LMFDB - Embedded newform 200.10.c.e.49.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [200,10,Mod(49,200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("200.49"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	200.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-26516] 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:	no
Analytic conductor:	$103.007167233$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6049})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3025x^{2} + 2286144 $ x^4 + 3025*x^2 + 2286144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$38.3877i$ of defining polynomial
Character	$\chi$	$=$	200.49
Dual form			200.10.c.e.49.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+109.551i q^{3} +9209.37i q^{7} +7681.66 q^{9} -29861.4 q^{11} +16775.2i q^{13} +312472. i q^{17} +36401.6 q^{19} -1.00889e6 q^{21} +951749. i q^{23} +2.99782e6i q^{27} +3.02575e6 q^{29} -9.43462e6 q^{31} -3.27134e6i q^{33} +1.16984e7i q^{37} -1.83774e6 q^{39} +3.49741e7 q^{41} -2.26136e7i q^{43} +5.58030e6i q^{47} -4.44590e7 q^{49} -3.42316e7 q^{51} +5.54605e7i q^{53} +3.98782e6i q^{57} -8.51084e7 q^{59} +4.85791e7 q^{61} +7.07433e7i q^{63} +5.20748e7i q^{67} -1.04265e8 q^{69} -1.14674e8 q^{71} +4.47760e8i q^{73} -2.75005e8i q^{77} +9.18141e7 q^{79} -1.77215e8 q^{81} -5.87254e8i q^{83} +3.31472e8i q^{87} +4.81494e8 q^{89} -1.54489e8 q^{91} -1.03357e9i q^{93} -1.37193e9i q^{97} -2.29385e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 26516 q^{9} - 16160 q^{11} + 2313360 q^{19} - 2945472 q^{21} + 8424520 q^{29} - 22722256 q^{31} + 34688784 q^{39} + 26060872 q^{41} - 18803332 q^{49} - 62392560 q^{51} + 94725072 q^{59} + 407268856 q^{61}+ \cdots - 1370960608 q^{99}+O(q^{100}) $ 4 * q - 26516 * q^9 - 16160 * q^11 + 2313360 * q^19 - 2945472 * q^21 + 8424520 * q^29 - 22722256 * q^31 + 34688784 * q^39 + 26060872 * q^41 - 18803332 * q^49 - 62392560 * q^51 + 94725072 * q^59 + 407268856 * q^61 - 1018360512 * q^69 - 699801584 * q^71 + 904174688 * q^79 - 1076642044 * q^81 + 512146520 * q^89 + 129077232 * q^91 - 1370960608 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$177$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	109.551i	0.780853i	0.920634	+	0.390426i	$0.127672\pi$
$3$	109.551i	0.780853i	−0.920634	+	0.390426i	$0.872328\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	9209.37i	1.44974i	0.688888	+	0.724868i	$0.258099\pi$
$7$	9209.37i	1.44974i	−0.688888	+	0.724868i	$0.741901\pi$
$8$	0	0
$9$	7681.66	0.390269
$10$	0	0
$11$	−29861.4	−0.614955	−0.307477	−	0.951555i	$-0.599485\pi$
$11$	−29861.4	−0.614955	−0.307477	+	0.951555i	$0.599485\pi$
$12$	0	0
$13$	16775.2i	0.162901i	0.996677	+	0.0814505i	$0.0259552\pi$
$13$	16775.2i	0.162901i	−0.996677	+	0.0814505i	$0.974045\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	312472.i	0.907385i	0.891158	+	0.453692i	$0.149894\pi$
$17$	312472.i	0.907385i	−0.891158	+	0.453692i	$0.850106\pi$
$18$	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.i	0.709165i	0.935025	+	0.354583i	$0.115377\pi$
$23$	951749.i	0.709165i	−0.935025	+	0.354583i	$0.884623\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.99782e6i	1.08560i
$28$	0	0
$29$	3.02575e6	0.794404	0.397202	−	0.917731i	$-0.369981\pi$
$29$	3.02575e6	0.794404	0.397202	+	0.917731i	$0.369981\pi$
$30$	0	0
$31$	−9.43462e6	−1.83483	−0.917417	−	0.397926i	$-0.869730\pi$
$31$	−9.43462e6	−1.83483	−0.917417	+	0.397926i	$0.869730\pi$
$32$	0	0
$33$	− 3.27134e6i	− 0.480189i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.16984e7i	1.02617i	0.858339	+	0.513084i	$0.171497\pi$
$37$	1.16984e7i	1.02617i	−0.858339	+	0.513084i	$0.828503\pi$
$38$	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
$43$	− 2.26136e7i	− 1.00870i	−0.863500	−	0.504349i	$-0.831732\pi$
$43$	− 2.26136e7i	− 1.00870i	0.863500	−	0.504349i	$-0.168268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.58030e6i	0.166808i	0.996516	+	0.0834041i	$0.0265792\pi$
$47$	5.58030e6i	0.166808i	−0.996516	+	0.0834041i	$0.973421\pi$
$48$	0	0
$49$	−4.44590e7	−1.10173
$50$	0	0
$51$	−3.42316e7	−0.708534
$52$	0	0
$53$	5.54605e7i	0.965477i	0.875765	+	0.482739i	$0.160358\pi$
$53$	5.54605e7i	0.965477i	−0.875765	+	0.482739i	$0.839642\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.98782e6i	0.0500378i
$58$	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
$63$	7.07433e7i	0.565787i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.20748e7i	0.315712i	0.987462	+	0.157856i	$0.0504582\pi$
$67$	5.20748e7i	0.315712i	−0.987462	+	0.157856i	$0.949542\pi$
$68$	0	0
$69$	−1.04265e8	−0.553754
$70$	0	0
$71$	−1.14674e8	−0.535551	−0.267776	−	0.963481i	$-0.586289\pi$
$71$	−1.14674e8	−0.535551	−0.267776	+	0.963481i	$0.586289\pi$
$72$	0	0
$73$	4.47760e8i	1.84541i	0.385508	+	0.922704i	$0.374026\pi$
$73$	4.47760e8i	1.84541i	−0.385508	+	0.922704i	$0.625974\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.75005e8i	− 0.891522i
$78$	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
$83$	− 5.87254e8i	− 1.35823i	−0.734030	−	0.679117i	$-0.762363\pi$
$83$	− 5.87254e8i	− 1.35823i	0.734030	−	0.679117i	$-0.237637\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.31472e8i	0.620313i
$88$	0	0
$89$	4.81494e8	0.813459	0.406729	−	0.913549i	$-0.366669\pi$
$89$	4.81494e8	0.813459	0.406729	+	0.913549i	$0.366669\pi$
$90$	0	0
$91$	−1.54489e8	−0.236163
$92$	0	0
$93$	− 1.03357e9i	− 1.43274i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.37193e9i	− 1.57347i	−0.617293	−	0.786733i	$-0.711771\pi$
$97$	− 1.37193e9i	− 1.57347i	0.617293	−	0.786733i	$-0.288229\pi$
$98$	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
$103$	− 8.07913e8i	− 0.707289i	−0.935380	−	0.353645i	$-0.884942\pi$
$103$	− 8.07913e8i	− 0.707289i	0.935380	−	0.353645i	$-0.115058\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.59875e9i	− 1.17911i	−0.807730	−	0.589553i	$-0.799304\pi$
$107$	− 1.59875e9i	− 1.17911i	0.807730	−	0.589553i	$-0.200696\pi$
$108$	0	0
$109$	1.08078e9	0.733361	0.366680	−	0.930347i	$-0.380494\pi$
$109$	1.08078e9	0.733361	0.366680	+	0.930347i	$0.380494\pi$
$110$	0	0
$111$	−1.28157e9	−0.801286
$112$	0	0
$113$	1.52041e9i	0.877217i	0.898678	+	0.438608i	$0.144528\pi$
$113$	1.52041e9i	0.877217i	−0.898678	+	0.438608i	$0.855472\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.28862e8i	0.0635751i
$118$	0	0
$119$	−2.87768e9	−1.31547
$120$	0	0
$121$	−1.46624e9	−0.621831
$122$	0	0
$123$	3.83144e9i	1.50935i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.75891e9i	− 1.62327i	−0.584166	−	0.811634i	$-0.698578\pi$
$127$	− 4.75891e9i	− 1.62327i	0.584166	−	0.811634i	$-0.301422\pi$
$128$	0	0
$129$	2.47733e9	0.787645
$130$	0	0
$131$	−3.43225e9	−1.01826	−0.509129	−	0.860690i	$-0.670032\pi$
$131$	−3.43225e9	−1.01826	−0.509129	+	0.860690i	$0.670032\pi$
$132$	0	0
$133$	3.35236e8i	0.0929005i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.21292e7i	0.0126427i	0.999980	+	0.00632133i	$0.00201215\pi$
$137$	5.21292e7i	0.0126427i	−0.999980	+	0.00632133i	$0.997988\pi$
$138$	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
$143$	− 5.00932e8i	− 0.100177i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4.87051e9i	− 0.860292i
$148$	0	0
$149$	−9.63272e9	−1.60107	−0.800535	−	0.599285i	$-0.795451\pi$
$149$	−9.63272e9	−1.60107	−0.800535	+	0.599285i	$0.795451\pi$
$150$	0	0
$151$	−6.41660e9	−1.00440	−0.502202	−	0.864750i	$-0.667477\pi$
$151$	−6.41660e9	−1.00440	−0.502202	+	0.864750i	$0.667477\pi$
$152$	0	0
$153$	2.40031e9i	0.354124i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1.12424e10i	− 1.47676i	−0.674384	−	0.738381i	$-0.735591\pi$
$157$	− 1.12424e10i	− 1.47676i	0.674384	−	0.738381i	$-0.264409\pi$
$158$	0	0
$159$	−6.07573e9	−0.753896
$160$	0	0
$161$	−8.76502e9	−1.02810
$162$	0	0
$163$	− 3.86388e9i	− 0.428726i	−0.976754	−	0.214363i	$-0.931232\pi$
$163$	− 3.86388e9i	− 0.428726i	0.976754	−	0.214363i	$-0.0687675\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.78459e10i	1.77547i	0.460353	+	0.887736i	$0.347723\pi$
$167$	1.78459e10i	1.77547i	−0.460353	+	0.887736i	$0.652277\pi$
$168$	0	0
$169$	1.03231e10	0.973463
$170$	0	0
$171$	2.79625e8	0.0250088
$172$	0	0
$173$	− 1.32509e10i	− 1.12470i	−0.826899	−	0.562351i	$-0.809897\pi$
$173$	− 1.32509e10i	− 1.12470i	0.826899	−	0.562351i	$-0.190103\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 9.32367e9i	− 0.714015i
$178$	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
$183$	5.32187e9i	0.350780i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 9.33087e9i	− 0.558001i
$188$	0	0
$189$	−2.76080e10	−1.57383
$190$	0	0
$191$	−2.90074e10	−1.57710	−0.788549	−	0.614972i	$-0.789167\pi$
$191$	−2.90074e10	−1.57710	−0.788549	+	0.614972i	$0.789167\pi$
$192$	0	0
$193$	2.00528e10i	1.04032i	0.854068	+	0.520161i	$0.174128\pi$
$193$	2.00528e10i	1.04032i	−0.854068	+	0.520161i	$0.825872\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.23440e10i	− 0.583928i	−0.956429	−	0.291964i	$-0.905691\pi$
$197$	− 1.23440e10i	− 0.583928i	0.956429	−	0.291964i	$-0.0943087\pi$
$198$	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
$203$	2.78652e10i	1.15168i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.31101e9i	0.276765i
$208$	0	0
$209$	−1.08700e9	−0.0394069
$210$	0	0
$211$	9.20982e9	0.319875	0.159937	−	0.987127i	$-0.448871\pi$
$211$	9.20982e9	0.319875	0.159937	+	0.987127i	$0.448871\pi$
$212$	0	0
$213$	− 1.25626e10i	− 0.418187i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 8.68870e10i	− 2.66003i
$218$	0	0
$219$	−4.90524e10	−1.44099
$220$	0	0
$221$	−5.24180e9	−0.147814
$222$	0	0
$223$	− 2.43587e10i	− 0.659601i	−0.944051	−	0.329801i	$-0.893018\pi$
$223$	− 2.43587e10i	− 0.659601i	0.944051	−	0.329801i	$-0.106982\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 5.95512e9i	− 0.148859i	−0.997226	−	0.0744294i	$-0.976286\pi$
$227$	− 5.95512e9i	− 0.148859i	0.997226	−	0.0744294i	$-0.0237135\pi$
$228$	0	0
$229$	1.97476e10	0.474521	0.237260	−	0.971446i	$-0.423751\pi$
$229$	1.97476e10	0.474521	0.237260	+	0.971446i	$0.423751\pi$
$230$	0	0
$231$	3.01270e10	0.696148
$232$	0	0
$233$	− 4.95653e10i	− 1.10173i	−0.834594	−	0.550866i	$-0.814298\pi$
$233$	− 4.95653e10i	− 1.10173i	0.834594	−	0.550866i	$-0.185702\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.00583e10i	0.207089i
$238$	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
$243$	3.95920e10i	0.728416i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.10645e8i	0.0104389i
$248$	0	0
$249$	6.43340e10	1.06058
$250$	0	0
$251$	3.60294e10	0.572962	0.286481	−	0.958086i	$-0.407515\pi$
$251$	3.60294e10	0.572962	0.286481	+	0.958086i	$0.407515\pi$
$252$	0	0
$253$	− 2.84206e10i	− 0.436104i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.80945e10i	1.11666i	0.829619	+	0.558330i	$0.188558\pi$
$257$	7.80945e10i	1.11666i	−0.829619	+	0.558330i	$0.811442\pi$
$258$	0	0
$259$	−1.07735e11	−1.48767
$260$	0	0
$261$	2.32427e10	0.310031
$262$	0	0
$263$	1.93335e10i	0.249178i	0.992208	+	0.124589i	$0.0397613\pi$
$263$	1.93335e10i	0.249178i	−0.992208	+	0.124589i	$0.960239\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.27479e10i	0.635192i
$268$	0	0
$269$	1.70381e11	1.98398	0.991989	−	0.126327i	$-0.0403187\pi$
$269$	1.70381e11	1.98398	0.991989	+	0.126327i	$0.0403187\pi$
$270$	0	0
$271$	−5.21132e10	−0.586929	−0.293465	−	0.955970i	$-0.594808\pi$
$271$	−5.21132e10	−0.586929	−0.293465	+	0.955970i	$0.594808\pi$
$272$	0	0
$273$	− 1.69244e10i	− 0.184409i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.96913e10i	0.711245i	0.934630	+	0.355623i	$0.115731\pi$
$277$	6.96913e10i	0.711245i	−0.934630	+	0.355623i	$0.884269\pi$
$278$	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
$283$	1.02892e11i	0.953544i	0.879027	+	0.476772i	$0.158193\pi$
$283$	1.02892e11i	0.953544i	−0.879027	+	0.476772i	$0.841807\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.22090e11i	2.80226i
$288$	0	0
$289$	2.09489e10	0.176653
$290$	0	0
$291$	1.50295e11	1.22865
$292$	0	0
$293$	− 4.52713e10i	− 0.358855i	−0.983771	−	0.179427i	$-0.942576\pi$
$293$	− 4.52713e10i	− 0.358855i	0.983771	−	0.179427i	$-0.0574245\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 8.95190e10i	− 0.667592i
$298$	0	0
$299$	−1.59658e10	−0.115524
$300$	0	0
$301$	2.08257e11	1.46235
$302$	0	0
$303$	− 1.34803e11i	− 0.918771i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.27144e11i	1.45941i	0.683760	+	0.729707i	$0.260343\pi$
$307$	2.27144e11i	1.45941i	−0.683760	+	0.729707i	$0.739657\pi$
$308$	0	0
$309$	8.85074e10	0.552289
$310$	0	0
$311$	1.15793e11	0.701877	0.350939	−	0.936398i	$-0.385863\pi$
$311$	1.15793e11	0.701877	0.350939	+	0.936398i	$0.385863\pi$
$312$	0	0
$313$	1.00580e11i	0.592329i	0.955137	+	0.296164i	$0.0957076\pi$
$313$	1.00580e11i	0.592329i	−0.955137	+	0.296164i	$0.904292\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.04758e10i	− 0.447609i	−0.974634	−	0.223805i	$-0.928152\pi$
$317$	− 8.04758e10i	− 0.447609i	0.974634	−	0.223805i	$-0.0718477\pi$
$318$	0	0
$319$	−9.03530e10	−0.488522
$320$	0	0
$321$	1.75144e11	0.920709
$322$	0	0
$323$	1.13745e10i	0.0581461i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.18400e11i	0.572647i
$328$	0	0
$329$	−5.13911e10	−0.241828
$330$	0	0
$331$	3.95335e11	1.81025	0.905127	−	0.425141i	$-0.139775\pi$
$331$	3.95335e11	1.81025	0.905127	+	0.425141i	$0.139775\pi$
$332$	0	0
$333$	8.98630e10i	0.400481i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 3.03418e11i	− 1.28146i	−0.767765	−	0.640732i	$-0.778631\pi$
$337$	− 3.03418e11i	− 1.28146i	0.767765	−	0.640732i	$-0.221369\pi$
$338$	0	0
$339$	−1.66562e11	−0.684977
$340$	0	0
$341$	2.81731e11	1.12834
$342$	0	0
$343$	− 3.78077e10i	− 0.147488i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.49981e11i	0.555332i	0.960678	+	0.277666i	$0.0895609\pi$
$347$	1.49981e11i	0.555332i	−0.960678	+	0.277666i	$0.910439\pi$
$348$	0	0
$349$	4.12719e11	1.48916	0.744578	−	0.667535i	$-0.232651\pi$
$349$	4.12719e11	1.48916	0.744578	+	0.667535i	$0.232651\pi$
$350$	0	0
$351$	−5.02891e10	−0.176845
$352$	0	0
$353$	− 3.72053e11i	− 1.27532i	−0.770319	−	0.637659i	$-0.779903\pi$
$353$	− 3.72053e11i	− 1.27532i	0.770319	−	0.637659i	$-0.220097\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3.15251e11i	− 1.02719i
$358$	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
$363$	− 1.60628e11i	− 0.485558i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.13597e11i	1.19009i	0.803692	+	0.595045i	$0.202866\pi$
$367$	4.13597e11i	1.19009i	−0.803692	+	0.595045i	$0.797134\pi$
$368$	0	0
$369$	2.68659e11	0.754368
$370$	0	0
$371$	−5.10756e11	−1.39969
$372$	0	0
$373$	4.96023e11i	1.32682i	0.748256	+	0.663410i	$0.230891\pi$
$373$	4.96023e11i	1.32682i	−0.748256	+	0.663410i	$0.769109\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.07576e10i	0.129409i
$378$	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
$383$	3.96068e11i	0.940535i	0.882524	+	0.470267i	$0.155843\pi$
$383$	3.96068e11i	0.940535i	−0.882524	+	0.470267i	$0.844157\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.73710e11i	− 0.393664i
$388$	0	0
$389$	4.64849e11	1.02929	0.514646	−	0.857403i	$-0.327923\pi$
$389$	4.64849e11	1.02929	0.514646	+	0.857403i	$0.327923\pi$
$390$	0	0
$391$	−2.97395e11	−0.643486
$392$	0	0
$393$	− 3.76005e11i	− 0.795110i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.04945e11i	0.818161i	0.912498	+	0.409080i	$0.134150\pi$
$397$	4.04945e11i	0.818161i	−0.912498	+	0.409080i	$0.865850\pi$
$398$	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
$403$	− 1.58268e11i	− 0.298896i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.49330e11i	− 0.631046i
$408$	0	0
$409$	−6.22611e11	−1.10017	−0.550087	−	0.835107i	$-0.685406\pi$
$409$	−6.22611e11	−1.10017	−0.550087	+	0.835107i	$0.685406\pi$
$410$	0	0
$411$	−5.71078e9	−0.00987205
$412$	0	0
$413$	− 7.83795e11i	− 1.32564i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 6.43525e11i	− 1.04220i
$418$	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
$423$	4.28660e10i	0.0651000i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.47383e11i	0.651259i
$428$	0	0
$429$	5.48774e10	0.0782233
$430$	0	0
$431$	7.43484e11	1.03782	0.518912	−	0.854828i	$-0.326337\pi$
$431$	7.43484e11	1.03782	0.518912	+	0.854828i	$0.326337\pi$
$432$	0	0
$433$	2.19028e11i	0.299436i	0.988729	+	0.149718i	$0.0478366\pi$
$433$	2.19028e11i	0.299436i	−0.988729	+	0.149718i	$0.952163\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.46452e10i	0.0454440i
$438$	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
$443$	9.53901e11i	1.17676i	0.808586	+	0.588378i	$0.200233\pi$
$443$	9.53901e11i	1.17676i	−0.808586	+	0.588378i	$0.799767\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 1.05527e12i	− 1.25020i
$448$	0	0
$449$	−9.16478e11	−1.06418	−0.532088	−	0.846689i	$-0.678593\pi$
$449$	−9.16478e11	−1.06418	−0.532088	+	0.846689i	$0.678593\pi$
$450$	0	0
$451$	−1.04438e12	−1.18867
$452$	0	0
$453$	− 7.02943e11i	− 0.784292i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 8.25903e11i	− 0.885740i	−0.896586	−	0.442870i	$-0.853960\pi$
$457$	− 8.25903e11i	− 0.885740i	0.896586	−	0.442870i	$-0.146040\pi$
$458$	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
$463$	1.37910e12i	1.39471i	0.716728	+	0.697353i	$0.245639\pi$
$463$	1.37910e12i	1.39471i	−0.716728	+	0.697353i	$0.754361\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9.55852e11i	− 0.929961i	−0.885321	−	0.464980i	$-0.846061\pi$
$467$	− 9.55852e11i	− 0.929961i	0.885321	−	0.464980i	$-0.153939\pi$
$468$	0	0
$469$	−4.79576e11	−0.457699
$470$	0	0
$471$	1.23161e12	1.15313
$472$	0	0
$473$	6.75274e11i	0.620304i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.26028e11i	0.376795i
$478$	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
$483$	− 9.60213e11i	− 0.802797i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.30662e11i	0.669182i	0.942364	+	0.334591i	$0.108598\pi$
$487$	8.30662e11i	0.669182i	−0.942364	+	0.334591i	$0.891402\pi$
$488$	0	0
$489$	4.23291e11	0.334772
$490$	0	0
$491$	3.17729e11	0.246712	0.123356	−	0.992362i	$-0.460634\pi$
$491$	3.17729e11	0.246712	0.123356	+	0.992362i	$0.460634\pi$
$492$	0	0
$493$	9.45462e11i	0.720830i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.05607e12i	− 0.776408i
$498$	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
$503$	1.24185e12i	0.864996i	0.901635	+	0.432498i	$0.142368\pi$
$503$	1.24185e12i	0.864996i	−0.901635	+	0.432498i	$0.857632\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.13090e12i	0.760132i
$508$	0	0
$509$	−2.49436e12	−1.64713	−0.823566	−	0.567220i	$-0.808019\pi$
$509$	−2.49436e12	−1.64713	−0.823566	+	0.567220i	$0.808019\pi$
$510$	0	0
$511$	−4.12359e12	−2.67536
$512$	0	0
$513$	1.09125e11i	0.0695660i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.66636e11i	− 0.102579i
$518$	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
$523$	− 3.03989e12i	− 1.77664i	−0.459223	−	0.888321i	$-0.651872\pi$
$523$	− 3.03989e12i	− 1.77664i	0.459223	−	0.888321i	$-0.348128\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.94806e12i	− 1.66490i
$528$	0	0
$529$	8.95326e11	0.497085
$530$	0	0
$531$	−6.53773e11	−0.356863
$532$	0	0
$533$	5.86700e11i	0.314879i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.55285e11i	0.340052i
$538$	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
$543$	− 2.45301e12i	− 1.21088i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 2.44503e12i	− 1.16772i	−0.811853	−	0.583862i	$-0.801541\pi$
$547$	− 2.44503e12i	− 1.16772i	0.811853	−	0.583862i	$-0.198459\pi$
$548$	0	0
$549$	3.73168e11	0.175319
$550$	0	0
$551$	1.10142e11	0.0509062
$552$	0	0
$553$	8.45550e11i	0.384482i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.40112e12i	0.616777i	0.951260	+	0.308389i	$0.0997897\pi$
$557$	1.40112e12i	0.616777i	−0.951260	+	0.308389i	$0.900210\pi$
$558$	0	0
$559$	3.79348e11	0.164318
$560$	0	0
$561$	1.02220e12	0.435716
$562$	0	0
$563$	− 3.12352e11i	− 0.131026i	−0.997852	−	0.0655129i	$-0.979132\pi$
$563$	− 3.12352e11i	− 0.131026i	0.997852	−	0.0655129i	$-0.0208683\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.63203e12i	− 0.663141i
$568$	0	0
$569$	1.50839e12	0.603266	0.301633	−	0.953424i	$-0.402468\pi$
$569$	1.50839e12	0.603266	0.301633	+	0.953424i	$0.402468\pi$
$570$	0	0
$571$	−1.77207e12	−0.697621	−0.348810	−	0.937193i	$-0.613414\pi$
$571$	−1.77207e12	−0.697621	−0.348810	+	0.937193i	$0.613414\pi$
$572$	0	0
$573$	− 3.17778e12i	− 1.23148i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 5.81248e10i	− 0.0218308i	−0.999940	−	0.0109154i	$-0.996525\pi$
$577$	− 5.81248e10i	− 0.0218308i	0.999940	−	0.0109154i	$-0.00347455\pi$
$578$	0	0
$579$	−2.19680e12	−0.812338
$580$	0	0
$581$	5.40824e12	1.96908
$582$	0	0
$583$	− 1.65613e12i	− 0.593725i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.72424e11i	0.233761i	0.993146	+	0.116880i	$0.0372894\pi$
$587$	6.72424e11i	0.233761i	−0.993146	+	0.116880i	$0.962711\pi$
$588$	0	0
$589$	−3.43435e11	−0.117578
$590$	0	0
$591$	1.35230e12	0.455962
$592$	0	0
$593$	4.33711e12i	1.44031i	0.693816	+	0.720153i	$0.255928\pi$
$593$	4.33711e12i	1.44031i	−0.693816	+	0.720153i	$0.744072\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.97527e12i	0.958611i
$598$	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
$603$	4.00021e11i	0.123212i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1.29060e12i	− 0.385872i	−0.981211	−	0.192936i	$-0.938199\pi$
$607$	− 1.29060e12i	− 0.385872i	0.981211	−	0.192936i	$-0.0618009\pi$
$608$	0	0
$609$	−3.05265e12	−0.899290
$610$	0	0
$611$	−9.36109e10	−0.0271732
$612$	0	0
$613$	3.58380e12i	1.02511i	0.858654	+	0.512556i	$0.171301\pi$
$613$	3.58380e12i	1.02511i	−0.858654	+	0.512556i	$0.828699\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.46900e12i	1.51923i	0.650370	+	0.759617i	$0.274614\pi$
$617$	5.46900e12i	1.51923i	−0.650370	+	0.759617i	$0.725386\pi$
$618$	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
$623$	4.43426e12i	1.17930i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.19082e11i	− 0.0307710i
$628$	0	0
$629$	−3.65542e12	−0.931129
$630$	0	0
$631$	4.43138e12	1.11277	0.556387	−	0.830923i	$-0.312187\pi$
$631$	4.43138e12	1.11277	0.556387	+	0.830923i	$0.312187\pi$
$632$	0	0
$633$	1.00894e12i	0.249775i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 7.45810e11i	− 0.179474i
$638$	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
$643$	− 1.42772e12i	− 0.329378i	−0.986346	−	0.164689i	$-0.947338\pi$
$643$	− 1.42772e12i	− 0.329378i	0.986346	−	0.164689i	$-0.0526620\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.43250e12i	0.994443i	0.867624	+	0.497221i	$0.165646\pi$
$647$	4.43250e12i	0.994443i	−0.867624	+	0.497221i	$0.834354\pi$
$648$	0	0
$649$	2.54146e12	0.562317
$650$	0	0
$651$	9.51852e12	2.07709
$652$	0	0
$653$	− 6.61930e11i	− 0.142463i	−0.997460	−	0.0712316i	$-0.977307\pi$
$653$	− 6.61930e11i	− 0.142463i	0.997460	−	0.0712316i	$-0.0226929\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.43954e12i	0.720205i
$658$	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
$663$	− 5.74242e11i	− 0.115421i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.87975e12i	0.563364i
$668$	0	0
$669$	2.66851e12	0.515052
$670$	0	0
$671$	−1.45064e12	−0.276254
$672$	0	0
$673$	− 3.67873e12i	− 0.691241i	−0.938374	−	0.345621i	$-0.887668\pi$
$673$	− 3.67873e12i	− 0.691241i	0.938374	−	0.345621i	$-0.112332\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.60384e12i	1.39118i	0.718438	+	0.695591i	$0.244857\pi$
$677$	7.60384e12i	1.39118i	−0.718438	+	0.695591i	$0.755143\pi$
$678$	0	0
$679$	1.26346e13	2.28111
$680$	0	0
$681$	6.52388e11	0.116237
$682$	0	0
$683$	− 6.89820e12i	− 1.21295i	−0.795103	−	0.606475i	$-0.792583\pi$
$683$	− 6.89820e12i	− 1.21295i	0.795103	−	0.606475i	$-0.207417\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.16336e12i	0.370531i
$688$	0	0
$689$	−9.30362e11	−0.157277
$690$	0	0
$691$	4.45838e12	0.743920	0.371960	−	0.928249i	$-0.378686\pi$
$691$	4.45838e12	0.743920	0.371960	+	0.928249i	$0.378686\pi$
$692$	0	0
$693$	− 2.11249e12i	− 0.347933i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.09285e13i	1.75393i
$698$	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
$703$	4.25840e11i	0.0657578i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.13322e13i	− 1.70579i
$708$	0	0
$709$	−5.10575e12	−0.758841	−0.379421	−	0.925224i	$-0.623877\pi$
$709$	−5.10575e12	−0.758841	−0.379421	+	0.925224i	$0.623877\pi$
$710$	0	0
$711$	7.05284e11	0.103503
$712$	0	0
$713$	− 8.97940e12i	− 1.30120i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.79467e12i	0.394907i
$718$	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
$723$	5.17503e12i	0.704353i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.87221e12i	0.381339i	0.981654	+	0.190669i	$0.0610659\pi$
$727$	2.87221e12i	0.381339i	−0.981654	+	0.190669i	$0.938934\pi$
$728$	0	0
$729$	−7.82545e12	−1.02621
$730$	0	0
$731$	7.06612e12	0.915278
$732$	0	0
$733$	3.91043e12i	0.500330i	0.968203	+	0.250165i	$0.0804849\pi$
$733$	3.91043e12i	0.500330i	−0.968203	+	0.250165i	$0.919515\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.55503e12i	− 0.194149i
$738$	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
$743$	− 5.19147e12i	− 0.624943i	−0.949927	−	0.312472i	$-0.898843\pi$
$743$	− 5.19147e12i	− 0.624943i	0.949927	−	0.312472i	$-0.101157\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4.51108e12i	− 0.530076i
$748$	0	0
$749$	1.47235e13	1.70939
$750$	0	0
$751$	−1.48763e13	−1.70654	−0.853268	−	0.521473i	$-0.825383\pi$
$751$	−1.48763e13	−1.70654	−0.853268	+	0.521473i	$0.825383\pi$
$752$	0	0
$753$	3.94705e12i	0.447399i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.03576e13i	− 1.14637i	−0.819425	−	0.573187i	$-0.805707\pi$
$757$	− 1.03576e13i	− 1.14637i	0.819425	−	0.573187i	$-0.194293\pi$
$758$	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
$763$	9.95330e12i	1.06318i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.42771e12i	− 0.148957i
$768$	0	0
$769$	−1.28801e13	−1.32816	−0.664081	−	0.747661i	$-0.731177\pi$
$769$	−1.28801e13	−1.32816	−0.664081	+	0.747661i	$0.731177\pi$
$770$	0	0
$771$	−8.55530e12	−0.871948
$772$	0	0
$773$	9.40486e12i	0.947425i	0.880680	+	0.473712i	$0.157086\pi$
$773$	9.40486e12i	0.947425i	−0.880680	+	0.473712i	$0.842914\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1.18024e13i	− 1.16165i
$778$	0	0
$779$	1.27311e12	0.123865
$780$	0	0
$781$	3.42431e12	0.329340
$782$	0	0
$783$	9.07063e12i	0.862401i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 8.24747e12i	− 0.766363i	−0.923673	−	0.383181i	$-0.874828\pi$
$787$	− 8.24747e12i	− 0.766363i	0.923673	−	0.383181i	$-0.125172\pi$
$788$	0	0
$789$	−2.11800e12	−0.194572
$790$	0	0
$791$	−1.40020e13	−1.27173
$792$	0	0
$793$	8.14926e11i	0.0731794i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3.35880e12i	− 0.294864i	−0.989072	−	0.147432i	$-0.952899\pi$
$797$	− 3.35880e12i	− 0.294864i	0.989072	−	0.147432i	$-0.0471008\pi$
$798$	0	0
$799$	−1.74369e12	−0.151359
$800$	0	0
$801$	3.69867e12	0.317468
$802$	0	0
$803$	− 1.33707e13i	− 1.13484i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.86654e13i	1.54919i
$808$	0	0
$809$	6.87559e12	0.564341	0.282171	−	0.959364i	$-0.408946\pi$
$809$	6.87559e12	0.564341	0.282171	+	0.959364i	$0.408946\pi$
$810$	0	0
$811$	2.11160e13	1.71402	0.857012	−	0.515296i	$-0.172318\pi$
$811$	2.11160e13	1.71402	0.857012	+	0.515296i	$0.172318\pi$
$812$	0	0
$813$	− 5.70903e12i	− 0.458305i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.23171e11i	− 0.0646384i
$818$	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
$823$	6.15696e12i	0.467807i	0.972260	+	0.233904i	$0.0751500\pi$
$823$	6.15696e12i	0.467807i	−0.972260	+	0.233904i	$0.924850\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.99295e12i	− 0.296838i	−0.988925	−	0.148419i	$-0.952582\pi$
$827$	− 3.99295e12i	− 0.296838i	0.988925	−	0.148419i	$-0.0474184\pi$
$828$	0	0
$829$	−1.16163e12	−0.0854229	−0.0427114	−	0.999087i	$-0.513600\pi$
$829$	−1.16163e12	−0.0854229	−0.0427114	+	0.999087i	$0.513600\pi$
$830$	0	0
$831$	−7.63472e12	−0.555378
$832$	0	0
$833$	− 1.38922e13i	− 0.999697i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.82833e13i	− 1.99189i
$838$	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
$843$	− 1.35565e13i	− 0.924537i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.35032e13i	− 0.901490i
$848$	0	0
$849$	−1.12718e13	−0.744578
$850$	0	0
$851$	−1.11339e13	−0.727722
$852$	0	0
$853$	1.18899e13i	0.768967i	0.923132	+	0.384484i	$0.125620\pi$
$853$	1.18899e13i	0.768967i	−0.923132	+	0.384484i	$0.874380\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.57517e12i	0.163077i	0.996670	+	0.0815385i	$0.0259834\pi$
$857$	2.57517e12i	0.163077i	−0.996670	+	0.0815385i	$0.974017\pi$
$858$	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
$863$	− 4.94014e12i	− 0.303173i	−0.988444	−	0.151586i	$-0.951562\pi$
$863$	− 4.94014e12i	− 0.303173i	0.988444	−	0.151586i	$-0.0484382\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.29496e12i	0.137940i
$868$	0	0
$869$	−2.74170e12	−0.163091
$870$	0	0
$871$	−8.73567e11	−0.0514298
$872$	0	0
$873$	− 1.05387e13i	− 0.614075i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.68472e12i	0.495744i	0.968793	+	0.247872i	$0.0797313\pi$
$877$	8.68472e12i	0.495744i	−0.968793	+	0.247872i	$0.920269\pi$
$878$	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
$883$	3.58719e12i	0.198578i	0.995059	+	0.0992891i	$0.0316569\pi$
$883$	3.58719e12i	0.198578i	−0.995059	+	0.0992891i	$0.968343\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.72905e13i	1.48032i	0.672431	+	0.740160i	$0.265250\pi$
$887$	2.72905e13i	1.48032i	−0.672431	+	0.740160i	$0.734750\pi$
$888$	0	0
$889$	4.38266e13	2.35331
$890$	0	0
$891$	5.29187e12	0.281294
$892$	0	0
$893$	2.03132e11i	0.0106892i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.74907e12i	− 0.0902070i
$898$	0	0
$899$	−2.85468e13	−1.45760
$900$	0	0
$901$	−1.73299e13	−0.876059
$902$	0	0
$903$	2.28147e13i	1.14188i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.79849e13i	0.882418i	0.897404	+	0.441209i	$0.145450\pi$
$907$	1.79849e13i	0.882418i	−0.897404	+	0.441209i	$0.854550\pi$
$908$	0	0
$909$	−9.45233e12	−0.459200
$910$	0	0
$911$	−1.80978e13	−0.870549	−0.435275	−	0.900298i	$-0.643349\pi$
$911$	−1.80978e13	−0.870549	−0.435275	+	0.900298i	$0.643349\pi$
$912$	0	0
$913$	1.75362e13i	0.835252i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3.16089e13i	− 1.47621i
$918$	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
$923$	− 1.92368e12i	− 0.0872418i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6.20611e12i	− 0.276033i
$928$	0	0
$929$	2.53410e13	1.11623	0.558113	−	0.829765i	$-0.311525\pi$
$929$	2.53410e13	1.11623	0.558113	+	0.829765i	$0.311525\pi$
$930$	0	0
$931$	−1.61838e12	−0.0706002
$932$	0	0
$933$	1.26852e13i	0.548063i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 7.05858e12i	− 0.299150i	−0.988750	−	0.149575i	$-0.952209\pi$
$937$	− 7.05858e12i	− 0.299150i	0.988750	−	0.149575i	$-0.0477905\pi$
$938$	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
$943$	3.32866e13i	1.37078i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 3.96392e12i	− 0.160158i	−0.996789	−	0.0800792i	$-0.974483\pi$
$947$	− 3.96392e12i	− 0.160158i	0.996789	−	0.0800792i	$-0.0255173\pi$
$948$	0	0
$949$	−7.51128e12	−0.300619
$950$	0	0
$951$	8.81618e12	0.349517
$952$	0	0
$953$	− 1.82636e12i	− 0.0717246i	−0.999357	−	0.0358623i	$-0.988582\pi$
$953$	− 1.82636e12i	− 0.0717246i	0.999357	−	0.0358623i	$-0.0114178\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 9.89823e12i	− 0.381464i
$958$	0	0
$959$	−4.80077e11	−0.0183285
$960$	0	0
$961$	6.25725e13	2.36662
$962$	0	0
$963$	− 1.22810e13i	− 0.460168i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1.71062e13i	− 0.629122i	−0.949237	−	0.314561i	$-0.898143\pi$
$967$	− 1.71062e13i	− 0.629122i	0.949237	−	0.314561i	$-0.101857\pi$
$968$	0	0
$969$	−1.24608e12	−0.0454036
$970$	0	0
$971$	−6.43910e12	−0.232455	−0.116227	−	0.993223i	$-0.537080\pi$
$971$	−6.43910e12	−0.232455	−0.116227	+	0.993223i	$0.537080\pi$
$972$	0	0
$973$	− 5.40979e13i	− 1.93496i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.72264e13i	− 0.604880i	−0.953168	−	0.302440i	$-0.902199\pi$
$977$	− 1.72264e13i	− 0.604880i	0.953168	−	0.302440i	$-0.0978011\pi$
$978$	0	0
$979$	−1.43781e13	−0.500240
$980$	0	0
$981$	8.30217e12	0.286208
$982$	0	0
$983$	1.22407e13i	0.418134i	0.977901	+	0.209067i	$0.0670426\pi$
$983$	1.22407e13i	0.418134i	−0.977901	+	0.209067i	$0.932957\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 5.62992e12i	− 0.188832i
$988$	0	0
$989$	2.15225e13	0.715334
$990$	0	0
$991$	−2.74736e13	−0.904867	−0.452434	−	0.891798i	$-0.649444\pi$
$991$	−2.74736e13	−0.904867	−0.452434	+	0.891798i	$0.649444\pi$
$992$	0	0
$993$	4.33092e13i	1.41354i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.32279e13i	− 0.423996i	−0.977270	−	0.211998i	$-0.932003\pi$
$997$	− 1.32279e13i	− 0.423996i	0.977270	−	0.211998i	$-0.0679970\pi$
$998$	0	0
$999$	−3.50696e13	−1.11400

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.10.a.b.1.2	✓	2	5.2	odd	4
80.10.a.h.1.1		2	20.7	even	4
200.10.a.d.1.1		2	5.3	odd	4
200.10.c.e.49.2		4	5.4	even	2	inner
200.10.c.e.49.3		4	1.1	even	1	trivial
320.10.a.o.1.2		2	40.27	even	4
320.10.a.p.1.1		2	40.37	odd	4
360.10.a.a.1.1		2	15.2	even	4
400.10.a.o.1.2		2	20.3	even	4
400.10.c.o.49.2		4	4.3	odd	2
400.10.c.o.49.3		4	20.19	odd	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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+109.551i q^{3} +9209.37i q^{7} +7681.66 q^{9} -29861.4 q^{11} +16775.2i q^{13} +312472. i q^{17} +36401.6 q^{19} -1.00889e6 q^{21} +951749. i q^{23} +2.99782e6i q^{27} +3.02575e6 q^{29} -9.43462e6 q^{31} -3.27134e6i q^{33} +1.16984e7i q^{37} -1.83774e6 q^{39} +3.49741e7 q^{41} -2.26136e7i q^{43} +5.58030e6i q^{47} -4.44590e7 q^{49} -3.42316e7 q^{51} +5.54605e7i q^{53} +3.98782e6i q^{57} -8.51084e7 q^{59} +4.85791e7 q^{61} +7.07433e7i q^{63} +5.20748e7i q^{67} -1.04265e8 q^{69} -1.14674e8 q^{71} +4.47760e8i q^{73} -2.75005e8i q^{77} +9.18141e7 q^{79} -1.77215e8 q^{81} -5.87254e8i q^{83} +3.31472e8i q^{87} +4.81494e8 q^{89} -1.54489e8 q^{91} -1.03357e9i q^{93} -1.37193e9i q^{97} -2.29385e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 26516 q^{9} - 16160 q^{11} + 2313360 q^{19} - 2945472 q^{21} + 8424520 q^{29} - 22722256 q^{31} + 34688784 q^{39} + 26060872 q^{41} - 18803332 q^{49} - 62392560 q^{51} + 94725072 q^{59} + 407268856 q^{61}+ \cdots - 1370960608 q^{99}+O(q^{100}) \) 4 * q - 26516 * q^9 - 16160 * q^11 + 2313360 * q^19 - 2945472 * q^21 + 8424520 * q^29 - 22722256 * q^31 + 34688784 * q^39 + 26060872 * q^41 - 18803332 * q^49 - 62392560 * q^51 + 94725072 * q^59 + 407268856 * q^61 - 1018360512 * q^69 - 699801584 * q^71 + 904174688 * q^79 - 1076642044 * q^81 + 512146520 * q^89 + 129077232 * q^91 - 1370960608 * q^99

Introduction

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Modular forms

Varieties

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Groups

Database

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