LMFDB - Embedded newform 9075.2.a.ba.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:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -0.618034 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -0.618034 q^{7} +2.23607 q^{8} +1.00000 q^{9} -0.618034 q^{12} +0.381966 q^{13} +1.00000 q^{14} -4.85410 q^{16} +0.236068 q^{17} -1.61803 q^{18} -4.23607 q^{19} +0.618034 q^{21} -2.85410 q^{23} -2.23607 q^{24} -0.618034 q^{26} -1.00000 q^{27} -0.381966 q^{28} -0.236068 q^{29} -1.14590 q^{31} +3.38197 q^{32} -0.381966 q^{34} +0.618034 q^{36} +8.94427 q^{37} +6.85410 q^{38} -0.381966 q^{39} +5.09017 q^{41} -1.00000 q^{42} +4.47214 q^{43} +4.61803 q^{46} +11.7082 q^{47} +4.85410 q^{48} -6.61803 q^{49} -0.236068 q^{51} +0.236068 q^{52} -0.0901699 q^{53} +1.61803 q^{54} -1.38197 q^{56} +4.23607 q^{57} +0.381966 q^{58} -4.70820 q^{59} +5.23607 q^{61} +1.85410 q^{62} -0.618034 q^{63} +4.23607 q^{64} -5.23607 q^{67} +0.145898 q^{68} +2.85410 q^{69} -6.23607 q^{71} +2.23607 q^{72} -5.61803 q^{73} -14.4721 q^{74} -2.61803 q^{76} +0.618034 q^{78} -3.70820 q^{79} +1.00000 q^{81} -8.23607 q^{82} -0.854102 q^{83} +0.381966 q^{84} -7.23607 q^{86} +0.236068 q^{87} -14.3262 q^{89} -0.236068 q^{91} -1.76393 q^{92} +1.14590 q^{93} -18.9443 q^{94} -3.38197 q^{96} -14.1803 q^{97} +10.7082 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 - 2 * q^3 - q^4 + q^6 + q^7 + 2 * q^9	$ 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} + q^{7} + 2 q^{9} + q^{12} + 3 q^{13} + 2 q^{14} - 3 q^{16} - 4 q^{17} - q^{18} - 4 q^{19} - q^{21} + q^{23} + q^{26} - 2 q^{27} - 3 q^{28} + 4 q^{29} - 9 q^{31} + 9 q^{32} - 3 q^{34} - q^{36} + 7 q^{38} - 3 q^{39} - q^{41} - 2 q^{42} + 7 q^{46} + 10 q^{47} + 3 q^{48} - 11 q^{49} + 4 q^{51} - 4 q^{52} + 11 q^{53} + q^{54} - 5 q^{56} + 4 q^{57} + 3 q^{58} + 4 q^{59} + 6 q^{61} - 3 q^{62} + q^{63} + 4 q^{64} - 6 q^{67} + 7 q^{68} - q^{69} - 8 q^{71} - 9 q^{73} - 20 q^{74} - 3 q^{76} - q^{78} + 6 q^{79} + 2 q^{81} - 12 q^{82} + 5 q^{83} + 3 q^{84} - 10 q^{86} - 4 q^{87} - 13 q^{89} + 4 q^{91} - 8 q^{92} + 9 q^{93} - 20 q^{94} - 9 q^{96} - 6 q^{97} + 8 q^{98}+O(q^{100}) $ 2 * q - q^2 - 2 * q^3 - q^4 + q^6 + q^7 + 2 * q^9 + q^12 + 3 * q^13 + 2 * q^14 - 3 * q^16 - 4 * q^17 - q^18 - 4 * q^19 - q^21 + q^23 + q^26 - 2 * q^27 - 3 * q^28 + 4 * q^29 - 9 * q^31 + 9 * q^32 - 3 * q^34 - q^36 + 7 * q^38 - 3 * q^39 - q^41 - 2 * q^42 + 7 * q^46 + 10 * q^47 + 3 * q^48 - 11 * q^49 + 4 * q^51 - 4 * q^52 + 11 * q^53 + q^54 - 5 * q^56 + 4 * q^57 + 3 * q^58 + 4 * q^59 + 6 * q^61 - 3 * q^62 + q^63 + 4 * q^64 - 6 * q^67 + 7 * q^68 - q^69 - 8 * q^71 - 9 * q^73 - 20 * q^74 - 3 * q^76 - q^78 + 6 * q^79 + 2 * q^81 - 12 * q^82 + 5 * q^83 + 3 * q^84 - 10 * q^86 - 4 * q^87 - 13 * q^89 + 4 * q^91 - 8 * q^92 + 9 * q^93 - 20 * q^94 - 9 * q^96 - 6 * q^97 + 8 * 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	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	−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
$7$	−0.618034	−0.233595	−0.116797	−	0.993156i	$-0.537263\pi$
$7$	−0.618034	−0.233595	−0.116797	+	0.993156i	$0.537263\pi$
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−0.618034	−0.178411
$13$	0.381966	0.105938	0.0529692	−	0.998596i	$-0.483131\pi$
$13$	0.381966	0.105938	0.0529692	+	0.998596i	$0.483131\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	−4.85410	−1.21353
$17$	0.236068	0.0572549	0.0286274	−	0.999590i	$-0.490886\pi$
$17$	0.236068	0.0572549	0.0286274	+	0.999590i	$0.490886\pi$
$18$	−1.61803	−0.381374
$19$	−4.23607	−0.971821	−0.485910	−	0.874009i	$-0.661512\pi$
$19$	−4.23607	−0.971821	−0.485910	+	0.874009i	$0.661512\pi$
$20$	0	0
$21$	0.618034	0.134866
$22$	0	0
$23$	−2.85410	−0.595121	−0.297561	−	0.954703i	$-0.596173\pi$
$23$	−2.85410	−0.595121	−0.297561	+	0.954703i	$0.596173\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	−0.618034	−0.121206
$27$	−1.00000	−0.192450
$28$	−0.381966	−0.0721848
$29$	−0.236068	−0.0438367	−0.0219184	−	0.999760i	$-0.506977\pi$
$29$	−0.236068	−0.0438367	−0.0219184	+	0.999760i	$0.506977\pi$
$30$	0	0
$31$	−1.14590	−0.205809	−0.102905	−	0.994691i	$-0.532814\pi$
$31$	−1.14590	−0.205809	−0.102905	+	0.994691i	$0.532814\pi$
$32$	3.38197	0.597853
$33$	0	0
$34$	−0.381966	−0.0655066
$35$	0	0
$36$	0.618034	0.103006
$37$	8.94427	1.47043	0.735215	−	0.677834i	$-0.237081\pi$
$37$	8.94427	1.47043	0.735215	+	0.677834i	$0.237081\pi$
$38$	6.85410	1.11188
$39$	−0.381966	−0.0611635
$40$	0	0
$41$	5.09017	0.794951	0.397475	−	0.917613i	$-0.369886\pi$
$41$	5.09017	0.794951	0.397475	+	0.917613i	$0.369886\pi$
$42$	−1.00000	−0.154303
$43$	4.47214	0.681994	0.340997	−	0.940064i	$-0.389235\pi$
$43$	4.47214	0.681994	0.340997	+	0.940064i	$0.389235\pi$
$44$	0	0
$45$	0	0
$46$	4.61803	0.680892
$47$	11.7082	1.70782	0.853909	−	0.520423i	$-0.174226\pi$
$47$	11.7082	1.70782	0.853909	+	0.520423i	$0.174226\pi$
$48$	4.85410	0.700629
$49$	−6.61803	−0.945433
$50$	0	0
$51$	−0.236068	−0.0330561
$52$	0.236068	0.0327367
$53$	−0.0901699	−0.0123858	−0.00619290	−	0.999981i	$-0.501971\pi$
$53$	−0.0901699	−0.0123858	−0.00619290	+	0.999981i	$0.501971\pi$
$54$	1.61803	0.220187
$55$	0	0
$56$	−1.38197	−0.184673
$57$	4.23607	0.561081
$58$	0.381966	0.0501546
$59$	−4.70820	−0.612956	−0.306478	−	0.951878i	$-0.599151\pi$
$59$	−4.70820	−0.612956	−0.306478	+	0.951878i	$0.599151\pi$
$60$	0	0
$61$	5.23607	0.670410	0.335205	−	0.942145i	$-0.391194\pi$
$61$	5.23607	0.670410	0.335205	+	0.942145i	$0.391194\pi$
$62$	1.85410	0.235471
$63$	−0.618034	−0.0778650
$64$	4.23607	0.529508
$65$	0	0
$66$	0	0
$67$	−5.23607	−0.639688	−0.319844	−	0.947470i	$-0.603630\pi$
$67$	−5.23607	−0.639688	−0.319844	+	0.947470i	$0.603630\pi$
$68$	0.145898	0.0176927
$69$	2.85410	0.343594
$70$	0	0
$71$	−6.23607	−0.740085	−0.370043	−	0.929015i	$-0.620657\pi$
$71$	−6.23607	−0.740085	−0.370043	+	0.929015i	$0.620657\pi$
$72$	2.23607	0.263523
$73$	−5.61803	−0.657541	−0.328771	−	0.944410i	$-0.606634\pi$
$73$	−5.61803	−0.657541	−0.328771	+	0.944410i	$0.606634\pi$
$74$	−14.4721	−1.68235
$75$	0	0
$76$	−2.61803	−0.300309
$77$	0	0
$78$	0.618034	0.0699786
$79$	−3.70820	−0.417206	−0.208603	−	0.978000i	$-0.566892\pi$
$79$	−3.70820	−0.417206	−0.208603	+	0.978000i	$0.566892\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−8.23607	−0.909522
$83$	−0.854102	−0.0937499	−0.0468749	−	0.998901i	$-0.514926\pi$
$83$	−0.854102	−0.0937499	−0.0468749	+	0.998901i	$0.514926\pi$
$84$	0.381966	0.0416759
$85$	0	0
$86$	−7.23607	−0.780285
$87$	0.236068	0.0253091
$88$	0	0
$89$	−14.3262	−1.51858	−0.759289	−	0.650753i	$-0.774453\pi$
$89$	−14.3262	−1.51858	−0.759289	+	0.650753i	$0.774453\pi$
$90$	0	0
$91$	−0.236068	−0.0247466
$92$	−1.76393	−0.183903
$93$	1.14590	0.118824
$94$	−18.9443	−1.95395
$95$	0	0
$96$	−3.38197	−0.345170
$97$	−14.1803	−1.43980	−0.719898	−	0.694080i	$-0.755811\pi$
$97$	−14.1803	−1.43980	−0.719898	+	0.694080i	$0.755811\pi$
$98$	10.7082	1.08169
$99$	0	0
$100$	0	0
$101$	−12.0902	−1.20302	−0.601508	−	0.798866i	$-0.705433\pi$
$101$	−12.0902	−1.20302	−0.601508	+	0.798866i	$0.705433\pi$
$102$	0.381966	0.0378203
$103$	3.52786	0.347611	0.173805	−	0.984780i	$-0.444394\pi$
$103$	3.52786	0.347611	0.173805	+	0.984780i	$0.444394\pi$
$104$	0.854102	0.0837516
$105$	0	0
$106$	0.145898	0.0141709
$107$	16.7984	1.62396	0.811980	−	0.583685i	$-0.198390\pi$
$107$	16.7984	1.62396	0.811980	+	0.583685i	$0.198390\pi$
$108$	−0.618034	−0.0594703
$109$	−5.18034	−0.496187	−0.248093	−	0.968736i	$-0.579804\pi$
$109$	−5.18034	−0.496187	−0.248093	+	0.968736i	$0.579804\pi$
$110$	0	0
$111$	−8.94427	−0.848953
$112$	3.00000	0.283473
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	−6.85410	−0.641945
$115$	0	0
$116$	−0.145898	−0.0135463
$117$	0.381966	0.0353128
$118$	7.61803	0.701297
$119$	−0.145898	−0.0133745
$120$	0	0
$121$	0	0
$122$	−8.47214	−0.767031
$123$	−5.09017	−0.458965
$124$	−0.708204	−0.0635986
$125$	0	0
$126$	1.00000	0.0890871
$127$	−9.00000	−0.798621	−0.399310	−	0.916816i	$-0.630750\pi$
$127$	−9.00000	−0.798621	−0.399310	+	0.916816i	$0.630750\pi$
$128$	−13.6180	−1.20368
$129$	−4.47214	−0.393750
$130$	0	0
$131$	16.5066	1.44219	0.721093	−	0.692838i	$-0.243640\pi$
$131$	16.5066	1.44219	0.721093	+	0.692838i	$0.243640\pi$
$132$	0	0
$133$	2.61803	0.227012
$134$	8.47214	0.731881
$135$	0	0
$136$	0.527864	0.0452640
$137$	−18.4721	−1.57818	−0.789091	−	0.614277i	$-0.789448\pi$
$137$	−18.4721	−1.57818	−0.789091	+	0.614277i	$0.789448\pi$
$138$	−4.61803	−0.393113
$139$	9.14590	0.775745	0.387872	−	0.921713i	$-0.373210\pi$
$139$	9.14590	0.775745	0.387872	+	0.921713i	$0.373210\pi$
$140$	0	0
$141$	−11.7082	−0.986009
$142$	10.0902	0.846748
$143$	0	0
$144$	−4.85410	−0.404508
$145$	0	0
$146$	9.09017	0.752308
$147$	6.61803	0.545846
$148$	5.52786	0.454388
$149$	−11.1459	−0.913108	−0.456554	−	0.889696i	$-0.650916\pi$
$149$	−11.1459	−0.913108	−0.456554	+	0.889696i	$0.650916\pi$
$150$	0	0
$151$	19.2705	1.56821	0.784106	−	0.620627i	$-0.213122\pi$
$151$	19.2705	1.56821	0.784106	+	0.620627i	$0.213122\pi$
$152$	−9.47214	−0.768292
$153$	0.236068	0.0190850
$154$	0	0
$155$	0	0
$156$	−0.236068	−0.0189006
$157$	18.7082	1.49308	0.746539	−	0.665342i	$-0.231714\pi$
$157$	18.7082	1.49308	0.746539	+	0.665342i	$0.231714\pi$
$158$	6.00000	0.477334
$159$	0.0901699	0.00715094
$160$	0	0
$161$	1.76393	0.139017
$162$	−1.61803	−0.127125
$163$	14.0902	1.10363	0.551814	−	0.833967i	$-0.313936\pi$
$163$	14.0902	1.10363	0.551814	+	0.833967i	$0.313936\pi$
$164$	3.14590	0.245653
$165$	0	0
$166$	1.38197	0.107261
$167$	9.03444	0.699106	0.349553	−	0.936917i	$-0.386333\pi$
$167$	9.03444	0.699106	0.349553	+	0.936917i	$0.386333\pi$
$168$	1.38197	0.106621
$169$	−12.8541	−0.988777
$170$	0	0
$171$	−4.23607	−0.323940
$172$	2.76393	0.210748
$173$	17.9443	1.36428	0.682139	−	0.731223i	$-0.261050\pi$
$173$	17.9443	1.36428	0.682139	+	0.731223i	$0.261050\pi$
$174$	−0.381966	−0.0289568
$175$	0	0
$176$	0	0
$177$	4.70820	0.353890
$178$	23.1803	1.73744
$179$	0.0557281	0.00416531	0.00208266	−	0.999998i	$-0.499337\pi$
$179$	0.0557281	0.00416531	0.00208266	+	0.999998i	$0.499337\pi$
$180$	0	0
$181$	−21.4164	−1.59187	−0.795935	−	0.605383i	$-0.793020\pi$
$181$	−21.4164	−1.59187	−0.795935	+	0.605383i	$0.793020\pi$
$182$	0.381966	0.0283132
$183$	−5.23607	−0.387061
$184$	−6.38197	−0.470485
$185$	0	0
$186$	−1.85410	−0.135949
$187$	0	0
$188$	7.23607	0.527744
$189$	0.618034	0.0449554
$190$	0	0
$191$	1.79837	0.130126	0.0650629	−	0.997881i	$-0.479275\pi$
$191$	1.79837	0.130126	0.0650629	+	0.997881i	$0.479275\pi$
$192$	−4.23607	−0.305712
$193$	21.8885	1.57557	0.787786	−	0.615949i	$-0.211227\pi$
$193$	21.8885	1.57557	0.787786	+	0.615949i	$0.211227\pi$
$194$	22.9443	1.64730
$195$	0	0
$196$	−4.09017	−0.292155
$197$	16.1459	1.15035	0.575174	−	0.818031i	$-0.304934\pi$
$197$	16.1459	1.15035	0.575174	+	0.818031i	$0.304934\pi$
$198$	0	0
$199$	4.18034	0.296336	0.148168	−	0.988962i	$-0.452662\pi$
$199$	4.18034	0.296336	0.148168	+	0.988962i	$0.452662\pi$
$200$	0	0
$201$	5.23607	0.369324
$202$	19.5623	1.37640
$203$	0.145898	0.0102400
$204$	−0.145898	−0.0102149
$205$	0	0
$206$	−5.70820	−0.397709
$207$	−2.85410	−0.198374
$208$	−1.85410	−0.128559
$209$	0	0
$210$	0	0
$211$	13.1803	0.907372	0.453686	−	0.891162i	$-0.350109\pi$
$211$	13.1803	0.907372	0.453686	+	0.891162i	$0.350109\pi$
$212$	−0.0557281	−0.00382742
$213$	6.23607	0.427288
$214$	−27.1803	−1.85801
$215$	0	0
$216$	−2.23607	−0.152145
$217$	0.708204	0.0480760
$218$	8.38197	0.567698
$219$	5.61803	0.379632
$220$	0	0
$221$	0.0901699	0.00606549
$222$	14.4721	0.971306
$223$	15.6525	1.04817	0.524084	−	0.851667i	$-0.324408\pi$
$223$	15.6525	1.04817	0.524084	+	0.851667i	$0.324408\pi$
$224$	−2.09017	−0.139655
$225$	0	0
$226$	−6.47214	−0.430520
$227$	−4.70820	−0.312494	−0.156247	−	0.987718i	$-0.549940\pi$
$227$	−4.70820	−0.312494	−0.156247	+	0.987718i	$0.549940\pi$
$228$	2.61803	0.173384
$229$	16.9787	1.12198	0.560992	−	0.827821i	$-0.310420\pi$
$229$	16.9787	1.12198	0.560992	+	0.827821i	$0.310420\pi$
$230$	0	0
$231$	0	0
$232$	−0.527864	−0.0346560
$233$	−4.67376	−0.306188	−0.153094	−	0.988212i	$-0.548924\pi$
$233$	−4.67376	−0.306188	−0.153094	+	0.988212i	$0.548924\pi$
$234$	−0.618034	−0.0404021
$235$	0	0
$236$	−2.90983	−0.189414
$237$	3.70820	0.240874
$238$	0.236068	0.0153020
$239$	−19.5279	−1.26315	−0.631576	−	0.775314i	$-0.717592\pi$
$239$	−19.5279	−1.26315	−0.631576	+	0.775314i	$0.717592\pi$
$240$	0	0
$241$	−2.23607	−0.144038	−0.0720189	−	0.997403i	$-0.522944\pi$
$241$	−2.23607	−0.144038	−0.0720189	+	0.997403i	$0.522944\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	3.23607	0.207168
$245$	0	0
$246$	8.23607	0.525113
$247$	−1.61803	−0.102953
$248$	−2.56231	−0.162707
$249$	0.854102	0.0541265
$250$	0	0
$251$	−24.4164	−1.54115	−0.770575	−	0.637349i	$-0.780031\pi$
$251$	−24.4164	−1.54115	−0.770575	+	0.637349i	$0.780031\pi$
$252$	−0.381966	−0.0240616
$253$	0	0
$254$	14.5623	0.913720
$255$	0	0
$256$	13.5623	0.847644
$257$	−8.90983	−0.555780	−0.277890	−	0.960613i	$-0.589635\pi$
$257$	−8.90983	−0.555780	−0.277890	+	0.960613i	$0.589635\pi$
$258$	7.23607	0.450498
$259$	−5.52786	−0.343485
$260$	0	0
$261$	−0.236068	−0.0146122
$262$	−26.7082	−1.65004
$263$	−16.4164	−1.01228	−0.506140	−	0.862452i	$-0.668928\pi$
$263$	−16.4164	−1.01228	−0.506140	+	0.862452i	$0.668928\pi$
$264$	0	0
$265$	0	0
$266$	−4.23607	−0.259730
$267$	14.3262	0.876752
$268$	−3.23607	−0.197674
$269$	−6.20163	−0.378120	−0.189060	−	0.981966i	$-0.560544\pi$
$269$	−6.20163	−0.378120	−0.189060	+	0.981966i	$0.560544\pi$
$270$	0	0
$271$	27.6180	1.67768	0.838838	−	0.544381i	$-0.183235\pi$
$271$	27.6180	1.67768	0.838838	+	0.544381i	$0.183235\pi$
$272$	−1.14590	−0.0694803
$273$	0.236068	0.0142875
$274$	29.8885	1.80563
$275$	0	0
$276$	1.76393	0.106176
$277$	−7.65248	−0.459793	−0.229896	−	0.973215i	$-0.573839\pi$
$277$	−7.65248	−0.459793	−0.229896	+	0.973215i	$0.573839\pi$
$278$	−14.7984	−0.887547
$279$	−1.14590	−0.0686031
$280$	0	0
$281$	12.0902	0.721239	0.360620	−	0.932713i	$-0.382565\pi$
$281$	12.0902	0.721239	0.360620	+	0.932713i	$0.382565\pi$
$282$	18.9443	1.12811
$283$	21.3262	1.26771	0.633857	−	0.773451i	$-0.281471\pi$
$283$	21.3262	1.26771	0.633857	+	0.773451i	$0.281471\pi$
$284$	−3.85410	−0.228699
$285$	0	0
$286$	0	0
$287$	−3.14590	−0.185696
$288$	3.38197	0.199284
$289$	−16.9443	−0.996722
$290$	0	0
$291$	14.1803	0.831266
$292$	−3.47214	−0.203191
$293$	−13.2361	−0.773259	−0.386630	−	0.922235i	$-0.626361\pi$
$293$	−13.2361	−0.773259	−0.386630	+	0.922235i	$0.626361\pi$
$294$	−10.7082	−0.624515
$295$	0	0
$296$	20.0000	1.16248
$297$	0	0
$298$	18.0344	1.04471
$299$	−1.09017	−0.0630462
$300$	0	0
$301$	−2.76393	−0.159310
$302$	−31.1803	−1.79423
$303$	12.0902	0.694562
$304$	20.5623	1.17933
$305$	0	0
$306$	−0.381966	−0.0218355
$307$	14.9443	0.852915	0.426457	−	0.904508i	$-0.359761\pi$
$307$	14.9443	0.852915	0.426457	+	0.904508i	$0.359761\pi$
$308$	0	0
$309$	−3.52786	−0.200693
$310$	0	0
$311$	−6.61803	−0.375274	−0.187637	−	0.982238i	$-0.560083\pi$
$311$	−6.61803	−0.375274	−0.187637	+	0.982238i	$0.560083\pi$
$312$	−0.854102	−0.0483540
$313$	−14.5967	−0.825057	−0.412528	−	0.910945i	$-0.635354\pi$
$313$	−14.5967	−0.825057	−0.412528	+	0.910945i	$0.635354\pi$
$314$	−30.2705	−1.70826
$315$	0	0
$316$	−2.29180	−0.128924
$317$	−9.18034	−0.515619	−0.257810	−	0.966196i	$-0.583001\pi$
$317$	−9.18034	−0.515619	−0.257810	+	0.966196i	$0.583001\pi$
$318$	−0.145898	−0.00818156
$319$	0	0
$320$	0	0
$321$	−16.7984	−0.937594
$322$	−2.85410	−0.159053
$323$	−1.00000	−0.0556415
$324$	0.618034	0.0343352
$325$	0	0
$326$	−22.7984	−1.26269
$327$	5.18034	0.286473
$328$	11.3820	0.628464
$329$	−7.23607	−0.398937
$330$	0	0
$331$	26.8885	1.47793	0.738964	−	0.673745i	$-0.235315\pi$
$331$	26.8885	1.47793	0.738964	+	0.673745i	$0.235315\pi$
$332$	−0.527864	−0.0289703
$333$	8.94427	0.490143
$334$	−14.6180	−0.799863
$335$	0	0
$336$	−3.00000	−0.163663
$337$	−0.888544	−0.0484021	−0.0242010	−	0.999707i	$-0.507704\pi$
$337$	−0.888544	−0.0484021	−0.0242010	+	0.999707i	$0.507704\pi$
$338$	20.7984	1.13128
$339$	−4.00000	−0.217250
$340$	0	0
$341$	0	0
$342$	6.85410	0.370627
$343$	8.41641	0.454443
$344$	10.0000	0.539164
$345$	0	0
$346$	−29.0344	−1.56090
$347$	−1.96556	−0.105517	−0.0527583	−	0.998607i	$-0.516801\pi$
$347$	−1.96556	−0.105517	−0.0527583	+	0.998607i	$0.516801\pi$
$348$	0.145898	0.00782096
$349$	−29.5623	−1.58243	−0.791217	−	0.611536i	$-0.790552\pi$
$349$	−29.5623	−1.58243	−0.791217	+	0.611536i	$0.790552\pi$
$350$	0	0
$351$	−0.381966	−0.0203878
$352$	0	0
$353$	−27.8328	−1.48139	−0.740696	−	0.671841i	$-0.765504\pi$
$353$	−27.8328	−1.48139	−0.740696	+	0.671841i	$0.765504\pi$
$354$	−7.61803	−0.404894
$355$	0	0
$356$	−8.85410	−0.469266
$357$	0.145898	0.00772174
$358$	−0.0901699	−0.00476563
$359$	36.7082	1.93738	0.968692	−	0.248264i	$-0.0798600\pi$
$359$	36.7082	1.93738	0.968692	+	0.248264i	$0.0798600\pi$
$360$	0	0
$361$	−1.05573	−0.0555646
$362$	34.6525	1.82129
$363$	0	0
$364$	−0.145898	−0.00764713
$365$	0	0
$366$	8.47214	0.442846
$367$	8.52786	0.445151	0.222575	−	0.974915i	$-0.428554\pi$
$367$	8.52786	0.445151	0.222575	+	0.974915i	$0.428554\pi$
$368$	13.8541	0.722195
$369$	5.09017	0.264984
$370$	0	0
$371$	0.0557281	0.00289326
$372$	0.708204	0.0367187
$373$	−34.0902	−1.76512	−0.882561	−	0.470198i	$-0.844183\pi$
$373$	−34.0902	−1.76512	−0.882561	+	0.470198i	$0.844183\pi$
$374$	0	0
$375$	0	0
$376$	26.1803	1.35015
$377$	−0.0901699	−0.00464399
$378$	−1.00000	−0.0514344
$379$	−20.6180	−1.05908	−0.529539	−	0.848286i	$-0.677635\pi$
$379$	−20.6180	−1.05908	−0.529539	+	0.848286i	$0.677635\pi$
$380$	0	0
$381$	9.00000	0.461084
$382$	−2.90983	−0.148880
$383$	−10.5623	−0.539709	−0.269854	−	0.962901i	$-0.586976\pi$
$383$	−10.5623	−0.539709	−0.269854	+	0.962901i	$0.586976\pi$
$384$	13.6180	0.694942
$385$	0	0
$386$	−35.4164	−1.80265
$387$	4.47214	0.227331
$388$	−8.76393	−0.444921
$389$	4.47214	0.226746	0.113373	−	0.993552i	$-0.463834\pi$
$389$	4.47214	0.226746	0.113373	+	0.993552i	$0.463834\pi$
$390$	0	0
$391$	−0.673762	−0.0340736
$392$	−14.7984	−0.747431
$393$	−16.5066	−0.832647
$394$	−26.1246	−1.31614
$395$	0	0
$396$	0	0
$397$	−10.7082	−0.537429	−0.268715	−	0.963220i	$-0.586599\pi$
$397$	−10.7082	−0.537429	−0.268715	+	0.963220i	$0.586599\pi$
$398$	−6.76393	−0.339045
$399$	−2.61803	−0.131066
$400$	0	0
$401$	12.5967	0.629052	0.314526	−	0.949249i	$-0.398155\pi$
$401$	12.5967	0.629052	0.314526	+	0.949249i	$0.398155\pi$
$402$	−8.47214	−0.422552
$403$	−0.437694	−0.0218031
$404$	−7.47214	−0.371753
$405$	0	0
$406$	−0.236068	−0.0117159
$407$	0	0
$408$	−0.527864	−0.0261332
$409$	16.5066	0.816198	0.408099	−	0.912938i	$-0.366192\pi$
$409$	16.5066	0.816198	0.408099	+	0.912938i	$0.366192\pi$
$410$	0	0
$411$	18.4721	0.911163
$412$	2.18034	0.107418
$413$	2.90983	0.143183
$414$	4.61803	0.226964
$415$	0	0
$416$	1.29180	0.0633355
$417$	−9.14590	−0.447877
$418$	0	0
$419$	−14.2705	−0.697160	−0.348580	−	0.937279i	$-0.613336\pi$
$419$	−14.2705	−0.697160	−0.348580	+	0.937279i	$0.613336\pi$
$420$	0	0
$421$	−6.70820	−0.326938	−0.163469	−	0.986548i	$-0.552268\pi$
$421$	−6.70820	−0.326938	−0.163469	+	0.986548i	$0.552268\pi$
$422$	−21.3262	−1.03815
$423$	11.7082	0.569272
$424$	−0.201626	−0.00979183
$425$	0	0
$426$	−10.0902	−0.488870
$427$	−3.23607	−0.156604
$428$	10.3820	0.501831
$429$	0	0
$430$	0	0
$431$	−11.5279	−0.555278	−0.277639	−	0.960686i	$-0.589552\pi$
$431$	−11.5279	−0.555278	−0.277639	+	0.960686i	$0.589552\pi$
$432$	4.85410	0.233543
$433$	9.03444	0.434168	0.217084	−	0.976153i	$-0.430346\pi$
$433$	9.03444	0.434168	0.217084	+	0.976153i	$0.430346\pi$
$434$	−1.14590	−0.0550049
$435$	0	0
$436$	−3.20163	−0.153330
$437$	12.0902	0.578351
$438$	−9.09017	−0.434345
$439$	−7.41641	−0.353966	−0.176983	−	0.984214i	$-0.556634\pi$
$439$	−7.41641	−0.353966	−0.176983	+	0.984214i	$0.556634\pi$
$440$	0	0
$441$	−6.61803	−0.315144
$442$	−0.145898	−0.00693966
$443$	−4.58359	−0.217773	−0.108887	−	0.994054i	$-0.534729\pi$
$443$	−4.58359	−0.217773	−0.108887	+	0.994054i	$0.534729\pi$
$444$	−5.52786	−0.262341
$445$	0	0
$446$	−25.3262	−1.19923
$447$	11.1459	0.527183
$448$	−2.61803	−0.123690
$449$	−31.0000	−1.46298	−0.731490	−	0.681852i	$-0.761175\pi$
$449$	−31.0000	−1.46298	−0.731490	+	0.681852i	$0.761175\pi$
$450$	0	0
$451$	0	0
$452$	2.47214	0.116279
$453$	−19.2705	−0.905408
$454$	7.61803	0.357532
$455$	0	0
$456$	9.47214	0.443573
$457$	−20.2361	−0.946603	−0.473302	−	0.880900i	$-0.656938\pi$
$457$	−20.2361	−0.946603	−0.473302	+	0.880900i	$0.656938\pi$
$458$	−27.4721	−1.28369
$459$	−0.236068	−0.0110187
$460$	0	0
$461$	8.05573	0.375193	0.187596	−	0.982246i	$-0.439930\pi$
$461$	8.05573	0.375193	0.187596	+	0.982246i	$0.439930\pi$
$462$	0	0
$463$	24.7082	1.14829	0.574144	−	0.818754i	$-0.305335\pi$
$463$	24.7082	1.14829	0.574144	+	0.818754i	$0.305335\pi$
$464$	1.14590	0.0531970
$465$	0	0
$466$	7.56231	0.350317
$467$	−32.2361	−1.49171	−0.745854	−	0.666110i	$-0.767958\pi$
$467$	−32.2361	−1.49171	−0.745854	+	0.666110i	$0.767958\pi$
$468$	0.236068	0.0109122
$469$	3.23607	0.149428
$470$	0	0
$471$	−18.7082	−0.862029
$472$	−10.5279	−0.484584
$473$	0	0
$474$	−6.00000	−0.275589
$475$	0	0
$476$	−0.0901699	−0.00413293
$477$	−0.0901699	−0.00412860
$478$	31.5967	1.44520
$479$	−15.4164	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$479$	−15.4164	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$480$	0	0
$481$	3.41641	0.155775
$482$	3.61803	0.164797
$483$	−1.76393	−0.0802617
$484$	0	0
$485$	0	0
$486$	1.61803	0.0733955
$487$	22.7082	1.02901	0.514503	−	0.857488i	$-0.327976\pi$
$487$	22.7082	1.02901	0.514503	+	0.857488i	$0.327976\pi$
$488$	11.7082	0.530005
$489$	−14.0902	−0.637180
$490$	0	0
$491$	15.5066	0.699802	0.349901	−	0.936787i	$-0.386215\pi$
$491$	15.5066	0.699802	0.349901	+	0.936787i	$0.386215\pi$
$492$	−3.14590	−0.141828
$493$	−0.0557281	−0.00250987
$494$	2.61803	0.117791
$495$	0	0
$496$	5.56231	0.249755
$497$	3.85410	0.172880
$498$	−1.38197	−0.0619274
$499$	−20.4721	−0.916459	−0.458229	−	0.888834i	$-0.651516\pi$
$499$	−20.4721	−0.916459	−0.458229	+	0.888834i	$0.651516\pi$
$500$	0	0
$501$	−9.03444	−0.403629
$502$	39.5066	1.76326
$503$	−23.8885	−1.06514	−0.532569	−	0.846387i	$-0.678773\pi$
$503$	−23.8885	−1.06514	−0.532569	+	0.846387i	$0.678773\pi$
$504$	−1.38197	−0.0615577
$505$	0	0
$506$	0	0
$507$	12.8541	0.570871
$508$	−5.56231	−0.246787
$509$	5.94427	0.263475	0.131738	−	0.991285i	$-0.457944\pi$
$509$	5.94427	0.263475	0.131738	+	0.991285i	$0.457944\pi$
$510$	0	0
$511$	3.47214	0.153598
$512$	5.29180	0.233867
$513$	4.23607	0.187027
$514$	14.4164	0.635880
$515$	0	0
$516$	−2.76393	−0.121675
$517$	0	0
$518$	8.94427	0.392989
$519$	−17.9443	−0.787666
$520$	0	0
$521$	5.59675	0.245198	0.122599	−	0.992456i	$-0.460877\pi$
$521$	5.59675	0.245198	0.122599	+	0.992456i	$0.460877\pi$
$522$	0.381966	0.0167182
$523$	37.8541	1.65524	0.827622	−	0.561286i	$-0.189693\pi$
$523$	37.8541	1.65524	0.827622	+	0.561286i	$0.189693\pi$
$524$	10.2016	0.445660
$525$	0	0
$526$	26.5623	1.15817
$527$	−0.270510	−0.0117836
$528$	0	0
$529$	−14.8541	−0.645831
$530$	0	0
$531$	−4.70820	−0.204319
$532$	1.61803	0.0701507
$533$	1.94427	0.0842158
$534$	−23.1803	−1.00311
$535$	0	0
$536$	−11.7082	−0.505717
$537$	−0.0557281	−0.00240484
$538$	10.0344	0.432616
$539$	0	0
$540$	0	0
$541$	43.6312	1.87585	0.937926	−	0.346836i	$-0.112744\pi$
$541$	43.6312	1.87585	0.937926	+	0.346836i	$0.112744\pi$
$542$	−44.6869	−1.91947
$543$	21.4164	0.919066
$544$	0.798374	0.0342300
$545$	0	0
$546$	−0.381966	−0.0163466
$547$	−42.5967	−1.82131	−0.910653	−	0.413173i	$-0.864421\pi$
$547$	−42.5967	−1.82131	−0.910653	+	0.413173i	$0.864421\pi$
$548$	−11.4164	−0.487685
$549$	5.23607	0.223470
$550$	0	0
$551$	1.00000	0.0426014
$552$	6.38197	0.271635
$553$	2.29180	0.0974571
$554$	12.3820	0.526059
$555$	0	0
$556$	5.65248	0.239718
$557$	5.03444	0.213316	0.106658	−	0.994296i	$-0.465985\pi$
$557$	5.03444	0.213316	0.106658	+	0.994296i	$0.465985\pi$
$558$	1.85410	0.0784904
$559$	1.70820	0.0722493
$560$	0	0
$561$	0	0
$562$	−19.5623	−0.825186
$563$	−30.7984	−1.29800	−0.648998	−	0.760790i	$-0.724812\pi$
$563$	−30.7984	−1.29800	−0.648998	+	0.760790i	$0.724812\pi$
$564$	−7.23607	−0.304693
$565$	0	0
$566$	−34.5066	−1.45042
$567$	−0.618034	−0.0259550
$568$	−13.9443	−0.585089
$569$	−19.2918	−0.808754	−0.404377	−	0.914592i	$-0.632512\pi$
$569$	−19.2918	−0.808754	−0.404377	+	0.914592i	$0.632512\pi$
$570$	0	0
$571$	−27.1246	−1.13513	−0.567565	−	0.823329i	$-0.692114\pi$
$571$	−27.1246	−1.13513	−0.567565	+	0.823329i	$0.692114\pi$
$572$	0	0
$573$	−1.79837	−0.0751281
$574$	5.09017	0.212460
$575$	0	0
$576$	4.23607	0.176503
$577$	35.9787	1.49781	0.748907	−	0.662675i	$-0.230579\pi$
$577$	35.9787	1.49781	0.748907	+	0.662675i	$0.230579\pi$
$578$	27.4164	1.14037
$579$	−21.8885	−0.909657
$580$	0	0
$581$	0.527864	0.0218995
$582$	−22.9443	−0.951071
$583$	0	0
$584$	−12.5623	−0.519832
$585$	0	0
$586$	21.4164	0.884704
$587$	−43.6180	−1.80031	−0.900154	−	0.435571i	$-0.856547\pi$
$587$	−43.6180	−1.80031	−0.900154	+	0.435571i	$0.856547\pi$
$588$	4.09017	0.168676
$589$	4.85410	0.200010
$590$	0	0
$591$	−16.1459	−0.664153
$592$	−43.4164	−1.78440
$593$	−17.7984	−0.730892	−0.365446	−	0.930833i	$-0.619083\pi$
$593$	−17.7984	−0.730892	−0.365446	+	0.930833i	$0.619083\pi$
$594$	0	0
$595$	0	0
$596$	−6.88854	−0.282166
$597$	−4.18034	−0.171090
$598$	1.76393	0.0721325
$599$	−16.6738	−0.681271	−0.340636	−	0.940195i	$-0.610642\pi$
$599$	−16.6738	−0.681271	−0.340636	+	0.940195i	$0.610642\pi$
$600$	0	0
$601$	37.7426	1.53955	0.769777	−	0.638313i	$-0.220367\pi$
$601$	37.7426	1.53955	0.769777	+	0.638313i	$0.220367\pi$
$602$	4.47214	0.182271
$603$	−5.23607	−0.213229
$604$	11.9098	0.484604
$605$	0	0
$606$	−19.5623	−0.794664
$607$	−33.8328	−1.37323	−0.686616	−	0.727020i	$-0.740905\pi$
$607$	−33.8328	−1.37323	−0.686616	+	0.727020i	$0.740905\pi$
$608$	−14.3262	−0.581006
$609$	−0.145898	−0.00591209
$610$	0	0
$611$	4.47214	0.180923
$612$	0.145898	0.00589758
$613$	24.0000	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$614$	−24.1803	−0.975839
$615$	0	0
$616$	0	0
$617$	−26.5066	−1.06711	−0.533557	−	0.845764i	$-0.679145\pi$
$617$	−26.5066	−1.06711	−0.533557	+	0.845764i	$0.679145\pi$
$618$	5.70820	0.229618
$619$	17.3607	0.697785	0.348892	−	0.937163i	$-0.386558\pi$
$619$	17.3607	0.697785	0.348892	+	0.937163i	$0.386558\pi$
$620$	0	0
$621$	2.85410	0.114531
$622$	10.7082	0.429360
$623$	8.85410	0.354732
$624$	1.85410	0.0742235
$625$	0	0
$626$	23.6180	0.943966
$627$	0	0
$628$	11.5623	0.461386
$629$	2.11146	0.0841893
$630$	0	0
$631$	−32.1591	−1.28023	−0.640116	−	0.768278i	$-0.721114\pi$
$631$	−32.1591	−1.28023	−0.640116	+	0.768278i	$0.721114\pi$
$632$	−8.29180	−0.329830
$633$	−13.1803	−0.523871
$634$	14.8541	0.589932
$635$	0	0
$636$	0.0557281	0.00220976
$637$	−2.52786	−0.100158
$638$	0	0
$639$	−6.23607	−0.246695
$640$	0	0
$641$	−7.43769	−0.293771	−0.146886	−	0.989153i	$-0.546925\pi$
$641$	−7.43769	−0.293771	−0.146886	+	0.989153i	$0.546925\pi$
$642$	27.1803	1.07272
$643$	−19.1246	−0.754201	−0.377101	−	0.926172i	$-0.623079\pi$
$643$	−19.1246	−0.754201	−0.377101	+	0.926172i	$0.623079\pi$
$644$	1.09017	0.0429587
$645$	0	0
$646$	1.61803	0.0636607
$647$	39.0689	1.53596	0.767978	−	0.640476i	$-0.221263\pi$
$647$	39.0689	1.53596	0.767978	+	0.640476i	$0.221263\pi$
$648$	2.23607	0.0878410
$649$	0	0
$650$	0	0
$651$	−0.708204	−0.0277567
$652$	8.70820	0.341040
$653$	−31.0000	−1.21312	−0.606562	−	0.795036i	$-0.707452\pi$
$653$	−31.0000	−1.21312	−0.606562	+	0.795036i	$0.707452\pi$
$654$	−8.38197	−0.327761
$655$	0	0
$656$	−24.7082	−0.964693
$657$	−5.61803	−0.219180
$658$	11.7082	0.456433
$659$	−24.7984	−0.966007	−0.483004	−	0.875618i	$-0.660454\pi$
$659$	−24.7984	−0.966007	−0.483004	+	0.875618i	$0.660454\pi$
$660$	0	0
$661$	13.3262	0.518331	0.259165	−	0.965833i	$-0.416553\pi$
$661$	13.3262	0.518331	0.259165	+	0.965833i	$0.416553\pi$
$662$	−43.5066	−1.69093
$663$	−0.0901699	−0.00350191
$664$	−1.90983	−0.0741158
$665$	0	0
$666$	−14.4721	−0.560784
$667$	0.673762	0.0260882
$668$	5.58359	0.216036
$669$	−15.6525	−0.605160
$670$	0	0
$671$	0	0
$672$	2.09017	0.0806301
$673$	7.09017	0.273306	0.136653	−	0.990619i	$-0.456365\pi$
$673$	7.09017	0.273306	0.136653	+	0.990619i	$0.456365\pi$
$674$	1.43769	0.0553779
$675$	0	0
$676$	−7.94427	−0.305549
$677$	−28.9787	−1.11374	−0.556871	−	0.830599i	$-0.687998\pi$
$677$	−28.9787	−1.11374	−0.556871	+	0.830599i	$0.687998\pi$
$678$	6.47214	0.248561
$679$	8.76393	0.336329
$680$	0	0
$681$	4.70820	0.180419
$682$	0	0
$683$	−10.4164	−0.398573	−0.199286	−	0.979941i	$-0.563862\pi$
$683$	−10.4164	−0.398573	−0.199286	+	0.979941i	$0.563862\pi$
$684$	−2.61803	−0.100103
$685$	0	0
$686$	−13.6180	−0.519939
$687$	−16.9787	−0.647778
$688$	−21.7082	−0.827618
$689$	−0.0344419	−0.00131213
$690$	0	0
$691$	20.9787	0.798068	0.399034	−	0.916936i	$-0.369346\pi$
$691$	20.9787	0.798068	0.399034	+	0.916936i	$0.369346\pi$
$692$	11.0902	0.421585
$693$	0	0
$694$	3.18034	0.120724
$695$	0	0
$696$	0.527864	0.0200086
$697$	1.20163	0.0455148
$698$	47.8328	1.81050
$699$	4.67376	0.176778
$700$	0	0
$701$	−13.6525	−0.515647	−0.257823	−	0.966192i	$-0.583005\pi$
$701$	−13.6525	−0.515647	−0.257823	+	0.966192i	$0.583005\pi$
$702$	0.618034	0.0233262
$703$	−37.8885	−1.42899
$704$	0	0
$705$	0	0
$706$	45.0344	1.69489
$707$	7.47214	0.281019
$708$	2.90983	0.109358
$709$	−38.6180	−1.45033	−0.725165	−	0.688575i	$-0.758237\pi$
$709$	−38.6180	−1.45033	−0.725165	+	0.688575i	$0.758237\pi$
$710$	0	0
$711$	−3.70820	−0.139069
$712$	−32.0344	−1.20054
$713$	3.27051	0.122482
$714$	−0.236068	−0.00883462
$715$	0	0
$716$	0.0344419	0.00128715
$717$	19.5279	0.729281
$718$	−59.3951	−2.21661
$719$	47.3050	1.76418	0.882089	−	0.471084i	$-0.156137\pi$
$719$	47.3050	1.76418	0.882089	+	0.471084i	$0.156137\pi$
$720$	0	0
$721$	−2.18034	−0.0812001
$722$	1.70820	0.0635728
$723$	2.23607	0.0831603
$724$	−13.2361	−0.491915
$725$	0	0
$726$	0	0
$727$	−9.97871	−0.370090	−0.185045	−	0.982730i	$-0.559243\pi$
$727$	−9.97871	−0.370090	−0.185045	+	0.982730i	$0.559243\pi$
$728$	−0.527864	−0.0195639
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.05573	0.0390475
$732$	−3.23607	−0.119609
$733$	16.4721	0.608412	0.304206	−	0.952606i	$-0.401609\pi$
$733$	16.4721	0.608412	0.304206	+	0.952606i	$0.401609\pi$
$734$	−13.7984	−0.509307
$735$	0	0
$736$	−9.65248	−0.355795
$737$	0	0
$738$	−8.23607	−0.303174
$739$	−16.0557	−0.590620	−0.295310	−	0.955402i	$-0.595423\pi$
$739$	−16.0557	−0.590620	−0.295310	+	0.955402i	$0.595423\pi$
$740$	0	0
$741$	1.61803	0.0594400
$742$	−0.0901699	−0.00331024
$743$	11.6738	0.428269	0.214134	−	0.976804i	$-0.431307\pi$
$743$	11.6738	0.428269	0.214134	+	0.976804i	$0.431307\pi$
$744$	2.56231	0.0939387
$745$	0	0
$746$	55.1591	2.01952
$747$	−0.854102	−0.0312500
$748$	0	0
$749$	−10.3820	−0.379349
$750$	0	0
$751$	−19.5066	−0.711805	−0.355903	−	0.934523i	$-0.615827\pi$
$751$	−19.5066	−0.711805	−0.355903	+	0.934523i	$0.615827\pi$
$752$	−56.8328	−2.07248
$753$	24.4164	0.889783
$754$	0.145898	0.00531329
$755$	0	0
$756$	0.381966	0.0138920
$757$	40.1591	1.45961	0.729803	−	0.683658i	$-0.239612\pi$
$757$	40.1591	1.45961	0.729803	+	0.683658i	$0.239612\pi$
$758$	33.3607	1.21171
$759$	0	0
$760$	0	0
$761$	−49.6525	−1.79990	−0.899950	−	0.435992i	$-0.856398\pi$
$761$	−49.6525	−1.79990	−0.899950	+	0.435992i	$0.856398\pi$
$762$	−14.5623	−0.527537
$763$	3.20163	0.115907
$764$	1.11146	0.0402111
$765$	0	0
$766$	17.0902	0.617493
$767$	−1.79837	−0.0649355
$768$	−13.5623	−0.489388
$769$	−12.2361	−0.441244	−0.220622	−	0.975359i	$-0.570809\pi$
$769$	−12.2361	−0.441244	−0.220622	+	0.975359i	$0.570809\pi$
$770$	0	0
$771$	8.90983	0.320880
$772$	13.5279	0.486878
$773$	22.2148	0.799010	0.399505	−	0.916731i	$-0.369182\pi$
$773$	22.2148	0.799010	0.399505	+	0.916731i	$0.369182\pi$
$774$	−7.23607	−0.260095
$775$	0	0
$776$	−31.7082	−1.13826
$777$	5.52786	0.198311
$778$	−7.23607	−0.259426
$779$	−21.5623	−0.772550
$780$	0	0
$781$	0	0
$782$	1.09017	0.0389844
$783$	0.236068	0.00843638
$784$	32.1246	1.14731
$785$	0	0
$786$	26.7082	0.952650
$787$	−5.34752	−0.190619	−0.0953093	−	0.995448i	$-0.530384\pi$
$787$	−5.34752	−0.190619	−0.0953093	+	0.995448i	$0.530384\pi$
$788$	9.97871	0.355477
$789$	16.4164	0.584440
$790$	0	0
$791$	−2.47214	−0.0878990
$792$	0	0
$793$	2.00000	0.0710221
$794$	17.3262	0.614885
$795$	0	0
$796$	2.58359	0.0915730
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	4.23607	0.149955
$799$	2.76393	0.0977809
$800$	0	0
$801$	−14.3262	−0.506193
$802$	−20.3820	−0.719712
$803$	0	0
$804$	3.23607	0.114127
$805$	0	0
$806$	0.708204	0.0249454
$807$	6.20163	0.218308
$808$	−27.0344	−0.951068
$809$	49.3951	1.73664	0.868320	−	0.496004i	$-0.165200\pi$
$809$	49.3951	1.73664	0.868320	+	0.496004i	$0.165200\pi$
$810$	0	0
$811$	−16.9787	−0.596203	−0.298102	−	0.954534i	$-0.596353\pi$
$811$	−16.9787	−0.596203	−0.298102	+	0.954534i	$0.596353\pi$
$812$	0.0901699	0.00316434
$813$	−27.6180	−0.968607
$814$	0	0
$815$	0	0
$816$	1.14590	0.0401145
$817$	−18.9443	−0.662776
$818$	−26.7082	−0.933830
$819$	−0.236068	−0.00824888
$820$	0	0
$821$	−22.5066	−0.785485	−0.392742	−	0.919648i	$-0.628474\pi$
$821$	−22.5066	−0.785485	−0.392742	+	0.919648i	$0.628474\pi$
$822$	−29.8885	−1.04248
$823$	2.41641	0.0842307	0.0421153	−	0.999113i	$-0.486590\pi$
$823$	2.41641	0.0842307	0.0421153	+	0.999113i	$0.486590\pi$
$824$	7.88854	0.274810
$825$	0	0
$826$	−4.70820	−0.163819
$827$	−44.6180	−1.55152	−0.775761	−	0.631027i	$-0.782634\pi$
$827$	−44.6180	−1.55152	−0.775761	+	0.631027i	$0.782634\pi$
$828$	−1.76393	−0.0613009
$829$	−17.2705	−0.599830	−0.299915	−	0.953966i	$-0.596958\pi$
$829$	−17.2705	−0.599830	−0.299915	+	0.953966i	$0.596958\pi$
$830$	0	0
$831$	7.65248	0.265461
$832$	1.61803	0.0560952
$833$	−1.56231	−0.0541307
$834$	14.7984	0.512426
$835$	0	0
$836$	0	0
$837$	1.14590	0.0396080
$838$	23.0902	0.797637
$839$	49.7426	1.71731	0.858653	−	0.512557i	$-0.171302\pi$
$839$	49.7426	1.71731	0.858653	+	0.512557i	$0.171302\pi$
$840$	0	0
$841$	−28.9443	−0.998078
$842$	10.8541	0.374057
$843$	−12.0902	−0.416408
$844$	8.14590	0.280393
$845$	0	0
$846$	−18.9443	−0.651317
$847$	0	0
$848$	0.437694	0.0150305
$849$	−21.3262	−0.731915
$850$	0	0
$851$	−25.5279	−0.875084
$852$	3.85410	0.132039
$853$	−16.5967	−0.568262	−0.284131	−	0.958785i	$-0.591705\pi$
$853$	−16.5967	−0.568262	−0.284131	+	0.958785i	$0.591705\pi$
$854$	5.23607	0.179175
$855$	0	0
$856$	37.5623	1.28385
$857$	13.7295	0.468990	0.234495	−	0.972117i	$-0.424656\pi$
$857$	13.7295	0.468990	0.234495	+	0.972117i	$0.424656\pi$
$858$	0	0
$859$	16.1459	0.550891	0.275445	−	0.961317i	$-0.411175\pi$
$859$	16.1459	0.550891	0.275445	+	0.961317i	$0.411175\pi$
$860$	0	0
$861$	3.14590	0.107212
$862$	18.6525	0.635306
$863$	−13.2361	−0.450561	−0.225280	−	0.974294i	$-0.572330\pi$
$863$	−13.2361	−0.450561	−0.225280	+	0.974294i	$0.572330\pi$
$864$	−3.38197	−0.115057
$865$	0	0
$866$	−14.6180	−0.496741
$867$	16.9443	0.575458
$868$	0.437694	0.0148563
$869$	0	0
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	−11.5836	−0.392270
$873$	−14.1803	−0.479932
$874$	−19.5623	−0.661705
$875$	0	0
$876$	3.47214	0.117313
$877$	−52.9230	−1.78708	−0.893541	−	0.448981i	$-0.851787\pi$
$877$	−52.9230	−1.78708	−0.893541	+	0.448981i	$0.851787\pi$
$878$	12.0000	0.404980
$879$	13.2361	0.446441
$880$	0	0
$881$	6.87539	0.231638	0.115819	−	0.993270i	$-0.463051\pi$
$881$	6.87539	0.231638	0.115819	+	0.993270i	$0.463051\pi$
$882$	10.7082	0.360564
$883$	−12.6180	−0.424631	−0.212315	−	0.977201i	$-0.568100\pi$
$883$	−12.6180	−0.424631	−0.212315	+	0.977201i	$0.568100\pi$
$884$	0.0557281	0.00187434
$885$	0	0
$886$	7.41641	0.249159
$887$	5.36068	0.179994	0.0899970	−	0.995942i	$-0.471314\pi$
$887$	5.36068	0.179994	0.0899970	+	0.995942i	$0.471314\pi$
$888$	−20.0000	−0.671156
$889$	5.56231	0.186554
$890$	0	0
$891$	0	0
$892$	9.67376	0.323902
$893$	−49.5967	−1.65969
$894$	−18.0344	−0.603162
$895$	0	0
$896$	8.41641	0.281172
$897$	1.09017	0.0363997
$898$	50.1591	1.67383
$899$	0.270510	0.00902201
$900$	0	0
$901$	−0.0212862	−0.000709147 0
$902$	0	0
$903$	2.76393	0.0919779
$904$	8.94427	0.297482
$905$	0	0
$906$	31.1803	1.03590
$907$	23.5623	0.782374	0.391187	−	0.920311i	$-0.372065\pi$
$907$	23.5623	0.782374	0.391187	+	0.920311i	$0.372065\pi$
$908$	−2.90983	−0.0965661
$909$	−12.0902	−0.401006
$910$	0	0
$911$	−21.1459	−0.700595	−0.350297	−	0.936639i	$-0.613919\pi$
$911$	−21.1459	−0.700595	−0.350297	+	0.936639i	$0.613919\pi$
$912$	−20.5623	−0.680886
$913$	0	0
$914$	32.7426	1.08303
$915$	0	0
$916$	10.4934	0.346712
$917$	−10.2016	−0.336887
$918$	0.381966	0.0126068
$919$	42.8328	1.41292	0.706462	−	0.707751i	$-0.250290\pi$
$919$	42.8328	1.41292	0.706462	+	0.707751i	$0.250290\pi$
$920$	0	0
$921$	−14.9443	−0.492431
$922$	−13.0344	−0.429266
$923$	−2.38197	−0.0784034
$924$	0	0
$925$	0	0
$926$	−39.9787	−1.31378
$927$	3.52786	0.115870
$928$	−0.798374	−0.0262079
$929$	−32.3394	−1.06102	−0.530511	−	0.847678i	$-0.678000\pi$
$929$	−32.3394	−1.06102	−0.530511	+	0.847678i	$0.678000\pi$
$930$	0	0
$931$	28.0344	0.918792
$932$	−2.88854	−0.0946174
$933$	6.61803	0.216665
$934$	52.1591	1.70670
$935$	0	0
$936$	0.854102	0.0279172
$937$	15.8541	0.517931	0.258965	−	0.965887i	$-0.416618\pi$
$937$	15.8541	0.517931	0.258965	+	0.965887i	$0.416618\pi$
$938$	−5.23607	−0.170964
$939$	14.5967	0.476347
$940$	0	0
$941$	−2.65248	−0.0864682	−0.0432341	−	0.999065i	$-0.513766\pi$
$941$	−2.65248	−0.0864682	−0.0432341	+	0.999065i	$0.513766\pi$
$942$	30.2705	0.986267
$943$	−14.5279	−0.473092
$944$	22.8541	0.743838
$945$	0	0
$946$	0	0
$947$	31.3394	1.01839	0.509197	−	0.860650i	$-0.329943\pi$
$947$	31.3394	1.01839	0.509197	+	0.860650i	$0.329943\pi$
$948$	2.29180	0.0744341
$949$	−2.14590	−0.0696588
$950$	0	0
$951$	9.18034	0.297693
$952$	−0.326238	−0.0105734
$953$	−14.2148	−0.460462	−0.230231	−	0.973136i	$-0.573948\pi$
$953$	−14.2148	−0.460462	−0.230231	+	0.973136i	$0.573948\pi$
$954$	0.145898	0.00472362
$955$	0	0
$956$	−12.0689	−0.390336
$957$	0	0
$958$	24.9443	0.805913
$959$	11.4164	0.368655
$960$	0	0
$961$	−29.6869	−0.957643
$962$	−5.52786	−0.178225
$963$	16.7984	0.541320
$964$	−1.38197	−0.0445101
$965$	0	0
$966$	2.85410	0.0918292
$967$	−52.8115	−1.69830	−0.849152	−	0.528148i	$-0.822887\pi$
$967$	−52.8115	−1.69830	−0.849152	+	0.528148i	$0.822887\pi$
$968$	0	0
$969$	1.00000	0.0321246
$970$	0	0
$971$	−17.9443	−0.575859	−0.287930	−	0.957652i	$-0.592967\pi$
$971$	−17.9443	−0.575859	−0.287930	+	0.957652i	$0.592967\pi$
$972$	−0.618034	−0.0198234
$973$	−5.65248	−0.181210
$974$	−36.7426	−1.17731
$975$	0	0
$976$	−25.4164	−0.813559
$977$	50.2148	1.60651	0.803257	−	0.595633i	$-0.203099\pi$
$977$	50.2148	1.60651	0.803257	+	0.595633i	$0.203099\pi$
$978$	22.7984	0.729012
$979$	0	0
$980$	0	0
$981$	−5.18034	−0.165396
$982$	−25.0902	−0.800659
$983$	−22.9443	−0.731809	−0.365904	−	0.930652i	$-0.619240\pi$
$983$	−22.9443	−0.731809	−0.365904	+	0.930652i	$0.619240\pi$
$984$	−11.3820	−0.362844
$985$	0	0
$986$	0.0901699	0.00287160
$987$	7.23607	0.230327
$988$	−1.00000	−0.0318142
$989$	−12.7639	−0.405869
$990$	0	0
$991$	30.4508	0.967303	0.483652	−	0.875261i	$-0.339310\pi$
$991$	30.4508	0.967303	0.483652	+	0.875261i	$0.339310\pi$
$992$	−3.87539	−0.123044
$993$	−26.8885	−0.853282
$994$	−6.23607	−0.197796
$995$	0	0
$996$	0.527864	0.0167260
$997$	1.05573	0.0334352	0.0167176	−	0.999860i	$-0.494678\pi$
$997$	1.05573	0.0334352	0.0167176	+	0.999860i	$0.494678\pi$
$998$	33.1246	1.04854
$999$	−8.94427	−0.282984

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.ba.1.1	✓	2
5.4	even	2		9075.2.a.bw.1.2	yes	2
11.10	odd	2		9075.2.a.bs.1.2	yes	2
55.54	odd	2		9075.2.a.be.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9075.2.a.ba.1.1	✓	2	1.1	even	1	trivial
9075.2.a.be.1.1	yes	2	55.54	odd	2
9075.2.a.bs.1.2	yes	2	11.10	odd	2
9075.2.a.bw.1.2	yes	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)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -0.618034 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -0.618034 q^{7} +2.23607 q^{8} +1.00000 q^{9} -0.618034 q^{12} +0.381966 q^{13} +1.00000 q^{14} -4.85410 q^{16} +0.236068 q^{17} -1.61803 q^{18} -4.23607 q^{19} +0.618034 q^{21} -2.85410 q^{23} -2.23607 q^{24} -0.618034 q^{26} -1.00000 q^{27} -0.381966 q^{28} -0.236068 q^{29} -1.14590 q^{31} +3.38197 q^{32} -0.381966 q^{34} +0.618034 q^{36} +8.94427 q^{37} +6.85410 q^{38} -0.381966 q^{39} +5.09017 q^{41} -1.00000 q^{42} +4.47214 q^{43} +4.61803 q^{46} +11.7082 q^{47} +4.85410 q^{48} -6.61803 q^{49} -0.236068 q^{51} +0.236068 q^{52} -0.0901699 q^{53} +1.61803 q^{54} -1.38197 q^{56} +4.23607 q^{57} +0.381966 q^{58} -4.70820 q^{59} +5.23607 q^{61} +1.85410 q^{62} -0.618034 q^{63} +4.23607 q^{64} -5.23607 q^{67} +0.145898 q^{68} +2.85410 q^{69} -6.23607 q^{71} +2.23607 q^{72} -5.61803 q^{73} -14.4721 q^{74} -2.61803 q^{76} +0.618034 q^{78} -3.70820 q^{79} +1.00000 q^{81} -8.23607 q^{82} -0.854102 q^{83} +0.381966 q^{84} -7.23607 q^{86} +0.236068 q^{87} -14.3262 q^{89} -0.236068 q^{91} -1.76393 q^{92} +1.14590 q^{93} -18.9443 q^{94} -3.38197 q^{96} -14.1803 q^{97} +10.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 - 2 * q^3 - q^4 + q^6 + q^7 + 2 * q^9	\( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} + q^{7} + 2 q^{9} + q^{12} + 3 q^{13} + 2 q^{14} - 3 q^{16} - 4 q^{17} - q^{18} - 4 q^{19} - q^{21} + q^{23} + q^{26} - 2 q^{27} - 3 q^{28} + 4 q^{29} - 9 q^{31} + 9 q^{32} - 3 q^{34} - q^{36} + 7 q^{38} - 3 q^{39} - q^{41} - 2 q^{42} + 7 q^{46} + 10 q^{47} + 3 q^{48} - 11 q^{49} + 4 q^{51} - 4 q^{52} + 11 q^{53} + q^{54} - 5 q^{56} + 4 q^{57} + 3 q^{58} + 4 q^{59} + 6 q^{61} - 3 q^{62} + q^{63} + 4 q^{64} - 6 q^{67} + 7 q^{68} - q^{69} - 8 q^{71} - 9 q^{73} - 20 q^{74} - 3 q^{76} - q^{78} + 6 q^{79} + 2 q^{81} - 12 q^{82} + 5 q^{83} + 3 q^{84} - 10 q^{86} - 4 q^{87} - 13 q^{89} + 4 q^{91} - 8 q^{92} + 9 q^{93} - 20 q^{94} - 9 q^{96} - 6 q^{97} + 8 q^{98}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 - q^4 + q^6 + q^7 + 2 * q^9 + q^12 + 3 * q^13 + 2 * q^14 - 3 * q^16 - 4 * q^17 - q^18 - 4 * q^19 - q^21 + q^23 + q^26 - 2 * q^27 - 3 * q^28 + 4 * q^29 - 9 * q^31 + 9 * q^32 - 3 * q^34 - q^36 + 7 * q^38 - 3 * q^39 - q^41 - 2 * q^42 + 7 * q^46 + 10 * q^47 + 3 * q^48 - 11 * q^49 + 4 * q^51 - 4 * q^52 + 11 * q^53 + q^54 - 5 * q^56 + 4 * q^57 + 3 * q^58 + 4 * q^59 + 6 * q^61 - 3 * q^62 + q^63 + 4 * q^64 - 6 * q^67 + 7 * q^68 - q^69 - 8 * q^71 - 9 * q^73 - 20 * q^74 - 3 * q^76 - q^78 + 6 * q^79 + 2 * q^81 - 12 * q^82 + 5 * q^83 + 3 * q^84 - 10 * q^86 - 4 * q^87 - 13 * q^89 + 4 * q^91 - 8 * q^92 + 9 * q^93 - 20 * q^94 - 9 * q^96 - 6 * q^97 + 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more