LMFDB - Embedded newform 7569.2.a.bl.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7569,2,Mod(1,7569)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7569.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [9,1,0,11,0,0,5,12,0,-4,3,0,5,15,0,-5,16,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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} - x^{8} - 14x^{7} + 9x^{6} + 70x^{5} - 23x^{4} - 141x^{3} + 14x^{2} + 84x - 7 $ x^9 - x^8 - 14x^7 + 9x^6 + 70x^5 - 23x^4 - 141x^3 + 14x^2 + 84*x - 7
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			1.3
Root			$-1.79470$ 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)

$f(q)$	$=$	$q-1.79470 q^{2} +1.22096 q^{4} -1.34531 q^{5} -1.11399 q^{7} +1.39815 q^{8} +2.41444 q^{10} +2.46928 q^{11} -2.81760 q^{13} +1.99929 q^{14} -4.95118 q^{16} -5.50816 q^{17} -0.353676 q^{19} -1.64257 q^{20} -4.43162 q^{22} -5.31509 q^{23} -3.19013 q^{25} +5.05675 q^{26} -1.36014 q^{28} -4.45044 q^{31} +6.08959 q^{32} +9.88551 q^{34} +1.49867 q^{35} -9.62608 q^{37} +0.634743 q^{38} -1.88095 q^{40} +9.05582 q^{41} -9.04936 q^{43} +3.01488 q^{44} +9.53901 q^{46} -3.59491 q^{47} -5.75902 q^{49} +5.72533 q^{50} -3.44016 q^{52} +6.58238 q^{53} -3.32196 q^{55} -1.55753 q^{56} +6.26677 q^{59} +0.118224 q^{61} +7.98721 q^{62} -1.02665 q^{64} +3.79055 q^{65} +14.3033 q^{67} -6.72523 q^{68} -2.68967 q^{70} -7.50622 q^{71} -15.9239 q^{73} +17.2759 q^{74} -0.431823 q^{76} -2.75076 q^{77} +0.257006 q^{79} +6.66089 q^{80} -16.2525 q^{82} +6.12997 q^{83} +7.41021 q^{85} +16.2409 q^{86} +3.45242 q^{88} -7.50256 q^{89} +3.13879 q^{91} -6.48950 q^{92} +6.45179 q^{94} +0.475805 q^{95} +7.07800 q^{97} +10.3357 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31}+ \cdots - 12 q^{98}+O(q^{100}) $ 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8 - 4 * q^10 + 3 * q^11 + 5 * q^13 + 15 * q^14 - 5 * q^16 + 16 * q^17 + q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 + 12 * q^26 - 24 * q^28 + 4 * q^31 + 25 * q^32 + 24 * q^34 + 44 * q^35 - 25 * q^37 + 10 * q^38 - 5 * q^40 + 34 * q^41 - 12 * q^43 - 23 * q^44 - 6 * q^46 + 8 * q^47 + 26 * q^49 + 27 * q^50 - 23 * q^52 + 32 * q^53 + 5 * q^55 - 14 * q^56 - 10 * q^59 - 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 + 11 * q^68 + 14 * q^70 - 7 * q^71 - 17 * q^73 + 62 * q^74 + 6 * q^76 + 64 * q^77 - 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 - 42 * q^85 + 70 * q^86 - 29 * q^88 - 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 - 20 * q^95 - 16 * q^97 - 12 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.79470	−1.26905	−0.634523	−	0.772904i	$-0.718803\pi$
$2$	−1.79470	−1.26905	−0.634523	+	0.772904i	$0.718803\pi$
$3$	0	0
$3$	0	0
$4$	1.22096	0.610478
$5$	−1.34531	−0.601643	−0.300822	−	0.953680i	$-0.597261\pi$
$5$	−1.34531	−0.601643	−0.300822	+	0.953680i	$0.597261\pi$
$6$	0	0
$7$	−1.11399	−0.421050	−0.210525	−	0.977588i	$-0.567517\pi$
$7$	−1.11399	−0.421050	−0.210525	+	0.977588i	$0.567517\pi$
$8$	1.39815	0.494321
$9$	0	0
$10$	2.41444	0.763513
$11$	2.46928	0.744515	0.372258	−	0.928129i	$-0.378584\pi$
$11$	2.46928	0.744515	0.372258	+	0.928129i	$0.378584\pi$
$12$	0	0
$13$	−2.81760	−0.781461	−0.390730	−	0.920505i	$-0.627778\pi$
$13$	−2.81760	−0.781461	−0.390730	+	0.920505i	$0.627778\pi$
$14$	1.99929	0.534332
$15$	0	0
$16$	−4.95118	−1.23779
$17$	−5.50816	−1.33593	−0.667963	−	0.744195i	$-0.732833\pi$
$17$	−5.50816	−1.33593	−0.667963	+	0.744195i	$0.732833\pi$
$18$	0	0
$19$	−0.353676	−0.0811388	−0.0405694	−	0.999177i	$-0.512917\pi$
$19$	−0.353676	−0.0811388	−0.0405694	+	0.999177i	$0.512917\pi$
$20$	−1.64257	−0.367290
$21$	0	0
$22$	−4.43162	−0.944824
$23$	−5.31509	−1.10827	−0.554137	−	0.832426i	$-0.686951\pi$
$23$	−5.31509	−1.10827	−0.554137	+	0.832426i	$0.686951\pi$
$24$	0	0
$25$	−3.19013	−0.638026
$26$	5.05675	0.991710
$27$	0	0
$28$	−1.36014	−0.257042
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−4.45044	−0.799322	−0.399661	−	0.916663i	$-0.630872\pi$
$31$	−4.45044	−0.799322	−0.399661	+	0.916663i	$0.630872\pi$
$32$	6.08959	1.07650
$33$	0	0
$34$	9.88551	1.69535
$35$	1.49867	0.253322
$36$	0	0
$37$	−9.62608	−1.58252	−0.791259	−	0.611481i	$-0.790574\pi$
$37$	−9.62608	−1.58252	−0.791259	+	0.611481i	$0.790574\pi$
$38$	0.634743	0.102969
$39$	0	0
$40$	−1.88095	−0.297405
$41$	9.05582	1.41428	0.707141	−	0.707073i	$-0.249985\pi$
$41$	9.05582	1.41428	0.707141	+	0.707073i	$0.249985\pi$
$42$	0	0
$43$	−9.04936	−1.38001	−0.690007	−	0.723802i	$-0.742393\pi$
$43$	−9.04936	−1.38001	−0.690007	+	0.723802i	$0.742393\pi$
$44$	3.01488	0.454510
$45$	0	0
$46$	9.53901	1.40645
$47$	−3.59491	−0.524371	−0.262186	−	0.965017i	$-0.584443\pi$
$47$	−3.59491	−0.524371	−0.262186	+	0.965017i	$0.584443\pi$
$48$	0	0
$49$	−5.75902	−0.822717
$50$	5.72533	0.809684
$51$	0	0
$52$	−3.44016	−0.477065
$53$	6.58238	0.904159	0.452080	−	0.891978i	$-0.350682\pi$
$53$	6.58238	0.904159	0.452080	+	0.891978i	$0.350682\pi$
$54$	0	0
$55$	−3.32196	−0.447932
$56$	−1.55753	−0.208134
$57$	0	0
$58$	0	0
$59$	6.26677	0.815864	0.407932	−	0.913012i	$-0.366250\pi$
$59$	6.26677	0.815864	0.407932	+	0.913012i	$0.366250\pi$
$60$	0	0
$61$	0.118224	0.0151370	0.00756849	−	0.999971i	$-0.497591\pi$
$61$	0.118224	0.0151370	0.00756849	+	0.999971i	$0.497591\pi$
$62$	7.98721	1.01438
$63$	0	0
$64$	−1.02665	−0.128331
$65$	3.79055	0.470160
$66$	0	0
$67$	14.3033	1.74743	0.873714	−	0.486440i	$-0.161705\pi$
$67$	14.3033	1.74743	0.873714	+	0.486440i	$0.161705\pi$
$68$	−6.72523	−0.815554
$69$	0	0
$70$	−2.68967	−0.321477
$71$	−7.50622	−0.890825	−0.445412	−	0.895326i	$-0.646943\pi$
$71$	−7.50622	−0.890825	−0.445412	+	0.895326i	$0.646943\pi$
$72$	0	0
$73$	−15.9239	−1.86375	−0.931873	−	0.362783i	$-0.881826\pi$
$73$	−15.9239	−1.86375	−0.931873	+	0.362783i	$0.881826\pi$
$74$	17.2759	2.00829
$75$	0	0
$76$	−0.431823	−0.0495335
$77$	−2.75076	−0.313478
$78$	0	0
$79$	0.257006	0.0289155	0.0144577	−	0.999895i	$-0.495398\pi$
$79$	0.257006	0.0289155	0.0144577	+	0.999895i	$0.495398\pi$
$80$	6.66089	0.744710
$81$	0	0
$82$	−16.2525	−1.79479
$83$	6.12997	0.672852	0.336426	−	0.941710i	$-0.390782\pi$
$83$	6.12997	0.672852	0.336426	+	0.941710i	$0.390782\pi$
$84$	0	0
$85$	7.41021	0.803750
$86$	16.2409	1.75130
$87$	0	0
$88$	3.45242	0.368029
$89$	−7.50256	−0.795270	−0.397635	−	0.917544i	$-0.630169\pi$
$89$	−7.50256	−0.795270	−0.397635	+	0.917544i	$0.630169\pi$
$90$	0	0
$91$	3.13879	0.329034
$92$	−6.48950	−0.676577
$93$	0	0
$94$	6.45179	0.665452
$95$	0.475805	0.0488166
$96$	0	0
$97$	7.07800	0.718662	0.359331	−	0.933210i	$-0.383005\pi$
$97$	7.07800	0.718662	0.359331	+	0.933210i	$0.383005\pi$
$98$	10.3357	1.04407
$99$	0	0
$100$	−3.89501	−0.389501
$101$	2.59786	0.258496	0.129248	−	0.991612i	$-0.458744\pi$
$101$	2.59786	0.258496	0.129248	+	0.991612i	$0.458744\pi$
$102$	0	0
$103$	−6.96303	−0.686088	−0.343044	−	0.939319i	$-0.611458\pi$
$103$	−6.96303	−0.686088	−0.343044	+	0.939319i	$0.611458\pi$
$104$	−3.93942	−0.386292
$105$	0	0
$106$	−11.8134	−1.14742
$107$	−15.1489	−1.46450	−0.732250	−	0.681036i	$-0.761530\pi$
$107$	−15.1489	−1.46450	−0.732250	+	0.681036i	$0.761530\pi$
$108$	0	0
$109$	16.1581	1.54766	0.773831	−	0.633393i	$-0.218338\pi$
$109$	16.1581	1.54766	0.773831	+	0.633393i	$0.218338\pi$
$110$	5.96192	0.568447
$111$	0	0
$112$	5.51558	0.521174
$113$	15.0604	1.41676	0.708381	−	0.705830i	$-0.249426\pi$
$113$	15.0604	1.41676	0.708381	+	0.705830i	$0.249426\pi$
$114$	0	0
$115$	7.15047	0.666785
$116$	0	0
$117$	0	0
$118$	−11.2470	−1.03537
$119$	6.13606	0.562492
$120$	0	0
$121$	−4.90267	−0.445697
$122$	−0.212176	−0.0192095
$123$	0	0
$124$	−5.43379	−0.487969
$125$	11.0183	0.985507
$126$	0	0
$127$	0.654012	0.0580342	0.0290171	−	0.999579i	$-0.490762\pi$
$127$	0.654012	0.0580342	0.0290171	+	0.999579i	$0.490762\pi$
$128$	−10.3367	−0.913640
$129$	0	0
$130$	−6.80292	−0.596655
$131$	0.0536221	0.00468498	0.00234249	−	0.999997i	$-0.499254\pi$
$131$	0.0536221	0.00468498	0.00234249	+	0.999997i	$0.499254\pi$
$132$	0	0
$133$	0.393993	0.0341635
$134$	−25.6702	−2.21757
$135$	0	0
$136$	−7.70124	−0.660376
$137$	−11.7817	−1.00658	−0.503288	−	0.864119i	$-0.667877\pi$
$137$	−11.7817	−1.00658	−0.503288	+	0.864119i	$0.667877\pi$
$138$	0	0
$139$	−7.09266	−0.601592	−0.300796	−	0.953689i	$-0.597252\pi$
$139$	−7.09266	−0.601592	−0.300796	+	0.953689i	$0.597252\pi$
$140$	1.82982	0.154648
$141$	0	0
$142$	13.4714	1.13050
$143$	−6.95743	−0.581809
$144$	0	0
$145$	0	0
$146$	28.5786	2.36518
$147$	0	0
$148$	−11.7530	−0.966093
$149$	3.92807	0.321800	0.160900	−	0.986971i	$-0.448560\pi$
$149$	3.92807	0.321800	0.160900	+	0.986971i	$0.448560\pi$
$150$	0	0
$151$	21.8506	1.77818	0.889088	−	0.457736i	$-0.151340\pi$
$151$	21.8506	1.77818	0.889088	+	0.457736i	$0.151340\pi$
$152$	−0.494492	−0.0401086
$153$	0	0
$154$	4.93680	0.397819
$155$	5.98724	0.480907
$156$	0	0
$157$	−10.3803	−0.828438	−0.414219	−	0.910177i	$-0.635945\pi$
$157$	−10.3803	−0.828438	−0.414219	+	0.910177i	$0.635945\pi$
$158$	−0.461250	−0.0366951
$159$	0	0
$160$	−8.19242	−0.647667
$161$	5.92098	0.466639
$162$	0	0
$163$	−8.92721	−0.699233	−0.349617	−	0.936893i	$-0.613688\pi$
$163$	−8.92721	−0.699233	−0.349617	+	0.936893i	$0.613688\pi$
$164$	11.0568	0.863388
$165$	0	0
$166$	−11.0015	−0.853880
$167$	−20.0070	−1.54819	−0.774094	−	0.633071i	$-0.781794\pi$
$167$	−20.0070	−1.54819	−0.774094	+	0.633071i	$0.781794\pi$
$168$	0	0
$169$	−5.06115	−0.389319
$170$	−13.2991	−1.02000
$171$	0	0
$172$	−11.0489	−0.842469
$173$	3.99833	0.303987	0.151994	−	0.988381i	$-0.451431\pi$
$173$	3.99833	0.303987	0.151994	+	0.988381i	$0.451431\pi$
$174$	0	0
$175$	3.55379	0.268641
$176$	−12.2258	−0.921557
$177$	0	0
$178$	13.4649	1.00923
$179$	10.3561	0.774049	0.387025	−	0.922069i	$-0.373503\pi$
$179$	10.3561	0.774049	0.387025	+	0.922069i	$0.373503\pi$
$180$	0	0
$181$	−9.69715	−0.720783	−0.360392	−	0.932801i	$-0.617357\pi$
$181$	−9.69715	−0.720783	−0.360392	+	0.932801i	$0.617357\pi$
$182$	−5.63319	−0.417560
$183$	0	0
$184$	−7.43130	−0.547843
$185$	12.9501	0.952111
$186$	0	0
$187$	−13.6012	−0.994617
$188$	−4.38923	−0.320117
$189$	0	0
$190$	−0.853929	−0.0619505
$191$	−1.15201	−0.0833568	−0.0416784	−	0.999131i	$-0.513270\pi$
$191$	−1.15201	−0.0833568	−0.0416784	+	0.999131i	$0.513270\pi$
$192$	0	0
$193$	−27.6684	−1.99161	−0.995806	−	0.0914906i	$-0.970837\pi$
$193$	−27.6684	−1.99161	−0.995806	+	0.0914906i	$0.970837\pi$
$194$	−12.7029	−0.912015
$195$	0	0
$196$	−7.03151	−0.502251
$197$	14.8936	1.06113	0.530564	−	0.847645i	$-0.321980\pi$
$197$	14.8936	1.06113	0.530564	+	0.847645i	$0.321980\pi$
$198$	0	0
$199$	−9.17257	−0.650227	−0.325113	−	0.945675i	$-0.605402\pi$
$199$	−9.17257	−0.650227	−0.325113	+	0.945675i	$0.605402\pi$
$200$	−4.46028	−0.315389
$201$	0	0
$202$	−4.66238	−0.328044
$203$	0	0
$204$	0	0
$205$	−12.1829	−0.850893
$206$	12.4966	0.870677
$207$	0	0
$208$	13.9504	0.967288
$209$	−0.873324	−0.0604091
$210$	0	0
$211$	−0.253657	−0.0174625	−0.00873124	−	0.999962i	$-0.502779\pi$
$211$	−0.253657	−0.0174625	−0.00873124	+	0.999962i	$0.502779\pi$
$212$	8.03680	0.551970
$213$	0	0
$214$	27.1878	1.85852
$215$	12.1742	0.830276
$216$	0	0
$217$	4.95776	0.336555
$218$	−28.9989	−1.96405
$219$	0	0
$220$	−4.05596	−0.273453
$221$	15.5198	1.04397
$222$	0	0
$223$	22.1228	1.48145	0.740725	−	0.671809i	$-0.234482\pi$
$223$	22.1228	1.48145	0.740725	+	0.671809i	$0.234482\pi$
$224$	−6.78377	−0.453260
$225$	0	0
$226$	−27.0289	−1.79794
$227$	−19.3877	−1.28681	−0.643404	−	0.765527i	$-0.722478\pi$
$227$	−19.3877	−1.28681	−0.643404	+	0.765527i	$0.722478\pi$
$228$	0	0
$229$	20.7055	1.36826	0.684128	−	0.729362i	$-0.260183\pi$
$229$	20.7055	1.36826	0.684128	+	0.729362i	$0.260183\pi$
$230$	−12.8330	−0.846181
$231$	0	0
$232$	0	0
$233$	−21.9063	−1.43513	−0.717566	−	0.696491i	$-0.754744\pi$
$233$	−21.9063	−1.43513	−0.717566	+	0.696491i	$0.754744\pi$
$234$	0	0
$235$	4.83628	0.315484
$236$	7.65146	0.498068
$237$	0	0
$238$	−11.0124	−0.713828
$239$	−0.467106	−0.0302145	−0.0151073	−	0.999886i	$-0.504809\pi$
$239$	−0.467106	−0.0302145	−0.0151073	+	0.999886i	$0.504809\pi$
$240$	0	0
$241$	8.76731	0.564752	0.282376	−	0.959304i	$-0.408877\pi$
$241$	8.76731	0.564752	0.282376	+	0.959304i	$0.408877\pi$
$242$	8.79883	0.565610
$243$	0	0
$244$	0.144346	0.00924080
$245$	7.74769	0.494982
$246$	0	0
$247$	0.996516	0.0634068
$248$	−6.22238	−0.395122
$249$	0	0
$250$	−19.7746	−1.25065
$251$	4.87995	0.308019	0.154010	−	0.988069i	$-0.450781\pi$
$251$	4.87995	0.308019	0.154010	+	0.988069i	$0.450781\pi$
$252$	0	0
$253$	−13.1244	−0.825126
$254$	−1.17376	−0.0736480
$255$	0	0
$256$	20.6045	1.28778
$257$	3.17838	0.198262	0.0991309	−	0.995074i	$-0.468394\pi$
$257$	3.17838	0.198262	0.0991309	+	0.995074i	$0.468394\pi$
$258$	0	0
$259$	10.7234	0.666320
$260$	4.62810	0.287023
$261$	0	0
$262$	−0.0962357	−0.00594546
$263$	−24.2946	−1.49807	−0.749036	−	0.662529i	$-0.769483\pi$
$263$	−24.2946	−1.49807	−0.749036	+	0.662529i	$0.769483\pi$
$264$	0	0
$265$	−8.85537	−0.543981
$266$	−0.707100	−0.0433551
$267$	0	0
$268$	17.4637	1.06677
$269$	−8.65347	−0.527611	−0.263806	−	0.964576i	$-0.584978\pi$
$269$	−8.65347	−0.527611	−0.263806	+	0.964576i	$0.584978\pi$
$270$	0	0
$271$	−14.2633	−0.866433	−0.433217	−	0.901290i	$-0.642622\pi$
$271$	−14.2633	−0.866433	−0.433217	+	0.901290i	$0.642622\pi$
$272$	27.2719	1.65360
$273$	0	0
$274$	21.1446	1.27739
$275$	−7.87731	−0.475020
$276$	0	0
$277$	7.59596	0.456397	0.228199	−	0.973615i	$-0.426716\pi$
$277$	7.59596	0.456397	0.228199	+	0.973615i	$0.426716\pi$
$278$	12.7292	0.763447
$279$	0	0
$280$	2.09537	0.125222
$281$	−1.30929	−0.0781056	−0.0390528	−	0.999237i	$-0.512434\pi$
$281$	−1.30929	−0.0781056	−0.0390528	+	0.999237i	$0.512434\pi$
$282$	0	0
$283$	23.5261	1.39848	0.699242	−	0.714885i	$-0.253521\pi$
$283$	23.5261	1.39848	0.699242	+	0.714885i	$0.253521\pi$
$284$	−9.16478	−0.543829
$285$	0	0
$286$	12.4865	0.738343
$287$	−10.0881	−0.595484
$288$	0	0
$289$	13.3399	0.784697
$290$	0	0
$291$	0	0
$292$	−19.4423	−1.13778
$293$	15.8262	0.924576	0.462288	−	0.886730i	$-0.347029\pi$
$293$	15.8262	0.924576	0.462288	+	0.886730i	$0.347029\pi$
$294$	0	0
$295$	−8.43078	−0.490859
$296$	−13.4587	−0.782272
$297$	0	0
$298$	−7.04971	−0.408379
$299$	14.9758	0.866072
$300$	0	0
$301$	10.0809	0.581056
$302$	−39.2153	−2.25659
$303$	0	0
$304$	1.75111	0.100433
$305$	−0.159048	−0.00910706
$306$	0	0
$307$	−7.55550	−0.431215	−0.215608	−	0.976480i	$-0.569173\pi$
$307$	−7.55550	−0.431215	−0.215608	+	0.976480i	$0.569173\pi$
$308$	−3.35856	−0.191372
$309$	0	0
$310$	−10.7453	−0.610293
$311$	−14.1674	−0.803359	−0.401680	−	0.915780i	$-0.631573\pi$
$311$	−14.1674	−0.803359	−0.401680	+	0.915780i	$0.631573\pi$
$312$	0	0
$313$	16.9609	0.958684	0.479342	−	0.877628i	$-0.340875\pi$
$313$	16.9609	0.958684	0.479342	+	0.877628i	$0.340875\pi$
$314$	18.6295	1.05133
$315$	0	0
$316$	0.313793	0.0176523
$317$	4.73068	0.265701	0.132851	−	0.991136i	$-0.457587\pi$
$317$	4.73068	0.265701	0.132851	+	0.991136i	$0.457587\pi$
$318$	0	0
$319$	0	0
$320$	1.38116	0.0772093
$321$	0	0
$322$	−10.6264	−0.592186
$323$	1.94810	0.108395
$324$	0	0
$325$	8.98850	0.498592
$326$	16.0217	0.887359
$327$	0	0
$328$	12.6614	0.699109
$329$	4.00471	0.220787
$330$	0	0
$331$	−1.77434	−0.0975263	−0.0487631	−	0.998810i	$-0.515528\pi$
$331$	−1.77434	−0.0975263	−0.0487631	+	0.998810i	$0.515528\pi$
$332$	7.48443	0.410761
$333$	0	0
$334$	35.9066	1.96472
$335$	−19.2425	−1.05133
$336$	0	0
$337$	1.24785	0.0679747	0.0339873	−	0.999422i	$-0.489179\pi$
$337$	1.24785	0.0679747	0.0339873	+	0.999422i	$0.489179\pi$
$338$	9.08325	0.494064
$339$	0	0
$340$	9.04755	0.490672
$341$	−10.9894	−0.595108
$342$	0	0
$343$	14.2135	0.767455
$344$	−12.6524	−0.682170
$345$	0	0
$346$	−7.17581	−0.385774
$347$	26.1773	1.40527	0.702636	−	0.711550i	$-0.252006\pi$
$347$	26.1773	1.40527	0.702636	+	0.711550i	$0.252006\pi$
$348$	0	0
$349$	23.8884	1.27872	0.639359	−	0.768908i	$-0.279200\pi$
$349$	23.8884	1.27872	0.639359	+	0.768908i	$0.279200\pi$
$350$	−6.37799	−0.340918
$351$	0	0
$352$	15.0369	0.801469
$353$	2.12694	0.113206	0.0566028	−	0.998397i	$-0.481973\pi$
$353$	2.12694	0.113206	0.0566028	+	0.998397i	$0.481973\pi$
$354$	0	0
$355$	10.0982	0.535959
$356$	−9.16031	−0.485495
$357$	0	0
$358$	−18.5861	−0.982305
$359$	27.4101	1.44665	0.723324	−	0.690509i	$-0.242613\pi$
$359$	27.4101	1.44665	0.723324	+	0.690509i	$0.242613\pi$
$360$	0	0
$361$	−18.8749	−0.993416
$362$	17.4035	0.914707
$363$	0	0
$364$	3.83232	0.200868
$365$	21.4226	1.12131
$366$	0	0
$367$	27.2659	1.42327	0.711634	−	0.702550i	$-0.247955\pi$
$367$	27.2659	1.42327	0.711634	+	0.702550i	$0.247955\pi$
$368$	26.3160	1.37181
$369$	0	0
$370$	−23.2416	−1.20827
$371$	−7.33273	−0.380697
$372$	0	0
$373$	6.45619	0.334289	0.167144	−	0.985932i	$-0.446545\pi$
$373$	6.45619	0.334289	0.167144	+	0.985932i	$0.446545\pi$
$374$	24.4101	1.26221
$375$	0	0
$376$	−5.02622	−0.259208
$377$	0	0
$378$	0	0
$379$	−19.1607	−0.984217	−0.492108	−	0.870534i	$-0.663774\pi$
$379$	−19.1607	−0.984217	−0.492108	+	0.870534i	$0.663774\pi$
$380$	0.580938	0.0298015
$381$	0	0
$382$	2.06752	0.105784
$383$	26.2817	1.34293	0.671466	−	0.741035i	$-0.265665\pi$
$383$	26.2817	1.34293	0.671466	+	0.741035i	$0.265665\pi$
$384$	0	0
$385$	3.70064	0.188602
$386$	49.6565	2.52745
$387$	0	0
$388$	8.64193	0.438728
$389$	15.8638	0.804324	0.402162	−	0.915568i	$-0.368259\pi$
$389$	15.8638	0.804324	0.402162	+	0.915568i	$0.368259\pi$
$390$	0	0
$391$	29.2764	1.48057
$392$	−8.05197	−0.406686
$393$	0	0
$394$	−26.7296	−1.34662
$395$	−0.345754	−0.0173968
$396$	0	0
$397$	−3.83677	−0.192562	−0.0962809	−	0.995354i	$-0.530695\pi$
$397$	−3.83677	−0.192562	−0.0962809	+	0.995354i	$0.530695\pi$
$398$	16.4620	0.825168
$399$	0	0
$400$	15.7949	0.789745
$401$	−24.8215	−1.23953	−0.619764	−	0.784789i	$-0.712771\pi$
$401$	−24.8215	−1.23953	−0.619764	+	0.784789i	$0.712771\pi$
$402$	0	0
$403$	12.5395	0.624639
$404$	3.17187	0.157806
$405$	0	0
$406$	0	0
$407$	−23.7695	−1.17821
$408$	0	0
$409$	24.3029	1.20170	0.600850	−	0.799362i	$-0.294829\pi$
$409$	24.3029	1.20170	0.600850	+	0.799362i	$0.294829\pi$
$410$	21.8647	1.07982
$411$	0	0
$412$	−8.50156	−0.418842
$413$	−6.98115	−0.343520
$414$	0	0
$415$	−8.24674	−0.404817
$416$	−17.1580	−0.841241
$417$	0	0
$418$	1.56736	0.0766619
$419$	−1.85751	−0.0907455	−0.0453727	−	0.998970i	$-0.514448\pi$
$419$	−1.85751	−0.0907455	−0.0453727	+	0.998970i	$0.514448\pi$
$420$	0	0
$421$	−17.5422	−0.854955	−0.427478	−	0.904026i	$-0.640598\pi$
$421$	−17.5422	−0.854955	−0.427478	+	0.904026i	$0.640598\pi$
$422$	0.455239	0.0221607
$423$	0	0
$424$	9.20316	0.446945
$425$	17.5717	0.852355
$426$	0	0
$427$	−0.131700	−0.00637343
$428$	−18.4961	−0.894045
$429$	0	0
$430$	−21.8491	−1.05366
$431$	4.17302	0.201007	0.100504	−	0.994937i	$-0.467955\pi$
$431$	4.17302	0.201007	0.100504	+	0.994937i	$0.467955\pi$
$432$	0	0
$433$	19.8023	0.951636	0.475818	−	0.879544i	$-0.342152\pi$
$433$	19.8023	0.951636	0.475818	+	0.879544i	$0.342152\pi$
$434$	−8.89771	−0.427104
$435$	0	0
$436$	19.7283	0.944814
$437$	1.87982	0.0899240
$438$	0	0
$439$	−27.1080	−1.29379	−0.646897	−	0.762578i	$-0.723934\pi$
$439$	−27.1080	−1.29379	−0.646897	+	0.762578i	$0.723934\pi$
$440$	−4.64459	−0.221422
$441$	0	0
$442$	−27.8534	−1.32485
$443$	−13.5845	−0.645417	−0.322709	−	0.946498i	$-0.604593\pi$
$443$	−13.5845	−0.645417	−0.322709	+	0.946498i	$0.604593\pi$
$444$	0	0
$445$	10.0933	0.478469
$446$	−39.7038	−1.88003
$447$	0	0
$448$	1.14368	0.0540337
$449$	−13.7546	−0.649119	−0.324559	−	0.945865i	$-0.605216\pi$
$449$	−13.7546	−0.649119	−0.324559	+	0.945865i	$0.605216\pi$
$450$	0	0
$451$	22.3613	1.05295
$452$	18.3881	0.864903
$453$	0	0
$454$	34.7952	1.63302
$455$	−4.22266	−0.197961
$456$	0	0
$457$	2.45948	0.115050	0.0575248	−	0.998344i	$-0.481679\pi$
$457$	2.45948	0.115050	0.0575248	+	0.998344i	$0.481679\pi$
$458$	−37.1602	−1.73638
$459$	0	0
$460$	8.73042	0.407058
$461$	36.0349	1.67831	0.839156	−	0.543891i	$-0.183050\pi$
$461$	36.0349	1.67831	0.839156	+	0.543891i	$0.183050\pi$
$462$	0	0
$463$	25.8918	1.20329	0.601646	−	0.798763i	$-0.294512\pi$
$463$	25.8918	1.20329	0.601646	+	0.798763i	$0.294512\pi$
$464$	0	0
$465$	0	0
$466$	39.3154	1.82125
$467$	−14.5287	−0.672307	−0.336154	−	0.941807i	$-0.609126\pi$
$467$	−14.5287	−0.672307	−0.336154	+	0.941807i	$0.609126\pi$
$468$	0	0
$469$	−15.9338	−0.735755
$470$	−8.67969	−0.400364
$471$	0	0
$472$	8.76189	0.403299
$473$	−22.3454	−1.02744
$474$	0	0
$475$	1.12827	0.0517686
$476$	7.49187	0.343389
$477$	0	0
$478$	0.838315	0.0383437
$479$	−2.18658	−0.0999074	−0.0499537	−	0.998752i	$-0.515907\pi$
$479$	−2.18658	−0.0999074	−0.0499537	+	0.998752i	$0.515907\pi$
$480$	0	0
$481$	27.1224	1.23668
$482$	−15.7347	−0.716696
$483$	0	0
$484$	−5.98595	−0.272088
$485$	−9.52214	−0.432378
$486$	0	0
$487$	−4.80186	−0.217593	−0.108797	−	0.994064i	$-0.534700\pi$
$487$	−4.80186	−0.217593	−0.108797	+	0.994064i	$0.534700\pi$
$488$	0.165294	0.00748252
$489$	0	0
$490$	−13.9048	−0.628155
$491$	17.1430	0.773653	0.386827	−	0.922152i	$-0.373571\pi$
$491$	17.1430	0.773653	0.386827	+	0.922152i	$0.373571\pi$
$492$	0	0
$493$	0	0
$494$	−1.78845	−0.0804662
$495$	0	0
$496$	22.0349	0.989397
$497$	8.36189	0.375082
$498$	0	0
$499$	−3.48677	−0.156089	−0.0780446	−	0.996950i	$-0.524868\pi$
$499$	−3.48677	−0.156089	−0.0780446	+	0.996950i	$0.524868\pi$
$500$	13.4529	0.601631
$501$	0	0
$502$	−8.75805	−0.390891
$503$	−32.9881	−1.47087	−0.735433	−	0.677597i	$-0.763021\pi$
$503$	−32.9881	−1.47087	−0.735433	+	0.677597i	$0.763021\pi$
$504$	0	0
$505$	−3.49493	−0.155523
$506$	23.5545	1.04712
$507$	0	0
$508$	0.798520	0.0354286
$509$	31.0258	1.37520	0.687598	−	0.726092i	$-0.258665\pi$
$509$	31.0258	1.37520	0.687598	+	0.726092i	$0.258665\pi$
$510$	0	0
$511$	17.7391	0.784731
$512$	−16.3057	−0.720615
$513$	0	0
$514$	−5.70425	−0.251603
$515$	9.36747	0.412780
$516$	0	0
$517$	−8.87683	−0.390402
$518$	−19.2453	−0.845590
$519$	0	0
$520$	5.29977	0.232410
$521$	−22.9227	−1.00426	−0.502130	−	0.864792i	$-0.667450\pi$
$521$	−22.9227	−1.00426	−0.502130	+	0.864792i	$0.667450\pi$
$522$	0	0
$523$	−40.2928	−1.76188	−0.880940	−	0.473229i	$-0.843088\pi$
$523$	−40.2928	−1.76188	−0.880940	+	0.473229i	$0.843088\pi$
$524$	0.0654703	0.00286008
$525$	0	0
$526$	43.6017	1.90112
$527$	24.5137	1.06784
$528$	0	0
$529$	5.25020	0.228270
$530$	15.8928	0.690337
$531$	0	0
$532$	0.481048	0.0208561
$533$	−25.5157	−1.10521
$534$	0	0
$535$	20.3800	0.881106
$536$	19.9982	0.863790
$537$	0	0
$538$	15.5304	0.669563
$539$	−14.2206	−0.612525
$540$	0	0
$541$	21.7485	0.935040	0.467520	−	0.883982i	$-0.345148\pi$
$541$	21.7485	0.935040	0.467520	+	0.883982i	$0.345148\pi$
$542$	25.5984	1.09954
$543$	0	0
$544$	−33.5425	−1.43812
$545$	−21.7377	−0.931139
$546$	0	0
$547$	−37.9410	−1.62224	−0.811121	−	0.584878i	$-0.801142\pi$
$547$	−37.9410	−1.62224	−0.811121	+	0.584878i	$0.801142\pi$
$548$	−14.3849	−0.614493
$549$	0	0
$550$	14.1374	0.602822
$551$	0	0
$552$	0	0
$553$	−0.286303	−0.0121749
$554$	−13.6325	−0.579189
$555$	0	0
$556$	−8.65983	−0.367259
$557$	7.41485	0.314177	0.157089	−	0.987585i	$-0.449789\pi$
$557$	7.41485	0.314177	0.157089	+	0.987585i	$0.449789\pi$
$558$	0	0
$559$	25.4975	1.07843
$560$	−7.42020	−0.313561
$561$	0	0
$562$	2.34978	0.0991196
$563$	−33.6129	−1.41661	−0.708307	−	0.705904i	$-0.750541\pi$
$563$	−33.6129	−1.41661	−0.708307	+	0.705904i	$0.750541\pi$
$564$	0	0
$565$	−20.2610	−0.852385
$566$	−42.2224	−1.77474
$567$	0	0
$568$	−10.4948	−0.440353
$569$	17.3381	0.726851	0.363425	−	0.931623i	$-0.381607\pi$
$569$	17.3381	0.726851	0.363425	+	0.931623i	$0.381607\pi$
$570$	0	0
$571$	−17.0429	−0.713224	−0.356612	−	0.934253i	$-0.616068\pi$
$571$	−17.0429	−0.713224	−0.356612	+	0.934253i	$0.616068\pi$
$572$	−8.49472	−0.355182
$573$	0	0
$574$	18.1052	0.755696
$575$	16.9558	0.707107
$576$	0	0
$577$	−22.2980	−0.928276	−0.464138	−	0.885763i	$-0.653636\pi$
$577$	−22.2980	−0.928276	−0.464138	+	0.885763i	$0.653636\pi$
$578$	−23.9411	−0.995817
$579$	0	0
$580$	0	0
$581$	−6.82875	−0.283304
$582$	0	0
$583$	16.2537	0.673160
$584$	−22.2640	−0.921289
$585$	0	0
$586$	−28.4033	−1.17333
$587$	12.0497	0.497344	0.248672	−	0.968588i	$-0.420006\pi$
$587$	12.0497	0.497344	0.248672	+	0.968588i	$0.420006\pi$
$588$	0	0
$589$	1.57401	0.0648561
$590$	15.1307	0.622923
$591$	0	0
$592$	47.6604	1.95883
$593$	−2.22448	−0.0913486	−0.0456743	−	0.998956i	$-0.514544\pi$
$593$	−2.22448	−0.0913486	−0.0456743	+	0.998956i	$0.514544\pi$
$594$	0	0
$595$	−8.25493	−0.338419
$596$	4.79600	0.196452
$597$	0	0
$598$	−26.8771	−1.09909
$599$	−1.37767	−0.0562902	−0.0281451	−	0.999604i	$-0.508960\pi$
$599$	−1.37767	−0.0562902	−0.0281451	+	0.999604i	$0.508960\pi$
$600$	0	0
$601$	−17.8146	−0.726672	−0.363336	−	0.931658i	$-0.618362\pi$
$601$	−17.8146	−0.726672	−0.363336	+	0.931658i	$0.618362\pi$
$602$	−18.0923	−0.737386
$603$	0	0
$604$	26.6786	1.08554
$605$	6.59563	0.268151
$606$	0	0
$607$	−39.7372	−1.61288	−0.806441	−	0.591315i	$-0.798609\pi$
$607$	−39.7372	−1.61288	−0.806441	+	0.591315i	$0.798609\pi$
$608$	−2.15374	−0.0873457
$609$	0	0
$610$	0.285444	0.0115573
$611$	10.1290	0.409776
$612$	0	0
$613$	−21.9879	−0.888085	−0.444042	−	0.896006i	$-0.646456\pi$
$613$	−21.9879	−0.888085	−0.444042	+	0.896006i	$0.646456\pi$
$614$	13.5599	0.547232
$615$	0	0
$616$	−3.84598	−0.154959
$617$	33.7175	1.35742	0.678708	−	0.734408i	$-0.262540\pi$
$617$	33.7175	1.35742	0.678708	+	0.734408i	$0.262540\pi$
$618$	0	0
$619$	40.5422	1.62953	0.814764	−	0.579792i	$-0.196866\pi$
$619$	40.5422	1.62953	0.814764	+	0.579792i	$0.196866\pi$
$620$	7.31016	0.293583
$621$	0	0
$622$	25.4263	1.01950
$623$	8.35781	0.334849
$624$	0	0
$625$	1.12756	0.0451025
$626$	−30.4397	−1.21661
$627$	0	0
$628$	−12.6739	−0.505743
$629$	53.0220	2.11413
$630$	0	0
$631$	−8.29320	−0.330147	−0.165074	−	0.986281i	$-0.552786\pi$
$631$	−8.29320	−0.330147	−0.165074	+	0.986281i	$0.552786\pi$
$632$	0.359333	0.0142935
$633$	0	0
$634$	−8.49016	−0.337187
$635$	−0.879852	−0.0349158
$636$	0	0
$637$	16.2266	0.642921
$638$	0	0
$639$	0	0
$640$	13.9061	0.549685
$641$	−12.2903	−0.485438	−0.242719	−	0.970097i	$-0.578039\pi$
$641$	−12.2903	−0.485438	−0.242719	+	0.970097i	$0.578039\pi$
$642$	0	0
$643$	11.3156	0.446243	0.223122	−	0.974791i	$-0.428375\pi$
$643$	11.3156	0.446243	0.223122	+	0.974791i	$0.428375\pi$
$644$	7.22926	0.284873
$645$	0	0
$646$	−3.49627	−0.137559
$647$	29.1117	1.14450	0.572250	−	0.820079i	$-0.306071\pi$
$647$	29.1117	1.14450	0.572250	+	0.820079i	$0.306071\pi$
$648$	0	0
$649$	15.4744	0.607423
$650$	−16.1317	−0.632736
$651$	0	0
$652$	−10.8997	−0.426867
$653$	23.8213	0.932199	0.466099	−	0.884732i	$-0.345659\pi$
$653$	23.8213	0.932199	0.466099	+	0.884732i	$0.345659\pi$
$654$	0	0
$655$	−0.0721386	−0.00281869
$656$	−44.8370	−1.75059
$657$	0	0
$658$	−7.18726	−0.280189
$659$	−11.3771	−0.443188	−0.221594	−	0.975139i	$-0.571126\pi$
$659$	−11.3771	−0.443188	−0.221594	+	0.975139i	$0.571126\pi$
$660$	0	0
$661$	37.1578	1.44527	0.722635	−	0.691230i	$-0.242931\pi$
$661$	37.1578	1.44527	0.722635	+	0.691230i	$0.242931\pi$
$662$	3.18440	0.123765
$663$	0	0
$664$	8.57062	0.332605
$665$	−0.530044	−0.0205542
$666$	0	0
$667$	0	0
$668$	−24.4277	−0.945135
$669$	0	0
$670$	34.5345	1.33418
$671$	0.291927	0.0112697
$672$	0	0
$673$	−8.13671	−0.313647	−0.156824	−	0.987627i	$-0.550125\pi$
$673$	−8.13671	−0.313647	−0.156824	+	0.987627i	$0.550125\pi$
$674$	−2.23952	−0.0862630
$675$	0	0
$676$	−6.17944	−0.237671
$677$	−23.4784	−0.902347	−0.451173	−	0.892436i	$-0.648994\pi$
$677$	−23.4784	−0.902347	−0.451173	+	0.892436i	$0.648994\pi$
$678$	0	0
$679$	−7.88485	−0.302593
$680$	10.3606	0.397311
$681$	0	0
$682$	19.7226	0.755219
$683$	47.0890	1.80181	0.900906	−	0.434014i	$-0.142903\pi$
$683$	47.0890	1.80181	0.900906	+	0.434014i	$0.142903\pi$
$684$	0	0
$685$	15.8501	0.605600
$686$	−25.5090	−0.973936
$687$	0	0
$688$	44.8050	1.70817
$689$	−18.5465	−0.706565
$690$	0	0
$691$	24.3963	0.928077	0.464039	−	0.885815i	$-0.346400\pi$
$691$	24.3963	0.928077	0.464039	+	0.885815i	$0.346400\pi$
$692$	4.88179	0.185578
$693$	0	0
$694$	−46.9805	−1.78335
$695$	9.54186	0.361943
$696$	0	0
$697$	−49.8809	−1.88937
$698$	−42.8726	−1.62275
$699$	0	0
$700$	4.33902	0.163999
$701$	6.76882	0.255655	0.127827	−	0.991796i	$-0.459200\pi$
$701$	6.76882	0.255655	0.127827	+	0.991796i	$0.459200\pi$
$702$	0	0
$703$	3.40451	0.128404
$704$	−2.53508	−0.0955442
$705$	0	0
$706$	−3.81722	−0.143663
$707$	−2.89400	−0.108840
$708$	0	0
$709$	−2.22958	−0.0837335	−0.0418667	−	0.999123i	$-0.513330\pi$
$709$	−2.22958	−0.0837335	−0.0418667	+	0.999123i	$0.513330\pi$
$710$	−18.1233	−0.680156
$711$	0	0
$712$	−10.4897	−0.393119
$713$	23.6545	0.885868
$714$	0	0
$715$	9.35993	0.350042
$716$	12.6443	0.472541
$717$	0	0
$718$	−49.1929	−1.83586
$719$	30.7520	1.14686	0.573429	−	0.819255i	$-0.305613\pi$
$719$	30.7520	1.14686	0.573429	+	0.819255i	$0.305613\pi$
$720$	0	0
$721$	7.75678	0.288878
$722$	33.8749	1.26069
$723$	0	0
$724$	−11.8398	−0.440023
$725$	0	0
$726$	0	0
$727$	19.9953	0.741585	0.370793	−	0.928716i	$-0.379086\pi$
$727$	19.9953	0.741585	0.370793	+	0.928716i	$0.379086\pi$
$728$	4.38850	0.162649
$729$	0	0
$730$	−38.4472	−1.42299
$731$	49.8453	1.84360
$732$	0	0
$733$	7.52317	0.277875	0.138937	−	0.990301i	$-0.455631\pi$
$733$	7.52317	0.277875	0.138937	+	0.990301i	$0.455631\pi$
$734$	−48.9342	−1.80619
$735$	0	0
$736$	−32.3667	−1.19305
$737$	35.3189	1.30099
$738$	0	0
$739$	17.3560	0.638452	0.319226	−	0.947679i	$-0.396577\pi$
$739$	17.3560	0.638452	0.319226	+	0.947679i	$0.396577\pi$
$740$	15.8115	0.581243
$741$	0	0
$742$	13.1601	0.483122
$743$	53.5602	1.96493	0.982466	−	0.186440i	$-0.0596949\pi$
$743$	53.5602	1.96493	0.982466	+	0.186440i	$0.0596949\pi$
$744$	0	0
$745$	−5.28448	−0.193609
$746$	−11.5869	−0.424228
$747$	0	0
$748$	−16.6065	−0.607192
$749$	16.8758	0.616628
$750$	0	0
$751$	−40.0498	−1.46144	−0.730719	−	0.682678i	$-0.760815\pi$
$751$	−40.0498	−1.46144	−0.730719	+	0.682678i	$0.760815\pi$
$752$	17.7990	0.649064
$753$	0	0
$754$	0	0
$755$	−29.3959	−1.06983
$756$	0	0
$757$	−3.25296	−0.118231	−0.0591155	−	0.998251i	$-0.518828\pi$
$757$	−3.25296	−0.118231	−0.0591155	+	0.998251i	$0.518828\pi$
$758$	34.3877	1.24902
$759$	0	0
$760$	0.665247	0.0241311
$761$	42.0559	1.52453	0.762263	−	0.647267i	$-0.224088\pi$
$761$	42.0559	1.52453	0.762263	+	0.647267i	$0.224088\pi$
$762$	0	0
$763$	−18.0000	−0.651643
$764$	−1.40656	−0.0508876
$765$	0	0
$766$	−47.1678	−1.70424
$767$	−17.6572	−0.637566
$768$	0	0
$769$	7.38181	0.266195	0.133098	−	0.991103i	$-0.457508\pi$
$769$	7.38181	0.266195	0.133098	+	0.991103i	$0.457508\pi$
$770$	−6.64155	−0.239345
$771$	0	0
$772$	−33.7819	−1.21584
$773$	24.2079	0.870698	0.435349	−	0.900262i	$-0.356625\pi$
$773$	24.2079	0.870698	0.435349	+	0.900262i	$0.356625\pi$
$774$	0	0
$775$	14.1975	0.509988
$776$	9.89611	0.355250
$777$	0	0
$778$	−28.4707	−1.02072
$779$	−3.20283	−0.114753
$780$	0	0
$781$	−18.5350	−0.663233
$782$	−52.5424	−1.87891
$783$	0	0
$784$	28.5139	1.01835
$785$	13.9648	0.498424
$786$	0	0
$787$	−14.9627	−0.533361	−0.266680	−	0.963785i	$-0.585927\pi$
$787$	−14.9627	−0.533361	−0.266680	+	0.963785i	$0.585927\pi$
$788$	18.1845	0.647796
$789$	0	0
$790$	0.620526	0.0220773
$791$	−16.7772	−0.596528
$792$	0	0
$793$	−0.333106	−0.0118290
$794$	6.88586	0.244370
$795$	0	0
$796$	−11.1993	−0.396949
$797$	42.3296	1.49939	0.749695	−	0.661783i	$-0.230200\pi$
$797$	42.3296	1.49939	0.749695	+	0.661783i	$0.230200\pi$
$798$	0	0
$799$	19.8013	0.700521
$800$	−19.4266	−0.686833
$801$	0	0
$802$	44.5472	1.57302
$803$	−39.3204	−1.38759
$804$	0	0
$805$	−7.96558	−0.280750
$806$	−22.5047	−0.792696
$807$	0	0
$808$	3.63220	0.127780
$809$	−51.1578	−1.79861	−0.899307	−	0.437318i	$-0.855928\pi$
$809$	−51.1578	−1.79861	−0.899307	+	0.437318i	$0.855928\pi$
$810$	0	0
$811$	−54.7356	−1.92203	−0.961014	−	0.276500i	$-0.910826\pi$
$811$	−54.7356	−1.92203	−0.961014	+	0.276500i	$0.910826\pi$
$812$	0	0
$813$	0	0
$814$	42.6591	1.49520
$815$	12.0099	0.420689
$816$	0	0
$817$	3.20054	0.111973
$818$	−43.6164	−1.52501
$819$	0	0
$820$	−14.8748	−0.519452
$821$	12.2305	0.426846	0.213423	−	0.976960i	$-0.431539\pi$
$821$	12.2305	0.426846	0.213423	+	0.976960i	$0.431539\pi$
$822$	0	0
$823$	−46.6846	−1.62732	−0.813662	−	0.581338i	$-0.802529\pi$
$823$	−46.6846	−1.62732	−0.813662	+	0.581338i	$0.802529\pi$
$824$	−9.73537	−0.339148
$825$	0	0
$826$	12.5291	0.435943
$827$	−15.1935	−0.528329	−0.264165	−	0.964478i	$-0.585096\pi$
$827$	−15.1935	−0.528329	−0.264165	+	0.964478i	$0.585096\pi$
$828$	0	0
$829$	9.62161	0.334172	0.167086	−	0.985942i	$-0.446564\pi$
$829$	9.62161	0.334172	0.167086	+	0.985942i	$0.446564\pi$
$830$	14.8004	0.513731
$831$	0	0
$832$	2.89268	0.100286
$833$	31.7216	1.09909
$834$	0	0
$835$	26.9157	0.931456
$836$	−1.06629	−0.0368784
$837$	0	0
$838$	3.33368	0.115160
$839$	−21.3585	−0.737376	−0.368688	−	0.929553i	$-0.620193\pi$
$839$	−21.3585	−0.737376	−0.368688	+	0.929553i	$0.620193\pi$
$840$	0	0
$841$	0	0
$842$	31.4831	1.08498
$843$	0	0
$844$	−0.309704	−0.0106605
$845$	6.80883	0.234231
$846$	0	0
$847$	5.46154	0.187661
$848$	−32.5905	−1.11916
$849$	0	0
$850$	−31.5361	−1.08168
$851$	51.1635	1.75386
$852$	0	0
$853$	45.6926	1.56449	0.782243	−	0.622973i	$-0.214075\pi$
$853$	45.6926	1.56449	0.782243	+	0.622973i	$0.214075\pi$
$854$	0.236363	0.00808818
$855$	0	0
$856$	−21.1804	−0.723932
$857$	27.0824	0.925116	0.462558	−	0.886589i	$-0.346932\pi$
$857$	27.0824	0.925116	0.462558	+	0.886589i	$0.346932\pi$
$858$	0	0
$859$	44.6910	1.52484	0.762418	−	0.647084i	$-0.224012\pi$
$859$	44.6910	1.52484	0.762418	+	0.647084i	$0.224012\pi$
$860$	14.8642	0.506866
$861$	0	0
$862$	−7.48933	−0.255088
$863$	41.7719	1.42193	0.710966	−	0.703226i	$-0.248258\pi$
$863$	41.7719	1.42193	0.710966	+	0.703226i	$0.248258\pi$
$864$	0	0
$865$	−5.37901	−0.182892
$866$	−35.5392	−1.20767
$867$	0	0
$868$	6.05322	0.205460
$869$	0.634620	0.0215280
$870$	0	0
$871$	−40.3010	−1.36555
$872$	22.5914	0.765041
$873$	0	0
$874$	−3.37372	−0.114118
$875$	−12.2743	−0.414948
$876$	0	0
$877$	27.4752	0.927770	0.463885	−	0.885895i	$-0.346455\pi$
$877$	27.4752	0.927770	0.463885	+	0.885895i	$0.346455\pi$
$878$	48.6508	1.64188
$879$	0	0
$880$	16.4476	0.554448
$881$	−47.5627	−1.60243	−0.801214	−	0.598378i	$-0.795812\pi$
$881$	−47.5627	−1.60243	−0.801214	+	0.598378i	$0.795812\pi$
$882$	0	0
$883$	24.9844	0.840791	0.420396	−	0.907341i	$-0.361891\pi$
$883$	24.9844	0.840791	0.420396	+	0.907341i	$0.361891\pi$
$884$	18.9490	0.637323
$885$	0	0
$886$	24.3801	0.819064
$887$	48.5275	1.62940	0.814698	−	0.579886i	$-0.196903\pi$
$887$	48.5275	1.62940	0.814698	+	0.579886i	$0.196903\pi$
$888$	0	0
$889$	−0.728565	−0.0244353
$890$	−18.1145	−0.607199
$891$	0	0
$892$	27.0109	0.904393
$893$	1.27143	0.0425469
$894$	0	0
$895$	−13.9322	−0.465701
$896$	11.5150	0.384688
$897$	0	0
$898$	24.6854	0.823762
$899$	0	0
$900$	0	0
$901$	−36.2568	−1.20789
$902$	−40.1319	−1.33625
$903$	0	0
$904$	21.0567	0.700335
$905$	13.0457	0.433654
$906$	0	0
$907$	17.1117	0.568184	0.284092	−	0.958797i	$-0.408308\pi$
$907$	17.1117	0.568184	0.284092	+	0.958797i	$0.408308\pi$
$908$	−23.6716	−0.785568
$909$	0	0
$910$	7.57841	0.251222
$911$	30.9538	1.02555	0.512773	−	0.858524i	$-0.328618\pi$
$911$	30.9538	1.02555	0.512773	+	0.858524i	$0.328618\pi$
$912$	0	0
$913$	15.1366	0.500948
$914$	−4.41403	−0.146003
$915$	0	0
$916$	25.2805	0.835291
$917$	−0.0597347	−0.00197261
$918$	0	0
$919$	34.7391	1.14594	0.572969	−	0.819577i	$-0.305792\pi$
$919$	34.7391	1.14594	0.572969	+	0.819577i	$0.305792\pi$
$920$	9.99744	0.329606
$921$	0	0
$922$	−64.6719	−2.12985
$923$	21.1495	0.696145
$924$	0	0
$925$	30.7084	1.00969
$926$	−46.4680	−1.52703
$927$	0	0
$928$	0	0
$929$	14.2102	0.466221	0.233111	−	0.972450i	$-0.425110\pi$
$929$	14.2102	0.466221	0.233111	+	0.972450i	$0.425110\pi$
$930$	0	0
$931$	2.03683	0.0667542
$932$	−26.7467	−0.876117
$933$	0	0
$934$	26.0747	0.853189
$935$	18.2979	0.598404
$936$	0	0
$937$	14.8639	0.485582	0.242791	−	0.970079i	$-0.421937\pi$
$937$	14.8639	0.485582	0.242791	+	0.970079i	$0.421937\pi$
$938$	28.5965	0.933707
$939$	0	0
$940$	5.90489	0.192596
$941$	20.4145	0.665494	0.332747	−	0.943016i	$-0.392025\pi$
$941$	20.4145	0.665494	0.332747	+	0.943016i	$0.392025\pi$
$942$	0	0
$943$	−48.1325	−1.56741
$944$	−31.0279	−1.00987
$945$	0	0
$946$	40.1033	1.30387
$947$	0.243770	0.00792146	0.00396073	−	0.999992i	$-0.498739\pi$
$947$	0.243770	0.00792146	0.00396073	+	0.999992i	$0.498739\pi$
$948$	0	0
$949$	44.8670	1.45645
$950$	−2.02491	−0.0656968
$951$	0	0
$952$	8.57914	0.278051
$953$	9.83584	0.318614	0.159307	−	0.987229i	$-0.449074\pi$
$953$	9.83584	0.318614	0.159307	+	0.987229i	$0.449074\pi$
$954$	0	0
$955$	1.54982	0.0501511
$956$	−0.570316	−0.0184453
$957$	0	0
$958$	3.92426	0.126787
$959$	13.1247	0.423819
$960$	0	0
$961$	−11.1936	−0.361084
$962$	−48.6767	−1.56940
$963$	0	0
$964$	10.7045	0.344769
$965$	37.2226	1.19824
$966$	0	0
$967$	−19.0021	−0.611065	−0.305532	−	0.952182i	$-0.598834\pi$
$967$	−19.0021	−0.611065	−0.305532	+	0.952182i	$0.598834\pi$
$968$	−6.85467	−0.220317
$969$	0	0
$970$	17.0894	0.548708
$971$	−46.0030	−1.47631	−0.738153	−	0.674633i	$-0.764302\pi$
$971$	−46.0030	−1.47631	−0.738153	+	0.674633i	$0.764302\pi$
$972$	0	0
$973$	7.90118	0.253300
$974$	8.61791	0.276136
$975$	0	0
$976$	−0.585346	−0.0187365
$977$	3.91662	0.125304	0.0626520	−	0.998035i	$-0.480044\pi$
$977$	3.91662	0.125304	0.0626520	+	0.998035i	$0.480044\pi$
$978$	0	0
$979$	−18.5259	−0.592091
$980$	9.45959	0.302176
$981$	0	0
$982$	−30.7666	−0.981802
$983$	−27.9391	−0.891118	−0.445559	−	0.895252i	$-0.646995\pi$
$983$	−27.9391	−0.891118	−0.445559	+	0.895252i	$0.646995\pi$
$984$	0	0
$985$	−20.0366	−0.638420
$986$	0	0
$987$	0	0
$988$	1.21670	0.0387085
$989$	48.0982	1.52943
$990$	0	0
$991$	−29.0573	−0.923036	−0.461518	−	0.887131i	$-0.652695\pi$
$991$	−29.0573	−0.923036	−0.461518	+	0.887131i	$0.652695\pi$
$992$	−27.1013	−0.860469
$993$	0	0
$994$	−15.0071	−0.475997
$995$	12.3400	0.391204
$996$	0	0
$997$	−29.5035	−0.934385	−0.467193	−	0.884156i	$-0.654735\pi$
$997$	−29.5035	−0.934385	−0.467193	+	0.884156i	$0.654735\pi$
$998$	6.25771	0.198084
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bl.1.3		9
3.2	odd	2		2523.2.a.p.1.7		9
29.23	even	7		261.2.k.b.181.3		18
29.24	even	7		261.2.k.b.199.3		18
29.28	even	2		7569.2.a.bk.1.7		9
87.23	odd	14		87.2.g.b.7.1	✓	18
87.53	odd	14		87.2.g.b.25.1	yes	18
87.86	odd	2		2523.2.a.q.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.b.7.1	✓	18	87.23	odd	14
87.2.g.b.25.1	yes	18	87.53	odd	14
261.2.k.b.181.3		18	29.23	even	7
261.2.k.b.199.3		18	29.24	even	7
2523.2.a.p.1.7		9	3.2	odd	2
2523.2.a.q.1.3		9	87.86	odd	2
7569.2.a.bk.1.7		9	29.28	even	2
7569.2.a.bl.1.3		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.79470 q^{2} +1.22096 q^{4} -1.34531 q^{5} -1.11399 q^{7} +1.39815 q^{8} +2.41444 q^{10} +2.46928 q^{11} -2.81760 q^{13} +1.99929 q^{14} -4.95118 q^{16} -5.50816 q^{17} -0.353676 q^{19} -1.64257 q^{20} -4.43162 q^{22} -5.31509 q^{23} -3.19013 q^{25} +5.05675 q^{26} -1.36014 q^{28} -4.45044 q^{31} +6.08959 q^{32} +9.88551 q^{34} +1.49867 q^{35} -9.62608 q^{37} +0.634743 q^{38} -1.88095 q^{40} +9.05582 q^{41} -9.04936 q^{43} +3.01488 q^{44} +9.53901 q^{46} -3.59491 q^{47} -5.75902 q^{49} +5.72533 q^{50} -3.44016 q^{52} +6.58238 q^{53} -3.32196 q^{55} -1.55753 q^{56} +6.26677 q^{59} +0.118224 q^{61} +7.98721 q^{62} -1.02665 q^{64} +3.79055 q^{65} +14.3033 q^{67} -6.72523 q^{68} -2.68967 q^{70} -7.50622 q^{71} -15.9239 q^{73} +17.2759 q^{74} -0.431823 q^{76} -2.75076 q^{77} +0.257006 q^{79} +6.66089 q^{80} -16.2525 q^{82} +6.12997 q^{83} +7.41021 q^{85} +16.2409 q^{86} +3.45242 q^{88} -7.50256 q^{89} +3.13879 q^{91} -6.48950 q^{92} +6.45179 q^{94} +0.475805 q^{95} +7.07800 q^{97} +10.3357 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31}+ \cdots - 12 q^{98}+O(q^{100}) \) 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8 - 4 * q^10 + 3 * q^11 + 5 * q^13 + 15 * q^14 - 5 * q^16 + 16 * q^17 + q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 + 12 * q^26 - 24 * q^28 + 4 * q^31 + 25 * q^32 + 24 * q^34 + 44 * q^35 - 25 * q^37 + 10 * q^38 - 5 * q^40 + 34 * q^41 - 12 * q^43 - 23 * q^44 - 6 * q^46 + 8 * q^47 + 26 * q^49 + 27 * q^50 - 23 * q^52 + 32 * q^53 + 5 * q^55 - 14 * q^56 - 10 * q^59 - 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 + 11 * q^68 + 14 * q^70 - 7 * q^71 - 17 * q^73 + 62 * q^74 + 6 * q^76 + 64 * q^77 - 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 - 42 * q^85 + 70 * q^86 - 29 * q^88 - 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 - 20 * q^95 - 16 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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