LMFDB - Embedded newform 400.10.a.o.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,10,Mod(1,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-92,0,0,0,-6908,0,13258] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$206.014334466$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6049}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1512 $ x^2 - x - 1512
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-38.3877$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+109.551 q^{3} -9209.37 q^{7} -7681.66 q^{9} +29861.4 q^{11} -16775.2 q^{13} +312472. q^{17} +36401.6 q^{19} -1.00889e6 q^{21} +951749. q^{23} -2.99782e6 q^{27} -3.02575e6 q^{29} +9.43462e6 q^{31} +3.27134e6 q^{33} +1.16984e7 q^{37} -1.83774e6 q^{39} +3.49741e7 q^{41} -2.26136e7 q^{43} -5.58030e6 q^{47} +4.44590e7 q^{49} +3.42316e7 q^{51} -5.54605e7 q^{53} +3.98782e6 q^{57} -8.51084e7 q^{59} +4.85791e7 q^{61} +7.07433e7 q^{63} -5.20748e7 q^{67} +1.04265e8 q^{69} +1.14674e8 q^{71} -4.47760e8 q^{73} -2.75005e8 q^{77} +9.18141e7 q^{79} -1.77215e8 q^{81} -5.87254e8 q^{83} -3.31472e8 q^{87} -4.81494e8 q^{89} +1.54489e8 q^{91} +1.03357e9 q^{93} -1.37193e9 q^{97} -2.29385e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 92 q^{3} - 6908 q^{7} + 13258 q^{9} + 8080 q^{11} - 111948 q^{13} + 327532 q^{17} + 1156680 q^{19} - 1472736 q^{21} - 1057252 q^{23} - 3251096 q^{27} - 4212260 q^{29} + 11361128 q^{31} + 7661392 q^{33}+ \cdots - 685480304 q^{99}+O(q^{100}) $ 2 * q - 92 * q^3 - 6908 * q^7 + 13258 * q^9 + 8080 * q^11 - 111948 * q^13 + 327532 * q^17 + 1156680 * q^19 - 1472736 * q^21 - 1057252 * q^23 - 3251096 * q^27 - 4212260 * q^29 + 11361128 * q^31 + 7661392 * q^33 + 7251860 * q^37 + 17344392 * q^39 + 13030436 * q^41 - 47934076 * q^43 + 30914292 * q^47 + 9401666 * q^49 + 31196280 * q^51 - 100922108 * q^53 - 221805008 * q^57 + 47362536 * q^59 + 203634428 * q^61 + 118933236 * q^63 - 58872852 * q^67 + 509180256 * q^69 + 349900792 * q^71 - 71160388 * q^73 - 325131984 * q^77 + 452087344 * q^79 - 538321022 * q^81 - 244865436 * q^83 - 92329544 * q^87 - 256073260 * q^89 - 64538616 * q^91 + 645280640 * q^93 + 184950572 * q^97 - 685480304 * 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
$3$	109.551	0.780853	0.390426	−	0.920634i	$-0.372328\pi$
$3$	109.551	0.780853	0.390426	+	0.920634i	$0.372328\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−9209.37	−1.44974	−0.724868	−	0.688888i	$-0.758099\pi$
$7$	−9209.37	−1.44974	−0.724868	+	0.688888i	$0.758099\pi$
$8$	0	0
$9$	−7681.66	−0.390269
$10$	0	0
$11$	29861.4	0.614955	0.307477	−	0.951555i	$-0.400515\pi$
$11$	29861.4	0.614955	0.307477	+	0.951555i	$0.400515\pi$
$12$	0	0
$13$	−16775.2	−0.162901	−0.0814505	−	0.996677i	$-0.525955\pi$
$13$	−16775.2	−0.162901	−0.0814505	+	0.996677i	$0.525955\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	312472.	0.907385	0.453692	−	0.891158i	$-0.350106\pi$
$17$	312472.	0.907385	0.453692	+	0.891158i	$0.350106\pi$
$18$	0	0
$19$	36401.6	0.0640810	0.0320405	−	0.999487i	$-0.489799\pi$
$19$	36401.6	0.0640810	0.0320405	+	0.999487i	$0.489799\pi$
$20$	0	0
$21$	−1.00889e6	−1.13203
$22$	0	0
$23$	951749.	0.709165	0.354583	−	0.935025i	$-0.384623\pi$
$23$	951749.	0.709165	0.354583	+	0.935025i	$0.384623\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.99782e6	−1.08560
$28$	0	0
$29$	−3.02575e6	−0.794404	−0.397202	−	0.917731i	$-0.630019\pi$
$29$	−3.02575e6	−0.794404	−0.397202	+	0.917731i	$0.630019\pi$
$30$	0	0
$31$	9.43462e6	1.83483	0.917417	−	0.397926i	$-0.130270\pi$
$31$	9.43462e6	1.83483	0.917417	+	0.397926i	$0.130270\pi$
$32$	0	0
$33$	3.27134e6	0.480189
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.16984e7	1.02617	0.513084	−	0.858339i	$-0.328503\pi$
$37$	1.16984e7	1.02617	0.513084	+	0.858339i	$0.328503\pi$
$38$	0	0
$39$	−1.83774e6	−0.127202
$40$	0	0
$41$	3.49741e7	1.93295	0.966473	−	0.256769i	$-0.0826578\pi$
$41$	3.49741e7	1.93295	0.966473	+	0.256769i	$0.0826578\pi$
$42$	0	0
$43$	−2.26136e7	−1.00870	−0.504349	−	0.863500i	$-0.668268\pi$
$43$	−2.26136e7	−1.00870	−0.504349	+	0.863500i	$0.668268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.58030e6	−0.166808	−0.0834041	−	0.996516i	$-0.526579\pi$
$47$	−5.58030e6	−0.166808	−0.0834041	+	0.996516i	$0.526579\pi$
$48$	0	0
$49$	4.44590e7	1.10173
$50$	0	0
$51$	3.42316e7	0.708534
$52$	0	0
$53$	−5.54605e7	−0.965477	−0.482739	−	0.875765i	$-0.660358\pi$
$53$	−5.54605e7	−0.965477	−0.482739	+	0.875765i	$0.660358\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.98782e6	0.0500378
$58$	0	0
$59$	−8.51084e7	−0.914404	−0.457202	−	0.889363i	$-0.651148\pi$
$59$	−8.51084e7	−0.914404	−0.457202	+	0.889363i	$0.651148\pi$
$60$	0	0
$61$	4.85791e7	0.449226	0.224613	−	0.974448i	$-0.427888\pi$
$61$	4.85791e7	0.449226	0.224613	+	0.974448i	$0.427888\pi$
$62$	0	0
$63$	7.07433e7	0.565787
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.20748e7	−0.315712	−0.157856	−	0.987462i	$-0.550458\pi$
$67$	−5.20748e7	−0.315712	−0.157856	+	0.987462i	$0.550458\pi$
$68$	0	0
$69$	1.04265e8	0.553754
$70$	0	0
$71$	1.14674e8	0.535551	0.267776	−	0.963481i	$-0.413711\pi$
$71$	1.14674e8	0.535551	0.267776	+	0.963481i	$0.413711\pi$
$72$	0	0
$73$	−4.47760e8	−1.84541	−0.922704	−	0.385508i	$-0.874026\pi$
$73$	−4.47760e8	−1.84541	−0.922704	+	0.385508i	$0.874026\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.75005e8	−0.891522
$78$	0	0
$79$	9.18141e7	0.265208	0.132604	−	0.991169i	$-0.457666\pi$
$79$	9.18141e7	0.265208	0.132604	+	0.991169i	$0.457666\pi$
$80$	0	0
$81$	−1.77215e8	−0.457422
$82$	0	0
$83$	−5.87254e8	−1.35823	−0.679117	−	0.734030i	$-0.737637\pi$
$83$	−5.87254e8	−1.35823	−0.679117	+	0.734030i	$0.737637\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.31472e8	−0.620313
$88$	0	0
$89$	−4.81494e8	−0.813459	−0.406729	−	0.913549i	$-0.633331\pi$
$89$	−4.81494e8	−0.813459	−0.406729	+	0.913549i	$0.633331\pi$
$90$	0	0
$91$	1.54489e8	0.236163
$92$	0	0
$93$	1.03357e9	1.43274
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.37193e9	−1.57347	−0.786733	−	0.617293i	$-0.788229\pi$
$97$	−1.37193e9	−1.57347	−0.786733	+	0.617293i	$0.788229\pi$
$98$	0	0
$99$	−2.29385e8	−0.239998
$100$	0	0
$101$	−1.23051e9	−1.17662	−0.588312	−	0.808634i	$-0.700207\pi$
$101$	−1.23051e9	−1.17662	−0.588312	+	0.808634i	$0.700207\pi$
$102$	0	0
$103$	−8.07913e8	−0.707289	−0.353645	−	0.935380i	$-0.615058\pi$
$103$	−8.07913e8	−0.707289	−0.353645	+	0.935380i	$0.615058\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.59875e9	1.17911	0.589553	−	0.807730i	$-0.299304\pi$
$107$	1.59875e9	1.17911	0.589553	+	0.807730i	$0.299304\pi$
$108$	0	0
$109$	−1.08078e9	−0.733361	−0.366680	−	0.930347i	$-0.619506\pi$
$109$	−1.08078e9	−0.733361	−0.366680	+	0.930347i	$0.619506\pi$
$110$	0	0
$111$	1.28157e9	0.801286
$112$	0	0
$113$	−1.52041e9	−0.877217	−0.438608	−	0.898678i	$-0.644528\pi$
$113$	−1.52041e9	−0.877217	−0.438608	+	0.898678i	$0.644528\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.28862e8	0.0635751
$118$	0	0
$119$	−2.87768e9	−1.31547
$120$	0	0
$121$	−1.46624e9	−0.621831
$122$	0	0
$123$	3.83144e9	1.50935
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.75891e9	1.62327	0.811634	−	0.584166i	$-0.198578\pi$
$127$	4.75891e9	1.62327	0.811634	+	0.584166i	$0.198578\pi$
$128$	0	0
$129$	−2.47733e9	−0.787645
$130$	0	0
$131$	3.43225e9	1.01826	0.509129	−	0.860690i	$-0.329968\pi$
$131$	3.43225e9	1.01826	0.509129	+	0.860690i	$0.329968\pi$
$132$	0	0
$133$	−3.35236e8	−0.0929005
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.21292e7	0.0126427	0.00632133	−	0.999980i	$-0.497988\pi$
$137$	5.21292e7	0.0126427	0.00632133	+	0.999980i	$0.497988\pi$
$138$	0	0
$139$	−5.87422e9	−1.33470	−0.667350	−	0.744744i	$-0.732571\pi$
$139$	−5.87422e9	−1.33470	−0.667350	+	0.744744i	$0.732571\pi$
$140$	0	0
$141$	−6.11325e8	−0.130253
$142$	0	0
$143$	−5.00932e8	−0.100177
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.87051e9	0.860292
$148$	0	0
$149$	9.63272e9	1.60107	0.800535	−	0.599285i	$-0.204549\pi$
$149$	9.63272e9	1.60107	0.800535	+	0.599285i	$0.204549\pi$
$150$	0	0
$151$	6.41660e9	1.00440	0.502202	−	0.864750i	$-0.332523\pi$
$151$	6.41660e9	1.00440	0.502202	+	0.864750i	$0.332523\pi$
$152$	0	0
$153$	−2.40031e9	−0.354124
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.12424e10	−1.47676	−0.738381	−	0.674384i	$-0.764409\pi$
$157$	−1.12424e10	−1.47676	−0.738381	+	0.674384i	$0.764409\pi$
$158$	0	0
$159$	−6.07573e9	−0.753896
$160$	0	0
$161$	−8.76502e9	−1.02810
$162$	0	0
$163$	−3.86388e9	−0.428726	−0.214363	−	0.976754i	$-0.568768\pi$
$163$	−3.86388e9	−0.428726	−0.214363	+	0.976754i	$0.568768\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.78459e10	−1.77547	−0.887736	−	0.460353i	$-0.847723\pi$
$167$	−1.78459e10	−1.77547	−0.887736	+	0.460353i	$0.847723\pi$
$168$	0	0
$169$	−1.03231e10	−0.973463
$170$	0	0
$171$	−2.79625e8	−0.0250088
$172$	0	0
$173$	1.32509e10	1.12470	0.562351	−	0.826899i	$-0.309897\pi$
$173$	1.32509e10	1.12470	0.562351	+	0.826899i	$0.309897\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.32367e9	−0.714015
$178$	0	0
$179$	5.98157e9	0.435488	0.217744	−	0.976006i	$-0.430130\pi$
$179$	5.98157e9	0.435488	0.217744	+	0.976006i	$0.430130\pi$
$180$	0	0
$181$	−2.23916e10	−1.55071	−0.775355	−	0.631525i	$-0.782429\pi$
$181$	−2.23916e10	−1.55071	−0.775355	+	0.631525i	$0.782429\pi$
$182$	0	0
$183$	5.32187e9	0.350780
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.33087e9	0.558001
$188$	0	0
$189$	2.76080e10	1.57383
$190$	0	0
$191$	2.90074e10	1.57710	0.788549	−	0.614972i	$-0.210833\pi$
$191$	2.90074e10	1.57710	0.788549	+	0.614972i	$0.210833\pi$
$192$	0	0
$193$	−2.00528e10	−1.04032	−0.520161	−	0.854068i	$-0.674128\pi$
$193$	−2.00528e10	−1.04032	−0.520161	+	0.854068i	$0.674128\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.23440e10	−0.583928	−0.291964	−	0.956429i	$-0.594309\pi$
$197$	−1.23440e10	−0.583928	−0.291964	+	0.956429i	$0.594309\pi$
$198$	0	0
$199$	2.71589e10	1.22765	0.613823	−	0.789443i	$-0.289631\pi$
$199$	2.71589e10	1.22765	0.613823	+	0.789443i	$0.289631\pi$
$200$	0	0
$201$	−5.70483e9	−0.246525
$202$	0	0
$203$	2.78652e10	1.15168
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.31101e9	−0.276765
$208$	0	0
$209$	1.08700e9	0.0394069
$210$	0	0
$211$	−9.20982e9	−0.319875	−0.159937	−	0.987127i	$-0.551129\pi$
$211$	−9.20982e9	−0.319875	−0.159937	+	0.987127i	$0.551129\pi$
$212$	0	0
$213$	1.25626e10	0.418187
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−8.68870e10	−2.66003
$218$	0	0
$219$	−4.90524e10	−1.44099
$220$	0	0
$221$	−5.24180e9	−0.147814
$222$	0	0
$223$	−2.43587e10	−0.659601	−0.329801	−	0.944051i	$-0.606982\pi$
$223$	−2.43587e10	−0.659601	−0.329801	+	0.944051i	$0.606982\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.95512e9	0.148859	0.0744294	−	0.997226i	$-0.476286\pi$
$227$	5.95512e9	0.148859	0.0744294	+	0.997226i	$0.476286\pi$
$228$	0	0
$229$	−1.97476e10	−0.474521	−0.237260	−	0.971446i	$-0.576249\pi$
$229$	−1.97476e10	−0.474521	−0.237260	+	0.971446i	$0.576249\pi$
$230$	0	0
$231$	−3.01270e10	−0.696148
$232$	0	0
$233$	4.95653e10	1.10173	0.550866	−	0.834594i	$-0.314298\pi$
$233$	4.95653e10	1.10173	0.550866	+	0.834594i	$0.314298\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.00583e10	0.207089
$238$	0	0
$239$	2.55103e10	0.505738	0.252869	−	0.967501i	$-0.418626\pi$
$239$	2.55103e10	0.505738	0.252869	+	0.967501i	$0.418626\pi$
$240$	0	0
$241$	4.72387e10	0.902031	0.451015	−	0.892516i	$-0.351062\pi$
$241$	4.72387e10	0.902031	0.451015	+	0.892516i	$0.351062\pi$
$242$	0	0
$243$	3.95920e10	0.728416
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.10645e8	−0.0104389
$248$	0	0
$249$	−6.43340e10	−1.06058
$250$	0	0
$251$	−3.60294e10	−0.572962	−0.286481	−	0.958086i	$-0.592485\pi$
$251$	−3.60294e10	−0.572962	−0.286481	+	0.958086i	$0.592485\pi$
$252$	0	0
$253$	2.84206e10	0.436104
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.80945e10	1.11666	0.558330	−	0.829619i	$-0.311442\pi$
$257$	7.80945e10	1.11666	0.558330	+	0.829619i	$0.311442\pi$
$258$	0	0
$259$	−1.07735e11	−1.48767
$260$	0	0
$261$	2.32427e10	0.310031
$262$	0	0
$263$	1.93335e10	0.249178	0.124589	−	0.992208i	$-0.460239\pi$
$263$	1.93335e10	0.249178	0.124589	+	0.992208i	$0.460239\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.27479e10	−0.635192
$268$	0	0
$269$	−1.70381e11	−1.98398	−0.991989	−	0.126327i	$-0.959681\pi$
$269$	−1.70381e11	−1.98398	−0.991989	+	0.126327i	$0.959681\pi$
$270$	0	0
$271$	5.21132e10	0.586929	0.293465	−	0.955970i	$-0.405192\pi$
$271$	5.21132e10	0.586929	0.293465	+	0.955970i	$0.405192\pi$
$272$	0	0
$273$	1.69244e10	0.184409
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.96913e10	0.711245	0.355623	−	0.934630i	$-0.384269\pi$
$277$	6.96913e10	0.711245	0.355623	+	0.934630i	$0.384269\pi$
$278$	0	0
$279$	−7.24736e10	−0.716078
$280$	0	0
$281$	−1.23747e11	−1.18401	−0.592005	−	0.805935i	$-0.701663\pi$
$281$	−1.23747e11	−1.18401	−0.592005	+	0.805935i	$0.701663\pi$
$282$	0	0
$283$	1.02892e11	0.953544	0.476772	−	0.879027i	$-0.341807\pi$
$283$	1.02892e11	0.953544	0.476772	+	0.879027i	$0.341807\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.22090e11	−2.80226
$288$	0	0
$289$	−2.09489e10	−0.176653
$290$	0	0
$291$	−1.50295e11	−1.22865
$292$	0	0
$293$	4.52713e10	0.358855	0.179427	−	0.983771i	$-0.442576\pi$
$293$	4.52713e10	0.358855	0.179427	+	0.983771i	$0.442576\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−8.95190e10	−0.667592
$298$	0	0
$299$	−1.59658e10	−0.115524
$300$	0	0
$301$	2.08257e11	1.46235
$302$	0	0
$303$	−1.34803e11	−0.918771
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.27144e11	−1.45941	−0.729707	−	0.683760i	$-0.760343\pi$
$307$	−2.27144e11	−1.45941	−0.729707	+	0.683760i	$0.760343\pi$
$308$	0	0
$309$	−8.85074e10	−0.552289
$310$	0	0
$311$	−1.15793e11	−0.701877	−0.350939	−	0.936398i	$-0.614137\pi$
$311$	−1.15793e11	−0.701877	−0.350939	+	0.936398i	$0.614137\pi$
$312$	0	0
$313$	−1.00580e11	−0.592329	−0.296164	−	0.955137i	$-0.595708\pi$
$313$	−1.00580e11	−0.592329	−0.296164	+	0.955137i	$0.595708\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.04758e10	−0.447609	−0.223805	−	0.974634i	$-0.571848\pi$
$317$	−8.04758e10	−0.447609	−0.223805	+	0.974634i	$0.571848\pi$
$318$	0	0
$319$	−9.03530e10	−0.488522
$320$	0	0
$321$	1.75144e11	0.920709
$322$	0	0
$323$	1.13745e10	0.0581461
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.18400e11	−0.572647
$328$	0	0
$329$	5.13911e10	0.241828
$330$	0	0
$331$	−3.95335e11	−1.81025	−0.905127	−	0.425141i	$-0.860225\pi$
$331$	−3.95335e11	−1.81025	−0.905127	+	0.425141i	$0.860225\pi$
$332$	0	0
$333$	−8.98630e10	−0.400481
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−3.03418e11	−1.28146	−0.640732	−	0.767765i	$-0.721369\pi$
$337$	−3.03418e11	−1.28146	−0.640732	+	0.767765i	$0.721369\pi$
$338$	0	0
$339$	−1.66562e11	−0.684977
$340$	0	0
$341$	2.81731e11	1.12834
$342$	0	0
$343$	−3.78077e10	−0.147488
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.49981e11	−0.555332	−0.277666	−	0.960678i	$-0.589561\pi$
$347$	−1.49981e11	−0.555332	−0.277666	+	0.960678i	$0.589561\pi$
$348$	0	0
$349$	−4.12719e11	−1.48916	−0.744578	−	0.667535i	$-0.767349\pi$
$349$	−4.12719e11	−1.48916	−0.744578	+	0.667535i	$0.767349\pi$
$350$	0	0
$351$	5.02891e10	0.176845
$352$	0	0
$353$	3.72053e11	1.27532	0.637659	−	0.770319i	$-0.279903\pi$
$353$	3.72053e11	1.27532	0.637659	+	0.770319i	$0.279903\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.15251e11	−1.02719
$358$	0	0
$359$	1.29954e11	0.412920	0.206460	−	0.978455i	$-0.433806\pi$
$359$	1.29954e11	0.412920	0.206460	+	0.978455i	$0.433806\pi$
$360$	0	0
$361$	−3.21363e11	−0.995894
$362$	0	0
$363$	−1.60628e11	−0.485558
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.13597e11	−1.19009	−0.595045	−	0.803692i	$-0.702866\pi$
$367$	−4.13597e11	−1.19009	−0.595045	+	0.803692i	$0.702866\pi$
$368$	0	0
$369$	−2.68659e11	−0.754368
$370$	0	0
$371$	5.10756e11	1.39969
$372$	0	0
$373$	−4.96023e11	−1.32682	−0.663410	−	0.748256i	$-0.730891\pi$
$373$	−4.96023e11	−1.32682	−0.663410	+	0.748256i	$0.730891\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.07576e10	0.129409
$378$	0	0
$379$	−1.86834e11	−0.465135	−0.232567	−	0.972580i	$-0.574713\pi$
$379$	−1.86834e11	−0.465135	−0.232567	+	0.972580i	$0.574713\pi$
$380$	0	0
$381$	5.21341e11	1.26753
$382$	0	0
$383$	3.96068e11	0.940535	0.470267	−	0.882524i	$-0.344157\pi$
$383$	3.96068e11	0.940535	0.470267	+	0.882524i	$0.344157\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.73710e11	0.393664
$388$	0	0
$389$	−4.64849e11	−1.02929	−0.514646	−	0.857403i	$-0.672077\pi$
$389$	−4.64849e11	−1.02929	−0.514646	+	0.857403i	$0.672077\pi$
$390$	0	0
$391$	2.97395e11	0.643486
$392$	0	0
$393$	3.76005e11	0.795110
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.04945e11	0.818161	0.409080	−	0.912498i	$-0.365850\pi$
$397$	4.04945e11	0.818161	0.409080	+	0.912498i	$0.365850\pi$
$398$	0	0
$399$	−3.67253e10	−0.0725416
$400$	0	0
$401$	−9.10722e11	−1.75888	−0.879440	−	0.476010i	$-0.842083\pi$
$401$	−9.10722e11	−1.75888	−0.879440	+	0.476010i	$0.842083\pi$
$402$	0	0
$403$	−1.58268e11	−0.298896
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.49330e11	0.631046
$408$	0	0
$409$	6.22611e11	1.10017	0.550087	−	0.835107i	$-0.314594\pi$
$409$	6.22611e11	1.10017	0.550087	+	0.835107i	$0.314594\pi$
$410$	0	0
$411$	5.71078e9	0.00987205
$412$	0	0
$413$	7.83795e11	1.32564
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.43525e11	−1.04220
$418$	0	0
$419$	5.77692e11	0.915657	0.457829	−	0.889040i	$-0.348627\pi$
$419$	5.77692e11	0.915657	0.457829	+	0.889040i	$0.348627\pi$
$420$	0	0
$421$	2.19223e11	0.340109	0.170054	−	0.985435i	$-0.445606\pi$
$421$	2.19223e11	0.340109	0.170054	+	0.985435i	$0.445606\pi$
$422$	0	0
$423$	4.28660e10	0.0651000
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.47383e11	−0.651259
$428$	0	0
$429$	−5.48774e10	−0.0782233
$430$	0	0
$431$	−7.43484e11	−1.03782	−0.518912	−	0.854828i	$-0.673663\pi$
$431$	−7.43484e11	−1.03782	−0.518912	+	0.854828i	$0.673663\pi$
$432$	0	0
$433$	−2.19028e11	−0.299436	−0.149718	−	0.988729i	$-0.547837\pi$
$433$	−2.19028e11	−0.299436	−0.149718	+	0.988729i	$0.547837\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.46452e10	0.0454440
$438$	0	0
$439$	−9.31553e11	−1.19706	−0.598532	−	0.801099i	$-0.704249\pi$
$439$	−9.31553e11	−1.19706	−0.598532	+	0.801099i	$0.704249\pi$
$440$	0	0
$441$	−3.41518e11	−0.429972
$442$	0	0
$443$	9.53901e11	1.17676	0.588378	−	0.808586i	$-0.299767\pi$
$443$	9.53901e11	1.17676	0.588378	+	0.808586i	$0.299767\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.05527e12	1.25020
$448$	0	0
$449$	9.16478e11	1.06418	0.532088	−	0.846689i	$-0.321407\pi$
$449$	9.16478e11	1.06418	0.532088	+	0.846689i	$0.321407\pi$
$450$	0	0
$451$	1.04438e12	1.18867
$452$	0	0
$453$	7.02943e11	0.784292
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.25903e11	−0.885740	−0.442870	−	0.896586i	$-0.646040\pi$
$457$	−8.25903e11	−0.885740	−0.442870	+	0.896586i	$0.646040\pi$
$458$	0	0
$459$	−9.36735e11	−0.985053
$460$	0	0
$461$	−1.60225e12	−1.65225	−0.826125	−	0.563487i	$-0.809459\pi$
$461$	−1.60225e12	−1.65225	−0.826125	+	0.563487i	$0.809459\pi$
$462$	0	0
$463$	1.37910e12	1.39471	0.697353	−	0.716728i	$-0.254361\pi$
$463$	1.37910e12	1.39471	0.697353	+	0.716728i	$0.254361\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.55852e11	0.929961	0.464980	−	0.885321i	$-0.346061\pi$
$467$	9.55852e11	0.929961	0.464980	+	0.885321i	$0.346061\pi$
$468$	0	0
$469$	4.79576e11	0.457699
$470$	0	0
$471$	−1.23161e12	−1.15313
$472$	0	0
$473$	−6.75274e11	−0.620304
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.26028e11	0.376795
$478$	0	0
$479$	−1.16208e12	−1.00861	−0.504307	−	0.863524i	$-0.668252\pi$
$479$	−1.16208e12	−1.00861	−0.504307	+	0.863524i	$0.668252\pi$
$480$	0	0
$481$	−1.96243e11	−0.167164
$482$	0	0
$483$	−9.60213e11	−0.802797
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−8.30662e11	−0.669182	−0.334591	−	0.942364i	$-0.608598\pi$
$487$	−8.30662e11	−0.669182	−0.334591	+	0.942364i	$0.608598\pi$
$488$	0	0
$489$	−4.23291e11	−0.334772
$490$	0	0
$491$	−3.17729e11	−0.246712	−0.123356	−	0.992362i	$-0.539366\pi$
$491$	−3.17729e11	−0.246712	−0.123356	+	0.992362i	$0.539366\pi$
$492$	0	0
$493$	−9.45462e11	−0.720830
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.05607e12	−0.776408
$498$	0	0
$499$	2.15696e12	1.55736	0.778680	−	0.627421i	$-0.215889\pi$
$499$	2.15696e12	1.55736	0.778680	+	0.627421i	$0.215889\pi$
$500$	0	0
$501$	−1.95503e12	−1.38638
$502$	0	0
$503$	1.24185e12	0.864996	0.432498	−	0.901635i	$-0.357632\pi$
$503$	1.24185e12	0.864996	0.432498	+	0.901635i	$0.357632\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.13090e12	−0.760132
$508$	0	0
$509$	2.49436e12	1.64713	0.823566	−	0.567220i	$-0.191981\pi$
$509$	2.49436e12	1.64713	0.823566	+	0.567220i	$0.191981\pi$
$510$	0	0
$511$	4.12359e12	2.67536
$512$	0	0
$513$	−1.09125e11	−0.0695660
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.66636e11	−0.102579
$518$	0	0
$519$	1.45164e12	0.878227
$520$	0	0
$521$	−6.12898e11	−0.364434	−0.182217	−	0.983258i	$-0.558327\pi$
$521$	−6.12898e11	−0.364434	−0.182217	+	0.983258i	$0.558327\pi$
$522$	0	0
$523$	−3.03989e12	−1.77664	−0.888321	−	0.459223i	$-0.848128\pi$
$523$	−3.03989e12	−1.77664	−0.888321	+	0.459223i	$0.848128\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.94806e12	1.66490
$528$	0	0
$529$	−8.95326e11	−0.497085
$530$	0	0
$531$	6.53773e11	0.356863
$532$	0	0
$533$	−5.86700e11	−0.314879
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.55285e11	0.340052
$538$	0	0
$539$	1.32761e12	0.677517
$540$	0	0
$541$	2.86670e12	1.43878	0.719390	−	0.694606i	$-0.244421\pi$
$541$	2.86670e12	1.43878	0.719390	+	0.694606i	$0.244421\pi$
$542$	0	0
$543$	−2.45301e12	−1.21088
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.44503e12	1.16772	0.583862	−	0.811853i	$-0.301541\pi$
$547$	2.44503e12	1.16772	0.583862	+	0.811853i	$0.301541\pi$
$548$	0	0
$549$	−3.73168e11	−0.175319
$550$	0	0
$551$	−1.10142e11	−0.0509062
$552$	0	0
$553$	−8.45550e11	−0.384482
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.40112e12	0.616777	0.308389	−	0.951260i	$-0.400210\pi$
$557$	1.40112e12	0.616777	0.308389	+	0.951260i	$0.400210\pi$
$558$	0	0
$559$	3.79348e11	0.164318
$560$	0	0
$561$	1.02220e12	0.435716
$562$	0	0
$563$	−3.12352e11	−0.131026	−0.0655129	−	0.997852i	$-0.520868\pi$
$563$	−3.12352e11	−0.131026	−0.0655129	+	0.997852i	$0.520868\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.63203e12	0.663141
$568$	0	0
$569$	−1.50839e12	−0.603266	−0.301633	−	0.953424i	$-0.597532\pi$
$569$	−1.50839e12	−0.603266	−0.301633	+	0.953424i	$0.597532\pi$
$570$	0	0
$571$	1.77207e12	0.697621	0.348810	−	0.937193i	$-0.386586\pi$
$571$	1.77207e12	0.697621	0.348810	+	0.937193i	$0.386586\pi$
$572$	0	0
$573$	3.17778e12	1.23148
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5.81248e10	−0.0218308	−0.0109154	−	0.999940i	$-0.503475\pi$
$577$	−5.81248e10	−0.0218308	−0.0109154	+	0.999940i	$0.503475\pi$
$578$	0	0
$579$	−2.19680e12	−0.812338
$580$	0	0
$581$	5.40824e12	1.96908
$582$	0	0
$583$	−1.65613e12	−0.593725
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.72424e11	−0.233761	−0.116880	−	0.993146i	$-0.537289\pi$
$587$	−6.72424e11	−0.233761	−0.116880	+	0.993146i	$0.537289\pi$
$588$	0	0
$589$	3.43435e11	0.117578
$590$	0	0
$591$	−1.35230e12	−0.455962
$592$	0	0
$593$	−4.33711e12	−1.44031	−0.720153	−	0.693816i	$-0.755928\pi$
$593$	−4.33711e12	−1.44031	−0.720153	+	0.693816i	$0.755928\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.97527e12	0.958611
$598$	0	0
$599$	3.37288e11	0.107048	0.0535242	−	0.998567i	$-0.482955\pi$
$599$	3.37288e11	0.107048	0.0535242	+	0.998567i	$0.482955\pi$
$600$	0	0
$601$	−3.13643e11	−0.0980619	−0.0490309	−	0.998797i	$-0.515613\pi$
$601$	−3.13643e11	−0.0980619	−0.0490309	+	0.998797i	$0.515613\pi$
$602$	0	0
$603$	4.00021e11	0.123212
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.29060e12	0.385872	0.192936	−	0.981211i	$-0.438199\pi$
$607$	1.29060e12	0.385872	0.192936	+	0.981211i	$0.438199\pi$
$608$	0	0
$609$	3.05265e12	0.899290
$610$	0	0
$611$	9.36109e10	0.0271732
$612$	0	0
$613$	−3.58380e12	−1.02511	−0.512556	−	0.858654i	$-0.671301\pi$
$613$	−3.58380e12	−1.02511	−0.512556	+	0.858654i	$0.671301\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.46900e12	1.51923	0.759617	−	0.650370i	$-0.225386\pi$
$617$	5.46900e12	1.51923	0.759617	+	0.650370i	$0.225386\pi$
$618$	0	0
$619$	5.83483e12	1.59743	0.798713	−	0.601713i	$-0.205515\pi$
$619$	5.83483e12	1.59743	0.798713	+	0.601713i	$0.205515\pi$
$620$	0	0
$621$	−2.85317e12	−0.769866
$622$	0	0
$623$	4.43426e12	1.17930
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.19082e11	0.0307710
$628$	0	0
$629$	3.65542e12	0.931129
$630$	0	0
$631$	−4.43138e12	−1.11277	−0.556387	−	0.830923i	$-0.687813\pi$
$631$	−4.43138e12	−1.11277	−0.556387	+	0.830923i	$0.687813\pi$
$632$	0	0
$633$	−1.00894e12	−0.249775
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.45810e11	−0.179474
$638$	0	0
$639$	−8.80883e11	−0.209009
$640$	0	0
$641$	2.38785e12	0.558659	0.279329	−	0.960195i	$-0.409888\pi$
$641$	2.38785e12	0.558659	0.279329	+	0.960195i	$0.409888\pi$
$642$	0	0
$643$	−1.42772e12	−0.329378	−0.164689	−	0.986346i	$-0.552662\pi$
$643$	−1.42772e12	−0.329378	−0.164689	+	0.986346i	$0.552662\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.43250e12	−0.994443	−0.497221	−	0.867624i	$-0.665646\pi$
$647$	−4.43250e12	−0.994443	−0.497221	+	0.867624i	$0.665646\pi$
$648$	0	0
$649$	−2.54146e12	−0.562317
$650$	0	0
$651$	−9.51852e12	−2.07709
$652$	0	0
$653$	6.61930e11	0.142463	0.0712316	−	0.997460i	$-0.477307\pi$
$653$	6.61930e11	0.142463	0.0712316	+	0.997460i	$0.477307\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.43954e12	0.720205
$658$	0	0
$659$	−7.94755e12	−1.64153	−0.820765	−	0.571266i	$-0.806452\pi$
$659$	−7.94755e12	−1.64153	−0.820765	+	0.571266i	$0.806452\pi$
$660$	0	0
$661$	8.06071e11	0.164235	0.0821177	−	0.996623i	$-0.473832\pi$
$661$	8.06071e11	0.164235	0.0821177	+	0.996623i	$0.473832\pi$
$662$	0	0
$663$	−5.74242e11	−0.115421
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.87975e12	−0.563364
$668$	0	0
$669$	−2.66851e12	−0.515052
$670$	0	0
$671$	1.45064e12	0.276254
$672$	0	0
$673$	3.67873e12	0.691241	0.345621	−	0.938374i	$-0.387668\pi$
$673$	3.67873e12	0.691241	0.345621	+	0.938374i	$0.387668\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.60384e12	1.39118	0.695591	−	0.718438i	$-0.255143\pi$
$677$	7.60384e12	1.39118	0.695591	+	0.718438i	$0.255143\pi$
$678$	0	0
$679$	1.26346e13	2.28111
$680$	0	0
$681$	6.52388e11	0.116237
$682$	0	0
$683$	−6.89820e12	−1.21295	−0.606475	−	0.795103i	$-0.707417\pi$
$683$	−6.89820e12	−1.21295	−0.606475	+	0.795103i	$0.707417\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.16336e12	−0.370531
$688$	0	0
$689$	9.30362e11	0.157277
$690$	0	0
$691$	−4.45838e12	−0.743920	−0.371960	−	0.928249i	$-0.621314\pi$
$691$	−4.45838e12	−0.743920	−0.371960	+	0.928249i	$0.621314\pi$
$692$	0	0
$693$	2.11249e12	0.347933
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.09285e13	1.75393
$698$	0	0
$699$	5.42991e12	0.860291
$700$	0	0
$701$	−2.34128e12	−0.366203	−0.183102	−	0.983094i	$-0.558614\pi$
$701$	−2.34128e12	−0.366203	−0.183102	+	0.983094i	$0.558614\pi$
$702$	0	0
$703$	4.25840e11	0.0657578
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.13322e13	1.70579
$708$	0	0
$709$	5.10575e12	0.758841	0.379421	−	0.925224i	$-0.376123\pi$
$709$	5.10575e12	0.758841	0.379421	+	0.925224i	$0.376123\pi$
$710$	0	0
$711$	−7.05284e11	−0.103503
$712$	0	0
$713$	8.97940e12	1.30120
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.79467e12	0.394907
$718$	0	0
$719$	1.39438e12	0.194581	0.0972907	−	0.995256i	$-0.468982\pi$
$719$	1.39438e12	0.194581	0.0972907	+	0.995256i	$0.468982\pi$
$720$	0	0
$721$	7.44037e12	1.02538
$722$	0	0
$723$	5.17503e12	0.704353
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.87221e12	−0.381339	−0.190669	−	0.981654i	$-0.561066\pi$
$727$	−2.87221e12	−0.381339	−0.190669	+	0.981654i	$0.561066\pi$
$728$	0	0
$729$	7.82545e12	1.02621
$730$	0	0
$731$	−7.06612e12	−0.915278
$732$	0	0
$733$	−3.91043e12	−0.500330	−0.250165	−	0.968203i	$-0.580485\pi$
$733$	−3.91043e12	−0.500330	−0.250165	+	0.968203i	$0.580485\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.55503e12	−0.194149
$738$	0	0
$739$	1.00098e12	0.123460	0.0617298	−	0.998093i	$-0.480338\pi$
$739$	1.00098e12	0.123460	0.0617298	+	0.998093i	$0.480338\pi$
$740$	0	0
$741$	−6.68966e10	−0.00815121
$742$	0	0
$743$	−5.19147e12	−0.624943	−0.312472	−	0.949927i	$-0.601157\pi$
$743$	−5.19147e12	−0.624943	−0.312472	+	0.949927i	$0.601157\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.51108e12	0.530076
$748$	0	0
$749$	−1.47235e13	−1.70939
$750$	0	0
$751$	1.48763e13	1.70654	0.853268	−	0.521473i	$-0.174617\pi$
$751$	1.48763e13	1.70654	0.853268	+	0.521473i	$0.174617\pi$
$752$	0	0
$753$	−3.94705e12	−0.447399
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.03576e13	−1.14637	−0.573187	−	0.819425i	$-0.694293\pi$
$757$	−1.03576e13	−1.14637	−0.573187	+	0.819425i	$0.694293\pi$
$758$	0	0
$759$	3.11349e12	0.340533
$760$	0	0
$761$	2.88282e11	0.0311592	0.0155796	−	0.999879i	$-0.495041\pi$
$761$	2.88282e11	0.0311592	0.0155796	+	0.999879i	$0.495041\pi$
$762$	0	0
$763$	9.95330e12	1.06318
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.42771e12	0.148957
$768$	0	0
$769$	1.28801e13	1.32816	0.664081	−	0.747661i	$-0.268823\pi$
$769$	1.28801e13	1.32816	0.664081	+	0.747661i	$0.268823\pi$
$770$	0	0
$771$	8.55530e12	0.871948
$772$	0	0
$773$	−9.40486e12	−0.947425	−0.473712	−	0.880680i	$-0.657086\pi$
$773$	−9.40486e12	−0.947425	−0.473712	+	0.880680i	$0.657086\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.18024e13	−1.16165
$778$	0	0
$779$	1.27311e12	0.123865
$780$	0	0
$781$	3.42431e12	0.329340
$782$	0	0
$783$	9.07063e12	0.862401
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.24747e12	0.766363	0.383181	−	0.923673i	$-0.374828\pi$
$787$	8.24747e12	0.766363	0.383181	+	0.923673i	$0.374828\pi$
$788$	0	0
$789$	2.11800e12	0.194572
$790$	0	0
$791$	1.40020e13	1.27173
$792$	0	0
$793$	−8.14926e11	−0.0731794
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.35880e12	−0.294864	−0.147432	−	0.989072i	$-0.547101\pi$
$797$	−3.35880e12	−0.294864	−0.147432	+	0.989072i	$0.547101\pi$
$798$	0	0
$799$	−1.74369e12	−0.151359
$800$	0	0
$801$	3.69867e12	0.317468
$802$	0	0
$803$	−1.33707e13	−1.13484
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.86654e13	−1.54919
$808$	0	0
$809$	−6.87559e12	−0.564341	−0.282171	−	0.959364i	$-0.591054\pi$
$809$	−6.87559e12	−0.564341	−0.282171	+	0.959364i	$0.591054\pi$
$810$	0	0
$811$	−2.11160e13	−1.71402	−0.857012	−	0.515296i	$-0.827682\pi$
$811$	−2.11160e13	−1.71402	−0.857012	+	0.515296i	$0.827682\pi$
$812$	0	0
$813$	5.70903e12	0.458305
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−8.23171e11	−0.0646384
$818$	0	0
$819$	−1.18674e12	−0.0921672
$820$	0	0
$821$	−3.67955e12	−0.282651	−0.141325	−	0.989963i	$-0.545136\pi$
$821$	−3.67955e12	−0.282651	−0.141325	+	0.989963i	$0.545136\pi$
$822$	0	0
$823$	6.15696e12	0.467807	0.233904	−	0.972260i	$-0.424850\pi$
$823$	6.15696e12	0.467807	0.233904	+	0.972260i	$0.424850\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.99295e12	0.296838	0.148419	−	0.988925i	$-0.452582\pi$
$827$	3.99295e12	0.296838	0.148419	+	0.988925i	$0.452582\pi$
$828$	0	0
$829$	1.16163e12	0.0854229	0.0427114	−	0.999087i	$-0.486400\pi$
$829$	1.16163e12	0.0854229	0.0427114	+	0.999087i	$0.486400\pi$
$830$	0	0
$831$	7.63472e12	0.555378
$832$	0	0
$833$	1.38922e13	0.999697
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.82833e13	−1.99189
$838$	0	0
$839$	−1.05218e13	−0.733099	−0.366550	−	0.930398i	$-0.619461\pi$
$839$	−1.05218e13	−0.733099	−0.366550	+	0.930398i	$0.619461\pi$
$840$	0	0
$841$	−5.35201e12	−0.368922
$842$	0	0
$843$	−1.35565e13	−0.924537
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.35032e13	0.901490
$848$	0	0
$849$	1.12718e13	0.744578
$850$	0	0
$851$	1.11339e13	0.727722
$852$	0	0
$853$	−1.18899e13	−0.768967	−0.384484	−	0.923132i	$-0.625620\pi$
$853$	−1.18899e13	−0.768967	−0.384484	+	0.923132i	$0.625620\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.57517e12	0.163077	0.0815385	−	0.996670i	$-0.474017\pi$
$857$	2.57517e12	0.163077	0.0815385	+	0.996670i	$0.474017\pi$
$858$	0	0
$859$	−1.61942e13	−1.01482	−0.507411	−	0.861704i	$-0.669398\pi$
$859$	−1.61942e13	−1.01482	−0.507411	+	0.861704i	$0.669398\pi$
$860$	0	0
$861$	−3.52852e13	−2.18815
$862$	0	0
$863$	−4.94014e12	−0.303173	−0.151586	−	0.988444i	$-0.548438\pi$
$863$	−4.94014e12	−0.303173	−0.151586	+	0.988444i	$0.548438\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.29496e12	−0.137940
$868$	0	0
$869$	2.74170e12	0.163091
$870$	0	0
$871$	8.73567e11	0.0514298
$872$	0	0
$873$	1.05387e13	0.614075
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.68472e12	0.495744	0.247872	−	0.968793i	$-0.420269\pi$
$877$	8.68472e12	0.495744	0.247872	+	0.968793i	$0.420269\pi$
$878$	0	0
$879$	4.95950e12	0.280213
$880$	0	0
$881$	−1.02139e13	−0.571214	−0.285607	−	0.958347i	$-0.592195\pi$
$881$	−1.02139e13	−0.571214	−0.285607	+	0.958347i	$0.592195\pi$
$882$	0	0
$883$	3.58719e12	0.198578	0.0992891	−	0.995059i	$-0.468343\pi$
$883$	3.58719e12	0.198578	0.0992891	+	0.995059i	$0.468343\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.72905e13	−1.48032	−0.740160	−	0.672431i	$-0.765250\pi$
$887$	−2.72905e13	−1.48032	−0.740160	+	0.672431i	$0.765250\pi$
$888$	0	0
$889$	−4.38266e13	−2.35331
$890$	0	0
$891$	−5.29187e12	−0.281294
$892$	0	0
$893$	−2.03132e11	−0.0106892
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.74907e12	−0.0902070
$898$	0	0
$899$	−2.85468e13	−1.45760
$900$	0	0
$901$	−1.73299e13	−0.876059
$902$	0	0
$903$	2.28147e13	1.14188
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1.79849e13	−0.882418	−0.441209	−	0.897404i	$-0.645450\pi$
$907$	−1.79849e13	−0.882418	−0.441209	+	0.897404i	$0.645450\pi$
$908$	0	0
$909$	9.45233e12	0.459200
$910$	0	0
$911$	1.80978e13	0.870549	0.435275	−	0.900298i	$-0.356651\pi$
$911$	1.80978e13	0.870549	0.435275	+	0.900298i	$0.356651\pi$
$912$	0	0
$913$	−1.75362e13	−0.835252
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.16089e13	−1.47621
$918$	0	0
$919$	−3.09898e13	−1.43317	−0.716586	−	0.697498i	$-0.754296\pi$
$919$	−3.09898e13	−1.43317	−0.716586	+	0.697498i	$0.754296\pi$
$920$	0	0
$921$	−2.48838e13	−1.13959
$922$	0	0
$923$	−1.92368e12	−0.0872418
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.20611e12	0.276033
$928$	0	0
$929$	−2.53410e13	−1.11623	−0.558113	−	0.829765i	$-0.688475\pi$
$929$	−2.53410e13	−1.11623	−0.558113	+	0.829765i	$0.688475\pi$
$930$	0	0
$931$	1.61838e12	0.0706002
$932$	0	0
$933$	−1.26852e13	−0.548063
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−7.05858e12	−0.299150	−0.149575	−	0.988750i	$-0.547791\pi$
$937$	−7.05858e12	−0.299150	−0.149575	+	0.988750i	$0.547791\pi$
$938$	0	0
$939$	−1.10186e13	−0.462522
$940$	0	0
$941$	4.06352e12	0.168947	0.0844733	−	0.996426i	$-0.473079\pi$
$941$	4.06352e12	0.168947	0.0844733	+	0.996426i	$0.473079\pi$
$942$	0	0
$943$	3.32866e13	1.37078
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.96392e12	0.160158	0.0800792	−	0.996789i	$-0.474483\pi$
$947$	3.96392e12	0.160158	0.0800792	+	0.996789i	$0.474483\pi$
$948$	0	0
$949$	7.51128e12	0.300619
$950$	0	0
$951$	−8.81618e12	−0.349517
$952$	0	0
$953$	1.82636e12	0.0717246	0.0358623	−	0.999357i	$-0.488582\pi$
$953$	1.82636e12	0.0717246	0.0358623	+	0.999357i	$0.488582\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−9.89823e12	−0.381464
$958$	0	0
$959$	−4.80077e11	−0.0183285
$960$	0	0
$961$	6.25725e13	2.36662
$962$	0	0
$963$	−1.22810e13	−0.460168
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.71062e13	0.629122	0.314561	−	0.949237i	$-0.398143\pi$
$967$	1.71062e13	0.629122	0.314561	+	0.949237i	$0.398143\pi$
$968$	0	0
$969$	1.24608e12	0.0454036
$970$	0	0
$971$	6.43910e12	0.232455	0.116227	−	0.993223i	$-0.462920\pi$
$971$	6.43910e12	0.232455	0.116227	+	0.993223i	$0.462920\pi$
$972$	0	0
$973$	5.40979e13	1.93496
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.72264e13	−0.604880	−0.302440	−	0.953168i	$-0.597801\pi$
$977$	−1.72264e13	−0.604880	−0.302440	+	0.953168i	$0.597801\pi$
$978$	0	0
$979$	−1.43781e13	−0.500240
$980$	0	0
$981$	8.30217e12	0.286208
$982$	0	0
$983$	1.22407e13	0.418134	0.209067	−	0.977901i	$-0.432957\pi$
$983$	1.22407e13	0.418134	0.209067	+	0.977901i	$0.432957\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.62992e12	0.188832
$988$	0	0
$989$	−2.15225e13	−0.715334
$990$	0	0
$991$	2.74736e13	0.904867	0.452434	−	0.891798i	$-0.350556\pi$
$991$	2.74736e13	0.904867	0.452434	+	0.891798i	$0.350556\pi$
$992$	0	0
$993$	−4.33092e13	−1.41354
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.32279e13	−0.423996	−0.211998	−	0.977270i	$-0.567997\pi$
$997$	−1.32279e13	−0.423996	−0.211998	+	0.977270i	$0.567997\pi$
$998$	0	0
$999$	−3.50696e13	−1.11400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.10.a.o.1.2		2
4.3	odd	2		200.10.a.d.1.1		2
5.2	odd	4		400.10.c.o.49.2		4
5.3	odd	4		400.10.c.o.49.3		4
5.4	even	2		80.10.a.h.1.1		2
20.3	even	4		200.10.c.e.49.2		4
20.7	even	4		200.10.c.e.49.3		4
20.19	odd	2		40.10.a.b.1.2	✓	2
40.19	odd	2		320.10.a.p.1.1		2
40.29	even	2		320.10.a.o.1.2		2
60.59	even	2		360.10.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.10.a.b.1.2	✓	2	20.19	odd	2
80.10.a.h.1.1		2	5.4	even	2
200.10.a.d.1.1		2	4.3	odd	2
200.10.c.e.49.2		4	20.3	even	4
200.10.c.e.49.3		4	20.7	even	4
320.10.a.o.1.2		2	40.29	even	2
320.10.a.p.1.1		2	40.19	odd	2
360.10.a.a.1.1		2	60.59	even	2
400.10.a.o.1.2		2	1.1	even	1	trivial
400.10.c.o.49.2		4	5.2	odd	4
400.10.c.o.49.3		4	5.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q+109.551 q^{3} -9209.37 q^{7} -7681.66 q^{9} +29861.4 q^{11} -16775.2 q^{13} +312472. q^{17} +36401.6 q^{19} -1.00889e6 q^{21} +951749. q^{23} -2.99782e6 q^{27} -3.02575e6 q^{29} +9.43462e6 q^{31} +3.27134e6 q^{33} +1.16984e7 q^{37} -1.83774e6 q^{39} +3.49741e7 q^{41} -2.26136e7 q^{43} -5.58030e6 q^{47} +4.44590e7 q^{49} +3.42316e7 q^{51} -5.54605e7 q^{53} +3.98782e6 q^{57} -8.51084e7 q^{59} +4.85791e7 q^{61} +7.07433e7 q^{63} -5.20748e7 q^{67} +1.04265e8 q^{69} +1.14674e8 q^{71} -4.47760e8 q^{73} -2.75005e8 q^{77} +9.18141e7 q^{79} -1.77215e8 q^{81} -5.87254e8 q^{83} -3.31472e8 q^{87} -4.81494e8 q^{89} +1.54489e8 q^{91} +1.03357e9 q^{93} -1.37193e9 q^{97} -2.29385e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 92 q^{3} - 6908 q^{7} + 13258 q^{9} + 8080 q^{11} - 111948 q^{13} + 327532 q^{17} + 1156680 q^{19} - 1472736 q^{21} - 1057252 q^{23} - 3251096 q^{27} - 4212260 q^{29} + 11361128 q^{31} + 7661392 q^{33}+ \cdots - 685480304 q^{99}+O(q^{100}) \) 2 * q - 92 * q^3 - 6908 * q^7 + 13258 * q^9 + 8080 * q^11 - 111948 * q^13 + 327532 * q^17 + 1156680 * q^19 - 1472736 * q^21 - 1057252 * q^23 - 3251096 * q^27 - 4212260 * q^29 + 11361128 * q^31 + 7661392 * q^33 + 7251860 * q^37 + 17344392 * q^39 + 13030436 * q^41 - 47934076 * q^43 + 30914292 * q^47 + 9401666 * q^49 + 31196280 * q^51 - 100922108 * q^53 - 221805008 * q^57 + 47362536 * q^59 + 203634428 * q^61 + 118933236 * q^63 - 58872852 * q^67 + 509180256 * q^69 + 349900792 * q^71 - 71160388 * q^73 - 325131984 * q^77 + 452087344 * q^79 - 538321022 * q^81 - 244865436 * q^83 - 92329544 * q^87 - 256073260 * q^89 - 64538616 * q^91 + 645280640 * q^93 + 184950572 * q^97 - 685480304 * q^99

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