LMFDB - Embedded newform 16.50.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,50,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, 50, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 50);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 50 $
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:	$243.305928158$
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} - x^{2} - 27962089502x + 71708842875120 $ x^3 - x^2 - 27962089502*x + 71708842875120
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-168486.$ 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)

$f(q)$	$=$	$q+8.64745e11 q^{3} +1.67413e17 q^{5} +3.42668e20 q^{7} +5.08484e23 q^{9} +O(q^{10})$	$q+8.64745e11 q^{3} +1.67413e17 q^{5} +3.42668e20 q^{7} +5.08484e23 q^{9} +2.47212e24 q^{11} -1.86505e26 q^{13} +1.44770e29 q^{15} +8.08684e29 q^{17} +1.31622e31 q^{19} +2.96321e32 q^{21} -3.96809e33 q^{23} +1.02637e34 q^{25} +2.32776e35 q^{27} +4.73835e35 q^{29} +5.52226e36 q^{31} +2.13775e36 q^{33} +5.73673e37 q^{35} +6.02154e37 q^{37} -1.61279e38 q^{39} -4.00631e39 q^{41} +1.57103e39 q^{43} +8.51271e40 q^{45} +3.77976e40 q^{47} -1.39502e41 q^{49} +6.99305e41 q^{51} -7.38411e41 q^{53} +4.13866e41 q^{55} +1.13819e43 q^{57} +3.19771e43 q^{59} +2.48724e43 q^{61} +1.74241e44 q^{63} -3.12234e43 q^{65} -5.76819e44 q^{67} -3.43138e45 q^{69} +1.66436e45 q^{71} +4.80904e45 q^{73} +8.87547e45 q^{75} +8.47118e44 q^{77} +4.21047e45 q^{79} +7.96119e46 q^{81} +1.20169e47 q^{83} +1.35385e47 q^{85} +4.09747e47 q^{87} +8.70490e47 q^{89} -6.39094e46 q^{91} +4.77534e48 q^{93} +2.20352e48 q^{95} -7.30863e48 q^{97} +1.25703e48 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 326954692404 q^{3} + 63\!\cdots\!50 q^{5}+ \cdots + 34\!\cdots\!19 q^{9}+O(q^{10}) $ 3 * q + 326954692404 * q^3 + 63884035717079250 * q^5 - 509391477498711192 * q^7 + 341625690512280455369319 * q^9	$ 3 q + 326954692404 q^{3} + 63\!\cdots\!50 q^{5}+ \cdots + 11\!\cdots\!52 q^{99}+O(q^{100}) $ 3 * q + 326954692404 * q^3 + 63884035717079250 * q^5 - 509391477498711192 * q^7 + 341625690512280455369319 * q^9 + 20839986192711815613370524 * q^11 - 187524857197340618282341014 * q^13 + 204230679539215002884128467000 * q^15 - 3305035636247475259184114894538 * q^17 + 45882942802931464757155696518660 * q^19 + 617794442803555252870956383262816 * q^21 - 4533392548023086653463590407813576 * q^23 - 13921151905580403838435182832171875 * q^25 + 316454625615073819697154158592097800 * q^27 + 1118089919316361970349176934175494810 * q^29 - 934226816157578575389156342339525216 * q^31 - 24487068877327908023713382739550873968 * q^33 + 118496887985378726635932152589238374000 * q^35 + 242135148519373130928922851818564983602 * q^37 - 1291017145684876237790670030697385259048 * q^39 - 6211739675976490815696297371880432723714 * q^41 + 7795173962887472637660666233812713782844 * q^43 + 76758284157261249391010855229819118430250 * q^45 - 72757717838871152990888073123214445891472 * q^47 - 280157966384090888246597869444400847657429 * q^49 + 1538734957331600122304206020917586572488104 * q^51 - 155850890670432045962015052011776669693854 * q^53 - 4640026560861316230546160030865773419591000 * q^55 - 7909742361043655643598112535376781525014800 * q^57 + 62596927936822013761436024025606581543386380 * q^59 - 24037263967789435219196348055077866809754374 * q^61 + 79613772299337773967480392108297858821089864 * q^63 - 244975900670806240216695171564505814930314500 * q^65 + 1019040852324305100821779153549119164107459188 * q^67 - 4867866269191569039747093560336346164459611872 * q^69 + 3208129335291950593367863544185773469328334504 * q^71 + 5668108076325412165814435883536677392865317726 * q^73 + 12104818074938487392123628092507024931595687500 * q^75 - 32043339793247959180079389483660528277811891936 * q^77 + 7871365395380306420112251724245018715125364240 * q^79 + 67270927730095597004530465543730351634592280043 * q^81 - 94164681901361229355849854653903786480577085116 * q^83 + 297863286656785614551226799130947646752077538500 * q^85 + 175500245979701689466461020645367234129141215000 * q^87 + 367807201561165354579457777200332871192114777230 * q^89 - 1613372397893666548508325640004129525440352221776 * q^91 + 5954871901405488350696717407304686000200433533312 * q^93 - 1475636199748052355599968141572624301097796705000 * q^95 - 10431108583780665799950508389192126734793760399578 * q^97 + 11387023179914295087805432025124514927226421923052 * 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
$3$	8.64745e11	1.76773	0.883867	−	0.467737i	$-0.154931\pi$
$3$	8.64745e11	1.76773	0.883867	+	0.467737i	$0.154931\pi$
$4$	0	0
$5$	1.67413e17	1.25610	0.628051	−	0.778172i	$-0.283853\pi$
$5$	1.67413e17	1.25610	0.628051	+	0.778172i	$0.283853\pi$
$6$	0	0
$7$	3.42668e20	0.676040	0.338020	−	0.941139i	$-0.390243\pi$
$7$	3.42668e20	0.676040	0.338020	+	0.941139i	$0.390243\pi$
$8$	0	0
$9$	5.08484e23	2.12489
$10$	0	0
$11$	2.47212e24	0.0756744	0.0378372	−	0.999284i	$-0.487953\pi$
$11$	2.47212e24	0.0756744	0.0378372	+	0.999284i	$0.487953\pi$
$12$	0	0
$13$	−1.86505e26	−0.0952968	−0.0476484	−	0.998864i	$-0.515173\pi$
$13$	−1.86505e26	−0.0952968	−0.0476484	+	0.998864i	$0.515173\pi$
$14$	0	0
$15$	1.44770e29	2.22046
$16$	0	0
$17$	8.08684e29	0.577803	0.288902	−	0.957359i	$-0.406710\pi$
$17$	8.08684e29	0.577803	0.288902	+	0.957359i	$0.406710\pi$
$18$	0	0
$19$	1.31622e31	0.616402	0.308201	−	0.951321i	$-0.400273\pi$
$19$	1.31622e31	0.616402	0.308201	+	0.951321i	$0.400273\pi$
$20$	0	0
$21$	2.96321e32	1.19506
$22$	0	0
$23$	−3.96809e33	−1.72286	−0.861428	−	0.507879i	$-0.830430\pi$
$23$	−3.96809e33	−1.72286	−0.861428	+	0.507879i	$0.830430\pi$
$24$	0	0
$25$	1.02637e34	0.577794
$26$	0	0
$27$	2.32776e35	1.98850
$28$	0	0
$29$	4.73835e35	0.702872	0.351436	−	0.936212i	$-0.385693\pi$
$29$	4.73835e35	0.702872	0.351436	+	0.936212i	$0.385693\pi$
$30$	0	0
$31$	5.52226e36	1.59866	0.799328	−	0.600895i	$-0.205189\pi$
$31$	5.52226e36	1.59866	0.799328	+	0.600895i	$0.205189\pi$
$32$	0	0
$33$	2.13775e36	0.133772
$34$	0	0
$35$	5.73673e37	0.849175
$36$	0	0
$37$	6.02154e37	0.228436	0.114218	−	0.993456i	$-0.463564\pi$
$37$	6.02154e37	0.228436	0.114218	+	0.993456i	$0.463564\pi$
$38$	0	0
$39$	−1.61279e38	−0.168459
$40$	0	0
$41$	−4.00631e39	−1.22897	−0.614483	−	0.788930i	$-0.710635\pi$
$41$	−4.00631e39	−1.22897	−0.614483	+	0.788930i	$0.710635\pi$
$42$	0	0
$43$	1.57103e39	0.150040	0.0750198	−	0.997182i	$-0.476098\pi$
$43$	1.57103e39	0.150040	0.0750198	+	0.997182i	$0.476098\pi$
$44$	0	0
$45$	8.51271e40	2.66908
$46$	0	0
$47$	3.77976e40	0.408382	0.204191	−	0.978931i	$-0.434544\pi$
$47$	3.77976e40	0.408382	0.204191	+	0.978931i	$0.434544\pi$
$48$	0	0
$49$	−1.39502e41	−0.542970
$50$	0	0
$51$	6.99305e41	1.02140
$52$	0	0
$53$	−7.38411e41	−0.420281	−0.210140	−	0.977671i	$-0.567392\pi$
$53$	−7.38411e41	−0.420281	−0.210140	+	0.977671i	$0.567392\pi$
$54$	0	0
$55$	4.13866e41	0.0950549
$56$	0	0
$57$	1.13819e43	1.08963
$58$	0	0
$59$	3.19771e43	1.31512	0.657560	−	0.753403i	$-0.271589\pi$
$59$	3.19771e43	1.31512	0.657560	+	0.753403i	$0.271589\pi$
$60$	0	0
$61$	2.48724e43	0.451998	0.225999	−	0.974127i	$-0.427435\pi$
$61$	2.48724e43	0.451998	0.225999	+	0.974127i	$0.427435\pi$
$62$	0	0
$63$	1.74241e44	1.43651
$64$	0	0
$65$	−3.12234e43	−0.119703
$66$	0	0
$67$	−5.76819e44	−1.05246	−0.526232	−	0.850341i	$-0.676396\pi$
$67$	−5.76819e44	−1.05246	−0.526232	+	0.850341i	$0.676396\pi$
$68$	0	0
$69$	−3.43138e45	−3.04555
$70$	0	0
$71$	1.66436e45	0.733530	0.366765	−	0.930314i	$-0.380465\pi$
$71$	1.66436e45	0.733530	0.366765	+	0.930314i	$0.380465\pi$
$72$	0	0
$73$	4.80904e45	1.07312	0.536561	−	0.843862i	$-0.319723\pi$
$73$	4.80904e45	1.07312	0.536561	+	0.843862i	$0.319723\pi$
$74$	0	0
$75$	8.87547e45	1.02139
$76$	0	0
$77$	8.47118e44	0.0511589
$78$	0	0
$79$	4.21047e45	0.135665	0.0678323	−	0.997697i	$-0.478392\pi$
$79$	4.21047e45	0.135665	0.0678323	+	0.997697i	$0.478392\pi$
$80$	0	0
$81$	7.96119e46	1.39026
$82$	0	0
$83$	1.20169e47	1.15446	0.577232	−	0.816581i	$-0.304133\pi$
$83$	1.20169e47	1.15446	0.577232	+	0.816581i	$0.304133\pi$
$84$	0	0
$85$	1.35385e47	0.725781
$86$	0	0
$87$	4.09747e47	1.24249
$88$	0	0
$89$	8.70490e47	1.51255	0.756273	−	0.654256i	$-0.227018\pi$
$89$	8.70490e47	1.51255	0.756273	+	0.654256i	$0.227018\pi$
$90$	0	0
$91$	−6.39094e46	−0.0644244
$92$	0	0
$93$	4.77534e48	2.82600
$94$	0	0
$95$	2.20352e48	0.774264
$96$	0	0
$97$	−7.30863e48	−1.54145	−0.770723	−	0.637171i	$-0.780105\pi$
$97$	−7.30863e48	−1.54145	−0.770723	+	0.637171i	$0.780105\pi$
$98$	0	0
$99$	1.25703e48	0.160800
$100$	0	0
$101$	1.06900e48	0.0837733	0.0418867	−	0.999122i	$-0.486663\pi$
$101$	1.06900e48	0.0837733	0.0418867	+	0.999122i	$0.486663\pi$
$102$	0	0
$103$	3.01827e49	1.46301	0.731503	−	0.681839i	$-0.238819\pi$
$103$	3.01827e49	1.46301	0.731503	+	0.681839i	$0.238819\pi$
$104$	0	0
$105$	4.96081e49	1.50112
$106$	0	0
$107$	4.53746e48	0.0864790	0.0432395	−	0.999065i	$-0.486232\pi$
$107$	4.53746e48	0.0864790	0.0432395	+	0.999065i	$0.486232\pi$
$108$	0	0
$109$	−1.12199e50	−1.35844	−0.679221	−	0.733933i	$-0.737682\pi$
$109$	−1.12199e50	−1.35844	−0.679221	+	0.733933i	$0.737682\pi$
$110$	0	0
$111$	5.20709e49	0.403815
$112$	0	0
$113$	−4.95437e49	−0.248066	−0.124033	−	0.992278i	$-0.539583\pi$
$113$	−4.95437e49	−0.248066	−0.124033	+	0.992278i	$0.539583\pi$
$114$	0	0
$115$	−6.64311e50	−2.16409
$116$	0	0
$117$	−9.48348e49	−0.202495
$118$	0	0
$119$	2.77110e50	0.390618
$120$	0	0
$121$	−1.06108e51	−0.994273
$122$	0	0
$123$	−3.46444e51	−2.17249
$124$	0	0
$125$	−1.25558e51	−0.530334
$126$	0	0
$127$	5.79879e50	0.166015	0.0830073	−	0.996549i	$-0.473548\pi$
$127$	5.79879e50	0.166015	0.0830073	+	0.996549i	$0.473548\pi$
$128$	0	0
$129$	1.35854e51	0.265230
$130$	0	0
$131$	−5.77681e51	−0.773645	−0.386822	−	0.922154i	$-0.626427\pi$
$131$	−5.77681e51	−0.773645	−0.386822	+	0.922154i	$0.626427\pi$
$132$	0	0
$133$	4.51026e51	0.416712
$134$	0	0
$135$	3.89698e52	2.49776
$136$	0	0
$137$	9.99643e51	0.446880	0.223440	−	0.974718i	$-0.428271\pi$
$137$	9.99643e51	0.446880	0.223440	+	0.974718i	$0.428271\pi$
$138$	0	0
$139$	5.47789e52	1.71692	0.858461	−	0.512878i	$-0.171421\pi$
$139$	5.47789e52	1.71692	0.858461	+	0.512878i	$0.171421\pi$
$140$	0	0
$141$	3.26853e52	0.721911
$142$	0	0
$143$	−4.61063e50	−0.00721153
$144$	0	0
$145$	7.93264e52	0.882880
$146$	0	0
$147$	−1.20634e53	−0.959828
$148$	0	0
$149$	−3.17720e53	−1.81543	−0.907717	−	0.419584i	$-0.862176\pi$
$149$	−3.17720e53	−1.81543	−0.907717	+	0.419584i	$0.862176\pi$
$150$	0	0
$151$	−4.36140e53	−1.79759	−0.898796	−	0.438367i	$-0.855557\pi$
$151$	−4.36140e53	−1.79759	−0.898796	+	0.438367i	$0.855557\pi$
$152$	0	0
$153$	4.11203e53	1.22777
$154$	0	0
$155$	9.24500e53	2.00808
$156$	0	0
$157$	6.03339e52	0.0957238	0.0478619	−	0.998854i	$-0.484759\pi$
$157$	6.03339e52	0.0957238	0.0478619	+	0.998854i	$0.484759\pi$
$158$	0	0
$159$	−6.38537e53	−0.742945
$160$	0	0
$161$	−1.35974e54	−1.16472
$162$	0	0
$163$	−1.51529e54	−0.959179	−0.479590	−	0.877493i	$-0.659214\pi$
$163$	−1.51529e54	−0.959179	−0.479590	+	0.877493i	$0.659214\pi$
$164$	0	0
$165$	3.57889e53	0.168032
$166$	0	0
$167$	−2.55570e54	−0.893217	−0.446608	−	0.894730i	$-0.647368\pi$
$167$	−2.55570e54	−0.893217	−0.446608	+	0.894730i	$0.647368\pi$
$168$	0	0
$169$	−3.79544e54	−0.990919
$170$	0	0
$171$	6.69275e54	1.30978
$172$	0	0
$173$	−8.31345e54	−1.22363	−0.611817	−	0.790999i	$-0.709561\pi$
$173$	−8.31345e54	−1.22363	−0.611817	+	0.790999i	$0.709561\pi$
$174$	0	0
$175$	3.51704e54	0.390612
$176$	0	0
$177$	2.76520e55	2.32478
$178$	0	0
$179$	−5.23266e54	−0.334059	−0.167030	−	0.985952i	$-0.553418\pi$
$179$	−5.23266e54	−0.334059	−0.167030	+	0.985952i	$0.553418\pi$
$180$	0	0
$181$	1.38390e54	0.0672946	0.0336473	−	0.999434i	$-0.489288\pi$
$181$	1.38390e54	0.0672946	0.0336473	+	0.999434i	$0.489288\pi$
$182$	0	0
$183$	2.15083e55	0.799013
$184$	0	0
$185$	1.00809e55	0.286940
$186$	0	0
$187$	1.99916e54	0.0437249
$188$	0	0
$189$	7.97650e55	1.34431
$190$	0	0
$191$	4.46194e54	0.0581040	0.0290520	−	0.999578i	$-0.490751\pi$
$191$	4.46194e54	0.0581040	0.0290520	+	0.999578i	$0.490751\pi$
$192$	0	0
$193$	−3.91405e55	−0.394886	−0.197443	−	0.980314i	$-0.563264\pi$
$193$	−3.91405e55	−0.394886	−0.197443	+	0.980314i	$0.563264\pi$
$194$	0	0
$195$	−2.70003e55	−0.211602
$196$	0	0
$197$	8.64318e55	0.527535	0.263767	−	0.964586i	$-0.415035\pi$
$197$	8.64318e55	0.527535	0.263767	+	0.964586i	$0.415035\pi$
$198$	0	0
$199$	1.38403e56	0.659545	0.329773	−	0.944060i	$-0.393028\pi$
$199$	1.38403e56	0.659545	0.329773	+	0.944060i	$0.393028\pi$
$200$	0	0
$201$	−4.98802e56	−1.86048
$202$	0	0
$203$	1.62368e56	0.475170
$204$	0	0
$205$	−6.70710e56	−1.54371
$206$	0	0
$207$	−2.01771e57	−3.66088
$208$	0	0
$209$	3.25385e55	0.0466458
$210$	0	0
$211$	1.39140e57	1.57954	0.789770	−	0.613404i	$-0.210200\pi$
$211$	1.39140e57	1.57954	0.789770	+	0.613404i	$0.210200\pi$
$212$	0	0
$213$	1.43925e57	1.29669
$214$	0	0
$215$	2.63011e56	0.188465
$216$	0	0
$217$	1.89230e57	1.08076
$218$	0	0
$219$	4.15860e57	1.89699
$220$	0	0
$221$	−1.50824e56	−0.0550628
$222$	0	0
$223$	−2.03036e57	−0.594434	−0.297217	−	0.954810i	$-0.596059\pi$
$223$	−2.03036e57	−0.594434	−0.297217	+	0.954810i	$0.596059\pi$
$224$	0	0
$225$	5.21892e57	1.22775
$226$	0	0
$227$	4.69172e57	0.888585	0.444292	−	0.895882i	$-0.353455\pi$
$227$	4.69172e57	0.888585	0.444292	+	0.895882i	$0.353455\pi$
$228$	0	0
$229$	5.97819e57	0.913273	0.456636	−	0.889653i	$-0.349054\pi$
$229$	5.97819e57	0.913273	0.456636	+	0.889653i	$0.349054\pi$
$230$	0	0
$231$	7.32541e56	0.0904354
$232$	0	0
$233$	−4.46843e57	−0.446617	−0.223308	−	0.974748i	$-0.571686\pi$
$233$	−4.46843e57	−0.446617	−0.223308	+	0.974748i	$0.571686\pi$
$234$	0	0
$235$	6.32782e57	0.512970
$236$	0	0
$237$	3.64098e57	0.239819
$238$	0	0
$239$	4.13711e57	0.221793	0.110897	−	0.993832i	$-0.464628\pi$
$239$	4.13711e57	0.221793	0.110897	+	0.993832i	$0.464628\pi$
$240$	0	0
$241$	−2.18261e58	−0.954022	−0.477011	−	0.878897i	$-0.658280\pi$
$241$	−2.18261e58	−0.954022	−0.477011	+	0.878897i	$0.658280\pi$
$242$	0	0
$243$	1.31409e58	0.469105
$244$	0	0
$245$	−2.33545e58	−0.682027
$246$	0	0
$247$	−2.45481e57	−0.0587411
$248$	0	0
$249$	1.03916e59	2.04078
$250$	0	0
$251$	−5.93439e58	−0.958009	−0.479005	−	0.877812i	$-0.659002\pi$
$251$	−5.93439e58	−0.958009	−0.479005	+	0.877812i	$0.659002\pi$
$252$	0	0
$253$	−9.80960e57	−0.130376
$254$	0	0
$255$	1.17073e59	1.28299
$256$	0	0
$257$	9.93030e58	0.898671	0.449336	−	0.893363i	$-0.351661\pi$
$257$	9.93030e58	0.898671	0.449336	+	0.893363i	$0.351661\pi$
$258$	0	0
$259$	2.06339e58	0.154432
$260$	0	0
$261$	2.40938e59	1.49352
$262$	0	0
$263$	6.39723e58	0.328909	0.164455	−	0.986385i	$-0.447414\pi$
$263$	6.39723e58	0.328909	0.164455	+	0.986385i	$0.447414\pi$
$264$	0	0
$265$	−1.23620e59	−0.527916
$266$	0	0
$267$	7.52752e59	2.67378
$268$	0	0
$269$	3.93934e59	1.16545	0.582723	−	0.812671i	$-0.301987\pi$
$269$	3.93934e59	1.16545	0.582723	+	0.812671i	$0.301987\pi$
$270$	0	0
$271$	−2.92581e59	−0.721935	−0.360968	−	0.932578i	$-0.617554\pi$
$271$	−2.92581e59	−0.721935	−0.360968	+	0.932578i	$0.617554\pi$
$272$	0	0
$273$	−5.52653e58	−0.113885
$274$	0	0
$275$	2.53731e58	0.0437243
$276$	0	0
$277$	8.20576e59	1.18403	0.592017	−	0.805925i	$-0.298332\pi$
$277$	8.20576e59	1.18403	0.592017	+	0.805925i	$0.298332\pi$
$278$	0	0
$279$	2.80798e60	3.39696
$280$	0	0
$281$	1.23805e60	1.25729	0.628643	−	0.777694i	$-0.283611\pi$
$281$	1.23805e60	1.25729	0.628643	+	0.777694i	$0.283611\pi$
$282$	0	0
$283$	1.21811e60	1.03972	0.519862	−	0.854250i	$-0.325983\pi$
$283$	1.21811e60	1.03972	0.519862	+	0.854250i	$0.325983\pi$
$284$	0	0
$285$	1.90548e60	1.36869
$286$	0	0
$287$	−1.37284e60	−0.830830
$288$	0	0
$289$	−1.30486e60	−0.666143
$290$	0	0
$291$	−6.32009e60	−2.72487
$292$	0	0
$293$	−2.58132e60	−0.940992	−0.470496	−	0.882402i	$-0.655925\pi$
$293$	−2.58132e60	−0.940992	−0.470496	+	0.882402i	$0.655925\pi$
$294$	0	0
$295$	5.35340e60	1.65192
$296$	0	0
$297$	5.75451e59	0.150479
$298$	0	0
$299$	7.40068e59	0.164183
$300$	0	0
$301$	5.38341e59	0.101433
$302$	0	0
$303$	9.24416e59	0.148089
$304$	0	0
$305$	4.16398e60	0.567756
$306$	0	0
$307$	−2.63375e60	−0.305974	−0.152987	−	0.988228i	$-0.548889\pi$
$307$	−2.63375e60	−0.305974	−0.152987	+	0.988228i	$0.548889\pi$
$308$	0	0
$309$	2.61003e61	2.58621
$310$	0	0
$311$	1.91232e61	1.61783	0.808913	−	0.587928i	$-0.200056\pi$
$311$	1.91232e61	1.61783	0.808913	+	0.587928i	$0.200056\pi$
$312$	0	0
$313$	−1.92491e61	−1.39179	−0.695896	−	0.718143i	$-0.744992\pi$
$313$	−1.92491e61	−1.39179	−0.695896	+	0.718143i	$0.744992\pi$
$314$	0	0
$315$	2.91704e61	1.80440
$316$	0	0
$317$	4.85204e60	0.257023	0.128512	−	0.991708i	$-0.458980\pi$
$317$	4.85204e60	0.257023	0.128512	+	0.991708i	$0.458980\pi$
$318$	0	0
$319$	1.17138e60	0.0531895
$320$	0	0
$321$	3.92375e60	0.152872
$322$	0	0
$323$	1.06440e61	0.356159
$324$	0	0
$325$	−1.91423e60	−0.0550619
$326$	0	0
$327$	−9.70238e61	−2.40137
$328$	0	0
$329$	1.29520e61	0.276082
$330$	0	0
$331$	3.46456e61	0.636594	0.318297	−	0.947991i	$-0.396889\pi$
$331$	3.46456e61	0.636594	0.318297	+	0.947991i	$0.396889\pi$
$332$	0	0
$333$	3.06186e61	0.485401
$334$	0	0
$335$	−9.65673e61	−1.32200
$336$	0	0
$337$	−1.47496e62	−1.74521	−0.872605	−	0.488427i	$-0.837571\pi$
$337$	−1.47496e62	−1.74521	−0.872605	+	0.488427i	$0.837571\pi$
$338$	0	0
$339$	−4.28427e61	−0.438515
$340$	0	0
$341$	1.36517e61	0.120977
$342$	0	0
$343$	−1.35843e62	−1.04311
$344$	0	0
$345$	−5.74459e62	−3.82553
$346$	0	0
$347$	−1.49446e62	−0.863799	−0.431900	−	0.901922i	$-0.642157\pi$
$347$	−1.49446e62	−0.863799	−0.431900	+	0.901922i	$0.642157\pi$
$348$	0	0
$349$	2.27300e62	1.14124	0.570620	−	0.821214i	$-0.306703\pi$
$349$	2.27300e62	1.14124	0.570620	+	0.821214i	$0.306703\pi$
$350$	0	0
$351$	−4.34139e61	−0.189498
$352$	0	0
$353$	−2.58609e61	−0.0982119	−0.0491059	−	0.998794i	$-0.515637\pi$
$353$	−2.58609e61	−0.0982119	−0.0491059	+	0.998794i	$0.515637\pi$
$354$	0	0
$355$	2.78636e62	0.921389
$356$	0	0
$357$	2.39630e62	0.690509
$358$	0	0
$359$	5.12401e62	1.28765	0.643823	−	0.765175i	$-0.277347\pi$
$359$	5.12401e62	1.28765	0.643823	+	0.765175i	$0.277347\pi$
$360$	0	0
$361$	−2.82717e62	−0.620049
$362$	0	0
$363$	−9.17562e62	−1.75761
$364$	0	0
$365$	8.05099e62	1.34795
$366$	0	0
$367$	−1.03293e61	−0.0151270	−0.00756348	−	0.999971i	$-0.502408\pi$
$367$	−1.03293e61	−0.0151270	−0.00756348	+	0.999971i	$0.502408\pi$
$368$	0	0
$369$	−2.03714e63	−2.61141
$370$	0	0
$371$	−2.53030e62	−0.284127
$372$	0	0
$373$	6.36683e62	0.626696	0.313348	−	0.949638i	$-0.398549\pi$
$373$	6.36683e62	0.626696	0.313348	+	0.949638i	$0.398549\pi$
$374$	0	0
$375$	−1.08576e63	−0.937490
$376$	0	0
$377$	−8.83726e61	−0.0669815
$378$	0	0
$379$	−1.19758e63	−0.797340	−0.398670	−	0.917094i	$-0.630528\pi$
$379$	−1.19758e63	−0.797340	−0.398670	+	0.917094i	$0.630528\pi$
$380$	0	0
$381$	5.01447e62	0.293470
$382$	0	0
$383$	1.72054e63	0.885717	0.442859	−	0.896591i	$-0.353964\pi$
$383$	1.72054e63	0.885717	0.442859	+	0.896591i	$0.353964\pi$
$384$	0	0
$385$	1.41819e62	0.0642609
$386$	0	0
$387$	7.98841e62	0.318817
$388$	0	0
$389$	−1.24290e63	−0.437189	−0.218595	−	0.975816i	$-0.570147\pi$
$389$	−1.24290e63	−0.437189	−0.218595	+	0.975816i	$0.570147\pi$
$390$	0	0
$391$	−3.20893e63	−0.995473
$392$	0	0
$393$	−4.99547e63	−1.36760
$394$	0	0
$395$	7.04889e62	0.170409
$396$	0	0
$397$	5.98417e63	1.27831	0.639155	−	0.769078i	$-0.279284\pi$
$397$	5.98417e63	1.27831	0.639155	+	0.769078i	$0.279284\pi$
$398$	0	0
$399$	3.90022e63	0.736636
$400$	0	0
$401$	−9.27508e63	−1.54982	−0.774909	−	0.632073i	$-0.782204\pi$
$401$	−9.27508e63	−1.54982	−0.774909	+	0.632073i	$0.782204\pi$
$402$	0	0
$403$	−1.02993e63	−0.152347
$404$	0	0
$405$	1.33281e64	1.74631
$406$	0	0
$407$	1.48860e62	0.0172868
$408$	0	0
$409$	−9.43587e63	−0.971764	−0.485882	−	0.874024i	$-0.661502\pi$
$409$	−9.43587e63	−0.971764	−0.485882	+	0.874024i	$0.661502\pi$
$410$	0	0
$411$	8.64436e63	0.789966
$412$	0	0
$413$	1.09576e64	0.889073
$414$	0	0
$415$	2.01179e64	1.45012
$416$	0	0
$417$	4.73698e64	3.03506
$418$	0	0
$419$	−7.04245e63	−0.401310	−0.200655	−	0.979662i	$-0.564307\pi$
$419$	−7.04245e63	−0.401310	−0.200655	+	0.979662i	$0.564307\pi$
$420$	0	0
$421$	1.20251e64	0.609788	0.304894	−	0.952386i	$-0.401379\pi$
$421$	1.20251e64	0.609788	0.304894	+	0.952386i	$0.401379\pi$
$422$	0	0
$423$	1.92195e64	0.867765
$424$	0	0
$425$	8.30008e63	0.333852
$426$	0	0
$427$	8.52299e63	0.305569
$428$	0	0
$429$	−3.98702e62	−0.0127481
$430$	0	0
$431$	−4.44020e63	−0.126680	−0.0633402	−	0.997992i	$-0.520175\pi$
$431$	−4.44020e63	−0.126680	−0.0633402	+	0.997992i	$0.520175\pi$
$432$	0	0
$433$	−1.83425e63	−0.0467201	−0.0233601	−	0.999727i	$-0.507436\pi$
$433$	−1.83425e63	−0.0467201	−0.0233601	+	0.999727i	$0.507436\pi$
$434$	0	0
$435$	6.85971e64	1.56070
$436$	0	0
$437$	−5.22286e64	−1.06197
$438$	0	0
$439$	1.95740e64	0.355876	0.177938	−	0.984042i	$-0.443057\pi$
$439$	1.95740e64	0.355876	0.177938	+	0.984042i	$0.443057\pi$
$440$	0	0
$441$	−7.09345e64	−1.15375
$442$	0	0
$443$	3.98745e63	0.0580502	0.0290251	−	0.999579i	$-0.490760\pi$
$443$	3.98745e63	0.0580502	0.0290251	+	0.999579i	$0.490760\pi$
$444$	0	0
$445$	1.45732e65	1.89991
$446$	0	0
$447$	−2.74746e65	−3.20921
$448$	0	0
$449$	1.31465e64	0.137649	0.0688246	−	0.997629i	$-0.478075\pi$
$449$	1.31465e64	0.137649	0.0688246	+	0.997629i	$0.478075\pi$
$450$	0	0
$451$	−9.90409e63	−0.0930013
$452$	0	0
$453$	−3.77150e65	−3.17767
$454$	0	0
$455$	−1.06993e64	−0.0809237
$456$	0	0
$457$	1.24818e65	0.847873	0.423937	−	0.905692i	$-0.360648\pi$
$457$	1.24818e65	0.847873	0.423937	+	0.905692i	$0.360648\pi$
$458$	0	0
$459$	1.88242e65	1.14896
$460$	0	0
$461$	−6.76961e64	−0.371442	−0.185721	−	0.982603i	$-0.559462\pi$
$461$	−6.76961e64	−0.371442	−0.185721	+	0.982603i	$0.559462\pi$
$462$	0	0
$463$	2.06723e65	1.02013	0.510063	−	0.860137i	$-0.329622\pi$
$463$	2.06723e65	1.02013	0.510063	+	0.860137i	$0.329622\pi$
$464$	0	0
$465$	7.99456e65	3.54975
$466$	0	0
$467$	−2.20986e65	−0.883284	−0.441642	−	0.897191i	$-0.645604\pi$
$467$	−2.20986e65	−0.883284	−0.441642	+	0.897191i	$0.645604\pi$
$468$	0	0
$469$	−1.97658e65	−0.711507
$470$	0	0
$471$	5.21734e64	0.169214
$472$	0	0
$473$	3.88377e63	0.0113542
$474$	0	0
$475$	1.35092e65	0.356153
$476$	0	0
$477$	−3.75470e65	−0.893049
$478$	0	0
$479$	−7.25016e64	−0.155642	−0.0778211	−	0.996967i	$-0.524796\pi$
$479$	−7.25016e64	−0.155642	−0.0778211	+	0.996967i	$0.524796\pi$
$480$	0	0
$481$	−1.12305e64	−0.0217693
$482$	0	0
$483$	−1.17583e66	−2.05892
$484$	0	0
$485$	−1.22356e66	−1.93621
$486$	0	0
$487$	5.28007e65	0.755405	0.377702	−	0.925927i	$-0.376714\pi$
$487$	5.28007e65	0.755405	0.377702	+	0.925927i	$0.376714\pi$
$488$	0	0
$489$	−1.31034e66	−1.69557
$490$	0	0
$491$	−1.31061e66	−1.53454	−0.767271	−	0.641324i	$-0.778386\pi$
$491$	−1.31061e66	−1.53454	−0.767271	+	0.641324i	$0.778386\pi$
$492$	0	0
$493$	3.83183e65	0.406122
$494$	0	0
$495$	2.10444e65	0.201981
$496$	0	0
$497$	5.70324e65	0.495895
$498$	0	0
$499$	2.40608e65	0.189603	0.0948016	−	0.995496i	$-0.469778\pi$
$499$	2.40608e65	0.189603	0.0948016	+	0.995496i	$0.469778\pi$
$500$	0	0
$501$	−2.21003e66	−1.57897
$502$	0	0
$503$	−1.13819e66	−0.737560	−0.368780	−	0.929517i	$-0.620224\pi$
$503$	−1.13819e66	−0.737560	−0.368780	+	0.929517i	$0.620224\pi$
$504$	0	0
$505$	1.78966e65	0.105228
$506$	0	0
$507$	−3.28209e66	−1.75168
$508$	0	0
$509$	1.25631e66	0.608849	0.304425	−	0.952536i	$-0.401536\pi$
$509$	1.25631e66	0.608849	0.304425	+	0.952536i	$0.401536\pi$
$510$	0	0
$511$	1.64791e66	0.725472
$512$	0	0
$513$	3.06383e66	1.22572
$514$	0	0
$515$	5.05299e66	1.83769
$516$	0	0
$517$	9.34402e64	0.0309041
$518$	0	0
$519$	−7.18901e66	−2.16306
$520$	0	0
$521$	−5.06272e66	−1.38631	−0.693154	−	0.720789i	$-0.743780\pi$
$521$	−5.06272e66	−1.38631	−0.693154	+	0.720789i	$0.743780\pi$
$522$	0	0
$523$	2.50752e66	0.625105	0.312552	−	0.949901i	$-0.398816\pi$
$523$	2.50752e66	0.625105	0.312552	+	0.949901i	$0.398816\pi$
$524$	0	0
$525$	3.04134e66	0.690498
$526$	0	0
$527$	4.46576e66	0.923709
$528$	0	0
$529$	1.04410e67	1.96824
$530$	0	0
$531$	1.62599e67	2.79448
$532$	0	0
$533$	7.47197e65	0.117117
$534$	0	0
$535$	7.59632e65	0.108627
$536$	0	0
$537$	−4.52492e66	−0.590528
$538$	0	0
$539$	−3.44866e65	−0.0410890
$540$	0	0
$541$	9.85403e66	1.07221	0.536106	−	0.844151i	$-0.319895\pi$
$541$	9.85403e66	1.07221	0.536106	+	0.844151i	$0.319895\pi$
$542$	0	0
$543$	1.19672e66	0.118959
$544$	0	0
$545$	−1.87837e67	−1.70634
$546$	0	0
$547$	−2.04995e67	−1.70237	−0.851185	−	0.524866i	$-0.824115\pi$
$547$	−2.04995e67	−1.70237	−0.851185	+	0.524866i	$0.824115\pi$
$548$	0	0
$549$	1.26472e67	0.960445
$550$	0	0
$551$	6.23669e66	0.433252
$552$	0	0
$553$	1.44279e66	0.0917147
$554$	0	0
$555$	8.71737e66	0.507233
$556$	0	0
$557$	2.90683e67	1.54870	0.774351	−	0.632757i	$-0.218077\pi$
$557$	2.90683e67	1.54870	0.774351	+	0.632757i	$0.218077\pi$
$558$	0	0
$559$	−2.93004e65	−0.0142983
$560$	0	0
$561$	1.72877e66	0.0772941
$562$	0	0
$563$	−6.00209e65	−0.0245950	−0.0122975	−	0.999924i	$-0.503915\pi$
$563$	−6.00209e65	−0.0245950	−0.0122975	+	0.999924i	$0.503915\pi$
$564$	0	0
$565$	−8.29429e66	−0.311596
$566$	0	0
$567$	2.72805e67	0.939869
$568$	0	0
$569$	−2.48462e67	−0.785252	−0.392626	−	0.919698i	$-0.628433\pi$
$569$	−2.48462e67	−0.785252	−0.392626	+	0.919698i	$0.628433\pi$
$570$	0	0
$571$	8.26527e66	0.239702	0.119851	−	0.992792i	$-0.461758\pi$
$571$	8.26527e66	0.239702	0.119851	+	0.992792i	$0.461758\pi$
$572$	0	0
$573$	3.85844e66	0.102712
$574$	0	0
$575$	−4.07272e67	−0.995457
$576$	0	0
$577$	−1.45109e67	−0.325751	−0.162875	−	0.986647i	$-0.552077\pi$
$577$	−1.45109e67	−0.325751	−0.162875	+	0.986647i	$0.552077\pi$
$578$	0	0
$579$	−3.38465e67	−0.698054
$580$	0	0
$581$	4.11782e67	0.780463
$582$	0	0
$583$	−1.82544e66	−0.0318045
$584$	0	0
$585$	−1.58766e67	−0.254354
$586$	0	0
$587$	8.02541e67	1.18259	0.591295	−	0.806456i	$-0.298617\pi$
$587$	8.02541e67	1.18259	0.591295	+	0.806456i	$0.298617\pi$
$588$	0	0
$589$	7.26848e67	0.985414
$590$	0	0
$591$	7.47415e67	0.932542
$592$	0	0
$593$	−1.46805e68	−1.68617	−0.843086	−	0.537779i	$-0.819263\pi$
$593$	−1.46805e68	−1.68617	−0.843086	+	0.537779i	$0.819263\pi$
$594$	0	0
$595$	4.63920e67	0.490656
$596$	0	0
$597$	1.19683e68	1.16590
$598$	0	0
$599$	9.17136e67	0.823145	0.411573	−	0.911377i	$-0.364980\pi$
$599$	9.17136e67	0.823145	0.411573	+	0.911377i	$0.364980\pi$
$600$	0	0
$601$	−2.27601e67	−0.188257	−0.0941283	−	0.995560i	$-0.530006\pi$
$601$	−2.27601e67	−0.188257	−0.0941283	+	0.995560i	$0.530006\pi$
$602$	0	0
$603$	−2.93303e68	−2.23637
$604$	0	0
$605$	−1.77639e68	−1.24891
$606$	0	0
$607$	−1.39704e68	−0.905914	−0.452957	−	0.891532i	$-0.649631\pi$
$607$	−1.39704e68	−0.905914	−0.452957	+	0.891532i	$0.649631\pi$
$608$	0	0
$609$	1.40407e68	0.839974
$610$	0	0
$611$	−7.04944e66	−0.0389175
$612$	0	0
$613$	−1.66422e68	−0.848066	−0.424033	−	0.905647i	$-0.639386\pi$
$613$	−1.66422e68	−0.848066	−0.424033	+	0.905647i	$0.639386\pi$
$614$	0	0
$615$	−5.79993e68	−2.72887
$616$	0	0
$617$	7.60071e67	0.330269	0.165134	−	0.986271i	$-0.447194\pi$
$617$	7.60071e67	0.330269	0.165134	+	0.986271i	$0.447194\pi$
$618$	0	0
$619$	−3.95260e68	−1.58658	−0.793291	−	0.608843i	$-0.791634\pi$
$619$	−3.95260e68	−1.58658	−0.793291	+	0.608843i	$0.791634\pi$
$620$	0	0
$621$	−9.23676e68	−3.42590
$622$	0	0
$623$	2.98290e68	1.02254
$624$	0	0
$625$	−3.92521e68	−1.24395
$626$	0	0
$627$	2.81375e67	0.0824575
$628$	0	0
$629$	4.86952e67	0.131991
$630$	0	0
$631$	2.28574e68	0.573201	0.286600	−	0.958050i	$-0.407475\pi$
$631$	2.28574e68	0.573201	0.286600	+	0.958050i	$0.407475\pi$
$632$	0	0
$633$	1.20320e69	2.79221
$634$	0	0
$635$	9.70796e67	0.208531
$636$	0	0
$637$	2.60178e67	0.0517433
$638$	0	0
$639$	8.46300e68	1.55867
$640$	0	0
$641$	−1.91449e68	−0.326611	−0.163305	−	0.986576i	$-0.552216\pi$
$641$	−1.91449e68	−0.326611	−0.163305	+	0.986576i	$0.552216\pi$
$642$	0	0
$643$	2.09210e68	0.330684	0.165342	−	0.986236i	$-0.447127\pi$
$643$	2.09210e68	0.330684	0.165342	+	0.986236i	$0.447127\pi$
$644$	0	0
$645$	2.27437e68	0.333157
$646$	0	0
$647$	−9.01802e68	−1.22449	−0.612247	−	0.790667i	$-0.709734\pi$
$647$	−9.01802e68	−1.22449	−0.612247	+	0.790667i	$0.709734\pi$
$648$	0	0
$649$	7.90514e67	0.0995209
$650$	0	0
$651$	1.63636e69	1.91049
$652$	0	0
$653$	−8.06774e68	−0.873734	−0.436867	−	0.899526i	$-0.643912\pi$
$653$	−8.06774e68	−0.873734	−0.436867	+	0.899526i	$0.643912\pi$
$654$	0	0
$655$	−9.67116e68	−0.971777
$656$	0	0
$657$	2.44532e69	2.28026
$658$	0	0
$659$	1.45122e69	1.25615	0.628075	−	0.778153i	$-0.283843\pi$
$659$	1.45122e69	1.25615	0.628075	+	0.778153i	$0.283843\pi$
$660$	0	0
$661$	1.72306e69	1.38473	0.692365	−	0.721548i	$-0.256569\pi$
$661$	1.72306e69	1.38473	0.692365	+	0.721548i	$0.256569\pi$
$662$	0	0
$663$	−1.30424e68	−0.0973364
$664$	0	0
$665$	7.55077e68	0.523433
$666$	0	0
$667$	−1.88022e69	−1.21095
$668$	0	0
$669$	−1.75574e69	−1.05080
$670$	0	0
$671$	6.14876e67	0.0342047
$672$	0	0
$673$	−3.25113e69	−1.68138	−0.840690	−	0.541517i	$-0.817850\pi$
$673$	−3.25113e69	−1.68138	−0.840690	+	0.541517i	$0.817850\pi$
$674$	0	0
$675$	2.38914e69	1.14895
$676$	0	0
$677$	−2.07144e69	−0.926512	−0.463256	−	0.886225i	$-0.653319\pi$
$677$	−2.07144e69	−0.926512	−0.463256	+	0.886225i	$0.653319\pi$
$678$	0	0
$679$	−2.50444e69	−1.04208
$680$	0	0
$681$	4.05714e69	1.57078
$682$	0	0
$683$	−1.47023e69	−0.529760	−0.264880	−	0.964281i	$-0.585332\pi$
$683$	−1.47023e69	−0.529760	−0.264880	+	0.964281i	$0.585332\pi$
$684$	0	0
$685$	1.67354e69	0.561327
$686$	0	0
$687$	5.16961e69	1.61442
$688$	0	0
$689$	1.37717e68	0.0400514
$690$	0	0
$691$	−1.33225e69	−0.360890	−0.180445	−	0.983585i	$-0.557754\pi$
$691$	−1.33225e69	−0.360890	−0.180445	+	0.983585i	$0.557754\pi$
$692$	0	0
$693$	4.30746e68	0.108707
$694$	0	0
$695$	9.17073e69	2.15663
$696$	0	0
$697$	−3.23984e69	−0.710101
$698$	0	0
$699$	−3.86405e69	−0.789500
$700$	0	0
$701$	7.12825e69	1.35797	0.678987	−	0.734150i	$-0.262419\pi$
$701$	7.12825e69	1.35797	0.678987	+	0.734150i	$0.262419\pi$
$702$	0	0
$703$	7.92564e68	0.140809
$704$	0	0
$705$	5.47195e69	0.906795
$706$	0	0
$707$	3.66314e68	0.0566341
$708$	0	0
$709$	4.28430e69	0.618083	0.309042	−	0.951049i	$-0.399992\pi$
$709$	4.28430e69	0.618083	0.309042	+	0.951049i	$0.399992\pi$
$710$	0	0
$711$	2.14096e69	0.288272
$712$	0	0
$713$	−2.19128e70	−2.75426
$714$	0	0
$715$	−7.71881e67	−0.00905842
$716$	0	0
$717$	3.57754e69	0.392072
$718$	0	0
$719$	−3.53639e69	−0.361995	−0.180997	−	0.983484i	$-0.557933\pi$
$719$	−3.53639e69	−0.361995	−0.180997	+	0.983484i	$0.557933\pi$
$720$	0	0
$721$	1.03427e70	0.989050
$722$	0	0
$723$	−1.88740e70	−1.68646
$724$	0	0
$725$	4.86330e69	0.406116
$726$	0	0
$727$	3.98958e69	0.311411	0.155706	−	0.987804i	$-0.450235\pi$
$727$	3.98958e69	0.311411	0.155706	+	0.987804i	$0.450235\pi$
$728$	0	0
$729$	−7.68760e69	−0.561004
$730$	0	0
$731$	1.27046e69	0.0866934
$732$	0	0
$733$	−8.35656e69	−0.533310	−0.266655	−	0.963792i	$-0.585918\pi$
$733$	−8.35656e69	−0.533310	−0.266655	+	0.963792i	$0.585918\pi$
$734$	0	0
$735$	−2.01957e70	−1.20564
$736$	0	0
$737$	−1.42597e69	−0.0796446
$738$	0	0
$739$	1.56314e70	0.816974	0.408487	−	0.912764i	$-0.366056\pi$
$739$	1.56314e70	0.816974	0.408487	+	0.912764i	$0.366056\pi$
$740$	0	0
$741$	−2.12278e69	−0.103839
$742$	0	0
$743$	−3.92359e70	−1.79662	−0.898312	−	0.439357i	$-0.855206\pi$
$743$	−3.92359e70	−1.79662	−0.898312	+	0.439357i	$0.855206\pi$
$744$	0	0
$745$	−5.31905e70	−2.28037
$746$	0	0
$747$	6.11041e70	2.45310
$748$	0	0
$749$	1.55485e69	0.0584632
$750$	0	0
$751$	−1.76438e70	−0.621462	−0.310731	−	0.950498i	$-0.600574\pi$
$751$	−1.76438e70	−0.621462	−0.310731	+	0.950498i	$0.600574\pi$
$752$	0	0
$753$	−5.13173e70	−1.69351
$754$	0	0
$755$	−7.30158e70	−2.25796
$756$	0	0
$757$	−1.73845e70	−0.503864	−0.251932	−	0.967745i	$-0.581066\pi$
$757$	−1.73845e70	−0.503864	−0.251932	+	0.967745i	$0.581066\pi$
$758$	0	0
$759$	−8.48280e69	−0.230471
$760$	0	0
$761$	−1.36361e70	−0.347349	−0.173675	−	0.984803i	$-0.555564\pi$
$761$	−1.36361e70	−0.347349	−0.173675	+	0.984803i	$0.555564\pi$
$762$	0	0
$763$	−3.84472e70	−0.918361
$764$	0	0
$765$	6.88409e70	1.54220
$766$	0	0
$767$	−5.96389e69	−0.125327
$768$	0	0
$769$	8.59382e70	1.69430	0.847152	−	0.531351i	$-0.178316\pi$
$769$	8.59382e70	1.69430	0.847152	+	0.531351i	$0.178316\pi$
$770$	0	0
$771$	8.58718e70	1.58861
$772$	0	0
$773$	4.66524e69	0.0809983	0.0404992	−	0.999180i	$-0.487105\pi$
$773$	4.66524e69	0.0809983	0.0404992	+	0.999180i	$0.487105\pi$
$774$	0	0
$775$	5.66787e70	0.923695
$776$	0	0
$777$	1.78431e70	0.272995
$778$	0	0
$779$	−5.27317e70	−0.757537
$780$	0	0
$781$	4.11450e69	0.0555094
$782$	0	0
$783$	1.10297e71	1.39766
$784$	0	0
$785$	1.01007e70	0.120239
$786$	0	0
$787$	6.73785e70	0.753599	0.376799	−	0.926295i	$-0.377025\pi$
$787$	6.73785e70	0.753599	0.376799	+	0.926295i	$0.377025\pi$
$788$	0	0
$789$	5.53197e70	0.581425
$790$	0	0
$791$	−1.69771e70	−0.167702
$792$	0	0
$793$	−4.63883e69	−0.0430740
$794$	0	0
$795$	−1.06900e71	−0.933215
$796$	0	0
$797$	−1.67868e71	−1.37797	−0.688985	−	0.724775i	$-0.741944\pi$
$797$	−1.67868e71	−1.37797	−0.688985	+	0.724775i	$0.741944\pi$
$798$	0	0
$799$	3.05663e70	0.235964
$800$	0	0
$801$	4.42630e71	3.21399
$802$	0	0
$803$	1.18885e70	0.0812078
$804$	0	0
$805$	−2.27638e71	−1.46301
$806$	0	0
$807$	3.40652e71	2.06020
$808$	0	0
$809$	7.08887e70	0.403495	0.201748	−	0.979438i	$-0.435338\pi$
$809$	7.08887e70	0.403495	0.201748	+	0.979438i	$0.435338\pi$
$810$	0	0
$811$	1.30313e71	0.698193	0.349097	−	0.937087i	$-0.386488\pi$
$811$	1.30313e71	0.698193	0.349097	+	0.937087i	$0.386488\pi$
$812$	0	0
$813$	−2.53008e71	−1.27619
$814$	0	0
$815$	−2.53680e71	−1.20483
$816$	0	0
$817$	2.06781e70	0.0924847
$818$	0	0
$819$	−3.24969e70	−0.136895
$820$	0	0
$821$	6.12434e70	0.243026	0.121513	−	0.992590i	$-0.461225\pi$
$821$	6.12434e70	0.243026	0.121513	+	0.992590i	$0.461225\pi$
$822$	0	0
$823$	−4.66189e71	−1.74288	−0.871438	−	0.490506i	$-0.836812\pi$
$823$	−4.66189e71	−1.74288	−0.871438	+	0.490506i	$0.836812\pi$
$824$	0	0
$825$	2.19412e70	0.0772929
$826$	0	0
$827$	−1.89180e71	−0.628043	−0.314022	−	0.949416i	$-0.601676\pi$
$827$	−1.89180e71	−0.628043	−0.314022	+	0.949416i	$0.601676\pi$
$828$	0	0
$829$	−1.94307e71	−0.607998	−0.303999	−	0.952672i	$-0.598322\pi$
$829$	−1.94307e71	−0.607998	−0.303999	+	0.952672i	$0.598322\pi$
$830$	0	0
$831$	7.09589e71	2.09306
$832$	0	0
$833$	−1.12813e71	−0.313730
$834$	0	0
$835$	−4.27859e71	−1.12197
$836$	0	0
$837$	1.28545e72	3.17893
$838$	0	0
$839$	1.81889e71	0.424266	0.212133	−	0.977241i	$-0.431959\pi$
$839$	1.81889e71	0.424266	0.212133	+	0.977241i	$0.431959\pi$
$840$	0	0
$841$	−2.29947e71	−0.505970
$842$	0	0
$843$	1.07060e72	2.22255
$844$	0	0
$845$	−6.35408e71	−1.24470
$846$	0	0
$847$	−3.63598e71	−0.672168
$848$	0	0
$849$	1.05335e72	1.83796
$850$	0	0
$851$	−2.38940e71	−0.393563
$852$	0	0
$853$	−8.16161e70	−0.126918	−0.0634592	−	0.997984i	$-0.520213\pi$
$853$	−8.16161e70	−0.126918	−0.0634592	+	0.997984i	$0.520213\pi$
$854$	0	0
$855$	1.12046e72	1.64522
$856$	0	0
$857$	1.39103e72	1.92889	0.964445	−	0.264285i	$-0.0851358\pi$
$857$	1.39103e72	1.92889	0.964445	+	0.264285i	$0.0851358\pi$
$858$	0	0
$859$	−3.97272e71	−0.520301	−0.260151	−	0.965568i	$-0.583772\pi$
$859$	−3.97272e71	−0.520301	−0.260151	+	0.965568i	$0.583772\pi$
$860$	0	0
$861$	−1.18715e72	−1.46869
$862$	0	0
$863$	−4.45935e71	−0.521203	−0.260601	−	0.965446i	$-0.583921\pi$
$863$	−4.45935e71	−0.521203	−0.260601	+	0.965446i	$0.583921\pi$
$864$	0	0
$865$	−1.39178e72	−1.53701
$866$	0	0
$867$	−1.12837e72	−1.17756
$868$	0	0
$869$	1.04088e70	0.0102663
$870$	0	0
$871$	1.07580e71	0.100296
$872$	0	0
$873$	−3.71632e72	−3.27540
$874$	0	0
$875$	−4.30248e71	−0.358527
$876$	0	0
$877$	−1.15171e72	−0.907512	−0.453756	−	0.891126i	$-0.649916\pi$
$877$	−1.15171e72	−0.907512	−0.453756	+	0.891126i	$0.649916\pi$
$878$	0	0
$879$	−2.23219e72	−1.66342
$880$	0	0
$881$	−2.42451e72	−1.70889	−0.854447	−	0.519539i	$-0.826104\pi$
$881$	−2.42451e72	−1.70889	−0.854447	+	0.519539i	$0.826104\pi$
$882$	0	0
$883$	−1.55419e72	−1.03626	−0.518128	−	0.855303i	$-0.673371\pi$
$883$	−1.55419e72	−1.03626	−0.518128	+	0.855303i	$0.673371\pi$
$884$	0	0
$885$	4.62932e72	2.92017
$886$	0	0
$887$	−5.49889e71	−0.328206	−0.164103	−	0.986443i	$-0.552473\pi$
$887$	−5.49889e71	−0.328206	−0.164103	+	0.986443i	$0.552473\pi$
$888$	0	0
$889$	1.98706e71	0.112232
$890$	0	0
$891$	1.96810e71	0.105207
$892$	0	0
$893$	4.97498e71	0.251727
$894$	0	0
$895$	−8.76018e71	−0.419613
$896$	0	0
$897$	6.39970e71	0.290232
$898$	0	0
$899$	2.61664e72	1.12365
$900$	0	0
$901$	−5.97141e71	−0.242840
$902$	0	0
$903$	4.65527e71	0.179306
$904$	0	0
$905$	2.31684e71	0.0845289
$906$	0	0
$907$	−2.00638e72	−0.693482	−0.346741	−	0.937961i	$-0.612712\pi$
$907$	−2.00638e72	−0.693482	−0.346741	+	0.937961i	$0.612712\pi$
$908$	0	0
$909$	5.43572e71	0.178009
$910$	0	0
$911$	6.79753e70	0.0210936	0.0105468	−	0.999944i	$-0.496643\pi$
$911$	6.79753e70	0.0210936	0.0105468	+	0.999944i	$0.496643\pi$
$912$	0	0
$913$	2.97073e71	0.0873633
$914$	0	0
$915$	3.60078e72	1.00364
$916$	0	0
$917$	−1.97953e72	−0.523014
$918$	0	0
$919$	−7.92073e71	−0.198397	−0.0991985	−	0.995068i	$-0.531628\pi$
$919$	−7.92073e71	−0.198397	−0.0991985	+	0.995068i	$0.531628\pi$
$920$	0	0
$921$	−2.27752e72	−0.540881
$922$	0	0
$923$	−3.10411e71	−0.0699030
$924$	0	0
$925$	6.18032e71	0.131989
$926$	0	0
$927$	1.53474e73	3.10872
$928$	0	0
$929$	4.36095e72	0.837908	0.418954	−	0.908008i	$-0.362397\pi$
$929$	4.36095e72	0.837908	0.418954	+	0.908008i	$0.362397\pi$
$930$	0	0
$931$	−1.83615e72	−0.334688
$932$	0	0
$933$	1.65367e73	2.85989
$934$	0	0
$935$	3.34687e71	0.0549230
$936$	0	0
$937$	−8.38565e72	−1.30592	−0.652960	−	0.757392i	$-0.726473\pi$
$937$	−8.38565e72	−1.30592	−0.652960	+	0.757392i	$0.726473\pi$
$938$	0	0
$939$	−1.66456e73	−2.46032
$940$	0	0
$941$	−5.95096e72	−0.834913	−0.417456	−	0.908697i	$-0.637078\pi$
$941$	−5.95096e72	−0.834913	−0.417456	+	0.908697i	$0.637078\pi$
$942$	0	0
$943$	1.58974e73	2.11733
$944$	0	0
$945$	1.33537e73	1.68859
$946$	0	0
$947$	−5.80008e72	−0.696402	−0.348201	−	0.937420i	$-0.613207\pi$
$947$	−5.80008e72	−0.696402	−0.348201	+	0.937420i	$0.613207\pi$
$948$	0	0
$949$	−8.96911e71	−0.102265
$950$	0	0
$951$	4.19577e72	0.454349
$952$	0	0
$953$	−1.78856e73	−1.83962	−0.919810	−	0.392365i	$-0.871657\pi$
$953$	−1.78856e73	−1.83962	−0.919810	+	0.392365i	$0.871657\pi$
$954$	0	0
$955$	7.46988e71	0.0729846
$956$	0	0
$957$	1.01294e72	0.0940249
$958$	0	0
$959$	3.42546e72	0.302109
$960$	0	0
$961$	1.85631e73	1.55570
$962$	0	0
$963$	2.30723e72	0.183758
$964$	0	0
$965$	−6.55264e72	−0.496018
$966$	0	0
$967$	−8.21075e72	−0.590792	−0.295396	−	0.955375i	$-0.595452\pi$
$967$	−8.21075e72	−0.590792	−0.295396	+	0.955375i	$0.595452\pi$
$968$	0	0
$969$	9.20436e72	0.629595
$970$	0	0
$971$	−1.52584e73	−0.992288	−0.496144	−	0.868240i	$-0.665251\pi$
$971$	−1.52584e73	−0.992288	−0.496144	+	0.868240i	$0.665251\pi$
$972$	0	0
$973$	1.87710e73	1.16071
$974$	0	0
$975$	−1.65532e72	−0.0973349
$976$	0	0
$977$	5.10455e72	0.285457	0.142728	−	0.989762i	$-0.454412\pi$
$977$	5.10455e72	0.285457	0.142728	+	0.989762i	$0.454412\pi$
$978$	0	0
$979$	2.15196e72	0.114461
$980$	0	0
$981$	−5.70516e73	−2.88654
$982$	0	0
$983$	3.35654e73	1.61559	0.807793	−	0.589466i	$-0.200662\pi$
$983$	3.35654e73	1.61559	0.807793	+	0.589466i	$0.200662\pi$
$984$	0	0
$985$	1.44699e73	0.662638
$986$	0	0
$987$	1.12002e73	0.488041
$988$	0	0
$989$	−6.23396e72	−0.258497
$990$	0	0
$991$	−3.04908e73	−1.20327	−0.601637	−	0.798770i	$-0.705484\pi$
$991$	−3.04908e73	−1.20327	−0.601637	+	0.798770i	$0.705484\pi$
$992$	0	0
$993$	2.99596e73	1.12533
$994$	0	0
$995$	2.31705e73	0.828457
$996$	0	0
$997$	−5.00437e70	−0.00170341	−0.000851703	−	1.00000i	$-0.500271\pi$
$997$	−5.00437e70	−0.00170341	−0.000851703		1.00000i	$0.500271\pi$
$998$	0	0
$999$	1.40167e73	0.454246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.50.a.c.1.3		3
4.3	odd	2		1.50.a.a.1.1	✓	3
12.11	even	2		9.50.a.a.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.50.a.a.1.1	✓	3	4.3	odd	2
9.50.a.a.1.3		3	12.11	even	2
16.50.a.c.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 50 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+8.64745e11 q^{3} +1.67413e17 q^{5} +3.42668e20 q^{7} +5.08484e23 q^{9} +O(q^{10})\)	\(q+8.64745e11 q^{3} +1.67413e17 q^{5} +3.42668e20 q^{7} +5.08484e23 q^{9} +2.47212e24 q^{11} -1.86505e26 q^{13} +1.44770e29 q^{15} +8.08684e29 q^{17} +1.31622e31 q^{19} +2.96321e32 q^{21} -3.96809e33 q^{23} +1.02637e34 q^{25} +2.32776e35 q^{27} +4.73835e35 q^{29} +5.52226e36 q^{31} +2.13775e36 q^{33} +5.73673e37 q^{35} +6.02154e37 q^{37} -1.61279e38 q^{39} -4.00631e39 q^{41} +1.57103e39 q^{43} +8.51271e40 q^{45} +3.77976e40 q^{47} -1.39502e41 q^{49} +6.99305e41 q^{51} -7.38411e41 q^{53} +4.13866e41 q^{55} +1.13819e43 q^{57} +3.19771e43 q^{59} +2.48724e43 q^{61} +1.74241e44 q^{63} -3.12234e43 q^{65} -5.76819e44 q^{67} -3.43138e45 q^{69} +1.66436e45 q^{71} +4.80904e45 q^{73} +8.87547e45 q^{75} +8.47118e44 q^{77} +4.21047e45 q^{79} +7.96119e46 q^{81} +1.20169e47 q^{83} +1.35385e47 q^{85} +4.09747e47 q^{87} +8.70490e47 q^{89} -6.39094e46 q^{91} +4.77534e48 q^{93} +2.20352e48 q^{95} -7.30863e48 q^{97} +1.25703e48 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 326954692404 q^{3} + 63\!\cdots\!50 q^{5}+ \cdots + 34\!\cdots\!19 q^{9}+O(q^{10}) \) 3 * q + 326954692404 * q^3 + 63884035717079250 * q^5 - 509391477498711192 * q^7 + 341625690512280455369319 * q^9	\( 3 q + 326954692404 q^{3} + 63\!\cdots\!50 q^{5}+ \cdots + 11\!\cdots\!52 q^{99}+O(q^{100}) \) 3 * q + 326954692404 * q^3 + 63884035717079250 * q^5 - 509391477498711192 * q^7 + 341625690512280455369319 * q^9 + 20839986192711815613370524 * q^11 - 187524857197340618282341014 * q^13 + 204230679539215002884128467000 * q^15 - 3305035636247475259184114894538 * q^17 + 45882942802931464757155696518660 * q^19 + 617794442803555252870956383262816 * q^21 - 4533392548023086653463590407813576 * q^23 - 13921151905580403838435182832171875 * q^25 + 316454625615073819697154158592097800 * q^27 + 1118089919316361970349176934175494810 * q^29 - 934226816157578575389156342339525216 * q^31 - 24487068877327908023713382739550873968 * q^33 + 118496887985378726635932152589238374000 * q^35 + 242135148519373130928922851818564983602 * q^37 - 1291017145684876237790670030697385259048 * q^39 - 6211739675976490815696297371880432723714 * q^41 + 7795173962887472637660666233812713782844 * q^43 + 76758284157261249391010855229819118430250 * q^45 - 72757717838871152990888073123214445891472 * q^47 - 280157966384090888246597869444400847657429 * q^49 + 1538734957331600122304206020917586572488104 * q^51 - 155850890670432045962015052011776669693854 * q^53 - 4640026560861316230546160030865773419591000 * q^55 - 7909742361043655643598112535376781525014800 * q^57 + 62596927936822013761436024025606581543386380 * q^59 - 24037263967789435219196348055077866809754374 * q^61 + 79613772299337773967480392108297858821089864 * q^63 - 244975900670806240216695171564505814930314500 * q^65 + 1019040852324305100821779153549119164107459188 * q^67 - 4867866269191569039747093560336346164459611872 * q^69 + 3208129335291950593367863544185773469328334504 * q^71 + 5668108076325412165814435883536677392865317726 * q^73 + 12104818074938487392123628092507024931595687500 * q^75 - 32043339793247959180079389483660528277811891936 * q^77 + 7871365395380306420112251724245018715125364240 * q^79 + 67270927730095597004530465543730351634592280043 * q^81 - 94164681901361229355849854653903786480577085116 * q^83 + 297863286656785614551226799130947646752077538500 * q^85 + 175500245979701689466461020645367234129141215000 * q^87 + 367807201561165354579457777200332871192114777230 * q^89 - 1613372397893666548508325640004129525440352221776 * q^91 + 5954871901405488350696717407304686000200433533312 * q^93 - 1475636199748052355599968141572624301097796705000 * q^95 - 10431108583780665799950508389192126734793760399578 * q^97 + 11387023179914295087805432025124514927226421923052 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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