LMFDB - Embedded newform 104.10.a.d.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [104,10,Mod(1,104)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("104.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 104 = 2^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	104.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,141] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$53.5637269610$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3 x^{7} - 135356 x^{6} - 24398 x^{5} + 5213582205 x^{4} + 598076469 x^{3} + \cdots + 15\!\cdots\!64 $ x^8 - 3x^7 - 135356x^6 - 24398x^5 + 5213582205x^4 + 598076469x^3 - 44131402568826x^2 + 60107523849756*x + 1558753465556664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-5.30867$ of defining polynomial
Character	$\chi$	$=$	104.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+23.3087 q^{3} -760.329 q^{5} -8557.89 q^{7} -19139.7 q^{9} -72978.2 q^{11} +28561.0 q^{13} -17722.3 q^{15} +266694. q^{17} -161914. q^{19} -199473. q^{21} +1.43017e6 q^{23} -1.37503e6 q^{25} -904906. q^{27} +5.84621e6 q^{29} -962113. q^{31} -1.70103e6 q^{33} +6.50681e6 q^{35} -3.09958e6 q^{37} +665719. q^{39} -1.35817e7 q^{41} +1.14462e7 q^{43} +1.45525e7 q^{45} -2.29931e6 q^{47} +3.28838e7 q^{49} +6.21629e6 q^{51} -3.07448e7 q^{53} +5.54874e7 q^{55} -3.77399e6 q^{57} +3.05327e7 q^{59} +6.53064e7 q^{61} +1.63795e8 q^{63} -2.17158e7 q^{65} -1.81368e8 q^{67} +3.33354e7 q^{69} -1.23157e8 q^{71} +4.31054e8 q^{73} -3.20500e7 q^{75} +6.24539e8 q^{77} +4.22314e8 q^{79} +3.55635e8 q^{81} +1.40247e8 q^{83} -2.02775e8 q^{85} +1.36267e8 q^{87} -5.82829e7 q^{89} -2.44422e8 q^{91} -2.24256e7 q^{93} +1.23108e8 q^{95} +5.66836e8 q^{97} +1.39678e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 141 q^{3} + 2051 q^{5} - 2417 q^{7} + 115741 q^{9} - 53118 q^{11} + 228488 q^{13} - 464555 q^{15} + 433095 q^{17} - 434954 q^{19} + 906875 q^{21} - 1124296 q^{23} + 5966065 q^{25} + 7820643 q^{27}+ \cdots - 641626736 q^{99}+O(q^{100}) $ 8 * q + 141 * q^3 + 2051 * q^5 - 2417 * q^7 + 115741 * q^9 - 53118 * q^11 + 228488 * q^13 - 464555 * q^15 + 433095 * q^17 - 434954 * q^19 + 906875 * q^21 - 1124296 * q^23 + 5966065 * q^25 + 7820643 * q^27 - 1913432 * q^29 + 5220148 * q^31 - 1626882 * q^33 + 950707 * q^35 - 419409 * q^37 + 4027101 * q^39 + 12148046 * q^41 + 32143187 * q^43 + 18893544 * q^45 + 57898411 * q^47 + 104302637 * q^49 + 198521477 * q^51 + 273106866 * q^53 + 141214662 * q^55 + 253032058 * q^57 + 50734502 * q^59 + 139520018 * q^61 - 20344158 * q^63 + 58578611 * q^65 + 528144786 * q^67 + 483506576 * q^69 + 334814973 * q^71 + 630864344 * q^73 + 2265796264 * q^75 + 939664146 * q^77 + 865037608 * q^79 + 3194281040 * q^81 + 1520229952 * q^83 + 2130158385 * q^85 + 1167867006 * q^87 + 894700240 * q^89 - 69031937 * q^91 + 3803209548 * q^93 - 2695926542 * q^95 + 980232164 * q^97 - 641626736 * 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
$3$	23.3087	0.166139	0.0830696	−	0.996544i	$-0.473528\pi$
$3$	23.3087	0.166139	0.0830696	+	0.996544i	$0.473528\pi$
$4$	0	0
$5$	−760.329	−0.544047	−0.272024	−	0.962291i	$-0.587693\pi$
$5$	−760.329	−0.544047	−0.272024	+	0.962291i	$0.587693\pi$
$6$	0	0
$7$	−8557.89	−1.34718	−0.673590	−	0.739106i	$-0.735248\pi$
$7$	−8557.89	−1.34718	−0.673590	+	0.739106i	$0.735248\pi$
$8$	0	0
$9$	−19139.7	−0.972398
$10$	0	0
$11$	−72978.2	−1.50289	−0.751443	−	0.659798i	$-0.770642\pi$
$11$	−72978.2	−1.50289	−0.751443	+	0.659798i	$0.770642\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	−17722.3	−0.0903875
$16$	0	0
$17$	266694.	0.774450	0.387225	−	0.921985i	$-0.373434\pi$
$17$	266694.	0.774450	0.387225	+	0.921985i	$0.373434\pi$
$18$	0	0
$19$	−161914.	−0.285031	−0.142516	−	0.989793i	$-0.545519\pi$
$19$	−161914.	−0.285031	−0.142516	+	0.989793i	$0.545519\pi$
$20$	0	0
$21$	−199473.	−0.223819
$22$	0	0
$23$	1.43017e6	1.06565	0.532823	−	0.846227i	$-0.321131\pi$
$23$	1.43017e6	1.06565	0.532823	+	0.846227i	$0.321131\pi$
$24$	0	0
$25$	−1.37503e6	−0.704013
$26$	0	0
$27$	−904906.	−0.327692
$28$	0	0
$29$	5.84621e6	1.53491	0.767456	−	0.641102i	$-0.221522\pi$
$29$	5.84621e6	1.53491	0.767456	+	0.641102i	$0.221522\pi$
$30$	0	0
$31$	−962113.	−0.187111	−0.0935553	−	0.995614i	$-0.529823\pi$
$31$	−962113.	−0.187111	−0.0935553	+	0.995614i	$0.529823\pi$
$32$	0	0
$33$	−1.70103e6	−0.249688
$34$	0	0
$35$	6.50681e6	0.732929
$36$	0	0
$37$	−3.09958e6	−0.271892	−0.135946	−	0.990716i	$-0.543407\pi$
$37$	−3.09958e6	−0.271892	−0.135946	+	0.990716i	$0.543407\pi$
$38$	0	0
$39$	665719.	0.0460787
$40$	0	0
$41$	−1.35817e7	−0.750634	−0.375317	−	0.926896i	$-0.622466\pi$
$41$	−1.35817e7	−0.750634	−0.375317	+	0.926896i	$0.622466\pi$
$42$	0	0
$43$	1.14462e7	0.510568	0.255284	−	0.966866i	$-0.417831\pi$
$43$	1.14462e7	0.510568	0.255284	+	0.966866i	$0.417831\pi$
$44$	0	0
$45$	1.45525e7	0.529030
$46$	0	0
$47$	−2.29931e6	−0.0687316	−0.0343658	−	0.999409i	$-0.510941\pi$
$47$	−2.29931e6	−0.0687316	−0.0343658	+	0.999409i	$0.510941\pi$
$48$	0	0
$49$	3.28838e7	0.814892
$50$	0	0
$51$	6.21629e6	0.128666
$52$	0	0
$53$	−3.07448e7	−0.535218	−0.267609	−	0.963528i	$-0.586234\pi$
$53$	−3.07448e7	−0.535218	−0.267609	+	0.963528i	$0.586234\pi$
$54$	0	0
$55$	5.54874e7	0.817641
$56$	0	0
$57$	−3.77399e6	−0.0473548
$58$	0	0
$59$	3.05327e7	0.328043	0.164022	−	0.986457i	$-0.447553\pi$
$59$	3.05327e7	0.328043	0.164022	+	0.986457i	$0.447553\pi$
$60$	0	0
$61$	6.53064e7	0.603909	0.301955	−	0.953322i	$-0.402361\pi$
$61$	6.53064e7	0.603909	0.301955	+	0.953322i	$0.402361\pi$
$62$	0	0
$63$	1.63795e8	1.30999
$64$	0	0
$65$	−2.17158e7	−0.150891
$66$	0	0
$67$	−1.81368e8	−1.09957	−0.549787	−	0.835305i	$-0.685291\pi$
$67$	−1.81368e8	−1.09957	−0.549787	+	0.835305i	$0.685291\pi$
$68$	0	0
$69$	3.33354e7	0.177045
$70$	0	0
$71$	−1.23157e8	−0.575172	−0.287586	−	0.957755i	$-0.592853\pi$
$71$	−1.23157e8	−0.575172	−0.287586	+	0.957755i	$0.592853\pi$
$72$	0	0
$73$	4.31054e8	1.77655	0.888277	−	0.459308i	$-0.151903\pi$
$73$	4.31054e8	1.77655	0.888277	+	0.459308i	$0.151903\pi$
$74$	0	0
$75$	−3.20500e7	−0.116964
$76$	0	0
$77$	6.24539e8	2.02466
$78$	0	0
$79$	4.22314e8	1.21987	0.609935	−	0.792451i	$-0.291195\pi$
$79$	4.22314e8	1.21987	0.609935	+	0.792451i	$0.291195\pi$
$80$	0	0
$81$	3.55635e8	0.917955
$82$	0	0
$83$	1.40247e8	0.324370	0.162185	−	0.986760i	$-0.448146\pi$
$83$	1.40247e8	0.324370	0.162185	+	0.986760i	$0.448146\pi$
$84$	0	0
$85$	−2.02775e8	−0.421337
$86$	0	0
$87$	1.36267e8	0.255009
$88$	0	0
$89$	−5.82829e7	−0.0984659	−0.0492330	−	0.998787i	$-0.515678\pi$
$89$	−5.82829e7	−0.0984659	−0.0492330	+	0.998787i	$0.515678\pi$
$90$	0	0
$91$	−2.44422e8	−0.373640
$92$	0	0
$93$	−2.24256e7	−0.0310864
$94$	0	0
$95$	1.23108e8	0.155070
$96$	0	0
$97$	5.66836e8	0.650107	0.325054	−	0.945696i	$-0.394618\pi$
$97$	5.66836e8	0.650107	0.325054	+	0.945696i	$0.394618\pi$
$98$	0	0
$99$	1.39678e9	1.46140
$100$	0	0
$101$	−9.25870e8	−0.885327	−0.442663	−	0.896688i	$-0.645966\pi$
$101$	−9.25870e8	−0.885327	−0.442663	+	0.896688i	$0.645966\pi$
$102$	0	0
$103$	−5.98439e8	−0.523905	−0.261952	−	0.965081i	$-0.584366\pi$
$103$	−5.98439e8	−0.523905	−0.261952	+	0.965081i	$0.584366\pi$
$104$	0	0
$105$	1.51665e8	0.121768
$106$	0	0
$107$	1.70825e9	1.25987	0.629933	−	0.776649i	$-0.283082\pi$
$107$	1.70825e9	1.25987	0.629933	+	0.776649i	$0.283082\pi$
$108$	0	0
$109$	−1.25896e9	−0.854264	−0.427132	−	0.904189i	$-0.640476\pi$
$109$	−1.25896e9	−0.854264	−0.427132	+	0.904189i	$0.640476\pi$
$110$	0	0
$111$	−7.22472e7	−0.0451718
$112$	0	0
$113$	3.28543e9	1.89557	0.947783	−	0.318916i	$-0.103319\pi$
$113$	3.28543e9	1.89557	0.947783	+	0.318916i	$0.103319\pi$
$114$	0	0
$115$	−1.08740e9	−0.579761
$116$	0	0
$117$	−5.46649e8	−0.269695
$118$	0	0
$119$	−2.28234e9	−1.04332
$120$	0	0
$121$	2.96787e9	1.25867
$122$	0	0
$123$	−3.16573e8	−0.124710
$124$	0	0
$125$	2.53049e9	0.927063
$126$	0	0
$127$	−4.05548e9	−1.38333	−0.691665	−	0.722218i	$-0.743123\pi$
$127$	−4.05548e9	−1.38333	−0.691665	+	0.722218i	$0.743123\pi$
$128$	0	0
$129$	2.66796e8	0.0848252
$130$	0	0
$131$	−3.77943e9	−1.12126	−0.560629	−	0.828067i	$-0.689441\pi$
$131$	−3.77943e9	−1.12126	−0.560629	+	0.828067i	$0.689441\pi$
$132$	0	0
$133$	1.38564e9	0.383988
$134$	0	0
$135$	6.88026e8	0.178280
$136$	0	0
$137$	4.81328e9	1.16734	0.583672	−	0.811990i	$-0.301616\pi$
$137$	4.81328e9	1.16734	0.583672	+	0.811990i	$0.301616\pi$
$138$	0	0
$139$	−4.46497e9	−1.01450	−0.507250	−	0.861799i	$-0.669338\pi$
$139$	−4.46497e9	−1.01450	−0.507250	+	0.861799i	$0.669338\pi$
$140$	0	0
$141$	−5.35938e7	−0.0114190
$142$	0	0
$143$	−2.08433e9	−0.416826
$144$	0	0
$145$	−4.44504e9	−0.835064
$146$	0	0
$147$	7.66478e8	0.135385
$148$	0	0
$149$	−6.05916e9	−1.00710	−0.503552	−	0.863965i	$-0.667974\pi$
$149$	−6.05916e9	−1.00710	−0.503552	+	0.863965i	$0.667974\pi$
$150$	0	0
$151$	−1.18321e10	−1.85211	−0.926054	−	0.377391i	$-0.876821\pi$
$151$	−1.18321e10	−1.85211	−0.926054	+	0.377391i	$0.876821\pi$
$152$	0	0
$153$	−5.10445e9	−0.753074
$154$	0	0
$155$	7.31522e8	0.101797
$156$	0	0
$157$	5.90973e9	0.776281	0.388140	−	0.921600i	$-0.373118\pi$
$157$	5.90973e9	0.776281	0.388140	+	0.921600i	$0.373118\pi$
$158$	0	0
$159$	−7.16621e8	−0.0889206
$160$	0	0
$161$	−1.22392e10	−1.43562
$162$	0	0
$163$	5.52891e9	0.613473	0.306736	−	0.951795i	$-0.400763\pi$
$163$	5.52891e9	0.613473	0.306736	+	0.951795i	$0.400763\pi$
$164$	0	0
$165$	1.29334e9	0.135842
$166$	0	0
$167$	−9.99601e9	−0.994494	−0.497247	−	0.867609i	$-0.665656\pi$
$167$	−9.99601e9	−0.994494	−0.497247	+	0.867609i	$0.665656\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	3.09898e9	0.277164
$172$	0	0
$173$	9.33064e9	0.791961	0.395980	−	0.918259i	$-0.370405\pi$
$173$	9.33064e9	0.791961	0.395980	+	0.918259i	$0.370405\pi$
$174$	0	0
$175$	1.17673e10	0.948431
$176$	0	0
$177$	7.11676e8	0.0545008
$178$	0	0
$179$	−1.22801e10	−0.894051	−0.447026	−	0.894521i	$-0.647517\pi$
$179$	−1.22801e10	−0.894051	−0.447026	+	0.894521i	$0.647517\pi$
$180$	0	0
$181$	1.05153e10	0.728227	0.364114	−	0.931355i	$-0.381372\pi$
$181$	1.05153e10	0.728227	0.364114	+	0.931355i	$0.381372\pi$
$182$	0	0
$183$	1.52221e9	0.100333
$184$	0	0
$185$	2.35670e9	0.147922
$186$	0	0
$187$	−1.94629e10	−1.16391
$188$	0	0
$189$	7.74408e9	0.441460
$190$	0	0
$191$	−3.37965e10	−1.83748	−0.918739	−	0.394866i	$-0.870791\pi$
$191$	−3.37965e10	−1.83748	−0.918739	+	0.394866i	$0.870791\pi$
$192$	0	0
$193$	−1.31326e10	−0.681306	−0.340653	−	0.940189i	$-0.610648\pi$
$193$	−1.31326e10	−0.681306	−0.340653	+	0.940189i	$0.610648\pi$
$194$	0	0
$195$	−5.06165e8	−0.0250690
$196$	0	0
$197$	5.65962e9	0.267725	0.133863	−	0.991000i	$-0.457262\pi$
$197$	5.65962e9	0.267725	0.133863	+	0.991000i	$0.457262\pi$
$198$	0	0
$199$	1.44927e10	0.655102	0.327551	−	0.944833i	$-0.393777\pi$
$199$	1.44927e10	0.655102	0.327551	+	0.944833i	$0.393777\pi$
$200$	0	0
$201$	−4.22745e9	−0.182682
$202$	0	0
$203$	−5.00312e10	−2.06780
$204$	0	0
$205$	1.03266e10	0.408380
$206$	0	0
$207$	−2.73731e10	−1.03623
$208$	0	0
$209$	1.18162e10	0.428369
$210$	0	0
$211$	2.59902e10	0.902689	0.451345	−	0.892350i	$-0.350945\pi$
$211$	2.59902e10	0.902689	0.451345	+	0.892350i	$0.350945\pi$
$212$	0	0
$213$	−2.87063e9	−0.0955586
$214$	0	0
$215$	−8.70287e9	−0.277773
$216$	0	0
$217$	8.23365e9	0.252071
$218$	0	0
$219$	1.00473e10	0.295155
$220$	0	0
$221$	7.61705e9	0.214794
$222$	0	0
$223$	6.57125e10	1.77941	0.889706	−	0.456535i	$-0.150910\pi$
$223$	6.57125e10	1.77941	0.889706	+	0.456535i	$0.150910\pi$
$224$	0	0
$225$	2.63176e10	0.684581
$226$	0	0
$227$	4.49432e10	1.12343	0.561716	−	0.827330i	$-0.310141\pi$
$227$	4.49432e10	1.12343	0.561716	+	0.827330i	$0.310141\pi$
$228$	0	0
$229$	3.42551e10	0.823124	0.411562	−	0.911382i	$-0.364983\pi$
$229$	3.42551e10	0.823124	0.411562	+	0.911382i	$0.364983\pi$
$230$	0	0
$231$	1.45572e10	0.336375
$232$	0	0
$233$	5.33250e10	1.18530	0.592651	−	0.805459i	$-0.298081\pi$
$233$	5.33250e10	1.18530	0.592651	+	0.805459i	$0.298081\pi$
$234$	0	0
$235$	1.74823e9	0.0373932
$236$	0	0
$237$	9.84358e9	0.202668
$238$	0	0
$239$	3.81658e10	0.756631	0.378315	−	0.925677i	$-0.376503\pi$
$239$	3.81658e10	0.756631	0.378315	+	0.925677i	$0.376503\pi$
$240$	0	0
$241$	−1.28089e9	−0.0244587	−0.0122294	−	0.999925i	$-0.503893\pi$
$241$	−1.28089e9	−0.0244587	−0.0122294	+	0.999925i	$0.503893\pi$
$242$	0	0
$243$	2.61006e10	0.480201
$244$	0	0
$245$	−2.50025e10	−0.443339
$246$	0	0
$247$	−4.62441e9	−0.0790534
$248$	0	0
$249$	3.26896e9	0.0538906
$250$	0	0
$251$	−8.97468e10	−1.42721	−0.713604	−	0.700549i	$-0.752938\pi$
$251$	−8.97468e10	−1.42721	−0.713604	+	0.700549i	$0.752938\pi$
$252$	0	0
$253$	−1.04371e11	−1.60154
$254$	0	0
$255$	−4.72642e9	−0.0700006
$256$	0	0
$257$	1.03761e11	1.48366	0.741829	−	0.670589i	$-0.233959\pi$
$257$	1.03761e11	1.48366	0.741829	+	0.670589i	$0.233959\pi$
$258$	0	0
$259$	2.65259e10	0.366287
$260$	0	0
$261$	−1.11895e11	−1.49254
$262$	0	0
$263$	9.31045e9	0.119997	0.0599984	−	0.998198i	$-0.480890\pi$
$263$	9.31045e9	0.119997	0.0599984	+	0.998198i	$0.480890\pi$
$264$	0	0
$265$	2.33762e10	0.291184
$266$	0	0
$267$	−1.35850e9	−0.0163590
$268$	0	0
$269$	1.67871e11	1.95475	0.977374	−	0.211520i	$-0.0678413\pi$
$269$	1.67871e11	1.95475	0.977374	+	0.211520i	$0.0678413\pi$
$270$	0	0
$271$	4.68715e10	0.527894	0.263947	−	0.964537i	$-0.414976\pi$
$271$	4.68715e10	0.527894	0.263947	+	0.964537i	$0.414976\pi$
$272$	0	0
$273$	−5.69715e9	−0.0620763
$274$	0	0
$275$	1.00347e11	1.05805
$276$	0	0
$277$	−9.62593e9	−0.0982389	−0.0491195	−	0.998793i	$-0.515642\pi$
$277$	−9.62593e9	−0.0982389	−0.0491195	+	0.998793i	$0.515642\pi$
$278$	0	0
$279$	1.84146e10	0.181946
$280$	0	0
$281$	−1.85107e10	−0.177111	−0.0885553	−	0.996071i	$-0.528225\pi$
$281$	−1.85107e10	−0.177111	−0.0885553	+	0.996071i	$0.528225\pi$
$282$	0	0
$283$	−2.04503e10	−0.189523	−0.0947613	−	0.995500i	$-0.530209\pi$
$283$	−2.04503e10	−0.189523	−0.0947613	+	0.995500i	$0.530209\pi$
$284$	0	0
$285$	2.86947e9	0.0257632
$286$	0	0
$287$	1.16231e11	1.01124
$288$	0	0
$289$	−4.74621e10	−0.400227
$290$	0	0
$291$	1.32122e10	0.108008
$292$	0	0
$293$	−2.17741e11	−1.72598	−0.862990	−	0.505220i	$-0.831411\pi$
$293$	−2.17741e11	−1.72598	−0.862990	+	0.505220i	$0.831411\pi$
$294$	0	0
$295$	−2.32149e10	−0.178471
$296$	0	0
$297$	6.60384e10	0.492484
$298$	0	0
$299$	4.08471e10	0.295557
$300$	0	0
$301$	−9.79553e10	−0.687826
$302$	0	0
$303$	−2.15808e10	−0.147087
$304$	0	0
$305$	−4.96544e10	−0.328555
$306$	0	0
$307$	2.42492e11	1.55802	0.779012	−	0.627009i	$-0.215721\pi$
$307$	2.42492e11	1.55802	0.779012	+	0.627009i	$0.215721\pi$
$308$	0	0
$309$	−1.39488e10	−0.0870410
$310$	0	0
$311$	3.92653e10	0.238005	0.119003	−	0.992894i	$-0.462030\pi$
$311$	3.92653e10	0.238005	0.119003	+	0.992894i	$0.462030\pi$
$312$	0	0
$313$	−2.32562e11	−1.36958	−0.684792	−	0.728738i	$-0.740107\pi$
$313$	−2.32562e11	−1.36958	−0.684792	+	0.728738i	$0.740107\pi$
$314$	0	0
$315$	−1.24538e11	−0.712698
$316$	0	0
$317$	−9.44806e10	−0.525504	−0.262752	−	0.964863i	$-0.584630\pi$
$317$	−9.44806e10	−0.525504	−0.262752	+	0.964863i	$0.584630\pi$
$318$	0	0
$319$	−4.26646e11	−2.30680
$320$	0	0
$321$	3.98171e10	0.209313
$322$	0	0
$323$	−4.31814e10	−0.220742
$324$	0	0
$325$	−3.92721e10	−0.195258
$326$	0	0
$327$	−2.93446e10	−0.141927
$328$	0	0
$329$	1.96772e10	0.0925938
$330$	0	0
$331$	3.11905e11	1.42822	0.714111	−	0.700032i	$-0.246831\pi$
$331$	3.11905e11	1.42822	0.714111	+	0.700032i	$0.246831\pi$
$332$	0	0
$333$	5.93251e10	0.264387
$334$	0	0
$335$	1.37899e11	0.598220
$336$	0	0
$337$	−3.56759e11	−1.50675	−0.753375	−	0.657592i	$-0.771575\pi$
$337$	−3.56759e11	−1.50675	−0.753375	+	0.657592i	$0.771575\pi$
$338$	0	0
$339$	7.65790e10	0.314928
$340$	0	0
$341$	7.02133e10	0.281206
$342$	0	0
$343$	6.39256e10	0.249374
$344$	0	0
$345$	−2.53459e10	−0.0963211
$346$	0	0
$347$	−2.88336e11	−1.06762	−0.533809	−	0.845605i	$-0.679240\pi$
$347$	−2.88336e11	−1.06762	−0.533809	+	0.845605i	$0.679240\pi$
$348$	0	0
$349$	2.28837e11	0.825680	0.412840	−	0.910803i	$-0.364537\pi$
$349$	2.28837e11	0.825680	0.412840	+	0.910803i	$0.364537\pi$
$350$	0	0
$351$	−2.58450e10	−0.0908855
$352$	0	0
$353$	−4.56093e11	−1.56339	−0.781695	−	0.623661i	$-0.785644\pi$
$353$	−4.56093e11	−1.56339	−0.781695	+	0.623661i	$0.785644\pi$
$354$	0	0
$355$	9.36401e10	0.312921
$356$	0	0
$357$	−5.31983e10	−0.173337
$358$	0	0
$359$	−1.33745e11	−0.424966	−0.212483	−	0.977165i	$-0.568155\pi$
$359$	−1.33745e11	−0.424966	−0.212483	+	0.977165i	$0.568155\pi$
$360$	0	0
$361$	−2.96472e11	−0.918757
$362$	0	0
$363$	6.91772e10	0.209114
$364$	0	0
$365$	−3.27742e11	−0.966529
$366$	0	0
$367$	−3.54059e11	−1.01878	−0.509388	−	0.860537i	$-0.670128\pi$
$367$	−3.54059e11	−1.01878	−0.509388	+	0.860537i	$0.670128\pi$
$368$	0	0
$369$	2.59951e11	0.729915
$370$	0	0
$371$	2.63111e11	0.721034
$372$	0	0
$373$	−7.64050e10	−0.204377	−0.102189	−	0.994765i	$-0.532585\pi$
$373$	−7.64050e10	−0.204377	−0.102189	+	0.994765i	$0.532585\pi$
$374$	0	0
$375$	5.89823e10	0.154021
$376$	0	0
$377$	1.66974e11	0.425708
$378$	0	0
$379$	2.67191e11	0.665190	0.332595	−	0.943070i	$-0.392076\pi$
$379$	2.67191e11	0.665190	0.332595	+	0.943070i	$0.392076\pi$
$380$	0	0
$381$	−9.45279e10	−0.229825
$382$	0	0
$383$	−4.56769e11	−1.08468	−0.542340	−	0.840159i	$-0.682462\pi$
$383$	−4.56769e11	−1.08468	−0.542340	+	0.840159i	$0.682462\pi$
$384$	0	0
$385$	−4.74855e11	−1.10151
$386$	0	0
$387$	−2.19077e11	−0.496475
$388$	0	0
$389$	−2.16479e10	−0.0479338	−0.0239669	−	0.999713i	$-0.507630\pi$
$389$	−2.16479e10	−0.0479338	−0.0239669	+	0.999713i	$0.507630\pi$
$390$	0	0
$391$	3.81419e11	0.825290
$392$	0	0
$393$	−8.80935e10	−0.186285
$394$	0	0
$395$	−3.21098e11	−0.663667
$396$	0	0
$397$	8.80505e11	1.77899	0.889497	−	0.456941i	$-0.151055\pi$
$397$	8.80505e11	1.77899	0.889497	+	0.456941i	$0.151055\pi$
$398$	0	0
$399$	3.22974e10	0.0637954
$400$	0	0
$401$	4.42207e10	0.0854035	0.0427018	−	0.999088i	$-0.486403\pi$
$401$	4.42207e10	0.0854035	0.0427018	+	0.999088i	$0.486403\pi$
$402$	0	0
$403$	−2.74789e10	−0.0518951
$404$	0	0
$405$	−2.70399e11	−0.499411
$406$	0	0
$407$	2.26202e11	0.408622
$408$	0	0
$409$	7.71666e11	1.36356	0.681780	−	0.731557i	$-0.261206\pi$
$409$	7.71666e11	1.36356	0.681780	+	0.731557i	$0.261206\pi$
$410$	0	0
$411$	1.12191e11	0.193941
$412$	0	0
$413$	−2.61295e11	−0.441933
$414$	0	0
$415$	−1.06634e11	−0.176473
$416$	0	0
$417$	−1.04073e11	−0.168548
$418$	0	0
$419$	5.30237e11	0.840441	0.420220	−	0.907422i	$-0.361953\pi$
$419$	5.30237e11	0.840441	0.420220	+	0.907422i	$0.361953\pi$
$420$	0	0
$421$	7.88979e11	1.22404	0.612021	−	0.790842i	$-0.290357\pi$
$421$	7.88979e11	1.22404	0.612021	+	0.790842i	$0.290357\pi$
$422$	0	0
$423$	4.40081e10	0.0668345
$424$	0	0
$425$	−3.66711e11	−0.545223
$426$	0	0
$427$	−5.58885e11	−0.813574
$428$	0	0
$429$	−4.85830e10	−0.0692511
$430$	0	0
$431$	−3.60743e11	−0.503558	−0.251779	−	0.967785i	$-0.581016\pi$
$431$	−3.60743e11	−0.503558	−0.251779	+	0.967785i	$0.581016\pi$
$432$	0	0
$433$	1.42994e12	1.95489	0.977446	−	0.211187i	$-0.0677330\pi$
$433$	1.42994e12	1.95489	0.977446	+	0.211187i	$0.0677330\pi$
$434$	0	0
$435$	−1.03608e11	−0.138737
$436$	0	0
$437$	−2.31564e11	−0.303742
$438$	0	0
$439$	−1.04184e11	−0.133878	−0.0669391	−	0.997757i	$-0.521323\pi$
$439$	−1.04184e11	−0.133878	−0.0669391	+	0.997757i	$0.521323\pi$
$440$	0	0
$441$	−6.29387e11	−0.792399
$442$	0	0
$443$	1.51201e12	1.86526	0.932629	−	0.360838i	$-0.117509\pi$
$443$	1.51201e12	1.86526	0.932629	+	0.360838i	$0.117509\pi$
$444$	0	0
$445$	4.43141e10	0.0535701
$446$	0	0
$447$	−1.41231e11	−0.167319
$448$	0	0
$449$	7.12255e11	0.827041	0.413520	−	0.910495i	$-0.364299\pi$
$449$	7.12255e11	0.827041	0.413520	+	0.910495i	$0.364299\pi$
$450$	0	0
$451$	9.91172e11	1.12812
$452$	0	0
$453$	−2.75791e11	−0.307708
$454$	0	0
$455$	1.85841e11	0.203278
$456$	0	0
$457$	6.97074e11	0.747577	0.373788	−	0.927514i	$-0.378059\pi$
$457$	6.97074e11	0.747577	0.373788	+	0.927514i	$0.378059\pi$
$458$	0	0
$459$	−2.41333e11	−0.253781
$460$	0	0
$461$	2.70683e11	0.279130	0.139565	−	0.990213i	$-0.455430\pi$
$461$	2.70683e11	0.279130	0.139565	+	0.990213i	$0.455430\pi$
$462$	0	0
$463$	−9.93801e10	−0.100504	−0.0502522	−	0.998737i	$-0.516003\pi$
$463$	−9.93801e10	−0.100504	−0.0502522	+	0.998737i	$0.516003\pi$
$464$	0	0
$465$	1.70508e10	0.0169125
$466$	0	0
$467$	3.19076e10	0.0310433	0.0155216	−	0.999880i	$-0.495059\pi$
$467$	3.19076e10	0.0310433	0.0155216	+	0.999880i	$0.495059\pi$
$468$	0	0
$469$	1.55213e12	1.48132
$470$	0	0
$471$	1.37748e11	0.128971
$472$	0	0
$473$	−8.35323e11	−0.767325
$474$	0	0
$475$	2.22635e11	0.200666
$476$	0	0
$477$	5.88447e11	0.520445
$478$	0	0
$479$	−1.70914e12	−1.48344	−0.741718	−	0.670712i	$-0.765989\pi$
$479$	−1.70914e12	−1.48344	−0.741718	+	0.670712i	$0.765989\pi$
$480$	0	0
$481$	−8.85272e10	−0.0754092
$482$	0	0
$483$	−2.85281e11	−0.238512
$484$	0	0
$485$	−4.30982e11	−0.353689
$486$	0	0
$487$	3.60245e11	0.290213	0.145107	−	0.989416i	$-0.453647\pi$
$487$	3.60245e11	0.290213	0.145107	+	0.989416i	$0.453647\pi$
$488$	0	0
$489$	1.28872e11	0.101922
$490$	0	0
$491$	−5.61754e11	−0.436194	−0.218097	−	0.975927i	$-0.569985\pi$
$491$	−5.61754e11	−0.436194	−0.218097	+	0.975927i	$0.569985\pi$
$492$	0	0
$493$	1.55915e12	1.18871
$494$	0	0
$495$	−1.06201e12	−0.795072
$496$	0	0
$497$	1.05397e12	0.774860
$498$	0	0
$499$	8.84030e11	0.638285	0.319143	−	0.947707i	$-0.396605\pi$
$499$	8.84030e11	0.638285	0.319143	+	0.947707i	$0.396605\pi$
$500$	0	0
$501$	−2.32994e11	−0.165224
$502$	0	0
$503$	−1.62861e11	−0.113439	−0.0567195	−	0.998390i	$-0.518064\pi$
$503$	−1.62861e11	−0.113439	−0.0567195	+	0.998390i	$0.518064\pi$
$504$	0	0
$505$	7.03965e11	0.481659
$506$	0	0
$507$	1.90136e10	0.0127799
$508$	0	0
$509$	−2.29095e11	−0.151281	−0.0756407	−	0.997135i	$-0.524100\pi$
$509$	−2.29095e11	−0.151281	−0.0756407	+	0.997135i	$0.524100\pi$
$510$	0	0
$511$	−3.68891e12	−2.39334
$512$	0	0
$513$	1.46517e11	0.0934025
$514$	0	0
$515$	4.55010e11	0.285029
$516$	0	0
$517$	1.67799e11	0.103296
$518$	0	0
$519$	2.17485e11	0.131576
$520$	0	0
$521$	2.26430e12	1.34637	0.673184	−	0.739475i	$-0.264926\pi$
$521$	2.26430e12	1.34637	0.673184	+	0.739475i	$0.264926\pi$
$522$	0	0
$523$	3.34805e12	1.95674	0.978371	−	0.206856i	$-0.0663231\pi$
$523$	3.34805e12	1.95674	0.978371	+	0.206856i	$0.0663231\pi$
$524$	0	0
$525$	2.74280e11	0.157572
$526$	0	0
$527$	−2.56590e11	−0.144908
$528$	0	0
$529$	2.44239e11	0.135601
$530$	0	0
$531$	−5.84387e11	−0.318988
$532$	0	0
$533$	−3.87908e11	−0.208188
$534$	0	0
$535$	−1.29883e12	−0.685427
$536$	0	0
$537$	−2.86232e11	−0.148537
$538$	0	0
$539$	−2.39980e12	−1.22469
$540$	0	0
$541$	−5.11816e11	−0.256877	−0.128439	−	0.991717i	$-0.540997\pi$
$541$	−5.11816e11	−0.256877	−0.128439	+	0.991717i	$0.540997\pi$
$542$	0	0
$543$	2.45097e11	0.120987
$544$	0	0
$545$	9.57222e11	0.464760
$546$	0	0
$547$	−1.95245e12	−0.932474	−0.466237	−	0.884660i	$-0.654391\pi$
$547$	−1.95245e12	−0.932474	−0.466237	+	0.884660i	$0.654391\pi$
$548$	0	0
$549$	−1.24995e12	−0.587240
$550$	0	0
$551$	−9.46581e11	−0.437497
$552$	0	0
$553$	−3.61412e12	−1.64338
$554$	0	0
$555$	5.49316e10	0.0245756
$556$	0	0
$557$	9.52980e10	0.0419503	0.0209752	−	0.999780i	$-0.493323\pi$
$557$	9.52980e10	0.0419503	0.0209752	+	0.999780i	$0.493323\pi$
$558$	0	0
$559$	3.26915e11	0.141606
$560$	0	0
$561$	−4.53654e11	−0.193371
$562$	0	0
$563$	1.40128e12	0.587812	0.293906	−	0.955834i	$-0.405045\pi$
$563$	1.40128e12	0.587812	0.293906	+	0.955834i	$0.405045\pi$
$564$	0	0
$565$	−2.49801e12	−1.03128
$566$	0	0
$567$	−3.04348e12	−1.23665
$568$	0	0
$569$	−2.77647e12	−1.11042	−0.555211	−	0.831709i	$-0.687363\pi$
$569$	−2.77647e12	−1.11042	−0.555211	+	0.831709i	$0.687363\pi$
$570$	0	0
$571$	1.78010e12	0.700779	0.350389	−	0.936604i	$-0.386049\pi$
$571$	1.78010e12	0.700779	0.350389	+	0.936604i	$0.386049\pi$
$572$	0	0
$573$	−7.87752e11	−0.305277
$574$	0	0
$575$	−1.96652e12	−0.750228
$576$	0	0
$577$	−2.97560e10	−0.0111759	−0.00558796	−	0.999984i	$-0.501779\pi$
$577$	−2.97560e10	−0.0111759	−0.00558796	+	0.999984i	$0.501779\pi$
$578$	0	0
$579$	−3.06103e11	−0.113192
$580$	0	0
$581$	−1.20022e12	−0.436985
$582$	0	0
$583$	2.24370e12	0.804372
$584$	0	0
$585$	4.15633e11	0.146727
$586$	0	0
$587$	−2.59430e12	−0.901879	−0.450940	−	0.892554i	$-0.648911\pi$
$587$	−2.59430e12	−0.901879	−0.450940	+	0.892554i	$0.648911\pi$
$588$	0	0
$589$	1.55779e11	0.0533323
$590$	0	0
$591$	1.31918e11	0.0444796
$592$	0	0
$593$	2.65204e12	0.880713	0.440356	−	0.897823i	$-0.354852\pi$
$593$	2.65204e12	0.880713	0.440356	+	0.897823i	$0.354852\pi$
$594$	0	0
$595$	1.73533e12	0.567617
$596$	0	0
$597$	3.37805e11	0.108838
$598$	0	0
$599$	−4.87808e12	−1.54821	−0.774103	−	0.633060i	$-0.781799\pi$
$599$	−4.87808e12	−1.54821	−0.774103	+	0.633060i	$0.781799\pi$
$600$	0	0
$601$	−1.99353e12	−0.623287	−0.311643	−	0.950199i	$-0.600879\pi$
$601$	−1.99353e12	−0.623287	−0.311643	+	0.950199i	$0.600879\pi$
$602$	0	0
$603$	3.47133e12	1.06922
$604$	0	0
$605$	−2.25656e12	−0.684774
$606$	0	0
$607$	5.03532e12	1.50549	0.752745	−	0.658312i	$-0.228729\pi$
$607$	5.03532e12	1.50549	0.752745	+	0.658312i	$0.228729\pi$
$608$	0	0
$609$	−1.16616e12	−0.343543
$610$	0	0
$611$	−6.56705e10	−0.0190627
$612$	0	0
$613$	3.95315e12	1.13076	0.565381	−	0.824830i	$-0.308729\pi$
$613$	3.95315e12	1.13076	0.565381	+	0.824830i	$0.308729\pi$
$614$	0	0
$615$	2.40699e11	0.0678479
$616$	0	0
$617$	1.59226e12	0.442313	0.221156	−	0.975238i	$-0.429017\pi$
$617$	1.59226e12	0.442313	0.221156	+	0.975238i	$0.429017\pi$
$618$	0	0
$619$	−3.02552e12	−0.828310	−0.414155	−	0.910206i	$-0.635923\pi$
$619$	−3.02552e12	−0.828310	−0.414155	+	0.910206i	$0.635923\pi$
$620$	0	0
$621$	−1.29417e12	−0.349204
$622$	0	0
$623$	4.98778e11	0.132651
$624$	0	0
$625$	7.61593e11	0.199647
$626$	0	0
$627$	2.75419e11	0.0711689
$628$	0	0
$629$	−8.26641e11	−0.210566
$630$	0	0
$631$	−4.22075e12	−1.05988	−0.529942	−	0.848034i	$-0.677786\pi$
$631$	−4.22075e12	−1.05988	−0.529942	+	0.848034i	$0.677786\pi$
$632$	0	0
$633$	6.05797e11	0.149972
$634$	0	0
$635$	3.08350e12	0.752597
$636$	0	0
$637$	9.39195e11	0.226010
$638$	0	0
$639$	2.35720e12	0.559296
$640$	0	0
$641$	4.24400e12	0.992920	0.496460	−	0.868060i	$-0.334633\pi$
$641$	4.24400e12	0.992920	0.496460	+	0.868060i	$0.334633\pi$
$642$	0	0
$643$	6.57522e11	0.151691	0.0758456	−	0.997120i	$-0.475834\pi$
$643$	6.57522e11	0.151691	0.0758456	+	0.997120i	$0.475834\pi$
$644$	0	0
$645$	−2.02852e11	−0.0461489
$646$	0	0
$647$	−4.83473e12	−1.08468	−0.542341	−	0.840158i	$-0.682462\pi$
$647$	−4.83473e12	−1.08468	−0.542341	+	0.840158i	$0.682462\pi$
$648$	0	0
$649$	−2.22822e12	−0.493011
$650$	0	0
$651$	1.91915e11	0.0418789
$652$	0	0
$653$	−7.26539e12	−1.56369	−0.781843	−	0.623475i	$-0.785720\pi$
$653$	−7.26539e12	−1.56369	−0.781843	+	0.623475i	$0.785720\pi$
$654$	0	0
$655$	2.87361e12	0.610017
$656$	0	0
$657$	−8.25024e12	−1.72752
$658$	0	0
$659$	−5.36348e11	−0.110780	−0.0553901	−	0.998465i	$-0.517640\pi$
$659$	−5.36348e11	−0.110780	−0.0553901	+	0.998465i	$0.517640\pi$
$660$	0	0
$661$	6.77323e12	1.38003	0.690016	−	0.723794i	$-0.257604\pi$
$661$	6.77323e12	1.38003	0.690016	+	0.723794i	$0.257604\pi$
$662$	0	0
$663$	1.77543e11	0.0356857
$664$	0	0
$665$	−1.05354e12	−0.208907
$666$	0	0
$667$	8.36108e12	1.63567
$668$	0	0
$669$	1.53167e12	0.295630
$670$	0	0
$671$	−4.76595e12	−0.907607
$672$	0	0
$673$	−1.08002e12	−0.202938	−0.101469	−	0.994839i	$-0.532354\pi$
$673$	−1.08002e12	−0.202938	−0.101469	+	0.994839i	$0.532354\pi$
$674$	0	0
$675$	1.24427e12	0.230700
$676$	0	0
$677$	3.29260e12	0.602406	0.301203	−	0.953560i	$-0.402612\pi$
$677$	3.29260e12	0.602406	0.301203	+	0.953560i	$0.402612\pi$
$678$	0	0
$679$	−4.85092e12	−0.875811
$680$	0	0
$681$	1.04757e12	0.186646
$682$	0	0
$683$	7.53068e12	1.32416	0.662080	−	0.749433i	$-0.269674\pi$
$683$	7.53068e12	1.32416	0.662080	+	0.749433i	$0.269674\pi$
$684$	0	0
$685$	−3.65967e12	−0.635089
$686$	0	0
$687$	7.98440e11	0.136753
$688$	0	0
$689$	−8.78103e11	−0.148443
$690$	0	0
$691$	1.05073e13	1.75323	0.876613	−	0.481196i	$-0.159797\pi$
$691$	1.05073e13	1.75323	0.876613	+	0.481196i	$0.159797\pi$
$692$	0	0
$693$	−1.19535e13	−1.96877
$694$	0	0
$695$	3.39485e12	0.551936
$696$	0	0
$697$	−3.62217e12	−0.581329
$698$	0	0
$699$	1.24293e12	0.196925
$700$	0	0
$701$	1.05023e13	1.64268	0.821340	−	0.570439i	$-0.193227\pi$
$701$	1.05023e13	1.64268	0.821340	+	0.570439i	$0.193227\pi$
$702$	0	0
$703$	5.01865e11	0.0774975
$704$	0	0
$705$	4.07489e10	0.00621248
$706$	0	0
$707$	7.92349e12	1.19269
$708$	0	0
$709$	6.60507e12	0.981678	0.490839	−	0.871250i	$-0.336690\pi$
$709$	6.60507e12	0.981678	0.490839	+	0.871250i	$0.336690\pi$
$710$	0	0
$711$	−8.08297e12	−1.18620
$712$	0	0
$713$	−1.37599e12	−0.199394
$714$	0	0
$715$	1.58478e12	0.226773
$716$	0	0
$717$	8.89594e11	0.125706
$718$	0	0
$719$	9.15000e12	1.27685	0.638427	−	0.769683i	$-0.279585\pi$
$719$	9.15000e12	1.27685	0.638427	+	0.769683i	$0.279585\pi$
$720$	0	0
$721$	5.12137e12	0.705793
$722$	0	0
$723$	−2.98558e10	−0.00406355
$724$	0	0
$725$	−8.03868e12	−1.08060
$726$	0	0
$727$	1.14659e13	1.52232	0.761158	−	0.648567i	$-0.224631\pi$
$727$	1.14659e13	1.52232	0.761158	+	0.648567i	$0.224631\pi$
$728$	0	0
$729$	−6.39159e12	−0.838175
$730$	0	0
$731$	3.05264e12	0.395409
$732$	0	0
$733$	1.05626e13	1.35145	0.675727	−	0.737152i	$-0.263830\pi$
$733$	1.05626e13	1.35145	0.675727	+	0.737152i	$0.263830\pi$
$734$	0	0
$735$	−5.82775e11	−0.0736560
$736$	0	0
$737$	1.32359e13	1.65253
$738$	0	0
$739$	1.05234e13	1.29795	0.648973	−	0.760811i	$-0.275199\pi$
$739$	1.05234e13	1.29795	0.648973	+	0.760811i	$0.275199\pi$
$740$	0	0
$741$	−1.07789e11	−0.0131339
$742$	0	0
$743$	−1.15961e13	−1.39592	−0.697961	−	0.716136i	$-0.745909\pi$
$743$	−1.15961e13	−1.39592	−0.697961	+	0.716136i	$0.745909\pi$
$744$	0	0
$745$	4.60695e12	0.547912
$746$	0	0
$747$	−2.68428e12	−0.315417
$748$	0	0
$749$	−1.46190e13	−1.69727
$750$	0	0
$751$	−6.47624e11	−0.0742922	−0.0371461	−	0.999310i	$-0.511827\pi$
$751$	−6.47624e11	−0.0742922	−0.0371461	+	0.999310i	$0.511827\pi$
$752$	0	0
$753$	−2.09188e12	−0.237115
$754$	0	0
$755$	8.99630e12	1.00763
$756$	0	0
$757$	9.55339e12	1.05737	0.528684	−	0.848819i	$-0.322686\pi$
$757$	9.55339e12	1.05737	0.528684	+	0.848819i	$0.322686\pi$
$758$	0	0
$759$	−2.43276e12	−0.266079
$760$	0	0
$761$	−1.16939e13	−1.26395	−0.631974	−	0.774989i	$-0.717755\pi$
$761$	−1.16939e13	−1.26395	−0.631974	+	0.774989i	$0.717755\pi$
$762$	0	0
$763$	1.07740e13	1.15085
$764$	0	0
$765$	3.88106e12	0.409707
$766$	0	0
$767$	8.72044e11	0.0909828
$768$	0	0
$769$	−1.01302e13	−1.04460	−0.522300	−	0.852762i	$-0.674926\pi$
$769$	−1.01302e13	−1.04460	−0.522300	+	0.852762i	$0.674926\pi$
$770$	0	0
$771$	2.41852e12	0.246494
$772$	0	0
$773$	−1.42240e13	−1.43290	−0.716448	−	0.697640i	$-0.754233\pi$
$773$	−1.42240e13	−1.43290	−0.716448	+	0.697640i	$0.754233\pi$
$774$	0	0
$775$	1.32293e12	0.131728
$776$	0	0
$777$	6.18283e11	0.0608545
$778$	0	0
$779$	2.19907e12	0.213954
$780$	0	0
$781$	8.98780e12	0.864418
$782$	0	0
$783$	−5.29027e12	−0.502979
$784$	0	0
$785$	−4.49333e12	−0.422333
$786$	0	0
$787$	−1.93779e12	−0.180062	−0.0900308	−	0.995939i	$-0.528697\pi$
$787$	−1.93779e12	−0.180062	−0.0900308	+	0.995939i	$0.528697\pi$
$788$	0	0
$789$	2.17014e11	0.0199362
$790$	0	0
$791$	−2.81163e13	−2.55367
$792$	0	0
$793$	1.86522e12	0.167494
$794$	0	0
$795$	5.44868e11	0.0483770
$796$	0	0
$797$	−9.79968e12	−0.860299	−0.430149	−	0.902758i	$-0.641539\pi$
$797$	−9.79968e12	−0.860299	−0.430149	+	0.902758i	$0.641539\pi$
$798$	0	0
$799$	−6.13212e11	−0.0532292
$800$	0	0
$801$	1.11552e12	0.0957480
$802$	0	0
$803$	−3.14575e13	−2.66996
$804$	0	0
$805$	9.30585e12	0.781043
$806$	0	0
$807$	3.91285e12	0.324760
$808$	0	0
$809$	2.37770e13	1.95159	0.975795	−	0.218685i	$-0.0701768\pi$
$809$	2.37770e13	1.95159	0.975795	+	0.218685i	$0.0701768\pi$
$810$	0	0
$811$	−2.09072e12	−0.169708	−0.0848541	−	0.996393i	$-0.527042\pi$
$811$	−2.09072e12	−0.169708	−0.0848541	+	0.996393i	$0.527042\pi$
$812$	0	0
$813$	1.09251e12	0.0877039
$814$	0	0
$815$	−4.20379e12	−0.333758
$816$	0	0
$817$	−1.85330e12	−0.145528
$818$	0	0
$819$	4.67816e12	0.363327
$820$	0	0
$821$	4.74309e11	0.0364349	0.0182175	−	0.999834i	$-0.494201\pi$
$821$	4.74309e11	0.0364349	0.0182175	+	0.999834i	$0.494201\pi$
$822$	0	0
$823$	−1.13971e13	−0.865952	−0.432976	−	0.901406i	$-0.642536\pi$
$823$	−1.13971e13	−0.865952	−0.432976	+	0.901406i	$0.642536\pi$
$824$	0	0
$825$	2.33895e12	0.175784
$826$	0	0
$827$	2.79487e12	0.207772	0.103886	−	0.994589i	$-0.466872\pi$
$827$	2.79487e12	0.207772	0.103886	+	0.994589i	$0.466872\pi$
$828$	0	0
$829$	9.07639e12	0.667449	0.333724	−	0.942671i	$-0.391695\pi$
$829$	9.07639e12	0.667449	0.333724	+	0.942671i	$0.391695\pi$
$830$	0	0
$831$	−2.24368e11	−0.0163213
$832$	0	0
$833$	8.76993e12	0.631093
$834$	0	0
$835$	7.60025e12	0.541052
$836$	0	0
$837$	8.70621e11	0.0613147
$838$	0	0
$839$	−1.92299e12	−0.133982	−0.0669912	−	0.997754i	$-0.521340\pi$
$839$	−1.92299e12	−0.133982	−0.0669912	+	0.997754i	$0.521340\pi$
$840$	0	0
$841$	1.96710e13	1.35595
$842$	0	0
$843$	−4.31460e11	−0.0294250
$844$	0	0
$845$	−6.20224e11	−0.0418498
$846$	0	0
$847$	−2.53987e13	−1.69565
$848$	0	0
$849$	−4.76670e11	−0.0314871
$850$	0	0
$851$	−4.43294e12	−0.289740
$852$	0	0
$853$	−2.41358e13	−1.56096	−0.780478	−	0.625183i	$-0.785024\pi$
$853$	−2.41358e13	−1.56096	−0.780478	+	0.625183i	$0.785024\pi$
$854$	0	0
$855$	−2.35624e12	−0.150790
$856$	0	0
$857$	9.09389e11	0.0575886	0.0287943	−	0.999585i	$-0.490833\pi$
$857$	9.09389e11	0.0575886	0.0287943	+	0.999585i	$0.490833\pi$
$858$	0	0
$859$	2.51030e13	1.57310	0.786551	−	0.617525i	$-0.211865\pi$
$859$	2.51030e13	1.57310	0.786551	+	0.617525i	$0.211865\pi$
$860$	0	0
$861$	2.70919e12	0.168006
$862$	0	0
$863$	−9.49620e11	−0.0582775	−0.0291388	−	0.999575i	$-0.509276\pi$
$863$	−9.49620e11	−0.0582775	−0.0291388	+	0.999575i	$0.509276\pi$
$864$	0	0
$865$	−7.09435e12	−0.430864
$866$	0	0
$867$	−1.10628e12	−0.0664933
$868$	0	0
$869$	−3.08197e13	−1.83333
$870$	0	0
$871$	−5.18005e12	−0.304967
$872$	0	0
$873$	−1.08491e13	−0.632163
$874$	0	0
$875$	−2.16556e13	−1.24892
$876$	0	0
$877$	−3.00749e13	−1.71674	−0.858372	−	0.513027i	$-0.828524\pi$
$877$	−3.00749e13	−1.71674	−0.858372	+	0.513027i	$0.828524\pi$
$878$	0	0
$879$	−5.07525e12	−0.286753
$880$	0	0
$881$	−3.02907e13	−1.69402	−0.847008	−	0.531581i	$-0.821598\pi$
$881$	−3.02907e13	−1.69402	−0.847008	+	0.531581i	$0.821598\pi$
$882$	0	0
$883$	1.09408e13	0.605654	0.302827	−	0.953046i	$-0.402070\pi$
$883$	1.09408e13	0.605654	0.302827	+	0.953046i	$0.402070\pi$
$884$	0	0
$885$	−5.41108e11	−0.0296510
$886$	0	0
$887$	−5.45555e12	−0.295926	−0.147963	−	0.988993i	$-0.547272\pi$
$887$	−5.45555e12	−0.295926	−0.147963	+	0.988993i	$0.547272\pi$
$888$	0	0
$889$	3.47064e13	1.86359
$890$	0	0
$891$	−2.59536e13	−1.37958
$892$	0	0
$893$	3.72289e11	0.0195906
$894$	0	0
$895$	9.33690e12	0.486406
$896$	0	0
$897$	9.52092e11	0.0491036
$898$	0	0
$899$	−5.62471e12	−0.287198
$900$	0	0
$901$	−8.19947e12	−0.414500
$902$	0	0
$903$	−2.28321e12	−0.114275
$904$	0	0
$905$	−7.99507e12	−0.396190
$906$	0	0
$907$	−3.28622e13	−1.61237	−0.806183	−	0.591666i	$-0.798470\pi$
$907$	−3.28622e13	−1.61237	−0.806183	+	0.591666i	$0.798470\pi$
$908$	0	0
$909$	1.77209e13	0.860890
$910$	0	0
$911$	3.47679e13	1.67242	0.836211	−	0.548408i	$-0.184766\pi$
$911$	3.47679e13	1.67242	0.836211	+	0.548408i	$0.184766\pi$
$912$	0	0
$913$	−1.02350e13	−0.487492
$914$	0	0
$915$	−1.15738e12	−0.0545859
$916$	0	0
$917$	3.23439e13	1.51054
$918$	0	0
$919$	2.19562e13	1.01540	0.507701	−	0.861533i	$-0.330495\pi$
$919$	2.19562e13	1.01540	0.507701	+	0.861533i	$0.330495\pi$
$920$	0	0
$921$	5.65216e12	0.258849
$922$	0	0
$923$	−3.51750e12	−0.159524
$924$	0	0
$925$	4.26201e12	0.191415
$926$	0	0
$927$	1.14539e13	0.509444
$928$	0	0
$929$	1.74259e13	0.767582	0.383791	−	0.923420i	$-0.374618\pi$
$929$	1.74259e13	0.767582	0.383791	+	0.923420i	$0.374618\pi$
$930$	0	0
$931$	−5.32434e12	−0.232269
$932$	0	0
$933$	9.15221e11	0.0395420
$934$	0	0
$935$	1.47982e13	0.633222
$936$	0	0
$937$	−4.43313e12	−0.187881	−0.0939405	−	0.995578i	$-0.529946\pi$
$937$	−4.43313e12	−0.187881	−0.0939405	+	0.995578i	$0.529946\pi$
$938$	0	0
$939$	−5.42071e12	−0.227542
$940$	0	0
$941$	3.54725e13	1.47482	0.737409	−	0.675447i	$-0.236049\pi$
$941$	3.54725e13	1.47482	0.737409	+	0.675447i	$0.236049\pi$
$942$	0	0
$943$	−1.94242e13	−0.799910
$944$	0	0
$945$	−5.88805e12	−0.240175
$946$	0	0
$947$	−9.18897e12	−0.371272	−0.185636	−	0.982619i	$-0.559434\pi$
$947$	−9.18897e12	−0.371272	−0.185636	+	0.982619i	$0.559434\pi$
$948$	0	0
$949$	1.23113e13	0.492727
$950$	0	0
$951$	−2.20222e12	−0.0873067
$952$	0	0
$953$	3.29318e13	1.29329	0.646646	−	0.762790i	$-0.276171\pi$
$953$	3.29318e13	1.29329	0.646646	+	0.762790i	$0.276171\pi$
$954$	0	0
$955$	2.56965e13	0.999674
$956$	0	0
$957$	−9.94455e12	−0.383249
$958$	0	0
$959$	−4.11915e13	−1.57262
$960$	0	0
$961$	−2.55140e13	−0.964990
$962$	0	0
$963$	−3.26954e13	−1.22509
$964$	0	0
$965$	9.98509e12	0.370663
$966$	0	0
$967$	1.21272e13	0.446007	0.223004	−	0.974818i	$-0.428414\pi$
$967$	1.21272e13	0.446007	0.223004	+	0.974818i	$0.428414\pi$
$968$	0	0
$969$	−1.00650e12	−0.0366739
$970$	0	0
$971$	−1.78158e12	−0.0643159	−0.0321579	−	0.999483i	$-0.510238\pi$
$971$	−1.78158e12	−0.0643159	−0.0321579	+	0.999483i	$0.510238\pi$
$972$	0	0
$973$	3.82107e13	1.36671
$974$	0	0
$975$	−9.15380e11	−0.0324400
$976$	0	0
$977$	−2.04326e13	−0.717461	−0.358731	−	0.933441i	$-0.616790\pi$
$977$	−2.04326e13	−0.717461	−0.358731	+	0.933441i	$0.616790\pi$
$978$	0	0
$979$	4.25338e12	0.147983
$980$	0	0
$981$	2.40961e13	0.830684
$982$	0	0
$983$	−4.47553e13	−1.52881	−0.764406	−	0.644735i	$-0.776968\pi$
$983$	−4.47553e13	−1.52881	−0.764406	+	0.644735i	$0.776968\pi$
$984$	0	0
$985$	−4.30317e12	−0.145655
$986$	0	0
$987$	4.58650e11	0.0153835
$988$	0	0
$989$	1.63700e13	0.544084
$990$	0	0
$991$	−1.66489e12	−0.0548346	−0.0274173	−	0.999624i	$-0.508728\pi$
$991$	−1.66489e12	−0.0548346	−0.0274173	+	0.999624i	$0.508728\pi$
$992$	0	0
$993$	7.27008e12	0.237284
$994$	0	0
$995$	−1.10192e13	−0.356406
$996$	0	0
$997$	4.18323e13	1.34086	0.670430	−	0.741973i	$-0.266110\pi$
$997$	4.18323e13	1.34086	0.670430	+	0.741973i	$0.266110\pi$
$998$	0	0
$999$	2.80483e12	0.0890968

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	104.10.a.d.1.5	✓	8
4.3	odd	2		208.10.a.m.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.a.d.1.5	✓	8	1.1	even	1	trivial
208.10.a.m.1.4		8	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 104 = 2^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	104.a (trivial)

\(f(q)\)	\(=\)	\(q+23.3087 q^{3} -760.329 q^{5} -8557.89 q^{7} -19139.7 q^{9} -72978.2 q^{11} +28561.0 q^{13} -17722.3 q^{15} +266694. q^{17} -161914. q^{19} -199473. q^{21} +1.43017e6 q^{23} -1.37503e6 q^{25} -904906. q^{27} +5.84621e6 q^{29} -962113. q^{31} -1.70103e6 q^{33} +6.50681e6 q^{35} -3.09958e6 q^{37} +665719. q^{39} -1.35817e7 q^{41} +1.14462e7 q^{43} +1.45525e7 q^{45} -2.29931e6 q^{47} +3.28838e7 q^{49} +6.21629e6 q^{51} -3.07448e7 q^{53} +5.54874e7 q^{55} -3.77399e6 q^{57} +3.05327e7 q^{59} +6.53064e7 q^{61} +1.63795e8 q^{63} -2.17158e7 q^{65} -1.81368e8 q^{67} +3.33354e7 q^{69} -1.23157e8 q^{71} +4.31054e8 q^{73} -3.20500e7 q^{75} +6.24539e8 q^{77} +4.22314e8 q^{79} +3.55635e8 q^{81} +1.40247e8 q^{83} -2.02775e8 q^{85} +1.36267e8 q^{87} -5.82829e7 q^{89} -2.44422e8 q^{91} -2.24256e7 q^{93} +1.23108e8 q^{95} +5.66836e8 q^{97} +1.39678e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 141 q^{3} + 2051 q^{5} - 2417 q^{7} + 115741 q^{9} - 53118 q^{11} + 228488 q^{13} - 464555 q^{15} + 433095 q^{17} - 434954 q^{19} + 906875 q^{21} - 1124296 q^{23} + 5966065 q^{25} + 7820643 q^{27}+ \cdots - 641626736 q^{99}+O(q^{100}) \) 8 * q + 141 * q^3 + 2051 * q^5 - 2417 * q^7 + 115741 * q^9 - 53118 * q^11 + 228488 * q^13 - 464555 * q^15 + 433095 * q^17 - 434954 * q^19 + 906875 * q^21 - 1124296 * q^23 + 5966065 * q^25 + 7820643 * q^27 - 1913432 * q^29 + 5220148 * q^31 - 1626882 * q^33 + 950707 * q^35 - 419409 * q^37 + 4027101 * q^39 + 12148046 * q^41 + 32143187 * q^43 + 18893544 * q^45 + 57898411 * q^47 + 104302637 * q^49 + 198521477 * q^51 + 273106866 * q^53 + 141214662 * q^55 + 253032058 * q^57 + 50734502 * q^59 + 139520018 * q^61 - 20344158 * q^63 + 58578611 * q^65 + 528144786 * q^67 + 483506576 * q^69 + 334814973 * q^71 + 630864344 * q^73 + 2265796264 * q^75 + 939664146 * q^77 + 865037608 * q^79 + 3194281040 * q^81 + 1520229952 * q^83 + 2130158385 * q^85 + 1167867006 * q^87 + 894700240 * q^89 - 69031937 * q^91 + 3803209548 * q^93 - 2695926542 * q^95 + 980232164 * q^97 - 641626736 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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