LMFDB - Embedded newform 9522.2.a.bq.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9522,2,Mod(1,9522)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{5}]$
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			1.4
Root			$1.30972$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.23648 q^{5} +1.47889 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.23648 q^{5} +1.47889 q^{7} -1.00000 q^{8} +1.23648 q^{10} -0.886752 q^{11} +0.0673089 q^{13} -1.47889 q^{14} +1.00000 q^{16} -4.29177 q^{17} +4.66759 q^{19} -1.23648 q^{20} +0.886752 q^{22} -3.47112 q^{25} -0.0673089 q^{26} +1.47889 q^{28} +7.67816 q^{29} -1.59145 q^{31} -1.00000 q^{32} +4.29177 q^{34} -1.82862 q^{35} +5.22871 q^{37} -4.66759 q^{38} +1.23648 q^{40} +0.135933 q^{41} -6.57798 q^{43} -0.886752 q^{44} -8.31271 q^{47} -4.81288 q^{49} +3.47112 q^{50} +0.0673089 q^{52} -2.99558 q^{53} +1.09645 q^{55} -1.47889 q^{56} -7.67816 q^{58} +6.83576 q^{59} +4.60015 q^{61} +1.59145 q^{62} +1.00000 q^{64} -0.0832260 q^{65} -12.4051 q^{67} -4.29177 q^{68} +1.82862 q^{70} +7.31821 q^{71} +13.1978 q^{73} -5.22871 q^{74} +4.66759 q^{76} -1.31141 q^{77} -7.89446 q^{79} -1.23648 q^{80} -0.135933 q^{82} -8.40858 q^{83} +5.30668 q^{85} +6.57798 q^{86} +0.886752 q^{88} +11.7235 q^{89} +0.0995426 q^{91} +8.31271 q^{94} -5.77138 q^{95} +17.1665 q^{97} +4.81288 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} - 7 q^{5} + 7 q^{7} - 5 q^{8}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 - 7 * q^5 + 7 * q^7 - 5 * q^8	$ 5 q - 5 q^{2} + 5 q^{4} - 7 q^{5} + 7 q^{7} - 5 q^{8} + 7 q^{10} - 13 q^{11} - 4 q^{13} - 7 q^{14} + 5 q^{16} - 9 q^{17} + 11 q^{19} - 7 q^{20} + 13 q^{22} - 2 q^{25} + 4 q^{26} + 7 q^{28} + 7 q^{29} - 8 q^{31} - 5 q^{32} + 9 q^{34} - q^{35} + 12 q^{37} - 11 q^{38} + 7 q^{40} + 10 q^{41} + 4 q^{43} - 13 q^{44} + 24 q^{47} - 12 q^{49} + 2 q^{50} - 4 q^{52} - 9 q^{53} + 16 q^{55} - 7 q^{56} - 7 q^{58} + 14 q^{59} + 5 q^{61} + 8 q^{62} + 5 q^{64} - 12 q^{65} + 13 q^{67} - 9 q^{68} + q^{70} + 19 q^{71} + 4 q^{73} - 12 q^{74} + 11 q^{76} - 5 q^{77} + 4 q^{79} - 7 q^{80} - 10 q^{82} - 24 q^{83} + 17 q^{85} - 4 q^{86} + 13 q^{88} - 4 q^{89} - 21 q^{91} - 24 q^{94} + 11 q^{95} - 9 q^{97} + 12 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 - 7 * q^5 + 7 * q^7 - 5 * q^8 + 7 * q^10 - 13 * q^11 - 4 * q^13 - 7 * q^14 + 5 * q^16 - 9 * q^17 + 11 * q^19 - 7 * q^20 + 13 * q^22 - 2 * q^25 + 4 * q^26 + 7 * q^28 + 7 * q^29 - 8 * q^31 - 5 * q^32 + 9 * q^34 - q^35 + 12 * q^37 - 11 * q^38 + 7 * q^40 + 10 * q^41 + 4 * q^43 - 13 * q^44 + 24 * q^47 - 12 * q^49 + 2 * q^50 - 4 * q^52 - 9 * q^53 + 16 * q^55 - 7 * q^56 - 7 * q^58 + 14 * q^59 + 5 * q^61 + 8 * q^62 + 5 * q^64 - 12 * q^65 + 13 * q^67 - 9 * q^68 + q^70 + 19 * q^71 + 4 * q^73 - 12 * q^74 + 11 * q^76 - 5 * q^77 + 4 * q^79 - 7 * q^80 - 10 * q^82 - 24 * q^83 + 17 * q^85 - 4 * q^86 + 13 * q^88 - 4 * q^89 - 21 * q^91 - 24 * q^94 + 11 * q^95 - 9 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

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
$5$	−1.23648	−0.552970	−0.276485	−	0.961018i	$-0.589170\pi$
$5$	−1.23648	−0.552970	−0.276485	+	0.961018i	$0.589170\pi$
$6$	0	0
$7$	1.47889	0.558968	0.279484	−	0.960150i	$-0.409837\pi$
$7$	1.47889	0.558968	0.279484	+	0.960150i	$0.409837\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.23648	0.391009
$11$	−0.886752	−0.267366	−0.133683	−	0.991024i	$-0.542680\pi$
$11$	−0.886752	−0.267366	−0.133683	+	0.991024i	$0.542680\pi$
$12$	0	0
$13$	0.0673089	0.0186681	0.00933407	−	0.999956i	$-0.497029\pi$
$13$	0.0673089	0.0186681	0.00933407	+	0.999956i	$0.497029\pi$
$14$	−1.47889	−0.395250
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.29177	−1.04091	−0.520454	−	0.853890i	$-0.674237\pi$
$17$	−4.29177	−1.04091	−0.520454	+	0.853890i	$0.674237\pi$
$18$	0	0
$19$	4.66759	1.07082	0.535410	−	0.844592i	$-0.320157\pi$
$19$	4.66759	1.07082	0.535410	+	0.844592i	$0.320157\pi$
$20$	−1.23648	−0.276485
$21$	0	0
$22$	0.886752	0.189056
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.47112	−0.694224
$26$	−0.0673089	−0.0132004
$27$	0	0
$28$	1.47889	0.279484
$29$	7.67816	1.42580	0.712899	−	0.701267i	$-0.247382\pi$
$29$	7.67816	1.42580	0.712899	+	0.701267i	$0.247382\pi$
$30$	0	0
$31$	−1.59145	−0.285834	−0.142917	−	0.989735i	$-0.545648\pi$
$31$	−1.59145	−0.285834	−0.142917	+	0.989735i	$0.545648\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.29177	0.736033
$35$	−1.82862	−0.309093
$36$	0	0
$37$	5.22871	0.859594	0.429797	−	0.902925i	$-0.358585\pi$
$37$	5.22871	0.859594	0.429797	+	0.902925i	$0.358585\pi$
$38$	−4.66759	−0.757184
$39$	0	0
$40$	1.23648	0.195504
$41$	0.135933	0.0212291	0.0106145	−	0.999944i	$-0.496621\pi$
$41$	0.135933	0.0212291	0.0106145	+	0.999944i	$0.496621\pi$
$42$	0	0
$43$	−6.57798	−1.00313	−0.501567	−	0.865119i	$-0.667243\pi$
$43$	−6.57798	−1.00313	−0.501567	+	0.865119i	$0.667243\pi$
$44$	−0.886752	−0.133683
$45$	0	0
$46$	0	0
$47$	−8.31271	−1.21253	−0.606266	−	0.795262i	$-0.707334\pi$
$47$	−8.31271	−1.21253	−0.606266	+	0.795262i	$0.707334\pi$
$48$	0	0
$49$	−4.81288	−0.687554
$50$	3.47112	0.490890
$51$	0	0
$52$	0.0673089	0.00933407
$53$	−2.99558	−0.411474	−0.205737	−	0.978607i	$-0.565959\pi$
$53$	−2.99558	−0.411474	−0.205737	+	0.978607i	$0.565959\pi$
$54$	0	0
$55$	1.09645	0.147845
$56$	−1.47889	−0.197625
$57$	0	0
$58$	−7.67816	−1.00819
$59$	6.83576	0.889940	0.444970	−	0.895545i	$-0.353214\pi$
$59$	6.83576	0.889940	0.444970	+	0.895545i	$0.353214\pi$
$60$	0	0
$61$	4.60015	0.588988	0.294494	−	0.955653i	$-0.404849\pi$
$61$	4.60015	0.588988	0.294494	+	0.955653i	$0.404849\pi$
$62$	1.59145	0.202115
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.0832260	−0.0103229
$66$	0	0
$67$	−12.4051	−1.51553	−0.757763	−	0.652530i	$-0.773707\pi$
$67$	−12.4051	−1.51553	−0.757763	+	0.652530i	$0.773707\pi$
$68$	−4.29177	−0.520454
$69$	0	0
$70$	1.82862	0.218562
$71$	7.31821	0.868512	0.434256	−	0.900790i	$-0.357011\pi$
$71$	7.31821	0.868512	0.434256	+	0.900790i	$0.357011\pi$
$72$	0	0
$73$	13.1978	1.54469	0.772343	−	0.635206i	$-0.219085\pi$
$73$	13.1978	1.54469	0.772343	+	0.635206i	$0.219085\pi$
$74$	−5.22871	−0.607825
$75$	0	0
$76$	4.66759	0.535410
$77$	−1.31141	−0.149449
$78$	0	0
$79$	−7.89446	−0.888197	−0.444098	−	0.895978i	$-0.646476\pi$
$79$	−7.89446	−0.888197	−0.444098	+	0.895978i	$0.646476\pi$
$80$	−1.23648	−0.138243
$81$	0	0
$82$	−0.135933	−0.0150112
$83$	−8.40858	−0.922961	−0.461481	−	0.887150i	$-0.652682\pi$
$83$	−8.40858	−0.922961	−0.461481	+	0.887150i	$0.652682\pi$
$84$	0	0
$85$	5.30668	0.575591
$86$	6.57798	0.709322
$87$	0	0
$88$	0.886752	0.0945281
$89$	11.7235	1.24268	0.621342	−	0.783539i	$-0.286588\pi$
$89$	11.7235	1.24268	0.621342	+	0.783539i	$0.286588\pi$
$90$	0	0
$91$	0.0995426	0.0104349
$92$	0	0
$93$	0	0
$94$	8.31271	0.857390
$95$	−5.77138	−0.592131
$96$	0	0
$97$	17.1665	1.74299	0.871496	−	0.490402i	$-0.163150\pi$
$97$	17.1665	1.74299	0.871496	+	0.490402i	$0.163150\pi$
$98$	4.81288	0.486174
$99$	0	0
$100$	−3.47112	−0.347112
$101$	−5.98712	−0.595741	−0.297870	−	0.954606i	$-0.596276\pi$
$101$	−5.98712	−0.595741	−0.297870	+	0.954606i	$0.596276\pi$
$102$	0	0
$103$	15.6415	1.54120	0.770602	−	0.637317i	$-0.219956\pi$
$103$	15.6415	1.54120	0.770602	+	0.637317i	$0.219956\pi$
$104$	−0.0673089	−0.00660018
$105$	0	0
$106$	2.99558	0.290956
$107$	−13.8185	−1.33589	−0.667944	−	0.744212i	$-0.732825\pi$
$107$	−13.8185	−1.33589	−0.667944	+	0.744212i	$0.732825\pi$
$108$	0	0
$109$	−10.7865	−1.03316	−0.516581	−	0.856238i	$-0.672796\pi$
$109$	−10.7865	−1.03316	−0.516581	+	0.856238i	$0.672796\pi$
$110$	−1.09645	−0.104542
$111$	0	0
$112$	1.47889	0.139742
$113$	−4.57032	−0.429939	−0.214970	−	0.976621i	$-0.568965\pi$
$113$	−4.57032	−0.429939	−0.214970	+	0.976621i	$0.568965\pi$
$114$	0	0
$115$	0	0
$116$	7.67816	0.712899
$117$	0	0
$118$	−6.83576	−0.629283
$119$	−6.34706	−0.581834
$120$	0	0
$121$	−10.2137	−0.928515
$122$	−4.60015	−0.416478
$123$	0	0
$124$	−1.59145	−0.142917
$125$	10.4744	0.936855
$126$	0	0
$127$	−14.4076	−1.27846	−0.639232	−	0.769014i	$-0.720748\pi$
$127$	−14.4076	−1.27846	−0.639232	+	0.769014i	$0.720748\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0.0832260	0.00729941
$131$	4.57998	0.400154	0.200077	−	0.979780i	$-0.435881\pi$
$131$	4.57998	0.400154	0.200077	+	0.979780i	$0.435881\pi$
$132$	0	0
$133$	6.90286	0.598554
$134$	12.4051	1.07164
$135$	0	0
$136$	4.29177	0.368016
$137$	−9.22788	−0.788391	−0.394196	−	0.919027i	$-0.628977\pi$
$137$	−9.22788	−0.788391	−0.394196	+	0.919027i	$0.628977\pi$
$138$	0	0
$139$	7.82143	0.663405	0.331702	−	0.943384i	$-0.392377\pi$
$139$	7.82143	0.663405	0.331702	+	0.943384i	$0.392377\pi$
$140$	−1.82862	−0.154546
$141$	0	0
$142$	−7.31821	−0.614131
$143$	−0.0596863	−0.00499122
$144$	0	0
$145$	−9.49388	−0.788424
$146$	−13.1978	−1.09226
$147$	0	0
$148$	5.22871	0.429797
$149$	−4.95242	−0.405718	−0.202859	−	0.979208i	$-0.565023\pi$
$149$	−4.95242	−0.405718	−0.202859	+	0.979208i	$0.565023\pi$
$150$	0	0
$151$	15.3636	1.25028	0.625138	−	0.780514i	$-0.285043\pi$
$151$	15.3636	1.25028	0.625138	+	0.780514i	$0.285043\pi$
$152$	−4.66759	−0.378592
$153$	0	0
$154$	1.31141	0.105676
$155$	1.96780	0.158057
$156$	0	0
$157$	−12.5073	−0.998188	−0.499094	−	0.866548i	$-0.666334\pi$
$157$	−12.5073	−0.998188	−0.499094	+	0.866548i	$0.666334\pi$
$158$	7.89446	0.628050
$159$	0	0
$160$	1.23648	0.0977522
$161$	0	0
$162$	0	0
$163$	−10.4300	−0.816943	−0.408472	−	0.912771i	$-0.633938\pi$
$163$	−10.4300	−0.816943	−0.408472	+	0.912771i	$0.633938\pi$
$164$	0.135933	0.0106145
$165$	0	0
$166$	8.40858	0.652632
$167$	25.3684	1.96306	0.981532	−	0.191299i	$-0.0612699\pi$
$167$	25.3684	1.96306	0.981532	+	0.191299i	$0.0612699\pi$
$168$	0	0
$169$	−12.9955	−0.999652
$170$	−5.30668	−0.407004
$171$	0	0
$172$	−6.57798	−0.501567
$173$	−17.1947	−1.30729	−0.653644	−	0.756802i	$-0.726761\pi$
$173$	−17.1947	−1.30729	−0.653644	+	0.756802i	$0.726761\pi$
$174$	0	0
$175$	−5.13341	−0.388049
$176$	−0.886752	−0.0668415
$177$	0	0
$178$	−11.7235	−0.878711
$179$	6.10800	0.456533	0.228267	−	0.973599i	$-0.426694\pi$
$179$	6.10800	0.456533	0.228267	+	0.973599i	$0.426694\pi$
$180$	0	0
$181$	−21.4901	−1.59734	−0.798672	−	0.601766i	$-0.794464\pi$
$181$	−21.4901	−1.59734	−0.798672	+	0.601766i	$0.794464\pi$
$182$	−0.0995426	−0.00737859
$183$	0	0
$184$	0	0
$185$	−6.46519	−0.475330
$186$	0	0
$187$	3.80574	0.278303
$188$	−8.31271	−0.606266
$189$	0	0
$190$	5.77138	0.418700
$191$	−2.83473	−0.205114	−0.102557	−	0.994727i	$-0.532702\pi$
$191$	−2.83473	−0.205114	−0.102557	+	0.994727i	$0.532702\pi$
$192$	0	0
$193$	−24.0877	−1.73387	−0.866936	−	0.498420i	$-0.833914\pi$
$193$	−24.0877	−1.73387	−0.866936	+	0.498420i	$0.833914\pi$
$194$	−17.1665	−1.23248
$195$	0	0
$196$	−4.81288	−0.343777
$197$	24.6054	1.75306	0.876532	−	0.481343i	$-0.159851\pi$
$197$	24.6054	1.75306	0.876532	+	0.481343i	$0.159851\pi$
$198$	0	0
$199$	−6.27783	−0.445024	−0.222512	−	0.974930i	$-0.571426\pi$
$199$	−6.27783	−0.445024	−0.222512	+	0.974930i	$0.571426\pi$
$200$	3.47112	0.245445
$201$	0	0
$202$	5.98712	0.421252
$203$	11.3552	0.796976
$204$	0	0
$205$	−0.168078	−0.0117391
$206$	−15.6415	−1.08980
$207$	0	0
$208$	0.0673089	0.00466703
$209$	−4.13900	−0.286301
$210$	0	0
$211$	19.3895	1.33483	0.667414	−	0.744686i	$-0.267401\pi$
$211$	19.3895	1.33483	0.667414	+	0.744686i	$0.267401\pi$
$212$	−2.99558	−0.205737
$213$	0	0
$214$	13.8185	0.944615
$215$	8.13354	0.554703
$216$	0	0
$217$	−2.35359	−0.159772
$218$	10.7865	0.730556
$219$	0	0
$220$	1.09645	0.0739227
$221$	−0.288874	−0.0194318
$222$	0	0
$223$	20.5735	1.37770	0.688852	−	0.724902i	$-0.258115\pi$
$223$	20.5735	1.37770	0.688852	+	0.724902i	$0.258115\pi$
$224$	−1.47889	−0.0988126
$225$	0	0
$226$	4.57032	0.304013
$227$	−23.0232	−1.52811	−0.764053	−	0.645154i	$-0.776793\pi$
$227$	−23.0232	−1.52811	−0.764053	+	0.645154i	$0.776793\pi$
$228$	0	0
$229$	−7.50367	−0.495857	−0.247928	−	0.968778i	$-0.579750\pi$
$229$	−7.50367	−0.495857	−0.247928	+	0.968778i	$0.579750\pi$
$230$	0	0
$231$	0	0
$232$	−7.67816	−0.504096
$233$	−24.3682	−1.59642	−0.798208	−	0.602382i	$-0.794218\pi$
$233$	−24.3682	−1.59642	−0.798208	+	0.602382i	$0.794218\pi$
$234$	0	0
$235$	10.2785	0.670495
$236$	6.83576	0.444970
$237$	0	0
$238$	6.34706	0.411419
$239$	−8.69645	−0.562527	−0.281263	−	0.959631i	$-0.590753\pi$
$239$	−8.69645	−0.562527	−0.281263	+	0.959631i	$0.590753\pi$
$240$	0	0
$241$	5.67714	0.365697	0.182848	−	0.983141i	$-0.441468\pi$
$241$	5.67714	0.365697	0.182848	+	0.983141i	$0.441468\pi$
$242$	10.2137	0.656560
$243$	0	0
$244$	4.60015	0.294494
$245$	5.95102	0.380197
$246$	0	0
$247$	0.314171	0.0199902
$248$	1.59145	0.101057
$249$	0	0
$250$	−10.4744	−0.662457
$251$	5.98412	0.377714	0.188857	−	0.982005i	$-0.439522\pi$
$251$	5.98412	0.377714	0.188857	+	0.982005i	$0.439522\pi$
$252$	0	0
$253$	0	0
$254$	14.4076	0.904010
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.5582	−0.970496	−0.485248	−	0.874376i	$-0.661271\pi$
$257$	−15.5582	−0.970496	−0.485248	+	0.874376i	$0.661271\pi$
$258$	0	0
$259$	7.73269	0.480486
$260$	−0.0832260	−0.00516146
$261$	0	0
$262$	−4.57998	−0.282952
$263$	10.2667	0.633069	0.316535	−	0.948581i	$-0.397481\pi$
$263$	10.2667	0.633069	0.316535	+	0.948581i	$0.397481\pi$
$264$	0	0
$265$	3.70397	0.227533
$266$	−6.90286	−0.423242
$267$	0	0
$268$	−12.4051	−0.757763
$269$	3.24314	0.197738	0.0988688	−	0.995100i	$-0.468478\pi$
$269$	3.24314	0.197738	0.0988688	+	0.995100i	$0.468478\pi$
$270$	0	0
$271$	15.5064	0.941944	0.470972	−	0.882148i	$-0.343903\pi$
$271$	15.5064	0.941944	0.470972	+	0.882148i	$0.343903\pi$
$272$	−4.29177	−0.260227
$273$	0	0
$274$	9.22788	0.557477
$275$	3.07802	0.185612
$276$	0	0
$277$	2.16313	0.129970	0.0649849	−	0.997886i	$-0.479300\pi$
$277$	2.16313	0.129970	0.0649849	+	0.997886i	$0.479300\pi$
$278$	−7.82143	−0.469098
$279$	0	0
$280$	1.82862	0.109281
$281$	20.5561	1.22627	0.613137	−	0.789977i	$-0.289907\pi$
$281$	20.5561	1.22627	0.613137	+	0.789977i	$0.289907\pi$
$282$	0	0
$283$	−16.9363	−1.00676	−0.503381	−	0.864065i	$-0.667911\pi$
$283$	−16.9363	−1.00676	−0.503381	+	0.864065i	$0.667911\pi$
$284$	7.31821	0.434256
$285$	0	0
$286$	0.0596863	0.00352933
$287$	0.201029	0.0118664
$288$	0	0
$289$	1.41930	0.0834884
$290$	9.49388	0.557500
$291$	0	0
$292$	13.1978	0.772343
$293$	2.06234	0.120483	0.0602416	−	0.998184i	$-0.480813\pi$
$293$	2.06234	0.120483	0.0602416	+	0.998184i	$0.480813\pi$
$294$	0	0
$295$	−8.45227	−0.492110
$296$	−5.22871	−0.303912
$297$	0	0
$298$	4.95242	0.286886
$299$	0	0
$300$	0	0
$301$	−9.72812	−0.560720
$302$	−15.3636	−0.884079
$303$	0	0
$304$	4.66759	0.267705
$305$	−5.68798	−0.325693
$306$	0	0
$307$	−4.08297	−0.233027	−0.116514	−	0.993189i	$-0.537172\pi$
$307$	−4.08297	−0.233027	−0.116514	+	0.993189i	$0.537172\pi$
$308$	−1.31141	−0.0747245
$309$	0	0
$310$	−1.96780	−0.111763
$311$	12.7493	0.722945	0.361472	−	0.932383i	$-0.382274\pi$
$311$	12.7493	0.722945	0.361472	+	0.932383i	$0.382274\pi$
$312$	0	0
$313$	7.78228	0.439881	0.219940	−	0.975513i	$-0.429414\pi$
$313$	7.78228	0.439881	0.219940	+	0.975513i	$0.429414\pi$
$314$	12.5073	0.705825
$315$	0	0
$316$	−7.89446	−0.444098
$317$	−19.7507	−1.10931	−0.554655	−	0.832080i	$-0.687150\pi$
$317$	−19.7507	−1.10931	−0.554655	+	0.832080i	$0.687150\pi$
$318$	0	0
$319$	−6.80862	−0.381210
$320$	−1.23648	−0.0691213
$321$	0	0
$322$	0	0
$323$	−20.0322	−1.11462
$324$	0	0
$325$	−0.233637	−0.0129599
$326$	10.4300	0.577666
$327$	0	0
$328$	−0.135933	−0.00750562
$329$	−12.2936	−0.677768
$330$	0	0
$331$	−4.29179	−0.235898	−0.117949	−	0.993020i	$-0.537632\pi$
$331$	−4.29179	−0.235898	−0.117949	+	0.993020i	$0.537632\pi$
$332$	−8.40858	−0.461481
$333$	0	0
$334$	−25.3684	−1.38810
$335$	15.3387	0.838040
$336$	0	0
$337$	−28.9585	−1.57747	−0.788735	−	0.614733i	$-0.789264\pi$
$337$	−28.9585	−1.57747	−0.788735	+	0.614733i	$0.789264\pi$
$338$	12.9955	0.706860
$339$	0	0
$340$	5.30668	0.287795
$341$	1.41123	0.0764221
$342$	0	0
$343$	−17.4700	−0.943290
$344$	6.57798	0.354661
$345$	0	0
$346$	17.1947	0.924392
$347$	18.7893	1.00866	0.504332	−	0.863510i	$-0.331739\pi$
$347$	18.7893	1.00866	0.504332	+	0.863510i	$0.331739\pi$
$348$	0	0
$349$	−15.5823	−0.834102	−0.417051	−	0.908883i	$-0.636936\pi$
$349$	−15.5823	−0.834102	−0.417051	+	0.908883i	$0.636936\pi$
$350$	5.13341	0.274392
$351$	0	0
$352$	0.886752	0.0472641
$353$	23.2408	1.23698	0.618492	−	0.785791i	$-0.287744\pi$
$353$	23.2408	1.23698	0.618492	+	0.785791i	$0.287744\pi$
$354$	0	0
$355$	−9.04881	−0.480261
$356$	11.7235	0.621342
$357$	0	0
$358$	−6.10800	−0.322818
$359$	14.4495	0.762614	0.381307	−	0.924449i	$-0.375474\pi$
$359$	14.4495	0.762614	0.381307	+	0.924449i	$0.375474\pi$
$360$	0	0
$361$	2.78643	0.146654
$362$	21.4901	1.12949
$363$	0	0
$364$	0.0995426	0.00521745
$365$	−16.3188	−0.854165
$366$	0	0
$367$	−26.6056	−1.38880	−0.694400	−	0.719589i	$-0.744330\pi$
$367$	−26.6056	−1.38880	−0.694400	+	0.719589i	$0.744330\pi$
$368$	0	0
$369$	0	0
$370$	6.46519	0.336109
$371$	−4.43013	−0.230001
$372$	0	0
$373$	4.88055	0.252705	0.126353	−	0.991985i	$-0.459673\pi$
$373$	4.88055	0.252705	0.126353	+	0.991985i	$0.459673\pi$
$374$	−3.80574	−0.196790
$375$	0	0
$376$	8.31271	0.428695
$377$	0.516808	0.0266170
$378$	0	0
$379$	−3.41084	−0.175203	−0.0876015	−	0.996156i	$-0.527920\pi$
$379$	−3.41084	−0.175203	−0.0876015	+	0.996156i	$0.527920\pi$
$380$	−5.77138	−0.296066
$381$	0	0
$382$	2.83473	0.145037
$383$	−28.3725	−1.44977	−0.724883	−	0.688872i	$-0.758106\pi$
$383$	−28.3725	−1.44977	−0.724883	+	0.688872i	$0.758106\pi$
$384$	0	0
$385$	1.62153	0.0826409
$386$	24.0877	1.22603
$387$	0	0
$388$	17.1665	0.871496
$389$	−10.6537	−0.540166	−0.270083	−	0.962837i	$-0.587051\pi$
$389$	−10.6537	−0.540166	−0.270083	+	0.962837i	$0.587051\pi$
$390$	0	0
$391$	0	0
$392$	4.81288	0.243087
$393$	0	0
$394$	−24.6054	−1.23960
$395$	9.76134	0.491146
$396$	0	0
$397$	20.9156	1.04972	0.524862	−	0.851187i	$-0.324117\pi$
$397$	20.9156	1.04972	0.524862	+	0.851187i	$0.324117\pi$
$398$	6.27783	0.314679
$399$	0	0
$400$	−3.47112	−0.173556
$401$	4.05114	0.202304	0.101152	−	0.994871i	$-0.467747\pi$
$401$	4.05114	0.202304	0.101152	+	0.994871i	$0.467747\pi$
$402$	0	0
$403$	−0.107119	−0.00533598
$404$	−5.98712	−0.297870
$405$	0	0
$406$	−11.3552	−0.563547
$407$	−4.63657	−0.229826
$408$	0	0
$409$	24.3805	1.20554	0.602770	−	0.797915i	$-0.294064\pi$
$409$	24.3805	1.20554	0.602770	+	0.797915i	$0.294064\pi$
$410$	0.168078	0.00830077
$411$	0	0
$412$	15.6415	0.770602
$413$	10.1093	0.497448
$414$	0	0
$415$	10.3970	0.510370
$416$	−0.0673089	−0.00330009
$417$	0	0
$418$	4.13900	0.202445
$419$	14.1181	0.689715	0.344858	−	0.938655i	$-0.387927\pi$
$419$	14.1181	0.689715	0.344858	+	0.938655i	$0.387927\pi$
$420$	0	0
$421$	−22.3142	−1.08753	−0.543763	−	0.839239i	$-0.683001\pi$
$421$	−22.3142	−1.08753	−0.543763	+	0.839239i	$0.683001\pi$
$422$	−19.3895	−0.943867
$423$	0	0
$424$	2.99558	0.145478
$425$	14.8973	0.722623
$426$	0	0
$427$	6.80312	0.329226
$428$	−13.8185	−0.667944
$429$	0	0
$430$	−8.13354	−0.392234
$431$	−18.0169	−0.867842	−0.433921	−	0.900951i	$-0.642870\pi$
$431$	−18.0169	−0.867842	−0.433921	+	0.900951i	$0.642870\pi$
$432$	0	0
$433$	−20.7161	−0.995554	−0.497777	−	0.867305i	$-0.665850\pi$
$433$	−20.7161	−0.995554	−0.497777	+	0.867305i	$0.665850\pi$
$434$	2.35359	0.112976
$435$	0	0
$436$	−10.7865	−0.516581
$437$	0	0
$438$	0	0
$439$	−13.5915	−0.648686	−0.324343	−	0.945940i	$-0.605143\pi$
$439$	−13.5915	−0.648686	−0.324343	+	0.945940i	$0.605143\pi$
$440$	−1.09645	−0.0522712
$441$	0	0
$442$	0.288874	0.0137404
$443$	−10.4025	−0.494238	−0.247119	−	0.968985i	$-0.579484\pi$
$443$	−10.4025	−0.494238	−0.247119	+	0.968985i	$0.579484\pi$
$444$	0	0
$445$	−14.4958	−0.687168
$446$	−20.5735	−0.974184
$447$	0	0
$448$	1.47889	0.0698711
$449$	−10.7318	−0.506465	−0.253232	−	0.967405i	$-0.581494\pi$
$449$	−10.7318	−0.506465	−0.253232	+	0.967405i	$0.581494\pi$
$450$	0	0
$451$	−0.120538	−0.00567593
$452$	−4.57032	−0.214970
$453$	0	0
$454$	23.0232	1.08053
$455$	−0.123082	−0.00577019
$456$	0	0
$457$	−0.845263	−0.0395397	−0.0197699	−	0.999805i	$-0.506293\pi$
$457$	−0.845263	−0.0395397	−0.0197699	+	0.999805i	$0.506293\pi$
$458$	7.50367	0.350624
$459$	0	0
$460$	0	0
$461$	17.2385	0.802875	0.401437	−	0.915886i	$-0.368511\pi$
$461$	17.2385	0.802875	0.401437	+	0.915886i	$0.368511\pi$
$462$	0	0
$463$	−1.81253	−0.0842354	−0.0421177	−	0.999113i	$-0.513410\pi$
$463$	−1.81253	−0.0842354	−0.0421177	+	0.999113i	$0.513410\pi$
$464$	7.67816	0.356449
$465$	0	0
$466$	24.3682	1.12884
$467$	−32.8225	−1.51884	−0.759422	−	0.650598i	$-0.774518\pi$
$467$	−32.8225	−1.51884	−0.759422	+	0.650598i	$0.774518\pi$
$468$	0	0
$469$	−18.3458	−0.847131
$470$	−10.2785	−0.474111
$471$	0	0
$472$	−6.83576	−0.314641
$473$	5.83304	0.268204
$474$	0	0
$475$	−16.2018	−0.743389
$476$	−6.34706	−0.290917
$477$	0	0
$478$	8.69645	0.397766
$479$	−20.2241	−0.924062	−0.462031	−	0.886864i	$-0.652879\pi$
$479$	−20.2241	−0.924062	−0.462031	+	0.886864i	$0.652879\pi$
$480$	0	0
$481$	0.351939	0.0160470
$482$	−5.67714	−0.258587
$483$	0	0
$484$	−10.2137	−0.464258
$485$	−21.2260	−0.963823
$486$	0	0
$487$	−39.5598	−1.79262	−0.896312	−	0.443424i	$-0.853764\pi$
$487$	−39.5598	−1.79262	−0.896312	+	0.443424i	$0.853764\pi$
$488$	−4.60015	−0.208239
$489$	0	0
$490$	−5.95102	−0.268840
$491$	−24.3279	−1.09790	−0.548951	−	0.835854i	$-0.684973\pi$
$491$	−24.3279	−1.09790	−0.548951	+	0.835854i	$0.684973\pi$
$492$	0	0
$493$	−32.9529	−1.48412
$494$	−0.314171	−0.0141352
$495$	0	0
$496$	−1.59145	−0.0714584
$497$	10.8228	0.485471
$498$	0	0
$499$	−15.8756	−0.710689	−0.355344	−	0.934735i	$-0.615636\pi$
$499$	−15.8756	−0.710689	−0.355344	+	0.934735i	$0.615636\pi$
$500$	10.4744	0.468428
$501$	0	0
$502$	−5.98412	−0.267084
$503$	6.54382	0.291774	0.145887	−	0.989301i	$-0.453396\pi$
$503$	6.54382	0.291774	0.145887	+	0.989301i	$0.453396\pi$
$504$	0	0
$505$	7.40295	0.329427
$506$	0	0
$507$	0	0
$508$	−14.4076	−0.639232
$509$	17.8626	0.791745	0.395873	−	0.918305i	$-0.370442\pi$
$509$	17.8626	0.791745	0.395873	+	0.918305i	$0.370442\pi$
$510$	0	0
$511$	19.5181	0.863431
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	15.5582	0.686244
$515$	−19.3404	−0.852240
$516$	0	0
$517$	7.37131	0.324190
$518$	−7.73269	−0.339755
$519$	0	0
$520$	0.0832260	0.00364970
$521$	11.8337	0.518445	0.259222	−	0.965818i	$-0.416534\pi$
$521$	11.8337	0.518445	0.259222	+	0.965818i	$0.416534\pi$
$522$	0	0
$523$	16.0125	0.700177	0.350088	−	0.936717i	$-0.386152\pi$
$523$	16.0125	0.700177	0.350088	+	0.936717i	$0.386152\pi$
$524$	4.57998	0.200077
$525$	0	0
$526$	−10.2667	−0.447648
$527$	6.83016	0.297526
$528$	0	0
$529$	0	0
$530$	−3.70397	−0.160890
$531$	0	0
$532$	6.90286	0.299277
$533$	0.00914947	0.000396307 0
$534$	0	0
$535$	17.0863	0.738706
$536$	12.4051	0.535819
$537$	0	0
$538$	−3.24314	−0.139822
$539$	4.26783	0.183829
$540$	0	0
$541$	−19.5524	−0.840621	−0.420311	−	0.907380i	$-0.638079\pi$
$541$	−19.5524	−0.840621	−0.420311	+	0.907380i	$0.638079\pi$
$542$	−15.5064	−0.666055
$543$	0	0
$544$	4.29177	0.184008
$545$	13.3373	0.571308
$546$	0	0
$547$	−1.13521	−0.0485380	−0.0242690	−	0.999705i	$-0.507726\pi$
$547$	−1.13521	−0.0485380	−0.0242690	+	0.999705i	$0.507726\pi$
$548$	−9.22788	−0.394196
$549$	0	0
$550$	−3.07802	−0.131247
$551$	35.8385	1.52677
$552$	0	0
$553$	−11.6751	−0.496474
$554$	−2.16313	−0.0919026
$555$	0	0
$556$	7.82143	0.331702
$557$	41.2506	1.74784	0.873922	−	0.486066i	$-0.161569\pi$
$557$	41.2506	1.74784	0.873922	+	0.486066i	$0.161569\pi$
$558$	0	0
$559$	−0.442757	−0.0187266
$560$	−1.82862	−0.0772732
$561$	0	0
$562$	−20.5561	−0.867106
$563$	−28.2407	−1.19020	−0.595101	−	0.803651i	$-0.702888\pi$
$563$	−28.2407	−1.19020	−0.595101	+	0.803651i	$0.702888\pi$
$564$	0	0
$565$	5.65110	0.237744
$566$	16.9363	0.711888
$567$	0	0
$568$	−7.31821	−0.307065
$569$	−41.5315	−1.74109	−0.870546	−	0.492086i	$-0.836234\pi$
$569$	−41.5315	−1.74109	−0.870546	+	0.492086i	$0.836234\pi$
$570$	0	0
$571$	31.5189	1.31902	0.659512	−	0.751694i	$-0.270763\pi$
$571$	31.5189	1.31902	0.659512	+	0.751694i	$0.270763\pi$
$572$	−0.0596863	−0.00249561
$573$	0	0
$574$	−0.201029	−0.00839081
$575$	0	0
$576$	0	0
$577$	20.5890	0.857132	0.428566	−	0.903511i	$-0.359019\pi$
$577$	20.5890	0.857132	0.428566	+	0.903511i	$0.359019\pi$
$578$	−1.41930	−0.0590352
$579$	0	0
$580$	−9.49388	−0.394212
$581$	−12.4354	−0.515906
$582$	0	0
$583$	2.65633	0.110014
$584$	−13.1978	−0.546129
$585$	0	0
$586$	−2.06234	−0.0851945
$587$	−8.51084	−0.351280	−0.175640	−	0.984454i	$-0.556199\pi$
$587$	−8.51084	−0.351280	−0.175640	+	0.984454i	$0.556199\pi$
$588$	0	0
$589$	−7.42826	−0.306076
$590$	8.45227	0.347975
$591$	0	0
$592$	5.22871	0.214899
$593$	9.47438	0.389066	0.194533	−	0.980896i	$-0.437681\pi$
$593$	9.47438	0.389066	0.194533	+	0.980896i	$0.437681\pi$
$594$	0	0
$595$	7.84801	0.321737
$596$	−4.95242	−0.202859
$597$	0	0
$598$	0	0
$599$	2.52422	0.103137	0.0515684	−	0.998669i	$-0.483578\pi$
$599$	2.52422	0.103137	0.0515684	+	0.998669i	$0.483578\pi$
$600$	0	0
$601$	−18.8422	−0.768588	−0.384294	−	0.923211i	$-0.625555\pi$
$601$	−18.8422	−0.768588	−0.384294	+	0.923211i	$0.625555\pi$
$602$	9.72812	0.396489
$603$	0	0
$604$	15.3636	0.625138
$605$	12.6290	0.513441
$606$	0	0
$607$	3.25339	0.132051	0.0660256	−	0.997818i	$-0.478968\pi$
$607$	3.25339	0.132051	0.0660256	+	0.997818i	$0.478968\pi$
$608$	−4.66759	−0.189296
$609$	0	0
$610$	5.68798	0.230300
$611$	−0.559519	−0.0226357
$612$	0	0
$613$	−16.1279	−0.651398	−0.325699	−	0.945474i	$-0.605600\pi$
$613$	−16.1279	−0.651398	−0.325699	+	0.945474i	$0.605600\pi$
$614$	4.08297	0.164775
$615$	0	0
$616$	1.31141	0.0528382
$617$	2.31927	0.0933702	0.0466851	−	0.998910i	$-0.485134\pi$
$617$	2.31927	0.0933702	0.0466851	+	0.998910i	$0.485134\pi$
$618$	0	0
$619$	−32.9947	−1.32617	−0.663085	−	0.748544i	$-0.730753\pi$
$619$	−32.9947	−1.32617	−0.663085	+	0.748544i	$0.730753\pi$
$620$	1.96780	0.0790287
$621$	0	0
$622$	−12.7493	−0.511199
$623$	17.3377	0.694622
$624$	0	0
$625$	4.40427	0.176171
$626$	−7.78228	−0.311043
$627$	0	0
$628$	−12.5073	−0.499094
$629$	−22.4404	−0.894758
$630$	0	0
$631$	30.4452	1.21200	0.606001	−	0.795464i	$-0.292773\pi$
$631$	30.4452	1.21200	0.606001	+	0.795464i	$0.292773\pi$
$632$	7.89446	0.314025
$633$	0	0
$634$	19.7507	0.784401
$635$	17.8146	0.706952
$636$	0	0
$637$	−0.323950	−0.0128354
$638$	6.80862	0.269556
$639$	0	0
$640$	1.23648	0.0488761
$641$	−44.7383	−1.76706	−0.883528	−	0.468378i	$-0.844839\pi$
$641$	−44.7383	−1.76706	−0.883528	+	0.468378i	$0.844839\pi$
$642$	0	0
$643$	−26.8514	−1.05892	−0.529459	−	0.848336i	$-0.677605\pi$
$643$	−26.8514	−1.05892	−0.529459	+	0.848336i	$0.677605\pi$
$644$	0	0
$645$	0	0
$646$	20.0322	0.788158
$647$	22.9140	0.900841	0.450420	−	0.892817i	$-0.351274\pi$
$647$	22.9140	0.900841	0.450420	+	0.892817i	$0.351274\pi$
$648$	0	0
$649$	−6.06163	−0.237940
$650$	0.233637	0.00916401
$651$	0	0
$652$	−10.4300	−0.408472
$653$	46.0369	1.80156	0.900782	−	0.434271i	$-0.142994\pi$
$653$	46.0369	1.80156	0.900782	+	0.434271i	$0.142994\pi$
$654$	0	0
$655$	−5.66304	−0.221273
$656$	0.135933	0.00530727
$657$	0	0
$658$	12.2936	0.479254
$659$	−27.6386	−1.07665	−0.538323	−	0.842739i	$-0.680942\pi$
$659$	−27.6386	−1.07665	−0.538323	+	0.842739i	$0.680942\pi$
$660$	0	0
$661$	−5.29014	−0.205763	−0.102881	−	0.994694i	$-0.532806\pi$
$661$	−5.29014	−0.205763	−0.102881	+	0.994694i	$0.532806\pi$
$662$	4.29179	0.166805
$663$	0	0
$664$	8.40858	0.326316
$665$	−8.53525	−0.330983
$666$	0	0
$667$	0	0
$668$	25.3684	0.981532
$669$	0	0
$670$	−15.3387	−0.592584
$671$	−4.07919	−0.157475
$672$	0	0
$673$	−40.6460	−1.56679	−0.783394	−	0.621525i	$-0.786513\pi$
$673$	−40.6460	−1.56679	−0.783394	+	0.621525i	$0.786513\pi$
$674$	28.9585	1.11544
$675$	0	0
$676$	−12.9955	−0.499826
$677$	−15.4320	−0.593100	−0.296550	−	0.955017i	$-0.595836\pi$
$677$	−15.4320	−0.593100	−0.296550	+	0.955017i	$0.595836\pi$
$678$	0	0
$679$	25.3874	0.974278
$680$	−5.30668	−0.203502
$681$	0	0
$682$	−1.41123	−0.0540386
$683$	−0.323786	−0.0123893	−0.00619467	−	0.999981i	$-0.501972\pi$
$683$	−0.323786	−0.0123893	−0.00619467	+	0.999981i	$0.501972\pi$
$684$	0	0
$685$	11.4101	0.435957
$686$	17.4700	0.667006
$687$	0	0
$688$	−6.57798	−0.250783
$689$	−0.201629	−0.00768145
$690$	0	0
$691$	−11.8762	−0.451791	−0.225896	−	0.974152i	$-0.572531\pi$
$691$	−11.8762	−0.451791	−0.225896	+	0.974152i	$0.572531\pi$
$692$	−17.1947	−0.653644
$693$	0	0
$694$	−18.7893	−0.713233
$695$	−9.67103	−0.366843
$696$	0	0
$697$	−0.583391	−0.0220975
$698$	15.5823	0.589799
$699$	0	0
$700$	−5.13341	−0.194025
$701$	29.1388	1.10056	0.550278	−	0.834981i	$-0.314522\pi$
$701$	29.1388	1.10056	0.550278	+	0.834981i	$0.314522\pi$
$702$	0	0
$703$	24.4055	0.920470
$704$	−0.886752	−0.0334207
$705$	0	0
$706$	−23.2408	−0.874680
$707$	−8.85430	−0.333000
$708$	0	0
$709$	13.3204	0.500259	0.250130	−	0.968212i	$-0.419527\pi$
$709$	13.3204	0.500259	0.250130	+	0.968212i	$0.419527\pi$
$710$	9.04881	0.339596
$711$	0	0
$712$	−11.7235	−0.439355
$713$	0	0
$714$	0	0
$715$	0.0738009	0.00276000
$716$	6.10800	0.228267
$717$	0	0
$718$	−14.4495	−0.539249
$719$	−40.7753	−1.52066	−0.760330	−	0.649537i	$-0.774963\pi$
$719$	−40.7753	−1.52066	−0.760330	+	0.649537i	$0.774963\pi$
$720$	0	0
$721$	23.1321	0.861484
$722$	−2.78643	−0.103700
$723$	0	0
$724$	−21.4901	−0.798672
$725$	−26.6518	−0.989823
$726$	0	0
$727$	32.8678	1.21900	0.609500	−	0.792786i	$-0.291370\pi$
$727$	32.8678	1.21900	0.609500	+	0.792786i	$0.291370\pi$
$728$	−0.0995426	−0.00368929
$729$	0	0
$730$	16.3188	0.603986
$731$	28.2312	1.04417
$732$	0	0
$733$	−49.1083	−1.81386	−0.906929	−	0.421284i	$-0.861579\pi$
$733$	−49.1083	−1.81386	−0.906929	+	0.421284i	$0.861579\pi$
$734$	26.6056	0.982030
$735$	0	0
$736$	0	0
$737$	11.0003	0.405200
$738$	0	0
$739$	−42.3548	−1.55805	−0.779023	−	0.626995i	$-0.784285\pi$
$739$	−42.3548	−1.55805	−0.779023	+	0.626995i	$0.784285\pi$
$740$	−6.46519	−0.237665
$741$	0	0
$742$	4.43013	0.162635
$743$	−2.66106	−0.0976250	−0.0488125	−	0.998808i	$-0.515544\pi$
$743$	−2.66106	−0.0976250	−0.0488125	+	0.998808i	$0.515544\pi$
$744$	0	0
$745$	6.12356	0.224350
$746$	−4.88055	−0.178689
$747$	0	0
$748$	3.80574	0.139152
$749$	−20.4361	−0.746719
$750$	0	0
$751$	−37.1251	−1.35471	−0.677357	−	0.735654i	$-0.736875\pi$
$751$	−37.1251	−1.35471	−0.677357	+	0.735654i	$0.736875\pi$
$752$	−8.31271	−0.303133
$753$	0	0
$754$	−0.516808	−0.0188210
$755$	−18.9968	−0.691365
$756$	0	0
$757$	23.2476	0.844950	0.422475	−	0.906375i	$-0.361162\pi$
$757$	23.2476	0.844950	0.422475	+	0.906375i	$0.361162\pi$
$758$	3.41084	0.123887
$759$	0	0
$760$	5.77138	0.209350
$761$	−17.6451	−0.639633	−0.319816	−	0.947480i	$-0.603621\pi$
$761$	−17.6451	−0.639633	−0.319816	+	0.947480i	$0.603621\pi$
$762$	0	0
$763$	−15.9521	−0.577505
$764$	−2.83473	−0.102557
$765$	0	0
$766$	28.3725	1.02514
$767$	0.460108	0.0166135
$768$	0	0
$769$	−36.5665	−1.31862	−0.659311	−	0.751871i	$-0.729152\pi$
$769$	−36.5665	−1.31862	−0.659311	+	0.751871i	$0.729152\pi$
$770$	−1.62153	−0.0584359
$771$	0	0
$772$	−24.0877	−0.866936
$773$	9.85872	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$773$	9.85872	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$774$	0	0
$775$	5.52413	0.198433
$776$	−17.1665	−0.616241
$777$	0	0
$778$	10.6537	0.381955
$779$	0.634478	0.0227325
$780$	0	0
$781$	−6.48944	−0.232210
$782$	0	0
$783$	0	0
$784$	−4.81288	−0.171889
$785$	15.4650	0.551968
$786$	0	0
$787$	−38.9210	−1.38738	−0.693692	−	0.720272i	$-0.744017\pi$
$787$	−38.9210	−1.38738	−0.693692	+	0.720272i	$0.744017\pi$
$788$	24.6054	0.876532
$789$	0	0
$790$	−9.76134	−0.347293
$791$	−6.75901	−0.240323
$792$	0	0
$793$	0.309631	0.0109953
$794$	−20.9156	−0.742268
$795$	0	0
$796$	−6.27783	−0.222512
$797$	1.36928	0.0485024	0.0242512	−	0.999706i	$-0.492280\pi$
$797$	1.36928	0.0485024	0.0242512	+	0.999706i	$0.492280\pi$
$798$	0	0
$799$	35.6762	1.26213
$800$	3.47112	0.122723
$801$	0	0
$802$	−4.05114	−0.143051
$803$	−11.7032	−0.412996
$804$	0	0
$805$	0	0
$806$	0.107119	0.00377311
$807$	0	0
$808$	5.98712	0.210626
$809$	43.2994	1.52233	0.761163	−	0.648561i	$-0.224629\pi$
$809$	43.2994	1.52233	0.761163	+	0.648561i	$0.224629\pi$
$810$	0	0
$811$	9.57980	0.336392	0.168196	−	0.985754i	$-0.446206\pi$
$811$	9.57980	0.336392	0.168196	+	0.985754i	$0.446206\pi$
$812$	11.3552	0.398488
$813$	0	0
$814$	4.63657	0.162512
$815$	12.8965	0.451745
$816$	0	0
$817$	−30.7034	−1.07417
$818$	−24.3805	−0.852445
$819$	0	0
$820$	−0.168078	−0.00586953
$821$	−23.4537	−0.818541	−0.409271	−	0.912413i	$-0.634217\pi$
$821$	−23.4537	−0.818541	−0.409271	+	0.912413i	$0.634217\pi$
$822$	0	0
$823$	−9.33665	−0.325455	−0.162728	−	0.986671i	$-0.552029\pi$
$823$	−9.33665	−0.325455	−0.162728	+	0.986671i	$0.552029\pi$
$824$	−15.6415	−0.544898
$825$	0	0
$826$	−10.1093	−0.351749
$827$	−54.7053	−1.90229	−0.951145	−	0.308745i	$-0.900091\pi$
$827$	−54.7053	−1.90229	−0.951145	+	0.308745i	$0.900091\pi$
$828$	0	0
$829$	−55.3720	−1.92315	−0.961574	−	0.274545i	$-0.911473\pi$
$829$	−55.3720	−1.92315	−0.961574	+	0.274545i	$0.911473\pi$
$830$	−10.3970	−0.360886
$831$	0	0
$832$	0.0673089	0.00233352
$833$	20.6558	0.715680
$834$	0	0
$835$	−31.3675	−1.08552
$836$	−4.13900	−0.143150
$837$	0	0
$838$	−14.1181	−0.487702
$839$	11.6657	0.402744	0.201372	−	0.979515i	$-0.435460\pi$
$839$	11.6657	0.402744	0.201372	+	0.979515i	$0.435460\pi$
$840$	0	0
$841$	29.9541	1.03290
$842$	22.3142	0.768998
$843$	0	0
$844$	19.3895	0.667414
$845$	16.0686	0.552777
$846$	0	0
$847$	−15.1049	−0.519011
$848$	−2.99558	−0.102869
$849$	0	0
$850$	−14.8973	−0.510972
$851$	0	0
$852$	0	0
$853$	39.0443	1.33685	0.668425	−	0.743779i	$-0.266969\pi$
$853$	39.0443	1.33685	0.668425	+	0.743779i	$0.266969\pi$
$854$	−6.80312	−0.232798
$855$	0	0
$856$	13.8185	0.472308
$857$	−36.8064	−1.25728	−0.628641	−	0.777696i	$-0.716388\pi$
$857$	−36.8064	−1.25728	−0.628641	+	0.777696i	$0.716388\pi$
$858$	0	0
$859$	−3.04908	−0.104033	−0.0520167	−	0.998646i	$-0.516565\pi$
$859$	−3.04908	−0.104033	−0.0520167	+	0.998646i	$0.516565\pi$
$860$	8.13354	0.277351
$861$	0	0
$862$	18.0169	0.613657
$863$	13.7177	0.466958	0.233479	−	0.972362i	$-0.424989\pi$
$863$	13.7177	0.466958	0.233479	+	0.972362i	$0.424989\pi$
$864$	0	0
$865$	21.2609	0.722891
$866$	20.7161	0.703963
$867$	0	0
$868$	−2.35359	−0.0798860
$869$	7.00043	0.237473
$870$	0	0
$871$	−0.834974	−0.0282920
$872$	10.7865	0.365278
$873$	0	0
$874$	0	0
$875$	15.4904	0.523673
$876$	0	0
$877$	21.0860	0.712023	0.356012	−	0.934481i	$-0.384136\pi$
$877$	21.0860	0.712023	0.356012	+	0.934481i	$0.384136\pi$
$878$	13.5915	0.458690
$879$	0	0
$880$	1.09645	0.0369613
$881$	−9.28725	−0.312895	−0.156448	−	0.987686i	$-0.550004\pi$
$881$	−9.28725	−0.312895	−0.156448	+	0.987686i	$0.550004\pi$
$882$	0	0
$883$	45.4416	1.52923	0.764616	−	0.644486i	$-0.222929\pi$
$883$	45.4416	1.52923	0.764616	+	0.644486i	$0.222929\pi$
$884$	−0.288874	−0.00971590
$885$	0	0
$886$	10.4025	0.349479
$887$	−5.72147	−0.192108	−0.0960540	−	0.995376i	$-0.530622\pi$
$887$	−5.72147	−0.192108	−0.0960540	+	0.995376i	$0.530622\pi$
$888$	0	0
$889$	−21.3072	−0.714621
$890$	14.4958	0.485901
$891$	0	0
$892$	20.5735	0.688852
$893$	−38.8003	−1.29840
$894$	0	0
$895$	−7.55241	−0.252449
$896$	−1.47889	−0.0494063
$897$	0	0
$898$	10.7318	0.358125
$899$	−12.2194	−0.407541
$900$	0	0
$901$	12.8563	0.428306
$902$	0.120538	0.00401349
$903$	0	0
$904$	4.57032	0.152007
$905$	26.5720	0.883284
$906$	0	0
$907$	26.2917	0.873003	0.436502	−	0.899704i	$-0.356217\pi$
$907$	26.2917	0.873003	0.436502	+	0.899704i	$0.356217\pi$
$908$	−23.0232	−0.764053
$909$	0	0
$910$	0.123082	0.00408014
$911$	−49.3931	−1.63647	−0.818233	−	0.574887i	$-0.805046\pi$
$911$	−49.3931	−1.63647	−0.818233	+	0.574887i	$0.805046\pi$
$912$	0	0
$913$	7.45633	0.246768
$914$	0.845263	0.0279588
$915$	0	0
$916$	−7.50367	−0.247928
$917$	6.77329	0.223674
$918$	0	0
$919$	−28.9737	−0.955753	−0.477876	−	0.878427i	$-0.658593\pi$
$919$	−28.9737	−0.955753	−0.477876	+	0.878427i	$0.658593\pi$
$920$	0	0
$921$	0	0
$922$	−17.2385	−0.567718
$923$	0.492581	0.0162135
$924$	0	0
$925$	−18.1495	−0.596751
$926$	1.81253	0.0595634
$927$	0	0
$928$	−7.67816	−0.252048
$929$	48.9460	1.60587	0.802934	−	0.596068i	$-0.203271\pi$
$929$	48.9460	1.60587	0.802934	+	0.596068i	$0.203271\pi$
$930$	0	0
$931$	−22.4646	−0.736247
$932$	−24.3682	−0.798208
$933$	0	0
$934$	32.8225	1.07399
$935$	−4.70571	−0.153893
$936$	0	0
$937$	−45.9165	−1.50003	−0.750014	−	0.661422i	$-0.769953\pi$
$937$	−45.9165	−1.50003	−0.750014	+	0.661422i	$0.769953\pi$
$938$	18.3458	0.599012
$939$	0	0
$940$	10.2785	0.335247
$941$	−37.6436	−1.22715	−0.613573	−	0.789638i	$-0.710268\pi$
$941$	−37.6436	−1.22715	−0.613573	+	0.789638i	$0.710268\pi$
$942$	0	0
$943$	0	0
$944$	6.83576	0.222485
$945$	0	0
$946$	−5.83304	−0.189649
$947$	8.57147	0.278535	0.139268	−	0.990255i	$-0.455525\pi$
$947$	8.57147	0.278535	0.139268	+	0.990255i	$0.455525\pi$
$948$	0	0
$949$	0.888330	0.0288364
$950$	16.2018	0.525655
$951$	0	0
$952$	6.34706	0.205710
$953$	38.8498	1.25847	0.629234	−	0.777216i	$-0.283369\pi$
$953$	38.8498	1.25847	0.629234	+	0.777216i	$0.283369\pi$
$954$	0	0
$955$	3.50508	0.113422
$956$	−8.69645	−0.281263
$957$	0	0
$958$	20.2241	0.653410
$959$	−13.6470	−0.440686
$960$	0	0
$961$	−28.4673	−0.918299
$962$	−0.351939	−0.0113470
$963$	0	0
$964$	5.67714	0.182848
$965$	29.7840	0.958779
$966$	0	0
$967$	17.0473	0.548205	0.274103	−	0.961700i	$-0.411619\pi$
$967$	17.0473	0.548205	0.274103	+	0.961700i	$0.411619\pi$
$968$	10.2137	0.328280
$969$	0	0
$970$	21.2260	0.681526
$971$	−46.5548	−1.49402	−0.747008	−	0.664815i	$-0.768510\pi$
$971$	−46.5548	−1.49402	−0.747008	+	0.664815i	$0.768510\pi$
$972$	0	0
$973$	11.5670	0.370822
$974$	39.5598	1.26758
$975$	0	0
$976$	4.60015	0.147247
$977$	−4.79148	−0.153293	−0.0766465	−	0.997058i	$-0.524421\pi$
$977$	−4.79148	−0.153293	−0.0766465	+	0.997058i	$0.524421\pi$
$978$	0	0
$979$	−10.3958	−0.332252
$980$	5.95102	0.190099
$981$	0	0
$982$	24.3279	0.776334
$983$	−25.8138	−0.823332	−0.411666	−	0.911335i	$-0.635053\pi$
$983$	−25.8138	−0.823332	−0.411666	+	0.911335i	$0.635053\pi$
$984$	0	0
$985$	−30.4241	−0.969392
$986$	32.9529	1.04943
$987$	0	0
$988$	0.314171	0.00999510
$989$	0	0
$990$	0	0
$991$	18.1853	0.577674	0.288837	−	0.957378i	$-0.406731\pi$
$991$	18.1853	0.577674	0.288837	+	0.957378i	$0.406731\pi$
$992$	1.59145	0.0505287
$993$	0	0
$994$	−10.8228	−0.343280
$995$	7.76241	0.246085
$996$	0	0
$997$	4.31292	0.136592	0.0682958	−	0.997665i	$-0.478244\pi$
$997$	4.31292	0.136592	0.0682958	+	0.997665i	$0.478244\pi$
$998$	15.8756	0.502533
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.bq.1.4		5
3.2	odd	2		3174.2.a.bd.1.2		5
23.5	odd	22		414.2.i.d.163.1		10
23.14	odd	22		414.2.i.d.127.1		10
23.22	odd	2		9522.2.a.bt.1.2		5
69.5	even	22		138.2.e.a.25.1	✓	10
69.14	even	22		138.2.e.a.127.1	yes	10
69.68	even	2		3174.2.a.bc.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.a.25.1	✓	10	69.5	even	22
138.2.e.a.127.1	yes	10	69.14	even	22
414.2.i.d.127.1		10	23.14	odd	22
414.2.i.d.163.1		10	23.5	odd	22
3174.2.a.bc.1.4		5	69.68	even	2
3174.2.a.bd.1.2		5	3.2	odd	2
9522.2.a.bq.1.4		5	1.1	even	1	trivial
9522.2.a.bt.1.2		5	23.22	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.23648 q^{5} +1.47889 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.23648 q^{5} +1.47889 q^{7} -1.00000 q^{8} +1.23648 q^{10} -0.886752 q^{11} +0.0673089 q^{13} -1.47889 q^{14} +1.00000 q^{16} -4.29177 q^{17} +4.66759 q^{19} -1.23648 q^{20} +0.886752 q^{22} -3.47112 q^{25} -0.0673089 q^{26} +1.47889 q^{28} +7.67816 q^{29} -1.59145 q^{31} -1.00000 q^{32} +4.29177 q^{34} -1.82862 q^{35} +5.22871 q^{37} -4.66759 q^{38} +1.23648 q^{40} +0.135933 q^{41} -6.57798 q^{43} -0.886752 q^{44} -8.31271 q^{47} -4.81288 q^{49} +3.47112 q^{50} +0.0673089 q^{52} -2.99558 q^{53} +1.09645 q^{55} -1.47889 q^{56} -7.67816 q^{58} +6.83576 q^{59} +4.60015 q^{61} +1.59145 q^{62} +1.00000 q^{64} -0.0832260 q^{65} -12.4051 q^{67} -4.29177 q^{68} +1.82862 q^{70} +7.31821 q^{71} +13.1978 q^{73} -5.22871 q^{74} +4.66759 q^{76} -1.31141 q^{77} -7.89446 q^{79} -1.23648 q^{80} -0.135933 q^{82} -8.40858 q^{83} +5.30668 q^{85} +6.57798 q^{86} +0.886752 q^{88} +11.7235 q^{89} +0.0995426 q^{91} +8.31271 q^{94} -5.77138 q^{95} +17.1665 q^{97} +4.81288 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} - 7 q^{5} + 7 q^{7} - 5 q^{8}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 - 7 * q^5 + 7 * q^7 - 5 * q^8	\( 5 q - 5 q^{2} + 5 q^{4} - 7 q^{5} + 7 q^{7} - 5 q^{8} + 7 q^{10} - 13 q^{11} - 4 q^{13} - 7 q^{14} + 5 q^{16} - 9 q^{17} + 11 q^{19} - 7 q^{20} + 13 q^{22} - 2 q^{25} + 4 q^{26} + 7 q^{28} + 7 q^{29} - 8 q^{31} - 5 q^{32} + 9 q^{34} - q^{35} + 12 q^{37} - 11 q^{38} + 7 q^{40} + 10 q^{41} + 4 q^{43} - 13 q^{44} + 24 q^{47} - 12 q^{49} + 2 q^{50} - 4 q^{52} - 9 q^{53} + 16 q^{55} - 7 q^{56} - 7 q^{58} + 14 q^{59} + 5 q^{61} + 8 q^{62} + 5 q^{64} - 12 q^{65} + 13 q^{67} - 9 q^{68} + q^{70} + 19 q^{71} + 4 q^{73} - 12 q^{74} + 11 q^{76} - 5 q^{77} + 4 q^{79} - 7 q^{80} - 10 q^{82} - 24 q^{83} + 17 q^{85} - 4 q^{86} + 13 q^{88} - 4 q^{89} - 21 q^{91} - 24 q^{94} + 11 q^{95} - 9 q^{97} + 12 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 - 7 * q^5 + 7 * q^7 - 5 * q^8 + 7 * q^10 - 13 * q^11 - 4 * q^13 - 7 * q^14 + 5 * q^16 - 9 * q^17 + 11 * q^19 - 7 * q^20 + 13 * q^22 - 2 * q^25 + 4 * q^26 + 7 * q^28 + 7 * q^29 - 8 * q^31 - 5 * q^32 + 9 * q^34 - q^35 + 12 * q^37 - 11 * q^38 + 7 * q^40 + 10 * q^41 + 4 * q^43 - 13 * q^44 + 24 * q^47 - 12 * q^49 + 2 * q^50 - 4 * q^52 - 9 * q^53 + 16 * q^55 - 7 * q^56 - 7 * q^58 + 14 * q^59 + 5 * q^61 + 8 * q^62 + 5 * q^64 - 12 * q^65 + 13 * q^67 - 9 * q^68 + q^70 + 19 * q^71 + 4 * q^73 - 12 * q^74 + 11 * q^76 - 5 * q^77 + 4 * q^79 - 7 * q^80 - 10 * q^82 - 24 * q^83 + 17 * q^85 - 4 * q^86 + 13 * q^88 - 4 * q^89 - 21 * q^91 - 24 * q^94 + 11 * q^95 - 9 * q^97 + 12 * q^98

Introduction

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