LMFDB - Embedded newform 64.24.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,24,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, 24, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 24 $
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:	$214.530583901$
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} - 1841764x - 103489260 $ x^3 - 1841764*x - 103489260
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{21}\cdot 3\cdot 5 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-56.2871$ 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)

$f(q)$	$=$	$q+442003. q^{3} +2.22890e7 q^{5} -8.79699e8 q^{7} +1.01223e11 q^{9} +O(q^{10})$	$q+442003. q^{3} +2.22890e7 q^{5} -8.79699e8 q^{7} +1.01223e11 q^{9} +1.55363e12 q^{11} -1.97761e12 q^{13} +9.85181e12 q^{15} -2.09169e14 q^{17} +4.91986e14 q^{19} -3.88830e14 q^{21} -8.38926e15 q^{23} -1.14241e16 q^{25} +3.12947e15 q^{27} -9.20414e16 q^{29} -1.51495e17 q^{31} +6.86711e17 q^{33} -1.96076e16 q^{35} -1.71122e17 q^{37} -8.74110e17 q^{39} -4.58022e17 q^{41} +5.33564e18 q^{43} +2.25617e18 q^{45} +2.08351e19 q^{47} -2.65949e19 q^{49} -9.24534e19 q^{51} -5.59140e19 q^{53} +3.46290e19 q^{55} +2.17459e20 q^{57} -9.00041e19 q^{59} -1.75379e20 q^{61} -8.90461e19 q^{63} -4.40790e19 q^{65} +3.44632e20 q^{67} -3.70808e21 q^{69} +1.58288e21 q^{71} -2.05963e20 q^{73} -5.04950e21 q^{75} -1.36673e21 q^{77} +1.20889e22 q^{79} -8.14626e21 q^{81} +2.39773e21 q^{83} -4.66218e21 q^{85} -4.06826e22 q^{87} -4.69027e22 q^{89} +1.73970e21 q^{91} -6.69610e22 q^{93} +1.09659e22 q^{95} -6.79984e22 q^{97} +1.57264e23 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 32708 q^{3} - 31480650 q^{5} - 993025320 q^{7} + 1389317071 q^{9}+O(q^{10}) $ 3 * q + 32708 * q^3 - 31480650 * q^5 - 993025320 * q^7 + 1389317071 * q^9	$ 3 q + 32708 q^{3} - 31480650 q^{5} - 993025320 q^{7} + 1389317071 q^{9} - 23441525844 q^{11} - 2019379246962 q^{13} + 4994553094600 q^{15} - 2160517821354 q^{17} + 312191787410964 q^{19} - 825707464014048 q^{21} - 47\!\cdots\!40 q^{23}+ \cdots + 25\!\cdots\!40 q^{99}+O(q^{100}) $ 3 * q + 32708 * q^3 - 31480650 * q^5 - 993025320 * q^7 + 1389317071 * q^9 - 23441525844 * q^11 - 2019379246962 * q^13 + 4994553094600 * q^15 - 2160517821354 * q^17 + 312191787410964 * q^19 - 825707464014048 * q^21 - 4776103684514040 * q^23 + 19807036356305925 * q^25 + 60172513040762216 * q^27 - 172150256810092098 * q^29 - 424442363135920032 * q^31 + 956775180295468688 * q^33 + 1538966670967748400 * q^35 - 2543857445080432362 * q^37 - 1740226054689893144 * q^39 + 2761653172426159758 * q^41 - 5964473505832923060 * q^43 + 11432054542232660750 * q^45 + 52578892233811303824 * q^47 - 36206020957260388773 * q^49 - 140919484512501823160 * q^51 - 15683594666449490106 * q^53 + 255144484970133759000 * q^55 + 312667957724352284272 * q^57 - 104790862916257022052 * q^59 - 1281299617116320867298 * q^61 + 104914943644727114808 * q^63 + 2914343573270694440700 * q^65 - 805429647134646351612 * q^67 - 4263874380293473266848 * q^69 + 3282309189857685782424 * q^71 + 2509034598106619331630 * q^73 - 10588030323195962675300 * q^75 + 3889098717006269829216 * q^77 + 17558070599920224291312 * q^79 - 10704146103369100114373 * q^81 - 15528616638328360005804 * q^83 + 10404913584518742482700 * q^85 - 20926087057105934811864 * q^87 - 13507192466673039723906 * q^89 + 87522067328056376581104 * q^91 - 13527526723927613073280 * q^93 - 181703400482520412239000 * q^95 - 1279961410088837059866 * q^97 + 257548906525338333755740 * q^99

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
$3$	442003.	1.44056	0.720279	−	0.693685i	$-0.244014\pi$
$3$	442003.	1.44056	0.720279	+	0.693685i	$0.244014\pi$
$4$	0	0
$5$	2.22890e7	0.204144	0.102072	−	0.994777i	$-0.467453\pi$
$5$	2.22890e7	0.204144	0.102072	+	0.994777i	$0.467453\pi$
$6$	0	0
$7$	−8.79699e8	−0.168154	−0.0840769	−	0.996459i	$-0.526794\pi$
$7$	−8.79699e8	−0.168154	−0.0840769	+	0.996459i	$0.526794\pi$
$8$	0	0
$9$	1.01223e11	1.07521
$10$	0	0
$11$	1.55363e12	1.64185	0.820924	−	0.571037i	$-0.193459\pi$
$11$	1.55363e12	1.64185	0.820924	+	0.571037i	$0.193459\pi$
$12$	0	0
$13$	−1.97761e12	−0.306050	−0.153025	−	0.988222i	$-0.548901\pi$
$13$	−1.97761e12	−0.306050	−0.153025	+	0.988222i	$0.548901\pi$
$14$	0	0
$15$	9.85181e12	0.294081
$16$	0	0
$17$	−2.09169e14	−1.48025	−0.740125	−	0.672470i	$-0.765234\pi$
$17$	−2.09169e14	−1.48025	−0.740125	+	0.672470i	$0.765234\pi$
$18$	0	0
$19$	4.91986e14	0.968916	0.484458	−	0.874814i	$-0.339017\pi$
$19$	4.91986e14	0.968916	0.484458	+	0.874814i	$0.339017\pi$
$20$	0	0
$21$	−3.88830e14	−0.242235
$22$	0	0
$23$	−8.38926e15	−1.83592	−0.917959	−	0.396675i	$-0.870164\pi$
$23$	−8.38926e15	−1.83592	−0.917959	+	0.396675i	$0.870164\pi$
$24$	0	0
$25$	−1.14241e16	−0.958325
$26$	0	0
$27$	3.12947e15	0.108340
$28$	0	0
$29$	−9.20414e16	−1.40090	−0.700449	−	0.713703i	$-0.747017\pi$
$29$	−9.20414e16	−1.40090	−0.700449	+	0.713703i	$0.747017\pi$
$30$	0	0
$31$	−1.51495e17	−1.07087	−0.535436	−	0.844576i	$-0.679853\pi$
$31$	−1.51495e17	−1.07087	−0.535436	+	0.844576i	$0.679853\pi$
$32$	0	0
$33$	6.86711e17	2.36518
$34$	0	0
$35$	−1.96076e16	−0.0343275
$36$	0	0
$37$	−1.71122e17	−0.158120	−0.0790598	−	0.996870i	$-0.525192\pi$
$37$	−1.71122e17	−0.158120	−0.0790598	+	0.996870i	$0.525192\pi$
$38$	0	0
$39$	−8.74110e17	−0.440883
$40$	0	0
$41$	−4.58022e17	−0.129979	−0.0649893	−	0.997886i	$-0.520701\pi$
$41$	−4.58022e17	−0.129979	−0.0649893	+	0.997886i	$0.520701\pi$
$42$	0	0
$43$	5.33564e18	0.875587	0.437794	−	0.899075i	$-0.355760\pi$
$43$	5.33564e18	0.875587	0.437794	+	0.899075i	$0.355760\pi$
$44$	0	0
$45$	2.25617e18	0.219497
$46$	0	0
$47$	2.08351e19	1.22933	0.614666	−	0.788787i	$-0.289291\pi$
$47$	2.08351e19	1.22933	0.614666	+	0.788787i	$0.289291\pi$
$48$	0	0
$49$	−2.65949e19	−0.971724
$50$	0	0
$51$	−9.24534e19	−2.13239
$52$	0	0
$53$	−5.59140e19	−0.828607	−0.414303	−	0.910139i	$-0.635975\pi$
$53$	−5.59140e19	−0.828607	−0.414303	+	0.910139i	$0.635975\pi$
$54$	0	0
$55$	3.46290e19	0.335173
$56$	0	0
$57$	2.17459e20	1.39578
$58$	0	0
$59$	−9.00041e19	−0.388565	−0.194283	−	0.980946i	$-0.562238\pi$
$59$	−9.00041e19	−0.388565	−0.194283	+	0.980946i	$0.562238\pi$
$60$	0	0
$61$	−1.75379e20	−0.516041	−0.258021	−	0.966139i	$-0.583070\pi$
$61$	−1.75379e20	−0.516041	−0.258021	+	0.966139i	$0.583070\pi$
$62$	0	0
$63$	−8.90461e19	−0.180800
$64$	0	0
$65$	−4.40790e19	−0.0624782
$66$	0	0
$67$	3.44632e20	0.344743	0.172372	−	0.985032i	$-0.444857\pi$
$67$	3.44632e20	0.344743	0.172372	+	0.985032i	$0.444857\pi$
$68$	0	0
$69$	−3.70808e21	−2.64475
$70$	0	0
$71$	1.58288e21	0.812787	0.406394	−	0.913698i	$-0.366786\pi$
$71$	1.58288e21	0.812787	0.406394	+	0.913698i	$0.366786\pi$
$72$	0	0
$73$	−2.05963e20	−0.0768380	−0.0384190	−	0.999262i	$-0.512232\pi$
$73$	−2.05963e20	−0.0768380	−0.0384190	+	0.999262i	$0.512232\pi$
$74$	0	0
$75$	−5.04950e21	−1.38052
$76$	0	0
$77$	−1.36673e21	−0.276083
$78$	0	0
$79$	1.20889e22	1.81834	0.909172	−	0.416420i	$-0.136716\pi$
$79$	1.20889e22	1.81834	0.909172	+	0.416420i	$0.136716\pi$
$80$	0	0
$81$	−8.14626e21	−0.919137
$82$	0	0
$83$	2.39773e21	0.204363	0.102181	−	0.994766i	$-0.467418\pi$
$83$	2.39773e21	0.204363	0.102181	+	0.994766i	$0.467418\pi$
$84$	0	0
$85$	−4.66218e21	−0.302184
$86$	0	0
$87$	−4.06826e22	−2.01807
$88$	0	0
$89$	−4.69027e22	−1.79148	−0.895742	−	0.444574i	$-0.853355\pi$
$89$	−4.69027e22	−1.79148	−0.895742	+	0.444574i	$0.853355\pi$
$90$	0	0
$91$	1.73970e21	0.0514635
$92$	0	0
$93$	−6.69610e22	−1.54265
$94$	0	0
$95$	1.09659e22	0.197798
$96$	0	0
$97$	−6.79984e22	−0.965214	−0.482607	−	0.875837i	$-0.660310\pi$
$97$	−6.79984e22	−0.965214	−0.482607	+	0.875837i	$0.660310\pi$
$98$	0	0
$99$	1.57264e23	1.76533
$100$	0	0
$101$	−9.82224e22	−0.876021	−0.438011	−	0.898970i	$-0.644317\pi$
$101$	−9.82224e22	−0.876021	−0.438011	+	0.898970i	$0.644317\pi$
$102$	0	0
$103$	−6.62575e22	−0.471636	−0.235818	−	0.971797i	$-0.575777\pi$
$103$	−6.62575e22	−0.471636	−0.235818	+	0.971797i	$0.575777\pi$
$104$	0	0
$105$	−8.66663e21	−0.0494508
$106$	0	0
$107$	−1.60597e23	−0.737603	−0.368802	−	0.929508i	$-0.620232\pi$
$107$	−1.60597e23	−0.737603	−0.368802	+	0.929508i	$0.620232\pi$
$108$	0	0
$109$	5.32123e23	1.97518	0.987592	−	0.157041i	$-0.0501954\pi$
$109$	5.32123e23	1.97518	0.987592	+	0.157041i	$0.0501954\pi$
$110$	0	0
$111$	−7.56363e22	−0.227780
$112$	0	0
$113$	−5.44469e23	−1.33528	−0.667638	−	0.744486i	$-0.732695\pi$
$113$	−5.44469e23	−1.33528	−0.667638	+	0.744486i	$0.732695\pi$
$114$	0	0
$115$	−1.86988e23	−0.374791
$116$	0	0
$117$	−2.00180e23	−0.329067
$118$	0	0
$119$	1.84006e23	0.248910
$120$	0	0
$121$	1.51835e24	1.69567
$122$	0	0
$123$	−2.02447e23	−0.187242
$124$	0	0
$125$	−5.20339e23	−0.399780
$126$	0	0
$127$	1.31363e24	0.840876	0.420438	−	0.907321i	$-0.361876\pi$
$127$	1.31363e24	0.840876	0.420438	+	0.907321i	$0.361876\pi$
$128$	0	0
$129$	2.35837e24	1.26133
$130$	0	0
$131$	−7.11943e23	−0.319025	−0.159513	−	0.987196i	$-0.550992\pi$
$131$	−7.11943e23	−0.319025	−0.159513	+	0.987196i	$0.550992\pi$
$132$	0	0
$133$	−4.32800e23	−0.162927
$134$	0	0
$135$	6.97528e22	0.0221168
$136$	0	0
$137$	5.48587e24	1.46879	0.734394	−	0.678723i	$-0.237466\pi$
$137$	5.48587e24	1.46879	0.734394	+	0.678723i	$0.237466\pi$
$138$	0	0
$139$	4.32254e24	0.979645	0.489823	−	0.871822i	$-0.337062\pi$
$139$	4.32254e24	0.979645	0.489823	+	0.871822i	$0.337062\pi$
$140$	0	0
$141$	9.20916e24	1.77092
$142$	0	0
$143$	−3.07248e24	−0.502488
$144$	0	0
$145$	−2.05151e24	−0.285984
$146$	0	0
$147$	−1.17550e25	−1.39982
$148$	0	0
$149$	4.47461e24	0.456155	0.228078	−	0.973643i	$-0.426756\pi$
$149$	4.47461e24	0.456155	0.228078	+	0.973643i	$0.426756\pi$
$150$	0	0
$151$	6.15479e24	0.538243	0.269122	−	0.963106i	$-0.413267\pi$
$151$	6.15479e24	0.538243	0.269122	+	0.963106i	$0.413267\pi$
$152$	0	0
$153$	−2.11728e25	−1.59157
$154$	0	0
$155$	−3.37667e24	−0.218612
$156$	0	0
$157$	−2.23074e25	−1.24624	−0.623122	−	0.782124i	$-0.714136\pi$
$157$	−2.23074e25	−1.24624	−0.623122	+	0.782124i	$0.714136\pi$
$158$	0	0
$159$	−2.47142e25	−1.19366
$160$	0	0
$161$	7.38002e24	0.308717
$162$	0	0
$163$	2.75547e25	1.00009	0.500045	−	0.866000i	$-0.333317\pi$
$163$	2.75547e25	1.00009	0.500045	+	0.866000i	$0.333317\pi$
$164$	0	0
$165$	1.53061e25	0.482836
$166$	0	0
$167$	−1.67099e25	−0.458918	−0.229459	−	0.973318i	$-0.573696\pi$
$167$	−1.67099e25	−0.458918	−0.229459	+	0.973318i	$0.573696\pi$
$168$	0	0
$169$	−3.78430e25	−0.906333
$170$	0	0
$171$	4.98005e25	1.04179
$172$	0	0
$173$	−2.32641e25	−0.425751	−0.212876	−	0.977079i	$-0.568283\pi$
$173$	−2.32641e25	−0.425751	−0.212876	+	0.977079i	$0.568283\pi$
$174$	0	0
$175$	1.00498e25	0.161146
$176$	0	0
$177$	−3.97821e25	−0.559751
$178$	0	0
$179$	−5.22044e25	−0.645501	−0.322751	−	0.946484i	$-0.604608\pi$
$179$	−5.22044e25	−0.645501	−0.322751	+	0.946484i	$0.604608\pi$
$180$	0	0
$181$	−1.48751e26	−1.61866	−0.809331	−	0.587353i	$-0.800170\pi$
$181$	−1.48751e26	−1.61866	−0.809331	+	0.587353i	$0.800170\pi$
$182$	0	0
$183$	−7.75180e25	−0.743387
$184$	0	0
$185$	−3.81414e24	−0.0322791
$186$	0	0
$187$	−3.24972e26	−2.43035
$188$	0	0
$189$	−2.75299e24	−0.0182177
$190$	0	0
$191$	2.99968e26	1.75870	0.879348	−	0.476179i	$-0.157979\pi$
$191$	2.99968e26	1.75870	0.879348	+	0.476179i	$0.157979\pi$
$192$	0	0
$193$	5.17307e25	0.269054	0.134527	−	0.990910i	$-0.457049\pi$
$193$	5.17307e25	0.269054	0.134527	+	0.990910i	$0.457049\pi$
$194$	0	0
$195$	−1.94831e25	−0.0900034
$196$	0	0
$197$	1.75715e26	0.721850	0.360925	−	0.932595i	$-0.382461\pi$
$197$	1.75715e26	0.721850	0.360925	+	0.932595i	$0.382461\pi$
$198$	0	0
$199$	−1.30329e26	−0.476682	−0.238341	−	0.971181i	$-0.576604\pi$
$199$	−1.30329e26	−0.476682	−0.238341	+	0.971181i	$0.576604\pi$
$200$	0	0
$201$	1.52328e26	0.496623
$202$	0	0
$203$	8.09688e25	0.235566
$204$	0	0
$205$	−1.02089e25	−0.0265343
$206$	0	0
$207$	−8.49189e26	−1.97399
$208$	0	0
$209$	7.64366e26	1.59081
$210$	0	0
$211$	−7.01421e25	−0.130837	−0.0654185	−	0.997858i	$-0.520838\pi$
$211$	−7.01421e25	−0.130837	−0.0654185	+	0.997858i	$0.520838\pi$
$212$	0	0
$213$	6.99637e26	1.17087
$214$	0	0
$215$	1.18926e26	0.178746
$216$	0	0
$217$	1.33270e26	0.180071
$218$	0	0
$219$	−9.10362e25	−0.110690
$220$	0	0
$221$	4.13655e26	0.453030
$222$	0	0
$223$	−1.19087e27	−1.17587	−0.587935	−	0.808908i	$-0.700059\pi$
$223$	−1.19087e27	−1.17587	−0.587935	+	0.808908i	$0.700059\pi$
$224$	0	0
$225$	−1.15639e27	−1.03040
$226$	0	0
$227$	−1.79297e27	−1.44303	−0.721515	−	0.692399i	$-0.756554\pi$
$227$	−1.79297e27	−1.44303	−0.721515	+	0.692399i	$0.756554\pi$
$228$	0	0
$229$	−1.64166e27	−1.19447	−0.597236	−	0.802065i	$-0.703735\pi$
$229$	−1.64166e27	−1.19447	−0.597236	+	0.802065i	$0.703735\pi$
$230$	0	0
$231$	−6.04099e26	−0.397714
$232$	0	0
$233$	−4.45039e26	−0.265341	−0.132671	−	0.991160i	$-0.542355\pi$
$233$	−4.45039e26	−0.265341	−0.132671	+	0.991160i	$0.542355\pi$
$234$	0	0
$235$	4.64393e26	0.250961
$236$	0	0
$237$	5.34334e27	2.61943
$238$	0	0
$239$	−2.49375e27	−1.10988	−0.554942	−	0.831889i	$-0.687259\pi$
$239$	−2.49375e27	−1.10988	−0.554942	+	0.831889i	$0.687259\pi$
$240$	0	0
$241$	−1.12410e27	−0.454577	−0.227288	−	0.973827i	$-0.572986\pi$
$241$	−1.12410e27	−0.454577	−0.227288	+	0.973827i	$0.572986\pi$
$242$	0	0
$243$	−3.89529e27	−1.43241
$244$	0	0
$245$	−5.92774e26	−0.198371
$246$	0	0
$247$	−9.72957e26	−0.296537
$248$	0	0
$249$	1.05980e27	0.294396
$250$	0	0
$251$	−6.86006e26	−0.173812	−0.0869060	−	0.996217i	$-0.527698\pi$
$251$	−6.86006e26	−0.173812	−0.0869060	+	0.996217i	$0.527698\pi$
$252$	0	0
$253$	−1.30338e28	−3.01430
$254$	0	0
$255$	−2.06070e27	−0.435313
$256$	0	0
$257$	−5.93562e26	−0.114613	−0.0573067	−	0.998357i	$-0.518251\pi$
$257$	−5.93562e26	−0.114613	−0.0573067	+	0.998357i	$0.518251\pi$
$258$	0	0
$259$	1.50536e26	0.0265884
$260$	0	0
$261$	−9.31674e27	−1.50625
$262$	0	0
$263$	4.32323e26	0.0640203	0.0320101	−	0.999488i	$-0.489809\pi$
$263$	4.32323e26	0.0640203	0.0320101	+	0.999488i	$0.489809\pi$
$264$	0	0
$265$	−1.24627e27	−0.169155
$266$	0	0
$267$	−2.07311e28	−2.58074
$268$	0	0
$269$	1.35113e27	0.154364	0.0771821	−	0.997017i	$-0.475408\pi$
$269$	1.35113e27	0.154364	0.0771821	+	0.997017i	$0.475408\pi$
$270$	0	0
$271$	−1.43116e28	−1.50155	−0.750775	−	0.660557i	$-0.770320\pi$
$271$	−1.43116e28	−1.50155	−0.750775	+	0.660557i	$0.770320\pi$
$272$	0	0
$273$	7.68954e26	0.0741361
$274$	0	0
$275$	−1.77489e28	−1.57342
$276$	0	0
$277$	1.79763e28	1.46616	0.733081	−	0.680142i	$-0.238082\pi$
$277$	1.79763e28	1.46616	0.733081	+	0.680142i	$0.238082\pi$
$278$	0	0
$279$	−1.53348e28	−1.15141
$280$	0	0
$281$	−1.43544e28	−0.992804	−0.496402	−	0.868093i	$-0.665346\pi$
$281$	−1.43544e28	−0.992804	−0.496402	+	0.868093i	$0.665346\pi$
$282$	0	0
$283$	1.84144e28	1.17385	0.586926	−	0.809641i	$-0.300338\pi$
$283$	1.84144e28	1.17385	0.586926	+	0.809641i	$0.300338\pi$
$284$	0	0
$285$	4.84695e27	0.284940
$286$	0	0
$287$	4.02921e26	0.0218564
$288$	0	0
$289$	2.37841e28	1.19114
$290$	0	0
$291$	−3.00555e28	−1.39045
$292$	0	0
$293$	3.29668e28	1.40961	0.704805	−	0.709401i	$-0.251034\pi$
$293$	3.29668e28	1.40961	0.704805	+	0.709401i	$0.251034\pi$
$294$	0	0
$295$	−2.00610e27	−0.0793232
$296$	0	0
$297$	4.86205e27	0.177877
$298$	0	0
$299$	1.65907e28	0.561883
$300$	0	0
$301$	−4.69376e27	−0.147233
$302$	0	0
$303$	−4.34146e28	−1.26196
$304$	0	0
$305$	−3.90903e27	−0.105347
$306$	0	0
$307$	3.97445e27	0.0993540	0.0496770	−	0.998765i	$-0.484181\pi$
$307$	3.97445e27	0.0993540	0.0496770	+	0.998765i	$0.484181\pi$
$308$	0	0
$309$	−2.92860e28	−0.679419
$310$	0	0
$311$	−2.36866e28	−0.510221	−0.255110	−	0.966912i	$-0.582112\pi$
$311$	−2.36866e28	−0.510221	−0.255110	+	0.966912i	$0.582112\pi$
$312$	0	0
$313$	7.03189e28	1.40706	0.703529	−	0.710666i	$-0.251606\pi$
$313$	7.03189e28	1.40706	0.703529	+	0.710666i	$0.251606\pi$
$314$	0	0
$315$	−1.98475e27	−0.0369092
$316$	0	0
$317$	5.79114e28	1.00134	0.500671	−	0.865638i	$-0.333087\pi$
$317$	5.79114e28	1.00134	0.500671	+	0.865638i	$0.333087\pi$
$318$	0	0
$319$	−1.42999e29	−2.30006
$320$	0	0
$321$	−7.09842e28	−1.06256
$322$	0	0
$323$	−1.02908e29	−1.43424
$324$	0	0
$325$	2.25925e28	0.293295
$326$	0	0
$327$	2.35200e29	2.84537
$328$	0	0
$329$	−1.83286e28	−0.206717
$330$	0	0
$331$	6.05435e28	0.636862	0.318431	−	0.947946i	$-0.396844\pi$
$331$	6.05435e28	0.636862	0.318431	+	0.947946i	$0.396844\pi$
$332$	0	0
$333$	−1.73215e28	−0.170011
$334$	0	0
$335$	7.68152e27	0.0703772
$336$	0	0
$337$	5.95933e28	0.509863	0.254932	−	0.966959i	$-0.417947\pi$
$337$	5.95933e28	0.509863	0.254932	+	0.966959i	$0.417947\pi$
$338$	0	0
$339$	−2.40657e29	−1.92354
$340$	0	0
$341$	−2.35367e29	−1.75821
$342$	0	0
$343$	4.74718e28	0.331553
$344$	0	0
$345$	−8.26494e28	−0.539908
$346$	0	0
$347$	1.44892e29	0.885636	0.442818	−	0.896611i	$-0.353979\pi$
$347$	1.44892e29	0.885636	0.442818	+	0.896611i	$0.353979\pi$
$348$	0	0
$349$	−2.30176e29	−1.31694	−0.658471	−	0.752606i	$-0.728797\pi$
$349$	−2.30176e29	−1.31694	−0.658471	+	0.752606i	$0.728797\pi$
$350$	0	0
$351$	−6.18887e27	−0.0331573
$352$	0	0
$353$	−1.43989e29	−0.722639	−0.361320	−	0.932442i	$-0.617674\pi$
$353$	−1.43989e29	−0.722639	−0.361320	+	0.932442i	$0.617674\pi$
$354$	0	0
$355$	3.52808e28	0.165925
$356$	0	0
$357$	8.13311e28	0.358569
$358$	0	0
$359$	2.99138e29	1.23676	0.618380	−	0.785879i	$-0.287789\pi$
$359$	2.99138e29	1.23676	0.618380	+	0.785879i	$0.287789\pi$
$360$	0	0
$361$	−1.57795e28	−0.0612013
$362$	0	0
$363$	6.71115e29	2.44270
$364$	0	0
$365$	−4.59071e27	−0.0156860
$366$	0	0
$367$	−4.29167e29	−1.37710	−0.688552	−	0.725187i	$-0.741753\pi$
$367$	−4.29167e29	−1.37710	−0.688552	+	0.725187i	$0.741753\pi$
$368$	0	0
$369$	−4.63625e28	−0.139754
$370$	0	0
$371$	4.91875e28	0.139333
$372$	0	0
$373$	1.70669e29	0.454467	0.227234	−	0.973840i	$-0.427032\pi$
$373$	1.70669e29	0.454467	0.227234	+	0.973840i	$0.427032\pi$
$374$	0	0
$375$	−2.29991e29	−0.575906
$376$	0	0
$377$	1.82022e29	0.428745
$378$	0	0
$379$	−1.75008e29	−0.387888	−0.193944	−	0.981013i	$-0.562128\pi$
$379$	−1.75008e29	−0.387888	−0.193944	+	0.981013i	$0.562128\pi$
$380$	0	0
$381$	5.80630e29	1.21133
$382$	0	0
$383$	5.85729e29	1.15056	0.575282	−	0.817955i	$-0.304892\pi$
$383$	5.85729e29	1.15056	0.575282	+	0.817955i	$0.304892\pi$
$384$	0	0
$385$	−3.04631e28	−0.0563606
$386$	0	0
$387$	5.40092e29	0.941437
$388$	0	0
$389$	4.71703e29	0.774904	0.387452	−	0.921890i	$-0.373355\pi$
$389$	4.71703e29	0.774904	0.387452	+	0.921890i	$0.373355\pi$
$390$	0	0
$391$	1.75477e30	2.71762
$392$	0	0
$393$	−3.14681e29	−0.459574
$394$	0	0
$395$	2.69450e29	0.371204
$396$	0	0
$397$	−2.44516e28	−0.0317845	−0.0158923	−	0.999874i	$-0.505059\pi$
$397$	−2.44516e28	−0.0317845	−0.0158923	+	0.999874i	$0.505059\pi$
$398$	0	0
$399$	−1.91299e29	−0.234706
$400$	0	0
$401$	8.25721e29	0.956473	0.478237	−	0.878231i	$-0.341276\pi$
$401$	8.25721e29	0.956473	0.478237	+	0.878231i	$0.341276\pi$
$402$	0	0
$403$	2.99597e29	0.327740
$404$	0	0
$405$	−1.81572e29	−0.187636
$406$	0	0
$407$	−2.65861e29	−0.259608
$408$	0	0
$409$	−3.10549e29	−0.286624	−0.143312	−	0.989678i	$-0.545775\pi$
$409$	−3.10549e29	−0.286624	−0.143312	+	0.989678i	$0.545775\pi$
$410$	0	0
$411$	2.42477e30	2.11588
$412$	0	0
$413$	7.91765e28	0.0653388
$414$	0	0
$415$	5.34430e28	0.0417194
$416$	0	0
$417$	1.91057e30	1.41124
$418$	0	0
$419$	−1.91545e30	−1.33909	−0.669545	−	0.742772i	$-0.733511\pi$
$419$	−1.91545e30	−1.33909	−0.669545	+	0.742772i	$0.733511\pi$
$420$	0	0
$421$	7.50234e29	0.496538	0.248269	−	0.968691i	$-0.420138\pi$
$421$	7.50234e29	0.496538	0.248269	+	0.968691i	$0.420138\pi$
$422$	0	0
$423$	2.10899e30	1.32179
$424$	0	0
$425$	2.38957e30	1.41856
$426$	0	0
$427$	1.54281e29	0.0867743
$428$	0	0
$429$	−1.35805e30	−0.723862
$430$	0	0
$431$	−7.63814e29	−0.385921	−0.192961	−	0.981206i	$-0.561809\pi$
$431$	−7.63814e29	−0.385921	−0.192961	+	0.981206i	$0.561809\pi$
$432$	0	0
$433$	−2.07528e30	−0.994183	−0.497092	−	0.867698i	$-0.665599\pi$
$433$	−2.07528e30	−0.994183	−0.497092	+	0.867698i	$0.665599\pi$
$434$	0	0
$435$	−9.06775e29	−0.411977
$436$	0	0
$437$	−4.12740e30	−1.77885
$438$	0	0
$439$	2.11766e30	0.865992	0.432996	−	0.901396i	$-0.357456\pi$
$439$	2.11766e30	0.865992	0.432996	+	0.901396i	$0.357456\pi$
$440$	0	0
$441$	−2.69202e30	−1.04480
$442$	0	0
$443$	2.32990e30	0.858408	0.429204	−	0.903208i	$-0.358794\pi$
$443$	2.32990e30	0.858408	0.429204	+	0.903208i	$0.358794\pi$
$444$	0	0
$445$	−1.04542e30	−0.365720
$446$	0	0
$447$	1.97779e30	0.657118
$448$	0	0
$449$	−2.72238e30	−0.859242	−0.429621	−	0.903009i	$-0.641353\pi$
$449$	−2.72238e30	−0.859242	−0.429621	+	0.903009i	$0.641353\pi$
$450$	0	0
$451$	−7.11598e29	−0.213405
$452$	0	0
$453$	2.72044e30	0.775370
$454$	0	0
$455$	3.87763e28	0.0105059
$456$	0	0
$457$	−7.03884e30	−1.81328	−0.906640	−	0.421904i	$-0.861362\pi$
$457$	−7.03884e30	−1.81328	−0.906640	+	0.421904i	$0.861362\pi$
$458$	0	0
$459$	−6.54588e29	−0.160370
$460$	0	0
$461$	3.75877e30	0.875963	0.437981	−	0.898984i	$-0.355694\pi$
$461$	3.75877e30	0.875963	0.437981	+	0.898984i	$0.355694\pi$
$462$	0	0
$463$	−3.65759e30	−0.810987	−0.405494	−	0.914098i	$-0.632900\pi$
$463$	−3.65759e30	−0.810987	−0.405494	+	0.914098i	$0.632900\pi$
$464$	0	0
$465$	−1.49250e30	−0.314923
$466$	0	0
$467$	7.55323e30	1.51701	0.758505	−	0.651667i	$-0.225930\pi$
$467$	7.55323e30	1.51701	0.758505	+	0.651667i	$0.225930\pi$
$468$	0	0
$469$	−3.03173e29	−0.0579699
$470$	0	0
$471$	−9.85994e30	−1.79529
$472$	0	0
$473$	8.28964e30	1.43758
$474$	0	0
$475$	−5.62051e30	−0.928537
$476$	0	0
$477$	−5.65981e30	−0.890923
$478$	0	0
$479$	3.85059e30	0.577656	0.288828	−	0.957381i	$-0.406734\pi$
$479$	3.85059e30	0.577656	0.288828	+	0.957381i	$0.406734\pi$
$480$	0	0
$481$	3.38412e29	0.0483925
$482$	0	0
$483$	3.26199e30	0.444724
$484$	0	0
$485$	−1.51562e30	−0.197042
$486$	0	0
$487$	−1.09137e31	−1.35328	−0.676642	−	0.736312i	$-0.736566\pi$
$487$	−1.09137e31	−1.35328	−0.676642	+	0.736312i	$0.736566\pi$
$488$	0	0
$489$	1.21793e31	1.44069
$490$	0	0
$491$	−1.57634e31	−1.77915	−0.889576	−	0.456788i	$-0.849000\pi$
$491$	−1.57634e31	−1.77915	−0.889576	+	0.456788i	$0.849000\pi$
$492$	0	0
$493$	1.92522e31	2.07368
$494$	0	0
$495$	3.50526e30	0.360380
$496$	0	0
$497$	−1.39246e30	−0.136673
$498$	0	0
$499$	−5.17947e30	−0.485432	−0.242716	−	0.970097i	$-0.578038\pi$
$499$	−5.17947e30	−0.485432	−0.242716	+	0.970097i	$0.578038\pi$
$500$	0	0
$501$	−7.38584e30	−0.661099
$502$	0	0
$503$	−6.55616e30	−0.560554	−0.280277	−	0.959919i	$-0.590426\pi$
$503$	−6.55616e30	−0.560554	−0.280277	+	0.959919i	$0.590426\pi$
$504$	0	0
$505$	−2.18928e30	−0.178834
$506$	0	0
$507$	−1.67267e31	−1.30563
$508$	0	0
$509$	−1.69709e31	−1.26605	−0.633024	−	0.774132i	$-0.718187\pi$
$509$	−1.69709e31	−1.26605	−0.633024	+	0.774132i	$0.718187\pi$
$510$	0	0
$511$	1.81185e29	0.0129206
$512$	0	0
$513$	1.53965e30	0.104972
$514$	0	0
$515$	−1.47681e30	−0.0962815
$516$	0	0
$517$	3.23701e31	2.01838
$518$	0	0
$519$	−1.02828e31	−0.613319
$520$	0	0
$521$	−7.10737e30	−0.405579	−0.202789	−	0.979222i	$-0.565001\pi$
$521$	−7.10737e30	−0.405579	−0.202789	+	0.979222i	$0.565001\pi$
$522$	0	0
$523$	−5.52438e28	−0.00301657	−0.00150829	−	0.999999i	$-0.500480\pi$
$523$	−5.52438e28	−0.00301657	−0.00150829	+	0.999999i	$0.500480\pi$
$524$	0	0
$525$	4.44204e30	0.232140
$526$	0	0
$527$	3.16880e31	1.58516
$528$	0	0
$529$	4.94992e31	2.37060
$530$	0	0
$531$	−9.11052e30	−0.417788
$532$	0	0
$533$	9.05789e29	0.0397799
$534$	0	0
$535$	−3.57954e30	−0.150577
$536$	0	0
$537$	−2.30745e31	−0.929882
$538$	0	0
$539$	−4.13187e31	−1.59542
$540$	0	0
$541$	−3.45602e31	−1.27882	−0.639408	−	0.768868i	$-0.720821\pi$
$541$	−3.45602e31	−1.27882	−0.639408	+	0.768868i	$0.720821\pi$
$542$	0	0
$543$	−6.57484e31	−2.33178
$544$	0	0
$545$	1.18605e31	0.403221
$546$	0	0
$547$	3.03938e31	0.990673	0.495337	−	0.868701i	$-0.335045\pi$
$547$	3.03938e31	0.990673	0.495337	+	0.868701i	$0.335045\pi$
$548$	0	0
$549$	−1.77525e31	−0.554851
$550$	0	0
$551$	−4.52831e31	−1.35735
$552$	0	0
$553$	−1.06346e31	−0.305762
$554$	0	0
$555$	−1.68586e30	−0.0464999
$556$	0	0
$557$	−3.79791e31	−1.00510	−0.502550	−	0.864548i	$-0.667605\pi$
$557$	−3.79791e31	−1.00510	−0.502550	+	0.864548i	$0.667605\pi$
$558$	0	0
$559$	−1.05518e31	−0.267973
$560$	0	0
$561$	−1.43639e32	−3.50105
$562$	0	0
$563$	−4.53676e29	−0.0106145	−0.00530724	−	0.999986i	$-0.501689\pi$
$563$	−4.53676e29	−0.0106145	−0.00530724	+	0.999986i	$0.501689\pi$
$564$	0	0
$565$	−1.21357e31	−0.272588
$566$	0	0
$567$	7.16626e30	0.154556
$568$	0	0
$569$	1.98059e31	0.410207	0.205103	−	0.978740i	$-0.434247\pi$
$569$	1.98059e31	0.410207	0.205103	+	0.978740i	$0.434247\pi$
$570$	0	0
$571$	4.96995e31	0.988637	0.494319	−	0.869281i	$-0.335418\pi$
$571$	4.96995e31	0.988637	0.494319	+	0.869281i	$0.335418\pi$
$572$	0	0
$573$	1.32587e32	2.53350
$574$	0	0
$575$	9.58399e31	1.75941
$576$	0	0
$577$	−2.06724e31	−0.364643	−0.182322	−	0.983239i	$-0.558361\pi$
$577$	−2.06724e31	−0.364643	−0.182322	+	0.983239i	$0.558361\pi$
$578$	0	0
$579$	2.28651e31	0.387588
$580$	0	0
$581$	−2.10928e30	−0.0343644
$582$	0	0
$583$	−8.68700e31	−1.36045
$584$	0	0
$585$	−4.46183e30	−0.0671770
$586$	0	0
$587$	−8.24154e31	−1.19308	−0.596541	−	0.802583i	$-0.703459\pi$
$587$	−8.24154e31	−1.19308	−0.596541	+	0.802583i	$0.703459\pi$
$588$	0	0
$589$	−7.45332e31	−1.03759
$590$	0	0
$591$	7.76665e31	1.03987
$592$	0	0
$593$	1.19972e32	1.54508	0.772541	−	0.634965i	$-0.218985\pi$
$593$	1.19972e32	1.54508	0.772541	+	0.634965i	$0.218985\pi$
$594$	0	0
$595$	4.10131e30	0.0508133
$596$	0	0
$597$	−5.76056e31	−0.686689
$598$	0	0
$599$	1.32202e32	1.51646	0.758228	−	0.651989i	$-0.226065\pi$
$599$	1.32202e32	1.51646	0.758228	+	0.651989i	$0.226065\pi$
$600$	0	0
$601$	−1.57431e31	−0.173794	−0.0868968	−	0.996217i	$-0.527695\pi$
$601$	−1.57431e31	−0.173794	−0.0868968	+	0.996217i	$0.527695\pi$
$602$	0	0
$603$	3.48848e31	0.370670
$604$	0	0
$605$	3.38425e31	0.346159
$606$	0	0
$607$	−5.51758e31	−0.543349	−0.271675	−	0.962389i	$-0.587577\pi$
$607$	−5.51758e31	−0.543349	−0.271675	+	0.962389i	$0.587577\pi$
$608$	0	0
$609$	3.57884e31	0.339347
$610$	0	0
$611$	−4.12036e31	−0.376237
$612$	0	0
$613$	4.69373e31	0.412784	0.206392	−	0.978469i	$-0.433828\pi$
$613$	4.69373e31	0.412784	0.206392	+	0.978469i	$0.433828\pi$
$614$	0	0
$615$	−4.51234e30	−0.0382242
$616$	0	0
$617$	1.52865e32	1.24747	0.623734	−	0.781637i	$-0.285615\pi$
$617$	1.52865e32	1.24747	0.623734	+	0.781637i	$0.285615\pi$
$618$	0	0
$619$	6.61600e31	0.520179	0.260089	−	0.965585i	$-0.416248\pi$
$619$	6.61600e31	0.520179	0.260089	+	0.965585i	$0.416248\pi$
$620$	0	0
$621$	−2.62539e31	−0.198903
$622$	0	0
$623$	4.12603e31	0.301245
$624$	0	0
$625$	1.24588e32	0.876713
$626$	0	0
$627$	3.37852e32	2.29166
$628$	0	0
$629$	3.57934e31	0.234056
$630$	0	0
$631$	1.02517e32	0.646338	0.323169	−	0.946341i	$-0.395252\pi$
$631$	1.02517e32	0.646338	0.323169	+	0.946341i	$0.395252\pi$
$632$	0	0
$633$	−3.10030e31	−0.188478
$634$	0	0
$635$	2.92796e31	0.171660
$636$	0	0
$637$	5.25943e31	0.297396
$638$	0	0
$639$	1.60224e32	0.873914
$640$	0	0
$641$	−2.92828e29	−0.00154080	−0.000770398	−	1.00000i	$-0.500245\pi$
$641$	−2.92828e29	−0.00154080	−0.000770398		1.00000i	$0.500245\pi$
$642$	0	0
$643$	−1.16226e32	−0.590035	−0.295017	−	0.955492i	$-0.595325\pi$
$643$	−1.16226e32	−0.590035	−0.295017	+	0.955492i	$0.595325\pi$
$644$	0	0
$645$	5.25658e31	0.257493
$646$	0	0
$647$	2.44530e32	1.15593	0.577967	−	0.816060i	$-0.303846\pi$
$647$	2.44530e32	1.15593	0.577967	+	0.816060i	$0.303846\pi$
$648$	0	0
$649$	−1.39833e32	−0.637965
$650$	0	0
$651$	5.89056e31	0.259403
$652$	0	0
$653$	1.62834e32	0.692221	0.346111	−	0.938194i	$-0.387502\pi$
$653$	1.62834e32	0.692221	0.346111	+	0.938194i	$0.387502\pi$
$654$	0	0
$655$	−1.58685e31	−0.0651270
$656$	0	0
$657$	−2.08483e31	−0.0826168
$658$	0	0
$659$	−2.38588e32	−0.912991	−0.456496	−	0.889726i	$-0.650896\pi$
$659$	−2.38588e32	−0.912991	−0.456496	+	0.889726i	$0.650896\pi$
$660$	0	0
$661$	4.10205e32	1.51595	0.757974	−	0.652284i	$-0.226189\pi$
$661$	4.10205e32	1.51595	0.757974	+	0.652284i	$0.226189\pi$
$662$	0	0
$663$	1.82837e32	0.652616
$664$	0	0
$665$	−9.64668e30	−0.0332605
$666$	0	0
$667$	7.72159e32	2.57193
$668$	0	0
$669$	−5.26369e32	−1.69391
$670$	0	0
$671$	−2.72475e32	−0.847262
$672$	0	0
$673$	−2.95459e32	−0.887817	−0.443908	−	0.896072i	$-0.646408\pi$
$673$	−2.95459e32	−0.887817	−0.443908	+	0.896072i	$0.646408\pi$
$674$	0	0
$675$	−3.57514e31	−0.103825
$676$	0	0
$677$	−5.39459e32	−1.51422	−0.757110	−	0.653288i	$-0.773389\pi$
$677$	−5.39459e32	−1.51422	−0.757110	+	0.653288i	$0.773389\pi$
$678$	0	0
$679$	5.98181e31	0.162304
$680$	0	0
$681$	−7.92497e32	−2.07877
$682$	0	0
$683$	4.75470e32	1.20583	0.602913	−	0.797807i	$-0.294006\pi$
$683$	4.75470e32	1.20583	0.602913	+	0.797807i	$0.294006\pi$
$684$	0	0
$685$	1.22275e32	0.299844
$686$	0	0
$687$	−7.25620e32	−1.72071
$688$	0	0
$689$	1.10576e32	0.253595
$690$	0	0
$691$	−4.89241e32	−1.08524	−0.542619	−	0.839979i	$-0.682567\pi$
$691$	−4.89241e32	−1.08524	−0.542619	+	0.839979i	$0.682567\pi$
$692$	0	0
$693$	−1.38345e32	−0.296846
$694$	0	0
$695$	9.63452e31	0.199988
$696$	0	0
$697$	9.58040e31	0.192401
$698$	0	0
$699$	−1.96709e32	−0.382240
$700$	0	0
$701$	−2.18501e32	−0.410863	−0.205431	−	0.978672i	$-0.565860\pi$
$701$	−2.18501e32	−0.410863	−0.205431	+	0.978672i	$0.565860\pi$
$702$	0	0
$703$	−8.41895e31	−0.153205
$704$	0	0
$705$	2.05263e32	0.361523
$706$	0	0
$707$	8.64062e31	0.147306
$708$	0	0
$709$	1.75315e32	0.289325	0.144663	−	0.989481i	$-0.453790\pi$
$709$	1.75315e32	0.289325	0.144663	+	0.989481i	$0.453790\pi$
$710$	0	0
$711$	1.22368e33	1.95510
$712$	0	0
$713$	1.27093e33	1.96603
$714$	0	0
$715$	−6.84827e31	−0.102580
$716$	0	0
$717$	−1.10225e33	−1.59885
$718$	0	0
$719$	−1.11019e32	−0.155960	−0.0779801	−	0.996955i	$-0.524847\pi$
$719$	−1.11019e32	−0.155960	−0.0779801	+	0.996955i	$0.524847\pi$
$720$	0	0
$721$	5.82867e31	0.0793074
$722$	0	0
$723$	−4.96854e32	−0.654844
$724$	0	0
$725$	1.05149e33	1.34252
$726$	0	0
$727$	5.18094e32	0.640860	0.320430	−	0.947272i	$-0.396173\pi$
$727$	5.18094e32	0.640860	0.320430	+	0.947272i	$0.396173\pi$
$728$	0	0
$729$	−9.54814e32	−1.14433
$730$	0	0
$731$	−1.11605e33	−1.29609
$732$	0	0
$733$	1.16138e33	1.30701	0.653506	−	0.756921i	$-0.273297\pi$
$733$	1.16138e33	1.30701	0.653506	+	0.756921i	$0.273297\pi$
$734$	0	0
$735$	−2.62008e32	−0.285765
$736$	0	0
$737$	5.35433e32	0.566016
$738$	0	0
$739$	−8.13707e32	−0.833791	−0.416896	−	0.908954i	$-0.636882\pi$
$739$	−8.13707e32	−0.833791	−0.416896	+	0.908954i	$0.636882\pi$
$740$	0	0
$741$	−4.30050e32	−0.427178
$742$	0	0
$743$	−7.46898e32	−0.719267	−0.359633	−	0.933094i	$-0.617098\pi$
$743$	−7.46898e32	−0.719267	−0.359633	+	0.933094i	$0.617098\pi$
$744$	0	0
$745$	9.97346e31	0.0931212
$746$	0	0
$747$	2.42706e32	0.219732
$748$	0	0
$749$	1.41277e32	0.124031
$750$	0	0
$751$	1.48612e32	0.126530	0.0632651	−	0.997997i	$-0.479849\pi$
$751$	1.48612e32	0.126530	0.0632651	+	0.997997i	$0.479849\pi$
$752$	0	0
$753$	−3.03217e32	−0.250386
$754$	0	0
$755$	1.37184e32	0.109879
$756$	0	0
$757$	8.30005e32	0.644879	0.322439	−	0.946590i	$-0.395497\pi$
$757$	8.30005e32	0.644879	0.322439	+	0.946590i	$0.395497\pi$
$758$	0	0
$759$	−5.76099e33	−4.34227
$760$	0	0
$761$	3.53203e32	0.258286	0.129143	−	0.991626i	$-0.458777\pi$
$761$	3.53203e32	0.258286	0.129143	+	0.991626i	$0.458777\pi$
$762$	0	0
$763$	−4.68109e32	−0.332135
$764$	0	0
$765$	−4.71921e32	−0.324910
$766$	0	0
$767$	1.77993e32	0.118920
$768$	0	0
$769$	2.22373e33	1.44188	0.720940	−	0.692997i	$-0.243710\pi$
$769$	2.22373e33	1.44188	0.720940	+	0.692997i	$0.243710\pi$
$770$	0	0
$771$	−2.62356e32	−0.165107
$772$	0	0
$773$	−3.83424e32	−0.234215	−0.117108	−	0.993119i	$-0.537362\pi$
$773$	−3.83424e32	−0.234215	−0.117108	+	0.993119i	$0.537362\pi$
$774$	0	0
$775$	1.73069e33	1.02624
$776$	0	0
$777$	6.65372e31	0.0383021
$778$	0	0
$779$	−2.25340e32	−0.125938
$780$	0	0
$781$	2.45922e33	1.33447
$782$	0	0
$783$	−2.88041e32	−0.151773
$784$	0	0
$785$	−4.97211e32	−0.254413
$786$	0	0
$787$	1.79827e32	0.0893606	0.0446803	−	0.999001i	$-0.485773\pi$
$787$	1.79827e32	0.0893606	0.0446803	+	0.999001i	$0.485773\pi$
$788$	0	0
$789$	1.91088e32	0.0922249
$790$	0	0
$791$	4.78969e32	0.224532
$792$	0	0
$793$	3.46831e32	0.157934
$794$	0	0
$795$	−5.50855e32	−0.243677
$796$	0	0
$797$	−6.12043e32	−0.263034	−0.131517	−	0.991314i	$-0.541985\pi$
$797$	−6.12043e32	−0.263034	−0.131517	+	0.991314i	$0.541985\pi$
$798$	0	0
$799$	−4.35805e33	−1.81972
$800$	0	0
$801$	−4.74765e33	−1.92622
$802$	0	0
$803$	−3.19991e32	−0.126156
$804$	0	0
$805$	1.64494e32	0.0630226
$806$	0	0
$807$	5.97205e32	0.222371
$808$	0	0
$809$	−3.90564e33	−1.41346	−0.706731	−	0.707482i	$-0.749831\pi$
$809$	−3.90564e33	−1.41346	−0.706731	+	0.707482i	$0.749831\pi$
$810$	0	0
$811$	−2.59224e33	−0.911874	−0.455937	−	0.890012i	$-0.650696\pi$
$811$	−2.59224e33	−0.911874	−0.455937	+	0.890012i	$0.650696\pi$
$812$	0	0
$813$	−6.32575e33	−2.16307
$814$	0	0
$815$	6.14168e32	0.204162
$816$	0	0
$817$	2.62506e33	0.848371
$818$	0	0
$819$	1.76099e32	0.0553339
$820$	0	0
$821$	4.02689e33	1.23034	0.615168	−	0.788396i	$-0.289088\pi$
$821$	4.02689e33	1.23034	0.615168	+	0.788396i	$0.289088\pi$
$822$	0	0
$823$	1.71925e33	0.510788	0.255394	−	0.966837i	$-0.417795\pi$
$823$	1.71925e33	0.510788	0.255394	+	0.966837i	$0.417795\pi$
$824$	0	0
$825$	−7.84507e33	−2.26661
$826$	0	0
$827$	1.32325e33	0.371815	0.185908	−	0.982567i	$-0.440478\pi$
$827$	1.32325e33	0.371815	0.185908	+	0.982567i	$0.440478\pi$
$828$	0	0
$829$	−5.11670e33	−1.39834	−0.699171	−	0.714955i	$-0.746447\pi$
$829$	−5.11670e33	−1.39834	−0.699171	+	0.714955i	$0.746447\pi$
$830$	0	0
$831$	7.94556e33	2.11209
$832$	0	0
$833$	5.56283e33	1.43839
$834$	0	0
$835$	−3.72448e32	−0.0936853
$836$	0	0
$837$	−4.74097e32	−0.116018
$838$	0	0
$839$	−6.47105e33	−1.54068	−0.770340	−	0.637633i	$-0.779913\pi$
$839$	−6.47105e33	−1.54068	−0.770340	+	0.637633i	$0.779913\pi$
$840$	0	0
$841$	4.15491e33	0.962514
$842$	0	0
$843$	−6.34470e33	−1.43019
$844$	0	0
$845$	−8.43483e32	−0.185022
$846$	0	0
$847$	−1.33569e33	−0.285133
$848$	0	0
$849$	8.13921e33	1.69100
$850$	0	0
$851$	1.43558e33	0.290295
$852$	0	0
$853$	−8.71987e33	−1.71631	−0.858156	−	0.513389i	$-0.828390\pi$
$853$	−8.71987e33	−1.71631	−0.858156	+	0.513389i	$0.828390\pi$
$854$	0	0
$855$	1.11000e33	0.212674
$856$	0	0
$857$	1.98587e33	0.370400	0.185200	−	0.982701i	$-0.440707\pi$
$857$	1.98587e33	0.370400	0.185200	+	0.982701i	$0.440707\pi$
$858$	0	0
$859$	4.86533e33	0.883469	0.441734	−	0.897146i	$-0.354363\pi$
$859$	4.86533e33	0.883469	0.441734	+	0.897146i	$0.354363\pi$
$860$	0	0
$861$	1.78092e32	0.0314854
$862$	0	0
$863$	4.12273e33	0.709677	0.354838	−	0.934928i	$-0.384536\pi$
$863$	4.12273e33	0.709677	0.354838	+	0.934928i	$0.384536\pi$
$864$	0	0
$865$	−5.18534e32	−0.0869145
$866$	0	0
$867$	1.05127e34	1.71590
$868$	0	0
$869$	1.87818e34	2.98545
$870$	0	0
$871$	−6.81549e32	−0.105509
$872$	0	0
$873$	−6.88303e33	−1.03780
$874$	0	0
$875$	4.57741e32	0.0672245
$876$	0	0
$877$	7.81242e32	0.111761	0.0558805	−	0.998437i	$-0.482203\pi$
$877$	7.81242e32	0.111761	0.0558805	+	0.998437i	$0.482203\pi$
$878$	0	0
$879$	1.45714e34	2.03063
$880$	0	0
$881$	−4.21119e33	−0.571718	−0.285859	−	0.958272i	$-0.592279\pi$
$881$	−4.21119e33	−0.571718	−0.285859	+	0.958272i	$0.592279\pi$
$882$	0	0
$883$	8.52971e33	1.12820	0.564100	−	0.825707i	$-0.309223\pi$
$883$	8.52971e33	1.12820	0.564100	+	0.825707i	$0.309223\pi$
$884$	0	0
$885$	−8.86704e32	−0.114270
$886$	0	0
$887$	−5.03404e33	−0.632113	−0.316056	−	0.948740i	$-0.602359\pi$
$887$	−5.03404e33	−0.632113	−0.316056	+	0.948740i	$0.602359\pi$
$888$	0	0
$889$	−1.15560e33	−0.141396
$890$	0	0
$891$	−1.26563e34	−1.50908
$892$	0	0
$893$	1.02506e34	1.19112
$894$	0	0
$895$	−1.16358e33	−0.131775
$896$	0	0
$897$	7.33313e33	0.809425
$898$	0	0
$899$	1.39438e34	1.50018
$900$	0	0
$901$	1.16955e34	1.22654
$902$	0	0
$903$	−2.07466e33	−0.212098
$904$	0	0
$905$	−3.31551e33	−0.330440
$906$	0	0
$907$	8.42383e33	0.818513	0.409256	−	0.912419i	$-0.365788\pi$
$907$	8.42383e33	0.818513	0.409256	+	0.912419i	$0.365788\pi$
$908$	0	0
$909$	−9.94240e33	−0.941904
$910$	0	0
$911$	6.82109e33	0.630075	0.315037	−	0.949079i	$-0.397983\pi$
$911$	6.82109e33	0.630075	0.315037	+	0.949079i	$0.397983\pi$
$912$	0	0
$913$	3.72519e33	0.335533
$914$	0	0
$915$	−1.72780e33	−0.151758
$916$	0	0
$917$	6.26295e32	0.0536453
$918$	0	0
$919$	1.26528e34	1.05696	0.528480	−	0.848946i	$-0.322762\pi$
$919$	1.26528e34	1.05696	0.528480	+	0.848946i	$0.322762\pi$
$920$	0	0
$921$	1.75672e33	0.143125
$922$	0	0
$923$	−3.13032e33	−0.248754
$924$	0	0
$925$	1.95492e33	0.151530
$926$	0	0
$927$	−6.70681e33	−0.507106
$928$	0	0
$929$	1.01913e33	0.0751707	0.0375854	−	0.999293i	$-0.488033\pi$
$929$	1.01913e33	0.0751707	0.0375854	+	0.999293i	$0.488033\pi$
$930$	0	0
$931$	−1.30843e34	−0.941519
$932$	0	0
$933$	−1.04695e34	−0.735002
$934$	0	0
$935$	−7.24332e33	−0.496140
$936$	0	0
$937$	1.62385e34	1.08528	0.542638	−	0.839966i	$-0.317425\pi$
$937$	1.62385e34	1.08528	0.542638	+	0.839966i	$0.317425\pi$
$938$	0	0
$939$	3.10811e34	2.02695
$940$	0	0
$941$	8.29419e32	0.0527829	0.0263914	−	0.999652i	$-0.491598\pi$
$941$	8.29419e32	0.0527829	0.0263914	+	0.999652i	$0.491598\pi$
$942$	0	0
$943$	3.84246e33	0.238630
$944$	0	0
$945$	−6.13615e31	−0.00371903
$946$	0	0
$947$	2.25597e34	1.33447	0.667235	−	0.744847i	$-0.267477\pi$
$947$	2.25597e34	1.33447	0.667235	+	0.744847i	$0.267477\pi$
$948$	0	0
$949$	4.07315e32	0.0235163
$950$	0	0
$951$	2.55970e34	1.44249
$952$	0	0
$953$	−3.48251e34	−1.91569	−0.957843	−	0.287293i	$-0.907245\pi$
$953$	−3.48251e34	−1.91569	−0.957843	+	0.287293i	$0.907245\pi$
$954$	0	0
$955$	6.68599e33	0.359027
$956$	0	0
$957$	−6.32059e34	−3.31337
$958$	0	0
$959$	−4.82592e33	−0.246982
$960$	0	0
$961$	2.93728e33	0.146766
$962$	0	0
$963$	−1.62561e34	−0.793076
$964$	0	0
$965$	1.15303e33	0.0549257
$966$	0	0
$967$	9.73913e33	0.453018	0.226509	−	0.974009i	$-0.427269\pi$
$967$	9.73913e33	0.453018	0.226509	+	0.974009i	$0.427269\pi$
$968$	0	0
$969$	−4.54857e34	−2.06610
$970$	0	0
$971$	−1.53491e34	−0.680867	−0.340433	−	0.940269i	$-0.610574\pi$
$971$	−1.53491e34	−0.680867	−0.340433	+	0.940269i	$0.610574\pi$
$972$	0	0
$973$	−3.80253e33	−0.164731
$974$	0	0
$975$	9.98594e33	0.422509
$976$	0	0
$977$	2.84722e34	1.17661	0.588307	−	0.808638i	$-0.299795\pi$
$977$	2.84722e34	1.17661	0.588307	+	0.808638i	$0.299795\pi$
$978$	0	0
$979$	−7.28697e34	−2.94134
$980$	0	0
$981$	5.38633e34	2.12373
$982$	0	0
$983$	2.94912e33	0.113587	0.0567933	−	0.998386i	$-0.481912\pi$
$983$	2.94912e33	0.113587	0.0567933	+	0.998386i	$0.481912\pi$
$984$	0	0
$985$	3.91651e33	0.147361
$986$	0	0
$987$	−8.10129e33	−0.297788
$988$	0	0
$989$	−4.47621e34	−1.60751
$990$	0	0
$991$	−3.48270e34	−1.22199	−0.610996	−	0.791634i	$-0.709231\pi$
$991$	−3.48270e34	−1.22199	−0.610996	+	0.791634i	$0.709231\pi$
$992$	0	0
$993$	2.67604e34	0.917437
$994$	0	0
$995$	−2.90490e33	−0.0973117
$996$	0	0
$997$	3.94510e34	1.29141	0.645704	−	0.763588i	$-0.276564\pi$
$997$	3.94510e34	1.29141	0.645704	+	0.763588i	$0.276564\pi$
$998$	0	0
$999$	−5.35520e32	−0.0171306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.24.a.j.1.3		3
4.3	odd	2		64.24.a.i.1.1		3
8.3	odd	2		8.24.a.b.1.3	✓	3
8.5	even	2		16.24.a.d.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.24.a.b.1.3	✓	3	8.3	odd	2
16.24.a.d.1.1		3	8.5	even	2
64.24.a.i.1.1		3	4.3	odd	2
64.24.a.j.1.3		3	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:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

\(f(q)\)	\(=\)	\(q+442003. q^{3} +2.22890e7 q^{5} -8.79699e8 q^{7} +1.01223e11 q^{9} +O(q^{10})\)	\(q+442003. q^{3} +2.22890e7 q^{5} -8.79699e8 q^{7} +1.01223e11 q^{9} +1.55363e12 q^{11} -1.97761e12 q^{13} +9.85181e12 q^{15} -2.09169e14 q^{17} +4.91986e14 q^{19} -3.88830e14 q^{21} -8.38926e15 q^{23} -1.14241e16 q^{25} +3.12947e15 q^{27} -9.20414e16 q^{29} -1.51495e17 q^{31} +6.86711e17 q^{33} -1.96076e16 q^{35} -1.71122e17 q^{37} -8.74110e17 q^{39} -4.58022e17 q^{41} +5.33564e18 q^{43} +2.25617e18 q^{45} +2.08351e19 q^{47} -2.65949e19 q^{49} -9.24534e19 q^{51} -5.59140e19 q^{53} +3.46290e19 q^{55} +2.17459e20 q^{57} -9.00041e19 q^{59} -1.75379e20 q^{61} -8.90461e19 q^{63} -4.40790e19 q^{65} +3.44632e20 q^{67} -3.70808e21 q^{69} +1.58288e21 q^{71} -2.05963e20 q^{73} -5.04950e21 q^{75} -1.36673e21 q^{77} +1.20889e22 q^{79} -8.14626e21 q^{81} +2.39773e21 q^{83} -4.66218e21 q^{85} -4.06826e22 q^{87} -4.69027e22 q^{89} +1.73970e21 q^{91} -6.69610e22 q^{93} +1.09659e22 q^{95} -6.79984e22 q^{97} +1.57264e23 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 32708 q^{3} - 31480650 q^{5} - 993025320 q^{7} + 1389317071 q^{9}+O(q^{10}) \) 3 * q + 32708 * q^3 - 31480650 * q^5 - 993025320 * q^7 + 1389317071 * q^9	\( 3 q + 32708 q^{3} - 31480650 q^{5} - 993025320 q^{7} + 1389317071 q^{9} - 23441525844 q^{11} - 2019379246962 q^{13} + 4994553094600 q^{15} - 2160517821354 q^{17} + 312191787410964 q^{19} - 825707464014048 q^{21} - 47\!\cdots\!40 q^{23}+ \cdots + 25\!\cdots\!40 q^{99}+O(q^{100}) \) 3 * q + 32708 * q^3 - 31480650 * q^5 - 993025320 * q^7 + 1389317071 * q^9 - 23441525844 * q^11 - 2019379246962 * q^13 + 4994553094600 * q^15 - 2160517821354 * q^17 + 312191787410964 * q^19 - 825707464014048 * q^21 - 4776103684514040 * q^23 + 19807036356305925 * q^25 + 60172513040762216 * q^27 - 172150256810092098 * q^29 - 424442363135920032 * q^31 + 956775180295468688 * q^33 + 1538966670967748400 * q^35 - 2543857445080432362 * q^37 - 1740226054689893144 * q^39 + 2761653172426159758 * q^41 - 5964473505832923060 * q^43 + 11432054542232660750 * q^45 + 52578892233811303824 * q^47 - 36206020957260388773 * q^49 - 140919484512501823160 * q^51 - 15683594666449490106 * q^53 + 255144484970133759000 * q^55 + 312667957724352284272 * q^57 - 104790862916257022052 * q^59 - 1281299617116320867298 * q^61 + 104914943644727114808 * q^63 + 2914343573270694440700 * q^65 - 805429647134646351612 * q^67 - 4263874380293473266848 * q^69 + 3282309189857685782424 * q^71 + 2509034598106619331630 * q^73 - 10588030323195962675300 * q^75 + 3889098717006269829216 * q^77 + 17558070599920224291312 * q^79 - 10704146103369100114373 * q^81 - 15528616638328360005804 * q^83 + 10404913584518742482700 * q^85 - 20926087057105934811864 * q^87 - 13507192466673039723906 * q^89 + 87522067328056376581104 * q^91 - 13527526723927613073280 * q^93 - 181703400482520412239000 * q^95 - 1279961410088837059866 * q^97 + 257548906525338333755740 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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