LMFDB - Embedded newform 608.6.b.b.303.96

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,6,Mod(303,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

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

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

chi := DirichletCharacter("608.303");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	608.b (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:	$97.5133624463$
Analytic rank:	$0$
Dimension:	$96$
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			303.96
Character	$\chi$	$=$	608.303
Dual form			608.6.b.b.303.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+28.5036i q^{3} +107.944i q^{5} -87.6251i q^{7} -569.454 q^{9} +O(q^{10})$	$q+28.5036i q^{3} +107.944i q^{5} -87.6251i q^{7} -569.454 q^{9} -486.899 q^{11} -416.772 q^{13} -3076.78 q^{15} +384.824 q^{17} +(888.357 + 1298.81i) q^{19} +2497.63 q^{21} +4435.79i q^{23} -8526.81 q^{25} -9305.09i q^{27} -4617.64 q^{29} +1290.41 q^{31} -13878.4i q^{33} +9458.56 q^{35} +4994.11 q^{37} -11879.5i q^{39} -7495.86i q^{41} +3332.16 q^{43} -61468.8i q^{45} -4616.83i q^{47} +9128.84 q^{49} +10968.8i q^{51} -21437.8 q^{53} -52557.6i q^{55} +(-37020.9 + 25321.4i) q^{57} +28840.3i q^{59} -10468.1i q^{61} +49898.4i q^{63} -44987.9i q^{65} +31728.8i q^{67} -126436. q^{69} +8413.83 q^{71} +15979.4 q^{73} -243044. i q^{75} +42664.6i q^{77} +33428.3 q^{79} +126851. q^{81} +59792.9 q^{83} +41539.2i q^{85} -131619. i q^{87} +95051.2i q^{89} +36519.7i q^{91} +36781.3i q^{93} +(-140199. + 95892.4i) q^{95} -48320.2i q^{97} +277266. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q - 6168 q^{9}+O(q^{10}) $ 96 * q - 6168 * q^9	$ 96 q - 6168 q^{9} - 944 q^{11} - 3832 q^{17} - 5240 q^{19} - 62504 q^{25} - 7720 q^{35} - 45096 q^{43} - 210840 q^{49} - 36336 q^{57} - 4336 q^{73} - 20624 q^{81} - 52152 q^{83} + 752768 q^{99}+O(q^{100}) $ 96 * q - 6168 * q^9 - 944 * q^11 - 3832 * q^17 - 5240 * q^19 - 62504 * q^25 - 7720 * q^35 - 45096 * q^43 - 210840 * q^49 - 36336 * q^57 - 4336 * q^73 - 20624 * q^81 - 52152 * q^83 + 752768 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/608\mathbb{Z}\right)^\times$.

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

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
$3$			28.5036i			1.82850i	0.405145	+	0.914252i	$0.367221\pi$
$3$			28.5036i			1.82850i	−0.405145	+	0.914252i	$0.632779\pi$
$4$		0			0
$5$			107.944i			1.93095i	0.260491	+	0.965476i	$0.416116\pi$
$5$			107.944i			1.93095i	−0.260491	+	0.965476i	$0.583884\pi$
$6$		0			0
$7$		−	87.6251i		−	0.675902i	−0.941164	−	0.337951i	$-0.890266\pi$
$7$		−	87.6251i		−	0.675902i	0.941164	−	0.337951i	$-0.109734\pi$
$8$		0			0
$9$	−569.454			−2.34343
$10$		0			0
$11$	−486.899			−1.21327			−0.606635	−	0.794981i	$-0.707481\pi$
$11$	−486.899			−1.21327			−0.606635	+	0.794981i	$0.707481\pi$
$12$		0			0
$13$	−416.772			−0.683976			−0.341988	−	0.939704i	$-0.611100\pi$
$13$	−416.772			−0.683976			−0.341988	+	0.939704i	$0.611100\pi$
$14$		0			0
$15$	−3076.78			−3.53076
$16$		0			0
$17$	384.824			0.322953			0.161477	−	0.986877i	$-0.448374\pi$
$17$	384.824			0.322953			0.161477	+	0.986877i	$0.448374\pi$
$18$		0			0
$19$	888.357	+	1298.81i	0.564552	+	0.825398i
$19$	888.357	+	1298.81i	0.564552	+	0.825398i
$20$		0			0
$21$	2497.63			1.23589
$22$		0			0
$23$			4435.79i			1.74844i	0.485526	+	0.874222i	$0.338628\pi$
$23$			4435.79i			1.74844i	−0.485526	+	0.874222i	$0.661372\pi$
$24$		0			0
$25$	−8526.81			−2.72858
$26$		0			0
$27$		−	9305.09i		−	2.45647i
$28$		0			0
$29$	−4617.64			−1.01959			−0.509794	−	0.860296i	$-0.670279\pi$
$29$	−4617.64			−1.01959			−0.509794	+	0.860296i	$0.670279\pi$
$30$		0			0
$31$	1290.41			0.241170			0.120585	−	0.992703i	$-0.461523\pi$
$31$	1290.41			0.241170			0.120585	+	0.992703i	$0.461523\pi$
$32$		0			0
$33$		−	13878.4i		−	2.21847i
$34$		0			0
$35$	9458.56			1.30513
$36$		0			0
$37$	4994.11			0.599728			0.299864	−	0.953982i	$-0.403059\pi$
$37$	4994.11			0.599728			0.299864	+	0.953982i	$0.403059\pi$
$38$		0			0
$39$		−	11879.5i		−	1.25065i
$40$		0			0
$41$		−	7495.86i		−	0.696405i	−0.937419	−	0.348203i	$-0.886792\pi$
$41$		−	7495.86i		−	0.696405i	0.937419	−	0.348203i	$-0.113208\pi$
$42$		0			0
$43$	3332.16			0.274824			0.137412	−	0.990514i	$-0.456122\pi$
$43$	3332.16			0.274824			0.137412	+	0.990514i	$0.456122\pi$
$44$		0			0
$45$		−	61468.8i		−	4.52505i
$46$		0			0
$47$		−	4616.83i		−	0.304859i	−0.988314	−	0.152429i	$-0.951290\pi$
$47$		−	4616.83i		−	0.304859i	0.988314	−	0.152429i	$-0.0487097\pi$
$48$		0			0
$49$	9128.84			0.543157
$50$		0			0
$51$			10968.8i			0.590521i
$52$		0			0
$53$	−21437.8			−1.04831			−0.524155	−	0.851623i	$-0.675619\pi$
$53$	−21437.8			−1.04831			−0.524155	+	0.851623i	$0.675619\pi$
$54$		0			0
$55$		−	52557.6i		−	2.34277i
$56$		0			0
$57$	−37020.9	+	25321.4i	−1.50924	+	1.03229i
$58$		0			0
$59$			28840.3i			1.07862i	0.842106	+	0.539312i	$0.181316\pi$
$59$			28840.3i			1.07862i	−0.842106	+	0.539312i	$0.818684\pi$
$60$		0			0
$61$		−	10468.1i		−	0.360199i	−0.983648	−	0.180099i	$-0.942358\pi$
$61$		−	10468.1i		−	0.360199i	0.983648	−	0.180099i	$-0.0576419\pi$
$62$		0			0
$63$			49898.4i			1.58393i
$64$		0			0
$65$		−	44987.9i		−	1.32072i
$66$		0			0
$67$			31728.8i			0.863509i	0.901991	+	0.431755i	$0.142105\pi$
$67$			31728.8i			0.863509i	−0.901991	+	0.431755i	$0.857895\pi$
$68$		0			0
$69$	−126436.			−3.19704
$70$		0			0
$71$	8413.83			0.198083			0.0990417	−	0.995083i	$-0.468422\pi$
$71$	8413.83			0.198083			0.0990417	+	0.995083i	$0.468422\pi$
$72$		0			0
$73$	15979.4			0.350956			0.175478	−	0.984483i	$-0.443853\pi$
$73$	15979.4			0.350956			0.175478	+	0.984483i	$0.443853\pi$
$74$		0			0
$75$		−	243044.i		−	4.98922i
$76$		0			0
$77$			42664.6i			0.820051i
$78$		0			0
$79$	33428.3			0.602625			0.301313	−	0.953525i	$-0.402575\pi$
$79$	33428.3			0.602625			0.301313	+	0.953525i	$0.402575\pi$
$80$		0			0
$81$	126851.			2.14824
$82$		0			0
$83$	59792.9			0.952695			0.476348	−	0.879257i	$-0.341960\pi$
$83$	59792.9			0.952695			0.476348	+	0.879257i	$0.341960\pi$
$84$		0			0
$85$			41539.2i			0.623607i
$86$		0			0
$87$		−	131619.i		−	1.86432i
$88$		0			0
$89$			95051.2i			1.27199i	0.771695	+	0.635993i	$0.219409\pi$
$89$			95051.2i			1.27199i	−0.771695	+	0.635993i	$0.780591\pi$
$90$		0			0
$91$			36519.7i			0.462300i
$92$		0			0
$93$			36781.3i			0.440981i
$94$		0			0
$95$	−140199.	+	95892.4i	−1.59380	+	1.09012i
$96$		0			0
$97$		−	48320.2i		−	0.521434i	−0.965415	−	0.260717i	$-0.916041\pi$
$97$		−	48320.2i		−	0.521434i	0.965415	−	0.260717i	$-0.0839590\pi$
$98$		0			0
$99$	277266.			2.84321
$100$		0			0
$101$		−	107308.i		−	1.04671i	−0.852114	−	0.523356i	$-0.824680\pi$
$101$		−	107308.i		−	1.04671i	0.852114	−	0.523356i	$-0.175320\pi$
$102$		0			0
$103$	104875.			0.974044			0.487022	−	0.873390i	$-0.338083\pi$
$103$	104875.			0.974044			0.487022	+	0.873390i	$0.338083\pi$
$104$		0			0
$105$			269603.i			2.38644i
$106$		0			0
$107$			199273.i			1.68264i	0.540541	+	0.841318i	$0.318220\pi$
$107$			199273.i			1.68264i	−0.540541	+	0.841318i	$0.681780\pi$
$108$		0			0
$109$	−126980.			−1.02369			−0.511846	−	0.859077i	$-0.671038\pi$
$109$	−126980.			−1.02369			−0.511846	+	0.859077i	$0.671038\pi$
$110$		0			0
$111$			142350.i			1.09661i
$112$		0			0
$113$			114899.i			0.846486i	0.906016	+	0.423243i	$0.139108\pi$
$113$			114899.i			0.846486i	−0.906016	+	0.423243i	$0.860892\pi$
$114$		0			0
$115$	−478815.			−3.37616
$116$		0			0
$117$	237333.			1.60285
$118$		0			0
$119$		−	33720.2i		−	0.218284i
$120$		0			0
$121$	76019.7			0.472022
$122$		0			0
$123$	213659.			1.27338
$124$		0			0
$125$		−	583090.i		−	3.33780i
$126$		0			0
$127$	−80625.9			−0.443573			−0.221787	−	0.975095i	$-0.571189\pi$
$127$	−80625.9			−0.443573			−0.221787	+	0.975095i	$0.571189\pi$
$128$		0			0
$129$			94978.4i			0.502517i
$130$		0			0
$131$	−23779.3			−0.121066			−0.0605329	−	0.998166i	$-0.519280\pi$
$131$	−23779.3			−0.121066			−0.0605329	+	0.998166i	$0.519280\pi$
$132$		0			0
$133$	113809.	−	77842.4i	0.557888	−	0.381581i
$134$		0			0
$135$	1.00442e6			4.74333
$136$		0			0
$137$	44390.8			0.202065			0.101033	−	0.994883i	$-0.467785\pi$
$137$	44390.8			0.202065			0.101033	+	0.994883i	$0.467785\pi$
$138$		0			0
$139$	265338.			1.16483			0.582415	−	0.812892i	$-0.302108\pi$
$139$	265338.			1.16483			0.582415	+	0.812892i	$0.302108\pi$
$140$		0			0
$141$	131596.			0.557436
$142$		0			0
$143$	202926.			0.829847
$144$		0			0
$145$		−	498444.i		−	1.96878i
$146$		0			0
$147$			260205.i			0.993165i
$148$		0			0
$149$			189532.i			0.699385i	0.936864	+	0.349693i	$0.113714\pi$
$149$			189532.i			0.699385i	−0.936864	+	0.349693i	$0.886286\pi$
$150$		0			0
$151$	−231618.			−0.826666			−0.413333	−	0.910580i	$-0.635636\pi$
$151$	−231618.			−0.826666			−0.413333	+	0.910580i	$0.635636\pi$
$152$		0			0
$153$	−219139.			−0.756818
$154$		0			0
$155$			139291.i			0.465688i
$156$		0			0
$157$		−	14527.3i		−	0.0470365i	−0.999723	−	0.0235182i	$-0.992513\pi$
$157$		−	14527.3i		−	0.0470365i	0.999723	−	0.0235182i	$-0.00748678\pi$
$158$		0			0
$159$		−	611052.i		−	1.91684i
$160$		0			0
$161$	388687.			1.18178
$162$		0			0
$163$	−365221.			−1.07668			−0.538340	−	0.842728i	$-0.680948\pi$
$163$	−365221.			−1.07668			−0.538340	+	0.842728i	$0.680948\pi$
$164$		0			0
$165$	1.49808e6			4.28376
$166$		0			0
$167$	−64713.3			−0.179557			−0.0897785	−	0.995962i	$-0.528616\pi$
$167$	−64713.3			−0.179557			−0.0897785	+	0.995962i	$0.528616\pi$
$168$		0			0
$169$	−197594.			−0.532177
$170$		0			0
$171$	−505878.	−	739615.i	−1.32299	−	1.93426i
$172$		0			0
$173$	240415.			0.610725			0.305362	−	0.952236i	$-0.401222\pi$
$173$	240415.			0.610725			0.305362	+	0.952236i	$0.401222\pi$
$174$		0			0
$175$			747162.i			1.84425i
$176$		0			0
$177$	−822052.			−1.97227
$178$		0			0
$179$			274685.i			0.640770i	0.947287	+	0.320385i	$0.103812\pi$
$179$			274685.i			0.640770i	−0.947287	+	0.320385i	$0.896188\pi$
$180$		0			0
$181$	−553197.			−1.25512			−0.627558	−	0.778570i	$-0.715945\pi$
$181$	−553197.			−1.25512			−0.627558	+	0.778570i	$0.715945\pi$
$182$		0			0
$183$	298377.			0.658625
$184$		0			0
$185$			539082.i			1.15805i
$186$		0			0
$187$	−187370.			−0.391829
$188$		0			0
$189$	−815360.			−1.66033
$190$		0			0
$191$		−	566777.i		−	1.12416i	−0.827082	−	0.562081i	$-0.810001\pi$
$191$		−	566777.i		−	1.12416i	0.827082	−	0.562081i	$-0.189999\pi$
$192$		0			0
$193$			682223.i			1.31836i	0.751986	+	0.659179i	$0.229096\pi$
$193$			682223.i			1.31836i	−0.751986	+	0.659179i	$0.770904\pi$
$194$		0			0
$195$	1.28232e6			2.41495
$196$		0			0
$197$			202201.i			0.371208i	0.982625	+	0.185604i	$0.0594242\pi$
$197$			202201.i			0.371208i	−0.982625	+	0.185604i	$0.940576\pi$
$198$		0			0
$199$			56553.1i			0.101233i	0.998718	+	0.0506167i	$0.0161187\pi$
$199$			56553.1i			0.101233i	−0.998718	+	0.0506167i	$0.983881\pi$
$200$		0			0
$201$	−904385.			−1.57893
$202$		0			0
$203$			404621.i			0.689142i
$204$		0			0
$205$	809130.			1.34473
$206$		0			0
$207$		−	2.52598e6i		−	4.09736i
$208$		0			0
$209$	−432540.	−	632392.i	−0.684953	−	1.00143i
$210$		0			0
$211$			218513.i			0.337887i	0.985626	+	0.168943i	$0.0540355\pi$
$211$			218513.i			0.337887i	−0.985626	+	0.168943i	$0.945965\pi$
$212$		0			0
$213$			239824.i			0.362196i
$214$		0			0
$215$			359685.i			0.530672i
$216$		0			0
$217$		−	113072.i		−	0.163007i
$218$		0			0
$219$			455469.i			0.641725i
$220$		0			0
$221$	−160384.			−0.220892
$222$		0			0
$223$	645531.			0.869270			0.434635	−	0.900607i	$-0.356877\pi$
$223$	645531.			0.869270			0.434635	+	0.900607i	$0.356877\pi$
$224$		0			0
$225$	4.85562e6			6.39423
$226$		0			0
$227$			484459.i			0.624012i	0.950080	+	0.312006i	$0.101001\pi$
$227$			484459.i			0.624012i	−0.950080	+	0.312006i	$0.898999\pi$
$228$		0			0
$229$		−	1.03118e6i		−	1.29940i	−0.760190	−	0.649701i	$-0.774894\pi$
$229$		−	1.03118e6i		−	1.29940i	0.760190	−	0.649701i	$-0.225106\pi$
$230$		0			0
$231$	−1.21609e6			−1.49947
$232$		0			0
$233$	1.18151e6			1.42576			0.712881	−	0.701285i	$-0.247390\pi$
$233$	1.18151e6			1.42576			0.712881	+	0.701285i	$0.247390\pi$
$234$		0			0
$235$	498357.			0.588668
$236$		0			0
$237$			952827.i			1.10190i
$238$		0			0
$239$		−	146530.i		−	0.165933i	−0.996552	−	0.0829664i	$-0.973561\pi$
$239$		−	146530.i		−	0.165933i	0.996552	−	0.0829664i	$-0.0264394\pi$
$240$		0			0
$241$		−	1.14545e6i		−	1.27038i	−0.772355	−	0.635191i	$-0.780922\pi$
$241$		−	1.14545e6i		−	1.27038i	0.772355	−	0.635191i	$-0.219078\pi$
$242$		0			0
$243$			1.35457e6i			1.47159i
$244$		0			0
$245$			985399.i			1.04881i
$246$		0			0
$247$	−370243.	−	541310.i	−0.386140	−	0.564552i
$248$		0			0
$249$			1.70431e6i			1.74201i
$250$		0			0
$251$	1.93804e6			1.94168			0.970841	−	0.239725i	$-0.0770573\pi$
$251$	1.93804e6			1.94168			0.970841	+	0.239725i	$0.0770573\pi$
$252$		0			0
$253$		−	2.15978e6i		−	2.12133i
$254$		0			0
$255$	−1.18402e6			−1.14027
$256$		0			0
$257$		−	1.43835e6i		−	1.35841i	−0.733949	−	0.679204i	$-0.762325\pi$
$257$		−	1.43835e6i		−	1.35841i	0.733949	−	0.679204i	$-0.237675\pi$
$258$		0			0
$259$		−	437610.i		−	0.405357i
$260$		0			0
$261$	2.62953e6			2.38934
$262$		0			0
$263$			935536.i			0.834010i	0.908904	+	0.417005i	$0.136920\pi$
$263$			935536.i			0.834010i	−0.908904	+	0.417005i	$0.863080\pi$
$264$		0			0
$265$		−	2.31407e6i		−	2.02424i
$266$		0			0
$267$	−2.70930e6			−2.32583
$268$		0			0
$269$	−1.71576e6			−1.44569			−0.722845	−	0.691010i	$-0.757166\pi$
$269$	−1.71576e6			−1.44569			−0.722845	+	0.691010i	$0.757166\pi$
$270$		0			0
$271$		−	1.11326e6i		−	0.920817i	−0.887707	−	0.460409i	$-0.847703\pi$
$271$		−	1.11326e6i		−	0.920817i	0.887707	−	0.460409i	$-0.152297\pi$
$272$		0			0
$273$	−1.04094e6			−0.845318
$274$		0			0
$275$	4.15169e6			3.31050
$276$		0			0
$277$		−	576274.i		−	0.451263i	−0.974213	−	0.225631i	$-0.927556\pi$
$277$		−	576274.i		−	0.451263i	0.974213	−	0.225631i	$-0.0724445\pi$
$278$		0			0
$279$	−734829.			−0.565166
$280$		0			0
$281$			1.16738e6i			0.881957i	0.897518	+	0.440978i	$0.145368\pi$
$281$			1.16738e6i			0.881957i	−0.897518	+	0.440978i	$0.854632\pi$
$282$		0			0
$283$	−1.44951e6			−1.07586			−0.537931	−	0.842989i	$-0.680794\pi$
$283$	−1.44951e6			−1.07586			−0.537931	+	0.842989i	$0.680794\pi$
$284$		0			0
$285$	−2.73328e6	−	3.99616e6i	−1.99329	−	2.91428i
$286$		0			0
$287$	−656826.			−0.470701
$288$		0			0
$289$	−1.27177e6			−0.895701
$290$		0			0
$291$	1.37730e6			0.953445
$292$		0			0
$293$	448214.			0.305012			0.152506	−	0.988303i	$-0.451266\pi$
$293$	448214.			0.305012			0.152506	+	0.988303i	$0.451266\pi$
$294$		0			0
$295$	−3.11313e6			−2.08277
$296$		0			0
$297$			4.53064e6i			2.98036i
$298$		0			0
$299$		−	1.84872e6i		−	1.19589i
$300$		0			0
$301$		−	291981.i		−	0.185754i
$302$		0			0
$303$	3.05865e6			1.91392
$304$		0			0
$305$	1.12996e6			0.695527
$306$		0			0
$307$		−	2.61571e6i		−	1.58396i	−0.610550	−	0.791978i	$-0.709052\pi$
$307$		−	2.61571e6i		−	1.58396i	0.610550	−	0.791978i	$-0.290948\pi$
$308$		0			0
$309$			2.98931e6i			1.78104i
$310$		0			0
$311$			2.86448e6i			1.67936i	0.543078	+	0.839682i	$0.317259\pi$
$311$			2.86448e6i			1.67936i	−0.543078	+	0.839682i	$0.682741\pi$
$312$		0			0
$313$	1.94436e6			1.12180			0.560902	−	0.827882i	$-0.310455\pi$
$313$	1.94436e6			1.12180			0.560902	+	0.827882i	$0.310455\pi$
$314$		0			0
$315$	−5.38621e6			−3.05849
$316$		0			0
$317$	98182.3			0.0548763			0.0274382	−	0.999624i	$-0.491265\pi$
$317$	98182.3			0.0548763			0.0274382	+	0.999624i	$0.491265\pi$
$318$		0			0
$319$	2.24832e6			1.23704
$320$		0			0
$321$	−5.68001e6			−3.07671
$322$		0			0
$323$	341861.	+	499815.i	0.182324	+	0.266565i
$324$		0			0
$325$	3.55374e6			1.86628
$326$		0			0
$327$		−	3.61939e6i		−	1.87183i
$328$		0			0
$329$	−404550.			−0.206055
$330$		0			0
$331$		−	2.71655e6i		−	1.36285i	−0.731889	−	0.681424i	$-0.761361\pi$
$331$		−	2.71655e6i		−	1.36285i	0.731889	−	0.681424i	$-0.238639\pi$
$332$		0			0
$333$	−2.84392e6			−1.40542
$334$		0			0
$335$	−3.42492e6			−1.66740
$336$		0			0
$337$			1.78296e6i			0.855198i	0.903968	+	0.427599i	$0.140641\pi$
$337$			1.78296e6i			0.855198i	−0.903968	+	0.427599i	$0.859359\pi$
$338$		0			0
$339$	−3.27503e6			−1.54780
$340$		0			0
$341$	−628300.			−0.292604
$342$		0			0
$343$		−	2.27263e6i		−	1.04302i
$344$		0			0
$345$		−	1.36479e7i		−	6.17333i
$346$		0			0
$347$	137970.			0.0615124			0.0307562	−	0.999527i	$-0.490208\pi$
$347$	137970.			0.0615124			0.0307562	+	0.999527i	$0.490208\pi$
$348$		0			0
$349$			3.14082e6i			1.38032i	0.723658	+	0.690159i	$0.242459\pi$
$349$			3.14082e6i			1.38032i	−0.723658	+	0.690159i	$0.757541\pi$
$350$		0			0
$351$			3.87811e6i			1.68016i
$352$		0			0
$353$	2.61200e6			1.11567			0.557836	−	0.829951i	$-0.311632\pi$
$353$	2.61200e6			1.11567			0.557836	+	0.829951i	$0.311632\pi$
$354$		0			0
$355$			908219.i			0.382490i
$356$		0			0
$357$	961146.			0.399134
$358$		0			0
$359$			1.36617e6i			0.559460i	0.960079	+	0.279730i	$0.0902450\pi$
$359$			1.36617e6i			0.559460i	−0.960079	+	0.279730i	$0.909755\pi$
$360$		0			0
$361$	−897741.	+	2.30762e6i	−0.362563	+	0.931959i
$362$		0			0
$363$			2.16683e6i			0.863095i
$364$		0			0
$365$			1.72487e6i			0.677680i
$366$		0			0
$367$		−	1.39167e6i		−	0.539349i	−0.962952	−	0.269674i	$-0.913084\pi$
$367$		−	1.39167e6i		−	0.539349i	0.962952	−	0.269674i	$-0.0869161\pi$
$368$		0			0
$369$			4.26855e6i			1.63198i
$370$		0			0
$371$			1.87849e6i			0.708554i
$372$		0			0
$373$	−1.45159e6			−0.540221			−0.270110	−	0.962829i	$-0.587060\pi$
$373$	−1.45159e6			−0.540221			−0.270110	+	0.962829i	$0.587060\pi$
$374$		0			0
$375$	1.66202e7			6.10319
$376$		0			0
$377$	1.92451e6			0.697374
$378$		0			0
$379$		−	4.49243e6i		−	1.60651i	−0.595635	−	0.803255i	$-0.703099\pi$
$379$		−	4.49243e6i		−	1.60651i	0.595635	−	0.803255i	$-0.296901\pi$
$380$		0			0
$381$		−	2.29813e6i		−	0.811076i
$382$		0			0
$383$	−3.85576e6			−1.34311			−0.671557	−	0.740953i	$-0.734374\pi$
$383$	−3.85576e6			−1.34311			−0.671557	+	0.740953i	$0.734374\pi$
$384$		0			0
$385$	−4.60537e6			−1.58348
$386$		0			0
$387$	−1.89751e6			−0.644031
$388$		0			0
$389$		−	5.52442e6i		−	1.85103i	−0.378717	−	0.925513i	$-0.623635\pi$
$389$		−	5.52442e6i		−	1.85103i	0.378717	−	0.925513i	$-0.376365\pi$
$390$		0			0
$391$			1.70700e6i			0.564665i
$392$		0			0
$393$		−	677796.i		−	0.221370i
$394$		0			0
$395$			3.60837e6i			1.16364i
$396$		0			0
$397$		−	2.47956e6i		−	0.789584i	−0.918770	−	0.394792i	$-0.870817\pi$
$397$		−	2.47956e6i		−	0.789584i	0.918770	−	0.394792i	$-0.129183\pi$
$398$		0			0
$399$	2.21879e6	+	3.24396e6i	0.697723	+	1.02010i
$400$		0			0
$401$			2.59539e6i			0.806011i	0.915197	+	0.403006i	$0.132035\pi$
$401$			2.59539e6i			0.806011i	−0.915197	+	0.403006i	$0.867965\pi$
$402$		0			0
$403$	−537808.			−0.164955
$404$		0			0
$405$			1.36928e7i			4.14814i
$406$		0			0
$407$	−2.43163e6			−0.727631
$408$		0			0
$409$			3.04967e6i			0.901457i	0.892661	+	0.450729i	$0.148836\pi$
$409$			3.04967e6i			0.901457i	−0.892661	+	0.450729i	$0.851164\pi$
$410$		0			0
$411$			1.26530e6i			0.369478i
$412$		0			0
$413$	2.52714e6			0.729044
$414$		0			0
$415$			6.45425e6i			1.83961i
$416$		0			0
$417$			7.56308e6i			2.12990i
$418$		0			0
$419$	1.67462e6			0.465994			0.232997	−	0.972477i	$-0.425147\pi$
$419$	1.67462e6			0.465994			0.232997	+	0.972477i	$0.425147\pi$
$420$		0			0
$421$	−648849.			−0.178418			−0.0892090	−	0.996013i	$-0.528434\pi$
$421$	−648849.			−0.178418			−0.0892090	+	0.996013i	$0.528434\pi$
$422$		0			0
$423$			2.62907e6i			0.714416i
$424$		0			0
$425$	−3.28132e6			−0.881203
$426$		0			0
$427$	−917266.			−0.243459
$428$		0			0
$429$			5.78412e6i			1.51738i
$430$		0			0
$431$	769722.			0.199591			0.0997955	−	0.995008i	$-0.468181\pi$
$431$	769722.			0.199591			0.0997955	+	0.995008i	$0.468181\pi$
$432$		0			0
$433$			3.02139e6i			0.774438i	0.921988	+	0.387219i	$0.126564\pi$
$433$			3.02139e6i			0.774438i	−0.921988	+	0.387219i	$0.873436\pi$
$434$		0			0
$435$	1.42074e7			3.59992
$436$		0			0
$437$	−5.76128e6	+	3.94057e6i	−1.44316	+	0.987087i
$438$		0			0
$439$	−1.29727e6			−0.321269			−0.160634	−	0.987014i	$-0.551354\pi$
$439$	−1.29727e6			−0.321269			−0.160634	+	0.987014i	$0.551354\pi$
$440$		0			0
$441$	−5.19845e6			−1.27285
$442$		0			0
$443$	−1.21945e6			−0.295226			−0.147613	−	0.989045i	$-0.547159\pi$
$443$	−1.21945e6			−0.295226			−0.147613	+	0.989045i	$0.547159\pi$
$444$		0			0
$445$	−1.02602e7			−2.45615
$446$		0			0
$447$	−5.40233e6			−1.27883
$448$		0			0
$449$		−	5.36707e6i		−	1.25638i	−0.778059	−	0.628191i	$-0.783796\pi$
$449$		−	5.36707e6i		−	1.25638i	0.778059	−	0.628191i	$-0.216204\pi$
$450$		0			0
$451$			3.64973e6i			0.844927i
$452$		0			0
$453$		−	6.60195e6i		−	1.51156i
$454$		0			0
$455$	−3.94207e6			−0.892680
$456$		0			0
$457$	3.00983e6			0.674142			0.337071	−	0.941479i	$-0.390564\pi$
$457$	3.00983e6			0.674142			0.337071	+	0.941479i	$0.390564\pi$
$458$		0			0
$459$		−	3.58082e6i		−	0.793324i
$460$		0			0
$461$			1.70263e6i			0.373137i	0.982442	+	0.186568i	$0.0597366\pi$
$461$			1.70263e6i			0.373137i	−0.982442	+	0.186568i	$0.940263\pi$
$462$		0			0
$463$		−	5.09539e6i		−	1.10465i	−0.833629	−	0.552325i	$-0.813741\pi$
$463$		−	5.09539e6i		−	1.10465i	0.833629	−	0.552325i	$-0.186259\pi$
$464$		0			0
$465$	−3.97030e6			−0.851513
$466$		0			0
$467$	8.38901e6			1.77999			0.889997	−	0.455967i	$-0.150707\pi$
$467$	8.38901e6			1.77999			0.889997	+	0.455967i	$0.150707\pi$
$468$		0			0
$469$	2.78024e6			0.583647
$470$		0			0
$471$	414079.			0.0860064
$472$		0			0
$473$	−1.62243e6			−0.333435
$474$		0			0
$475$	−7.57485e6	−	1.10747e7i	−1.54042	−	2.25216i
$476$		0			0
$477$	1.22078e7			2.45664
$478$		0			0
$479$			6.30836e6i			1.25626i	0.778110	+	0.628128i	$0.216178\pi$
$479$			6.30836e6i			1.25626i	−0.778110	+	0.628128i	$0.783822\pi$
$480$		0			0
$481$	−2.08141e6			−0.410199
$482$		0			0
$483$			1.10790e7i			2.16088i
$484$		0			0
$485$	5.21586e6			1.00687
$486$		0			0
$487$	852875.			0.162953			0.0814766	−	0.996675i	$-0.474036\pi$
$487$	852875.			0.162953			0.0814766	+	0.996675i	$0.474036\pi$
$488$		0			0
$489$		−	1.04101e7i		−	1.96871i
$490$		0			0
$491$	−1.03276e7			−1.93328			−0.966639	−	0.256141i	$-0.917549\pi$
$491$	−1.03276e7			−1.93328			−0.966639	+	0.256141i	$0.917549\pi$
$492$		0			0
$493$	−1.77698e6			−0.329279
$494$		0			0
$495$			2.99291e7i			5.49011i
$496$		0			0
$497$		−	737263.i		−	0.133885i
$498$		0			0
$499$	1.67531e6			0.301192			0.150596	−	0.988595i	$-0.451881\pi$
$499$	1.67531e6			0.301192			0.150596	+	0.988595i	$0.451881\pi$
$500$		0			0
$501$		−	1.84456e6i		−	0.328321i
$502$		0			0
$503$		−	7.29065e6i		−	1.28483i	−0.766356	−	0.642416i	$-0.777932\pi$
$503$		−	7.29065e6i		−	1.28483i	0.766356	−	0.642416i	$-0.222068\pi$
$504$		0			0
$505$	1.15832e7			2.02115
$506$		0			0
$507$		−	5.63213e6i		−	0.973089i
$508$		0			0
$509$	−4.39175e6			−0.751352			−0.375676	−	0.926751i	$-0.622589\pi$
$509$	−4.39175e6			−0.751352			−0.375676	+	0.926751i	$0.622589\pi$
$510$		0			0
$511$		−	1.40019e6i		−	0.237212i
$512$		0			0
$513$	1.20856e7	−	8.26625e6i	2.02756	−	1.38680i
$514$		0			0
$515$			1.13206e7i			1.88083i
$516$		0			0
$517$			2.24793e6i			0.369876i
$518$		0			0
$519$			6.85267e6i			1.11671i
$520$		0			0
$521$			5.36102e6i			0.865272i	0.901569	+	0.432636i	$0.142417\pi$
$521$			5.36102e6i			0.865272i	−0.901569	+	0.432636i	$0.857583\pi$
$522$		0			0
$523$			2.33893e6i			0.373906i	0.982369	+	0.186953i	$0.0598612\pi$
$523$			2.33893e6i			0.373906i	−0.982369	+	0.186953i	$0.940139\pi$
$524$		0			0
$525$	−2.12968e7			−3.37222
$526$		0			0
$527$	496580.			0.0778867
$528$		0			0
$529$	−1.32399e7			−2.05706
$530$		0			0
$531$		−	1.64232e7i		−	2.52768i
$532$		0			0
$533$			3.12407e6i			0.476324i
$534$		0			0
$535$	−2.15103e7			−3.24909
$536$		0			0
$537$	−7.82950e6			−1.17165
$538$		0			0
$539$	−4.44482e6			−0.658996
$540$		0			0
$541$		−	7.80271e6i		−	1.14618i	−0.819493	−	0.573089i	$-0.805745\pi$
$541$		−	7.80271e6i		−	1.14618i	0.819493	−	0.573089i	$-0.194255\pi$
$542$		0			0
$543$		−	1.57681e7i		−	2.29499i
$544$		0			0
$545$		−	1.37067e7i		−	1.97670i
$546$		0			0
$547$			2.08885e6i			0.298496i	0.988800	+	0.149248i	$0.0476853\pi$
$547$			2.08885e6i			0.298496i	−0.988800	+	0.149248i	$0.952315\pi$
$548$		0			0
$549$			5.96108e6i			0.844100i
$550$		0			0
$551$	−4.10211e6	−	5.99746e6i	−0.575611	−	0.841566i
$552$		0			0
$553$		−	2.92916e6i		−	0.407315i
$554$		0			0
$555$	−1.53658e7			−2.11749
$556$		0			0
$557$		−	2.39511e6i		−	0.327105i	−0.986535	−	0.163552i	$-0.947705\pi$
$557$		−	2.39511e6i		−	0.327105i	0.986535	−	0.163552i	$-0.0522953\pi$
$558$		0			0
$559$	−1.38875e6			−0.187973
$560$		0			0
$561$		−	5.34072e6i		−	0.716461i
$562$		0			0
$563$		−	3.08044e6i		−	0.409583i	−0.978806	−	0.204792i	$-0.934348\pi$
$563$		−	3.08044e6i		−	0.409583i	0.978806	−	0.204792i	$-0.0656517\pi$
$564$		0			0
$565$	−1.24026e7			−1.63452
$566$		0			0
$567$		−	1.11153e7i		−	1.45200i
$568$		0			0
$569$		−	1.47624e6i		−	0.191151i	−0.995422	−	0.0955756i	$-0.969531\pi$
$569$		−	1.47624e6i		−	0.191151i	0.995422	−	0.0955756i	$-0.0304692\pi$
$570$		0			0
$571$	3.25915e6			0.418326			0.209163	−	0.977881i	$-0.432926\pi$
$571$	3.25915e6			0.418326			0.209163	+	0.977881i	$0.432926\pi$
$572$		0			0
$573$	1.61552e7			2.05553
$574$		0			0
$575$		−	3.78232e7i		−	4.77077i
$576$		0			0
$577$	5.09234e6			0.636763			0.318382	−	0.947963i	$-0.396861\pi$
$577$	5.09234e6			0.636763			0.318382	+	0.947963i	$0.396861\pi$
$578$		0			0
$579$	−1.94458e7			−2.41062
$580$		0			0
$581$		−	5.23936e6i		−	0.643928i
$582$		0			0
$583$	1.04380e7			1.27188
$584$		0			0
$585$			2.56185e7i			3.09503i
$586$		0			0
$587$	5.57945e6			0.668338			0.334169	−	0.942513i	$-0.391544\pi$
$587$	5.57945e6			0.668338			0.334169	+	0.942513i	$0.391544\pi$
$588$		0			0
$589$	1.14635e6	+	1.67600e6i	0.136153	+	0.199061i
$590$		0			0
$591$	−5.76345e6			−0.678756
$592$		0			0
$593$	9.80184e6			1.14464			0.572322	−	0.820029i	$-0.306043\pi$
$593$	9.80184e6			1.14464			0.572322	+	0.820029i	$0.306043\pi$
$594$		0			0
$595$	3.63988e6			0.421497
$596$		0			0
$597$	−1.61196e6			−0.185106
$598$		0			0
$599$	−1.68384e7			−1.91749			−0.958746	−	0.284263i	$-0.908251\pi$
$599$	−1.68384e7			−1.91749			−0.958746	+	0.284263i	$0.908251\pi$
$600$		0			0
$601$			6.92827e6i			0.782417i	0.920302	+	0.391209i	$0.127943\pi$
$601$			6.92827e6i			0.782417i	−0.920302	+	0.391209i	$0.872057\pi$
$602$		0			0
$603$		−	1.80681e7i		−	2.02357i
$604$		0			0
$605$			8.20583e6i			0.911453i
$606$		0			0
$607$	9.90735e6			1.09140			0.545702	−	0.837979i	$-0.316263\pi$
$607$	9.90735e6			1.09140			0.545702	+	0.837979i	$0.316263\pi$
$608$		0			0
$609$	−1.15331e7			−1.26010
$610$		0			0
$611$			1.92417e6i			0.208516i
$612$		0			0
$613$		−	9.14242e6i		−	0.982676i	−0.870969	−	0.491338i	$-0.836508\pi$
$613$		−	9.14242e6i		−	0.982676i	0.870969	−	0.491338i	$-0.163492\pi$
$614$		0			0
$615$			2.30631e7i			2.45884i
$616$		0			0
$617$	−4.19657e6			−0.443795			−0.221897	−	0.975070i	$-0.571225\pi$
$617$	−4.19657e6			−0.443795			−0.221897	+	0.975070i	$0.571225\pi$
$618$		0			0
$619$	2.05006e6			0.215050			0.107525	−	0.994202i	$-0.465707\pi$
$619$	2.05006e6			0.215050			0.107525	+	0.994202i	$0.465707\pi$
$620$		0			0
$621$	4.12755e7			4.29500
$622$		0			0
$623$	8.32887e6			0.859737
$624$		0			0
$625$	3.62946e7			3.71656
$626$		0			0
$627$	1.80254e7	−	1.23289e7i	1.83112	−	1.25244i
$628$		0			0
$629$	1.92185e6			0.193684
$630$		0			0
$631$		−	1.07445e6i		−	0.107427i	−0.998556	−	0.0537134i	$-0.982894\pi$
$631$		−	1.07445e6i		−	0.107427i	0.998556	−	0.0537134i	$-0.0171058\pi$
$632$		0			0
$633$	−6.22840e6			−0.617827
$634$		0			0
$635$		−	8.70305e6i		−	0.856519i
$636$		0			0
$637$	−3.80465e6			−0.371506
$638$		0			0
$639$	−4.79129e6			−0.464194
$640$		0			0
$641$		−	1.12011e7i		−	1.07675i	−0.842705	−	0.538376i	$-0.819038\pi$
$641$		−	1.12011e7i		−	1.07675i	0.842705	−	0.538376i	$-0.180962\pi$
$642$		0			0
$643$	−1.00653e6			−0.0960061			−0.0480030	−	0.998847i	$-0.515286\pi$
$643$	−1.00653e6			−0.0960061			−0.0480030	+	0.998847i	$0.515286\pi$
$644$		0			0
$645$	−1.02523e7			−0.970336
$646$		0			0
$647$		−	1.43017e7i		−	1.34316i	−0.740931	−	0.671581i	$-0.765616\pi$
$647$		−	1.43017e7i		−	1.34316i	0.740931	−	0.671581i	$-0.234384\pi$
$648$		0			0
$649$		−	1.40423e7i		−	1.30866i
$650$		0			0
$651$	3.22297e6			0.298060
$652$		0			0
$653$			1.06249e7i			0.975083i	0.873100	+	0.487541i	$0.162106\pi$
$653$			1.06249e7i			0.975083i	−0.873100	+	0.487541i	$0.837894\pi$
$654$		0			0
$655$		−	2.56683e6i		−	0.233772i
$656$		0			0
$657$	−9.09951e6			−0.822441
$658$		0			0
$659$		−	435445.i		−	0.0390589i	−0.999809	−	0.0195295i	$-0.993783\pi$
$659$		−	435445.i		−	0.0390589i	0.999809	−	0.0195295i	$-0.00621681\pi$
$660$		0			0
$661$	−2.13248e7			−1.89837			−0.949186	−	0.314715i	$-0.898091\pi$
$661$	−2.13248e7			−1.89837			−0.949186	+	0.314715i	$0.898091\pi$
$662$		0			0
$663$		−	4.57151e6i		−	0.403902i
$664$		0			0
$665$	8.40259e6	+	1.22849e7i	0.736816	+	1.07725i
$666$		0			0
$667$		−	2.04829e7i		−	1.78269i
$668$		0			0
$669$			1.83999e7i			1.58947i
$670$		0			0
$671$			5.09689e6i			0.437018i
$672$		0			0
$673$		−	1.39908e7i		−	1.19071i	−0.803464	−	0.595353i	$-0.797012\pi$
$673$		−	1.39908e7i		−	1.19071i	0.803464	−	0.595353i	$-0.202988\pi$
$674$		0			0
$675$			7.93427e7i			6.70267i
$676$		0			0
$677$	−1.68979e7			−1.41697			−0.708485	−	0.705726i	$-0.750621\pi$
$677$	−1.68979e7			−1.41697			−0.708485	+	0.705726i	$0.750621\pi$
$678$		0			0
$679$	−4.23406e6			−0.352438
$680$		0			0
$681$	−1.38088e7			−1.14101
$682$		0			0
$683$		−	5.55844e6i		−	0.455933i	−0.973669	−	0.227967i	$-0.926792\pi$
$683$		−	5.55844e6i		−	0.455933i	0.973669	−	0.227967i	$-0.0732077\pi$
$684$		0			0
$685$			4.79171e6i			0.390179i
$686$		0			0
$687$	2.93922e7			2.37596
$688$		0			0
$689$	8.93466e6			0.717018
$690$		0			0
$691$	−1.02770e6			−0.0818789			−0.0409394	−	0.999162i	$-0.513035\pi$
$691$	−1.02770e6			−0.0818789			−0.0409394	+	0.999162i	$0.513035\pi$
$692$		0			0
$693$		−	2.42955e7i		−	1.92173i
$694$		0			0
$695$			2.86415e7i			2.24923i
$696$		0			0
$697$		−	2.88459e6i		−	0.224906i
$698$		0			0
$699$			3.36772e7i			2.60701i
$700$		0			0
$701$		−	1.36429e7i		−	1.04860i	−0.851532	−	0.524302i	$-0.824326\pi$
$701$		−	1.36429e7i		−	1.04860i	0.851532	−	0.524302i	$-0.175674\pi$
$702$		0			0
$703$	4.43656e6	+	6.48643e6i	0.338577	+	0.495014i
$704$		0			0
$705$			1.42049e7i			1.07638i
$706$		0			0
$707$	−9.40284e6			−0.707474
$708$		0			0
$709$		−	1.76666e6i		−	0.131989i	−0.997820	−	0.0659946i	$-0.978978\pi$
$709$		−	1.76666e6i		−	0.131989i	0.997820	−	0.0659946i	$-0.0210220\pi$
$710$		0			0
$711$	−1.90359e7			−1.41221
$712$		0			0
$713$			5.72400e6i			0.421673i
$714$		0			0
$715$			2.19046e7i			1.60239i
$716$		0			0
$717$	4.17663e6			0.303409
$718$		0			0
$719$		−	1.18432e7i		−	0.854373i	−0.904164	−	0.427186i	$-0.859505\pi$
$719$		−	1.18432e7i		−	0.854373i	0.904164	−	0.427186i	$-0.140495\pi$
$720$		0			0
$721$		−	9.18967e6i		−	0.658358i
$722$		0			0
$723$	3.26495e7			2.32290
$724$		0			0
$725$	3.93737e7			2.78203
$726$		0			0
$727$			2.23911e7i			1.57123i	0.618718	+	0.785613i	$0.287652\pi$
$727$			2.23911e7i			1.57123i	−0.618718	+	0.785613i	$0.712348\pi$
$728$		0			0
$729$	−7.78534e6			−0.542574
$730$		0			0
$731$	1.28229e6			0.0887552
$732$		0			0
$733$			2.15794e7i			1.48347i	0.670690	+	0.741737i	$0.265998\pi$
$733$			2.15794e7i			1.48347i	−0.670690	+	0.741737i	$0.734002\pi$
$734$		0			0
$735$	−2.80874e7			−1.91776
$736$		0			0
$737$		−	1.54487e7i		−	1.04767i
$738$		0			0
$739$	−9.51318e6			−0.640788			−0.320394	−	0.947284i	$-0.603815\pi$
$739$	−9.51318e6			−0.640788			−0.320394	+	0.947284i	$0.603815\pi$
$740$		0			0
$741$	1.54293e7	−	1.05532e7i	1.03229	−	0.706058i
$742$		0			0
$743$	−4.02857e6			−0.267719			−0.133859	−	0.991000i	$-0.542737\pi$
$743$	−4.02857e6			−0.267719			−0.133859	+	0.991000i	$0.542737\pi$
$744$		0			0
$745$	−2.04587e7			−1.35048
$746$		0			0
$747$	−3.40493e7			−2.23258
$748$		0			0
$749$	1.74614e7			1.13730
$750$		0			0
$751$	1.54309e7			0.998370			0.499185	−	0.866495i	$-0.333633\pi$
$751$	1.54309e7			0.998370			0.499185	+	0.866495i	$0.333633\pi$
$752$		0			0
$753$			5.52410e7i			3.55037i
$754$		0			0
$755$		−	2.50017e7i		−	1.59625i
$756$		0			0
$757$		−	5.44099e6i		−	0.345095i	−0.985001	−	0.172547i	$-0.944800\pi$
$757$		−	5.44099e6i		−	0.345095i	0.985001	−	0.172547i	$-0.0551998\pi$
$758$		0			0
$759$	6.15616e7			3.87887
$760$		0			0
$761$	3.57369e6			0.223694			0.111847	−	0.993725i	$-0.464323\pi$
$761$	3.57369e6			0.223694			0.111847	+	0.993725i	$0.464323\pi$
$762$		0			0
$763$			1.11266e7i			0.691915i
$764$		0			0
$765$		−	2.36547e7i		−	1.46138i
$766$		0			0
$767$		−	1.20199e7i		−	0.737753i
$768$		0			0
$769$	1.22202e7			0.745184			0.372592	−	0.927995i	$-0.378469\pi$
$769$	1.22202e7			0.745184			0.372592	+	0.927995i	$0.378469\pi$
$770$		0			0
$771$	4.09980e7			2.48386
$772$		0			0
$773$	1.90679e7			1.14777			0.573885	−	0.818936i	$-0.305436\pi$
$773$	1.90679e7			1.14777			0.573885	+	0.818936i	$0.305436\pi$
$774$		0			0
$775$	−1.10031e7			−0.658052
$776$		0			0
$777$	1.24734e7			0.741197
$778$		0			0
$779$	9.73574e6	−	6.65901e6i	0.574811	−	0.393157i
$780$		0			0
$781$	−4.09669e6			−0.240328
$782$		0			0
$783$			4.29676e7i			2.50459i
$784$		0			0
$785$	1.56812e6			0.0908252
$786$		0			0
$787$		−	3.45195e6i		−	0.198668i	−0.995054	−	0.0993339i	$-0.968329\pi$
$787$		−	3.45195e6i		−	0.198668i	0.995054	−	0.0993339i	$-0.0316712\pi$
$788$		0			0
$789$	−2.66661e7			−1.52499
$790$		0			0
$791$	1.00680e7			0.572141
$792$		0			0
$793$			4.36280e6i			0.246367i
$794$		0			0
$795$	6.59592e7			3.70133
$796$		0			0
$797$	1.76442e6			0.0983909			0.0491955	−	0.998789i	$-0.484334\pi$
$797$	1.76442e6			0.0983909			0.0491955	+	0.998789i	$0.484334\pi$
$798$		0			0
$799$		−	1.77666e6i		−	0.0984551i
$800$		0			0
$801$		−	5.41272e7i		−	2.98081i
$802$		0			0
$803$	−7.78034e6			−0.425804
$804$		0			0
$805$			4.19562e7i			2.28195i
$806$		0			0
$807$		−	4.89052e7i		−	2.64345i
$808$		0			0
$809$	−2.93512e7			−1.57672			−0.788360	−	0.615214i	$-0.789069\pi$
$809$	−2.93512e7			−1.57672			−0.788360	+	0.615214i	$0.789069\pi$
$810$		0			0
$811$		−	657773.i		−	0.0351175i	−0.999846	−	0.0175588i	$-0.994411\pi$
$811$		−	657773.i		−	0.0351175i	0.999846	−	0.0175588i	$-0.00558942\pi$
$812$		0			0
$813$	3.17319e7			1.68372
$814$		0			0
$815$		−	3.94232e7i		−	2.07902i
$816$		0			0
$817$	2.96015e6	+	4.32786e6i	0.155152	+	0.226839i
$818$		0			0
$819$		−	2.07963e7i		−	1.08337i
$820$		0			0
$821$		−	2.24178e7i		−	1.16074i	−0.814352	−	0.580371i	$-0.802907\pi$
$821$		−	2.24178e7i		−	1.16074i	0.814352	−	0.580371i	$-0.197093\pi$
$822$		0			0
$823$			2.34801e7i			1.20837i	0.796843	+	0.604187i	$0.206502\pi$
$823$			2.34801e7i			1.20837i	−0.796843	+	0.604187i	$0.793498\pi$
$824$		0			0
$825$			1.18338e8i			6.05327i
$826$		0			0
$827$		−	1.52128e7i		−	0.773475i	−0.922190	−	0.386738i	$-0.873602\pi$
$827$		−	1.52128e7i		−	0.773475i	0.922190	−	0.386738i	$-0.126398\pi$
$828$		0			0
$829$	8.99072e6			0.454368			0.227184	−	0.973852i	$-0.427048\pi$
$829$	8.99072e6			0.454368			0.227184	+	0.973852i	$0.427048\pi$
$830$		0			0
$831$	1.64259e7			0.825136
$832$		0			0
$833$	3.51299e6			0.175414
$834$		0			0
$835$		−	6.98538e6i		−	0.346716i
$836$		0			0
$837$		−	1.20074e7i		−	0.592427i
$838$		0			0
$839$	2.86242e7			1.40387			0.701937	−	0.712239i	$-0.252319\pi$
$839$	2.86242e7			1.40387			0.701937	+	0.712239i	$0.252319\pi$
$840$		0			0
$841$	811449.			0.0395613
$842$		0			0
$843$	−3.32746e7			−1.61266
$844$		0			0
$845$		−	2.13290e7i		−	1.02761i
$846$		0			0
$847$		−	6.66123e6i		−	0.319041i
$848$		0			0
$849$		−	4.13163e7i		−	1.96722i
$850$		0			0
$851$			2.21529e7i			1.04859i
$852$		0			0
$853$		−	2.64820e6i		−	0.124617i	−0.998057	−	0.0623087i	$-0.980154\pi$
$853$		−	2.64820e6i		−	0.124617i	0.998057	−	0.0623087i	$-0.0198463\pi$
$854$		0			0
$855$	7.98366e7	−	5.46063e7i	3.73497	−	2.55463i
$856$		0			0
$857$		−	1.75279e7i		−	0.815227i	−0.913154	−	0.407614i	$-0.866361\pi$
$857$		−	1.75279e7i		−	0.815227i	0.913154	−	0.407614i	$-0.133639\pi$
$858$		0			0
$859$	1.40420e7			0.649300			0.324650	−	0.945834i	$-0.394753\pi$
$859$	1.40420e7			0.649300			0.324650	+	0.945834i	$0.394753\pi$
$860$		0			0
$861$		−	1.87219e7i		−	0.860680i
$862$		0			0
$863$	1.99615e6			0.0912358			0.0456179	−	0.998959i	$-0.485474\pi$
$863$	1.99615e6			0.0912358			0.0456179	+	0.998959i	$0.485474\pi$
$864$		0			0
$865$			2.59512e7i			1.17928i
$866$		0			0
$867$		−	3.62499e7i		−	1.63779i
$868$		0			0
$869$	−1.62762e7			−0.731146
$870$		0			0
$871$		−	1.32237e7i		−	0.590619i
$872$		0			0
$873$			2.75161e7i			1.22195i
$874$		0			0
$875$	−5.10933e7			−2.25603
$876$		0			0
$877$	−3.73211e7			−1.63854			−0.819268	−	0.573411i	$-0.805620\pi$
$877$	−3.73211e7			−1.63854			−0.819268	+	0.573411i	$0.805620\pi$
$878$		0			0
$879$			1.27757e7i			0.557715i
$880$		0			0
$881$	−3.37709e6			−0.146589			−0.0732947	−	0.997310i	$-0.523351\pi$
$881$	−3.37709e6			−0.146589			−0.0732947	+	0.997310i	$0.523351\pi$
$882$		0			0
$883$	3.62547e6			0.156481			0.0782407	−	0.996934i	$-0.475070\pi$
$883$	3.62547e6			0.156481			0.0782407	+	0.996934i	$0.475070\pi$
$884$		0			0
$885$		−	8.87352e7i		−	3.80836i
$886$		0			0
$887$	−3.24080e7			−1.38307			−0.691533	−	0.722345i	$-0.743064\pi$
$887$	−3.24080e7			−1.38307			−0.691533	+	0.722345i	$0.743064\pi$
$888$		0			0
$889$			7.06485e6i			0.299812i
$890$		0			0
$891$	−6.17637e7			−2.60639
$892$		0			0
$893$	5.99640e6	−	4.10139e6i	0.251630	−	0.172109i
$894$		0			0
$895$	−2.96504e7			−1.23730
$896$		0			0
$897$	5.26950e7			2.18670
$898$		0			0
$899$	−5.95865e6			−0.245894
$900$		0			0
$901$	−8.24975e6			−0.338555
$902$		0			0
$903$	8.32249e6			0.339652
$904$		0			0
$905$		−	5.97141e7i		−	2.42357i
$906$		0			0
$907$		−	9.21912e6i		−	0.372110i	−0.982539	−	0.186055i	$-0.940430\pi$
$907$		−	9.21912e6i		−	0.372110i	0.982539	−	0.186055i	$-0.0595703\pi$
$908$		0			0
$909$			6.11067e7i			2.45290i
$910$		0			0
$911$	−2.58114e7			−1.03042			−0.515212	−	0.857063i	$-0.672287\pi$
$911$	−2.58114e7			−1.03042			−0.515212	+	0.857063i	$0.672287\pi$
$912$		0			0
$913$	−2.91131e7			−1.15588
$914$		0			0
$915$			3.22079e7i			1.27177i
$916$		0			0
$917$			2.08367e6i			0.0818286i
$918$		0			0
$919$			4.04625e7i			1.58039i	0.612856	+	0.790194i	$0.290020\pi$
$919$			4.04625e7i			1.58039i	−0.612856	+	0.790194i	$0.709980\pi$
$920$		0			0
$921$	7.45569e7			2.89627
$922$		0			0
$923$	−3.50665e6			−0.135484
$924$		0			0
$925$	−4.25839e7			−1.63640
$926$		0			0
$927$	−5.97214e7			−2.28260
$928$		0			0
$929$	−191349.			−0.00727424			−0.00363712	−	0.999993i	$-0.501158\pi$
$929$	−191349.			−0.00727424			−0.00363712	+	0.999993i	$0.501158\pi$
$930$		0			0
$931$	8.10967e6	+	1.18567e7i	0.306640	+	0.448321i
$932$		0			0
$933$	−8.16480e7			−3.07073
$934$		0			0
$935$		−	2.02254e7i		−	0.756603i
$936$		0			0
$937$	−2.88130e7			−1.07211			−0.536056	−	0.844183i	$-0.680086\pi$
$937$	−2.88130e7			−1.07211			−0.536056	+	0.844183i	$0.680086\pi$
$938$		0			0
$939$			5.54213e7i			2.05122i
$940$		0			0
$941$	5.02395e7			1.84957			0.924785	−	0.380489i	$-0.124244\pi$
$941$	5.02395e7			1.84957			0.924785	+	0.380489i	$0.124244\pi$
$942$		0			0
$943$	3.32501e7			1.21763
$944$		0			0
$945$		−	8.80128e7i		−	3.20602i
$946$		0			0
$947$	−1.96661e7			−0.712597			−0.356298	−	0.934372i	$-0.615961\pi$
$947$	−1.96661e7			−0.712597			−0.356298	+	0.934372i	$0.615961\pi$
$948$		0			0
$949$	−6.65977e6			−0.240045
$950$		0			0
$951$			2.79855e6i			0.100342i
$952$		0			0
$953$			4.08757e7i			1.45792i	0.684558	+	0.728958i	$0.259995\pi$
$953$			4.08757e7i			1.45792i	−0.684558	+	0.728958i	$0.740005\pi$
$954$		0			0
$955$	6.11799e7			2.17070
$956$		0			0
$957$			6.40853e7i			2.26193i
$958$		0			0
$959$		−	3.88975e6i		−	0.136576i
$960$		0			0
$961$	−2.69640e7			−0.941837
$962$		0			0
$963$		−	1.13477e8i		−	3.94314i
$964$		0			0
$965$	−7.36416e7			−2.54569
$966$		0			0
$967$		−	3.72816e7i		−	1.28212i	−0.767492	−	0.641059i	$-0.778495\pi$
$967$		−	3.72816e7i		−	1.28212i	0.767492	−	0.641059i	$-0.221505\pi$
$968$		0			0
$969$	−1.42465e7	+	9.74426e6i	−0.487415	+	0.333380i
$970$		0			0
$971$		−	3.48090e7i		−	1.18479i	−0.805646	−	0.592397i	$-0.798182\pi$
$971$		−	3.48090e7i		−	1.18479i	0.805646	−	0.592397i	$-0.201818\pi$
$972$		0			0
$973$		−	2.32503e7i		−	0.787310i
$974$		0			0
$975$			1.01294e8i			3.41250i
$976$		0			0
$977$		−	1.92230e7i		−	0.644296i	−0.946689	−	0.322148i	$-0.895595\pi$
$977$		−	1.92230e7i		−	0.644296i	0.946689	−	0.322148i	$-0.104405\pi$
$978$		0			0
$979$		−	4.62803e7i		−	1.54326i
$980$		0			0
$981$	7.23093e7			2.39895
$982$		0			0
$983$	3.86955e7			1.27725			0.638626	−	0.769517i	$-0.279503\pi$
$983$	3.86955e7			1.27725			0.638626	+	0.769517i	$0.279503\pi$
$984$		0			0
$985$	−2.18263e7			−0.716786
$986$		0			0
$987$		−	1.15311e7i		−	0.376772i
$988$		0			0
$989$			1.47808e7i			0.480514i
$990$		0			0
$991$	−2.40556e7			−0.778093			−0.389047	−	0.921218i	$-0.627195\pi$
$991$	−2.40556e7			−0.778093			−0.389047	+	0.921218i	$0.627195\pi$
$992$		0			0
$993$	7.74313e7			2.49197
$994$		0			0
$995$	−6.10454e6			−0.195477
$996$		0			0
$997$		−	3.45626e6i		−	0.110121i	−0.998483	−	0.0550604i	$-0.982465\pi$
$997$		−	3.45626e6i		−	0.110121i	0.998483	−	0.0550604i	$-0.0175351\pi$
$998$		0			0
$999$		−	4.64707e7i		−	1.47321i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.6.b.b.303.96		96
4.3	odd	2		152.6.b.b.75.18	yes	96
8.3	odd	2	inner	608.6.b.b.303.95		96
8.5	even	2		152.6.b.b.75.80	yes	96
19.18	odd	2	inner	608.6.b.b.303.2		96
76.75	even	2		152.6.b.b.75.79	yes	96
152.37	odd	2		152.6.b.b.75.17	✓	96
152.75	even	2	inner	608.6.b.b.303.1		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.6.b.b.75.17	✓	96	152.37	odd	2
152.6.b.b.75.18	yes	96	4.3	odd	2
152.6.b.b.75.79	yes	96	76.75	even	2
152.6.b.b.75.80	yes	96	8.5	even	2
608.6.b.b.303.1		96	152.75	even	2	inner
608.6.b.b.303.2		96	19.18	odd	2	inner
608.6.b.b.303.95		96	8.3	odd	2	inner
608.6.b.b.303.96		96	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	608.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+28.5036i q^{3} +107.944i q^{5} -87.6251i q^{7} -569.454 q^{9} +O(q^{10})\)	\(q+28.5036i q^{3} +107.944i q^{5} -87.6251i q^{7} -569.454 q^{9} -486.899 q^{11} -416.772 q^{13} -3076.78 q^{15} +384.824 q^{17} +(888.357 + 1298.81i) q^{19} +2497.63 q^{21} +4435.79i q^{23} -8526.81 q^{25} -9305.09i q^{27} -4617.64 q^{29} +1290.41 q^{31} -13878.4i q^{33} +9458.56 q^{35} +4994.11 q^{37} -11879.5i q^{39} -7495.86i q^{41} +3332.16 q^{43} -61468.8i q^{45} -4616.83i q^{47} +9128.84 q^{49} +10968.8i q^{51} -21437.8 q^{53} -52557.6i q^{55} +(-37020.9 + 25321.4i) q^{57} +28840.3i q^{59} -10468.1i q^{61} +49898.4i q^{63} -44987.9i q^{65} +31728.8i q^{67} -126436. q^{69} +8413.83 q^{71} +15979.4 q^{73} -243044. i q^{75} +42664.6i q^{77} +33428.3 q^{79} +126851. q^{81} +59792.9 q^{83} +41539.2i q^{85} -131619. i q^{87} +95051.2i q^{89} +36519.7i q^{91} +36781.3i q^{93} +(-140199. + 95892.4i) q^{95} -48320.2i q^{97} +277266. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6168 q^{9}+O(q^{10}) \) 96 * q - 6168 * q^9	\( 96 q - 6168 q^{9} - 944 q^{11} - 3832 q^{17} - 5240 q^{19} - 62504 q^{25} - 7720 q^{35} - 45096 q^{43} - 210840 q^{49} - 36336 q^{57} - 4336 q^{73} - 20624 q^{81} - 52152 q^{83} + 752768 q^{99}+O(q^{100}) \) 96 * q - 6168 * q^9 - 944 * q^11 - 3832 * q^17 - 5240 * q^19 - 62504 * q^25 - 7720 * q^35 - 45096 * q^43 - 210840 * q^49 - 36336 * q^57 - 4336 * q^73 - 20624 * q^81 - 52152 * q^83 + 752768 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more