LMFDB - Embedded newform 112.10.a.h.1.3

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			1.3
Root			$18.2745$ 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)

$f(q)$	$=$	$q+79.6469 q^{3} +1423.70 q^{5} -2401.00 q^{7} -13339.4 q^{9} -69354.4 q^{11} +105959. q^{13} +113393. q^{15} +568267. q^{17} +396405. q^{19} -191232. q^{21} +620765. q^{23} +73796.7 q^{25} -2.63013e6 q^{27} +4.87652e6 q^{29} +1.42482e6 q^{31} -5.52386e6 q^{33} -3.41830e6 q^{35} +1.31092e7 q^{37} +8.43931e6 q^{39} -2.03049e7 q^{41} +1.11768e7 q^{43} -1.89913e7 q^{45} +1.99352e7 q^{47} +5.76480e6 q^{49} +4.52607e7 q^{51} +5.65007e7 q^{53} -9.87398e7 q^{55} +3.15725e7 q^{57} +1.09340e8 q^{59} +3.20008e7 q^{61} +3.20278e7 q^{63} +1.50854e8 q^{65} -8.02869e7 q^{67} +4.94420e7 q^{69} -2.07893e8 q^{71} -2.70274e8 q^{73} +5.87768e6 q^{75} +1.66520e8 q^{77} +5.16196e8 q^{79} +5.30772e7 q^{81} +6.82693e8 q^{83} +8.09042e8 q^{85} +3.88400e8 q^{87} -1.47150e8 q^{89} -2.54408e8 q^{91} +1.13482e8 q^{93} +5.64362e8 q^{95} +1.09643e9 q^{97} +9.25144e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	79.6469	0.567706	0.283853	−	0.958868i	$-0.408387\pi$
$3$	79.6469	0.567706	0.283853	+	0.958868i	$0.408387\pi$
$4$	0	0
$5$	1423.70	1.01872	0.509358	−	0.860554i	$-0.329883\pi$
$5$	1423.70	1.01872	0.509358	+	0.860554i	$0.329883\pi$
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	0	0
$9$	−13339.4	−0.677710
$10$	0	0
$11$	−69354.4	−1.42826	−0.714129	−	0.700014i	$-0.753177\pi$
$11$	−69354.4	−1.42826	−0.714129	+	0.700014i	$0.753177\pi$
$12$	0	0
$13$	105959.	1.02895	0.514473	−	0.857506i	$-0.327987\pi$
$13$	105959.	1.02895	0.514473	+	0.857506i	$0.327987\pi$
$14$	0	0
$15$	113393.	0.578331
$16$	0	0
$17$	568267.	1.65018	0.825092	−	0.564998i	$-0.191123\pi$
$17$	568267.	1.65018	0.825092	+	0.564998i	$0.191123\pi$
$18$	0	0
$19$	396405.	0.697828	0.348914	−	0.937155i	$-0.386551\pi$
$19$	396405.	0.697828	0.348914	+	0.937155i	$0.386551\pi$
$20$	0	0
$21$	−191232.	−0.214573
$22$	0	0
$23$	620765.	0.462543	0.231271	−	0.972889i	$-0.425712\pi$
$23$	620765.	0.462543	0.231271	+	0.972889i	$0.425712\pi$
$24$	0	0
$25$	73796.7	0.0377839
$26$	0	0
$27$	−2.63013e6	−0.952446
$28$	0	0
$29$	4.87652e6	1.28032	0.640161	−	0.768241i	$-0.278868\pi$
$29$	4.87652e6	1.28032	0.640161	+	0.768241i	$0.278868\pi$
$30$	0	0
$31$	1.42482e6	0.277096	0.138548	−	0.990356i	$-0.455756\pi$
$31$	1.42482e6	0.277096	0.138548	+	0.990356i	$0.455756\pi$
$32$	0	0
$33$	−5.52386e6	−0.810830
$34$	0	0
$35$	−3.41830e6	−0.385039
$36$	0	0
$37$	1.31092e7	1.14992	0.574960	−	0.818182i	$-0.305018\pi$
$37$	1.31092e7	1.14992	0.574960	+	0.818182i	$0.305018\pi$
$38$	0	0
$39$	8.43931e6	0.584139
$40$	0	0
$41$	−2.03049e7	−1.12221	−0.561105	−	0.827744i	$-0.689624\pi$
$41$	−2.03049e7	−1.12221	−0.561105	+	0.827744i	$0.689624\pi$
$42$	0	0
$43$	1.11768e7	0.498551	0.249276	−	0.968433i	$-0.419807\pi$
$43$	1.11768e7	0.498551	0.249276	+	0.968433i	$0.419807\pi$
$44$	0	0
$45$	−1.89913e7	−0.690395
$46$	0	0
$47$	1.99352e7	0.595909	0.297955	−	0.954580i	$-0.403696\pi$
$47$	1.99352e7	0.595909	0.297955	+	0.954580i	$0.403696\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	0	0
$51$	4.52607e7	0.936819
$52$	0	0
$53$	5.65007e7	0.983586	0.491793	−	0.870712i	$-0.336342\pi$
$53$	5.65007e7	0.983586	0.491793	+	0.870712i	$0.336342\pi$
$54$	0	0
$55$	−9.87398e7	−1.45499
$56$	0	0
$57$	3.15725e7	0.396161
$58$	0	0
$59$	1.09340e8	1.17475	0.587375	−	0.809315i	$-0.300161\pi$
$59$	1.09340e8	1.17475	0.587375	+	0.809315i	$0.300161\pi$
$60$	0	0
$61$	3.20008e7	0.295921	0.147961	−	0.988993i	$-0.452729\pi$
$61$	3.20008e7	0.295921	0.147961	+	0.988993i	$0.452729\pi$
$62$	0	0
$63$	3.20278e7	0.256150
$64$	0	0
$65$	1.50854e8	1.04821
$66$	0	0
$67$	−8.02869e7	−0.486752	−0.243376	−	0.969932i	$-0.578255\pi$
$67$	−8.02869e7	−0.486752	−0.243376	+	0.969932i	$0.578255\pi$
$68$	0	0
$69$	4.94420e7	0.262588
$70$	0	0
$71$	−2.07893e8	−0.970906	−0.485453	−	0.874263i	$-0.661345\pi$
$71$	−2.07893e8	−0.970906	−0.485453	+	0.874263i	$0.661345\pi$
$72$	0	0
$73$	−2.70274e8	−1.11391	−0.556957	−	0.830541i	$-0.688031\pi$
$73$	−2.70274e8	−1.11391	−0.556957	+	0.830541i	$0.688031\pi$
$74$	0	0
$75$	5.87768e6	0.0214501
$76$	0	0
$77$	1.66520e8	0.539831
$78$	0	0
$79$	5.16196e8	1.49105	0.745526	−	0.666477i	$-0.232198\pi$
$79$	5.16196e8	1.49105	0.745526	+	0.666477i	$0.232198\pi$
$80$	0	0
$81$	5.30772e7	0.137002
$82$	0	0
$83$	6.82693e8	1.57897	0.789486	−	0.613769i	$-0.210347\pi$
$83$	6.82693e8	1.57897	0.789486	+	0.613769i	$0.210347\pi$
$84$	0	0
$85$	8.09042e8	1.68107
$86$	0	0
$87$	3.88400e8	0.726846
$88$	0	0
$89$	−1.47150e8	−0.248602	−0.124301	−	0.992245i	$-0.539669\pi$
$89$	−1.47150e8	−0.248602	−0.124301	+	0.992245i	$0.539669\pi$
$90$	0	0
$91$	−2.54408e8	−0.388905
$92$	0	0
$93$	1.13482e8	0.157309
$94$	0	0
$95$	5.64362e8	0.710889
$96$	0	0
$97$	1.09643e9	1.25750	0.628750	−	0.777608i	$-0.283567\pi$
$97$	1.09643e9	1.25750	0.628750	+	0.777608i	$0.283567\pi$
$98$	0	0
$99$	9.25144e8	0.967945
$100$	0	0
$101$	2.08683e8	0.199545	0.0997727	−	0.995010i	$-0.468188\pi$
$101$	2.08683e8	0.199545	0.0997727	+	0.995010i	$0.468188\pi$
$102$	0	0
$103$	−6.78194e8	−0.593727	−0.296863	−	0.954920i	$-0.595941\pi$
$103$	−6.78194e8	−0.593727	−0.296863	+	0.954920i	$0.595941\pi$
$104$	0	0
$105$	−2.72257e8	−0.218589
$106$	0	0
$107$	4.59542e8	0.338921	0.169461	−	0.985537i	$-0.445797\pi$
$107$	4.59542e8	0.338921	0.169461	+	0.985537i	$0.445797\pi$
$108$	0	0
$109$	5.21086e8	0.353582	0.176791	−	0.984248i	$-0.443428\pi$
$109$	5.21086e8	0.353582	0.176791	+	0.984248i	$0.443428\pi$
$110$	0	0
$111$	1.04410e9	0.652816
$112$	0	0
$113$	4.45612e8	0.257101	0.128551	−	0.991703i	$-0.458968\pi$
$113$	4.45612e8	0.257101	0.128551	+	0.991703i	$0.458968\pi$
$114$	0	0
$115$	8.83783e8	0.471200
$116$	0	0
$117$	−1.41343e9	−0.697328
$118$	0	0
$119$	−1.36441e9	−0.623711
$120$	0	0
$121$	2.45208e9	1.03992
$122$	0	0
$123$	−1.61723e9	−0.637086
$124$	0	0
$125$	−2.67560e9	−0.980226
$126$	0	0
$127$	−9.28626e8	−0.316755	−0.158378	−	0.987379i	$-0.550626\pi$
$127$	−9.28626e8	−0.316755	−0.158378	+	0.987379i	$0.550626\pi$
$128$	0	0
$129$	8.90198e8	0.283030
$130$	0	0
$131$	−2.57694e9	−0.764509	−0.382255	−	0.924057i	$-0.624852\pi$
$131$	−2.57694e9	−0.764509	−0.382255	+	0.924057i	$0.624852\pi$
$132$	0	0
$133$	−9.51769e8	−0.263754
$134$	0	0
$135$	−3.74452e9	−0.970272
$136$	0	0
$137$	−4.44116e9	−1.07710	−0.538548	−	0.842595i	$-0.681027\pi$
$137$	−4.44116e9	−1.07710	−0.538548	+	0.842595i	$0.681027\pi$
$138$	0	0
$139$	7.28389e9	1.65499	0.827497	−	0.561470i	$-0.189764\pi$
$139$	7.28389e9	1.65499	0.827497	+	0.561470i	$0.189764\pi$
$140$	0	0
$141$	1.58778e9	0.338301
$142$	0	0
$143$	−7.34872e9	−1.46960
$144$	0	0
$145$	6.94270e9	1.30428
$146$	0	0
$147$	4.59149e8	0.0811008
$148$	0	0
$149$	4.87355e9	0.810042	0.405021	−	0.914307i	$-0.367264\pi$
$149$	4.87355e9	0.810042	0.405021	+	0.914307i	$0.367264\pi$
$150$	0	0
$151$	8.63776e9	1.35209	0.676044	−	0.736861i	$-0.263693\pi$
$151$	8.63776e9	1.35209	0.676044	+	0.736861i	$0.263693\pi$
$152$	0	0
$153$	−7.58033e9	−1.11835
$154$	0	0
$155$	2.02851e9	0.282283
$156$	0	0
$157$	−9.34170e9	−1.22709	−0.613546	−	0.789659i	$-0.710258\pi$
$157$	−9.34170e9	−1.22709	−0.613546	+	0.789659i	$0.710258\pi$
$158$	0	0
$159$	4.50010e9	0.558387
$160$	0	0
$161$	−1.49046e9	−0.174825
$162$	0	0
$163$	2.10680e9	0.233765	0.116883	−	0.993146i	$-0.462710\pi$
$163$	2.10680e9	0.233765	0.116883	+	0.993146i	$0.462710\pi$
$164$	0	0
$165$	−7.86432e9	−0.826006
$166$	0	0
$167$	−1.39800e10	−1.39086	−0.695431	−	0.718593i	$-0.744787\pi$
$167$	−1.39800e10	−1.39086	−0.695431	+	0.718593i	$0.744787\pi$
$168$	0	0
$169$	6.22820e8	0.0587316
$170$	0	0
$171$	−5.28780e9	−0.472925
$172$	0	0
$173$	−1.24435e10	−1.05618	−0.528088	−	0.849190i	$-0.677091\pi$
$173$	−1.24435e10	−1.05618	−0.528088	+	0.849190i	$0.677091\pi$
$174$	0	0
$175$	−1.77186e8	−0.0142810
$176$	0	0
$177$	8.70861e9	0.666913
$178$	0	0
$179$	−7.30178e9	−0.531606	−0.265803	−	0.964027i	$-0.585637\pi$
$179$	−7.30178e9	−0.531606	−0.265803	+	0.964027i	$0.585637\pi$
$180$	0	0
$181$	−1.27074e10	−0.880038	−0.440019	−	0.897988i	$-0.645028\pi$
$181$	−1.27074e10	−0.880038	−0.440019	+	0.897988i	$0.645028\pi$
$182$	0	0
$183$	2.54876e9	0.167996
$184$	0	0
$185$	1.86635e10	1.17144
$186$	0	0
$187$	−3.94118e10	−2.35689
$188$	0	0
$189$	6.31494e9	0.359991
$190$	0	0
$191$	−1.61547e10	−0.878311	−0.439155	−	0.898411i	$-0.644722\pi$
$191$	−1.61547e10	−0.878311	−0.439155	+	0.898411i	$0.644722\pi$
$192$	0	0
$193$	−1.52841e10	−0.792924	−0.396462	−	0.918051i	$-0.629762\pi$
$193$	−1.52841e10	−0.792924	−0.396462	+	0.918051i	$0.629762\pi$
$194$	0	0
$195$	1.20150e10	0.595072
$196$	0	0
$197$	2.33886e10	1.10639	0.553193	−	0.833053i	$-0.313409\pi$
$197$	2.33886e10	1.10639	0.553193	+	0.833053i	$0.313409\pi$
$198$	0	0
$199$	2.53610e10	1.14638	0.573189	−	0.819423i	$-0.305706\pi$
$199$	2.53610e10	1.14638	0.573189	+	0.819423i	$0.305706\pi$
$200$	0	0
$201$	−6.39460e9	−0.276332
$202$	0	0
$203$	−1.17085e10	−0.483916
$204$	0	0
$205$	−2.89082e10	−1.14322
$206$	0	0
$207$	−8.28061e9	−0.313470
$208$	0	0
$209$	−2.74924e10	−0.996678
$210$	0	0
$211$	9.23757e8	0.0320838	0.0160419	−	0.999871i	$-0.494893\pi$
$211$	9.23757e8	0.0320838	0.0160419	+	0.999871i	$0.494893\pi$
$212$	0	0
$213$	−1.65580e10	−0.551189
$214$	0	0
$215$	1.59124e10	0.507883
$216$	0	0
$217$	−3.42098e9	−0.104733
$218$	0	0
$219$	−2.15265e10	−0.632375
$220$	0	0
$221$	6.02130e10	1.69795
$222$	0	0
$223$	−6.68635e9	−0.181058	−0.0905290	−	0.995894i	$-0.528856\pi$
$223$	−6.68635e9	−0.181058	−0.0905290	+	0.995894i	$0.528856\pi$
$224$	0	0
$225$	−9.84401e8	−0.0256065
$226$	0	0
$227$	4.82876e10	1.20703	0.603516	−	0.797351i	$-0.293766\pi$
$227$	4.82876e10	1.20703	0.603516	+	0.797351i	$0.293766\pi$
$228$	0	0
$229$	−2.34264e10	−0.562918	−0.281459	−	0.959573i	$-0.590818\pi$
$229$	−2.34264e10	−0.562918	−0.281459	+	0.959573i	$0.590818\pi$
$230$	0	0
$231$	1.32628e10	0.306465
$232$	0	0
$233$	−3.29140e10	−0.731609	−0.365805	−	0.930692i	$-0.619206\pi$
$233$	−3.29140e10	−0.731609	−0.365805	+	0.930692i	$0.619206\pi$
$234$	0	0
$235$	2.83817e10	0.607063
$236$	0	0
$237$	4.11134e10	0.846479
$238$	0	0
$239$	−2.51973e10	−0.499533	−0.249766	−	0.968306i	$-0.580354\pi$
$239$	−2.51973e10	−0.499533	−0.249766	+	0.968306i	$0.580354\pi$
$240$	0	0
$241$	7.00815e10	1.33822	0.669109	−	0.743165i	$-0.266676\pi$
$241$	7.00815e10	1.33822	0.669109	+	0.743165i	$0.266676\pi$
$242$	0	0
$243$	5.59963e10	1.03022
$244$	0	0
$245$	8.20735e9	0.145531
$246$	0	0
$247$	4.20027e10	0.718028
$248$	0	0
$249$	5.43744e10	0.896391
$250$	0	0
$251$	−4.19681e10	−0.667401	−0.333701	−	0.942679i	$-0.608297\pi$
$251$	−4.19681e10	−0.667401	−0.333701	+	0.942679i	$0.608297\pi$
$252$	0	0
$253$	−4.30527e10	−0.660630
$254$	0	0
$255$	6.44377e10	0.954353
$256$	0	0
$257$	8.37595e10	1.19766	0.598832	−	0.800875i	$-0.295632\pi$
$257$	8.37595e10	1.19766	0.598832	+	0.800875i	$0.295632\pi$
$258$	0	0
$259$	−3.14751e10	−0.434629
$260$	0	0
$261$	−6.50497e10	−0.867687
$262$	0	0
$263$	2.34604e9	0.0302367	0.0151184	−	0.999886i	$-0.495187\pi$
$263$	2.34604e9	0.0302367	0.0151184	+	0.999886i	$0.495187\pi$
$264$	0	0
$265$	8.04400e10	1.00200
$266$	0	0
$267$	−1.17200e10	−0.141133
$268$	0	0
$269$	−6.44659e10	−0.750663	−0.375332	−	0.926891i	$-0.622471\pi$
$269$	−6.44659e10	−0.750663	−0.375332	+	0.926891i	$0.622471\pi$
$270$	0	0
$271$	1.40530e10	0.158273	0.0791365	−	0.996864i	$-0.474784\pi$
$271$	1.40530e10	0.158273	0.0791365	+	0.996864i	$0.474784\pi$
$272$	0	0
$273$	−2.02628e10	−0.220784
$274$	0	0
$275$	−5.11812e9	−0.0539652
$276$	0	0
$277$	−7.52768e10	−0.768249	−0.384124	−	0.923281i	$-0.625497\pi$
$277$	−7.52768e10	−0.768249	−0.384124	+	0.923281i	$0.625497\pi$
$278$	0	0
$279$	−1.90061e10	−0.187791
$280$	0	0
$281$	9.56085e10	0.914783	0.457391	−	0.889266i	$-0.348784\pi$
$281$	9.56085e10	0.914783	0.457391	+	0.889266i	$0.348784\pi$
$282$	0	0
$283$	−4.82806e10	−0.447439	−0.223719	−	0.974654i	$-0.571820\pi$
$283$	−4.82806e10	−0.447439	−0.223719	+	0.974654i	$0.571820\pi$
$284$	0	0
$285$	4.49497e10	0.403576
$286$	0	0
$287$	4.87522e10	0.424156
$288$	0	0
$289$	2.04340e11	1.72311
$290$	0	0
$291$	8.73272e10	0.713890
$292$	0	0
$293$	−7.07439e10	−0.560770	−0.280385	−	0.959888i	$-0.590462\pi$
$293$	−7.07439e10	−0.560770	−0.280385	+	0.959888i	$0.590462\pi$
$294$	0	0
$295$	1.55668e11	1.19674
$296$	0	0
$297$	1.82411e11	1.36034
$298$	0	0
$299$	6.57756e10	0.475932
$300$	0	0
$301$	−2.68355e10	−0.188435
$302$	0	0
$303$	1.66210e10	0.113283
$304$	0	0
$305$	4.55595e10	0.301460
$306$	0	0
$307$	−1.30493e11	−0.838429	−0.419214	−	0.907887i	$-0.637694\pi$
$307$	−1.30493e11	−0.838429	−0.419214	+	0.907887i	$0.637694\pi$
$308$	0	0
$309$	−5.40161e10	−0.337062
$310$	0	0
$311$	−8.51715e10	−0.516265	−0.258132	−	0.966110i	$-0.583107\pi$
$311$	−8.51715e10	−0.516265	−0.258132	+	0.966110i	$0.583107\pi$
$312$	0	0
$313$	−1.21745e11	−0.716969	−0.358484	−	0.933536i	$-0.616706\pi$
$313$	−1.21745e11	−0.716969	−0.358484	+	0.933536i	$0.616706\pi$
$314$	0	0
$315$	4.55980e10	0.260945
$316$	0	0
$317$	−2.14595e11	−1.19358	−0.596791	−	0.802397i	$-0.703558\pi$
$317$	−2.14595e11	−1.19358	−0.596791	+	0.802397i	$0.703558\pi$
$318$	0	0
$319$	−3.38208e11	−1.82863
$320$	0	0
$321$	3.66011e10	0.192407
$322$	0	0
$323$	2.25264e11	1.15154
$324$	0	0
$325$	7.81943e9	0.0388776
$326$	0	0
$327$	4.15029e10	0.200731
$328$	0	0
$329$	−4.78644e10	−0.225233
$330$	0	0
$331$	5.48000e10	0.250931	0.125466	−	0.992098i	$-0.459958\pi$
$331$	5.48000e10	0.250931	0.125466	+	0.992098i	$0.459958\pi$
$332$	0	0
$333$	−1.74868e11	−0.779312
$334$	0	0
$335$	−1.14304e11	−0.495863
$336$	0	0
$337$	−2.34297e11	−0.989538	−0.494769	−	0.869025i	$-0.664747\pi$
$337$	−2.34297e11	−0.989538	−0.494769	+	0.869025i	$0.664747\pi$
$338$	0	0
$339$	3.54916e10	0.145958
$340$	0	0
$341$	−9.88172e10	−0.395765
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	0	0
$345$	7.03905e10	0.267503
$346$	0	0
$347$	−3.43449e10	−0.127169	−0.0635843	−	0.997976i	$-0.520253\pi$
$347$	−3.43449e10	−0.127169	−0.0635843	+	0.997976i	$0.520253\pi$
$348$	0	0
$349$	−2.13485e11	−0.770288	−0.385144	−	0.922856i	$-0.625848\pi$
$349$	−2.13485e11	−0.770288	−0.385144	+	0.922856i	$0.625848\pi$
$350$	0	0
$351$	−2.78686e11	−0.980016
$352$	0	0
$353$	2.75882e11	0.945664	0.472832	−	0.881153i	$-0.343232\pi$
$353$	2.75882e11	0.945664	0.472832	+	0.881153i	$0.343232\pi$
$354$	0	0
$355$	−2.95977e11	−0.989079
$356$	0	0
$357$	−1.08671e11	−0.354084
$358$	0	0
$359$	3.46238e11	1.10015	0.550073	−	0.835117i	$-0.314600\pi$
$359$	3.46238e11	1.10015	0.550073	+	0.835117i	$0.314600\pi$
$360$	0	0
$361$	−1.65550e11	−0.513036
$362$	0	0
$363$	1.95300e11	0.590369
$364$	0	0
$365$	−3.84789e11	−1.13476
$366$	0	0
$367$	−3.56842e11	−1.02678	−0.513391	−	0.858155i	$-0.671611\pi$
$367$	−3.56842e11	−1.02678	−0.513391	+	0.858155i	$0.671611\pi$
$368$	0	0
$369$	2.70855e11	0.760534
$370$	0	0
$371$	−1.35658e11	−0.371760
$372$	0	0
$373$	6.73833e11	1.80245	0.901224	−	0.433354i	$-0.142670\pi$
$373$	6.73833e11	1.80245	0.901224	+	0.433354i	$0.142670\pi$
$374$	0	0
$375$	−2.13103e11	−0.556480
$376$	0	0
$377$	5.16711e11	1.31738
$378$	0	0
$379$	−5.90163e11	−1.46925	−0.734625	−	0.678473i	$-0.762642\pi$
$379$	−5.90163e11	−1.46925	−0.734625	+	0.678473i	$0.762642\pi$
$380$	0	0
$381$	−7.39622e10	−0.179824
$382$	0	0
$383$	1.58931e11	0.377412	0.188706	−	0.982034i	$-0.439571\pi$
$383$	1.58931e11	0.377412	0.188706	+	0.982034i	$0.439571\pi$
$384$	0	0
$385$	2.37074e11	0.549935
$386$	0	0
$387$	−1.49092e11	−0.337873
$388$	0	0
$389$	−3.75434e11	−0.831304	−0.415652	−	0.909524i	$-0.636447\pi$
$389$	−3.75434e11	−0.831304	−0.415652	+	0.909524i	$0.636447\pi$
$390$	0	0
$391$	3.52760e11	0.763280
$392$	0	0
$393$	−2.05245e11	−0.434016
$394$	0	0
$395$	7.34908e11	1.51896
$396$	0	0
$397$	−4.33507e11	−0.875869	−0.437935	−	0.899007i	$-0.644290\pi$
$397$	−4.33507e11	−0.875869	−0.437935	+	0.899007i	$0.644290\pi$
$398$	0	0
$399$	−7.58055e10	−0.149735
$400$	0	0
$401$	−1.77805e11	−0.343395	−0.171698	−	0.985150i	$-0.554925\pi$
$401$	−1.77805e11	−0.343395	−0.171698	+	0.985150i	$0.554925\pi$
$402$	0	0
$403$	1.50972e11	0.285118
$404$	0	0
$405$	7.55660e10	0.139566
$406$	0	0
$407$	−9.09178e11	−1.64238
$408$	0	0
$409$	−7.67870e11	−1.35685	−0.678427	−	0.734668i	$-0.737338\pi$
$409$	−7.67870e11	−1.35685	−0.678427	+	0.734668i	$0.737338\pi$
$410$	0	0
$411$	−3.53725e11	−0.611473
$412$	0	0
$413$	−2.62526e11	−0.444014
$414$	0	0
$415$	9.71951e11	1.60852
$416$	0	0
$417$	5.80139e11	0.939550
$418$	0	0
$419$	−4.96552e11	−0.787048	−0.393524	−	0.919314i	$-0.628744\pi$
$419$	−4.96552e11	−0.787048	−0.393524	+	0.919314i	$0.628744\pi$
$420$	0	0
$421$	−4.48514e11	−0.695835	−0.347917	−	0.937525i	$-0.613111\pi$
$421$	−4.48514e11	−0.695835	−0.347917	+	0.937525i	$0.613111\pi$
$422$	0	0
$423$	−2.65923e11	−0.403854
$424$	0	0
$425$	4.19362e10	0.0623504
$426$	0	0
$427$	−7.68338e10	−0.111848
$428$	0	0
$429$	−5.85303e11	−0.834301
$430$	0	0
$431$	1.65940e11	0.231635	0.115817	−	0.993271i	$-0.463051\pi$
$431$	1.65940e11	0.231635	0.115817	+	0.993271i	$0.463051\pi$
$432$	0	0
$433$	−4.10674e11	−0.561438	−0.280719	−	0.959790i	$-0.590573\pi$
$433$	−4.10674e11	−0.561438	−0.280719	+	0.959790i	$0.590573\pi$
$434$	0	0
$435$	5.52965e11	0.740450
$436$	0	0
$437$	2.46074e11	0.322775
$438$	0	0
$439$	3.29593e11	0.423534	0.211767	−	0.977320i	$-0.432078\pi$
$439$	3.29593e11	0.423534	0.211767	+	0.977320i	$0.432078\pi$
$440$	0	0
$441$	−7.68988e10	−0.0968158
$442$	0	0
$443$	−5.09091e11	−0.628027	−0.314014	−	0.949418i	$-0.601674\pi$
$443$	−5.09091e11	−0.628027	−0.314014	+	0.949418i	$0.601674\pi$
$444$	0	0
$445$	−2.09497e11	−0.253255
$446$	0	0
$447$	3.88163e11	0.459865
$448$	0	0
$449$	1.49596e12	1.73705	0.868525	−	0.495646i	$-0.165069\pi$
$449$	1.49596e12	1.73705	0.868525	+	0.495646i	$0.165069\pi$
$450$	0	0
$451$	1.40824e12	1.60281
$452$	0	0
$453$	6.87971e11	0.767588
$454$	0	0
$455$	−3.62200e11	−0.396184
$456$	0	0
$457$	1.43920e12	1.54347	0.771735	−	0.635944i	$-0.219389\pi$
$457$	1.43920e12	1.54347	0.771735	+	0.635944i	$0.219389\pi$
$458$	0	0
$459$	−1.49462e12	−1.57171
$460$	0	0
$461$	−1.37741e12	−1.42039	−0.710195	−	0.704005i	$-0.751393\pi$
$461$	−1.37741e12	−1.42039	−0.710195	+	0.704005i	$0.751393\pi$
$462$	0	0
$463$	−1.76612e12	−1.78610	−0.893049	−	0.449960i	$-0.851438\pi$
$463$	−1.76612e12	−1.78610	−0.893049	+	0.449960i	$0.851438\pi$
$464$	0	0
$465$	1.61565e11	0.160254
$466$	0	0
$467$	1.17323e12	1.14145	0.570727	−	0.821140i	$-0.306661\pi$
$467$	1.17323e12	1.14145	0.570727	+	0.821140i	$0.306661\pi$
$468$	0	0
$469$	1.92769e11	0.183975
$470$	0	0
$471$	−7.44037e11	−0.696627
$472$	0	0
$473$	−7.75161e11	−0.712060
$474$	0	0
$475$	2.92534e10	0.0263667
$476$	0	0
$477$	−7.53683e11	−0.666586
$478$	0	0
$479$	1.98723e12	1.72480	0.862401	−	0.506225i	$-0.168960\pi$
$479$	1.98723e12	1.72480	0.862401	+	0.506225i	$0.168960\pi$
$480$	0	0
$481$	1.38903e12	1.18321
$482$	0	0
$483$	−1.18710e11	−0.0992489
$484$	0	0
$485$	1.56099e12	1.28104
$486$	0	0
$487$	1.23240e12	0.992825	0.496413	−	0.868087i	$-0.334650\pi$
$487$	1.23240e12	0.992825	0.496413	+	0.868087i	$0.334650\pi$
$488$	0	0
$489$	1.67800e11	0.132710
$490$	0	0
$491$	2.03763e12	1.58219	0.791095	−	0.611693i	$-0.209511\pi$
$491$	2.03763e12	1.58219	0.791095	+	0.611693i	$0.209511\pi$
$492$	0	0
$493$	2.77117e12	2.11277
$494$	0	0
$495$	1.31713e12	0.986062
$496$	0	0
$497$	4.99151e11	0.366968
$498$	0	0
$499$	3.26299e11	0.235594	0.117797	−	0.993038i	$-0.462417\pi$
$499$	3.26299e11	0.235594	0.117797	+	0.993038i	$0.462417\pi$
$500$	0	0
$501$	−1.11347e12	−0.789601
$502$	0	0
$503$	4.46869e11	0.311261	0.155630	−	0.987815i	$-0.450259\pi$
$503$	4.46869e11	0.311261	0.155630	+	0.987815i	$0.450259\pi$
$504$	0	0
$505$	2.97103e11	0.203280
$506$	0	0
$507$	4.96056e10	0.0333423
$508$	0	0
$509$	1.34100e12	0.885523	0.442761	−	0.896639i	$-0.353999\pi$
$509$	1.34100e12	0.885523	0.442761	+	0.896639i	$0.353999\pi$
$510$	0	0
$511$	6.48928e11	0.421020
$512$	0	0
$513$	−1.04260e12	−0.664643
$514$	0	0
$515$	−9.65545e11	−0.604839
$516$	0	0
$517$	−1.38259e12	−0.851112
$518$	0	0
$519$	−9.91089e11	−0.599597
$520$	0	0
$521$	2.98523e12	1.77504	0.887520	−	0.460769i	$-0.152426\pi$
$521$	2.98523e12	1.77504	0.887520	+	0.460769i	$0.152426\pi$
$522$	0	0
$523$	−1.64651e12	−0.962289	−0.481145	−	0.876641i	$-0.659779\pi$
$523$	−1.64651e12	−0.962289	−0.481145	+	0.876641i	$0.659779\pi$
$524$	0	0
$525$	−1.41123e10	−0.00810739
$526$	0	0
$527$	8.09676e11	0.457260
$528$	0	0
$529$	−1.41580e12	−0.786054
$530$	0	0
$531$	−1.45853e12	−0.796141
$532$	0	0
$533$	−2.15149e12	−1.15470
$534$	0	0
$535$	6.54250e11	0.345265
$536$	0	0
$537$	−5.81564e11	−0.301796
$538$	0	0
$539$	−3.99814e11	−0.204037
$540$	0	0
$541$	−4.57968e11	−0.229852	−0.114926	−	0.993374i	$-0.536663\pi$
$541$	−4.57968e11	−0.229852	−0.114926	+	0.993374i	$0.536663\pi$
$542$	0	0
$543$	−1.01210e12	−0.499603
$544$	0	0
$545$	7.41871e11	0.360200
$546$	0	0
$547$	−2.39624e12	−1.14443	−0.572213	−	0.820105i	$-0.693915\pi$
$547$	−2.39624e12	−1.14443	−0.572213	+	0.820105i	$0.693915\pi$
$548$	0	0
$549$	−4.26870e11	−0.200549
$550$	0	0
$551$	1.93308e12	0.893444
$552$	0	0
$553$	−1.23939e12	−0.563565
$554$	0	0
$555$	1.48649e12	0.665034
$556$	0	0
$557$	1.07863e12	0.474814	0.237407	−	0.971410i	$-0.423702\pi$
$557$	1.07863e12	0.474814	0.237407	+	0.971410i	$0.423702\pi$
$558$	0	0
$559$	1.18428e12	0.512983
$560$	0	0
$561$	−3.13903e12	−1.33802
$562$	0	0
$563$	−2.33140e11	−0.0977976	−0.0488988	−	0.998804i	$-0.515571\pi$
$563$	−2.33140e11	−0.0977976	−0.0488988	+	0.998804i	$0.515571\pi$
$564$	0	0
$565$	6.34418e11	0.261913
$566$	0	0
$567$	−1.27438e11	−0.0517817
$568$	0	0
$569$	4.50535e11	0.180187	0.0900934	−	0.995933i	$-0.471283\pi$
$569$	4.50535e11	0.180187	0.0900934	+	0.995933i	$0.471283\pi$
$570$	0	0
$571$	4.38839e12	1.72760	0.863800	−	0.503835i	$-0.168078\pi$
$571$	4.38839e12	1.72760	0.863800	+	0.503835i	$0.168078\pi$
$572$	0	0
$573$	−1.28667e12	−0.498622
$574$	0	0
$575$	4.58104e10	0.0174767
$576$	0	0
$577$	3.13994e12	1.17932	0.589658	−	0.807653i	$-0.299263\pi$
$577$	3.13994e12	1.17932	0.589658	+	0.807653i	$0.299263\pi$
$578$	0	0
$579$	−1.21733e12	−0.450148
$580$	0	0
$581$	−1.63915e12	−0.596795
$582$	0	0
$583$	−3.91857e12	−1.40481
$584$	0	0
$585$	−2.01230e12	−0.710380
$586$	0	0
$587$	−2.52512e12	−0.877829	−0.438915	−	0.898529i	$-0.644637\pi$
$587$	−2.52512e12	−0.877829	−0.438915	+	0.898529i	$0.644637\pi$
$588$	0	0
$589$	5.64805e11	0.193366
$590$	0	0
$591$	1.86283e12	0.628102
$592$	0	0
$593$	−9.35417e11	−0.310641	−0.155321	−	0.987864i	$-0.549641\pi$
$593$	−9.35417e11	−0.310641	−0.155321	+	0.987864i	$0.549641\pi$
$594$	0	0
$595$	−1.94251e12	−0.635385
$596$	0	0
$597$	2.01993e12	0.650805
$598$	0	0
$599$	−4.73586e12	−1.50307	−0.751534	−	0.659694i	$-0.770686\pi$
$599$	−4.73586e12	−1.50307	−0.751534	+	0.659694i	$0.770686\pi$
$600$	0	0
$601$	−5.99855e12	−1.87548	−0.937738	−	0.347344i	$-0.887084\pi$
$601$	−5.99855e12	−1.87548	−0.937738	+	0.347344i	$0.887084\pi$
$602$	0	0
$603$	1.07098e12	0.329877
$604$	0	0
$605$	3.49103e12	1.05938
$606$	0	0
$607$	−3.43715e12	−1.02766	−0.513830	−	0.857892i	$-0.671774\pi$
$607$	−3.43715e12	−1.02766	−0.513830	+	0.857892i	$0.671774\pi$
$608$	0	0
$609$	−9.32548e11	−0.274722
$610$	0	0
$611$	2.11231e12	0.613159
$612$	0	0
$613$	4.15273e12	1.18785	0.593925	−	0.804520i	$-0.297577\pi$
$613$	4.15273e12	1.18785	0.593925	+	0.804520i	$0.297577\pi$
$614$	0	0
$615$	−2.30244e12	−0.649010
$616$	0	0
$617$	−1.12196e12	−0.311669	−0.155834	−	0.987783i	$-0.549807\pi$
$617$	−1.12196e12	−0.311669	−0.155834	+	0.987783i	$0.549807\pi$
$618$	0	0
$619$	5.98940e12	1.63974	0.819871	−	0.572549i	$-0.194045\pi$
$619$	5.98940e12	1.63974	0.819871	+	0.572549i	$0.194045\pi$
$620$	0	0
$621$	−1.63269e12	−0.440547
$622$	0	0
$623$	3.53307e11	0.0939628
$624$	0	0
$625$	−3.95339e12	−1.03636
$626$	0	0
$627$	−2.18969e12	−0.565820
$628$	0	0
$629$	7.44951e12	1.89758
$630$	0	0
$631$	−4.97135e12	−1.24837	−0.624184	−	0.781277i	$-0.714568\pi$
$631$	−4.97135e12	−1.24837	−0.624184	+	0.781277i	$0.714568\pi$
$632$	0	0
$633$	7.35743e10	0.0182142
$634$	0	0
$635$	−1.32208e12	−0.322684
$636$	0	0
$637$	6.10833e11	0.146992
$638$	0	0
$639$	2.77316e12	0.657993
$640$	0	0
$641$	−2.41181e12	−0.564263	−0.282131	−	0.959376i	$-0.591041\pi$
$641$	−2.41181e12	−0.564263	−0.282131	+	0.959376i	$0.591041\pi$
$642$	0	0
$643$	−6.86804e12	−1.58447	−0.792234	−	0.610217i	$-0.791082\pi$
$643$	−6.86804e12	−1.58447	−0.792234	+	0.610217i	$0.791082\pi$
$644$	0	0
$645$	1.26738e12	0.288328
$646$	0	0
$647$	1.73394e12	0.389013	0.194506	−	0.980901i	$-0.437690\pi$
$647$	1.73394e12	0.389013	0.194506	+	0.980901i	$0.437690\pi$
$648$	0	0
$649$	−7.58322e12	−1.67785
$650$	0	0
$651$	−2.72471e11	−0.0594573
$652$	0	0
$653$	2.86440e10	0.00616488	0.00308244	−	0.999995i	$-0.499019\pi$
$653$	2.86440e10	0.00616488	0.00308244	+	0.999995i	$0.499019\pi$
$654$	0	0
$655$	−3.66878e12	−0.778818
$656$	0	0
$657$	3.60529e12	0.754911
$658$	0	0
$659$	6.31728e12	1.30481	0.652403	−	0.757872i	$-0.273761\pi$
$659$	6.31728e12	1.30481	0.652403	+	0.757872i	$0.273761\pi$
$660$	0	0
$661$	−3.49558e12	−0.712217	−0.356109	−	0.934445i	$-0.615897\pi$
$661$	−3.49558e12	−0.712217	−0.356109	+	0.934445i	$0.615897\pi$
$662$	0	0
$663$	4.79578e12	0.963937
$664$	0	0
$665$	−1.35503e12	−0.268691
$666$	0	0
$667$	3.02717e12	0.592203
$668$	0	0
$669$	−5.32547e11	−0.102788
$670$	0	0
$671$	−2.21939e12	−0.422652
$672$	0	0
$673$	−8.02535e12	−1.50798	−0.753991	−	0.656885i	$-0.771874\pi$
$673$	−8.02535e12	−1.50798	−0.753991	+	0.656885i	$0.771874\pi$
$674$	0	0
$675$	−1.94095e11	−0.0359871
$676$	0	0
$677$	1.17163e12	0.214358	0.107179	−	0.994240i	$-0.465818\pi$
$677$	1.17163e12	0.214358	0.107179	+	0.994240i	$0.465818\pi$
$678$	0	0
$679$	−2.63253e12	−0.475290
$680$	0	0
$681$	3.84596e12	0.685239
$682$	0	0
$683$	−4.69754e12	−0.825995	−0.412997	−	0.910732i	$-0.635518\pi$
$683$	−4.69754e12	−0.825995	−0.412997	+	0.910732i	$0.635518\pi$
$684$	0	0
$685$	−6.32288e12	−1.09726
$686$	0	0
$687$	−1.86584e12	−0.319572
$688$	0	0
$689$	5.98676e12	1.01206
$690$	0	0
$691$	7.83193e12	1.30683	0.653413	−	0.757001i	$-0.273336\pi$
$691$	7.83193e12	1.30683	0.653413	+	0.757001i	$0.273336\pi$
$692$	0	0
$693$	−2.22127e12	−0.365849
$694$	0	0
$695$	1.03701e13	1.68597
$696$	0	0
$697$	−1.15386e13	−1.85185
$698$	0	0
$699$	−2.62150e12	−0.415339
$700$	0	0
$701$	7.13243e12	1.11559	0.557797	−	0.829977i	$-0.311647\pi$
$701$	7.13243e12	1.11559	0.557797	+	0.829977i	$0.311647\pi$
$702$	0	0
$703$	5.19655e12	0.802446
$704$	0	0
$705$	2.26052e12	0.344633
$706$	0	0
$707$	−5.01049e11	−0.0754211
$708$	0	0
$709$	8.65274e12	1.28601	0.643007	−	0.765861i	$-0.277687\pi$
$709$	8.65274e12	1.28601	0.643007	+	0.765861i	$0.277687\pi$
$710$	0	0
$711$	−6.88573e12	−1.01050
$712$	0	0
$713$	8.84475e11	0.128169
$714$	0	0
$715$	−1.04624e13	−1.49711
$716$	0	0
$717$	−2.00689e12	−0.283588
$718$	0	0
$719$	−4.58446e11	−0.0639747	−0.0319873	−	0.999488i	$-0.510184\pi$
$719$	−4.58446e11	−0.0639747	−0.0319873	+	0.999488i	$0.510184\pi$
$720$	0	0
$721$	1.62834e12	0.224408
$722$	0	0
$723$	5.58177e12	0.759713
$724$	0	0
$725$	3.59871e11	0.0483755
$726$	0	0
$727$	2.53850e12	0.337033	0.168517	−	0.985699i	$-0.446102\pi$
$727$	2.53850e12	0.337033	0.168517	+	0.985699i	$0.446102\pi$
$728$	0	0
$729$	3.41521e12	0.447861
$730$	0	0
$731$	6.35141e12	0.822701
$732$	0	0
$733$	1.09361e13	1.39925	0.699624	−	0.714511i	$-0.253351\pi$
$733$	1.09361e13	1.39925	0.699624	+	0.714511i	$0.253351\pi$
$734$	0	0
$735$	6.53690e11	0.0826188
$736$	0	0
$737$	5.56824e12	0.695208
$738$	0	0
$739$	−7.34996e12	−0.906536	−0.453268	−	0.891374i	$-0.649742\pi$
$739$	−7.34996e12	−0.906536	−0.453268	+	0.891374i	$0.649742\pi$
$740$	0	0
$741$	3.34539e12	0.407628
$742$	0	0
$743$	1.60813e12	0.193584	0.0967922	−	0.995305i	$-0.469142\pi$
$743$	1.60813e12	0.193584	0.0967922	+	0.995305i	$0.469142\pi$
$744$	0	0
$745$	6.93848e12	0.825203
$746$	0	0
$747$	−9.10670e12	−1.07009
$748$	0	0
$749$	−1.10336e12	−0.128100
$750$	0	0
$751$	2.95903e12	0.339446	0.169723	−	0.985492i	$-0.445713\pi$
$751$	2.95903e12	0.339446	0.169723	+	0.985492i	$0.445713\pi$
$752$	0	0
$753$	−3.34263e12	−0.378888
$754$	0	0
$755$	1.22976e13	1.37739
$756$	0	0
$757$	−4.83223e12	−0.534830	−0.267415	−	0.963581i	$-0.586170\pi$
$757$	−4.83223e12	−0.534830	−0.267415	+	0.963581i	$0.586170\pi$
$758$	0	0
$759$	−3.42902e12	−0.375043
$760$	0	0
$761$	3.15213e12	0.340701	0.170350	−	0.985384i	$-0.445510\pi$
$761$	3.15213e12	0.340701	0.170350	+	0.985384i	$0.445510\pi$
$762$	0	0
$763$	−1.25113e12	−0.133642
$764$	0	0
$765$	−1.07921e13	−1.13928
$766$	0	0
$767$	1.15856e13	1.20876
$768$	0	0
$769$	−6.87650e12	−0.709086	−0.354543	−	0.935040i	$-0.615364\pi$
$769$	−6.87650e12	−0.709086	−0.354543	+	0.935040i	$0.615364\pi$
$770$	0	0
$771$	6.67118e12	0.679920
$772$	0	0
$773$	1.16083e13	1.16939	0.584696	−	0.811253i	$-0.301214\pi$
$773$	1.16083e13	1.16939	0.584696	+	0.811253i	$0.301214\pi$
$774$	0	0
$775$	1.05147e11	0.0104698
$776$	0	0
$777$	−2.50690e12	−0.246741
$778$	0	0
$779$	−8.04899e12	−0.783110
$780$	0	0
$781$	1.44183e13	1.38670
$782$	0	0
$783$	−1.28259e13	−1.21944
$784$	0	0
$785$	−1.32998e13	−1.25006
$786$	0	0
$787$	2.50005e12	0.232307	0.116154	−	0.993231i	$-0.462943\pi$
$787$	2.50005e12	0.232307	0.116154	+	0.993231i	$0.462943\pi$
$788$	0	0
$789$	1.86855e11	0.0171656
$790$	0	0
$791$	−1.06991e12	−0.0971751
$792$	0	0
$793$	3.39077e12	0.304487
$794$	0	0
$795$	6.40680e12	0.568838
$796$	0	0
$797$	2.77023e12	0.243194	0.121597	−	0.992580i	$-0.461198\pi$
$797$	2.77023e12	0.243194	0.121597	+	0.992580i	$0.461198\pi$
$798$	0	0
$799$	1.13285e13	0.983360
$800$	0	0
$801$	1.96289e12	0.168480
$802$	0	0
$803$	1.87447e13	1.59096
$804$	0	0
$805$	−2.12196e12	−0.178097
$806$	0	0
$807$	−5.13451e12	−0.426156
$808$	0	0
$809$	−2.96640e12	−0.243479	−0.121739	−	0.992562i	$-0.538847\pi$
$809$	−2.96640e12	−0.243479	−0.121739	+	0.992562i	$0.538847\pi$
$810$	0	0
$811$	9.01447e12	0.731722	0.365861	−	0.930669i	$-0.380775\pi$
$811$	9.01447e12	0.731722	0.365861	+	0.930669i	$0.380775\pi$
$812$	0	0
$813$	1.11928e12	0.0898524
$814$	0	0
$815$	2.99946e12	0.238140
$816$	0	0
$817$	4.43055e12	0.347903
$818$	0	0
$819$	3.39364e12	0.263565
$820$	0	0
$821$	1.25176e13	0.961563	0.480782	−	0.876840i	$-0.340353\pi$
$821$	1.25176e13	0.961563	0.480782	+	0.876840i	$0.340353\pi$
$822$	0	0
$823$	1.50348e13	1.14234	0.571172	−	0.820830i	$-0.306489\pi$
$823$	1.50348e13	1.14234	0.571172	+	0.820830i	$0.306489\pi$
$824$	0	0
$825$	−4.07642e11	−0.0306363
$826$	0	0
$827$	−2.31522e13	−1.72115	−0.860574	−	0.509325i	$-0.829895\pi$
$827$	−2.31522e13	−1.72115	−0.860574	+	0.509325i	$0.829895\pi$
$828$	0	0
$829$	4.88071e12	0.358912	0.179456	−	0.983766i	$-0.442566\pi$
$829$	4.88071e12	0.358912	0.179456	+	0.983766i	$0.442566\pi$
$830$	0	0
$831$	−5.99556e12	−0.436139
$832$	0	0
$833$	3.27595e12	0.235741
$834$	0	0
$835$	−1.99034e13	−1.41690
$836$	0	0
$837$	−3.74745e12	−0.263919
$838$	0	0
$839$	1.18015e13	0.822258	0.411129	−	0.911577i	$-0.365135\pi$
$839$	1.18015e13	0.822258	0.411129	+	0.911577i	$0.365135\pi$
$840$	0	0
$841$	9.27330e12	0.639223
$842$	0	0
$843$	7.61492e12	0.519327
$844$	0	0
$845$	8.86708e11	0.0598309
$846$	0	0
$847$	−5.88744e12	−0.393053
$848$	0	0
$849$	−3.84540e12	−0.254014
$850$	0	0
$851$	8.13771e12	0.531887
$852$	0	0
$853$	−2.54707e13	−1.64729	−0.823646	−	0.567104i	$-0.808064\pi$
$853$	−2.54707e13	−1.64729	−0.823646	+	0.567104i	$0.808064\pi$
$854$	0	0
$855$	−7.52824e12	−0.481777
$856$	0	0
$857$	1.26584e13	0.801615	0.400807	−	0.916162i	$-0.368730\pi$
$857$	1.26584e13	0.801615	0.400807	+	0.916162i	$0.368730\pi$
$858$	0	0
$859$	−2.01387e13	−1.26201	−0.631004	−	0.775779i	$-0.717357\pi$
$859$	−2.01387e13	−1.26201	−0.631004	+	0.775779i	$0.717357\pi$
$860$	0	0
$861$	3.88296e12	0.240796
$862$	0	0
$863$	1.44977e13	0.889716	0.444858	−	0.895601i	$-0.353254\pi$
$863$	1.44977e13	0.889716	0.444858	+	0.895601i	$0.353254\pi$
$864$	0	0
$865$	−1.77159e13	−1.07594
$866$	0	0
$867$	1.62750e13	0.978217
$868$	0	0
$869$	−3.58004e13	−2.12961
$870$	0	0
$871$	−8.50712e12	−0.500842
$872$	0	0
$873$	−1.46257e13	−0.852221
$874$	0	0
$875$	6.42411e12	0.370490
$876$	0	0
$877$	1.83297e13	1.04630	0.523150	−	0.852241i	$-0.324757\pi$
$877$	1.83297e13	1.04630	0.523150	+	0.852241i	$0.324757\pi$
$878$	0	0
$879$	−5.63453e12	−0.318352
$880$	0	0
$881$	4.57369e12	0.255785	0.127892	−	0.991788i	$-0.459179\pi$
$881$	4.57369e12	0.255785	0.127892	+	0.991788i	$0.459179\pi$
$882$	0	0
$883$	−3.07854e13	−1.70420	−0.852101	−	0.523377i	$-0.824672\pi$
$883$	−3.07854e13	−1.70420	−0.852101	+	0.523377i	$0.824672\pi$
$884$	0	0
$885$	1.23984e13	0.679395
$886$	0	0
$887$	2.58803e13	1.40382	0.701911	−	0.712264i	$-0.252330\pi$
$887$	2.58803e13	1.40382	0.701911	+	0.712264i	$0.252330\pi$
$888$	0	0
$889$	2.22963e12	0.119722
$890$	0	0
$891$	−3.68114e12	−0.195674
$892$	0	0
$893$	7.90242e12	0.415842
$894$	0	0
$895$	−1.03955e13	−0.541556
$896$	0	0
$897$	5.23882e12	0.270189
$898$	0	0
$899$	6.94814e12	0.354773
$900$	0	0
$901$	3.21075e13	1.62310
$902$	0	0
$903$	−2.13737e12	−0.106975
$904$	0	0
$905$	−1.80915e13	−0.896510
$906$	0	0
$907$	−9.64118e12	−0.473040	−0.236520	−	0.971627i	$-0.576007\pi$
$907$	−9.64118e12	−0.473040	−0.236520	+	0.971627i	$0.576007\pi$
$908$	0	0
$909$	−2.78371e12	−0.135234
$910$	0	0
$911$	−7.28746e12	−0.350545	−0.175272	−	0.984520i	$-0.556081\pi$
$911$	−7.28746e12	−0.350545	−0.175272	+	0.984520i	$0.556081\pi$
$912$	0	0
$913$	−4.73478e13	−2.25518
$914$	0	0
$915$	3.62867e12	0.171140
$916$	0	0
$917$	6.18722e12	0.288957
$918$	0	0
$919$	−2.73276e13	−1.26381	−0.631906	−	0.775045i	$-0.717727\pi$
$919$	−2.73276e13	−1.26381	−0.631906	+	0.775045i	$0.717727\pi$
$920$	0	0
$921$	−1.03934e13	−0.475981
$922$	0	0
$923$	−2.20281e13	−0.999011
$924$	0	0
$925$	9.67413e11	0.0434484
$926$	0	0
$927$	9.04668e12	0.402375
$928$	0	0
$929$	−7.21805e12	−0.317943	−0.158971	−	0.987283i	$-0.550818\pi$
$929$	−7.21805e12	−0.317943	−0.158971	+	0.987283i	$0.550818\pi$
$930$	0	0
$931$	2.28520e12	0.0996897
$932$	0	0
$933$	−6.78365e12	−0.293087
$934$	0	0
$935$	−5.61106e13	−2.40100
$936$	0	0
$937$	1.41988e13	0.601761	0.300880	−	0.953662i	$-0.402719\pi$
$937$	1.41988e13	0.601761	0.300880	+	0.953662i	$0.402719\pi$
$938$	0	0
$939$	−9.69658e12	−0.407027
$940$	0	0
$941$	−2.53242e13	−1.05289	−0.526445	−	0.850209i	$-0.676475\pi$
$941$	−2.53242e13	−1.05289	−0.526445	+	0.850209i	$0.676475\pi$
$942$	0	0
$943$	−1.26046e13	−0.519070
$944$	0	0
$945$	8.99058e12	0.366728
$946$	0	0
$947$	1.37160e13	0.554182	0.277091	−	0.960844i	$-0.410630\pi$
$947$	1.37160e13	0.554182	0.277091	+	0.960844i	$0.410630\pi$
$948$	0	0
$949$	−2.86380e13	−1.14616
$950$	0	0
$951$	−1.70918e13	−0.677604
$952$	0	0
$953$	2.44096e13	0.958610	0.479305	−	0.877648i	$-0.340889\pi$
$953$	2.44096e13	0.958610	0.479305	+	0.877648i	$0.340889\pi$
$954$	0	0
$955$	−2.29994e13	−0.894750
$956$	0	0
$957$	−2.69372e13	−1.03812
$958$	0	0
$959$	1.06632e13	0.407104
$960$	0	0
$961$	−2.44095e13	−0.923218
$962$	0	0
$963$	−6.13001e12	−0.229690
$964$	0	0
$965$	−2.17600e13	−0.807765
$966$	0	0
$967$	4.19850e13	1.54410	0.772049	−	0.635563i	$-0.219232\pi$
$967$	4.19850e13	1.54410	0.772049	+	0.635563i	$0.219232\pi$
$968$	0	0
$969$	1.79416e13	0.653738
$970$	0	0
$971$	−3.13254e12	−0.113086	−0.0565432	−	0.998400i	$-0.518008\pi$
$971$	−3.13254e12	−0.113086	−0.0565432	+	0.998400i	$0.518008\pi$
$972$	0	0
$973$	−1.74886e13	−0.625529
$974$	0	0
$975$	6.22793e11	0.0220710
$976$	0	0
$977$	4.53124e13	1.59108	0.795539	−	0.605902i	$-0.207188\pi$
$977$	4.53124e13	1.59108	0.795539	+	0.605902i	$0.207188\pi$
$978$	0	0
$979$	1.02055e13	0.355068
$980$	0	0
$981$	−6.95097e12	−0.239626
$982$	0	0
$983$	2.54481e13	0.869290	0.434645	−	0.900602i	$-0.356874\pi$
$983$	2.54481e13	0.869290	0.434645	+	0.900602i	$0.356874\pi$
$984$	0	0
$985$	3.32984e13	1.12709
$986$	0	0
$987$	−3.81225e12	−0.127866
$988$	0	0
$989$	6.93817e12	0.230601
$990$	0	0
$991$	5.41691e13	1.78410	0.892052	−	0.451933i	$-0.149266\pi$
$991$	5.41691e13	1.78410	0.892052	+	0.451933i	$0.149266\pi$
$992$	0	0
$993$	4.36465e12	0.142455
$994$	0	0
$995$	3.61065e13	1.16783
$996$	0	0
$997$	−4.29329e13	−1.37614	−0.688069	−	0.725646i	$-0.741541\pi$
$997$	−4.29329e13	−1.37614	−0.688069	+	0.725646i	$0.741541\pi$
$998$	0	0
$999$	−3.44788e13	−1.09524

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.a.h.1.3		3
4.3	odd	2		7.10.a.b.1.1	✓	3
12.11	even	2		63.10.a.e.1.3		3
20.3	even	4		175.10.b.d.99.5		6
20.7	even	4		175.10.b.d.99.2		6
20.19	odd	2		175.10.a.d.1.3		3
28.3	even	6		49.10.c.e.30.3		6
28.11	odd	6		49.10.c.d.30.3		6
28.19	even	6		49.10.c.e.18.3		6
28.23	odd	6		49.10.c.d.18.3		6
28.27	even	2		49.10.a.c.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.a.b.1.1	✓	3	4.3	odd	2
49.10.a.c.1.1		3	28.27	even	2
49.10.c.d.18.3		6	28.23	odd	6
49.10.c.d.30.3		6	28.11	odd	6
49.10.c.e.18.3		6	28.19	even	6
49.10.c.e.30.3		6	28.3	even	6
63.10.a.e.1.3		3	12.11	even	2
112.10.a.h.1.3		3	1.1	even	1	trivial
175.10.a.d.1.3		3	20.19	odd	2
175.10.b.d.99.2		6	20.7	even	4
175.10.b.d.99.5		6	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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

\(f(q)\)	\(=\)	\(q+79.6469 q^{3} +1423.70 q^{5} -2401.00 q^{7} -13339.4 q^{9} -69354.4 q^{11} +105959. q^{13} +113393. q^{15} +568267. q^{17} +396405. q^{19} -191232. q^{21} +620765. q^{23} +73796.7 q^{25} -2.63013e6 q^{27} +4.87652e6 q^{29} +1.42482e6 q^{31} -5.52386e6 q^{33} -3.41830e6 q^{35} +1.31092e7 q^{37} +8.43931e6 q^{39} -2.03049e7 q^{41} +1.11768e7 q^{43} -1.89913e7 q^{45} +1.99352e7 q^{47} +5.76480e6 q^{49} +4.52607e7 q^{51} +5.65007e7 q^{53} -9.87398e7 q^{55} +3.15725e7 q^{57} +1.09340e8 q^{59} +3.20008e7 q^{61} +3.20278e7 q^{63} +1.50854e8 q^{65} -8.02869e7 q^{67} +4.94420e7 q^{69} -2.07893e8 q^{71} -2.70274e8 q^{73} +5.87768e6 q^{75} +1.66520e8 q^{77} +5.16196e8 q^{79} +5.30772e7 q^{81} +6.82693e8 q^{83} +8.09042e8 q^{85} +3.88400e8 q^{87} -1.47150e8 q^{89} -2.54408e8 q^{91} +1.13482e8 q^{93} +5.64362e8 q^{95} +1.09643e9 q^{97} +9.25144e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 84 q^{3} + 1554 q^{5} - 7203 q^{7} - 26001 q^{9} + 3444 q^{11} - 19782 q^{13} - 200304 q^{15} + 1016694 q^{17} - 222852 q^{19} + 201684 q^{21} - 1885632 q^{23} + 3073221 q^{25} - 551880 q^{27} + 4081818 q^{29}+ \cdots + 1900979172 q^{99}+O(q^{100}) \) 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

Modular forms

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Database

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