LMFDB - Embedded newform 112.10.a.j.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(1,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-84,0,1540] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$57.6840136504$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 63x^{2} - 176x + 28 $ x^4 - 63x^2 - 176x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{3}\cdot 17 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.74786$ 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)

$f(q)$	$=$	$q-96.1756 q^{3} -594.465 q^{5} +2401.00 q^{7} -10433.3 q^{9} -69095.5 q^{11} +46210.6 q^{13} +57173.0 q^{15} +243586. q^{17} -147349. q^{19} -230918. q^{21} -2.45805e6 q^{23} -1.59974e6 q^{25} +2.89645e6 q^{27} -611750. q^{29} -5.64287e6 q^{31} +6.64530e6 q^{33} -1.42731e6 q^{35} +5.68230e6 q^{37} -4.44434e6 q^{39} +2.54760e7 q^{41} -1.31131e7 q^{43} +6.20220e6 q^{45} -2.83428e7 q^{47} +5.76480e6 q^{49} -2.34270e7 q^{51} +8.12628e7 q^{53} +4.10748e7 q^{55} +1.41713e7 q^{57} +1.20116e8 q^{59} +3.80693e7 q^{61} -2.50503e7 q^{63} -2.74706e7 q^{65} -2.37967e6 q^{67} +2.36404e8 q^{69} +3.07615e8 q^{71} +2.88540e7 q^{73} +1.53856e8 q^{75} -1.65898e8 q^{77} +3.73894e7 q^{79} -7.32098e7 q^{81} +5.42856e8 q^{83} -1.44803e8 q^{85} +5.88354e7 q^{87} +5.78562e8 q^{89} +1.10952e8 q^{91} +5.42707e8 q^{93} +8.75935e7 q^{95} -1.52799e9 q^{97} +7.20891e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 84 q^{3} + 1540 q^{5} + 9604 q^{7} + 48060 q^{9} + 53016 q^{11} + 172732 q^{13} - 228096 q^{15} - 128352 q^{17} - 85148 q^{19} - 201684 q^{21} + 342368 q^{23} + 4030196 q^{25} - 11284056 q^{27} + 6392064 q^{29}+ \cdots + 3054793464 q^{99}+O(q^{100}) $ 4 * q - 84 * q^3 + 1540 * q^5 + 9604 * q^7 + 48060 * q^9 + 53016 * q^11 + 172732 * q^13 - 228096 * q^15 - 128352 * q^17 - 85148 * q^19 - 201684 * q^21 + 342368 * q^23 + 4030196 * q^25 - 11284056 * q^27 + 6392064 * q^29 - 3692920 * q^31 + 2660448 * q^33 + 3697540 * q^35 + 20408192 * q^37 - 54281520 * q^39 + 51466576 * q^41 - 35333016 * q^43 + 140298228 * q^45 - 70879592 * q^47 + 23059204 * q^49 + 14919624 * q^51 + 127260312 * q^53 + 43963920 * q^55 + 244217976 * q^57 + 80548804 * q^59 + 92246084 * q^61 + 115392060 * q^63 + 419393416 * q^65 + 522443376 * q^67 - 365219904 * q^69 + 104419600 * q^71 - 60133192 * q^73 + 567926436 * q^75 + 127291416 * q^77 + 1069501840 * q^79 + 816541884 * q^81 + 1270627540 * q^83 - 199027992 * q^85 + 2194287480 * q^87 - 356859384 * q^89 + 414729532 * q^91 - 1074588912 * q^93 + 2083565536 * q^95 + 1288837312 * q^97 + 3054793464 * 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
$3$	−96.1756	−0.685518	−0.342759	−	0.939423i	$-0.611362\pi$
$3$	−96.1756	−0.685518	−0.342759	+	0.939423i	$0.611362\pi$
$4$	0	0
$5$	−594.465	−0.425364	−0.212682	−	0.977121i	$-0.568220\pi$
$5$	−594.465	−0.425364	−0.212682	+	0.977121i	$0.568220\pi$
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	0	0
$9$	−10433.3	−0.530064
$10$	0	0
$11$	−69095.5	−1.42293	−0.711463	−	0.702723i	$-0.751967\pi$
$11$	−69095.5	−1.42293	−0.711463	+	0.702723i	$0.751967\pi$
$12$	0	0
$13$	46210.6	0.448742	0.224371	−	0.974504i	$-0.427967\pi$
$13$	46210.6	0.448742	0.224371	+	0.974504i	$0.427967\pi$
$14$	0	0
$15$	57173.0	0.291595
$16$	0	0
$17$	243586.	0.707347	0.353674	−	0.935369i	$-0.384932\pi$
$17$	243586.	0.707347	0.353674	+	0.935369i	$0.384932\pi$
$18$	0	0
$19$	−147349.	−0.259391	−0.129695	−	0.991554i	$-0.541400\pi$
$19$	−147349.	−0.259391	−0.129695	+	0.991554i	$0.541400\pi$
$20$	0	0
$21$	−230918.	−0.259102
$22$	0	0
$23$	−2.45805e6	−1.83153	−0.915767	−	0.401711i	$-0.868416\pi$
$23$	−2.45805e6	−1.83153	−0.915767	+	0.401711i	$0.868416\pi$
$24$	0	0
$25$	−1.59974e6	−0.819065
$26$	0	0
$27$	2.89645e6	1.04889
$28$	0	0
$29$	−611750.	−0.160614	−0.0803069	−	0.996770i	$-0.525590\pi$
$29$	−611750.	−0.160614	−0.0803069	+	0.996770i	$0.525590\pi$
$30$	0	0
$31$	−5.64287e6	−1.09742	−0.548710	−	0.836013i	$-0.684881\pi$
$31$	−5.64287e6	−1.09742	−0.548710	+	0.836013i	$0.684881\pi$
$32$	0	0
$33$	6.64530e6	0.975442
$34$	0	0
$35$	−1.42731e6	−0.160773
$36$	0	0
$37$	5.68230e6	0.498444	0.249222	−	0.968446i	$-0.419825\pi$
$37$	5.68230e6	0.498444	0.249222	+	0.968446i	$0.419825\pi$
$38$	0	0
$39$	−4.44434e6	−0.307621
$40$	0	0
$41$	2.54760e7	1.40801	0.704003	−	0.710197i	$-0.251394\pi$
$41$	2.54760e7	1.40801	0.704003	+	0.710197i	$0.251394\pi$
$42$	0	0
$43$	−1.31131e7	−0.584923	−0.292461	−	0.956277i	$-0.594474\pi$
$43$	−1.31131e7	−0.584923	−0.292461	+	0.956277i	$0.594474\pi$
$44$	0	0
$45$	6.20220e6	0.225470
$46$	0	0
$47$	−2.83428e7	−0.847232	−0.423616	−	0.905842i	$-0.639239\pi$
$47$	−2.83428e7	−0.847232	−0.423616	+	0.905842i	$0.639239\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	0	0
$51$	−2.34270e7	−0.484900
$52$	0	0
$53$	8.12628e7	1.41465	0.707327	−	0.706886i	$-0.249901\pi$
$53$	8.12628e7	1.41465	0.707327	+	0.706886i	$0.249901\pi$
$54$	0	0
$55$	4.10748e7	0.605262
$56$	0	0
$57$	1.41713e7	0.177817
$58$	0	0
$59$	1.20116e8	1.29053	0.645265	−	0.763958i	$-0.276747\pi$
$59$	1.20116e8	1.29053	0.645265	+	0.763958i	$0.276747\pi$
$60$	0	0
$61$	3.80693e7	0.352039	0.176019	−	0.984387i	$-0.443678\pi$
$61$	3.80693e7	0.352039	0.176019	+	0.984387i	$0.443678\pi$
$62$	0	0
$63$	−2.50503e7	−0.200346
$64$	0	0
$65$	−2.74706e7	−0.190879
$66$	0	0
$67$	−2.37967e6	−0.0144271	−0.00721357	−	0.999974i	$-0.502296\pi$
$67$	−2.37967e6	−0.0144271	−0.00721357	+	0.999974i	$0.502296\pi$
$68$	0	0
$69$	2.36404e8	1.25555
$70$	0	0
$71$	3.07615e8	1.43663	0.718315	−	0.695718i	$-0.244914\pi$
$71$	3.07615e8	1.43663	0.718315	+	0.695718i	$0.244914\pi$
$72$	0	0
$73$	2.88540e7	0.118919	0.0594597	−	0.998231i	$-0.481062\pi$
$73$	2.88540e7	0.118919	0.0594597	+	0.998231i	$0.481062\pi$
$74$	0	0
$75$	1.53856e8	0.561484
$76$	0	0
$77$	−1.65898e8	−0.537816
$78$	0	0
$79$	3.73894e7	0.108001	0.0540004	−	0.998541i	$-0.482803\pi$
$79$	3.73894e7	0.108001	0.0540004	+	0.998541i	$0.482803\pi$
$80$	0	0
$81$	−7.32098e7	−0.188967
$82$	0	0
$83$	5.42856e8	1.25555	0.627774	−	0.778395i	$-0.283966\pi$
$83$	5.42856e8	1.25555	0.627774	+	0.778395i	$0.283966\pi$
$84$	0	0
$85$	−1.44803e8	−0.300880
$86$	0	0
$87$	5.88354e7	0.110104
$88$	0	0
$89$	5.78562e8	0.977451	0.488726	−	0.872438i	$-0.337462\pi$
$89$	5.78562e8	0.977451	0.488726	+	0.872438i	$0.337462\pi$
$90$	0	0
$91$	1.10952e8	0.169609
$92$	0	0
$93$	5.42707e8	0.752302
$94$	0	0
$95$	8.75935e7	0.110336
$96$	0	0
$97$	−1.52799e9	−1.75246	−0.876229	−	0.481895i	$-0.839949\pi$
$97$	−1.52799e9	−1.75246	−0.876229	+	0.481895i	$0.839949\pi$
$98$	0	0
$99$	7.20891e8	0.754243
$100$	0	0
$101$	−8.18511e8	−0.782670	−0.391335	−	0.920248i	$-0.627987\pi$
$101$	−8.18511e8	−0.782670	−0.391335	+	0.920248i	$0.627987\pi$
$102$	0	0
$103$	−2.24948e7	−0.0196931	−0.00984654	−	0.999952i	$-0.503134\pi$
$103$	−2.24948e7	−0.0196931	−0.00984654	+	0.999952i	$0.503134\pi$
$104$	0	0
$105$	1.37272e8	0.110213
$106$	0	0
$107$	1.93505e9	1.42714	0.713569	−	0.700585i	$-0.247077\pi$
$107$	1.93505e9	1.42714	0.713569	+	0.700585i	$0.247077\pi$
$108$	0	0
$109$	1.89146e9	1.28345	0.641723	−	0.766936i	$-0.278220\pi$
$109$	1.89146e9	1.28345	0.641723	+	0.766936i	$0.278220\pi$
$110$	0	0
$111$	−5.46498e8	−0.341693
$112$	0	0
$113$	2.10443e9	1.21418	0.607088	−	0.794634i	$-0.292337\pi$
$113$	2.10443e9	1.21418	0.607088	+	0.794634i	$0.292337\pi$
$114$	0	0
$115$	1.46122e9	0.779069
$116$	0	0
$117$	−4.82128e8	−0.237862
$118$	0	0
$119$	5.84851e8	0.267352
$120$	0	0
$121$	2.41623e9	1.02472
$122$	0	0
$123$	−2.45017e9	−0.965214
$124$	0	0
$125$	2.11205e9	0.773765
$126$	0	0
$127$	−1.05219e9	−0.358902	−0.179451	−	0.983767i	$-0.557432\pi$
$127$	−1.05219e9	−0.358902	−0.179451	+	0.983767i	$0.557432\pi$
$128$	0	0
$129$	1.26116e9	0.400975
$130$	0	0
$131$	−3.74529e9	−1.11113	−0.555565	−	0.831473i	$-0.687498\pi$
$131$	−3.74529e9	−1.11113	−0.555565	+	0.831473i	$0.687498\pi$
$132$	0	0
$133$	−3.53784e8	−0.0980405
$134$	0	0
$135$	−1.72184e9	−0.446159
$136$	0	0
$137$	−7.39345e8	−0.179310	−0.0896550	−	0.995973i	$-0.528576\pi$
$137$	−7.39345e8	−0.179310	−0.0896550	+	0.995973i	$0.528576\pi$
$138$	0	0
$139$	6.76179e8	0.153637	0.0768184	−	0.997045i	$-0.475524\pi$
$139$	6.76179e8	0.153637	0.0768184	+	0.997045i	$0.475524\pi$
$140$	0	0
$141$	2.72589e9	0.580793
$142$	0	0
$143$	−3.19295e9	−0.638527
$144$	0	0
$145$	3.63664e8	0.0683194
$146$	0	0
$147$	−5.54433e8	−0.0979312
$148$	0	0
$149$	−5.98176e9	−0.994240	−0.497120	−	0.867682i	$-0.665609\pi$
$149$	−5.98176e9	−0.994240	−0.497120	+	0.867682i	$0.665609\pi$
$150$	0	0
$151$	1.85428e9	0.290254	0.145127	−	0.989413i	$-0.453641\pi$
$151$	1.85428e9	0.290254	0.145127	+	0.989413i	$0.453641\pi$
$152$	0	0
$153$	−2.54140e9	−0.374940
$154$	0	0
$155$	3.35449e9	0.466803
$156$	0	0
$157$	−9.51640e8	−0.125004	−0.0625020	−	0.998045i	$-0.519908\pi$
$157$	−9.51640e8	−0.125004	−0.0625020	+	0.998045i	$0.519908\pi$
$158$	0	0
$159$	−7.81550e9	−0.969772
$160$	0	0
$161$	−5.90177e9	−0.692254
$162$	0	0
$163$	−4.68783e9	−0.520149	−0.260075	−	0.965589i	$-0.583747\pi$
$163$	−4.68783e9	−0.520149	−0.260075	+	0.965589i	$0.583747\pi$
$164$	0	0
$165$	−3.95039e9	−0.414918
$166$	0	0
$167$	−8.14061e9	−0.809903	−0.404952	−	0.914338i	$-0.632712\pi$
$167$	−8.14061e9	−0.809903	−0.404952	+	0.914338i	$0.632712\pi$
$168$	0	0
$169$	−8.46908e9	−0.798630
$170$	0	0
$171$	1.53733e9	0.137494
$172$	0	0
$173$	−8.53803e9	−0.724687	−0.362343	−	0.932045i	$-0.618023\pi$
$173$	−8.53803e9	−0.724687	−0.362343	+	0.932045i	$0.618023\pi$
$174$	0	0
$175$	−3.84097e9	−0.309578
$176$	0	0
$177$	−1.15523e10	−0.884683
$178$	0	0
$179$	1.36010e10	0.990223	0.495111	−	0.868829i	$-0.335127\pi$
$179$	1.36010e10	0.990223	0.495111	+	0.868829i	$0.335127\pi$
$180$	0	0
$181$	−8.22877e8	−0.0569878	−0.0284939	−	0.999594i	$-0.509071\pi$
$181$	−8.22877e8	−0.0569878	−0.0284939	+	0.999594i	$0.509071\pi$
$182$	0	0
$183$	−3.66134e9	−0.241329
$184$	0	0
$185$	−3.37793e9	−0.212020
$186$	0	0
$187$	−1.68307e10	−1.00650
$188$	0	0
$189$	6.95437e9	0.396442
$190$	0	0
$191$	1.09353e10	0.594538	0.297269	−	0.954794i	$-0.403924\pi$
$191$	1.09353e10	0.594538	0.297269	+	0.954794i	$0.403924\pi$
$192$	0	0
$193$	−2.85178e10	−1.47948	−0.739740	−	0.672893i	$-0.765051\pi$
$193$	−2.85178e10	−1.47948	−0.739740	+	0.672893i	$0.765051\pi$
$194$	0	0
$195$	2.64200e9	0.130851
$196$	0	0
$197$	2.78862e10	1.31914	0.659570	−	0.751643i	$-0.270738\pi$
$197$	2.78862e10	1.31914	0.659570	+	0.751643i	$0.270738\pi$
$198$	0	0
$199$	2.39154e10	1.08103	0.540516	−	0.841334i	$-0.318229\pi$
$199$	2.39154e10	1.08103	0.540516	+	0.841334i	$0.318229\pi$
$200$	0	0
$201$	2.28866e8	0.00989007
$202$	0	0
$203$	−1.46881e9	−0.0607063
$204$	0	0
$205$	−1.51446e10	−0.598915
$206$	0	0
$207$	2.56454e10	0.970831
$208$	0	0
$209$	1.01811e10	0.369094
$210$	0	0
$211$	−1.11219e10	−0.386286	−0.193143	−	0.981171i	$-0.561868\pi$
$211$	−1.11219e10	−0.386286	−0.193143	+	0.981171i	$0.561868\pi$
$212$	0	0
$213$	−2.95850e10	−0.984836
$214$	0	0
$215$	7.79530e9	0.248805
$216$	0	0
$217$	−1.35485e10	−0.414786
$218$	0	0
$219$	−2.77505e9	−0.0815215
$220$	0	0
$221$	1.12563e10	0.317417
$222$	0	0
$223$	−5.43946e10	−1.47294	−0.736469	−	0.676472i	$-0.763508\pi$
$223$	−5.43946e10	−1.47294	−0.736469	+	0.676472i	$0.763508\pi$
$224$	0	0
$225$	1.66905e10	0.434157
$226$	0	0
$227$	−3.41199e10	−0.852888	−0.426444	−	0.904514i	$-0.640234\pi$
$227$	−3.41199e10	−0.852888	−0.426444	+	0.904514i	$0.640234\pi$
$228$	0	0
$229$	8.30367e10	1.99531	0.997655	−	0.0684494i	$-0.0218052\pi$
$229$	8.30367e10	1.99531	0.997655	+	0.0684494i	$0.0218052\pi$
$230$	0	0
$231$	1.59554e10	0.368683
$232$	0	0
$233$	−3.62840e10	−0.806517	−0.403258	−	0.915086i	$-0.632122\pi$
$233$	−3.62840e10	−0.806517	−0.403258	+	0.915086i	$0.632122\pi$
$234$	0	0
$235$	1.68488e10	0.360382
$236$	0	0
$237$	−3.59595e9	−0.0740365
$238$	0	0
$239$	6.35940e10	1.26074	0.630370	−	0.776295i	$-0.282903\pi$
$239$	6.35940e10	1.26074	0.630370	+	0.776295i	$0.282903\pi$
$240$	0	0
$241$	8.74376e10	1.66963	0.834817	−	0.550527i	$-0.185573\pi$
$241$	8.74376e10	1.66963	0.834817	+	0.550527i	$0.185573\pi$
$242$	0	0
$243$	−4.99698e10	−0.919347
$244$	0	0
$245$	−3.42697e9	−0.0607663
$246$	0	0
$247$	−6.80907e9	−0.116400
$248$	0	0
$249$	−5.22095e10	−0.860702
$250$	0	0
$251$	−3.84566e10	−0.611560	−0.305780	−	0.952102i	$-0.598917\pi$
$251$	−3.84566e10	−0.611560	−0.305780	+	0.952102i	$0.598917\pi$
$252$	0	0
$253$	1.69840e11	2.60614
$254$	0	0
$255$	1.39265e10	0.206259
$256$	0	0
$257$	3.95178e10	0.565059	0.282530	−	0.959259i	$-0.408826\pi$
$257$	3.95178e10	0.565059	0.282530	+	0.959259i	$0.408826\pi$
$258$	0	0
$259$	1.36432e10	0.188394
$260$	0	0
$261$	6.38254e9	0.0851357
$262$	0	0
$263$	3.33651e10	0.430023	0.215012	−	0.976612i	$-0.431021\pi$
$263$	3.33651e10	0.430023	0.215012	+	0.976612i	$0.431021\pi$
$264$	0	0
$265$	−4.83079e10	−0.601743
$266$	0	0
$267$	−5.56436e10	−0.670061
$268$	0	0
$269$	−2.93430e10	−0.341679	−0.170840	−	0.985299i	$-0.554648\pi$
$269$	−2.93430e10	−0.341679	−0.170840	+	0.985299i	$0.554648\pi$
$270$	0	0
$271$	−1.32333e11	−1.49042	−0.745209	−	0.666831i	$-0.767650\pi$
$271$	−1.32333e11	−1.49042	−0.745209	+	0.666831i	$0.767650\pi$
$272$	0	0
$273$	−1.06709e10	−0.116270
$274$	0	0
$275$	1.10535e11	1.16547
$276$	0	0
$277$	8.13179e10	0.829903	0.414951	−	0.909844i	$-0.363799\pi$
$277$	8.13179e10	0.829903	0.414951	+	0.909844i	$0.363799\pi$
$278$	0	0
$279$	5.88736e10	0.581703
$280$	0	0
$281$	−4.82697e10	−0.461845	−0.230922	−	0.972972i	$-0.574174\pi$
$281$	−4.82697e10	−0.461845	−0.230922	+	0.972972i	$0.574174\pi$
$282$	0	0
$283$	1.80311e11	1.67103	0.835514	−	0.549469i	$-0.185170\pi$
$283$	1.80311e11	1.67103	0.835514	+	0.549469i	$0.185170\pi$
$284$	0	0
$285$	−8.42435e9	−0.0756371
$286$	0	0
$287$	6.11680e10	0.532176
$288$	0	0
$289$	−5.92536e10	−0.499660
$290$	0	0
$291$	1.46955e11	1.20134
$292$	0	0
$293$	2.18759e11	1.73405	0.867026	−	0.498262i	$-0.166028\pi$
$293$	2.18759e11	1.73405	0.867026	+	0.498262i	$0.166028\pi$
$294$	0	0
$295$	−7.14050e10	−0.548946
$296$	0	0
$297$	−2.00131e11	−1.49249
$298$	0	0
$299$	−1.13588e11	−0.821886
$300$	0	0
$301$	−3.14846e10	−0.221080
$302$	0	0
$303$	7.87208e10	0.536534
$304$	0	0
$305$	−2.26309e10	−0.149745
$306$	0	0
$307$	1.73635e11	1.11561	0.557807	−	0.829971i	$-0.311643\pi$
$307$	1.73635e11	1.11561	0.557807	+	0.829971i	$0.311643\pi$
$308$	0	0
$309$	2.16345e9	0.0135000
$310$	0	0
$311$	1.11473e11	0.675688	0.337844	−	0.941202i	$-0.390302\pi$
$311$	1.11473e11	0.675688	0.337844	+	0.941202i	$0.390302\pi$
$312$	0	0
$313$	2.38241e11	1.40303	0.701514	−	0.712656i	$-0.252508\pi$
$313$	2.38241e11	1.40303	0.701514	+	0.712656i	$0.252508\pi$
$314$	0	0
$315$	1.48915e10	0.0852198
$316$	0	0
$317$	−1.80726e11	−1.00520	−0.502602	−	0.864518i	$-0.667624\pi$
$317$	−1.80726e11	−1.00520	−0.502602	+	0.864518i	$0.667624\pi$
$318$	0	0
$319$	4.22691e10	0.228542
$320$	0	0
$321$	−1.86105e11	−0.978330
$322$	0	0
$323$	−3.58921e10	−0.183479
$324$	0	0
$325$	−7.39249e10	−0.367549
$326$	0	0
$327$	−1.81912e11	−0.879826
$328$	0	0
$329$	−6.80511e10	−0.320224
$330$	0	0
$331$	−1.72924e11	−0.791826	−0.395913	−	0.918288i	$-0.629572\pi$
$331$	−1.72924e11	−0.791826	−0.395913	+	0.918288i	$0.629572\pi$
$332$	0	0
$333$	−5.92849e10	−0.264207
$334$	0	0
$335$	1.41463e9	0.00613679
$336$	0	0
$337$	−1.92313e10	−0.0812220	−0.0406110	−	0.999175i	$-0.512930\pi$
$337$	−1.92313e10	−0.0812220	−0.0406110	+	0.999175i	$0.512930\pi$
$338$	0	0
$339$	−2.02395e11	−0.832340
$340$	0	0
$341$	3.89897e11	1.56155
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	0	0
$345$	−1.40534e11	−0.534066
$346$	0	0
$347$	1.90347e11	0.704797	0.352399	−	0.935850i	$-0.385366\pi$
$347$	1.90347e11	0.704797	0.352399	+	0.935850i	$0.385366\pi$
$348$	0	0
$349$	3.18032e11	1.14751	0.573755	−	0.819027i	$-0.305486\pi$
$349$	3.18032e11	1.14751	0.573755	+	0.819027i	$0.305486\pi$
$350$	0	0
$351$	1.33847e11	0.470680
$352$	0	0
$353$	−4.99824e11	−1.71329	−0.856645	−	0.515907i	$-0.827455\pi$
$353$	−4.99824e11	−1.71329	−0.856645	+	0.515907i	$0.827455\pi$
$354$	0	0
$355$	−1.82866e11	−0.611091
$356$	0	0
$357$	−5.62483e10	−0.183275
$358$	0	0
$359$	2.40479e11	0.764104	0.382052	−	0.924141i	$-0.375218\pi$
$359$	2.40479e11	0.764104	0.382052	+	0.924141i	$0.375218\pi$
$360$	0	0
$361$	−3.00976e11	−0.932716
$362$	0	0
$363$	−2.32383e11	−0.702464
$364$	0	0
$365$	−1.71527e10	−0.0505841
$366$	0	0
$367$	−6.37649e11	−1.83478	−0.917390	−	0.397989i	$-0.869708\pi$
$367$	−6.37649e11	−1.83478	−0.917390	+	0.397989i	$0.869708\pi$
$368$	0	0
$369$	−2.65798e11	−0.746334
$370$	0	0
$371$	1.95112e11	0.534689
$372$	0	0
$373$	1.15352e11	0.308557	0.154278	−	0.988027i	$-0.450695\pi$
$373$	1.15352e11	0.308557	0.154278	+	0.988027i	$0.450695\pi$
$374$	0	0
$375$	−2.03128e11	−0.530430
$376$	0	0
$377$	−2.82694e10	−0.0720742
$378$	0	0
$379$	−3.66454e11	−0.912310	−0.456155	−	0.889900i	$-0.650774\pi$
$379$	−3.66454e11	−0.912310	−0.456155	+	0.889900i	$0.650774\pi$
$380$	0	0
$381$	1.01195e11	0.246034
$382$	0	0
$383$	−6.31066e11	−1.49858	−0.749290	−	0.662242i	$-0.769605\pi$
$383$	−6.31066e11	−1.49858	−0.749290	+	0.662242i	$0.769605\pi$
$384$	0	0
$385$	9.86206e10	0.228767
$386$	0	0
$387$	1.36813e11	0.310047
$388$	0	0
$389$	−2.05717e11	−0.455508	−0.227754	−	0.973719i	$-0.573138\pi$
$389$	−2.05717e11	−0.455508	−0.227754	+	0.973719i	$0.573138\pi$
$390$	0	0
$391$	−5.98746e11	−1.29553
$392$	0	0
$393$	3.60205e11	0.761700
$394$	0	0
$395$	−2.22267e10	−0.0459396
$396$	0	0
$397$	6.71192e11	1.35609	0.678047	−	0.735019i	$-0.262827\pi$
$397$	6.71192e11	1.35609	0.678047	+	0.735019i	$0.262827\pi$
$398$	0	0
$399$	3.40254e10	0.0672086
$400$	0	0
$401$	−3.16015e11	−0.610320	−0.305160	−	0.952301i	$-0.598710\pi$
$401$	−3.16015e11	−0.610320	−0.305160	+	0.952301i	$0.598710\pi$
$402$	0	0
$403$	−2.60761e11	−0.492459
$404$	0	0
$405$	4.35206e10	0.0803799
$406$	0	0
$407$	−3.92621e11	−0.709249
$408$	0	0
$409$	−1.07807e12	−1.90498	−0.952492	−	0.304562i	$-0.901490\pi$
$409$	−1.07807e12	−1.90498	−0.952492	+	0.304562i	$0.901490\pi$
$410$	0	0
$411$	7.11069e10	0.122920
$412$	0	0
$413$	2.88400e11	0.487775
$414$	0	0
$415$	−3.22709e11	−0.534065
$416$	0	0
$417$	−6.50319e10	−0.105321
$418$	0	0
$419$	7.79914e11	1.23619	0.618093	−	0.786105i	$-0.287906\pi$
$419$	7.79914e11	1.23619	0.618093	+	0.786105i	$0.287906\pi$
$420$	0	0
$421$	8.76084e11	1.35918	0.679589	−	0.733593i	$-0.262158\pi$
$421$	8.76084e11	1.35918	0.679589	+	0.733593i	$0.262158\pi$
$422$	0	0
$423$	2.95708e11	0.449088
$424$	0	0
$425$	−3.89674e11	−0.579363
$426$	0	0
$427$	9.14044e10	0.133058
$428$	0	0
$429$	3.07083e11	0.437722
$430$	0	0
$431$	1.23778e12	1.72780	0.863902	−	0.503660i	$-0.168013\pi$
$431$	1.23778e12	1.72780	0.863902	+	0.503660i	$0.168013\pi$
$432$	0	0
$433$	1.00463e12	1.37344	0.686722	−	0.726921i	$-0.259049\pi$
$433$	1.00463e12	1.37344	0.686722	+	0.726921i	$0.259049\pi$
$434$	0	0
$435$	−3.49756e10	−0.0468342
$436$	0	0
$437$	3.62190e11	0.475083
$438$	0	0
$439$	9.38698e11	1.20625	0.603123	−	0.797648i	$-0.293923\pi$
$439$	9.38698e11	1.20625	0.603123	+	0.797648i	$0.293923\pi$
$440$	0	0
$441$	−6.01457e10	−0.0757235
$442$	0	0
$443$	7.88802e11	0.973086	0.486543	−	0.873657i	$-0.338258\pi$
$443$	7.88802e11	0.973086	0.486543	+	0.873657i	$0.338258\pi$
$444$	0	0
$445$	−3.43935e11	−0.415773
$446$	0	0
$447$	5.75300e11	0.681570
$448$	0	0
$449$	−4.43110e11	−0.514521	−0.257260	−	0.966342i	$-0.582820\pi$
$449$	−4.43110e11	−0.514521	−0.257260	+	0.966342i	$0.582820\pi$
$450$	0	0
$451$	−1.76028e12	−2.00349
$452$	0	0
$453$	−1.78336e11	−0.198974
$454$	0	0
$455$	−6.59569e10	−0.0721454
$456$	0	0
$457$	−1.36081e12	−1.45940	−0.729698	−	0.683770i	$-0.760339\pi$
$457$	−1.36081e12	−1.45940	−0.729698	+	0.683770i	$0.760339\pi$
$458$	0	0
$459$	7.05535e11	0.741927
$460$	0	0
$461$	−4.56477e11	−0.470722	−0.235361	−	0.971908i	$-0.575627\pi$
$461$	−4.56477e11	−0.470722	−0.235361	+	0.971908i	$0.575627\pi$
$462$	0	0
$463$	9.47962e11	0.958686	0.479343	−	0.877628i	$-0.340875\pi$
$463$	9.47962e11	0.958686	0.479343	+	0.877628i	$0.340875\pi$
$464$	0	0
$465$	−3.22620e11	−0.320002
$466$	0	0
$467$	5.14404e11	0.500470	0.250235	−	0.968185i	$-0.419492\pi$
$467$	5.14404e11	0.500470	0.250235	+	0.968185i	$0.419492\pi$
$468$	0	0
$469$	−5.71359e9	−0.00545295
$470$	0	0
$471$	9.15245e10	0.0856926
$472$	0	0
$473$	9.06058e11	0.832302
$474$	0	0
$475$	2.35719e11	0.212458
$476$	0	0
$477$	−8.47836e11	−0.749858
$478$	0	0
$479$	−4.30057e11	−0.373264	−0.186632	−	0.982430i	$-0.559757\pi$
$479$	−4.30057e11	−0.373264	−0.186632	+	0.982430i	$0.559757\pi$
$480$	0	0
$481$	2.62583e11	0.223673
$482$	0	0
$483$	5.67606e11	0.474553
$484$	0	0
$485$	9.08336e11	0.745433
$486$	0	0
$487$	6.65219e11	0.535901	0.267950	−	0.963433i	$-0.413654\pi$
$487$	6.65219e11	0.535901	0.267950	+	0.963433i	$0.413654\pi$
$488$	0	0
$489$	4.50855e11	0.356572
$490$	0	0
$491$	2.50114e11	0.194210	0.0971049	−	0.995274i	$-0.469042\pi$
$491$	2.50114e11	0.194210	0.0971049	+	0.995274i	$0.469042\pi$
$492$	0	0
$493$	−1.49014e11	−0.113610
$494$	0	0
$495$	−4.28544e11	−0.320828
$496$	0	0
$497$	7.38583e11	0.542995
$498$	0	0
$499$	−2.29287e12	−1.65549	−0.827746	−	0.561103i	$-0.810377\pi$
$499$	−2.29287e12	−1.65549	−0.827746	+	0.561103i	$0.810377\pi$
$500$	0	0
$501$	7.82928e11	0.555204
$502$	0	0
$503$	1.83989e12	1.28155	0.640775	−	0.767729i	$-0.278613\pi$
$503$	1.83989e12	1.28155	0.640775	+	0.767729i	$0.278613\pi$
$504$	0	0
$505$	4.86576e11	0.332920
$506$	0	0
$507$	8.14518e11	0.547476
$508$	0	0
$509$	−1.84905e12	−1.22101	−0.610504	−	0.792013i	$-0.709033\pi$
$509$	−1.84905e12	−1.22101	−0.610504	+	0.792013i	$0.709033\pi$
$510$	0	0
$511$	6.92784e10	0.0449473
$512$	0	0
$513$	−4.26787e11	−0.272072
$514$	0	0
$515$	1.33723e10	0.00837673
$516$	0	0
$517$	1.95836e12	1.20555
$518$	0	0
$519$	8.21150e11	0.496786
$520$	0	0
$521$	−8.17742e11	−0.486235	−0.243118	−	0.969997i	$-0.578170\pi$
$521$	−8.17742e11	−0.486235	−0.243118	+	0.969997i	$0.578170\pi$
$522$	0	0
$523$	−3.66718e11	−0.214326	−0.107163	−	0.994241i	$-0.534177\pi$
$523$	−3.66718e11	−0.214326	−0.107163	+	0.994241i	$0.534177\pi$
$524$	0	0
$525$	3.69407e11	0.212221
$526$	0	0
$527$	−1.37453e12	−0.776257
$528$	0	0
$529$	4.24084e12	2.35451
$530$	0	0
$531$	−1.25321e12	−0.684064
$532$	0	0
$533$	1.17726e12	0.631832
$534$	0	0
$535$	−1.15032e12	−0.607054
$536$	0	0
$537$	−1.30809e12	−0.678816
$538$	0	0
$539$	−3.98322e11	−0.203275
$540$	0	0
$541$	−1.02025e12	−0.512056	−0.256028	−	0.966669i	$-0.582414\pi$
$541$	−1.02025e12	−0.512056	−0.256028	+	0.966669i	$0.582414\pi$
$542$	0	0
$543$	7.91407e10	0.0390662
$544$	0	0
$545$	−1.12441e12	−0.545932
$546$	0	0
$547$	2.66643e12	1.27347	0.636733	−	0.771085i	$-0.280285\pi$
$547$	2.66643e12	1.27347	0.636733	+	0.771085i	$0.280285\pi$
$548$	0	0
$549$	−3.97187e11	−0.186603
$550$	0	0
$551$	9.01405e10	0.0416618
$552$	0	0
$553$	8.97720e10	0.0408204
$554$	0	0
$555$	3.24874e11	0.145344
$556$	0	0
$557$	2.46250e12	1.08400	0.541998	−	0.840380i	$-0.317668\pi$
$557$	2.46250e12	1.08400	0.541998	+	0.840380i	$0.317668\pi$
$558$	0	0
$559$	−6.05967e11	−0.262480
$560$	0	0
$561$	1.61870e12	0.689976
$562$	0	0
$563$	−2.45114e12	−1.02821	−0.514104	−	0.857728i	$-0.671875\pi$
$563$	−2.45114e12	−1.02821	−0.514104	+	0.857728i	$0.671875\pi$
$564$	0	0
$565$	−1.25101e12	−0.516467
$566$	0	0
$567$	−1.75777e11	−0.0714229
$568$	0	0
$569$	3.12385e12	1.24935	0.624677	−	0.780883i	$-0.285231\pi$
$569$	3.12385e12	1.24935	0.624677	+	0.780883i	$0.285231\pi$
$570$	0	0
$571$	−2.63520e12	−1.03741	−0.518707	−	0.854952i	$-0.673586\pi$
$571$	−2.63520e12	−1.03741	−0.518707	+	0.854952i	$0.673586\pi$
$572$	0	0
$573$	−1.05171e12	−0.407567
$574$	0	0
$575$	3.93223e12	1.50015
$576$	0	0
$577$	−2.96498e12	−1.11360	−0.556802	−	0.830645i	$-0.687972\pi$
$577$	−2.96498e12	−1.11360	−0.556802	+	0.830645i	$0.687972\pi$
$578$	0	0
$579$	2.74272e12	1.01421
$580$	0	0
$581$	1.30340e12	0.474553
$582$	0	0
$583$	−5.61489e12	−2.01295
$584$	0	0
$585$	2.86608e11	0.101178
$586$	0	0
$587$	2.08868e12	0.726106	0.363053	−	0.931768i	$-0.381734\pi$
$587$	2.08868e12	0.726106	0.363053	+	0.931768i	$0.381734\pi$
$588$	0	0
$589$	8.31469e11	0.284661
$590$	0	0
$591$	−2.68197e12	−0.904295
$592$	0	0
$593$	2.53298e12	0.841174	0.420587	−	0.907252i	$-0.361824\pi$
$593$	2.53298e12	0.841174	0.420587	+	0.907252i	$0.361824\pi$
$594$	0	0
$595$	−3.47673e11	−0.113722
$596$	0	0
$597$	−2.30007e12	−0.741067
$598$	0	0
$599$	−3.17127e12	−1.00650	−0.503249	−	0.864142i	$-0.667862\pi$
$599$	−3.17127e12	−1.00650	−0.503249	+	0.864142i	$0.667862\pi$
$600$	0	0
$601$	−5.12857e12	−1.60347	−0.801735	−	0.597680i	$-0.796089\pi$
$601$	−5.12857e12	−1.60347	−0.801735	+	0.597680i	$0.796089\pi$
$602$	0	0
$603$	2.48277e10	0.00764731
$604$	0	0
$605$	−1.43637e12	−0.435879
$606$	0	0
$607$	8.33405e11	0.249177	0.124588	−	0.992209i	$-0.460239\pi$
$607$	8.33405e11	0.249177	0.124588	+	0.992209i	$0.460239\pi$
$608$	0	0
$609$	1.41264e11	0.0416153
$610$	0	0
$611$	−1.30974e12	−0.380189
$612$	0	0
$613$	−3.03610e12	−0.868448	−0.434224	−	0.900805i	$-0.642977\pi$
$613$	−3.03610e12	−0.868448	−0.434224	+	0.900805i	$0.642977\pi$
$614$	0	0
$615$	1.45654e12	0.410567
$616$	0	0
$617$	−4.94154e11	−0.137271	−0.0686355	−	0.997642i	$-0.521865\pi$
$617$	−4.94154e11	−0.137271	−0.0686355	+	0.997642i	$0.521865\pi$
$618$	0	0
$619$	−5.49105e12	−1.50331	−0.751653	−	0.659559i	$-0.770743\pi$
$619$	−5.49105e12	−1.50331	−0.751653	+	0.659559i	$0.770743\pi$
$620$	0	0
$621$	−7.11960e12	−1.92107
$622$	0	0
$623$	1.38913e12	0.369442
$624$	0	0
$625$	1.86895e12	0.489933
$626$	0	0
$627$	−9.79175e11	−0.253021
$628$	0	0
$629$	1.38413e12	0.352573
$630$	0	0
$631$	5.75017e12	1.44394	0.721969	−	0.691925i	$-0.243237\pi$
$631$	5.75017e12	1.44394	0.721969	+	0.691925i	$0.243237\pi$
$632$	0	0
$633$	1.06966e12	0.264806
$634$	0	0
$635$	6.25487e11	0.152664
$636$	0	0
$637$	2.66395e11	0.0641060
$638$	0	0
$639$	−3.20942e12	−0.761506
$640$	0	0
$641$	3.91727e12	0.916478	0.458239	−	0.888829i	$-0.348480\pi$
$641$	3.91727e12	0.916478	0.458239	+	0.888829i	$0.348480\pi$
$642$	0	0
$643$	1.43518e12	0.331099	0.165549	−	0.986202i	$-0.447060\pi$
$643$	1.43518e12	0.331099	0.165549	+	0.986202i	$0.447060\pi$
$644$	0	0
$645$	−7.49717e11	−0.170561
$646$	0	0
$647$	−1.01796e12	−0.228382	−0.114191	−	0.993459i	$-0.536428\pi$
$647$	−1.01796e12	−0.228382	−0.114191	+	0.993459i	$0.536428\pi$
$648$	0	0
$649$	−8.29950e12	−1.83633
$650$	0	0
$651$	1.30304e12	0.284343
$652$	0	0
$653$	−8.70065e12	−1.87259	−0.936295	−	0.351215i	$-0.885769\pi$
$653$	−8.70065e12	−1.87259	−0.936295	+	0.351215i	$0.885769\pi$
$654$	0	0
$655$	2.22644e12	0.472635
$656$	0	0
$657$	−3.01041e11	−0.0630350
$658$	0	0
$659$	5.63435e12	1.16375	0.581875	−	0.813278i	$-0.302319\pi$
$659$	5.63435e12	1.16375	0.581875	+	0.813278i	$0.302319\pi$
$660$	0	0
$661$	−8.28154e12	−1.68735	−0.843673	−	0.536857i	$-0.819611\pi$
$661$	−8.28154e12	−1.68735	−0.843673	+	0.536857i	$0.819611\pi$
$662$	0	0
$663$	−1.08258e12	−0.217595
$664$	0	0
$665$	2.10312e11	0.0417029
$666$	0	0
$667$	1.50371e12	0.294170
$668$	0	0
$669$	5.23144e12	1.00973
$670$	0	0
$671$	−2.63042e12	−0.500925
$672$	0	0
$673$	−4.53783e11	−0.0852669	−0.0426335	−	0.999091i	$-0.513575\pi$
$673$	−4.53783e11	−0.0852669	−0.0426335	+	0.999091i	$0.513575\pi$
$674$	0	0
$675$	−4.63356e12	−0.859107
$676$	0	0
$677$	−9.30547e12	−1.70251	−0.851254	−	0.524753i	$-0.824158\pi$
$677$	−9.30547e12	−1.70251	−0.851254	+	0.524753i	$0.824158\pi$
$678$	0	0
$679$	−3.66870e12	−0.662367
$680$	0	0
$681$	3.28151e12	0.584670
$682$	0	0
$683$	9.97313e12	1.75363	0.876816	−	0.480827i	$-0.159663\pi$
$683$	9.97313e12	1.75363	0.876816	+	0.480827i	$0.159663\pi$
$684$	0	0
$685$	4.39514e11	0.0762721
$686$	0	0
$687$	−7.98610e12	−1.36782
$688$	0	0
$689$	3.75521e12	0.634815
$690$	0	0
$691$	2.37926e12	0.396999	0.198500	−	0.980101i	$-0.436393\pi$
$691$	2.37926e12	0.396999	0.198500	+	0.980101i	$0.436393\pi$
$692$	0	0
$693$	1.73086e12	0.285077
$694$	0	0
$695$	−4.01964e11	−0.0653516
$696$	0	0
$697$	6.20561e12	0.995949
$698$	0	0
$699$	3.48963e12	0.552882
$700$	0	0
$701$	9.19884e12	1.43880	0.719402	−	0.694594i	$-0.244416\pi$
$701$	9.19884e12	1.43880	0.719402	+	0.694594i	$0.244416\pi$
$702$	0	0
$703$	−8.37279e11	−0.129292
$704$	0	0
$705$	−1.62044e12	−0.247049
$706$	0	0
$707$	−1.96525e12	−0.295821
$708$	0	0
$709$	1.30512e12	0.193973	0.0969867	−	0.995286i	$-0.469080\pi$
$709$	1.30512e12	0.193973	0.0969867	+	0.995286i	$0.469080\pi$
$710$	0	0
$711$	−3.90093e11	−0.0572473
$712$	0	0
$713$	1.38704e13	2.00996
$714$	0	0
$715$	1.89809e12	0.271607
$716$	0	0
$717$	−6.11619e12	−0.864261
$718$	0	0
$719$	−1.15600e12	−0.161316	−0.0806579	−	0.996742i	$-0.525702\pi$
$719$	−1.15600e12	−0.161316	−0.0806579	+	0.996742i	$0.525702\pi$
$720$	0	0
$721$	−5.40099e10	−0.00744329
$722$	0	0
$723$	−8.40936e12	−1.14457
$724$	0	0
$725$	9.78639e11	0.131553
$726$	0	0
$727$	6.57013e12	0.872306	0.436153	−	0.899872i	$-0.356341\pi$
$727$	6.57013e12	0.872306	0.436153	+	0.899872i	$0.356341\pi$
$728$	0	0
$729$	6.24686e12	0.819197
$730$	0	0
$731$	−3.19418e12	−0.413743
$732$	0	0
$733$	3.86571e12	0.494609	0.247304	−	0.968938i	$-0.420455\pi$
$733$	3.86571e12	0.494609	0.247304	+	0.968938i	$0.420455\pi$
$734$	0	0
$735$	3.29591e11	0.0416564
$736$	0	0
$737$	1.64424e11	0.0205288
$738$	0	0
$739$	−4.25576e11	−0.0524901	−0.0262451	−	0.999656i	$-0.508355\pi$
$739$	−4.25576e11	−0.0524901	−0.0262451	+	0.999656i	$0.508355\pi$
$740$	0	0
$741$	6.54866e11	0.0797941
$742$	0	0
$743$	1.50308e13	1.80940	0.904698	−	0.426054i	$-0.140097\pi$
$743$	1.50308e13	1.80940	0.904698	+	0.426054i	$0.140097\pi$
$744$	0	0
$745$	3.55595e12	0.422914
$746$	0	0
$747$	−5.66376e12	−0.665522
$748$	0	0
$749$	4.64607e12	0.539408
$750$	0	0
$751$	6.71667e12	0.770503	0.385252	−	0.922812i	$-0.374115\pi$
$751$	6.71667e12	0.770503	0.385252	+	0.922812i	$0.374115\pi$
$752$	0	0
$753$	3.69859e12	0.419236
$754$	0	0
$755$	−1.10230e12	−0.123464
$756$	0	0
$757$	−5.38917e12	−0.596472	−0.298236	−	0.954492i	$-0.596398\pi$
$757$	−5.38917e12	−0.596472	−0.298236	+	0.954492i	$0.596398\pi$
$758$	0	0
$759$	−1.63344e13	−1.78655
$760$	0	0
$761$	1.62051e13	1.75154	0.875769	−	0.482730i	$-0.160355\pi$
$761$	1.62051e13	1.75154	0.875769	+	0.482730i	$0.160355\pi$
$762$	0	0
$763$	4.54139e12	0.485097
$764$	0	0
$765$	1.51077e12	0.159486
$766$	0	0
$767$	5.55066e12	0.579116
$768$	0	0
$769$	3.79853e11	0.0391694	0.0195847	−	0.999808i	$-0.493766\pi$
$769$	3.79853e11	0.0391694	0.0195847	+	0.999808i	$0.493766\pi$
$770$	0	0
$771$	−3.80065e12	−0.387359
$772$	0	0
$773$	3.14551e12	0.316872	0.158436	−	0.987369i	$-0.449355\pi$
$773$	3.14551e12	0.316872	0.158436	+	0.987369i	$0.449355\pi$
$774$	0	0
$775$	9.02711e12	0.898858
$776$	0	0
$777$	−1.31214e12	−0.129148
$778$	0	0
$779$	−3.75386e12	−0.365224
$780$	0	0
$781$	−2.12548e13	−2.04422
$782$	0	0
$783$	−1.77190e12	−0.168466
$784$	0	0
$785$	5.65716e11	0.0531723
$786$	0	0
$787$	7.69122e12	0.714675	0.357338	−	0.933975i	$-0.383685\pi$
$787$	7.69122e12	0.714675	0.357338	+	0.933975i	$0.383685\pi$
$788$	0	0
$789$	−3.20891e12	−0.294789
$790$	0	0
$791$	5.05274e12	0.458916
$792$	0	0
$793$	1.75921e12	0.157975
$794$	0	0
$795$	4.64604e12	0.412506
$796$	0	0
$797$	8.17419e12	0.717599	0.358800	−	0.933415i	$-0.383186\pi$
$797$	8.17419e12	0.717599	0.358800	+	0.933415i	$0.383186\pi$
$798$	0	0
$799$	−6.90392e12	−0.599287
$800$	0	0
$801$	−6.03629e12	−0.518112
$802$	0	0
$803$	−1.99368e12	−0.169214
$804$	0	0
$805$	3.50839e12	0.294460
$806$	0	0
$807$	2.82208e12	0.234227
$808$	0	0
$809$	1.85735e13	1.52450	0.762248	−	0.647285i	$-0.224096\pi$
$809$	1.85735e13	1.52450	0.762248	+	0.647285i	$0.224096\pi$
$810$	0	0
$811$	−1.77241e12	−0.143870	−0.0719349	−	0.997409i	$-0.522917\pi$
$811$	−1.77241e12	−0.143870	−0.0719349	+	0.997409i	$0.522917\pi$
$812$	0	0
$813$	1.27272e13	1.02171
$814$	0	0
$815$	2.78675e12	0.221253
$816$	0	0
$817$	1.93220e12	0.151724
$818$	0	0
$819$	−1.15759e12	−0.0899035
$820$	0	0
$821$	4.47706e12	0.343913	0.171957	−	0.985105i	$-0.444991\pi$
$821$	4.47706e12	0.343913	0.171957	+	0.985105i	$0.444991\pi$
$822$	0	0
$823$	1.32576e13	1.00731	0.503656	−	0.863904i	$-0.331988\pi$
$823$	1.32576e13	1.00731	0.503656	+	0.863904i	$0.331988\pi$
$824$	0	0
$825$	−1.06307e13	−0.798951
$826$	0	0
$827$	3.03450e12	0.225586	0.112793	−	0.993618i	$-0.464020\pi$
$827$	3.03450e12	0.225586	0.112793	+	0.993618i	$0.464020\pi$
$828$	0	0
$829$	−1.04519e13	−0.768598	−0.384299	−	0.923209i	$-0.625557\pi$
$829$	−1.04519e13	−0.768598	−0.384299	+	0.923209i	$0.625557\pi$
$830$	0	0
$831$	−7.82080e12	−0.568914
$832$	0	0
$833$	1.40423e12	0.101050
$834$	0	0
$835$	4.83931e12	0.344504
$836$	0	0
$837$	−1.63443e13	−1.15107
$838$	0	0
$839$	−3.74536e11	−0.0260954	−0.0130477	−	0.999915i	$-0.504153\pi$
$839$	−3.74536e11	−0.0260954	−0.0130477	+	0.999915i	$0.504153\pi$
$840$	0	0
$841$	−1.41329e13	−0.974203
$842$	0	0
$843$	4.64237e12	0.316603
$844$	0	0
$845$	5.03457e12	0.339709
$846$	0	0
$847$	5.80138e12	0.387307
$848$	0	0
$849$	−1.73415e13	−1.14552
$850$	0	0
$851$	−1.39674e13	−0.912917
$852$	0	0
$853$	8.62211e10	0.00557626	0.00278813	−	0.999996i	$-0.499113\pi$
$853$	8.62211e10	0.00557626	0.00278813	+	0.999996i	$0.499113\pi$
$854$	0	0
$855$	−9.13885e11	−0.0584850
$856$	0	0
$857$	−2.62843e13	−1.66450	−0.832248	−	0.554404i	$-0.812946\pi$
$857$	−2.62843e13	−1.66450	−0.832248	+	0.554404i	$0.812946\pi$
$858$	0	0
$859$	−1.64181e13	−1.02885	−0.514426	−	0.857535i	$-0.671995\pi$
$859$	−1.64181e13	−1.02885	−0.514426	+	0.857535i	$0.671995\pi$
$860$	0	0
$861$	−5.88286e12	−0.364817
$862$	0	0
$863$	1.74094e13	1.06840	0.534202	−	0.845357i	$-0.320612\pi$
$863$	1.74094e13	1.06840	0.534202	+	0.845357i	$0.320612\pi$
$864$	0	0
$865$	5.07556e12	0.308256
$866$	0	0
$867$	5.69875e12	0.342526
$868$	0	0
$869$	−2.58344e12	−0.153677
$870$	0	0
$871$	−1.09966e11	−0.00647407
$872$	0	0
$873$	1.59419e13	0.928916
$874$	0	0
$875$	5.07103e12	0.292456
$876$	0	0
$877$	−2.60150e13	−1.48500	−0.742500	−	0.669846i	$-0.766360\pi$
$877$	−2.60150e13	−1.48500	−0.742500	+	0.669846i	$0.766360\pi$
$878$	0	0
$879$	−2.10393e13	−1.18873
$880$	0	0
$881$	8.13378e11	0.0454884	0.0227442	−	0.999741i	$-0.492760\pi$
$881$	8.13378e11	0.0454884	0.0227442	+	0.999741i	$0.492760\pi$
$882$	0	0
$883$	−9.12098e12	−0.504915	−0.252458	−	0.967608i	$-0.581239\pi$
$883$	−9.12098e12	−0.504915	−0.252458	+	0.967608i	$0.581239\pi$
$884$	0	0
$885$	6.86741e12	0.376312
$886$	0	0
$887$	2.26227e13	1.22712	0.613560	−	0.789648i	$-0.289737\pi$
$887$	2.26227e13	1.22712	0.613560	+	0.789648i	$0.289737\pi$
$888$	0	0
$889$	−2.52630e12	−0.135652
$890$	0	0
$891$	5.05847e12	0.268887
$892$	0	0
$893$	4.17627e12	0.219764
$894$	0	0
$895$	−8.08533e12	−0.421205
$896$	0	0
$897$	1.09244e13	0.563418
$898$	0	0
$899$	3.45203e12	0.176261
$900$	0	0
$901$	1.97945e13	1.00065
$902$	0	0
$903$	3.02805e12	0.151554
$904$	0	0
$905$	4.89171e11	0.0242406
$906$	0	0
$907$	2.93607e13	1.44057	0.720284	−	0.693679i	$-0.244011\pi$
$907$	2.93607e13	1.44057	0.720284	+	0.693679i	$0.244011\pi$
$908$	0	0
$909$	8.53974e12	0.414865
$910$	0	0
$911$	−1.66340e13	−0.800136	−0.400068	−	0.916485i	$-0.631014\pi$
$911$	−1.66340e13	−0.800136	−0.400068	+	0.916485i	$0.631014\pi$
$912$	0	0
$913$	−3.75089e13	−1.78655
$914$	0	0
$915$	2.17654e12	0.102653
$916$	0	0
$917$	−8.99244e12	−0.419967
$918$	0	0
$919$	−2.48587e13	−1.14963	−0.574816	−	0.818282i	$-0.694926\pi$
$919$	−2.48587e13	−1.14963	−0.574816	+	0.818282i	$0.694926\pi$
$920$	0	0
$921$	−1.66994e13	−0.764774
$922$	0	0
$923$	1.42151e13	0.644676
$924$	0	0
$925$	−9.09018e12	−0.408258
$926$	0	0
$927$	2.34694e11	0.0104386
$928$	0	0
$929$	2.97476e13	1.31033	0.655165	−	0.755486i	$-0.272599\pi$
$929$	2.97476e13	1.31033	0.655165	+	0.755486i	$0.272599\pi$
$930$	0	0
$931$	−8.49435e11	−0.0370558
$932$	0	0
$933$	−1.07209e13	−0.463197
$934$	0	0
$935$	1.00053e13	0.428130
$936$	0	0
$937$	−2.22444e13	−0.942740	−0.471370	−	0.881935i	$-0.656240\pi$
$937$	−2.22444e13	−0.942740	−0.471370	+	0.881935i	$0.656240\pi$
$938$	0	0
$939$	−2.29129e13	−0.961801
$940$	0	0
$941$	4.00117e13	1.66354	0.831770	−	0.555121i	$-0.187328\pi$
$941$	4.00117e13	1.66354	0.831770	+	0.555121i	$0.187328\pi$
$942$	0	0
$943$	−6.26213e13	−2.57881
$944$	0	0
$945$	−4.13413e12	−0.168632
$946$	0	0
$947$	−1.18567e13	−0.479058	−0.239529	−	0.970889i	$-0.576993\pi$
$947$	−1.18567e13	−0.479058	−0.239529	+	0.970889i	$0.576993\pi$
$948$	0	0
$949$	1.33336e12	0.0533642
$950$	0	0
$951$	1.73814e13	0.689086
$952$	0	0
$953$	1.97468e13	0.775493	0.387747	−	0.921766i	$-0.373254\pi$
$953$	1.97468e13	0.775493	0.387747	+	0.921766i	$0.373254\pi$
$954$	0	0
$955$	−6.50064e12	−0.252895
$956$	0	0
$957$	−4.06526e12	−0.156670
$958$	0	0
$959$	−1.77517e12	−0.0677728
$960$	0	0
$961$	5.40241e12	0.204330
$962$	0	0
$963$	−2.01889e13	−0.756475
$964$	0	0
$965$	1.69529e13	0.629318
$966$	0	0
$967$	−1.24596e13	−0.458232	−0.229116	−	0.973399i	$-0.573583\pi$
$967$	−1.24596e13	−0.458232	−0.229116	+	0.973399i	$0.573583\pi$
$968$	0	0
$969$	3.45194e12	0.125778
$970$	0	0
$971$	−4.93595e13	−1.78190	−0.890952	−	0.454098i	$-0.849962\pi$
$971$	−4.93595e13	−1.78190	−0.890952	+	0.454098i	$0.849962\pi$
$972$	0	0
$973$	1.62351e12	0.0580692
$974$	0	0
$975$	7.10977e12	0.251962
$976$	0	0
$977$	−3.23161e13	−1.13473	−0.567366	−	0.823466i	$-0.692037\pi$
$977$	−3.23161e13	−1.13473	−0.567366	+	0.823466i	$0.692037\pi$
$978$	0	0
$979$	−3.99760e13	−1.39084
$980$	0	0
$981$	−1.97341e13	−0.680309
$982$	0	0
$983$	1.45695e13	0.497685	0.248843	−	0.968544i	$-0.419950\pi$
$983$	1.45695e13	0.497685	0.248843	+	0.968544i	$0.419950\pi$
$984$	0	0
$985$	−1.65773e13	−0.561115
$986$	0	0
$987$	6.54485e12	0.219519
$988$	0	0
$989$	3.22327e13	1.07131
$990$	0	0
$991$	2.10443e12	0.0693113	0.0346556	−	0.999399i	$-0.488967\pi$
$991$	2.10443e12	0.0693113	0.0346556	+	0.999399i	$0.488967\pi$
$992$	0	0
$993$	1.66311e13	0.542811
$994$	0	0
$995$	−1.42168e13	−0.459832
$996$	0	0
$997$	2.98914e13	0.958115	0.479057	−	0.877784i	$-0.340979\pi$
$997$	2.98914e13	0.958115	0.479057	+	0.877784i	$0.340979\pi$
$998$	0	0
$999$	1.64585e13	0.522812

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.a.j.1.2		4
4.3	odd	2		56.10.a.d.1.3	✓	4
28.27	even	2		392.10.a.e.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.10.a.d.1.3	✓	4	4.3	odd	2
112.10.a.j.1.2		4	1.1	even	1	trivial
392.10.a.e.1.2		4	28.27	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)

\(f(q)\)	\(=\)	\(q-96.1756 q^{3} -594.465 q^{5} +2401.00 q^{7} -10433.3 q^{9} -69095.5 q^{11} +46210.6 q^{13} +57173.0 q^{15} +243586. q^{17} -147349. q^{19} -230918. q^{21} -2.45805e6 q^{23} -1.59974e6 q^{25} +2.89645e6 q^{27} -611750. q^{29} -5.64287e6 q^{31} +6.64530e6 q^{33} -1.42731e6 q^{35} +5.68230e6 q^{37} -4.44434e6 q^{39} +2.54760e7 q^{41} -1.31131e7 q^{43} +6.20220e6 q^{45} -2.83428e7 q^{47} +5.76480e6 q^{49} -2.34270e7 q^{51} +8.12628e7 q^{53} +4.10748e7 q^{55} +1.41713e7 q^{57} +1.20116e8 q^{59} +3.80693e7 q^{61} -2.50503e7 q^{63} -2.74706e7 q^{65} -2.37967e6 q^{67} +2.36404e8 q^{69} +3.07615e8 q^{71} +2.88540e7 q^{73} +1.53856e8 q^{75} -1.65898e8 q^{77} +3.73894e7 q^{79} -7.32098e7 q^{81} +5.42856e8 q^{83} -1.44803e8 q^{85} +5.88354e7 q^{87} +5.78562e8 q^{89} +1.10952e8 q^{91} +5.42707e8 q^{93} +8.75935e7 q^{95} -1.52799e9 q^{97} +7.20891e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 84 q^{3} + 1540 q^{5} + 9604 q^{7} + 48060 q^{9} + 53016 q^{11} + 172732 q^{13} - 228096 q^{15} - 128352 q^{17} - 85148 q^{19} - 201684 q^{21} + 342368 q^{23} + 4030196 q^{25} - 11284056 q^{27} + 6392064 q^{29}+ \cdots + 3054793464 q^{99}+O(q^{100}) \) 4 * q - 84 * q^3 + 1540 * q^5 + 9604 * q^7 + 48060 * q^9 + 53016 * q^11 + 172732 * q^13 - 228096 * q^15 - 128352 * q^17 - 85148 * q^19 - 201684 * q^21 + 342368 * q^23 + 4030196 * q^25 - 11284056 * q^27 + 6392064 * q^29 - 3692920 * q^31 + 2660448 * q^33 + 3697540 * q^35 + 20408192 * q^37 - 54281520 * q^39 + 51466576 * q^41 - 35333016 * q^43 + 140298228 * q^45 - 70879592 * q^47 + 23059204 * q^49 + 14919624 * q^51 + 127260312 * q^53 + 43963920 * q^55 + 244217976 * q^57 + 80548804 * q^59 + 92246084 * q^61 + 115392060 * q^63 + 419393416 * q^65 + 522443376 * q^67 - 365219904 * q^69 + 104419600 * q^71 - 60133192 * q^73 + 567926436 * q^75 + 127291416 * q^77 + 1069501840 * q^79 + 816541884 * q^81 + 1270627540 * q^83 - 199027992 * q^85 + 2194287480 * q^87 - 356859384 * q^89 + 414729532 * q^91 - 1074588912 * q^93 + 2083565536 * q^95 + 1288837312 * q^97 + 3054793464 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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