LMFDB - Embedded newform 342.6.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,6,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("342.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	342.a (trivial)

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:	yes
Analytic conductor:	$54.8512663760$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1441}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 360 $ x^2 - x - 360
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-18.4803$ of defining polynomial
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} +79.4408 q^{5} +132.921 q^{7} +64.0000 q^{8} +O(q^{10})$	\(q+4.00000 q^{2} +16.0000 q^{4} +79.4408 q^{5} +132.921 q^{7} +64.0000 q^{8} +317.763 q^{10} -311.520 q^{11} +901.401 q^{13} +531.684 q^{14} +256.000 q^{16} +157.803 q^{17} +361.000 q^{19} +1271.05 q^{20} -1246.08 q^{22} +2522.53 q^{23} +3185.83 q^{25} +3605.61 q^{26} +2126.74 q^{28} -4738.28 q^{29} -6587.76 q^{31} +1024.00 q^{32} +631.210 q^{34} +10559.3 q^{35} +8508.60 q^{37} +1444.00 q^{38} +5084.21 q^{40} -19741.1 q^{41} +10985.0 q^{43} -4984.32 q^{44} +10090.1 q^{46} -15085.5 q^{47} +860.995 q^{49} +12743.3 q^{50} +14422.4 q^{52} -21699.6 q^{53} -24747.4 q^{55} +8506.94 q^{56} -18953.1 q^{58} +40676.1 q^{59} +6151.79 q^{61} -26351.0 q^{62} +4096.00 q^{64} +71608.0 q^{65} +62760.3 q^{67} +2524.84 q^{68} +42237.4 q^{70} +55311.0 q^{71} -48528.1 q^{73} +34034.4 q^{74} +5776.00 q^{76} -41407.5 q^{77} +31017.6 q^{79} +20336.8 q^{80} -78964.4 q^{82} -41068.7 q^{83} +12536.0 q^{85} +43939.8 q^{86} -19937.3 q^{88} +17065.6 q^{89} +119815. q^{91} +40360.5 q^{92} -60341.9 q^{94} +28678.1 q^{95} +139045. q^{97} +3443.98 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 32 q^{4} + 45 q^{5} + 114 q^{7} + 128 q^{8}+O(q^{10}) $ 2 * q + 8 * q^2 + 32 * q^4 + 45 * q^5 + 114 * q^7 + 128 * q^8	$ 2 q + 8 q^{2} + 32 q^{4} + 45 q^{5} + 114 q^{7} + 128 q^{8} + 180 q^{10} - 661 q^{11} + 1613 q^{13} + 456 q^{14} + 512 q^{16} - 64 q^{17} + 722 q^{19} + 720 q^{20} - 2644 q^{22} + 3185 q^{23} + 1247 q^{25} + 6452 q^{26} + 1824 q^{28} + 2481 q^{29} - 1180 q^{31} + 2048 q^{32} - 256 q^{34} + 11211 q^{35} + 10488 q^{37} + 2888 q^{38} + 2880 q^{40} - 16630 q^{41} + 11303 q^{43} - 10576 q^{44} + 12740 q^{46} + 12155 q^{47} - 15588 q^{49} + 4988 q^{50} + 25808 q^{52} - 20585 q^{53} - 12711 q^{55} + 7296 q^{56} + 9924 q^{58} + 78581 q^{59} + 43621 q^{61} - 4720 q^{62} + 8192 q^{64} + 47100 q^{65} + 7805 q^{67} - 1024 q^{68} + 44844 q^{70} + 62488 q^{71} + 16218 q^{73} + 41952 q^{74} + 11552 q^{76} - 34795 q^{77} + 67122 q^{79} + 11520 q^{80} - 66520 q^{82} + 10714 q^{83} + 20175 q^{85} + 45212 q^{86} - 42304 q^{88} - 128188 q^{89} + 106351 q^{91} + 50960 q^{92} + 48620 q^{94} + 16245 q^{95} + 178558 q^{97} - 62352 q^{98}+O(q^{100}) $ 2 * q + 8 * q^2 + 32 * q^4 + 45 * q^5 + 114 * q^7 + 128 * q^8 + 180 * q^10 - 661 * q^11 + 1613 * q^13 + 456 * q^14 + 512 * q^16 - 64 * q^17 + 722 * q^19 + 720 * q^20 - 2644 * q^22 + 3185 * q^23 + 1247 * q^25 + 6452 * q^26 + 1824 * q^28 + 2481 * q^29 - 1180 * q^31 + 2048 * q^32 - 256 * q^34 + 11211 * q^35 + 10488 * q^37 + 2888 * q^38 + 2880 * q^40 - 16630 * q^41 + 11303 * q^43 - 10576 * q^44 + 12740 * q^46 + 12155 * q^47 - 15588 * q^49 + 4988 * q^50 + 25808 * q^52 - 20585 * q^53 - 12711 * q^55 + 7296 * q^56 + 9924 * q^58 + 78581 * q^59 + 43621 * q^61 - 4720 * q^62 + 8192 * q^64 + 47100 * q^65 + 7805 * q^67 - 1024 * q^68 + 44844 * q^70 + 62488 * q^71 + 16218 * q^73 + 41952 * q^74 + 11552 * q^76 - 34795 * q^77 + 67122 * q^79 + 11520 * q^80 - 66520 * q^82 + 10714 * q^83 + 20175 * q^85 + 45212 * q^86 - 42304 * q^88 - 128188 * q^89 + 106351 * q^91 + 50960 * q^92 + 48620 * q^94 + 16245 * q^95 + 178558 * q^97 - 62352 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	79.4408	1.42108	0.710540	−	0.703657i	$-0.248451\pi$
$5$	79.4408	1.42108	0.710540	+	0.703657i	$0.248451\pi$
$6$	0	0
$7$	132.921	1.02529	0.512647	−	0.858599i	$-0.328665\pi$
$7$	132.921	1.02529	0.512647	+	0.858599i	$0.328665\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	317.763	1.00485
$11$	−311.520	−0.776254	−0.388127	−	0.921606i	$-0.626878\pi$
$11$	−311.520	−0.776254	−0.388127	+	0.921606i	$0.626878\pi$
$12$	0	0
$13$	901.401	1.47931	0.739656	−	0.672985i	$-0.234988\pi$
$13$	901.401	1.47931	0.739656	+	0.672985i	$0.234988\pi$
$14$	531.684	0.724993
$15$	0	0
$16$	256.000	0.250000
$17$	157.803	0.132432	0.0662158	−	0.997805i	$-0.478907\pi$
$17$	157.803	0.132432	0.0662158	+	0.997805i	$0.478907\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	1271.05	0.710540
$21$	0	0
$22$	−1246.08	−0.548894
$23$	2522.53	0.994299	0.497150	−	0.867665i	$-0.334380\pi$
$23$	2522.53	0.994299	0.497150	+	0.867665i	$0.334380\pi$
$24$	0	0
$25$	3185.83	1.01947
$26$	3605.61	1.04603
$27$	0	0
$28$	2126.74	0.512647
$29$	−4738.28	−1.04623	−0.523113	−	0.852263i	$-0.675230\pi$
$29$	−4738.28	−1.04623	−0.523113	+	0.852263i	$0.675230\pi$
$30$	0	0
$31$	−6587.76	−1.23121	−0.615607	−	0.788053i	$-0.711089\pi$
$31$	−6587.76	−1.23121	−0.615607	+	0.788053i	$0.711089\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	631.210	0.0936433
$35$	10559.3	1.45702
$36$	0	0
$37$	8508.60	1.02177	0.510886	−	0.859648i	$-0.329317\pi$
$37$	8508.60	1.02177	0.510886	+	0.859648i	$0.329317\pi$
$38$	1444.00	0.162221
$39$	0	0
$40$	5084.21	0.502427
$41$	−19741.1	−1.83405	−0.917027	−	0.398826i	$-0.869418\pi$
$41$	−19741.1	−1.83405	−0.917027	+	0.398826i	$0.869418\pi$
$42$	0	0
$43$	10985.0	0.905997	0.452999	−	0.891511i	$-0.350354\pi$
$43$	10985.0	0.905997	0.452999	+	0.891511i	$0.350354\pi$
$44$	−4984.32	−0.388127
$45$	0	0
$46$	10090.1	0.703076
$47$	−15085.5	−0.996127	−0.498063	−	0.867141i	$-0.665955\pi$
$47$	−15085.5	−0.996127	−0.498063	+	0.867141i	$0.665955\pi$
$48$	0	0
$49$	860.995	0.0512284
$50$	12743.3	0.720872
$51$	0	0
$52$	14422.4	0.739656
$53$	−21699.6	−1.06112	−0.530558	−	0.847649i	$-0.678018\pi$
$53$	−21699.6	−1.06112	−0.530558	+	0.847649i	$0.678018\pi$
$54$	0	0
$55$	−24747.4	−1.10312
$56$	8506.94	0.362496
$57$	0	0
$58$	−18953.1	−0.739794
$59$	40676.1	1.52128	0.760639	−	0.649175i	$-0.224886\pi$
$59$	40676.1	1.52128	0.760639	+	0.649175i	$0.224886\pi$
$60$	0	0
$61$	6151.79	0.211679	0.105839	−	0.994383i	$-0.466247\pi$
$61$	6151.79	0.211679	0.105839	+	0.994383i	$0.466247\pi$
$62$	−26351.0	−0.870600
$63$	0	0
$64$	4096.00	0.125000
$65$	71608.0	2.10222
$66$	0	0
$67$	62760.3	1.70804	0.854019	−	0.520241i	$-0.174158\pi$
$67$	62760.3	1.70804	0.854019	+	0.520241i	$0.174158\pi$
$68$	2524.84	0.0662158
$69$	0	0
$70$	42237.4	1.03027
$71$	55311.0	1.30216	0.651081	−	0.759008i	$-0.274316\pi$
$71$	55311.0	1.30216	0.651081	+	0.759008i	$0.274316\pi$
$72$	0	0
$73$	−48528.1	−1.06583	−0.532913	−	0.846170i	$-0.678903\pi$
$73$	−48528.1	−1.06583	−0.532913	+	0.846170i	$0.678903\pi$
$74$	34034.4	0.722502
$75$	0	0
$76$	5776.00	0.114708
$77$	−41407.5	−0.795889
$78$	0	0
$79$	31017.6	0.559166	0.279583	−	0.960121i	$-0.409804\pi$
$79$	31017.6	0.559166	0.279583	+	0.960121i	$0.409804\pi$
$80$	20336.8	0.355270
$81$	0	0
$82$	−78964.4	−1.29687
$83$	−41068.7	−0.654358	−0.327179	−	0.944962i	$-0.606098\pi$
$83$	−41068.7	−0.654358	−0.327179	+	0.944962i	$0.606098\pi$
$84$	0	0
$85$	12536.0	0.188196
$86$	43939.8	0.640637
$87$	0	0
$88$	−19937.3	−0.274447
$89$	17065.6	0.228373	0.114187	−	0.993459i	$-0.463574\pi$
$89$	17065.6	0.228373	0.114187	+	0.993459i	$0.463574\pi$
$90$	0	0
$91$	119815.	1.51673
$92$	40360.5	0.497150
$93$	0	0
$94$	−60341.9	−0.704368
$95$	28678.1	0.326018
$96$	0	0
$97$	139045.	1.50047	0.750234	−	0.661172i	$-0.229941\pi$
$97$	139045.	1.50047	0.750234	+	0.661172i	$0.229941\pi$
$98$	3443.98	0.0362239
$99$	0	0
$100$	50973.3	0.509733
$101$	122253.	1.19249	0.596247	−	0.802801i	$-0.296658\pi$
$101$	122253.	1.19249	0.596247	+	0.802801i	$0.296658\pi$
$102$	0	0
$103$	−71932.4	−0.668084	−0.334042	−	0.942558i	$-0.608413\pi$
$103$	−71932.4	−0.668084	−0.334042	+	0.942558i	$0.608413\pi$
$104$	57689.7	0.523016
$105$	0	0
$106$	−86798.5	−0.750322
$107$	−14833.3	−0.125250	−0.0626249	−	0.998037i	$-0.519947\pi$
$107$	−14833.3	−0.125250	−0.0626249	+	0.998037i	$0.519947\pi$
$108$	0	0
$109$	140025.	1.12886	0.564429	−	0.825482i	$-0.309096\pi$
$109$	140025.	1.12886	0.564429	+	0.825482i	$0.309096\pi$
$110$	−98989.5	−0.780023
$111$	0	0
$112$	34027.8	0.256324
$113$	−235172.	−1.73256	−0.866282	−	0.499555i	$-0.833497\pi$
$113$	−235172.	−1.73256	−0.866282	+	0.499555i	$0.833497\pi$
$114$	0	0
$115$	200392.	1.41298
$116$	−75812.5	−0.523113
$117$	0	0
$118$	162704.	1.07571
$119$	20975.3	0.135781
$120$	0	0
$121$	−64006.4	−0.397430
$122$	24607.2	0.149679
$123$	0	0
$124$	−105404.	−0.615607
$125$	4832.71	0.0276640
$126$	0	0
$127$	−24783.9	−0.136351	−0.0681757	−	0.997673i	$-0.521718\pi$
$127$	−24783.9	−0.136351	−0.0681757	+	0.997673i	$0.521718\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	286432.	1.48649
$131$	152549.	0.776661	0.388331	−	0.921520i	$-0.373052\pi$
$131$	152549.	0.776661	0.388331	+	0.921520i	$0.373052\pi$
$132$	0	0
$133$	47984.5	0.235219
$134$	251041.	1.20777
$135$	0	0
$136$	10099.4	0.0468216
$137$	−192265.	−0.875184	−0.437592	−	0.899174i	$-0.644169\pi$
$137$	−192265.	−0.875184	−0.437592	+	0.899174i	$0.644169\pi$
$138$	0	0
$139$	−342833.	−1.50503	−0.752515	−	0.658575i	$-0.771159\pi$
$139$	−342833.	−1.50503	−0.752515	+	0.658575i	$0.771159\pi$
$140$	168950.	0.728512
$141$	0	0
$142$	221244.	0.920768
$143$	−280804.	−1.14832
$144$	0	0
$145$	−376413.	−1.48677
$146$	−194112.	−0.753652
$147$	0	0
$148$	136138.	0.510886
$149$	−335859.	−1.23934	−0.619671	−	0.784862i	$-0.712734\pi$
$149$	−335859.	−1.23934	−0.619671	+	0.784862i	$0.712734\pi$
$150$	0	0
$151$	266683.	0.951816	0.475908	−	0.879495i	$-0.342120\pi$
$151$	266683.	0.951816	0.475908	+	0.879495i	$0.342120\pi$
$152$	23104.0	0.0811107
$153$	0	0
$154$	−165630.	−0.562778
$155$	−523337.	−1.74965
$156$	0	0
$157$	−173779.	−0.562664	−0.281332	−	0.959611i	$-0.590776\pi$
$157$	−173779.	−0.562664	−0.281332	+	0.959611i	$0.590776\pi$
$158$	124071.	0.395390
$159$	0	0
$160$	81347.3	0.251214
$161$	335298.	1.01945
$162$	0	0
$163$	−406317.	−1.19783	−0.598917	−	0.800811i	$-0.704402\pi$
$163$	−406317.	−1.19783	−0.598917	+	0.800811i	$0.704402\pi$
$164$	−315858.	−0.917027
$165$	0	0
$166$	−164275.	−0.462701
$167$	319695.	0.887043	0.443522	−	0.896264i	$-0.353729\pi$
$167$	319695.	0.887043	0.443522	+	0.896264i	$0.353729\pi$
$168$	0	0
$169$	441231.	1.18836
$170$	50143.8	0.133075
$171$	0	0
$172$	175759.	0.452999
$173$	427313.	1.08550	0.542752	−	0.839893i	$-0.317382\pi$
$173$	427313.	1.08550	0.542752	+	0.839893i	$0.317382\pi$
$174$	0	0
$175$	423464.	1.04525
$176$	−79749.1	−0.194064
$177$	0	0
$178$	68262.2	0.161484
$179$	−361946.	−0.844329	−0.422164	−	0.906519i	$-0.638729\pi$
$179$	−361946.	−0.844329	−0.422164	+	0.906519i	$0.638729\pi$
$180$	0	0
$181$	−416686.	−0.945393	−0.472697	−	0.881225i	$-0.656719\pi$
$181$	−416686.	−0.945393	−0.472697	+	0.881225i	$0.656719\pi$
$182$	479261.	1.07249
$183$	0	0
$184$	161442.	0.351538
$185$	675930.	1.45202
$186$	0	0
$187$	−49158.6	−0.102801
$188$	−241368.	−0.498063
$189$	0	0
$190$	114712.	0.230530
$191$	−581586.	−1.15353	−0.576767	−	0.816909i	$-0.695686\pi$
$191$	−581586.	−1.15353	−0.576767	+	0.816909i	$0.695686\pi$
$192$	0	0
$193$	−182832.	−0.353312	−0.176656	−	0.984273i	$-0.556528\pi$
$193$	−182832.	−0.353312	−0.176656	+	0.984273i	$0.556528\pi$
$194$	556181.	1.06099
$195$	0	0
$196$	13775.9	0.0256142
$197$	93344.9	0.171366	0.0856831	−	0.996322i	$-0.472693\pi$
$197$	93344.9	0.171366	0.0856831	+	0.996322i	$0.472693\pi$
$198$	0	0
$199$	723166.	1.29451	0.647255	−	0.762274i	$-0.275917\pi$
$199$	723166.	1.29451	0.647255	+	0.762274i	$0.275917\pi$
$200$	203893.	0.360436
$201$	0	0
$202$	489012.	0.843221
$203$	−629817.	−1.07269
$204$	0	0
$205$	−1.56825e6	−2.60634
$206$	−287730.	−0.472407
$207$	0	0
$208$	230759.	0.369828
$209$	−112459.	−0.178085
$210$	0	0
$211$	85741.4	0.132582	0.0662910	−	0.997800i	$-0.478883\pi$
$211$	85741.4	0.132582	0.0662910	+	0.997800i	$0.478883\pi$
$212$	−347194.	−0.530558
$213$	0	0
$214$	−59333.0	−0.0885650
$215$	872653.	1.28749
$216$	0	0
$217$	−875652.	−1.26236
$218$	560100.	0.798223
$219$	0	0
$220$	−395958.	−0.551559
$221$	142243.	0.195908
$222$	0	0
$223$	111106.	0.149615	0.0748076	−	0.997198i	$-0.476166\pi$
$223$	111106.	0.149615	0.0748076	+	0.997198i	$0.476166\pi$
$224$	136111.	0.181248
$225$	0	0
$226$	−940688.	−1.22511
$227$	314299.	0.404835	0.202418	−	0.979299i	$-0.435120\pi$
$227$	314299.	0.404835	0.202418	+	0.979299i	$0.435120\pi$
$228$	0	0
$229$	1.45815e6	1.83744	0.918718	−	0.394913i	$-0.129225\pi$
$229$	1.45815e6	1.83744	0.918718	+	0.394913i	$0.129225\pi$
$230$	801568.	0.999127
$231$	0	0
$232$	−303250.	−0.369897
$233$	−1.38626e6	−1.67284	−0.836418	−	0.548091i	$-0.815355\pi$
$233$	−1.38626e6	−1.67284	−0.836418	+	0.548091i	$0.815355\pi$
$234$	0	0
$235$	−1.19840e6	−1.41558
$236$	650817.	0.760639
$237$	0	0
$238$	83901.1	0.0960119
$239$	−365117.	−0.413464	−0.206732	−	0.978398i	$-0.566283\pi$
$239$	−365117.	−0.413464	−0.206732	+	0.978398i	$0.566283\pi$
$240$	0	0
$241$	−312312.	−0.346374	−0.173187	−	0.984889i	$-0.555407\pi$
$241$	−312312.	−0.346374	−0.173187	+	0.984889i	$0.555407\pi$
$242$	−256026.	−0.281025
$243$	0	0
$244$	98428.7	0.105839
$245$	68398.1	0.0727996
$246$	0	0
$247$	325406.	0.339377
$248$	−421617.	−0.435300
$249$	0	0
$250$	19330.8	0.0195614
$251$	970095.	0.971919	0.485959	−	0.873981i	$-0.338470\pi$
$251$	970095.	0.971919	0.485959	+	0.873981i	$0.338470\pi$
$252$	0	0
$253$	−785819.	−0.771829
$254$	−99135.5	−0.0964151
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.28988e6	−1.21819	−0.609096	−	0.793096i	$-0.708468\pi$
$257$	−1.28988e6	−1.21819	−0.609096	+	0.793096i	$0.708468\pi$
$258$	0	0
$259$	1.13097e6	1.04762
$260$	1.14573e6	1.05111
$261$	0	0
$262$	610197.	0.549183
$263$	501426.	0.447011	0.223505	−	0.974703i	$-0.428250\pi$
$263$	501426.	0.447011	0.223505	+	0.974703i	$0.428250\pi$
$264$	0	0
$265$	−1.72384e6	−1.50793
$266$	191938.	0.166325
$267$	0	0
$268$	1.00416e6	0.854019
$269$	−1.42986e6	−1.20479	−0.602397	−	0.798197i	$-0.705787\pi$
$269$	−1.42986e6	−1.20479	−0.602397	+	0.798197i	$0.705787\pi$
$270$	0	0
$271$	−709506.	−0.586857	−0.293429	−	0.955981i	$-0.594796\pi$
$271$	−709506.	−0.586857	−0.293429	+	0.955981i	$0.594796\pi$
$272$	40397.4	0.0331079
$273$	0	0
$274$	−769061.	−0.618849
$275$	−992450.	−0.791365
$276$	0	0
$277$	−766740.	−0.600411	−0.300205	−	0.953875i	$-0.597055\pi$
$277$	−766740.	−0.600411	−0.300205	+	0.953875i	$0.597055\pi$
$278$	−1.37133e6	−1.06422
$279$	0	0
$280$	675798.	0.515136
$281$	−5975.32	−0.00451435	−0.00225718	−	0.999997i	$-0.500718\pi$
$281$	−5975.32	−0.00451435	−0.00225718	+	0.999997i	$0.500718\pi$
$282$	0	0
$283$	−189246.	−0.140463	−0.0702314	−	0.997531i	$-0.522374\pi$
$283$	−189246.	−0.140463	−0.0702314	+	0.997531i	$0.522374\pi$
$284$	884975.	0.651081
$285$	0	0
$286$	−1.12322e6	−0.811986
$287$	−2.62401e6	−1.88044
$288$	0	0
$289$	−1.39496e6	−0.982462
$290$	−1.50565e6	−1.05131
$291$	0	0
$292$	−776449.	−0.532913
$293$	−986180.	−0.671100	−0.335550	−	0.942022i	$-0.608922\pi$
$293$	−986180.	−0.671100	−0.335550	+	0.942022i	$0.608922\pi$
$294$	0	0
$295$	3.23134e6	2.16186
$296$	544551.	0.361251
$297$	0	0
$298$	−1.34343e6	−0.876347
$299$	2.27381e6	1.47088
$300$	0	0
$301$	1.46013e6	0.928914
$302$	1.06673e6	0.673035
$303$	0	0
$304$	92416.0	0.0573539
$305$	488703.	0.300812
$306$	0	0
$307$	−1.45722e6	−0.882428	−0.441214	−	0.897402i	$-0.645452\pi$
$307$	−1.45722e6	−0.882428	−0.441214	+	0.897402i	$0.645452\pi$
$308$	−662520.	−0.397944
$309$	0	0
$310$	−2.09335e6	−1.23719
$311$	−2.40526e6	−1.41014	−0.705069	−	0.709138i	$-0.749084\pi$
$311$	−2.40526e6	−1.41014	−0.705069	+	0.709138i	$0.749084\pi$
$312$	0	0
$313$	−3.20380e6	−1.84843	−0.924217	−	0.381867i	$-0.875281\pi$
$313$	−3.20380e6	−1.84843	−0.924217	+	0.381867i	$0.875281\pi$
$314$	−695117.	−0.397863
$315$	0	0
$316$	496282.	0.279583
$317$	603046.	0.337056	0.168528	−	0.985697i	$-0.446099\pi$
$317$	603046.	0.337056	0.168528	+	0.985697i	$0.446099\pi$
$318$	0	0
$319$	1.47607e6	0.812137
$320$	325389.	0.177635
$321$	0	0
$322$	1.34119e6	0.720860
$323$	56966.7	0.0303819
$324$	0	0
$325$	2.87171e6	1.50811
$326$	−1.62527e6	−0.846996
$327$	0	0
$328$	−1.26343e6	−0.648436
$329$	−2.00518e6	−1.02132
$330$	0	0
$331$	−3.77416e6	−1.89344	−0.946719	−	0.322062i	$-0.895624\pi$
$331$	−3.77416e6	−1.89344	−0.946719	+	0.322062i	$0.895624\pi$
$332$	−657099.	−0.327179
$333$	0	0
$334$	1.27878e6	0.627234
$335$	4.98572e6	2.42726
$336$	0	0
$337$	−2.12850e6	−1.02094	−0.510470	−	0.859896i	$-0.670528\pi$
$337$	−2.12850e6	−1.02094	−0.510470	+	0.859896i	$0.670528\pi$
$338$	1.76492e6	0.840300
$339$	0	0
$340$	200575.	0.0940979
$341$	2.05222e6	0.955735
$342$	0	0
$343$	−2.11956e6	−0.972770
$344$	703037.	0.320318
$345$	0	0
$346$	1.70925e6	0.767567
$347$	−2.40305e6	−1.07137	−0.535685	−	0.844418i	$-0.679947\pi$
$347$	−2.40305e6	−1.07137	−0.535685	+	0.844418i	$0.679947\pi$
$348$	0	0
$349$	3.71870e6	1.63428	0.817141	−	0.576438i	$-0.195558\pi$
$349$	3.71870e6	1.63428	0.817141	+	0.576438i	$0.195558\pi$
$350$	1.69386e6	0.739106
$351$	0	0
$352$	−318996.	−0.137224
$353$	2.04192e6	0.872174	0.436087	−	0.899905i	$-0.356364\pi$
$353$	2.04192e6	0.872174	0.436087	+	0.899905i	$0.356364\pi$
$354$	0	0
$355$	4.39394e6	1.85048
$356$	273049.	0.114187
$357$	0	0
$358$	−1.44778e6	−0.597030
$359$	949493.	0.388826	0.194413	−	0.980920i	$-0.437720\pi$
$359$	949493.	0.388826	0.194413	+	0.980920i	$0.437720\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	−1.66674e6	−0.668494
$363$	0	0
$364$	1.91704e6	0.758365
$365$	−3.85511e6	−1.51462
$366$	0	0
$367$	1.74418e6	0.675967	0.337984	−	0.941152i	$-0.390255\pi$
$367$	1.74418e6	0.675967	0.337984	+	0.941152i	$0.390255\pi$
$368$	645768.	0.248575
$369$	0	0
$370$	2.70372e6	1.02673
$371$	−2.88434e6	−1.08796
$372$	0	0
$373$	4.59987e6	1.71188	0.855940	−	0.517076i	$-0.172980\pi$
$373$	4.59987e6	1.71188	0.855940	+	0.517076i	$0.172980\pi$
$374$	−196634.	−0.0726910
$375$	0	0
$376$	−965471.	−0.352184
$377$	−4.27109e6	−1.54770
$378$	0	0
$379$	−6993.69	−0.00250097	−0.00125048	−	0.999999i	$-0.500398\pi$
$379$	−6993.69	−0.00250097	−0.00125048	+	0.999999i	$0.500398\pi$
$380$	458850.	0.163009
$381$	0	0
$382$	−2.32634e6	−0.815672
$383$	−1.37680e6	−0.479594	−0.239797	−	0.970823i	$-0.577081\pi$
$383$	−1.37680e6	−0.479594	−0.239797	+	0.970823i	$0.577081\pi$
$384$	0	0
$385$	−3.28944e6	−1.13102
$386$	−731327.	−0.249829
$387$	0	0
$388$	2.22472e6	0.750234
$389$	−1.40115e6	−0.469473	−0.234736	−	0.972059i	$-0.575423\pi$
$389$	−1.40115e6	−0.469473	−0.234736	+	0.972059i	$0.575423\pi$
$390$	0	0
$391$	398062.	0.131677
$392$	55103.7	0.0181120
$393$	0	0
$394$	373380.	0.121174
$395$	2.46407e6	0.794620
$396$	0	0
$397$	3.33402e6	1.06168	0.530839	−	0.847473i	$-0.321877\pi$
$397$	3.33402e6	1.06168	0.530839	+	0.847473i	$0.321877\pi$
$398$	2.89266e6	0.915357
$399$	0	0
$400$	815574.	0.254867
$401$	5.32612e6	1.65406	0.827028	−	0.562161i	$-0.190030\pi$
$401$	5.32612e6	1.65406	0.827028	+	0.562161i	$0.190030\pi$
$402$	0	0
$403$	−5.93822e6	−1.82135
$404$	1.95605e6	0.596247
$405$	0	0
$406$	−2.51927e6	−0.758506
$407$	−2.65060e6	−0.793155
$408$	0	0
$409$	−3.00861e6	−0.889319	−0.444659	−	0.895700i	$-0.646675\pi$
$409$	−3.00861e6	−0.889319	−0.444659	+	0.895700i	$0.646675\pi$
$410$	−6.27300e6	−1.84296
$411$	0	0
$412$	−1.15092e6	−0.334042
$413$	5.40670e6	1.55976
$414$	0	0
$415$	−3.26253e6	−0.929895
$416$	923035.	0.261508
$417$	0	0
$418$	−449835.	−0.125925
$419$	−2.47470e6	−0.688633	−0.344317	−	0.938854i	$-0.611889\pi$
$419$	−2.47470e6	−0.688633	−0.344317	+	0.938854i	$0.611889\pi$
$420$	0	0
$421$	−3.25587e6	−0.895286	−0.447643	−	0.894212i	$-0.647737\pi$
$421$	−3.25587e6	−0.895286	−0.447643	+	0.894212i	$0.647737\pi$
$422$	342966.	0.0937496
$423$	0	0
$424$	−1.38878e6	−0.375161
$425$	502733.	0.135010
$426$	0	0
$427$	817702.	0.217033
$428$	−237332.	−0.0626249
$429$	0	0
$430$	3.49061e6	0.910396
$431$	3.54756e6	0.919891	0.459945	−	0.887947i	$-0.347869\pi$
$431$	3.54756e6	0.919891	0.459945	+	0.887947i	$0.347869\pi$
$432$	0	0
$433$	2.58960e6	0.663763	0.331881	−	0.943321i	$-0.392317\pi$
$433$	2.58960e6	0.663763	0.331881	+	0.943321i	$0.392317\pi$
$434$	−3.50261e6	−0.892621
$435$	0	0
$436$	2.24040e6	0.564429
$437$	910634.	0.228108
$438$	0	0
$439$	−3.47682e6	−0.861036	−0.430518	−	0.902582i	$-0.641669\pi$
$439$	−3.47682e6	−0.861036	−0.430518	+	0.902582i	$0.641669\pi$
$440$	−1.58383e6	−0.390011
$441$	0	0
$442$	568974.	0.138528
$443$	3.22496e6	0.780756	0.390378	−	0.920655i	$-0.372344\pi$
$443$	3.22496e6	0.780756	0.390378	+	0.920655i	$0.372344\pi$
$444$	0	0
$445$	1.35570e6	0.324537
$446$	444424.	0.105794
$447$	0	0
$448$	544444.	0.128162
$449$	4.47721e6	1.04807	0.524036	−	0.851696i	$-0.324426\pi$
$449$	4.47721e6	1.04807	0.524036	+	0.851696i	$0.324426\pi$
$450$	0	0
$451$	6.14975e6	1.42369
$452$	−3.76275e6	−0.866282
$453$	0	0
$454$	1.25720e6	0.286262
$455$	9.51821e6	2.15539
$456$	0	0
$457$	563938.	0.126311	0.0631555	−	0.998004i	$-0.479884\pi$
$457$	563938.	0.126311	0.0631555	+	0.998004i	$0.479884\pi$
$458$	5.83259e6	1.29926
$459$	0	0
$460$	3.20627e6	0.706489
$461$	3.05325e6	0.669130	0.334565	−	0.942373i	$-0.391411\pi$
$461$	3.05325e6	0.669130	0.334565	+	0.942373i	$0.391411\pi$
$462$	0	0
$463$	7.31911e6	1.58674	0.793370	−	0.608740i	$-0.208325\pi$
$463$	7.31911e6	1.58674	0.793370	+	0.608740i	$0.208325\pi$
$464$	−1.21300e6	−0.261557
$465$	0	0
$466$	−5.54502e6	−1.18287
$467$	3.83007e6	0.812671	0.406336	−	0.913724i	$-0.366806\pi$
$467$	3.83007e6	0.812671	0.406336	+	0.913724i	$0.366806\pi$
$468$	0	0
$469$	8.34216e6	1.75124
$470$	−4.79361e6	−1.00096
$471$	0	0
$472$	2.60327e6	0.537853
$473$	−3.42203e6	−0.703284
$474$	0	0
$475$	1.15009e6	0.233882
$476$	335604.	0.0678907
$477$	0	0
$478$	−1.46047e6	−0.292363
$479$	852140.	0.169696	0.0848481	−	0.996394i	$-0.472960\pi$
$479$	852140.	0.169696	0.0848481	+	0.996394i	$0.472960\pi$
$480$	0	0
$481$	7.66967e6	1.51152
$482$	−1.24925e6	−0.244924
$483$	0	0
$484$	−1.02410e6	−0.198715
$485$	1.10459e7	2.13228
$486$	0	0
$487$	−1.76953e6	−0.338093	−0.169046	−	0.985608i	$-0.554069\pi$
$487$	−1.76953e6	−0.338093	−0.169046	+	0.985608i	$0.554069\pi$
$488$	393715.	0.0748397
$489$	0	0
$490$	273592.	0.0514771
$491$	1.12702e6	0.210973	0.105487	−	0.994421i	$-0.466360\pi$
$491$	1.12702e6	0.210973	0.105487	+	0.994421i	$0.466360\pi$
$492$	0	0
$493$	−747713.	−0.138553
$494$	1.30162e6	0.239976
$495$	0	0
$496$	−1.68647e6	−0.307803
$497$	7.35199e6	1.33510
$498$	0	0
$499$	−5.96123e6	−1.07173	−0.535864	−	0.844304i	$-0.680014\pi$
$499$	−5.96123e6	−1.07173	−0.535864	+	0.844304i	$0.680014\pi$
$500$	77323.4	0.0138320
$501$	0	0
$502$	3.88038e6	0.687250
$503$	5.66845e6	0.998951	0.499476	−	0.866328i	$-0.333526\pi$
$503$	5.66845e6	0.998951	0.499476	+	0.866328i	$0.333526\pi$
$504$	0	0
$505$	9.71188e6	1.69463
$506$	−3.14327e6	−0.545765
$507$	0	0
$508$	−396542.	−0.0681757
$509$	−8.42220e6	−1.44089	−0.720446	−	0.693512i	$-0.756063\pi$
$509$	−8.42220e6	−1.44089	−0.720446	+	0.693512i	$0.756063\pi$
$510$	0	0
$511$	−6.45040e6	−1.09278
$512$	262144.	0.0441942
$513$	0	0
$514$	−5.15951e6	−0.861392
$515$	−5.71436e6	−0.949401
$516$	0	0
$517$	4.69943e6	0.773247
$518$	4.52389e6	0.740777
$519$	0	0
$520$	4.58291e6	0.743247
$521$	−5.81861e6	−0.939128	−0.469564	−	0.882899i	$-0.655589\pi$
$521$	−5.81861e6	−0.939128	−0.469564	+	0.882899i	$0.655589\pi$
$522$	0	0
$523$	4.81037e6	0.768996	0.384498	−	0.923126i	$-0.374375\pi$
$523$	4.81037e6	0.768996	0.384498	+	0.923126i	$0.374375\pi$
$524$	2.44079e6	0.388331
$525$	0	0
$526$	2.00571e6	0.316084
$527$	−1.03957e6	−0.163052
$528$	0	0
$529$	−73173.3	−0.0113688
$530$	−6.89534e6	−1.06627
$531$	0	0
$532$	767752.	0.117609
$533$	−1.77947e7	−2.71314
$534$	0	0
$535$	−1.17836e6	−0.177990
$536$	4.01666e6	0.603883
$537$	0	0
$538$	−5.71944e6	−0.851918
$539$	−268217.	−0.0397662
$540$	0	0
$541$	3.11994e6	0.458304	0.229152	−	0.973391i	$-0.426405\pi$
$541$	3.11994e6	0.458304	0.229152	+	0.973391i	$0.426405\pi$
$542$	−2.83802e6	−0.414971
$543$	0	0
$544$	161590.	0.0234108
$545$	1.11237e7	1.60420
$546$	0	0
$547$	2.61616e6	0.373848	0.186924	−	0.982374i	$-0.440148\pi$
$547$	2.61616e6	0.373848	0.186924	+	0.982374i	$0.440148\pi$
$548$	−3.07624e6	−0.437592
$549$	0	0
$550$	−3.96980e6	−0.559580
$551$	−1.71052e6	−0.240021
$552$	0	0
$553$	4.12290e6	0.573310
$554$	−3.06696e6	−0.424555
$555$	0	0
$556$	−5.48532e6	−0.752515
$557$	−1.91062e6	−0.260937	−0.130469	−	0.991452i	$-0.541648\pi$
$557$	−1.91062e6	−0.260937	−0.130469	+	0.991452i	$0.541648\pi$
$558$	0	0
$559$	9.90185e6	1.34025
$560$	2.70319e6	0.364256
$561$	0	0
$562$	−23901.3	−0.00319213
$563$	−84059.8	−0.0111768	−0.00558840	−	0.999984i	$-0.501779\pi$
$563$	−84059.8	−0.0111768	−0.00558840	+	0.999984i	$0.501779\pi$
$564$	0	0
$565$	−1.86822e7	−2.46211
$566$	−756986.	−0.0993222
$567$	0	0
$568$	3.53990e6	0.460384
$569$	2.22862e6	0.288572	0.144286	−	0.989536i	$-0.453911\pi$
$569$	2.22862e6	0.288572	0.144286	+	0.989536i	$0.453911\pi$
$570$	0	0
$571$	8.28284e6	1.06314	0.531568	−	0.847015i	$-0.321603\pi$
$571$	8.28284e6	1.06314	0.531568	+	0.847015i	$0.321603\pi$
$572$	−4.49287e6	−0.574161
$573$	0	0
$574$	−1.04960e7	−1.32968
$575$	8.03637e6	1.01366
$576$	0	0
$577$	−1.50122e7	−1.87717	−0.938585	−	0.345047i	$-0.887863\pi$
$577$	−1.50122e7	−1.87717	−0.938585	+	0.345047i	$0.887863\pi$
$578$	−5.57982e6	−0.694705
$579$	0	0
$580$	−6.02260e6	−0.743385
$581$	−5.45889e6	−0.670910
$582$	0	0
$583$	6.75986e6	0.823695
$584$	−3.10580e6	−0.376826
$585$	0	0
$586$	−3.94472e6	−0.474539
$587$	4.07944e6	0.488658	0.244329	−	0.969692i	$-0.421432\pi$
$587$	4.07944e6	0.488658	0.244329	+	0.969692i	$0.421432\pi$
$588$	0	0
$589$	−2.37818e6	−0.282460
$590$	1.29253e7	1.52866
$591$	0	0
$592$	2.17820e6	0.255443
$593$	545602.	0.0637147	0.0318573	−	0.999492i	$-0.489858\pi$
$593$	545602.	0.0637147	0.0318573	+	0.999492i	$0.489858\pi$
$594$	0	0
$595$	1.66629e6	0.192956
$596$	−5.37374e6	−0.619671
$597$	0	0
$598$	9.09526e6	1.04007
$599$	8.45396e6	0.962704	0.481352	−	0.876527i	$-0.340146\pi$
$599$	8.45396e6	0.962704	0.481352	+	0.876527i	$0.340146\pi$
$600$	0	0
$601$	−5.39862e6	−0.609673	−0.304836	−	0.952405i	$-0.598602\pi$
$601$	−5.39862e6	−0.609673	−0.304836	+	0.952405i	$0.598602\pi$
$602$	5.84052e6	0.656841
$603$	0	0
$604$	4.26693e6	0.475908
$605$	−5.08472e6	−0.564779
$606$	0	0
$607$	−1.14275e7	−1.25886	−0.629432	−	0.777055i	$-0.716712\pi$
$607$	−1.14275e7	−1.25886	−0.629432	+	0.777055i	$0.716712\pi$
$608$	369664.	0.0405554
$609$	0	0
$610$	1.95481e6	0.212706
$611$	−1.35981e7	−1.47358
$612$	0	0
$613$	−894639.	−0.0961605	−0.0480802	−	0.998843i	$-0.515310\pi$
$613$	−894639.	−0.0961605	−0.0480802	+	0.998843i	$0.515310\pi$
$614$	−5.82888e6	−0.623971
$615$	0	0
$616$	−2.65008e6	−0.281389
$617$	−1.18925e7	−1.25765	−0.628826	−	0.777546i	$-0.716464\pi$
$617$	−1.18925e7	−1.25765	−0.628826	+	0.777546i	$0.716464\pi$
$618$	0	0
$619$	1.03513e7	1.08585	0.542924	−	0.839782i	$-0.317317\pi$
$619$	1.03513e7	1.08585	0.542924	+	0.839782i	$0.317317\pi$
$620$	−8.37339e6	−0.874826
$621$	0	0
$622$	−9.62106e6	−0.997118
$623$	2.26837e6	0.234150
$624$	0	0
$625$	−9.57182e6	−0.980154
$626$	−1.28152e7	−1.30704
$627$	0	0
$628$	−2.78047e6	−0.281332
$629$	1.34268e6	0.135315
$630$	0	0
$631$	−1.17088e6	−0.117068	−0.0585342	−	0.998285i	$-0.518643\pi$
$631$	−1.17088e6	−0.117068	−0.0585342	+	0.998285i	$0.518643\pi$
$632$	1.98513e6	0.197695
$633$	0	0
$634$	2.41219e6	0.238335
$635$	−1.96885e6	−0.193766
$636$	0	0
$637$	776102.	0.0757828
$638$	5.90427e6	0.574268
$639$	0	0
$640$	1.30156e6	0.125607
$641$	4.72990e6	0.454681	0.227341	−	0.973815i	$-0.426997\pi$
$641$	4.72990e6	0.454681	0.227341	+	0.973815i	$0.426997\pi$
$642$	0	0
$643$	−1.49075e7	−1.42193	−0.710963	−	0.703229i	$-0.751741\pi$
$643$	−1.49075e7	−1.42193	−0.710963	+	0.703229i	$0.751741\pi$
$644$	5.36476e6	0.509725
$645$	0	0
$646$	227867.	0.0214832
$647$	4.27732e6	0.401708	0.200854	−	0.979621i	$-0.435628\pi$
$647$	4.27732e6	0.401708	0.200854	+	0.979621i	$0.435628\pi$
$648$	0	0
$649$	−1.26714e7	−1.18090
$650$	1.14869e7	1.06639
$651$	0	0
$652$	−6.50108e6	−0.598917
$653$	−2.25370e6	−0.206829	−0.103415	−	0.994638i	$-0.532977\pi$
$653$	−2.25370e6	−0.206829	−0.103415	+	0.994638i	$0.532977\pi$
$654$	0	0
$655$	1.21186e7	1.10370
$656$	−5.05372e6	−0.458513
$657$	0	0
$658$	−8.02071e6	−0.722184
$659$	2.93133e6	0.262937	0.131468	−	0.991320i	$-0.458031\pi$
$659$	2.93133e6	0.262937	0.131468	+	0.991320i	$0.458031\pi$
$660$	0	0
$661$	6.56037e6	0.584016	0.292008	−	0.956416i	$-0.405677\pi$
$661$	6.56037e6	0.584016	0.292008	+	0.956416i	$0.405677\pi$
$662$	−1.50967e7	−1.33886
$663$	0	0
$664$	−2.62840e6	−0.231351
$665$	3.81192e6	0.334264
$666$	0	0
$667$	−1.19525e7	−1.04026
$668$	5.11512e6	0.443522
$669$	0	0
$670$	1.99429e7	1.71633
$671$	−1.91640e6	−0.164316
$672$	0	0
$673$	−3.19596e6	−0.271996	−0.135998	−	0.990709i	$-0.543424\pi$
$673$	−3.19596e6	−0.271996	−0.135998	+	0.990709i	$0.543424\pi$
$674$	−8.51402e6	−0.721913
$675$	0	0
$676$	7.05970e6	0.594182
$677$	2.06961e6	0.173547	0.0867734	−	0.996228i	$-0.472344\pi$
$677$	2.06961e6	0.173547	0.0867734	+	0.996228i	$0.472344\pi$
$678$	0	0
$679$	1.84820e7	1.53842
$680$	802301.	0.0665373
$681$	0	0
$682$	8.20887e6	0.675807
$683$	2.61594e6	0.214574	0.107287	−	0.994228i	$-0.465784\pi$
$683$	2.61594e6	0.214574	0.107287	+	0.994228i	$0.465784\pi$
$684$	0	0
$685$	−1.52737e7	−1.24371
$686$	−8.47824e6	−0.687852
$687$	0	0
$688$	2.81215e6	0.226499
$689$	−1.95601e7	−1.56972
$690$	0	0
$691$	−1.40203e7	−1.11703	−0.558513	−	0.829496i	$-0.688628\pi$
$691$	−1.40203e7	−1.11703	−0.558513	+	0.829496i	$0.688628\pi$
$692$	6.83701e6	0.542752
$693$	0	0
$694$	−9.61221e6	−0.757573
$695$	−2.72349e7	−2.13877
$696$	0	0
$697$	−3.11520e6	−0.242887
$698$	1.48748e7	1.15561
$699$	0	0
$700$	6.77543e6	0.522627
$701$	−3.42664e6	−0.263375	−0.131687	−	0.991291i	$-0.542039\pi$
$701$	−3.42664e6	−0.263375	−0.131687	+	0.991291i	$0.542039\pi$
$702$	0	0
$703$	3.07161e6	0.234411
$704$	−1.27598e6	−0.0970318
$705$	0	0
$706$	8.16770e6	0.616720
$707$	1.62500e7	1.22266
$708$	0	0
$709$	8.04969e6	0.601400	0.300700	−	0.953719i	$-0.402780\pi$
$709$	8.04969e6	0.601400	0.300700	+	0.953719i	$0.402780\pi$
$710$	1.75758e7	1.30848
$711$	0	0
$712$	1.09220e6	0.0807422
$713$	−1.66178e7	−1.22420
$714$	0	0
$715$	−2.23073e7	−1.63186
$716$	−5.79114e6	−0.422164
$717$	0	0
$718$	3.79797e6	0.274942
$719$	2.66646e7	1.92359	0.961795	−	0.273771i	$-0.0882711\pi$
$719$	2.66646e7	1.92359	0.961795	+	0.273771i	$0.0882711\pi$
$720$	0	0
$721$	−9.56132e6	−0.684983
$722$	521284.	0.0372161
$723$	0	0
$724$	−6.66698e6	−0.472697
$725$	−1.50954e7	−1.06659
$726$	0	0
$727$	1.25706e7	0.882102	0.441051	−	0.897482i	$-0.354606\pi$
$727$	1.25706e7	0.882102	0.441051	+	0.897482i	$0.354606\pi$
$728$	7.66817e6	0.536245
$729$	0	0
$730$	−1.54204e7	−1.07100
$731$	1.73345e6	0.119983
$732$	0	0
$733$	−1.12272e7	−0.771814	−0.385907	−	0.922538i	$-0.626111\pi$
$733$	−1.12272e7	−0.771814	−0.385907	+	0.922538i	$0.626111\pi$
$734$	6.97671e6	0.477981
$735$	0	0
$736$	2.58307e6	0.175769
$737$	−1.95511e7	−1.32587
$738$	0	0
$739$	1.05677e7	0.711820	0.355910	−	0.934520i	$-0.384171\pi$
$739$	1.05677e7	0.711820	0.355910	+	0.934520i	$0.384171\pi$
$740$	1.08149e7	0.726010
$741$	0	0
$742$	−1.15373e7	−0.769301
$743$	5.83579e6	0.387817	0.193909	−	0.981020i	$-0.437883\pi$
$743$	5.83579e6	0.387817	0.193909	+	0.981020i	$0.437883\pi$
$744$	0	0
$745$	−2.66809e7	−1.76120
$746$	1.83995e7	1.21048
$747$	0	0
$748$	−786538.	−0.0514003
$749$	−1.97165e6	−0.128418
$750$	0	0
$751$	4.48767e6	0.290349	0.145175	−	0.989406i	$-0.453626\pi$
$751$	4.48767e6	0.290349	0.145175	+	0.989406i	$0.453626\pi$
$752$	−3.86188e6	−0.249032
$753$	0	0
$754$	−1.70844e7	−1.09439
$755$	2.11855e7	1.35261
$756$	0	0
$757$	7.81327e6	0.495557	0.247778	−	0.968817i	$-0.420300\pi$
$757$	7.81327e6	0.495557	0.247778	+	0.968817i	$0.420300\pi$
$758$	−27974.8	−0.00176845
$759$	0	0
$760$	1.83540e6	0.115265
$761$	−1.81639e7	−1.13697	−0.568483	−	0.822695i	$-0.692469\pi$
$761$	−1.81639e7	−1.13697	−0.568483	+	0.822695i	$0.692469\pi$
$762$	0	0
$763$	1.86123e7	1.15741
$764$	−9.30537e6	−0.576767
$765$	0	0
$766$	−5.50720e6	−0.339124
$767$	3.66655e7	2.25045
$768$	0	0
$769$	−4.47189e6	−0.272694	−0.136347	−	0.990661i	$-0.543536\pi$
$769$	−4.47189e6	−0.272694	−0.136347	+	0.990661i	$0.543536\pi$
$770$	−1.31578e7	−0.799753
$771$	0	0
$772$	−2.92531e6	−0.176656
$773$	2.53207e7	1.52415	0.762075	−	0.647489i	$-0.224181\pi$
$773$	2.53207e7	1.52415	0.762075	+	0.647489i	$0.224181\pi$
$774$	0	0
$775$	−2.09875e7	−1.25518
$776$	8.89889e6	0.530496
$777$	0	0
$778$	−5.60460e6	−0.331967
$779$	−7.12654e6	−0.420761
$780$	0	0
$781$	−1.72305e7	−1.01081
$782$	1.59225e6	0.0931095
$783$	0	0
$784$	220415.	0.0128071
$785$	−1.38052e7	−0.799590
$786$	0	0
$787$	4.73524e6	0.272524	0.136262	−	0.990673i	$-0.456491\pi$
$787$	4.73524e6	0.272524	0.136262	+	0.990673i	$0.456491\pi$
$788$	1.49352e6	0.0856831
$789$	0	0
$790$	9.85626e6	0.561881
$791$	−3.12593e7	−1.77639
$792$	0	0
$793$	5.54523e6	0.313139
$794$	1.33361e7	0.750719
$795$	0	0
$796$	1.15707e7	0.647255
$797$	−1.54780e7	−0.863116	−0.431558	−	0.902085i	$-0.642036\pi$
$797$	−1.54780e7	−0.863116	−0.431558	+	0.902085i	$0.642036\pi$
$798$	0	0
$799$	−2.38053e6	−0.131919
$800$	3.26229e6	0.180218
$801$	0	0
$802$	2.13045e7	1.16959
$803$	1.51175e7	0.827351
$804$	0	0
$805$	2.66363e7	1.44872
$806$	−2.37529e7	−1.28789
$807$	0	0
$808$	7.82420e6	0.421611
$809$	−5.54794e6	−0.298031	−0.149015	−	0.988835i	$-0.547610\pi$
$809$	−5.54794e6	−0.298031	−0.149015	+	0.988835i	$0.547610\pi$
$810$	0	0
$811$	9.82955e6	0.524785	0.262392	−	0.964961i	$-0.415489\pi$
$811$	9.82955e6	0.524785	0.262392	+	0.964961i	$0.415489\pi$
$812$	−1.00771e7	−0.536345
$813$	0	0
$814$	−1.06024e7	−0.560845
$815$	−3.22782e7	−1.70222
$816$	0	0
$817$	3.96557e6	0.207850
$818$	−1.20344e7	−0.628843
$819$	0	0
$820$	−2.50920e7	−1.30317
$821$	−1.78229e7	−0.922830	−0.461415	−	0.887185i	$-0.652658\pi$
$821$	−1.78229e7	−0.922830	−0.461415	+	0.887185i	$0.652658\pi$
$822$	0	0
$823$	−1.29509e7	−0.666502	−0.333251	−	0.942838i	$-0.608146\pi$
$823$	−1.29509e7	−0.666502	−0.333251	+	0.942838i	$0.608146\pi$
$824$	−4.60367e6	−0.236204
$825$	0	0
$826$	2.16268e7	1.10292
$827$	2.57276e7	1.30808	0.654042	−	0.756458i	$-0.273072\pi$
$827$	2.57276e7	1.30808	0.654042	+	0.756458i	$0.273072\pi$
$828$	0	0
$829$	2.26984e7	1.14712	0.573559	−	0.819164i	$-0.305562\pi$
$829$	2.26984e7	1.14712	0.573559	+	0.819164i	$0.305562\pi$
$830$	−1.30501e7	−0.657535
$831$	0	0
$832$	3.69214e6	0.184914
$833$	135867.	0.00678426
$834$	0	0
$835$	2.53968e7	1.26056
$836$	−1.79934e6	−0.0890424
$837$	0	0
$838$	−9.89881e6	−0.486937
$839$	9.22885e6	0.452629	0.226315	−	0.974054i	$-0.427332\pi$
$839$	9.22885e6	0.452629	0.226315	+	0.974054i	$0.427332\pi$
$840$	0	0
$841$	1.94015e6	0.0945898
$842$	−1.30235e7	−0.633063
$843$	0	0
$844$	1.37186e6	0.0662910
$845$	3.50517e7	1.68876
$846$	0	0
$847$	−8.50780e6	−0.407482
$848$	−5.55511e6	−0.265279
$849$	0	0
$850$	2.01093e6	0.0954662
$851$	2.14632e7	1.01595
$852$	0	0
$853$	−3.17055e7	−1.49198	−0.745989	−	0.665959i	$-0.768023\pi$
$853$	−3.17055e7	−1.49198	−0.745989	+	0.665959i	$0.768023\pi$
$854$	3.27081e6	0.153465
$855$	0	0
$856$	−949328.	−0.0442825
$857$	−3.01353e7	−1.40160	−0.700799	−	0.713359i	$-0.747173\pi$
$857$	−3.01353e7	−1.40160	−0.700799	+	0.713359i	$0.747173\pi$
$858$	0	0
$859$	−1.58417e7	−0.732520	−0.366260	−	0.930513i	$-0.619362\pi$
$859$	−1.58417e7	−0.732520	−0.366260	+	0.930513i	$0.619362\pi$
$860$	1.39624e7	0.643747
$861$	0	0
$862$	1.41902e7	0.650461
$863$	2.32831e7	1.06418	0.532089	−	0.846689i	$-0.321407\pi$
$863$	2.32831e7	1.06418	0.532089	+	0.846689i	$0.321407\pi$
$864$	0	0
$865$	3.39461e7	1.54259
$866$	1.03584e7	0.469351
$867$	0	0
$868$	−1.40104e7	−0.631178
$869$	−9.66261e6	−0.434055
$870$	0	0
$871$	5.65722e7	2.52672
$872$	8.96160e6	0.399111
$873$	0	0
$874$	3.64254e6	0.161297
$875$	642369.	0.0283638
$876$	0	0
$877$	2.89258e7	1.26995	0.634975	−	0.772532i	$-0.281010\pi$
$877$	2.89258e7	1.26995	0.634975	+	0.772532i	$0.281010\pi$
$878$	−1.39073e7	−0.608844
$879$	0	0
$880$	−6.33533e6	−0.275780
$881$	2.35636e7	1.02283	0.511413	−	0.859335i	$-0.329122\pi$
$881$	2.35636e7	1.02283	0.511413	+	0.859335i	$0.329122\pi$
$882$	0	0
$883$	−1.91319e6	−0.0825764	−0.0412882	−	0.999147i	$-0.513146\pi$
$883$	−1.91319e6	−0.0825764	−0.0412882	+	0.999147i	$0.513146\pi$
$884$	2.27589e6	0.0979538
$885$	0	0
$886$	1.28998e7	0.552078
$887$	−3.48791e7	−1.48852	−0.744262	−	0.667888i	$-0.767199\pi$
$887$	−3.48791e7	−1.48852	−0.744262	+	0.667888i	$0.767199\pi$
$888$	0	0
$889$	−3.29430e6	−0.139800
$890$	5.42280e6	0.229482
$891$	0	0
$892$	1.77770e6	0.0748076
$893$	−5.44586e6	−0.228527
$894$	0	0
$895$	−2.87533e7	−1.19986
$896$	2.17778e6	0.0906241
$897$	0	0
$898$	1.79088e7	0.741099
$899$	3.12146e7	1.28813
$900$	0	0
$901$	−3.42426e6	−0.140525
$902$	2.45990e7	1.00670
$903$	0	0
$904$	−1.50510e7	−0.612554
$905$	−3.31018e7	−1.34348
$906$	0	0
$907$	1.34601e7	0.543287	0.271644	−	0.962398i	$-0.412433\pi$
$907$	1.34601e7	0.543287	0.271644	+	0.962398i	$0.412433\pi$
$908$	5.02878e6	0.202418
$909$	0	0
$910$	3.80728e7	1.52409
$911$	1.42316e7	0.568144	0.284072	−	0.958803i	$-0.408315\pi$
$911$	1.42316e7	0.568144	0.284072	+	0.958803i	$0.408315\pi$
$912$	0	0
$913$	1.27937e7	0.507948
$914$	2.25575e6	0.0893154
$915$	0	0
$916$	2.33303e7	0.918718
$917$	2.02770e7	0.796307
$918$	0	0
$919$	1.48289e7	0.579188	0.289594	−	0.957150i	$-0.406480\pi$
$919$	1.48289e7	0.579188	0.289594	+	0.957150i	$0.406480\pi$
$920$	1.28251e7	0.499563
$921$	0	0
$922$	1.22130e7	0.473146
$923$	4.98574e7	1.92631
$924$	0	0
$925$	2.71070e7	1.04166
$926$	2.92764e7	1.12199
$927$	0	0
$928$	−4.85200e6	−0.184948
$929$	−1.48557e7	−0.564748	−0.282374	−	0.959304i	$-0.591122\pi$
$929$	−1.48557e7	−0.564748	−0.282374	+	0.959304i	$0.591122\pi$
$930$	0	0
$931$	310819.	0.0117526
$932$	−2.21801e7	−0.836418
$933$	0	0
$934$	1.53203e7	0.574645
$935$	−3.90520e6	−0.146088
$936$	0	0
$937$	9.74272e6	0.362519	0.181260	−	0.983435i	$-0.441983\pi$
$937$	9.74272e6	0.362519	0.181260	+	0.983435i	$0.441983\pi$
$938$	3.33686e7	1.23832
$939$	0	0
$940$	−1.91744e7	−0.707788
$941$	2.16418e6	0.0796745	0.0398373	−	0.999206i	$-0.487316\pi$
$941$	2.16418e6	0.0796745	0.0398373	+	0.999206i	$0.487316\pi$
$942$	0	0
$943$	−4.97976e7	−1.82360
$944$	1.04131e7	0.380320
$945$	0	0
$946$	−1.36881e7	−0.497297
$947$	−1.60138e7	−0.580254	−0.290127	−	0.956988i	$-0.593698\pi$
$947$	−1.60138e7	−0.580254	−0.290127	+	0.956988i	$0.593698\pi$
$948$	0	0
$949$	−4.37433e7	−1.57669
$950$	4.60034e6	0.165379
$951$	0	0
$952$	1.34242e6	0.0480060
$953$	2.67154e6	0.0952860	0.0476430	−	0.998864i	$-0.484829\pi$
$953$	2.67154e6	0.0952860	0.0476430	+	0.998864i	$0.484829\pi$
$954$	0	0
$955$	−4.62016e7	−1.63926
$956$	−5.84188e6	−0.206732
$957$	0	0
$958$	3.40856e6	0.119993
$959$	−2.55561e7	−0.897321
$960$	0	0
$961$	1.47694e7	0.515888
$962$	3.06787e7	1.06881
$963$	0	0
$964$	−4.99699e6	−0.173187
$965$	−1.45243e7	−0.502085
$966$	0	0
$967$	−7.08291e6	−0.243582	−0.121791	−	0.992556i	$-0.538864\pi$
$967$	−7.08291e6	−0.243582	−0.121791	+	0.992556i	$0.538864\pi$
$968$	−4.09641e6	−0.140513
$969$	0	0
$970$	4.41834e7	1.50775
$971$	−2.85104e7	−0.970411	−0.485205	−	0.874400i	$-0.661255\pi$
$971$	−2.85104e7	−0.970411	−0.485205	+	0.874400i	$0.661255\pi$
$972$	0	0
$973$	−4.55697e7	−1.54310
$974$	−7.07813e6	−0.239068
$975$	0	0
$976$	1.57486e6	0.0529197
$977$	2.48752e7	0.833739	0.416869	−	0.908966i	$-0.363127\pi$
$977$	2.48752e7	0.833739	0.416869	+	0.908966i	$0.363127\pi$
$978$	0	0
$979$	−5.31626e6	−0.177276
$980$	1.09437e6	0.0363998
$981$	0	0
$982$	4.50808e6	0.149181
$983$	2.66082e6	0.0878277	0.0439138	−	0.999035i	$-0.486017\pi$
$983$	2.66082e6	0.0878277	0.0439138	+	0.999035i	$0.486017\pi$
$984$	0	0
$985$	7.41539e6	0.243525
$986$	−2.99085e6	−0.0979721
$987$	0	0
$988$	5.20649e6	0.169689
$989$	2.77099e7	0.900833
$990$	0	0
$991$	5.19709e7	1.68103	0.840516	−	0.541786i	$-0.182252\pi$
$991$	5.19709e7	1.68103	0.840516	+	0.541786i	$0.182252\pi$
$992$	−6.74587e6	−0.217650
$993$	0	0
$994$	2.94080e7	0.944058
$995$	5.74489e7	1.83960
$996$	0	0
$997$	−1.86644e7	−0.594672	−0.297336	−	0.954773i	$-0.596098\pi$
$997$	−1.86644e7	−0.594672	−0.297336	+	0.954773i	$0.596098\pi$
$998$	−2.38449e7	−0.757826
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.6.a.i.1.2		2
3.2	odd	2		38.6.a.c.1.1	✓	2
12.11	even	2		304.6.a.f.1.2		2
15.14	odd	2		950.6.a.d.1.2		2
57.56	even	2		722.6.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.a.c.1.1	✓	2	3.2	odd	2
304.6.a.f.1.2		2	12.11	even	2
342.6.a.i.1.2		2	1.1	even	1	trivial
722.6.a.c.1.2		2	57.56	even	2
950.6.a.d.1.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} +79.4408 q^{5} +132.921 q^{7} +64.0000 q^{8} +O(q^{10})\)	\(q+4.00000 q^{2} +16.0000 q^{4} +79.4408 q^{5} +132.921 q^{7} +64.0000 q^{8} +317.763 q^{10} -311.520 q^{11} +901.401 q^{13} +531.684 q^{14} +256.000 q^{16} +157.803 q^{17} +361.000 q^{19} +1271.05 q^{20} -1246.08 q^{22} +2522.53 q^{23} +3185.83 q^{25} +3605.61 q^{26} +2126.74 q^{28} -4738.28 q^{29} -6587.76 q^{31} +1024.00 q^{32} +631.210 q^{34} +10559.3 q^{35} +8508.60 q^{37} +1444.00 q^{38} +5084.21 q^{40} -19741.1 q^{41} +10985.0 q^{43} -4984.32 q^{44} +10090.1 q^{46} -15085.5 q^{47} +860.995 q^{49} +12743.3 q^{50} +14422.4 q^{52} -21699.6 q^{53} -24747.4 q^{55} +8506.94 q^{56} -18953.1 q^{58} +40676.1 q^{59} +6151.79 q^{61} -26351.0 q^{62} +4096.00 q^{64} +71608.0 q^{65} +62760.3 q^{67} +2524.84 q^{68} +42237.4 q^{70} +55311.0 q^{71} -48528.1 q^{73} +34034.4 q^{74} +5776.00 q^{76} -41407.5 q^{77} +31017.6 q^{79} +20336.8 q^{80} -78964.4 q^{82} -41068.7 q^{83} +12536.0 q^{85} +43939.8 q^{86} -19937.3 q^{88} +17065.6 q^{89} +119815. q^{91} +40360.5 q^{92} -60341.9 q^{94} +28678.1 q^{95} +139045. q^{97} +3443.98 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 32 q^{4} + 45 q^{5} + 114 q^{7} + 128 q^{8}+O(q^{10}) \) 2 * q + 8 * q^2 + 32 * q^4 + 45 * q^5 + 114 * q^7 + 128 * q^8	\( 2 q + 8 q^{2} + 32 q^{4} + 45 q^{5} + 114 q^{7} + 128 q^{8} + 180 q^{10} - 661 q^{11} + 1613 q^{13} + 456 q^{14} + 512 q^{16} - 64 q^{17} + 722 q^{19} + 720 q^{20} - 2644 q^{22} + 3185 q^{23} + 1247 q^{25} + 6452 q^{26} + 1824 q^{28} + 2481 q^{29} - 1180 q^{31} + 2048 q^{32} - 256 q^{34} + 11211 q^{35} + 10488 q^{37} + 2888 q^{38} + 2880 q^{40} - 16630 q^{41} + 11303 q^{43} - 10576 q^{44} + 12740 q^{46} + 12155 q^{47} - 15588 q^{49} + 4988 q^{50} + 25808 q^{52} - 20585 q^{53} - 12711 q^{55} + 7296 q^{56} + 9924 q^{58} + 78581 q^{59} + 43621 q^{61} - 4720 q^{62} + 8192 q^{64} + 47100 q^{65} + 7805 q^{67} - 1024 q^{68} + 44844 q^{70} + 62488 q^{71} + 16218 q^{73} + 41952 q^{74} + 11552 q^{76} - 34795 q^{77} + 67122 q^{79} + 11520 q^{80} - 66520 q^{82} + 10714 q^{83} + 20175 q^{85} + 45212 q^{86} - 42304 q^{88} - 128188 q^{89} + 106351 q^{91} + 50960 q^{92} + 48620 q^{94} + 16245 q^{95} + 178558 q^{97} - 62352 q^{98}+O(q^{100}) \) 2 * q + 8 * q^2 + 32 * q^4 + 45 * q^5 + 114 * q^7 + 128 * q^8 + 180 * q^10 - 661 * q^11 + 1613 * q^13 + 456 * q^14 + 512 * q^16 - 64 * q^17 + 722 * q^19 + 720 * q^20 - 2644 * q^22 + 3185 * q^23 + 1247 * q^25 + 6452 * q^26 + 1824 * q^28 + 2481 * q^29 - 1180 * q^31 + 2048 * q^32 - 256 * q^34 + 11211 * q^35 + 10488 * q^37 + 2888 * q^38 + 2880 * q^40 - 16630 * q^41 + 11303 * q^43 - 10576 * q^44 + 12740 * q^46 + 12155 * q^47 - 15588 * q^49 + 4988 * q^50 + 25808 * q^52 - 20585 * q^53 - 12711 * q^55 + 7296 * q^56 + 9924 * q^58 + 78581 * q^59 + 43621 * q^61 - 4720 * q^62 + 8192 * q^64 + 47100 * q^65 + 7805 * q^67 - 1024 * q^68 + 44844 * q^70 + 62488 * q^71 + 16218 * q^73 + 41952 * q^74 + 11552 * q^76 - 34795 * q^77 + 67122 * q^79 + 11520 * q^80 - 66520 * q^82 + 10714 * q^83 + 20175 * q^85 + 45212 * q^86 - 42304 * q^88 - 128188 * q^89 + 106351 * q^91 + 50960 * q^92 + 48620 * q^94 + 16245 * q^95 + 178558 * q^97 - 62352 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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