LMFDB - Embedded newform 112.12.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,12,Mod(1,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("112.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	112.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:	$86.0544362227$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3369}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 842 $ x^2 - x - 842
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$29.5215$ of defining polynomial
Character	$\chi$	$=$	112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-408.259 q^{3} -7330.43 q^{5} -16807.0 q^{7} -10472.0 q^{9} +O(q^{10})$	$q-408.259 q^{3} -7330.43 q^{5} -16807.0 q^{7} -10472.0 q^{9} +616867. q^{11} -1.73806e6 q^{13} +2.99271e6 q^{15} +2.25996e6 q^{17} +4.46399e6 q^{19} +6.86160e6 q^{21} +3.07878e7 q^{23} +4.90709e6 q^{25} +7.65970e7 q^{27} +1.42668e8 q^{29} -7.19272e7 q^{31} -2.51841e8 q^{33} +1.23203e8 q^{35} +6.71313e7 q^{37} +7.09576e8 q^{39} -1.76664e7 q^{41} -3.80136e8 q^{43} +7.67641e7 q^{45} +1.92943e9 q^{47} +2.82475e8 q^{49} -9.22648e8 q^{51} +1.67671e9 q^{53} -4.52190e9 q^{55} -1.82246e9 q^{57} -7.61950e9 q^{59} +4.21825e9 q^{61} +1.76003e8 q^{63} +1.27407e10 q^{65} +6.85501e9 q^{67} -1.25694e10 q^{69} +1.56926e10 q^{71} -2.70063e10 q^{73} -2.00336e9 q^{75} -1.03677e10 q^{77} -2.62444e10 q^{79} -2.94163e10 q^{81} -2.20820e10 q^{83} -1.65665e10 q^{85} -5.82454e10 q^{87} -2.81097e10 q^{89} +2.92115e10 q^{91} +2.93649e10 q^{93} -3.27229e10 q^{95} -1.07918e11 q^{97} -6.45982e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 120 q^{3} - 13500 q^{5} - 33614 q^{7} - 104526 q^{9}+O(q^{10}) $ 2 * q - 120 * q^3 - 13500 * q^5 - 33614 * q^7 - 104526 * q^9	$ 2 q - 120 q^{3} - 13500 q^{5} - 33614 q^{7} - 104526 q^{9} + 750816 q^{11} - 9548 q^{13} + 1214280 q^{15} + 4160052 q^{17} + 17998712 q^{19} + 2016840 q^{21} + 66161016 q^{23} - 5857450 q^{25} - 1578960 q^{27} + 61515612 q^{29} + 15281552 q^{31} - 213229440 q^{33} + 226894500 q^{35} - 527218340 q^{37} + 1207833816 q^{39} - 178276140 q^{41} - 1826745232 q^{43} + 657036900 q^{45} - 568240704 q^{47} + 564950498 q^{49} - 374929920 q^{51} - 4185816372 q^{53} - 5348308800 q^{55} + 2079040344 q^{57} - 3111345000 q^{59} + 15042595060 q^{61} + 1756768482 q^{63} + 2076550560 q^{65} - 9856523968 q^{67} - 2372754960 q^{69} + 24312011328 q^{71} - 30890001932 q^{73} - 5106333000 q^{75} - 12618964512 q^{77} - 1992804256 q^{79} - 35289830358 q^{81} - 5277014568 q^{83} - 28289229000 q^{85} - 81638222976 q^{87} - 101541312828 q^{89} + 160473236 q^{91} + 54503588112 q^{93} - 116226367560 q^{95} - 192228621116 q^{97} - 19058239008 q^{99}+O(q^{100}) $ 2 * q - 120 * q^3 - 13500 * q^5 - 33614 * q^7 - 104526 * q^9 + 750816 * q^11 - 9548 * q^13 + 1214280 * q^15 + 4160052 * q^17 + 17998712 * q^19 + 2016840 * q^21 + 66161016 * q^23 - 5857450 * q^25 - 1578960 * q^27 + 61515612 * q^29 + 15281552 * q^31 - 213229440 * q^33 + 226894500 * q^35 - 527218340 * q^37 + 1207833816 * q^39 - 178276140 * q^41 - 1826745232 * q^43 + 657036900 * q^45 - 568240704 * q^47 + 564950498 * q^49 - 374929920 * q^51 - 4185816372 * q^53 - 5348308800 * q^55 + 2079040344 * q^57 - 3111345000 * q^59 + 15042595060 * q^61 + 1756768482 * q^63 + 2076550560 * q^65 - 9856523968 * q^67 - 2372754960 * q^69 + 24312011328 * q^71 - 30890001932 * q^73 - 5106333000 * q^75 - 12618964512 * q^77 - 1992804256 * q^79 - 35289830358 * q^81 - 5277014568 * q^83 - 28289229000 * q^85 - 81638222976 * q^87 - 101541312828 * q^89 + 160473236 * q^91 + 54503588112 * q^93 - 116226367560 * q^95 - 192228621116 * q^97 - 19058239008 * 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
$3$	−408.259	−0.969992	−0.484996	−	0.874516i	$-0.661179\pi$
$3$	−408.259	−0.969992	−0.484996	+	0.874516i	$0.661179\pi$
$4$	0	0
$5$	−7330.43	−1.04905	−0.524523	−	0.851396i	$-0.675756\pi$
$5$	−7330.43	−1.04905	−0.524523	+	0.851396i	$0.675756\pi$
$6$	0	0
$7$	−16807.0	−0.377964
$7$	−16807.0	−0.377964
$8$	0	0
$9$	−10472.0	−0.0591146
$10$	0	0
$11$	616867.	1.15487	0.577433	−	0.816438i	$-0.304054\pi$
$11$	616867.	1.15487	0.577433	+	0.816438i	$0.304054\pi$
$12$	0	0
$13$	−1.73806e6	−1.29830	−0.649151	−	0.760660i	$-0.724876\pi$
$13$	−1.73806e6	−1.29830	−0.649151	+	0.760660i	$0.724876\pi$
$14$	0	0
$15$	2.99271e6	1.01757
$16$	0	0
$17$	2.25996e6	0.386039	0.193020	−	0.981195i	$-0.438172\pi$
$17$	2.25996e6	0.386039	0.193020	+	0.981195i	$0.438172\pi$
$18$	0	0
$19$	4.46399e6	0.413598	0.206799	−	0.978383i	$-0.433695\pi$
$19$	4.46399e6	0.413598	0.206799	+	0.978383i	$0.433695\pi$
$20$	0	0
$21$	6.86160e6	0.366623
$22$	0	0
$23$	3.07878e7	0.997414	0.498707	−	0.866771i	$-0.333808\pi$
$23$	3.07878e7	0.997414	0.498707	+	0.866771i	$0.333808\pi$
$24$	0	0
$25$	4.90709e6	0.100497
$26$	0	0
$27$	7.65970e7	1.02733
$28$	0	0
$29$	1.42668e8	1.29163	0.645814	−	0.763495i	$-0.276518\pi$
$29$	1.42668e8	1.29163	0.645814	+	0.763495i	$0.276518\pi$
$30$	0	0
$31$	−7.19272e7	−0.451236	−0.225618	−	0.974216i	$-0.572440\pi$
$31$	−7.19272e7	−0.451236	−0.225618	+	0.974216i	$0.572440\pi$
$32$	0	0
$33$	−2.51841e8	−1.12021
$34$	0	0
$35$	1.23203e8	0.396502
$36$	0	0
$37$	6.71313e7	0.159153	0.0795766	−	0.996829i	$-0.474643\pi$
$37$	6.71313e7	0.159153	0.0795766	+	0.996829i	$0.474643\pi$
$38$	0	0
$39$	7.09576e8	1.25934
$40$	0	0
$41$	−1.76664e7	−0.0238142	−0.0119071	−	0.999929i	$-0.503790\pi$
$41$	−1.76664e7	−0.0238142	−0.0119071	+	0.999929i	$0.503790\pi$
$42$	0	0
$43$	−3.80136e8	−0.394332	−0.197166	−	0.980370i	$-0.563174\pi$
$43$	−3.80136e8	−0.394332	−0.197166	+	0.980370i	$0.563174\pi$
$44$	0	0
$45$	7.67641e7	0.0620139
$46$	0	0
$47$	1.92943e9	1.22713	0.613566	−	0.789643i	$-0.289734\pi$
$47$	1.92943e9	1.22713	0.613566	+	0.789643i	$0.289734\pi$
$48$	0	0
$49$	2.82475e8	0.142857
$50$	0	0
$51$	−9.22648e8	−0.374455
$52$	0	0
$53$	1.67671e9	0.550732	0.275366	−	0.961339i	$-0.411201\pi$
$53$	1.67671e9	0.550732	0.275366	+	0.961339i	$0.411201\pi$
$54$	0	0
$55$	−4.52190e9	−1.21151
$56$	0	0
$57$	−1.82246e9	−0.401186
$58$	0	0
$59$	−7.61950e9	−1.38752	−0.693762	−	0.720204i	$-0.744048\pi$
$59$	−7.61950e9	−1.38752	−0.693762	+	0.720204i	$0.744048\pi$
$60$	0	0
$61$	4.21825e9	0.639467	0.319733	−	0.947508i	$-0.396407\pi$
$61$	4.21825e9	0.639467	0.319733	+	0.947508i	$0.396407\pi$
$62$	0	0
$63$	1.76003e8	0.0223432
$64$	0	0
$65$	1.27407e10	1.36198
$66$	0	0
$67$	6.85501e9	0.620292	0.310146	−	0.950689i	$-0.399622\pi$
$67$	6.85501e9	0.620292	0.310146	+	0.950689i	$0.399622\pi$
$68$	0	0
$69$	−1.25694e10	−0.967484
$70$	0	0
$71$	1.56926e10	1.03223	0.516113	−	0.856521i	$-0.327379\pi$
$71$	1.56926e10	1.03223	0.516113	+	0.856521i	$0.327379\pi$
$72$	0	0
$73$	−2.70063e10	−1.52472	−0.762358	−	0.647155i	$-0.775959\pi$
$73$	−2.70063e10	−1.52472	−0.762358	+	0.647155i	$0.775959\pi$
$74$	0	0
$75$	−2.00336e9	−0.0974816
$76$	0	0
$77$	−1.03677e10	−0.436499
$78$	0	0
$79$	−2.62444e10	−0.959595	−0.479797	−	0.877379i	$-0.659290\pi$
$79$	−2.62444e10	−0.959595	−0.479797	+	0.877379i	$0.659290\pi$
$80$	0	0
$81$	−2.94163e10	−0.937391
$82$	0	0
$83$	−2.20820e10	−0.615331	−0.307665	−	0.951495i	$-0.599548\pi$
$83$	−2.20820e10	−0.615331	−0.307665	+	0.951495i	$0.599548\pi$
$84$	0	0
$85$	−1.65665e10	−0.404973
$86$	0	0
$87$	−5.82454e10	−1.25287
$88$	0	0
$89$	−2.81097e10	−0.533594	−0.266797	−	0.963753i	$-0.585965\pi$
$89$	−2.81097e10	−0.533594	−0.266797	+	0.963753i	$0.585965\pi$
$90$	0	0
$91$	2.92115e10	0.490712
$92$	0	0
$93$	2.93649e10	0.437696
$94$	0	0
$95$	−3.27229e10	−0.433883
$96$	0	0
$97$	−1.07918e11	−1.27600	−0.638000	−	0.770036i	$-0.720238\pi$
$97$	−1.07918e11	−1.27600	−0.638000	+	0.770036i	$0.720238\pi$
$98$	0	0
$99$	−6.45982e9	−0.0682695
$100$	0	0
$101$	2.02228e11	1.91459	0.957293	−	0.289121i	$-0.0933630\pi$
$101$	2.02228e11	1.91459	0.957293	+	0.289121i	$0.0933630\pi$
$102$	0	0
$103$	−8.39818e10	−0.713806	−0.356903	−	0.934141i	$-0.616167\pi$
$103$	−8.39818e10	−0.713806	−0.356903	+	0.934141i	$0.616167\pi$
$104$	0	0
$105$	−5.02985e10	−0.384604
$106$	0	0
$107$	1.00996e11	0.696138	0.348069	−	0.937469i	$-0.386837\pi$
$107$	1.00996e11	0.696138	0.348069	+	0.937469i	$0.386837\pi$
$108$	0	0
$109$	−2.20579e11	−1.37315	−0.686575	−	0.727059i	$-0.740887\pi$
$109$	−2.20579e11	−1.37315	−0.686575	+	0.727059i	$0.740887\pi$
$110$	0	0
$111$	−2.74069e10	−0.154377
$112$	0	0
$113$	−1.40380e11	−0.716760	−0.358380	−	0.933576i	$-0.616671\pi$
$113$	−1.40380e11	−0.716760	−0.358380	+	0.933576i	$0.616671\pi$
$114$	0	0
$115$	−2.25688e11	−1.04633
$116$	0	0
$117$	1.82009e10	0.0767486
$118$	0	0
$119$	−3.79831e10	−0.145909
$120$	0	0
$121$	9.52135e10	0.333718
$122$	0	0
$123$	7.21244e9	0.0230996
$124$	0	0
$125$	3.21960e11	0.943620
$126$	0	0
$127$	2.97267e11	0.798411	0.399205	−	0.916862i	$-0.369286\pi$
$127$	2.97267e11	0.798411	0.399205	+	0.916862i	$0.369286\pi$
$128$	0	0
$129$	1.55194e11	0.382499
$130$	0	0
$131$	−5.11248e11	−1.15782	−0.578908	−	0.815393i	$-0.696521\pi$
$131$	−5.11248e11	−1.15782	−0.578908	+	0.815393i	$0.696521\pi$
$132$	0	0
$133$	−7.50262e10	−0.156325
$134$	0	0
$135$	−5.61489e11	−1.07772
$136$	0	0
$137$	−4.51597e11	−0.799443	−0.399722	−	0.916637i	$-0.630893\pi$
$137$	−4.51597e11	−0.799443	−0.399722	+	0.916637i	$0.630893\pi$
$138$	0	0
$139$	5.22803e11	0.854588	0.427294	−	0.904113i	$-0.359467\pi$
$139$	5.22803e11	0.854588	0.427294	+	0.904113i	$0.359467\pi$
$140$	0	0
$141$	−7.87708e11	−1.19031
$142$	0	0
$143$	−1.07215e12	−1.49937
$144$	0	0
$145$	−1.04582e12	−1.35498
$146$	0	0
$147$	−1.15323e11	−0.138570
$148$	0	0
$149$	−1.30064e12	−1.45088	−0.725441	−	0.688285i	$-0.758364\pi$
$149$	−1.30064e12	−1.45088	−0.725441	+	0.688285i	$0.758364\pi$
$150$	0	0
$151$	−1.21019e12	−1.25453	−0.627266	−	0.778805i	$-0.715826\pi$
$151$	−1.21019e12	−1.25453	−0.627266	+	0.778805i	$0.715826\pi$
$152$	0	0
$153$	−2.36662e10	−0.0228206
$154$	0	0
$155$	5.27258e11	0.473368
$156$	0	0
$157$	−1.50952e12	−1.26296	−0.631481	−	0.775391i	$-0.717553\pi$
$157$	−1.50952e12	−1.26296	−0.631481	+	0.775391i	$0.717553\pi$
$158$	0	0
$159$	−6.84531e11	−0.534206
$160$	0	0
$161$	−5.17451e11	−0.376987
$162$	0	0
$163$	2.20464e12	1.50074	0.750371	−	0.661017i	$-0.229875\pi$
$163$	2.20464e12	1.50074	0.750371	+	0.661017i	$0.229875\pi$
$164$	0	0
$165$	1.84611e12	1.17515
$166$	0	0
$167$	−1.27576e12	−0.760027	−0.380013	−	0.924981i	$-0.624081\pi$
$167$	−1.27576e12	−0.760027	−0.380013	+	0.924981i	$0.624081\pi$
$168$	0	0
$169$	1.22868e12	0.685586
$170$	0	0
$171$	−4.67468e10	−0.0244497
$172$	0	0
$173$	−1.46187e12	−0.717225	−0.358612	−	0.933487i	$-0.616750\pi$
$173$	−1.46187e12	−0.717225	−0.358612	+	0.933487i	$0.616750\pi$
$174$	0	0
$175$	−8.24735e10	−0.0379844
$176$	0	0
$177$	3.11073e12	1.34589
$178$	0	0
$179$	8.38880e11	0.341199	0.170600	−	0.985340i	$-0.445429\pi$
$179$	8.38880e11	0.341199	0.170600	+	0.985340i	$0.445429\pi$
$180$	0	0
$181$	3.15736e12	1.20807	0.604034	−	0.796958i	$-0.293559\pi$
$181$	3.15736e12	1.20807	0.604034	+	0.796958i	$0.293559\pi$
$182$	0	0
$183$	−1.72214e12	−0.620278
$184$	0	0
$185$	−4.92101e11	−0.166959
$186$	0	0
$187$	1.39410e12	0.445824
$188$	0	0
$189$	−1.28737e12	−0.388295
$190$	0	0
$191$	3.19466e12	0.909370	0.454685	−	0.890652i	$-0.349752\pi$
$191$	3.19466e12	0.909370	0.454685	+	0.890652i	$0.349752\pi$
$192$	0	0
$193$	3.23302e12	0.869046	0.434523	−	0.900661i	$-0.356917\pi$
$193$	3.23302e12	0.869046	0.434523	+	0.900661i	$0.356917\pi$
$194$	0	0
$195$	−5.20150e12	−1.32111
$196$	0	0
$197$	5.87031e12	1.40960	0.704802	−	0.709404i	$-0.251036\pi$
$197$	5.87031e12	1.40960	0.704802	+	0.709404i	$0.251036\pi$
$198$	0	0
$199$	6.06637e12	1.37796	0.688981	−	0.724779i	$-0.258058\pi$
$199$	6.06637e12	1.37796	0.688981	+	0.724779i	$0.258058\pi$
$200$	0	0
$201$	−2.79862e12	−0.601679
$202$	0	0
$203$	−2.39782e12	−0.488189
$204$	0	0
$205$	1.29502e11	0.0249822
$206$	0	0
$207$	−3.22409e11	−0.0589618
$208$	0	0
$209$	2.75369e12	0.477650
$210$	0	0
$211$	1.11171e13	1.82995	0.914976	−	0.403509i	$-0.132209\pi$
$211$	1.11171e13	1.82995	0.914976	+	0.403509i	$0.132209\pi$
$212$	0	0
$213$	−6.40665e12	−1.00125
$214$	0	0
$215$	2.78656e12	0.413673
$216$	0	0
$217$	1.20888e12	0.170551
$218$	0	0
$219$	1.10255e13	1.47896
$220$	0	0
$221$	−3.92794e12	−0.501195
$222$	0	0
$223$	−1.03209e13	−1.25326	−0.626630	−	0.779317i	$-0.715566\pi$
$223$	−1.03209e13	−1.25326	−0.626630	+	0.779317i	$0.715566\pi$
$224$	0	0
$225$	−5.13870e10	−0.00594086
$226$	0	0
$227$	2.97703e12	0.327824	0.163912	−	0.986475i	$-0.447589\pi$
$227$	2.97703e12	0.327824	0.163912	+	0.986475i	$0.447589\pi$
$228$	0	0
$229$	3.60534e12	0.378313	0.189156	−	0.981947i	$-0.439425\pi$
$229$	3.60534e12	0.378313	0.189156	+	0.981947i	$0.439425\pi$
$230$	0	0
$231$	4.23270e12	0.423400
$232$	0	0
$233$	−8.41895e12	−0.803157	−0.401578	−	0.915825i	$-0.631538\pi$
$233$	−8.41895e12	−0.803157	−0.401578	+	0.915825i	$0.631538\pi$
$234$	0	0
$235$	−1.41436e13	−1.28732
$236$	0	0
$237$	1.07145e13	0.930800
$238$	0	0
$239$	1.05441e13	0.874624	0.437312	−	0.899310i	$-0.355930\pi$
$239$	1.05441e13	0.874624	0.437312	+	0.899310i	$0.355930\pi$
$240$	0	0
$241$	2.04221e13	1.61811	0.809053	−	0.587736i	$-0.199981\pi$
$241$	2.04221e13	1.61811	0.809053	+	0.587736i	$0.199981\pi$
$242$	0	0
$243$	−1.55947e12	−0.118071
$244$	0	0
$245$	−2.07067e12	−0.149864
$246$	0	0
$247$	−7.75866e12	−0.536974
$248$	0	0
$249$	9.01516e12	0.596866
$250$	0	0
$251$	9.93571e12	0.629497	0.314748	−	0.949175i	$-0.398080\pi$
$251$	9.93571e12	0.629497	0.314748	+	0.949175i	$0.398080\pi$
$252$	0	0
$253$	1.89920e13	1.15188
$254$	0	0
$255$	6.76341e12	0.392821
$256$	0	0
$257$	−2.52092e13	−1.40258	−0.701290	−	0.712876i	$-0.747392\pi$
$257$	−2.52092e13	−1.40258	−0.701290	+	0.712876i	$0.747392\pi$
$258$	0	0
$259$	−1.12827e12	−0.0601542
$260$	0	0
$261$	−1.49401e12	−0.0763541
$262$	0	0
$263$	5.72806e12	0.280705	0.140353	−	0.990102i	$-0.455176\pi$
$263$	5.72806e12	0.280705	0.140353	+	0.990102i	$0.455176\pi$
$264$	0	0
$265$	−1.22910e13	−0.577743
$266$	0	0
$267$	1.14760e13	0.517582
$268$	0	0
$269$	1.55236e13	0.671979	0.335989	−	0.941866i	$-0.390929\pi$
$269$	1.55236e13	0.671979	0.335989	+	0.941866i	$0.390929\pi$
$270$	0	0
$271$	−1.01341e13	−0.421168	−0.210584	−	0.977576i	$-0.567537\pi$
$271$	−1.01341e13	−0.421168	−0.210584	+	0.977576i	$0.567537\pi$
$272$	0	0
$273$	−1.19259e13	−0.475987
$274$	0	0
$275$	3.02702e12	0.116061
$276$	0	0
$277$	3.20344e13	1.18026	0.590131	−	0.807307i	$-0.299076\pi$
$277$	3.20344e13	1.18026	0.590131	+	0.807307i	$0.299076\pi$
$278$	0	0
$279$	7.53220e11	0.0266747
$280$	0	0
$281$	−2.74006e13	−0.932986	−0.466493	−	0.884525i	$-0.654483\pi$
$281$	−2.74006e13	−0.932986	−0.466493	+	0.884525i	$0.654483\pi$
$282$	0	0
$283$	−5.02818e13	−1.64659	−0.823294	−	0.567615i	$-0.807866\pi$
$283$	−5.02818e13	−1.64659	−0.823294	+	0.567615i	$0.807866\pi$
$284$	0	0
$285$	1.33594e13	0.420863
$286$	0	0
$287$	2.96919e11	0.00900092
$288$	0	0
$289$	−2.91645e13	−0.850974
$290$	0	0
$291$	4.40586e13	1.23771
$292$	0	0
$293$	−9.42991e12	−0.255115	−0.127557	−	0.991831i	$-0.540714\pi$
$293$	−9.42991e12	−0.255115	−0.127557	+	0.991831i	$0.540714\pi$
$294$	0	0
$295$	5.58542e13	1.45558
$296$	0	0
$297$	4.72502e13	1.18643
$298$	0	0
$299$	−5.35110e13	−1.29494
$300$	0	0
$301$	6.38894e12	0.149044
$302$	0	0
$303$	−8.25615e13	−1.85713
$304$	0	0
$305$	−3.09216e13	−0.670830
$306$	0	0
$307$	−4.62480e13	−0.967904	−0.483952	−	0.875095i	$-0.660799\pi$
$307$	−4.62480e13	−0.967904	−0.483952	+	0.875095i	$0.660799\pi$
$308$	0	0
$309$	3.42863e13	0.692387
$310$	0	0
$311$	3.52679e13	0.687382	0.343691	−	0.939083i	$-0.388323\pi$
$311$	3.52679e13	0.687382	0.343691	+	0.939083i	$0.388323\pi$
$312$	0	0
$313$	9.39030e12	0.176679	0.0883396	−	0.996090i	$-0.471844\pi$
$313$	9.39030e12	0.176679	0.0883396	+	0.996090i	$0.471844\pi$
$314$	0	0
$315$	−1.29017e12	−0.0234391
$316$	0	0
$317$	−7.92424e13	−1.39037	−0.695187	−	0.718829i	$-0.744679\pi$
$317$	−7.92424e13	−1.39037	−0.695187	+	0.718829i	$0.744679\pi$
$318$	0	0
$319$	8.80072e13	1.49166
$320$	0	0
$321$	−4.12327e13	−0.675249
$322$	0	0
$323$	1.00884e13	0.159665
$324$	0	0
$325$	−8.52880e12	−0.130476
$326$	0	0
$327$	9.00532e13	1.33195
$328$	0	0
$329$	−3.24280e13	−0.463813
$330$	0	0
$331$	−7.49687e13	−1.03711	−0.518557	−	0.855043i	$-0.673531\pi$
$331$	−7.49687e13	−1.03711	−0.518557	+	0.855043i	$0.673531\pi$
$332$	0	0
$333$	−7.02997e11	−0.00940828
$334$	0	0
$335$	−5.02502e13	−0.650715
$336$	0	0
$337$	−7.26608e13	−0.910617	−0.455309	−	0.890334i	$-0.650471\pi$
$337$	−7.26608e13	−0.910617	−0.455309	+	0.890334i	$0.650471\pi$
$338$	0	0
$339$	5.73113e13	0.695252
$340$	0	0
$341$	−4.43696e13	−0.521118
$342$	0	0
$343$	−4.74756e12	−0.0539949
$344$	0	0
$345$	9.21390e13	1.01494
$346$	0	0
$347$	−9.38496e13	−1.00143	−0.500715	−	0.865612i	$-0.666929\pi$
$347$	−9.38496e13	−1.00143	−0.500715	+	0.865612i	$0.666929\pi$
$348$	0	0
$349$	−9.33832e13	−0.965448	−0.482724	−	0.875772i	$-0.660353\pi$
$349$	−9.33832e13	−0.965448	−0.482724	+	0.875772i	$0.660353\pi$
$350$	0	0
$351$	−1.33130e14	−1.33379
$352$	0	0
$353$	−9.67068e11	−0.00939066	−0.00469533	−	0.999989i	$-0.501495\pi$
$353$	−9.67068e11	−0.00939066	−0.00469533	+	0.999989i	$0.501495\pi$
$354$	0	0
$355$	−1.15034e14	−1.08285
$356$	0	0
$357$	1.55069e13	0.141531
$358$	0	0
$359$	−3.43634e13	−0.304142	−0.152071	−	0.988370i	$-0.548594\pi$
$359$	−3.43634e13	−0.304142	−0.152071	+	0.988370i	$0.548594\pi$
$360$	0	0
$361$	−9.65631e13	−0.828937
$362$	0	0
$363$	−3.88717e13	−0.323704
$364$	0	0
$365$	1.97968e14	1.59950
$366$	0	0
$367$	−2.25987e13	−0.177182	−0.0885912	−	0.996068i	$-0.528236\pi$
$367$	−2.25987e13	−0.177182	−0.0885912	+	0.996068i	$0.528236\pi$
$368$	0	0
$369$	1.85002e11	0.00140777
$370$	0	0
$371$	−2.81804e13	−0.208157
$372$	0	0
$373$	−3.24297e13	−0.232565	−0.116283	−	0.993216i	$-0.537098\pi$
$373$	−3.24297e13	−0.232565	−0.116283	+	0.993216i	$0.537098\pi$
$374$	0	0
$375$	−1.31443e14	−0.915304
$376$	0	0
$377$	−2.47965e14	−1.67692
$378$	0	0
$379$	8.85164e12	0.0581444	0.0290722	−	0.999577i	$-0.490745\pi$
$379$	8.85164e12	0.0581444	0.0290722	+	0.999577i	$0.490745\pi$
$380$	0	0
$381$	−1.21362e14	−0.774453
$382$	0	0
$383$	−7.69548e13	−0.477136	−0.238568	−	0.971126i	$-0.576678\pi$
$383$	−7.69548e13	−0.477136	−0.238568	+	0.971126i	$0.576678\pi$
$384$	0	0
$385$	7.59996e13	0.457907
$386$	0	0
$387$	3.98077e12	0.0233108
$388$	0	0
$389$	2.38408e14	1.35706	0.678528	−	0.734575i	$-0.262618\pi$
$389$	2.38408e14	1.35706	0.678528	+	0.734575i	$0.262618\pi$
$390$	0	0
$391$	6.95792e13	0.385041
$392$	0	0
$393$	2.08721e14	1.12307
$394$	0	0
$395$	1.92383e14	1.00666
$396$	0	0
$397$	−9.65653e13	−0.491443	−0.245722	−	0.969340i	$-0.579025\pi$
$397$	−9.65653e13	−0.491443	−0.245722	+	0.969340i	$0.579025\pi$
$398$	0	0
$399$	3.06301e13	0.151634
$400$	0	0
$401$	−1.66286e14	−0.800868	−0.400434	−	0.916326i	$-0.631141\pi$
$401$	−1.66286e14	−0.800868	−0.400434	+	0.916326i	$0.631141\pi$
$402$	0	0
$403$	1.25014e14	0.585841
$404$	0	0
$405$	2.15634e14	0.983366
$406$	0	0
$407$	4.14111e13	0.183801
$408$	0	0
$409$	−3.11178e14	−1.34441	−0.672203	−	0.740367i	$-0.734652\pi$
$409$	−3.11178e14	−1.34441	−0.672203	+	0.740367i	$0.734652\pi$
$410$	0	0
$411$	1.84368e14	0.775454
$412$	0	0
$413$	1.28061e14	0.524435
$414$	0	0
$415$	1.61870e14	0.645510
$416$	0	0
$417$	−2.13439e14	−0.828944
$418$	0	0
$419$	2.62930e14	0.994633	0.497316	−	0.867569i	$-0.334319\pi$
$419$	2.62930e14	0.994633	0.497316	+	0.867569i	$0.334319\pi$
$420$	0	0
$421$	−2.36898e13	−0.0872993	−0.0436496	−	0.999047i	$-0.513899\pi$
$421$	−2.36898e13	−0.0872993	−0.0436496	+	0.999047i	$0.513899\pi$
$422$	0	0
$423$	−2.02050e13	−0.0725415
$424$	0	0
$425$	1.10898e13	0.0387959
$426$	0	0
$427$	−7.08961e13	−0.241696
$428$	0	0
$429$	4.37714e14	1.45437
$430$	0	0
$431$	6.02794e13	0.195229	0.0976145	−	0.995224i	$-0.468879\pi$
$431$	6.02794e13	0.195229	0.0976145	+	0.995224i	$0.468879\pi$
$432$	0	0
$433$	3.38893e14	1.06999	0.534995	−	0.844856i	$-0.320314\pi$
$433$	3.38893e14	1.06999	0.534995	+	0.844856i	$0.320314\pi$
$434$	0	0
$435$	4.26964e14	1.31432
$436$	0	0
$437$	1.37436e14	0.412528
$438$	0	0
$439$	−2.58057e14	−0.755373	−0.377686	−	0.925934i	$-0.623280\pi$
$439$	−2.58057e14	−0.755373	−0.377686	+	0.925934i	$0.623280\pi$
$440$	0	0
$441$	−2.95807e12	−0.00844495
$442$	0	0
$443$	−2.38645e13	−0.0664557	−0.0332278	−	0.999448i	$-0.510579\pi$
$443$	−2.38645e13	−0.0664557	−0.0332278	+	0.999448i	$0.510579\pi$
$444$	0	0
$445$	2.06056e14	0.559765
$446$	0	0
$447$	5.30996e14	1.40734
$448$	0	0
$449$	2.55207e14	0.659990	0.329995	−	0.943983i	$-0.392953\pi$
$449$	2.55207e14	0.659990	0.329995	+	0.943983i	$0.392953\pi$
$450$	0	0
$451$	−1.08978e13	−0.0275022
$452$	0	0
$453$	4.94072e14	1.21689
$454$	0	0
$455$	−2.14133e14	−0.514779
$456$	0	0
$457$	−3.09569e14	−0.726471	−0.363236	−	0.931697i	$-0.618328\pi$
$457$	−3.09569e14	−0.726471	−0.363236	+	0.931697i	$0.618328\pi$
$458$	0	0
$459$	1.73106e14	0.396591
$460$	0	0
$461$	2.55261e14	0.570992	0.285496	−	0.958380i	$-0.407842\pi$
$461$	2.55261e14	0.570992	0.285496	+	0.958380i	$0.407842\pi$
$462$	0	0
$463$	−6.91247e14	−1.50986	−0.754932	−	0.655804i	$-0.772330\pi$
$463$	−6.91247e14	−1.50986	−0.754932	+	0.655804i	$0.772330\pi$
$464$	0	0
$465$	−2.15257e14	−0.459163
$466$	0	0
$467$	9.35481e13	0.194891	0.0974456	−	0.995241i	$-0.468933\pi$
$467$	9.35481e13	0.194891	0.0974456	+	0.995241i	$0.468933\pi$
$468$	0	0
$469$	−1.15212e14	−0.234449
$470$	0	0
$471$	6.16274e14	1.22506
$472$	0	0
$473$	−2.34493e14	−0.455401
$474$	0	0
$475$	2.19052e13	0.0415654
$476$	0	0
$477$	−1.75585e13	−0.0325563
$478$	0	0
$479$	−9.19182e14	−1.66554	−0.832772	−	0.553616i	$-0.813247\pi$
$479$	−9.19182e14	−1.66554	−0.832772	+	0.553616i	$0.813247\pi$
$480$	0	0
$481$	−1.16678e14	−0.206629
$482$	0	0
$483$	2.11254e14	0.365675
$484$	0	0
$485$	7.91088e14	1.33858
$486$	0	0
$487$	2.54129e14	0.420384	0.210192	−	0.977660i	$-0.432591\pi$
$487$	2.54129e14	0.420384	0.210192	+	0.977660i	$0.432591\pi$
$488$	0	0
$489$	−9.00064e14	−1.45571
$490$	0	0
$491$	−4.20049e14	−0.664282	−0.332141	−	0.943230i	$-0.607771\pi$
$491$	−4.20049e14	−0.664282	−0.332141	+	0.943230i	$0.607771\pi$
$492$	0	0
$493$	3.22424e14	0.498619
$494$	0	0
$495$	4.73533e13	0.0716179
$496$	0	0
$497$	−2.63746e14	−0.390145
$498$	0	0
$499$	−5.07590e14	−0.734446	−0.367223	−	0.930133i	$-0.619691\pi$
$499$	−5.07590e14	−0.734446	−0.367223	+	0.930133i	$0.619691\pi$
$500$	0	0
$501$	5.20840e14	0.737220
$502$	0	0
$503$	2.94182e14	0.407373	0.203686	−	0.979036i	$-0.434708\pi$
$503$	2.94182e14	0.407373	0.203686	+	0.979036i	$0.434708\pi$
$504$	0	0
$505$	−1.48242e15	−2.00849
$506$	0	0
$507$	−5.01619e14	−0.665014
$508$	0	0
$509$	−8.90073e13	−0.115472	−0.0577361	−	0.998332i	$-0.518388\pi$
$509$	−8.90073e13	−0.115472	−0.0577361	+	0.998332i	$0.518388\pi$
$510$	0	0
$511$	4.53895e14	0.576289
$512$	0	0
$513$	3.41928e14	0.424902
$514$	0	0
$515$	6.15623e14	0.748816
$516$	0	0
$517$	1.19020e15	1.41718
$518$	0	0
$519$	5.96821e14	0.695703
$520$	0	0
$521$	−8.61760e14	−0.983510	−0.491755	−	0.870734i	$-0.663644\pi$
$521$	−8.61760e14	−0.983510	−0.491755	+	0.870734i	$0.663644\pi$
$522$	0	0
$523$	2.34857e14	0.262449	0.131224	−	0.991353i	$-0.458109\pi$
$523$	2.34857e14	0.262449	0.131224	+	0.991353i	$0.458109\pi$
$524$	0	0
$525$	3.36705e13	0.0368446
$526$	0	0
$527$	−1.62553e14	−0.174195
$528$	0	0
$529$	−4.92076e12	−0.00516447
$530$	0	0
$531$	7.97912e13	0.0820230
$532$	0	0
$533$	3.07051e13	0.0309180
$534$	0	0
$535$	−7.40348e14	−0.730281
$536$	0	0
$537$	−3.42480e14	−0.330961
$538$	0	0
$539$	1.74250e14	0.164981
$540$	0	0
$541$	1.03679e15	0.961845	0.480922	−	0.876763i	$-0.340302\pi$
$541$	1.03679e15	0.961845	0.480922	+	0.876763i	$0.340302\pi$
$542$	0	0
$543$	−1.28902e15	−1.17182
$544$	0	0
$545$	1.61694e15	1.44050
$546$	0	0
$547$	1.03887e15	0.907047	0.453523	−	0.891244i	$-0.350167\pi$
$547$	1.03887e15	0.907047	0.453523	+	0.891244i	$0.350167\pi$
$548$	0	0
$549$	−4.41734e13	−0.0378018
$550$	0	0
$551$	6.36867e14	0.534214
$552$	0	0
$553$	4.41090e14	0.362693
$554$	0	0
$555$	2.00904e14	0.161949
$556$	0	0
$557$	−1.77674e15	−1.40417	−0.702087	−	0.712092i	$-0.747748\pi$
$557$	−1.77674e15	−1.40417	−0.702087	+	0.712092i	$0.747748\pi$
$558$	0	0
$559$	6.60698e14	0.511962
$560$	0	0
$561$	−5.69151e14	−0.432446
$562$	0	0
$563$	−5.17003e14	−0.385210	−0.192605	−	0.981276i	$-0.561694\pi$
$563$	−5.17003e14	−0.385210	−0.192605	+	0.981276i	$0.561694\pi$
$564$	0	0
$565$	1.02905e15	0.751914
$566$	0	0
$567$	4.94400e14	0.354300
$568$	0	0
$569$	−1.91376e15	−1.34515	−0.672573	−	0.740031i	$-0.734811\pi$
$569$	−1.91376e15	−1.34515	−0.672573	+	0.740031i	$0.734811\pi$
$570$	0	0
$571$	1.01850e14	0.0702201	0.0351100	−	0.999383i	$-0.488822\pi$
$571$	1.01850e14	0.0702201	0.0351100	+	0.999383i	$0.488822\pi$
$572$	0	0
$573$	−1.30425e15	−0.882082
$574$	0	0
$575$	1.51079e14	0.100237
$576$	0	0
$577$	−6.05823e14	−0.394347	−0.197174	−	0.980369i	$-0.563176\pi$
$577$	−6.05823e14	−0.394347	−0.197174	+	0.980369i	$0.563176\pi$
$578$	0	0
$579$	−1.31991e15	−0.842968
$580$	0	0
$581$	3.71132e14	0.232573
$582$	0	0
$583$	1.03431e15	0.636022
$584$	0	0
$585$	−1.33420e14	−0.0805128
$586$	0	0
$587$	−3.07868e15	−1.82329	−0.911644	−	0.410982i	$-0.865186\pi$
$587$	−3.07868e15	−1.82329	−0.911644	+	0.410982i	$0.865186\pi$
$588$	0	0
$589$	−3.21082e14	−0.186630
$590$	0	0
$591$	−2.39660e15	−1.36730
$592$	0	0
$593$	−2.12463e15	−1.18982	−0.594912	−	0.803791i	$-0.702813\pi$
$593$	−2.12463e15	−1.18982	−0.594912	+	0.803791i	$0.702813\pi$
$594$	0	0
$595$	2.78433e14	0.153065
$596$	0	0
$597$	−2.47665e15	−1.33661
$598$	0	0
$599$	−1.10000e15	−0.582833	−0.291416	−	0.956596i	$-0.594126\pi$
$599$	−1.10000e15	−0.582833	−0.291416	+	0.956596i	$0.594126\pi$
$600$	0	0
$601$	1.35394e15	0.704354	0.352177	−	0.935933i	$-0.385441\pi$
$601$	1.35394e15	0.704354	0.352177	+	0.935933i	$0.385441\pi$
$602$	0	0
$603$	−7.17855e13	−0.0366684
$604$	0	0
$605$	−6.97956e14	−0.350085
$606$	0	0
$607$	2.52388e15	1.24317	0.621587	−	0.783345i	$-0.286488\pi$
$607$	2.52388e15	1.24317	0.621587	+	0.783345i	$0.286488\pi$
$608$	0	0
$609$	9.78930e14	0.473540
$610$	0	0
$611$	−3.35346e15	−1.59319
$612$	0	0
$613$	2.76657e15	1.29095	0.645474	−	0.763782i	$-0.276660\pi$
$613$	2.76657e15	1.29095	0.645474	+	0.763782i	$0.276660\pi$
$614$	0	0
$615$	−5.28703e13	−0.0242325
$616$	0	0
$617$	2.77227e15	1.24815	0.624075	−	0.781365i	$-0.285476\pi$
$617$	2.77227e15	1.24815	0.624075	+	0.781365i	$0.285476\pi$
$618$	0	0
$619$	4.10980e11	0.000181770 0	9.08851e−5	−	1.00000i	$-0.499971\pi$
$619$	4.10980e11	0.000181770 0	9.08851e−5		1.00000i	$0.499971\pi$
$620$	0	0
$621$	2.35826e15	1.02468
$622$	0	0
$623$	4.72440e14	0.201680
$624$	0	0
$625$	−2.59971e15	−1.09040
$626$	0	0
$627$	−1.12422e15	−0.463317
$628$	0	0
$629$	1.51714e14	0.0614394
$630$	0	0
$631$	−4.30802e15	−1.71441	−0.857207	−	0.514972i	$-0.827802\pi$
$631$	−4.30802e15	−1.71441	−0.857207	+	0.514972i	$0.827802\pi$
$632$	0	0
$633$	−4.53867e15	−1.77504
$634$	0	0
$635$	−2.17910e15	−0.837570
$636$	0	0
$637$	−4.90958e14	−0.185472
$638$	0	0
$639$	−1.64333e14	−0.0610196
$640$	0	0
$641$	3.18195e15	1.16138	0.580690	−	0.814125i	$-0.302783\pi$
$641$	3.18195e15	1.16138	0.580690	+	0.814125i	$0.302783\pi$
$642$	0	0
$643$	−1.61122e15	−0.578087	−0.289044	−	0.957316i	$-0.593337\pi$
$643$	−1.61122e15	−0.578087	−0.289044	+	0.957316i	$0.593337\pi$
$644$	0	0
$645$	−1.13764e15	−0.401259
$646$	0	0
$647$	1.66716e15	0.578100	0.289050	−	0.957314i	$-0.406661\pi$
$647$	1.66716e15	0.578100	0.289050	+	0.957314i	$0.406661\pi$
$648$	0	0
$649$	−4.70022e15	−1.60241
$650$	0	0
$651$	−4.93536e14	−0.165433
$652$	0	0
$653$	−4.62553e15	−1.52454	−0.762270	−	0.647259i	$-0.775915\pi$
$653$	−4.62553e15	−1.52454	−0.762270	+	0.647259i	$0.775915\pi$
$654$	0	0
$655$	3.74767e15	1.21460
$656$	0	0
$657$	2.82809e14	0.0901331
$658$	0	0
$659$	3.74915e15	1.17507	0.587533	−	0.809200i	$-0.300099\pi$
$659$	3.74915e15	1.17507	0.587533	+	0.809200i	$0.300099\pi$
$660$	0	0
$661$	1.21748e15	0.375280	0.187640	−	0.982238i	$-0.439916\pi$
$661$	1.21748e15	0.375280	0.187640	+	0.982238i	$0.439916\pi$
$662$	0	0
$663$	1.60361e15	0.486156
$664$	0	0
$665$	5.49974e14	0.163992
$666$	0	0
$667$	4.39243e15	1.28829
$668$	0	0
$669$	4.21360e15	1.21565
$670$	0	0
$671$	2.60210e15	0.738499
$672$	0	0
$673$	−1.92073e15	−0.536270	−0.268135	−	0.963381i	$-0.586407\pi$
$673$	−1.92073e15	−0.536270	−0.268135	+	0.963381i	$0.586407\pi$
$674$	0	0
$675$	3.75869e14	0.103244
$676$	0	0
$677$	2.47962e15	0.670113	0.335056	−	0.942198i	$-0.391245\pi$
$677$	2.47962e15	0.670113	0.335056	+	0.942198i	$0.391245\pi$
$678$	0	0
$679$	1.81378e15	0.482283
$680$	0	0
$681$	−1.21540e15	−0.317987
$682$	0	0
$683$	2.32076e15	0.597470	0.298735	−	0.954336i	$-0.403435\pi$
$683$	2.32076e15	0.597470	0.298735	+	0.954336i	$0.403435\pi$
$684$	0	0
$685$	3.31040e15	0.838653
$686$	0	0
$687$	−1.47191e15	−0.366961
$688$	0	0
$689$	−2.91421e15	−0.715016
$690$	0	0
$691$	−7.53652e14	−0.181987	−0.0909937	−	0.995851i	$-0.529004\pi$
$691$	−7.53652e14	−0.181987	−0.0909937	+	0.995851i	$0.529004\pi$
$692$	0	0
$693$	1.08570e14	0.0258035
$694$	0	0
$695$	−3.83237e15	−0.896502
$696$	0	0
$697$	−3.99253e13	−0.00919322
$698$	0	0
$699$	3.43711e15	0.779056
$700$	0	0
$701$	4.29080e15	0.957390	0.478695	−	0.877981i	$-0.341110\pi$
$701$	4.29080e15	0.957390	0.478695	+	0.877981i	$0.341110\pi$
$702$	0	0
$703$	2.99673e14	0.0658254
$704$	0	0
$705$	5.77424e15	1.24869
$706$	0	0
$707$	−3.39885e15	−0.723645
$708$	0	0
$709$	5.59252e15	1.17234	0.586169	−	0.810188i	$-0.300635\pi$
$709$	5.59252e15	1.17234	0.586169	+	0.810188i	$0.300635\pi$
$710$	0	0
$711$	2.74831e14	0.0567261
$712$	0	0
$713$	−2.21448e15	−0.450070
$714$	0	0
$715$	7.85932e15	1.57290
$716$	0	0
$717$	−4.30472e15	−0.848379
$718$	0	0
$719$	−9.34565e14	−0.181385	−0.0906923	−	0.995879i	$-0.528908\pi$
$719$	−9.34565e14	−0.181385	−0.0906923	+	0.995879i	$0.528908\pi$
$720$	0	0
$721$	1.41148e15	0.269793
$722$	0	0
$723$	−8.33750e15	−1.56955
$724$	0	0
$725$	7.00084e14	0.129805
$726$	0	0
$727$	−4.67306e15	−0.853418	−0.426709	−	0.904389i	$-0.640327\pi$
$727$	−4.67306e15	−0.853418	−0.426709	+	0.904389i	$0.640327\pi$
$728$	0	0
$729$	5.84768e15	1.05192
$730$	0	0
$731$	−8.59092e14	−0.152228
$732$	0	0
$733$	−8.79200e15	−1.53467	−0.767336	−	0.641245i	$-0.778418\pi$
$733$	−8.79200e15	−1.53467	−0.767336	+	0.641245i	$0.778418\pi$
$734$	0	0
$735$	8.45367e14	0.145367
$736$	0	0
$737$	4.22863e15	0.716355
$738$	0	0
$739$	4.30796e15	0.718998	0.359499	−	0.933146i	$-0.382948\pi$
$739$	4.30796e15	0.718998	0.359499	+	0.933146i	$0.382948\pi$
$740$	0	0
$741$	3.16754e15	0.520861
$742$	0	0
$743$	4.01936e15	0.651207	0.325603	−	0.945506i	$-0.394433\pi$
$743$	4.01936e15	0.651207	0.325603	+	0.945506i	$0.394433\pi$
$744$	0	0
$745$	9.53423e15	1.52204
$746$	0	0
$747$	2.31242e14	0.0363750
$748$	0	0
$749$	−1.69745e15	−0.263115
$750$	0	0
$751$	−2.88632e15	−0.440885	−0.220443	−	0.975400i	$-0.570750\pi$
$751$	−2.88632e15	−0.440885	−0.220443	+	0.975400i	$0.570750\pi$
$752$	0	0
$753$	−4.05634e15	−0.610607
$754$	0	0
$755$	8.87125e15	1.31606
$756$	0	0
$757$	1.00258e16	1.46586	0.732931	−	0.680303i	$-0.238152\pi$
$757$	1.00258e16	1.46586	0.732931	+	0.680303i	$0.238152\pi$
$758$	0	0
$759$	−7.75364e15	−1.11732
$760$	0	0
$761$	−1.42341e15	−0.202169	−0.101084	−	0.994878i	$-0.532231\pi$
$761$	−1.42341e15	−0.202169	−0.101084	+	0.994878i	$0.532231\pi$
$762$	0	0
$763$	3.70727e15	0.519002
$764$	0	0
$765$	1.73484e14	0.0239398
$766$	0	0
$767$	1.32431e16	1.80142
$768$	0	0
$769$	2.49417e15	0.334450	0.167225	−	0.985919i	$-0.446519\pi$
$769$	2.49417e15	0.334450	0.167225	+	0.985919i	$0.446519\pi$
$770$	0	0
$771$	1.02919e16	1.36049
$772$	0	0
$773$	−7.83570e15	−1.02115	−0.510576	−	0.859832i	$-0.670568\pi$
$773$	−7.83570e15	−1.02115	−0.510576	+	0.859832i	$0.670568\pi$
$774$	0	0
$775$	−3.52954e14	−0.0453480
$776$	0	0
$777$	4.60628e14	0.0583492
$778$	0	0
$779$	−7.88624e13	−0.00984949
$780$	0	0
$781$	9.68027e15	1.19208
$782$	0	0
$783$	1.09279e16	1.32693
$784$	0	0
$785$	1.10654e16	1.32490
$786$	0	0
$787$	−1.42638e16	−1.68413	−0.842063	−	0.539379i	$-0.818659\pi$
$787$	−1.42638e16	−1.68413	−0.842063	+	0.539379i	$0.818659\pi$
$788$	0	0
$789$	−2.33853e15	−0.272282
$790$	0	0
$791$	2.35937e15	0.270910
$792$	0	0
$793$	−7.33155e15	−0.830220
$794$	0	0
$795$	5.01790e15	0.560406
$796$	0	0
$797$	−1.76450e16	−1.94357	−0.971785	−	0.235869i	$-0.924206\pi$
$797$	−1.76450e16	−1.94357	−0.971785	+	0.235869i	$0.924206\pi$
$798$	0	0
$799$	4.36044e15	0.473722
$800$	0	0
$801$	2.94364e14	0.0315432
$802$	0	0
$803$	−1.66593e16	−1.76085
$804$	0	0
$805$	3.79314e15	0.395477
$806$	0	0
$807$	−6.33765e15	−0.651814
$808$	0	0
$809$	−1.11716e16	−1.13344	−0.566722	−	0.823909i	$-0.691788\pi$
$809$	−1.11716e16	−1.13344	−0.566722	+	0.823909i	$0.691788\pi$
$810$	0	0
$811$	−5.32562e15	−0.533034	−0.266517	−	0.963830i	$-0.585873\pi$
$811$	−5.32562e15	−0.533034	−0.266517	+	0.963830i	$0.585873\pi$
$812$	0	0
$813$	4.13735e15	0.408530
$814$	0	0
$815$	−1.61610e16	−1.57435
$816$	0	0
$817$	−1.69692e15	−0.163095
$818$	0	0
$819$	−3.05902e14	−0.0290082
$820$	0	0
$821$	−1.56619e15	−0.146540	−0.0732701	−	0.997312i	$-0.523344\pi$
$821$	−1.56619e15	−0.146540	−0.0732701	+	0.997312i	$0.523344\pi$
$822$	0	0
$823$	−1.19387e16	−1.10219	−0.551095	−	0.834442i	$-0.685790\pi$
$823$	−1.19387e16	−1.10219	−0.551095	+	0.834442i	$0.685790\pi$
$824$	0	0
$825$	−1.23581e15	−0.112578
$826$	0	0
$827$	1.61047e16	1.44768	0.723839	−	0.689969i	$-0.242376\pi$
$827$	1.61047e16	1.44768	0.723839	+	0.689969i	$0.242376\pi$
$828$	0	0
$829$	−2.03151e16	−1.80206	−0.901029	−	0.433760i	$-0.857187\pi$
$829$	−2.03151e16	−1.80206	−0.901029	+	0.433760i	$0.857187\pi$
$830$	0	0
$831$	−1.30783e16	−1.14485
$832$	0	0
$833$	6.38383e14	0.0551485
$834$	0	0
$835$	9.35188e15	0.797303
$836$	0	0
$837$	−5.50941e15	−0.463570
$838$	0	0
$839$	6.36294e15	0.528405	0.264203	−	0.964467i	$-0.414891\pi$
$839$	6.36294e15	0.528405	0.264203	+	0.964467i	$0.414891\pi$
$840$	0	0
$841$	8.15362e15	0.668302
$842$	0	0
$843$	1.11865e16	0.904989
$844$	0	0
$845$	−9.00676e15	−0.719211
$846$	0	0
$847$	−1.60025e15	−0.126133
$848$	0	0
$849$	2.05280e16	1.59718
$850$	0	0
$851$	2.06682e15	0.158742
$852$	0	0
$853$	−3.44689e15	−0.261341	−0.130670	−	0.991426i	$-0.541713\pi$
$853$	−3.44689e15	−0.261341	−0.130670	+	0.991426i	$0.541713\pi$
$854$	0	0
$855$	3.42674e14	0.0256488
$856$	0	0
$857$	3.06220e15	0.226276	0.113138	−	0.993579i	$-0.463910\pi$
$857$	3.06220e15	0.226276	0.113138	+	0.993579i	$0.463910\pi$
$858$	0	0
$859$	−1.78449e16	−1.30182	−0.650911	−	0.759154i	$-0.725613\pi$
$859$	−1.78449e16	−1.30182	−0.650911	+	0.759154i	$0.725613\pi$
$860$	0	0
$861$	−1.21220e14	−0.00873083
$862$	0	0
$863$	−7.12682e15	−0.506800	−0.253400	−	0.967362i	$-0.581549\pi$
$863$	−7.12682e15	−0.506800	−0.253400	+	0.967362i	$0.581549\pi$
$864$	0	0
$865$	1.07161e16	0.752402
$866$	0	0
$867$	1.19066e16	0.825438
$868$	0	0
$869$	−1.61893e16	−1.10820
$870$	0	0
$871$	−1.19144e16	−0.805327
$872$	0	0
$873$	1.13012e15	0.0754302
$874$	0	0
$875$	−5.41118e15	−0.356655
$876$	0	0
$877$	−1.35692e16	−0.883192	−0.441596	−	0.897214i	$-0.645587\pi$
$877$	−1.35692e16	−0.883192	−0.441596	+	0.897214i	$0.645587\pi$
$878$	0	0
$879$	3.84984e15	0.247459
$880$	0	0
$881$	1.26009e16	0.799896	0.399948	−	0.916538i	$-0.369028\pi$
$881$	1.26009e16	0.799896	0.399948	+	0.916538i	$0.369028\pi$
$882$	0	0
$883$	2.58733e15	0.162206	0.0811031	−	0.996706i	$-0.474156\pi$
$883$	2.58733e15	0.162206	0.0811031	+	0.996706i	$0.474156\pi$
$884$	0	0
$885$	−2.28030e16	−1.41190
$886$	0	0
$887$	1.52043e16	0.929794	0.464897	−	0.885365i	$-0.346091\pi$
$887$	1.52043e16	0.929794	0.464897	+	0.885365i	$0.346091\pi$
$888$	0	0
$889$	−4.99617e15	−0.301771
$890$	0	0
$891$	−1.81460e16	−1.08256
$892$	0	0
$893$	8.61296e15	0.507539
$894$	0	0
$895$	−6.14935e15	−0.357934
$896$	0	0
$897$	2.18463e16	1.25609
$898$	0	0
$899$	−1.02617e16	−0.582829
$900$	0	0
$901$	3.78929e15	0.212604
$902$	0	0
$903$	−2.60834e15	−0.144571
$904$	0	0
$905$	−2.31448e16	−1.26732
$906$	0	0
$907$	2.12864e16	1.15149	0.575747	−	0.817628i	$-0.304711\pi$
$907$	2.12864e16	1.15149	0.575747	+	0.817628i	$0.304711\pi$
$908$	0	0
$909$	−2.11773e15	−0.113180
$910$	0	0
$911$	−2.76268e16	−1.45875	−0.729374	−	0.684115i	$-0.760188\pi$
$911$	−2.76268e16	−1.45875	−0.729374	+	0.684115i	$0.760188\pi$
$912$	0	0
$913$	−1.36217e16	−0.710625
$914$	0	0
$915$	1.26240e16	0.650700
$916$	0	0
$917$	8.59255e15	0.437614
$918$	0	0
$919$	2.77806e16	1.39799	0.698997	−	0.715124i	$-0.253630\pi$
$919$	2.77806e16	1.39799	0.698997	+	0.715124i	$0.253630\pi$
$920$	0	0
$921$	1.88812e16	0.938860
$922$	0	0
$923$	−2.72747e16	−1.34014
$924$	0	0
$925$	3.29419e14	0.0159945
$926$	0	0
$927$	8.79455e14	0.0421964
$928$	0	0
$929$	3.63541e15	0.172372	0.0861860	−	0.996279i	$-0.472532\pi$
$929$	3.63541e15	0.172372	0.0861860	+	0.996279i	$0.472532\pi$
$930$	0	0
$931$	1.26097e15	0.0590854
$932$	0	0
$933$	−1.43984e16	−0.666755
$934$	0	0
$935$	−1.02193e16	−0.467690
$936$	0	0
$937$	4.32101e16	1.95442	0.977209	−	0.212280i	$-0.0680889\pi$
$937$	4.32101e16	1.95442	0.977209	+	0.212280i	$0.0680889\pi$
$938$	0	0
$939$	−3.83367e15	−0.171378
$940$	0	0
$941$	−3.29889e16	−1.45756	−0.728778	−	0.684750i	$-0.759911\pi$
$941$	−3.29889e16	−1.45756	−0.728778	+	0.684750i	$0.759911\pi$
$942$	0	0
$943$	−5.43909e14	−0.0237526
$944$	0	0
$945$	9.43695e15	0.407340
$946$	0	0
$947$	−3.33846e16	−1.42436	−0.712182	−	0.701995i	$-0.752293\pi$
$947$	−3.33846e16	−1.42436	−0.712182	+	0.701995i	$0.752293\pi$
$948$	0	0
$949$	4.69385e16	1.97954
$950$	0	0
$951$	3.23514e16	1.34865
$952$	0	0
$953$	1.76816e16	0.728636	0.364318	−	0.931275i	$-0.381302\pi$
$953$	1.76816e16	0.728636	0.364318	+	0.931275i	$0.381302\pi$
$954$	0	0
$955$	−2.34182e16	−0.953971
$956$	0	0
$957$	−3.59297e16	−1.44690
$958$	0	0
$959$	7.58999e15	0.302161
$960$	0	0
$961$	−2.02349e16	−0.796386
$962$	0	0
$963$	−1.05763e15	−0.0411519
$964$	0	0
$965$	−2.36994e16	−0.911669
$966$	0	0
$967$	3.83019e16	1.45671	0.728357	−	0.685198i	$-0.240284\pi$
$967$	3.83019e16	1.45671	0.728357	+	0.685198i	$0.240284\pi$
$968$	0	0
$969$	−4.11869e15	−0.154874
$970$	0	0
$971$	2.82557e15	0.105051	0.0525256	−	0.998620i	$-0.483273\pi$
$971$	2.82557e15	0.105051	0.0525256	+	0.998620i	$0.483273\pi$
$972$	0	0
$973$	−8.78676e15	−0.323004
$974$	0	0
$975$	3.48196e15	0.126560
$976$	0	0
$977$	−5.12420e16	−1.84165	−0.920823	−	0.389981i	$-0.872482\pi$
$977$	−5.12420e16	−1.84165	−0.920823	+	0.389981i	$0.872482\pi$
$978$	0	0
$979$	−1.73400e16	−0.616230
$980$	0	0
$981$	2.30990e15	0.0811732
$982$	0	0
$983$	−2.34135e16	−0.813620	−0.406810	−	0.913513i	$-0.633359\pi$
$983$	−2.34135e16	−0.813620	−0.406810	+	0.913513i	$0.633359\pi$
$984$	0	0
$985$	−4.30319e16	−1.47874
$986$	0	0
$987$	1.32390e16	0.449895
$988$	0	0
$989$	−1.17035e16	−0.393313
$990$	0	0
$991$	1.61300e16	0.536080	0.268040	−	0.963408i	$-0.413624\pi$
$991$	1.61300e16	0.536080	0.268040	+	0.963408i	$0.413624\pi$
$992$	0	0
$993$	3.06066e16	1.00599
$994$	0	0
$995$	−4.44691e16	−1.44555
$996$	0	0
$997$	−4.38138e16	−1.40860	−0.704300	−	0.709903i	$-0.748739\pi$
$997$	−4.38138e16	−1.40860	−0.704300	+	0.709903i	$0.748739\pi$
$998$	0	0
$999$	5.14206e15	0.163503

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.12.a.d.1.1		2
4.3	odd	2		7.12.a.a.1.1	✓	2
12.11	even	2		63.12.a.c.1.2		2
20.3	even	4		175.12.b.a.99.4		4
20.7	even	4		175.12.b.a.99.1		4
20.19	odd	2		175.12.a.a.1.2		2
28.3	even	6		49.12.c.e.30.2		4
28.11	odd	6		49.12.c.d.30.2		4
28.19	even	6		49.12.c.e.18.2		4
28.23	odd	6		49.12.c.d.18.2		4
28.27	even	2		49.12.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.12.a.a.1.1	✓	2	4.3	odd	2
49.12.a.c.1.1		2	28.27	even	2
49.12.c.d.18.2		4	28.23	odd	6
49.12.c.d.30.2		4	28.11	odd	6
49.12.c.e.18.2		4	28.19	even	6
49.12.c.e.30.2		4	28.3	even	6
63.12.a.c.1.2		2	12.11	even	2
112.12.a.d.1.1		2	1.1	even	1	trivial
175.12.a.a.1.2		2	20.19	odd	2
175.12.b.a.99.1		4	20.7	even	4
175.12.b.a.99.4		4	20.3	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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

\(f(q)\)	\(=\)	\(q-408.259 q^{3} -7330.43 q^{5} -16807.0 q^{7} -10472.0 q^{9} +O(q^{10})\)	\(q-408.259 q^{3} -7330.43 q^{5} -16807.0 q^{7} -10472.0 q^{9} +616867. q^{11} -1.73806e6 q^{13} +2.99271e6 q^{15} +2.25996e6 q^{17} +4.46399e6 q^{19} +6.86160e6 q^{21} +3.07878e7 q^{23} +4.90709e6 q^{25} +7.65970e7 q^{27} +1.42668e8 q^{29} -7.19272e7 q^{31} -2.51841e8 q^{33} +1.23203e8 q^{35} +6.71313e7 q^{37} +7.09576e8 q^{39} -1.76664e7 q^{41} -3.80136e8 q^{43} +7.67641e7 q^{45} +1.92943e9 q^{47} +2.82475e8 q^{49} -9.22648e8 q^{51} +1.67671e9 q^{53} -4.52190e9 q^{55} -1.82246e9 q^{57} -7.61950e9 q^{59} +4.21825e9 q^{61} +1.76003e8 q^{63} +1.27407e10 q^{65} +6.85501e9 q^{67} -1.25694e10 q^{69} +1.56926e10 q^{71} -2.70063e10 q^{73} -2.00336e9 q^{75} -1.03677e10 q^{77} -2.62444e10 q^{79} -2.94163e10 q^{81} -2.20820e10 q^{83} -1.65665e10 q^{85} -5.82454e10 q^{87} -2.81097e10 q^{89} +2.92115e10 q^{91} +2.93649e10 q^{93} -3.27229e10 q^{95} -1.07918e11 q^{97} -6.45982e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 120 q^{3} - 13500 q^{5} - 33614 q^{7} - 104526 q^{9}+O(q^{10}) \) 2 * q - 120 * q^3 - 13500 * q^5 - 33614 * q^7 - 104526 * q^9	\( 2 q - 120 q^{3} - 13500 q^{5} - 33614 q^{7} - 104526 q^{9} + 750816 q^{11} - 9548 q^{13} + 1214280 q^{15} + 4160052 q^{17} + 17998712 q^{19} + 2016840 q^{21} + 66161016 q^{23} - 5857450 q^{25} - 1578960 q^{27} + 61515612 q^{29} + 15281552 q^{31} - 213229440 q^{33} + 226894500 q^{35} - 527218340 q^{37} + 1207833816 q^{39} - 178276140 q^{41} - 1826745232 q^{43} + 657036900 q^{45} - 568240704 q^{47} + 564950498 q^{49} - 374929920 q^{51} - 4185816372 q^{53} - 5348308800 q^{55} + 2079040344 q^{57} - 3111345000 q^{59} + 15042595060 q^{61} + 1756768482 q^{63} + 2076550560 q^{65} - 9856523968 q^{67} - 2372754960 q^{69} + 24312011328 q^{71} - 30890001932 q^{73} - 5106333000 q^{75} - 12618964512 q^{77} - 1992804256 q^{79} - 35289830358 q^{81} - 5277014568 q^{83} - 28289229000 q^{85} - 81638222976 q^{87} - 101541312828 q^{89} + 160473236 q^{91} + 54503588112 q^{93} - 116226367560 q^{95} - 192228621116 q^{97} - 19058239008 q^{99}+O(q^{100}) \) 2 * q - 120 * q^3 - 13500 * q^5 - 33614 * q^7 - 104526 * q^9 + 750816 * q^11 - 9548 * q^13 + 1214280 * q^15 + 4160052 * q^17 + 17998712 * q^19 + 2016840 * q^21 + 66161016 * q^23 - 5857450 * q^25 - 1578960 * q^27 + 61515612 * q^29 + 15281552 * q^31 - 213229440 * q^33 + 226894500 * q^35 - 527218340 * q^37 + 1207833816 * q^39 - 178276140 * q^41 - 1826745232 * q^43 + 657036900 * q^45 - 568240704 * q^47 + 564950498 * q^49 - 374929920 * q^51 - 4185816372 * q^53 - 5348308800 * q^55 + 2079040344 * q^57 - 3111345000 * q^59 + 15042595060 * q^61 + 1756768482 * q^63 + 2076550560 * q^65 - 9856523968 * q^67 - 2372754960 * q^69 + 24312011328 * q^71 - 30890001932 * q^73 - 5106333000 * q^75 - 12618964512 * q^77 - 1992804256 * q^79 - 35289830358 * q^81 - 5277014568 * q^83 - 28289229000 * q^85 - 81638222976 * q^87 - 101541312828 * q^89 + 160473236 * q^91 + 54503588112 * q^93 - 116226367560 * q^95 - 192228621116 * q^97 - 19058239008 * q^99

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