LMFDB - Embedded newform 9522.2.a.ca.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9522,2,Mod(1,9522)] 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("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,5,0,5,11,0,1,5,0,11,11,0,10,1,0,5,11,0,1,11,0,11,0,0,30,10, 0,1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.284630$ of defining polynomial
Character	$\chi$	$=$	9522.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.95185 q^{5} +0.458044 q^{7} +1.00000 q^{8} +1.95185 q^{10} +1.25954 q^{11} +5.60149 q^{13} +0.458044 q^{14} +1.00000 q^{16} +6.54436 q^{17} -2.11970 q^{19} +1.95185 q^{20} +1.25954 q^{22} -1.19028 q^{25} +5.60149 q^{26} +0.458044 q^{28} -5.06465 q^{29} -0.109003 q^{31} +1.00000 q^{32} +6.54436 q^{34} +0.894034 q^{35} +0.926128 q^{37} -2.11970 q^{38} +1.95185 q^{40} +9.83955 q^{41} -6.77428 q^{43} +1.25954 q^{44} +2.50740 q^{47} -6.79020 q^{49} -1.19028 q^{50} +5.60149 q^{52} -2.64612 q^{53} +2.45843 q^{55} +0.458044 q^{56} -5.06465 q^{58} +4.27686 q^{59} +8.58595 q^{61} -0.109003 q^{62} +1.00000 q^{64} +10.9333 q^{65} +15.0332 q^{67} +6.54436 q^{68} +0.894034 q^{70} +0.303444 q^{71} +3.34620 q^{73} +0.926128 q^{74} -2.11970 q^{76} +0.576924 q^{77} +8.04741 q^{79} +1.95185 q^{80} +9.83955 q^{82} +10.4920 q^{83} +12.7736 q^{85} -6.77428 q^{86} +1.25954 q^{88} +5.95710 q^{89} +2.56573 q^{91} +2.50740 q^{94} -4.13734 q^{95} -18.2589 q^{97} -6.79020 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} + 11 q^{5} + q^{7} + 5 q^{8} + 11 q^{10} + 11 q^{11} + 10 q^{13} + q^{14} + 5 q^{16} + 11 q^{17} + q^{19} + 11 q^{20} + 11 q^{22} + 30 q^{25} + 10 q^{26} + q^{28} - 3 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 + 11 * q^5 + q^7 + 5 * q^8 + 11 * q^10 + 11 * q^11 + 10 * q^13 + q^14 + 5 * q^16 + 11 * q^17 + q^19 + 11 * q^20 + 11 * q^22 + 30 * q^25 + 10 * q^26 + q^28 - 3 * q^29 + 2 * q^31 + 5 * q^32 + 11 * q^34 + 11 * q^35 - 16 * q^37 + q^38 + 11 * q^40 + 8 * q^41 + 12 * q^43 + 11 * q^44 - 4 * q^49 + 30 * q^50 + 10 * q^52 + 27 * q^53 + q^56 - 3 * q^58 + 6 * q^59 - 5 * q^61 + 2 * q^62 + 5 * q^64 + 22 * q^65 + 7 * q^67 + 11 * q^68 + 11 * q^70 + 11 * q^71 - 12 * q^73 - 16 * q^74 + q^76 - 11 * q^77 - 16 * q^79 + 11 * q^80 + 8 * q^82 + 22 * q^83 - 11 * q^85 + 12 * q^86 + 11 * q^88 + 38 * q^89 - 9 * q^91 - 11 * q^95 - 5 * q^97 - 4 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.95185	0.872894	0.436447	−	0.899730i	$-0.356237\pi$
$5$	1.95185	0.872894	0.436447	+	0.899730i	$0.356237\pi$
$6$	0	0
$7$	0.458044	0.173125	0.0865623	−	0.996246i	$-0.472412\pi$
$7$	0.458044	0.173125	0.0865623	+	0.996246i	$0.472412\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.95185	0.617229
$11$	1.25954	0.379765	0.189882	−	0.981807i	$-0.439189\pi$
$11$	1.25954	0.379765	0.189882	+	0.981807i	$0.439189\pi$
$12$	0	0
$13$	5.60149	1.55357	0.776787	−	0.629763i	$-0.216848\pi$
$13$	5.60149	1.55357	0.776787	+	0.629763i	$0.216848\pi$
$14$	0.458044	0.122418
$15$	0	0
$16$	1.00000	0.250000
$17$	6.54436	1.58724	0.793621	−	0.608413i	$-0.208194\pi$
$17$	6.54436	1.58724	0.793621	+	0.608413i	$0.208194\pi$
$18$	0	0
$19$	−2.11970	−0.486294	−0.243147	−	0.969990i	$-0.578180\pi$
$19$	−2.11970	−0.486294	−0.243147	+	0.969990i	$0.578180\pi$
$20$	1.95185	0.436447
$21$	0	0
$22$	1.25954	0.268534
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−1.19028	−0.238057
$26$	5.60149	1.09854
$27$	0	0
$28$	0.458044	0.0865623
$29$	−5.06465	−0.940481	−0.470241	−	0.882538i	$-0.655833\pi$
$29$	−5.06465	−0.940481	−0.470241	+	0.882538i	$0.655833\pi$
$30$	0	0
$31$	−0.109003	−0.0195775	−0.00978875	−	0.999952i	$-0.503116\pi$
$31$	−0.109003	−0.0195775	−0.00978875	+	0.999952i	$0.503116\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.54436	1.12235
$35$	0.894034	0.151119
$36$	0	0
$37$	0.926128	0.152254	0.0761272	−	0.997098i	$-0.475744\pi$
$37$	0.926128	0.152254	0.0761272	+	0.997098i	$0.475744\pi$
$38$	−2.11970	−0.343861
$39$	0	0
$40$	1.95185	0.308614
$41$	9.83955	1.53668	0.768340	−	0.640042i	$-0.221083\pi$
$41$	9.83955	1.53668	0.768340	+	0.640042i	$0.221083\pi$
$42$	0	0
$43$	−6.77428	−1.03307	−0.516534	−	0.856267i	$-0.672778\pi$
$43$	−6.77428	−1.03307	−0.516534	+	0.856267i	$0.672778\pi$
$44$	1.25954	0.189882
$45$	0	0
$46$	0	0
$47$	2.50740	0.365742	0.182871	−	0.983137i	$-0.441461\pi$
$47$	2.50740	0.365742	0.182871	+	0.983137i	$0.441461\pi$
$48$	0	0
$49$	−6.79020	−0.970028
$50$	−1.19028	−0.168332
$51$	0	0
$52$	5.60149	0.776787
$53$	−2.64612	−0.363472	−0.181736	−	0.983347i	$-0.558172\pi$
$53$	−2.64612	−0.363472	−0.181736	+	0.983347i	$0.558172\pi$
$54$	0	0
$55$	2.45843	0.331494
$56$	0.458044	0.0612088
$57$	0	0
$58$	−5.06465	−0.665021
$59$	4.27686	0.556800	0.278400	−	0.960465i	$-0.410196\pi$
$59$	4.27686	0.556800	0.278400	+	0.960465i	$0.410196\pi$
$60$	0	0
$61$	8.58595	1.09932	0.549659	−	0.835389i	$-0.314758\pi$
$61$	8.58595	1.09932	0.549659	+	0.835389i	$0.314758\pi$
$62$	−0.109003	−0.0138434
$63$	0	0
$64$	1.00000	0.125000
$65$	10.9333	1.35611
$66$	0	0
$67$	15.0332	1.83660	0.918298	−	0.395890i	$-0.129564\pi$
$67$	15.0332	1.83660	0.918298	+	0.395890i	$0.129564\pi$
$68$	6.54436	0.793621
$69$	0	0
$70$	0.894034	0.106857
$71$	0.303444	0.0360122	0.0180061	−	0.999838i	$-0.494268\pi$
$71$	0.303444	0.0360122	0.0180061	+	0.999838i	$0.494268\pi$
$72$	0	0
$73$	3.34620	0.391643	0.195822	−	0.980640i	$-0.437263\pi$
$73$	3.34620	0.391643	0.195822	+	0.980640i	$0.437263\pi$
$74$	0.926128	0.107660
$75$	0	0
$76$	−2.11970	−0.243147
$77$	0.576924	0.0657466
$78$	0	0
$79$	8.04741	0.905405	0.452702	−	0.891662i	$-0.350460\pi$
$79$	8.04741	0.905405	0.452702	+	0.891662i	$0.350460\pi$
$80$	1.95185	0.218223
$81$	0	0
$82$	9.83955	1.08660
$83$	10.4920	1.15165	0.575825	−	0.817573i	$-0.304681\pi$
$83$	10.4920	1.15165	0.575825	+	0.817573i	$0.304681\pi$
$84$	0	0
$85$	12.7736	1.38549
$86$	−6.77428	−0.730489
$87$	0	0
$88$	1.25954	0.134267
$89$	5.95710	0.631451	0.315726	−	0.948851i	$-0.397752\pi$
$89$	5.95710	0.631451	0.315726	+	0.948851i	$0.397752\pi$
$90$	0	0
$91$	2.56573	0.268962
$92$	0	0
$93$	0	0
$94$	2.50740	0.258619
$95$	−4.13734	−0.424482
$96$	0	0
$97$	−18.2589	−1.85391	−0.926953	−	0.375177i	$-0.877582\pi$
$97$	−18.2589	−1.85391	−0.926953	+	0.375177i	$0.877582\pi$
$98$	−6.79020	−0.685913
$99$	0	0
$100$	−1.19028	−0.119028
$101$	−10.6597	−1.06068	−0.530341	−	0.847785i	$-0.677936\pi$
$101$	−10.6597	−1.06068	−0.530341	+	0.847785i	$0.677936\pi$
$102$	0	0
$103$	−7.07465	−0.697086	−0.348543	−	0.937293i	$-0.613323\pi$
$103$	−7.07465	−0.697086	−0.348543	+	0.937293i	$0.613323\pi$
$104$	5.60149	0.549272
$105$	0	0
$106$	−2.64612	−0.257014
$107$	−2.76376	−0.267182	−0.133591	−	0.991037i	$-0.542651\pi$
$107$	−2.76376	−0.267182	−0.133591	+	0.991037i	$0.542651\pi$
$108$	0	0
$109$	−16.4960	−1.58003	−0.790013	−	0.613090i	$-0.789926\pi$
$109$	−16.4960	−1.58003	−0.790013	+	0.613090i	$0.789926\pi$
$110$	2.45843	0.234402
$111$	0	0
$112$	0.458044	0.0432811
$113$	15.3359	1.44268	0.721342	−	0.692579i	$-0.243526\pi$
$113$	15.3359	1.44268	0.721342	+	0.692579i	$0.243526\pi$
$114$	0	0
$115$	0	0
$116$	−5.06465	−0.470241
$117$	0	0
$118$	4.27686	0.393717
$119$	2.99761	0.274790
$120$	0	0
$121$	−9.41356	−0.855779
$122$	8.58595	0.777335
$123$	0	0
$124$	−0.109003	−0.00978875
$125$	−12.0825	−1.08069
$126$	0	0
$127$	−21.6633	−1.92231	−0.961155	−	0.276010i	$-0.910988\pi$
$127$	−21.6633	−1.92231	−0.961155	+	0.276010i	$0.910988\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	10.9333	0.958911
$131$	−17.1468	−1.49812	−0.749062	−	0.662500i	$-0.769495\pi$
$131$	−17.1468	−1.49812	−0.749062	+	0.662500i	$0.769495\pi$
$132$	0	0
$133$	−0.970919	−0.0841893
$134$	15.0332	1.29867
$135$	0	0
$136$	6.54436	0.561174
$137$	−3.10334	−0.265136	−0.132568	−	0.991174i	$-0.542322\pi$
$137$	−3.10334	−0.265136	−0.132568	+	0.991174i	$0.542322\pi$
$138$	0	0
$139$	−22.4485	−1.90406	−0.952028	−	0.306011i	$-0.901006\pi$
$139$	−22.4485	−1.90406	−0.952028	+	0.306011i	$0.901006\pi$
$140$	0.894034	0.0755596
$141$	0	0
$142$	0.303444	0.0254644
$143$	7.05529	0.589993
$144$	0	0
$145$	−9.88543	−0.820940
$146$	3.34620	0.276933
$147$	0	0
$148$	0.926128	0.0761272
$149$	15.7100	1.28701	0.643507	−	0.765440i	$-0.277479\pi$
$149$	15.7100	1.28701	0.643507	+	0.765440i	$0.277479\pi$
$150$	0	0
$151$	−5.57516	−0.453700	−0.226850	−	0.973930i	$-0.572843\pi$
$151$	−5.57516	−0.453700	−0.226850	+	0.973930i	$0.572843\pi$
$152$	−2.11970	−0.171931
$153$	0	0
$154$	0.576924	0.0464899
$155$	−0.212757	−0.0170891
$156$	0	0
$157$	8.16701	0.651799	0.325899	−	0.945404i	$-0.394333\pi$
$157$	8.16701	0.651799	0.325899	+	0.945404i	$0.394333\pi$
$158$	8.04741	0.640218
$159$	0	0
$160$	1.95185	0.154307
$161$	0	0
$162$	0	0
$163$	−17.0615	−1.33636	−0.668181	−	0.743999i	$-0.732927\pi$
$163$	−17.0615	−1.33636	−0.668181	+	0.743999i	$0.732927\pi$
$164$	9.83955	0.768340
$165$	0	0
$166$	10.4920	0.814340
$167$	−18.0693	−1.39825	−0.699123	−	0.715001i	$-0.746426\pi$
$167$	−18.0693	−1.39825	−0.699123	+	0.715001i	$0.746426\pi$
$168$	0	0
$169$	18.3767	1.41359
$170$	12.7736	0.979691
$171$	0	0
$172$	−6.77428	−0.516534
$173$	15.9612	1.21350	0.606752	−	0.794891i	$-0.292472\pi$
$173$	15.9612	1.21350	0.606752	+	0.794891i	$0.292472\pi$
$174$	0	0
$175$	−0.545203	−0.0412135
$176$	1.25954	0.0949412
$177$	0	0
$178$	5.95710	0.446503
$179$	15.6916	1.17284	0.586421	−	0.810006i	$-0.300536\pi$
$179$	15.6916	1.17284	0.586421	+	0.810006i	$0.300536\pi$
$180$	0	0
$181$	1.81930	0.135228	0.0676139	−	0.997712i	$-0.478461\pi$
$181$	1.81930	0.135228	0.0676139	+	0.997712i	$0.478461\pi$
$182$	2.56573	0.190185
$183$	0	0
$184$	0	0
$185$	1.80766	0.132902
$186$	0	0
$187$	8.24287	0.602779
$188$	2.50740	0.182871
$189$	0	0
$190$	−4.13734	−0.300154
$191$	4.24849	0.307410	0.153705	−	0.988117i	$-0.450880\pi$
$191$	4.24849	0.307410	0.153705	+	0.988117i	$0.450880\pi$
$192$	0	0
$193$	13.1121	0.943828	0.471914	−	0.881645i	$-0.343563\pi$
$193$	13.1121	0.943828	0.471914	+	0.881645i	$0.343563\pi$
$194$	−18.2589	−1.31091
$195$	0	0
$196$	−6.79020	−0.485014
$197$	9.92381	0.707042	0.353521	−	0.935427i	$-0.384984\pi$
$197$	9.92381	0.707042	0.353521	+	0.935427i	$0.384984\pi$
$198$	0	0
$199$	−6.30555	−0.446989	−0.223494	−	0.974705i	$-0.571746\pi$
$199$	−6.30555	−0.446989	−0.223494	+	0.974705i	$0.571746\pi$
$200$	−1.19028	−0.0841658
$201$	0	0
$202$	−10.6597	−0.750015
$203$	−2.31983	−0.162820
$204$	0	0
$205$	19.2053	1.34136
$206$	−7.07465	−0.492914
$207$	0	0
$208$	5.60149	0.388394
$209$	−2.66985	−0.184677
$210$	0	0
$211$	25.0546	1.72483	0.862415	−	0.506202i	$-0.168951\pi$
$211$	25.0546	1.72483	0.862415	+	0.506202i	$0.168951\pi$
$212$	−2.64612	−0.181736
$213$	0	0
$214$	−2.76376	−0.188926
$215$	−13.2224	−0.901758
$216$	0	0
$217$	−0.0499282	−0.00338935
$218$	−16.4960	−1.11725
$219$	0	0
$220$	2.45843	0.165747
$221$	36.6582	2.46590
$222$	0	0
$223$	1.78108	0.119270	0.0596348	−	0.998220i	$-0.481006\pi$
$223$	1.78108	0.119270	0.0596348	+	0.998220i	$0.481006\pi$
$224$	0.458044	0.0306044
$225$	0	0
$226$	15.3359	1.02013
$227$	−20.7093	−1.37453	−0.687264	−	0.726408i	$-0.741188\pi$
$227$	−20.7093	−1.37453	−0.687264	+	0.726408i	$0.741188\pi$
$228$	0	0
$229$	−8.62288	−0.569816	−0.284908	−	0.958555i	$-0.591963\pi$
$229$	−8.62288	−0.569816	−0.284908	+	0.958555i	$0.591963\pi$
$230$	0	0
$231$	0	0
$232$	−5.06465	−0.332510
$233$	−4.77637	−0.312910	−0.156455	−	0.987685i	$-0.550007\pi$
$233$	−4.77637	−0.312910	−0.156455	+	0.987685i	$0.550007\pi$
$234$	0	0
$235$	4.89407	0.319254
$236$	4.27686	0.278400
$237$	0	0
$238$	2.99761	0.194306
$239$	23.9090	1.54655	0.773273	−	0.634073i	$-0.218618\pi$
$239$	23.9090	1.54655	0.773273	+	0.634073i	$0.218618\pi$
$240$	0	0
$241$	23.0500	1.48478	0.742389	−	0.669969i	$-0.233693\pi$
$241$	23.0500	1.48478	0.742389	+	0.669969i	$0.233693\pi$
$242$	−9.41356	−0.605127
$243$	0	0
$244$	8.58595	0.549659
$245$	−13.2534	−0.846731
$246$	0	0
$247$	−11.8735	−0.755493
$248$	−0.109003	−0.00692169
$249$	0	0
$250$	−12.0825	−0.764165
$251$	−0.221255	−0.0139655	−0.00698273	−	0.999976i	$-0.502223\pi$
$251$	−0.221255	−0.0139655	−0.00698273	+	0.999976i	$0.502223\pi$
$252$	0	0
$253$	0	0
$254$	−21.6633	−1.35928
$255$	0	0
$256$	1.00000	0.0625000
$257$	0.544992	0.0339957	0.0169978	−	0.999856i	$-0.494589\pi$
$257$	0.544992	0.0339957	0.0169978	+	0.999856i	$0.494589\pi$
$258$	0	0
$259$	0.424208	0.0263590
$260$	10.9333	0.678053
$261$	0	0
$262$	−17.1468	−1.05933
$263$	14.7270	0.908104	0.454052	−	0.890975i	$-0.349978\pi$
$263$	14.7270	0.908104	0.454052	+	0.890975i	$0.349978\pi$
$264$	0	0
$265$	−5.16482	−0.317272
$266$	−0.970919	−0.0595309
$267$	0	0
$268$	15.0332	0.918298
$269$	−1.00716	−0.0614078	−0.0307039	−	0.999529i	$-0.509775\pi$
$269$	−1.00716	−0.0614078	−0.0307039	+	0.999529i	$0.509775\pi$
$270$	0	0
$271$	−19.9145	−1.20972	−0.604860	−	0.796332i	$-0.706771\pi$
$271$	−19.9145	−1.20972	−0.604860	+	0.796332i	$0.706771\pi$
$272$	6.54436	0.396810
$273$	0	0
$274$	−3.10334	−0.187479
$275$	−1.49921	−0.0904057
$276$	0	0
$277$	−24.4934	−1.47167	−0.735834	−	0.677162i	$-0.763209\pi$
$277$	−24.4934	−1.47167	−0.735834	+	0.677162i	$0.763209\pi$
$278$	−22.4485	−1.34637
$279$	0	0
$280$	0.894034	0.0534287
$281$	22.2104	1.32496	0.662480	−	0.749080i	$-0.269504\pi$
$281$	22.2104	1.32496	0.662480	+	0.749080i	$0.269504\pi$
$282$	0	0
$283$	−16.4220	−0.976185	−0.488093	−	0.872792i	$-0.662307\pi$
$283$	−16.4220	−0.976185	−0.488093	+	0.872792i	$0.662307\pi$
$284$	0.303444	0.0180061
$285$	0	0
$286$	7.05529	0.417188
$287$	4.50695	0.266037
$288$	0	0
$289$	25.8287	1.51933
$290$	−9.88543	−0.580492
$291$	0	0
$292$	3.34620	0.195822
$293$	24.4969	1.43112	0.715561	−	0.698550i	$-0.246171\pi$
$293$	24.4969	1.43112	0.715561	+	0.698550i	$0.246171\pi$
$294$	0	0
$295$	8.34778	0.486027
$296$	0.926128	0.0538301
$297$	0	0
$298$	15.7100	0.910057
$299$	0	0
$300$	0	0
$301$	−3.10292	−0.178849
$302$	−5.57516	−0.320815
$303$	0	0
$304$	−2.11970	−0.121573
$305$	16.7585	0.959588
$306$	0	0
$307$	14.6749	0.837539	0.418769	−	0.908093i	$-0.362462\pi$
$307$	14.6749	0.837539	0.418769	+	0.908093i	$0.362462\pi$
$308$	0.576924	0.0328733
$309$	0	0
$310$	−0.212757	−0.0120838
$311$	−33.2393	−1.88483	−0.942414	−	0.334449i	$-0.891450\pi$
$311$	−33.2393	−1.88483	−0.942414	+	0.334449i	$0.891450\pi$
$312$	0	0
$313$	20.4935	1.15836	0.579181	−	0.815199i	$-0.303373\pi$
$313$	20.4935	1.15836	0.579181	+	0.815199i	$0.303373\pi$
$314$	8.16701	0.460891
$315$	0	0
$316$	8.04741	0.452702
$317$	−22.9004	−1.28622	−0.643108	−	0.765775i	$-0.722355\pi$
$317$	−22.9004	−1.28622	−0.643108	+	0.765775i	$0.722355\pi$
$318$	0	0
$319$	−6.37911	−0.357162
$320$	1.95185	0.109112
$321$	0	0
$322$	0	0
$323$	−13.8721	−0.771865
$324$	0	0
$325$	−6.66737	−0.369839
$326$	−17.0615	−0.944950
$327$	0	0
$328$	9.83955	0.543298
$329$	1.14850	0.0633190
$330$	0	0
$331$	−10.2762	−0.564831	−0.282416	−	0.959292i	$-0.591136\pi$
$331$	−10.2762	−0.564831	−0.282416	+	0.959292i	$0.591136\pi$
$332$	10.4920	0.575825
$333$	0	0
$334$	−18.0693	−0.988709
$335$	29.3425	1.60315
$336$	0	0
$337$	−11.6162	−0.632773	−0.316386	−	0.948630i	$-0.602470\pi$
$337$	−11.6162	−0.632773	−0.316386	+	0.948630i	$0.602470\pi$
$338$	18.3767	0.999562
$339$	0	0
$340$	12.7736	0.692746
$341$	−0.137293	−0.00743485
$342$	0	0
$343$	−6.31652	−0.341060
$344$	−6.77428	−0.365245
$345$	0	0
$346$	15.9612	0.858077
$347$	12.8984	0.692424	0.346212	−	0.938156i	$-0.387468\pi$
$347$	12.8984	0.692424	0.346212	+	0.938156i	$0.387468\pi$
$348$	0	0
$349$	15.8619	0.849070	0.424535	−	0.905412i	$-0.360438\pi$
$349$	15.8619	0.849070	0.424535	+	0.905412i	$0.360438\pi$
$350$	−0.545203	−0.0291423
$351$	0	0
$352$	1.25954	0.0671336
$353$	1.79503	0.0955397	0.0477698	−	0.998858i	$-0.484789\pi$
$353$	1.79503	0.0955397	0.0477698	+	0.998858i	$0.484789\pi$
$354$	0	0
$355$	0.592277	0.0314348
$356$	5.95710	0.315726
$357$	0	0
$358$	15.6916	0.829325
$359$	19.0479	1.00531	0.502655	−	0.864487i	$-0.332357\pi$
$359$	19.0479	1.00531	0.502655	+	0.864487i	$0.332357\pi$
$360$	0	0
$361$	−14.5069	−0.763519
$362$	1.81930	0.0956205
$363$	0	0
$364$	2.56573	0.134481
$365$	6.53128	0.341863
$366$	0	0
$367$	3.88864	0.202985	0.101493	−	0.994836i	$-0.467638\pi$
$367$	3.88864	0.202985	0.101493	+	0.994836i	$0.467638\pi$
$368$	0	0
$369$	0	0
$370$	1.80766	0.0939759
$371$	−1.21204	−0.0629259
$372$	0	0
$373$	31.2555	1.61835	0.809175	−	0.587567i	$-0.199914\pi$
$373$	31.2555	1.61835	0.809175	+	0.587567i	$0.199914\pi$
$374$	8.24287	0.426229
$375$	0	0
$376$	2.50740	0.129309
$377$	−28.3696	−1.46111
$378$	0	0
$379$	7.32134	0.376072	0.188036	−	0.982162i	$-0.439788\pi$
$379$	7.32134	0.376072	0.188036	+	0.982162i	$0.439788\pi$
$380$	−4.13734	−0.212241
$381$	0	0
$382$	4.24849	0.217371
$383$	19.2426	0.983253	0.491627	−	0.870806i	$-0.336402\pi$
$383$	19.2426	0.983253	0.491627	+	0.870806i	$0.336402\pi$
$384$	0	0
$385$	1.12607	0.0573898
$386$	13.1121	0.667387
$387$	0	0
$388$	−18.2589	−0.926953
$389$	8.25813	0.418704	0.209352	−	0.977840i	$-0.432865\pi$
$389$	8.25813	0.418704	0.209352	+	0.977840i	$0.432865\pi$
$390$	0	0
$391$	0	0
$392$	−6.79020	−0.342957
$393$	0	0
$394$	9.92381	0.499954
$395$	15.7073	0.790322
$396$	0	0
$397$	−11.3695	−0.570620	−0.285310	−	0.958435i	$-0.592096\pi$
$397$	−11.3695	−0.570620	−0.285310	+	0.958435i	$0.592096\pi$
$398$	−6.30555	−0.316069
$399$	0	0
$400$	−1.19028	−0.0595142
$401$	15.2135	0.759728	0.379864	−	0.925042i	$-0.375971\pi$
$401$	15.2135	0.759728	0.379864	+	0.925042i	$0.375971\pi$
$402$	0	0
$403$	−0.610579	−0.0304151
$404$	−10.6597	−0.530341
$405$	0	0
$406$	−2.31983	−0.115131
$407$	1.16649	0.0578209
$408$	0	0
$409$	1.53365	0.0758340	0.0379170	−	0.999281i	$-0.487928\pi$
$409$	1.53365	0.0758340	0.0379170	+	0.999281i	$0.487928\pi$
$410$	19.2053	0.948483
$411$	0	0
$412$	−7.07465	−0.348543
$413$	1.95899	0.0963956
$414$	0	0
$415$	20.4789	1.00527
$416$	5.60149	0.274636
$417$	0	0
$418$	−2.66985	−0.130587
$419$	4.63094	0.226236	0.113118	−	0.993582i	$-0.463916\pi$
$419$	4.63094	0.226236	0.113118	+	0.993582i	$0.463916\pi$
$420$	0	0
$421$	2.59683	0.126562	0.0632809	−	0.997996i	$-0.479844\pi$
$421$	2.59683	0.126562	0.0632809	+	0.997996i	$0.479844\pi$
$422$	25.0546	1.21964
$423$	0	0
$424$	−2.64612	−0.128507
$425$	−7.78965	−0.377854
$426$	0	0
$427$	3.93275	0.190319
$428$	−2.76376	−0.133591
$429$	0	0
$430$	−13.2224	−0.637639
$431$	−9.94166	−0.478873	−0.239436	−	0.970912i	$-0.576963\pi$
$431$	−9.94166	−0.478873	−0.239436	+	0.970912i	$0.576963\pi$
$432$	0	0
$433$	21.5133	1.03386	0.516931	−	0.856027i	$-0.327074\pi$
$433$	21.5133	1.03386	0.516931	+	0.856027i	$0.327074\pi$
$434$	−0.0499282	−0.00239663
$435$	0	0
$436$	−16.4960	−0.790013
$437$	0	0
$438$	0	0
$439$	−25.3531	−1.21004	−0.605019	−	0.796211i	$-0.706834\pi$
$439$	−25.3531	−1.21004	−0.605019	+	0.796211i	$0.706834\pi$
$440$	2.45843	0.117201
$441$	0	0
$442$	36.6582	1.74365
$443$	6.75984	0.321170	0.160585	−	0.987022i	$-0.448662\pi$
$443$	6.75984	0.321170	0.160585	+	0.987022i	$0.448662\pi$
$444$	0	0
$445$	11.6274	0.551190
$446$	1.78108	0.0843364
$447$	0	0
$448$	0.458044	0.0216406
$449$	−7.80490	−0.368336	−0.184168	−	0.982895i	$-0.558959\pi$
$449$	−7.80490	−0.368336	−0.184168	+	0.982895i	$0.558959\pi$
$450$	0	0
$451$	12.3933	0.583577
$452$	15.3359	0.721342
$453$	0	0
$454$	−20.7093	−0.971938
$455$	5.00792	0.234775
$456$	0	0
$457$	−9.29867	−0.434973	−0.217487	−	0.976063i	$-0.569786\pi$
$457$	−9.29867	−0.434973	−0.217487	+	0.976063i	$0.569786\pi$
$458$	−8.62288	−0.402921
$459$	0	0
$460$	0	0
$461$	17.2422	0.803049	0.401525	−	0.915848i	$-0.368480\pi$
$461$	17.2422	0.803049	0.401525	+	0.915848i	$0.368480\pi$
$462$	0	0
$463$	−9.03520	−0.419901	−0.209951	−	0.977712i	$-0.567330\pi$
$463$	−9.03520	−0.419901	−0.209951	+	0.977712i	$0.567330\pi$
$464$	−5.06465	−0.235120
$465$	0	0
$466$	−4.77637	−0.221261
$467$	−10.5457	−0.487996	−0.243998	−	0.969776i	$-0.578459\pi$
$467$	−10.5457	−0.487996	−0.243998	+	0.969776i	$0.578459\pi$
$468$	0	0
$469$	6.88587	0.317960
$470$	4.89407	0.225747
$471$	0	0
$472$	4.27686	0.196858
$473$	−8.53246	−0.392323
$474$	0	0
$475$	2.52305	0.115766
$476$	2.99761	0.137395
$477$	0	0
$478$	23.9090	1.09357
$479$	−13.1682	−0.601670	−0.300835	−	0.953676i	$-0.597265\pi$
$479$	−13.1682	−0.601670	−0.300835	+	0.953676i	$0.597265\pi$
$480$	0	0
$481$	5.18770	0.236539
$482$	23.0500	1.04990
$483$	0	0
$484$	−9.41356	−0.427889
$485$	−35.6385	−1.61826
$486$	0	0
$487$	−3.03122	−0.137358	−0.0686789	−	0.997639i	$-0.521878\pi$
$487$	−3.03122	−0.137358	−0.0686789	+	0.997639i	$0.521878\pi$
$488$	8.58595	0.388668
$489$	0	0
$490$	−13.2534	−0.598729
$491$	−6.30567	−0.284571	−0.142285	−	0.989826i	$-0.545445\pi$
$491$	−6.30567	−0.284571	−0.142285	+	0.989826i	$0.545445\pi$
$492$	0	0
$493$	−33.1449	−1.49277
$494$	−11.8735	−0.534214
$495$	0	0
$496$	−0.109003	−0.00489438
$497$	0.138991	0.00623459
$498$	0	0
$499$	−6.78493	−0.303735	−0.151868	−	0.988401i	$-0.548529\pi$
$499$	−6.78493	−0.303735	−0.151868	+	0.988401i	$0.548529\pi$
$500$	−12.0825	−0.540346
$501$	0	0
$502$	−0.221255	−0.00987508
$503$	−35.9779	−1.60418	−0.802088	−	0.597206i	$-0.796277\pi$
$503$	−35.9779	−1.60418	−0.802088	+	0.597206i	$0.796277\pi$
$504$	0	0
$505$	−20.8062	−0.925862
$506$	0	0
$507$	0	0
$508$	−21.6633	−0.961155
$509$	−12.0660	−0.534817	−0.267409	−	0.963583i	$-0.586167\pi$
$509$	−12.0660	−0.534817	−0.267409	+	0.963583i	$0.586167\pi$
$510$	0	0
$511$	1.53271	0.0678030
$512$	1.00000	0.0441942
$513$	0	0
$514$	0.544992	0.0240386
$515$	−13.8086	−0.608481
$516$	0	0
$517$	3.15817	0.138896
$518$	0.424208	0.0186386
$519$	0	0
$520$	10.9333	0.479456
$521$	−0.654256	−0.0286635	−0.0143317	−	0.999897i	$-0.504562\pi$
$521$	−0.654256	−0.0286635	−0.0143317	+	0.999897i	$0.504562\pi$
$522$	0	0
$523$	37.4241	1.63644	0.818220	−	0.574905i	$-0.194961\pi$
$523$	37.4241	1.63644	0.818220	+	0.574905i	$0.194961\pi$
$524$	−17.1468	−0.749062
$525$	0	0
$526$	14.7270	0.642127
$527$	−0.713355	−0.0310742
$528$	0	0
$529$	0	0
$530$	−5.16482	−0.224346
$531$	0	0
$532$	−0.970919	−0.0420947
$533$	55.1162	2.38735
$534$	0	0
$535$	−5.39443	−0.233222
$536$	15.0332	0.649335
$537$	0	0
$538$	−1.00716	−0.0434219
$539$	−8.55251	−0.368383
$540$	0	0
$541$	26.4762	1.13830	0.569151	−	0.822233i	$-0.307272\pi$
$541$	26.4762	1.13830	0.569151	+	0.822233i	$0.307272\pi$
$542$	−19.9145	−0.855401
$543$	0	0
$544$	6.54436	0.280587
$545$	−32.1976	−1.37919
$546$	0	0
$547$	−6.74099	−0.288224	−0.144112	−	0.989561i	$-0.546033\pi$
$547$	−6.74099	−0.288224	−0.144112	+	0.989561i	$0.546033\pi$
$548$	−3.10334	−0.132568
$549$	0	0
$550$	−1.49921	−0.0639265
$551$	10.7356	0.457350
$552$	0	0
$553$	3.68607	0.156748
$554$	−24.4934	−1.04063
$555$	0	0
$556$	−22.4485	−0.952028
$557$	−18.8827	−0.800087	−0.400043	−	0.916496i	$-0.631005\pi$
$557$	−18.8827	−0.800087	−0.400043	+	0.916496i	$0.631005\pi$
$558$	0	0
$559$	−37.9461	−1.60495
$560$	0.894034	0.0377798
$561$	0	0
$562$	22.2104	0.936888
$563$	−3.81476	−0.160773	−0.0803864	−	0.996764i	$-0.525615\pi$
$563$	−3.81476	−0.160773	−0.0803864	+	0.996764i	$0.525615\pi$
$564$	0	0
$565$	29.9335	1.25931
$566$	−16.4220	−0.690267
$567$	0	0
$568$	0.303444	0.0127322
$569$	−18.6062	−0.780014	−0.390007	−	0.920812i	$-0.627527\pi$
$569$	−18.6062	−0.780014	−0.390007	+	0.920812i	$0.627527\pi$
$570$	0	0
$571$	42.9221	1.79623	0.898117	−	0.439757i	$-0.144935\pi$
$571$	42.9221	1.79623	0.898117	+	0.439757i	$0.144935\pi$
$572$	7.05529	0.294997
$573$	0	0
$574$	4.50695	0.188117
$575$	0	0
$576$	0	0
$577$	−11.4723	−0.477599	−0.238799	−	0.971069i	$-0.576754\pi$
$577$	−11.4723	−0.477599	−0.238799	+	0.971069i	$0.576754\pi$
$578$	25.8287	1.07433
$579$	0	0
$580$	−9.88543	−0.410470
$581$	4.80582	0.199379
$582$	0	0
$583$	−3.33288	−0.138034
$584$	3.34620	0.138467
$585$	0	0
$586$	24.4969	1.01196
$587$	−11.0101	−0.454436	−0.227218	−	0.973844i	$-0.572963\pi$
$587$	−11.0101	−0.454436	−0.227218	+	0.973844i	$0.572963\pi$
$588$	0	0
$589$	0.231054	0.00952041
$590$	8.34778	0.343673
$591$	0	0
$592$	0.926128	0.0380636
$593$	−17.4334	−0.715904	−0.357952	−	0.933740i	$-0.616525\pi$
$593$	−17.4334	−0.715904	−0.357952	+	0.933740i	$0.616525\pi$
$594$	0	0
$595$	5.85088	0.239863
$596$	15.7100	0.643507
$597$	0	0
$598$	0	0
$599$	11.8571	0.484470	0.242235	−	0.970218i	$-0.422120\pi$
$599$	11.8571	0.484470	0.242235	+	0.970218i	$0.422120\pi$
$600$	0	0
$601$	5.00627	0.204210	0.102105	−	0.994774i	$-0.467442\pi$
$601$	5.00627	0.204210	0.102105	+	0.994774i	$0.467442\pi$
$602$	−3.10292	−0.126466
$603$	0	0
$604$	−5.57516	−0.226850
$605$	−18.3739	−0.747004
$606$	0	0
$607$	31.9481	1.29673	0.648367	−	0.761328i	$-0.275452\pi$
$607$	31.9481	1.29673	0.648367	+	0.761328i	$0.275452\pi$
$608$	−2.11970	−0.0859654
$609$	0	0
$610$	16.7585	0.678531
$611$	14.0452	0.568208
$612$	0	0
$613$	30.3307	1.22504	0.612522	−	0.790454i	$-0.290155\pi$
$613$	30.3307	1.22504	0.612522	+	0.790454i	$0.290155\pi$
$614$	14.6749	0.592229
$615$	0	0
$616$	0.576924	0.0232449
$617$	10.9864	0.442294	0.221147	−	0.975241i	$-0.429020\pi$
$617$	10.9864	0.442294	0.221147	+	0.975241i	$0.429020\pi$
$618$	0	0
$619$	11.7786	0.473422	0.236711	−	0.971580i	$-0.423930\pi$
$619$	11.7786	0.473422	0.236711	+	0.971580i	$0.423930\pi$
$620$	−0.212757	−0.00854454
$621$	0	0
$622$	−33.2393	−1.33277
$623$	2.72862	0.109320
$624$	0	0
$625$	−17.6318	−0.705272
$626$	20.4935	0.819085
$627$	0	0
$628$	8.16701	0.325899
$629$	6.06092	0.241665
$630$	0	0
$631$	11.6774	0.464870	0.232435	−	0.972612i	$-0.425331\pi$
$631$	11.6774	0.464870	0.232435	+	0.972612i	$0.425331\pi$
$632$	8.04741	0.320109
$633$	0	0
$634$	−22.9004	−0.909492
$635$	−42.2836	−1.67797
$636$	0	0
$637$	−38.0352	−1.50701
$638$	−6.37911	−0.252552
$639$	0	0
$640$	1.95185	0.0771536
$641$	−4.70626	−0.185886	−0.0929431	−	0.995671i	$-0.529627\pi$
$641$	−4.70626	−0.185886	−0.0929431	+	0.995671i	$0.529627\pi$
$642$	0	0
$643$	−38.9574	−1.53633	−0.768165	−	0.640252i	$-0.778830\pi$
$643$	−38.9574	−1.53633	−0.768165	+	0.640252i	$0.778830\pi$
$644$	0	0
$645$	0	0
$646$	−13.8721	−0.545791
$647$	−42.8026	−1.68274	−0.841371	−	0.540457i	$-0.818251\pi$
$647$	−42.8026	−1.68274	−0.841371	+	0.540457i	$0.818251\pi$
$648$	0	0
$649$	5.38686	0.211453
$650$	−6.66737	−0.261516
$651$	0	0
$652$	−17.0615	−0.668181
$653$	21.2091	0.829976	0.414988	−	0.909827i	$-0.363786\pi$
$653$	21.2091	0.829976	0.414988	+	0.909827i	$0.363786\pi$
$654$	0	0
$655$	−33.4680	−1.30770
$656$	9.83955	0.384170
$657$	0	0
$658$	1.14850	0.0447733
$659$	−10.2226	−0.398215	−0.199107	−	0.979978i	$-0.563804\pi$
$659$	−10.2226	−0.398215	−0.199107	+	0.979978i	$0.563804\pi$
$660$	0	0
$661$	−39.9796	−1.55503	−0.777513	−	0.628867i	$-0.783519\pi$
$661$	−39.9796	−1.55503	−0.777513	+	0.628867i	$0.783519\pi$
$662$	−10.2762	−0.399396
$663$	0	0
$664$	10.4920	0.407170
$665$	−1.89509	−0.0734883
$666$	0	0
$667$	0	0
$668$	−18.0693	−0.699123
$669$	0	0
$670$	29.3425	1.13360
$671$	10.8143	0.417483
$672$	0	0
$673$	−6.90220	−0.266060	−0.133030	−	0.991112i	$-0.542471\pi$
$673$	−6.90220	−0.266060	−0.133030	+	0.991112i	$0.542471\pi$
$674$	−11.6162	−0.447438
$675$	0	0
$676$	18.3767	0.706797
$677$	17.8035	0.684245	0.342122	−	0.939655i	$-0.388854\pi$
$677$	17.8035	0.684245	0.342122	+	0.939655i	$0.388854\pi$
$678$	0	0
$679$	−8.36337	−0.320957
$680$	12.7736	0.489846
$681$	0	0
$682$	−0.137293	−0.00525723
$683$	51.9690	1.98854	0.994268	−	0.106913i	$-0.0340966\pi$
$683$	51.9690	1.98854	0.994268	+	0.106913i	$0.0340966\pi$
$684$	0	0
$685$	−6.05725	−0.231435
$686$	−6.31652	−0.241166
$687$	0	0
$688$	−6.77428	−0.258267
$689$	−14.8222	−0.564681
$690$	0	0
$691$	−26.8796	−1.02255	−0.511274	−	0.859418i	$-0.670826\pi$
$691$	−26.8796	−1.02255	−0.511274	+	0.859418i	$0.670826\pi$
$692$	15.9612	0.606752
$693$	0	0
$694$	12.8984	0.489618
$695$	−43.8161	−1.66204
$696$	0	0
$697$	64.3936	2.43908
$698$	15.8619	0.600383
$699$	0	0
$700$	−0.545203	−0.0206067
$701$	5.09731	0.192523	0.0962614	−	0.995356i	$-0.469312\pi$
$701$	5.09731	0.192523	0.0962614	+	0.995356i	$0.469312\pi$
$702$	0	0
$703$	−1.96312	−0.0740404
$704$	1.25954	0.0474706
$705$	0	0
$706$	1.79503	0.0675567
$707$	−4.88262	−0.183630
$708$	0	0
$709$	−0.828261	−0.0311060	−0.0155530	−	0.999879i	$-0.504951\pi$
$709$	−0.828261	−0.0311060	−0.0155530	+	0.999879i	$0.504951\pi$
$710$	0.592277	0.0222277
$711$	0	0
$712$	5.95710	0.223252
$713$	0	0
$714$	0	0
$715$	13.7709	0.515001
$716$	15.6916	0.586421
$717$	0	0
$718$	19.0479	0.710862
$719$	−4.75003	−0.177146	−0.0885731	−	0.996070i	$-0.528231\pi$
$719$	−4.75003	−0.177146	−0.0885731	+	0.996070i	$0.528231\pi$
$720$	0	0
$721$	−3.24050	−0.120683
$722$	−14.5069	−0.539889
$723$	0	0
$724$	1.81930	0.0676139
$725$	6.02837	0.223888
$726$	0	0
$727$	31.3593	1.16305	0.581526	−	0.813528i	$-0.302456\pi$
$727$	31.3593	1.16305	0.581526	+	0.813528i	$0.302456\pi$
$728$	2.56573	0.0950924
$729$	0	0
$730$	6.53128	0.241733
$731$	−44.3333	−1.63973
$732$	0	0
$733$	−11.3462	−0.419081	−0.209541	−	0.977800i	$-0.567197\pi$
$733$	−11.3462	−0.419081	−0.209541	+	0.977800i	$0.567197\pi$
$734$	3.88864	0.143532
$735$	0	0
$736$	0	0
$737$	18.9349	0.697475
$738$	0	0
$739$	27.4253	1.00885	0.504427	−	0.863454i	$-0.331704\pi$
$739$	27.4253	1.00885	0.504427	+	0.863454i	$0.331704\pi$
$740$	1.80766	0.0664510
$741$	0	0
$742$	−1.21204	−0.0444954
$743$	−52.7833	−1.93643	−0.968216	−	0.250117i	$-0.919531\pi$
$743$	−52.7833	−1.93643	−0.968216	+	0.250117i	$0.919531\pi$
$744$	0	0
$745$	30.6636	1.12343
$746$	31.2555	1.14435
$747$	0	0
$748$	8.24287	0.301389
$749$	−1.26592	−0.0462558
$750$	0	0
$751$	−17.4346	−0.636197	−0.318098	−	0.948058i	$-0.603044\pi$
$751$	−17.4346	−0.636197	−0.318098	+	0.948058i	$0.603044\pi$
$752$	2.50740	0.0914356
$753$	0	0
$754$	−28.3696	−1.03316
$755$	−10.8819	−0.396032
$756$	0	0
$757$	−18.1347	−0.659118	−0.329559	−	0.944135i	$-0.606900\pi$
$757$	−18.1347	−0.659118	−0.329559	+	0.944135i	$0.606900\pi$
$758$	7.32134	0.265923
$759$	0	0
$760$	−4.13734	−0.150077
$761$	−4.37269	−0.158510	−0.0792550	−	0.996854i	$-0.525254\pi$
$761$	−4.37269	−0.158510	−0.0792550	+	0.996854i	$0.525254\pi$
$762$	0	0
$763$	−7.55588	−0.273541
$764$	4.24849	0.153705
$765$	0	0
$766$	19.2426	0.695265
$767$	23.9568	0.865030
$768$	0	0
$769$	−27.7635	−1.00118	−0.500588	−	0.865686i	$-0.666883\pi$
$769$	−27.7635	−1.00118	−0.500588	+	0.865686i	$0.666883\pi$
$770$	1.12607	0.0405807
$771$	0	0
$772$	13.1121	0.471914
$773$	0.160762	0.00578221	0.00289111	−	0.999996i	$-0.499080\pi$
$773$	0.160762	0.00578221	0.00289111	+	0.999996i	$0.499080\pi$
$774$	0	0
$775$	0.129745	0.00466056
$776$	−18.2589	−0.655455
$777$	0	0
$778$	8.25813	0.296068
$779$	−20.8569	−0.747278
$780$	0	0
$781$	0.382199	0.0136762
$782$	0	0
$783$	0	0
$784$	−6.79020	−0.242507
$785$	15.9408	0.568951
$786$	0	0
$787$	−37.6412	−1.34176	−0.670882	−	0.741564i	$-0.734084\pi$
$787$	−37.6412	−1.34176	−0.670882	+	0.741564i	$0.734084\pi$
$788$	9.92381	0.353521
$789$	0	0
$790$	15.7073	0.558842
$791$	7.02454	0.249764
$792$	0	0
$793$	48.0941	1.70787
$794$	−11.3695	−0.403489
$795$	0	0
$796$	−6.30555	−0.223494
$797$	−25.8591	−0.915976	−0.457988	−	0.888958i	$-0.651430\pi$
$797$	−25.8591	−0.915976	−0.457988	+	0.888958i	$0.651430\pi$
$798$	0	0
$799$	16.4094	0.580521
$800$	−1.19028	−0.0420829
$801$	0	0
$802$	15.2135	0.537209
$803$	4.21467	0.148732
$804$	0	0
$805$	0	0
$806$	−0.610579	−0.0215067
$807$	0	0
$808$	−10.6597	−0.375007
$809$	11.4089	0.401115	0.200558	−	0.979682i	$-0.435725\pi$
$809$	11.4089	0.401115	0.200558	+	0.979682i	$0.435725\pi$
$810$	0	0
$811$	33.0446	1.16035	0.580176	−	0.814491i	$-0.302984\pi$
$811$	33.0446	1.16035	0.580176	+	0.814491i	$0.302984\pi$
$812$	−2.31983	−0.0814102
$813$	0	0
$814$	1.16649	0.0408856
$815$	−33.3015	−1.16650
$816$	0	0
$817$	14.3595	0.502374
$818$	1.53365	0.0536227
$819$	0	0
$820$	19.2053	0.670679
$821$	5.57488	0.194565	0.0972824	−	0.995257i	$-0.468985\pi$
$821$	5.57488	0.194565	0.0972824	+	0.995257i	$0.468985\pi$
$822$	0	0
$823$	33.5946	1.17104	0.585518	−	0.810660i	$-0.300891\pi$
$823$	33.5946	1.17104	0.585518	+	0.810660i	$0.300891\pi$
$824$	−7.07465	−0.246457
$825$	0	0
$826$	1.95899	0.0681620
$827$	−50.3483	−1.75078	−0.875391	−	0.483416i	$-0.839396\pi$
$827$	−50.3483	−1.75078	−0.875391	+	0.483416i	$0.839396\pi$
$828$	0	0
$829$	43.2992	1.50384	0.751922	−	0.659252i	$-0.229127\pi$
$829$	43.2992	1.50384	0.751922	+	0.659252i	$0.229127\pi$
$830$	20.4789	0.710832
$831$	0	0
$832$	5.60149	0.194197
$833$	−44.4375	−1.53967
$834$	0	0
$835$	−35.2686	−1.22052
$836$	−2.66985	−0.0923386
$837$	0	0
$838$	4.63094	0.159973
$839$	47.6218	1.64409	0.822044	−	0.569425i	$-0.192834\pi$
$839$	47.6218	1.64409	0.822044	+	0.569425i	$0.192834\pi$
$840$	0	0
$841$	−3.34936	−0.115495
$842$	2.59683	0.0894927
$843$	0	0
$844$	25.0546	0.862415
$845$	35.8686	1.23392
$846$	0	0
$847$	−4.31183	−0.148156
$848$	−2.64612	−0.0908680
$849$	0	0
$850$	−7.78965	−0.267183
$851$	0	0
$852$	0	0
$853$	−48.8622	−1.67301	−0.836504	−	0.547960i	$-0.815404\pi$
$853$	−48.8622	−1.67301	−0.836504	+	0.547960i	$0.815404\pi$
$854$	3.93275	0.134576
$855$	0	0
$856$	−2.76376	−0.0944632
$857$	−1.52668	−0.0521505	−0.0260752	−	0.999660i	$-0.508301\pi$
$857$	−1.52668	−0.0521505	−0.0260752	+	0.999660i	$0.508301\pi$
$858$	0	0
$859$	−1.19558	−0.0407928	−0.0203964	−	0.999792i	$-0.506493\pi$
$859$	−1.19558	−0.0407928	−0.0203964	+	0.999792i	$0.506493\pi$
$860$	−13.2224	−0.450879
$861$	0	0
$862$	−9.94166	−0.338614
$863$	−49.0154	−1.66851	−0.834253	−	0.551383i	$-0.814100\pi$
$863$	−49.0154	−1.66851	−0.834253	+	0.551383i	$0.814100\pi$
$864$	0	0
$865$	31.1538	1.05926
$866$	21.5133	0.731051
$867$	0	0
$868$	−0.0499282	−0.00169467
$869$	10.1360	0.343841
$870$	0	0
$871$	84.2083	2.85329
$872$	−16.4960	−0.558624
$873$	0	0
$874$	0	0
$875$	−5.53432	−0.187094
$876$	0	0
$877$	1.08702	0.0367061	0.0183531	−	0.999832i	$-0.494158\pi$
$877$	1.08702	0.0367061	0.0183531	+	0.999832i	$0.494158\pi$
$878$	−25.3531	−0.855626
$879$	0	0
$880$	2.45843	0.0828736
$881$	−44.7765	−1.50856	−0.754279	−	0.656555i	$-0.772013\pi$
$881$	−44.7765	−1.50856	−0.754279	+	0.656555i	$0.772013\pi$
$882$	0	0
$883$	−48.4265	−1.62968	−0.814841	−	0.579685i	$-0.803176\pi$
$883$	−48.4265	−1.62968	−0.814841	+	0.579685i	$0.803176\pi$
$884$	36.6582	1.23295
$885$	0	0
$886$	6.75984	0.227101
$887$	15.0113	0.504029	0.252015	−	0.967723i	$-0.418907\pi$
$887$	15.0113	0.504029	0.252015	+	0.967723i	$0.418907\pi$
$888$	0	0
$889$	−9.92277	−0.332799
$890$	11.6274	0.389750
$891$	0	0
$892$	1.78108	0.0596348
$893$	−5.31495	−0.177858
$894$	0	0
$895$	30.6276	1.02377
$896$	0.458044	0.0153022
$897$	0	0
$898$	−7.80490	−0.260453
$899$	0.552061	0.0184123
$900$	0	0
$901$	−17.3171	−0.576918
$902$	12.3933	0.412651
$903$	0	0
$904$	15.3359	0.510066
$905$	3.55101	0.118039
$906$	0	0
$907$	23.6578	0.785545	0.392773	−	0.919636i	$-0.371516\pi$
$907$	23.6578	0.785545	0.392773	+	0.919636i	$0.371516\pi$
$908$	−20.7093	−0.687264
$909$	0	0
$910$	5.00792	0.166011
$911$	−29.2129	−0.967866	−0.483933	−	0.875105i	$-0.660792\pi$
$911$	−29.2129	−0.967866	−0.483933	+	0.875105i	$0.660792\pi$
$912$	0	0
$913$	13.2151	0.437357
$914$	−9.29867	−0.307573
$915$	0	0
$916$	−8.62288	−0.284908
$917$	−7.85400	−0.259362
$918$	0	0
$919$	12.1150	0.399637	0.199818	−	0.979833i	$-0.435965\pi$
$919$	12.1150	0.399637	0.199818	+	0.979833i	$0.435965\pi$
$920$	0	0
$921$	0	0
$922$	17.2422	0.567842
$923$	1.69974	0.0559476
$924$	0	0
$925$	−1.10236	−0.0362452
$926$	−9.03520	−0.296915
$927$	0	0
$928$	−5.06465	−0.166255
$929$	10.1569	0.333236	0.166618	−	0.986022i	$-0.446715\pi$
$929$	10.1569	0.333236	0.166618	+	0.986022i	$0.446715\pi$
$930$	0	0
$931$	14.3932	0.471718
$932$	−4.77637	−0.156455
$933$	0	0
$934$	−10.5457	−0.345065
$935$	16.0888	0.526161
$936$	0	0
$937$	21.9518	0.717133	0.358567	−	0.933504i	$-0.383266\pi$
$937$	21.9518	0.717133	0.358567	+	0.933504i	$0.383266\pi$
$938$	6.88587	0.224832
$939$	0	0
$940$	4.89407	0.159627
$941$	−37.9048	−1.23566	−0.617831	−	0.786311i	$-0.711988\pi$
$941$	−37.9048	−1.23566	−0.617831	+	0.786311i	$0.711988\pi$
$942$	0	0
$943$	0	0
$944$	4.27686	0.139200
$945$	0	0
$946$	−8.53246	−0.277414
$947$	−34.5347	−1.12223	−0.561114	−	0.827739i	$-0.689627\pi$
$947$	−34.5347	−1.12223	−0.561114	+	0.827739i	$0.689627\pi$
$948$	0	0
$949$	18.7437	0.608447
$950$	2.52305	0.0818586
$951$	0	0
$952$	2.99761	0.0971531
$953$	−46.1294	−1.49428	−0.747139	−	0.664668i	$-0.768573\pi$
$953$	−46.1294	−1.49428	−0.747139	+	0.664668i	$0.768573\pi$
$954$	0	0
$955$	8.29240	0.268336
$956$	23.9090	0.773273
$957$	0	0
$958$	−13.1682	−0.425445
$959$	−1.42147	−0.0459015
$960$	0	0
$961$	−30.9881	−0.999617
$962$	5.18770	0.167258
$963$	0	0
$964$	23.0500	0.742389
$965$	25.5928	0.823861
$966$	0	0
$967$	32.2668	1.03763	0.518815	−	0.854886i	$-0.326373\pi$
$967$	32.2668	1.03763	0.518815	+	0.854886i	$0.326373\pi$
$968$	−9.41356	−0.302563
$969$	0	0
$970$	−35.6385	−1.14428
$971$	−30.5548	−0.980550	−0.490275	−	0.871568i	$-0.663104\pi$
$971$	−30.5548	−0.980550	−0.490275	+	0.871568i	$0.663104\pi$
$972$	0	0
$973$	−10.2824	−0.329639
$974$	−3.03122	−0.0971266
$975$	0	0
$976$	8.58595	0.274830
$977$	−24.3056	−0.777604	−0.388802	−	0.921321i	$-0.627111\pi$
$977$	−24.3056	−0.777604	−0.388802	+	0.921321i	$0.627111\pi$
$978$	0	0
$979$	7.50319	0.239803
$980$	−13.2534	−0.423366
$981$	0	0
$982$	−6.30567	−0.201222
$983$	−58.2648	−1.85836	−0.929180	−	0.369628i	$-0.879485\pi$
$983$	−58.2648	−1.85836	−0.929180	+	0.369628i	$0.879485\pi$
$984$	0	0
$985$	19.3698	0.617172
$986$	−33.1449	−1.05555
$987$	0	0
$988$	−11.8735	−0.377747
$989$	0	0
$990$	0	0
$991$	35.2587	1.12003	0.560014	−	0.828483i	$-0.310796\pi$
$991$	35.2587	1.12003	0.560014	+	0.828483i	$0.310796\pi$
$992$	−0.109003	−0.00346085
$993$	0	0
$994$	0.138991	0.00440852
$995$	−12.3075	−0.390173
$996$	0	0
$997$	−44.2899	−1.40268	−0.701338	−	0.712829i	$-0.747413\pi$
$997$	−44.2899	−1.40268	−0.701338	+	0.712829i	$0.747413\pi$
$998$	−6.78493	−0.214773
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.ca.1.2		5
3.2	odd	2		3174.2.a.y.1.4		5
23.17	odd	22		414.2.i.b.289.1		10
23.19	odd	22		414.2.i.b.361.1		10
23.22	odd	2		9522.2.a.bv.1.4		5
69.17	even	22		138.2.e.c.13.1	✓	10
69.65	even	22		138.2.e.c.85.1	yes	10
69.68	even	2		3174.2.a.z.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.c.13.1	✓	10	69.17	even	22
138.2.e.c.85.1	yes	10	69.65	even	22
414.2.i.b.289.1		10	23.17	odd	22
414.2.i.b.361.1		10	23.19	odd	22
3174.2.a.y.1.4		5	3.2	odd	2
3174.2.a.z.1.2		5	69.68	even	2
9522.2.a.bv.1.4		5	23.22	odd	2
9522.2.a.ca.1.2		5	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:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.95185 q^{5} +0.458044 q^{7} +1.00000 q^{8} +1.95185 q^{10} +1.25954 q^{11} +5.60149 q^{13} +0.458044 q^{14} +1.00000 q^{16} +6.54436 q^{17} -2.11970 q^{19} +1.95185 q^{20} +1.25954 q^{22} -1.19028 q^{25} +5.60149 q^{26} +0.458044 q^{28} -5.06465 q^{29} -0.109003 q^{31} +1.00000 q^{32} +6.54436 q^{34} +0.894034 q^{35} +0.926128 q^{37} -2.11970 q^{38} +1.95185 q^{40} +9.83955 q^{41} -6.77428 q^{43} +1.25954 q^{44} +2.50740 q^{47} -6.79020 q^{49} -1.19028 q^{50} +5.60149 q^{52} -2.64612 q^{53} +2.45843 q^{55} +0.458044 q^{56} -5.06465 q^{58} +4.27686 q^{59} +8.58595 q^{61} -0.109003 q^{62} +1.00000 q^{64} +10.9333 q^{65} +15.0332 q^{67} +6.54436 q^{68} +0.894034 q^{70} +0.303444 q^{71} +3.34620 q^{73} +0.926128 q^{74} -2.11970 q^{76} +0.576924 q^{77} +8.04741 q^{79} +1.95185 q^{80} +9.83955 q^{82} +10.4920 q^{83} +12.7736 q^{85} -6.77428 q^{86} +1.25954 q^{88} +5.95710 q^{89} +2.56573 q^{91} +2.50740 q^{94} -4.13734 q^{95} -18.2589 q^{97} -6.79020 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} + 11 q^{5} + q^{7} + 5 q^{8} + 11 q^{10} + 11 q^{11} + 10 q^{13} + q^{14} + 5 q^{16} + 11 q^{17} + q^{19} + 11 q^{20} + 11 q^{22} + 30 q^{25} + 10 q^{26} + q^{28} - 3 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 + 11 * q^5 + q^7 + 5 * q^8 + 11 * q^10 + 11 * q^11 + 10 * q^13 + q^14 + 5 * q^16 + 11 * q^17 + q^19 + 11 * q^20 + 11 * q^22 + 30 * q^25 + 10 * q^26 + q^28 - 3 * q^29 + 2 * q^31 + 5 * q^32 + 11 * q^34 + 11 * q^35 - 16 * q^37 + q^38 + 11 * q^40 + 8 * q^41 + 12 * q^43 + 11 * q^44 - 4 * q^49 + 30 * q^50 + 10 * q^52 + 27 * q^53 + q^56 - 3 * q^58 + 6 * q^59 - 5 * q^61 + 2 * q^62 + 5 * q^64 + 22 * q^65 + 7 * q^67 + 11 * q^68 + 11 * q^70 + 11 * q^71 - 12 * q^73 - 16 * q^74 + q^76 - 11 * q^77 - 16 * q^79 + 11 * q^80 + 8 * q^82 + 22 * q^83 - 11 * q^85 + 12 * q^86 + 11 * q^88 + 38 * q^89 - 9 * q^91 - 11 * q^95 - 5 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more