LMFDB - Embedded newform 688.5.b.d.257.7

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			257.7
Root			$3.65497i$ of defining polynomial
Character	$\chi$	$=$	688.257
Dual form			688.5.b.d.257.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.18965i q^{3} -45.6695i q^{5} +34.3337i q^{7} +63.4469 q^{9} +O(q^{10})$	$q+4.18965i q^{3} -45.6695i q^{5} +34.3337i q^{7} +63.4469 q^{9} +103.099 q^{11} +134.875 q^{13} +191.339 q^{15} +240.390 q^{17} -100.800i q^{19} -143.846 q^{21} +475.879 q^{23} -1460.70 q^{25} +605.181i q^{27} +159.825i q^{29} -1118.26 q^{31} +431.949i q^{33} +1568.00 q^{35} +2451.77i q^{37} +565.080i q^{39} +1069.81 q^{41} +(-1742.01 - 619.833i) q^{43} -2897.59i q^{45} +2933.56 q^{47} +1222.20 q^{49} +1007.15i q^{51} -825.685 q^{53} -4708.49i q^{55} +422.318 q^{57} +1241.46 q^{59} +6202.15i q^{61} +2178.37i q^{63} -6159.69i q^{65} -265.952 q^{67} +1993.77i q^{69} -3861.16i q^{71} -8274.70i q^{73} -6119.84i q^{75} +3539.78i q^{77} -4953.26 q^{79} +2603.70 q^{81} +13512.1 q^{83} -10978.5i q^{85} -669.609 q^{87} -5359.31i q^{89} +4630.77i q^{91} -4685.12i q^{93} -4603.51 q^{95} +3191.32 q^{97} +6541.32 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			4.18965i			0.465516i	0.972535	+	0.232758i	$0.0747751\pi$
$3$			4.18965i			0.465516i	−0.972535	+	0.232758i	$0.925225\pi$
$4$		0			0
$5$		−	45.6695i		−	1.82678i	−0.407085	−	0.913390i	$-0.633455\pi$
$5$		−	45.6695i		−	1.82678i	0.407085	−	0.913390i	$-0.366545\pi$
$6$		0			0
$7$			34.3337i			0.700688i	0.936621	+	0.350344i	$0.113935\pi$
$7$			34.3337i			0.700688i	−0.936621	+	0.350344i	$0.886065\pi$
$8$		0			0
$9$	63.4469			0.783295
$10$		0			0
$11$	103.099			0.852059			0.426030	−	0.904709i	$-0.359912\pi$
$11$	103.099			0.852059			0.426030	+	0.904709i	$0.359912\pi$
$12$		0			0
$13$	134.875			0.798079			0.399039	−	0.916934i	$-0.369344\pi$
$13$	134.875			0.798079			0.399039	+	0.916934i	$0.369344\pi$
$14$		0			0
$15$	191.339			0.850396
$16$		0			0
$17$	240.390			0.831800			0.415900	−	0.909410i	$-0.363467\pi$
$17$	240.390			0.831800			0.415900	+	0.909410i	$0.363467\pi$
$18$		0			0
$19$		−	100.800i		−	0.279225i	−0.990206	−	0.139613i	$-0.955414\pi$
$19$		−	100.800i		−	0.279225i	0.990206	−	0.139613i	$-0.0445858\pi$
$20$		0			0
$21$	−143.846			−0.326182
$22$		0			0
$23$	475.879			0.899582			0.449791	−	0.893134i	$-0.351498\pi$
$23$	475.879			0.899582			0.449791	+	0.893134i	$0.351498\pi$
$24$		0			0
$25$	−1460.70			−2.33713
$26$		0			0
$27$			605.181i			0.830153i
$28$		0			0
$29$			159.825i			0.190041i	0.995475	+	0.0950207i	$0.0302917\pi$
$29$			159.825i			0.190041i	−0.995475	+	0.0950207i	$0.969708\pi$
$30$		0			0
$31$	−1118.26			−1.16364			−0.581822	−	0.813316i	$-0.697660\pi$
$31$	−1118.26			−1.16364			−0.581822	+	0.813316i	$0.697660\pi$
$32$		0			0
$33$			431.949i			0.396648i
$34$		0			0
$35$	1568.00			1.28000
$36$		0			0
$37$			2451.77i			1.79092i	0.445145	+	0.895459i	$0.353152\pi$
$37$			2451.77i			1.79092i	−0.445145	+	0.895459i	$0.646848\pi$
$38$		0			0
$39$			565.080i			0.371519i
$40$		0			0
$41$	1069.81			0.636416			0.318208	−	0.948021i	$-0.396919\pi$
$41$	1069.81			0.636416			0.318208	+	0.948021i	$0.396919\pi$
$42$		0			0
$43$	−1742.01	−	619.833i	−0.942138	−	0.335226i
$43$	−1742.01	−	619.833i	−0.942138	−	0.335226i
$44$		0			0
$45$		−	2897.59i		−	1.43091i
$46$		0			0
$47$	2933.56			1.32800			0.664002	−	0.747731i	$-0.268857\pi$
$47$	2933.56			1.32800			0.664002	+	0.747731i	$0.268857\pi$
$48$		0			0
$49$	1222.20			0.509036
$50$		0			0
$51$			1007.15i			0.387216i
$52$		0			0
$53$	−825.685			−0.293943			−0.146971	−	0.989141i	$-0.546953\pi$
$53$	−825.685			−0.293943			−0.146971	+	0.989141i	$0.546953\pi$
$54$		0			0
$55$		−	4708.49i		−	1.55653i
$56$		0			0
$57$	422.318			0.129984
$58$		0			0
$59$	1241.46			0.356639			0.178320	−	0.983973i	$-0.442934\pi$
$59$	1241.46			0.356639			0.178320	+	0.983973i	$0.442934\pi$
$60$		0			0
$61$			6202.15i			1.66680i	0.552673	+	0.833398i	$0.313608\pi$
$61$			6202.15i			1.66680i	−0.552673	+	0.833398i	$0.686392\pi$
$62$		0			0
$63$			2178.37i			0.548845i
$64$		0			0
$65$		−	6159.69i		−	1.45791i
$66$		0			0
$67$	−265.952			−0.0592452			−0.0296226	−	0.999561i	$-0.509431\pi$
$67$	−265.952			−0.0592452			−0.0296226	+	0.999561i	$0.509431\pi$
$68$		0			0
$69$			1993.77i			0.418770i
$70$		0			0
$71$		−	3861.16i		−	0.765952i	−0.923758	−	0.382976i	$-0.874899\pi$
$71$		−	3861.16i		−	0.765952i	0.923758	−	0.382976i	$-0.125101\pi$
$72$		0			0
$73$		−	8274.70i		−	1.55277i	−0.630260	−	0.776384i	$-0.717052\pi$
$73$		−	8274.70i		−	1.55277i	0.630260	−	0.776384i	$-0.282948\pi$
$74$		0			0
$75$		−	6119.84i		−	1.08797i
$76$		0			0
$77$			3539.78i			0.597028i
$78$		0			0
$79$	−4953.26			−0.793665			−0.396833	−	0.917891i	$-0.629891\pi$
$79$	−4953.26			−0.793665			−0.396833	+	0.917891i	$0.629891\pi$
$80$		0			0
$81$	2603.70			0.396845
$82$		0			0
$83$	13512.1			1.96140			0.980700	−	0.195519i	$-0.0626392\pi$
$83$	13512.1			1.96140			0.980700	+	0.195519i	$0.0626392\pi$
$84$		0			0
$85$		−	10978.5i		−	1.51952i
$86$		0			0
$87$	−669.609			−0.0884673
$88$		0			0
$89$		−	5359.31i		−	0.676595i	−0.941039	−	0.338298i	$-0.890149\pi$
$89$		−	5359.31i		−	0.676595i	0.941039	−	0.338298i	$-0.109851\pi$
$90$		0			0
$91$			4630.77i			0.559204i
$92$		0			0
$93$		−	4685.12i		−	0.541695i
$94$		0			0
$95$	−4603.51			−0.510084
$96$		0			0
$97$	3191.32			0.339177			0.169588	−	0.985515i	$-0.445756\pi$
$97$	3191.32			0.339177			0.169588	+	0.985515i	$0.445756\pi$
$98$		0			0
$99$	6541.32			0.667413
$100$		0			0
$101$	6784.27			0.665059			0.332529	−	0.943093i	$-0.392098\pi$
$101$	6784.27			0.665059			0.332529	+	0.943093i	$0.392098\pi$
$102$		0			0
$103$	650.667			0.0613316			0.0306658	−	0.999530i	$-0.490237\pi$
$103$	650.667			0.0613316			0.0306658	+	0.999530i	$0.490237\pi$
$104$		0			0
$105$			6569.39i			0.595863i
$106$		0			0
$107$	−15902.5			−1.38899			−0.694495	−	0.719497i	$-0.744372\pi$
$107$	−15902.5			−1.38899			−0.694495	+	0.719497i	$0.744372\pi$
$108$		0			0
$109$	17290.7			1.45533			0.727663	−	0.685935i	$-0.240607\pi$
$109$	17290.7			1.45533			0.727663	+	0.685935i	$0.240607\pi$
$110$		0			0
$111$	−10272.0			−0.833701
$112$		0			0
$113$		−	15504.7i		−	1.21425i	−0.794607	−	0.607124i	$-0.792323\pi$
$113$		−	15504.7i		−	1.21425i	0.794607	−	0.607124i	$-0.207677\pi$
$114$		0			0
$115$		−	21733.2i		−	1.64334i
$116$		0			0
$117$	8557.41			0.625131
$118$		0			0
$119$			8253.49i			0.582832i
$120$		0			0
$121$	−4011.56			−0.273995
$122$		0			0
$123$			4482.15i			0.296262i
$124$		0			0
$125$			38166.2i			2.44264i
$126$		0			0
$127$	24303.2			1.50680			0.753400	−	0.657562i	$-0.228412\pi$
$127$	24303.2			1.50680			0.753400	+	0.657562i	$0.228412\pi$
$128$		0			0
$129$	2596.88	−	7298.42i	0.156053	−	0.438581i
$130$		0			0
$131$			6058.45i			0.353036i	0.984297	+	0.176518i	$0.0564834\pi$
$131$			6058.45i			0.353036i	−0.984297	+	0.176518i	$0.943517\pi$
$132$		0			0
$133$	3460.85			0.195650
$134$		0			0
$135$	27638.3			1.51651
$136$		0			0
$137$		−	20546.4i		−	1.09470i	−0.836905	−	0.547349i	$-0.815637\pi$
$137$		−	20546.4i		−	1.09470i	0.836905	−	0.547349i	$-0.184363\pi$
$138$		0			0
$139$	7143.71			0.369738			0.184869	−	0.982763i	$-0.440814\pi$
$139$	7143.71			0.369738			0.184869	+	0.982763i	$0.440814\pi$
$140$		0			0
$141$			12290.6i			0.618208i
$142$		0			0
$143$	13905.5			0.680010
$144$		0			0
$145$	7299.12			0.347164
$146$		0			0
$147$			5120.57i			0.236965i
$148$		0			0
$149$			29115.5i			1.31145i	0.754999	+	0.655726i	$0.227637\pi$
$149$			29115.5i			1.31145i	−0.754999	+	0.655726i	$0.772363\pi$
$150$		0			0
$151$			11488.6i			0.503864i	0.967745	+	0.251932i	$0.0810659\pi$
$151$			11488.6i			0.503864i	−0.967745	+	0.251932i	$0.918934\pi$
$152$		0			0
$153$	15252.0			0.651544
$154$		0			0
$155$			51070.5i			2.12572i
$156$		0			0
$157$		−	2810.63i		−	0.114026i	−0.998373	−	0.0570131i	$-0.981842\pi$
$157$		−	2810.63i		−	0.114026i	0.998373	−	0.0570131i	$-0.0181577\pi$
$158$		0			0
$159$		−	3459.33i		−	0.136835i
$160$		0			0
$161$			16338.7i			0.630327i
$162$		0			0
$163$			506.838i			0.0190763i	0.999955	+	0.00953815i	$0.00303613\pi$
$163$			506.838i			0.0190763i	−0.999955	+	0.00953815i	$0.996964\pi$
$164$		0			0
$165$	19726.9			0.724588
$166$		0			0
$167$	45960.0			1.64796			0.823980	−	0.566619i	$-0.191749\pi$
$167$	45960.0			1.64796			0.823980	+	0.566619i	$0.191749\pi$
$168$		0			0
$169$	−10369.6			−0.363070
$170$		0			0
$171$		−	6395.47i		−	0.218716i
$172$		0			0
$173$	−23016.4			−0.769033			−0.384516	−	0.923118i	$-0.625632\pi$
$173$	−23016.4			−0.769033			−0.384516	+	0.923118i	$0.625632\pi$
$174$		0			0
$175$		−	50151.4i		−	1.63760i
$176$		0			0
$177$			5201.28i			0.166021i
$178$		0			0
$179$		−	54490.4i		−	1.70065i	−0.526260	−	0.850324i	$-0.676406\pi$
$179$		−	54490.4i		−	1.70065i	0.526260	−	0.850324i	$-0.323594\pi$
$180$		0			0
$181$	24427.1			0.745616			0.372808	−	0.927909i	$-0.378395\pi$
$181$	24427.1			0.745616			0.372808	+	0.927909i	$0.378395\pi$
$182$		0			0
$183$	−25984.8			−0.775921
$184$		0			0
$185$	111971.			3.27161
$186$		0			0
$187$	24784.0			0.708743
$188$		0			0
$189$	−20778.1			−0.581678
$190$		0			0
$191$			25639.4i			0.702815i	0.936223	+	0.351408i	$0.114297\pi$
$191$			25639.4i			0.702815i	−0.936223	+	0.351408i	$0.885703\pi$
$192$		0			0
$193$	−45243.0			−1.21461			−0.607305	−	0.794469i	$-0.707749\pi$
$193$	−45243.0			−1.21461			−0.607305	+	0.794469i	$0.707749\pi$
$194$		0			0
$195$	25806.9			0.678683
$196$		0			0
$197$	−19109.6			−0.492401			−0.246200	−	0.969219i	$-0.579182\pi$
$197$	−19109.6			−0.492401			−0.246200	+	0.969219i	$0.579182\pi$
$198$		0			0
$199$			41414.6i			1.04580i	0.852395	+	0.522899i	$0.175149\pi$
$199$			41414.6i			1.04580i	−0.852395	+	0.522899i	$0.824851\pi$
$200$		0			0
$201$		−	1114.24i		−	0.0275796i
$202$		0			0
$203$	−5487.38			−0.133160
$204$		0			0
$205$		−	48857.9i		−	1.16259i
$206$		0			0
$207$	30193.0			0.704638
$208$		0			0
$209$		−	10392.4i		−	0.237917i
$210$		0			0
$211$		−	40579.3i		−	0.911464i	−0.890117	−	0.455732i	$-0.849377\pi$
$211$		−	40579.3i		−	0.911464i	0.890117	−	0.455732i	$-0.150623\pi$
$212$		0			0
$213$	16176.9			0.356563
$214$		0			0
$215$	−28307.5	+	79556.9i	−0.612384	+	1.72108i
$216$		0			0
$217$		−	38394.1i		−	0.815351i
$218$		0			0
$219$	34668.1			0.722839
$220$		0			0
$221$	32422.7			0.663842
$222$		0			0
$223$		−	76426.0i		−	1.53685i	−0.639940	−	0.768425i	$-0.721041\pi$
$223$		−	76426.0i		−	1.53685i	0.639940	−	0.768425i	$-0.278959\pi$
$224$		0			0
$225$	−92677.1			−1.83066
$226$		0			0
$227$		−	44850.6i		−	0.870395i	−0.900335	−	0.435197i	$-0.856679\pi$
$227$		−	44850.6i		−	0.870395i	0.900335	−	0.435197i	$-0.143321\pi$
$228$		0			0
$229$	−68025.3			−1.29718			−0.648589	−	0.761139i	$-0.724640\pi$
$229$	−68025.3			−1.29718			−0.648589	+	0.761139i	$0.724640\pi$
$230$		0			0
$231$	−14830.4			−0.277926
$232$		0			0
$233$		−	61038.6i		−	1.12433i	−0.827026	−	0.562164i	$-0.809969\pi$
$233$		−	61038.6i		−	1.12433i	0.827026	−	0.562164i	$-0.190031\pi$
$234$		0			0
$235$		−	133974.i		−	2.42597i
$236$		0			0
$237$		−	20752.4i		−	0.369464i
$238$		0			0
$239$	18090.2			0.316699			0.158349	−	0.987383i	$-0.449383\pi$
$239$	18090.2			0.316699			0.158349	+	0.987383i	$0.449383\pi$
$240$		0			0
$241$		−	63866.5i		−	1.09961i	−0.835293	−	0.549805i	$-0.814702\pi$
$241$		−	63866.5i		−	1.09961i	0.835293	−	0.549805i	$-0.185298\pi$
$242$		0			0
$243$			59928.3i			1.01489i
$244$		0			0
$245$		−	55817.1i		−	0.929897i
$246$		0			0
$247$		−	13595.5i		−	0.222844i
$248$		0			0
$249$			56610.9i			0.913064i
$250$		0			0
$251$	−29813.8			−0.473227			−0.236613	−	0.971604i	$-0.576037\pi$
$251$	−29813.8			−0.473227			−0.236613	+	0.971604i	$0.576037\pi$
$252$		0			0
$253$	49062.7			0.766497
$254$		0			0
$255$	45996.0			0.707359
$256$		0			0
$257$			56144.1i			0.850037i	0.905185	+	0.425018i	$0.139732\pi$
$257$			56144.1i			0.850037i	−0.905185	+	0.425018i	$0.860268\pi$
$258$		0			0
$259$	−84178.3			−1.25487
$260$		0			0
$261$			10140.4i			0.148858i
$262$		0			0
$263$			7995.51i			0.115594i	0.998328	+	0.0577969i	$0.0184076\pi$
$263$			7995.51i			0.115594i	−0.998328	+	0.0577969i	$0.981592\pi$
$264$		0			0
$265$			37708.6i			0.536969i
$266$		0			0
$267$	22453.6			0.314966
$268$		0			0
$269$	6211.47			0.0858401			0.0429200	−	0.999079i	$-0.486334\pi$
$269$	6211.47			0.0858401			0.0429200	+	0.999079i	$0.486334\pi$
$270$		0			0
$271$	15284.9			0.208125			0.104063	−	0.994571i	$-0.466816\pi$
$271$	15284.9			0.208125			0.104063	+	0.994571i	$0.466816\pi$
$272$		0			0
$273$	−19401.3			−0.260319
$274$		0			0
$275$	−150597.			−1.99137
$276$		0			0
$277$		−	44696.5i		−	0.582525i	−0.956643	−	0.291262i	$-0.905925\pi$
$277$		−	44696.5i		−	0.582525i	0.956643	−	0.291262i	$-0.0940753\pi$
$278$		0			0
$279$	−70950.2			−0.911476
$280$		0			0
$281$	91345.4			1.15684			0.578421	−	0.815739i	$-0.303669\pi$
$281$	91345.4			1.15684			0.578421	+	0.815739i	$0.303669\pi$
$282$		0			0
$283$	−25305.7			−0.315970			−0.157985	−	0.987442i	$-0.550500\pi$
$283$	−25305.7			−0.315970			−0.157985	+	0.987442i	$0.550500\pi$
$284$		0			0
$285$		−	19287.1i		−	0.237452i
$286$		0			0
$287$			36730.7i			0.445929i
$288$		0			0
$289$	−25733.6			−0.308109
$290$		0			0
$291$			13370.5i			0.157892i
$292$		0			0
$293$	5876.11			0.0684470			0.0342235	−	0.999414i	$-0.489104\pi$
$293$	5876.11			0.0684470			0.0342235	+	0.999414i	$0.489104\pi$
$294$		0			0
$295$		−	56696.9i		−	0.651501i
$296$		0			0
$297$			62393.7i			0.707339i
$298$		0			0
$299$	64184.3			0.717938
$300$		0			0
$301$	21281.2	−	59809.8i	0.234889	−	0.660145i
$302$		0			0
$303$			28423.7i			0.309596i
$304$		0			0
$305$	283249.			3.04487
$306$		0			0
$307$	82720.6			0.877681			0.438841	−	0.898565i	$-0.355389\pi$
$307$	82720.6			0.877681			0.438841	+	0.898565i	$0.355389\pi$
$308$		0			0
$309$			2726.07i			0.0285509i
$310$		0			0
$311$	−160383.			−1.65820			−0.829102	−	0.559097i	$-0.811148\pi$
$311$	−160383.			−1.65820			−0.829102	+	0.559097i	$0.811148\pi$
$312$		0			0
$313$			182768.i			1.86557i	0.360429	+	0.932787i	$0.382630\pi$
$313$			182768.i			1.86557i	−0.360429	+	0.932787i	$0.617370\pi$
$314$		0			0
$315$	99485.0			1.00262
$316$		0			0
$317$	−99340.5			−0.988571			−0.494286	−	0.869300i	$-0.664570\pi$
$317$	−99340.5			−0.988571			−0.494286	+	0.869300i	$0.664570\pi$
$318$		0			0
$319$			16477.8i			0.161926i
$320$		0			0
$321$		−	66626.1i		−	0.646598i
$322$		0			0
$323$		−	24231.4i		−	0.232260i
$324$		0			0
$325$	−197013.			−1.86521
$326$		0			0
$327$			72442.0i			0.677478i
$328$		0			0
$329$			100720.i			0.930517i
$330$		0			0
$331$			128433.i			1.17225i	0.810221	+	0.586124i	$0.199347\pi$
$331$			128433.i			1.17225i	−0.810221	+	0.586124i	$0.800653\pi$
$332$		0			0
$333$			155557.i			1.40282i
$334$		0			0
$335$			12145.9i			0.108228i
$336$		0			0
$337$	−78727.5			−0.693213			−0.346606	−	0.938011i	$-0.612666\pi$
$337$	−78727.5			−0.693213			−0.346606	+	0.938011i	$0.612666\pi$
$338$		0			0
$339$	64959.4			0.565252
$340$		0			0
$341$	−115292.			−0.991493
$342$		0			0
$343$			124398.i			1.05736i
$344$		0			0
$345$	91054.3			0.765001
$346$		0			0
$347$			136310.i			1.13205i	0.824387	+	0.566027i	$0.191520\pi$
$347$			136310.i			1.13205i	−0.824387	+	0.566027i	$0.808480\pi$
$348$		0			0
$349$		−	48465.9i		−	0.397910i	−0.980009	−	0.198955i	$-0.936245\pi$
$349$		−	48465.9i		−	0.397910i	0.980009	−	0.198955i	$-0.0637548\pi$
$350$		0			0
$351$			81624.0i			0.662527i
$352$		0			0
$353$	25854.2			0.207483			0.103741	−	0.994604i	$-0.466919\pi$
$353$	25854.2			0.207483			0.103741	+	0.994604i	$0.466919\pi$
$354$		0			0
$355$	−176337.			−1.39923
$356$		0			0
$357$	−34579.2			−0.271318
$358$		0			0
$359$	125163.			0.971155			0.485577	−	0.874194i	$-0.338609\pi$
$359$	125163.			0.971155			0.485577	+	0.874194i	$0.338609\pi$
$360$		0			0
$361$	120160.			0.922033
$362$		0			0
$363$		−	16807.0i		−	0.127549i
$364$		0			0
$365$	−377901.			−2.83657
$366$		0			0
$367$	−112207.			−0.833082			−0.416541	−	0.909117i	$-0.636758\pi$
$367$	−112207.			−0.833082			−0.416541	+	0.909117i	$0.636758\pi$
$368$		0			0
$369$	67876.4			0.498501
$370$		0			0
$371$		−	28348.9i		−	0.205962i
$372$		0			0
$373$		−	35635.8i		−	0.256135i	−0.991765	−	0.128068i	$-0.959123\pi$
$373$		−	35635.8i		−	0.256135i	0.991765	−	0.128068i	$-0.0408775\pi$
$374$		0			0
$375$	−159903.			−1.13709
$376$		0			0
$377$			21556.4i			0.151668i
$378$		0			0
$379$	205315.			1.42936			0.714679	−	0.699452i	$-0.246573\pi$
$379$	205315.			1.42936			0.714679	+	0.699452i	$0.246573\pi$
$380$		0			0
$381$			101822.i			0.701440i
$382$		0			0
$383$		−	164110.i		−	1.11876i	−0.828912	−	0.559380i	$-0.811039\pi$
$383$		−	164110.i		−	1.11876i	0.828912	−	0.559380i	$-0.188961\pi$
$384$		0			0
$385$	161660.			1.09064
$386$		0			0
$387$	−110525.	−	39326.5i	−0.737971	−	0.262581i
$388$		0			0
$389$			190984.i			1.26211i	0.775738	+	0.631055i	$0.217378\pi$
$389$			190984.i			1.26211i	−0.775738	+	0.631055i	$0.782622\pi$
$390$		0			0
$391$	114397.			0.748272
$392$		0			0
$393$	−25382.8			−0.164344
$394$		0			0
$395$			226213.i			1.44985i
$396$		0			0
$397$	−170575.			−1.08226			−0.541132	−	0.840938i	$-0.682004\pi$
$397$	−170575.			−1.08226			−0.541132	+	0.840938i	$0.682004\pi$
$398$		0			0
$399$			14499.8i			0.0910783i
$400$		0			0
$401$	52348.9			0.325551			0.162775	−	0.986663i	$-0.447955\pi$
$401$	52348.9			0.325551			0.162775	+	0.986663i	$0.447955\pi$
$402$		0			0
$403$	−150826.			−0.928679
$404$		0			0
$405$		−	118910.i		−	0.724949i
$406$		0			0
$407$			252775.i			1.52597i
$408$		0			0
$409$		−	164844.i		−	0.985430i	−0.870191	−	0.492715i	$-0.836004\pi$
$409$		−	164844.i		−	0.985430i	0.870191	−	0.492715i	$-0.163996\pi$
$410$		0			0
$411$	86082.0			0.509599
$412$		0			0
$413$			42624.0i			0.249893i
$414$		0			0
$415$		−	617090.i		−	3.58305i
$416$		0			0
$417$			29929.6i			0.172119i
$418$		0			0
$419$		−	37268.6i		−	0.212283i	−0.994351	−	0.106141i	$-0.966150\pi$
$419$		−	37268.6i		−	0.212283i	0.994351	−	0.106141i	$-0.0338496\pi$
$420$		0			0
$421$		−	99646.0i		−	0.562206i	−0.959678	−	0.281103i	$-0.909300\pi$
$421$		−	99646.0i		−	0.562206i	0.959678	−	0.281103i	$-0.0907003\pi$
$422$		0			0
$423$	186125.			1.04022
$424$		0			0
$425$	−351139.			−1.94402
$426$		0			0
$427$	−212943.			−1.16790
$428$		0			0
$429$			58259.3i			0.316556i
$430$		0			0
$431$	−144175.			−0.776133			−0.388067	−	0.921631i	$-0.626857\pi$
$431$	−144175.			−0.776133			−0.388067	+	0.921631i	$0.626857\pi$
$432$		0			0
$433$		−	220583.i		−	1.17651i	−0.808675	−	0.588256i	$-0.799815\pi$
$433$		−	220583.i		−	1.17651i	0.808675	−	0.588256i	$-0.200185\pi$
$434$		0			0
$435$			30580.7i			0.161610i
$436$		0			0
$437$		−	47968.8i		−	0.251186i
$438$		0			0
$439$	−68794.6			−0.356965			−0.178482	−	0.983943i	$-0.557119\pi$
$439$	−68794.6			−0.356965			−0.178482	+	0.983943i	$0.557119\pi$
$440$		0			0
$441$	77544.4			0.398725
$442$		0			0
$443$	−321595.			−1.63871			−0.819355	−	0.573286i	$-0.805668\pi$
$443$	−321595.			−1.63871			−0.819355	+	0.573286i	$0.805668\pi$
$444$		0			0
$445$	−244757.			−1.23599
$446$		0			0
$447$	−121984.			−0.610502
$448$		0			0
$449$			2687.03i			0.0133285i	0.999978	+	0.00666423i	$0.00212131\pi$
$449$			2687.03i			0.0133285i	−0.999978	+	0.00666423i	$0.997879\pi$
$450$		0			0
$451$	110297.			0.542264
$452$		0			0
$453$	−48133.2			−0.234557
$454$		0			0
$455$	211485.			1.02154
$456$		0			0
$457$			222198.i			1.06392i	0.846770	+	0.531959i	$0.178544\pi$
$457$			222198.i			1.06392i	−0.846770	+	0.531959i	$0.821456\pi$
$458$		0			0
$459$			145480.i			0.690521i
$460$		0			0
$461$	−63915.9			−0.300751			−0.150375	−	0.988629i	$-0.548048\pi$
$461$	−63915.9			−0.300751			−0.150375	+	0.988629i	$0.548048\pi$
$462$		0			0
$463$			94323.6i			0.440006i	0.975499	+	0.220003i	$0.0706067\pi$
$463$			94323.6i			0.440006i	−0.975499	+	0.220003i	$0.929393\pi$
$464$		0			0
$465$	−213967.			−0.989558
$466$		0			0
$467$		−	288915.i		−	1.32476i	−0.749170	−	0.662378i	$-0.769547\pi$
$467$		−	288915.i		−	1.32476i	0.749170	−	0.662378i	$-0.230453\pi$
$468$		0			0
$469$		−	9131.11i		−	0.0415124i
$470$		0			0
$471$	11775.5			0.0530810
$472$		0			0
$473$	−179600.	−	63904.3i	−0.802757	−	0.285632i
$474$		0			0
$475$			147240.i			0.652586i
$476$		0			0
$477$	−52387.1			−0.230244
$478$		0			0
$479$	−328139.			−1.43017			−0.715084	−	0.699039i	$-0.753612\pi$
$479$	−328139.			−1.43017			−0.715084	+	0.699039i	$0.753612\pi$
$480$		0			0
$481$			330683.i			1.42929i
$482$		0			0
$483$	−68453.4			−0.293427
$484$		0			0
$485$		−	145746.i		−	0.619602i
$486$		0			0
$487$	225179.			0.949447			0.474723	−	0.880135i	$-0.342548\pi$
$487$	225179.			0.949447			0.474723	+	0.880135i	$0.342548\pi$
$488$		0			0
$489$	−2123.47			−0.00888033
$490$		0			0
$491$		−	119348.i		−	0.495053i	−0.968881	−	0.247526i	$-0.920382\pi$
$491$		−	119348.i		−	0.495053i	0.968881	−	0.247526i	$-0.0796177\pi$
$492$		0			0
$493$			38420.3i			0.158076i
$494$		0			0
$495$		−	298739.i		−	1.21922i
$496$		0			0
$497$	132568.			0.536693
$498$		0			0
$499$		−	184960.i		−	0.742807i	−0.928471	−	0.371404i	$-0.878877\pi$
$499$		−	184960.i		−	0.742807i	0.928471	−	0.371404i	$-0.121123\pi$
$500$		0			0
$501$			192556.i			0.767153i
$502$		0			0
$503$		−	340779.i		−	1.34691i	−0.739230	−	0.673453i	$-0.764810\pi$
$503$		−	340779.i		−	1.34691i	0.739230	−	0.673453i	$-0.235190\pi$
$504$		0			0
$505$		−	309834.i		−	1.21492i
$506$		0			0
$507$		−	43445.2i		−	0.169015i
$508$		0			0
$509$	35186.6			0.135813			0.0679066	−	0.997692i	$-0.478368\pi$
$509$	35186.6			0.135813			0.0679066	+	0.997692i	$0.478368\pi$
$510$		0			0
$511$	284101.			1.08801
$512$		0			0
$513$	61002.5			0.231800
$514$		0			0
$515$		−	29715.7i		−	0.112039i
$516$		0			0
$517$	302448.			1.13154
$518$		0			0
$519$		−	96430.5i		−	0.357997i
$520$		0			0
$521$			167672.i			0.617712i	0.951109	+	0.308856i	$0.0999460\pi$
$521$			167672.i			0.617712i	−0.951109	+	0.308856i	$0.900054\pi$
$522$		0			0
$523$			388697.i			1.42105i	0.703674	+	0.710523i	$0.251541\pi$
$523$			388697.i			1.42105i	−0.703674	+	0.710523i	$0.748459\pi$
$524$		0			0
$525$	210117.			0.762329
$526$		0			0
$527$	−268819.			−0.967918
$528$		0			0
$529$	−53380.2			−0.190752
$530$		0			0
$531$	78766.8			0.279353
$532$		0			0
$533$	144292.			0.507910
$534$		0			0
$535$			726262.i			2.53738i
$536$		0			0
$537$	228296.			0.791679
$538$		0			0
$539$	126007.			0.433729
$540$		0			0
$541$	258688.			0.883856			0.441928	−	0.897050i	$-0.354295\pi$
$541$	258688.			0.883856			0.441928	+	0.897050i	$0.354295\pi$
$542$		0			0
$543$			102341.i			0.347096i
$544$		0			0
$545$		−	789659.i		−	2.65856i
$546$		0			0
$547$	11333.8			0.0378791			0.0189395	−	0.999821i	$-0.493971\pi$
$547$	11333.8			0.0378791			0.0189395	+	0.999821i	$0.493971\pi$
$548$		0			0
$549$			393507.i			1.30559i
$550$		0			0
$551$	16110.4			0.0530644
$552$		0			0
$553$		−	170064.i		−	0.556112i
$554$		0			0
$555$			469119.i			1.52299i
$556$		0			0
$557$	−90762.0			−0.292546			−0.146273	−	0.989244i	$-0.546728\pi$
$557$	−90762.0			−0.292546			−0.146273	+	0.989244i	$0.546728\pi$
$558$		0			0
$559$	−234955.	−	83600.2i	−0.751900	−	0.267537i
$560$		0			0
$561$			103836.i			0.329931i
$562$		0			0
$563$	−198527.			−0.626329			−0.313165	−	0.949699i	$-0.601389\pi$
$563$	−198527.			−0.626329			−0.313165	+	0.949699i	$0.601389\pi$
$564$		0			0
$565$	−708094.			−2.21817
$566$		0			0
$567$			89394.7i			0.278065i
$568$		0			0
$569$	204563.			0.631832			0.315916	−	0.948787i	$-0.397688\pi$
$569$	204563.			0.631832			0.315916	+	0.948787i	$0.397688\pi$
$570$		0			0
$571$			103248.i			0.316670i	0.987385	+	0.158335i	$0.0506127\pi$
$571$			103248.i			0.316670i	−0.987385	+	0.158335i	$0.949387\pi$
$572$		0			0
$573$	−107420.			−0.327172
$574$		0			0
$575$	−695119.			−2.10244
$576$		0			0
$577$		−	214909.i		−	0.645510i	−0.946482	−	0.322755i	$-0.895391\pi$
$577$		−	214909.i		−	0.645510i	0.946482	−	0.322755i	$-0.104609\pi$
$578$		0			0
$579$		−	189552.i		−	0.565421i
$580$		0			0
$581$			463920.i			1.37433i
$582$		0			0
$583$	−85127.5			−0.250457
$584$		0			0
$585$		−	390813.i		−	1.14198i
$586$		0			0
$587$			508043.i			1.47443i	0.675657	+	0.737216i	$0.263860\pi$
$587$			508043.i			1.47443i	−0.675657	+	0.737216i	$0.736140\pi$
$588$		0			0
$589$			112721.i			0.324919i
$590$		0			0
$591$		−	80062.4i		−	0.229221i
$592$		0			0
$593$		−	182995.i		−	0.520390i	−0.965556	−	0.260195i	$-0.916213\pi$
$593$		−	182995.i		−	0.520390i	0.965556	−	0.260195i	$-0.0837869\pi$
$594$		0			0
$595$	376933.			1.06471
$596$		0			0
$597$	−173513.			−0.486836
$598$		0			0
$599$	173564.			0.483733			0.241867	−	0.970310i	$-0.422240\pi$
$599$	173564.			0.483733			0.241867	+	0.970310i	$0.422240\pi$
$600$		0			0
$601$			370900.i			1.02685i	0.858134	+	0.513426i	$0.171624\pi$
$601$			370900.i			1.02685i	−0.858134	+	0.513426i	$0.828376\pi$
$602$		0			0
$603$	−16873.8			−0.0464064
$604$		0			0
$605$			183206.i			0.500529i
$606$		0			0
$607$			274917.i			0.746147i	0.927802	+	0.373074i	$0.121696\pi$
$607$			274917.i			0.746147i	−0.927802	+	0.373074i	$0.878304\pi$
$608$		0			0
$609$		−	22990.2i		−	0.0619880i
$610$		0			0
$611$	395665.			1.05985
$612$		0			0
$613$	358829.			0.954919			0.477459	−	0.878654i	$-0.341558\pi$
$613$	358829.			0.954919			0.477459	+	0.878654i	$0.341558\pi$
$614$		0			0
$615$	204697.			0.541205
$616$		0			0
$617$	275204.			0.722909			0.361455	−	0.932390i	$-0.382280\pi$
$617$	275204.			0.722909			0.361455	+	0.932390i	$0.382280\pi$
$618$		0			0
$619$	303567.			0.792270			0.396135	−	0.918192i	$-0.370351\pi$
$619$	303567.			0.792270			0.396135	+	0.918192i	$0.370351\pi$
$620$		0			0
$621$			287993.i			0.746791i
$622$		0			0
$623$	184005.			0.474082
$624$		0			0
$625$	830093.			2.12504
$626$		0			0
$627$	43540.6			0.110754
$628$		0			0
$629$			589380.i			1.48968i
$630$		0			0
$631$		−	350000.i		−	0.879041i	−0.898233	−	0.439520i	$-0.855148\pi$
$631$		−	350000.i		−	0.879041i	0.898233	−	0.439520i	$-0.144852\pi$
$632$		0			0
$633$	170013.			0.424302
$634$		0			0
$635$		−	1.10991e6i		−	2.75259i
$636$		0			0
$637$	164844.			0.406251
$638$		0			0
$639$		−	244979.i		−	0.599966i
$640$		0			0
$641$		−	404169.i		−	0.983665i	−0.870690	−	0.491832i	$-0.836327\pi$
$641$		−	404169.i		−	0.983665i	0.870690	−	0.491832i	$-0.163673\pi$
$642$		0			0
$643$	−488598.			−1.18176			−0.590881	−	0.806759i	$-0.701220\pi$
$643$	−488598.			−1.18176			−0.590881	+	0.806759i	$0.701220\pi$
$644$		0			0
$645$	−333315.	−	118598.i	−0.801190	−	0.285075i
$646$		0			0
$647$			111849.i			0.267191i	0.991036	+	0.133595i	$0.0426523\pi$
$647$			111849.i			0.267191i	−0.991036	+	0.133595i	$0.957348\pi$
$648$		0			0
$649$	127994.			0.303878
$650$		0			0
$651$	160858.			0.379559
$652$		0			0
$653$		−	92008.4i		−	0.215775i	−0.994163	−	0.107887i	$-0.965591\pi$
$653$		−	92008.4i		−	0.215775i	0.994163	−	0.107887i	$-0.0344086\pi$
$654$		0			0
$655$	276686.			0.644919
$656$		0			0
$657$		−	525004.i		−	1.21627i
$658$		0			0
$659$	532415.			1.22597			0.612985	−	0.790095i	$-0.289969\pi$
$659$	532415.			1.22597			0.612985	+	0.790095i	$0.289969\pi$
$660$		0			0
$661$	−770144.			−1.76266			−0.881331	−	0.472499i	$-0.843352\pi$
$661$	−770144.			−1.76266			−0.881331	+	0.472499i	$0.843352\pi$
$662$		0			0
$663$			135840.i			0.309029i
$664$		0			0
$665$		−	158055.i		−	0.357410i
$666$		0			0
$667$			76057.3i			0.170958i
$668$		0			0
$669$	320198.			0.715429
$670$		0			0
$671$			639437.i			1.42021i
$672$		0			0
$673$			296438.i			0.654492i	0.944939	+	0.327246i	$0.106121\pi$
$673$			296438.i			0.654492i	−0.944939	+	0.327246i	$0.893879\pi$
$674$		0			0
$675$		−	883991.i		−	1.94017i
$676$		0			0
$677$		−	255234.i		−	0.556880i	−0.960454	−	0.278440i	$-0.910183\pi$
$677$		−	255234.i		−	0.556880i	0.960454	−	0.278440i	$-0.0898174\pi$
$678$		0			0
$679$			109570.i			0.237657i
$680$		0			0
$681$	187908.			0.405183
$682$		0			0
$683$	−659236.			−1.41319			−0.706594	−	0.707620i	$-0.749769\pi$
$683$	−659236.			−1.41319			−0.706594	+	0.707620i	$0.749769\pi$
$684$		0			0
$685$	−938343.			−1.99977
$686$		0			0
$687$		−	285002.i		−	0.603857i
$688$		0			0
$689$	−111365.			−0.234589
$690$		0			0
$691$			428774.i			0.897993i	0.893534	+	0.448996i	$0.148218\pi$
$691$			428774.i			0.897993i	−0.893534	+	0.448996i	$0.851782\pi$
$692$		0			0
$693$			224588.i			0.467649i
$694$		0			0
$695$		−	326250.i		−	0.675430i
$696$		0			0
$697$	257173.			0.529370
$698$		0			0
$699$	255730.			0.523393
$700$		0			0
$701$	446529.			0.908686			0.454343	−	0.890827i	$-0.349874\pi$
$701$	446529.			0.908686			0.454343	+	0.890827i	$0.349874\pi$
$702$		0			0
$703$	247139.			0.500070
$704$		0			0
$705$	561305.			1.12933
$706$		0			0
$707$			232929.i			0.465999i
$708$		0			0
$709$	−43598.6			−0.0867321			−0.0433660	−	0.999059i	$-0.513808\pi$
$709$	−43598.6			−0.0867321			−0.0433660	+	0.999059i	$0.513808\pi$
$710$		0			0
$711$	−314269.			−0.621673
$712$		0			0
$713$	−532157.			−1.04679
$714$		0			0
$715$		−	635059.i		−	1.24223i
$716$		0			0
$717$			75791.4i			0.147429i
$718$		0			0
$719$	−46047.8			−0.0890741			−0.0445370	−	0.999008i	$-0.514181\pi$
$719$	−46047.8			−0.0890741			−0.0445370	+	0.999008i	$0.514181\pi$
$720$		0			0
$721$			22339.8i			0.0429743i
$722$		0			0
$723$	267578.			0.511887
$724$		0			0
$725$		−	233457.i		−	0.444151i
$726$		0			0
$727$		−	275601.i		−	0.521450i	−0.965413	−	0.260725i	$-0.916038\pi$
$727$		−	275601.i		−	0.521450i	0.965413	−	0.260725i	$-0.0839615\pi$
$728$		0			0
$729$	−40178.7			−0.0756032
$730$		0			0
$731$	−418763.	−	149002.i	−0.783670	−	0.278841i
$732$		0			0
$733$		−	324631.i		−	0.604201i	−0.953276	−	0.302101i	$-0.902312\pi$
$733$		−	324631.i		−	0.604201i	0.953276	−	0.302101i	$-0.0976879\pi$
$734$		0			0
$735$	233854.			0.432882
$736$		0			0
$737$	−27419.4			−0.0504804
$738$		0			0
$739$			359031.i			0.657421i	0.944431	+	0.328710i	$0.106614\pi$
$739$			359031.i			0.657421i	−0.944431	+	0.328710i	$0.893386\pi$
$740$		0			0
$741$	56960.3			0.103737
$742$		0			0
$743$			164101.i			0.297259i	0.988893	+	0.148629i	$0.0474862\pi$
$743$			164101.i			0.297259i	−0.988893	+	0.148629i	$0.952514\pi$
$744$		0			0
$745$	1.32969e6			2.39573
$746$		0			0
$747$	857299.			1.53635
$748$		0			0
$749$		−	545994.i		−	0.973249i
$750$		0			0
$751$		−	862173.i		−	1.52867i	−0.644818	−	0.764336i	$-0.723067\pi$
$751$		−	862173.i		−	1.52867i	0.644818	−	0.764336i	$-0.276933\pi$
$752$		0			0
$753$		−	124909.i		−	0.220295i
$754$		0			0
$755$	524679.			0.920449
$756$		0			0
$757$			878771.i			1.53350i	0.641946	+	0.766750i	$0.278127\pi$
$757$			878771.i			1.53350i	−0.641946	+	0.766750i	$0.721873\pi$
$758$		0			0
$759$			205556.i			0.356817i
$760$		0			0
$761$		−	452736.i		−	0.781763i	−0.920441	−	0.390882i	$-0.872170\pi$
$761$		−	452736.i		−	0.781763i	0.920441	−	0.390882i	$-0.127830\pi$
$762$		0			0
$763$			593655.i			1.01973i
$764$		0			0
$765$		−	696551.i		−	1.19023i
$766$		0			0
$767$	167442.			0.284626
$768$		0			0
$769$	379858.			0.642346			0.321173	−	0.947021i	$-0.395923\pi$
$769$	379858.			0.642346			0.321173	+	0.947021i	$0.395923\pi$
$770$		0			0
$771$	−235224.			−0.395706
$772$		0			0
$773$			466697.i			0.781044i	0.920594	+	0.390522i	$0.127705\pi$
$773$			466697.i			0.781044i	−0.920594	+	0.390522i	$0.872295\pi$
$774$		0			0
$775$	1.63345e6			2.71958
$776$		0			0
$777$		−	352677.i		−	0.584165i
$778$		0			0
$779$		−	107838.i		−	0.177703i
$780$		0			0
$781$		−	398083.i		−	0.652636i
$782$		0			0
$783$	−96723.0			−0.157763
$784$		0			0
$785$	−128360.			−0.208301
$786$		0			0
$787$	−946342.			−1.52791			−0.763956	−	0.645268i	$-0.776746\pi$
$787$	−946342.			−1.52791			−0.763956	+	0.645268i	$0.776746\pi$
$788$		0			0
$789$	−33498.3			−0.0538108
$790$		0			0
$791$	532335.			0.850810
$792$		0			0
$793$			836517.i			1.33024i
$794$		0			0
$795$	−157986.			−0.249968
$796$		0			0
$797$	−561047.			−0.883248			−0.441624	−	0.897200i	$-0.645597\pi$
$797$	−561047.			−0.883248			−0.441624	+	0.897200i	$0.645597\pi$
$798$		0			0
$799$	705199.			1.10463
$800$		0			0
$801$		−	340031.i		−	0.529973i
$802$		0			0
$803$		−	853115.i		−	1.32305i
$804$		0			0
$805$	746181.			1.15147
$806$		0			0
$807$			26023.9i			0.0399600i
$808$		0			0
$809$	−938377.			−1.43377			−0.716886	−	0.697190i	$-0.754433\pi$
$809$	−938377.			−1.43377			−0.716886	+	0.697190i	$0.754433\pi$
$810$		0			0
$811$			526766.i			0.800896i	0.916319	+	0.400448i	$0.131146\pi$
$811$			526766.i			0.800896i	−0.916319	+	0.400448i	$0.868854\pi$
$812$		0			0
$813$			64038.4i			0.0968856i
$814$		0			0
$815$	23147.1			0.0348482
$816$		0			0
$817$	−62479.4	+	175596.i	−0.0936036	+	0.263069i
$818$		0			0
$819$			293808.i			0.438022i
$820$		0			0
$821$	−1.19525e6			−1.77325			−0.886626	−	0.462486i	$-0.846957\pi$
$821$	−1.19525e6			−1.77325			−0.886626	+	0.462486i	$0.846957\pi$
$822$		0			0
$823$	−230617.			−0.340480			−0.170240	−	0.985403i	$-0.554454\pi$
$823$	−230617.			−0.340480			−0.170240	+	0.985403i	$0.554454\pi$
$824$		0			0
$825$		−	630950.i		−	0.927016i
$826$		0			0
$827$	353042.			0.516197			0.258099	−	0.966119i	$-0.416904\pi$
$827$	353042.			0.516197			0.258099	+	0.966119i	$0.416904\pi$
$828$		0			0
$829$			1.18220e6i			1.72021i	0.510116	+	0.860106i	$0.329603\pi$
$829$			1.18220e6i			1.72021i	−0.510116	+	0.860106i	$0.670397\pi$
$830$		0			0
$831$	187263.			0.271175
$832$		0			0
$833$	293804.			0.423416
$834$		0			0
$835$		−	2.09897e6i		−	3.01046i
$836$		0			0
$837$		−	676751.i		−	0.966002i
$838$		0			0
$839$			229327.i			0.325785i	0.986644	+	0.162892i	$0.0520823\pi$
$839$			229327.i			0.325785i	−0.986644	+	0.162892i	$0.947918\pi$
$840$		0			0
$841$	681737.			0.963884
$842$		0			0
$843$			382705.i			0.538529i
$844$		0			0
$845$			473577.i			0.663250i
$846$		0			0
$847$		−	137732.i		−	0.191985i
$848$		0			0
$849$		−	106022.i		−	0.147089i
$850$		0			0
$851$			1.16674e6i			1.61108i
$852$		0			0
$853$	736313.			1.01196			0.505981	−	0.862545i	$-0.331131\pi$
$853$	736313.			1.01196			0.505981	+	0.862545i	$0.331131\pi$
$854$		0			0
$855$	−292078.			−0.399546
$856$		0			0
$857$	1.18206e6			1.60945			0.804726	−	0.593646i	$-0.202312\pi$
$857$	1.18206e6			1.60945			0.804726	+	0.593646i	$0.202312\pi$
$858$		0			0
$859$			1.14109e6i			1.54645i	0.634133	+	0.773224i	$0.281357\pi$
$859$			1.14109e6i			1.54645i	−0.634133	+	0.773224i	$0.718643\pi$
$860$		0			0
$861$	−153889.			−0.207587
$862$		0			0
$863$		−	81940.4i		−	0.110021i	−0.998486	−	0.0550106i	$-0.982481\pi$
$863$		−	81940.4i		−	0.110021i	0.998486	−	0.0550106i	$-0.0175193\pi$
$864$		0			0
$865$			1.05115e6i			1.40485i
$866$		0			0
$867$		−	107815.i		−	0.143430i
$868$		0			0
$869$	−510677.			−0.676250
$870$		0			0
$871$	−35870.3			−0.0472823
$872$		0			0
$873$	202479.			0.265675
$874$		0			0
$875$	−1.31039e6			−1.71153
$876$		0			0
$877$	−347511.			−0.451824			−0.225912	−	0.974148i	$-0.572536\pi$
$877$	−347511.			−0.451824			−0.225912	+	0.974148i	$0.572536\pi$
$878$		0			0
$879$			24618.8i			0.0318632i
$880$		0			0
$881$	150733.			0.194204			0.0971018	−	0.995274i	$-0.469043\pi$
$881$	150733.			0.194204			0.0971018	+	0.995274i	$0.469043\pi$
$882$		0			0
$883$	1.40226e6			1.79849			0.899246	−	0.437443i	$-0.144116\pi$
$883$	1.40226e6			1.79849			0.899246	+	0.437443i	$0.144116\pi$
$884$		0			0
$885$	237540.			0.303285
$886$		0			0
$887$			900825.i			1.14497i	0.819916	+	0.572484i	$0.194020\pi$
$887$			900825.i			1.14497i	−0.819916	+	0.572484i	$0.805980\pi$
$888$		0			0
$889$			834419.i			1.05580i
$890$		0			0
$891$	268439.			0.338135
$892$		0			0
$893$		−	295704.i		−	0.370813i
$894$		0			0
$895$	−2.48855e6			−3.10671
$896$		0			0
$897$			268910.i			0.334212i
$898$		0			0
$899$		−	178726.i		−	0.221140i
$900$		0			0
$901$	−198487.			−0.244502
$902$		0			0
$903$	250582.	+	89160.6i	0.307308	+	0.109345i
$904$		0			0
$905$		−	1.11557e6i		−	1.36208i
$906$		0			0
$907$	−754265.			−0.916874			−0.458437	−	0.888727i	$-0.651591\pi$
$907$	−754265.			−0.916874			−0.458437	+	0.888727i	$0.651591\pi$
$908$		0			0
$909$	430440.			0.520937
$910$		0			0
$911$			834583.i			1.00562i	0.864398	+	0.502809i	$0.167700\pi$
$911$			834583.i			1.00562i	−0.864398	+	0.502809i	$0.832300\pi$
$912$		0			0
$913$	1.39308e6			1.67123
$914$		0			0
$915$			1.18671e6i			1.41744i
$916$		0			0
$917$	−208009.			−0.247368
$918$		0			0
$919$	558842.			0.661696			0.330848	−	0.943684i	$-0.392665\pi$
$919$	558842.			0.661696			0.330848	+	0.943684i	$0.392665\pi$
$920$		0			0
$921$			346570.i			0.408575i
$922$		0			0
$923$		−	520776.i		−	0.611290i
$924$		0			0
$925$		−	3.58131e6i		−	4.18560i
$926$		0			0
$927$	41282.8			0.0480407
$928$		0			0
$929$			911473.i			1.05612i	0.849208	+	0.528059i	$0.177080\pi$
$929$			911473.i			1.05612i	−0.849208	+	0.528059i	$0.822920\pi$
$930$		0			0
$931$		−	123198.i		−	0.142136i
$932$		0			0
$933$		−	671949.i		−	0.771921i
$934$		0			0
$935$		−	1.13187e6i		−	1.29472i
$936$		0			0
$937$			297178.i			0.338484i	0.985575	+	0.169242i	$0.0541319\pi$
$937$			297178.i			0.338484i	−0.985575	+	0.169242i	$0.945868\pi$
$938$		0			0
$939$	−765735.			−0.868455
$940$		0			0
$941$	−44565.6			−0.0503292			−0.0251646	−	0.999683i	$-0.508011\pi$
$941$	−44565.6			−0.0503292			−0.0251646	+	0.999683i	$0.508011\pi$
$942$		0			0
$943$	509102.			0.572508
$944$		0			0
$945$			948927.i			1.06260i
$946$		0			0
$947$	1.06777e6			1.19064			0.595319	−	0.803490i	$-0.297026\pi$
$947$	1.06777e6			1.19064			0.595319	+	0.803490i	$0.297026\pi$
$948$		0			0
$949$		−	1.11605e6i		−	1.23923i
$950$		0			0
$951$		−	416202.i		−	0.460196i
$952$		0			0
$953$			1.12239e6i			1.23583i	0.786246	+	0.617913i	$0.212022\pi$
$953$			1.12239e6i			1.23583i	−0.786246	+	0.617913i	$0.787978\pi$
$954$		0			0
$955$	1.17094e6			1.28389
$956$		0			0
$957$	−69036.2			−0.0753794
$958$		0			0
$959$	705433.			0.767041
$960$		0			0
$961$	326988.			0.354066
$962$		0			0
$963$	−1.00897e6			−1.08799
$964$		0			0
$965$			2.06623e6i			2.21883i
$966$		0			0
$967$	−769222.			−0.822619			−0.411310	−	0.911496i	$-0.634928\pi$
$967$	−769222.			−0.822619			−0.411310	+	0.911496i	$0.634928\pi$
$968$		0			0
$969$	101521.			0.108121
$970$		0			0
$971$	−1.12938e6			−1.19785			−0.598926	−	0.800804i	$-0.704406\pi$
$971$	−1.12938e6			−1.19785			−0.598926	+	0.800804i	$0.704406\pi$
$972$		0			0
$973$			245270.i			0.259071i
$974$		0			0
$975$		−	825415.i		−	0.868287i
$976$		0			0
$977$	−986156.			−1.03313			−0.516567	−	0.856247i	$-0.672790\pi$
$977$	−986156.			−1.03313			−0.516567	+	0.856247i	$0.672790\pi$
$978$		0			0
$979$		−	552540.i		−	0.576499i
$980$		0			0
$981$	1.09704e6			1.13995
$982$		0			0
$983$		−	777551.i		−	0.804677i	−0.915491	−	0.402339i	$-0.868197\pi$
$983$		−	777551.i		−	0.804677i	0.915491	−	0.402339i	$-0.131803\pi$
$984$		0			0
$985$			872725.i			0.899508i
$986$		0			0
$987$	−421982.			−0.433171
$988$		0			0
$989$	−828987.	−	294965.i	−0.847530	−	0.301563i
$990$		0			0
$991$			1.84394e6i			1.87759i	0.344478	+	0.938794i	$0.388056\pi$
$991$			1.84394e6i			1.87759i	−0.344478	+	0.938794i	$0.611944\pi$
$992$		0			0
$993$	−538088.			−0.545701
$994$		0			0
$995$	1.89139e6			1.91044
$996$		0			0
$997$			105165.i			0.105799i	0.998600	+	0.0528996i	$0.0168463\pi$
$997$			105165.i			0.105799i	−0.998600	+	0.0528996i	$0.983154\pi$
$998$		0			0
$999$	−1.48376e6			−1.48674

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.5.b.d.257.7		12
4.3	odd	2		43.5.b.b.42.9	yes	12
12.11	even	2		387.5.b.c.343.4		12
43.42	odd	2	inner	688.5.b.d.257.6		12
172.171	even	2		43.5.b.b.42.4	✓	12
516.515	odd	2		387.5.b.c.343.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.b.b.42.4	✓	12	172.171	even	2
43.5.b.b.42.9	yes	12	4.3	odd	2
387.5.b.c.343.4		12	12.11	even	2
387.5.b.c.343.9		12	516.515	odd	2
688.5.b.d.257.6		12	43.42	odd	2	inner
688.5.b.d.257.7		12	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:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	688.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.18965i q^{3} -45.6695i q^{5} +34.3337i q^{7} +63.4469 q^{9} +O(q^{10})\)	\(q+4.18965i q^{3} -45.6695i q^{5} +34.3337i q^{7} +63.4469 q^{9} +103.099 q^{11} +134.875 q^{13} +191.339 q^{15} +240.390 q^{17} -100.800i q^{19} -143.846 q^{21} +475.879 q^{23} -1460.70 q^{25} +605.181i q^{27} +159.825i q^{29} -1118.26 q^{31} +431.949i q^{33} +1568.00 q^{35} +2451.77i q^{37} +565.080i q^{39} +1069.81 q^{41} +(-1742.01 - 619.833i) q^{43} -2897.59i q^{45} +2933.56 q^{47} +1222.20 q^{49} +1007.15i q^{51} -825.685 q^{53} -4708.49i q^{55} +422.318 q^{57} +1241.46 q^{59} +6202.15i q^{61} +2178.37i q^{63} -6159.69i q^{65} -265.952 q^{67} +1993.77i q^{69} -3861.16i q^{71} -8274.70i q^{73} -6119.84i q^{75} +3539.78i q^{77} -4953.26 q^{79} +2603.70 q^{81} +13512.1 q^{83} -10978.5i q^{85} -669.609 q^{87} -5359.31i q^{89} +4630.77i q^{91} -4685.12i q^{93} -4603.51 q^{95} +3191.32 q^{97} +6541.32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 462 q^{9}+O(q^{10}) \) 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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