LMFDB - Embedded newform 64.22.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,22,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, 22, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 22 $
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:	$178.865500344$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 796953x^{2} + 26634907x + 142212501002 $ x^4 - x^3 - 796953x^2 + 26634907x + 142212501002
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{4}\cdot 5 $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-477.158$ 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)

$f(q)$	$=$	$q+104797. q^{3} +3.34494e7 q^{5} -1.11987e9 q^{7} +5.22119e8 q^{9} +O(q^{10})$	$q+104797. q^{3} +3.34494e7 q^{5} -1.11987e9 q^{7} +5.22119e8 q^{9} -5.16032e10 q^{11} -8.51385e10 q^{13} +3.50541e12 q^{15} +8.07667e12 q^{17} -2.57882e13 q^{19} -1.17360e14 q^{21} -2.78558e14 q^{23} +6.42025e14 q^{25} -1.04150e15 q^{27} +1.18307e15 q^{29} +3.28170e15 q^{31} -5.40787e15 q^{33} -3.74591e16 q^{35} +4.58151e16 q^{37} -8.92229e15 q^{39} +1.40811e17 q^{41} +2.97854e16 q^{43} +1.74646e16 q^{45} +6.94078e17 q^{47} +6.95571e17 q^{49} +8.46413e17 q^{51} +1.69987e18 q^{53} -1.72609e18 q^{55} -2.70253e18 q^{57} +5.82064e18 q^{59} -1.75947e18 q^{61} -5.84708e17 q^{63} -2.84783e18 q^{65} -1.50890e19 q^{67} -2.91921e19 q^{69} +3.58026e19 q^{71} -2.53946e19 q^{73} +6.72825e19 q^{75} +5.77890e19 q^{77} +1.33413e20 q^{79} -1.14608e20 q^{81} +9.09690e19 q^{83} +2.70160e20 q^{85} +1.23983e20 q^{87} +1.52183e20 q^{89} +9.53444e19 q^{91} +3.43913e20 q^{93} -8.62599e20 q^{95} +3.90849e18 q^{97} -2.69430e19 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 37124680 q^{5} + 16226262324 q^{9}+O(q^{10}) $ 4 * q + 37124680 * q^5 + 16226262324 * q^9	$ 4 q + 37124680 q^{5} + 16226262324 q^{9} + 1438891028904 q^{13} + 27067357021960 q^{17} - 58494636232704 q^{21} + 773625552400700 q^{25} + 15\!\cdots\!08 q^{29}+ \cdots - 14\!\cdots\!88 q^{97}+O(q^{100}) $ 4 * q + 37124680 * q^5 + 16226262324 * q^9 + 1438891028904 * q^13 + 27067357021960 * q^17 - 58494636232704 * q^21 + 773625552400700 * q^25 + 1535657143500008 * q^29 - 2740052722894848 * q^33 + 51861754826473544 * q^37 + 76119164178377384 * q^41 - 191086569820209240 * q^45 + 1134240365253238628 * q^49 + 4626949431882959624 * q^53 - 2265200566382131200 * q^57 - 13244181711188448536 * q^61 - 29651450007494917680 * q^65 + 7851231862535245824 * q^69 + 39323341470423072360 * q^73 + 154997157739134799872 * q^77 - 491616252151343842140 * q^81 + 377841710648841826960 * q^85 - 296515048889049879640 * q^89 + 2114246149921519091712 * q^93 - 1497970122472846632888 * q^97

Display 100 coefficients 10 coefficients

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
$3$	104797.	1.02465	0.512327	−	0.858791i	$-0.328784\pi$
$3$	104797.	1.02465	0.512327	+	0.858791i	$0.328784\pi$
$4$	0	0
$5$	3.34494e7	1.53180	0.765902	−	0.642957i	$-0.222293\pi$
$5$	3.34494e7	1.53180	0.765902	+	0.642957i	$0.222293\pi$
$6$	0	0
$7$	−1.11987e9	−1.49844	−0.749220	−	0.662321i	$-0.769572\pi$
$7$	−1.11987e9	−1.49844	−0.749220	+	0.662321i	$0.769572\pi$
$8$	0	0
$9$	5.22119e8	0.0499141
$10$	0	0
$11$	−5.16032e10	−0.599864	−0.299932	−	0.953961i	$-0.596964\pi$
$11$	−5.16032e10	−0.599864	−0.299932	+	0.953961i	$0.596964\pi$
$12$	0	0
$13$	−8.51385e10	−0.171286	−0.0856428	−	0.996326i	$-0.527294\pi$
$13$	−8.51385e10	−0.171286	−0.0856428	+	0.996326i	$0.527294\pi$
$14$	0	0
$15$	3.50541e12	1.56957
$16$	0	0
$17$	8.07667e12	0.971669	0.485835	−	0.874051i	$-0.338516\pi$
$17$	8.07667e12	0.971669	0.485835	+	0.874051i	$0.338516\pi$
$18$	0	0
$19$	−2.57882e13	−0.964956	−0.482478	−	0.875908i	$-0.660263\pi$
$19$	−2.57882e13	−0.964956	−0.482478	+	0.875908i	$0.660263\pi$
$20$	0	0
$21$	−1.17360e14	−1.53538
$22$	0	0
$23$	−2.78558e14	−1.40208	−0.701041	−	0.713121i	$-0.747281\pi$
$23$	−2.78558e14	−1.40208	−0.701041	+	0.713121i	$0.747281\pi$
$24$	0	0
$25$	6.42025e14	1.34642
$26$	0	0
$27$	−1.04150e15	−0.973509
$28$	0	0
$29$	1.18307e15	0.522194	0.261097	−	0.965313i	$-0.415916\pi$
$29$	1.18307e15	0.522194	0.261097	+	0.965313i	$0.415916\pi$
$30$	0	0
$31$	3.28170e15	0.719120	0.359560	−	0.933122i	$-0.382927\pi$
$31$	3.28170e15	0.719120	0.359560	+	0.933122i	$0.382927\pi$
$32$	0	0
$33$	−5.40787e15	−0.614653
$34$	0	0
$35$	−3.74591e16	−2.29532
$36$	0	0
$37$	4.58151e16	1.56636	0.783178	−	0.621797i	$-0.213597\pi$
$37$	4.58151e16	1.56636	0.783178	+	0.621797i	$0.213597\pi$
$38$	0	0
$39$	−8.92229e15	−0.175508
$40$	0	0
$41$	1.40811e17	1.63835	0.819177	−	0.573541i	$-0.194431\pi$
$41$	1.40811e17	1.63835	0.819177	+	0.573541i	$0.194431\pi$
$42$	0	0
$43$	2.97854e16	0.210177	0.105088	−	0.994463i	$-0.466487\pi$
$43$	2.97854e16	0.210177	0.105088	+	0.994463i	$0.466487\pi$
$44$	0	0
$45$	1.74646e16	0.0764587
$46$	0	0
$47$	6.94078e17	1.92478	0.962389	−	0.271674i	$-0.0875771\pi$
$47$	6.94078e17	1.92478	0.962389	+	0.271674i	$0.0875771\pi$
$48$	0	0
$49$	6.95571e17	1.24533
$50$	0	0
$51$	8.46413e17	0.995624
$52$	0	0
$53$	1.69987e18	1.33512	0.667559	−	0.744557i	$-0.267339\pi$
$53$	1.69987e18	1.33512	0.667559	+	0.744557i	$0.267339\pi$
$54$	0	0
$55$	−1.72609e18	−0.918875
$56$	0	0
$57$	−2.70253e18	−0.988746
$58$	0	0
$59$	5.82064e18	1.48260	0.741301	−	0.671172i	$-0.234209\pi$
$59$	5.82064e18	1.48260	0.741301	+	0.671172i	$0.234209\pi$
$60$	0	0
$61$	−1.75947e18	−0.315804	−0.157902	−	0.987455i	$-0.550473\pi$
$61$	−1.75947e18	−0.315804	−0.157902	+	0.987455i	$0.550473\pi$
$62$	0	0
$63$	−5.84708e17	−0.0747934
$64$	0	0
$65$	−2.84783e18	−0.262376
$66$	0	0
$67$	−1.50890e19	−1.01129	−0.505646	−	0.862741i	$-0.668746\pi$
$67$	−1.50890e19	−1.01129	−0.505646	+	0.862741i	$0.668746\pi$
$68$	0	0
$69$	−2.91921e19	−1.43665
$70$	0	0
$71$	3.58026e19	1.30527	0.652637	−	0.757671i	$-0.273663\pi$
$71$	3.58026e19	1.30527	0.652637	+	0.757671i	$0.273663\pi$
$72$	0	0
$73$	−2.53946e19	−0.691595	−0.345797	−	0.938309i	$-0.612392\pi$
$73$	−2.53946e19	−0.691595	−0.345797	+	0.938309i	$0.612392\pi$
$74$	0	0
$75$	6.72825e19	1.37962
$76$	0	0
$77$	5.77890e19	0.898861
$78$	0	0
$79$	1.33413e20	1.58530	0.792652	−	0.609674i	$-0.208700\pi$
$79$	1.33413e20	1.58530	0.792652	+	0.609674i	$0.208700\pi$
$80$	0	0
$81$	−1.14608e20	−1.04742
$82$	0	0
$83$	9.09690e19	0.643537	0.321768	−	0.946818i	$-0.395723\pi$
$83$	9.09690e19	0.643537	0.321768	+	0.946818i	$0.395723\pi$
$84$	0	0
$85$	2.70160e20	1.48841
$86$	0	0
$87$	1.23983e20	0.535068
$88$	0	0
$89$	1.52183e20	0.517334	0.258667	−	0.965967i	$-0.416717\pi$
$89$	1.52183e20	0.517334	0.258667	+	0.965967i	$0.416717\pi$
$90$	0	0
$91$	9.53444e19	0.256661
$92$	0	0
$93$	3.43913e20	0.736849
$94$	0	0
$95$	−8.62599e20	−1.47812
$96$	0	0
$97$	3.90849e18	0.00538153	0.00269077	−	0.999996i	$-0.499144\pi$
$97$	3.90849e18	0.00538153	0.00269077	+	0.999996i	$0.499144\pi$
$98$	0	0
$99$	−2.69430e19	−0.0299417
$100$	0	0
$101$	1.01088e21	0.910599	0.455300	−	0.890338i	$-0.349532\pi$
$101$	1.01088e21	0.910599	0.455300	+	0.890338i	$0.349532\pi$
$102$	0	0
$103$	−1.15748e21	−0.848635	−0.424317	−	0.905514i	$-0.639486\pi$
$103$	−1.15748e21	−0.848635	−0.424317	+	0.905514i	$0.639486\pi$
$104$	0	0
$105$	−3.92561e21	−2.35191
$106$	0	0
$107$	−7.04501e20	−0.346220	−0.173110	−	0.984903i	$-0.555382\pi$
$107$	−7.04501e20	−0.346220	−0.173110	+	0.984903i	$0.555382\pi$
$108$	0	0
$109$	3.43578e21	1.39011	0.695053	−	0.718959i	$-0.255381\pi$
$109$	3.43578e21	1.39011	0.695053	+	0.718959i	$0.255381\pi$
$110$	0	0
$111$	4.80130e21	1.60497
$112$	0	0
$113$	−5.71437e21	−1.58360	−0.791798	−	0.610783i	$-0.790855\pi$
$113$	−5.71437e21	−1.58360	−0.791798	+	0.610783i	$0.790855\pi$
$114$	0	0
$115$	−9.31761e21	−2.14772
$116$	0	0
$117$	−4.44525e19	−0.00854957
$118$	0	0
$119$	−9.04485e21	−1.45599
$120$	0	0
$121$	−4.73736e21	−0.640163
$122$	0	0
$123$	1.47567e22	1.67874
$124$	0	0
$125$	5.52545e21	0.530655
$126$	0	0
$127$	8.97196e21	0.729370	0.364685	−	0.931131i	$-0.381177\pi$
$127$	8.97196e21	0.729370	0.364685	+	0.931131i	$0.381177\pi$
$128$	0	0
$129$	3.12143e21	0.215358
$130$	0	0
$131$	−1.22407e22	−0.718553	−0.359276	−	0.933231i	$-0.616976\pi$
$131$	−1.22407e22	−0.718553	−0.359276	+	0.933231i	$0.616976\pi$
$132$	0	0
$133$	2.88795e22	1.44593
$134$	0	0
$135$	−3.48376e22	−1.49122
$136$	0	0
$137$	−8.07808e21	−0.296307	−0.148154	−	0.988964i	$-0.547333\pi$
$137$	−8.07808e21	−0.296307	−0.148154	+	0.988964i	$0.547333\pi$
$138$	0	0
$139$	−9.75870e21	−0.307423	−0.153712	−	0.988116i	$-0.549123\pi$
$139$	−9.75870e21	−0.307423	−0.153712	+	0.988116i	$0.549123\pi$
$140$	0	0
$141$	7.27375e22	1.97223
$142$	0	0
$143$	4.39342e21	0.102748
$144$	0	0
$145$	3.95730e22	0.799900
$146$	0	0
$147$	7.28940e22	1.27603
$148$	0	0
$149$	−1.20240e23	−1.82638	−0.913191	−	0.407531i	$-0.866390\pi$
$149$	−1.20240e23	−1.82638	−0.913191	+	0.407531i	$0.866390\pi$
$150$	0	0
$151$	1.26202e23	1.66651	0.833253	−	0.552892i	$-0.186476\pi$
$151$	1.26202e23	1.66651	0.833253	+	0.552892i	$0.186476\pi$
$152$	0	0
$153$	4.21698e21	0.0485000
$154$	0	0
$155$	1.09771e23	1.10155
$156$	0	0
$157$	−5.17595e22	−0.453987	−0.226994	−	0.973896i	$-0.572890\pi$
$157$	−5.17595e22	−0.453987	−0.226994	+	0.973896i	$0.572890\pi$
$158$	0	0
$159$	1.78142e23	1.36803
$160$	0	0
$161$	3.11950e23	2.10094
$162$	0	0
$163$	−2.39405e23	−1.41633	−0.708164	−	0.706048i	$-0.750476\pi$
$163$	−2.39405e23	−1.41633	−0.708164	+	0.706048i	$0.750476\pi$
$164$	0	0
$165$	−1.80890e23	−0.941528
$166$	0	0
$167$	−5.70278e22	−0.261555	−0.130778	−	0.991412i	$-0.541747\pi$
$167$	−5.70278e22	−0.261555	−0.130778	+	0.991412i	$0.541747\pi$
$168$	0	0
$169$	−2.39816e23	−0.970661
$170$	0	0
$171$	−1.34645e22	−0.0481649
$172$	0	0
$173$	8.39251e22	0.265710	0.132855	−	0.991135i	$-0.457586\pi$
$173$	8.39251e22	0.265710	0.132855	+	0.991135i	$0.457586\pi$
$174$	0	0
$175$	−7.18987e23	−2.01754
$176$	0	0
$177$	6.09988e23	1.51915
$178$	0	0
$179$	2.47523e23	0.547846	0.273923	−	0.961752i	$-0.411679\pi$
$179$	2.47523e23	0.547846	0.273923	+	0.961752i	$0.411679\pi$
$180$	0	0
$181$	−2.64993e22	−0.0521927	−0.0260964	−	0.999659i	$-0.508308\pi$
$181$	−2.64993e22	−0.0521927	−0.0260964	+	0.999659i	$0.508308\pi$
$182$	0	0
$183$	−1.84387e23	−0.323590
$184$	0	0
$185$	1.53249e24	2.39935
$186$	0	0
$187$	−4.16782e23	−0.582870
$188$	0	0
$189$	1.16635e24	1.45875
$190$	0	0
$191$	−2.31217e23	−0.258922	−0.129461	−	0.991584i	$-0.541325\pi$
$191$	−2.31217e23	−0.258922	−0.129461	+	0.991584i	$0.541325\pi$
$192$	0	0
$193$	1.80562e24	1.81249	0.906245	−	0.422753i	$-0.138936\pi$
$193$	1.80562e24	1.81249	0.906245	+	0.422753i	$0.138936\pi$
$194$	0	0
$195$	−2.98445e23	−0.268844
$196$	0	0
$197$	−1.17627e24	−0.951947	−0.475973	−	0.879460i	$-0.657904\pi$
$197$	−1.17627e24	−0.951947	−0.475973	+	0.879460i	$0.657904\pi$
$198$	0	0
$199$	1.42888e24	1.04001	0.520007	−	0.854162i	$-0.325929\pi$
$199$	1.42888e24	1.04001	0.520007	+	0.854162i	$0.325929\pi$
$200$	0	0
$201$	−1.58129e24	−1.03622
$202$	0	0
$203$	−1.32489e24	−0.782477
$204$	0	0
$205$	4.71006e24	2.50964
$206$	0	0
$207$	−1.45441e23	−0.0699837
$208$	0	0
$209$	1.33075e24	0.578843
$210$	0	0
$211$	3.29336e24	1.29620	0.648102	−	0.761554i	$-0.275563\pi$
$211$	3.29336e24	1.29620	0.648102	+	0.761554i	$0.275563\pi$
$212$	0	0
$213$	3.75201e24	1.33745
$214$	0	0
$215$	9.96304e23	0.321950
$216$	0	0
$217$	−3.67509e24	−1.07756
$218$	0	0
$219$	−2.66129e24	−0.708645
$220$	0	0
$221$	−6.87636e23	−0.166433
$222$	0	0
$223$	−3.31316e24	−0.729528	−0.364764	−	0.931100i	$-0.618850\pi$
$223$	−3.31316e24	−0.729528	−0.364764	+	0.931100i	$0.618850\pi$
$224$	0	0
$225$	3.35214e23	0.0672056
$226$	0	0
$227$	−4.38255e24	−0.800673	−0.400337	−	0.916368i	$-0.631107\pi$
$227$	−4.38255e24	−0.800673	−0.400337	+	0.916368i	$0.631107\pi$
$228$	0	0
$229$	7.68190e24	1.27996	0.639979	−	0.768392i	$-0.278943\pi$
$229$	7.68190e24	1.27996	0.639979	+	0.768392i	$0.278943\pi$
$230$	0	0
$231$	6.05613e24	0.921021
$232$	0	0
$233$	9.90588e24	1.37612	0.688060	−	0.725654i	$-0.258463\pi$
$233$	9.90588e24	1.37612	0.688060	+	0.725654i	$0.258463\pi$
$234$	0	0
$235$	2.32165e25	2.94838
$236$	0	0
$237$	1.39813e25	1.62439
$238$	0	0
$239$	−4.92239e24	−0.523599	−0.261799	−	0.965122i	$-0.584316\pi$
$239$	−4.92239e24	−0.523599	−0.261799	+	0.965122i	$0.584316\pi$
$240$	0	0
$241$	1.12256e24	0.109404	0.0547018	−	0.998503i	$-0.482579\pi$
$241$	1.12256e24	0.109404	0.0547018	+	0.998503i	$0.482579\pi$
$242$	0	0
$243$	−1.11614e24	−0.0997365
$244$	0	0
$245$	2.32664e25	1.90759
$246$	0	0
$247$	2.19557e24	0.165283
$248$	0	0
$249$	9.53330e24	0.659402
$250$	0	0
$251$	−1.99144e24	−0.126647	−0.0633233	−	0.997993i	$-0.520170\pi$
$251$	−1.99144e24	−0.126647	−0.0633233	+	0.997993i	$0.520170\pi$
$252$	0	0
$253$	1.43745e25	0.841060
$254$	0	0
$255$	2.83120e25	1.52510
$256$	0	0
$257$	1.48178e24	0.0735335	0.0367667	−	0.999324i	$-0.488294\pi$
$257$	1.48178e24	0.0735335	0.0367667	+	0.999324i	$0.488294\pi$
$258$	0	0
$259$	−5.13071e25	−2.34709
$260$	0	0
$261$	6.17705e23	0.0260649
$262$	0	0
$263$	−2.92758e25	−1.14018	−0.570090	−	0.821582i	$-0.693092\pi$
$263$	−2.92758e25	−1.14018	−0.570090	+	0.821582i	$0.693092\pi$
$264$	0	0
$265$	5.68597e25	2.04514
$266$	0	0
$267$	1.59484e25	0.530088
$268$	0	0
$269$	−5.56826e25	−1.71128	−0.855640	−	0.517572i	$-0.826836\pi$
$269$	−5.56826e25	−1.71128	−0.855640	+	0.517572i	$0.826836\pi$
$270$	0	0
$271$	−2.66609e25	−0.758049	−0.379024	−	0.925387i	$-0.623740\pi$
$271$	−2.66609e25	−0.758049	−0.379024	+	0.925387i	$0.623740\pi$
$272$	0	0
$273$	9.99183e24	0.262989
$274$	0	0
$275$	−3.31305e25	−0.807672
$276$	0	0
$277$	−2.36190e25	−0.533611	−0.266805	−	0.963750i	$-0.585968\pi$
$277$	−2.36190e25	−0.533611	−0.266805	+	0.963750i	$0.585968\pi$
$278$	0	0
$279$	1.71344e24	0.0358942
$280$	0	0
$281$	−5.39109e24	−0.104776	−0.0523878	−	0.998627i	$-0.516683\pi$
$281$	−5.39109e24	−0.104776	−0.0523878	+	0.998627i	$0.516683\pi$
$282$	0	0
$283$	4.10596e25	0.740725	0.370363	−	0.928887i	$-0.379233\pi$
$283$	4.10596e25	0.740725	0.370363	+	0.928887i	$0.379233\pi$
$284$	0	0
$285$	−9.03980e25	−1.51456
$286$	0	0
$287$	−1.57691e26	−2.45498
$288$	0	0
$289$	−3.85937e24	−0.0558585
$290$	0	0
$291$	4.09599e23	0.00551421
$292$	0	0
$293$	8.55806e25	1.07217	0.536087	−	0.844163i	$-0.319902\pi$
$293$	8.55806e25	1.07217	0.536087	+	0.844163i	$0.319902\pi$
$294$	0	0
$295$	1.94697e26	2.27106
$296$	0	0
$297$	5.37447e25	0.583973
$298$	0	0
$299$	2.37160e25	0.240157
$300$	0	0
$301$	−3.33559e25	−0.314938
$302$	0	0
$303$	1.05938e26	0.933048
$304$	0	0
$305$	−5.88531e25	−0.483750
$306$	0	0
$307$	1.60284e26	1.23009	0.615047	−	0.788491i	$-0.289137\pi$
$307$	1.60284e26	1.23009	0.615047	+	0.788491i	$0.289137\pi$
$308$	0	0
$309$	−1.21300e26	−0.869556
$310$	0	0
$311$	1.56836e25	0.105066	0.0525329	−	0.998619i	$-0.483271\pi$
$311$	1.56836e25	0.105066	0.0525329	+	0.998619i	$0.483271\pi$
$312$	0	0
$313$	2.04456e26	1.28052	0.640258	−	0.768160i	$-0.278827\pi$
$313$	2.04456e26	1.28052	0.640258	+	0.768160i	$0.278827\pi$
$314$	0	0
$315$	−1.95581e25	−0.114569
$316$	0	0
$317$	5.44096e25	0.298231	0.149116	−	0.988820i	$-0.452357\pi$
$317$	5.44096e25	0.298231	0.149116	+	0.988820i	$0.452357\pi$
$318$	0	0
$319$	−6.10502e25	−0.313246
$320$	0	0
$321$	−7.38298e25	−0.354755
$322$	0	0
$323$	−2.08282e26	−0.937619
$324$	0	0
$325$	−5.46611e25	−0.230623
$326$	0	0
$327$	3.60061e26	1.42438
$328$	0	0
$329$	−7.77280e26	−2.88417
$330$	0	0
$331$	9.68262e25	0.337131	0.168566	−	0.985690i	$-0.446086\pi$
$331$	9.68262e25	0.337131	0.168566	+	0.985690i	$0.446086\pi$
$332$	0	0
$333$	2.39210e25	0.0781833
$334$	0	0
$335$	−5.04719e26	−1.54910
$336$	0	0
$337$	−3.56970e26	−1.02924	−0.514621	−	0.857418i	$-0.672067\pi$
$337$	−3.56970e26	−1.02924	−0.514621	+	0.857418i	$0.672067\pi$
$338$	0	0
$339$	−5.98850e26	−1.62264
$340$	0	0
$341$	−1.69346e26	−0.431375
$342$	0	0
$343$	−1.53451e26	−0.367605
$344$	0	0
$345$	−9.76460e26	−2.20066
$346$	0	0
$347$	6.52002e26	1.38290	0.691448	−	0.722426i	$-0.256973\pi$
$347$	6.52002e26	1.38290	0.691448	+	0.722426i	$0.256973\pi$
$348$	0	0
$349$	3.65851e26	0.730529	0.365264	−	0.930904i	$-0.380979\pi$
$349$	3.65851e26	0.730529	0.365264	+	0.930904i	$0.380979\pi$
$350$	0	0
$351$	8.86718e25	0.166748
$352$	0	0
$353$	−3.50026e26	−0.620106	−0.310053	−	0.950719i	$-0.600347\pi$
$353$	−3.50026e26	−0.620106	−0.310053	+	0.950719i	$0.600347\pi$
$354$	0	0
$355$	1.19757e27	1.99942
$356$	0	0
$357$	−9.47876e26	−1.49188
$358$	0	0
$359$	2.23431e26	0.331628	0.165814	−	0.986157i	$-0.446975\pi$
$359$	2.23431e26	0.331628	0.165814	+	0.986157i	$0.446975\pi$
$360$	0	0
$361$	−4.91799e25	−0.0688593
$362$	0	0
$363$	−4.96463e26	−0.655945
$364$	0	0
$365$	−8.49435e26	−1.05939
$366$	0	0
$367$	−1.26863e27	−1.49397	−0.746985	−	0.664841i	$-0.768499\pi$
$367$	−1.26863e27	−1.49397	−0.746985	+	0.664841i	$0.768499\pi$
$368$	0	0
$369$	7.35204e25	0.0817769
$370$	0	0
$371$	−1.90364e27	−2.00060
$372$	0	0
$373$	7.53713e26	0.748623	0.374312	−	0.927303i	$-0.377879\pi$
$373$	7.53713e26	0.748623	0.374312	+	0.927303i	$0.377879\pi$
$374$	0	0
$375$	5.79052e26	0.543737
$376$	0	0
$377$	−1.00725e26	−0.0894444
$378$	0	0
$379$	−1.86391e27	−1.56572	−0.782858	−	0.622201i	$-0.786239\pi$
$379$	−1.86391e27	−1.56572	−0.782858	+	0.622201i	$0.786239\pi$
$380$	0	0
$381$	9.40237e26	0.747352
$382$	0	0
$383$	1.32204e27	0.994621	0.497311	−	0.867573i	$-0.334321\pi$
$383$	1.32204e27	0.994621	0.497311	+	0.867573i	$0.334321\pi$
$384$	0	0
$385$	1.93301e27	1.37688
$386$	0	0
$387$	1.55515e25	0.0104908
$388$	0	0
$389$	−1.47698e27	−0.943850	−0.471925	−	0.881639i	$-0.656441\pi$
$389$	−1.47698e27	−0.943850	−0.471925	+	0.881639i	$0.656441\pi$
$390$	0	0
$391$	−2.24982e27	−1.36236
$392$	0	0
$393$	−1.28280e27	−0.736268
$394$	0	0
$395$	4.46257e27	2.42838
$396$	0	0
$397$	1.68855e27	0.871390	0.435695	−	0.900094i	$-0.356503\pi$
$397$	1.68855e27	0.871390	0.435695	+	0.900094i	$0.356503\pi$
$398$	0	0
$399$	3.02649e27	1.48158
$400$	0	0
$401$	1.76009e27	0.817557	0.408778	−	0.912634i	$-0.365955\pi$
$401$	1.76009e27	0.817557	0.408778	+	0.912634i	$0.365955\pi$
$402$	0	0
$403$	−2.79399e26	−0.123175
$404$	0	0
$405$	−3.83357e27	−1.60445
$406$	0	0
$407$	−2.36420e27	−0.939602
$408$	0	0
$409$	3.70780e27	1.39966	0.699829	−	0.714311i	$-0.253260\pi$
$409$	3.70780e27	1.39966	0.699829	+	0.714311i	$0.253260\pi$
$410$	0	0
$411$	−8.46561e26	−0.303612
$412$	0	0
$413$	−6.51838e27	−2.22159
$414$	0	0
$415$	3.04286e27	0.985772
$416$	0	0
$417$	−1.02269e27	−0.315002
$418$	0	0
$419$	3.98223e27	1.16648	0.583242	−	0.812299i	$-0.301784\pi$
$419$	3.98223e27	1.16648	0.583242	+	0.812299i	$0.301784\pi$
$420$	0	0
$421$	3.49021e27	0.972500	0.486250	−	0.873820i	$-0.338365\pi$
$421$	3.49021e27	0.972500	0.486250	+	0.873820i	$0.338365\pi$
$422$	0	0
$423$	3.62392e26	0.0960736
$424$	0	0
$425$	5.18543e27	1.30828
$426$	0	0
$427$	1.97038e27	0.473214
$428$	0	0
$429$	4.60418e26	0.105281
$430$	0	0
$431$	1.31467e27	0.286289	0.143145	−	0.989702i	$-0.454279\pi$
$431$	1.31467e27	0.286289	0.143145	+	0.989702i	$0.454279\pi$
$432$	0	0
$433$	−1.50356e27	−0.311888	−0.155944	−	0.987766i	$-0.549842\pi$
$433$	−1.50356e27	−0.311888	−0.155944	+	0.987766i	$0.549842\pi$
$434$	0	0
$435$	4.14715e27	0.819620
$436$	0	0
$437$	7.18351e27	1.35295
$438$	0	0
$439$	1.82537e27	0.327699	0.163849	−	0.986485i	$-0.447609\pi$
$439$	1.82537e27	0.327699	0.163849	+	0.986485i	$0.447609\pi$
$440$	0	0
$441$	3.63171e26	0.0621593
$442$	0	0
$443$	−1.18392e28	−1.93235	−0.966173	−	0.257895i	$-0.916971\pi$
$443$	−1.18392e28	−1.93235	−0.966173	+	0.257895i	$0.916971\pi$
$444$	0	0
$445$	5.09043e27	0.792454
$446$	0	0
$447$	−1.26008e28	−1.87141
$448$	0	0
$449$	−2.63665e27	−0.373650	−0.186825	−	0.982393i	$-0.559820\pi$
$449$	−2.63665e27	−0.373650	−0.186825	+	0.982393i	$0.559820\pi$
$450$	0	0
$451$	−7.26632e27	−0.982790
$452$	0	0
$453$	1.32256e28	1.70759
$454$	0	0
$455$	3.18921e27	0.393155
$456$	0	0
$457$	1.05250e28	1.23908	0.619542	−	0.784964i	$-0.287318\pi$
$457$	1.05250e28	1.23908	0.619542	+	0.784964i	$0.287318\pi$
$458$	0	0
$459$	−8.41185e27	−0.945928
$460$	0	0
$461$	5.59350e27	0.600930	0.300465	−	0.953793i	$-0.402858\pi$
$461$	5.59350e27	0.600930	0.300465	+	0.953793i	$0.402858\pi$
$462$	0	0
$463$	3.40544e27	0.349601	0.174801	−	0.984604i	$-0.444072\pi$
$463$	3.40544e27	0.349601	0.174801	+	0.984604i	$0.444072\pi$
$464$	0	0
$465$	1.15037e28	1.12871
$466$	0	0
$467$	7.42236e27	0.696169	0.348084	−	0.937463i	$-0.386832\pi$
$467$	7.42236e27	0.696169	0.348084	+	0.937463i	$0.386832\pi$
$468$	0	0
$469$	1.68978e28	1.51536
$470$	0	0
$471$	−5.42425e27	−0.465179
$472$	0	0
$473$	−1.53702e27	−0.126078
$474$	0	0
$475$	−1.65567e28	−1.29924
$476$	0	0
$477$	8.87536e26	0.0666412
$478$	0	0
$479$	7.67474e27	0.551494	0.275747	−	0.961230i	$-0.411075\pi$
$479$	7.67474e27	0.551494	0.275747	+	0.961230i	$0.411075\pi$
$480$	0	0
$481$	−3.90063e27	−0.268294
$482$	0	0
$483$	3.26915e28	2.15273
$484$	0	0
$485$	1.30737e26	0.00824346
$486$	0	0
$487$	−2.05168e28	−1.23896	−0.619479	−	0.785013i	$-0.712656\pi$
$487$	−2.05168e28	−1.23896	−0.619479	+	0.785013i	$0.712656\pi$
$488$	0	0
$489$	−2.50890e28	−1.45125
$490$	0	0
$491$	2.22388e28	1.23241	0.616207	−	0.787585i	$-0.288669\pi$
$491$	2.22388e28	1.23241	0.616207	+	0.787585i	$0.288669\pi$
$492$	0	0
$493$	9.55528e27	0.507400
$494$	0	0
$495$	−9.01227e26	−0.0458648
$496$	0	0
$497$	−4.00943e28	−1.95587
$498$	0	0
$499$	−4.77253e26	−0.0223199	−0.0111600	−	0.999938i	$-0.503552\pi$
$499$	−4.77253e26	−0.0223199	−0.0111600	+	0.999938i	$0.503552\pi$
$500$	0	0
$501$	−5.97636e27	−0.268003
$502$	0	0
$503$	7.56864e27	0.325502	0.162751	−	0.986667i	$-0.447963\pi$
$503$	7.56864e27	0.325502	0.162751	+	0.986667i	$0.447963\pi$
$504$	0	0
$505$	3.38135e28	1.39486
$506$	0	0
$507$	−2.51321e28	−0.994591
$508$	0	0
$509$	1.40098e28	0.531981	0.265991	−	0.963976i	$-0.414301\pi$
$509$	1.40098e28	0.531981	0.265991	+	0.963976i	$0.414301\pi$
$510$	0	0
$511$	2.84388e28	1.03631
$512$	0	0
$513$	2.68584e28	0.939393
$514$	0	0
$515$	−3.87169e28	−1.29994
$516$	0	0
$517$	−3.58166e28	−1.15461
$518$	0	0
$519$	8.79513e27	0.272261
$520$	0	0
$521$	−3.15008e28	−0.936538	−0.468269	−	0.883586i	$-0.655122\pi$
$521$	−3.15008e28	−0.936538	−0.468269	+	0.883586i	$0.655122\pi$
$522$	0	0
$523$	5.01448e28	1.43205	0.716025	−	0.698075i	$-0.245960\pi$
$523$	5.01448e28	1.43205	0.716025	+	0.698075i	$0.245960\pi$
$524$	0	0
$525$	−7.53479e28	−2.06728
$526$	0	0
$527$	2.65052e28	0.698747
$528$	0	0
$529$	3.81231e28	0.965836
$530$	0	0
$531$	3.03907e27	0.0740028
$532$	0	0
$533$	−1.19885e28	−0.280626
$534$	0	0
$535$	−2.35651e28	−0.530341
$536$	0	0
$537$	2.59397e28	0.561352
$538$	0	0
$539$	−3.58937e28	−0.747026
$540$	0	0
$541$	2.49456e27	0.0499370	0.0249685	−	0.999688i	$-0.492051\pi$
$541$	2.49456e27	0.0499370	0.0249685	+	0.999688i	$0.492051\pi$
$542$	0	0
$543$	−2.77706e27	−0.0534794
$544$	0	0
$545$	1.14925e29	2.12937
$546$	0	0
$547$	−8.60694e27	−0.153455	−0.0767276	−	0.997052i	$-0.524447\pi$
$547$	−8.60694e27	−0.153455	−0.0767276	+	0.997052i	$0.524447\pi$
$548$	0	0
$549$	−9.18652e26	−0.0157631
$550$	0	0
$551$	−3.05093e28	−0.503895
$552$	0	0
$553$	−1.49405e29	−2.37548
$554$	0	0
$555$	1.60601e29	2.45850
$556$	0	0
$557$	2.16380e28	0.318962	0.159481	−	0.987201i	$-0.449018\pi$
$557$	2.16380e28	0.318962	0.159481	+	0.987201i	$0.449018\pi$
$558$	0	0
$559$	−2.53589e27	−0.0360003
$560$	0	0
$561$	−4.36776e28	−0.597240
$562$	0	0
$563$	−1.11049e29	−1.46277	−0.731383	−	0.681966i	$-0.761125\pi$
$563$	−1.11049e29	−1.46277	−0.731383	+	0.681966i	$0.761125\pi$
$564$	0	0
$565$	−1.91142e29	−2.42576
$566$	0	0
$567$	1.28346e29	1.56950
$568$	0	0
$569$	−2.16903e27	−0.0255615	−0.0127808	−	0.999918i	$-0.504068\pi$
$569$	−2.16903e27	−0.0255615	−0.0127808	+	0.999918i	$0.504068\pi$
$570$	0	0
$571$	−9.33570e28	−1.06039	−0.530197	−	0.847874i	$-0.677882\pi$
$571$	−9.33570e28	−1.06039	−0.530197	+	0.847874i	$0.677882\pi$
$572$	0	0
$573$	−2.42309e28	−0.265306
$574$	0	0
$575$	−1.78841e29	−1.88780
$576$	0	0
$577$	−1.08356e29	−1.10282	−0.551411	−	0.834234i	$-0.685910\pi$
$577$	−1.08356e29	−1.10282	−0.551411	+	0.834234i	$0.685910\pi$
$578$	0	0
$579$	1.89225e29	1.85717
$580$	0	0
$581$	−1.01874e29	−0.964302
$582$	0	0
$583$	−8.77188e28	−0.800890
$584$	0	0
$585$	−1.48691e27	−0.0130963
$586$	0	0
$587$	−9.13012e28	−0.775848	−0.387924	−	0.921691i	$-0.626808\pi$
$587$	−9.13012e28	−0.775848	−0.387924	+	0.921691i	$0.626808\pi$
$588$	0	0
$589$	−8.46291e28	−0.693919
$590$	0	0
$591$	−1.23270e29	−0.975415
$592$	0	0
$593$	−2.71231e28	−0.207141	−0.103570	−	0.994622i	$-0.533027\pi$
$593$	−2.71231e28	−0.207141	−0.103570	+	0.994622i	$0.533027\pi$
$594$	0	0
$595$	−3.02545e29	−2.23029
$596$	0	0
$597$	1.49743e29	1.06565
$598$	0	0
$599$	−4.80899e28	−0.330425	−0.165212	−	0.986258i	$-0.552831\pi$
$599$	−4.80899e28	−0.330425	−0.165212	+	0.986258i	$0.552831\pi$
$600$	0	0
$601$	−1.95880e29	−1.29960	−0.649798	−	0.760107i	$-0.725147\pi$
$601$	−1.95880e29	−1.29960	−0.649798	+	0.760107i	$0.725147\pi$
$602$	0	0
$603$	−7.87828e27	−0.0504777
$604$	0	0
$605$	−1.58462e29	−0.980604
$606$	0	0
$607$	2.12296e29	1.26900	0.634498	−	0.772925i	$-0.281207\pi$
$607$	2.12296e29	1.26900	0.634498	+	0.772925i	$0.281207\pi$
$608$	0	0
$609$	−1.38845e29	−0.801768
$610$	0	0
$611$	−5.90928e28	−0.329687
$612$	0	0
$613$	3.24221e29	1.74786	0.873929	−	0.486053i	$-0.161564\pi$
$613$	3.24221e29	1.74786	0.873929	+	0.486053i	$0.161564\pi$
$614$	0	0
$615$	4.93601e29	2.57151
$616$	0	0
$617$	−1.51597e29	−0.763303	−0.381651	−	0.924306i	$-0.624644\pi$
$617$	−1.51597e29	−0.763303	−0.381651	+	0.924306i	$0.624644\pi$
$618$	0	0
$619$	2.81570e29	1.37036	0.685181	−	0.728373i	$-0.259723\pi$
$619$	2.81570e29	1.37036	0.685181	+	0.728373i	$0.259723\pi$
$620$	0	0
$621$	2.90118e29	1.36494
$622$	0	0
$623$	−1.70426e29	−0.775194
$624$	0	0
$625$	−1.21319e29	−0.533565
$626$	0	0
$627$	1.39459e29	0.593113
$628$	0	0
$629$	3.70033e29	1.52198
$630$	0	0
$631$	6.70437e28	0.266716	0.133358	−	0.991068i	$-0.457424\pi$
$631$	6.70437e28	0.266716	0.133358	+	0.991068i	$0.457424\pi$
$632$	0	0
$633$	3.45135e29	1.32816
$634$	0	0
$635$	3.00107e29	1.11725
$636$	0	0
$637$	−5.92199e28	−0.213306
$638$	0	0
$639$	1.86932e28	0.0651516
$640$	0	0
$641$	4.81731e29	1.62478	0.812391	−	0.583113i	$-0.198166\pi$
$641$	4.81731e29	1.62478	0.812391	+	0.583113i	$0.198166\pi$
$642$	0	0
$643$	3.76113e29	1.22773	0.613865	−	0.789411i	$-0.289614\pi$
$643$	3.76113e29	1.22773	0.613865	+	0.789411i	$0.289614\pi$
$644$	0	0
$645$	1.04410e29	0.329887
$646$	0	0
$647$	−7.45316e28	−0.227953	−0.113977	−	0.993483i	$-0.536359\pi$
$647$	−7.45316e28	−0.227953	−0.113977	+	0.993483i	$0.536359\pi$
$648$	0	0
$649$	−3.00364e29	−0.889361
$650$	0	0
$651$	−3.85140e29	−1.10412
$652$	0	0
$653$	−2.86960e29	−0.796586	−0.398293	−	0.917258i	$-0.630397\pi$
$653$	−2.86960e29	−0.796586	−0.398293	+	0.917258i	$0.630397\pi$
$654$	0	0
$655$	−4.09445e29	−1.10068
$656$	0	0
$657$	−1.32590e28	−0.0345203
$658$	0	0
$659$	2.02987e29	0.511884	0.255942	−	0.966692i	$-0.417614\pi$
$659$	2.02987e29	0.511884	0.255942	+	0.966692i	$0.417614\pi$
$660$	0	0
$661$	1.89913e29	0.463917	0.231959	−	0.972726i	$-0.425487\pi$
$661$	1.89913e29	0.463917	0.231959	+	0.972726i	$0.425487\pi$
$662$	0	0
$663$	−7.20623e28	−0.170536
$664$	0	0
$665$	9.66002e29	2.21488
$666$	0	0
$667$	−3.29554e29	−0.732160
$668$	0	0
$669$	−3.47210e29	−0.747513
$670$	0	0
$671$	9.07941e28	0.189440
$672$	0	0
$673$	−6.79583e28	−0.137431	−0.0687154	−	0.997636i	$-0.521890\pi$
$673$	−6.79583e28	−0.137431	−0.0687154	+	0.997636i	$0.521890\pi$
$674$	0	0
$675$	−6.68669e29	−1.31076
$676$	0	0
$677$	3.61456e29	0.686870	0.343435	−	0.939177i	$-0.388410\pi$
$677$	3.61456e29	0.686870	0.343435	+	0.939177i	$0.388410\pi$
$678$	0	0
$679$	−4.37702e27	−0.00806391
$680$	0	0
$681$	−4.59279e29	−0.820413
$682$	0	0
$683$	−2.71379e29	−0.470067	−0.235033	−	0.971987i	$-0.575520\pi$
$683$	−2.71379e29	−0.470067	−0.235033	+	0.971987i	$0.575520\pi$
$684$	0	0
$685$	−2.70207e29	−0.453885
$686$	0	0
$687$	8.05042e29	1.31151
$688$	0	0
$689$	−1.44725e29	−0.228686
$690$	0	0
$691$	−1.00367e29	−0.153841	−0.0769205	−	0.997037i	$-0.524509\pi$
$691$	−1.00367e29	−0.153841	−0.0769205	+	0.997037i	$0.524509\pi$
$692$	0	0
$693$	3.01728e28	0.0448659
$694$	0	0
$695$	−3.26423e29	−0.470912
$696$	0	0
$697$	1.13729e30	1.59194
$698$	0	0
$699$	1.03811e30	1.41005
$700$	0	0
$701$	8.73247e28	0.115106	0.0575530	−	0.998342i	$-0.481670\pi$
$701$	8.73247e28	0.115106	0.0575530	+	0.998342i	$0.481670\pi$
$702$	0	0
$703$	−1.18149e30	−1.51147
$704$	0	0
$705$	2.43303e30	3.02107
$706$	0	0
$707$	−1.13206e30	−1.36448
$708$	0	0
$709$	−6.44798e28	−0.0754464	−0.0377232	−	0.999288i	$-0.512011\pi$
$709$	−6.44798e28	−0.0754464	−0.0377232	+	0.999288i	$0.512011\pi$
$710$	0	0
$711$	6.96573e28	0.0791291
$712$	0	0
$713$	−9.14145e29	−1.00827
$714$	0	0
$715$	1.46957e29	0.157390
$716$	0	0
$717$	−5.15854e29	−0.536507
$718$	0	0
$719$	1.34474e30	1.35827	0.679135	−	0.734013i	$-0.262355\pi$
$719$	1.34474e30	1.35827	0.679135	+	0.734013i	$0.262355\pi$
$720$	0	0
$721$	1.29623e30	1.27163
$722$	0	0
$723$	1.17641e29	0.112101
$724$	0	0
$725$	7.59562e29	0.703095
$726$	0	0
$727$	−1.76319e30	−1.58558	−0.792789	−	0.609496i	$-0.791372\pi$
$727$	−1.76319e30	−1.58558	−0.792789	+	0.609496i	$0.791372\pi$
$728$	0	0
$729$	1.08187e30	0.945227
$730$	0	0
$731$	2.40567e29	0.204222
$732$	0	0
$733$	6.52500e29	0.538256	0.269128	−	0.963104i	$-0.413265\pi$
$733$	6.52500e29	0.538256	0.269128	+	0.963104i	$0.413265\pi$
$734$	0	0
$735$	2.43826e30	1.95462
$736$	0	0
$737$	7.78642e29	0.606638
$738$	0	0
$739$	4.16483e29	0.315378	0.157689	−	0.987489i	$-0.449596\pi$
$739$	4.16483e29	0.315378	0.157689	+	0.987489i	$0.449596\pi$
$740$	0	0
$741$	2.30089e29	0.169358
$742$	0	0
$743$	−2.31511e30	−1.65649	−0.828245	−	0.560367i	$-0.810660\pi$
$743$	−2.31511e30	−1.65649	−0.828245	+	0.560367i	$0.810660\pi$
$744$	0	0
$745$	−4.02194e30	−2.79766
$746$	0	0
$747$	4.74966e28	0.0321216
$748$	0	0
$749$	7.88952e29	0.518790
$750$	0	0
$751$	−8.45099e29	−0.540366	−0.270183	−	0.962809i	$-0.587084\pi$
$751$	−8.45099e29	−0.540366	−0.270183	+	0.962809i	$0.587084\pi$
$752$	0	0
$753$	−2.08698e29	−0.129769
$754$	0	0
$755$	4.22137e30	2.55276
$756$	0	0
$757$	1.46904e30	0.864027	0.432014	−	0.901867i	$-0.357803\pi$
$757$	1.46904e30	0.864027	0.432014	+	0.901867i	$0.357803\pi$
$758$	0	0
$759$	1.50641e30	0.861794
$760$	0	0
$761$	−1.97165e30	−1.09721	−0.548605	−	0.836082i	$-0.684841\pi$
$761$	−1.97165e30	−1.09721	−0.548605	+	0.836082i	$0.684841\pi$
$762$	0	0
$763$	−3.84765e30	−2.08299
$764$	0	0
$765$	1.41056e29	0.0742925
$766$	0	0
$767$	−4.95561e29	−0.253949
$768$	0	0
$769$	1.41752e29	0.0706809	0.0353405	−	0.999375i	$-0.488748\pi$
$769$	1.41752e29	0.0706809	0.0353405	+	0.999375i	$0.488748\pi$
$770$	0	0
$771$	1.55286e29	0.0753463
$772$	0	0
$773$	−1.70521e30	−0.805182	−0.402591	−	0.915380i	$-0.631890\pi$
$773$	−1.70521e30	−0.805182	−0.402591	+	0.915380i	$0.631890\pi$
$774$	0	0
$775$	2.10694e30	0.968241
$776$	0	0
$777$	−5.37685e30	−2.40496
$778$	0	0
$779$	−3.63127e30	−1.58094
$780$	0	0
$781$	−1.84753e30	−0.782987
$782$	0	0
$783$	−1.23217e30	−0.508361
$784$	0	0
$785$	−1.73132e30	−0.695420
$786$	0	0
$787$	1.02893e30	0.402393	0.201197	−	0.979551i	$-0.435517\pi$
$787$	1.02893e30	0.402393	0.201197	+	0.979551i	$0.435517\pi$
$788$	0	0
$789$	−3.06803e30	−1.16829
$790$	0	0
$791$	6.39937e30	2.37293
$792$	0	0
$793$	1.49798e29	0.0540927
$794$	0	0
$795$	5.95874e30	2.09556
$796$	0	0
$797$	−4.28099e30	−1.46633	−0.733166	−	0.680050i	$-0.761958\pi$
$797$	−4.28099e30	−1.46633	−0.733166	+	0.680050i	$0.761958\pi$
$798$	0	0
$799$	5.60584e30	1.87025
$800$	0	0
$801$	7.94576e28	0.0258223
$802$	0	0
$803$	1.31044e30	0.414863
$804$	0	0
$805$	1.04345e31	3.21823
$806$	0	0
$807$	−5.83539e30	−1.75347
$808$	0	0
$809$	−1.79480e30	−0.525480	−0.262740	−	0.964867i	$-0.584626\pi$
$809$	−1.79480e30	−0.525480	−0.262740	+	0.964867i	$0.584626\pi$
$810$	0	0
$811$	−5.66004e30	−1.61473	−0.807366	−	0.590052i	$-0.799107\pi$
$811$	−5.66004e30	−1.61473	−0.807366	+	0.590052i	$0.799107\pi$
$812$	0	0
$813$	−2.79399e30	−0.776737
$814$	0	0
$815$	−8.00796e30	−2.16954
$816$	0	0
$817$	−7.68111e29	−0.202812
$818$	0	0
$819$	4.97811e28	0.0128110
$820$	0	0
$821$	−1.74066e30	−0.436627	−0.218313	−	0.975879i	$-0.570056\pi$
$821$	−1.74066e30	−0.436627	−0.218313	+	0.975879i	$0.570056\pi$
$822$	0	0
$823$	−7.23459e30	−1.76895	−0.884475	−	0.466588i	$-0.845483\pi$
$823$	−7.23459e30	−1.76895	−0.884475	+	0.466588i	$0.845483\pi$
$824$	0	0
$825$	−3.47199e30	−0.827584
$826$	0	0
$827$	1.43144e30	0.332633	0.166317	−	0.986072i	$-0.446813\pi$
$827$	1.43144e30	0.332633	0.166317	+	0.986072i	$0.446813\pi$
$828$	0	0
$829$	1.83497e30	0.415726	0.207863	−	0.978158i	$-0.433349\pi$
$829$	1.83497e30	0.415726	0.207863	+	0.978158i	$0.433349\pi$
$830$	0	0
$831$	−2.47521e30	−0.546766
$832$	0	0
$833$	5.61790e30	1.21004
$834$	0	0
$835$	−1.90755e30	−0.400652
$836$	0	0
$837$	−3.41789e30	−0.700069
$838$	0	0
$839$	3.95467e30	0.789968	0.394984	−	0.918688i	$-0.370750\pi$
$839$	3.95467e30	0.789968	0.394984	+	0.918688i	$0.370750\pi$
$840$	0	0
$841$	−3.73318e30	−0.727313
$842$	0	0
$843$	−5.64972e29	−0.107359
$844$	0	0
$845$	−8.02170e30	−1.48686
$846$	0	0
$847$	5.30525e30	0.959246
$848$	0	0
$849$	4.30293e30	0.758986
$850$	0	0
$851$	−1.27622e31	−2.19616
$852$	0	0
$853$	6.13633e30	1.03025	0.515126	−	0.857114i	$-0.327745\pi$
$853$	6.13633e30	1.03025	0.515126	+	0.857114i	$0.327745\pi$
$854$	0	0
$855$	−4.50379e29	−0.0737793
$856$	0	0
$857$	8.41098e30	1.34446	0.672230	−	0.740342i	$-0.265337\pi$
$857$	8.41098e30	1.34446	0.672230	+	0.740342i	$0.265337\pi$
$858$	0	0
$859$	1.13172e31	1.76526	0.882632	−	0.470064i	$-0.155769\pi$
$859$	1.13172e31	1.76526	0.882632	+	0.470064i	$0.155769\pi$
$860$	0	0
$861$	−1.65256e31	−2.51550
$862$	0	0
$863$	−6.81213e30	−1.01197	−0.505987	−	0.862541i	$-0.668872\pi$
$863$	−6.81213e30	−1.01197	−0.505987	+	0.862541i	$0.668872\pi$
$864$	0	0
$865$	2.80725e30	0.407016
$866$	0	0
$867$	−4.04451e29	−0.0572356
$868$	0	0
$869$	−6.88452e30	−0.950968
$870$	0	0
$871$	1.28466e30	0.173220
$872$	0	0
$873$	2.04070e27	0.000268614 0
$874$	0	0
$875$	−6.18780e30	−0.795155
$876$	0	0
$877$	−7.50120e30	−0.941098	−0.470549	−	0.882374i	$-0.655944\pi$
$877$	−7.50120e30	−0.941098	−0.470549	+	0.882374i	$0.655944\pi$
$878$	0	0
$879$	8.96862e30	1.09861
$880$	0	0
$881$	9.11985e29	0.109079	0.0545394	−	0.998512i	$-0.482631\pi$
$881$	9.11985e29	0.109079	0.0545394	+	0.998512i	$0.482631\pi$
$882$	0	0
$883$	7.41053e30	0.865489	0.432745	−	0.901517i	$-0.357545\pi$
$883$	7.41053e30	0.865489	0.432745	+	0.901517i	$0.357545\pi$
$884$	0	0
$885$	2.04037e31	2.32705
$886$	0	0
$887$	1.16195e31	1.29417	0.647083	−	0.762420i	$-0.275989\pi$
$887$	1.16195e31	1.29417	0.647083	+	0.762420i	$0.275989\pi$
$888$	0	0
$889$	−1.00475e31	−1.09292
$890$	0	0
$891$	5.91413e30	0.628312
$892$	0	0
$893$	−1.78990e31	−1.85733
$894$	0	0
$895$	8.27949e30	0.839193
$896$	0	0
$897$	2.48538e30	0.246077
$898$	0	0
$899$	3.88249e30	0.375520
$900$	0	0
$901$	1.37293e31	1.29729
$902$	0	0
$903$	−3.49561e30	−0.322702
$904$	0	0
$905$	−8.86387e29	−0.0799490
$906$	0	0
$907$	1.10105e31	0.970357	0.485178	−	0.874415i	$-0.338755\pi$
$907$	1.10105e31	0.970357	0.485178	+	0.874415i	$0.338755\pi$
$908$	0	0
$909$	5.27802e29	0.0454517
$910$	0	0
$911$	−5.85430e30	−0.492643	−0.246321	−	0.969188i	$-0.579222\pi$
$911$	−5.85430e30	−0.492643	−0.246321	+	0.969188i	$0.579222\pi$
$912$	0	0
$913$	−4.69429e30	−0.386035
$914$	0	0
$915$	−6.16765e30	−0.495676
$916$	0	0
$917$	1.37081e31	1.07671
$918$	0	0
$919$	2.04737e30	0.157175	0.0785875	−	0.996907i	$-0.474959\pi$
$919$	2.04737e30	0.157175	0.0785875	+	0.996907i	$0.474959\pi$
$920$	0	0
$921$	1.67974e31	1.26042
$922$	0	0
$923$	−3.04818e30	−0.223575
$924$	0	0
$925$	2.94145e31	2.10898
$926$	0	0
$927$	−6.04340e29	−0.0423589
$928$	0	0
$929$	2.65595e31	1.81993	0.909966	−	0.414683i	$-0.136107\pi$
$929$	2.65595e31	1.81993	0.909966	+	0.414683i	$0.136107\pi$
$930$	0	0
$931$	−1.79375e31	−1.20168
$932$	0	0
$933$	1.64360e30	0.107656
$934$	0	0
$935$	−1.39411e31	−0.892843
$936$	0	0
$937$	−1.63091e31	−1.02132	−0.510662	−	0.859781i	$-0.670600\pi$
$937$	−1.63091e31	−1.02132	−0.510662	+	0.859781i	$0.670600\pi$
$938$	0	0
$939$	2.14265e31	1.31208
$940$	0	0
$941$	1.56346e31	0.936256	0.468128	−	0.883661i	$-0.344929\pi$
$941$	1.56346e31	0.936256	0.468128	+	0.883661i	$0.344929\pi$
$942$	0	0
$943$	−3.92242e31	−2.29711
$944$	0	0
$945$	3.90137e31	2.23451
$946$	0	0
$947$	−1.26330e30	−0.0707674	−0.0353837	−	0.999374i	$-0.511265\pi$
$947$	−1.26330e30	−0.0707674	−0.0353837	+	0.999374i	$0.511265\pi$
$948$	0	0
$949$	2.16206e30	0.118460
$950$	0	0
$951$	5.70198e30	0.305584
$952$	0	0
$953$	−3.00788e31	−1.57683	−0.788416	−	0.615143i	$-0.789098\pi$
$953$	−3.00788e31	−1.57683	−0.788416	+	0.615143i	$0.789098\pi$
$954$	0	0
$955$	−7.73407e30	−0.396618
$956$	0	0
$957$	−6.39790e30	−0.320968
$958$	0	0
$959$	9.04643e30	0.443999
$960$	0	0
$961$	−1.00559e31	−0.482866
$962$	0	0
$963$	−3.67833e29	−0.0172812
$964$	0	0
$965$	6.03970e31	2.77638
$966$	0	0
$967$	4.30677e31	1.93719	0.968597	−	0.248634i	$-0.0799818\pi$
$967$	4.30677e31	1.93719	0.968597	+	0.248634i	$0.0799818\pi$
$968$	0	0
$969$	−2.18274e31	−0.960734
$970$	0	0
$971$	4.75784e29	0.0204931	0.0102466	−	0.999948i	$-0.496738\pi$
$971$	4.75784e29	0.0204931	0.0102466	+	0.999948i	$0.496738\pi$
$972$	0	0
$973$	1.09285e31	0.460655
$974$	0	0
$975$	−5.72833e30	−0.236309
$976$	0	0
$977$	2.92155e31	1.17956	0.589781	−	0.807563i	$-0.299214\pi$
$977$	2.92155e31	1.17956	0.589781	+	0.807563i	$0.299214\pi$
$978$	0	0
$979$	−7.85312e30	−0.310330
$980$	0	0
$981$	1.79389e30	0.0693859
$982$	0	0
$983$	1.56094e31	0.590983	0.295492	−	0.955345i	$-0.404517\pi$
$983$	1.56094e31	0.590983	0.295492	+	0.955345i	$0.404517\pi$
$984$	0	0
$985$	−3.93456e31	−1.45820
$986$	0	0
$987$	−8.14568e31	−2.95527
$988$	0	0
$989$	−8.29697e30	−0.294685
$990$	0	0
$991$	3.30429e31	1.14896	0.574480	−	0.818518i	$-0.305204\pi$
$991$	3.30429e31	1.14896	0.574480	+	0.818518i	$0.305204\pi$
$992$	0	0
$993$	1.01471e31	0.345443
$994$	0	0
$995$	4.77953e31	1.59310
$996$	0	0
$997$	−8.84293e29	−0.0288600	−0.0144300	−	0.999896i	$-0.504593\pi$
$997$	−8.84293e29	−0.0288600	−0.0144300	+	0.999896i	$0.504593\pi$
$998$	0	0
$999$	−4.77164e31	−1.52486

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.22.a.n.1.3		4
4.3	odd	2	inner	64.22.a.n.1.2		4
8.3	odd	2		32.22.a.b.1.3	yes	4
8.5	even	2		32.22.a.b.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.22.a.b.1.2	✓	4	8.5	even	2
32.22.a.b.1.3	yes	4	8.3	odd	2
64.22.a.n.1.2		4	4.3	odd	2	inner
64.22.a.n.1.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

\(f(q)\)	\(=\)	\(q+104797. q^{3} +3.34494e7 q^{5} -1.11987e9 q^{7} +5.22119e8 q^{9} +O(q^{10})\)	\(q+104797. q^{3} +3.34494e7 q^{5} -1.11987e9 q^{7} +5.22119e8 q^{9} -5.16032e10 q^{11} -8.51385e10 q^{13} +3.50541e12 q^{15} +8.07667e12 q^{17} -2.57882e13 q^{19} -1.17360e14 q^{21} -2.78558e14 q^{23} +6.42025e14 q^{25} -1.04150e15 q^{27} +1.18307e15 q^{29} +3.28170e15 q^{31} -5.40787e15 q^{33} -3.74591e16 q^{35} +4.58151e16 q^{37} -8.92229e15 q^{39} +1.40811e17 q^{41} +2.97854e16 q^{43} +1.74646e16 q^{45} +6.94078e17 q^{47} +6.95571e17 q^{49} +8.46413e17 q^{51} +1.69987e18 q^{53} -1.72609e18 q^{55} -2.70253e18 q^{57} +5.82064e18 q^{59} -1.75947e18 q^{61} -5.84708e17 q^{63} -2.84783e18 q^{65} -1.50890e19 q^{67} -2.91921e19 q^{69} +3.58026e19 q^{71} -2.53946e19 q^{73} +6.72825e19 q^{75} +5.77890e19 q^{77} +1.33413e20 q^{79} -1.14608e20 q^{81} +9.09690e19 q^{83} +2.70160e20 q^{85} +1.23983e20 q^{87} +1.52183e20 q^{89} +9.53444e19 q^{91} +3.43913e20 q^{93} -8.62599e20 q^{95} +3.90849e18 q^{97} -2.69430e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 37124680 q^{5} + 16226262324 q^{9}+O(q^{10}) \) 4 * q + 37124680 * q^5 + 16226262324 * q^9	\( 4 q + 37124680 q^{5} + 16226262324 q^{9} + 1438891028904 q^{13} + 27067357021960 q^{17} - 58494636232704 q^{21} + 773625552400700 q^{25} + 15\!\cdots\!08 q^{29}+ \cdots - 14\!\cdots\!88 q^{97}+O(q^{100}) \) 4 * q + 37124680 * q^5 + 16226262324 * q^9 + 1438891028904 * q^13 + 27067357021960 * q^17 - 58494636232704 * q^21 + 773625552400700 * q^25 + 1535657143500008 * q^29 - 2740052722894848 * q^33 + 51861754826473544 * q^37 + 76119164178377384 * q^41 - 191086569820209240 * q^45 + 1134240365253238628 * q^49 + 4626949431882959624 * q^53 - 2265200566382131200 * q^57 - 13244181711188448536 * q^61 - 29651450007494917680 * q^65 + 7851231862535245824 * q^69 + 39323341470423072360 * q^73 + 154997157739134799872 * q^77 - 491616252151343842140 * q^81 + 377841710648841826960 * q^85 - 296515048889049879640 * q^89 + 2114246149921519091712 * q^93 - 1497970122472846632888 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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