LMFDB - Embedded newform 304.7.e.f.113.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,7,Mod(113,304)]

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, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	304.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$69.9364414204$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.15
Character	$\chi$	$=$	304.113
Dual form			304.7.e.f.113.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.05423i q^{3} +140.601 q^{5} -450.706 q^{7} +679.238 q^{9} +O(q^{10})$	$q-7.05423i q^{3} +140.601 q^{5} -450.706 q^{7} +679.238 q^{9} -746.944 q^{11} -485.335i q^{13} -991.829i q^{15} -1739.52 q^{17} +(411.663 + 6846.64i) q^{19} +3179.38i q^{21} +2379.12 q^{23} +4143.57 q^{25} -9934.03i q^{27} -2537.34i q^{29} +29635.4i q^{31} +5269.11i q^{33} -63369.6 q^{35} +31931.5i q^{37} -3423.67 q^{39} +25891.1i q^{41} -23064.3 q^{43} +95501.3 q^{45} -55760.5 q^{47} +85487.2 q^{49} +12271.0i q^{51} +22270.9i q^{53} -105021. q^{55} +(48297.7 - 2903.96i) q^{57} -27406.2i q^{59} +85056.7 q^{61} -306137. q^{63} -68238.5i q^{65} -289353. i q^{67} -16782.8i q^{69} +224277. i q^{71} +13777.9 q^{73} -29229.7i q^{75} +336652. q^{77} +497489. i q^{79} +425088. q^{81} -163314. q^{83} -244578. q^{85} -17899.0 q^{87} +1.05311e6i q^{89} +218744. i q^{91} +209055. q^{93} +(57880.1 + 962642. i) q^{95} +1.12183e6i q^{97} -507352. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) $ 30 * q + 720 * q^7 - 8670 * q^9	$ 30 q + 720 q^{7} - 8670 q^{9} + 2524 q^{11} + 9700 q^{17} - 4014 q^{19} + 39376 q^{23} + 110742 q^{25} + 19976 q^{35} + 266500 q^{39} + 106788 q^{43} - 91360 q^{45} - 222756 q^{47} + 593586 q^{49} - 540936 q^{55} - 545972 q^{57} - 242640 q^{61} + 377716 q^{63} + 545964 q^{73} - 272356 q^{77} + 2189926 q^{81} - 1542652 q^{83} - 826908 q^{85} + 2729572 q^{87} - 2139912 q^{93} - 2142716 q^{95} + 293012 q^{99}+O(q^{100}) $ 30 * q + 720 * q^7 - 8670 * q^9 + 2524 * q^11 + 9700 * q^17 - 4014 * q^19 + 39376 * q^23 + 110742 * q^25 + 19976 * q^35 + 266500 * q^39 + 106788 * q^43 - 91360 * q^45 - 222756 * q^47 + 593586 * q^49 - 540936 * q^55 - 545972 * q^57 - 242640 * q^61 + 377716 * q^63 + 545964 * q^73 - 272356 * q^77 + 2189926 * q^81 - 1542652 * q^83 - 826908 * q^85 + 2729572 * q^87 - 2139912 * q^93 - 2142716 * q^95 + 293012 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	7.05423i		−	0.261268i	−0.991431	−	0.130634i	$-0.958299\pi$
$3$		−	7.05423i		−	0.261268i	0.991431	−	0.130634i	$-0.0417012\pi$
$4$		0			0
$5$	140.601			1.12481			0.562403	−	0.826863i	$-0.309877\pi$
$5$	140.601			1.12481			0.562403	+	0.826863i	$0.309877\pi$
$6$		0			0
$7$	−450.706			−1.31401			−0.657006	−	0.753885i	$-0.728177\pi$
$7$	−450.706			−1.31401			−0.657006	+	0.753885i	$0.728177\pi$
$8$		0			0
$9$	679.238			0.931739
$10$		0			0
$11$	−746.944			−0.561190			−0.280595	−	0.959826i	$-0.590532\pi$
$11$	−746.944			−0.561190			−0.280595	+	0.959826i	$0.590532\pi$
$12$		0			0
$13$		−	485.335i		−	0.220908i	−0.993881	−	0.110454i	$-0.964769\pi$
$13$		−	485.335i		−	0.220908i	0.993881	−	0.110454i	$-0.0352305\pi$
$14$		0			0
$15$		−	991.829i		−	0.293875i
$16$		0			0
$17$	−1739.52			−0.354065			−0.177033	−	0.984205i	$-0.556650\pi$
$17$	−1739.52			−0.354065			−0.177033	+	0.984205i	$0.556650\pi$
$18$		0			0
$19$	411.663	+	6846.64i	0.0600179	+	0.998197i
$19$	411.663	+	6846.64i	0.0600179	+	0.998197i
$20$		0			0
$21$			3179.38i			0.343309i
$22$		0			0
$23$	2379.12			0.195538			0.0977692	−	0.995209i	$-0.468829\pi$
$23$	2379.12			0.195538			0.0977692	+	0.995209i	$0.468829\pi$
$24$		0			0
$25$	4143.57			0.265188
$26$		0			0
$27$		−	9934.03i		−	0.504701i
$28$		0			0
$29$		−	2537.34i		−	0.104036i	−0.998646	−	0.0520181i	$-0.983435\pi$
$29$		−	2537.34i		−	0.104036i	0.998646	−	0.0520181i	$-0.0165654\pi$
$30$		0			0
$31$			29635.4i			0.994778i	0.867528	+	0.497389i	$0.165708\pi$
$31$			29635.4i			0.994778i	−0.867528	+	0.497389i	$0.834292\pi$
$32$		0			0
$33$			5269.11i			0.146621i
$34$		0			0
$35$	−63369.6			−1.47801
$36$		0			0
$37$			31931.5i			0.630398i	0.949026	+	0.315199i	$0.102071\pi$
$37$			31931.5i			0.630398i	−0.949026	+	0.315199i	$0.897929\pi$
$38$		0			0
$39$	−3423.67			−0.0577162
$40$		0			0
$41$			25891.1i			0.375664i	0.982201	+	0.187832i	$0.0601460\pi$
$41$			25891.1i			0.375664i	−0.982201	+	0.187832i	$0.939854\pi$
$42$		0			0
$43$	−23064.3			−0.290092			−0.145046	−	0.989425i	$-0.546333\pi$
$43$	−23064.3			−0.290092			−0.145046	+	0.989425i	$0.546333\pi$
$44$		0			0
$45$	95501.3			1.04803
$46$		0			0
$47$	−55760.5			−0.537073			−0.268536	−	0.963270i	$-0.586540\pi$
$47$	−55760.5			−0.537073			−0.268536	+	0.963270i	$0.586540\pi$
$48$		0			0
$49$	85487.2			0.726629
$50$		0			0
$51$			12271.0i			0.0925058i
$52$		0			0
$53$			22270.9i			0.149593i	0.997199	+	0.0747964i	$0.0238307\pi$
$53$			22270.9i			0.149593i	−0.997199	+	0.0747964i	$0.976169\pi$
$54$		0			0
$55$	−105021.			−0.631230
$56$		0			0
$57$	48297.7	−	2903.96i	0.260797	−	0.0156807i
$58$		0			0
$59$		−	27406.2i		−	0.133442i	−0.997772	−	0.0667211i	$-0.978746\pi$
$59$		−	27406.2i		−	0.133442i	0.997772	−	0.0667211i	$-0.0212538\pi$
$60$		0			0
$61$	85056.7			0.374730			0.187365	−	0.982290i	$-0.440005\pi$
$61$	85056.7			0.374730			0.187365	+	0.982290i	$0.440005\pi$
$62$		0			0
$63$	−306137.			−1.22432
$64$		0			0
$65$		−	68238.5i		−	0.248479i
$66$		0			0
$67$		−	289353.i		−	0.962062i	−0.876704	−	0.481031i	$-0.840262\pi$
$67$		−	289353.i		−	0.962062i	0.876704	−	0.481031i	$-0.159738\pi$
$68$		0			0
$69$		−	16782.8i		−	0.0510879i
$70$		0			0
$71$			224277.i			0.626628i	0.949650	+	0.313314i	$0.101439\pi$
$71$			224277.i			0.626628i	−0.949650	+	0.313314i	$0.898561\pi$
$72$		0			0
$73$	13777.9			0.0354173			0.0177086	−	0.999843i	$-0.494363\pi$
$73$	13777.9			0.0354173			0.0177086	+	0.999843i	$0.494363\pi$
$74$		0			0
$75$		−	29229.7i		−	0.0692851i
$76$		0			0
$77$	336652.			0.737410
$78$		0			0
$79$			497489.i			1.00902i	0.863404	+	0.504512i	$0.168328\pi$
$79$			497489.i			1.00902i	−0.863404	+	0.504512i	$0.831672\pi$
$80$		0			0
$81$	425088.			0.799877
$82$		0			0
$83$	−163314.			−0.285621			−0.142810	−	0.989750i	$-0.545614\pi$
$83$	−163314.			−0.285621			−0.142810	+	0.989750i	$0.545614\pi$
$84$		0			0
$85$	−244578.			−0.398255
$86$		0			0
$87$	−17899.0			−0.0271813
$88$		0			0
$89$			1.05311e6i			1.49383i	0.664918	+	0.746917i	$0.268467\pi$
$89$			1.05311e6i			1.49383i	−0.664918	+	0.746917i	$0.731533\pi$
$90$		0			0
$91$			218744.i			0.290276i
$92$		0			0
$93$	209055.			0.259903
$94$		0			0
$95$	57880.1	+	962642.i	0.0675085	+	1.12278i
$96$		0			0
$97$			1.12183e6i			1.22917i	0.788851	+	0.614584i	$0.210676\pi$
$97$			1.12183e6i			1.22917i	−0.788851	+	0.614584i	$0.789324\pi$
$98$		0			0
$99$	−507352.			−0.522883
$100$		0			0
$101$	−130756.			−0.126910			−0.0634552	−	0.997985i	$-0.520212\pi$
$101$	−130756.			−0.126910			−0.0634552	+	0.997985i	$0.520212\pi$
$102$		0			0
$103$			1.11387e6i			1.01935i	0.860368	+	0.509673i	$0.170234\pi$
$103$			1.11387e6i			1.01935i	−0.860368	+	0.509673i	$0.829766\pi$
$104$		0			0
$105$			447024.i			0.386156i
$106$		0			0
$107$			1.56309e6i			1.27595i	0.770059	+	0.637973i	$0.220227\pi$
$107$			1.56309e6i			1.27595i	−0.770059	+	0.637973i	$0.779773\pi$
$108$		0			0
$109$			798690.i			0.616735i	0.951267	+	0.308367i	$0.0997826\pi$
$109$			798690.i			0.616735i	−0.951267	+	0.308367i	$0.900217\pi$
$110$		0			0
$111$	225252.			0.164702
$112$		0			0
$113$			2.07379e6i			1.43724i	0.695403	+	0.718620i	$0.255226\pi$
$113$			2.07379e6i			1.43724i	−0.695403	+	0.718620i	$0.744774\pi$
$114$		0			0
$115$	334505.			0.219943
$116$		0			0
$117$		−	329658.i		−	0.205829i
$118$		0			0
$119$	784013.			0.465246
$120$		0			0
$121$	−1.21364e6			−0.685066
$122$		0			0
$123$	182642.			0.0981488
$124$		0			0
$125$	−1.61430e6			−0.826520
$126$		0			0
$127$		−	143739.i		−	0.0701717i	−0.999384	−	0.0350859i	$-0.988830\pi$
$127$		−	143739.i		−	0.0701717i	0.999384	−	0.0350859i	$-0.0111705\pi$
$128$		0			0
$129$			162701.i			0.0757916i
$130$		0			0
$131$	3.08969e6			1.37436			0.687182	−	0.726486i	$-0.258848\pi$
$131$	3.08969e6			1.37436			0.687182	+	0.726486i	$0.258848\pi$
$132$		0			0
$133$	−185539.	−	3.08582e6i	−0.0788643	−	1.31164i
$134$		0			0
$135$		−	1.39673e6i		−	0.567691i
$136$		0			0
$137$	−3.10883e6			−1.20903			−0.604513	−	0.796596i	$-0.706632\pi$
$137$	−3.10883e6			−1.20903			−0.604513	+	0.796596i	$0.706632\pi$
$138$		0			0
$139$	1.87884e6			0.699594			0.349797	−	0.936826i	$-0.386251\pi$
$139$	1.87884e6			0.699594			0.349797	+	0.936826i	$0.386251\pi$
$140$		0			0
$141$			393347.i			0.140320i
$142$		0			0
$143$			362518.i			0.123971i
$144$		0			0
$145$		−	356752.i		−	0.117021i
$146$		0			0
$147$		−	603046.i		−	0.189845i
$148$		0			0
$149$	438305.			0.132501			0.0662503	−	0.997803i	$-0.478896\pi$
$149$	438305.			0.132501			0.0662503	+	0.997803i	$0.478896\pi$
$150$		0			0
$151$		−	1.93403e6i		−	0.561737i	−0.959746	−	0.280869i	$-0.909378\pi$
$151$		−	1.93403e6i		−	0.561737i	0.959746	−	0.280869i	$-0.0906225\pi$
$152$		0			0
$153$	−1.18155e6			−0.329896
$154$		0			0
$155$			4.16677e6i			1.11893i
$156$		0			0
$157$	−2.42266e6			−0.626027			−0.313014	−	0.949749i	$-0.601339\pi$
$157$	−2.42266e6			−0.626027			−0.313014	+	0.949749i	$0.601339\pi$
$158$		0			0
$159$	157104.			0.0390837
$160$		0			0
$161$	−1.07228e6			−0.256940
$162$		0			0
$163$	−2.30189e6			−0.531522			−0.265761	−	0.964039i	$-0.585623\pi$
$163$	−2.30189e6			−0.531522			−0.265761	+	0.964039i	$0.585623\pi$
$164$		0			0
$165$			740841.i			0.164920i
$166$		0			0
$167$			1.20925e6i			0.259636i	0.991538	+	0.129818i	$0.0414393\pi$
$167$			1.20925e6i			0.259636i	−0.991538	+	0.129818i	$0.958561\pi$
$168$		0			0
$169$	4.59126e6			0.951200
$170$		0			0
$171$	279617.	+	4.65049e6i	0.0559210	+	0.930060i
$172$		0			0
$173$			9.28539e6i			1.79334i	0.442703	+	0.896669i	$0.354020\pi$
$173$			9.28539e6i			1.79334i	−0.442703	+	0.896669i	$0.645980\pi$
$174$		0			0
$175$	−1.86753e6			−0.348461
$176$		0			0
$177$	−193330.			−0.0348641
$178$		0			0
$179$			1.03484e7i			1.80433i	0.431392	+	0.902165i	$0.358023\pi$
$179$			1.03484e7i			1.80433i	−0.431392	+	0.902165i	$0.641977\pi$
$180$		0			0
$181$			3.79280e6i			0.639623i	0.947481	+	0.319811i	$0.103620\pi$
$181$			3.79280e6i			0.639623i	−0.947481	+	0.319811i	$0.896380\pi$
$182$		0			0
$183$		−	600009.i		−	0.0979049i
$184$		0			0
$185$			4.48960e6i			0.709075i
$186$		0			0
$187$	1.29932e6			0.198698
$188$		0			0
$189$			4.47733e6i			0.663183i
$190$		0			0
$191$	2.41561e6			0.346678			0.173339	−	0.984862i	$-0.444544\pi$
$191$	2.41561e6			0.346678			0.173339	+	0.984862i	$0.444544\pi$
$192$		0			0
$193$		−	7.65205e6i		−	1.06440i	−0.846618	−	0.532201i	$-0.821365\pi$
$193$		−	7.65205e6i		−	1.06440i	0.846618	−	0.532201i	$-0.178635\pi$
$194$		0			0
$195$	−481370.			−0.0649195
$196$		0			0
$197$	−3.41845e6			−0.447127			−0.223563	−	0.974689i	$-0.571769\pi$
$197$	−3.41845e6			−0.447127			−0.223563	+	0.974689i	$0.571769\pi$
$198$		0			0
$199$	−1.41073e6			−0.179012			−0.0895062	−	0.995986i	$-0.528529\pi$
$199$	−1.41073e6			−0.179012			−0.0895062	+	0.995986i	$0.528529\pi$
$200$		0			0
$201$	−2.04116e6			−0.251356
$202$		0			0
$203$			1.14360e6i			0.136705i
$204$		0			0
$205$			3.64031e6i			0.422549i
$206$		0			0
$207$	1.61599e6			0.182191
$208$		0			0
$209$	−307489.	−	5.11405e6i	−0.0336814	−	0.560178i
$210$		0			0
$211$		−	1.33027e7i		−	1.41609i	−0.706165	−	0.708047i	$-0.749576\pi$
$211$		−	1.33027e7i		−	1.41609i	0.706165	−	0.708047i	$-0.250424\pi$
$212$		0			0
$213$	1.58210e6			0.163718
$214$		0			0
$215$	−3.24286e6			−0.326297
$216$		0			0
$217$		−	1.33569e7i		−	1.30715i
$218$		0			0
$219$		−	97192.6i		−	0.00925339i
$220$		0			0
$221$			844252.i			0.0782159i
$222$		0			0
$223$		−	3.38391e6i		−	0.305144i	−0.988292	−	0.152572i	$-0.951244\pi$
$223$		−	3.38391e6i		−	0.305144i	0.988292	−	0.152572i	$-0.0487556\pi$
$224$		0			0
$225$	2.81447e6			0.247086
$226$		0			0
$227$		−	1.54275e7i		−	1.31892i	−0.751741	−	0.659458i	$-0.770786\pi$
$227$		−	1.54275e7i		−	1.31892i	0.751741	−	0.659458i	$-0.229214\pi$
$228$		0			0
$229$	1.52168e7			1.26712			0.633560	−	0.773694i	$-0.281593\pi$
$229$	1.52168e7			1.26712			0.633560	+	0.773694i	$0.281593\pi$
$230$		0			0
$231$		−	2.37482e6i		−	0.192661i
$232$		0			0
$233$	2.36662e7			1.87094			0.935471	−	0.353404i	$-0.114976\pi$
$233$	2.36662e7			1.87094			0.935471	+	0.353404i	$0.114976\pi$
$234$		0			0
$235$	−7.83997e6			−0.604103
$236$		0			0
$237$	3.50940e6			0.263626
$238$		0			0
$239$	−2.48397e7			−1.81950			−0.909752	−	0.415152i	$-0.863728\pi$
$239$	−2.48397e7			−1.81950			−0.909752	+	0.415152i	$0.863728\pi$
$240$		0			0
$241$		−	2.53572e7i		−	1.81155i	−0.423760	−	0.905775i	$-0.639290\pi$
$241$		−	2.53572e7i		−	1.81155i	0.423760	−	0.905775i	$-0.360710\pi$
$242$		0			0
$243$		−	1.02406e7i		−	0.713683i
$244$		0			0
$245$	1.20196e7			0.817316
$246$		0			0
$247$	3.32291e6	−	199795.i	0.220510	−	0.0132585i
$248$		0			0
$249$			1.15206e6i			0.0746235i
$250$		0			0
$251$	−1.57470e7			−0.995813			−0.497907	−	0.867231i	$-0.665898\pi$
$251$	−1.57470e7			−0.995813			−0.497907	+	0.867231i	$0.665898\pi$
$252$		0			0
$253$	−1.77707e6			−0.109734
$254$		0			0
$255$			1.72531e6i			0.104051i
$256$		0			0
$257$		−	2.16152e7i		−	1.27339i	−0.771117	−	0.636694i	$-0.780302\pi$
$257$		−	2.16152e7i		−	1.27339i	0.771117	−	0.636694i	$-0.219698\pi$
$258$		0			0
$259$		−	1.43917e7i		−	0.828350i
$260$		0			0
$261$		−	1.72346e6i		−	0.0969347i
$262$		0			0
$263$	−9.70229e6			−0.533344			−0.266672	−	0.963787i	$-0.585924\pi$
$263$	−9.70229e6			−0.533344			−0.266672	+	0.963787i	$0.585924\pi$
$264$		0			0
$265$			3.13131e6i			0.168263i
$266$		0			0
$267$	7.42885e6			0.390290
$268$		0			0
$269$			2.71353e7i			1.39405i	0.717049	+	0.697023i	$0.245493\pi$
$269$			2.71353e7i			1.39405i	−0.717049	+	0.697023i	$0.754507\pi$
$270$		0			0
$271$	2.54823e7			1.28035			0.640177	−	0.768228i	$-0.278861\pi$
$271$	2.54823e7			1.28035			0.640177	+	0.768228i	$0.278861\pi$
$272$		0			0
$273$	1.54307e6			0.0758398
$274$		0			0
$275$	−3.09501e6			−0.148821
$276$		0			0
$277$	−3.57565e7			−1.68235			−0.841173	−	0.540766i	$-0.818134\pi$
$277$	−3.57565e7			−1.68235			−0.841173	+	0.540766i	$0.818134\pi$
$278$		0			0
$279$			2.01295e7i			0.926874i
$280$		0			0
$281$		−	1.53746e7i		−	0.692923i	−0.938064	−	0.346462i	$-0.887383\pi$
$281$		−	1.53746e7i		−	0.692923i	0.938064	−	0.346462i	$-0.112617\pi$
$282$		0			0
$283$	2.66243e7			1.17468			0.587339	−	0.809341i	$-0.300176\pi$
$283$	2.66243e7			1.17468			0.587339	+	0.809341i	$0.300176\pi$
$284$		0			0
$285$	6.79069e6	−	408299.i	0.293346	−	0.0176378i
$286$		0			0
$287$		−	1.16693e7i		−	0.493627i
$288$		0			0
$289$	−2.11116e7			−0.874638
$290$		0			0
$291$	7.91364e6			0.321142
$292$		0			0
$293$		−	3.31522e7i		−	1.31798i	−0.752151	−	0.658991i	$-0.770984\pi$
$293$		−	3.31522e7i		−	1.31798i	0.752151	−	0.658991i	$-0.229016\pi$
$294$		0			0
$295$		−	3.85334e6i		−	0.150097i
$296$		0			0
$297$			7.42016e6i			0.283233i
$298$		0			0
$299$		−	1.15467e6i		−	0.0431960i
$300$		0			0
$301$	1.03952e7			0.381184
$302$		0			0
$303$			922382.i			0.0331576i
$304$		0			0
$305$	1.19590e7			0.421499
$306$		0			0
$307$		−	2.60630e7i		−	0.900761i	−0.892837	−	0.450381i	$-0.851288\pi$
$307$		−	2.60630e7i		−	0.900761i	0.892837	−	0.450381i	$-0.148712\pi$
$308$		0			0
$309$	7.85747e6			0.266322
$310$		0			0
$311$	−2.17615e6			−0.0723449			−0.0361725	−	0.999346i	$-0.511517\pi$
$311$	−2.17615e6			−0.0723449			−0.0361725	+	0.999346i	$0.511517\pi$
$312$		0			0
$313$	−3.53608e7			−1.15316			−0.576579	−	0.817042i	$-0.695613\pi$
$313$	−3.53608e7			−1.15316			−0.576579	+	0.817042i	$0.695613\pi$
$314$		0			0
$315$	−4.30431e7			−1.37712
$316$		0			0
$317$			3.71042e6i			0.116478i	0.998303	+	0.0582391i	$0.0185486\pi$
$317$			3.71042e6i			0.116478i	−0.998303	+	0.0582391i	$0.981451\pi$
$318$		0			0
$319$			1.89525e6i			0.0583841i
$320$		0			0
$321$	1.10264e7			0.333363
$322$		0			0
$323$	−716097.	−	1.19099e7i	−0.0212502	−	0.353427i
$324$		0			0
$325$		−	2.01102e6i		−	0.0585823i
$326$		0			0
$327$	5.63414e6			0.161133
$328$		0			0
$329$	2.51316e7			0.705720
$330$		0			0
$331$			63934.2i			0.00176299i	1.00000		0.000881494i	$0.000280588\pi$
$331$			63934.2i			0.00176299i	−1.00000		0.000881494i	$0.999719\pi$
$332$		0			0
$333$			2.16891e7i			0.587366i
$334$		0			0
$335$		−	4.06832e7i		−	1.08213i
$336$		0			0
$337$		−	4.38978e7i		−	1.14697i	−0.819215	−	0.573487i	$-0.805590\pi$
$337$		−	4.38978e7i		−	1.14697i	0.819215	−	0.573487i	$-0.194410\pi$
$338$		0			0
$339$	1.46290e7			0.375504
$340$		0			0
$341$		−	2.21360e7i		−	0.558259i
$342$		0			0
$343$	1.44955e7			0.359213
$344$		0			0
$345$		−	2.35968e6i		−	0.0574639i
$346$		0			0
$347$	4.20561e7			1.00656			0.503281	−	0.864123i	$-0.332126\pi$
$347$	4.20561e7			1.00656			0.503281	+	0.864123i	$0.332126\pi$
$348$		0			0
$349$	−7.60208e7			−1.78836			−0.894182	−	0.447703i	$-0.852242\pi$
$349$	−7.60208e7			−1.78836			−0.894182	+	0.447703i	$0.852242\pi$
$350$		0			0
$351$	−4.82134e6			−0.111493
$352$		0			0
$353$	−4.38529e7			−0.996951			−0.498475	−	0.866904i	$-0.666107\pi$
$353$	−4.38529e7			−0.996951			−0.498475	+	0.866904i	$0.666107\pi$
$354$		0			0
$355$			3.15335e7i			0.704835i
$356$		0			0
$357$		−	5.53061e6i		−	0.121554i
$358$		0			0
$359$	3.13674e7			0.677946			0.338973	−	0.940796i	$-0.389920\pi$
$359$	3.13674e7			0.677946			0.338973	+	0.940796i	$0.389920\pi$
$360$		0			0
$361$	−4.67069e7	+	5.63701e6i	−0.992796	+	0.119819i
$362$		0			0
$363$			8.56126e6i			0.178986i
$364$		0			0
$365$	1.93719e6			0.0398376
$366$		0			0
$367$	7.36341e7			1.48964			0.744819	−	0.667266i	$-0.232536\pi$
$367$	7.36341e7			1.48964			0.744819	+	0.667266i	$0.232536\pi$
$368$		0			0
$369$			1.75862e7i			0.350021i
$370$		0			0
$371$		−	1.00376e7i		−	0.196567i
$372$		0			0
$373$			4.19433e7i			0.808232i	0.914708	+	0.404116i	$0.132421\pi$
$373$			4.19433e7i			0.808232i	−0.914708	+	0.404116i	$0.867579\pi$
$374$		0			0
$375$			1.13876e7i			0.215943i
$376$		0			0
$377$	−1.23146e6			−0.0229825
$378$		0			0
$379$			4.18547e7i			0.768823i	0.923162	+	0.384412i	$0.125596\pi$
$379$			4.18547e7i			0.768823i	−0.923162	+	0.384412i	$0.874404\pi$
$380$		0			0
$381$	−1.01396e6			−0.0183336
$382$		0			0
$383$		−	5.82347e7i		−	1.03654i	−0.855218	−	0.518269i	$-0.826577\pi$
$383$		−	5.82347e7i		−	1.03654i	0.855218	−	0.518269i	$-0.173423\pi$
$384$		0			0
$385$	4.73335e7			0.829444
$386$		0			0
$387$	−1.56662e7			−0.270290
$388$		0			0
$389$	−4.66771e7			−0.792967			−0.396484	−	0.918042i	$-0.629770\pi$
$389$	−4.66771e7			−0.792967			−0.396484	+	0.918042i	$0.629770\pi$
$390$		0			0
$391$	−4.13852e6			−0.0692333
$392$		0			0
$393$		−	2.17954e7i		−	0.359077i
$394$		0			0
$395$			6.99473e7i			1.13496i
$396$		0			0
$397$	2.73862e7			0.437683			0.218842	−	0.975760i	$-0.429772\pi$
$397$	2.73862e7			0.437683			0.218842	+	0.975760i	$0.429772\pi$
$398$		0			0
$399$	−2.17681e7	+	1.30883e6i	−0.342690	+	0.0206047i
$400$		0			0
$401$		−	6.36758e7i		−	0.987509i	−0.869601	−	0.493755i	$-0.835624\pi$
$401$		−	6.36758e7i		−	0.987509i	0.869601	−	0.493755i	$-0.164376\pi$
$402$		0			0
$403$	1.43831e7			0.219755
$404$		0			0
$405$	5.97676e7			0.899707
$406$		0			0
$407$		−	2.38510e7i		−	0.353773i
$408$		0			0
$409$			2.91430e7i			0.425956i	0.977057	+	0.212978i	$0.0683163\pi$
$409$			2.91430e7i			0.425956i	−0.977057	+	0.212978i	$0.931684\pi$
$410$		0			0
$411$			2.19304e7i			0.315879i
$412$		0			0
$413$			1.23522e7i			0.175345i
$414$		0			0
$415$	−2.29621e7			−0.321268
$416$		0			0
$417$		−	1.32538e7i		−	0.182781i
$418$		0			0
$419$	1.39455e6			0.0189580			0.00947898	−	0.999955i	$-0.496983\pi$
$419$	1.39455e6			0.0189580			0.00947898	+	0.999955i	$0.496983\pi$
$420$		0			0
$421$			1.06439e8i			1.42645i	0.700936	+	0.713225i	$0.252766\pi$
$421$			1.06439e8i			1.42645i	−0.700936	+	0.713225i	$0.747234\pi$
$422$		0			0
$423$	−3.78747e7			−0.500412
$424$		0			0
$425$	−7.20783e6			−0.0938939
$426$		0			0
$427$	−3.83356e7			−0.492400
$428$		0			0
$429$	2.55729e6			0.0323897
$430$		0			0
$431$		−	5.26003e7i		−	0.656986i	−0.944506	−	0.328493i	$-0.893459\pi$
$431$		−	5.26003e7i		−	0.656986i	0.944506	−	0.328493i	$-0.106541\pi$
$432$		0			0
$433$		−	4.94789e7i		−	0.609476i	−0.952436	−	0.304738i	$-0.901431\pi$
$433$		−	4.94789e7i		−	0.609476i	0.952436	−	0.304738i	$-0.0985689\pi$
$434$		0			0
$435$	−2.51661e6			−0.0305737
$436$		0			0
$437$	979393.	+	1.62889e7i	0.0117358	+	0.195186i
$438$		0			0
$439$			1.67332e8i			1.97782i	0.148528	+	0.988908i	$0.452547\pi$
$439$			1.67332e8i			1.97782i	−0.148528	+	0.988908i	$0.547453\pi$
$440$		0			0
$441$	5.80661e7			0.677029
$442$		0			0
$443$	4.28721e7			0.493133			0.246566	−	0.969126i	$-0.420698\pi$
$443$	4.28721e7			0.493133			0.246566	+	0.969126i	$0.420698\pi$
$444$		0			0
$445$			1.48068e8i			1.68027i
$446$		0			0
$447$		−	3.09191e6i		−	0.0346181i
$448$		0			0
$449$		−	4.18651e7i		−	0.462501i	−0.972894	−	0.231250i	$-0.925718\pi$
$449$		−	4.18651e7i		−	0.462501i	0.972894	−	0.231250i	$-0.0742817\pi$
$450$		0			0
$451$		−	1.93392e7i		−	0.210819i
$452$		0			0
$453$	−1.36431e7			−0.146764
$454$		0			0
$455$			3.07555e7i			0.326504i
$456$		0			0
$457$	4.84244e7			0.507359			0.253680	−	0.967288i	$-0.418359\pi$
$457$	4.84244e7			0.507359			0.253680	+	0.967288i	$0.418359\pi$
$458$		0			0
$459$			1.72805e7i			0.178697i
$460$		0			0
$461$	7.76046e7			0.792109			0.396055	−	0.918227i	$-0.370379\pi$
$461$	7.76046e7			0.792109			0.396055	+	0.918227i	$0.370379\pi$
$462$		0			0
$463$	−5.50972e7			−0.555119			−0.277560	−	0.960708i	$-0.589526\pi$
$463$	−5.50972e7			−0.555119			−0.277560	+	0.960708i	$0.589526\pi$
$464$		0			0
$465$	2.93933e7			0.292341
$466$		0			0
$467$	−1.64895e8			−1.61904			−0.809520	−	0.587092i	$-0.800273\pi$
$467$	−1.64895e8			−1.61904			−0.809520	+	0.587092i	$0.800273\pi$
$468$		0			0
$469$			1.30413e8i			1.26416i
$470$		0			0
$471$			1.70900e7i			0.163561i
$472$		0			0
$473$	1.72278e7			0.162797
$474$		0			0
$475$	1.70575e6	+	2.83695e7i	0.0159160	+	0.264710i
$476$		0			0
$477$			1.51273e7i			0.139381i
$478$		0			0
$479$	5.07449e7			0.461728			0.230864	−	0.972986i	$-0.425845\pi$
$479$	5.07449e7			0.461728			0.230864	+	0.972986i	$0.425845\pi$
$480$		0			0
$481$	1.54975e7			0.139260
$482$		0			0
$483$			7.56412e6i			0.0671301i
$484$		0			0
$485$			1.57730e8i			1.38258i
$486$		0			0
$487$			1.12286e8i			0.972163i	0.873913	+	0.486082i	$0.161574\pi$
$487$			1.12286e8i			0.972163i	−0.873913	+	0.486082i	$0.838426\pi$
$488$		0			0
$489$			1.62380e7i			0.138869i
$490$		0			0
$491$	−1.68625e8			−1.42455			−0.712277	−	0.701899i	$-0.752336\pi$
$491$	−1.68625e8			−1.42455			−0.712277	+	0.701899i	$0.752336\pi$
$492$		0			0
$493$			4.41376e6i			0.0368356i
$494$		0			0
$495$	−7.13341e7			−0.588141
$496$		0			0
$497$		−	1.01083e8i		−	0.823397i
$498$		0			0
$499$	1.41676e8			1.14024			0.570119	−	0.821562i	$-0.306897\pi$
$499$	1.41676e8			1.14024			0.570119	+	0.821562i	$0.306897\pi$
$500$		0			0
$501$	8.53029e6			0.0678345
$502$		0			0
$503$	−8.44068e7			−0.663244			−0.331622	−	0.943412i	$-0.607596\pi$
$503$	−8.44068e7			−0.663244			−0.331622	+	0.943412i	$0.607596\pi$
$504$		0			0
$505$	−1.83844e7			−0.142750
$506$		0			0
$507$		−	3.23878e7i		−	0.248518i
$508$		0			0
$509$			1.73197e8i			1.31337i	0.754165	+	0.656685i	$0.228042\pi$
$509$			1.73197e8i			1.31337i	−0.754165	+	0.656685i	$0.771958\pi$
$510$		0			0
$511$	−6.20980e6			−0.0465387
$512$		0			0
$513$	6.80147e7	−	4.08947e6i	0.503791	−	0.0302911i
$514$		0			0
$515$			1.56611e8i			1.14657i
$516$		0			0
$517$	4.16500e7			0.301400
$518$		0			0
$519$	6.55013e7			0.468541
$520$		0			0
$521$		−	1.00959e8i		−	0.713889i	−0.934126	−	0.356944i	$-0.883819\pi$
$521$		−	1.00959e8i		−	0.713889i	0.934126	−	0.356944i	$-0.116181\pi$
$522$		0			0
$523$		−	7.80472e7i		−	0.545572i	−0.962075	−	0.272786i	$-0.912055\pi$
$523$		−	7.80472e7i		−	0.545572i	0.962075	−	0.272786i	$-0.0879452\pi$
$524$		0			0
$525$			1.31740e7i			0.0910415i
$526$		0			0
$527$		−	5.15515e7i		−	0.352216i
$528$		0			0
$529$	−1.42376e8			−0.961765
$530$		0			0
$531$		−	1.86154e7i		−	0.124333i
$532$		0			0
$533$	1.25659e7			0.0829872
$534$		0			0
$535$			2.19771e8i			1.43519i
$536$		0			0
$537$	7.30002e7			0.471413
$538$		0			0
$539$	−6.38541e7			−0.407777
$540$		0			0
$541$	−1.90151e8			−1.20090			−0.600449	−	0.799663i	$-0.705011\pi$
$541$	−1.90151e8			−1.20090			−0.600449	+	0.799663i	$0.705011\pi$
$542$		0			0
$543$	2.67553e7			0.167113
$544$		0			0
$545$			1.12296e8i			0.693707i
$546$		0			0
$547$		−	1.15295e8i		−	0.704445i	−0.935916	−	0.352223i	$-0.885426\pi$
$547$		−	1.15295e8i		−	0.704445i	0.935916	−	0.352223i	$-0.114574\pi$
$548$		0			0
$549$	5.77737e7			0.349151
$550$		0			0
$551$	1.73722e7	−	1.04453e6i	0.103849	−	0.00624404i
$552$		0			0
$553$		−	2.24221e8i		−	1.32587i
$554$		0			0
$555$	3.16706e7			0.185258
$556$		0			0
$557$	1.62145e8			0.938292			0.469146	−	0.883121i	$-0.344562\pi$
$557$	1.62145e8			0.938292			0.469146	+	0.883121i	$0.344562\pi$
$558$		0			0
$559$			1.11939e7i			0.0640837i
$560$		0			0
$561$		−	9.16573e6i		−	0.0519133i
$562$		0			0
$563$		−	2.97662e7i		−	0.166801i	−0.996516	−	0.0834005i	$-0.973422\pi$
$563$		−	2.97662e7i		−	0.166801i	0.996516	−	0.0834005i	$-0.0265781\pi$
$564$		0			0
$565$			2.91576e8i			1.61662i
$566$		0			0
$567$	−1.91590e8			−1.05105
$568$		0			0
$569$		−	2.12116e8i		−	1.15143i	−0.817651	−	0.575714i	$-0.804724\pi$
$569$		−	2.12116e8i		−	1.15143i	0.817651	−	0.575714i	$-0.195276\pi$
$570$		0			0
$571$	1.56406e8			0.840130			0.420065	−	0.907494i	$-0.362007\pi$
$571$	1.56406e8			0.840130			0.420065	+	0.907494i	$0.362007\pi$
$572$		0			0
$573$		−	1.70403e7i		−	0.0905759i
$574$		0			0
$575$	9.85803e6			0.0518545
$576$		0			0
$577$	−2.60620e8			−1.35669			−0.678345	−	0.734743i	$-0.737303\pi$
$577$	−2.60620e8			−1.35669			−0.678345	+	0.734743i	$0.737303\pi$
$578$		0			0
$579$	−5.39793e7			−0.278094
$580$		0			0
$581$	7.36067e7			0.375309
$582$		0			0
$583$		−	1.66351e7i		−	0.0839499i
$584$		0			0
$585$		−	4.63502e7i		−	0.231518i
$586$		0			0
$587$	3.53809e7			0.174926			0.0874630	−	0.996168i	$-0.472124\pi$
$587$	3.53809e7			0.174926			0.0874630	+	0.996168i	$0.472124\pi$
$588$		0			0
$589$	−2.02903e8	+	1.21998e7i	−0.992985	+	0.0597045i
$590$		0			0
$591$			2.41145e7i			0.116820i
$592$		0			0
$593$	2.88008e8			1.38115			0.690573	−	0.723262i	$-0.257358\pi$
$593$	2.88008e8			1.38115			0.690573	+	0.723262i	$0.257358\pi$
$594$		0			0
$595$	1.10233e8			0.523311
$596$		0			0
$597$			9.95158e6i			0.0467702i
$598$		0			0
$599$		−	7.36741e7i		−	0.342795i	−0.985202	−	0.171397i	$-0.945172\pi$
$599$		−	7.36741e7i		−	0.342795i	0.985202	−	0.171397i	$-0.0548282\pi$
$600$		0			0
$601$		−	3.29511e8i		−	1.51791i	−0.651141	−	0.758957i	$-0.725709\pi$
$601$		−	3.29511e8i		−	1.51791i	0.651141	−	0.758957i	$-0.274291\pi$
$602$		0			0
$603$		−	1.96539e8i		−	0.896391i
$604$		0			0
$605$	−1.70638e8			−0.770566
$606$		0			0
$607$			8.95189e7i			0.400266i	0.979769	+	0.200133i	$0.0641374\pi$
$607$			8.95189e7i			0.400266i	−0.979769	+	0.200133i	$0.935863\pi$
$608$		0			0
$609$	8.06718e6			0.0357166
$610$		0			0
$611$			2.70626e7i			0.118644i
$612$		0			0
$613$	−1.92537e8			−0.835858			−0.417929	−	0.908480i	$-0.637244\pi$
$613$	−1.92537e8			−0.835858			−0.417929	+	0.908480i	$0.637244\pi$
$614$		0			0
$615$	2.56796e7			0.110398
$616$		0			0
$617$	1.07529e8			0.457796			0.228898	−	0.973450i	$-0.426488\pi$
$617$	1.07529e8			0.457796			0.228898	+	0.973450i	$0.426488\pi$
$618$		0			0
$619$	3.19035e8			1.34513			0.672567	−	0.740036i	$-0.265192\pi$
$619$	3.19035e8			1.34513			0.672567	+	0.740036i	$0.265192\pi$
$620$		0			0
$621$		−	2.36342e7i		−	0.0986884i
$622$		0			0
$623$		−	4.74642e8i		−	1.96292i
$624$		0			0
$625$	−2.91715e8			−1.19486
$626$		0			0
$627$	−3.60757e7	+	2.16910e6i	−0.146356	+	0.00879987i
$628$		0			0
$629$		−	5.55456e7i		−	0.223202i
$630$		0			0
$631$	3.80336e8			1.51384			0.756918	−	0.653509i	$-0.226704\pi$
$631$	3.80336e8			1.51384			0.756918	+	0.653509i	$0.226704\pi$
$632$		0			0
$633$	−9.38402e7			−0.369980
$634$		0			0
$635$		−	2.02098e7i		−	0.0789296i
$636$		0			0
$637$		−	4.14899e7i		−	0.160518i
$638$		0			0
$639$			1.52338e8i			0.583854i
$640$		0			0
$641$			5.24526e7i			0.199156i	0.995030	+	0.0995779i	$0.0317493\pi$
$641$			5.24526e7i			0.199156i	−0.995030	+	0.0995779i	$0.968251\pi$
$642$		0			0
$643$	2.31991e8			0.872645			0.436322	−	0.899790i	$-0.356281\pi$
$643$	2.31991e8			0.872645			0.436322	+	0.899790i	$0.356281\pi$
$644$		0			0
$645$			2.28759e7i			0.0852509i
$646$		0			0
$647$	−1.98789e8			−0.733971			−0.366986	−	0.930227i	$-0.619610\pi$
$647$	−1.98789e8			−0.733971			−0.366986	+	0.930227i	$0.619610\pi$
$648$		0			0
$649$			2.04709e7i			0.0748864i
$650$		0			0
$651$	−9.42225e7			−0.341516
$652$		0			0
$653$	−1.04703e6			−0.00376029			−0.00188014	−	0.999998i	$-0.500598\pi$
$653$	−1.04703e6			−0.00376029			−0.00188014	+	0.999998i	$0.500598\pi$
$654$		0			0
$655$	4.34413e8			1.54589
$656$		0			0
$657$	9.35849e6			0.0329997
$658$		0			0
$659$			1.43873e8i			0.502717i	0.967894	+	0.251359i	$0.0808773\pi$
$659$			1.43873e8i			0.502717i	−0.967894	+	0.251359i	$0.919123\pi$
$660$		0			0
$661$			3.03081e8i			1.04943i	0.851278	+	0.524715i	$0.175828\pi$
$661$			3.03081e8i			1.04943i	−0.851278	+	0.524715i	$0.824172\pi$
$662$		0			0
$663$	5.95554e6			0.0204353
$664$		0			0
$665$	−2.60869e7	−	4.33869e8i	−0.0887070	−	1.47534i
$666$		0			0
$667$		−	6.03663e6i		−	0.0203431i
$668$		0			0
$669$	−2.38709e7			−0.0797242
$670$		0			0
$671$	−6.35325e7			−0.210295
$672$		0			0
$673$		−	2.28558e8i		−	0.749810i	−0.927063	−	0.374905i	$-0.877675\pi$
$673$		−	2.28558e8i		−	0.749810i	0.927063	−	0.374905i	$-0.122325\pi$
$674$		0			0
$675$		−	4.11623e7i		−	0.133841i
$676$		0			0
$677$			7.03968e6i			0.0226875i	0.999936	+	0.0113438i	$0.00361091\pi$
$677$			7.03968e6i			0.0226875i	−0.999936	+	0.0113438i	$0.996389\pi$
$678$		0			0
$679$		−	5.05615e8i		−	1.61514i
$680$		0			0
$681$	−1.08829e8			−0.344590
$682$		0			0
$683$			1.31171e8i			0.411695i	0.978584	+	0.205847i	$0.0659950\pi$
$683$			1.31171e8i			0.411695i	−0.978584	+	0.205847i	$0.934005\pi$
$684$		0			0
$685$	−4.37104e8			−1.35992
$686$		0			0
$687$		−	1.07343e8i		−	0.331057i
$688$		0			0
$689$	1.08089e7			0.0330463
$690$		0			0
$691$	−1.78758e8			−0.541789			−0.270895	−	0.962609i	$-0.587319\pi$
$691$	−1.78758e8			−0.541789			−0.270895	+	0.962609i	$0.587319\pi$
$692$		0			0
$693$	2.28667e8			0.687074
$694$		0			0
$695$	2.64167e8			0.786907
$696$		0			0
$697$		−	4.50382e7i		−	0.133009i
$698$		0			0
$699$		−	1.66947e8i		−	0.488816i
$700$		0			0
$701$	3.12715e8			0.907810			0.453905	−	0.891050i	$-0.350030\pi$
$701$	3.12715e8			0.907810			0.453905	+	0.891050i	$0.350030\pi$
$702$		0			0
$703$	−2.18624e8	+	1.31450e7i	−0.629261	+	0.0378351i
$704$		0			0
$705$			5.53049e7i			0.157833i
$706$		0			0
$707$	5.89325e7			0.166762
$708$		0			0
$709$	−1.66800e7			−0.0468014			−0.0234007	−	0.999726i	$-0.507449\pi$
$709$	−1.66800e7			−0.0468014			−0.0234007	+	0.999726i	$0.507449\pi$
$710$		0			0
$711$			3.37913e8i			0.940148i
$712$		0			0
$713$			7.05061e7i			0.194517i
$714$		0			0
$715$			5.09703e7i			0.139444i
$716$		0			0
$717$			1.75225e8i			0.475378i
$718$		0			0
$719$	−3.99235e8			−1.07409			−0.537047	−	0.843553i	$-0.680460\pi$
$719$	−3.99235e8			−1.07409			−0.537047	+	0.843553i	$0.680460\pi$
$720$		0			0
$721$		−	5.02027e8i		−	1.33943i
$722$		0			0
$723$	−1.78875e8			−0.473299
$724$		0			0
$725$		−	1.05136e7i		−	0.0275892i
$726$		0			0
$727$	−1.50012e8			−0.390411			−0.195205	−	0.980762i	$-0.562537\pi$
$727$	−1.50012e8			−0.390411			−0.195205	+	0.980762i	$0.562537\pi$
$728$		0			0
$729$	2.37650e8			0.613415
$730$		0			0
$731$	4.01209e7			0.102711
$732$		0			0
$733$	5.55074e8			1.40941			0.704707	−	0.709498i	$-0.251078\pi$
$733$	5.55074e8			1.40941			0.704707	+	0.709498i	$0.251078\pi$
$734$		0			0
$735$		−	8.47887e7i		−	0.213538i
$736$		0			0
$737$			2.16130e8i			0.539899i
$738$		0			0
$739$	8.07071e7			0.199976			0.0999881	−	0.994989i	$-0.468120\pi$
$739$	8.07071e7			0.199976			0.0999881	+	0.994989i	$0.468120\pi$
$740$		0			0
$741$	−1.40940e6	−	2.34406e7i	−0.00346400	−	0.0576121i
$742$		0			0
$743$		−	1.72685e8i		−	0.421007i	−0.977593	−	0.210503i	$-0.932490\pi$
$743$		−	1.72685e8i		−	0.421007i	0.977593	−	0.210503i	$-0.0675103\pi$
$744$		0			0
$745$	6.16261e7			0.149038
$746$		0			0
$747$	−1.10929e8			−0.266124
$748$		0			0
$749$		−	7.04493e8i		−	1.67661i
$750$		0			0
$751$		−	1.92284e8i		−	0.453966i	−0.973899	−	0.226983i	$-0.927114\pi$
$751$		−	1.92284e8i		−	0.453966i	0.973899	−	0.226983i	$-0.0728862\pi$
$752$		0			0
$753$			1.11083e8i			0.260174i
$754$		0			0
$755$		−	2.71927e8i		−	0.631845i
$756$		0			0
$757$	2.66482e7			0.0614299			0.0307149	−	0.999528i	$-0.490222\pi$
$757$	2.66482e7			0.0614299			0.0307149	+	0.999528i	$0.490222\pi$
$758$		0			0
$759$			1.25358e7i			0.0286700i
$760$		0			0
$761$	2.01010e8			0.456103			0.228052	−	0.973649i	$-0.426764\pi$
$761$	2.01010e8			0.456103			0.228052	+	0.973649i	$0.426764\pi$
$762$		0			0
$763$		−	3.59974e8i		−	0.810397i
$764$		0			0
$765$	−1.66127e8			−0.371069
$766$		0			0
$767$	−1.33012e7			−0.0294785
$768$		0			0
$769$	3.63424e8			0.799161			0.399580	−	0.916698i	$-0.369156\pi$
$769$	3.63424e8			0.799161			0.399580	+	0.916698i	$0.369156\pi$
$770$		0			0
$771$	−1.52479e8			−0.332695
$772$		0			0
$773$		−	4.33863e8i		−	0.939321i	−0.882847	−	0.469660i	$-0.844376\pi$
$773$		−	4.33863e8i		−	0.939321i	0.882847	−	0.469660i	$-0.155624\pi$
$774$		0			0
$775$			1.22796e8i			0.263804i
$776$		0			0
$777$	−1.01523e8			−0.216421
$778$		0			0
$779$	−1.77267e8	+	1.06584e7i	−0.374986	+	0.0225465i
$780$		0			0
$781$		−	1.67522e8i		−	0.351657i
$782$		0			0
$783$	−2.52060e7			−0.0525072
$784$		0			0
$785$	−3.40628e8			−0.704159
$786$		0			0
$787$			5.48524e8i			1.12531i	0.826692	+	0.562654i	$0.190220\pi$
$787$			5.48524e8i			1.12531i	−0.826692	+	0.562654i	$0.809780\pi$
$788$		0			0
$789$			6.84422e7i			0.139345i
$790$		0			0
$791$		−	9.34670e8i		−	1.88855i
$792$		0			0
$793$		−	4.12810e7i		−	0.0827810i
$794$		0			0
$795$	2.20890e7			0.0439616
$796$		0			0
$797$		−	4.23765e8i		−	0.837047i	−0.908206	−	0.418523i	$-0.862548\pi$
$797$		−	4.23765e8i		−	0.837047i	0.908206	−	0.418523i	$-0.137452\pi$
$798$		0			0
$799$	9.69966e7			0.190159
$800$		0			0
$801$			7.15310e8i			1.39186i
$802$		0			0
$803$	−1.02913e7			−0.0198758
$804$		0			0
$805$	−1.50764e8			−0.289007
$806$		0			0
$807$	1.91418e8			0.364219
$808$		0			0
$809$	−3.43785e7			−0.0649295			−0.0324647	−	0.999473i	$-0.510336\pi$
$809$	−3.43785e7			−0.0649295			−0.0324647	+	0.999473i	$0.510336\pi$
$810$		0			0
$811$			1.19679e8i			0.224364i	0.993688	+	0.112182i	$0.0357840\pi$
$811$			1.19679e8i			0.224364i	−0.993688	+	0.112182i	$0.964216\pi$
$812$		0			0
$813$		−	1.79758e8i		−	0.334515i
$814$		0			0
$815$	−3.23647e8			−0.597859
$816$		0			0
$817$	−9.49473e6	−	1.57913e8i	−0.0174107	−	0.289569i
$818$		0			0
$819$			1.48579e8i			0.270462i
$820$		0			0
$821$	−1.43435e8			−0.259195			−0.129598	−	0.991567i	$-0.541369\pi$
$821$	−1.43435e8			−0.259195			−0.129598	+	0.991567i	$0.541369\pi$
$822$		0			0
$823$	−1.03287e9			−1.85287			−0.926436	−	0.376452i	$-0.877144\pi$
$823$	−1.03287e9			−1.85287			−0.926436	+	0.376452i	$0.877144\pi$
$824$		0			0
$825$			2.18329e7i			0.0388821i
$826$		0			0
$827$		−	6.45982e8i		−	1.14210i	−0.820915	−	0.571050i	$-0.806536\pi$
$827$		−	6.45982e8i		−	1.14210i	0.820915	−	0.571050i	$-0.193464\pi$
$828$		0			0
$829$			6.83941e7i			0.120048i	0.998197	+	0.0600240i	$0.0191177\pi$
$829$			6.83941e7i			0.120048i	−0.998197	+	0.0600240i	$0.980882\pi$
$830$		0			0
$831$			2.52234e8i			0.439542i
$832$		0			0
$833$	−1.48707e8			−0.257274
$834$		0			0
$835$			1.70021e8i			0.292040i
$836$		0			0
$837$	2.94399e8			0.502066
$838$		0			0
$839$		−	8.87325e8i		−	1.50244i	−0.660053	−	0.751219i	$-0.729466\pi$
$839$		−	8.87325e8i		−	1.50244i	0.660053	−	0.751219i	$-0.270534\pi$
$840$		0			0
$841$	5.88385e8			0.989176
$842$		0			0
$843$	−1.08456e8			−0.181038
$844$		0			0
$845$	6.45534e8			1.06991
$846$		0			0
$847$	5.46993e8			0.900185
$848$		0			0
$849$		−	1.87814e8i		−	0.306905i
$850$		0			0
$851$			7.59688e7i			0.123267i
$852$		0			0
$853$	4.16960e8			0.671812			0.335906	−	0.941896i	$-0.390958\pi$
$853$	4.16960e8			0.671812			0.335906	+	0.941896i	$0.390958\pi$
$854$		0			0
$855$	3.93144e7	+	6.53863e8i	0.0629003	+	1.04614i
$856$		0			0
$857$		−	9.17263e7i		−	0.145731i	−0.997342	−	0.0728654i	$-0.976786\pi$
$857$		−	9.17263e7i		−	0.145731i	0.997342	−	0.0728654i	$-0.0232143\pi$
$858$		0			0
$859$	−8.31747e8			−1.31223			−0.656117	−	0.754659i	$-0.727802\pi$
$859$	−8.31747e8			−1.31223			−0.656117	+	0.754659i	$0.727802\pi$
$860$		0			0
$861$	−8.23178e7			−0.128969
$862$		0			0
$863$			6.32943e8i			0.984764i	0.870379	+	0.492382i	$0.163874\pi$
$863$			6.32943e8i			0.984764i	−0.870379	+	0.492382i	$0.836126\pi$
$864$		0			0
$865$			1.30553e9i			2.01716i
$866$		0			0
$867$			1.48926e8i			0.228515i
$868$		0			0
$869$		−	3.71596e8i		−	0.566254i
$870$		0			0
$871$	−1.40433e8			−0.212527
$872$		0			0
$873$			7.61989e8i			1.14526i
$874$		0			0
$875$	7.27574e8			1.08606
$876$		0			0
$877$			1.09008e9i			1.61607i	0.589137	+	0.808034i	$0.299468\pi$
$877$			1.09008e9i			1.61607i	−0.589137	+	0.808034i	$0.700532\pi$
$878$		0			0
$879$	−2.33863e8			−0.344346
$880$		0			0
$881$	7.00834e8			1.02491			0.512457	−	0.858713i	$-0.328736\pi$
$881$	7.00834e8			1.02491			0.512457	+	0.858713i	$0.328736\pi$
$882$		0			0
$883$	−4.71162e8			−0.684365			−0.342183	−	0.939634i	$-0.611166\pi$
$883$	−4.71162e8			−0.684365			−0.342183	+	0.939634i	$0.611166\pi$
$884$		0			0
$885$	−2.71823e7			−0.0392154
$886$		0			0
$887$			9.15741e8i			1.31221i	0.754671	+	0.656103i	$0.227796\pi$
$887$			9.15741e8i			1.31221i	−0.754671	+	0.656103i	$0.772204\pi$
$888$		0			0
$889$			6.47839e7i			0.0922065i
$890$		0			0
$891$	−3.17516e8			−0.448883
$892$		0			0
$893$	−2.29545e7	−	3.81772e8i	−0.0322340	−	0.536105i
$894$		0			0
$895$			1.45500e9i			2.02952i
$896$		0			0
$897$	−8.14530e6			−0.0112857
$898$		0			0
$899$	7.51952e7			0.103493
$900$		0			0
$901$		−	3.87407e7i		−	0.0529656i
$902$		0			0
$903$		−	7.33304e7i		−	0.0995912i
$904$		0			0
$905$			5.33270e8i			0.719451i
$906$		0			0
$907$		−	4.88619e8i		−	0.654860i	−0.944875	−	0.327430i	$-0.893817\pi$
$907$		−	4.88619e8i		−	0.654860i	0.944875	−	0.327430i	$-0.106183\pi$
$908$		0			0
$909$	−8.88144e7			−0.118247
$910$		0			0
$911$		−	1.81370e8i		−	0.239889i	−0.992781	−	0.119944i	$-0.961728\pi$
$911$		−	1.81370e8i		−	0.239889i	0.992781	−	0.119944i	$-0.0382716\pi$
$912$		0			0
$913$	1.21987e8			0.160287
$914$		0			0
$915$		−	8.43617e7i		−	0.110124i
$916$		0			0
$917$	−1.39254e9			−1.80593
$918$		0			0
$919$	1.05377e9			1.35769			0.678844	−	0.734283i	$-0.262481\pi$
$919$	1.05377e9			1.35769			0.678844	+	0.734283i	$0.262481\pi$
$920$		0			0
$921$	−1.83854e8			−0.235340
$922$		0			0
$923$	1.08850e8			0.138427
$924$		0			0
$925$			1.32310e8i			0.167174i
$926$		0			0
$927$			7.56581e8i			0.949765i
$928$		0			0
$929$	−8.77698e8			−1.09471			−0.547354	−	0.836901i	$-0.684365\pi$
$929$	−8.77698e8			−1.09471			−0.547354	+	0.836901i	$0.684365\pi$
$930$		0			0
$931$	3.51919e7	+	5.85299e8i	0.0436107	+	0.725319i
$932$		0			0
$933$			1.53511e7i			0.0189014i
$934$		0			0
$935$	1.82686e8			0.223496
$936$		0			0
$937$	−1.01073e9			−1.22861			−0.614307	−	0.789067i	$-0.710564\pi$
$937$	−1.01073e9			−1.22861			−0.614307	+	0.789067i	$0.710564\pi$
$938$		0			0
$939$			2.49443e8i			0.301283i
$940$		0			0
$941$			6.81777e8i			0.818227i	0.912484	+	0.409113i	$0.134162\pi$
$941$			6.81777e8i			0.818227i	−0.912484	+	0.409113i	$0.865838\pi$
$942$		0			0
$943$			6.15980e7i			0.0734567i
$944$		0			0
$945$			6.29516e8i			0.745953i
$946$		0			0
$947$	9.99641e8			1.17705			0.588524	−	0.808480i	$-0.299709\pi$
$947$	9.99641e8			1.17705			0.588524	+	0.808480i	$0.299709\pi$
$948$		0			0
$949$		−	6.68691e6i		−	0.00782397i
$950$		0			0
$951$	2.61741e7			0.0304320
$952$		0			0
$953$			1.10138e9i			1.27250i	0.771481	+	0.636252i	$0.219516\pi$
$953$			1.10138e9i			1.27250i	−0.771481	+	0.636252i	$0.780484\pi$
$954$		0			0
$955$	3.39637e8			0.389946
$956$		0			0
$957$	1.33695e7			0.0152539
$958$		0			0
$959$	1.40117e9			1.58867
$960$		0			0
$961$	9.24428e6			0.0104160
$962$		0			0
$963$			1.06171e9i			1.18885i
$964$		0			0
$965$		−	1.07588e9i		−	1.19725i
$966$		0			0
$967$	4.09726e8			0.453121			0.226560	−	0.973997i	$-0.427252\pi$
$967$	4.09726e8			0.453121			0.226560	+	0.973997i	$0.427252\pi$
$968$		0			0
$969$	−8.40149e7	+	5.05151e6i	−0.0923390	+	0.00555200i
$970$		0			0
$971$			6.54667e8i			0.715093i	0.933895	+	0.357547i	$0.116387\pi$
$971$			6.54667e8i			0.715093i	−0.933895	+	0.357547i	$0.883613\pi$
$972$		0			0
$973$	−8.46806e8			−0.919275
$974$		0			0
$975$	−1.41862e7			−0.0153057
$976$		0			0
$977$		−	1.54073e9i		−	1.65213i	−0.563575	−	0.826065i	$-0.690574\pi$
$977$		−	1.54073e9i		−	1.65213i	0.563575	−	0.826065i	$-0.309426\pi$
$978$		0			0
$979$		−	7.86611e8i		−	0.838324i
$980$		0			0
$981$			5.42500e8i			0.574636i
$982$		0			0
$983$		−	4.01024e8i		−	0.422192i	−0.977465	−	0.211096i	$-0.932297\pi$
$983$		−	4.01024e8i		−	0.422192i	0.977465	−	0.211096i	$-0.0677033\pi$
$984$		0			0
$985$	−4.80637e8			−0.502931
$986$		0			0
$987$		−	1.77284e8i		−	0.184382i
$988$		0			0
$989$	−5.48727e7			−0.0567241
$990$		0			0
$991$		−	1.84076e8i		−	0.189137i	−0.995518	−	0.0945685i	$-0.969853\pi$
$991$		−	1.84076e8i		−	0.189137i	0.995518	−	0.0945685i	$-0.0301471\pi$
$992$		0			0
$993$	451007.			0.000460612
$994$		0			0
$995$	−1.98349e8			−0.201354
$996$		0			0
$997$	5.79233e8			0.584477			0.292239	−	0.956345i	$-0.405600\pi$
$997$	5.79233e8			0.584477			0.292239	+	0.956345i	$0.405600\pi$
$998$		0			0
$999$	3.17209e8			0.318162

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.f.113.15		30
4.3	odd	2		152.7.e.a.113.16	yes	30
19.18	odd	2	inner	304.7.e.f.113.16		30
76.75	even	2		152.7.e.a.113.15	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.7.e.a.113.15	✓	30	76.75	even	2
152.7.e.a.113.16	yes	30	4.3	odd	2
304.7.e.f.113.15		30	1.1	even	1	trivial
304.7.e.f.113.16		30	19.18	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.05423i q^{3} +140.601 q^{5} -450.706 q^{7} +679.238 q^{9} +O(q^{10})\)	\(q-7.05423i q^{3} +140.601 q^{5} -450.706 q^{7} +679.238 q^{9} -746.944 q^{11} -485.335i q^{13} -991.829i q^{15} -1739.52 q^{17} +(411.663 + 6846.64i) q^{19} +3179.38i q^{21} +2379.12 q^{23} +4143.57 q^{25} -9934.03i q^{27} -2537.34i q^{29} +29635.4i q^{31} +5269.11i q^{33} -63369.6 q^{35} +31931.5i q^{37} -3423.67 q^{39} +25891.1i q^{41} -23064.3 q^{43} +95501.3 q^{45} -55760.5 q^{47} +85487.2 q^{49} +12271.0i q^{51} +22270.9i q^{53} -105021. q^{55} +(48297.7 - 2903.96i) q^{57} -27406.2i q^{59} +85056.7 q^{61} -306137. q^{63} -68238.5i q^{65} -289353. i q^{67} -16782.8i q^{69} +224277. i q^{71} +13777.9 q^{73} -29229.7i q^{75} +336652. q^{77} +497489. i q^{79} +425088. q^{81} -163314. q^{83} -244578. q^{85} -17899.0 q^{87} +1.05311e6i q^{89} +218744. i q^{91} +209055. q^{93} +(57880.1 + 962642. i) q^{95} +1.12183e6i q^{97} -507352. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) \) 30 * q + 720 * q^7 - 8670 * q^9	\( 30 q + 720 q^{7} - 8670 q^{9} + 2524 q^{11} + 9700 q^{17} - 4014 q^{19} + 39376 q^{23} + 110742 q^{25} + 19976 q^{35} + 266500 q^{39} + 106788 q^{43} - 91360 q^{45} - 222756 q^{47} + 593586 q^{49} - 540936 q^{55} - 545972 q^{57} - 242640 q^{61} + 377716 q^{63} + 545964 q^{73} - 272356 q^{77} + 2189926 q^{81} - 1542652 q^{83} - 826908 q^{85} + 2729572 q^{87} - 2139912 q^{93} - 2142716 q^{95} + 293012 q^{99}+O(q^{100}) \) 30 * q + 720 * q^7 - 8670 * q^9 + 2524 * q^11 + 9700 * q^17 - 4014 * q^19 + 39376 * q^23 + 110742 * q^25 + 19976 * q^35 + 266500 * q^39 + 106788 * q^43 - 91360 * q^45 - 222756 * q^47 + 593586 * q^49 - 540936 * q^55 - 545972 * q^57 - 242640 * q^61 + 377716 * q^63 + 545964 * q^73 - 272356 * q^77 + 2189926 * q^81 - 1542652 * q^83 - 826908 * q^85 + 2729572 * q^87 - 2139912 * q^93 - 2142716 * q^95 + 293012 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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