LMFDB - Embedded newform 208.10.a.l.1.4

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 112810 x^{5} + 1645934 x^{4} + 3493976849 x^{3} - 83049726457 x^{2} + \cdots + 864293655586764 $ x^7 - x^6 - 112810x^5 + 1645934x^4 + 3493976849x^3 - 83049726457x^2 - 21841140807080*x + 864293655586764
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19}\cdot 3 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$46.9663$ 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)

$f(q)$	$=$	$q+44.9663 q^{3} +193.817 q^{5} +763.904 q^{7} -17661.0 q^{9} -7561.06 q^{11} -28561.0 q^{13} +8715.25 q^{15} +535351. q^{17} +716609. q^{19} +34349.9 q^{21} -627632. q^{23} -1.91556e6 q^{25} -1.67922e6 q^{27} -1.84961e6 q^{29} +296342. q^{31} -339993. q^{33} +148058. q^{35} -6.20567e6 q^{37} -1.28428e6 q^{39} +3.01484e7 q^{41} -1.43641e6 q^{43} -3.42302e6 q^{45} -5.48782e7 q^{47} -3.97701e7 q^{49} +2.40728e7 q^{51} +2.37822e7 q^{53} -1.46547e6 q^{55} +3.22232e7 q^{57} -7.12523e7 q^{59} +1.75021e8 q^{61} -1.34913e7 q^{63} -5.53562e6 q^{65} -2.91001e8 q^{67} -2.82223e7 q^{69} +1.75425e8 q^{71} -8.41153e7 q^{73} -8.61356e7 q^{75} -5.77592e6 q^{77} -1.29299e8 q^{79} +2.72114e8 q^{81} -8.12518e8 q^{83} +1.03760e8 q^{85} -8.31700e7 q^{87} -2.87146e8 q^{89} -2.18179e7 q^{91} +1.33254e7 q^{93} +1.38891e8 q^{95} +1.18180e9 q^{97} +1.33536e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 13 q^{3} - 801 q^{5} + 1091 q^{7} + 87864 q^{9} - 32218 q^{11} - 199927 q^{13} - 58323 q^{15} - 870531 q^{17} + 128950 q^{19} - 719663 q^{21} - 2198844 q^{23} + 992010 q^{25} - 5441383 q^{27} + 6327710 q^{29}+ \cdots - 5204667600 q^{99}+O(q^{100}) $ 7 * q - 13 * q^3 - 801 * q^5 + 1091 * q^7 + 87864 * q^9 - 32218 * q^11 - 199927 * q^13 - 58323 * q^15 - 870531 * q^17 + 128950 * q^19 - 719663 * q^21 - 2198844 * q^23 + 992010 * q^25 - 5441383 * q^27 + 6327710 * q^29 + 1660376 * q^31 + 14199598 * q^33 + 19231769 * q^35 + 5058787 * q^37 + 371293 * q^39 + 14779748 * q^41 - 9405075 * q^43 + 56644022 * q^45 - 133415721 * q^47 + 44544988 * q^49 - 132066169 * q^51 + 43962424 * q^53 - 191344746 * q^55 + 138963078 * q^57 - 22370274 * q^59 + 23851748 * q^61 - 780720706 * q^63 + 22877361 * q^65 - 12058786 * q^67 + 1036453420 * q^69 - 864303791 * q^71 + 670491202 * q^73 - 693324044 * q^75 + 841052238 * q^77 - 1019547572 * q^79 + 900194079 * q^81 - 1345041700 * q^83 + 824066375 * q^85 - 2589172074 * q^87 + 1409158546 * q^89 - 31160051 * q^91 - 445200892 * q^93 - 3285213058 * q^95 + 320206906 * q^97 - 5204667600 * 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
$3$	44.9663	0.320510	0.160255	−	0.987076i	$-0.448768\pi$
$3$	44.9663	0.320510	0.160255	+	0.987076i	$0.448768\pi$
$4$	0	0
$5$	193.817	0.138684	0.0693422	−	0.997593i	$-0.477910\pi$
$5$	193.817	0.138684	0.0693422	+	0.997593i	$0.477910\pi$
$6$	0	0
$7$	763.904	0.120253	0.0601267	−	0.998191i	$-0.480850\pi$
$7$	763.904	0.120253	0.0601267	+	0.998191i	$0.480850\pi$
$8$	0	0
$9$	−17661.0	−0.897273
$10$	0	0
$11$	−7561.06	−0.155710	−0.0778548	−	0.996965i	$-0.524807\pi$
$11$	−7561.06	−0.155710	−0.0778548	+	0.996965i	$0.524807\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	0	0
$15$	8715.25	0.0444497
$16$	0	0
$17$	535351.	1.55460	0.777300	−	0.629130i	$-0.216589\pi$
$17$	535351.	1.55460	0.777300	+	0.629130i	$0.216589\pi$
$18$	0	0
$19$	716609.	1.26151	0.630755	−	0.775982i	$-0.282745\pi$
$19$	716609.	1.26151	0.630755	+	0.775982i	$0.282745\pi$
$20$	0	0
$21$	34349.9	0.0385424
$22$	0	0
$23$	−627632.	−0.467659	−0.233830	−	0.972278i	$-0.575126\pi$
$23$	−627632.	−0.467659	−0.233830	+	0.972278i	$0.575126\pi$
$24$	0	0
$25$	−1.91556e6	−0.980767
$26$	0	0
$27$	−1.67922e6	−0.608095
$28$	0	0
$29$	−1.84961e6	−0.485611	−0.242806	−	0.970075i	$-0.578068\pi$
$29$	−1.84961e6	−0.485611	−0.242806	+	0.970075i	$0.578068\pi$
$30$	0	0
$31$	296342.	0.0576323	0.0288162	−	0.999585i	$-0.490826\pi$
$31$	296342.	0.0576323	0.0288162	+	0.999585i	$0.490826\pi$
$32$	0	0
$33$	−339993.	−0.0499065
$34$	0	0
$35$	148058.	0.0166773
$36$	0	0
$37$	−6.20567e6	−0.544353	−0.272177	−	0.962247i	$-0.587744\pi$
$37$	−6.20567e6	−0.544353	−0.272177	+	0.962247i	$0.587744\pi$
$38$	0	0
$39$	−1.28428e6	−0.0888934
$40$	0	0
$41$	3.01484e7	1.66624	0.833119	−	0.553094i	$-0.186553\pi$
$41$	3.01484e7	1.66624	0.833119	+	0.553094i	$0.186553\pi$
$42$	0	0
$43$	−1.43641e6	−0.0640721	−0.0320361	−	0.999487i	$-0.510199\pi$
$43$	−1.43641e6	−0.0640721	−0.0320361	+	0.999487i	$0.510199\pi$
$44$	0	0
$45$	−3.42302e6	−0.124438
$46$	0	0
$47$	−5.48782e7	−1.64044	−0.820218	−	0.572051i	$-0.806148\pi$
$47$	−5.48782e7	−1.64044	−0.820218	+	0.572051i	$0.806148\pi$
$48$	0	0
$49$	−3.97701e7	−0.985539
$50$	0	0
$51$	2.40728e7	0.498265
$52$	0	0
$53$	2.37822e7	0.414010	0.207005	−	0.978340i	$-0.433628\pi$
$53$	2.37822e7	0.414010	0.207005	+	0.978340i	$0.433628\pi$
$54$	0	0
$55$	−1.46547e6	−0.0215945
$56$	0	0
$57$	3.22232e7	0.404327
$58$	0	0
$59$	−7.12523e7	−0.765535	−0.382767	−	0.923845i	$-0.625029\pi$
$59$	−7.12523e7	−0.765535	−0.382767	+	0.923845i	$0.625029\pi$
$60$	0	0
$61$	1.75021e8	1.61847	0.809236	−	0.587484i	$-0.199882\pi$
$61$	1.75021e8	1.61847	0.809236	+	0.587484i	$0.199882\pi$
$62$	0	0
$63$	−1.34913e7	−0.107900
$64$	0	0
$65$	−5.53562e6	−0.0384641
$66$	0	0
$67$	−2.91001e8	−1.76424	−0.882120	−	0.471024i	$-0.843885\pi$
$67$	−2.91001e8	−1.76424	−0.882120	+	0.471024i	$0.843885\pi$
$68$	0	0
$69$	−2.82223e7	−0.149889
$70$	0	0
$71$	1.75425e8	0.819273	0.409636	−	0.912249i	$-0.365656\pi$
$71$	1.75425e8	0.819273	0.409636	+	0.912249i	$0.365656\pi$
$72$	0	0
$73$	−8.41153e7	−0.346675	−0.173337	−	0.984863i	$-0.555455\pi$
$73$	−8.41153e7	−0.346675	−0.173337	+	0.984863i	$0.555455\pi$
$74$	0	0
$75$	−8.61356e7	−0.314345
$76$	0	0
$77$	−5.77592e6	−0.0187246
$78$	0	0
$79$	−1.29299e8	−0.373486	−0.186743	−	0.982409i	$-0.559793\pi$
$79$	−1.29299e8	−0.373486	−0.186743	+	0.982409i	$0.559793\pi$
$80$	0	0
$81$	2.72114e8	0.702373
$82$	0	0
$83$	−8.12518e8	−1.87924	−0.939618	−	0.342225i	$-0.888820\pi$
$83$	−8.12518e8	−1.87924	−0.939618	+	0.342225i	$0.888820\pi$
$84$	0	0
$85$	1.03760e8	0.215599
$86$	0	0
$87$	−8.31700e7	−0.155643
$88$	0	0
$89$	−2.87146e8	−0.485118	−0.242559	−	0.970137i	$-0.577987\pi$
$89$	−2.87146e8	−0.485118	−0.242559	+	0.970137i	$0.577987\pi$
$90$	0	0
$91$	−2.18179e7	−0.0333523
$92$	0	0
$93$	1.33254e7	0.0184717
$94$	0	0
$95$	1.38891e8	0.174952
$96$	0	0
$97$	1.18180e9	1.35541	0.677706	−	0.735333i	$-0.262974\pi$
$97$	1.18180e9	1.35541	0.677706	+	0.735333i	$0.262974\pi$
$98$	0	0
$99$	1.33536e8	0.139714
$100$	0	0
$101$	1.18751e9	1.13551	0.567755	−	0.823198i	$-0.307812\pi$
$101$	1.18751e9	1.13551	0.567755	+	0.823198i	$0.307812\pi$
$102$	0	0
$103$	−1.27767e9	−1.11854	−0.559271	−	0.828985i	$-0.688919\pi$
$103$	−1.27767e9	−1.11854	−0.559271	+	0.828985i	$0.688919\pi$
$104$	0	0
$105$	6.65761e6	0.00534523
$106$	0	0
$107$	−5.95864e8	−0.439461	−0.219730	−	0.975561i	$-0.570518\pi$
$107$	−5.95864e8	−0.439461	−0.219730	+	0.975561i	$0.570518\pi$
$108$	0	0
$109$	−2.66555e9	−1.80871	−0.904353	−	0.426784i	$-0.859646\pi$
$109$	−2.66555e9	−1.80871	−0.904353	+	0.426784i	$0.859646\pi$
$110$	0	0
$111$	−2.79046e8	−0.174471
$112$	0	0
$113$	8.61591e8	0.497105	0.248552	−	0.968618i	$-0.420045\pi$
$113$	8.61591e8	0.497105	0.248552	+	0.968618i	$0.420045\pi$
$114$	0	0
$115$	−1.21646e8	−0.0648571
$116$	0	0
$117$	5.04417e8	0.248859
$118$	0	0
$119$	4.08957e8	0.186946
$120$	0	0
$121$	−2.30078e9	−0.975754
$122$	0	0
$123$	1.35566e9	0.534045
$124$	0	0
$125$	−7.49819e8	−0.274702
$126$	0	0
$127$	−4.76529e9	−1.62545	−0.812723	−	0.582650i	$-0.802016\pi$
$127$	−4.76529e9	−1.62545	−0.812723	+	0.582650i	$0.802016\pi$
$128$	0	0
$129$	−6.45898e7	−0.0205357
$130$	0	0
$131$	−3.80511e9	−1.12888	−0.564438	−	0.825476i	$-0.690907\pi$
$131$	−3.80511e9	−1.12888	−0.564438	+	0.825476i	$0.690907\pi$
$132$	0	0
$133$	5.47420e8	0.151701
$134$	0	0
$135$	−3.25463e8	−0.0843333
$136$	0	0
$137$	−5.02656e9	−1.21907	−0.609535	−	0.792759i	$-0.708644\pi$
$137$	−5.02656e9	−1.21907	−0.609535	+	0.792759i	$0.708644\pi$
$138$	0	0
$139$	6.38922e9	1.45171	0.725857	−	0.687845i	$-0.241443\pi$
$139$	6.38922e9	1.45171	0.725857	+	0.687845i	$0.241443\pi$
$140$	0	0
$141$	−2.46767e9	−0.525776
$142$	0	0
$143$	2.15951e8	0.0431861
$144$	0	0
$145$	−3.58486e8	−0.0673467
$146$	0	0
$147$	−1.78831e9	−0.315875
$148$	0	0
$149$	−4.24336e9	−0.705296	−0.352648	−	0.935756i	$-0.614719\pi$
$149$	−4.24336e9	−0.705296	−0.352648	+	0.935756i	$0.614719\pi$
$150$	0	0
$151$	3.01475e9	0.471905	0.235953	−	0.971765i	$-0.424179\pi$
$151$	3.01475e9	0.471905	0.235953	+	0.971765i	$0.424179\pi$
$152$	0	0
$153$	−9.45486e9	−1.39490
$154$	0	0
$155$	5.74363e7	0.00799271
$156$	0	0
$157$	−7.29377e9	−0.958084	−0.479042	−	0.877792i	$-0.659016\pi$
$157$	−7.29377e9	−0.958084	−0.479042	+	0.877792i	$0.659016\pi$
$158$	0	0
$159$	1.06940e9	0.132694
$160$	0	0
$161$	−4.79450e8	−0.0562376
$162$	0	0
$163$	−2.54785e9	−0.282702	−0.141351	−	0.989960i	$-0.545145\pi$
$163$	−2.54785e9	−0.282702	−0.141351	+	0.989960i	$0.545145\pi$
$164$	0	0
$165$	−6.58965e7	−0.00692125
$166$	0	0
$167$	2.62273e9	0.260933	0.130467	−	0.991453i	$-0.458352\pi$
$167$	2.62273e9	0.260933	0.130467	+	0.991453i	$0.458352\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	−1.26561e10	−1.13192
$172$	0	0
$173$	−1.42300e10	−1.20781	−0.603903	−	0.797058i	$-0.706389\pi$
$173$	−1.42300e10	−1.20781	−0.603903	+	0.797058i	$0.706389\pi$
$174$	0	0
$175$	−1.46330e9	−0.117941
$176$	0	0
$177$	−3.20395e9	−0.245361
$178$	0	0
$179$	7.10504e9	0.517283	0.258641	−	0.965973i	$-0.416725\pi$
$179$	7.10504e9	0.517283	0.258641	+	0.965973i	$0.416725\pi$
$180$	0	0
$181$	−1.67076e10	−1.15708	−0.578538	−	0.815656i	$-0.696376\pi$
$181$	−1.67076e10	−1.15708	−0.578538	+	0.815656i	$0.696376\pi$
$182$	0	0
$183$	7.87003e9	0.518736
$184$	0	0
$185$	−1.20277e9	−0.0754933
$186$	0	0
$187$	−4.04782e9	−0.242066
$188$	0	0
$189$	−1.28276e9	−0.0731255
$190$	0	0
$191$	−2.43295e10	−1.32276	−0.661382	−	0.750049i	$-0.730030\pi$
$191$	−2.43295e10	−1.32276	−0.661382	+	0.750049i	$0.730030\pi$
$192$	0	0
$193$	1.51656e9	0.0786779	0.0393390	−	0.999226i	$-0.487475\pi$
$193$	1.51656e9	0.0786779	0.0393390	+	0.999226i	$0.487475\pi$
$194$	0	0
$195$	−2.48916e8	−0.0123281
$196$	0	0
$197$	−6.69974e9	−0.316927	−0.158464	−	0.987365i	$-0.550654\pi$
$197$	−6.69974e9	−0.316927	−0.158464	+	0.987365i	$0.550654\pi$
$198$	0	0
$199$	−2.89503e10	−1.30862	−0.654312	−	0.756225i	$-0.727042\pi$
$199$	−2.89503e10	−1.30862	−0.654312	+	0.756225i	$0.727042\pi$
$200$	0	0
$201$	−1.30852e10	−0.565456
$202$	0	0
$203$	−1.41292e9	−0.0583964
$204$	0	0
$205$	5.84328e9	0.231081
$206$	0	0
$207$	1.10846e10	0.419618
$208$	0	0
$209$	−5.41832e9	−0.196429
$210$	0	0
$211$	1.32094e10	0.458787	0.229394	−	0.973334i	$-0.426326\pi$
$211$	1.32094e10	0.458787	0.229394	+	0.973334i	$0.426326\pi$
$212$	0	0
$213$	7.88820e9	0.262585
$214$	0	0
$215$	−2.78400e8	−0.00888581
$216$	0	0
$217$	2.26377e8	0.00693048
$218$	0	0
$219$	−3.78235e9	−0.111113
$220$	0	0
$221$	−1.52902e10	−0.431169
$222$	0	0
$223$	3.34948e9	0.0906997	0.0453499	−	0.998971i	$-0.485560\pi$
$223$	3.34948e9	0.0906997	0.0453499	+	0.998971i	$0.485560\pi$
$224$	0	0
$225$	3.38308e10	0.880016
$226$	0	0
$227$	−7.11141e10	−1.77762	−0.888811	−	0.458275i	$-0.848468\pi$
$227$	−7.11141e10	−1.77762	−0.888811	+	0.458275i	$0.848468\pi$
$228$	0	0
$229$	4.27622e10	1.02754	0.513772	−	0.857927i	$-0.328248\pi$
$229$	4.27622e10	1.02754	0.513772	+	0.857927i	$0.328248\pi$
$230$	0	0
$231$	−2.59722e8	−0.00600142
$232$	0	0
$233$	5.90867e10	1.31337	0.656687	−	0.754163i	$-0.271957\pi$
$233$	5.90867e10	1.31337	0.656687	+	0.754163i	$0.271957\pi$
$234$	0	0
$235$	−1.06363e10	−0.227503
$236$	0	0
$237$	−5.81412e9	−0.119706
$238$	0	0
$239$	3.67583e10	0.728726	0.364363	−	0.931257i	$-0.381287\pi$
$239$	3.67583e10	0.728726	0.364363	+	0.931257i	$0.381287\pi$
$240$	0	0
$241$	−1.27679e10	−0.243805	−0.121902	−	0.992542i	$-0.538899\pi$
$241$	−1.27679e10	−0.243805	−0.121902	+	0.992542i	$0.538899\pi$
$242$	0	0
$243$	4.52881e10	0.833212
$244$	0	0
$245$	−7.70813e9	−0.136679
$246$	0	0
$247$	−2.04671e10	−0.349880
$248$	0	0
$249$	−3.65359e10	−0.602314
$250$	0	0
$251$	−2.44043e10	−0.388092	−0.194046	−	0.980992i	$-0.562161\pi$
$251$	−2.44043e10	−0.388092	−0.194046	+	0.980992i	$0.562161\pi$
$252$	0	0
$253$	4.74556e9	0.0728191
$254$	0	0
$255$	4.66572e9	0.0691016
$256$	0	0
$257$	6.02903e10	0.862081	0.431040	−	0.902333i	$-0.358147\pi$
$257$	6.02903e10	0.862081	0.431040	+	0.902333i	$0.358147\pi$
$258$	0	0
$259$	−4.74053e9	−0.0654603
$260$	0	0
$261$	3.26660e10	0.435726
$262$	0	0
$263$	−2.02168e10	−0.260563	−0.130281	−	0.991477i	$-0.541588\pi$
$263$	−2.02168e10	−0.260563	−0.130281	+	0.991477i	$0.541588\pi$
$264$	0	0
$265$	4.60940e9	0.0574167
$266$	0	0
$267$	−1.29119e10	−0.155485
$268$	0	0
$269$	4.60447e10	0.536160	0.268080	−	0.963397i	$-0.413611\pi$
$269$	4.60447e10	0.536160	0.268080	+	0.963397i	$0.413611\pi$
$270$	0	0
$271$	−1.29942e11	−1.46348	−0.731741	−	0.681583i	$-0.761292\pi$
$271$	−1.29942e11	−1.46348	−0.731741	+	0.681583i	$0.761292\pi$
$272$	0	0
$273$	−9.81068e8	−0.0106897
$274$	0	0
$275$	1.44837e10	0.152715
$276$	0	0
$277$	3.59452e10	0.366845	0.183422	−	0.983034i	$-0.441282\pi$
$277$	3.59452e10	0.366845	0.183422	+	0.983034i	$0.441282\pi$
$278$	0	0
$279$	−5.23371e9	−0.0517120
$280$	0	0
$281$	−8.72897e9	−0.0835188	−0.0417594	−	0.999128i	$-0.513296\pi$
$281$	−8.72897e9	−0.0835188	−0.0417594	+	0.999128i	$0.513296\pi$
$282$	0	0
$283$	7.13022e10	0.660791	0.330396	−	0.943843i	$-0.392818\pi$
$283$	7.13022e10	0.660791	0.330396	+	0.943843i	$0.392818\pi$
$284$	0	0
$285$	6.24542e9	0.0560738
$286$	0	0
$287$	2.30305e10	0.200371
$288$	0	0
$289$	1.68013e11	1.41678
$290$	0	0
$291$	5.31412e10	0.434423
$292$	0	0
$293$	−7.53611e10	−0.597369	−0.298685	−	0.954352i	$-0.596548\pi$
$293$	−7.53611e10	−0.597369	−0.298685	+	0.954352i	$0.596548\pi$
$294$	0	0
$295$	−1.38099e10	−0.106168
$296$	0	0
$297$	1.26967e10	0.0946862
$298$	0	0
$299$	1.79258e10	0.129705
$300$	0	0
$301$	−1.09728e9	−0.00770489
$302$	0	0
$303$	5.33979e10	0.363942
$304$	0	0
$305$	3.39220e10	0.224457
$306$	0	0
$307$	1.81582e10	0.116667	0.0583337	−	0.998297i	$-0.481421\pi$
$307$	1.81582e10	0.116667	0.0583337	+	0.998297i	$0.481421\pi$
$308$	0	0
$309$	−5.74522e10	−0.358504
$310$	0	0
$311$	2.05503e11	1.24565	0.622825	−	0.782361i	$-0.285985\pi$
$311$	2.05503e11	1.24565	0.622825	+	0.782361i	$0.285985\pi$
$312$	0	0
$313$	−5.23346e10	−0.308205	−0.154102	−	0.988055i	$-0.549249\pi$
$313$	−5.23346e10	−0.308205	−0.154102	+	0.988055i	$0.549249\pi$
$314$	0	0
$315$	−2.61485e9	−0.0149641
$316$	0	0
$317$	8.85228e10	0.492366	0.246183	−	0.969223i	$-0.420824\pi$
$317$	8.85228e10	0.492366	0.246183	+	0.969223i	$0.420824\pi$
$318$	0	0
$319$	1.39850e10	0.0756144
$320$	0	0
$321$	−2.67938e10	−0.140851
$322$	0	0
$323$	3.83638e11	1.96115
$324$	0	0
$325$	5.47103e10	0.272016
$326$	0	0
$327$	−1.19860e11	−0.579708
$328$	0	0
$329$	−4.19216e10	−0.197268
$330$	0	0
$331$	2.72421e11	1.24743	0.623713	−	0.781653i	$-0.285623\pi$
$331$	2.72421e11	1.24743	0.623713	+	0.781653i	$0.285623\pi$
$332$	0	0
$333$	1.09599e11	0.488434
$334$	0	0
$335$	−5.64011e10	−0.244673
$336$	0	0
$337$	−2.45525e11	−1.03696	−0.518478	−	0.855091i	$-0.673501\pi$
$337$	−2.45525e11	−1.03696	−0.518478	+	0.855091i	$0.673501\pi$
$338$	0	0
$339$	3.87425e10	0.159327
$340$	0	0
$341$	−2.24066e9	−0.00897391
$342$	0	0
$343$	−6.12068e10	−0.238768
$344$	0	0
$345$	−5.46996e9	−0.0207873
$346$	0	0
$347$	1.20829e11	0.447391	0.223695	−	0.974659i	$-0.428188\pi$
$347$	1.20829e11	0.447391	0.223695	+	0.974659i	$0.428188\pi$
$348$	0	0
$349$	−3.24249e11	−1.16994	−0.584971	−	0.811054i	$-0.698894\pi$
$349$	−3.24249e11	−1.16994	−0.584971	+	0.811054i	$0.698894\pi$
$350$	0	0
$351$	4.79603e10	0.168655
$352$	0	0
$353$	3.01775e11	1.03442	0.517209	−	0.855859i	$-0.326971\pi$
$353$	3.01775e11	1.03442	0.517209	+	0.855859i	$0.326971\pi$
$354$	0	0
$355$	3.40004e10	0.113620
$356$	0	0
$357$	1.83893e10	0.0599180
$358$	0	0
$359$	3.12893e11	0.994194	0.497097	−	0.867695i	$-0.334399\pi$
$359$	3.12893e11	0.994194	0.497097	+	0.867695i	$0.334399\pi$
$360$	0	0
$361$	1.90841e11	0.591409
$362$	0	0
$363$	−1.03457e11	−0.312739
$364$	0	0
$365$	−1.63030e10	−0.0480784
$366$	0	0
$367$	4.62832e11	1.33176	0.665880	−	0.746059i	$-0.268056\pi$
$367$	4.62832e11	1.33176	0.665880	+	0.746059i	$0.268056\pi$
$368$	0	0
$369$	−5.32452e11	−1.49507
$370$	0	0
$371$	1.81673e10	0.0497861
$372$	0	0
$373$	5.81695e11	1.55599	0.777993	−	0.628274i	$-0.216238\pi$
$373$	5.81695e11	1.55599	0.777993	+	0.628274i	$0.216238\pi$
$374$	0	0
$375$	−3.37165e10	−0.0880445
$376$	0	0
$377$	5.28267e10	0.134684
$378$	0	0
$379$	4.51535e11	1.12413	0.562063	−	0.827095i	$-0.310008\pi$
$379$	4.51535e11	1.12413	0.562063	+	0.827095i	$0.310008\pi$
$380$	0	0
$381$	−2.14277e11	−0.520971
$382$	0	0
$383$	−2.86456e11	−0.680242	−0.340121	−	0.940382i	$-0.610468\pi$
$383$	−2.86456e11	−0.680242	−0.340121	+	0.940382i	$0.610468\pi$
$384$	0	0
$385$	−1.11947e9	−0.00259681
$386$	0	0
$387$	2.53684e10	0.0574902
$388$	0	0
$389$	−2.00404e11	−0.443745	−0.221872	−	0.975076i	$-0.571217\pi$
$389$	−2.00404e11	−0.443745	−0.221872	+	0.975076i	$0.571217\pi$
$390$	0	0
$391$	−3.36003e11	−0.727023
$392$	0	0
$393$	−1.71102e11	−0.361816
$394$	0	0
$395$	−2.50605e10	−0.0517968
$396$	0	0
$397$	3.69392e11	0.746329	0.373165	−	0.927765i	$-0.378273\pi$
$397$	3.69392e11	0.746329	0.373165	+	0.927765i	$0.378273\pi$
$398$	0	0
$399$	2.46154e10	0.0486216
$400$	0	0
$401$	1.80193e11	0.348008	0.174004	−	0.984745i	$-0.444329\pi$
$401$	1.80193e11	0.348008	0.174004	+	0.984745i	$0.444329\pi$
$402$	0	0
$403$	−8.46383e9	−0.0159843
$404$	0	0
$405$	5.27404e10	0.0974082
$406$	0	0
$407$	4.69214e10	0.0847611
$408$	0	0
$409$	4.95350e11	0.875301	0.437651	−	0.899145i	$-0.355811\pi$
$409$	4.95350e11	0.875301	0.437651	+	0.899145i	$0.355811\pi$
$410$	0	0
$411$	−2.26026e11	−0.390724
$412$	0	0
$413$	−5.44299e10	−0.0920581
$414$	0	0
$415$	−1.57480e11	−0.260621
$416$	0	0
$417$	2.87299e11	0.465289
$418$	0	0
$419$	−1.16304e12	−1.84346	−0.921728	−	0.387837i	$-0.873222\pi$
$419$	−1.16304e12	−1.84346	−0.921728	+	0.387837i	$0.873222\pi$
$420$	0	0
$421$	3.47281e11	0.538780	0.269390	−	0.963031i	$-0.413178\pi$
$421$	3.47281e11	0.538780	0.269390	+	0.963031i	$0.413178\pi$
$422$	0	0
$423$	9.69206e11	1.47192
$424$	0	0
$425$	−1.02550e12	−1.52470
$426$	0	0
$427$	1.33699e11	0.194627
$428$	0	0
$429$	9.71053e9	0.0138416
$430$	0	0
$431$	−3.66122e11	−0.511067	−0.255534	−	0.966800i	$-0.582251\pi$
$431$	−3.66122e11	−0.511067	−0.255534	+	0.966800i	$0.582251\pi$
$432$	0	0
$433$	3.58487e11	0.490093	0.245046	−	0.969511i	$-0.421197\pi$
$433$	3.58487e11	0.490093	0.245046	+	0.969511i	$0.421197\pi$
$434$	0	0
$435$	−1.61198e10	−0.0215853
$436$	0	0
$437$	−4.49766e11	−0.589957
$438$	0	0
$439$	−1.39866e12	−1.79731	−0.898653	−	0.438660i	$-0.855453\pi$
$439$	−1.39866e12	−1.79731	−0.898653	+	0.438660i	$0.855453\pi$
$440$	0	0
$441$	7.02380e11	0.884298
$442$	0	0
$443$	5.38017e11	0.663712	0.331856	−	0.943330i	$-0.392325\pi$
$443$	5.38017e11	0.663712	0.331856	+	0.943330i	$0.392325\pi$
$444$	0	0
$445$	−5.56539e10	−0.0672784
$446$	0	0
$447$	−1.90808e11	−0.226054
$448$	0	0
$449$	5.19645e10	0.0603390	0.0301695	−	0.999545i	$-0.490395\pi$
$449$	5.19645e10	0.0603390	0.0301695	+	0.999545i	$0.490395\pi$
$450$	0	0
$451$	−2.27954e11	−0.259449
$452$	0	0
$453$	1.35562e11	0.151250
$454$	0	0
$455$	−4.22868e9	−0.00462544
$456$	0	0
$457$	−1.12990e12	−1.21176	−0.605882	−	0.795554i	$-0.707180\pi$
$457$	−1.12990e12	−1.21176	−0.605882	+	0.795554i	$0.707180\pi$
$458$	0	0
$459$	−8.98974e11	−0.945344
$460$	0	0
$461$	8.06097e10	0.0831252	0.0415626	−	0.999136i	$-0.486766\pi$
$461$	8.06097e10	0.0831252	0.0415626	+	0.999136i	$0.486766\pi$
$462$	0	0
$463$	−1.83902e12	−1.85982	−0.929911	−	0.367784i	$-0.880117\pi$
$463$	−1.83902e12	−1.85982	−0.929911	+	0.367784i	$0.880117\pi$
$464$	0	0
$465$	2.58270e9	0.00256174
$466$	0	0
$467$	1.55579e12	1.51365	0.756826	−	0.653617i	$-0.226749\pi$
$467$	1.55579e12	1.51365	0.756826	+	0.653617i	$0.226749\pi$
$468$	0	0
$469$	−2.22297e11	−0.212156
$470$	0	0
$471$	−3.27974e11	−0.307075
$472$	0	0
$473$	1.08608e10	0.00997665
$474$	0	0
$475$	−1.37271e12	−1.23725
$476$	0	0
$477$	−4.20018e11	−0.371480
$478$	0	0
$479$	−2.05558e12	−1.78412	−0.892061	−	0.451914i	$-0.850741\pi$
$479$	−2.05558e12	−1.78412	−0.892061	+	0.451914i	$0.850741\pi$
$480$	0	0
$481$	1.77240e11	0.150976
$482$	0	0
$483$	−2.15591e10	−0.0180247
$484$	0	0
$485$	2.29054e11	0.187975
$486$	0	0
$487$	−4.90785e11	−0.395376	−0.197688	−	0.980265i	$-0.563343\pi$
$487$	−4.90785e11	−0.395376	−0.197688	+	0.980265i	$0.563343\pi$
$488$	0	0
$489$	−1.14567e11	−0.0906088
$490$	0	0
$491$	1.80560e12	1.40202	0.701010	−	0.713152i	$-0.252733\pi$
$491$	1.80560e12	1.40202	0.701010	+	0.713152i	$0.252733\pi$
$492$	0	0
$493$	−9.90190e11	−0.754931
$494$	0	0
$495$	2.58816e10	0.0193762
$496$	0	0
$497$	1.34008e11	0.0985203
$498$	0	0
$499$	1.46846e12	1.06025	0.530125	−	0.847920i	$-0.322145\pi$
$499$	1.46846e12	1.06025	0.530125	+	0.847920i	$0.322145\pi$
$500$	0	0
$501$	1.17934e11	0.0836316
$502$	0	0
$503$	−1.89165e12	−1.31761	−0.658803	−	0.752315i	$-0.728937\pi$
$503$	−1.89165e12	−1.31761	−0.658803	+	0.752315i	$0.728937\pi$
$504$	0	0
$505$	2.30160e11	0.157478
$506$	0	0
$507$	3.66804e10	0.0246546
$508$	0	0
$509$	1.78314e12	1.17748	0.588742	−	0.808321i	$-0.299624\pi$
$509$	1.78314e12	1.17748	0.588742	+	0.808321i	$0.299624\pi$
$510$	0	0
$511$	−6.42560e10	−0.0416888
$512$	0	0
$513$	−1.20335e12	−0.767118
$514$	0	0
$515$	−2.47635e11	−0.155124
$516$	0	0
$517$	4.14937e11	0.255432
$518$	0	0
$519$	−6.39870e11	−0.387114
$520$	0	0
$521$	−1.87038e12	−1.11214	−0.556069	−	0.831136i	$-0.687691\pi$
$521$	−1.87038e12	−1.11214	−0.556069	+	0.831136i	$0.687691\pi$
$522$	0	0
$523$	7.75520e11	0.453248	0.226624	−	0.973982i	$-0.427231\pi$
$523$	7.75520e11	0.453248	0.226624	+	0.973982i	$0.427231\pi$
$524$	0	0
$525$	−6.57993e10	−0.0378011
$526$	0	0
$527$	1.58647e11	0.0895952
$528$	0	0
$529$	−1.40723e12	−0.781295
$530$	0	0
$531$	1.25839e12	0.686894
$532$	0	0
$533$	−8.61068e11	−0.462131
$534$	0	0
$535$	−1.15489e11	−0.0609464
$536$	0	0
$537$	3.19487e11	0.165794
$538$	0	0
$539$	3.00704e11	0.153458
$540$	0	0
$541$	2.36216e12	1.18555	0.592776	−	0.805367i	$-0.298032\pi$
$541$	2.36216e12	1.18555	0.592776	+	0.805367i	$0.298032\pi$
$542$	0	0
$543$	−7.51281e11	−0.370854
$544$	0	0
$545$	−5.16631e11	−0.250840
$546$	0	0
$547$	−2.05243e12	−0.980224	−0.490112	−	0.871659i	$-0.663044\pi$
$547$	−2.05243e12	−0.980224	−0.490112	+	0.871659i	$0.663044\pi$
$548$	0	0
$549$	−3.09105e12	−1.45221
$550$	0	0
$551$	−1.32545e12	−0.612604
$552$	0	0
$553$	−9.87723e10	−0.0449130
$554$	0	0
$555$	−5.40839e10	−0.0241964
$556$	0	0
$557$	−3.18520e12	−1.40213	−0.701066	−	0.713097i	$-0.747292\pi$
$557$	−3.18520e12	−1.40213	−0.701066	+	0.713097i	$0.747292\pi$
$558$	0	0
$559$	4.10252e10	0.0177704
$560$	0	0
$561$	−1.82016e11	−0.0775846
$562$	0	0
$563$	2.23578e12	0.937867	0.468934	−	0.883233i	$-0.344638\pi$
$563$	2.23578e12	0.937867	0.468934	+	0.883233i	$0.344638\pi$
$564$	0	0
$565$	1.66991e11	0.0689407
$566$	0	0
$567$	2.07869e11	0.0844628
$568$	0	0
$569$	3.46124e11	0.138429	0.0692144	−	0.997602i	$-0.477951\pi$
$569$	3.46124e11	0.138429	0.0692144	+	0.997602i	$0.477951\pi$
$570$	0	0
$571$	1.06244e11	0.0418255	0.0209128	−	0.999781i	$-0.493343\pi$
$571$	1.06244e11	0.0418255	0.0209128	+	0.999781i	$0.493343\pi$
$572$	0	0
$573$	−1.09401e12	−0.423959
$574$	0	0
$575$	1.20227e12	0.458665
$576$	0	0
$577$	9.88176e11	0.371145	0.185572	−	0.982631i	$-0.440586\pi$
$577$	9.88176e11	0.371145	0.185572	+	0.982631i	$0.440586\pi$
$578$	0	0
$579$	6.81942e10	0.0252170
$580$	0	0
$581$	−6.20685e11	−0.225985
$582$	0	0
$583$	−1.79819e11	−0.0644653
$584$	0	0
$585$	9.77648e10	0.0345129
$586$	0	0
$587$	2.45712e12	0.854189	0.427095	−	0.904207i	$-0.359537\pi$
$587$	2.45712e12	0.854189	0.427095	+	0.904207i	$0.359537\pi$
$588$	0	0
$589$	2.12362e11	0.0727038
$590$	0	0
$591$	−3.01262e11	−0.101578
$592$	0	0
$593$	−1.54983e12	−0.514682	−0.257341	−	0.966321i	$-0.582846\pi$
$593$	−1.54983e12	−0.514682	−0.257341	+	0.966321i	$0.582846\pi$
$594$	0	0
$595$	7.92630e10	0.0259265
$596$	0	0
$597$	−1.30179e12	−0.419427
$598$	0	0
$599$	4.31316e12	1.36891	0.684456	−	0.729055i	$-0.260040\pi$
$599$	4.31316e12	1.36891	0.684456	+	0.729055i	$0.260040\pi$
$600$	0	0
$601$	1.71612e12	0.536554	0.268277	−	0.963342i	$-0.413546\pi$
$601$	1.71612e12	0.536554	0.268277	+	0.963342i	$0.413546\pi$
$602$	0	0
$603$	5.13938e12	1.58301
$604$	0	0
$605$	−4.45931e11	−0.135322
$606$	0	0
$607$	1.07880e12	0.322547	0.161273	−	0.986910i	$-0.448440\pi$
$607$	1.07880e12	0.322547	0.161273	+	0.986910i	$0.448440\pi$
$608$	0	0
$609$	−6.35339e10	−0.0187166
$610$	0	0
$611$	1.56738e12	0.454975
$612$	0	0
$613$	6.89026e11	0.197090	0.0985448	−	0.995133i	$-0.468581\pi$
$613$	6.89026e11	0.197090	0.0985448	+	0.995133i	$0.468581\pi$
$614$	0	0
$615$	2.62751e11	0.0740638
$616$	0	0
$617$	2.22658e12	0.618523	0.309262	−	0.950977i	$-0.399918\pi$
$617$	2.22658e12	0.618523	0.309262	+	0.950977i	$0.399918\pi$
$618$	0	0
$619$	3.03265e12	0.830261	0.415131	−	0.909762i	$-0.363736\pi$
$619$	3.03265e12	0.830261	0.415131	+	0.909762i	$0.363736\pi$
$620$	0	0
$621$	1.05393e12	0.284381
$622$	0	0
$623$	−2.19352e11	−0.0583371
$624$	0	0
$625$	3.59600e12	0.942670
$626$	0	0
$627$	−2.43642e11	−0.0629576
$628$	0	0
$629$	−3.32221e12	−0.846252
$630$	0	0
$631$	−4.37390e12	−1.09834	−0.549170	−	0.835710i	$-0.685056\pi$
$631$	−4.37390e12	−1.09834	−0.549170	+	0.835710i	$0.685056\pi$
$632$	0	0
$633$	5.93976e11	0.147046
$634$	0	0
$635$	−9.23596e11	−0.225424
$636$	0	0
$637$	1.13587e12	0.273339
$638$	0	0
$639$	−3.09818e12	−0.735111
$640$	0	0
$641$	−5.17133e12	−1.20988	−0.604938	−	0.796272i	$-0.706802\pi$
$641$	−5.17133e12	−1.20988	−0.604938	+	0.796272i	$0.706802\pi$
$642$	0	0
$643$	3.17173e12	0.731723	0.365861	−	0.930669i	$-0.380774\pi$
$643$	3.17173e12	0.731723	0.365861	+	0.930669i	$0.380774\pi$
$644$	0	0
$645$	−1.25186e10	−0.00284799
$646$	0	0
$647$	−8.59340e12	−1.92795	−0.963975	−	0.265992i	$-0.914300\pi$
$647$	−8.59340e12	−1.92795	−0.963975	+	0.265992i	$0.914300\pi$
$648$	0	0
$649$	5.38743e11	0.119201
$650$	0	0
$651$	1.01793e10	0.00222129
$652$	0	0
$653$	7.34215e12	1.58021	0.790103	−	0.612974i	$-0.210027\pi$
$653$	7.34215e12	1.58021	0.790103	+	0.612974i	$0.210027\pi$
$654$	0	0
$655$	−7.37496e11	−0.156558
$656$	0	0
$657$	1.48556e12	0.311062
$658$	0	0
$659$	8.53358e12	1.76257	0.881286	−	0.472584i	$-0.156679\pi$
$659$	8.53358e12	1.76257	0.881286	+	0.472584i	$0.156679\pi$
$660$	0	0
$661$	3.89705e12	0.794016	0.397008	−	0.917815i	$-0.370049\pi$
$661$	3.89705e12	0.794016	0.397008	+	0.917815i	$0.370049\pi$
$662$	0	0
$663$	−6.87542e11	−0.138194
$664$	0	0
$665$	1.06100e11	0.0210386
$666$	0	0
$667$	1.16087e12	0.227101
$668$	0	0
$669$	1.50614e11	0.0290701
$670$	0	0
$671$	−1.32334e12	−0.252012
$672$	0	0
$673$	4.93969e12	0.928180	0.464090	−	0.885788i	$-0.346381\pi$
$673$	4.93969e12	0.928180	0.464090	+	0.885788i	$0.346381\pi$
$674$	0	0
$675$	3.21665e12	0.596399
$676$	0	0
$677$	−4.27408e12	−0.781977	−0.390988	−	0.920396i	$-0.627867\pi$
$677$	−4.27408e12	−0.781977	−0.390988	+	0.920396i	$0.627867\pi$
$678$	0	0
$679$	9.02782e11	0.162993
$680$	0	0
$681$	−3.19774e12	−0.569745
$682$	0	0
$683$	−2.74685e12	−0.482994	−0.241497	−	0.970402i	$-0.577638\pi$
$683$	−2.74685e12	−0.482994	−0.241497	+	0.970402i	$0.577638\pi$
$684$	0	0
$685$	−9.74236e11	−0.169066
$686$	0	0
$687$	1.92286e12	0.329338
$688$	0	0
$689$	−6.79243e11	−0.114826
$690$	0	0
$691$	−7.31354e12	−1.22033	−0.610164	−	0.792275i	$-0.708896\pi$
$691$	−7.31354e12	−1.22033	−0.610164	+	0.792275i	$0.708896\pi$
$692$	0	0
$693$	1.02009e11	0.0168011
$694$	0	0
$695$	1.23834e12	0.201330
$696$	0	0
$697$	1.61400e13	2.59033
$698$	0	0
$699$	2.65691e12	0.420949
$700$	0	0
$701$	4.54373e12	0.710693	0.355346	−	0.934735i	$-0.384363\pi$
$701$	4.54373e12	0.710693	0.355346	+	0.934735i	$0.384363\pi$
$702$	0	0
$703$	−4.44704e12	−0.686708
$704$	0	0
$705$	−4.78277e11	−0.0729170
$706$	0	0
$707$	9.07143e11	0.136549
$708$	0	0
$709$	6.96821e12	1.03565	0.517825	−	0.855487i	$-0.326742\pi$
$709$	6.96821e12	1.03565	0.517825	+	0.855487i	$0.326742\pi$
$710$	0	0
$711$	2.28356e12	0.335119
$712$	0	0
$713$	−1.85994e11	−0.0269523
$714$	0	0
$715$	4.18552e10	0.00598924
$716$	0	0
$717$	1.65288e12	0.233564
$718$	0	0
$719$	−6.53484e12	−0.911917	−0.455958	−	0.890001i	$-0.650703\pi$
$719$	−6.53484e12	−0.911917	−0.455958	+	0.890001i	$0.650703\pi$
$720$	0	0
$721$	−9.76020e11	−0.134509
$722$	0	0
$723$	−5.74124e11	−0.0781418
$724$	0	0
$725$	3.54303e12	0.476271
$726$	0	0
$727$	−7.44685e11	−0.0988708	−0.0494354	−	0.998777i	$-0.515742\pi$
$727$	−7.44685e11	−0.0988708	−0.0494354	+	0.998777i	$0.515742\pi$
$728$	0	0
$729$	−3.31958e12	−0.435320
$730$	0	0
$731$	−7.68982e11	−0.0996065
$732$	0	0
$733$	1.05042e12	0.134399	0.0671993	−	0.997740i	$-0.478594\pi$
$733$	1.05042e12	0.134399	0.0671993	+	0.997740i	$0.478594\pi$
$734$	0	0
$735$	−3.46606e11	−0.0438069
$736$	0	0
$737$	2.20028e12	0.274709
$738$	0	0
$739$	−3.05643e12	−0.376977	−0.188488	−	0.982075i	$-0.560359\pi$
$739$	−3.05643e12	−0.376977	−0.188488	+	0.982075i	$0.560359\pi$
$740$	0	0
$741$	−9.20328e11	−0.112140
$742$	0	0
$743$	3.27513e12	0.394256	0.197128	−	0.980378i	$-0.436838\pi$
$743$	3.27513e12	0.394256	0.197128	+	0.980378i	$0.436838\pi$
$744$	0	0
$745$	−8.22437e11	−0.0978136
$746$	0	0
$747$	1.43499e13	1.68619
$748$	0	0
$749$	−4.55183e11	−0.0528467
$750$	0	0
$751$	−8.12818e12	−0.932425	−0.466212	−	0.884673i	$-0.654382\pi$
$751$	−8.12818e12	−0.932425	−0.466212	+	0.884673i	$0.654382\pi$
$752$	0	0
$753$	−1.09737e12	−0.124387
$754$	0	0
$755$	5.84311e11	0.0654459
$756$	0	0
$757$	−5.65281e12	−0.625652	−0.312826	−	0.949810i	$-0.601276\pi$
$757$	−5.65281e12	−0.625652	−0.312826	+	0.949810i	$0.601276\pi$
$758$	0	0
$759$	2.13390e11	0.0233392
$760$	0	0
$761$	8.94783e12	0.967134	0.483567	−	0.875307i	$-0.339341\pi$
$761$	8.94783e12	0.967134	0.483567	+	0.875307i	$0.339341\pi$
$762$	0	0
$763$	−2.03623e12	−0.217503
$764$	0	0
$765$	−1.83252e12	−0.193451
$766$	0	0
$767$	2.03504e12	0.212321
$768$	0	0
$769$	1.35334e13	1.39553	0.697766	−	0.716326i	$-0.254178\pi$
$769$	1.35334e13	1.39553	0.697766	+	0.716326i	$0.254178\pi$
$770$	0	0
$771$	2.71103e12	0.276305
$772$	0	0
$773$	6.35436e12	0.640124	0.320062	−	0.947397i	$-0.396296\pi$
$773$	6.35436e12	0.640124	0.320062	+	0.947397i	$0.396296\pi$
$774$	0	0
$775$	−5.67662e11	−0.0565239
$776$	0	0
$777$	−2.13164e11	−0.0209807
$778$	0	0
$779$	2.16046e13	2.10198
$780$	0	0
$781$	−1.32640e12	−0.127569
$782$	0	0
$783$	3.10590e12	0.295298
$784$	0	0
$785$	−1.41366e12	−0.132871
$786$	0	0
$787$	−9.99821e11	−0.0929044	−0.0464522	−	0.998921i	$-0.514792\pi$
$787$	−9.99821e11	−0.0929044	−0.0464522	+	0.998921i	$0.514792\pi$
$788$	0	0
$789$	−9.09076e11	−0.0835130
$790$	0	0
$791$	6.58172e11	0.0597785
$792$	0	0
$793$	−4.99876e12	−0.448883
$794$	0	0
$795$	2.07268e11	0.0184026
$796$	0	0
$797$	−7.69462e12	−0.675499	−0.337750	−	0.941236i	$-0.609666\pi$
$797$	−7.69462e12	−0.675499	−0.337750	+	0.941236i	$0.609666\pi$
$798$	0	0
$799$	−2.93791e13	−2.55022
$800$	0	0
$801$	5.07130e12	0.435284
$802$	0	0
$803$	6.36001e11	0.0539806
$804$	0	0
$805$	−9.29258e10	−0.00779928
$806$	0	0
$807$	2.07046e12	0.171844
$808$	0	0
$809$	−1.49499e13	−1.22708	−0.613538	−	0.789666i	$-0.710254\pi$
$809$	−1.49499e13	−1.22708	−0.613538	+	0.789666i	$0.710254\pi$
$810$	0	0
$811$	9.73391e12	0.790121	0.395060	−	0.918655i	$-0.370724\pi$
$811$	9.73391e12	0.790121	0.395060	+	0.918655i	$0.370724\pi$
$812$	0	0
$813$	−5.84300e12	−0.469060
$814$	0	0
$815$	−4.93817e11	−0.0392064
$816$	0	0
$817$	−1.02934e12	−0.0808277
$818$	0	0
$819$	3.85326e11	0.0299261
$820$	0	0
$821$	−9.54265e12	−0.733035	−0.366517	−	0.930411i	$-0.619450\pi$
$821$	−9.54265e12	−0.733035	−0.366517	+	0.930411i	$0.619450\pi$
$822$	0	0
$823$	1.34813e12	0.102431	0.0512155	−	0.998688i	$-0.483690\pi$
$823$	1.34813e12	0.102431	0.0512155	+	0.998688i	$0.483690\pi$
$824$	0	0
$825$	6.51277e11	0.0489466
$826$	0	0
$827$	−1.05233e13	−0.782307	−0.391153	−	0.920326i	$-0.627924\pi$
$827$	−1.05233e13	−0.782307	−0.391153	+	0.920326i	$0.627924\pi$
$828$	0	0
$829$	1.10484e13	0.812461	0.406231	−	0.913771i	$-0.366843\pi$
$829$	1.10484e13	0.812461	0.406231	+	0.913771i	$0.366843\pi$
$830$	0	0
$831$	1.61632e12	0.117577
$832$	0	0
$833$	−2.12910e13	−1.53212
$834$	0	0
$835$	5.08330e11	0.0361874
$836$	0	0
$837$	−4.97625e11	−0.0350459
$838$	0	0
$839$	−2.06097e13	−1.43596	−0.717980	−	0.696063i	$-0.754933\pi$
$839$	−2.06097e13	−1.43596	−0.717980	+	0.696063i	$0.754933\pi$
$840$	0	0
$841$	−1.10861e13	−0.764182
$842$	0	0
$843$	−3.92509e11	−0.0267686
$844$	0	0
$845$	1.58103e11	0.0106680
$846$	0	0
$847$	−1.75757e12	−0.117338
$848$	0	0
$849$	3.20620e12	0.211790
$850$	0	0
$851$	3.89487e12	0.254572
$852$	0	0
$853$	2.14975e13	1.39033	0.695166	−	0.718850i	$-0.255331\pi$
$853$	2.14975e13	1.39033	0.695166	+	0.718850i	$0.255331\pi$
$854$	0	0
$855$	−2.45296e12	−0.156980
$856$	0	0
$857$	−1.28165e13	−0.811627	−0.405814	−	0.913956i	$-0.633012\pi$
$857$	−1.28165e13	−0.811627	−0.405814	+	0.913956i	$0.633012\pi$
$858$	0	0
$859$	2.40613e13	1.50782	0.753911	−	0.656976i	$-0.228165\pi$
$859$	2.40613e13	1.50782	0.753911	+	0.656976i	$0.228165\pi$
$860$	0	0
$861$	1.03559e12	0.0642208
$862$	0	0
$863$	2.14086e13	1.31383	0.656917	−	0.753963i	$-0.271860\pi$
$863$	2.14086e13	1.31383	0.656917	+	0.753963i	$0.271860\pi$
$864$	0	0
$865$	−2.75802e12	−0.167504
$866$	0	0
$867$	7.55493e12	0.454093
$868$	0	0
$869$	9.77641e11	0.0581555
$870$	0	0
$871$	8.31128e12	0.489312
$872$	0	0
$873$	−2.08718e13	−1.21618
$874$	0	0
$875$	−5.72789e11	−0.0330338
$876$	0	0
$877$	−2.05602e13	−1.17362	−0.586812	−	0.809723i	$-0.699617\pi$
$877$	−2.05602e13	−1.17362	−0.586812	+	0.809723i	$0.699617\pi$
$878$	0	0
$879$	−3.38871e12	−0.191463
$880$	0	0
$881$	1.48422e13	0.830055	0.415028	−	0.909809i	$-0.363772\pi$
$881$	1.48422e13	0.830055	0.415028	+	0.909809i	$0.363772\pi$
$882$	0	0
$883$	7.99446e12	0.442553	0.221277	−	0.975211i	$-0.428978\pi$
$883$	7.99446e12	0.442553	0.221277	+	0.975211i	$0.428978\pi$
$884$	0	0
$885$	−6.20981e11	−0.0340278
$886$	0	0
$887$	−3.36659e12	−0.182614	−0.0913070	−	0.995823i	$-0.529104\pi$
$887$	−3.36659e12	−0.182614	−0.0913070	+	0.995823i	$0.529104\pi$
$888$	0	0
$889$	−3.64022e12	−0.195465
$890$	0	0
$891$	−2.05747e12	−0.109366
$892$	0	0
$893$	−3.93262e13	−2.06943
$894$	0	0
$895$	1.37708e12	0.0717391
$896$	0	0
$897$	8.06056e11	0.0415718
$898$	0	0
$899$	−5.48117e11	−0.0279869
$900$	0	0
$901$	1.27318e13	0.643620
$902$	0	0
$903$	−4.93404e10	−0.00246949
$904$	0	0
$905$	−3.23823e12	−0.160468
$906$	0	0
$907$	−3.60908e13	−1.77078	−0.885388	−	0.464853i	$-0.846107\pi$
$907$	−3.60908e13	−1.77078	−0.885388	+	0.464853i	$0.846107\pi$
$908$	0	0
$909$	−2.09726e13	−1.01886
$910$	0	0
$911$	−2.48291e13	−1.19434	−0.597170	−	0.802115i	$-0.703708\pi$
$911$	−2.48291e13	−1.19434	−0.597170	+	0.802115i	$0.703708\pi$
$912$	0	0
$913$	6.14350e12	0.292615
$914$	0	0
$915$	1.52535e12	0.0719406
$916$	0	0
$917$	−2.90674e12	−0.135751
$918$	0	0
$919$	−6.20332e12	−0.286883	−0.143441	−	0.989659i	$-0.545817\pi$
$919$	−6.20332e12	−0.286883	−0.143441	+	0.989659i	$0.545817\pi$
$920$	0	0
$921$	8.16506e11	0.0373930
$922$	0	0
$923$	−5.01031e12	−0.227225
$924$	0	0
$925$	1.18873e13	0.533884
$926$	0	0
$927$	2.25650e13	1.00364
$928$	0	0
$929$	−1.83085e13	−0.806459	−0.403230	−	0.915099i	$-0.632112\pi$
$929$	−1.83085e13	−0.806459	−0.403230	+	0.915099i	$0.632112\pi$
$930$	0	0
$931$	−2.84996e13	−1.24327
$932$	0	0
$933$	9.24070e12	0.399243
$934$	0	0
$935$	−7.84539e11	−0.0335708
$936$	0	0
$937$	−1.62316e13	−0.687912	−0.343956	−	0.938986i	$-0.611767\pi$
$937$	−1.62316e13	−0.687912	−0.343956	+	0.938986i	$0.611767\pi$
$938$	0	0
$939$	−2.35329e12	−0.0987826
$940$	0	0
$941$	−3.97614e13	−1.65313	−0.826567	−	0.562839i	$-0.809709\pi$
$941$	−3.97614e13	−1.65313	−0.826567	+	0.562839i	$0.809709\pi$
$942$	0	0
$943$	−1.89221e13	−0.779231
$944$	0	0
$945$	−2.48622e11	−0.0101414
$946$	0	0
$947$	−2.21556e13	−0.895176	−0.447588	−	0.894240i	$-0.647717\pi$
$947$	−2.21556e13	−0.895176	−0.447588	+	0.894240i	$0.647717\pi$
$948$	0	0
$949$	2.40242e12	0.0961503
$950$	0	0
$951$	3.98054e12	0.157808
$952$	0	0
$953$	−1.39008e12	−0.0545911	−0.0272956	−	0.999627i	$-0.508690\pi$
$953$	−1.39008e12	−0.0545911	−0.0272956	+	0.999627i	$0.508690\pi$
$954$	0	0
$955$	−4.71548e12	−0.183447
$956$	0	0
$957$	6.28853e11	0.0242351
$958$	0	0
$959$	−3.83981e12	−0.146597
$960$	0	0
$961$	−2.63518e13	−0.996679
$962$	0	0
$963$	1.05236e13	0.394316
$964$	0	0
$965$	2.93936e11	0.0109114
$966$	0	0
$967$	3.92185e13	1.44236	0.721178	−	0.692750i	$-0.243601\pi$
$967$	3.92185e13	1.44236	0.721178	+	0.692750i	$0.243601\pi$
$968$	0	0
$969$	1.72508e13	0.628566
$970$	0	0
$971$	1.67024e13	0.602964	0.301482	−	0.953472i	$-0.402519\pi$
$971$	1.67024e13	0.602964	0.301482	+	0.953472i	$0.402519\pi$
$972$	0	0
$973$	4.88075e12	0.174574
$974$	0	0
$975$	2.46012e12	0.0871837
$976$	0	0
$977$	4.71451e13	1.65543	0.827715	−	0.561148i	$-0.189640\pi$
$977$	4.71451e13	1.65543	0.827715	+	0.561148i	$0.189640\pi$
$978$	0	0
$979$	2.17113e12	0.0755376
$980$	0	0
$981$	4.70764e13	1.62290
$982$	0	0
$983$	−4.28954e13	−1.46528	−0.732638	−	0.680618i	$-0.761711\pi$
$983$	−4.28954e13	−1.46528	−0.732638	+	0.680618i	$0.761711\pi$
$984$	0	0
$985$	−1.29853e12	−0.0439529
$986$	0	0
$987$	−1.88506e12	−0.0632264
$988$	0	0
$989$	9.01533e11	0.0299639
$990$	0	0
$991$	−3.17587e13	−1.04600	−0.523000	−	0.852333i	$-0.675187\pi$
$991$	−3.17587e13	−1.04600	−0.523000	+	0.852333i	$0.675187\pi$
$992$	0	0
$993$	1.22498e13	0.399813
$994$	0	0
$995$	−5.61108e12	−0.181486
$996$	0	0
$997$	−8.44481e12	−0.270683	−0.135342	−	0.990799i	$-0.543213\pi$
$997$	−8.44481e12	−0.270683	−0.135342	+	0.990799i	$0.543213\pi$
$998$	0	0
$999$	1.04207e13	0.331018

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.l.1.4		7
4.3	odd	2		104.10.a.c.1.4	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.c.1.4	✓	7	4.3	odd	2
208.10.a.l.1.4		7	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:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

\(f(q)\)	\(=\)	\(q+44.9663 q^{3} +193.817 q^{5} +763.904 q^{7} -17661.0 q^{9} -7561.06 q^{11} -28561.0 q^{13} +8715.25 q^{15} +535351. q^{17} +716609. q^{19} +34349.9 q^{21} -627632. q^{23} -1.91556e6 q^{25} -1.67922e6 q^{27} -1.84961e6 q^{29} +296342. q^{31} -339993. q^{33} +148058. q^{35} -6.20567e6 q^{37} -1.28428e6 q^{39} +3.01484e7 q^{41} -1.43641e6 q^{43} -3.42302e6 q^{45} -5.48782e7 q^{47} -3.97701e7 q^{49} +2.40728e7 q^{51} +2.37822e7 q^{53} -1.46547e6 q^{55} +3.22232e7 q^{57} -7.12523e7 q^{59} +1.75021e8 q^{61} -1.34913e7 q^{63} -5.53562e6 q^{65} -2.91001e8 q^{67} -2.82223e7 q^{69} +1.75425e8 q^{71} -8.41153e7 q^{73} -8.61356e7 q^{75} -5.77592e6 q^{77} -1.29299e8 q^{79} +2.72114e8 q^{81} -8.12518e8 q^{83} +1.03760e8 q^{85} -8.31700e7 q^{87} -2.87146e8 q^{89} -2.18179e7 q^{91} +1.33254e7 q^{93} +1.38891e8 q^{95} +1.18180e9 q^{97} +1.33536e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 13 q^{3} - 801 q^{5} + 1091 q^{7} + 87864 q^{9} - 32218 q^{11} - 199927 q^{13} - 58323 q^{15} - 870531 q^{17} + 128950 q^{19} - 719663 q^{21} - 2198844 q^{23} + 992010 q^{25} - 5441383 q^{27} + 6327710 q^{29}+ \cdots - 5204667600 q^{99}+O(q^{100}) \) 7 * q - 13 * q^3 - 801 * q^5 + 1091 * q^7 + 87864 * q^9 - 32218 * q^11 - 199927 * q^13 - 58323 * q^15 - 870531 * q^17 + 128950 * q^19 - 719663 * q^21 - 2198844 * q^23 + 992010 * q^25 - 5441383 * q^27 + 6327710 * q^29 + 1660376 * q^31 + 14199598 * q^33 + 19231769 * q^35 + 5058787 * q^37 + 371293 * q^39 + 14779748 * q^41 - 9405075 * q^43 + 56644022 * q^45 - 133415721 * q^47 + 44544988 * q^49 - 132066169 * q^51 + 43962424 * q^53 - 191344746 * q^55 + 138963078 * q^57 - 22370274 * q^59 + 23851748 * q^61 - 780720706 * q^63 + 22877361 * q^65 - 12058786 * q^67 + 1036453420 * q^69 - 864303791 * q^71 + 670491202 * q^73 - 693324044 * q^75 + 841052238 * q^77 - 1019547572 * q^79 + 900194079 * q^81 - 1345041700 * q^83 + 824066375 * q^85 - 2589172074 * q^87 + 1409158546 * q^89 - 31160051 * q^91 - 445200892 * q^93 - 3285213058 * q^95 + 320206906 * q^97 - 5204667600 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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