LMFDB - Embedded newform 7569.2.a.bm.1.4

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			1.4
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)

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

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	$-0.516255\pi$
$2$	−0.144378	−0.102091	−0.0510453	+	0.998696i	$0.516255\pi$
$3$	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	$-0.536929\pi$
$11$	−0.767834	−0.231511	−0.115755	+	0.993278i	$0.536929\pi$
$12$	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	$-0.640140\pi$
$17$	−3.51434	−0.852353	−0.426176	+	0.904640i	$0.640140\pi$
$18$	0	0
$19$	−5.24800	−1.20397	−0.601987	−	0.798506i	$-0.705624\pi$
$19$	−5.24800	−1.20397	−0.601987	+	0.798506i	$0.705624\pi$
$20$	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	$-0.459424\pi$
$31$	1.41562	0.254254	0.127127	+	0.991886i	$0.459424\pi$
$32$	−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	$-0.420530\pi$
$37$	3.00581	0.494152	0.247076	+	0.968996i	$0.420530\pi$
$38$	0.757696	0.122915
$39$	0	0
$40$	−0.892912	−0.141182
$41$	5.79055	0.904332	0.452166	−	0.891934i	$-0.350651\pi$
$41$	5.79055	0.904332	0.452166	+	0.891934i	$0.350651\pi$
$42$	0	0
$43$	−4.29616	−0.655159	−0.327579	−	0.944824i	$-0.606233\pi$
$43$	−4.29616	−0.655159	−0.327579	+	0.944824i	$0.606233\pi$
$44$	1.51966	0.229098
$45$	0	0
$46$	1.22682	0.180885
$47$	1.77439	0.258821	0.129411	−	0.991591i	$-0.458691\pi$
$47$	1.77439	0.258821	0.129411	+	0.991591i	$0.458691\pi$
$48$	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	$-0.841211\pi$
$61$	−13.7169	−1.75627	−0.878133	+	0.478416i	$0.841211\pi$
$62$	−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	$-0.680953\pi$
$73$	−9.19937	−1.07670	−0.538352	+	0.842720i	$0.680953\pi$
$74$	−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	$-0.509776\pi$
$79$	−0.545882	−0.0614165	−0.0307083	+	0.999528i	$0.509776\pi$
$80$	−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	$-0.239414\pi$
$89$	13.7779	1.46046	0.730228	+	0.683203i	$0.239414\pi$
$90$	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	$-0.471257\pi$
$97$	1.77624	0.180350	0.0901751	+	0.995926i	$0.471257\pi$
$98$	0.673766	0.0680606
$99$	0	0
$100$	5.11483	0.511483
$101$	−4.46601	−0.444384	−0.222192	−	0.975003i	$-0.571321\pi$
$101$	−4.46601	−0.444384	−0.222192	+	0.975003i	$0.571321\pi$
$102$	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	$-0.703002\pi$
$113$	−12.6581	−1.19078	−0.595389	+	0.803437i	$0.703002\pi$
$114$	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	$-0.828300\pi$
$127$	−19.3386	−1.71602	−0.858012	+	0.513630i	$0.828300\pi$
$128$	4.50046	0.397788
$129$	0	0
$130$	−0.587500	−0.0515271
$131$	16.8082	1.46854	0.734271	−	0.678857i	$-0.237524\pi$
$131$	16.8082	1.46854	0.734271	+	0.678857i	$0.237524\pi$
$132$	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	$-0.215396\pi$
$137$	18.2512	1.55930	0.779652	+	0.626213i	$0.215396\pi$
$138$	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	$-0.813553\pi$
$157$	−20.8825	−1.66661	−0.833304	+	0.552815i	$0.813553\pi$
$158$	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	$-0.496886\pi$
$163$	0.249811	0.0195667	0.00978335	+	0.999952i	$0.496886\pi$
$164$	−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	$-0.770965\pi$
$191$	−20.7888	−1.50422	−0.752112	+	0.659035i	$0.770965\pi$
$192$	0	0
$193$	−20.2432	−1.45714	−0.728569	−	0.684973i	$-0.759814\pi$
$193$	−20.2432	−1.45714	−0.728569	+	0.684973i	$0.759814\pi$
$194$	−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	$-0.545744\pi$
$211$	−4.16064	−0.286430	−0.143215	+	0.989692i	$0.545744\pi$
$212$	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	$-0.732525\pi$
$229$	−20.1944	−1.33448	−0.667241	+	0.744842i	$0.732525\pi$
$230$	−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	$-0.635171\pi$
$251$	−13.0548	−0.824009	−0.412004	+	0.911182i	$0.635171\pi$
$252$	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	$-0.278990\pi$
$263$	20.7538	1.27973	0.639866	+	0.768487i	$0.278990\pi$
$264$	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	$-0.490698\pi$
$269$	0.958446	0.0584375	0.0292187	+	0.999573i	$0.490698\pi$
$270$	0	0
$271$	21.1163	1.28272	0.641361	−	0.767240i	$-0.278370\pi$
$271$	21.1163	1.28272	0.641361	+	0.767240i	$0.278370\pi$
$272$	−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	$-0.652785\pi$
$293$	−15.8084	−0.923538	−0.461769	+	0.887000i	$0.652785\pi$
$294$	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	$-0.581125\pi$
$307$	−8.83474	−0.504225	−0.252113	+	0.967698i	$0.581125\pi$
$308$	−2.32132	−0.132269
$309$	0	0
$310$	0.317662	0.0180420
$311$	−12.9878	−0.736469	−0.368235	−	0.929733i	$-0.620038\pi$
$311$	−12.9878	−0.736469	−0.368235	+	0.929733i	$0.620038\pi$
$312$	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	$-0.230906\pi$
$317$	26.6435	1.49645	0.748225	+	0.663445i	$0.230906\pi$
$318$	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	$-0.480610\pi$
$331$	2.21519	0.121758	0.0608789	+	0.998145i	$0.480610\pi$
$332$	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	$-0.570070\pi$
$337$	−8.01703	−0.436715	−0.218358	+	0.975869i	$0.570070\pi$
$338$	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	$-0.688994\pi$
$359$	−21.2008	−1.11894	−0.559468	+	0.828852i	$0.688994\pi$
$360$	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	$-0.329147\pi$
$367$	19.5920	1.02269	0.511347	+	0.859374i	$0.329147\pi$
$368$	−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	$-0.303972\pi$
$379$	22.4911	1.15529	0.577645	+	0.816288i	$0.303972\pi$
$380$	−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	$-0.686638\pi$
$389$	−21.8263	−1.10664	−0.553318	+	0.832970i	$0.686638\pi$
$390$	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	$-0.221004\pi$
$409$	31.0838	1.53700	0.768498	+	0.639852i	$0.221004\pi$
$410$	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	$-0.433383\pi$
$421$	8.52568	0.415516	0.207758	+	0.978180i	$0.433383\pi$
$422$	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	$-0.532373\pi$
$433$	−4.22530	−0.203055	−0.101527	+	0.994833i	$0.532373\pi$
$434$	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	$-0.494123\pi$
$443$	0.777164	0.0369242	0.0184621	+	0.999830i	$0.494123\pi$
$444$	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	$-0.523508\pi$
$449$	−3.12702	−0.147573	−0.0737865	+	0.997274i	$0.523508\pi$
$450$	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	$-0.725007\pi$
$461$	−27.8892	−1.29893	−0.649464	+	0.760392i	$0.725007\pi$
$462$	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	$-0.668966\pi$
$467$	−21.8800	−1.01249	−0.506243	+	0.862391i	$0.668966\pi$
$468$	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	$-0.320015\pi$
$479$	23.4525	1.07157	0.535786	+	0.844354i	$0.320015\pi$
$480$	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	$-0.479829\pi$
$491$	2.80650	0.126656	0.0633278	+	0.997993i	$0.479829\pi$
$492$	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	$-0.635613\pi$
$503$	−18.5373	−0.826538	−0.413269	+	0.910609i	$0.635613\pi$
$504$	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	$-0.657638\pi$
$541$	−22.1075	−0.950477	−0.475239	+	0.879857i	$0.657638\pi$
$542$	−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	$-0.406221\pi$
$563$	13.7797	0.580744	0.290372	+	0.956914i	$0.406221\pi$
$564$	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	$-0.228564\pi$
$569$	35.9279	1.50618	0.753088	+	0.657920i	$0.228564\pi$
$570$	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	$-0.446290\pi$
$577$	8.06782	0.335868	0.167934	+	0.985798i	$0.446290\pi$
$578$	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	$-0.724011\pi$
$599$	−31.6740	−1.29416	−0.647082	+	0.762421i	$0.724011\pi$
$600$	0	0
$601$	−3.86546	−0.157675	−0.0788376	−	0.996887i	$-0.525121\pi$
$601$	−3.86546	−0.157675	−0.0788376	+	0.996887i	$0.525121\pi$
$602$	−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	$-0.508590\pi$
$607$	−1.32954	−0.0539645	−0.0269823	+	0.999636i	$0.508590\pi$
$608$	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	$-0.487656\pi$
$617$	1.92609	0.0775415	0.0387708	+	0.999248i	$0.487656\pi$
$618$	0	0
$619$	−0.295624	−0.0118821	−0.00594107	−	0.999982i	$-0.501891\pi$
$619$	−0.295624	−0.0118821	−0.00594107	+	0.999982i	$0.501891\pi$
$620$	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	$-0.639609\pi$
$641$	−21.5035	−0.849336	−0.424668	+	0.905349i	$0.639609\pi$
$642$	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	$-0.804907\pi$
$653$	−41.8052	−1.63596	−0.817982	+	0.575244i	$0.804907\pi$
$654$	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	$-0.539063\pi$
$659$	−6.28494	−0.244827	−0.122413	+	0.992479i	$0.539063\pi$
$660$	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	$-0.401812\pi$
$677$	15.7988	0.607197	0.303599	+	0.952800i	$0.401812\pi$
$678$	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	$-0.543375\pi$
$727$	−7.32560	−0.271692	−0.135846	+	0.990730i	$0.543375\pi$
$728$	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	$-0.530880\pi$
$733$	−5.24481	−0.193722	−0.0968608	+	0.995298i	$0.530880\pi$
$734$	−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	$-0.642757\pi$
$739$	−23.5744	−0.867198	−0.433599	+	0.901106i	$0.642757\pi$
$740$	9.24610	0.339893
$741$	0	0
$742$	−1.49345	−0.0548262
$743$	−19.5239	−0.716263	−0.358131	−	0.933671i	$-0.616586\pi$
$743$	−19.5239	−0.716263	−0.358131	+	0.933671i	$0.616586\pi$
$744$	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	$-0.508226\pi$
$751$	−1.41626	−0.0516802	−0.0258401	+	0.999666i	$0.508226\pi$
$752$	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	$-0.788500\pi$
$757$	−43.3206	−1.57451	−0.787257	+	0.616625i	$0.788500\pi$
$758$	−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	$-0.405169\pi$
$769$	16.2798	0.587065	0.293532	+	0.955949i	$0.405169\pi$
$770$	0.263192	0.00948477
$771$	0	0
$772$	40.0644	1.44195
$773$	17.3203	0.622968	0.311484	−	0.950251i	$-0.399174\pi$
$773$	17.3203	0.622968	0.311484	+	0.950251i	$0.399174\pi$
$774$	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	$-0.207792\pi$
$797$	44.8530	1.58878	0.794388	+	0.607411i	$0.207792\pi$
$798$	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	$-0.327013\pi$
$809$	29.4154	1.03419	0.517095	+	0.855928i	$0.327013\pi$
$810$	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	$-0.631598\pi$
$823$	−23.0508	−0.803499	−0.401750	+	0.915750i	$0.631598\pi$
$824$	5.92880	0.206539
$825$	0	0
$826$	−3.29939	−0.114800
$827$	−44.5689	−1.54981	−0.774907	−	0.632076i	$-0.782203\pi$
$827$	−44.5689	−1.54981	−0.774907	+	0.632076i	$0.782203\pi$
$828$	0	0
$829$	−44.8796	−1.55873	−0.779367	−	0.626568i	$-0.784459\pi$
$829$	−44.8796	−1.55873	−0.779367	+	0.626568i	$0.784459\pi$
$830$	−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	$-0.427899\pi$
$839$	13.0101	0.449157	0.224579	+	0.974456i	$0.427899\pi$
$840$	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	$-0.657499\pi$
$853$	−27.7373	−0.949706	−0.474853	+	0.880065i	$0.657499\pi$
$854$	−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	$-0.630333\pi$
$859$	−23.3360	−0.796214	−0.398107	+	0.917339i	$0.630333\pi$
$860$	−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	$-0.632322\pi$
$881$	−23.9728	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$882$	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	$-0.445484\pi$
$887$	10.1518	0.340864	0.170432	+	0.985369i	$0.445484\pi$
$888$	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	$-0.205244\pi$
$907$	48.1396	1.59845	0.799225	+	0.601033i	$0.205244\pi$
$908$	−27.0876	−0.898932
$909$	0	0
$910$	0.897418	0.0297491
$911$	−18.1251	−0.600510	−0.300255	−	0.953859i	$-0.597072\pi$
$911$	−18.1251	−0.600510	−0.300255	+	0.953859i	$0.597072\pi$
$912$	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	$-0.571189\pi$
$947$	−13.6503	−0.443575	−0.221787	+	0.975095i	$0.571189\pi$
$948$	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	$-0.560552\pi$
$967$	−11.7598	−0.378169	−0.189085	+	0.981961i	$0.560552\pi$
$968$	−5.98081	−0.192231
$969$	0	0
$970$	0.398584	0.0127978
$971$	−38.2018	−1.22595	−0.612976	−	0.790101i	$-0.710028\pi$
$971$	−38.2018	−1.22595	−0.612976	+	0.790101i	$0.710028\pi$
$972$	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	$-0.244436\pi$
$983$	45.1079	1.43872	0.719359	+	0.694638i	$0.244436\pi$
$984$	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	$-0.317704\pi$
$997$	34.2215	1.08381	0.541903	+	0.840441i	$0.317704\pi$
$998$	−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.bm.1.4		9
3.2	odd	2		2523.2.a.o.1.6		9
29.5	even	14		261.2.k.c.199.2		18
29.6	even	14		261.2.k.c.181.2		18
29.28	even	2		7569.2.a.bj.1.6		9
87.5	odd	14		87.2.g.a.25.2	yes	18
87.35	odd	14		87.2.g.a.7.2	✓	18
87.86	odd	2		2523.2.a.r.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.a.7.2	✓	18	87.35	odd	14
87.2.g.a.25.2	yes	18	87.5	odd	14
261.2.k.c.181.2		18	29.6	even	14
261.2.k.c.199.2		18	29.5	even	14
2523.2.a.o.1.6		9	3.2	odd	2
2523.2.a.r.1.4		9	87.86	odd	2
7569.2.a.bj.1.6		9	29.28	even	2
7569.2.a.bm.1.4		9	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 $\F_{q}$

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

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