LMFDB - Embedded newform 16.32.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,32,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 32, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 32);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 32 $
Character orbit:	$[\chi]$	$=$	16.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:	$97.4034125104$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4573872 $ x^2 - x - 4573872
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2139.16$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.08552e7 q^{3} -8.25073e10 q^{5} -1.14368e13 q^{7} +3.34371e14 q^{9} +O(q^{10})$	$q-3.08552e7 q^{3} -8.25073e10 q^{5} -1.14368e13 q^{7} +3.34371e14 q^{9} +2.40262e15 q^{11} -2.01886e17 q^{13} +2.54578e18 q^{15} +1.09264e19 q^{17} +1.42851e19 q^{19} +3.52885e20 q^{21} +3.85062e18 q^{23} +2.15084e21 q^{25} +8.74135e21 q^{27} +7.63087e22 q^{29} -1.86701e23 q^{31} -7.41334e22 q^{33} +9.43620e23 q^{35} +1.23709e24 q^{37} +6.22922e24 q^{39} +1.38199e25 q^{41} +2.67871e25 q^{43} -2.75881e25 q^{45} -7.40922e25 q^{47} -2.69748e25 q^{49} -3.37138e26 q^{51} +3.56092e25 q^{53} -1.98234e26 q^{55} -4.40770e26 q^{57} +2.36122e27 q^{59} -5.44842e27 q^{61} -3.82414e27 q^{63} +1.66570e28 q^{65} +9.41082e27 q^{67} -1.18812e26 q^{69} +2.10678e28 q^{71} +3.92731e28 q^{73} -6.63646e28 q^{75} -2.74783e28 q^{77} -1.79850e29 q^{79} -4.76249e29 q^{81} +4.54329e29 q^{83} -9.01511e29 q^{85} -2.35452e30 q^{87} +2.60812e29 q^{89} +2.30893e30 q^{91} +5.76072e30 q^{93} -1.17863e30 q^{95} -5.38067e30 q^{97} +8.03368e29 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 17363160 q^{3} - 19391218020 q^{5} - 30257527577200 q^{7} - 101266456303926 q^{9}+O(q^{10}) $ 2 * q - 17363160 * q^3 - 19391218020 * q^5 - 30257527577200 * q^7 - 101266456303926 * q^9	$ 2 q - 17363160 q^{3} - 19391218020 q^{5} - 30257527577200 q^{7} - 101266456303926 q^{9} + 77\!\cdots\!76 q^{11}+ \cdots - 15\!\cdots\!88 q^{99}+O(q^{100}) $ 2 * q - 17363160 * q^3 - 19391218020 * q^5 - 30257527577200 * q^7 - 101266456303926 * q^9 + 7782353745118776 * q^11 + 74708953050260620 * q^13 + 3397345822674581040 * q^15 + 17224607828987089380 * q^17 + 12370563328022164040 * q^19 + 98954957416071161664 * q^21 - 1897344841989911219280 * q^23 + 1477861250884539239150 * q^25 - 5469986602319662295280 * q^27 + 128576144217217055807340 * q^29 - 125733527517961838793664 * q^31 - 1549761835334435045280 * q^33 - 244269703121729304206880 * q^35 - 833815207054016911025060 * q^37 + 9961054164411363568284912 * q^39 + 8724924335662925840671284 * q^41 + 18397105293779438708372600 * q^43 - 55083793466681039024257140 * q^45 - 95450963964856190793148320 * q^47 + 169469294372320202530407186 * q^49 - 252162404739970320085380144 * q^51 + 194822508473721098983660860 * q^53 + 141313671368671671047892240 * q^55 - 466601717043760934967596640 * q^57 + 198723263547765513990746520 * q^59 - 12056218201113004361157656276 * q^61 + 4374875036601718967620305360 * q^63 + 34114584794425827928886318760 * q^65 + 9688140802872256994032247720 * q^67 - 25769856286394326078494576192 * q^69 - 55784576625034657512878287344 * q^71 + 62234095932086098750552955380 * q^73 - 75444402265645086720647965800 * q^75 - 128728748793695217194494555200 * q^77 + 119164191093299704964212710560 * q^79 - 398906922646891269649062634158 * q^81 + 264863373681569145625357596360 * q^83 - 503995105657776450242931421320 * q^85 - 1649324800814841914442821796240 * q^87 - 2154414614246670291602721895980 * q^89 - 2896781368068903069956049509024 * q^91 + 6583298380663842229690665227520 * q^93 - 1299465173198290044974084355600 * q^95 - 9060767994874032615529957205180 * q^97 - 1540246307744094989457731085288 * 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
$3$	−3.08552e7	−1.24151	−0.620754	−	0.784006i	$-0.713173\pi$
$3$	−3.08552e7	−1.24151	−0.620754	+	0.784006i	$0.713173\pi$
$4$	0	0
$5$	−8.25073e10	−1.20909	−0.604543	−	0.796573i	$-0.706644\pi$
$5$	−8.25073e10	−1.20909	−0.604543	+	0.796573i	$0.706644\pi$
$6$	0	0
$7$	−1.14368e13	−0.910511	−0.455256	−	0.890361i	$-0.650452\pi$
$7$	−1.14368e13	−0.910511	−0.455256	+	0.890361i	$0.650452\pi$
$8$	0	0
$9$	3.34371e14	0.541340
$10$	0	0
$11$	2.40262e15	0.173420	0.0867099	−	0.996234i	$-0.472365\pi$
$11$	2.40262e15	0.173420	0.0867099	+	0.996234i	$0.472365\pi$
$12$	0	0
$13$	−2.01886e17	−1.09391	−0.546957	−	0.837161i	$-0.684214\pi$
$13$	−2.01886e17	−1.09391	−0.546957	+	0.837161i	$0.684214\pi$
$14$	0	0
$15$	2.54578e18	1.50109
$16$	0	0
$17$	1.09264e19	0.925807	0.462903	−	0.886409i	$-0.346808\pi$
$17$	1.09264e19	0.925807	0.462903	+	0.886409i	$0.346808\pi$
$18$	0	0
$19$	1.42851e19	0.215875	0.107938	−	0.994158i	$-0.465575\pi$
$19$	1.42851e19	0.215875	0.107938	+	0.994158i	$0.465575\pi$
$20$	0	0
$21$	3.52885e20	1.13041
$22$	0	0
$23$	3.85062e18	0.00301127	0.00150564	−	0.999999i	$-0.499521\pi$
$23$	3.85062e18	0.00301127	0.00150564	+	0.999999i	$0.499521\pi$
$24$	0	0
$25$	2.15084e21	0.461889
$26$	0	0
$27$	8.74135e21	0.569430
$28$	0	0
$29$	7.63087e22	1.64212	0.821060	−	0.570842i	$-0.193383\pi$
$29$	7.63087e22	1.64212	0.821060	+	0.570842i	$0.193383\pi$
$30$	0	0
$31$	−1.86701e23	−1.42903	−0.714515	−	0.699620i	$-0.753353\pi$
$31$	−1.86701e23	−1.42903	−0.714515	+	0.699620i	$0.753353\pi$
$32$	0	0
$33$	−7.41334e22	−0.215302
$34$	0	0
$35$	9.43620e23	1.10089
$36$	0	0
$37$	1.23709e24	0.609924	0.304962	−	0.952364i	$-0.401356\pi$
$37$	1.23709e24	0.609924	0.304962	+	0.952364i	$0.401356\pi$
$38$	0	0
$39$	6.22922e24	1.35810
$40$	0	0
$41$	1.38199e25	1.38789	0.693945	−	0.720028i	$-0.255871\pi$
$41$	1.38199e25	1.38789	0.693945	+	0.720028i	$0.255871\pi$
$42$	0	0
$43$	2.67871e25	1.28578	0.642888	−	0.765960i	$-0.277736\pi$
$43$	2.67871e25	1.28578	0.642888	+	0.765960i	$0.277736\pi$
$44$	0	0
$45$	−2.75881e25	−0.654526
$46$	0	0
$47$	−7.40922e25	−0.895892	−0.447946	−	0.894061i	$-0.647844\pi$
$47$	−7.40922e25	−0.895892	−0.447946	+	0.894061i	$0.647844\pi$
$48$	0	0
$49$	−2.69748e25	−0.170970
$50$	0	0
$51$	−3.37138e26	−1.14940
$52$	0	0
$53$	3.56092e25	0.0668784	0.0334392	−	0.999441i	$-0.489354\pi$
$53$	3.56092e25	0.0668784	0.0334392	+	0.999441i	$0.489354\pi$
$54$	0	0
$55$	−1.98234e26	−0.209680
$56$	0	0
$57$	−4.40770e26	−0.268011
$58$	0	0
$59$	2.36122e27	0.841260	0.420630	−	0.907232i	$-0.361809\pi$
$59$	2.36122e27	0.841260	0.420630	+	0.907232i	$0.361809\pi$
$60$	0	0
$61$	−5.44842e27	−1.15786	−0.578932	−	0.815376i	$-0.696530\pi$
$61$	−5.44842e27	−1.15786	−0.578932	+	0.815376i	$0.696530\pi$
$62$	0	0
$63$	−3.82414e27	−0.492896
$64$	0	0
$65$	1.66570e28	1.32264
$66$	0	0
$67$	9.41082e27	0.467163	0.233581	−	0.972337i	$-0.424956\pi$
$67$	9.41082e27	0.467163	0.233581	+	0.972337i	$0.424956\pi$
$68$	0	0
$69$	−1.18812e26	−0.00373851
$70$	0	0
$71$	2.10678e28	0.425712	0.212856	−	0.977084i	$-0.431724\pi$
$71$	2.10678e28	0.425712	0.212856	+	0.977084i	$0.431724\pi$
$72$	0	0
$73$	3.92731e28	0.515929	0.257965	−	0.966154i	$-0.416948\pi$
$73$	3.92731e28	0.515929	0.257965	+	0.966154i	$0.416948\pi$
$74$	0	0
$75$	−6.63646e28	−0.573438
$76$	0	0
$77$	−2.74783e28	−0.157901
$78$	0	0
$79$	−1.79850e29	−0.694529	−0.347264	−	0.937767i	$-0.612889\pi$
$79$	−1.79850e29	−0.694529	−0.347264	+	0.937767i	$0.612889\pi$
$80$	0	0
$81$	−4.76249e29	−1.24829
$82$	0	0
$83$	4.54329e29	0.815944	0.407972	−	0.912995i	$-0.366236\pi$
$83$	4.54329e29	0.815944	0.407972	+	0.912995i	$0.366236\pi$
$84$	0	0
$85$	−9.01511e29	−1.11938
$86$	0	0
$87$	−2.35452e30	−2.03870
$88$	0	0
$89$	2.60812e29	0.158775	0.0793875	−	0.996844i	$-0.474704\pi$
$89$	2.60812e29	0.158775	0.0793875	+	0.996844i	$0.474704\pi$
$90$	0	0
$91$	2.30893e30	0.996021
$92$	0	0
$93$	5.76072e30	1.77415
$94$	0	0
$95$	−1.17863e30	−0.261012
$96$	0	0
$97$	−5.38067e30	−0.862729	−0.431365	−	0.902178i	$-0.641968\pi$
$97$	−5.38067e30	−0.862729	−0.431365	+	0.902178i	$0.641968\pi$
$98$	0	0
$99$	8.03368e29	0.0938791
$100$	0	0
$101$	−1.28863e31	−1.10445	−0.552225	−	0.833695i	$-0.686221\pi$
$101$	−1.28863e31	−1.10445	−0.552225	+	0.833695i	$0.686221\pi$
$102$	0	0
$103$	−1.91208e31	−1.20928	−0.604642	−	0.796497i	$-0.706684\pi$
$103$	−1.91208e31	−1.20928	−0.604642	+	0.796497i	$0.706684\pi$
$104$	0	0
$105$	−2.91156e31	−1.36676
$106$	0	0
$107$	−3.15334e31	−1.10490	−0.552449	−	0.833547i	$-0.686307\pi$
$107$	−3.15334e31	−1.10490	−0.552449	+	0.833547i	$0.686307\pi$
$108$	0	0
$109$	−3.61029e31	−0.949360	−0.474680	−	0.880158i	$-0.657436\pi$
$109$	−3.61029e31	−0.949360	−0.474680	+	0.880158i	$0.657436\pi$
$110$	0	0
$111$	−3.81708e31	−0.757226
$112$	0	0
$113$	1.52549e31	0.229454	0.114727	−	0.993397i	$-0.463401\pi$
$113$	1.52549e31	0.229454	0.114727	+	0.993397i	$0.463401\pi$
$114$	0	0
$115$	−3.17705e29	−0.00364088
$116$	0	0
$117$	−6.75047e31	−0.592179
$118$	0	0
$119$	−1.24964e32	−0.842957
$120$	0	0
$121$	−1.86171e32	−0.969926
$122$	0	0
$123$	−4.26416e32	−1.72308
$124$	0	0
$125$	2.06745e32	0.650623
$126$	0	0
$127$	5.69060e32	1.40023	0.700116	−	0.714029i	$-0.253131\pi$
$127$	5.69060e32	1.40023	0.700116	+	0.714029i	$0.253131\pi$
$128$	0	0
$129$	−8.26523e32	−1.59630
$130$	0	0
$131$	−1.80373e31	−0.0274452	−0.0137226	−	0.999906i	$-0.504368\pi$
$131$	−1.80373e31	−0.0274452	−0.0137226	+	0.999906i	$0.504368\pi$
$132$	0	0
$133$	−1.63376e32	−0.196557
$134$	0	0
$135$	−7.21225e32	−0.688490
$136$	0	0
$137$	−1.53340e32	−0.116543	−0.0582716	−	0.998301i	$-0.518559\pi$
$137$	−1.53340e32	−0.116543	−0.0582716	+	0.998301i	$0.518559\pi$
$138$	0	0
$139$	−1.88034e33	−1.14158	−0.570791	−	0.821095i	$-0.693364\pi$
$139$	−1.88034e33	−1.14158	−0.570791	+	0.821095i	$0.693364\pi$
$140$	0	0
$141$	2.28613e33	1.11226
$142$	0	0
$143$	−4.85055e32	−0.189706
$144$	0	0
$145$	−6.29602e33	−1.98546
$146$	0	0
$147$	8.32313e32	0.212260
$148$	0	0
$149$	6.64703e33	1.37480	0.687401	−	0.726278i	$-0.258751\pi$
$149$	6.64703e33	1.37480	0.687401	+	0.726278i	$0.258751\pi$
$150$	0	0
$151$	6.58887e33	1.10833	0.554164	−	0.832408i	$-0.313038\pi$
$151$	6.58887e33	1.10833	0.554164	+	0.832408i	$0.313038\pi$
$152$	0	0
$153$	3.65349e33	0.501176
$154$	0	0
$155$	1.54042e34	1.72782
$156$	0	0
$157$	1.60319e34	1.47414	0.737072	−	0.675814i	$-0.236208\pi$
$157$	1.60319e34	1.47414	0.737072	+	0.675814i	$0.236208\pi$
$158$	0	0
$159$	−1.09873e33	−0.0830300
$160$	0	0
$161$	−4.40389e31	−0.00274179
$162$	0	0
$163$	1.63484e34	0.840555	0.420277	−	0.907396i	$-0.361933\pi$
$163$	1.63484e34	0.840555	0.420277	+	0.907396i	$0.361933\pi$
$164$	0	0
$165$	6.11655e33	0.260319
$166$	0	0
$167$	−1.68065e34	−0.593433	−0.296717	−	0.954966i	$-0.595892\pi$
$167$	−1.68065e34	−0.593433	−0.296717	+	0.954966i	$0.595892\pi$
$168$	0	0
$169$	6.69784e33	0.196649
$170$	0	0
$171$	4.77653e33	0.116862
$172$	0	0
$173$	1.67734e34	0.342695	0.171347	−	0.985211i	$-0.445188\pi$
$173$	1.67734e34	0.342695	0.171347	+	0.985211i	$0.445188\pi$
$174$	0	0
$175$	−2.45987e34	−0.420555
$176$	0	0
$177$	−7.28560e34	−1.04443
$178$	0	0
$179$	−5.67856e33	−0.0683937	−0.0341968	−	0.999415i	$-0.510887\pi$
$179$	−5.67856e33	−0.0683937	−0.0341968	+	0.999415i	$0.510887\pi$
$180$	0	0
$181$	1.47595e35	1.49642	0.748210	−	0.663462i	$-0.230914\pi$
$181$	1.47595e35	1.49642	0.748210	+	0.663462i	$0.230914\pi$
$182$	0	0
$183$	1.68112e35	1.43750
$184$	0	0
$185$	−1.02069e35	−0.737451
$186$	0	0
$187$	2.62521e34	0.160553
$188$	0	0
$189$	−9.99731e34	−0.518472
$190$	0	0
$191$	4.38203e34	0.193045	0.0965224	−	0.995331i	$-0.469228\pi$
$191$	4.38203e34	0.193045	0.0965224	+	0.995331i	$0.469228\pi$
$192$	0	0
$193$	2.63656e35	0.988323	0.494161	−	0.869370i	$-0.335475\pi$
$193$	2.63656e35	0.988323	0.494161	+	0.869370i	$0.335475\pi$
$194$	0	0
$195$	−5.13956e35	−1.64206
$196$	0	0
$197$	−2.72680e35	−0.743749	−0.371875	−	0.928283i	$-0.621285\pi$
$197$	−2.72680e35	−0.743749	−0.371875	+	0.928283i	$0.621285\pi$
$198$	0	0
$199$	−3.17714e35	−0.740991	−0.370496	−	0.928834i	$-0.620812\pi$
$199$	−3.17714e35	−0.740991	−0.370496	+	0.928834i	$0.620812\pi$
$200$	0	0
$201$	−2.90373e35	−0.579986
$202$	0	0
$203$	−8.72728e35	−1.49517
$204$	0	0
$205$	−1.14024e36	−1.67808
$206$	0	0
$207$	1.28754e33	0.00163012
$208$	0	0
$209$	3.43217e34	0.0374371
$210$	0	0
$211$	2.10102e36	1.97721	0.988604	−	0.150541i	$-0.0481017\pi$
$211$	2.10102e36	1.97721	0.988604	+	0.150541i	$0.0481017\pi$
$212$	0	0
$213$	−6.50053e35	−0.528524
$214$	0	0
$215$	−2.21013e36	−1.55461
$216$	0	0
$217$	2.13527e36	1.30115
$218$	0	0
$219$	−1.21178e36	−0.640530
$220$	0	0
$221$	−2.20589e36	−1.01275
$222$	0	0
$223$	2.62490e36	1.04806	0.524032	−	0.851699i	$-0.324427\pi$
$223$	2.62490e36	1.04806	0.524032	+	0.851699i	$0.324427\pi$
$224$	0	0
$225$	7.19178e35	0.250039
$226$	0	0
$227$	−1.16088e35	−0.0351874	−0.0175937	−	0.999845i	$-0.505601\pi$
$227$	−1.16088e35	−0.0351874	−0.0175937	+	0.999845i	$0.505601\pi$
$228$	0	0
$229$	1.39980e36	0.370353	0.185177	−	0.982705i	$-0.440714\pi$
$229$	1.39980e36	0.370353	0.185177	+	0.982705i	$0.440714\pi$
$230$	0	0
$231$	8.47850e35	0.196035
$232$	0	0
$233$	7.15120e36	1.44665	0.723323	−	0.690510i	$-0.242614\pi$
$233$	7.15120e36	1.44665	0.723323	+	0.690510i	$0.242614\pi$
$234$	0	0
$235$	6.11315e36	1.08321
$236$	0	0
$237$	5.54931e36	0.862262
$238$	0	0
$239$	−2.96558e36	−0.404522	−0.202261	−	0.979332i	$-0.564829\pi$
$239$	−2.96558e36	−0.404522	−0.202261	+	0.979332i	$0.564829\pi$
$240$	0	0
$241$	−7.20536e36	−0.863757	−0.431879	−	0.901932i	$-0.642149\pi$
$241$	−7.20536e36	−0.863757	−0.431879	+	0.901932i	$0.642149\pi$
$242$	0	0
$243$	9.29545e36	0.980332
$244$	0	0
$245$	2.22562e36	0.206717
$246$	0	0
$247$	−2.88396e36	−0.236149
$248$	0	0
$249$	−1.40184e37	−1.01300
$250$	0	0
$251$	−9.47842e36	−0.605053	−0.302526	−	0.953141i	$-0.597830\pi$
$251$	−9.47842e36	−0.605053	−0.302526	+	0.953141i	$0.597830\pi$
$252$	0	0
$253$	9.25159e33	0.000522214 0
$254$	0	0
$255$	2.78163e37	1.38972
$256$	0	0
$257$	3.15504e37	1.39650	0.698252	−	0.715852i	$-0.253962\pi$
$257$	3.15504e37	1.39650	0.698252	+	0.715852i	$0.253962\pi$
$258$	0	0
$259$	−1.41484e37	−0.555343
$260$	0	0
$261$	2.55154e37	0.888945
$262$	0	0
$263$	1.25731e37	0.389159	0.194580	−	0.980887i	$-0.437666\pi$
$263$	1.25731e37	0.389159	0.194580	+	0.980887i	$0.437666\pi$
$264$	0	0
$265$	−2.93802e36	−0.0808618
$266$	0	0
$267$	−8.04741e36	−0.197120
$268$	0	0
$269$	3.34018e37	0.728801	0.364401	−	0.931242i	$-0.381274\pi$
$269$	3.34018e37	0.728801	0.364401	+	0.931242i	$0.381274\pi$
$270$	0	0
$271$	4.00526e36	0.0779125	0.0389562	−	0.999241i	$-0.487597\pi$
$271$	4.00526e36	0.0779125	0.0389562	+	0.999241i	$0.487597\pi$
$272$	0	0
$273$	−7.12424e37	−1.23657
$274$	0	0
$275$	5.16765e36	0.0801007
$276$	0	0
$277$	−7.94796e37	−1.10108	−0.550541	−	0.834808i	$-0.685578\pi$
$277$	−7.94796e37	−1.10108	−0.550541	+	0.834808i	$0.685578\pi$
$278$	0	0
$279$	−6.24276e37	−0.773591
$280$	0	0
$281$	−1.69314e38	−1.87822	−0.939108	−	0.343621i	$-0.888346\pi$
$281$	−1.69314e38	−1.87822	−0.939108	+	0.343621i	$0.888346\pi$
$282$	0	0
$283$	−1.18735e38	−1.18002	−0.590008	−	0.807397i	$-0.700875\pi$
$283$	−1.18735e38	−1.18002	−0.590008	+	0.807397i	$0.700875\pi$
$284$	0	0
$285$	3.63668e37	0.324048
$286$	0	0
$287$	−1.58056e38	−1.26369
$288$	0	0
$289$	−1.99019e37	−0.142882
$290$	0	0
$291$	1.66022e38	1.07108
$292$	0	0
$293$	9.32191e37	0.540825	0.270413	−	0.962745i	$-0.412840\pi$
$293$	9.32191e37	0.540825	0.270413	+	0.962745i	$0.412840\pi$
$294$	0	0
$295$	−1.94818e38	−1.01716
$296$	0	0
$297$	2.10022e37	0.0987504
$298$	0	0
$299$	−7.77386e35	−0.00329407
$300$	0	0
$301$	−3.06359e38	−1.17071
$302$	0	0
$303$	3.97609e38	1.37118
$304$	0	0
$305$	4.49534e38	1.39996
$306$	0	0
$307$	2.10829e38	0.593314	0.296657	−	0.954984i	$-0.404128\pi$
$307$	2.10829e38	0.593314	0.296657	+	0.954984i	$0.404128\pi$
$308$	0	0
$309$	5.89976e38	1.50134
$310$	0	0
$311$	−4.24765e38	−0.978053	−0.489027	−	0.872269i	$-0.662648\pi$
$311$	−4.24765e38	−0.978053	−0.489027	+	0.872269i	$0.662648\pi$
$312$	0	0
$313$	−3.07637e38	−0.641360	−0.320680	−	0.947188i	$-0.603911\pi$
$313$	−3.07637e38	−0.641360	−0.320680	+	0.947188i	$0.603911\pi$
$314$	0	0
$315$	3.15519e38	0.595954
$316$	0	0
$317$	−9.76522e38	−1.67210	−0.836052	−	0.548650i	$-0.815142\pi$
$317$	−9.76522e38	−1.67210	−0.836052	+	0.548650i	$0.815142\pi$
$318$	0	0
$319$	1.83341e38	0.284776
$320$	0	0
$321$	9.72970e38	1.37174
$322$	0	0
$323$	1.56085e38	0.199859
$324$	0	0
$325$	−4.34223e38	−0.505267
$326$	0	0
$327$	1.11396e39	1.17864
$328$	0	0
$329$	8.47379e38	0.815720
$330$	0	0
$331$	−8.30576e38	−0.727856	−0.363928	−	0.931427i	$-0.618565\pi$
$331$	−8.30576e38	−0.727856	−0.363928	+	0.931427i	$0.618565\pi$
$332$	0	0
$333$	4.13649e38	0.330176
$334$	0	0
$335$	−7.76461e38	−0.564840
$336$	0	0
$337$	1.84495e39	1.22383	0.611916	−	0.790923i	$-0.290399\pi$
$337$	1.84495e39	1.22383	0.611916	+	0.790923i	$0.290399\pi$
$338$	0	0
$339$	−4.70695e38	−0.284868
$340$	0	0
$341$	−4.48573e38	−0.247822
$342$	0	0
$343$	2.11295e39	1.06618
$344$	0	0
$345$	9.80284e36	0.00452018
$346$	0	0
$347$	3.59852e39	1.51711	0.758555	−	0.651609i	$-0.225906\pi$
$347$	3.59852e39	1.51711	0.758555	+	0.651609i	$0.225906\pi$
$348$	0	0
$349$	4.09325e38	0.157861	0.0789303	−	0.996880i	$-0.474850\pi$
$349$	4.09325e38	0.157861	0.0789303	+	0.996880i	$0.474850\pi$
$350$	0	0
$351$	−1.76475e39	−0.622907
$352$	0	0
$353$	−5.32510e39	−1.72116	−0.860578	−	0.509319i	$-0.829897\pi$
$353$	−5.32510e39	−1.72116	−0.860578	+	0.509319i	$0.829897\pi$
$354$	0	0
$355$	−1.73825e39	−0.514722
$356$	0	0
$357$	3.85578e39	1.04654
$358$	0	0
$359$	−4.10824e39	−1.02257	−0.511283	−	0.859412i	$-0.670830\pi$
$359$	−4.10824e39	−1.02257	−0.511283	+	0.859412i	$0.670830\pi$
$360$	0	0
$361$	−4.17480e39	−0.953398
$362$	0	0
$363$	5.74434e39	1.20417
$364$	0	0
$365$	−3.24031e39	−0.623803
$366$	0	0
$367$	−9.50139e39	−1.68060	−0.840298	−	0.542125i	$-0.817620\pi$
$367$	−9.50139e39	−1.68060	−0.840298	+	0.542125i	$0.817620\pi$
$368$	0	0
$369$	4.62098e39	0.751320
$370$	0	0
$371$	−4.07255e38	−0.0608935
$372$	0	0
$373$	5.31105e39	0.730624	0.365312	−	0.930885i	$-0.380962\pi$
$373$	5.31105e39	0.730624	0.365312	+	0.930885i	$0.380962\pi$
$374$	0	0
$375$	−6.37915e39	−0.807753
$376$	0	0
$377$	−1.54056e40	−1.79634
$378$	0	0
$379$	−6.56433e39	−0.705152	−0.352576	−	0.935783i	$-0.614694\pi$
$379$	−6.56433e39	−0.705152	−0.352576	+	0.935783i	$0.614694\pi$
$380$	0	0
$381$	−1.75585e40	−1.73840
$382$	0	0
$383$	−4.90622e39	−0.447884	−0.223942	−	0.974602i	$-0.571893\pi$
$383$	−4.90622e39	−0.447884	−0.223942	+	0.974602i	$0.571893\pi$
$384$	0	0
$385$	2.26716e39	0.190916
$386$	0	0
$387$	8.95685e39	0.696041
$388$	0	0
$389$	1.47786e40	1.06026	0.530132	−	0.847915i	$-0.322142\pi$
$389$	1.47786e40	1.06026	0.530132	+	0.847915i	$0.322142\pi$
$390$	0	0
$391$	4.20736e37	0.00278785
$392$	0	0
$393$	5.56545e38	0.0340734
$394$	0	0
$395$	1.48389e40	0.839745
$396$	0	0
$397$	9.79733e39	0.512690	0.256345	−	0.966585i	$-0.417482\pi$
$397$	9.79733e39	0.512690	0.256345	+	0.966585i	$0.417482\pi$
$398$	0	0
$399$	5.04101e39	0.244027
$400$	0	0
$401$	1.11974e40	0.501624	0.250812	−	0.968036i	$-0.419302\pi$
$401$	1.11974e40	0.501624	0.250812	+	0.968036i	$0.419302\pi$
$402$	0	0
$403$	3.76923e40	1.56324
$404$	0	0
$405$	3.92940e40	1.50929
$406$	0	0
$407$	2.97227e39	0.105773
$408$	0	0
$409$	−2.34914e40	−0.774815	−0.387408	−	0.921909i	$-0.626629\pi$
$409$	−2.34914e40	−0.774815	−0.387408	+	0.921909i	$0.626629\pi$
$410$	0	0
$411$	4.73134e39	0.144689
$412$	0	0
$413$	−2.70048e40	−0.765977
$414$	0	0
$415$	−3.74855e40	−0.986546
$416$	0	0
$417$	5.80184e40	1.41728
$418$	0	0
$419$	−4.42578e40	−1.00386	−0.501929	−	0.864909i	$-0.667376\pi$
$419$	−4.42578e40	−1.00386	−0.501929	+	0.864909i	$0.667376\pi$
$420$	0	0
$421$	−4.58467e40	−0.965907	−0.482954	−	0.875646i	$-0.660436\pi$
$421$	−4.58467e40	−0.965907	−0.482954	+	0.875646i	$0.660436\pi$
$422$	0	0
$423$	−2.47743e40	−0.484982
$424$	0	0
$425$	2.35010e40	0.427620
$426$	0	0
$427$	6.23125e40	1.05425
$428$	0	0
$429$	1.49665e40	0.235522
$430$	0	0
$431$	1.63894e40	0.239975	0.119987	−	0.992775i	$-0.461715\pi$
$431$	1.63894e40	0.239975	0.119987	+	0.992775i	$0.461715\pi$
$432$	0	0
$433$	7.21840e39	0.0983737	0.0491868	−	0.998790i	$-0.484337\pi$
$433$	7.21840e39	0.0983737	0.0491868	+	0.998790i	$0.484337\pi$
$434$	0	0
$435$	1.94265e41	2.46497
$436$	0	0
$437$	5.50066e37	0.000650059 0
$438$	0	0
$439$	−1.72040e40	−0.189422	−0.0947109	−	0.995505i	$-0.530193\pi$
$439$	−1.72040e40	−0.189422	−0.0947109	+	0.995505i	$0.530193\pi$
$440$	0	0
$441$	−9.01960e39	−0.0925527
$442$	0	0
$443$	9.22567e39	0.0882552	0.0441276	−	0.999026i	$-0.485949\pi$
$443$	9.22567e39	0.0882552	0.0441276	+	0.999026i	$0.485949\pi$
$444$	0	0
$445$	−2.15189e40	−0.191973
$446$	0	0
$447$	−2.05095e41	−1.70683
$448$	0	0
$449$	1.80147e41	1.39897	0.699484	−	0.714648i	$-0.253413\pi$
$449$	1.80147e41	1.39897	0.699484	+	0.714648i	$0.253413\pi$
$450$	0	0
$451$	3.32040e40	0.240688
$452$	0	0
$453$	−2.03301e41	−1.37600
$454$	0	0
$455$	−1.90503e41	−1.20428
$456$	0	0
$457$	7.03528e40	0.415509	0.207754	−	0.978181i	$-0.433385\pi$
$457$	7.03528e40	0.415509	0.207754	+	0.978181i	$0.433385\pi$
$458$	0	0
$459$	9.55118e40	0.527182
$460$	0	0
$461$	−2.15013e40	−0.110943	−0.0554717	−	0.998460i	$-0.517666\pi$
$461$	−2.15013e40	−0.110943	−0.0554717	+	0.998460i	$0.517666\pi$
$462$	0	0
$463$	−3.68798e41	−1.77944	−0.889720	−	0.456506i	$-0.849101\pi$
$463$	−3.68798e41	−1.77944	−0.889720	+	0.456506i	$0.849101\pi$
$464$	0	0
$465$	−4.75301e41	−2.14510
$466$	0	0
$467$	3.35225e41	1.41555	0.707775	−	0.706438i	$-0.249699\pi$
$467$	3.35225e41	1.41555	0.707775	+	0.706438i	$0.249699\pi$
$468$	0	0
$469$	−1.07630e41	−0.425357
$470$	0	0
$471$	−4.94668e41	−1.83016
$472$	0	0
$473$	6.43593e40	0.222979
$474$	0	0
$475$	3.07250e40	0.0997104
$476$	0	0
$477$	1.19067e40	0.0362040
$478$	0	0
$479$	1.78256e41	0.507976	0.253988	−	0.967207i	$-0.418258\pi$
$479$	1.78256e41	0.507976	0.253988	+	0.967207i	$0.418258\pi$
$480$	0	0
$481$	−2.49751e41	−0.667205
$482$	0	0
$483$	1.35883e39	0.00340396
$484$	0	0
$485$	4.43944e41	1.04311
$486$	0	0
$487$	2.96115e41	0.652772	0.326386	−	0.945237i	$-0.394169\pi$
$487$	2.96115e41	0.652772	0.326386	+	0.945237i	$0.394169\pi$
$488$	0	0
$489$	−5.04432e41	−1.04355
$490$	0	0
$491$	−2.87172e41	−0.557671	−0.278836	−	0.960339i	$-0.589948\pi$
$491$	−2.87172e41	−0.557671	−0.278836	+	0.960339i	$0.589948\pi$
$492$	0	0
$493$	8.33782e41	1.52029
$494$	0	0
$495$	−6.62837e40	−0.113508
$496$	0	0
$497$	−2.40949e41	−0.387615
$498$	0	0
$499$	1.19160e42	1.80125	0.900623	−	0.434601i	$-0.143111\pi$
$499$	1.19160e42	1.80125	0.900623	+	0.434601i	$0.143111\pi$
$500$	0	0
$501$	5.18568e41	0.736752
$502$	0	0
$503$	−1.92359e41	−0.256926	−0.128463	−	0.991714i	$-0.541004\pi$
$503$	−1.92359e41	−0.256926	−0.128463	+	0.991714i	$0.541004\pi$
$504$	0	0
$505$	1.06321e42	1.33538
$506$	0	0
$507$	−2.06663e41	−0.244141
$508$	0	0
$509$	−4.08219e41	−0.453699	−0.226849	−	0.973930i	$-0.572843\pi$
$509$	−4.08219e41	−0.453699	−0.226849	+	0.973930i	$0.572843\pi$
$510$	0	0
$511$	−4.49158e41	−0.469759
$512$	0	0
$513$	1.24871e41	0.122926
$514$	0	0
$515$	1.57760e42	1.46213
$516$	0	0
$517$	−1.78016e41	−0.155366
$518$	0	0
$519$	−5.17548e41	−0.425458
$520$	0	0
$521$	8.76030e41	0.678476	0.339238	−	0.940701i	$-0.389831\pi$
$521$	8.76030e41	0.678476	0.339238	+	0.940701i	$0.389831\pi$
$522$	0	0
$523$	1.52051e42	1.10972	0.554859	−	0.831944i	$-0.312772\pi$
$523$	1.52051e42	1.10972	0.554859	+	0.831944i	$0.312772\pi$
$524$	0	0
$525$	7.58999e41	0.522122
$526$	0	0
$527$	−2.03998e42	−1.32301
$528$	0	0
$529$	−1.63516e42	−0.999991
$530$	0	0
$531$	7.89524e41	0.455408
$532$	0	0
$533$	−2.79004e42	−1.51823
$534$	0	0
$535$	2.60173e42	1.33592
$536$	0	0
$537$	1.75213e41	0.0849113
$538$	0	0
$539$	−6.48102e40	−0.0296495
$540$	0	0
$541$	2.17683e42	0.940301	0.470150	−	0.882586i	$-0.344200\pi$
$541$	2.17683e42	0.940301	0.470150	+	0.882586i	$0.344200\pi$
$542$	0	0
$543$	−4.55407e42	−1.85782
$544$	0	0
$545$	2.97875e42	1.14786
$546$	0	0
$547$	−2.57006e42	−0.935707	−0.467854	−	0.883806i	$-0.654973\pi$
$547$	−2.57006e42	−0.935707	−0.467854	+	0.883806i	$0.654973\pi$
$548$	0	0
$549$	−1.82179e42	−0.626798
$550$	0	0
$551$	1.09008e42	0.354493
$552$	0	0
$553$	2.05691e42	0.632376
$554$	0	0
$555$	3.14937e42	0.915551
$556$	0	0
$557$	−2.88974e42	−0.794517	−0.397258	−	0.917707i	$-0.630038\pi$
$557$	−2.88974e42	−0.794517	−0.397258	+	0.917707i	$0.630038\pi$
$558$	0	0
$559$	−5.40793e42	−1.40653
$560$	0	0
$561$	−8.10014e41	−0.199328
$562$	0	0
$563$	5.57652e42	1.29862	0.649312	−	0.760522i	$-0.275057\pi$
$563$	5.57652e42	1.29862	0.649312	+	0.760522i	$0.275057\pi$
$564$	0	0
$565$	−1.25864e42	−0.277429
$566$	0	0
$567$	5.44676e42	1.13658
$568$	0	0
$569$	−3.00301e42	−0.593357	−0.296679	−	0.954977i	$-0.595879\pi$
$569$	−3.00301e42	−0.593357	−0.296679	+	0.954977i	$0.595879\pi$
$570$	0	0
$571$	−1.00812e43	−1.88649	−0.943245	−	0.332099i	$-0.892243\pi$
$571$	−1.00812e43	−1.88649	−0.943245	+	0.332099i	$0.892243\pi$
$572$	0	0
$573$	−1.35208e42	−0.239666
$574$	0	0
$575$	8.28207e39	0.00139087
$576$	0	0
$577$	−7.02075e42	−1.11727	−0.558635	−	0.829413i	$-0.688675\pi$
$577$	−7.02075e42	−1.11727	−0.558635	+	0.829413i	$0.688675\pi$
$578$	0	0
$579$	−8.13515e42	−1.22701
$580$	0	0
$581$	−5.19607e42	−0.742926
$582$	0	0
$583$	8.55554e40	0.0115980
$584$	0	0
$585$	5.56963e42	0.715996
$586$	0	0
$587$	−1.56627e43	−1.90974	−0.954872	−	0.297018i	$-0.904008\pi$
$587$	−1.56627e43	−1.90974	−0.954872	+	0.297018i	$0.904008\pi$
$588$	0	0
$589$	−2.66705e42	−0.308492
$590$	0	0
$591$	8.41360e42	0.923370
$592$	0	0
$593$	−1.51645e42	−0.157936	−0.0789678	−	0.996877i	$-0.525162\pi$
$593$	−1.51645e42	−0.157936	−0.0789678	+	0.996877i	$0.525162\pi$
$594$	0	0
$595$	1.03104e43	1.01921
$596$	0	0
$597$	9.80312e42	0.919946
$598$	0	0
$599$	8.56958e42	0.763561	0.381781	−	0.924253i	$-0.375311\pi$
$599$	8.56958e42	0.763561	0.381781	+	0.924253i	$0.375311\pi$
$600$	0	0
$601$	−2.71633e42	−0.229841	−0.114921	−	0.993375i	$-0.536661\pi$
$601$	−2.71633e42	−0.229841	−0.114921	+	0.993375i	$0.536661\pi$
$602$	0	0
$603$	3.14671e42	0.252894
$604$	0	0
$605$	1.53604e43	1.17272
$606$	0	0
$607$	−5.39744e42	−0.391528	−0.195764	−	0.980651i	$-0.562719\pi$
$607$	−5.39744e42	−0.391528	−0.195764	+	0.980651i	$0.562719\pi$
$608$	0	0
$609$	2.69282e43	1.85626
$610$	0	0
$611$	1.49582e43	0.980030
$612$	0	0
$613$	6.77481e42	0.421949	0.210975	−	0.977492i	$-0.432336\pi$
$613$	6.77481e42	0.421949	0.210975	+	0.977492i	$0.432336\pi$
$614$	0	0
$615$	3.51824e43	2.08335
$616$	0	0
$617$	−8.39156e41	−0.0472523	−0.0236261	−	0.999721i	$-0.507521\pi$
$617$	−8.39156e41	−0.0472523	−0.0236261	+	0.999721i	$0.507521\pi$
$618$	0	0
$619$	−1.34876e43	−0.722318	−0.361159	−	0.932504i	$-0.617619\pi$
$619$	−1.34876e43	−0.722318	−0.361159	+	0.932504i	$0.617619\pi$
$620$	0	0
$621$	3.36597e40	0.00171471
$622$	0	0
$623$	−2.98286e42	−0.144566
$624$	0	0
$625$	−2.70736e43	−1.24855
$626$	0	0
$627$	−1.05900e42	−0.0464784
$628$	0	0
$629$	1.35170e43	0.564672
$630$	0	0
$631$	4.65263e42	0.185031	0.0925153	−	0.995711i	$-0.470509\pi$
$631$	4.65263e42	0.185031	0.0925153	+	0.995711i	$0.470509\pi$
$632$	0	0
$633$	−6.48274e43	−2.45472
$634$	0	0
$635$	−4.69516e43	−1.69300
$636$	0	0
$637$	5.44582e42	0.187026
$638$	0	0
$639$	7.04448e42	0.230455
$640$	0	0
$641$	4.24972e43	1.32453	0.662263	−	0.749271i	$-0.269596\pi$
$641$	4.24972e43	1.32453	0.662263	+	0.749271i	$0.269596\pi$
$642$	0	0
$643$	5.33186e43	1.58347	0.791733	−	0.610868i	$-0.209179\pi$
$643$	5.33186e43	1.58347	0.791733	+	0.610868i	$0.209179\pi$
$644$	0	0
$645$	6.81941e43	1.93006
$646$	0	0
$647$	1.00492e43	0.271092	0.135546	−	0.990771i	$-0.456721\pi$
$647$	1.00492e43	0.271092	0.135546	+	0.990771i	$0.456721\pi$
$648$	0	0
$649$	5.67312e42	0.145891
$650$	0	0
$651$	−6.58842e43	−1.61538
$652$	0	0
$653$	−7.98263e43	−1.86634	−0.933172	−	0.359430i	$-0.882971\pi$
$653$	−7.98263e43	−1.86634	−0.933172	+	0.359430i	$0.882971\pi$
$654$	0	0
$655$	1.48821e42	0.0331836
$656$	0	0
$657$	1.31318e43	0.279293
$658$	0	0
$659$	−7.54017e43	−1.52988	−0.764940	−	0.644102i	$-0.777231\pi$
$659$	−7.54017e43	−1.52988	−0.764940	+	0.644102i	$0.777231\pi$
$660$	0	0
$661$	−3.00409e42	−0.0581555	−0.0290777	−	0.999577i	$-0.509257\pi$
$661$	−3.00409e42	−0.0581555	−0.0290777	+	0.999577i	$0.509257\pi$
$662$	0	0
$663$	6.80632e43	1.25734
$664$	0	0
$665$	1.34797e43	0.237654
$666$	0	0
$667$	2.93836e41	0.00494487
$668$	0	0
$669$	−8.09919e43	−1.30118
$670$	0	0
$671$	−1.30905e43	−0.200797
$672$	0	0
$673$	2.05730e43	0.301346	0.150673	−	0.988584i	$-0.451856\pi$
$673$	2.05730e43	0.301346	0.150673	+	0.988584i	$0.451856\pi$
$674$	0	0
$675$	1.88012e43	0.263013
$676$	0	0
$677$	−1.03692e44	−1.38554	−0.692772	−	0.721157i	$-0.743611\pi$
$677$	−1.03692e44	−1.38554	−0.692772	+	0.721157i	$0.743611\pi$
$678$	0	0
$679$	6.15377e43	0.785524
$680$	0	0
$681$	3.58193e42	0.0436855
$682$	0	0
$683$	−1.01818e44	−1.18660	−0.593302	−	0.804980i	$-0.702176\pi$
$683$	−1.01818e44	−1.18660	−0.593302	+	0.804980i	$0.702176\pi$
$684$	0	0
$685$	1.26517e43	0.140911
$686$	0	0
$687$	−4.31911e43	−0.459796
$688$	0	0
$689$	−7.18898e42	−0.0731593
$690$	0	0
$691$	−1.12626e44	−1.09579	−0.547895	−	0.836547i	$-0.684571\pi$
$691$	−1.12626e44	−1.09579	−0.547895	+	0.836547i	$0.684571\pi$
$692$	0	0
$693$	−9.18796e42	−0.0854779
$694$	0	0
$695$	1.55142e44	1.38027
$696$	0	0
$697$	1.51002e44	1.28492
$698$	0	0
$699$	−2.20652e44	−1.79602
$700$	0	0
$701$	3.06210e43	0.238446	0.119223	−	0.992867i	$-0.461960\pi$
$701$	3.06210e43	0.238446	0.119223	+	0.992867i	$0.461960\pi$
$702$	0	0
$703$	1.76720e43	0.131668
$704$	0	0
$705$	−1.88623e44	−1.34481
$706$	0	0
$707$	1.47378e44	1.00561
$708$	0	0
$709$	−9.24433e43	−0.603752	−0.301876	−	0.953347i	$-0.597613\pi$
$709$	−9.24433e43	−0.603752	−0.301876	+	0.953347i	$0.597613\pi$
$710$	0	0
$711$	−6.01366e43	−0.375976
$712$	0	0
$713$	−7.18917e41	−0.00430320
$714$	0	0
$715$	4.00205e43	0.229371
$716$	0	0
$717$	9.15036e43	0.502217
$718$	0	0
$719$	9.29670e43	0.488687	0.244344	−	0.969689i	$-0.421428\pi$
$719$	9.29670e43	0.488687	0.244344	+	0.969689i	$0.421428\pi$
$720$	0	0
$721$	2.18681e44	1.10107
$722$	0	0
$723$	2.22323e44	1.07236
$724$	0	0
$725$	1.64128e44	0.758477
$726$	0	0
$727$	−1.34756e44	−0.596710	−0.298355	−	0.954455i	$-0.596438\pi$
$727$	−1.34756e44	−0.596710	−0.298355	+	0.954455i	$0.596438\pi$
$728$	0	0
$729$	7.35277e42	0.0312014
$730$	0	0
$731$	2.92688e44	1.19038
$732$	0	0
$733$	1.96770e44	0.767093	0.383546	−	0.923522i	$-0.374703\pi$
$733$	1.96770e44	0.767093	0.383546	+	0.923522i	$0.374703\pi$
$734$	0	0
$735$	−6.86719e43	−0.256641
$736$	0	0
$737$	2.26106e43	0.0810153
$738$	0	0
$739$	3.41064e44	1.17178	0.585891	−	0.810390i	$-0.300745\pi$
$739$	3.41064e44	1.17178	0.585891	+	0.810390i	$0.300745\pi$
$740$	0	0
$741$	8.89852e43	0.293181
$742$	0	0
$743$	2.64820e44	0.836805	0.418403	−	0.908262i	$-0.362590\pi$
$743$	2.64820e44	0.836805	0.418403	+	0.908262i	$0.362590\pi$
$744$	0	0
$745$	−5.48428e44	−1.66225
$746$	0	0
$747$	1.51915e44	0.441703
$748$	0	0
$749$	3.60641e44	1.00602
$750$	0	0
$751$	−3.88190e44	−1.03902	−0.519511	−	0.854464i	$-0.673886\pi$
$751$	−3.88190e44	−1.03902	−0.519511	+	0.854464i	$0.673886\pi$
$752$	0	0
$753$	2.92459e44	0.751177
$754$	0	0
$755$	−5.43630e44	−1.34006
$756$	0	0
$757$	2.10907e44	0.499004	0.249502	−	0.968374i	$-0.419733\pi$
$757$	2.10907e44	0.499004	0.249502	+	0.968374i	$0.419733\pi$
$758$	0	0
$759$	−2.85460e41	−0.000648332 0
$760$	0	0
$761$	3.24151e43	0.0706782	0.0353391	−	0.999375i	$-0.488749\pi$
$761$	3.24151e43	0.0706782	0.0353391	+	0.999375i	$0.488749\pi$
$762$	0	0
$763$	4.12902e44	0.864403
$764$	0	0
$765$	−3.01439e44	−0.605965
$766$	0	0
$767$	−4.76696e44	−0.920266
$768$	0	0
$769$	−6.38736e44	−1.18430	−0.592152	−	0.805827i	$-0.701721\pi$
$769$	−6.38736e44	−1.18430	−0.592152	+	0.805827i	$0.701721\pi$
$770$	0	0
$771$	−9.73496e44	−1.73377
$772$	0	0
$773$	−8.28645e44	−1.41771	−0.708853	−	0.705356i	$-0.750787\pi$
$773$	−8.28645e44	−1.41771	−0.708853	+	0.705356i	$0.750787\pi$
$774$	0	0
$775$	−4.01565e44	−0.660053
$776$	0	0
$777$	4.36552e44	0.689462
$778$	0	0
$779$	1.97419e44	0.299611
$780$	0	0
$781$	5.06180e43	0.0738268
$782$	0	0
$783$	6.67041e44	0.935072
$784$	0	0
$785$	−1.32275e45	−1.78237
$786$	0	0
$787$	8.07362e44	1.04583	0.522913	−	0.852386i	$-0.324845\pi$
$787$	8.07362e44	1.04583	0.522913	+	0.852386i	$0.324845\pi$
$788$	0	0
$789$	−3.87945e44	−0.483144
$790$	0	0
$791$	−1.74468e44	−0.208920
$792$	0	0
$793$	1.09996e45	1.26660
$794$	0	0
$795$	9.06532e43	0.100390
$796$	0	0
$797$	8.20375e44	0.873792	0.436896	−	0.899512i	$-0.356078\pi$
$797$	8.20375e44	0.873792	0.436896	+	0.899512i	$0.356078\pi$
$798$	0	0
$799$	−8.09564e44	−0.829423
$800$	0	0
$801$	8.72080e43	0.0859513
$802$	0	0
$803$	9.43583e43	0.0894724
$804$	0	0
$805$	3.63353e42	0.00331507
$806$	0	0
$807$	−1.03062e45	−0.904812
$808$	0	0
$809$	−2.25872e44	−0.190836	−0.0954182	−	0.995437i	$-0.530419\pi$
$809$	−2.25872e44	−0.190836	−0.0954182	+	0.995437i	$0.530419\pi$
$810$	0	0
$811$	1.42546e45	1.15913	0.579565	−	0.814926i	$-0.303222\pi$
$811$	1.42546e45	1.15913	0.579565	+	0.814926i	$0.303222\pi$
$812$	0	0
$813$	−1.23583e44	−0.0967289
$814$	0	0
$815$	−1.34886e45	−1.01630
$816$	0	0
$817$	3.82657e44	0.277567
$818$	0	0
$819$	7.72039e44	0.539186
$820$	0	0
$821$	−2.34861e44	−0.157940	−0.0789699	−	0.996877i	$-0.525163\pi$
$821$	−2.34861e44	−0.157940	−0.0789699	+	0.996877i	$0.525163\pi$
$822$	0	0
$823$	−2.26372e45	−1.46597	−0.732984	−	0.680246i	$-0.761873\pi$
$823$	−2.26372e45	−1.46597	−0.732984	+	0.680246i	$0.761873\pi$
$824$	0	0
$825$	−1.59449e44	−0.0994456
$826$	0	0
$827$	−1.11856e45	−0.671929	−0.335965	−	0.941875i	$-0.609062\pi$
$827$	−1.11856e45	−0.671929	−0.335965	+	0.941875i	$0.609062\pi$
$828$	0	0
$829$	−2.19670e45	−1.27109	−0.635544	−	0.772064i	$-0.719224\pi$
$829$	−2.19670e45	−1.27109	−0.635544	+	0.772064i	$0.719224\pi$
$830$	0	0
$831$	2.45236e45	1.36700
$832$	0	0
$833$	−2.94738e44	−0.158285
$834$	0	0
$835$	1.38666e45	0.717512
$836$	0	0
$837$	−1.63202e45	−0.813733
$838$	0	0
$839$	−3.71878e45	−1.78686	−0.893429	−	0.449204i	$-0.851707\pi$
$839$	−3.71878e45	−1.78686	−0.893429	+	0.449204i	$0.851707\pi$
$840$	0	0
$841$	3.66359e45	1.69656
$842$	0	0
$843$	5.22423e45	2.33182
$844$	0	0
$845$	−5.52621e44	−0.237765
$846$	0	0
$847$	2.12920e45	0.883128
$848$	0	0
$849$	3.66359e45	1.46500
$850$	0	0
$851$	4.76359e42	0.00183665
$852$	0	0
$853$	2.48938e45	0.925509	0.462754	−	0.886487i	$-0.346861\pi$
$853$	2.48938e45	0.925509	0.462754	+	0.886487i	$0.346861\pi$
$854$	0	0
$855$	−3.94099e44	−0.141296
$856$	0	0
$857$	−8.68133e44	−0.300181	−0.150091	−	0.988672i	$-0.547957\pi$
$857$	−8.68133e44	−0.300181	−0.150091	+	0.988672i	$0.547957\pi$
$858$	0	0
$859$	1.43587e45	0.478875	0.239437	−	0.970912i	$-0.423037\pi$
$859$	1.43587e45	0.478875	0.239437	+	0.970912i	$0.423037\pi$
$860$	0	0
$861$	4.87684e45	1.56888
$862$	0	0
$863$	3.91776e45	1.21582	0.607911	−	0.794005i	$-0.292008\pi$
$863$	3.91776e45	1.21582	0.607911	+	0.794005i	$0.292008\pi$
$864$	0	0
$865$	−1.38393e45	−0.414347
$866$	0	0
$867$	6.14076e44	0.177389
$868$	0	0
$869$	−4.32111e44	−0.120445
$870$	0	0
$871$	−1.89991e45	−0.511036
$872$	0	0
$873$	−1.79914e45	−0.467030
$874$	0	0
$875$	−2.36450e45	−0.592399
$876$	0	0
$877$	−2.59910e45	−0.628535	−0.314267	−	0.949335i	$-0.601759\pi$
$877$	−2.59910e45	−0.628535	−0.314267	+	0.949335i	$0.601759\pi$
$878$	0	0
$879$	−2.87630e45	−0.671438
$880$	0	0
$881$	1.04135e45	0.234676	0.117338	−	0.993092i	$-0.462564\pi$
$881$	1.04135e45	0.234676	0.117338	+	0.993092i	$0.462564\pi$
$882$	0	0
$883$	9.31982e44	0.202776	0.101388	−	0.994847i	$-0.467672\pi$
$883$	9.31982e44	0.202776	0.101388	+	0.994847i	$0.467672\pi$
$884$	0	0
$885$	6.01115e45	1.26281
$886$	0	0
$887$	−1.93996e45	−0.393529	−0.196764	−	0.980451i	$-0.563043\pi$
$887$	−1.93996e45	−0.393529	−0.196764	+	0.980451i	$0.563043\pi$
$888$	0	0
$889$	−6.50823e45	−1.27493
$890$	0	0
$891$	−1.14425e45	−0.216478
$892$	0	0
$893$	−1.05842e45	−0.193401
$894$	0	0
$895$	4.68522e44	0.0826939
$896$	0	0
$897$	2.39864e43	0.00408961
$898$	0	0
$899$	−1.42469e46	−2.34664
$900$	0	0
$901$	3.89082e44	0.0619165
$902$	0	0
$903$	9.45278e45	1.45345
$904$	0	0
$905$	−1.21777e46	−1.80930
$906$	0	0
$907$	8.10222e45	1.16330	0.581649	−	0.813440i	$-0.302408\pi$
$907$	8.10222e45	1.16330	0.581649	+	0.813440i	$0.302408\pi$
$908$	0	0
$909$	−4.30880e45	−0.597883
$910$	0	0
$911$	1.13425e46	1.52115	0.760575	−	0.649250i	$-0.224917\pi$
$911$	1.13425e46	1.52115	0.760575	+	0.649250i	$0.224917\pi$
$912$	0	0
$913$	1.09158e45	0.141501
$914$	0	0
$915$	−1.38705e46	−1.73806
$916$	0	0
$917$	2.06289e44	0.0249892
$918$	0	0
$919$	−1.39631e46	−1.63527	−0.817637	−	0.575734i	$-0.804717\pi$
$919$	−1.39631e46	−1.63527	−0.817637	+	0.575734i	$0.804717\pi$
$920$	0	0
$921$	−6.50516e45	−0.736604
$922$	0	0
$923$	−4.25329e45	−0.465692
$924$	0	0
$925$	2.66079e45	0.281717
$926$	0	0
$927$	−6.39344e45	−0.654634
$928$	0	0
$929$	−3.21761e45	−0.318632	−0.159316	−	0.987228i	$-0.550929\pi$
$929$	−3.21761e45	−0.318632	−0.159316	+	0.987228i	$0.550929\pi$
$930$	0	0
$931$	−3.85338e44	−0.0369081
$932$	0	0
$933$	1.31062e46	1.21426
$934$	0	0
$935$	−2.16599e45	−0.194123
$936$	0	0
$937$	1.44001e46	1.24853	0.624267	−	0.781211i	$-0.285398\pi$
$937$	1.44001e46	1.24853	0.624267	+	0.781211i	$0.285398\pi$
$938$	0	0
$939$	9.49221e45	0.796252
$940$	0	0
$941$	−1.38781e46	−1.12640	−0.563198	−	0.826322i	$-0.690429\pi$
$941$	−1.38781e46	−1.12640	−0.563198	+	0.826322i	$0.690429\pi$
$942$	0	0
$943$	5.32153e43	0.00417931
$944$	0	0
$945$	8.24851e45	0.626877
$946$	0	0
$947$	3.58686e45	0.263809	0.131905	−	0.991262i	$-0.457891\pi$
$947$	3.58686e45	0.263809	0.131905	+	0.991262i	$0.457891\pi$
$948$	0	0
$949$	−7.92866e45	−0.564383
$950$	0	0
$951$	3.01308e46	2.07593
$952$	0	0
$953$	8.63539e45	0.575893	0.287947	−	0.957646i	$-0.407027\pi$
$953$	8.63539e45	0.575893	0.287947	+	0.957646i	$0.407027\pi$
$954$	0	0
$955$	−3.61549e45	−0.233408
$956$	0	0
$957$	−5.65702e45	−0.353552
$958$	0	0
$959$	1.75372e45	0.106114
$960$	0	0
$961$	1.77883e46	1.04213
$962$	0	0
$963$	−1.05439e46	−0.598125
$964$	0	0
$965$	−2.17535e46	−1.19497
$966$	0	0
$967$	1.07239e46	0.570481	0.285241	−	0.958456i	$-0.407927\pi$
$967$	1.07239e46	0.570481	0.285241	+	0.958456i	$0.407927\pi$
$968$	0	0
$969$	−4.81605e45	−0.248126
$970$	0	0
$971$	−1.69197e46	−0.844295	−0.422148	−	0.906527i	$-0.638724\pi$
$971$	−1.69197e46	−0.844295	−0.422148	+	0.906527i	$0.638724\pi$
$972$	0	0
$973$	2.15051e46	1.03942
$974$	0	0
$975$	1.33980e46	0.627292
$976$	0	0
$977$	−1.63831e46	−0.743069	−0.371534	−	0.928419i	$-0.621168\pi$
$977$	−1.63831e46	−0.743069	−0.371534	+	0.928419i	$0.621168\pi$
$978$	0	0
$979$	6.26632e44	0.0275347
$980$	0	0
$981$	−1.20718e46	−0.513927
$982$	0	0
$983$	−1.26750e46	−0.522839	−0.261419	−	0.965225i	$-0.584191\pi$
$983$	−1.26750e46	−0.522839	−0.261419	+	0.965225i	$0.584191\pi$
$984$	0	0
$985$	2.24981e46	0.899257
$986$	0	0
$987$	−2.61461e46	−1.01272
$988$	0	0
$989$	1.03147e44	0.00387182
$990$	0	0
$991$	3.41939e46	1.24396	0.621980	−	0.783033i	$-0.286328\pi$
$991$	3.41939e46	1.24396	0.621980	+	0.783033i	$0.286328\pi$
$992$	0	0
$993$	2.56276e46	0.903638
$994$	0	0
$995$	2.62137e46	0.895922
$996$	0	0
$997$	3.16608e46	1.04893	0.524466	−	0.851432i	$-0.324265\pi$
$997$	3.16608e46	1.04893	0.524466	+	0.851432i	$0.324265\pi$
$998$	0	0
$999$	1.08139e46	0.347309

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.32.a.b.1.1		2
4.3	odd	2		1.32.a.a.1.2	✓	2
12.11	even	2		9.32.a.a.1.1		2
20.3	even	4		25.32.b.a.24.1		4
20.7	even	4		25.32.b.a.24.4		4
20.19	odd	2		25.32.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.32.a.a.1.2	✓	2	4.3	odd	2
9.32.a.a.1.1		2	12.11	even	2
16.32.a.b.1.1		2	1.1	even	1	trivial
25.32.a.a.1.1		2	20.19	odd	2
25.32.b.a.24.1		4	20.3	even	4
25.32.b.a.24.4		4	20.7	even	4

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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 32 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-3.08552e7 q^{3} -8.25073e10 q^{5} -1.14368e13 q^{7} +3.34371e14 q^{9} +O(q^{10})\)	\(q-3.08552e7 q^{3} -8.25073e10 q^{5} -1.14368e13 q^{7} +3.34371e14 q^{9} +2.40262e15 q^{11} -2.01886e17 q^{13} +2.54578e18 q^{15} +1.09264e19 q^{17} +1.42851e19 q^{19} +3.52885e20 q^{21} +3.85062e18 q^{23} +2.15084e21 q^{25} +8.74135e21 q^{27} +7.63087e22 q^{29} -1.86701e23 q^{31} -7.41334e22 q^{33} +9.43620e23 q^{35} +1.23709e24 q^{37} +6.22922e24 q^{39} +1.38199e25 q^{41} +2.67871e25 q^{43} -2.75881e25 q^{45} -7.40922e25 q^{47} -2.69748e25 q^{49} -3.37138e26 q^{51} +3.56092e25 q^{53} -1.98234e26 q^{55} -4.40770e26 q^{57} +2.36122e27 q^{59} -5.44842e27 q^{61} -3.82414e27 q^{63} +1.66570e28 q^{65} +9.41082e27 q^{67} -1.18812e26 q^{69} +2.10678e28 q^{71} +3.92731e28 q^{73} -6.63646e28 q^{75} -2.74783e28 q^{77} -1.79850e29 q^{79} -4.76249e29 q^{81} +4.54329e29 q^{83} -9.01511e29 q^{85} -2.35452e30 q^{87} +2.60812e29 q^{89} +2.30893e30 q^{91} +5.76072e30 q^{93} -1.17863e30 q^{95} -5.38067e30 q^{97} +8.03368e29 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 17363160 q^{3} - 19391218020 q^{5} - 30257527577200 q^{7} - 101266456303926 q^{9}+O(q^{10}) \) 2 * q - 17363160 * q^3 - 19391218020 * q^5 - 30257527577200 * q^7 - 101266456303926 * q^9	\( 2 q - 17363160 q^{3} - 19391218020 q^{5} - 30257527577200 q^{7} - 101266456303926 q^{9} + 77\!\cdots\!76 q^{11}+ \cdots - 15\!\cdots\!88 q^{99}+O(q^{100}) \) 2 * q - 17363160 * q^3 - 19391218020 * q^5 - 30257527577200 * q^7 - 101266456303926 * q^9 + 7782353745118776 * q^11 + 74708953050260620 * q^13 + 3397345822674581040 * q^15 + 17224607828987089380 * q^17 + 12370563328022164040 * q^19 + 98954957416071161664 * q^21 - 1897344841989911219280 * q^23 + 1477861250884539239150 * q^25 - 5469986602319662295280 * q^27 + 128576144217217055807340 * q^29 - 125733527517961838793664 * q^31 - 1549761835334435045280 * q^33 - 244269703121729304206880 * q^35 - 833815207054016911025060 * q^37 + 9961054164411363568284912 * q^39 + 8724924335662925840671284 * q^41 + 18397105293779438708372600 * q^43 - 55083793466681039024257140 * q^45 - 95450963964856190793148320 * q^47 + 169469294372320202530407186 * q^49 - 252162404739970320085380144 * q^51 + 194822508473721098983660860 * q^53 + 141313671368671671047892240 * q^55 - 466601717043760934967596640 * q^57 + 198723263547765513990746520 * q^59 - 12056218201113004361157656276 * q^61 + 4374875036601718967620305360 * q^63 + 34114584794425827928886318760 * q^65 + 9688140802872256994032247720 * q^67 - 25769856286394326078494576192 * q^69 - 55784576625034657512878287344 * q^71 + 62234095932086098750552955380 * q^73 - 75444402265645086720647965800 * q^75 - 128728748793695217194494555200 * q^77 + 119164191093299704964212710560 * q^79 - 398906922646891269649062634158 * q^81 + 264863373681569145625357596360 * q^83 - 503995105657776450242931421320 * q^85 - 1649324800814841914442821796240 * q^87 - 2154414614246670291602721895980 * q^89 - 2896781368068903069956049509024 * q^91 + 6583298380663842229690665227520 * q^93 - 1299465173198290044974084355600 * q^95 - 9060767994874032615529957205180 * q^97 - 1540246307744094989457731085288 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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