LMFDB - Embedded newform 52.10.a.b.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [52,10,Mod(1,52)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(52, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("52.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 52 = 2^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	52.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.7818634805$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5049x^{3} - 87027x^{2} + 2867800x + 48215700 $ x^5 - x^4 - 5049x^3 - 87027x^2 + 2867800*x + 48215700
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$75.2404$ of defining polynomial
Character	$\chi$	$=$	52.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+195.721 q^{3} +1887.16 q^{5} +6689.32 q^{7} +18623.8 q^{9} +33813.8 q^{11} -28561.0 q^{13} +369357. q^{15} -100152. q^{17} -528083. q^{19} +1.30924e6 q^{21} -635662. q^{23} +1.60825e6 q^{25} -207311. q^{27} -5.60786e6 q^{29} +4.13899e6 q^{31} +6.61807e6 q^{33} +1.26238e7 q^{35} -3.49585e6 q^{37} -5.58999e6 q^{39} +3.30438e7 q^{41} +3.58060e7 q^{43} +3.51461e7 q^{45} +2.81522e7 q^{47} +4.39336e6 q^{49} -1.96019e7 q^{51} -3.87720e7 q^{53} +6.38120e7 q^{55} -1.03357e8 q^{57} -7.53492e7 q^{59} -1.53785e8 q^{61} +1.24580e8 q^{63} -5.38992e7 q^{65} +1.69378e7 q^{67} -1.24413e8 q^{69} +9.46824e7 q^{71} +6.01966e7 q^{73} +3.14768e8 q^{75} +2.26191e8 q^{77} +1.39281e8 q^{79} -4.07147e8 q^{81} -8.05240e8 q^{83} -1.89003e8 q^{85} -1.09758e9 q^{87} -7.35041e8 q^{89} -1.91054e8 q^{91} +8.10089e8 q^{93} -9.96577e8 q^{95} +1.43090e9 q^{97} +6.29740e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 147 q^{3} + 1721 q^{5} + 3317 q^{7} - 3204 q^{9} + 6562 q^{11} - 142805 q^{13} + 80283 q^{15} - 146493 q^{17} + 110250 q^{19} + 2250015 q^{21} + 1675652 q^{23} + 6048610 q^{25} + 4937895 q^{27} + 7019714 q^{29}+ \cdots + 302611248 q^{99}+O(q^{100}) $ 5 * q - 147 * q^3 + 1721 * q^5 + 3317 * q^7 - 3204 * q^9 + 6562 * q^11 - 142805 * q^13 + 80283 * q^15 - 146493 * q^17 + 110250 * q^19 + 2250015 * q^21 + 1675652 * q^23 + 6048610 * q^25 + 4937895 * q^27 + 7019714 * q^29 + 14112296 * q^31 + 21400890 * q^33 + 19283879 * q^35 + 28285653 * q^37 + 4198467 * q^39 + 55512492 * q^41 + 61336547 * q^43 + 90036738 * q^45 + 78843745 * q^47 + 88081704 * q^49 + 10216377 * q^51 + 105957576 * q^53 + 71100802 * q^55 + 21427866 * q^57 - 59686630 * q^59 - 41439220 * q^61 - 49300830 * q^63 - 49153481 * q^65 - 214570942 * q^67 - 432347172 * q^69 - 477669785 * q^71 - 201698114 * q^73 - 727786788 * q^75 - 521247014 * q^77 - 582737180 * q^79 - 855021987 * q^81 - 1134244092 * q^83 - 552245895 * q^85 - 2198696622 * q^87 - 476858778 * q^89 - 94736837 * q^91 + 1038474204 * q^93 - 866887902 * q^95 + 1473121030 * q^97 + 302611248 * 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
$3$	195.721	1.39506	0.697529	−	0.716557i	$-0.254283\pi$
$3$	195.721	1.39506	0.697529	+	0.716557i	$0.254283\pi$
$4$	0	0
$5$	1887.16	1.35034	0.675171	−	0.737661i	$-0.264070\pi$
$5$	1887.16	1.35034	0.675171	+	0.737661i	$0.264070\pi$
$6$	0	0
$7$	6689.32	1.05303	0.526515	−	0.850166i	$-0.323498\pi$
$7$	6689.32	1.05303	0.526515	+	0.850166i	$0.323498\pi$
$8$	0	0
$9$	18623.8	0.946186
$10$	0	0
$11$	33813.8	0.696348	0.348174	−	0.937430i	$-0.386802\pi$
$11$	33813.8	0.696348	0.348174	+	0.937430i	$0.386802\pi$
$12$	0	0
$13$	−28561.0	−0.277350
$13$	−28561.0	−0.277350
$14$	0	0
$15$	369357.	1.88380
$16$	0	0
$17$	−100152.	−0.290831	−0.145416	−	0.989371i	$-0.546452\pi$
$17$	−100152.	−0.290831	−0.145416	+	0.989371i	$0.546452\pi$
$18$	0	0
$19$	−528083.	−0.929632	−0.464816	−	0.885407i	$-0.653880\pi$
$19$	−528083.	−0.929632	−0.464816	+	0.885407i	$0.653880\pi$
$20$	0	0
$21$	1.30924e6	1.46904
$22$	0	0
$23$	−635662.	−0.473643	−0.236822	−	0.971553i	$-0.576106\pi$
$23$	−635662.	−0.473643	−0.236822	+	0.971553i	$0.576106\pi$
$24$	0	0
$25$	1.60825e6	0.823423
$26$	0	0
$27$	−207311.	−0.0750732
$28$	0	0
$29$	−5.60786e6	−1.47233	−0.736167	−	0.676800i	$-0.763366\pi$
$29$	−5.60786e6	−1.47233	−0.736167	+	0.676800i	$0.763366\pi$
$30$	0	0
$31$	4.13899e6	0.804947	0.402473	−	0.915432i	$-0.368151\pi$
$31$	4.13899e6	0.804947	0.402473	+	0.915432i	$0.368151\pi$
$32$	0	0
$33$	6.61807e6	0.971446
$34$	0	0
$35$	1.26238e7	1.42195
$36$	0	0
$37$	−3.49585e6	−0.306651	−0.153326	−	0.988176i	$-0.548998\pi$
$37$	−3.49585e6	−0.306651	−0.153326	+	0.988176i	$0.548998\pi$
$38$	0	0
$39$	−5.58999e6	−0.386919
$40$	0	0
$41$	3.30438e7	1.82626	0.913130	−	0.407669i	$-0.133658\pi$
$41$	3.30438e7	1.82626	0.913130	+	0.407669i	$0.133658\pi$
$42$	0	0
$43$	3.58060e7	1.59716	0.798579	−	0.601890i	$-0.205585\pi$
$43$	3.58060e7	1.59716	0.798579	+	0.601890i	$0.205585\pi$
$44$	0	0
$45$	3.51461e7	1.27767
$46$	0	0
$47$	2.81522e7	0.841534	0.420767	−	0.907169i	$-0.361761\pi$
$47$	2.81522e7	0.841534	0.420767	+	0.907169i	$0.361761\pi$
$48$	0	0
$49$	4.39336e6	0.108872
$50$	0	0
$51$	−1.96019e7	−0.405726
$52$	0	0
$53$	−3.87720e7	−0.674959	−0.337479	−	0.941333i	$-0.609574\pi$
$53$	−3.87720e7	−0.674959	−0.337479	+	0.941333i	$0.609574\pi$
$54$	0	0
$55$	6.38120e7	0.940308
$56$	0	0
$57$	−1.03357e8	−1.29689
$58$	0	0
$59$	−7.53492e7	−0.809551	−0.404776	−	0.914416i	$-0.632650\pi$
$59$	−7.53492e7	−0.809551	−0.404776	+	0.914416i	$0.632650\pi$
$60$	0	0
$61$	−1.53785e8	−1.42210	−0.711048	−	0.703143i	$-0.751779\pi$
$61$	−1.53785e8	−1.42210	−0.711048	+	0.703143i	$0.751779\pi$
$62$	0	0
$63$	1.24580e8	0.996362
$64$	0	0
$65$	−5.38992e7	−0.374517
$66$	0	0
$67$	1.69378e7	0.102688	0.0513440	−	0.998681i	$-0.483650\pi$
$67$	1.69378e7	0.102688	0.0513440	+	0.998681i	$0.483650\pi$
$68$	0	0
$69$	−1.24413e8	−0.660759
$70$	0	0
$71$	9.46824e7	0.442188	0.221094	−	0.975253i	$-0.429037\pi$
$71$	9.46824e7	0.442188	0.221094	+	0.975253i	$0.429037\pi$
$72$	0	0
$73$	6.01966e7	0.248095	0.124048	−	0.992276i	$-0.460412\pi$
$73$	6.01966e7	0.248095	0.124048	+	0.992276i	$0.460412\pi$
$74$	0	0
$75$	3.14768e8	1.14872
$76$	0	0
$77$	2.26191e8	0.733275
$78$	0	0
$79$	1.39281e8	0.402319	0.201159	−	0.979559i	$-0.435529\pi$
$79$	1.39281e8	0.402319	0.201159	+	0.979559i	$0.435529\pi$
$80$	0	0
$81$	−4.07147e8	−1.05092
$82$	0	0
$83$	−8.05240e8	−1.86240	−0.931202	−	0.364504i	$-0.881238\pi$
$83$	−8.05240e8	−1.86240	−0.931202	+	0.364504i	$0.881238\pi$
$84$	0	0
$85$	−1.89003e8	−0.392721
$86$	0	0
$87$	−1.09758e9	−2.05399
$88$	0	0
$89$	−7.35041e8	−1.24181	−0.620907	−	0.783884i	$-0.713236\pi$
$89$	−7.35041e8	−1.24181	−0.620907	+	0.783884i	$0.713236\pi$
$90$	0	0
$91$	−1.91054e8	−0.292058
$92$	0	0
$93$	8.10089e8	1.12295
$94$	0	0
$95$	−9.96577e8	−1.25532
$96$	0	0
$97$	1.43090e9	1.64111	0.820555	−	0.571567i	$-0.193664\pi$
$97$	1.43090e9	1.64111	0.820555	+	0.571567i	$0.193664\pi$
$98$	0	0
$99$	6.29740e8	0.658875
$100$	0	0
$101$	−4.00832e8	−0.383280	−0.191640	−	0.981465i	$-0.561381\pi$
$101$	−4.00832e8	−0.383280	−0.191640	+	0.981465i	$0.561381\pi$
$102$	0	0
$103$	−1.95959e9	−1.71553	−0.857764	−	0.514044i	$-0.828147\pi$
$103$	−1.95959e9	−1.71553	−0.857764	+	0.514044i	$0.828147\pi$
$104$	0	0
$105$	2.47075e9	1.98370
$106$	0	0
$107$	−8.19599e8	−0.604469	−0.302235	−	0.953234i	$-0.597733\pi$
$107$	−8.19599e8	−0.604469	−0.302235	+	0.953234i	$0.597733\pi$
$108$	0	0
$109$	2.47004e9	1.67604	0.838021	−	0.545639i	$-0.183713\pi$
$109$	2.47004e9	1.67604	0.838021	+	0.545639i	$0.183713\pi$
$110$	0	0
$111$	−6.84211e8	−0.427796
$112$	0	0
$113$	2.62429e9	1.51412	0.757058	−	0.653348i	$-0.226636\pi$
$113$	2.62429e9	1.51412	0.757058	+	0.653348i	$0.226636\pi$
$114$	0	0
$115$	−1.19960e9	−0.639580
$116$	0	0
$117$	−5.31914e8	−0.262425
$118$	0	0
$119$	−6.69950e8	−0.306254
$120$	0	0
$121$	−1.21458e9	−0.515099
$122$	0	0
$123$	6.46737e9	2.54774
$124$	0	0
$125$	−6.50840e8	−0.238440
$126$	0	0
$127$	2.46880e9	0.842111	0.421056	−	0.907035i	$-0.361660\pi$
$127$	2.46880e9	0.842111	0.421056	+	0.907035i	$0.361660\pi$
$128$	0	0
$129$	7.00800e9	2.22813
$130$	0	0
$131$	3.58925e9	1.06484	0.532418	−	0.846481i	$-0.321283\pi$
$131$	3.58925e9	1.06484	0.532418	+	0.846481i	$0.321283\pi$
$132$	0	0
$133$	−3.53251e9	−0.978930
$134$	0	0
$135$	−3.91228e8	−0.101374
$136$	0	0
$137$	−2.85083e9	−0.691399	−0.345699	−	0.938345i	$-0.612358\pi$
$137$	−2.85083e9	−0.691399	−0.345699	+	0.938345i	$0.612358\pi$
$138$	0	0
$139$	−6.65459e9	−1.51201	−0.756005	−	0.654565i	$-0.772852\pi$
$139$	−6.65459e9	−1.51201	−0.756005	+	0.654565i	$0.772852\pi$
$140$	0	0
$141$	5.50998e9	1.17399
$142$	0	0
$143$	−9.65755e8	−0.193132
$144$	0	0
$145$	−1.05829e10	−1.98815
$146$	0	0
$147$	8.59873e8	0.151882
$148$	0	0
$149$	3.53264e9	0.587166	0.293583	−	0.955934i	$-0.405152\pi$
$149$	3.53264e9	0.587166	0.293583	+	0.955934i	$0.405152\pi$
$150$	0	0
$151$	−8.49892e9	−1.33036	−0.665178	−	0.746685i	$-0.731644\pi$
$151$	−8.49892e9	−1.33036	−0.665178	+	0.746685i	$0.731644\pi$
$152$	0	0
$153$	−1.86521e9	−0.275180
$154$	0	0
$155$	7.81094e9	1.08695
$156$	0	0
$157$	4.16789e9	0.547479	0.273739	−	0.961804i	$-0.411739\pi$
$157$	4.16789e9	0.547479	0.273739	+	0.961804i	$0.411739\pi$
$158$	0	0
$159$	−7.58851e9	−0.941607
$160$	0	0
$161$	−4.25215e9	−0.498760
$162$	0	0
$163$	−6.39222e9	−0.709263	−0.354632	−	0.935006i	$-0.615394\pi$
$163$	−6.39222e9	−0.709263	−0.354632	+	0.935006i	$0.615394\pi$
$164$	0	0
$165$	1.24894e10	1.31178
$166$	0	0
$167$	−1.68883e10	−1.68020	−0.840100	−	0.542432i	$-0.817504\pi$
$167$	−1.68883e10	−1.68020	−0.840100	+	0.542432i	$0.817504\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	−9.83491e9	−0.879605
$172$	0	0
$173$	1.22379e10	1.03872	0.519361	−	0.854555i	$-0.326170\pi$
$173$	1.22379e10	1.03872	0.519361	+	0.854555i	$0.326170\pi$
$174$	0	0
$175$	1.07581e10	0.867088
$176$	0	0
$177$	−1.47474e10	−1.12937
$178$	0	0
$179$	1.83860e10	1.33859	0.669295	−	0.742996i	$-0.266596\pi$
$179$	1.83860e10	1.33859	0.669295	+	0.742996i	$0.266596\pi$
$180$	0	0
$181$	1.76860e10	1.22483	0.612414	−	0.790537i	$-0.290198\pi$
$181$	1.76860e10	1.22483	0.612414	+	0.790537i	$0.290198\pi$
$182$	0	0
$183$	−3.00989e10	−1.98391
$184$	0	0
$185$	−6.59722e9	−0.414084
$186$	0	0
$187$	−3.38653e9	−0.202520
$188$	0	0
$189$	−1.38677e9	−0.0790543
$190$	0	0
$191$	2.54411e10	1.38320	0.691602	−	0.722279i	$-0.256905\pi$
$191$	2.54411e10	1.38320	0.691602	+	0.722279i	$0.256905\pi$
$192$	0	0
$193$	−3.17582e10	−1.64759	−0.823793	−	0.566891i	$-0.808146\pi$
$193$	−3.17582e10	−1.64759	−0.823793	+	0.566891i	$0.808146\pi$
$194$	0	0
$195$	−1.05492e10	−0.522473
$196$	0	0
$197$	−1.56437e10	−0.740015	−0.370008	−	0.929029i	$-0.620645\pi$
$197$	−1.56437e10	−0.740015	−0.370008	+	0.929029i	$0.620645\pi$
$198$	0	0
$199$	−1.45182e10	−0.656258	−0.328129	−	0.944633i	$-0.606418\pi$
$199$	−1.45182e10	−0.656258	−0.328129	+	0.944633i	$0.606418\pi$
$200$	0	0
$201$	3.31508e9	0.143256
$202$	0	0
$203$	−3.75127e10	−1.55041
$204$	0	0
$205$	6.23589e10	2.46607
$206$	0	0
$207$	−1.18384e10	−0.448155
$208$	0	0
$209$	−1.78565e10	−0.647348
$210$	0	0
$211$	−3.70320e10	−1.28619	−0.643097	−	0.765785i	$-0.722351\pi$
$211$	−3.70320e10	−1.28619	−0.643097	+	0.765785i	$0.722351\pi$
$212$	0	0
$213$	1.85314e10	0.616877
$214$	0	0
$215$	6.75717e10	2.15671
$216$	0	0
$217$	2.76870e10	0.847633
$218$	0	0
$219$	1.17817e10	0.346108
$220$	0	0
$221$	2.86045e9	0.0806620
$222$	0	0
$223$	5.40531e9	0.146369	0.0731844	−	0.997318i	$-0.476684\pi$
$223$	5.40531e9	0.146369	0.0731844	+	0.997318i	$0.476684\pi$
$224$	0	0
$225$	2.99516e10	0.779111
$226$	0	0
$227$	−4.90433e9	−0.122592	−0.0612962	−	0.998120i	$-0.519523\pi$
$227$	−4.90433e9	−0.122592	−0.0612962	+	0.998120i	$0.519523\pi$
$228$	0	0
$229$	6.58728e10	1.58287	0.791437	−	0.611251i	$-0.209333\pi$
$229$	6.58728e10	1.58287	0.791437	+	0.611251i	$0.209333\pi$
$230$	0	0
$231$	4.42704e10	1.02296
$232$	0	0
$233$	3.31701e10	0.737302	0.368651	−	0.929568i	$-0.379820\pi$
$233$	3.31701e10	0.737302	0.368651	+	0.929568i	$0.379820\pi$
$234$	0	0
$235$	5.31276e10	1.13636
$236$	0	0
$237$	2.72603e10	0.561258
$238$	0	0
$239$	−3.32294e9	−0.0658767	−0.0329383	−	0.999457i	$-0.510486\pi$
$239$	−3.32294e9	−0.0658767	−0.0329383	+	0.999457i	$0.510486\pi$
$240$	0	0
$241$	3.39919e10	0.649081	0.324540	−	0.945872i	$-0.394790\pi$
$241$	3.39919e10	0.649081	0.324540	+	0.945872i	$0.394790\pi$
$242$	0	0
$243$	−7.56068e10	−1.39102
$244$	0	0
$245$	8.29097e9	0.147014
$246$	0	0
$247$	1.50826e10	0.257833
$248$	0	0
$249$	−1.57602e11	−2.59816
$250$	0	0
$251$	2.40320e10	0.382172	0.191086	−	0.981573i	$-0.438799\pi$
$251$	2.40320e10	0.382172	0.191086	+	0.981573i	$0.438799\pi$
$252$	0	0
$253$	−2.14941e10	−0.329821
$254$	0	0
$255$	−3.69920e10	−0.547869
$256$	0	0
$257$	8.49875e10	1.21522	0.607612	−	0.794234i	$-0.292128\pi$
$257$	8.49875e10	1.21522	0.607612	+	0.794234i	$0.292128\pi$
$258$	0	0
$259$	−2.33848e10	−0.322913
$260$	0	0
$261$	−1.04440e11	−1.39310
$262$	0	0
$263$	2.72793e10	0.351586	0.175793	−	0.984427i	$-0.443751\pi$
$263$	2.72793e10	0.351586	0.175793	+	0.984427i	$0.443751\pi$
$264$	0	0
$265$	−7.31690e10	−0.911425
$266$	0	0
$267$	−1.43863e11	−1.73240
$268$	0	0
$269$	1.45306e9	0.0169200	0.00845998	−	0.999964i	$-0.497307\pi$
$269$	1.45306e9	0.0169200	0.00845998	+	0.999964i	$0.497307\pi$
$270$	0	0
$271$	−3.35590e10	−0.377961	−0.188981	−	0.981981i	$-0.560518\pi$
$271$	−3.35590e10	−0.377961	−0.188981	+	0.981981i	$0.560518\pi$
$272$	0	0
$273$	−3.73932e10	−0.407438
$274$	0	0
$275$	5.43809e10	0.573389
$276$	0	0
$277$	2.51248e10	0.256415	0.128208	−	0.991747i	$-0.459078\pi$
$277$	2.51248e10	0.256415	0.128208	+	0.991747i	$0.459078\pi$
$278$	0	0
$279$	7.70837e10	0.761629
$280$	0	0
$281$	1.10842e11	1.06054	0.530268	−	0.847830i	$-0.322091\pi$
$281$	1.10842e11	1.06054	0.530268	+	0.847830i	$0.322091\pi$
$282$	0	0
$283$	−1.74705e11	−1.61907	−0.809536	−	0.587070i	$-0.800281\pi$
$283$	−1.74705e11	−1.61907	−0.809536	+	0.587070i	$0.800281\pi$
$284$	0	0
$285$	−1.95051e11	−1.75124
$286$	0	0
$287$	2.21040e11	1.92311
$288$	0	0
$289$	−1.08557e11	−0.915417
$290$	0	0
$291$	2.80058e11	2.28944
$292$	0	0
$293$	−3.76354e10	−0.298327	−0.149163	−	0.988813i	$-0.547658\pi$
$293$	−3.76354e10	−0.298327	−0.149163	+	0.988813i	$0.547658\pi$
$294$	0	0
$295$	−1.42196e11	−1.09317
$296$	0	0
$297$	−7.00996e9	−0.0522771
$298$	0	0
$299$	1.81551e10	0.131365
$300$	0	0
$301$	2.39518e11	1.68186
$302$	0	0
$303$	−7.84513e10	−0.534698
$304$	0	0
$305$	−2.90216e11	−1.92032
$306$	0	0
$307$	−1.16535e11	−0.748747	−0.374373	−	0.927278i	$-0.622142\pi$
$307$	−1.16535e11	−0.748747	−0.374373	+	0.927278i	$0.622142\pi$
$308$	0	0
$309$	−3.83533e11	−2.39326
$310$	0	0
$311$	1.86355e11	1.12958	0.564792	−	0.825233i	$-0.308956\pi$
$311$	1.86355e11	1.12958	0.564792	+	0.825233i	$0.308956\pi$
$312$	0	0
$313$	−1.00540e11	−0.592093	−0.296046	−	0.955174i	$-0.595668\pi$
$313$	−1.00540e11	−0.592093	−0.296046	+	0.955174i	$0.595668\pi$
$314$	0	0
$315$	2.35103e11	1.34543
$316$	0	0
$317$	2.01223e11	1.11921	0.559605	−	0.828760i	$-0.310953\pi$
$317$	2.01223e11	1.11921	0.559605	+	0.828760i	$0.310953\pi$
$318$	0	0
$319$	−1.89623e11	−1.02526
$320$	0	0
$321$	−1.60413e11	−0.843270
$322$	0	0
$323$	5.28887e10	0.270366
$324$	0	0
$325$	−4.59331e10	−0.228376
$326$	0	0
$327$	4.83439e11	2.33817
$328$	0	0
$329$	1.88319e11	0.886160
$330$	0	0
$331$	−2.42987e11	−1.11265	−0.556323	−	0.830966i	$-0.687788\pi$
$331$	−2.42987e11	−1.11265	−0.556323	+	0.830966i	$0.687788\pi$
$332$	0	0
$333$	−6.51059e10	−0.290149
$334$	0	0
$335$	3.19643e10	0.138664
$336$	0	0
$337$	−2.02728e10	−0.0856207	−0.0428103	−	0.999083i	$-0.513631\pi$
$337$	−2.02728e10	−0.0856207	−0.0428103	+	0.999083i	$0.513631\pi$
$338$	0	0
$339$	5.13629e11	2.11228
$340$	0	0
$341$	1.39955e11	0.560523
$342$	0	0
$343$	−2.40550e11	−0.938385
$344$	0	0
$345$	−2.34786e11	−0.892251
$346$	0	0
$347$	−6.94989e10	−0.257333	−0.128667	−	0.991688i	$-0.541070\pi$
$347$	−6.94989e10	−0.257333	−0.128667	+	0.991688i	$0.541070\pi$
$348$	0	0
$349$	1.76281e10	0.0636048	0.0318024	−	0.999494i	$-0.489875\pi$
$349$	1.76281e10	0.0636048	0.0318024	+	0.999494i	$0.489875\pi$
$350$	0	0
$351$	5.92100e9	0.0208216
$352$	0	0
$353$	−1.65313e10	−0.0566659	−0.0283330	−	0.999599i	$-0.509020\pi$
$353$	−1.65313e10	−0.0566659	−0.0283330	+	0.999599i	$0.509020\pi$
$354$	0	0
$355$	1.78681e11	0.597104
$356$	0	0
$357$	−1.31123e11	−0.427242
$358$	0	0
$359$	4.90587e11	1.55880	0.779401	−	0.626525i	$-0.215523\pi$
$359$	4.90587e11	1.55880	0.779401	+	0.626525i	$0.215523\pi$
$360$	0	0
$361$	−4.38160e10	−0.135785
$362$	0	0
$363$	−2.37718e11	−0.718593
$364$	0	0
$365$	1.13601e11	0.335014
$366$	0	0
$367$	4.06208e11	1.16883	0.584414	−	0.811455i	$-0.301324\pi$
$367$	4.06208e11	1.16883	0.584414	+	0.811455i	$0.301324\pi$
$368$	0	0
$369$	6.15401e11	1.72798
$370$	0	0
$371$	−2.59359e11	−0.710752
$372$	0	0
$373$	−6.00800e11	−1.60709	−0.803545	−	0.595244i	$-0.797056\pi$
$373$	−6.00800e11	−1.60709	−0.803545	+	0.595244i	$0.797056\pi$
$374$	0	0
$375$	−1.27383e11	−0.332637
$376$	0	0
$377$	1.60166e11	0.408352
$378$	0	0
$379$	1.72621e11	0.429752	0.214876	−	0.976641i	$-0.431065\pi$
$379$	1.72621e11	0.429752	0.214876	+	0.976641i	$0.431065\pi$
$380$	0	0
$381$	4.83197e11	1.17479
$382$	0	0
$383$	−4.57405e11	−1.08619	−0.543096	−	0.839671i	$-0.682748\pi$
$383$	−4.57405e11	−1.08619	−0.543096	+	0.839671i	$0.682748\pi$
$384$	0	0
$385$	4.26859e11	0.990172
$386$	0	0
$387$	6.66844e11	1.51121
$388$	0	0
$389$	2.60035e10	0.0575782	0.0287891	−	0.999586i	$-0.490835\pi$
$389$	2.60035e10	0.0575782	0.0287891	+	0.999586i	$0.490835\pi$
$390$	0	0
$391$	6.36630e10	0.137750
$392$	0	0
$393$	7.02493e11	1.48551
$394$	0	0
$395$	2.62846e11	0.543268
$396$	0	0
$397$	2.03597e11	0.411353	0.205677	−	0.978620i	$-0.434060\pi$
$397$	2.03597e11	0.411353	0.205677	+	0.978620i	$0.434060\pi$
$398$	0	0
$399$	−6.91388e11	−1.36566
$400$	0	0
$401$	6.40571e11	1.23714	0.618569	−	0.785731i	$-0.287713\pi$
$401$	6.40571e11	1.23714	0.618569	+	0.785731i	$0.287713\pi$
$402$	0	0
$403$	−1.18214e11	−0.223252
$404$	0	0
$405$	−7.68352e11	−1.41910
$406$	0	0
$407$	−1.18208e11	−0.213536
$408$	0	0
$409$	−6.75615e11	−1.19384	−0.596918	−	0.802303i	$-0.703608\pi$
$409$	−6.75615e11	−1.19384	−0.596918	+	0.802303i	$0.703608\pi$
$410$	0	0
$411$	−5.57967e11	−0.964541
$412$	0	0
$413$	−5.04035e11	−0.852482
$414$	0	0
$415$	−1.51962e12	−2.51488
$416$	0	0
$417$	−1.30244e12	−2.10934
$418$	0	0
$419$	−7.61552e11	−1.20708	−0.603540	−	0.797333i	$-0.706244\pi$
$419$	−7.61552e11	−1.20708	−0.603540	+	0.797333i	$0.706244\pi$
$420$	0	0
$421$	7.10331e11	1.10202	0.551012	−	0.834497i	$-0.314242\pi$
$421$	7.10331e11	1.10202	0.551012	+	0.834497i	$0.314242\pi$
$422$	0	0
$423$	5.24300e11	0.796248
$424$	0	0
$425$	−1.61070e11	−0.239477
$426$	0	0
$427$	−1.02872e12	−1.49751
$428$	0	0
$429$	−1.89019e11	−0.269431
$430$	0	0
$431$	3.13989e11	0.438296	0.219148	−	0.975692i	$-0.429672\pi$
$431$	3.13989e11	0.438296	0.219148	+	0.975692i	$0.429672\pi$
$432$	0	0
$433$	1.20439e12	1.64653	0.823267	−	0.567655i	$-0.192149\pi$
$433$	1.20439e12	1.64653	0.823267	+	0.567655i	$0.192149\pi$
$434$	0	0
$435$	−2.07130e12	−2.77359
$436$	0	0
$437$	3.35682e11	0.440314
$438$	0	0
$439$	1.45395e12	1.86836	0.934179	−	0.356804i	$-0.116134\pi$
$439$	1.45395e12	1.86836	0.934179	+	0.356804i	$0.116134\pi$
$440$	0	0
$441$	8.18210e10	0.103013
$442$	0	0
$443$	2.03318e11	0.250819	0.125409	−	0.992105i	$-0.459976\pi$
$443$	2.03318e11	0.250819	0.125409	+	0.992105i	$0.459976\pi$
$444$	0	0
$445$	−1.38714e12	−1.67687
$446$	0	0
$447$	6.91412e11	0.819130
$448$	0	0
$449$	8.26254e11	0.959411	0.479706	−	0.877429i	$-0.340743\pi$
$449$	8.26254e11	0.959411	0.479706	+	0.877429i	$0.340743\pi$
$450$	0	0
$451$	1.11734e12	1.27171
$452$	0	0
$453$	−1.66342e12	−1.85592
$454$	0	0
$455$	−3.60549e11	−0.394378
$456$	0	0
$457$	−1.40840e12	−1.51044	−0.755218	−	0.655474i	$-0.772469\pi$
$457$	−1.40840e12	−1.51044	−0.755218	+	0.655474i	$0.772469\pi$
$458$	0	0
$459$	2.07626e10	0.0218336
$460$	0	0
$461$	9.68042e11	0.998252	0.499126	−	0.866530i	$-0.333655\pi$
$461$	9.68042e11	0.998252	0.499126	+	0.866530i	$0.333655\pi$
$462$	0	0
$463$	3.09545e11	0.313047	0.156523	−	0.987674i	$-0.449971\pi$
$463$	3.09545e11	0.313047	0.156523	+	0.987674i	$0.449971\pi$
$464$	0	0
$465$	1.52877e12	1.51636
$466$	0	0
$467$	8.91404e11	0.867259	0.433629	−	0.901091i	$-0.357233\pi$
$467$	8.91404e11	0.867259	0.433629	+	0.901091i	$0.357233\pi$
$468$	0	0
$469$	1.13302e11	0.108134
$470$	0	0
$471$	8.15743e11	0.763765
$472$	0	0
$473$	1.21074e12	1.11218
$474$	0	0
$475$	−8.49288e11	−0.765480
$476$	0	0
$477$	−7.22082e11	−0.638637
$478$	0	0
$479$	−1.95587e11	−0.169758	−0.0848789	−	0.996391i	$-0.527050\pi$
$479$	−1.95587e11	−0.169758	−0.0848789	+	0.996391i	$0.527050\pi$
$480$	0	0
$481$	9.98449e10	0.0850497
$482$	0	0
$483$	−8.32235e11	−0.695799
$484$	0	0
$485$	2.70035e12	2.21606
$486$	0	0
$487$	−9.30929e11	−0.749957	−0.374978	−	0.927034i	$-0.622350\pi$
$487$	−9.30929e11	−0.749957	−0.374978	+	0.927034i	$0.622350\pi$
$488$	0	0
$489$	−1.25109e12	−0.989463
$490$	0	0
$491$	1.91366e12	1.48593	0.742964	−	0.669331i	$-0.233419\pi$
$491$	1.91366e12	1.48593	0.742964	+	0.669331i	$0.233419\pi$
$492$	0	0
$493$	5.61640e11	0.428200
$494$	0	0
$495$	1.18842e12	0.889707
$496$	0	0
$497$	6.33361e11	0.465637
$498$	0	0
$499$	−1.72803e11	−0.124767	−0.0623834	−	0.998052i	$-0.519870\pi$
$499$	−1.72803e11	−0.124767	−0.0623834	+	0.998052i	$0.519870\pi$
$500$	0	0
$501$	−3.30539e12	−2.34398
$502$	0	0
$503$	−6.24705e11	−0.435130	−0.217565	−	0.976046i	$-0.569811\pi$
$503$	−6.24705e11	−0.435130	−0.217565	+	0.976046i	$0.569811\pi$
$504$	0	0
$505$	−7.56434e11	−0.517559
$506$	0	0
$507$	1.59656e11	0.107312
$508$	0	0
$509$	6.05040e11	0.399534	0.199767	−	0.979843i	$-0.435981\pi$
$509$	6.05040e11	0.399534	0.199767	+	0.979843i	$0.435981\pi$
$510$	0	0
$511$	4.02674e11	0.261252
$512$	0	0
$513$	1.09477e11	0.0697904
$514$	0	0
$515$	−3.69806e12	−2.31655
$516$	0	0
$517$	9.51931e11	0.586001
$518$	0	0
$519$	2.39521e12	1.44908
$520$	0	0
$521$	4.85788e11	0.288853	0.144427	−	0.989516i	$-0.453866\pi$
$521$	4.85788e11	0.288853	0.144427	+	0.989516i	$0.453866\pi$
$522$	0	0
$523$	7.82169e11	0.457133	0.228567	−	0.973528i	$-0.426596\pi$
$523$	7.82169e11	0.457133	0.228567	+	0.973528i	$0.426596\pi$
$524$	0	0
$525$	2.10558e12	1.20964
$526$	0	0
$527$	−4.14530e11	−0.234103
$528$	0	0
$529$	−1.39709e12	−0.775662
$530$	0	0
$531$	−1.40329e12	−0.765986
$532$	0	0
$533$	−9.43764e11	−0.506513
$534$	0	0
$535$	−1.54671e12	−0.816240
$536$	0	0
$537$	3.59852e12	1.86741
$538$	0	0
$539$	1.48556e11	0.0758125
$540$	0	0
$541$	1.26847e12	0.636637	0.318319	−	0.947984i	$-0.396882\pi$
$541$	1.26847e12	0.636637	0.318319	+	0.947984i	$0.396882\pi$
$542$	0	0
$543$	3.46152e12	1.70871
$544$	0	0
$545$	4.66136e12	2.26323
$546$	0	0
$547$	5.05021e11	0.241194	0.120597	−	0.992702i	$-0.461519\pi$
$547$	5.05021e11	0.241194	0.120597	+	0.992702i	$0.461519\pi$
$548$	0	0
$549$	−2.86405e12	−1.34557
$550$	0	0
$551$	2.96142e12	1.36873
$552$	0	0
$553$	9.31695e11	0.423654
$554$	0	0
$555$	−1.29122e12	−0.577671
$556$	0	0
$557$	−1.12861e12	−0.496816	−0.248408	−	0.968655i	$-0.579907\pi$
$557$	−1.12861e12	−0.496816	−0.248408	+	0.968655i	$0.579907\pi$
$558$	0	0
$559$	−1.02266e12	−0.442972
$560$	0	0
$561$	−6.62815e11	−0.282527
$562$	0	0
$563$	−3.06092e12	−1.28400	−0.641998	−	0.766706i	$-0.721894\pi$
$563$	−3.06092e12	−1.28400	−0.641998	+	0.766706i	$0.721894\pi$
$564$	0	0
$565$	4.95246e12	2.04457
$566$	0	0
$567$	−2.72354e12	−1.10665
$568$	0	0
$569$	−8.86327e11	−0.354478	−0.177239	−	0.984168i	$-0.556717\pi$
$569$	−8.86327e11	−0.354478	−0.177239	+	0.984168i	$0.556717\pi$
$570$	0	0
$571$	−2.03959e12	−0.802935	−0.401467	−	0.915873i	$-0.631500\pi$
$571$	−2.03959e12	−0.802935	−0.401467	+	0.915873i	$0.631500\pi$
$572$	0	0
$573$	4.97937e12	1.92965
$574$	0	0
$575$	−1.02230e12	−0.390008
$576$	0	0
$577$	3.66489e12	1.37648	0.688239	−	0.725484i	$-0.258384\pi$
$577$	3.66489e12	1.37648	0.688239	+	0.725484i	$0.258384\pi$
$578$	0	0
$579$	−6.21575e12	−2.29848
$580$	0	0
$581$	−5.38650e12	−1.96117
$582$	0	0
$583$	−1.31103e12	−0.470006
$584$	0	0
$585$	−1.00381e12	−0.354363
$586$	0	0
$587$	−4.50691e12	−1.56678	−0.783389	−	0.621532i	$-0.786511\pi$
$587$	−4.50691e12	−1.56678	−0.783389	+	0.621532i	$0.786511\pi$
$588$	0	0
$589$	−2.18573e12	−0.748304
$590$	0	0
$591$	−3.06180e12	−1.03236
$592$	0	0
$593$	3.44539e12	1.14418	0.572088	−	0.820192i	$-0.306134\pi$
$593$	3.44539e12	1.14418	0.572088	+	0.820192i	$0.306134\pi$
$594$	0	0
$595$	−1.26430e12	−0.413547
$596$	0	0
$597$	−2.84153e12	−0.915519
$598$	0	0
$599$	4.48065e12	1.42207	0.711033	−	0.703158i	$-0.248227\pi$
$599$	4.48065e12	1.42207	0.711033	+	0.703158i	$0.248227\pi$
$600$	0	0
$601$	−3.56039e12	−1.11317	−0.556587	−	0.830790i	$-0.687889\pi$
$601$	−3.56039e12	−1.11317	−0.556587	+	0.830790i	$0.687889\pi$
$602$	0	0
$603$	3.15446e11	0.0971620
$604$	0	0
$605$	−2.29210e12	−0.695560
$606$	0	0
$607$	5.31638e12	1.58952	0.794762	−	0.606921i	$-0.207596\pi$
$607$	5.31638e12	1.58952	0.794762	+	0.606921i	$0.207596\pi$
$608$	0	0
$609$	−7.34204e12	−2.16291
$610$	0	0
$611$	−8.04054e11	−0.233399
$612$	0	0
$613$	5.37069e12	1.53624	0.768118	−	0.640308i	$-0.221193\pi$
$613$	5.37069e12	1.53624	0.768118	+	0.640308i	$0.221193\pi$
$614$	0	0
$615$	1.22050e13	3.44032
$616$	0	0
$617$	−4.42993e12	−1.23059	−0.615296	−	0.788296i	$-0.710963\pi$
$617$	−4.42993e12	−1.23059	−0.615296	+	0.788296i	$0.710963\pi$
$618$	0	0
$619$	−2.94637e12	−0.806639	−0.403319	−	0.915059i	$-0.632144\pi$
$619$	−2.94637e12	−0.806639	−0.403319	+	0.915059i	$0.632144\pi$
$620$	0	0
$621$	1.31780e11	0.0355579
$622$	0	0
$623$	−4.91692e12	−1.30767
$624$	0	0
$625$	−4.36935e12	−1.14540
$626$	0	0
$627$	−3.49489e12	−0.903087
$628$	0	0
$629$	3.50117e11	0.0891837
$630$	0	0
$631$	−3.02633e11	−0.0759949	−0.0379974	−	0.999278i	$-0.512098\pi$
$631$	−3.02633e11	−0.0759949	−0.0379974	+	0.999278i	$0.512098\pi$
$632$	0	0
$633$	−7.24795e12	−1.79431
$634$	0	0
$635$	4.65902e12	1.13714
$636$	0	0
$637$	−1.25479e11	−0.0301955
$638$	0	0
$639$	1.76334e12	0.418392
$640$	0	0
$641$	−6.55797e12	−1.53429	−0.767146	−	0.641472i	$-0.778324\pi$
$641$	−6.55797e12	−1.53429	−0.767146	+	0.641472i	$0.778324\pi$
$642$	0	0
$643$	1.94130e12	0.447862	0.223931	−	0.974605i	$-0.428111\pi$
$643$	1.94130e12	0.447862	0.223931	+	0.974605i	$0.428111\pi$
$644$	0	0
$645$	1.32252e13	3.00873
$646$	0	0
$647$	5.24666e12	1.17710	0.588550	−	0.808461i	$-0.299699\pi$
$647$	5.24666e12	1.17710	0.588550	+	0.808461i	$0.299699\pi$
$648$	0	0
$649$	−2.54784e12	−0.563730
$650$	0	0
$651$	5.41894e12	1.18250
$652$	0	0
$653$	4.70789e12	1.01325	0.506626	−	0.862166i	$-0.330893\pi$
$653$	4.70789e12	1.01325	0.506626	+	0.862166i	$0.330893\pi$
$654$	0	0
$655$	6.77349e12	1.43789
$656$	0	0
$657$	1.12109e12	0.234745
$658$	0	0
$659$	−7.26742e12	−1.50105	−0.750527	−	0.660840i	$-0.770200\pi$
$659$	−7.26742e12	−1.50105	−0.750527	+	0.660840i	$0.770200\pi$
$660$	0	0
$661$	−5.03101e12	−1.02506	−0.512530	−	0.858670i	$-0.671292\pi$
$661$	−5.03101e12	−1.02506	−0.512530	+	0.858670i	$0.671292\pi$
$662$	0	0
$663$	5.59851e11	0.112528
$664$	0	0
$665$	−6.66642e12	−1.32189
$666$	0	0
$667$	3.56470e12	0.697360
$668$	0	0
$669$	1.05793e12	0.204193
$670$	0	0
$671$	−5.20004e12	−0.990275
$672$	0	0
$673$	6.34252e12	1.19177	0.595887	−	0.803068i	$-0.296800\pi$
$673$	6.34252e12	1.19177	0.595887	+	0.803068i	$0.296800\pi$
$674$	0	0
$675$	−3.33407e11	−0.0618169
$676$	0	0
$677$	−4.08727e12	−0.747798	−0.373899	−	0.927469i	$-0.621979\pi$
$677$	−4.08727e12	−0.747798	−0.373899	+	0.927469i	$0.621979\pi$
$678$	0	0
$679$	9.57177e12	1.72814
$680$	0	0
$681$	−9.59881e11	−0.171023
$682$	0	0
$683$	8.27796e12	1.45556	0.727780	−	0.685811i	$-0.240552\pi$
$683$	8.27796e12	1.45556	0.727780	+	0.685811i	$0.240552\pi$
$684$	0	0
$685$	−5.37997e12	−0.933624
$686$	0	0
$687$	1.28927e13	2.20820
$688$	0	0
$689$	1.10737e12	0.187200
$690$	0	0
$691$	7.37742e12	1.23099	0.615493	−	0.788142i	$-0.288957\pi$
$691$	7.37742e12	1.23099	0.615493	+	0.788142i	$0.288957\pi$
$692$	0	0
$693$	4.21253e12	0.693815
$694$	0	0
$695$	−1.25583e13	−2.04173
$696$	0	0
$697$	−3.30941e12	−0.531133
$698$	0	0
$699$	6.49210e12	1.02858
$700$	0	0
$701$	9.00916e12	1.40914	0.704568	−	0.709636i	$-0.251141\pi$
$701$	9.00916e12	1.40914	0.704568	+	0.709636i	$0.251141\pi$
$702$	0	0
$703$	1.84610e12	0.285073
$704$	0	0
$705$	1.03982e13	1.58528
$706$	0	0
$707$	−2.68129e12	−0.403605
$708$	0	0
$709$	−5.99607e12	−0.891167	−0.445583	−	0.895240i	$-0.647004\pi$
$709$	−5.99607e12	−0.891167	−0.445583	+	0.895240i	$0.647004\pi$
$710$	0	0
$711$	2.59394e12	0.380668
$712$	0	0
$713$	−2.63100e12	−0.381257
$714$	0	0
$715$	−1.82253e12	−0.260795
$716$	0	0
$717$	−6.50369e11	−0.0919018
$718$	0	0
$719$	−1.87453e12	−0.261585	−0.130792	−	0.991410i	$-0.541752\pi$
$719$	−1.87453e12	−0.261585	−0.130792	+	0.991410i	$0.541752\pi$
$720$	0	0
$721$	−1.31083e13	−1.80650
$722$	0	0
$723$	6.65293e12	0.905505
$724$	0	0
$725$	−9.01882e12	−1.21235
$726$	0	0
$727$	−8.58334e12	−1.13960	−0.569799	−	0.821784i	$-0.692979\pi$
$727$	−8.58334e12	−1.13960	−0.569799	+	0.821784i	$0.692979\pi$
$728$	0	0
$729$	−6.78398e12	−0.889633
$730$	0	0
$731$	−3.58606e12	−0.464503
$732$	0	0
$733$	8.69180e12	1.11209	0.556047	−	0.831151i	$-0.312317\pi$
$733$	8.69180e12	1.11209	0.556047	+	0.831151i	$0.312317\pi$
$734$	0	0
$735$	1.62272e12	0.205093
$736$	0	0
$737$	5.72730e11	0.0715066
$738$	0	0
$739$	−6.47515e12	−0.798638	−0.399319	−	0.916812i	$-0.630753\pi$
$739$	−6.47515e12	−0.798638	−0.399319	+	0.916812i	$0.630753\pi$
$740$	0	0
$741$	2.95198e12	0.359693
$742$	0	0
$743$	1.01411e13	1.22078	0.610389	−	0.792102i	$-0.291013\pi$
$743$	1.01411e13	1.22078	0.610389	+	0.792102i	$0.291013\pi$
$744$	0	0
$745$	6.66665e12	0.792874
$746$	0	0
$747$	−1.49966e13	−1.76218
$748$	0	0
$749$	−5.48256e12	−0.636524
$750$	0	0
$751$	−1.31246e13	−1.50558	−0.752792	−	0.658258i	$-0.771294\pi$
$751$	−1.31246e13	−1.50558	−0.752792	+	0.658258i	$0.771294\pi$
$752$	0	0
$753$	4.70358e12	0.533152
$754$	0	0
$755$	−1.60388e13	−1.79643
$756$	0	0
$757$	7.46928e12	0.826699	0.413349	−	0.910573i	$-0.364359\pi$
$757$	7.46928e12	0.826699	0.413349	+	0.910573i	$0.364359\pi$
$758$	0	0
$759$	−4.20686e12	−0.460119
$760$	0	0
$761$	2.20992e12	0.238861	0.119431	−	0.992843i	$-0.461893\pi$
$761$	2.20992e12	0.238861	0.119431	+	0.992843i	$0.461893\pi$
$762$	0	0
$763$	1.65229e13	1.76492
$764$	0	0
$765$	−3.51996e12	−0.371587
$766$	0	0
$767$	2.15205e12	0.224529
$768$	0	0
$769$	1.32928e13	1.37072	0.685359	−	0.728205i	$-0.259645\pi$
$769$	1.32928e13	1.37072	0.685359	+	0.728205i	$0.259645\pi$
$770$	0	0
$771$	1.66339e13	1.69531
$772$	0	0
$773$	−8.23975e12	−0.830054	−0.415027	−	0.909809i	$-0.636228\pi$
$773$	−8.23975e12	−0.830054	−0.415027	+	0.909809i	$0.636228\pi$
$774$	0	0
$775$	6.65652e12	0.662811
$776$	0	0
$777$	−4.57691e12	−0.450482
$778$	0	0
$779$	−1.74499e13	−1.69775
$780$	0	0
$781$	3.20157e12	0.307917
$782$	0	0
$783$	1.16257e12	0.110533
$784$	0	0
$785$	7.86547e12	0.739283
$786$	0	0
$787$	7.95024e11	0.0738744	0.0369372	−	0.999318i	$-0.488240\pi$
$787$	7.95024e11	0.0738744	0.0369372	+	0.999318i	$0.488240\pi$
$788$	0	0
$789$	5.33913e12	0.490483
$790$	0	0
$791$	1.75547e13	1.59441
$792$	0	0
$793$	4.39225e12	0.394419
$794$	0	0
$795$	−1.43207e13	−1.27149
$796$	0	0
$797$	−4.62851e12	−0.406330	−0.203165	−	0.979145i	$-0.565123\pi$
$797$	−4.62851e12	−0.406330	−0.203165	+	0.979145i	$0.565123\pi$
$798$	0	0
$799$	−2.81950e12	−0.244744
$800$	0	0
$801$	−1.36892e13	−1.17499
$802$	0	0
$803$	2.03547e12	0.172761
$804$	0	0
$805$	−8.02448e12	−0.673497
$806$	0	0
$807$	2.84395e11	0.0236043
$808$	0	0
$809$	−5.61069e12	−0.460519	−0.230260	−	0.973129i	$-0.573958\pi$
$809$	−5.61069e12	−0.460519	−0.230260	+	0.973129i	$0.573958\pi$
$810$	0	0
$811$	−2.20994e12	−0.179385	−0.0896926	−	0.995969i	$-0.528588\pi$
$811$	−2.20994e12	−0.179385	−0.0896926	+	0.995969i	$0.528588\pi$
$812$	0	0
$813$	−6.56821e12	−0.527278
$814$	0	0
$815$	−1.20631e13	−0.957748
$816$	0	0
$817$	−1.89086e13	−1.48477
$818$	0	0
$819$	−3.55814e12	−0.276341
$820$	0	0
$821$	−1.24618e13	−0.957275	−0.478637	−	0.878013i	$-0.658869\pi$
$821$	−1.24618e13	−0.957275	−0.478637	+	0.878013i	$0.658869\pi$
$822$	0	0
$823$	1.06275e11	0.00807476	0.00403738	−	0.999992i	$-0.498715\pi$
$823$	1.06275e11	0.00807476	0.00403738	+	0.999992i	$0.498715\pi$
$824$	0	0
$825$	1.06435e13	0.799911
$826$	0	0
$827$	−4.78634e11	−0.0355819	−0.0177909	−	0.999842i	$-0.505663\pi$
$827$	−4.78634e11	−0.0355819	−0.0177909	+	0.999842i	$0.505663\pi$
$828$	0	0
$829$	−2.31568e13	−1.70288	−0.851439	−	0.524454i	$-0.824270\pi$
$829$	−2.31568e13	−1.70288	−0.851439	+	0.524454i	$0.824270\pi$
$830$	0	0
$831$	4.91746e12	0.357714
$832$	0	0
$833$	−4.40005e11	−0.0316632
$834$	0	0
$835$	−3.18708e13	−2.26884
$836$	0	0
$837$	−8.58058e11	−0.0604299
$838$	0	0
$839$	−3.71923e12	−0.259134	−0.129567	−	0.991571i	$-0.541359\pi$
$839$	−3.71923e12	−0.259134	−0.129567	+	0.991571i	$0.541359\pi$
$840$	0	0
$841$	1.69409e13	1.16776
$842$	0	0
$843$	2.16941e13	1.47951
$844$	0	0
$845$	1.53941e12	0.103872
$846$	0	0
$847$	−8.12469e12	−0.542415
$848$	0	0
$849$	−3.41935e13	−2.25870
$850$	0	0
$851$	2.22218e12	0.145243
$852$	0	0
$853$	1.32172e13	0.854806	0.427403	−	0.904061i	$-0.359429\pi$
$853$	1.32172e13	0.854806	0.427403	+	0.904061i	$0.359429\pi$
$854$	0	0
$855$	−1.85600e13	−1.18777
$856$	0	0
$857$	2.92197e13	1.85039	0.925194	−	0.379495i	$-0.123902\pi$
$857$	2.92197e13	1.85039	0.925194	+	0.379495i	$0.123902\pi$
$858$	0	0
$859$	2.32836e13	1.45909	0.729544	−	0.683934i	$-0.239732\pi$
$859$	2.32836e13	1.45909	0.729544	+	0.683934i	$0.239732\pi$
$860$	0	0
$861$	4.32623e13	2.68284
$862$	0	0
$863$	−5.15139e12	−0.316137	−0.158069	−	0.987428i	$-0.550527\pi$
$863$	−5.15139e12	−0.316137	−0.158069	+	0.987428i	$0.550527\pi$
$864$	0	0
$865$	2.30949e13	1.40263
$866$	0	0
$867$	−2.12470e13	−1.27706
$868$	0	0
$869$	4.70962e12	0.280154
$870$	0	0
$871$	−4.83760e11	−0.0284805
$872$	0	0
$873$	2.66489e13	1.55280
$874$	0	0
$875$	−4.35367e12	−0.251084
$876$	0	0
$877$	1.72783e13	0.986284	0.493142	−	0.869949i	$-0.335848\pi$
$877$	1.72783e13	0.986284	0.493142	+	0.869949i	$0.335848\pi$
$878$	0	0
$879$	−7.36605e12	−0.416183
$880$	0	0
$881$	2.34967e13	1.31406	0.657029	−	0.753865i	$-0.271813\pi$
$881$	2.34967e13	1.31406	0.657029	+	0.753865i	$0.271813\pi$
$882$	0	0
$883$	−2.13677e12	−0.118286	−0.0591432	−	0.998250i	$-0.518837\pi$
$883$	−2.13677e12	−0.118286	−0.0591432	+	0.998250i	$0.518837\pi$
$884$	0	0
$885$	−2.78308e13	−1.52504
$886$	0	0
$887$	1.78709e13	0.969373	0.484686	−	0.874688i	$-0.338934\pi$
$887$	1.78709e13	0.969373	0.484686	+	0.874688i	$0.338934\pi$
$888$	0	0
$889$	1.65146e13	0.886768
$890$	0	0
$891$	−1.37672e13	−0.731805
$892$	0	0
$893$	−1.48667e13	−0.782316
$894$	0	0
$895$	3.46973e13	1.80755
$896$	0	0
$897$	3.55335e12	0.183262
$898$	0	0
$899$	−2.32109e13	−1.18515
$900$	0	0
$901$	3.88311e12	0.196299
$902$	0	0
$903$	4.68787e13	2.34629
$904$	0	0
$905$	3.33763e13	1.65394
$906$	0	0
$907$	1.05287e13	0.516587	0.258294	−	0.966066i	$-0.416840\pi$
$907$	1.05287e13	0.516587	0.258294	+	0.966066i	$0.416840\pi$
$908$	0	0
$909$	−7.46501e12	−0.362654
$910$	0	0
$911$	3.65775e13	1.75947	0.879734	−	0.475466i	$-0.157720\pi$
$911$	3.65775e13	1.75947	0.879734	+	0.475466i	$0.157720\pi$
$912$	0	0
$913$	−2.72282e13	−1.29688
$914$	0	0
$915$	−5.68015e13	−2.67895
$916$	0	0
$917$	2.40096e13	1.12130
$918$	0	0
$919$	2.88133e13	1.33252	0.666258	−	0.745721i	$-0.267895\pi$
$919$	2.88133e13	1.33252	0.666258	+	0.745721i	$0.267895\pi$
$920$	0	0
$921$	−2.28084e13	−1.04455
$922$	0	0
$923$	−2.70422e12	−0.122641
$924$	0	0
$925$	−5.62218e12	−0.252503
$926$	0	0
$927$	−3.64950e13	−1.62321
$928$	0	0
$929$	2.06330e13	0.908849	0.454425	−	0.890785i	$-0.349845\pi$
$929$	2.06330e13	0.908849	0.454425	+	0.890785i	$0.349845\pi$
$930$	0	0
$931$	−2.32006e12	−0.101210
$932$	0	0
$933$	3.64735e13	1.57583
$934$	0	0
$935$	−6.39092e12	−0.273471
$936$	0	0
$937$	2.84504e13	1.20576	0.602879	−	0.797832i	$-0.294020\pi$
$937$	2.84504e13	1.20576	0.602879	+	0.797832i	$0.294020\pi$
$938$	0	0
$939$	−1.96778e13	−0.826003
$940$	0	0
$941$	−4.21717e13	−1.75335	−0.876674	−	0.481084i	$-0.840243\pi$
$941$	−4.21717e13	−1.75335	−0.876674	+	0.481084i	$0.840243\pi$
$942$	0	0
$943$	−2.10047e13	−0.864995
$944$	0	0
$945$	−2.61705e12	−0.106750
$946$	0	0
$947$	−1.10763e13	−0.447529	−0.223765	−	0.974643i	$-0.571835\pi$
$947$	−1.10763e13	−0.447529	−0.223765	+	0.974643i	$0.571835\pi$
$948$	0	0
$949$	−1.71927e12	−0.0688093
$950$	0	0
$951$	3.93836e13	1.56136
$952$	0	0
$953$	−1.68223e13	−0.660642	−0.330321	−	0.943869i	$-0.607157\pi$
$953$	−1.68223e13	−0.660642	−0.330321	+	0.943869i	$0.607157\pi$
$954$	0	0
$955$	4.80115e13	1.86780
$956$	0	0
$957$	−3.71132e13	−1.43029
$958$	0	0
$959$	−1.90701e13	−0.728063
$960$	0	0
$961$	−9.30836e12	−0.352061
$962$	0	0
$963$	−1.52640e13	−0.571941
$964$	0	0
$965$	−5.99328e13	−2.22480
$966$	0	0
$967$	3.22369e12	0.118559	0.0592795	−	0.998241i	$-0.481120\pi$
$967$	3.22369e12	0.118559	0.0592795	+	0.998241i	$0.481120\pi$
$968$	0	0
$969$	1.03514e13	0.377176
$970$	0	0
$971$	3.63067e13	1.31069	0.655346	−	0.755329i	$-0.272523\pi$
$971$	3.63067e13	1.31069	0.655346	+	0.755329i	$0.272523\pi$
$972$	0	0
$973$	−4.45147e13	−1.59219
$974$	0	0
$975$	−8.99009e12	−0.318598
$976$	0	0
$977$	2.12642e13	0.746662	0.373331	−	0.927698i	$-0.378216\pi$
$977$	2.12642e13	0.746662	0.373331	+	0.927698i	$0.378216\pi$
$978$	0	0
$979$	−2.48545e13	−0.864735
$980$	0	0
$981$	4.60015e13	1.58585
$982$	0	0
$983$	1.06078e13	0.362354	0.181177	−	0.983451i	$-0.442009\pi$
$983$	1.06078e13	0.362354	0.181177	+	0.983451i	$0.442009\pi$
$984$	0	0
$985$	−2.95221e13	−0.999274
$986$	0	0
$987$	3.68580e13	1.23624
$988$	0	0
$989$	−2.27605e13	−0.756483
$990$	0	0
$991$	−4.09012e13	−1.34711	−0.673557	−	0.739135i	$-0.735234\pi$
$991$	−4.09012e13	−1.34711	−0.673557	+	0.739135i	$0.735234\pi$
$992$	0	0
$993$	−4.75577e13	−1.55220
$994$	0	0
$995$	−2.73982e13	−0.886173
$996$	0	0
$997$	3.47353e13	1.11338	0.556689	−	0.830721i	$-0.312072\pi$
$997$	3.47353e13	1.11338	0.556689	+	0.830721i	$0.312072\pi$
$998$	0	0
$999$	7.24726e11	0.0230213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	52.10.a.b.1.5	✓	5
4.3	odd	2		208.10.a.i.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.10.a.b.1.5	✓	5	1.1	even	1	trivial
208.10.a.i.1.1		5	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:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	52.a (trivial)

\(f(q)\)	\(=\)	\(q+195.721 q^{3} +1887.16 q^{5} +6689.32 q^{7} +18623.8 q^{9} +33813.8 q^{11} -28561.0 q^{13} +369357. q^{15} -100152. q^{17} -528083. q^{19} +1.30924e6 q^{21} -635662. q^{23} +1.60825e6 q^{25} -207311. q^{27} -5.60786e6 q^{29} +4.13899e6 q^{31} +6.61807e6 q^{33} +1.26238e7 q^{35} -3.49585e6 q^{37} -5.58999e6 q^{39} +3.30438e7 q^{41} +3.58060e7 q^{43} +3.51461e7 q^{45} +2.81522e7 q^{47} +4.39336e6 q^{49} -1.96019e7 q^{51} -3.87720e7 q^{53} +6.38120e7 q^{55} -1.03357e8 q^{57} -7.53492e7 q^{59} -1.53785e8 q^{61} +1.24580e8 q^{63} -5.38992e7 q^{65} +1.69378e7 q^{67} -1.24413e8 q^{69} +9.46824e7 q^{71} +6.01966e7 q^{73} +3.14768e8 q^{75} +2.26191e8 q^{77} +1.39281e8 q^{79} -4.07147e8 q^{81} -8.05240e8 q^{83} -1.89003e8 q^{85} -1.09758e9 q^{87} -7.35041e8 q^{89} -1.91054e8 q^{91} +8.10089e8 q^{93} -9.96577e8 q^{95} +1.43090e9 q^{97} +6.29740e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 147 q^{3} + 1721 q^{5} + 3317 q^{7} - 3204 q^{9} + 6562 q^{11} - 142805 q^{13} + 80283 q^{15} - 146493 q^{17} + 110250 q^{19} + 2250015 q^{21} + 1675652 q^{23} + 6048610 q^{25} + 4937895 q^{27} + 7019714 q^{29}+ \cdots + 302611248 q^{99}+O(q^{100}) \) 5 * q - 147 * q^3 + 1721 * q^5 + 3317 * q^7 - 3204 * q^9 + 6562 * q^11 - 142805 * q^13 + 80283 * q^15 - 146493 * q^17 + 110250 * q^19 + 2250015 * q^21 + 1675652 * q^23 + 6048610 * q^25 + 4937895 * q^27 + 7019714 * q^29 + 14112296 * q^31 + 21400890 * q^33 + 19283879 * q^35 + 28285653 * q^37 + 4198467 * q^39 + 55512492 * q^41 + 61336547 * q^43 + 90036738 * q^45 + 78843745 * q^47 + 88081704 * q^49 + 10216377 * q^51 + 105957576 * q^53 + 71100802 * q^55 + 21427866 * q^57 - 59686630 * q^59 - 41439220 * q^61 - 49300830 * q^63 - 49153481 * q^65 - 214570942 * q^67 - 432347172 * q^69 - 477669785 * q^71 - 201698114 * q^73 - 727786788 * q^75 - 521247014 * q^77 - 582737180 * q^79 - 855021987 * q^81 - 1134244092 * q^83 - 552245895 * q^85 - 2198696622 * q^87 - 476858778 * q^89 - 94736837 * q^91 + 1038474204 * q^93 - 866887902 * q^95 + 1473121030 * q^97 + 302611248 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more