LMFDB - Embedded newform 688.5.b.d.257.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,5,Mod(257,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("688.257");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	688.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:	$71.1185346017$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 142x^{10} + 7173x^{8} + 157368x^{6} + 1510016x^{4} + 5098688x^{2} + 90352 $ x^12 + 142x^10 + 7173x^8 + 157368x^6 + 1510016x^4 + 5098688*x^2 + 90352
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			257.5
Root			$-4.43775i$ of defining polynomial
Character	$\chi$	$=$	688.257
Dual form			688.5.b.d.257.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.93950i q^{3} -22.3554i q^{5} +51.9837i q^{7} +32.8434 q^{9} +O(q^{10})$	$q-6.93950i q^{3} -22.3554i q^{5} +51.9837i q^{7} +32.8434 q^{9} +25.0158 q^{11} -38.7510 q^{13} -155.135 q^{15} +111.381 q^{17} -238.930i q^{19} +360.741 q^{21} -823.523 q^{23} +125.236 q^{25} -790.016i q^{27} +424.840i q^{29} +1447.58 q^{31} -173.597i q^{33} +1162.12 q^{35} +626.677i q^{37} +268.912i q^{39} +580.694 q^{41} +(528.728 - 1771.79i) q^{43} -734.227i q^{45} +170.609 q^{47} -301.309 q^{49} -772.930i q^{51} +4308.39 q^{53} -559.237i q^{55} -1658.05 q^{57} +65.4372 q^{59} -3896.58i q^{61} +1707.32i q^{63} +866.294i q^{65} +5449.76 q^{67} +5714.84i q^{69} -5845.23i q^{71} -4773.87i q^{73} -869.073i q^{75} +1300.41i q^{77} -10139.3 q^{79} -2822.00 q^{81} -4438.95 q^{83} -2489.97i q^{85} +2948.17 q^{87} -9870.51i q^{89} -2014.42i q^{91} -10045.5i q^{93} -5341.37 q^{95} -7761.47 q^{97} +821.602 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 462 q^{9}+O(q^{10}) $ 12 * q - 462 * q^9	$ 12 q - 462 q^{9} + 180 q^{11} - 216 q^{13} + 92 q^{15} + 678 q^{17} - 2392 q^{21} - 1566 q^{23} - 174 q^{25} - 5710 q^{31} - 936 q^{35} + 4878 q^{41} + 1108 q^{43} + 5526 q^{47} - 8544 q^{49} + 1212 q^{53} - 7692 q^{57} - 14016 q^{59} + 1088 q^{67} - 24302 q^{79} - 23660 q^{81} + 7032 q^{83} - 17850 q^{87} - 606 q^{95} - 5842 q^{97} + 25924 q^{99}+O(q^{100}) $ 12 * q - 462 * q^9 + 180 * q^11 - 216 * q^13 + 92 * q^15 + 678 * q^17 - 2392 * q^21 - 1566 * q^23 - 174 * q^25 - 5710 * q^31 - 936 * q^35 + 4878 * q^41 + 1108 * q^43 + 5526 * q^47 - 8544 * q^49 + 1212 * q^53 - 7692 * q^57 - 14016 * q^59 + 1088 * q^67 - 24302 * q^79 - 23660 * q^81 + 7032 * q^83 - 17850 * q^87 - 606 * q^95 - 5842 * q^97 + 25924 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$.

$n$	$431$	$433$	$517$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	6.93950i		−	0.771055i	−0.922696	−	0.385528i	$-0.874019\pi$
$3$		−	6.93950i		−	0.771055i	0.922696	−	0.385528i	$-0.125981\pi$
$4$		0			0
$5$		−	22.3554i		−	0.894216i	−0.894480	−	0.447108i	$-0.852454\pi$
$5$		−	22.3554i		−	0.894216i	0.894480	−	0.447108i	$-0.147546\pi$
$6$		0			0
$7$			51.9837i			1.06089i	0.847719	+	0.530446i	$0.177976\pi$
$7$			51.9837i			1.06089i	−0.847719	+	0.530446i	$0.822024\pi$
$8$		0			0
$9$	32.8434			0.405474
$10$		0			0
$11$	25.0158			0.206742			0.103371	−	0.994643i	$-0.467037\pi$
$11$	25.0158			0.206742			0.103371	+	0.994643i	$0.467037\pi$
$12$		0			0
$13$	−38.7510			−0.229296			−0.114648	−	0.993406i	$-0.536574\pi$
$13$	−38.7510			−0.229296			−0.114648	+	0.993406i	$0.536574\pi$
$14$		0			0
$15$	−155.135			−0.689490
$16$		0			0
$17$	111.381			0.385402			0.192701	−	0.981257i	$-0.438275\pi$
$17$	111.381			0.385402			0.192701	+	0.981257i	$0.438275\pi$
$18$		0			0
$19$		−	238.930i		−	0.661856i	−0.943656	−	0.330928i	$-0.892638\pi$
$19$		−	238.930i		−	0.661856i	0.943656	−	0.330928i	$-0.107362\pi$
$20$		0			0
$21$	360.741			0.818007
$22$		0			0
$23$	−823.523			−1.55675			−0.778377	−	0.627797i	$-0.783957\pi$
$23$	−823.523			−1.55675			−0.778377	+	0.627797i	$0.783957\pi$
$24$		0			0
$25$	125.236			0.200377
$26$		0			0
$27$		−	790.016i		−	1.08370i
$28$		0			0
$29$			424.840i			0.505160i	0.967576	+	0.252580i	$0.0812791\pi$
$29$			424.840i			0.505160i	−0.967576	+	0.252580i	$0.918721\pi$
$30$		0			0
$31$	1447.58			1.50632			0.753162	−	0.657835i	$-0.228528\pi$
$31$	1447.58			1.50632			0.753162	+	0.657835i	$0.228528\pi$
$32$		0			0
$33$		−	173.597i		−	0.159409i
$34$		0			0
$35$	1162.12			0.948668
$36$		0			0
$37$			626.677i			0.457763i	0.973454	+	0.228881i	$0.0735068\pi$
$37$			626.677i			0.457763i	−0.973454	+	0.228881i	$0.926493\pi$
$38$		0			0
$39$			268.912i			0.176800i
$40$		0			0
$41$	580.694			0.345446			0.172723	−	0.984970i	$-0.444744\pi$
$41$	580.694			0.345446			0.172723	+	0.984970i	$0.444744\pi$
$42$		0			0
$43$	528.728	−	1771.79i	0.285953	−	0.958244i
$43$	528.728	−	1771.79i	0.285953	−	0.958244i
$44$		0			0
$45$		−	734.227i		−	0.362581i
$46$		0			0
$47$	170.609			0.0772336			0.0386168	−	0.999254i	$-0.487705\pi$
$47$	170.609			0.0772336			0.0386168	+	0.999254i	$0.487705\pi$
$48$		0			0
$49$	−301.309			−0.125493
$50$		0			0
$51$		−	772.930i		−	0.297167i
$52$		0			0
$53$	4308.39			1.53378			0.766890	−	0.641779i	$-0.221803\pi$
$53$	4308.39			1.53378			0.766890	+	0.641779i	$0.221803\pi$
$54$		0			0
$55$		−	559.237i		−	0.184872i
$56$		0			0
$57$	−1658.05			−0.510327
$58$		0			0
$59$	65.4372			0.0187984			0.00939919	−	0.999956i	$-0.497008\pi$
$59$	65.4372			0.0187984			0.00939919	+	0.999956i	$0.497008\pi$
$60$		0			0
$61$		−	3896.58i		−	1.04719i	−0.851968	−	0.523594i	$-0.824591\pi$
$61$		−	3896.58i		−	1.04719i	0.851968	−	0.523594i	$-0.175409\pi$
$62$		0			0
$63$			1707.32i			0.430164i
$64$		0			0
$65$			866.294i			0.205040i
$66$		0			0
$67$	5449.76			1.21403			0.607013	−	0.794692i	$-0.292368\pi$
$67$	5449.76			1.21403			0.607013	+	0.794692i	$0.292368\pi$
$68$		0			0
$69$			5714.84i			1.20034i
$70$		0			0
$71$		−	5845.23i		−	1.15954i	−0.814781	−	0.579769i	$-0.803143\pi$
$71$		−	5845.23i		−	1.15954i	0.814781	−	0.579769i	$-0.196857\pi$
$72$		0			0
$73$		−	4773.87i		−	0.895828i	−0.894077	−	0.447914i	$-0.852167\pi$
$73$		−	4773.87i		−	0.895828i	0.894077	−	0.447914i	$-0.147833\pi$
$74$		0			0
$75$		−	869.073i		−	0.154502i
$76$		0			0
$77$			1300.41i			0.219331i
$78$		0			0
$79$	−10139.3			−1.62463			−0.812316	−	0.583217i	$-0.801794\pi$
$79$	−10139.3			−1.62463			−0.812316	+	0.583217i	$0.801794\pi$
$80$		0			0
$81$	−2822.00			−0.430117
$82$		0			0
$83$	−4438.95			−0.644353			−0.322176	−	0.946680i	$-0.604414\pi$
$83$	−4438.95			−0.644353			−0.322176	+	0.946680i	$0.604414\pi$
$84$		0			0
$85$		−	2489.97i		−	0.344633i
$86$		0			0
$87$	2948.17			0.389506
$88$		0			0
$89$		−	9870.51i		−	1.24612i	−0.782174	−	0.623060i	$-0.785889\pi$
$89$		−	9870.51i		−	1.24612i	0.782174	−	0.623060i	$-0.214111\pi$
$90$		0			0
$91$		−	2014.42i		−	0.243258i
$92$		0			0
$93$		−	10045.5i		−	1.16146i
$94$		0			0
$95$	−5341.37			−0.591842
$96$		0			0
$97$	−7761.47			−0.824899			−0.412449	−	0.910981i	$-0.635327\pi$
$97$	−7761.47			−0.824899			−0.412449	+	0.910981i	$0.635327\pi$
$98$		0			0
$99$	821.602			0.0838284
$100$		0			0
$101$	−7907.71			−0.775190			−0.387595	−	0.921830i	$-0.626694\pi$
$101$	−7907.71			−0.775190			−0.387595	+	0.921830i	$0.626694\pi$
$102$		0			0
$103$	−11579.3			−1.09146			−0.545728	−	0.837962i	$-0.683747\pi$
$103$	−11579.3			−1.09146			−0.545728	+	0.837962i	$0.683747\pi$
$104$		0			0
$105$		−	8064.51i		−	0.731475i
$106$		0			0
$107$	6463.42			0.564540			0.282270	−	0.959335i	$-0.408913\pi$
$107$	6463.42			0.564540			0.282270	+	0.959335i	$0.408913\pi$
$108$		0			0
$109$	3408.56			0.286892			0.143446	−	0.989658i	$-0.454182\pi$
$109$	3408.56			0.286892			0.143446	+	0.989658i	$0.454182\pi$
$110$		0			0
$111$	4348.82			0.352960
$112$		0			0
$113$			11268.0i			0.882446i	0.897397	+	0.441223i	$0.145455\pi$
$113$			11268.0i			0.882446i	−0.897397	+	0.441223i	$0.854545\pi$
$114$		0			0
$115$			18410.2i			1.39208i
$116$		0			0
$117$	−1272.71			−0.0929734
$118$		0			0
$119$			5790.02i			0.408871i
$120$		0			0
$121$	−14015.2			−0.957258
$122$		0			0
$123$		−	4029.73i		−	0.266358i
$124$		0			0
$125$		−	16771.8i		−	1.07340i
$126$		0			0
$127$	−13207.4			−0.818860			−0.409430	−	0.912342i	$-0.634272\pi$
$127$	−13207.4			−0.818860			−0.409430	+	0.912342i	$0.634272\pi$
$128$		0			0
$129$	−12295.3	−	3669.10i	−0.738859	−	0.220486i
$130$		0			0
$131$		−	29457.5i		−	1.71653i	−0.513203	−	0.858267i	$-0.671541\pi$
$131$		−	29457.5i		−	1.71653i	0.513203	−	0.858267i	$-0.328459\pi$
$132$		0			0
$133$	12420.5			0.702158
$134$		0			0
$135$	−17661.1			−0.969060
$136$		0			0
$137$			14241.6i			0.758783i	0.925236	+	0.379392i	$0.123867\pi$
$137$			14241.6i			0.758783i	−0.925236	+	0.379392i	$0.876133\pi$
$138$		0			0
$139$	18549.0			0.960042			0.480021	−	0.877257i	$-0.340629\pi$
$139$	18549.0			0.960042			0.480021	+	0.877257i	$0.340629\pi$
$140$		0			0
$141$		−	1183.94i		−	0.0595513i
$142$		0			0
$143$	−969.385			−0.0474050
$144$		0			0
$145$	9497.46			0.451722
$146$		0			0
$147$			2090.94i			0.0967623i
$148$		0			0
$149$		−	31018.5i		−	1.39717i	−0.715528	−	0.698584i	$-0.753814\pi$
$149$		−	31018.5i		−	1.39717i	0.715528	−	0.698584i	$-0.246186\pi$
$150$		0			0
$151$		−	10079.5i		−	0.442063i	−0.975267	−	0.221032i	$-0.929058\pi$
$151$		−	10079.5i		−	0.442063i	0.975267	−	0.221032i	$-0.0709424\pi$
$152$		0			0
$153$	3658.14			0.156271
$154$		0			0
$155$		−	32361.2i		−	1.34698i
$156$		0			0
$157$			27782.4i			1.12712i	0.826075	+	0.563560i	$0.190569\pi$
$157$			27782.4i			1.12712i	−0.826075	+	0.563560i	$0.809431\pi$
$158$		0			0
$159$		−	29898.0i		−	1.18263i
$160$		0			0
$161$		−	42809.8i		−	1.65155i
$162$		0			0
$163$		−	23274.3i		−	0.875993i	−0.898977	−	0.437996i	$-0.855688\pi$
$163$		−	23274.3i		−	0.875993i	0.898977	−	0.437996i	$-0.144312\pi$
$164$		0			0
$165$	−3880.83			−0.142546
$166$		0			0
$167$	38477.5			1.37967			0.689833	−	0.723968i	$-0.257684\pi$
$167$	38477.5			1.37967			0.689833	+	0.723968i	$0.257684\pi$
$168$		0			0
$169$	−27059.4			−0.947423
$170$		0			0
$171$		−	7847.26i		−	0.268365i
$172$		0			0
$173$	26454.5			0.883909			0.441954	−	0.897038i	$-0.354285\pi$
$173$	26454.5			0.883909			0.441954	+	0.897038i	$0.354285\pi$
$174$		0			0
$175$			6510.22i			0.212579i
$176$		0			0
$177$		−	454.101i		−	0.0144946i
$178$		0			0
$179$			5807.25i			0.181244i	0.995885	+	0.0906222i	$0.0288856\pi$
$179$			5807.25i			0.181244i	−0.995885	+	0.0906222i	$0.971114\pi$
$180$		0			0
$181$	−63912.2			−1.95086			−0.975431	−	0.220304i	$-0.929295\pi$
$181$	−63912.2			−1.95086			−0.975431	+	0.220304i	$0.929295\pi$
$182$		0			0
$183$	−27040.3			−0.807439
$184$		0			0
$185$	14009.6			0.409339
$186$		0			0
$187$	2786.29			0.0796788
$188$		0			0
$189$	41068.0			1.14969
$190$		0			0
$191$			58852.6i			1.61324i	0.591071	+	0.806620i	$0.298705\pi$
$191$			58852.6i			1.61324i	−0.591071	+	0.806620i	$0.701295\pi$
$192$		0			0
$193$	−4293.94			−0.115277			−0.0576384	−	0.998338i	$-0.518357\pi$
$193$	−4293.94			−0.115277			−0.0576384	+	0.998338i	$0.518357\pi$
$194$		0			0
$195$	6011.65			0.158097
$196$		0			0
$197$	60627.5			1.56220			0.781101	−	0.624405i	$-0.214658\pi$
$197$	60627.5			1.56220			0.781101	+	0.624405i	$0.214658\pi$
$198$		0			0
$199$		−	75910.5i		−	1.91688i	−0.285290	−	0.958441i	$-0.592090\pi$
$199$		−	75910.5i		−	1.91688i	0.285290	−	0.958441i	$-0.407910\pi$
$200$		0			0
$201$		−	37818.6i		−	0.936081i
$202$		0			0
$203$	−22084.8			−0.535921
$204$		0			0
$205$		−	12981.7i		−	0.308903i
$206$		0			0
$207$	−27047.3			−0.631223
$208$		0			0
$209$		−	5977.01i		−	0.136833i
$210$		0			0
$211$			34858.4i			0.782964i	0.920186	+	0.391482i	$0.128038\pi$
$211$			34858.4i			0.782964i	−0.920186	+	0.391482i	$0.871962\pi$
$212$		0			0
$213$	−40563.0			−0.894068
$214$		0			0
$215$	−39609.1	−	11819.9i	−0.856877	−	0.255704i
$216$		0			0
$217$			75250.5i			1.59805i
$218$		0			0
$219$	−33128.3			−0.690733
$220$		0			0
$221$	−4316.13			−0.0883711
$222$		0			0
$223$		−	69871.4i		−	1.40504i	−0.711662	−	0.702522i	$-0.752057\pi$
$223$		−	69871.4i		−	1.40504i	0.711662	−	0.702522i	$-0.247943\pi$
$224$		0			0
$225$	4113.16			0.0812476
$226$		0			0
$227$			5952.36i			0.115515i	0.998331	+	0.0577574i	$0.0183950\pi$
$227$			5952.36i			0.115515i	−0.998331	+	0.0577574i	$0.981605\pi$
$228$		0			0
$229$	−31808.4			−0.606556			−0.303278	−	0.952902i	$-0.598081\pi$
$229$	−31808.4			−0.606556			−0.303278	+	0.952902i	$0.598081\pi$
$230$		0			0
$231$	9024.21			0.169116
$232$		0			0
$233$		−	55507.2i		−	1.02244i	−0.859450	−	0.511220i	$-0.829194\pi$
$233$		−	55507.2i		−	1.02244i	0.859450	−	0.511220i	$-0.170806\pi$
$234$		0			0
$235$		−	3814.03i		−	0.0690635i
$236$		0			0
$237$			70361.9i			1.25268i
$238$		0			0
$239$	62886.2			1.10093			0.550465	−	0.834859i	$-0.314451\pi$
$239$	62886.2			1.10093			0.550465	+	0.834859i	$0.314451\pi$
$240$		0			0
$241$		−	19314.0i		−	0.332536i	−0.986081	−	0.166268i	$-0.946828\pi$
$241$		−	19314.0i		−	0.332536i	0.986081	−	0.166268i	$-0.0531717\pi$
$242$		0			0
$243$		−	44408.0i		−	0.752054i
$244$		0			0
$245$			6735.90i			0.112218i
$246$		0			0
$247$			9258.77i			0.151761i
$248$		0			0
$249$			30804.1i			0.496832i
$250$		0			0
$251$	62983.5			0.999723			0.499862	−	0.866105i	$-0.333384\pi$
$251$	62983.5			0.999723			0.499862	+	0.866105i	$0.333384\pi$
$252$		0			0
$253$	−20601.0			−0.321846
$254$		0			0
$255$	−17279.2			−0.265731
$256$		0			0
$257$		−	53736.0i		−	0.813579i	−0.913522	−	0.406789i	$-0.866648\pi$
$257$		−	53736.0i		−	0.813579i	0.913522	−	0.406789i	$-0.133352\pi$
$258$		0			0
$259$	−32577.0			−0.485637
$260$		0			0
$261$			13953.2i			0.204829i
$262$		0			0
$263$			133357.i			1.92798i	0.265932	+	0.963992i	$0.414320\pi$
$263$			133357.i			1.92798i	−0.265932	+	0.963992i	$0.585680\pi$
$264$		0			0
$265$		−	96315.7i		−	1.37153i
$266$		0			0
$267$	−68496.4			−0.960827
$268$		0			0
$269$	61320.3			0.847421			0.423711	−	0.905798i	$-0.360727\pi$
$269$	61320.3			0.847421			0.423711	+	0.905798i	$0.360727\pi$
$270$		0			0
$271$	−66120.9			−0.900326			−0.450163	−	0.892946i	$-0.648634\pi$
$271$	−66120.9			−0.900326			−0.450163	+	0.892946i	$0.648634\pi$
$272$		0			0
$273$	−13979.1			−0.187565
$274$		0			0
$275$	3132.86			0.0414263
$276$		0			0
$277$		−	62649.4i		−	0.816502i	−0.912870	−	0.408251i	$-0.866139\pi$
$277$		−	62649.4i		−	0.816502i	0.912870	−	0.408251i	$-0.133861\pi$
$278$		0			0
$279$	47543.3			0.610775
$280$		0			0
$281$	110235.			1.39606			0.698032	−	0.716066i	$-0.254059\pi$
$281$	110235.			1.39606			0.698032	+	0.716066i	$0.254059\pi$
$282$		0			0
$283$	109263.			1.36427			0.682135	−	0.731226i	$-0.261052\pi$
$283$	109263.			1.36427			0.682135	+	0.731226i	$0.261052\pi$
$284$		0			0
$285$			37066.5i			0.456343i
$286$		0			0
$287$			30186.7i			0.366481i
$288$		0			0
$289$	−71115.2			−0.851465
$290$		0			0
$291$			53860.7i			0.636042i
$292$		0			0
$293$	−137382.			−1.60028			−0.800139	−	0.599814i	$-0.795241\pi$
$293$	−137382.			−1.60028			−0.800139	+	0.599814i	$0.795241\pi$
$294$		0			0
$295$		−	1462.88i		−	0.0168098i
$296$		0			0
$297$		−	19762.8i		−	0.224046i
$298$		0			0
$299$	31912.3			0.356957
$300$		0			0
$301$	92104.4	+	27485.2i	1.01659	+	0.303366i
$302$		0			0
$303$			54875.5i			0.597714i
$304$		0			0
$305$	−87109.7			−0.936412
$306$		0			0
$307$	13056.1			0.138528			0.0692639	−	0.997598i	$-0.477935\pi$
$307$	13056.1			0.138528			0.0692639	+	0.997598i	$0.477935\pi$
$308$		0			0
$309$			80354.2i			0.841573i
$310$		0			0
$311$	190885.			1.97356			0.986781	−	0.162061i	$-0.0518140\pi$
$311$	190885.			1.97356			0.986781	+	0.162061i	$0.0518140\pi$
$312$		0			0
$313$			31940.8i			0.326030i	0.986624	+	0.163015i	$0.0521219\pi$
$313$			31940.8i			0.326030i	−0.986624	+	0.163015i	$0.947878\pi$
$314$		0			0
$315$	38167.9			0.384660
$316$		0			0
$317$	10721.3			0.106692			0.0533459	−	0.998576i	$-0.483011\pi$
$317$	10721.3			0.106692			0.0533459	+	0.998576i	$0.483011\pi$
$318$		0			0
$319$			10627.7i			0.104438i
$320$		0			0
$321$		−	44852.9i		−	0.435291i
$322$		0			0
$323$		−	26612.3i		−	0.255081i
$324$		0			0
$325$	−4853.00			−0.0459456
$326$		0			0
$327$		−	23653.7i		−	0.221209i
$328$		0			0
$329$			8868.89i			0.0819365i
$330$		0			0
$331$		−	49144.6i		−	0.448560i	−0.974525	−	0.224280i	$-0.927997\pi$
$331$		−	49144.6i		−	0.448560i	0.974525	−	0.224280i	$-0.0720030\pi$
$332$		0			0
$333$			20582.2i			0.185611i
$334$		0			0
$335$		−	121832.i		−	1.08560i
$336$		0			0
$337$	113598.			1.00025			0.500127	−	0.865952i	$-0.333287\pi$
$337$	113598.			1.00025			0.500127	+	0.865952i	$0.333287\pi$
$338$		0			0
$339$	78194.0			0.680415
$340$		0			0
$341$	36212.2			0.311420
$342$		0			0
$343$			109150.i			0.927758i
$344$		0			0
$345$	127757.			1.07337
$346$		0			0
$347$		−	25603.4i		−	0.212637i	−0.994332	−	0.106319i	$-0.966094\pi$
$347$		−	25603.4i		−	0.212637i	0.994332	−	0.106319i	$-0.0339063\pi$
$348$		0			0
$349$			4308.53i			0.0353735i	0.999844	+	0.0176867i	$0.00563016\pi$
$349$			4308.53i			0.0353735i	−0.999844	+	0.0176867i	$0.994370\pi$
$350$		0			0
$351$			30613.9i			0.248487i
$352$		0			0
$353$	101165.			0.811856			0.405928	−	0.913905i	$-0.366948\pi$
$353$	101165.			0.811856			0.405928	+	0.913905i	$0.366948\pi$
$354$		0			0
$355$	−130673.			−1.03688
$356$		0			0
$357$	40179.8			0.315262
$358$		0			0
$359$	−3103.18			−0.0240779			−0.0120389	−	0.999928i	$-0.503832\pi$
$359$	−3103.18			−0.0240779			−0.0120389	+	0.999928i	$0.503832\pi$
$360$		0			0
$361$	73233.5			0.561947
$362$		0			0
$363$			97258.5i			0.738099i
$364$		0			0
$365$	−106722.			−0.801064
$366$		0			0
$367$	−158319.			−1.17544			−0.587722	−	0.809063i	$-0.699975\pi$
$367$	−158319.			−1.17544			−0.587722	+	0.809063i	$0.699975\pi$
$368$		0			0
$369$	19072.0			0.140069
$370$		0			0
$371$			223966.i			1.62718i
$372$		0			0
$373$			186262.i			1.33877i	0.742914	+	0.669387i	$0.233443\pi$
$373$			186262.i			1.33877i	−0.742914	+	0.669387i	$0.766557\pi$
$374$		0			0
$375$	−116388.			−0.827648
$376$		0			0
$377$		−	16463.0i		−	0.115831i
$378$		0			0
$379$	−31309.9			−0.217973			−0.108987	−	0.994043i	$-0.534761\pi$
$379$	−31309.9			−0.217973			−0.108987	+	0.994043i	$0.534761\pi$
$380$		0			0
$381$			91652.6i			0.631386i
$382$		0			0
$383$		−	18588.7i		−	0.126722i	−0.997991	−	0.0633609i	$-0.979818\pi$
$383$		−	18588.7i		−	0.126722i	0.997991	−	0.0633609i	$-0.0201819\pi$
$384$		0			0
$385$	29071.3			0.196129
$386$		0			0
$387$	17365.2	−	58191.6i	0.115947	−	0.388543i
$388$		0			0
$389$			158939.i			1.05034i	0.850997	+	0.525170i	$0.175998\pi$
$389$			158939.i			1.05034i	−0.850997	+	0.525170i	$0.824002\pi$
$390$		0			0
$391$	−91725.1			−0.599977
$392$		0			0
$393$	−204420.			−1.32354
$394$		0			0
$395$			226669.i			1.45277i
$396$		0			0
$397$	97945.4			0.621445			0.310723	−	0.950501i	$-0.399429\pi$
$397$	97945.4			0.621445			0.310723	+	0.950501i	$0.399429\pi$
$398$		0			0
$399$		−	86191.8i		−	0.541402i
$400$		0			0
$401$	−252905.			−1.57278			−0.786392	−	0.617727i	$-0.788054\pi$
$401$	−252905.			−1.57278			−0.786392	+	0.617727i	$0.788054\pi$
$402$		0			0
$403$	−56095.0			−0.345394
$404$		0			0
$405$			63086.9i			0.384618i
$406$		0			0
$407$			15676.8i			0.0946386i
$408$		0			0
$409$		−	134853.i		−	0.806149i	−0.915167	−	0.403074i	$-0.867942\pi$
$409$		−	134853.i		−	0.806149i	0.915167	−	0.403074i	$-0.132058\pi$
$410$		0			0
$411$	98829.6			0.585064
$412$		0			0
$413$			3401.67i			0.0199431i
$414$		0			0
$415$			99234.5i			0.576191i
$416$		0			0
$417$		−	128720.i		−	0.740245i
$418$		0			0
$419$			10482.5i			0.0597086i	0.999554	+	0.0298543i	$0.00950433\pi$
$419$			10482.5i			0.0597086i	−0.999554	+	0.0298543i	$0.990496\pi$
$420$		0			0
$421$			241369.i			1.36181i	0.732371	+	0.680906i	$0.238414\pi$
$421$			241369.i			1.36181i	−0.732371	+	0.680906i	$0.761586\pi$
$422$		0			0
$423$	5603.37			0.0313162
$424$		0			0
$425$	13948.9			0.0772258
$426$		0			0
$427$	202559.			1.11095
$428$		0			0
$429$			6727.04i			0.0365519i
$430$		0			0
$431$	−59107.4			−0.318191			−0.159095	−	0.987263i	$-0.550858\pi$
$431$	−59107.4			−0.318191			−0.159095	+	0.987263i	$0.550858\pi$
$432$		0			0
$433$			193520.i			1.03216i	0.856539	+	0.516082i	$0.172610\pi$
$433$			193520.i			1.03216i	−0.856539	+	0.516082i	$0.827390\pi$
$434$		0			0
$435$		−	65907.6i		−	0.348303i
$436$		0			0
$437$			196764.i			1.03035i
$438$		0			0
$439$	4663.78			0.0241997			0.0120998	−	0.999927i	$-0.496148\pi$
$439$	4663.78			0.0241997			0.0120998	+	0.999927i	$0.496148\pi$
$440$		0			0
$441$	−9896.02			−0.0508843
$442$		0			0
$443$	−102777.			−0.523708			−0.261854	−	0.965107i	$-0.584334\pi$
$443$	−102777.			−0.523708			−0.261854	+	0.965107i	$0.584334\pi$
$444$		0			0
$445$	−220659.			−1.11430
$446$		0			0
$447$	−215253.			−1.07729
$448$		0			0
$449$			222984.i			1.10606i	0.833160	+	0.553032i	$0.186529\pi$
$449$			222984.i			1.10606i	−0.833160	+	0.553032i	$0.813471\pi$
$450$		0			0
$451$	14526.5			0.0714180
$452$		0			0
$453$	−69946.6			−0.340855
$454$		0			0
$455$	−45033.2			−0.217525
$456$		0			0
$457$		−	239772.i		−	1.14806i	−0.818833	−	0.574032i	$-0.805378\pi$
$457$		−	239772.i		−	1.14806i	0.818833	−	0.574032i	$-0.194622\pi$
$458$		0			0
$459$		−	87993.0i		−	0.417660i
$460$		0			0
$461$	234784.			1.10476			0.552378	−	0.833593i	$-0.313720\pi$
$461$	234784.			1.10476			0.552378	+	0.833593i	$0.313720\pi$
$462$		0			0
$463$			323705.i			1.51004i	0.655704	+	0.755018i	$0.272372\pi$
$463$			323705.i			1.51004i	−0.655704	+	0.755018i	$0.727628\pi$
$464$		0			0
$465$	−224570.			−1.03860
$466$		0			0
$467$		−	158850.i		−	0.728372i	−0.931326	−	0.364186i	$-0.881347\pi$
$467$		−	158850.i		−	0.728372i	0.931326	−	0.364186i	$-0.118653\pi$
$468$		0			0
$469$			283299.i			1.28795i
$470$		0			0
$471$	192796.			0.869072
$472$		0			0
$473$	13226.5	−	44322.7i	0.0591185	−	0.198109i
$474$		0			0
$475$		−	29922.5i		−	0.132621i
$476$		0			0
$477$	141502.			0.621907
$478$		0			0
$479$	50616.7			0.220609			0.110304	−	0.993898i	$-0.464817\pi$
$479$	50616.7			0.220609			0.110304	+	0.993898i	$0.464817\pi$
$480$		0			0
$481$		−	24284.3i		−	0.104963i
$482$		0			0
$483$	−297079.			−1.27344
$484$		0			0
$485$			173511.i			0.737638i
$486$		0			0
$487$	414555.			1.74793			0.873966	−	0.485987i	$-0.161540\pi$
$487$	414555.			1.74793			0.873966	+	0.485987i	$0.161540\pi$
$488$		0			0
$489$	−161512.			−0.675439
$490$		0			0
$491$		−	162691.i		−	0.674841i	−0.941354	−	0.337421i	$-0.890446\pi$
$491$		−	162691.i		−	0.674841i	0.941354	−	0.337421i	$-0.109554\pi$
$492$		0			0
$493$			47319.2i			0.194690i
$494$		0			0
$495$		−	18367.2i		−	0.0749607i
$496$		0			0
$497$	303857.			1.23015
$498$		0			0
$499$			202146.i			0.811829i	0.913911	+	0.405914i	$0.133047\pi$
$499$			202146.i			0.811829i	−0.913911	+	0.405914i	$0.866953\pi$
$500$		0			0
$501$		−	267015.i		−	1.06380i
$502$		0			0
$503$			142571.i			0.563501i	0.959488	+	0.281751i	$0.0909151\pi$
$503$			142571.i			0.563501i	−0.959488	+	0.281751i	$0.909085\pi$
$504$		0			0
$505$			176780.i			0.693187i
$506$		0			0
$507$			187778.i			0.730516i
$508$		0			0
$509$	−132370.			−0.510922			−0.255461	−	0.966819i	$-0.582227\pi$
$509$	−132370.			−0.510922			−0.255461	+	0.966819i	$0.582227\pi$
$510$		0			0
$511$	248164.			0.950378
$512$		0			0
$513$	−188758.			−0.717251
$514$		0			0
$515$			258859.i			0.975998i
$516$		0			0
$517$	4267.91			0.0159674
$518$		0			0
$519$		−	183581.i		−	0.681542i
$520$		0			0
$521$		−	222159.i		−	0.818444i	−0.912435	−	0.409222i	$-0.865800\pi$
$521$		−	222159.i		−	0.818444i	0.912435	−	0.409222i	$-0.134200\pi$
$522$		0			0
$523$		−	426785.i		−	1.56029i	−0.625599	−	0.780145i	$-0.715145\pi$
$523$		−	426785.i		−	1.56029i	0.625599	−	0.780145i	$-0.284855\pi$
$524$		0			0
$525$	45177.6			0.163910
$526$		0			0
$527$	161233.			0.580541
$528$		0			0
$529$	398349.			1.42348
$530$		0			0
$531$	2149.18			0.00762225
$532$		0			0
$533$	−22502.5			−0.0792092
$534$		0			0
$535$		−	144492.i		−	0.504821i
$536$		0			0
$537$	40299.4			0.139750
$538$		0			0
$539$	−7537.48			−0.0259447
$540$		0			0
$541$	225636.			0.770927			0.385463	−	0.922723i	$-0.374042\pi$
$541$	225636.			0.770927			0.385463	+	0.922723i	$0.374042\pi$
$542$		0			0
$543$			443519.i			1.50422i
$544$		0			0
$545$		−	76199.8i		−	0.256543i
$546$		0			0
$547$	−45128.1			−0.150825			−0.0754124	−	0.997152i	$-0.524027\pi$
$547$	−45128.1			−0.150825			−0.0754124	+	0.997152i	$0.524027\pi$
$548$		0			0
$549$		−	127977.i		−	0.424607i
$550$		0			0
$551$	101507.			0.334343
$552$		0			0
$553$		−	527080.i		−	1.72356i
$554$		0			0
$555$		−	97219.7i		−	0.315623i
$556$		0			0
$557$	561545.			1.80998			0.904991	−	0.425430i	$-0.139877\pi$
$557$	561545.			1.80998			0.904991	+	0.425430i	$0.139877\pi$
$558$		0			0
$559$	−20488.7	+	68658.7i	−0.0655679	+	0.219721i
$560$		0			0
$561$		−	19335.4i		−	0.0614367i
$562$		0			0
$563$	−140409.			−0.442975			−0.221487	−	0.975163i	$-0.571091\pi$
$563$	−140409.			−0.442975			−0.221487	+	0.975163i	$0.571091\pi$
$564$		0			0
$565$	251900.			0.789098
$566$		0			0
$567$		−	146698.i		−	0.456308i
$568$		0			0
$569$	−434476.			−1.34197			−0.670983	−	0.741473i	$-0.734128\pi$
$569$	−434476.			−1.34197			−0.670983	+	0.741473i	$0.734128\pi$
$570$		0			0
$571$			63191.7i			0.193815i	0.995293	+	0.0969076i	$0.0308951\pi$
$571$			63191.7i			0.193815i	−0.995293	+	0.0969076i	$0.969105\pi$
$572$		0			0
$573$	408407.			1.24390
$574$		0			0
$575$	−103134.			−0.311938
$576$		0			0
$577$		−	524185.i		−	1.57447i	−0.616656	−	0.787233i	$-0.711513\pi$
$577$		−	524185.i		−	1.57447i	0.616656	−	0.787233i	$-0.288487\pi$
$578$		0			0
$579$			29797.8i			0.0888847i
$580$		0			0
$581$		−	230753.i		−	0.683589i
$582$		0			0
$583$	107778.			0.317096
$584$		0			0
$585$			28452.0i			0.0831383i
$586$		0			0
$587$			653761.i			1.89733i	0.316284	+	0.948665i	$0.397565\pi$
$587$			653761.i			1.89733i	−0.316284	+	0.948665i	$0.602435\pi$
$588$		0			0
$589$		−	345869.i		−	0.996969i
$590$		0			0
$591$		−	420724.i		−	1.20454i
$592$		0			0
$593$		−	73594.7i		−	0.209285i	−0.994510	−	0.104642i	$-0.966630\pi$
$593$		−	73594.7i		−	0.209285i	0.994510	−	0.104642i	$-0.0333698\pi$
$594$		0			0
$595$	129438.			0.365619
$596$		0			0
$597$	−526781.			−1.47802
$598$		0			0
$599$	219023.			0.610430			0.305215	−	0.952283i	$-0.401272\pi$
$599$	219023.			0.610430			0.305215	+	0.952283i	$0.401272\pi$
$600$		0			0
$601$			350767.i			0.971112i	0.874206	+	0.485556i	$0.161383\pi$
$601$			350767.i			0.971112i	−0.874206	+	0.485556i	$0.838617\pi$
$602$		0			0
$603$	178989.			0.492256
$604$		0			0
$605$			313316.i			0.855996i
$606$		0			0
$607$		−	93560.7i		−	0.253931i	−0.991907	−	0.126966i	$-0.959476\pi$
$607$		−	93560.7i		−	0.253931i	0.991907	−	0.126966i	$-0.0405238\pi$
$608$		0			0
$609$			153257.i			0.413224i
$610$		0			0
$611$	−6611.26			−0.0177093
$612$		0			0
$613$	385616.			1.02620			0.513102	−	0.858327i	$-0.328496\pi$
$613$	385616.			1.02620			0.513102	+	0.858327i	$0.328496\pi$
$614$		0			0
$615$	−90086.2			−0.238181
$616$		0			0
$617$	−262907.			−0.690609			−0.345304	−	0.938491i	$-0.612224\pi$
$617$	−262907.			−0.690609			−0.345304	+	0.938491i	$0.612224\pi$
$618$		0			0
$619$	−105010.			−0.274063			−0.137031	−	0.990567i	$-0.543756\pi$
$619$	−105010.			−0.274063			−0.137031	+	0.990567i	$0.543756\pi$
$620$		0			0
$621$			650596.i			1.68705i
$622$		0			0
$623$	513106.			1.32200
$624$		0			0
$625$	−296669.			−0.759472
$626$		0			0
$627$	−41477.4			−0.105506
$628$		0			0
$629$			69800.1i			0.176423i
$630$		0			0
$631$		−	584197.i		−	1.46724i	−0.679561	−	0.733619i	$-0.737830\pi$
$631$		−	584197.i		−	1.46724i	0.679561	−	0.733619i	$-0.262170\pi$
$632$		0			0
$633$	241899.			0.603709
$634$		0			0
$635$			295257.i			0.732238i
$636$		0			0
$637$	11676.0			0.0287751
$638$		0			0
$639$		−	191977.i		−	0.470162i
$640$		0			0
$641$		−	230667.i		−	0.561397i	−0.959796	−	0.280699i	$-0.909434\pi$
$641$		−	230667.i		−	0.561397i	0.959796	−	0.280699i	$-0.0905661\pi$
$642$		0			0
$643$	−205024.			−0.495886			−0.247943	−	0.968775i	$-0.579755\pi$
$643$	−205024.			−0.495886			−0.247943	+	0.968775i	$0.579755\pi$
$644$		0			0
$645$	−82024.3	+	274868.i	−0.197162	+	0.660700i
$646$		0			0
$647$			151853.i			0.362757i	0.983413	+	0.181379i	$0.0580560\pi$
$647$			151853.i			0.362757i	−0.983413	+	0.181379i	$0.941944\pi$
$648$		0			0
$649$	1636.96			0.00388641
$650$		0			0
$651$	522200.			1.23218
$652$		0			0
$653$		−	165779.i		−	0.388780i	−0.980924	−	0.194390i	$-0.937727\pi$
$653$		−	165779.i		−	0.388780i	0.980924	−	0.194390i	$-0.0622727\pi$
$654$		0			0
$655$	−658533.			−1.53495
$656$		0			0
$657$		−	156790.i		−	0.363235i
$658$		0			0
$659$	−461726.			−1.06320			−0.531599	−	0.846996i	$-0.678408\pi$
$659$	−461726.			−1.06320			−0.531599	+	0.846996i	$0.678408\pi$
$660$		0			0
$661$	−206270.			−0.472099			−0.236050	−	0.971741i	$-0.575853\pi$
$661$	−206270.			−0.472099			−0.236050	+	0.971741i	$0.575853\pi$
$662$		0			0
$663$			29951.8i			0.0681390i
$664$		0			0
$665$		−	277665.i		−	0.627881i
$666$		0			0
$667$		−	349865.i		−	0.786410i
$668$		0			0
$669$	−484873.			−1.08337
$670$		0			0
$671$		−	97476.0i		−	0.216497i
$672$		0			0
$673$		−	739539.i		−	1.63279i	−0.577492	−	0.816397i	$-0.695968\pi$
$673$		−	739539.i		−	1.63279i	0.577492	−	0.816397i	$-0.304032\pi$
$674$		0			0
$675$		−	98938.2i		−	0.217148i
$676$		0			0
$677$			248420.i			0.542012i	0.962578	+	0.271006i	$0.0873563\pi$
$677$			248420.i			0.542012i	−0.962578	+	0.271006i	$0.912644\pi$
$678$		0			0
$679$		−	403470.i		−	0.875129i
$680$		0			0
$681$	41306.4			0.0890682
$682$		0			0
$683$	−536792.			−1.15071			−0.575354	−	0.817905i	$-0.695136\pi$
$683$	−536792.			−1.15071			−0.575354	+	0.817905i	$0.695136\pi$
$684$		0			0
$685$	318377.			0.678517
$686$		0			0
$687$			220734.i			0.467688i
$688$		0			0
$689$	−166954.			−0.351689
$690$		0			0
$691$		−	253638.i		−	0.531199i	−0.964083	−	0.265600i	$-0.914430\pi$
$691$		−	253638.i		−	0.531199i	0.964083	−	0.265600i	$-0.0855699\pi$
$692$		0			0
$693$			42709.9i			0.0889329i
$694$		0			0
$695$		−	414670.i		−	0.858485i
$696$		0			0
$697$	64678.5			0.133136
$698$		0			0
$699$	−385192.			−0.788358
$700$		0			0
$701$	−64810.3			−0.131889			−0.0659444	−	0.997823i	$-0.521006\pi$
$701$	−64810.3			−0.131889			−0.0659444	+	0.997823i	$0.521006\pi$
$702$		0			0
$703$	149732.			0.302973
$704$		0			0
$705$	−26467.5			−0.0532518
$706$		0			0
$707$		−	411072.i		−	0.822393i
$708$		0			0
$709$	−120036.			−0.238792			−0.119396	−	0.992847i	$-0.538096\pi$
$709$	−120036.			−0.238792			−0.119396	+	0.992847i	$0.538096\pi$
$710$		0			0
$711$	−333010.			−0.658746
$712$		0			0
$713$	−1.19211e6			−2.34498
$714$		0			0
$715$			21671.0i			0.0423903i
$716$		0			0
$717$		−	436398.i		−	0.848877i
$718$		0			0
$719$	332719.			0.643605			0.321802	−	0.946807i	$-0.395711\pi$
$719$	332719.			0.643605			0.321802	+	0.946807i	$0.395711\pi$
$720$		0			0
$721$		−	601933.i		−	1.15792i
$722$		0			0
$723$	−134030.			−0.256404
$724$		0			0
$725$			53205.1i			0.101222i
$726$		0			0
$727$			582297.i			1.10173i	0.834594	+	0.550866i	$0.185702\pi$
$727$			582297.i			1.10173i	−0.834594	+	0.550866i	$0.814298\pi$
$728$		0			0
$729$	−536751.			−1.00999
$730$		0			0
$731$	58890.4	−	197345.i	0.110207	−	0.369309i
$732$		0			0
$733$			469633.i			0.874079i	0.899442	+	0.437039i	$0.143973\pi$
$733$			469633.i			0.874079i	−0.899442	+	0.437039i	$0.856027\pi$
$734$		0			0
$735$	46743.7			0.0865264
$736$		0			0
$737$	136330.			0.250990
$738$		0			0
$739$		−	755334.i		−	1.38309i	−0.722334	−	0.691545i	$-0.756930\pi$
$739$		−	755334.i		−	1.38309i	0.722334	−	0.691545i	$-0.243070\pi$
$740$		0			0
$741$	64251.2			0.117016
$742$		0			0
$743$			18481.9i			0.0334788i	0.999860	+	0.0167394i	$0.00532857\pi$
$743$			18481.9i			0.0334788i	−0.999860	+	0.0167394i	$0.994671\pi$
$744$		0			0
$745$	−693432.			−1.24937
$746$		0			0
$747$	−145790.			−0.261268
$748$		0			0
$749$			335993.i			0.598916i
$750$		0			0
$751$			62072.3i			0.110057i	0.998485	+	0.0550286i	$0.0175250\pi$
$751$			62072.3i			0.110057i	−0.998485	+	0.0550286i	$0.982475\pi$
$752$		0			0
$753$		−	437074.i		−	0.770842i
$754$		0			0
$755$	−225331.			−0.395300
$756$		0			0
$757$			452452.i			0.789551i	0.918778	+	0.394776i	$0.129178\pi$
$757$			452452.i			0.789551i	−0.918778	+	0.394776i	$0.870822\pi$
$758$		0			0
$759$			142961.i			0.248161i
$760$		0			0
$761$		−	239204.i		−	0.413047i	−0.978442	−	0.206523i	$-0.933785\pi$
$761$		−	239204.i		−	0.413047i	0.978442	−	0.206523i	$-0.0662150\pi$
$762$		0			0
$763$			177190.i			0.304361i
$764$		0			0
$765$		−	81779.2i		−	0.139740i
$766$		0			0
$767$	−2535.76			−0.00431039
$768$		0			0
$769$	838340.			1.41765			0.708823	−	0.705386i	$-0.249227\pi$
$769$	838340.			1.41765			0.708823	+	0.705386i	$0.249227\pi$
$770$		0			0
$771$	−372901.			−0.627314
$772$		0			0
$773$			520027.i			0.870295i	0.900359	+	0.435148i	$0.143304\pi$
$773$			520027.i			0.870295i	−0.900359	+	0.435148i	$0.856696\pi$
$774$		0			0
$775$	181288.			0.301833
$776$		0			0
$777$			226068.i			0.374453i
$778$		0			0
$779$		−	138745.i		−	0.228635i
$780$		0			0
$781$		−	146223.i		−	0.239725i
$782$		0			0
$783$	335630.			0.547441
$784$		0			0
$785$	621087.			1.00789
$786$		0			0
$787$	297689.			0.480633			0.240316	−	0.970695i	$-0.422749\pi$
$787$	297689.			0.480633			0.240316	+	0.970695i	$0.422749\pi$
$788$		0			0
$789$	925429.			1.48658
$790$		0			0
$791$	−585751.			−0.936181
$792$		0			0
$793$			150996.i			0.240116i
$794$		0			0
$795$	−668383.			−1.05753
$796$		0			0
$797$	943410.			1.48520			0.742598	−	0.669737i	$-0.233593\pi$
$797$	943410.			1.48520			0.742598	+	0.669737i	$0.233593\pi$
$798$		0			0
$799$	19002.6			0.0297660
$800$		0			0
$801$		−	324181.i		−	0.505269i
$802$		0			0
$803$		−	119422.i		−	0.185205i
$804$		0			0
$805$	−957031.			−1.47684
$806$		0			0
$807$		−	425532.i		−	0.653409i
$808$		0			0
$809$	365446.			0.558375			0.279187	−	0.960237i	$-0.409935\pi$
$809$	365446.			0.558375			0.279187	+	0.960237i	$0.409935\pi$
$810$		0			0
$811$			465367.i			0.707544i	0.935332	+	0.353772i	$0.115101\pi$
$811$			465367.i			0.707544i	−0.935332	+	0.353772i	$0.884899\pi$
$812$		0			0
$813$			458846.i			0.694201i
$814$		0			0
$815$	−520305.			−0.783327
$816$		0			0
$817$	−423334.	−	126329.i	−0.634219	−	0.189260i
$818$		0			0
$819$		−	66160.4i		−	0.0986348i
$820$		0			0
$821$	−391246.			−0.580449			−0.290224	−	0.956959i	$-0.593730\pi$
$821$	−391246.			−0.580449			−0.290224	+	0.956959i	$0.593730\pi$
$822$		0			0
$823$	329039.			0.485789			0.242895	−	0.970053i	$-0.421903\pi$
$823$	329039.			0.485789			0.242895	+	0.970053i	$0.421903\pi$
$824$		0			0
$825$		−	21740.5i		−	0.0319420i
$826$		0			0
$827$	42044.1			0.0614744			0.0307372	−	0.999528i	$-0.490215\pi$
$827$	42044.1			0.0614744			0.0307372	+	0.999528i	$0.490215\pi$
$828$		0			0
$829$		−	591458.i		−	0.860626i	−0.902680	−	0.430313i	$-0.858403\pi$
$829$		−	591458.i		−	0.860626i	0.902680	−	0.430313i	$-0.141597\pi$
$830$		0			0
$831$	−434755.			−0.629568
$832$		0			0
$833$	−33560.2			−0.0483654
$834$		0			0
$835$		−	860181.i		−	1.23372i
$836$		0			0
$837$		−	1.14361e6i		−	1.63240i
$838$		0			0
$839$		−	1.21494e6i		−	1.72597i	−0.505233	−	0.862983i	$-0.668593\pi$
$839$		−	1.21494e6i		−	1.72597i	0.505233	−	0.862983i	$-0.331407\pi$
$840$		0			0
$841$	526792.			0.744813
$842$		0			0
$843$		−	764973.i		−	1.07644i
$844$		0			0
$845$			604923.i			0.847202i
$846$		0			0
$847$		−	728563.i		−	1.01555i
$848$		0			0
$849$		−	758230.i		−	1.05193i
$850$		0			0
$851$		−	516083.i		−	0.712624i
$852$		0			0
$853$	−1.22261e6			−1.68031			−0.840154	−	0.542347i	$-0.817536\pi$
$853$	−1.22261e6			−1.68031			−0.840154	+	0.542347i	$0.817536\pi$
$854$		0			0
$855$	−175429.			−0.239976
$856$		0			0
$857$	−25443.6			−0.0346432			−0.0173216	−	0.999850i	$-0.505514\pi$
$857$	−25443.6			−0.0346432			−0.0173216	+	0.999850i	$0.505514\pi$
$858$		0			0
$859$			853100.i			1.15615i	0.815984	+	0.578074i	$0.196196\pi$
$859$			853100.i			1.15615i	−0.815984	+	0.578074i	$0.803804\pi$
$860$		0			0
$861$	209480.			0.282577
$862$		0			0
$863$			657351.i			0.882624i	0.897354	+	0.441312i	$0.145487\pi$
$863$			657351.i			0.882624i	−0.897354	+	0.441312i	$0.854513\pi$
$864$		0			0
$865$		−	591401.i		−	0.790406i
$866$		0			0
$867$			493504.i			0.656527i
$868$		0			0
$869$	−253643.			−0.335879
$870$		0			0
$871$	−211184.			−0.278371
$872$		0			0
$873$	−254913.			−0.334475
$874$		0			0
$875$	871862.			1.13876
$876$		0			0
$877$	−853878.			−1.11019			−0.555094	−	0.831788i	$-0.687318\pi$
$877$	−853878.			−1.11019			−0.555094	+	0.831788i	$0.687318\pi$
$878$		0			0
$879$			953364.i			1.23390i
$880$		0			0
$881$	170801.			0.220059			0.110030	−	0.993928i	$-0.464905\pi$
$881$	170801.			0.220059			0.110030	+	0.993928i	$0.464905\pi$
$882$		0			0
$883$	608395.			0.780304			0.390152	−	0.920750i	$-0.372422\pi$
$883$	608395.			0.780304			0.390152	+	0.920750i	$0.372422\pi$
$884$		0			0
$885$	−10151.6			−0.0129613
$886$		0			0
$887$			433884.i			0.551476i	0.961233	+	0.275738i	$0.0889222\pi$
$887$			433884.i			0.551476i	−0.961233	+	0.275738i	$0.911078\pi$
$888$		0			0
$889$		−	686570.i		−	0.868722i
$890$		0			0
$891$	−70594.4			−0.0889232
$892$		0			0
$893$		−	40763.6i		−	0.0511175i
$894$		0			0
$895$	129824.			0.162072
$896$		0			0
$897$		−	221455.i		−	0.275234i
$898$		0			0
$899$			614988.i			0.760935i
$900$		0			0
$901$	479874.			0.591122
$902$		0			0
$903$	190734.	−	639158.i	0.233912	−	0.783850i
$904$		0			0
$905$			1.42878e6i			1.74449i
$906$		0			0
$907$	1.51273e6			1.83885			0.919425	−	0.393265i	$-0.128655\pi$
$907$	1.51273e6			1.83885			0.919425	+	0.393265i	$0.128655\pi$
$908$		0			0
$909$	−259716.			−0.314319
$910$		0			0
$911$			1.18401e6i			1.42665i	0.700831	+	0.713327i	$0.252813\pi$
$911$			1.18401e6i			1.42665i	−0.700831	+	0.713327i	$0.747187\pi$
$912$		0			0
$913$	−111044.			−0.133215
$914$		0			0
$915$			604498.i			0.722025i
$916$		0			0
$917$	1.53131e6			1.82106
$918$		0			0
$919$	−367164.			−0.434740			−0.217370	−	0.976089i	$-0.569748\pi$
$919$	−367164.			−0.434740			−0.217370	+	0.976089i	$0.569748\pi$
$920$		0			0
$921$		−	90602.9i		−	0.106813i
$922$		0			0
$923$			226508.i			0.265877i
$924$		0			0
$925$			78482.3i			0.0917251i
$926$		0			0
$927$	−380302.			−0.442557
$928$		0			0
$929$			1.60017e6i			1.85410i	0.374936	+	0.927051i	$0.377665\pi$
$929$			1.60017e6i			1.85410i	−0.374936	+	0.927051i	$0.622335\pi$
$930$		0			0
$931$			71991.8i			0.0830585i
$932$		0			0
$933$		−	1.32464e6i		−	1.52173i
$934$		0			0
$935$		−	62288.6i		−	0.0712501i
$936$		0			0
$937$		−	241683.i		−	0.275275i	−0.990483	−	0.137638i	$-0.956049\pi$
$937$		−	241683.i		−	0.275275i	0.990483	−	0.137638i	$-0.0439510\pi$
$938$		0			0
$939$	221653.			0.251387
$940$		0			0
$941$	−586773.			−0.662660			−0.331330	−	0.943515i	$-0.607497\pi$
$941$	−586773.			−0.662660			−0.331330	+	0.943515i	$0.607497\pi$
$942$		0			0
$943$	−478215.			−0.537774
$944$		0			0
$945$		−	918091.i		−	1.02807i
$946$		0			0
$947$	1.14388e6			1.27550			0.637748	−	0.770245i	$-0.279866\pi$
$947$	1.14388e6			1.27550			0.637748	+	0.770245i	$0.279866\pi$
$948$		0			0
$949$			184992.i			0.205410i
$950$		0			0
$951$		−	74400.7i		−	0.0822652i
$952$		0			0
$953$		−	1.25121e6i		−	1.37766i	−0.724922	−	0.688831i	$-0.758124\pi$
$953$		−	1.25121e6i		−	1.37766i	0.724922	−	0.688831i	$-0.241876\pi$
$954$		0			0
$955$	1.31567e6			1.44259
$956$		0			0
$957$	73750.8			0.0805272
$958$		0			0
$959$	−740332.			−0.804988
$960$		0			0
$961$	1.17196e6			1.26901
$962$		0			0
$963$	212280.			0.228906
$964$		0			0
$965$			95992.8i			0.103082i
$966$		0			0
$967$	−687143.			−0.734843			−0.367421	−	0.930055i	$-0.619759\pi$
$967$	−687143.			−0.734843			−0.367421	+	0.930055i	$0.619759\pi$
$968$		0			0
$969$	−184676.			−0.196681
$970$		0			0
$971$	288966.			0.306484			0.153242	−	0.988189i	$-0.451029\pi$
$971$	288966.			0.306484			0.153242	+	0.988189i	$0.451029\pi$
$972$		0			0
$973$			964245.i			1.01850i
$974$		0			0
$975$			33677.4i			0.0354266i
$976$		0			0
$977$	1.40399e6			1.47087			0.735434	−	0.677597i	$-0.236978\pi$
$977$	1.40399e6			1.47087			0.735434	+	0.677597i	$0.236978\pi$
$978$		0			0
$979$		−	246918.i		−	0.257625i
$980$		0			0
$981$	111949.			0.116327
$982$		0			0
$983$			308569.i			0.319334i	0.987171	+	0.159667i	$0.0510421\pi$
$983$			308569.i			0.319334i	−0.987171	+	0.159667i	$0.948958\pi$
$984$		0			0
$985$		−	1.35535e6i		−	1.39695i
$986$		0			0
$987$	61545.6			0.0631776
$988$		0			0
$989$	−435419.	+	1.45911e6i	−0.445159	+	1.49175i
$990$		0			0
$991$		−	630206.i		−	0.641705i	−0.947129	−	0.320853i	$-0.896031\pi$
$991$		−	630206.i		−	0.641705i	0.947129	−	0.320853i	$-0.103969\pi$
$992$		0			0
$993$	−341039.			−0.345864
$994$		0			0
$995$	−1.69701e6			−1.71411
$996$		0			0
$997$			1.44327e6i			1.45197i	0.687713	+	0.725983i	$0.258615\pi$
$997$			1.44327e6i			1.45197i	−0.687713	+	0.725983i	$0.741385\pi$
$998$		0			0
$999$	495085.			0.496076

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.5.b.d.257.5		12
4.3	odd	2		43.5.b.b.42.3	✓	12
12.11	even	2		387.5.b.c.343.10		12
43.42	odd	2	inner	688.5.b.d.257.8		12
172.171	even	2		43.5.b.b.42.10	yes	12
516.515	odd	2		387.5.b.c.343.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.b.b.42.3	✓	12	4.3	odd	2
43.5.b.b.42.10	yes	12	172.171	even	2
387.5.b.c.343.3		12	516.515	odd	2
387.5.b.c.343.10		12	12.11	even	2
688.5.b.d.257.5		12	1.1	even	1	trivial
688.5.b.d.257.8		12	43.42	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:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	688.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.93950i q^{3} -22.3554i q^{5} +51.9837i q^{7} +32.8434 q^{9} +O(q^{10})\)	\(q-6.93950i q^{3} -22.3554i q^{5} +51.9837i q^{7} +32.8434 q^{9} +25.0158 q^{11} -38.7510 q^{13} -155.135 q^{15} +111.381 q^{17} -238.930i q^{19} +360.741 q^{21} -823.523 q^{23} +125.236 q^{25} -790.016i q^{27} +424.840i q^{29} +1447.58 q^{31} -173.597i q^{33} +1162.12 q^{35} +626.677i q^{37} +268.912i q^{39} +580.694 q^{41} +(528.728 - 1771.79i) q^{43} -734.227i q^{45} +170.609 q^{47} -301.309 q^{49} -772.930i q^{51} +4308.39 q^{53} -559.237i q^{55} -1658.05 q^{57} +65.4372 q^{59} -3896.58i q^{61} +1707.32i q^{63} +866.294i q^{65} +5449.76 q^{67} +5714.84i q^{69} -5845.23i q^{71} -4773.87i q^{73} -869.073i q^{75} +1300.41i q^{77} -10139.3 q^{79} -2822.00 q^{81} -4438.95 q^{83} -2489.97i q^{85} +2948.17 q^{87} -9870.51i q^{89} -2014.42i q^{91} -10045.5i q^{93} -5341.37 q^{95} -7761.47 q^{97} +821.602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 462 q^{9}+O(q^{10}) \) 12 * q - 462 * q^9	\( 12 q - 462 q^{9} + 180 q^{11} - 216 q^{13} + 92 q^{15} + 678 q^{17} - 2392 q^{21} - 1566 q^{23} - 174 q^{25} - 5710 q^{31} - 936 q^{35} + 4878 q^{41} + 1108 q^{43} + 5526 q^{47} - 8544 q^{49} + 1212 q^{53} - 7692 q^{57} - 14016 q^{59} + 1088 q^{67} - 24302 q^{79} - 23660 q^{81} + 7032 q^{83} - 17850 q^{87} - 606 q^{95} - 5842 q^{97} + 25924 q^{99}+O(q^{100}) \) 12 * q - 462 * q^9 + 180 * q^11 - 216 * q^13 + 92 * q^15 + 678 * q^17 - 2392 * q^21 - 1566 * q^23 - 174 * q^25 - 5710 * q^31 - 936 * q^35 + 4878 * q^41 + 1108 * q^43 + 5526 * q^47 - 8544 * q^49 + 1212 * q^53 - 7692 * q^57 - 14016 * q^59 + 1088 * q^67 - 24302 * q^79 - 23660 * q^81 + 7032 * q^83 - 17850 * q^87 - 606 * q^95 - 5842 * q^97 + 25924 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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