LMFDB - Embedded newform 9075.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.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:	$72.4642398343$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +5.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +5.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} -0.618034 q^{12} -3.23607 q^{13} +8.47214 q^{14} -4.85410 q^{16} -2.00000 q^{17} +1.61803 q^{18} -5.00000 q^{19} -5.23607 q^{21} +4.61803 q^{23} +2.23607 q^{24} -5.23607 q^{26} -1.00000 q^{27} +3.23607 q^{28} -0.854102 q^{29} +7.00000 q^{31} -3.38197 q^{32} -3.23607 q^{34} +0.618034 q^{36} +1.47214 q^{37} -8.09017 q^{38} +3.23607 q^{39} -11.4721 q^{41} -8.47214 q^{42} -9.09017 q^{43} +7.47214 q^{46} -6.09017 q^{47} +4.85410 q^{48} +20.4164 q^{49} +2.00000 q^{51} -2.00000 q^{52} -5.38197 q^{53} -1.61803 q^{54} -11.7082 q^{56} +5.00000 q^{57} -1.38197 q^{58} +6.70820 q^{59} -7.00000 q^{61} +11.3262 q^{62} +5.23607 q^{63} +4.23607 q^{64} -11.6180 q^{67} -1.23607 q^{68} -4.61803 q^{69} -8.00000 q^{71} -2.23607 q^{72} +4.52786 q^{73} +2.38197 q^{74} -3.09017 q^{76} +5.23607 q^{78} -3.09017 q^{79} +1.00000 q^{81} -18.5623 q^{82} -7.70820 q^{83} -3.23607 q^{84} -14.7082 q^{86} +0.854102 q^{87} +4.14590 q^{89} -16.9443 q^{91} +2.85410 q^{92} -7.00000 q^{93} -9.85410 q^{94} +3.38197 q^{96} +2.52786 q^{97} +33.0344 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9	$ 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9} + q^{12} - 2 q^{13} + 8 q^{14} - 3 q^{16} - 4 q^{17} + q^{18} - 10 q^{19} - 6 q^{21} + 7 q^{23} - 6 q^{26} - 2 q^{27} + 2 q^{28} + 5 q^{29} + 14 q^{31} - 9 q^{32} - 2 q^{34} - q^{36} - 6 q^{37} - 5 q^{38} + 2 q^{39} - 14 q^{41} - 8 q^{42} - 7 q^{43} + 6 q^{46} - q^{47} + 3 q^{48} + 14 q^{49} + 4 q^{51} - 4 q^{52} - 13 q^{53} - q^{54} - 10 q^{56} + 10 q^{57} - 5 q^{58} - 14 q^{61} + 7 q^{62} + 6 q^{63} + 4 q^{64} - 21 q^{67} + 2 q^{68} - 7 q^{69} - 16 q^{71} + 18 q^{73} + 7 q^{74} + 5 q^{76} + 6 q^{78} + 5 q^{79} + 2 q^{81} - 17 q^{82} - 2 q^{83} - 2 q^{84} - 16 q^{86} - 5 q^{87} + 15 q^{89} - 16 q^{91} - q^{92} - 14 q^{93} - 13 q^{94} + 9 q^{96} + 14 q^{97} + 37 q^{98}+O(q^{100}) $ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 + q^12 - 2 * q^13 + 8 * q^14 - 3 * q^16 - 4 * q^17 + q^18 - 10 * q^19 - 6 * q^21 + 7 * q^23 - 6 * q^26 - 2 * q^27 + 2 * q^28 + 5 * q^29 + 14 * q^31 - 9 * q^32 - 2 * q^34 - q^36 - 6 * q^37 - 5 * q^38 + 2 * q^39 - 14 * q^41 - 8 * q^42 - 7 * q^43 + 6 * q^46 - q^47 + 3 * q^48 + 14 * q^49 + 4 * q^51 - 4 * q^52 - 13 * q^53 - q^54 - 10 * q^56 + 10 * q^57 - 5 * q^58 - 14 * q^61 + 7 * q^62 + 6 * q^63 + 4 * q^64 - 21 * q^67 + 2 * q^68 - 7 * q^69 - 16 * q^71 + 18 * q^73 + 7 * q^74 + 5 * q^76 + 6 * q^78 + 5 * q^79 + 2 * q^81 - 17 * q^82 - 2 * q^83 - 2 * q^84 - 16 * q^86 - 5 * q^87 + 15 * q^89 - 16 * q^91 - q^92 - 14 * q^93 - 13 * q^94 + 9 * q^96 + 14 * q^97 + 37 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	5.23607	1.97905	0.989524	−	0.144370i	$-0.0461154\pi$
$7$	5.23607	1.97905	0.989524	+	0.144370i	$0.0461154\pi$
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−0.618034	−0.178411
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	8.47214	2.26427
$15$	0	0
$16$	−4.85410	−1.21353
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.61803	0.381374
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−5.23607	−1.14260
$22$	0	0
$23$	4.61803	0.962927	0.481463	−	0.876466i	$-0.340105\pi$
$23$	4.61803	0.962927	0.481463	+	0.876466i	$0.340105\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	−5.23607	−1.02688
$27$	−1.00000	−0.192450
$28$	3.23607	0.611559
$29$	−0.854102	−0.158603	−0.0793014	−	0.996851i	$-0.525269\pi$
$29$	−0.854102	−0.158603	−0.0793014	+	0.996851i	$0.525269\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	−3.23607	−0.554981
$35$	0	0
$36$	0.618034	0.103006
$37$	1.47214	0.242018	0.121009	−	0.992651i	$-0.461387\pi$
$37$	1.47214	0.242018	0.121009	+	0.992651i	$0.461387\pi$
$38$	−8.09017	−1.31240
$39$	3.23607	0.518186
$40$	0	0
$41$	−11.4721	−1.79165	−0.895823	−	0.444410i	$-0.853413\pi$
$41$	−11.4721	−1.79165	−0.895823	+	0.444410i	$0.853413\pi$
$42$	−8.47214	−1.30728
$43$	−9.09017	−1.38624	−0.693119	−	0.720823i	$-0.743764\pi$
$43$	−9.09017	−1.38624	−0.693119	+	0.720823i	$0.743764\pi$
$44$	0	0
$45$	0	0
$46$	7.47214	1.10171
$47$	−6.09017	−0.888343	−0.444171	−	0.895942i	$-0.646502\pi$
$47$	−6.09017	−0.888343	−0.444171	+	0.895942i	$0.646502\pi$
$48$	4.85410	0.700629
$49$	20.4164	2.91663
$50$	0	0
$51$	2.00000	0.280056
$52$	−2.00000	−0.277350
$53$	−5.38197	−0.739270	−0.369635	−	0.929177i	$-0.620517\pi$
$53$	−5.38197	−0.739270	−0.369635	+	0.929177i	$0.620517\pi$
$54$	−1.61803	−0.220187
$55$	0	0
$56$	−11.7082	−1.56457
$57$	5.00000	0.662266
$58$	−1.38197	−0.181461
$59$	6.70820	0.873334	0.436667	−	0.899623i	$-0.356159\pi$
$59$	6.70820	0.873334	0.436667	+	0.899623i	$0.356159\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	11.3262	1.43843
$63$	5.23607	0.659683
$64$	4.23607	0.529508
$65$	0	0
$66$	0	0
$67$	−11.6180	−1.41937	−0.709684	−	0.704520i	$-0.751162\pi$
$67$	−11.6180	−1.41937	−0.709684	+	0.704520i	$0.751162\pi$
$68$	−1.23607	−0.149895
$69$	−4.61803	−0.555946
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−2.23607	−0.263523
$73$	4.52786	0.529946	0.264973	−	0.964256i	$-0.414637\pi$
$73$	4.52786	0.529946	0.264973	+	0.964256i	$0.414637\pi$
$74$	2.38197	0.276898
$75$	0	0
$76$	−3.09017	−0.354467
$77$	0	0
$78$	5.23607	0.592868
$79$	−3.09017	−0.347671	−0.173836	−	0.984775i	$-0.555616\pi$
$79$	−3.09017	−0.347671	−0.173836	+	0.984775i	$0.555616\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−18.5623	−2.04986
$83$	−7.70820	−0.846085	−0.423043	−	0.906110i	$-0.639038\pi$
$83$	−7.70820	−0.846085	−0.423043	+	0.906110i	$0.639038\pi$
$84$	−3.23607	−0.353084
$85$	0	0
$86$	−14.7082	−1.58603
$87$	0.854102	0.0915693
$88$	0	0
$89$	4.14590	0.439464	0.219732	−	0.975560i	$-0.429482\pi$
$89$	4.14590	0.439464	0.219732	+	0.975560i	$0.429482\pi$
$90$	0	0
$91$	−16.9443	−1.77624
$92$	2.85410	0.297561
$93$	−7.00000	−0.725866
$94$	−9.85410	−1.01637
$95$	0	0
$96$	3.38197	0.345170
$97$	2.52786	0.256666	0.128333	−	0.991731i	$-0.459037\pi$
$97$	2.52786	0.256666	0.128333	+	0.991731i	$0.459037\pi$
$98$	33.0344	3.33698
$99$	0	0
$100$	0	0
$101$	−19.5623	−1.94652	−0.973261	−	0.229702i	$-0.926225\pi$
$101$	−19.5623	−1.94652	−0.973261	+	0.229702i	$0.926225\pi$
$102$	3.23607	0.320418
$103$	4.61803	0.455028	0.227514	−	0.973775i	$-0.426940\pi$
$103$	4.61803	0.455028	0.227514	+	0.973775i	$0.426940\pi$
$104$	7.23607	0.709555
$105$	0	0
$106$	−8.70820	−0.845816
$107$	3.85410	0.372590	0.186295	−	0.982494i	$-0.440352\pi$
$107$	3.85410	0.372590	0.186295	+	0.982494i	$0.440352\pi$
$108$	−0.618034	−0.0594703
$109$	−10.8541	−1.03963	−0.519817	−	0.854278i	$-0.674000\pi$
$109$	−10.8541	−1.03963	−0.519817	+	0.854278i	$0.674000\pi$
$110$	0	0
$111$	−1.47214	−0.139729
$112$	−25.4164	−2.40162
$113$	−3.47214	−0.326631	−0.163316	−	0.986574i	$-0.552219\pi$
$113$	−3.47214	−0.326631	−0.163316	+	0.986574i	$0.552219\pi$
$114$	8.09017	0.757714
$115$	0	0
$116$	−0.527864	−0.0490109
$117$	−3.23607	−0.299175
$118$	10.8541	0.999201
$119$	−10.4721	−0.959979
$120$	0	0
$121$	0	0
$122$	−11.3262	−1.02543
$123$	11.4721	1.03441
$124$	4.32624	0.388508
$125$	0	0
$126$	8.47214	0.754758
$127$	1.94427	0.172526	0.0862631	−	0.996272i	$-0.472507\pi$
$127$	1.94427	0.172526	0.0862631	+	0.996272i	$0.472507\pi$
$128$	13.6180	1.20368
$129$	9.09017	0.800345
$130$	0	0
$131$	9.18034	0.802090	0.401045	−	0.916058i	$-0.368647\pi$
$131$	9.18034	0.802090	0.401045	+	0.916058i	$0.368647\pi$
$132$	0	0
$133$	−26.1803	−2.27012
$134$	−18.7984	−1.62393
$135$	0	0
$136$	4.47214	0.383482
$137$	5.29180	0.452109	0.226054	−	0.974115i	$-0.427417\pi$
$137$	5.29180	0.452109	0.226054	+	0.974115i	$0.427417\pi$
$138$	−7.47214	−0.636070
$139$	1.18034	0.100115	0.0500576	−	0.998746i	$-0.484060\pi$
$139$	1.18034	0.100115	0.0500576	+	0.998746i	$0.484060\pi$
$140$	0	0
$141$	6.09017	0.512885
$142$	−12.9443	−1.08626
$143$	0	0
$144$	−4.85410	−0.404508
$145$	0	0
$146$	7.32624	0.606324
$147$	−20.4164	−1.68392
$148$	0.909830	0.0747876
$149$	20.1246	1.64867	0.824336	−	0.566101i	$-0.191549\pi$
$149$	20.1246	1.64867	0.824336	+	0.566101i	$0.191549\pi$
$150$	0	0
$151$	15.2361	1.23989	0.619947	−	0.784644i	$-0.287154\pi$
$151$	15.2361	1.23989	0.619947	+	0.784644i	$0.287154\pi$
$152$	11.1803	0.906845
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	12.3262	0.983741	0.491870	−	0.870668i	$-0.336313\pi$
$157$	12.3262	0.983741	0.491870	+	0.870668i	$0.336313\pi$
$158$	−5.00000	−0.397779
$159$	5.38197	0.426818
$160$	0	0
$161$	24.1803	1.90568
$162$	1.61803	0.127125
$163$	−6.56231	−0.513999	−0.257000	−	0.966411i	$-0.582734\pi$
$163$	−6.56231	−0.513999	−0.257000	+	0.966411i	$0.582734\pi$
$164$	−7.09017	−0.553649
$165$	0	0
$166$	−12.4721	−0.968025
$167$	−9.03444	−0.699106	−0.349553	−	0.936917i	$-0.613667\pi$
$167$	−9.03444	−0.699106	−0.349553	+	0.936917i	$0.613667\pi$
$168$	11.7082	0.903308
$169$	−2.52786	−0.194451
$170$	0	0
$171$	−5.00000	−0.382360
$172$	−5.61803	−0.428371
$173$	16.5623	1.25921	0.629604	−	0.776916i	$-0.283217\pi$
$173$	16.5623	1.25921	0.629604	+	0.776916i	$0.283217\pi$
$174$	1.38197	0.104767
$175$	0	0
$176$	0	0
$177$	−6.70820	−0.504219
$178$	6.70820	0.502801
$179$	−10.5279	−0.786890	−0.393445	−	0.919348i	$-0.628717\pi$
$179$	−10.5279	−0.786890	−0.393445	+	0.919348i	$0.628717\pi$
$180$	0	0
$181$	−0.763932	−0.0567826	−0.0283913	−	0.999597i	$-0.509038\pi$
$181$	−0.763932	−0.0567826	−0.0283913	+	0.999597i	$0.509038\pi$
$182$	−27.4164	−2.03224
$183$	7.00000	0.517455
$184$	−10.3262	−0.761260
$185$	0	0
$186$	−11.3262	−0.830480
$187$	0	0
$188$	−3.76393	−0.274513
$189$	−5.23607	−0.380868
$190$	0	0
$191$	−16.4164	−1.18785	−0.593925	−	0.804521i	$-0.702422\pi$
$191$	−16.4164	−1.18785	−0.593925	+	0.804521i	$0.702422\pi$
$192$	−4.23607	−0.305712
$193$	−8.03444	−0.578332	−0.289166	−	0.957279i	$-0.593378\pi$
$193$	−8.03444	−0.578332	−0.289166	+	0.957279i	$0.593378\pi$
$194$	4.09017	0.293657
$195$	0	0
$196$	12.6180	0.901288
$197$	−9.76393	−0.695651	−0.347826	−	0.937559i	$-0.613080\pi$
$197$	−9.76393	−0.695651	−0.347826	+	0.937559i	$0.613080\pi$
$198$	0	0
$199$	−4.79837	−0.340148	−0.170074	−	0.985431i	$-0.554401\pi$
$199$	−4.79837	−0.340148	−0.170074	+	0.985431i	$0.554401\pi$
$200$	0	0
$201$	11.6180	0.819473
$202$	−31.6525	−2.22706
$203$	−4.47214	−0.313882
$204$	1.23607	0.0865421
$205$	0	0
$206$	7.47214	0.520608
$207$	4.61803	0.320976
$208$	15.7082	1.08917
$209$	0	0
$210$	0	0
$211$	−2.85410	−0.196484	−0.0982422	−	0.995163i	$-0.531322\pi$
$211$	−2.85410	−0.196484	−0.0982422	+	0.995163i	$0.531322\pi$
$212$	−3.32624	−0.228447
$213$	8.00000	0.548151
$214$	6.23607	0.426289
$215$	0	0
$216$	2.23607	0.152145
$217$	36.6525	2.48813
$218$	−17.5623	−1.18947
$219$	−4.52786	−0.305965
$220$	0	0
$221$	6.47214	0.435363
$222$	−2.38197	−0.159867
$223$	−21.0344	−1.40857	−0.704285	−	0.709917i	$-0.748732\pi$
$223$	−21.0344	−1.40857	−0.704285	+	0.709917i	$0.748732\pi$
$224$	−17.7082	−1.18318
$225$	0	0
$226$	−5.61803	−0.373706
$227$	−8.05573	−0.534677	−0.267339	−	0.963603i	$-0.586144\pi$
$227$	−8.05573	−0.534677	−0.267339	+	0.963603i	$0.586144\pi$
$228$	3.09017	0.204652
$229$	6.90983	0.456614	0.228307	−	0.973589i	$-0.426681\pi$
$229$	6.90983	0.456614	0.228307	+	0.973589i	$0.426681\pi$
$230$	0	0
$231$	0	0
$232$	1.90983	0.125386
$233$	−10.2705	−0.672843	−0.336422	−	0.941711i	$-0.609217\pi$
$233$	−10.2705	−0.672843	−0.336422	+	0.941711i	$0.609217\pi$
$234$	−5.23607	−0.342292
$235$	0	0
$236$	4.14590	0.269875
$237$	3.09017	0.200728
$238$	−16.9443	−1.09833
$239$	−22.0344	−1.42529	−0.712645	−	0.701525i	$-0.752503\pi$
$239$	−22.0344	−1.42529	−0.712645	+	0.701525i	$0.752503\pi$
$240$	0	0
$241$	−0.618034	−0.0398111	−0.0199055	−	0.999802i	$-0.506337\pi$
$241$	−0.618034	−0.0398111	−0.0199055	+	0.999802i	$0.506337\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−4.32624	−0.276959
$245$	0	0
$246$	18.5623	1.18349
$247$	16.1803	1.02953
$248$	−15.6525	−0.993933
$249$	7.70820	0.488488
$250$	0	0
$251$	16.4721	1.03971	0.519856	−	0.854254i	$-0.325986\pi$
$251$	16.4721	1.03971	0.519856	+	0.854254i	$0.325986\pi$
$252$	3.23607	0.203853
$253$	0	0
$254$	3.14590	0.197391
$255$	0	0
$256$	13.5623	0.847644
$257$	9.56231	0.596480	0.298240	−	0.954491i	$-0.403600\pi$
$257$	9.56231	0.596480	0.298240	+	0.954491i	$0.403600\pi$
$258$	14.7082	0.915693
$259$	7.70820	0.478964
$260$	0	0
$261$	−0.854102	−0.0528676
$262$	14.8541	0.917689
$263$	−24.2148	−1.49315	−0.746574	−	0.665303i	$-0.768302\pi$
$263$	−24.2148	−1.49315	−0.746574	+	0.665303i	$0.768302\pi$
$264$	0	0
$265$	0	0
$266$	−42.3607	−2.59730
$267$	−4.14590	−0.253725
$268$	−7.18034	−0.438609
$269$	0.854102	0.0520755	0.0260378	−	0.999661i	$-0.491711\pi$
$269$	0.854102	0.0520755	0.0260378	+	0.999661i	$0.491711\pi$
$270$	0	0
$271$	−18.3820	−1.11662	−0.558312	−	0.829631i	$-0.688551\pi$
$271$	−18.3820	−1.11662	−0.558312	+	0.829631i	$0.688551\pi$
$272$	9.70820	0.588646
$273$	16.9443	1.02551
$274$	8.56231	0.517268
$275$	0	0
$276$	−2.85410	−0.171797
$277$	−22.6525	−1.36106	−0.680528	−	0.732722i	$-0.738249\pi$
$277$	−22.6525	−1.36106	−0.680528	+	0.732722i	$0.738249\pi$
$278$	1.90983	0.114544
$279$	7.00000	0.419079
$280$	0	0
$281$	1.09017	0.0650341	0.0325170	−	0.999471i	$-0.489648\pi$
$281$	1.09017	0.0650341	0.0325170	+	0.999471i	$0.489648\pi$
$282$	9.85410	0.586803
$283$	16.0344	0.953149	0.476574	−	0.879134i	$-0.341878\pi$
$283$	16.0344	0.953149	0.476574	+	0.879134i	$0.341878\pi$
$284$	−4.94427	−0.293389
$285$	0	0
$286$	0	0
$287$	−60.0689	−3.54575
$288$	−3.38197	−0.199284
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−2.52786	−0.148186
$292$	2.79837	0.163762
$293$	−25.4721	−1.48810	−0.744049	−	0.668125i	$-0.767097\pi$
$293$	−25.4721	−1.48810	−0.744049	+	0.668125i	$0.767097\pi$
$294$	−33.0344	−1.92661
$295$	0	0
$296$	−3.29180	−0.191332
$297$	0	0
$298$	32.5623	1.88628
$299$	−14.9443	−0.864250
$300$	0	0
$301$	−47.5967	−2.74343
$302$	24.6525	1.41859
$303$	19.5623	1.12383
$304$	24.2705	1.39201
$305$	0	0
$306$	−3.23607	−0.184994
$307$	28.1246	1.60516	0.802578	−	0.596547i	$-0.203461\pi$
$307$	28.1246	1.60516	0.802578	+	0.596547i	$0.203461\pi$
$308$	0	0
$309$	−4.61803	−0.262711
$310$	0	0
$311$	12.5279	0.710390	0.355195	−	0.934792i	$-0.384414\pi$
$311$	12.5279	0.710390	0.355195	+	0.934792i	$0.384414\pi$
$312$	−7.23607	−0.409662
$313$	−8.47214	−0.478873	−0.239437	−	0.970912i	$-0.576963\pi$
$313$	−8.47214	−0.478873	−0.239437	+	0.970912i	$0.576963\pi$
$314$	19.9443	1.12552
$315$	0	0
$316$	−1.90983	−0.107436
$317$	3.58359	0.201275	0.100637	−	0.994923i	$-0.467912\pi$
$317$	3.58359	0.201275	0.100637	+	0.994923i	$0.467912\pi$
$318$	8.70820	0.488332
$319$	0	0
$320$	0	0
$321$	−3.85410	−0.215115
$322$	39.1246	2.18033
$323$	10.0000	0.556415
$324$	0.618034	0.0343352
$325$	0	0
$326$	−10.6180	−0.588079
$327$	10.8541	0.600233
$328$	25.6525	1.41642
$329$	−31.8885	−1.75807
$330$	0	0
$331$	−27.5967	−1.51685	−0.758427	−	0.651758i	$-0.774032\pi$
$331$	−27.5967	−1.51685	−0.758427	+	0.651758i	$0.774032\pi$
$332$	−4.76393	−0.261455
$333$	1.47214	0.0806726
$334$	−14.6180	−0.799863
$335$	0	0
$336$	25.4164	1.38658
$337$	28.8541	1.57178	0.785892	−	0.618364i	$-0.212204\pi$
$337$	28.8541	1.57178	0.785892	+	0.618364i	$0.212204\pi$
$338$	−4.09017	−0.222476
$339$	3.47214	0.188581
$340$	0	0
$341$	0	0
$342$	−8.09017	−0.437466
$343$	70.2492	3.79310
$344$	20.3262	1.09592
$345$	0	0
$346$	26.7984	1.44069
$347$	−28.1803	−1.51280	−0.756400	−	0.654109i	$-0.773044\pi$
$347$	−28.1803	−1.51280	−0.756400	+	0.654109i	$0.773044\pi$
$348$	0.527864	0.0282965
$349$	−32.2361	−1.72556	−0.862779	−	0.505582i	$-0.831278\pi$
$349$	−32.2361	−1.72556	−0.862779	+	0.505582i	$0.831278\pi$
$350$	0	0
$351$	3.23607	0.172729
$352$	0	0
$353$	17.8328	0.949145	0.474573	−	0.880216i	$-0.342603\pi$
$353$	17.8328	0.949145	0.474573	+	0.880216i	$0.342603\pi$
$354$	−10.8541	−0.576889
$355$	0	0
$356$	2.56231	0.135802
$357$	10.4721	0.554244
$358$	−17.0344	−0.900298
$359$	−3.29180	−0.173734	−0.0868672	−	0.996220i	$-0.527686\pi$
$359$	−3.29180	−0.173734	−0.0868672	+	0.996220i	$0.527686\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−1.23607	−0.0649663
$363$	0	0
$364$	−10.4721	−0.548889
$365$	0	0
$366$	11.3262	0.592032
$367$	22.1246	1.15490	0.577448	−	0.816428i	$-0.304049\pi$
$367$	22.1246	1.15490	0.577448	+	0.816428i	$0.304049\pi$
$368$	−22.4164	−1.16854
$369$	−11.4721	−0.597216
$370$	0	0
$371$	−28.1803	−1.46305
$372$	−4.32624	−0.224305
$373$	32.7426	1.69535	0.847675	−	0.530516i	$-0.178002\pi$
$373$	32.7426	1.69535	0.847675	+	0.530516i	$0.178002\pi$
$374$	0	0
$375$	0	0
$376$	13.6180	0.702296
$377$	2.76393	0.142350
$378$	−8.47214	−0.435760
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	−1.94427	−0.0996081
$382$	−26.5623	−1.35905
$383$	13.3607	0.682699	0.341349	−	0.939936i	$-0.389116\pi$
$383$	13.3607	0.682699	0.341349	+	0.939936i	$0.389116\pi$
$384$	−13.6180	−0.694942
$385$	0	0
$386$	−13.0000	−0.661683
$387$	−9.09017	−0.462079
$388$	1.56231	0.0793141
$389$	−21.0557	−1.06757	−0.533784	−	0.845621i	$-0.679230\pi$
$389$	−21.0557	−1.06757	−0.533784	+	0.845621i	$0.679230\pi$
$390$	0	0
$391$	−9.23607	−0.467088
$392$	−45.6525	−2.30580
$393$	−9.18034	−0.463087
$394$	−15.7984	−0.795911
$395$	0	0
$396$	0	0
$397$	2.72949	0.136989	0.0684946	−	0.997651i	$-0.478180\pi$
$397$	2.72949	0.136989	0.0684946	+	0.997651i	$0.478180\pi$
$398$	−7.76393	−0.389171
$399$	26.1803	1.31066
$400$	0	0
$401$	−30.8885	−1.54250	−0.771250	−	0.636532i	$-0.780368\pi$
$401$	−30.8885	−1.54250	−0.771250	+	0.636532i	$0.780368\pi$
$402$	18.7984	0.937578
$403$	−22.6525	−1.12840
$404$	−12.0902	−0.601508
$405$	0	0
$406$	−7.23607	−0.359120
$407$	0	0
$408$	−4.47214	−0.221404
$409$	26.3050	1.30070	0.650348	−	0.759636i	$-0.274623\pi$
$409$	26.3050	1.30070	0.650348	+	0.759636i	$0.274623\pi$
$410$	0	0
$411$	−5.29180	−0.261025
$412$	2.85410	0.140612
$413$	35.1246	1.72837
$414$	7.47214	0.367235
$415$	0	0
$416$	10.9443	0.536587
$417$	−1.18034	−0.0578015
$418$	0	0
$419$	−24.5967	−1.20163	−0.600815	−	0.799388i	$-0.705157\pi$
$419$	−24.5967	−1.20163	−0.600815	+	0.799388i	$0.705157\pi$
$420$	0	0
$421$	−3.85410	−0.187837	−0.0939187	−	0.995580i	$-0.529939\pi$
$421$	−3.85410	−0.187837	−0.0939187	+	0.995580i	$0.529939\pi$
$422$	−4.61803	−0.224802
$423$	−6.09017	−0.296114
$424$	12.0344	0.584444
$425$	0	0
$426$	12.9443	0.627152
$427$	−36.6525	−1.77374
$428$	2.38197	0.115137
$429$	0	0
$430$	0	0
$431$	15.8885	0.765324	0.382662	−	0.923888i	$-0.375007\pi$
$431$	15.8885	0.765324	0.382662	+	0.923888i	$0.375007\pi$
$432$	4.85410	0.233543
$433$	−1.03444	−0.0497121	−0.0248561	−	0.999691i	$-0.507913\pi$
$433$	−1.03444	−0.0497121	−0.0248561	+	0.999691i	$0.507913\pi$
$434$	59.3050	2.84673
$435$	0	0
$436$	−6.70820	−0.321265
$437$	−23.0902	−1.10455
$438$	−7.32624	−0.350061
$439$	−25.3262	−1.20876	−0.604378	−	0.796698i	$-0.706578\pi$
$439$	−25.3262	−1.20876	−0.604378	+	0.796698i	$0.706578\pi$
$440$	0	0
$441$	20.4164	0.972210
$442$	10.4721	0.498109
$443$	−8.27051	−0.392944	−0.196472	−	0.980509i	$-0.562948\pi$
$443$	−8.27051	−0.392944	−0.196472	+	0.980509i	$0.562948\pi$
$444$	−0.909830	−0.0431786
$445$	0	0
$446$	−34.0344	−1.61158
$447$	−20.1246	−0.951861
$448$	22.1803	1.04792
$449$	−34.2705	−1.61733	−0.808663	−	0.588273i	$-0.799808\pi$
$449$	−34.2705	−1.61733	−0.808663	+	0.588273i	$0.799808\pi$
$450$	0	0
$451$	0	0
$452$	−2.14590	−0.100935
$453$	−15.2361	−0.715853
$454$	−13.0344	−0.611737
$455$	0	0
$456$	−11.1803	−0.523567
$457$	7.47214	0.349532	0.174766	−	0.984610i	$-0.444083\pi$
$457$	7.47214	0.349532	0.174766	+	0.984610i	$0.444083\pi$
$458$	11.1803	0.522423
$459$	2.00000	0.0933520
$460$	0	0
$461$	−8.05573	−0.375193	−0.187596	−	0.982246i	$-0.560070\pi$
$461$	−8.05573	−0.375193	−0.187596	+	0.982246i	$0.560070\pi$
$462$	0	0
$463$	0.270510	0.0125717	0.00628583	−	0.999980i	$-0.497999\pi$
$463$	0.270510	0.0125717	0.00628583	+	0.999980i	$0.497999\pi$
$464$	4.14590	0.192468
$465$	0	0
$466$	−16.6180	−0.769816
$467$	−30.8885	−1.42935	−0.714676	−	0.699456i	$-0.753426\pi$
$467$	−30.8885	−1.42935	−0.714676	+	0.699456i	$0.753426\pi$
$468$	−2.00000	−0.0924500
$469$	−60.8328	−2.80900
$470$	0	0
$471$	−12.3262	−0.567963
$472$	−15.0000	−0.690431
$473$	0	0
$474$	5.00000	0.229658
$475$	0	0
$476$	−6.47214	−0.296650
$477$	−5.38197	−0.246423
$478$	−35.6525	−1.63071
$479$	−6.58359	−0.300812	−0.150406	−	0.988624i	$-0.548058\pi$
$479$	−6.58359	−0.300812	−0.150406	+	0.988624i	$0.548058\pi$
$480$	0	0
$481$	−4.76393	−0.217217
$482$	−1.00000	−0.0455488
$483$	−24.1803	−1.10024
$484$	0	0
$485$	0	0
$486$	−1.61803	−0.0733955
$487$	8.18034	0.370687	0.185343	−	0.982674i	$-0.440660\pi$
$487$	8.18034	0.370687	0.185343	+	0.982674i	$0.440660\pi$
$488$	15.6525	0.708554
$489$	6.56231	0.296758
$490$	0	0
$491$	−7.20163	−0.325005	−0.162502	−	0.986708i	$-0.551957\pi$
$491$	−7.20163	−0.325005	−0.162502	+	0.986708i	$0.551957\pi$
$492$	7.09017	0.319650
$493$	1.70820	0.0769336
$494$	26.1803	1.17791
$495$	0	0
$496$	−33.9787	−1.52569
$497$	−41.8885	−1.87896
$498$	12.4721	0.558890
$499$	11.1803	0.500501	0.250250	−	0.968181i	$-0.419487\pi$
$499$	11.1803	0.500501	0.250250	+	0.968181i	$0.419487\pi$
$500$	0	0
$501$	9.03444	0.403629
$502$	26.6525	1.18956
$503$	−13.8885	−0.619260	−0.309630	−	0.950857i	$-0.600205\pi$
$503$	−13.8885	−0.619260	−0.309630	+	0.950857i	$0.600205\pi$
$504$	−11.7082	−0.521525
$505$	0	0
$506$	0	0
$507$	2.52786	0.112266
$508$	1.20163	0.0533135
$509$	9.47214	0.419845	0.209923	−	0.977718i	$-0.432679\pi$
$509$	9.47214	0.419845	0.209923	+	0.977718i	$0.432679\pi$
$510$	0	0
$511$	23.7082	1.04879
$512$	−5.29180	−0.233867
$513$	5.00000	0.220755
$514$	15.4721	0.682447
$515$	0	0
$516$	5.61803	0.247320
$517$	0	0
$518$	12.4721	0.547994
$519$	−16.5623	−0.727005
$520$	0	0
$521$	29.3607	1.28631	0.643157	−	0.765734i	$-0.277624\pi$
$521$	29.3607	1.28631	0.643157	+	0.765734i	$0.277624\pi$
$522$	−1.38197	−0.0604870
$523$	−3.76393	−0.164585	−0.0822926	−	0.996608i	$-0.526224\pi$
$523$	−3.76393	−0.164585	−0.0822926	+	0.996608i	$0.526224\pi$
$524$	5.67376	0.247859
$525$	0	0
$526$	−39.1803	−1.70834
$527$	−14.0000	−0.609850
$528$	0	0
$529$	−1.67376	−0.0727723
$530$	0	0
$531$	6.70820	0.291111
$532$	−16.1803	−0.701507
$533$	37.1246	1.60805
$534$	−6.70820	−0.290292
$535$	0	0
$536$	25.9787	1.12211
$537$	10.5279	0.454311
$538$	1.38197	0.0595808
$539$	0	0
$540$	0	0
$541$	1.61803	0.0695647	0.0347824	−	0.999395i	$-0.488926\pi$
$541$	1.61803	0.0695647	0.0347824	+	0.999395i	$0.488926\pi$
$542$	−29.7426	−1.27756
$543$	0.763932	0.0327835
$544$	6.76393	0.290001
$545$	0	0
$546$	27.4164	1.17331
$547$	−19.3607	−0.827803	−0.413901	−	0.910322i	$-0.635834\pi$
$547$	−19.3607	−0.827803	−0.413901	+	0.910322i	$0.635834\pi$
$548$	3.27051	0.139709
$549$	−7.00000	−0.298753
$550$	0	0
$551$	4.27051	0.181930
$552$	10.3262	0.439514
$553$	−16.1803	−0.688058
$554$	−36.6525	−1.55721
$555$	0	0
$556$	0.729490	0.0309373
$557$	42.3951	1.79634	0.898169	−	0.439649i	$-0.144897\pi$
$557$	42.3951	1.79634	0.898169	+	0.439649i	$0.144897\pi$
$558$	11.3262	0.479478
$559$	29.4164	1.24418
$560$	0	0
$561$	0	0
$562$	1.76393	0.0744070
$563$	−20.6738	−0.871295	−0.435648	−	0.900117i	$-0.643481\pi$
$563$	−20.6738	−0.871295	−0.435648	+	0.900117i	$0.643481\pi$
$564$	3.76393	0.158490
$565$	0	0
$566$	25.9443	1.09052
$567$	5.23607	0.219894
$568$	17.8885	0.750587
$569$	22.5623	0.945861	0.472931	−	0.881100i	$-0.343196\pi$
$569$	22.5623	0.945861	0.472931	+	0.881100i	$0.343196\pi$
$570$	0	0
$571$	11.0902	0.464109	0.232055	−	0.972703i	$-0.425455\pi$
$571$	11.0902	0.464109	0.232055	+	0.972703i	$0.425455\pi$
$572$	0	0
$573$	16.4164	0.685805
$574$	−97.1935	−4.05678
$575$	0	0
$576$	4.23607	0.176503
$577$	30.0132	1.24946	0.624732	−	0.780839i	$-0.285208\pi$
$577$	30.0132	1.24946	0.624732	+	0.780839i	$0.285208\pi$
$578$	−21.0344	−0.874917
$579$	8.03444	0.333900
$580$	0	0
$581$	−40.3607	−1.67444
$582$	−4.09017	−0.169543
$583$	0	0
$584$	−10.1246	−0.418959
$585$	0	0
$586$	−41.2148	−1.70257
$587$	−21.6180	−0.892272	−0.446136	−	0.894965i	$-0.647200\pi$
$587$	−21.6180	−0.892272	−0.446136	+	0.894965i	$0.647200\pi$
$588$	−12.6180	−0.520359
$589$	−35.0000	−1.44215
$590$	0	0
$591$	9.76393	0.401634
$592$	−7.14590	−0.293695
$593$	−3.11146	−0.127772	−0.0638861	−	0.997957i	$-0.520349\pi$
$593$	−3.11146	−0.127772	−0.0638861	+	0.997957i	$0.520349\pi$
$594$	0	0
$595$	0	0
$596$	12.4377	0.509468
$597$	4.79837	0.196384
$598$	−24.1803	−0.988808
$599$	−17.7639	−0.725815	−0.362907	−	0.931825i	$-0.618216\pi$
$599$	−17.7639	−0.725815	−0.362907	+	0.931825i	$0.618216\pi$
$600$	0	0
$601$	10.3607	0.422621	0.211310	−	0.977419i	$-0.432227\pi$
$601$	10.3607	0.422621	0.211310	+	0.977419i	$0.432227\pi$
$602$	−77.0132	−3.13882
$603$	−11.6180	−0.473123
$604$	9.41641	0.383148
$605$	0	0
$606$	31.6525	1.28579
$607$	33.0000	1.33943	0.669714	−	0.742619i	$-0.266417\pi$
$607$	33.0000	1.33943	0.669714	+	0.742619i	$0.266417\pi$
$608$	16.9098	0.685784
$609$	4.47214	0.181220
$610$	0	0
$611$	19.7082	0.797309
$612$	−1.23607	−0.0499651
$613$	−3.76393	−0.152024	−0.0760119	−	0.997107i	$-0.524219\pi$
$613$	−3.76393	−0.152024	−0.0760119	+	0.997107i	$0.524219\pi$
$614$	45.5066	1.83650
$615$	0	0
$616$	0	0
$617$	42.2492	1.70089	0.850445	−	0.526064i	$-0.176333\pi$
$617$	42.2492	1.70089	0.850445	+	0.526064i	$0.176333\pi$
$618$	−7.47214	−0.300573
$619$	42.8885	1.72384	0.861918	−	0.507048i	$-0.169263\pi$
$619$	42.8885	1.72384	0.861918	+	0.507048i	$0.169263\pi$
$620$	0	0
$621$	−4.61803	−0.185315
$622$	20.2705	0.812773
$623$	21.7082	0.869721
$624$	−15.7082	−0.628831
$625$	0	0
$626$	−13.7082	−0.547890
$627$	0	0
$628$	7.61803	0.303993
$629$	−2.94427	−0.117396
$630$	0	0
$631$	−39.6312	−1.57769	−0.788846	−	0.614590i	$-0.789321\pi$
$631$	−39.6312	−1.57769	−0.788846	+	0.614590i	$0.789321\pi$
$632$	6.90983	0.274858
$633$	2.85410	0.113440
$634$	5.79837	0.230283
$635$	0	0
$636$	3.32624	0.131894
$637$	−66.0689	−2.61774
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	4.23607	0.167315	0.0836573	−	0.996495i	$-0.473340\pi$
$641$	4.23607	0.167315	0.0836573	+	0.996495i	$0.473340\pi$
$642$	−6.23607	−0.246118
$643$	−13.6738	−0.539241	−0.269620	−	0.962967i	$-0.586898\pi$
$643$	−13.6738	−0.539241	−0.269620	+	0.962967i	$0.586898\pi$
$644$	14.9443	0.588887
$645$	0	0
$646$	16.1803	0.636607
$647$	31.5967	1.24220	0.621098	−	0.783733i	$-0.286687\pi$
$647$	31.5967	1.24220	0.621098	+	0.783733i	$0.286687\pi$
$648$	−2.23607	−0.0878410
$649$	0	0
$650$	0	0
$651$	−36.6525	−1.43652
$652$	−4.05573	−0.158835
$653$	43.5623	1.70472	0.852362	−	0.522952i	$-0.175169\pi$
$653$	43.5623	1.70472	0.852362	+	0.522952i	$0.175169\pi$
$654$	17.5623	0.686741
$655$	0	0
$656$	55.6869	2.17421
$657$	4.52786	0.176649
$658$	−51.5967	−2.01145
$659$	−42.0344	−1.63743	−0.818715	−	0.574201i	$-0.805313\pi$
$659$	−42.0344	−1.63743	−0.818715	+	0.574201i	$0.805313\pi$
$660$	0	0
$661$	15.0902	0.586940	0.293470	−	0.955968i	$-0.405190\pi$
$661$	15.0902	0.586940	0.293470	+	0.955968i	$0.405190\pi$
$662$	−44.6525	−1.73547
$663$	−6.47214	−0.251357
$664$	17.2361	0.668889
$665$	0	0
$666$	2.38197	0.0922993
$667$	−3.94427	−0.152723
$668$	−5.58359	−0.216036
$669$	21.0344	0.813239
$670$	0	0
$671$	0	0
$672$	17.7082	0.683109
$673$	35.3050	1.36091	0.680453	−	0.732792i	$-0.261783\pi$
$673$	35.3050	1.36091	0.680453	+	0.732792i	$0.261783\pi$
$674$	46.6869	1.79831
$675$	0	0
$676$	−1.56231	−0.0600887
$677$	−5.94427	−0.228457	−0.114228	−	0.993455i	$-0.536440\pi$
$677$	−5.94427	−0.228457	−0.114228	+	0.993455i	$0.536440\pi$
$678$	5.61803	0.215759
$679$	13.2361	0.507954
$680$	0	0
$681$	8.05573	0.308696
$682$	0	0
$683$	−15.7082	−0.601058	−0.300529	−	0.953773i	$-0.597163\pi$
$683$	−15.7082	−0.601058	−0.300529	+	0.953773i	$0.597163\pi$
$684$	−3.09017	−0.118156
$685$	0	0
$686$	113.666	4.33977
$687$	−6.90983	−0.263626
$688$	44.1246	1.68224
$689$	17.4164	0.663512
$690$	0	0
$691$	−29.3050	−1.11481	−0.557406	−	0.830240i	$-0.688203\pi$
$691$	−29.3050	−1.11481	−0.557406	+	0.830240i	$0.688203\pi$
$692$	10.2361	0.389117
$693$	0	0
$694$	−45.5967	−1.73083
$695$	0	0
$696$	−1.90983	−0.0723919
$697$	22.9443	0.869076
$698$	−52.1591	−1.97425
$699$	10.2705	0.388466
$700$	0	0
$701$	−40.4164	−1.52651	−0.763253	−	0.646099i	$-0.776399\pi$
$701$	−40.4164	−1.52651	−0.763253	+	0.646099i	$0.776399\pi$
$702$	5.23607	0.197623
$703$	−7.36068	−0.277613
$704$	0	0
$705$	0	0
$706$	28.8541	1.08594
$707$	−102.430	−3.85226
$708$	−4.14590	−0.155812
$709$	18.8197	0.706787	0.353394	−	0.935475i	$-0.385028\pi$
$709$	18.8197	0.706787	0.353394	+	0.935475i	$0.385028\pi$
$710$	0	0
$711$	−3.09017	−0.115890
$712$	−9.27051	−0.347427
$713$	32.3262	1.21063
$714$	16.9443	0.634123
$715$	0	0
$716$	−6.50658	−0.243162
$717$	22.0344	0.822891
$718$	−5.32624	−0.198773
$719$	26.9098	1.00357	0.501784	−	0.864993i	$-0.332677\pi$
$719$	26.9098	1.00357	0.501784	+	0.864993i	$0.332677\pi$
$720$	0	0
$721$	24.1803	0.900523
$722$	9.70820	0.361302
$723$	0.618034	0.0229849
$724$	−0.472136	−0.0175468
$725$	0	0
$726$	0	0
$727$	21.6738	0.803835	0.401918	−	0.915676i	$-0.368344\pi$
$727$	21.6738	0.803835	0.401918	+	0.915676i	$0.368344\pi$
$728$	37.8885	1.40424
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.1803	0.672424
$732$	4.32624	0.159902
$733$	4.85410	0.179290	0.0896452	−	0.995974i	$-0.471427\pi$
$733$	4.85410	0.179290	0.0896452	+	0.995974i	$0.471427\pi$
$734$	35.7984	1.32134
$735$	0	0
$736$	−15.6180	−0.575688
$737$	0	0
$738$	−18.5623	−0.683288
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−16.1803	−0.594400
$742$	−45.5967	−1.67391
$743$	−31.6525	−1.16122	−0.580608	−	0.814183i	$-0.697185\pi$
$743$	−31.6525	−1.16122	−0.580608	+	0.814183i	$0.697185\pi$
$744$	15.6525	0.573848
$745$	0	0
$746$	52.9787	1.93969
$747$	−7.70820	−0.282028
$748$	0	0
$749$	20.1803	0.737374
$750$	0	0
$751$	−10.1115	−0.368972	−0.184486	−	0.982835i	$-0.559062\pi$
$751$	−10.1115	−0.368972	−0.184486	+	0.982835i	$0.559062\pi$
$752$	29.5623	1.07803
$753$	−16.4721	−0.600278
$754$	4.47214	0.162866
$755$	0	0
$756$	−3.23607	−0.117695
$757$	−1.81966	−0.0661367	−0.0330683	−	0.999453i	$-0.510528\pi$
$757$	−1.81966	−0.0661367	−0.0330683	+	0.999453i	$0.510528\pi$
$758$	−8.09017	−0.293848
$759$	0	0
$760$	0	0
$761$	−9.96556	−0.361251	−0.180626	−	0.983552i	$-0.557812\pi$
$761$	−9.96556	−0.361251	−0.180626	+	0.983552i	$0.557812\pi$
$762$	−3.14590	−0.113964
$763$	−56.8328	−2.05749
$764$	−10.1459	−0.367066
$765$	0	0
$766$	21.6180	0.781091
$767$	−21.7082	−0.783838
$768$	−13.5623	−0.489388
$769$	−17.2361	−0.621549	−0.310774	−	0.950484i	$-0.600588\pi$
$769$	−17.2361	−0.621549	−0.310774	+	0.950484i	$0.600588\pi$
$770$	0	0
$771$	−9.56231	−0.344378
$772$	−4.96556	−0.178714
$773$	27.5066	0.989343	0.494671	−	0.869080i	$-0.335288\pi$
$773$	27.5066	0.989343	0.494671	+	0.869080i	$0.335288\pi$
$774$	−14.7082	−0.528675
$775$	0	0
$776$	−5.65248	−0.202912
$777$	−7.70820	−0.276530
$778$	−34.0689	−1.22143
$779$	57.3607	2.05516
$780$	0	0
$781$	0	0
$782$	−14.9443	−0.534406
$783$	0.854102	0.0305231
$784$	−99.1033	−3.53940
$785$	0	0
$786$	−14.8541	−0.529828
$787$	32.7984	1.16914	0.584568	−	0.811345i	$-0.301264\pi$
$787$	32.7984	1.16914	0.584568	+	0.811345i	$0.301264\pi$
$788$	−6.03444	−0.214968
$789$	24.2148	0.862069
$790$	0	0
$791$	−18.1803	−0.646418
$792$	0	0
$793$	22.6525	0.804413
$794$	4.41641	0.156732
$795$	0	0
$796$	−2.96556	−0.105111
$797$	15.2918	0.541663	0.270832	−	0.962627i	$-0.412701\pi$
$797$	15.2918	0.541663	0.270832	+	0.962627i	$0.412701\pi$
$798$	42.3607	1.49955
$799$	12.1803	0.430909
$800$	0	0
$801$	4.14590	0.146488
$802$	−49.9787	−1.76481
$803$	0	0
$804$	7.18034	0.253231
$805$	0	0
$806$	−36.6525	−1.29103
$807$	−0.854102	−0.0300658
$808$	43.7426	1.53886
$809$	−29.6738	−1.04327	−0.521637	−	0.853168i	$-0.674678\pi$
$809$	−29.6738	−1.04327	−0.521637	+	0.853168i	$0.674678\pi$
$810$	0	0
$811$	−43.6312	−1.53210	−0.766049	−	0.642782i	$-0.777780\pi$
$811$	−43.6312	−1.53210	−0.766049	+	0.642782i	$0.777780\pi$
$812$	−2.76393	−0.0969950
$813$	18.3820	0.644684
$814$	0	0
$815$	0	0
$816$	−9.70820	−0.339855
$817$	45.4508	1.59012
$818$	42.5623	1.48816
$819$	−16.9443	−0.592081
$820$	0	0
$821$	−20.8197	−0.726611	−0.363306	−	0.931670i	$-0.618352\pi$
$821$	−20.8197	−0.726611	−0.363306	+	0.931670i	$0.618352\pi$
$822$	−8.56231	−0.298645
$823$	16.5279	0.576125	0.288063	−	0.957612i	$-0.406989\pi$
$823$	16.5279	0.576125	0.288063	+	0.957612i	$0.406989\pi$
$824$	−10.3262	−0.359732
$825$	0	0
$826$	56.8328	1.97747
$827$	51.4164	1.78792	0.893962	−	0.448143i	$-0.147914\pi$
$827$	51.4164	1.78792	0.893962	+	0.448143i	$0.147914\pi$
$828$	2.85410	0.0991869
$829$	−32.8885	−1.14227	−0.571133	−	0.820857i	$-0.693496\pi$
$829$	−32.8885	−1.14227	−0.571133	+	0.820857i	$0.693496\pi$
$830$	0	0
$831$	22.6525	0.785806
$832$	−13.7082	−0.475246
$833$	−40.8328	−1.41477
$834$	−1.90983	−0.0661320
$835$	0	0
$836$	0	0
$837$	−7.00000	−0.241955
$838$	−39.7984	−1.37481
$839$	36.3050	1.25339	0.626693	−	0.779266i	$-0.284408\pi$
$839$	36.3050	1.25339	0.626693	+	0.779266i	$0.284408\pi$
$840$	0	0
$841$	−28.2705	−0.974845
$842$	−6.23607	−0.214909
$843$	−1.09017	−0.0375474
$844$	−1.76393	−0.0607170
$845$	0	0
$846$	−9.85410	−0.338791
$847$	0	0
$848$	26.1246	0.897123
$849$	−16.0344	−0.550301
$850$	0	0
$851$	6.79837	0.233045
$852$	4.94427	0.169388
$853$	−23.3607	−0.799854	−0.399927	−	0.916547i	$-0.630965\pi$
$853$	−23.3607	−0.799854	−0.399927	+	0.916547i	$0.630965\pi$
$854$	−59.3050	−2.02937
$855$	0	0
$856$	−8.61803	−0.294558
$857$	41.0132	1.40098	0.700491	−	0.713661i	$-0.252964\pi$
$857$	41.0132	1.40098	0.700491	+	0.713661i	$0.252964\pi$
$858$	0	0
$859$	−39.2705	−1.33989	−0.669946	−	0.742410i	$-0.733683\pi$
$859$	−39.2705	−1.33989	−0.669946	+	0.742410i	$0.733683\pi$
$860$	0	0
$861$	60.0689	2.04714
$862$	25.7082	0.875625
$863$	23.0344	0.784102	0.392051	−	0.919944i	$-0.371766\pi$
$863$	23.0344	0.784102	0.392051	+	0.919944i	$0.371766\pi$
$864$	3.38197	0.115057
$865$	0	0
$866$	−1.67376	−0.0568768
$867$	13.0000	0.441503
$868$	22.6525	0.768875
$869$	0	0
$870$	0	0
$871$	37.5967	1.27392
$872$	24.2705	0.821903
$873$	2.52786	0.0855552
$874$	−37.3607	−1.26374
$875$	0	0
$876$	−2.79837	−0.0945483
$877$	−7.52786	−0.254198	−0.127099	−	0.991890i	$-0.540567\pi$
$877$	−7.52786	−0.254198	−0.127099	+	0.991890i	$0.540567\pi$
$878$	−40.9787	−1.38296
$879$	25.4721	0.859154
$880$	0	0
$881$	−17.1459	−0.577660	−0.288830	−	0.957380i	$-0.593266\pi$
$881$	−17.1459	−0.577660	−0.288830	+	0.957380i	$0.593266\pi$
$882$	33.0344	1.11233
$883$	23.2361	0.781956	0.390978	−	0.920400i	$-0.372137\pi$
$883$	23.2361	0.781956	0.390978	+	0.920400i	$0.372137\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−13.3820	−0.449576
$887$	−0.167184	−0.00561350	−0.00280675	−	0.999996i	$-0.500893\pi$
$887$	−0.167184	−0.00561350	−0.00280675	+	0.999996i	$0.500893\pi$
$888$	3.29180	0.110465
$889$	10.1803	0.341438
$890$	0	0
$891$	0	0
$892$	−13.0000	−0.435272
$893$	30.4508	1.01900
$894$	−32.5623	−1.08905
$895$	0	0
$896$	71.3050	2.38213
$897$	14.9443	0.498975
$898$	−55.4508	−1.85042
$899$	−5.97871	−0.199401
$900$	0	0
$901$	10.7639	0.358599
$902$	0	0
$903$	47.5967	1.58392
$904$	7.76393	0.258225
$905$	0	0
$906$	−24.6525	−0.819024
$907$	44.8885	1.49050	0.745250	−	0.666785i	$-0.232330\pi$
$907$	44.8885	1.49050	0.745250	+	0.666785i	$0.232330\pi$
$908$	−4.97871	−0.165224
$909$	−19.5623	−0.648841
$910$	0	0
$911$	52.9017	1.75271	0.876356	−	0.481664i	$-0.159968\pi$
$911$	52.9017	1.75271	0.876356	+	0.481664i	$0.159968\pi$
$912$	−24.2705	−0.803677
$913$	0	0
$914$	12.0902	0.399907
$915$	0	0
$916$	4.27051	0.141102
$917$	48.0689	1.58737
$918$	3.23607	0.106806
$919$	22.2361	0.733500	0.366750	−	0.930319i	$-0.380470\pi$
$919$	22.2361	0.733500	0.366750	+	0.930319i	$0.380470\pi$
$920$	0	0
$921$	−28.1246	−0.926737
$922$	−13.0344	−0.429266
$923$	25.8885	0.852132
$924$	0	0
$925$	0	0
$926$	0.437694	0.0143835
$927$	4.61803	0.151676
$928$	2.88854	0.0948211
$929$	−10.8541	−0.356112	−0.178056	−	0.984020i	$-0.556981\pi$
$929$	−10.8541	−0.356112	−0.178056	+	0.984020i	$0.556981\pi$
$930$	0	0
$931$	−102.082	−3.34560
$932$	−6.34752	−0.207920
$933$	−12.5279	−0.410144
$934$	−49.9787	−1.63535
$935$	0	0
$936$	7.23607	0.236518
$937$	−17.2016	−0.561953	−0.280976	−	0.959715i	$-0.590658\pi$
$937$	−17.2016	−0.561953	−0.280976	+	0.959715i	$0.590658\pi$
$938$	−98.4296	−3.21384
$939$	8.47214	0.276478
$940$	0	0
$941$	9.70820	0.316478	0.158239	−	0.987401i	$-0.449418\pi$
$941$	9.70820	0.316478	0.158239	+	0.987401i	$0.449418\pi$
$942$	−19.9443	−0.649819
$943$	−52.9787	−1.72522
$944$	−32.5623	−1.05981
$945$	0	0
$946$	0	0
$947$	4.56231	0.148255	0.0741275	−	0.997249i	$-0.476383\pi$
$947$	4.56231	0.148255	0.0741275	+	0.997249i	$0.476383\pi$
$948$	1.90983	0.0620284
$949$	−14.6525	−0.475639
$950$	0	0
$951$	−3.58359	−0.116206
$952$	23.4164	0.758930
$953$	−49.0902	−1.59019	−0.795093	−	0.606487i	$-0.792578\pi$
$953$	−49.0902	−1.59019	−0.795093	+	0.606487i	$0.792578\pi$
$954$	−8.70820	−0.281939
$955$	0	0
$956$	−13.6180	−0.440439
$957$	0	0
$958$	−10.6525	−0.344166
$959$	27.7082	0.894745
$960$	0	0
$961$	18.0000	0.580645
$962$	−7.70820	−0.248522
$963$	3.85410	0.124197
$964$	−0.381966	−0.0123023
$965$	0	0
$966$	−39.1246	−1.25881
$967$	32.6738	1.05072	0.525359	−	0.850881i	$-0.323931\pi$
$967$	32.6738	1.05072	0.525359	+	0.850881i	$0.323931\pi$
$968$	0	0
$969$	−10.0000	−0.321246
$970$	0	0
$971$	−31.5410	−1.01220	−0.506100	−	0.862475i	$-0.668913\pi$
$971$	−31.5410	−1.01220	−0.506100	+	0.862475i	$0.668913\pi$
$972$	−0.618034	−0.0198234
$973$	6.18034	0.198133
$974$	13.2361	0.424111
$975$	0	0
$976$	33.9787	1.08763
$977$	−46.0902	−1.47456	−0.737278	−	0.675590i	$-0.763889\pi$
$977$	−46.0902	−1.47456	−0.737278	+	0.675590i	$0.763889\pi$
$978$	10.6180	0.339527
$979$	0	0
$980$	0	0
$981$	−10.8541	−0.346545
$982$	−11.6525	−0.371845
$983$	−19.0000	−0.606006	−0.303003	−	0.952990i	$-0.597989\pi$
$983$	−19.0000	−0.606006	−0.303003	+	0.952990i	$0.597989\pi$
$984$	−25.6525	−0.817771
$985$	0	0
$986$	2.76393	0.0880215
$987$	31.8885	1.01502
$988$	10.0000	0.318142
$989$	−41.9787	−1.33485
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	−23.6738	−0.751643
$993$	27.5967	0.875756
$994$	−67.7771	−2.14976
$995$	0	0
$996$	4.76393	0.150951
$997$	11.2918	0.357615	0.178807	−	0.983884i	$-0.442776\pi$
$997$	11.2918	0.357615	0.178807	+	0.983884i	$0.442776\pi$
$998$	18.0902	0.572634
$999$	−1.47214	−0.0465763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.bt.1.2		2
5.4	even	2		9075.2.a.bc.1.1		2
11.7	odd	10		825.2.n.b.676.1	yes	4
11.8	odd	10		825.2.n.b.526.1	✓	4
11.10	odd	2		9075.2.a.z.1.1		2
55.7	even	20		825.2.bx.c.49.2		8
55.8	even	20		825.2.bx.c.724.2		8
55.18	even	20		825.2.bx.c.49.1		8
55.19	odd	10		825.2.n.d.526.1	yes	4
55.29	odd	10		825.2.n.d.676.1	yes	4
55.52	even	20		825.2.bx.c.724.1		8
55.54	odd	2		9075.2.a.by.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.n.b.526.1	✓	4	11.8	odd	10
825.2.n.b.676.1	yes	4	11.7	odd	10
825.2.n.d.526.1	yes	4	55.19	odd	10
825.2.n.d.676.1	yes	4	55.29	odd	10
825.2.bx.c.49.1		8	55.18	even	20
825.2.bx.c.49.2		8	55.7	even	20
825.2.bx.c.724.1		8	55.52	even	20
825.2.bx.c.724.2		8	55.8	even	20
9075.2.a.z.1.1		2	11.10	odd	2
9075.2.a.bc.1.1		2	5.4	even	2
9075.2.a.bt.1.2		2	1.1	even	1	trivial
9075.2.a.by.1.2		2	55.54	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +5.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +5.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} -0.618034 q^{12} -3.23607 q^{13} +8.47214 q^{14} -4.85410 q^{16} -2.00000 q^{17} +1.61803 q^{18} -5.00000 q^{19} -5.23607 q^{21} +4.61803 q^{23} +2.23607 q^{24} -5.23607 q^{26} -1.00000 q^{27} +3.23607 q^{28} -0.854102 q^{29} +7.00000 q^{31} -3.38197 q^{32} -3.23607 q^{34} +0.618034 q^{36} +1.47214 q^{37} -8.09017 q^{38} +3.23607 q^{39} -11.4721 q^{41} -8.47214 q^{42} -9.09017 q^{43} +7.47214 q^{46} -6.09017 q^{47} +4.85410 q^{48} +20.4164 q^{49} +2.00000 q^{51} -2.00000 q^{52} -5.38197 q^{53} -1.61803 q^{54} -11.7082 q^{56} +5.00000 q^{57} -1.38197 q^{58} +6.70820 q^{59} -7.00000 q^{61} +11.3262 q^{62} +5.23607 q^{63} +4.23607 q^{64} -11.6180 q^{67} -1.23607 q^{68} -4.61803 q^{69} -8.00000 q^{71} -2.23607 q^{72} +4.52786 q^{73} +2.38197 q^{74} -3.09017 q^{76} +5.23607 q^{78} -3.09017 q^{79} +1.00000 q^{81} -18.5623 q^{82} -7.70820 q^{83} -3.23607 q^{84} -14.7082 q^{86} +0.854102 q^{87} +4.14590 q^{89} -16.9443 q^{91} +2.85410 q^{92} -7.00000 q^{93} -9.85410 q^{94} +3.38197 q^{96} +2.52786 q^{97} +33.0344 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9	\( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9} + q^{12} - 2 q^{13} + 8 q^{14} - 3 q^{16} - 4 q^{17} + q^{18} - 10 q^{19} - 6 q^{21} + 7 q^{23} - 6 q^{26} - 2 q^{27} + 2 q^{28} + 5 q^{29} + 14 q^{31} - 9 q^{32} - 2 q^{34} - q^{36} - 6 q^{37} - 5 q^{38} + 2 q^{39} - 14 q^{41} - 8 q^{42} - 7 q^{43} + 6 q^{46} - q^{47} + 3 q^{48} + 14 q^{49} + 4 q^{51} - 4 q^{52} - 13 q^{53} - q^{54} - 10 q^{56} + 10 q^{57} - 5 q^{58} - 14 q^{61} + 7 q^{62} + 6 q^{63} + 4 q^{64} - 21 q^{67} + 2 q^{68} - 7 q^{69} - 16 q^{71} + 18 q^{73} + 7 q^{74} + 5 q^{76} + 6 q^{78} + 5 q^{79} + 2 q^{81} - 17 q^{82} - 2 q^{83} - 2 q^{84} - 16 q^{86} - 5 q^{87} + 15 q^{89} - 16 q^{91} - q^{92} - 14 q^{93} - 13 q^{94} + 9 q^{96} + 14 q^{97} + 37 q^{98}+O(q^{100}) \) 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 + q^12 - 2 * q^13 + 8 * q^14 - 3 * q^16 - 4 * q^17 + q^18 - 10 * q^19 - 6 * q^21 + 7 * q^23 - 6 * q^26 - 2 * q^27 + 2 * q^28 + 5 * q^29 + 14 * q^31 - 9 * q^32 - 2 * q^34 - q^36 - 6 * q^37 - 5 * q^38 + 2 * q^39 - 14 * q^41 - 8 * q^42 - 7 * q^43 + 6 * q^46 - q^47 + 3 * q^48 + 14 * q^49 + 4 * q^51 - 4 * q^52 - 13 * q^53 - q^54 - 10 * q^56 + 10 * q^57 - 5 * q^58 - 14 * q^61 + 7 * q^62 + 6 * q^63 + 4 * q^64 - 21 * q^67 + 2 * q^68 - 7 * q^69 - 16 * q^71 + 18 * q^73 + 7 * q^74 + 5 * q^76 + 6 * q^78 + 5 * q^79 + 2 * q^81 - 17 * q^82 - 2 * q^83 - 2 * q^84 - 16 * q^86 - 5 * q^87 + 15 * q^89 - 16 * q^91 - q^92 - 14 * q^93 - 13 * q^94 + 9 * q^96 + 14 * q^97 + 37 * q^98

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