LMFDB - Embedded newform 7569.2.a.bj.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7569,2,Mod(1,7569)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7569, 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("7569.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.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:	$60.4387692899$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 6x^{7} + 33x^{6} + 6x^{5} - 90x^{4} + 21x^{3} + 84x^{2} - 36x - 8 $ x^9 - 4x^8 - 6x^7 + 33x^6 + 6x^5 - 90x^4 + 21x^3 + 84x^2 - 36x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 87)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.14438$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.144378 q^{2} -1.97916 q^{4} -1.55424 q^{5} -1.52752 q^{7} -0.574502 q^{8} +O(q^{10})$	\(q+0.144378 q^{2} -1.97916 q^{4} -1.55424 q^{5} -1.52752 q^{7} -0.574502 q^{8} -0.224397 q^{10} +0.767834 q^{11} -2.61812 q^{13} -0.220540 q^{14} +3.87536 q^{16} +3.51434 q^{17} +5.24800 q^{19} +3.07607 q^{20} +0.110858 q^{22} -8.49729 q^{23} -2.58435 q^{25} -0.377999 q^{26} +3.02320 q^{28} -1.41562 q^{31} +1.70852 q^{32} +0.507393 q^{34} +2.37413 q^{35} -3.00581 q^{37} +0.757696 q^{38} +0.892912 q^{40} -5.79055 q^{41} +4.29616 q^{43} -1.51966 q^{44} -1.22682 q^{46} -1.77439 q^{47} -4.66668 q^{49} -0.373123 q^{50} +5.18167 q^{52} -6.77178 q^{53} -1.19340 q^{55} +0.877564 q^{56} -14.9605 q^{59} +13.7169 q^{61} -0.204385 q^{62} -7.50406 q^{64} +4.06918 q^{65} -7.29495 q^{67} -6.95542 q^{68} +0.342772 q^{70} +4.81074 q^{71} +9.19937 q^{73} -0.433973 q^{74} -10.3866 q^{76} -1.17288 q^{77} +0.545882 q^{79} -6.02323 q^{80} -0.836028 q^{82} -11.6477 q^{83} -5.46211 q^{85} +0.620271 q^{86} -0.441122 q^{88} -13.7779 q^{89} +3.99924 q^{91} +16.8175 q^{92} -0.256183 q^{94} -8.15664 q^{95} -1.77624 q^{97} -0.673766 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} - 24 q^{8}+O(q^{10}) $ 9 * q - 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 - 24 * q^8	$ 9 q - 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} - 24 q^{8} + q^{11} + q^{13} - 9 q^{14} + 35 q^{16} - 2 q^{17} + 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} + 8 q^{26} + 40 q^{28} + 8 q^{31} - 43 q^{32} - 4 q^{34} - 22 q^{35} + 27 q^{37} + 30 q^{38} - 29 q^{40} - 12 q^{41} + 16 q^{43} - 37 q^{44} - 22 q^{46} + 8 q^{47} - 6 q^{49} + 7 q^{50} + 33 q^{52} + 8 q^{53} + 9 q^{55} - 40 q^{56} + 16 q^{59} + 21 q^{61} + 32 q^{62} + 36 q^{64} + 31 q^{65} + 3 q^{67} - 33 q^{68} - 6 q^{70} + 33 q^{71} + 3 q^{73} - 28 q^{74} - 26 q^{76} - 24 q^{77} + 3 q^{79} + 64 q^{80} + 13 q^{82} - 13 q^{83} + 6 q^{85} - 58 q^{86} + 27 q^{88} - 6 q^{89} + q^{91} + 29 q^{92} - 18 q^{94} - 48 q^{95} + 4 q^{97} + 30 q^{98}+O(q^{100}) $ 9 * q - 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 - 24 * q^8 + q^11 + q^13 - 9 * q^14 + 35 * q^16 - 2 * q^17 + 9 * q^19 + 18 * q^20 - 4 * q^22 + 4 * q^23 + q^25 + 8 * q^26 + 40 * q^28 + 8 * q^31 - 43 * q^32 - 4 * q^34 - 22 * q^35 + 27 * q^37 + 30 * q^38 - 29 * q^40 - 12 * q^41 + 16 * q^43 - 37 * q^44 - 22 * q^46 + 8 * q^47 - 6 * q^49 + 7 * q^50 + 33 * q^52 + 8 * q^53 + 9 * q^55 - 40 * q^56 + 16 * q^59 + 21 * q^61 + 32 * q^62 + 36 * q^64 + 31 * q^65 + 3 * q^67 - 33 * q^68 - 6 * q^70 + 33 * q^71 + 3 * q^73 - 28 * q^74 - 26 * q^76 - 24 * q^77 + 3 * q^79 + 64 * q^80 + 13 * q^82 - 13 * q^83 + 6 * q^85 - 58 * q^86 + 27 * q^88 - 6 * q^89 + q^91 + 29 * q^92 - 18 * q^94 - 48 * q^95 + 4 * q^97 + 30 * 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 $

$2$	0.144378	0.102091	0.0510453	−	0.998696i	$-0.483745\pi$
$2$	0.144378	0.102091	0.0510453	+	0.998696i	$0.483745\pi$
$3$	0	0
$3$	0	0
$4$	−1.97916	−0.989578
$5$	−1.55424	−0.695076	−0.347538	−	0.937666i	$-0.612982\pi$
$5$	−1.55424	−0.695076	−0.347538	+	0.937666i	$0.612982\pi$
$6$	0	0
$7$	−1.52752	−0.577349	−0.288674	−	0.957427i	$-0.593214\pi$
$7$	−1.52752	−0.577349	−0.288674	+	0.957427i	$0.593214\pi$
$8$	−0.574502	−0.203117
$9$	0	0
$10$	−0.224397	−0.0709607
$11$	0.767834	0.231511	0.115755	−	0.993278i	$-0.463071\pi$
$11$	0.767834	0.231511	0.115755	+	0.993278i	$0.463071\pi$
$12$	0	0
$13$	−2.61812	−0.726137	−0.363068	−	0.931763i	$-0.618271\pi$
$13$	−2.61812	−0.726137	−0.363068	+	0.931763i	$0.618271\pi$
$14$	−0.220540	−0.0589419
$15$	0	0
$16$	3.87536	0.968841
$17$	3.51434	0.852353	0.426176	−	0.904640i	$-0.359860\pi$
$17$	3.51434	0.852353	0.426176	+	0.904640i	$0.359860\pi$
$18$	0	0
$19$	5.24800	1.20397	0.601987	−	0.798506i	$-0.294376\pi$
$19$	5.24800	1.20397	0.601987	+	0.798506i	$0.294376\pi$
$20$	3.07607	0.687831
$21$	0	0
$22$	0.110858	0.0236351
$23$	−8.49729	−1.77181	−0.885904	−	0.463869i	$-0.846461\pi$
$23$	−8.49729	−1.77181	−0.885904	+	0.463869i	$0.846461\pi$
$24$	0	0
$25$	−2.58435	−0.516870
$26$	−0.377999	−0.0741317
$27$	0	0
$28$	3.02320	0.571331
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−1.41562	−0.254254	−0.127127	−	0.991886i	$-0.540576\pi$
$31$	−1.41562	−0.254254	−0.127127	+	0.991886i	$0.540576\pi$
$32$	1.70852	0.302027
$33$	0	0
$34$	0.507393	0.0870172
$35$	2.37413	0.401301
$36$	0	0
$37$	−3.00581	−0.494152	−0.247076	−	0.968996i	$-0.579470\pi$
$37$	−3.00581	−0.494152	−0.247076	+	0.968996i	$0.579470\pi$
$38$	0.757696	0.122915
$39$	0	0
$40$	0.892912	0.141182
$41$	−5.79055	−0.904332	−0.452166	−	0.891934i	$-0.649349\pi$
$41$	−5.79055	−0.904332	−0.452166	+	0.891934i	$0.649349\pi$
$42$	0	0
$43$	4.29616	0.655159	0.327579	−	0.944824i	$-0.393767\pi$
$43$	4.29616	0.655159	0.327579	+	0.944824i	$0.393767\pi$
$44$	−1.51966	−0.229098
$45$	0	0
$46$	−1.22682	−0.180885
$47$	−1.77439	−0.258821	−0.129411	−	0.991591i	$-0.541309\pi$
$47$	−1.77439	−0.258821	−0.129411	+	0.991591i	$0.541309\pi$
$48$	0	0
$49$	−4.66668	−0.666669
$50$	−0.373123	−0.0527676
$51$	0	0
$52$	5.18167	0.718568
$53$	−6.77178	−0.930175	−0.465087	−	0.885265i	$-0.653977\pi$
$53$	−6.77178	−0.930175	−0.465087	+	0.885265i	$0.653977\pi$
$54$	0	0
$55$	−1.19340	−0.160917
$56$	0.877564	0.117269
$57$	0	0
$58$	0	0
$59$	−14.9605	−1.94769	−0.973845	−	0.227213i	$-0.927039\pi$
$59$	−14.9605	−1.94769	−0.973845	+	0.227213i	$0.927039\pi$
$60$	0	0
$61$	13.7169	1.75627	0.878133	−	0.478416i	$-0.158789\pi$
$61$	13.7169	1.75627	0.878133	+	0.478416i	$0.158789\pi$
$62$	−0.204385	−0.0259569
$63$	0	0
$64$	−7.50406	−0.938007
$65$	4.06918	0.504720
$66$	0	0
$67$	−7.29495	−0.891220	−0.445610	−	0.895227i	$-0.647013\pi$
$67$	−7.29495	−0.891220	−0.445610	+	0.895227i	$0.647013\pi$
$68$	−6.95542	−0.843469
$69$	0	0
$70$	0.342772	0.0409690
$71$	4.81074	0.570930	0.285465	−	0.958389i	$-0.407852\pi$
$71$	4.81074	0.570930	0.285465	+	0.958389i	$0.407852\pi$
$72$	0	0
$73$	9.19937	1.07670	0.538352	−	0.842720i	$-0.319047\pi$
$73$	9.19937	1.07670	0.538352	+	0.842720i	$0.319047\pi$
$74$	−0.433973	−0.0504483
$75$	0	0
$76$	−10.3866	−1.19143
$77$	−1.17288	−0.133662
$78$	0	0
$79$	0.545882	0.0614165	0.0307083	−	0.999528i	$-0.490224\pi$
$79$	0.545882	0.0614165	0.0307083	+	0.999528i	$0.490224\pi$
$80$	−6.02323	−0.673418
$81$	0	0
$82$	−0.836028	−0.0923238
$83$	−11.6477	−1.27851	−0.639253	−	0.768997i	$-0.720756\pi$
$83$	−11.6477	−1.27851	−0.639253	+	0.768997i	$0.720756\pi$
$84$	0	0
$85$	−5.46211	−0.592449
$86$	0.620271	0.0668856
$87$	0	0
$88$	−0.441122	−0.0470238
$89$	−13.7779	−1.46046	−0.730228	−	0.683203i	$-0.760586\pi$
$89$	−13.7779	−1.46046	−0.730228	+	0.683203i	$0.760586\pi$
$90$	0	0
$91$	3.99924	0.419234
$92$	16.8175	1.75334
$93$	0	0
$94$	−0.256183	−0.0264232
$95$	−8.15664	−0.836853
$96$	0	0
$97$	−1.77624	−0.180350	−0.0901751	−	0.995926i	$-0.528743\pi$
$97$	−1.77624	−0.180350	−0.0901751	+	0.995926i	$0.528743\pi$
$98$	−0.673766	−0.0680606
$99$	0	0
$100$	5.11483	0.511483
$101$	4.46601	0.444384	0.222192	−	0.975003i	$-0.428679\pi$
$101$	4.46601	0.444384	0.222192	+	0.975003i	$0.428679\pi$
$102$	0	0
$103$	10.3199	1.01685	0.508424	−	0.861107i	$-0.330228\pi$
$103$	10.3199	1.01685	0.508424	+	0.861107i	$0.330228\pi$
$104$	1.50412	0.147491
$105$	0	0
$106$	−0.977695	−0.0949621
$107$	15.6906	1.51687	0.758435	−	0.651749i	$-0.225964\pi$
$107$	15.6906	1.51687	0.758435	+	0.651749i	$0.225964\pi$
$108$	0	0
$109$	−9.83683	−0.942197	−0.471099	−	0.882081i	$-0.656142\pi$
$109$	−9.83683	−0.942197	−0.471099	+	0.882081i	$0.656142\pi$
$110$	−0.172300	−0.0164282
$111$	0	0
$112$	−5.91970	−0.559359
$113$	12.6581	1.19078	0.595389	−	0.803437i	$-0.296998\pi$
$113$	12.6581	1.19078	0.595389	+	0.803437i	$0.296998\pi$
$114$	0	0
$115$	13.2068	1.23154
$116$	0	0
$117$	0	0
$118$	−2.15996	−0.198841
$119$	−5.36823	−0.492105
$120$	0	0
$121$	−10.4104	−0.946403
$122$	1.98041	0.179298
$123$	0	0
$124$	2.80174	0.251604
$125$	11.7879	1.05434
$126$	0	0
$127$	19.3386	1.71602	0.858012	−	0.513630i	$-0.171700\pi$
$127$	19.3386	1.71602	0.858012	+	0.513630i	$0.171700\pi$
$128$	−4.50046	−0.397788
$129$	0	0
$130$	0.587500	0.0515271
$131$	−16.8082	−1.46854	−0.734271	−	0.678857i	$-0.762476\pi$
$131$	−16.8082	−1.46854	−0.734271	+	0.678857i	$0.762476\pi$
$132$	0	0
$133$	−8.01644	−0.695113
$134$	−1.05323	−0.0909852
$135$	0	0
$136$	−2.01900	−0.173127
$137$	−18.2512	−1.55930	−0.779652	−	0.626213i	$-0.784604\pi$
$137$	−18.2512	−1.55930	−0.779652	+	0.626213i	$0.784604\pi$
$138$	0	0
$139$	−6.86591	−0.582359	−0.291180	−	0.956668i	$-0.594048\pi$
$139$	−6.86591	−0.582359	−0.291180	+	0.956668i	$0.594048\pi$
$140$	−4.69877	−0.397118
$141$	0	0
$142$	0.694565	0.0582866
$143$	−2.01028	−0.168108
$144$	0	0
$145$	0	0
$146$	1.32819	0.109921
$147$	0	0
$148$	5.94897	0.489002
$149$	1.82628	0.149615	0.0748074	−	0.997198i	$-0.476166\pi$
$149$	1.82628	0.149615	0.0748074	+	0.997198i	$0.476166\pi$
$150$	0	0
$151$	6.56174	0.533987	0.266993	−	0.963698i	$-0.413970\pi$
$151$	6.56174	0.533987	0.266993	+	0.963698i	$0.413970\pi$
$152$	−3.01499	−0.244548
$153$	0	0
$154$	−0.169338	−0.0136457
$155$	2.20021	0.176725
$156$	0	0
$157$	20.8825	1.66661	0.833304	−	0.552815i	$-0.186447\pi$
$157$	20.8825	1.66661	0.833304	+	0.552815i	$0.186447\pi$
$158$	0.0788133	0.00627005
$159$	0	0
$160$	−2.65545	−0.209931
$161$	12.9798	1.02295
$162$	0	0
$163$	−0.249811	−0.0195667	−0.00978335	−	0.999952i	$-0.503114\pi$
$163$	−0.249811	−0.0195667	−0.00978335	+	0.999952i	$0.503114\pi$
$164$	11.4604	0.894907
$165$	0	0
$166$	−1.68168	−0.130523
$167$	−13.4150	−1.03808	−0.519042	−	0.854749i	$-0.673711\pi$
$167$	−13.4150	−1.03808	−0.519042	+	0.854749i	$0.673711\pi$
$168$	0	0
$169$	−6.14543	−0.472726
$170$	−0.788609	−0.0604835
$171$	0	0
$172$	−8.50277	−0.648330
$173$	3.51856	0.267511	0.133755	−	0.991014i	$-0.457296\pi$
$173$	3.51856	0.267511	0.133755	+	0.991014i	$0.457296\pi$
$174$	0	0
$175$	3.94765	0.298414
$176$	2.97564	0.224297
$177$	0	0
$178$	−1.98923	−0.149099
$179$	2.44064	0.182422	0.0912109	−	0.995832i	$-0.470926\pi$
$179$	2.44064	0.182422	0.0912109	+	0.995832i	$0.470926\pi$
$180$	0	0
$181$	−0.647356	−0.0481176	−0.0240588	−	0.999711i	$-0.507659\pi$
$181$	−0.647356	−0.0481176	−0.0240588	+	0.999711i	$0.507659\pi$
$182$	0.577401	0.0427998
$183$	0	0
$184$	4.88171	0.359885
$185$	4.67174	0.343473
$186$	0	0
$187$	2.69843	0.197329
$188$	3.51179	0.256124
$189$	0	0
$190$	−1.17764	−0.0854349
$191$	20.7888	1.50422	0.752112	−	0.659035i	$-0.229035\pi$
$191$	20.7888	1.50422	0.752112	+	0.659035i	$0.229035\pi$
$192$	0	0
$193$	20.2432	1.45714	0.728569	−	0.684973i	$-0.240186\pi$
$193$	20.2432	1.45714	0.728569	+	0.684973i	$0.240186\pi$
$194$	−0.256450	−0.0184121
$195$	0	0
$196$	9.23609	0.659720
$197$	−3.44438	−0.245402	−0.122701	−	0.992444i	$-0.539156\pi$
$197$	−3.44438	−0.245402	−0.122701	+	0.992444i	$0.539156\pi$
$198$	0	0
$199$	12.0270	0.852574	0.426287	−	0.904588i	$-0.359821\pi$
$199$	12.0270	0.852574	0.426287	+	0.904588i	$0.359821\pi$
$200$	1.48471	0.104985
$201$	0	0
$202$	0.644793	0.0453675
$203$	0	0
$204$	0	0
$205$	8.99988	0.628579
$206$	1.48996	0.103811
$207$	0	0
$208$	−10.1462	−0.703511
$209$	4.02960	0.278733
$210$	0	0
$211$	4.16064	0.286430	0.143215	−	0.989692i	$-0.454256\pi$
$211$	4.16064	0.286430	0.143215	+	0.989692i	$0.454256\pi$
$212$	13.4024	0.920480
$213$	0	0
$214$	2.26538	0.154858
$215$	−6.67725	−0.455385
$216$	0	0
$217$	2.16240	0.146793
$218$	−1.42022	−0.0961895
$219$	0	0
$220$	2.36191	0.159240
$221$	−9.20097	−0.618924
$222$	0	0
$223$	−4.70372	−0.314984	−0.157492	−	0.987520i	$-0.550341\pi$
$223$	−4.70372	−0.314984	−0.157492	+	0.987520i	$0.550341\pi$
$224$	−2.60980	−0.174375
$225$	0	0
$226$	1.82756	0.121567
$227$	13.6864	0.908400	0.454200	−	0.890900i	$-0.349925\pi$
$227$	13.6864	0.908400	0.454200	+	0.890900i	$0.349925\pi$
$228$	0	0
$229$	20.1944	1.33448	0.667241	−	0.744842i	$-0.267475\pi$
$229$	20.1944	1.33448	0.667241	+	0.744842i	$0.267475\pi$
$230$	1.90677	0.125729
$231$	0	0
$232$	0	0
$233$	7.28500	0.477257	0.238628	−	0.971111i	$-0.423302\pi$
$233$	7.28500	0.477257	0.238628	+	0.971111i	$0.423302\pi$
$234$	0	0
$235$	2.75782	0.179900
$236$	29.6091	1.92739
$237$	0	0
$238$	−0.775053	−0.0502393
$239$	23.3694	1.51164	0.755822	−	0.654777i	$-0.227238\pi$
$239$	23.3694	1.51164	0.755822	+	0.654777i	$0.227238\pi$
$240$	0	0
$241$	25.4593	1.63998	0.819989	−	0.572379i	$-0.193980\pi$
$241$	25.4593	1.63998	0.819989	+	0.572379i	$0.193980\pi$
$242$	−1.50304	−0.0966188
$243$	0	0
$244$	−27.1478	−1.73796
$245$	7.25312	0.463385
$246$	0	0
$247$	−13.7399	−0.874250
$248$	0.813279	0.0516433
$249$	0	0
$250$	1.70191	0.107638
$251$	13.0548	0.824009	0.412004	−	0.911182i	$-0.364829\pi$
$251$	13.0548	0.824009	0.412004	+	0.911182i	$0.364829\pi$
$252$	0	0
$253$	−6.52451	−0.410192
$254$	2.79207	0.175190
$255$	0	0
$256$	14.3583	0.897397
$257$	−1.37201	−0.0855836	−0.0427918	−	0.999084i	$-0.513625\pi$
$257$	−1.37201	−0.0855836	−0.0427918	+	0.999084i	$0.513625\pi$
$258$	0	0
$259$	4.59144	0.285298
$260$	−8.05354	−0.499459
$261$	0	0
$262$	−2.42674	−0.149924
$263$	−20.7538	−1.27973	−0.639866	−	0.768487i	$-0.721010\pi$
$263$	−20.7538	−1.27973	−0.639866	+	0.768487i	$0.721010\pi$
$264$	0	0
$265$	10.5249	0.646542
$266$	−1.15740	−0.0709645
$267$	0	0
$268$	14.4378	0.881931
$269$	−0.958446	−0.0584375	−0.0292187	−	0.999573i	$-0.509302\pi$
$269$	−0.958446	−0.0584375	−0.0292187	+	0.999573i	$0.509302\pi$
$270$	0	0
$271$	−21.1163	−1.28272	−0.641361	−	0.767240i	$-0.721630\pi$
$271$	−21.1163	−1.28272	−0.641361	+	0.767240i	$0.721630\pi$
$272$	13.6193	0.825794
$273$	0	0
$274$	−2.63507	−0.159190
$275$	−1.98435	−0.119661
$276$	0	0
$277$	10.4181	0.625963	0.312982	−	0.949759i	$-0.398672\pi$
$277$	10.4181	0.625963	0.312982	+	0.949759i	$0.398672\pi$
$278$	−0.991286	−0.0594534
$279$	0	0
$280$	−1.36394	−0.0815111
$281$	−3.34474	−0.199530	−0.0997651	−	0.995011i	$-0.531809\pi$
$281$	−3.34474	−0.199530	−0.0997651	+	0.995011i	$0.531809\pi$
$282$	0	0
$283$	−15.7748	−0.937713	−0.468856	−	0.883274i	$-0.655334\pi$
$283$	−15.7748	−0.937713	−0.468856	+	0.883274i	$0.655334\pi$
$284$	−9.52120	−0.564979
$285$	0	0
$286$	−0.290240	−0.0171623
$287$	8.84518	0.522115
$288$	0	0
$289$	−4.64941	−0.273495
$290$	0	0
$291$	0	0
$292$	−18.2070	−1.06548
$293$	15.8084	0.923538	0.461769	−	0.887000i	$-0.347215\pi$
$293$	15.8084	0.923538	0.461769	+	0.887000i	$0.347215\pi$
$294$	0	0
$295$	23.2521	1.35379
$296$	1.72685	0.100371
$297$	0	0
$298$	0.263675	0.0152743
$299$	22.2470	1.28657
$300$	0	0
$301$	−6.56248	−0.378255
$302$	0.947370	0.0545150
$303$	0	0
$304$	20.3379	1.16646
$305$	−21.3193	−1.22074
$306$	0	0
$307$	8.83474	0.504225	0.252113	−	0.967698i	$-0.418875\pi$
$307$	8.83474	0.504225	0.252113	+	0.967698i	$0.418875\pi$
$308$	2.32132	0.132269
$309$	0	0
$310$	0.317662	0.0180420
$311$	12.9878	0.736469	0.368235	−	0.929733i	$-0.379962\pi$
$311$	12.9878	0.736469	0.368235	+	0.929733i	$0.379962\pi$
$312$	0	0
$313$	19.1428	1.08201	0.541007	−	0.841018i	$-0.318043\pi$
$313$	19.1428	1.08201	0.541007	+	0.841018i	$0.318043\pi$
$314$	3.01498	0.170145
$315$	0	0
$316$	−1.08039	−0.0607764
$317$	−26.6435	−1.49645	−0.748225	−	0.663445i	$-0.769094\pi$
$317$	−26.6435	−1.49645	−0.748225	+	0.663445i	$0.769094\pi$
$318$	0	0
$319$	0	0
$320$	11.6631	0.651986
$321$	0	0
$322$	1.87400	0.104434
$323$	18.4433	1.02621
$324$	0	0
$325$	6.76615	0.375318
$326$	−0.0360672	−0.00199758
$327$	0	0
$328$	3.32668	0.183685
$329$	2.71042	0.149430
$330$	0	0
$331$	−2.21519	−0.121758	−0.0608789	−	0.998145i	$-0.519390\pi$
$331$	−2.21519	−0.121758	−0.0608789	+	0.998145i	$0.519390\pi$
$332$	23.0527	1.26518
$333$	0	0
$334$	−1.93683	−0.105979
$335$	11.3381	0.619465
$336$	0	0
$337$	8.01703	0.436715	0.218358	−	0.975869i	$-0.429930\pi$
$337$	8.01703	0.436715	0.218358	+	0.975869i	$0.429930\pi$
$338$	−0.887265	−0.0482609
$339$	0	0
$340$	10.8104	0.586275
$341$	−1.08696	−0.0588624
$342$	0	0
$343$	17.8211	0.962249
$344$	−2.46815	−0.133074
$345$	0	0
$346$	0.508002	0.0273104
$347$	26.2715	1.41033	0.705165	−	0.709044i	$-0.250873\pi$
$347$	26.2715	1.41033	0.705165	+	0.709044i	$0.250873\pi$
$348$	0	0
$349$	−3.86308	−0.206786	−0.103393	−	0.994641i	$-0.532970\pi$
$349$	−3.86308	−0.206786	−0.103393	+	0.994641i	$0.532970\pi$
$350$	0.569953	0.0304653
$351$	0	0
$352$	1.31186	0.0699224
$353$	−20.0667	−1.06804	−0.534021	−	0.845471i	$-0.679320\pi$
$353$	−20.0667	−1.06804	−0.534021	+	0.845471i	$0.679320\pi$
$354$	0	0
$355$	−7.47703	−0.396839
$356$	27.2686	1.44524
$357$	0	0
$358$	0.352374	0.0186236
$359$	21.2008	1.11894	0.559468	−	0.828852i	$-0.311006\pi$
$359$	21.2008	1.11894	0.559468	+	0.828852i	$0.311006\pi$
$360$	0	0
$361$	8.54155	0.449555
$362$	−0.0934640	−0.00491236
$363$	0	0
$364$	−7.91511	−0.414864
$365$	−14.2980	−0.748391
$366$	0	0
$367$	−19.5920	−1.02269	−0.511347	−	0.859374i	$-0.670853\pi$
$367$	−19.5920	−1.02269	−0.511347	+	0.859374i	$0.670853\pi$
$368$	−32.9301	−1.71660
$369$	0	0
$370$	0.674496	0.0350654
$371$	10.3440	0.537035
$372$	0	0
$373$	0.197465	0.0102244	0.00511218	−	0.999987i	$-0.498373\pi$
$373$	0.197465	0.0102244	0.00511218	+	0.999987i	$0.498373\pi$
$374$	0.389594	0.0201454
$375$	0	0
$376$	1.01939	0.0525711
$377$	0	0
$378$	0	0
$379$	−22.4911	−1.15529	−0.577645	−	0.816288i	$-0.696028\pi$
$379$	−22.4911	−1.15529	−0.577645	+	0.816288i	$0.696028\pi$
$380$	16.1433	0.828131
$381$	0	0
$382$	3.00144	0.153567
$383$	0.603334	0.0308289	0.0154145	−	0.999881i	$-0.495093\pi$
$383$	0.603334	0.0308289	0.0154145	+	0.999881i	$0.495093\pi$
$384$	0	0
$385$	1.82294	0.0929054
$386$	2.92267	0.148760
$387$	0	0
$388$	3.51546	0.178471
$389$	21.8263	1.10664	0.553318	−	0.832970i	$-0.313362\pi$
$389$	21.8263	1.10664	0.553318	+	0.832970i	$0.313362\pi$
$390$	0	0
$391$	−29.8624	−1.51021
$392$	2.68102	0.135412
$393$	0	0
$394$	−0.497292	−0.0250532
$395$	−0.848430	−0.0426891
$396$	0	0
$397$	14.4481	0.725132	0.362566	−	0.931958i	$-0.381901\pi$
$397$	14.4481	0.725132	0.362566	+	0.931958i	$0.381901\pi$
$398$	1.73644	0.0870398
$399$	0	0
$400$	−10.0153	−0.500765
$401$	26.3790	1.31730	0.658652	−	0.752448i	$-0.271127\pi$
$401$	26.3790	1.31730	0.658652	+	0.752448i	$0.271127\pi$
$402$	0	0
$403$	3.70628	0.184623
$404$	−8.83892	−0.439753
$405$	0	0
$406$	0	0
$407$	−2.30796	−0.114402
$408$	0	0
$409$	−31.0838	−1.53700	−0.768498	−	0.639852i	$-0.778996\pi$
$409$	−31.0838	−1.53700	−0.768498	+	0.639852i	$0.778996\pi$
$410$	1.29938	0.0641720
$411$	0	0
$412$	−20.4247	−1.00625
$413$	22.8525	1.12450
$414$	0	0
$415$	18.1033	0.888658
$416$	−4.47312	−0.219313
$417$	0	0
$418$	0.581785	0.0284560
$419$	−20.3478	−0.994053	−0.497026	−	0.867735i	$-0.665575\pi$
$419$	−20.3478	−0.994053	−0.497026	+	0.867735i	$0.665575\pi$
$420$	0	0
$421$	−8.52568	−0.415516	−0.207758	−	0.978180i	$-0.566617\pi$
$421$	−8.52568	−0.415516	−0.207758	+	0.978180i	$0.566617\pi$
$422$	0.600705	0.0292419
$423$	0	0
$424$	3.89040	0.188934
$425$	−9.08228	−0.440556
$426$	0	0
$427$	−20.9528	−1.01398
$428$	−31.0542	−1.50106
$429$	0	0
$430$	−0.964048	−0.0464905
$431$	4.76237	0.229395	0.114698	−	0.993400i	$-0.463410\pi$
$431$	4.76237	0.229395	0.114698	+	0.993400i	$0.463410\pi$
$432$	0	0
$433$	4.22530	0.203055	0.101527	−	0.994833i	$-0.467627\pi$
$433$	4.22530	0.203055	0.101527	+	0.994833i	$0.467627\pi$
$434$	0.312202	0.0149862
$435$	0	0
$436$	19.4686	0.932377
$437$	−44.5938	−2.13321
$438$	0	0
$439$	27.3895	1.30723	0.653615	−	0.756827i	$-0.273252\pi$
$439$	27.3895	1.30723	0.653615	+	0.756827i	$0.273252\pi$
$440$	0.685608	0.0326851
$441$	0	0
$442$	−1.32842	−0.0631864
$443$	−0.777164	−0.0369242	−0.0184621	−	0.999830i	$-0.505877\pi$
$443$	−0.777164	−0.0369242	−0.0184621	+	0.999830i	$0.505877\pi$
$444$	0	0
$445$	21.4141	1.01513
$446$	−0.679113	−0.0321569
$447$	0	0
$448$	11.4626	0.541557
$449$	3.12702	0.147573	0.0737865	−	0.997274i	$-0.476492\pi$
$449$	3.12702	0.147573	0.0737865	+	0.997274i	$0.476492\pi$
$450$	0	0
$451$	−4.44618	−0.209362
$452$	−25.0524	−1.17837
$453$	0	0
$454$	1.97602	0.0927391
$455$	−6.21576	−0.291399
$456$	0	0
$457$	−28.6290	−1.33921	−0.669604	−	0.742718i	$-0.733536\pi$
$457$	−28.6290	−1.33921	−0.669604	+	0.742718i	$0.733536\pi$
$458$	2.91562	0.136238
$459$	0	0
$460$	−26.1383	−1.21870
$461$	27.8892	1.29893	0.649464	−	0.760392i	$-0.274993\pi$
$461$	27.8892	1.29893	0.649464	+	0.760392i	$0.274993\pi$
$462$	0	0
$463$	−18.3196	−0.851382	−0.425691	−	0.904869i	$-0.639969\pi$
$463$	−18.3196	−0.851382	−0.425691	+	0.904869i	$0.639969\pi$
$464$	0	0
$465$	0	0
$466$	1.05179	0.0487234
$467$	21.8800	1.01249	0.506243	−	0.862391i	$-0.331034\pi$
$467$	21.8800	1.01249	0.506243	+	0.862391i	$0.331034\pi$
$468$	0	0
$469$	11.1432	0.514544
$470$	0.398168	0.0183661
$471$	0	0
$472$	8.59483	0.395609
$473$	3.29874	0.151676
$474$	0	0
$475$	−13.5627	−0.622298
$476$	10.6246	0.486976
$477$	0	0
$478$	3.37403	0.154325
$479$	−23.4525	−1.07157	−0.535786	−	0.844354i	$-0.679985\pi$
$479$	−23.4525	−1.07157	−0.535786	+	0.844354i	$0.679985\pi$
$480$	0	0
$481$	7.86958	0.358822
$482$	3.67576	0.167426
$483$	0	0
$484$	20.6039	0.936539
$485$	2.76070	0.125357
$486$	0	0
$487$	−35.6248	−1.61431	−0.807157	−	0.590336i	$-0.798995\pi$
$487$	−35.6248	−1.61431	−0.807157	+	0.590336i	$0.798995\pi$
$488$	−7.88038	−0.356728
$489$	0	0
$490$	1.04719	0.0473073
$491$	−2.80650	−0.126656	−0.0633278	−	0.997993i	$-0.520171\pi$
$491$	−2.80650	−0.126656	−0.0633278	+	0.997993i	$0.520171\pi$
$492$	0	0
$493$	0	0
$494$	−1.98374	−0.0892527
$495$	0	0
$496$	−5.48606	−0.246331
$497$	−7.34851	−0.329626
$498$	0	0
$499$	40.6599	1.82019	0.910094	−	0.414401i	$-0.136009\pi$
$499$	40.6599	1.82019	0.910094	+	0.414401i	$0.136009\pi$
$500$	−23.3300	−1.04335
$501$	0	0
$502$	1.88482	0.0841236
$503$	18.5373	0.826538	0.413269	−	0.910609i	$-0.364387\pi$
$503$	18.5373	0.826538	0.413269	+	0.910609i	$0.364387\pi$
$504$	0	0
$505$	−6.94123	−0.308881
$506$	−0.941995	−0.0418768
$507$	0	0
$508$	−38.2741	−1.69814
$509$	42.5525	1.88611	0.943054	−	0.332640i	$-0.107939\pi$
$509$	42.5525	1.88611	0.943054	+	0.332640i	$0.107939\pi$
$510$	0	0
$511$	−14.0522	−0.621634
$512$	11.0740	0.489404
$513$	0	0
$514$	−0.198088	−0.00873728
$515$	−16.0395	−0.706786
$516$	0	0
$517$	−1.36244	−0.0599199
$518$	0.662902	0.0291263
$519$	0	0
$520$	−2.33775	−0.102517
$521$	27.6372	1.21081	0.605404	−	0.795919i	$-0.293012\pi$
$521$	27.6372	1.21081	0.605404	+	0.795919i	$0.293012\pi$
$522$	0	0
$523$	−21.3122	−0.931919	−0.465959	−	0.884806i	$-0.654291\pi$
$523$	−21.3122	−0.931919	−0.465959	+	0.884806i	$0.654291\pi$
$524$	33.2661	1.45324
$525$	0	0
$526$	−2.99638	−0.130649
$527$	−4.97499	−0.216714
$528$	0	0
$529$	49.2040	2.13930
$530$	1.51957	0.0660058
$531$	0	0
$532$	15.8658	0.687868
$533$	15.1604	0.656668
$534$	0	0
$535$	−24.3869	−1.05434
$536$	4.19096	0.181022
$537$	0	0
$538$	−0.138378	−0.00596592
$539$	−3.58324	−0.154341
$540$	0	0
$541$	22.1075	0.950477	0.475239	−	0.879857i	$-0.342362\pi$
$541$	22.1075	0.950477	0.475239	+	0.879857i	$0.342362\pi$
$542$	−3.04872	−0.130954
$543$	0	0
$544$	6.00433	0.257433
$545$	15.2888	0.654898
$546$	0	0
$547$	16.5033	0.705630	0.352815	−	0.935693i	$-0.385224\pi$
$547$	16.5033	0.705630	0.352815	+	0.935693i	$0.385224\pi$
$548$	36.1219	1.54305
$549$	0	0
$550$	−0.286497	−0.0122163
$551$	0	0
$552$	0	0
$553$	−0.833846	−0.0354587
$554$	1.50414	0.0639050
$555$	0	0
$556$	13.5887	0.576290
$557$	−5.47950	−0.232174	−0.116087	−	0.993239i	$-0.537035\pi$
$557$	−5.47950	−0.232174	−0.116087	+	0.993239i	$0.537035\pi$
$558$	0	0
$559$	−11.2479	−0.475735
$560$	9.20061	0.388797
$561$	0	0
$562$	−0.482906	−0.0203702
$563$	−13.7797	−0.580744	−0.290372	−	0.956914i	$-0.593779\pi$
$563$	−13.7797	−0.580744	−0.290372	+	0.956914i	$0.593779\pi$
$564$	0	0
$565$	−19.6738	−0.827681
$566$	−2.27753	−0.0957317
$567$	0	0
$568$	−2.76378	−0.115966
$569$	−35.9279	−1.50618	−0.753088	−	0.657920i	$-0.771436\pi$
$569$	−35.9279	−1.50618	−0.753088	+	0.657920i	$0.771436\pi$
$570$	0	0
$571$	−6.56420	−0.274703	−0.137352	−	0.990522i	$-0.543859\pi$
$571$	−6.56420	−0.274703	−0.137352	+	0.990522i	$0.543859\pi$
$572$	3.97866	0.166356
$573$	0	0
$574$	1.27705	0.0533030
$575$	21.9600	0.915794
$576$	0	0
$577$	−8.06782	−0.335868	−0.167934	−	0.985798i	$-0.553710\pi$
$577$	−8.06782	−0.335868	−0.167934	+	0.985798i	$0.553710\pi$
$578$	−0.671273	−0.0279213
$579$	0	0
$580$	0	0
$581$	17.7922	0.738143
$582$	0	0
$583$	−5.19960	−0.215345
$584$	−5.28506	−0.218697
$585$	0	0
$586$	2.28239	0.0942845
$587$	8.05202	0.332343	0.166171	−	0.986097i	$-0.446860\pi$
$587$	8.05202	0.332343	0.166171	+	0.986097i	$0.446860\pi$
$588$	0	0
$589$	−7.42920	−0.306115
$590$	3.35710	0.138209
$591$	0	0
$592$	−11.6486	−0.478755
$593$	−34.4563	−1.41495	−0.707474	−	0.706739i	$-0.750165\pi$
$593$	−34.4563	−1.41495	−0.707474	+	0.706739i	$0.750165\pi$
$594$	0	0
$595$	8.34349	0.342050
$596$	−3.61449	−0.148055
$597$	0	0
$598$	3.21197	0.131347
$599$	31.6740	1.29416	0.647082	−	0.762421i	$-0.275989\pi$
$599$	31.6740	1.29416	0.647082	+	0.762421i	$0.275989\pi$
$600$	0	0
$601$	3.86546	0.157675	0.0788376	−	0.996887i	$-0.474879\pi$
$601$	3.86546	0.157675	0.0788376	+	0.996887i	$0.474879\pi$
$602$	−0.947477	−0.0386163
$603$	0	0
$604$	−12.9867	−0.528421
$605$	16.1803	0.657821
$606$	0	0
$607$	1.32954	0.0539645	0.0269823	−	0.999636i	$-0.491410\pi$
$607$	1.32954	0.0539645	0.0269823	+	0.999636i	$0.491410\pi$
$608$	8.96633	0.363633
$609$	0	0
$610$	−3.07803	−0.124626
$611$	4.64557	0.187940
$612$	0	0
$613$	3.02255	0.122080	0.0610399	−	0.998135i	$-0.480558\pi$
$613$	3.02255	0.122080	0.0610399	+	0.998135i	$0.480558\pi$
$614$	1.27554	0.0514767
$615$	0	0
$616$	0.673823	0.0271491
$617$	−1.92609	−0.0775415	−0.0387708	−	0.999248i	$-0.512344\pi$
$617$	−1.92609	−0.0775415	−0.0387708	+	0.999248i	$0.512344\pi$
$618$	0	0
$619$	0.295624	0.0118821	0.00594107	−	0.999982i	$-0.498109\pi$
$619$	0.295624	0.0118821	0.00594107	+	0.999982i	$0.498109\pi$
$620$	−4.35457	−0.174884
$621$	0	0
$622$	1.87515	0.0751866
$623$	21.0461	0.843192
$624$	0	0
$625$	−5.39938	−0.215975
$626$	2.76380	0.110464
$627$	0	0
$628$	−41.3298	−1.64924
$629$	−10.5634	−0.421192
$630$	0	0
$631$	33.0085	1.31405	0.657024	−	0.753869i	$-0.271815\pi$
$631$	33.0085	1.31405	0.657024	+	0.753869i	$0.271815\pi$
$632$	−0.313610	−0.0124748
$633$	0	0
$634$	−3.84674	−0.152774
$635$	−30.0568	−1.19277
$636$	0	0
$637$	12.2179	0.484093
$638$	0	0
$639$	0	0
$640$	6.99478	0.276493
$641$	21.5035	0.849336	0.424668	−	0.905349i	$-0.360391\pi$
$641$	21.5035	0.849336	0.424668	+	0.905349i	$0.360391\pi$
$642$	0	0
$643$	−16.6571	−0.656891	−0.328445	−	0.944523i	$-0.606525\pi$
$643$	−16.6571	−0.656891	−0.328445	+	0.944523i	$0.606525\pi$
$644$	−25.6890	−1.01229
$645$	0	0
$646$	2.66280	0.104767
$647$	0.641821	0.0252326	0.0126163	−	0.999920i	$-0.495984\pi$
$647$	0.641821	0.0252326	0.0126163	+	0.999920i	$0.495984\pi$
$648$	0	0
$649$	−11.4872	−0.450911
$650$	0.976882	0.0383165
$651$	0	0
$652$	0.494414	0.0193628
$653$	41.8052	1.63596	0.817982	−	0.575244i	$-0.195093\pi$
$653$	41.8052	1.63596	0.817982	+	0.575244i	$0.195093\pi$
$654$	0	0
$655$	26.1239	1.02075
$656$	−22.4405	−0.876154
$657$	0	0
$658$	0.391324	0.0152554
$659$	6.28494	0.244827	0.122413	−	0.992479i	$-0.460937\pi$
$659$	6.28494	0.244827	0.122413	+	0.992479i	$0.460937\pi$
$660$	0	0
$661$	−20.2713	−0.788461	−0.394231	−	0.919012i	$-0.628989\pi$
$661$	−20.2713	−0.788461	−0.394231	+	0.919012i	$0.628989\pi$
$662$	−0.319824	−0.0124303
$663$	0	0
$664$	6.69165	0.259686
$665$	12.4594	0.483156
$666$	0	0
$667$	0	0
$668$	26.5504	1.02726
$669$	0	0
$670$	1.63697	0.0632416
$671$	10.5323	0.406594
$672$	0	0
$673$	−21.7317	−0.837694	−0.418847	−	0.908057i	$-0.637566\pi$
$673$	−21.7317	−0.837694	−0.418847	+	0.908057i	$0.637566\pi$
$674$	1.15748	0.0445845
$675$	0	0
$676$	12.1628	0.467799
$677$	−15.7988	−0.607197	−0.303599	−	0.952800i	$-0.598188\pi$
$677$	−15.7988	−0.607197	−0.303599	+	0.952800i	$0.598188\pi$
$678$	0	0
$679$	2.71325	0.104125
$680$	3.13800	0.120337
$681$	0	0
$682$	−0.156934	−0.00600930
$683$	34.1996	1.30861	0.654306	−	0.756230i	$-0.272961\pi$
$683$	34.1996	1.30861	0.654306	+	0.756230i	$0.272961\pi$
$684$	0	0
$685$	28.3666	1.08383
$686$	2.57297	0.0982366
$687$	0	0
$688$	16.6492	0.634745
$689$	17.7293	0.675434
$690$	0	0
$691$	−34.3566	−1.30699	−0.653494	−	0.756931i	$-0.726698\pi$
$691$	−34.3566	−1.30699	−0.653494	+	0.756931i	$0.726698\pi$
$692$	−6.96377	−0.264723
$693$	0	0
$694$	3.79303	0.143981
$695$	10.6713	0.404784
$696$	0	0
$697$	−20.3500	−0.770810
$698$	−0.557743	−0.0211109
$699$	0	0
$700$	−7.81301	−0.295304
$701$	−1.71987	−0.0649588	−0.0324794	−	0.999472i	$-0.510340\pi$
$701$	−1.71987	−0.0649588	−0.0324794	+	0.999472i	$0.510340\pi$
$702$	0	0
$703$	−15.7745	−0.594947
$704$	−5.76187	−0.217159
$705$	0	0
$706$	−2.89719	−0.109037
$707$	−6.82192	−0.256565
$708$	0	0
$709$	42.0692	1.57994	0.789970	−	0.613145i	$-0.210096\pi$
$709$	42.0692	1.57994	0.789970	+	0.613145i	$0.210096\pi$
$710$	−1.07952	−0.0405136
$711$	0	0
$712$	7.91545	0.296644
$713$	12.0290	0.450489
$714$	0	0
$715$	3.12445	0.116848
$716$	−4.83040	−0.180521
$717$	0	0
$718$	3.06093	0.114233
$719$	−2.01196	−0.0750335	−0.0375168	−	0.999296i	$-0.511945\pi$
$719$	−2.01196	−0.0750335	−0.0375168	+	0.999296i	$0.511945\pi$
$720$	0	0
$721$	−15.7638	−0.587076
$722$	1.23321	0.0458954
$723$	0	0
$724$	1.28122	0.0476161
$725$	0	0
$726$	0	0
$727$	7.32560	0.271692	0.135846	−	0.990730i	$-0.456625\pi$
$727$	7.32560	0.271692	0.135846	+	0.990730i	$0.456625\pi$
$728$	−2.29757	−0.0851536
$729$	0	0
$730$	−2.06431	−0.0764037
$731$	15.0982	0.558426
$732$	0	0
$733$	5.24481	0.193722	0.0968608	−	0.995298i	$-0.469120\pi$
$733$	5.24481	0.193722	0.0968608	+	0.995298i	$0.469120\pi$
$734$	−2.82865	−0.104407
$735$	0	0
$736$	−14.5178	−0.535133
$737$	−5.60131	−0.206327
$738$	0	0
$739$	23.5744	0.867198	0.433599	−	0.901106i	$-0.357243\pi$
$739$	23.5744	0.867198	0.433599	+	0.901106i	$0.357243\pi$
$740$	−9.24610	−0.339893
$741$	0	0
$742$	1.49345	0.0548262
$743$	19.5239	0.716263	0.358131	−	0.933671i	$-0.383414\pi$
$743$	19.5239	0.716263	0.358131	+	0.933671i	$0.383414\pi$
$744$	0	0
$745$	−2.83847	−0.103994
$746$	0.0285096	0.00104381
$747$	0	0
$748$	−5.34061	−0.195272
$749$	−23.9678	−0.875763
$750$	0	0
$751$	1.41626	0.0516802	0.0258401	−	0.999666i	$-0.491774\pi$
$751$	1.41626	0.0516802	0.0258401	+	0.999666i	$0.491774\pi$
$752$	−6.87641	−0.250757
$753$	0	0
$754$	0	0
$755$	−10.1985	−0.371161
$756$	0	0
$757$	43.3206	1.57451	0.787257	−	0.616625i	$-0.211500\pi$
$757$	43.3206	1.57451	0.787257	+	0.616625i	$0.211500\pi$
$758$	−3.24722	−0.117944
$759$	0	0
$760$	4.68601	0.169979
$761$	−0.481297	−0.0174470	−0.00872351	−	0.999962i	$-0.502777\pi$
$761$	−0.481297	−0.0174470	−0.00872351	+	0.999962i	$0.502777\pi$
$762$	0	0
$763$	15.0260	0.543976
$764$	−41.1443	−1.48855
$765$	0	0
$766$	0.0871081	0.00314734
$767$	39.1684	1.41429
$768$	0	0
$769$	−16.2798	−0.587065	−0.293532	−	0.955949i	$-0.594831\pi$
$769$	−16.2798	−0.587065	−0.293532	+	0.955949i	$0.594831\pi$
$770$	0.263192	0.00948477
$771$	0	0
$772$	−40.0644	−1.44195
$773$	−17.3203	−0.622968	−0.311484	−	0.950251i	$-0.600826\pi$
$773$	−17.3203	−0.622968	−0.311484	+	0.950251i	$0.600826\pi$
$774$	0	0
$775$	3.65847	0.131416
$776$	1.02046	0.0366322
$777$	0	0
$778$	3.15123	0.112977
$779$	−30.3888	−1.08879
$780$	0	0
$781$	3.69385	0.132176
$782$	−4.31147	−0.154178
$783$	0	0
$784$	−18.0851	−0.645896
$785$	−32.4564	−1.15842
$786$	0	0
$787$	0.680377	0.0242528	0.0121264	−	0.999926i	$-0.496140\pi$
$787$	0.680377	0.0242528	0.0121264	+	0.999926i	$0.496140\pi$
$788$	6.81695	0.242844
$789$	0	0
$790$	−0.122495	−0.00435816
$791$	−19.3356	−0.687494
$792$	0	0
$793$	−35.9125	−1.27529
$794$	2.08599	0.0740291
$795$	0	0
$796$	−23.8034	−0.843688
$797$	−44.8530	−1.58878	−0.794388	−	0.607411i	$-0.792208\pi$
$797$	−44.8530	−1.58878	−0.794388	+	0.607411i	$0.792208\pi$
$798$	0	0
$799$	−6.23581	−0.220607
$800$	−4.41542	−0.156109
$801$	0	0
$802$	3.80854	0.134484
$803$	7.06359	0.249269
$804$	0	0
$805$	−20.1737	−0.711028
$806$	0.535105	0.0188483
$807$	0	0
$808$	−2.56573	−0.0902621
$809$	−29.4154	−1.03419	−0.517095	−	0.855928i	$-0.672987\pi$
$809$	−29.4154	−1.03419	−0.517095	+	0.855928i	$0.672987\pi$
$810$	0	0
$811$	−20.6973	−0.726781	−0.363391	−	0.931637i	$-0.618381\pi$
$811$	−20.6973	−0.726781	−0.363391	+	0.931637i	$0.618381\pi$
$812$	0	0
$813$	0	0
$814$	−0.333219	−0.0116793
$815$	0.388265	0.0136003
$816$	0	0
$817$	22.5463	0.788795
$818$	−4.48782	−0.156913
$819$	0	0
$820$	−17.8122	−0.622028
$821$	−27.5661	−0.962063	−0.481031	−	0.876703i	$-0.659738\pi$
$821$	−27.5661	−0.962063	−0.481031	+	0.876703i	$0.659738\pi$
$822$	0	0
$823$	23.0508	0.803499	0.401750	−	0.915750i	$-0.368402\pi$
$823$	23.0508	0.803499	0.401750	+	0.915750i	$0.368402\pi$
$824$	−5.92880	−0.206539
$825$	0	0
$826$	3.29939	0.114800
$827$	44.5689	1.54981	0.774907	−	0.632076i	$-0.217797\pi$
$827$	44.5689	1.54981	0.774907	+	0.632076i	$0.217797\pi$
$828$	0	0
$829$	44.8796	1.55873	0.779367	−	0.626568i	$-0.215541\pi$
$829$	44.8796	1.55873	0.779367	+	0.626568i	$0.215541\pi$
$830$	2.61372	0.0907236
$831$	0	0
$832$	19.6465	0.681121
$833$	−16.4003	−0.568237
$834$	0	0
$835$	20.8501	0.721546
$836$	−7.97519	−0.275828
$837$	0	0
$838$	−2.93777	−0.101483
$839$	−13.0101	−0.449157	−0.224579	−	0.974456i	$-0.572101\pi$
$839$	−13.0101	−0.449157	−0.224579	+	0.974456i	$0.572101\pi$
$840$	0	0
$841$	0	0
$842$	−1.23092	−0.0424203
$843$	0	0
$844$	−8.23456	−0.283445
$845$	9.55146	0.328580
$846$	0	0
$847$	15.9021	0.546404
$848$	−26.2431	−0.901192
$849$	0	0
$850$	−1.31128	−0.0449766
$851$	25.5413	0.875543
$852$	0	0
$853$	27.7373	0.949706	0.474853	−	0.880065i	$-0.342501\pi$
$853$	27.7373	0.949706	0.474853	+	0.880065i	$0.342501\pi$
$854$	−3.02512	−0.103518
$855$	0	0
$856$	−9.01430	−0.308102
$857$	−18.6936	−0.638563	−0.319281	−	0.947660i	$-0.603442\pi$
$857$	−18.6936	−0.638563	−0.319281	+	0.947660i	$0.603442\pi$
$858$	0	0
$859$	23.3360	0.796214	0.398107	−	0.917339i	$-0.369667\pi$
$859$	23.3360	0.796214	0.398107	+	0.917339i	$0.369667\pi$
$860$	13.2153	0.450639
$861$	0	0
$862$	0.687581	0.0234191
$863$	10.7677	0.366538	0.183269	−	0.983063i	$-0.441332\pi$
$863$	10.7677	0.366538	0.183269	+	0.983063i	$0.441332\pi$
$864$	0	0
$865$	−5.46867	−0.185940
$866$	0.610039	0.0207300
$867$	0	0
$868$	−4.27972	−0.145263
$869$	0.419147	0.0142186
$870$	0	0
$871$	19.0991	0.647147
$872$	5.65128	0.191376
$873$	0	0
$874$	−6.43836	−0.217781
$875$	−18.0062	−0.608721
$876$	0	0
$877$	−42.6677	−1.44079	−0.720393	−	0.693567i	$-0.756038\pi$
$877$	−42.6677	−1.44079	−0.720393	+	0.693567i	$0.756038\pi$
$878$	3.95444	0.133456
$879$	0	0
$880$	−4.62484	−0.155903
$881$	23.9728	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$881$	23.9728	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$882$	0	0
$883$	−31.0935	−1.04638	−0.523191	−	0.852216i	$-0.675258\pi$
$883$	−31.0935	−1.04638	−0.523191	+	0.852216i	$0.675258\pi$
$884$	18.2102	0.612474
$885$	0	0
$886$	−0.112205	−0.00376961
$887$	−10.1518	−0.340864	−0.170432	−	0.985369i	$-0.554516\pi$
$887$	−10.1518	−0.340864	−0.170432	+	0.985369i	$0.554516\pi$
$888$	0	0
$889$	−29.5401	−0.990744
$890$	3.09173	0.103635
$891$	0	0
$892$	9.30938	0.311701
$893$	−9.31201	−0.311614
$894$	0	0
$895$	−3.79333	−0.126797
$896$	6.87455	0.229663
$897$	0	0
$898$	0.451472	0.0150658
$899$	0	0
$900$	0	0
$901$	−23.7983	−0.792837
$902$	−0.641930	−0.0213739
$903$	0	0
$904$	−7.27213	−0.241868
$905$	1.00614	0.0334454
$906$	0	0
$907$	−48.1396	−1.59845	−0.799225	−	0.601033i	$-0.794756\pi$
$907$	−48.1396	−1.59845	−0.799225	+	0.601033i	$0.794756\pi$
$908$	−27.0876	−0.898932
$909$	0	0
$910$	−0.897418	−0.0297491
$911$	18.1251	0.600510	0.300255	−	0.953859i	$-0.402928\pi$
$911$	18.1251	0.600510	0.300255	+	0.953859i	$0.402928\pi$
$912$	0	0
$913$	−8.94353	−0.295988
$914$	−4.13340	−0.136721
$915$	0	0
$916$	−39.9678	−1.32057
$917$	25.6749	0.847860
$918$	0	0
$919$	−36.8715	−1.21628	−0.608139	−	0.793830i	$-0.708084\pi$
$919$	−36.8715	−1.21628	−0.608139	+	0.793830i	$0.708084\pi$
$920$	−7.58733	−0.250147
$921$	0	0
$922$	4.02658	0.132608
$923$	−12.5951	−0.414573
$924$	0	0
$925$	7.76807	0.255413
$926$	−2.64494	−0.0869181
$927$	0	0
$928$	0	0
$929$	−25.3154	−0.830571	−0.415285	−	0.909691i	$-0.636318\pi$
$929$	−25.3154	−0.830571	−0.415285	+	0.909691i	$0.636318\pi$
$930$	0	0
$931$	−24.4908	−0.802652
$932$	−14.4182	−0.472282
$933$	0	0
$934$	3.15899	0.103365
$935$	−4.19400	−0.137158
$936$	0	0
$937$	−50.7043	−1.65644	−0.828219	−	0.560404i	$-0.810646\pi$
$937$	−50.7043	−1.65644	−0.828219	+	0.560404i	$0.810646\pi$
$938$	1.60883	0.0525301
$939$	0	0
$940$	−5.45816	−0.178025
$941$	−1.77666	−0.0579175	−0.0289587	−	0.999581i	$-0.509219\pi$
$941$	−1.77666	−0.0579175	−0.0289587	+	0.999581i	$0.509219\pi$
$942$	0	0
$943$	49.2040	1.60230
$944$	−57.9774	−1.88700
$945$	0	0
$946$	0.476265	0.0154847
$947$	13.6503	0.443575	0.221787	−	0.975095i	$-0.428811\pi$
$947$	13.6503	0.443575	0.221787	+	0.975095i	$0.428811\pi$
$948$	0	0
$949$	−24.0851	−0.781835
$950$	−1.95815	−0.0635308
$951$	0	0
$952$	3.08406	0.0999549
$953$	−28.3260	−0.917571	−0.458785	−	0.888547i	$-0.651715\pi$
$953$	−28.3260	−0.917571	−0.458785	+	0.888547i	$0.651715\pi$
$954$	0	0
$955$	−32.3107	−1.04555
$956$	−46.2517	−1.49589
$957$	0	0
$958$	−3.38602	−0.109397
$959$	27.8791	0.900262
$960$	0	0
$961$	−28.9960	−0.935355
$962$	1.13619	0.0366324
$963$	0	0
$964$	−50.3879	−1.62289
$965$	−31.4627	−1.01282
$966$	0	0
$967$	11.7598	0.378169	0.189085	−	0.981961i	$-0.439448\pi$
$967$	11.7598	0.378169	0.189085	+	0.981961i	$0.439448\pi$
$968$	5.98081	0.192231
$969$	0	0
$970$	0.398584	0.0127978
$971$	38.2018	1.22595	0.612976	−	0.790101i	$-0.289972\pi$
$971$	38.2018	1.22595	0.612976	+	0.790101i	$0.289972\pi$
$972$	0	0
$973$	10.4878	0.336224
$974$	−5.14344	−0.164806
$975$	0	0
$976$	53.1579	1.70154
$977$	32.1135	1.02740	0.513701	−	0.857969i	$-0.328274\pi$
$977$	32.1135	1.02740	0.513701	+	0.857969i	$0.328274\pi$
$978$	0	0
$979$	−10.5792	−0.338111
$980$	−14.3551	−0.458555
$981$	0	0
$982$	−0.405197	−0.0129303
$983$	−45.1079	−1.43872	−0.719359	−	0.694638i	$-0.755564\pi$
$983$	−45.1079	−1.43872	−0.719359	+	0.694638i	$0.755564\pi$
$984$	0	0
$985$	5.35337	0.170573
$986$	0	0
$987$	0	0
$988$	27.1934	0.865138
$989$	−36.5058	−1.16082
$990$	0	0
$991$	6.12567	0.194588	0.0972942	−	0.995256i	$-0.468981\pi$
$991$	6.12567	0.194588	0.0972942	+	0.995256i	$0.468981\pi$
$992$	−2.41862	−0.0767914
$993$	0	0
$994$	−1.06096	−0.0336517
$995$	−18.6929	−0.592603
$996$	0	0
$997$	−34.2215	−1.08381	−0.541903	−	0.840441i	$-0.682296\pi$
$997$	−34.2215	−1.08381	−0.541903	+	0.840441i	$0.682296\pi$
$998$	5.87040	0.185824
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bj.1.6		9
3.2	odd	2		2523.2.a.r.1.4		9
29.23	even	7		261.2.k.c.181.2		18
29.24	even	7		261.2.k.c.199.2		18
29.28	even	2		7569.2.a.bm.1.4		9
87.23	odd	14		87.2.g.a.7.2	✓	18
87.53	odd	14		87.2.g.a.25.2	yes	18
87.86	odd	2		2523.2.a.o.1.6		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.a.7.2	✓	18	87.23	odd	14
87.2.g.a.25.2	yes	18	87.53	odd	14
261.2.k.c.181.2		18	29.23	even	7
261.2.k.c.199.2		18	29.24	even	7
2523.2.a.o.1.6		9	87.86	odd	2
2523.2.a.r.1.4		9	3.2	odd	2
7569.2.a.bj.1.6		9	1.1	even	1	trivial
7569.2.a.bm.1.4		9	29.28	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q+0.144378 q^{2} -1.97916 q^{4} -1.55424 q^{5} -1.52752 q^{7} -0.574502 q^{8} +O(q^{10})\)	\(q+0.144378 q^{2} -1.97916 q^{4} -1.55424 q^{5} -1.52752 q^{7} -0.574502 q^{8} -0.224397 q^{10} +0.767834 q^{11} -2.61812 q^{13} -0.220540 q^{14} +3.87536 q^{16} +3.51434 q^{17} +5.24800 q^{19} +3.07607 q^{20} +0.110858 q^{22} -8.49729 q^{23} -2.58435 q^{25} -0.377999 q^{26} +3.02320 q^{28} -1.41562 q^{31} +1.70852 q^{32} +0.507393 q^{34} +2.37413 q^{35} -3.00581 q^{37} +0.757696 q^{38} +0.892912 q^{40} -5.79055 q^{41} +4.29616 q^{43} -1.51966 q^{44} -1.22682 q^{46} -1.77439 q^{47} -4.66668 q^{49} -0.373123 q^{50} +5.18167 q^{52} -6.77178 q^{53} -1.19340 q^{55} +0.877564 q^{56} -14.9605 q^{59} +13.7169 q^{61} -0.204385 q^{62} -7.50406 q^{64} +4.06918 q^{65} -7.29495 q^{67} -6.95542 q^{68} +0.342772 q^{70} +4.81074 q^{71} +9.19937 q^{73} -0.433973 q^{74} -10.3866 q^{76} -1.17288 q^{77} +0.545882 q^{79} -6.02323 q^{80} -0.836028 q^{82} -11.6477 q^{83} -5.46211 q^{85} +0.620271 q^{86} -0.441122 q^{88} -13.7779 q^{89} +3.99924 q^{91} +16.8175 q^{92} -0.256183 q^{94} -8.15664 q^{95} -1.77624 q^{97} -0.673766 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} - 24 q^{8}+O(q^{10}) \) 9 * q - 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 - 24 * q^8	\( 9 q - 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} - 24 q^{8} + q^{11} + q^{13} - 9 q^{14} + 35 q^{16} - 2 q^{17} + 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} + 8 q^{26} + 40 q^{28} + 8 q^{31} - 43 q^{32} - 4 q^{34} - 22 q^{35} + 27 q^{37} + 30 q^{38} - 29 q^{40} - 12 q^{41} + 16 q^{43} - 37 q^{44} - 22 q^{46} + 8 q^{47} - 6 q^{49} + 7 q^{50} + 33 q^{52} + 8 q^{53} + 9 q^{55} - 40 q^{56} + 16 q^{59} + 21 q^{61} + 32 q^{62} + 36 q^{64} + 31 q^{65} + 3 q^{67} - 33 q^{68} - 6 q^{70} + 33 q^{71} + 3 q^{73} - 28 q^{74} - 26 q^{76} - 24 q^{77} + 3 q^{79} + 64 q^{80} + 13 q^{82} - 13 q^{83} + 6 q^{85} - 58 q^{86} + 27 q^{88} - 6 q^{89} + q^{91} + 29 q^{92} - 18 q^{94} - 48 q^{95} + 4 q^{97} + 30 q^{98}+O(q^{100}) \) 9 * q - 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 - 24 * q^8 + q^11 + q^13 - 9 * q^14 + 35 * q^16 - 2 * q^17 + 9 * q^19 + 18 * q^20 - 4 * q^22 + 4 * q^23 + q^25 + 8 * q^26 + 40 * q^28 + 8 * q^31 - 43 * q^32 - 4 * q^34 - 22 * q^35 + 27 * q^37 + 30 * q^38 - 29 * q^40 - 12 * q^41 + 16 * q^43 - 37 * q^44 - 22 * q^46 + 8 * q^47 - 6 * q^49 + 7 * q^50 + 33 * q^52 + 8 * q^53 + 9 * q^55 - 40 * q^56 + 16 * q^59 + 21 * q^61 + 32 * q^62 + 36 * q^64 + 31 * q^65 + 3 * q^67 - 33 * q^68 - 6 * q^70 + 33 * q^71 + 3 * q^73 - 28 * q^74 - 26 * q^76 - 24 * q^77 + 3 * q^79 + 64 * q^80 + 13 * q^82 - 13 * q^83 + 6 * q^85 - 58 * q^86 + 27 * q^88 - 6 * q^89 + q^91 + 29 * q^92 - 18 * q^94 - 48 * q^95 + 4 * q^97 + 30 * q^98

Introduction

L-functions

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