LMFDB - Embedded newform 32.10.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,10,Mod(1,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("32.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	32.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:	$16.4811467572$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{106}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 106 $ x^2 - 106
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$10.2956$ of defining polynomial
Character	$\chi$	$=$	32.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+252.730 q^{3} +2019.84 q^{5} +3039.30 q^{7} +44189.5 q^{9} +O(q^{10})$	$q+252.730 q^{3} +2019.84 q^{5} +3039.30 q^{7} +44189.5 q^{9} -42489.0 q^{11} -74173.8 q^{13} +510474. q^{15} -607725. q^{17} -164849. q^{19} +768123. q^{21} +2.08911e6 q^{23} +2.12663e6 q^{25} +6.19353e6 q^{27} +1.87705e6 q^{29} -669635. q^{31} -1.07383e7 q^{33} +6.13890e6 q^{35} -5.06145e6 q^{37} -1.87459e7 q^{39} -1.46245e7 q^{41} -1.15906e7 q^{43} +8.92557e7 q^{45} +3.35490e7 q^{47} -3.11163e7 q^{49} -1.53590e8 q^{51} -2.03950e7 q^{53} -8.58211e7 q^{55} -4.16623e7 q^{57} +1.19399e8 q^{59} -9.81307e7 q^{61} +1.34305e8 q^{63} -1.49819e8 q^{65} +1.01247e8 q^{67} +5.27981e8 q^{69} -3.11299e8 q^{71} -6.82495e6 q^{73} +5.37464e8 q^{75} -1.29137e8 q^{77} +5.23080e8 q^{79} +6.95509e8 q^{81} -2.37668e8 q^{83} -1.22751e9 q^{85} +4.74388e8 q^{87} -6.21070e8 q^{89} -2.25436e8 q^{91} -1.69237e8 q^{93} -3.32969e8 q^{95} +1.11708e9 q^{97} -1.87757e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 176 q^{3} + 1404 q^{5} + 2784 q^{7} + 30394 q^{9}+O(q^{10}) $ 2 * q + 176 * q^3 + 1404 * q^5 + 2784 * q^7 + 30394 * q^9	$ 2 q + 176 q^{3} + 1404 q^{5} + 2784 q^{7} + 30394 q^{9} - 6896 q^{11} - 45556 q^{13} + 557728 q^{15} - 472188 q^{17} + 780912 q^{19} + 787712 q^{21} + 3883680 q^{23} + 552766 q^{25} + 8762336 q^{27} + 3909612 q^{29} + 4857216 q^{31} - 13469312 q^{33} + 6296128 q^{35} + 8608892 q^{37} - 20941792 q^{39} - 32359020 q^{41} - 32230512 q^{43} + 97751564 q^{45} - 20943040 q^{47} - 71404686 q^{49} - 163990176 q^{51} + 51108252 q^{53} - 107740704 q^{55} - 114230656 q^{57} + 63098960 q^{59} + 25048812 q^{61} + 137827168 q^{63} - 167443224 q^{65} + 224200368 q^{67} + 390283008 q^{69} + 6805472 q^{71} - 23217420 q^{73} + 658226512 q^{75} - 138223872 q^{77} + 726598848 q^{79} + 769941010 q^{81} + 27482736 q^{83} - 1310977032 q^{85} + 318429472 q^{87} - 71429100 q^{89} - 232742592 q^{91} - 593312768 q^{93} - 915407072 q^{95} + 1850743460 q^{97} - 2368592176 q^{99}+O(q^{100}) $ 2 * q + 176 * q^3 + 1404 * q^5 + 2784 * q^7 + 30394 * q^9 - 6896 * q^11 - 45556 * q^13 + 557728 * q^15 - 472188 * q^17 + 780912 * q^19 + 787712 * q^21 + 3883680 * q^23 + 552766 * q^25 + 8762336 * q^27 + 3909612 * q^29 + 4857216 * q^31 - 13469312 * q^33 + 6296128 * q^35 + 8608892 * q^37 - 20941792 * q^39 - 32359020 * q^41 - 32230512 * q^43 + 97751564 * q^45 - 20943040 * q^47 - 71404686 * q^49 - 163990176 * q^51 + 51108252 * q^53 - 107740704 * q^55 - 114230656 * q^57 + 63098960 * q^59 + 25048812 * q^61 + 137827168 * q^63 - 167443224 * q^65 + 224200368 * q^67 + 390283008 * q^69 + 6805472 * q^71 - 23217420 * q^73 + 658226512 * q^75 - 138223872 * q^77 + 726598848 * q^79 + 769941010 * q^81 + 27482736 * q^83 - 1310977032 * q^85 + 318429472 * q^87 - 71429100 * q^89 - 232742592 * q^91 - 593312768 * q^93 - 915407072 * q^95 + 1850743460 * q^97 - 2368592176 * 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
$3$	252.730	1.80140	0.900702	−	0.434437i	$-0.143053\pi$
$3$	252.730	1.80140	0.900702	+	0.434437i	$0.143053\pi$
$4$	0	0
$5$	2019.84	1.44528	0.722640	−	0.691224i	$-0.242928\pi$
$5$	2019.84	1.44528	0.722640	+	0.691224i	$0.242928\pi$
$6$	0	0
$7$	3039.30	0.478446	0.239223	−	0.970965i	$-0.423107\pi$
$7$	3039.30	0.478446	0.239223	+	0.970965i	$0.423107\pi$
$8$	0	0
$9$	44189.5	2.24506
$10$	0	0
$11$	−42489.0	−0.875003	−0.437502	−	0.899218i	$-0.644137\pi$
$11$	−42489.0	−0.875003	−0.437502	+	0.899218i	$0.644137\pi$
$12$	0	0
$13$	−74173.8	−0.720287	−0.360143	−	0.932897i	$-0.617272\pi$
$13$	−74173.8	−0.720287	−0.360143	+	0.932897i	$0.617272\pi$
$14$	0	0
$15$	510474.	2.60353
$16$	0	0
$17$	−607725.	−1.76477	−0.882383	−	0.470532i	$-0.844062\pi$
$17$	−607725.	−1.76477	−0.882383	+	0.470532i	$0.844062\pi$
$18$	0	0
$19$	−164849.	−0.290199	−0.145099	−	0.989417i	$-0.546350\pi$
$19$	−164849.	−0.290199	−0.145099	+	0.989417i	$0.546350\pi$
$20$	0	0
$21$	768123.	0.861874
$22$	0	0
$23$	2.08911e6	1.55663	0.778316	−	0.627873i	$-0.216074\pi$
$23$	2.08911e6	1.55663	0.778316	+	0.627873i	$0.216074\pi$
$24$	0	0
$25$	2.12663e6	1.08884
$26$	0	0
$27$	6.19353e6	2.24285
$28$	0	0
$29$	1.87705e6	0.492817	0.246408	−	0.969166i	$-0.420750\pi$
$29$	1.87705e6	0.492817	0.246408	+	0.969166i	$0.420750\pi$
$30$	0	0
$31$	−669635.	−0.130230	−0.0651150	−	0.997878i	$-0.520741\pi$
$31$	−669635.	−0.130230	−0.0651150	+	0.997878i	$0.520741\pi$
$32$	0	0
$33$	−1.07383e7	−1.57624
$34$	0	0
$35$	6.13890e6	0.691488
$36$	0	0
$37$	−5.06145e6	−0.443984	−0.221992	−	0.975049i	$-0.571256\pi$
$37$	−5.06145e6	−0.443984	−0.221992	+	0.975049i	$0.571256\pi$
$38$	0	0
$39$	−1.87459e7	−1.29753
$40$	0	0
$41$	−1.46245e7	−0.808262	−0.404131	−	0.914701i	$-0.632426\pi$
$41$	−1.46245e7	−0.808262	−0.404131	+	0.914701i	$0.632426\pi$
$42$	0	0
$43$	−1.15906e7	−0.517009	−0.258505	−	0.966010i	$-0.583230\pi$
$43$	−1.15906e7	−0.517009	−0.258505	+	0.966010i	$0.583230\pi$
$44$	0	0
$45$	8.92557e7	3.24474
$46$	0	0
$47$	3.35490e7	1.00286	0.501428	−	0.865199i	$-0.332808\pi$
$47$	3.35490e7	1.00286	0.501428	+	0.865199i	$0.332808\pi$
$48$	0	0
$49$	−3.11163e7	−0.771090
$50$	0	0
$51$	−1.53590e8	−3.17906
$52$	0	0
$53$	−2.03950e7	−0.355045	−0.177522	−	0.984117i	$-0.556808\pi$
$53$	−2.03950e7	−0.355045	−0.177522	+	0.984117i	$0.556808\pi$
$54$	0	0
$55$	−8.58211e7	−1.26463
$56$	0	0
$57$	−4.16623e7	−0.522765
$58$	0	0
$59$	1.19399e8	1.28282	0.641412	−	0.767196i	$-0.278349\pi$
$59$	1.19399e8	1.28282	0.641412	+	0.767196i	$0.278349\pi$
$60$	0	0
$61$	−9.81307e7	−0.907446	−0.453723	−	0.891143i	$-0.649904\pi$
$61$	−9.81307e7	−0.907446	−0.453723	+	0.891143i	$0.649904\pi$
$62$	0	0
$63$	1.34305e8	1.07414
$64$	0	0
$65$	−1.49819e8	−1.04102
$66$	0	0
$67$	1.01247e8	0.613824	0.306912	−	0.951738i	$-0.400704\pi$
$67$	1.01247e8	0.613824	0.306912	+	0.951738i	$0.400704\pi$
$68$	0	0
$69$	5.27981e8	2.80412
$70$	0	0
$71$	−3.11299e8	−1.45383	−0.726917	−	0.686726i	$-0.759047\pi$
$71$	−3.11299e8	−1.45383	−0.726917	+	0.686726i	$0.759047\pi$
$72$	0	0
$73$	−6.82495e6	−0.0281285	−0.0140642	−	0.999901i	$-0.504477\pi$
$73$	−6.82495e6	−0.0281285	−0.0140642	+	0.999901i	$0.504477\pi$
$74$	0	0
$75$	5.37464e8	1.96143
$76$	0	0
$77$	−1.29137e8	−0.418641
$78$	0	0
$79$	5.23080e8	1.51094	0.755469	−	0.655185i	$-0.227409\pi$
$79$	5.23080e8	1.51094	0.755469	+	0.655185i	$0.227409\pi$
$80$	0	0
$81$	6.95509e8	1.79523
$82$	0	0
$83$	−2.37668e8	−0.549691	−0.274845	−	0.961488i	$-0.588627\pi$
$83$	−2.37668e8	−0.549691	−0.274845	+	0.961488i	$0.588627\pi$
$84$	0	0
$85$	−1.22751e9	−2.55058
$86$	0	0
$87$	4.74388e8	0.887763
$88$	0	0
$89$	−6.21070e8	−1.04927	−0.524633	−	0.851328i	$-0.675798\pi$
$89$	−6.21070e8	−1.04927	−0.524633	+	0.851328i	$0.675798\pi$
$90$	0	0
$91$	−2.25436e8	−0.344618
$92$	0	0
$93$	−1.69237e8	−0.234597
$94$	0	0
$95$	−3.32969e8	−0.419418
$96$	0	0
$97$	1.11708e9	1.28118	0.640591	−	0.767882i	$-0.278690\pi$
$97$	1.11708e9	1.28118	0.640591	+	0.767882i	$0.278690\pi$
$98$	0	0
$99$	−1.87757e9	−1.96443
$100$	0	0
$101$	2.84480e8	0.272023	0.136011	−	0.990707i	$-0.456572\pi$
$101$	2.84480e8	0.272023	0.136011	+	0.990707i	$0.456572\pi$
$102$	0	0
$103$	−5.93311e8	−0.519416	−0.259708	−	0.965687i	$-0.583626\pi$
$103$	−5.93311e8	−0.519416	−0.259708	+	0.965687i	$0.583626\pi$
$104$	0	0
$105$	1.55149e9	1.24565
$106$	0	0
$107$	−6.44578e8	−0.475388	−0.237694	−	0.971340i	$-0.576392\pi$
$107$	−6.44578e8	−0.475388	−0.237694	+	0.971340i	$0.576392\pi$
$108$	0	0
$109$	5.64540e8	0.383067	0.191534	−	0.981486i	$-0.438654\pi$
$109$	5.64540e8	0.383067	0.191534	+	0.981486i	$0.438654\pi$
$110$	0	0
$111$	−1.27918e9	−0.799794
$112$	0	0
$113$	−3.95050e8	−0.227929	−0.113964	−	0.993485i	$-0.536355\pi$
$113$	−3.95050e8	−0.227929	−0.113964	+	0.993485i	$0.536355\pi$
$114$	0	0
$115$	4.21967e9	2.24977
$116$	0	0
$117$	−3.27770e9	−1.61709
$118$	0	0
$119$	−1.84706e9	−0.844344
$120$	0	0
$121$	−5.52630e8	−0.234369
$122$	0	0
$123$	−3.69604e9	−1.45601
$124$	0	0
$125$	3.50455e8	0.128392
$126$	0	0
$127$	−3.41306e9	−1.16420	−0.582099	−	0.813118i	$-0.697769\pi$
$127$	−3.41306e9	−1.16420	−0.582099	+	0.813118i	$0.697769\pi$
$128$	0	0
$129$	−2.92930e9	−0.931343
$130$	0	0
$131$	5.95732e9	1.76738	0.883691	−	0.468071i	$-0.155051\pi$
$131$	5.95732e9	1.76738	0.883691	+	0.468071i	$0.155051\pi$
$132$	0	0
$133$	−5.01026e8	−0.138844
$134$	0	0
$135$	1.25099e10	3.24155
$136$	0	0
$137$	3.18592e9	0.772666	0.386333	−	0.922359i	$-0.373742\pi$
$137$	3.18592e9	0.772666	0.386333	+	0.922359i	$0.373742\pi$
$138$	0	0
$139$	1.31882e9	0.299652	0.149826	−	0.988712i	$-0.452129\pi$
$139$	1.31882e9	0.299652	0.149826	+	0.988712i	$0.452129\pi$
$140$	0	0
$141$	8.47883e9	1.80655
$142$	0	0
$143$	3.15157e9	0.630253
$144$	0	0
$145$	3.79135e9	0.712259
$146$	0	0
$147$	−7.86401e9	−1.38904
$148$	0	0
$149$	6.79472e9	1.12936	0.564681	−	0.825309i	$-0.308999\pi$
$149$	6.79472e9	1.12936	0.564681	+	0.825309i	$0.308999\pi$
$150$	0	0
$151$	4.19125e9	0.656065	0.328032	−	0.944666i	$-0.393614\pi$
$151$	4.19125e9	0.656065	0.328032	+	0.944666i	$0.393614\pi$
$152$	0	0
$153$	−2.68551e10	−3.96200
$154$	0	0
$155$	−1.35256e9	−0.188219
$156$	0	0
$157$	8.30545e9	1.09097	0.545487	−	0.838119i	$-0.316345\pi$
$157$	8.30545e9	1.09097	0.545487	+	0.838119i	$0.316345\pi$
$158$	0	0
$159$	−5.15444e9	−0.639579
$160$	0	0
$161$	6.34943e9	0.744763
$162$	0	0
$163$	1.02083e10	1.13269	0.566343	−	0.824170i	$-0.308358\pi$
$163$	1.02083e10	1.13269	0.566343	+	0.824170i	$0.308358\pi$
$164$	0	0
$165$	−2.16896e10	−2.27810
$166$	0	0
$167$	−1.78819e10	−1.77906	−0.889529	−	0.456880i	$-0.848967\pi$
$167$	−1.78819e10	−1.77906	−0.889529	+	0.456880i	$0.848967\pi$
$168$	0	0
$169$	−5.10275e9	−0.481187
$170$	0	0
$171$	−7.28460e9	−0.651513
$172$	0	0
$173$	−4.35787e9	−0.369885	−0.184943	−	0.982749i	$-0.559210\pi$
$173$	−4.35787e9	−0.369885	−0.184943	+	0.982749i	$0.559210\pi$
$174$	0	0
$175$	6.46347e9	0.520948
$176$	0	0
$177$	3.01758e10	2.31089
$178$	0	0
$179$	−1.08673e10	−0.791193	−0.395597	−	0.918424i	$-0.629462\pi$
$179$	−1.08673e10	−0.791193	−0.395597	+	0.918424i	$0.629462\pi$
$180$	0	0
$181$	2.14725e10	1.48706	0.743530	−	0.668703i	$-0.233150\pi$
$181$	2.14725e10	1.48706	0.743530	+	0.668703i	$0.233150\pi$
$182$	0	0
$183$	−2.48006e10	−1.63468
$184$	0	0
$185$	−1.02233e10	−0.641681
$186$	0	0
$187$	2.58216e10	1.54418
$188$	0	0
$189$	1.88240e10	1.07308
$190$	0	0
$191$	−2.77603e10	−1.50930	−0.754648	−	0.656130i	$-0.772192\pi$
$191$	−2.77603e10	−1.50930	−0.754648	+	0.656130i	$0.772192\pi$
$192$	0	0
$193$	−1.74020e10	−0.902801	−0.451400	−	0.892322i	$-0.649075\pi$
$193$	−1.74020e10	−0.902801	−0.451400	+	0.892322i	$0.649075\pi$
$194$	0	0
$195$	−3.78638e10	−1.87529
$196$	0	0
$197$	−2.25126e10	−1.06495	−0.532473	−	0.846447i	$-0.678737\pi$
$197$	−2.25126e10	−1.06495	−0.532473	+	0.846447i	$0.678737\pi$
$198$	0	0
$199$	3.06974e10	1.38760	0.693798	−	0.720170i	$-0.255936\pi$
$199$	3.06974e10	1.38760	0.693798	+	0.720170i	$0.255936\pi$
$200$	0	0
$201$	2.55881e10	1.10575
$202$	0	0
$203$	5.70493e9	0.235786
$204$	0	0
$205$	−2.95391e10	−1.16817
$206$	0	0
$207$	9.23167e10	3.49473
$208$	0	0
$209$	7.00428e9	0.253925
$210$	0	0
$211$	2.20646e10	0.766348	0.383174	−	0.923676i	$-0.374831\pi$
$211$	2.20646e10	0.766348	0.383174	+	0.923676i	$0.374831\pi$
$212$	0	0
$213$	−7.86745e10	−2.61894
$214$	0	0
$215$	−2.34112e10	−0.747223
$216$	0	0
$217$	−2.03522e9	−0.0623079
$218$	0	0
$219$	−1.72487e9	−0.0506708
$220$	0	0
$221$	4.50773e10	1.27114
$222$	0	0
$223$	1.51667e10	0.410695	0.205348	−	0.978689i	$-0.434167\pi$
$223$	1.51667e10	0.410695	0.205348	+	0.978689i	$0.434167\pi$
$224$	0	0
$225$	9.39748e10	2.44450
$226$	0	0
$227$	−2.19810e10	−0.549455	−0.274727	−	0.961522i	$-0.588588\pi$
$227$	−2.19810e10	−0.549455	−0.274727	+	0.961522i	$0.588588\pi$
$228$	0	0
$229$	−3.45174e10	−0.829427	−0.414714	−	0.909952i	$-0.636118\pi$
$229$	−3.45174e10	−0.829427	−0.414714	+	0.909952i	$0.636118\pi$
$230$	0	0
$231$	−3.26368e10	−0.754143
$232$	0	0
$233$	−4.06497e10	−0.903558	−0.451779	−	0.892130i	$-0.649210\pi$
$233$	−4.06497e10	−0.903558	−0.451779	+	0.892130i	$0.649210\pi$
$234$	0	0
$235$	6.77636e10	1.44941
$236$	0	0
$237$	1.32198e11	2.72181
$238$	0	0
$239$	3.09648e10	0.613872	0.306936	−	0.951730i	$-0.400696\pi$
$239$	3.09648e10	0.613872	0.306936	+	0.951730i	$0.400696\pi$
$240$	0	0
$241$	3.55827e10	0.679457	0.339728	−	0.940524i	$-0.389665\pi$
$241$	3.55827e10	0.679457	0.339728	+	0.940524i	$0.389665\pi$
$242$	0	0
$243$	5.38689e10	0.991082
$244$	0	0
$245$	−6.28499e10	−1.11444
$246$	0	0
$247$	1.22275e10	0.209026
$248$	0	0
$249$	−6.00657e10	−0.990216
$250$	0	0
$251$	−2.11617e10	−0.336525	−0.168263	−	0.985742i	$-0.553816\pi$
$251$	−2.11617e10	−0.336525	−0.168263	+	0.985742i	$0.553816\pi$
$252$	0	0
$253$	−8.87642e10	−1.36206
$254$	0	0
$255$	−3.10228e11	−4.59463
$256$	0	0
$257$	−1.03338e11	−1.47761	−0.738804	−	0.673921i	$-0.764609\pi$
$257$	−1.03338e11	−1.47761	−0.738804	+	0.673921i	$0.764609\pi$
$258$	0	0
$259$	−1.53833e10	−0.212422
$260$	0	0
$261$	8.29460e10	1.10640
$262$	0	0
$263$	9.63331e10	1.24158	0.620790	−	0.783977i	$-0.286812\pi$
$263$	9.63331e10	1.24158	0.620790	+	0.783977i	$0.286812\pi$
$264$	0	0
$265$	−4.11947e10	−0.513139
$266$	0	0
$267$	−1.56963e11	−1.89015
$268$	0	0
$269$	7.17446e10	0.835419	0.417709	−	0.908581i	$-0.362833\pi$
$269$	7.17446e10	0.835419	0.417709	+	0.908581i	$0.362833\pi$
$270$	0	0
$271$	1.31827e11	1.48471	0.742356	−	0.670006i	$-0.233708\pi$
$271$	1.31827e11	1.48471	0.742356	+	0.670006i	$0.233708\pi$
$272$	0	0
$273$	−5.69746e10	−0.620796
$274$	0	0
$275$	−9.03585e10	−0.952734
$276$	0	0
$277$	9.13454e10	0.932240	0.466120	−	0.884722i	$-0.345652\pi$
$277$	9.13454e10	0.932240	0.466120	+	0.884722i	$0.345652\pi$
$278$	0	0
$279$	−2.95908e10	−0.292374
$280$	0	0
$281$	7.50850e10	0.718414	0.359207	−	0.933258i	$-0.383047\pi$
$281$	7.50850e10	0.718414	0.359207	+	0.933258i	$0.383047\pi$
$282$	0	0
$283$	−9.96369e10	−0.923381	−0.461691	−	0.887041i	$-0.652757\pi$
$283$	−9.96369e10	−0.923381	−0.461691	+	0.887041i	$0.652757\pi$
$284$	0	0
$285$	−8.41513e10	−0.755542
$286$	0	0
$287$	−4.44481e10	−0.386710
$288$	0	0
$289$	2.50742e11	2.11440
$290$	0	0
$291$	2.82319e11	2.30793
$292$	0	0
$293$	2.18507e11	1.73205	0.866026	−	0.499999i	$-0.166666\pi$
$293$	2.18507e11	1.73205	0.866026	+	0.499999i	$0.166666\pi$
$294$	0	0
$295$	2.41167e11	1.85404
$296$	0	0
$297$	−2.63157e11	−1.96251
$298$	0	0
$299$	−1.54957e11	−1.12122
$300$	0	0
$301$	−3.52274e10	−0.247361
$302$	0	0
$303$	7.18966e10	0.490023
$304$	0	0
$305$	−1.98208e11	−1.31151
$306$	0	0
$307$	4.39947e10	0.282669	0.141334	−	0.989962i	$-0.454861\pi$
$307$	4.39947e10	0.282669	0.141334	+	0.989962i	$0.454861\pi$
$308$	0	0
$309$	−1.49948e11	−0.935678
$310$	0	0
$311$	−1.89071e11	−1.14605	−0.573023	−	0.819539i	$-0.694230\pi$
$311$	−1.89071e11	−1.14605	−0.573023	+	0.819539i	$0.694230\pi$
$312$	0	0
$313$	−2.35414e10	−0.138638	−0.0693191	−	0.997595i	$-0.522083\pi$
$313$	−2.35414e10	−0.138638	−0.0693191	+	0.997595i	$0.522083\pi$
$314$	0	0
$315$	2.71275e11	1.55243
$316$	0	0
$317$	1.91969e11	1.06774	0.533869	−	0.845567i	$-0.320738\pi$
$317$	1.91969e11	1.06774	0.533869	+	0.845567i	$0.320738\pi$
$318$	0	0
$319$	−7.97542e10	−0.431216
$320$	0	0
$321$	−1.62904e11	−0.856366
$322$	0	0
$323$	1.00183e11	0.512133
$324$	0	0
$325$	−1.57740e11	−0.784273
$326$	0	0
$327$	1.42676e11	0.690059
$328$	0	0
$329$	1.01965e11	0.479812
$330$	0	0
$331$	4.23565e11	1.93952	0.969759	−	0.244063i	$-0.0784805\pi$
$331$	4.23565e11	1.93952	0.969759	+	0.244063i	$0.0784805\pi$
$332$	0	0
$333$	−2.23663e11	−0.996769
$334$	0	0
$335$	2.04502e11	0.887148
$336$	0	0
$337$	3.28269e11	1.38642	0.693212	−	0.720734i	$-0.256195\pi$
$337$	3.28269e11	1.38642	0.693212	+	0.720734i	$0.256195\pi$
$338$	0	0
$339$	−9.98410e10	−0.410592
$340$	0	0
$341$	2.84522e10	0.113952
$342$	0	0
$343$	−2.17218e11	−0.847370
$344$	0	0
$345$	1.06644e12	4.05274
$346$	0	0
$347$	−2.25945e11	−0.836604	−0.418302	−	0.908308i	$-0.637375\pi$
$347$	−2.25945e11	−0.836604	−0.418302	+	0.908308i	$0.637375\pi$
$348$	0	0
$349$	−5.81052e10	−0.209653	−0.104826	−	0.994491i	$-0.533429\pi$
$349$	−5.81052e10	−0.209653	−0.104826	+	0.994491i	$0.533429\pi$
$350$	0	0
$351$	−4.59397e11	−1.61550
$352$	0	0
$353$	−1.00599e11	−0.344832	−0.172416	−	0.985024i	$-0.555157\pi$
$353$	−1.00599e11	−0.344832	−0.172416	+	0.985024i	$0.555157\pi$
$354$	0	0
$355$	−6.28774e11	−2.10120
$356$	0	0
$357$	−4.66807e11	−1.52101
$358$	0	0
$359$	−1.42379e11	−0.452397	−0.226198	−	0.974081i	$-0.572630\pi$
$359$	−1.42379e11	−0.452397	−0.226198	+	0.974081i	$0.572630\pi$
$360$	0	0
$361$	−2.95512e11	−0.915785
$362$	0	0
$363$	−1.39666e11	−0.422194
$364$	0	0
$365$	−1.37853e10	−0.0406536
$366$	0	0
$367$	5.83998e11	1.68041	0.840203	−	0.542272i	$-0.182436\pi$
$367$	5.83998e11	1.68041	0.840203	+	0.542272i	$0.182436\pi$
$368$	0	0
$369$	−6.46247e11	−1.81460
$370$	0	0
$371$	−6.19866e10	−0.169869
$372$	0	0
$373$	7.04616e11	1.88479	0.942394	−	0.334505i	$-0.108569\pi$
$373$	7.04616e11	1.88479	0.942394	+	0.334505i	$0.108569\pi$
$374$	0	0
$375$	8.85705e10	0.231286
$376$	0	0
$377$	−1.39228e11	−0.354969
$378$	0	0
$379$	−4.54881e11	−1.13246	−0.566228	−	0.824248i	$-0.691598\pi$
$379$	−4.54881e11	−1.13246	−0.566228	+	0.824248i	$0.691598\pi$
$380$	0	0
$381$	−8.62583e11	−2.09719
$382$	0	0
$383$	6.11355e11	1.45177	0.725887	−	0.687814i	$-0.241429\pi$
$383$	6.11355e11	1.45177	0.725887	+	0.687814i	$0.241429\pi$
$384$	0	0
$385$	−2.60836e11	−0.605054
$386$	0	0
$387$	−5.12183e11	−1.16072
$388$	0	0
$389$	9.28801e10	0.205660	0.102830	−	0.994699i	$-0.467210\pi$
$389$	9.28801e10	0.205660	0.102830	+	0.994699i	$0.467210\pi$
$390$	0	0
$391$	−1.26960e12	−2.74709
$392$	0	0
$393$	1.50559e12	3.18377
$394$	0	0
$395$	1.05654e12	2.18373
$396$	0	0
$397$	−6.58291e11	−1.33003	−0.665014	−	0.746831i	$-0.731574\pi$
$397$	−6.58291e11	−1.33003	−0.665014	+	0.746831i	$0.731574\pi$
$398$	0	0
$399$	−1.26624e11	−0.250115
$400$	0	0
$401$	−1.93836e11	−0.374356	−0.187178	−	0.982326i	$-0.559934\pi$
$401$	−1.93836e11	−0.374356	−0.187178	+	0.982326i	$0.559934\pi$
$402$	0	0
$403$	4.96694e10	0.0938029
$404$	0	0
$405$	1.40482e12	2.59461
$406$	0	0
$407$	2.15056e11	0.388487
$408$	0	0
$409$	−3.09099e11	−0.546188	−0.273094	−	0.961987i	$-0.588047\pi$
$409$	−3.09099e11	−0.546188	−0.273094	+	0.961987i	$0.588047\pi$
$410$	0	0
$411$	8.05177e11	1.39188
$412$	0	0
$413$	3.62890e11	0.613762
$414$	0	0
$415$	−4.80051e11	−0.794457
$416$	0	0
$417$	3.33304e11	0.539795
$418$	0	0
$419$	6.55357e11	1.03876	0.519380	−	0.854544i	$-0.326163\pi$
$419$	6.55357e11	1.03876	0.519380	+	0.854544i	$0.326163\pi$
$420$	0	0
$421$	−2.00415e11	−0.310929	−0.155465	−	0.987841i	$-0.549687\pi$
$421$	−2.00415e11	−0.310929	−0.155465	+	0.987841i	$0.549687\pi$
$422$	0	0
$423$	1.48251e12	2.25147
$424$	0	0
$425$	−1.29241e12	−1.92154
$426$	0	0
$427$	−2.98249e11	−0.434163
$428$	0	0
$429$	7.96497e11	1.13534
$430$	0	0
$431$	−5.21200e11	−0.727540	−0.363770	−	0.931489i	$-0.618511\pi$
$431$	−5.21200e11	−0.727540	−0.363770	+	0.931489i	$0.618511\pi$
$432$	0	0
$433$	3.23838e11	0.442723	0.221361	−	0.975192i	$-0.428950\pi$
$433$	3.23838e11	0.442723	0.221361	+	0.975192i	$0.428950\pi$
$434$	0	0
$435$	9.58188e11	1.28307
$436$	0	0
$437$	−3.44388e11	−0.451732
$438$	0	0
$439$	−1.59945e11	−0.205532	−0.102766	−	0.994706i	$-0.532769\pi$
$439$	−1.59945e11	−0.205532	−0.102766	+	0.994706i	$0.532769\pi$
$440$	0	0
$441$	−1.37501e12	−1.73114
$442$	0	0
$443$	2.33953e11	0.288610	0.144305	−	0.989533i	$-0.453905\pi$
$443$	2.33953e11	0.288610	0.144305	+	0.989533i	$0.453905\pi$
$444$	0	0
$445$	−1.25446e12	−1.51648
$446$	0	0
$447$	1.71723e12	2.03444
$448$	0	0
$449$	−1.44728e12	−1.68052	−0.840262	−	0.542180i	$-0.817599\pi$
$449$	−1.44728e12	−1.68052	−0.840262	+	0.542180i	$0.817599\pi$
$450$	0	0
$451$	6.21379e11	0.707232
$452$	0	0
$453$	1.05925e12	1.18184
$454$	0	0
$455$	−4.55346e11	−0.498070
$456$	0	0
$457$	1.09012e12	1.16909	0.584547	−	0.811359i	$-0.301272\pi$
$457$	1.09012e12	1.16909	0.584547	+	0.811359i	$0.301272\pi$
$458$	0	0
$459$	−3.76396e12	−3.95811
$460$	0	0
$461$	1.66397e12	1.71589	0.857947	−	0.513738i	$-0.171740\pi$
$461$	1.66397e12	1.71589	0.857947	+	0.513738i	$0.171740\pi$
$462$	0	0
$463$	−1.55834e12	−1.57597	−0.787985	−	0.615695i	$-0.788875\pi$
$463$	−1.55834e12	−1.57597	−0.787985	+	0.615695i	$0.788875\pi$
$464$	0	0
$465$	−3.41832e11	−0.339058
$466$	0	0
$467$	−1.17880e12	−1.14687	−0.573436	−	0.819251i	$-0.694390\pi$
$467$	−1.17880e12	−1.14687	−0.573436	+	0.819251i	$0.694390\pi$
$468$	0	0
$469$	3.07719e11	0.293681
$470$	0	0
$471$	2.09904e12	1.96529
$472$	0	0
$473$	4.92474e11	0.452385
$474$	0	0
$475$	−3.50573e11	−0.315979
$476$	0	0
$477$	−9.01246e11	−0.797096
$478$	0	0
$479$	−1.05266e11	−0.0913645	−0.0456822	−	0.998956i	$-0.514546\pi$
$479$	−1.05266e11	−0.0913645	−0.0456822	+	0.998956i	$0.514546\pi$
$480$	0	0
$481$	3.75427e11	0.319795
$482$	0	0
$483$	1.60469e12	1.34162
$484$	0	0
$485$	2.25632e12	1.85167
$486$	0	0
$487$	−2.92582e11	−0.235704	−0.117852	−	0.993031i	$-0.537601\pi$
$487$	−2.92582e11	−0.235704	−0.117852	+	0.993031i	$0.537601\pi$
$488$	0	0
$489$	2.57995e12	2.04043
$490$	0	0
$491$	1.72770e12	1.34154	0.670768	−	0.741667i	$-0.265965\pi$
$491$	1.72770e12	1.34154	0.670768	+	0.741667i	$0.265965\pi$
$492$	0	0
$493$	−1.14073e12	−0.869706
$494$	0	0
$495$	−3.79239e12	−2.83916
$496$	0	0
$497$	−9.46130e11	−0.695580
$498$	0	0
$499$	−2.08491e12	−1.50534	−0.752672	−	0.658396i	$-0.771235\pi$
$499$	−2.08491e12	−1.50534	−0.752672	+	0.658396i	$0.771235\pi$
$500$	0	0
$501$	−4.51930e12	−3.20480
$502$	0	0
$503$	2.67459e11	0.186295	0.0931474	−	0.995652i	$-0.470307\pi$
$503$	2.67459e11	0.186295	0.0931474	+	0.995652i	$0.470307\pi$
$504$	0	0
$505$	5.74604e11	0.393149
$506$	0	0
$507$	−1.28962e12	−0.866813
$508$	0	0
$509$	−7.85816e11	−0.518909	−0.259454	−	0.965755i	$-0.583543\pi$
$509$	−7.85816e11	−0.518909	−0.259454	+	0.965755i	$0.583543\pi$
$510$	0	0
$511$	−2.07431e10	−0.0134580
$512$	0	0
$513$	−1.02100e12	−0.650873
$514$	0	0
$515$	−1.19839e12	−0.750701
$516$	0	0
$517$	−1.42546e12	−0.877503
$518$	0	0
$519$	−1.10137e12	−0.666313
$520$	0	0
$521$	−4.69385e11	−0.279100	−0.139550	−	0.990215i	$-0.544566\pi$
$521$	−4.69385e11	−0.279100	−0.139550	+	0.990215i	$0.544566\pi$
$522$	0	0
$523$	2.40838e12	1.40756	0.703780	−	0.710418i	$-0.251494\pi$
$523$	2.40838e12	1.40756	0.703780	+	0.710418i	$0.251494\pi$
$524$	0	0
$525$	1.63351e12	0.938439
$526$	0	0
$527$	4.06954e11	0.229825
$528$	0	0
$529$	2.56322e12	1.42310
$530$	0	0
$531$	5.27619e12	2.88002
$532$	0	0
$533$	1.08475e12	0.582181
$534$	0	0
$535$	−1.30194e12	−0.687069
$536$	0	0
$537$	−2.74649e12	−1.42526
$538$	0	0
$539$	1.32210e12	0.674706
$540$	0	0
$541$	2.43978e12	1.22451	0.612255	−	0.790660i	$-0.290263\pi$
$541$	2.43978e12	1.22451	0.612255	+	0.790660i	$0.290263\pi$
$542$	0	0
$543$	5.42674e12	2.67880
$544$	0	0
$545$	1.14028e12	0.553640
$546$	0	0
$547$	−8.95402e11	−0.427637	−0.213818	−	0.976873i	$-0.568590\pi$
$547$	−8.95402e11	−0.427637	−0.213818	+	0.976873i	$0.568590\pi$
$548$	0	0
$549$	−4.33635e12	−2.03727
$550$	0	0
$551$	−3.09431e11	−0.143015
$552$	0	0
$553$	1.58980e12	0.722901
$554$	0	0
$555$	−2.58374e12	−1.15593
$556$	0	0
$557$	3.89951e12	1.71657	0.858286	−	0.513171i	$-0.171529\pi$
$557$	3.89951e12	1.71657	0.858286	+	0.513171i	$0.171529\pi$
$558$	0	0
$559$	8.59720e11	0.372395
$560$	0	0
$561$	6.52591e12	2.78169
$562$	0	0
$563$	−3.25127e11	−0.136384	−0.0681922	−	0.997672i	$-0.521723\pi$
$563$	−3.25127e11	−0.136384	−0.0681922	+	0.997672i	$0.521723\pi$
$564$	0	0
$565$	−7.97938e11	−0.329421
$566$	0	0
$567$	2.11386e12	0.858920
$568$	0	0
$569$	−7.30167e11	−0.292023	−0.146012	−	0.989283i	$-0.546644\pi$
$569$	−7.30167e11	−0.292023	−0.146012	+	0.989283i	$0.546644\pi$
$570$	0	0
$571$	−1.59452e11	−0.0627723	−0.0313861	−	0.999507i	$-0.509992\pi$
$571$	−1.59452e11	−0.0627723	−0.0313861	+	0.999507i	$0.509992\pi$
$572$	0	0
$573$	−7.01587e12	−2.71885
$574$	0	0
$575$	4.44276e12	1.69492
$576$	0	0
$577$	1.82973e12	0.687218	0.343609	−	0.939113i	$-0.388350\pi$
$577$	1.82973e12	0.687218	0.343609	+	0.939113i	$0.388350\pi$
$578$	0	0
$579$	−4.39802e12	−1.62631
$580$	0	0
$581$	−7.22343e11	−0.262997
$582$	0	0
$583$	8.66565e11	0.310665
$584$	0	0
$585$	−6.62044e12	−2.33714
$586$	0	0
$587$	3.74725e12	1.30269	0.651345	−	0.758782i	$-0.274205\pi$
$587$	3.74725e12	1.30269	0.651345	+	0.758782i	$0.274205\pi$
$588$	0	0
$589$	1.10389e11	0.0377925
$590$	0	0
$591$	−5.68961e12	−1.91840
$592$	0	0
$593$	−4.51432e12	−1.49916	−0.749578	−	0.661916i	$-0.769743\pi$
$593$	−4.51432e12	−1.49916	−0.749578	+	0.661916i	$0.769743\pi$
$594$	0	0
$595$	−3.73077e12	−1.22031
$596$	0	0
$597$	7.75816e12	2.49962
$598$	0	0
$599$	−2.95283e12	−0.937168	−0.468584	−	0.883419i	$-0.655236\pi$
$599$	−2.95283e12	−0.937168	−0.468584	+	0.883419i	$0.655236\pi$
$600$	0	0
$601$	−3.34262e12	−1.04509	−0.522543	−	0.852613i	$-0.675016\pi$
$601$	−3.34262e12	−1.04509	−0.522543	+	0.852613i	$0.675016\pi$
$602$	0	0
$603$	4.47404e12	1.37807
$604$	0	0
$605$	−1.11622e12	−0.338729
$606$	0	0
$607$	4.79377e12	1.43327	0.716636	−	0.697448i	$-0.245681\pi$
$607$	4.79377e12	1.43327	0.716636	+	0.697448i	$0.245681\pi$
$608$	0	0
$609$	1.44181e12	0.424746
$610$	0	0
$611$	−2.48845e12	−0.722344
$612$	0	0
$613$	−3.04095e12	−0.869837	−0.434918	−	0.900470i	$-0.643223\pi$
$613$	−3.04095e12	−0.869837	−0.434918	+	0.900470i	$0.643223\pi$
$614$	0	0
$615$	−7.46541e12	−2.10434
$616$	0	0
$617$	−1.62097e12	−0.450288	−0.225144	−	0.974325i	$-0.572285\pi$
$617$	−1.62097e12	−0.450288	−0.225144	+	0.974325i	$0.572285\pi$
$618$	0	0
$619$	1.34898e12	0.369315	0.184658	−	0.982803i	$-0.440882\pi$
$619$	1.34898e12	0.369315	0.184658	+	0.982803i	$0.440882\pi$
$620$	0	0
$621$	1.29390e13	3.49130
$622$	0	0
$623$	−1.88762e12	−0.502017
$624$	0	0
$625$	−3.44571e12	−0.903273
$626$	0	0
$627$	1.77019e12	0.457421
$628$	0	0
$629$	3.07597e12	0.783527
$630$	0	0
$631$	−8.33673e11	−0.209346	−0.104673	−	0.994507i	$-0.533380\pi$
$631$	−8.33673e11	−0.209346	−0.104673	+	0.994507i	$0.533380\pi$
$632$	0	0
$633$	5.57640e12	1.38050
$634$	0	0
$635$	−6.89384e12	−1.68259
$636$	0	0
$637$	2.30801e12	0.555406
$638$	0	0
$639$	−1.37561e13	−3.26394
$640$	0	0
$641$	5.03354e12	1.17764	0.588820	−	0.808264i	$-0.299593\pi$
$641$	5.03354e12	1.17764	0.588820	+	0.808264i	$0.299593\pi$
$642$	0	0
$643$	5.08235e12	1.17251	0.586253	−	0.810128i	$-0.300602\pi$
$643$	5.08235e12	1.17251	0.586253	+	0.810128i	$0.300602\pi$
$644$	0	0
$645$	−5.91671e12	−1.34605
$646$	0	0
$647$	−3.34349e12	−0.750120	−0.375060	−	0.927001i	$-0.622378\pi$
$647$	−3.34349e12	−0.750120	−0.375060	+	0.927001i	$0.622378\pi$
$648$	0	0
$649$	−5.07316e12	−1.12248
$650$	0	0
$651$	−5.14362e11	−0.112242
$652$	0	0
$653$	−5.29726e12	−1.14010	−0.570049	−	0.821611i	$-0.693076\pi$
$653$	−5.29726e12	−1.14010	−0.570049	+	0.821611i	$0.693076\pi$
$654$	0	0
$655$	1.20328e13	2.55436
$656$	0	0
$657$	−3.01591e11	−0.0631501
$658$	0	0
$659$	−7.48186e12	−1.54534	−0.772672	−	0.634805i	$-0.781080\pi$
$659$	−7.48186e12	−1.54534	−0.772672	+	0.634805i	$0.781080\pi$
$660$	0	0
$661$	1.03230e12	0.210330	0.105165	−	0.994455i	$-0.466463\pi$
$661$	1.03230e12	0.210330	0.105165	+	0.994455i	$0.466463\pi$
$662$	0	0
$663$	1.13924e13	2.28983
$664$	0	0
$665$	−1.01199e12	−0.200669
$666$	0	0
$667$	3.92137e12	0.767134
$668$	0	0
$669$	3.83309e12	0.739828
$670$	0	0
$671$	4.16948e12	0.794018
$672$	0	0
$673$	−8.13048e12	−1.52774	−0.763868	−	0.645373i	$-0.776702\pi$
$673$	−8.13048e12	−1.52774	−0.763868	+	0.645373i	$0.776702\pi$
$674$	0	0
$675$	1.31714e13	2.44210
$676$	0	0
$677$	−7.61157e12	−1.39260	−0.696298	−	0.717753i	$-0.745171\pi$
$677$	−7.61157e12	−1.39260	−0.696298	+	0.717753i	$0.745171\pi$
$678$	0	0
$679$	3.39514e12	0.612976
$680$	0	0
$681$	−5.55527e12	−0.989790
$682$	0	0
$683$	−7.40992e12	−1.30293	−0.651464	−	0.758680i	$-0.725845\pi$
$683$	−7.40992e12	−1.30293	−0.651464	+	0.758680i	$0.725845\pi$
$684$	0	0
$685$	6.43504e12	1.11672
$686$	0	0
$687$	−8.72358e12	−1.49413
$688$	0	0
$689$	1.51278e12	0.255734
$690$	0	0
$691$	−3.64791e11	−0.0608685	−0.0304343	−	0.999537i	$-0.509689\pi$
$691$	−3.64791e11	−0.0608685	−0.0304343	+	0.999537i	$0.509689\pi$
$692$	0	0
$693$	−5.70650e12	−0.939875
$694$	0	0
$695$	2.66380e12	0.433081
$696$	0	0
$697$	8.88765e12	1.42639
$698$	0	0
$699$	−1.02734e13	−1.62767
$700$	0	0
$701$	−2.33617e12	−0.365404	−0.182702	−	0.983168i	$-0.558484\pi$
$701$	−2.33617e12	−0.365404	−0.182702	+	0.983168i	$0.558484\pi$
$702$	0	0
$703$	8.34375e11	0.128843
$704$	0	0
$705$	1.71259e13	2.61097
$706$	0	0
$707$	8.64620e11	0.130148
$708$	0	0
$709$	−2.17052e12	−0.322594	−0.161297	−	0.986906i	$-0.551568\pi$
$709$	−2.17052e12	−0.322594	−0.161297	+	0.986906i	$0.551568\pi$
$710$	0	0
$711$	2.31147e13	3.39214
$712$	0	0
$713$	−1.39894e12	−0.202720
$714$	0	0
$715$	6.36567e12	0.910893
$716$	0	0
$717$	7.82574e12	1.10583
$718$	0	0
$719$	2.66823e10	0.00372342	0.00186171	−	0.999998i	$-0.499407\pi$
$719$	2.66823e10	0.00372342	0.00186171	+	0.999998i	$0.499407\pi$
$720$	0	0
$721$	−1.80325e12	−0.248512
$722$	0	0
$723$	8.99281e12	1.22398
$724$	0	0
$725$	3.99180e12	0.536596
$726$	0	0
$727$	−8.81984e12	−1.17100	−0.585499	−	0.810673i	$-0.699101\pi$
$727$	−8.81984e12	−1.17100	−0.585499	+	0.810673i	$0.699101\pi$
$728$	0	0
$729$	−7.54253e10	−0.00989107
$730$	0	0
$731$	7.04391e12	0.912400
$732$	0	0
$733$	−6.94718e12	−0.888874	−0.444437	−	0.895810i	$-0.646596\pi$
$733$	−6.94718e12	−0.888874	−0.444437	+	0.895810i	$0.646596\pi$
$734$	0	0
$735$	−1.58841e13	−2.00756
$736$	0	0
$737$	−4.30187e12	−0.537098
$738$	0	0
$739$	1.65991e12	0.204732	0.102366	−	0.994747i	$-0.467359\pi$
$739$	1.65991e12	0.204732	0.102366	+	0.994747i	$0.467359\pi$
$740$	0	0
$741$	3.09025e12	0.376541
$742$	0	0
$743$	−6.92810e12	−0.833997	−0.416998	−	0.908907i	$-0.636918\pi$
$743$	−6.92810e12	−0.833997	−0.416998	+	0.908907i	$0.636918\pi$
$744$	0	0
$745$	1.37242e13	1.63224
$746$	0	0
$747$	−1.05024e13	−1.23409
$748$	0	0
$749$	−1.95907e12	−0.227447
$750$	0	0
$751$	1.09479e13	1.25589	0.627943	−	0.778259i	$-0.283897\pi$
$751$	1.09479e13	1.25589	0.627943	+	0.778259i	$0.283897\pi$
$752$	0	0
$753$	−5.34819e12	−0.606219
$754$	0	0
$755$	8.46565e12	0.948198
$756$	0	0
$757$	−1.11681e13	−1.23608	−0.618039	−	0.786147i	$-0.712073\pi$
$757$	−1.11681e13	−1.23608	−0.618039	+	0.786147i	$0.712073\pi$
$758$	0	0
$759$	−2.24334e13	−2.45362
$760$	0	0
$761$	6.16538e12	0.666391	0.333195	−	0.942858i	$-0.391873\pi$
$761$	6.16538e12	0.666391	0.333195	+	0.942858i	$0.391873\pi$
$762$	0	0
$763$	1.71581e12	0.183277
$764$	0	0
$765$	−5.42429e13	−5.72620
$766$	0	0
$767$	−8.85629e12	−0.924002
$768$	0	0
$769$	−4.00721e12	−0.413213	−0.206606	−	0.978424i	$-0.566242\pi$
$769$	−4.00721e12	−0.413213	−0.206606	+	0.978424i	$0.566242\pi$
$770$	0	0
$771$	−2.61165e13	−2.66177
$772$	0	0
$773$	1.57080e13	1.58239	0.791193	−	0.611567i	$-0.209460\pi$
$773$	1.57080e13	1.58239	0.791193	+	0.611567i	$0.209460\pi$
$774$	0	0
$775$	−1.42407e12	−0.141799
$776$	0	0
$777$	−3.88781e12	−0.382658
$778$	0	0
$779$	2.41083e12	0.234557
$780$	0	0
$781$	1.32268e13	1.27211
$782$	0	0
$783$	1.16256e13	1.10532
$784$	0	0
$785$	1.67757e13	1.57676
$786$	0	0
$787$	4.11748e12	0.382600	0.191300	−	0.981532i	$-0.438730\pi$
$787$	4.11748e12	0.382600	0.191300	+	0.981532i	$0.438730\pi$
$788$	0	0
$789$	2.43463e13	2.23659
$790$	0	0
$791$	−1.20068e12	−0.109052
$792$	0	0
$793$	7.27873e12	0.653621
$794$	0	0
$795$	−1.04111e13	−0.924371
$796$	0	0
$797$	−6.62590e12	−0.581677	−0.290839	−	0.956772i	$-0.593934\pi$
$797$	−6.62590e12	−0.581677	−0.290839	+	0.956772i	$0.593934\pi$
$798$	0	0
$799$	−2.03886e13	−1.76981
$800$	0	0
$801$	−2.74448e13	−2.35566
$802$	0	0
$803$	2.89985e11	0.0246125
$804$	0	0
$805$	1.28248e13	1.07639
$806$	0	0
$807$	1.81320e13	1.50493
$808$	0	0
$809$	1.19920e12	0.0984290	0.0492145	−	0.998788i	$-0.484328\pi$
$809$	1.19920e12	0.0984290	0.0492145	+	0.998788i	$0.484328\pi$
$810$	0	0
$811$	2.43378e12	0.197555	0.0987775	−	0.995110i	$-0.468507\pi$
$811$	2.43378e12	0.197555	0.0987775	+	0.995110i	$0.468507\pi$
$812$	0	0
$813$	3.33166e13	2.67457
$814$	0	0
$815$	2.06191e13	1.63705
$816$	0	0
$817$	1.91070e12	0.150035
$818$	0	0
$819$	−9.96192e12	−0.773688
$820$	0	0
$821$	−5.54529e12	−0.425971	−0.212986	−	0.977055i	$-0.568319\pi$
$821$	−5.54529e12	−0.425971	−0.212986	+	0.977055i	$0.568319\pi$
$822$	0	0
$823$	1.48256e13	1.12646	0.563228	−	0.826302i	$-0.309559\pi$
$823$	1.48256e13	1.12646	0.563228	+	0.826302i	$0.309559\pi$
$824$	0	0
$825$	−2.28363e13	−1.71626
$826$	0	0
$827$	1.41723e13	1.05358	0.526788	−	0.849997i	$-0.323396\pi$
$827$	1.41723e13	1.05358	0.526788	+	0.849997i	$0.323396\pi$
$828$	0	0
$829$	9.67424e11	0.0711412	0.0355706	−	0.999367i	$-0.488675\pi$
$829$	9.67424e11	0.0711412	0.0355706	+	0.999367i	$0.488675\pi$
$830$	0	0
$831$	2.30857e13	1.67934
$832$	0	0
$833$	1.89101e13	1.36079
$834$	0	0
$835$	−3.61186e13	−2.57124
$836$	0	0
$837$	−4.14741e12	−0.292087
$838$	0	0
$839$	−1.63583e13	−1.13975	−0.569875	−	0.821732i	$-0.693008\pi$
$839$	−1.63583e13	−1.13975	−0.569875	+	0.821732i	$0.693008\pi$
$840$	0	0
$841$	−1.09838e13	−0.757131
$842$	0	0
$843$	1.89762e13	1.29415
$844$	0	0
$845$	−1.03067e13	−0.695450
$846$	0	0
$847$	−1.67961e12	−0.112133
$848$	0	0
$849$	−2.51812e13	−1.66338
$850$	0	0
$851$	−1.05739e13	−0.691119
$852$	0	0
$853$	−8.51975e12	−0.551006	−0.275503	−	0.961300i	$-0.588844\pi$
$853$	−8.51975e12	−0.551006	−0.275503	+	0.961300i	$0.588844\pi$
$854$	0	0
$855$	−1.47137e13	−0.941619
$856$	0	0
$857$	2.49683e12	0.158116	0.0790578	−	0.996870i	$-0.474809\pi$
$857$	2.49683e12	0.158116	0.0790578	+	0.996870i	$0.474809\pi$
$858$	0	0
$859$	2.99678e13	1.87796	0.938979	−	0.343974i	$-0.111773\pi$
$859$	2.99678e13	1.87796	0.938979	+	0.343974i	$0.111773\pi$
$860$	0	0
$861$	−1.12334e13	−0.696620
$862$	0	0
$863$	−9.11663e12	−0.559482	−0.279741	−	0.960076i	$-0.590249\pi$
$863$	−9.11663e12	−0.559482	−0.279741	+	0.960076i	$0.590249\pi$
$864$	0	0
$865$	−8.80221e12	−0.534588
$866$	0	0
$867$	6.33700e13	3.80889
$868$	0	0
$869$	−2.22252e13	−1.32208
$870$	0	0
$871$	−7.50984e12	−0.442129
$872$	0	0
$873$	4.93631e13	2.87633
$874$	0	0
$875$	1.06514e12	0.0614285
$876$	0	0
$877$	1.62554e13	0.927897	0.463948	−	0.885862i	$-0.346432\pi$
$877$	1.62554e13	0.927897	0.463948	+	0.885862i	$0.346432\pi$
$878$	0	0
$879$	5.52233e13	3.12013
$880$	0	0
$881$	−3.34287e13	−1.86951	−0.934755	−	0.355294i	$-0.884381\pi$
$881$	−3.34287e13	−1.86951	−0.934755	+	0.355294i	$0.884381\pi$
$882$	0	0
$883$	1.62698e13	0.900655	0.450328	−	0.892863i	$-0.351307\pi$
$883$	1.62698e13	0.900655	0.450328	+	0.892863i	$0.351307\pi$
$884$	0	0
$885$	6.09503e13	3.33988
$886$	0	0
$887$	−1.60289e13	−0.869454	−0.434727	−	0.900562i	$-0.643155\pi$
$887$	−1.60289e13	−0.869454	−0.434727	+	0.900562i	$0.643155\pi$
$888$	0	0
$889$	−1.03733e13	−0.557006
$890$	0	0
$891$	−2.95515e13	−1.57083
$892$	0	0
$893$	−5.53052e12	−0.291028
$894$	0	0
$895$	−2.19502e13	−1.14350
$896$	0	0
$897$	−3.91623e13	−2.01977
$898$	0	0
$899$	−1.25694e12	−0.0641795
$900$	0	0
$901$	1.23946e13	0.626570
$902$	0	0
$903$	−8.90301e12	−0.445597
$904$	0	0
$905$	4.33710e13	2.14922
$906$	0	0
$907$	−1.59734e13	−0.783726	−0.391863	−	0.920024i	$-0.628169\pi$
$907$	−1.59734e13	−0.783726	−0.391863	+	0.920024i	$0.628169\pi$
$908$	0	0
$909$	1.25710e13	0.610707
$910$	0	0
$911$	−1.21013e13	−0.582104	−0.291052	−	0.956707i	$-0.594005\pi$
$911$	−1.21013e13	−0.582104	−0.291052	+	0.956707i	$0.594005\pi$
$912$	0	0
$913$	1.00983e13	0.480981
$914$	0	0
$915$	−5.00932e13	−2.36257
$916$	0	0
$917$	1.81061e13	0.845596
$918$	0	0
$919$	−1.97502e13	−0.913379	−0.456689	−	0.889626i	$-0.650965\pi$
$919$	−1.97502e13	−0.913379	−0.456689	+	0.889626i	$0.650965\pi$
$920$	0	0
$921$	1.11188e13	0.509201
$922$	0	0
$923$	2.30902e13	1.04718
$924$	0	0
$925$	−1.07638e13	−0.483425
$926$	0	0
$927$	−2.62181e13	−1.16612
$928$	0	0
$929$	7.13965e12	0.314490	0.157245	−	0.987560i	$-0.449739\pi$
$929$	7.13965e12	0.314490	0.157245	+	0.987560i	$0.449739\pi$
$930$	0	0
$931$	5.12949e12	0.223769
$932$	0	0
$933$	−4.77838e13	−2.06449
$934$	0	0
$935$	5.21556e13	2.23177
$936$	0	0
$937$	−3.17448e13	−1.34538	−0.672689	−	0.739925i	$-0.734861\pi$
$937$	−3.17448e13	−1.34538	−0.672689	+	0.739925i	$0.734861\pi$
$938$	0	0
$939$	−5.94962e12	−0.249743
$940$	0	0
$941$	−4.59442e13	−1.91019	−0.955096	−	0.296296i	$-0.904248\pi$
$941$	−4.59442e13	−1.91019	−0.955096	+	0.296296i	$0.904248\pi$
$942$	0	0
$943$	−3.05521e13	−1.25817
$944$	0	0
$945$	3.80215e13	1.55091
$946$	0	0
$947$	−4.19613e12	−0.169541	−0.0847705	−	0.996401i	$-0.527016\pi$
$947$	−4.19613e12	−0.169541	−0.0847705	+	0.996401i	$0.527016\pi$
$948$	0	0
$949$	5.06232e11	0.0202606
$950$	0	0
$951$	4.85164e13	1.92343
$952$	0	0
$953$	9.66942e11	0.0379737	0.0189868	−	0.999820i	$-0.493956\pi$
$953$	9.66942e11	0.0379737	0.0189868	+	0.999820i	$0.493956\pi$
$954$	0	0
$955$	−5.60715e13	−2.18136
$956$	0	0
$957$	−2.01563e13	−0.776795
$958$	0	0
$959$	9.68296e12	0.369679
$960$	0	0
$961$	−2.59912e13	−0.983040
$962$	0	0
$963$	−2.84836e13	−1.06727
$964$	0	0
$965$	−3.51493e13	−1.30480
$966$	0	0
$967$	−9.39038e12	−0.345354	−0.172677	−	0.984979i	$-0.555242\pi$
$967$	−9.39038e12	−0.345354	−0.172677	+	0.984979i	$0.555242\pi$
$968$	0	0
$969$	2.53192e13	0.922558
$970$	0	0
$971$	−2.23874e13	−0.808196	−0.404098	−	0.914716i	$-0.632414\pi$
$971$	−2.23874e13	−0.808196	−0.404098	+	0.914716i	$0.632414\pi$
$972$	0	0
$973$	4.00828e12	0.143367
$974$	0	0
$975$	−3.98657e13	−1.41279
$976$	0	0
$977$	1.91366e13	0.671953	0.335977	−	0.941870i	$-0.390934\pi$
$977$	1.91366e13	0.671953	0.335977	+	0.941870i	$0.390934\pi$
$978$	0	0
$979$	2.63887e13	0.918112
$980$	0	0
$981$	2.49467e13	0.860009
$982$	0	0
$983$	3.12487e13	1.06744	0.533718	−	0.845663i	$-0.320795\pi$
$983$	3.12487e13	1.06744	0.533718	+	0.845663i	$0.320795\pi$
$984$	0	0
$985$	−4.54719e13	−1.53915
$986$	0	0
$987$	2.57697e13	0.864336
$988$	0	0
$989$	−2.42141e13	−0.804793
$990$	0	0
$991$	7.43111e12	0.244750	0.122375	−	0.992484i	$-0.460949\pi$
$991$	7.43111e12	0.244750	0.122375	+	0.992484i	$0.460949\pi$
$992$	0	0
$993$	1.07048e14	3.49386
$994$	0	0
$995$	6.20039e13	2.00546
$996$	0	0
$997$	2.49813e13	0.800730	0.400365	−	0.916356i	$-0.368883\pi$
$997$	2.49813e13	0.800730	0.400365	+	0.916356i	$0.368883\pi$
$998$	0	0
$999$	−3.13482e13	−0.995791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.10.a.e.1.2	yes	2
3.2	odd	2		288.10.a.e.1.1		2
4.3	odd	2		32.10.a.b.1.1	✓	2
8.3	odd	2		64.10.a.m.1.2		2
8.5	even	2		64.10.a.j.1.1		2
12.11	even	2		288.10.a.d.1.1		2
16.3	odd	4		256.10.b.o.129.1		4
16.5	even	4		256.10.b.l.129.1		4
16.11	odd	4		256.10.b.o.129.4		4
16.13	even	4		256.10.b.l.129.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.10.a.b.1.1	✓	2	4.3	odd	2
32.10.a.e.1.2	yes	2	1.1	even	1	trivial
64.10.a.j.1.1		2	8.5	even	2
64.10.a.m.1.2		2	8.3	odd	2
256.10.b.l.129.1		4	16.5	even	4
256.10.b.l.129.4		4	16.13	even	4
256.10.b.o.129.1		4	16.3	odd	4
256.10.b.o.129.4		4	16.11	odd	4
288.10.a.d.1.1		2	12.11	even	2
288.10.a.e.1.1		2	3.2	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:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	32.a (trivial)

\(f(q)\)	\(=\)	\(q+252.730 q^{3} +2019.84 q^{5} +3039.30 q^{7} +44189.5 q^{9} +O(q^{10})\)	\(q+252.730 q^{3} +2019.84 q^{5} +3039.30 q^{7} +44189.5 q^{9} -42489.0 q^{11} -74173.8 q^{13} +510474. q^{15} -607725. q^{17} -164849. q^{19} +768123. q^{21} +2.08911e6 q^{23} +2.12663e6 q^{25} +6.19353e6 q^{27} +1.87705e6 q^{29} -669635. q^{31} -1.07383e7 q^{33} +6.13890e6 q^{35} -5.06145e6 q^{37} -1.87459e7 q^{39} -1.46245e7 q^{41} -1.15906e7 q^{43} +8.92557e7 q^{45} +3.35490e7 q^{47} -3.11163e7 q^{49} -1.53590e8 q^{51} -2.03950e7 q^{53} -8.58211e7 q^{55} -4.16623e7 q^{57} +1.19399e8 q^{59} -9.81307e7 q^{61} +1.34305e8 q^{63} -1.49819e8 q^{65} +1.01247e8 q^{67} +5.27981e8 q^{69} -3.11299e8 q^{71} -6.82495e6 q^{73} +5.37464e8 q^{75} -1.29137e8 q^{77} +5.23080e8 q^{79} +6.95509e8 q^{81} -2.37668e8 q^{83} -1.22751e9 q^{85} +4.74388e8 q^{87} -6.21070e8 q^{89} -2.25436e8 q^{91} -1.69237e8 q^{93} -3.32969e8 q^{95} +1.11708e9 q^{97} -1.87757e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 176 q^{3} + 1404 q^{5} + 2784 q^{7} + 30394 q^{9}+O(q^{10}) \) 2 * q + 176 * q^3 + 1404 * q^5 + 2784 * q^7 + 30394 * q^9	\( 2 q + 176 q^{3} + 1404 q^{5} + 2784 q^{7} + 30394 q^{9} - 6896 q^{11} - 45556 q^{13} + 557728 q^{15} - 472188 q^{17} + 780912 q^{19} + 787712 q^{21} + 3883680 q^{23} + 552766 q^{25} + 8762336 q^{27} + 3909612 q^{29} + 4857216 q^{31} - 13469312 q^{33} + 6296128 q^{35} + 8608892 q^{37} - 20941792 q^{39} - 32359020 q^{41} - 32230512 q^{43} + 97751564 q^{45} - 20943040 q^{47} - 71404686 q^{49} - 163990176 q^{51} + 51108252 q^{53} - 107740704 q^{55} - 114230656 q^{57} + 63098960 q^{59} + 25048812 q^{61} + 137827168 q^{63} - 167443224 q^{65} + 224200368 q^{67} + 390283008 q^{69} + 6805472 q^{71} - 23217420 q^{73} + 658226512 q^{75} - 138223872 q^{77} + 726598848 q^{79} + 769941010 q^{81} + 27482736 q^{83} - 1310977032 q^{85} + 318429472 q^{87} - 71429100 q^{89} - 232742592 q^{91} - 593312768 q^{93} - 915407072 q^{95} + 1850743460 q^{97} - 2368592176 q^{99}+O(q^{100}) \) 2 * q + 176 * q^3 + 1404 * q^5 + 2784 * q^7 + 30394 * q^9 - 6896 * q^11 - 45556 * q^13 + 557728 * q^15 - 472188 * q^17 + 780912 * q^19 + 787712 * q^21 + 3883680 * q^23 + 552766 * q^25 + 8762336 * q^27 + 3909612 * q^29 + 4857216 * q^31 - 13469312 * q^33 + 6296128 * q^35 + 8608892 * q^37 - 20941792 * q^39 - 32359020 * q^41 - 32230512 * q^43 + 97751564 * q^45 - 20943040 * q^47 - 71404686 * q^49 - 163990176 * q^51 + 51108252 * q^53 - 107740704 * q^55 - 114230656 * q^57 + 63098960 * q^59 + 25048812 * q^61 + 137827168 * q^63 - 167443224 * q^65 + 224200368 * q^67 + 390283008 * q^69 + 6805472 * q^71 - 23217420 * q^73 + 658226512 * q^75 - 138223872 * q^77 + 726598848 * q^79 + 769941010 * q^81 + 27482736 * q^83 - 1310977032 * q^85 + 318429472 * q^87 - 71429100 * q^89 - 232742592 * q^91 - 593312768 * q^93 - 915407072 * q^95 + 1850743460 * q^97 - 2368592176 * q^99

Introduction

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