LMFDB - Embedded newform 9075.2.a.z.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:	$0$
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			$-0.618034$ 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)

$f(q)$	$=$	$q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -0.763932 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -0.763932 q^{7} -2.23607 q^{8} +1.00000 q^{9} +1.61803 q^{12} -1.23607 q^{13} -0.472136 q^{14} +1.85410 q^{16} +2.00000 q^{17} +0.618034 q^{18} +5.00000 q^{19} +0.763932 q^{21} +2.38197 q^{23} +2.23607 q^{24} -0.763932 q^{26} -1.00000 q^{27} +1.23607 q^{28} -5.85410 q^{29} +7.00000 q^{31} +5.61803 q^{32} +1.23607 q^{34} -1.61803 q^{36} -7.47214 q^{37} +3.09017 q^{38} +1.23607 q^{39} +2.52786 q^{41} +0.472136 q^{42} -2.09017 q^{43} +1.47214 q^{46} +5.09017 q^{47} -1.85410 q^{48} -6.41641 q^{49} -2.00000 q^{51} +2.00000 q^{52} -7.61803 q^{53} -0.618034 q^{54} +1.70820 q^{56} -5.00000 q^{57} -3.61803 q^{58} -6.70820 q^{59} +7.00000 q^{61} +4.32624 q^{62} -0.763932 q^{63} -0.236068 q^{64} -9.38197 q^{67} -3.23607 q^{68} -2.38197 q^{69} -8.00000 q^{71} -2.23607 q^{72} -13.4721 q^{73} -4.61803 q^{74} -8.09017 q^{76} +0.763932 q^{78} -8.09017 q^{79} +1.00000 q^{81} +1.56231 q^{82} -5.70820 q^{83} -1.23607 q^{84} -1.29180 q^{86} +5.85410 q^{87} +10.8541 q^{89} +0.944272 q^{91} -3.85410 q^{92} -7.00000 q^{93} +3.14590 q^{94} -5.61803 q^{96} +11.4721 q^{97} -3.96556 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	−0.618034	−0.252311
$7$	−0.763932	−0.288739	−0.144370	−	0.989524i	$-0.546115\pi$
$7$	−0.763932	−0.288739	−0.144370	+	0.989524i	$0.546115\pi$
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	1.61803	0.467086
$13$	−1.23607	−0.342824	−0.171412	−	0.985199i	$-0.554833\pi$
$13$	−1.23607	−0.342824	−0.171412	+	0.985199i	$0.554833\pi$
$14$	−0.472136	−0.126184
$15$	0	0
$16$	1.85410	0.463525
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0.618034	0.145672
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	0.763932	0.166704
$22$	0	0
$23$	2.38197	0.496674	0.248337	−	0.968674i	$-0.420116\pi$
$23$	2.38197	0.496674	0.248337	+	0.968674i	$0.420116\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	−0.763932	−0.149819
$27$	−1.00000	−0.192450
$28$	1.23607	0.233595
$29$	−5.85410	−1.08708	−0.543540	−	0.839383i	$-0.682916\pi$
$29$	−5.85410	−1.08708	−0.543540	+	0.839383i	$0.682916\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	1.23607	0.211984
$35$	0	0
$36$	−1.61803	−0.269672
$37$	−7.47214	−1.22841	−0.614206	−	0.789146i	$-0.710524\pi$
$37$	−7.47214	−1.22841	−0.614206	+	0.789146i	$0.710524\pi$
$38$	3.09017	0.501292
$39$	1.23607	0.197929
$40$	0	0
$41$	2.52786	0.394786	0.197393	−	0.980324i	$-0.436752\pi$
$41$	2.52786	0.394786	0.197393	+	0.980324i	$0.436752\pi$
$42$	0.472136	0.0728522
$43$	−2.09017	−0.318748	−0.159374	−	0.987218i	$-0.550948\pi$
$43$	−2.09017	−0.318748	−0.159374	+	0.987218i	$0.550948\pi$
$44$	0	0
$45$	0	0
$46$	1.47214	0.217055
$47$	5.09017	0.742478	0.371239	−	0.928537i	$-0.378933\pi$
$47$	5.09017	0.742478	0.371239	+	0.928537i	$0.378933\pi$
$48$	−1.85410	−0.267617
$49$	−6.41641	−0.916630
$50$	0	0
$51$	−2.00000	−0.280056
$52$	2.00000	0.277350
$53$	−7.61803	−1.04642	−0.523209	−	0.852205i	$-0.675265\pi$
$53$	−7.61803	−1.04642	−0.523209	+	0.852205i	$0.675265\pi$
$54$	−0.618034	−0.0841038
$55$	0	0
$56$	1.70820	0.228268
$57$	−5.00000	−0.662266
$58$	−3.61803	−0.475071
$59$	−6.70820	−0.873334	−0.436667	−	0.899623i	$-0.643841\pi$
$59$	−6.70820	−0.873334	−0.436667	+	0.899623i	$0.643841\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	4.32624	0.549433
$63$	−0.763932	−0.0962464
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0	0
$67$	−9.38197	−1.14619	−0.573095	−	0.819489i	$-0.694257\pi$
$67$	−9.38197	−1.14619	−0.573095	+	0.819489i	$0.694257\pi$
$68$	−3.23607	−0.392431
$69$	−2.38197	−0.286755
$70$	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
$73$	−13.4721	−1.57679	−0.788397	−	0.615167i	$-0.789089\pi$
$73$	−13.4721	−1.57679	−0.788397	+	0.615167i	$0.789089\pi$
$74$	−4.61803	−0.536836
$75$	0	0
$76$	−8.09017	−0.928006
$77$	0	0
$78$	0.763932	0.0864983
$79$	−8.09017	−0.910215	−0.455108	−	0.890436i	$-0.650399\pi$
$79$	−8.09017	−0.910215	−0.455108	+	0.890436i	$0.650399\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.56231	0.172528
$83$	−5.70820	−0.626557	−0.313278	−	0.949661i	$-0.601427\pi$
$83$	−5.70820	−0.626557	−0.313278	+	0.949661i	$0.601427\pi$
$84$	−1.23607	−0.134866
$85$	0	0
$86$	−1.29180	−0.139298
$87$	5.85410	0.627626
$88$	0	0
$89$	10.8541	1.15053	0.575266	−	0.817966i	$-0.304898\pi$
$89$	10.8541	1.15053	0.575266	+	0.817966i	$0.304898\pi$
$90$	0	0
$91$	0.944272	0.0989866
$92$	−3.85410	−0.401818
$93$	−7.00000	−0.725866
$94$	3.14590	0.324475
$95$	0	0
$96$	−5.61803	−0.573388
$97$	11.4721	1.16482	0.582409	−	0.812896i	$-0.302110\pi$
$97$	11.4721	1.16482	0.582409	+	0.812896i	$0.302110\pi$
$98$	−3.96556	−0.400582
$99$	0	0
$100$	0	0
$101$	−0.562306	−0.0559515	−0.0279758	−	0.999609i	$-0.508906\pi$
$101$	−0.562306	−0.0559515	−0.0279758	+	0.999609i	$0.508906\pi$
$102$	−1.23607	−0.122389
$103$	2.38197	0.234702	0.117351	−	0.993090i	$-0.462560\pi$
$103$	2.38197	0.234702	0.117351	+	0.993090i	$0.462560\pi$
$104$	2.76393	0.271026
$105$	0	0
$106$	−4.70820	−0.457301
$107$	2.85410	0.275916	0.137958	−	0.990438i	$-0.455946\pi$
$107$	2.85410	0.275916	0.137958	+	0.990438i	$0.455946\pi$
$108$	1.61803	0.155695
$109$	4.14590	0.397105	0.198553	−	0.980090i	$-0.436376\pi$
$109$	4.14590	0.397105	0.198553	+	0.980090i	$0.436376\pi$
$110$	0	0
$111$	7.47214	0.709224
$112$	−1.41641	−0.133838
$113$	5.47214	0.514775	0.257388	−	0.966308i	$-0.417138\pi$
$113$	5.47214	0.514775	0.257388	+	0.966308i	$0.417138\pi$
$114$	−3.09017	−0.289421
$115$	0	0
$116$	9.47214	0.879466
$117$	−1.23607	−0.114275
$118$	−4.14590	−0.381661
$119$	−1.52786	−0.140059
$120$	0	0
$121$	0	0
$122$	4.32624	0.391679
$123$	−2.52786	−0.227930
$124$	−11.3262	−1.01713
$125$	0	0
$126$	−0.472136	−0.0420612
$127$	15.9443	1.41483	0.707413	−	0.706801i	$-0.249862\pi$
$127$	15.9443	1.41483	0.707413	+	0.706801i	$0.249862\pi$
$128$	−11.3820	−1.00603
$129$	2.09017	0.184029
$130$	0	0
$131$	13.1803	1.15157	0.575786	−	0.817601i	$-0.304696\pi$
$131$	13.1803	1.15157	0.575786	+	0.817601i	$0.304696\pi$
$132$	0	0
$133$	−3.81966	−0.331207
$134$	−5.79837	−0.500903
$135$	0	0
$136$	−4.47214	−0.383482
$137$	18.7082	1.59835	0.799175	−	0.601099i	$-0.205270\pi$
$137$	18.7082	1.59835	0.799175	+	0.601099i	$0.205270\pi$
$138$	−1.47214	−0.125317
$139$	21.1803	1.79649	0.898246	−	0.439492i	$-0.144842\pi$
$139$	21.1803	1.79649	0.898246	+	0.439492i	$0.144842\pi$
$140$	0	0
$141$	−5.09017	−0.428670
$142$	−4.94427	−0.414914
$143$	0	0
$144$	1.85410	0.154508
$145$	0	0
$146$	−8.32624	−0.689084
$147$	6.41641	0.529216
$148$	12.0902	0.993806
$149$	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
$151$	−10.7639	−0.875956	−0.437978	−	0.898986i	$-0.644305\pi$
$151$	−10.7639	−0.875956	−0.437978	+	0.898986i	$0.644305\pi$
$152$	−11.1803	−0.906845
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−3.32624	−0.265463	−0.132731	−	0.991152i	$-0.542375\pi$
$157$	−3.32624	−0.265463	−0.132731	+	0.991152i	$0.542375\pi$
$158$	−5.00000	−0.397779
$159$	7.61803	0.604149
$160$	0	0
$161$	−1.81966	−0.143409
$162$	0.618034	0.0485573
$163$	13.5623	1.06228	0.531141	−	0.847284i	$-0.321763\pi$
$163$	13.5623	1.06228	0.531141	+	0.847284i	$0.321763\pi$
$164$	−4.09017	−0.319389
$165$	0	0
$166$	−3.52786	−0.273815
$167$	−20.0344	−1.55031	−0.775156	−	0.631770i	$-0.782329\pi$
$167$	−20.0344	−1.55031	−0.775156	+	0.631770i	$0.782329\pi$
$168$	−1.70820	−0.131791
$169$	−11.4721	−0.882472
$170$	0	0
$171$	5.00000	0.382360
$172$	3.38197	0.257872
$173$	3.56231	0.270837	0.135419	−	0.990788i	$-0.456762\pi$
$173$	3.56231	0.270837	0.135419	+	0.990788i	$0.456762\pi$
$174$	3.61803	0.274282
$175$	0	0
$176$	0	0
$177$	6.70820	0.504219
$178$	6.70820	0.502801
$179$	−19.4721	−1.45542	−0.727708	−	0.685887i	$-0.759414\pi$
$179$	−19.4721	−1.45542	−0.727708	+	0.685887i	$0.759414\pi$
$180$	0	0
$181$	−5.23607	−0.389194	−0.194597	−	0.980883i	$-0.562340\pi$
$181$	−5.23607	−0.389194	−0.194597	+	0.980883i	$0.562340\pi$
$182$	0.583592	0.0432587
$183$	−7.00000	−0.517455
$184$	−5.32624	−0.392655
$185$	0	0
$186$	−4.32624	−0.317215
$187$	0	0
$188$	−8.23607	−0.600677
$189$	0.763932	0.0555679
$190$	0	0
$191$	10.4164	0.753705	0.376852	−	0.926273i	$-0.377006\pi$
$191$	10.4164	0.753705	0.376852	+	0.926273i	$0.377006\pi$
$192$	0.236068	0.0170367
$193$	−21.0344	−1.51409	−0.757046	−	0.653361i	$-0.773358\pi$
$193$	−21.0344	−1.51409	−0.757046	+	0.653361i	$0.773358\pi$
$194$	7.09017	0.509045
$195$	0	0
$196$	10.3820	0.741569
$197$	14.2361	1.01428	0.507139	−	0.861864i	$-0.330703\pi$
$197$	14.2361	1.01428	0.507139	+	0.861864i	$0.330703\pi$
$198$	0	0
$199$	19.7984	1.40347	0.701735	−	0.712438i	$-0.252409\pi$
$199$	19.7984	1.40347	0.701735	+	0.712438i	$0.252409\pi$
$200$	0	0
$201$	9.38197	0.661753
$202$	−0.347524	−0.0244517
$203$	4.47214	0.313882
$204$	3.23607	0.226570
$205$	0	0
$206$	1.47214	0.102569
$207$	2.38197	0.165558
$208$	−2.29180	−0.158907
$209$	0	0
$210$	0	0
$211$	−3.85410	−0.265327	−0.132664	−	0.991161i	$-0.542353\pi$
$211$	−3.85410	−0.265327	−0.132664	+	0.991161i	$0.542353\pi$
$212$	12.3262	0.846569
$213$	8.00000	0.548151
$214$	1.76393	0.120580
$215$	0	0
$216$	2.23607	0.152145
$217$	−5.34752	−0.363014
$218$	2.56231	0.173541
$219$	13.4721	0.910363
$220$	0	0
$221$	−2.47214	−0.166294
$222$	4.61803	0.309942
$223$	8.03444	0.538026	0.269013	−	0.963137i	$-0.413303\pi$
$223$	8.03444	0.538026	0.269013	+	0.963137i	$0.413303\pi$
$224$	−4.29180	−0.286758
$225$	0	0
$226$	3.38197	0.224965
$227$	25.9443	1.72198	0.860991	−	0.508620i	$-0.169844\pi$
$227$	25.9443	1.72198	0.860991	+	0.508620i	$0.169844\pi$
$228$	8.09017	0.535785
$229$	18.0902	1.19543	0.597716	−	0.801708i	$-0.296075\pi$
$229$	18.0902	1.19543	0.597716	+	0.801708i	$0.296075\pi$
$230$	0	0
$231$	0	0
$232$	13.0902	0.859412
$233$	−23.2705	−1.52450	−0.762251	−	0.647282i	$-0.775906\pi$
$233$	−23.2705	−1.52450	−0.762251	+	0.647282i	$0.775906\pi$
$234$	−0.763932	−0.0499398
$235$	0	0
$236$	10.8541	0.706542
$237$	8.09017	0.525513
$238$	−0.944272	−0.0612081
$239$	−7.03444	−0.455020	−0.227510	−	0.973776i	$-0.573058\pi$
$239$	−7.03444	−0.455020	−0.227510	+	0.973776i	$0.573058\pi$
$240$	0	0
$241$	−1.61803	−0.104227	−0.0521134	−	0.998641i	$-0.516596\pi$
$241$	−1.61803	−0.104227	−0.0521134	+	0.998641i	$0.516596\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−11.3262	−0.725088
$245$	0	0
$246$	−1.56231	−0.0996090
$247$	−6.18034	−0.393246
$248$	−15.6525	−0.993933
$249$	5.70820	0.361743
$250$	0	0
$251$	7.52786	0.475155	0.237577	−	0.971369i	$-0.423647\pi$
$251$	7.52786	0.475155	0.237577	+	0.971369i	$0.423647\pi$
$252$	1.23607	0.0778650
$253$	0	0
$254$	9.85410	0.618301
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−10.5623	−0.658859	−0.329429	−	0.944180i	$-0.606856\pi$
$257$	−10.5623	−0.658859	−0.329429	+	0.944180i	$0.606856\pi$
$258$	1.29180	0.0804237
$259$	5.70820	0.354691
$260$	0	0
$261$	−5.85410	−0.362360
$262$	8.14590	0.503255
$263$	−27.2148	−1.67814	−0.839068	−	0.544027i	$-0.816899\pi$
$263$	−27.2148	−1.67814	−0.839068	+	0.544027i	$0.816899\pi$
$264$	0	0
$265$	0	0
$266$	−2.36068	−0.144743
$267$	−10.8541	−0.664260
$268$	15.1803	0.927287
$269$	−5.85410	−0.356931	−0.178465	−	0.983946i	$-0.557113\pi$
$269$	−5.85410	−0.356931	−0.178465	+	0.983946i	$0.557113\pi$
$270$	0	0
$271$	20.6180	1.25246	0.626228	−	0.779640i	$-0.284598\pi$
$271$	20.6180	1.25246	0.626228	+	0.779640i	$0.284598\pi$
$272$	3.70820	0.224843
$273$	−0.944272	−0.0571499
$274$	11.5623	0.698504
$275$	0	0
$276$	3.85410	0.231990
$277$	−8.65248	−0.519877	−0.259938	−	0.965625i	$-0.583702\pi$
$277$	−8.65248	−0.519877	−0.259938	+	0.965625i	$0.583702\pi$
$278$	13.0902	0.785096
$279$	7.00000	0.419079
$280$	0	0
$281$	10.0902	0.601929	0.300965	−	0.953635i	$-0.402691\pi$
$281$	10.0902	0.601929	0.300965	+	0.953635i	$0.402691\pi$
$282$	−3.14590	−0.187336
$283$	13.0344	0.774817	0.387409	−	0.921908i	$-0.373370\pi$
$283$	13.0344	0.774817	0.387409	+	0.921908i	$0.373370\pi$
$284$	12.9443	0.768101
$285$	0	0
$286$	0	0
$287$	−1.93112	−0.113990
$288$	5.61803	0.331046
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−11.4721	−0.672509
$292$	21.7984	1.27565
$293$	16.5279	0.965568	0.482784	−	0.875739i	$-0.339626\pi$
$293$	16.5279	0.965568	0.482784	+	0.875739i	$0.339626\pi$
$294$	3.96556	0.231276
$295$	0	0
$296$	16.7082	0.971145
$297$	0	0
$298$	12.4377	0.720496
$299$	−2.94427	−0.170272
$300$	0	0
$301$	1.59675	0.0920350
$302$	−6.65248	−0.382807
$303$	0.562306	0.0323036
$304$	9.27051	0.531700
$305$	0	0
$306$	1.23607	0.0706613
$307$	12.1246	0.691988	0.345994	−	0.938237i	$-0.387542\pi$
$307$	12.1246	0.691988	0.345994	+	0.938237i	$0.387542\pi$
$308$	0	0
$309$	−2.38197	−0.135505
$310$	0	0
$311$	21.4721	1.21757	0.608787	−	0.793334i	$-0.291657\pi$
$311$	21.4721	1.21757	0.608787	+	0.793334i	$0.291657\pi$
$312$	−2.76393	−0.156477
$313$	0.472136	0.0266867	0.0133434	−	0.999911i	$-0.495753\pi$
$313$	0.472136	0.0266867	0.0133434	+	0.999911i	$0.495753\pi$
$314$	−2.05573	−0.116011
$315$	0	0
$316$	13.0902	0.736380
$317$	30.4164	1.70836	0.854178	−	0.519981i	$-0.174061\pi$
$317$	30.4164	1.70836	0.854178	+	0.519981i	$0.174061\pi$
$318$	4.70820	0.264023
$319$	0	0
$320$	0	0
$321$	−2.85410	−0.159300
$322$	−1.12461	−0.0626722
$323$	10.0000	0.556415
$324$	−1.61803	−0.0898908
$325$	0	0
$326$	8.38197	0.464234
$327$	−4.14590	−0.229269
$328$	−5.65248	−0.312106
$329$	−3.88854	−0.214382
$330$	0	0
$331$	21.5967	1.18706	0.593532	−	0.804810i	$-0.297733\pi$
$331$	21.5967	1.18706	0.593532	+	0.804810i	$0.297733\pi$
$332$	9.23607	0.506895
$333$	−7.47214	−0.409471
$334$	−12.3820	−0.677511
$335$	0	0
$336$	1.41641	0.0772714
$337$	−22.1459	−1.20636	−0.603182	−	0.797604i	$-0.706101\pi$
$337$	−22.1459	−1.20636	−0.603182	+	0.797604i	$0.706101\pi$
$338$	−7.09017	−0.385654
$339$	−5.47214	−0.297206
$340$	0	0
$341$	0	0
$342$	3.09017	0.167097
$343$	10.2492	0.553406
$344$	4.67376	0.251992
$345$	0	0
$346$	2.20163	0.118360
$347$	5.81966	0.312416	0.156208	−	0.987724i	$-0.450073\pi$
$347$	5.81966	0.312416	0.156208	+	0.987724i	$0.450073\pi$
$348$	−9.47214	−0.507760
$349$	27.7639	1.48617	0.743085	−	0.669197i	$-0.233362\pi$
$349$	27.7639	1.48617	0.743085	+	0.669197i	$0.233362\pi$
$350$	0	0
$351$	1.23607	0.0659764
$352$	0	0
$353$	−35.8328	−1.90719	−0.953594	−	0.301095i	$-0.902648\pi$
$353$	−35.8328	−1.90719	−0.953594	+	0.301095i	$0.902648\pi$
$354$	4.14590	0.220352
$355$	0	0
$356$	−17.5623	−0.930800
$357$	1.52786	0.0808631
$358$	−12.0344	−0.636040
$359$	16.7082	0.881825	0.440913	−	0.897550i	$-0.354655\pi$
$359$	16.7082	0.881825	0.440913	+	0.897550i	$0.354655\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−3.23607	−0.170084
$363$	0	0
$364$	−1.52786	−0.0800818
$365$	0	0
$366$	−4.32624	−0.226136
$367$	−18.1246	−0.946097	−0.473049	−	0.881036i	$-0.656846\pi$
$367$	−18.1246	−0.946097	−0.473049	+	0.881036i	$0.656846\pi$
$368$	4.41641	0.230221
$369$	2.52786	0.131595
$370$	0	0
$371$	5.81966	0.302142
$372$	11.3262	0.587238
$373$	9.74265	0.504455	0.252228	−	0.967668i	$-0.418837\pi$
$373$	9.74265	0.504455	0.252228	+	0.967668i	$0.418837\pi$
$374$	0	0
$375$	0	0
$376$	−11.3820	−0.586980
$377$	7.23607	0.372676
$378$	0.472136	0.0242841
$379$	−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
$381$	−15.9443	−0.816850
$382$	6.43769	0.329381
$383$	−31.3607	−1.60246	−0.801228	−	0.598359i	$-0.795820\pi$
$383$	−31.3607	−1.60246	−0.801228	+	0.598359i	$0.795820\pi$
$384$	11.3820	0.580834
$385$	0	0
$386$	−13.0000	−0.661683
$387$	−2.09017	−0.106249
$388$	−18.5623	−0.942358
$389$	−38.9443	−1.97455	−0.987276	−	0.159013i	$-0.949169\pi$
$389$	−38.9443	−1.97455	−0.987276	+	0.159013i	$0.949169\pi$
$390$	0	0
$391$	4.76393	0.240922
$392$	14.3475	0.724659
$393$	−13.1803	−0.664860
$394$	8.79837	0.443256
$395$	0	0
$396$	0	0
$397$	36.2705	1.82036	0.910182	−	0.414208i	$-0.135941\pi$
$397$	36.2705	1.82036	0.910182	+	0.414208i	$0.135941\pi$
$398$	12.2361	0.613339
$399$	3.81966	0.191222
$400$	0	0
$401$	4.88854	0.244122	0.122061	−	0.992523i	$-0.461050\pi$
$401$	4.88854	0.244122	0.122061	+	0.992523i	$0.461050\pi$
$402$	5.79837	0.289197
$403$	−8.65248	−0.431011
$404$	0.909830	0.0452657
$405$	0	0
$406$	2.76393	0.137172
$407$	0	0
$408$	4.47214	0.221404
$409$	36.3050	1.79516	0.897582	−	0.440847i	$-0.145322\pi$
$409$	36.3050	1.79516	0.897582	+	0.440847i	$0.145322\pi$
$410$	0	0
$411$	−18.7082	−0.922808
$412$	−3.85410	−0.189878
$413$	5.12461	0.252166
$414$	1.47214	0.0723515
$415$	0	0
$416$	−6.94427	−0.340471
$417$	−21.1803	−1.03721
$418$	0	0
$419$	24.5967	1.20163	0.600815	−	0.799388i	$-0.294843\pi$
$419$	24.5967	1.20163	0.600815	+	0.799388i	$0.294843\pi$
$420$	0	0
$421$	2.85410	0.139100	0.0695502	−	0.997578i	$-0.477844\pi$
$421$	2.85410	0.139100	0.0695502	+	0.997578i	$0.477844\pi$
$422$	−2.38197	−0.115952
$423$	5.09017	0.247493
$424$	17.0344	0.827266
$425$	0	0
$426$	4.94427	0.239551
$427$	−5.34752	−0.258785
$428$	−4.61803	−0.223221
$429$	0	0
$430$	0	0
$431$	19.8885	0.957997	0.478999	−	0.877816i	$-0.341000\pi$
$431$	19.8885	0.957997	0.478999	+	0.877816i	$0.341000\pi$
$432$	−1.85410	−0.0892055
$433$	28.0344	1.34725	0.673625	−	0.739074i	$-0.264736\pi$
$433$	28.0344	1.34725	0.673625	+	0.739074i	$0.264736\pi$
$434$	−3.30495	−0.158643
$435$	0	0
$436$	−6.70820	−0.321265
$437$	11.9098	0.569724
$438$	8.32624	0.397843
$439$	9.67376	0.461703	0.230852	−	0.972989i	$-0.425849\pi$
$439$	9.67376	0.461703	0.230852	+	0.972989i	$0.425849\pi$
$440$	0	0
$441$	−6.41641	−0.305543
$442$	−1.52786	−0.0726731
$443$	25.2705	1.20064	0.600319	−	0.799761i	$-0.295040\pi$
$443$	25.2705	1.20064	0.600319	+	0.799761i	$0.295040\pi$
$444$	−12.0902	−0.573774
$445$	0	0
$446$	4.96556	0.235126
$447$	−20.1246	−0.951861
$448$	0.180340	0.00852026
$449$	−0.729490	−0.0344268	−0.0172134	−	0.999852i	$-0.505479\pi$
$449$	−0.729490	−0.0344268	−0.0172134	+	0.999852i	$0.505479\pi$
$450$	0	0
$451$	0	0
$452$	−8.85410	−0.416462
$453$	10.7639	0.505734
$454$	16.0344	0.752534
$455$	0	0
$456$	11.1803	0.523567
$457$	1.47214	0.0688636	0.0344318	−	0.999407i	$-0.489038\pi$
$457$	1.47214	0.0688636	0.0344318	+	0.999407i	$0.489038\pi$
$458$	11.1803	0.522423
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	25.9443	1.20835	0.604173	−	0.796853i	$-0.293504\pi$
$461$	25.9443	1.20835	0.604173	+	0.796853i	$0.293504\pi$
$462$	0	0
$463$	−33.2705	−1.54621	−0.773106	−	0.634277i	$-0.781298\pi$
$463$	−33.2705	−1.54621	−0.773106	+	0.634277i	$0.781298\pi$
$464$	−10.8541	−0.503889
$465$	0	0
$466$	−14.3820	−0.666232
$467$	4.88854	0.226215	0.113107	−	0.993583i	$-0.463920\pi$
$467$	4.88854	0.226215	0.113107	+	0.993583i	$0.463920\pi$
$468$	2.00000	0.0924500
$469$	7.16718	0.330950
$470$	0	0
$471$	3.32624	0.153265
$472$	15.0000	0.690431
$473$	0	0
$474$	5.00000	0.229658
$475$	0	0
$476$	2.47214	0.113310
$477$	−7.61803	−0.348806
$478$	−4.34752	−0.198851
$479$	33.4164	1.52683	0.763417	−	0.645906i	$-0.223520\pi$
$479$	33.4164	1.52683	0.763417	+	0.645906i	$0.223520\pi$
$480$	0	0
$481$	9.23607	0.421128
$482$	−1.00000	−0.0455488
$483$	1.81966	0.0827974
$484$	0	0
$485$	0	0
$486$	−0.618034	−0.0280346
$487$	−14.1803	−0.642573	−0.321286	−	0.946982i	$-0.604115\pi$
$487$	−14.1803	−0.642573	−0.321286	+	0.946982i	$0.604115\pi$
$488$	−15.6525	−0.708554
$489$	−13.5623	−0.613309
$490$	0	0
$491$	31.7984	1.43504	0.717520	−	0.696538i	$-0.245277\pi$
$491$	31.7984	1.43504	0.717520	+	0.696538i	$0.245277\pi$
$492$	4.09017	0.184399
$493$	−11.7082	−0.527311
$494$	−3.81966	−0.171855
$495$	0	0
$496$	12.9787	0.582761
$497$	6.11146	0.274136
$498$	3.52786	0.158087
$499$	−11.1803	−0.500501	−0.250250	−	0.968181i	$-0.580513\pi$
$499$	−11.1803	−0.500501	−0.250250	+	0.968181i	$0.580513\pi$
$500$	0	0
$501$	20.0344	0.895073
$502$	4.65248	0.207650
$503$	−21.8885	−0.975962	−0.487981	−	0.872854i	$-0.662266\pi$
$503$	−21.8885	−0.975962	−0.487981	+	0.872854i	$0.662266\pi$
$504$	1.70820	0.0760895
$505$	0	0
$506$	0	0
$507$	11.4721	0.509495
$508$	−25.7984	−1.14462
$509$	0.527864	0.0233972	0.0116986	−	0.999932i	$-0.496276\pi$
$509$	0.527864	0.0233972	0.0116986	+	0.999932i	$0.496276\pi$
$510$	0	0
$511$	10.2918	0.455282
$512$	18.7082	0.826794
$513$	−5.00000	−0.220755
$514$	−6.52786	−0.287932
$515$	0	0
$516$	−3.38197	−0.148883
$517$	0	0
$518$	3.52786	0.155005
$519$	−3.56231	−0.156368
$520$	0	0
$521$	−15.3607	−0.672964	−0.336482	−	0.941690i	$-0.609237\pi$
$521$	−15.3607	−0.672964	−0.336482	+	0.941690i	$0.609237\pi$
$522$	−3.61803	−0.158357
$523$	8.23607	0.360138	0.180069	−	0.983654i	$-0.442368\pi$
$523$	8.23607	0.360138	0.180069	+	0.983654i	$0.442368\pi$
$524$	−21.3262	−0.931641
$525$	0	0
$526$	−16.8197	−0.733372
$527$	14.0000	0.609850
$528$	0	0
$529$	−17.3262	−0.753315
$530$	0	0
$531$	−6.70820	−0.291111
$532$	6.18034	0.267952
$533$	−3.12461	−0.135342
$534$	−6.70820	−0.290292
$535$	0	0
$536$	20.9787	0.906142
$537$	19.4721	0.840285
$538$	−3.61803	−0.155985
$539$	0	0
$540$	0	0
$541$	0.618034	0.0265714	0.0132857	−	0.999912i	$-0.495771\pi$
$541$	0.618034	0.0265714	0.0132857	+	0.999912i	$0.495771\pi$
$542$	12.7426	0.547344
$543$	5.23607	0.224701
$544$	11.2361	0.481742
$545$	0	0
$546$	−0.583592	−0.0249754
$547$	−25.3607	−1.08434	−0.542172	−	0.840267i	$-0.682398\pi$
$547$	−25.3607	−1.08434	−0.542172	+	0.840267i	$0.682398\pi$
$548$	−30.2705	−1.29309
$549$	7.00000	0.298753
$550$	0	0
$551$	−29.2705	−1.24697
$552$	5.32624	0.226700
$553$	6.18034	0.262815
$554$	−5.34752	−0.227195
$555$	0	0
$556$	−34.2705	−1.45339
$557$	31.3951	1.33025	0.665127	−	0.746730i	$-0.268377\pi$
$557$	31.3951	1.33025	0.665127	+	0.746730i	$0.268377\pi$
$558$	4.32624	0.183144
$559$	2.58359	0.109274
$560$	0	0
$561$	0	0
$562$	6.23607	0.263053
$563$	36.3262	1.53097	0.765484	−	0.643455i	$-0.222500\pi$
$563$	36.3262	1.53097	0.765484	+	0.643455i	$0.222500\pi$
$564$	8.23607	0.346801
$565$	0	0
$566$	8.05573	0.338608
$567$	−0.763932	−0.0320821
$568$	17.8885	0.750587
$569$	−2.43769	−0.102193	−0.0510967	−	0.998694i	$-0.516272\pi$
$569$	−2.43769	−0.102193	−0.0510967	+	0.998694i	$0.516272\pi$
$570$	0	0
$571$	0.0901699	0.00377349	0.00188675	−	0.999998i	$-0.499399\pi$
$571$	0.0901699	0.00377349	0.00188675	+	0.999998i	$0.499399\pi$
$572$	0	0
$573$	−10.4164	−0.435152
$574$	−1.19350	−0.0498155
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	−46.0132	−1.91555	−0.957776	−	0.287514i	$-0.907171\pi$
$577$	−46.0132	−1.91555	−0.957776	+	0.287514i	$0.907171\pi$
$578$	−8.03444	−0.334189
$579$	21.0344	0.874162
$580$	0	0
$581$	4.36068	0.180911
$582$	−7.09017	−0.293897
$583$	0	0
$584$	30.1246	1.24657
$585$	0	0
$586$	10.2148	0.421969
$587$	−19.3820	−0.799979	−0.399990	−	0.916520i	$-0.630986\pi$
$587$	−19.3820	−0.799979	−0.399990	+	0.916520i	$0.630986\pi$
$588$	−10.3820	−0.428145
$589$	35.0000	1.44215
$590$	0	0
$591$	−14.2361	−0.585594
$592$	−13.8541	−0.569400
$593$	38.8885	1.59696	0.798481	−	0.602021i	$-0.205638\pi$
$593$	38.8885	1.59696	0.798481	+	0.602021i	$0.205638\pi$
$594$	0	0
$595$	0	0
$596$	−32.5623	−1.33380
$597$	−19.7984	−0.810294
$598$	−1.81966	−0.0744114
$599$	−22.2361	−0.908541	−0.454271	−	0.890864i	$-0.650100\pi$
$599$	−22.2361	−0.908541	−0.454271	+	0.890864i	$0.650100\pi$
$600$	0	0
$601$	34.3607	1.40160	0.700801	−	0.713357i	$-0.252826\pi$
$601$	34.3607	1.40160	0.700801	+	0.713357i	$0.252826\pi$
$602$	0.986844	0.0402208
$603$	−9.38197	−0.382063
$604$	17.4164	0.708664
$605$	0	0
$606$	0.347524	0.0141172
$607$	−33.0000	−1.33943	−0.669714	−	0.742619i	$-0.733583\pi$
$607$	−33.0000	−1.33943	−0.669714	+	0.742619i	$0.733583\pi$
$608$	28.0902	1.13921
$609$	−4.47214	−0.181220
$610$	0	0
$611$	−6.29180	−0.254539
$612$	−3.23607	−0.130810
$613$	8.23607	0.332652	0.166326	−	0.986071i	$-0.446810\pi$
$613$	8.23607	0.332652	0.166326	+	0.986071i	$0.446810\pi$
$614$	7.49342	0.302410
$615$	0	0
$616$	0	0
$617$	−38.2492	−1.53986	−0.769928	−	0.638131i	$-0.779708\pi$
$617$	−38.2492	−1.53986	−0.769928	+	0.638131i	$0.779708\pi$
$618$	−1.47214	−0.0592180
$619$	7.11146	0.285834	0.142917	−	0.989735i	$-0.454352\pi$
$619$	7.11146	0.285834	0.142917	+	0.989735i	$0.454352\pi$
$620$	0	0
$621$	−2.38197	−0.0955850
$622$	13.2705	0.532099
$623$	−8.29180	−0.332204
$624$	2.29180	0.0917453
$625$	0	0
$626$	0.291796	0.0116625
$627$	0	0
$628$	5.38197	0.214764
$629$	−14.9443	−0.595867
$630$	0	0
$631$	38.6312	1.53788	0.768942	−	0.639319i	$-0.220784\pi$
$631$	38.6312	1.53788	0.768942	+	0.639319i	$0.220784\pi$
$632$	18.0902	0.719588
$633$	3.85410	0.153187
$634$	18.7984	0.746579
$635$	0	0
$636$	−12.3262	−0.488767
$637$	7.93112	0.314242
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−0.236068	−0.00932412	−0.00466206	−	0.999989i	$-0.501484\pi$
$641$	−0.236068	−0.00932412	−0.00466206	+	0.999989i	$0.501484\pi$
$642$	−1.76393	−0.0696168
$643$	−29.3262	−1.15651	−0.578257	−	0.815855i	$-0.696267\pi$
$643$	−29.3262	−1.15651	−0.578257	+	0.815855i	$0.696267\pi$
$644$	2.94427	0.116021
$645$	0	0
$646$	6.18034	0.243162
$647$	−17.5967	−0.691800	−0.345900	−	0.938271i	$-0.612426\pi$
$647$	−17.5967	−0.691800	−0.345900	+	0.938271i	$0.612426\pi$
$648$	−2.23607	−0.0878410
$649$	0	0
$650$	0	0
$651$	5.34752	0.209586
$652$	−21.9443	−0.859404
$653$	23.4377	0.917188	0.458594	−	0.888646i	$-0.348353\pi$
$653$	23.4377	0.917188	0.458594	+	0.888646i	$0.348353\pi$
$654$	−2.56231	−0.100194
$655$	0	0
$656$	4.68692	0.182993
$657$	−13.4721	−0.525598
$658$	−2.40325	−0.0936885
$659$	12.9656	0.505066	0.252533	−	0.967588i	$-0.418736\pi$
$659$	12.9656	0.505066	0.252533	+	0.967588i	$0.418736\pi$
$660$	0	0
$661$	3.90983	0.152075	0.0760374	−	0.997105i	$-0.475773\pi$
$661$	3.90983	0.152075	0.0760374	+	0.997105i	$0.475773\pi$
$662$	13.3475	0.518766
$663$	2.47214	0.0960098
$664$	12.7639	0.495337
$665$	0	0
$666$	−4.61803	−0.178945
$667$	−13.9443	−0.539924
$668$	32.4164	1.25423
$669$	−8.03444	−0.310629
$670$	0	0
$671$	0	0
$672$	4.29180	0.165560
$673$	27.3050	1.05253	0.526264	−	0.850321i	$-0.323592\pi$
$673$	27.3050	1.05253	0.526264	+	0.850321i	$0.323592\pi$
$674$	−13.6869	−0.527200
$675$	0	0
$676$	18.5623	0.713935
$677$	−11.9443	−0.459056	−0.229528	−	0.973302i	$-0.573718\pi$
$677$	−11.9443	−0.459056	−0.229528	+	0.973302i	$0.573718\pi$
$678$	−3.38197	−0.129884
$679$	−8.76393	−0.336329
$680$	0	0
$681$	−25.9443	−0.994187
$682$	0	0
$683$	−2.29180	−0.0876931	−0.0438466	−	0.999038i	$-0.513961\pi$
$683$	−2.29180	−0.0876931	−0.0438466	+	0.999038i	$0.513961\pi$
$684$	−8.09017	−0.309335
$685$	0	0
$686$	6.33437	0.241847
$687$	−18.0902	−0.690183
$688$	−3.87539	−0.147748
$689$	9.41641	0.358737
$690$	0	0
$691$	33.3050	1.26698	0.633490	−	0.773751i	$-0.281622\pi$
$691$	33.3050	1.26698	0.633490	+	0.773751i	$0.281622\pi$
$692$	−5.76393	−0.219112
$693$	0	0
$694$	3.59675	0.136531
$695$	0	0
$696$	−13.0902	−0.496182
$697$	5.05573	0.191499
$698$	17.1591	0.649480
$699$	23.2705	0.880172
$700$	0	0
$701$	13.5836	0.513045	0.256523	−	0.966538i	$-0.417423\pi$
$701$	13.5836	0.513045	0.256523	+	0.966538i	$0.417423\pi$
$702$	0.763932	0.0288328
$703$	−37.3607	−1.40908
$704$	0	0
$705$	0	0
$706$	−22.1459	−0.833472
$707$	0.429563	0.0161554
$708$	−10.8541	−0.407922
$709$	41.1803	1.54656	0.773280	−	0.634065i	$-0.218615\pi$
$709$	41.1803	1.54656	0.773280	+	0.634065i	$0.218615\pi$
$710$	0	0
$711$	−8.09017	−0.303405
$712$	−24.2705	−0.909576
$713$	16.6738	0.624437
$714$	0.944272	0.0353385
$715$	0	0
$716$	31.5066	1.17746
$717$	7.03444	0.262706
$718$	10.3262	0.385372
$719$	38.0902	1.42052	0.710262	−	0.703938i	$-0.248577\pi$
$719$	38.0902	1.42052	0.710262	+	0.703938i	$0.248577\pi$
$720$	0	0
$721$	−1.81966	−0.0677677
$722$	3.70820	0.138005
$723$	1.61803	0.0601753
$724$	8.47214	0.314864
$725$	0	0
$726$	0	0
$727$	37.3262	1.38435	0.692177	−	0.721728i	$-0.256652\pi$
$727$	37.3262	1.38435	0.692177	+	0.721728i	$0.256652\pi$
$728$	−2.11146	−0.0782558
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.18034	−0.154615
$732$	11.3262	0.418630
$733$	1.85410	0.0684828	0.0342414	−	0.999414i	$-0.489098\pi$
$733$	1.85410	0.0684828	0.0342414	+	0.999414i	$0.489098\pi$
$734$	−11.2016	−0.413460
$735$	0	0
$736$	13.3820	0.493266
$737$	0	0
$738$	1.56231	0.0575093
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	6.18034	0.227040
$742$	3.59675	0.132041
$743$	0.347524	0.0127494	0.00637471	−	0.999980i	$-0.497971\pi$
$743$	0.347524	0.0127494	0.00637471	+	0.999980i	$0.497971\pi$
$744$	15.6525	0.573848
$745$	0	0
$746$	6.02129	0.220455
$747$	−5.70820	−0.208852
$748$	0	0
$749$	−2.18034	−0.0796679
$750$	0	0
$751$	−45.8885	−1.67450	−0.837248	−	0.546823i	$-0.815837\pi$
$751$	−45.8885	−1.67450	−0.837248	+	0.546823i	$0.815837\pi$
$752$	9.43769	0.344157
$753$	−7.52786	−0.274331
$754$	4.47214	0.162866
$755$	0	0
$756$	−1.23607	−0.0449554
$757$	−24.1803	−0.878849	−0.439425	−	0.898279i	$-0.644818\pi$
$757$	−24.1803	−0.878849	−0.439425	+	0.898279i	$0.644818\pi$
$758$	−3.09017	−0.112240
$759$	0	0
$760$	0	0
$761$	39.0344	1.41500	0.707499	−	0.706715i	$-0.249824\pi$
$761$	39.0344	1.41500	0.707499	+	0.706715i	$0.249824\pi$
$762$	−9.85410	−0.356976
$763$	−3.16718	−0.114660
$764$	−16.8541	−0.609760
$765$	0	0
$766$	−19.3820	−0.700299
$767$	8.29180	0.299399
$768$	6.56231	0.236797
$769$	12.7639	0.460279	0.230140	−	0.973158i	$-0.426082\pi$
$769$	12.7639	0.460279	0.230140	+	0.973158i	$0.426082\pi$
$770$	0	0
$771$	10.5623	0.380392
$772$	34.0344	1.22493
$773$	−10.5066	−0.377895	−0.188948	−	0.981987i	$-0.560508\pi$
$773$	−10.5066	−0.377895	−0.188948	+	0.981987i	$0.560508\pi$
$774$	−1.29180	−0.0464327
$775$	0	0
$776$	−25.6525	−0.920870
$777$	−5.70820	−0.204781
$778$	−24.0689	−0.862911
$779$	12.6393	0.452851
$780$	0	0
$781$	0	0
$782$	2.94427	0.105287
$783$	5.85410	0.209209
$784$	−11.8967	−0.424881
$785$	0	0
$786$	−8.14590	−0.290555
$787$	−8.20163	−0.292356	−0.146178	−	0.989258i	$-0.546697\pi$
$787$	−8.20163	−0.292356	−0.146178	+	0.989258i	$0.546697\pi$
$788$	−23.0344	−0.820568
$789$	27.2148	0.968872
$790$	0	0
$791$	−4.18034	−0.148636
$792$	0	0
$793$	−8.65248	−0.307258
$794$	22.4164	0.795529
$795$	0	0
$796$	−32.0344	−1.13543
$797$	28.7082	1.01690	0.508448	−	0.861092i	$-0.330219\pi$
$797$	28.7082	1.01690	0.508448	+	0.861092i	$0.330219\pi$
$798$	2.36068	0.0835672
$799$	10.1803	0.360155
$800$	0	0
$801$	10.8541	0.383511
$802$	3.02129	0.106685
$803$	0	0
$804$	−15.1803	−0.535369
$805$	0	0
$806$	−5.34752	−0.188359
$807$	5.85410	0.206074
$808$	1.25735	0.0442336
$809$	45.3262	1.59359	0.796793	−	0.604253i	$-0.206528\pi$
$809$	45.3262	1.59359	0.796793	+	0.604253i	$0.206528\pi$
$810$	0	0
$811$	−34.6312	−1.21607	−0.608033	−	0.793912i	$-0.708041\pi$
$811$	−34.6312	−1.21607	−0.608033	+	0.793912i	$0.708041\pi$
$812$	−7.23607	−0.253936
$813$	−20.6180	−0.723106
$814$	0	0
$815$	0	0
$816$	−3.70820	−0.129813
$817$	−10.4508	−0.365629
$818$	22.4377	0.784516
$819$	0.944272	0.0329955
$820$	0	0
$821$	43.1803	1.50700	0.753502	−	0.657445i	$-0.228363\pi$
$821$	43.1803	1.50700	0.753502	+	0.657445i	$0.228363\pi$
$822$	−11.5623	−0.403282
$823$	25.4721	0.887903	0.443951	−	0.896051i	$-0.353576\pi$
$823$	25.4721	0.887903	0.443951	+	0.896051i	$0.353576\pi$
$824$	−5.32624	−0.185548
$825$	0	0
$826$	3.16718	0.110200
$827$	−24.5836	−0.854855	−0.427428	−	0.904050i	$-0.640580\pi$
$827$	−24.5836	−0.854855	−0.427428	+	0.904050i	$0.640580\pi$
$828$	−3.85410	−0.133939
$829$	2.88854	0.100323	0.0501616	−	0.998741i	$-0.484026\pi$
$829$	2.88854	0.100323	0.0501616	+	0.998741i	$0.484026\pi$
$830$	0	0
$831$	8.65248	0.300151
$832$	0.291796	0.0101162
$833$	−12.8328	−0.444631
$834$	−13.0902	−0.453276
$835$	0	0
$836$	0	0
$837$	−7.00000	−0.241955
$838$	15.2016	0.525131
$839$	−26.3050	−0.908148	−0.454074	−	0.890964i	$-0.650030\pi$
$839$	−26.3050	−0.908148	−0.454074	+	0.890964i	$0.650030\pi$
$840$	0	0
$841$	5.27051	0.181742
$842$	1.76393	0.0607891
$843$	−10.0902	−0.347524
$844$	6.23607	0.214654
$845$	0	0
$846$	3.14590	0.108158
$847$	0	0
$848$	−14.1246	−0.485041
$849$	−13.0344	−0.447341
$850$	0	0
$851$	−17.7984	−0.610120
$852$	−12.9443	−0.443463
$853$	−21.3607	−0.731376	−0.365688	−	0.930738i	$-0.619166\pi$
$853$	−21.3607	−0.731376	−0.365688	+	0.930738i	$0.619166\pi$
$854$	−3.30495	−0.113093
$855$	0	0
$856$	−6.38197	−0.218131
$857$	35.0132	1.19603	0.598013	−	0.801486i	$-0.295957\pi$
$857$	35.0132	1.19603	0.598013	+	0.801486i	$0.295957\pi$
$858$	0	0
$859$	−5.72949	−0.195488	−0.0977438	−	0.995212i	$-0.531163\pi$
$859$	−5.72949	−0.195488	−0.0977438	+	0.995212i	$0.531163\pi$
$860$	0	0
$861$	1.93112	0.0658123
$862$	12.2918	0.418660
$863$	−6.03444	−0.205415	−0.102707	−	0.994712i	$-0.532751\pi$
$863$	−6.03444	−0.205415	−0.102707	+	0.994712i	$0.532751\pi$
$864$	−5.61803	−0.191129
$865$	0	0
$866$	17.3262	0.588770
$867$	13.0000	0.441503
$868$	8.65248	0.293684
$869$	0	0
$870$	0	0
$871$	11.5967	0.392941
$872$	−9.27051	−0.313939
$873$	11.4721	0.388273
$874$	7.36068	0.248979
$875$	0	0
$876$	−21.7984	−0.736499
$877$	16.4721	0.556225	0.278112	−	0.960549i	$-0.410291\pi$
$877$	16.4721	0.556225	0.278112	+	0.960549i	$0.410291\pi$
$878$	5.97871	0.201772
$879$	−16.5279	−0.557471
$880$	0	0
$881$	−23.8541	−0.803665	−0.401833	−	0.915713i	$-0.631627\pi$
$881$	−23.8541	−0.803665	−0.401833	+	0.915713i	$0.631627\pi$
$882$	−3.96556	−0.133527
$883$	18.7639	0.631457	0.315728	−	0.948850i	$-0.397751\pi$
$883$	18.7639	0.631457	0.315728	+	0.948850i	$0.397751\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	15.6180	0.524698
$887$	53.8328	1.80753	0.903765	−	0.428030i	$-0.140792\pi$
$887$	53.8328	1.80753	0.903765	+	0.428030i	$0.140792\pi$
$888$	−16.7082	−0.560691
$889$	−12.1803	−0.408515
$890$	0	0
$891$	0	0
$892$	−13.0000	−0.435272
$893$	25.4508	0.851680
$894$	−12.4377	−0.415979
$895$	0	0
$896$	8.69505	0.290481
$897$	2.94427	0.0983064
$898$	−0.450850	−0.0150450
$899$	−40.9787	−1.36672
$900$	0	0
$901$	−15.2361	−0.507587
$902$	0	0
$903$	−1.59675	−0.0531364
$904$	−12.2361	−0.406966
$905$	0	0
$906$	6.65248	0.221014
$907$	9.11146	0.302541	0.151270	−	0.988492i	$-0.451664\pi$
$907$	9.11146	0.302541	0.151270	+	0.988492i	$0.451664\pi$
$908$	−41.9787	−1.39311
$909$	−0.562306	−0.0186505
$910$	0	0
$911$	−58.9017	−1.95150	−0.975750	−	0.218887i	$-0.929757\pi$
$911$	−58.9017	−1.95150	−0.975750	+	0.218887i	$0.929757\pi$
$912$	−9.27051	−0.306977
$913$	0	0
$914$	0.909830	0.0300945
$915$	0	0
$916$	−29.2705	−0.967125
$917$	−10.0689	−0.332504
$918$	−1.23607	−0.0407963
$919$	−17.7639	−0.585978	−0.292989	−	0.956116i	$-0.594650\pi$
$919$	−17.7639	−0.585978	−0.292989	+	0.956116i	$0.594650\pi$
$920$	0	0
$921$	−12.1246	−0.399520
$922$	16.0344	0.528066
$923$	9.88854	0.325485
$924$	0	0
$925$	0	0
$926$	−20.5623	−0.675719
$927$	2.38197	0.0782340
$928$	−32.8885	−1.07962
$929$	−4.14590	−0.136023	−0.0680113	−	0.997685i	$-0.521665\pi$
$929$	−4.14590	−0.136023	−0.0680113	+	0.997685i	$0.521665\pi$
$930$	0	0
$931$	−32.0820	−1.05145
$932$	37.6525	1.23335
$933$	−21.4721	−0.702966
$934$	3.02129	0.0988595
$935$	0	0
$936$	2.76393	0.0903419
$937$	41.7984	1.36549	0.682747	−	0.730655i	$-0.260785\pi$
$937$	41.7984	1.36549	0.682747	+	0.730655i	$0.260785\pi$
$938$	4.42956	0.144630
$939$	−0.472136	−0.0154076
$940$	0	0
$941$	3.70820	0.120884	0.0604420	−	0.998172i	$-0.480749\pi$
$941$	3.70820	0.120884	0.0604420	+	0.998172i	$0.480749\pi$
$942$	2.05573	0.0669792
$943$	6.02129	0.196080
$944$	−12.4377	−0.404812
$945$	0	0
$946$	0	0
$947$	−15.5623	−0.505707	−0.252853	−	0.967505i	$-0.581369\pi$
$947$	−15.5623	−0.505707	−0.252853	+	0.967505i	$0.581369\pi$
$948$	−13.0902	−0.425149
$949$	16.6525	0.540562
$950$	0	0
$951$	−30.4164	−0.986320
$952$	3.41641	0.110726
$953$	37.9098	1.22802	0.614010	−	0.789298i	$-0.289555\pi$
$953$	37.9098	1.22802	0.614010	+	0.789298i	$0.289555\pi$
$954$	−4.70820	−0.152434
$955$	0	0
$956$	11.3820	0.368119
$957$	0	0
$958$	20.6525	0.667251
$959$	−14.2918	−0.461506
$960$	0	0
$961$	18.0000	0.580645
$962$	5.70820	0.184040
$963$	2.85410	0.0919722
$964$	2.61803	0.0843212
$965$	0	0
$966$	1.12461	0.0361838
$967$	−48.3262	−1.55407	−0.777034	−	0.629459i	$-0.783276\pi$
$967$	−48.3262	−1.55407	−0.777034	+	0.629459i	$0.783276\pi$
$968$	0	0
$969$	−10.0000	−0.321246
$970$	0	0
$971$	35.5410	1.14057	0.570283	−	0.821448i	$-0.306834\pi$
$971$	35.5410	1.14057	0.570283	+	0.821448i	$0.306834\pi$
$972$	1.61803	0.0518985
$973$	−16.1803	−0.518718
$974$	−8.76393	−0.280814
$975$	0	0
$976$	12.9787	0.415439
$977$	−34.9098	−1.11686	−0.558432	−	0.829550i	$-0.688597\pi$
$977$	−34.9098	−1.11686	−0.558432	+	0.829550i	$0.688597\pi$
$978$	−8.38197	−0.268026
$979$	0	0
$980$	0	0
$981$	4.14590	0.132368
$982$	19.6525	0.627136
$983$	−19.0000	−0.606006	−0.303003	−	0.952990i	$-0.597989\pi$
$983$	−19.0000	−0.606006	−0.303003	+	0.952990i	$0.597989\pi$
$984$	5.65248	0.180194
$985$	0	0
$986$	−7.23607	−0.230443
$987$	3.88854	0.123774
$988$	10.0000	0.318142
$989$	−4.97871	−0.158314
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	39.3262	1.24861
$993$	−21.5967	−0.685352
$994$	3.77709	0.119802
$995$	0	0
$996$	−9.23607	−0.292656
$997$	−24.7082	−0.782517	−0.391258	−	0.920281i	$-0.627960\pi$
$997$	−24.7082	−0.782517	−0.391258	+	0.920281i	$0.627960\pi$
$998$	−6.90983	−0.218727
$999$	7.47214	0.236408

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.z.1.2		2
5.4	even	2		9075.2.a.by.1.1		2
11.5	even	5		825.2.n.b.751.1	yes	4
11.9	even	5		825.2.n.b.301.1	✓	4
11.10	odd	2		9075.2.a.bt.1.1		2
55.9	even	10		825.2.n.d.301.1	yes	4
55.27	odd	20		825.2.bx.c.124.1		8
55.38	odd	20		825.2.bx.c.124.2		8
55.42	odd	20		825.2.bx.c.499.2		8
55.49	even	10		825.2.n.d.751.1	yes	4
55.53	odd	20		825.2.bx.c.499.1		8
55.54	odd	2		9075.2.a.bc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.n.b.301.1	✓	4	11.9	even	5
825.2.n.b.751.1	yes	4	11.5	even	5
825.2.n.d.301.1	yes	4	55.9	even	10
825.2.n.d.751.1	yes	4	55.49	even	10
825.2.bx.c.124.1		8	55.27	odd	20
825.2.bx.c.124.2		8	55.38	odd	20
825.2.bx.c.499.1		8	55.53	odd	20
825.2.bx.c.499.2		8	55.42	odd	20
9075.2.a.z.1.2		2	1.1	even	1	trivial
9075.2.a.bc.1.2		2	55.54	odd	2
9075.2.a.bt.1.1		2	11.10	odd	2
9075.2.a.by.1.1		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -0.763932 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -0.763932 q^{7} -2.23607 q^{8} +1.00000 q^{9} +1.61803 q^{12} -1.23607 q^{13} -0.472136 q^{14} +1.85410 q^{16} +2.00000 q^{17} +0.618034 q^{18} +5.00000 q^{19} +0.763932 q^{21} +2.38197 q^{23} +2.23607 q^{24} -0.763932 q^{26} -1.00000 q^{27} +1.23607 q^{28} -5.85410 q^{29} +7.00000 q^{31} +5.61803 q^{32} +1.23607 q^{34} -1.61803 q^{36} -7.47214 q^{37} +3.09017 q^{38} +1.23607 q^{39} +2.52786 q^{41} +0.472136 q^{42} -2.09017 q^{43} +1.47214 q^{46} +5.09017 q^{47} -1.85410 q^{48} -6.41641 q^{49} -2.00000 q^{51} +2.00000 q^{52} -7.61803 q^{53} -0.618034 q^{54} +1.70820 q^{56} -5.00000 q^{57} -3.61803 q^{58} -6.70820 q^{59} +7.00000 q^{61} +4.32624 q^{62} -0.763932 q^{63} -0.236068 q^{64} -9.38197 q^{67} -3.23607 q^{68} -2.38197 q^{69} -8.00000 q^{71} -2.23607 q^{72} -13.4721 q^{73} -4.61803 q^{74} -8.09017 q^{76} +0.763932 q^{78} -8.09017 q^{79} +1.00000 q^{81} +1.56231 q^{82} -5.70820 q^{83} -1.23607 q^{84} -1.29180 q^{86} +5.85410 q^{87} +10.8541 q^{89} +0.944272 q^{91} -3.85410 q^{92} -7.00000 q^{93} +3.14590 q^{94} -5.61803 q^{96} +11.4721 q^{97} -3.96556 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 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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