LMFDB - Embedded newform 9075.2.a.bu.1.1

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			1.1
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)

$f(q)$	$=$	$q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +3.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +3.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +1.61803 q^{12} +4.85410 q^{13} -1.85410 q^{14} +1.85410 q^{16} -1.47214 q^{17} -0.618034 q^{18} +7.23607 q^{19} -3.00000 q^{21} -5.38197 q^{23} -2.23607 q^{24} -3.00000 q^{26} -1.00000 q^{27} -4.85410 q^{28} -1.38197 q^{29} -2.14590 q^{31} -5.61803 q^{32} +0.909830 q^{34} -1.61803 q^{36} -2.14590 q^{37} -4.47214 q^{38} -4.85410 q^{39} -4.23607 q^{41} +1.85410 q^{42} +6.23607 q^{43} +3.32624 q^{46} -11.6180 q^{47} -1.85410 q^{48} +2.00000 q^{49} +1.47214 q^{51} -7.85410 q^{52} -8.47214 q^{53} +0.618034 q^{54} +6.70820 q^{56} -7.23607 q^{57} +0.854102 q^{58} -10.8541 q^{59} +10.2361 q^{61} +1.32624 q^{62} +3.00000 q^{63} -0.236068 q^{64} -4.70820 q^{67} +2.38197 q^{68} +5.38197 q^{69} -7.47214 q^{71} +2.23607 q^{72} -0.145898 q^{73} +1.32624 q^{74} -11.7082 q^{76} +3.00000 q^{78} -12.5623 q^{79} +1.00000 q^{81} +2.61803 q^{82} +12.6180 q^{83} +4.85410 q^{84} -3.85410 q^{86} +1.38197 q^{87} -17.5623 q^{89} +14.5623 q^{91} +8.70820 q^{92} +2.14590 q^{93} +7.18034 q^{94} +5.61803 q^{96} -16.4164 q^{97} -1.23607 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 3 q^{13} + 3 q^{14} - 3 q^{16} + 6 q^{17} + q^{18} + 10 q^{19} - 6 q^{21} - 13 q^{23} - 6 q^{26} - 2 q^{27} - 3 q^{28} - 5 q^{29} - 11 q^{31} - 9 q^{32} + 13 q^{34} - q^{36} - 11 q^{37} - 3 q^{39} - 4 q^{41} - 3 q^{42} + 8 q^{43} - 9 q^{46} - 21 q^{47} + 3 q^{48} + 4 q^{49} - 6 q^{51} - 9 q^{52} - 8 q^{53} - q^{54} - 10 q^{57} - 5 q^{58} - 15 q^{59} + 16 q^{61} - 13 q^{62} + 6 q^{63} + 4 q^{64} + 4 q^{67} + 7 q^{68} + 13 q^{69} - 6 q^{71} - 7 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{78} - 5 q^{79} + 2 q^{81} + 3 q^{82} + 23 q^{83} + 3 q^{84} - q^{86} + 5 q^{87} - 15 q^{89} + 9 q^{91} + 4 q^{92} + 11 q^{93} - 8 q^{94} + 9 q^{96} - 6 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 + q^12 + 3 * q^13 + 3 * q^14 - 3 * q^16 + 6 * q^17 + q^18 + 10 * q^19 - 6 * q^21 - 13 * q^23 - 6 * q^26 - 2 * q^27 - 3 * q^28 - 5 * q^29 - 11 * q^31 - 9 * q^32 + 13 * q^34 - q^36 - 11 * q^37 - 3 * q^39 - 4 * q^41 - 3 * q^42 + 8 * q^43 - 9 * q^46 - 21 * q^47 + 3 * q^48 + 4 * q^49 - 6 * q^51 - 9 * q^52 - 8 * q^53 - q^54 - 10 * q^57 - 5 * q^58 - 15 * q^59 + 16 * q^61 - 13 * q^62 + 6 * q^63 + 4 * q^64 + 4 * q^67 + 7 * q^68 + 13 * q^69 - 6 * q^71 - 7 * q^73 - 13 * q^74 - 10 * q^76 + 6 * q^78 - 5 * q^79 + 2 * q^81 + 3 * q^82 + 23 * q^83 + 3 * q^84 - q^86 + 5 * q^87 - 15 * q^89 + 9 * q^91 + 4 * q^92 + 11 * q^93 - 8 * q^94 + 9 * q^96 - 6 * q^97 + 2 * 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	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	−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
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	1.61803	0.467086
$13$	4.85410	1.34629	0.673143	−	0.739512i	$-0.264944\pi$
$13$	4.85410	1.34629	0.673143	+	0.739512i	$0.264944\pi$
$14$	−1.85410	−0.495530
$15$	0	0
$16$	1.85410	0.463525
$17$	−1.47214	−0.357045	−0.178523	−	0.983936i	$-0.557132\pi$
$17$	−1.47214	−0.357045	−0.178523	+	0.983936i	$0.557132\pi$
$18$	−0.618034	−0.145672
$19$	7.23607	1.66007	0.830034	−	0.557713i	$-0.188321\pi$
$19$	7.23607	1.66007	0.830034	+	0.557713i	$0.188321\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	−5.38197	−1.12222	−0.561109	−	0.827742i	$-0.689625\pi$
$23$	−5.38197	−1.12222	−0.561109	+	0.827742i	$0.689625\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	−3.00000	−0.588348
$27$	−1.00000	−0.192450
$28$	−4.85410	−0.917339
$29$	−1.38197	−0.256625	−0.128312	−	0.991734i	$-0.540956\pi$
$29$	−1.38197	−0.256625	−0.128312	+	0.991734i	$0.540956\pi$
$30$	0	0
$31$	−2.14590	−0.385415	−0.192707	−	0.981256i	$-0.561727\pi$
$31$	−2.14590	−0.385415	−0.192707	+	0.981256i	$0.561727\pi$
$32$	−5.61803	−0.993137
$33$	0	0
$34$	0.909830	0.156035
$35$	0	0
$36$	−1.61803	−0.269672
$37$	−2.14590	−0.352783	−0.176392	−	0.984320i	$-0.556443\pi$
$37$	−2.14590	−0.352783	−0.176392	+	0.984320i	$0.556443\pi$
$38$	−4.47214	−0.725476
$39$	−4.85410	−0.777278
$40$	0	0
$41$	−4.23607	−0.661563	−0.330781	−	0.943707i	$-0.607312\pi$
$41$	−4.23607	−0.661563	−0.330781	+	0.943707i	$0.607312\pi$
$42$	1.85410	0.286094
$43$	6.23607	0.950991	0.475496	−	0.879718i	$-0.342269\pi$
$43$	6.23607	0.950991	0.475496	+	0.879718i	$0.342269\pi$
$44$	0	0
$45$	0	0
$46$	3.32624	0.490427
$47$	−11.6180	−1.69466	−0.847332	−	0.531063i	$-0.821793\pi$
$47$	−11.6180	−1.69466	−0.847332	+	0.531063i	$0.821793\pi$
$48$	−1.85410	−0.267617
$49$	2.00000	0.285714
$50$	0	0
$51$	1.47214	0.206140
$52$	−7.85410	−1.08917
$53$	−8.47214	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$53$	−8.47214	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$54$	0.618034	0.0841038
$55$	0	0
$56$	6.70820	0.896421
$57$	−7.23607	−0.958441
$58$	0.854102	0.112149
$59$	−10.8541	−1.41308	−0.706542	−	0.707671i	$-0.749746\pi$
$59$	−10.8541	−1.41308	−0.706542	+	0.707671i	$0.749746\pi$
$60$	0	0
$61$	10.2361	1.31059	0.655297	−	0.755371i	$-0.272543\pi$
$61$	10.2361	1.31059	0.655297	+	0.755371i	$0.272543\pi$
$62$	1.32624	0.168432
$63$	3.00000	0.377964
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0	0
$67$	−4.70820	−0.575199	−0.287599	−	0.957751i	$-0.592857\pi$
$67$	−4.70820	−0.575199	−0.287599	+	0.957751i	$0.592857\pi$
$68$	2.38197	0.288856
$69$	5.38197	0.647913
$70$	0	0
$71$	−7.47214	−0.886779	−0.443390	−	0.896329i	$-0.646224\pi$
$71$	−7.47214	−0.886779	−0.443390	+	0.896329i	$0.646224\pi$
$72$	2.23607	0.263523
$73$	−0.145898	−0.0170761	−0.00853804	−	0.999964i	$-0.502718\pi$
$73$	−0.145898	−0.0170761	−0.00853804	+	0.999964i	$0.502718\pi$
$74$	1.32624	0.154172
$75$	0	0
$76$	−11.7082	−1.34302
$77$	0	0
$78$	3.00000	0.339683
$79$	−12.5623	−1.41337	−0.706685	−	0.707528i	$-0.749810\pi$
$79$	−12.5623	−1.41337	−0.706685	+	0.707528i	$0.749810\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.61803	0.289113
$83$	12.6180	1.38501	0.692505	−	0.721413i	$-0.256507\pi$
$83$	12.6180	1.38501	0.692505	+	0.721413i	$0.256507\pi$
$84$	4.85410	0.529626
$85$	0	0
$86$	−3.85410	−0.415599
$87$	1.38197	0.148162
$88$	0	0
$89$	−17.5623	−1.86160	−0.930800	−	0.365528i	$-0.880888\pi$
$89$	−17.5623	−1.86160	−0.930800	+	0.365528i	$0.880888\pi$
$90$	0	0
$91$	14.5623	1.52654
$92$	8.70820	0.907893
$93$	2.14590	0.222519
$94$	7.18034	0.740596
$95$	0	0
$96$	5.61803	0.573388
$97$	−16.4164	−1.66683	−0.833417	−	0.552645i	$-0.813619\pi$
$97$	−16.4164	−1.66683	−0.833417	+	0.552645i	$0.813619\pi$
$98$	−1.23607	−0.124862
$99$	0	0
$100$	0	0
$101$	−10.0902	−1.00401	−0.502005	−	0.864865i	$-0.667404\pi$
$101$	−10.0902	−1.00401	−0.502005	+	0.864865i	$0.667404\pi$
$102$	−0.909830	−0.0900866
$103$	7.18034	0.707500	0.353750	−	0.935340i	$-0.384906\pi$
$103$	7.18034	0.707500	0.353750	+	0.935340i	$0.384906\pi$
$104$	10.8541	1.06433
$105$	0	0
$106$	5.23607	0.508572
$107$	−10.4164	−1.00699	−0.503496	−	0.863998i	$-0.667953\pi$
$107$	−10.4164	−1.00699	−0.503496	+	0.863998i	$0.667953\pi$
$108$	1.61803	0.155695
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	0	0
$111$	2.14590	0.203680
$112$	5.56231	0.525589
$113$	0.472136	0.0444148	0.0222074	−	0.999753i	$-0.492931\pi$
$113$	0.472136	0.0444148	0.0222074	+	0.999753i	$0.492931\pi$
$114$	4.47214	0.418854
$115$	0	0
$116$	2.23607	0.207614
$117$	4.85410	0.448762
$118$	6.70820	0.617540
$119$	−4.41641	−0.404851
$120$	0	0
$121$	0	0
$122$	−6.32624	−0.572751
$123$	4.23607	0.381953
$124$	3.47214	0.311807
$125$	0	0
$126$	−1.85410	−0.165177
$127$	4.38197	0.388837	0.194418	−	0.980919i	$-0.437718\pi$
$127$	4.38197	0.388837	0.194418	+	0.980919i	$0.437718\pi$
$128$	11.3820	1.00603
$129$	−6.23607	−0.549055
$130$	0	0
$131$	8.32624	0.727467	0.363733	−	0.931503i	$-0.381502\pi$
$131$	8.32624	0.727467	0.363733	+	0.931503i	$0.381502\pi$
$132$	0	0
$133$	21.7082	1.88234
$134$	2.90983	0.251371
$135$	0	0
$136$	−3.29180	−0.282269
$137$	11.7984	1.00800	0.504002	−	0.863703i	$-0.331861\pi$
$137$	11.7984	1.00800	0.504002	+	0.863703i	$0.331861\pi$
$138$	−3.32624	−0.283148
$139$	0.854102	0.0724440	0.0362220	−	0.999344i	$-0.488468\pi$
$139$	0.854102	0.0724440	0.0362220	+	0.999344i	$0.488468\pi$
$140$	0	0
$141$	11.6180	0.978415
$142$	4.61803	0.387537
$143$	0	0
$144$	1.85410	0.154508
$145$	0	0
$146$	0.0901699	0.00746252
$147$	−2.00000	−0.164957
$148$	3.47214	0.285408
$149$	6.18034	0.506313	0.253157	−	0.967425i	$-0.418531\pi$
$149$	6.18034	0.506313	0.253157	+	0.967425i	$0.418531\pi$
$150$	0	0
$151$	−13.1803	−1.07260	−0.536300	−	0.844027i	$-0.680179\pi$
$151$	−13.1803	−1.07260	−0.536300	+	0.844027i	$0.680179\pi$
$152$	16.1803	1.31240
$153$	−1.47214	−0.119015
$154$	0	0
$155$	0	0
$156$	7.85410	0.628831
$157$	6.79837	0.542569	0.271285	−	0.962499i	$-0.412552\pi$
$157$	6.79837	0.542569	0.271285	+	0.962499i	$0.412552\pi$
$158$	7.76393	0.617665
$159$	8.47214	0.671884
$160$	0	0
$161$	−16.1459	−1.27248
$162$	−0.618034	−0.0485573
$163$	−5.90983	−0.462894	−0.231447	−	0.972848i	$-0.574346\pi$
$163$	−5.90983	−0.462894	−0.231447	+	0.972848i	$0.574346\pi$
$164$	6.85410	0.535215
$165$	0	0
$166$	−7.79837	−0.605271
$167$	−3.05573	−0.236459	−0.118230	−	0.992986i	$-0.537722\pi$
$167$	−3.05573	−0.236459	−0.118230	+	0.992986i	$0.537722\pi$
$168$	−6.70820	−0.517549
$169$	10.5623	0.812485
$170$	0	0
$171$	7.23607	0.553356
$172$	−10.0902	−0.769368
$173$	3.47214	0.263982	0.131991	−	0.991251i	$-0.457863\pi$
$173$	3.47214	0.263982	0.131991	+	0.991251i	$0.457863\pi$
$174$	−0.854102	−0.0647493
$175$	0	0
$176$	0	0
$177$	10.8541	0.815844
$178$	10.8541	0.813549
$179$	20.1246	1.50418	0.752092	−	0.659058i	$-0.229045\pi$
$179$	20.1246	1.50418	0.752092	+	0.659058i	$0.229045\pi$
$180$	0	0
$181$	11.7984	0.876966	0.438483	−	0.898739i	$-0.355516\pi$
$181$	11.7984	0.876966	0.438483	+	0.898739i	$0.355516\pi$
$182$	−9.00000	−0.667124
$183$	−10.2361	−0.756672
$184$	−12.0344	−0.887191
$185$	0	0
$186$	−1.32624	−0.0972445
$187$	0	0
$188$	18.7984	1.37101
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−26.7426	−1.93503	−0.967515	−	0.252814i	$-0.918644\pi$
$191$	−26.7426	−1.93503	−0.967515	+	0.252814i	$0.918644\pi$
$192$	0.236068	0.0170367
$193$	9.52786	0.685831	0.342915	−	0.939366i	$-0.388586\pi$
$193$	9.52786	0.685831	0.342915	+	0.939366i	$0.388586\pi$
$194$	10.1459	0.728433
$195$	0	0
$196$	−3.23607	−0.231148
$197$	−3.70820	−0.264199	−0.132099	−	0.991236i	$-0.542172\pi$
$197$	−3.70820	−0.264199	−0.132099	+	0.991236i	$0.542172\pi$
$198$	0	0
$199$	−21.7082	−1.53885	−0.769427	−	0.638735i	$-0.779458\pi$
$199$	−21.7082	−1.53885	−0.769427	+	0.638735i	$0.779458\pi$
$200$	0	0
$201$	4.70820	0.332091
$202$	6.23607	0.438768
$203$	−4.14590	−0.290985
$204$	−2.38197	−0.166771
$205$	0	0
$206$	−4.43769	−0.309189
$207$	−5.38197	−0.374072
$208$	9.00000	0.624038
$209$	0	0
$210$	0	0
$211$	9.18034	0.632001	0.316000	−	0.948759i	$-0.397660\pi$
$211$	9.18034	0.632001	0.316000	+	0.948759i	$0.397660\pi$
$212$	13.7082	0.941483
$213$	7.47214	0.511982
$214$	6.43769	0.440072
$215$	0	0
$216$	−2.23607	−0.152145
$217$	−6.43769	−0.437019
$218$	9.27051	0.627878
$219$	0.145898	0.00985888
$220$	0	0
$221$	−7.14590	−0.480685
$222$	−1.32624	−0.0890113
$223$	−0.708204	−0.0474248	−0.0237124	−	0.999719i	$-0.507549\pi$
$223$	−0.708204	−0.0474248	−0.0237124	+	0.999719i	$0.507549\pi$
$224$	−16.8541	−1.12611
$225$	0	0
$226$	−0.291796	−0.0194100
$227$	5.56231	0.369183	0.184592	−	0.982815i	$-0.440904\pi$
$227$	5.56231	0.369183	0.184592	+	0.982815i	$0.440904\pi$
$228$	11.7082	0.775395
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	−3.09017	−0.202880
$233$	9.32624	0.610982	0.305491	−	0.952195i	$-0.401179\pi$
$233$	9.32624	0.610982	0.305491	+	0.952195i	$0.401179\pi$
$234$	−3.00000	−0.196116
$235$	0	0
$236$	17.5623	1.14321
$237$	12.5623	0.816009
$238$	2.72949	0.176927
$239$	−10.6525	−0.689051	−0.344526	−	0.938777i	$-0.611960\pi$
$239$	−10.6525	−0.689051	−0.344526	+	0.938777i	$0.611960\pi$
$240$	0	0
$241$	−1.14590	−0.0738138	−0.0369069	−	0.999319i	$-0.511750\pi$
$241$	−1.14590	−0.0738138	−0.0369069	+	0.999319i	$0.511750\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−16.5623	−1.06029
$245$	0	0
$246$	−2.61803	−0.166920
$247$	35.1246	2.23493
$248$	−4.79837	−0.304697
$249$	−12.6180	−0.799635
$250$	0	0
$251$	−0.437694	−0.0276270	−0.0138135	−	0.999905i	$-0.504397\pi$
$251$	−0.437694	−0.0276270	−0.0138135	+	0.999905i	$0.504397\pi$
$252$	−4.85410	−0.305780
$253$	0	0
$254$	−2.70820	−0.169928
$255$	0	0
$256$	−6.56231	−0.410144
$257$	19.0344	1.18734	0.593668	−	0.804710i	$-0.297679\pi$
$257$	19.0344	1.18734	0.593668	+	0.804710i	$0.297679\pi$
$258$	3.85410	0.239946
$259$	−6.43769	−0.400019
$260$	0	0
$261$	−1.38197	−0.0855415
$262$	−5.14590	−0.317915
$263$	29.1246	1.79590	0.897950	−	0.440097i	$-0.145056\pi$
$263$	29.1246	1.79590	0.897950	+	0.440097i	$0.145056\pi$
$264$	0	0
$265$	0	0
$266$	−13.4164	−0.822613
$267$	17.5623	1.07480
$268$	7.61803	0.465345
$269$	−9.47214	−0.577526	−0.288763	−	0.957401i	$-0.593244\pi$
$269$	−9.47214	−0.577526	−0.288763	+	0.957401i	$0.593244\pi$
$270$	0	0
$271$	−11.2705	−0.684635	−0.342317	−	0.939584i	$-0.611212\pi$
$271$	−11.2705	−0.684635	−0.342317	+	0.939584i	$0.611212\pi$
$272$	−2.72949	−0.165500
$273$	−14.5623	−0.881351
$274$	−7.29180	−0.440514
$275$	0	0
$276$	−8.70820	−0.524172
$277$	−22.9787	−1.38066	−0.690329	−	0.723496i	$-0.742534\pi$
$277$	−22.9787	−1.38066	−0.690329	+	0.723496i	$0.742534\pi$
$278$	−0.527864	−0.0316592
$279$	−2.14590	−0.128472
$280$	0	0
$281$	−12.3262	−0.735322	−0.367661	−	0.929960i	$-0.619841\pi$
$281$	−12.3262	−0.735322	−0.367661	+	0.929960i	$0.619841\pi$
$282$	−7.18034	−0.427583
$283$	15.9098	0.945741	0.472871	−	0.881132i	$-0.343218\pi$
$283$	15.9098	0.945741	0.472871	+	0.881132i	$0.343218\pi$
$284$	12.0902	0.717420
$285$	0	0
$286$	0	0
$287$	−12.7082	−0.750142
$288$	−5.61803	−0.331046
$289$	−14.8328	−0.872519
$290$	0	0
$291$	16.4164	0.962347
$292$	0.236068	0.0138148
$293$	7.41641	0.433271	0.216636	−	0.976253i	$-0.430492\pi$
$293$	7.41641	0.433271	0.216636	+	0.976253i	$0.430492\pi$
$294$	1.23607	0.0720889
$295$	0	0
$296$	−4.79837	−0.278900
$297$	0	0
$298$	−3.81966	−0.221267
$299$	−26.1246	−1.51083
$300$	0	0
$301$	18.7082	1.07832
$302$	8.14590	0.468744
$303$	10.0902	0.579665
$304$	13.4164	0.769484
$305$	0	0
$306$	0.909830	0.0520115
$307$	−19.8885	−1.13510	−0.567550	−	0.823339i	$-0.692108\pi$
$307$	−19.8885	−1.13510	−0.567550	+	0.823339i	$0.692108\pi$
$308$	0	0
$309$	−7.18034	−0.408475
$310$	0	0
$311$	−27.4721	−1.55780	−0.778901	−	0.627147i	$-0.784223\pi$
$311$	−27.4721	−1.55780	−0.778901	+	0.627147i	$0.784223\pi$
$312$	−10.8541	−0.614493
$313$	−17.2918	−0.977390	−0.488695	−	0.872455i	$-0.662527\pi$
$313$	−17.2918	−0.977390	−0.488695	+	0.872455i	$0.662527\pi$
$314$	−4.20163	−0.237111
$315$	0	0
$316$	20.3262	1.14344
$317$	13.1803	0.740282	0.370141	−	0.928976i	$-0.379309\pi$
$317$	13.1803	0.740282	0.370141	+	0.928976i	$0.379309\pi$
$318$	−5.23607	−0.293624
$319$	0	0
$320$	0	0
$321$	10.4164	0.581387
$322$	9.97871	0.556092
$323$	−10.6525	−0.592720
$324$	−1.61803	−0.0898908
$325$	0	0
$326$	3.65248	0.202292
$327$	15.0000	0.829502
$328$	−9.47214	−0.523011
$329$	−34.8541	−1.92157
$330$	0	0
$331$	10.4164	0.572538	0.286269	−	0.958149i	$-0.407585\pi$
$331$	10.4164	0.572538	0.286269	+	0.958149i	$0.407585\pi$
$332$	−20.4164	−1.12050
$333$	−2.14590	−0.117594
$334$	1.88854	0.103337
$335$	0	0
$336$	−5.56231	−0.303449
$337$	−10.2918	−0.560630	−0.280315	−	0.959908i	$-0.590439\pi$
$337$	−10.2918	−0.560630	−0.280315	+	0.959908i	$0.590439\pi$
$338$	−6.52786	−0.355069
$339$	−0.472136	−0.0256429
$340$	0	0
$341$	0	0
$342$	−4.47214	−0.241825
$343$	−15.0000	−0.809924
$344$	13.9443	0.751825
$345$	0	0
$346$	−2.14590	−0.115364
$347$	3.32624	0.178562	0.0892809	−	0.996006i	$-0.471543\pi$
$347$	3.32624	0.178562	0.0892809	+	0.996006i	$0.471543\pi$
$348$	−2.23607	−0.119866
$349$	−11.0557	−0.591800	−0.295900	−	0.955219i	$-0.595619\pi$
$349$	−11.0557	−0.591800	−0.295900	+	0.955219i	$0.595619\pi$
$350$	0	0
$351$	−4.85410	−0.259093
$352$	0	0
$353$	−26.8885	−1.43113	−0.715566	−	0.698545i	$-0.753831\pi$
$353$	−26.8885	−1.43113	−0.715566	+	0.698545i	$0.753831\pi$
$354$	−6.70820	−0.356537
$355$	0	0
$356$	28.4164	1.50607
$357$	4.41641	0.233741
$358$	−12.4377	−0.657353
$359$	33.2148	1.75301	0.876505	−	0.481394i	$-0.159869\pi$
$359$	33.2148	1.75301	0.876505	+	0.481394i	$0.159869\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	−7.29180	−0.383248
$363$	0	0
$364$	−23.5623	−1.23500
$365$	0	0
$366$	6.32624	0.330678
$367$	−34.8328	−1.81826	−0.909129	−	0.416514i	$-0.863252\pi$
$367$	−34.8328	−1.81826	−0.909129	+	0.416514i	$0.863252\pi$
$368$	−9.97871	−0.520176
$369$	−4.23607	−0.220521
$370$	0	0
$371$	−25.4164	−1.31955
$372$	−3.47214	−0.180022
$373$	−9.41641	−0.487563	−0.243782	−	0.969830i	$-0.578388\pi$
$373$	−9.41641	−0.487563	−0.243782	+	0.969830i	$0.578388\pi$
$374$	0	0
$375$	0	0
$376$	−25.9787	−1.33975
$377$	−6.70820	−0.345490
$378$	1.85410	0.0953647
$379$	28.2148	1.44930	0.724648	−	0.689119i	$-0.242002\pi$
$379$	28.2148	1.44930	0.724648	+	0.689119i	$0.242002\pi$
$380$	0	0
$381$	−4.38197	−0.224495
$382$	16.5279	0.845639
$383$	30.4721	1.55705	0.778527	−	0.627611i	$-0.215967\pi$
$383$	30.4721	1.55705	0.778527	+	0.627611i	$0.215967\pi$
$384$	−11.3820	−0.580834
$385$	0	0
$386$	−5.88854	−0.299719
$387$	6.23607	0.316997
$388$	26.5623	1.34850
$389$	−6.90983	−0.350342	−0.175171	−	0.984538i	$-0.556048\pi$
$389$	−6.90983	−0.350342	−0.175171	+	0.984538i	$0.556048\pi$
$390$	0	0
$391$	7.92299	0.400683
$392$	4.47214	0.225877
$393$	−8.32624	−0.420003
$394$	2.29180	0.115459
$395$	0	0
$396$	0	0
$397$	−0.562306	−0.0282213	−0.0141107	−	0.999900i	$-0.504492\pi$
$397$	−0.562306	−0.0282213	−0.0141107	+	0.999900i	$0.504492\pi$
$398$	13.4164	0.672504
$399$	−21.7082	−1.08677
$400$	0	0
$401$	23.1803	1.15757	0.578785	−	0.815480i	$-0.303527\pi$
$401$	23.1803	1.15757	0.578785	+	0.815480i	$0.303527\pi$
$402$	−2.90983	−0.145129
$403$	−10.4164	−0.518878
$404$	16.3262	0.812261
$405$	0	0
$406$	2.56231	0.127165
$407$	0	0
$408$	3.29180	0.162968
$409$	−30.1246	−1.48957	−0.744783	−	0.667307i	$-0.767447\pi$
$409$	−30.1246	−1.48957	−0.744783	+	0.667307i	$0.767447\pi$
$410$	0	0
$411$	−11.7984	−0.581971
$412$	−11.6180	−0.572379
$413$	−32.5623	−1.60229
$414$	3.32624	0.163476
$415$	0	0
$416$	−27.2705	−1.33705
$417$	−0.854102	−0.0418256
$418$	0	0
$419$	1.05573	0.0515757	0.0257878	−	0.999667i	$-0.491791\pi$
$419$	1.05573	0.0515757	0.0257878	+	0.999667i	$0.491791\pi$
$420$	0	0
$421$	26.2705	1.28035	0.640173	−	0.768231i	$-0.278863\pi$
$421$	26.2705	1.28035	0.640173	+	0.768231i	$0.278863\pi$
$422$	−5.67376	−0.276194
$423$	−11.6180	−0.564888
$424$	−18.9443	−0.920015
$425$	0	0
$426$	−4.61803	−0.223744
$427$	30.7082	1.48607
$428$	16.8541	0.814674
$429$	0	0
$430$	0	0
$431$	−22.8541	−1.10084	−0.550422	−	0.834887i	$-0.685533\pi$
$431$	−22.8541	−1.10084	−0.550422	+	0.834887i	$0.685533\pi$
$432$	−1.85410	−0.0892055
$433$	4.81966	0.231618	0.115809	−	0.993271i	$-0.463054\pi$
$433$	4.81966	0.231618	0.115809	+	0.993271i	$0.463054\pi$
$434$	3.97871	0.190984
$435$	0	0
$436$	24.2705	1.16235
$437$	−38.9443	−1.86296
$438$	−0.0901699	−0.00430849
$439$	−27.5623	−1.31548	−0.657739	−	0.753246i	$-0.728487\pi$
$439$	−27.5623	−1.31548	−0.657739	+	0.753246i	$0.728487\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	4.41641	0.210067
$443$	−10.5066	−0.499183	−0.249591	−	0.968351i	$-0.580296\pi$
$443$	−10.5066	−0.499183	−0.249591	+	0.968351i	$0.580296\pi$
$444$	−3.47214	−0.164780
$445$	0	0
$446$	0.437694	0.0207254
$447$	−6.18034	−0.292320
$448$	−0.708204	−0.0334595
$449$	5.52786	0.260876	0.130438	−	0.991456i	$-0.458362\pi$
$449$	5.52786	0.260876	0.130438	+	0.991456i	$0.458362\pi$
$450$	0	0
$451$	0	0
$452$	−0.763932	−0.0359323
$453$	13.1803	0.619266
$454$	−3.43769	−0.161339
$455$	0	0
$456$	−16.1803	−0.757714
$457$	32.4721	1.51898	0.759491	−	0.650518i	$-0.225448\pi$
$457$	32.4721	1.51898	0.759491	+	0.650518i	$0.225448\pi$
$458$	6.18034	0.288788
$459$	1.47214	0.0687134
$460$	0	0
$461$	−13.1803	−0.613870	−0.306935	−	0.951731i	$-0.599303\pi$
$461$	−13.1803	−0.613870	−0.306935	+	0.951731i	$0.599303\pi$
$462$	0	0
$463$	22.7082	1.05534	0.527670	−	0.849450i	$-0.323066\pi$
$463$	22.7082	1.05534	0.527670	+	0.849450i	$0.323066\pi$
$464$	−2.56231	−0.118952
$465$	0	0
$466$	−5.76393	−0.267009
$467$	34.0344	1.57493	0.787463	−	0.616362i	$-0.211394\pi$
$467$	34.0344	1.57493	0.787463	+	0.616362i	$0.211394\pi$
$468$	−7.85410	−0.363056
$469$	−14.1246	−0.652214
$470$	0	0
$471$	−6.79837	−0.313253
$472$	−24.2705	−1.11714
$473$	0	0
$474$	−7.76393	−0.356609
$475$	0	0
$476$	7.14590	0.327532
$477$	−8.47214	−0.387912
$478$	6.58359	0.301126
$479$	37.3607	1.70705	0.853527	−	0.521049i	$-0.174459\pi$
$479$	37.3607	1.70705	0.853527	+	0.521049i	$0.174459\pi$
$480$	0	0
$481$	−10.4164	−0.474947
$482$	0.708204	0.0322578
$483$	16.1459	0.734664
$484$	0	0
$485$	0	0
$486$	0.618034	0.0280346
$487$	−32.2705	−1.46232	−0.731158	−	0.682208i	$-0.761020\pi$
$487$	−32.2705	−1.46232	−0.731158	+	0.682208i	$0.761020\pi$
$488$	22.8885	1.03612
$489$	5.90983	0.267252
$490$	0	0
$491$	7.14590	0.322490	0.161245	−	0.986914i	$-0.448449\pi$
$491$	7.14590	0.322490	0.161245	+	0.986914i	$0.448449\pi$
$492$	−6.85410	−0.309007
$493$	2.03444	0.0916267
$494$	−21.7082	−0.976698
$495$	0	0
$496$	−3.97871	−0.178650
$497$	−22.4164	−1.00551
$498$	7.79837	0.349453
$499$	−41.8328	−1.87269	−0.936347	−	0.351076i	$-0.885816\pi$
$499$	−41.8328	−1.87269	−0.936347	+	0.351076i	$0.885816\pi$
$500$	0	0
$501$	3.05573	0.136520
$502$	0.270510	0.0120734
$503$	−23.5623	−1.05059	−0.525296	−	0.850920i	$-0.676045\pi$
$503$	−23.5623	−1.05059	−0.525296	+	0.850920i	$0.676045\pi$
$504$	6.70820	0.298807
$505$	0	0
$506$	0	0
$507$	−10.5623	−0.469088
$508$	−7.09017	−0.314575
$509$	9.79837	0.434305	0.217153	−	0.976138i	$-0.430323\pi$
$509$	9.79837	0.434305	0.217153	+	0.976138i	$0.430323\pi$
$510$	0	0
$511$	−0.437694	−0.0193624
$512$	−18.7082	−0.826794
$513$	−7.23607	−0.319480
$514$	−11.7639	−0.518885
$515$	0	0
$516$	10.0902	0.444195
$517$	0	0
$518$	3.97871	0.174815
$519$	−3.47214	−0.152410
$520$	0	0
$521$	37.2492	1.63192	0.815959	−	0.578110i	$-0.196209\pi$
$521$	37.2492	1.63192	0.815959	+	0.578110i	$0.196209\pi$
$522$	0.854102	0.0373830
$523$	14.2016	0.620994	0.310497	−	0.950574i	$-0.399505\pi$
$523$	14.2016	0.620994	0.310497	+	0.950574i	$0.399505\pi$
$524$	−13.4721	−0.588533
$525$	0	0
$526$	−18.0000	−0.784837
$527$	3.15905	0.137611
$528$	0	0
$529$	5.96556	0.259372
$530$	0	0
$531$	−10.8541	−0.471028
$532$	−35.1246	−1.52285
$533$	−20.5623	−0.890652
$534$	−10.8541	−0.469703
$535$	0	0
$536$	−10.5279	−0.454734
$537$	−20.1246	−0.868441
$538$	5.85410	0.252388
$539$	0	0
$540$	0	0
$541$	−28.8328	−1.23962	−0.619810	−	0.784752i	$-0.712790\pi$
$541$	−28.8328	−1.23962	−0.619810	+	0.784752i	$0.712790\pi$
$542$	6.96556	0.299196
$543$	−11.7984	−0.506317
$544$	8.27051	0.354595
$545$	0	0
$546$	9.00000	0.385164
$547$	34.7082	1.48402	0.742008	−	0.670391i	$-0.233874\pi$
$547$	34.7082	1.48402	0.742008	+	0.670391i	$0.233874\pi$
$548$	−19.0902	−0.815492
$549$	10.2361	0.436865
$550$	0	0
$551$	−10.0000	−0.426014
$552$	12.0344	0.512220
$553$	−37.6869	−1.60261
$554$	14.2016	0.603369
$555$	0	0
$556$	−1.38197	−0.0586084
$557$	36.5410	1.54829	0.774146	−	0.633007i	$-0.218179\pi$
$557$	36.5410	1.54829	0.774146	+	0.633007i	$0.218179\pi$
$558$	1.32624	0.0561441
$559$	30.2705	1.28031
$560$	0	0
$561$	0	0
$562$	7.61803	0.321347
$563$	35.8328	1.51017	0.755087	−	0.655625i	$-0.227595\pi$
$563$	35.8328	1.51017	0.755087	+	0.655625i	$0.227595\pi$
$564$	−18.7984	−0.791554
$565$	0	0
$566$	−9.83282	−0.413304
$567$	3.00000	0.125988
$568$	−16.7082	−0.701061
$569$	−24.4721	−1.02593	−0.512963	−	0.858411i	$-0.671452\pi$
$569$	−24.4721	−1.02593	−0.512963	+	0.858411i	$0.671452\pi$
$570$	0	0
$571$	−40.4164	−1.69137	−0.845687	−	0.533679i	$-0.820809\pi$
$571$	−40.4164	−1.69137	−0.845687	+	0.533679i	$0.820809\pi$
$572$	0	0
$573$	26.7426	1.11719
$574$	7.85410	0.327824
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	−31.2148	−1.29949	−0.649744	−	0.760153i	$-0.725124\pi$
$577$	−31.2148	−1.29949	−0.649744	+	0.760153i	$0.725124\pi$
$578$	9.16718	0.381305
$579$	−9.52786	−0.395965
$580$	0	0
$581$	37.8541	1.57045
$582$	−10.1459	−0.420561
$583$	0	0
$584$	−0.326238	−0.0134998
$585$	0	0
$586$	−4.58359	−0.189346
$587$	28.3050	1.16827	0.584135	−	0.811656i	$-0.301434\pi$
$587$	28.3050	1.16827	0.584135	+	0.811656i	$0.301434\pi$
$588$	3.23607	0.133453
$589$	−15.5279	−0.639814
$590$	0	0
$591$	3.70820	0.152535
$592$	−3.97871	−0.163524
$593$	−12.5066	−0.513584	−0.256792	−	0.966467i	$-0.582665\pi$
$593$	−12.5066	−0.513584	−0.256792	+	0.966467i	$0.582665\pi$
$594$	0	0
$595$	0	0
$596$	−10.0000	−0.409616
$597$	21.7082	0.888458
$598$	16.1459	0.660255
$599$	25.5279	1.04304	0.521520	−	0.853239i	$-0.325365\pi$
$599$	25.5279	1.04304	0.521520	+	0.853239i	$0.325365\pi$
$600$	0	0
$601$	31.6180	1.28973	0.644863	−	0.764298i	$-0.276914\pi$
$601$	31.6180	1.28973	0.644863	+	0.764298i	$0.276914\pi$
$602$	−11.5623	−0.471244
$603$	−4.70820	−0.191733
$604$	21.3262	0.867752
$605$	0	0
$606$	−6.23607	−0.253323
$607$	22.1459	0.898874	0.449437	−	0.893312i	$-0.351625\pi$
$607$	22.1459	0.898874	0.449437	+	0.893312i	$0.351625\pi$
$608$	−40.6525	−1.64868
$609$	4.14590	0.168000
$610$	0	0
$611$	−56.3951	−2.28150
$612$	2.38197	0.0962853
$613$	−21.9787	−0.887712	−0.443856	−	0.896098i	$-0.646390\pi$
$613$	−21.9787	−0.887712	−0.443856	+	0.896098i	$0.646390\pi$
$614$	12.2918	0.496057
$615$	0	0
$616$	0	0
$617$	−26.9443	−1.08474	−0.542368	−	0.840141i	$-0.682472\pi$
$617$	−26.9443	−1.08474	−0.542368	+	0.840141i	$0.682472\pi$
$618$	4.43769	0.178510
$619$	47.3607	1.90359	0.951793	−	0.306740i	$-0.0992383\pi$
$619$	47.3607	1.90359	0.951793	+	0.306740i	$0.0992383\pi$
$620$	0	0
$621$	5.38197	0.215971
$622$	16.9787	0.680784
$623$	−52.6869	−2.11086
$624$	−9.00000	−0.360288
$625$	0	0
$626$	10.6869	0.427135
$627$	0	0
$628$	−11.0000	−0.438948
$629$	3.15905	0.125960
$630$	0	0
$631$	−26.8197	−1.06767	−0.533837	−	0.845587i	$-0.679250\pi$
$631$	−26.8197	−1.06767	−0.533837	+	0.845587i	$0.679250\pi$
$632$	−28.0902	−1.11737
$633$	−9.18034	−0.364886
$634$	−8.14590	−0.323515
$635$	0	0
$636$	−13.7082	−0.543566
$637$	9.70820	0.384653
$638$	0	0
$639$	−7.47214	−0.295593
$640$	0	0
$641$	−30.1591	−1.19121	−0.595605	−	0.803277i	$-0.703088\pi$
$641$	−30.1591	−1.19121	−0.595605	+	0.803277i	$0.703088\pi$
$642$	−6.43769	−0.254076
$643$	−22.6180	−0.891968	−0.445984	−	0.895041i	$-0.647146\pi$
$643$	−22.6180	−0.891968	−0.445984	+	0.895041i	$0.647146\pi$
$644$	26.1246	1.02945
$645$	0	0
$646$	6.58359	0.259028
$647$	−37.7984	−1.48601	−0.743004	−	0.669287i	$-0.766600\pi$
$647$	−37.7984	−1.48601	−0.743004	+	0.669287i	$0.766600\pi$
$648$	2.23607	0.0878410
$649$	0	0
$650$	0	0
$651$	6.43769	0.252313
$652$	9.56231	0.374489
$653$	−12.9443	−0.506549	−0.253274	−	0.967395i	$-0.581508\pi$
$653$	−12.9443	−0.506549	−0.253274	+	0.967395i	$0.581508\pi$
$654$	−9.27051	−0.362506
$655$	0	0
$656$	−7.85410	−0.306651
$657$	−0.145898	−0.00569202
$658$	21.5410	0.839756
$659$	−14.0689	−0.548046	−0.274023	−	0.961723i	$-0.588354\pi$
$659$	−14.0689	−0.548046	−0.274023	+	0.961723i	$0.588354\pi$
$660$	0	0
$661$	−3.00000	−0.116686	−0.0583432	−	0.998297i	$-0.518582\pi$
$661$	−3.00000	−0.116686	−0.0583432	+	0.998297i	$0.518582\pi$
$662$	−6.43769	−0.250208
$663$	7.14590	0.277524
$664$	28.2148	1.09495
$665$	0	0
$666$	1.32624	0.0513907
$667$	7.43769	0.287989
$668$	4.94427	0.191300
$669$	0.708204	0.0273807
$670$	0	0
$671$	0	0
$672$	16.8541	0.650161
$673$	15.5066	0.597735	0.298867	−	0.954295i	$-0.403391\pi$
$673$	15.5066	0.597735	0.298867	+	0.954295i	$0.403391\pi$
$674$	6.36068	0.245004
$675$	0	0
$676$	−17.0902	−0.657314
$677$	−16.4721	−0.633076	−0.316538	−	0.948580i	$-0.602520\pi$
$677$	−16.4721	−0.633076	−0.316538	+	0.948580i	$0.602520\pi$
$678$	0.291796	0.0112064
$679$	−49.2492	−1.89001
$680$	0	0
$681$	−5.56231	−0.213148
$682$	0	0
$683$	−0.708204	−0.0270987	−0.0135493	−	0.999908i	$-0.504313\pi$
$683$	−0.708204	−0.0270987	−0.0135493	+	0.999908i	$0.504313\pi$
$684$	−11.7082	−0.447674
$685$	0	0
$686$	9.27051	0.353950
$687$	10.0000	0.381524
$688$	11.5623	0.440809
$689$	−41.1246	−1.56672
$690$	0	0
$691$	−11.2918	−0.429560	−0.214780	−	0.976662i	$-0.568903\pi$
$691$	−11.2918	−0.429560	−0.214780	+	0.976662i	$0.568903\pi$
$692$	−5.61803	−0.213566
$693$	0	0
$694$	−2.05573	−0.0780344
$695$	0	0
$696$	3.09017	0.117133
$697$	6.23607	0.236208
$698$	6.83282	0.258626
$699$	−9.32624	−0.352751
$700$	0	0
$701$	−9.76393	−0.368779	−0.184389	−	0.982853i	$-0.559031\pi$
$701$	−9.76393	−0.368779	−0.184389	+	0.982853i	$0.559031\pi$
$702$	3.00000	0.113228
$703$	−15.5279	−0.585644
$704$	0	0
$705$	0	0
$706$	16.6180	0.625428
$707$	−30.2705	−1.13844
$708$	−17.5623	−0.660032
$709$	−0.527864	−0.0198244	−0.00991218	−	0.999951i	$-0.503155\pi$
$709$	−0.527864	−0.0198244	−0.00991218	+	0.999951i	$0.503155\pi$
$710$	0	0
$711$	−12.5623	−0.471123
$712$	−39.2705	−1.47172
$713$	11.5492	0.432519
$714$	−2.72949	−0.102149
$715$	0	0
$716$	−32.5623	−1.21691
$717$	10.6525	0.397824
$718$	−20.5279	−0.766093
$719$	−43.9443	−1.63884	−0.819422	−	0.573190i	$-0.805706\pi$
$719$	−43.9443	−1.63884	−0.819422	+	0.573190i	$0.805706\pi$
$720$	0	0
$721$	21.5410	0.802229
$722$	−20.6180	−0.767324
$723$	1.14590	0.0426164
$724$	−19.0902	−0.709481
$725$	0	0
$726$	0	0
$727$	4.23607	0.157107	0.0785535	−	0.996910i	$-0.474970\pi$
$727$	4.23607	0.157107	0.0785535	+	0.996910i	$0.474970\pi$
$728$	32.5623	1.20684
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.18034	−0.339547
$732$	16.5623	0.612160
$733$	26.5623	0.981101	0.490551	−	0.871413i	$-0.336796\pi$
$733$	26.5623	0.981101	0.490551	+	0.871413i	$0.336796\pi$
$734$	21.5279	0.794608
$735$	0	0
$736$	30.2361	1.11452
$737$	0	0
$738$	2.61803	0.0963712
$739$	−10.7295	−0.394691	−0.197345	−	0.980334i	$-0.563232\pi$
$739$	−10.7295	−0.394691	−0.197345	+	0.980334i	$0.563232\pi$
$740$	0	0
$741$	−35.1246	−1.29033
$742$	15.7082	0.576666
$743$	−14.8197	−0.543681	−0.271840	−	0.962342i	$-0.587632\pi$
$743$	−14.8197	−0.543681	−0.271840	+	0.962342i	$0.587632\pi$
$744$	4.79837	0.175917
$745$	0	0
$746$	5.81966	0.213073
$747$	12.6180	0.461670
$748$	0	0
$749$	−31.2492	−1.14182
$750$	0	0
$751$	−8.52786	−0.311186	−0.155593	−	0.987821i	$-0.549729\pi$
$751$	−8.52786	−0.311186	−0.155593	+	0.987821i	$0.549729\pi$
$752$	−21.5410	−0.785520
$753$	0.437694	0.0159505
$754$	4.14590	0.150985
$755$	0	0
$756$	4.85410	0.176542
$757$	9.96556	0.362204	0.181102	−	0.983464i	$-0.442034\pi$
$757$	9.96556	0.362204	0.181102	+	0.983464i	$0.442034\pi$
$758$	−17.4377	−0.633366
$759$	0	0
$760$	0	0
$761$	35.8885	1.30096	0.650479	−	0.759524i	$-0.274568\pi$
$761$	35.8885	1.30096	0.650479	+	0.759524i	$0.274568\pi$
$762$	2.70820	0.0981079
$763$	−45.0000	−1.62911
$764$	43.2705	1.56547
$765$	0	0
$766$	−18.8328	−0.680457
$767$	−52.6869	−1.90241
$768$	6.56231	0.236797
$769$	30.8541	1.11263	0.556314	−	0.830972i	$-0.312215\pi$
$769$	30.8541	1.11263	0.556314	+	0.830972i	$0.312215\pi$
$770$	0	0
$771$	−19.0344	−0.685509
$772$	−15.4164	−0.554849
$773$	−9.65248	−0.347175	−0.173588	−	0.984818i	$-0.555536\pi$
$773$	−9.65248	−0.347175	−0.173588	+	0.984818i	$0.555536\pi$
$774$	−3.85410	−0.138533
$775$	0	0
$776$	−36.7082	−1.31775
$777$	6.43769	0.230951
$778$	4.27051	0.153105
$779$	−30.6525	−1.09824
$780$	0	0
$781$	0	0
$782$	−4.89667	−0.175105
$783$	1.38197	0.0493874
$784$	3.70820	0.132436
$785$	0	0
$786$	5.14590	0.183548
$787$	−23.3820	−0.833477	−0.416739	−	0.909026i	$-0.636827\pi$
$787$	−23.3820	−0.833477	−0.416739	+	0.909026i	$0.636827\pi$
$788$	6.00000	0.213741
$789$	−29.1246	−1.03686
$790$	0	0
$791$	1.41641	0.0503617
$792$	0	0
$793$	49.6869	1.76443
$794$	0.347524	0.0123332
$795$	0	0
$796$	35.1246	1.24496
$797$	−32.3951	−1.14749	−0.573747	−	0.819033i	$-0.694511\pi$
$797$	−32.3951	−1.14749	−0.573747	+	0.819033i	$0.694511\pi$
$798$	13.4164	0.474936
$799$	17.1033	0.605072
$800$	0	0
$801$	−17.5623	−0.620534
$802$	−14.3262	−0.505877
$803$	0	0
$804$	−7.61803	−0.268667
$805$	0	0
$806$	6.43769	0.226758
$807$	9.47214	0.333435
$808$	−22.5623	−0.793739
$809$	2.23607	0.0786160	0.0393080	−	0.999227i	$-0.487485\pi$
$809$	2.23607	0.0786160	0.0393080	+	0.999227i	$0.487485\pi$
$810$	0	0
$811$	6.29180	0.220935	0.110467	−	0.993880i	$-0.464765\pi$
$811$	6.29180	0.220935	0.110467	+	0.993880i	$0.464765\pi$
$812$	6.70820	0.235412
$813$	11.2705	0.395274
$814$	0	0
$815$	0	0
$816$	2.72949	0.0955513
$817$	45.1246	1.57871
$818$	18.6180	0.650964
$819$	14.5623	0.508848
$820$	0	0
$821$	17.3951	0.607094	0.303547	−	0.952816i	$-0.401829\pi$
$821$	17.3951	0.607094	0.303547	+	0.952816i	$0.401829\pi$
$822$	7.29180	0.254331
$823$	5.47214	0.190747	0.0953733	−	0.995442i	$-0.469596\pi$
$823$	5.47214	0.190747	0.0953733	+	0.995442i	$0.469596\pi$
$824$	16.0557	0.559328
$825$	0	0
$826$	20.1246	0.700225
$827$	−35.8673	−1.24723	−0.623613	−	0.781733i	$-0.714336\pi$
$827$	−35.8673	−1.24723	−0.623613	+	0.781733i	$0.714336\pi$
$828$	8.70820	0.302631
$829$	−9.67376	−0.335984	−0.167992	−	0.985788i	$-0.553728\pi$
$829$	−9.67376	−0.335984	−0.167992	+	0.985788i	$0.553728\pi$
$830$	0	0
$831$	22.9787	0.797123
$832$	−1.14590	−0.0397269
$833$	−2.94427	−0.102013
$834$	0.527864	0.0182784
$835$	0	0
$836$	0	0
$837$	2.14590	0.0741731
$838$	−0.652476	−0.0225394
$839$	0.201626	0.00696091	0.00348045	−	0.999994i	$-0.498892\pi$
$839$	0.201626	0.00696091	0.00348045	+	0.999994i	$0.498892\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	−16.2361	−0.559532
$843$	12.3262	0.424538
$844$	−14.8541	−0.511299
$845$	0	0
$846$	7.18034	0.246865
$847$	0	0
$848$	−15.7082	−0.539422
$849$	−15.9098	−0.546024
$850$	0	0
$851$	11.5492	0.395900
$852$	−12.0902	−0.414202
$853$	6.68692	0.228956	0.114478	−	0.993426i	$-0.463481\pi$
$853$	6.68692	0.228956	0.114478	+	0.993426i	$0.463481\pi$
$854$	−18.9787	−0.649438
$855$	0	0
$856$	−23.2918	−0.796097
$857$	1.09017	0.0372395	0.0186197	−	0.999827i	$-0.494073\pi$
$857$	1.09017	0.0372395	0.0186197	+	0.999827i	$0.494073\pi$
$858$	0	0
$859$	17.5623	0.599218	0.299609	−	0.954062i	$-0.403144\pi$
$859$	17.5623	0.599218	0.299609	+	0.954062i	$0.403144\pi$
$860$	0	0
$861$	12.7082	0.433094
$862$	14.1246	0.481086
$863$	32.3820	1.10229	0.551147	−	0.834408i	$-0.314190\pi$
$863$	32.3820	1.10229	0.551147	+	0.834408i	$0.314190\pi$
$864$	5.61803	0.191129
$865$	0	0
$866$	−2.97871	−0.101221
$867$	14.8328	0.503749
$868$	10.4164	0.353556
$869$	0	0
$870$	0	0
$871$	−22.8541	−0.774382
$872$	−33.5410	−1.13584
$873$	−16.4164	−0.555611
$874$	24.0689	0.814142
$875$	0	0
$876$	−0.236068	−0.00797600
$877$	56.4164	1.90505	0.952523	−	0.304465i	$-0.0984778\pi$
$877$	56.4164	1.90505	0.952523	+	0.304465i	$0.0984778\pi$
$878$	17.0344	0.574885
$879$	−7.41641	−0.250149
$880$	0	0
$881$	−11.8197	−0.398214	−0.199107	−	0.979978i	$-0.563804\pi$
$881$	−11.8197	−0.398214	−0.199107	+	0.979978i	$0.563804\pi$
$882$	−1.23607	−0.0416206
$883$	43.6869	1.47018	0.735091	−	0.677969i	$-0.237139\pi$
$883$	43.6869	1.47018	0.735091	+	0.677969i	$0.237139\pi$
$884$	11.5623	0.388882
$885$	0	0
$886$	6.49342	0.218151
$887$	4.18034	0.140362	0.0701810	−	0.997534i	$-0.477642\pi$
$887$	4.18034	0.140362	0.0701810	+	0.997534i	$0.477642\pi$
$888$	4.79837	0.161023
$889$	13.1459	0.440899
$890$	0	0
$891$	0	0
$892$	1.14590	0.0383675
$893$	−84.0689	−2.81326
$894$	3.81966	0.127749
$895$	0	0
$896$	34.1459	1.14073
$897$	26.1246	0.872275
$898$	−3.41641	−0.114007
$899$	2.96556	0.0989069
$900$	0	0
$901$	12.4721	0.415507
$902$	0	0
$903$	−18.7082	−0.622570
$904$	1.05573	0.0351130
$905$	0	0
$906$	−8.14590	−0.270629
$907$	0.167184	0.00555126	0.00277563	−	0.999996i	$-0.499116\pi$
$907$	0.167184	0.00555126	0.00277563	+	0.999996i	$0.499116\pi$
$908$	−9.00000	−0.298675
$909$	−10.0902	−0.334670
$910$	0	0
$911$	24.9656	0.827146	0.413573	−	0.910471i	$-0.364281\pi$
$911$	24.9656	0.827146	0.413573	+	0.910471i	$0.364281\pi$
$912$	−13.4164	−0.444262
$913$	0	0
$914$	−20.0689	−0.663820
$915$	0	0
$916$	16.1803	0.534613
$917$	24.9787	0.824870
$918$	−0.909830	−0.0300289
$919$	−9.14590	−0.301695	−0.150848	−	0.988557i	$-0.548200\pi$
$919$	−9.14590	−0.301695	−0.150848	+	0.988557i	$0.548200\pi$
$920$	0	0
$921$	19.8885	0.655350
$922$	8.14590	0.268271
$923$	−36.2705	−1.19386
$924$	0	0
$925$	0	0
$926$	−14.0344	−0.461200
$927$	7.18034	0.235833
$928$	7.76393	0.254864
$929$	−45.5279	−1.49372	−0.746860	−	0.664981i	$-0.768440\pi$
$929$	−45.5279	−1.49372	−0.746860	+	0.664981i	$0.768440\pi$
$930$	0	0
$931$	14.4721	0.474305
$932$	−15.0902	−0.494295
$933$	27.4721	0.899397
$934$	−21.0344	−0.688268
$935$	0	0
$936$	10.8541	0.354777
$937$	9.58359	0.313082	0.156541	−	0.987671i	$-0.449966\pi$
$937$	9.58359	0.313082	0.156541	+	0.987671i	$0.449966\pi$
$938$	8.72949	0.285028
$939$	17.2918	0.564296
$940$	0	0
$941$	3.00000	0.0977972	0.0488986	−	0.998804i	$-0.484429\pi$
$941$	3.00000	0.0977972	0.0488986	+	0.998804i	$0.484429\pi$
$942$	4.20163	0.136896
$943$	22.7984	0.742417
$944$	−20.1246	−0.655000
$945$	0	0
$946$	0	0
$947$	−59.5066	−1.93370	−0.966852	−	0.255338i	$-0.917813\pi$
$947$	−59.5066	−1.93370	−0.966852	+	0.255338i	$0.917813\pi$
$948$	−20.3262	−0.660166
$949$	−0.708204	−0.0229893
$950$	0	0
$951$	−13.1803	−0.427402
$952$	−9.87539	−0.320063
$953$	33.7214	1.09234	0.546171	−	0.837674i	$-0.316085\pi$
$953$	33.7214	1.09234	0.546171	+	0.837674i	$0.316085\pi$
$954$	5.23607	0.169524
$955$	0	0
$956$	17.2361	0.557454
$957$	0	0
$958$	−23.0902	−0.746010
$959$	35.3951	1.14297
$960$	0	0
$961$	−26.3951	−0.851456
$962$	6.43769	0.207560
$963$	−10.4164	−0.335664
$964$	1.85410	0.0597166
$965$	0	0
$966$	−9.97871	−0.321060
$967$	46.2148	1.48617	0.743084	−	0.669199i	$-0.233362\pi$
$967$	46.2148	1.48617	0.743084	+	0.669199i	$0.233362\pi$
$968$	0	0
$969$	10.6525	0.342207
$970$	0	0
$971$	−25.4853	−0.817862	−0.408931	−	0.912565i	$-0.634098\pi$
$971$	−25.4853	−0.817862	−0.408931	+	0.912565i	$0.634098\pi$
$972$	1.61803	0.0518985
$973$	2.56231	0.0821438
$974$	19.9443	0.639056
$975$	0	0
$976$	18.9787	0.607494
$977$	2.72949	0.0873241	0.0436621	−	0.999046i	$-0.486098\pi$
$977$	2.72949	0.0873241	0.0436621	+	0.999046i	$0.486098\pi$
$978$	−3.65248	−0.116793
$979$	0	0
$980$	0	0
$981$	−15.0000	−0.478913
$982$	−4.41641	−0.140933
$983$	41.1246	1.31167	0.655836	−	0.754904i	$-0.272316\pi$
$983$	41.1246	1.31167	0.655836	+	0.754904i	$0.272316\pi$
$984$	9.47214	0.301961
$985$	0	0
$986$	−1.25735	−0.0400423
$987$	34.8541	1.10942
$988$	−56.8328	−1.80809
$989$	−33.5623	−1.06722
$990$	0	0
$991$	7.45085	0.236684	0.118342	−	0.992973i	$-0.462242\pi$
$991$	7.45085	0.236684	0.118342	+	0.992973i	$0.462242\pi$
$992$	12.0557	0.382770
$993$	−10.4164	−0.330555
$994$	13.8541	0.439425
$995$	0	0
$996$	20.4164	0.646919
$997$	1.29180	0.0409116	0.0204558	−	0.999791i	$-0.493488\pi$
$997$	1.29180	0.0409116	0.0204558	+	0.999791i	$0.493488\pi$
$998$	25.8541	0.818397
$999$	2.14590	0.0678932

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.bu.1.1		2
5.4	even	2		9075.2.a.bb.1.2		2
11.7	odd	10		825.2.n.e.676.1	yes	4
11.8	odd	10		825.2.n.e.526.1	yes	4
11.10	odd	2		9075.2.a.y.1.2		2
55.7	even	20		825.2.bx.a.49.1		8
55.8	even	20		825.2.bx.a.724.1		8
55.18	even	20		825.2.bx.a.49.2		8
55.19	odd	10		825.2.n.a.526.1	✓	4
55.29	odd	10		825.2.n.a.676.1	yes	4
55.52	even	20		825.2.bx.a.724.2		8
55.54	odd	2		9075.2.a.bz.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.n.a.526.1	✓	4	55.19	odd	10
825.2.n.a.676.1	yes	4	55.29	odd	10
825.2.n.e.526.1	yes	4	11.8	odd	10
825.2.n.e.676.1	yes	4	11.7	odd	10
825.2.bx.a.49.1		8	55.7	even	20
825.2.bx.a.49.2		8	55.18	even	20
825.2.bx.a.724.1		8	55.8	even	20
825.2.bx.a.724.2		8	55.52	even	20
9075.2.a.y.1.2		2	11.10	odd	2
9075.2.a.bb.1.2		2	5.4	even	2
9075.2.a.bu.1.1		2	1.1	even	1	trivial
9075.2.a.bz.1.1		2	55.54	odd	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)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +3.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +0.618034 q^{6} +3.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +1.61803 q^{12} +4.85410 q^{13} -1.85410 q^{14} +1.85410 q^{16} -1.47214 q^{17} -0.618034 q^{18} +7.23607 q^{19} -3.00000 q^{21} -5.38197 q^{23} -2.23607 q^{24} -3.00000 q^{26} -1.00000 q^{27} -4.85410 q^{28} -1.38197 q^{29} -2.14590 q^{31} -5.61803 q^{32} +0.909830 q^{34} -1.61803 q^{36} -2.14590 q^{37} -4.47214 q^{38} -4.85410 q^{39} -4.23607 q^{41} +1.85410 q^{42} +6.23607 q^{43} +3.32624 q^{46} -11.6180 q^{47} -1.85410 q^{48} +2.00000 q^{49} +1.47214 q^{51} -7.85410 q^{52} -8.47214 q^{53} +0.618034 q^{54} +6.70820 q^{56} -7.23607 q^{57} +0.854102 q^{58} -10.8541 q^{59} +10.2361 q^{61} +1.32624 q^{62} +3.00000 q^{63} -0.236068 q^{64} -4.70820 q^{67} +2.38197 q^{68} +5.38197 q^{69} -7.47214 q^{71} +2.23607 q^{72} -0.145898 q^{73} +1.32624 q^{74} -11.7082 q^{76} +3.00000 q^{78} -12.5623 q^{79} +1.00000 q^{81} +2.61803 q^{82} +12.6180 q^{83} +4.85410 q^{84} -3.85410 q^{86} +1.38197 q^{87} -17.5623 q^{89} +14.5623 q^{91} +8.70820 q^{92} +2.14590 q^{93} +7.18034 q^{94} +5.61803 q^{96} -16.4164 q^{97} -1.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9	\( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9} + q^{12} + 3 q^{13} + 3 q^{14} - 3 q^{16} + 6 q^{17} + q^{18} + 10 q^{19} - 6 q^{21} - 13 q^{23} - 6 q^{26} - 2 q^{27} - 3 q^{28} - 5 q^{29} - 11 q^{31} - 9 q^{32} + 13 q^{34} - q^{36} - 11 q^{37} - 3 q^{39} - 4 q^{41} - 3 q^{42} + 8 q^{43} - 9 q^{46} - 21 q^{47} + 3 q^{48} + 4 q^{49} - 6 q^{51} - 9 q^{52} - 8 q^{53} - q^{54} - 10 q^{57} - 5 q^{58} - 15 q^{59} + 16 q^{61} - 13 q^{62} + 6 q^{63} + 4 q^{64} + 4 q^{67} + 7 q^{68} + 13 q^{69} - 6 q^{71} - 7 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{78} - 5 q^{79} + 2 q^{81} + 3 q^{82} + 23 q^{83} + 3 q^{84} - q^{86} + 5 q^{87} - 15 q^{89} + 9 q^{91} + 4 q^{92} + 11 q^{93} - 8 q^{94} + 9 q^{96} - 6 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 + q^12 + 3 * q^13 + 3 * q^14 - 3 * q^16 + 6 * q^17 + q^18 + 10 * q^19 - 6 * q^21 - 13 * q^23 - 6 * q^26 - 2 * q^27 - 3 * q^28 - 5 * q^29 - 11 * q^31 - 9 * q^32 + 13 * q^34 - q^36 - 11 * q^37 - 3 * q^39 - 4 * q^41 - 3 * q^42 + 8 * q^43 - 9 * q^46 - 21 * q^47 + 3 * q^48 + 4 * q^49 - 6 * q^51 - 9 * q^52 - 8 * q^53 - q^54 - 10 * q^57 - 5 * q^58 - 15 * q^59 + 16 * q^61 - 13 * q^62 + 6 * q^63 + 4 * q^64 + 4 * q^67 + 7 * q^68 + 13 * q^69 - 6 * q^71 - 7 * q^73 - 13 * q^74 - 10 * q^76 + 6 * q^78 - 5 * q^79 + 2 * q^81 + 3 * q^82 + 23 * q^83 + 3 * q^84 - q^86 + 5 * q^87 - 15 * q^89 + 9 * q^91 + 4 * q^92 + 11 * q^93 - 8 * q^94 + 9 * q^96 - 6 * q^97 + 2 * q^98

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