LMFDB - Embedded newform 64.14.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,14,Mod(1,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 14, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	64.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:	$68.6277945292$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2195x - 37995 $ x^3 - x^2 - 2195*x - 37995
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-23.3851$ of defining polynomial
Character	$\chi$	$=$	64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-932.324 q^{3} +41284.4 q^{5} +324537. q^{7} -725095. q^{9} -5.64388e6 q^{11} +2.79852e6 q^{13} -3.84904e7 q^{15} -1.76212e8 q^{17} +1.19940e8 q^{19} -3.02574e8 q^{21} +8.11512e8 q^{23} +4.83695e8 q^{25} +2.16245e9 q^{27} +2.33918e8 q^{29} -4.37444e9 q^{31} +5.26192e9 q^{33} +1.33983e10 q^{35} +1.01530e10 q^{37} -2.60913e9 q^{39} -3.06079e10 q^{41} +4.90073e10 q^{43} -2.99351e10 q^{45} -1.39240e11 q^{47} +8.43522e9 q^{49} +1.64286e11 q^{51} -2.02631e11 q^{53} -2.33004e11 q^{55} -1.11823e11 q^{57} -3.65225e11 q^{59} +4.13236e11 q^{61} -2.35320e11 q^{63} +1.15535e11 q^{65} +7.06874e11 q^{67} -7.56592e11 q^{69} +1.17586e12 q^{71} -2.21587e12 q^{73} -4.50961e11 q^{75} -1.83165e12 q^{77} +3.45937e12 q^{79} -8.60067e11 q^{81} -5.28507e12 q^{83} -7.27479e12 q^{85} -2.18087e11 q^{87} -4.65365e12 q^{89} +9.08223e11 q^{91} +4.07840e12 q^{93} +4.95164e12 q^{95} -4.76860e12 q^{97} +4.09235e12 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 520 q^{3} - 11594 q^{5} - 96304 q^{7} - 196793 q^{9} + 1923816 q^{11} + 11211006 q^{13} + 10944368 q^{15} - 16892538 q^{17} + 74665048 q^{19} + 115146880 q^{21} - 486642576 q^{23} + 442696461 q^{25}+ \cdots + 2082116453576 q^{99}+O(q^{100}) $ 3 * q - 520 * q^3 - 11594 * q^5 - 96304 * q^7 - 196793 * q^9 + 1923816 * q^11 + 11211006 * q^13 + 10944368 * q^15 - 16892538 * q^17 + 74665048 * q^19 + 115146880 * q^21 - 486642576 * q^23 + 442696461 * q^25 + 3111524144 * q^27 - 1747468946 * q^29 - 6471356864 * q^31 - 2901436352 * q^33 + 32905782432 * q^35 + 1189828278 * q^37 - 73495249232 * q^39 - 12181275970 * q^41 + 151151885096 * q^43 - 19024866434 * q^45 - 273552663072 * q^47 - 26716022709 * q^49 + 361685604336 * q^51 - 151381970362 * q^53 - 594621149104 * q^55 - 253183022656 * q^57 + 534333367560 * q^59 - 598414222482 * q^61 - 138475736176 * q^63 - 1313304118564 * q^65 - 703511405320 * q^67 - 1246094112384 * q^69 + 1867178467152 * q^71 - 2138161799730 * q^73 - 3649896495992 * q^75 - 4774654840448 * q^77 + 6521880375328 * q^79 - 6029061199109 * q^81 - 5725896739752 * q^83 - 8753386407828 * q^85 + 9313876797488 * q^87 - 11930629520834 * q^89 - 10949018828768 * q^91 - 15654956180992 * q^93 + 3955366679088 * q^95 - 23076828823050 * q^97 + 2082116453576 * 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
$3$	−932.324	−0.738378	−0.369189	−	0.929354i	$-0.620364\pi$
$3$	−932.324	−0.738378	−0.369189	+	0.929354i	$0.620364\pi$
$4$	0	0
$5$	41284.4	1.18163	0.590814	−	0.806808i	$-0.298807\pi$
$5$	41284.4	1.18163	0.590814	+	0.806808i	$0.298807\pi$
$6$	0	0
$7$	324537.	1.04262	0.521311	−	0.853367i	$-0.325443\pi$
$7$	324537.	1.04262	0.521311	+	0.853367i	$0.325443\pi$
$8$	0	0
$9$	−725095.	−0.454798
$10$	0	0
$11$	−5.64388e6	−0.960561	−0.480281	−	0.877115i	$-0.659465\pi$
$11$	−5.64388e6	−0.960561	−0.480281	+	0.877115i	$0.659465\pi$
$12$	0	0
$13$	2.79852e6	0.160804	0.0804019	−	0.996763i	$-0.474380\pi$
$13$	2.79852e6	0.160804	0.0804019	+	0.996763i	$0.474380\pi$
$14$	0	0
$15$	−3.84904e7	−0.872488
$16$	0	0
$17$	−1.76212e8	−1.77059	−0.885293	−	0.465034i	$-0.846042\pi$
$17$	−1.76212e8	−1.77059	−0.885293	+	0.465034i	$0.846042\pi$
$18$	0	0
$19$	1.19940e8	0.584878	0.292439	−	0.956284i	$-0.405533\pi$
$19$	1.19940e8	0.584878	0.292439	+	0.956284i	$0.405533\pi$
$20$	0	0
$21$	−3.02574e8	−0.769849
$22$	0	0
$23$	8.11512e8	1.14305	0.571524	−	0.820586i	$-0.306353\pi$
$23$	8.11512e8	1.14305	0.571524	+	0.820586i	$0.306353\pi$
$24$	0	0
$25$	4.83695e8	0.396243
$26$	0	0
$27$	2.16245e9	1.07419
$28$	0	0
$29$	2.33918e8	0.0730258	0.0365129	−	0.999333i	$-0.488375\pi$
$29$	2.33918e8	0.0730258	0.0365129	+	0.999333i	$0.488375\pi$
$30$	0	0
$31$	−4.37444e9	−0.885262	−0.442631	−	0.896704i	$-0.645955\pi$
$31$	−4.37444e9	−0.885262	−0.442631	+	0.896704i	$0.645955\pi$
$32$	0	0
$33$	5.26192e9	0.709257
$34$	0	0
$35$	1.33983e10	1.23199
$36$	0	0
$37$	1.01530e10	0.650556	0.325278	−	0.945618i	$-0.394542\pi$
$37$	1.01530e10	0.650556	0.325278	+	0.945618i	$0.394542\pi$
$38$	0	0
$39$	−2.60913e9	−0.118734
$40$	0	0
$41$	−3.06079e10	−1.00632	−0.503162	−	0.864192i	$-0.667830\pi$
$41$	−3.06079e10	−1.00632	−0.503162	+	0.864192i	$0.667830\pi$
$42$	0	0
$43$	4.90073e10	1.18227	0.591134	−	0.806573i	$-0.298680\pi$
$43$	4.90073e10	1.18227	0.591134	+	0.806573i	$0.298680\pi$
$44$	0	0
$45$	−2.99351e10	−0.537402
$46$	0	0
$47$	−1.39240e11	−1.88421	−0.942105	−	0.335318i	$-0.891156\pi$
$47$	−1.39240e11	−1.88421	−0.942105	+	0.335318i	$0.891156\pi$
$48$	0	0
$49$	8.43522e9	0.0870606
$50$	0	0
$51$	1.64286e11	1.30736
$52$	0	0
$53$	−2.02631e11	−1.25578	−0.627890	−	0.778302i	$-0.716081\pi$
$53$	−2.02631e11	−1.25578	−0.627890	+	0.778302i	$0.716081\pi$
$54$	0	0
$55$	−2.33004e11	−1.13503
$56$	0	0
$57$	−1.11823e11	−0.431861
$58$	0	0
$59$	−3.65225e11	−1.12725	−0.563627	−	0.826029i	$-0.690595\pi$
$59$	−3.65225e11	−1.12725	−0.563627	+	0.826029i	$0.690595\pi$
$60$	0	0
$61$	4.13236e11	1.02696	0.513481	−	0.858101i	$-0.328356\pi$
$61$	4.13236e11	1.02696	0.513481	+	0.858101i	$0.328356\pi$
$62$	0	0
$63$	−2.35320e11	−0.474182
$64$	0	0
$65$	1.15535e11	0.190010
$66$	0	0
$67$	7.06874e11	0.954676	0.477338	−	0.878720i	$-0.341602\pi$
$67$	7.06874e11	0.954676	0.477338	+	0.878720i	$0.341602\pi$
$68$	0	0
$69$	−7.56592e11	−0.844001
$70$	0	0
$71$	1.17586e12	1.08937	0.544686	−	0.838640i	$-0.316649\pi$
$71$	1.17586e12	1.08937	0.544686	+	0.838640i	$0.316649\pi$
$72$	0	0
$73$	−2.21587e12	−1.71374	−0.856871	−	0.515530i	$-0.827595\pi$
$73$	−2.21587e12	−1.71374	−0.856871	+	0.515530i	$0.827595\pi$
$74$	0	0
$75$	−4.50961e11	−0.292577
$76$	0	0
$77$	−1.83165e12	−1.00150
$78$	0	0
$79$	3.45937e12	1.60111	0.800555	−	0.599259i	$-0.204538\pi$
$79$	3.45937e12	1.60111	0.800555	+	0.599259i	$0.204538\pi$
$80$	0	0
$81$	−8.60067e11	−0.338361
$82$	0	0
$83$	−5.28507e12	−1.77436	−0.887182	−	0.461419i	$-0.847341\pi$
$83$	−5.28507e12	−1.77436	−0.887182	+	0.461419i	$0.847341\pi$
$84$	0	0
$85$	−7.27479e12	−2.09217
$86$	0	0
$87$	−2.18087e11	−0.0539207
$88$	0	0
$89$	−4.65365e12	−0.992563	−0.496282	−	0.868162i	$-0.665302\pi$
$89$	−4.65365e12	−0.992563	−0.496282	+	0.868162i	$0.665302\pi$
$90$	0	0
$91$	9.08223e11	0.167658
$92$	0	0
$93$	4.07840e12	0.653658
$94$	0	0
$95$	4.95164e12	0.691108
$96$	0	0
$97$	−4.76860e12	−0.581266	−0.290633	−	0.956835i	$-0.593866\pi$
$97$	−4.76860e12	−0.581266	−0.290633	+	0.956835i	$0.593866\pi$
$98$	0	0
$99$	4.09235e12	0.436861
$100$	0	0
$101$	−2.83778e12	−0.266005	−0.133002	−	0.991116i	$-0.542462\pi$
$101$	−2.83778e12	−0.266005	−0.133002	+	0.991116i	$0.542462\pi$
$102$	0	0
$103$	−1.09776e13	−0.905871	−0.452935	−	0.891543i	$-0.649623\pi$
$103$	−1.09776e13	−0.905871	−0.452935	+	0.891543i	$0.649623\pi$
$104$	0	0
$105$	−1.24916e13	−0.909675
$106$	0	0
$107$	−1.51150e13	−0.973674	−0.486837	−	0.873493i	$-0.661850\pi$
$107$	−1.51150e13	−0.973674	−0.486837	+	0.873493i	$0.661850\pi$
$108$	0	0
$109$	−3.31531e13	−1.89344	−0.946721	−	0.322054i	$-0.895627\pi$
$109$	−3.31531e13	−1.89344	−0.946721	+	0.322054i	$0.895627\pi$
$110$	0	0
$111$	−9.46591e12	−0.480356
$112$	0	0
$113$	−1.04483e13	−0.472101	−0.236050	−	0.971741i	$-0.575853\pi$
$113$	−1.04483e13	−0.472101	−0.236050	+	0.971741i	$0.575853\pi$
$114$	0	0
$115$	3.35028e13	1.35066
$116$	0	0
$117$	−2.02919e12	−0.0731333
$118$	0	0
$119$	−5.71872e13	−1.84605
$120$	0	0
$121$	−2.66938e12	−0.0773224
$122$	0	0
$123$	2.85365e13	0.743048
$124$	0	0
$125$	−3.04269e13	−0.713416
$126$	0	0
$127$	2.38379e13	0.504132	0.252066	−	0.967710i	$-0.418890\pi$
$127$	2.38379e13	0.504132	0.252066	+	0.967710i	$0.418890\pi$
$128$	0	0
$129$	−4.56907e13	−0.872961
$130$	0	0
$131$	5.97675e13	1.03324	0.516621	−	0.856214i	$-0.327190\pi$
$131$	5.97675e13	1.03324	0.516621	+	0.856214i	$0.327190\pi$
$132$	0	0
$133$	3.89249e13	0.609807
$134$	0	0
$135$	8.92753e13	1.26929
$136$	0	0
$137$	1.01680e14	1.31387	0.656935	−	0.753947i	$-0.271853\pi$
$137$	1.01680e14	1.31387	0.656935	+	0.753947i	$0.271853\pi$
$138$	0	0
$139$	−8.23992e13	−0.969006	−0.484503	−	0.874790i	$-0.661000\pi$
$139$	−8.23992e13	−0.969006	−0.484503	+	0.874790i	$0.661000\pi$
$140$	0	0
$141$	1.29817e14	1.39126
$142$	0	0
$143$	−1.57945e13	−0.154462
$144$	0	0
$145$	9.65715e12	0.0862893
$146$	0	0
$147$	−7.86436e12	−0.0642836
$148$	0	0
$149$	7.24250e13	0.542223	0.271111	−	0.962548i	$-0.412609\pi$
$149$	7.24250e13	0.542223	0.271111	+	0.962548i	$0.412609\pi$
$150$	0	0
$151$	−1.47164e13	−0.101030	−0.0505150	−	0.998723i	$-0.516086\pi$
$151$	−1.47164e13	−0.101030	−0.0505150	+	0.998723i	$0.516086\pi$
$152$	0	0
$153$	1.27770e14	0.805259
$154$	0	0
$155$	−1.80596e14	−1.04605
$156$	0	0
$157$	−1.94720e14	−1.03768	−0.518840	−	0.854871i	$-0.673636\pi$
$157$	−1.94720e14	−1.03768	−0.518840	+	0.854871i	$0.673636\pi$
$158$	0	0
$159$	1.88918e14	0.927240
$160$	0	0
$161$	2.63366e14	1.19177
$162$	0	0
$163$	−2.97079e14	−1.24066	−0.620329	−	0.784342i	$-0.713001\pi$
$163$	−2.97079e14	−1.24066	−0.620329	+	0.784342i	$0.713001\pi$
$164$	0	0
$165$	2.17235e14	0.838078
$166$	0	0
$167$	−5.12321e11	−0.00182762	−0.000913808	−	1.00000i	$-0.500291\pi$
$167$	−5.12321e11	−0.00182762	−0.000913808		1.00000i	$0.500291\pi$
$168$	0	0
$169$	−2.95043e14	−0.974142
$170$	0	0
$171$	−8.69679e13	−0.266001
$172$	0	0
$173$	4.77299e14	1.35360	0.676800	−	0.736167i	$-0.263366\pi$
$173$	4.77299e14	1.35360	0.676800	+	0.736167i	$0.263366\pi$
$174$	0	0
$175$	1.56977e14	0.413132
$176$	0	0
$177$	3.40508e14	0.832340
$178$	0	0
$179$	2.62846e14	0.597250	0.298625	−	0.954371i	$-0.403472\pi$
$179$	2.62846e14	0.597250	0.298625	+	0.954371i	$0.403472\pi$
$180$	0	0
$181$	1.81052e13	0.0382731	0.0191365	−	0.999817i	$-0.493908\pi$
$181$	1.81052e13	0.0382731	0.0191365	+	0.999817i	$0.493908\pi$
$182$	0	0
$183$	−3.85270e14	−0.758286
$184$	0	0
$185$	4.19161e14	0.768714
$186$	0	0
$187$	9.94517e14	1.70076
$188$	0	0
$189$	7.01795e14	1.11997
$190$	0	0
$191$	−3.16940e14	−0.472345	−0.236173	−	0.971711i	$-0.575893\pi$
$191$	−3.16940e14	−0.472345	−0.236173	+	0.971711i	$0.575893\pi$
$192$	0	0
$193$	6.15909e14	0.857816	0.428908	−	0.903348i	$-0.358898\pi$
$193$	6.15909e14	0.857816	0.428908	+	0.903348i	$0.358898\pi$
$194$	0	0
$195$	−1.07716e14	−0.140299
$196$	0	0
$197$	−2.07479e14	−0.252897	−0.126448	−	0.991973i	$-0.540358\pi$
$197$	−2.07479e14	−0.252897	−0.126448	+	0.991973i	$0.540358\pi$
$198$	0	0
$199$	−3.96818e14	−0.452946	−0.226473	−	0.974017i	$-0.572720\pi$
$199$	−3.96818e14	−0.452946	−0.226473	+	0.974017i	$0.572720\pi$
$200$	0	0
$201$	−6.59036e14	−0.704912
$202$	0	0
$203$	7.59150e13	0.0761383
$204$	0	0
$205$	−1.26363e15	−1.18910
$206$	0	0
$207$	−5.88424e14	−0.519856
$208$	0	0
$209$	−6.76926e14	−0.561811
$210$	0	0
$211$	9.25973e14	0.722374	0.361187	−	0.932493i	$-0.382372\pi$
$211$	9.25973e14	0.722374	0.361187	+	0.932493i	$0.382372\pi$
$212$	0	0
$213$	−1.09628e15	−0.804368
$214$	0	0
$215$	2.02324e15	1.39700
$216$	0	0
$217$	−1.41967e15	−0.922993
$218$	0	0
$219$	2.06591e15	1.26539
$220$	0	0
$221$	−4.93132e14	−0.284717
$222$	0	0
$223$	−3.12454e15	−1.70139	−0.850696	−	0.525657i	$-0.823819\pi$
$223$	−3.12454e15	−1.70139	−0.850696	+	0.525657i	$0.823819\pi$
$224$	0	0
$225$	−3.50725e14	−0.180211
$226$	0	0
$227$	−2.31376e15	−1.12241	−0.561203	−	0.827679i	$-0.689661\pi$
$227$	−2.31376e15	−1.12241	−0.561203	+	0.827679i	$0.689661\pi$
$228$	0	0
$229$	1.48501e15	0.680455	0.340227	−	0.940343i	$-0.389496\pi$
$229$	1.48501e15	0.680455	0.340227	+	0.940343i	$0.389496\pi$
$230$	0	0
$231$	1.70769e15	0.739487
$232$	0	0
$233$	−2.35057e15	−0.962407	−0.481204	−	0.876609i	$-0.659800\pi$
$233$	−2.35057e15	−0.962407	−0.481204	+	0.876609i	$0.659800\pi$
$234$	0	0
$235$	−5.74845e15	−2.22643
$236$	0	0
$237$	−3.22526e15	−1.18222
$238$	0	0
$239$	1.29200e15	0.448411	0.224205	−	0.974542i	$-0.428021\pi$
$239$	1.29200e15	0.448411	0.224205	+	0.974542i	$0.428021\pi$
$240$	0	0
$241$	−1.79077e15	−0.588747	−0.294374	−	0.955690i	$-0.595111\pi$
$241$	−1.79077e15	−0.588747	−0.294374	+	0.955690i	$0.595111\pi$
$242$	0	0
$243$	−2.64578e15	−0.824353
$244$	0	0
$245$	3.48243e14	0.102873
$246$	0	0
$247$	3.35654e14	0.0940507
$248$	0	0
$249$	4.92739e15	1.31015
$250$	0	0
$251$	8.75475e14	0.220986	0.110493	−	0.993877i	$-0.464757\pi$
$251$	8.75475e14	0.220986	0.110493	+	0.993877i	$0.464757\pi$
$252$	0	0
$253$	−4.58008e15	−1.09797
$254$	0	0
$255$	6.78246e15	1.54481
$256$	0	0
$257$	−1.22938e15	−0.266147	−0.133073	−	0.991106i	$-0.542485\pi$
$257$	−1.22938e15	−0.266147	−0.133073	+	0.991106i	$0.542485\pi$
$258$	0	0
$259$	3.29503e15	0.678284
$260$	0	0
$261$	−1.69613e14	−0.0332120
$262$	0	0
$263$	−2.40399e15	−0.447941	−0.223971	−	0.974596i	$-0.571902\pi$
$263$	−2.40399e15	−0.447941	−0.223971	+	0.974596i	$0.571902\pi$
$264$	0	0
$265$	−8.36551e15	−1.48386
$266$	0	0
$267$	4.33871e15	0.732887
$268$	0	0
$269$	−4.27020e15	−0.687161	−0.343581	−	0.939123i	$-0.611640\pi$
$269$	−4.27020e15	−0.687161	−0.343581	+	0.939123i	$0.611640\pi$
$270$	0	0
$271$	1.62300e14	0.0248896	0.0124448	−	0.999923i	$-0.496039\pi$
$271$	1.62300e14	0.0248896	0.0124448	+	0.999923i	$0.496039\pi$
$272$	0	0
$273$	−8.46758e14	−0.123795
$274$	0	0
$275$	−2.72992e15	−0.380616
$276$	0	0
$277$	1.17855e16	1.56758	0.783792	−	0.621023i	$-0.213283\pi$
$277$	1.17855e16	1.56758	0.783792	+	0.621023i	$0.213283\pi$
$278$	0	0
$279$	3.17189e15	0.402615
$280$	0	0
$281$	−1.73366e15	−0.210074	−0.105037	−	0.994468i	$-0.533496\pi$
$281$	−1.73366e15	−0.210074	−0.105037	+	0.994468i	$0.533496\pi$
$282$	0	0
$283$	3.84405e15	0.444813	0.222406	−	0.974954i	$-0.428609\pi$
$283$	3.84405e15	0.444813	0.222406	+	0.974954i	$0.428609\pi$
$284$	0	0
$285$	−4.61654e15	−0.510299
$286$	0	0
$287$	−9.93338e15	−1.04922
$288$	0	0
$289$	2.11460e16	2.13497
$290$	0	0
$291$	4.44588e15	0.429194
$292$	0	0
$293$	3.20731e15	0.296143	0.148071	−	0.988977i	$-0.452693\pi$
$293$	3.20731e15	0.296143	0.148071	+	0.988977i	$0.452693\pi$
$294$	0	0
$295$	−1.50781e16	−1.33199
$296$	0	0
$297$	−1.22046e16	−1.03183
$298$	0	0
$299$	2.27103e15	0.183806
$300$	0	0
$301$	1.59047e16	1.23266
$302$	0	0
$303$	2.64573e15	0.196412
$304$	0	0
$305$	1.70602e16	1.21349
$306$	0	0
$307$	1.90574e16	1.29916	0.649581	−	0.760292i	$-0.274944\pi$
$307$	1.90574e16	1.29916	0.649581	+	0.760292i	$0.274944\pi$
$308$	0	0
$309$	1.02347e16	0.668875
$310$	0	0
$311$	−1.81260e16	−1.13595	−0.567976	−	0.823045i	$-0.692273\pi$
$311$	−1.81260e16	−1.13595	−0.567976	+	0.823045i	$0.692273\pi$
$312$	0	0
$313$	3.11747e16	1.87398	0.936990	−	0.349357i	$-0.113600\pi$
$313$	3.11747e16	1.87398	0.936990	+	0.349357i	$0.113600\pi$
$314$	0	0
$315$	−9.71504e15	−0.560307
$316$	0	0
$317$	3.06718e15	0.169767	0.0848837	−	0.996391i	$-0.472948\pi$
$317$	3.06718e15	0.169767	0.0848837	+	0.996391i	$0.472948\pi$
$318$	0	0
$319$	−1.32020e15	−0.0701458
$320$	0	0
$321$	1.40921e16	0.718940
$322$	0	0
$323$	−2.11348e16	−1.03558
$324$	0	0
$325$	1.35363e15	0.0637174
$326$	0	0
$327$	3.09095e16	1.39808
$328$	0	0
$329$	−4.51887e16	−1.96452
$330$	0	0
$331$	−1.05739e16	−0.441930	−0.220965	−	0.975282i	$-0.570921\pi$
$331$	−1.05739e16	−0.441930	−0.220965	+	0.975282i	$0.570921\pi$
$332$	0	0
$333$	−7.36191e15	−0.295871
$334$	0	0
$335$	2.91829e16	1.12807
$336$	0	0
$337$	5.91789e15	0.220076	0.110038	−	0.993927i	$-0.464903\pi$
$337$	5.91789e15	0.220076	0.110038	+	0.993927i	$0.464903\pi$
$338$	0	0
$339$	9.74118e15	0.348589
$340$	0	0
$341$	2.46888e16	0.850348
$342$	0	0
$343$	−2.87065e16	−0.951851
$344$	0	0
$345$	−3.12354e16	−0.997295
$346$	0	0
$347$	−5.50660e16	−1.69333	−0.846666	−	0.532125i	$-0.821393\pi$
$347$	−5.50660e16	−1.69333	−0.846666	+	0.532125i	$0.821393\pi$
$348$	0	0
$349$	−1.96517e16	−0.582151	−0.291075	−	0.956700i	$-0.594013\pi$
$349$	−1.96517e16	−0.582151	−0.291075	+	0.956700i	$0.594013\pi$
$350$	0	0
$351$	6.05166e15	0.172734
$352$	0	0
$353$	6.92289e16	1.90437	0.952186	−	0.305520i	$-0.0988302\pi$
$353$	6.92289e16	1.90437	0.952186	+	0.305520i	$0.0988302\pi$
$354$	0	0
$355$	4.85446e16	1.28723
$356$	0	0
$357$	5.33170e16	1.36308
$358$	0	0
$359$	7.65291e16	1.88674	0.943371	−	0.331740i	$-0.107636\pi$
$359$	7.65291e16	1.88674	0.943371	+	0.331740i	$0.107636\pi$
$360$	0	0
$361$	−2.76674e16	−0.657918
$362$	0	0
$363$	2.48873e15	0.0570932
$364$	0	0
$365$	−9.14808e16	−2.02501
$366$	0	0
$367$	−8.08198e15	−0.172659	−0.0863293	−	0.996267i	$-0.527514\pi$
$367$	−8.08198e15	−0.172659	−0.0863293	+	0.996267i	$0.527514\pi$
$368$	0	0
$369$	2.21936e16	0.457674
$370$	0	0
$371$	−6.57614e16	−1.30930
$372$	0	0
$373$	6.86553e15	0.131998	0.0659989	−	0.997820i	$-0.478977\pi$
$373$	6.86553e15	0.131998	0.0659989	+	0.997820i	$0.478977\pi$
$374$	0	0
$375$	2.83677e16	0.526770
$376$	0	0
$377$	6.54624e14	0.0117428
$378$	0	0
$379$	5.95589e16	1.03227	0.516133	−	0.856508i	$-0.327371\pi$
$379$	5.95589e16	1.03227	0.516133	+	0.856508i	$0.327371\pi$
$380$	0	0
$381$	−2.22247e16	−0.372240
$382$	0	0
$383$	4.30499e15	0.0696915	0.0348457	−	0.999393i	$-0.488906\pi$
$383$	4.30499e15	0.0696915	0.0348457	+	0.999393i	$0.488906\pi$
$384$	0	0
$385$	−7.56183e16	−1.18340
$386$	0	0
$387$	−3.55350e16	−0.537693
$388$	0	0
$389$	4.41199e16	0.645597	0.322799	−	0.946468i	$-0.395376\pi$
$389$	4.41199e16	0.645597	0.322799	+	0.946468i	$0.395376\pi$
$390$	0	0
$391$	−1.42998e17	−2.02386
$392$	0	0
$393$	−5.57227e16	−0.762922
$394$	0	0
$395$	1.42818e17	1.89192
$396$	0	0
$397$	−1.24787e17	−1.59967	−0.799833	−	0.600222i	$-0.795079\pi$
$397$	−1.24787e17	−1.59967	−0.799833	+	0.600222i	$0.795079\pi$
$398$	0	0
$399$	−3.62907e16	−0.450268
$400$	0	0
$401$	−8.00680e16	−0.961658	−0.480829	−	0.876814i	$-0.659664\pi$
$401$	−8.00680e16	−0.961658	−0.480829	+	0.876814i	$0.659664\pi$
$402$	0	0
$403$	−1.22420e16	−0.142354
$404$	0	0
$405$	−3.55073e16	−0.399816
$406$	0	0
$407$	−5.73024e16	−0.624898
$408$	0	0
$409$	−1.05304e17	−1.11235	−0.556176	−	0.831064i	$-0.687732\pi$
$409$	−1.05304e17	−1.11235	−0.556176	+	0.831064i	$0.687732\pi$
$410$	0	0
$411$	−9.47988e16	−0.970132
$412$	0	0
$413$	−1.18529e17	−1.17530
$414$	0	0
$415$	−2.18191e17	−2.09664
$416$	0	0
$417$	7.68227e16	0.715493
$418$	0	0
$419$	1.37042e16	0.123726	0.0618632	−	0.998085i	$-0.480296\pi$
$419$	1.37042e16	0.123726	0.0618632	+	0.998085i	$0.480296\pi$
$420$	0	0
$421$	8.00501e16	0.700694	0.350347	−	0.936620i	$-0.386064\pi$
$421$	8.00501e16	0.700694	0.350347	+	0.936620i	$0.386064\pi$
$422$	0	0
$423$	1.00963e17	0.856935
$424$	0	0
$425$	−8.52328e16	−0.701583
$426$	0	0
$427$	1.34110e17	1.07073
$428$	0	0
$429$	1.47256e16	0.114051
$430$	0	0
$431$	2.25261e17	1.69272	0.846359	−	0.532613i	$-0.178790\pi$
$431$	2.25261e17	1.69272	0.846359	+	0.532613i	$0.178790\pi$
$432$	0	0
$433$	1.69110e17	1.23310	0.616550	−	0.787315i	$-0.288530\pi$
$433$	1.69110e17	1.23310	0.616550	+	0.787315i	$0.288530\pi$
$434$	0	0
$435$	−9.00359e15	−0.0637141
$436$	0	0
$437$	9.73327e16	0.668543
$438$	0	0
$439$	1.03678e17	0.691299	0.345649	−	0.938364i	$-0.387659\pi$
$439$	1.03678e17	0.691299	0.345649	+	0.938364i	$0.387659\pi$
$440$	0	0
$441$	−6.11634e15	−0.0395950
$442$	0	0
$443$	2.87099e16	0.180471	0.0902357	−	0.995920i	$-0.471238\pi$
$443$	2.87099e16	0.180471	0.0902357	+	0.995920i	$0.471238\pi$
$444$	0	0
$445$	−1.92123e17	−1.17284
$446$	0	0
$447$	−6.75236e16	−0.400365
$448$	0	0
$449$	1.52452e17	0.878074	0.439037	−	0.898469i	$-0.355320\pi$
$449$	1.52452e17	0.878074	0.439037	+	0.898469i	$0.355320\pi$
$450$	0	0
$451$	1.72747e17	0.966636
$452$	0	0
$453$	1.37204e16	0.0745983
$454$	0	0
$455$	3.74954e16	0.198109
$456$	0	0
$457$	−5.20481e16	−0.267270	−0.133635	−	0.991031i	$-0.542665\pi$
$457$	−5.20481e16	−0.267270	−0.133635	+	0.991031i	$0.542665\pi$
$458$	0	0
$459$	−3.81049e17	−1.90195
$460$	0	0
$461$	2.88623e17	1.40047	0.700236	−	0.713911i	$-0.253078\pi$
$461$	2.88623e17	1.40047	0.700236	+	0.713911i	$0.253078\pi$
$462$	0	0
$463$	2.13822e16	0.100873	0.0504365	−	0.998727i	$-0.483939\pi$
$463$	2.13822e16	0.100873	0.0504365	+	0.998727i	$0.483939\pi$
$464$	0	0
$465$	1.68374e17	0.772380
$466$	0	0
$467$	−2.29228e17	−1.02261	−0.511303	−	0.859400i	$-0.670837\pi$
$467$	−2.29228e17	−1.02261	−0.511303	+	0.859400i	$0.670837\pi$
$468$	0	0
$469$	2.29407e17	0.995366
$470$	0	0
$471$	1.81542e17	0.766200
$472$	0	0
$473$	−2.76591e17	−1.13564
$474$	0	0
$475$	5.80144e16	0.231754
$476$	0	0
$477$	1.46927e17	0.571126
$478$	0	0
$479$	9.79968e16	0.370707	0.185354	−	0.982672i	$-0.440657\pi$
$479$	9.79968e16	0.370707	0.185354	+	0.982672i	$0.440657\pi$
$480$	0	0
$481$	2.84135e16	0.104612
$482$	0	0
$483$	−2.45542e17	−0.879974
$484$	0	0
$485$	−1.96869e17	−0.686840
$486$	0	0
$487$	2.19669e17	0.746158	0.373079	−	0.927800i	$-0.378302\pi$
$487$	2.19669e17	0.746158	0.373079	+	0.927800i	$0.378302\pi$
$488$	0	0
$489$	2.76974e17	0.916074
$490$	0	0
$491$	−2.11096e17	−0.679908	−0.339954	−	0.940442i	$-0.610411\pi$
$491$	−2.11096e17	−0.679908	−0.339954	+	0.940442i	$0.610411\pi$
$492$	0	0
$493$	−4.12191e16	−0.129298
$494$	0	0
$495$	1.68950e17	0.516207
$496$	0	0
$497$	3.81610e17	1.13580
$498$	0	0
$499$	5.29964e17	1.53671	0.768357	−	0.640021i	$-0.221075\pi$
$499$	5.29964e17	1.53671	0.768357	+	0.640021i	$0.221075\pi$
$500$	0	0
$501$	4.77649e14	0.00134947
$502$	0	0
$503$	−4.29152e17	−1.18146	−0.590730	−	0.806870i	$-0.701160\pi$
$503$	−4.29152e17	−1.18146	−0.590730	+	0.806870i	$0.701160\pi$
$504$	0	0
$505$	−1.17156e17	−0.314318
$506$	0	0
$507$	2.75076e17	0.719285
$508$	0	0
$509$	9.16821e16	0.233679	0.116839	−	0.993151i	$-0.462724\pi$
$509$	9.16821e16	0.233679	0.116839	+	0.993151i	$0.462724\pi$
$510$	0	0
$511$	−7.19132e17	−1.78679
$512$	0	0
$513$	2.59364e17	0.628271
$514$	0	0
$515$	−4.53204e17	−1.07040
$516$	0	0
$517$	7.85856e17	1.80990
$518$	0	0
$519$	−4.44997e17	−0.999469
$520$	0	0
$521$	4.41292e17	0.966675	0.483338	−	0.875434i	$-0.339424\pi$
$521$	4.41292e17	0.966675	0.483338	+	0.875434i	$0.339424\pi$
$522$	0	0
$523$	2.67498e17	0.571556	0.285778	−	0.958296i	$-0.407748\pi$
$523$	2.67498e17	0.571556	0.285778	+	0.958296i	$0.407748\pi$
$524$	0	0
$525$	−1.46353e17	−0.305047
$526$	0	0
$527$	7.70828e17	1.56743
$528$	0	0
$529$	1.54516e17	0.306557
$530$	0	0
$531$	2.64823e17	0.512673
$532$	0	0
$533$	−8.56567e16	−0.161821
$534$	0	0
$535$	−6.24013e17	−1.15052
$536$	0	0
$537$	−2.45057e17	−0.440996
$538$	0	0
$539$	−4.76073e16	−0.0836271
$540$	0	0
$541$	3.34334e17	0.573321	0.286660	−	0.958032i	$-0.407455\pi$
$541$	3.34334e17	0.573321	0.286660	+	0.958032i	$0.407455\pi$
$542$	0	0
$543$	−1.68800e16	−0.0282600
$544$	0	0
$545$	−1.36871e18	−2.23734
$546$	0	0
$547$	8.23548e17	1.31453	0.657266	−	0.753659i	$-0.271713\pi$
$547$	8.23548e17	1.31453	0.657266	+	0.753659i	$0.271713\pi$
$548$	0	0
$549$	−2.99636e17	−0.467061
$550$	0	0
$551$	2.80561e16	0.0427112
$552$	0	0
$553$	1.12269e18	1.66935
$554$	0	0
$555$	−3.90794e17	−0.567602
$556$	0	0
$557$	−8.21693e17	−1.16587	−0.582936	−	0.812518i	$-0.698096\pi$
$557$	−8.21693e17	−1.16587	−0.582936	+	0.812518i	$0.698096\pi$
$558$	0	0
$559$	1.37148e17	0.190113
$560$	0	0
$561$	−9.27212e17	−1.25580
$562$	0	0
$563$	8.88601e17	1.17599	0.587993	−	0.808866i	$-0.299918\pi$
$563$	8.88601e17	1.17599	0.587993	+	0.808866i	$0.299918\pi$
$564$	0	0
$565$	−4.31351e17	−0.557847
$566$	0	0
$567$	−2.79124e17	−0.352782
$568$	0	0
$569$	−9.25189e17	−1.14288	−0.571440	−	0.820644i	$-0.693615\pi$
$569$	−9.25189e17	−1.14288	−0.571440	+	0.820644i	$0.693615\pi$
$570$	0	0
$571$	2.85287e17	0.344467	0.172233	−	0.985056i	$-0.444902\pi$
$571$	2.85287e17	0.344467	0.172233	+	0.985056i	$0.444902\pi$
$572$	0	0
$573$	2.95490e17	0.348769
$574$	0	0
$575$	3.92525e17	0.452925
$576$	0	0
$577$	−3.62620e17	−0.409081	−0.204540	−	0.978858i	$-0.565570\pi$
$577$	−3.62620e17	−0.409081	−0.204540	+	0.978858i	$0.565570\pi$
$578$	0	0
$579$	−5.74227e17	−0.633392
$580$	0	0
$581$	−1.71520e18	−1.84999
$582$	0	0
$583$	1.14363e18	1.20625
$584$	0	0
$585$	−8.37739e16	−0.0864163
$586$	0	0
$587$	−4.88536e16	−0.0492889	−0.0246444	−	0.999696i	$-0.507845\pi$
$587$	−4.88536e16	−0.0492889	−0.0246444	+	0.999696i	$0.507845\pi$
$588$	0	0
$589$	−5.24670e17	−0.517770
$590$	0	0
$591$	1.93437e17	0.186733
$592$	0	0
$593$	−1.15938e17	−0.109489	−0.0547446	−	0.998500i	$-0.517434\pi$
$593$	−1.15938e17	−0.109489	−0.0547446	+	0.998500i	$0.517434\pi$
$594$	0	0
$595$	−2.36094e18	−2.18135
$596$	0	0
$597$	3.69963e17	0.334445
$598$	0	0
$599$	4.75963e17	0.421016	0.210508	−	0.977592i	$-0.432488\pi$
$599$	4.75963e17	0.421016	0.210508	+	0.977592i	$0.432488\pi$
$600$	0	0
$601$	1.49802e18	1.29668	0.648341	−	0.761350i	$-0.275463\pi$
$601$	1.49802e18	1.29668	0.648341	+	0.761350i	$0.275463\pi$
$602$	0	0
$603$	−5.12551e17	−0.434185
$604$	0	0
$605$	−1.10204e17	−0.0913663
$606$	0	0
$607$	−4.84664e17	−0.393292	−0.196646	−	0.980475i	$-0.563005\pi$
$607$	−4.84664e17	−0.393292	−0.196646	+	0.980475i	$0.563005\pi$
$608$	0	0
$609$	−7.07774e16	−0.0562189
$610$	0	0
$611$	−3.89667e17	−0.302988
$612$	0	0
$613$	5.15858e17	0.392678	0.196339	−	0.980536i	$-0.437095\pi$
$613$	5.15858e17	0.392678	0.196339	+	0.980536i	$0.437095\pi$
$614$	0	0
$615$	1.17811e18	0.878006
$616$	0	0
$617$	−1.21887e18	−0.889415	−0.444707	−	0.895676i	$-0.646692\pi$
$617$	−1.21887e18	−0.889415	−0.444707	+	0.895676i	$0.646692\pi$
$618$	0	0
$619$	−1.70605e18	−1.21900	−0.609499	−	0.792787i	$-0.708630\pi$
$619$	−1.70605e18	−1.21900	−0.609499	+	0.792787i	$0.708630\pi$
$620$	0	0
$621$	1.75485e18	1.22785
$622$	0	0
$623$	−1.51028e18	−1.03487
$624$	0	0
$625$	−1.84660e18	−1.23923
$626$	0	0
$627$	6.31114e17	0.414829
$628$	0	0
$629$	−1.78908e18	−1.15186
$630$	0	0
$631$	−1.04361e17	−0.0658186	−0.0329093	−	0.999458i	$-0.510477\pi$
$631$	−1.04361e17	−0.0658186	−0.0329093	+	0.999458i	$0.510477\pi$
$632$	0	0
$633$	−8.63307e17	−0.533385
$634$	0	0
$635$	9.84134e17	0.595697
$636$	0	0
$637$	2.36061e16	0.0139997
$638$	0	0
$639$	−8.52609e17	−0.495444
$640$	0	0
$641$	9.82007e16	0.0559162	0.0279581	−	0.999609i	$-0.491100\pi$
$641$	9.82007e16	0.0559162	0.0279581	+	0.999609i	$0.491100\pi$
$642$	0	0
$643$	8.77886e17	0.489854	0.244927	−	0.969542i	$-0.421236\pi$
$643$	8.77886e17	0.489854	0.244927	+	0.969542i	$0.421236\pi$
$644$	0	0
$645$	−1.88631e18	−1.03151
$646$	0	0
$647$	3.75124e16	0.0201047	0.0100523	−	0.999949i	$-0.496800\pi$
$647$	3.75124e16	0.0201047	0.0100523	+	0.999949i	$0.496800\pi$
$648$	0	0
$649$	2.06128e18	1.08280
$650$	0	0
$651$	1.32359e18	0.681518
$652$	0	0
$653$	2.37735e18	1.19993	0.599967	−	0.800025i	$-0.295180\pi$
$653$	2.37735e18	1.19993	0.599967	+	0.800025i	$0.295180\pi$
$654$	0	0
$655$	2.46746e18	1.22091
$656$	0	0
$657$	1.60672e18	0.779407
$658$	0	0
$659$	3.79077e17	0.180290	0.0901451	−	0.995929i	$-0.471267\pi$
$659$	3.79077e17	0.180290	0.0901451	+	0.995929i	$0.471267\pi$
$660$	0	0
$661$	−2.24798e18	−1.04829	−0.524146	−	0.851629i	$-0.675615\pi$
$661$	−2.24798e18	−1.04829	−0.524146	+	0.851629i	$0.675615\pi$
$662$	0	0
$663$	4.59759e17	0.210229
$664$	0	0
$665$	1.60699e18	0.720564
$666$	0	0
$667$	1.89827e17	0.0834720
$668$	0	0
$669$	2.91308e18	1.25627
$670$	0	0
$671$	−2.33225e18	−0.986460
$672$	0	0
$673$	−2.83190e18	−1.17484	−0.587421	−	0.809282i	$-0.699857\pi$
$673$	−2.83190e18	−1.17484	−0.587421	+	0.809282i	$0.699857\pi$
$674$	0	0
$675$	1.04597e18	0.425641
$676$	0	0
$677$	−2.10587e18	−0.840633	−0.420316	−	0.907378i	$-0.638081\pi$
$677$	−2.10587e18	−0.840633	−0.420316	+	0.907378i	$0.638081\pi$
$678$	0	0
$679$	−1.54759e18	−0.606041
$680$	0	0
$681$	2.15717e18	0.828759
$682$	0	0
$683$	4.67469e18	1.76205	0.881026	−	0.473068i	$-0.156854\pi$
$683$	4.67469e18	1.76205	0.881026	+	0.473068i	$0.156854\pi$
$684$	0	0
$685$	4.19780e18	1.55250
$686$	0	0
$687$	−1.38451e18	−0.502433
$688$	0	0
$689$	−5.67068e17	−0.201934
$690$	0	0
$691$	−2.38768e18	−0.834391	−0.417195	−	0.908817i	$-0.636987\pi$
$691$	−2.38768e18	−0.834391	−0.417195	+	0.908817i	$0.636987\pi$
$692$	0	0
$693$	1.32812e18	0.455481
$694$	0	0
$695$	−3.40180e18	−1.14500
$696$	0	0
$697$	5.39347e18	1.78178
$698$	0	0
$699$	2.19149e18	0.710620
$700$	0	0
$701$	1.20101e18	0.382279	0.191140	−	0.981563i	$-0.438782\pi$
$701$	1.20101e18	0.382279	0.191140	+	0.981563i	$0.438782\pi$
$702$	0	0
$703$	1.21775e18	0.380496
$704$	0	0
$705$	5.35942e18	1.64395
$706$	0	0
$707$	−9.20963e17	−0.277342
$708$	0	0
$709$	6.07854e18	1.79721	0.898605	−	0.438758i	$-0.144581\pi$
$709$	6.07854e18	1.79721	0.898605	+	0.438758i	$0.144581\pi$
$710$	0	0
$711$	−2.50837e18	−0.728182
$712$	0	0
$713$	−3.54991e18	−1.01190
$714$	0	0
$715$	−6.52066e17	−0.182516
$716$	0	0
$717$	−1.20456e18	−0.331096
$718$	0	0
$719$	3.60498e18	0.973118	0.486559	−	0.873648i	$-0.338252\pi$
$719$	3.60498e18	0.973118	0.486559	+	0.873648i	$0.338252\pi$
$720$	0	0
$721$	−3.56264e18	−0.944481
$722$	0	0
$723$	1.66958e18	0.434718
$724$	0	0
$725$	1.13145e17	0.0289360
$726$	0	0
$727$	7.42892e18	1.86617	0.933086	−	0.359653i	$-0.117105\pi$
$727$	7.42892e18	1.86617	0.933086	+	0.359653i	$0.117105\pi$
$728$	0	0
$729$	3.83795e18	0.947044
$730$	0	0
$731$	−8.63567e18	−2.09331
$732$	0	0
$733$	−1.94485e17	−0.0463139	−0.0231569	−	0.999732i	$-0.507372\pi$
$733$	−1.94485e17	−0.0463139	−0.0231569	+	0.999732i	$0.507372\pi$
$734$	0	0
$735$	−3.24675e17	−0.0759593
$736$	0	0
$737$	−3.98951e18	−0.917025
$738$	0	0
$739$	−3.07766e18	−0.695074	−0.347537	−	0.937666i	$-0.612982\pi$
$739$	−3.07766e18	−0.695074	−0.347537	+	0.937666i	$0.612982\pi$
$740$	0	0
$741$	−3.12938e17	−0.0694449
$742$	0	0
$743$	−1.87603e17	−0.0409083	−0.0204542	−	0.999791i	$-0.506511\pi$
$743$	−1.87603e17	−0.0409083	−0.0204542	+	0.999791i	$0.506511\pi$
$744$	0	0
$745$	2.99002e18	0.640705
$746$	0	0
$747$	3.83218e18	0.806978
$748$	0	0
$749$	−4.90538e18	−1.01517
$750$	0	0
$751$	−3.58767e18	−0.729713	−0.364857	−	0.931064i	$-0.618882\pi$
$751$	−3.58767e18	−0.729713	−0.364857	+	0.931064i	$0.618882\pi$
$752$	0	0
$753$	−8.16226e17	−0.163171
$754$	0	0
$755$	−6.07556e17	−0.119380
$756$	0	0
$757$	−4.57646e18	−0.883906	−0.441953	−	0.897038i	$-0.645714\pi$
$757$	−4.57646e18	−0.883906	−0.441953	+	0.897038i	$0.645714\pi$
$758$	0	0
$759$	4.27011e18	0.810714
$760$	0	0
$761$	2.74253e18	0.511859	0.255930	−	0.966695i	$-0.417618\pi$
$761$	2.74253e18	0.511859	0.255930	+	0.966695i	$0.417618\pi$
$762$	0	0
$763$	−1.07594e19	−1.97415
$764$	0	0
$765$	5.27491e18	0.951516
$766$	0	0
$767$	−1.02209e18	−0.181267
$768$	0	0
$769$	−2.95322e18	−0.514961	−0.257480	−	0.966284i	$-0.582892\pi$
$769$	−2.95322e18	−0.514961	−0.257480	+	0.966284i	$0.582892\pi$
$770$	0	0
$771$	1.14618e18	0.196517
$772$	0	0
$773$	−5.92667e18	−0.999181	−0.499590	−	0.866262i	$-0.666516\pi$
$773$	−5.92667e18	−0.999181	−0.499590	+	0.866262i	$0.666516\pi$
$774$	0	0
$775$	−2.11590e18	−0.350779
$776$	0	0
$777$	−3.07204e18	−0.500830
$778$	0	0
$779$	−3.67111e18	−0.588577
$780$	0	0
$781$	−6.63640e18	−1.04641
$782$	0	0
$783$	5.05836e17	0.0784437
$784$	0	0
$785$	−8.03890e18	−1.22615
$786$	0	0
$787$	2.47707e18	0.371623	0.185811	−	0.982585i	$-0.440509\pi$
$787$	2.47707e18	0.371623	0.185811	+	0.982585i	$0.440509\pi$
$788$	0	0
$789$	2.24130e18	0.330750
$790$	0	0
$791$	−3.39085e18	−0.492223
$792$	0	0
$793$	1.15645e18	0.165140
$794$	0	0
$795$	7.79936e18	1.09565
$796$	0	0
$797$	3.50125e18	0.483886	0.241943	−	0.970290i	$-0.422215\pi$
$797$	3.50125e18	0.483886	0.241943	+	0.970290i	$0.422215\pi$
$798$	0	0
$799$	2.45358e19	3.33616
$800$	0	0
$801$	3.37434e18	0.451416
$802$	0	0
$803$	1.25061e19	1.64615
$804$	0	0
$805$	1.08729e19	1.40822
$806$	0	0
$807$	3.98121e18	0.507385
$808$	0	0
$809$	−1.34541e19	−1.68728	−0.843641	−	0.536908i	$-0.819592\pi$
$809$	−1.34541e19	−1.68728	−0.843641	+	0.536908i	$0.819592\pi$
$810$	0	0
$811$	−5.51711e18	−0.680889	−0.340445	−	0.940265i	$-0.610578\pi$
$811$	−5.51711e18	−0.680889	−0.340445	+	0.940265i	$0.610578\pi$
$812$	0	0
$813$	−1.51316e17	−0.0183779
$814$	0	0
$815$	−1.22647e19	−1.46600
$816$	0	0
$817$	5.87794e18	0.691483
$818$	0	0
$819$	−6.58548e17	−0.0762504
$820$	0	0
$821$	−1.22485e19	−1.39590	−0.697949	−	0.716147i	$-0.745904\pi$
$821$	−1.22485e19	−1.39590	−0.697949	+	0.716147i	$0.745904\pi$
$822$	0	0
$823$	−1.47057e19	−1.64963	−0.824813	−	0.565406i	$-0.808720\pi$
$823$	−1.47057e19	−1.64963	−0.824813	+	0.565406i	$0.808720\pi$
$824$	0	0
$825$	2.54517e18	0.281038
$826$	0	0
$827$	1.17822e19	1.28068	0.640340	−	0.768092i	$-0.278794\pi$
$827$	1.17822e19	1.28068	0.640340	+	0.768092i	$0.278794\pi$
$828$	0	0
$829$	−1.07291e19	−1.14805	−0.574023	−	0.818839i	$-0.694618\pi$
$829$	−1.07291e19	−1.14805	−0.574023	+	0.818839i	$0.694618\pi$
$830$	0	0
$831$	−1.09879e19	−1.15747
$832$	0	0
$833$	−1.48638e18	−0.154148
$834$	0	0
$835$	−2.11508e16	−0.00215956
$836$	0	0
$837$	−9.45951e18	−0.950940
$838$	0	0
$839$	−3.64163e18	−0.360449	−0.180224	−	0.983626i	$-0.557682\pi$
$839$	−3.64163e18	−0.360449	−0.180224	+	0.983626i	$0.557682\pi$
$840$	0	0
$841$	−1.02059e19	−0.994667
$842$	0	0
$843$	1.61633e18	0.155114
$844$	0	0
$845$	−1.21807e19	−1.15107
$846$	0	0
$847$	−8.66312e17	−0.0806181
$848$	0	0
$849$	−3.58390e18	−0.328440
$850$	0	0
$851$	8.23931e18	0.743616
$852$	0	0
$853$	8.66074e18	0.769815	0.384907	−	0.922955i	$-0.374233\pi$
$853$	8.66074e18	0.769815	0.384907	+	0.922955i	$0.374233\pi$
$854$	0	0
$855$	−3.59041e18	−0.314315
$856$	0	0
$857$	−1.94010e19	−1.67282	−0.836408	−	0.548107i	$-0.815349\pi$
$857$	−1.94010e19	−1.67282	−0.836408	+	0.548107i	$0.815349\pi$
$858$	0	0
$859$	5.21902e18	0.443234	0.221617	−	0.975134i	$-0.428866\pi$
$859$	5.21902e18	0.443234	0.221617	+	0.975134i	$0.428866\pi$
$860$	0	0
$861$	9.26113e18	0.774718
$862$	0	0
$863$	−3.14037e18	−0.258768	−0.129384	−	0.991595i	$-0.541300\pi$
$863$	−3.14037e18	−0.258768	−0.129384	+	0.991595i	$0.541300\pi$
$864$	0	0
$865$	1.97050e19	1.59945
$866$	0	0
$867$	−1.97149e19	−1.57642
$868$	0	0
$869$	−1.95243e19	−1.53796
$870$	0	0
$871$	1.97820e18	0.153516
$872$	0	0
$873$	3.45769e18	0.264359
$874$	0	0
$875$	−9.87465e18	−0.743823
$876$	0	0
$877$	1.64678e19	1.22219	0.611093	−	0.791559i	$-0.290730\pi$
$877$	1.64678e19	1.22219	0.611093	+	0.791559i	$0.290730\pi$
$878$	0	0
$879$	−2.99025e18	−0.218665
$880$	0	0
$881$	3.41428e18	0.246011	0.123006	−	0.992406i	$-0.460747\pi$
$881$	3.41428e18	0.246011	0.123006	+	0.992406i	$0.460747\pi$
$882$	0	0
$883$	−7.98967e18	−0.567262	−0.283631	−	0.958933i	$-0.591539\pi$
$883$	−7.98967e18	−0.567262	−0.283631	+	0.958933i	$0.591539\pi$
$884$	0	0
$885$	1.40576e19	0.983515
$886$	0	0
$887$	1.42206e19	0.980422	0.490211	−	0.871604i	$-0.336920\pi$
$887$	1.42206e19	0.980422	0.490211	+	0.871604i	$0.336920\pi$
$888$	0	0
$889$	7.73629e18	0.525619
$890$	0	0
$891$	4.85411e18	0.325016
$892$	0	0
$893$	−1.67005e19	−1.10203
$894$	0	0
$895$	1.08514e19	0.705727
$896$	0	0
$897$	−2.11734e18	−0.135719
$898$	0	0
$899$	−1.02326e18	−0.0646470
$900$	0	0
$901$	3.57060e19	2.22347
$902$	0	0
$903$	−1.48283e19	−0.910168
$904$	0	0
$905$	7.47463e17	0.0452245
$906$	0	0
$907$	1.30557e19	0.778667	0.389334	−	0.921097i	$-0.372705\pi$
$907$	1.30557e19	0.778667	0.389334	+	0.921097i	$0.372705\pi$
$908$	0	0
$909$	2.05766e18	0.120978
$910$	0	0
$911$	−8.09933e16	−0.00469439	−0.00234720	−	0.999997i	$-0.500747\pi$
$911$	−8.09933e16	−0.00469439	−0.00234720	+	0.999997i	$0.500747\pi$
$912$	0	0
$913$	2.98283e19	1.70439
$914$	0	0
$915$	−1.59056e19	−0.896012
$916$	0	0
$917$	1.93968e19	1.07728
$918$	0	0
$919$	2.64226e19	1.44685	0.723426	−	0.690401i	$-0.242566\pi$
$919$	2.64226e19	1.44685	0.723426	+	0.690401i	$0.242566\pi$
$920$	0	0
$921$	−1.77676e19	−0.959273
$922$	0	0
$923$	3.29066e18	0.175175
$924$	0	0
$925$	4.91097e18	0.257778
$926$	0	0
$927$	7.95982e18	0.411988
$928$	0	0
$929$	6.73591e18	0.343791	0.171895	−	0.985115i	$-0.445011\pi$
$929$	6.73591e18	0.343791	0.171895	+	0.985115i	$0.445011\pi$
$930$	0	0
$931$	1.01172e18	0.0509199
$932$	0	0
$933$	1.68993e19	0.838761
$934$	0	0
$935$	4.10580e19	2.00966
$936$	0	0
$937$	2.28647e19	1.10372	0.551859	−	0.833937i	$-0.313919\pi$
$937$	2.28647e19	1.10372	0.551859	+	0.833937i	$0.313919\pi$
$938$	0	0
$939$	−2.90650e19	−1.38370
$940$	0	0
$941$	−4.10540e19	−1.92762	−0.963812	−	0.266581i	$-0.914106\pi$
$941$	−4.10540e19	−1.92762	−0.963812	+	0.266581i	$0.914106\pi$
$942$	0	0
$943$	−2.48387e19	−1.15028
$944$	0	0
$945$	2.89731e19	1.32339
$946$	0	0
$947$	4.57566e18	0.206148	0.103074	−	0.994674i	$-0.467132\pi$
$947$	4.57566e18	0.206148	0.103074	+	0.994674i	$0.467132\pi$
$948$	0	0
$949$	−6.20115e18	−0.275576
$950$	0	0
$951$	−2.85961e18	−0.125353
$952$	0	0
$953$	−2.21585e19	−0.958158	−0.479079	−	0.877772i	$-0.659029\pi$
$953$	−2.21585e19	−0.958158	−0.479079	+	0.877772i	$0.659029\pi$
$954$	0	0
$955$	−1.30846e19	−0.558136
$956$	0	0
$957$	1.23086e18	0.0517941
$958$	0	0
$959$	3.29989e19	1.36987
$960$	0	0
$961$	−5.28179e18	−0.216311
$962$	0	0
$963$	1.09598e19	0.442825
$964$	0	0
$965$	2.54274e19	1.01362
$966$	0	0
$967$	−3.62724e19	−1.42661	−0.713304	−	0.700855i	$-0.752802\pi$
$967$	−3.62724e19	−1.42661	−0.713304	+	0.700855i	$0.752802\pi$
$968$	0	0
$969$	1.97045e19	0.764647
$970$	0	0
$971$	−1.22360e19	−0.468504	−0.234252	−	0.972176i	$-0.575264\pi$
$971$	−1.22360e19	−0.468504	−0.234252	+	0.972176i	$0.575264\pi$
$972$	0	0
$973$	−2.67416e19	−1.01031
$974$	0	0
$975$	−1.26202e18	−0.0470476
$976$	0	0
$977$	−1.82463e19	−0.671211	−0.335606	−	0.942003i	$-0.608941\pi$
$977$	−1.82463e19	−0.671211	−0.335606	+	0.942003i	$0.608941\pi$
$978$	0	0
$979$	2.62646e19	0.953418
$980$	0	0
$981$	2.40392e19	0.861134
$982$	0	0
$983$	−9.19383e18	−0.325011	−0.162506	−	0.986708i	$-0.551958\pi$
$983$	−9.19383e18	−0.325011	−0.162506	+	0.986708i	$0.551958\pi$
$984$	0	0
$985$	−8.56562e18	−0.298830
$986$	0	0
$987$	4.21305e19	1.45056
$988$	0	0
$989$	3.97701e19	1.35139
$990$	0	0
$991$	−4.59757e19	−1.54188	−0.770938	−	0.636910i	$-0.780212\pi$
$991$	−4.59757e19	−1.54188	−0.770938	+	0.636910i	$0.780212\pi$
$992$	0	0
$993$	9.85830e18	0.326311
$994$	0	0
$995$	−1.63824e19	−0.535214
$996$	0	0
$997$	1.09104e19	0.351820	0.175910	−	0.984406i	$-0.443713\pi$
$997$	1.09104e19	0.351820	0.175910	+	0.984406i	$0.443713\pi$
$998$	0	0
$999$	2.19554e19	0.698821

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.14.a.m.1.2		3
4.3	odd	2		64.14.a.n.1.2		3
8.3	odd	2		32.14.a.c.1.2	✓	3
8.5	even	2		32.14.a.d.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.14.a.c.1.2	✓	3	8.3	odd	2
32.14.a.d.1.2	yes	3	8.5	even	2
64.14.a.m.1.2		3	1.1	even	1	trivial
64.14.a.n.1.2		3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

\(f(q)\)	\(=\)	\(q-932.324 q^{3} +41284.4 q^{5} +324537. q^{7} -725095. q^{9} -5.64388e6 q^{11} +2.79852e6 q^{13} -3.84904e7 q^{15} -1.76212e8 q^{17} +1.19940e8 q^{19} -3.02574e8 q^{21} +8.11512e8 q^{23} +4.83695e8 q^{25} +2.16245e9 q^{27} +2.33918e8 q^{29} -4.37444e9 q^{31} +5.26192e9 q^{33} +1.33983e10 q^{35} +1.01530e10 q^{37} -2.60913e9 q^{39} -3.06079e10 q^{41} +4.90073e10 q^{43} -2.99351e10 q^{45} -1.39240e11 q^{47} +8.43522e9 q^{49} +1.64286e11 q^{51} -2.02631e11 q^{53} -2.33004e11 q^{55} -1.11823e11 q^{57} -3.65225e11 q^{59} +4.13236e11 q^{61} -2.35320e11 q^{63} +1.15535e11 q^{65} +7.06874e11 q^{67} -7.56592e11 q^{69} +1.17586e12 q^{71} -2.21587e12 q^{73} -4.50961e11 q^{75} -1.83165e12 q^{77} +3.45937e12 q^{79} -8.60067e11 q^{81} -5.28507e12 q^{83} -7.27479e12 q^{85} -2.18087e11 q^{87} -4.65365e12 q^{89} +9.08223e11 q^{91} +4.07840e12 q^{93} +4.95164e12 q^{95} -4.76860e12 q^{97} +4.09235e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 520 q^{3} - 11594 q^{5} - 96304 q^{7} - 196793 q^{9} + 1923816 q^{11} + 11211006 q^{13} + 10944368 q^{15} - 16892538 q^{17} + 74665048 q^{19} + 115146880 q^{21} - 486642576 q^{23} + 442696461 q^{25}+ \cdots + 2082116453576 q^{99}+O(q^{100}) \) 3 * q - 520 * q^3 - 11594 * q^5 - 96304 * q^7 - 196793 * q^9 + 1923816 * q^11 + 11211006 * q^13 + 10944368 * q^15 - 16892538 * q^17 + 74665048 * q^19 + 115146880 * q^21 - 486642576 * q^23 + 442696461 * q^25 + 3111524144 * q^27 - 1747468946 * q^29 - 6471356864 * q^31 - 2901436352 * q^33 + 32905782432 * q^35 + 1189828278 * q^37 - 73495249232 * q^39 - 12181275970 * q^41 + 151151885096 * q^43 - 19024866434 * q^45 - 273552663072 * q^47 - 26716022709 * q^49 + 361685604336 * q^51 - 151381970362 * q^53 - 594621149104 * q^55 - 253183022656 * q^57 + 534333367560 * q^59 - 598414222482 * q^61 - 138475736176 * q^63 - 1313304118564 * q^65 - 703511405320 * q^67 - 1246094112384 * q^69 + 1867178467152 * q^71 - 2138161799730 * q^73 - 3649896495992 * q^75 - 4774654840448 * q^77 + 6521880375328 * q^79 - 6029061199109 * q^81 - 5725896739752 * q^83 - 8753386407828 * q^85 + 9313876797488 * q^87 - 11930629520834 * q^89 - 10949018828768 * q^91 - 15654956180992 * q^93 + 3955366679088 * q^95 - 23076828823050 * q^97 + 2082116453576 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more