LMFDB - Embedded newform 9747.2.a.bm.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9747,2,Mod(1,9747)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9747, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9747.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9747 = 3^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9747.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-1,0,3,5,0,1,-6,0,1,2,0,-5,-2,0,-3,10,0,0,1,0,-4,3,0,5,1,0, 2,6,0,2,-8,0,4,3,0,-13,0,0,-6,13,0,3,-20,0,5,18,0,13,3,0,-22,1,0,6,24, 0,14,-11,0,13,22,0,0,-36,0,9,5,0,-30,6,0,15,43,0,0,29,0,-13,1,0,-4,8,0, 3,7,0,15,-29,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

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

Self dual:	yes
Analytic conductor:	$77.8301868501$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.54265221.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 7x^{4} + 4x^{3} + 12x^{2} - 3x - 3 $ x^6 - x^5 - 7x^4 + 4x^3 + 12x^2 - 3x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 513)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.433013$ of defining polynomial
Character	$\chi$	$=$	9747.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.433013 q^{2} -1.81250 q^{4} -2.03183 q^{5} -1.15054 q^{7} -1.65086 q^{8} -0.879808 q^{10} +3.69231 q^{11} +1.38097 q^{13} -0.498199 q^{14} +2.91016 q^{16} -2.34317 q^{17} +3.68269 q^{20} +1.59882 q^{22} -2.53003 q^{23} -0.871671 q^{25} +0.597977 q^{26} +2.08535 q^{28} +0.619872 q^{29} -0.936857 q^{31} +4.56186 q^{32} -1.01462 q^{34} +2.33770 q^{35} -3.82480 q^{37} +3.35427 q^{40} +5.03183 q^{41} +10.1890 q^{43} -6.69231 q^{44} -1.09553 q^{46} +2.55737 q^{47} -5.67626 q^{49} -0.377445 q^{50} -2.50300 q^{52} -10.7807 q^{53} -7.50214 q^{55} +1.89938 q^{56} +0.268412 q^{58} +7.53451 q^{59} -7.10705 q^{61} -0.405671 q^{62} -3.84497 q^{64} -2.80589 q^{65} -3.25052 q^{67} +4.24699 q^{68} +1.01225 q^{70} -7.94477 q^{71} -4.75513 q^{73} -1.65619 q^{74} -4.24815 q^{77} -5.81592 q^{79} -5.91294 q^{80} +2.17885 q^{82} -16.8621 q^{83} +4.76092 q^{85} +4.41196 q^{86} -6.09549 q^{88} -11.1598 q^{89} -1.58886 q^{91} +4.58568 q^{92} +1.10737 q^{94} -11.0318 q^{97} -2.45789 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} + 3 q^{4} + 5 q^{5} + q^{7} - 6 q^{8} + q^{10} + 2 q^{11} - 5 q^{13} - 2 q^{14} - 3 q^{16} + 10 q^{17} + q^{20} - 4 q^{22} + 3 q^{23} + 5 q^{25} + q^{26} + 2 q^{28} + 6 q^{29} + 2 q^{31}+ \cdots + 59 q^{98}+O(q^{100}) $ 6 * q - q^2 + 3 * q^4 + 5 * q^5 + q^7 - 6 * q^8 + q^10 + 2 * q^11 - 5 * q^13 - 2 * q^14 - 3 * q^16 + 10 * q^17 + q^20 - 4 * q^22 + 3 * q^23 + 5 * q^25 + q^26 + 2 * q^28 + 6 * q^29 + 2 * q^31 - 8 * q^32 + 4 * q^34 + 3 * q^35 - 13 * q^37 - 6 * q^40 + 13 * q^41 + 3 * q^43 - 20 * q^44 + 5 * q^46 + 18 * q^47 + 13 * q^49 + 3 * q^50 - 22 * q^52 + q^53 + 6 * q^55 + 24 * q^56 + 14 * q^58 - 11 * q^59 + 13 * q^61 + 22 * q^62 - 36 * q^65 + 9 * q^67 + 5 * q^68 - 30 * q^70 + 6 * q^71 + 15 * q^73 + 43 * q^74 + 29 * q^77 - 13 * q^79 + q^80 - 4 * q^82 + 8 * q^83 + 3 * q^85 + 7 * q^86 + 15 * q^88 - 29 * q^89 + 22 * q^91 + 21 * q^92 - 18 * q^94 + 20 * q^97 + 59 * 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 $

$2$	0.433013	0.306186	0.153093	−	0.988212i	$-0.451077\pi$
$2$	0.433013	0.306186	0.153093	+	0.988212i	$0.451077\pi$
$3$	0	0
$3$	0	0
$4$	−1.81250	−0.906250
$5$	−2.03183	−0.908662	−0.454331	−	0.890833i	$-0.650122\pi$
$5$	−2.03183	−0.908662	−0.454331	+	0.890833i	$0.650122\pi$
$6$	0	0
$7$	−1.15054	−0.434863	−0.217432	−	0.976076i	$-0.569768\pi$
$7$	−1.15054	−0.434863	−0.217432	+	0.976076i	$0.569768\pi$
$8$	−1.65086	−0.583668
$9$	0	0
$10$	−0.879808	−0.278220
$11$	3.69231	1.11327	0.556636	−	0.830756i	$-0.312092\pi$
$11$	3.69231	1.11327	0.556636	+	0.830756i	$0.312092\pi$
$12$	0	0
$13$	1.38097	0.383012	0.191506	−	0.981491i	$-0.438663\pi$
$13$	1.38097	0.383012	0.191506	+	0.981491i	$0.438663\pi$
$14$	−0.498199	−0.133149
$15$	0	0
$16$	2.91016	0.727539
$17$	−2.34317	−0.568302	−0.284151	−	0.958780i	$-0.591712\pi$
$17$	−2.34317	−0.568302	−0.284151	+	0.958780i	$0.591712\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	3.68269	0.823475
$21$	0	0
$22$	1.59882	0.340869
$23$	−2.53003	−0.527547	−0.263774	−	0.964585i	$-0.584967\pi$
$23$	−2.53003	−0.527547	−0.263774	+	0.964585i	$0.584967\pi$
$24$	0	0
$25$	−0.871671	−0.174334
$26$	0.597977	0.117273
$27$	0	0
$28$	2.08535	0.394095
$29$	0.619872	0.115107	0.0575536	−	0.998342i	$-0.481670\pi$
$29$	0.619872	0.115107	0.0575536	+	0.998342i	$0.481670\pi$
$30$	0	0
$31$	−0.936857	−0.168264	−0.0841322	−	0.996455i	$-0.526812\pi$
$31$	−0.936857	−0.168264	−0.0841322	+	0.996455i	$0.526812\pi$
$32$	4.56186	0.806430
$33$	0	0
$34$	−1.01462	−0.174006
$35$	2.33770	0.395144
$36$	0	0
$37$	−3.82480	−0.628794	−0.314397	−	0.949292i	$-0.601802\pi$
$37$	−3.82480	−0.628794	−0.314397	+	0.949292i	$0.601802\pi$
$38$	0	0
$39$	0	0
$40$	3.35427	0.530356
$41$	5.03183	0.785840	0.392920	−	0.919573i	$-0.371465\pi$
$41$	5.03183	0.785840	0.392920	+	0.919573i	$0.371465\pi$
$42$	0	0
$43$	10.1890	1.55380	0.776902	−	0.629621i	$-0.216790\pi$
$43$	10.1890	1.55380	0.776902	+	0.629621i	$0.216790\pi$
$44$	−6.69231	−1.00890
$45$	0	0
$46$	−1.09553	−0.161528
$47$	2.55737	0.373031	0.186515	−	0.982452i	$-0.440281\pi$
$47$	2.55737	0.373031	0.186515	+	0.982452i	$0.440281\pi$
$48$	0	0
$49$	−5.67626	−0.810894
$50$	−0.377445	−0.0533787
$51$	0	0
$52$	−2.50300	−0.347104
$53$	−10.7807	−1.48084	−0.740419	−	0.672145i	$-0.765373\pi$
$53$	−10.7807	−1.48084	−0.740419	+	0.672145i	$0.765373\pi$
$54$	0	0
$55$	−7.50214	−1.01159
$56$	1.89938	0.253816
$57$	0	0
$58$	0.268412	0.0352443
$59$	7.53451	0.980910	0.490455	−	0.871466i	$-0.336831\pi$
$59$	7.53451	0.980910	0.490455	+	0.871466i	$0.336831\pi$
$60$	0	0
$61$	−7.10705	−0.909964	−0.454982	−	0.890501i	$-0.650354\pi$
$61$	−7.10705	−0.909964	−0.454982	+	0.890501i	$0.650354\pi$
$62$	−0.405671	−0.0515203
$63$	0	0
$64$	−3.84497	−0.480621
$65$	−2.80589	−0.348028
$66$	0	0
$67$	−3.25052	−0.397114	−0.198557	−	0.980089i	$-0.563625\pi$
$67$	−3.25052	−0.397114	−0.198557	+	0.980089i	$0.563625\pi$
$68$	4.24699	0.515024
$69$	0	0
$70$	1.01225	0.120988
$71$	−7.94477	−0.942871	−0.471435	−	0.881901i	$-0.656264\pi$
$71$	−7.94477	−0.942871	−0.471435	+	0.881901i	$0.656264\pi$
$72$	0	0
$73$	−4.75513	−0.556545	−0.278273	−	0.960502i	$-0.589762\pi$
$73$	−4.75513	−0.556545	−0.278273	+	0.960502i	$0.589762\pi$
$74$	−1.65619	−0.192528
$75$	0	0
$76$	0	0
$77$	−4.24815	−0.484122
$78$	0	0
$79$	−5.81592	−0.654343	−0.327171	−	0.944965i	$-0.606095\pi$
$79$	−5.81592	−0.654343	−0.327171	+	0.944965i	$0.606095\pi$
$80$	−5.91294	−0.661087
$81$	0	0
$82$	2.17885	0.240613
$83$	−16.8621	−1.85086	−0.925428	−	0.378924i	$-0.876294\pi$
$83$	−16.8621	−1.85086	−0.925428	+	0.378924i	$0.876294\pi$
$84$	0	0
$85$	4.76092	0.516394
$86$	4.41196	0.475754
$87$	0	0
$88$	−6.09549	−0.649781
$89$	−11.1598	−1.18294	−0.591470	−	0.806327i	$-0.701452\pi$
$89$	−11.1598	−1.18294	−0.591470	+	0.806327i	$0.701452\pi$
$90$	0	0
$91$	−1.58886	−0.166558
$92$	4.58568	0.478090
$93$	0	0
$94$	1.10737	0.114217
$95$	0	0
$96$	0	0
$97$	−11.0318	−1.12011	−0.560056	−	0.828455i	$-0.689220\pi$
$97$	−11.0318	−1.12011	−0.560056	+	0.828455i	$0.689220\pi$
$98$	−2.45789	−0.248285
$99$	0	0
$100$	1.57990	0.157990
$101$	−8.00449	−0.796476	−0.398238	−	0.917282i	$-0.630378\pi$
$101$	−8.00449	−0.796476	−0.398238	+	0.917282i	$0.630378\pi$
$102$	0	0
$103$	13.5972	1.33977	0.669884	−	0.742466i	$-0.266344\pi$
$103$	13.5972	1.33977	0.669884	+	0.742466i	$0.266344\pi$
$104$	−2.27979	−0.223551
$105$	0	0
$106$	−4.66817	−0.453412
$107$	11.7261	1.13360	0.566802	−	0.823854i	$-0.308181\pi$
$107$	11.7261	1.13360	0.566802	+	0.823854i	$0.308181\pi$
$108$	0	0
$109$	13.7239	1.31451	0.657256	−	0.753667i	$-0.271717\pi$
$109$	13.7239	1.31451	0.657256	+	0.753667i	$0.271717\pi$
$110$	−3.24852	−0.309734
$111$	0	0
$112$	−3.34825	−0.316380
$113$	16.9792	1.59727	0.798635	−	0.601815i	$-0.205556\pi$
$113$	16.9792	1.59727	0.798635	+	0.601815i	$0.205556\pi$
$114$	0	0
$115$	5.14058	0.479362
$116$	−1.12352	−0.104316
$117$	0	0
$118$	3.26254	0.300341
$119$	2.69591	0.247134
$120$	0	0
$121$	2.63314	0.239376
$122$	−3.07744	−0.278619
$123$	0	0
$124$	1.69805	0.152490
$125$	11.9302	1.06707
$126$	0	0
$127$	10.9449	0.971200	0.485600	−	0.874181i	$-0.338601\pi$
$127$	10.9449	0.971200	0.485600	+	0.874181i	$0.338601\pi$
$128$	−10.7886	−0.953590
$129$	0	0
$130$	−1.21499	−0.106561
$131$	22.4705	1.96326	0.981628	−	0.190806i	$-0.0611103\pi$
$131$	22.4705	1.96326	0.981628	+	0.190806i	$0.0611103\pi$
$132$	0	0
$133$	0	0
$134$	−1.40752	−0.121591
$135$	0	0
$136$	3.86825	0.331699
$137$	6.60344	0.564170	0.282085	−	0.959389i	$-0.408974\pi$
$137$	6.60344	0.564170	0.282085	+	0.959389i	$0.408974\pi$
$138$	0	0
$139$	−15.3840	−1.30485	−0.652426	−	0.757853i	$-0.726249\pi$
$139$	−15.3840	−1.30485	−0.652426	+	0.757853i	$0.726249\pi$
$140$	−4.23708	−0.358099
$141$	0	0
$142$	−3.44019	−0.288694
$143$	5.09896	0.426396
$144$	0	0
$145$	−1.25947	−0.104594
$146$	−2.05903	−0.170407
$147$	0	0
$148$	6.93245	0.569844
$149$	−2.03622	−0.166813	−0.0834067	−	0.996516i	$-0.526580\pi$
$149$	−2.03622	−0.166813	−0.0834067	+	0.996516i	$0.526580\pi$
$150$	0	0
$151$	13.4553	1.09498	0.547490	−	0.836812i	$-0.315583\pi$
$151$	13.4553	1.09498	0.547490	+	0.836812i	$0.315583\pi$
$152$	0	0
$153$	0	0
$154$	−1.83950	−0.148231
$155$	1.90353	0.152895
$156$	0	0
$157$	−7.85535	−0.626925	−0.313463	−	0.949601i	$-0.601489\pi$
$157$	−7.85535	−0.626925	−0.313463	+	0.949601i	$0.601489\pi$
$158$	−2.51837	−0.200351
$159$	0	0
$160$	−9.26891	−0.732772
$161$	2.91090	0.229411
$162$	0	0
$163$	6.47140	0.506879	0.253440	−	0.967351i	$-0.418438\pi$
$163$	6.47140	0.506879	0.253440	+	0.967351i	$0.418438\pi$
$164$	−9.12019	−0.712167
$165$	0	0
$166$	−7.30150	−0.566706
$167$	−17.1882	−1.33006	−0.665032	−	0.746815i	$-0.731582\pi$
$167$	−17.1882	−1.33006	−0.665032	+	0.746815i	$0.731582\pi$
$168$	0	0
$169$	−11.0929	−0.853302
$170$	2.06154	0.158113
$171$	0	0
$172$	−18.4675	−1.40814
$173$	9.34682	0.710625	0.355313	−	0.934748i	$-0.384374\pi$
$173$	9.34682	0.710625	0.355313	+	0.934748i	$0.384374\pi$
$174$	0	0
$175$	1.00289	0.0758115
$176$	10.7452	0.809949
$177$	0	0
$178$	−4.83235	−0.362200
$179$	−13.0265	−0.973643	−0.486822	−	0.873501i	$-0.661844\pi$
$179$	−13.0265	−0.973643	−0.486822	+	0.873501i	$0.661844\pi$
$180$	0	0
$181$	12.6690	0.941676	0.470838	−	0.882220i	$-0.343952\pi$
$181$	12.6690	0.941676	0.470838	+	0.882220i	$0.343952\pi$
$182$	−0.687996	−0.0509977
$183$	0	0
$184$	4.17672	0.307912
$185$	7.77134	0.571360
$186$	0	0
$187$	−8.65170	−0.632675
$188$	−4.63523	−0.338059
$189$	0	0
$190$	0	0
$191$	19.6919	1.42486	0.712429	−	0.701744i	$-0.247595\pi$
$191$	19.6919	1.42486	0.712429	+	0.701744i	$0.247595\pi$
$192$	0	0
$193$	−1.53289	−0.110340	−0.0551699	−	0.998477i	$-0.517570\pi$
$193$	−1.53289	−0.110340	−0.0551699	+	0.998477i	$0.517570\pi$
$194$	−4.77692	−0.342963
$195$	0	0
$196$	10.2882	0.734873
$197$	−4.01965	−0.286388	−0.143194	−	0.989695i	$-0.545737\pi$
$197$	−4.01965	−0.286388	−0.143194	+	0.989695i	$0.545737\pi$
$198$	0	0
$199$	7.53526	0.534160	0.267080	−	0.963674i	$-0.413941\pi$
$199$	7.53526	0.534160	0.267080	+	0.963674i	$0.413941\pi$
$200$	1.43901	0.101753
$201$	0	0
$202$	−3.46604	−0.243870
$203$	−0.713187	−0.0500559
$204$	0	0
$205$	−10.2238	−0.714062
$206$	5.88774	0.410218
$207$	0	0
$208$	4.01883	0.278656
$209$	0	0
$210$	0	0
$211$	3.19135	0.219701	0.109851	−	0.993948i	$-0.464963\pi$
$211$	3.19135	0.219701	0.109851	+	0.993948i	$0.464963\pi$
$212$	19.5400	1.34201
$213$	0	0
$214$	5.07754	0.347094
$215$	−20.7023	−1.41188
$216$	0	0
$217$	1.07789	0.0731720
$218$	5.94263	0.402486
$219$	0	0
$220$	13.5976	0.916752
$221$	−3.23584	−0.217666
$222$	0	0
$223$	14.3950	0.963960	0.481980	−	0.876182i	$-0.339918\pi$
$223$	14.3950	0.963960	0.481980	+	0.876182i	$0.339918\pi$
$224$	−5.24860	−0.350687
$225$	0	0
$226$	7.35222	0.489062
$227$	8.88794	0.589914	0.294957	−	0.955511i	$-0.404695\pi$
$227$	8.88794	0.589914	0.294957	+	0.955511i	$0.404695\pi$
$228$	0	0
$229$	−6.83849	−0.451900	−0.225950	−	0.974139i	$-0.572549\pi$
$229$	−6.83849	−0.451900	−0.225950	+	0.974139i	$0.572549\pi$
$230$	2.22594	0.146774
$231$	0	0
$232$	−1.02332	−0.0671844
$233$	−1.03738	−0.0679607	−0.0339804	−	0.999423i	$-0.510818\pi$
$233$	−1.03738	−0.0679607	−0.0339804	+	0.999423i	$0.510818\pi$
$234$	0	0
$235$	−5.19614	−0.338959
$236$	−13.6563	−0.888950
$237$	0	0
$238$	1.16736	0.0756689
$239$	18.5956	1.20285	0.601426	−	0.798929i	$-0.294600\pi$
$239$	18.5956	1.20285	0.601426	+	0.798929i	$0.294600\pi$
$240$	0	0
$241$	−26.6875	−1.71909	−0.859545	−	0.511060i	$-0.829253\pi$
$241$	−26.6875	−1.71909	−0.859545	+	0.511060i	$0.829253\pi$
$242$	1.14018	0.0732937
$243$	0	0
$244$	12.8815	0.824655
$245$	11.5332	0.736828
$246$	0	0
$247$	0	0
$248$	1.54662	0.0982105
$249$	0	0
$250$	5.16594	0.326723
$251$	−2.80194	−0.176857	−0.0884286	−	0.996083i	$-0.528185\pi$
$251$	−2.80194	−0.176857	−0.0884286	+	0.996083i	$0.528185\pi$
$252$	0	0
$253$	−9.34164	−0.587304
$254$	4.73927	0.297368
$255$	0	0
$256$	3.01832	0.188645
$257$	7.61432	0.474969	0.237484	−	0.971391i	$-0.423677\pi$
$257$	7.61432	0.474969	0.237484	+	0.971391i	$0.423677\pi$
$258$	0	0
$259$	4.40059	0.273439
$260$	5.08568	0.315400
$261$	0	0
$262$	9.73001	0.601122
$263$	15.2930	0.943006	0.471503	−	0.881864i	$-0.343712\pi$
$263$	15.2930	0.943006	0.471503	+	0.881864i	$0.343712\pi$
$264$	0	0
$265$	21.9045	1.34558
$266$	0	0
$267$	0	0
$268$	5.89156	0.359884
$269$	30.0220	1.83047	0.915237	−	0.402916i	$-0.132003\pi$
$269$	30.0220	1.83047	0.915237	+	0.402916i	$0.132003\pi$
$270$	0	0
$271$	−12.4564	−0.756670	−0.378335	−	0.925669i	$-0.623503\pi$
$271$	−12.4564	−0.756670	−0.378335	+	0.925669i	$0.623503\pi$
$272$	−6.81899	−0.413462
$273$	0	0
$274$	2.85938	0.172741
$275$	−3.21848	−0.194081
$276$	0	0
$277$	11.0756	0.665466	0.332733	−	0.943021i	$-0.392029\pi$
$277$	11.0756	0.665466	0.332733	+	0.943021i	$0.392029\pi$
$278$	−6.66146	−0.399528
$279$	0	0
$280$	−3.85922	−0.230633
$281$	16.6963	0.996020	0.498010	−	0.867171i	$-0.334064\pi$
$281$	16.6963	0.996020	0.498010	+	0.867171i	$0.334064\pi$
$282$	0	0
$283$	25.8258	1.53519	0.767593	−	0.640937i	$-0.221454\pi$
$283$	25.8258	1.53519	0.767593	+	0.640937i	$0.221454\pi$
$284$	14.3999	0.854476
$285$	0	0
$286$	2.20791	0.130557
$287$	−5.78932	−0.341733
$288$	0	0
$289$	−11.5096	−0.677033
$290$	−0.545368	−0.0320251
$291$	0	0
$292$	8.61867	0.504369
$293$	−27.9386	−1.63219	−0.816094	−	0.577919i	$-0.803865\pi$
$293$	−27.9386	−1.63219	−0.816094	+	0.577919i	$0.803865\pi$
$294$	0	0
$295$	−15.3088	−0.891315
$296$	6.31422	0.367006
$297$	0	0
$298$	−0.881708	−0.0510760
$299$	−3.49389	−0.202057
$300$	0	0
$301$	−11.7228	−0.675693
$302$	5.82634	0.335268
$303$	0	0
$304$	0	0
$305$	14.4403	0.826850
$306$	0	0
$307$	−12.5241	−0.714789	−0.357394	−	0.933954i	$-0.616335\pi$
$307$	−12.5241	−0.714789	−0.357394	+	0.933954i	$0.616335\pi$
$308$	7.69977	0.438735
$309$	0	0
$310$	0.824254	0.0468145
$311$	24.4906	1.38873	0.694366	−	0.719622i	$-0.255685\pi$
$311$	24.4906	1.38873	0.694366	+	0.719622i	$0.255685\pi$
$312$	0	0
$313$	−20.4667	−1.15685	−0.578424	−	0.815736i	$-0.696332\pi$
$313$	−20.4667	−1.15685	−0.578424	+	0.815736i	$0.696332\pi$
$314$	−3.40147	−0.191956
$315$	0	0
$316$	10.5414	0.592998
$317$	−21.9961	−1.23543	−0.617713	−	0.786404i	$-0.711941\pi$
$317$	−21.9961	−1.23543	−0.617713	+	0.786404i	$0.711941\pi$
$318$	0	0
$319$	2.28876	0.128146
$320$	7.81232	0.436722
$321$	0	0
$322$	1.26046	0.0702425
$323$	0	0
$324$	0	0
$325$	−1.20375	−0.0667720
$326$	2.80220	0.155199
$327$	0	0
$328$	−8.30685	−0.458669
$329$	−2.94236	−0.162217
$330$	0	0
$331$	−32.5077	−1.78679	−0.893393	−	0.449275i	$-0.851682\pi$
$331$	−32.5077	−1.78679	−0.893393	+	0.449275i	$0.851682\pi$
$332$	30.5625	1.67734
$333$	0	0
$334$	−7.44271	−0.407247
$335$	6.60449	0.360842
$336$	0	0
$337$	−2.56368	−0.139653	−0.0698264	−	0.997559i	$-0.522245\pi$
$337$	−2.56368	−0.139653	−0.0698264	+	0.997559i	$0.522245\pi$
$338$	−4.80338	−0.261269
$339$	0	0
$340$	−8.62916	−0.467982
$341$	−3.45916	−0.187324
$342$	0	0
$343$	14.5845	0.787491
$344$	−16.8206	−0.906905
$345$	0	0
$346$	4.04729	0.217584
$347$	9.34905	0.501883	0.250942	−	0.968002i	$-0.419260\pi$
$347$	9.34905	0.501883	0.250942	+	0.968002i	$0.419260\pi$
$348$	0	0
$349$	9.97826	0.534124	0.267062	−	0.963679i	$-0.413947\pi$
$349$	9.97826	0.534124	0.267062	+	0.963679i	$0.413947\pi$
$350$	0.434265	0.0232125
$351$	0	0
$352$	16.8438	0.897776
$353$	27.7171	1.47523	0.737617	−	0.675219i	$-0.235951\pi$
$353$	27.7171	1.47523	0.737617	+	0.675219i	$0.235951\pi$
$354$	0	0
$355$	16.1424	0.856750
$356$	20.2272	1.07204
$357$	0	0
$358$	−5.64062	−0.298116
$359$	34.3935	1.81522	0.907609	−	0.419817i	$-0.137906\pi$
$359$	34.3935	1.81522	0.907609	+	0.419817i	$0.137906\pi$
$360$	0	0
$361$	0	0
$362$	5.48582	0.288328
$363$	0	0
$364$	2.87981	0.150943
$365$	9.66160	0.505711
$366$	0	0
$367$	32.2953	1.68580	0.842901	−	0.538068i	$-0.180846\pi$
$367$	32.2953	1.68580	0.842901	+	0.538068i	$0.180846\pi$
$368$	−7.36278	−0.383811
$369$	0	0
$370$	3.36509	0.174943
$371$	12.4036	0.643963
$372$	0	0
$373$	35.2967	1.82759	0.913796	−	0.406173i	$-0.133137\pi$
$373$	35.2967	1.82759	0.913796	+	0.406173i	$0.133137\pi$
$374$	−3.74630	−0.193716
$375$	0	0
$376$	−4.22186	−0.217726
$377$	0.856023	0.0440874
$378$	0	0
$379$	−24.0421	−1.23496	−0.617479	−	0.786587i	$-0.711846\pi$
$379$	−24.0421	−1.23496	−0.617479	+	0.786587i	$0.711846\pi$
$380$	0	0
$381$	0	0
$382$	8.52686	0.436272
$383$	6.71864	0.343307	0.171653	−	0.985157i	$-0.445089\pi$
$383$	6.71864	0.343307	0.171653	+	0.985157i	$0.445089\pi$
$384$	0	0
$385$	8.63151	0.439903
$386$	−0.663761	−0.0337845
$387$	0	0
$388$	19.9952	1.01510
$389$	36.4486	1.84802	0.924010	−	0.382369i	$-0.124892\pi$
$389$	36.4486	1.84802	0.924010	+	0.382369i	$0.124892\pi$
$390$	0	0
$391$	5.92828	0.299806
$392$	9.37071	0.473292
$393$	0	0
$394$	−1.74056	−0.0876882
$395$	11.8170	0.594576
$396$	0	0
$397$	5.04933	0.253419	0.126709	−	0.991940i	$-0.459558\pi$
$397$	5.04933	0.253419	0.126709	+	0.991940i	$0.459558\pi$
$398$	3.26286	0.163553
$399$	0	0
$400$	−2.53670	−0.126835
$401$	−11.1438	−0.556494	−0.278247	−	0.960510i	$-0.589753\pi$
$401$	−11.1438	−0.556494	−0.278247	+	0.960510i	$0.589753\pi$
$402$	0	0
$403$	−1.29377	−0.0644472
$404$	14.5081	0.721807
$405$	0	0
$406$	−0.308819	−0.0153264
$407$	−14.1223	−0.700019
$408$	0	0
$409$	−4.92406	−0.243479	−0.121740	−	0.992562i	$-0.538847\pi$
$409$	−4.92406	−0.243479	−0.121740	+	0.992562i	$0.538847\pi$
$410$	−4.42704	−0.218636
$411$	0	0
$412$	−24.6448	−1.21416
$413$	−8.66876	−0.426562
$414$	0	0
$415$	34.2609	1.68180
$416$	6.29978	0.308872
$417$	0	0
$418$	0	0
$419$	−14.1877	−0.693116	−0.346558	−	0.938029i	$-0.612650\pi$
$419$	−14.1877	−0.693116	−0.346558	+	0.938029i	$0.612650\pi$
$420$	0	0
$421$	−17.0145	−0.829238	−0.414619	−	0.909995i	$-0.636085\pi$
$421$	−17.0145	−0.829238	−0.414619	+	0.909995i	$0.636085\pi$
$422$	1.38189	0.0672695
$423$	0	0
$424$	17.7974	0.864317
$425$	2.04247	0.0990744
$426$	0	0
$427$	8.17695	0.395710
$428$	−21.2535	−1.02733
$429$	0	0
$430$	−8.96434	−0.432299
$431$	−1.94991	−0.0939237	−0.0469618	−	0.998897i	$-0.514954\pi$
$431$	−1.94991	−0.0939237	−0.0469618	+	0.998897i	$0.514954\pi$
$432$	0	0
$433$	29.3603	1.41097	0.705483	−	0.708727i	$-0.250730\pi$
$433$	29.3603	1.41097	0.705483	+	0.708727i	$0.250730\pi$
$434$	0.466741	0.0224043
$435$	0	0
$436$	−24.8746	−1.19128
$437$	0	0
$438$	0	0
$439$	10.7153	0.511415	0.255708	−	0.966754i	$-0.417692\pi$
$439$	10.7153	0.511415	0.255708	+	0.966754i	$0.417692\pi$
$440$	12.3850	0.590431
$441$	0	0
$442$	−1.40116	−0.0666464
$443$	−19.8348	−0.942379	−0.471190	−	0.882032i	$-0.656175\pi$
$443$	−19.8348	−0.942379	−0.471190	+	0.882032i	$0.656175\pi$
$444$	0	0
$445$	22.6749	1.07489
$446$	6.23321	0.295151
$447$	0	0
$448$	4.42379	0.209005
$449$	25.1068	1.18486	0.592431	−	0.805621i	$-0.298168\pi$
$449$	25.1068	1.18486	0.592431	+	0.805621i	$0.298168\pi$
$450$	0	0
$451$	18.5791	0.874854
$452$	−30.7748	−1.44753
$453$	0	0
$454$	3.84859	0.180623
$455$	3.22829	0.151345
$456$	0	0
$457$	10.1416	0.474404	0.237202	−	0.971460i	$-0.423770\pi$
$457$	10.1416	0.474404	0.237202	+	0.971460i	$0.423770\pi$
$458$	−2.96115	−0.138366
$459$	0	0
$460$	−9.31731	−0.434422
$461$	20.5207	0.955746	0.477873	−	0.878429i	$-0.341408\pi$
$461$	20.5207	0.955746	0.477873	+	0.878429i	$0.341408\pi$
$462$	0	0
$463$	20.1006	0.934155	0.467077	−	0.884216i	$-0.345307\pi$
$463$	20.1006	0.934155	0.467077	+	0.884216i	$0.345307\pi$
$464$	1.80392	0.0837450
$465$	0	0
$466$	−0.449197	−0.0208086
$467$	−13.4426	−0.622048	−0.311024	−	0.950402i	$-0.600672\pi$
$467$	−13.4426	−0.622048	−0.311024	+	0.950402i	$0.600672\pi$
$468$	0	0
$469$	3.73985	0.172690
$470$	−2.24999	−0.103784
$471$	0	0
$472$	−12.4384	−0.572525
$473$	37.6208	1.72981
$474$	0	0
$475$	0	0
$476$	−4.88634	−0.223965
$477$	0	0
$478$	8.05214	0.368297
$479$	−1.32570	−0.0605727	−0.0302864	−	0.999541i	$-0.509642\pi$
$479$	−1.32570	−0.0605727	−0.0302864	+	0.999541i	$0.509642\pi$
$480$	0	0
$481$	−5.28193	−0.240835
$482$	−11.5560	−0.526362
$483$	0	0
$484$	−4.77256	−0.216935
$485$	22.4148	1.01780
$486$	0	0
$487$	5.65193	0.256113	0.128057	−	0.991767i	$-0.459126\pi$
$487$	5.65193	0.256113	0.128057	+	0.991767i	$0.459126\pi$
$488$	11.7327	0.531117
$489$	0	0
$490$	4.99402	0.225607
$491$	41.4642	1.87125	0.935626	−	0.352993i	$-0.114836\pi$
$491$	41.4642	1.87125	0.935626	+	0.352993i	$0.114836\pi$
$492$	0	0
$493$	−1.45246	−0.0654157
$494$	0	0
$495$	0	0
$496$	−2.72640	−0.122419
$497$	9.14078	0.410020
$498$	0	0
$499$	−0.329164	−0.0147354	−0.00736769	−	0.999973i	$-0.502345\pi$
$499$	−0.329164	−0.0147354	−0.00736769	+	0.999973i	$0.502345\pi$
$500$	−21.6235	−0.967034
$501$	0	0
$502$	−1.21328	−0.0541512
$503$	18.1748	0.810375	0.405188	−	0.914234i	$-0.367206\pi$
$503$	18.1748	0.810375	0.405188	+	0.914234i	$0.367206\pi$
$504$	0	0
$505$	16.2637	0.723727
$506$	−4.04505	−0.179824
$507$	0	0
$508$	−19.8376	−0.880150
$509$	−4.30950	−0.191015	−0.0955076	−	0.995429i	$-0.530447\pi$
$509$	−4.30950	−0.191015	−0.0955076	+	0.995429i	$0.530447\pi$
$510$	0	0
$511$	5.47097	0.242021
$512$	22.8842	1.01135
$513$	0	0
$514$	3.29710	0.145429
$515$	−27.6271	−1.21740
$516$	0	0
$517$	9.44260	0.415285
$518$	1.90551	0.0837233
$519$	0	0
$520$	4.63214	0.203133
$521$	−36.6688	−1.60649	−0.803245	−	0.595649i	$-0.796895\pi$
$521$	−36.6688	−1.60649	−0.803245	+	0.595649i	$0.796895\pi$
$522$	0	0
$523$	−3.73409	−0.163280	−0.0816402	−	0.996662i	$-0.526016\pi$
$523$	−3.73409	−0.163280	−0.0816402	+	0.996662i	$0.526016\pi$
$524$	−40.7278	−1.77920
$525$	0	0
$526$	6.62206	0.288735
$527$	2.19521	0.0956250
$528$	0	0
$529$	−16.5990	−0.721694
$530$	9.48492	0.411998
$531$	0	0
$532$	0	0
$533$	6.94879	0.300986
$534$	0	0
$535$	−23.8254	−1.03006
$536$	5.36615	0.231782
$537$	0	0
$538$	12.9999	0.560466
$539$	−20.9585	−0.902746
$540$	0	0
$541$	20.5902	0.885240	0.442620	−	0.896709i	$-0.354049\pi$
$541$	20.5902	0.885240	0.442620	+	0.896709i	$0.354049\pi$
$542$	−5.39376	−0.231682
$543$	0	0
$544$	−10.6892	−0.458296
$545$	−27.8846	−1.19445
$546$	0	0
$547$	−14.8575	−0.635262	−0.317631	−	0.948214i	$-0.602887\pi$
$547$	−14.8575	−0.635262	−0.317631	+	0.948214i	$0.602887\pi$
$548$	−11.9687	−0.511279
$549$	0	0
$550$	−1.39364	−0.0594251
$551$	0	0
$552$	0	0
$553$	6.69146	0.284550
$554$	4.79586	0.203756
$555$	0	0
$556$	27.8834	1.18252
$557$	12.6378	0.535479	0.267740	−	0.963491i	$-0.413723\pi$
$557$	12.6378	0.535479	0.267740	+	0.963491i	$0.413723\pi$
$558$	0	0
$559$	14.0707	0.595125
$560$	6.80308	0.287482
$561$	0	0
$562$	7.22973	0.304968
$563$	13.9053	0.586041	0.293020	−	0.956106i	$-0.405340\pi$
$563$	13.9053	0.586041	0.293020	+	0.956106i	$0.405340\pi$
$564$	0	0
$565$	−34.4989	−1.45138
$566$	11.1829	0.470053
$567$	0	0
$568$	13.1157	0.550323
$569$	−11.5682	−0.484963	−0.242482	−	0.970156i	$-0.577961\pi$
$569$	−11.5682	−0.484963	−0.242482	+	0.970156i	$0.577961\pi$
$570$	0	0
$571$	20.2650	0.848062	0.424031	−	0.905648i	$-0.360615\pi$
$571$	20.2650	0.848062	0.424031	+	0.905648i	$0.360615\pi$
$572$	−9.24186	−0.386422
$573$	0	0
$574$	−2.50685	−0.104634
$575$	2.20535	0.0919695
$576$	0	0
$577$	−19.0894	−0.794702	−0.397351	−	0.917667i	$-0.630070\pi$
$577$	−19.0894	−0.794702	−0.397351	+	0.917667i	$0.630070\pi$
$578$	−4.98379	−0.207298
$579$	0	0
$580$	2.28279	0.0947879
$581$	19.4005	0.804869
$582$	0	0
$583$	−39.8055	−1.64858
$584$	7.85005	0.324837
$585$	0	0
$586$	−12.0978	−0.499754
$587$	23.4105	0.966255	0.483127	−	0.875550i	$-0.339501\pi$
$587$	23.4105	0.966255	0.483127	+	0.875550i	$0.339501\pi$
$588$	0	0
$589$	0	0
$590$	−6.62893	−0.272909
$591$	0	0
$592$	−11.1308	−0.457472
$593$	16.3786	0.672588	0.336294	−	0.941757i	$-0.390826\pi$
$593$	16.3786	0.672588	0.336294	+	0.941757i	$0.390826\pi$
$594$	0	0
$595$	−5.47763	−0.224561
$596$	3.69064	0.151175
$597$	0	0
$598$	−1.51290	−0.0618670
$599$	18.1659	0.742240	0.371120	−	0.928585i	$-0.378974\pi$
$599$	18.1659	0.742240	0.371120	+	0.928585i	$0.378974\pi$
$600$	0	0
$601$	7.00570	0.285768	0.142884	−	0.989739i	$-0.454362\pi$
$601$	7.00570	0.285768	0.142884	+	0.989739i	$0.454362\pi$
$602$	−5.07614	−0.206888
$603$	0	0
$604$	−24.3878	−0.992326
$605$	−5.35008	−0.217512
$606$	0	0
$607$	−10.8546	−0.440576	−0.220288	−	0.975435i	$-0.570700\pi$
$607$	−10.8546	−0.440576	−0.220288	+	0.975435i	$0.570700\pi$
$608$	0	0
$609$	0	0
$610$	6.25284	0.253170
$611$	3.53165	0.142875
$612$	0	0
$613$	−48.9731	−1.97800	−0.989002	−	0.147903i	$-0.952748\pi$
$613$	−48.9731	−1.97800	−0.989002	+	0.147903i	$0.952748\pi$
$614$	−5.42310	−0.218858
$615$	0	0
$616$	7.01310	0.282566
$617$	36.9461	1.48739	0.743697	−	0.668517i	$-0.233071\pi$
$617$	36.9461	1.48739	0.743697	+	0.668517i	$0.233071\pi$
$618$	0	0
$619$	−38.4572	−1.54573	−0.772863	−	0.634573i	$-0.781176\pi$
$619$	−38.4572	−1.54573	−0.772863	+	0.634573i	$0.781176\pi$
$620$	−3.45015	−0.138561
$621$	0	0
$622$	10.6047	0.425211
$623$	12.8398	0.514417
$624$	0	0
$625$	−19.8818	−0.795273
$626$	−8.86236	−0.354211
$627$	0	0
$628$	14.2378	0.568151
$629$	8.96216	0.357345
$630$	0	0
$631$	8.10426	0.322625	0.161313	−	0.986903i	$-0.448427\pi$
$631$	8.10426	0.322625	0.161313	+	0.986903i	$0.448427\pi$
$632$	9.60128	0.381919
$633$	0	0
$634$	−9.52460	−0.378270
$635$	−22.2381	−0.882492
$636$	0	0
$637$	−7.83873	−0.310582
$638$	0.991061	0.0392365
$639$	0	0
$640$	21.9207	0.866490
$641$	9.82271	0.387974	0.193987	−	0.981004i	$-0.437858\pi$
$641$	9.82271	0.387974	0.193987	+	0.981004i	$0.437858\pi$
$642$	0	0
$643$	27.0257	1.06579	0.532895	−	0.846181i	$-0.321104\pi$
$643$	27.0257	1.06579	0.532895	+	0.846181i	$0.321104\pi$
$644$	−5.27600	−0.207904
$645$	0	0
$646$	0	0
$647$	43.8308	1.72316	0.861582	−	0.507618i	$-0.169474\pi$
$647$	43.8308	1.72316	0.861582	+	0.507618i	$0.169474\pi$
$648$	0	0
$649$	27.8197	1.09202
$650$	−0.521239	−0.0204447
$651$	0	0
$652$	−11.7294	−0.459359
$653$	26.9486	1.05458	0.527291	−	0.849685i	$-0.323208\pi$
$653$	26.9486	1.05458	0.527291	+	0.849685i	$0.323208\pi$
$654$	0	0
$655$	−45.6562	−1.78393
$656$	14.6434	0.571729
$657$	0	0
$658$	−1.27408	−0.0496687
$659$	−37.9335	−1.47768	−0.738840	−	0.673881i	$-0.764626\pi$
$659$	−37.9335	−1.47768	−0.738840	+	0.673881i	$0.764626\pi$
$660$	0	0
$661$	12.3793	0.481499	0.240749	−	0.970587i	$-0.422607\pi$
$661$	12.3793	0.481499	0.240749	+	0.970587i	$0.422607\pi$
$662$	−14.0763	−0.547089
$663$	0	0
$664$	27.8370	1.08028
$665$	0	0
$666$	0	0
$667$	−1.56829	−0.0607245
$668$	31.1536	1.20537
$669$	0	0
$670$	2.85983	0.110485
$671$	−26.2414	−1.01304
$672$	0	0
$673$	−46.7887	−1.80357	−0.901786	−	0.432183i	$-0.857744\pi$
$673$	−46.7887	−1.80357	−0.901786	+	0.432183i	$0.857744\pi$
$674$	−1.11011	−0.0427598
$675$	0	0
$676$	20.1059	0.773305
$677$	34.4443	1.32380	0.661901	−	0.749591i	$-0.269750\pi$
$677$	34.4443	1.32380	0.661901	+	0.749591i	$0.269750\pi$
$678$	0	0
$679$	12.6926	0.487096
$680$	−7.85961	−0.301402
$681$	0	0
$682$	−1.49786	−0.0573561
$683$	15.5415	0.594679	0.297339	−	0.954772i	$-0.403901\pi$
$683$	15.5415	0.594679	0.297339	+	0.954772i	$0.403901\pi$
$684$	0	0
$685$	−13.4171	−0.512640
$686$	6.31529	0.241119
$687$	0	0
$688$	29.6515	1.13045
$689$	−14.8878	−0.567178
$690$	0	0
$691$	−11.0740	−0.421273	−0.210637	−	0.977564i	$-0.567554\pi$
$691$	−11.0740	−0.421273	−0.210637	+	0.977564i	$0.567554\pi$
$692$	−16.9411	−0.644004
$693$	0	0
$694$	4.04826	0.153670
$695$	31.2576	1.18567
$696$	0	0
$697$	−11.7904	−0.446594
$698$	4.32071	0.163541
$699$	0	0
$700$	−1.81774	−0.0687042
$701$	21.4708	0.810941	0.405471	−	0.914108i	$-0.367108\pi$
$701$	21.4708	0.810941	0.405471	+	0.914108i	$0.367108\pi$
$702$	0	0
$703$	0	0
$704$	−14.1968	−0.535063
$705$	0	0
$706$	12.0019	0.451697
$707$	9.20949	0.346358
$708$	0	0
$709$	−48.2308	−1.81135	−0.905674	−	0.423976i	$-0.860634\pi$
$709$	−48.2308	−1.81135	−0.905674	+	0.423976i	$0.860634\pi$
$710$	6.98987	0.262325
$711$	0	0
$712$	18.4233	0.690444
$713$	2.37027	0.0887674
$714$	0	0
$715$	−10.3602	−0.387450
$716$	23.6105	0.882364
$717$	0	0
$718$	14.8928	0.555795
$719$	−38.4792	−1.43503	−0.717517	−	0.696541i	$-0.754721\pi$
$719$	−38.4792	−1.43503	−0.717517	+	0.696541i	$0.754721\pi$
$720$	0	0
$721$	−15.6441	−0.582616
$722$	0	0
$723$	0	0
$724$	−22.9625	−0.853394
$725$	−0.540324	−0.0200671
$726$	0	0
$727$	43.2956	1.60574	0.802872	−	0.596152i	$-0.203304\pi$
$727$	43.2956	1.60574	0.802872	+	0.596152i	$0.203304\pi$
$728$	2.62299	0.0972143
$729$	0	0
$730$	4.18360	0.154842
$731$	−23.8745	−0.883030
$732$	0	0
$733$	−13.5020	−0.498708	−0.249354	−	0.968412i	$-0.580218\pi$
$733$	−13.5020	−0.498708	−0.249354	+	0.968412i	$0.580218\pi$
$734$	13.9843	0.516170
$735$	0	0
$736$	−11.5416	−0.425430
$737$	−12.0019	−0.442096
$738$	0	0
$739$	−9.79526	−0.360324	−0.180162	−	0.983637i	$-0.557662\pi$
$739$	−9.79526	−0.360324	−0.180162	+	0.983637i	$0.557662\pi$
$740$	−14.0856	−0.517795
$741$	0	0
$742$	5.37091	0.197172
$743$	−9.12970	−0.334936	−0.167468	−	0.985877i	$-0.553559\pi$
$743$	−9.12970	−0.334936	−0.167468	+	0.985877i	$0.553559\pi$
$744$	0	0
$745$	4.13724	0.151577
$746$	15.2839	0.559584
$747$	0	0
$748$	15.6812	0.573362
$749$	−13.4913	−0.492962
$750$	0	0
$751$	10.4251	0.380417	0.190208	−	0.981744i	$-0.439084\pi$
$751$	10.4251	0.380417	0.190208	+	0.981744i	$0.439084\pi$
$752$	7.44234	0.271394
$753$	0	0
$754$	0.370669	0.0134990
$755$	−27.3390	−0.994967
$756$	0	0
$757$	7.79432	0.283289	0.141645	−	0.989918i	$-0.454761\pi$
$757$	7.79432	0.283289	0.141645	+	0.989918i	$0.454761\pi$
$758$	−10.4105	−0.378127
$759$	0	0
$760$	0	0
$761$	36.7419	1.33189	0.665947	−	0.745999i	$-0.268028\pi$
$761$	36.7419	1.33189	0.665947	+	0.745999i	$0.268028\pi$
$762$	0	0
$763$	−15.7899	−0.571633
$764$	−35.6916	−1.29128
$765$	0	0
$766$	2.90926	0.105116
$767$	10.4049	0.375700
$768$	0	0
$769$	8.15284	0.293999	0.146999	−	0.989137i	$-0.453038\pi$
$769$	8.15284	0.293999	0.146999	+	0.989137i	$0.453038\pi$
$770$	3.73756	0.134692
$771$	0	0
$772$	2.77836	0.0999955
$773$	18.4971	0.665295	0.332648	−	0.943051i	$-0.392058\pi$
$773$	18.4971	0.665295	0.332648	+	0.943051i	$0.392058\pi$
$774$	0	0
$775$	0.816631	0.0293342
$776$	18.2120	0.653773
$777$	0	0
$778$	15.7827	0.565838
$779$	0	0
$780$	0	0
$781$	−29.3345	−1.04967
$782$	2.56702	0.0917965
$783$	0	0
$784$	−16.5188	−0.589957
$785$	15.9607	0.569663
$786$	0	0
$787$	−16.0647	−0.572643	−0.286322	−	0.958134i	$-0.592433\pi$
$787$	−16.0647	−0.572643	−0.286322	+	0.958134i	$0.592433\pi$
$788$	7.28562	0.259539
$789$	0	0
$790$	5.11690	0.182051
$791$	−19.5353	−0.694595
$792$	0	0
$793$	−9.81461	−0.348527
$794$	2.18643	0.0775933
$795$	0	0
$796$	−13.6577	−0.484083
$797$	−6.01294	−0.212989	−0.106495	−	0.994313i	$-0.533963\pi$
$797$	−6.01294	−0.212989	−0.106495	+	0.994313i	$0.533963\pi$
$798$	0	0
$799$	−5.99235	−0.211994
$800$	−3.97644	−0.140588
$801$	0	0
$802$	−4.82540	−0.170391
$803$	−17.5574	−0.619587
$804$	0	0
$805$	−5.91445	−0.208457
$806$	−0.560219	−0.0197329
$807$	0	0
$808$	13.2143	0.464877
$809$	−0.0546010	−0.00191967	−0.000959834	−	1.00000i	$-0.500306\pi$
$809$	−0.0546010	−0.00191967	−0.000959834		1.00000i	$0.500306\pi$
$810$	0	0
$811$	26.0940	0.916286	0.458143	−	0.888879i	$-0.348515\pi$
$811$	26.0940	0.916286	0.458143	+	0.888879i	$0.348515\pi$
$812$	1.29265	0.0453632
$813$	0	0
$814$	−6.11516	−0.214336
$815$	−13.1488	−0.460582
$816$	0	0
$817$	0	0
$818$	−2.13218	−0.0745500
$819$	0	0
$820$	18.5307	0.647119
$821$	−24.5397	−0.856440	−0.428220	−	0.903674i	$-0.640859\pi$
$821$	−24.5397	−0.856440	−0.428220	+	0.903674i	$0.640859\pi$
$822$	0	0
$823$	3.90219	0.136022	0.0680108	−	0.997685i	$-0.478335\pi$
$823$	3.90219	0.136022	0.0680108	+	0.997685i	$0.478335\pi$
$824$	−22.4470	−0.781979
$825$	0	0
$826$	−3.75369	−0.130607
$827$	−22.2092	−0.772289	−0.386145	−	0.922438i	$-0.626193\pi$
$827$	−22.2092	−0.772289	−0.386145	+	0.922438i	$0.626193\pi$
$828$	0	0
$829$	−40.1536	−1.39459	−0.697296	−	0.716784i	$-0.745613\pi$
$829$	−40.1536	−1.39459	−0.697296	+	0.716784i	$0.745613\pi$
$830$	14.8354	0.514944
$831$	0	0
$832$	−5.30978	−0.184084
$833$	13.3004	0.460833
$834$	0	0
$835$	34.9235	1.20858
$836$	0	0
$837$	0	0
$838$	−6.14347	−0.212223
$839$	17.6139	0.608101	0.304050	−	0.952656i	$-0.401661\pi$
$839$	17.6139	0.608101	0.304050	+	0.952656i	$0.401661\pi$
$840$	0	0
$841$	−28.6158	−0.986750
$842$	−7.36751	−0.253901
$843$	0	0
$844$	−5.78431	−0.199104
$845$	22.5389	0.775363
$846$	0	0
$847$	−3.02953	−0.104096
$848$	−31.3734	−1.07737
$849$	0	0
$850$	0.884416	0.0303352
$851$	9.67685	0.331718
$852$	0	0
$853$	−3.11904	−0.106794	−0.0533970	−	0.998573i	$-0.517005\pi$
$853$	−3.11904	−0.106794	−0.0533970	+	0.998573i	$0.517005\pi$
$854$	3.54072	0.121161
$855$	0	0
$856$	−19.3581	−0.661647
$857$	19.0436	0.650518	0.325259	−	0.945625i	$-0.394549\pi$
$857$	19.0436	0.650518	0.325259	+	0.945625i	$0.394549\pi$
$858$	0	0
$859$	−19.5111	−0.665710	−0.332855	−	0.942978i	$-0.608012\pi$
$859$	−19.5111	−0.665710	−0.332855	+	0.942978i	$0.608012\pi$
$860$	37.5229	1.27952
$861$	0	0
$862$	−0.844334	−0.0287581
$863$	−28.7796	−0.979670	−0.489835	−	0.871815i	$-0.662943\pi$
$863$	−28.7796	−0.979670	−0.489835	+	0.871815i	$0.662943\pi$
$864$	0	0
$865$	−18.9911	−0.645718
$866$	12.7134	0.432018
$867$	0	0
$868$	−1.95368	−0.0663122
$869$	−21.4742	−0.728462
$870$	0	0
$871$	−4.48886	−0.152099
$872$	−22.6563	−0.767238
$873$	0	0
$874$	0	0
$875$	−13.7262	−0.464031
$876$	0	0
$877$	52.4228	1.77019	0.885096	−	0.465409i	$-0.154093\pi$
$877$	52.4228	1.77019	0.885096	+	0.465409i	$0.154093\pi$
$878$	4.63988	0.156588
$879$	0	0
$880$	−21.8324	−0.735970
$881$	10.4947	0.353575	0.176788	−	0.984249i	$-0.443429\pi$
$881$	10.4947	0.353575	0.176788	+	0.984249i	$0.443429\pi$
$882$	0	0
$883$	40.0003	1.34612	0.673059	−	0.739589i	$-0.264980\pi$
$883$	40.0003	1.34612	0.673059	+	0.739589i	$0.264980\pi$
$884$	5.86496	0.197260
$885$	0	0
$886$	−8.58872	−0.288544
$887$	−12.5542	−0.421528	−0.210764	−	0.977537i	$-0.567595\pi$
$887$	−12.5542	−0.421528	−0.210764	+	0.977537i	$0.567595\pi$
$888$	0	0
$889$	−12.5925	−0.422339
$890$	9.81851	0.329117
$891$	0	0
$892$	−26.0909	−0.873588
$893$	0	0
$894$	0	0
$895$	26.4675	0.884712
$896$	12.4128	0.414681
$897$	0	0
$898$	10.8716	0.362788
$899$	−0.580731	−0.0193685
$900$	0	0
$901$	25.2609	0.841563
$902$	8.04497	0.267868
$903$	0	0
$904$	−28.0303	−0.932275
$905$	−25.7411	−0.855665
$906$	0	0
$907$	−33.0482	−1.09735	−0.548673	−	0.836037i	$-0.684867\pi$
$907$	−33.0482	−1.09735	−0.548673	+	0.836037i	$0.684867\pi$
$908$	−16.1094	−0.534609
$909$	0	0
$910$	1.39789	0.0463396
$911$	45.8125	1.51784	0.758918	−	0.651186i	$-0.225728\pi$
$911$	45.8125	1.51784	0.758918	+	0.651186i	$0.225728\pi$
$912$	0	0
$913$	−62.2600	−2.06051
$914$	4.39145	0.145256
$915$	0	0
$916$	12.3948	0.409534
$917$	−25.8532	−0.853748
$918$	0	0
$919$	−41.5838	−1.37172	−0.685861	−	0.727733i	$-0.740574\pi$
$919$	−41.5838	−1.37172	−0.685861	+	0.727733i	$0.740574\pi$
$920$	−8.48639	−0.279788
$921$	0	0
$922$	8.88574	0.292636
$923$	−10.9715	−0.361130
$924$	0	0
$925$	3.33397	0.109620
$926$	8.70382	0.286025
$927$	0	0
$928$	2.82777	0.0928259
$929$	8.58031	0.281511	0.140755	−	0.990044i	$-0.455047\pi$
$929$	8.58031	0.281511	0.140755	+	0.990044i	$0.455047\pi$
$930$	0	0
$931$	0	0
$932$	1.88024	0.0615894
$933$	0	0
$934$	−5.82081	−0.190463
$935$	17.5788	0.574887
$936$	0	0
$937$	−6.68899	−0.218520	−0.109260	−	0.994013i	$-0.534848\pi$
$937$	−6.68899	−0.218520	−0.109260	+	0.994013i	$0.534848\pi$
$938$	1.61940	0.0528754
$939$	0	0
$940$	9.41800	0.307181
$941$	−13.5643	−0.442183	−0.221091	−	0.975253i	$-0.570962\pi$
$941$	−13.5643	−0.442183	−0.221091	+	0.975253i	$0.570962\pi$
$942$	0	0
$943$	−12.7307	−0.414568
$944$	21.9266	0.713651
$945$	0	0
$946$	16.2903	0.529643
$947$	−26.2948	−0.854467	−0.427234	−	0.904141i	$-0.640512\pi$
$947$	−26.2948	−0.854467	−0.427234	+	0.904141i	$0.640512\pi$
$948$	0	0
$949$	−6.56668	−0.213163
$950$	0	0
$951$	0	0
$952$	−4.45057	−0.144244
$953$	−7.58446	−0.245685	−0.122842	−	0.992426i	$-0.539201\pi$
$953$	−7.58446	−0.245685	−0.122842	+	0.992426i	$0.539201\pi$
$954$	0	0
$955$	−40.0106	−1.29471
$956$	−33.7046	−1.09008
$957$	0	0
$958$	−0.574044	−0.0185465
$959$	−7.59753	−0.245337
$960$	0	0
$961$	−30.1223	−0.971687
$962$	−2.28714	−0.0737404
$963$	0	0
$964$	48.3710	1.55793
$965$	3.11457	0.100262
$966$	0	0
$967$	2.14576	0.0690031	0.0345016	−	0.999405i	$-0.489016\pi$
$967$	2.14576	0.0690031	0.0345016	+	0.999405i	$0.489016\pi$
$968$	−4.34694	−0.139716
$969$	0	0
$970$	9.70589	0.311637
$971$	19.1139	0.613396	0.306698	−	0.951807i	$-0.400776\pi$
$971$	19.1139	0.613396	0.306698	+	0.951807i	$0.400776\pi$
$972$	0	0
$973$	17.6999	0.567432
$974$	2.44736	0.0784184
$975$	0	0
$976$	−20.6826	−0.662035
$977$	45.5157	1.45618	0.728089	−	0.685483i	$-0.240409\pi$
$977$	45.5157	1.45618	0.728089	+	0.685483i	$0.240409\pi$
$978$	0	0
$979$	−41.2055	−1.31693
$980$	−20.9039	−0.667750
$981$	0	0
$982$	17.9545	0.572952
$983$	17.0414	0.543538	0.271769	−	0.962363i	$-0.412391\pi$
$983$	17.0414	0.543538	0.271769	+	0.962363i	$0.412391\pi$
$984$	0	0
$985$	8.16724	0.260230
$986$	−0.628935	−0.0200294
$987$	0	0
$988$	0	0
$989$	−25.7784	−0.819705
$990$	0	0
$991$	11.2810	0.358353	0.179176	−	0.983817i	$-0.442657\pi$
$991$	11.2810	0.358353	0.179176	+	0.983817i	$0.442657\pi$
$992$	−4.27381	−0.135693
$993$	0	0
$994$	3.95807	0.125542
$995$	−15.3104	−0.485371
$996$	0	0
$997$	−52.3584	−1.65821	−0.829104	−	0.559095i	$-0.811149\pi$
$997$	−52.3584	−1.65821	−0.829104	+	0.559095i	$0.811149\pi$
$998$	−0.142532	−0.00451177
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9747.2.a.bm.1.4		6
3.2	odd	2		9747.2.a.br.1.3		6
19.8	odd	6		513.2.f.f.406.4	yes	12
19.12	odd	6		513.2.f.f.163.4	✓	12
19.18	odd	2		9747.2.a.bs.1.3		6
57.8	even	6		513.2.f.h.406.3	yes	12
57.50	even	6		513.2.f.h.163.3	yes	12
57.56	even	2		9747.2.a.bl.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.f.f.163.4	✓	12	19.12	odd	6
513.2.f.f.406.4	yes	12	19.8	odd	6
513.2.f.h.163.3	yes	12	57.50	even	6
513.2.f.h.406.3	yes	12	57.8	even	6
9747.2.a.bl.1.4		6	57.56	even	2
9747.2.a.bm.1.4		6	1.1	even	1	trivial
9747.2.a.br.1.3		6	3.2	odd	2
9747.2.a.bs.1.3		6	19.18	odd	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:	\( N \)	\(=\)	\( 9747 = 3^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9747.a (trivial)

\(f(q)\)	\(=\)	\(q+0.433013 q^{2} -1.81250 q^{4} -2.03183 q^{5} -1.15054 q^{7} -1.65086 q^{8} -0.879808 q^{10} +3.69231 q^{11} +1.38097 q^{13} -0.498199 q^{14} +2.91016 q^{16} -2.34317 q^{17} +3.68269 q^{20} +1.59882 q^{22} -2.53003 q^{23} -0.871671 q^{25} +0.597977 q^{26} +2.08535 q^{28} +0.619872 q^{29} -0.936857 q^{31} +4.56186 q^{32} -1.01462 q^{34} +2.33770 q^{35} -3.82480 q^{37} +3.35427 q^{40} +5.03183 q^{41} +10.1890 q^{43} -6.69231 q^{44} -1.09553 q^{46} +2.55737 q^{47} -5.67626 q^{49} -0.377445 q^{50} -2.50300 q^{52} -10.7807 q^{53} -7.50214 q^{55} +1.89938 q^{56} +0.268412 q^{58} +7.53451 q^{59} -7.10705 q^{61} -0.405671 q^{62} -3.84497 q^{64} -2.80589 q^{65} -3.25052 q^{67} +4.24699 q^{68} +1.01225 q^{70} -7.94477 q^{71} -4.75513 q^{73} -1.65619 q^{74} -4.24815 q^{77} -5.81592 q^{79} -5.91294 q^{80} +2.17885 q^{82} -16.8621 q^{83} +4.76092 q^{85} +4.41196 q^{86} -6.09549 q^{88} -11.1598 q^{89} -1.58886 q^{91} +4.58568 q^{92} +1.10737 q^{94} -11.0318 q^{97} -2.45789 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 3 q^{4} + 5 q^{5} + q^{7} - 6 q^{8} + q^{10} + 2 q^{11} - 5 q^{13} - 2 q^{14} - 3 q^{16} + 10 q^{17} + q^{20} - 4 q^{22} + 3 q^{23} + 5 q^{25} + q^{26} + 2 q^{28} + 6 q^{29} + 2 q^{31}+ \cdots + 59 q^{98}+O(q^{100}) \) 6 * q - q^2 + 3 * q^4 + 5 * q^5 + q^7 - 6 * q^8 + q^10 + 2 * q^11 - 5 * q^13 - 2 * q^14 - 3 * q^16 + 10 * q^17 + q^20 - 4 * q^22 + 3 * q^23 + 5 * q^25 + q^26 + 2 * q^28 + 6 * q^29 + 2 * q^31 - 8 * q^32 + 4 * q^34 + 3 * q^35 - 13 * q^37 - 6 * q^40 + 13 * q^41 + 3 * q^43 - 20 * q^44 + 5 * q^46 + 18 * q^47 + 13 * q^49 + 3 * q^50 - 22 * q^52 + q^53 + 6 * q^55 + 24 * q^56 + 14 * q^58 - 11 * q^59 + 13 * q^61 + 22 * q^62 - 36 * q^65 + 9 * q^67 + 5 * q^68 - 30 * q^70 + 6 * q^71 + 15 * q^73 + 43 * q^74 + 29 * q^77 - 13 * q^79 + q^80 - 4 * q^82 + 8 * q^83 + 3 * q^85 + 7 * q^86 + 15 * q^88 - 29 * q^89 + 22 * q^91 + 21 * q^92 - 18 * q^94 + 20 * q^97 + 59 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more