LMFDB - Embedded newform 1200.4.f.k.49.2

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 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -13.0000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} -13.0000i q^{7} -9.00000 q^{9} -6.00000 q^{11} +5.00000i q^{13} +78.0000i q^{17} +65.0000 q^{19} +39.0000 q^{21} -138.000i q^{23} -27.0000i q^{27} -66.0000 q^{29} -299.000 q^{31} -18.0000i q^{33} +214.000i q^{37} -15.0000 q^{39} +360.000 q^{41} -203.000i q^{43} +78.0000i q^{47} +174.000 q^{49} -234.000 q^{51} +636.000i q^{53} +195.000i q^{57} +786.000 q^{59} +467.000 q^{61} +117.000i q^{63} -217.000i q^{67} +414.000 q^{69} +360.000 q^{71} -286.000i q^{73} +78.0000i q^{77} +272.000 q^{79} +81.0000 q^{81} -498.000i q^{83} -198.000i q^{87} +65.0000 q^{91} -897.000i q^{93} +511.000i q^{97} +54.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} - 12 q^{11} + 130 q^{19} + 78 q^{21} - 132 q^{29} - 598 q^{31} - 30 q^{39} + 720 q^{41} + 348 q^{49} - 468 q^{51} + 1572 q^{59} + 934 q^{61} + 828 q^{69} + 720 q^{71} + 544 q^{79} + 162 q^{81} + 130 q^{91} + 108 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 - 12 * q^11 + 130 * q^19 + 78 * q^21 - 132 * q^29 - 598 * q^31 - 30 * q^39 + 720 * q^41 + 348 * q^49 - 468 * q^51 + 1572 * q^59 + 934 * q^61 + 828 * q^69 + 720 * q^71 + 544 * q^79 + 162 * q^81 + 130 * q^91 + 108 * 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
$7$	− 13.0000i	− 0.701934i	−0.936388	−	0.350967i	$-0.885853\pi$
$7$	− 13.0000i	− 0.701934i	0.936388	−	0.350967i	$-0.114147\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−6.00000	−0.164461	−0.0822304	−	0.996613i	$-0.526204\pi$
$11$	−6.00000	−0.164461	−0.0822304	+	0.996613i	$0.526204\pi$
$12$	0	0
$13$	5.00000i	0.106673i	0.998577	+	0.0533366i	$0.0169856\pi$
$13$	5.00000i	0.106673i	−0.998577	+	0.0533366i	$0.983014\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	78.0000i	1.11281i	0.830911	+	0.556405i	$0.187820\pi$
$17$	78.0000i	1.11281i	−0.830911	+	0.556405i	$0.812180\pi$
$18$	0	0
$19$	65.0000	0.784843	0.392422	−	0.919785i	$-0.371637\pi$
$19$	65.0000	0.784843	0.392422	+	0.919785i	$0.371637\pi$
$20$	0	0
$21$	39.0000	0.405262
$22$	0	0
$23$	− 138.000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 138.000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	−66.0000	−0.422617	−0.211308	−	0.977419i	$-0.567772\pi$
$29$	−66.0000	−0.422617	−0.211308	+	0.977419i	$0.567772\pi$
$30$	0	0
$31$	−299.000	−1.73232	−0.866161	−	0.499765i	$-0.833420\pi$
$31$	−299.000	−1.73232	−0.866161	+	0.499765i	$0.833420\pi$
$32$	0	0
$33$	− 18.0000i	− 0.0949514i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	214.000i	0.950848i	0.879757	+	0.475424i	$0.157705\pi$
$37$	214.000i	0.950848i	−0.879757	+	0.475424i	$0.842295\pi$
$38$	0	0
$39$	−15.0000	−0.0615878
$40$	0	0
$41$	360.000	1.37128	0.685641	−	0.727940i	$-0.259522\pi$
$41$	360.000	1.37128	0.685641	+	0.727940i	$0.259522\pi$
$42$	0	0
$43$	− 203.000i	− 0.719935i	−0.932965	−	0.359968i	$-0.882788\pi$
$43$	− 203.000i	− 0.719935i	0.932965	−	0.359968i	$-0.117212\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	78.0000i	0.242074i	0.992648	+	0.121037i	$0.0386219\pi$
$47$	78.0000i	0.242074i	−0.992648	+	0.121037i	$0.961378\pi$
$48$	0	0
$49$	174.000	0.507289
$50$	0	0
$51$	−234.000	−0.642481
$52$	0	0
$53$	636.000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	636.000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	195.000i	0.453129i
$58$	0	0
$59$	786.000	1.73438	0.867191	−	0.497976i	$-0.165923\pi$
$59$	786.000	1.73438	0.867191	+	0.497976i	$0.165923\pi$
$60$	0	0
$61$	467.000	0.980217	0.490108	−	0.871662i	$-0.336957\pi$
$61$	467.000	0.980217	0.490108	+	0.871662i	$0.336957\pi$
$62$	0	0
$63$	117.000i	0.233978i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 217.000i	− 0.395683i	−0.980234	−	0.197842i	$-0.936607\pi$
$67$	− 217.000i	− 0.395683i	0.980234	−	0.197842i	$-0.0633932\pi$
$68$	0	0
$69$	414.000	0.722315
$70$	0	0
$71$	360.000	0.601748	0.300874	−	0.953664i	$-0.402722\pi$
$71$	360.000	0.601748	0.300874	+	0.953664i	$0.402722\pi$
$72$	0	0
$73$	− 286.000i	− 0.458545i	−0.973362	−	0.229272i	$-0.926365\pi$
$73$	− 286.000i	− 0.458545i	0.973362	−	0.229272i	$-0.0736347\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	78.0000i	0.115441i
$78$	0	0
$79$	272.000	0.387372	0.193686	−	0.981064i	$-0.437956\pi$
$79$	272.000	0.387372	0.193686	+	0.981064i	$0.437956\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 498.000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 498.000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 198.000i	− 0.243998i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	65.0000	0.0748775
$92$	0	0
$93$	− 897.000i	− 1.00016i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	511.000i	0.534889i	0.963573	+	0.267444i	$0.0861791\pi$
$97$	511.000i	0.534889i	−0.963573	+	0.267444i	$0.913821\pi$
$98$	0	0
$99$	54.0000	0.0548202
$100$	0	0
$101$	−1812.00	−1.78516	−0.892578	−	0.450893i	$-0.851106\pi$
$101$	−1812.00	−1.78516	−0.892578	+	0.450893i	$0.851106\pi$
$102$	0	0
$103$	1708.00i	1.63392i	0.576691	+	0.816962i	$0.304344\pi$
$103$	1708.00i	1.63392i	−0.576691	+	0.816962i	$0.695656\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1236.00i	1.11672i	0.829600	+	0.558358i	$0.188568\pi$
$107$	1236.00i	1.11672i	−0.829600	+	0.558358i	$0.811432\pi$
$108$	0	0
$109$	1543.00	1.35590	0.677948	−	0.735110i	$-0.262870\pi$
$109$	1543.00	1.35590	0.677948	+	0.735110i	$0.262870\pi$
$110$	0	0
$111$	−642.000	−0.548972
$112$	0	0
$113$	1884.00i	1.56842i	0.620494	+	0.784212i	$0.286932\pi$
$113$	1884.00i	1.56842i	−0.620494	+	0.784212i	$0.713068\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 45.0000i	− 0.0355577i
$118$	0	0
$119$	1014.00	0.781120
$120$	0	0
$121$	−1295.00	−0.972953
$122$	0	0
$123$	1080.00i	0.791710i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2072.00i	1.44772i	0.689948	+	0.723859i	$0.257634\pi$
$127$	2072.00i	1.44772i	−0.689948	+	0.723859i	$0.742366\pi$
$128$	0	0
$129$	609.000	0.415655
$130$	0	0
$131$	2508.00	1.67271	0.836355	−	0.548188i	$-0.184682\pi$
$131$	2508.00	1.67271	0.836355	+	0.548188i	$0.184682\pi$
$132$	0	0
$133$	− 845.000i	− 0.550908i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1566.00i	− 0.976587i	−0.872679	−	0.488293i	$-0.837620\pi$
$137$	− 1566.00i	− 0.976587i	0.872679	−	0.488293i	$-0.162380\pi$
$138$	0	0
$139$	−196.000	−0.119601	−0.0598004	−	0.998210i	$-0.519046\pi$
$139$	−196.000	−0.119601	−0.0598004	+	0.998210i	$0.519046\pi$
$140$	0	0
$141$	−234.000	−0.139761
$142$	0	0
$143$	− 30.0000i	− 0.0175435i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	522.000i	0.292883i
$148$	0	0
$149$	1278.00	0.702670	0.351335	−	0.936250i	$-0.385728\pi$
$149$	1278.00	0.702670	0.351335	+	0.936250i	$0.385728\pi$
$150$	0	0
$151$	−1385.00	−0.746422	−0.373211	−	0.927747i	$-0.621743\pi$
$151$	−1385.00	−0.746422	−0.373211	+	0.927747i	$0.621743\pi$
$152$	0	0
$153$	− 702.000i	− 0.370937i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	481.000i	0.244509i	0.992499	+	0.122255i	$0.0390125\pi$
$157$	481.000i	0.244509i	−0.992499	+	0.122255i	$0.960988\pi$
$158$	0	0
$159$	−1908.00	−0.951662
$160$	0	0
$161$	−1794.00	−0.878180
$162$	0	0
$163$	2815.00i	1.35269i	0.736587	+	0.676343i	$0.236436\pi$
$163$	2815.00i	1.35269i	−0.736587	+	0.676343i	$0.763564\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1956.00i	0.906346i	0.891423	+	0.453173i	$0.149708\pi$
$167$	1956.00i	0.906346i	−0.891423	+	0.453173i	$0.850292\pi$
$168$	0	0
$169$	2172.00	0.988621
$170$	0	0
$171$	−585.000	−0.261614
$172$	0	0
$173$	2382.00i	1.04682i	0.852081	+	0.523411i	$0.175341\pi$
$173$	2382.00i	1.04682i	−0.852081	+	0.523411i	$0.824659\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2358.00i	1.00135i
$178$	0	0
$179$	4386.00	1.83142	0.915712	−	0.401834i	$-0.131627\pi$
$179$	4386.00	1.83142	0.915712	+	0.401834i	$0.131627\pi$
$180$	0	0
$181$	−2275.00	−0.934251	−0.467125	−	0.884191i	$-0.654710\pi$
$181$	−2275.00	−0.934251	−0.467125	+	0.884191i	$0.654710\pi$
$182$	0	0
$183$	1401.00i	0.565928i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 468.000i	− 0.183014i
$188$	0	0
$189$	−351.000	−0.135087
$190$	0	0
$191$	−714.000	−0.270488	−0.135244	−	0.990812i	$-0.543182\pi$
$191$	−714.000	−0.270488	−0.135244	+	0.990812i	$0.543182\pi$
$192$	0	0
$193$	1547.00i	0.576971i	0.957484	+	0.288486i	$0.0931518\pi$
$193$	1547.00i	0.576971i	−0.957484	+	0.288486i	$0.906848\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 438.000i	− 0.158407i	−0.996858	−	0.0792036i	$-0.974762\pi$
$197$	− 438.000i	− 0.158407i	0.996858	−	0.0792036i	$-0.0252377\pi$
$198$	0	0
$199$	437.000	0.155669	0.0778344	−	0.996966i	$-0.475199\pi$
$199$	437.000	0.155669	0.0778344	+	0.996966i	$0.475199\pi$
$200$	0	0
$201$	651.000	0.228448
$202$	0	0
$203$	858.000i	0.296649i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1242.00i	0.417029i
$208$	0	0
$209$	−390.000	−0.129076
$210$	0	0
$211$	−1625.00	−0.530188	−0.265094	−	0.964223i	$-0.585403\pi$
$211$	−1625.00	−0.530188	−0.265094	+	0.964223i	$0.585403\pi$
$212$	0	0
$213$	1080.00i	0.347420i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3887.00i	1.21598i
$218$	0	0
$219$	858.000	0.264741
$220$	0	0
$221$	−390.000	−0.118707
$222$	0	0
$223$	− 875.000i	− 0.262755i	−0.991332	−	0.131377i	$-0.958060\pi$
$223$	− 875.000i	− 0.262755i	0.991332	−	0.131377i	$-0.0419400\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4680.00i	− 1.36838i	−0.729303	−	0.684191i	$-0.760156\pi$
$227$	− 4680.00i	− 1.36838i	0.729303	−	0.684191i	$-0.239844\pi$
$228$	0	0
$229$	−1469.00	−0.423905	−0.211953	−	0.977280i	$-0.567982\pi$
$229$	−1469.00	−0.423905	−0.211953	+	0.977280i	$0.567982\pi$
$230$	0	0
$231$	−234.000	−0.0666497
$232$	0	0
$233$	3960.00i	1.11343i	0.830705	+	0.556713i	$0.187938\pi$
$233$	3960.00i	1.11343i	−0.830705	+	0.556713i	$0.812062\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	816.000i	0.223649i
$238$	0	0
$239$	2652.00	0.717756	0.358878	−	0.933385i	$-0.383159\pi$
$239$	2652.00	0.717756	0.358878	+	0.933385i	$0.383159\pi$
$240$	0	0
$241$	5753.00	1.53769	0.768845	−	0.639435i	$-0.220832\pi$
$241$	5753.00	1.53769	0.768845	+	0.639435i	$0.220832\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	325.000i	0.0837217i
$248$	0	0
$249$	1494.00	0.380235
$250$	0	0
$251$	−3588.00	−0.902281	−0.451141	−	0.892453i	$-0.648983\pi$
$251$	−3588.00	−0.902281	−0.451141	+	0.892453i	$0.648983\pi$
$252$	0	0
$253$	828.000i	0.205755i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	564.000i	0.136892i	0.997655	+	0.0684462i	$0.0218042\pi$
$257$	564.000i	0.136892i	−0.997655	+	0.0684462i	$0.978196\pi$
$258$	0	0
$259$	2782.00	0.667433
$260$	0	0
$261$	594.000	0.140872
$262$	0	0
$263$	− 1248.00i	− 0.292604i	−0.989240	−	0.146302i	$-0.953263\pi$
$263$	− 1248.00i	− 0.292604i	0.989240	−	0.146302i	$-0.0467372\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7266.00	1.64690	0.823450	−	0.567390i	$-0.192047\pi$
$269$	7266.00	1.64690	0.823450	+	0.567390i	$0.192047\pi$
$270$	0	0
$271$	−3224.00	−0.722672	−0.361336	−	0.932436i	$-0.617679\pi$
$271$	−3224.00	−0.722672	−0.361336	+	0.932436i	$0.617679\pi$
$272$	0	0
$273$	195.000i	0.0432305i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	151.000i	0.0327535i	0.999866	+	0.0163767i	$0.00521311\pi$
$277$	151.000i	0.0327535i	−0.999866	+	0.0163767i	$0.994787\pi$
$278$	0	0
$279$	2691.00	0.577441
$280$	0	0
$281$	7566.00	1.60623	0.803113	−	0.595826i	$-0.203175\pi$
$281$	7566.00	1.60623	0.803113	+	0.595826i	$0.203175\pi$
$282$	0	0
$283$	− 1469.00i	− 0.308562i	−0.988027	−	0.154281i	$-0.950694\pi$
$283$	− 1469.00i	− 0.308562i	0.988027	−	0.154281i	$-0.0493061\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4680.00i	− 0.962549i
$288$	0	0
$289$	−1171.00	−0.238347
$290$	0	0
$291$	−1533.00	−0.308818
$292$	0	0
$293$	− 306.000i	− 0.0610127i	−0.999535	−	0.0305063i	$-0.990288\pi$
$293$	− 306.000i	− 0.0610127i	0.999535	−	0.0305063i	$-0.00971197\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	162.000i	0.0316505i
$298$	0	0
$299$	690.000	0.133457
$300$	0	0
$301$	−2639.00	−0.505347
$302$	0	0
$303$	− 5436.00i	− 1.03066i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4661.00i	0.866506i	0.901272	+	0.433253i	$0.142634\pi$
$307$	4661.00i	0.866506i	−0.901272	+	0.433253i	$0.857366\pi$
$308$	0	0
$309$	−5124.00	−0.943347
$310$	0	0
$311$	−2514.00	−0.458379	−0.229189	−	0.973382i	$-0.573608\pi$
$311$	−2514.00	−0.458379	−0.229189	+	0.973382i	$0.573608\pi$
$312$	0	0
$313$	6707.00i	1.21119i	0.795774	+	0.605594i	$0.207065\pi$
$313$	6707.00i	1.21119i	−0.795774	+	0.605594i	$0.792935\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4632.00i	− 0.820691i	−0.911930	−	0.410345i	$-0.865408\pi$
$317$	− 4632.00i	− 0.820691i	0.911930	−	0.410345i	$-0.134592\pi$
$318$	0	0
$319$	396.000	0.0695039
$320$	0	0
$321$	−3708.00	−0.644736
$322$	0	0
$323$	5070.00i	0.873382i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4629.00i	0.782827i
$328$	0	0
$329$	1014.00	0.169920
$330$	0	0
$331$	988.000	0.164065	0.0820323	−	0.996630i	$-0.473859\pi$
$331$	988.000	0.164065	0.0820323	+	0.996630i	$0.473859\pi$
$332$	0	0
$333$	− 1926.00i	− 0.316949i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9079.00i	1.46755i	0.679392	+	0.733775i	$0.262244\pi$
$337$	9079.00i	1.46755i	−0.679392	+	0.733775i	$0.737756\pi$
$338$	0	0
$339$	−5652.00	−0.905530
$340$	0	0
$341$	1794.00	0.284899
$342$	0	0
$343$	− 6721.00i	− 1.05802i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 5994.00i	− 0.927305i	−0.886017	−	0.463652i	$-0.846539\pi$
$347$	− 5994.00i	− 0.927305i	0.886017	−	0.463652i	$-0.153461\pi$
$348$	0	0
$349$	286.000	0.0438660	0.0219330	−	0.999759i	$-0.493018\pi$
$349$	286.000	0.0438660	0.0219330	+	0.999759i	$0.493018\pi$
$350$	0	0
$351$	135.000	0.0205293
$352$	0	0
$353$	− 11466.0i	− 1.72882i	−0.502787	−	0.864410i	$-0.667692\pi$
$353$	− 11466.0i	− 1.72882i	0.502787	−	0.864410i	$-0.332308\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3042.00i	0.450980i
$358$	0	0
$359$	−624.000	−0.0917367	−0.0458683	−	0.998947i	$-0.514605\pi$
$359$	−624.000	−0.0917367	−0.0458683	+	0.998947i	$0.514605\pi$
$360$	0	0
$361$	−2634.00	−0.384021
$362$	0	0
$363$	− 3885.00i	− 0.561734i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8477.00i	1.20571i	0.797851	+	0.602855i	$0.205970\pi$
$367$	8477.00i	1.20571i	−0.797851	+	0.602855i	$0.794030\pi$
$368$	0	0
$369$	−3240.00	−0.457094
$370$	0	0
$371$	8268.00	1.15702
$372$	0	0
$373$	− 11257.0i	− 1.56264i	−0.624130	−	0.781321i	$-0.714546\pi$
$373$	− 11257.0i	− 1.56264i	0.624130	−	0.781321i	$-0.285454\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 330.000i	− 0.0450819i
$378$	0	0
$379$	−1213.00	−0.164400	−0.0822000	−	0.996616i	$-0.526195\pi$
$379$	−1213.00	−0.164400	−0.0822000	+	0.996616i	$0.526195\pi$
$380$	0	0
$381$	−6216.00	−0.835841
$382$	0	0
$383$	11076.0i	1.47769i	0.673873	+	0.738847i	$0.264630\pi$
$383$	11076.0i	1.47769i	−0.673873	+	0.738847i	$0.735370\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1827.00i	0.239978i
$388$	0	0
$389$	8016.00	1.04480	0.522400	−	0.852700i	$-0.325037\pi$
$389$	8016.00	1.04480	0.522400	+	0.852700i	$0.325037\pi$
$390$	0	0
$391$	10764.0	1.39222
$392$	0	0
$393$	7524.00i	0.965739i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 6851.00i	− 0.866100i	−0.901370	−	0.433050i	$-0.857437\pi$
$397$	− 6851.00i	− 0.866100i	0.901370	−	0.433050i	$-0.142563\pi$
$398$	0	0
$399$	2535.00	0.318067
$400$	0	0
$401$	12636.0	1.57360	0.786798	−	0.617211i	$-0.211738\pi$
$401$	12636.0	1.57360	0.786798	+	0.617211i	$0.211738\pi$
$402$	0	0
$403$	− 1495.00i	− 0.184792i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1284.00i	− 0.156377i
$408$	0	0
$409$	−10829.0	−1.30919	−0.654596	−	0.755979i	$-0.727161\pi$
$409$	−10829.0	−1.30919	−0.654596	+	0.755979i	$0.727161\pi$
$410$	0	0
$411$	4698.00	0.563833
$412$	0	0
$413$	− 10218.0i	− 1.21742i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 588.000i	− 0.0690515i
$418$	0	0
$419$	−6252.00	−0.728950	−0.364475	−	0.931213i	$-0.618752\pi$
$419$	−6252.00	−0.728950	−0.364475	+	0.931213i	$0.618752\pi$
$420$	0	0
$421$	−15730.0	−1.82098	−0.910491	−	0.413529i	$-0.864296\pi$
$421$	−15730.0	−1.82098	−0.910491	+	0.413529i	$0.864296\pi$
$422$	0	0
$423$	− 702.000i	− 0.0806913i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 6071.00i	− 0.688047i
$428$	0	0
$429$	90.0000	0.0101288
$430$	0	0
$431$	10062.0	1.12452	0.562262	−	0.826959i	$-0.309931\pi$
$431$	10062.0	1.12452	0.562262	+	0.826959i	$0.309931\pi$
$432$	0	0
$433$	707.000i	0.0784671i	0.999230	+	0.0392335i	$0.0124916\pi$
$433$	707.000i	0.0784671i	−0.999230	+	0.0392335i	$0.987508\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8970.00i	− 0.981907i
$438$	0	0
$439$	−12493.0	−1.35822	−0.679110	−	0.734037i	$-0.737634\pi$
$439$	−12493.0	−1.35822	−0.679110	+	0.734037i	$0.737634\pi$
$440$	0	0
$441$	−1566.00	−0.169096
$442$	0	0
$443$	5928.00i	0.635774i	0.948129	+	0.317887i	$0.102973\pi$
$443$	5928.00i	0.635774i	−0.948129	+	0.317887i	$0.897027\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3834.00i	0.405687i
$448$	0	0
$449$	2028.00	0.213156	0.106578	−	0.994304i	$-0.466011\pi$
$449$	2028.00	0.213156	0.106578	+	0.994304i	$0.466011\pi$
$450$	0	0
$451$	−2160.00	−0.225522
$452$	0	0
$453$	− 4155.00i	− 0.430947i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 17030.0i	− 1.74317i	−0.490242	−	0.871586i	$-0.663092\pi$
$457$	− 17030.0i	− 1.74317i	0.490242	−	0.871586i	$-0.336908\pi$
$458$	0	0
$459$	2106.00	0.214160
$460$	0	0
$461$	−1044.00	−0.105475	−0.0527374	−	0.998608i	$-0.516795\pi$
$461$	−1044.00	−0.105475	−0.0527374	+	0.998608i	$0.516795\pi$
$462$	0	0
$463$	− 9704.00i	− 0.974046i	−0.873389	−	0.487023i	$-0.838083\pi$
$463$	− 9704.00i	− 0.974046i	0.873389	−	0.487023i	$-0.161917\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4194.00i	− 0.415579i	−0.978174	−	0.207789i	$-0.933373\pi$
$467$	− 4194.00i	− 0.415579i	0.978174	−	0.207789i	$-0.0666268\pi$
$468$	0	0
$469$	−2821.00	−0.277743
$470$	0	0
$471$	−1443.00	−0.141168
$472$	0	0
$473$	1218.00i	0.118401i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5724.00i	− 0.549442i
$478$	0	0
$479$	−12174.0	−1.16126	−0.580631	−	0.814167i	$-0.697194\pi$
$479$	−12174.0	−1.16126	−0.580631	+	0.814167i	$0.697194\pi$
$480$	0	0
$481$	−1070.00	−0.101430
$482$	0	0
$483$	− 5382.00i	− 0.507018i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1651.00i	− 0.153622i	−0.997046	−	0.0768110i	$-0.975526\pi$
$487$	− 1651.00i	− 0.153622i	0.997046	−	0.0768110i	$-0.0244738\pi$
$488$	0	0
$489$	−8445.00	−0.780974
$490$	0	0
$491$	6456.00	0.593391	0.296696	−	0.954972i	$-0.404115\pi$
$491$	6456.00	0.593391	0.296696	+	0.954972i	$0.404115\pi$
$492$	0	0
$493$	− 5148.00i	− 0.470293i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4680.00i	− 0.422388i
$498$	0	0
$499$	−559.000	−0.0501489	−0.0250744	−	0.999686i	$-0.507982\pi$
$499$	−559.000	−0.0501489	−0.0250744	+	0.999686i	$0.507982\pi$
$500$	0	0
$501$	−5868.00	−0.523279
$502$	0	0
$503$	14208.0i	1.25945i	0.776818	+	0.629725i	$0.216832\pi$
$503$	14208.0i	1.25945i	−0.776818	+	0.629725i	$0.783168\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6516.00i	0.570781i
$508$	0	0
$509$	8082.00	0.703789	0.351894	−	0.936040i	$-0.385538\pi$
$509$	8082.00	0.703789	0.351894	+	0.936040i	$0.385538\pi$
$510$	0	0
$511$	−3718.00	−0.321868
$512$	0	0
$513$	− 1755.00i	− 0.151043i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 468.000i	− 0.0398116i
$518$	0	0
$519$	−7146.00	−0.604383
$520$	0	0
$521$	20502.0	1.72401	0.862005	−	0.506900i	$-0.169209\pi$
$521$	20502.0	1.72401	0.862005	+	0.506900i	$0.169209\pi$
$522$	0	0
$523$	− 2069.00i	− 0.172985i	−0.996253	−	0.0864924i	$-0.972434\pi$
$523$	− 2069.00i	− 0.172985i	0.996253	−	0.0864924i	$-0.0275658\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 23322.0i	− 1.92775i
$528$	0	0
$529$	−6877.00	−0.565217
$530$	0	0
$531$	−7074.00	−0.578127
$532$	0	0
$533$	1800.00i	0.146279i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	13158.0i	1.05737i
$538$	0	0
$539$	−1044.00	−0.0834291
$540$	0	0
$541$	1553.00	0.123417	0.0617086	−	0.998094i	$-0.480345\pi$
$541$	1553.00	0.123417	0.0617086	+	0.998094i	$0.480345\pi$
$542$	0	0
$543$	− 6825.00i	− 0.539390i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12584.0i	0.983643i	0.870696	+	0.491822i	$0.163669\pi$
$547$	12584.0i	0.983643i	−0.870696	+	0.491822i	$0.836331\pi$
$548$	0	0
$549$	−4203.00	−0.326739
$550$	0	0
$551$	−4290.00	−0.331688
$552$	0	0
$553$	− 3536.00i	− 0.271910i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 20754.0i	− 1.57877i	−0.613898	−	0.789385i	$-0.710399\pi$
$557$	− 20754.0i	− 1.57877i	0.613898	−	0.789385i	$-0.289601\pi$
$558$	0	0
$559$	1015.00	0.0767977
$560$	0	0
$561$	1404.00	0.105663
$562$	0	0
$563$	− 14370.0i	− 1.07571i	−0.843038	−	0.537854i	$-0.819235\pi$
$563$	− 14370.0i	− 1.07571i	0.843038	−	0.537854i	$-0.180765\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1053.00i	− 0.0779927i
$568$	0	0
$569$	−15774.0	−1.16218	−0.581090	−	0.813839i	$-0.697374\pi$
$569$	−15774.0	−1.16218	−0.581090	+	0.813839i	$0.697374\pi$
$570$	0	0
$571$	3055.00	0.223902	0.111951	−	0.993714i	$-0.464290\pi$
$571$	3055.00	0.223902	0.111951	+	0.993714i	$0.464290\pi$
$572$	0	0
$573$	− 2142.00i	− 0.156166i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	25021.0i	1.80526i	0.430412	+	0.902632i	$0.358368\pi$
$577$	25021.0i	1.80526i	−0.430412	+	0.902632i	$0.641632\pi$
$578$	0	0
$579$	−4641.00	−0.333115
$580$	0	0
$581$	−6474.00	−0.462284
$582$	0	0
$583$	− 3816.00i	− 0.271085i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 18828.0i	− 1.32388i	−0.749559	−	0.661938i	$-0.769734\pi$
$587$	− 18828.0i	− 1.32388i	0.749559	−	0.661938i	$-0.230266\pi$
$588$	0	0
$589$	−19435.0	−1.35960
$590$	0	0
$591$	1314.00	0.0914564
$592$	0	0
$593$	19968.0i	1.38278i	0.722483	+	0.691389i	$0.243001\pi$
$593$	19968.0i	1.38278i	−0.722483	+	0.691389i	$0.756999\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1311.00i	0.0898755i
$598$	0	0
$599$	9684.00	0.660563	0.330282	−	0.943882i	$-0.392856\pi$
$599$	9684.00	0.660563	0.330282	+	0.943882i	$0.392856\pi$
$600$	0	0
$601$	23243.0	1.57754	0.788770	−	0.614688i	$-0.210718\pi$
$601$	23243.0	1.57754	0.788770	+	0.614688i	$0.210718\pi$
$602$	0	0
$603$	1953.00i	0.131894i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 22984.0i	− 1.53689i	−0.639916	−	0.768445i	$-0.721031\pi$
$607$	− 22984.0i	− 1.53689i	0.639916	−	0.768445i	$-0.278969\pi$
$608$	0	0
$609$	−2574.00	−0.171271
$610$	0	0
$611$	−390.000	−0.0258228
$612$	0	0
$613$	13754.0i	0.906230i	0.891452	+	0.453115i	$0.149687\pi$
$613$	13754.0i	0.906230i	−0.891452	+	0.453115i	$0.850313\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9468.00i	− 0.617775i	−0.951099	−	0.308888i	$-0.900043\pi$
$617$	− 9468.00i	− 0.617775i	0.951099	−	0.308888i	$-0.0999567\pi$
$618$	0	0
$619$	−28099.0	−1.82455	−0.912273	−	0.409582i	$-0.865674\pi$
$619$	−28099.0	−1.82455	−0.912273	+	0.409582i	$0.865674\pi$
$620$	0	0
$621$	−3726.00	−0.240772
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1170.00i	− 0.0745220i
$628$	0	0
$629$	−16692.0	−1.05811
$630$	0	0
$631$	−14807.0	−0.934164	−0.467082	−	0.884214i	$-0.654695\pi$
$631$	−14807.0	−0.934164	−0.467082	+	0.884214i	$0.654695\pi$
$632$	0	0
$633$	− 4875.00i	− 0.306104i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	870.000i	0.0541141i
$638$	0	0
$639$	−3240.00	−0.200583
$640$	0	0
$641$	20844.0	1.28438	0.642191	−	0.766545i	$-0.278026\pi$
$641$	20844.0	1.28438	0.642191	+	0.766545i	$0.278026\pi$
$642$	0	0
$643$	− 3692.00i	− 0.226436i	−0.993570	−	0.113218i	$-0.963884\pi$
$643$	− 3692.00i	− 0.226436i	0.993570	−	0.113218i	$-0.0361158\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 20004.0i	− 1.21552i	−0.794123	−	0.607758i	$-0.792069\pi$
$647$	− 20004.0i	− 1.21552i	0.794123	−	0.607758i	$-0.207931\pi$
$648$	0	0
$649$	−4716.00	−0.285238
$650$	0	0
$651$	−11661.0	−0.702044
$652$	0	0
$653$	− 17862.0i	− 1.07044i	−0.844714	−	0.535218i	$-0.820230\pi$
$653$	− 17862.0i	− 1.07044i	0.844714	−	0.535218i	$-0.179770\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2574.00i	0.152848i
$658$	0	0
$659$	18552.0	1.09664	0.548318	−	0.836270i	$-0.315268\pi$
$659$	18552.0	1.09664	0.548318	+	0.836270i	$0.315268\pi$
$660$	0	0
$661$	−12382.0	−0.728599	−0.364300	−	0.931282i	$-0.618692\pi$
$661$	−12382.0	−0.728599	−0.364300	+	0.931282i	$0.618692\pi$
$662$	0	0
$663$	− 1170.00i	− 0.0685355i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9108.00i	0.528730i
$668$	0	0
$669$	2625.00	0.151702
$670$	0	0
$671$	−2802.00	−0.161207
$672$	0	0
$673$	− 1690.00i	− 0.0967975i	−0.998828	−	0.0483987i	$-0.984588\pi$
$673$	− 1690.00i	− 0.0967975i	0.998828	−	0.0483987i	$-0.0154118\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 5682.00i	− 0.322566i	−0.986908	−	0.161283i	$-0.948437\pi$
$677$	− 5682.00i	− 0.322566i	0.986908	−	0.161283i	$-0.0515631\pi$
$678$	0	0
$679$	6643.00	0.375456
$680$	0	0
$681$	14040.0	0.790035
$682$	0	0
$683$	− 31512.0i	− 1.76541i	−0.469930	−	0.882704i	$-0.655721\pi$
$683$	− 31512.0i	− 1.76541i	0.469930	−	0.882704i	$-0.344279\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 4407.00i	− 0.244742i
$688$	0	0
$689$	−3180.00	−0.175832
$690$	0	0
$691$	−20648.0	−1.13674	−0.568370	−	0.822773i	$-0.692426\pi$
$691$	−20648.0	−1.13674	−0.568370	+	0.822773i	$0.692426\pi$
$692$	0	0
$693$	− 702.000i	− 0.0384802i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	28080.0i	1.52598i
$698$	0	0
$699$	−11880.0	−0.642837
$700$	0	0
$701$	−11874.0	−0.639764	−0.319882	−	0.947457i	$-0.603643\pi$
$701$	−11874.0	−0.639764	−0.319882	+	0.947457i	$0.603643\pi$
$702$	0	0
$703$	13910.0i	0.746267i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	23556.0i	1.25306i
$708$	0	0
$709$	10699.0	0.566727	0.283363	−	0.959013i	$-0.408550\pi$
$709$	10699.0	0.566727	0.283363	+	0.959013i	$0.408550\pi$
$710$	0	0
$711$	−2448.00	−0.129124
$712$	0	0
$713$	41262.0i	2.16728i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7956.00i	0.414396i
$718$	0	0
$719$	−5598.00	−0.290362	−0.145181	−	0.989405i	$-0.546376\pi$
$719$	−5598.00	−0.290362	−0.145181	+	0.989405i	$0.546376\pi$
$720$	0	0
$721$	22204.0	1.14691
$722$	0	0
$723$	17259.0i	0.887786i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	22859.0i	1.16615i	0.812417	+	0.583077i	$0.198151\pi$
$727$	22859.0i	1.16615i	−0.812417	+	0.583077i	$0.801849\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	15834.0	0.801151
$732$	0	0
$733$	− 31642.0i	− 1.59444i	−0.603689	−	0.797220i	$-0.706303\pi$
$733$	− 31642.0i	− 1.59444i	0.603689	−	0.797220i	$-0.293697\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1302.00i	0.0650743i
$738$	0	0
$739$	−12220.0	−0.608281	−0.304141	−	0.952627i	$-0.598369\pi$
$739$	−12220.0	−0.608281	−0.304141	+	0.952627i	$0.598369\pi$
$740$	0	0
$741$	−975.000	−0.0483367
$742$	0	0
$743$	− 17892.0i	− 0.883437i	−0.897154	−	0.441719i	$-0.854369\pi$
$743$	− 17892.0i	− 0.883437i	0.897154	−	0.441719i	$-0.145631\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4482.00i	0.219529i
$748$	0	0
$749$	16068.0	0.783861
$750$	0	0
$751$	16648.0	0.808914	0.404457	−	0.914557i	$-0.367461\pi$
$751$	16648.0	0.808914	0.404457	+	0.914557i	$0.367461\pi$
$752$	0	0
$753$	− 10764.0i	− 0.520932i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 31733.0i	− 1.52359i	−0.647820	−	0.761794i	$-0.724319\pi$
$757$	− 31733.0i	− 1.52359i	0.647820	−	0.761794i	$-0.275681\pi$
$758$	0	0
$759$	−2484.00	−0.118792
$760$	0	0
$761$	−10068.0	−0.479586	−0.239793	−	0.970824i	$-0.577080\pi$
$761$	−10068.0	−0.479586	−0.239793	+	0.970824i	$0.577080\pi$
$762$	0	0
$763$	− 20059.0i	− 0.951749i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3930.00i	0.185012i
$768$	0	0
$769$	−24323.0	−1.14058	−0.570292	−	0.821442i	$-0.693170\pi$
$769$	−24323.0	−1.14058	−0.570292	+	0.821442i	$0.693170\pi$
$770$	0	0
$771$	−1692.00	−0.0790349
$772$	0	0
$773$	− 2184.00i	− 0.101621i	−0.998708	−	0.0508105i	$-0.983820\pi$
$773$	− 2184.00i	− 0.101621i	0.998708	−	0.0508105i	$-0.0161804\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8346.00i	0.385342i
$778$	0	0
$779$	23400.0	1.07624
$780$	0	0
$781$	−2160.00	−0.0989640
$782$	0	0
$783$	1782.00i	0.0813327i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 37531.0i	− 1.69992i	−0.526849	−	0.849959i	$-0.676627\pi$
$787$	− 37531.0i	− 1.69992i	0.526849	−	0.849959i	$-0.323373\pi$
$788$	0	0
$789$	3744.00	0.168935
$790$	0	0
$791$	24492.0	1.10093
$792$	0	0
$793$	2335.00i	0.104563i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 28644.0i	− 1.27305i	−0.771255	−	0.636526i	$-0.780371\pi$
$797$	− 28644.0i	− 1.27305i	0.771255	−	0.636526i	$-0.219629\pi$
$798$	0	0
$799$	−6084.00	−0.269382
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1716.00i	0.0754126i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21798.0i	0.950838i
$808$	0	0
$809$	17316.0	0.752532	0.376266	−	0.926512i	$-0.377208\pi$
$809$	17316.0	0.752532	0.376266	+	0.926512i	$0.377208\pi$
$810$	0	0
$811$	−425.000	−0.0184017	−0.00920084	−	0.999958i	$-0.502929\pi$
$811$	−425.000	−0.0184017	−0.00920084	+	0.999958i	$0.502929\pi$
$812$	0	0
$813$	− 9672.00i	− 0.417235i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 13195.0i	− 0.565036i
$818$	0	0
$819$	−585.000	−0.0249592
$820$	0	0
$821$	−25194.0	−1.07098	−0.535491	−	0.844541i	$-0.679874\pi$
$821$	−25194.0	−1.07098	−0.535491	+	0.844541i	$0.679874\pi$
$822$	0	0
$823$	33961.0i	1.43840i	0.694801	+	0.719202i	$0.255492\pi$
$823$	33961.0i	1.43840i	−0.694801	+	0.719202i	$0.744508\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1956.00i	− 0.0822452i	−0.999154	−	0.0411226i	$-0.986907\pi$
$827$	− 1956.00i	− 0.0822452i	0.999154	−	0.0411226i	$-0.0130934\pi$
$828$	0	0
$829$	47302.0	1.98174	0.990872	−	0.134803i	$-0.0430403\pi$
$829$	47302.0	1.98174	0.990872	+	0.134803i	$0.0430403\pi$
$830$	0	0
$831$	−453.000	−0.0189102
$832$	0	0
$833$	13572.0i	0.564516i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8073.00i	0.333386i
$838$	0	0
$839$	30726.0	1.26434	0.632169	−	0.774831i	$-0.282165\pi$
$839$	30726.0	1.26434	0.632169	+	0.774831i	$0.282165\pi$
$840$	0	0
$841$	−20033.0	−0.821395
$842$	0	0
$843$	22698.0i	0.927355i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	16835.0i	0.682949i
$848$	0	0
$849$	4407.00	0.178148
$850$	0	0
$851$	29532.0	1.18959
$852$	0	0
$853$	− 34489.0i	− 1.38439i	−0.721713	−	0.692193i	$-0.756645\pi$
$853$	− 34489.0i	− 1.38439i	0.721713	−	0.692193i	$-0.243355\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 8484.00i	− 0.338166i	−0.985602	−	0.169083i	$-0.945919\pi$
$857$	− 8484.00i	− 0.338166i	0.985602	−	0.169083i	$-0.0540805\pi$
$858$	0	0
$859$	−520.000	−0.0206544	−0.0103272	−	0.999947i	$-0.503287\pi$
$859$	−520.000	−0.0206544	−0.0103272	+	0.999947i	$0.503287\pi$
$860$	0	0
$861$	14040.0	0.555728
$862$	0	0
$863$	6864.00i	0.270745i	0.990795	+	0.135373i	$0.0432232\pi$
$863$	6864.00i	0.270745i	−0.990795	+	0.135373i	$0.956777\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 3513.00i	− 0.137610i
$868$	0	0
$869$	−1632.00	−0.0637075
$870$	0	0
$871$	1085.00	0.0422088
$872$	0	0
$873$	− 4599.00i	− 0.178296i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38857.0i	1.49613i	0.663624	+	0.748066i	$0.269017\pi$
$877$	38857.0i	1.49613i	−0.663624	+	0.748066i	$0.730983\pi$
$878$	0	0
$879$	918.000	0.0352257
$880$	0	0
$881$	−10080.0	−0.385475	−0.192738	−	0.981250i	$-0.561737\pi$
$881$	−10080.0	−0.385475	−0.192738	+	0.981250i	$0.561737\pi$
$882$	0	0
$883$	34549.0i	1.31672i	0.752702	+	0.658362i	$0.228750\pi$
$883$	34549.0i	1.31672i	−0.752702	+	0.658362i	$0.771250\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 18798.0i	− 0.711584i	−0.934565	−	0.355792i	$-0.884211\pi$
$887$	− 18798.0i	− 0.711584i	0.934565	−	0.355792i	$-0.115789\pi$
$888$	0	0
$889$	26936.0	1.01620
$890$	0	0
$891$	−486.000	−0.0182734
$892$	0	0
$893$	5070.00i	0.189990i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2070.00i	0.0770516i
$898$	0	0
$899$	19734.0	0.732109
$900$	0	0
$901$	−49608.0	−1.83428
$902$	0	0
$903$	− 7917.00i	− 0.291762i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	50276.0i	1.84056i	0.391261	+	0.920280i	$0.372039\pi$
$907$	50276.0i	1.84056i	−0.391261	+	0.920280i	$0.627961\pi$
$908$	0	0
$909$	16308.0	0.595052
$910$	0	0
$911$	20502.0	0.745622	0.372811	−	0.927907i	$-0.378394\pi$
$911$	20502.0	0.745622	0.372811	+	0.927907i	$0.378394\pi$
$912$	0	0
$913$	2988.00i	0.108311i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 32604.0i	− 1.17413i
$918$	0	0
$919$	37817.0	1.35742	0.678709	−	0.734407i	$-0.262540\pi$
$919$	37817.0	1.35742	0.678709	+	0.734407i	$0.262540\pi$
$920$	0	0
$921$	−13983.0	−0.500277
$922$	0	0
$923$	1800.00i	0.0641904i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 15372.0i	− 0.544642i
$928$	0	0
$929$	11778.0	0.415957	0.207978	−	0.978133i	$-0.433312\pi$
$929$	11778.0	0.415957	0.207978	+	0.978133i	$0.433312\pi$
$930$	0	0
$931$	11310.0	0.398142
$932$	0	0
$933$	− 7542.00i	− 0.264645i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	18409.0i	0.641831i	0.947108	+	0.320916i	$0.103991\pi$
$937$	18409.0i	0.641831i	−0.947108	+	0.320916i	$0.896009\pi$
$938$	0	0
$939$	−20121.0	−0.699280
$940$	0	0
$941$	−9330.00	−0.323219	−0.161610	−	0.986855i	$-0.551669\pi$
$941$	−9330.00	−0.323219	−0.161610	+	0.986855i	$0.551669\pi$
$942$	0	0
$943$	− 49680.0i	− 1.71559i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	49146.0i	1.68641i	0.537592	+	0.843205i	$0.319334\pi$
$947$	49146.0i	1.68641i	−0.537592	+	0.843205i	$0.680666\pi$
$948$	0	0
$949$	1430.00	0.0489144
$950$	0	0
$951$	13896.0	0.473826
$952$	0	0
$953$	− 5928.00i	− 0.201497i	−0.994912	−	0.100749i	$-0.967876\pi$
$953$	− 5928.00i	− 0.201497i	0.994912	−	0.100749i	$-0.0321238\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1188.00i	0.0401281i
$958$	0	0
$959$	−20358.0	−0.685500
$960$	0	0
$961$	59610.0	2.00094
$962$	0	0
$963$	− 11124.0i	− 0.372239i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 9520.00i	− 0.316590i	−0.987392	−	0.158295i	$-0.949400\pi$
$967$	− 9520.00i	− 0.316590i	0.987392	−	0.158295i	$-0.0505997\pi$
$968$	0	0
$969$	−15210.0	−0.504247
$970$	0	0
$971$	−12558.0	−0.415042	−0.207521	−	0.978231i	$-0.566539\pi$
$971$	−12558.0	−0.415042	−0.207521	+	0.978231i	$0.566539\pi$
$972$	0	0
$973$	2548.00i	0.0839518i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46206.0i	1.51306i	0.653958	+	0.756531i	$0.273107\pi$
$977$	46206.0i	1.51306i	−0.653958	+	0.756531i	$0.726893\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−13887.0	−0.451965
$982$	0	0
$983$	32676.0i	1.06023i	0.847927	+	0.530113i	$0.177851\pi$
$983$	32676.0i	1.06023i	−0.847927	+	0.530113i	$0.822149\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3042.00i	0.0981033i
$988$	0	0
$989$	−28014.0	−0.900701
$990$	0	0
$991$	8137.00	0.260828	0.130414	−	0.991460i	$-0.458369\pi$
$991$	8137.00	0.260828	0.130414	+	0.991460i	$0.458369\pi$
$992$	0	0
$993$	2964.00i	0.0947228i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 18578.0i	− 0.590142i	−0.955475	−	0.295071i	$-0.904657\pi$
$997$	− 18578.0i	− 0.590142i	0.955475	−	0.295071i	$-0.0953432\pi$
$998$	0	0
$999$	5778.00	0.182991

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.k.49.2		2
4.3	odd	2		300.4.d.c.49.1		2
5.2	odd	4		1200.4.a.bi.1.1		1
5.3	odd	4		1200.4.a.d.1.1		1
5.4	even	2	inner	1200.4.f.k.49.1		2
12.11	even	2		900.4.d.e.649.2		2
20.3	even	4		300.4.a.h.1.1	yes	1
20.7	even	4		300.4.a.a.1.1	✓	1
20.19	odd	2		300.4.d.c.49.2		2
60.23	odd	4		900.4.a.l.1.1		1
60.47	odd	4		900.4.a.f.1.1		1
60.59	even	2		900.4.d.e.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.4.a.a.1.1	✓	1	20.7	even	4
300.4.a.h.1.1	yes	1	20.3	even	4
300.4.d.c.49.1		2	4.3	odd	2
300.4.d.c.49.2		2	20.19	odd	2
900.4.a.f.1.1		1	60.47	odd	4
900.4.a.l.1.1		1	60.23	odd	4
900.4.d.e.649.1		2	60.59	even	2
900.4.d.e.649.2		2	12.11	even	2
1200.4.a.d.1.1		1	5.3	odd	4
1200.4.a.bi.1.1		1	5.2	odd	4
1200.4.f.k.49.1		2	5.4	even	2	inner
1200.4.f.k.49.2		2	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -13.0000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} -13.0000i q^{7} -9.00000 q^{9} -6.00000 q^{11} +5.00000i q^{13} +78.0000i q^{17} +65.0000 q^{19} +39.0000 q^{21} -138.000i q^{23} -27.0000i q^{27} -66.0000 q^{29} -299.000 q^{31} -18.0000i q^{33} +214.000i q^{37} -15.0000 q^{39} +360.000 q^{41} -203.000i q^{43} +78.0000i q^{47} +174.000 q^{49} -234.000 q^{51} +636.000i q^{53} +195.000i q^{57} +786.000 q^{59} +467.000 q^{61} +117.000i q^{63} -217.000i q^{67} +414.000 q^{69} +360.000 q^{71} -286.000i q^{73} +78.0000i q^{77} +272.000 q^{79} +81.0000 q^{81} -498.000i q^{83} -198.000i q^{87} +65.0000 q^{91} -897.000i q^{93} +511.000i q^{97} +54.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} - 12 q^{11} + 130 q^{19} + 78 q^{21} - 132 q^{29} - 598 q^{31} - 30 q^{39} + 720 q^{41} + 348 q^{49} - 468 q^{51} + 1572 q^{59} + 934 q^{61} + 828 q^{69} + 720 q^{71} + 544 q^{79} + 162 q^{81} + 130 q^{91} + 108 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 - 12 * q^11 + 130 * q^19 + 78 * q^21 - 132 * q^29 - 598 * q^31 - 30 * q^39 + 720 * q^41 + 348 * q^49 - 468 * q^51 + 1572 * q^59 + 934 * q^61 + 828 * q^69 + 720 * q^71 + 544 * q^79 + 162 * q^81 + 130 * q^91 + 108 * q^99

Introduction

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Modular forms

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