LMFDB - Embedded newform 208.10.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,10,Mod(1,208)]

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, 0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	208.a (trivial)

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:	yes
Analytic conductor:	$107.127453922$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6144x - 66096 $ x^3 - x^2 - 6144*x - 66096
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-10.9937$ of defining polynomial
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-159.509 q^{3} +1767.44 q^{5} -6243.46 q^{7} +5760.16 q^{9} +41072.0 q^{11} +28561.0 q^{13} -281924. q^{15} +98274.9 q^{17} -507054. q^{19} +995890. q^{21} -1.89860e6 q^{23} +1.17074e6 q^{25} +2.22082e6 q^{27} -321475. q^{29} +6.09551e6 q^{31} -6.55136e6 q^{33} -1.10350e7 q^{35} +1.77236e7 q^{37} -4.55574e6 q^{39} -1.79328e7 q^{41} +1.22817e7 q^{43} +1.01808e7 q^{45} -4.23821e7 q^{47} -1.37275e6 q^{49} -1.56757e7 q^{51} +3.92537e6 q^{53} +7.25925e7 q^{55} +8.08798e7 q^{57} +1.72026e8 q^{59} +1.36776e8 q^{61} -3.59633e7 q^{63} +5.04800e7 q^{65} -1.21248e8 q^{67} +3.02844e8 q^{69} +3.66817e8 q^{71} -1.13266e8 q^{73} -1.86743e8 q^{75} -2.56432e8 q^{77} -2.79788e8 q^{79} -4.67618e8 q^{81} +7.50041e8 q^{83} +1.73695e8 q^{85} +5.12782e7 q^{87} -7.04433e8 q^{89} -1.78320e8 q^{91} -9.72289e8 q^{93} -8.96190e8 q^{95} -1.21751e9 q^{97} +2.36581e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 248 q^{5} + 2956 q^{7} - 20863 q^{9} + 31324 q^{11} + 85683 q^{13} - 392140 q^{15} + 905228 q^{17} - 1726316 q^{19} + 1771246 q^{21} - 2135256 q^{23} + 4074295 q^{25} - 3038724 q^{27} + 372426 q^{29}+ \cdots + 304026580 q^{99}+O(q^{100}) $ 3 * q + 248 * q^5 + 2956 * q^7 - 20863 * q^9 + 31324 * q^11 + 85683 * q^13 - 392140 * q^15 + 905228 * q^17 - 1726316 * q^19 + 1771246 * q^21 - 2135256 * q^23 + 4074295 * q^25 - 3038724 * q^27 + 372426 * q^29 + 1444776 * q^31 - 7719476 * q^33 + 3459436 * q^35 - 2809672 * q^37 - 7424806 * q^41 - 36070040 * q^43 + 32156162 * q^45 - 7032884 * q^47 + 41844597 * q^49 + 47158124 * q^51 - 15011674 * q^53 - 121479136 * q^55 - 14804300 * q^57 + 40286396 * q^59 + 230710478 * q^61 - 151778040 * q^63 + 7083128 * q^65 - 201453084 * q^67 + 268135008 * q^69 + 99165572 * q^71 - 350942946 * q^73 + 28166788 * q^75 - 1066624588 * q^77 - 138794592 * q^79 - 363498349 * q^81 + 1364469516 * q^83 - 1225485962 * q^85 + 93372576 * q^87 - 709437058 * q^89 + 84426316 * q^91 - 1382471232 * q^93 + 833387824 * q^95 - 362189698 * q^97 + 304026580 * q^99

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
$3$	−159.509	−1.13695	−0.568473	−	0.822702i	$-0.692466\pi$
$3$	−159.509	−1.13695	−0.568473	+	0.822702i	$0.692466\pi$
$4$	0	0
$5$	1767.44	1.26468	0.632340	−	0.774691i	$-0.282095\pi$
$5$	1767.44	1.26468	0.632340	+	0.774691i	$0.282095\pi$
$6$	0	0
$7$	−6243.46	−0.982844	−0.491422	−	0.870922i	$-0.663523\pi$
$7$	−6243.46	−0.982844	−0.491422	+	0.870922i	$0.663523\pi$
$8$	0	0
$9$	5760.16	0.292646
$10$	0	0
$11$	41072.0	0.845821	0.422911	−	0.906171i	$-0.361008\pi$
$11$	41072.0	0.845821	0.422911	+	0.906171i	$0.361008\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	−281924.	−1.43787
$16$	0	0
$17$	98274.9	0.285379	0.142690	−	0.989767i	$-0.454425\pi$
$17$	98274.9	0.285379	0.142690	+	0.989767i	$0.454425\pi$
$18$	0	0
$19$	−507054.	−0.892613	−0.446307	−	0.894880i	$-0.647261\pi$
$19$	−507054.	−0.892613	−0.446307	+	0.894880i	$0.647261\pi$
$20$	0	0
$21$	995890.	1.11744
$22$	0	0
$23$	−1.89860e6	−1.41468	−0.707340	−	0.706874i	$-0.750105\pi$
$23$	−1.89860e6	−1.41468	−0.707340	+	0.706874i	$0.750105\pi$
$24$	0	0
$25$	1.17074e6	0.599416
$26$	0	0
$27$	2.22082e6	0.804223
$28$	0	0
$29$	−321475.	−0.0844027	−0.0422013	−	0.999109i	$-0.513437\pi$
$29$	−321475.	−0.0844027	−0.0422013	+	0.999109i	$0.513437\pi$
$30$	0	0
$31$	6.09551e6	1.18545	0.592724	−	0.805406i	$-0.298053\pi$
$31$	6.09551e6	1.18545	0.592724	+	0.805406i	$0.298053\pi$
$32$	0	0
$33$	−6.55136e6	−0.961653
$34$	0	0
$35$	−1.10350e7	−1.24298
$36$	0	0
$37$	1.77236e7	1.55469	0.777344	−	0.629076i	$-0.216566\pi$
$37$	1.77236e7	1.55469	0.777344	+	0.629076i	$0.216566\pi$
$38$	0	0
$39$	−4.55574e6	−0.315332
$40$	0	0
$41$	−1.79328e7	−0.991110	−0.495555	−	0.868577i	$-0.665035\pi$
$41$	−1.79328e7	−0.991110	−0.495555	+	0.868577i	$0.665035\pi$
$42$	0	0
$43$	1.22817e7	0.547834	0.273917	−	0.961753i	$-0.411681\pi$
$43$	1.22817e7	0.547834	0.273917	+	0.961753i	$0.411681\pi$
$44$	0	0
$45$	1.01808e7	0.370104
$46$	0	0
$47$	−4.23821e7	−1.26690	−0.633449	−	0.773784i	$-0.718361\pi$
$47$	−4.23821e7	−1.26690	−0.633449	+	0.773784i	$0.718361\pi$
$48$	0	0
$49$	−1.37275e6	−0.0340181
$50$	0	0
$51$	−1.56757e7	−0.324461
$52$	0	0
$53$	3.92537e6	0.0683344	0.0341672	−	0.999416i	$-0.489122\pi$
$53$	3.92537e6	0.0683344	0.0341672	+	0.999416i	$0.489122\pi$
$54$	0	0
$55$	7.25925e7	1.06969
$56$	0	0
$57$	8.08798e7	1.01485
$58$	0	0
$59$	1.72026e8	1.84824	0.924122	−	0.382098i	$-0.124798\pi$
$59$	1.72026e8	1.84824	0.924122	+	0.382098i	$0.124798\pi$
$60$	0	0
$61$	1.36776e8	1.26481	0.632407	−	0.774636i	$-0.282067\pi$
$61$	1.36776e8	1.26481	0.632407	+	0.774636i	$0.282067\pi$
$62$	0	0
$63$	−3.59633e7	−0.287626
$64$	0	0
$65$	5.04800e7	0.350759
$66$	0	0
$67$	−1.21248e8	−0.735086	−0.367543	−	0.930006i	$-0.619801\pi$
$67$	−1.21248e8	−0.735086	−0.367543	+	0.930006i	$0.619801\pi$
$68$	0	0
$69$	3.02844e8	1.60841
$70$	0	0
$71$	3.66817e8	1.71311	0.856557	−	0.516052i	$-0.172599\pi$
$71$	3.66817e8	1.71311	0.856557	+	0.516052i	$0.172599\pi$
$72$	0	0
$73$	−1.13266e8	−0.466816	−0.233408	−	0.972379i	$-0.574988\pi$
$73$	−1.13266e8	−0.466816	−0.233408	+	0.972379i	$0.574988\pi$
$74$	0	0
$75$	−1.86743e8	−0.681504
$76$	0	0
$77$	−2.56432e8	−0.831310
$78$	0	0
$79$	−2.79788e8	−0.808179	−0.404089	−	0.914720i	$-0.632412\pi$
$79$	−2.79788e8	−0.808179	−0.404089	+	0.914720i	$0.632412\pi$
$80$	0	0
$81$	−4.67618e8	−1.20700
$82$	0	0
$83$	7.50041e8	1.73474	0.867368	−	0.497667i	$-0.165810\pi$
$83$	7.50041e8	1.73474	0.867368	+	0.497667i	$0.165810\pi$
$84$	0	0
$85$	1.73695e8	0.360914
$86$	0	0
$87$	5.12782e7	0.0959613
$88$	0	0
$89$	−7.04433e8	−1.19010	−0.595052	−	0.803688i	$-0.702868\pi$
$89$	−7.04433e8	−1.19010	−0.595052	+	0.803688i	$0.702868\pi$
$90$	0	0
$91$	−1.78320e8	−0.272592
$92$	0	0
$93$	−9.72289e8	−1.34779
$94$	0	0
$95$	−8.96190e8	−1.12887
$96$	0	0
$97$	−1.21751e9	−1.39637	−0.698183	−	0.715920i	$-0.746008\pi$
$97$	−1.21751e9	−1.39637	−0.698183	+	0.715920i	$0.746008\pi$
$98$	0	0
$99$	2.36581e8	0.247527
$100$	0	0
$101$	−9.52618e8	−0.910904	−0.455452	−	0.890260i	$-0.650522\pi$
$101$	−9.52618e8	−0.910904	−0.455452	+	0.890260i	$0.650522\pi$
$102$	0	0
$103$	−1.27341e9	−1.11481	−0.557405	−	0.830240i	$-0.688203\pi$
$103$	−1.27341e9	−1.11481	−0.557405	+	0.830240i	$0.688203\pi$
$104$	0	0
$105$	1.76018e9	1.41320
$106$	0	0
$107$	−9.71126e8	−0.716223	−0.358112	−	0.933679i	$-0.616579\pi$
$107$	−9.71126e8	−0.716223	−0.358112	+	0.933679i	$0.616579\pi$
$108$	0	0
$109$	−1.35575e9	−0.919940	−0.459970	−	0.887935i	$-0.652140\pi$
$109$	−1.35575e9	−0.919940	−0.459970	+	0.887935i	$0.652140\pi$
$110$	0	0
$111$	−2.82707e9	−1.76760
$112$	0	0
$113$	2.19491e9	1.26638	0.633189	−	0.773997i	$-0.281746\pi$
$113$	2.19491e9	1.26638	0.633189	+	0.773997i	$0.281746\pi$
$114$	0	0
$115$	−3.35567e9	−1.78912
$116$	0	0
$117$	1.64516e8	0.0811655
$118$	0	0
$119$	−6.13576e8	−0.280483
$120$	0	0
$121$	−6.71039e8	−0.284586
$122$	0	0
$123$	2.86045e9	1.12684
$124$	0	0
$125$	−1.38283e9	−0.506610
$126$	0	0
$127$	−2.77758e9	−0.947437	−0.473718	−	0.880676i	$-0.657089\pi$
$127$	−2.77758e9	−0.947437	−0.473718	+	0.880676i	$0.657089\pi$
$128$	0	0
$129$	−1.95904e9	−0.622858
$130$	0	0
$131$	5.91870e8	0.175592	0.0877961	−	0.996138i	$-0.472018\pi$
$131$	5.91870e8	0.175592	0.0877961	+	0.996138i	$0.472018\pi$
$132$	0	0
$133$	3.16578e9	0.877299
$134$	0	0
$135$	3.92518e9	1.01708
$136$	0	0
$137$	−4.78496e9	−1.16048	−0.580238	−	0.814447i	$-0.697040\pi$
$137$	−4.78496e9	−1.16048	−0.580238	+	0.814447i	$0.697040\pi$
$138$	0	0
$139$	9.91846e8	0.225360	0.112680	−	0.993631i	$-0.464056\pi$
$139$	9.91846e8	0.225360	0.112680	+	0.993631i	$0.464056\pi$
$140$	0	0
$141$	6.76033e9	1.44039
$142$	0	0
$143$	1.17306e9	0.234589
$144$	0	0
$145$	−5.68189e8	−0.106742
$146$	0	0
$147$	2.18967e8	0.0386767
$148$	0	0
$149$	−7.40400e9	−1.23063	−0.615316	−	0.788280i	$-0.710972\pi$
$149$	−7.40400e9	−1.23063	−0.615316	+	0.788280i	$0.710972\pi$
$150$	0	0
$151$	−8.11389e9	−1.27009	−0.635043	−	0.772477i	$-0.719017\pi$
$151$	−8.11389e9	−1.27009	−0.635043	+	0.772477i	$0.719017\pi$
$152$	0	0
$153$	5.66079e8	0.0835152
$154$	0	0
$155$	1.07735e10	1.49921
$156$	0	0
$157$	−1.10905e10	−1.45681	−0.728407	−	0.685145i	$-0.759739\pi$
$157$	−1.10905e10	−1.45681	−0.728407	+	0.685145i	$0.759739\pi$
$158$	0	0
$159$	−6.26132e8	−0.0776925
$160$	0	0
$161$	1.18538e10	1.39041
$162$	0	0
$163$	−7.83983e8	−0.0869886	−0.0434943	−	0.999054i	$-0.513849\pi$
$163$	−7.83983e8	−0.0869886	−0.0434943	+	0.999054i	$0.513849\pi$
$164$	0	0
$165$	−1.15792e10	−1.21618
$166$	0	0
$167$	−1.26374e10	−1.25729	−0.628643	−	0.777694i	$-0.716389\pi$
$167$	−1.26374e10	−1.25729	−0.628643	+	0.777694i	$0.716389\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	−2.92071e9	−0.261220
$172$	0	0
$173$	−3.65563e9	−0.310281	−0.155140	−	0.987892i	$-0.549583\pi$
$173$	−3.65563e9	−0.310281	−0.155140	+	0.987892i	$0.549583\pi$
$174$	0	0
$175$	−7.30944e9	−0.589133
$176$	0	0
$177$	−2.74397e10	−2.10135
$178$	0	0
$179$	−1.00526e10	−0.731879	−0.365939	−	0.930639i	$-0.619252\pi$
$179$	−1.00526e10	−0.731879	−0.365939	+	0.930639i	$0.619252\pi$
$180$	0	0
$181$	−6.35857e9	−0.440358	−0.220179	−	0.975459i	$-0.570664\pi$
$181$	−6.35857e9	−0.440358	−0.220179	+	0.975459i	$0.570664\pi$
$182$	0	0
$183$	−2.18171e10	−1.43803
$184$	0	0
$185$	3.13254e10	1.96618
$186$	0	0
$187$	4.03635e9	0.241380
$188$	0	0
$189$	−1.38656e10	−0.790426
$190$	0	0
$191$	2.56451e10	1.39429	0.697146	−	0.716929i	$-0.254453\pi$
$191$	2.56451e10	1.39429	0.697146	+	0.716929i	$0.254453\pi$
$192$	0	0
$193$	3.05861e10	1.58678	0.793389	−	0.608715i	$-0.208315\pi$
$193$	3.05861e10	1.58678	0.793389	+	0.608715i	$0.208315\pi$
$194$	0	0
$195$	−8.05202e9	−0.398794
$196$	0	0
$197$	−1.73305e10	−0.819811	−0.409905	−	0.912128i	$-0.634438\pi$
$197$	−1.73305e10	−0.819811	−0.409905	+	0.912128i	$0.634438\pi$
$198$	0	0
$199$	1.44467e10	0.653025	0.326512	−	0.945193i	$-0.394126\pi$
$199$	1.44467e10	0.653025	0.326512	+	0.945193i	$0.394126\pi$
$200$	0	0
$201$	1.93402e10	0.835753
$202$	0	0
$203$	2.00712e9	0.0829546
$204$	0	0
$205$	−3.16953e10	−1.25344
$206$	0	0
$207$	−1.09362e10	−0.414001
$208$	0	0
$209$	−2.08257e10	−0.754991
$210$	0	0
$211$	−5.08715e10	−1.76686	−0.883432	−	0.468559i	$-0.844774\pi$
$211$	−5.08715e10	−1.76686	−0.883432	+	0.468559i	$0.844774\pi$
$212$	0	0
$213$	−5.85106e10	−1.94772
$214$	0	0
$215$	2.17072e10	0.692835
$216$	0	0
$217$	−3.80571e10	−1.16511
$218$	0	0
$219$	1.80669e10	0.530745
$220$	0	0
$221$	2.80683e9	0.0791500
$222$	0	0
$223$	−2.27613e10	−0.616347	−0.308174	−	0.951330i	$-0.599718\pi$
$223$	−2.27613e10	−0.616347	−0.308174	+	0.951330i	$0.599718\pi$
$224$	0	0
$225$	6.74362e9	0.175417
$226$	0	0
$227$	−2.24918e10	−0.562222	−0.281111	−	0.959675i	$-0.590703\pi$
$227$	−2.24918e10	−0.562222	−0.281111	+	0.959675i	$0.590703\pi$
$228$	0	0
$229$	7.10153e10	1.70644	0.853222	−	0.521548i	$-0.174645\pi$
$229$	7.10153e10	1.70644	0.853222	+	0.521548i	$0.174645\pi$
$230$	0	0
$231$	4.09032e10	0.945155
$232$	0	0
$233$	5.11111e10	1.13609	0.568046	−	0.822997i	$-0.307699\pi$
$233$	5.11111e10	1.13609	0.568046	+	0.822997i	$0.307699\pi$
$234$	0	0
$235$	−7.49079e10	−1.60222
$236$	0	0
$237$	4.46288e10	0.918856
$238$	0	0
$239$	2.14909e10	0.426054	0.213027	−	0.977046i	$-0.431668\pi$
$239$	2.14909e10	0.426054	0.213027	+	0.977046i	$0.431668\pi$
$240$	0	0
$241$	5.08676e10	0.971326	0.485663	−	0.874146i	$-0.338578\pi$
$241$	5.08676e10	0.971326	0.485663	+	0.874146i	$0.338578\pi$
$242$	0	0
$243$	3.08770e10	0.568076
$244$	0	0
$245$	−2.42626e9	−0.0430220
$246$	0	0
$247$	−1.44820e10	−0.247566
$248$	0	0
$249$	−1.19638e11	−1.97230
$250$	0	0
$251$	−4.23328e10	−0.673201	−0.336601	−	0.941647i	$-0.609277\pi$
$251$	−4.23328e10	−0.673201	−0.336601	+	0.941647i	$0.609277\pi$
$252$	0	0
$253$	−7.79792e10	−1.19657
$254$	0	0
$255$	−2.77060e10	−0.410339
$256$	0	0
$257$	6.92463e10	0.990142	0.495071	−	0.868852i	$-0.335142\pi$
$257$	6.92463e10	0.990142	0.495071	+	0.868852i	$0.335142\pi$
$258$	0	0
$259$	−1.10656e11	−1.52802
$260$	0	0
$261$	−1.85175e9	−0.0247001
$262$	0	0
$263$	4.17509e10	0.538102	0.269051	−	0.963126i	$-0.413290\pi$
$263$	4.17509e10	0.538102	0.269051	+	0.963126i	$0.413290\pi$
$264$	0	0
$265$	6.93787e9	0.0864211
$266$	0	0
$267$	1.12363e11	1.35308
$268$	0	0
$269$	−6.76023e10	−0.787184	−0.393592	−	0.919285i	$-0.628768\pi$
$269$	−6.76023e10	−0.787184	−0.393592	+	0.919285i	$0.628768\pi$
$270$	0	0
$271$	−7.29675e10	−0.821803	−0.410902	−	0.911680i	$-0.634786\pi$
$271$	−7.29675e10	−0.821803	−0.410902	+	0.911680i	$0.634786\pi$
$272$	0	0
$273$	2.84436e10	0.309922
$274$	0	0
$275$	4.80844e10	0.506999
$276$	0	0
$277$	−1.13424e10	−0.115757	−0.0578785	−	0.998324i	$-0.518434\pi$
$277$	−1.13424e10	−0.115757	−0.0578785	+	0.998324i	$0.518434\pi$
$278$	0	0
$279$	3.51111e10	0.346917
$280$	0	0
$281$	9.00245e10	0.861355	0.430678	−	0.902506i	$-0.358275\pi$
$281$	9.00245e10	0.861355	0.430678	+	0.902506i	$0.358275\pi$
$282$	0	0
$283$	−4.33672e10	−0.401904	−0.200952	−	0.979601i	$-0.564404\pi$
$283$	−4.33672e10	−0.401904	−0.200952	+	0.979601i	$0.564404\pi$
$284$	0	0
$285$	1.42951e11	1.28346
$286$	0	0
$287$	1.11963e11	0.974106
$288$	0	0
$289$	−1.08930e11	−0.918559
$290$	0	0
$291$	1.94204e11	1.58759
$292$	0	0
$293$	−4.06751e10	−0.322422	−0.161211	−	0.986920i	$-0.551540\pi$
$293$	−4.06751e10	−0.322422	−0.161211	+	0.986920i	$0.551540\pi$
$294$	0	0
$295$	3.04046e11	2.33744
$296$	0	0
$297$	9.12135e10	0.680229
$298$	0	0
$299$	−5.42259e10	−0.392361
$300$	0	0
$301$	−7.66801e10	−0.538435
$302$	0	0
$303$	1.51951e11	1.03565
$304$	0	0
$305$	2.41745e11	1.59959
$306$	0	0
$307$	2.35037e11	1.51013	0.755063	−	0.655652i	$-0.227606\pi$
$307$	2.35037e11	1.51013	0.755063	+	0.655652i	$0.227606\pi$
$308$	0	0
$309$	2.03121e11	1.26748
$310$	0	0
$311$	−1.60150e11	−0.970745	−0.485373	−	0.874307i	$-0.661316\pi$
$311$	−1.60150e11	−0.970745	−0.485373	+	0.874307i	$0.661316\pi$
$312$	0	0
$313$	−2.82360e11	−1.66285	−0.831426	−	0.555635i	$-0.812475\pi$
$313$	−2.82360e11	−1.66285	−0.831426	+	0.555635i	$0.812475\pi$
$314$	0	0
$315$	−6.35632e10	−0.363754
$316$	0	0
$317$	−1.27999e11	−0.711936	−0.355968	−	0.934498i	$-0.615849\pi$
$317$	−1.27999e11	−0.711936	−0.355968	+	0.934498i	$0.615849\pi$
$318$	0	0
$319$	−1.32036e10	−0.0713896
$320$	0	0
$321$	1.54903e11	0.814307
$322$	0	0
$323$	−4.98307e10	−0.254733
$324$	0	0
$325$	3.34374e10	0.166248
$326$	0	0
$327$	2.16254e11	1.04592
$328$	0	0
$329$	2.64611e11	1.24516
$330$	0	0
$331$	−8.41594e10	−0.385369	−0.192684	−	0.981261i	$-0.561719\pi$
$331$	−8.41594e10	−0.385369	−0.192684	+	0.981261i	$0.561719\pi$
$332$	0	0
$333$	1.02091e11	0.454974
$334$	0	0
$335$	−2.14299e11	−0.929649
$336$	0	0
$337$	−3.35056e11	−1.41509	−0.707543	−	0.706671i	$-0.750196\pi$
$337$	−3.35056e11	−1.41509	−0.707543	+	0.706671i	$0.750196\pi$
$338$	0	0
$339$	−3.50108e11	−1.43980
$340$	0	0
$341$	2.50355e11	1.00268
$342$	0	0
$343$	2.60517e11	1.01628
$344$	0	0
$345$	5.35260e11	2.03413
$346$	0	0
$347$	−1.52065e11	−0.563050	−0.281525	−	0.959554i	$-0.590840\pi$
$347$	−1.52065e11	−0.563050	−0.281525	+	0.959554i	$0.590840\pi$
$348$	0	0
$349$	3.54508e11	1.27912	0.639561	−	0.768741i	$-0.279116\pi$
$349$	3.54508e11	1.27912	0.639561	+	0.768741i	$0.279116\pi$
$350$	0	0
$351$	6.34288e10	0.223051
$352$	0	0
$353$	−2.46472e11	−0.844852	−0.422426	−	0.906397i	$-0.638821\pi$
$353$	−2.46472e11	−0.844852	−0.422426	+	0.906397i	$0.638821\pi$
$354$	0	0
$355$	6.48328e11	2.16654
$356$	0	0
$357$	9.78710e10	0.318894
$358$	0	0
$359$	2.29573e11	0.729451	0.364726	−	0.931115i	$-0.381163\pi$
$359$	2.29573e11	0.729451	0.364726	+	0.931115i	$0.381163\pi$
$360$	0	0
$361$	−6.55836e10	−0.203242
$362$	0	0
$363$	1.07037e11	0.323559
$364$	0	0
$365$	−2.00191e11	−0.590374
$366$	0	0
$367$	−5.22127e11	−1.50238	−0.751189	−	0.660087i	$-0.770519\pi$
$367$	−5.22127e11	−1.50238	−0.751189	+	0.660087i	$0.770519\pi$
$368$	0	0
$369$	−1.03296e11	−0.290045
$370$	0	0
$371$	−2.45079e10	−0.0671620
$372$	0	0
$373$	2.20771e11	0.590543	0.295272	−	0.955413i	$-0.404590\pi$
$373$	2.20771e11	0.590543	0.295272	+	0.955413i	$0.404590\pi$
$374$	0	0
$375$	2.20574e11	0.575989
$376$	0	0
$377$	−9.18165e9	−0.0234091
$378$	0	0
$379$	−1.87709e11	−0.467314	−0.233657	−	0.972319i	$-0.575069\pi$
$379$	−1.87709e11	−0.467314	−0.233657	+	0.972319i	$0.575069\pi$
$380$	0	0
$381$	4.43050e11	1.07718
$382$	0	0
$383$	−6.27959e11	−1.49120	−0.745601	−	0.666392i	$-0.767838\pi$
$383$	−6.27959e11	−1.49120	−0.745601	+	0.666392i	$0.767838\pi$
$384$	0	0
$385$	−4.53229e11	−1.05134
$386$	0	0
$387$	7.07443e10	0.160322
$388$	0	0
$389$	8.24996e11	1.82675	0.913374	−	0.407122i	$-0.133467\pi$
$389$	8.24996e11	1.82675	0.913374	+	0.407122i	$0.133467\pi$
$390$	0	0
$391$	−1.86585e11	−0.403720
$392$	0	0
$393$	−9.44086e10	−0.199639
$394$	0	0
$395$	−4.94510e11	−1.02209
$396$	0	0
$397$	−8.55547e11	−1.72857	−0.864284	−	0.503005i	$-0.832228\pi$
$397$	−8.55547e11	−1.72857	−0.864284	+	0.503005i	$0.832228\pi$
$398$	0	0
$399$	−5.04970e11	−0.997442
$400$	0	0
$401$	−2.36025e11	−0.455835	−0.227917	−	0.973680i	$-0.573192\pi$
$401$	−2.36025e11	−0.455835	−0.227917	+	0.973680i	$0.573192\pi$
$402$	0	0
$403$	1.74094e11	0.328784
$404$	0	0
$405$	−8.26489e11	−1.52647
$406$	0	0
$407$	7.27942e11	1.31499
$408$	0	0
$409$	−2.21041e11	−0.390587	−0.195293	−	0.980745i	$-0.562566\pi$
$409$	−2.21041e11	−0.390587	−0.195293	+	0.980745i	$0.562566\pi$
$410$	0	0
$411$	7.63245e11	1.31940
$412$	0	0
$413$	−1.07404e12	−1.81653
$414$	0	0
$415$	1.32566e12	2.19389
$416$	0	0
$417$	−1.58208e11	−0.256223
$418$	0	0
$419$	6.43251e10	0.101957	0.0509786	−	0.998700i	$-0.483766\pi$
$419$	6.43251e10	0.101957	0.0509786	+	0.998700i	$0.483766\pi$
$420$	0	0
$421$	−4.06630e11	−0.630855	−0.315428	−	0.948950i	$-0.602148\pi$
$421$	−4.06630e11	−0.630855	−0.315428	+	0.948950i	$0.602148\pi$
$422$	0	0
$423$	−2.44127e11	−0.370753
$424$	0	0
$425$	1.15054e11	0.171061
$426$	0	0
$427$	−8.53958e11	−1.24311
$428$	0	0
$429$	−1.87113e11	−0.266715
$430$	0	0
$431$	−1.07838e12	−1.50530	−0.752651	−	0.658420i	$-0.771225\pi$
$431$	−1.07838e12	−1.50530	−0.752651	+	0.658420i	$0.771225\pi$
$432$	0	0
$433$	2.02656e11	0.277053	0.138527	−	0.990359i	$-0.455763\pi$
$433$	2.02656e11	0.277053	0.138527	+	0.990359i	$0.455763\pi$
$434$	0	0
$435$	9.06314e10	0.121360
$436$	0	0
$437$	9.62693e11	1.26276
$438$	0	0
$439$	−9.26805e11	−1.19096	−0.595481	−	0.803369i	$-0.703039\pi$
$439$	−9.26805e11	−1.19096	−0.595481	+	0.803369i	$0.703039\pi$
$440$	0	0
$441$	−7.90727e9	−0.00995527
$442$	0	0
$443$	−3.93461e11	−0.485383	−0.242692	−	0.970103i	$-0.578030\pi$
$443$	−3.93461e11	−0.485383	−0.242692	+	0.970103i	$0.578030\pi$
$444$	0	0
$445$	−1.24505e12	−1.50510
$446$	0	0
$447$	1.18101e12	1.39916
$448$	0	0
$449$	6.50584e10	0.0755431	0.0377716	−	0.999286i	$-0.487974\pi$
$449$	6.50584e10	0.0755431	0.0377716	+	0.999286i	$0.487974\pi$
$450$	0	0
$451$	−7.36538e11	−0.838302
$452$	0	0
$453$	1.29424e12	1.44402
$454$	0	0
$455$	−3.15170e11	−0.344742
$456$	0	0
$457$	1.01701e12	1.09069	0.545344	−	0.838213i	$-0.316399\pi$
$457$	1.01701e12	1.09069	0.545344	+	0.838213i	$0.316399\pi$
$458$	0	0
$459$	2.18251e11	0.229509
$460$	0	0
$461$	1.74309e12	1.79748	0.898741	−	0.438480i	$-0.144483\pi$
$461$	1.74309e12	1.79748	0.898741	+	0.438480i	$0.144483\pi$
$462$	0	0
$463$	−3.39417e11	−0.343257	−0.171629	−	0.985162i	$-0.554903\pi$
$463$	−3.39417e11	−0.343257	−0.171629	+	0.985162i	$0.554903\pi$
$464$	0	0
$465$	−1.71847e12	−1.70452
$466$	0	0
$467$	−4.76376e11	−0.463473	−0.231736	−	0.972779i	$-0.574441\pi$
$467$	−4.76376e11	−0.463473	−0.231736	+	0.972779i	$0.574441\pi$
$468$	0	0
$469$	7.57008e11	0.722475
$470$	0	0
$471$	1.76904e12	1.65632
$472$	0	0
$473$	5.04432e11	0.463370
$474$	0	0
$475$	−5.93626e11	−0.535047
$476$	0	0
$477$	2.26107e10	0.0199978
$478$	0	0
$479$	7.63824e11	0.662954	0.331477	−	0.943463i	$-0.392453\pi$
$479$	7.63824e11	0.662954	0.331477	+	0.943463i	$0.392453\pi$
$480$	0	0
$481$	5.06203e11	0.431193
$482$	0	0
$483$	−1.89079e12	−1.58082
$484$	0	0
$485$	−2.15188e12	−1.76596
$486$	0	0
$487$	1.59493e12	1.28488	0.642440	−	0.766336i	$-0.277922\pi$
$487$	1.59493e12	1.28488	0.642440	+	0.766336i	$0.277922\pi$
$488$	0	0
$489$	1.25052e11	0.0989013
$490$	0	0
$491$	9.38889e11	0.729034	0.364517	−	0.931197i	$-0.381234\pi$
$491$	9.38889e11	0.729034	0.364517	+	0.931197i	$0.381234\pi$
$492$	0	0
$493$	−3.15929e10	−0.0240868
$494$	0	0
$495$	4.18144e11	0.313042
$496$	0	0
$497$	−2.29021e12	−1.68372
$498$	0	0
$499$	8.21282e11	0.592979	0.296490	−	0.955036i	$-0.404184\pi$
$499$	8.21282e11	0.592979	0.296490	+	0.955036i	$0.404184\pi$
$500$	0	0
$501$	2.01578e12	1.42947
$502$	0	0
$503$	−2.06643e12	−1.43934	−0.719671	−	0.694315i	$-0.755707\pi$
$503$	−2.06643e12	−1.43934	−0.719671	+	0.694315i	$0.755707\pi$
$504$	0	0
$505$	−1.68370e12	−1.15200
$506$	0	0
$507$	−1.30116e11	−0.0874574
$508$	0	0
$509$	1.63442e12	1.07928	0.539638	−	0.841897i	$-0.318561\pi$
$509$	1.63442e12	1.07928	0.539638	+	0.841897i	$0.318561\pi$
$510$	0	0
$511$	7.07171e11	0.458808
$512$	0	0
$513$	−1.12608e12	−0.717860
$514$	0	0
$515$	−2.25068e12	−1.40988
$516$	0	0
$517$	−1.74072e12	−1.07157
$518$	0	0
$519$	5.83107e11	0.352773
$520$	0	0
$521$	−2.99302e12	−1.77967	−0.889835	−	0.456283i	$-0.849180\pi$
$521$	−2.99302e12	−1.77967	−0.889835	+	0.456283i	$0.849180\pi$
$522$	0	0
$523$	−1.24516e12	−0.727723	−0.363862	−	0.931453i	$-0.618542\pi$
$523$	−1.24516e12	−0.727723	−0.363862	+	0.931453i	$0.618542\pi$
$524$	0	0
$525$	1.16592e12	0.669812
$526$	0	0
$527$	5.99036e11	0.338302
$528$	0	0
$529$	1.80353e12	1.00132
$530$	0	0
$531$	9.90895e11	0.540882
$532$	0	0
$533$	−5.12180e11	−0.274884
$534$	0	0
$535$	−1.71641e12	−0.905793
$536$	0	0
$537$	1.60348e12	0.832107
$538$	0	0
$539$	−5.63817e10	−0.0287732
$540$	0	0
$541$	2.83767e12	1.42421	0.712104	−	0.702074i	$-0.247742\pi$
$541$	2.83767e12	1.42421	0.712104	+	0.702074i	$0.247742\pi$
$542$	0	0
$543$	1.01425e12	0.500663
$544$	0	0
$545$	−2.39621e12	−1.16343
$546$	0	0
$547$	−2.73600e12	−1.30669	−0.653345	−	0.757060i	$-0.726635\pi$
$547$	−2.73600e12	−1.30669	−0.653345	+	0.757060i	$0.726635\pi$
$548$	0	0
$549$	7.87853e11	0.370143
$550$	0	0
$551$	1.63005e11	0.0753389
$552$	0	0
$553$	1.74685e12	0.794313
$554$	0	0
$555$	−4.99669e12	−2.23544
$556$	0	0
$557$	−3.63597e12	−1.60056	−0.800279	−	0.599627i	$-0.795315\pi$
$557$	−3.63597e12	−1.60056	−0.800279	+	0.599627i	$0.795315\pi$
$558$	0	0
$559$	3.50777e11	0.151942
$560$	0	0
$561$	−6.43834e11	−0.274436
$562$	0	0
$563$	−1.81268e12	−0.760385	−0.380192	−	0.924907i	$-0.624142\pi$
$563$	−1.81268e12	−0.760385	−0.380192	+	0.924907i	$0.624142\pi$
$564$	0	0
$565$	3.87938e12	1.60156
$566$	0	0
$567$	2.91956e12	1.18630
$568$	0	0
$569$	−2.53400e12	−1.01345	−0.506723	−	0.862109i	$-0.669143\pi$
$569$	−2.53400e12	−1.01345	−0.506723	+	0.862109i	$0.669143\pi$
$570$	0	0
$571$	−1.44649e12	−0.569446	−0.284723	−	0.958610i	$-0.591902\pi$
$571$	−1.44649e12	−0.569446	−0.284723	+	0.958610i	$0.591902\pi$
$572$	0	0
$573$	−4.09062e12	−1.58523
$574$	0	0
$575$	−2.22276e12	−0.847982
$576$	0	0
$577$	−1.45725e12	−0.547321	−0.273661	−	0.961826i	$-0.588234\pi$
$577$	−1.45725e12	−0.547321	−0.273661	+	0.961826i	$0.588234\pi$
$578$	0	0
$579$	−4.87876e12	−1.80408
$580$	0	0
$581$	−4.68285e12	−1.70498
$582$	0	0
$583$	1.61223e11	0.0577987
$584$	0	0
$585$	2.90773e11	0.102648
$586$	0	0
$587$	1.98221e12	0.689095	0.344547	−	0.938769i	$-0.388032\pi$
$587$	1.98221e12	0.689095	0.344547	+	0.938769i	$0.388032\pi$
$588$	0	0
$589$	−3.09075e12	−1.05815
$590$	0	0
$591$	2.76438e12	0.932081
$592$	0	0
$593$	5.30729e12	1.76249	0.881246	−	0.472658i	$-0.156705\pi$
$593$	5.30729e12	1.76249	0.881246	+	0.472658i	$0.156705\pi$
$594$	0	0
$595$	−1.08446e12	−0.354722
$596$	0	0
$597$	−2.30438e12	−0.742454
$598$	0	0
$599$	2.54920e12	0.809064	0.404532	−	0.914524i	$-0.367434\pi$
$599$	2.54920e12	0.809064	0.404532	+	0.914524i	$0.367434\pi$
$600$	0	0
$601$	1.77817e12	0.555952	0.277976	−	0.960588i	$-0.410336\pi$
$601$	1.77817e12	0.555952	0.277976	+	0.960588i	$0.410336\pi$
$602$	0	0
$603$	−6.98408e11	−0.215120
$604$	0	0
$605$	−1.18602e12	−0.359910
$606$	0	0
$607$	6.45909e11	0.193118	0.0965588	−	0.995327i	$-0.469216\pi$
$607$	6.45909e11	0.193118	0.0965588	+	0.995327i	$0.469216\pi$
$608$	0	0
$609$	−3.20154e11	−0.0943150
$610$	0	0
$611$	−1.21047e12	−0.351374
$612$	0	0
$613$	2.63173e12	0.752782	0.376391	−	0.926461i	$-0.377165\pi$
$613$	2.63173e12	0.752782	0.376391	+	0.926461i	$0.377165\pi$
$614$	0	0
$615$	5.05569e12	1.42509
$616$	0	0
$617$	3.85338e12	1.07043	0.535216	−	0.844716i	$-0.320230\pi$
$617$	3.85338e12	1.07043	0.535216	+	0.844716i	$0.320230\pi$
$618$	0	0
$619$	6.18414e12	1.69306	0.846529	−	0.532343i	$-0.178688\pi$
$619$	6.18414e12	1.69306	0.846529	+	0.532343i	$0.178688\pi$
$620$	0	0
$621$	−4.21645e12	−1.13772
$622$	0	0
$623$	4.39810e12	1.16969
$624$	0	0
$625$	−4.73067e12	−1.24012
$626$	0	0
$627$	3.32189e12	0.858384
$628$	0	0
$629$	1.74178e12	0.443676
$630$	0	0
$631$	−1.63522e8	−4.10623e−5 0	−2.05311e−5	−	1.00000i	$-0.500007\pi$
$631$	−1.63522e8	−4.10623e−5 0	−2.05311e−5		1.00000i	$0.500007\pi$
$632$	0	0
$633$	8.11447e12	2.00883
$634$	0	0
$635$	−4.90922e12	−1.19820
$636$	0	0
$637$	−3.92072e10	−0.00943492
$638$	0	0
$639$	2.11292e12	0.501337
$640$	0	0
$641$	2.70983e11	0.0633988	0.0316994	−	0.999497i	$-0.489908\pi$
$641$	2.70983e11	0.0633988	0.0316994	+	0.999497i	$0.489908\pi$
$642$	0	0
$643$	−1.80868e12	−0.417265	−0.208632	−	0.977994i	$-0.566901\pi$
$643$	−1.80868e12	−0.417265	−0.208632	+	0.977994i	$0.566901\pi$
$644$	0	0
$645$	−3.46249e12	−0.787716
$646$	0	0
$647$	4.85979e12	1.09031	0.545153	−	0.838337i	$-0.316472\pi$
$647$	4.85979e12	1.09031	0.545153	+	0.838337i	$0.316472\pi$
$648$	0	0
$649$	7.06544e12	1.56328
$650$	0	0
$651$	6.07045e12	1.32467
$652$	0	0
$653$	−5.49983e10	−0.0118369	−0.00591847	−	0.999982i	$-0.501884\pi$
$653$	−5.49983e10	−0.0118369	−0.00591847	+	0.999982i	$0.501884\pi$
$654$	0	0
$655$	1.04610e12	0.222068
$656$	0	0
$657$	−6.52429e11	−0.136612
$658$	0	0
$659$	−5.88872e12	−1.21629	−0.608144	−	0.793827i	$-0.708086\pi$
$659$	−5.88872e12	−1.21629	−0.608144	+	0.793827i	$0.708086\pi$
$660$	0	0
$661$	5.52398e12	1.12550	0.562750	−	0.826627i	$-0.309743\pi$
$661$	5.52398e12	1.12550	0.562750	+	0.826627i	$0.309743\pi$
$662$	0	0
$663$	−4.47715e11	−0.0899893
$664$	0	0
$665$	5.59533e12	1.10950
$666$	0	0
$667$	6.10352e11	0.119403
$668$	0	0
$669$	3.63064e12	0.700754
$670$	0	0
$671$	5.61768e12	1.06981
$672$	0	0
$673$	6.15531e12	1.15660	0.578298	−	0.815825i	$-0.303717\pi$
$673$	6.15531e12	1.15660	0.578298	+	0.815825i	$0.303717\pi$
$674$	0	0
$675$	2.59999e12	0.482064
$676$	0	0
$677$	5.73319e12	1.04893	0.524466	−	0.851432i	$-0.324265\pi$
$677$	5.73319e12	1.04893	0.524466	+	0.851432i	$0.324265\pi$
$678$	0	0
$679$	7.60147e12	1.37241
$680$	0	0
$681$	3.58765e12	0.639216
$682$	0	0
$683$	3.72368e12	0.654756	0.327378	−	0.944894i	$-0.393835\pi$
$683$	3.72368e12	0.654756	0.327378	+	0.944894i	$0.393835\pi$
$684$	0	0
$685$	−8.45715e12	−1.46763
$686$	0	0
$687$	−1.13276e13	−1.94013
$688$	0	0
$689$	1.12112e11	0.0189525
$690$	0	0
$691$	1.21804e11	0.0203240	0.0101620	−	0.999948i	$-0.496765\pi$
$691$	1.21804e11	0.0203240	0.0101620	+	0.999948i	$0.496765\pi$
$692$	0	0
$693$	−1.47709e12	−0.243280
$694$	0	0
$695$	1.75303e12	0.285009
$696$	0	0
$697$	−1.76235e12	−0.282842
$698$	0	0
$699$	−8.15269e12	−1.29168
$700$	0	0
$701$	7.48411e12	1.17060	0.585300	−	0.810816i	$-0.300977\pi$
$701$	7.48411e12	1.17060	0.585300	+	0.810816i	$0.300977\pi$
$702$	0	0
$703$	−8.98681e12	−1.38774
$704$	0	0
$705$	1.19485e13	1.82164
$706$	0	0
$707$	5.94764e12	0.895277
$708$	0	0
$709$	2.03611e12	0.302616	0.151308	−	0.988487i	$-0.451651\pi$
$709$	2.03611e12	0.302616	0.151308	+	0.988487i	$0.451651\pi$
$710$	0	0
$711$	−1.61162e12	−0.236511
$712$	0	0
$713$	−1.15729e13	−1.67703
$714$	0	0
$715$	2.07331e12	0.296680
$716$	0	0
$717$	−3.42800e12	−0.484400
$718$	0	0
$719$	−1.45945e12	−0.203661	−0.101831	−	0.994802i	$-0.532470\pi$
$719$	−1.45945e12	−0.203661	−0.101831	+	0.994802i	$0.532470\pi$
$720$	0	0
$721$	7.95050e12	1.09569
$722$	0	0
$723$	−8.11385e12	−1.10435
$724$	0	0
$725$	−3.76362e11	−0.0505923
$726$	0	0
$727$	8.61854e10	0.0114427	0.00572135	−	0.999984i	$-0.498179\pi$
$727$	8.61854e10	0.0114427	0.00572135	+	0.999984i	$0.498179\pi$
$728$	0	0
$729$	4.27897e12	0.561133
$730$	0	0
$731$	1.20698e12	0.156341
$732$	0	0
$733$	−6.94903e12	−0.889111	−0.444556	−	0.895751i	$-0.646638\pi$
$733$	−6.94903e12	−0.889111	−0.444556	+	0.895751i	$0.646638\pi$
$734$	0	0
$735$	3.87011e11	0.0489137
$736$	0	0
$737$	−4.97990e12	−0.621752
$738$	0	0
$739$	−2.94805e12	−0.363609	−0.181804	−	0.983335i	$-0.558194\pi$
$739$	−2.94805e12	−0.363609	−0.181804	+	0.983335i	$0.558194\pi$
$740$	0	0
$741$	2.31001e12	0.281470
$742$	0	0
$743$	1.30043e13	1.56544	0.782719	−	0.622375i	$-0.213832\pi$
$743$	1.30043e13	1.56544	0.782719	+	0.622375i	$0.213832\pi$
$744$	0	0
$745$	−1.30862e13	−1.55636
$746$	0	0
$747$	4.32035e12	0.507664
$748$	0	0
$749$	6.06319e12	0.703936
$750$	0	0
$751$	1.57055e12	0.180166	0.0900829	−	0.995934i	$-0.471287\pi$
$751$	1.57055e12	0.180166	0.0900829	+	0.995934i	$0.471287\pi$
$752$	0	0
$753$	6.75246e12	0.765394
$754$	0	0
$755$	−1.43408e13	−1.60625
$756$	0	0
$757$	−9.58378e12	−1.06073	−0.530366	−	0.847769i	$-0.677945\pi$
$757$	−9.58378e12	−1.06073	−0.530366	+	0.847769i	$0.677945\pi$
$758$	0	0
$759$	1.24384e13	1.36043
$760$	0	0
$761$	−7.11276e12	−0.768789	−0.384395	−	0.923169i	$-0.625590\pi$
$761$	−7.11276e12	−0.768789	−0.384395	+	0.923169i	$0.625590\pi$
$762$	0	0
$763$	8.46456e12	0.904157
$764$	0	0
$765$	1.00051e12	0.105620
$766$	0	0
$767$	4.91322e12	0.512611
$768$	0	0
$769$	−1.73274e13	−1.78676	−0.893379	−	0.449304i	$-0.851672\pi$
$769$	−1.73274e13	−1.78676	−0.893379	+	0.449304i	$0.851672\pi$
$770$	0	0
$771$	−1.10454e13	−1.12574
$772$	0	0
$773$	−9.87303e12	−0.994587	−0.497294	−	0.867582i	$-0.665673\pi$
$773$	−9.87303e12	−0.994587	−0.497294	+	0.867582i	$0.665673\pi$
$774$	0	0
$775$	7.13623e12	0.710577
$776$	0	0
$777$	1.76507e13	1.73727
$778$	0	0
$779$	9.09293e12	0.884678
$780$	0	0
$781$	1.50659e13	1.44899
$782$	0	0
$783$	−7.13938e11	−0.0678786
$784$	0	0
$785$	−1.96019e13	−1.84240
$786$	0	0
$787$	6.54652e12	0.608309	0.304155	−	0.952623i	$-0.401626\pi$
$787$	6.54652e12	0.608309	0.304155	+	0.952623i	$0.401626\pi$
$788$	0	0
$789$	−6.65964e12	−0.611793
$790$	0	0
$791$	−1.37038e13	−1.24465
$792$	0	0
$793$	3.90647e12	0.350796
$794$	0	0
$795$	−1.10665e12	−0.0982562
$796$	0	0
$797$	4.91818e12	0.431760	0.215880	−	0.976420i	$-0.430738\pi$
$797$	4.91818e12	0.431760	0.215880	+	0.976420i	$0.430738\pi$
$798$	0	0
$799$	−4.16509e12	−0.361547
$800$	0	0
$801$	−4.05764e12	−0.348279
$802$	0	0
$803$	−4.65205e12	−0.394843
$804$	0	0
$805$	2.09510e13	1.75842
$806$	0	0
$807$	1.07832e13	0.894986
$808$	0	0
$809$	8.28568e12	0.680080	0.340040	−	0.940411i	$-0.389559\pi$
$809$	8.28568e12	0.680080	0.340040	+	0.940411i	$0.389559\pi$
$810$	0	0
$811$	2.06397e13	1.67537	0.837684	−	0.546155i	$-0.183909\pi$
$811$	2.06397e13	1.67537	0.837684	+	0.546155i	$0.183909\pi$
$812$	0	0
$813$	1.16390e13	0.934346
$814$	0	0
$815$	−1.38565e12	−0.110013
$816$	0	0
$817$	−6.22747e12	−0.489004
$818$	0	0
$819$	−1.02715e12	−0.0797730
$820$	0	0
$821$	−1.24767e9	−9.58422e−5 0	−4.79211e−5	−	1.00000i	$-0.500015\pi$
$821$	−1.24767e9	−9.58422e−5 0	−4.79211e−5		1.00000i	$0.500015\pi$
$822$	0	0
$823$	−1.23070e13	−0.935090	−0.467545	−	0.883969i	$-0.654861\pi$
$823$	−1.23070e13	−0.935090	−0.467545	+	0.883969i	$0.654861\pi$
$824$	0	0
$825$	−7.66990e12	−0.576431
$826$	0	0
$827$	−5.07786e12	−0.377490	−0.188745	−	0.982026i	$-0.560442\pi$
$827$	−5.07786e12	−0.377490	−0.188745	+	0.982026i	$0.560442\pi$
$828$	0	0
$829$	−1.95629e13	−1.43859	−0.719295	−	0.694705i	$-0.755535\pi$
$829$	−1.95629e13	−1.43859	−0.719295	+	0.694705i	$0.755535\pi$
$830$	0	0
$831$	1.80922e12	0.131610
$832$	0	0
$833$	−1.34907e11	−0.00970806
$834$	0	0
$835$	−2.23359e13	−1.59006
$836$	0	0
$837$	1.35370e13	0.953364
$838$	0	0
$839$	1.50732e13	1.05021	0.525107	−	0.851036i	$-0.324025\pi$
$839$	1.50732e13	1.05021	0.525107	+	0.851036i	$0.324025\pi$
$840$	0	0
$841$	−1.44038e13	−0.992876
$842$	0	0
$843$	−1.43597e13	−0.979315
$844$	0	0
$845$	1.44176e12	0.0972831
$846$	0	0
$847$	4.18961e12	0.279704
$848$	0	0
$849$	6.91747e12	0.456944
$850$	0	0
$851$	−3.36499e13	−2.19939
$852$	0	0
$853$	5.33191e12	0.344835	0.172418	−	0.985024i	$-0.444842\pi$
$853$	5.33191e12	0.344835	0.172418	+	0.985024i	$0.444842\pi$
$854$	0	0
$855$	−5.16220e12	−0.330360
$856$	0	0
$857$	−7.95292e12	−0.503631	−0.251816	−	0.967775i	$-0.581028\pi$
$857$	−7.95292e12	−0.503631	−0.251816	+	0.967775i	$0.581028\pi$
$858$	0	0
$859$	−4.62716e12	−0.289965	−0.144982	−	0.989434i	$-0.546313\pi$
$859$	−4.62716e12	−0.289965	−0.144982	+	0.989434i	$0.546313\pi$
$860$	0	0
$861$	−1.78591e13	−1.10751
$862$	0	0
$863$	−6.60844e12	−0.405556	−0.202778	−	0.979225i	$-0.564997\pi$
$863$	−6.60844e12	−0.405556	−0.202778	+	0.979225i	$0.564997\pi$
$864$	0	0
$865$	−6.46113e12	−0.392406
$866$	0	0
$867$	1.73753e13	1.04435
$868$	0	0
$869$	−1.14915e13	−0.683575
$870$	0	0
$871$	−3.46297e12	−0.203876
$872$	0	0
$873$	−7.01304e12	−0.408641
$874$	0	0
$875$	8.63366e12	0.497919
$876$	0	0
$877$	3.39068e12	0.193548	0.0967739	−	0.995306i	$-0.469148\pi$
$877$	3.39068e12	0.193548	0.0967739	+	0.995306i	$0.469148\pi$
$878$	0	0
$879$	6.48805e12	0.366576
$880$	0	0
$881$	1.20785e13	0.675495	0.337748	−	0.941237i	$-0.390335\pi$
$881$	1.20785e13	0.675495	0.337748	+	0.941237i	$0.390335\pi$
$882$	0	0
$883$	−5.33631e12	−0.295405	−0.147702	−	0.989032i	$-0.547188\pi$
$883$	−5.33631e12	−0.295405	−0.147702	+	0.989032i	$0.547188\pi$
$884$	0	0
$885$	−4.84981e13	−2.65754
$886$	0	0
$887$	−1.35083e13	−0.732731	−0.366366	−	0.930471i	$-0.619398\pi$
$887$	−1.35083e13	−0.732731	−0.366366	+	0.930471i	$0.619398\pi$
$888$	0	0
$889$	1.73417e13	0.931182
$890$	0	0
$891$	−1.92060e13	−1.02091
$892$	0	0
$893$	2.14900e13	1.13085
$894$	0	0
$895$	−1.77674e13	−0.925593
$896$	0	0
$897$	8.64952e12	0.446094
$898$	0	0
$899$	−1.95955e12	−0.100055
$900$	0	0
$901$	3.85765e11	0.0195012
$902$	0	0
$903$	1.22312e13	0.612172
$904$	0	0
$905$	−1.12384e13	−0.556912
$906$	0	0
$907$	−2.95151e13	−1.44814	−0.724071	−	0.689726i	$-0.757731\pi$
$907$	−2.95151e13	−1.44814	−0.724071	+	0.689726i	$0.757731\pi$
$908$	0	0
$909$	−5.48723e12	−0.266573
$910$	0	0
$911$	2.98046e12	0.143368	0.0716838	−	0.997427i	$-0.477163\pi$
$911$	2.98046e12	0.143368	0.0716838	+	0.997427i	$0.477163\pi$
$912$	0	0
$913$	3.08057e13	1.46728
$914$	0	0
$915$	−3.85605e13	−1.81864
$916$	0	0
$917$	−3.69532e12	−0.172580
$918$	0	0
$919$	3.51732e13	1.62664	0.813321	−	0.581815i	$-0.197657\pi$
$919$	3.51732e13	1.62664	0.813321	+	0.581815i	$0.197657\pi$
$920$	0	0
$921$	−3.74905e13	−1.71693
$922$	0	0
$923$	1.04766e13	0.475132
$924$	0	0
$925$	2.07496e13	0.931906
$926$	0	0
$927$	−7.33505e12	−0.326245
$928$	0	0
$929$	−2.56701e13	−1.13072	−0.565361	−	0.824843i	$-0.691263\pi$
$929$	−2.56701e13	−1.13072	−0.565361	+	0.824843i	$0.691263\pi$
$930$	0	0
$931$	6.96060e11	0.0303650
$932$	0	0
$933$	2.55454e13	1.10368
$934$	0	0
$935$	7.13402e12	0.305268
$936$	0	0
$937$	3.44586e11	0.0146039	0.00730197	−	0.999973i	$-0.497676\pi$
$937$	3.44586e11	0.0146039	0.00730197	+	0.999973i	$0.497676\pi$
$938$	0	0
$939$	4.50390e13	1.89057
$940$	0	0
$941$	1.04276e13	0.433542	0.216771	−	0.976223i	$-0.430448\pi$
$941$	1.04276e13	0.433542	0.216771	+	0.976223i	$0.430448\pi$
$942$	0	0
$943$	3.40473e13	1.40210
$944$	0	0
$945$	−2.45067e13	−0.999636
$946$	0	0
$947$	−3.05879e13	−1.23588	−0.617939	−	0.786226i	$-0.712032\pi$
$947$	−3.05879e13	−1.23588	−0.617939	+	0.786226i	$0.712032\pi$
$948$	0	0
$949$	−3.23499e12	−0.129472
$950$	0	0
$951$	2.04171e13	0.809433
$952$	0	0
$953$	−3.43552e13	−1.34919	−0.674596	−	0.738187i	$-0.735682\pi$
$953$	−3.43552e13	−1.34919	−0.674596	+	0.738187i	$0.735682\pi$
$954$	0	0
$955$	4.53262e13	1.76333
$956$	0	0
$957$	2.10610e12	0.0811661
$958$	0	0
$959$	2.98747e13	1.14057
$960$	0	0
$961$	1.07156e13	0.405286
$962$	0	0
$963$	−5.59384e12	−0.209600
$964$	0	0
$965$	5.40592e13	2.00677
$966$	0	0
$967$	−5.29016e13	−1.94558	−0.972791	−	0.231685i	$-0.925576\pi$
$967$	−5.29016e13	−1.94558	−0.972791	+	0.231685i	$0.925576\pi$
$968$	0	0
$969$	7.94845e12	0.289618
$970$	0	0
$971$	−3.46593e13	−1.25122	−0.625610	−	0.780136i	$-0.715150\pi$
$971$	−3.46593e13	−1.25122	−0.625610	+	0.780136i	$0.715150\pi$
$972$	0	0
$973$	−6.19255e12	−0.221494
$974$	0	0
$975$	−5.33356e12	−0.189015
$976$	0	0
$977$	1.48680e13	0.522069	0.261035	−	0.965329i	$-0.415936\pi$
$977$	1.48680e13	0.522069	0.261035	+	0.965329i	$0.415936\pi$
$978$	0	0
$979$	−2.89325e13	−1.00661
$980$	0	0
$981$	−7.80932e12	−0.269217
$982$	0	0
$983$	−1.71439e13	−0.585624	−0.292812	−	0.956170i	$-0.594591\pi$
$983$	−1.71439e13	−0.585624	−0.292812	+	0.956170i	$0.594591\pi$
$984$	0	0
$985$	−3.06307e13	−1.03680
$986$	0	0
$987$	−4.22079e13	−1.41568
$988$	0	0
$989$	−2.33180e13	−0.775010
$990$	0	0
$991$	−3.48129e13	−1.14659	−0.573296	−	0.819348i	$-0.694336\pi$
$991$	−3.48129e13	−1.14659	−0.573296	+	0.819348i	$0.694336\pi$
$992$	0	0
$993$	1.34242e13	0.438144
$994$	0	0
$995$	2.55337e13	0.825868
$996$	0	0
$997$	−3.63962e13	−1.16662	−0.583308	−	0.812251i	$-0.698242\pi$
$997$	−3.63962e13	−1.16662	−0.583308	+	0.812251i	$0.698242\pi$
$998$	0	0
$999$	3.93609e13	1.25032

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.e.1.1		3
4.3	odd	2		26.10.a.d.1.3	✓	3
12.11	even	2		234.10.a.l.1.1		3
52.51	odd	2		338.10.a.f.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.10.a.d.1.3	✓	3	4.3	odd	2
208.10.a.e.1.1		3	1.1	even	1	trivial
234.10.a.l.1.1		3	12.11	even	2
338.10.a.f.1.3		3	52.51	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

\(f(q)\)	\(=\)	\(q-159.509 q^{3} +1767.44 q^{5} -6243.46 q^{7} +5760.16 q^{9} +41072.0 q^{11} +28561.0 q^{13} -281924. q^{15} +98274.9 q^{17} -507054. q^{19} +995890. q^{21} -1.89860e6 q^{23} +1.17074e6 q^{25} +2.22082e6 q^{27} -321475. q^{29} +6.09551e6 q^{31} -6.55136e6 q^{33} -1.10350e7 q^{35} +1.77236e7 q^{37} -4.55574e6 q^{39} -1.79328e7 q^{41} +1.22817e7 q^{43} +1.01808e7 q^{45} -4.23821e7 q^{47} -1.37275e6 q^{49} -1.56757e7 q^{51} +3.92537e6 q^{53} +7.25925e7 q^{55} +8.08798e7 q^{57} +1.72026e8 q^{59} +1.36776e8 q^{61} -3.59633e7 q^{63} +5.04800e7 q^{65} -1.21248e8 q^{67} +3.02844e8 q^{69} +3.66817e8 q^{71} -1.13266e8 q^{73} -1.86743e8 q^{75} -2.56432e8 q^{77} -2.79788e8 q^{79} -4.67618e8 q^{81} +7.50041e8 q^{83} +1.73695e8 q^{85} +5.12782e7 q^{87} -7.04433e8 q^{89} -1.78320e8 q^{91} -9.72289e8 q^{93} -8.96190e8 q^{95} -1.21751e9 q^{97} +2.36581e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 248 q^{5} + 2956 q^{7} - 20863 q^{9} + 31324 q^{11} + 85683 q^{13} - 392140 q^{15} + 905228 q^{17} - 1726316 q^{19} + 1771246 q^{21} - 2135256 q^{23} + 4074295 q^{25} - 3038724 q^{27} + 372426 q^{29}+ \cdots + 304026580 q^{99}+O(q^{100}) \) 3 * q + 248 * q^5 + 2956 * q^7 - 20863 * q^9 + 31324 * q^11 + 85683 * q^13 - 392140 * q^15 + 905228 * q^17 - 1726316 * q^19 + 1771246 * q^21 - 2135256 * q^23 + 4074295 * q^25 - 3038724 * q^27 + 372426 * q^29 + 1444776 * q^31 - 7719476 * q^33 + 3459436 * q^35 - 2809672 * q^37 - 7424806 * q^41 - 36070040 * q^43 + 32156162 * q^45 - 7032884 * q^47 + 41844597 * q^49 + 47158124 * q^51 - 15011674 * q^53 - 121479136 * q^55 - 14804300 * q^57 + 40286396 * q^59 + 230710478 * q^61 - 151778040 * q^63 + 7083128 * q^65 - 201453084 * q^67 + 268135008 * q^69 + 99165572 * q^71 - 350942946 * q^73 + 28166788 * q^75 - 1066624588 * q^77 - 138794592 * q^79 - 363498349 * q^81 + 1364469516 * q^83 - 1225485962 * q^85 + 93372576 * q^87 - 709437058 * q^89 + 84426316 * q^91 - 1382471232 * q^93 + 833387824 * q^95 - 362189698 * q^97 + 304026580 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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