LMFDB - Embedded newform 304.6.a.f.1.1

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			1.1
Root			$19.4803$ 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)

$f(q)$	$=$	$q-20.4803 q^{3} +34.4408 q^{5} +18.9210 q^{7} +176.441 q^{9} +O(q^{10})$	$q-20.4803 q^{3} +34.4408 q^{5} +18.9210 q^{7} +176.441 q^{9} -349.480 q^{11} +711.599 q^{13} -705.355 q^{15} +221.803 q^{17} -361.000 q^{19} -387.507 q^{21} +662.468 q^{23} -1938.83 q^{25} +1363.15 q^{27} -7219.28 q^{29} -5407.76 q^{31} +7157.44 q^{33} +651.654 q^{35} +1979.40 q^{37} -14573.7 q^{39} -3111.11 q^{41} -318.049 q^{43} +6076.75 q^{45} +27240.5 q^{47} -16449.0 q^{49} -4542.57 q^{51} -1114.63 q^{53} -12036.4 q^{55} +7393.37 q^{57} +37904.9 q^{59} +37469.2 q^{61} +3338.44 q^{63} +24508.0 q^{65} +54955.3 q^{67} -13567.5 q^{69} +7177.04 q^{71} +64746.1 q^{73} +39707.8 q^{75} -6612.52 q^{77} -36104.4 q^{79} -70792.8 q^{81} +51782.7 q^{83} +7639.05 q^{85} +147853. q^{87} +145254. q^{89} +13464.2 q^{91} +110752. q^{93} -12433.1 q^{95} +39512.8 q^{97} -61662.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−20.4803	−1.31381	−0.656904	−	0.753974i	$-0.728135\pi$
$3$	−20.4803	−1.31381	−0.656904	+	0.753974i	$0.728135\pi$
$4$	0	0
$5$	34.4408	0.616095	0.308048	−	0.951371i	$-0.400324\pi$
$5$	34.4408	0.616095	0.308048	+	0.951371i	$0.400324\pi$
$6$	0	0
$7$	18.9210	0.145948	0.0729742	−	0.997334i	$-0.476751\pi$
$7$	18.9210	0.145948	0.0729742	+	0.997334i	$0.476751\pi$
$8$	0	0
$9$	176.441	0.726094
$10$	0	0
$11$	−349.480	−0.870845	−0.435423	−	0.900226i	$-0.643401\pi$
$11$	−349.480	−0.870845	−0.435423	+	0.900226i	$0.643401\pi$
$12$	0	0
$13$	711.599	1.16782	0.583911	−	0.811818i	$-0.301522\pi$
$13$	711.599	1.16782	0.583911	+	0.811818i	$0.301522\pi$
$14$	0	0
$15$	−705.355	−0.809431
$16$	0	0
$17$	221.803	0.186142	0.0930710	−	0.995659i	$-0.470332\pi$
$17$	221.803	0.186142	0.0930710	+	0.995659i	$0.470332\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	0	0
$21$	−387.507	−0.191748
$22$	0	0
$23$	662.468	0.261123	0.130561	−	0.991440i	$-0.458322\pi$
$23$	662.468	0.261123	0.130561	+	0.991440i	$0.458322\pi$
$24$	0	0
$25$	−1938.83	−0.620427
$26$	0	0
$27$	1363.15	0.359861
$28$	0	0
$29$	−7219.28	−1.59404	−0.797019	−	0.603954i	$-0.793591\pi$
$29$	−7219.28	−1.59404	−0.797019	+	0.603954i	$0.793591\pi$
$30$	0	0
$31$	−5407.76	−1.01068	−0.505339	−	0.862921i	$-0.668633\pi$
$31$	−5407.76	−1.01068	−0.505339	+	0.862921i	$0.668633\pi$
$32$	0	0
$33$	7157.44	1.14412
$34$	0	0
$35$	651.654	0.0899181
$36$	0	0
$37$	1979.40	0.237700	0.118850	−	0.992912i	$-0.462079\pi$
$37$	1979.40	0.237700	0.118850	+	0.992912i	$0.462079\pi$
$38$	0	0
$39$	−14573.7	−1.53430
$40$	0	0
$41$	−3111.11	−0.289039	−0.144519	−	0.989502i	$-0.546164\pi$
$41$	−3111.11	−0.289039	−0.144519	+	0.989502i	$0.546164\pi$
$42$	0	0
$43$	−318.049	−0.0262315	−0.0131157	−	0.999914i	$-0.504175\pi$
$43$	−318.049	−0.0262315	−0.0131157	+	0.999914i	$0.504175\pi$
$44$	0	0
$45$	6076.75	0.447343
$46$	0	0
$47$	27240.5	1.79875	0.899374	−	0.437181i	$-0.144023\pi$
$47$	27240.5	1.79875	0.899374	+	0.437181i	$0.144023\pi$
$48$	0	0
$49$	−16449.0	−0.978699
$50$	0	0
$51$	−4542.57	−0.244555
$52$	0	0
$53$	−1114.63	−0.0545057	−0.0272528	−	0.999629i	$-0.508676\pi$
$53$	−1114.63	−0.0545057	−0.0272528	+	0.999629i	$0.508676\pi$
$54$	0	0
$55$	−12036.4	−0.536523
$56$	0	0
$57$	7393.37	0.301408
$58$	0	0
$59$	37904.9	1.41764	0.708820	−	0.705390i	$-0.249228\pi$
$59$	37904.9	1.41764	0.708820	+	0.705390i	$0.249228\pi$
$60$	0	0
$61$	37469.2	1.28929	0.644644	−	0.764483i	$-0.277006\pi$
$61$	37469.2	1.28929	0.644644	+	0.764483i	$0.277006\pi$
$62$	0	0
$63$	3338.44	0.105972
$64$	0	0
$65$	24508.0	0.719490
$66$	0	0
$67$	54955.3	1.49562	0.747812	−	0.663911i	$-0.231105\pi$
$67$	54955.3	1.49562	0.747812	+	0.663911i	$0.231105\pi$
$68$	0	0
$69$	−13567.5	−0.343066
$70$	0	0
$71$	7177.04	0.168966	0.0844830	−	0.996425i	$-0.473076\pi$
$71$	7177.04	0.168966	0.0844830	+	0.996425i	$0.473076\pi$
$72$	0	0
$73$	64746.1	1.42202	0.711011	−	0.703181i	$-0.248238\pi$
$73$	64746.1	1.42202	0.711011	+	0.703181i	$0.248238\pi$
$74$	0	0
$75$	39707.8	0.815122
$76$	0	0
$77$	−6612.52	−0.127098
$78$	0	0
$79$	−36104.4	−0.650866	−0.325433	−	0.945565i	$-0.605510\pi$
$79$	−36104.4	−0.650866	−0.325433	+	0.945565i	$0.605510\pi$
$80$	0	0
$81$	−70792.8	−1.19888
$82$	0	0
$83$	51782.7	0.825067	0.412534	−	0.910942i	$-0.364644\pi$
$83$	51782.7	0.825067	0.412534	+	0.910942i	$0.364644\pi$
$84$	0	0
$85$	7639.05	0.114681
$86$	0	0
$87$	147853.	2.09426
$88$	0	0
$89$	145254.	1.94380	0.971900	−	0.235392i	$-0.0756375\pi$
$89$	145254.	1.94380	0.971900	+	0.235392i	$0.0756375\pi$
$90$	0	0
$91$	13464.2	0.170442
$92$	0	0
$93$	110752.	1.32784
$94$	0	0
$95$	−12433.1	−0.141342
$96$	0	0
$97$	39512.8	0.426391	0.213196	−	0.977010i	$-0.431613\pi$
$97$	39512.8	0.426391	0.213196	+	0.977010i	$0.431613\pi$
$98$	0	0
$99$	−61662.6	−0.632315
$100$	0	0
$101$	−16722.9	−0.163120	−0.0815602	−	0.996668i	$-0.525990\pi$
$101$	−16722.9	−0.163120	−0.0815602	+	0.996668i	$0.525990\pi$
$102$	0	0
$103$	−92664.4	−0.860636	−0.430318	−	0.902677i	$-0.641599\pi$
$103$	−92664.4	−0.860636	−0.430318	+	0.902677i	$0.641599\pi$
$104$	0	0
$105$	−13346.0	−0.118135
$106$	0	0
$107$	47156.3	0.398180	0.199090	−	0.979981i	$-0.436201\pi$
$107$	47156.3	0.398180	0.199090	+	0.979981i	$0.436201\pi$
$108$	0	0
$109$	52450.1	0.422844	0.211422	−	0.977395i	$-0.432191\pi$
$109$	52450.1	0.422844	0.211422	+	0.977395i	$0.432191\pi$
$110$	0	0
$111$	−40538.5	−0.312292
$112$	0	0
$113$	150596.	1.10947	0.554737	−	0.832026i	$-0.312819\pi$
$113$	150596.	1.10947	0.554737	+	0.832026i	$0.312819\pi$
$114$	0	0
$115$	22815.9	0.160877
$116$	0	0
$117$	125555.	0.847948
$118$	0	0
$119$	4196.73	0.0271671
$120$	0	0
$121$	−38914.6	−0.241629
$122$	0	0
$123$	63716.4	0.379742
$124$	0	0
$125$	−174402.	−0.998337
$126$	0	0
$127$	−352240.	−1.93789	−0.968945	−	0.247276i	$-0.920465\pi$
$127$	−352240.	−1.93789	−0.968945	+	0.247276i	$0.920465\pi$
$128$	0	0
$129$	6513.72	0.0344632
$130$	0	0
$131$	−19070.2	−0.0970907	−0.0485453	−	0.998821i	$-0.515459\pi$
$131$	−19070.2	−0.0970907	−0.0485453	+	0.998821i	$0.515459\pi$
$132$	0	0
$133$	−6830.49	−0.0334829
$134$	0	0
$135$	46947.9	0.221708
$136$	0	0
$137$	−266677.	−1.21390	−0.606952	−	0.794738i	$-0.707608\pi$
$137$	−266677.	−1.21390	−0.606952	+	0.794738i	$0.707608\pi$
$138$	0	0
$139$	294888.	1.29456	0.647278	−	0.762254i	$-0.275907\pi$
$139$	294888.	1.29456	0.647278	+	0.762254i	$0.275907\pi$
$140$	0	0
$141$	−557892.	−2.36321
$142$	0	0
$143$	−248690.	−1.01699
$144$	0	0
$145$	−248637.	−0.982079
$146$	0	0
$147$	336880.	1.28582
$148$	0	0
$149$	−103990.	−0.383729	−0.191864	−	0.981421i	$-0.561453\pi$
$149$	−103990.	−0.383729	−0.191864	+	0.981421i	$0.561453\pi$
$150$	0	0
$151$	477343.	1.70368	0.851840	−	0.523802i	$-0.175487\pi$
$151$	477343.	1.70368	0.851840	+	0.523802i	$0.175487\pi$
$152$	0	0
$153$	39135.0	0.135156
$154$	0	0
$155$	−186247.	−0.622674
$156$	0	0
$157$	434955.	1.40830	0.704151	−	0.710051i	$-0.251328\pi$
$157$	434955.	1.40830	0.704151	+	0.710051i	$0.251328\pi$
$158$	0	0
$159$	22827.9	0.0716101
$160$	0	0
$161$	12534.6	0.0381105
$162$	0	0
$163$	255083.	0.751990	0.375995	−	0.926622i	$-0.377301\pi$
$163$	255083.	0.751990	0.375995	+	0.926622i	$0.377301\pi$
$164$	0	0
$165$	246508.	0.704889
$166$	0	0
$167$	−510501.	−1.41646	−0.708232	−	0.705980i	$-0.750507\pi$
$167$	−510501.	−1.41646	−0.708232	+	0.705980i	$0.750507\pi$
$168$	0	0
$169$	135080.	0.363809
$170$	0	0
$171$	−63695.1	−0.166577
$172$	0	0
$173$	773453.	1.96480	0.982401	−	0.186783i	$-0.0598062\pi$
$173$	773453.	1.96480	0.982401	+	0.186783i	$0.0598062\pi$
$174$	0	0
$175$	−36684.7	−0.0905503
$176$	0	0
$177$	−776303.	−1.86251
$178$	0	0
$179$	477664.	1.11427	0.557135	−	0.830422i	$-0.311901\pi$
$179$	477664.	1.11427	0.557135	+	0.830422i	$0.311901\pi$
$180$	0	0
$181$	729114.	1.65424	0.827121	−	0.562024i	$-0.189977\pi$
$181$	729114.	1.65424	0.827121	+	0.562024i	$0.189977\pi$
$182$	0	0
$183$	−767379.	−1.69388
$184$	0	0
$185$	68171.9	0.146446
$186$	0	0
$187$	−77515.6	−0.162101
$188$	0	0
$189$	25792.2	0.0525211
$190$	0	0
$191$	285584.	0.566435	0.283218	−	0.959056i	$-0.408598\pi$
$191$	285584.	0.566435	0.283218	+	0.959056i	$0.408598\pi$
$192$	0	0
$193$	−53690.2	−0.103753	−0.0518766	−	0.998654i	$-0.516520\pi$
$193$	−53690.2	−0.103753	−0.0518766	+	0.998654i	$0.516520\pi$
$194$	0	0
$195$	−501930.	−0.945272
$196$	0	0
$197$	14538.9	0.0266910	0.0133455	−	0.999911i	$-0.495752\pi$
$197$	14538.9	0.0266910	0.0133455	+	0.999911i	$0.495752\pi$
$198$	0	0
$199$	143700.	0.257232	0.128616	−	0.991695i	$-0.458947\pi$
$199$	143700.	0.257232	0.128616	+	0.991695i	$0.458947\pi$
$200$	0	0
$201$	−1.12550e6	−1.96496
$202$	0	0
$203$	−136596.	−0.232647
$204$	0	0
$205$	−107149.	−0.178075
$206$	0	0
$207$	116886.	0.189600
$208$	0	0
$209$	126162.	0.199786
$210$	0	0
$211$	−83653.6	−0.129354	−0.0646768	−	0.997906i	$-0.520602\pi$
$211$	−83653.6	−0.129354	−0.0646768	+	0.997906i	$0.520602\pi$
$212$	0	0
$213$	−146988.	−0.221989
$214$	0	0
$215$	−10953.8	−0.0161611
$216$	0	0
$217$	−102320.	−0.147507
$218$	0	0
$219$	−1.32602e6	−1.86826
$220$	0	0
$221$	157834.	0.217381
$222$	0	0
$223$	665414.	0.896045	0.448022	−	0.894022i	$-0.352128\pi$
$223$	665414.	0.896045	0.448022	+	0.894022i	$0.352128\pi$
$224$	0	0
$225$	−342089.	−0.450488
$226$	0	0
$227$	−1.15382e6	−1.48619	−0.743096	−	0.669185i	$-0.766643\pi$
$227$	−1.15382e6	−1.48619	−0.743096	+	0.669185i	$0.766643\pi$
$228$	0	0
$229$	−433691.	−0.546502	−0.273251	−	0.961943i	$-0.588099\pi$
$229$	−433691.	−0.546502	−0.273251	+	0.961943i	$0.588099\pi$
$230$	0	0
$231$	135426.	0.166983
$232$	0	0
$233$	387173.	0.467214	0.233607	−	0.972331i	$-0.424947\pi$
$233$	387173.	0.467214	0.233607	+	0.972331i	$0.424947\pi$
$234$	0	0
$235$	938183.	1.10820
$236$	0	0
$237$	739426.	0.855114
$238$	0	0
$239$	622463.	0.704886	0.352443	−	0.935833i	$-0.385351\pi$
$239$	622463.	0.704886	0.352443	+	0.935833i	$0.385351\pi$
$240$	0	0
$241$	−371454.	−0.411967	−0.205984	−	0.978555i	$-0.566039\pi$
$241$	−371454.	−0.411967	−0.205984	+	0.978555i	$0.566039\pi$
$242$	0	0
$243$	1.11861e6	1.21524
$244$	0	0
$245$	−566516.	−0.602972
$246$	0	0
$247$	−256887.	−0.267917
$248$	0	0
$249$	−1.06052e6	−1.08398
$250$	0	0
$251$	−376098.	−0.376805	−0.188403	−	0.982092i	$-0.560331\pi$
$251$	−376098.	−0.376805	−0.188403	+	0.982092i	$0.560331\pi$
$252$	0	0
$253$	−231519.	−0.227398
$254$	0	0
$255$	−156450.	−0.150669
$256$	0	0
$257$	1.64382e6	1.55247	0.776233	−	0.630446i	$-0.217128\pi$
$257$	1.64382e6	1.55247	0.776233	+	0.630446i	$0.217128\pi$
$258$	0	0
$259$	37452.2	0.0346919
$260$	0	0
$261$	−1.27378e6	−1.15742
$262$	0	0
$263$	−775299.	−0.691162	−0.345581	−	0.938389i	$-0.612318\pi$
$263$	−775299.	−0.691162	−0.345581	+	0.938389i	$0.612318\pi$
$264$	0	0
$265$	−38388.8	−0.0335807
$266$	0	0
$267$	−2.97483e6	−2.55378
$268$	0	0
$269$	334851.	0.282144	0.141072	−	0.989999i	$-0.454945\pi$
$269$	334851.	0.282144	0.141072	+	0.989999i	$0.454945\pi$
$270$	0	0
$271$	−893377.	−0.738944	−0.369472	−	0.929242i	$-0.620461\pi$
$271$	−893377.	−0.738944	−0.369472	+	0.929242i	$0.620461\pi$
$272$	0	0
$273$	−275750.	−0.223928
$274$	0	0
$275$	677584.	0.540296
$276$	0	0
$277$	−1.13462e6	−0.888483	−0.444242	−	0.895907i	$-0.646527\pi$
$277$	−1.13462e6	−0.888483	−0.444242	+	0.895907i	$0.646527\pi$
$278$	0	0
$279$	−954149.	−0.733847
$280$	0	0
$281$	−1.98111e6	−1.49672	−0.748362	−	0.663290i	$-0.769159\pi$
$281$	−1.98111e6	−1.49672	−0.748362	+	0.663290i	$0.769159\pi$
$282$	0	0
$283$	−1.32192e6	−0.981161	−0.490580	−	0.871396i	$-0.663215\pi$
$283$	−1.32192e6	−0.981161	−0.490580	+	0.871396i	$0.663215\pi$
$284$	0	0
$285$	254633.	0.185696
$286$	0	0
$287$	−58865.4	−0.0421847
$288$	0	0
$289$	−1.37066e6	−0.965351
$290$	0	0
$291$	−809232.	−0.560197
$292$	0	0
$293$	1.35808e6	0.924179	0.462089	−	0.886833i	$-0.347100\pi$
$293$	1.35808e6	0.924179	0.462089	+	0.886833i	$0.347100\pi$
$294$	0	0
$295$	1.30547e6	0.873401
$296$	0	0
$297$	−476394.	−0.313383
$298$	0	0
$299$	471411.	0.304945
$300$	0	0
$301$	−6017.81	−0.00382844
$302$	0	0
$303$	342489.	0.214309
$304$	0	0
$305$	1.29047e6	0.794324
$306$	0	0
$307$	1.23842e6	0.749930	0.374965	−	0.927039i	$-0.377655\pi$
$307$	1.23842e6	0.749930	0.374965	+	0.927039i	$0.377655\pi$
$308$	0	0
$309$	1.89779e6	1.13071
$310$	0	0
$311$	1.78565e6	1.04688	0.523439	−	0.852063i	$-0.324649\pi$
$311$	1.78565e6	1.04688	0.523439	+	0.852063i	$0.324649\pi$
$312$	0	0
$313$	−3.01608e6	−1.74013	−0.870066	−	0.492935i	$-0.835924\pi$
$313$	−3.01608e6	−1.74013	−0.870066	+	0.492935i	$0.835924\pi$
$314$	0	0
$315$	114978.	0.0652889
$316$	0	0
$317$	515839.	0.288314	0.144157	−	0.989555i	$-0.453953\pi$
$317$	515839.	0.288314	0.144157	+	0.989555i	$0.453953\pi$
$318$	0	0
$319$	2.52300e6	1.38816
$320$	0	0
$321$	−965772.	−0.523133
$322$	0	0
$323$	−80070.7	−0.0427039
$324$	0	0
$325$	−1.37967e6	−0.724548
$326$	0	0
$327$	−1.07419e6	−0.555536
$328$	0	0
$329$	515417.	0.262524
$330$	0	0
$331$	2.25449e6	1.13104	0.565521	−	0.824734i	$-0.308675\pi$
$331$	2.25449e6	1.13104	0.565521	+	0.824734i	$0.308675\pi$
$332$	0	0
$333$	349246.	0.172592
$334$	0	0
$335$	1.89270e6	0.921446
$336$	0	0
$337$	2.40261e6	1.15242	0.576208	−	0.817303i	$-0.304532\pi$
$337$	2.40261e6	1.15242	0.576208	+	0.817303i	$0.304532\pi$
$338$	0	0
$339$	−3.08424e6	−1.45764
$340$	0	0
$341$	1.88991e6	0.880145
$342$	0	0
$343$	−629237.	−0.288788
$344$	0	0
$345$	−467275.	−0.211361
$346$	0	0
$347$	684693.	0.305262	0.152631	−	0.988283i	$-0.451225\pi$
$347$	684693.	0.305262	0.152631	+	0.988283i	$0.451225\pi$
$348$	0	0
$349$	2.19857e6	0.966220	0.483110	−	0.875560i	$-0.339507\pi$
$349$	2.19857e6	0.966220	0.483110	+	0.875560i	$0.339507\pi$
$350$	0	0
$351$	970016.	0.420253
$352$	0	0
$353$	2.03446e6	0.868987	0.434493	−	0.900675i	$-0.356927\pi$
$353$	2.03446e6	0.868987	0.434493	+	0.900675i	$0.356927\pi$
$354$	0	0
$355$	247183.	0.104099
$356$	0	0
$357$	−85950.1	−0.0356924
$358$	0	0
$359$	−2.30592e6	−0.944298	−0.472149	−	0.881519i	$-0.656522\pi$
$359$	−2.30592e6	−0.944298	−0.472149	+	0.881519i	$0.656522\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	796980.	0.317454
$364$	0	0
$365$	2.22990e6	0.876101
$366$	0	0
$367$	−2.64475e6	−1.02499	−0.512495	−	0.858690i	$-0.671279\pi$
$367$	−2.64475e6	−1.02499	−0.512495	+	0.858690i	$0.671279\pi$
$368$	0	0
$369$	−548927.	−0.209869
$370$	0	0
$371$	−21090.0	−0.00795502
$372$	0	0
$373$	173405.	0.0645342	0.0322671	−	0.999479i	$-0.489727\pi$
$373$	173405.	0.0645342	0.0322671	+	0.999479i	$0.489727\pi$
$374$	0	0
$375$	3.57180e6	1.31162
$376$	0	0
$377$	−5.13723e6	−1.86155
$378$	0	0
$379$	−2.23408e6	−0.798915	−0.399458	−	0.916752i	$-0.630802\pi$
$379$	−2.23408e6	−0.798915	−0.399458	+	0.916752i	$0.630802\pi$
$380$	0	0
$381$	7.21396e6	2.54602
$382$	0	0
$383$	4.44976e6	1.55003	0.775014	−	0.631944i	$-0.217743\pi$
$383$	4.44976e6	1.55003	0.775014	+	0.631944i	$0.217743\pi$
$384$	0	0
$385$	−227740.	−0.0783047
$386$	0	0
$387$	−56116.8	−0.0190465
$388$	0	0
$389$	2.43083e6	0.814480	0.407240	−	0.913321i	$-0.366491\pi$
$389$	2.43083e6	0.814480	0.407240	+	0.913321i	$0.366491\pi$
$390$	0	0
$391$	146937.	0.0486059
$392$	0	0
$393$	390563.	0.127559
$394$	0	0
$395$	−1.24346e6	−0.400996
$396$	0	0
$397$	2.61205e6	0.831775	0.415887	−	0.909416i	$-0.363471\pi$
$397$	2.61205e6	0.831775	0.415887	+	0.909416i	$0.363471\pi$
$398$	0	0
$399$	139890.	0.0439901
$400$	0	0
$401$	1.42660e6	0.443038	0.221519	−	0.975156i	$-0.428899\pi$
$401$	1.42660e6	0.443038	0.221519	+	0.975156i	$0.428899\pi$
$402$	0	0
$403$	−3.84816e6	−1.18029
$404$	0	0
$405$	−2.43816e6	−0.738625
$406$	0	0
$407$	−691760.	−0.207000
$408$	0	0
$409$	4.73321e6	1.39910	0.699548	−	0.714586i	$-0.253385\pi$
$409$	4.73321e6	1.39910	0.699548	+	0.714586i	$0.253385\pi$
$410$	0	0
$411$	5.46162e6	1.59484
$412$	0	0
$413$	717200.	0.206902
$414$	0	0
$415$	1.78344e6	0.508320
$416$	0	0
$417$	−6.03939e6	−1.70080
$418$	0	0
$419$	357759.	0.0995531	0.0497766	−	0.998760i	$-0.484149\pi$
$419$	357759.	0.0995531	0.0497766	+	0.998760i	$0.484149\pi$
$420$	0	0
$421$	652504.	0.179423	0.0897115	−	0.995968i	$-0.471405\pi$
$421$	652504.	0.179423	0.0897115	+	0.995968i	$0.471405\pi$
$422$	0	0
$423$	4.80633e6	1.30606
$424$	0	0
$425$	−430038.	−0.115487
$426$	0	0
$427$	708955.	0.188169
$428$	0	0
$429$	5.09323e6	1.33613
$430$	0	0
$431$	6.25148e6	1.62103	0.810513	−	0.585721i	$-0.199188\pi$
$431$	6.25148e6	1.62103	0.810513	+	0.585721i	$0.199188\pi$
$432$	0	0
$433$	4.45832e6	1.14275	0.571375	−	0.820689i	$-0.306410\pi$
$433$	4.45832e6	1.14275	0.571375	+	0.820689i	$0.306410\pi$
$434$	0	0
$435$	5.09216e6	1.29026
$436$	0	0
$437$	−239151.	−0.0599057
$438$	0	0
$439$	−5.00652e6	−1.23986	−0.619932	−	0.784655i	$-0.712840\pi$
$439$	−5.00652e6	−1.23986	−0.619932	+	0.784655i	$0.712840\pi$
$440$	0	0
$441$	−2.90227e6	−0.710627
$442$	0	0
$443$	−715908.	−0.173320	−0.0866599	−	0.996238i	$-0.527619\pi$
$443$	−715908.	−0.173320	−0.0866599	+	0.996238i	$0.527619\pi$
$444$	0	0
$445$	5.00264e6	1.19757
$446$	0	0
$447$	2.12974e6	0.504147
$448$	0	0
$449$	−4.83183e6	−1.13109	−0.565544	−	0.824718i	$-0.691334\pi$
$449$	−4.83183e6	−1.13109	−0.565544	+	0.824718i	$0.691334\pi$
$450$	0	0
$451$	1.08727e6	0.251708
$452$	0	0
$453$	−9.77610e6	−2.23831
$454$	0	0
$455$	463716.	0.105008
$456$	0	0
$457$	−6.44410e6	−1.44335	−0.721675	−	0.692232i	$-0.756628\pi$
$457$	−6.44410e6	−1.44335	−0.721675	+	0.692232i	$0.756628\pi$
$458$	0	0
$459$	302350.	0.0669851
$460$	0	0
$461$	−4.10108e6	−0.898764	−0.449382	−	0.893340i	$-0.648356\pi$
$461$	−4.10108e6	−0.898764	−0.449382	+	0.893340i	$0.648356\pi$
$462$	0	0
$463$	−8.98683e6	−1.94829	−0.974146	−	0.225919i	$-0.927461\pi$
$463$	−8.98683e6	−1.94829	−0.974146	+	0.225919i	$0.927461\pi$
$464$	0	0
$465$	3.81439e6	0.818075
$466$	0	0
$467$	8.84409e6	1.87655	0.938276	−	0.345886i	$-0.112422\pi$
$467$	8.84409e6	1.87655	0.938276	+	0.345886i	$0.112422\pi$
$468$	0	0
$469$	1.03981e6	0.218284
$470$	0	0
$471$	−8.90800e6	−1.85024
$472$	0	0
$473$	111152.	0.0228436
$474$	0	0
$475$	699919.	0.142336
$476$	0	0
$477$	−196667.	−0.0395762
$478$	0	0
$479$	−2.89320e6	−0.576155	−0.288077	−	0.957607i	$-0.593016\pi$
$479$	−2.89320e6	−0.576155	−0.288077	+	0.957607i	$0.593016\pi$
$480$	0	0
$481$	1.40854e6	0.277591
$482$	0	0
$483$	−256711.	−0.0500699
$484$	0	0
$485$	1.36085e6	0.262697
$486$	0	0
$487$	3.18166e6	0.607900	0.303950	−	0.952688i	$-0.401694\pi$
$487$	3.18166e6	0.607900	0.303950	+	0.952688i	$0.401694\pi$
$488$	0	0
$489$	−5.22416e6	−0.987971
$490$	0	0
$491$	4.45509e6	0.833975	0.416987	−	0.908912i	$-0.363086\pi$
$491$	4.45509e6	0.833975	0.416987	+	0.908912i	$0.363086\pi$
$492$	0	0
$493$	−1.60125e6	−0.296717
$494$	0	0
$495$	−2.12371e6	−0.389566
$496$	0	0
$497$	135797.	0.0246603
$498$	0	0
$499$	−9.31472e6	−1.67463	−0.837315	−	0.546721i	$-0.815876\pi$
$499$	−9.31472e6	−1.67463	−0.837315	+	0.546721i	$0.815876\pi$
$500$	0	0
$501$	1.04552e7	1.86096
$502$	0	0
$503$	2.04835e6	0.360980	0.180490	−	0.983577i	$-0.442232\pi$
$503$	2.04835e6	0.360980	0.180490	+	0.983577i	$0.442232\pi$
$504$	0	0
$505$	−575949.	−0.100498
$506$	0	0
$507$	−2.76647e6	−0.477976
$508$	0	0
$509$	−166211.	−0.0284358	−0.0142179	−	0.999899i	$-0.504526\pi$
$509$	−166211.	−0.0284358	−0.0142179	+	0.999899i	$0.504526\pi$
$510$	0	0
$511$	1.22506e6	0.207542
$512$	0	0
$513$	−492097.	−0.0825577
$514$	0	0
$515$	−3.19143e6	−0.530234
$516$	0	0
$517$	−9.52001e6	−1.56643
$518$	0	0
$519$	−1.58405e7	−2.58137
$520$	0	0
$521$	2.64602e6	0.427069	0.213535	−	0.976935i	$-0.431502\pi$
$521$	2.64602e6	0.427069	0.213535	+	0.976935i	$0.431502\pi$
$522$	0	0
$523$	7.02287e6	1.12269	0.561346	−	0.827581i	$-0.310284\pi$
$523$	7.02287e6	1.12269	0.561346	+	0.827581i	$0.310284\pi$
$524$	0	0
$525$	751312.	0.118966
$526$	0	0
$527$	−1.19945e6	−0.188130
$528$	0	0
$529$	−5.99748e6	−0.931815
$530$	0	0
$531$	6.68798e6	1.02934
$532$	0	0
$533$	−2.21386e6	−0.337546
$534$	0	0
$535$	1.62410e6	0.245317
$536$	0	0
$537$	−9.78268e6	−1.46394
$538$	0	0
$539$	5.74860e6	0.852295
$540$	0	0
$541$	4.35066e6	0.639090	0.319545	−	0.947571i	$-0.396470\pi$
$541$	4.35066e6	0.639090	0.319545	+	0.947571i	$0.396470\pi$
$542$	0	0
$543$	−1.49324e7	−2.17336
$544$	0	0
$545$	1.80642e6	0.260512
$546$	0	0
$547$	−9.77794e6	−1.39727	−0.698633	−	0.715480i	$-0.746208\pi$
$547$	−9.77794e6	−1.39727	−0.698633	+	0.715480i	$0.746208\pi$
$548$	0	0
$549$	6.61110e6	0.936144
$550$	0	0
$551$	2.60616e6	0.365698
$552$	0	0
$553$	−683131.	−0.0949929
$554$	0	0
$555$	−1.39618e6	−0.192401
$556$	0	0
$557$	6.69323e6	0.914108	0.457054	−	0.889439i	$-0.348905\pi$
$557$	6.69323e6	0.914108	0.457054	+	0.889439i	$0.348905\pi$
$558$	0	0
$559$	−226323.	−0.0306337
$560$	0	0
$561$	1.58754e6	0.212969
$562$	0	0
$563$	−7.54034e6	−1.00258	−0.501291	−	0.865279i	$-0.667141\pi$
$563$	−7.54034e6	−1.00258	−0.501291	+	0.865279i	$0.667141\pi$
$564$	0	0
$565$	5.18664e6	0.683542
$566$	0	0
$567$	−1.33947e6	−0.174975
$568$	0	0
$569$	1.34726e6	0.174450	0.0872252	−	0.996189i	$-0.472200\pi$
$569$	1.34726e6	0.174450	0.0872252	+	0.996189i	$0.472200\pi$
$570$	0	0
$571$	−2.05762e6	−0.264104	−0.132052	−	0.991243i	$-0.542157\pi$
$571$	−2.05762e6	−0.264104	−0.132052	+	0.991243i	$0.542157\pi$
$572$	0	0
$573$	−5.84883e6	−0.744188
$574$	0	0
$575$	−1.28441e6	−0.162008
$576$	0	0
$577$	7.88561e6	0.986043	0.493021	−	0.870017i	$-0.335892\pi$
$577$	7.88561e6	0.986043	0.493021	+	0.870017i	$0.335892\pi$
$578$	0	0
$579$	1.09959e6	0.136312
$580$	0	0
$581$	979781.	0.120417
$582$	0	0
$583$	389542.	0.0474660
$584$	0	0
$585$	4.32421e6	0.522417
$586$	0	0
$587$	−1.90439e6	−0.228119	−0.114059	−	0.993474i	$-0.536385\pi$
$587$	−1.90439e6	−0.228119	−0.114059	+	0.993474i	$0.536385\pi$
$588$	0	0
$589$	1.95220e6	0.231866
$590$	0	0
$591$	−297760.	−0.0350669
$592$	0	0
$593$	−1.35835e7	−1.58626	−0.793132	−	0.609050i	$-0.791551\pi$
$593$	−1.35835e7	−1.58626	−0.793132	+	0.609050i	$0.791551\pi$
$594$	0	0
$595$	144539.	0.0167375
$596$	0	0
$597$	−2.94301e6	−0.337953
$598$	0	0
$599$	1.04361e7	1.18842	0.594212	−	0.804309i	$-0.297464\pi$
$599$	1.04361e7	1.18842	0.594212	+	0.804309i	$0.297464\pi$
$600$	0	0
$601$	−1.17196e7	−1.32351	−0.661753	−	0.749722i	$-0.730187\pi$
$601$	−1.17196e7	−1.32351	−0.661753	+	0.749722i	$0.730187\pi$
$602$	0	0
$603$	9.69635e6	1.08596
$604$	0	0
$605$	−1.34025e6	−0.148866
$606$	0	0
$607$	7.53524e6	0.830091	0.415045	−	0.909801i	$-0.363766\pi$
$607$	7.53524e6	0.830091	0.415045	+	0.909801i	$0.363766\pi$
$608$	0	0
$609$	2.79752e6	0.305654
$610$	0	0
$611$	1.93843e7	2.10062
$612$	0	0
$613$	−4.37292e6	−0.470025	−0.235012	−	0.971992i	$-0.575513\pi$
$613$	−4.37292e6	−0.470025	−0.235012	+	0.971992i	$0.575513\pi$
$614$	0	0
$615$	2.19444e6	0.233957
$616$	0	0
$617$	−7.50595e6	−0.793767	−0.396883	−	0.917869i	$-0.629908\pi$
$617$	−7.50595e6	−0.793767	−0.396883	+	0.917869i	$0.629908\pi$
$618$	0	0
$619$	1.30877e7	1.37289	0.686447	−	0.727180i	$-0.259169\pi$
$619$	1.30877e7	1.37289	0.686447	+	0.727180i	$0.259169\pi$
$620$	0	0
$621$	903043.	0.0939679
$622$	0	0
$623$	2.74834e6	0.283695
$624$	0	0
$625$	52309.5	0.00535649
$626$	0	0
$627$	−2.58384e6	−0.262480
$628$	0	0
$629$	439035.	0.0442459
$630$	0	0
$631$	−8.92096e6	−0.891945	−0.445972	−	0.895047i	$-0.647142\pi$
$631$	−8.92096e6	−0.891945	−0.445972	+	0.895047i	$0.647142\pi$
$632$	0	0
$633$	1.71325e6	0.169946
$634$	0	0
$635$	−1.21314e7	−1.19392
$636$	0	0
$637$	−1.17051e7	−1.14295
$638$	0	0
$639$	1.26632e6	0.122685
$640$	0	0
$641$	1.45438e7	1.39808	0.699041	−	0.715082i	$-0.253611\pi$
$641$	1.45438e7	1.39808	0.699041	+	0.715082i	$0.253611\pi$
$642$	0	0
$643$	−1.45757e7	−1.39028	−0.695140	−	0.718874i	$-0.744658\pi$
$643$	−1.45757e7	−1.39028	−0.695140	+	0.718874i	$0.744658\pi$
$644$	0	0
$645$	224338.	0.0212326
$646$	0	0
$647$	2.24522e6	0.210862	0.105431	−	0.994427i	$-0.466378\pi$
$647$	2.24522e6	0.210862	0.105431	+	0.994427i	$0.466378\pi$
$648$	0	0
$649$	−1.32470e7	−1.23454
$650$	0	0
$651$	2.09555e6	0.193796
$652$	0	0
$653$	1.54461e7	1.41754	0.708770	−	0.705440i	$-0.249251\pi$
$653$	1.54461e7	1.41754	0.708770	+	0.705440i	$0.249251\pi$
$654$	0	0
$655$	−656793.	−0.0598171
$656$	0	0
$657$	1.14238e7	1.03252
$658$	0	0
$659$	−8.68940e6	−0.779429	−0.389714	−	0.920936i	$-0.627426\pi$
$659$	−8.68940e6	−0.779429	−0.389714	+	0.920936i	$0.627426\pi$
$660$	0	0
$661$	−2.03442e7	−1.81108	−0.905538	−	0.424266i	$-0.860532\pi$
$661$	−2.03442e7	−1.81108	−0.905538	+	0.424266i	$0.860532\pi$
$662$	0	0
$663$	−3.23249e6	−0.285597
$664$	0	0
$665$	−235247.	−0.0206286
$666$	0	0
$667$	−4.78254e6	−0.416240
$668$	0	0
$669$	−1.36278e7	−1.17723
$670$	0	0
$671$	−1.30947e7	−1.12277
$672$	0	0
$673$	1.71139e7	1.45650	0.728251	−	0.685310i	$-0.240333\pi$
$673$	1.71139e7	1.45650	0.728251	+	0.685310i	$0.240333\pi$
$674$	0	0
$675$	−2.64292e6	−0.223267
$676$	0	0
$677$	2.25299e6	0.188924	0.0944621	−	0.995528i	$-0.469887\pi$
$677$	2.25299e6	0.188924	0.0944621	+	0.995528i	$0.469887\pi$
$678$	0	0
$679$	747622.	0.0622311
$680$	0	0
$681$	2.36306e7	1.95257
$682$	0	0
$683$	−5.33481e6	−0.437590	−0.218795	−	0.975771i	$-0.570213\pi$
$683$	−5.33481e6	−0.437590	−0.218795	+	0.975771i	$0.570213\pi$
$684$	0	0
$685$	−9.18457e6	−0.747881
$686$	0	0
$687$	8.88211e6	0.718000
$688$	0	0
$689$	−793171.	−0.0636530
$690$	0	0
$691$	−8.08495e6	−0.644143	−0.322071	−	0.946715i	$-0.604379\pi$
$691$	−8.08495e6	−0.644143	−0.322071	+	0.946715i	$0.604379\pi$
$692$	0	0
$693$	−1.16672e6	−0.0922854
$694$	0	0
$695$	1.01562e7	0.797569
$696$	0	0
$697$	−690053.	−0.0538022
$698$	0	0
$699$	−7.92941e6	−0.613829
$700$	0	0
$701$	7.88116e6	0.605752	0.302876	−	0.953030i	$-0.402053\pi$
$701$	7.88116e6	0.605752	0.302876	+	0.953030i	$0.402053\pi$
$702$	0	0
$703$	−714562.	−0.0545320
$704$	0	0
$705$	−1.92142e7	−1.45596
$706$	0	0
$707$	−316414.	−0.0238071
$708$	0	0
$709$	−2.07701e7	−1.55175	−0.775876	−	0.630885i	$-0.782692\pi$
$709$	−2.07701e7	−1.55175	−0.775876	+	0.630885i	$0.782692\pi$
$710$	0	0
$711$	−6.37028e6	−0.472590
$712$	0	0
$713$	−3.58247e6	−0.263911
$714$	0	0
$715$	−8.56506e6	−0.626564
$716$	0	0
$717$	−1.27482e7	−0.926086
$718$	0	0
$719$	−1.03826e7	−0.749004	−0.374502	−	0.927226i	$-0.622186\pi$
$719$	−1.03826e7	−0.749004	−0.374502	+	0.927226i	$0.622186\pi$
$720$	0	0
$721$	−1.75330e6	−0.125608
$722$	0	0
$723$	7.60748e6	0.541246
$724$	0	0
$725$	1.39970e7	0.988985
$726$	0	0
$727$	6.52486e6	0.457863	0.228931	−	0.973443i	$-0.426477\pi$
$727$	6.52486e6	0.457863	0.228931	+	0.973443i	$0.426477\pi$
$728$	0	0
$729$	−5.70674e6	−0.397712
$730$	0	0
$731$	−70544.1	−0.00488278
$732$	0	0
$733$	−2.46101e6	−0.169182	−0.0845908	−	0.996416i	$-0.526958\pi$
$733$	−2.46101e6	−0.169182	−0.0845908	+	0.996416i	$0.526958\pi$
$734$	0	0
$735$	1.16024e7	0.792189
$736$	0	0
$737$	−1.92058e7	−1.30246
$738$	0	0
$739$	7.33586e6	0.494128	0.247064	−	0.968999i	$-0.420534\pi$
$739$	7.33586e6	0.494128	0.247064	+	0.968999i	$0.420534\pi$
$740$	0	0
$741$	5.26111e6	0.351991
$742$	0	0
$743$	−2.44394e7	−1.62412	−0.812060	−	0.583574i	$-0.801654\pi$
$743$	−2.44394e7	−1.62412	−0.812060	+	0.583574i	$0.801654\pi$
$744$	0	0
$745$	−3.58148e6	−0.236414
$746$	0	0
$747$	9.13658e6	0.599076
$748$	0	0
$749$	892244.	0.0581138
$750$	0	0
$751$	2.11169e6	0.136625	0.0683126	−	0.997664i	$-0.478238\pi$
$751$	2.11169e6	0.136625	0.0683126	+	0.997664i	$0.478238\pi$
$752$	0	0
$753$	7.70259e6	0.495050
$754$	0	0
$755$	1.64401e7	1.04963
$756$	0	0
$757$	−1.47627e7	−0.936325	−0.468162	−	0.883642i	$-0.655084\pi$
$757$	−1.47627e7	−0.936325	−0.468162	+	0.883642i	$0.655084\pi$
$758$	0	0
$759$	4.74157e6	0.298757
$760$	0	0
$761$	2.67074e7	1.67174	0.835871	−	0.548926i	$-0.184963\pi$
$761$	2.67074e7	1.67174	0.835871	+	0.548926i	$0.184963\pi$
$762$	0	0
$763$	992408.	0.0617133
$764$	0	0
$765$	1.34784e6	0.0832692
$766$	0	0
$767$	2.69731e7	1.65555
$768$	0	0
$769$	3.05104e6	0.186051	0.0930256	−	0.995664i	$-0.470346\pi$
$769$	3.05104e6	0.186051	0.0930256	+	0.995664i	$0.470346\pi$
$770$	0	0
$771$	−3.36659e7	−2.03964
$772$	0	0
$773$	−2.18723e7	−1.31657	−0.658287	−	0.752767i	$-0.728719\pi$
$773$	−2.18723e7	−1.31657	−0.658287	+	0.752767i	$0.728719\pi$
$774$	0	0
$775$	1.04847e7	0.627052
$776$	0	0
$777$	−767030.	−0.0455785
$778$	0	0
$779$	1.12311e6	0.0663100
$780$	0	0
$781$	−2.50823e6	−0.147143
$782$	0	0
$783$	−9.84096e6	−0.573632
$784$	0	0
$785$	1.49802e7	0.867647
$786$	0	0
$787$	1.26837e7	0.729975	0.364987	−	0.931013i	$-0.381073\pi$
$787$	1.26837e7	0.729975	0.364987	+	0.931013i	$0.381073\pi$
$788$	0	0
$789$	1.58783e7	0.908055
$790$	0	0
$791$	2.84943e6	0.161926
$792$	0	0
$793$	2.66630e7	1.50566
$794$	0	0
$795$	786212.	0.0441186
$796$	0	0
$797$	7.28989e6	0.406514	0.203257	−	0.979125i	$-0.434847\pi$
$797$	7.28989e6	0.406514	0.203257	+	0.979125i	$0.434847\pi$
$798$	0	0
$799$	6.04201e6	0.334822
$800$	0	0
$801$	2.56286e7	1.41138
$802$	0	0
$803$	−2.26275e7	−1.23836
$804$	0	0
$805$	431700.	0.0234797
$806$	0	0
$807$	−6.85783e6	−0.370683
$808$	0	0
$809$	−2.72506e7	−1.46388	−0.731939	−	0.681370i	$-0.761385\pi$
$809$	−2.72506e7	−1.46388	−0.731939	+	0.681370i	$0.761385\pi$
$810$	0	0
$811$	−1.45954e7	−0.779229	−0.389615	−	0.920978i	$-0.627392\pi$
$811$	−1.45954e7	−0.779229	−0.389615	+	0.920978i	$0.627392\pi$
$812$	0	0
$813$	1.82966e7	0.970831
$814$	0	0
$815$	8.78524e6	0.463297
$816$	0	0
$817$	114816.	0.00601791
$818$	0	0
$819$	2.37563e6	0.123757
$820$	0	0
$821$	4.16570e6	0.215690	0.107845	−	0.994168i	$-0.465605\pi$
$821$	4.16570e6	0.215690	0.107845	+	0.994168i	$0.465605\pi$
$822$	0	0
$823$	9.74487e6	0.501506	0.250753	−	0.968051i	$-0.419322\pi$
$823$	9.74487e6	0.501506	0.250753	+	0.968051i	$0.419322\pi$
$824$	0	0
$825$	−1.38771e7	−0.709845
$826$	0	0
$827$	−1.04970e6	−0.0533707	−0.0266854	−	0.999644i	$-0.508495\pi$
$827$	−1.04970e6	−0.0533707	−0.0266854	+	0.999644i	$0.508495\pi$
$828$	0	0
$829$	1.87001e7	0.945056	0.472528	−	0.881316i	$-0.343342\pi$
$829$	1.87001e7	0.945056	0.472528	+	0.881316i	$0.343342\pi$
$830$	0	0
$831$	2.32372e7	1.16730
$832$	0	0
$833$	−3.64843e6	−0.182177
$834$	0	0
$835$	−1.75820e7	−0.872676
$836$	0	0
$837$	−7.37159e6	−0.363703
$838$	0	0
$839$	−1.96094e7	−0.961742	−0.480871	−	0.876791i	$-0.659680\pi$
$839$	−1.96094e7	−0.961742	−0.480871	+	0.876791i	$0.659680\pi$
$840$	0	0
$841$	3.16068e7	1.54096
$842$	0	0
$843$	4.05735e7	1.96641
$844$	0	0
$845$	4.65225e6	0.224141
$846$	0	0
$847$	−736303.	−0.0352653
$848$	0	0
$849$	2.70733e7	1.28906
$850$	0	0
$851$	1.31129e6	0.0620688
$852$	0	0
$853$	−5.00288e6	−0.235422	−0.117711	−	0.993048i	$-0.537556\pi$
$853$	−5.00288e6	−0.235422	−0.117711	+	0.993048i	$0.537556\pi$
$854$	0	0
$855$	−2.19371e6	−0.102627
$856$	0	0
$857$	1.36218e7	0.633553	0.316776	−	0.948500i	$-0.397400\pi$
$857$	1.36218e7	0.633553	0.316776	+	0.948500i	$0.397400\pi$
$858$	0	0
$859$	−1.88600e7	−0.872084	−0.436042	−	0.899926i	$-0.643620\pi$
$859$	−1.88600e7	−0.872084	−0.436042	+	0.899926i	$0.643620\pi$
$860$	0	0
$861$	1.20558e6	0.0554227
$862$	0	0
$863$	3.62744e7	1.65796	0.828979	−	0.559280i	$-0.188922\pi$
$863$	3.62744e7	1.65796	0.828979	+	0.559280i	$0.188922\pi$
$864$	0	0
$865$	2.66383e7	1.21050
$866$	0	0
$867$	2.80715e7	1.26829
$868$	0	0
$869$	1.26178e7	0.566804
$870$	0	0
$871$	3.91061e7	1.74662
$872$	0	0
$873$	6.97166e6	0.309600
$874$	0	0
$875$	−3.29987e6	−0.145706
$876$	0	0
$877$	−1.47531e7	−0.647715	−0.323857	−	0.946106i	$-0.604980\pi$
$877$	−1.47531e7	−0.647715	−0.323857	+	0.946106i	$0.604980\pi$
$878$	0	0
$879$	−2.78138e7	−1.21419
$880$	0	0
$881$	1.41746e7	0.615277	0.307638	−	0.951503i	$-0.400461\pi$
$881$	1.41746e7	0.615277	0.307638	+	0.951503i	$0.400461\pi$
$882$	0	0
$883$	−2.05570e7	−0.887276	−0.443638	−	0.896206i	$-0.646312\pi$
$883$	−2.05570e7	−0.887276	−0.443638	+	0.896206i	$0.646312\pi$
$884$	0	0
$885$	−2.67365e7	−1.14748
$886$	0	0
$887$	9.48962e6	0.404986	0.202493	−	0.979284i	$-0.435096\pi$
$887$	9.48962e6	0.404986	0.202493	+	0.979284i	$0.435096\pi$
$888$	0	0
$889$	−6.66473e6	−0.282832
$890$	0	0
$891$	2.47407e7	1.04404
$892$	0	0
$893$	−9.83381e6	−0.412661
$894$	0	0
$895$	1.64511e7	0.686496
$896$	0	0
$897$	−9.65462e6	−0.400640
$898$	0	0
$899$	3.90401e7	1.61106
$900$	0	0
$901$	−247228.	−0.0101458
$902$	0	0
$903$	123246.	0.00502984
$904$	0	0
$905$	2.51112e7	1.01917
$906$	0	0
$907$	−1.30672e7	−0.527428	−0.263714	−	0.964601i	$-0.584947\pi$
$907$	−1.30672e7	−0.527428	−0.263714	+	0.964601i	$0.584947\pi$
$908$	0	0
$909$	−2.95060e6	−0.118441
$910$	0	0
$911$	4.60549e7	1.83857	0.919284	−	0.393594i	$-0.128768\pi$
$911$	4.60549e7	1.83857	0.919284	+	0.393594i	$0.128768\pi$
$912$	0	0
$913$	−1.80970e7	−0.718506
$914$	0	0
$915$	−2.64291e7	−1.04359
$916$	0	0
$917$	−360828.	−0.0141702
$918$	0	0
$919$	6.67836e6	0.260844	0.130422	−	0.991459i	$-0.458367\pi$
$919$	6.67836e6	0.260844	0.130422	+	0.991459i	$0.458367\pi$
$920$	0	0
$921$	−2.53631e7	−0.985264
$922$	0	0
$923$	5.10717e6	0.197322
$924$	0	0
$925$	−3.83772e6	−0.147475
$926$	0	0
$927$	−1.63498e7	−0.624903
$928$	0	0
$929$	−84737.5	−0.00322134	−0.00161067	−	0.999999i	$-0.500513\pi$
$929$	−84737.5	−0.00322134	−0.00161067	+	0.999999i	$0.500513\pi$
$930$	0	0
$931$	5.93809e6	0.224529
$932$	0	0
$933$	−3.65706e7	−1.37540
$934$	0	0
$935$	−2.66970e6	−0.0998695
$936$	0	0
$937$	−3.12820e7	−1.16398	−0.581989	−	0.813196i	$-0.697725\pi$
$937$	−3.12820e7	−1.16398	−0.581989	+	0.813196i	$0.697725\pi$
$938$	0	0
$939$	6.17701e7	2.28620
$940$	0	0
$941$	2.88841e7	1.06337	0.531685	−	0.846942i	$-0.321559\pi$
$941$	2.88841e7	1.06337	0.531685	+	0.846942i	$0.321559\pi$
$942$	0	0
$943$	−2.06101e6	−0.0754746
$944$	0	0
$945$	888302.	0.0323580
$946$	0	0
$947$	−1.99925e7	−0.724422	−0.362211	−	0.932096i	$-0.617978\pi$
$947$	−1.99925e7	−0.724422	−0.362211	+	0.932096i	$0.617978\pi$
$948$	0	0
$949$	4.60732e7	1.66067
$950$	0	0
$951$	−1.05645e7	−0.378790
$952$	0	0
$953$	5.22187e7	1.86249	0.931244	−	0.364396i	$-0.118724\pi$
$953$	5.22187e7	1.86249	0.931244	+	0.364396i	$0.118724\pi$
$954$	0	0
$955$	9.83573e6	0.348978
$956$	0	0
$957$	−5.16716e7	−1.82378
$958$	0	0
$959$	−5.04580e6	−0.177167
$960$	0	0
$961$	614716.	0.0214717
$962$	0	0
$963$	8.32029e6	0.289116
$964$	0	0
$965$	−1.84913e6	−0.0639218
$966$	0	0
$967$	−2.66303e7	−0.915818	−0.457909	−	0.888999i	$-0.651402\pi$
$967$	−2.66303e7	−0.915818	−0.457909	+	0.888999i	$0.651402\pi$
$968$	0	0
$969$	1.63987e6	0.0561047
$970$	0	0
$971$	−1.04206e7	−0.354686	−0.177343	−	0.984149i	$-0.556750\pi$
$971$	−1.04206e7	−0.354686	−0.177343	+	0.984149i	$0.556750\pi$
$972$	0	0
$973$	5.57959e6	0.188938
$974$	0	0
$975$	2.82560e7	0.951918
$976$	0	0
$977$	1.82795e7	0.612673	0.306336	−	0.951923i	$-0.400897\pi$
$977$	1.82795e7	0.612673	0.306336	+	0.951923i	$0.400897\pi$
$978$	0	0
$979$	−5.07633e7	−1.69275
$980$	0	0
$981$	9.25433e6	0.307024
$982$	0	0
$983$	−2.25159e7	−0.743199	−0.371600	−	0.928393i	$-0.621191\pi$
$983$	−2.25159e7	−0.743199	−0.371600	+	0.928393i	$0.621191\pi$
$984$	0	0
$985$	500730.	0.0164442
$986$	0	0
$987$	−1.05559e7	−0.344907
$988$	0	0
$989$	−210697.	−0.00684964
$990$	0	0
$991$	−3.42068e7	−1.10644	−0.553221	−	0.833034i	$-0.686602\pi$
$991$	−3.42068e7	−1.10644	−0.553221	+	0.833034i	$0.686602\pi$
$992$	0	0
$993$	−4.61726e7	−1.48597
$994$	0	0
$995$	4.94914e6	0.158479
$996$	0	0
$997$	−1.12682e7	−0.359018	−0.179509	−	0.983756i	$-0.557451\pi$
$997$	−1.12682e7	−0.359018	−0.179509	+	0.983756i	$0.557451\pi$
$998$	0	0
$999$	2.69821e6	0.0855387

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.a.c.1.2	✓	2	4.3	odd	2
304.6.a.f.1.1		2	1.1	even	1	trivial
342.6.a.i.1.1		2	12.11	even	2
722.6.a.c.1.1		2	76.75	even	2
950.6.a.d.1.1		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)

\(f(q)\)	\(=\)	\(q-20.4803 q^{3} +34.4408 q^{5} +18.9210 q^{7} +176.441 q^{9} +O(q^{10})\)	\(q-20.4803 q^{3} +34.4408 q^{5} +18.9210 q^{7} +176.441 q^{9} -349.480 q^{11} +711.599 q^{13} -705.355 q^{15} +221.803 q^{17} -361.000 q^{19} -387.507 q^{21} +662.468 q^{23} -1938.83 q^{25} +1363.15 q^{27} -7219.28 q^{29} -5407.76 q^{31} +7157.44 q^{33} +651.654 q^{35} +1979.40 q^{37} -14573.7 q^{39} -3111.11 q^{41} -318.049 q^{43} +6076.75 q^{45} +27240.5 q^{47} -16449.0 q^{49} -4542.57 q^{51} -1114.63 q^{53} -12036.4 q^{55} +7393.37 q^{57} +37904.9 q^{59} +37469.2 q^{61} +3338.44 q^{63} +24508.0 q^{65} +54955.3 q^{67} -13567.5 q^{69} +7177.04 q^{71} +64746.1 q^{73} +39707.8 q^{75} -6612.52 q^{77} -36104.4 q^{79} -70792.8 q^{81} +51782.7 q^{83} +7639.05 q^{85} +147853. q^{87} +145254. q^{89} +13464.2 q^{91} +110752. q^{93} -12433.1 q^{95} +39512.8 q^{97} -61662.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 45 q^{5} - 114 q^{7} + 239 q^{9}+O(q^{10}) \) 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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Database

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