LMFDB - Embedded newform 64.24.a.i.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:	$0$
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			$-1328.10$ 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)

$f(q)$	$=$	$q+253079. q^{3} +1.36864e8 q^{5} -4.69307e9 q^{7} -3.00944e10 q^{9} +O(q^{10})$	$q+253079. q^{3} +1.36864e8 q^{5} -4.69307e9 q^{7} -3.00944e10 q^{9} +2.44669e11 q^{11} +9.00910e12 q^{13} +3.46374e13 q^{15} +1.66505e14 q^{17} +6.92964e14 q^{19} -1.18772e15 q^{21} +8.91449e13 q^{23} +6.81085e15 q^{25} -3.14419e16 q^{27} -7.47679e16 q^{29} +1.11443e17 q^{31} +6.19204e16 q^{33} -6.42313e17 q^{35} -1.38869e18 q^{37} +2.28001e18 q^{39} -5.25975e17 q^{41} +1.20778e19 q^{43} -4.11885e18 q^{45} -8.11663e18 q^{47} -5.34384e18 q^{49} +4.21388e19 q^{51} +9.12192e19 q^{53} +3.34864e19 q^{55} +1.75374e20 q^{57} +4.14154e20 q^{59} -6.24417e20 q^{61} +1.41235e20 q^{63} +1.23302e21 q^{65} -3.72566e20 q^{67} +2.25607e19 q^{69} -2.77403e21 q^{71} +1.39128e21 q^{73} +1.72368e21 q^{75} -1.14825e21 q^{77} -4.37393e21 q^{79} -5.12408e21 q^{81} +1.10744e21 q^{83} +2.27885e22 q^{85} -1.89222e22 q^{87} +5.05165e22 q^{89} -4.22803e22 q^{91} +2.82037e22 q^{93} +9.48418e22 q^{95} -7.16266e21 q^{97} -7.36316e21 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	253079.	0.824823	0.412412	−	0.910998i	$-0.364687\pi$
$3$	253079.	0.824823	0.412412	+	0.910998i	$0.364687\pi$
$4$	0	0
$5$	1.36864e8	1.25353	0.626765	−	0.779209i	$-0.284379\pi$
$5$	1.36864e8	1.25353	0.626765	+	0.779209i	$0.284379\pi$
$6$	0	0
$7$	−4.69307e9	−0.897077	−0.448538	−	0.893764i	$-0.648055\pi$
$7$	−4.69307e9	−0.897077	−0.448538	+	0.893764i	$0.648055\pi$
$8$	0	0
$9$	−3.00944e10	−0.319667
$10$	0	0
$11$	2.44669e11	0.258561	0.129280	−	0.991608i	$-0.458733\pi$
$11$	2.44669e11	0.258561	0.129280	+	0.991608i	$0.458733\pi$
$12$	0	0
$13$	9.00910e12	1.39422	0.697112	−	0.716962i	$-0.254468\pi$
$13$	9.00910e12	1.39422	0.697112	+	0.716962i	$0.254468\pi$
$14$	0	0
$15$	3.46374e13	1.03394
$16$	0	0
$17$	1.66505e14	1.17832	0.589161	−	0.808016i	$-0.299458\pi$
$17$	1.66505e14	1.17832	0.589161	+	0.808016i	$0.299458\pi$
$18$	0	0
$19$	6.92964e14	1.36472	0.682361	−	0.731016i	$-0.260953\pi$
$19$	6.92964e14	1.36472	0.682361	+	0.731016i	$0.260953\pi$
$20$	0	0
$21$	−1.18772e15	−0.739930
$22$	0	0
$23$	8.91449e13	0.0195086	0.00975431	−	0.999952i	$-0.496895\pi$
$23$	8.91449e13	0.0195086	0.00975431	+	0.999952i	$0.496895\pi$
$24$	0	0
$25$	6.81085e15	0.571336
$26$	0	0
$27$	−3.14419e16	−1.08849
$28$	0	0
$29$	−7.47679e16	−1.13799	−0.568995	−	0.822341i	$-0.692668\pi$
$29$	−7.47679e16	−1.13799	−0.568995	+	0.822341i	$0.692668\pi$
$30$	0	0
$31$	1.11443e17	0.787757	0.393878	−	0.919163i	$-0.371133\pi$
$31$	1.11443e17	0.787757	0.393878	+	0.919163i	$0.371133\pi$
$32$	0	0
$33$	6.19204e16	0.213267
$34$	0	0
$35$	−6.42313e17	−1.12451
$36$	0	0
$37$	−1.38869e18	−1.28318	−0.641588	−	0.767050i	$-0.721724\pi$
$37$	−1.38869e18	−1.28318	−0.641588	+	0.767050i	$0.721724\pi$
$38$	0	0
$39$	2.28001e18	1.14999
$40$	0	0
$41$	−5.25975e17	−0.149262	−0.0746312	−	0.997211i	$-0.523778\pi$
$41$	−5.25975e17	−0.149262	−0.0746312	+	0.997211i	$0.523778\pi$
$42$	0	0
$43$	1.20778e19	1.98198	0.990992	−	0.133924i	$-0.0427579\pi$
$43$	1.20778e19	1.98198	0.990992	+	0.133924i	$0.0427579\pi$
$44$	0	0
$45$	−4.11885e18	−0.400711
$46$	0	0
$47$	−8.11663e18	−0.478906	−0.239453	−	0.970908i	$-0.576968\pi$
$47$	−8.11663e18	−0.478906	−0.239453	+	0.970908i	$0.576968\pi$
$48$	0	0
$49$	−5.34384e18	−0.195253
$50$	0	0
$51$	4.21388e19	0.971907
$52$	0	0
$53$	9.12192e19	1.35180	0.675902	−	0.736991i	$-0.263754\pi$
$53$	9.12192e19	1.35180	0.675902	+	0.736991i	$0.263754\pi$
$54$	0	0
$55$	3.34864e19	0.324114
$56$	0	0
$57$	1.75374e20	1.12565
$58$	0	0
$59$	4.14154e20	1.78798	0.893992	−	0.448083i	$-0.147893\pi$
$59$	4.14154e20	1.78798	0.893992	+	0.448083i	$0.147893\pi$
$60$	0	0
$61$	−6.24417e20	−1.83731	−0.918653	−	0.395065i	$-0.870722\pi$
$61$	−6.24417e20	−1.83731	−0.918653	+	0.395065i	$0.870722\pi$
$62$	0	0
$63$	1.41235e20	0.286765
$64$	0	0
$65$	1.23302e21	1.74770
$66$	0	0
$67$	−3.72566e20	−0.372686	−0.186343	−	0.982485i	$-0.559664\pi$
$67$	−3.72566e20	−0.372686	−0.186343	+	0.982485i	$0.559664\pi$
$68$	0	0
$69$	2.25607e19	0.0160912
$70$	0	0
$71$	−2.77403e21	−1.42443	−0.712213	−	0.701964i	$-0.752307\pi$
$71$	−2.77403e21	−1.42443	−0.712213	+	0.701964i	$0.752307\pi$
$72$	0	0
$73$	1.39128e21	0.519040	0.259520	−	0.965738i	$-0.416436\pi$
$73$	1.39128e21	0.519040	0.259520	+	0.965738i	$0.416436\pi$
$74$	0	0
$75$	1.72368e21	0.471251
$76$	0	0
$77$	−1.14825e21	−0.231949
$78$	0	0
$79$	−4.37393e21	−0.657901	−0.328950	−	0.944347i	$-0.606695\pi$
$79$	−4.37393e21	−0.657901	−0.328950	+	0.944347i	$0.606695\pi$
$80$	0	0
$81$	−5.12408e21	−0.578147
$82$	0	0
$83$	1.10744e21	0.0943892	0.0471946	−	0.998886i	$-0.484972\pi$
$83$	1.10744e21	0.0943892	0.0471946	+	0.998886i	$0.484972\pi$
$84$	0	0
$85$	2.27885e22	1.47706
$86$	0	0
$87$	−1.89222e22	−0.938640
$88$	0	0
$89$	5.05165e22	1.92952	0.964758	−	0.263140i	$-0.0847583\pi$
$89$	5.05165e22	1.92952	0.964758	+	0.263140i	$0.0847583\pi$
$90$	0	0
$91$	−4.22803e22	−1.25073
$92$	0	0
$93$	2.82037e22	0.649760
$94$	0	0
$95$	9.48418e22	1.71072
$96$	0	0
$97$	−7.16266e21	−0.101672	−0.0508358	−	0.998707i	$-0.516189\pi$
$97$	−7.16266e21	−0.101672	−0.0508358	+	0.998707i	$0.516189\pi$
$98$	0	0
$99$	−7.36316e21	−0.0826532
$100$	0	0
$101$	8.17185e22	0.728828	0.364414	−	0.931237i	$-0.381269\pi$
$101$	8.17185e22	0.728828	0.364414	+	0.931237i	$0.381269\pi$
$102$	0	0
$103$	2.52980e23	1.80077	0.900386	−	0.435091i	$-0.143284\pi$
$103$	2.52980e23	1.80077	0.900386	+	0.435091i	$0.143284\pi$
$104$	0	0
$105$	−1.62556e23	−0.927524
$106$	0	0
$107$	1.72254e23	0.791144	0.395572	−	0.918435i	$-0.370546\pi$
$107$	1.72254e23	0.791144	0.395572	+	0.918435i	$0.370546\pi$
$108$	0	0
$109$	8.74196e22	0.324492	0.162246	−	0.986750i	$-0.448126\pi$
$109$	8.74196e22	0.324492	0.162246	+	0.986750i	$0.448126\pi$
$110$	0	0
$111$	−3.51448e23	−1.05839
$112$	0	0
$113$	1.03919e23	0.254855	0.127428	−	0.991848i	$-0.459328\pi$
$113$	1.03919e23	0.254855	0.127428	+	0.991848i	$0.459328\pi$
$114$	0	0
$115$	1.22007e22	0.0244546
$116$	0	0
$117$	−2.71124e23	−0.445687
$118$	0	0
$119$	−7.81418e23	−1.05704
$120$	0	0
$121$	−8.35567e23	−0.933146
$122$	0	0
$123$	−1.33113e23	−0.123115
$124$	0	0
$125$	−6.99386e23	−0.537343
$126$	0	0
$127$	1.22313e24	0.782943	0.391472	−	0.920190i	$-0.371966\pi$
$127$	1.22313e24	0.782943	0.391472	+	0.920190i	$0.371966\pi$
$128$	0	0
$129$	3.05663e24	1.63479
$130$	0	0
$131$	−1.34489e24	−0.602650	−0.301325	−	0.953521i	$-0.597429\pi$
$131$	−1.34489e24	−0.602650	−0.301325	+	0.953521i	$0.597429\pi$
$132$	0	0
$133$	−3.25213e24	−1.22426
$134$	0	0
$135$	−4.30326e24	−1.36446
$136$	0	0
$137$	−3.99835e23	−0.107052	−0.0535259	−	0.998566i	$-0.517046\pi$
$137$	−3.99835e23	−0.107052	−0.0535259	+	0.998566i	$0.517046\pi$
$138$	0	0
$139$	4.05683e24	0.919426	0.459713	−	0.888068i	$-0.347952\pi$
$139$	4.05683e24	0.919426	0.459713	+	0.888068i	$0.347952\pi$
$140$	0	0
$141$	−2.05414e24	−0.395013
$142$	0	0
$143$	2.20424e24	0.360492
$144$	0	0
$145$	−1.02330e25	−1.42650
$146$	0	0
$147$	−1.35241e24	−0.161050
$148$	0	0
$149$	3.91668e24	0.399279	0.199639	−	0.979869i	$-0.436023\pi$
$149$	3.91668e24	0.399279	0.199639	+	0.979869i	$0.436023\pi$
$150$	0	0
$151$	4.67769e24	0.409069	0.204534	−	0.978859i	$-0.434432\pi$
$151$	4.67769e24	0.409069	0.204534	+	0.978859i	$0.434432\pi$
$152$	0	0
$153$	−5.01086e24	−0.376670
$154$	0	0
$155$	1.52525e25	0.987476
$156$	0	0
$157$	1.71378e25	0.957433	0.478716	−	0.877970i	$-0.341102\pi$
$157$	1.71378e25	0.957433	0.478716	+	0.877970i	$0.341102\pi$
$158$	0	0
$159$	2.30856e25	1.11500
$160$	0	0
$161$	−4.18363e23	−0.0175007
$162$	0	0
$163$	−2.46913e25	−0.896163	−0.448081	−	0.893993i	$-0.647893\pi$
$163$	−2.46913e25	−0.896163	−0.448081	+	0.893993i	$0.647893\pi$
$164$	0	0
$165$	8.47468e24	0.267336
$166$	0	0
$167$	3.58388e25	0.984269	0.492135	−	0.870519i	$-0.336217\pi$
$167$	3.58388e25	0.984269	0.492135	+	0.870519i	$0.336217\pi$
$168$	0	0
$169$	3.94099e25	0.943862
$170$	0	0
$171$	−2.08543e25	−0.436256
$172$	0	0
$173$	2.45437e25	0.449170	0.224585	−	0.974455i	$-0.427897\pi$
$173$	2.45437e25	0.449170	0.224585	+	0.974455i	$0.427897\pi$
$174$	0	0
$175$	−3.19638e25	−0.512532
$176$	0	0
$177$	1.04813e26	1.47477
$178$	0	0
$179$	1.76539e25	0.218289	0.109144	−	0.994026i	$-0.465189\pi$
$179$	1.76539e25	0.218289	0.109144	+	0.994026i	$0.465189\pi$
$180$	0	0
$181$	−4.21958e25	−0.459162	−0.229581	−	0.973290i	$-0.573736\pi$
$181$	−4.21958e25	−0.459162	−0.229581	+	0.973290i	$0.573736\pi$
$182$	0	0
$183$	−1.58027e26	−1.51545
$184$	0	0
$185$	−1.90062e26	−1.60850
$186$	0	0
$187$	4.07385e25	0.304668
$188$	0	0
$189$	1.47559e26	0.976461
$190$	0	0
$191$	1.28030e26	0.750631	0.375315	−	0.926897i	$-0.377534\pi$
$191$	1.28030e26	0.750631	0.375315	+	0.926897i	$0.377534\pi$
$192$	0	0
$193$	1.55233e26	0.807373	0.403686	−	0.914897i	$-0.367729\pi$
$193$	1.55233e26	0.807373	0.403686	+	0.914897i	$0.367729\pi$
$194$	0	0
$195$	3.12051e26	1.44154
$196$	0	0
$197$	−3.75660e26	−1.54324	−0.771619	−	0.636084i	$-0.780553\pi$
$197$	−3.75660e26	−1.54324	−0.771619	+	0.636084i	$0.780553\pi$
$198$	0	0
$199$	8.16474e25	0.298629	0.149315	−	0.988790i	$-0.452293\pi$
$199$	8.16474e25	0.298629	0.149315	+	0.988790i	$0.452293\pi$
$200$	0	0
$201$	−9.42884e25	−0.307400
$202$	0	0
$203$	3.50891e26	1.02086
$204$	0	0
$205$	−7.19871e25	−0.187105
$206$	0	0
$207$	−2.68277e24	−0.00623625
$208$	0	0
$209$	1.69547e26	0.352863
$210$	0	0
$211$	−2.94030e26	−0.548458	−0.274229	−	0.961664i	$-0.588423\pi$
$211$	−2.94030e26	−0.548458	−0.274229	+	0.961664i	$0.588423\pi$
$212$	0	0
$213$	−7.02047e26	−1.17490
$214$	0	0
$215$	1.65301e27	2.48447
$216$	0	0
$217$	−5.23008e26	−0.706678
$218$	0	0
$219$	3.52102e26	0.428116
$220$	0	0
$221$	1.50006e27	1.64284
$222$	0	0
$223$	1.11598e27	1.10192	0.550960	−	0.834531i	$-0.314262\pi$
$223$	1.11598e27	1.10192	0.550960	+	0.834531i	$0.314262\pi$
$224$	0	0
$225$	−2.04969e26	−0.182637
$226$	0	0
$227$	1.35769e27	1.09270	0.546352	−	0.837556i	$-0.316016\pi$
$227$	1.35769e27	1.09270	0.546352	+	0.837556i	$0.316016\pi$
$228$	0	0
$229$	8.29881e26	0.603821	0.301910	−	0.953336i	$-0.402376\pi$
$229$	8.29881e26	0.603821	0.301910	+	0.953336i	$0.402376\pi$
$230$	0	0
$231$	−2.90597e26	−0.191317
$232$	0	0
$233$	−2.12415e27	−1.26646	−0.633231	−	0.773963i	$-0.718272\pi$
$233$	−2.12415e27	−1.26646	−0.633231	+	0.773963i	$0.718272\pi$
$234$	0	0
$235$	−1.11087e27	−0.600323
$236$	0	0
$237$	−1.10695e27	−0.542652
$238$	0	0
$239$	−3.15849e27	−1.40574	−0.702868	−	0.711321i	$-0.748097\pi$
$239$	−3.15849e27	−1.40574	−0.702868	+	0.711321i	$0.748097\pi$
$240$	0	0
$241$	1.18603e27	0.479623	0.239811	−	0.970819i	$-0.422914\pi$
$241$	1.18603e27	0.479623	0.239811	+	0.970819i	$0.422914\pi$
$242$	0	0
$243$	1.66324e27	0.611623
$244$	0	0
$245$	−7.31380e26	−0.244756
$246$	0	0
$247$	6.24298e27	1.90273
$248$	0	0
$249$	2.80269e26	0.0778544
$250$	0	0
$251$	4.00342e27	1.01434	0.507170	−	0.861846i	$-0.330692\pi$
$251$	4.00342e27	1.01434	0.507170	+	0.861846i	$0.330692\pi$
$252$	0	0
$253$	2.18110e25	0.00504416
$254$	0	0
$255$	5.76728e27	1.21831
$256$	0	0
$257$	8.08017e27	1.56023	0.780117	−	0.625634i	$-0.215160\pi$
$257$	8.08017e27	1.56023	0.780117	+	0.625634i	$0.215160\pi$
$258$	0	0
$259$	6.51723e27	1.15111
$260$	0	0
$261$	2.25010e27	0.363777
$262$	0	0
$263$	2.71077e27	0.401422	0.200711	−	0.979650i	$-0.435675\pi$
$263$	2.71077e27	0.401422	0.200711	+	0.979650i	$0.435675\pi$
$264$	0	0
$265$	1.24846e28	1.69453
$266$	0	0
$267$	1.27846e28	1.59151
$268$	0	0
$269$	2.12093e27	0.242312	0.121156	−	0.992634i	$-0.461340\pi$
$269$	2.12093e27	0.242312	0.121156	+	0.992634i	$0.461340\pi$
$270$	0	0
$271$	1.12613e28	1.18152	0.590760	−	0.806847i	$-0.298828\pi$
$271$	1.12613e28	1.18152	0.590760	+	0.806847i	$0.298828\pi$
$272$	0	0
$273$	−1.07002e28	−1.03163
$274$	0	0
$275$	1.66640e27	0.147725
$276$	0	0
$277$	−1.30069e28	−1.06085	−0.530426	−	0.847731i	$-0.677968\pi$
$277$	−1.30069e28	−1.06085	−0.530426	+	0.847731i	$0.677968\pi$
$278$	0	0
$279$	−3.35380e27	−0.251819
$280$	0	0
$281$	−1.94937e28	−1.34825	−0.674125	−	0.738617i	$-0.735479\pi$
$281$	−1.94937e28	−1.34825	−0.674125	+	0.738617i	$0.735479\pi$
$282$	0	0
$283$	2.82562e28	1.80123	0.900616	−	0.434615i	$-0.143115\pi$
$283$	2.82562e28	1.80123	0.900616	+	0.434615i	$0.143115\pi$
$284$	0	0
$285$	2.40024e28	1.41104
$286$	0	0
$287$	2.46844e27	0.133900
$288$	0	0
$289$	7.75623e27	0.388441
$290$	0	0
$291$	−1.81272e27	−0.0838611
$292$	0	0
$293$	−1.92215e27	−0.0821881	−0.0410940	−	0.999155i	$-0.513084\pi$
$293$	−1.92215e27	−0.0821881	−0.0410940	+	0.999155i	$0.513084\pi$
$294$	0	0
$295$	5.66828e28	2.24129
$296$	0	0
$297$	−7.69284e27	−0.281441
$298$	0	0
$299$	8.03115e26	0.0271994
$300$	0	0
$301$	−5.66819e28	−1.77799
$302$	0	0
$303$	2.06812e28	0.601154
$304$	0	0
$305$	−8.54603e28	−2.30312
$306$	0	0
$307$	−7.21136e28	−1.80271	−0.901354	−	0.433082i	$-0.857426\pi$
$307$	−7.21136e28	−1.80271	−0.901354	+	0.433082i	$0.857426\pi$
$308$	0	0
$309$	6.40239e28	1.48532
$310$	0	0
$311$	−1.56980e28	−0.338142	−0.169071	−	0.985604i	$-0.554077\pi$
$311$	−1.56980e28	−0.338142	−0.169071	+	0.985604i	$0.554077\pi$
$312$	0	0
$313$	7.71165e28	1.54308	0.771538	−	0.636183i	$-0.219488\pi$
$313$	7.71165e28	1.54308	0.771538	+	0.636183i	$0.219488\pi$
$314$	0	0
$315$	1.93300e28	0.359469
$316$	0	0
$317$	−8.30984e28	−1.43685	−0.718424	−	0.695605i	$-0.755136\pi$
$317$	−8.30984e28	−1.43685	−0.718424	+	0.695605i	$0.755136\pi$
$318$	0	0
$319$	−1.82934e28	−0.294239
$320$	0	0
$321$	4.35937e28	0.652554
$322$	0	0
$323$	1.15382e29	1.60808
$324$	0	0
$325$	6.13597e28	0.796571
$326$	0	0
$327$	2.21240e28	0.267649
$328$	0	0
$329$	3.80919e28	0.429615
$330$	0	0
$331$	9.43113e28	0.992068	0.496034	−	0.868303i	$-0.334789\pi$
$331$	9.43113e28	0.992068	0.496034	+	0.868303i	$0.334789\pi$
$332$	0	0
$333$	4.17919e28	0.410188
$334$	0	0
$335$	−5.09909e28	−0.467173
$336$	0	0
$337$	−7.22053e28	−0.617767	−0.308884	−	0.951100i	$-0.599955\pi$
$337$	−7.22053e28	−0.617767	−0.308884	+	0.951100i	$0.599955\pi$
$338$	0	0
$339$	2.62997e28	0.210211
$340$	0	0
$341$	2.72665e28	0.203683
$342$	0	0
$343$	1.53522e29	1.07223
$344$	0	0
$345$	3.08775e27	0.0201707
$346$	0	0
$347$	1.33840e29	0.818083	0.409042	−	0.912516i	$-0.365863\pi$
$347$	1.33840e29	0.818083	0.409042	+	0.912516i	$0.365863\pi$
$348$	0	0
$349$	−6.75028e28	−0.386215	−0.193107	−	0.981178i	$-0.561857\pi$
$349$	−6.75028e28	−0.386215	−0.193107	+	0.981178i	$0.561857\pi$
$350$	0	0
$351$	−2.83263e29	−1.51760
$352$	0	0
$353$	1.65413e29	0.830156	0.415078	−	0.909786i	$-0.363754\pi$
$353$	1.65413e29	0.830156	0.415078	+	0.909786i	$0.363754\pi$
$354$	0	0
$355$	−3.79665e29	−1.78556
$356$	0	0
$357$	−1.97760e29	−0.871875
$358$	0	0
$359$	6.44513e28	0.266468	0.133234	−	0.991085i	$-0.457464\pi$
$359$	6.44513e28	0.266468	0.133234	+	0.991085i	$0.457464\pi$
$360$	0	0
$361$	2.22369e29	0.862465
$362$	0	0
$363$	−2.11464e29	−0.769681
$364$	0	0
$365$	1.90416e29	0.650631
$366$	0	0
$367$	−1.58165e29	−0.507516	−0.253758	−	0.967268i	$-0.581667\pi$
$367$	−1.58165e29	−0.507516	−0.253758	+	0.967268i	$0.581667\pi$
$368$	0	0
$369$	1.58289e28	0.0477142
$370$	0	0
$371$	−4.28098e29	−1.21267
$372$	0	0
$373$	2.04559e29	0.544712	0.272356	−	0.962197i	$-0.412197\pi$
$373$	2.04559e29	0.544712	0.272356	+	0.962197i	$0.412197\pi$
$374$	0	0
$375$	−1.77000e29	−0.443213
$376$	0	0
$377$	−6.73591e29	−1.58661
$378$	0	0
$379$	−3.01761e29	−0.668823	−0.334412	−	0.942427i	$-0.608538\pi$
$379$	−3.01761e29	−0.668823	−0.334412	+	0.942427i	$0.608538\pi$
$380$	0	0
$381$	3.09548e29	0.645790
$382$	0	0
$383$	−4.68508e29	−0.920304	−0.460152	−	0.887840i	$-0.652205\pi$
$383$	−4.68508e29	−0.920304	−0.460152	+	0.887840i	$0.652205\pi$
$384$	0	0
$385$	−1.57154e29	−0.290755
$386$	0	0
$387$	−3.63474e29	−0.633574
$388$	0	0
$389$	−2.03807e29	−0.334810	−0.167405	−	0.985888i	$-0.553539\pi$
$389$	−2.03807e29	−0.334810	−0.167405	+	0.985888i	$0.553539\pi$
$390$	0	0
$391$	1.48430e28	0.0229874
$392$	0	0
$393$	−3.40362e29	−0.497080
$394$	0	0
$395$	−5.98634e29	−0.824698
$396$	0	0
$397$	4.19036e29	0.544704	0.272352	−	0.962198i	$-0.412198\pi$
$397$	4.19036e29	0.544704	0.272352	+	0.962198i	$0.412198\pi$
$398$	0	0
$399$	−8.23044e29	−1.00980
$400$	0	0
$401$	1.12283e28	0.0130063	0.00650317	−	0.999979i	$-0.497930\pi$
$401$	1.12283e28	0.0130063	0.00650317	+	0.999979i	$0.497930\pi$
$402$	0	0
$403$	1.00400e30	1.09831
$404$	0	0
$405$	−7.01303e29	−0.724724
$406$	0	0
$407$	−3.39769e29	−0.331779
$408$	0	0
$409$	−2.62630e29	−0.242396	−0.121198	−	0.992628i	$-0.538674\pi$
$409$	−2.62630e29	−0.242396	−0.121198	+	0.992628i	$0.538674\pi$
$410$	0	0
$411$	−1.01190e29	−0.0882988
$412$	0	0
$413$	−1.94365e30	−1.60396
$414$	0	0
$415$	1.51569e29	0.118320
$416$	0	0
$417$	1.02670e30	0.758364
$418$	0	0
$419$	−2.56273e30	−1.79160	−0.895801	−	0.444455i	$-0.853397\pi$
$419$	−2.56273e30	−1.79160	−0.895801	+	0.444455i	$0.853397\pi$
$420$	0	0
$421$	−1.02097e30	−0.675721	−0.337860	−	0.941196i	$-0.609703\pi$
$421$	−1.02097e30	−0.675721	−0.337860	+	0.941196i	$0.609703\pi$
$422$	0	0
$423$	2.44265e29	0.153090
$424$	0	0
$425$	1.13404e30	0.673217
$426$	0	0
$427$	2.93043e30	1.64820
$428$	0	0
$429$	5.57847e29	0.297342
$430$	0	0
$431$	−2.56620e30	−1.29659	−0.648293	−	0.761391i	$-0.724517\pi$
$431$	−2.56620e30	−1.29659	−0.648293	+	0.761391i	$0.724517\pi$
$432$	0	0
$433$	7.83936e28	0.0375552	0.0187776	−	0.999824i	$-0.494023\pi$
$433$	7.83936e28	0.0375552	0.0187776	+	0.999824i	$0.494023\pi$
$434$	0	0
$435$	−2.58976e30	−1.17661
$436$	0	0
$437$	6.17742e28	0.0266238
$438$	0	0
$439$	−1.20855e30	−0.494221	−0.247111	−	0.968987i	$-0.579481\pi$
$439$	−1.20855e30	−0.494221	−0.247111	+	0.968987i	$0.579481\pi$
$440$	0	0
$441$	1.60820e29	0.0624160
$442$	0	0
$443$	−1.77452e29	−0.0653790	−0.0326895	−	0.999466i	$-0.510407\pi$
$443$	−1.77452e29	−0.0653790	−0.0326895	+	0.999466i	$0.510407\pi$
$444$	0	0
$445$	6.91390e30	2.41870
$446$	0	0
$447$	9.91229e29	0.329335
$448$	0	0
$449$	1.27997e30	0.403987	0.201994	−	0.979387i	$-0.435258\pi$
$449$	1.27997e30	0.403987	0.201994	+	0.979387i	$0.435258\pi$
$450$	0	0
$451$	−1.28690e29	−0.0385934
$452$	0	0
$453$	1.18382e30	0.337409
$454$	0	0
$455$	−5.78666e30	−1.56782
$456$	0	0
$457$	−3.96412e30	−1.02120	−0.510600	−	0.859819i	$-0.670577\pi$
$457$	−3.96412e30	−1.02120	−0.510600	+	0.859819i	$0.670577\pi$
$458$	0	0
$459$	−5.23522e30	−1.28259
$460$	0	0
$461$	−2.13527e30	−0.497614	−0.248807	−	0.968553i	$-0.580038\pi$
$461$	−2.13527e30	−0.497614	−0.248807	+	0.968553i	$0.580038\pi$
$462$	0	0
$463$	3.79521e30	0.841500	0.420750	−	0.907177i	$-0.361767\pi$
$463$	3.79521e30	0.841500	0.420750	+	0.907177i	$0.361767\pi$
$464$	0	0
$465$	3.86008e30	0.814493
$466$	0	0
$467$	3.54902e30	0.712795	0.356397	−	0.934335i	$-0.384005\pi$
$467$	3.54902e30	0.712795	0.356397	+	0.934335i	$0.384005\pi$
$468$	0	0
$469$	1.74848e30	0.334328
$470$	0	0
$471$	4.33720e30	0.789713
$472$	0	0
$473$	2.95506e30	0.512463
$474$	0	0
$475$	4.71967e30	0.779714
$476$	0	0
$477$	−2.74519e30	−0.432127
$478$	0	0
$479$	1.02769e31	1.54172	0.770858	−	0.637007i	$-0.219828\pi$
$479$	1.02769e31	1.54172	0.770858	+	0.637007i	$0.219828\pi$
$480$	0	0
$481$	−1.25109e31	−1.78903
$482$	0	0
$483$	−1.05879e29	−0.0144350
$484$	0	0
$485$	−9.80311e29	−0.127448
$486$	0	0
$487$	1.44242e31	1.78858	0.894290	−	0.447489i	$-0.147682\pi$
$487$	1.44242e31	1.78858	0.894290	+	0.447489i	$0.147682\pi$
$488$	0	0
$489$	−6.24885e30	−0.739176
$490$	0	0
$491$	−5.88611e30	−0.664341	−0.332171	−	0.943219i	$-0.607781\pi$
$491$	−5.88611e30	−0.664341	−0.332171	+	0.943219i	$0.607781\pi$
$492$	0	0
$493$	−1.24492e31	−1.34092
$494$	0	0
$495$	−1.00775e30	−0.103608
$496$	0	0
$497$	1.30187e31	1.27782
$498$	0	0
$499$	1.97106e31	1.84732	0.923662	−	0.383208i	$-0.125181\pi$
$499$	1.97106e31	1.84732	0.923662	+	0.383208i	$0.125181\pi$
$500$	0	0
$501$	9.07003e30	0.811848
$502$	0	0
$503$	−1.17974e31	−1.00869	−0.504343	−	0.863503i	$-0.668265\pi$
$503$	−1.17974e31	−1.00869	−0.504343	+	0.863503i	$0.668265\pi$
$504$	0	0
$505$	1.11843e31	0.913607
$506$	0	0
$507$	9.97381e30	0.778519
$508$	0	0
$509$	−2.25204e30	−0.168005	−0.0840024	−	0.996466i	$-0.526770\pi$
$509$	−2.25204e30	−0.168005	−0.0840024	+	0.996466i	$0.526770\pi$
$510$	0	0
$511$	−6.52936e30	−0.465618
$512$	0	0
$513$	−2.17881e31	−1.48549
$514$	0	0
$515$	3.46239e31	2.25732
$516$	0	0
$517$	−1.98589e30	−0.123826
$518$	0	0
$519$	6.21149e30	0.370486
$520$	0	0
$521$	1.95659e31	1.11652	0.558259	−	0.829666i	$-0.311469\pi$
$521$	1.95659e31	1.11652	0.558259	+	0.829666i	$0.311469\pi$
$522$	0	0
$523$	8.24893e29	0.0450431	0.0225216	−	0.999746i	$-0.492831\pi$
$523$	8.24893e29	0.0450431	0.0225216	+	0.999746i	$0.492831\pi$
$524$	0	0
$525$	−8.08936e30	−0.422748
$526$	0	0
$527$	1.85557e31	0.928230
$528$	0	0
$529$	−2.08725e31	−0.999619
$530$	0	0
$531$	−1.24637e31	−0.571559
$532$	0	0
$533$	−4.73856e30	−0.208105
$534$	0	0
$535$	2.35754e31	0.991722
$536$	0	0
$537$	4.46783e30	0.180050
$538$	0	0
$539$	−1.30747e30	−0.0504849
$540$	0	0
$541$	−4.29121e30	−0.158786	−0.0793928	−	0.996843i	$-0.525298\pi$
$541$	−4.29121e30	−0.158786	−0.0793928	+	0.996843i	$0.525298\pi$
$542$	0	0
$543$	−1.06789e31	−0.378727
$544$	0	0
$545$	1.19646e31	0.406760
$546$	0	0
$547$	−2.22362e31	−0.724780	−0.362390	−	0.932027i	$-0.618039\pi$
$547$	−2.22362e31	−0.724780	−0.362390	+	0.932027i	$0.618039\pi$
$548$	0	0
$549$	1.87915e31	0.587325
$550$	0	0
$551$	−5.18114e31	−1.55304
$552$	0	0
$553$	2.05272e31	0.590187
$554$	0	0
$555$	−4.81006e31	−1.32673
$556$	0	0
$557$	1.44997e31	0.383728	0.191864	−	0.981421i	$-0.438547\pi$
$557$	1.44997e31	0.383728	0.191864	+	0.981421i	$0.438547\pi$
$558$	0	0
$559$	1.08810e32	2.76333
$560$	0	0
$561$	1.03100e31	0.251297
$562$	0	0
$563$	5.36230e31	1.25460	0.627299	−	0.778779i	$-0.284161\pi$
$563$	5.36230e31	1.25460	0.627299	+	0.778779i	$0.284161\pi$
$564$	0	0
$565$	1.42228e31	0.319469
$566$	0	0
$567$	2.40477e31	0.518642
$568$	0	0
$569$	−7.34282e31	−1.52080	−0.760400	−	0.649455i	$-0.774997\pi$
$569$	−7.34282e31	−1.52080	−0.760400	+	0.649455i	$0.774997\pi$
$570$	0	0
$571$	5.66166e31	1.12623	0.563116	−	0.826378i	$-0.309602\pi$
$571$	5.66166e31	1.12623	0.563116	+	0.826378i	$0.309602\pi$
$572$	0	0
$573$	3.24015e31	0.619138
$574$	0	0
$575$	6.07153e29	0.0111460
$576$	0	0
$577$	3.70400e31	0.653354	0.326677	−	0.945136i	$-0.394071\pi$
$577$	3.70400e31	0.653354	0.326677	+	0.945136i	$0.394071\pi$
$578$	0	0
$579$	3.92861e31	0.665940
$580$	0	0
$581$	−5.19730e30	−0.0846744
$582$	0	0
$583$	2.23185e31	0.349524
$584$	0	0
$585$	−3.71071e31	−0.558682
$586$	0	0
$587$	1.16143e30	0.0168134	0.00840670	−	0.999965i	$-0.497324\pi$
$587$	1.16143e30	0.0168134	0.00840670	+	0.999965i	$0.497324\pi$
$588$	0	0
$589$	7.72257e31	1.07507
$590$	0	0
$591$	−9.50715e31	−1.27290
$592$	0	0
$593$	7.56126e31	0.973790	0.486895	−	0.873461i	$-0.338129\pi$
$593$	7.56126e31	0.973790	0.486895	+	0.873461i	$0.338129\pi$
$594$	0	0
$595$	−1.06948e32	−1.32504
$596$	0	0
$597$	2.06632e31	0.246316
$598$	0	0
$599$	−1.07219e32	−1.22988	−0.614941	−	0.788573i	$-0.710820\pi$
$599$	−1.07219e32	−1.22988	−0.614941	+	0.788573i	$0.710820\pi$
$600$	0	0
$601$	−5.42835e31	−0.599255	−0.299628	−	0.954056i	$-0.596862\pi$
$601$	−5.42835e31	−0.599255	−0.299628	+	0.954056i	$0.596862\pi$
$602$	0	0
$603$	1.12122e31	0.119135
$604$	0	0
$605$	−1.14359e32	−1.16973
$606$	0	0
$607$	3.13447e31	0.308669	0.154335	−	0.988019i	$-0.450677\pi$
$607$	3.13447e31	0.308669	0.154335	+	0.988019i	$0.450677\pi$
$608$	0	0
$609$	8.88030e31	0.842032
$610$	0	0
$611$	−7.31235e31	−0.667702
$612$	0	0
$613$	−2.02510e32	−1.78094	−0.890472	−	0.455039i	$-0.849625\pi$
$613$	−2.02510e32	−1.78094	−0.890472	+	0.455039i	$0.849625\pi$
$614$	0	0
$615$	−1.82184e31	−0.154328
$616$	0	0
$617$	7.84071e31	0.639847	0.319923	−	0.947443i	$-0.396343\pi$
$617$	7.84071e31	0.639847	0.319923	+	0.947443i	$0.396343\pi$
$618$	0	0
$619$	−2.22045e32	−1.74582	−0.872910	−	0.487882i	$-0.837770\pi$
$619$	−2.22045e32	−1.74582	−0.872910	+	0.487882i	$0.837770\pi$
$620$	0	0
$621$	−2.80288e30	−0.0212350
$622$	0	0
$623$	−2.37077e32	−1.73092
$624$	0	0
$625$	−1.76913e32	−1.24491
$626$	0	0
$627$	4.29086e31	0.291050
$628$	0	0
$629$	−2.31224e32	−1.51199
$630$	0	0
$631$	−1.68752e32	−1.06392	−0.531962	−	0.846768i	$-0.678545\pi$
$631$	−1.68752e32	−1.06392	−0.531962	+	0.846768i	$0.678545\pi$
$632$	0	0
$633$	−7.44127e31	−0.452381
$634$	0	0
$635$	1.67403e32	0.981442
$636$	0	0
$637$	−4.81432e31	−0.272227
$638$	0	0
$639$	8.34828e31	0.455341
$640$	0	0
$641$	6.62363e29	0.00348521	0.00174260	−	0.999998i	$-0.499445\pi$
$641$	6.62363e29	0.00348521	0.00174260	+	0.999998i	$0.499445\pi$
$642$	0	0
$643$	−2.98853e32	−1.51716	−0.758580	−	0.651580i	$-0.774106\pi$
$643$	−2.98853e32	−1.51716	−0.758580	+	0.651580i	$0.774106\pi$
$644$	0	0
$645$	4.18343e32	2.04925
$646$	0	0
$647$	−1.06002e32	−0.501088	−0.250544	−	0.968105i	$-0.580609\pi$
$647$	−1.06002e32	−0.501088	−0.250544	+	0.968105i	$0.580609\pi$
$648$	0	0
$649$	1.01331e32	0.462303
$650$	0	0
$651$	−1.32362e32	−0.582885
$652$	0	0
$653$	2.53610e31	0.107811	0.0539056	−	0.998546i	$-0.482833\pi$
$653$	2.53610e31	0.107811	0.0539056	+	0.998546i	$0.482833\pi$
$654$	0	0
$655$	−1.84067e32	−0.755440
$656$	0	0
$657$	−4.18697e31	−0.165920
$658$	0	0
$659$	−3.68431e31	−0.140985	−0.0704927	−	0.997512i	$-0.522457\pi$
$659$	−3.68431e31	−0.140985	−0.0704927	+	0.997512i	$0.522457\pi$
$660$	0	0
$661$	3.90389e32	1.44272	0.721358	−	0.692563i	$-0.243518\pi$
$661$	3.90389e32	1.44272	0.721358	+	0.692563i	$0.243518\pi$
$662$	0	0
$663$	3.79632e32	1.35506
$664$	0	0
$665$	−4.45099e32	−1.53465
$666$	0	0
$667$	−6.66518e30	−0.0222006
$668$	0	0
$669$	2.82431e32	0.908890
$670$	0	0
$671$	−1.52775e32	−0.475055
$672$	0	0
$673$	8.36950e31	0.251493	0.125747	−	0.992062i	$-0.459867\pi$
$673$	8.36950e31	0.251493	0.125747	+	0.992062i	$0.459867\pi$
$674$	0	0
$675$	−2.14146e32	−0.621894
$676$	0	0
$677$	−4.81868e32	−1.35257	−0.676283	−	0.736642i	$-0.736410\pi$
$677$	−4.81868e32	−1.35257	−0.676283	+	0.736642i	$0.736410\pi$
$678$	0	0
$679$	3.36149e31	0.0912072
$680$	0	0
$681$	3.43602e32	0.901287
$682$	0	0
$683$	1.39396e32	0.353518	0.176759	−	0.984254i	$-0.443439\pi$
$683$	1.39396e32	0.353518	0.176759	+	0.984254i	$0.443439\pi$
$684$	0	0
$685$	−5.47230e31	−0.134193
$686$	0	0
$687$	2.10025e32	0.498045
$688$	0	0
$689$	8.21803e32	1.88472
$690$	0	0
$691$	2.98916e32	0.663059	0.331529	−	0.943445i	$-0.392435\pi$
$691$	2.98916e32	0.663059	0.331529	+	0.943445i	$0.392435\pi$
$692$	0	0
$693$	3.45558e31	0.0741463
$694$	0	0
$695$	5.55235e32	1.15253
$696$	0	0
$697$	−8.75772e31	−0.175879
$698$	0	0
$699$	−5.37577e32	−1.04461
$700$	0	0
$701$	−1.25757e32	−0.236470	−0.118235	−	0.992986i	$-0.537724\pi$
$701$	−1.25757e32	−0.236470	−0.118235	+	0.992986i	$0.537724\pi$
$702$	0	0
$703$	−9.62313e32	−1.75118
$704$	0	0
$705$	−2.81139e32	−0.495160
$706$	0	0
$707$	−3.83511e32	−0.653814
$708$	0	0
$709$	6.82043e32	1.12559	0.562795	−	0.826597i	$-0.309726\pi$
$709$	6.82043e32	1.12559	0.562795	+	0.826597i	$0.309726\pi$
$710$	0	0
$711$	1.31631e32	0.210309
$712$	0	0
$713$	9.93455e30	0.0153680
$714$	0	0
$715$	3.01682e32	0.451887
$716$	0	0
$717$	−7.99346e32	−1.15948
$718$	0	0
$719$	−3.47901e32	−0.488735	−0.244367	−	0.969683i	$-0.578580\pi$
$719$	−3.47901e32	−0.488735	−0.244367	+	0.969683i	$0.578580\pi$
$720$	0	0
$721$	−1.18725e33	−1.61543
$722$	0	0
$723$	3.00159e32	0.395604
$724$	0	0
$725$	−5.09233e32	−0.650174
$726$	0	0
$727$	−6.73378e32	−0.832939	−0.416470	−	0.909150i	$-0.636733\pi$
$727$	−6.73378e32	−0.832939	−0.416470	+	0.909150i	$0.636733\pi$
$728$	0	0
$729$	9.03328e32	1.08263
$730$	0	0
$731$	2.01101e33	2.33541
$732$	0	0
$733$	−5.58897e32	−0.628979	−0.314490	−	0.949261i	$-0.601833\pi$
$733$	−5.58897e32	−0.628979	−0.314490	+	0.949261i	$0.601833\pi$
$734$	0	0
$735$	−1.85097e32	−0.201880
$736$	0	0
$737$	−9.11552e31	−0.0963619
$738$	0	0
$739$	−7.83985e31	−0.0803335	−0.0401668	−	0.999193i	$-0.512789\pi$
$739$	−7.83985e31	−0.0803335	−0.0401668	+	0.999193i	$0.512789\pi$
$740$	0	0
$741$	1.57996e33	1.56941
$742$	0	0
$743$	1.06893e33	1.02939	0.514693	−	0.857374i	$-0.327906\pi$
$743$	1.06893e33	1.02939	0.514693	+	0.857374i	$0.327906\pi$
$744$	0	0
$745$	5.36053e32	0.500508
$746$	0	0
$747$	−3.33278e31	−0.0301731
$748$	0	0
$749$	−8.08399e32	−0.709717
$750$	0	0
$751$	1.31482e33	1.11946	0.559729	−	0.828676i	$-0.310905\pi$
$751$	1.31482e33	1.11946	0.559729	+	0.828676i	$0.310905\pi$
$752$	0	0
$753$	1.01318e33	0.836651
$754$	0	0
$755$	6.40207e32	0.512780
$756$	0	0
$757$	1.90780e32	0.148228	0.0741140	−	0.997250i	$-0.476387\pi$
$757$	1.90780e32	0.148228	0.0741140	+	0.997250i	$0.476387\pi$
$758$	0	0
$759$	5.51989e30	0.00416054
$760$	0	0
$761$	−2.43130e33	−1.77793	−0.888967	−	0.457971i	$-0.848576\pi$
$761$	−2.43130e33	−1.77793	−0.888967	+	0.457971i	$0.848576\pi$
$762$	0	0
$763$	−4.10266e32	−0.291094
$764$	0	0
$765$	−6.85807e32	−0.472167
$766$	0	0
$767$	3.73115e33	2.49285
$768$	0	0
$769$	−1.43591e33	−0.931051	−0.465526	−	0.885034i	$-0.654135\pi$
$769$	−1.43591e33	−0.931051	−0.465526	+	0.885034i	$0.654135\pi$
$770$	0	0
$771$	2.04492e33	1.28692
$772$	0	0
$773$	−6.94569e31	−0.0424279	−0.0212139	−	0.999775i	$-0.506753\pi$
$773$	−6.94569e31	−0.0424279	−0.0212139	+	0.999775i	$0.506753\pi$
$774$	0	0
$775$	7.59020e32	0.450074
$776$	0	0
$777$	1.64937e33	0.949460
$778$	0	0
$779$	−3.64481e32	−0.203702
$780$	0	0
$781$	−6.78718e32	−0.368301
$782$	0	0
$783$	2.35084e33	1.23869
$784$	0	0
$785$	2.34554e33	1.20017
$786$	0	0
$787$	−2.75934e33	−1.37119	−0.685593	−	0.727985i	$-0.740457\pi$
$787$	−2.75934e33	−1.37119	−0.685593	+	0.727985i	$0.740457\pi$
$788$	0	0
$789$	6.86038e32	0.331102
$790$	0	0
$791$	−4.87700e32	−0.228625
$792$	0	0
$793$	−5.62543e33	−2.56162
$794$	0	0
$795$	3.15959e33	1.39769
$796$	0	0
$797$	1.78610e33	0.767599	0.383799	−	0.923417i	$-0.374615\pi$
$797$	1.78610e33	0.767599	0.383799	+	0.923417i	$0.374615\pi$
$798$	0	0
$799$	−1.35146e33	−0.564305
$800$	0	0
$801$	−1.52026e33	−0.616801
$802$	0	0
$803$	3.40402e32	0.134203
$804$	0	0
$805$	−5.72589e31	−0.0219377
$806$	0	0
$807$	5.36761e32	0.199864
$808$	0	0
$809$	9.86830e32	0.357137	0.178568	−	0.983928i	$-0.442853\pi$
$809$	9.86830e32	0.357137	0.178568	+	0.983928i	$0.442853\pi$
$810$	0	0
$811$	3.34475e33	1.17659	0.588293	−	0.808648i	$-0.299800\pi$
$811$	3.34475e33	1.17659	0.588293	+	0.808648i	$0.299800\pi$
$812$	0	0
$813$	2.84999e33	0.974546
$814$	0	0
$815$	−3.37936e33	−1.12337
$816$	0	0
$817$	8.36946e33	2.70485
$818$	0	0
$819$	1.27240e33	0.399815
$820$	0	0
$821$	2.56144e33	0.782597	0.391298	−	0.920264i	$-0.372026\pi$
$821$	2.56144e33	0.782597	0.391298	+	0.920264i	$0.372026\pi$
$822$	0	0
$823$	−9.65592e31	−0.0286877	−0.0143438	−	0.999897i	$-0.504566\pi$
$823$	−9.65592e31	−0.0286877	−0.0143438	+	0.999897i	$0.504566\pi$
$824$	0	0
$825$	4.21731e32	0.121847
$826$	0	0
$827$	−6.81231e33	−1.91417	−0.957087	−	0.289802i	$-0.906411\pi$
$827$	−6.81231e33	−1.91417	−0.957087	+	0.289802i	$0.906411\pi$
$828$	0	0
$829$	−4.67934e33	−1.27881	−0.639407	−	0.768868i	$-0.720820\pi$
$829$	−4.67934e33	−1.27881	−0.639407	+	0.768868i	$0.720820\pi$
$830$	0	0
$831$	−3.29176e33	−0.875016
$832$	0	0
$833$	−8.89774e32	−0.230071
$834$	0	0
$835$	4.90504e33	1.23381
$836$	0	0
$837$	−3.50397e33	−0.857466
$838$	0	0
$839$	−1.96056e33	−0.466786	−0.233393	−	0.972382i	$-0.574983\pi$
$839$	−1.96056e33	−0.466786	−0.233393	+	0.972382i	$0.574983\pi$
$840$	0	0
$841$	1.27352e33	0.295020
$842$	0	0
$843$	−4.93343e33	−1.11207
$844$	0	0
$845$	5.39380e33	1.18316
$846$	0	0
$847$	3.92138e33	0.837104
$848$	0	0
$849$	7.15104e33	1.48570
$850$	0	0
$851$	−1.23795e32	−0.0250330
$852$	0	0
$853$	−4.97796e33	−0.979801	−0.489901	−	0.871778i	$-0.662967\pi$
$853$	−4.97796e33	−0.979801	−0.489901	+	0.871778i	$0.662967\pi$
$854$	0	0
$855$	−2.85421e33	−0.546859
$856$	0	0
$857$	−9.21396e33	−1.71857	−0.859284	−	0.511498i	$-0.829091\pi$
$857$	−9.21396e33	−1.71857	−0.859284	+	0.511498i	$0.829091\pi$
$858$	0	0
$859$	−5.83006e33	−1.05865	−0.529324	−	0.848420i	$-0.677554\pi$
$859$	−5.83006e33	−1.05865	−0.529324	+	0.848420i	$0.677554\pi$
$860$	0	0
$861$	6.24708e32	0.110444
$862$	0	0
$863$	3.17042e33	0.545749	0.272874	−	0.962050i	$-0.412026\pi$
$863$	3.17042e33	0.545749	0.272874	+	0.962050i	$0.412026\pi$
$864$	0	0
$865$	3.35916e33	0.563047
$866$	0	0
$867$	1.96293e33	0.320395
$868$	0	0
$869$	−1.07016e33	−0.170107
$870$	0	0
$871$	−3.35648e33	−0.519608
$872$	0	0
$873$	2.15556e32	0.0325010
$874$	0	0
$875$	3.28227e33	0.482038
$876$	0	0
$877$	−1.37771e34	−1.97089	−0.985444	−	0.170003i	$-0.945622\pi$
$877$	−1.37771e34	−1.97089	−0.985444	+	0.170003i	$0.945622\pi$
$878$	0	0
$879$	−4.86454e32	−0.0677906
$880$	0	0
$881$	−1.15469e34	−1.56763	−0.783815	−	0.620995i	$-0.786729\pi$
$881$	−1.15469e34	−1.56763	−0.783815	+	0.620995i	$0.786729\pi$
$882$	0	0
$883$	4.61935e33	0.610989	0.305494	−	0.952194i	$-0.401178\pi$
$883$	4.61935e33	0.610989	0.305494	+	0.952194i	$0.401178\pi$
$884$	0	0
$885$	1.43452e34	1.84867
$886$	0	0
$887$	1.82011e33	0.228547	0.114274	−	0.993449i	$-0.463546\pi$
$887$	1.82011e33	0.228547	0.114274	+	0.993449i	$0.463546\pi$
$888$	0	0
$889$	−5.74024e33	−0.702360
$890$	0	0
$891$	−1.25370e33	−0.149486
$892$	0	0
$893$	−5.62453e33	−0.653573
$894$	0	0
$895$	2.41619e33	0.273631
$896$	0	0
$897$	2.03251e32	0.0224347
$898$	0	0
$899$	−8.33234e33	−0.896459
$900$	0	0
$901$	1.51884e34	1.59286
$902$	0	0
$903$	−1.43450e34	−1.46653
$904$	0	0
$905$	−5.77509e33	−0.575573
$906$	0	0
$907$	−7.63493e33	−0.741859	−0.370929	−	0.928661i	$-0.620961\pi$
$907$	−7.63493e33	−0.741859	−0.370929	+	0.928661i	$0.620961\pi$
$908$	0	0
$909$	−2.45927e33	−0.232982
$910$	0	0
$911$	−1.34653e34	−1.24381	−0.621904	−	0.783093i	$-0.713641\pi$
$911$	−1.34653e34	−1.24381	−0.621904	+	0.783093i	$0.713641\pi$
$912$	0	0
$913$	2.70956e32	0.0244053
$914$	0	0
$915$	−2.16282e34	−1.89966
$916$	0	0
$917$	6.31164e33	0.540624
$918$	0	0
$919$	4.71664e33	0.394008	0.197004	−	0.980403i	$-0.436879\pi$
$919$	4.71664e33	0.394008	0.197004	+	0.980403i	$0.436879\pi$
$920$	0	0
$921$	−1.82504e34	−1.48692
$922$	0	0
$923$	−2.49915e34	−1.98597
$924$	0	0
$925$	−9.45818e33	−0.733124
$926$	0	0
$927$	−7.61330e33	−0.575647
$928$	0	0
$929$	2.34003e34	1.72600	0.862999	−	0.505206i	$-0.168583\pi$
$929$	2.34003e34	1.72600	0.862999	+	0.505206i	$0.168583\pi$
$930$	0	0
$931$	−3.70309e33	−0.266466
$932$	0	0
$933$	−3.97282e33	−0.278907
$934$	0	0
$935$	5.57564e33	0.381910
$936$	0	0
$937$	2.36472e34	1.58043	0.790213	−	0.612832i	$-0.209970\pi$
$937$	2.36472e34	1.58043	0.790213	+	0.612832i	$0.209970\pi$
$938$	0	0
$939$	1.95165e34	1.27277
$940$	0	0
$941$	−1.30959e34	−0.833400	−0.416700	−	0.909044i	$-0.636813\pi$
$941$	−1.30959e34	−0.833400	−0.416700	+	0.909044i	$0.636813\pi$
$942$	0	0
$943$	−4.68880e31	−0.00291190
$944$	0	0
$945$	2.01955e34	1.22402
$946$	0	0
$947$	1.07513e34	0.635970	0.317985	−	0.948096i	$-0.396994\pi$
$947$	1.07513e34	0.635970	0.317985	+	0.948096i	$0.396994\pi$
$948$	0	0
$949$	1.25341e34	0.723658
$950$	0	0
$951$	−2.10304e34	−1.18515
$952$	0	0
$953$	−1.05168e34	−0.578515	−0.289257	−	0.957251i	$-0.593408\pi$
$953$	−1.05168e34	−0.578515	−0.289257	+	0.957251i	$0.593408\pi$
$954$	0	0
$955$	1.75226e34	0.940938
$956$	0	0
$957$	−4.62966e33	−0.242696
$958$	0	0
$959$	1.87645e33	0.0960337
$960$	0	0
$961$	−7.59384e33	−0.379440
$962$	0	0
$963$	−5.18388e33	−0.252902
$964$	0	0
$965$	2.12458e34	1.01207
$966$	0	0
$967$	−3.44461e33	−0.160227	−0.0801134	−	0.996786i	$-0.525528\pi$
$967$	−3.44461e33	−0.160227	−0.0801134	+	0.996786i	$0.525528\pi$
$968$	0	0
$969$	2.92006e34	1.32638
$970$	0	0
$971$	−6.52185e33	−0.289301	−0.144650	−	0.989483i	$-0.546206\pi$
$971$	−6.52185e33	−0.289301	−0.144650	+	0.989483i	$0.546206\pi$
$972$	0	0
$973$	−1.90390e34	−0.824796
$974$	0	0
$975$	1.55288e34	0.657030
$976$	0	0
$977$	−1.94629e34	−0.804304	−0.402152	−	0.915573i	$-0.631738\pi$
$977$	−1.94629e34	−0.804304	−0.402152	+	0.915573i	$0.631738\pi$
$978$	0	0
$979$	1.23598e34	0.498897
$980$	0	0
$981$	−2.63084e33	−0.103729
$982$	0	0
$983$	−2.36330e33	−0.0910233	−0.0455117	−	0.998964i	$-0.514492\pi$
$983$	−2.36330e33	−0.0910233	−0.0455117	+	0.998964i	$0.514492\pi$
$984$	0	0
$985$	−5.14143e34	−1.93450
$986$	0	0
$987$	9.64024e33	0.354357
$988$	0	0
$989$	1.07667e33	0.0386658
$990$	0	0
$991$	−4.23876e34	−1.48728	−0.743638	−	0.668583i	$-0.766901\pi$
$991$	−4.23876e34	−1.48728	−0.743638	+	0.668583i	$0.766901\pi$
$992$	0	0
$993$	2.38682e34	0.818281
$994$	0	0
$995$	1.11746e34	0.374340
$996$	0	0
$997$	2.50269e34	0.819242	0.409621	−	0.912256i	$-0.365661\pi$
$997$	2.50269e34	0.819242	0.409621	+	0.912256i	$0.365661\pi$
$998$	0	0
$999$	4.36631e34	1.39673

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.24.a.b.1.1	✓	3	8.5	even	2
16.24.a.d.1.3		3	8.3	odd	2
64.24.a.i.1.3		3	1.1	even	1	trivial
64.24.a.j.1.1		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:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

\(f(q)\)	\(=\)	\(q+253079. q^{3} +1.36864e8 q^{5} -4.69307e9 q^{7} -3.00944e10 q^{9} +O(q^{10})\)	\(q+253079. q^{3} +1.36864e8 q^{5} -4.69307e9 q^{7} -3.00944e10 q^{9} +2.44669e11 q^{11} +9.00910e12 q^{13} +3.46374e13 q^{15} +1.66505e14 q^{17} +6.92964e14 q^{19} -1.18772e15 q^{21} +8.91449e13 q^{23} +6.81085e15 q^{25} -3.14419e16 q^{27} -7.47679e16 q^{29} +1.11443e17 q^{31} +6.19204e16 q^{33} -6.42313e17 q^{35} -1.38869e18 q^{37} +2.28001e18 q^{39} -5.25975e17 q^{41} +1.20778e19 q^{43} -4.11885e18 q^{45} -8.11663e18 q^{47} -5.34384e18 q^{49} +4.21388e19 q^{51} +9.12192e19 q^{53} +3.34864e19 q^{55} +1.75374e20 q^{57} +4.14154e20 q^{59} -6.24417e20 q^{61} +1.41235e20 q^{63} +1.23302e21 q^{65} -3.72566e20 q^{67} +2.25607e19 q^{69} -2.77403e21 q^{71} +1.39128e21 q^{73} +1.72368e21 q^{75} -1.14825e21 q^{77} -4.37393e21 q^{79} -5.12408e21 q^{81} +1.10744e21 q^{83} +2.27885e22 q^{85} -1.89222e22 q^{87} +5.05165e22 q^{89} -4.22803e22 q^{91} +2.82037e22 q^{93} +9.48418e22 q^{95} -7.16266e21 q^{97} -7.36316e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 32708 q^{3} - 31480650 q^{5} + 993025320 q^{7} + 1389317071 q^{9}+O(q^{10}) \) 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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