$950$	14900.3i	0.0165100i
$951$	0	0
$952$	48152.8	0.0531309
$953$	− 555766.i	− 0.611937i	−0.952042	−	0.305968i	$-0.901020\pi$
$953$	− 555766.i	− 0.611937i	0.952042	−	0.305968i	$-0.0989801\pi$
$954$	0	0
$955$	−38414.7	−0.0421202
$956$	− 349010.i	− 0.381876i
$957$	0	0
$958$	1.12078e6	1.22121
$959$	26983.6i	0.0293402i
$960$	0	0
$961$	−659259.	−0.713854
$962$	565805.i	0.611387i
$963$	0	0
$964$	−99577.9	−0.107154
$965$	470715.i	0.505479i
$966$	0	0
$967$	−862383.	−0.922247	−0.461124	−	0.887336i	$-0.652554\pi$
$967$	−862383.	−0.922247	−0.461124	+	0.887336i	$0.652554\pi$
$968$	284622.i	0.303751i
$969$	0	0
$970$	−56045.8	−0.0595661
$971$	− 1.69499e6i	− 1.79775i	−0.438209	−	0.898873i	$-0.644387\pi$
$971$	− 1.69499e6i	− 1.79775i	0.438209	−	0.898873i	$-0.355613\pi$
$972$	0	0
$973$	53403.8	0.0564087
$974$	581572.i	0.613035i
$975$	0	0
$976$	−28092.7	−0.0294913
$977$	− 1.20992e6i	− 1.26755i	−0.773516	−	0.633777i	$-0.781504\pi$
$977$	− 1.20992e6i	− 1.26755i	0.773516	−	0.633777i	$-0.218496\pi$
$978$	0	0
$979$	−324161.	−0.338217
$980$	− 49028.1i	− 0.0510497i
$981$	0	0
$982$	−414205.	−0.429529
$983$	− 1.28838e6i	− 1.33332i	−0.745360	−	0.666662i	$-0.767723\pi$
$983$	− 1.28838e6i	− 1.33332i	0.745360	−	0.666662i	$-0.232277\pi$
$984$	0	0
$985$	−1.12109e6	−1.15549
$986$	− 188154.i	− 0.193535i
$987$	0	0
$988$	−21424.2	−0.0219478
$989$	129449.i	0.132345i
$990$	0	0
$991$	−1.23803e6	−1.26062	−0.630312	−	0.776342i	$-0.717073\pi$
$991$	−1.23803e6	−1.26062	−0.630312	+	0.776342i	$0.717073\pi$
$992$	− 93055.5i	− 0.0945625i
$993$	0	0
$994$	−72197.4	−0.0730716
$995$	− 739073.i	− 0.746520i
$996$	0	0
$997$	−288690.	−0.290430	−0.145215	−	0.989400i	$-0.546387\pi$
$997$	−288690.	−0.290430	−0.145215	+	0.989400i	$0.546387\pi$
$998$	753313.i	0.756335i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.5.b.b.323.3	✓	8
3.2	odd	2	inner	378.5.b.b.323.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.5.b.b.323.3	✓	8	1.1	even	1	trivial
378.5.b.b.323.6	yes	8	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	378.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} +17.8674i q^{5} -18.5203 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} +17.8674i q^{5} -18.5203 q^{7} +22.6274i q^{8} +50.5366 q^{10} -45.4134i q^{11} -155.432 q^{13} +52.3832i q^{14} +64.0000 q^{16} +114.905i q^{17} -17.2296 q^{19} -142.939i q^{20} -128.449 q^{22} +55.0943i q^{23} +305.756 q^{25} +439.629i q^{26} +148.162 q^{28} -578.935i q^{29} +514.064 q^{31} -181.019i q^{32} +325.001 q^{34} -330.909i q^{35} +1287.01 q^{37} +48.7326i q^{38} -404.293 q^{40} -2151.69i q^{41} +2349.59 q^{43} +363.307i q^{44} +155.830 q^{46} -1965.66i q^{47} +343.000 q^{49} -864.809i q^{50} +1243.46 q^{52} -3259.56i q^{53} +811.419 q^{55} -419.066i q^{56} -1637.48 q^{58} +276.141i q^{59} -438.948 q^{61} -1453.99i q^{62} -512.000 q^{64} -2777.17i q^{65} -1761.95 q^{67} -919.241i q^{68} -935.951 q^{70} +1378.26i q^{71} +1873.02 q^{73} -3640.20i q^{74} +137.837 q^{76} +841.068i q^{77} +4585.41 q^{79} +1143.51i q^{80} -6085.89 q^{82} -7058.33i q^{83} -2053.05 q^{85} -6645.64i q^{86} +1027.59 q^{88} -7138.01i q^{89} +2878.64 q^{91} -440.755i q^{92} -5559.72 q^{94} -307.848i q^{95} -1109.01 q^{97} -970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4}+O(q^{10}) \) 8 * q - 64 * q^4	\( 8 q - 64 q^{4} - 32 q^{10} + 296 q^{13} + 512 q^{16} - 80 q^{19} + 800 q^{22} - 1464 q^{25} - 1024 q^{31} + 896 q^{34} + 904 q^{37} + 256 q^{40} - 3224 q^{43} - 4384 q^{46} + 2744 q^{49} - 2368 q^{52} + 8312 q^{55} + 2944 q^{58} + 128 q^{61} - 4096 q^{64} - 10152 q^{67} + 1568 q^{70} + 10632 q^{73} + 640 q^{76} - 4072 q^{79} - 3552 q^{82} + 13448 q^{85} - 6400 q^{88} - 6664 q^{91} - 384 q^{94} + 35616 q^{97}+O(q^{100}) \) 8 * q - 64 * q^4 - 32 * q^10 + 296 * q^13 + 512 * q^16 - 80 * q^19 + 800 * q^22 - 1464 * q^25 - 1024 * q^31 + 896 * q^34 + 904 * q^37 + 256 * q^40 - 3224 * q^43 - 4384 * q^46 + 2744 * q^49 - 2368 * q^52 + 8312 * q^55 + 2944 * q^58 + 128 * q^61 - 4096 * q^64 - 10152 * q^67 + 1568 * q^70 + 10632 * q^73 + 640 * q^76 - 4072 * q^79 - 3552 * q^82 + 13448 * q^85 - 6400 * q^88 - 6664 * q^91 - 384 * q^94 + 35616 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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