LMFDB - Embedded newform 378.5.b.b.323.4

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			323.4
Root			$-5.99257i$ of defining polynomial
Character	$\chi$	$=$	378.323
Dual form			378.5.b.b.323.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} +34.7917i q^{5} +18.5203 q^{7} +22.6274i q^{8} +O(q^{10})$	\(q-2.82843i q^{2} -8.00000 q^{4} +34.7917i q^{5} +18.5203 q^{7} +22.6274i q^{8} +98.4059 q^{10} +30.1276i q^{11} +143.140 q^{13} -52.3832i q^{14} +64.0000 q^{16} -116.175i q^{17} -89.0628 q^{19} -278.334i q^{20} +85.2137 q^{22} -23.6671i q^{23} -585.464 q^{25} -404.860i q^{26} -148.162 q^{28} +1180.46i q^{29} +171.591 q^{31} -181.019i q^{32} -328.593 q^{34} +644.352i q^{35} -974.714 q^{37} +251.908i q^{38} -787.247 q^{40} +417.362i q^{41} -3148.03 q^{43} -241.021i q^{44} -66.9407 q^{46} +2961.00i q^{47} +343.000 q^{49} +1655.94i q^{50} -1145.12 q^{52} +2248.40i q^{53} -1048.19 q^{55} +419.066i q^{56} +3338.85 q^{58} -827.250i q^{59} -6889.71 q^{61} -485.332i q^{62} -512.000 q^{64} +4980.08i q^{65} +4866.31 q^{67} +929.402i q^{68} +1822.50 q^{70} +5488.08i q^{71} +777.420 q^{73} +2756.91i q^{74} +712.502 q^{76} +557.971i q^{77} +3899.43 q^{79} +2226.67i q^{80} +1180.48 q^{82} -826.579i q^{83} +4041.94 q^{85} +8903.96i q^{86} -681.709 q^{88} -1055.44i q^{89} +2650.99 q^{91} +189.337i q^{92} +8374.98 q^{94} -3098.65i q^{95} +5713.52 q^{97} -970.151i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	34.7917i	1.39167i	0.718202	+	0.695834i	$0.244965\pi$
$5$	34.7917i	1.39167i	−0.718202	+	0.695834i	$0.755035\pi$
$6$	0	0
$7$	18.5203	0.377964
$7$	18.5203	0.377964
$8$	22.6274i	0.353553i
$9$	0	0
$10$	98.4059	0.984059
$11$	30.1276i	0.248988i	0.992220	+	0.124494i	$0.0397308\pi$
$11$	30.1276i	0.248988i	−0.992220	+	0.124494i	$0.960269\pi$
$12$	0	0
$13$	143.140	0.846981	0.423491	−	0.905901i	$-0.360805\pi$
$13$	143.140	0.846981	0.423491	+	0.905901i	$0.360805\pi$
$14$	− 52.3832i	− 0.267261i
$15$	0	0
$16$	64.0000	0.250000
$17$	− 116.175i	− 0.401991i	−0.979592	−	0.200995i	$-0.935582\pi$
$17$	− 116.175i	− 0.401991i	0.979592	−	0.200995i	$-0.0644176\pi$
$18$	0	0
$19$	−89.0628	−0.246711	−0.123356	−	0.992363i	$-0.539366\pi$
$19$	−89.0628	−0.246711	−0.123356	+	0.992363i	$0.539366\pi$
$20$	− 278.334i	− 0.695834i
$21$	0	0
$22$	85.2137	0.176061
$23$	− 23.6671i	− 0.0447393i	−0.999750	−	0.0223697i	$-0.992879\pi$
$23$	− 23.6671i	− 0.0447393i	0.999750	−	0.0223697i	$-0.00712108\pi$
$24$	0	0
$25$	−585.464	−0.936742
$26$	− 404.860i	− 0.598906i
$27$	0	0
$28$	−148.162	−0.188982
$29$	1180.46i	1.40364i	0.712353	+	0.701821i	$0.247629\pi$
$29$	1180.46i	1.40364i	−0.712353	+	0.701821i	$0.752371\pi$
$30$	0	0
$31$	171.591	0.178554	0.0892772	−	0.996007i	$-0.471544\pi$
$31$	171.591	0.178554	0.0892772	+	0.996007i	$0.471544\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	−328.593	−0.284250
$35$	644.352i	0.526001i
$36$	0	0
$37$	−974.714	−0.711989	−0.355995	−	0.934488i	$-0.615858\pi$
$37$	−974.714	−0.711989	−0.355995	+	0.934488i	$0.615858\pi$
$38$	251.908i	0.174451i
$39$	0	0
$40$	−787.247	−0.492029
$41$	417.362i	0.248282i	0.992265	+	0.124141i	$0.0396175\pi$
$41$	417.362i	0.248282i	−0.992265	+	0.124141i	$0.960382\pi$
$42$	0	0
$43$	−3148.03	−1.70256	−0.851278	−	0.524715i	$-0.824172\pi$
$43$	−3148.03	−1.70256	−0.851278	+	0.524715i	$0.824172\pi$
$44$	− 241.021i	− 0.124494i
$45$	0	0
$46$	−66.9407	−0.0316355
$47$	2961.00i	1.34043i	0.742169	+	0.670213i	$0.233797\pi$
$47$	2961.00i	1.34043i	−0.742169	+	0.670213i	$0.766203\pi$
$48$	0	0
$49$	343.000	0.142857
$50$	1655.94i	0.662377i
$51$	0	0
$52$	−1145.12	−0.423491
$53$	2248.40i	0.800427i	0.916422	+	0.400214i	$0.131064\pi$
$53$	2248.40i	0.800427i	−0.916422	+	0.400214i	$0.868936\pi$
$54$	0	0
$55$	−1048.19	−0.346509
$56$	419.066i	0.133631i
$57$	0	0
$58$	3338.85	0.992525
$59$	− 827.250i	− 0.237647i	−0.992915	−	0.118824i	$-0.962088\pi$
$59$	− 827.250i	− 0.237647i	0.992915	−	0.118824i	$-0.0379123\pi$
$60$	0	0
$61$	−6889.71	−1.85157	−0.925787	−	0.378046i	$-0.876596\pi$
$61$	−6889.71	−1.85157	−0.925787	+	0.378046i	$0.876596\pi$
$62$	− 485.332i	− 0.126257i
$63$	0	0
$64$	−512.000	−0.125000
$65$	4980.08i	1.17872i
$66$	0	0
$67$	4866.31	1.08405	0.542026	−	0.840361i	$-0.317657\pi$
$67$	4866.31	1.08405	0.542026	+	0.840361i	$0.317657\pi$
$68$	929.402i	0.200995i
$69$	0	0
$70$	1822.50	0.371939
$71$	5488.08i	1.08869i	0.838862	+	0.544344i	$0.183221\pi$
$71$	5488.08i	1.08869i	−0.838862	+	0.544344i	$0.816779\pi$
$72$	0	0
$73$	777.420	0.145885	0.0729424	−	0.997336i	$-0.476761\pi$
$73$	777.420	0.145885	0.0729424	+	0.997336i	$0.476761\pi$
$74$	2756.91i	0.503453i
$75$	0	0
$76$	712.502	0.123356
$77$	557.971i	0.0941087i
$78$	0	0
$79$	3899.43	0.624808	0.312404	−	0.949949i	$-0.398866\pi$
$79$	3899.43	0.624808	0.312404	+	0.949949i	$0.398866\pi$
$80$	2226.67i	0.347917i
$81$	0	0
$82$	1180.48	0.175562
$83$	− 826.579i	− 0.119985i	−0.998199	−	0.0599927i	$-0.980892\pi$
$83$	− 826.579i	− 0.119985i	0.998199	−	0.0599927i	$-0.0191077\pi$
$84$	0	0
$85$	4041.94	0.559438
$86$	8903.96i	1.20389i
$87$	0	0
$88$	−681.709	−0.0880307
$89$	− 1055.44i	− 0.133246i	−0.997778	−	0.0666231i	$-0.978777\pi$
$89$	− 1055.44i	− 0.133246i	0.997778	−	0.0666231i	$-0.0212225\pi$
$90$	0	0
$91$	2650.99	0.320129
$92$	189.337i	0.0223697i
$93$	0	0
$94$	8374.98	0.947825
$95$	− 3098.65i	− 0.343341i
$96$	0	0
$97$	5713.52	0.607240	0.303620	−	0.952793i	$-0.401805\pi$
$97$	5713.52	0.607240	0.303620	+	0.952793i	$0.401805\pi$
$98$	− 970.151i	− 0.101015i
$99$	0	0
$100$	4683.71	0.468371
$101$	10670.0i	1.04598i	0.852339	+	0.522990i	$0.175183\pi$
$101$	10670.0i	1.04598i	−0.852339	+	0.522990i	$0.824817\pi$
$102$	0	0
$103$	−16761.8	−1.57996	−0.789981	−	0.613132i	$-0.789910\pi$
$103$	−16761.8	−1.57996	−0.789981	+	0.613132i	$0.789910\pi$
$104$	3238.88i	0.299453i
$105$	0	0
$106$	6359.44	0.565987
$107$	− 12800.3i	− 1.11803i	−0.829159	−	0.559013i	$-0.811180\pi$
$107$	− 12800.3i	− 1.11803i	0.829159	−	0.559013i	$-0.188820\pi$
$108$	0	0
$109$	−1931.09	−0.162536	−0.0812680	−	0.996692i	$-0.525897\pi$
$109$	−1931.09	−0.162536	−0.0812680	+	0.996692i	$0.525897\pi$
$110$	2964.73i	0.245019i
$111$	0	0
$112$	1185.30	0.0944911
$113$	19494.9i	1.52674i	0.645964	+	0.763368i	$0.276456\pi$
$113$	19494.9i	1.52674i	−0.645964	+	0.763368i	$0.723544\pi$
$114$	0	0
$115$	823.419	0.0622623
$116$	− 9443.71i	− 0.701821i
$117$	0	0
$118$	−2339.82	−0.168042
$119$	− 2151.60i	− 0.151938i
$120$	0	0
$121$	13733.3	0.938005
$122$	19487.0i	1.30926i
$123$	0	0
$124$	−1372.73	−0.0892772
$125$	1375.53i	0.0880338i
$126$	0	0
$127$	−27254.3	−1.68977	−0.844885	−	0.534949i	$-0.820331\pi$
$127$	−27254.3	−1.68977	−0.844885	+	0.534949i	$0.820331\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	14085.8	0.833479
$131$	− 17965.1i	− 1.04686i	−0.852069	−	0.523429i	$-0.824653\pi$
$131$	− 17965.1i	− 1.04686i	0.852069	−	0.523429i	$-0.175347\pi$
$132$	0	0
$133$	−1649.47	−0.0932481
$134$	− 13764.0i	− 0.766541i
$135$	0	0
$136$	2628.75	0.142125
$137$	29725.1i	1.58373i	0.610693	+	0.791867i	$0.290891\pi$
$137$	29725.1i	1.58373i	−0.610693	+	0.791867i	$0.709109\pi$
$138$	0	0
$139$	−12266.3	−0.634871	−0.317435	−	0.948280i	$-0.602822\pi$
$139$	−12266.3	−0.634871	−0.317435	+	0.948280i	$0.602822\pi$
$140$	− 5154.81i	− 0.263001i
$141$	0	0
$142$	15522.6	0.769819
$143$	4312.46i	0.210888i
$144$	0	0
$145$	−41070.3	−1.95341
$146$	− 2198.88i	− 0.103156i
$147$	0	0
$148$	7797.71	0.355995
$149$	− 2566.34i	− 0.115596i	−0.998328	−	0.0577978i	$-0.981592\pi$
$149$	− 2566.34i	− 0.115596i	0.998328	−	0.0577978i	$-0.0184079\pi$
$150$	0	0
$151$	6919.18	0.303459	0.151730	−	0.988422i	$-0.451516\pi$
$151$	6919.18	0.303459	0.151730	+	0.988422i	$0.451516\pi$
$152$	− 2015.26i	− 0.0872256i
$153$	0	0
$154$	1578.18	0.0665449
$155$	5969.94i	0.248488i
$156$	0	0
$157$	−1569.92	−0.0636912	−0.0318456	−	0.999493i	$-0.510138\pi$
$157$	−1569.92	−0.0636912	−0.0318456	+	0.999493i	$0.510138\pi$
$158$	− 11029.2i	− 0.441806i
$159$	0	0
$160$	6297.97	0.246015
$161$	− 438.321i	− 0.0169099i
$162$	0	0
$163$	27851.1	1.04826	0.524128	−	0.851639i	$-0.324391\pi$
$163$	27851.1	1.04826	0.524128	+	0.851639i	$0.324391\pi$
$164$	− 3338.90i	− 0.124141i
$165$	0	0
$166$	−2337.92	−0.0848425
$167$	40958.1i	1.46861i	0.678819	+	0.734306i	$0.262492\pi$
$167$	40958.1i	1.46861i	−0.678819	+	0.734306i	$0.737508\pi$
$168$	0	0
$169$	−8072.00	−0.282623
$170$	− 11432.3i	− 0.395582i
$171$	0	0
$172$	25184.2	0.851278
$173$	13024.4i	0.435175i	0.976041	+	0.217588i	$0.0698188\pi$
$173$	13024.4i	0.435175i	−0.976041	+	0.217588i	$0.930181\pi$
$174$	0	0
$175$	−10842.9	−0.354055
$176$	1928.17i	0.0622471i
$177$	0	0
$178$	−2985.24	−0.0942193
$179$	− 35142.5i	− 1.09680i	−0.836217	−	0.548399i	$-0.815238\pi$
$179$	− 35142.5i	− 1.09680i	0.836217	−	0.548399i	$-0.184762\pi$
$180$	0	0
$181$	−14046.6	−0.428760	−0.214380	−	0.976750i	$-0.568773\pi$
$181$	−14046.6	−0.428760	−0.214380	+	0.976750i	$0.568773\pi$
$182$	− 7498.12i	− 0.226365i
$183$	0	0
$184$	535.525	0.0158177
$185$	− 33912.0i	− 0.990854i
$186$	0	0
$187$	3500.08	0.100091
$188$	− 23688.0i	− 0.670213i
$189$	0	0
$190$	−8764.30	−0.242778
$191$	− 38790.0i	− 1.06329i	−0.846966	−	0.531647i	$-0.821574\pi$
$191$	− 38790.0i	− 1.06329i	0.846966	−	0.531647i	$-0.178426\pi$
$192$	0	0
$193$	3790.94	0.101773	0.0508864	−	0.998704i	$-0.483795\pi$
$193$	3790.94	0.101773	0.0508864	+	0.998704i	$0.483795\pi$
$194$	− 16160.3i	− 0.429383i
$195$	0	0
$196$	−2744.00	−0.0714286
$197$	− 17172.3i	− 0.442481i	−0.975219	−	0.221241i	$-0.928989\pi$
$197$	− 17172.3i	− 0.442481i	0.975219	−	0.221241i	$-0.0710106\pi$
$198$	0	0
$199$	−33411.7	−0.843708	−0.421854	−	0.906664i	$-0.638621\pi$
$199$	−33411.7	−0.843708	−0.421854	+	0.906664i	$0.638621\pi$
$200$	− 13247.5i	− 0.331188i
$201$	0	0
$202$	30179.4	0.739619
$203$	21862.5i	0.530527i
$204$	0	0
$205$	−14520.7	−0.345526
$206$	47409.6i	1.11720i
$207$	0	0
$208$	9160.95	0.211745
$209$	− 2683.25i	− 0.0614282i
$210$	0	0
$211$	44323.4	0.995563	0.497781	−	0.867303i	$-0.334148\pi$
$211$	44323.4	0.995563	0.497781	+	0.867303i	$0.334148\pi$
$212$	− 17987.2i	− 0.400214i
$213$	0	0
$214$	−36204.7	−0.790564
$215$	− 109525.i	− 2.36939i
$216$	0	0
$217$	3177.90	0.0674872
$218$	5461.95i	0.114930i
$219$	0	0
$220$	8385.53	0.173255
$221$	− 16629.3i	− 0.340478i
$222$	0	0
$223$	12601.5	0.253404	0.126702	−	0.991941i	$-0.459561\pi$
$223$	12601.5	0.253404	0.126702	+	0.991941i	$0.459561\pi$
$224$	− 3352.53i	− 0.0668153i
$225$	0	0
$226$	55139.9	1.07957
$227$	− 38690.2i	− 0.750844i	−0.926854	−	0.375422i	$-0.877498\pi$
$227$	− 38690.2i	− 0.750844i	0.926854	−	0.375422i	$-0.122502\pi$
$228$	0	0
$229$	−72276.5	−1.37824	−0.689122	−	0.724645i	$-0.742004\pi$
$229$	−72276.5	−1.37824	−0.689122	+	0.724645i	$0.742004\pi$
$230$	− 2328.98i	− 0.0440261i
$231$	0	0
$232$	−26710.8	−0.496262
$233$	− 63243.2i	− 1.16494i	−0.812853	−	0.582468i	$-0.802087\pi$
$233$	− 63243.2i	− 1.16494i	0.812853	−	0.582468i	$-0.197913\pi$
$234$	0	0
$235$	−103018.	−1.86543
$236$	6618.00i	0.118824i
$237$	0	0
$238$	−6085.63	−0.107437
$239$	48102.6i	0.842118i	0.907033	+	0.421059i	$0.138341\pi$
$239$	48102.6i	0.842118i	−0.907033	+	0.421059i	$0.861659\pi$
$240$	0	0
$241$	−103824.	−1.78757	−0.893786	−	0.448494i	$-0.851961\pi$
$241$	−103824.	−1.78757	−0.893786	+	0.448494i	$0.851961\pi$
$242$	− 38843.7i	− 0.663270i
$243$	0	0
$244$	55117.6	0.925787
$245$	11933.6i	0.198810i
$246$	0	0
$247$	−12748.4	−0.208960
$248$	3882.65i	0.0631285i
$249$	0	0
$250$	3890.58	0.0622493
$251$	− 55160.0i	− 0.875542i	−0.899087	−	0.437771i	$-0.855768\pi$
$251$	− 55160.0i	− 0.875542i	0.899087	−	0.437771i	$-0.144232\pi$
$252$	0	0
$253$	713.032	0.0111396
$254$	77086.8i	1.19485i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 22410.6i	− 0.339303i	−0.985504	−	0.169652i	$-0.945736\pi$
$257$	− 22410.6i	− 0.339303i	0.985504	−	0.169652i	$-0.0542643\pi$
$258$	0	0
$259$	−18051.9	−0.269107
$260$	− 39840.6i	− 0.589359i
$261$	0	0
$262$	−50813.0	−0.740240
$263$	5733.50i	0.0828911i	0.999141	+	0.0414456i	$0.0131963\pi$
$263$	5733.50i	0.0828911i	−0.999141	+	0.0414456i	$0.986804\pi$
$264$	0	0
$265$	−78225.7	−1.11393
$266$	4665.39i	0.0659364i
$267$	0	0
$268$	−38930.5	−0.542026
$269$	− 122046.i	− 1.68663i	−0.537422	−	0.843314i	$-0.680602\pi$
$269$	− 122046.i	− 1.68663i	0.537422	−	0.843314i	$-0.319398\pi$
$270$	0	0
$271$	138859.	1.89076	0.945378	−	0.325975i	$-0.105693\pi$
$271$	138859.	1.89076	0.945378	+	0.325975i	$0.105693\pi$
$272$	− 7435.22i	− 0.100498i
$273$	0	0
$274$	84075.3	1.11987
$275$	− 17638.6i	− 0.233238i
$276$	0	0
$277$	−6187.42	−0.0806400	−0.0403200	−	0.999187i	$-0.512838\pi$
$277$	−6187.42	−0.0806400	−0.0403200	+	0.999187i	$0.512838\pi$
$278$	34694.4i	0.448921i
$279$	0	0
$280$	−14580.0	−0.185970
$281$	99067.2i	1.25463i	0.778764	+	0.627317i	$0.215847\pi$
$281$	99067.2i	1.25463i	−0.778764	+	0.627317i	$0.784153\pi$
$282$	0	0
$283$	20879.0	0.260697	0.130349	−	0.991468i	$-0.458390\pi$
$283$	20879.0	0.260697	0.130349	+	0.991468i	$0.458390\pi$
$284$	− 43904.6i	− 0.544344i
$285$	0	0
$286$	12197.5	0.149121
$287$	7729.66i	0.0938418i
$288$	0	0
$289$	70024.3	0.838404
$290$	116164.i	1.38127i
$291$	0	0
$292$	−6219.36	−0.0729424
$293$	33461.8i	0.389775i	0.980826	+	0.194888i	$0.0624342\pi$
$293$	33461.8i	0.389775i	−0.980826	+	0.194888i	$0.937566\pi$
$294$	0	0
$295$	28781.5	0.330726
$296$	− 22055.2i	− 0.251726i
$297$	0	0
$298$	−7258.70	−0.0817384
$299$	− 3387.70i	− 0.0378933i
$300$	0	0
$301$	−58302.3	−0.643506
$302$	− 19570.4i	− 0.214578i
$303$	0	0
$304$	−5700.02	−0.0616778
$305$	− 239705.i	− 2.57678i
$306$	0	0
$307$	151671.	1.60926	0.804630	−	0.593776i	$-0.202364\pi$
$307$	151671.	1.60926	0.804630	+	0.593776i	$0.202364\pi$
$308$	− 4463.77i	− 0.0470544i
$309$	0	0
$310$	16885.5	0.175708
$311$	88401.3i	0.913982i	0.889471	+	0.456991i	$0.151073\pi$
$311$	88401.3i	0.913982i	−0.889471	+	0.456991i	$0.848927\pi$
$312$	0	0
$313$	152618.	1.55782	0.778911	−	0.627135i	$-0.215772\pi$
$313$	152618.	1.55782	0.778911	+	0.627135i	$0.215772\pi$
$314$	4440.41i	0.0450364i
$315$	0	0
$316$	−31195.4	−0.312404
$317$	33405.1i	0.332426i	0.986090	+	0.166213i	$0.0531539\pi$
$317$	33405.1i	0.332426i	−0.986090	+	0.166213i	$0.946846\pi$
$318$	0	0
$319$	−35564.5	−0.349491
$320$	− 17813.4i	− 0.173959i
$321$	0	0
$322$	−1239.76	−0.0119571
$323$	10346.9i	0.0991756i
$324$	0	0
$325$	−83803.2	−0.793403
$326$	− 78774.9i	− 0.741229i
$327$	0	0
$328$	−9443.83	−0.0877810
$329$	54838.5i	0.506634i
$330$	0	0
$331$	208993.	1.90755	0.953773	−	0.300529i	$-0.0971631\pi$
$331$	208993.	1.90755	0.953773	+	0.300529i	$0.0971631\pi$
$332$	6612.63i	0.0599927i
$333$	0	0
$334$	115847.	1.03847
$335$	169307.i	1.50864i
$336$	0	0
$337$	203893.	1.79532	0.897659	−	0.440690i	$-0.145266\pi$
$337$	203893.	1.79532	0.897659	+	0.440690i	$0.145266\pi$
$338$	22831.1i	0.199845i
$339$	0	0
$340$	−32335.5	−0.279719
$341$	5169.61i	0.0444579i
$342$	0	0
$343$	6352.45	0.0539949
$344$	− 71231.7i	− 0.601945i
$345$	0	0
$346$	36838.4	0.307715
$347$	43169.0i	0.358520i	0.983802	+	0.179260i	$0.0573703\pi$
$347$	43169.0i	0.358520i	−0.983802	+	0.179260i	$0.942630\pi$
$348$	0	0
$349$	203110.	1.66755	0.833777	−	0.552101i	$-0.186174\pi$
$349$	203110.	1.66755	0.833777	+	0.552101i	$0.186174\pi$
$350$	30668.5i	0.250355i
$351$	0	0
$352$	5453.68	0.0440153
$353$	173776.i	1.39457i	0.716796	+	0.697283i	$0.245608\pi$
$353$	173776.i	1.39457i	−0.716796	+	0.697283i	$0.754392\pi$
$354$	0	0
$355$	−190940.	−1.51509
$356$	8443.54i	0.0666231i
$357$	0	0
$358$	−99398.0	−0.775553
$359$	190257.i	1.47622i	0.674680	+	0.738110i	$0.264282\pi$
$359$	190257.i	1.47622i	−0.674680	+	0.738110i	$0.735718\pi$
$360$	0	0
$361$	−122389.	−0.939134
$362$	39729.8i	0.303179i
$363$	0	0
$364$	−21207.9	−0.160064
$365$	27047.8i	0.203023i
$366$	0	0
$367$	147637.	1.09613	0.548067	−	0.836434i	$-0.315364\pi$
$367$	147637.	1.09613	0.548067	+	0.836434i	$0.315364\pi$
$368$	− 1514.69i	− 0.0111848i
$369$	0	0
$370$	−95917.5	−0.700639
$371$	41640.9i	0.302533i
$372$	0	0
$373$	−221469.	−1.59182	−0.795912	−	0.605413i	$-0.793008\pi$
$373$	−221469.	−1.59182	−0.795912	+	0.605413i	$0.793008\pi$
$374$	− 9899.72i	− 0.0707750i
$375$	0	0
$376$	−66999.8	−0.473912
$377$	168971.i	1.18886i
$378$	0	0
$379$	100936.	0.702693	0.351347	−	0.936245i	$-0.385724\pi$
$379$	100936.	0.702693	0.351347	+	0.936245i	$0.385724\pi$
$380$	24789.2i	0.171670i
$381$	0	0
$382$	−109715.	−0.751862
$383$	111462.i	0.759854i	0.925016	+	0.379927i	$0.124051\pi$
$383$	111462.i	0.759854i	−0.925016	+	0.379927i	$0.875949\pi$
$384$	0	0
$385$	−19412.8	−0.130968
$386$	− 10722.4i	− 0.0719643i
$387$	0	0
$388$	−45708.1	−0.303620
$389$	− 208045.i	− 1.37486i	−0.726250	−	0.687430i	$-0.758739\pi$
$389$	− 208045.i	− 1.37486i	0.726250	−	0.687430i	$-0.241261\pi$
$390$	0	0
$391$	−2749.53	−0.0179848
$392$	7761.20i	0.0505076i
$393$	0	0
$394$	−48570.5	−0.312882
$395$	135668.i	0.869526i
$396$	0	0
$397$	−83732.9	−0.531270	−0.265635	−	0.964074i	$-0.585582\pi$
$397$	−83732.9	−0.531270	−0.265635	+	0.964074i	$0.585582\pi$
$398$	94502.5i	0.596592i
$399$	0	0
$400$	−37469.7	−0.234186
$401$	− 196215.i	− 1.22023i	−0.792312	−	0.610117i	$-0.791123\pi$
$401$	− 196215.i	− 1.22023i	0.792312	−	0.610117i	$-0.208877\pi$
$402$	0	0
$403$	24561.5	0.151232
$404$	− 85360.3i	− 0.522990i
$405$	0	0
$406$	61836.4	0.375139
$407$	− 29365.8i	− 0.177277i
$408$	0	0
$409$	−109093.	−0.652153	−0.326077	−	0.945343i	$-0.605727\pi$
$409$	−109093.	−0.652153	−0.326077	+	0.945343i	$0.605727\pi$
$410$	41070.9i	0.244324i
$411$	0	0
$412$	134094.	0.789981
$413$	− 15320.9i	− 0.0898222i
$414$	0	0
$415$	28758.1	0.166980
$416$	− 25911.1i	− 0.149727i
$417$	0	0
$418$	−7589.37	−0.0434363
$419$	115118.i	0.655715i	0.944727	+	0.327858i	$0.106327\pi$
$419$	115118.i	0.655715i	−0.944727	+	0.327858i	$0.893673\pi$
$420$	0	0
$421$	41327.3	0.233170	0.116585	−	0.993181i	$-0.462805\pi$
$421$	41327.3	0.233170	0.116585	+	0.993181i	$0.462805\pi$
$422$	− 125366.i	− 0.703969i
$423$	0	0
$424$	−50875.5	−0.282994
$425$	68016.4i	0.376562i
$426$	0	0
$427$	−127599.	−0.699829
$428$	102402.i	0.559013i
$429$	0	0
$430$	−309784.	−1.67541
$431$	69512.7i	0.374205i	0.982340	+	0.187103i	$0.0599097\pi$
$431$	69512.7i	0.374205i	−0.982340	+	0.187103i	$0.940090\pi$
$432$	0	0
$433$	−93109.8	−0.496615	−0.248307	−	0.968681i	$-0.579874\pi$
$433$	−93109.8	−0.496615	−0.248307	+	0.968681i	$0.579874\pi$
$434$	− 8988.47i	− 0.0477206i
$435$	0	0
$436$	15448.7	0.0812680
$437$	2107.86i	0.0110377i
$438$	0	0
$439$	77844.3	0.403922	0.201961	−	0.979394i	$-0.435269\pi$
$439$	77844.3	0.403922	0.201961	+	0.979394i	$0.435269\pi$
$440$	− 23717.8i	− 0.122510i
$441$	0	0
$442$	−47034.8	−0.240755
$443$	− 166627.i	− 0.849058i	−0.905414	−	0.424529i	$-0.860440\pi$
$443$	− 166627.i	− 0.849058i	0.905414	−	0.424529i	$-0.139560\pi$
$444$	0	0
$445$	36720.7	0.185435
$446$	− 35642.5i	− 0.179184i
$447$	0	0
$448$	−9482.37	−0.0472456
$449$	9590.42i	0.0475713i	0.999717	+	0.0237856i	$0.00757192\pi$
$449$	9590.42i	0.0475713i	−0.999717	+	0.0237856i	$0.992428\pi$
$450$	0	0
$451$	−12574.1	−0.0618193
$452$	− 155959.i	− 0.763368i
$453$	0	0
$454$	−109432.	−0.530927
$455$	92232.4i	0.445513i
$456$	0	0
$457$	20636.1	0.0988086	0.0494043	−	0.998779i	$-0.484268\pi$
$457$	20636.1	0.0988086	0.0494043	+	0.998779i	$0.484268\pi$
$458$	204429.i	0.974566i
$459$	0	0
$460$	−6587.35	−0.0311312
$461$	367647.i	1.72993i	0.501831	+	0.864966i	$0.332660\pi$
$461$	367647.i	1.72993i	−0.501831	+	0.864966i	$0.667340\pi$
$462$	0	0
$463$	155758.	0.726588	0.363294	−	0.931675i	$-0.381652\pi$
$463$	155758.	0.726588	0.363294	+	0.931675i	$0.381652\pi$
$464$	75549.6i	0.350911i
$465$	0	0
$466$	−178879.	−0.823734
$467$	− 297597.i	− 1.36457i	−0.731087	−	0.682284i	$-0.760987\pi$
$467$	− 297597.i	− 1.36457i	0.731087	−	0.682284i	$-0.239013\pi$
$468$	0	0
$469$	90125.4	0.409733
$470$	291380.i	1.31906i
$471$	0	0
$472$	18718.5	0.0840210
$473$	− 94842.4i	− 0.423917i
$474$	0	0
$475$	52143.1	0.231105
$476$	17212.8i	0.0759691i
$477$	0	0
$478$	136055.	0.595467
$479$	203202.i	0.885639i	0.896611	+	0.442819i	$0.146022\pi$
$479$	203202.i	0.885639i	−0.896611	+	0.442819i	$0.853978\pi$
$480$	0	0
$481$	−139520.	−0.603042
$482$	293658.i	1.26400i
$483$	0	0
$484$	−109867.	−0.469002
$485$	198783.i	0.845076i
$486$	0	0
$487$	312755.	1.31870	0.659351	−	0.751835i	$-0.270831\pi$
$487$	312755.	1.31870	0.659351	+	0.751835i	$0.270831\pi$
$488$	− 155896.i	− 0.654630i
$489$	0	0
$490$	33753.2	0.140580
$491$	26245.9i	0.108867i	0.998517	+	0.0544337i	$0.0173354\pi$
$491$	26245.9i	0.108867i	−0.998517	+	0.0544337i	$0.982665\pi$
$492$	0	0
$493$	137141.	0.564251
$494$	36058.0i	0.147757i
$495$	0	0
$496$	10981.8	0.0446386
$497$	101641.i	0.411485i
$498$	0	0
$499$	137671.	0.552891	0.276446	−	0.961030i	$-0.410843\pi$
$499$	137671.	0.552891	0.276446	+	0.961030i	$0.410843\pi$
$500$	− 11004.2i	− 0.0440169i
$501$	0	0
$502$	−156016.	−0.619102
$503$	336996.i	1.33195i	0.745973	+	0.665977i	$0.231985\pi$
$503$	336996.i	1.33195i	−0.745973	+	0.665977i	$0.768015\pi$
$504$	0	0
$505$	−371229.	−1.45566
$506$	− 2016.76i	− 0.00787686i
$507$	0	0
$508$	218034.	0.844885
$509$	− 299188.i	− 1.15480i	−0.816460	−	0.577402i	$-0.804067\pi$
$509$	− 299188.i	− 1.15480i	0.816460	−	0.577402i	$-0.195933\pi$
$510$	0	0
$511$	14398.0	0.0551393
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	−63386.9	−0.239924
$515$	− 583172.i	− 2.19878i
$516$	0	0
$517$	−89207.9	−0.333751
$518$	51058.6i	0.190287i
$519$	0	0
$520$	−112686.	−0.416739
$521$	248631.i	0.915967i	0.888961	+	0.457983i	$0.151428\pi$
$521$	248631.i	0.915967i	−0.888961	+	0.457983i	$0.848572\pi$
$522$	0	0
$523$	45984.3	0.168115	0.0840574	−	0.996461i	$-0.473212\pi$
$523$	45984.3	0.168115	0.0840574	+	0.996461i	$0.473212\pi$
$524$	143721.i	0.523429i
$525$	0	0
$526$	16216.8	0.0586129
$527$	− 19934.6i	− 0.0717772i
$528$	0	0
$529$	279281.	0.997998
$530$	221256.i	0.787667i
$531$	0	0
$532$	13195.7	0.0466241
$533$	59741.1i	0.210290i
$534$	0	0
$535$	445344.	1.55592
$536$	110112.i	0.383271i
$537$	0	0
$538$	−345198.	−1.19263
$539$	10333.8i	0.0355698i
$540$	0	0
$541$	306453.	1.04705	0.523527	−	0.852009i	$-0.324616\pi$
$541$	306453.	1.04705	0.523527	+	0.852009i	$0.324616\pi$
$542$	− 392753.i	− 1.33697i
$543$	0	0
$544$	−21030.0	−0.0710626
$545$	− 67185.9i	− 0.226196i
$546$	0	0
$547$	−17480.8	−0.0584234	−0.0292117	−	0.999573i	$-0.509300\pi$
$547$	−17480.8	−0.0584234	−0.0292117	+	0.999573i	$0.509300\pi$
$548$	− 237801.i	− 0.791867i
$549$	0	0
$550$	−49889.5	−0.164924
$551$	− 105135.i	− 0.346294i
$552$	0	0
$553$	72218.4	0.236155
$554$	17500.7i	0.0570211i
$555$	0	0
$556$	98130.7	0.317435
$557$	− 500038.i	− 1.61173i	−0.592099	−	0.805866i	$-0.701700\pi$
$557$	− 500038.i	− 1.61173i	0.592099	−	0.805866i	$-0.298300\pi$
$558$	0	0
$559$	−450608.	−1.44203
$560$	41238.5i	0.131500i
$561$	0	0
$562$	280204.	0.887160
$563$	− 40744.8i	− 0.128545i	−0.997932	−	0.0642725i	$-0.979527\pi$
$563$	− 40744.8i	− 0.128545i	0.997932	−	0.0642725i	$-0.0204727\pi$
$564$	0	0
$565$	−678261.	−2.12471
$566$	− 59054.7i	− 0.184341i
$567$	0	0
$568$	−124181.	−0.384909
$569$	− 527036.i	− 1.62786i	−0.580966	−	0.813928i	$-0.697325\pi$
$569$	− 527036.i	− 1.62786i	0.580966	−	0.813928i	$-0.302675\pi$
$570$	0	0
$571$	393264.	1.20618	0.603090	−	0.797673i	$-0.293936\pi$
$571$	393264.	1.20618	0.603090	+	0.797673i	$0.293936\pi$
$572$	− 34499.7i	− 0.105444i
$573$	0	0
$574$	21862.8	0.0663562
$575$	13856.2i	0.0419092i
$576$	0	0
$577$	−92157.9	−0.276810	−0.138405	−	0.990376i	$-0.544197\pi$
$577$	−92157.9	−0.276810	−0.138405	+	0.990376i	$0.544197\pi$
$578$	− 198059.i	− 0.592841i
$579$	0	0
$580$	328563.	0.976703
$581$	− 15308.5i	− 0.0453502i
$582$	0	0
$583$	−67738.9	−0.199297
$584$	17591.0i	0.0515781i
$585$	0	0
$586$	94644.3	0.275613
$587$	− 643433.i	− 1.86736i	−0.358112	−	0.933679i	$-0.616579\pi$
$587$	− 643433.i	− 1.86736i	0.358112	−	0.933679i	$-0.383421\pi$
$588$	0	0
$589$	−15282.3	−0.0440514
$590$	− 81406.3i	− 0.233859i
$591$	0	0
$592$	−62381.7	−0.177997
$593$	351311.i	0.999038i	0.866303	+	0.499519i	$0.166490\pi$
$593$	351311.i	0.999038i	−0.866303	+	0.499519i	$0.833510\pi$
$594$	0	0
$595$	74857.7	0.211448
$596$	20530.7i	0.0577978i
$597$	0	0
$598$	−9581.87	−0.0267946
$599$	− 251755.i	− 0.701656i	−0.936440	−	0.350828i	$-0.885900\pi$
$599$	− 251755.i	− 0.701656i	0.936440	−	0.350828i	$-0.114100\pi$
$600$	0	0
$601$	195702.	0.541810	0.270905	−	0.962606i	$-0.412677\pi$
$601$	195702.	0.541810	0.270905	+	0.962606i	$0.412677\pi$
$602$	164904.i	0.455027i
$603$	0	0
$604$	−55353.4	−0.151730
$605$	477806.i	1.30539i
$606$	0	0
$607$	15312.0	0.0415581	0.0207790	−	0.999784i	$-0.493385\pi$
$607$	15312.0	0.0415581	0.0207790	+	0.999784i	$0.493385\pi$
$608$	16122.1i	0.0436128i
$609$	0	0
$610$	−677987.	−1.82206
$611$	423837.i	1.13532i
$612$	0	0
$613$	−517619.	−1.37749	−0.688747	−	0.725002i	$-0.741839\pi$
$613$	−517619.	−1.37749	−0.688747	+	0.725002i	$0.741839\pi$
$614$	− 428991.i	− 1.13792i
$615$	0	0
$616$	−12625.4	−0.0332725
$617$	79270.8i	0.208230i	0.994565	+	0.104115i	$0.0332010\pi$
$617$	79270.8i	0.208230i	−0.994565	+	0.104115i	$0.966799\pi$
$618$	0	0
$619$	454097.	1.18513	0.592567	−	0.805521i	$-0.298115\pi$
$619$	454097.	1.18513	0.592567	+	0.805521i	$0.298115\pi$
$620$	− 47759.5i	− 0.124244i
$621$	0	0
$622$	250037.	0.646283
$623$	− 19547.1i	− 0.0503623i
$624$	0	0
$625$	−413772.	−1.05926
$626$	− 431670.i	− 1.10155i
$627$	0	0
$628$	12559.4	0.0318456
$629$	113238.i	0.286213i
$630$	0	0
$631$	589600.	1.48081	0.740404	−	0.672162i	$-0.234634\pi$
$631$	589600.	1.48081	0.740404	+	0.672162i	$0.234634\pi$
$632$	88234.0i	0.220903i
$633$	0	0
$634$	94483.9	0.235060
$635$	− 948224.i	− 2.35160i
$636$	0	0
$637$	49097.0	0.120997
$638$	100592.i	0.247127i
$639$	0	0
$640$	−50383.8	−0.123007
$641$	− 784231.i	− 1.90866i	−0.298759	−	0.954329i	$-0.596573\pi$
$641$	− 784231.i	− 1.90866i	0.298759	−	0.954329i	$-0.403427\pi$
$642$	0	0
$643$	−2503.23	−0.00605452	−0.00302726	−	0.999995i	$-0.500964\pi$
$643$	−2503.23	−0.00605452	−0.00302726	+	0.999995i	$0.500964\pi$
$644$	3506.57i	0.00845493i
$645$	0	0
$646$	29265.4	0.0701278
$647$	661672.i	1.58064i	0.612691	+	0.790322i	$0.290087\pi$
$647$	661672.i	1.58064i	−0.612691	+	0.790322i	$0.709913\pi$
$648$	0	0
$649$	24923.1	0.0591714
$650$	237031.i	0.561021i
$651$	0	0
$652$	−222809.	−0.524128
$653$	− 3986.16i	− 0.00934821i	−0.999989	−	0.00467410i	$-0.998512\pi$
$653$	− 3986.16i	− 0.00934821i	0.999989	−	0.00467410i	$-0.00148782\pi$
$654$	0	0
$655$	625038.	1.45688
$656$	26711.2i	0.0620705i
$657$	0	0
$658$	155107.	0.358244
$659$	− 124318.i	− 0.286261i	−0.989704	−	0.143130i	$-0.954283\pi$
$659$	− 124318.i	− 0.286261i	0.989704	−	0.143130i	$-0.0457168\pi$
$660$	0	0
$661$	112154.	0.256691	0.128346	−	0.991729i	$-0.459033\pi$
$661$	112154.	0.256691	0.128346	+	0.991729i	$0.459033\pi$
$662$	− 591120.i	− 1.34884i
$663$	0	0
$664$	18703.4	0.0424212
$665$	− 57387.8i	− 0.129771i
$666$	0	0
$667$	27938.1	0.0627980
$668$	− 327665.i	− 0.734306i
$669$	0	0
$670$	478874.	1.06677
$671$	− 207570.i	− 0.461020i
$672$	0	0
$673$	1349.83	0.00298023	0.00149012	−	0.999999i	$-0.499526\pi$
$673$	1349.83	0.00298023	0.00149012	+	0.999999i	$0.499526\pi$
$674$	− 576695.i	− 1.26948i
$675$	0	0
$676$	64576.0	0.141312
$677$	468915.i	1.02310i	0.859254	+	0.511549i	$0.170928\pi$
$677$	468915.i	1.02310i	−0.859254	+	0.511549i	$0.829072\pi$
$678$	0	0
$679$	105816.	0.229515
$680$	91458.6i	0.197791i
$681$	0	0
$682$	14621.9	0.0314365
$683$	− 485991.i	− 1.04181i	−0.853616	−	0.520903i	$-0.825595\pi$
$683$	− 485991.i	− 1.04181i	0.853616	−	0.520903i	$-0.174405\pi$
$684$	0	0
$685$	−1.03419e6	−2.20403
$686$	− 17967.4i	− 0.0381802i
$687$	0	0
$688$	−201474.	−0.425639
$689$	321836.i	0.677947i
$690$	0	0
$691$	−7451.91	−0.0156067	−0.00780335	−	0.999970i	$-0.502484\pi$
$691$	−7451.91	−0.0156067	−0.00780335	+	0.999970i	$0.502484\pi$
$692$	− 104195.i	− 0.217588i
$693$	0	0
$694$	122100.	0.253512
$695$	− 426767.i	− 0.883530i
$696$	0	0
$697$	48487.2	0.0998071
$698$	− 574481.i	− 1.17914i
$699$	0	0
$700$	86743.6	0.177028
$701$	685495.i	1.39498i	0.716594	+	0.697491i	$0.245700\pi$
$701$	685495.i	1.39498i	−0.716594	+	0.697491i	$0.754300\pi$
$702$	0	0
$703$	86810.7	0.175656
$704$	− 15425.3i	− 0.0311235i
$705$	0	0
$706$	491511.	0.986107
$707$	197612.i	0.395343i
$708$	0	0
$709$	−372506.	−0.741039	−0.370520	−	0.928825i	$-0.620820\pi$
$709$	−372506.	−0.741039	−0.370520	+	0.928825i	$0.620820\pi$
$710$	540059.i	1.07133i
$711$	0	0
$712$	23881.9	0.0471096
$713$	− 4061.05i	− 0.00798840i
$714$	0	0
$715$	−150038.	−0.293487
$716$	281140.i	0.548399i
$717$	0	0
$718$	538127.	1.04385
$719$	− 981227.i	− 1.89807i	−0.315174	−	0.949034i	$-0.602063\pi$
$719$	− 981227.i	− 1.89807i	0.315174	−	0.949034i	$-0.397937\pi$
$720$	0	0
$721$	−310433.	−0.597169
$722$	346168.i	0.664068i
$723$	0	0
$724$	112373.	0.214380
$725$	− 691119.i	− 1.31485i
$726$	0	0
$727$	290468.	0.549578	0.274789	−	0.961505i	$-0.411392\pi$
$727$	290468.	0.549578	0.274789	+	0.961505i	$0.411392\pi$
$728$	59985.0i	0.113183i
$729$	0	0
$730$	76502.7	0.143559
$731$	365723.i	0.684412i
$732$	0	0
$733$	−243787.	−0.453736	−0.226868	−	0.973926i	$-0.572849\pi$
$733$	−243787.	−0.453736	−0.226868	+	0.973926i	$0.572849\pi$
$734$	− 417581.i	− 0.775084i
$735$	0	0
$736$	−4284.20	−0.00790887
$737$	146610.i	0.269916i
$738$	0	0
$739$	133565.	0.244570	0.122285	−	0.992495i	$-0.460978\pi$
$739$	133565.	0.244570	0.122285	+	0.992495i	$0.460978\pi$
$740$	271296.i	0.495427i
$741$	0	0
$742$	117778.	0.213923
$743$	− 96297.3i	− 0.174436i	−0.996189	−	0.0872181i	$-0.972202\pi$
$743$	− 96297.3i	− 0.174436i	0.996189	−	0.0872181i	$-0.0277977\pi$
$744$	0	0
$745$	89287.3	0.160871
$746$	626408.i	1.12559i
$747$	0	0
$748$	−28000.6	−0.0500455
$749$	− 237065.i	− 0.422574i
$750$	0	0
$751$	−123750.	−0.219415	−0.109707	−	0.993964i	$-0.534991\pi$
$751$	−123750.	−0.219415	−0.109707	+	0.993964i	$0.534991\pi$
$752$	189504.i	0.335107i
$753$	0	0
$754$	477923.	0.840650
$755$	240730.i	0.422315i
$756$	0	0
$757$	644846.	1.12529	0.562645	−	0.826699i	$-0.309784\pi$
$757$	644846.	1.12529	0.562645	+	0.826699i	$0.309784\pi$
$758$	− 285489.i	− 0.496879i
$759$	0	0
$760$	70114.4	0.121389
$761$	359205.i	0.620259i	0.950694	+	0.310129i	$0.100372\pi$
$761$	359205.i	0.620259i	−0.950694	+	0.310129i	$0.899628\pi$
$762$	0	0
$763$	−35764.3	−0.0614328
$764$	310320.i	0.531647i
$765$	0	0
$766$	315263.	0.537298
$767$	− 118412.i	− 0.201283i
$768$	0	0
$769$	−390626.	−0.660554	−0.330277	−	0.943884i	$-0.607142\pi$
$769$	−390626.	−0.660554	−0.330277	+	0.943884i	$0.607142\pi$
$770$	54907.6i	0.0926085i
$771$	0	0
$772$	−30327.5	−0.0508864
$773$	− 314838.i	− 0.526900i	−0.964673	−	0.263450i	$-0.915140\pi$
$773$	− 314838.i	− 0.526900i	0.964673	−	0.263450i	$-0.0848604\pi$
$774$	0	0
$775$	−100460.	−0.167259
$776$	129282.i	0.214692i
$777$	0	0
$778$	−588441.	−0.972174
$779$	− 37171.4i	− 0.0612540i
$780$	0	0
$781$	−165343.	−0.271071
$782$	7776.85i	0.0127172i
$783$	0	0
$784$	21952.0	0.0357143
$785$	− 54620.3i	− 0.0886370i
$786$	0	0
$787$	−691316.	−1.11616	−0.558081	−	0.829787i	$-0.688462\pi$
$787$	−691316.	−1.11616	−0.558081	+	0.829787i	$0.688462\pi$
$788$	137378.i	0.221241i
$789$	0	0
$790$	383727.	0.614848
$791$	361050.i	0.577052i
$792$	0	0
$793$	−986191.	−1.56825
$794$	236832.i	0.375664i
$795$	0	0
$796$	267294.	0.421854
$797$	667905.i	1.05147i	0.850647	+	0.525737i	$0.176210\pi$
$797$	667905.i	1.05147i	−0.850647	+	0.525737i	$0.823790\pi$
$798$	0	0
$799$	343995.	0.538839
$800$	105980.i	0.165594i
$801$	0	0
$802$	−554979.	−0.862835
$803$	23421.8i	0.0363236i
$804$	0	0
$805$	15249.9	0.0235329
$806$	− 69470.3i	− 0.106937i
$807$	0	0
$808$	−241435.	−0.369810
$809$	185296.i	0.283120i	0.989930	+	0.141560i	$0.0452118\pi$
$809$	185296.i	0.283120i	−0.989930	+	0.141560i	$0.954788\pi$
$810$	0	0
$811$	−902916.	−1.37279	−0.686397	−	0.727227i	$-0.740809\pi$
$811$	−902916.	−1.37279	−0.686397	+	0.727227i	$0.740809\pi$
$812$	− 174900.i	− 0.265263i
$813$	0	0
$814$	−83058.9	−0.125354
$815$	968989.i	1.45883i
$816$	0	0
$817$	280372.	0.420040
$818$	308561.i	0.461142i
$819$	0	0
$820$	116166.	0.172763
$821$	− 697811.i	− 1.03526i	−0.855603	−	0.517632i	$-0.826813\pi$
$821$	− 697811.i	− 1.03526i	0.855603	−	0.517632i	$-0.173187\pi$
$822$	0	0
$823$	−1.07628e6	−1.58900	−0.794501	−	0.607263i	$-0.792268\pi$
$823$	−1.07628e6	−1.58900	−0.794501	+	0.607263i	$0.792268\pi$
$824$	− 379276.i	− 0.558601i
$825$	0	0
$826$	−43334.0	−0.0635139
$827$	− 224433.i	− 0.328153i	−0.986448	−	0.164076i	$-0.947536\pi$
$827$	− 224433.i	− 0.328153i	0.986448	−	0.164076i	$-0.0524643\pi$
$828$	0	0
$829$	211296.	0.307455	0.153727	−	0.988113i	$-0.450872\pi$
$829$	211296.	0.307455	0.153727	+	0.988113i	$0.450872\pi$
$830$	− 81340.2i	− 0.118073i
$831$	0	0
$832$	−73287.6	−0.105873
$833$	− 39848.1i	− 0.0574272i
$834$	0	0
$835$	−1.42500e6	−2.04382
$836$	21466.0i	0.0307141i
$837$	0	0
$838$	325603.	0.463661
$839$	244959.i	0.347992i	0.984746	+	0.173996i	$0.0556679\pi$
$839$	244959.i	0.347992i	−0.984746	+	0.173996i	$0.944332\pi$
$840$	0	0
$841$	−686212.	−0.970212
$842$	− 116891.i	− 0.164876i
$843$	0	0
$844$	−354588.	−0.497781
$845$	− 280839.i	− 0.393318i
$846$	0	0
$847$	254345.	0.354532
$848$	143898.i	0.200107i
$849$	0	0
$850$	192380.	0.266269
$851$	23068.6i	0.0318539i
$852$	0	0
$853$	−608980.	−0.836961	−0.418480	−	0.908226i	$-0.637437\pi$
$853$	−608980.	−0.836961	−0.418480	+	0.908226i	$0.637437\pi$
$854$	360905.i	0.494854i
$855$	0	0
$856$	289638.	0.395282
$857$	− 1.24963e6i	− 1.70145i	−0.525613	−	0.850724i	$-0.676164\pi$
$857$	− 1.24963e6i	− 1.70145i	0.525613	−	0.850724i	$-0.323836\pi$
$858$	0	0
$859$	510641.	0.692037	0.346019	−	0.938228i	$-0.387533\pi$
$859$	510641.	0.692037	0.346019	+	0.938228i	$0.387533\pi$
$860$	876202.i	1.18470i
$861$	0	0
$862$	196612.	0.264603
$863$	828820.i	1.11286i	0.830896	+	0.556428i	$0.187828\pi$
$863$	828820.i	1.11286i	−0.830896	+	0.556428i	$0.812172\pi$
$864$	0	0
$865$	−453140.	−0.605620
$866$	263354.i	0.351160i
$867$	0	0
$868$	−25423.2	−0.0337436
$869$	117480.i	0.155570i
$870$	0	0
$871$	696563.	0.918172
$872$	− 43695.6i	− 0.0574651i
$873$	0	0
$874$	5961.92	0.00780483
$875$	25475.1i	0.0332736i
$876$	0	0
$877$	−1.16328e6	−1.51247	−0.756233	−	0.654302i	$-0.772963\pi$
$877$	−1.16328e6	−1.51247	−0.756233	+	0.654302i	$0.772963\pi$
$878$	− 220177.i	− 0.285616i
$879$	0	0
$880$	−67084.2	−0.0866273
$881$	1.16834e6i	1.50529i	0.658429	+	0.752643i	$0.271221\pi$
$881$	1.16834e6i	1.50529i	−0.658429	+	0.752643i	$0.728779\pi$
$882$	0	0
$883$	879010.	1.12739	0.563693	−	0.825985i	$-0.309380\pi$
$883$	879010.	1.12739	0.563693	+	0.825985i	$0.309380\pi$
$884$	133034.i	0.170239i
$885$	0	0
$886$	−471292.	−0.600375
$887$	− 715518.i	− 0.909438i	−0.890635	−	0.454719i	$-0.849740\pi$
$887$	− 715518.i	− 0.909438i	0.890635	−	0.454719i	$-0.150260\pi$
$888$	0	0
$889$	−504756.	−0.638673
$890$	− 103862.i	− 0.131122i
$891$	0	0
$892$	−100812.	−0.126702
$893$	− 263715.i	− 0.330698i
$894$	0	0
$895$	1.22267e6	1.52638
$896$	26820.2i	0.0334077i
$897$	0	0
$898$	27125.8	0.0336380
$899$	202556.i	0.250626i
$900$	0	0
$901$	261208.	0.321764
$902$	35565.0i	0.0437129i
$903$	0	0
$904$	−441119.	−0.539783
$905$	− 488706.i	− 0.596692i
$906$	0	0
$907$	−319774.	−0.388713	−0.194357	−	0.980931i	$-0.562262\pi$
$907$	−319774.	−0.388713	−0.194357	+	0.980931i	$0.562262\pi$
$908$	309522.i	0.375422i
$909$	0	0
$910$	260873.	0.315025
$911$	1.15028e6i	1.38601i	0.720934	+	0.693003i	$0.243713\pi$
$911$	1.15028e6i	1.38601i	−0.720934	+	0.693003i	$0.756287\pi$
$912$	0	0
$913$	24902.8	0.0298750
$914$	− 58367.7i	− 0.0698683i
$915$	0	0
$916$	578212.	0.689122
$917$	− 332719.i	− 0.395675i
$918$	0	0
$919$	276537.	0.327433	0.163716	−	0.986507i	$-0.447652\pi$
$919$	276537.	0.327433	0.163716	+	0.986507i	$0.447652\pi$
$920$	18631.8i	0.0220130i
$921$	0	0
$922$	1.03986e6	1.22325
$923$	785562.i	0.922098i
$924$	0	0
$925$	570660.	0.666951
$926$	− 440550.i	− 0.513775i
$927$	0	0
$928$	213687.	0.248131
$929$	1.25259e6i	1.45137i	0.688025	+	0.725687i	$0.258478\pi$
$929$	1.25259e6i	1.45137i	−0.688025	+	0.725687i	$0.741522\pi$
$930$	0	0
$931$	−30548.5	−0.0352445
$932$	505946.i	0.582468i
$933$	0	0
$934$	−841732.	−0.964895
$935$	121774.i	0.139293i
$936$	0	0
$937$	−1.25465e6	−1.42904	−0.714520	−	0.699615i	$-0.753355\pi$
$937$	−1.25465e6	−1.42904	−0.714520	+	0.699615i	$0.753355\pi$
$938$	− 254913.i	− 0.289725i
$939$	0	0
$940$	824147.	0.932715
$941$	1.16300e6i	1.31341i	0.754148	+	0.656704i	$0.228050\pi$
$941$	1.16300e6i	1.31341i	−0.754148	+	0.656704i	$0.771950\pi$
$942$	0	0
$943$	9877.75	0.0111080
$944$	− 52944.0i	− 0.0594118i
$945$	0	0
$946$	−268255.	−0.299754
$947$	− 63206.8i	− 0.0704797i	−0.999379	−	0.0352399i	$-0.988780\pi$
$947$	− 63206.8i	− 0.0704797i	0.999379	−	0.0352399i	$-0.0112195\pi$
$948$	0	0
$949$	111280.	0.123562
$950$	− 147483.i	− 0.163416i
$951$	0	0
$952$	48685.1	0.0537183
$953$	933092.i	1.02740i	0.857971	+	0.513699i	$0.171725\pi$
$953$	933092.i	1.02740i	−0.857971	+	0.513699i	$0.828275\pi$
$954$	0	0
$955$	1.34957e6	1.47975
$956$	− 384821.i	− 0.421059i
$957$	0	0
$958$	574741.	0.626241
$959$	550517.i	0.598595i
$960$	0	0
$961$	−894078.	−0.968118
$962$	394623.i	0.426415i
$963$	0	0
$964$	830592.	0.893786
$965$	131893.i	0.141634i
$966$	0	0
$967$	124623.	0.133274	0.0666370	−	0.997777i	$-0.478773\pi$
$967$	124623.	0.133274	0.0666370	+	0.997777i	$0.478773\pi$
$968$	310750.i	0.331635i
$969$	0	0
$970$	562244.	0.597559
$971$	− 514717.i	− 0.545922i	−0.962025	−	0.272961i	$-0.911997\pi$
$971$	− 514717.i	− 0.545922i	0.962025	−	0.272961i	$-0.0880029\pi$
$972$	0	0
$973$	−227176.	−0.239959
$974$	− 884605.i	− 0.932463i
$975$	0	0
$976$	−440941.	−0.462893
$977$	1.00025e6i	1.04790i	0.851749	+	0.523950i	$0.175542\pi$
$977$	1.00025e6i	1.04790i	−0.851749	+	0.523950i	$0.824458\pi$
$978$	0	0
$979$	31797.9	0.0331767
$980$	− 95468.5i	− 0.0994049i
$981$	0	0
$982$	74234.6	0.0769809
$983$	− 337021.i	− 0.348779i	−0.984677	−	0.174389i	$-0.944205\pi$
$983$	− 337021.i	− 0.348779i	0.984677	−	0.174389i	$-0.0557951\pi$
$984$	0	0
$985$	597452.	0.615788
$986$	− 387892.i	− 0.398986i
$987$	0	0
$988$	101987.	0.104480
$989$	74504.6i	0.0761712i
$990$	0	0
$991$	1.26960e6	1.29277	0.646385	−	0.763012i	$-0.276280\pi$
$991$	1.26960e6	1.29277	0.646385	+	0.763012i	$0.276280\pi$
$992$	− 31061.2i	− 0.0315642i
$993$	0	0
$994$	287483.	0.290964
$995$	− 1.16245e6i	− 1.17416i
$996$	0	0
$997$	750408.	0.754931	0.377465	−	0.926024i	$-0.376796\pi$
$997$	750408.	0.754931	0.377465	+	0.926024i	$0.376796\pi$
$998$	− 389391.i	− 0.390953i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.5.b.b.323.4	✓	8
3.2	odd	2	inner	378.5.b.b.323.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.5.b.b.323.4	✓	8	1.1	even	1	trivial
378.5.b.b.323.5	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)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} +34.7917i q^{5} +18.5203 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} +34.7917i q^{5} +18.5203 q^{7} +22.6274i q^{8} +98.4059 q^{10} +30.1276i q^{11} +143.140 q^{13} -52.3832i q^{14} +64.0000 q^{16} -116.175i q^{17} -89.0628 q^{19} -278.334i q^{20} +85.2137 q^{22} -23.6671i q^{23} -585.464 q^{25} -404.860i q^{26} -148.162 q^{28} +1180.46i q^{29} +171.591 q^{31} -181.019i q^{32} -328.593 q^{34} +644.352i q^{35} -974.714 q^{37} +251.908i q^{38} -787.247 q^{40} +417.362i q^{41} -3148.03 q^{43} -241.021i q^{44} -66.9407 q^{46} +2961.00i q^{47} +343.000 q^{49} +1655.94i q^{50} -1145.12 q^{52} +2248.40i q^{53} -1048.19 q^{55} +419.066i q^{56} +3338.85 q^{58} -827.250i q^{59} -6889.71 q^{61} -485.332i q^{62} -512.000 q^{64} +4980.08i q^{65} +4866.31 q^{67} +929.402i q^{68} +1822.50 q^{70} +5488.08i q^{71} +777.420 q^{73} +2756.91i q^{74} +712.502 q^{76} +557.971i q^{77} +3899.43 q^{79} +2226.67i q^{80} +1180.48 q^{82} -826.579i q^{83} +4041.94 q^{85} +8903.96i q^{86} -681.709 q^{88} -1055.44i q^{89} +2650.99 q^{91} +189.337i q^{92} +8374.98 q^{94} -3098.65i q^{95} +5713.52 q^{97} -970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{4}+O(q^{10}) \) 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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