LMFDB - Embedded newform 1197.2.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1197,2,Mod(1,1197)] 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("1197.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1197, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1197 = 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1197.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,-4,0,2,-6,0,4,-4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.55809312195$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1197.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.41421 q^{2} +3.82843 q^{4} -2.00000 q^{5} +1.00000 q^{7} -4.41421 q^{8} +4.82843 q^{10} +0.828427 q^{11} +2.00000 q^{13} -2.41421 q^{14} +3.00000 q^{16} +2.00000 q^{17} -1.00000 q^{19} -7.65685 q^{20} -2.00000 q^{22} -0.828427 q^{23} -1.00000 q^{25} -4.82843 q^{26} +3.82843 q^{28} +1.17157 q^{29} +4.00000 q^{31} +1.58579 q^{32} -4.82843 q^{34} -2.00000 q^{35} -3.65685 q^{37} +2.41421 q^{38} +8.82843 q^{40} -3.17157 q^{41} -1.65685 q^{43} +3.17157 q^{44} +2.00000 q^{46} +2.82843 q^{47} +1.00000 q^{49} +2.41421 q^{50} +7.65685 q^{52} +8.48528 q^{53} -1.65685 q^{55} -4.41421 q^{56} -2.82843 q^{58} -1.65685 q^{59} +0.343146 q^{61} -9.65685 q^{62} -9.82843 q^{64} -4.00000 q^{65} +9.65685 q^{67} +7.65685 q^{68} +4.82843 q^{70} +2.00000 q^{71} +3.65685 q^{73} +8.82843 q^{74} -3.82843 q^{76} +0.828427 q^{77} -11.3137 q^{79} -6.00000 q^{80} +7.65685 q^{82} -12.4853 q^{83} -4.00000 q^{85} +4.00000 q^{86} -3.65685 q^{88} +4.82843 q^{89} +2.00000 q^{91} -3.17157 q^{92} -6.82843 q^{94} +2.00000 q^{95} +0.343146 q^{97} -2.41421 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} - 6 q^{8} + 4 q^{10} - 4 q^{11} + 4 q^{13} - 2 q^{14} + 6 q^{16} + 4 q^{17} - 2 q^{19} - 4 q^{20} - 4 q^{22} + 4 q^{23} - 2 q^{25} - 4 q^{26} + 2 q^{28} + 8 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^7 - 6 * q^8 + 4 * q^10 - 4 * q^11 + 4 * q^13 - 2 * q^14 + 6 * q^16 + 4 * q^17 - 2 * q^19 - 4 * q^20 - 4 * q^22 + 4 * q^23 - 2 * q^25 - 4 * q^26 + 2 * q^28 + 8 * q^29 + 8 * q^31 + 6 * q^32 - 4 * q^34 - 4 * q^35 + 4 * q^37 + 2 * q^38 + 12 * q^40 - 12 * q^41 + 8 * q^43 + 12 * q^44 + 4 * q^46 + 2 * q^49 + 2 * q^50 + 4 * q^52 + 8 * q^55 - 6 * q^56 + 8 * q^59 + 12 * q^61 - 8 * q^62 - 14 * q^64 - 8 * q^65 + 8 * q^67 + 4 * q^68 + 4 * q^70 + 4 * q^71 - 4 * q^73 + 12 * q^74 - 2 * q^76 - 4 * q^77 - 12 * q^80 + 4 * q^82 - 8 * q^83 - 8 * q^85 + 8 * q^86 + 4 * q^88 + 4 * q^89 + 4 * q^91 - 12 * q^92 - 8 * q^94 + 4 * q^95 + 12 * q^97 - 2 * q^98

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 $

$2$	−2.41421	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−4.41421	−1.56066
$9$	0	0
$10$	4.82843	1.52688
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−2.41421	−0.645226
$15$	0	0
$16$	3.00000	0.750000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−7.65685	−1.71212
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−0.828427	−0.172739	−0.0863695	−	0.996263i	$-0.527527\pi$
$23$	−0.828427	−0.172739	−0.0863695	+	0.996263i	$0.527527\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−4.82843	−0.946932
$27$	0	0
$28$	3.82843	0.723505
$29$	1.17157	0.217556	0.108778	−	0.994066i	$-0.465306\pi$
$29$	1.17157	0.217556	0.108778	+	0.994066i	$0.465306\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.58579	0.280330
$33$	0	0
$34$	−4.82843	−0.828068
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−3.65685	−0.601183	−0.300592	−	0.953753i	$-0.597184\pi$
$37$	−3.65685	−0.601183	−0.300592	+	0.953753i	$0.597184\pi$
$38$	2.41421	0.391637
$39$	0	0
$40$	8.82843	1.39590
$41$	−3.17157	−0.495316	−0.247658	−	0.968847i	$-0.579661\pi$
$41$	−3.17157	−0.495316	−0.247658	+	0.968847i	$0.579661\pi$
$42$	0	0
$43$	−1.65685	−0.252668	−0.126334	−	0.991988i	$-0.540321\pi$
$43$	−1.65685	−0.252668	−0.126334	+	0.991988i	$0.540321\pi$
$44$	3.17157	0.478133
$45$	0	0
$46$	2.00000	0.294884
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.41421	0.341421
$51$	0	0
$52$	7.65685	1.06181
$53$	8.48528	1.16554	0.582772	−	0.812636i	$-0.301968\pi$
$53$	8.48528	1.16554	0.582772	+	0.812636i	$0.301968\pi$
$54$	0	0
$55$	−1.65685	−0.223410
$56$	−4.41421	−0.589874
$57$	0	0
$58$	−2.82843	−0.371391
$59$	−1.65685	−0.215704	−0.107852	−	0.994167i	$-0.534397\pi$
$59$	−1.65685	−0.215704	−0.107852	+	0.994167i	$0.534397\pi$
$60$	0	0
$61$	0.343146	0.0439353	0.0219677	−	0.999759i	$-0.493007\pi$
$61$	0.343146	0.0439353	0.0219677	+	0.999759i	$0.493007\pi$
$62$	−9.65685	−1.22642
$63$	0	0
$64$	−9.82843	−1.22855
$65$	−4.00000	−0.496139
$66$	0	0
$67$	9.65685	1.17977	0.589886	−	0.807486i	$-0.299173\pi$
$67$	9.65685	1.17977	0.589886	+	0.807486i	$0.299173\pi$
$68$	7.65685	0.928530
$69$	0	0
$70$	4.82843	0.577107
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	3.65685	0.428002	0.214001	−	0.976833i	$-0.431350\pi$
$73$	3.65685	0.428002	0.214001	+	0.976833i	$0.431350\pi$
$74$	8.82843	1.02628
$75$	0	0
$76$	−3.82843	−0.439151
$77$	0.828427	0.0944080
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	−6.00000	−0.670820
$81$	0	0
$82$	7.65685	0.845558
$83$	−12.4853	−1.37044	−0.685219	−	0.728337i	$-0.740293\pi$
$83$	−12.4853	−1.37044	−0.685219	+	0.728337i	$0.740293\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	4.00000	0.431331
$87$	0	0
$88$	−3.65685	−0.389822
$89$	4.82843	0.511812	0.255906	−	0.966702i	$-0.417626\pi$
$89$	4.82843	0.511812	0.255906	+	0.966702i	$0.417626\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−3.17157	−0.330659
$93$	0	0
$94$	−6.82843	−0.704298
$95$	2.00000	0.205196
$96$	0	0
$97$	0.343146	0.0348412	0.0174206	−	0.999848i	$-0.494455\pi$
$97$	0.343146	0.0348412	0.0174206	+	0.999848i	$0.494455\pi$
$98$	−2.41421	−0.243872
$99$	0	0
$100$	−3.82843	−0.382843
$101$	4.34315	0.432159	0.216080	−	0.976376i	$-0.430673\pi$
$101$	4.34315	0.432159	0.216080	+	0.976376i	$0.430673\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	−8.82843	−0.865699
$105$	0	0
$106$	−20.4853	−1.98971
$107$	13.3137	1.28708	0.643542	−	0.765410i	$-0.277464\pi$
$107$	13.3137	1.28708	0.643542	+	0.765410i	$0.277464\pi$
$108$	0	0
$109$	13.3137	1.27522	0.637611	−	0.770358i	$-0.279923\pi$
$109$	13.3137	1.27522	0.637611	+	0.770358i	$0.279923\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	3.00000	0.283473
$113$	12.4853	1.17452	0.587258	−	0.809400i	$-0.300207\pi$
$113$	12.4853	1.17452	0.587258	+	0.809400i	$0.300207\pi$
$114$	0	0
$115$	1.65685	0.154502
$116$	4.48528	0.416448
$117$	0	0
$118$	4.00000	0.368230
$119$	2.00000	0.183340
$120$	0	0
$121$	−10.3137	−0.937610
$122$	−0.828427	−0.0750023
$123$	0	0
$124$	15.3137	1.37521
$125$	12.0000	1.07331
$126$	0	0
$127$	16.9706	1.50589	0.752947	−	0.658081i	$-0.228632\pi$
$127$	16.9706	1.50589	0.752947	+	0.658081i	$0.228632\pi$
$128$	20.5563	1.81694
$129$	0	0
$130$	9.65685	0.846962
$131$	20.4853	1.78981	0.894904	−	0.446259i	$-0.147244\pi$
$131$	20.4853	1.78981	0.894904	+	0.446259i	$0.147244\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	−23.3137	−2.01400
$135$	0	0
$136$	−8.82843	−0.757031
$137$	9.65685	0.825041	0.412520	−	0.910948i	$-0.364649\pi$
$137$	9.65685	0.825041	0.412520	+	0.910948i	$0.364649\pi$
$138$	0	0
$139$	21.6569	1.83691	0.918455	−	0.395525i	$-0.129437\pi$
$139$	21.6569	1.83691	0.918455	+	0.395525i	$0.129437\pi$
$140$	−7.65685	−0.647122
$141$	0	0
$142$	−4.82843	−0.405193
$143$	1.65685	0.138553
$144$	0	0
$145$	−2.34315	−0.194588
$146$	−8.82843	−0.730646
$147$	0	0
$148$	−14.0000	−1.15079
$149$	−0.686292	−0.0562232	−0.0281116	−	0.999605i	$-0.508949\pi$
$149$	−0.686292	−0.0562232	−0.0281116	+	0.999605i	$0.508949\pi$
$150$	0	0
$151$	−11.3137	−0.920697	−0.460348	−	0.887738i	$-0.652275\pi$
$151$	−11.3137	−0.920697	−0.460348	+	0.887738i	$0.652275\pi$
$152$	4.41421	0.358040
$153$	0	0
$154$	−2.00000	−0.161165
$155$	−8.00000	−0.642575
$156$	0	0
$157$	11.6569	0.930318	0.465159	−	0.885227i	$-0.345997\pi$
$157$	11.6569	0.930318	0.465159	+	0.885227i	$0.345997\pi$
$158$	27.3137	2.17296
$159$	0	0
$160$	−3.17157	−0.250735
$161$	−0.828427	−0.0652892
$162$	0	0
$163$	1.65685	0.129775	0.0648874	−	0.997893i	$-0.479331\pi$
$163$	1.65685	0.129775	0.0648874	+	0.997893i	$0.479331\pi$
$164$	−12.1421	−0.948141
$165$	0	0
$166$	30.1421	2.33948
$167$	−0.686292	−0.0531068	−0.0265534	−	0.999647i	$-0.508453\pi$
$167$	−0.686292	−0.0531068	−0.0265534	+	0.999647i	$0.508453\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	9.65685	0.740647
$171$	0	0
$172$	−6.34315	−0.483660
$173$	19.1716	1.45759	0.728794	−	0.684733i	$-0.240081\pi$
$173$	19.1716	1.45759	0.728794	+	0.684733i	$0.240081\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	2.48528	0.187335
$177$	0	0
$178$	−11.6569	−0.873718
$179$	9.31371	0.696139	0.348070	−	0.937469i	$-0.386837\pi$
$179$	9.31371	0.696139	0.348070	+	0.937469i	$0.386837\pi$
$180$	0	0
$181$	−0.343146	−0.0255058	−0.0127529	−	0.999919i	$-0.504059\pi$
$181$	−0.343146	−0.0255058	−0.0127529	+	0.999919i	$0.504059\pi$
$182$	−4.82843	−0.357907
$183$	0	0
$184$	3.65685	0.269587
$185$	7.31371	0.537715
$186$	0	0
$187$	1.65685	0.121161
$188$	10.8284	0.789744
$189$	0	0
$190$	−4.82843	−0.350291
$191$	−21.7990	−1.57732	−0.788660	−	0.614830i	$-0.789225\pi$
$191$	−21.7990	−1.57732	−0.788660	+	0.614830i	$0.789225\pi$
$192$	0	0
$193$	3.65685	0.263226	0.131613	−	0.991301i	$-0.457984\pi$
$193$	3.65685	0.263226	0.131613	+	0.991301i	$0.457984\pi$
$194$	−0.828427	−0.0594776
$195$	0	0
$196$	3.82843	0.273459
$197$	−12.9706	−0.924114	−0.462057	−	0.886850i	$-0.652889\pi$
$197$	−12.9706	−0.924114	−0.462057	+	0.886850i	$0.652889\pi$
$198$	0	0
$199$	11.3137	0.802008	0.401004	−	0.916076i	$-0.368661\pi$
$199$	11.3137	0.802008	0.401004	+	0.916076i	$0.368661\pi$
$200$	4.41421	0.312132
$201$	0	0
$202$	−10.4853	−0.737742
$203$	1.17157	0.0822283
$204$	0	0
$205$	6.34315	0.443025
$206$	−28.9706	−2.01847
$207$	0	0
$208$	6.00000	0.416025
$209$	−0.828427	−0.0573035
$210$	0	0
$211$	−9.65685	−0.664805	−0.332403	−	0.943138i	$-0.607859\pi$
$211$	−9.65685	−0.664805	−0.332403	+	0.943138i	$0.607859\pi$
$212$	32.4853	2.23110
$213$	0	0
$214$	−32.1421	−2.19719
$215$	3.31371	0.225993
$216$	0	0
$217$	4.00000	0.271538
$218$	−32.1421	−2.17694
$219$	0	0
$220$	−6.34315	−0.427655
$221$	4.00000	0.269069
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	1.58579	0.105955
$225$	0	0
$226$	−30.1421	−2.00503
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	−5.17157	−0.339530
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	0	0
$235$	−5.65685	−0.369012
$236$	−6.34315	−0.412904
$237$	0	0
$238$	−4.82843	−0.312980
$239$	−6.48528	−0.419498	−0.209749	−	0.977755i	$-0.567265\pi$
$239$	−6.48528	−0.419498	−0.209749	+	0.977755i	$0.567265\pi$
$240$	0	0
$241$	3.65685	0.235559	0.117779	−	0.993040i	$-0.462422\pi$
$241$	3.65685	0.235559	0.117779	+	0.993040i	$0.462422\pi$
$242$	24.8995	1.60060
$243$	0	0
$244$	1.31371	0.0841016
$245$	−2.00000	−0.127775
$246$	0	0
$247$	−2.00000	−0.127257
$248$	−17.6569	−1.12121
$249$	0	0
$250$	−28.9706	−1.83226
$251$	−0.485281	−0.0306307	−0.0153153	−	0.999883i	$-0.504875\pi$
$251$	−0.485281	−0.0306307	−0.0153153	+	0.999883i	$0.504875\pi$
$252$	0	0
$253$	−0.686292	−0.0431468
$254$	−40.9706	−2.57072
$255$	0	0
$256$	−29.9706	−1.87316
$257$	29.7990	1.85881	0.929405	−	0.369062i	$-0.120321\pi$
$257$	29.7990	1.85881	0.929405	+	0.369062i	$0.120321\pi$
$258$	0	0
$259$	−3.65685	−0.227226
$260$	−15.3137	−0.949716
$261$	0	0
$262$	−49.4558	−3.05539
$263$	−6.48528	−0.399900	−0.199950	−	0.979806i	$-0.564078\pi$
$263$	−6.48528	−0.399900	−0.199950	+	0.979806i	$0.564078\pi$
$264$	0	0
$265$	−16.9706	−1.04249
$266$	2.41421	0.148025
$267$	0	0
$268$	36.9706	2.25834
$269$	−23.4558	−1.43013	−0.715064	−	0.699059i	$-0.753602\pi$
$269$	−23.4558	−1.43013	−0.715064	+	0.699059i	$0.753602\pi$
$270$	0	0
$271$	−12.9706	−0.787906	−0.393953	−	0.919131i	$-0.628893\pi$
$271$	−12.9706	−0.787906	−0.393953	+	0.919131i	$0.628893\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−23.3137	−1.40843
$275$	−0.828427	−0.0499560
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	−52.2843	−3.13580
$279$	0	0
$280$	8.82843	0.527599
$281$	−8.48528	−0.506189	−0.253095	−	0.967442i	$-0.581448\pi$
$281$	−8.48528	−0.506189	−0.253095	+	0.967442i	$0.581448\pi$
$282$	0	0
$283$	−26.6274	−1.58284	−0.791418	−	0.611276i	$-0.790657\pi$
$283$	−26.6274	−1.58284	−0.791418	+	0.611276i	$0.790657\pi$
$284$	7.65685	0.454351
$285$	0	0
$286$	−4.00000	−0.236525
$287$	−3.17157	−0.187212
$288$	0	0
$289$	−13.0000	−0.764706
$290$	5.65685	0.332182
$291$	0	0
$292$	14.0000	0.819288
$293$	3.17157	0.185285	0.0926426	−	0.995699i	$-0.470469\pi$
$293$	3.17157	0.185285	0.0926426	+	0.995699i	$0.470469\pi$
$294$	0	0
$295$	3.31371	0.192932
$296$	16.1421	0.938243
$297$	0	0
$298$	1.65685	0.0959790
$299$	−1.65685	−0.0958184
$300$	0	0
$301$	−1.65685	−0.0954995
$302$	27.3137	1.57173
$303$	0	0
$304$	−3.00000	−0.172062
$305$	−0.686292	−0.0392969
$306$	0	0
$307$	23.3137	1.33058	0.665292	−	0.746583i	$-0.268307\pi$
$307$	23.3137	1.33058	0.665292	+	0.746583i	$0.268307\pi$
$308$	3.17157	0.180717
$309$	0	0
$310$	19.3137	1.09694
$311$	−1.85786	−0.105350	−0.0526749	−	0.998612i	$-0.516775\pi$
$311$	−1.85786	−0.105350	−0.0526749	+	0.998612i	$0.516775\pi$
$312$	0	0
$313$	−3.65685	−0.206698	−0.103349	−	0.994645i	$-0.532956\pi$
$313$	−3.65685	−0.206698	−0.103349	+	0.994645i	$0.532956\pi$
$314$	−28.1421	−1.58815
$315$	0	0
$316$	−43.3137	−2.43659
$317$	−24.4853	−1.37523	−0.687615	−	0.726075i	$-0.741342\pi$
$317$	−24.4853	−1.37523	−0.687615	+	0.726075i	$0.741342\pi$
$318$	0	0
$319$	0.970563	0.0543411
$320$	19.6569	1.09885
$321$	0	0
$322$	2.00000	0.111456
$323$	−2.00000	−0.111283
$324$	0	0
$325$	−2.00000	−0.110940
$326$	−4.00000	−0.221540
$327$	0	0
$328$	14.0000	0.773021
$329$	2.82843	0.155936
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	−47.7990	−2.62331
$333$	0	0
$334$	1.65685	0.0906590
$335$	−19.3137	−1.05522
$336$	0	0
$337$	9.31371	0.507350	0.253675	−	0.967290i	$-0.418361\pi$
$337$	9.31371	0.507350	0.253675	+	0.967290i	$0.418361\pi$
$338$	21.7279	1.18184
$339$	0	0
$340$	−15.3137	−0.830502
$341$	3.31371	0.179447
$342$	0	0
$343$	1.00000	0.0539949
$344$	7.31371	0.394329
$345$	0	0
$346$	−46.2843	−2.48826
$347$	20.8284	1.11813	0.559064	−	0.829124i	$-0.311160\pi$
$347$	20.8284	1.11813	0.559064	+	0.829124i	$0.311160\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	2.41421	0.129045
$351$	0	0
$352$	1.31371	0.0700209
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	18.4853	0.979718
$357$	0	0
$358$	−22.4853	−1.18838
$359$	−10.4853	−0.553392	−0.276696	−	0.960958i	$-0.589239\pi$
$359$	−10.4853	−0.553392	−0.276696	+	0.960958i	$0.589239\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0.828427	0.0435412
$363$	0	0
$364$	7.65685	0.401328
$365$	−7.31371	−0.382817
$366$	0	0
$367$	22.6274	1.18114	0.590571	−	0.806986i	$-0.298903\pi$
$367$	22.6274	1.18114	0.590571	+	0.806986i	$0.298903\pi$
$368$	−2.48528	−0.129554
$369$	0	0
$370$	−17.6569	−0.917936
$371$	8.48528	0.440534
$372$	0	0
$373$	17.3137	0.896470	0.448235	−	0.893916i	$-0.352053\pi$
$373$	17.3137	0.896470	0.448235	+	0.893916i	$0.352053\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	−12.4853	−0.643879
$377$	2.34315	0.120678
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	7.65685	0.392788
$381$	0	0
$382$	52.6274	2.69265
$383$	−16.9706	−0.867155	−0.433578	−	0.901116i	$-0.642749\pi$
$383$	−16.9706	−0.867155	−0.433578	+	0.901116i	$0.642749\pi$
$384$	0	0
$385$	−1.65685	−0.0844411
$386$	−8.82843	−0.449355
$387$	0	0
$388$	1.31371	0.0666934
$389$	−0.970563	−0.0492095	−0.0246047	−	0.999697i	$-0.507833\pi$
$389$	−0.970563	−0.0492095	−0.0246047	+	0.999697i	$0.507833\pi$
$390$	0	0
$391$	−1.65685	−0.0837907
$392$	−4.41421	−0.222951
$393$	0	0
$394$	31.3137	1.57756
$395$	22.6274	1.13851
$396$	0	0
$397$	−32.6274	−1.63752	−0.818762	−	0.574134i	$-0.805339\pi$
$397$	−32.6274	−1.63752	−0.818762	+	0.574134i	$0.805339\pi$
$398$	−27.3137	−1.36911
$399$	0	0
$400$	−3.00000	−0.150000
$401$	−33.4558	−1.67071	−0.835353	−	0.549715i	$-0.814736\pi$
$401$	−33.4558	−1.67071	−0.835353	+	0.549715i	$0.814736\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	16.6274	0.827245
$405$	0	0
$406$	−2.82843	−0.140372
$407$	−3.02944	−0.150164
$408$	0	0
$409$	9.31371	0.460533	0.230267	−	0.973128i	$-0.426040\pi$
$409$	9.31371	0.460533	0.230267	+	0.973128i	$0.426040\pi$
$410$	−15.3137	−0.756290
$411$	0	0
$412$	45.9411	2.26336
$413$	−1.65685	−0.0815285
$414$	0	0
$415$	24.9706	1.22576
$416$	3.17157	0.155499
$417$	0	0
$418$	2.00000	0.0978232
$419$	−30.1421	−1.47254	−0.736270	−	0.676688i	$-0.763415\pi$
$419$	−30.1421	−1.47254	−0.736270	+	0.676688i	$0.763415\pi$
$420$	0	0
$421$	6.97056	0.339724	0.169862	−	0.985468i	$-0.445668\pi$
$421$	6.97056	0.339724	0.169862	+	0.985468i	$0.445668\pi$
$422$	23.3137	1.13489
$423$	0	0
$424$	−37.4558	−1.81902
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0.343146	0.0166060
$428$	50.9706	2.46376
$429$	0	0
$430$	−8.00000	−0.385794
$431$	−33.3137	−1.60466	−0.802332	−	0.596877i	$-0.796408\pi$
$431$	−33.3137	−1.60466	−0.802332	+	0.596877i	$0.796408\pi$
$432$	0	0
$433$	2.97056	0.142756	0.0713781	−	0.997449i	$-0.477260\pi$
$433$	2.97056	0.142756	0.0713781	+	0.997449i	$0.477260\pi$
$434$	−9.65685	−0.463544
$435$	0	0
$436$	50.9706	2.44105
$437$	0.828427	0.0396290
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	7.31371	0.348667
$441$	0	0
$442$	−9.65685	−0.459330
$443$	−25.7990	−1.22575	−0.612873	−	0.790181i	$-0.709986\pi$
$443$	−25.7990	−1.22575	−0.612873	+	0.790181i	$0.709986\pi$
$444$	0	0
$445$	−9.65685	−0.457779
$446$	−38.6274	−1.82906
$447$	0	0
$448$	−9.82843	−0.464350
$449$	−15.7990	−0.745600	−0.372800	−	0.927912i	$-0.621602\pi$
$449$	−15.7990	−0.745600	−0.372800	+	0.927912i	$0.621602\pi$
$450$	0	0
$451$	−2.62742	−0.123720
$452$	47.7990	2.24828
$453$	0	0
$454$	9.65685	0.453219
$455$	−4.00000	−0.187523
$456$	0	0
$457$	20.6274	0.964910	0.482455	−	0.875921i	$-0.339745\pi$
$457$	20.6274	0.964910	0.482455	+	0.875921i	$0.339745\pi$
$458$	−24.1421	−1.12809
$459$	0	0
$460$	6.34315	0.295751
$461$	−6.97056	−0.324651	−0.162326	−	0.986737i	$-0.551900\pi$
$461$	−6.97056	−0.324651	−0.162326	+	0.986737i	$0.551900\pi$
$462$	0	0
$463$	−11.3137	−0.525793	−0.262896	−	0.964824i	$-0.584678\pi$
$463$	−11.3137	−0.525793	−0.262896	+	0.964824i	$0.584678\pi$
$464$	3.51472	0.163167
$465$	0	0
$466$	−9.65685	−0.447345
$467$	34.8284	1.61167	0.805834	−	0.592142i	$-0.201718\pi$
$467$	34.8284	1.61167	0.805834	+	0.592142i	$0.201718\pi$
$468$	0	0
$469$	9.65685	0.445912
$470$	13.6569	0.629944
$471$	0	0
$472$	7.31371	0.336641
$473$	−1.37258	−0.0631114
$474$	0	0
$475$	1.00000	0.0458831
$476$	7.65685	0.350951
$477$	0	0
$478$	15.6569	0.716128
$479$	3.79899	0.173580	0.0867901	−	0.996227i	$-0.472339\pi$
$479$	3.79899	0.173580	0.0867901	+	0.996227i	$0.472339\pi$
$480$	0	0
$481$	−7.31371	−0.333476
$482$	−8.82843	−0.402124
$483$	0	0
$484$	−39.4853	−1.79479
$485$	−0.686292	−0.0311629
$486$	0	0
$487$	5.65685	0.256337	0.128168	−	0.991752i	$-0.459090\pi$
$487$	5.65685	0.256337	0.128168	+	0.991752i	$0.459090\pi$
$488$	−1.51472	−0.0685681
$489$	0	0
$490$	4.82843	0.218126
$491$	36.8284	1.66204	0.831022	−	0.556240i	$-0.187756\pi$
$491$	36.8284	1.66204	0.831022	+	0.556240i	$0.187756\pi$
$492$	0	0
$493$	2.34315	0.105530
$494$	4.82843	0.217241
$495$	0	0
$496$	12.0000	0.538816
$497$	2.00000	0.0897123
$498$	0	0
$499$	39.3137	1.75992	0.879962	−	0.475045i	$-0.157568\pi$
$499$	39.3137	1.75992	0.879962	+	0.475045i	$0.157568\pi$
$500$	45.9411	2.05455
$501$	0	0
$502$	1.17157	0.0522899
$503$	1.17157	0.0522379	0.0261189	−	0.999659i	$-0.491685\pi$
$503$	1.17157	0.0522379	0.0261189	+	0.999659i	$0.491685\pi$
$504$	0	0
$505$	−8.68629	−0.386535
$506$	1.65685	0.0736562
$507$	0	0
$508$	64.9706	2.88260
$509$	−13.5147	−0.599029	−0.299515	−	0.954092i	$-0.596825\pi$
$509$	−13.5147	−0.599029	−0.299515	+	0.954092i	$0.596825\pi$
$510$	0	0
$511$	3.65685	0.161770
$512$	31.2426	1.38074
$513$	0	0
$514$	−71.9411	−3.17319
$515$	−24.0000	−1.05757
$516$	0	0
$517$	2.34315	0.103051
$518$	8.82843	0.387899
$519$	0	0
$520$	17.6569	0.774304
$521$	40.8284	1.78873	0.894363	−	0.447342i	$-0.147629\pi$
$521$	40.8284	1.78873	0.894363	+	0.447342i	$0.147629\pi$
$522$	0	0
$523$	12.6863	0.554733	0.277366	−	0.960764i	$-0.410538\pi$
$523$	12.6863	0.554733	0.277366	+	0.960764i	$0.410538\pi$
$524$	78.4264	3.42607
$525$	0	0
$526$	15.6569	0.682671
$527$	8.00000	0.348485
$528$	0	0
$529$	−22.3137	−0.970161
$530$	40.9706	1.77965
$531$	0	0
$532$	−3.82843	−0.165983
$533$	−6.34315	−0.274752
$534$	0	0
$535$	−26.6274	−1.15120
$536$	−42.6274	−1.84122
$537$	0	0
$538$	56.6274	2.44138
$539$	0.828427	0.0356829
$540$	0	0
$541$	−6.68629	−0.287466	−0.143733	−	0.989616i	$-0.545911\pi$
$541$	−6.68629	−0.287466	−0.143733	+	0.989616i	$0.545911\pi$
$542$	31.3137	1.34504
$543$	0	0
$544$	3.17157	0.135980
$545$	−26.6274	−1.14059
$546$	0	0
$547$	42.6274	1.82262	0.911308	−	0.411724i	$-0.135073\pi$
$547$	42.6274	1.82262	0.911308	+	0.411724i	$0.135073\pi$
$548$	36.9706	1.57930
$549$	0	0
$550$	2.00000	0.0852803
$551$	−1.17157	−0.0499107
$552$	0	0
$553$	−11.3137	−0.481108
$554$	43.4558	1.84626
$555$	0	0
$556$	82.9117	3.51624
$557$	39.3137	1.66578	0.832888	−	0.553442i	$-0.186686\pi$
$557$	39.3137	1.66578	0.832888	+	0.553442i	$0.186686\pi$
$558$	0	0
$559$	−3.31371	−0.140155
$560$	−6.00000	−0.253546
$561$	0	0
$562$	20.4853	0.864119
$563$	8.68629	0.366084	0.183042	−	0.983105i	$-0.441406\pi$
$563$	8.68629	0.366084	0.183042	+	0.983105i	$0.441406\pi$
$564$	0	0
$565$	−24.9706	−1.05052
$566$	64.2843	2.70207
$567$	0	0
$568$	−8.82843	−0.370433
$569$	−40.4853	−1.69723	−0.848616	−	0.529010i	$-0.822563\pi$
$569$	−40.4853	−1.69723	−0.848616	+	0.529010i	$0.822563\pi$
$570$	0	0
$571$	10.6274	0.444744	0.222372	−	0.974962i	$-0.428620\pi$
$571$	10.6274	0.444744	0.222372	+	0.974962i	$0.428620\pi$
$572$	6.34315	0.265220
$573$	0	0
$574$	7.65685	0.319591
$575$	0.828427	0.0345478
$576$	0	0
$577$	−14.9706	−0.623233	−0.311616	−	0.950208i	$-0.600870\pi$
$577$	−14.9706	−0.623233	−0.311616	+	0.950208i	$0.600870\pi$
$578$	31.3848	1.30543
$579$	0	0
$580$	−8.97056	−0.372482
$581$	−12.4853	−0.517977
$582$	0	0
$583$	7.02944	0.291130
$584$	−16.1421	−0.667966
$585$	0	0
$586$	−7.65685	−0.316302
$587$	9.45584	0.390284	0.195142	−	0.980775i	$-0.437483\pi$
$587$	9.45584	0.390284	0.195142	+	0.980775i	$0.437483\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	−8.00000	−0.329355
$591$	0	0
$592$	−10.9706	−0.450887
$593$	−11.6569	−0.478690	−0.239345	−	0.970935i	$-0.576933\pi$
$593$	−11.6569	−0.478690	−0.239345	+	0.970935i	$0.576933\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	−2.62742	−0.107623
$597$	0	0
$598$	4.00000	0.163572
$599$	26.9706	1.10199	0.550994	−	0.834509i	$-0.314249\pi$
$599$	26.9706	1.10199	0.550994	+	0.834509i	$0.314249\pi$
$600$	0	0
$601$	−14.9706	−0.610662	−0.305331	−	0.952246i	$-0.598767\pi$
$601$	−14.9706	−0.610662	−0.305331	+	0.952246i	$0.598767\pi$
$602$	4.00000	0.163028
$603$	0	0
$604$	−43.3137	−1.76241
$605$	20.6274	0.838624
$606$	0	0
$607$	12.6863	0.514921	0.257460	−	0.966289i	$-0.417114\pi$
$607$	12.6863	0.514921	0.257460	+	0.966289i	$0.417114\pi$
$608$	−1.58579	−0.0643121
$609$	0	0
$610$	1.65685	0.0670841
$611$	5.65685	0.228852
$612$	0	0
$613$	−47.2548	−1.90860	−0.954302	−	0.298843i	$-0.903399\pi$
$613$	−47.2548	−1.90860	−0.954302	+	0.298843i	$0.903399\pi$
$614$	−56.2843	−2.27145
$615$	0	0
$616$	−3.65685	−0.147339
$617$	−3.02944	−0.121961	−0.0609803	−	0.998139i	$-0.519423\pi$
$617$	−3.02944	−0.121961	−0.0609803	+	0.998139i	$0.519423\pi$
$618$	0	0
$619$	20.2843	0.815294	0.407647	−	0.913140i	$-0.366349\pi$
$619$	20.2843	0.815294	0.407647	+	0.913140i	$0.366349\pi$
$620$	−30.6274	−1.23003
$621$	0	0
$622$	4.48528	0.179843
$623$	4.82843	0.193447
$624$	0	0
$625$	−19.0000	−0.760000
$626$	8.82843	0.352855
$627$	0	0
$628$	44.6274	1.78083
$629$	−7.31371	−0.291617
$630$	0	0
$631$	5.65685	0.225196	0.112598	−	0.993641i	$-0.464083\pi$
$631$	5.65685	0.225196	0.112598	+	0.993641i	$0.464083\pi$
$632$	49.9411	1.98655
$633$	0	0
$634$	59.1127	2.34767
$635$	−33.9411	−1.34691
$636$	0	0
$637$	2.00000	0.0792429
$638$	−2.34315	−0.0927660
$639$	0	0
$640$	−41.1127	−1.62512
$641$	5.17157	0.204265	0.102132	−	0.994771i	$-0.467433\pi$
$641$	5.17157	0.204265	0.102132	+	0.994771i	$0.467433\pi$
$642$	0	0
$643$	−18.6274	−0.734594	−0.367297	−	0.930104i	$-0.619717\pi$
$643$	−18.6274	−0.734594	−0.367297	+	0.930104i	$0.619717\pi$
$644$	−3.17157	−0.124977
$645$	0	0
$646$	4.82843	0.189972
$647$	15.5147	0.609947	0.304973	−	0.952361i	$-0.401352\pi$
$647$	15.5147	0.609947	0.304973	+	0.952361i	$0.401352\pi$
$648$	0	0
$649$	−1.37258	−0.0538786
$650$	4.82843	0.189386
$651$	0	0
$652$	6.34315	0.248417
$653$	−38.6274	−1.51161	−0.755804	−	0.654798i	$-0.772754\pi$
$653$	−38.6274	−1.51161	−0.755804	+	0.654798i	$0.772754\pi$
$654$	0	0
$655$	−40.9706	−1.60085
$656$	−9.51472	−0.371487
$657$	0	0
$658$	−6.82843	−0.266200
$659$	4.34315	0.169185	0.0845925	−	0.996416i	$-0.473041\pi$
$659$	4.34315	0.169185	0.0845925	+	0.996416i	$0.473041\pi$
$660$	0	0
$661$	−43.9411	−1.70911	−0.854556	−	0.519360i	$-0.826171\pi$
$661$	−43.9411	−1.70911	−0.854556	+	0.519360i	$0.826171\pi$
$662$	−28.9706	−1.12597
$663$	0	0
$664$	55.1127	2.13879
$665$	2.00000	0.0775567
$666$	0	0
$667$	−0.970563	−0.0375803
$668$	−2.62742	−0.101658
$669$	0	0
$670$	46.6274	1.80137
$671$	0.284271	0.0109742
$672$	0	0
$673$	27.6569	1.06609	0.533047	−	0.846086i	$-0.321047\pi$
$673$	27.6569	1.06609	0.533047	+	0.846086i	$0.321047\pi$
$674$	−22.4853	−0.866101
$675$	0	0
$676$	−34.4558	−1.32522
$677$	−39.1716	−1.50549	−0.752743	−	0.658315i	$-0.771270\pi$
$677$	−39.1716	−1.50549	−0.752743	+	0.658315i	$0.771270\pi$
$678$	0	0
$679$	0.343146	0.0131687
$680$	17.6569	0.677109
$681$	0	0
$682$	−8.00000	−0.306336
$683$	−43.9411	−1.68136	−0.840680	−	0.541532i	$-0.817845\pi$
$683$	−43.9411	−1.68136	−0.840680	+	0.541532i	$0.817845\pi$
$684$	0	0
$685$	−19.3137	−0.737939
$686$	−2.41421	−0.0921751
$687$	0	0
$688$	−4.97056	−0.189501
$689$	16.9706	0.646527
$690$	0	0
$691$	−8.97056	−0.341256	−0.170628	−	0.985335i	$-0.554580\pi$
$691$	−8.97056	−0.341256	−0.170628	+	0.985335i	$0.554580\pi$
$692$	73.3970	2.79013
$693$	0	0
$694$	−50.2843	−1.90876
$695$	−43.3137	−1.64298
$696$	0	0
$697$	−6.34315	−0.240264
$698$	−33.7990	−1.27931
$699$	0	0
$700$	−3.82843	−0.144701
$701$	38.6274	1.45894	0.729469	−	0.684014i	$-0.239767\pi$
$701$	38.6274	1.45894	0.729469	+	0.684014i	$0.239767\pi$
$702$	0	0
$703$	3.65685	0.137921
$704$	−8.14214	−0.306868
$705$	0	0
$706$	33.7990	1.27204
$707$	4.34315	0.163341
$708$	0	0
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	9.65685	0.362415
$711$	0	0
$712$	−21.3137	−0.798765
$713$	−3.31371	−0.124099
$714$	0	0
$715$	−3.31371	−0.123926
$716$	35.6569	1.33256
$717$	0	0
$718$	25.3137	0.944699
$719$	42.1421	1.57164	0.785818	−	0.618458i	$-0.212242\pi$
$719$	42.1421	1.57164	0.785818	+	0.618458i	$0.212242\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	−2.41421	−0.0898477
$723$	0	0
$724$	−1.31371	−0.0488236
$725$	−1.17157	−0.0435111
$726$	0	0
$727$	22.3431	0.828661	0.414331	−	0.910126i	$-0.364016\pi$
$727$	22.3431	0.828661	0.414331	+	0.910126i	$0.364016\pi$
$728$	−8.82843	−0.327203
$729$	0	0
$730$	17.6569	0.653509
$731$	−3.31371	−0.122562
$732$	0	0
$733$	−5.02944	−0.185767	−0.0928833	−	0.995677i	$-0.529608\pi$
$733$	−5.02944	−0.185767	−0.0928833	+	0.995677i	$0.529608\pi$
$734$	−54.6274	−2.01633
$735$	0	0
$736$	−1.31371	−0.0484239
$737$	8.00000	0.294684
$738$	0	0
$739$	−14.3431	−0.527621	−0.263811	−	0.964575i	$-0.584979\pi$
$739$	−14.3431	−0.527621	−0.263811	+	0.964575i	$0.584979\pi$
$740$	28.0000	1.02930
$741$	0	0
$742$	−20.4853	−0.752038
$743$	−18.6863	−0.685534	−0.342767	−	0.939421i	$-0.611364\pi$
$743$	−18.6863	−0.685534	−0.342767	+	0.939421i	$0.611364\pi$
$744$	0	0
$745$	1.37258	0.0502876
$746$	−41.7990	−1.53037
$747$	0	0
$748$	6.34315	0.231928
$749$	13.3137	0.486472
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	8.48528	0.309426
$753$	0	0
$754$	−5.65685	−0.206010
$755$	22.6274	0.823496
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	−28.9706	−1.05226
$759$	0	0
$760$	−8.82843	−0.320241
$761$	−16.6274	−0.602743	−0.301372	−	0.953507i	$-0.597444\pi$
$761$	−16.6274	−0.602743	−0.301372	+	0.953507i	$0.597444\pi$
$762$	0	0
$763$	13.3137	0.481989
$764$	−83.4558	−3.01933
$765$	0	0
$766$	40.9706	1.48033
$767$	−3.31371	−0.119651
$768$	0	0
$769$	−5.02944	−0.181366	−0.0906831	−	0.995880i	$-0.528905\pi$
$769$	−5.02944	−0.181366	−0.0906831	+	0.995880i	$0.528905\pi$
$770$	4.00000	0.144150
$771$	0	0
$772$	14.0000	0.503871
$773$	−33.5147	−1.20544	−0.602720	−	0.797953i	$-0.705917\pi$
$773$	−33.5147	−1.20544	−0.602720	+	0.797953i	$0.705917\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	−1.51472	−0.0543752
$777$	0	0
$778$	2.34315	0.0840058
$779$	3.17157	0.113633
$780$	0	0
$781$	1.65685	0.0592869
$782$	4.00000	0.143040
$783$	0	0
$784$	3.00000	0.107143
$785$	−23.3137	−0.832102
$786$	0	0
$787$	35.3137	1.25880	0.629399	−	0.777082i	$-0.283301\pi$
$787$	35.3137	1.25880	0.629399	+	0.777082i	$0.283301\pi$
$788$	−49.6569	−1.76895
$789$	0	0
$790$	−54.6274	−1.94356
$791$	12.4853	0.443925
$792$	0	0
$793$	0.686292	0.0243709
$794$	78.7696	2.79543
$795$	0	0
$796$	43.3137	1.53521
$797$	6.48528	0.229720	0.114860	−	0.993382i	$-0.463358\pi$
$797$	6.48528	0.229720	0.114860	+	0.993382i	$0.463358\pi$
$798$	0	0
$799$	5.65685	0.200125
$800$	−1.58579	−0.0560660
$801$	0	0
$802$	80.7696	2.85207
$803$	3.02944	0.106907
$804$	0	0
$805$	1.65685	0.0583964
$806$	−19.3137	−0.680296
$807$	0	0
$808$	−19.1716	−0.674454
$809$	−27.3137	−0.960299	−0.480149	−	0.877187i	$-0.659418\pi$
$809$	−27.3137	−0.960299	−0.480149	+	0.877187i	$0.659418\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	4.48528	0.157403
$813$	0	0
$814$	7.31371	0.256345
$815$	−3.31371	−0.116074
$816$	0	0
$817$	1.65685	0.0579660
$818$	−22.4853	−0.786179
$819$	0	0
$820$	24.2843	0.848044
$821$	10.6274	0.370899	0.185450	−	0.982654i	$-0.440626\pi$
$821$	10.6274	0.370899	0.185450	+	0.982654i	$0.440626\pi$
$822$	0	0
$823$	−40.9706	−1.42814	−0.714072	−	0.700072i	$-0.753151\pi$
$823$	−40.9706	−1.42814	−0.714072	+	0.700072i	$0.753151\pi$
$824$	−52.9706	−1.84532
$825$	0	0
$826$	4.00000	0.139178
$827$	13.3137	0.462963	0.231482	−	0.972839i	$-0.425643\pi$
$827$	13.3137	0.462963	0.231482	+	0.972839i	$0.425643\pi$
$828$	0	0
$829$	33.5980	1.16691	0.583453	−	0.812147i	$-0.301701\pi$
$829$	33.5980	1.16691	0.583453	+	0.812147i	$0.301701\pi$
$830$	−60.2843	−2.09250
$831$	0	0
$832$	−19.6569	−0.681479
$833$	2.00000	0.0692959
$834$	0	0
$835$	1.37258	0.0475002
$836$	−3.17157	−0.109691
$837$	0	0
$838$	72.7696	2.51378
$839$	−40.2843	−1.39077	−0.695384	−	0.718639i	$-0.744766\pi$
$839$	−40.2843	−1.39077	−0.695384	+	0.718639i	$0.744766\pi$
$840$	0	0
$841$	−27.6274	−0.952670
$842$	−16.8284	−0.579946
$843$	0	0
$844$	−36.9706	−1.27258
$845$	18.0000	0.619219
$846$	0	0
$847$	−10.3137	−0.354383
$848$	25.4558	0.874157
$849$	0	0
$850$	4.82843	0.165614
$851$	3.02944	0.103848
$852$	0	0
$853$	39.9411	1.36756	0.683779	−	0.729689i	$-0.260335\pi$
$853$	39.9411	1.36756	0.683779	+	0.729689i	$0.260335\pi$
$854$	−0.828427	−0.0283482
$855$	0	0
$856$	−58.7696	−2.00870
$857$	1.51472	0.0517418	0.0258709	−	0.999665i	$-0.491764\pi$
$857$	1.51472	0.0517418	0.0258709	+	0.999665i	$0.491764\pi$
$858$	0	0
$859$	2.62742	0.0896463	0.0448232	−	0.998995i	$-0.485728\pi$
$859$	2.62742	0.0896463	0.0448232	+	0.998995i	$0.485728\pi$
$860$	12.6863	0.432599
$861$	0	0
$862$	80.4264	2.73933
$863$	7.37258	0.250966	0.125483	−	0.992096i	$-0.459952\pi$
$863$	7.37258	0.250966	0.125483	+	0.992096i	$0.459952\pi$
$864$	0	0
$865$	−38.3431	−1.30371
$866$	−7.17157	−0.243700
$867$	0	0
$868$	15.3137	0.519781
$869$	−9.37258	−0.317943
$870$	0	0
$871$	19.3137	0.654420
$872$	−58.7696	−1.99019
$873$	0	0
$874$	−2.00000	−0.0676510
$875$	12.0000	0.405674
$876$	0	0
$877$	−40.6274	−1.37189	−0.685945	−	0.727653i	$-0.740611\pi$
$877$	−40.6274	−1.37189	−0.685945	+	0.727653i	$0.740611\pi$
$878$	−19.3137	−0.651806
$879$	0	0
$880$	−4.97056	−0.167558
$881$	40.6274	1.36877	0.684386	−	0.729120i	$-0.260070\pi$
$881$	40.6274	1.36877	0.684386	+	0.729120i	$0.260070\pi$
$882$	0	0
$883$	23.3137	0.784569	0.392284	−	0.919844i	$-0.371685\pi$
$883$	23.3137	0.784569	0.392284	+	0.919844i	$0.371685\pi$
$884$	15.3137	0.515056
$885$	0	0
$886$	62.2843	2.09248
$887$	8.68629	0.291657	0.145829	−	0.989310i	$-0.453415\pi$
$887$	8.68629	0.291657	0.145829	+	0.989310i	$0.453415\pi$
$888$	0	0
$889$	16.9706	0.569174
$890$	23.3137	0.781477
$891$	0	0
$892$	61.2548	2.05096
$893$	−2.82843	−0.0946497
$894$	0	0
$895$	−18.6274	−0.622646
$896$	20.5563	0.686739
$897$	0	0
$898$	38.1421	1.27282
$899$	4.68629	0.156297
$900$	0	0
$901$	16.9706	0.565371
$902$	6.34315	0.211204
$903$	0	0
$904$	−55.1127	−1.83302
$905$	0.686292	0.0228131
$906$	0	0
$907$	−16.2843	−0.540710	−0.270355	−	0.962761i	$-0.587141\pi$
$907$	−16.2843	−0.540710	−0.270355	+	0.962761i	$0.587141\pi$
$908$	−15.3137	−0.508203
$909$	0	0
$910$	9.65685	0.320122
$911$	17.0294	0.564210	0.282105	−	0.959383i	$-0.408967\pi$
$911$	17.0294	0.564210	0.282105	+	0.959383i	$0.408967\pi$
$912$	0	0
$913$	−10.3431	−0.342308
$914$	−49.7990	−1.64720
$915$	0	0
$916$	38.2843	1.26495
$917$	20.4853	0.676484
$918$	0	0
$919$	6.62742	0.218618	0.109309	−	0.994008i	$-0.465136\pi$
$919$	6.62742	0.218618	0.109309	+	0.994008i	$0.465136\pi$
$920$	−7.31371	−0.241126
$921$	0	0
$922$	16.8284	0.554215
$923$	4.00000	0.131662
$924$	0	0
$925$	3.65685	0.120237
$926$	27.3137	0.897584
$927$	0	0
$928$	1.85786	0.0609874
$929$	46.9706	1.54105	0.770527	−	0.637407i	$-0.219993\pi$
$929$	46.9706	1.54105	0.770527	+	0.637407i	$0.219993\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	15.3137	0.501617
$933$	0	0
$934$	−84.0833	−2.75129
$935$	−3.31371	−0.108370
$936$	0	0
$937$	−52.6274	−1.71926	−0.859631	−	0.510915i	$-0.829307\pi$
$937$	−52.6274	−1.71926	−0.859631	+	0.510915i	$0.829307\pi$
$938$	−23.3137	−0.761220
$939$	0	0
$940$	−21.6569	−0.706369
$941$	19.8579	0.647348	0.323674	−	0.946169i	$-0.395082\pi$
$941$	19.8579	0.647348	0.323674	+	0.946169i	$0.395082\pi$
$942$	0	0
$943$	2.62742	0.0855605
$944$	−4.97056	−0.161778
$945$	0	0
$946$	3.31371	0.107738
$947$	−21.7990	−0.708372	−0.354186	−	0.935175i	$-0.615242\pi$
$947$	−21.7990	−0.708372	−0.354186	+	0.935175i	$0.615242\pi$
$948$	0	0
$949$	7.31371	0.237413
$950$	−2.41421	−0.0783274
$951$	0	0
$952$	−8.82843	−0.286131
$953$	−41.1716	−1.33368	−0.666839	−	0.745202i	$-0.732353\pi$
$953$	−41.1716	−1.33368	−0.666839	+	0.745202i	$0.732353\pi$
$954$	0	0
$955$	43.5980	1.41080
$956$	−24.8284	−0.803009
$957$	0	0
$958$	−9.17157	−0.296320
$959$	9.65685	0.311836
$960$	0	0
$961$	−15.0000	−0.483871
$962$	17.6569	0.569280
$963$	0	0
$964$	14.0000	0.450910
$965$	−7.31371	−0.235437
$966$	0	0
$967$	31.5980	1.01612	0.508061	−	0.861321i	$-0.330362\pi$
$967$	31.5980	1.01612	0.508061	+	0.861321i	$0.330362\pi$
$968$	45.5269	1.46329
$969$	0	0
$970$	1.65685	0.0531984
$971$	22.6274	0.726148	0.363074	−	0.931760i	$-0.381727\pi$
$971$	22.6274	0.726148	0.363074	+	0.931760i	$0.381727\pi$
$972$	0	0
$973$	21.6569	0.694287
$974$	−13.6569	−0.437594
$975$	0	0
$976$	1.02944	0.0329515
$977$	23.1127	0.739441	0.369720	−	0.929143i	$-0.379453\pi$
$977$	23.1127	0.739441	0.369720	+	0.929143i	$0.379453\pi$
$978$	0	0
$979$	4.00000	0.127841
$980$	−7.65685	−0.244589
$981$	0	0
$982$	−88.9117	−2.83729
$983$	41.9411	1.33771	0.668857	−	0.743391i	$-0.266784\pi$
$983$	41.9411	1.33771	0.668857	+	0.743391i	$0.266784\pi$
$984$	0	0
$985$	25.9411	0.826553
$986$	−5.65685	−0.180151
$987$	0	0
$988$	−7.65685	−0.243597
$989$	1.37258	0.0436456
$990$	0	0
$991$	12.2843	0.390223	0.195111	−	0.980781i	$-0.437493\pi$
$991$	12.2843	0.390223	0.195111	+	0.980781i	$0.437493\pi$
$992$	6.34315	0.201395
$993$	0	0
$994$	−4.82843	−0.153148
$995$	−22.6274	−0.717337
$996$	0	0
$997$	45.5980	1.44410	0.722051	−	0.691840i	$-0.243199\pi$
$997$	45.5980	1.44410	0.722051	+	0.691840i	$0.243199\pi$
$998$	−94.9117	−3.00438
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1197.2.a.f.1.1	✓	2
3.2	odd	2		1197.2.a.i.1.2	yes	2
7.6	odd	2		8379.2.a.v.1.1		2
21.20	even	2		8379.2.a.bg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1197.2.a.f.1.1	✓	2	1.1	even	1	trivial
1197.2.a.i.1.2	yes	2	3.2	odd	2
8379.2.a.v.1.1		2	7.6	odd	2
8379.2.a.bg.1.2		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1197 = 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1197.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +3.82843 q^{4} -2.00000 q^{5} +1.00000 q^{7} -4.41421 q^{8} +4.82843 q^{10} +0.828427 q^{11} +2.00000 q^{13} -2.41421 q^{14} +3.00000 q^{16} +2.00000 q^{17} -1.00000 q^{19} -7.65685 q^{20} -2.00000 q^{22} -0.828427 q^{23} -1.00000 q^{25} -4.82843 q^{26} +3.82843 q^{28} +1.17157 q^{29} +4.00000 q^{31} +1.58579 q^{32} -4.82843 q^{34} -2.00000 q^{35} -3.65685 q^{37} +2.41421 q^{38} +8.82843 q^{40} -3.17157 q^{41} -1.65685 q^{43} +3.17157 q^{44} +2.00000 q^{46} +2.82843 q^{47} +1.00000 q^{49} +2.41421 q^{50} +7.65685 q^{52} +8.48528 q^{53} -1.65685 q^{55} -4.41421 q^{56} -2.82843 q^{58} -1.65685 q^{59} +0.343146 q^{61} -9.65685 q^{62} -9.82843 q^{64} -4.00000 q^{65} +9.65685 q^{67} +7.65685 q^{68} +4.82843 q^{70} +2.00000 q^{71} +3.65685 q^{73} +8.82843 q^{74} -3.82843 q^{76} +0.828427 q^{77} -11.3137 q^{79} -6.00000 q^{80} +7.65685 q^{82} -12.4853 q^{83} -4.00000 q^{85} +4.00000 q^{86} -3.65685 q^{88} +4.82843 q^{89} +2.00000 q^{91} -3.17157 q^{92} -6.82843 q^{94} +2.00000 q^{95} +0.343146 q^{97} -2.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} - 6 q^{8} + 4 q^{10} - 4 q^{11} + 4 q^{13} - 2 q^{14} + 6 q^{16} + 4 q^{17} - 2 q^{19} - 4 q^{20} - 4 q^{22} + 4 q^{23} - 2 q^{25} - 4 q^{26} + 2 q^{28} + 8 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^7 - 6 * q^8 + 4 * q^10 - 4 * q^11 + 4 * q^13 - 2 * q^14 + 6 * q^16 + 4 * q^17 - 2 * q^19 - 4 * q^20 - 4 * q^22 + 4 * q^23 - 2 * q^25 - 4 * q^26 + 2 * q^28 + 8 * q^29 + 8 * q^31 + 6 * q^32 - 4 * q^34 - 4 * q^35 + 4 * q^37 + 2 * q^38 + 12 * q^40 - 12 * q^41 + 8 * q^43 + 12 * q^44 + 4 * q^46 + 2 * q^49 + 2 * q^50 + 4 * q^52 + 8 * q^55 - 6 * q^56 + 8 * q^59 + 12 * q^61 - 8 * q^62 - 14 * q^64 - 8 * q^65 + 8 * q^67 + 4 * q^68 + 4 * q^70 + 4 * q^71 - 4 * q^73 + 12 * q^74 - 2 * q^76 - 4 * q^77 - 12 * q^80 + 4 * q^82 - 8 * q^83 - 8 * q^85 + 8 * q^86 + 4 * q^88 + 4 * q^89 + 4 * q^91 - 12 * q^92 - 8 * q^94 + 4 * q^95 + 12 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more