LMFDB - Embedded newform 208.4.f.d.129.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,4,Mod(129,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("208.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	208.f (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:	$12.2723972812$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{217})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 109x^{2} + 2916 $ x^4 + 109*x^2 + 2916
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.3
Root			$-6.86546i$ of defining polynomial
Character	$\chi$	$=$	208.129
Dual form			208.4.f.d.129.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.86546 q^{3} -16.8655i q^{5} -10.8655i q^{7} +51.5964 q^{9} +O(q^{10})$	$q+8.86546 q^{3} -16.8655i q^{5} -10.8655i q^{7} +51.5964 q^{9} +35.1928i q^{11} +(-20.1345 - 42.3273i) q^{13} -149.520i q^{15} -30.3273 q^{17} -28.3855i q^{19} -96.3273i q^{21} -24.6546 q^{23} -159.444 q^{25} +218.058 q^{27} +290.116 q^{29} +219.731i q^{31} +312.000i q^{33} -183.251 q^{35} +118.713i q^{37} +(-178.502 - 375.251i) q^{39} -83.6947i q^{41} +293.636 q^{43} -870.197i q^{45} +166.211i q^{47} +224.942 q^{49} -268.865 q^{51} -76.3855 q^{53} +593.542 q^{55} -251.651i q^{57} -184.691i q^{59} +197.811 q^{61} -560.618i q^{63} +(-713.869 + 339.578i) q^{65} +321.273i q^{67} -218.574 q^{69} -368.946i q^{71} +843.273i q^{73} -1413.54 q^{75} +382.386 q^{77} -184.044 q^{79} +540.084 q^{81} +1274.16i q^{83} +511.484i q^{85} +2572.02 q^{87} -1367.43i q^{89} +(-459.906 + 218.771i) q^{91} +1948.02i q^{93} -478.735 q^{95} +690.241i q^{97} +1815.82i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{3} + 118 q^{9}+O(q^{10}) $ 4 * q + 6 * q^3 + 118 * q^9	$ 4 q + 6 q^{3} + 118 q^{9} - 110 q^{13} + 26 q^{17} + 196 q^{23} - 78 q^{25} + 666 q^{27} + 748 q^{29} - 350 q^{35} + 52 q^{39} + 438 q^{43} + 1106 q^{49} - 1046 q^{51} + 48 q^{53} + 960 q^{55} - 564 q^{61} - 1294 q^{65} - 1876 q^{69} - 4240 q^{75} + 1176 q^{77} + 1444 q^{79} - 668 q^{81} + 4160 q^{87} - 1162 q^{91} - 324 q^{95}+O(q^{100}) $ 4 * q + 6 * q^3 + 118 * q^9 - 110 * q^13 + 26 * q^17 + 196 * q^23 - 78 * q^25 + 666 * q^27 + 748 * q^29 - 350 * q^35 + 52 * q^39 + 438 * q^43 + 1106 * q^49 - 1046 * q^51 + 48 * q^53 + 960 * q^55 - 564 * q^61 - 1294 * q^65 - 1876 * q^69 - 4240 * q^75 + 1176 * q^77 + 1444 * q^79 - 668 * q^81 + 4160 * q^87 - 1162 * q^91 - 324 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	8.86546			1.70616			0.853079	−	0.521781i	$-0.174732\pi$
$3$	8.86546			1.70616			0.853079	+	0.521781i	$0.174732\pi$
$4$		0			0
$5$		−	16.8655i		−	1.50849i	−0.656592	−	0.754246i	$-0.728002\pi$
$5$		−	16.8655i		−	1.50849i	0.656592	−	0.754246i	$-0.271998\pi$
$6$		0			0
$7$		−	10.8655i		−	0.586680i	−0.956008	−	0.293340i	$-0.905233\pi$
$7$		−	10.8655i		−	0.586680i	0.956008	−	0.293340i	$-0.0947667\pi$
$8$		0			0
$9$	51.5964			1.91098
$10$		0			0
$11$			35.1928i			0.964638i	0.875996	+	0.482319i	$0.160205\pi$
$11$			35.1928i			0.964638i	−0.875996	+	0.482319i	$0.839795\pi$
$12$		0			0
$13$	−20.1345	−	42.3273i	−0.429563	−	0.903037i
$13$	−20.1345	−	42.3273i	−0.429563	−	0.903037i
$14$		0			0
$15$		−	149.520i		−	2.57373i
$16$		0			0
$17$	−30.3273			−0.432674			−0.216337	−	0.976319i	$-0.569411\pi$
$17$	−30.3273			−0.432674			−0.216337	+	0.976319i	$0.569411\pi$
$18$		0			0
$19$		−	28.3855i		−	0.342741i	−0.985207	−	0.171371i	$-0.945180\pi$
$19$		−	28.3855i		−	0.342741i	0.985207	−	0.171371i	$-0.0548196\pi$
$20$		0			0
$21$		−	96.3273i		−	1.00097i
$22$		0			0
$23$	−24.6546			−0.223515			−0.111757	−	0.993736i	$-0.535648\pi$
$23$	−24.6546			−0.223515			−0.111757	+	0.993736i	$0.535648\pi$
$24$		0			0
$25$	−159.444			−1.27555
$26$		0			0
$27$	218.058			1.55427
$28$		0			0
$29$	290.116			1.85770			0.928849	−	0.370457i	$-0.120799\pi$
$29$	290.116			1.85770			0.928849	+	0.370457i	$0.120799\pi$
$30$		0			0
$31$			219.731i			1.27306i	0.771252	+	0.636530i	$0.219631\pi$
$31$			219.731i			1.27306i	−0.771252	+	0.636530i	$0.780369\pi$
$32$		0			0
$33$			312.000i			1.64583i
$34$		0			0
$35$	−183.251			−0.885002
$36$		0			0
$37$			118.713i			0.527467i	0.964596	+	0.263733i	$0.0849539\pi$
$37$			118.713i			0.527467i	−0.964596	+	0.263733i	$0.915046\pi$
$38$		0			0
$39$	−178.502	−	375.251i	−0.732902	−	1.54072i
$40$		0			0
$41$		−	83.6947i		−	0.318803i	−0.987214	−	0.159401i	$-0.949044\pi$
$41$		−	83.6947i		−	0.318803i	0.987214	−	0.159401i	$-0.0509564\pi$
$42$		0			0
$43$	293.636			1.04138			0.520688	−	0.853747i	$-0.325676\pi$
$43$	293.636			1.04138			0.520688	+	0.853747i	$0.325676\pi$
$44$		0			0
$45$		−	870.197i		−	2.88269i
$46$		0			0
$47$			166.211i			0.515837i	0.966167	+	0.257919i	$0.0830366\pi$
$47$			166.211i			0.515837i	−0.966167	+	0.257919i	$0.916963\pi$
$48$		0			0
$49$	224.942			0.655807
$50$		0			0
$51$	−268.865			−0.738210
$52$		0			0
$53$	−76.3855			−0.197969			−0.0989845	−	0.995089i	$-0.531559\pi$
$53$	−76.3855			−0.197969			−0.0989845	+	0.995089i	$0.531559\pi$
$54$		0			0
$55$	593.542			1.45515
$56$		0			0
$57$		−	251.651i		−	0.584771i
$58$		0			0
$59$		−	184.691i		−	0.407537i	−0.979019	−	0.203769i	$-0.934681\pi$
$59$		−	184.691i		−	0.407537i	0.979019	−	0.203769i	$-0.0653190\pi$
$60$		0			0
$61$	197.811			0.415199			0.207599	−	0.978214i	$-0.433435\pi$
$61$	197.811			0.415199			0.207599	+	0.978214i	$0.433435\pi$
$62$		0			0
$63$		−	560.618i		−	1.12113i
$64$		0			0
$65$	−713.869	+	339.578i	−1.36222	+	0.647992i
$66$		0			0
$67$			321.273i			0.585817i	0.956140	+	0.292909i	$0.0946231\pi$
$67$			321.273i			0.585817i	−0.956140	+	0.292909i	$0.905377\pi$
$68$		0			0
$69$	−218.574			−0.381352
$70$		0			0
$71$		−	368.946i		−	0.616701i	−0.951273	−	0.308351i	$-0.900223\pi$
$71$		−	368.946i		−	0.616701i	0.951273	−	0.308351i	$-0.0997770\pi$
$72$		0			0
$73$			843.273i			1.35202i	0.736891	+	0.676011i	$0.236293\pi$
$73$			843.273i			1.35202i	−0.736891	+	0.676011i	$0.763707\pi$
$74$		0			0
$75$	−1413.54			−2.17629
$76$		0			0
$77$	382.386			0.565933
$78$		0			0
$79$	−184.044			−0.262109			−0.131054	−	0.991375i	$-0.541836\pi$
$79$	−184.044			−0.262109			−0.131054	+	0.991375i	$0.541836\pi$
$80$		0			0
$81$	540.084			0.740856
$82$		0			0
$83$			1274.16i			1.68503i	0.538675	+	0.842514i	$0.318925\pi$
$83$			1274.16i			1.68503i	−0.538675	+	0.842514i	$0.681075\pi$
$84$		0			0
$85$			511.484i			0.652685i
$86$		0			0
$87$	2572.02			3.16953
$88$		0			0
$89$		−	1367.43i		−	1.62863i	−0.580426	−	0.814313i	$-0.697114\pi$
$89$		−	1367.43i		−	1.62863i	0.580426	−	0.814313i	$-0.302886\pi$
$90$		0			0
$91$	−459.906	+	218.771i	−0.529793	+	0.252016i
$92$		0			0
$93$			1948.02i			2.17204i
$94$		0			0
$95$	−478.735			−0.517023
$96$		0			0
$97$			690.241i			0.722509i	0.932467	+	0.361254i	$0.117651\pi$
$97$			690.241i			0.722509i	−0.932467	+	0.361254i	$0.882349\pi$
$98$		0			0
$99$			1815.82i			1.84340i
$100$		0			0
$101$	−1395.21			−1.37454			−0.687269	−	0.726403i	$-0.741191\pi$
$101$	−1395.21			−1.37454			−0.687269	+	0.726403i	$0.741191\pi$
$102$		0			0
$103$	−1416.28			−1.35485			−0.677427	−	0.735590i	$-0.736905\pi$
$103$	−1416.28			−1.35485			−0.677427	+	0.735590i	$0.736905\pi$
$104$		0			0
$105$	−1624.60			−1.50995
$106$		0			0
$107$	−451.068			−0.407537			−0.203768	−	0.979019i	$-0.565319\pi$
$107$	−451.068			−0.407537			−0.203768	+	0.979019i	$0.565319\pi$
$108$		0			0
$109$			1386.13i			1.21805i	0.793152	+	0.609024i	$0.208439\pi$
$109$			1386.13i			1.21805i	−0.793152	+	0.609024i	$0.791561\pi$
$110$		0			0
$111$			1052.44i			0.899942i
$112$		0			0
$113$	1528.78			1.27270			0.636351	−	0.771399i	$-0.280443\pi$
$113$	1528.78			1.27270			0.636351	+	0.771399i	$0.280443\pi$
$114$		0			0
$115$			415.811i			0.337170i
$116$		0			0
$117$	−1038.87	−	2183.94i	−0.820885	−	1.72568i
$118$		0			0
$119$			329.520i			0.253841i
$120$		0			0
$121$	92.4697			0.0694738
$122$		0			0
$123$		−	741.992i		−	0.543928i
$124$		0			0
$125$			580.909i			0.415665i
$126$		0			0
$127$	−1447.67			−1.01149			−0.505747	−	0.862682i	$-0.668783\pi$
$127$	−1447.67			−1.01149			−0.505747	+	0.862682i	$0.668783\pi$
$128$		0			0
$129$	2603.22			1.77675
$130$		0			0
$131$	−631.877			−0.421430			−0.210715	−	0.977548i	$-0.567579\pi$
$131$	−631.877			−0.421430			−0.210715	+	0.977548i	$0.567579\pi$
$132$		0			0
$133$	−308.422			−0.201079
$134$		0			0
$135$		−	3677.65i		−	2.34461i
$136$		0			0
$137$		−	547.418i		−	0.341380i	−0.985325	−	0.170690i	$-0.945400\pi$
$137$		−	547.418i		−	0.341380i	0.985325	−	0.170690i	$-0.0545997\pi$
$138$		0			0
$139$	677.070			0.413153			0.206577	−	0.978430i	$-0.433768\pi$
$139$	677.070			0.413153			0.206577	+	0.978430i	$0.433768\pi$
$140$		0			0
$141$			1473.54i			0.880100i
$142$		0			0
$143$	1489.61	−	708.590i	0.871104	−	0.414373i
$144$		0			0
$145$		−	4892.95i		−	2.80233i
$146$		0			0
$147$	1994.21			1.11891
$148$		0			0
$149$			284.334i			0.156332i	0.996940	+	0.0781662i	$0.0249065\pi$
$149$			284.334i			0.156332i	−0.996940	+	0.0781662i	$0.975094\pi$
$150$		0			0
$151$		−	2708.68i		−	1.45980i	−0.683555	−	0.729899i	$-0.739567\pi$
$151$		−	2708.68i		−	1.45980i	0.683555	−	0.729899i	$-0.260433\pi$
$152$		0			0
$153$	−1564.78			−0.826829
$154$		0			0
$155$	3705.86			1.92040
$156$		0			0
$157$	−883.775			−0.449254			−0.224627	−	0.974445i	$-0.572116\pi$
$157$	−883.775			−0.449254			−0.224627	+	0.974445i	$0.572116\pi$
$158$		0			0
$159$	−677.193			−0.337767
$160$		0			0
$161$			267.884i			0.131132i
$162$		0			0
$163$			3340.78i			1.60534i	0.596425	+	0.802669i	$0.296587\pi$
$163$			3340.78i			1.60534i	−0.596425	+	0.802669i	$0.703413\pi$
$164$		0			0
$165$	5262.02			2.48271
$166$		0			0
$167$		−	1816.52i		−	0.841715i	−0.907127	−	0.420857i	$-0.861729\pi$
$167$		−	1816.52i		−	0.841715i	0.907127	−	0.420857i	$-0.138271\pi$
$168$		0			0
$169$	−1386.20	+	1704.48i	−0.630952	+	0.775822i
$170$		0			0
$171$		−	1464.59i		−	0.654971i
$172$		0			0
$173$	984.241			0.432546			0.216273	−	0.976333i	$-0.430610\pi$
$173$	984.241			0.432546			0.216273	+	0.976333i	$0.430610\pi$
$174$		0			0
$175$			1732.43i			0.748339i
$176$		0			0
$177$		−	1637.37i		−	0.695323i
$178$		0			0
$179$	−3777.32			−1.57726			−0.788631	−	0.614866i	$-0.789210\pi$
$179$	−3777.32			−1.57726			−0.788631	+	0.614866i	$0.789210\pi$
$180$		0			0
$181$	−2329.80			−0.956754			−0.478377	−	0.878154i	$-0.658775\pi$
$181$	−2329.80			−0.956754			−0.478377	+	0.878154i	$0.658775\pi$
$182$		0			0
$183$	1753.69			0.708395
$184$		0			0
$185$	2002.15			0.795680
$186$		0			0
$187$		−	1067.30i		−	0.417373i
$188$		0			0
$189$		−	2369.30i		−	0.911859i
$190$		0			0
$191$	4611.93			1.74716			0.873579	−	0.486681i	$-0.161793\pi$
$191$	4611.93			1.74716			0.873579	+	0.486681i	$0.161793\pi$
$192$		0			0
$193$			952.799i			0.355358i	0.984089	+	0.177679i	$0.0568588\pi$
$193$			952.799i			0.355358i	−0.984089	+	0.177679i	$0.943141\pi$
$194$		0			0
$195$	−6328.78	+	3010.52i	−2.32417	+	1.10558i
$196$		0			0
$197$		−	2050.54i		−	0.741598i	−0.928713	−	0.370799i	$-0.879084\pi$
$197$		−	2050.54i		−	0.741598i	0.928713	−	0.370799i	$-0.120916\pi$
$198$		0			0
$199$	−2585.31			−0.920943			−0.460472	−	0.887674i	$-0.652320\pi$
$199$	−2585.31			−0.920943			−0.460472	+	0.887674i	$0.652320\pi$
$200$		0			0
$201$			2848.23i			0.999497i
$202$		0			0
$203$		−	3152.25i		−	1.08987i
$204$		0			0
$205$	−1411.55			−0.480912
$206$		0			0
$207$	−1272.09			−0.427132
$208$		0			0
$209$	998.965			0.330621
$210$		0			0
$211$	1756.23			0.573003			0.286501	−	0.958080i	$-0.407508\pi$
$211$	1756.23			0.573003			0.286501	+	0.958080i	$0.407508\pi$
$212$		0			0
$213$		−	3270.87i		−	1.05219i
$214$		0			0
$215$		−	4952.31i		−	1.57091i
$216$		0			0
$217$	2387.48			0.746878
$218$		0			0
$219$			7476.00i			2.30676i
$220$		0			0
$221$	610.626	+	1283.67i	0.185860	+	0.390720i
$222$		0			0
$223$			2038.12i			0.612030i	0.952027	+	0.306015i	$0.0989959\pi$
$223$			2038.12i			0.612030i	−0.952027	+	0.306015i	$0.901004\pi$
$224$		0			0
$225$	−8226.72			−2.43755
$226$		0			0
$227$		−	4691.85i		−	1.37185i	−0.727674	−	0.685923i	$-0.759399\pi$
$227$		−	4691.85i		−	1.37185i	0.727674	−	0.685923i	$-0.240601\pi$
$228$		0			0
$229$			576.154i			0.166259i	0.996539	+	0.0831296i	$0.0264915\pi$
$229$			576.154i			0.166259i	−0.996539	+	0.0831296i	$0.973508\pi$
$230$		0			0
$231$	3390.02			0.965572
$232$		0			0
$233$	−724.319			−0.203656			−0.101828	−	0.994802i	$-0.532469\pi$
$233$	−724.319			−0.203656			−0.101828	+	0.994802i	$0.532469\pi$
$234$		0			0
$235$	2803.22			0.778137
$236$		0			0
$237$	−1631.64			−0.447199
$238$		0			0
$239$			3137.80i			0.849237i	0.905372	+	0.424619i	$0.139592\pi$
$239$			3137.80i			0.849237i	−0.905372	+	0.424619i	$0.860408\pi$
$240$		0			0
$241$		−	1763.74i		−	0.471421i	−0.971823	−	0.235710i	$-0.924258\pi$
$241$		−	1763.74i		−	0.471421i	0.971823	−	0.235710i	$-0.0757417\pi$
$242$		0			0
$243$	−1099.48			−0.290253
$244$		0			0
$245$		−	3793.75i		−	0.989280i
$246$		0			0
$247$	−1201.48	+	571.529i	−0.309508	+	0.147229i
$248$		0			0
$249$			11296.0i			2.87492i
$250$		0			0
$251$	−1646.01			−0.413925			−0.206962	−	0.978349i	$-0.566358\pi$
$251$	−1646.01			−0.413925			−0.206962	+	0.978349i	$0.566358\pi$
$252$		0			0
$253$		−	867.663i		−	0.215611i
$254$		0			0
$255$			4534.54i			1.11358i
$256$		0			0
$257$	6415.73			1.55721			0.778603	−	0.627516i	$-0.215928\pi$
$257$	6415.73			1.55721			0.778603	+	0.627516i	$0.215928\pi$
$258$		0			0
$259$	1289.87			0.309454
$260$		0			0
$261$	14969.0			3.55002
$262$		0			0
$263$	3660.48			0.858232			0.429116	−	0.903249i	$-0.358825\pi$
$263$	3660.48			0.858232			0.429116	+	0.903249i	$0.358825\pi$
$264$		0			0
$265$			1288.28i			0.298635i
$266$		0			0
$267$		−	12122.9i		−	2.77869i
$268$		0			0
$269$	−5645.30			−1.27955			−0.639777	−	0.768560i	$-0.720973\pi$
$269$	−5645.30			−1.27955			−0.639777	+	0.768560i	$0.720973\pi$
$270$		0			0
$271$		−	6671.87i		−	1.49552i	−0.663966	−	0.747762i	$-0.731128\pi$
$271$		−	6671.87i		−	1.49552i	0.663966	−	0.747762i	$-0.268872\pi$
$272$		0			0
$273$	−4077.27	+	1939.51i	−0.903912	+	0.429979i
$274$		0			0
$275$		−	5611.27i		−	1.23044i
$276$		0			0
$277$	2905.39			0.630210			0.315105	−	0.949057i	$-0.397960\pi$
$277$	2905.39			0.630210			0.315105	+	0.949057i	$0.397960\pi$
$278$		0			0
$279$			11337.3i			2.43279i
$280$		0			0
$281$		−	8009.76i		−	1.70044i	−0.526431	−	0.850218i	$-0.676470\pi$
$281$		−	8009.76i		−	1.70044i	0.526431	−	0.850218i	$-0.323530\pi$
$282$		0			0
$283$	3707.71			0.778800			0.389400	−	0.921069i	$-0.372682\pi$
$283$	3707.71			0.778800			0.389400	+	0.921069i	$0.372682\pi$
$284$		0			0
$285$	−4244.20			−0.882123
$286$		0			0
$287$	−909.382			−0.187035
$288$		0			0
$289$	−3993.25			−0.812794
$290$		0			0
$291$			6119.30i			1.23271i
$292$		0			0
$293$			1674.00i			0.333775i	0.985976	+	0.166887i	$0.0533717\pi$
$293$			1674.00i			0.333775i	−0.985976	+	0.166887i	$0.946628\pi$
$294$		0			0
$295$	−3114.90			−0.614767
$296$		0			0
$297$			7674.07i			1.49931i
$298$		0			0
$299$	496.409	+	1043.56i	0.0960136	+	0.201842i
$300$		0			0
$301$		−	3190.50i		−	0.610954i
$302$		0			0
$303$	−12369.2			−2.34518
$304$		0			0
$305$		−	3336.18i		−	0.626324i
$306$		0			0
$307$			299.935i			0.0557597i	0.999611	+	0.0278798i	$0.00887558\pi$
$307$			299.935i			0.0557597i	−0.999611	+	0.0278798i	$0.991124\pi$
$308$		0			0
$309$	−12555.9			−2.31159
$310$		0			0
$311$	−890.140			−0.162300			−0.0811498	−	0.996702i	$-0.525859\pi$
$311$	−890.140			−0.162300			−0.0811498	+	0.996702i	$0.525859\pi$
$312$		0			0
$313$	−1154.20			−0.208432			−0.104216	−	0.994555i	$-0.533233\pi$
$313$	−1154.20			−0.208432			−0.104216	+	0.994555i	$0.533233\pi$
$314$		0			0
$315$	−9455.09			−1.69122
$316$		0			0
$317$		−	2708.86i		−	0.479951i	−0.970779	−	0.239976i	$-0.922861\pi$
$317$		−	2708.86i		−	0.479951i	0.970779	−	0.239976i	$-0.0771394\pi$
$318$		0			0
$319$			10210.0i			1.79201i
$320$		0			0
$321$	−3998.93			−0.695322
$322$		0			0
$323$			860.856i			0.148295i
$324$		0			0
$325$	3210.33	+	6748.82i	0.547929	+	1.15187i
$326$		0			0
$327$			12288.7i			2.07818i
$328$		0			0
$329$	1805.96			0.302631
$330$		0			0
$331$			8187.12i			1.35953i	0.733429	+	0.679766i	$0.237919\pi$
$331$			8187.12i			1.35953i	−0.733429	+	0.679766i	$0.762081\pi$
$332$		0			0
$333$			6125.15i			1.00798i
$334$		0			0
$335$	5418.42			0.883701
$336$		0			0
$337$	8770.94			1.41776			0.708878	−	0.705331i	$-0.249202\pi$
$337$	8770.94			1.41776			0.708878	+	0.705331i	$0.249202\pi$
$338$		0			0
$339$	13553.3			2.17143
$340$		0			0
$341$	−7732.94			−1.22804
$342$		0			0
$343$		−	6170.95i		−	0.971428i
$344$		0			0
$345$			3686.36i			0.575266i
$346$		0			0
$347$	4278.96			0.661978			0.330989	−	0.943635i	$-0.392618\pi$
$347$	4278.96			0.661978			0.330989	+	0.943635i	$0.392618\pi$
$348$		0			0
$349$			6181.46i			0.948097i	0.880499	+	0.474049i	$0.157208\pi$
$349$			6181.46i			0.948097i	−0.880499	+	0.474049i	$0.842792\pi$
$350$		0			0
$351$	−4390.50	−	9229.82i	−0.667657	−	1.40356i
$352$		0			0
$353$			5471.61i			0.824998i	0.910958	+	0.412499i	$0.135344\pi$
$353$			5471.61i			0.824998i	−0.910958	+	0.412499i	$0.864656\pi$
$354$		0			0
$355$	−6222.44			−0.930289
$356$		0			0
$357$			2921.35i			0.433093i
$358$		0			0
$359$		−	6398.77i		−	0.940709i	−0.882478	−	0.470354i	$-0.844126\pi$
$359$		−	6398.77i		−	0.940709i	0.882478	−	0.470354i	$-0.155874\pi$
$360$		0			0
$361$	6053.26			0.882528
$362$		0			0
$363$	819.786			0.118533
$364$		0			0
$365$	14222.2			2.03952
$366$		0			0
$367$	−9636.19			−1.37059			−0.685293	−	0.728267i	$-0.740326\pi$
$367$	−9636.19			−1.37059			−0.685293	+	0.728267i	$0.740326\pi$
$368$		0			0
$369$		−	4318.34i		−	0.609225i
$370$		0			0
$371$			829.964i			0.116144i
$372$		0			0
$373$	8807.88			1.22267			0.611333	−	0.791373i	$-0.290634\pi$
$373$	8807.88			1.22267			0.611333	+	0.791373i	$0.290634\pi$
$374$		0			0
$375$			5150.03i			0.709190i
$376$		0			0
$377$	−5841.36	−	12279.8i	−0.797998	−	1.67757i
$378$		0			0
$379$		−	8678.95i		−	1.17627i	−0.808761	−	0.588137i	$-0.799862\pi$
$379$		−	8678.95i		−	1.17627i	0.808761	−	0.588137i	$-0.200138\pi$
$380$		0			0
$381$	−12834.2			−1.72577
$382$		0			0
$383$		−	2712.30i		−	0.361860i	−0.983496	−	0.180930i	$-0.942089\pi$
$383$		−	2712.30i		−	0.361860i	0.983496	−	0.180930i	$-0.0579107\pi$
$384$		0			0
$385$		−	6449.11i		−	0.853706i
$386$		0			0
$387$	15150.6			1.99004
$388$		0			0
$389$	290.941			0.0379211			0.0189605	−	0.999820i	$-0.493964\pi$
$389$	290.941			0.0379211			0.0189605	+	0.999820i	$0.493964\pi$
$390$		0			0
$391$	747.707			0.0967089
$392$		0			0
$393$	−5601.88			−0.719027
$394$		0			0
$395$			3103.99i			0.395389i
$396$		0			0
$397$			14114.0i			1.78429i	0.451751	+	0.892144i	$0.350799\pi$
$397$			14114.0i			1.78429i	−0.451751	+	0.892144i	$0.649201\pi$
$398$		0			0
$399$	−2734.30			−0.343073
$400$		0			0
$401$		−	11342.0i		−	1.41245i	−0.707987	−	0.706226i	$-0.750396\pi$
$401$		−	11342.0i		−	1.41245i	0.707987	−	0.706226i	$-0.249604\pi$
$402$		0			0
$403$	9300.62	−	4424.18i	1.14962	−	0.546859i
$404$		0			0
$405$		−	9108.77i		−	1.11758i
$406$		0			0
$407$	−4177.83			−0.508814
$408$		0			0
$409$		−	4686.94i		−	0.566636i	−0.959026	−	0.283318i	$-0.908565\pi$
$409$		−	4686.94i		−	0.566636i	0.959026	−	0.283318i	$-0.0914352\pi$
$410$		0			0
$411$		−	4853.11i		−	0.582448i
$412$		0			0
$413$	−2006.75			−0.239094
$414$		0			0
$415$	21489.3			2.54185
$416$		0			0
$417$	6002.54			0.704905
$418$		0			0
$419$	−11889.7			−1.38627			−0.693137	−	0.720806i	$-0.743772\pi$
$419$	−11889.7			−1.38627			−0.693137	+	0.720806i	$0.743772\pi$
$420$		0			0
$421$			4755.64i			0.550536i	0.961367	+	0.275268i	$0.0887666\pi$
$421$			4755.64i			0.550536i	−0.961367	+	0.275268i	$0.911233\pi$
$422$		0			0
$423$			8575.88i			0.985753i
$424$		0			0
$425$	4835.50			0.551897
$426$		0			0
$427$		−	2149.31i		−	0.243589i
$428$		0			0
$429$	13206.1	−	6281.98i	1.48624	−	0.706985i
$430$		0			0
$431$		−	4378.13i		−	0.489297i	−0.969612	−	0.244649i	$-0.921327\pi$
$431$		−	4378.13i		−	0.489297i	0.969612	−	0.244649i	$-0.0786726\pi$
$432$		0			0
$433$	−15576.3			−1.72875			−0.864377	−	0.502844i	$-0.832287\pi$
$433$	−15576.3			−1.72875			−0.864377	+	0.502844i	$0.832287\pi$
$434$		0			0
$435$		−	43378.2i		−	4.78121i
$436$		0			0
$437$			699.834i			0.0766077i
$438$		0			0
$439$	−13474.8			−1.46495			−0.732477	−	0.680792i	$-0.761636\pi$
$439$	−13474.8			−1.46495			−0.732477	+	0.680792i	$0.761636\pi$
$440$		0			0
$441$	11606.2			1.25323
$442$		0			0
$443$	24.4564			0.00262294			0.00131147	−	0.999999i	$-0.499583\pi$
$443$	24.4564			0.00262294			0.00131147	+	0.999999i	$0.499583\pi$
$444$		0			0
$445$	−23062.4			−2.45677
$446$		0			0
$447$			2520.75i			0.266728i
$448$		0			0
$449$		−	13636.7i		−	1.43331i	−0.697427	−	0.716655i	$-0.745672\pi$
$449$		−	13636.7i		−	1.43331i	0.697427	−	0.716655i	$-0.254328\pi$
$450$		0			0
$451$	2945.45			0.307529
$452$		0			0
$453$		−	24013.7i		−	2.49065i
$454$		0			0
$455$	3689.67	+	7756.52i	0.380164	+	0.799190i
$456$		0			0
$457$		−	13380.5i		−	1.36962i	−0.728723	−	0.684808i	$-0.759886\pi$
$457$		−	13380.5i		−	1.36962i	0.728723	−	0.684808i	$-0.240114\pi$
$458$		0			0
$459$	−6613.12			−0.672492
$460$		0			0
$461$			6782.55i			0.685238i	0.939474	+	0.342619i	$0.111314\pi$
$461$			6782.55i			0.685238i	−0.939474	+	0.342619i	$0.888686\pi$
$462$		0			0
$463$			10966.2i			1.10074i	0.834922	+	0.550368i	$0.185512\pi$
$463$			10966.2i			1.10074i	−0.834922	+	0.550368i	$0.814488\pi$
$464$		0			0
$465$	32854.2			3.27651
$466$		0			0
$467$	−609.640			−0.0604085			−0.0302043	−	0.999544i	$-0.509616\pi$
$467$	−609.640			−0.0604085			−0.0302043	+	0.999544i	$0.509616\pi$
$468$		0			0
$469$	3490.78			0.343687
$470$		0			0
$471$	−7835.07			−0.766499
$472$		0			0
$473$			10333.9i			1.00455i
$474$		0			0
$475$			4525.89i			0.437184i
$476$		0			0
$477$	−3941.22			−0.378314
$478$		0			0
$479$		−	8405.28i		−	0.801769i	−0.916129	−	0.400884i	$-0.868703\pi$
$479$		−	8405.28i		−	0.801769i	0.916129	−	0.400884i	$-0.131297\pi$
$480$		0			0
$481$	5024.79	−	2390.23i	0.476322	−	0.226580i
$482$		0			0
$483$			2374.91i			0.223731i
$484$		0			0
$485$	11641.2			1.08990
$486$		0			0
$487$			16160.4i			1.50370i	0.659337	+	0.751848i	$0.270837\pi$
$487$			16160.4i			1.50370i	−0.659337	+	0.751848i	$0.729163\pi$
$488$		0			0
$489$			29617.5i			2.73896i
$490$		0			0
$491$	6426.96			0.590722			0.295361	−	0.955386i	$-0.404560\pi$
$491$	6426.96			0.590722			0.295361	+	0.955386i	$0.404560\pi$
$492$		0			0
$493$	−8798.45			−0.803777
$494$		0			0
$495$	30624.6			2.78076
$496$		0			0
$497$	−4008.76			−0.361806
$498$		0			0
$499$		−	15240.3i		−	1.36724i	−0.729840	−	0.683618i	$-0.760405\pi$
$499$		−	15240.3i		−	1.36724i	0.729840	−	0.683618i	$-0.239595\pi$
$500$		0			0
$501$		−	16104.3i		−	1.43610i
$502$		0			0
$503$	−10404.0			−0.922250			−0.461125	−	0.887335i	$-0.652554\pi$
$503$	−10404.0			−0.922250			−0.461125	+	0.887335i	$0.652554\pi$
$504$		0			0
$505$			23530.8i			2.07348i
$506$		0			0
$507$	−12289.3	+	15111.0i	−1.07650	+	1.32368i
$508$		0			0
$509$			14623.3i			1.27341i	0.771106	+	0.636707i	$0.219704\pi$
$509$			14623.3i			1.27341i	−0.771106	+	0.636707i	$0.780296\pi$
$510$		0			0
$511$	9162.55			0.793204
$512$		0			0
$513$		−	6189.70i		−	0.532713i
$514$		0			0
$515$			23886.2i			2.04379i
$516$		0			0
$517$	−5849.42			−0.497596
$518$		0			0
$519$	8725.75			0.737992
$520$		0			0
$521$	−19937.9			−1.67657			−0.838285	−	0.545232i	$-0.816442\pi$
$521$	−19937.9			−1.67657			−0.838285	+	0.545232i	$0.816442\pi$
$522$		0			0
$523$	−4327.00			−0.361771			−0.180886	−	0.983504i	$-0.557896\pi$
$523$	−4327.00			−0.361771			−0.180886	+	0.983504i	$0.557896\pi$
$524$		0			0
$525$			15358.8i			1.27679i
$526$		0			0
$527$		−	6663.85i		−	0.550819i
$528$		0			0
$529$	−11559.2			−0.950041
$530$		0			0
$531$		−	9529.38i		−	0.778794i
$532$		0			0
$533$	−3542.57	+	1685.15i	−0.287891	+	0.136946i
$534$		0			0
$535$			7607.48i			0.614766i
$536$		0			0
$537$	−33487.7			−2.69106
$538$		0			0
$539$			7916.32i			0.632616i
$540$		0			0
$541$		−	2602.93i		−	0.206855i	−0.994637	−	0.103428i	$-0.967019\pi$
$541$		−	2602.93i		−	0.206855i	0.994637	−	0.103428i	$-0.0329810\pi$
$542$		0			0
$543$	−20654.7			−1.63237
$544$		0			0
$545$	23377.7			1.83742
$546$		0			0
$547$	−11225.3			−0.877441			−0.438720	−	0.898624i	$-0.644568\pi$
$547$	−11225.3			−0.877441			−0.438720	+	0.898624i	$0.644568\pi$
$548$		0			0
$549$	10206.3			0.793435
$550$		0			0
$551$		−	8235.11i		−	0.636710i
$552$		0			0
$553$			1999.72i			0.153774i
$554$		0			0
$555$	17749.9			1.35756
$556$		0			0
$557$			8248.83i			0.627494i	0.949507	+	0.313747i	$0.101584\pi$
$557$			8248.83i			0.627494i	−0.949507	+	0.313747i	$0.898416\pi$
$558$		0			0
$559$	−5912.24	−	12428.8i	−0.447336	−	0.940401i
$560$		0			0
$561$		−	9462.12i		−	0.712105i
$562$		0			0
$563$	−1693.06			−0.126739			−0.0633696	−	0.997990i	$-0.520185\pi$
$563$	−1693.06			−0.126739			−0.0633696	+	0.997990i	$0.520185\pi$
$564$		0			0
$565$		−	25783.6i		−	1.91986i
$566$		0			0
$567$		−	5868.26i		−	0.434645i
$568$		0			0
$569$	−8251.55			−0.607949			−0.303974	−	0.952680i	$-0.598314\pi$
$569$	−8251.55			−0.607949			−0.303974	+	0.952680i	$0.598314\pi$
$570$		0			0
$571$	−23153.2			−1.69691			−0.848453	−	0.529271i	$-0.822466\pi$
$571$	−23153.2			−1.69691			−0.848453	+	0.529271i	$0.822466\pi$
$572$		0			0
$573$	40886.9			2.98093
$574$		0			0
$575$	3931.02			0.285104
$576$		0			0
$577$		−	2058.46i		−	0.148518i	−0.997239	−	0.0742590i	$-0.976341\pi$
$577$		−	2058.46i		−	0.148518i	0.997239	−	0.0742590i	$-0.0236591\pi$
$578$		0			0
$579$			8447.00i			0.606296i
$580$		0			0
$581$	13844.3			0.988571
$582$		0			0
$583$		−	2688.22i		−	0.190968i
$584$		0			0
$585$	−36833.1	+	17521.0i	−2.60318	+	1.23830i
$586$		0			0
$587$			12861.7i			0.904361i	0.891926	+	0.452180i	$0.149354\pi$
$587$			12861.7i			0.904361i	−0.891926	+	0.452180i	$0.850646\pi$
$588$		0			0
$589$	6237.18			0.436330
$590$		0			0
$591$		−	18179.0i		−	1.26528i
$592$		0			0
$593$			18098.4i			1.25331i	0.779298	+	0.626653i	$0.215576\pi$
$593$			18098.4i			1.25331i	−0.779298	+	0.626653i	$0.784424\pi$
$594$		0			0
$595$	5557.51			0.382917
$596$		0			0
$597$	−22920.0			−1.57128
$598$		0			0
$599$	15338.0			1.04623			0.523115	−	0.852262i	$-0.324770\pi$
$599$	15338.0			1.04623			0.523115	+	0.852262i	$0.324770\pi$
$600$		0			0
$601$	22289.4			1.51282			0.756409	−	0.654099i	$-0.226952\pi$
$601$	22289.4			1.51282			0.756409	+	0.654099i	$0.226952\pi$
$602$		0			0
$603$			16576.5i			1.11948i
$604$		0			0
$605$		−	1559.54i		−	0.104801i
$606$		0			0
$607$	−7191.81			−0.480900			−0.240450	−	0.970662i	$-0.577295\pi$
$607$	−7191.81			−0.480900			−0.240450	+	0.970662i	$0.577295\pi$
$608$		0			0
$609$		−	27946.1i		−	1.85950i
$610$		0			0
$611$	7035.26	−	3346.58i	0.465820	−	0.221584i
$612$		0			0
$613$		−	797.058i		−	0.0525169i	−0.999655	−	0.0262584i	$-0.991641\pi$
$613$		−	797.058i		−	0.0525169i	0.999655	−	0.0262584i	$-0.00835928\pi$
$614$		0			0
$615$	−12514.0			−0.820512
$616$		0			0
$617$			20908.3i			1.36424i	0.731240	+	0.682120i	$0.238942\pi$
$617$			20908.3i			1.36424i	−0.731240	+	0.682120i	$0.761058\pi$
$618$		0			0
$619$		−	12309.9i		−	0.799315i	−0.916665	−	0.399658i	$-0.869129\pi$
$619$		−	12309.9i		−	0.799315i	0.916665	−	0.399658i	$-0.130871\pi$
$620$		0			0
$621$	−5376.14			−0.347403
$622$		0			0
$623$	−14857.8			−0.955481
$624$		0			0
$625$	−10133.2			−0.648522
$626$		0			0
$627$	8856.28			0.564092
$628$		0			0
$629$		−	3600.24i		−	0.228221i
$630$		0			0
$631$			10332.7i			0.651881i	0.945390	+	0.325940i	$0.105681\pi$
$631$			10332.7i			0.651881i	−0.945390	+	0.325940i	$0.894319\pi$
$632$		0			0
$633$	15569.8			0.977634
$634$		0			0
$635$			24415.6i			1.52583i
$636$		0			0
$637$	−4529.10	−	9521.18i	−0.281710	−	0.592218i
$638$		0			0
$639$		−	19036.3i		−	1.17850i
$640$		0			0
$641$	−676.867			−0.0417077			−0.0208539	−	0.999783i	$-0.506638\pi$
$641$	−676.867			−0.0417077			−0.0208539	+	0.999783i	$0.506638\pi$
$642$		0			0
$643$			26312.7i			1.61379i	0.590692	+	0.806897i	$0.298855\pi$
$643$			26312.7i			1.61379i	−0.590692	+	0.806897i	$0.701145\pi$
$644$		0			0
$645$		−	43904.5i		−	2.68022i
$646$		0			0
$647$	169.876			0.0103223			0.00516113	−	0.999987i	$-0.498357\pi$
$647$	169.876			0.0103223			0.00516113	+	0.999987i	$0.498357\pi$
$648$		0			0
$649$	6499.78			0.393126
$650$		0			0
$651$	21166.1			1.27429
$652$		0			0
$653$	−14426.6			−0.864558			−0.432279	−	0.901740i	$-0.642290\pi$
$653$	−14426.6			−0.864558			−0.432279	+	0.901740i	$0.642290\pi$
$654$		0			0
$655$			10656.9i			0.635724i
$656$		0			0
$657$			43509.8i			2.58368i
$658$		0			0
$659$	25211.8			1.49031			0.745155	−	0.666892i	$-0.232376\pi$
$659$	25211.8			1.49031			0.745155	+	0.666892i	$0.232376\pi$
$660$		0			0
$661$			22839.0i			1.34393i	0.740585	+	0.671963i	$0.234548\pi$
$661$			22839.0i			1.34393i	−0.740585	+	0.671963i	$0.765452\pi$
$662$		0			0
$663$	5413.48	+	11380.3i	0.317107	+	0.666631i
$664$		0			0
$665$			5201.67i			0.303327i
$666$		0			0
$667$	−7152.70			−0.415223
$668$		0			0
$669$			18068.9i			1.04422i
$670$		0			0
$671$			6961.52i			0.400516i
$672$		0			0
$673$	16708.2			0.956989			0.478495	−	0.878090i	$-0.341183\pi$
$673$	16708.2			0.956989			0.478495	+	0.878090i	$0.341183\pi$
$674$		0			0
$675$	−34768.0			−1.98255
$676$		0			0
$677$	−15842.6			−0.899378			−0.449689	−	0.893185i	$-0.648465\pi$
$677$	−15842.6			−0.899378			−0.449689	+	0.893185i	$0.648465\pi$
$678$		0			0
$679$	7499.78			0.423881
$680$		0			0
$681$		−	41595.4i		−	2.34059i
$682$		0			0
$683$		−	16467.3i		−	0.922550i	−0.887257	−	0.461275i	$-0.847392\pi$
$683$		−	16467.3i		−	0.922550i	0.887257	−	0.461275i	$-0.152608\pi$
$684$		0			0
$685$	−9232.45			−0.514969
$686$		0			0
$687$			5107.87i			0.283664i
$688$		0			0
$689$	1537.99	+	3233.19i	0.0850401	+	0.178773i
$690$		0			0
$691$		−	16531.0i		−	0.910086i	−0.890470	−	0.455043i	$-0.849624\pi$
$691$		−	16531.0i		−	0.910086i	0.890470	−	0.455043i	$-0.150376\pi$
$692$		0			0
$693$	19729.7			1.08149
$694$		0			0
$695$		−	11419.1i		−	0.623239i
$696$		0			0
$697$			2538.23i			0.137938i
$698$		0			0
$699$	−6421.43			−0.347469
$700$		0			0
$701$	11032.2			0.594409			0.297205	−	0.954814i	$-0.403946\pi$
$701$	11032.2			0.594409			0.297205	+	0.954814i	$0.403946\pi$
$702$		0			0
$703$	3369.72			0.180785
$704$		0			0
$705$	24851.9			1.32762
$706$		0			0
$707$			15159.6i			0.806414i
$708$		0			0
$709$			20979.1i			1.11127i	0.831427	+	0.555633i	$0.187524\pi$
$709$			20979.1i			1.11127i	−0.831427	+	0.555633i	$0.812476\pi$
$710$		0			0
$711$	−9496.01			−0.500883
$712$		0			0
$713$		−	5417.38i		−	0.284548i
$714$		0			0
$715$	−11950.7	−	25123.0i	−0.625078	−	1.31405i
$716$		0			0
$717$			27818.1i			1.44893i
$718$		0			0
$719$	−17699.8			−0.918068			−0.459034	−	0.888419i	$-0.651804\pi$
$719$	−17699.8			−0.918068			−0.459034	+	0.888419i	$0.651804\pi$
$720$		0			0
$721$			15388.5i			0.794865i
$722$		0			0
$723$		−	15636.4i		−	0.804319i
$724$		0			0
$725$	−46257.2			−2.36959
$726$		0			0
$727$	33899.2			1.72937			0.864686	−	0.502313i	$-0.167518\pi$
$727$	33899.2			1.72937			0.864686	+	0.502313i	$0.167518\pi$
$728$		0			0
$729$	−24329.6			−1.23607
$730$		0			0
$731$	−8905.20			−0.450576
$732$		0			0
$733$		−	29937.3i		−	1.50854i	−0.656564	−	0.754270i	$-0.727991\pi$
$733$		−	29937.3i		−	1.50854i	0.656564	−	0.754270i	$-0.272009\pi$
$734$		0			0
$735$		−	33633.3i		−	1.68787i
$736$		0			0
$737$	−11306.5			−0.565101
$738$		0			0
$739$			16595.2i			0.826068i	0.910716	+	0.413034i	$0.135531\pi$
$739$			16595.2i			0.826068i	−0.910716	+	0.413034i	$0.864469\pi$
$740$		0			0
$741$	−10651.7	+	5066.87i	−0.528070	+	0.251196i
$742$		0			0
$743$		−	4929.07i		−	0.243378i	−0.992568	−	0.121689i	$-0.961169\pi$
$743$		−	4929.07i		−	0.243378i	0.992568	−	0.121689i	$-0.0388311\pi$
$744$		0			0
$745$	4795.42			0.235826
$746$		0			0
$747$			65742.1i			3.22005i
$748$		0			0
$749$			4901.07i			0.239094i
$750$		0			0
$751$	−10687.7			−0.519305			−0.259653	−	0.965702i	$-0.583608\pi$
$751$	−10687.7			−0.519305			−0.259653	+	0.965702i	$0.583608\pi$
$752$		0			0
$753$	−14592.6			−0.706221
$754$		0			0
$755$	−45683.2			−2.20210
$756$		0			0
$757$	28909.2			1.38801			0.694004	−	0.719971i	$-0.255845\pi$
$757$	28909.2			1.38801			0.694004	+	0.719971i	$0.255845\pi$
$758$		0			0
$759$		−	7692.23i		−	0.367866i
$760$		0			0
$761$		−	25399.7i		−	1.20991i	−0.796261	−	0.604954i	$-0.793192\pi$
$761$		−	25399.7i		−	1.20991i	0.796261	−	0.604954i	$-0.206808\pi$
$762$		0			0
$763$	15060.9			0.714604
$764$		0			0
$765$			26390.7i			1.24727i
$766$		0			0
$767$	−7817.46	+	3718.66i	−0.368021	+	0.175063i
$768$		0			0
$769$		−	23740.3i		−	1.11326i	−0.830761	−	0.556630i	$-0.812094\pi$
$769$		−	23740.3i		−	1.11326i	0.830761	−	0.556630i	$-0.187906\pi$
$770$		0			0
$771$	56878.4			2.65684
$772$		0			0
$773$		−	37.2842i		−	0.00173483i	−1.00000		0.000867413i	$-0.999724\pi$
$773$		−	37.2842i		−	0.00173483i	1.00000		0.000867413i	$-0.000276106\pi$
$774$		0			0
$775$		−	35034.7i		−	1.62385i
$776$		0			0
$777$	11435.3			0.527978
$778$		0			0
$779$	−2375.72			−0.109267
$780$		0			0
$781$	12984.2			0.594893
$782$		0			0
$783$	63262.3			2.88737
$784$		0			0
$785$			14905.3i			0.677697i
$786$		0			0
$787$			6972.34i			0.315803i	0.987455	+	0.157902i	$0.0504729\pi$
$787$			6972.34i			0.315803i	−0.987455	+	0.157902i	$0.949527\pi$
$788$		0			0
$789$	32451.9			1.46428
$790$		0			0
$791$		−	16610.9i		−	0.746669i
$792$		0			0
$793$	−3982.84	−	8372.81i	−0.178354	−	0.374940i
$794$		0			0
$795$			11421.2i			0.509518i
$796$		0			0
$797$	−13689.7			−0.608425			−0.304212	−	0.952604i	$-0.598393\pi$
$797$	−13689.7			−0.608425			−0.304212	+	0.952604i	$0.598393\pi$
$798$		0			0
$799$		−	5040.73i		−	0.223189i
$800$		0			0
$801$		−	70554.6i		−	3.11227i
$802$		0			0
$803$	−29677.1			−1.30421
$804$		0			0
$805$	4517.98			0.197811
$806$		0			0
$807$	−50048.2			−2.18312
$808$		0			0
$809$	−7098.88			−0.308509			−0.154254	−	0.988031i	$-0.549298\pi$
$809$	−7098.88			−0.308509			−0.154254	+	0.988031i	$0.549298\pi$
$810$		0			0
$811$		−	24631.2i		−	1.06649i	−0.845962	−	0.533243i	$-0.820973\pi$
$811$		−	24631.2i		−	1.06649i	0.845962	−	0.533243i	$-0.179027\pi$
$812$		0			0
$813$		−	59149.2i		−	2.55160i
$814$		0			0
$815$	56343.8			2.42164
$816$		0			0
$817$		−	8335.02i		−	0.356922i
$818$		0			0
$819$	−23729.5	+	11287.8i	−1.01242	+	0.481596i
$820$		0			0
$821$			2290.09i			0.0973506i	0.998815	+	0.0486753i	$0.0154999\pi$
$821$			2290.09i			0.0973506i	−0.998815	+	0.0486753i	$0.984500\pi$
$822$		0			0
$823$	36484.9			1.54530			0.772651	−	0.634831i	$-0.218930\pi$
$823$	36484.9			1.54530			0.772651	+	0.634831i	$0.218930\pi$
$824$		0			0
$825$		−	49746.4i		−	2.09933i
$826$		0			0
$827$			3929.01i			0.165206i	0.996583	+	0.0826028i	$0.0263233\pi$
$827$			3929.01i			0.165206i	−0.996583	+	0.0826028i	$0.973677\pi$
$828$		0			0
$829$	−17401.7			−0.729055			−0.364527	−	0.931193i	$-0.618769\pi$
$829$	−17401.7			−0.729055			−0.364527	+	0.931193i	$0.618769\pi$
$830$		0			0
$831$	25757.6			1.07524
$832$		0			0
$833$	−6821.88			−0.283750
$834$		0			0
$835$	−30636.4			−1.26972
$836$		0			0
$837$			47914.1i			1.97868i
$838$		0			0
$839$		−	39966.0i		−	1.64455i	−0.569089	−	0.822276i	$-0.692704\pi$
$839$		−	39966.0i		−	1.64455i	0.569089	−	0.822276i	$-0.307296\pi$
$840$		0			0
$841$	59778.5			2.45105
$842$		0			0
$843$		−	71010.2i		−	2.90121i
$844$		0			0
$845$	28746.9	+	23378.9i	1.17032	+	0.951786i
$846$		0			0
$847$		−	1004.73i		−	0.0407589i
$848$		0			0
$849$	32870.6			1.32876
$850$		0			0
$851$		−	2926.82i		−	0.117897i
$852$		0			0
$853$		−	7753.33i		−	0.311218i	−0.987819	−	0.155609i	$-0.950266\pi$
$853$		−	7753.33i		−	0.311218i	0.987819	−	0.155609i	$-0.0497340\pi$
$854$		0			0
$855$	−24701.0			−0.988019
$856$		0			0
$857$	34346.7			1.36903			0.684515	−	0.728998i	$-0.260014\pi$
$857$	34346.7			1.36903			0.684515	+	0.728998i	$0.260014\pi$
$858$		0			0
$859$	23048.5			0.915489			0.457744	−	0.889084i	$-0.348658\pi$
$859$	23048.5			0.915489			0.457744	+	0.889084i	$0.348658\pi$
$860$		0			0
$861$	−8062.09			−0.319112
$862$		0			0
$863$			8452.37i			0.333398i	0.986008	+	0.166699i	$0.0533107\pi$
$863$			8452.37i			0.333398i	−0.986008	+	0.166699i	$0.946689\pi$
$864$		0			0
$865$		−	16599.7i		−	0.652492i
$866$		0			0
$867$	−35402.0			−1.38675
$868$		0			0
$869$		−	6477.02i		−	0.252840i
$870$		0			0
$871$	13598.6	−	6468.68i	0.529014	−	0.251645i
$872$		0			0
$873$			35613.9i			1.38070i
$874$		0			0
$875$	6311.85			0.243862
$876$		0			0
$877$		−	3371.07i		−	0.129798i	−0.997892	−	0.0648990i	$-0.979327\pi$
$877$		−	3371.07i		−	0.129798i	0.997892	−	0.0648990i	$-0.0206725\pi$
$878$		0			0
$879$			14840.8i			0.569473i
$880$		0			0
$881$	−9930.37			−0.379753			−0.189877	−	0.981808i	$-0.560809\pi$
$881$	−9930.37			−0.379753			−0.189877	+	0.981808i	$0.560809\pi$
$882$		0			0
$883$	38422.6			1.46435			0.732176	−	0.681116i	$-0.238505\pi$
$883$	38422.6			1.46435			0.732176	+	0.681116i	$0.238505\pi$
$884$		0			0
$885$	−27615.0			−1.04889
$886$		0			0
$887$	26078.1			0.987168			0.493584	−	0.869698i	$-0.335686\pi$
$887$	26078.1			0.987168			0.493584	+	0.869698i	$0.335686\pi$
$888$		0			0
$889$			15729.6i			0.593422i
$890$		0			0
$891$			19007.1i			0.714658i
$892$		0			0
$893$	4717.98			0.176799
$894$		0			0
$895$			63706.2i			2.37929i
$896$		0			0
$897$	4400.89	+	9251.66i	0.163814	+	0.344375i
$898$		0			0
$899$			63747.6i			2.36496i
$900$		0			0
$901$	2316.57			0.0856560
$902$		0			0
$903$		−	28285.2i		−	1.04238i
$904$		0			0
$905$			39293.1i			1.44326i
$906$		0			0
$907$	8988.90			0.329076			0.164538	−	0.986371i	$-0.447387\pi$
$907$	8988.90			0.329076			0.164538	+	0.986371i	$0.447387\pi$
$908$		0			0
$909$	−71987.7			−2.62671
$910$		0			0
$911$	−28318.0			−1.02988			−0.514938	−	0.857227i	$-0.672185\pi$
$911$	−28318.0			−1.02988			−0.514938	+	0.857227i	$0.672185\pi$
$912$		0			0
$913$	−44841.2			−1.62544
$914$		0			0
$915$		−	29576.7i		−	1.06861i
$916$		0			0
$917$			6865.64i			0.247245i
$918$		0			0
$919$	12863.6			0.461730			0.230865	−	0.972986i	$-0.425845\pi$
$919$	12863.6			0.461730			0.230865	+	0.972986i	$0.425845\pi$
$920$		0			0
$921$			2659.07i			0.0951348i
$922$		0			0
$923$	−15616.5	+	7428.55i	−0.556904	+	0.264912i
$924$		0			0
$925$		−	18928.0i		−	0.672810i
$926$		0			0
$927$	−73074.8			−2.58909
$928$		0			0
$929$			1899.49i			0.0670833i	0.999437	+	0.0335416i	$0.0106786\pi$
$929$			1899.49i			0.0670833i	−0.999437	+	0.0335416i	$0.989321\pi$
$930$		0			0
$931$		−	6385.09i		−	0.224772i
$932$		0			0
$933$	−7891.50			−0.276909
$934$		0			0
$935$	−18000.5			−0.629605
$936$		0			0
$937$	−35209.9			−1.22760			−0.613798	−	0.789463i	$-0.710359\pi$
$937$	−35209.9			−1.22760			−0.613798	+	0.789463i	$0.710359\pi$
$938$		0			0
$939$	−10232.5			−0.355618
$940$		0			0
$941$		−	56666.1i		−	1.96308i	−0.191247	−	0.981542i	$-0.561253\pi$
$941$		−	56666.1i		−	1.96308i	0.191247	−	0.981542i	$-0.438747\pi$
$942$		0			0
$943$			2063.46i			0.0712572i
$944$		0			0
$945$	−39959.4			−1.37553
$946$		0			0
$947$			27336.3i			0.938026i	0.883191	+	0.469013i	$0.155390\pi$
$947$			27336.3i			0.938026i	−0.883191	+	0.469013i	$0.844610\pi$
$948$		0			0
$949$	35693.5	−	16978.9i	1.22093	−	0.580779i
$950$		0			0
$951$		−	24015.3i		−	0.818873i
$952$		0			0
$953$	14435.1			0.490659			0.245329	−	0.969440i	$-0.421104\pi$
$953$	14435.1			0.490659			0.245329	+	0.969440i	$0.421104\pi$
$954$		0			0
$955$		−	77782.3i		−	2.63558i
$956$		0			0
$957$			90516.3i			3.05745i
$958$		0			0
$959$	−5947.95			−0.200281
$960$		0			0
$961$	−18490.7			−0.620680
$962$		0			0
$963$	−23273.5			−0.778793
$964$		0			0
$965$	16069.4			0.536054
$966$		0			0
$967$			21556.9i			0.716879i	0.933553	+	0.358440i	$0.116691\pi$
$967$			21556.9i			0.716879i	−0.933553	+	0.358440i	$0.883309\pi$
$968$		0			0
$969$			7631.89i			0.253015i
$970$		0			0
$971$	3153.54			0.104225			0.0521123	−	0.998641i	$-0.483405\pi$
$971$	3153.54			0.104225			0.0521123	+	0.998641i	$0.483405\pi$
$972$		0			0
$973$		−	7356.68i		−	0.242389i
$974$		0			0
$975$	28461.0	+	59831.4i	0.934853	+	1.96527i
$976$		0			0
$977$			4929.61i			0.161425i	0.996737	+	0.0807125i	$0.0257196\pi$
$977$			4929.61i			0.161425i	−0.996737	+	0.0807125i	$0.974280\pi$
$978$		0			0
$979$	48123.8			1.57103
$980$		0			0
$981$			71519.3i			2.32766i
$982$		0			0
$983$			15526.2i			0.503774i	0.967757	+	0.251887i	$0.0810511\pi$
$983$			15526.2i			0.503774i	−0.967757	+	0.251887i	$0.918949\pi$
$984$		0			0
$985$	−34583.3			−1.11870
$986$		0			0
$987$	16010.6			0.516337
$988$		0			0
$989$	−7239.49			−0.232763
$990$		0			0
$991$	−29876.8			−0.957687			−0.478844	−	0.877900i	$-0.658944\pi$
$991$	−29876.8			−0.957687			−0.478844	+	0.877900i	$0.658944\pi$
$992$		0			0
$993$			72582.6i			2.31958i
$994$		0			0
$995$			43602.4i			1.38924i
$996$		0			0
$997$	2600.46			0.0826052			0.0413026	−	0.999147i	$-0.486849\pi$
$997$	2600.46			0.0826052			0.0413026	+	0.999147i	$0.486849\pi$
$998$		0			0
$999$			25886.3i			0.819826i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.4.f.d.129.3		4
4.3	odd	2		26.4.b.a.25.1	✓	4
8.3	odd	2		832.4.f.j.129.4		4
8.5	even	2		832.4.f.h.129.2		4
12.11	even	2		234.4.b.b.181.4		4
13.12	even	2	inner	208.4.f.d.129.4		4
20.3	even	4		650.4.c.e.649.1		4
20.7	even	4		650.4.c.f.649.4		4
20.19	odd	2		650.4.d.d.51.4		4
52.3	odd	6		338.4.e.g.147.4		8
52.7	even	12		338.4.c.h.315.2		4
52.11	even	12		338.4.c.h.191.2		4
52.15	even	12		338.4.c.i.191.2		4
52.19	even	12		338.4.c.i.315.2		4
52.23	odd	6		338.4.e.g.147.2		8
52.31	even	4		338.4.a.f.1.1		2
52.35	odd	6		338.4.e.g.23.2		8
52.43	odd	6		338.4.e.g.23.4		8
52.47	even	4		338.4.a.i.1.1		2
52.51	odd	2		26.4.b.a.25.3	yes	4
104.51	odd	2		832.4.f.j.129.3		4
104.77	even	2		832.4.f.h.129.1		4
156.155	even	2		234.4.b.b.181.1		4
260.103	even	4		650.4.c.f.649.1		4
260.207	even	4		650.4.c.e.649.4		4
260.259	odd	2		650.4.d.d.51.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.4.b.a.25.1	✓	4	4.3	odd	2
26.4.b.a.25.3	yes	4	52.51	odd	2
208.4.f.d.129.3		4	1.1	even	1	trivial
208.4.f.d.129.4		4	13.12	even	2	inner
234.4.b.b.181.1		4	156.155	even	2
234.4.b.b.181.4		4	12.11	even	2
338.4.a.f.1.1		2	52.31	even	4
338.4.a.i.1.1		2	52.47	even	4
338.4.c.h.191.2		4	52.11	even	12
338.4.c.h.315.2		4	52.7	even	12
338.4.c.i.191.2		4	52.15	even	12
338.4.c.i.315.2		4	52.19	even	12
338.4.e.g.23.2		8	52.35	odd	6
338.4.e.g.23.4		8	52.43	odd	6
338.4.e.g.147.2		8	52.23	odd	6
338.4.e.g.147.4		8	52.3	odd	6
650.4.c.e.649.1		4	20.3	even	4
650.4.c.e.649.4		4	260.207	even	4
650.4.c.f.649.1		4	260.103	even	4
650.4.c.f.649.4		4	20.7	even	4
650.4.d.d.51.2		4	260.259	odd	2
650.4.d.d.51.4		4	20.19	odd	2
832.4.f.h.129.1		4	104.77	even	2
832.4.f.h.129.2		4	8.5	even	2
832.4.f.j.129.3		4	104.51	odd	2
832.4.f.j.129.4		4	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.86546 q^{3} -16.8655i q^{5} -10.8655i q^{7} +51.5964 q^{9} +O(q^{10})\)	\(q+8.86546 q^{3} -16.8655i q^{5} -10.8655i q^{7} +51.5964 q^{9} +35.1928i q^{11} +(-20.1345 - 42.3273i) q^{13} -149.520i q^{15} -30.3273 q^{17} -28.3855i q^{19} -96.3273i q^{21} -24.6546 q^{23} -159.444 q^{25} +218.058 q^{27} +290.116 q^{29} +219.731i q^{31} +312.000i q^{33} -183.251 q^{35} +118.713i q^{37} +(-178.502 - 375.251i) q^{39} -83.6947i q^{41} +293.636 q^{43} -870.197i q^{45} +166.211i q^{47} +224.942 q^{49} -268.865 q^{51} -76.3855 q^{53} +593.542 q^{55} -251.651i q^{57} -184.691i q^{59} +197.811 q^{61} -560.618i q^{63} +(-713.869 + 339.578i) q^{65} +321.273i q^{67} -218.574 q^{69} -368.946i q^{71} +843.273i q^{73} -1413.54 q^{75} +382.386 q^{77} -184.044 q^{79} +540.084 q^{81} +1274.16i q^{83} +511.484i q^{85} +2572.02 q^{87} -1367.43i q^{89} +(-459.906 + 218.771i) q^{91} +1948.02i q^{93} -478.735 q^{95} +690.241i q^{97} +1815.82i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{3} + 118 q^{9}+O(q^{10}) \) 4 * q + 6 * q^3 + 118 * q^9	\( 4 q + 6 q^{3} + 118 q^{9} - 110 q^{13} + 26 q^{17} + 196 q^{23} - 78 q^{25} + 666 q^{27} + 748 q^{29} - 350 q^{35} + 52 q^{39} + 438 q^{43} + 1106 q^{49} - 1046 q^{51} + 48 q^{53} + 960 q^{55} - 564 q^{61} - 1294 q^{65} - 1876 q^{69} - 4240 q^{75} + 1176 q^{77} + 1444 q^{79} - 668 q^{81} + 4160 q^{87} - 1162 q^{91} - 324 q^{95}+O(q^{100}) \) 4 * q + 6 * q^3 + 118 * q^9 - 110 * q^13 + 26 * q^17 + 196 * q^23 - 78 * q^25 + 666 * q^27 + 748 * q^29 - 350 * q^35 + 52 * q^39 + 438 * q^43 + 1106 * q^49 - 1046 * q^51 + 48 * q^53 + 960 * q^55 - 564 * q^61 - 1294 * q^65 - 1876 * q^69 - 4240 * q^75 + 1176 * q^77 + 1444 * q^79 - 668 * q^81 + 4160 * q^87 - 1162 * q^91 - 324 * q^95

Introduction

L-functions

Modular forms

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Database

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