LMFDB - Embedded newform 200.8.c.g.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [200,8,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, 8, 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, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
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,-6644] 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:	$62.4770050968$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{601})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 301x^{2} + 22500 $ x^4 + 301*x^2 + 22500
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
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			49.1
Root			$-12.7577i$ of defining polynomial
Character	$\chi$	$=$	200.49
Dual form			200.8.c.g.49.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-87.0306i q^{3} +741.214i q^{7} -5387.33 q^{9} -6065.61 q^{11} +4234.04i q^{13} +18488.3i q^{17} +28071.6 q^{19} +64508.3 q^{21} -104956. i q^{23} +278526. i q^{27} +204768. q^{29} +5109.59 q^{31} +527894. i q^{33} +186181. i q^{37} +368491. q^{39} -618450. q^{41} +329301. i q^{43} +56218.8i q^{47} +274144. q^{49} +1.60905e6 q^{51} -143301. i q^{53} -2.44309e6i q^{57} +3.09530e6 q^{59} -701736. q^{61} -3.99316e6i q^{63} -1.15239e6i q^{67} -9.13438e6 q^{69} +5.03809e6 q^{71} +4.64023e6i q^{73} -4.49592e6i q^{77} -2.67305e6 q^{79} +1.24582e7 q^{81} -2.64570e6i q^{83} -1.78211e7i q^{87} -3.11153e6 q^{89} -3.13833e6 q^{91} -444690. i q^{93} -4.05990e6i q^{97} +3.26774e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6644 q^{9} - 1120 q^{11} - 39120 q^{19} + 127808 q^{21} + 422120 q^{29} + 330704 q^{31} + 929136 q^{39} - 1702648 q^{41} + 2189372 q^{49} + 2974640 q^{51} + 4177008 q^{59} + 2819416 q^{61} - 17720768 q^{69}+ \cdots + 88096608 q^{99}+O(q^{100}) $ 4 * q - 6644 * q^9 - 1120 * q^11 - 39120 * q^19 + 127808 * q^21 + 422120 * q^29 + 330704 * q^31 + 929136 * q^39 - 1702648 * q^41 + 2189372 * q^49 + 2974640 * q^51 + 4177008 * q^59 + 2819416 * q^61 - 17720768 * q^69 + 19638896 * q^71 + 4056992 * q^79 + 32915396 * q^81 - 11377640 * q^89 - 5322288 * q^91 + 88096608 * 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
$3$	− 87.0306i	− 1.86101i	−0.366285	−	0.930503i	$-0.619371\pi$
$3$	− 87.0306i	− 1.86101i	0.366285	−	0.930503i	$-0.380629\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	741.214i	0.816772i	0.912809	+	0.408386i	$0.133908\pi$
$7$	741.214i	0.816772i	−0.912809	+	0.408386i	$0.866092\pi$
$8$	0	0
$9$	−5387.33	−2.46334
$10$	0	0
$11$	−6065.61	−1.37404	−0.687021	−	0.726637i	$-0.741082\pi$
$11$	−6065.61	−1.37404	−0.687021	+	0.726637i	$0.741082\pi$
$12$	0	0
$13$	4234.04i	0.534507i	0.963626	+	0.267253i	$0.0861161\pi$
$13$	4234.04i	0.534507i	−0.963626	+	0.267253i	$0.913884\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18488.3i	0.912696i	0.889801	+	0.456348i	$0.150843\pi$
$17$	18488.3i	0.912696i	−0.889801	+	0.456348i	$0.849157\pi$
$18$	0	0
$19$	28071.6	0.938923	0.469461	−	0.882953i	$-0.344448\pi$
$19$	28071.6	0.938923	0.469461	+	0.882953i	$0.344448\pi$
$20$	0	0
$21$	64508.3	1.52002
$22$	0	0
$23$	− 104956.i	− 1.79870i	−0.437225	−	0.899352i	$-0.644039\pi$
$23$	− 104956.i	− 1.79870i	0.437225	−	0.899352i	$-0.355961\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	278526.i	2.72328i
$28$	0	0
$29$	204768.	1.55908	0.779541	−	0.626351i	$-0.215452\pi$
$29$	204768.	1.55908	0.779541	+	0.626351i	$0.215452\pi$
$30$	0	0
$31$	5109.59	0.0308049	0.0154025	−	0.999881i	$-0.495097\pi$
$31$	5109.59	0.0308049	0.0154025	+	0.999881i	$0.495097\pi$
$32$	0	0
$33$	527894.i	2.55710i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	186181.i	0.604269i	0.953265	+	0.302134i	$0.0976991\pi$
$37$	186181.i	0.604269i	−0.953265	+	0.302134i	$0.902301\pi$
$38$	0	0
$39$	368491.	0.994720
$40$	0	0
$41$	−618450.	−1.40140	−0.700699	−	0.713457i	$-0.747128\pi$
$41$	−618450.	−1.40140	−0.700699	+	0.713457i	$0.747128\pi$
$42$	0	0
$43$	329301.i	0.631616i	0.948823	+	0.315808i	$0.102276\pi$
$43$	329301.i	0.631616i	−0.948823	+	0.315808i	$0.897724\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	56218.8i	0.0789839i	0.999220	+	0.0394920i	$0.0125740\pi$
$47$	56218.8i	0.0789839i	−0.999220	+	0.0394920i	$0.987426\pi$
$48$	0	0
$49$	274144.	0.332884
$50$	0	0
$51$	1.60905e6	1.69853
$52$	0	0
$53$	− 143301.i	− 0.132216i	−0.997812	−	0.0661079i	$-0.978942\pi$
$53$	− 143301.i	− 0.132216i	0.997812	−	0.0661079i	$-0.0210582\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.44309e6i	− 1.74734i
$58$	0	0
$59$	3.09530e6	1.96210	0.981049	−	0.193759i	$-0.0620681\pi$
$59$	3.09530e6	1.96210	0.981049	+	0.193759i	$0.0620681\pi$
$60$	0	0
$61$	−701736.	−0.395840	−0.197920	−	0.980218i	$-0.563419\pi$
$61$	−701736.	−0.395840	−0.197920	+	0.980218i	$0.563419\pi$
$62$	0	0
$63$	− 3.99316e6i	− 2.01199i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.15239e6i	− 0.468101i	−0.972224	−	0.234050i	$-0.924802\pi$
$67$	− 1.15239e6i	− 0.468101i	0.972224	−	0.234050i	$-0.0751981\pi$
$68$	0	0
$69$	−9.13438e6	−3.34740
$70$	0	0
$71$	5.03809e6	1.67056	0.835279	−	0.549827i	$-0.185306\pi$
$71$	5.03809e6	1.67056	0.835279	+	0.549827i	$0.185306\pi$
$72$	0	0
$73$	4.64023e6i	1.39608i	0.716060	+	0.698039i	$0.245944\pi$
$73$	4.64023e6i	1.39608i	−0.716060	+	0.698039i	$0.754056\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.49592e6i	− 1.12228i
$78$	0	0
$79$	−2.67305e6	−0.609975	−0.304988	−	0.952356i	$-0.598652\pi$
$79$	−2.67305e6	−0.609975	−0.304988	+	0.952356i	$0.598652\pi$
$80$	0	0
$81$	1.24582e7	2.60471
$82$	0	0
$83$	− 2.64570e6i	− 0.507887i	−0.967219	−	0.253944i	$-0.918272\pi$
$83$	− 2.64570e6i	− 0.507887i	0.967219	−	0.253944i	$-0.0817278\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1.78211e7i	− 2.90146i
$88$	0	0
$89$	−3.11153e6	−0.467852	−0.233926	−	0.972254i	$-0.575157\pi$
$89$	−3.11153e6	−0.467852	−0.233926	+	0.972254i	$0.575157\pi$
$90$	0	0
$91$	−3.13833e6	−0.436570
$92$	0	0
$93$	− 444690.i	− 0.0573281i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.05990e6i	− 0.451662i	−0.974166	−	0.225831i	$-0.927490\pi$
$97$	− 4.05990e6i	− 0.451662i	0.974166	−	0.225831i	$-0.0725098\pi$
$98$	0	0
$99$	3.26774e7	3.38473
$100$	0	0
$101$	−550732.	−0.0531883	−0.0265941	−	0.999646i	$-0.508466\pi$
$101$	−550732.	−0.0531883	−0.0265941	+	0.999646i	$0.508466\pi$
$102$	0	0
$103$	4.05931e6i	0.366034i	0.983110	+	0.183017i	$0.0585863\pi$
$103$	4.05931e6i	0.366034i	−0.983110	+	0.183017i	$0.941414\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.79134e7i	1.41363i	0.707399	+	0.706814i	$0.249868\pi$
$107$	1.79134e7i	1.41363i	−0.707399	+	0.706814i	$0.750132\pi$
$108$	0	0
$109$	1.85504e7	1.37202	0.686011	−	0.727592i	$-0.259360\pi$
$109$	1.85504e7	1.37202	0.686011	+	0.727592i	$0.259360\pi$
$110$	0	0
$111$	1.62035e7	1.12455
$112$	0	0
$113$	− 1.19073e7i	− 0.776314i	−0.921593	−	0.388157i	$-0.873112\pi$
$113$	− 1.19073e7i	− 0.776314i	0.921593	−	0.388157i	$-0.126888\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.28102e7i	− 1.31667i
$118$	0	0
$119$	−1.37038e7	−0.745464
$120$	0	0
$121$	1.73045e7	0.887993
$122$	0	0
$123$	5.38241e7i	2.60801i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.43595e7i	1.48845i	0.667931	+	0.744223i	$0.267180\pi$
$127$	3.43595e7i	1.48845i	−0.667931	+	0.744223i	$0.732820\pi$
$128$	0	0
$129$	2.86592e7	1.17544
$130$	0	0
$131$	2.21781e7	0.861936	0.430968	−	0.902367i	$-0.358172\pi$
$131$	2.21781e7	0.861936	0.430968	+	0.902367i	$0.358172\pi$
$132$	0	0
$133$	2.08071e7i	0.766886i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.16559e7i	− 0.387279i	−0.981073	−	0.193639i	$-0.937971\pi$
$137$	− 1.16559e7i	− 0.387279i	0.981073	−	0.193639i	$-0.0620292\pi$
$138$	0	0
$139$	−5.62296e6	−0.177588	−0.0887939	−	0.996050i	$-0.528301\pi$
$139$	−5.62296e6	−0.177588	−0.0887939	+	0.996050i	$0.528301\pi$
$140$	0	0
$141$	4.89275e6	0.146989
$142$	0	0
$143$	− 2.56820e7i	− 0.734435i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.38590e7i	− 0.619499i
$148$	0	0
$149$	−2.74946e7	−0.680918	−0.340459	−	0.940259i	$-0.610582\pi$
$149$	−2.74946e7	−0.680918	−0.340459	+	0.940259i	$0.610582\pi$
$150$	0	0
$151$	1.37682e7	0.325430	0.162715	−	0.986673i	$-0.447975\pi$
$151$	1.37682e7	0.325430	0.162715	+	0.986673i	$0.447975\pi$
$152$	0	0
$153$	− 9.96026e7i	− 2.24828i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.97641e7i	0.613825i	0.951738	+	0.306912i	$0.0992959\pi$
$157$	2.97641e7i	0.613825i	−0.951738	+	0.306912i	$0.900704\pi$
$158$	0	0
$159$	−1.24716e7	−0.246054
$160$	0	0
$161$	7.77949e7	1.46913
$162$	0	0
$163$	− 1.91229e7i	− 0.345858i	−0.984934	−	0.172929i	$-0.944677\pi$
$163$	− 1.91229e7i	− 0.345858i	0.984934	−	0.172929i	$-0.0553232\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.55629e7i	0.258574i	0.991607	+	0.129287i	$0.0412688\pi$
$167$	1.55629e7i	0.258574i	−0.991607	+	0.129287i	$0.958731\pi$
$168$	0	0
$169$	4.48214e7	0.714303
$170$	0	0
$171$	−1.51231e8	−2.31289
$172$	0	0
$173$	7.98560e6i	0.117259i	0.998280	+	0.0586295i	$0.0186731\pi$
$173$	7.98560e6i	0.117259i	−0.998280	+	0.0586295i	$0.981327\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 2.69386e8i	− 3.65147i
$178$	0	0
$179$	−9.75110e7	−1.27077	−0.635386	−	0.772194i	$-0.719159\pi$
$179$	−9.75110e7	−1.27077	−0.635386	+	0.772194i	$0.719159\pi$
$180$	0	0
$181$	−9.94948e7	−1.24717	−0.623585	−	0.781756i	$-0.714324\pi$
$181$	−9.94948e7	−1.24717	−0.623585	+	0.781756i	$0.714324\pi$
$182$	0	0
$183$	6.10725e7i	0.736660i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.12143e8i	− 1.25408i
$188$	0	0
$189$	−2.06448e8	−2.22430
$190$	0	0
$191$	6.12311e7	0.635851	0.317925	−	0.948116i	$-0.397014\pi$
$191$	6.12311e7	0.635851	0.317925	+	0.948116i	$0.397014\pi$
$192$	0	0
$193$	8.89562e7i	0.890688i	0.895360	+	0.445344i	$0.146919\pi$
$193$	8.89562e7i	0.890688i	−0.895360	+	0.445344i	$0.853081\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.14248e8i	1.06467i	0.846534	+	0.532335i	$0.178685\pi$
$197$	1.14248e8i	1.06467i	−0.846534	+	0.532335i	$0.821315\pi$
$198$	0	0
$199$	1.12801e8	1.01468	0.507338	−	0.861747i	$-0.330630\pi$
$199$	1.12801e8	1.01468	0.507338	+	0.861747i	$0.330630\pi$
$200$	0	0
$201$	−1.00294e8	−0.871138
$202$	0	0
$203$	1.51777e8i	1.27341i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.65432e8i	4.43082i
$208$	0	0
$209$	−1.70272e8	−1.29012
$210$	0	0
$211$	−4.35111e7	−0.318868	−0.159434	−	0.987209i	$-0.550967\pi$
$211$	−4.35111e7	−0.318868	−0.159434	+	0.987209i	$0.550967\pi$
$212$	0	0
$213$	− 4.38468e8i	− 3.10892i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.78730e6i	0.0251606i
$218$	0	0
$219$	4.03842e8	2.59811
$220$	0	0
$221$	−7.82803e7	−0.487842
$222$	0	0
$223$	4.72521e7i	0.285334i	0.989771	+	0.142667i	$0.0455679\pi$
$223$	4.72521e7i	0.285334i	−0.989771	+	0.142667i	$0.954432\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.84159e8i	1.04497i	0.852650	+	0.522483i	$0.174994\pi$
$227$	1.84159e8i	1.04497i	−0.852650	+	0.522483i	$0.825006\pi$
$228$	0	0
$229$	−1.20722e8	−0.664297	−0.332149	−	0.943227i	$-0.607774\pi$
$229$	−1.20722e8	−0.664297	−0.332149	+	0.943227i	$0.607774\pi$
$230$	0	0
$231$	−3.91282e8	−2.08857
$232$	0	0
$233$	1.99119e8i	1.03126i	0.856812	+	0.515629i	$0.172442\pi$
$233$	1.99119e8i	1.03126i	−0.856812	+	0.515629i	$0.827558\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.32637e8i	1.13517i
$238$	0	0
$239$	2.95160e8	1.39851	0.699255	−	0.714872i	$-0.253515\pi$
$239$	2.95160e8	1.39851	0.699255	+	0.714872i	$0.253515\pi$
$240$	0	0
$241$	3.22537e8	1.48429	0.742146	−	0.670238i	$-0.233808\pi$
$241$	3.22537e8	1.48429	0.742146	+	0.670238i	$0.233808\pi$
$242$	0	0
$243$	− 4.75110e8i	− 2.12409i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.18856e8i	0.501861i
$248$	0	0
$249$	−2.30257e8	−0.945181
$250$	0	0
$251$	1.24978e8	0.498856	0.249428	−	0.968393i	$-0.419757\pi$
$251$	1.24978e8	0.498856	0.249428	+	0.968393i	$0.419757\pi$
$252$	0	0
$253$	6.36622e8i	2.47150i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.95980e8i	0.720188i	0.932916	+	0.360094i	$0.117255\pi$
$257$	1.95980e8i	0.720188i	−0.932916	+	0.360094i	$0.882745\pi$
$258$	0	0
$259$	−1.38000e8	−0.493550
$260$	0	0
$261$	−1.10315e9	−3.84055
$262$	0	0
$263$	− 2.03297e8i	− 0.689105i	−0.938767	−	0.344552i	$-0.888031\pi$
$263$	− 2.03297e8i	− 0.689105i	0.938767	−	0.344552i	$-0.111969\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.70798e8i	0.870676i
$268$	0	0
$269$	−2.29992e8	−0.720409	−0.360205	−	0.932873i	$-0.617293\pi$
$269$	−2.29992e8	−0.720409	−0.360205	+	0.932873i	$0.617293\pi$
$270$	0	0
$271$	2.98723e8	0.911752	0.455876	−	0.890043i	$-0.349326\pi$
$271$	2.98723e8	0.911752	0.455876	+	0.890043i	$0.349326\pi$
$272$	0	0
$273$	2.73131e8i	0.812459i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2.55584e8i	− 0.722528i	−0.932464	−	0.361264i	$-0.882345\pi$
$277$	− 2.55584e8i	− 0.722528i	0.932464	−	0.361264i	$-0.117655\pi$
$278$	0	0
$279$	−2.75270e7	−0.0758830
$280$	0	0
$281$	4.50862e8	1.21219	0.606096	−	0.795392i	$-0.292735\pi$
$281$	4.50862e8	1.21219	0.606096	+	0.795392i	$0.292735\pi$
$282$	0	0
$283$	3.23467e8i	0.848356i	0.905579	+	0.424178i	$0.139437\pi$
$283$	3.23467e8i	0.848356i	−0.905579	+	0.424178i	$0.860563\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.58404e8i	− 1.14462i
$288$	0	0
$289$	6.85205e7	0.166985
$290$	0	0
$291$	−3.53335e8	−0.840546
$292$	0	0
$293$	− 5.52109e8i	− 1.28230i	−0.767418	−	0.641148i	$-0.778459\pi$
$293$	− 5.52109e8i	− 1.28230i	0.767418	−	0.641148i	$-0.221541\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.68943e9i	− 3.74191i
$298$	0	0
$299$	4.44388e8	0.961420
$300$	0	0
$301$	−2.44082e8	−0.515886
$302$	0	0
$303$	4.79306e7i	0.0989836i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.38962e8i	1.06310i	0.847027	+	0.531550i	$0.178390\pi$
$307$	5.38962e8i	1.06310i	−0.847027	+	0.531550i	$0.821610\pi$
$308$	0	0
$309$	3.53284e8	0.681191
$310$	0	0
$311$	6.11434e8	1.15263	0.576313	−	0.817229i	$-0.304491\pi$
$311$	6.11434e8	1.15263	0.576313	+	0.817229i	$0.304491\pi$
$312$	0	0
$313$	− 1.01255e9i	− 1.86642i	−0.359328	−	0.933211i	$-0.616994\pi$
$313$	− 1.01255e9i	− 1.86642i	0.359328	−	0.933211i	$-0.383006\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.22278e8i	0.744545i	0.928123	+	0.372273i	$0.121421\pi$
$317$	4.22278e8i	0.744545i	−0.928123	+	0.372273i	$0.878579\pi$
$318$	0	0
$319$	−1.24204e9	−2.14224
$320$	0	0
$321$	1.55902e9	2.63077
$322$	0	0
$323$	5.18997e8i	0.856952i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.61445e9i	− 2.55334i
$328$	0	0
$329$	−4.16701e7	−0.0645118
$330$	0	0
$331$	6.58640e8	0.998275	0.499138	−	0.866523i	$-0.333650\pi$
$331$	6.58640e8	0.998275	0.499138	+	0.866523i	$0.333650\pi$
$332$	0	0
$333$	− 1.00302e9i	− 1.48852i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 4.51883e8i	− 0.643164i	−0.946882	−	0.321582i	$-0.895785\pi$
$337$	− 4.51883e8i	− 0.643164i	0.946882	−	0.321582i	$-0.104215\pi$
$338$	0	0
$339$	−1.03630e9	−1.44472
$340$	0	0
$341$	−3.09928e7	−0.0423272
$342$	0	0
$343$	8.13622e8i	1.08866i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.77854e8i	0.356996i	0.983940	+	0.178498i	$0.0571239\pi$
$347$	2.77854e8i	0.356996i	−0.983940	+	0.178498i	$0.942876\pi$
$348$	0	0
$349$	−1.66487e8	−0.209648	−0.104824	−	0.994491i	$-0.533428\pi$
$349$	−1.66487e8	−0.209648	−0.104824	+	0.994491i	$0.533428\pi$
$350$	0	0
$351$	−1.17929e9	−1.45561
$352$	0	0
$353$	1.40229e9i	1.69678i	0.529369	+	0.848392i	$0.322429\pi$
$353$	1.40229e9i	1.69678i	−0.529369	+	0.848392i	$0.677571\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.19265e9i	1.38731i
$358$	0	0
$359$	6.43009e8	0.733477	0.366739	−	0.930324i	$-0.380474\pi$
$359$	6.43009e8	0.733477	0.366739	+	0.930324i	$0.380474\pi$
$360$	0	0
$361$	−1.05856e8	−0.118424
$362$	0	0
$363$	− 1.50602e9i	− 1.65256i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 9.38196e7i	− 0.0990746i	−0.998772	−	0.0495373i	$-0.984225\pi$
$367$	− 9.38196e7i	− 0.0990746i	0.998772	−	0.0495373i	$-0.0157747\pi$
$368$	0	0
$369$	3.33179e9	3.45212
$370$	0	0
$371$	1.06217e8	0.107990
$372$	0	0
$373$	1.01907e9i	1.01677i	0.861129	+	0.508386i	$0.169758\pi$
$373$	1.01907e9i	1.01677i	−0.861129	+	0.508386i	$0.830242\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.66996e8i	0.833340i
$378$	0	0
$379$	6.10359e7	0.0575902	0.0287951	−	0.999585i	$-0.490833\pi$
$379$	6.10359e7	0.0575902	0.0287951	+	0.999585i	$0.490833\pi$
$380$	0	0
$381$	2.99032e9	2.77001
$382$	0	0
$383$	1.85097e9i	1.68346i	0.539895	+	0.841732i	$0.318464\pi$
$383$	1.85097e9i	1.68346i	−0.539895	+	0.841732i	$0.681536\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.77405e9i	− 1.55589i
$388$	0	0
$389$	6.80195e8	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$389$	6.80195e8	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$390$	0	0
$391$	1.94046e9	1.64167
$392$	0	0
$393$	− 1.93017e9i	− 1.60407i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.86725e9i	1.49774i	0.662720	+	0.748868i	$0.269402\pi$
$397$	1.86725e9i	1.49774i	−0.662720	+	0.748868i	$0.730598\pi$
$398$	0	0
$399$	1.81085e9	1.42718
$400$	0	0
$401$	7.41095e7	0.0573943	0.0286971	−	0.999588i	$-0.490864\pi$
$401$	7.41095e7	0.0573943	0.0286971	+	0.999588i	$0.490864\pi$
$402$	0	0
$403$	2.16342e7i	0.0164654i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.12930e9i	− 0.830291i
$408$	0	0
$409$	1.24701e9	0.901237	0.450618	−	0.892717i	$-0.351204\pi$
$409$	1.24701e9	0.901237	0.450618	+	0.892717i	$0.351204\pi$
$410$	0	0
$411$	−1.01442e9	−0.720728
$412$	0	0
$413$	2.29428e9i	1.60259i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.89370e8i	0.330492i
$418$	0	0
$419$	5.87311e8	0.390049	0.195025	−	0.980798i	$-0.437521\pi$
$419$	5.87311e8	0.390049	0.195025	+	0.980798i	$0.437521\pi$
$420$	0	0
$421$	−2.72214e9	−1.77796	−0.888982	−	0.457942i	$-0.848587\pi$
$421$	−2.72214e9	−1.77796	−0.888982	+	0.457942i	$0.848587\pi$
$422$	0	0
$423$	− 3.02869e8i	− 0.194564i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 5.20137e8i	− 0.323311i
$428$	0	0
$429$	−2.23512e9	−1.36679
$430$	0	0
$431$	−1.33258e9	−0.801721	−0.400861	−	0.916139i	$-0.631289\pi$
$431$	−1.33258e9	−0.801721	−0.400861	+	0.916139i	$0.631289\pi$
$432$	0	0
$433$	1.39398e9i	0.825179i	0.910917	+	0.412590i	$0.135376\pi$
$433$	1.39398e9i	0.825179i	−0.910917	+	0.412590i	$0.864624\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.94629e9i	− 1.68885i
$438$	0	0
$439$	−1.96180e9	−1.10670	−0.553348	−	0.832950i	$-0.686650\pi$
$439$	−1.96180e9	−1.10670	−0.553348	+	0.832950i	$0.686650\pi$
$440$	0	0
$441$	−1.47691e9	−0.820007
$442$	0	0
$443$	− 1.36082e9i	− 0.743684i	−0.928296	−	0.371842i	$-0.878726\pi$
$443$	− 1.36082e9i	− 0.743684i	0.928296	−	0.371842i	$-0.121274\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.39287e9i	1.26719i
$448$	0	0
$449$	−2.43167e9	−1.26778	−0.633889	−	0.773424i	$-0.718542\pi$
$449$	−2.43167e9	−1.26778	−0.633889	+	0.773424i	$0.718542\pi$
$450$	0	0
$451$	3.75128e9	1.92558
$452$	0	0
$453$	− 1.19826e9i	− 0.605628i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3.09420e8i	− 0.151650i	−0.997121	−	0.0758250i	$-0.975841\pi$
$457$	− 3.09420e8i	− 0.151650i	0.997121	−	0.0758250i	$-0.0241590\pi$
$458$	0	0
$459$	−5.14948e9	−2.48553
$460$	0	0
$461$	−2.43256e9	−1.15640	−0.578202	−	0.815894i	$-0.696245\pi$
$461$	−2.43256e9	−1.15640	−0.578202	+	0.815894i	$0.696245\pi$
$462$	0	0
$463$	− 1.49638e9i	− 0.700664i	−0.936626	−	0.350332i	$-0.886069\pi$
$463$	− 1.49638e9i	− 0.700664i	0.936626	−	0.350332i	$-0.113931\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.09165e9i	− 1.40469i	−0.711836	−	0.702346i	$-0.752136\pi$
$467$	− 3.09165e9i	− 1.40469i	0.711836	−	0.702346i	$-0.247864\pi$
$468$	0	0
$469$	8.54171e8	0.382332
$470$	0	0
$471$	2.59039e9	1.14233
$472$	0	0
$473$	− 1.99741e9i	− 0.867867i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.72009e8i	0.325693i
$478$	0	0
$479$	−2.24028e8	−0.0931384	−0.0465692	−	0.998915i	$-0.514829\pi$
$479$	−2.24028e8	−0.0931384	−0.0465692	+	0.998915i	$0.514829\pi$
$480$	0	0
$481$	−7.88299e8	−0.322986
$482$	0	0
$483$	− 6.77054e9i	− 2.73406i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.38421e9i	0.935390i	0.883890	+	0.467695i	$0.154915\pi$
$487$	2.38421e9i	0.935390i	−0.883890	+	0.467695i	$0.845085\pi$
$488$	0	0
$489$	−1.66428e9	−0.643644
$490$	0	0
$491$	1.28112e8	0.0488432	0.0244216	−	0.999702i	$-0.492226\pi$
$491$	1.28112e8	0.0488432	0.0244216	+	0.999702i	$0.492226\pi$
$492$	0	0
$493$	3.78582e9i	1.42297i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.73430e9i	1.36446i
$498$	0	0
$499$	2.42499e9	0.873690	0.436845	−	0.899537i	$-0.356096\pi$
$499$	2.42499e9	0.873690	0.436845	+	0.899537i	$0.356096\pi$
$500$	0	0
$501$	1.35445e9	0.481207
$502$	0	0
$503$	− 2.22661e9i	− 0.780110i	−0.920792	−	0.390055i	$-0.872456\pi$
$503$	− 2.22661e9i	− 0.780110i	0.920792	−	0.390055i	$-0.127544\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 3.90084e9i	− 1.32932i
$508$	0	0
$509$	3.67616e9	1.23561	0.617807	−	0.786330i	$-0.288021\pi$
$509$	3.67616e9	1.23561	0.617807	+	0.786330i	$0.288021\pi$
$510$	0	0
$511$	−3.43941e9	−1.14028
$512$	0	0
$513$	7.81869e9i	2.55695i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 3.41001e8i	− 0.108527i
$518$	0	0
$519$	6.94992e8	0.218220
$520$	0	0
$521$	9.21625e8	0.285511	0.142755	−	0.989758i	$-0.454404\pi$
$521$	9.21625e8	0.285511	0.142755	+	0.989758i	$0.454404\pi$
$522$	0	0
$523$	1.57199e8i	0.0480502i	0.999711	+	0.0240251i	$0.00764816\pi$
$523$	1.57199e8i	0.0480502i	−0.999711	+	0.0240251i	$0.992352\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.44677e7i	0.0281155i
$528$	0	0
$529$	−7.61094e9	−2.23534
$530$	0	0
$531$	−1.66754e10	−4.83332
$532$	0	0
$533$	− 2.61854e9i	− 0.749056i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.48644e9i	2.36491i
$538$	0	0
$539$	−1.66285e9	−0.457397
$540$	0	0
$541$	−2.80816e8	−0.0762486	−0.0381243	−	0.999273i	$-0.512138\pi$
$541$	−2.80816e8	−0.0762486	−0.0381243	+	0.999273i	$0.512138\pi$
$542$	0	0
$543$	8.65909e9i	2.32099i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.15456e9i	0.301622i	0.988563	+	0.150811i	$0.0481884\pi$
$547$	1.15456e9i	0.301622i	−0.988563	+	0.150811i	$0.951812\pi$
$548$	0	0
$549$	3.78048e9	0.975088
$550$	0	0
$551$	5.74817e9	1.46386
$552$	0	0
$553$	− 1.98130e9i	− 0.498210i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.36841e9i	0.580715i	0.956918	+	0.290357i	$0.0937742\pi$
$557$	2.36841e9i	0.580715i	−0.956918	+	0.290357i	$0.906226\pi$
$558$	0	0
$559$	−1.39427e9	−0.337603
$560$	0	0
$561$	−9.75987e9	−2.33386
$562$	0	0
$563$	2.50492e9i	0.591580i	0.955253	+	0.295790i	$0.0955829\pi$
$563$	2.50492e9i	0.591580i	−0.955253	+	0.295790i	$0.904417\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.23422e9i	2.12745i
$568$	0	0
$569$	4.42619e9	1.00725	0.503625	−	0.863922i	$-0.331999\pi$
$569$	4.42619e9	1.00725	0.503625	+	0.863922i	$0.331999\pi$
$570$	0	0
$571$	−1.32550e9	−0.297956	−0.148978	−	0.988841i	$-0.547598\pi$
$571$	−1.32550e9	−0.297956	−0.148978	+	0.988841i	$0.547598\pi$
$572$	0	0
$573$	− 5.32898e9i	− 1.18332i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.76604e9i	0.599436i	0.954028	+	0.299718i	$0.0968926\pi$
$577$	2.76604e9i	0.599436i	−0.954028	+	0.299718i	$0.903107\pi$
$578$	0	0
$579$	7.74191e9	1.65757
$580$	0	0
$581$	1.96103e9	0.414828
$582$	0	0
$583$	8.69208e8i	0.181670i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.15575e9i	0.439912i	0.975510	+	0.219956i	$0.0705914\pi$
$587$	2.15575e9i	0.439912i	−0.975510	+	0.219956i	$0.929409\pi$
$588$	0	0
$589$	1.43434e8	0.0289234
$590$	0	0
$591$	9.94303e9	1.98136
$592$	0	0
$593$	− 9.50086e9i	− 1.87099i	−0.353341	−	0.935495i	$-0.614954\pi$
$593$	− 9.50086e9i	− 1.87099i	0.353341	−	0.935495i	$-0.385046\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 9.81715e9i	− 1.88832i
$598$	0	0
$599$	9.55306e8	0.181614	0.0908068	−	0.995869i	$-0.471055\pi$
$599$	9.55306e8	0.181614	0.0908068	+	0.995869i	$0.471055\pi$
$600$	0	0
$601$	−8.41945e9	−1.58206	−0.791030	−	0.611777i	$-0.790455\pi$
$601$	−8.41945e9	−1.58206	−0.791030	+	0.611777i	$0.790455\pi$
$602$	0	0
$603$	6.20833e9i	1.15309i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 9.80243e9i	− 1.77899i	−0.456945	−	0.889495i	$-0.651056\pi$
$607$	− 9.80243e9i	− 1.77899i	0.456945	−	0.889495i	$-0.348944\pi$
$608$	0	0
$609$	1.32092e10	2.36983
$610$	0	0
$611$	−2.38032e8	−0.0422174
$612$	0	0
$613$	1.33970e9i	0.234907i	0.993078	+	0.117454i	$0.0374732\pi$
$613$	1.33970e9i	0.234907i	−0.993078	+	0.117454i	$0.962527\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.03037e10i	1.76602i	0.469351	+	0.883012i	$0.344488\pi$
$617$	1.03037e10i	1.76602i	−0.469351	+	0.883012i	$0.655512\pi$
$618$	0	0
$619$	−2.87364e9	−0.486984	−0.243492	−	0.969903i	$-0.578293\pi$
$619$	−2.87364e9	−0.486984	−0.243492	+	0.969903i	$0.578293\pi$
$620$	0	0
$621$	2.92330e10	4.89838
$622$	0	0
$623$	− 2.30631e9i	− 0.382128i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.48188e10i	2.40092i
$628$	0	0
$629$	−3.44218e9	−0.551514
$630$	0	0
$631$	−6.17623e9	−0.978635	−0.489317	−	0.872106i	$-0.662754\pi$
$631$	−6.17623e9	−0.978635	−0.489317	+	0.872106i	$0.662754\pi$
$632$	0	0
$633$	3.78680e9i	0.593416i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.16074e9i	0.177929i
$638$	0	0
$639$	−2.71418e10	−4.11515
$640$	0	0
$641$	−9.14422e9	−1.37134	−0.685668	−	0.727915i	$-0.740490\pi$
$641$	−9.14422e9	−1.37134	−0.685668	+	0.727915i	$0.740490\pi$
$642$	0	0
$643$	7.48915e8i	0.111095i	0.998456	+	0.0555475i	$0.0176904\pi$
$643$	7.48915e8i	0.111095i	−0.998456	+	0.0555475i	$0.982310\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 6.13359e9i	− 0.890327i	−0.895449	−	0.445163i	$-0.853146\pi$
$647$	− 6.13359e9i	− 0.890327i	0.895449	−	0.445163i	$-0.146854\pi$
$648$	0	0
$649$	−1.87749e10	−2.69601
$650$	0	0
$651$	3.29611e8	0.0468239
$652$	0	0
$653$	− 1.37749e10i	− 1.93594i	−0.251059	−	0.967972i	$-0.580779\pi$
$653$	− 1.37749e10i	− 1.93594i	0.251059	−	0.967972i	$-0.419221\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.49984e10i	− 3.43901i
$658$	0	0
$659$	8.08420e9	1.10037	0.550184	−	0.835044i	$-0.314558\pi$
$659$	8.08420e9	1.10037	0.550184	+	0.835044i	$0.314558\pi$
$660$	0	0
$661$	−2.44260e9	−0.328964	−0.164482	−	0.986380i	$-0.552595\pi$
$661$	−2.44260e9	−0.328964	−0.164482	+	0.986380i	$0.552595\pi$
$662$	0	0
$663$	6.81278e9i	0.907877i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2.14916e10i	− 2.80433i
$668$	0	0
$669$	4.11238e9	0.531009
$670$	0	0
$671$	4.25646e9	0.543900
$672$	0	0
$673$	1.23905e8i	0.0156689i	0.999969	+	0.00783443i	$0.00249380\pi$
$673$	1.23905e8i	0.0156689i	−0.999969	+	0.00783443i	$0.997506\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.14942e10i	− 1.42369i	−0.702335	−	0.711847i	$-0.747859\pi$
$677$	− 1.14942e10i	− 1.42369i	0.702335	−	0.711847i	$-0.252141\pi$
$678$	0	0
$679$	3.00925e9	0.368905
$680$	0	0
$681$	1.60275e10	1.94469
$682$	0	0
$683$	− 8.92504e9i	− 1.07186i	−0.844263	−	0.535929i	$-0.819961\pi$
$683$	− 8.92504e9i	− 1.07186i	0.844263	−	0.535929i	$-0.180039\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.05065e10i	1.23626i
$688$	0	0
$689$	6.06742e8	0.0706703
$690$	0	0
$691$	−2.95336e9	−0.340521	−0.170260	−	0.985399i	$-0.554461\pi$
$691$	−2.95336e9	−0.340521	−0.170260	+	0.985399i	$0.554461\pi$
$692$	0	0
$693$	2.42210e10i	2.76455i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.14341e10i	− 1.27905i
$698$	0	0
$699$	1.73295e10	1.91918
$700$	0	0
$701$	−1.43095e9	−0.156896	−0.0784479	−	0.996918i	$-0.524996\pi$
$701$	−1.43095e9	−0.156896	−0.0784479	+	0.996918i	$0.524996\pi$
$702$	0	0
$703$	5.22641e9i	0.567362i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.08211e8i	− 0.0434427i
$708$	0	0
$709$	1.25151e10	1.31878	0.659392	−	0.751799i	$-0.270814\pi$
$709$	1.25151e10	1.31878	0.659392	+	0.751799i	$0.270814\pi$
$710$	0	0
$711$	1.44006e10	1.50258
$712$	0	0
$713$	− 5.36282e8i	− 0.0554089i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 2.56880e10i	− 2.60263i
$718$	0	0
$719$	8.89943e9	0.892917	0.446459	−	0.894804i	$-0.352685\pi$
$719$	8.89943e9	0.892917	0.446459	+	0.894804i	$0.352685\pi$
$720$	0	0
$721$	−3.00881e9	−0.298966
$722$	0	0
$723$	− 2.80705e10i	− 2.76227i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.56000e9i	0.343621i	0.985130	+	0.171810i	$0.0549616\pi$
$727$	3.56000e9i	0.343621i	−0.985130	+	0.171810i	$0.945038\pi$
$728$	0	0
$729$	−1.41030e10	−1.34823
$730$	0	0
$731$	−6.08822e9	−0.576474
$732$	0	0
$733$	− 5.05266e9i	− 0.473867i	−0.971526	−	0.236934i	$-0.923858\pi$
$733$	− 5.05266e9i	− 0.473867i	0.971526	−	0.236934i	$-0.0761424\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.98998e9i	0.643191i
$738$	0	0
$739$	−3.15300e9	−0.287387	−0.143694	−	0.989622i	$-0.545898\pi$
$739$	−3.15300e9	−0.287387	−0.143694	+	0.989622i	$0.545898\pi$
$740$	0	0
$741$	1.03441e10	0.933965
$742$	0	0
$743$	− 1.93745e9i	− 0.173288i	−0.996239	−	0.0866442i	$-0.972386\pi$
$743$	− 1.93745e9i	− 0.173288i	0.996239	−	0.0866442i	$-0.0276143\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.42533e10i	1.25110i
$748$	0	0
$749$	−1.32777e10	−1.15461
$750$	0	0
$751$	1.58399e10	1.36462	0.682311	−	0.731062i	$-0.260975\pi$
$751$	1.58399e10	1.36462	0.682311	+	0.731062i	$0.260975\pi$
$752$	0	0
$753$	− 1.08769e10i	− 0.928374i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.57162e10i	1.31678i	0.752678	+	0.658389i	$0.228762\pi$
$757$	1.57162e10i	1.31678i	−0.752678	+	0.658389i	$0.771238\pi$
$758$	0	0
$759$	5.54056e10	4.59947
$760$	0	0
$761$	−2.50123e9	−0.205735	−0.102867	−	0.994695i	$-0.532802\pi$
$761$	−2.50123e9	−0.205735	−0.102867	+	0.994695i	$0.532802\pi$
$762$	0	0
$763$	1.37498e10i	1.12063i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.31056e10i	1.04875i
$768$	0	0
$769$	−8.48850e9	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$769$	−8.48850e9	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$770$	0	0
$771$	1.70563e10	1.34027
$772$	0	0
$773$	2.42832e10i	1.89094i	0.325707	+	0.945471i	$0.394398\pi$
$773$	2.42832e10i	1.89094i	−0.325707	+	0.945471i	$0.605602\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.20102e10i	0.918498i
$778$	0	0
$779$	−1.73609e10	−1.31580
$780$	0	0
$781$	−3.05591e10	−2.29542
$782$	0	0
$783$	5.70333e10i	4.24582i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 6.20737e9i	− 0.453938i	−0.973902	−	0.226969i	$-0.927118\pi$
$787$	− 6.20737e9i	− 0.453938i	0.973902	−	0.226969i	$-0.0728816\pi$
$788$	0	0
$789$	−1.76930e10	−1.28243
$790$	0	0
$791$	8.82583e9	0.634071
$792$	0	0
$793$	− 2.97118e9i	− 0.211579i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.88550e7i	0.00131924i	1.00000		0.000659619i	$0.000209963\pi$
$797$	1.88550e7i	0.00131924i	−1.00000		0.000659619i	$0.999790\pi$
$798$	0	0
$799$	−1.03939e9	−0.0720883
$800$	0	0
$801$	1.67628e10	1.15248
$802$	0	0
$803$	− 2.81458e10i	− 1.91827i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.00163e10i	1.34069i
$808$	0	0
$809$	1.59140e10	1.05672	0.528360	−	0.849021i	$-0.322807\pi$
$809$	1.59140e10	1.05672	0.528360	+	0.849021i	$0.322807\pi$
$810$	0	0
$811$	−2.62774e10	−1.72986	−0.864928	−	0.501896i	$-0.832636\pi$
$811$	−2.62774e10	−1.72986	−0.864928	+	0.501896i	$0.832636\pi$
$812$	0	0
$813$	− 2.59981e10i	− 1.69678i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.24401e9i	0.593039i
$818$	0	0
$819$	1.69072e10	1.07542
$820$	0	0
$821$	9.37571e9	0.591293	0.295647	−	0.955297i	$-0.404465\pi$
$821$	9.37571e9	0.591293	0.295647	+	0.955297i	$0.404465\pi$
$822$	0	0
$823$	1.03455e10i	0.646920i	0.946242	+	0.323460i	$0.104846\pi$
$823$	1.03455e10i	0.646920i	−0.946242	+	0.323460i	$0.895154\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.19356e10i	− 1.34859i	−0.738461	−	0.674296i	$-0.764447\pi$
$827$	− 2.19356e10i	− 1.34859i	0.738461	−	0.674296i	$-0.235553\pi$
$828$	0	0
$829$	−3.05413e10	−1.86186	−0.930929	−	0.365199i	$-0.881001\pi$
$829$	−3.05413e10	−1.86186	−0.930929	+	0.365199i	$0.881001\pi$
$830$	0	0
$831$	−2.22436e10	−1.34463
$832$	0	0
$833$	5.06847e9i	0.303822i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.42315e9i	0.0838905i
$838$	0	0
$839$	2.52012e10	1.47318	0.736588	−	0.676342i	$-0.236436\pi$
$839$	2.52012e10	1.47318	0.736588	+	0.676342i	$0.236436\pi$
$840$	0	0
$841$	2.46800e10	1.43074
$842$	0	0
$843$	− 3.92388e10i	− 2.25589i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.28263e10i	0.725287i
$848$	0	0
$849$	2.81516e10	1.57879
$850$	0	0
$851$	1.95409e10	1.08690
$852$	0	0
$853$	− 3.04039e9i	− 0.167729i	−0.996477	−	0.0838644i	$-0.973274\pi$
$853$	− 3.04039e9i	− 0.167729i	0.996477	−	0.0838644i	$-0.0267262\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 7.30094e9i	− 0.396229i	−0.980179	−	0.198114i	$-0.936518\pi$
$857$	− 7.30094e9i	− 0.396229i	0.980179	−	0.198114i	$-0.0634818\pi$
$858$	0	0
$859$	−2.16215e10	−1.16388	−0.581941	−	0.813231i	$-0.697707\pi$
$859$	−2.16215e10	−1.16388	−0.581941	+	0.813231i	$0.697707\pi$
$860$	0	0
$861$	−3.98952e10	−2.13015
$862$	0	0
$863$	− 3.05503e10i	− 1.61800i	−0.587810	−	0.808999i	$-0.700010\pi$
$863$	− 3.05503e10i	− 1.61800i	0.587810	−	0.808999i	$-0.299990\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 5.96338e9i	− 0.310760i
$868$	0	0
$869$	1.62137e10	0.838132
$870$	0	0
$871$	4.87929e9	0.250203
$872$	0	0
$873$	2.18720e10i	1.11260i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.48353e8i	0.0124329i	0.999981	+	0.00621644i	$0.00197877\pi$
$877$	2.48353e8i	0.0124329i	−0.999981	+	0.00621644i	$0.998021\pi$
$878$	0	0
$879$	−4.80504e10	−2.38636
$880$	0	0
$881$	−2.96667e10	−1.46169	−0.730843	−	0.682546i	$-0.760873\pi$
$881$	−2.96667e10	−1.46169	−0.730843	+	0.682546i	$0.760873\pi$
$882$	0	0
$883$	7.57977e9i	0.370505i	0.982691	+	0.185252i	$0.0593102\pi$
$883$	7.57977e9i	0.370505i	−0.982691	+	0.185252i	$0.940690\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.01060e10i	− 0.486235i	−0.969997	−	0.243117i	$-0.921830\pi$
$887$	− 1.01060e10i	− 0.486235i	0.969997	−	0.243117i	$-0.0781700\pi$
$888$	0	0
$889$	−2.54677e10	−1.21572
$890$	0	0
$891$	−7.55668e10	−3.57898
$892$	0	0
$893$	1.57815e9i	0.0741598i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3.86753e10i	− 1.78921i
$898$	0	0
$899$	1.04628e9	0.0480274
$900$	0	0
$901$	2.64939e9	0.120673
$902$	0	0
$903$	2.12426e10i	0.960067i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.98649e10i	0.884016i	0.897011	+	0.442008i	$0.145734\pi$
$907$	1.98649e10i	0.884016i	−0.897011	+	0.442008i	$0.854266\pi$
$908$	0	0
$909$	2.96698e9	0.131021
$910$	0	0
$911$	−2.90396e10	−1.27256	−0.636278	−	0.771460i	$-0.719527\pi$
$911$	−2.90396e10	−1.27256	−0.636278	+	0.771460i	$0.719527\pi$
$912$	0	0
$913$	1.60478e10i	0.697859i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.64387e10i	0.704005i
$918$	0	0
$919$	−3.06973e10	−1.30465	−0.652327	−	0.757938i	$-0.726207\pi$
$919$	−3.06973e10	−1.30465	−0.652327	+	0.757938i	$0.726207\pi$
$920$	0	0
$921$	4.69062e10	1.97843
$922$	0	0
$923$	2.13315e10i	0.892924i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.18688e10i	− 0.901666i
$928$	0	0
$929$	1.29984e10	0.531904	0.265952	−	0.963986i	$-0.414314\pi$
$929$	1.29984e10	0.531904	0.265952	+	0.963986i	$0.414314\pi$
$930$	0	0
$931$	7.69568e9	0.312553
$932$	0	0
$933$	− 5.32135e10i	− 2.14504i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 3.51243e10i	− 1.39482i	−0.716670	−	0.697412i	$-0.754335\pi$
$937$	− 3.51243e10i	− 1.39482i	0.716670	−	0.697412i	$-0.245665\pi$
$938$	0	0
$939$	−8.81226e10	−3.47342
$940$	0	0
$941$	4.51801e9	0.176760	0.0883800	−	0.996087i	$-0.471831\pi$
$941$	4.51801e9	0.176760	0.0883800	+	0.996087i	$0.471831\pi$
$942$	0	0
$943$	6.49101e10i	2.52070i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.07244e9i	0.270610i	0.990804	+	0.135305i	$0.0432015\pi$
$947$	7.07244e9i	0.270610i	−0.990804	+	0.135305i	$0.956799\pi$
$948$	0	0
$949$	−1.96469e10	−0.746213
$950$	0	0
$951$	3.67511e10	1.38560
$952$	0	0
$953$	− 7.75183e9i	− 0.290121i	−0.989423	−	0.145060i	$-0.953662\pi$
$953$	− 7.75183e9i	− 0.290121i	0.989423	−	0.145060i	$-0.0463377\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.08096e11i	3.98673i
$958$	0	0
$959$	8.63951e9	0.316318
$960$	0	0
$961$	−2.74865e10	−0.999051
$962$	0	0
$963$	− 9.65054e10i	− 3.48225i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.07254e9i	− 0.0737071i	−0.999321	−	0.0368536i	$-0.988266\pi$
$967$	− 2.07254e9i	− 0.0737071i	0.999321	−	0.0368536i	$-0.0117335\pi$
$968$	0	0
$969$	4.51687e10	1.59479
$970$	0	0
$971$	3.00236e10	1.05244	0.526218	−	0.850350i	$-0.323610\pi$
$971$	3.00236e10	1.05244	0.526218	+	0.850350i	$0.323610\pi$
$972$	0	0
$973$	− 4.16782e9i	− 0.145049i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.62484e9i	− 0.0900476i	−0.998986	−	0.0450238i	$-0.985664\pi$
$977$	− 2.62484e9i	− 0.0900476i	0.998986	−	0.0450238i	$-0.0143364\pi$
$978$	0	0
$979$	1.88733e10	0.642849
$980$	0	0
$981$	−9.99371e10	−3.37976
$982$	0	0
$983$	− 3.23601e9i	− 0.108661i	−0.998523	−	0.0543304i	$-0.982698\pi$
$983$	− 3.23601e9i	− 0.108661i	0.998523	−	0.0543304i	$-0.0173024\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.62658e9i	0.120057i
$988$	0	0
$989$	3.45621e10	1.13609
$990$	0	0
$991$	3.91654e10	1.27834	0.639168	−	0.769067i	$-0.279279\pi$
$991$	3.91654e10	1.27834	0.639168	+	0.769067i	$0.279279\pi$
$992$	0	0
$993$	− 5.73219e10i	− 1.85780i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.65367e10i	1.48718i	0.668638	+	0.743588i	$0.266877\pi$
$997$	4.65367e10i	1.48718i	−0.668638	+	0.743588i	$0.733123\pi$
$998$	0	0
$999$	−5.18564e10	−1.64560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.8.c.g.49.1		4
4.3	odd	2		400.8.c.p.49.4		4
5.2	odd	4		200.8.a.j.1.1		2
5.3	odd	4		40.8.a.d.1.2	✓	2
5.4	even	2	inner	200.8.c.g.49.4		4
15.8	even	4		360.8.a.f.1.2		2
20.3	even	4		80.8.a.e.1.1		2
20.7	even	4		400.8.a.bg.1.2		2
20.19	odd	2		400.8.c.p.49.1		4
40.3	even	4		320.8.a.w.1.2		2
40.13	odd	4		320.8.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.8.a.d.1.2	✓	2	5.3	odd	4
80.8.a.e.1.1		2	20.3	even	4
200.8.a.j.1.1		2	5.2	odd	4
200.8.c.g.49.1		4	1.1	even	1	trivial
200.8.c.g.49.4		4	5.4	even	2	inner
320.8.a.i.1.1		2	40.13	odd	4
320.8.a.w.1.2		2	40.3	even	4
360.8.a.f.1.2		2	15.8	even	4
400.8.a.bg.1.2		2	20.7	even	4
400.8.c.p.49.1		4	20.19	odd	2
400.8.c.p.49.4		4	4.3	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-87.0306i q^{3} +741.214i q^{7} -5387.33 q^{9} -6065.61 q^{11} +4234.04i q^{13} +18488.3i q^{17} +28071.6 q^{19} +64508.3 q^{21} -104956. i q^{23} +278526. i q^{27} +204768. q^{29} +5109.59 q^{31} +527894. i q^{33} +186181. i q^{37} +368491. q^{39} -618450. q^{41} +329301. i q^{43} +56218.8i q^{47} +274144. q^{49} +1.60905e6 q^{51} -143301. i q^{53} -2.44309e6i q^{57} +3.09530e6 q^{59} -701736. q^{61} -3.99316e6i q^{63} -1.15239e6i q^{67} -9.13438e6 q^{69} +5.03809e6 q^{71} +4.64023e6i q^{73} -4.49592e6i q^{77} -2.67305e6 q^{79} +1.24582e7 q^{81} -2.64570e6i q^{83} -1.78211e7i q^{87} -3.11153e6 q^{89} -3.13833e6 q^{91} -444690. i q^{93} -4.05990e6i q^{97} +3.26774e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6644 q^{9} - 1120 q^{11} - 39120 q^{19} + 127808 q^{21} + 422120 q^{29} + 330704 q^{31} + 929136 q^{39} - 1702648 q^{41} + 2189372 q^{49} + 2974640 q^{51} + 4177008 q^{59} + 2819416 q^{61} - 17720768 q^{69}+ \cdots + 88096608 q^{99}+O(q^{100}) \) 4 * q - 6644 * q^9 - 1120 * q^11 - 39120 * q^19 + 127808 * q^21 + 422120 * q^29 + 330704 * q^31 + 929136 * q^39 - 1702648 * q^41 + 2189372 * q^49 + 2974640 * q^51 + 4177008 * q^59 + 2819416 * q^61 - 17720768 * q^69 + 19638896 * q^71 + 4056992 * q^79 + 32915396 * q^81 - 11377640 * q^89 - 5322288 * q^91 + 88096608 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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