LMFDB - Embedded newform 112.10.a.d.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11209}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 2802 $ x^2 - x - 2802
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$53.4363$ 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)

$f(q)$	$=$	$q-70.8726 q^{3} +2788.58 q^{5} +2401.00 q^{7} -14660.1 q^{9} +O(q^{10})$	$q-70.8726 q^{3} +2788.58 q^{5} +2401.00 q^{7} -14660.1 q^{9} -50462.8 q^{11} +52172.1 q^{13} -197634. q^{15} -279343. q^{17} -799356. q^{19} -170165. q^{21} -1.91973e6 q^{23} +5.82305e6 q^{25} +2.43398e6 q^{27} -3.59558e6 q^{29} +3.40682e6 q^{31} +3.57643e6 q^{33} +6.69538e6 q^{35} +7.70232e6 q^{37} -3.69757e6 q^{39} -1.66041e7 q^{41} +5.41081e6 q^{43} -4.08808e7 q^{45} +1.84584e7 q^{47} +5.76480e6 q^{49} +1.97978e7 q^{51} -8.32973e7 q^{53} -1.40720e8 q^{55} +5.66524e7 q^{57} -1.45420e8 q^{59} -1.82139e8 q^{61} -3.51989e7 q^{63} +1.45486e8 q^{65} +6.88897e7 q^{67} +1.36056e8 q^{69} -1.18923e8 q^{71} +4.56639e7 q^{73} -4.12694e8 q^{75} -1.21161e8 q^{77} +3.93842e8 q^{79} +1.16052e8 q^{81} +1.00224e8 q^{83} -7.78971e8 q^{85} +2.54828e8 q^{87} -5.65770e8 q^{89} +1.25265e8 q^{91} -2.41450e8 q^{93} -2.22907e9 q^{95} -1.82831e8 q^{97} +7.39789e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 70 q^{3} + 1554 q^{5} + 4802 q^{7} - 14498 q^{9}+O(q^{10}) $ 2 * q + 70 * q^3 + 1554 * q^5 + 4802 * q^7 - 14498 * q^9	$ 2 q + 70 q^{3} + 1554 q^{5} + 4802 q^{7} - 14498 q^{9} - 62388 q^{11} + 122766 q^{13} - 371552 q^{15} + 73584 q^{17} - 1171198 q^{19} + 168070 q^{21} - 2262384 q^{23} + 5394106 q^{25} - 315980 q^{27} - 1923360 q^{29} - 2977884 q^{31} + 1896496 q^{33} + 3731154 q^{35} - 13418528 q^{37} + 6247176 q^{39} - 36367800 q^{41} + 21964916 q^{43} - 41080886 q^{45} + 1362732 q^{47} + 11529602 q^{49} + 69515588 q^{51} - 17898612 q^{53} - 125996920 q^{55} + 4270012 q^{57} - 224710542 q^{59} - 85847118 q^{61} - 34809698 q^{63} + 58332228 q^{65} - 179568872 q^{67} + 87785824 q^{69} - 231378168 q^{71} + 88098332 q^{73} - 473120158 q^{75} - 149793588 q^{77} + 184274184 q^{79} - 274532642 q^{81} - 624641094 q^{83} - 1214688044 q^{85} + 490397404 q^{87} - 1574777148 q^{89} + 294761166 q^{91} - 1140879096 q^{93} - 1769997744 q^{95} + 213665984 q^{97} + 737855932 q^{99}+O(q^{100}) $ 2 * q + 70 * q^3 + 1554 * q^5 + 4802 * q^7 - 14498 * q^9 - 62388 * q^11 + 122766 * q^13 - 371552 * q^15 + 73584 * q^17 - 1171198 * q^19 + 168070 * q^21 - 2262384 * q^23 + 5394106 * q^25 - 315980 * q^27 - 1923360 * q^29 - 2977884 * q^31 + 1896496 * q^33 + 3731154 * q^35 - 13418528 * q^37 + 6247176 * q^39 - 36367800 * q^41 + 21964916 * q^43 - 41080886 * q^45 + 1362732 * q^47 + 11529602 * q^49 + 69515588 * q^51 - 17898612 * q^53 - 125996920 * q^55 + 4270012 * q^57 - 224710542 * q^59 - 85847118 * q^61 - 34809698 * q^63 + 58332228 * q^65 - 179568872 * q^67 + 87785824 * q^69 - 231378168 * q^71 + 88098332 * q^73 - 473120158 * q^75 - 149793588 * q^77 + 184274184 * q^79 - 274532642 * q^81 - 624641094 * q^83 - 1214688044 * q^85 + 490397404 * q^87 - 1574777148 * q^89 + 294761166 * q^91 - 1140879096 * q^93 - 1769997744 * q^95 + 213665984 * q^97 + 737855932 * 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
$3$	−70.8726	−0.505164	−0.252582	−	0.967575i	$-0.581280\pi$
$3$	−70.8726	−0.505164	−0.252582	+	0.967575i	$0.581280\pi$
$4$	0	0
$5$	2788.58	1.99534	0.997672	−	0.0681914i	$-0.0217229\pi$
$5$	2788.58	1.99534	0.997672	+	0.0681914i	$0.0217229\pi$
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	0	0
$9$	−14660.1	−0.744809
$10$	0	0
$11$	−50462.8	−1.03921	−0.519606	−	0.854406i	$-0.673921\pi$
$11$	−50462.8	−1.03921	−0.519606	+	0.854406i	$0.673921\pi$
$12$	0	0
$13$	52172.1	0.506633	0.253316	−	0.967384i	$-0.418479\pi$
$13$	52172.1	0.506633	0.253316	+	0.967384i	$0.418479\pi$
$14$	0	0
$15$	−197634.	−1.00798
$16$	0	0
$17$	−279343.	−0.811182	−0.405591	−	0.914055i	$-0.632934\pi$
$17$	−279343.	−0.811182	−0.405591	+	0.914055i	$0.632934\pi$
$18$	0	0
$19$	−799356.	−1.40718	−0.703589	−	0.710607i	$-0.748420\pi$
$19$	−799356.	−1.40718	−0.703589	+	0.710607i	$0.748420\pi$
$20$	0	0
$21$	−170165.	−0.190934
$22$	0	0
$23$	−1.91973e6	−1.43042	−0.715212	−	0.698907i	$-0.753670\pi$
$23$	−1.91973e6	−1.43042	−0.715212	+	0.698907i	$0.753670\pi$
$24$	0	0
$25$	5.82305e6	2.98140
$26$	0	0
$27$	2.43398e6	0.881415
$28$	0	0
$29$	−3.59558e6	−0.944012	−0.472006	−	0.881595i	$-0.656470\pi$
$29$	−3.59558e6	−0.944012	−0.472006	+	0.881595i	$0.656470\pi$
$30$	0	0
$31$	3.40682e6	0.662554	0.331277	−	0.943534i	$-0.392521\pi$
$31$	3.40682e6	0.662554	0.331277	+	0.943534i	$0.392521\pi$
$32$	0	0
$33$	3.57643e6	0.524973
$34$	0	0
$35$	6.69538e6	0.754169
$36$	0	0
$37$	7.70232e6	0.675638	0.337819	−	0.941211i	$-0.390311\pi$
$37$	7.70232e6	0.675638	0.337819	+	0.941211i	$0.390311\pi$
$38$	0	0
$39$	−3.69757e6	−0.255933
$40$	0	0
$41$	−1.66041e7	−0.917671	−0.458836	−	0.888521i	$-0.651733\pi$
$41$	−1.66041e7	−0.917671	−0.458836	+	0.888521i	$0.651733\pi$
$42$	0	0
$43$	5.41081e6	0.241354	0.120677	−	0.992692i	$-0.461494\pi$
$43$	5.41081e6	0.241354	0.120677	+	0.992692i	$0.461494\pi$
$44$	0	0
$45$	−4.08808e7	−1.48615
$46$	0	0
$47$	1.84584e7	0.551765	0.275883	−	0.961191i	$-0.411030\pi$
$47$	1.84584e7	0.551765	0.275883	+	0.961191i	$0.411030\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	0	0
$51$	1.97978e7	0.409780
$52$	0	0
$53$	−8.32973e7	−1.45007	−0.725036	−	0.688711i	$-0.758177\pi$
$53$	−8.32973e7	−1.45007	−0.725036	+	0.688711i	$0.758177\pi$
$54$	0	0
$55$	−1.40720e8	−2.07359
$56$	0	0
$57$	5.66524e7	0.710856
$58$	0	0
$59$	−1.45420e8	−1.56239	−0.781196	−	0.624286i	$-0.785390\pi$
$59$	−1.45420e8	−1.56239	−0.781196	+	0.624286i	$0.785390\pi$
$60$	0	0
$61$	−1.82139e8	−1.68430	−0.842148	−	0.539246i	$-0.818709\pi$
$61$	−1.82139e8	−1.68430	−0.842148	+	0.539246i	$0.818709\pi$
$62$	0	0
$63$	−3.51989e7	−0.281511
$64$	0	0
$65$	1.45486e8	1.01091
$66$	0	0
$67$	6.88897e7	0.417655	0.208828	−	0.977952i	$-0.433035\pi$
$67$	6.88897e7	0.417655	0.208828	+	0.977952i	$0.433035\pi$
$68$	0	0
$69$	1.36056e8	0.722599
$70$	0	0
$71$	−1.18923e8	−0.555398	−0.277699	−	0.960668i	$-0.589572\pi$
$71$	−1.18923e8	−0.555398	−0.277699	+	0.960668i	$0.589572\pi$
$72$	0	0
$73$	4.56639e7	0.188200	0.0941002	−	0.995563i	$-0.470003\pi$
$73$	4.56639e7	0.188200	0.0941002	+	0.995563i	$0.470003\pi$
$74$	0	0
$75$	−4.12694e8	−1.50610
$76$	0	0
$77$	−1.21161e8	−0.392785
$78$	0	0
$79$	3.93842e8	1.13763	0.568814	−	0.822466i	$-0.307402\pi$
$79$	3.93842e8	1.13763	0.568814	+	0.822466i	$0.307402\pi$
$80$	0	0
$81$	1.16052e8	0.299550
$82$	0	0
$83$	1.00224e8	0.231804	0.115902	−	0.993261i	$-0.463024\pi$
$83$	1.00224e8	0.231804	0.115902	+	0.993261i	$0.463024\pi$
$84$	0	0
$85$	−7.78971e8	−1.61859
$86$	0	0
$87$	2.54828e8	0.476881
$88$	0	0
$89$	−5.65770e8	−0.955840	−0.477920	−	0.878403i	$-0.658609\pi$
$89$	−5.65770e8	−0.955840	−0.477920	+	0.878403i	$0.658609\pi$
$90$	0	0
$91$	1.25265e8	0.191489
$92$	0	0
$93$	−2.41450e8	−0.334698
$94$	0	0
$95$	−2.22907e9	−2.80780
$96$	0	0
$97$	−1.82831e8	−0.209690	−0.104845	−	0.994489i	$-0.533435\pi$
$97$	−1.82831e8	−0.209690	−0.104845	+	0.994489i	$0.533435\pi$
$98$	0	0
$99$	7.39789e8	0.774015
$100$	0	0
$101$	−1.25737e9	−1.20232	−0.601158	−	0.799131i	$-0.705294\pi$
$101$	−1.25737e9	−1.20232	−0.601158	+	0.799131i	$0.705294\pi$
$102$	0	0
$103$	−5.45347e8	−0.477425	−0.238713	−	0.971090i	$-0.576725\pi$
$103$	−5.45347e8	−0.477425	−0.238713	+	0.971090i	$0.576725\pi$
$104$	0	0
$105$	−4.74519e8	−0.380979
$106$	0	0
$107$	1.86684e9	1.37683	0.688415	−	0.725317i	$-0.258307\pi$
$107$	1.86684e9	1.37683	0.688415	+	0.725317i	$0.258307\pi$
$108$	0	0
$109$	−3.97024e8	−0.269400	−0.134700	−	0.990886i	$-0.543007\pi$
$109$	−3.97024e8	−0.269400	−0.134700	+	0.990886i	$0.543007\pi$
$110$	0	0
$111$	−5.45883e8	−0.341308
$112$	0	0
$113$	−3.15703e8	−0.182148	−0.0910742	−	0.995844i	$-0.529030\pi$
$113$	−3.15703e8	−0.182148	−0.0910742	+	0.995844i	$0.529030\pi$
$114$	0	0
$115$	−5.35332e9	−2.85419
$116$	0	0
$117$	−7.64847e8	−0.377345
$118$	0	0
$119$	−6.70704e8	−0.306598
$120$	0	0
$121$	1.88547e8	0.0799624
$122$	0	0
$123$	1.17677e9	0.463575
$124$	0	0
$125$	1.07916e10	3.95358
$126$	0	0
$127$	2.21139e8	0.0754309	0.0377155	−	0.999289i	$-0.487992\pi$
$127$	2.21139e8	0.0754309	0.0377155	+	0.999289i	$0.487992\pi$
$128$	0	0
$129$	−3.83478e8	−0.121923
$130$	0	0
$131$	1.29798e9	0.385076	0.192538	−	0.981290i	$-0.438328\pi$
$131$	1.29798e9	0.385076	0.192538	+	0.981290i	$0.438328\pi$
$132$	0	0
$133$	−1.91925e9	−0.531863
$134$	0	0
$135$	6.78735e9	1.75873
$136$	0	0
$137$	−1.47759e9	−0.358354	−0.179177	−	0.983817i	$-0.557343\pi$
$137$	−1.47759e9	−0.358354	−0.179177	+	0.983817i	$0.557343\pi$
$138$	0	0
$139$	−2.47776e9	−0.562979	−0.281490	−	0.959564i	$-0.590829\pi$
$139$	−2.47776e9	−0.562979	−0.281490	+	0.959564i	$0.590829\pi$
$140$	0	0
$141$	−1.30820e9	−0.278732
$142$	0	0
$143$	−2.63275e9	−0.526499
$144$	0	0
$145$	−1.00266e10	−1.88363
$146$	0	0
$147$	−4.08566e8	−0.0721663
$148$	0	0
$149$	4.39299e9	0.730167	0.365083	−	0.930975i	$-0.381041\pi$
$149$	4.39299e9	0.730167	0.365083	+	0.930975i	$0.381041\pi$
$150$	0	0
$151$	−6.62801e9	−1.03750	−0.518749	−	0.854927i	$-0.673602\pi$
$151$	−6.62801e9	−1.03750	−0.518749	+	0.854927i	$0.673602\pi$
$152$	0	0
$153$	4.09520e9	0.604176
$154$	0	0
$155$	9.50018e9	1.32202
$156$	0	0
$157$	−6.08159e9	−0.798856	−0.399428	−	0.916765i	$-0.630791\pi$
$157$	−6.08159e9	−0.798856	−0.399428	+	0.916765i	$0.630791\pi$
$158$	0	0
$159$	5.90349e9	0.732524
$160$	0	0
$161$	−4.60927e9	−0.540650
$162$	0	0
$163$	1.61851e10	1.79586	0.897928	−	0.440143i	$-0.145072\pi$
$163$	1.61851e10	1.79586	0.897928	+	0.440143i	$0.145072\pi$
$164$	0	0
$165$	9.97315e9	1.04750
$166$	0	0
$167$	−1.09231e10	−1.08673	−0.543365	−	0.839497i	$-0.682850\pi$
$167$	−1.09231e10	−1.08673	−0.543365	+	0.839497i	$0.682850\pi$
$168$	0	0
$169$	−7.88257e9	−0.743323
$170$	0	0
$171$	1.17186e10	1.04808
$172$	0	0
$173$	2.12555e10	1.80412	0.902059	−	0.431613i	$-0.142055\pi$
$173$	2.12555e10	1.80412	0.902059	+	0.431613i	$0.142055\pi$
$174$	0	0
$175$	1.39811e10	1.12686
$176$	0	0
$177$	1.03063e10	0.789265
$178$	0	0
$179$	−1.94715e10	−1.41762	−0.708812	−	0.705398i	$-0.750768\pi$
$179$	−1.94715e10	−1.41762	−0.708812	+	0.705398i	$0.750768\pi$
$180$	0	0
$181$	6.90903e9	0.478480	0.239240	−	0.970961i	$-0.423102\pi$
$181$	6.90903e9	0.478480	0.239240	+	0.970961i	$0.423102\pi$
$182$	0	0
$183$	1.29087e10	0.850846
$184$	0	0
$185$	2.14785e10	1.34813
$186$	0	0
$187$	1.40965e10	0.842991
$188$	0	0
$189$	5.84399e9	0.333144
$190$	0	0
$191$	−4.14517e8	−0.0225368	−0.0112684	−	0.999937i	$-0.503587\pi$
$191$	−4.14517e8	−0.0225368	−0.0112684	+	0.999937i	$0.503587\pi$
$192$	0	0
$193$	1.11524e10	0.578574	0.289287	−	0.957242i	$-0.406582\pi$
$193$	1.11524e10	0.578574	0.289287	+	0.957242i	$0.406582\pi$
$194$	0	0
$195$	−1.03110e10	−0.510674
$196$	0	0
$197$	−1.91105e9	−0.0904013	−0.0452006	−	0.998978i	$-0.514393\pi$
$197$	−1.91105e9	−0.0904013	−0.0452006	+	0.998978i	$0.514393\pi$
$198$	0	0
$199$	−1.96216e10	−0.886942	−0.443471	−	0.896289i	$-0.646253\pi$
$199$	−1.96216e10	−0.886942	−0.443471	+	0.896289i	$0.646253\pi$
$200$	0	0
$201$	−4.88239e9	−0.210984
$202$	0	0
$203$	−8.63298e9	−0.356803
$204$	0	0
$205$	−4.63018e10	−1.83107
$206$	0	0
$207$	2.81434e10	1.06539
$208$	0	0
$209$	4.03377e10	1.46236
$210$	0	0
$211$	−5.53204e10	−1.92138	−0.960692	−	0.277617i	$-0.910455\pi$
$211$	−5.53204e10	−1.92138	−0.960692	+	0.277617i	$0.910455\pi$
$212$	0	0
$213$	8.42840e9	0.280567
$214$	0	0
$215$	1.50885e10	0.481584
$216$	0	0
$217$	8.17977e9	0.250422
$218$	0	0
$219$	−3.23632e9	−0.0950721
$220$	0	0
$221$	−1.45739e10	−0.410971
$222$	0	0
$223$	6.55585e9	0.177524	0.0887619	−	0.996053i	$-0.471709\pi$
$223$	6.55585e9	0.177524	0.0887619	+	0.996053i	$0.471709\pi$
$224$	0	0
$225$	−8.53663e10	−2.22057
$226$	0	0
$227$	1.33363e10	0.333365	0.166682	−	0.986011i	$-0.446695\pi$
$227$	1.33363e10	0.333365	0.166682	+	0.986011i	$0.446695\pi$
$228$	0	0
$229$	−3.91393e10	−0.940489	−0.470245	−	0.882536i	$-0.655834\pi$
$229$	−3.91393e10	−0.940489	−0.470245	+	0.882536i	$0.655834\pi$
$230$	0	0
$231$	8.58700e9	0.198421
$232$	0	0
$233$	−4.66798e10	−1.03759	−0.518797	−	0.854897i	$-0.673620\pi$
$233$	−4.66798e10	−1.03759	−0.518797	+	0.854897i	$0.673620\pi$
$234$	0	0
$235$	5.14728e10	1.10096
$236$	0	0
$237$	−2.79126e10	−0.574689
$238$	0	0
$239$	−1.83798e10	−0.364377	−0.182189	−	0.983264i	$-0.558318\pi$
$239$	−1.83798e10	−0.364377	−0.182189	+	0.983264i	$0.558318\pi$
$240$	0	0
$241$	2.06244e10	0.393827	0.196914	−	0.980421i	$-0.436908\pi$
$241$	2.06244e10	0.393827	0.196914	+	0.980421i	$0.436908\pi$
$242$	0	0
$243$	−5.61330e10	−1.03274
$244$	0	0
$245$	1.60756e10	0.285049
$246$	0	0
$247$	−4.17041e10	−0.712922
$248$	0	0
$249$	−7.10315e9	−0.117099
$250$	0	0
$251$	2.53048e10	0.402413	0.201206	−	0.979549i	$-0.435514\pi$
$251$	2.53048e10	0.402413	0.201206	+	0.979549i	$0.435514\pi$
$252$	0	0
$253$	9.68750e10	1.48652
$254$	0	0
$255$	5.52077e10	0.817653
$256$	0	0
$257$	1.21286e11	1.73425	0.867123	−	0.498093i	$-0.165966\pi$
$257$	1.21286e11	1.73425	0.867123	+	0.498093i	$0.165966\pi$
$258$	0	0
$259$	1.84933e10	0.255367
$260$	0	0
$261$	5.27115e10	0.703109
$262$	0	0
$263$	−1.02516e11	−1.32127	−0.660635	−	0.750707i	$-0.729713\pi$
$263$	−1.02516e11	−1.32127	−0.660635	+	0.750707i	$0.729713\pi$
$264$	0	0
$265$	−2.32281e11	−2.89339
$266$	0	0
$267$	4.00976e10	0.482856
$268$	0	0
$269$	−2.36100e10	−0.274922	−0.137461	−	0.990507i	$-0.543894\pi$
$269$	−2.36100e10	−0.274922	−0.137461	+	0.990507i	$0.543894\pi$
$270$	0	0
$271$	1.38680e11	1.56189	0.780946	−	0.624598i	$-0.214737\pi$
$271$	1.38680e11	1.56189	0.780946	+	0.624598i	$0.214737\pi$
$272$	0	0
$273$	−8.87786e9	−0.0967334
$274$	0	0
$275$	−2.93847e11	−3.09831
$276$	0	0
$277$	1.24410e11	1.26969	0.634843	−	0.772641i	$-0.281065\pi$
$277$	1.24410e11	1.26969	0.634843	+	0.772641i	$0.281065\pi$
$278$	0	0
$279$	−4.99442e10	−0.493476
$280$	0	0
$281$	−1.00922e11	−0.965622	−0.482811	−	0.875725i	$-0.660384\pi$
$281$	−1.00922e11	−0.965622	−0.482811	+	0.875725i	$0.660384\pi$
$282$	0	0
$283$	6.95077e10	0.644161	0.322080	−	0.946712i	$-0.395618\pi$
$283$	6.95077e10	0.644161	0.322080	+	0.946712i	$0.395618\pi$
$284$	0	0
$285$	1.57980e11	1.41840
$286$	0	0
$287$	−3.98664e10	−0.346847
$288$	0	0
$289$	−4.05551e10	−0.341984
$290$	0	0
$291$	1.29577e10	0.105928
$292$	0	0
$293$	−3.03850e9	−0.0240854	−0.0120427	−	0.999927i	$-0.503833\pi$
$293$	−3.03850e9	−0.0240854	−0.0120427	+	0.999927i	$0.503833\pi$
$294$	0	0
$295$	−4.05515e11	−3.11751
$296$	0	0
$297$	−1.22826e11	−0.915977
$298$	0	0
$299$	−1.00156e11	−0.724700
$300$	0	0
$301$	1.29914e10	0.0912232
$302$	0	0
$303$	8.91133e10	0.607366
$304$	0	0
$305$	−5.07909e11	−3.36075
$306$	0	0
$307$	1.88388e11	1.21041	0.605203	−	0.796071i	$-0.293092\pi$
$307$	1.88388e11	1.21041	0.605203	+	0.796071i	$0.293092\pi$
$308$	0	0
$309$	3.86501e10	0.241178
$310$	0	0
$311$	1.94821e10	0.118090	0.0590452	−	0.998255i	$-0.481194\pi$
$311$	1.94821e10	0.118090	0.0590452	+	0.998255i	$0.481194\pi$
$312$	0	0
$313$	−1.63467e11	−0.962677	−0.481338	−	0.876535i	$-0.659849\pi$
$313$	−1.63467e11	−0.962677	−0.481338	+	0.876535i	$0.659849\pi$
$314$	0	0
$315$	−9.81548e10	−0.561712
$316$	0	0
$317$	2.17518e11	1.20984	0.604922	−	0.796285i	$-0.293204\pi$
$317$	2.17518e11	1.20984	0.604922	+	0.796285i	$0.293204\pi$
$318$	0	0
$319$	1.81443e11	0.981029
$320$	0	0
$321$	−1.32308e11	−0.695525
$322$	0	0
$323$	2.23295e11	1.14148
$324$	0	0
$325$	3.03800e11	1.51047
$326$	0	0
$327$	2.81381e10	0.136091
$328$	0	0
$329$	4.43187e10	0.208548
$330$	0	0
$331$	1.93724e11	0.887068	0.443534	−	0.896257i	$-0.353724\pi$
$331$	1.93724e11	0.887068	0.443534	+	0.896257i	$0.353724\pi$
$332$	0	0
$333$	−1.12917e11	−0.503221
$334$	0	0
$335$	1.92104e11	0.833366
$336$	0	0
$337$	3.54200e11	1.49594	0.747970	−	0.663732i	$-0.231029\pi$
$337$	3.54200e11	1.49594	0.747970	+	0.663732i	$0.231029\pi$
$338$	0	0
$339$	2.23747e10	0.0920149
$340$	0	0
$341$	−1.71918e11	−0.688534
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	0	0
$345$	3.79404e11	1.44183
$346$	0	0
$347$	1.24036e11	0.459267	0.229634	−	0.973277i	$-0.426247\pi$
$347$	1.24036e11	0.459267	0.229634	+	0.973277i	$0.426247\pi$
$348$	0	0
$349$	9.16465e10	0.330675	0.165337	−	0.986237i	$-0.447129\pi$
$349$	9.16465e10	0.330675	0.165337	+	0.986237i	$0.447129\pi$
$350$	0	0
$351$	1.26986e11	0.446554
$352$	0	0
$353$	−2.06271e11	−0.707051	−0.353526	−	0.935425i	$-0.615017\pi$
$353$	−2.06271e11	−0.707051	−0.353526	+	0.935425i	$0.615017\pi$
$354$	0	0
$355$	−3.31627e11	−1.10821
$356$	0	0
$357$	4.75345e10	0.154882
$358$	0	0
$359$	5.77558e11	1.83515	0.917574	−	0.397566i	$-0.130145\pi$
$359$	5.77558e11	1.83515	0.917574	+	0.397566i	$0.130145\pi$
$360$	0	0
$361$	3.16282e11	0.980148
$362$	0	0
$363$	−1.33628e10	−0.0403941
$364$	0	0
$365$	1.27337e11	0.375525
$366$	0	0
$367$	−3.15091e11	−0.906649	−0.453325	−	0.891345i	$-0.649762\pi$
$367$	−3.15091e11	−0.906649	−0.453325	+	0.891345i	$0.649762\pi$
$368$	0	0
$369$	2.43417e11	0.683490
$370$	0	0
$371$	−1.99997e11	−0.548075
$372$	0	0
$373$	5.06915e11	1.35596	0.677978	−	0.735082i	$-0.262857\pi$
$373$	5.06915e11	1.35596	0.677978	+	0.735082i	$0.262857\pi$
$374$	0	0
$375$	−7.64827e11	−1.99720
$376$	0	0
$377$	−1.87589e11	−0.478267
$378$	0	0
$379$	2.56015e11	0.637366	0.318683	−	0.947861i	$-0.396759\pi$
$379$	2.56015e11	0.637366	0.318683	+	0.947861i	$0.396759\pi$
$380$	0	0
$381$	−1.56727e10	−0.0381050
$382$	0	0
$383$	−3.42881e11	−0.814235	−0.407117	−	0.913376i	$-0.633466\pi$
$383$	−3.42881e11	−0.814235	−0.407117	+	0.913376i	$0.633466\pi$
$384$	0	0
$385$	−3.37868e11	−0.783742
$386$	0	0
$387$	−7.93229e10	−0.179763
$388$	0	0
$389$	3.68633e11	0.816247	0.408123	−	0.912927i	$-0.366183\pi$
$389$	3.68633e11	0.816247	0.408123	+	0.912927i	$0.366183\pi$
$390$	0	0
$391$	5.36264e11	1.16034
$392$	0	0
$393$	−9.19910e10	−0.194526
$394$	0	0
$395$	1.09826e12	2.26996
$396$	0	0
$397$	2.33918e11	0.472614	0.236307	−	0.971678i	$-0.424063\pi$
$397$	2.33918e11	0.472614	0.236307	+	0.971678i	$0.424063\pi$
$398$	0	0
$399$	1.36022e11	0.268678
$400$	0	0
$401$	−5.05945e11	−0.977132	−0.488566	−	0.872527i	$-0.662480\pi$
$401$	−5.05945e11	−0.977132	−0.488566	+	0.872527i	$0.662480\pi$
$402$	0	0
$403$	1.77741e11	0.335671
$404$	0	0
$405$	3.23620e11	0.597705
$406$	0	0
$407$	−3.88681e11	−0.702131
$408$	0	0
$409$	4.94465e11	0.873737	0.436868	−	0.899525i	$-0.356088\pi$
$409$	4.94465e11	0.873737	0.436868	+	0.899525i	$0.356088\pi$
$410$	0	0
$411$	1.04721e11	0.181028
$412$	0	0
$413$	−3.49153e11	−0.590529
$414$	0	0
$415$	2.79483e11	0.462530
$416$	0	0
$417$	1.75605e11	0.284397
$418$	0	0
$419$	−1.04160e12	−1.65096	−0.825479	−	0.564433i	$-0.809095\pi$
$419$	−1.04160e12	−1.65096	−0.825479	+	0.564433i	$0.809095\pi$
$420$	0	0
$421$	−1.07184e12	−1.66287	−0.831436	−	0.555621i	$-0.812480\pi$
$421$	−1.07184e12	−1.66287	−0.831436	+	0.555621i	$0.812480\pi$
$422$	0	0
$423$	−2.70602e11	−0.410960
$424$	0	0
$425$	−1.62663e12	−2.41846
$426$	0	0
$427$	−4.37315e11	−0.636604
$428$	0	0
$429$	1.86590e11	0.265968
$430$	0	0
$431$	3.59644e11	0.502025	0.251013	−	0.967984i	$-0.419236\pi$
$431$	3.59644e11	0.502025	0.251013	+	0.967984i	$0.419236\pi$
$432$	0	0
$433$	8.92983e11	1.22081	0.610404	−	0.792090i	$-0.291007\pi$
$433$	8.92983e11	1.22081	0.610404	+	0.792090i	$0.291007\pi$
$434$	0	0
$435$	7.10607e11	0.951542
$436$	0	0
$437$	1.53455e12	2.01286
$438$	0	0
$439$	1.15608e12	1.48559	0.742795	−	0.669519i	$-0.233500\pi$
$439$	1.15608e12	1.48559	0.742795	+	0.669519i	$0.233500\pi$
$440$	0	0
$441$	−8.45124e10	−0.106401
$442$	0	0
$443$	6.57442e11	0.811037	0.405519	−	0.914087i	$-0.367091\pi$
$443$	6.57442e11	0.811037	0.405519	+	0.914087i	$0.367091\pi$
$444$	0	0
$445$	−1.57769e12	−1.90723
$446$	0	0
$447$	−3.11342e11	−0.368854
$448$	0	0
$449$	−3.62252e11	−0.420632	−0.210316	−	0.977633i	$-0.567449\pi$
$449$	−3.62252e11	−0.420632	−0.210316	+	0.977633i	$0.567449\pi$
$450$	0	0
$451$	8.37888e11	0.953655
$452$	0	0
$453$	4.69744e11	0.524107
$454$	0	0
$455$	3.49312e11	0.382087
$456$	0	0
$457$	−3.08135e11	−0.330459	−0.165230	−	0.986255i	$-0.552837\pi$
$457$	−3.08135e11	−0.330459	−0.165230	+	0.986255i	$0.552837\pi$
$458$	0	0
$459$	−6.79917e11	−0.714988
$460$	0	0
$461$	3.01632e11	0.311045	0.155523	−	0.987832i	$-0.450294\pi$
$461$	3.01632e11	0.311045	0.155523	+	0.987832i	$0.450294\pi$
$462$	0	0
$463$	−1.56062e12	−1.57828	−0.789139	−	0.614215i	$-0.789473\pi$
$463$	−1.56062e12	−1.57828	−0.789139	+	0.614215i	$0.789473\pi$
$464$	0	0
$465$	−6.73302e11	−0.667839
$466$	0	0
$467$	−2.39028e11	−0.232554	−0.116277	−	0.993217i	$-0.537096\pi$
$467$	−2.39028e11	−0.232554	−0.116277	+	0.993217i	$0.537096\pi$
$468$	0	0
$469$	1.65404e11	0.157859
$470$	0	0
$471$	4.31018e11	0.403553
$472$	0	0
$473$	−2.73045e11	−0.250818
$474$	0	0
$475$	−4.65469e12	−4.19536
$476$	0	0
$477$	1.22114e12	1.08003
$478$	0	0
$479$	−8.91781e11	−0.774013	−0.387007	−	0.922077i	$-0.626491\pi$
$479$	−8.91781e11	−0.774013	−0.387007	+	0.922077i	$0.626491\pi$
$480$	0	0
$481$	4.01846e11	0.342300
$482$	0	0
$483$	3.26671e11	0.273117
$484$	0	0
$485$	−5.09839e11	−0.418404
$486$	0	0
$487$	−2.09961e12	−1.69145	−0.845725	−	0.533619i	$-0.820831\pi$
$487$	−2.09961e12	−1.69145	−0.845725	+	0.533619i	$0.820831\pi$
$488$	0	0
$489$	−1.14708e12	−0.907202
$490$	0	0
$491$	1.47453e12	1.14495	0.572474	−	0.819923i	$-0.305984\pi$
$491$	1.47453e12	1.14495	0.572474	+	0.819923i	$0.305984\pi$
$492$	0	0
$493$	1.00440e12	0.765766
$494$	0	0
$495$	2.06296e12	1.54443
$496$	0	0
$497$	−2.85535e11	−0.209921
$498$	0	0
$499$	−9.57627e11	−0.691423	−0.345712	−	0.938341i	$-0.612362\pi$
$499$	−9.57627e11	−0.691423	−0.345712	+	0.938341i	$0.612362\pi$
$500$	0	0
$501$	7.74147e11	0.548977
$502$	0	0
$503$	1.67922e11	0.116964	0.0584821	−	0.998288i	$-0.481374\pi$
$503$	1.67922e11	0.116964	0.0584821	+	0.998288i	$0.481374\pi$
$504$	0	0
$505$	−3.50629e12	−2.39903
$506$	0	0
$507$	5.58658e11	0.375500
$508$	0	0
$509$	1.26770e12	0.837119	0.418560	−	0.908189i	$-0.362535\pi$
$509$	1.26770e12	0.837119	0.418560	+	0.908189i	$0.362535\pi$
$510$	0	0
$511$	1.09639e11	0.0711331
$512$	0	0
$513$	−1.94562e12	−1.24031
$514$	0	0
$515$	−1.52074e12	−0.952627
$516$	0	0
$517$	−9.31464e11	−0.573401
$518$	0	0
$519$	−1.50644e12	−0.911375
$520$	0	0
$521$	1.65260e12	0.982646	0.491323	−	0.870978i	$-0.336514\pi$
$521$	1.65260e12	0.982646	0.491323	+	0.870978i	$0.336514\pi$
$522$	0	0
$523$	−1.71480e12	−1.00221	−0.501103	−	0.865388i	$-0.667072\pi$
$523$	−1.71480e12	−1.00221	−0.501103	+	0.865388i	$0.667072\pi$
$524$	0	0
$525$	−9.90879e11	−0.569251
$526$	0	0
$527$	−9.51672e11	−0.537452
$528$	0	0
$529$	1.88421e12	1.04612
$530$	0	0
$531$	2.13187e12	1.16368
$532$	0	0
$533$	−8.66269e11	−0.464922
$534$	0	0
$535$	5.20583e12	2.74725
$536$	0	0
$537$	1.38000e12	0.716133
$538$	0	0
$539$	−2.90908e11	−0.148459
$540$	0	0
$541$	−1.75381e12	−0.880226	−0.440113	−	0.897942i	$-0.645062\pi$
$541$	−1.75381e12	−0.880226	−0.440113	+	0.897942i	$0.645062\pi$
$542$	0	0
$543$	−4.89661e11	−0.241711
$544$	0	0
$545$	−1.10713e12	−0.537546
$546$	0	0
$547$	3.46952e12	1.65701	0.828507	−	0.559979i	$-0.189191\pi$
$547$	3.46952e12	1.65701	0.828507	+	0.559979i	$0.189191\pi$
$548$	0	0
$549$	2.67017e12	1.25448
$550$	0	0
$551$	2.87415e12	1.32839
$552$	0	0
$553$	9.45615e11	0.429983
$554$	0	0
$555$	−1.52224e12	−0.681027
$556$	0	0
$557$	−1.37686e12	−0.606098	−0.303049	−	0.952975i	$-0.598005\pi$
$557$	−1.37686e12	−0.606098	−0.303049	+	0.952975i	$0.598005\pi$
$558$	0	0
$559$	2.82293e11	0.122278
$560$	0	0
$561$	−9.99052e11	−0.425849
$562$	0	0
$563$	9.89242e11	0.414968	0.207484	−	0.978238i	$-0.433472\pi$
$563$	9.89242e11	0.414968	0.207484	+	0.978238i	$0.433472\pi$
$564$	0	0
$565$	−8.80362e11	−0.363449
$566$	0	0
$567$	2.78640e11	0.113219
$568$	0	0
$569$	2.63692e11	0.105461	0.0527305	−	0.998609i	$-0.483208\pi$
$569$	2.63692e11	0.105461	0.0527305	+	0.998609i	$0.483208\pi$
$570$	0	0
$571$	4.59135e12	1.80750	0.903750	−	0.428060i	$-0.140803\pi$
$571$	4.59135e12	1.80750	0.903750	+	0.428060i	$0.140803\pi$
$572$	0	0
$573$	2.93779e10	0.0113848
$574$	0	0
$575$	−1.11787e13	−4.26467
$576$	0	0
$577$	2.77381e12	1.04180	0.520901	−	0.853617i	$-0.325596\pi$
$577$	2.77381e12	1.04180	0.520901	+	0.853617i	$0.325596\pi$
$578$	0	0
$579$	−7.90396e11	−0.292275
$580$	0	0
$581$	2.40639e11	0.0876138
$582$	0	0
$583$	4.20341e12	1.50693
$584$	0	0
$585$	−2.13284e12	−0.752933
$586$	0	0
$587$	−3.91289e12	−1.36027	−0.680136	−	0.733086i	$-0.738079\pi$
$587$	−3.91289e12	−1.36027	−0.680136	+	0.733086i	$0.738079\pi$
$588$	0	0
$589$	−2.72326e12	−0.932331
$590$	0	0
$591$	1.35441e11	0.0456675
$592$	0	0
$593$	3.28673e12	1.09148	0.545742	−	0.837953i	$-0.316248\pi$
$593$	3.28673e12	1.09148	0.545742	+	0.837953i	$0.316248\pi$
$594$	0	0
$595$	−1.87031e12	−0.611769
$596$	0	0
$597$	1.39063e12	0.448051
$598$	0	0
$599$	−1.06111e11	−0.0336775	−0.0168387	−	0.999858i	$-0.505360\pi$
$599$	−1.06111e11	−0.0336775	−0.0168387	+	0.999858i	$0.505360\pi$
$600$	0	0
$601$	−4.35038e11	−0.136017	−0.0680084	−	0.997685i	$-0.521664\pi$
$601$	−4.35038e11	−0.136017	−0.0680084	+	0.997685i	$0.521664\pi$
$602$	0	0
$603$	−1.00993e12	−0.311073
$604$	0	0
$605$	5.25779e11	0.159553
$606$	0	0
$607$	−2.04322e11	−0.0610893	−0.0305447	−	0.999533i	$-0.509724\pi$
$607$	−2.04322e11	−0.0610893	−0.0305447	+	0.999533i	$0.509724\pi$
$608$	0	0
$609$	6.11842e11	0.180244
$610$	0	0
$611$	9.63015e11	0.279542
$612$	0	0
$613$	−3.67798e12	−1.05205	−0.526025	−	0.850469i	$-0.676318\pi$
$613$	−3.67798e12	−1.05205	−0.526025	+	0.850469i	$0.676318\pi$
$614$	0	0
$615$	3.28152e12	0.924991
$616$	0	0
$617$	3.43041e12	0.952933	0.476466	−	0.879193i	$-0.341917\pi$
$617$	3.43041e12	0.952933	0.476466	+	0.879193i	$0.341917\pi$
$618$	0	0
$619$	3.11037e12	0.851537	0.425768	−	0.904832i	$-0.360004\pi$
$619$	3.11037e12	0.851537	0.425768	+	0.904832i	$0.360004\pi$
$620$	0	0
$621$	−4.67259e12	−1.26080
$622$	0	0
$623$	−1.35841e12	−0.361273
$624$	0	0
$625$	1.87200e13	4.90734
$626$	0	0
$627$	−2.85884e12	−0.738730
$628$	0	0
$629$	−2.15159e12	−0.548065
$630$	0	0
$631$	−1.29444e12	−0.325049	−0.162524	−	0.986705i	$-0.551964\pi$
$631$	−1.29444e12	−0.325049	−0.162524	+	0.986705i	$0.551964\pi$
$632$	0	0
$633$	3.92070e12	0.970614
$634$	0	0
$635$	6.16665e11	0.150511
$636$	0	0
$637$	3.00762e11	0.0723761
$638$	0	0
$639$	1.74342e12	0.413666
$640$	0	0
$641$	−4.02693e12	−0.942135	−0.471067	−	0.882097i	$-0.656131\pi$
$641$	−4.02693e12	−0.942135	−0.471067	+	0.882097i	$0.656131\pi$
$642$	0	0
$643$	6.51248e12	1.50244	0.751219	−	0.660053i	$-0.229466\pi$
$643$	6.51248e12	1.50244	0.751219	+	0.660053i	$0.229466\pi$
$644$	0	0
$645$	−1.06936e12	−0.243279
$646$	0	0
$647$	−1.12144e12	−0.251599	−0.125799	−	0.992056i	$-0.540150\pi$
$647$	−1.12144e12	−0.251599	−0.125799	+	0.992056i	$0.540150\pi$
$648$	0	0
$649$	7.33830e12	1.62366
$650$	0	0
$651$	−5.79721e11	−0.126504
$652$	0	0
$653$	2.44317e12	0.525829	0.262914	−	0.964819i	$-0.415316\pi$
$653$	2.44317e12	0.525829	0.262914	+	0.964819i	$0.415316\pi$
$654$	0	0
$655$	3.61951e12	0.768359
$656$	0	0
$657$	−6.69437e11	−0.140173
$658$	0	0
$659$	−7.04685e11	−0.145550	−0.0727748	−	0.997348i	$-0.523185\pi$
$659$	−7.04685e11	−0.145550	−0.0727748	+	0.997348i	$0.523185\pi$
$660$	0	0
$661$	−5.40645e12	−1.10155	−0.550776	−	0.834653i	$-0.685668\pi$
$661$	−5.40645e12	−1.10155	−0.550776	+	0.834653i	$0.685668\pi$
$662$	0	0
$663$	1.03289e12	0.207608
$664$	0	0
$665$	−5.35199e12	−1.06125
$666$	0	0
$667$	6.90254e12	1.35034
$668$	0	0
$669$	−4.64630e11	−0.0896787
$670$	0	0
$671$	9.19124e12	1.75034
$672$	0	0
$673$	8.69051e11	0.163297	0.0816484	−	0.996661i	$-0.473982\pi$
$673$	8.69051e11	0.163297	0.0816484	+	0.996661i	$0.473982\pi$
$674$	0	0
$675$	1.41732e13	2.62785
$676$	0	0
$677$	4.55159e12	0.832749	0.416375	−	0.909193i	$-0.363301\pi$
$677$	4.55159e12	0.832749	0.416375	+	0.909193i	$0.363301\pi$
$678$	0	0
$679$	−4.38978e11	−0.0792553
$680$	0	0
$681$	−9.45180e11	−0.168404
$682$	0	0
$683$	−1.01532e13	−1.78530	−0.892650	−	0.450751i	$-0.851156\pi$
$683$	−1.01532e13	−1.78530	−0.892650	+	0.450751i	$0.851156\pi$
$684$	0	0
$685$	−4.12038e12	−0.715040
$686$	0	0
$687$	2.77391e12	0.475101
$688$	0	0
$689$	−4.34579e12	−0.734653
$690$	0	0
$691$	−1.75034e12	−0.292059	−0.146029	−	0.989280i	$-0.546649\pi$
$691$	−1.75034e12	−0.292059	−0.146029	+	0.989280i	$0.546649\pi$
$692$	0	0
$693$	1.77623e12	0.292550
$694$	0	0
$695$	−6.90943e12	−1.12334
$696$	0	0
$697$	4.63824e12	0.744399
$698$	0	0
$699$	3.30832e12	0.524155
$700$	0	0
$701$	−1.38237e12	−0.216219	−0.108110	−	0.994139i	$-0.534480\pi$
$701$	−1.38237e12	−0.216219	−0.108110	+	0.994139i	$0.534480\pi$
$702$	0	0
$703$	−6.15689e12	−0.950742
$704$	0	0
$705$	−3.64801e12	−0.556166
$706$	0	0
$707$	−3.01896e12	−0.454432
$708$	0	0
$709$	3.27785e12	0.487170	0.243585	−	0.969880i	$-0.421676\pi$
$709$	3.27785e12	0.487170	0.243585	+	0.969880i	$0.421676\pi$
$710$	0	0
$711$	−5.77376e12	−0.847316
$712$	0	0
$713$	−6.54017e12	−0.947734
$714$	0	0
$715$	−7.34163e12	−1.05055
$716$	0	0
$717$	1.30263e12	0.184070
$718$	0	0
$719$	−9.16639e12	−1.27914	−0.639570	−	0.768732i	$-0.720888\pi$
$719$	−9.16639e12	−1.27914	−0.639570	+	0.768732i	$0.720888\pi$
$720$	0	0
$721$	−1.30938e12	−0.180450
$722$	0	0
$723$	−1.46171e12	−0.198947
$724$	0	0
$725$	−2.09372e13	−2.81448
$726$	0	0
$727$	−7.70818e12	−1.02340	−0.511702	−	0.859163i	$-0.670985\pi$
$727$	−7.70818e12	−1.02340	−0.511702	+	0.859163i	$0.670985\pi$
$728$	0	0
$729$	1.69404e12	0.222152
$730$	0	0
$731$	−1.51147e12	−0.195782
$732$	0	0
$733$	−9.99397e12	−1.27870	−0.639352	−	0.768914i	$-0.720797\pi$
$733$	−9.99397e12	−1.27870	−0.639352	+	0.768914i	$0.720797\pi$
$734$	0	0
$735$	−1.13932e12	−0.143997
$736$	0	0
$737$	−3.47637e12	−0.434033
$738$	0	0
$739$	1.81509e12	0.223871	0.111936	−	0.993715i	$-0.464295\pi$
$739$	1.81509e12	0.223871	0.111936	+	0.993715i	$0.464295\pi$
$740$	0	0
$741$	2.95567e12	0.360143
$742$	0	0
$743$	3.24224e12	0.390297	0.195148	−	0.980774i	$-0.437481\pi$
$743$	3.24224e12	0.390297	0.195148	+	0.980774i	$0.437481\pi$
$744$	0	0
$745$	1.22502e13	1.45693
$746$	0	0
$747$	−1.46930e12	−0.172650
$748$	0	0
$749$	4.48228e12	0.520393
$750$	0	0
$751$	1.26604e13	1.45234	0.726171	−	0.687514i	$-0.241298\pi$
$751$	1.26604e13	1.45234	0.726171	+	0.687514i	$0.241298\pi$
$752$	0	0
$753$	−1.79342e12	−0.203285
$754$	0	0
$755$	−1.84827e13	−2.07017
$756$	0	0
$757$	1.80729e12	0.200031	0.100015	−	0.994986i	$-0.468111\pi$
$757$	1.80729e12	0.200031	0.100015	+	0.994986i	$0.468111\pi$
$758$	0	0
$759$	−6.86578e12	−0.750934
$760$	0	0
$761$	6.46799e12	0.699099	0.349549	−	0.936918i	$-0.386335\pi$
$761$	6.46799e12	0.699099	0.349549	+	0.936918i	$0.386335\pi$
$762$	0	0
$763$	−9.53255e11	−0.101824
$764$	0	0
$765$	1.14198e13	1.20554
$766$	0	0
$767$	−7.58687e12	−0.791559
$768$	0	0
$769$	−4.62358e12	−0.476771	−0.238385	−	0.971171i	$-0.576618\pi$
$769$	−4.62358e12	−0.476771	−0.238385	+	0.971171i	$0.576618\pi$
$770$	0	0
$771$	−8.59584e12	−0.876079
$772$	0	0
$773$	4.79199e12	0.482735	0.241367	−	0.970434i	$-0.422404\pi$
$773$	4.79199e12	0.482735	0.241367	+	0.970434i	$0.422404\pi$
$774$	0	0
$775$	1.98381e13	1.97534
$776$	0	0
$777$	−1.31067e12	−0.129002
$778$	0	0
$779$	1.32726e13	1.29133
$780$	0	0
$781$	6.00120e12	0.577176
$782$	0	0
$783$	−8.75157e12	−0.832067
$784$	0	0
$785$	−1.69590e13	−1.59399
$786$	0	0
$787$	−1.27043e13	−1.18050	−0.590249	−	0.807221i	$-0.700970\pi$
$787$	−1.27043e13	−1.18050	−0.590249	+	0.807221i	$0.700970\pi$
$788$	0	0
$789$	7.26559e12	0.667459
$790$	0	0
$791$	−7.58003e11	−0.0688456
$792$	0	0
$793$	−9.50257e12	−0.853319
$794$	0	0
$795$	1.64623e13	1.46164
$796$	0	0
$797$	5.54048e12	0.486390	0.243195	−	0.969977i	$-0.421804\pi$
$797$	5.54048e12	0.486390	0.243195	+	0.969977i	$0.421804\pi$
$798$	0	0
$799$	−5.15624e12	−0.447582
$800$	0	0
$801$	8.29424e12	0.711918
$802$	0	0
$803$	−2.30433e12	−0.195580
$804$	0	0
$805$	−1.28533e13	−1.07878
$806$	0	0
$807$	1.67330e12	0.138881
$808$	0	0
$809$	−5.01623e12	−0.411727	−0.205863	−	0.978581i	$-0.566000\pi$
$809$	−5.01623e12	−0.411727	−0.205863	+	0.978581i	$0.566000\pi$
$810$	0	0
$811$	8.73000e12	0.708632	0.354316	−	0.935126i	$-0.384714\pi$
$811$	8.73000e12	0.708632	0.354316	+	0.935126i	$0.384714\pi$
$812$	0	0
$813$	−9.82859e12	−0.789012
$814$	0	0
$815$	4.51334e13	3.58335
$816$	0	0
$817$	−4.32516e12	−0.339628
$818$	0	0
$819$	−1.83640e12	−0.142623
$820$	0	0
$821$	−2.54078e13	−1.95175	−0.975874	−	0.218336i	$-0.929937\pi$
$821$	−2.54078e13	−1.95175	−0.975874	+	0.218336i	$0.929937\pi$
$822$	0	0
$823$	−1.45577e13	−1.10609	−0.553047	−	0.833150i	$-0.686535\pi$
$823$	−1.45577e13	−1.10609	−0.553047	+	0.833150i	$0.686535\pi$
$824$	0	0
$825$	2.08257e13	1.56515
$826$	0	0
$827$	1.39537e12	0.103732	0.0518660	−	0.998654i	$-0.483483\pi$
$827$	1.39537e12	0.103732	0.0518660	+	0.998654i	$0.483483\pi$
$828$	0	0
$829$	2.11015e12	0.155174	0.0775868	−	0.996986i	$-0.475278\pi$
$829$	2.11015e12	0.155174	0.0775868	+	0.996986i	$0.475278\pi$
$830$	0	0
$831$	−8.81726e12	−0.641400
$832$	0	0
$833$	−1.61036e12	−0.115883
$834$	0	0
$835$	−3.04599e13	−2.16840
$836$	0	0
$837$	8.29213e12	0.583985
$838$	0	0
$839$	−9.85480e12	−0.686624	−0.343312	−	0.939221i	$-0.611549\pi$
$839$	−9.85480e12	−0.686624	−0.343312	+	0.939221i	$0.611549\pi$
$840$	0	0
$841$	−1.57897e12	−0.108841
$842$	0	0
$843$	7.15259e12	0.487797
$844$	0	0
$845$	−2.19812e13	−1.48319
$846$	0	0
$847$	4.52702e11	0.0302230
$848$	0	0
$849$	−4.92619e12	−0.325407
$850$	0	0
$851$	−1.47864e13	−0.966449
$852$	0	0
$853$	−1.38050e13	−0.892823	−0.446412	−	0.894828i	$-0.647298\pi$
$853$	−1.38050e13	−0.892823	−0.446412	+	0.894828i	$0.647298\pi$
$854$	0	0
$855$	3.26783e13	2.09128
$856$	0	0
$857$	−1.08692e12	−0.0688309	−0.0344155	−	0.999408i	$-0.510957\pi$
$857$	−1.08692e12	−0.0688309	−0.0344155	+	0.999408i	$0.510957\pi$
$858$	0	0
$859$	2.19124e13	1.37316	0.686578	−	0.727056i	$-0.259112\pi$
$859$	2.19124e13	1.37316	0.686578	+	0.727056i	$0.259112\pi$
$860$	0	0
$861$	2.82543e12	0.175215
$862$	0	0
$863$	−6.97060e12	−0.427781	−0.213890	−	0.976858i	$-0.568614\pi$
$863$	−6.97060e12	−0.427781	−0.213890	+	0.976858i	$0.568614\pi$
$864$	0	0
$865$	5.92728e13	3.59984
$866$	0	0
$867$	2.87424e12	0.172758
$868$	0	0
$869$	−1.98744e13	−1.18224
$870$	0	0
$871$	3.59412e12	0.211598
$872$	0	0
$873$	2.68032e12	0.156179
$874$	0	0
$875$	2.59106e13	1.49431
$876$	0	0
$877$	−2.38011e13	−1.35863	−0.679313	−	0.733849i	$-0.737722\pi$
$877$	−2.38011e13	−1.35863	−0.679313	+	0.733849i	$0.737722\pi$
$878$	0	0
$879$	2.15346e11	0.0121671
$880$	0	0
$881$	2.82612e12	0.158052	0.0790259	−	0.996873i	$-0.474819\pi$
$881$	2.82612e12	0.158052	0.0790259	+	0.996873i	$0.474819\pi$
$882$	0	0
$883$	−2.31631e13	−1.28225	−0.641127	−	0.767435i	$-0.721533\pi$
$883$	−2.31631e13	−1.28225	−0.641127	+	0.767435i	$0.721533\pi$
$884$	0	0
$885$	2.87399e13	1.57485
$886$	0	0
$887$	−3.08401e13	−1.67286	−0.836430	−	0.548074i	$-0.815361\pi$
$887$	−3.08401e13	−1.67286	−0.836430	+	0.548074i	$0.815361\pi$
$888$	0	0
$889$	5.30956e11	0.0285102
$890$	0	0
$891$	−5.85630e12	−0.311296
$892$	0	0
$893$	−1.47549e13	−0.776432
$894$	0	0
$895$	−5.42978e13	−2.82865
$896$	0	0
$897$	7.09834e12	0.366092
$898$	0	0
$899$	−1.22495e13	−0.625459
$900$	0	0
$901$	2.32685e13	1.17627
$902$	0	0
$903$	−9.20730e11	−0.0460827
$904$	0	0
$905$	1.92664e13	0.954732
$906$	0	0
$907$	2.07395e13	1.01757	0.508787	−	0.860893i	$-0.330094\pi$
$907$	2.07395e13	1.01757	0.508787	+	0.860893i	$0.330094\pi$
$908$	0	0
$909$	1.84332e13	0.895495
$910$	0	0
$911$	1.72994e13	0.832142	0.416071	−	0.909332i	$-0.363407\pi$
$911$	1.72994e13	0.832142	0.416071	+	0.909332i	$0.363407\pi$
$912$	0	0
$913$	−5.05760e12	−0.240894
$914$	0	0
$915$	3.59968e13	1.69773
$916$	0	0
$917$	3.11644e12	0.145545
$918$	0	0
$919$	7.15172e12	0.330743	0.165372	−	0.986231i	$-0.447118\pi$
$919$	7.15172e12	0.330743	0.165372	+	0.986231i	$0.447118\pi$
$920$	0	0
$921$	−1.33516e13	−0.611454
$922$	0	0
$923$	−6.20448e12	−0.281383
$924$	0	0
$925$	4.48510e13	2.01435
$926$	0	0
$927$	7.99483e12	0.355591
$928$	0	0
$929$	6.35919e11	0.0280112	0.0140056	−	0.999902i	$-0.495542\pi$
$929$	6.35919e11	0.0280112	0.0140056	+	0.999902i	$0.495542\pi$
$930$	0	0
$931$	−4.60813e12	−0.201025
$932$	0	0
$933$	−1.38075e12	−0.0596551
$934$	0	0
$935$	3.93091e13	1.68206
$936$	0	0
$937$	−2.74356e13	−1.16275	−0.581376	−	0.813635i	$-0.697485\pi$
$937$	−2.74356e13	−1.16275	−0.581376	+	0.813635i	$0.697485\pi$
$938$	0	0
$939$	1.15853e13	0.486310
$940$	0	0
$941$	2.99283e13	1.24431	0.622156	−	0.782893i	$-0.286257\pi$
$941$	2.99283e13	1.24431	0.622156	+	0.782893i	$0.286257\pi$
$942$	0	0
$943$	3.18753e13	1.31266
$944$	0	0
$945$	1.62964e13	0.664736
$946$	0	0
$947$	3.56965e13	1.44229	0.721143	−	0.692787i	$-0.243617\pi$
$947$	3.56965e13	1.44229	0.721143	+	0.692787i	$0.243617\pi$
$948$	0	0
$949$	2.38238e12	0.0953484
$950$	0	0
$951$	−1.54161e13	−0.611170
$952$	0	0
$953$	9.51801e12	0.373790	0.186895	−	0.982380i	$-0.440158\pi$
$953$	9.51801e12	0.373790	0.186895	+	0.982380i	$0.440158\pi$
$954$	0	0
$955$	−1.15591e12	−0.0449687
$956$	0	0
$957$	−1.28593e13	−0.495581
$958$	0	0
$959$	−3.54770e12	−0.135445
$960$	0	0
$961$	−1.48332e13	−0.561022
$962$	0	0
$963$	−2.73680e13	−1.02548
$964$	0	0
$965$	3.10992e13	1.15445
$966$	0	0
$967$	4.44303e13	1.63403	0.817016	−	0.576615i	$-0.195627\pi$
$967$	4.44303e13	1.63403	0.817016	+	0.576615i	$0.195627\pi$
$968$	0	0
$969$	−1.58255e13	−0.576633
$970$	0	0
$971$	−3.78655e13	−1.36696	−0.683481	−	0.729968i	$-0.739535\pi$
$971$	−3.78655e13	−1.36696	−0.683481	+	0.729968i	$0.739535\pi$
$972$	0	0
$973$	−5.94910e12	−0.212786
$974$	0	0
$975$	−2.15311e13	−0.763037
$976$	0	0
$977$	−2.47961e13	−0.870678	−0.435339	−	0.900267i	$-0.643371\pi$
$977$	−2.47961e13	−0.870678	−0.435339	+	0.900267i	$0.643371\pi$
$978$	0	0
$979$	2.85504e13	0.993320
$980$	0	0
$981$	5.82041e12	0.200652
$982$	0	0
$983$	−1.41740e13	−0.484175	−0.242088	−	0.970254i	$-0.577832\pi$
$983$	−1.41740e13	−0.484175	−0.242088	+	0.970254i	$0.577832\pi$
$984$	0	0
$985$	−5.32912e12	−0.180382
$986$	0	0
$987$	−3.14098e12	−0.105351
$988$	0	0
$989$	−1.03873e13	−0.345238
$990$	0	0
$991$	1.00580e13	0.331268	0.165634	−	0.986187i	$-0.447033\pi$
$991$	1.00580e13	0.331268	0.165634	+	0.986187i	$0.447033\pi$
$992$	0	0
$993$	−1.37297e13	−0.448115
$994$	0	0
$995$	−5.47163e13	−1.76975
$996$	0	0
$997$	−1.43003e13	−0.458372	−0.229186	−	0.973383i	$-0.573606\pi$
$997$	−1.43003e13	−0.458372	−0.229186	+	0.973383i	$0.573606\pi$
$998$	0	0
$999$	1.87473e13	0.595517

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.a.d.1.1		2
4.3	odd	2		28.10.a.b.1.2	✓	2
12.11	even	2		252.10.a.b.1.1		2
28.3	even	6		196.10.e.d.177.2		4
28.11	odd	6		196.10.e.e.177.1		4
28.19	even	6		196.10.e.d.165.2		4
28.23	odd	6		196.10.e.e.165.1		4
28.27	even	2		196.10.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.10.a.b.1.2	✓	2	4.3	odd	2
112.10.a.d.1.1		2	1.1	even	1	trivial
196.10.a.b.1.1		2	28.27	even	2
196.10.e.d.165.2		4	28.19	even	6
196.10.e.d.177.2		4	28.3	even	6
196.10.e.e.165.1		4	28.23	odd	6
196.10.e.e.177.1		4	28.11	odd	6
252.10.a.b.1.1		2	12.11	even	2

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)

\(f(q)\)	\(=\)	\(q-70.8726 q^{3} +2788.58 q^{5} +2401.00 q^{7} -14660.1 q^{9} +O(q^{10})\)	\(q-70.8726 q^{3} +2788.58 q^{5} +2401.00 q^{7} -14660.1 q^{9} -50462.8 q^{11} +52172.1 q^{13} -197634. q^{15} -279343. q^{17} -799356. q^{19} -170165. q^{21} -1.91973e6 q^{23} +5.82305e6 q^{25} +2.43398e6 q^{27} -3.59558e6 q^{29} +3.40682e6 q^{31} +3.57643e6 q^{33} +6.69538e6 q^{35} +7.70232e6 q^{37} -3.69757e6 q^{39} -1.66041e7 q^{41} +5.41081e6 q^{43} -4.08808e7 q^{45} +1.84584e7 q^{47} +5.76480e6 q^{49} +1.97978e7 q^{51} -8.32973e7 q^{53} -1.40720e8 q^{55} +5.66524e7 q^{57} -1.45420e8 q^{59} -1.82139e8 q^{61} -3.51989e7 q^{63} +1.45486e8 q^{65} +6.88897e7 q^{67} +1.36056e8 q^{69} -1.18923e8 q^{71} +4.56639e7 q^{73} -4.12694e8 q^{75} -1.21161e8 q^{77} +3.93842e8 q^{79} +1.16052e8 q^{81} +1.00224e8 q^{83} -7.78971e8 q^{85} +2.54828e8 q^{87} -5.65770e8 q^{89} +1.25265e8 q^{91} -2.41450e8 q^{93} -2.22907e9 q^{95} -1.82831e8 q^{97} +7.39789e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 70 q^{3} + 1554 q^{5} + 4802 q^{7} - 14498 q^{9}+O(q^{10}) \) 2 * q + 70 * q^3 + 1554 * q^5 + 4802 * q^7 - 14498 * q^9	\( 2 q + 70 q^{3} + 1554 q^{5} + 4802 q^{7} - 14498 q^{9} - 62388 q^{11} + 122766 q^{13} - 371552 q^{15} + 73584 q^{17} - 1171198 q^{19} + 168070 q^{21} - 2262384 q^{23} + 5394106 q^{25} - 315980 q^{27} - 1923360 q^{29} - 2977884 q^{31} + 1896496 q^{33} + 3731154 q^{35} - 13418528 q^{37} + 6247176 q^{39} - 36367800 q^{41} + 21964916 q^{43} - 41080886 q^{45} + 1362732 q^{47} + 11529602 q^{49} + 69515588 q^{51} - 17898612 q^{53} - 125996920 q^{55} + 4270012 q^{57} - 224710542 q^{59} - 85847118 q^{61} - 34809698 q^{63} + 58332228 q^{65} - 179568872 q^{67} + 87785824 q^{69} - 231378168 q^{71} + 88098332 q^{73} - 473120158 q^{75} - 149793588 q^{77} + 184274184 q^{79} - 274532642 q^{81} - 624641094 q^{83} - 1214688044 q^{85} + 490397404 q^{87} - 1574777148 q^{89} + 294761166 q^{91} - 1140879096 q^{93} - 1769997744 q^{95} + 213665984 q^{97} + 737855932 q^{99}+O(q^{100}) \) 2 * q + 70 * q^3 + 1554 * q^5 + 4802 * q^7 - 14498 * q^9 - 62388 * q^11 + 122766 * q^13 - 371552 * q^15 + 73584 * q^17 - 1171198 * q^19 + 168070 * q^21 - 2262384 * q^23 + 5394106 * q^25 - 315980 * q^27 - 1923360 * q^29 - 2977884 * q^31 + 1896496 * q^33 + 3731154 * q^35 - 13418528 * q^37 + 6247176 * q^39 - 36367800 * q^41 + 21964916 * q^43 - 41080886 * q^45 + 1362732 * q^47 + 11529602 * q^49 + 69515588 * q^51 - 17898612 * q^53 - 125996920 * q^55 + 4270012 * q^57 - 224710542 * q^59 - 85847118 * q^61 - 34809698 * q^63 + 58332228 * q^65 - 179568872 * q^67 + 87785824 * q^69 - 231378168 * q^71 + 88098332 * q^73 - 473120158 * q^75 - 149793588 * q^77 + 184274184 * q^79 - 274532642 * q^81 - 624641094 * q^83 - 1214688044 * q^85 + 490397404 * q^87 - 1574777148 * q^89 + 294761166 * q^91 - 1140879096 * q^93 - 1769997744 * q^95 + 213665984 * q^97 + 737855932 * q^99

Introduction

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