LMFDB - Embedded newform 304.6.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,6,Mod(1,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("304.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	304.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:	$48.7566812231$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+17.4803 q^{3} -79.4408 q^{5} -132.921 q^{7} +62.5592 q^{9} +O(q^{10})$	$q+17.4803 q^{3} -79.4408 q^{5} -132.921 q^{7} +62.5592 q^{9} -311.520 q^{11} +901.401 q^{13} -1388.64 q^{15} -157.803 q^{17} -361.000 q^{19} -2323.49 q^{21} +2522.53 q^{23} +3185.83 q^{25} -3154.15 q^{27} +4738.28 q^{29} +6587.76 q^{31} -5445.44 q^{33} +10559.3 q^{35} +8508.60 q^{37} +15756.7 q^{39} +19741.1 q^{41} -10985.0 q^{43} -4969.75 q^{45} -15085.5 q^{47} +860.995 q^{49} -2758.43 q^{51} +21699.6 q^{53} +24747.4 q^{55} -6310.37 q^{57} +40676.1 q^{59} +6151.79 q^{61} -8315.44 q^{63} -71608.0 q^{65} -62760.3 q^{67} +44094.5 q^{69} +55311.0 q^{71} -48528.1 q^{73} +55689.2 q^{75} +41407.5 q^{77} -31017.6 q^{79} -70337.2 q^{81} -41068.7 q^{83} +12536.0 q^{85} +82826.3 q^{87} -17065.6 q^{89} -119815. q^{91} +115156. q^{93} +28678.1 q^{95} +139045. q^{97} -19488.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 45 q^{5} - 114 q^{7} + 239 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 45 * q^5 - 114 * q^7 + 239 * q^9	$ 2 q - 3 q^{3} - 45 q^{5} - 114 q^{7} + 239 q^{9} - 661 q^{11} + 1613 q^{13} - 2094 q^{15} + 64 q^{17} - 722 q^{19} - 2711 q^{21} + 3185 q^{23} + 1247 q^{25} - 1791 q^{27} - 2481 q^{29} + 1180 q^{31} + 1712 q^{33} + 11211 q^{35} + 10488 q^{37} + 1183 q^{39} + 16630 q^{41} - 11303 q^{43} + 1107 q^{45} + 12155 q^{47} - 15588 q^{49} - 7301 q^{51} + 20585 q^{53} + 12711 q^{55} + 1083 q^{57} + 78581 q^{59} + 43621 q^{61} - 4977 q^{63} - 47100 q^{65} - 7805 q^{67} + 30527 q^{69} + 62488 q^{71} + 16218 q^{73} + 95397 q^{75} + 34795 q^{77} - 67122 q^{79} - 141130 q^{81} + 10714 q^{83} + 20175 q^{85} + 230679 q^{87} + 128188 q^{89} - 106351 q^{91} + 225908 q^{93} + 16245 q^{95} + 178558 q^{97} - 81151 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 45 * q^5 - 114 * q^7 + 239 * q^9 - 661 * q^11 + 1613 * q^13 - 2094 * q^15 + 64 * q^17 - 722 * q^19 - 2711 * q^21 + 3185 * q^23 + 1247 * q^25 - 1791 * q^27 - 2481 * q^29 + 1180 * q^31 + 1712 * q^33 + 11211 * q^35 + 10488 * q^37 + 1183 * q^39 + 16630 * q^41 - 11303 * q^43 + 1107 * q^45 + 12155 * q^47 - 15588 * q^49 - 7301 * q^51 + 20585 * q^53 + 12711 * q^55 + 1083 * q^57 + 78581 * q^59 + 43621 * q^61 - 4977 * q^63 - 47100 * q^65 - 7805 * q^67 + 30527 * q^69 + 62488 * q^71 + 16218 * q^73 + 95397 * q^75 + 34795 * q^77 - 67122 * q^79 - 141130 * q^81 + 10714 * q^83 + 20175 * q^85 + 230679 * q^87 + 128188 * q^89 - 106351 * q^91 + 225908 * q^93 + 16245 * q^95 + 178558 * q^97 - 81151 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	17.4803	1.12136	0.560679	−	0.828033i	$-0.310540\pi$
$3$	17.4803	1.12136	0.560679	+	0.828033i	$0.310540\pi$
$4$	0	0
$5$	−79.4408	−1.42108	−0.710540	−	0.703657i	$-0.751549\pi$
$5$	−79.4408	−1.42108	−0.710540	+	0.703657i	$0.751549\pi$
$6$	0	0
$7$	−132.921	−1.02529	−0.512647	−	0.858599i	$-0.671335\pi$
$7$	−132.921	−1.02529	−0.512647	+	0.858599i	$0.671335\pi$
$8$	0	0
$9$	62.5592	0.257445
$10$	0	0
$11$	−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	$0.234988\pi$
$14$	0	0
$15$	−1388.64	−1.59354
$16$	0	0
$17$	−157.803	−0.132432	−0.0662158	−	0.997805i	$-0.521093\pi$
$17$	−157.803	−0.132432	−0.0662158	+	0.997805i	$0.521093\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	0	0
$21$	−2323.49	−1.14972
$22$	0	0
$23$	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
$26$	0	0
$27$	−3154.15	−0.832670
$28$	0	0
$29$	4738.28	1.04623	0.523113	−	0.852263i	$-0.324770\pi$
$29$	4738.28	1.04623	0.523113	+	0.852263i	$0.324770\pi$
$30$	0	0
$31$	6587.76	1.23121	0.615607	−	0.788053i	$-0.288911\pi$
$31$	6587.76	1.23121	0.615607	+	0.788053i	$0.288911\pi$
$32$	0	0
$33$	−5445.44	−0.870459
$34$	0	0
$35$	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	$0.329317\pi$
$38$	0	0
$39$	15756.7	1.65884
$40$	0	0
$41$	19741.1	1.83405	0.917027	−	0.398826i	$-0.130582\pi$
$41$	19741.1	1.83405	0.917027	+	0.398826i	$0.130582\pi$
$42$	0	0
$43$	−10985.0	−0.905997	−0.452999	−	0.891511i	$-0.649646\pi$
$43$	−10985.0	−0.905997	−0.452999	+	0.891511i	$0.649646\pi$
$44$	0	0
$45$	−4969.75	−0.365850
$46$	0	0
$47$	−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
$50$	0	0
$51$	−2758.43	−0.148503
$52$	0	0
$53$	21699.6	1.06112	0.530558	−	0.847649i	$-0.321982\pi$
$53$	21699.6	1.06112	0.530558	+	0.847649i	$0.321982\pi$
$54$	0	0
$55$	24747.4	1.10312
$56$	0	0
$57$	−6310.37	−0.257257
$58$	0	0
$59$	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	$0.466247\pi$
$62$	0	0
$63$	−8315.44	−0.263957
$64$	0	0
$65$	−71608.0	−2.10222
$66$	0	0
$67$	−62760.3	−1.70804	−0.854019	−	0.520241i	$-0.825842\pi$
$67$	−62760.3	−1.70804	−0.854019	+	0.520241i	$0.825842\pi$
$68$	0	0
$69$	44094.5	1.11497
$70$	0	0
$71$	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	$0.678903\pi$
$74$	0	0
$75$	55689.2	1.14319
$76$	0	0
$77$	41407.5	0.795889
$78$	0	0
$79$	−31017.6	−0.559166	−0.279583	−	0.960121i	$-0.590196\pi$
$79$	−31017.6	−0.559166	−0.279583	+	0.960121i	$0.590196\pi$
$80$	0	0
$81$	−70337.2	−1.19117
$82$	0	0
$83$	−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
$86$	0	0
$87$	82826.3	1.17320
$88$	0	0
$89$	−17065.6	−0.228373	−0.114187	−	0.993459i	$-0.536426\pi$
$89$	−17065.6	−0.228373	−0.114187	+	0.993459i	$0.536426\pi$
$90$	0	0
$91$	−119815.	−1.51673
$92$	0	0
$93$	115156.	1.38063
$94$	0	0
$95$	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	$0.229941\pi$
$98$	0	0
$99$	−19488.4	−0.199843
$100$	0	0
$101$	−122253.	−1.19249	−0.596247	−	0.802801i	$-0.703342\pi$
$101$	−122253.	−1.19249	−0.596247	+	0.802801i	$0.703342\pi$
$102$	0	0
$103$	71932.4	0.668084	0.334042	−	0.942558i	$-0.391587\pi$
$103$	71932.4	0.668084	0.334042	+	0.942558i	$0.391587\pi$
$104$	0	0
$105$	184580.	1.63385
$106$	0	0
$107$	−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	$0.309096\pi$
$110$	0	0
$111$	148733.	1.14577
$112$	0	0
$113$	235172.	1.73256	0.866282	−	0.499555i	$-0.166503\pi$
$113$	235172.	1.73256	0.866282	+	0.499555i	$0.166503\pi$
$114$	0	0
$115$	−200392.	−1.41298
$116$	0	0
$117$	56391.0	0.380842
$118$	0	0
$119$	20975.3	0.135781
$120$	0	0
$121$	−64006.4	−0.397430
$122$	0	0
$123$	345080.	2.05663
$124$	0	0
$125$	−4832.71	−0.0276640
$126$	0	0
$127$	24783.9	0.136351	0.0681757	−	0.997673i	$-0.478282\pi$
$127$	24783.9	0.136351	0.0681757	+	0.997673i	$0.478282\pi$
$128$	0	0
$129$	−192020.	−1.01595
$130$	0	0
$131$	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
$134$	0	0
$135$	250568.	1.18329
$136$	0	0
$137$	192265.	0.875184	0.437592	−	0.899174i	$-0.355831\pi$
$137$	192265.	0.875184	0.437592	+	0.899174i	$0.355831\pi$
$138$	0	0
$139$	342833.	1.50503	0.752515	−	0.658575i	$-0.228841\pi$
$139$	342833.	1.50503	0.752515	+	0.658575i	$0.228841\pi$
$140$	0	0
$141$	−263698.	−1.11702
$142$	0	0
$143$	−280804.	−1.14832
$144$	0	0
$145$	−376413.	−1.48677
$146$	0	0
$147$	15050.4	0.0574454
$148$	0	0
$149$	335859.	1.23934	0.619671	−	0.784862i	$-0.287266\pi$
$149$	335859.	1.23934	0.619671	+	0.784862i	$0.287266\pi$
$150$	0	0
$151$	−266683.	−0.951816	−0.475908	−	0.879495i	$-0.657880\pi$
$151$	−266683.	−0.951816	−0.475908	+	0.879495i	$0.657880\pi$
$152$	0	0
$153$	−9872.01	−0.0340939
$154$	0	0
$155$	−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	$0.590776\pi$
$158$	0	0
$159$	379315.	1.18989
$160$	0	0
$161$	−335298.	−1.01945
$162$	0	0
$163$	406317.	1.19783	0.598917	−	0.800811i	$-0.295598\pi$
$163$	406317.	1.19783	0.598917	+	0.800811i	$0.295598\pi$
$164$	0	0
$165$	432590.	1.23699
$166$	0	0
$167$	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
$170$	0	0
$171$	−22583.9	−0.0590620
$172$	0	0
$173$	−427313.	−1.08550	−0.542752	−	0.839893i	$-0.682618\pi$
$173$	−427313.	−1.08550	−0.542752	+	0.839893i	$0.682618\pi$
$174$	0	0
$175$	−423464.	−1.04525
$176$	0	0
$177$	711028.	1.70590
$178$	0	0
$179$	−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	$0.656719\pi$
$182$	0	0
$183$	107535.	0.237368
$184$	0	0
$185$	−675930.	−1.45202
$186$	0	0
$187$	49158.6	0.102801
$188$	0	0
$189$	419253.	0.853732
$190$	0	0
$191$	−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	$0.556528\pi$
$194$	0	0
$195$	−1.25173e6	−2.35734
$196$	0	0
$197$	−93344.9	−0.171366	−0.0856831	−	0.996322i	$-0.527307\pi$
$197$	−93344.9	−0.171366	−0.0856831	+	0.996322i	$0.527307\pi$
$198$	0	0
$199$	−723166.	−1.29451	−0.647255	−	0.762274i	$-0.724083\pi$
$199$	−723166.	−1.29451	−0.647255	+	0.762274i	$0.724083\pi$
$200$	0	0
$201$	−1.09707e6	−1.91532
$202$	0	0
$203$	−629817.	−1.07269
$204$	0	0
$205$	−1.56825e6	−2.60634
$206$	0	0
$207$	157808.	0.255978
$208$	0	0
$209$	112459.	0.178085
$210$	0	0
$211$	−85741.4	−0.132582	−0.0662910	−	0.997800i	$-0.521117\pi$
$211$	−85741.4	−0.132582	−0.0662910	+	0.997800i	$0.521117\pi$
$212$	0	0
$213$	966850.	1.46019
$214$	0	0
$215$	872653.	1.28749
$216$	0	0
$217$	−875652.	−1.26236
$218$	0	0
$219$	−848283.	−1.19517
$220$	0	0
$221$	−142243.	−0.195908
$222$	0	0
$223$	−111106.	−0.149615	−0.0748076	−	0.997198i	$-0.523834\pi$
$223$	−111106.	−0.149615	−0.0748076	+	0.997198i	$0.523834\pi$
$224$	0	0
$225$	199303.	0.262457
$226$	0	0
$227$	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	$0.129225\pi$
$230$	0	0
$231$	723814.	0.892477
$232$	0	0
$233$	1.38626e6	1.67284	0.836418	−	0.548091i	$-0.184645\pi$
$233$	1.38626e6	1.67284	0.836418	+	0.548091i	$0.184645\pi$
$234$	0	0
$235$	1.19840e6	1.41558
$236$	0	0
$237$	−542196.	−0.627026
$238$	0	0
$239$	−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	$0.555407\pi$
$242$	0	0
$243$	−463054.	−0.503056
$244$	0	0
$245$	−68398.1	−0.0727996
$246$	0	0
$247$	−325406.	−0.339377
$248$	0	0
$249$	−717891.	−0.733771
$250$	0	0
$251$	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
$254$	0	0
$255$	219132.	0.211035
$256$	0	0
$257$	1.28988e6	1.21819	0.609096	−	0.793096i	$-0.291532\pi$
$257$	1.28988e6	1.21819	0.609096	+	0.793096i	$0.291532\pi$
$258$	0	0
$259$	−1.13097e6	−1.04762
$260$	0	0
$261$	296423.	0.269346
$262$	0	0
$263$	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
$266$	0	0
$267$	−298310.	−0.256089
$268$	0	0
$269$	1.42986e6	1.20479	0.602397	−	0.798197i	$-0.294213\pi$
$269$	1.42986e6	1.20479	0.602397	+	0.798197i	$0.294213\pi$
$270$	0	0
$271$	709506.	0.586857	0.293429	−	0.955981i	$-0.405204\pi$
$271$	709506.	0.586857	0.293429	+	0.955981i	$0.405204\pi$
$272$	0	0
$273$	−2.09440e6	−1.70080
$274$	0	0
$275$	−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	$0.597055\pi$
$278$	0	0
$279$	412125.	0.316970
$280$	0	0
$281$	5975.32	0.00451435	0.00225718	−	0.999997i	$-0.499282\pi$
$281$	5975.32	0.00451435	0.00225718	+	0.999997i	$0.499282\pi$
$282$	0	0
$283$	189246.	0.140463	0.0702314	−	0.997531i	$-0.477626\pi$
$283$	189246.	0.140463	0.0702314	+	0.997531i	$0.477626\pi$
$284$	0	0
$285$	501301.	0.365583
$286$	0	0
$287$	−2.62401e6	−1.88044
$288$	0	0
$289$	−1.39496e6	−0.982462
$290$	0	0
$291$	2.43055e6	1.68256
$292$	0	0
$293$	986180.	0.671100	0.335550	−	0.942022i	$-0.391078\pi$
$293$	986180.	0.671100	0.335550	+	0.942022i	$0.391078\pi$
$294$	0	0
$295$	−3.23134e6	−2.16186
$296$	0	0
$297$	982580.	0.646363
$298$	0	0
$299$	2.27381e6	1.47088
$300$	0	0
$301$	1.46013e6	0.928914
$302$	0	0
$303$	−2.13702e6	−1.33721
$304$	0	0
$305$	−488703.	−0.300812
$306$	0	0
$307$	1.45722e6	0.882428	0.441214	−	0.897402i	$-0.354548\pi$
$307$	1.45722e6	0.882428	0.441214	+	0.897402i	$0.354548\pi$
$308$	0	0
$309$	1.25740e6	0.749162
$310$	0	0
$311$	−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	$0.875281\pi$
$314$	0	0
$315$	660585.	0.375104
$316$	0	0
$317$	−603046.	−0.337056	−0.168528	−	0.985697i	$-0.553901\pi$
$317$	−603046.	−0.337056	−0.168528	+	0.985697i	$0.553901\pi$
$318$	0	0
$319$	−1.47607e6	−0.812137
$320$	0	0
$321$	−259289.	−0.140450
$322$	0	0
$323$	56966.7	0.0303819
$324$	0	0
$325$	2.87171e6	1.50811
$326$	0	0
$327$	2.44767e6	1.26585
$328$	0	0
$329$	2.00518e6	1.02132
$330$	0	0
$331$	3.77416e6	1.89344	0.946719	−	0.322062i	$-0.104376\pi$
$331$	3.77416e6	1.89344	0.946719	+	0.322062i	$0.104376\pi$
$332$	0	0
$333$	532292.	0.263051
$334$	0	0
$335$	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	$0.670528\pi$
$338$	0	0
$339$	4.11087e6	1.94283
$340$	0	0
$341$	−2.05222e6	−0.955735
$342$	0	0
$343$	2.11956e6	0.972770
$344$	0	0
$345$	−3.50290e6	−1.58446
$346$	0	0
$347$	−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	$0.195558\pi$
$350$	0	0
$351$	−2.84315e6	−1.23178
$352$	0	0
$353$	−2.04192e6	−0.872174	−0.436087	−	0.899905i	$-0.643636\pi$
$353$	−2.04192e6	−0.872174	−0.436087	+	0.899905i	$0.643636\pi$
$354$	0	0
$355$	−4.39394e6	−1.85048
$356$	0	0
$357$	366653.	0.152260
$358$	0	0
$359$	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
$362$	0	0
$363$	−1.11885e6	−0.445661
$364$	0	0
$365$	3.85511e6	1.51462
$366$	0	0
$367$	−1.74418e6	−0.675967	−0.337984	−	0.941152i	$-0.609745\pi$
$367$	−1.74418e6	−0.675967	−0.337984	+	0.941152i	$0.609745\pi$
$368$	0	0
$369$	1.23499e6	0.472169
$370$	0	0
$371$	−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	$0.172980\pi$
$374$	0	0
$375$	−84477.0	−0.0310213
$376$	0	0
$377$	4.27109e6	1.54770
$378$	0	0
$379$	6993.69	0.00250097	0.00125048	−	0.999999i	$-0.499602\pi$
$379$	6993.69	0.00250097	0.00125048	+	0.999999i	$0.499602\pi$
$380$	0	0
$381$	433228.	0.152899
$382$	0	0
$383$	−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
$386$	0	0
$387$	−687210.	−0.233245
$388$	0	0
$389$	1.40115e6	0.469473	0.234736	−	0.972059i	$-0.424577\pi$
$389$	1.40115e6	0.469473	0.234736	+	0.972059i	$0.424577\pi$
$390$	0	0
$391$	−398062.	−0.131677
$392$	0	0
$393$	2.66660e6	0.870916
$394$	0	0
$395$	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	$0.321877\pi$
$398$	0	0
$399$	838781.	0.263764
$400$	0	0
$401$	−5.32612e6	−1.65406	−0.827028	−	0.562161i	$-0.809970\pi$
$401$	−5.32612e6	−1.65406	−0.827028	+	0.562161i	$0.809970\pi$
$402$	0	0
$403$	5.93822e6	1.82135
$404$	0	0
$405$	5.58764e6	1.69274
$406$	0	0
$407$	−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	$0.646675\pi$
$410$	0	0
$411$	3.36085e6	0.981395
$412$	0	0
$413$	−5.40670e6	−1.55976
$414$	0	0
$415$	3.26253e6	0.929895
$416$	0	0
$417$	5.99280e6	1.68768
$418$	0	0
$419$	−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	$0.647737\pi$
$422$	0	0
$423$	−943736.	−0.256448
$424$	0	0
$425$	−502733.	−0.135010
$426$	0	0
$427$	−817702.	−0.217033
$428$	0	0
$429$	−4.90853e6	−1.28768
$430$	0	0
$431$	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	$0.392317\pi$
$434$	0	0
$435$	−6.57979e6	−1.66720
$436$	0	0
$437$	−910634.	−0.228108
$438$	0	0
$439$	3.47682e6	0.861036	0.430518	−	0.902582i	$-0.358331\pi$
$439$	3.47682e6	0.861036	0.430518	+	0.902582i	$0.358331\pi$
$440$	0	0
$441$	53863.2	0.0131885
$442$	0	0
$443$	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
$446$	0	0
$447$	5.87089e6	1.38975
$448$	0	0
$449$	−4.47721e6	−1.04807	−0.524036	−	0.851696i	$-0.675574\pi$
$449$	−4.47721e6	−1.04807	−0.524036	+	0.851696i	$0.675574\pi$
$450$	0	0
$451$	−6.14975e6	−1.42369
$452$	0	0
$453$	−4.66169e6	−1.06733
$454$	0	0
$455$	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	$0.479884\pi$
$458$	0	0
$459$	497733.	0.110272
$460$	0	0
$461$	−3.05325e6	−0.669130	−0.334565	−	0.942373i	$-0.608589\pi$
$461$	−3.05325e6	−0.669130	−0.334565	+	0.942373i	$0.608589\pi$
$462$	0	0
$463$	−7.31911e6	−1.58674	−0.793370	−	0.608740i	$-0.791675\pi$
$463$	−7.31911e6	−1.58674	−0.793370	+	0.608740i	$0.791675\pi$
$464$	0	0
$465$	−9.14806e6	−1.96199
$466$	0	0
$467$	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
$470$	0	0
$471$	−3.03771e6	−0.630948
$472$	0	0
$473$	3.42203e6	0.703284
$474$	0	0
$475$	−1.15009e6	−0.233882
$476$	0	0
$477$	1.35751e6	0.273179
$478$	0	0
$479$	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
$482$	0	0
$483$	−5.86109e6	−1.14317
$484$	0	0
$485$	−1.10459e7	−2.13228
$486$	0	0
$487$	1.76953e6	0.338093	0.169046	−	0.985608i	$-0.445931\pi$
$487$	1.76953e6	0.338093	0.169046	+	0.985608i	$0.445931\pi$
$488$	0	0
$489$	7.10253e6	1.34320
$490$	0	0
$491$	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
$494$	0	0
$495$	1.54818e6	0.283993
$496$	0	0
$497$	−7.35199e6	−1.33510
$498$	0	0
$499$	5.96123e6	1.07173	0.535864	−	0.844304i	$-0.319986\pi$
$499$	5.96123e6	1.07173	0.535864	+	0.844304i	$0.319986\pi$
$500$	0	0
$501$	5.58835e6	0.994694
$502$	0	0
$503$	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
$506$	0	0
$507$	7.71283e6	1.33258
$508$	0	0
$509$	8.42220e6	1.44089	0.720446	−	0.693512i	$-0.243937\pi$
$509$	8.42220e6	1.44089	0.720446	+	0.693512i	$0.243937\pi$
$510$	0	0
$511$	6.45040e6	1.09278
$512$	0	0
$513$	1.13865e6	0.191028
$514$	0	0
$515$	−5.71436e6	−0.949401
$516$	0	0
$517$	4.69943e6	0.773247
$518$	0	0
$519$	−7.46955e6	−1.21724
$520$	0	0
$521$	5.81861e6	0.939128	0.469564	−	0.882899i	$-0.344411\pi$
$521$	5.81861e6	0.939128	0.469564	+	0.882899i	$0.344411\pi$
$522$	0	0
$523$	−4.81037e6	−0.768996	−0.384498	−	0.923126i	$-0.625625\pi$
$523$	−4.81037e6	−0.768996	−0.384498	+	0.923126i	$0.625625\pi$
$524$	0	0
$525$	−7.40226e6	−1.17210
$526$	0	0
$527$	−1.03957e6	−0.163052
$528$	0	0
$529$	−73173.3	−0.0113688
$530$	0	0
$531$	2.54466e6	0.391646
$532$	0	0
$533$	1.77947e7	2.71314
$534$	0	0
$535$	1.17836e6	0.177990
$536$	0	0
$537$	−6.32691e6	−0.946795
$538$	0	0
$539$	−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	$0.426405\pi$
$542$	0	0
$543$	−7.28378e6	−1.06012
$544$	0	0
$545$	−1.11237e7	−1.60420
$546$	0	0
$547$	−2.61616e6	−0.373848	−0.186924	−	0.982374i	$-0.559852\pi$
$547$	−2.61616e6	−0.373848	−0.186924	+	0.982374i	$0.559852\pi$
$548$	0	0
$549$	384851.	0.0544957
$550$	0	0
$551$	−1.71052e6	−0.240021
$552$	0	0
$553$	4.12290e6	0.573310
$554$	0	0
$555$	−1.18154e7	−1.62823
$556$	0	0
$557$	1.91062e6	0.260937	0.130469	−	0.991452i	$-0.458352\pi$
$557$	1.91062e6	0.260937	0.130469	+	0.991452i	$0.458352\pi$
$558$	0	0
$559$	−9.90185e6	−1.34025
$560$	0	0
$561$	859305.	0.115276
$562$	0	0
$563$	−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
$566$	0	0
$567$	9.34930e6	1.22130
$568$	0	0
$569$	−2.22862e6	−0.288572	−0.144286	−	0.989536i	$-0.546089\pi$
$569$	−2.22862e6	−0.288572	−0.144286	+	0.989536i	$0.546089\pi$
$570$	0	0
$571$	−8.28284e6	−1.06314	−0.531568	−	0.847015i	$-0.678397\pi$
$571$	−8.28284e6	−1.06314	−0.531568	+	0.847015i	$0.678397\pi$
$572$	0	0
$573$	−1.01663e7	−1.29353
$574$	0	0
$575$	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	$0.887863\pi$
$578$	0	0
$579$	−3.19595e6	−0.396190
$580$	0	0
$581$	5.45889e6	0.670910
$582$	0	0
$583$	−6.75986e6	−0.823695
$584$	0	0
$585$	−4.47974e6	−0.541207
$586$	0	0
$587$	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
$590$	0	0
$591$	−1.63169e6	−0.192163
$592$	0	0
$593$	−545602.	−0.0637147	−0.0318573	−	0.999492i	$-0.510142\pi$
$593$	−545602.	−0.0637147	−0.0318573	+	0.999492i	$0.510142\pi$
$594$	0	0
$595$	−1.66629e6	−0.192956
$596$	0	0
$597$	−1.26411e7	−1.45161
$598$	0	0
$599$	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	$0.598602\pi$
$602$	0	0
$603$	−3.92623e6	−0.439727
$604$	0	0
$605$	5.08472e6	0.564779
$606$	0	0
$607$	1.14275e7	1.25886	0.629432	−	0.777055i	$-0.283288\pi$
$607$	1.14275e7	1.25886	0.629432	+	0.777055i	$0.283288\pi$
$608$	0	0
$609$	−1.10094e7	−1.20287
$610$	0	0
$611$	−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	$0.515310\pi$
$614$	0	0
$615$	−2.74134e7	−2.92264
$616$	0	0
$617$	1.18925e7	1.25765	0.628826	−	0.777546i	$-0.283536\pi$
$617$	1.18925e7	1.25765	0.628826	+	0.777546i	$0.283536\pi$
$618$	0	0
$619$	−1.03513e7	−1.08585	−0.542924	−	0.839782i	$-0.682683\pi$
$619$	−1.03513e7	−1.08585	−0.542924	+	0.839782i	$0.682683\pi$
$620$	0	0
$621$	−7.95645e6	−0.827923
$622$	0	0
$623$	2.26837e6	0.234150
$624$	0	0
$625$	−9.57182e6	−0.980154
$626$	0	0
$627$	1.96581e6	0.199697
$628$	0	0
$629$	−1.34268e6	−0.135315
$630$	0	0
$631$	1.17088e6	0.117068	0.0585342	−	0.998285i	$-0.481357\pi$
$631$	1.17088e6	0.117068	0.0585342	+	0.998285i	$0.481357\pi$
$632$	0	0
$633$	−1.49878e6	−0.148672
$634$	0	0
$635$	−1.96885e6	−0.193766
$636$	0	0
$637$	776102.	0.0757828
$638$	0	0
$639$	3.46021e6	0.335236
$640$	0	0
$641$	−4.72990e6	−0.454681	−0.227341	−	0.973815i	$-0.573003\pi$
$641$	−4.72990e6	−0.454681	−0.227341	+	0.973815i	$0.573003\pi$
$642$	0	0
$643$	1.49075e7	1.42193	0.710963	−	0.703229i	$-0.248259\pi$
$643$	1.49075e7	1.42193	0.710963	+	0.703229i	$0.248259\pi$
$644$	0	0
$645$	1.52542e7	1.44374
$646$	0	0
$647$	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
$650$	0	0
$651$	−1.53066e7	−1.41555
$652$	0	0
$653$	2.25370e6	0.206829	0.103415	−	0.994638i	$-0.467023\pi$
$653$	2.25370e6	0.206829	0.103415	+	0.994638i	$0.467023\pi$
$654$	0	0
$655$	−1.21186e7	−1.10370
$656$	0	0
$657$	−3.03588e6	−0.274392
$658$	0	0
$659$	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	$0.405677\pi$
$662$	0	0
$663$	−2.48645e6	−0.219683
$664$	0	0
$665$	−3.81192e6	−0.334264
$666$	0	0
$667$	1.19525e7	1.04026
$668$	0	0
$669$	−1.94216e6	−0.167772
$670$	0	0
$671$	−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	$0.543424\pi$
$674$	0	0
$675$	−1.00486e7	−0.848880
$676$	0	0
$677$	−2.06961e6	−0.173547	−0.0867734	−	0.996228i	$-0.527656\pi$
$677$	−2.06961e6	−0.173547	−0.0867734	+	0.996228i	$0.527656\pi$
$678$	0	0
$679$	−1.84820e7	−1.53842
$680$	0	0
$681$	5.49402e6	0.453965
$682$	0	0
$683$	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
$686$	0	0
$687$	2.54888e7	2.06043
$688$	0	0
$689$	1.95601e7	1.56972
$690$	0	0
$691$	1.40203e7	1.11703	0.558513	−	0.829496i	$-0.311372\pi$
$691$	1.40203e7	1.11703	0.558513	+	0.829496i	$0.311372\pi$
$692$	0	0
$693$	2.59042e6	0.204898
$694$	0	0
$695$	−2.72349e7	−2.13877
$696$	0	0
$697$	−3.11520e6	−0.242887
$698$	0	0
$699$	2.42321e7	1.87585
$700$	0	0
$701$	3.42664e6	0.263375	0.131687	−	0.991291i	$-0.457961\pi$
$701$	3.42664e6	0.263375	0.131687	+	0.991291i	$0.457961\pi$
$702$	0	0
$703$	−3.07161e6	−0.234411
$704$	0	0
$705$	2.09484e7	1.58737
$706$	0	0
$707$	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	$0.402780\pi$
$710$	0	0
$711$	−1.94044e6	−0.143955
$712$	0	0
$713$	1.66178e7	1.22420
$714$	0	0
$715$	2.23073e7	1.63186
$716$	0	0
$717$	−6.38234e6	−0.463641
$718$	0	0
$719$	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
$722$	0	0
$723$	−5.45929e6	−0.388410
$724$	0	0
$725$	1.50954e7	1.06659
$726$	0	0
$727$	−1.25706e7	−0.882102	−0.441051	−	0.897482i	$-0.645394\pi$
$727$	−1.25706e7	−0.882102	−0.441051	+	0.897482i	$0.645394\pi$
$728$	0	0
$729$	8.99764e6	0.627061
$730$	0	0
$731$	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	$0.626111\pi$
$734$	0	0
$735$	−1.19562e6	−0.0816345
$736$	0	0
$737$	1.95511e7	1.32587
$738$	0	0
$739$	−1.05677e7	−0.711820	−0.355910	−	0.934520i	$-0.615829\pi$
$739$	−1.05677e7	−0.711820	−0.355910	+	0.934520i	$0.615829\pi$
$740$	0	0
$741$	−5.68818e6	−0.380564
$742$	0	0
$743$	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
$746$	0	0
$747$	−2.56923e6	−0.168462
$748$	0	0
$749$	1.97165e6	0.128418
$750$	0	0
$751$	−4.48767e6	−0.290349	−0.145175	−	0.989406i	$-0.546374\pi$
$751$	−4.48767e6	−0.290349	−0.145175	+	0.989406i	$0.546374\pi$
$752$	0	0
$753$	1.69575e7	1.08987
$754$	0	0
$755$	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	$0.420300\pi$
$758$	0	0
$759$	−1.37363e7	−0.865497
$760$	0	0
$761$	1.81639e7	1.13697	0.568483	−	0.822695i	$-0.307531\pi$
$761$	1.81639e7	1.13697	0.568483	+	0.822695i	$0.307531\pi$
$762$	0	0
$763$	−1.86123e7	−1.15741
$764$	0	0
$765$	784240.	0.0484502
$766$	0	0
$767$	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	$0.543536\pi$
$770$	0	0
$771$	2.25474e7	1.36603
$772$	0	0
$773$	−2.53207e7	−1.52415	−0.762075	−	0.647489i	$-0.775819\pi$
$773$	−2.53207e7	−1.52415	−0.762075	+	0.647489i	$0.775819\pi$
$774$	0	0
$775$	2.09875e7	1.25518
$776$	0	0
$777$	−1.97697e7	−1.17475
$778$	0	0
$779$	−7.12654e6	−0.420761
$780$	0	0
$781$	−1.72305e7	−1.01081
$782$	0	0
$783$	−1.49452e7	−0.871161
$784$	0	0
$785$	1.38052e7	0.799590
$786$	0	0
$787$	−4.73524e6	−0.272524	−0.136262	−	0.990673i	$-0.543509\pi$
$787$	−4.73524e6	−0.272524	−0.136262	+	0.990673i	$0.543509\pi$
$788$	0	0
$789$	8.76506e6	0.501259
$790$	0	0
$791$	−3.12593e7	−1.77639
$792$	0	0
$793$	5.54523e6	0.313139
$794$	0	0
$795$	−3.01331e7	−1.69093
$796$	0	0
$797$	1.54780e7	0.863116	0.431558	−	0.902085i	$-0.357964\pi$
$797$	1.54780e7	0.863116	0.431558	+	0.902085i	$0.357964\pi$
$798$	0	0
$799$	2.38053e6	0.131919
$800$	0	0
$801$	−1.06761e6	−0.0587937
$802$	0	0
$803$	1.51175e7	0.827351
$804$	0	0
$805$	2.66363e7	1.44872
$806$	0	0
$807$	2.49943e7	1.35101
$808$	0	0
$809$	5.54794e6	0.298031	0.149015	−	0.988835i	$-0.452390\pi$
$809$	5.54794e6	0.298031	0.149015	+	0.988835i	$0.452390\pi$
$810$	0	0
$811$	−9.82955e6	−0.524785	−0.262392	−	0.964961i	$-0.584511\pi$
$811$	−9.82955e6	−0.524785	−0.262392	+	0.964961i	$0.584511\pi$
$812$	0	0
$813$	1.24023e7	0.658078
$814$	0	0
$815$	−3.22782e7	−1.70222
$816$	0	0
$817$	3.96557e6	0.207850
$818$	0	0
$819$	−7.49555e6	−0.390475
$820$	0	0
$821$	1.78229e7	0.922830	0.461415	−	0.887185i	$-0.347342\pi$
$821$	1.78229e7	0.922830	0.461415	+	0.887185i	$0.347342\pi$
$822$	0	0
$823$	1.29509e7	0.666502	0.333251	−	0.942838i	$-0.391854\pi$
$823$	1.29509e7	0.666502	0.333251	+	0.942838i	$0.391854\pi$
$824$	0	0
$825$	−1.73483e7	−0.887404
$826$	0	0
$827$	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	$0.305562\pi$
$830$	0	0
$831$	−1.34028e7	−0.673276
$832$	0	0
$833$	−135867.	−0.00678426
$834$	0	0
$835$	−2.53968e7	−1.26056
$836$	0	0
$837$	−2.07788e7	−1.02519
$838$	0	0
$839$	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
$842$	0	0
$843$	104450.	0.00506221
$844$	0	0
$845$	−3.50517e7	−1.68876
$846$	0	0
$847$	8.50780e6	0.407482
$848$	0	0
$849$	3.30807e6	0.157509
$850$	0	0
$851$	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	$0.768023\pi$
$854$	0	0
$855$	1.79408e6	0.0839318
$856$	0	0
$857$	3.01353e7	1.40160	0.700799	−	0.713359i	$-0.252827\pi$
$857$	3.01353e7	1.40160	0.700799	+	0.713359i	$0.252827\pi$
$858$	0	0
$859$	1.58417e7	0.732520	0.366260	−	0.930513i	$-0.380638\pi$
$859$	1.58417e7	0.732520	0.366260	+	0.930513i	$0.380638\pi$
$860$	0	0
$861$	−4.58683e7	−2.10865
$862$	0	0
$863$	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
$866$	0	0
$867$	−2.43842e7	−1.10169
$868$	0	0
$869$	9.66261e6	0.434055
$870$	0	0
$871$	−5.65722e7	−2.52672
$872$	0	0
$873$	8.69856e6	0.386289
$874$	0	0
$875$	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	$0.281010\pi$
$878$	0	0
$879$	1.72387e7	0.752544
$880$	0	0
$881$	−2.35636e7	−1.02283	−0.511413	−	0.859335i	$-0.670878\pi$
$881$	−2.35636e7	−1.02283	−0.511413	+	0.859335i	$0.670878\pi$
$882$	0	0
$883$	1.91319e6	0.0825764	0.0412882	−	0.999147i	$-0.486854\pi$
$883$	1.91319e6	0.0825764	0.0412882	+	0.999147i	$0.486854\pi$
$884$	0	0
$885$	−5.64846e7	−2.42422
$886$	0	0
$887$	−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
$890$	0	0
$891$	2.19114e7	0.924648
$892$	0	0
$893$	5.44586e6	0.228527
$894$	0	0
$895$	2.87533e7	1.19986
$896$	0	0
$897$	3.97468e7	1.64938
$898$	0	0
$899$	3.12146e7	1.28813
$900$	0	0
$901$	−3.42426e6	−0.140525
$902$	0	0
$903$	2.55235e7	1.04165
$904$	0	0
$905$	3.31018e7	1.34348
$906$	0	0
$907$	−1.34601e7	−0.543287	−0.271644	−	0.962398i	$-0.587567\pi$
$907$	−1.34601e7	−0.543287	−0.271644	+	0.962398i	$0.587567\pi$
$908$	0	0
$909$	−7.64806e6	−0.307002
$910$	0	0
$911$	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
$914$	0	0
$915$	−8.54265e6	−0.337318
$916$	0	0
$917$	−2.02770e7	−0.796307
$918$	0	0
$919$	−1.48289e7	−0.579188	−0.289594	−	0.957150i	$-0.593520\pi$
$919$	−1.48289e7	−0.579188	−0.289594	+	0.957150i	$0.593520\pi$
$920$	0	0
$921$	2.54726e7	0.989518
$922$	0	0
$923$	4.98574e7	1.92631
$924$	0	0
$925$	2.71070e7	1.04166
$926$	0	0
$927$	4.50003e6	0.171995
$928$	0	0
$929$	1.48557e7	0.564748	0.282374	−	0.959304i	$-0.408878\pi$
$929$	1.48557e7	0.564748	0.282374	+	0.959304i	$0.408878\pi$
$930$	0	0
$931$	−310819.	−0.0117526
$932$	0	0
$933$	−4.20446e7	−1.58127
$934$	0	0
$935$	−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	$0.441983\pi$
$938$	0	0
$939$	−5.60032e7	−2.07276
$940$	0	0
$941$	−2.16418e6	−0.0796745	−0.0398373	−	0.999206i	$-0.512684\pi$
$941$	−2.16418e6	−0.0796745	−0.0398373	+	0.999206i	$0.512684\pi$
$942$	0	0
$943$	4.97976e7	1.82360
$944$	0	0
$945$	−3.33058e7	−1.21322
$946$	0	0
$947$	−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
$950$	0	0
$951$	−1.05414e7	−0.377961
$952$	0	0
$953$	−2.67154e6	−0.0952860	−0.0476430	−	0.998864i	$-0.515171\pi$
$953$	−2.67154e6	−0.0952860	−0.0476430	+	0.998864i	$0.515171\pi$
$954$	0	0
$955$	4.62016e7	1.63926
$956$	0	0
$957$	−2.58020e7	−0.910697
$958$	0	0
$959$	−2.55561e7	−0.897321
$960$	0	0
$961$	1.47694e7	0.515888
$962$	0	0
$963$	−927957.	−0.0322450
$964$	0	0
$965$	1.45243e7	0.502085
$966$	0	0
$967$	7.08291e6	0.243582	0.121791	−	0.992556i	$-0.461136\pi$
$967$	7.08291e6	0.243582	0.121791	+	0.992556i	$0.461136\pi$
$968$	0	0
$969$	995793.	0.0340690
$970$	0	0
$971$	−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
$974$	0	0
$975$	5.01983e7	1.69113
$976$	0	0
$977$	−2.48752e7	−0.833739	−0.416869	−	0.908966i	$-0.636873\pi$
$977$	−2.48752e7	−0.833739	−0.416869	+	0.908966i	$0.636873\pi$
$978$	0	0
$979$	5.31626e6	0.177276
$980$	0	0
$981$	8.75985e6	0.290619
$982$	0	0
$983$	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
$986$	0	0
$987$	3.50510e7	1.14527
$988$	0	0
$989$	−2.77099e7	−0.900833
$990$	0	0
$991$	−5.19709e7	−1.68103	−0.840516	−	0.541786i	$-0.817748\pi$
$991$	−5.19709e7	−1.68103	−0.840516	+	0.541786i	$0.817748\pi$
$992$	0	0
$993$	6.59734e7	2.12322
$994$	0	0
$995$	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	$0.596098\pi$
$998$	0	0
$999$	−2.68374e7	−0.850799

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.6.a.f.1.2		2
4.3	odd	2		38.6.a.c.1.1	✓	2
12.11	even	2		342.6.a.i.1.2		2
20.19	odd	2		950.6.a.d.1.2		2
76.75	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	4.3	odd	2
304.6.a.f.1.2		2	1.1	even	1	trivial
342.6.a.i.1.2		2	12.11	even	2
722.6.a.c.1.2		2	76.75	even	2
950.6.a.d.1.2		2	20.19	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q+17.4803 q^{3} -79.4408 q^{5} -132.921 q^{7} +62.5592 q^{9} +O(q^{10})\)	\(q+17.4803 q^{3} -79.4408 q^{5} -132.921 q^{7} +62.5592 q^{9} -311.520 q^{11} +901.401 q^{13} -1388.64 q^{15} -157.803 q^{17} -361.000 q^{19} -2323.49 q^{21} +2522.53 q^{23} +3185.83 q^{25} -3154.15 q^{27} +4738.28 q^{29} +6587.76 q^{31} -5445.44 q^{33} +10559.3 q^{35} +8508.60 q^{37} +15756.7 q^{39} +19741.1 q^{41} -10985.0 q^{43} -4969.75 q^{45} -15085.5 q^{47} +860.995 q^{49} -2758.43 q^{51} +21699.6 q^{53} +24747.4 q^{55} -6310.37 q^{57} +40676.1 q^{59} +6151.79 q^{61} -8315.44 q^{63} -71608.0 q^{65} -62760.3 q^{67} +44094.5 q^{69} +55311.0 q^{71} -48528.1 q^{73} +55689.2 q^{75} +41407.5 q^{77} -31017.6 q^{79} -70337.2 q^{81} -41068.7 q^{83} +12536.0 q^{85} +82826.3 q^{87} -17065.6 q^{89} -119815. q^{91} +115156. q^{93} +28678.1 q^{95} +139045. q^{97} -19488.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 45 q^{5} - 114 q^{7} + 239 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 45 * q^5 - 114 * q^7 + 239 * q^9	\( 2 q - 3 q^{3} - 45 q^{5} - 114 q^{7} + 239 q^{9} - 661 q^{11} + 1613 q^{13} - 2094 q^{15} + 64 q^{17} - 722 q^{19} - 2711 q^{21} + 3185 q^{23} + 1247 q^{25} - 1791 q^{27} - 2481 q^{29} + 1180 q^{31} + 1712 q^{33} + 11211 q^{35} + 10488 q^{37} + 1183 q^{39} + 16630 q^{41} - 11303 q^{43} + 1107 q^{45} + 12155 q^{47} - 15588 q^{49} - 7301 q^{51} + 20585 q^{53} + 12711 q^{55} + 1083 q^{57} + 78581 q^{59} + 43621 q^{61} - 4977 q^{63} - 47100 q^{65} - 7805 q^{67} + 30527 q^{69} + 62488 q^{71} + 16218 q^{73} + 95397 q^{75} + 34795 q^{77} - 67122 q^{79} - 141130 q^{81} + 10714 q^{83} + 20175 q^{85} + 230679 q^{87} + 128188 q^{89} - 106351 q^{91} + 225908 q^{93} + 16245 q^{95} + 178558 q^{97} - 81151 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 45 * q^5 - 114 * q^7 + 239 * q^9 - 661 * q^11 + 1613 * q^13 - 2094 * q^15 + 64 * q^17 - 722 * q^19 - 2711 * q^21 + 3185 * q^23 + 1247 * q^25 - 1791 * q^27 - 2481 * q^29 + 1180 * q^31 + 1712 * q^33 + 11211 * q^35 + 10488 * q^37 + 1183 * q^39 + 16630 * q^41 - 11303 * q^43 + 1107 * q^45 + 12155 * q^47 - 15588 * q^49 - 7301 * q^51 + 20585 * q^53 + 12711 * q^55 + 1083 * q^57 + 78581 * q^59 + 43621 * q^61 - 4977 * q^63 - 47100 * q^65 - 7805 * q^67 + 30527 * q^69 + 62488 * q^71 + 16218 * q^73 + 95397 * q^75 + 34795 * q^77 - 67122 * q^79 - 141130 * q^81 + 10714 * q^83 + 20175 * q^85 + 230679 * q^87 + 128188 * q^89 - 106351 * q^91 + 225908 * q^93 + 16245 * q^95 + 178558 * q^97 - 81151 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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