LMFDB - Embedded newform 1200.4.f.n.49.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,4,Mod(49,1200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,0,0,0,0,0,-18,0,48,0,0,0,0,0,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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 60)
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.n.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -28.0000i q^{7} -9.00000 q^{9} +24.0000 q^{11} -70.0000i q^{13} -102.000i q^{17} +20.0000 q^{19} +84.0000 q^{21} +72.0000i q^{23} -27.0000i q^{27} -306.000 q^{29} +136.000 q^{31} +72.0000i q^{33} +214.000i q^{37} +210.000 q^{39} -150.000 q^{41} +292.000i q^{43} -72.0000i q^{47} -441.000 q^{49} +306.000 q^{51} -414.000i q^{53} +60.0000i q^{57} -744.000 q^{59} -418.000 q^{61} +252.000i q^{63} +188.000i q^{67} -216.000 q^{69} -480.000 q^{71} +434.000i q^{73} -672.000i q^{77} +1352.00 q^{79} +81.0000 q^{81} +612.000i q^{83} -918.000i q^{87} +30.0000 q^{89} -1960.00 q^{91} +408.000i q^{93} +286.000i q^{97} -216.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} + 48 q^{11} + 40 q^{19} + 168 q^{21} - 612 q^{29} + 272 q^{31} + 420 q^{39} - 300 q^{41} - 882 q^{49} + 612 q^{51} - 1488 q^{59} - 836 q^{61} - 432 q^{69} - 960 q^{71} + 2704 q^{79} + 162 q^{81}+ \cdots - 432 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 + 48 * q^11 + 40 * q^19 + 168 * q^21 - 612 * q^29 + 272 * q^31 + 420 * q^39 - 300 * q^41 - 882 * q^49 + 612 * q^51 - 1488 * q^59 - 836 * q^61 - 432 * q^69 - 960 * q^71 + 2704 * q^79 + 162 * q^81 + 60 * q^89 - 3920 * q^91 - 432 * q^99

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
$7$	− 28.0000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 28.0000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	24.0000	0.657843	0.328921	−	0.944357i	$-0.393315\pi$
$11$	24.0000	0.657843	0.328921	+	0.944357i	$0.393315\pi$
$12$	0	0
$13$	− 70.0000i	− 1.49342i	−0.665148	−	0.746712i	$-0.731631\pi$
$13$	− 70.0000i	− 1.49342i	0.665148	−	0.746712i	$-0.268369\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 102.000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 102.000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	0	0
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	0	0
$21$	84.0000	0.872872
$22$	0	0
$23$	72.0000i	0.652741i	0.945242	+	0.326370i	$0.105826\pi$
$23$	72.0000i	0.652741i	−0.945242	+	0.326370i	$0.894174\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	−306.000	−1.95941	−0.979703	−	0.200455i	$-0.935758\pi$
$29$	−306.000	−1.95941	−0.979703	+	0.200455i	$0.935758\pi$
$30$	0	0
$31$	136.000	0.787946	0.393973	−	0.919122i	$-0.371100\pi$
$31$	136.000	0.787946	0.393973	+	0.919122i	$0.371100\pi$
$32$	0	0
$33$	72.0000i	0.379806i
$34$	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
$39$	210.000	0.862229
$40$	0	0
$41$	−150.000	−0.571367	−0.285684	−	0.958324i	$-0.592221\pi$
$41$	−150.000	−0.571367	−0.285684	+	0.958324i	$0.592221\pi$
$42$	0	0
$43$	292.000i	1.03557i	0.855510	+	0.517786i	$0.173244\pi$
$43$	292.000i	1.03557i	−0.855510	+	0.517786i	$0.826756\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 72.0000i	− 0.223453i	−0.993739	−	0.111726i	$-0.964362\pi$
$47$	− 72.0000i	− 0.223453i	0.993739	−	0.111726i	$-0.0356380\pi$
$48$	0	0
$49$	−441.000	−1.28571
$50$	0	0
$51$	306.000	0.840168
$52$	0	0
$53$	− 414.000i	− 1.07297i	−0.843911	−	0.536484i	$-0.819752\pi$
$53$	− 414.000i	− 1.07297i	0.843911	−	0.536484i	$-0.180248\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	60.0000i	0.139424i
$58$	0	0
$59$	−744.000	−1.64170	−0.820852	−	0.571141i	$-0.806501\pi$
$59$	−744.000	−1.64170	−0.820852	+	0.571141i	$0.806501\pi$
$60$	0	0
$61$	−418.000	−0.877367	−0.438684	−	0.898642i	$-0.644555\pi$
$61$	−418.000	−0.877367	−0.438684	+	0.898642i	$0.644555\pi$
$62$	0	0
$63$	252.000i	0.503953i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	188.000i	0.342804i	0.985201	+	0.171402i	$0.0548297\pi$
$67$	188.000i	0.342804i	−0.985201	+	0.171402i	$0.945170\pi$
$68$	0	0
$69$	−216.000	−0.376860
$70$	0	0
$71$	−480.000	−0.802331	−0.401166	−	0.916006i	$-0.631395\pi$
$71$	−480.000	−0.802331	−0.401166	+	0.916006i	$0.631395\pi$
$72$	0	0
$73$	434.000i	0.695834i	0.937525	+	0.347917i	$0.113111\pi$
$73$	434.000i	0.695834i	−0.937525	+	0.347917i	$0.886889\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 672.000i	− 0.994565i
$78$	0	0
$79$	1352.00	1.92547	0.962733	−	0.270452i	$-0.0871732\pi$
$79$	1352.00	1.92547	0.962733	+	0.270452i	$0.0871732\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	612.000i	0.809346i	0.914461	+	0.404673i	$0.132615\pi$
$83$	612.000i	0.809346i	−0.914461	+	0.404673i	$0.867385\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 918.000i	− 1.13126i
$88$	0	0
$89$	30.0000	0.0357303	0.0178651	−	0.999840i	$-0.494313\pi$
$89$	30.0000	0.0357303	0.0178651	+	0.999840i	$0.494313\pi$
$90$	0	0
$91$	−1960.00	−2.25784
$92$	0	0
$93$	408.000i	0.454921i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	286.000i	0.299370i	0.988734	+	0.149685i	$0.0478260\pi$
$97$	286.000i	0.299370i	−0.988734	+	0.149685i	$0.952174\pi$
$98$	0	0
$99$	−216.000	−0.219281
$100$	0	0
$101$	−1542.00	−1.51916	−0.759578	−	0.650416i	$-0.774594\pi$
$101$	−1542.00	−1.51916	−0.759578	+	0.650416i	$0.774594\pi$
$102$	0	0
$103$	− 1172.00i	− 1.12117i	−0.828097	−	0.560585i	$-0.810576\pi$
$103$	− 1172.00i	− 1.12117i	0.828097	−	0.560585i	$-0.189424\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1956.00i	1.76723i	0.468214	+	0.883615i	$0.344898\pi$
$107$	1956.00i	1.76723i	−0.468214	+	0.883615i	$0.655102\pi$
$108$	0	0
$109$	1858.00	1.63270	0.816349	−	0.577559i	$-0.195995\pi$
$109$	1858.00	1.63270	0.816349	+	0.577559i	$0.195995\pi$
$110$	0	0
$111$	−642.000	−0.548972
$112$	0	0
$113$	174.000i	0.144854i	0.997374	+	0.0724272i	$0.0230745\pi$
$113$	174.000i	0.144854i	−0.997374	+	0.0724272i	$0.976926\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	630.000i	0.497808i
$118$	0	0
$119$	−2856.00	−2.20008
$120$	0	0
$121$	−755.000	−0.567243
$122$	0	0
$123$	− 450.000i	− 0.329879i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2068.00i	− 1.44492i	−0.691411	−	0.722462i	$-0.743010\pi$
$127$	− 2068.00i	− 1.44492i	0.691411	−	0.722462i	$-0.256990\pi$
$128$	0	0
$129$	−876.000	−0.597888
$130$	0	0
$131$	−312.000	−0.208088	−0.104044	−	0.994573i	$-0.533178\pi$
$131$	−312.000	−0.208088	−0.104044	+	0.994573i	$0.533178\pi$
$132$	0	0
$133$	− 560.000i	− 0.365099i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2646.00i	− 1.65010i	−0.565063	−	0.825048i	$-0.691148\pi$
$137$	− 2646.00i	− 1.65010i	0.565063	−	0.825048i	$-0.308852\pi$
$138$	0	0
$139$	−1276.00	−0.778625	−0.389313	−	0.921106i	$-0.627287\pi$
$139$	−1276.00	−0.778625	−0.389313	+	0.921106i	$0.627287\pi$
$140$	0	0
$141$	216.000	0.129011
$142$	0	0
$143$	− 1680.00i	− 0.982438i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1323.00i	− 0.742307i
$148$	0	0
$149$	3198.00	1.75832	0.879162	−	0.476522i	$-0.158103\pi$
$149$	3198.00	1.75832	0.879162	+	0.476522i	$0.158103\pi$
$150$	0	0
$151$	760.000	0.409589	0.204794	−	0.978805i	$-0.434347\pi$
$151$	760.000	0.409589	0.204794	+	0.978805i	$0.434347\pi$
$152$	0	0
$153$	918.000i	0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	166.000i	0.0843837i	0.999110	+	0.0421919i	$0.0134341\pi$
$157$	166.000i	0.0843837i	−0.999110	+	0.0421919i	$0.986566\pi$
$158$	0	0
$159$	1242.00	0.619478
$160$	0	0
$161$	2016.00	0.986851
$162$	0	0
$163$	− 3020.00i	− 1.45119i	−0.688120	−	0.725597i	$-0.741564\pi$
$163$	− 3020.00i	− 1.45119i	0.688120	−	0.725597i	$-0.258436\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 984.000i	− 0.455953i	−0.973667	−	0.227977i	$-0.926789\pi$
$167$	− 984.000i	− 0.455953i	0.973667	−	0.227977i	$-0.0732110\pi$
$168$	0	0
$169$	−2703.00	−1.23031
$170$	0	0
$171$	−180.000	−0.0804967
$172$	0	0
$173$	1962.00i	0.862243i	0.902294	+	0.431122i	$0.141882\pi$
$173$	1962.00i	0.862243i	−0.902294	+	0.431122i	$0.858118\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 2232.00i	− 0.947838i
$178$	0	0
$179$	576.000	0.240515	0.120258	−	0.992743i	$-0.461628\pi$
$179$	576.000	0.240515	0.120258	+	0.992743i	$0.461628\pi$
$180$	0	0
$181$	−1210.00	−0.496898	−0.248449	−	0.968645i	$-0.579921\pi$
$181$	−1210.00	−0.496898	−0.248449	+	0.968645i	$0.579921\pi$
$182$	0	0
$183$	− 1254.00i	− 0.506548i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2448.00i	− 0.957302i
$188$	0	0
$189$	−756.000	−0.290957
$190$	0	0
$191$	−3384.00	−1.28198	−0.640989	−	0.767550i	$-0.721475\pi$
$191$	−3384.00	−1.28198	−0.640989	+	0.767550i	$0.721475\pi$
$192$	0	0
$193$	− 2038.00i	− 0.760096i	−0.924967	−	0.380048i	$-0.875908\pi$
$193$	− 2038.00i	− 0.760096i	0.924967	−	0.380048i	$-0.124092\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4098.00i	− 1.48208i	−0.671459	−	0.741042i	$-0.734332\pi$
$197$	− 4098.00i	− 1.48208i	0.671459	−	0.741042i	$-0.265668\pi$
$198$	0	0
$199$	−2248.00	−0.800786	−0.400393	−	0.916343i	$-0.631126\pi$
$199$	−2248.00	−0.800786	−0.400393	+	0.916343i	$0.631126\pi$
$200$	0	0
$201$	−564.000	−0.197918
$202$	0	0
$203$	8568.00i	2.96234i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 648.000i	− 0.217580i
$208$	0	0
$209$	480.000	0.158863
$210$	0	0
$211$	−3260.00	−1.06364	−0.531819	−	0.846858i	$-0.678491\pi$
$211$	−3260.00	−1.06364	−0.531819	+	0.846858i	$0.678491\pi$
$212$	0	0
$213$	− 1440.00i	− 0.463226i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3808.00i	− 1.19126i
$218$	0	0
$219$	−1302.00	−0.401740
$220$	0	0
$221$	−7140.00	−2.17325
$222$	0	0
$223$	2980.00i	0.894868i	0.894317	+	0.447434i	$0.147662\pi$
$223$	2980.00i	0.894868i	−0.894317	+	0.447434i	$0.852338\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3180.00i	− 0.929797i	−0.885364	−	0.464899i	$-0.846091\pi$
$227$	− 3180.00i	− 0.929797i	0.885364	−	0.464899i	$-0.153909\pi$
$228$	0	0
$229$	−3374.00	−0.973625	−0.486813	−	0.873506i	$-0.661841\pi$
$229$	−3374.00	−0.973625	−0.486813	+	0.873506i	$0.661841\pi$
$230$	0	0
$231$	2016.00	0.574212
$232$	0	0
$233$	1950.00i	0.548278i	0.961690	+	0.274139i	$0.0883928\pi$
$233$	1950.00i	0.548278i	−0.961690	+	0.274139i	$0.911607\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4056.00i	1.11167i
$238$	0	0
$239$	2232.00	0.604084	0.302042	−	0.953295i	$-0.402332\pi$
$239$	2232.00	0.604084	0.302042	+	0.953295i	$0.402332\pi$
$240$	0	0
$241$	−1822.00	−0.486993	−0.243497	−	0.969902i	$-0.578294\pi$
$241$	−1822.00	−0.486993	−0.243497	+	0.969902i	$0.578294\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1400.00i	− 0.360647i
$248$	0	0
$249$	−1836.00	−0.467276
$250$	0	0
$251$	−1488.00	−0.374190	−0.187095	−	0.982342i	$-0.559907\pi$
$251$	−1488.00	−0.374190	−0.187095	+	0.982342i	$0.559907\pi$
$252$	0	0
$253$	1728.00i	0.429401i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2994.00i	0.726695i	0.931654	+	0.363347i	$0.118366\pi$
$257$	2994.00i	0.726695i	−0.931654	+	0.363347i	$0.881634\pi$
$258$	0	0
$259$	5992.00	1.43755
$260$	0	0
$261$	2754.00	0.653135
$262$	0	0
$263$	2472.00i	0.579582i	0.957090	+	0.289791i	$0.0935858\pi$
$263$	2472.00i	0.579582i	−0.957090	+	0.289791i	$0.906414\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	90.0000i	0.0206289i
$268$	0	0
$269$	−3954.00	−0.896207	−0.448103	−	0.893982i	$-0.647900\pi$
$269$	−3954.00	−0.896207	−0.448103	+	0.893982i	$0.647900\pi$
$270$	0	0
$271$	2176.00	0.487759	0.243879	−	0.969806i	$-0.421580\pi$
$271$	2176.00	0.487759	0.243879	+	0.969806i	$0.421580\pi$
$272$	0	0
$273$	− 5880.00i	− 1.30357i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1034.00i	− 0.224285i	−0.993692	−	0.112143i	$-0.964229\pi$
$277$	− 1034.00i	− 0.224285i	0.993692	−	0.112143i	$-0.0357714\pi$
$278$	0	0
$279$	−1224.00	−0.262649
$280$	0	0
$281$	−6654.00	−1.41261	−0.706307	−	0.707906i	$-0.749640\pi$
$281$	−6654.00	−1.41261	−0.706307	+	0.707906i	$0.749640\pi$
$282$	0	0
$283$	1756.00i	0.368846i	0.982847	+	0.184423i	$0.0590416\pi$
$283$	1756.00i	0.368846i	−0.982847	+	0.184423i	$0.940958\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4200.00i	0.863826i
$288$	0	0
$289$	−5491.00	−1.11765
$290$	0	0
$291$	−858.000	−0.172841
$292$	0	0
$293$	3234.00i	0.644820i	0.946600	+	0.322410i	$0.104493\pi$
$293$	3234.00i	0.644820i	−0.946600	+	0.322410i	$0.895507\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 648.000i	− 0.126602i
$298$	0	0
$299$	5040.00	0.974818
$300$	0	0
$301$	8176.00	1.56564
$302$	0	0
$303$	− 4626.00i	− 0.877085i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2036.00i	0.378504i	0.981929	+	0.189252i	$0.0606063\pi$
$307$	2036.00i	0.378504i	−0.981929	+	0.189252i	$0.939394\pi$
$308$	0	0
$309$	3516.00	0.647308
$310$	0	0
$311$	96.0000	0.0175037	0.00875187	−	0.999962i	$-0.497214\pi$
$311$	96.0000	0.0175037	0.00875187	+	0.999962i	$0.497214\pi$
$312$	0	0
$313$	1202.00i	0.217064i	0.994093	+	0.108532i	$0.0346150\pi$
$313$	1202.00i	0.217064i	−0.994093	+	0.108532i	$0.965385\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3798.00i	0.672924i	0.941697	+	0.336462i	$0.109230\pi$
$317$	3798.00i	0.672924i	−0.941697	+	0.336462i	$0.890770\pi$
$318$	0	0
$319$	−7344.00	−1.28898
$320$	0	0
$321$	−5868.00	−1.02031
$322$	0	0
$323$	− 2040.00i	− 0.351420i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5574.00i	0.942639i
$328$	0	0
$329$	−2016.00	−0.337829
$330$	0	0
$331$	5668.00	0.941213	0.470606	−	0.882343i	$-0.344035\pi$
$331$	5668.00	0.941213	0.470606	+	0.882343i	$0.344035\pi$
$332$	0	0
$333$	− 1926.00i	− 0.316949i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	454.000i	0.0733856i	0.999327	+	0.0366928i	$0.0116823\pi$
$337$	454.000i	0.0733856i	−0.999327	+	0.0366928i	$0.988318\pi$
$338$	0	0
$339$	−522.000	−0.0836317
$340$	0	0
$341$	3264.00	0.518345
$342$	0	0
$343$	2744.00i	0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 5604.00i	− 0.866970i	−0.901161	−	0.433485i	$-0.857284\pi$
$347$	− 5604.00i	− 0.866970i	0.901161	−	0.433485i	$-0.142716\pi$
$348$	0	0
$349$	11266.0	1.72795	0.863976	−	0.503533i	$-0.167967\pi$
$349$	11266.0	1.72795	0.863976	+	0.503533i	$0.167967\pi$
$350$	0	0
$351$	−1890.00	−0.287410
$352$	0	0
$353$	− 6426.00i	− 0.968899i	−0.874819	−	0.484450i	$-0.839020\pi$
$353$	− 6426.00i	− 0.968899i	0.874819	−	0.484450i	$-0.160980\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 8568.00i	− 1.27021i
$358$	0	0
$359$	6936.00	1.01969	0.509844	−	0.860267i	$-0.329703\pi$
$359$	6936.00	1.01969	0.509844	+	0.860267i	$0.329703\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	0	0
$363$	− 2265.00i	− 0.327498i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 388.000i	− 0.0551865i	−0.999619	−	0.0275932i	$-0.991216\pi$
$367$	− 388.000i	− 0.0551865i	0.999619	−	0.0275932i	$-0.00878431\pi$
$368$	0	0
$369$	1350.00	0.190456
$370$	0	0
$371$	−11592.0	−1.62217
$372$	0	0
$373$	− 8062.00i	− 1.11913i	−0.828787	−	0.559564i	$-0.810969\pi$
$373$	− 8062.00i	− 1.11913i	0.828787	−	0.559564i	$-0.189031\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	21420.0i	2.92622i
$378$	0	0
$379$	−3388.00	−0.459182	−0.229591	−	0.973287i	$-0.573739\pi$
$379$	−3388.00	−0.459182	−0.229591	+	0.973287i	$0.573739\pi$
$380$	0	0
$381$	6204.00	0.834227
$382$	0	0
$383$	− 6984.00i	− 0.931764i	−0.884847	−	0.465882i	$-0.845737\pi$
$383$	− 6984.00i	− 0.931764i	0.884847	−	0.465882i	$-0.154263\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2628.00i	− 0.345191i
$388$	0	0
$389$	2526.00	0.329237	0.164619	−	0.986357i	$-0.447361\pi$
$389$	2526.00	0.329237	0.164619	+	0.986357i	$0.447361\pi$
$390$	0	0
$391$	7344.00	0.949877
$392$	0	0
$393$	− 936.000i	− 0.120140i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 6146.00i	− 0.776975i	−0.921454	−	0.388487i	$-0.872998\pi$
$397$	− 6146.00i	− 0.776975i	0.921454	−	0.388487i	$-0.127002\pi$
$398$	0	0
$399$	1680.00	0.210790
$400$	0	0
$401$	9786.00	1.21868	0.609339	−	0.792910i	$-0.291435\pi$
$401$	9786.00	1.21868	0.609339	+	0.792910i	$0.291435\pi$
$402$	0	0
$403$	− 9520.00i	− 1.17674i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5136.00i	0.625509i
$408$	0	0
$409$	886.000	0.107115	0.0535573	−	0.998565i	$-0.482944\pi$
$409$	886.000	0.107115	0.0535573	+	0.998565i	$0.482944\pi$
$410$	0	0
$411$	7938.00	0.952683
$412$	0	0
$413$	20832.0i	2.48202i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 3828.00i	− 0.449539i
$418$	0	0
$419$	−11352.0	−1.32358	−0.661792	−	0.749688i	$-0.730204\pi$
$419$	−11352.0	−1.32358	−0.661792	+	0.749688i	$0.730204\pi$
$420$	0	0
$421$	10190.0	1.17964	0.589822	−	0.807533i	$-0.299198\pi$
$421$	10190.0	1.17964	0.589822	+	0.807533i	$0.299198\pi$
$422$	0	0
$423$	648.000i	0.0744843i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11704.0i	1.32645i
$428$	0	0
$429$	5040.00	0.567211
$430$	0	0
$431$	−2448.00	−0.273587	−0.136794	−	0.990600i	$-0.543680\pi$
$431$	−2448.00	−0.273587	−0.136794	+	0.990600i	$0.543680\pi$
$432$	0	0
$433$	− 7078.00i	− 0.785559i	−0.919633	−	0.392779i	$-0.871514\pi$
$433$	− 7078.00i	− 0.785559i	0.919633	−	0.392779i	$-0.128486\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1440.00i	0.157631i
$438$	0	0
$439$	−18088.0	−1.96650	−0.983250	−	0.182264i	$-0.941657\pi$
$439$	−18088.0	−1.96650	−0.983250	+	0.182264i	$0.941657\pi$
$440$	0	0
$441$	3969.00	0.428571
$442$	0	0
$443$	− 3852.00i	− 0.413124i	−0.978433	−	0.206562i	$-0.933772\pi$
$443$	− 3852.00i	− 0.413124i	0.978433	−	0.206562i	$-0.0662276\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9594.00i	1.01517i
$448$	0	0
$449$	−6522.00	−0.685506	−0.342753	−	0.939426i	$-0.611359\pi$
$449$	−6522.00	−0.685506	−0.342753	+	0.939426i	$0.611359\pi$
$450$	0	0
$451$	−3600.00	−0.375870
$452$	0	0
$453$	2280.00i	0.236476i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2090.00i	− 0.213930i	−0.994263	−	0.106965i	$-0.965887\pi$
$457$	− 2090.00i	− 0.213930i	0.994263	−	0.106965i	$-0.0341133\pi$
$458$	0	0
$459$	−2754.00	−0.280056
$460$	0	0
$461$	−9894.00	−0.999587	−0.499793	−	0.866145i	$-0.666591\pi$
$461$	−9894.00	−0.999587	−0.499793	+	0.866145i	$0.666591\pi$
$462$	0	0
$463$	− 3044.00i	− 0.305544i	−0.988261	−	0.152772i	$-0.951180\pi$
$463$	− 3044.00i	− 0.305544i	0.988261	−	0.152772i	$-0.0488199\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10236.0i	1.01427i	0.861866	+	0.507137i	$0.169296\pi$
$467$	10236.0i	1.01427i	−0.861866	+	0.507137i	$0.830704\pi$
$468$	0	0
$469$	5264.00	0.518271
$470$	0	0
$471$	−498.000	−0.0487190
$472$	0	0
$473$	7008.00i	0.681244i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3726.00i	0.357656i
$478$	0	0
$479$	11496.0	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	11496.0	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	14980.0	1.42002
$482$	0	0
$483$	6048.00i	0.569759i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 15316.0i	− 1.42512i	−0.701610	−	0.712561i	$-0.747535\pi$
$487$	− 15316.0i	− 1.42512i	0.701610	−	0.712561i	$-0.252465\pi$
$488$	0	0
$489$	9060.00	0.837847
$490$	0	0
$491$	11616.0	1.06766	0.533832	−	0.845591i	$-0.320752\pi$
$491$	11616.0	1.06766	0.533832	+	0.845591i	$0.320752\pi$
$492$	0	0
$493$	31212.0i	2.85135i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13440.0i	1.21301i
$498$	0	0
$499$	14996.0	1.34532	0.672658	−	0.739953i	$-0.265152\pi$
$499$	14996.0	1.34532	0.672658	+	0.739953i	$0.265152\pi$
$500$	0	0
$501$	2952.00	0.263245
$502$	0	0
$503$	21648.0i	1.91896i	0.281778	+	0.959480i	$0.409076\pi$
$503$	21648.0i	1.91896i	−0.281778	+	0.959480i	$0.590924\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 8109.00i	− 0.710322i
$508$	0	0
$509$	−3378.00	−0.294160	−0.147080	−	0.989125i	$-0.546987\pi$
$509$	−3378.00	−0.294160	−0.147080	+	0.989125i	$0.546987\pi$
$510$	0	0
$511$	12152.0	1.05200
$512$	0	0
$513$	− 540.000i	− 0.0464748i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1728.00i	− 0.146997i
$518$	0	0
$519$	−5886.00	−0.497816
$520$	0	0
$521$	−16158.0	−1.35872	−0.679362	−	0.733804i	$-0.737743\pi$
$521$	−16158.0	−1.35872	−0.679362	+	0.733804i	$0.737743\pi$
$522$	0	0
$523$	76.0000i	0.00635420i	0.999995	+	0.00317710i	$0.00101130\pi$
$523$	76.0000i	0.00635420i	−0.999995	+	0.00317710i	$0.998989\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 13872.0i	− 1.14663i
$528$	0	0
$529$	6983.00	0.573929
$530$	0	0
$531$	6696.00	0.547235
$532$	0	0
$533$	10500.0i	0.853294i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1728.00i	0.138862i
$538$	0	0
$539$	−10584.0	−0.845798
$540$	0	0
$541$	9278.00	0.737324	0.368662	−	0.929563i	$-0.379816\pi$
$541$	9278.00	0.737324	0.368662	+	0.929563i	$0.379816\pi$
$542$	0	0
$543$	− 3630.00i	− 0.286884i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14564.0i	1.13841i	0.822195	+	0.569206i	$0.192749\pi$
$547$	14564.0i	1.13841i	−0.822195	+	0.569206i	$0.807251\pi$
$548$	0	0
$549$	3762.00	0.292456
$550$	0	0
$551$	−6120.00	−0.473177
$552$	0	0
$553$	− 37856.0i	− 2.91103i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2154.00i	− 0.163856i	−0.996638	−	0.0819281i	$-0.973892\pi$
$557$	− 2154.00i	− 0.163856i	0.996638	−	0.0819281i	$-0.0261078\pi$
$558$	0	0
$559$	20440.0	1.54655
$560$	0	0
$561$	7344.00	0.552699
$562$	0	0
$563$	8700.00i	0.651263i	0.945497	+	0.325632i	$0.105577\pi$
$563$	8700.00i	0.651263i	−0.945497	+	0.325632i	$0.894423\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2268.00i	− 0.167984i
$568$	0	0
$569$	−4194.00	−0.309001	−0.154501	−	0.987993i	$-0.549377\pi$
$569$	−4194.00	−0.309001	−0.154501	+	0.987993i	$0.549377\pi$
$570$	0	0
$571$	8020.00	0.587787	0.293894	−	0.955838i	$-0.405049\pi$
$571$	8020.00	0.587787	0.293894	+	0.955838i	$0.405049\pi$
$572$	0	0
$573$	− 10152.0i	− 0.740150i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2686.00i	0.193795i	0.995294	+	0.0968974i	$0.0308919\pi$
$577$	2686.00i	0.193795i	−0.995294	+	0.0968974i	$0.969108\pi$
$578$	0	0
$579$	6114.00	0.438841
$580$	0	0
$581$	17136.0	1.22362
$582$	0	0
$583$	− 9936.00i	− 0.705844i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3012.00i	0.211786i	0.994378	+	0.105893i	$0.0337701\pi$
$587$	3012.00i	0.211786i	−0.994378	+	0.105893i	$0.966230\pi$
$588$	0	0
$589$	2720.00	0.190281
$590$	0	0
$591$	12294.0	0.855681
$592$	0	0
$593$	− 15522.0i	− 1.07489i	−0.843298	−	0.537447i	$-0.819389\pi$
$593$	− 15522.0i	− 1.07489i	0.843298	−	0.537447i	$-0.180611\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6744.00i	− 0.462334i
$598$	0	0
$599$	19224.0	1.31130	0.655652	−	0.755063i	$-0.272394\pi$
$599$	19224.0	1.31130	0.655652	+	0.755063i	$0.272394\pi$
$600$	0	0
$601$	−6502.00	−0.441301	−0.220651	−	0.975353i	$-0.570818\pi$
$601$	−6502.00	−0.441301	−0.220651	+	0.975353i	$0.570818\pi$
$602$	0	0
$603$	− 1692.00i	− 0.114268i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	29396.0i	1.96565i	0.184552	+	0.982823i	$0.440917\pi$
$607$	29396.0i	1.96565i	−0.184552	+	0.982823i	$0.559083\pi$
$608$	0	0
$609$	−25704.0	−1.71031
$610$	0	0
$611$	−5040.00	−0.333710
$612$	0	0
$613$	− 10006.0i	− 0.659280i	−0.944107	−	0.329640i	$-0.893073\pi$
$613$	− 10006.0i	− 0.659280i	0.944107	−	0.329640i	$-0.106927\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 23118.0i	− 1.50842i	−0.656633	−	0.754210i	$-0.728020\pi$
$617$	− 23118.0i	− 1.50842i	0.656633	−	0.754210i	$-0.271980\pi$
$618$	0	0
$619$	14036.0	0.911397	0.455698	−	0.890134i	$-0.349390\pi$
$619$	14036.0	0.911397	0.455698	+	0.890134i	$0.349390\pi$
$620$	0	0
$621$	1944.00	0.125620
$622$	0	0
$623$	− 840.000i	− 0.0540191i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1440.00i	0.0917194i
$628$	0	0
$629$	21828.0	1.38369
$630$	0	0
$631$	4288.00	0.270527	0.135264	−	0.990810i	$-0.456812\pi$
$631$	4288.00	0.270527	0.135264	+	0.990810i	$0.456812\pi$
$632$	0	0
$633$	− 9780.00i	− 0.614092i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	30870.0i	1.92012i
$638$	0	0
$639$	4320.00	0.267444
$640$	0	0
$641$	1314.00	0.0809671	0.0404835	−	0.999180i	$-0.487110\pi$
$641$	1314.00	0.0809671	0.0404835	+	0.999180i	$0.487110\pi$
$642$	0	0
$643$	628.000i	0.0385162i	0.999815	+	0.0192581i	$0.00613042\pi$
$643$	628.000i	0.0385162i	−0.999815	+	0.0192581i	$0.993870\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 10944.0i	− 0.664997i	−0.943104	−	0.332498i	$-0.892108\pi$
$647$	− 10944.0i	− 0.664997i	0.943104	−	0.332498i	$-0.107892\pi$
$648$	0	0
$649$	−17856.0	−1.07998
$650$	0	0
$651$	11424.0	0.687776
$652$	0	0
$653$	1098.00i	0.0658010i	0.999459	+	0.0329005i	$0.0104744\pi$
$653$	1098.00i	0.0658010i	−0.999459	+	0.0329005i	$0.989526\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3906.00i	− 0.231945i
$658$	0	0
$659$	312.000	0.0184428	0.00922139	−	0.999957i	$-0.497065\pi$
$659$	312.000	0.0184428	0.00922139	+	0.999957i	$0.497065\pi$
$660$	0	0
$661$	8678.00	0.510643	0.255322	−	0.966856i	$-0.417819\pi$
$661$	8678.00	0.510643	0.255322	+	0.966856i	$0.417819\pi$
$662$	0	0
$663$	− 21420.0i	− 1.25473i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 22032.0i	− 1.27898i
$668$	0	0
$669$	−8940.00	−0.516652
$670$	0	0
$671$	−10032.0	−0.577170
$672$	0	0
$673$	− 14470.0i	− 0.828793i	−0.910097	−	0.414396i	$-0.863993\pi$
$673$	− 14470.0i	− 0.828793i	0.910097	−	0.414396i	$-0.136007\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11838.0i	0.672040i	0.941855	+	0.336020i	$0.109081\pi$
$677$	11838.0i	0.672040i	−0.941855	+	0.336020i	$0.890919\pi$
$678$	0	0
$679$	8008.00	0.452605
$680$	0	0
$681$	9540.00	0.536819
$682$	0	0
$683$	25548.0i	1.43128i	0.698467	+	0.715642i	$0.253866\pi$
$683$	25548.0i	1.43128i	−0.698467	+	0.715642i	$0.746134\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 10122.0i	− 0.562123i
$688$	0	0
$689$	−28980.0	−1.60239
$690$	0	0
$691$	18412.0	1.01364	0.506820	−	0.862052i	$-0.330821\pi$
$691$	18412.0	1.01364	0.506820	+	0.862052i	$0.330821\pi$
$692$	0	0
$693$	6048.00i	0.331522i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	15300.0i	0.831462i
$698$	0	0
$699$	−5850.00	−0.316548
$700$	0	0
$701$	−8814.00	−0.474893	−0.237447	−	0.971401i	$-0.576310\pi$
$701$	−8814.00	−0.474893	−0.237447	+	0.971401i	$0.576310\pi$
$702$	0	0
$703$	4280.00i	0.229621i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	43176.0i	2.29675i
$708$	0	0
$709$	17314.0	0.917124	0.458562	−	0.888662i	$-0.348365\pi$
$709$	17314.0	0.917124	0.458562	+	0.888662i	$0.348365\pi$
$710$	0	0
$711$	−12168.0	−0.641822
$712$	0	0
$713$	9792.00i	0.514324i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6696.00i	0.348768i
$718$	0	0
$719$	−768.000	−0.0398353	−0.0199176	−	0.999802i	$-0.506340\pi$
$719$	−768.000	−0.0398353	−0.0199176	+	0.999802i	$0.506340\pi$
$720$	0	0
$721$	−32816.0	−1.69505
$722$	0	0
$723$	− 5466.00i	− 0.281166i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 18196.0i	− 0.928270i	−0.885764	−	0.464135i	$-0.846365\pi$
$727$	− 18196.0i	− 0.928270i	0.885764	−	0.464135i	$-0.153635\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	29784.0	1.50698
$732$	0	0
$733$	− 18142.0i	− 0.914175i	−0.889422	−	0.457087i	$-0.848893\pi$
$733$	− 18142.0i	− 0.914175i	0.889422	−	0.457087i	$-0.151107\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4512.00i	0.225511i
$738$	0	0
$739$	−13660.0	−0.679961	−0.339981	−	0.940432i	$-0.610420\pi$
$739$	−13660.0	−0.679961	−0.339981	+	0.940432i	$0.610420\pi$
$740$	0	0
$741$	4200.00	0.208220
$742$	0	0
$743$	12768.0i	0.630434i	0.949020	+	0.315217i	$0.102077\pi$
$743$	12768.0i	0.630434i	−0.949020	+	0.315217i	$0.897923\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5508.00i	− 0.269782i
$748$	0	0
$749$	54768.0	2.67180
$750$	0	0
$751$	−22952.0	−1.11522	−0.557610	−	0.830103i	$-0.688282\pi$
$751$	−22952.0	−1.11522	−0.557610	+	0.830103i	$0.688282\pi$
$752$	0	0
$753$	− 4464.00i	− 0.216039i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 15818.0i	− 0.759465i	−0.925096	−	0.379732i	$-0.876016\pi$
$757$	− 15818.0i	− 0.759465i	0.925096	−	0.379732i	$-0.123984\pi$
$758$	0	0
$759$	−5184.00	−0.247915
$760$	0	0
$761$	−18558.0	−0.884004	−0.442002	−	0.897014i	$-0.645732\pi$
$761$	−18558.0	−0.884004	−0.442002	+	0.897014i	$0.645732\pi$
$762$	0	0
$763$	− 52024.0i	− 2.46841i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	52080.0i	2.45176i
$768$	0	0
$769$	−14978.0	−0.702367	−0.351184	−	0.936307i	$-0.614221\pi$
$769$	−14978.0	−0.702367	−0.351184	+	0.936307i	$0.614221\pi$
$770$	0	0
$771$	−8982.00	−0.419557
$772$	0	0
$773$	8946.00i	0.416255i	0.978102	+	0.208128i	$0.0667369\pi$
$773$	8946.00i	0.416255i	−0.978102	+	0.208128i	$0.933263\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	17976.0i	0.829968i
$778$	0	0
$779$	−3000.00	−0.137980
$780$	0	0
$781$	−11520.0	−0.527808
$782$	0	0
$783$	8262.00i	0.377088i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 18436.0i	− 0.835035i	−0.908669	−	0.417517i	$-0.862900\pi$
$787$	− 18436.0i	− 0.835035i	0.908669	−	0.417517i	$-0.137100\pi$
$788$	0	0
$789$	−7416.00	−0.334622
$790$	0	0
$791$	4872.00	0.218999
$792$	0	0
$793$	29260.0i	1.31028i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 16314.0i	− 0.725058i	−0.931972	−	0.362529i	$-0.881913\pi$
$797$	− 16314.0i	− 0.725058i	0.931972	−	0.362529i	$-0.118087\pi$
$798$	0	0
$799$	−7344.00	−0.325172
$800$	0	0
$801$	−270.000	−0.0119101
$802$	0	0
$803$	10416.0i	0.457749i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 11862.0i	− 0.517425i
$808$	0	0
$809$	25446.0	1.10585	0.552926	−	0.833231i	$-0.313511\pi$
$809$	25446.0	1.10585	0.552926	+	0.833231i	$0.313511\pi$
$810$	0	0
$811$	−42740.0	−1.85056	−0.925280	−	0.379284i	$-0.876170\pi$
$811$	−42740.0	−1.85056	−0.925280	+	0.379284i	$0.876170\pi$
$812$	0	0
$813$	6528.00i	0.281608i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5840.00i	0.250080i
$818$	0	0
$819$	17640.0	0.752615
$820$	0	0
$821$	29946.0	1.27299	0.636494	−	0.771282i	$-0.280384\pi$
$821$	29946.0	1.27299	0.636494	+	0.771282i	$0.280384\pi$
$822$	0	0
$823$	32596.0i	1.38059i	0.723528	+	0.690295i	$0.242519\pi$
$823$	32596.0i	1.38059i	−0.723528	+	0.690295i	$0.757481\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3804.00i	0.159949i	0.996797	+	0.0799746i	$0.0254839\pi$
$827$	3804.00i	0.159949i	−0.996797	+	0.0799746i	$0.974516\pi$
$828$	0	0
$829$	−3278.00	−0.137334	−0.0686669	−	0.997640i	$-0.521875\pi$
$829$	−3278.00	−0.137334	−0.0686669	+	0.997640i	$0.521875\pi$
$830$	0	0
$831$	3102.00	0.129491
$832$	0	0
$833$	44982.0i	1.87099i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3672.00i	− 0.151640i
$838$	0	0
$839$	−5784.00	−0.238005	−0.119002	−	0.992894i	$-0.537970\pi$
$839$	−5784.00	−0.238005	−0.119002	+	0.992894i	$0.537970\pi$
$840$	0	0
$841$	69247.0	2.83927
$842$	0	0
$843$	− 19962.0i	− 0.815573i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	21140.0i	0.857590i
$848$	0	0
$849$	−5268.00	−0.212953
$850$	0	0
$851$	−15408.0	−0.620657
$852$	0	0
$853$	17306.0i	0.694661i	0.937743	+	0.347331i	$0.112912\pi$
$853$	17306.0i	0.694661i	−0.937743	+	0.347331i	$0.887088\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 31134.0i	− 1.24098i	−0.784216	−	0.620488i	$-0.786934\pi$
$857$	− 31134.0i	− 1.24098i	0.784216	−	0.620488i	$-0.213066\pi$
$858$	0	0
$859$	−10780.0	−0.428183	−0.214091	−	0.976814i	$-0.568679\pi$
$859$	−10780.0	−0.428183	−0.214091	+	0.976814i	$0.568679\pi$
$860$	0	0
$861$	−12600.0	−0.498730
$862$	0	0
$863$	− 3456.00i	− 0.136319i	−0.997674	−	0.0681597i	$-0.978287\pi$
$863$	− 3456.00i	− 0.136319i	0.997674	−	0.0681597i	$-0.0217127\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 16473.0i	− 0.645274i
$868$	0	0
$869$	32448.0	1.26665
$870$	0	0
$871$	13160.0	0.511951
$872$	0	0
$873$	− 2574.00i	− 0.0997900i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2618.00i	− 0.100802i	−0.998729	−	0.0504011i	$-0.983950\pi$
$877$	− 2618.00i	− 0.100802i	0.998729	−	0.0504011i	$-0.0160500\pi$
$878$	0	0
$879$	−9702.00	−0.372287
$880$	0	0
$881$	−26550.0	−1.01531	−0.507657	−	0.861559i	$-0.669488\pi$
$881$	−26550.0	−1.01531	−0.507657	+	0.861559i	$0.669488\pi$
$882$	0	0
$883$	− 27596.0i	− 1.05173i	−0.850567	−	0.525866i	$-0.823741\pi$
$883$	− 27596.0i	− 1.05173i	0.850567	−	0.525866i	$-0.176259\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 37848.0i	− 1.43271i	−0.697737	−	0.716354i	$-0.745810\pi$
$887$	− 37848.0i	− 1.43271i	0.697737	−	0.716354i	$-0.254190\pi$
$888$	0	0
$889$	−57904.0	−2.18452
$890$	0	0
$891$	1944.00	0.0730937
$892$	0	0
$893$	− 1440.00i	− 0.0539617i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	15120.0i	0.562812i
$898$	0	0
$899$	−41616.0	−1.54391
$900$	0	0
$901$	−42228.0	−1.56140
$902$	0	0
$903$	24528.0i	0.903921i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4804.00i	− 0.175870i	−0.996126	−	0.0879351i	$-0.971973\pi$
$907$	− 4804.00i	− 0.175870i	0.996126	−	0.0879351i	$-0.0280268\pi$
$908$	0	0
$909$	13878.0	0.506385
$910$	0	0
$911$	−28608.0	−1.04042	−0.520211	−	0.854037i	$-0.674147\pi$
$911$	−28608.0	−1.04042	−0.520211	+	0.854037i	$0.674147\pi$
$912$	0	0
$913$	14688.0i	0.532423i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8736.00i	0.314600i
$918$	0	0
$919$	−40768.0	−1.46334	−0.731672	−	0.681657i	$-0.761259\pi$
$919$	−40768.0	−1.46334	−0.731672	+	0.681657i	$0.761259\pi$
$920$	0	0
$921$	−6108.00	−0.218529
$922$	0	0
$923$	33600.0i	1.19822i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10548.0i	0.373724i
$928$	0	0
$929$	−27642.0	−0.976216	−0.488108	−	0.872783i	$-0.662313\pi$
$929$	−27642.0	−0.976216	−0.488108	+	0.872783i	$0.662313\pi$
$930$	0	0
$931$	−8820.00	−0.310487
$932$	0	0
$933$	288.000i	0.0101058i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 28106.0i	− 0.979918i	−0.871746	−	0.489959i	$-0.837012\pi$
$937$	− 28106.0i	− 0.979918i	0.871746	−	0.489959i	$-0.162988\pi$
$938$	0	0
$939$	−3606.00	−0.125322
$940$	0	0
$941$	14730.0	0.510291	0.255146	−	0.966903i	$-0.417877\pi$
$941$	14730.0	0.510291	0.255146	+	0.966903i	$0.417877\pi$
$942$	0	0
$943$	− 10800.0i	− 0.372955i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9564.00i	− 0.328182i	−0.986445	−	0.164091i	$-0.947531\pi$
$947$	− 9564.00i	− 0.328182i	0.986445	−	0.164091i	$-0.0524691\pi$
$948$	0	0
$949$	30380.0	1.03917
$950$	0	0
$951$	−11394.0	−0.388513
$952$	0	0
$953$	− 53898.0i	− 1.83203i	−0.401141	−	0.916017i	$-0.631386\pi$
$953$	− 53898.0i	− 1.83203i	0.401141	−	0.916017i	$-0.368614\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 22032.0i	− 0.744194i
$958$	0	0
$959$	−74088.0	−2.49471
$960$	0	0
$961$	−11295.0	−0.379141
$962$	0	0
$963$	− 17604.0i	− 0.589077i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	15140.0i	0.503485i	0.967794	+	0.251742i	$0.0810035\pi$
$967$	15140.0i	0.503485i	−0.967794	+	0.251742i	$0.918996\pi$
$968$	0	0
$969$	6120.00	0.202892
$970$	0	0
$971$	−23808.0	−0.786854	−0.393427	−	0.919356i	$-0.628711\pi$
$971$	−23808.0	−0.786854	−0.393427	+	0.919356i	$0.628711\pi$
$972$	0	0
$973$	35728.0i	1.17717i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 23094.0i	− 0.756236i	−0.925757	−	0.378118i	$-0.876571\pi$
$977$	− 23094.0i	− 0.756236i	0.925757	−	0.378118i	$-0.123429\pi$
$978$	0	0
$979$	720.000	0.0235049
$980$	0	0
$981$	−16722.0	−0.544233
$982$	0	0
$983$	− 7584.00i	− 0.246075i	−0.992402	−	0.123038i	$-0.960736\pi$
$983$	− 7584.00i	− 0.246075i	0.992402	−	0.123038i	$-0.0392636\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6048.00i	− 0.195046i
$988$	0	0
$989$	−21024.0	−0.675960
$990$	0	0
$991$	26752.0	0.857523	0.428761	−	0.903418i	$-0.358950\pi$
$991$	26752.0	0.857523	0.428761	+	0.903418i	$0.358950\pi$
$992$	0	0
$993$	17004.0i	0.543409i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 7778.00i	− 0.247073i	−0.992340	−	0.123536i	$-0.960576\pi$
$997$	− 7778.00i	− 0.247073i	0.992340	−	0.123536i	$-0.0394236\pi$
$998$	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.n.49.2		2
4.3	odd	2		300.4.d.b.49.1		2
5.2	odd	4		240.4.a.i.1.1		1
5.3	odd	4		1200.4.a.a.1.1		1
5.4	even	2	inner	1200.4.f.n.49.1		2
12.11	even	2		900.4.d.h.649.2		2
15.2	even	4		720.4.a.bb.1.1		1
20.3	even	4		300.4.a.i.1.1		1
20.7	even	4		60.4.a.a.1.1	✓	1
20.19	odd	2		300.4.d.b.49.2		2
40.27	even	4		960.4.a.bc.1.1		1
40.37	odd	4		960.4.a.r.1.1		1
60.23	odd	4		900.4.a.q.1.1		1
60.47	odd	4		180.4.a.d.1.1		1
60.59	even	2		900.4.d.h.649.1		2
180.7	even	12		1620.4.i.l.1081.1		2
180.47	odd	12		1620.4.i.f.1081.1		2
180.67	even	12		1620.4.i.l.541.1		2
180.167	odd	12		1620.4.i.f.541.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.a.a.1.1	✓	1	20.7	even	4
180.4.a.d.1.1		1	60.47	odd	4
240.4.a.i.1.1		1	5.2	odd	4
300.4.a.i.1.1		1	20.3	even	4
300.4.d.b.49.1		2	4.3	odd	2
300.4.d.b.49.2		2	20.19	odd	2
720.4.a.bb.1.1		1	15.2	even	4
900.4.a.q.1.1		1	60.23	odd	4
900.4.d.h.649.1		2	60.59	even	2
900.4.d.h.649.2		2	12.11	even	2
960.4.a.r.1.1		1	40.37	odd	4
960.4.a.bc.1.1		1	40.27	even	4
1200.4.a.a.1.1		1	5.3	odd	4
1200.4.f.n.49.1		2	5.4	even	2	inner
1200.4.f.n.49.2		2	1.1	even	1	trivial
1620.4.i.f.541.1		2	180.167	odd	12
1620.4.i.f.1081.1		2	180.47	odd	12
1620.4.i.l.541.1		2	180.67	even	12
1620.4.i.l.1081.1		2	180.7	even	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -28.0000i q^{7} -9.00000 q^{9} +24.0000 q^{11} -70.0000i q^{13} -102.000i q^{17} +20.0000 q^{19} +84.0000 q^{21} +72.0000i q^{23} -27.0000i q^{27} -306.000 q^{29} +136.000 q^{31} +72.0000i q^{33} +214.000i q^{37} +210.000 q^{39} -150.000 q^{41} +292.000i q^{43} -72.0000i q^{47} -441.000 q^{49} +306.000 q^{51} -414.000i q^{53} +60.0000i q^{57} -744.000 q^{59} -418.000 q^{61} +252.000i q^{63} +188.000i q^{67} -216.000 q^{69} -480.000 q^{71} +434.000i q^{73} -672.000i q^{77} +1352.00 q^{79} +81.0000 q^{81} +612.000i q^{83} -918.000i q^{87} +30.0000 q^{89} -1960.00 q^{91} +408.000i q^{93} +286.000i q^{97} -216.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} + 48 q^{11} + 40 q^{19} + 168 q^{21} - 612 q^{29} + 272 q^{31} + 420 q^{39} - 300 q^{41} - 882 q^{49} + 612 q^{51} - 1488 q^{59} - 836 q^{61} - 432 q^{69} - 960 q^{71} + 2704 q^{79} + 162 q^{81}+ \cdots - 432 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 + 48 * q^11 + 40 * q^19 + 168 * q^21 - 612 * q^29 + 272 * q^31 + 420 * q^39 - 300 * q^41 - 882 * q^49 + 612 * q^51 - 1488 * q^59 - 836 * q^61 - 432 * q^69 - 960 * q^71 + 2704 * q^79 + 162 * q^81 + 60 * q^89 - 3920 * q^91 - 432 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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