LMFDB - Embedded newform 200.6.c.e.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,6,Mod(49,200)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("200.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	200.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$32.0767639626$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{129})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 65x^{2} + 1024 $ x^4 + 65*x^2 + 1024
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			49.1
Root			$-6.17891i$ of defining polynomial
Character	$\chi$	$=$	200.49
Dual form			200.6.c.e.49.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-28.7156i q^{3} +42.1469i q^{7} -581.588 q^{9} +O(q^{10})$	$q-28.7156i q^{3} +42.1469i q^{7} -581.588 q^{9} +416.294 q^{11} +966.588i q^{13} +1834.11i q^{17} -317.763 q^{19} +1210.27 q^{21} +1568.02i q^{23} +9722.76i q^{27} -7757.28 q^{29} +102.644 q^{31} -11954.1i q^{33} -1936.58i q^{37} +27756.2 q^{39} +7994.36 q^{41} +16542.6i q^{43} -18649.3i q^{47} +15030.6 q^{49} +52667.7 q^{51} -14972.4i q^{53} +9124.76i q^{57} -19843.3 q^{59} -18024.1 q^{61} -24512.1i q^{63} +55040.6i q^{67} +45026.8 q^{69} +11201.3 q^{71} -4013.95i q^{73} +17545.5i q^{77} -24018.8 q^{79} +137869. q^{81} +70512.8i q^{83} +222755. i q^{87} +60765.7 q^{89} -40738.7 q^{91} -2947.49i q^{93} +31112.2i q^{97} -242111. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 1236 q^{9}+O(q^{10}) $ 4 * q - 1236 * q^9	$ 4 q - 1236 q^{9} + 1120 q^{11} + 2000 q^{19} + 5568 q^{21} - 2680 q^{29} - 4496 q^{31} + 41424 q^{39} + 46152 q^{41} + 45948 q^{49} + 171600 q^{51} - 125168 q^{59} + 28216 q^{61} + 224448 q^{69} + 94416 q^{71} + 131808 q^{79} + 142596 q^{81} + 110040 q^{89} - 2128 q^{91} - 494688 q^{99}+O(q^{100}) $ 4 * q - 1236 * q^9 + 1120 * q^11 + 2000 * q^19 + 5568 * q^21 - 2680 * q^29 - 4496 * q^31 + 41424 * q^39 + 46152 * q^41 + 45948 * q^49 + 171600 * q^51 - 125168 * q^59 + 28216 * q^61 + 224448 * q^69 + 94416 * q^71 + 131808 * q^79 + 142596 * q^81 + 110040 * q^89 - 2128 * q^91 - 494688 * q^99

Display 100 coefficients 10 coefficients

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
$3$	− 28.7156i	− 1.84211i	−0.389434	−	0.921054i	$-0.627329\pi$
$3$	− 28.7156i	− 1.84211i	0.389434	−	0.921054i	$-0.372671\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	42.1469i	0.325103i	0.986700	+	0.162551i	$0.0519723\pi$
$7$	42.1469i	0.325103i	−0.986700	+	0.162551i	$0.948028\pi$
$8$	0	0
$9$	−581.588	−2.39336
$10$	0	0
$11$	416.294	1.03733	0.518667	−	0.854977i	$-0.326429\pi$
$11$	416.294	1.03733	0.518667	+	0.854977i	$0.326429\pi$
$12$	0	0
$13$	966.588i	1.58629i	0.609032	+	0.793145i	$0.291558\pi$
$13$	966.588i	1.58629i	−0.609032	+	0.793145i	$0.708442\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1834.11i	1.53923i	0.638508	+	0.769616i	$0.279552\pi$
$17$	1834.11i	1.53923i	−0.638508	+	0.769616i	$0.720448\pi$
$18$	0	0
$19$	−317.763	−0.201938	−0.100969	−	0.994890i	$-0.532194\pi$
$19$	−317.763	−0.201938	−0.100969	+	0.994890i	$0.532194\pi$
$20$	0	0
$21$	1210.27	0.598874
$22$	0	0
$23$	1568.02i	0.618063i	0.951052	+	0.309032i	$0.100005\pi$
$23$	1568.02i	0.618063i	−0.951052	+	0.309032i	$0.899995\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	9722.76i	2.56673i
$28$	0	0
$29$	−7757.28	−1.71283	−0.856415	−	0.516288i	$-0.827313\pi$
$29$	−7757.28	−1.71283	−0.856415	+	0.516288i	$0.827313\pi$
$30$	0	0
$31$	102.644	0.0191836	0.00959180	−	0.999954i	$-0.496947\pi$
$31$	102.644	0.0191836	0.00959180	+	0.999954i	$0.496947\pi$
$32$	0	0
$33$	− 11954.1i	− 1.91088i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1936.58i	− 0.232558i	−0.993217	−	0.116279i	$-0.962903\pi$
$37$	− 1936.58i	− 0.232558i	0.993217	−	0.116279i	$-0.0370966\pi$
$38$	0	0
$39$	27756.2	2.92212
$40$	0	0
$41$	7994.36	0.742718	0.371359	−	0.928489i	$-0.378892\pi$
$41$	7994.36	0.742718	0.371359	+	0.928489i	$0.378892\pi$
$42$	0	0
$43$	16542.6i	1.36437i	0.731179	+	0.682186i	$0.238970\pi$
$43$	16542.6i	1.36437i	−0.731179	+	0.682186i	$0.761030\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 18649.3i	− 1.23146i	−0.787959	−	0.615728i	$-0.788862\pi$
$47$	− 18649.3i	− 1.23146i	0.787959	−	0.615728i	$-0.211138\pi$
$48$	0	0
$49$	15030.6	0.894308
$50$	0	0
$51$	52667.7	2.83543
$52$	0	0
$53$	− 14972.4i	− 0.732155i	−0.930584	−	0.366077i	$-0.880701\pi$
$53$	− 14972.4i	− 0.732155i	0.930584	−	0.366077i	$-0.119299\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	9124.76i	0.371993i
$58$	0	0
$59$	−19843.3	−0.742137	−0.371069	−	0.928605i	$-0.621009\pi$
$59$	−19843.3	−0.742137	−0.371069	+	0.928605i	$0.621009\pi$
$60$	0	0
$61$	−18024.1	−0.620195	−0.310097	−	0.950705i	$-0.600362\pi$
$61$	−18024.1	−0.620195	−0.310097	+	0.950705i	$0.600362\pi$
$62$	0	0
$63$	− 24512.1i	− 0.778089i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	55040.6i	1.49795i	0.662601	+	0.748973i	$0.269453\pi$
$67$	55040.6i	1.49795i	−0.662601	+	0.748973i	$0.730547\pi$
$68$	0	0
$69$	45026.8	1.13854
$70$	0	0
$71$	11201.3	0.263707	0.131853	−	0.991269i	$-0.457907\pi$
$71$	11201.3	0.263707	0.131853	+	0.991269i	$0.457907\pi$
$72$	0	0
$73$	− 4013.95i	− 0.0881587i	−0.999028	−	0.0440793i	$-0.985965\pi$
$73$	− 4013.95i	− 0.0881587i	0.999028	−	0.0440793i	$-0.0140354\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	17545.5i	0.337240i
$78$	0	0
$79$	−24018.8	−0.432996	−0.216498	−	0.976283i	$-0.569463\pi$
$79$	−24018.8	−0.432996	−0.216498	+	0.976283i	$0.569463\pi$
$80$	0	0
$81$	137869.	2.33483
$82$	0	0
$83$	70512.8i	1.12350i	0.827307	+	0.561750i	$0.189872\pi$
$83$	70512.8i	1.12350i	−0.827307	+	0.561750i	$0.810128\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	222755.i	3.15522i
$88$	0	0
$89$	60765.7	0.813174	0.406587	−	0.913612i	$-0.366719\pi$
$89$	60765.7	0.813174	0.406587	+	0.913612i	$0.366719\pi$
$90$	0	0
$91$	−40738.7	−0.515707
$92$	0	0
$93$	− 2947.49i	− 0.0353383i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	31112.2i	0.335739i	0.985809	+	0.167869i	$0.0536887\pi$
$97$	31112.2i	0.335739i	−0.985809	+	0.167869i	$0.946311\pi$
$98$	0	0
$99$	−242111.	−2.48272
$100$	0	0
$101$	−49491.4	−0.482755	−0.241377	−	0.970431i	$-0.577599\pi$
$101$	−49491.4	−0.482755	−0.241377	+	0.970431i	$0.577599\pi$
$102$	0	0
$103$	− 95117.9i	− 0.883424i	−0.897157	−	0.441712i	$-0.854371\pi$
$103$	− 95117.9i	− 0.883424i	0.897157	−	0.441712i	$-0.145629\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 88429.8i	− 0.746688i	−0.927693	−	0.373344i	$-0.878211\pi$
$107$	− 88429.8i	− 0.746688i	0.927693	−	0.373344i	$-0.121789\pi$
$108$	0	0
$109$	10598.9	0.0854464	0.0427232	−	0.999087i	$-0.486397\pi$
$109$	10598.9	0.0854464	0.0427232	+	0.999087i	$0.486397\pi$
$110$	0	0
$111$	−55610.0	−0.428396
$112$	0	0
$113$	235124.i	1.73221i	0.499859	+	0.866107i	$0.333385\pi$
$113$	235124.i	1.73221i	−0.499859	+	0.866107i	$0.666615\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 562155.i	− 3.79657i
$118$	0	0
$119$	−77302.2	−0.500408
$120$	0	0
$121$	12249.5	0.0760599
$122$	0	0
$123$	− 229563.i	− 1.36817i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	102533.i	0.564098i	0.959400	+	0.282049i	$0.0910141\pi$
$127$	102533.i	0.564098i	−0.959400	+	0.282049i	$0.908986\pi$
$128$	0	0
$129$	475031.	2.51332
$130$	0	0
$131$	50603.0	0.257631	0.128815	−	0.991669i	$-0.458883\pi$
$131$	50603.0	0.257631	0.128815	+	0.991669i	$0.458883\pi$
$132$	0	0
$133$	− 13392.7i	− 0.0656507i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 47811.7i	− 0.217637i	−0.994062	−	0.108818i	$-0.965293\pi$
$137$	− 47811.7i	− 0.217637i	0.994062	−	0.108818i	$-0.0347067\pi$
$138$	0	0
$139$	220190.	0.966629	0.483314	−	0.875447i	$-0.339433\pi$
$139$	220190.	0.966629	0.483314	+	0.875447i	$0.339433\pi$
$140$	0	0
$141$	−535527.	−2.26847
$142$	0	0
$143$	402384.i	1.64551i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 431614.i	− 1.64741i
$148$	0	0
$149$	−154533.	−0.570236	−0.285118	−	0.958492i	$-0.592033\pi$
$149$	−154533.	−0.570236	−0.285118	+	0.958492i	$0.592033\pi$
$150$	0	0
$151$	−455395.	−1.62534	−0.812672	−	0.582721i	$-0.801988\pi$
$151$	−455395.	−1.62534	−0.812672	+	0.582721i	$0.801988\pi$
$152$	0	0
$153$	− 1.06670e6i	− 3.68394i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 361690.i	− 1.17108i	−0.810643	−	0.585541i	$-0.800882\pi$
$157$	− 361690.i	− 1.17108i	0.810643	−	0.585541i	$-0.199118\pi$
$158$	0	0
$159$	−429943.	−1.34871
$160$	0	0
$161$	−66087.3	−0.200934
$162$	0	0
$163$	490843.i	1.44702i	0.690316	+	0.723508i	$0.257472\pi$
$163$	490843.i	1.44702i	−0.690316	+	0.723508i	$0.742528\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	196273.i	0.544591i	0.962214	+	0.272295i	$0.0877827\pi$
$167$	196273.i	0.544591i	−0.962214	+	0.272295i	$0.912217\pi$
$168$	0	0
$169$	−562999.	−1.51632
$170$	0	0
$171$	184807.	0.483312
$172$	0	0
$173$	183395.i	0.465877i	0.972491	+	0.232938i	$0.0748341\pi$
$173$	183395.i	0.465877i	−0.972491	+	0.232938i	$0.925166\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	569814.i	1.36710i
$178$	0	0
$179$	38660.0	0.0901839	0.0450920	−	0.998983i	$-0.485642\pi$
$179$	38660.0	0.0901839	0.0450920	+	0.998983i	$0.485642\pi$
$180$	0	0
$181$	−561287.	−1.27347	−0.636735	−	0.771083i	$-0.719715\pi$
$181$	−561287.	−1.27347	−0.636735	+	0.771083i	$0.719715\pi$
$182$	0	0
$183$	517572.i	1.14247i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	763530.i	1.59670i
$188$	0	0
$189$	−409784.	−0.834450
$190$	0	0
$191$	265393.	0.526387	0.263194	−	0.964743i	$-0.415224\pi$
$191$	265393.	0.526387	0.263194	+	0.964743i	$0.415224\pi$
$192$	0	0
$193$	863148.i	1.66798i	0.551776	+	0.833992i	$0.313950\pi$
$193$	863148.i	1.66798i	−0.551776	+	0.833992i	$0.686050\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 281871.i	− 0.517469i	−0.965949	−	0.258734i	$-0.916695\pi$
$197$	− 281871.i	− 0.517469i	0.965949	−	0.258734i	$-0.0833054\pi$
$198$	0	0
$199$	192798.	0.345120	0.172560	−	0.984999i	$-0.444796\pi$
$199$	192798.	0.345120	0.172560	+	0.984999i	$0.444796\pi$
$200$	0	0
$201$	1.58053e6	2.75938
$202$	0	0
$203$	− 326945.i	− 0.556846i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 911943.i	− 1.47925i
$208$	0	0
$209$	−132283.	−0.209477
$210$	0	0
$211$	44631.4	0.0690136	0.0345068	−	0.999404i	$-0.489014\pi$
$211$	44631.4	0.0690136	0.0345068	+	0.999404i	$0.489014\pi$
$212$	0	0
$213$	− 321651.i	− 0.485776i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4326.13i	0.00623664i
$218$	0	0
$219$	−115263.	−0.162398
$220$	0	0
$221$	−1.77283e6	−2.44167
$222$	0	0
$223$	− 904469.i	− 1.21796i	−0.793187	−	0.608978i	$-0.791580\pi$
$223$	− 904469.i	− 1.21796i	0.793187	−	0.608978i	$-0.208420\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.02684e6i	1.32263i	0.750109	+	0.661314i	$0.230001\pi$
$227$	1.02684e6i	1.32263i	−0.750109	+	0.661314i	$0.769999\pi$
$228$	0	0
$229$	101521.	0.127928	0.0639641	−	0.997952i	$-0.479626\pi$
$229$	101521.	0.127928	0.0639641	+	0.997952i	$0.479626\pi$
$230$	0	0
$231$	503830.	0.621232
$232$	0	0
$233$	− 313713.i	− 0.378567i	−0.981923	−	0.189283i	$-0.939384\pi$
$233$	− 313713.i	− 0.378567i	0.981923	−	0.189283i	$-0.0606165\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	689715.i	0.797625i
$238$	0	0
$239$	1.53093e6	1.73365	0.866825	−	0.498612i	$-0.166157\pi$
$239$	1.53093e6	1.73365	0.866825	+	0.498612i	$0.166157\pi$
$240$	0	0
$241$	506999.	0.562296	0.281148	−	0.959664i	$-0.409285\pi$
$241$	506999.	0.562296	0.281148	+	0.959664i	$0.409285\pi$
$242$	0	0
$243$	− 1.59638e6i	− 1.73428i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 307146.i	− 0.320333i
$248$	0	0
$249$	2.02482e6	2.06961
$250$	0	0
$251$	−511659.	−0.512621	−0.256311	−	0.966594i	$-0.582507\pi$
$251$	−511659.	−0.512621	−0.256311	+	0.966594i	$0.582507\pi$
$252$	0	0
$253$	652758.i	0.641137i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	256082.i	0.241850i	0.992662	+	0.120925i	$0.0385861\pi$
$257$	256082.i	0.241850i	−0.992662	+	0.120925i	$0.961414\pi$
$258$	0	0
$259$	81620.7	0.0756051
$260$	0	0
$261$	4.51154e6	4.09943
$262$	0	0
$263$	1.32451e6i	1.18077i	0.807121	+	0.590386i	$0.201024\pi$
$263$	1.32451e6i	1.18077i	−0.807121	+	0.590386i	$0.798976\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1.74493e6i	− 1.49795i
$268$	0	0
$269$	1.42989e6	1.20482	0.602410	−	0.798187i	$-0.294207\pi$
$269$	1.42989e6	1.20482	0.602410	+	0.798187i	$0.294207\pi$
$270$	0	0
$271$	−706426.	−0.584310	−0.292155	−	0.956371i	$-0.594372\pi$
$271$	−706426.	−0.584310	−0.292155	+	0.956371i	$0.594372\pi$
$272$	0	0
$273$	1.16984e6i	0.949989i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 314677.i	− 0.246414i	−0.992381	−	0.123207i	$-0.960682\pi$
$277$	− 314677.i	− 0.246414i	0.992381	−	0.123207i	$-0.0393179\pi$
$278$	0	0
$279$	−59696.6	−0.0459134
$280$	0	0
$281$	−437793.	−0.330752	−0.165376	−	0.986231i	$-0.552884\pi$
$281$	−437793.	−0.330752	−0.165376	+	0.986231i	$0.552884\pi$
$282$	0	0
$283$	− 2.08248e6i	− 1.54566i	−0.634612	−	0.772831i	$-0.718840\pi$
$283$	− 2.08248e6i	− 1.54566i	0.634612	−	0.772831i	$-0.281160\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	336938.i	0.241460i
$288$	0	0
$289$	−1.94411e6	−1.36923
$290$	0	0
$291$	893407.	0.618468
$292$	0	0
$293$	90716.3i	0.0617329i	0.999524	+	0.0308664i	$0.00982665\pi$
$293$	90716.3i	0.0617329i	−0.999524	+	0.0308664i	$0.990173\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.04752e6i	2.66255i
$298$	0	0
$299$	−1.51563e6	−0.980428
$300$	0	0
$301$	−697219.	−0.443561
$302$	0	0
$303$	1.42118e6i	0.889286i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	571699.i	0.346196i	0.984905	+	0.173098i	$0.0553777\pi$
$307$	571699.i	0.346196i	−0.984905	+	0.173098i	$0.944622\pi$
$308$	0	0
$309$	−2.73137e6	−1.62736
$310$	0	0
$311$	2.59515e6	1.52147	0.760733	−	0.649065i	$-0.224840\pi$
$311$	2.59515e6	1.52147	0.760733	+	0.649065i	$0.224840\pi$
$312$	0	0
$313$	− 510659.i	− 0.294625i	−0.989090	−	0.147313i	$-0.952938\pi$
$313$	− 510659.i	− 0.294625i	0.989090	−	0.147313i	$-0.0470623\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.37030e6i	1.88374i	0.335984	+	0.941868i	$0.390931\pi$
$317$	3.37030e6i	1.88374i	−0.335984	+	0.941868i	$0.609069\pi$
$318$	0	0
$319$	−3.22931e6	−1.77678
$320$	0	0
$321$	−2.53932e6	−1.37548
$322$	0	0
$323$	− 582813.i	− 0.310830i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 304354.i	− 0.157402i
$328$	0	0
$329$	786012.	0.400349
$330$	0	0
$331$	3.80172e6	1.90726	0.953632	−	0.300976i	$-0.0973125\pi$
$331$	3.80172e6	1.90726	0.953632	+	0.300976i	$0.0973125\pi$
$332$	0	0
$333$	1.12629e6i	0.556595i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2.06627e6i	− 0.991088i	−0.868583	−	0.495544i	$-0.834969\pi$
$337$	− 2.06627e6i	− 0.991088i	0.868583	−	0.495544i	$-0.165031\pi$
$338$	0	0
$339$	6.75174e6	3.19093
$340$	0	0
$341$	42730.1	0.0198998
$342$	0	0
$343$	1.34186e6i	0.615845i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.35066e6i	− 1.04801i	−0.851715	−	0.524005i	$-0.824437\pi$
$347$	− 2.35066e6i	− 1.04801i	0.851715	−	0.524005i	$-0.175563\pi$
$348$	0	0
$349$	−1.67220e6	−0.734896	−0.367448	−	0.930044i	$-0.619768\pi$
$349$	−1.67220e6	−0.734896	−0.367448	+	0.930044i	$0.619768\pi$
$350$	0	0
$351$	−9.39790e6	−4.07158
$352$	0	0
$353$	1.71355e6i	0.731914i	0.930632	+	0.365957i	$0.119258\pi$
$353$	1.71355e6i	0.731914i	−0.930632	+	0.365957i	$0.880742\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.21978e6i	0.921806i
$358$	0	0
$359$	3.85773e6	1.57978	0.789888	−	0.613251i	$-0.210139\pi$
$359$	3.85773e6	1.57978	0.789888	+	0.613251i	$0.210139\pi$
$360$	0	0
$361$	−2.37513e6	−0.959221
$362$	0	0
$363$	− 351753.i	− 0.140111i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.18109e6i	1.23285i	0.787412	+	0.616427i	$0.211420\pi$
$367$	3.18109e6i	1.23285i	−0.787412	+	0.616427i	$0.788580\pi$
$368$	0	0
$369$	−4.64942e6	−1.77760
$370$	0	0
$371$	631042.	0.238026
$372$	0	0
$373$	− 4.79295e6i	− 1.78374i	−0.452295	−	0.891868i	$-0.649395\pi$
$373$	− 4.79295e6i	− 1.78374i	0.452295	−	0.891868i	$-0.350605\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 7.49809e6i	− 2.71705i
$378$	0	0
$379$	−7018.36	−0.00250979	−0.00125490	−	0.999999i	$-0.500399\pi$
$379$	−7018.36	−0.00250979	−0.00125490	+	0.999999i	$0.500399\pi$
$380$	0	0
$381$	2.94430e6	1.03913
$382$	0	0
$383$	− 694105.i	− 0.241784i	−0.992666	−	0.120892i	$-0.961424\pi$
$383$	− 694105.i	− 0.241784i	0.992666	−	0.120892i	$-0.0385755\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 9.62097e6i	− 3.26544i
$388$	0	0
$389$	53514.1	0.0179306	0.00896529	−	0.999960i	$-0.497146\pi$
$389$	53514.1	0.0179306	0.00896529	+	0.999960i	$0.497146\pi$
$390$	0	0
$391$	−2.87593e6	−0.951342
$392$	0	0
$393$	− 1.45310e6i	− 0.474584i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 907937.i	− 0.289121i	−0.989496	−	0.144560i	$-0.953823\pi$
$397$	− 907937.i	− 0.289121i	0.989496	−	0.144560i	$-0.0461768\pi$
$398$	0	0
$399$	−384580.	−0.120936
$400$	0	0
$401$	−514404.	−0.159751	−0.0798755	−	0.996805i	$-0.525452\pi$
$401$	−514404.	−0.159751	−0.0798755	+	0.996805i	$0.525452\pi$
$402$	0	0
$403$	99214.6i	0.0304308i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 806185.i	− 0.241240i
$408$	0	0
$409$	−5.61814e6	−1.66067	−0.830337	−	0.557262i	$-0.811852\pi$
$409$	−5.61814e6	−1.66067	−0.830337	+	0.557262i	$0.811852\pi$
$410$	0	0
$411$	−1.37294e6	−0.400911
$412$	0	0
$413$	− 836334.i	− 0.241271i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 6.32288e6i	− 1.78064i
$418$	0	0
$419$	−708382.	−0.197121	−0.0985605	−	0.995131i	$-0.531424\pi$
$419$	−708382.	−0.197121	−0.0985605	+	0.995131i	$0.531424\pi$
$420$	0	0
$421$	−3.91741e6	−1.07719	−0.538597	−	0.842563i	$-0.681046\pi$
$421$	−3.91741e6	−1.07719	−0.538597	+	0.842563i	$0.681046\pi$
$422$	0	0
$423$	1.08462e7i	2.94732i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 759658.i	− 0.201627i
$428$	0	0
$429$	1.15547e7	3.03121
$430$	0	0
$431$	4.42095e6	1.14636	0.573181	−	0.819429i	$-0.305709\pi$
$431$	4.42095e6	1.14636	0.573181	+	0.819429i	$0.305709\pi$
$432$	0	0
$433$	− 4.01914e6i	− 1.03018i	−0.857136	−	0.515090i	$-0.827758\pi$
$433$	− 4.01914e6i	− 1.03018i	0.857136	−	0.515090i	$-0.172242\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 498259.i	− 0.124811i
$438$	0	0
$439$	5.41795e6	1.34176	0.670879	−	0.741567i	$-0.265917\pi$
$439$	5.41795e6	1.34176	0.670879	+	0.741567i	$0.265917\pi$
$440$	0	0
$441$	−8.74163e6	−2.14041
$442$	0	0
$443$	− 7.21518e6i	− 1.74678i	−0.487023	−	0.873389i	$-0.661917\pi$
$443$	− 7.21518e6i	− 1.74678i	0.487023	−	0.873389i	$-0.338083\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.43751e6i	1.05044i
$448$	0	0
$449$	203792.	0.0477057	0.0238529	−	0.999715i	$-0.492407\pi$
$449$	203792.	0.0477057	0.0238529	+	0.999715i	$0.492407\pi$
$450$	0	0
$451$	3.32800e6	0.770446
$452$	0	0
$453$	1.30769e7i	2.99406i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.45642e6i	0.774169i	0.922044	+	0.387085i	$0.126518\pi$
$457$	3.45642e6i	0.774169i	−0.922044	+	0.387085i	$0.873482\pi$
$458$	0	0
$459$	−1.78326e7	−3.95079
$460$	0	0
$461$	6.40596e6	1.40389	0.701944	−	0.712233i	$-0.252316\pi$
$461$	6.40596e6	1.40389	0.701944	+	0.712233i	$0.252316\pi$
$462$	0	0
$463$	− 7.59550e6i	− 1.64666i	−0.567563	−	0.823330i	$-0.692113\pi$
$463$	− 7.59550e6i	− 1.64666i	0.567563	−	0.823330i	$-0.307887\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.58166e6i	0.972145i	0.873919	+	0.486072i	$0.161571\pi$
$467$	4.58166e6i	0.972145i	−0.873919	+	0.486072i	$0.838429\pi$
$468$	0	0
$469$	−2.31979e6	−0.486986
$470$	0	0
$471$	−1.03862e7	−2.15726
$472$	0	0
$473$	6.88658e6i	1.41531i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.70779e6i	1.75231i
$478$	0	0
$479$	−5.01195e6	−0.998085	−0.499043	−	0.866577i	$-0.666315\pi$
$479$	−5.01195e6	−0.998085	−0.499043	+	0.866577i	$0.666315\pi$
$480$	0	0
$481$	1.87187e6	0.368904
$482$	0	0
$483$	1.89774e6i	0.370142i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.10670e6i	0.402514i	0.979539	+	0.201257i	$0.0645026\pi$
$487$	2.10670e6i	0.402514i	−0.979539	+	0.201257i	$0.935497\pi$
$488$	0	0
$489$	1.40949e7	2.66556
$490$	0	0
$491$	5.43322e6	1.01708	0.508538	−	0.861040i	$-0.330186\pi$
$491$	5.43322e6	1.01708	0.508538	+	0.861040i	$0.330186\pi$
$492$	0	0
$493$	− 1.42277e7i	− 2.63644i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	472099.i	0.0857318i
$498$	0	0
$499$	−9.65183e6	−1.73523	−0.867617	−	0.497233i	$-0.834350\pi$
$499$	−9.65183e6	−1.73523	−0.867617	+	0.497233i	$0.834350\pi$
$500$	0	0
$501$	5.63611e6	1.00319
$502$	0	0
$503$	1.33954e6i	0.236067i	0.993010	+	0.118034i	$0.0376591\pi$
$503$	1.33954e6i	0.236067i	−0.993010	+	0.118034i	$0.962341\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.61669e7i	2.79322i
$508$	0	0
$509$	−4.60771e6	−0.788299	−0.394150	−	0.919046i	$-0.628961\pi$
$509$	−4.60771e6	−0.788299	−0.394150	+	0.919046i	$0.628961\pi$
$510$	0	0
$511$	169176.	0.0286606
$512$	0	0
$513$	− 3.08953e6i	− 0.518321i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 7.76360e6i	− 1.27743i
$518$	0	0
$519$	5.26629e6	0.858196
$520$	0	0
$521$	−5.80941e6	−0.937644	−0.468822	−	0.883293i	$-0.655321\pi$
$521$	−5.80941e6	−0.937644	−0.468822	+	0.883293i	$0.655321\pi$
$522$	0	0
$523$	− 3.83877e6i	− 0.613674i	−0.951762	−	0.306837i	$-0.900729\pi$
$523$	− 3.83877e6i	− 0.613674i	0.951762	−	0.306837i	$-0.0992707\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	188261.i	0.0295280i
$528$	0	0
$529$	3.97765e6	0.617998
$530$	0	0
$531$	1.15406e7	1.77621
$532$	0	0
$533$	7.72725e6i	1.17817i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.11015e6i	− 0.166129i
$538$	0	0
$539$	6.25716e6	0.927696
$540$	0	0
$541$	6.28830e6	0.923720	0.461860	−	0.886953i	$-0.347182\pi$
$541$	6.28830e6	0.923720	0.461860	+	0.886953i	$0.347182\pi$
$542$	0	0
$543$	1.61177e7i	2.34587i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4.54365e6i	− 0.649287i	−0.945836	−	0.324644i	$-0.894756\pi$
$547$	− 4.54365e6i	− 0.649287i	0.945836	−	0.324644i	$-0.105244\pi$
$548$	0	0
$549$	1.04826e7	1.48435
$550$	0	0
$551$	2.46497e6	0.345886
$552$	0	0
$553$	− 1.01232e6i	− 0.140768i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.20198e7i	− 1.64157i	−0.571239	−	0.820783i	$-0.693537\pi$
$557$	− 1.20198e7i	− 1.64157i	0.571239	−	0.820783i	$-0.306463\pi$
$558$	0	0
$559$	−1.59899e7	−2.16429
$560$	0	0
$561$	2.19252e7	2.94129
$562$	0	0
$563$	3.65741e6i	0.486298i	0.969989	+	0.243149i	$0.0781804\pi$
$563$	3.65741e6i	0.486298i	−0.969989	+	0.243149i	$0.921820\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.81077e6i	0.759059i
$568$	0	0
$569$	2.80214e6	0.362835	0.181418	−	0.983406i	$-0.441931\pi$
$569$	2.80214e6	0.362835	0.181418	+	0.983406i	$0.441931\pi$
$570$	0	0
$571$	−6.78472e6	−0.870846	−0.435423	−	0.900226i	$-0.643401\pi$
$571$	−6.78472e6	−0.870846	−0.435423	+	0.900226i	$0.643401\pi$
$572$	0	0
$573$	− 7.62091e6i	− 0.969662i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.25416e6i	0.156825i	0.996921	+	0.0784123i	$0.0249850\pi$
$577$	1.25416e6i	0.156825i	−0.996921	+	0.0784123i	$0.975015\pi$
$578$	0	0
$579$	2.47858e7	3.07261
$580$	0	0
$581$	−2.97190e6	−0.365253
$582$	0	0
$583$	− 6.23293e6i	− 0.759488i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 726476.i	− 0.0870213i	−0.999053	−	0.0435107i	$-0.986146\pi$
$587$	− 726476.i	− 0.0870213i	0.999053	−	0.0435107i	$-0.0138542\pi$
$588$	0	0
$589$	−32616.5	−0.00387391
$590$	0	0
$591$	−8.09409e6	−0.953234
$592$	0	0
$593$	933494.i	0.109012i	0.998513	+	0.0545060i	$0.0173584\pi$
$593$	933494.i	0.109012i	−0.998513	+	0.0545060i	$0.982642\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 5.53632e6i	− 0.635749i
$598$	0	0
$599$	1.27354e7	1.45025	0.725127	−	0.688615i	$-0.241781\pi$
$599$	1.27354e7	1.45025	0.725127	+	0.688615i	$0.241781\pi$
$600$	0	0
$601$	−6.87190e6	−0.776052	−0.388026	−	0.921648i	$-0.626843\pi$
$601$	−6.87190e6	−0.776052	−0.388026	+	0.921648i	$0.626843\pi$
$602$	0	0
$603$	− 3.20109e7i	− 3.58513i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.87130e6i	0.426467i	0.977001	+	0.213233i	$0.0683995\pi$
$607$	3.87130e6i	0.426467i	−0.977001	+	0.213233i	$0.931601\pi$
$608$	0	0
$609$	−9.38844e6	−1.02577
$610$	0	0
$611$	1.80262e7	1.95345
$612$	0	0
$613$	− 2.38824e6i	− 0.256701i	−0.991729	−	0.128350i	$-0.959032\pi$
$613$	− 2.38824e6i	− 0.256701i	0.991729	−	0.128350i	$-0.0409682\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.07299e6i	0.324974i	0.986711	+	0.162487i	$0.0519516\pi$
$617$	3.07299e6i	0.324974i	−0.986711	+	0.162487i	$0.948048\pi$
$618$	0	0
$619$	−8.52257e6	−0.894013	−0.447007	−	0.894531i	$-0.647510\pi$
$619$	−8.52257e6	−0.894013	−0.447007	+	0.894531i	$0.647510\pi$
$620$	0	0
$621$	−1.52455e7	−1.58640
$622$	0	0
$623$	2.56109e6i	0.264365i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.79858e6i	0.385880i
$628$	0	0
$629$	3.55190e6	0.357960
$630$	0	0
$631$	8.54170e6	0.854026	0.427013	−	0.904246i	$-0.359566\pi$
$631$	8.54170e6	0.854026	0.427013	+	0.904246i	$0.359566\pi$
$632$	0	0
$633$	− 1.28162e6i	− 0.127131i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.45284e7i	1.41863i
$638$	0	0
$639$	−6.51452e6	−0.631146
$640$	0	0
$641$	3.81006e6	0.366257	0.183129	−	0.983089i	$-0.441377\pi$
$641$	3.81006e6	0.366257	0.183129	+	0.983089i	$0.441377\pi$
$642$	0	0
$643$	1.40516e7i	1.34029i	0.742231	+	0.670144i	$0.233768\pi$
$643$	1.40516e7i	1.34029i	−0.742231	+	0.670144i	$0.766232\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 962366.i	− 0.0903815i	−0.998978	−	0.0451908i	$-0.985610\pi$
$647$	− 962366.i	− 0.0903815i	0.998978	−	0.0451908i	$-0.0143896\pi$
$648$	0	0
$649$	−8.26065e6	−0.769844
$650$	0	0
$651$	124228.	0.0114886
$652$	0	0
$653$	− 413989.i	− 0.0379932i	−0.999820	−	0.0189966i	$-0.993953\pi$
$653$	− 413989.i	− 0.0379932i	0.999820	−	0.0189966i	$-0.00604717\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.33446e6i	0.210996i
$658$	0	0
$659$	1.92401e7	1.72581	0.862905	−	0.505367i	$-0.168643\pi$
$659$	1.92401e7	1.72581	0.862905	+	0.505367i	$0.168643\pi$
$660$	0	0
$661$	−2.04652e7	−1.82185	−0.910924	−	0.412574i	$-0.864630\pi$
$661$	−2.04652e7	−1.82185	−0.910924	+	0.412574i	$0.864630\pi$
$662$	0	0
$663$	5.09080e7i	4.49782i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.21636e7i	− 1.05864i
$668$	0	0
$669$	−2.59724e7	−2.24361
$670$	0	0
$671$	−7.50330e6	−0.643348
$672$	0	0
$673$	− 9.12086e6i	− 0.776244i	−0.921608	−	0.388122i	$-0.873124\pi$
$673$	− 9.12086e6i	− 0.776244i	0.921608	−	0.388122i	$-0.126876\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.29457e7i	1.08556i	0.839876	+	0.542778i	$0.182628\pi$
$677$	1.29457e7i	1.08556i	−0.839876	+	0.542778i	$0.817372\pi$
$678$	0	0
$679$	−1.31128e6	−0.109150
$680$	0	0
$681$	2.94863e7	2.43642
$682$	0	0
$683$	− 8.56637e6i	− 0.702660i	−0.936252	−	0.351330i	$-0.885730\pi$
$683$	− 8.56637e6i	− 0.702660i	0.936252	−	0.351330i	$-0.114270\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 2.91523e6i	− 0.235658i
$688$	0	0
$689$	1.44722e7	1.16141
$690$	0	0
$691$	−1.01163e7	−0.805984	−0.402992	−	0.915204i	$-0.632030\pi$
$691$	−1.01163e7	−0.805984	−0.402992	+	0.915204i	$0.632030\pi$
$692$	0	0
$693$	− 1.02042e7i	− 0.807138i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.46626e7i	1.14322i
$698$	0	0
$699$	−9.00846e6	−0.697361
$700$	0	0
$701$	−1.95732e7	−1.50441	−0.752207	−	0.658927i	$-0.771011\pi$
$701$	−1.95732e7	−1.50441	−0.752207	+	0.658927i	$0.771011\pi$
$702$	0	0
$703$	615372.i	0.0469623i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.08591e6i	− 0.156945i
$708$	0	0
$709$	−2.36252e7	−1.76506	−0.882531	−	0.470254i	$-0.844162\pi$
$709$	−2.36252e7	−1.76506	−0.882531	+	0.470254i	$0.844162\pi$
$710$	0	0
$711$	1.39690e7	1.03632
$712$	0	0
$713$	160948.i	0.0118567i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 4.39617e7i	− 3.19357i
$718$	0	0
$719$	−2.44994e7	−1.76740	−0.883698	−	0.468058i	$-0.844954\pi$
$719$	−2.44994e7	−1.76740	−0.883698	+	0.468058i	$0.844954\pi$
$720$	0	0
$721$	4.00892e6	0.287204
$722$	0	0
$723$	− 1.45588e7i	− 1.03581i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.75199e7i	1.22941i	0.788759	+	0.614703i	$0.210724\pi$
$727$	1.75199e7i	1.22941i	−0.788759	+	0.614703i	$0.789276\pi$
$728$	0	0
$729$	−1.23387e7	−0.859904
$730$	0	0
$731$	−3.03410e7	−2.10008
$732$	0	0
$733$	2.29025e7i	1.57443i	0.616679	+	0.787215i	$0.288478\pi$
$733$	2.29025e7i	1.57443i	−0.616679	+	0.787215i	$0.711522\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.29131e7i	1.55387i
$738$	0	0
$739$	−1.31983e7	−0.889010	−0.444505	−	0.895776i	$-0.646620\pi$
$739$	−1.31983e7	−0.889010	−0.444505	+	0.895776i	$0.646620\pi$
$740$	0	0
$741$	−8.81988e6	−0.590089
$742$	0	0
$743$	− 1.89399e6i	− 0.125865i	−0.998018	−	0.0629326i	$-0.979955\pi$
$743$	− 1.89399e6i	− 0.125865i	0.998018	−	0.0629326i	$-0.0200453\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4.10094e7i	− 2.68894i
$748$	0	0
$749$	3.72704e6	0.242750
$750$	0	0
$751$	−1.71988e7	−1.11275	−0.556376	−	0.830930i	$-0.687809\pi$
$751$	−1.71988e7	−1.11275	−0.556376	+	0.830930i	$0.687809\pi$
$752$	0	0
$753$	1.46926e7i	0.944304i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 2.02697e7i	− 1.28561i	−0.766031	−	0.642804i	$-0.777771\pi$
$757$	− 2.02697e7i	− 1.28561i	0.766031	−	0.642804i	$-0.222229\pi$
$758$	0	0
$759$	1.87444e7	1.18104
$760$	0	0
$761$	−6.54894e6	−0.409929	−0.204965	−	0.978769i	$-0.565708\pi$
$761$	−6.54894e6	−0.409929	−0.204965	+	0.978769i	$0.565708\pi$
$762$	0	0
$763$	446710.i	0.0277789i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.91803e7i	− 1.17725i
$768$	0	0
$769$	2.48043e7	1.51256	0.756278	−	0.654251i	$-0.227016\pi$
$769$	2.48043e7	1.51256	0.756278	+	0.654251i	$0.227016\pi$
$770$	0	0
$771$	7.35357e6	0.445515
$772$	0	0
$773$	2.27616e7i	1.37010i	0.728495	+	0.685052i	$0.240220\pi$
$773$	2.27616e7i	1.37010i	−0.728495	+	0.685052i	$0.759780\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.34379e6i	− 0.139273i
$778$	0	0
$779$	−2.54031e6	−0.149983
$780$	0	0
$781$	4.66302e6	0.273552
$782$	0	0
$783$	− 7.54221e7i	− 4.39637i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.56494e7i	− 0.900658i	−0.892863	−	0.450329i	$-0.851307\pi$
$787$	− 1.56494e7i	− 0.900658i	0.892863	−	0.450329i	$-0.148693\pi$
$788$	0	0
$789$	3.80341e7	2.17511
$790$	0	0
$791$	−9.90976e6	−0.563147
$792$	0	0
$793$	− 1.74218e7i	− 0.983809i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.26205e7i	1.26141i	0.776023	+	0.630704i	$0.217234\pi$
$797$	2.26205e7i	1.26141i	−0.776023	+	0.630704i	$0.782766\pi$
$798$	0	0
$799$	3.42050e7	1.89549
$800$	0	0
$801$	−3.53406e7	−1.94622
$802$	0	0
$803$	− 1.67098e6i	− 0.0914499i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 4.10602e7i	− 2.21941i
$808$	0	0
$809$	2.25367e7	1.21065	0.605326	−	0.795977i	$-0.293043\pi$
$809$	2.25367e7	1.21065	0.605326	+	0.795977i	$0.293043\pi$
$810$	0	0
$811$	−410196.	−0.0218997	−0.0109499	−	0.999940i	$-0.503486\pi$
$811$	−410196.	−0.0218997	−0.0109499	+	0.999940i	$0.503486\pi$
$812$	0	0
$813$	2.02855e7i	1.07636i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 5.25662e6i	− 0.275519i
$818$	0	0
$819$	2.36931e7	1.23428
$820$	0	0
$821$	1.62749e7	0.842678	0.421339	−	0.906903i	$-0.361560\pi$
$821$	1.62749e7	0.842678	0.421339	+	0.906903i	$0.361560\pi$
$822$	0	0
$823$	− 628881.i	− 0.0323645i	−0.999869	−	0.0161823i	$-0.994849\pi$
$823$	− 628881.i	− 0.0323645i	0.999869	−	0.0161823i	$-0.00515120\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.94403e7i	0.988417i	0.869343	+	0.494209i	$0.164542\pi$
$827$	1.94403e7i	0.988417i	−0.869343	+	0.494209i	$0.835458\pi$
$828$	0	0
$829$	−3.39088e7	−1.71366	−0.856832	−	0.515595i	$-0.827571\pi$
$829$	−3.39088e7	−1.71366	−0.856832	+	0.515595i	$0.827571\pi$
$830$	0	0
$831$	−9.03614e6	−0.453921
$832$	0	0
$833$	2.75679e7i	1.37655i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	997985.i	0.0492391i
$838$	0	0
$839$	−6.84583e6	−0.335754	−0.167877	−	0.985808i	$-0.553691\pi$
$839$	−6.84583e6	−0.335754	−0.167877	+	0.985808i	$0.553691\pi$
$840$	0	0
$841$	3.96642e7	1.93379
$842$	0	0
$843$	1.25715e7i	0.609282i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	516280.i	0.0247273i
$848$	0	0
$849$	−5.97997e7	−2.84728
$850$	0	0
$851$	3.03660e6	0.143735
$852$	0	0
$853$	− 2.67852e7i	− 1.26044i	−0.776416	−	0.630221i	$-0.782964\pi$
$853$	− 2.67852e7i	− 1.26044i	0.776416	−	0.630221i	$-0.217036\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 9.32339e6i	− 0.433632i	−0.976212	−	0.216816i	$-0.930433\pi$
$857$	− 9.32339e6i	− 0.433632i	0.976212	−	0.216816i	$-0.0695672\pi$
$858$	0	0
$859$	−3.73055e7	−1.72500	−0.862502	−	0.506054i	$-0.831104\pi$
$859$	−3.73055e7	−1.72500	−0.862502	+	0.506054i	$0.831104\pi$
$860$	0	0
$861$	9.67537e6	0.444795
$862$	0	0
$863$	− 5.18636e6i	− 0.237048i	−0.992951	−	0.118524i	$-0.962184\pi$
$863$	− 5.18636e6i	− 0.237048i	0.992951	−	0.118524i	$-0.0378162\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.58265e7i	2.52228i
$868$	0	0
$869$	−9.99888e6	−0.449161
$870$	0	0
$871$	−5.32016e7	−2.37618
$872$	0	0
$873$	− 1.80945e7i	− 0.803546i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5.63396e6i	0.247352i	0.992323	+	0.123676i	$0.0394683\pi$
$877$	5.63396e6i	0.247352i	−0.992323	+	0.123676i	$0.960532\pi$
$878$	0	0
$879$	2.60498e6	0.113719
$880$	0	0
$881$	−5.76880e6	−0.250407	−0.125203	−	0.992131i	$-0.539958\pi$
$881$	−5.76880e6	−0.250407	−0.125203	+	0.992131i	$0.539958\pi$
$882$	0	0
$883$	2.60630e7i	1.12492i	0.826823	+	0.562462i	$0.190146\pi$
$883$	2.60630e7i	1.12492i	−0.826823	+	0.562462i	$0.809854\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 3.24889e7i	− 1.38652i	−0.720688	−	0.693259i	$-0.756174\pi$
$887$	− 3.24889e7i	− 1.38652i	0.720688	−	0.693259i	$-0.243826\pi$
$888$	0	0
$889$	−4.32145e6	−0.183390
$890$	0	0
$891$	5.73942e7	2.42200
$892$	0	0
$893$	5.92607e6i	0.248678i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.35223e7i	1.80605i
$898$	0	0
$899$	−796240.	−0.0328583
$900$	0	0
$901$	2.74612e7	1.12696
$902$	0	0
$903$	2.00211e7i	0.817087i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3.11745e7i	1.25829i	0.777288	+	0.629145i	$0.216595\pi$
$907$	3.11745e7i	1.25829i	−0.777288	+	0.629145i	$0.783405\pi$
$908$	0	0
$909$	2.87836e7	1.15541
$910$	0	0
$911$	3.58254e7	1.43019	0.715097	−	0.699025i	$-0.246382\pi$
$911$	3.58254e7	1.43019	0.715097	+	0.699025i	$0.246382\pi$
$912$	0	0
$913$	2.93540e7i	1.16544i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.13276e6i	0.0837565i
$918$	0	0
$919$	3.14710e7	1.22920	0.614598	−	0.788840i	$-0.289318\pi$
$919$	3.14710e7	1.22920	0.614598	+	0.788840i	$0.289318\pi$
$920$	0	0
$921$	1.64167e7	0.637730
$922$	0	0
$923$	1.08270e7i	0.418316i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.53194e7i	2.11436i
$928$	0	0
$929$	−532600.	−0.0202471	−0.0101235	−	0.999949i	$-0.503222\pi$
$929$	−532600.	−0.0202471	−0.0101235	+	0.999949i	$0.503222\pi$
$930$	0	0
$931$	−4.77618e6	−0.180595
$932$	0	0
$933$	− 7.45215e7i	− 2.80270i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.97320e7i	− 0.734214i	−0.930179	−	0.367107i	$-0.880348\pi$
$937$	− 1.97320e7i	− 0.734214i	0.930179	−	0.367107i	$-0.119652\pi$
$938$	0	0
$939$	−1.46639e7	−0.542732
$940$	0	0
$941$	−2.12060e7	−0.780702	−0.390351	−	0.920666i	$-0.627646\pi$
$941$	−2.12060e7	−0.780702	−0.390351	+	0.920666i	$0.627646\pi$
$942$	0	0
$943$	1.25353e7i	0.459047i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.60224e7i	− 1.66761i	−0.552060	−	0.833804i	$-0.686158\pi$
$947$	− 4.60224e7i	− 1.66761i	0.552060	−	0.833804i	$-0.313842\pi$
$948$	0	0
$949$	3.87984e6	0.139845
$950$	0	0
$951$	9.67802e7	3.47004
$952$	0	0
$953$	4.76525e7i	1.69963i	0.527085	+	0.849813i	$0.323285\pi$
$953$	4.76525e7i	1.69963i	−0.527085	+	0.849813i	$0.676715\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.27316e7i	3.27301i
$958$	0	0
$959$	2.01511e6	0.0707543
$960$	0	0
$961$	−2.86186e7	−0.999632
$962$	0	0
$963$	5.14297e7i	1.78710i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.29686e7i	0.789893i	0.918704	+	0.394946i	$0.129237\pi$
$967$	2.29686e7i	0.789893i	−0.918704	+	0.394946i	$0.870763\pi$
$968$	0	0
$969$	−1.67358e7	−0.572583
$970$	0	0
$971$	2.22231e7	0.756408	0.378204	−	0.925722i	$-0.376542\pi$
$971$	2.22231e7	0.756408	0.378204	+	0.925722i	$0.376542\pi$
$972$	0	0
$973$	9.28031e6i	0.314254i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.68581e7i	1.23537i	0.786426	+	0.617684i	$0.211929\pi$
$977$	3.68581e7i	1.23537i	−0.786426	+	0.617684i	$0.788071\pi$
$978$	0	0
$979$	2.52964e7	0.843532
$980$	0	0
$981$	−6.16418e6	−0.204504
$982$	0	0
$983$	− 4.53136e7i	− 1.49570i	−0.663868	−	0.747850i	$-0.731086\pi$
$983$	− 4.53136e7i	− 1.49570i	0.663868	−	0.747850i	$-0.268914\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.25708e7i	− 0.737487i
$988$	0	0
$989$	−2.59392e7	−0.843268
$990$	0	0
$991$	−3.68582e7	−1.19220	−0.596102	−	0.802909i	$-0.703284\pi$
$991$	−3.68582e7	−1.19220	−0.596102	+	0.802909i	$0.703284\pi$
$992$	0	0
$993$	− 1.09169e8i	− 3.51339i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.19245e7i	1.33577i	0.744266	+	0.667883i	$0.232799\pi$
$997$	4.19245e7i	1.33577i	−0.744266	+	0.667883i	$0.767201\pi$
$998$	0	0
$999$	1.88289e7	0.596912

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.6.c.e.49.1		4
4.3	odd	2		400.6.c.l.49.4		4
5.2	odd	4		40.6.a.d.1.1	✓	2
5.3	odd	4		200.6.a.g.1.2		2
5.4	even	2	inner	200.6.c.e.49.4		4
15.2	even	4		360.6.a.l.1.1		2
20.3	even	4		400.6.a.q.1.1		2
20.7	even	4		80.6.a.i.1.2		2
20.19	odd	2		400.6.c.l.49.1		4
40.27	even	4		320.6.a.q.1.1		2
40.37	odd	4		320.6.a.w.1.2		2
60.47	odd	4		720.6.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.a.d.1.1	✓	2	5.2	odd	4
80.6.a.i.1.2		2	20.7	even	4
200.6.a.g.1.2		2	5.3	odd	4
200.6.c.e.49.1		4	1.1	even	1	trivial
200.6.c.e.49.4		4	5.4	even	2	inner
320.6.a.q.1.1		2	40.27	even	4
320.6.a.w.1.2		2	40.37	odd	4
360.6.a.l.1.1		2	15.2	even	4
400.6.a.q.1.1		2	20.3	even	4
400.6.c.l.49.1		4	20.19	odd	2
400.6.c.l.49.4		4	4.3	odd	2
720.6.a.z.1.2		2	60.47	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-28.7156i q^{3} +42.1469i q^{7} -581.588 q^{9} +O(q^{10})\)	\(q-28.7156i q^{3} +42.1469i q^{7} -581.588 q^{9} +416.294 q^{11} +966.588i q^{13} +1834.11i q^{17} -317.763 q^{19} +1210.27 q^{21} +1568.02i q^{23} +9722.76i q^{27} -7757.28 q^{29} +102.644 q^{31} -11954.1i q^{33} -1936.58i q^{37} +27756.2 q^{39} +7994.36 q^{41} +16542.6i q^{43} -18649.3i q^{47} +15030.6 q^{49} +52667.7 q^{51} -14972.4i q^{53} +9124.76i q^{57} -19843.3 q^{59} -18024.1 q^{61} -24512.1i q^{63} +55040.6i q^{67} +45026.8 q^{69} +11201.3 q^{71} -4013.95i q^{73} +17545.5i q^{77} -24018.8 q^{79} +137869. q^{81} +70512.8i q^{83} +222755. i q^{87} +60765.7 q^{89} -40738.7 q^{91} -2947.49i q^{93} +31112.2i q^{97} -242111. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 1236 q^{9}+O(q^{10}) \) 4 * q - 1236 * q^9	\( 4 q - 1236 q^{9} + 1120 q^{11} + 2000 q^{19} + 5568 q^{21} - 2680 q^{29} - 4496 q^{31} + 41424 q^{39} + 46152 q^{41} + 45948 q^{49} + 171600 q^{51} - 125168 q^{59} + 28216 q^{61} + 224448 q^{69} + 94416 q^{71} + 131808 q^{79} + 142596 q^{81} + 110040 q^{89} - 2128 q^{91} - 494688 q^{99}+O(q^{100}) \) 4 * q - 1236 * q^9 + 1120 * q^11 + 2000 * q^19 + 5568 * q^21 - 2680 * q^29 - 4496 * q^31 + 41424 * q^39 + 46152 * q^41 + 45948 * q^49 + 171600 * q^51 - 125168 * q^59 + 28216 * q^61 + 224448 * q^69 + 94416 * q^71 + 131808 * q^79 + 142596 * q^81 + 110040 * q^89 - 2128 * q^91 - 494688 * q^99

Introduction

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