LMFDB - Embedded newform 1200.4.f.s.49.1

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,-18,0,108,0,0,0,0,0,0,0,-50] 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 300)
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.s.49.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} +54.0000 q^{11} -55.0000i q^{13} -18.0000i q^{17} -25.0000 q^{19} -21.0000 q^{21} +18.0000i q^{23} +27.0000i q^{27} +54.0000 q^{29} +271.000 q^{31} -162.000i q^{33} -314.000i q^{37} -165.000 q^{39} -360.000 q^{41} +163.000i q^{43} +522.000i q^{47} +294.000 q^{49} -54.0000 q^{51} -36.0000i q^{53} +75.0000i q^{57} +126.000 q^{59} +47.0000 q^{61} +63.0000i q^{63} -343.000i q^{67} +54.0000 q^{69} +1080.00 q^{71} -1054.00i q^{73} -378.000i q^{77} -568.000 q^{79} +81.0000 q^{81} -1422.00i q^{83} -162.000i q^{87} -1440.00 q^{89} -385.000 q^{91} -813.000i q^{93} +439.000i q^{97} -486.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} + 108 q^{11} - 50 q^{19} - 42 q^{21} + 108 q^{29} + 542 q^{31} - 330 q^{39} - 720 q^{41} + 588 q^{49} - 108 q^{51} + 252 q^{59} + 94 q^{61} + 108 q^{69} + 2160 q^{71} - 1136 q^{79} + 162 q^{81}+ \cdots - 972 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 + 108 * q^11 - 50 * q^19 - 42 * q^21 + 108 * q^29 + 542 * q^31 - 330 * q^39 - 720 * q^41 + 588 * q^49 - 108 * q^51 + 252 * q^59 + 94 * q^61 + 108 * q^69 + 2160 * q^71 - 1136 * q^79 + 162 * q^81 - 2880 * q^89 - 770 * q^91 - 972 * 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
$7$	− 7.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 7.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	54.0000	1.48015	0.740073	−	0.672526i	$-0.234791\pi$
$11$	54.0000	1.48015	0.740073	+	0.672526i	$0.234791\pi$
$12$	0	0
$13$	− 55.0000i	− 1.17340i	−0.809803	−	0.586702i	$-0.800426\pi$
$13$	− 55.0000i	− 1.17340i	0.809803	−	0.586702i	$-0.199574\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 18.0000i	− 0.256802i	−0.991722	−	0.128401i	$-0.959015\pi$
$17$	− 18.0000i	− 0.256802i	0.991722	−	0.128401i	$-0.0409845\pi$
$18$	0	0
$19$	−25.0000	−0.301863	−0.150931	−	0.988544i	$-0.548227\pi$
$19$	−25.0000	−0.301863	−0.150931	+	0.988544i	$0.548227\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	18.0000i	0.163185i	0.996666	+	0.0815926i	$0.0260006\pi$
$23$	18.0000i	0.163185i	−0.996666	+	0.0815926i	$0.973999\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	54.0000	0.345778	0.172889	−	0.984941i	$-0.444690\pi$
$29$	54.0000	0.345778	0.172889	+	0.984941i	$0.444690\pi$
$30$	0	0
$31$	271.000	1.57010	0.785049	−	0.619434i	$-0.212638\pi$
$31$	271.000	1.57010	0.785049	+	0.619434i	$0.212638\pi$
$32$	0	0
$33$	− 162.000i	− 0.854563i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 314.000i	− 1.39517i	−0.716502	−	0.697585i	$-0.754258\pi$
$37$	− 314.000i	− 1.39517i	0.716502	−	0.697585i	$-0.245742\pi$
$38$	0	0
$39$	−165.000	−0.677465
$40$	0	0
$41$	−360.000	−1.37128	−0.685641	−	0.727940i	$-0.740478\pi$
$41$	−360.000	−1.37128	−0.685641	+	0.727940i	$0.740478\pi$
$42$	0	0
$43$	163.000i	0.578076i	0.957318	+	0.289038i	$0.0933354\pi$
$43$	163.000i	0.578076i	−0.957318	+	0.289038i	$0.906665\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	522.000i	1.62003i	0.586407	+	0.810016i	$0.300542\pi$
$47$	522.000i	1.62003i	−0.586407	+	0.810016i	$0.699458\pi$
$48$	0	0
$49$	294.000	0.857143
$50$	0	0
$51$	−54.0000	−0.148265
$52$	0	0
$53$	− 36.0000i	− 0.0933015i	−0.998911	−	0.0466508i	$-0.985145\pi$
$53$	− 36.0000i	− 0.0933015i	0.998911	−	0.0466508i	$-0.0148548\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	75.0000i	0.174281i
$58$	0	0
$59$	126.000	0.278031	0.139015	−	0.990290i	$-0.455606\pi$
$59$	126.000	0.278031	0.139015	+	0.990290i	$0.455606\pi$
$60$	0	0
$61$	47.0000	0.0986514	0.0493257	−	0.998783i	$-0.484293\pi$
$61$	47.0000	0.0986514	0.0493257	+	0.998783i	$0.484293\pi$
$62$	0	0
$63$	63.0000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 343.000i	− 0.625435i	−0.949846	−	0.312717i	$-0.898761\pi$
$67$	− 343.000i	− 0.625435i	0.949846	−	0.312717i	$-0.101239\pi$
$68$	0	0
$69$	54.0000	0.0942150
$70$	0	0
$71$	1080.00	1.80525	0.902623	−	0.430433i	$-0.141639\pi$
$71$	1080.00	1.80525	0.902623	+	0.430433i	$0.141639\pi$
$72$	0	0
$73$	− 1054.00i	− 1.68988i	−0.534860	−	0.844941i	$-0.679636\pi$
$73$	− 1054.00i	− 1.68988i	0.534860	−	0.844941i	$-0.320364\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 378.000i	− 0.559443i
$78$	0	0
$79$	−568.000	−0.808924	−0.404462	−	0.914555i	$-0.632541\pi$
$79$	−568.000	−0.808924	−0.404462	+	0.914555i	$0.632541\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1422.00i	− 1.88054i	−0.340430	−	0.940270i	$-0.610573\pi$
$83$	− 1422.00i	− 1.88054i	0.340430	−	0.940270i	$-0.389427\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 162.000i	− 0.199635i
$88$	0	0
$89$	−1440.00	−1.71505	−0.857526	−	0.514440i	$-0.828000\pi$
$89$	−1440.00	−1.71505	−0.857526	+	0.514440i	$0.828000\pi$
$90$	0	0
$91$	−385.000	−0.443505
$92$	0	0
$93$	− 813.000i	− 0.906496i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	439.000i	0.459523i	0.973247	+	0.229761i	$0.0737946\pi$
$97$	439.000i	0.459523i	−0.973247	+	0.229761i	$0.926205\pi$
$98$	0	0
$99$	−486.000	−0.493382
$100$	0	0
$101$	828.000	0.815733	0.407867	−	0.913041i	$-0.366273\pi$
$101$	828.000	0.815733	0.407867	+	0.913041i	$0.366273\pi$
$102$	0	0
$103$	− 548.000i	− 0.524233i	−0.965036	−	0.262117i	$-0.915579\pi$
$103$	− 548.000i	− 0.524233i	0.965036	−	0.262117i	$-0.0844205\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1476.00i	− 1.33355i	−0.745257	−	0.666777i	$-0.767673\pi$
$107$	− 1476.00i	− 1.33355i	0.745257	−	0.666777i	$-0.232327\pi$
$108$	0	0
$109$	−1277.00	−1.12215	−0.561075	−	0.827765i	$-0.689612\pi$
$109$	−1277.00	−1.12215	−0.561075	+	0.827765i	$0.689612\pi$
$110$	0	0
$111$	−942.000	−0.805502
$112$	0	0
$113$	1836.00i	1.52846i	0.644942	+	0.764232i	$0.276882\pi$
$113$	1836.00i	1.52846i	−0.644942	+	0.764232i	$0.723118\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	495.000i	0.391135i
$118$	0	0
$119$	−126.000	−0.0970622
$120$	0	0
$121$	1585.00	1.19083
$122$	0	0
$123$	1080.00i	0.791710i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 592.000i	− 0.413634i	−0.978380	−	0.206817i	$-0.933690\pi$
$127$	− 592.000i	− 0.413634i	0.978380	−	0.206817i	$-0.0663105\pi$
$128$	0	0
$129$	489.000	0.333752
$130$	0	0
$131$	468.000	0.312132	0.156066	−	0.987747i	$-0.450119\pi$
$131$	468.000	0.312132	0.156066	+	0.987747i	$0.450119\pi$
$132$	0	0
$133$	175.000i	0.114093i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2574.00i	− 1.60519i	−0.596521	−	0.802597i	$-0.703451\pi$
$137$	− 2574.00i	− 1.60519i	0.596521	−	0.802597i	$-0.296549\pi$
$138$	0	0
$139$	−1756.00	−1.07153	−0.535763	−	0.844369i	$-0.679976\pi$
$139$	−1756.00	−1.07153	−0.535763	+	0.844369i	$0.679976\pi$
$140$	0	0
$141$	1566.00	0.935326
$142$	0	0
$143$	− 2970.00i	− 1.73681i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 882.000i	− 0.494872i
$148$	0	0
$149$	−2682.00	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−2682.00	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−3395.00	−1.82968	−0.914838	−	0.403820i	$-0.867682\pi$
$151$	−3395.00	−1.82968	−0.914838	+	0.403820i	$0.867682\pi$
$152$	0	0
$153$	162.000i	0.0856008i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1549.00i	0.787412i	0.919236	+	0.393706i	$0.128807\pi$
$157$	1549.00i	0.787412i	−0.919236	+	0.393706i	$0.871193\pi$
$158$	0	0
$159$	−108.000	−0.0538677
$160$	0	0
$161$	126.000	0.0616782
$162$	0	0
$163$	505.000i	0.242667i	0.992612	+	0.121333i	$0.0387170\pi$
$163$	505.000i	0.242667i	−0.992612	+	0.121333i	$0.961283\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1476.00i	− 0.683930i	−0.939713	−	0.341965i	$-0.888908\pi$
$167$	− 1476.00i	− 0.683930i	0.939713	−	0.341965i	$-0.111092\pi$
$168$	0	0
$169$	−828.000	−0.376878
$170$	0	0
$171$	225.000	0.100621
$172$	0	0
$173$	2358.00i	1.03627i	0.855298	+	0.518137i	$0.173374\pi$
$173$	2358.00i	1.03627i	−0.855298	+	0.518137i	$0.826626\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 378.000i	− 0.160521i
$178$	0	0
$179$	1566.00	0.653901	0.326951	−	0.945041i	$-0.393979\pi$
$179$	1566.00	0.653901	0.326951	+	0.945041i	$0.393979\pi$
$180$	0	0
$181$	305.000	0.125251	0.0626256	−	0.998037i	$-0.480053\pi$
$181$	305.000	0.125251	0.0626256	+	0.998037i	$0.480053\pi$
$182$	0	0
$183$	− 141.000i	− 0.0569564i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 972.000i	− 0.380105i
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	1746.00	0.661446	0.330723	−	0.943728i	$-0.392707\pi$
$191$	1746.00	0.661446	0.330723	+	0.943728i	$0.392707\pi$
$192$	0	0
$193$	− 3877.00i	− 1.44597i	−0.690863	−	0.722986i	$-0.742769\pi$
$193$	− 3877.00i	− 1.44597i	0.690863	−	0.722986i	$-0.257231\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2142.00i	− 0.774676i	−0.921938	−	0.387338i	$-0.873395\pi$
$197$	− 2142.00i	− 0.774676i	0.921938	−	0.387338i	$-0.126605\pi$
$198$	0	0
$199$	−4033.00	−1.43664	−0.718321	−	0.695712i	$-0.755089\pi$
$199$	−4033.00	−1.43664	−0.718321	+	0.695712i	$0.755089\pi$
$200$	0	0
$201$	−1029.00	−0.361095
$202$	0	0
$203$	− 378.000i	− 0.130692i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 162.000i	− 0.0543951i
$208$	0	0
$209$	−1350.00	−0.446801
$210$	0	0
$211$	4105.00	1.33934	0.669668	−	0.742661i	$-0.266436\pi$
$211$	4105.00	1.33934	0.669668	+	0.742661i	$0.266436\pi$
$212$	0	0
$213$	− 3240.00i	− 1.04226i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1897.00i	− 0.593441i
$218$	0	0
$219$	−3162.00	−0.975654
$220$	0	0
$221$	−990.000	−0.301333
$222$	0	0
$223$	− 1385.00i	− 0.415903i	−0.978139	−	0.207952i	$-0.933320\pi$
$223$	− 1385.00i	− 0.415903i	0.978139	−	0.207952i	$-0.0666797\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2520.00i	− 0.736821i	−0.929663	−	0.368410i	$-0.879902\pi$
$227$	− 2520.00i	− 0.736821i	0.929663	−	0.368410i	$-0.120098\pi$
$228$	0	0
$229$	−5129.00	−1.48006	−0.740030	−	0.672574i	$-0.765189\pi$
$229$	−5129.00	−1.48006	−0.740030	+	0.672574i	$0.765189\pi$
$230$	0	0
$231$	−1134.00	−0.322994
$232$	0	0
$233$	− 3240.00i	− 0.910985i	−0.890240	−	0.455492i	$-0.849463\pi$
$233$	− 3240.00i	− 0.910985i	0.890240	−	0.455492i	$-0.150537\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1704.00i	0.467032i
$238$	0	0
$239$	−2988.00	−0.808693	−0.404347	−	0.914606i	$-0.632501\pi$
$239$	−2988.00	−0.808693	−0.404347	+	0.914606i	$0.632501\pi$
$240$	0	0
$241$	−2647.00	−0.707503	−0.353752	−	0.935339i	$-0.615094\pi$
$241$	−2647.00	−0.707503	−0.353752	+	0.935339i	$0.615094\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1375.00i	0.354207i
$248$	0	0
$249$	−4266.00	−1.08573
$250$	0	0
$251$	4212.00	1.05920	0.529600	−	0.848248i	$-0.322342\pi$
$251$	4212.00	1.05920	0.529600	+	0.848248i	$0.322342\pi$
$252$	0	0
$253$	972.000i	0.241538i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5724.00i	− 1.38931i	−0.719342	−	0.694656i	$-0.755556\pi$
$257$	− 5724.00i	− 1.38931i	0.719342	−	0.694656i	$-0.244444\pi$
$258$	0	0
$259$	−2198.00	−0.527325
$260$	0	0
$261$	−486.000	−0.115259
$262$	0	0
$263$	4608.00i	1.08039i	0.841541	+	0.540193i	$0.181649\pi$
$263$	4608.00i	1.08039i	−0.841541	+	0.540193i	$0.818351\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4320.00i	0.990186i
$268$	0	0
$269$	6426.00	1.45651	0.728253	−	0.685308i	$-0.240333\pi$
$269$	6426.00	1.45651	0.728253	+	0.685308i	$0.240333\pi$
$270$	0	0
$271$	3376.00	0.756743	0.378372	−	0.925654i	$-0.376484\pi$
$271$	3376.00	0.756743	0.378372	+	0.925654i	$0.376484\pi$
$272$	0	0
$273$	1155.00i	0.256058i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 5381.00i	− 1.16719i	−0.812043	−	0.583597i	$-0.801645\pi$
$277$	− 5381.00i	− 1.16719i	0.812043	−	0.583597i	$-0.198355\pi$
$278$	0	0
$279$	−2439.00	−0.523366
$280$	0	0
$281$	−3474.00	−0.737514	−0.368757	−	0.929526i	$-0.620217\pi$
$281$	−3474.00	−0.737514	−0.368757	+	0.929526i	$0.620217\pi$
$282$	0	0
$283$	2269.00i	0.476601i	0.971191	+	0.238300i	$0.0765903\pi$
$283$	2269.00i	0.476601i	−0.971191	+	0.238300i	$0.923410\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2520.00i	0.518296i
$288$	0	0
$289$	4589.00	0.934053
$290$	0	0
$291$	1317.00	0.265306
$292$	0	0
$293$	− 1674.00i	− 0.333775i	−0.985976	−	0.166888i	$-0.946628\pi$
$293$	− 1674.00i	− 0.333775i	0.985976	−	0.166888i	$-0.0533717\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1458.00i	0.284854i
$298$	0	0
$299$	990.000	0.191482
$300$	0	0
$301$	1141.00	0.218492
$302$	0	0
$303$	− 2484.00i	− 0.470964i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	539.000i	0.100203i	0.998744	+	0.0501016i	$0.0159545\pi$
$307$	539.000i	0.100203i	−0.998744	+	0.0501016i	$0.984046\pi$
$308$	0	0
$309$	−1644.00	−0.302666
$310$	0	0
$311$	−1494.00	−0.272402	−0.136201	−	0.990681i	$-0.543489\pi$
$311$	−1494.00	−0.272402	−0.136201	+	0.990681i	$0.543489\pi$
$312$	0	0
$313$	− 3997.00i	− 0.721801i	−0.932604	−	0.360901i	$-0.882469\pi$
$313$	− 3997.00i	− 0.721801i	0.932604	−	0.360901i	$-0.117531\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3672.00i	0.650600i	0.945611	+	0.325300i	$0.105465\pi$
$317$	3672.00i	0.650600i	−0.945611	+	0.325300i	$0.894535\pi$
$318$	0	0
$319$	2916.00	0.511801
$320$	0	0
$321$	−4428.00	−0.769928
$322$	0	0
$323$	450.000i	0.0775191i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3831.00i	0.647874i
$328$	0	0
$329$	3654.00	0.612315
$330$	0	0
$331$	−1052.00	−0.174692	−0.0873461	−	0.996178i	$-0.527839\pi$
$331$	−1052.00	−0.174692	−0.0873461	+	0.996178i	$0.527839\pi$
$332$	0	0
$333$	2826.00i	0.465057i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3871.00i	0.625718i	0.949800	+	0.312859i	$0.101287\pi$
$337$	3871.00i	0.625718i	−0.949800	+	0.312859i	$0.898713\pi$
$338$	0	0
$339$	5508.00	0.882459
$340$	0	0
$341$	14634.0	2.32398
$342$	0	0
$343$	− 4459.00i	− 0.701934i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7686.00i	− 1.18907i	−0.804071	−	0.594533i	$-0.797337\pi$
$347$	− 7686.00i	− 1.18907i	0.804071	−	0.594533i	$-0.202663\pi$
$348$	0	0
$349$	46.0000	0.00705537	0.00352768	−	0.999994i	$-0.498877\pi$
$349$	46.0000	0.00705537	0.00352768	+	0.999994i	$0.498877\pi$
$350$	0	0
$351$	1485.00	0.225822
$352$	0	0
$353$	− 6714.00i	− 1.01232i	−0.862439	−	0.506162i	$-0.831064\pi$
$353$	− 6714.00i	− 1.01232i	0.862439	−	0.506162i	$-0.168936\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	378.000i	0.0560389i
$358$	0	0
$359$	1296.00	0.190530	0.0952650	−	0.995452i	$-0.469630\pi$
$359$	1296.00	0.190530	0.0952650	+	0.995452i	$0.469630\pi$
$360$	0	0
$361$	−6234.00	−0.908879
$362$	0	0
$363$	− 4755.00i	− 0.687528i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5903.00i	0.839602i	0.907616	+	0.419801i	$0.137900\pi$
$367$	5903.00i	0.839602i	−0.907616	+	0.419801i	$0.862100\pi$
$368$	0	0
$369$	3240.00	0.457094
$370$	0	0
$371$	−252.000	−0.0352647
$372$	0	0
$373$	8867.00i	1.23087i	0.788186	+	0.615437i	$0.211020\pi$
$373$	8867.00i	1.23087i	−0.788186	+	0.615437i	$0.788980\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2970.00i	− 0.405737i
$378$	0	0
$379$	8837.00	1.19769	0.598847	−	0.800863i	$-0.295626\pi$
$379$	8837.00	1.19769	0.598847	+	0.800863i	$0.295626\pi$
$380$	0	0
$381$	−1776.00	−0.238812
$382$	0	0
$383$	10044.0i	1.34001i	0.742356	+	0.670006i	$0.233708\pi$
$383$	10044.0i	1.34001i	−0.742356	+	0.670006i	$0.766292\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1467.00i	− 0.192692i
$388$	0	0
$389$	2736.00	0.356609	0.178304	−	0.983975i	$-0.442939\pi$
$389$	2736.00	0.356609	0.178304	+	0.983975i	$0.442939\pi$
$390$	0	0
$391$	324.000	0.0419064
$392$	0	0
$393$	− 1404.00i	− 0.180210i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12241.0i	1.54750i	0.633490	+	0.773751i	$0.281622\pi$
$397$	12241.0i	1.54750i	−0.633490	+	0.773751i	$0.718378\pi$
$398$	0	0
$399$	525.000	0.0658719
$400$	0	0
$401$	9036.00	1.12528	0.562639	−	0.826703i	$-0.309786\pi$
$401$	9036.00	1.12528	0.562639	+	0.826703i	$0.309786\pi$
$402$	0	0
$403$	− 14905.0i	− 1.84236i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 16956.0i	− 2.06506i
$408$	0	0
$409$	−8549.00	−1.03355	−0.516774	−	0.856122i	$-0.672867\pi$
$409$	−8549.00	−1.03355	−0.516774	+	0.856122i	$0.672867\pi$
$410$	0	0
$411$	−7722.00	−0.926760
$412$	0	0
$413$	− 882.000i	− 0.105086i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5268.00i	0.618645i
$418$	0	0
$419$	1548.00	0.180489	0.0902443	−	0.995920i	$-0.471235\pi$
$419$	1548.00	0.180489	0.0902443	+	0.995920i	$0.471235\pi$
$420$	0	0
$421$	6110.00	0.707323	0.353662	−	0.935373i	$-0.384936\pi$
$421$	6110.00	0.707323	0.353662	+	0.935373i	$0.384936\pi$
$422$	0	0
$423$	− 4698.00i	− 0.540011i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 329.000i	− 0.0372867i
$428$	0	0
$429$	−8910.00	−1.00275
$430$	0	0
$431$	−5958.00	−0.665863	−0.332931	−	0.942951i	$-0.608038\pi$
$431$	−5958.00	−0.665863	−0.332931	+	0.942951i	$0.608038\pi$
$432$	0	0
$433$	7163.00i	0.794993i	0.917604	+	0.397496i	$0.130121\pi$
$433$	7163.00i	0.794993i	−0.917604	+	0.397496i	$0.869879\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 450.000i	− 0.0492595i
$438$	0	0
$439$	17.0000	0.00184821	0.000924107	−	1.00000i	$-0.499706\pi$
$439$	17.0000	0.00184821	0.000924107		1.00000i	$0.499706\pi$
$440$	0	0
$441$	−2646.00	−0.285714
$442$	0	0
$443$	9432.00i	1.01158i	0.862658	+	0.505788i	$0.168798\pi$
$443$	9432.00i	1.01158i	−0.862658	+	0.505788i	$0.831202\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8046.00i	0.851371i
$448$	0	0
$449$	15228.0	1.60057	0.800283	−	0.599623i	$-0.204683\pi$
$449$	15228.0	1.60057	0.800283	+	0.599623i	$0.204683\pi$
$450$	0	0
$451$	−19440.0	−2.02970
$452$	0	0
$453$	10185.0i	1.05636i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	250.000i	0.0255897i	0.999918	+	0.0127949i	$0.00407284\pi$
$457$	250.000i	0.0255897i	−0.999918	+	0.0127949i	$0.995927\pi$
$458$	0	0
$459$	486.000	0.0494217
$460$	0	0
$461$	16956.0	1.71306	0.856529	−	0.516099i	$-0.172616\pi$
$461$	16956.0	1.71306	0.856529	+	0.516099i	$0.172616\pi$
$462$	0	0
$463$	4384.00i	0.440047i	0.975495	+	0.220023i	$0.0706134\pi$
$463$	4384.00i	0.440047i	−0.975495	+	0.220023i	$0.929387\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5166.00i	− 0.511893i	−0.966691	−	0.255946i	$-0.917613\pi$
$467$	− 5166.00i	− 0.511893i	0.966691	−	0.255946i	$-0.0823871\pi$
$468$	0	0
$469$	−2401.00	−0.236392
$470$	0	0
$471$	4647.00	0.454612
$472$	0	0
$473$	8802.00i	0.855637i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	324.000i	0.0311005i
$478$	0	0
$479$	6966.00	0.664477	0.332239	−	0.943195i	$-0.392196\pi$
$479$	6966.00	0.664477	0.332239	+	0.943195i	$0.392196\pi$
$480$	0	0
$481$	−17270.0	−1.63710
$482$	0	0
$483$	− 378.000i	− 0.0356099i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	18431.0i	1.71497i	0.514512	+	0.857483i	$0.327973\pi$
$487$	18431.0i	1.71497i	−0.514512	+	0.857483i	$0.672027\pi$
$488$	0	0
$489$	1515.00	0.140104
$490$	0	0
$491$	−1224.00	−0.112502	−0.0562509	−	0.998417i	$-0.517915\pi$
$491$	−1224.00	−0.112502	−0.0562509	+	0.998417i	$0.517915\pi$
$492$	0	0
$493$	− 972.000i	− 0.0887965i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7560.00i	− 0.682319i
$498$	0	0
$499$	−2449.00	−0.219704	−0.109852	−	0.993948i	$-0.535038\pi$
$499$	−2449.00	−0.219704	−0.109852	+	0.993948i	$0.535038\pi$
$500$	0	0
$501$	−4428.00	−0.394867
$502$	0	0
$503$	3312.00i	0.293588i	0.989167	+	0.146794i	$0.0468954\pi$
$503$	3312.00i	0.293588i	−0.989167	+	0.146794i	$0.953105\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2484.00i	0.217590i
$508$	0	0
$509$	9162.00	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	9162.00	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	−7378.00	−0.638715
$512$	0	0
$513$	− 675.000i	− 0.0580935i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	28188.0i	2.39789i
$518$	0	0
$519$	7074.00	0.598293
$520$	0	0
$521$	−5418.00	−0.455599	−0.227799	−	0.973708i	$-0.573153\pi$
$521$	−5418.00	−0.455599	−0.227799	+	0.973708i	$0.573153\pi$
$522$	0	0
$523$	6829.00i	0.570959i	0.958385	+	0.285479i	$0.0921528\pi$
$523$	6829.00i	0.570959i	−0.958385	+	0.285479i	$0.907847\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4878.00i	− 0.403205i
$528$	0	0
$529$	11843.0	0.973371
$530$	0	0
$531$	−1134.00	−0.0926769
$532$	0	0
$533$	19800.0i	1.60907i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4698.00i	− 0.377530i
$538$	0	0
$539$	15876.0	1.26870
$540$	0	0
$541$	3053.00	0.242622	0.121311	−	0.992615i	$-0.461290\pi$
$541$	3053.00	0.242622	0.121311	+	0.992615i	$0.461290\pi$
$542$	0	0
$543$	− 915.000i	− 0.0723138i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20456.0i	1.59897i	0.600687	+	0.799484i	$0.294894\pi$
$547$	20456.0i	1.59897i	−0.600687	+	0.799484i	$0.705106\pi$
$548$	0	0
$549$	−423.000	−0.0328838
$550$	0	0
$551$	−1350.00	−0.104377
$552$	0	0
$553$	3976.00i	0.305745i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10854.0i	0.825671i	0.910806	+	0.412835i	$0.135462\pi$
$557$	10854.0i	0.825671i	−0.910806	+	0.412835i	$0.864538\pi$
$558$	0	0
$559$	8965.00	0.678317
$560$	0	0
$561$	−2916.00	−0.219454
$562$	0	0
$563$	24930.0i	1.86621i	0.359609	+	0.933103i	$0.382910\pi$
$563$	24930.0i	1.86621i	−0.359609	+	0.933103i	$0.617090\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 567.000i	− 0.0419961i
$568$	0	0
$569$	24786.0	1.82616	0.913078	−	0.407784i	$-0.133698\pi$
$569$	24786.0	1.82616	0.913078	+	0.407784i	$0.133698\pi$
$570$	0	0
$571$	14785.0	1.08360	0.541798	−	0.840509i	$-0.317744\pi$
$571$	14785.0	1.08360	0.541798	+	0.840509i	$0.317744\pi$
$572$	0	0
$573$	− 5238.00i	− 0.381886i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 15851.0i	− 1.14365i	−0.820376	−	0.571825i	$-0.806236\pi$
$577$	− 15851.0i	− 1.14365i	0.820376	−	0.571825i	$-0.193764\pi$
$578$	0	0
$579$	−11631.0	−0.834832
$580$	0	0
$581$	−9954.00	−0.710777
$582$	0	0
$583$	− 1944.00i	− 0.138100i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23148.0i	1.62763i	0.581122	+	0.813816i	$0.302614\pi$
$587$	23148.0i	1.62763i	−0.581122	+	0.813816i	$0.697386\pi$
$588$	0	0
$589$	−6775.00	−0.473954
$590$	0	0
$591$	−6426.00	−0.447259
$592$	0	0
$593$	− 21888.0i	− 1.51574i	−0.652407	−	0.757869i	$-0.726241\pi$
$593$	− 21888.0i	− 1.51574i	0.652407	−	0.757869i	$-0.273759\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12099.0i	0.829446i
$598$	0	0
$599$	10764.0	0.734232	0.367116	−	0.930175i	$-0.380345\pi$
$599$	10764.0	0.734232	0.367116	+	0.930175i	$0.380345\pi$
$600$	0	0
$601$	−25597.0	−1.73731	−0.868655	−	0.495417i	$-0.835015\pi$
$601$	−25597.0	−1.73731	−0.868655	+	0.495417i	$0.835015\pi$
$602$	0	0
$603$	3087.00i	0.208478i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 24976.0i	− 1.67009i	−0.550182	−	0.835045i	$-0.685442\pi$
$607$	− 24976.0i	− 1.67009i	0.550182	−	0.835045i	$-0.314558\pi$
$608$	0	0
$609$	−1134.00	−0.0754548
$610$	0	0
$611$	28710.0	1.90095
$612$	0	0
$613$	− 2134.00i	− 0.140606i	−0.997526	−	0.0703030i	$-0.977603\pi$
$613$	− 2134.00i	− 0.140606i	0.997526	−	0.0703030i	$-0.0223966\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 13932.0i	− 0.909046i	−0.890735	−	0.454523i	$-0.849810\pi$
$617$	− 13932.0i	− 0.909046i	0.890735	−	0.454523i	$-0.150190\pi$
$618$	0	0
$619$	−10429.0	−0.677184	−0.338592	−	0.940933i	$-0.609951\pi$
$619$	−10429.0	−0.677184	−0.338592	+	0.940933i	$0.609951\pi$
$620$	0	0
$621$	−486.000	−0.0314050
$622$	0	0
$623$	10080.0i	0.648229i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4050.00i	0.257961i
$628$	0	0
$629$	−5652.00	−0.358283
$630$	0	0
$631$	6283.00	0.396390	0.198195	−	0.980163i	$-0.436492\pi$
$631$	6283.00	0.396390	0.198195	+	0.980163i	$0.436492\pi$
$632$	0	0
$633$	− 12315.0i	− 0.773266i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 16170.0i	− 1.00578i
$638$	0	0
$639$	−9720.00	−0.601748
$640$	0	0
$641$	−20916.0	−1.28882	−0.644409	−	0.764681i	$-0.722897\pi$
$641$	−20916.0	−1.28882	−0.644409	+	0.764681i	$0.722897\pi$
$642$	0	0
$643$	23452.0i	1.43835i	0.694831	+	0.719173i	$0.255479\pi$
$643$	23452.0i	1.43835i	−0.694831	+	0.719173i	$0.744521\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 11916.0i	− 0.724059i	−0.932167	−	0.362030i	$-0.882084\pi$
$647$	− 11916.0i	− 0.724059i	0.932167	−	0.362030i	$-0.117916\pi$
$648$	0	0
$649$	6804.00	0.411526
$650$	0	0
$651$	−5691.00	−0.342623
$652$	0	0
$653$	31842.0i	1.90823i	0.299441	+	0.954115i	$0.403200\pi$
$653$	31842.0i	1.90823i	−0.299441	+	0.954115i	$0.596800\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9486.00i	0.563294i
$658$	0	0
$659$	3672.00	0.217057	0.108529	−	0.994093i	$-0.465386\pi$
$659$	3672.00	0.217057	0.108529	+	0.994093i	$0.465386\pi$
$660$	0	0
$661$	5138.00	0.302337	0.151169	−	0.988508i	$-0.451696\pi$
$661$	5138.00	0.302337	0.151169	+	0.988508i	$0.451696\pi$
$662$	0	0
$663$	2970.00i	0.173975i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	972.000i	0.0564258i
$668$	0	0
$669$	−4155.00	−0.240122
$670$	0	0
$671$	2538.00	0.146018
$672$	0	0
$673$	− 5050.00i	− 0.289247i	−0.989487	−	0.144623i	$-0.953803\pi$
$673$	− 5050.00i	− 0.289247i	0.989487	−	0.144623i	$-0.0461971\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 28458.0i	− 1.61555i	−0.589488	−	0.807777i	$-0.700671\pi$
$677$	− 28458.0i	− 1.61555i	0.589488	−	0.807777i	$-0.299329\pi$
$678$	0	0
$679$	3073.00	0.173683
$680$	0	0
$681$	−7560.00	−0.425404
$682$	0	0
$683$	− 24408.0i	− 1.36742i	−0.729755	−	0.683709i	$-0.760366\pi$
$683$	− 24408.0i	− 1.36742i	0.729755	−	0.683709i	$-0.239634\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15387.0i	0.854513i
$688$	0	0
$689$	−1980.00	−0.109480
$690$	0	0
$691$	−16328.0	−0.898909	−0.449455	−	0.893303i	$-0.648382\pi$
$691$	−16328.0	−0.898909	−0.449455	+	0.893303i	$0.648382\pi$
$692$	0	0
$693$	3402.00i	0.186481i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6480.00i	0.352148i
$698$	0	0
$699$	−9720.00	−0.525957
$700$	0	0
$701$	6246.00	0.336531	0.168265	−	0.985742i	$-0.446183\pi$
$701$	6246.00	0.336531	0.168265	+	0.985742i	$0.446183\pi$
$702$	0	0
$703$	7850.00i	0.421150i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 5796.00i	− 0.308318i
$708$	0	0
$709$	30679.0	1.62507	0.812535	−	0.582913i	$-0.198087\pi$
$709$	30679.0	1.62507	0.812535	+	0.582913i	$0.198087\pi$
$710$	0	0
$711$	5112.00	0.269641
$712$	0	0
$713$	4878.00i	0.256217i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8964.00i	0.466899i
$718$	0	0
$719$	15462.0	0.801996	0.400998	−	0.916079i	$-0.368663\pi$
$719$	15462.0	0.801996	0.400998	+	0.916079i	$0.368663\pi$
$720$	0	0
$721$	−3836.00	−0.198142
$722$	0	0
$723$	7941.00i	0.408477i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26801.0i	1.36725i	0.729831	+	0.683627i	$0.239599\pi$
$727$	26801.0i	1.36725i	−0.729831	+	0.683627i	$0.760401\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	2934.00	0.148451
$732$	0	0
$733$	− 22858.0i	− 1.15181i	−0.817515	−	0.575907i	$-0.804649\pi$
$733$	− 22858.0i	− 1.15181i	0.817515	−	0.575907i	$-0.195351\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 18522.0i	− 0.925735i
$738$	0	0
$739$	−16900.0	−0.841240	−0.420620	−	0.907237i	$-0.638187\pi$
$739$	−16900.0	−0.841240	−0.420620	+	0.907237i	$0.638187\pi$
$740$	0	0
$741$	4125.00	0.204502
$742$	0	0
$743$	− 8028.00i	− 0.396391i	−0.980162	−	0.198196i	$-0.936492\pi$
$743$	− 8028.00i	− 0.396391i	0.980162	−	0.198196i	$-0.0635081\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12798.0i	0.626846i
$748$	0	0
$749$	−10332.0	−0.504036
$750$	0	0
$751$	12448.0	0.604839	0.302419	−	0.953175i	$-0.402206\pi$
$751$	12448.0	0.604839	0.302419	+	0.953175i	$0.402206\pi$
$752$	0	0
$753$	− 12636.0i	− 0.611529i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	30103.0i	1.44533i	0.691200	+	0.722663i	$0.257082\pi$
$757$	30103.0i	1.44533i	−0.691200	+	0.722663i	$0.742918\pi$
$758$	0	0
$759$	2916.00	0.139452
$760$	0	0
$761$	−17748.0	−0.845420	−0.422710	−	0.906265i	$-0.638921\pi$
$761$	−17748.0	−0.845420	−0.422710	+	0.906265i	$0.638921\pi$
$762$	0	0
$763$	8939.00i	0.424133i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 6930.00i	− 0.326242i
$768$	0	0
$769$	−13283.0	−0.622883	−0.311442	−	0.950265i	$-0.600812\pi$
$769$	−13283.0	−0.622883	−0.311442	+	0.950265i	$0.600812\pi$
$770$	0	0
$771$	−17172.0	−0.802120
$772$	0	0
$773$	26424.0i	1.22950i	0.788721	+	0.614751i	$0.210744\pi$
$773$	26424.0i	1.22950i	−0.788721	+	0.614751i	$0.789256\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6594.00i	0.304451i
$778$	0	0
$779$	9000.00	0.413939
$780$	0	0
$781$	58320.0	2.67203
$782$	0	0
$783$	1458.00i	0.0665449i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 9709.00i	− 0.439757i	−0.975527	−	0.219878i	$-0.929434\pi$
$787$	− 9709.00i	− 0.439757i	0.975527	−	0.219878i	$-0.0705660\pi$
$788$	0	0
$789$	13824.0	0.623761
$790$	0	0
$791$	12852.0	0.577705
$792$	0	0
$793$	− 2585.00i	− 0.115758i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7884.00i	0.350396i	0.984533	+	0.175198i	$0.0560566\pi$
$797$	7884.00i	0.350396i	−0.984533	+	0.175198i	$0.943943\pi$
$798$	0	0
$799$	9396.00	0.416028
$800$	0	0
$801$	12960.0	0.571684
$802$	0	0
$803$	− 56916.0i	− 2.50127i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 19278.0i	− 0.840914i
$808$	0	0
$809$	1476.00	0.0641451	0.0320726	−	0.999486i	$-0.489789\pi$
$809$	1476.00	0.0641451	0.0320726	+	0.999486i	$0.489789\pi$
$810$	0	0
$811$	−21455.0	−0.928960	−0.464480	−	0.885583i	$-0.653759\pi$
$811$	−21455.0	−0.928960	−0.464480	+	0.885583i	$0.653759\pi$
$812$	0	0
$813$	− 10128.0i	− 0.436906i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4075.00i	− 0.174500i
$818$	0	0
$819$	3465.00	0.147835
$820$	0	0
$821$	31086.0	1.32145	0.660724	−	0.750629i	$-0.270249\pi$
$821$	31086.0	1.32145	0.660724	+	0.750629i	$0.270249\pi$
$822$	0	0
$823$	− 23381.0i	− 0.990292i	−0.868810	−	0.495146i	$-0.835115\pi$
$823$	− 23381.0i	− 0.990292i	0.868810	−	0.495146i	$-0.164885\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3636.00i	0.152885i	0.997074	+	0.0764426i	$0.0243562\pi$
$827$	3636.00i	0.152885i	−0.997074	+	0.0764426i	$0.975644\pi$
$828$	0	0
$829$	−4058.00	−0.170012	−0.0850061	−	0.996380i	$-0.527091\pi$
$829$	−4058.00	−0.170012	−0.0850061	+	0.996380i	$0.527091\pi$
$830$	0	0
$831$	−16143.0	−0.673880
$832$	0	0
$833$	− 5292.00i	− 0.220116i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7317.00i	0.302165i
$838$	0	0
$839$	306.000	0.0125915	0.00629576	−	0.999980i	$-0.497996\pi$
$839$	306.000	0.0125915	0.00629576	+	0.999980i	$0.497996\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	0	0
$843$	10422.0i	0.425804i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 11095.0i	− 0.450093i
$848$	0	0
$849$	6807.00	0.275166
$850$	0	0
$851$	5652.00	0.227671
$852$	0	0
$853$	42299.0i	1.69788i	0.528491	+	0.848939i	$0.322758\pi$
$853$	42299.0i	1.69788i	−0.528491	+	0.848939i	$0.677242\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11484.0i	0.457743i	0.973457	+	0.228872i	$0.0735036\pi$
$857$	11484.0i	0.457743i	−0.973457	+	0.228872i	$0.926496\pi$
$858$	0	0
$859$	33560.0	1.33301	0.666503	−	0.745502i	$-0.267790\pi$
$859$	33560.0	1.33301	0.666503	+	0.745502i	$0.267790\pi$
$860$	0	0
$861$	7560.00	0.299238
$862$	0	0
$863$	14976.0i	0.590717i	0.955386	+	0.295359i	$0.0954391\pi$
$863$	14976.0i	0.590717i	−0.955386	+	0.295359i	$0.904561\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 13767.0i	− 0.539275i
$868$	0	0
$869$	−30672.0	−1.19733
$870$	0	0
$871$	−18865.0	−0.733888
$872$	0	0
$873$	− 3951.00i	− 0.153174i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41893.0i	1.61303i	0.591215	+	0.806514i	$0.298649\pi$
$877$	41893.0i	1.61303i	−0.591215	+	0.806514i	$0.701351\pi$
$878$	0	0
$879$	−5022.00	−0.192705
$880$	0	0
$881$	−720.000	−0.0275340	−0.0137670	−	0.999905i	$-0.504382\pi$
$881$	−720.000	−0.0275340	−0.0137670	+	0.999905i	$0.504382\pi$
$882$	0	0
$883$	− 17309.0i	− 0.659676i	−0.944037	−	0.329838i	$-0.893006\pi$
$883$	− 17309.0i	− 0.659676i	0.944037	−	0.329838i	$-0.106994\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 7002.00i	− 0.265055i	−0.991179	−	0.132528i	$-0.957691\pi$
$887$	− 7002.00i	− 0.265055i	0.991179	−	0.132528i	$-0.0423094\pi$
$888$	0	0
$889$	−4144.00	−0.156339
$890$	0	0
$891$	4374.00	0.164461
$892$	0	0
$893$	− 13050.0i	− 0.489028i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2970.00i	− 0.110552i
$898$	0	0
$899$	14634.0	0.542905
$900$	0	0
$901$	−648.000	−0.0239601
$902$	0	0
$903$	− 3423.00i	− 0.126147i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1484.00i	0.0543279i	0.999631	+	0.0271640i	$0.00864762\pi$
$907$	1484.00i	0.0543279i	−0.999631	+	0.0271640i	$0.991352\pi$
$908$	0	0
$909$	−7452.00	−0.271911
$910$	0	0
$911$	18882.0	0.686705	0.343353	−	0.939207i	$-0.388437\pi$
$911$	18882.0	0.686705	0.343353	+	0.939207i	$0.388437\pi$
$912$	0	0
$913$	− 76788.0i	− 2.78347i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3276.00i	− 0.117975i
$918$	0	0
$919$	−50653.0	−1.81816	−0.909080	−	0.416622i	$-0.863214\pi$
$919$	−50653.0	−1.81816	−0.909080	+	0.416622i	$0.863214\pi$
$920$	0	0
$921$	1617.00	0.0578523
$922$	0	0
$923$	− 59400.0i	− 2.11828i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4932.00i	0.174744i
$928$	0	0
$929$	−35262.0	−1.24533	−0.622663	−	0.782490i	$-0.713949\pi$
$929$	−35262.0	−1.24533	−0.622663	+	0.782490i	$0.713949\pi$
$930$	0	0
$931$	−7350.00	−0.258740
$932$	0	0
$933$	4482.00i	0.157271i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 20279.0i	− 0.707029i	−0.935429	−	0.353514i	$-0.884987\pi$
$937$	− 20279.0i	− 0.707029i	0.935429	−	0.353514i	$-0.115013\pi$
$938$	0	0
$939$	−11991.0	−0.416732
$940$	0	0
$941$	42390.0	1.46852	0.734259	−	0.678870i	$-0.237530\pi$
$941$	42390.0	1.46852	0.734259	+	0.678870i	$0.237530\pi$
$942$	0	0
$943$	− 6480.00i	− 0.223773i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 42426.0i	− 1.45582i	−0.685674	−	0.727909i	$-0.740492\pi$
$947$	− 42426.0i	− 1.45582i	0.685674	−	0.727909i	$-0.259508\pi$
$948$	0	0
$949$	−57970.0	−1.98291
$950$	0	0
$951$	11016.0	0.375624
$952$	0	0
$953$	48168.0i	1.63727i	0.574317	+	0.818633i	$0.305268\pi$
$953$	48168.0i	1.63727i	−0.574317	+	0.818633i	$0.694732\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 8748.00i	− 0.295489i
$958$	0	0
$959$	−18018.0	−0.606707
$960$	0	0
$961$	43650.0	1.46521
$962$	0	0
$963$	13284.0i	0.444518i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 51400.0i	− 1.70932i	−0.519188	−	0.854660i	$-0.673766\pi$
$967$	− 51400.0i	− 1.70932i	0.519188	−	0.854660i	$-0.326234\pi$
$968$	0	0
$969$	1350.00	0.0447557
$970$	0	0
$971$	−33858.0	−1.11901	−0.559503	−	0.828828i	$-0.689008\pi$
$971$	−33858.0	−1.11901	−0.559503	+	0.828828i	$0.689008\pi$
$972$	0	0
$973$	12292.0i	0.404998i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 47106.0i	− 1.54253i	−0.636513	−	0.771266i	$-0.719624\pi$
$977$	− 47106.0i	− 1.54253i	0.636513	−	0.771266i	$-0.280376\pi$
$978$	0	0
$979$	−77760.0	−2.53853
$980$	0	0
$981$	11493.0	0.374050
$982$	0	0
$983$	20844.0i	0.676318i	0.941089	+	0.338159i	$0.109804\pi$
$983$	20844.0i	0.676318i	−0.941089	+	0.338159i	$0.890196\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 10962.0i	− 0.353520i
$988$	0	0
$989$	−2934.00	−0.0943334
$990$	0	0
$991$	−4133.00	−0.132481	−0.0662407	−	0.997804i	$-0.521101\pi$
$991$	−4133.00	−0.132481	−0.0662407	+	0.997804i	$0.521101\pi$
$992$	0	0
$993$	3156.00i	0.100859i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 33122.0i	− 1.05214i	−0.850441	−	0.526070i	$-0.823665\pi$
$997$	− 33122.0i	− 1.05214i	0.850441	−	0.526070i	$-0.176335\pi$
$998$	0	0
$999$	8478.00	0.268501

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} +54.0000 q^{11} -55.0000i q^{13} -18.0000i q^{17} -25.0000 q^{19} -21.0000 q^{21} +18.0000i q^{23} +27.0000i q^{27} +54.0000 q^{29} +271.000 q^{31} -162.000i q^{33} -314.000i q^{37} -165.000 q^{39} -360.000 q^{41} +163.000i q^{43} +522.000i q^{47} +294.000 q^{49} -54.0000 q^{51} -36.0000i q^{53} +75.0000i q^{57} +126.000 q^{59} +47.0000 q^{61} +63.0000i q^{63} -343.000i q^{67} +54.0000 q^{69} +1080.00 q^{71} -1054.00i q^{73} -378.000i q^{77} -568.000 q^{79} +81.0000 q^{81} -1422.00i q^{83} -162.000i q^{87} -1440.00 q^{89} -385.000 q^{91} -813.000i q^{93} +439.000i q^{97} -486.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} + 108 q^{11} - 50 q^{19} - 42 q^{21} + 108 q^{29} + 542 q^{31} - 330 q^{39} - 720 q^{41} + 588 q^{49} - 108 q^{51} + 252 q^{59} + 94 q^{61} + 108 q^{69} + 2160 q^{71} - 1136 q^{79} + 162 q^{81}+ \cdots - 972 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 + 108 * q^11 - 50 * q^19 - 42 * q^21 + 108 * q^29 + 542 * q^31 - 330 * q^39 - 720 * q^41 + 588 * q^49 - 108 * q^51 + 252 * q^59 + 94 * q^61 + 108 * q^69 + 2160 * q^71 - 1136 * q^79 + 162 * q^81 - 2880 * q^89 - 770 * q^91 - 972 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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