LMFDB - Embedded newform 112.10.a.h.1.2

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			1.2
Root			$4.96128$ 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)

$f(q)$	$=$	$q-0.232339 q^{3} -1791.89 q^{5} -2401.00 q^{7} -19682.9 q^{9} -17401.5 q^{11} -122541. q^{13} +416.326 q^{15} +331933. q^{17} -761707. q^{19} +557.846 q^{21} -1.23249e6 q^{23} +1.25774e6 q^{25} +9146.25 q^{27} +634604. q^{29} +5.38069e6 q^{31} +4043.06 q^{33} +4.30232e6 q^{35} -3.03611e6 q^{37} +28471.1 q^{39} -7.37009e6 q^{41} +2.06990e7 q^{43} +3.52696e7 q^{45} -2.03632e7 q^{47} +5.76480e6 q^{49} -77120.9 q^{51} -5.97380e7 q^{53} +3.11816e7 q^{55} +176974. q^{57} -6.03461e7 q^{59} -9.44357e6 q^{61} +4.72588e7 q^{63} +2.19580e8 q^{65} +2.19187e8 q^{67} +286355. q^{69} +5.58741e7 q^{71} +4.54332e8 q^{73} -292222. q^{75} +4.17811e7 q^{77} -4.51057e7 q^{79} +3.87417e8 q^{81} +3.34665e8 q^{83} -5.94786e8 q^{85} -147443. q^{87} +6.51886e8 q^{89} +2.94221e8 q^{91} -1.25014e6 q^{93} +1.36489e9 q^{95} -1.42804e9 q^{97} +3.42514e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.232339	−0.00165606	−0.000828031	−	1.00000i	$-0.500264\pi$
$3$	−0.232339	−0.00165606	−0.000828031		1.00000i	$0.500264\pi$
$4$	0	0
$5$	−1791.89	−1.28217	−0.641086	−	0.767469i	$-0.721516\pi$
$5$	−1791.89	−1.28217	−0.641086	+	0.767469i	$0.721516\pi$
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	0	0
$9$	−19682.9	−0.999997
$10$	0	0
$11$	−17401.5	−0.358361	−0.179180	−	0.983816i	$-0.557345\pi$
$11$	−17401.5	−0.358361	−0.179180	+	0.983816i	$0.557345\pi$
$12$	0	0
$13$	−122541.	−1.18997	−0.594986	−	0.803736i	$-0.702842\pi$
$13$	−122541.	−1.18997	−0.594986	+	0.803736i	$0.702842\pi$
$14$	0	0
$15$	416.326	0.00212335
$16$	0	0
$17$	331933.	0.963895	0.481948	−	0.876200i	$-0.339930\pi$
$17$	331933.	0.963895	0.481948	+	0.876200i	$0.339930\pi$
$18$	0	0
$19$	−761707.	−1.34090	−0.670450	−	0.741954i	$-0.733899\pi$
$19$	−761707.	−1.34090	−0.670450	+	0.741954i	$0.733899\pi$
$20$	0	0
$21$	557.846	0.000625933 0
$22$	0	0
$23$	−1.23249e6	−0.918347	−0.459173	−	0.888347i	$-0.651854\pi$
$23$	−1.23249e6	−0.918347	−0.459173	+	0.888347i	$0.651854\pi$
$24$	0	0
$25$	1.25774e6	0.643963
$26$	0	0
$27$	9146.25	0.00331212
$28$	0	0
$29$	634604.	0.166614	0.0833071	−	0.996524i	$-0.473452\pi$
$29$	634604.	0.166614	0.0833071	+	0.996524i	$0.473452\pi$
$30$	0	0
$31$	5.38069e6	1.04643	0.523215	−	0.852201i	$-0.324732\pi$
$31$	5.38069e6	1.04643	0.523215	+	0.852201i	$0.324732\pi$
$32$	0	0
$33$	4043.06	0.000593468 0
$34$	0	0
$35$	4.30232e6	0.484615
$36$	0	0
$37$	−3.03611e6	−0.266324	−0.133162	−	0.991094i	$-0.542513\pi$
$37$	−3.03611e6	−0.266324	−0.133162	+	0.991094i	$0.542513\pi$
$38$	0	0
$39$	28471.1	0.00197067
$40$	0	0
$41$	−7.37009e6	−0.407329	−0.203665	−	0.979041i	$-0.565285\pi$
$41$	−7.37009e6	−0.407329	−0.203665	+	0.979041i	$0.565285\pi$
$42$	0	0
$43$	2.06990e7	0.923297	0.461649	−	0.887063i	$-0.347258\pi$
$43$	2.06990e7	0.923297	0.461649	+	0.887063i	$0.347258\pi$
$44$	0	0
$45$	3.52696e7	1.28217
$46$	0	0
$47$	−2.03632e7	−0.608702	−0.304351	−	0.952560i	$-0.598440\pi$
$47$	−2.03632e7	−0.608702	−0.304351	+	0.952560i	$0.598440\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	0	0
$51$	−77120.9	−0.00159627
$52$	0	0
$53$	−5.97380e7	−1.03994	−0.519971	−	0.854184i	$-0.674057\pi$
$53$	−5.97380e7	−1.03994	−0.519971	+	0.854184i	$0.674057\pi$
$54$	0	0
$55$	3.11816e7	0.459480
$56$	0	0
$57$	176974.	0.00222061
$58$	0	0
$59$	−6.03461e7	−0.648358	−0.324179	−	0.945996i	$-0.605088\pi$
$59$	−6.03461e7	−0.648358	−0.324179	+	0.945996i	$0.605088\pi$
$60$	0	0
$61$	−9.44357e6	−0.0873277	−0.0436639	−	0.999046i	$-0.513903\pi$
$61$	−9.44357e6	−0.0873277	−0.0436639	+	0.999046i	$0.513903\pi$
$62$	0	0
$63$	4.72588e7	0.377963
$64$	0	0
$65$	2.19580e8	1.52575
$66$	0	0
$67$	2.19187e8	1.32885	0.664427	−	0.747353i	$-0.268676\pi$
$67$	2.19187e8	1.32885	0.664427	+	0.747353i	$0.268676\pi$
$68$	0	0
$69$	286355.	0.00152084
$70$	0	0
$71$	5.58741e7	0.260944	0.130472	−	0.991452i	$-0.458351\pi$
$71$	5.58741e7	0.260944	0.130472	+	0.991452i	$0.458351\pi$
$72$	0	0
$73$	4.54332e8	1.87250	0.936248	−	0.351340i	$-0.114274\pi$
$73$	4.54332e8	1.87250	0.936248	+	0.351340i	$0.114274\pi$
$74$	0	0
$75$	−292222.	−0.00106644
$76$	0	0
$77$	4.17811e7	0.135448
$78$	0	0
$79$	−4.51057e7	−0.130289	−0.0651447	−	0.997876i	$-0.520751\pi$
$79$	−4.51057e7	−0.130289	−0.0651447	+	0.997876i	$0.520751\pi$
$80$	0	0
$81$	3.87417e8	0.999992
$82$	0	0
$83$	3.34665e8	0.774031	0.387016	−	0.922073i	$-0.373506\pi$
$83$	3.34665e8	0.774031	0.387016	+	0.922073i	$0.373506\pi$
$84$	0	0
$85$	−5.94786e8	−1.23588
$86$	0	0
$87$	−147443.	−0.000275923 0
$88$	0	0
$89$	6.51886e8	1.10133	0.550664	−	0.834727i	$-0.314375\pi$
$89$	6.51886e8	1.10133	0.550664	+	0.834727i	$0.314375\pi$
$90$	0	0
$91$	2.94221e8	0.449767
$92$	0	0
$93$	−1.25014e6	−0.00173295
$94$	0	0
$95$	1.36489e9	1.71926
$96$	0	0
$97$	−1.42804e9	−1.63783	−0.818914	−	0.573916i	$-0.805424\pi$
$97$	−1.42804e9	−1.63783	−0.818914	+	0.573916i	$0.805424\pi$
$98$	0	0
$99$	3.42514e8	0.358360
$100$	0	0
$101$	8.91532e8	0.852493	0.426247	−	0.904607i	$-0.359836\pi$
$101$	8.91532e8	0.852493	0.426247	+	0.904607i	$0.359836\pi$
$102$	0	0
$103$	7.12000e8	0.623322	0.311661	−	0.950193i	$-0.399115\pi$
$103$	7.12000e8	0.623322	0.311661	+	0.950193i	$0.399115\pi$
$104$	0	0
$105$	−999598.	−0.000802553 0
$106$	0	0
$107$	−2.48598e9	−1.83346	−0.916729	−	0.399510i	$-0.869180\pi$
$107$	−2.48598e9	−1.83346	−0.916729	+	0.399510i	$0.869180\pi$
$108$	0	0
$109$	3.83455e8	0.260193	0.130096	−	0.991501i	$-0.458471\pi$
$109$	3.83455e8	0.260193	0.130096	+	0.991501i	$0.458471\pi$
$110$	0	0
$111$	705407.	0.000441048 0
$112$	0	0
$113$	2.39582e9	1.38230	0.691148	−	0.722713i	$-0.257105\pi$
$113$	2.39582e9	1.38230	0.691148	+	0.722713i	$0.257105\pi$
$114$	0	0
$115$	2.20848e9	1.17748
$116$	0	0
$117$	2.41197e9	1.18997
$118$	0	0
$119$	−7.96970e8	−0.364318
$120$	0	0
$121$	−2.05513e9	−0.871578
$122$	0	0
$123$	1.71236e6	0.000674562 0
$124$	0	0
$125$	1.24605e9	0.456501
$126$	0	0
$127$	−1.88261e9	−0.642161	−0.321080	−	0.947052i	$-0.604046\pi$
$127$	−1.88261e9	−0.642161	−0.321080	+	0.947052i	$0.604046\pi$
$128$	0	0
$129$	−4.80919e6	−0.00152904
$130$	0	0
$131$	1.49241e9	0.442760	0.221380	−	0.975188i	$-0.428944\pi$
$131$	1.49241e9	0.442760	0.221380	+	0.975188i	$0.428944\pi$
$132$	0	0
$133$	1.82886e9	0.506813
$134$	0	0
$135$	−1.63891e7	−0.00424670
$136$	0	0
$137$	−3.21118e9	−0.778792	−0.389396	−	0.921070i	$-0.627316\pi$
$137$	−3.21118e9	−0.778792	−0.389396	+	0.921070i	$0.627316\pi$
$138$	0	0
$139$	3.03934e9	0.690577	0.345289	−	0.938497i	$-0.387781\pi$
$139$	3.03934e9	0.690577	0.345289	+	0.938497i	$0.387781\pi$
$140$	0	0
$141$	4.73116e6	0.00100805
$142$	0	0
$143$	2.13240e9	0.426439
$144$	0	0
$145$	−1.13714e9	−0.213628
$146$	0	0
$147$	−1.33939e6	−0.000236580 0
$148$	0	0
$149$	−3.64286e9	−0.605487	−0.302743	−	0.953072i	$-0.597902\pi$
$149$	−3.64286e9	−0.605487	−0.302743	+	0.953072i	$0.597902\pi$
$150$	0	0
$151$	−5.05862e9	−0.791837	−0.395919	−	0.918286i	$-0.629574\pi$
$151$	−5.05862e9	−0.791837	−0.395919	+	0.918286i	$0.629574\pi$
$152$	0	0
$153$	−6.53341e9	−0.963893
$154$	0	0
$155$	−9.64159e9	−1.34170
$156$	0	0
$157$	−1.19687e10	−1.57216	−0.786082	−	0.618122i	$-0.787894\pi$
$157$	−1.19687e10	−1.57216	−0.786082	+	0.618122i	$0.787894\pi$
$158$	0	0
$159$	1.38795e7	0.00172221
$160$	0	0
$161$	2.95920e9	0.347102
$162$	0	0
$163$	−1.40960e10	−1.56405	−0.782027	−	0.623245i	$-0.785814\pi$
$163$	−1.40960e10	−1.56405	−0.782027	+	0.623245i	$0.785814\pi$
$164$	0	0
$165$	−7.24471e6	−0.000760927 0
$166$	0	0
$167$	−3.56028e9	−0.354210	−0.177105	−	0.984192i	$-0.556673\pi$
$167$	−3.56028e9	−0.354210	−0.177105	+	0.984192i	$0.556673\pi$
$168$	0	0
$169$	4.41184e9	0.416034
$170$	0	0
$171$	1.49926e10	1.34090
$172$	0	0
$173$	2.64068e9	0.224134	0.112067	−	0.993701i	$-0.464253\pi$
$173$	2.64068e9	0.224134	0.112067	+	0.993701i	$0.464253\pi$
$174$	0	0
$175$	−3.01983e9	−0.243395
$176$	0	0
$177$	1.40208e7	0.00107372
$178$	0	0
$179$	−1.26973e9	−0.0924430	−0.0462215	−	0.998931i	$-0.514718\pi$
$179$	−1.26973e9	−0.0924430	−0.0462215	+	0.998931i	$0.514718\pi$
$180$	0	0
$181$	−2.17242e10	−1.50449	−0.752247	−	0.658881i	$-0.771030\pi$
$181$	−2.17242e10	−1.50449	−0.752247	+	0.658881i	$0.771030\pi$
$182$	0	0
$183$	2.19411e6	0.000144620 0
$184$	0	0
$185$	5.44037e9	0.341473
$186$	0	0
$187$	−5.77614e9	−0.345422
$188$	0	0
$189$	−2.19601e7	−0.00125186
$190$	0	0
$191$	1.07035e10	0.581937	0.290968	−	0.956733i	$-0.406023\pi$
$191$	1.07035e10	0.581937	0.290968	+	0.956733i	$0.406023\pi$
$192$	0	0
$193$	7.66063e9	0.397426	0.198713	−	0.980058i	$-0.436324\pi$
$193$	7.66063e9	0.397426	0.198713	+	0.980058i	$0.436324\pi$
$194$	0	0
$195$	−5.10170e7	−0.00252673
$196$	0	0
$197$	−4.26160e9	−0.201592	−0.100796	−	0.994907i	$-0.532139\pi$
$197$	−4.26160e9	−0.201592	−0.100796	+	0.994907i	$0.532139\pi$
$198$	0	0
$199$	−3.16291e10	−1.42971	−0.714856	−	0.699272i	$-0.753508\pi$
$199$	−3.16291e10	−1.42971	−0.714856	+	0.699272i	$0.753508\pi$
$200$	0	0
$201$	−5.09256e7	−0.00220066
$202$	0	0
$203$	−1.52368e9	−0.0629742
$204$	0	0
$205$	1.32064e10	0.522266
$206$	0	0
$207$	2.42590e10	0.918344
$208$	0	0
$209$	1.32549e10	0.480526
$210$	0	0
$211$	−1.76561e10	−0.613229	−0.306615	−	0.951834i	$-0.599196\pi$
$211$	−1.76561e10	−0.613229	−0.306615	+	0.951834i	$0.599196\pi$
$212$	0	0
$213$	−1.29817e7	−0.000432140 0
$214$	0	0
$215$	−3.70903e10	−1.18383
$216$	0	0
$217$	−1.29190e10	−0.395513
$218$	0	0
$219$	−1.05559e8	−0.00310097
$220$	0	0
$221$	−4.06754e10	−1.14701
$222$	0	0
$223$	1.04686e10	0.283476	0.141738	−	0.989904i	$-0.454731\pi$
$223$	1.04686e10	0.283476	0.141738	+	0.989904i	$0.454731\pi$
$224$	0	0
$225$	−2.47560e10	−0.643961
$226$	0	0
$227$	−1.95043e10	−0.487543	−0.243772	−	0.969833i	$-0.578385\pi$
$227$	−1.95043e10	−0.487543	−0.243772	+	0.969833i	$0.578385\pi$
$228$	0	0
$229$	−5.96135e10	−1.43247	−0.716234	−	0.697860i	$-0.754136\pi$
$229$	−5.96135e10	−1.43247	−0.716234	+	0.697860i	$0.754136\pi$
$230$	0	0
$231$	−9.70738e6	−0.000224310 0
$232$	0	0
$233$	8.42619e10	1.87296	0.936482	−	0.350716i	$-0.114062\pi$
$233$	8.42619e10	1.87296	0.936482	+	0.350716i	$0.114062\pi$
$234$	0	0
$235$	3.64885e10	0.780460
$236$	0	0
$237$	1.04798e7	0.000215767 0
$238$	0	0
$239$	7.28353e10	1.44395	0.721974	−	0.691920i	$-0.243235\pi$
$239$	7.28353e10	1.44395	0.721974	+	0.691920i	$0.243235\pi$
$240$	0	0
$241$	4.45555e10	0.850795	0.425397	−	0.905007i	$-0.360134\pi$
$241$	4.45555e10	0.850795	0.425397	+	0.905007i	$0.360134\pi$
$242$	0	0
$243$	−2.70038e8	−0.00496817
$244$	0	0
$245$	−1.03299e10	−0.183167
$246$	0	0
$247$	9.33404e10	1.59564
$248$	0	0
$249$	−7.77557e7	−0.00128184
$250$	0	0
$251$	9.55068e10	1.51881	0.759404	−	0.650620i	$-0.225491\pi$
$251$	9.55068e10	1.51881	0.759404	+	0.650620i	$0.225491\pi$
$252$	0	0
$253$	2.14472e10	0.329100
$254$	0	0
$255$	1.38192e8	0.00204669
$256$	0	0
$257$	9.12794e10	1.30519	0.652595	−	0.757707i	$-0.273680\pi$
$257$	9.12794e10	1.30519	0.652595	+	0.757707i	$0.273680\pi$
$258$	0	0
$259$	7.28970e9	0.100661
$260$	0	0
$261$	−1.24909e10	−0.166614
$262$	0	0
$263$	−1.50976e11	−1.94584	−0.972921	−	0.231138i	$-0.925755\pi$
$263$	−1.50976e11	−1.94584	−0.972921	+	0.231138i	$0.925755\pi$
$264$	0	0
$265$	1.07044e11	1.33338
$266$	0	0
$267$	−1.51458e8	−0.00182387
$268$	0	0
$269$	2.39622e10	0.279024	0.139512	−	0.990220i	$-0.455447\pi$
$269$	2.39622e10	0.279024	0.139512	+	0.990220i	$0.455447\pi$
$270$	0	0
$271$	5.51248e10	0.620848	0.310424	−	0.950598i	$-0.399529\pi$
$271$	5.51248e10	0.620848	0.310424	+	0.950598i	$0.399529\pi$
$272$	0	0
$273$	−6.83591e7	−0.000744842 0
$274$	0	0
$275$	−2.18866e10	−0.230771
$276$	0	0
$277$	2.19632e10	0.224149	0.112074	−	0.993700i	$-0.464251\pi$
$277$	2.19632e10	0.224149	0.112074	+	0.993700i	$0.464251\pi$
$278$	0	0
$279$	−1.05908e11	−1.04643
$280$	0	0
$281$	−1.62670e10	−0.155643	−0.0778213	−	0.996967i	$-0.524796\pi$
$281$	−1.62670e10	−0.155643	−0.0778213	+	0.996967i	$0.524796\pi$
$282$	0	0
$283$	−1.35590e10	−0.125657	−0.0628287	−	0.998024i	$-0.520012\pi$
$283$	−1.35590e10	−0.125657	−0.0628287	+	0.998024i	$0.520012\pi$
$284$	0	0
$285$	−3.17118e8	−0.00284721
$286$	0	0
$287$	1.76956e10	0.153956
$288$	0	0
$289$	−8.40858e9	−0.0709059
$290$	0	0
$291$	3.31790e8	0.00271235
$292$	0	0
$293$	5.41900e10	0.429551	0.214776	−	0.976663i	$-0.431098\pi$
$293$	5.41900e10	0.429551	0.214776	+	0.976663i	$0.431098\pi$
$294$	0	0
$295$	1.08133e11	0.831306
$296$	0	0
$297$	−1.59159e8	−0.00118693
$298$	0	0
$299$	1.51030e11	1.09281
$300$	0	0
$301$	−4.96983e10	−0.348974
$302$	0	0
$303$	−2.07138e8	−0.00141178
$304$	0	0
$305$	1.69218e10	0.111969
$306$	0	0
$307$	−7.32498e10	−0.470634	−0.235317	−	0.971919i	$-0.575613\pi$
$307$	−7.32498e10	−0.470634	−0.235317	+	0.971919i	$0.575613\pi$
$308$	0	0
$309$	−1.65425e8	−0.00103226
$310$	0	0
$311$	2.44502e11	1.48204	0.741021	−	0.671482i	$-0.234342\pi$
$311$	2.44502e11	1.48204	0.741021	+	0.671482i	$0.234342\pi$
$312$	0	0
$313$	−4.39895e10	−0.259060	−0.129530	−	0.991576i	$-0.541347\pi$
$313$	−4.39895e10	−0.259060	−0.129530	+	0.991576i	$0.541347\pi$
$314$	0	0
$315$	−8.46824e10	−0.484614
$316$	0	0
$317$	−2.17653e11	−1.21059	−0.605297	−	0.795999i	$-0.706946\pi$
$317$	−2.17653e11	−1.21059	−0.605297	+	0.795999i	$0.706946\pi$
$318$	0	0
$319$	−1.10431e10	−0.0597080
$320$	0	0
$321$	5.77591e8	0.00303632
$322$	0	0
$323$	−2.52835e11	−1.29249
$324$	0	0
$325$	−1.54125e11	−0.766298
$326$	0	0
$327$	−8.90917e7	−0.000430896 0
$328$	0	0
$329$	4.88920e10	0.230068
$330$	0	0
$331$	3.66945e11	1.68026	0.840128	−	0.542389i	$-0.182480\pi$
$331$	3.66945e11	1.68026	0.840128	+	0.542389i	$0.182480\pi$
$332$	0	0
$333$	5.97596e10	0.266323
$334$	0	0
$335$	−3.92758e11	−1.70382
$336$	0	0
$337$	1.99451e11	0.842367	0.421183	−	0.906976i	$-0.361615\pi$
$337$	1.99451e11	0.842367	0.421183	+	0.906976i	$0.361615\pi$
$338$	0	0
$339$	−5.56642e8	−0.00228917
$340$	0	0
$341$	−9.36322e10	−0.374999
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	0	0
$345$	−5.13116e8	−0.00194998
$346$	0	0
$347$	−6.63180e10	−0.245555	−0.122777	−	0.992434i	$-0.539180\pi$
$347$	−6.63180e10	−0.245555	−0.122777	+	0.992434i	$0.539180\pi$
$348$	0	0
$349$	−1.20301e11	−0.434065	−0.217033	−	0.976164i	$-0.569638\pi$
$349$	−1.20301e11	−0.434065	−0.217033	+	0.976164i	$0.569638\pi$
$350$	0	0
$351$	−1.12079e9	−0.00394133
$352$	0	0
$353$	2.43060e11	0.833159	0.416580	−	0.909099i	$-0.363229\pi$
$353$	2.43060e11	0.833159	0.416580	+	0.909099i	$0.363229\pi$
$354$	0	0
$355$	−1.00120e11	−0.334575
$356$	0	0
$357$	1.85167e8	0.000603333 0
$358$	0	0
$359$	−2.81654e11	−0.894933	−0.447466	−	0.894301i	$-0.647674\pi$
$359$	−2.81654e11	−0.894933	−0.447466	+	0.894301i	$0.647674\pi$
$360$	0	0
$361$	2.57510e11	0.798015
$362$	0	0
$363$	4.77488e8	0.00144339
$364$	0	0
$365$	−8.14113e11	−2.40086
$366$	0	0
$367$	2.41000e11	0.693458	0.346729	−	0.937965i	$-0.387292\pi$
$367$	2.41000e11	0.693458	0.346729	+	0.937965i	$0.387292\pi$
$368$	0	0
$369$	1.45065e11	0.407328
$370$	0	0
$371$	1.43431e11	0.393061
$372$	0	0
$373$	−1.33921e11	−0.358227	−0.179114	−	0.983828i	$-0.557323\pi$
$373$	−1.33921e11	−0.358227	−0.179114	+	0.983828i	$0.557323\pi$
$374$	0	0
$375$	−2.89507e8	−0.000755993 0
$376$	0	0
$377$	−7.77651e10	−0.198266
$378$	0	0
$379$	−5.67201e11	−1.41208	−0.706042	−	0.708170i	$-0.749521\pi$
$379$	−5.67201e11	−1.41208	−0.706042	+	0.708170i	$0.749521\pi$
$380$	0	0
$381$	4.37404e8	0.00106346
$382$	0	0
$383$	1.03509e11	0.245800	0.122900	−	0.992419i	$-0.460781\pi$
$383$	1.03509e11	0.245800	0.122900	+	0.992419i	$0.460781\pi$
$384$	0	0
$385$	−7.48671e10	−0.173667
$386$	0	0
$387$	−4.07417e11	−0.923295
$388$	0	0
$389$	−5.83600e11	−1.29224	−0.646119	−	0.763237i	$-0.723609\pi$
$389$	−5.83600e11	−1.29224	−0.646119	+	0.763237i	$0.723609\pi$
$390$	0	0
$391$	−4.09102e11	−0.885190
$392$	0	0
$393$	−3.46746e8	−0.000733238 0
$394$	0	0
$395$	8.08243e10	0.167053
$396$	0	0
$397$	−6.54542e10	−0.132245	−0.0661226	−	0.997812i	$-0.521063\pi$
$397$	−6.54542e10	−0.132245	−0.0661226	+	0.997812i	$0.521063\pi$
$398$	0	0
$399$	−4.24915e8	−0.000839313 0
$400$	0	0
$401$	4.06647e10	0.0785359	0.0392679	−	0.999229i	$-0.487497\pi$
$401$	4.06647e10	0.0785359	0.0392679	+	0.999229i	$0.487497\pi$
$402$	0	0
$403$	−6.59356e11	−1.24522
$404$	0	0
$405$	−6.94209e11	−1.28216
$406$	0	0
$407$	5.28330e10	0.0954400
$408$	0	0
$409$	1.06163e12	1.87594	0.937970	−	0.346716i	$-0.112703\pi$
$409$	1.06163e12	1.87594	0.937970	+	0.346716i	$0.112703\pi$
$410$	0	0
$411$	7.46081e8	0.00128973
$412$	0	0
$413$	1.44891e11	0.245056
$414$	0	0
$415$	−5.99682e11	−0.992441
$416$	0	0
$417$	−7.06157e8	−0.00114364
$418$	0	0
$419$	1.19335e12	1.89149	0.945744	−	0.324912i	$-0.105335\pi$
$419$	1.19335e12	1.89149	0.945744	+	0.324912i	$0.105335\pi$
$420$	0	0
$421$	−5.88397e11	−0.912853	−0.456427	−	0.889761i	$-0.650871\pi$
$421$	−5.88397e11	−0.912853	−0.456427	+	0.889761i	$0.650871\pi$
$422$	0	0
$423$	4.00807e11	0.608701
$424$	0	0
$425$	4.17485e11	0.620713
$426$	0	0
$427$	2.26740e10	0.0330068
$428$	0	0
$429$	−4.95441e8	−0.000706210 0
$430$	0	0
$431$	1.22932e12	1.71599	0.857997	−	0.513655i	$-0.171709\pi$
$431$	1.22932e12	1.71599	0.857997	+	0.513655i	$0.171709\pi$
$432$	0	0
$433$	−4.73819e10	−0.0647764	−0.0323882	−	0.999475i	$-0.510311\pi$
$433$	−4.73819e10	−0.0647764	−0.0323882	+	0.999475i	$0.510311\pi$
$434$	0	0
$435$	2.64202e8	0.000353781 0
$436$	0	0
$437$	9.38793e11	1.23141
$438$	0	0
$439$	−6.14236e11	−0.789305	−0.394652	−	0.918830i	$-0.629135\pi$
$439$	−6.14236e11	−0.789305	−0.394652	+	0.918830i	$0.629135\pi$
$440$	0	0
$441$	−1.13468e11	−0.142857
$442$	0	0
$443$	8.19199e11	1.01058	0.505292	−	0.862948i	$-0.331385\pi$
$443$	8.19199e11	1.01058	0.505292	+	0.862948i	$0.331385\pi$
$444$	0	0
$445$	−1.16811e12	−1.41209
$446$	0	0
$447$	8.46379e8	0.00100272
$448$	0	0
$449$	4.81029e11	0.558551	0.279276	−	0.960211i	$-0.409906\pi$
$449$	4.81029e11	0.558551	0.279276	+	0.960211i	$0.409906\pi$
$450$	0	0
$451$	1.28251e11	0.145971
$452$	0	0
$453$	1.17532e9	0.00131133
$454$	0	0
$455$	−5.27212e11	−0.576679
$456$	0	0
$457$	1.15328e12	1.23684	0.618419	−	0.785849i	$-0.287774\pi$
$457$	1.15328e12	1.23684	0.618419	+	0.785849i	$0.287774\pi$
$458$	0	0
$459$	3.03594e9	0.00319254
$460$	0	0
$461$	−6.06986e11	−0.625928	−0.312964	−	0.949765i	$-0.601322\pi$
$461$	−6.06986e11	−0.625928	−0.312964	+	0.949765i	$0.601322\pi$
$462$	0	0
$463$	8.87758e11	0.897801	0.448900	−	0.893582i	$-0.351816\pi$
$463$	8.87758e11	0.897801	0.448900	+	0.893582i	$0.351816\pi$
$464$	0	0
$465$	2.24012e9	0.00222194
$466$	0	0
$467$	1.09779e12	1.06806	0.534029	−	0.845466i	$-0.320677\pi$
$467$	1.09779e12	1.06806	0.534029	+	0.845466i	$0.320677\pi$
$468$	0	0
$469$	−5.26267e11	−0.502260
$470$	0	0
$471$	2.78079e9	0.00260360
$472$	0	0
$473$	−3.60195e11	−0.330874
$474$	0	0
$475$	−9.58029e11	−0.863490
$476$	0	0
$477$	1.17582e12	1.03994
$478$	0	0
$479$	3.12229e11	0.270996	0.135498	−	0.990778i	$-0.456737\pi$
$479$	3.12229e11	0.270996	0.135498	+	0.990778i	$0.456737\pi$
$480$	0	0
$481$	3.72048e11	0.316918
$482$	0	0
$483$	−6.87537e8	−0.000574823 0
$484$	0	0
$485$	2.55889e12	2.09998
$486$	0	0
$487$	−1.08377e12	−0.873088	−0.436544	−	0.899683i	$-0.643798\pi$
$487$	−1.08377e12	−0.873088	−0.436544	+	0.899683i	$0.643798\pi$
$488$	0	0
$489$	3.27505e9	0.00259017
$490$	0	0
$491$	−2.14702e12	−1.66713	−0.833567	−	0.552419i	$-0.813705\pi$
$491$	−2.14702e12	−1.66713	−0.833567	+	0.552419i	$0.813705\pi$
$492$	0	0
$493$	2.10646e11	0.160599
$494$	0	0
$495$	−6.13746e11	−0.459479
$496$	0	0
$497$	−1.34154e11	−0.0986277
$498$	0	0
$499$	1.85705e12	1.34082	0.670411	−	0.741990i	$-0.266118\pi$
$499$	1.85705e12	1.34082	0.670411	+	0.741990i	$0.266118\pi$
$500$	0	0
$501$	8.27193e8	0.000586593 0
$502$	0	0
$503$	1.59040e12	1.10777	0.553885	−	0.832593i	$-0.313145\pi$
$503$	1.59040e12	1.10777	0.553885	+	0.832593i	$0.313145\pi$
$504$	0	0
$505$	−1.59753e12	−1.09304
$506$	0	0
$507$	−1.02504e9	−0.000688979 0
$508$	0	0
$509$	2.32864e12	1.53771	0.768853	−	0.639426i	$-0.220828\pi$
$509$	2.32864e12	1.53771	0.768853	+	0.639426i	$0.220828\pi$
$510$	0	0
$511$	−1.09085e12	−0.707737
$512$	0	0
$513$	−6.96676e9	−0.00444122
$514$	0	0
$515$	−1.27582e12	−0.799205
$516$	0	0
$517$	3.54350e11	0.218135
$518$	0	0
$519$	−6.13532e8	−0.000371180 0
$520$	0	0
$521$	−1.48366e12	−0.882197	−0.441099	−	0.897459i	$-0.645411\pi$
$521$	−1.48366e12	−0.882197	−0.441099	+	0.897459i	$0.645411\pi$
$522$	0	0
$523$	1.95584e12	1.14308	0.571540	−	0.820574i	$-0.306346\pi$
$523$	1.95584e12	1.14308	0.571540	+	0.820574i	$0.306346\pi$
$524$	0	0
$525$	7.01625e8	0.000403077 0
$526$	0	0
$527$	1.78603e12	1.00865
$528$	0	0
$529$	−2.82131e11	−0.156639
$530$	0	0
$531$	1.18779e12	0.648357
$532$	0	0
$533$	9.03139e11	0.484710
$534$	0	0
$535$	4.45460e12	2.35081
$536$	0	0
$537$	2.95009e8	0.000153091 0
$538$	0	0
$539$	−1.00316e11	−0.0511944
$540$	0	0
$541$	−1.34802e12	−0.676563	−0.338281	−	0.941045i	$-0.609846\pi$
$541$	−1.34802e12	−0.676563	−0.338281	+	0.941045i	$0.609846\pi$
$542$	0	0
$543$	5.04738e9	0.00249154
$544$	0	0
$545$	−6.87109e11	−0.333612
$546$	0	0
$547$	−1.02503e12	−0.489548	−0.244774	−	0.969580i	$-0.578714\pi$
$547$	−1.02503e12	−0.489548	−0.244774	+	0.969580i	$0.578714\pi$
$548$	0	0
$549$	1.85877e11	0.0873275
$550$	0	0
$551$	−4.83382e11	−0.223413
$552$	0	0
$553$	1.08299e11	0.0492448
$554$	0	0
$555$	−1.26401e9	−0.000565500 0
$556$	0	0
$557$	1.75343e12	0.771864	0.385932	−	0.922527i	$-0.373880\pi$
$557$	1.75343e12	0.771864	0.385932	+	0.922527i	$0.373880\pi$
$558$	0	0
$559$	−2.53648e12	−1.09870
$560$	0	0
$561$	1.34202e9	0.000572041 0
$562$	0	0
$563$	2.17456e12	0.912185	0.456092	−	0.889932i	$-0.349249\pi$
$563$	2.17456e12	0.912185	0.456092	+	0.889932i	$0.349249\pi$
$564$	0	0
$565$	−4.29304e12	−1.77234
$566$	0	0
$567$	−9.30189e11	−0.377961
$568$	0	0
$569$	3.13812e12	1.25506	0.627531	−	0.778592i	$-0.284066\pi$
$569$	3.13812e12	1.25506	0.627531	+	0.778592i	$0.284066\pi$
$570$	0	0
$571$	−2.05979e12	−0.810887	−0.405444	−	0.914120i	$-0.632883\pi$
$571$	−2.05979e12	−0.810887	−0.405444	+	0.914120i	$0.632883\pi$
$572$	0	0
$573$	−2.48684e9	−0.000963723 0
$574$	0	0
$575$	−1.55015e12	−0.591381
$576$	0	0
$577$	3.94380e12	1.48123	0.740617	−	0.671927i	$-0.234533\pi$
$577$	3.94380e12	1.48123	0.740617	+	0.671927i	$0.234533\pi$
$578$	0	0
$579$	−1.77986e9	−0.000658162 0
$580$	0	0
$581$	−8.03530e11	−0.292556
$582$	0	0
$583$	1.03953e12	0.372675
$584$	0	0
$585$	−4.32198e12	−1.52574
$586$	0	0
$587$	−3.68239e12	−1.28014	−0.640071	−	0.768315i	$-0.721095\pi$
$587$	−3.68239e12	−1.28014	−0.640071	+	0.768315i	$0.721095\pi$
$588$	0	0
$589$	−4.09851e12	−1.40316
$590$	0	0
$591$	9.90135e8	0.000333850 0
$592$	0	0
$593$	−1.17114e12	−0.388921	−0.194460	−	0.980910i	$-0.562296\pi$
$593$	−1.17114e12	−0.388921	−0.194460	+	0.980910i	$0.562296\pi$
$594$	0	0
$595$	1.42808e12	0.467118
$596$	0	0
$597$	7.34868e9	0.00236769
$598$	0	0
$599$	−2.83757e12	−0.900586	−0.450293	−	0.892881i	$-0.648681\pi$
$599$	−2.83757e12	−0.900586	−0.450293	+	0.892881i	$0.648681\pi$
$600$	0	0
$601$	2.37066e12	0.741198	0.370599	−	0.928793i	$-0.379152\pi$
$601$	2.37066e12	0.741198	0.370599	+	0.928793i	$0.379152\pi$
$602$	0	0
$603$	−4.31424e12	−1.32885
$604$	0	0
$605$	3.68257e12	1.11751
$606$	0	0
$607$	−4.40080e12	−1.31578	−0.657889	−	0.753115i	$-0.728550\pi$
$607$	−4.40080e12	−1.31578	−0.657889	+	0.753115i	$0.728550\pi$
$608$	0	0
$609$	3.54011e8	0.000104289 0
$610$	0	0
$611$	2.49533e12	0.724339
$612$	0	0
$613$	−1.87570e12	−0.536527	−0.268264	−	0.963346i	$-0.586450\pi$
$613$	−1.87570e12	−0.536527	−0.268264	+	0.963346i	$0.586450\pi$
$614$	0	0
$615$	−3.06836e9	−0.000864904 0
$616$	0	0
$617$	−3.18170e12	−0.883845	−0.441922	−	0.897053i	$-0.645703\pi$
$617$	−3.18170e12	−0.883845	−0.441922	+	0.897053i	$0.645703\pi$
$618$	0	0
$619$	1.16476e11	0.0318880	0.0159440	−	0.999873i	$-0.494925\pi$
$619$	1.16476e11	0.0318880	0.0159440	+	0.999873i	$0.494925\pi$
$620$	0	0
$621$	−1.12726e10	−0.00304167
$622$	0	0
$623$	−1.56518e12	−0.416263
$624$	0	0
$625$	−4.68931e12	−1.22927
$626$	0	0
$627$	−3.07962e9	−0.000795781 0
$628$	0	0
$629$	−1.00778e12	−0.256708
$630$	0	0
$631$	3.71283e12	0.932338	0.466169	−	0.884696i	$-0.345634\pi$
$631$	3.71283e12	0.932338	0.466169	+	0.884696i	$0.345634\pi$
$632$	0	0
$633$	4.10219e9	0.00101555
$634$	0	0
$635$	3.37343e12	0.823360
$636$	0	0
$637$	−7.06425e11	−0.169996
$638$	0	0
$639$	−1.09977e12	−0.260944
$640$	0	0
$641$	2.93764e11	0.0687285	0.0343642	−	0.999409i	$-0.489059\pi$
$641$	2.93764e11	0.0687285	0.0343642	+	0.999409i	$0.489059\pi$
$642$	0	0
$643$	−5.27980e12	−1.21806	−0.609029	−	0.793148i	$-0.708441\pi$
$643$	−5.27980e12	−1.21806	−0.609029	+	0.793148i	$0.708441\pi$
$644$	0	0
$645$	8.61753e9	0.00196049
$646$	0	0
$647$	2.62000e12	0.587804	0.293902	−	0.955836i	$-0.405046\pi$
$647$	2.62000e12	0.587804	0.293902	+	0.955836i	$0.405046\pi$
$648$	0	0
$649$	1.05011e12	0.232346
$650$	0	0
$651$	3.00160e9	0.000654995 0
$652$	0	0
$653$	3.15918e12	0.679931	0.339965	−	0.940438i	$-0.389585\pi$
$653$	3.15918e12	0.679931	0.339965	+	0.940438i	$0.389585\pi$
$654$	0	0
$655$	−2.67424e12	−0.567694
$656$	0	0
$657$	−8.94260e12	−1.87249
$658$	0	0
$659$	−8.77292e11	−0.181201	−0.0906004	−	0.995887i	$-0.528879\pi$
$659$	−8.77292e11	−0.181201	−0.0906004	+	0.995887i	$0.528879\pi$
$660$	0	0
$661$	8.22108e12	1.67503	0.837514	−	0.546416i	$-0.184008\pi$
$661$	8.22108e12	1.67503	0.837514	+	0.546416i	$0.184008\pi$
$662$	0	0
$663$	9.45049e9	0.00189952
$664$	0	0
$665$	−3.27711e12	−0.649821
$666$	0	0
$667$	−7.82140e11	−0.153010
$668$	0	0
$669$	−2.43226e9	−0.000469454 0
$670$	0	0
$671$	1.64333e11	0.0312948
$672$	0	0
$673$	−7.47304e12	−1.40420	−0.702101	−	0.712077i	$-0.747754\pi$
$673$	−7.47304e12	−1.40420	−0.702101	+	0.712077i	$0.747754\pi$
$674$	0	0
$675$	1.15036e10	0.00213288
$676$	0	0
$677$	8.71309e12	1.59413	0.797064	−	0.603895i	$-0.206386\pi$
$677$	8.71309e12	1.59413	0.797064	+	0.603895i	$0.206386\pi$
$678$	0	0
$679$	3.42873e12	0.619041
$680$	0	0
$681$	4.53160e9	0.000807402 0
$682$	0	0
$683$	3.73903e12	0.657455	0.328727	−	0.944425i	$-0.393380\pi$
$683$	3.73903e12	0.657455	0.328727	+	0.944425i	$0.393380\pi$
$684$	0	0
$685$	5.75407e12	0.998545
$686$	0	0
$687$	1.38505e10	0.00237226
$688$	0	0
$689$	7.32036e12	1.23750
$690$	0	0
$691$	6.26971e12	1.04616	0.523078	−	0.852285i	$-0.324784\pi$
$691$	6.26971e12	1.04616	0.523078	+	0.852285i	$0.324784\pi$
$692$	0	0
$693$	−8.22375e11	−0.135447
$694$	0	0
$695$	−5.44616e12	−0.885439
$696$	0	0
$697$	−2.44637e12	−0.392623
$698$	0	0
$699$	−1.95773e10	−0.00310174
$700$	0	0
$701$	−5.13904e12	−0.803805	−0.401903	−	0.915682i	$-0.631651\pi$
$701$	−5.13904e12	−0.803805	−0.401903	+	0.915682i	$0.631651\pi$
$702$	0	0
$703$	2.31263e12	0.357114
$704$	0	0
$705$	−8.47771e9	−0.00129249
$706$	0	0
$707$	−2.14057e12	−0.322212
$708$	0	0
$709$	−2.95438e12	−0.439094	−0.219547	−	0.975602i	$-0.570458\pi$
$709$	−2.95438e12	−0.439094	−0.219547	+	0.975602i	$0.570458\pi$
$710$	0	0
$711$	8.87813e11	0.130289
$712$	0	0
$713$	−6.63162e12	−0.960986
$714$	0	0
$715$	−3.82103e12	−0.546768
$716$	0	0
$717$	−1.69225e10	−0.00239127
$718$	0	0
$719$	1.21819e13	1.69994	0.849971	−	0.526829i	$-0.176619\pi$
$719$	1.21819e13	1.69994	0.849971	+	0.526829i	$0.176619\pi$
$720$	0	0
$721$	−1.70951e12	−0.235594
$722$	0	0
$723$	−1.03520e10	−0.00140897
$724$	0	0
$725$	7.98167e11	0.107293
$726$	0	0
$727$	−1.26291e13	−1.67675	−0.838376	−	0.545093i	$-0.816494\pi$
$727$	−1.26291e13	−1.67675	−0.838376	+	0.545093i	$0.816494\pi$
$728$	0	0
$729$	−7.62547e12	−0.999984
$730$	0	0
$731$	6.87068e12	0.889962
$732$	0	0
$733$	9.16594e12	1.17276	0.586380	−	0.810036i	$-0.300553\pi$
$733$	9.16594e12	1.17276	0.586380	+	0.810036i	$0.300553\pi$
$734$	0	0
$735$	2.40003e9	0.000303336 0
$736$	0	0
$737$	−3.81418e12	−0.476209
$738$	0	0
$739$	−1.10658e13	−1.36484	−0.682420	−	0.730960i	$-0.739073\pi$
$739$	−1.10658e13	−1.36484	−0.682420	+	0.730960i	$0.739073\pi$
$740$	0	0
$741$	−2.16866e10	−0.00264247
$742$	0	0
$743$	−1.84642e12	−0.222270	−0.111135	−	0.993805i	$-0.535449\pi$
$743$	−1.84642e12	−0.222270	−0.111135	+	0.993805i	$0.535449\pi$
$744$	0	0
$745$	6.52760e12	0.776337
$746$	0	0
$747$	−6.58719e12	−0.774029
$748$	0	0
$749$	5.96884e12	0.692982
$750$	0	0
$751$	9.19672e12	1.05500	0.527501	−	0.849555i	$-0.323129\pi$
$751$	9.19672e12	1.05500	0.527501	+	0.849555i	$0.323129\pi$
$752$	0	0
$753$	−2.21900e10	−0.00251524
$754$	0	0
$755$	9.06449e12	1.01527
$756$	0	0
$757$	8.81778e12	0.975950	0.487975	−	0.872858i	$-0.337736\pi$
$757$	8.81778e12	0.975950	0.487975	+	0.872858i	$0.337736\pi$
$758$	0	0
$759$	−4.98301e9	−0.000545009 0
$760$	0	0
$761$	−2.42043e12	−0.261614	−0.130807	−	0.991408i	$-0.541757\pi$
$761$	−2.42043e12	−0.261614	−0.130807	+	0.991408i	$0.541757\pi$
$762$	0	0
$763$	−9.20676e11	−0.0983437
$764$	0	0
$765$	1.17071e13	1.23588
$766$	0	0
$767$	7.39488e12	0.771528
$768$	0	0
$769$	7.67945e12	0.791885	0.395942	−	0.918275i	$-0.370418\pi$
$769$	7.67945e12	0.791885	0.395942	+	0.918275i	$0.370418\pi$
$770$	0	0
$771$	−2.12078e10	−0.00216148
$772$	0	0
$773$	1.04693e13	1.05466	0.527328	−	0.849662i	$-0.323194\pi$
$773$	1.04693e13	1.05466	0.527328	+	0.849662i	$0.323194\pi$
$774$	0	0
$775$	6.76751e12	0.673862
$776$	0	0
$777$	−1.69368e9	−0.000166701 0
$778$	0	0
$779$	5.61385e12	0.546188
$780$	0	0
$781$	−9.72296e11	−0.0935123
$782$	0	0
$783$	5.80424e9	0.000551846 0
$784$	0	0
$785$	2.14465e13	2.01578
$786$	0	0
$787$	1.26490e13	1.17535	0.587677	−	0.809095i	$-0.300042\pi$
$787$	1.26490e13	1.17535	0.587677	+	0.809095i	$0.300042\pi$
$788$	0	0
$789$	3.50777e10	0.00322243
$790$	0	0
$791$	−5.75236e12	−0.522459
$792$	0	0
$793$	1.15723e12	0.103918
$794$	0	0
$795$	−2.48705e10	−0.00220817
$796$	0	0
$797$	−6.02337e11	−0.0528782	−0.0264391	−	0.999650i	$-0.508417\pi$
$797$	−6.02337e11	−0.0528782	−0.0264391	+	0.999650i	$0.508417\pi$
$798$	0	0
$799$	−6.75920e12	−0.586725
$800$	0	0
$801$	−1.28310e13	−1.10132
$802$	0	0
$803$	−7.90608e12	−0.671029
$804$	0	0
$805$	−5.30256e12	−0.445045
$806$	0	0
$807$	−5.56736e9	−0.000462081 0
$808$	0	0
$809$	−7.72967e12	−0.634443	−0.317222	−	0.948351i	$-0.602750\pi$
$809$	−7.72967e12	−0.634443	−0.317222	+	0.948351i	$0.602750\pi$
$810$	0	0
$811$	1.20157e13	0.975335	0.487668	−	0.873029i	$-0.337848\pi$
$811$	1.20157e13	0.975335	0.487668	+	0.873029i	$0.337848\pi$
$812$	0	0
$813$	−1.28077e10	−0.00102816
$814$	0	0
$815$	2.52585e13	2.00538
$816$	0	0
$817$	−1.57666e13	−1.23805
$818$	0	0
$819$	−5.79114e12	−0.449766
$820$	0	0
$821$	1.39952e13	1.07507	0.537533	−	0.843243i	$-0.319356\pi$
$821$	1.39952e13	1.07507	0.537533	+	0.843243i	$0.319356\pi$
$822$	0	0
$823$	−1.61982e13	−1.23074	−0.615371	−	0.788238i	$-0.710994\pi$
$823$	−1.61982e13	−1.23074	−0.615371	+	0.788238i	$0.710994\pi$
$824$	0	0
$825$	5.08511e9	0.000382171 0
$826$	0	0
$827$	1.30127e13	0.967374	0.483687	−	0.875241i	$-0.339297\pi$
$827$	1.30127e13	0.967374	0.483687	+	0.875241i	$0.339297\pi$
$828$	0	0
$829$	−1.31364e13	−0.966007	−0.483004	−	0.875618i	$-0.660454\pi$
$829$	−1.31364e13	−0.966007	−0.483004	+	0.875618i	$0.660454\pi$
$830$	0	0
$831$	−5.10290e9	−0.000371204 0
$832$	0	0
$833$	1.91353e12	0.137699
$834$	0	0
$835$	6.37963e12	0.454157
$836$	0	0
$837$	4.92131e10	0.00346590
$838$	0	0
$839$	5.22420e12	0.363991	0.181996	−	0.983299i	$-0.441744\pi$
$839$	5.22420e12	0.363991	0.181996	+	0.983299i	$0.441744\pi$
$840$	0	0
$841$	−1.41044e13	−0.972240
$842$	0	0
$843$	3.77945e9	0.000257754 0
$844$	0	0
$845$	−7.90552e12	−0.533427
$846$	0	0
$847$	4.93438e12	0.329425
$848$	0	0
$849$	3.15028e9	0.000208096 0
$850$	0	0
$851$	3.74196e12	0.244578
$852$	0	0
$853$	−5.65076e12	−0.365457	−0.182728	−	0.983163i	$-0.558493\pi$
$853$	−5.65076e12	−0.365457	−0.182728	+	0.983163i	$0.558493\pi$
$854$	0	0
$855$	−2.68651e13	−1.71926
$856$	0	0
$857$	1.42516e12	0.0902503	0.0451251	−	0.998981i	$-0.485631\pi$
$857$	1.42516e12	0.0902503	0.0451251	+	0.998981i	$0.485631\pi$
$858$	0	0
$859$	−2.77961e13	−1.74186	−0.870932	−	0.491404i	$-0.836484\pi$
$859$	−2.77961e13	−1.74186	−0.870932	+	0.491404i	$0.836484\pi$
$860$	0	0
$861$	−4.11137e9	−0.000254960 0
$862$	0	0
$863$	1.14117e13	0.700326	0.350163	−	0.936689i	$-0.386126\pi$
$863$	1.14117e13	0.700326	0.350163	+	0.936689i	$0.386126\pi$
$864$	0	0
$865$	−4.73180e12	−0.287378
$866$	0	0
$867$	1.95364e9	0.000117425 0
$868$	0	0
$869$	7.84908e11	0.0466906
$870$	0	0
$871$	−2.68594e13	−1.58130
$872$	0	0
$873$	2.81081e13	1.63782
$874$	0	0
$875$	−2.99177e12	−0.172541
$876$	0	0
$877$	−1.21188e13	−0.691769	−0.345884	−	0.938277i	$-0.612421\pi$
$877$	−1.21188e13	−0.691769	−0.345884	+	0.938277i	$0.612421\pi$
$878$	0	0
$879$	−1.25905e10	−0.000711364 0
$880$	0	0
$881$	−4.60741e12	−0.257671	−0.128836	−	0.991666i	$-0.541124\pi$
$881$	−4.60741e12	−0.257671	−0.128836	+	0.991666i	$0.541124\pi$
$882$	0	0
$883$	−1.04709e13	−0.579642	−0.289821	−	0.957081i	$-0.593596\pi$
$883$	−1.04709e13	−0.579642	−0.289821	+	0.957081i	$0.593596\pi$
$884$	0	0
$885$	−2.51236e10	−0.00137669
$886$	0	0
$887$	3.68711e12	0.200000	0.0999999	−	0.994987i	$-0.468116\pi$
$887$	3.68711e12	0.200000	0.0999999	+	0.994987i	$0.468116\pi$
$888$	0	0
$889$	4.52015e12	0.242714
$890$	0	0
$891$	−6.74166e12	−0.358358
$892$	0	0
$893$	1.55108e13	0.816209
$894$	0	0
$895$	2.27522e12	0.118528
$896$	0	0
$897$	−3.50902e10	−0.00180976
$898$	0	0
$899$	3.41461e12	0.174350
$900$	0	0
$901$	−1.98290e13	−1.00240
$902$	0	0
$903$	1.15469e10	0.000577922 0
$904$	0	0
$905$	3.89274e13	1.92902
$906$	0	0
$907$	6.05209e11	0.0296943	0.0148471	−	0.999890i	$-0.495274\pi$
$907$	6.05209e11	0.0296943	0.0148471	+	0.999890i	$0.495274\pi$
$908$	0	0
$909$	−1.75480e13	−0.852491
$910$	0	0
$911$	3.27450e12	0.157512	0.0787559	−	0.996894i	$-0.474905\pi$
$911$	3.27450e12	0.157512	0.0787559	+	0.996894i	$0.474905\pi$
$912$	0	0
$913$	−5.82368e12	−0.277383
$914$	0	0
$915$	−3.93160e9	−0.000185428 0
$916$	0	0
$917$	−3.58329e12	−0.167348
$918$	0	0
$919$	−2.21117e11	−0.0102259	−0.00511295	−	0.999987i	$-0.501628\pi$
$919$	−2.21117e11	−0.0102259	−0.00511295	+	0.999987i	$0.501628\pi$
$920$	0	0
$921$	1.70188e10	0.000779400 0
$922$	0	0
$923$	−6.84688e12	−0.310517
$924$	0	0
$925$	−3.81864e12	−0.171503
$926$	0	0
$927$	−1.40143e13	−0.623320
$928$	0	0
$929$	−4.00147e12	−0.176258	−0.0881290	−	0.996109i	$-0.528089\pi$
$929$	−4.00147e12	−0.176258	−0.0881290	+	0.996109i	$0.528089\pi$
$930$	0	0
$931$	−4.39109e12	−0.191557
$932$	0	0
$933$	−5.68073e10	−0.00245435
$934$	0	0
$935$	1.03502e13	0.442890
$936$	0	0
$937$	−3.76138e10	−0.00159411	−0.000797056	−	1.00000i	$-0.500254\pi$
$937$	−3.76138e10	−0.00159411	−0.000797056		1.00000i	$0.500254\pi$
$938$	0	0
$939$	1.02205e10	0.000429019 0
$940$	0	0
$941$	−2.21494e12	−0.0920894	−0.0460447	−	0.998939i	$-0.514662\pi$
$941$	−2.21494e12	−0.0920894	−0.0460447	+	0.998939i	$0.514662\pi$
$942$	0	0
$943$	9.08353e12	0.374069
$944$	0	0
$945$	3.93501e10	0.00160510
$946$	0	0
$947$	−2.36266e13	−0.954610	−0.477305	−	0.878738i	$-0.658386\pi$
$947$	−2.36266e13	−0.954610	−0.477305	+	0.878738i	$0.658386\pi$
$948$	0	0
$949$	−5.56744e13	−2.22822
$950$	0	0
$951$	5.05694e10	0.00200482
$952$	0	0
$953$	−1.15037e13	−0.451771	−0.225885	−	0.974154i	$-0.572527\pi$
$953$	−1.15037e13	−0.451771	−0.225885	+	0.974154i	$0.572527\pi$
$954$	0	0
$955$	−1.91795e13	−0.746142
$956$	0	0
$957$	2.56574e9	9.88801e−5 0
$958$	0	0
$959$	7.71003e12	0.294356
$960$	0	0
$961$	2.51218e12	0.0950156
$962$	0	0
$963$	4.89314e13	1.83345
$964$	0	0
$965$	−1.37270e13	−0.509568
$966$	0	0
$967$	1.36863e13	0.503346	0.251673	−	0.967812i	$-0.419019\pi$
$967$	1.36863e13	0.503346	0.251673	+	0.967812i	$0.419019\pi$
$968$	0	0
$969$	5.87435e10	0.00214044
$970$	0	0
$971$	−9.95259e12	−0.359294	−0.179647	−	0.983731i	$-0.557496\pi$
$971$	−9.95259e12	−0.359294	−0.179647	+	0.983731i	$0.557496\pi$
$972$	0	0
$973$	−7.29745e12	−0.261014
$974$	0	0
$975$	3.58092e10	0.00126904
$976$	0	0
$977$	1.51387e13	0.531575	0.265787	−	0.964032i	$-0.414368\pi$
$977$	1.51387e13	0.531575	0.265787	+	0.964032i	$0.414368\pi$
$978$	0	0
$979$	−1.13438e13	−0.394673
$980$	0	0
$981$	−7.54753e12	−0.260192
$982$	0	0
$983$	−5.47141e13	−1.86899	−0.934497	−	0.355970i	$-0.884151\pi$
$983$	−5.47141e13	−1.86899	−0.934497	+	0.355970i	$0.884151\pi$
$984$	0	0
$985$	7.63631e12	0.258476
$986$	0	0
$987$	−1.13595e10	−0.000381007 0
$988$	0	0
$989$	−2.55112e13	−0.847907
$990$	0	0
$991$	−4.86384e13	−1.60195	−0.800974	−	0.598699i	$-0.795684\pi$
$991$	−4.86384e13	−1.60195	−0.800974	+	0.598699i	$0.795684\pi$
$992$	0	0
$993$	−8.52557e10	−0.00278261
$994$	0	0
$995$	5.66759e13	1.83314
$996$	0	0
$997$	2.77758e13	0.890304	0.445152	−	0.895455i	$-0.353150\pi$
$997$	2.77758e13	0.890304	0.445152	+	0.895455i	$0.353150\pi$
$998$	0	0
$999$	−2.77690e10	−0.000882096 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.a.h.1.2		3
4.3	odd	2		7.10.a.b.1.3	✓	3
12.11	even	2		63.10.a.e.1.1		3
20.3	even	4		175.10.b.d.99.1		6
20.7	even	4		175.10.b.d.99.6		6
20.19	odd	2		175.10.a.d.1.1		3
28.3	even	6		49.10.c.e.30.1		6
28.11	odd	6		49.10.c.d.30.1		6
28.19	even	6		49.10.c.e.18.1		6
28.23	odd	6		49.10.c.d.18.1		6
28.27	even	2		49.10.a.c.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.a.b.1.3	✓	3	4.3	odd	2
49.10.a.c.1.3		3	28.27	even	2
49.10.c.d.18.1		6	28.23	odd	6
49.10.c.d.30.1		6	28.11	odd	6
49.10.c.e.18.1		6	28.19	even	6
49.10.c.e.30.1		6	28.3	even	6
63.10.a.e.1.1		3	12.11	even	2
112.10.a.h.1.2		3	1.1	even	1	trivial
175.10.a.d.1.1		3	20.19	odd	2
175.10.b.d.99.1		6	20.3	even	4
175.10.b.d.99.6		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)

\(f(q)\)	\(=\)	\(q-0.232339 q^{3} -1791.89 q^{5} -2401.00 q^{7} -19682.9 q^{9} -17401.5 q^{11} -122541. q^{13} +416.326 q^{15} +331933. q^{17} -761707. q^{19} +557.846 q^{21} -1.23249e6 q^{23} +1.25774e6 q^{25} +9146.25 q^{27} +634604. q^{29} +5.38069e6 q^{31} +4043.06 q^{33} +4.30232e6 q^{35} -3.03611e6 q^{37} +28471.1 q^{39} -7.37009e6 q^{41} +2.06990e7 q^{43} +3.52696e7 q^{45} -2.03632e7 q^{47} +5.76480e6 q^{49} -77120.9 q^{51} -5.97380e7 q^{53} +3.11816e7 q^{55} +176974. q^{57} -6.03461e7 q^{59} -9.44357e6 q^{61} +4.72588e7 q^{63} +2.19580e8 q^{65} +2.19187e8 q^{67} +286355. q^{69} +5.58741e7 q^{71} +4.54332e8 q^{73} -292222. q^{75} +4.17811e7 q^{77} -4.51057e7 q^{79} +3.87417e8 q^{81} +3.34665e8 q^{83} -5.94786e8 q^{85} -147443. q^{87} +6.51886e8 q^{89} +2.94221e8 q^{91} -1.25014e6 q^{93} +1.36489e9 q^{95} -1.42804e9 q^{97} +3.42514e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 84 q^{3} + 1554 q^{5} - 7203 q^{7} - 26001 q^{9} + 3444 q^{11} - 19782 q^{13} - 200304 q^{15} + 1016694 q^{17} - 222852 q^{19} + 201684 q^{21} - 1885632 q^{23} + 3073221 q^{25} - 551880 q^{27} + 4081818 q^{29}+ \cdots + 1900979172 q^{99}+O(q^{100}) \) 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

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