LMFDB - Embedded newform 112.10.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,10,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, 10, names="a")

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

chi := DirichletCharacter("112.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
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:	$57.6840136504$
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} - 426x + 2016 $ x^3 - x^2 - 426*x + 2016
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-22.2358$ 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)

$f(q)$	$=$	$q-163.415 q^{3} +1922.19 q^{5} -2401.00 q^{7} +7021.32 q^{9} +90199.9 q^{11} -3199.89 q^{13} -314114. q^{15} +116494. q^{17} +142449. q^{19} +392358. q^{21} -1.27391e6 q^{23} +1.74168e6 q^{25} +2.06910e6 q^{27} -1.42931e6 q^{29} -9.67494e6 q^{31} -1.47400e7 q^{33} -4.61518e6 q^{35} -8.67744e6 q^{37} +522908. q^{39} +1.32544e7 q^{41} +2.97554e7 q^{43} +1.34963e7 q^{45} +1.07969e7 q^{47} +5.76480e6 q^{49} -1.90369e7 q^{51} +7.07399e7 q^{53} +1.73381e8 q^{55} -2.32783e7 q^{57} -6.40400e6 q^{59} +1.69190e8 q^{61} -1.68582e7 q^{63} -6.15078e6 q^{65} +1.16276e8 q^{67} +2.08176e8 q^{69} -1.44496e8 q^{71} +1.60155e8 q^{73} -2.84617e8 q^{75} -2.16570e8 q^{77} +4.89322e8 q^{79} -4.76322e8 q^{81} +8.31590e7 q^{83} +2.23924e8 q^{85} +2.33569e8 q^{87} +2.08083e6 q^{89} +7.68292e6 q^{91} +1.58103e9 q^{93} +2.73815e8 q^{95} -3.15885e8 q^{97} +6.33322e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 84 q^{3} + 1554 q^{5} - 7203 q^{7} - 26001 q^{9} + 3444 q^{11} - 19782 q^{13} - 200304 q^{15} + 1016694 q^{17} - 222852 q^{19} + 201684 q^{21} - 1885632 q^{23} + 3073221 q^{25} - 551880 q^{27} + 4081818 q^{29}+ \cdots + 1900979172 q^{99}+O(q^{100}) $ 3 * q - 84 * q^3 + 1554 * q^5 - 7203 * q^7 - 26001 * q^9 + 3444 * q^11 - 19782 * q^13 - 200304 * q^15 + 1016694 * q^17 - 222852 * q^19 + 201684 * q^21 - 1885632 * q^23 + 3073221 * q^25 - 551880 * q^27 + 4081818 * q^29 - 2869440 * q^31 - 20259792 * q^33 - 3731154 * q^35 + 1395618 * q^37 + 8990688 * q^39 - 14420658 * q^41 + 61631172 * q^43 + 29774682 * q^45 + 10368960 * q^47 + 17294403 * q^49 + 26146728 * q^51 + 67502610 * q^53 + 105823032 * q^55 + 8471112 * q^57 + 42590100 * q^59 + 191746842 * q^61 + 62428401 * q^63 + 364283220 * q^65 + 255175788 * q^67 + 257903856 * q^69 - 296514504 * q^71 + 344213310 * q^73 - 279031116 * q^75 - 8269044 * q^77 + 960412656 * q^79 - 35827677 * q^81 + 1100517180 * q^83 + 438179412 * q^85 + 621821592 * q^87 + 506816478 * q^89 + 47496582 * q^91 + 1693258512 * q^93 + 2203071072 * q^95 - 647498250 * q^97 + 1900979172 * q^99

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
$3$	−163.415	−1.16478	−0.582392	−	0.812908i	$-0.697883\pi$
$3$	−163.415	−1.16478	−0.582392	+	0.812908i	$0.697883\pi$
$4$	0	0
$5$	1922.19	1.37541	0.687703	−	0.725992i	$-0.258619\pi$
$5$	1922.19	1.37541	0.687703	+	0.725992i	$0.258619\pi$
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	0	0
$9$	7021.32	0.356720
$10$	0	0
$11$	90199.9	1.85754	0.928772	−	0.370652i	$-0.120866\pi$
$11$	90199.9	1.85754	0.928772	+	0.370652i	$0.120866\pi$
$12$	0	0
$13$	−3199.89	−0.0310734	−0.0155367	−	0.999879i	$-0.504946\pi$
$13$	−3199.89	−0.0310734	−0.0155367	+	0.999879i	$0.504946\pi$
$14$	0	0
$15$	−314114.	−1.60205
$16$	0	0
$17$	116494.	0.338286	0.169143	−	0.985591i	$-0.445900\pi$
$17$	116494.	0.338286	0.169143	+	0.985591i	$0.445900\pi$
$18$	0	0
$19$	142449.	0.250767	0.125383	−	0.992108i	$-0.459984\pi$
$19$	142449.	0.250767	0.125383	+	0.992108i	$0.459984\pi$
$20$	0	0
$21$	392358.	0.440247
$22$	0	0
$23$	−1.27391e6	−0.949213	−0.474606	−	0.880198i	$-0.657410\pi$
$23$	−1.27391e6	−0.949213	−0.474606	+	0.880198i	$0.657410\pi$
$24$	0	0
$25$	1.74168e6	0.891742
$26$	0	0
$27$	2.06910e6	0.749282
$28$	0	0
$29$	−1.42931e6	−0.375262	−0.187631	−	0.982240i	$-0.560081\pi$
$29$	−1.42931e6	−0.375262	−0.187631	+	0.982240i	$0.560081\pi$
$30$	0	0
$31$	−9.67494e6	−1.88157	−0.940786	−	0.339001i	$-0.889911\pi$
$31$	−9.67494e6	−1.88157	−0.940786	+	0.339001i	$0.889911\pi$
$32$	0	0
$33$	−1.47400e7	−2.16364
$34$	0	0
$35$	−4.61518e6	−0.519855
$36$	0	0
$37$	−8.67744e6	−0.761174	−0.380587	−	0.924745i	$-0.624278\pi$
$37$	−8.67744e6	−0.761174	−0.380587	+	0.924745i	$0.624278\pi$
$38$	0	0
$39$	522908.	0.0361938
$40$	0	0
$41$	1.32544e7	0.732541	0.366271	−	0.930508i	$-0.380634\pi$
$41$	1.32544e7	0.732541	0.366271	+	0.930508i	$0.380634\pi$
$42$	0	0
$43$	2.97554e7	1.32726	0.663632	−	0.748060i	$-0.269014\pi$
$43$	2.97554e7	1.32726	0.663632	+	0.748060i	$0.269014\pi$
$44$	0	0
$45$	1.34963e7	0.490635
$46$	0	0
$47$	1.07969e7	0.322745	0.161373	−	0.986894i	$-0.448408\pi$
$47$	1.07969e7	0.322745	0.161373	+	0.986894i	$0.448408\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	0	0
$51$	−1.90369e7	−0.394030
$52$	0	0
$53$	7.07399e7	1.23147	0.615734	−	0.787954i	$-0.288860\pi$
$53$	7.07399e7	1.23147	0.615734	+	0.787954i	$0.288860\pi$
$54$	0	0
$55$	1.73381e8	2.55488
$56$	0	0
$57$	−2.32783e7	−0.292089
$58$	0	0
$59$	−6.40400e6	−0.0688046	−0.0344023	−	0.999408i	$-0.510953\pi$
$59$	−6.40400e6	−0.0688046	−0.0344023	+	0.999408i	$0.510953\pi$
$60$	0	0
$61$	1.69190e8	1.56455	0.782275	−	0.622933i	$-0.214059\pi$
$61$	1.69190e8	1.56455	0.782275	+	0.622933i	$0.214059\pi$
$62$	0	0
$63$	−1.68582e7	−0.134827
$64$	0	0
$65$	−6.15078e6	−0.0427386
$66$	0	0
$67$	1.16276e8	0.704943	0.352471	−	0.935823i	$-0.385341\pi$
$67$	1.16276e8	0.704943	0.352471	+	0.935823i	$0.385341\pi$
$68$	0	0
$69$	2.08176e8	1.10563
$70$	0	0
$71$	−1.44496e8	−0.674826	−0.337413	−	0.941357i	$-0.609552\pi$
$71$	−1.44496e8	−0.674826	−0.337413	+	0.941357i	$0.609552\pi$
$72$	0	0
$73$	1.60155e8	0.660066	0.330033	−	0.943969i	$-0.392940\pi$
$73$	1.60155e8	0.660066	0.330033	+	0.943969i	$0.392940\pi$
$74$	0	0
$75$	−2.84617e8	−1.03869
$76$	0	0
$77$	−2.16570e8	−0.702085
$78$	0	0
$79$	4.89322e8	1.41343	0.706713	−	0.707500i	$-0.250177\pi$
$79$	4.89322e8	1.41343	0.706713	+	0.707500i	$0.250177\pi$
$80$	0	0
$81$	−4.76322e8	−1.22947
$82$	0	0
$83$	8.31590e7	0.192335	0.0961674	−	0.995365i	$-0.469342\pi$
$83$	8.31590e7	0.192335	0.0961674	+	0.995365i	$0.469342\pi$
$84$	0	0
$85$	2.23924e8	0.465281
$86$	0	0
$87$	2.33569e8	0.437098
$88$	0	0
$89$	2.08083e6	0.00351546	0.00175773	−	0.999998i	$-0.499440\pi$
$89$	2.08083e6	0.00351546	0.00175773	+	0.999998i	$0.499440\pi$
$90$	0	0
$91$	7.68292e6	0.0117447
$92$	0	0
$93$	1.58103e9	2.19162
$94$	0	0
$95$	2.73815e8	0.344906
$96$	0	0
$97$	−3.15885e8	−0.362290	−0.181145	−	0.983456i	$-0.557980\pi$
$97$	−3.15885e8	−0.362290	−0.181145	+	0.983456i	$0.557980\pi$
$98$	0	0
$99$	6.33322e8	0.662623
$100$	0	0
$101$	−5.74841e8	−0.549669	−0.274835	−	0.961492i	$-0.588623\pi$
$101$	−5.74841e8	−0.549669	−0.274835	+	0.961492i	$0.588623\pi$
$102$	0	0
$103$	1.51870e9	1.32955	0.664775	−	0.747044i	$-0.268527\pi$
$103$	1.51870e9	1.32955	0.664775	+	0.747044i	$0.268527\pi$
$104$	0	0
$105$	7.54187e8	0.605518
$106$	0	0
$107$	2.01863e8	0.148878	0.0744390	−	0.997226i	$-0.476283\pi$
$107$	2.01863e8	0.148878	0.0744390	+	0.997226i	$0.476283\pi$
$108$	0	0
$109$	−8.73952e8	−0.593019	−0.296509	−	0.955030i	$-0.595823\pi$
$109$	−8.73952e8	−0.593019	−0.296509	+	0.955030i	$0.595823\pi$
$110$	0	0
$111$	1.41802e9	0.886603
$112$	0	0
$113$	1.52955e9	0.882491	0.441245	−	0.897386i	$-0.354537\pi$
$113$	1.52955e9	0.882491	0.441245	+	0.897386i	$0.354537\pi$
$114$	0	0
$115$	−2.44870e9	−1.30555
$116$	0	0
$117$	−2.24674e7	−0.0110845
$118$	0	0
$119$	−2.79703e8	−0.127860
$120$	0	0
$121$	5.77807e9	2.45047
$122$	0	0
$123$	−2.16596e9	−0.853252
$124$	0	0
$125$	−4.06429e8	−0.148898
$126$	0	0
$127$	8.71958e8	0.297426	0.148713	−	0.988880i	$-0.452487\pi$
$127$	8.71958e8	0.297426	0.148713	+	0.988880i	$0.452487\pi$
$128$	0	0
$129$	−4.86246e9	−1.54597
$130$	0	0
$131$	−2.24404e9	−0.665747	−0.332874	−	0.942971i	$-0.608018\pi$
$131$	−2.24404e9	−0.665747	−0.332874	+	0.942971i	$0.608018\pi$
$132$	0	0
$133$	−3.42021e8	−0.0947809
$134$	0	0
$135$	3.97721e9	1.03057
$136$	0	0
$137$	4.16141e9	1.00925	0.504624	−	0.863339i	$-0.331631\pi$
$137$	4.16141e9	1.00925	0.504624	+	0.863339i	$0.331631\pi$
$138$	0	0
$139$	6.03383e9	1.37097	0.685483	−	0.728089i	$-0.259591\pi$
$139$	6.03383e9	1.37097	0.685483	+	0.728089i	$0.259591\pi$
$140$	0	0
$141$	−1.76438e9	−0.375928
$142$	0	0
$143$	−2.88629e8	−0.0577203
$144$	0	0
$145$	−2.74740e9	−0.516137
$146$	0	0
$147$	−9.42052e8	−0.166398
$148$	0	0
$149$	−4.37832e9	−0.727728	−0.363864	−	0.931452i	$-0.618543\pi$
$149$	−4.37832e9	−0.727728	−0.363864	+	0.931452i	$0.618543\pi$
$150$	0	0
$151$	2.69365e9	0.421642	0.210821	−	0.977525i	$-0.432386\pi$
$151$	2.69365e9	0.421642	0.210821	+	0.977525i	$0.432386\pi$
$152$	0	0
$153$	8.17943e8	0.120673
$154$	0	0
$155$	−1.85971e10	−2.58793
$156$	0	0
$157$	−1.33044e9	−0.174762	−0.0873810	−	0.996175i	$-0.527850\pi$
$157$	−1.33044e9	−0.174762	−0.0873810	+	0.996175i	$0.527850\pi$
$158$	0	0
$159$	−1.15599e10	−1.43439
$160$	0	0
$161$	3.05866e9	0.358769
$162$	0	0
$163$	3.56094e9	0.395112	0.197556	−	0.980292i	$-0.436699\pi$
$163$	3.56094e9	0.395112	0.197556	+	0.980292i	$0.436699\pi$
$164$	0	0
$165$	−2.83330e10	−2.97588
$166$	0	0
$167$	1.04285e10	1.03752	0.518762	−	0.854919i	$-0.326393\pi$
$167$	1.04285e10	1.03752	0.518762	+	0.854919i	$0.326393\pi$
$168$	0	0
$169$	−1.05943e10	−0.999034
$170$	0	0
$171$	1.00018e9	0.0894534
$172$	0	0
$173$	2.04717e10	1.73759	0.868793	−	0.495176i	$-0.164896\pi$
$173$	2.04717e10	1.73759	0.868793	+	0.495176i	$0.164896\pi$
$174$	0	0
$175$	−4.18178e9	−0.337047
$176$	0	0
$177$	1.04651e9	0.0801424
$178$	0	0
$179$	−5.46705e9	−0.398029	−0.199014	−	0.979997i	$-0.563774\pi$
$179$	−5.46705e9	−0.398029	−0.199014	+	0.979997i	$0.563774\pi$
$180$	0	0
$181$	−2.11628e9	−0.146561	−0.0732807	−	0.997311i	$-0.523347\pi$
$181$	−2.11628e9	−0.146561	−0.0732807	+	0.997311i	$0.523347\pi$
$182$	0	0
$183$	−2.76481e10	−1.82236
$184$	0	0
$185$	−1.66797e10	−1.04692
$186$	0	0
$187$	1.05078e10	0.628381
$188$	0	0
$189$	−4.96792e9	−0.283202
$190$	0	0
$191$	−1.72421e10	−0.937431	−0.468715	−	0.883349i	$-0.655283\pi$
$191$	−1.72421e10	−0.937431	−0.468715	+	0.883349i	$0.655283\pi$
$192$	0	0
$193$	2.02030e10	1.04811	0.524055	−	0.851684i	$-0.324418\pi$
$193$	2.02030e10	1.04811	0.524055	+	0.851684i	$0.324418\pi$
$194$	0	0
$195$	1.00513e9	0.0497812
$196$	0	0
$197$	−2.22592e10	−1.05296	−0.526481	−	0.850187i	$-0.676489\pi$
$197$	−2.22592e10	−1.05296	−0.526481	+	0.850187i	$0.676489\pi$
$198$	0	0
$199$	−1.70588e10	−0.771098	−0.385549	−	0.922687i	$-0.625988\pi$
$199$	−1.70588e10	−0.771098	−0.385549	+	0.922687i	$0.625988\pi$
$200$	0	0
$201$	−1.90012e10	−0.821105
$202$	0	0
$203$	3.43176e9	0.141836
$204$	0	0
$205$	2.54774e10	1.00754
$206$	0	0
$207$	−8.94453e9	−0.338603
$208$	0	0
$209$	1.28489e10	0.465810
$210$	0	0
$211$	3.19873e10	1.11098	0.555490	−	0.831523i	$-0.312531\pi$
$211$	3.19873e10	1.11098	0.555490	+	0.831523i	$0.312531\pi$
$212$	0	0
$213$	2.36127e10	0.786026
$214$	0	0
$215$	5.71954e10	1.82553
$216$	0	0
$217$	2.32295e10	0.711167
$218$	0	0
$219$	−2.61717e10	−0.768834
$220$	0	0
$221$	−3.72768e8	−0.0105117
$222$	0	0
$223$	2.30967e10	0.625428	0.312714	−	0.949847i	$-0.398762\pi$
$223$	2.30967e10	0.625428	0.312714	+	0.949847i	$0.398762\pi$
$224$	0	0
$225$	1.22289e10	0.318102
$226$	0	0
$227$	2.30894e10	0.577160	0.288580	−	0.957456i	$-0.406817\pi$
$227$	2.30894e10	0.577160	0.288580	+	0.957456i	$0.406817\pi$
$228$	0	0
$229$	4.25496e10	1.02244	0.511218	−	0.859451i	$-0.329195\pi$
$229$	4.25496e10	1.02244	0.511218	+	0.859451i	$0.329195\pi$
$230$	0	0
$231$	3.53907e10	0.817777
$232$	0	0
$233$	−1.26679e10	−0.281582	−0.140791	−	0.990039i	$-0.544964\pi$
$233$	−1.26679e10	−0.281582	−0.140791	+	0.990039i	$0.544964\pi$
$234$	0	0
$235$	2.07537e10	0.443906
$236$	0	0
$237$	−7.99624e10	−1.64633
$238$	0	0
$239$	−6.37875e10	−1.26458	−0.632289	−	0.774733i	$-0.717884\pi$
$239$	−6.37875e10	−1.26458	−0.632289	+	0.774733i	$0.717884\pi$
$240$	0	0
$241$	−2.91604e10	−0.556823	−0.278412	−	0.960462i	$-0.589808\pi$
$241$	−2.91604e10	−0.556823	−0.278412	+	0.960462i	$0.589808\pi$
$242$	0	0
$243$	3.71118e10	0.682785
$244$	0	0
$245$	1.10810e10	0.196487
$246$	0	0
$247$	−4.55822e8	−0.00779218
$248$	0	0
$249$	−1.35894e10	−0.224028
$250$	0	0
$251$	−6.28939e10	−1.00018	−0.500088	−	0.865974i	$-0.666699\pi$
$251$	−6.28939e10	−1.00018	−0.500088	+	0.865974i	$0.666699\pi$
$252$	0	0
$253$	−1.14907e11	−1.76320
$254$	0	0
$255$	−3.65924e10	−0.541952
$256$	0	0
$257$	1.14480e11	1.63694	0.818469	−	0.574551i	$-0.194823\pi$
$257$	1.14480e11	1.63694	0.818469	+	0.574551i	$0.194823\pi$
$258$	0	0
$259$	2.08345e10	0.287697
$260$	0	0
$261$	−1.00356e10	−0.133863
$262$	0	0
$263$	−1.40705e10	−0.181346	−0.0906728	−	0.995881i	$-0.528902\pi$
$263$	−1.40705e10	−0.181346	−0.0906728	+	0.995881i	$0.528902\pi$
$264$	0	0
$265$	1.35975e11	1.69377
$266$	0	0
$267$	−3.40038e8	−0.00409475
$268$	0	0
$269$	8.39143e10	0.977127	0.488563	−	0.872528i	$-0.337521\pi$
$269$	8.39143e10	0.977127	0.488563	+	0.872528i	$0.337521\pi$
$270$	0	0
$271$	−1.98401e10	−0.223451	−0.111726	−	0.993739i	$-0.535638\pi$
$271$	−1.98401e10	−0.223451	−0.111726	+	0.993739i	$0.535638\pi$
$272$	0	0
$273$	−1.25550e9	−0.0136800
$274$	0	0
$275$	1.57100e11	1.65645
$276$	0	0
$277$	−7.17911e10	−0.732675	−0.366338	−	0.930482i	$-0.619389\pi$
$277$	−7.17911e10	−0.732675	−0.366338	+	0.930482i	$0.619389\pi$
$278$	0	0
$279$	−6.79308e10	−0.671194
$280$	0	0
$281$	1.02853e11	0.984101	0.492050	−	0.870567i	$-0.336248\pi$
$281$	1.02853e11	0.984101	0.492050	+	0.870567i	$0.336248\pi$
$282$	0	0
$283$	5.52883e10	0.512382	0.256191	−	0.966626i	$-0.417532\pi$
$283$	5.52883e10	0.512382	0.256191	+	0.966626i	$0.417532\pi$
$284$	0	0
$285$	−4.47453e10	−0.401741
$286$	0	0
$287$	−3.18238e10	−0.276875
$288$	0	0
$289$	−1.05017e11	−0.885562
$290$	0	0
$291$	5.16202e10	0.421989
$292$	0	0
$293$	1.05721e11	0.838028	0.419014	−	0.907980i	$-0.362376\pi$
$293$	1.05721e11	0.838028	0.419014	+	0.907980i	$0.362376\pi$
$294$	0	0
$295$	−1.23097e10	−0.0946343
$296$	0	0
$297$	1.86633e11	1.39182
$298$	0	0
$299$	4.07637e9	0.0294953
$300$	0	0
$301$	−7.14426e10	−0.501658
$302$	0	0
$303$	9.39374e10	0.640246
$304$	0	0
$305$	3.25214e11	2.15189
$306$	0	0
$307$	8.10064e10	0.520471	0.260236	−	0.965545i	$-0.416200\pi$
$307$	8.10064e10	0.520471	0.260236	+	0.965545i	$0.416200\pi$
$308$	0	0
$309$	−2.48178e11	−1.54864
$310$	0	0
$311$	3.18435e11	1.93018	0.965091	−	0.261913i	$-0.0843534\pi$
$311$	3.18435e11	1.93018	0.965091	+	0.261913i	$0.0843534\pi$
$312$	0	0
$313$	1.28876e11	0.758965	0.379483	−	0.925199i	$-0.376102\pi$
$313$	1.28876e11	0.758965	0.379483	+	0.925199i	$0.376102\pi$
$314$	0	0
$315$	−3.24046e10	−0.185443
$316$	0	0
$317$	−7.44722e10	−0.414217	−0.207108	−	0.978318i	$-0.566405\pi$
$317$	−7.44722e10	−0.414217	−0.207108	+	0.978318i	$0.566405\pi$
$318$	0	0
$319$	−1.28923e11	−0.697065
$320$	0	0
$321$	−3.29874e10	−0.173411
$322$	0	0
$323$	1.65945e10	0.0848309
$324$	0	0
$325$	−5.57319e9	−0.0277095
$326$	0	0
$327$	1.42816e11	0.690738
$328$	0	0
$329$	−2.59234e10	−0.121986
$330$	0	0
$331$	−2.83840e11	−1.29971	−0.649857	−	0.760057i	$-0.725171\pi$
$331$	−2.83840e11	−1.29971	−0.649857	+	0.760057i	$0.725171\pi$
$332$	0	0
$333$	−6.09271e10	−0.271526
$334$	0	0
$335$	2.23505e11	0.969583
$336$	0	0
$337$	−5.61414e9	−0.0237109	−0.0118555	−	0.999930i	$-0.503774\pi$
$337$	−5.61414e9	−0.0237109	−0.0118555	+	0.999930i	$0.503774\pi$
$338$	0	0
$339$	−2.49950e11	−1.02791
$340$	0	0
$341$	−8.72679e11	−3.49510
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	0	0
$345$	4.00153e11	1.52069
$346$	0	0
$347$	−3.39846e11	−1.25834	−0.629171	−	0.777267i	$-0.716606\pi$
$347$	−3.39846e11	−1.25834	−0.629171	+	0.777267i	$0.716606\pi$
$348$	0	0
$349$	3.46718e10	0.125101	0.0625506	−	0.998042i	$-0.480077\pi$
$349$	3.46718e10	0.125101	0.0625506	+	0.998042i	$0.480077\pi$
$350$	0	0
$351$	−6.62089e9	−0.0232828
$352$	0	0
$353$	−3.46900e11	−1.18910	−0.594550	−	0.804059i	$-0.702670\pi$
$353$	−3.46900e11	−1.18910	−0.594550	+	0.804059i	$0.702670\pi$
$354$	0	0
$355$	−2.77748e11	−0.928160
$356$	0	0
$357$	4.57075e10	0.148929
$358$	0	0
$359$	2.51011e11	0.797569	0.398785	−	0.917045i	$-0.369432\pi$
$359$	2.51011e11	0.797569	0.398785	+	0.917045i	$0.369432\pi$
$360$	0	0
$361$	−3.02396e11	−0.937116
$362$	0	0
$363$	−9.44221e11	−2.85426
$364$	0	0
$365$	3.07848e11	0.907859
$366$	0	0
$367$	−4.79871e11	−1.38079	−0.690394	−	0.723433i	$-0.742563\pi$
$367$	−4.79871e11	−1.38079	−0.690394	+	0.723433i	$0.742563\pi$
$368$	0	0
$369$	9.30632e10	0.261312
$370$	0	0
$371$	−1.69847e11	−0.465451
$372$	0	0
$373$	−1.84794e11	−0.494308	−0.247154	−	0.968976i	$-0.579495\pi$
$373$	−1.84794e11	−0.494308	−0.247154	+	0.968976i	$0.579495\pi$
$374$	0	0
$375$	6.64164e10	0.173434
$376$	0	0
$377$	4.57361e9	0.0116607
$378$	0	0
$379$	7.08908e11	1.76487	0.882437	−	0.470431i	$-0.155902\pi$
$379$	7.08908e11	1.76487	0.882437	+	0.470431i	$0.155902\pi$
$380$	0	0
$381$	−1.42491e11	−0.346437
$382$	0	0
$383$	−1.23789e11	−0.293959	−0.146980	−	0.989140i	$-0.546955\pi$
$383$	−1.23789e11	−0.293959	−0.146980	+	0.989140i	$0.546955\pi$
$384$	0	0
$385$	−4.16288e11	−0.965653
$386$	0	0
$387$	2.08922e11	0.473461
$388$	0	0
$389$	−2.97971e11	−0.659782	−0.329891	−	0.944019i	$-0.607012\pi$
$389$	−2.97971e11	−0.659782	−0.329891	+	0.944019i	$0.607012\pi$
$390$	0	0
$391$	−1.48403e11	−0.321106
$392$	0	0
$393$	3.66708e11	0.775451
$394$	0	0
$395$	9.40570e11	1.94404
$396$	0	0
$397$	5.33426e11	1.07775	0.538873	−	0.842387i	$-0.318850\pi$
$397$	5.33426e11	1.07775	0.538873	+	0.842387i	$0.318850\pi$
$398$	0	0
$399$	5.58912e10	0.110399
$400$	0	0
$401$	4.15640e11	0.802726	0.401363	−	0.915919i	$-0.368537\pi$
$401$	4.15640e11	0.802726	0.401363	+	0.915919i	$0.368537\pi$
$402$	0	0
$403$	3.09587e10	0.0584669
$404$	0	0
$405$	−9.15581e11	−1.69102
$406$	0	0
$407$	−7.82704e11	−1.41391
$408$	0	0
$409$	1.07978e12	1.90801	0.954006	−	0.299788i	$-0.0969159\pi$
$409$	1.07978e12	1.90801	0.954006	+	0.299788i	$0.0969159\pi$
$410$	0	0
$411$	−6.80035e11	−1.17556
$412$	0	0
$413$	1.53760e10	0.0260057
$414$	0	0
$415$	1.59847e11	0.264539
$416$	0	0
$417$	−9.86015e11	−1.59688
$418$	0	0
$419$	2.20998e11	0.350289	0.175144	−	0.984543i	$-0.443961\pi$
$419$	2.20998e11	0.350289	0.175144	+	0.984543i	$0.443961\pi$
$420$	0	0
$421$	3.47478e11	0.539086	0.269543	−	0.962988i	$-0.413127\pi$
$421$	3.47478e11	0.539086	0.269543	+	0.962988i	$0.413127\pi$
$422$	0	0
$423$	7.58087e10	0.115130
$424$	0	0
$425$	2.02896e11	0.301664
$426$	0	0
$427$	−4.06224e11	−0.591344
$428$	0	0
$429$	4.71662e10	0.0672316
$430$	0	0
$431$	−1.23574e11	−0.172496	−0.0862479	−	0.996274i	$-0.527488\pi$
$431$	−1.23574e11	−0.172496	−0.0862479	+	0.996274i	$0.527488\pi$
$432$	0	0
$433$	−7.27679e11	−0.994820	−0.497410	−	0.867516i	$-0.665716\pi$
$433$	−7.27679e11	−0.994820	−0.497410	+	0.867516i	$0.665716\pi$
$434$	0	0
$435$	4.48964e11	0.601188
$436$	0	0
$437$	−1.81468e11	−0.238031
$438$	0	0
$439$	5.85966e11	0.752977	0.376489	−	0.926421i	$-0.377131\pi$
$439$	5.85966e11	0.752977	0.376489	+	0.926421i	$0.377131\pi$
$440$	0	0
$441$	4.04765e10	0.0509600
$442$	0	0
$443$	2.11546e11	0.260969	0.130484	−	0.991450i	$-0.458347\pi$
$443$	2.11546e11	0.260969	0.130484	+	0.991450i	$0.458347\pi$
$444$	0	0
$445$	3.99975e9	0.00483519
$446$	0	0
$447$	7.15480e11	0.847645
$448$	0	0
$449$	−9.21048e11	−1.06948	−0.534741	−	0.845016i	$-0.679591\pi$
$449$	−9.21048e11	−1.06948	−0.534741	+	0.845016i	$0.679591\pi$
$450$	0	0
$451$	1.19554e12	1.36073
$452$	0	0
$453$	−4.40181e11	−0.491122
$454$	0	0
$455$	1.47680e10	0.0161537
$456$	0	0
$457$	8.11185e11	0.869955	0.434977	−	0.900441i	$-0.356756\pi$
$457$	8.11185e11	0.869955	0.434977	+	0.900441i	$0.356756\pi$
$458$	0	0
$459$	2.41039e11	0.253472
$460$	0	0
$461$	−1.90069e11	−0.196001	−0.0980004	−	0.995186i	$-0.531245\pi$
$461$	−1.90069e11	−0.196001	−0.0980004	+	0.995186i	$0.531245\pi$
$462$	0	0
$463$	−4.76945e11	−0.482341	−0.241170	−	0.970483i	$-0.577531\pi$
$463$	−4.76945e11	−0.482341	−0.241170	+	0.970483i	$0.577531\pi$
$464$	0	0
$465$	3.03903e12	3.01437
$466$	0	0
$467$	−1.06392e12	−1.03510	−0.517549	−	0.855653i	$-0.673156\pi$
$467$	−1.06392e12	−1.03510	−0.517549	+	0.855653i	$0.673156\pi$
$468$	0	0
$469$	−2.79179e11	−0.266443
$470$	0	0
$471$	2.17413e11	0.203560
$472$	0	0
$473$	2.68393e12	2.46545
$474$	0	0
$475$	2.48102e11	0.223619
$476$	0	0
$477$	4.96687e11	0.439289
$478$	0	0
$479$	8.43415e11	0.732034	0.366017	−	0.930608i	$-0.380721\pi$
$479$	8.43415e11	0.732034	0.366017	+	0.930608i	$0.380721\pi$
$480$	0	0
$481$	2.77668e10	0.0236523
$482$	0	0
$483$	−4.99829e11	−0.417888
$484$	0	0
$485$	−6.07191e11	−0.498296
$486$	0	0
$487$	−1.15202e12	−0.928065	−0.464032	−	0.885818i	$-0.653598\pi$
$487$	−1.15202e12	−0.928065	−0.464032	+	0.885818i	$0.653598\pi$
$488$	0	0
$489$	−5.81910e11	−0.460220
$490$	0	0
$491$	9.68703e11	0.752184	0.376092	−	0.926582i	$-0.377268\pi$
$491$	9.68703e11	0.752184	0.376092	+	0.926582i	$0.377268\pi$
$492$	0	0
$493$	−1.66506e11	−0.126946
$494$	0	0
$495$	1.21736e12	0.911375
$496$	0	0
$497$	3.46934e11	0.255060
$498$	0	0
$499$	−2.62821e12	−1.89761	−0.948805	−	0.315863i	$-0.897706\pi$
$499$	−2.62821e12	−1.89761	−0.948805	+	0.315863i	$0.897706\pi$
$500$	0	0
$501$	−1.70417e12	−1.20849
$502$	0	0
$503$	−3.95070e11	−0.275181	−0.137590	−	0.990489i	$-0.543936\pi$
$503$	−3.95070e11	−0.275181	−0.137590	+	0.990489i	$0.543936\pi$
$504$	0	0
$505$	−1.10495e12	−0.756019
$506$	0	0
$507$	1.73126e12	1.16366
$508$	0	0
$509$	6.64157e11	0.438572	0.219286	−	0.975661i	$-0.429627\pi$
$509$	6.64157e11	0.438572	0.219286	+	0.975661i	$0.429627\pi$
$510$	0	0
$511$	−3.84532e11	−0.249482
$512$	0	0
$513$	2.94743e11	0.187895
$514$	0	0
$515$	2.91923e12	1.82867
$516$	0	0
$517$	9.73882e11	0.599513
$518$	0	0
$519$	−3.34537e12	−2.02391
$520$	0	0
$521$	5.22766e11	0.310841	0.155420	−	0.987848i	$-0.450327\pi$
$521$	5.22766e11	0.310841	0.155420	+	0.987848i	$0.450327\pi$
$522$	0	0
$523$	3.09135e12	1.80672	0.903360	−	0.428882i	$-0.141093\pi$
$523$	3.09135e12	1.80672	0.903360	+	0.428882i	$0.141093\pi$
$524$	0	0
$525$	6.83364e11	0.392587
$526$	0	0
$527$	−1.12708e12	−0.636510
$528$	0	0
$529$	−1.78305e11	−0.0989947
$530$	0	0
$531$	−4.49645e10	−0.0245440
$532$	0	0
$533$	−4.24125e10	−0.0227626
$534$	0	0
$535$	3.88019e11	0.204768
$536$	0	0
$537$	8.93396e11	0.463617
$538$	0	0
$539$	5.19984e11	0.265363
$540$	0	0
$541$	−1.47490e12	−0.740243	−0.370121	−	0.928983i	$-0.620684\pi$
$541$	−1.47490e12	−0.740243	−0.370121	+	0.928983i	$0.620684\pi$
$542$	0	0
$543$	3.45831e11	0.170712
$544$	0	0
$545$	−1.67990e12	−0.815642
$546$	0	0
$547$	−2.12294e12	−1.01390	−0.506949	−	0.861976i	$-0.669227\pi$
$547$	−2.12294e12	−1.01390	−0.506949	+	0.861976i	$0.669227\pi$
$548$	0	0
$549$	1.18793e12	0.558106
$550$	0	0
$551$	−2.03604e11	−0.0941031
$552$	0	0
$553$	−1.17486e12	−0.534225
$554$	0	0
$555$	2.72570e12	1.21944
$556$	0	0
$557$	−3.52857e12	−1.55328	−0.776641	−	0.629943i	$-0.783078\pi$
$557$	−3.52857e12	−1.55328	−0.776641	+	0.629943i	$0.783078\pi$
$558$	0	0
$559$	−9.52137e10	−0.0412426
$560$	0	0
$561$	−1.71712e12	−0.731928
$562$	0	0
$563$	3.35280e12	1.40644	0.703218	−	0.710974i	$-0.251746\pi$
$563$	3.35280e12	1.40644	0.703218	+	0.710974i	$0.251746\pi$
$564$	0	0
$565$	2.94008e12	1.21378
$566$	0	0
$567$	1.14365e12	0.464696
$568$	0	0
$569$	2.99364e12	1.19727	0.598637	−	0.801020i	$-0.295709\pi$
$569$	2.99364e12	1.19727	0.598637	+	0.801020i	$0.295709\pi$
$570$	0	0
$571$	−4.67267e12	−1.83951	−0.919756	−	0.392490i	$-0.871614\pi$
$571$	−4.67267e12	−1.83951	−0.919756	+	0.392490i	$0.871614\pi$
$572$	0	0
$573$	2.81761e12	1.09190
$574$	0	0
$575$	−2.21875e12	−0.846453
$576$	0	0
$577$	−3.62799e12	−1.36262	−0.681310	−	0.731995i	$-0.738589\pi$
$577$	−3.62799e12	−1.36262	−0.681310	+	0.731995i	$0.738589\pi$
$578$	0	0
$579$	−3.30146e12	−1.22082
$580$	0	0
$581$	−1.99665e11	−0.0726957
$582$	0	0
$583$	6.38073e12	2.28751
$584$	0	0
$585$	−4.31866e10	−0.0152457
$586$	0	0
$587$	−3.97200e12	−1.38082	−0.690411	−	0.723417i	$-0.742570\pi$
$587$	−3.97200e12	−1.38082	−0.690411	+	0.723417i	$0.742570\pi$
$588$	0	0
$589$	−1.37819e12	−0.471835
$590$	0	0
$591$	3.63748e12	1.22647
$592$	0	0
$593$	2.67436e12	0.888123	0.444061	−	0.895996i	$-0.353537\pi$
$593$	2.67436e12	0.888123	0.444061	+	0.895996i	$0.353537\pi$
$594$	0	0
$595$	−5.37641e11	−0.175860
$596$	0	0
$597$	2.78765e12	0.898162
$598$	0	0
$599$	−4.84522e12	−1.53777	−0.768887	−	0.639384i	$-0.779189\pi$
$599$	−4.84522e12	−1.53777	−0.768887	+	0.639384i	$0.779189\pi$
$600$	0	0
$601$	−4.64764e12	−1.45311	−0.726553	−	0.687110i	$-0.758879\pi$
$601$	−4.64764e12	−1.45311	−0.726553	+	0.687110i	$0.758879\pi$
$602$	0	0
$603$	8.16411e11	0.251467
$604$	0	0
$605$	1.11065e13	3.37039
$606$	0	0
$607$	6.52447e12	1.95073	0.975363	−	0.220604i	$-0.0708030\pi$
$607$	6.52447e12	1.95073	0.975363	+	0.220604i	$0.0708030\pi$
$608$	0	0
$609$	−5.60800e11	−0.165208
$610$	0	0
$611$	−3.45489e10	−0.0100288
$612$	0	0
$613$	−9.54536e11	−0.273036	−0.136518	−	0.990638i	$-0.543591\pi$
$613$	−9.54536e11	−0.273036	−0.136518	+	0.990638i	$0.543591\pi$
$614$	0	0
$615$	−4.16338e12	−1.17357
$616$	0	0
$617$	−4.50764e12	−1.25218	−0.626088	−	0.779752i	$-0.715345\pi$
$617$	−4.50764e12	−1.25218	−0.626088	+	0.779752i	$0.715345\pi$
$618$	0	0
$619$	−3.64346e12	−0.997484	−0.498742	−	0.866750i	$-0.666204\pi$
$619$	−3.64346e12	−0.997484	−0.498742	+	0.866750i	$0.666204\pi$
$620$	0	0
$621$	−2.63585e12	−0.711228
$622$	0	0
$623$	−4.99608e9	−0.00132872
$624$	0	0
$625$	−4.18296e12	−1.09654
$626$	0	0
$627$	−2.09970e12	−0.542567
$628$	0	0
$629$	−1.01087e12	−0.257495
$630$	0	0
$631$	−3.61498e12	−0.907766	−0.453883	−	0.891061i	$-0.649962\pi$
$631$	−3.61498e12	−0.907766	−0.453883	+	0.891061i	$0.649962\pi$
$632$	0	0
$633$	−5.22718e12	−1.29405
$634$	0	0
$635$	1.67607e12	0.409081
$636$	0	0
$637$	−1.84467e10	−0.00443906
$638$	0	0
$639$	−1.01455e12	−0.240724
$640$	0	0
$641$	−2.66304e12	−0.623041	−0.311521	−	0.950239i	$-0.600838\pi$
$641$	−2.66304e12	−0.623041	−0.311521	+	0.950239i	$0.600838\pi$
$642$	0	0
$643$	−4.09899e12	−0.945644	−0.472822	−	0.881158i	$-0.656765\pi$
$643$	−4.09899e12	−0.945644	−0.472822	+	0.881158i	$0.656765\pi$
$644$	0	0
$645$	−9.34656e12	−2.12634
$646$	0	0
$647$	−6.11325e12	−1.37152	−0.685761	−	0.727827i	$-0.740531\pi$
$647$	−6.11325e12	−1.37152	−0.685761	+	0.727827i	$0.740531\pi$
$648$	0	0
$649$	−5.77640e11	−0.127807
$650$	0	0
$651$	−3.79604e12	−0.828356
$652$	0	0
$653$	7.66270e12	1.64920	0.824598	−	0.565719i	$-0.191401\pi$
$653$	7.66270e12	1.64920	0.824598	+	0.565719i	$0.191401\pi$
$654$	0	0
$655$	−4.31346e12	−0.915673
$656$	0	0
$657$	1.12450e12	0.235459
$658$	0	0
$659$	8.20110e9	0.00169390	0.000846950	−	1.00000i	$-0.499730\pi$
$659$	8.20110e9	0.00169390	0.000846950		1.00000i	$0.499730\pi$
$660$	0	0
$661$	3.20922e12	0.653872	0.326936	−	0.945046i	$-0.393984\pi$
$661$	3.20922e12	0.653872	0.326936	+	0.945046i	$0.393984\pi$
$662$	0	0
$663$	6.09158e10	0.0122439
$664$	0	0
$665$	−6.57429e11	−0.130362
$666$	0	0
$667$	1.82081e12	0.356203
$668$	0	0
$669$	−3.77433e12	−0.728488
$670$	0	0
$671$	1.52609e13	2.90622
$672$	0	0
$673$	−4.91229e12	−0.923031	−0.461515	−	0.887132i	$-0.652694\pi$
$673$	−4.91229e12	−0.923031	−0.461515	+	0.887132i	$0.652694\pi$
$674$	0	0
$675$	3.60372e12	0.668166
$676$	0	0
$677$	−3.11910e12	−0.570664	−0.285332	−	0.958429i	$-0.592104\pi$
$677$	−3.11910e12	−0.570664	−0.285332	+	0.958429i	$0.592104\pi$
$678$	0	0
$679$	7.58440e11	0.136933
$680$	0	0
$681$	−3.77315e12	−0.672267
$682$	0	0
$683$	−2.91806e12	−0.513099	−0.256550	−	0.966531i	$-0.582586\pi$
$683$	−2.91806e12	−0.513099	−0.256550	+	0.966531i	$0.582586\pi$
$684$	0	0
$685$	7.99902e12	1.38813
$686$	0	0
$687$	−6.95322e12	−1.19092
$688$	0	0
$689$	−2.26360e11	−0.0382660
$690$	0	0
$691$	−4.74697e12	−0.792073	−0.396037	−	0.918235i	$-0.629615\pi$
$691$	−4.74697e12	−0.792073	−0.396037	+	0.918235i	$0.629615\pi$
$692$	0	0
$693$	−1.52061e12	−0.250448
$694$	0	0
$695$	1.15982e13	1.88563
$696$	0	0
$697$	1.54406e12	0.247809
$698$	0	0
$699$	2.07013e12	0.327981
$700$	0	0
$701$	−2.24423e11	−0.0351023	−0.0175512	−	0.999846i	$-0.505587\pi$
$701$	−2.24423e11	−0.0351023	−0.0175512	+	0.999846i	$0.505587\pi$
$702$	0	0
$703$	−1.23610e12	−0.190877
$704$	0	0
$705$	−3.39146e12	−0.517054
$706$	0	0
$707$	1.38019e12	0.207755
$708$	0	0
$709$	4.25463e12	0.632344	0.316172	−	0.948702i	$-0.397602\pi$
$709$	4.25463e12	0.632344	0.316172	+	0.948702i	$0.397602\pi$
$710$	0	0
$711$	3.43569e12	0.504197
$712$	0	0
$713$	1.23250e13	1.78601
$714$	0	0
$715$	−5.54800e11	−0.0793888
$716$	0	0
$717$	1.04238e13	1.47296
$718$	0	0
$719$	4.28931e12	0.598559	0.299280	−	0.954165i	$-0.403254\pi$
$719$	4.28931e12	0.598559	0.299280	+	0.954165i	$0.403254\pi$
$720$	0	0
$721$	−3.64640e12	−0.502523
$722$	0	0
$723$	4.76524e12	0.648578
$724$	0	0
$725$	−2.48940e12	−0.334637
$726$	0	0
$727$	−1.28935e13	−1.71185	−0.855926	−	0.517098i	$-0.827012\pi$
$727$	−1.28935e13	−1.71185	−0.855926	+	0.517098i	$0.827012\pi$
$728$	0	0
$729$	3.31084e12	0.434174
$730$	0	0
$731$	3.46633e12	0.448995
$732$	0	0
$733$	2.71447e12	0.347310	0.173655	−	0.984807i	$-0.444442\pi$
$733$	2.71447e12	0.347310	0.173655	+	0.984807i	$0.444442\pi$
$734$	0	0
$735$	−1.81080e12	−0.228864
$736$	0	0
$737$	1.04881e13	1.30946
$738$	0	0
$739$	9.62579e12	1.18723	0.593617	−	0.804747i	$-0.297699\pi$
$739$	9.62579e12	1.18723	0.593617	+	0.804747i	$0.297699\pi$
$740$	0	0
$741$	7.44880e10	0.00907620
$742$	0	0
$743$	1.24362e13	1.49706	0.748529	−	0.663103i	$-0.230761\pi$
$743$	1.24362e13	1.49706	0.748529	+	0.663103i	$0.230761\pi$
$744$	0	0
$745$	−8.41595e12	−1.00092
$746$	0	0
$747$	5.83886e11	0.0686097
$748$	0	0
$749$	−4.84674e11	−0.0562706
$750$	0	0
$751$	−4.26959e12	−0.489786	−0.244893	−	0.969550i	$-0.578753\pi$
$751$	−4.26959e12	−0.489786	−0.244893	+	0.969550i	$0.578753\pi$
$752$	0	0
$753$	1.02778e13	1.16499
$754$	0	0
$755$	5.17770e12	0.579930
$756$	0	0
$757$	9.76840e12	1.08116	0.540582	−	0.841291i	$-0.318204\pi$
$757$	9.76840e12	1.08116	0.540582	+	0.841291i	$0.318204\pi$
$758$	0	0
$759$	1.87774e13	2.05375
$760$	0	0
$761$	1.23336e13	1.33309	0.666543	−	0.745466i	$-0.267773\pi$
$761$	1.23336e13	1.33309	0.666543	+	0.745466i	$0.267773\pi$
$762$	0	0
$763$	2.09836e12	0.224140
$764$	0	0
$765$	1.57224e12	0.165975
$766$	0	0
$767$	2.04921e10	0.00213800
$768$	0	0
$769$	−1.68919e13	−1.74185	−0.870926	−	0.491415i	$-0.836480\pi$
$769$	−1.68919e13	−1.74185	−0.870926	+	0.491415i	$0.836480\pi$
$770$	0	0
$771$	−1.87078e13	−1.90668
$772$	0	0
$773$	1.47186e13	1.48272	0.741359	−	0.671109i	$-0.234182\pi$
$773$	1.47186e13	1.48272	0.741359	+	0.671109i	$0.234182\pi$
$774$	0	0
$775$	−1.68507e13	−1.67788
$776$	0	0
$777$	−3.40467e12	−0.335104
$778$	0	0
$779$	1.88808e12	0.183697
$780$	0	0
$781$	−1.30335e13	−1.25352
$782$	0	0
$783$	−2.95738e12	−0.281177
$784$	0	0
$785$	−2.55736e12	−0.240369
$786$	0	0
$787$	1.82466e13	1.69549	0.847744	−	0.530406i	$-0.177960\pi$
$787$	1.82466e13	1.69549	0.847744	+	0.530406i	$0.177960\pi$
$788$	0	0
$789$	2.29932e12	0.211228
$790$	0	0
$791$	−3.67245e12	−0.333550
$792$	0	0
$793$	−5.41388e11	−0.0486160
$794$	0	0
$795$	−2.22204e13	−1.97287
$796$	0	0
$797$	−1.21558e13	−1.06714	−0.533568	−	0.845757i	$-0.679149\pi$
$797$	−1.21558e13	−1.06714	−0.533568	+	0.845757i	$0.679149\pi$
$798$	0	0
$799$	1.25778e12	0.109180
$800$	0	0
$801$	1.46102e10	0.00125403
$802$	0	0
$803$	1.44460e13	1.22610
$804$	0	0
$805$	5.87932e12	0.493453
$806$	0	0
$807$	−1.37128e13	−1.13814
$808$	0	0
$809$	−2.61893e12	−0.214959	−0.107479	−	0.994207i	$-0.534278\pi$
$809$	−2.61893e12	−0.214959	−0.107479	+	0.994207i	$0.534278\pi$
$810$	0	0
$811$	−1.16994e13	−0.949662	−0.474831	−	0.880077i	$-0.657491\pi$
$811$	−1.16994e13	−0.949662	−0.474831	+	0.880077i	$0.657491\pi$
$812$	0	0
$813$	3.24217e12	0.260272
$814$	0	0
$815$	6.84480e12	0.543440
$816$	0	0
$817$	4.23863e12	0.332833
$818$	0	0
$819$	5.39443e10	0.00418955
$820$	0	0
$821$	5.17673e12	0.397659	0.198830	−	0.980034i	$-0.436286\pi$
$821$	5.17673e12	0.397659	0.198830	+	0.980034i	$0.436286\pi$
$822$	0	0
$823$	7.84017e12	0.595698	0.297849	−	0.954613i	$-0.403731\pi$
$823$	7.84017e12	0.595698	0.297849	+	0.954613i	$0.403731\pi$
$824$	0	0
$825$	−2.56724e13	−1.92941
$826$	0	0
$827$	1.43263e13	1.06502	0.532512	−	0.846422i	$-0.321248\pi$
$827$	1.43263e13	1.06502	0.532512	+	0.846422i	$0.321248\pi$
$828$	0	0
$829$	4.95546e12	0.364408	0.182204	−	0.983261i	$-0.441677\pi$
$829$	4.95546e12	0.364408	0.182204	+	0.983261i	$0.441677\pi$
$830$	0	0
$831$	1.17317e13	0.853408
$832$	0	0
$833$	6.71566e11	0.0483266
$834$	0	0
$835$	2.00456e13	1.42702
$836$	0	0
$837$	−2.00185e13	−1.40983
$838$	0	0
$839$	−7.38235e12	−0.514358	−0.257179	−	0.966364i	$-0.582793\pi$
$839$	−7.38235e12	−0.514358	−0.257179	+	0.966364i	$0.582793\pi$
$840$	0	0
$841$	−1.24642e13	−0.859179
$842$	0	0
$843$	−1.68077e13	−1.14626
$844$	0	0
$845$	−2.03642e13	−1.37408
$846$	0	0
$847$	−1.38732e13	−0.926189
$848$	0	0
$849$	−9.03491e12	−0.596814
$850$	0	0
$851$	1.10543e13	0.722516
$852$	0	0
$853$	2.13356e13	1.37985	0.689927	−	0.723879i	$-0.257643\pi$
$853$	2.13356e13	1.37985	0.689927	+	0.723879i	$0.257643\pi$
$854$	0	0
$855$	1.92254e12	0.123035
$856$	0	0
$857$	1.65008e13	1.04494	0.522468	−	0.852659i	$-0.325011\pi$
$857$	1.65008e13	1.04494	0.522468	+	0.852659i	$0.325011\pi$
$858$	0	0
$859$	1.94817e13	1.22083	0.610417	−	0.792080i	$-0.291002\pi$
$859$	1.94817e13	1.22083	0.610417	+	0.792080i	$0.291002\pi$
$860$	0	0
$861$	5.20047e12	0.322499
$862$	0	0
$863$	−1.61059e13	−0.988411	−0.494205	−	0.869345i	$-0.664541\pi$
$863$	−1.61059e13	−0.988411	−0.494205	+	0.869345i	$0.664541\pi$
$864$	0	0
$865$	3.93504e13	2.38989
$866$	0	0
$867$	1.71613e13	1.03149
$868$	0	0
$869$	4.41368e13	2.62550
$870$	0	0
$871$	−3.72070e11	−0.0219050
$872$	0	0
$873$	−2.21793e12	−0.129236
$874$	0	0
$875$	9.75836e11	0.0562782
$876$	0	0
$877$	−1.84919e13	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$877$	−1.84919e13	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$878$	0	0
$879$	−1.72764e13	−0.976121
$880$	0	0
$881$	−2.83078e13	−1.58312	−0.791562	−	0.611089i	$-0.790732\pi$
$881$	−2.83078e13	−1.58312	−0.791562	+	0.611089i	$0.790732\pi$
$882$	0	0
$883$	7.32328e12	0.405399	0.202700	−	0.979241i	$-0.435029\pi$
$883$	7.32328e12	0.405399	0.202700	+	0.979241i	$0.435029\pi$
$884$	0	0
$885$	2.01158e12	0.110228
$886$	0	0
$887$	−1.51372e13	−0.821087	−0.410544	−	0.911841i	$-0.634661\pi$
$887$	−1.51372e13	−0.821087	−0.410544	+	0.911841i	$0.634661\pi$
$888$	0	0
$889$	−2.09357e12	−0.112416
$890$	0	0
$891$	−4.29642e13	−2.28380
$892$	0	0
$893$	1.53802e12	0.0809337
$894$	0	0
$895$	−1.05087e13	−0.547451
$896$	0	0
$897$	−6.66138e11	−0.0343556
$898$	0	0
$899$	1.38285e13	0.706082
$900$	0	0
$901$	8.24080e12	0.416589
$902$	0	0
$903$	1.16748e13	0.584323
$904$	0	0
$905$	−4.06789e12	−0.201582
$906$	0	0
$907$	2.24337e13	1.10070	0.550349	−	0.834935i	$-0.314495\pi$
$907$	2.24337e13	1.10070	0.550349	+	0.834935i	$0.314495\pi$
$908$	0	0
$909$	−4.03614e12	−0.196078
$910$	0	0
$911$	6.34178e12	0.305055	0.152528	−	0.988299i	$-0.451259\pi$
$911$	6.34178e12	0.305055	0.152528	+	0.988299i	$0.451259\pi$
$912$	0	0
$913$	7.50094e12	0.357270
$914$	0	0
$915$	−5.31448e13	−2.50649
$916$	0	0
$917$	5.38793e12	0.251629
$918$	0	0
$919$	−9.04471e12	−0.418288	−0.209144	−	0.977885i	$-0.567068\pi$
$919$	−9.04471e12	−0.418288	−0.209144	+	0.977885i	$0.567068\pi$
$920$	0	0
$921$	−1.32376e13	−0.606236
$922$	0	0
$923$	4.62369e11	0.0209692
$924$	0	0
$925$	−1.51134e13	−0.678771
$926$	0	0
$927$	1.06633e13	0.474277
$928$	0	0
$929$	8.50641e12	0.374693	0.187346	−	0.982294i	$-0.440011\pi$
$929$	8.50641e12	0.374693	0.187346	+	0.982294i	$0.440011\pi$
$930$	0	0
$931$	8.21193e11	0.0358238
$932$	0	0
$933$	−5.20368e13	−2.24824
$934$	0	0
$935$	2.01979e13	0.864280
$936$	0	0
$937$	−3.00276e13	−1.27260	−0.636300	−	0.771442i	$-0.719536\pi$
$937$	−3.00276e13	−1.27260	−0.636300	+	0.771442i	$0.719536\pi$
$938$	0	0
$939$	−2.10602e13	−0.884030
$940$	0	0
$941$	−3.15982e13	−1.31374	−0.656870	−	0.754004i	$-0.728120\pi$
$941$	−3.15982e13	−1.31374	−0.656870	+	0.754004i	$0.728120\pi$
$942$	0	0
$943$	−1.68849e13	−0.695338
$944$	0	0
$945$	−9.54927e12	−0.389518
$946$	0	0
$947$	8.84885e12	0.357529	0.178765	−	0.983892i	$-0.442790\pi$
$947$	8.84885e12	0.357529	0.178765	+	0.983892i	$0.442790\pi$
$948$	0	0
$949$	−5.12477e11	−0.0205105
$950$	0	0
$951$	1.21698e13	0.482473
$952$	0	0
$953$	−4.90260e12	−0.192534	−0.0962672	−	0.995356i	$-0.530690\pi$
$953$	−4.90260e12	−0.192534	−0.0962672	+	0.995356i	$0.530690\pi$
$954$	0	0
$955$	−3.31425e13	−1.28935
$956$	0	0
$957$	2.10679e13	0.811929
$958$	0	0
$959$	−9.99155e12	−0.381460
$960$	0	0
$961$	6.71649e13	2.54031
$962$	0	0
$963$	1.41735e12	0.0531077
$964$	0	0
$965$	3.88339e13	1.44158
$966$	0	0
$967$	−1.55633e13	−0.572378	−0.286189	−	0.958173i	$-0.592388\pi$
$967$	−1.55633e13	−0.572378	−0.286189	+	0.958173i	$0.592388\pi$
$968$	0	0
$969$	−2.71179e12	−0.0988096
$970$	0	0
$971$	2.49197e13	0.899613	0.449806	−	0.893126i	$-0.351493\pi$
$971$	2.49197e13	0.899613	0.449806	+	0.893126i	$0.351493\pi$
$972$	0	0
$973$	−1.44872e13	−0.518176
$974$	0	0
$975$	9.10740e11	0.0322756
$976$	0	0
$977$	4.47898e12	0.157273	0.0786364	−	0.996903i	$-0.474943\pi$
$977$	4.47898e12	0.157273	0.0786364	+	0.996903i	$0.474943\pi$
$978$	0	0
$979$	1.87691e11	0.00653012
$980$	0	0
$981$	−6.13629e12	−0.211542
$982$	0	0
$983$	1.43233e13	0.489275	0.244637	−	0.969615i	$-0.421331\pi$
$983$	1.43233e13	0.489275	0.244637	+	0.969615i	$0.421331\pi$
$984$	0	0
$985$	−4.27865e13	−1.44825
$986$	0	0
$987$	4.23626e12	0.142088
$988$	0	0
$989$	−3.79057e13	−1.25986
$990$	0	0
$991$	2.28575e13	0.752829	0.376415	−	0.926451i	$-0.377157\pi$
$991$	2.28575e13	0.752829	0.376415	+	0.926451i	$0.377157\pi$
$992$	0	0
$993$	4.63836e13	1.51388
$994$	0	0
$995$	−3.27902e13	−1.06057
$996$	0	0
$997$	−2.67132e13	−0.856243	−0.428121	−	0.903721i	$-0.640824\pi$
$997$	−2.67132e13	−0.856243	−0.428121	+	0.903721i	$0.640824\pi$
$998$	0	0
$999$	−1.79545e13	−0.570334

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.a.h.1.1		3
4.3	odd	2		7.10.a.b.1.2	✓	3
12.11	even	2		63.10.a.e.1.2		3
20.3	even	4		175.10.b.d.99.3		6
20.7	even	4		175.10.b.d.99.4		6
20.19	odd	2		175.10.a.d.1.2		3
28.3	even	6		49.10.c.e.30.2		6
28.11	odd	6		49.10.c.d.30.2		6
28.19	even	6		49.10.c.e.18.2		6
28.23	odd	6		49.10.c.d.18.2		6
28.27	even	2		49.10.a.c.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.a.b.1.2	✓	3	4.3	odd	2
49.10.a.c.1.2		3	28.27	even	2
49.10.c.d.18.2		6	28.23	odd	6
49.10.c.d.30.2		6	28.11	odd	6
49.10.c.e.18.2		6	28.19	even	6
49.10.c.e.30.2		6	28.3	even	6
63.10.a.e.1.2		3	12.11	even	2
112.10.a.h.1.1		3	1.1	even	1	trivial
175.10.a.d.1.2		3	20.19	odd	2
175.10.b.d.99.3		6	20.3	even	4
175.10.b.d.99.4		6	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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

\(f(q)\)	\(=\)	\(q-163.415 q^{3} +1922.19 q^{5} -2401.00 q^{7} +7021.32 q^{9} +90199.9 q^{11} -3199.89 q^{13} -314114. q^{15} +116494. q^{17} +142449. q^{19} +392358. q^{21} -1.27391e6 q^{23} +1.74168e6 q^{25} +2.06910e6 q^{27} -1.42931e6 q^{29} -9.67494e6 q^{31} -1.47400e7 q^{33} -4.61518e6 q^{35} -8.67744e6 q^{37} +522908. q^{39} +1.32544e7 q^{41} +2.97554e7 q^{43} +1.34963e7 q^{45} +1.07969e7 q^{47} +5.76480e6 q^{49} -1.90369e7 q^{51} +7.07399e7 q^{53} +1.73381e8 q^{55} -2.32783e7 q^{57} -6.40400e6 q^{59} +1.69190e8 q^{61} -1.68582e7 q^{63} -6.15078e6 q^{65} +1.16276e8 q^{67} +2.08176e8 q^{69} -1.44496e8 q^{71} +1.60155e8 q^{73} -2.84617e8 q^{75} -2.16570e8 q^{77} +4.89322e8 q^{79} -4.76322e8 q^{81} +8.31590e7 q^{83} +2.23924e8 q^{85} +2.33569e8 q^{87} +2.08083e6 q^{89} +7.68292e6 q^{91} +1.58103e9 q^{93} +2.73815e8 q^{95} -3.15885e8 q^{97} +6.33322e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 84 q^{3} + 1554 q^{5} - 7203 q^{7} - 26001 q^{9} + 3444 q^{11} - 19782 q^{13} - 200304 q^{15} + 1016694 q^{17} - 222852 q^{19} + 201684 q^{21} - 1885632 q^{23} + 3073221 q^{25} - 551880 q^{27} + 4081818 q^{29}+ \cdots + 1900979172 q^{99}+O(q^{100}) \) 3 * q - 84 * q^3 + 1554 * q^5 - 7203 * q^7 - 26001 * q^9 + 3444 * q^11 - 19782 * q^13 - 200304 * q^15 + 1016694 * q^17 - 222852 * q^19 + 201684 * q^21 - 1885632 * q^23 + 3073221 * q^25 - 551880 * q^27 + 4081818 * q^29 - 2869440 * q^31 - 20259792 * q^33 - 3731154 * q^35 + 1395618 * q^37 + 8990688 * q^39 - 14420658 * q^41 + 61631172 * q^43 + 29774682 * q^45 + 10368960 * q^47 + 17294403 * q^49 + 26146728 * q^51 + 67502610 * q^53 + 105823032 * q^55 + 8471112 * q^57 + 42590100 * q^59 + 191746842 * q^61 + 62428401 * q^63 + 364283220 * q^65 + 255175788 * q^67 + 257903856 * q^69 - 296514504 * q^71 + 344213310 * q^73 - 279031116 * q^75 - 8269044 * q^77 + 960412656 * q^79 - 35827677 * q^81 + 1100517180 * q^83 + 438179412 * q^85 + 621821592 * q^87 + 506816478 * q^89 + 47496582 * q^91 + 1693258512 * q^93 + 2203071072 * q^95 - 647498250 * q^97 + 1900979172 * q^99

Introduction

L-functions

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