LMFDB - Embedded newform 9300.2.a.bf.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(1,9300)] 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("9300.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,7,0,0,0,4,0,7,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$74.2608738798$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 15x^{5} + 49x^{4} + 13x^{3} - 69x^{2} - 35x - 1 $ x^7 - 3x^6 - 15x^5 + 49x^4 + 13x^3 - 69x^2 - 35x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.53062$ of defining polynomial
Character	$\chi$	$=$	9300.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.13530 q^{7} +1.00000 q^{9} -1.23000 q^{11} +5.04769 q^{13} -4.23525 q^{17} +3.14923 q^{19} +4.13530 q^{21} -6.74669 q^{23} +1.00000 q^{27} -6.24879 q^{29} -1.00000 q^{31} -1.23000 q^{33} +8.70424 q^{37} +5.04769 q^{39} +8.60270 q^{41} -10.8027 q^{43} +2.82599 q^{47} +10.1007 q^{49} -4.23525 q^{51} +4.85119 q^{53} +3.14923 q^{57} +7.88238 q^{59} +9.08644 q^{61} +4.13530 q^{63} +15.5348 q^{67} -6.74669 q^{69} -3.44301 q^{71} -3.49487 q^{73} -5.08644 q^{77} +4.50259 q^{79} +1.00000 q^{81} -9.87942 q^{83} -6.24879 q^{87} +4.64367 q^{89} +20.8737 q^{91} -1.00000 q^{93} +16.4824 q^{97} -1.23000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} + 4 q^{7} + 7 q^{9} + 4 q^{11} + 10 q^{13} + 6 q^{17} + 10 q^{19} + 4 q^{21} + 7 q^{27} + 10 q^{29} - 7 q^{31} + 4 q^{33} + 16 q^{37} + 10 q^{39} + 2 q^{41} - 8 q^{43} + 12 q^{47} + 9 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) $ 7 * q + 7 * q^3 + 4 * q^7 + 7 * q^9 + 4 * q^11 + 10 * q^13 + 6 * q^17 + 10 * q^19 + 4 * q^21 + 7 * q^27 + 10 * q^29 - 7 * q^31 + 4 * q^33 + 16 * q^37 + 10 * q^39 + 2 * q^41 - 8 * q^43 + 12 * q^47 + 9 * q^49 + 6 * q^51 + 16 * q^53 + 10 * q^57 + 14 * q^59 + 10 * q^61 + 4 * q^63 + 22 * q^67 - 2 * q^71 + 14 * q^73 + 18 * q^77 + 24 * q^79 + 7 * q^81 - 12 * q^83 + 10 * q^87 + 12 * q^89 + 8 * q^91 - 7 * q^93 + 24 * q^97 + 4 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.13530	1.56300	0.781499	−	0.623907i	$-0.214456\pi$
$7$	4.13530	1.56300	0.781499	+	0.623907i	$0.214456\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.23000	−0.370860	−0.185430	−	0.982657i	$-0.559368\pi$
$11$	−1.23000	−0.370860	−0.185430	+	0.982657i	$0.559368\pi$
$12$	0	0
$13$	5.04769	1.39998	0.699989	−	0.714154i	$-0.253188\pi$
$13$	5.04769	1.39998	0.699989	+	0.714154i	$0.253188\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.23525	−1.02720	−0.513599	−	0.858030i	$-0.671688\pi$
$17$	−4.23525	−1.02720	−0.513599	+	0.858030i	$0.671688\pi$
$18$	0	0
$19$	3.14923	0.722483	0.361242	−	0.932472i	$-0.382353\pi$
$19$	3.14923	0.722483	0.361242	+	0.932472i	$0.382353\pi$
$20$	0	0
$21$	4.13530	0.902397
$22$	0	0
$23$	−6.74669	−1.40678	−0.703391	−	0.710803i	$-0.748332\pi$
$23$	−6.74669	−1.40678	−0.703391	+	0.710803i	$0.748332\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.24879	−1.16037	−0.580185	−	0.814485i	$-0.697020\pi$
$29$	−6.24879	−1.16037	−0.580185	+	0.814485i	$0.697020\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−1.23000	−0.214116
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.70424	1.43097	0.715484	−	0.698629i	$-0.246206\pi$
$37$	8.70424	1.43097	0.715484	+	0.698629i	$0.246206\pi$
$38$	0	0
$39$	5.04769	0.808278
$40$	0	0
$41$	8.60270	1.34352	0.671758	−	0.740771i	$-0.265539\pi$
$41$	8.60270	1.34352	0.671758	+	0.740771i	$0.265539\pi$
$42$	0	0
$43$	−10.8027	−1.64739	−0.823697	−	0.567030i	$-0.808092\pi$
$43$	−10.8027	−1.64739	−0.823697	+	0.567030i	$0.808092\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.82599	0.412213	0.206106	−	0.978530i	$-0.433921\pi$
$47$	2.82599	0.412213	0.206106	+	0.978530i	$0.433921\pi$
$48$	0	0
$49$	10.1007	1.44296
$50$	0	0
$51$	−4.23525	−0.593053
$52$	0	0
$53$	4.85119	0.666363	0.333181	−	0.942863i	$-0.391878\pi$
$53$	4.85119	0.666363	0.333181	+	0.942863i	$0.391878\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.14923	0.417126
$58$	0	0
$59$	7.88238	1.02620	0.513099	−	0.858329i	$-0.328497\pi$
$59$	7.88238	1.02620	0.513099	+	0.858329i	$0.328497\pi$
$60$	0	0
$61$	9.08644	1.16340	0.581700	−	0.813404i	$-0.302388\pi$
$61$	9.08644	1.16340	0.581700	+	0.813404i	$0.302388\pi$
$62$	0	0
$63$	4.13530	0.520999
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.5348	1.89788	0.948938	−	0.315463i	$-0.102160\pi$
$67$	15.5348	1.89788	0.948938	+	0.315463i	$0.102160\pi$
$68$	0	0
$69$	−6.74669	−0.812206
$70$	0	0
$71$	−3.44301	−0.408610	−0.204305	−	0.978907i	$-0.565493\pi$
$71$	−3.44301	−0.408610	−0.204305	+	0.978907i	$0.565493\pi$
$72$	0	0
$73$	−3.49487	−0.409043	−0.204522	−	0.978862i	$-0.565564\pi$
$73$	−3.49487	−0.409043	−0.204522	+	0.978862i	$0.565564\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.08644	−0.579654
$78$	0	0
$79$	4.50259	0.506581	0.253291	−	0.967390i	$-0.418487\pi$
$79$	4.50259	0.506581	0.253291	+	0.967390i	$0.418487\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.87942	−1.08441	−0.542203	−	0.840247i	$-0.682410\pi$
$83$	−9.87942	−1.08441	−0.542203	+	0.840247i	$0.682410\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.24879	−0.669940
$88$	0	0
$89$	4.64367	0.492229	0.246114	−	0.969241i	$-0.420846\pi$
$89$	4.64367	0.492229	0.246114	+	0.969241i	$0.420846\pi$
$90$	0	0
$91$	20.8737	2.18816
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.4824	1.67354	0.836768	−	0.547558i	$-0.184442\pi$
$97$	16.4824	1.67354	0.836768	+	0.547558i	$0.184442\pi$
$98$	0	0
$99$	−1.23000	−0.123620
$100$	0	0
$101$	19.3883	1.92920	0.964602	−	0.263712i	$-0.0849467\pi$
$101$	19.3883	1.92920	0.964602	+	0.263712i	$0.0849467\pi$
$102$	0	0
$103$	12.0426	1.18660	0.593298	−	0.804983i	$-0.297826\pi$
$103$	12.0426	1.18660	0.593298	+	0.804983i	$0.297826\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.9510	−1.25202	−0.626010	−	0.779815i	$-0.715313\pi$
$107$	−12.9510	−1.25202	−0.626010	+	0.779815i	$0.715313\pi$
$108$	0	0
$109$	2.55686	0.244903	0.122451	−	0.992474i	$-0.460924\pi$
$109$	2.55686	0.244903	0.122451	+	0.992474i	$0.460924\pi$
$110$	0	0
$111$	8.70424	0.826170
$112$	0	0
$113$	−7.21730	−0.678946	−0.339473	−	0.940616i	$-0.610249\pi$
$113$	−7.21730	−0.678946	−0.339473	+	0.940616i	$0.610249\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.04769	0.466659
$118$	0	0
$119$	−17.5140	−1.60551
$120$	0	0
$121$	−9.48709	−0.862463
$122$	0	0
$123$	8.60270	0.775679
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.9567	1.14972	0.574862	−	0.818250i	$-0.305056\pi$
$127$	12.9567	1.14972	0.574862	+	0.818250i	$0.305056\pi$
$128$	0	0
$129$	−10.8027	−0.951123
$130$	0	0
$131$	22.1173	1.93240	0.966198	−	0.257803i	$-0.0829984\pi$
$131$	22.1173	1.93240	0.966198	+	0.257803i	$0.0829984\pi$
$132$	0	0
$133$	13.0230	1.12924
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.08842	0.178426	0.0892130	−	0.996013i	$-0.471565\pi$
$137$	2.08842	0.178426	0.0892130	+	0.996013i	$0.471565\pi$
$138$	0	0
$139$	−7.94778	−0.674122	−0.337061	−	0.941483i	$-0.609433\pi$
$139$	−7.94778	−0.674122	−0.337061	+	0.941483i	$0.609433\pi$
$140$	0	0
$141$	2.82599	0.237991
$142$	0	0
$143$	−6.20868	−0.519196
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.1007	0.833093
$148$	0	0
$149$	−17.6946	−1.44960	−0.724798	−	0.688962i	$-0.758067\pi$
$149$	−17.6946	−1.44960	−0.724798	+	0.688962i	$0.758067\pi$
$150$	0	0
$151$	−1.21393	−0.0987878	−0.0493939	−	0.998779i	$-0.515729\pi$
$151$	−1.21393	−0.0987878	−0.0493939	+	0.998779i	$0.515729\pi$
$152$	0	0
$153$	−4.23525	−0.342399
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.5148	−0.998792	−0.499396	−	0.866374i	$-0.666445\pi$
$157$	−12.5148	−0.998792	−0.499396	+	0.866374i	$0.666445\pi$
$158$	0	0
$159$	4.85119	0.384725
$160$	0	0
$161$	−27.8996	−2.19880
$162$	0	0
$163$	−1.93084	−0.151235	−0.0756175	−	0.997137i	$-0.524093\pi$
$163$	−1.93084	−0.151235	−0.0756175	+	0.997137i	$0.524093\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.00880	0.155445	0.0777227	−	0.996975i	$-0.475235\pi$
$167$	2.00880	0.155445	0.0777227	+	0.996975i	$0.475235\pi$
$168$	0	0
$169$	12.4792	0.959938
$170$	0	0
$171$	3.14923	0.240828
$172$	0	0
$173$	−10.7159	−0.814714	−0.407357	−	0.913269i	$-0.633549\pi$
$173$	−10.7159	−0.814714	−0.407357	+	0.913269i	$0.633549\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.88238	0.592476
$178$	0	0
$179$	7.93384	0.593003	0.296501	−	0.955032i	$-0.404180\pi$
$179$	7.93384	0.593003	0.296501	+	0.955032i	$0.404180\pi$
$180$	0	0
$181$	−21.7718	−1.61828	−0.809142	−	0.587613i	$-0.800068\pi$
$181$	−21.7718	−1.61828	−0.809142	+	0.587613i	$0.800068\pi$
$182$	0	0
$183$	9.08644	0.671689
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.20937	0.380947
$188$	0	0
$189$	4.13530	0.300799
$190$	0	0
$191$	−20.7280	−1.49982	−0.749912	−	0.661538i	$-0.769904\pi$
$191$	−20.7280	−1.49982	−0.749912	+	0.661538i	$0.769904\pi$
$192$	0	0
$193$	11.7694	0.847182	0.423591	−	0.905853i	$-0.360769\pi$
$193$	11.7694	0.847182	0.423591	+	0.905853i	$0.360769\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.85134	0.559385	0.279692	−	0.960090i	$-0.409767\pi$
$197$	7.85134	0.559385	0.279692	+	0.960090i	$0.409767\pi$
$198$	0	0
$199$	−15.8120	−1.12088	−0.560440	−	0.828195i	$-0.689368\pi$
$199$	−15.8120	−1.12088	−0.560440	+	0.828195i	$0.689368\pi$
$200$	0	0
$201$	15.5348	1.09574
$202$	0	0
$203$	−25.8406	−1.81366
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.74669	−0.468927
$208$	0	0
$209$	−3.87357	−0.267940
$210$	0	0
$211$	18.4243	1.26838	0.634190	−	0.773177i	$-0.281333\pi$
$211$	18.4243	1.26838	0.634190	+	0.773177i	$0.281333\pi$
$212$	0	0
$213$	−3.44301	−0.235911
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.13530	−0.280723
$218$	0	0
$219$	−3.49487	−0.236161
$220$	0	0
$221$	−21.3782	−1.43805
$222$	0	0
$223$	−12.7146	−0.851432	−0.425716	−	0.904857i	$-0.639978\pi$
$223$	−12.7146	−0.851432	−0.425716	+	0.904857i	$0.639978\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.06936	0.535582	0.267791	−	0.963477i	$-0.413706\pi$
$227$	8.06936	0.535582	0.267791	+	0.963477i	$0.413706\pi$
$228$	0	0
$229$	7.49603	0.495352	0.247676	−	0.968843i	$-0.420333\pi$
$229$	7.49603	0.495352	0.247676	+	0.968843i	$0.420333\pi$
$230$	0	0
$231$	−5.08644	−0.334663
$232$	0	0
$233$	23.2848	1.52544	0.762719	−	0.646730i	$-0.223864\pi$
$233$	23.2848	1.52544	0.762719	+	0.646730i	$0.223864\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.50259	0.292475
$238$	0	0
$239$	24.3491	1.57501	0.787507	−	0.616305i	$-0.211371\pi$
$239$	24.3491	1.57501	0.787507	+	0.616305i	$0.211371\pi$
$240$	0	0
$241$	−21.3399	−1.37462	−0.687310	−	0.726364i	$-0.741209\pi$
$241$	−21.3399	−1.37462	−0.687310	+	0.726364i	$0.741209\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	15.8963	1.01146
$248$	0	0
$249$	−9.87942	−0.626083
$250$	0	0
$251$	−8.42008	−0.531470	−0.265735	−	0.964046i	$-0.585615\pi$
$251$	−8.42008	−0.531470	−0.265735	+	0.964046i	$0.585615\pi$
$252$	0	0
$253$	8.29846	0.521720
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.4448	1.08818	0.544088	−	0.839028i	$-0.316876\pi$
$257$	17.4448	1.08818	0.544088	+	0.839028i	$0.316876\pi$
$258$	0	0
$259$	35.9947	2.23660
$260$	0	0
$261$	−6.24879	−0.386790
$262$	0	0
$263$	18.7676	1.15726	0.578630	−	0.815590i	$-0.303588\pi$
$263$	18.7676	1.15726	0.578630	+	0.815590i	$0.303588\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.64367	0.284188
$268$	0	0
$269$	−3.78062	−0.230509	−0.115254	−	0.993336i	$-0.536768\pi$
$269$	−3.78062	−0.230509	−0.115254	+	0.993336i	$0.536768\pi$
$270$	0	0
$271$	−27.4186	−1.66556	−0.832782	−	0.553601i	$-0.813253\pi$
$271$	−27.4186	−1.66556	−0.832782	+	0.553601i	$0.813253\pi$
$272$	0	0
$273$	20.8737	1.26334
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.06800	−0.424675	−0.212338	−	0.977196i	$-0.568108\pi$
$277$	−7.06800	−0.424675	−0.212338	+	0.977196i	$0.568108\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−3.46379	−0.206632	−0.103316	−	0.994649i	$-0.532945\pi$
$281$	−3.46379	−0.206632	−0.103316	+	0.994649i	$0.532945\pi$
$282$	0	0
$283$	−6.34958	−0.377444	−0.188722	−	0.982031i	$-0.560434\pi$
$283$	−6.34958	−0.377444	−0.188722	+	0.982031i	$0.560434\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	35.5748	2.09991
$288$	0	0
$289$	0.937303	0.0551355
$290$	0	0
$291$	16.4824	0.966216
$292$	0	0
$293$	28.0568	1.63910	0.819548	−	0.573010i	$-0.194224\pi$
$293$	28.0568	1.63910	0.819548	+	0.573010i	$0.194224\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.23000	−0.0713721
$298$	0	0
$299$	−34.0552	−1.96946
$300$	0	0
$301$	−44.6724	−2.57487
$302$	0	0
$303$	19.3883	1.11383
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−26.0895	−1.48901	−0.744504	−	0.667617i	$-0.767314\pi$
$307$	−26.0895	−1.48901	−0.744504	+	0.667617i	$0.767314\pi$
$308$	0	0
$309$	12.0426	0.685082
$310$	0	0
$311$	−6.01182	−0.340899	−0.170449	−	0.985366i	$-0.554522\pi$
$311$	−6.01182	−0.340899	−0.170449	+	0.985366i	$0.554522\pi$
$312$	0	0
$313$	−9.33471	−0.527629	−0.263814	−	0.964573i	$-0.584981\pi$
$313$	−9.33471	−0.527629	−0.263814	+	0.964573i	$0.584981\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.3996	1.25809	0.629043	−	0.777371i	$-0.283447\pi$
$317$	22.3996	1.25809	0.629043	+	0.777371i	$0.283447\pi$
$318$	0	0
$319$	7.68603	0.430335
$320$	0	0
$321$	−12.9510	−0.722855
$322$	0	0
$323$	−13.3378	−0.742133
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.55686	0.141395
$328$	0	0
$329$	11.6863	0.644287
$330$	0	0
$331$	15.6884	0.862315	0.431157	−	0.902277i	$-0.358105\pi$
$331$	15.6884	0.862315	0.431157	+	0.902277i	$0.358105\pi$
$332$	0	0
$333$	8.70424	0.476989
$334$	0	0
$335$	0	0
$336$	0	0
$337$	33.6766	1.83448	0.917241	−	0.398333i	$-0.130411\pi$
$337$	33.6766	1.83448	0.917241	+	0.398333i	$0.130411\pi$
$338$	0	0
$339$	−7.21730	−0.391990
$340$	0	0
$341$	1.23000	0.0666085
$342$	0	0
$343$	12.8224	0.692345
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.91259	0.263722	0.131861	−	0.991268i	$-0.457905\pi$
$347$	4.91259	0.263722	0.131861	+	0.991268i	$0.457905\pi$
$348$	0	0
$349$	−32.2056	−1.72392	−0.861962	−	0.506973i	$-0.830765\pi$
$349$	−32.2056	−1.72392	−0.861962	+	0.506973i	$0.830765\pi$
$350$	0	0
$351$	5.04769	0.269426
$352$	0	0
$353$	4.20283	0.223694	0.111847	−	0.993725i	$-0.464323\pi$
$353$	4.20283	0.223694	0.111847	+	0.993725i	$0.464323\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−17.5140	−0.926940
$358$	0	0
$359$	−1.59922	−0.0844036	−0.0422018	−	0.999109i	$-0.513437\pi$
$359$	−1.59922	−0.0844036	−0.0422018	+	0.999109i	$0.513437\pi$
$360$	0	0
$361$	−9.08235	−0.478018
$362$	0	0
$363$	−9.48709	−0.497943
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−23.4687	−1.22506	−0.612528	−	0.790449i	$-0.709847\pi$
$367$	−23.4687	−1.22506	−0.612528	+	0.790449i	$0.709847\pi$
$368$	0	0
$369$	8.60270	0.447839
$370$	0	0
$371$	20.0612	1.04152
$372$	0	0
$373$	−3.44239	−0.178240	−0.0891202	−	0.996021i	$-0.528406\pi$
$373$	−3.44239	−0.178240	−0.0891202	+	0.996021i	$0.528406\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−31.5419	−1.62449
$378$	0	0
$379$	16.2722	0.835845	0.417923	−	0.908483i	$-0.362758\pi$
$379$	16.2722	0.835845	0.417923	+	0.908483i	$0.362758\pi$
$380$	0	0
$381$	12.9567	0.663793
$382$	0	0
$383$	−11.8227	−0.604112	−0.302056	−	0.953290i	$-0.597673\pi$
$383$	−11.8227	−0.604112	−0.302056	+	0.953290i	$0.597673\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.8027	−0.549131
$388$	0	0
$389$	17.1646	0.870279	0.435139	−	0.900363i	$-0.356699\pi$
$389$	17.1646	0.870279	0.435139	+	0.900363i	$0.356699\pi$
$390$	0	0
$391$	28.5739	1.44504
$392$	0	0
$393$	22.1173	1.11567
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.39398	0.120150	0.0600752	−	0.998194i	$-0.480866\pi$
$397$	2.39398	0.120150	0.0600752	+	0.998194i	$0.480866\pi$
$398$	0	0
$399$	13.0230	0.651966
$400$	0	0
$401$	−8.14214	−0.406599	−0.203299	−	0.979117i	$-0.565166\pi$
$401$	−8.14214	−0.406599	−0.203299	+	0.979117i	$0.565166\pi$
$402$	0	0
$403$	−5.04769	−0.251443
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.7063	−0.530689
$408$	0	0
$409$	−4.48277	−0.221659	−0.110829	−	0.993839i	$-0.535351\pi$
$409$	−4.48277	−0.221659	−0.110829	+	0.993839i	$0.535351\pi$
$410$	0	0
$411$	2.08842	0.103014
$412$	0	0
$413$	32.5960	1.60395
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−7.94778	−0.389205
$418$	0	0
$419$	22.9387	1.12063	0.560315	−	0.828280i	$-0.310680\pi$
$419$	22.9387	1.12063	0.560315	+	0.828280i	$0.310680\pi$
$420$	0	0
$421$	10.4929	0.511391	0.255696	−	0.966757i	$-0.417696\pi$
$421$	10.4929	0.511391	0.255696	+	0.966757i	$0.417696\pi$
$422$	0	0
$423$	2.82599	0.137404
$424$	0	0
$425$	0	0
$426$	0	0
$427$	37.5752	1.81839
$428$	0	0
$429$	−6.20868	−0.299758
$430$	0	0
$431$	−10.8745	−0.523804	−0.261902	−	0.965094i	$-0.584350\pi$
$431$	−10.8745	−0.523804	−0.261902	+	0.965094i	$0.584350\pi$
$432$	0	0
$433$	−40.9494	−1.96790	−0.983952	−	0.178435i	$-0.942897\pi$
$433$	−40.9494	−1.96790	−0.983952	+	0.178435i	$0.942897\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−21.2469	−1.01638
$438$	0	0
$439$	−10.9518	−0.522699	−0.261350	−	0.965244i	$-0.584168\pi$
$439$	−10.9518	−0.522699	−0.261350	+	0.965244i	$0.584168\pi$
$440$	0	0
$441$	10.1007	0.480987
$442$	0	0
$443$	−0.358624	−0.0170387	−0.00851937	−	0.999964i	$-0.502712\pi$
$443$	−0.358624	−0.0170387	−0.00851937	+	0.999964i	$0.502712\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−17.6946	−0.836924
$448$	0	0
$449$	−30.2571	−1.42792	−0.713960	−	0.700187i	$-0.753100\pi$
$449$	−30.2571	−1.42792	−0.713960	+	0.700187i	$0.753100\pi$
$450$	0	0
$451$	−10.5814	−0.498257
$452$	0	0
$453$	−1.21393	−0.0570352
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.3292	1.51230	0.756148	−	0.654400i	$-0.227079\pi$
$457$	32.3292	1.51230	0.756148	+	0.654400i	$0.227079\pi$
$458$	0	0
$459$	−4.23525	−0.197684
$460$	0	0
$461$	−14.5464	−0.677493	−0.338747	−	0.940878i	$-0.610003\pi$
$461$	−14.5464	−0.677493	−0.338747	+	0.940878i	$0.610003\pi$
$462$	0	0
$463$	−19.4733	−0.905002	−0.452501	−	0.891764i	$-0.649468\pi$
$463$	−19.4733	−0.905002	−0.452501	+	0.891764i	$0.649468\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.1238	1.34769	0.673844	−	0.738873i	$-0.264642\pi$
$467$	29.1238	1.34769	0.673844	+	0.738873i	$0.264642\pi$
$468$	0	0
$469$	64.2410	2.96637
$470$	0	0
$471$	−12.5148	−0.576653
$472$	0	0
$473$	13.2873	0.610953
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.85119	0.222121
$478$	0	0
$479$	25.7301	1.17564	0.587819	−	0.808993i	$-0.299987\pi$
$479$	25.7301	1.17564	0.587819	+	0.808993i	$0.299987\pi$
$480$	0	0
$481$	43.9363	2.00332
$482$	0	0
$483$	−27.8996	−1.26948
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−18.3786	−0.832815	−0.416407	−	0.909178i	$-0.636711\pi$
$487$	−18.3786	−0.832815	−0.416407	+	0.909178i	$0.636711\pi$
$488$	0	0
$489$	−1.93084	−0.0873156
$490$	0	0
$491$	16.8807	0.761817	0.380909	−	0.924613i	$-0.375611\pi$
$491$	16.8807	0.761817	0.380909	+	0.924613i	$0.375611\pi$
$492$	0	0
$493$	26.4651	1.19193
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.2379	−0.638656
$498$	0	0
$499$	−0.792736	−0.0354877	−0.0177439	−	0.999843i	$-0.505648\pi$
$499$	−0.792736	−0.0354877	−0.0177439	+	0.999843i	$0.505648\pi$
$500$	0	0
$501$	2.00880	0.0897464
$502$	0	0
$503$	25.9962	1.15911	0.579556	−	0.814932i	$-0.303226\pi$
$503$	25.9962	1.15911	0.579556	+	0.814932i	$0.303226\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.4792	0.554221
$508$	0	0
$509$	23.2824	1.03197	0.515987	−	0.856596i	$-0.327425\pi$
$509$	23.2824	1.03197	0.515987	+	0.856596i	$0.327425\pi$
$510$	0	0
$511$	−14.4523	−0.639334
$512$	0	0
$513$	3.14923	0.139042
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3.47598	−0.152873
$518$	0	0
$519$	−10.7159	−0.470375
$520$	0	0
$521$	−42.3760	−1.85653	−0.928264	−	0.371923i	$-0.878699\pi$
$521$	−42.3760	−1.85653	−0.928264	+	0.371923i	$0.878699\pi$
$522$	0	0
$523$	20.6833	0.904416	0.452208	−	0.891912i	$-0.350636\pi$
$523$	20.6833	0.904416	0.452208	+	0.891912i	$0.350636\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.23525	0.184490
$528$	0	0
$529$	22.5178	0.979037
$530$	0	0
$531$	7.88238	0.342066
$532$	0	0
$533$	43.4238	1.88089
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.93384	0.342370
$538$	0	0
$539$	−12.4239	−0.535137
$540$	0	0
$541$	16.4730	0.708228	0.354114	−	0.935202i	$-0.384782\pi$
$541$	16.4730	0.708228	0.354114	+	0.935202i	$0.384782\pi$
$542$	0	0
$543$	−21.7718	−0.934317
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−18.7028	−0.799674	−0.399837	−	0.916586i	$-0.630933\pi$
$547$	−18.7028	−0.799674	−0.399837	+	0.916586i	$0.630933\pi$
$548$	0	0
$549$	9.08644	0.387800
$550$	0	0
$551$	−19.6789	−0.838348
$552$	0	0
$553$	18.6196	0.791785
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.5421	−0.785655	−0.392827	−	0.919612i	$-0.628503\pi$
$557$	−18.5421	−0.785655	−0.392827	+	0.919612i	$0.628503\pi$
$558$	0	0
$559$	−54.5286	−2.30631
$560$	0	0
$561$	5.20937	0.219940
$562$	0	0
$563$	−24.7985	−1.04513	−0.522567	−	0.852598i	$-0.675025\pi$
$563$	−24.7985	−1.04513	−0.522567	+	0.852598i	$0.675025\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.13530	0.173666
$568$	0	0
$569$	−20.1909	−0.846446	−0.423223	−	0.906026i	$-0.639101\pi$
$569$	−20.1909	−0.846446	−0.423223	+	0.906026i	$0.639101\pi$
$570$	0	0
$571$	−18.2047	−0.761844	−0.380922	−	0.924607i	$-0.624393\pi$
$571$	−18.2047	−0.761844	−0.380922	+	0.924607i	$0.624393\pi$
$572$	0	0
$573$	−20.7280	−0.865924
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.6361	0.484417	0.242209	−	0.970224i	$-0.422128\pi$
$577$	11.6361	0.484417	0.242209	+	0.970224i	$0.422128\pi$
$578$	0	0
$579$	11.7694	0.489121
$580$	0	0
$581$	−40.8544	−1.69492
$582$	0	0
$583$	−5.96699	−0.247128
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0.123414	0.00509386	0.00254693	−	0.999997i	$-0.499189\pi$
$587$	0.123414	0.00509386	0.00254693	+	0.999997i	$0.499189\pi$
$588$	0	0
$589$	−3.14923	−0.129762
$590$	0	0
$591$	7.85134	0.322961
$592$	0	0
$593$	7.47609	0.307006	0.153503	−	0.988148i	$-0.450945\pi$
$593$	7.47609	0.307006	0.153503	+	0.988148i	$0.450945\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.8120	−0.647140
$598$	0	0
$599$	−43.4323	−1.77460	−0.887298	−	0.461197i	$-0.847420\pi$
$599$	−43.4323	−1.77460	−0.887298	+	0.461197i	$0.847420\pi$
$600$	0	0
$601$	−0.271120	−0.0110592	−0.00552960	−	0.999985i	$-0.501760\pi$
$601$	−0.271120	−0.0110592	−0.00552960	+	0.999985i	$0.501760\pi$
$602$	0	0
$603$	15.5348	0.632625
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−17.1085	−0.694413	−0.347207	−	0.937789i	$-0.612870\pi$
$607$	−17.1085	−0.694413	−0.347207	+	0.937789i	$0.612870\pi$
$608$	0	0
$609$	−25.8406	−1.04711
$610$	0	0
$611$	14.2647	0.577088
$612$	0	0
$613$	6.86010	0.277077	0.138538	−	0.990357i	$-0.455760\pi$
$613$	6.86010	0.277077	0.138538	+	0.990357i	$0.455760\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19.3799	0.780206	0.390103	−	0.920771i	$-0.372439\pi$
$617$	19.3799	0.780206	0.390103	+	0.920771i	$0.372439\pi$
$618$	0	0
$619$	−10.5822	−0.425336	−0.212668	−	0.977125i	$-0.568215\pi$
$619$	−10.5822	−0.425336	−0.212668	+	0.977125i	$0.568215\pi$
$620$	0	0
$621$	−6.74669	−0.270735
$622$	0	0
$623$	19.2030	0.769352
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.87357	−0.154695
$628$	0	0
$629$	−36.8646	−1.46989
$630$	0	0
$631$	23.7825	0.946765	0.473383	−	0.880857i	$-0.343033\pi$
$631$	23.7825	0.946765	0.473383	+	0.880857i	$0.343033\pi$
$632$	0	0
$633$	18.4243	0.732300
$634$	0	0
$635$	0	0
$636$	0	0
$637$	50.9853	2.02011
$638$	0	0
$639$	−3.44301	−0.136203
$640$	0	0
$641$	33.9800	1.34213	0.671065	−	0.741399i	$-0.265837\pi$
$641$	33.9800	1.34213	0.671065	+	0.741399i	$0.265837\pi$
$642$	0	0
$643$	19.3024	0.761213	0.380607	−	0.924737i	$-0.375715\pi$
$643$	19.3024	0.761213	0.380607	+	0.924737i	$0.375715\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.7269	0.972115	0.486058	−	0.873927i	$-0.338435\pi$
$647$	24.7269	0.972115	0.486058	+	0.873927i	$0.338435\pi$
$648$	0	0
$649$	−9.69537	−0.380576
$650$	0	0
$651$	−4.13530	−0.162075
$652$	0	0
$653$	−18.5460	−0.725762	−0.362881	−	0.931836i	$-0.618207\pi$
$653$	−18.5460	−0.725762	−0.362881	+	0.931836i	$0.618207\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3.49487	−0.136348
$658$	0	0
$659$	−28.8949	−1.12559	−0.562794	−	0.826597i	$-0.690273\pi$
$659$	−28.8949	−1.12559	−0.562794	+	0.826597i	$0.690273\pi$
$660$	0	0
$661$	−28.4085	−1.10496	−0.552481	−	0.833525i	$-0.686319\pi$
$661$	−28.4085	−1.10496	−0.552481	+	0.833525i	$0.686319\pi$
$662$	0	0
$663$	−21.3782	−0.830261
$664$	0	0
$665$	0	0
$666$	0	0
$667$	42.1586	1.63239
$668$	0	0
$669$	−12.7146	−0.491574
$670$	0	0
$671$	−11.1764	−0.431459
$672$	0	0
$673$	−8.99249	−0.346635	−0.173318	−	0.984866i	$-0.555449\pi$
$673$	−8.99249	−0.346635	−0.173318	+	0.984866i	$0.555449\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−0.980771	−0.0376941	−0.0188471	−	0.999822i	$-0.506000\pi$
$677$	−0.980771	−0.0376941	−0.0188471	+	0.999822i	$0.506000\pi$
$678$	0	0
$679$	68.1598	2.61573
$680$	0	0
$681$	8.06936	0.309219
$682$	0	0
$683$	17.8700	0.683775	0.341888	−	0.939741i	$-0.388934\pi$
$683$	17.8700	0.683775	0.341888	+	0.939741i	$0.388934\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.49603	0.285992
$688$	0	0
$689$	24.4873	0.932893
$690$	0	0
$691$	−0.547525	−0.0208288	−0.0104144	−	0.999946i	$-0.503315\pi$
$691$	−0.547525	−0.0208288	−0.0104144	+	0.999946i	$0.503315\pi$
$692$	0	0
$693$	−5.08644	−0.193218
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−36.4346	−1.38006
$698$	0	0
$699$	23.2848	0.880712
$700$	0	0
$701$	22.6553	0.855677	0.427839	−	0.903855i	$-0.359275\pi$
$701$	22.6553	0.855677	0.427839	+	0.903855i	$0.359275\pi$
$702$	0	0
$703$	27.4117	1.03385
$704$	0	0
$705$	0	0
$706$	0	0
$707$	80.1763	3.01534
$708$	0	0
$709$	−23.8798	−0.896825	−0.448413	−	0.893827i	$-0.648011\pi$
$709$	−23.8798	−0.896825	−0.448413	+	0.893827i	$0.648011\pi$
$710$	0	0
$711$	4.50259	0.168860
$712$	0	0
$713$	6.74669	0.252666
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.3491	0.909335
$718$	0	0
$719$	−4.91869	−0.183436	−0.0917180	−	0.995785i	$-0.529236\pi$
$719$	−4.91869	−0.183436	−0.0917180	+	0.995785i	$0.529236\pi$
$720$	0	0
$721$	49.7999	1.85465
$722$	0	0
$723$	−21.3399	−0.793638
$724$	0	0
$725$	0	0
$726$	0	0
$727$	25.5003	0.945755	0.472878	−	0.881128i	$-0.343215\pi$
$727$	25.5003	0.945755	0.472878	+	0.881128i	$0.343215\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	45.7520	1.69220
$732$	0	0
$733$	6.15853	0.227470	0.113735	−	0.993511i	$-0.463718\pi$
$733$	6.15853	0.227470	0.113735	+	0.993511i	$0.463718\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.1079	−0.703847
$738$	0	0
$739$	−4.51439	−0.166065	−0.0830323	−	0.996547i	$-0.526460\pi$
$739$	−4.51439	−0.166065	−0.0830323	+	0.996547i	$0.526460\pi$
$740$	0	0
$741$	15.8963	0.583967
$742$	0	0
$743$	16.2577	0.596438	0.298219	−	0.954497i	$-0.403607\pi$
$743$	16.2577	0.596438	0.298219	+	0.954497i	$0.403607\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.87942	−0.361469
$748$	0	0
$749$	−53.5563	−1.95691
$750$	0	0
$751$	25.8769	0.944261	0.472130	−	0.881529i	$-0.343485\pi$
$751$	25.8769	0.944261	0.472130	+	0.881529i	$0.343485\pi$
$752$	0	0
$753$	−8.42008	−0.306845
$754$	0	0
$755$	0	0
$756$	0	0
$757$	30.9958	1.12656	0.563280	−	0.826266i	$-0.309539\pi$
$757$	30.9958	1.12656	0.563280	+	0.826266i	$0.309539\pi$
$758$	0	0
$759$	8.29846	0.301215
$760$	0	0
$761$	−5.98129	−0.216822	−0.108411	−	0.994106i	$-0.534576\pi$
$761$	−5.98129	−0.216822	−0.108411	+	0.994106i	$0.534576\pi$
$762$	0	0
$763$	10.5734	0.382783
$764$	0	0
$765$	0	0
$766$	0	0
$767$	39.7878	1.43666
$768$	0	0
$769$	−24.2357	−0.873963	−0.436981	−	0.899471i	$-0.643952\pi$
$769$	−24.2357	−0.873963	−0.436981	+	0.899471i	$0.643952\pi$
$770$	0	0
$771$	17.4448	0.628259
$772$	0	0
$773$	18.4427	0.663336	0.331668	−	0.943396i	$-0.392389\pi$
$773$	18.4427	0.663336	0.331668	+	0.943396i	$0.392389\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	35.9947	1.29130
$778$	0	0
$779$	27.0919	0.970668
$780$	0	0
$781$	4.23491	0.151537
$782$	0	0
$783$	−6.24879	−0.223313
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−0.479410	−0.0170891	−0.00854455	−	0.999963i	$-0.502720\pi$
$787$	−0.479410	−0.0170891	−0.00854455	+	0.999963i	$0.502720\pi$
$788$	0	0
$789$	18.7676	0.668144
$790$	0	0
$791$	−29.8457	−1.06119
$792$	0	0
$793$	45.8656	1.62873
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.8647	−0.951595	−0.475798	−	0.879555i	$-0.657841\pi$
$797$	−26.8647	−0.951595	−0.475798	+	0.879555i	$0.657841\pi$
$798$	0	0
$799$	−11.9687	−0.423424
$800$	0	0
$801$	4.64367	0.164076
$802$	0	0
$803$	4.29870	0.151698
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.78062	−0.133084
$808$	0	0
$809$	−33.6481	−1.18300	−0.591502	−	0.806304i	$-0.701465\pi$
$809$	−33.6481	−1.18300	−0.591502	+	0.806304i	$0.701465\pi$
$810$	0	0
$811$	−21.8752	−0.768142	−0.384071	−	0.923304i	$-0.625478\pi$
$811$	−21.8752	−0.768142	−0.384071	+	0.923304i	$0.625478\pi$
$812$	0	0
$813$	−27.4186	−0.961614
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−34.0201	−1.19021
$818$	0	0
$819$	20.8737	0.729387
$820$	0	0
$821$	−28.2928	−0.987424	−0.493712	−	0.869626i	$-0.664360\pi$
$821$	−28.2928	−0.987424	−0.493712	+	0.869626i	$0.664360\pi$
$822$	0	0
$823$	−48.2446	−1.68170	−0.840851	−	0.541266i	$-0.817945\pi$
$823$	−48.2446	−1.68170	−0.840851	+	0.541266i	$0.817945\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.85226	0.307823	0.153912	−	0.988085i	$-0.450813\pi$
$827$	8.85226	0.307823	0.153912	+	0.988085i	$0.450813\pi$
$828$	0	0
$829$	−35.1918	−1.22226	−0.611131	−	0.791530i	$-0.709285\pi$
$829$	−35.1918	−1.22226	−0.611131	+	0.791530i	$0.709285\pi$
$830$	0	0
$831$	−7.06800	−0.245186
$832$	0	0
$833$	−42.7790	−1.48221
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−23.0745	−0.796621	−0.398311	−	0.917251i	$-0.630403\pi$
$839$	−23.0745	−0.796621	−0.398311	+	0.917251i	$0.630403\pi$
$840$	0	0
$841$	10.0473	0.346459
$842$	0	0
$843$	−3.46379	−0.119299
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−39.2320	−1.34803
$848$	0	0
$849$	−6.34958	−0.217917
$850$	0	0
$851$	−58.7248	−2.01306
$852$	0	0
$853$	−23.1742	−0.793469	−0.396735	−	0.917933i	$-0.629857\pi$
$853$	−23.1742	−0.793469	−0.396735	+	0.917933i	$0.629857\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.5040	−0.392969	−0.196485	−	0.980507i	$-0.562953\pi$
$857$	−11.5040	−0.392969	−0.196485	+	0.980507i	$0.562953\pi$
$858$	0	0
$859$	−43.4382	−1.48209	−0.741046	−	0.671455i	$-0.765670\pi$
$859$	−43.4382	−1.48209	−0.741046	+	0.671455i	$0.765670\pi$
$860$	0	0
$861$	35.5748	1.21238
$862$	0	0
$863$	1.29609	0.0441194	0.0220597	−	0.999757i	$-0.492978\pi$
$863$	1.29609	0.0441194	0.0220597	+	0.999757i	$0.492978\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0.937303	0.0318325
$868$	0	0
$869$	−5.53821	−0.187871
$870$	0	0
$871$	78.4148	2.65698
$872$	0	0
$873$	16.4824	0.557845
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27.1306	−0.916134	−0.458067	−	0.888918i	$-0.651458\pi$
$877$	−27.1306	−0.916134	−0.458067	+	0.888918i	$0.651458\pi$
$878$	0	0
$879$	28.0568	0.946333
$880$	0	0
$881$	26.6011	0.896215	0.448108	−	0.893980i	$-0.352098\pi$
$881$	26.6011	0.896215	0.448108	+	0.893980i	$0.352098\pi$
$882$	0	0
$883$	7.76856	0.261433	0.130716	−	0.991420i	$-0.458272\pi$
$883$	7.76856	0.261433	0.130716	+	0.991420i	$0.458272\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.8686	−0.667123	−0.333561	−	0.942728i	$-0.608250\pi$
$887$	−19.8686	−0.667123	−0.333561	+	0.942728i	$0.608250\pi$
$888$	0	0
$889$	53.5800	1.79701
$890$	0	0
$891$	−1.23000	−0.0412067
$892$	0	0
$893$	8.89968	0.297817
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−34.0552	−1.13707
$898$	0	0
$899$	6.24879	0.208409
$900$	0	0
$901$	−20.5460	−0.684486
$902$	0	0
$903$	−44.6724	−1.48660
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−27.8043	−0.923227	−0.461614	−	0.887081i	$-0.652729\pi$
$907$	−27.8043	−0.923227	−0.461614	+	0.887081i	$0.652729\pi$
$908$	0	0
$909$	19.3883	0.643068
$910$	0	0
$911$	9.37340	0.310555	0.155277	−	0.987871i	$-0.450373\pi$
$911$	9.37340	0.310555	0.155277	+	0.987871i	$0.450373\pi$
$912$	0	0
$913$	12.1517	0.402164
$914$	0	0
$915$	0	0
$916$	0	0
$917$	91.4616	3.02033
$918$	0	0
$919$	−12.2498	−0.404082	−0.202041	−	0.979377i	$-0.564757\pi$
$919$	−12.2498	−0.404082	−0.202041	+	0.979377i	$0.564757\pi$
$920$	0	0
$921$	−26.0895	−0.859680
$922$	0	0
$923$	−17.3792	−0.572045
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.0426	0.395532
$928$	0	0
$929$	0.546093	0.0179167	0.00895836	−	0.999960i	$-0.497148\pi$
$929$	0.546093	0.0179167	0.00895836	+	0.999960i	$0.497148\pi$
$930$	0	0
$931$	31.8095	1.04251
$932$	0	0
$933$	−6.01182	−0.196818
$934$	0	0
$935$	0	0
$936$	0	0
$937$	43.5748	1.42353	0.711763	−	0.702420i	$-0.247897\pi$
$937$	43.5748	1.42353	0.711763	+	0.702420i	$0.247897\pi$
$938$	0	0
$939$	−9.33471	−0.304627
$940$	0	0
$941$	−35.4879	−1.15687	−0.578436	−	0.815728i	$-0.696337\pi$
$941$	−35.4879	−1.15687	−0.578436	+	0.815728i	$0.696337\pi$
$942$	0	0
$943$	−58.0398	−1.89004
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.5125	−1.34897	−0.674487	−	0.738287i	$-0.735635\pi$
$947$	−41.5125	−1.34897	−0.674487	+	0.738287i	$0.735635\pi$
$948$	0	0
$949$	−17.6410	−0.572652
$950$	0	0
$951$	22.3996	0.726356
$952$	0	0
$953$	−34.8250	−1.12809	−0.564046	−	0.825743i	$-0.690756\pi$
$953$	−34.8250	−1.12809	−0.564046	+	0.825743i	$0.690756\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.68603	0.248454
$958$	0	0
$959$	8.63626	0.278879
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−12.9510	−0.417340
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−38.0236	−1.22276	−0.611378	−	0.791338i	$-0.709385\pi$
$967$	−38.0236	−1.22276	−0.611378	+	0.791338i	$0.709385\pi$
$968$	0	0
$969$	−13.3378	−0.428471
$970$	0	0
$971$	−60.5081	−1.94180	−0.970898	−	0.239492i	$-0.923019\pi$
$971$	−60.5081	−1.94180	−0.970898	+	0.239492i	$0.923019\pi$
$972$	0	0
$973$	−32.8665	−1.05365
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.7646	0.920262	0.460131	−	0.887851i	$-0.347802\pi$
$977$	28.7646	0.920262	0.460131	+	0.887851i	$0.347802\pi$
$978$	0	0
$979$	−5.71174	−0.182548
$980$	0	0
$981$	2.55686	0.0816343
$982$	0	0
$983$	16.7628	0.534652	0.267326	−	0.963606i	$-0.413860\pi$
$983$	16.7628	0.534652	0.267326	+	0.963606i	$0.413860\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	11.6863	0.371979
$988$	0	0
$989$	72.8824	2.31752
$990$	0	0
$991$	12.3289	0.391641	0.195821	−	0.980640i	$-0.437263\pi$
$991$	12.3289	0.391641	0.195821	+	0.980640i	$0.437263\pi$
$992$	0	0
$993$	15.6884	0.497858
$994$	0	0
$995$	0	0
$996$	0	0
$997$	24.4187	0.773347	0.386674	−	0.922217i	$-0.373624\pi$
$997$	24.4187	0.773347	0.386674	+	0.922217i	$0.373624\pi$
$998$	0	0
$999$	8.70424	0.275390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.a.bf.1.7		7
5.2	odd	4		1860.2.g.a.1489.1	✓	14
5.3	odd	4		1860.2.g.a.1489.8	yes	14
5.4	even	2		9300.2.a.bc.1.1		7
15.2	even	4		5580.2.g.d.3349.14		14
15.8	even	4		5580.2.g.d.3349.13		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.g.a.1489.1	✓	14	5.2	odd	4
1860.2.g.a.1489.8	yes	14	5.3	odd	4
5580.2.g.d.3349.13		14	15.8	even	4
5580.2.g.d.3349.14		14	15.2	even	4
9300.2.a.bc.1.1		7	5.4	even	2
9300.2.a.bf.1.7		7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.13530 q^{7} +1.00000 q^{9} -1.23000 q^{11} +5.04769 q^{13} -4.23525 q^{17} +3.14923 q^{19} +4.13530 q^{21} -6.74669 q^{23} +1.00000 q^{27} -6.24879 q^{29} -1.00000 q^{31} -1.23000 q^{33} +8.70424 q^{37} +5.04769 q^{39} +8.60270 q^{41} -10.8027 q^{43} +2.82599 q^{47} +10.1007 q^{49} -4.23525 q^{51} +4.85119 q^{53} +3.14923 q^{57} +7.88238 q^{59} +9.08644 q^{61} +4.13530 q^{63} +15.5348 q^{67} -6.74669 q^{69} -3.44301 q^{71} -3.49487 q^{73} -5.08644 q^{77} +4.50259 q^{79} +1.00000 q^{81} -9.87942 q^{83} -6.24879 q^{87} +4.64367 q^{89} +20.8737 q^{91} -1.00000 q^{93} +16.4824 q^{97} -1.23000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} + 4 q^{7} + 7 q^{9} + 4 q^{11} + 10 q^{13} + 6 q^{17} + 10 q^{19} + 4 q^{21} + 7 q^{27} + 10 q^{29} - 7 q^{31} + 4 q^{33} + 16 q^{37} + 10 q^{39} + 2 q^{41} - 8 q^{43} + 12 q^{47} + 9 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) \) 7 * q + 7 * q^3 + 4 * q^7 + 7 * q^9 + 4 * q^11 + 10 * q^13 + 6 * q^17 + 10 * q^19 + 4 * q^21 + 7 * q^27 + 10 * q^29 - 7 * q^31 + 4 * q^33 + 16 * q^37 + 10 * q^39 + 2 * q^41 - 8 * q^43 + 12 * q^47 + 9 * q^49 + 6 * q^51 + 16 * q^53 + 10 * q^57 + 14 * q^59 + 10 * q^61 + 4 * q^63 + 22 * q^67 - 2 * q^71 + 14 * q^73 + 18 * q^77 + 24 * q^79 + 7 * q^81 - 12 * q^83 + 10 * q^87 + 12 * q^89 + 8 * q^91 - 7 * q^93 + 24 * q^97 + 4 * q^99

Introduction

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