LMFDB - Embedded newform 1200.4.f.o.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(49,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.f (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:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1200.49
Dual form			1200.4.f.o.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +16.0000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +16.0000i q^{7} -9.00000 q^{9} +28.0000 q^{11} +26.0000i q^{13} -62.0000i q^{17} -68.0000 q^{19} +48.0000 q^{21} -208.000i q^{23} +27.0000i q^{27} +58.0000 q^{29} -160.000 q^{31} -84.0000i q^{33} +270.000i q^{37} +78.0000 q^{39} +282.000 q^{41} +76.0000i q^{43} +280.000i q^{47} +87.0000 q^{49} -186.000 q^{51} +210.000i q^{53} +204.000i q^{57} +196.000 q^{59} +742.000 q^{61} -144.000i q^{63} -836.000i q^{67} -624.000 q^{69} +504.000 q^{71} +1062.00i q^{73} +448.000i q^{77} +768.000 q^{79} +81.0000 q^{81} -1052.00i q^{83} -174.000i q^{87} +726.000 q^{89} -416.000 q^{91} +480.000i q^{93} -1406.00i q^{97} -252.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} + 56 q^{11} - 136 q^{19} + 96 q^{21} + 116 q^{29} - 320 q^{31} + 156 q^{39} + 564 q^{41} + 174 q^{49} - 372 q^{51} + 392 q^{59} + 1484 q^{61} - 1248 q^{69} + 1008 q^{71} + 1536 q^{79} + 162 q^{81} + 1452 q^{89} - 832 q^{91} - 504 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 + 56 * q^11 - 136 * q^19 + 96 * q^21 + 116 * q^29 - 320 * q^31 + 156 * q^39 + 564 * q^41 + 174 * q^49 - 372 * q^51 + 392 * q^59 + 1484 * q^61 - 1248 * q^69 + 1008 * q^71 + 1536 * q^79 + 162 * q^81 + 1452 * q^89 - 832 * q^91 - 504 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	16.0000i	0.863919i	0.901893	+	0.431959i	$0.142178\pi$
$7$	16.0000i	0.863919i	−0.901893	+	0.431959i	$0.857822\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	28.0000	0.767483	0.383742	−	0.923440i	$-0.374635\pi$
$11$	28.0000	0.767483	0.383742	+	0.923440i	$0.374635\pi$
$12$	0	0
$13$	26.0000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	26.0000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 62.0000i	− 0.884542i	−0.896882	−	0.442271i	$-0.854173\pi$
$17$	− 62.0000i	− 0.884542i	0.896882	−	0.442271i	$-0.145827\pi$
$18$	0	0
$19$	−68.0000	−0.821067	−0.410533	−	0.911846i	$-0.634657\pi$
$19$	−68.0000	−0.821067	−0.410533	+	0.911846i	$0.634657\pi$
$20$	0	0
$21$	48.0000	0.498784
$22$	0	0
$23$	− 208.000i	− 1.88570i	−0.333224	−	0.942848i	$-0.608136\pi$
$23$	− 208.000i	− 1.88570i	0.333224	−	0.942848i	$-0.391864\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	58.0000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	58.0000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−160.000	−0.926995	−0.463498	−	0.886098i	$-0.653406\pi$
$31$	−160.000	−0.926995	−0.463498	+	0.886098i	$0.653406\pi$
$32$	0	0
$33$	− 84.0000i	− 0.443107i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	270.000i	1.19967i	0.800124	+	0.599834i	$0.204767\pi$
$37$	270.000i	1.19967i	−0.800124	+	0.599834i	$0.795233\pi$
$38$	0	0
$39$	78.0000	0.320256
$40$	0	0
$41$	282.000	1.07417	0.537085	−	0.843528i	$-0.319525\pi$
$41$	282.000	1.07417	0.537085	+	0.843528i	$0.319525\pi$
$42$	0	0
$43$	76.0000i	0.269532i	0.990877	+	0.134766i	$0.0430283\pi$
$43$	76.0000i	0.269532i	−0.990877	+	0.134766i	$0.956972\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	280.000i	0.868983i	0.900676	+	0.434491i	$0.143072\pi$
$47$	280.000i	0.868983i	−0.900676	+	0.434491i	$0.856928\pi$
$48$	0	0
$49$	87.0000	0.253644
$50$	0	0
$51$	−186.000	−0.510690
$52$	0	0
$53$	210.000i	0.544259i	0.962261	+	0.272129i	$0.0877279\pi$
$53$	210.000i	0.544259i	−0.962261	+	0.272129i	$0.912272\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	204.000i	0.474043i
$58$	0	0
$59$	196.000	0.432492	0.216246	−	0.976339i	$-0.430619\pi$
$59$	196.000	0.432492	0.216246	+	0.976339i	$0.430619\pi$
$60$	0	0
$61$	742.000	1.55743	0.778716	−	0.627376i	$-0.215871\pi$
$61$	742.000	1.55743	0.778716	+	0.627376i	$0.215871\pi$
$62$	0	0
$63$	− 144.000i	− 0.287973i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 836.000i	− 1.52438i	−0.647352	−	0.762191i	$-0.724123\pi$
$67$	− 836.000i	− 1.52438i	0.647352	−	0.762191i	$-0.275877\pi$
$68$	0	0
$69$	−624.000	−1.08871
$70$	0	0
$71$	504.000	0.842448	0.421224	−	0.906957i	$-0.361601\pi$
$71$	504.000	0.842448	0.421224	+	0.906957i	$0.361601\pi$
$72$	0	0
$73$	1062.00i	1.70271i	0.524591	+	0.851354i	$0.324218\pi$
$73$	1062.00i	1.70271i	−0.524591	+	0.851354i	$0.675782\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	448.000i	0.663043i
$78$	0	0
$79$	768.000	1.09376	0.546878	−	0.837212i	$-0.315816\pi$
$79$	768.000	1.09376	0.546878	+	0.837212i	$0.315816\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1052.00i	− 1.39123i	−0.718415	−	0.695614i	$-0.755132\pi$
$83$	− 1052.00i	− 1.39123i	0.718415	−	0.695614i	$-0.244868\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 174.000i	− 0.214423i
$88$	0	0
$89$	726.000	0.864672	0.432336	−	0.901712i	$-0.357689\pi$
$89$	726.000	0.864672	0.432336	+	0.901712i	$0.357689\pi$
$90$	0	0
$91$	−416.000	−0.479216
$92$	0	0
$93$	480.000i	0.535201i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1406.00i	− 1.47173i	−0.677129	−	0.735864i	$-0.736776\pi$
$97$	− 1406.00i	− 1.47173i	0.677129	−	0.735864i	$-0.263224\pi$
$98$	0	0
$99$	−252.000	−0.255828
$100$	0	0
$101$	990.000	0.975333	0.487667	−	0.873030i	$-0.337848\pi$
$101$	990.000	0.975333	0.487667	+	0.873030i	$0.337848\pi$
$102$	0	0
$103$	736.000i	0.704080i	0.935985	+	0.352040i	$0.114512\pi$
$103$	736.000i	0.704080i	−0.935985	+	0.352040i	$0.885488\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1212.00i	− 1.09503i	−0.836795	−	0.547516i	$-0.815573\pi$
$107$	− 1212.00i	− 1.09503i	0.836795	−	0.547516i	$-0.184427\pi$
$108$	0	0
$109$	1834.00	1.61161	0.805804	−	0.592182i	$-0.201733\pi$
$109$	1834.00	1.61161	0.805804	+	0.592182i	$0.201733\pi$
$110$	0	0
$111$	810.000	0.692629
$112$	0	0
$113$	2046.00i	1.70329i	0.524121	+	0.851644i	$0.324394\pi$
$113$	2046.00i	1.70329i	−0.524121	+	0.851644i	$0.675606\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 234.000i	− 0.184900i
$118$	0	0
$119$	992.000	0.764172
$120$	0	0
$121$	−547.000	−0.410969
$122$	0	0
$123$	− 846.000i	− 0.620173i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1176.00i	− 0.821678i	−0.911708	−	0.410839i	$-0.865236\pi$
$127$	− 1176.00i	− 0.821678i	0.911708	−	0.410839i	$-0.134764\pi$
$128$	0	0
$129$	228.000	0.155615
$130$	0	0
$131$	−12.0000	−0.00800340	−0.00400170	−	0.999992i	$-0.501274\pi$
$131$	−12.0000	−0.00800340	−0.00400170	+	0.999992i	$0.501274\pi$
$132$	0	0
$133$	− 1088.00i	− 0.709335i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 790.000i	− 0.492659i	−0.969186	−	0.246329i	$-0.920775\pi$
$137$	− 790.000i	− 0.492659i	0.969186	−	0.246329i	$-0.0792245\pi$
$138$	0	0
$139$	−924.000	−0.563832	−0.281916	−	0.959439i	$-0.590970\pi$
$139$	−924.000	−0.563832	−0.281916	+	0.959439i	$0.590970\pi$
$140$	0	0
$141$	840.000	0.501708
$142$	0	0
$143$	728.000i	0.425723i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 261.000i	− 0.146442i
$148$	0	0
$149$	−3022.00	−1.66156	−0.830778	−	0.556604i	$-0.812104\pi$
$149$	−3022.00	−1.66156	−0.830778	+	0.556604i	$0.812104\pi$
$150$	0	0
$151$	−1736.00	−0.935587	−0.467794	−	0.883838i	$-0.654951\pi$
$151$	−1736.00	−0.935587	−0.467794	+	0.883838i	$0.654951\pi$
$152$	0	0
$153$	558.000i	0.294847i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 1322.00i	− 0.672020i	−0.941858	−	0.336010i	$-0.890922\pi$
$157$	− 1322.00i	− 0.672020i	0.941858	−	0.336010i	$-0.109078\pi$
$158$	0	0
$159$	630.000	0.314228
$160$	0	0
$161$	3328.00	1.62909
$162$	0	0
$163$	− 908.000i	− 0.436319i	−0.975913	−	0.218160i	$-0.929995\pi$
$163$	− 908.000i	− 0.436319i	0.975913	−	0.218160i	$-0.0700054\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1296.00i	− 0.600524i	−0.953857	−	0.300262i	$-0.902926\pi$
$167$	− 1296.00i	− 0.600524i	0.953857	−	0.300262i	$-0.0970741\pi$
$168$	0	0
$169$	1521.00	0.692308
$170$	0	0
$171$	612.000	0.273689
$172$	0	0
$173$	− 2134.00i	− 0.937832i	−0.883243	−	0.468916i	$-0.844645\pi$
$173$	− 2134.00i	− 0.937832i	0.883243	−	0.468916i	$-0.155355\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 588.000i	− 0.249699i
$178$	0	0
$179$	1612.00	0.673109	0.336555	−	0.941664i	$-0.390738\pi$
$179$	1612.00	0.673109	0.336555	+	0.941664i	$0.390738\pi$
$180$	0	0
$181$	3086.00	1.26730	0.633648	−	0.773621i	$-0.281557\pi$
$181$	3086.00	1.26730	0.633648	+	0.773621i	$0.281557\pi$
$182$	0	0
$183$	− 2226.00i	− 0.899184i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1736.00i	− 0.678871i
$188$	0	0
$189$	−432.000	−0.166261
$190$	0	0
$191$	4208.00	1.59414	0.797069	−	0.603889i	$-0.206383\pi$
$191$	4208.00	1.59414	0.797069	+	0.603889i	$0.206383\pi$
$192$	0	0
$193$	− 2818.00i	− 1.05101i	−0.850792	−	0.525503i	$-0.823877\pi$
$193$	− 2818.00i	− 1.05101i	0.850792	−	0.525503i	$-0.176123\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 418.000i	− 0.151174i	−0.997139	−	0.0755870i	$-0.975917\pi$
$197$	− 418.000i	− 0.151174i	0.997139	−	0.0755870i	$-0.0240831\pi$
$198$	0	0
$199$	−3352.00	−1.19406	−0.597028	−	0.802221i	$-0.703652\pi$
$199$	−3352.00	−1.19406	−0.597028	+	0.802221i	$0.703652\pi$
$200$	0	0
$201$	−2508.00	−0.880103
$202$	0	0
$203$	928.000i	0.320851i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1872.00i	0.628565i
$208$	0	0
$209$	−1904.00	−0.630155
$210$	0	0
$211$	4276.00	1.39513	0.697564	−	0.716523i	$-0.254267\pi$
$211$	4276.00	1.39513	0.697564	+	0.716523i	$0.254267\pi$
$212$	0	0
$213$	− 1512.00i	− 0.486387i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2560.00i	− 0.800848i
$218$	0	0
$219$	3186.00	0.983059
$220$	0	0
$221$	1612.00	0.490655
$222$	0	0
$223$	4712.00i	1.41497i	0.706727	+	0.707486i	$0.250171\pi$
$223$	4712.00i	1.41497i	−0.706727	+	0.707486i	$0.749829\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	732.000i	0.214029i	0.994257	+	0.107014i	$0.0341291\pi$
$227$	732.000i	0.214029i	−0.994257	+	0.107014i	$0.965871\pi$
$228$	0	0
$229$	5186.00	1.49651	0.748254	−	0.663412i	$-0.230892\pi$
$229$	5186.00	1.49651	0.748254	+	0.663412i	$0.230892\pi$
$230$	0	0
$231$	1344.00	0.382808
$232$	0	0
$233$	3798.00i	1.06788i	0.845523	+	0.533938i	$0.179289\pi$
$233$	3798.00i	1.06788i	−0.845523	+	0.533938i	$0.820711\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2304.00i	− 0.631481i
$238$	0	0
$239$	−3120.00	−0.844419	−0.422209	−	0.906498i	$-0.638745\pi$
$239$	−3120.00	−0.844419	−0.422209	+	0.906498i	$0.638745\pi$
$240$	0	0
$241$	1490.00	0.398255	0.199127	−	0.979974i	$-0.436189\pi$
$241$	1490.00	0.398255	0.199127	+	0.979974i	$0.436189\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1768.00i	− 0.455446i
$248$	0	0
$249$	−3156.00	−0.803226
$250$	0	0
$251$	5292.00	1.33079	0.665395	−	0.746492i	$-0.268263\pi$
$251$	5292.00	1.33079	0.665395	+	0.746492i	$0.268263\pi$
$252$	0	0
$253$	− 5824.00i	− 1.44724i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3918.00i	− 0.950965i	−0.879725	−	0.475483i	$-0.842273\pi$
$257$	− 3918.00i	− 0.950965i	0.879725	−	0.475483i	$-0.157727\pi$
$258$	0	0
$259$	−4320.00	−1.03642
$260$	0	0
$261$	−522.000	−0.123797
$262$	0	0
$263$	6624.00i	1.55305i	0.630084	+	0.776527i	$0.283021\pi$
$263$	6624.00i	1.55305i	−0.630084	+	0.776527i	$0.716979\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2178.00i	− 0.499219i
$268$	0	0
$269$	2954.00	0.669549	0.334774	−	0.942298i	$-0.391340\pi$
$269$	2954.00	0.669549	0.334774	+	0.942298i	$0.391340\pi$
$270$	0	0
$271$	6576.00	1.47404	0.737018	−	0.675874i	$-0.236233\pi$
$271$	6576.00	1.47404	0.737018	+	0.675874i	$0.236233\pi$
$272$	0	0
$273$	1248.00i	0.276675i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4478.00i	0.971325i	0.874146	+	0.485662i	$0.161422\pi$
$277$	4478.00i	0.971325i	−0.874146	+	0.485662i	$0.838578\pi$
$278$	0	0
$279$	1440.00	0.308998
$280$	0	0
$281$	−6358.00	−1.34977	−0.674887	−	0.737921i	$-0.735808\pi$
$281$	−6358.00	−1.34977	−0.674887	+	0.737921i	$0.735808\pi$
$282$	0	0
$283$	860.000i	0.180642i	0.995913	+	0.0903210i	$0.0287893\pi$
$283$	860.000i	0.180642i	−0.995913	+	0.0903210i	$0.971211\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4512.00i	0.927996i
$288$	0	0
$289$	1069.00	0.217586
$290$	0	0
$291$	−4218.00	−0.849703
$292$	0	0
$293$	5794.00i	1.15525i	0.816301	+	0.577626i	$0.196021\pi$
$293$	5794.00i	1.15525i	−0.816301	+	0.577626i	$0.803979\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	756.000i	0.147702i
$298$	0	0
$299$	5408.00	1.04600
$300$	0	0
$301$	−1216.00	−0.232854
$302$	0	0
$303$	− 2970.00i	− 0.563109i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6860.00i	1.27531i	0.770321	+	0.637656i	$0.220096\pi$
$307$	6860.00i	1.27531i	−0.770321	+	0.637656i	$0.779904\pi$
$308$	0	0
$309$	2208.00	0.406501
$310$	0	0
$311$	6248.00	1.13920	0.569601	−	0.821922i	$-0.307098\pi$
$311$	6248.00	1.13920	0.569601	+	0.821922i	$0.307098\pi$
$312$	0	0
$313$	− 11018.0i	− 1.98969i	−0.101388	−	0.994847i	$-0.532328\pi$
$313$	− 11018.0i	− 1.98969i	0.101388	−	0.994847i	$-0.467672\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 954.000i	− 0.169028i	−0.996422	−	0.0845142i	$-0.973066\pi$
$317$	− 954.000i	− 0.169028i	0.996422	−	0.0845142i	$-0.0269338\pi$
$318$	0	0
$319$	1624.00	0.285036
$320$	0	0
$321$	−3636.00	−0.632217
$322$	0	0
$323$	4216.00i	0.726268i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5502.00i	− 0.930463i
$328$	0	0
$329$	−4480.00	−0.750731
$330$	0	0
$331$	−9396.00	−1.56027	−0.780137	−	0.625608i	$-0.784851\pi$
$331$	−9396.00	−1.56027	−0.780137	+	0.625608i	$0.784851\pi$
$332$	0	0
$333$	− 2430.00i	− 0.399889i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5074.00i	0.820173i	0.912047	+	0.410087i	$0.134502\pi$
$337$	5074.00i	0.820173i	−0.912047	+	0.410087i	$0.865498\pi$
$338$	0	0
$339$	6138.00	0.983394
$340$	0	0
$341$	−4480.00	−0.711453
$342$	0	0
$343$	6880.00i	1.08305i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3916.00i	− 0.605827i	−0.953018	−	0.302913i	$-0.902041\pi$
$347$	− 3916.00i	− 0.605827i	0.953018	−	0.302913i	$-0.0979593\pi$
$348$	0	0
$349$	1818.00	0.278840	0.139420	−	0.990233i	$-0.455476\pi$
$349$	1818.00	0.278840	0.139420	+	0.990233i	$0.455476\pi$
$350$	0	0
$351$	−702.000	−0.106752
$352$	0	0
$353$	7118.00i	1.07324i	0.843825	+	0.536619i	$0.180299\pi$
$353$	7118.00i	1.07324i	−0.843825	+	0.536619i	$0.819701\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2976.00i	− 0.441195i
$358$	0	0
$359$	5304.00	0.779762	0.389881	−	0.920865i	$-0.372516\pi$
$359$	5304.00	0.779762	0.389881	+	0.920865i	$0.372516\pi$
$360$	0	0
$361$	−2235.00	−0.325849
$362$	0	0
$363$	1641.00i	0.237273i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 5672.00i	− 0.806747i	−0.915036	−	0.403373i	$-0.867838\pi$
$367$	− 5672.00i	− 0.806747i	0.915036	−	0.403373i	$-0.132162\pi$
$368$	0	0
$369$	−2538.00	−0.358057
$370$	0	0
$371$	−3360.00	−0.470195
$372$	0	0
$373$	− 7774.00i	− 1.07915i	−0.841938	−	0.539574i	$-0.818585\pi$
$373$	− 7774.00i	− 1.07915i	0.841938	−	0.539574i	$-0.181415\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1508.00i	0.206010i
$378$	0	0
$379$	−5516.00	−0.747593	−0.373797	−	0.927511i	$-0.621944\pi$
$379$	−5516.00	−0.747593	−0.373797	+	0.927511i	$0.621944\pi$
$380$	0	0
$381$	−3528.00	−0.474396
$382$	0	0
$383$	− 7128.00i	− 0.950976i	−0.879722	−	0.475488i	$-0.842272\pi$
$383$	− 7128.00i	− 0.950976i	0.879722	−	0.475488i	$-0.157728\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 684.000i	− 0.0898441i
$388$	0	0
$389$	10722.0	1.39750	0.698749	−	0.715367i	$-0.253740\pi$
$389$	10722.0	1.39750	0.698749	+	0.715367i	$0.253740\pi$
$390$	0	0
$391$	−12896.0	−1.66798
$392$	0	0
$393$	36.0000i	0.00462076i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 12122.0i	− 1.53246i	−0.642568	−	0.766229i	$-0.722131\pi$
$397$	− 12122.0i	− 1.53246i	0.642568	−	0.766229i	$-0.277869\pi$
$398$	0	0
$399$	−3264.00	−0.409535
$400$	0	0
$401$	10482.0	1.30535	0.652676	−	0.757637i	$-0.273646\pi$
$401$	10482.0	1.30535	0.652676	+	0.757637i	$0.273646\pi$
$402$	0	0
$403$	− 4160.00i	− 0.514204i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7560.00i	0.920726i
$408$	0	0
$409$	−3850.00	−0.465453	−0.232726	−	0.972542i	$-0.574765\pi$
$409$	−3850.00	−0.465453	−0.232726	+	0.972542i	$0.574765\pi$
$410$	0	0
$411$	−2370.00	−0.284437
$412$	0	0
$413$	3136.00i	0.373638i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2772.00i	0.325529i
$418$	0	0
$419$	−5796.00	−0.675783	−0.337892	−	0.941185i	$-0.609714\pi$
$419$	−5796.00	−0.675783	−0.337892	+	0.941185i	$0.609714\pi$
$420$	0	0
$421$	3294.00	0.381330	0.190665	−	0.981655i	$-0.438936\pi$
$421$	3294.00	0.381330	0.190665	+	0.981655i	$0.438936\pi$
$422$	0	0
$423$	− 2520.00i	− 0.289661i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11872.0i	1.34549i
$428$	0	0
$429$	2184.00	0.245791
$430$	0	0
$431$	−1696.00	−0.189544	−0.0947720	−	0.995499i	$-0.530212\pi$
$431$	−1696.00	−0.189544	−0.0947720	+	0.995499i	$0.530212\pi$
$432$	0	0
$433$	12334.0i	1.36890i	0.729059	+	0.684451i	$0.239958\pi$
$433$	12334.0i	1.36890i	−0.729059	+	0.684451i	$0.760042\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14144.0i	1.54828i
$438$	0	0
$439$	376.000	0.0408781	0.0204391	−	0.999791i	$-0.493494\pi$
$439$	376.000	0.0408781	0.0204391	+	0.999791i	$0.493494\pi$
$440$	0	0
$441$	−783.000	−0.0845481
$442$	0	0
$443$	8028.00i	0.860997i	0.902591	+	0.430499i	$0.141662\pi$
$443$	8028.00i	0.860997i	−0.902591	+	0.430499i	$0.858338\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9066.00i	0.959300i
$448$	0	0
$449$	−8898.00	−0.935240	−0.467620	−	0.883930i	$-0.654888\pi$
$449$	−8898.00	−0.935240	−0.467620	+	0.883930i	$0.654888\pi$
$450$	0	0
$451$	7896.00	0.824408
$452$	0	0
$453$	5208.00i	0.540162i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10330.0i	1.05737i	0.848819	+	0.528684i	$0.177314\pi$
$457$	10330.0i	1.05737i	−0.848819	+	0.528684i	$0.822686\pi$
$458$	0	0
$459$	1674.00	0.170230
$460$	0	0
$461$	1878.00	0.189734	0.0948668	−	0.995490i	$-0.469757\pi$
$461$	1878.00	0.189734	0.0948668	+	0.995490i	$0.469757\pi$
$462$	0	0
$463$	− 13224.0i	− 1.32737i	−0.748013	−	0.663684i	$-0.768992\pi$
$463$	− 13224.0i	− 1.32737i	0.748013	−	0.663684i	$-0.231008\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8012.00i	0.793900i	0.917840	+	0.396950i	$0.129931\pi$
$467$	8012.00i	0.793900i	−0.917840	+	0.396950i	$0.870069\pi$
$468$	0	0
$469$	13376.0	1.31694
$470$	0	0
$471$	−3966.00	−0.387991
$472$	0	0
$473$	2128.00i	0.206862i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1890.00i	− 0.181420i
$478$	0	0
$479$	−1792.00	−0.170936	−0.0854682	−	0.996341i	$-0.527239\pi$
$479$	−1792.00	−0.170936	−0.0854682	+	0.996341i	$0.527239\pi$
$480$	0	0
$481$	−7020.00	−0.665456
$482$	0	0
$483$	− 9984.00i	− 0.940554i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 8272.00i	− 0.769692i	−0.922981	−	0.384846i	$-0.874255\pi$
$487$	− 8272.00i	− 0.769692i	0.922981	−	0.384846i	$-0.125745\pi$
$488$	0	0
$489$	−2724.00	−0.251909
$490$	0	0
$491$	−516.000	−0.0474272	−0.0237136	−	0.999719i	$-0.507549\pi$
$491$	−516.000	−0.0474272	−0.0237136	+	0.999719i	$0.507549\pi$
$492$	0	0
$493$	− 3596.00i	− 0.328511i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8064.00i	0.727807i
$498$	0	0
$499$	−14020.0	−1.25776	−0.628879	−	0.777503i	$-0.716486\pi$
$499$	−14020.0	−1.25776	−0.628879	+	0.777503i	$0.716486\pi$
$500$	0	0
$501$	−3888.00	−0.346713
$502$	0	0
$503$	1872.00i	0.165941i	0.996552	+	0.0829705i	$0.0264407\pi$
$503$	1872.00i	0.165941i	−0.996552	+	0.0829705i	$0.973559\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4563.00i	− 0.399704i
$508$	0	0
$509$	−8678.00	−0.755689	−0.377844	−	0.925869i	$-0.623335\pi$
$509$	−8678.00	−0.755689	−0.377844	+	0.925869i	$0.623335\pi$
$510$	0	0
$511$	−16992.0	−1.47100
$512$	0	0
$513$	− 1836.00i	− 0.158014i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7840.00i	0.666930i
$518$	0	0
$519$	−6402.00	−0.541458
$520$	0	0
$521$	18074.0	1.51984	0.759920	−	0.650017i	$-0.225238\pi$
$521$	18074.0	1.51984	0.759920	+	0.650017i	$0.225238\pi$
$522$	0	0
$523$	− 20852.0i	− 1.74339i	−0.490047	−	0.871696i	$-0.663020\pi$
$523$	− 20852.0i	− 1.74339i	0.490047	−	0.871696i	$-0.336980\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9920.00i	0.819966i
$528$	0	0
$529$	−31097.0	−2.55585
$530$	0	0
$531$	−1764.00	−0.144164
$532$	0	0
$533$	7332.00i	0.595843i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4836.00i	− 0.388620i
$538$	0	0
$539$	2436.00	0.194668
$540$	0	0
$541$	−12410.0	−0.986225	−0.493112	−	0.869966i	$-0.664141\pi$
$541$	−12410.0	−0.986225	−0.493112	+	0.869966i	$0.664141\pi$
$542$	0	0
$543$	− 9258.00i	− 0.731674i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 3620.00i	− 0.282962i	−0.989941	−	0.141481i	$-0.954814\pi$
$547$	− 3620.00i	− 0.282962i	0.989941	−	0.141481i	$-0.0451864\pi$
$548$	0	0
$549$	−6678.00	−0.519144
$550$	0	0
$551$	−3944.00	−0.304937
$552$	0	0
$553$	12288.0i	0.944917i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11734.0i	0.892613i	0.894880	+	0.446307i	$0.147261\pi$
$557$	11734.0i	0.892613i	−0.894880	+	0.446307i	$0.852739\pi$
$558$	0	0
$559$	−1976.00	−0.149510
$560$	0	0
$561$	−5208.00	−0.391946
$562$	0	0
$563$	− 1372.00i	− 0.102705i	−0.998681	−	0.0513525i	$-0.983647\pi$
$563$	− 1372.00i	− 0.102705i	0.998681	−	0.0513525i	$-0.0163532\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1296.00i	0.0959910i
$568$	0	0
$569$	−18922.0	−1.39412	−0.697058	−	0.717015i	$-0.745508\pi$
$569$	−18922.0	−1.39412	−0.697058	+	0.717015i	$0.745508\pi$
$570$	0	0
$571$	−14596.0	−1.06974	−0.534872	−	0.844933i	$-0.679640\pi$
$571$	−14596.0	−1.06974	−0.534872	+	0.844933i	$0.679640\pi$
$572$	0	0
$573$	− 12624.0i	− 0.920376i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2302.00i	− 0.166089i	−0.996546	−	0.0830446i	$-0.973536\pi$
$577$	− 2302.00i	− 0.166089i	0.996546	−	0.0830446i	$-0.0264644\pi$
$578$	0	0
$579$	−8454.00	−0.606798
$580$	0	0
$581$	16832.0	1.20191
$582$	0	0
$583$	5880.00i	0.417710i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 23292.0i	− 1.63776i	−0.573966	−	0.818879i	$-0.694596\pi$
$587$	− 23292.0i	− 1.63776i	0.573966	−	0.818879i	$-0.305404\pi$
$588$	0	0
$589$	10880.0	0.761125
$590$	0	0
$591$	−1254.00	−0.0872803
$592$	0	0
$593$	16542.0i	1.14553i	0.819720	+	0.572764i	$0.194129\pi$
$593$	16542.0i	1.14553i	−0.819720	+	0.572764i	$0.805871\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10056.0i	0.689388i
$598$	0	0
$599$	7464.00	0.509133	0.254567	−	0.967055i	$-0.418067\pi$
$599$	7464.00	0.509133	0.254567	+	0.967055i	$0.418067\pi$
$600$	0	0
$601$	−17270.0	−1.17214	−0.586072	−	0.810259i	$-0.699326\pi$
$601$	−17270.0	−1.17214	−0.586072	+	0.810259i	$0.699326\pi$
$602$	0	0
$603$	7524.00i	0.508128i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 984.000i	− 0.0657979i	−0.999459	−	0.0328990i	$-0.989526\pi$
$607$	− 984.000i	− 0.0657979i	0.999459	−	0.0328990i	$-0.0104740\pi$
$608$	0	0
$609$	2784.00	0.185244
$610$	0	0
$611$	−7280.00	−0.482025
$612$	0	0
$613$	− 7278.00i	− 0.479536i	−0.970830	−	0.239768i	$-0.922929\pi$
$613$	− 7278.00i	− 0.479536i	0.970830	−	0.239768i	$-0.0770714\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18090.0i	1.18035i	0.807275	+	0.590175i	$0.200941\pi$
$617$	18090.0i	1.18035i	−0.807275	+	0.590175i	$0.799059\pi$
$618$	0	0
$619$	24740.0	1.60644	0.803219	−	0.595684i	$-0.203119\pi$
$619$	24740.0	1.60644	0.803219	+	0.595684i	$0.203119\pi$
$620$	0	0
$621$	5616.00	0.362902
$622$	0	0
$623$	11616.0i	0.747007i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5712.00i	0.363820i
$628$	0	0
$629$	16740.0	1.06116
$630$	0	0
$631$	−19720.0	−1.24412	−0.622061	−	0.782969i	$-0.713704\pi$
$631$	−19720.0	−1.24412	−0.622061	+	0.782969i	$0.713704\pi$
$632$	0	0
$633$	− 12828.0i	− 0.805477i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2262.00i	0.140697i
$638$	0	0
$639$	−4536.00	−0.280816
$640$	0	0
$641$	−16542.0	−1.01930	−0.509649	−	0.860383i	$-0.670225\pi$
$641$	−16542.0	−1.01930	−0.509649	+	0.860383i	$0.670225\pi$
$642$	0	0
$643$	− 10092.0i	− 0.618957i	−0.950906	−	0.309479i	$-0.899845\pi$
$643$	− 10092.0i	− 0.618957i	0.950906	−	0.309479i	$-0.100155\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14544.0i	0.883746i	0.897078	+	0.441873i	$0.145686\pi$
$647$	14544.0i	0.883746i	−0.897078	+	0.441873i	$0.854314\pi$
$648$	0	0
$649$	5488.00	0.331930
$650$	0	0
$651$	−7680.00	−0.462370
$652$	0	0
$653$	− 23062.0i	− 1.38206i	−0.722826	−	0.691030i	$-0.757157\pi$
$653$	− 23062.0i	− 1.38206i	0.722826	−	0.691030i	$-0.242843\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 9558.00i	− 0.567569i
$658$	0	0
$659$	−28020.0	−1.65630	−0.828152	−	0.560504i	$-0.810608\pi$
$659$	−28020.0	−1.65630	−0.828152	+	0.560504i	$0.810608\pi$
$660$	0	0
$661$	−6738.00	−0.396487	−0.198243	−	0.980153i	$-0.563524\pi$
$661$	−6738.00	−0.396487	−0.198243	+	0.980153i	$0.563524\pi$
$662$	0	0
$663$	− 4836.00i	− 0.283280i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12064.0i	− 0.700330i
$668$	0	0
$669$	14136.0	0.816935
$670$	0	0
$671$	20776.0	1.19530
$672$	0	0
$673$	14430.0i	0.826502i	0.910617	+	0.413251i	$0.135607\pi$
$673$	14430.0i	0.826502i	−0.910617	+	0.413251i	$0.864393\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 17890.0i	− 1.01561i	−0.861472	−	0.507805i	$-0.830457\pi$
$677$	− 17890.0i	− 1.01561i	0.861472	−	0.507805i	$-0.169543\pi$
$678$	0	0
$679$	22496.0	1.27145
$680$	0	0
$681$	2196.00	0.123570
$682$	0	0
$683$	10860.0i	0.608413i	0.952606	+	0.304207i	$0.0983914\pi$
$683$	10860.0i	0.608413i	−0.952606	+	0.304207i	$0.901609\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 15558.0i	− 0.864010i
$688$	0	0
$689$	−5460.00	−0.301900
$690$	0	0
$691$	8692.00	0.478523	0.239261	−	0.970955i	$-0.423095\pi$
$691$	8692.00	0.478523	0.239261	+	0.970955i	$0.423095\pi$
$692$	0	0
$693$	− 4032.00i	− 0.221014i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 17484.0i	− 0.950149i
$698$	0	0
$699$	11394.0	0.616539
$700$	0	0
$701$	−698.000	−0.0376078	−0.0188039	−	0.999823i	$-0.505986\pi$
$701$	−698.000	−0.0376078	−0.0188039	+	0.999823i	$0.505986\pi$
$702$	0	0
$703$	− 18360.0i	− 0.985008i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15840.0i	0.842609i
$708$	0	0
$709$	−2654.00	−0.140583	−0.0702913	−	0.997527i	$-0.522393\pi$
$709$	−2654.00	−0.140583	−0.0702913	+	0.997527i	$0.522393\pi$
$710$	0	0
$711$	−6912.00	−0.364585
$712$	0	0
$713$	33280.0i	1.74803i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	9360.00i	0.487525i
$718$	0	0
$719$	−28240.0	−1.46478	−0.732388	−	0.680887i	$-0.761594\pi$
$719$	−28240.0	−1.46478	−0.732388	+	0.680887i	$0.761594\pi$
$720$	0	0
$721$	−11776.0	−0.608268
$722$	0	0
$723$	− 4470.00i	− 0.229932i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	8320.00i	0.424445i	0.977221	+	0.212223i	$0.0680702\pi$
$727$	8320.00i	0.424445i	−0.977221	+	0.212223i	$0.931930\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	4712.00	0.238413
$732$	0	0
$733$	2154.00i	0.108540i	0.998526	+	0.0542700i	$0.0172832\pi$
$733$	2154.00i	0.108540i	−0.998526	+	0.0542700i	$0.982717\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 23408.0i	− 1.16994i
$738$	0	0
$739$	22380.0	1.11402	0.557011	−	0.830505i	$-0.311948\pi$
$739$	22380.0	1.11402	0.557011	+	0.830505i	$0.311948\pi$
$740$	0	0
$741$	−5304.00	−0.262952
$742$	0	0
$743$	− 5760.00i	− 0.284406i	−0.989837	−	0.142203i	$-0.954581\pi$
$743$	− 5760.00i	− 0.284406i	0.989837	−	0.142203i	$-0.0454186\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9468.00i	0.463743i
$748$	0	0
$749$	19392.0	0.946019
$750$	0	0
$751$	6192.00	0.300865	0.150432	−	0.988620i	$-0.451933\pi$
$751$	6192.00	0.300865	0.150432	+	0.988620i	$0.451933\pi$
$752$	0	0
$753$	− 15876.0i	− 0.768331i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 13666.0i	− 0.656142i	−0.944653	−	0.328071i	$-0.893602\pi$
$757$	− 13666.0i	− 0.656142i	0.944653	−	0.328071i	$-0.106398\pi$
$758$	0	0
$759$	−17472.0	−0.835564
$760$	0	0
$761$	−32022.0	−1.52536	−0.762678	−	0.646778i	$-0.776116\pi$
$761$	−32022.0	−1.52536	−0.762678	+	0.646778i	$0.776116\pi$
$762$	0	0
$763$	29344.0i	1.39230i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5096.00i	0.239903i
$768$	0	0
$769$	−22786.0	−1.06851	−0.534255	−	0.845323i	$-0.679408\pi$
$769$	−22786.0	−1.06851	−0.534255	+	0.845323i	$0.679408\pi$
$770$	0	0
$771$	−11754.0	−0.549040
$772$	0	0
$773$	− 8286.00i	− 0.385546i	−0.981243	−	0.192773i	$-0.938252\pi$
$773$	− 8286.00i	− 0.385546i	0.981243	−	0.192773i	$-0.0617480\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	12960.0i	0.598375i
$778$	0	0
$779$	−19176.0	−0.881966
$780$	0	0
$781$	14112.0	0.646565
$782$	0	0
$783$	1566.00i	0.0714742i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	25804.0i	1.16876i	0.811481	+	0.584379i	$0.198662\pi$
$787$	25804.0i	1.16876i	−0.811481	+	0.584379i	$0.801338\pi$
$788$	0	0
$789$	19872.0	0.896656
$790$	0	0
$791$	−32736.0	−1.47150
$792$	0	0
$793$	19292.0i	0.863908i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	17670.0i	0.785324i	0.919683	+	0.392662i	$0.128446\pi$
$797$	17670.0i	0.785324i	−0.919683	+	0.392662i	$0.871554\pi$
$798$	0	0
$799$	17360.0	0.768652
$800$	0	0
$801$	−6534.00	−0.288224
$802$	0	0
$803$	29736.0i	1.30680i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 8862.00i	− 0.386564i
$808$	0	0
$809$	7398.00	0.321508	0.160754	−	0.986995i	$-0.448607\pi$
$809$	7398.00	0.321508	0.160754	+	0.986995i	$0.448607\pi$
$810$	0	0
$811$	28108.0	1.21702	0.608511	−	0.793545i	$-0.291767\pi$
$811$	28108.0	1.21702	0.608511	+	0.793545i	$0.291767\pi$
$812$	0	0
$813$	− 19728.0i	− 0.851035i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 5168.00i	− 0.221304i
$818$	0	0
$819$	3744.00	0.159739
$820$	0	0
$821$	30830.0	1.31057	0.655283	−	0.755384i	$-0.272549\pi$
$821$	30830.0	1.31057	0.655283	+	0.755384i	$0.272549\pi$
$822$	0	0
$823$	5872.00i	0.248706i	0.992238	+	0.124353i	$0.0396855\pi$
$823$	5872.00i	0.248706i	−0.992238	+	0.124353i	$0.960314\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16308.0i	0.685713i	0.939388	+	0.342857i	$0.111394\pi$
$827$	16308.0i	0.685713i	−0.939388	+	0.342857i	$0.888606\pi$
$828$	0	0
$829$	−28294.0	−1.18539	−0.592697	−	0.805426i	$-0.701937\pi$
$829$	−28294.0	−1.18539	−0.592697	+	0.805426i	$0.701937\pi$
$830$	0	0
$831$	13434.0	0.560795
$832$	0	0
$833$	− 5394.00i	− 0.224359i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4320.00i	− 0.178400i
$838$	0	0
$839$	20536.0	0.845032	0.422516	−	0.906356i	$-0.361147\pi$
$839$	20536.0	0.845032	0.422516	+	0.906356i	$0.361147\pi$
$840$	0	0
$841$	−21025.0	−0.862069
$842$	0	0
$843$	19074.0i	0.779292i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 8752.00i	− 0.355044i
$848$	0	0
$849$	2580.00	0.104294
$850$	0	0
$851$	56160.0	2.26221
$852$	0	0
$853$	− 27710.0i	− 1.11228i	−0.831090	−	0.556139i	$-0.812282\pi$
$853$	− 27710.0i	− 1.11228i	0.831090	−	0.556139i	$-0.187718\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12858.0i	0.512510i	0.966609	+	0.256255i	$0.0824887\pi$
$857$	12858.0i	0.512510i	−0.966609	+	0.256255i	$0.917511\pi$
$858$	0	0
$859$	−3148.00	−0.125039	−0.0625194	−	0.998044i	$-0.519914\pi$
$859$	−3148.00	−0.125039	−0.0625194	+	0.998044i	$0.519914\pi$
$860$	0	0
$861$	13536.0	0.535779
$862$	0	0
$863$	48456.0i	1.91131i	0.294487	+	0.955656i	$0.404851\pi$
$863$	48456.0i	1.91131i	−0.294487	+	0.955656i	$0.595149\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 3207.00i	− 0.125623i
$868$	0	0
$869$	21504.0	0.839440
$870$	0	0
$871$	21736.0	0.845576
$872$	0	0
$873$	12654.0i	0.490576i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	9478.00i	0.364937i	0.983212	+	0.182468i	$0.0584087\pi$
$877$	9478.00i	0.364937i	−0.983212	+	0.182468i	$0.941591\pi$
$878$	0	0
$879$	17382.0	0.666986
$880$	0	0
$881$	8178.00	0.312740	0.156370	−	0.987699i	$-0.450021\pi$
$881$	8178.00	0.312740	0.156370	+	0.987699i	$0.450021\pi$
$882$	0	0
$883$	− 316.000i	− 0.0120433i	−0.999982	−	0.00602166i	$-0.998083\pi$
$883$	− 316.000i	− 0.0120433i	0.999982	−	0.00602166i	$-0.00191676\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6304.00i	0.238633i	0.992856	+	0.119317i	$0.0380703\pi$
$887$	6304.00i	0.238633i	−0.992856	+	0.119317i	$0.961930\pi$
$888$	0	0
$889$	18816.0	0.709863
$890$	0	0
$891$	2268.00	0.0852759
$892$	0	0
$893$	− 19040.0i	− 0.713493i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 16224.0i	− 0.603906i
$898$	0	0
$899$	−9280.00	−0.344277
$900$	0	0
$901$	13020.0	0.481420
$902$	0	0
$903$	3648.00i	0.134438i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1596.00i	− 0.0584281i	−0.999573	−	0.0292141i	$-0.990700\pi$
$907$	− 1596.00i	− 0.0584281i	0.999573	−	0.0292141i	$-0.00930045\pi$
$908$	0	0
$909$	−8910.00	−0.325111
$910$	0	0
$911$	25792.0	0.938010	0.469005	−	0.883196i	$-0.344613\pi$
$911$	25792.0	0.938010	0.469005	+	0.883196i	$0.344613\pi$
$912$	0	0
$913$	− 29456.0i	− 1.06775i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 192.000i	− 0.00691428i
$918$	0	0
$919$	−9736.00	−0.349468	−0.174734	−	0.984616i	$-0.555907\pi$
$919$	−9736.00	−0.349468	−0.174734	+	0.984616i	$0.555907\pi$
$920$	0	0
$921$	20580.0	0.736302
$922$	0	0
$923$	13104.0i	0.467306i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6624.00i	− 0.234693i
$928$	0	0
$929$	94.0000	0.00331974	0.00165987	−	0.999999i	$-0.499472\pi$
$929$	94.0000	0.00331974	0.00165987	+	0.999999i	$0.499472\pi$
$930$	0	0
$931$	−5916.00	−0.208259
$932$	0	0
$933$	− 18744.0i	− 0.657718i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 8678.00i	− 0.302559i	−0.988491	−	0.151280i	$-0.951661\pi$
$937$	− 8678.00i	− 0.302559i	0.988491	−	0.151280i	$-0.0483394\pi$
$938$	0	0
$939$	−33054.0	−1.14875
$940$	0	0
$941$	28406.0	0.984069	0.492035	−	0.870576i	$-0.336253\pi$
$941$	28406.0	0.984069	0.492035	+	0.870576i	$0.336253\pi$
$942$	0	0
$943$	− 58656.0i	− 2.02556i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 31988.0i	− 1.09765i	−0.835939	−	0.548823i	$-0.815076\pi$
$947$	− 31988.0i	− 1.09765i	0.835939	−	0.548823i	$-0.184924\pi$
$948$	0	0
$949$	−27612.0	−0.944493
$950$	0	0
$951$	−2862.00	−0.0975885
$952$	0	0
$953$	− 6714.00i	− 0.228214i	−0.993468	−	0.114107i	$-0.963599\pi$
$953$	− 6714.00i	− 0.228214i	0.993468	−	0.114107i	$-0.0364006\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 4872.00i	− 0.164566i
$958$	0	0
$959$	12640.0	0.425617
$960$	0	0
$961$	−4191.00	−0.140680
$962$	0	0
$963$	10908.0i	0.365011i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 15312.0i	− 0.509204i	−0.967046	−	0.254602i	$-0.918055\pi$
$967$	− 15312.0i	− 0.509204i	0.967046	−	0.254602i	$-0.0819445\pi$
$968$	0	0
$969$	12648.0	0.419311
$970$	0	0
$971$	8540.00	0.282247	0.141123	−	0.989992i	$-0.454929\pi$
$971$	8540.00	0.282247	0.141123	+	0.989992i	$0.454929\pi$
$972$	0	0
$973$	− 14784.0i	− 0.487105i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 8126.00i	− 0.266094i	−0.991110	−	0.133047i	$-0.957524\pi$
$977$	− 8126.00i	− 0.266094i	0.991110	−	0.133047i	$-0.0424761\pi$
$978$	0	0
$979$	20328.0	0.663622
$980$	0	0
$981$	−16506.0	−0.537203
$982$	0	0
$983$	1392.00i	0.0451657i	0.999745	+	0.0225829i	$0.00718896\pi$
$983$	1392.00i	0.0451657i	−0.999745	+	0.0225829i	$0.992811\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13440.0i	0.433435i
$988$	0	0
$989$	15808.0	0.508256
$990$	0	0
$991$	48832.0	1.56529	0.782644	−	0.622470i	$-0.213871\pi$
$991$	48832.0	1.56529	0.782644	+	0.622470i	$0.213871\pi$
$992$	0	0
$993$	28188.0i	0.900825i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	46926.0i	1.49063i	0.666711	+	0.745317i	$0.267702\pi$
$997$	46926.0i	1.49063i	−0.666711	+	0.745317i	$0.732298\pi$
$998$	0	0
$999$	−7290.00	−0.230876

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.o.49.1		2
4.3	odd	2		600.4.f.c.49.2		2
5.2	odd	4		1200.4.a.c.1.1		1
5.3	odd	4		240.4.a.l.1.1		1
5.4	even	2	inner	1200.4.f.o.49.2		2
12.11	even	2		1800.4.f.r.649.1		2
15.8	even	4		720.4.a.l.1.1		1
20.3	even	4		120.4.a.c.1.1	✓	1
20.7	even	4		600.4.a.q.1.1		1
20.19	odd	2		600.4.f.c.49.1		2
40.3	even	4		960.4.a.u.1.1		1
40.13	odd	4		960.4.a.h.1.1		1
60.23	odd	4		360.4.a.b.1.1		1
60.47	odd	4		1800.4.a.bb.1.1		1
60.59	even	2		1800.4.f.r.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.a.c.1.1	✓	1	20.3	even	4
240.4.a.l.1.1		1	5.3	odd	4
360.4.a.b.1.1		1	60.23	odd	4
600.4.a.q.1.1		1	20.7	even	4
600.4.f.c.49.1		2	20.19	odd	2
600.4.f.c.49.2		2	4.3	odd	2
720.4.a.l.1.1		1	15.8	even	4
960.4.a.h.1.1		1	40.13	odd	4
960.4.a.u.1.1		1	40.3	even	4
1200.4.a.c.1.1		1	5.2	odd	4
1200.4.f.o.49.1		2	1.1	even	1	trivial
1200.4.f.o.49.2		2	5.4	even	2	inner
1800.4.a.bb.1.1		1	60.47	odd	4
1800.4.f.r.649.1		2	12.11	even	2
1800.4.f.r.649.2		2	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +16.0000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +16.0000i q^{7} -9.00000 q^{9} +28.0000 q^{11} +26.0000i q^{13} -62.0000i q^{17} -68.0000 q^{19} +48.0000 q^{21} -208.000i q^{23} +27.0000i q^{27} +58.0000 q^{29} -160.000 q^{31} -84.0000i q^{33} +270.000i q^{37} +78.0000 q^{39} +282.000 q^{41} +76.0000i q^{43} +280.000i q^{47} +87.0000 q^{49} -186.000 q^{51} +210.000i q^{53} +204.000i q^{57} +196.000 q^{59} +742.000 q^{61} -144.000i q^{63} -836.000i q^{67} -624.000 q^{69} +504.000 q^{71} +1062.00i q^{73} +448.000i q^{77} +768.000 q^{79} +81.0000 q^{81} -1052.00i q^{83} -174.000i q^{87} +726.000 q^{89} -416.000 q^{91} +480.000i q^{93} -1406.00i q^{97} -252.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} + 56 q^{11} - 136 q^{19} + 96 q^{21} + 116 q^{29} - 320 q^{31} + 156 q^{39} + 564 q^{41} + 174 q^{49} - 372 q^{51} + 392 q^{59} + 1484 q^{61} - 1248 q^{69} + 1008 q^{71} + 1536 q^{79} + 162 q^{81} + 1452 q^{89} - 832 q^{91} - 504 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 + 56 * q^11 - 136 * q^19 + 96 * q^21 + 116 * q^29 - 320 * q^31 + 156 * q^39 + 564 * q^41 + 174 * q^49 - 372 * q^51 + 392 * q^59 + 1484 * q^61 - 1248 * q^69 + 1008 * q^71 + 1536 * q^79 + 162 * q^81 + 1452 * q^89 - 832 * q^91 - 504 * q^99

Introduction

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Modular forms

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Database

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