LMFDB - Embedded newform 1530.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1530,2,Mod(1,1530)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1530, 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("1530.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1530.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:	$12.2171115093$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 170)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1530.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.12311 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.12311 q^{7} -1.00000 q^{8} +1.00000 q^{10} +4.00000 q^{11} +0.438447 q^{13} +3.12311 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.56155 q^{19} -1.00000 q^{20} -4.00000 q^{22} -3.12311 q^{23} +1.00000 q^{25} -0.438447 q^{26} -3.12311 q^{28} -6.68466 q^{29} -2.43845 q^{31} -1.00000 q^{32} +1.00000 q^{34} +3.12311 q^{35} +1.12311 q^{37} -1.56155 q^{38} +1.00000 q^{40} +12.2462 q^{41} +7.12311 q^{43} +4.00000 q^{44} +3.12311 q^{46} -2.43845 q^{47} +2.75379 q^{49} -1.00000 q^{50} +0.438447 q^{52} -3.56155 q^{53} -4.00000 q^{55} +3.12311 q^{56} +6.68466 q^{58} +12.6847 q^{59} +11.5616 q^{61} +2.43845 q^{62} +1.00000 q^{64} -0.438447 q^{65} -0.876894 q^{67} -1.00000 q^{68} -3.12311 q^{70} +16.6847 q^{71} +13.8078 q^{73} -1.12311 q^{74} +1.56155 q^{76} -12.4924 q^{77} -1.00000 q^{80} -12.2462 q^{82} -10.2462 q^{83} +1.00000 q^{85} -7.12311 q^{86} -4.00000 q^{88} -2.68466 q^{89} -1.36932 q^{91} -3.12311 q^{92} +2.43845 q^{94} -1.56155 q^{95} +10.6847 q^{97} -2.75379 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} + 8 q^{11} + 5 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} - q^{19} - 2 q^{20} - 8 q^{22} + 2 q^{23} + 2 q^{25} - 5 q^{26} + 2 q^{28} - q^{29} - 9 q^{31} - 2 q^{32} + 2 q^{34} - 2 q^{35} - 6 q^{37} + q^{38} + 2 q^{40} + 8 q^{41} + 6 q^{43} + 8 q^{44} - 2 q^{46} - 9 q^{47} + 22 q^{49} - 2 q^{50} + 5 q^{52} - 3 q^{53} - 8 q^{55} - 2 q^{56} + q^{58} + 13 q^{59} + 19 q^{61} + 9 q^{62} + 2 q^{64} - 5 q^{65} - 10 q^{67} - 2 q^{68} + 2 q^{70} + 21 q^{71} + 7 q^{73} + 6 q^{74} - q^{76} + 8 q^{77} - 2 q^{80} - 8 q^{82} - 4 q^{83} + 2 q^{85} - 6 q^{86} - 8 q^{88} + 7 q^{89} + 22 q^{91} + 2 q^{92} + 9 q^{94} + q^{95} + 9 q^{97} - 22 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 2 * q^10 + 8 * q^11 + 5 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 - q^19 - 2 * q^20 - 8 * q^22 + 2 * q^23 + 2 * q^25 - 5 * q^26 + 2 * q^28 - q^29 - 9 * q^31 - 2 * q^32 + 2 * q^34 - 2 * q^35 - 6 * q^37 + q^38 + 2 * q^40 + 8 * q^41 + 6 * q^43 + 8 * q^44 - 2 * q^46 - 9 * q^47 + 22 * q^49 - 2 * q^50 + 5 * q^52 - 3 * q^53 - 8 * q^55 - 2 * q^56 + q^58 + 13 * q^59 + 19 * q^61 + 9 * q^62 + 2 * q^64 - 5 * q^65 - 10 * q^67 - 2 * q^68 + 2 * q^70 + 21 * q^71 + 7 * q^73 + 6 * q^74 - q^76 + 8 * q^77 - 2 * q^80 - 8 * q^82 - 4 * q^83 + 2 * q^85 - 6 * q^86 - 8 * q^88 + 7 * q^89 + 22 * q^91 + 2 * q^92 + 9 * q^94 + q^95 + 9 * q^97 - 22 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.12311	−1.18042	−0.590211	−	0.807249i	$-0.700956\pi$
$7$	−3.12311	−1.18042	−0.590211	+	0.807249i	$0.700956\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	0.438447	0.121603	0.0608017	−	0.998150i	$-0.480634\pi$
$13$	0.438447	0.121603	0.0608017	+	0.998150i	$0.480634\pi$
$14$	3.12311	0.834685
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	1.56155	0.358245	0.179122	−	0.983827i	$-0.442674\pi$
$19$	1.56155	0.358245	0.179122	+	0.983827i	$0.442674\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−4.00000	−0.852803
$23$	−3.12311	−0.651213	−0.325606	−	0.945505i	$-0.605568\pi$
$23$	−3.12311	−0.651213	−0.325606	+	0.945505i	$0.605568\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.438447	−0.0859866
$27$	0	0
$28$	−3.12311	−0.590211
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	0	0
$31$	−2.43845	−0.437958	−0.218979	−	0.975730i	$-0.570273\pi$
$31$	−2.43845	−0.437958	−0.218979	+	0.975730i	$0.570273\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	3.12311	0.527901
$36$	0	0
$37$	1.12311	0.184637	0.0923187	−	0.995730i	$-0.470572\pi$
$37$	1.12311	0.184637	0.0923187	+	0.995730i	$0.470572\pi$
$38$	−1.56155	−0.253317
$39$	0	0
$40$	1.00000	0.158114
$41$	12.2462	1.91254	0.956268	−	0.292490i	$-0.0944840\pi$
$41$	12.2462	1.91254	0.956268	+	0.292490i	$0.0944840\pi$
$42$	0	0
$43$	7.12311	1.08626	0.543132	−	0.839648i	$-0.317238\pi$
$43$	7.12311	1.08626	0.543132	+	0.839648i	$0.317238\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	3.12311	0.460477
$47$	−2.43845	−0.355684	−0.177842	−	0.984059i	$-0.556912\pi$
$47$	−2.43845	−0.355684	−0.177842	+	0.984059i	$0.556912\pi$
$48$	0	0
$49$	2.75379	0.393398
$50$	−1.00000	−0.141421
$51$	0	0
$52$	0.438447	0.0608017
$53$	−3.56155	−0.489217	−0.244608	−	0.969622i	$-0.578659\pi$
$53$	−3.56155	−0.489217	−0.244608	+	0.969622i	$0.578659\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	3.12311	0.417343
$57$	0	0
$58$	6.68466	0.877739
$59$	12.6847	1.65140	0.825701	−	0.564108i	$-0.190780\pi$
$59$	12.6847	1.65140	0.825701	+	0.564108i	$0.190780\pi$
$60$	0	0
$61$	11.5616	1.48031	0.740153	−	0.672439i	$-0.234753\pi$
$61$	11.5616	1.48031	0.740153	+	0.672439i	$0.234753\pi$
$62$	2.43845	0.309683
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.438447	−0.0543827
$66$	0	0
$67$	−0.876894	−0.107130	−0.0535648	−	0.998564i	$-0.517058\pi$
$67$	−0.876894	−0.107130	−0.0535648	+	0.998564i	$0.517058\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	−3.12311	−0.373283
$71$	16.6847	1.98010	0.990052	−	0.140700i	$-0.0449352\pi$
$71$	16.6847	1.98010	0.990052	+	0.140700i	$0.0449352\pi$
$72$	0	0
$73$	13.8078	1.61608	0.808038	−	0.589130i	$-0.200529\pi$
$73$	13.8078	1.61608	0.808038	+	0.589130i	$0.200529\pi$
$74$	−1.12311	−0.130558
$75$	0	0
$76$	1.56155	0.179122
$77$	−12.4924	−1.42364
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−12.2462	−1.35237
$83$	−10.2462	−1.12467	−0.562334	−	0.826910i	$-0.690096\pi$
$83$	−10.2462	−1.12467	−0.562334	+	0.826910i	$0.690096\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	−7.12311	−0.768104
$87$	0	0
$88$	−4.00000	−0.426401
$89$	−2.68466	−0.284573	−0.142287	−	0.989825i	$-0.545445\pi$
$89$	−2.68466	−0.284573	−0.142287	+	0.989825i	$0.545445\pi$
$90$	0	0
$91$	−1.36932	−0.143543
$92$	−3.12311	−0.325606
$93$	0	0
$94$	2.43845	0.251507
$95$	−1.56155	−0.160212
$96$	0	0
$97$	10.6847	1.08486	0.542431	−	0.840100i	$-0.317504\pi$
$97$	10.6847	1.08486	0.542431	+	0.840100i	$0.317504\pi$
$98$	−2.75379	−0.278175
$99$	0	0
$100$	1.00000	0.100000
$101$	−18.4924	−1.84006	−0.920032	−	0.391842i	$-0.871838\pi$
$101$	−18.4924	−1.84006	−0.920032	+	0.391842i	$0.871838\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−0.438447	−0.0429933
$105$	0	0
$106$	3.56155	0.345929
$107$	16.4924	1.59438	0.797191	−	0.603727i	$-0.206318\pi$
$107$	16.4924	1.59438	0.797191	+	0.603727i	$0.206318\pi$
$108$	0	0
$109$	−4.43845	−0.425126	−0.212563	−	0.977147i	$-0.568181\pi$
$109$	−4.43845	−0.425126	−0.212563	+	0.977147i	$0.568181\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	−3.12311	−0.295106
$113$	17.8078	1.67521	0.837607	−	0.546274i	$-0.183954\pi$
$113$	17.8078	1.67521	0.837607	+	0.546274i	$0.183954\pi$
$114$	0	0
$115$	3.12311	0.291231
$116$	−6.68466	−0.620655
$117$	0	0
$118$	−12.6847	−1.16772
$119$	3.12311	0.286295
$120$	0	0
$121$	5.00000	0.454545
$122$	−11.5616	−1.04673
$123$	0	0
$124$	−2.43845	−0.218979
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.5616	1.20339	0.601697	−	0.798725i	$-0.294492\pi$
$127$	13.5616	1.20339	0.601697	+	0.798725i	$0.294492\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0.438447	0.0384544
$131$	7.12311	0.622349	0.311174	−	0.950353i	$-0.399278\pi$
$131$	7.12311	0.622349	0.311174	+	0.950353i	$0.399278\pi$
$132$	0	0
$133$	−4.87689	−0.422880
$134$	0.876894	0.0757521
$135$	0	0
$136$	1.00000	0.0857493
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	−16.4924	−1.39887	−0.699435	−	0.714697i	$-0.746565\pi$
$139$	−16.4924	−1.39887	−0.699435	+	0.714697i	$0.746565\pi$
$140$	3.12311	0.263951
$141$	0	0
$142$	−16.6847	−1.40015
$143$	1.75379	0.146659
$144$	0	0
$145$	6.68466	0.555131
$146$	−13.8078	−1.14274
$147$	0	0
$148$	1.12311	0.0923187
$149$	5.12311	0.419701	0.209851	−	0.977733i	$-0.432702\pi$
$149$	5.12311	0.419701	0.209851	+	0.977733i	$0.432702\pi$
$150$	0	0
$151$	9.36932	0.762464	0.381232	−	0.924479i	$-0.375500\pi$
$151$	9.36932	0.762464	0.381232	+	0.924479i	$0.375500\pi$
$152$	−1.56155	−0.126659
$153$	0	0
$154$	12.4924	1.00667
$155$	2.43845	0.195861
$156$	0	0
$157$	−14.4924	−1.15662	−0.578311	−	0.815817i	$-0.696288\pi$
$157$	−14.4924	−1.15662	−0.578311	+	0.815817i	$0.696288\pi$
$158$	0	0
$159$	0	0
$160$	1.00000	0.0790569
$161$	9.75379	0.768706
$162$	0	0
$163$	−2.24621	−0.175937	−0.0879684	−	0.996123i	$-0.528037\pi$
$163$	−2.24621	−0.175937	−0.0879684	+	0.996123i	$0.528037\pi$
$164$	12.2462	0.956268
$165$	0	0
$166$	10.2462	0.795260
$167$	3.12311	0.241673	0.120837	−	0.992672i	$-0.461442\pi$
$167$	3.12311	0.241673	0.120837	+	0.992672i	$0.461442\pi$
$168$	0	0
$169$	−12.8078	−0.985213
$170$	−1.00000	−0.0766965
$171$	0	0
$172$	7.12311	0.543132
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	−3.12311	−0.236085
$176$	4.00000	0.301511
$177$	0	0
$178$	2.68466	0.201224
$179$	24.4924	1.83065	0.915325	−	0.402716i	$-0.131934\pi$
$179$	24.4924	1.83065	0.915325	+	0.402716i	$0.131934\pi$
$180$	0	0
$181$	−0.246211	−0.0183007	−0.00915037	−	0.999958i	$-0.502913\pi$
$181$	−0.246211	−0.0183007	−0.00915037	+	0.999958i	$0.502913\pi$
$182$	1.36932	0.101501
$183$	0	0
$184$	3.12311	0.230238
$185$	−1.12311	−0.0825724
$186$	0	0
$187$	−4.00000	−0.292509
$188$	−2.43845	−0.177842
$189$	0	0
$190$	1.56155	0.113287
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−7.75379	−0.558130	−0.279065	−	0.960272i	$-0.590024\pi$
$193$	−7.75379	−0.558130	−0.279065	+	0.960272i	$0.590024\pi$
$194$	−10.6847	−0.767114
$195$	0	0
$196$	2.75379	0.196699
$197$	11.3693	0.810030	0.405015	−	0.914310i	$-0.367266\pi$
$197$	11.3693	0.810030	0.405015	+	0.914310i	$0.367266\pi$
$198$	0	0
$199$	10.4384	0.739962	0.369981	−	0.929039i	$-0.379364\pi$
$199$	10.4384	0.739962	0.369981	+	0.929039i	$0.379364\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	18.4924	1.30112
$203$	20.8769	1.46527
$204$	0	0
$205$	−12.2462	−0.855312
$206$	−8.00000	−0.557386
$207$	0	0
$208$	0.438447	0.0304008
$209$	6.24621	0.432059
$210$	0	0
$211$	21.3693	1.47112	0.735562	−	0.677457i	$-0.236918\pi$
$211$	21.3693	1.47112	0.735562	+	0.677457i	$0.236918\pi$
$212$	−3.56155	−0.244608
$213$	0	0
$214$	−16.4924	−1.12740
$215$	−7.12311	−0.485792
$216$	0	0
$217$	7.61553	0.516976
$218$	4.43845	0.300610
$219$	0	0
$220$	−4.00000	−0.269680
$221$	−0.438447	−0.0294931
$222$	0	0
$223$	−8.68466	−0.581568	−0.290784	−	0.956789i	$-0.593916\pi$
$223$	−8.68466	−0.581568	−0.290784	+	0.956789i	$0.593916\pi$
$224$	3.12311	0.208671
$225$	0	0
$226$	−17.8078	−1.18455
$227$	−28.6847	−1.90387	−0.951934	−	0.306304i	$-0.900908\pi$
$227$	−28.6847	−1.90387	−0.951934	+	0.306304i	$0.900908\pi$
$228$	0	0
$229$	18.4924	1.22201	0.611007	−	0.791625i	$-0.290765\pi$
$229$	18.4924	1.22201	0.611007	+	0.791625i	$0.290765\pi$
$230$	−3.12311	−0.205931
$231$	0	0
$232$	6.68466	0.438869
$233$	−18.6847	−1.22407	−0.612036	−	0.790830i	$-0.709649\pi$
$233$	−18.6847	−1.22407	−0.612036	+	0.790830i	$0.709649\pi$
$234$	0	0
$235$	2.43845	0.159067
$236$	12.6847	0.825701
$237$	0	0
$238$	−3.12311	−0.202441
$239$	−1.36932	−0.0885737	−0.0442869	−	0.999019i	$-0.514102\pi$
$239$	−1.36932	−0.0885737	−0.0442869	+	0.999019i	$0.514102\pi$
$240$	0	0
$241$	−4.24621	−0.273523	−0.136761	−	0.990604i	$-0.543669\pi$
$241$	−4.24621	−0.273523	−0.136761	+	0.990604i	$0.543669\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	11.5616	0.740153
$245$	−2.75379	−0.175933
$246$	0	0
$247$	0.684658	0.0435638
$248$	2.43845	0.154842
$249$	0	0
$250$	1.00000	0.0632456
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	−12.4924	−0.785392
$254$	−13.5616	−0.850928
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	−3.50758	−0.217950
$260$	−0.438447	−0.0271913
$261$	0	0
$262$	−7.12311	−0.440067
$263$	−4.19224	−0.258504	−0.129252	−	0.991612i	$-0.541258\pi$
$263$	−4.19224	−0.258504	−0.129252	+	0.991612i	$0.541258\pi$
$264$	0	0
$265$	3.56155	0.218784
$266$	4.87689	0.299022
$267$	0	0
$268$	−0.876894	−0.0535648
$269$	−17.8078	−1.08576	−0.542879	−	0.839811i	$-0.682666\pi$
$269$	−17.8078	−1.08576	−0.542879	+	0.839811i	$0.682666\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	10.0000	0.604122
$275$	4.00000	0.241209
$276$	0	0
$277$	−32.2462	−1.93749	−0.968744	−	0.248064i	$-0.920206\pi$
$277$	−32.2462	−1.93749	−0.968744	+	0.248064i	$0.920206\pi$
$278$	16.4924	0.989150
$279$	0	0
$280$	−3.12311	−0.186641
$281$	−12.4384	−0.742016	−0.371008	−	0.928630i	$-0.620988\pi$
$281$	−12.4384	−0.742016	−0.371008	+	0.928630i	$0.620988\pi$
$282$	0	0
$283$	20.6847	1.22958	0.614788	−	0.788693i	$-0.289242\pi$
$283$	20.6847	1.22958	0.614788	+	0.788693i	$0.289242\pi$
$284$	16.6847	0.990052
$285$	0	0
$286$	−1.75379	−0.103704
$287$	−38.2462	−2.25760
$288$	0	0
$289$	1.00000	0.0588235
$290$	−6.68466	−0.392537
$291$	0	0
$292$	13.8078	0.808038
$293$	12.4384	0.726662	0.363331	−	0.931660i	$-0.381639\pi$
$293$	12.4384	0.726662	0.363331	+	0.931660i	$0.381639\pi$
$294$	0	0
$295$	−12.6847	−0.738529
$296$	−1.12311	−0.0652792
$297$	0	0
$298$	−5.12311	−0.296774
$299$	−1.36932	−0.0791896
$300$	0	0
$301$	−22.2462	−1.28225
$302$	−9.36932	−0.539144
$303$	0	0
$304$	1.56155	0.0895612
$305$	−11.5616	−0.662013
$306$	0	0
$307$	−34.2462	−1.95453	−0.977267	−	0.212011i	$-0.931999\pi$
$307$	−34.2462	−1.95453	−0.977267	+	0.212011i	$0.931999\pi$
$308$	−12.4924	−0.711822
$309$	0	0
$310$	−2.43845	−0.138494
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	14.4924	0.817855
$315$	0	0
$316$	0	0
$317$	9.61553	0.540062	0.270031	−	0.962852i	$-0.412966\pi$
$317$	9.61553	0.540062	0.270031	+	0.962852i	$0.412966\pi$
$318$	0	0
$319$	−26.7386	−1.49708
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−9.75379	−0.543557
$323$	−1.56155	−0.0868871
$324$	0	0
$325$	0.438447	0.0243207
$326$	2.24621	0.124406
$327$	0	0
$328$	−12.2462	−0.676184
$329$	7.61553	0.419858
$330$	0	0
$331$	−14.0540	−0.772476	−0.386238	−	0.922399i	$-0.626226\pi$
$331$	−14.0540	−0.772476	−0.386238	+	0.922399i	$0.626226\pi$
$332$	−10.2462	−0.562334
$333$	0	0
$334$	−3.12311	−0.170889
$335$	0.876894	0.0479099
$336$	0	0
$337$	−22.6847	−1.23571	−0.617856	−	0.786291i	$-0.711999\pi$
$337$	−22.6847	−1.23571	−0.617856	+	0.786291i	$0.711999\pi$
$338$	12.8078	0.696651
$339$	0	0
$340$	1.00000	0.0542326
$341$	−9.75379	−0.528197
$342$	0	0
$343$	13.2614	0.716046
$344$	−7.12311	−0.384052
$345$	0	0
$346$	−18.0000	−0.967686
$347$	6.43845	0.345634	0.172817	−	0.984954i	$-0.444713\pi$
$347$	6.43845	0.345634	0.172817	+	0.984954i	$0.444713\pi$
$348$	0	0
$349$	−6.87689	−0.368112	−0.184056	−	0.982916i	$-0.558923\pi$
$349$	−6.87689	−0.368112	−0.184056	+	0.982916i	$0.558923\pi$
$350$	3.12311	0.166937
$351$	0	0
$352$	−4.00000	−0.213201
$353$	10.4924	0.558455	0.279228	−	0.960225i	$-0.409922\pi$
$353$	10.4924	0.558455	0.279228	+	0.960225i	$0.409922\pi$
$354$	0	0
$355$	−16.6847	−0.885530
$356$	−2.68466	−0.142287
$357$	0	0
$358$	−24.4924	−1.29446
$359$	19.1231	1.00928	0.504639	−	0.863330i	$-0.331625\pi$
$359$	19.1231	1.00928	0.504639	+	0.863330i	$0.331625\pi$
$360$	0	0
$361$	−16.5616	−0.871661
$362$	0.246211	0.0129406
$363$	0	0
$364$	−1.36932	−0.0717717
$365$	−13.8078	−0.722731
$366$	0	0
$367$	7.61553	0.397527	0.198764	−	0.980047i	$-0.436307\pi$
$367$	7.61553	0.397527	0.198764	+	0.980047i	$0.436307\pi$
$368$	−3.12311	−0.162803
$369$	0	0
$370$	1.12311	0.0583875
$371$	11.1231	0.577483
$372$	0	0
$373$	24.7386	1.28092	0.640459	−	0.767992i	$-0.278744\pi$
$373$	24.7386	1.28092	0.640459	+	0.767992i	$0.278744\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	2.43845	0.125753
$377$	−2.93087	−0.150947
$378$	0	0
$379$	2.24621	0.115380	0.0576901	−	0.998335i	$-0.481626\pi$
$379$	2.24621	0.115380	0.0576901	+	0.998335i	$0.481626\pi$
$380$	−1.56155	−0.0801060
$381$	0	0
$382$	0	0
$383$	−3.80776	−0.194568	−0.0972838	−	0.995257i	$-0.531015\pi$
$383$	−3.80776	−0.194568	−0.0972838	+	0.995257i	$0.531015\pi$
$384$	0	0
$385$	12.4924	0.636673
$386$	7.75379	0.394657
$387$	0	0
$388$	10.6847	0.542431
$389$	11.3693	0.576447	0.288224	−	0.957563i	$-0.406935\pi$
$389$	11.3693	0.576447	0.288224	+	0.957563i	$0.406935\pi$
$390$	0	0
$391$	3.12311	0.157942
$392$	−2.75379	−0.139087
$393$	0	0
$394$	−11.3693	−0.572778
$395$	0	0
$396$	0	0
$397$	−3.36932	−0.169101	−0.0845506	−	0.996419i	$-0.526945\pi$
$397$	−3.36932	−0.169101	−0.0845506	+	0.996419i	$0.526945\pi$
$398$	−10.4384	−0.523232
$399$	0	0
$400$	1.00000	0.0500000
$401$	−35.3693	−1.76626	−0.883130	−	0.469129i	$-0.844568\pi$
$401$	−35.3693	−1.76626	−0.883130	+	0.469129i	$0.844568\pi$
$402$	0	0
$403$	−1.06913	−0.0532572
$404$	−18.4924	−0.920032
$405$	0	0
$406$	−20.8769	−1.03610
$407$	4.49242	0.222681
$408$	0	0
$409$	−14.6847	−0.726110	−0.363055	−	0.931768i	$-0.618266\pi$
$409$	−14.6847	−0.726110	−0.363055	+	0.931768i	$0.618266\pi$
$410$	12.2462	0.604797
$411$	0	0
$412$	8.00000	0.394132
$413$	−39.6155	−1.94935
$414$	0	0
$415$	10.2462	0.502967
$416$	−0.438447	−0.0214966
$417$	0	0
$418$	−6.24621	−0.305512
$419$	16.8769	0.824490	0.412245	−	0.911073i	$-0.364745\pi$
$419$	16.8769	0.824490	0.412245	+	0.911073i	$0.364745\pi$
$420$	0	0
$421$	−33.6155	−1.63832	−0.819160	−	0.573565i	$-0.805560\pi$
$421$	−33.6155	−1.63832	−0.819160	+	0.573565i	$0.805560\pi$
$422$	−21.3693	−1.04024
$423$	0	0
$424$	3.56155	0.172964
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−36.1080	−1.74739
$428$	16.4924	0.797191
$429$	0	0
$430$	7.12311	0.343507
$431$	−12.4924	−0.601739	−0.300869	−	0.953665i	$-0.597277\pi$
$431$	−12.4924	−0.601739	−0.300869	+	0.953665i	$0.597277\pi$
$432$	0	0
$433$	30.4924	1.46537	0.732686	−	0.680567i	$-0.238266\pi$
$433$	30.4924	1.46537	0.732686	+	0.680567i	$0.238266\pi$
$434$	−7.61553	−0.365557
$435$	0	0
$436$	−4.43845	−0.212563
$437$	−4.87689	−0.233293
$438$	0	0
$439$	32.9848	1.57428	0.787140	−	0.616774i	$-0.211561\pi$
$439$	32.9848	1.57428	0.787140	+	0.616774i	$0.211561\pi$
$440$	4.00000	0.190693
$441$	0	0
$442$	0.438447	0.0208548
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	0	0
$445$	2.68466	0.127265
$446$	8.68466	0.411230
$447$	0	0
$448$	−3.12311	−0.147553
$449$	25.1231	1.18563	0.592816	−	0.805338i	$-0.298016\pi$
$449$	25.1231	1.18563	0.592816	+	0.805338i	$0.298016\pi$
$450$	0	0
$451$	48.9848	2.30661
$452$	17.8078	0.837607
$453$	0	0
$454$	28.6847	1.34624
$455$	1.36932	0.0641946
$456$	0	0
$457$	0.246211	0.0115173	0.00575864	−	0.999983i	$-0.498167\pi$
$457$	0.246211	0.0115173	0.00575864	+	0.999983i	$0.498167\pi$
$458$	−18.4924	−0.864094
$459$	0	0
$460$	3.12311	0.145616
$461$	30.4924	1.42017	0.710087	−	0.704114i	$-0.248656\pi$
$461$	30.4924	1.42017	0.710087	+	0.704114i	$0.248656\pi$
$462$	0	0
$463$	14.9309	0.693896	0.346948	−	0.937884i	$-0.387218\pi$
$463$	14.9309	0.693896	0.346948	+	0.937884i	$0.387218\pi$
$464$	−6.68466	−0.310327
$465$	0	0
$466$	18.6847	0.865550
$467$	−0.492423	−0.0227866	−0.0113933	−	0.999935i	$-0.503627\pi$
$467$	−0.492423	−0.0227866	−0.0113933	+	0.999935i	$0.503627\pi$
$468$	0	0
$469$	2.73863	0.126458
$470$	−2.43845	−0.112477
$471$	0	0
$472$	−12.6847	−0.583859
$473$	28.4924	1.31008
$474$	0	0
$475$	1.56155	0.0716490
$476$	3.12311	0.143147
$477$	0	0
$478$	1.36932	0.0626311
$479$	−34.4384	−1.57353	−0.786766	−	0.617251i	$-0.788246\pi$
$479$	−34.4384	−1.57353	−0.786766	+	0.617251i	$0.788246\pi$
$480$	0	0
$481$	0.492423	0.0224525
$482$	4.24621	0.193410
$483$	0	0
$484$	5.00000	0.227273
$485$	−10.6847	−0.485165
$486$	0	0
$487$	−15.6155	−0.707607	−0.353804	−	0.935320i	$-0.615112\pi$
$487$	−15.6155	−0.707607	−0.353804	+	0.935320i	$0.615112\pi$
$488$	−11.5616	−0.523367
$489$	0	0
$490$	2.75379	0.124403
$491$	1.56155	0.0704719	0.0352359	−	0.999379i	$-0.488782\pi$
$491$	1.56155	0.0704719	0.0352359	+	0.999379i	$0.488782\pi$
$492$	0	0
$493$	6.68466	0.301062
$494$	−0.684658	−0.0308042
$495$	0	0
$496$	−2.43845	−0.109490
$497$	−52.1080	−2.33736
$498$	0	0
$499$	−0.876894	−0.0392552	−0.0196276	−	0.999807i	$-0.506248\pi$
$499$	−0.876894	−0.0392552	−0.0196276	+	0.999807i	$0.506248\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−4.00000	−0.178529
$503$	26.7386	1.19222	0.596108	−	0.802904i	$-0.296713\pi$
$503$	26.7386	1.19222	0.596108	+	0.802904i	$0.296713\pi$
$504$	0	0
$505$	18.4924	0.822902
$506$	12.4924	0.555356
$507$	0	0
$508$	13.5616	0.601697
$509$	−25.1231	−1.11356	−0.556781	−	0.830659i	$-0.687964\pi$
$509$	−25.1231	−1.11356	−0.556781	+	0.830659i	$0.687964\pi$
$510$	0	0
$511$	−43.1231	−1.90765
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	2.00000	0.0882162
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−9.75379	−0.428971
$518$	3.50758	0.154114
$519$	0	0
$520$	0.438447	0.0192272
$521$	−2.38447	−0.104466	−0.0522328	−	0.998635i	$-0.516634\pi$
$521$	−2.38447	−0.104466	−0.0522328	+	0.998635i	$0.516634\pi$
$522$	0	0
$523$	−17.8617	−0.781039	−0.390520	−	0.920595i	$-0.627705\pi$
$523$	−17.8617	−0.781039	−0.390520	+	0.920595i	$0.627705\pi$
$524$	7.12311	0.311174
$525$	0	0
$526$	4.19224	0.182790
$527$	2.43845	0.106220
$528$	0	0
$529$	−13.2462	−0.575922
$530$	−3.56155	−0.154704
$531$	0	0
$532$	−4.87689	−0.211440
$533$	5.36932	0.232571
$534$	0	0
$535$	−16.4924	−0.713030
$536$	0.876894	0.0378761
$537$	0	0
$538$	17.8078	0.767747
$539$	11.0152	0.474456
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	1.00000	0.0428746
$545$	4.43845	0.190122
$546$	0	0
$547$	33.5616	1.43499	0.717494	−	0.696564i	$-0.245289\pi$
$547$	33.5616	1.43499	0.717494	+	0.696564i	$0.245289\pi$
$548$	−10.0000	−0.427179
$549$	0	0
$550$	−4.00000	−0.170561
$551$	−10.4384	−0.444693
$552$	0	0
$553$	0	0
$554$	32.2462	1.37001
$555$	0	0
$556$	−16.4924	−0.699435
$557$	−32.4384	−1.37446	−0.687231	−	0.726439i	$-0.741174\pi$
$557$	−32.4384	−1.37446	−0.687231	+	0.726439i	$0.741174\pi$
$558$	0	0
$559$	3.12311	0.132093
$560$	3.12311	0.131975
$561$	0	0
$562$	12.4384	0.524684
$563$	29.3693	1.23777	0.618885	−	0.785482i	$-0.287585\pi$
$563$	29.3693	1.23777	0.618885	+	0.785482i	$0.287585\pi$
$564$	0	0
$565$	−17.8078	−0.749178
$566$	−20.6847	−0.869441
$567$	0	0
$568$	−16.6847	−0.700073
$569$	−24.9309	−1.04516	−0.522578	−	0.852591i	$-0.675030\pi$
$569$	−24.9309	−1.04516	−0.522578	+	0.852591i	$0.675030\pi$
$570$	0	0
$571$	16.8769	0.706276	0.353138	−	0.935571i	$-0.385115\pi$
$571$	16.8769	0.706276	0.353138	+	0.935571i	$0.385115\pi$
$572$	1.75379	0.0733296
$573$	0	0
$574$	38.2462	1.59637
$575$	−3.12311	−0.130243
$576$	0	0
$577$	19.3693	0.806355	0.403178	−	0.915122i	$-0.367906\pi$
$577$	19.3693	0.806355	0.403178	+	0.915122i	$0.367906\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	6.68466	0.277565
$581$	32.0000	1.32758
$582$	0	0
$583$	−14.2462	−0.590018
$584$	−13.8078	−0.571369
$585$	0	0
$586$	−12.4384	−0.513828
$587$	15.1231	0.624197	0.312099	−	0.950050i	$-0.398968\pi$
$587$	15.1231	0.624197	0.312099	+	0.950050i	$0.398968\pi$
$588$	0	0
$589$	−3.80776	−0.156896
$590$	12.6847	0.522219
$591$	0	0
$592$	1.12311	0.0461594
$593$	−8.24621	−0.338631	−0.169316	−	0.985562i	$-0.554156\pi$
$593$	−8.24621	−0.338631	−0.169316	+	0.985562i	$0.554156\pi$
$594$	0	0
$595$	−3.12311	−0.128035
$596$	5.12311	0.209851
$597$	0	0
$598$	1.36932	0.0559955
$599$	−31.6155	−1.29178	−0.645888	−	0.763432i	$-0.723513\pi$
$599$	−31.6155	−1.29178	−0.645888	+	0.763432i	$0.723513\pi$
$600$	0	0
$601$	33.6155	1.37121	0.685603	−	0.727976i	$-0.259539\pi$
$601$	33.6155	1.37121	0.685603	+	0.727976i	$0.259539\pi$
$602$	22.2462	0.906688
$603$	0	0
$604$	9.36932	0.381232
$605$	−5.00000	−0.203279
$606$	0	0
$607$	13.8617	0.562631	0.281315	−	0.959615i	$-0.409229\pi$
$607$	13.8617	0.562631	0.281315	+	0.959615i	$0.409229\pi$
$608$	−1.56155	−0.0633293
$609$	0	0
$610$	11.5616	0.468114
$611$	−1.06913	−0.0432524
$612$	0	0
$613$	4.93087	0.199156	0.0995780	−	0.995030i	$-0.468251\pi$
$613$	4.93087	0.199156	0.0995780	+	0.995030i	$0.468251\pi$
$614$	34.2462	1.38206
$615$	0	0
$616$	12.4924	0.503334
$617$	−18.6847	−0.752216	−0.376108	−	0.926576i	$-0.622738\pi$
$617$	−18.6847	−0.752216	−0.376108	+	0.926576i	$0.622738\pi$
$618$	0	0
$619$	0.876894	0.0352454	0.0176227	−	0.999845i	$-0.494390\pi$
$619$	0.876894	0.0352454	0.0176227	+	0.999845i	$0.494390\pi$
$620$	2.43845	0.0979304
$621$	0	0
$622$	8.00000	0.320771
$623$	8.38447	0.335917
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	0	0
$628$	−14.4924	−0.578311
$629$	−1.12311	−0.0447812
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	0	0
$634$	−9.61553	−0.381881
$635$	−13.5616	−0.538174
$636$	0	0
$637$	1.20739	0.0478386
$638$	26.7386	1.05859
$639$	0	0
$640$	1.00000	0.0395285
$641$	7.75379	0.306256	0.153128	−	0.988206i	$-0.451065\pi$
$641$	7.75379	0.306256	0.153128	+	0.988206i	$0.451065\pi$
$642$	0	0
$643$	7.50758	0.296070	0.148035	−	0.988982i	$-0.452705\pi$
$643$	7.50758	0.296070	0.148035	+	0.988982i	$0.452705\pi$
$644$	9.75379	0.384353
$645$	0	0
$646$	1.56155	0.0614385
$647$	0.684658	0.0269167	0.0134584	−	0.999909i	$-0.495716\pi$
$647$	0.684658	0.0269167	0.0134584	+	0.999909i	$0.495716\pi$
$648$	0	0
$649$	50.7386	1.99167
$650$	−0.438447	−0.0171973
$651$	0	0
$652$	−2.24621	−0.0879684
$653$	5.50758	0.215528	0.107764	−	0.994176i	$-0.465631\pi$
$653$	5.50758	0.215528	0.107764	+	0.994176i	$0.465631\pi$
$654$	0	0
$655$	−7.12311	−0.278323
$656$	12.2462	0.478134
$657$	0	0
$658$	−7.61553	−0.296884
$659$	13.0691	0.509101	0.254551	−	0.967059i	$-0.418072\pi$
$659$	13.0691	0.509101	0.254551	+	0.967059i	$0.418072\pi$
$660$	0	0
$661$	−39.8617	−1.55044	−0.775221	−	0.631690i	$-0.782362\pi$
$661$	−39.8617	−1.55044	−0.775221	+	0.631690i	$0.782362\pi$
$662$	14.0540	0.546223
$663$	0	0
$664$	10.2462	0.397630
$665$	4.87689	0.189118
$666$	0	0
$667$	20.8769	0.808357
$668$	3.12311	0.120837
$669$	0	0
$670$	−0.876894	−0.0338774
$671$	46.2462	1.78532
$672$	0	0
$673$	−41.4233	−1.59675	−0.798375	−	0.602160i	$-0.794307\pi$
$673$	−41.4233	−1.59675	−0.798375	+	0.602160i	$0.794307\pi$
$674$	22.6847	0.873780
$675$	0	0
$676$	−12.8078	−0.492606
$677$	−8.73863	−0.335853	−0.167926	−	0.985800i	$-0.553707\pi$
$677$	−8.73863	−0.335853	−0.167926	+	0.985800i	$0.553707\pi$
$678$	0	0
$679$	−33.3693	−1.28060
$680$	−1.00000	−0.0383482
$681$	0	0
$682$	9.75379	0.373492
$683$	−12.3002	−0.470654	−0.235327	−	0.971916i	$-0.575616\pi$
$683$	−12.3002	−0.470654	−0.235327	+	0.971916i	$0.575616\pi$
$684$	0	0
$685$	10.0000	0.382080
$686$	−13.2614	−0.506321
$687$	0	0
$688$	7.12311	0.271566
$689$	−1.56155	−0.0594904
$690$	0	0
$691$	26.2462	0.998453	0.499226	−	0.866472i	$-0.333618\pi$
$691$	26.2462	0.998453	0.499226	+	0.866472i	$0.333618\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−6.43845	−0.244400
$695$	16.4924	0.625593
$696$	0	0
$697$	−12.2462	−0.463858
$698$	6.87689	0.260294
$699$	0	0
$700$	−3.12311	−0.118042
$701$	44.3542	1.67523	0.837617	−	0.546258i	$-0.183948\pi$
$701$	44.3542	1.67523	0.837617	+	0.546258i	$0.183948\pi$
$702$	0	0
$703$	1.75379	0.0661454
$704$	4.00000	0.150756
$705$	0	0
$706$	−10.4924	−0.394888
$707$	57.7538	2.17205
$708$	0	0
$709$	−32.5464	−1.22231	−0.611153	−	0.791513i	$-0.709294\pi$
$709$	−32.5464	−1.22231	−0.611153	+	0.791513i	$0.709294\pi$
$710$	16.6847	0.626164
$711$	0	0
$712$	2.68466	0.100612
$713$	7.61553	0.285204
$714$	0	0
$715$	−1.75379	−0.0655880
$716$	24.4924	0.915325
$717$	0	0
$718$	−19.1231	−0.713668
$719$	−35.8078	−1.33540	−0.667702	−	0.744429i	$-0.732722\pi$
$719$	−35.8078	−1.33540	−0.667702	+	0.744429i	$0.732722\pi$
$720$	0	0
$721$	−24.9848	−0.930484
$722$	16.5616	0.616357
$723$	0	0
$724$	−0.246211	−0.00915037
$725$	−6.68466	−0.248262
$726$	0	0
$727$	42.4384	1.57395	0.786977	−	0.616982i	$-0.211645\pi$
$727$	42.4384	1.57395	0.786977	+	0.616982i	$0.211645\pi$
$728$	1.36932	0.0507503
$729$	0	0
$730$	13.8078	0.511048
$731$	−7.12311	−0.263458
$732$	0	0
$733$	10.4924	0.387546	0.193773	−	0.981046i	$-0.437927\pi$
$733$	10.4924	0.387546	0.193773	+	0.981046i	$0.437927\pi$
$734$	−7.61553	−0.281094
$735$	0	0
$736$	3.12311	0.115119
$737$	−3.50758	−0.129203
$738$	0	0
$739$	25.1771	0.926154	0.463077	−	0.886318i	$-0.346745\pi$
$739$	25.1771	0.926154	0.463077	+	0.886318i	$0.346745\pi$
$740$	−1.12311	−0.0412862
$741$	0	0
$742$	−11.1231	−0.408342
$743$	12.8769	0.472407	0.236204	−	0.971704i	$-0.424097\pi$
$743$	12.8769	0.472407	0.236204	+	0.971704i	$0.424097\pi$
$744$	0	0
$745$	−5.12311	−0.187696
$746$	−24.7386	−0.905746
$747$	0	0
$748$	−4.00000	−0.146254
$749$	−51.5076	−1.88205
$750$	0	0
$751$	21.1771	0.772763	0.386381	−	0.922339i	$-0.373725\pi$
$751$	21.1771	0.772763	0.386381	+	0.922339i	$0.373725\pi$
$752$	−2.43845	−0.0889210
$753$	0	0
$754$	2.93087	0.106736
$755$	−9.36932	−0.340984
$756$	0	0
$757$	−38.7926	−1.40994	−0.704971	−	0.709236i	$-0.749040\pi$
$757$	−38.7926	−1.40994	−0.704971	+	0.709236i	$0.749040\pi$
$758$	−2.24621	−0.0815861
$759$	0	0
$760$	1.56155	0.0566435
$761$	−28.7386	−1.04177	−0.520887	−	0.853625i	$-0.674399\pi$
$761$	−28.7386	−1.04177	−0.520887	+	0.853625i	$0.674399\pi$
$762$	0	0
$763$	13.8617	0.501829
$764$	0	0
$765$	0	0
$766$	3.80776	0.137580
$767$	5.56155	0.200816
$768$	0	0
$769$	15.5616	0.561164	0.280582	−	0.959830i	$-0.409473\pi$
$769$	15.5616	0.561164	0.280582	+	0.959830i	$0.409473\pi$
$770$	−12.4924	−0.450196
$771$	0	0
$772$	−7.75379	−0.279065
$773$	28.7386	1.03366	0.516828	−	0.856089i	$-0.327113\pi$
$773$	28.7386	1.03366	0.516828	+	0.856089i	$0.327113\pi$
$774$	0	0
$775$	−2.43845	−0.0875916
$776$	−10.6847	−0.383557
$777$	0	0
$778$	−11.3693	−0.407610
$779$	19.1231	0.685156
$780$	0	0
$781$	66.7386	2.38810
$782$	−3.12311	−0.111682
$783$	0	0
$784$	2.75379	0.0983496
$785$	14.4924	0.517257
$786$	0	0
$787$	42.5464	1.51662	0.758308	−	0.651897i	$-0.226027\pi$
$787$	42.5464	1.51662	0.758308	+	0.651897i	$0.226027\pi$
$788$	11.3693	0.405015
$789$	0	0
$790$	0	0
$791$	−55.6155	−1.97746
$792$	0	0
$793$	5.06913	0.180010
$794$	3.36932	0.119573
$795$	0	0
$796$	10.4384	0.369981
$797$	14.4924	0.513348	0.256674	−	0.966498i	$-0.417373\pi$
$797$	14.4924	0.513348	0.256674	+	0.966498i	$0.417373\pi$
$798$	0	0
$799$	2.43845	0.0862661
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	35.3693	1.24893
$803$	55.2311	1.94906
$804$	0	0
$805$	−9.75379	−0.343776
$806$	1.06913	0.0376585
$807$	0	0
$808$	18.4924	0.650561
$809$	46.9848	1.65190	0.825950	−	0.563744i	$-0.190640\pi$
$809$	46.9848	1.65190	0.825950	+	0.563744i	$0.190640\pi$
$810$	0	0
$811$	−21.3693	−0.750378	−0.375189	−	0.926948i	$-0.622422\pi$
$811$	−21.3693	−0.750378	−0.375189	+	0.926948i	$0.622422\pi$
$812$	20.8769	0.732635
$813$	0	0
$814$	−4.49242	−0.157459
$815$	2.24621	0.0786813
$816$	0	0
$817$	11.1231	0.389148
$818$	14.6847	0.513437
$819$	0	0
$820$	−12.2462	−0.427656
$821$	−38.3002	−1.33669	−0.668343	−	0.743853i	$-0.732996\pi$
$821$	−38.3002	−1.33669	−0.668343	+	0.743853i	$0.732996\pi$
$822$	0	0
$823$	25.3693	0.884319	0.442159	−	0.896936i	$-0.354213\pi$
$823$	25.3693	0.884319	0.442159	+	0.896936i	$0.354213\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	39.6155	1.37840
$827$	32.4924	1.12987	0.564936	−	0.825135i	$-0.308901\pi$
$827$	32.4924	1.12987	0.564936	+	0.825135i	$0.308901\pi$
$828$	0	0
$829$	15.3693	0.533798	0.266899	−	0.963724i	$-0.414001\pi$
$829$	15.3693	0.533798	0.266899	+	0.963724i	$0.414001\pi$
$830$	−10.2462	−0.355651
$831$	0	0
$832$	0.438447	0.0152004
$833$	−2.75379	−0.0954131
$834$	0	0
$835$	−3.12311	−0.108080
$836$	6.24621	0.216030
$837$	0	0
$838$	−16.8769	−0.583003
$839$	2.05398	0.0709111	0.0354556	−	0.999371i	$-0.488712\pi$
$839$	2.05398	0.0709111	0.0354556	+	0.999371i	$0.488712\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	33.6155	1.15847
$843$	0	0
$844$	21.3693	0.735562
$845$	12.8078	0.440600
$846$	0	0
$847$	−15.6155	−0.536556
$848$	−3.56155	−0.122304
$849$	0	0
$850$	1.00000	0.0342997
$851$	−3.50758	−0.120238
$852$	0	0
$853$	31.7538	1.08723	0.543615	−	0.839335i	$-0.317055\pi$
$853$	31.7538	1.08723	0.543615	+	0.839335i	$0.317055\pi$
$854$	36.1080	1.23559
$855$	0	0
$856$	−16.4924	−0.563699
$857$	−47.1771	−1.61154	−0.805769	−	0.592230i	$-0.798248\pi$
$857$	−47.1771	−1.61154	−0.805769	+	0.592230i	$0.798248\pi$
$858$	0	0
$859$	34.5464	1.17871	0.589354	−	0.807875i	$-0.299382\pi$
$859$	34.5464	1.17871	0.589354	+	0.807875i	$0.299382\pi$
$860$	−7.12311	−0.242896
$861$	0	0
$862$	12.4924	0.425494
$863$	−44.4924	−1.51454	−0.757270	−	0.653102i	$-0.773467\pi$
$863$	−44.4924	−1.51454	−0.757270	+	0.653102i	$0.773467\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	−30.4924	−1.03617
$867$	0	0
$868$	7.61553	0.258488
$869$	0	0
$870$	0	0
$871$	−0.384472	−0.0130273
$872$	4.43845	0.150305
$873$	0	0
$874$	4.87689	0.164963
$875$	3.12311	0.105580
$876$	0	0
$877$	−10.3845	−0.350659	−0.175329	−	0.984510i	$-0.556099\pi$
$877$	−10.3845	−0.350659	−0.175329	+	0.984510i	$0.556099\pi$
$878$	−32.9848	−1.11318
$879$	0	0
$880$	−4.00000	−0.134840
$881$	−20.7386	−0.698702	−0.349351	−	0.936992i	$-0.613598\pi$
$881$	−20.7386	−0.698702	−0.349351	+	0.936992i	$0.613598\pi$
$882$	0	0
$883$	2.63068	0.0885295	0.0442648	−	0.999020i	$-0.485905\pi$
$883$	2.63068	0.0885295	0.0442648	+	0.999020i	$0.485905\pi$
$884$	−0.438447	−0.0147466
$885$	0	0
$886$	28.0000	0.940678
$887$	56.6004	1.90045	0.950227	−	0.311558i	$-0.100851\pi$
$887$	56.6004	1.90045	0.950227	+	0.311558i	$0.100851\pi$
$888$	0	0
$889$	−42.3542	−1.42051
$890$	−2.68466	−0.0899900
$891$	0	0
$892$	−8.68466	−0.290784
$893$	−3.80776	−0.127422
$894$	0	0
$895$	−24.4924	−0.818691
$896$	3.12311	0.104336
$897$	0	0
$898$	−25.1231	−0.838369
$899$	16.3002	0.543642
$900$	0	0
$901$	3.56155	0.118653
$902$	−48.9848	−1.63102
$903$	0	0
$904$	−17.8078	−0.592277
$905$	0.246211	0.00818434
$906$	0	0
$907$	14.4384	0.479421	0.239710	−	0.970844i	$-0.422948\pi$
$907$	14.4384	0.479421	0.239710	+	0.970844i	$0.422948\pi$
$908$	−28.6847	−0.951934
$909$	0	0
$910$	−1.36932	−0.0453924
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−40.9848	−1.35640
$914$	−0.246211	−0.00814394
$915$	0	0
$916$	18.4924	0.611007
$917$	−22.2462	−0.734635
$918$	0	0
$919$	−12.8769	−0.424770	−0.212385	−	0.977186i	$-0.568123\pi$
$919$	−12.8769	−0.424770	−0.212385	+	0.977186i	$0.568123\pi$
$920$	−3.12311	−0.102966
$921$	0	0
$922$	−30.4924	−1.00421
$923$	7.31534	0.240787
$924$	0	0
$925$	1.12311	0.0369275
$926$	−14.9309	−0.490659
$927$	0	0
$928$	6.68466	0.219435
$929$	−39.4773	−1.29521	−0.647604	−	0.761977i	$-0.724229\pi$
$929$	−39.4773	−1.29521	−0.647604	+	0.761977i	$0.724229\pi$
$930$	0	0
$931$	4.30019	0.140933
$932$	−18.6847	−0.612036
$933$	0	0
$934$	0.492423	0.0161126
$935$	4.00000	0.130814
$936$	0	0
$937$	30.1080	0.983584	0.491792	−	0.870713i	$-0.336342\pi$
$937$	30.1080	0.983584	0.491792	+	0.870713i	$0.336342\pi$
$938$	−2.73863	−0.0894196
$939$	0	0
$940$	2.43845	0.0795334
$941$	31.5616	1.02888	0.514439	−	0.857527i	$-0.328000\pi$
$941$	31.5616	1.02888	0.514439	+	0.857527i	$0.328000\pi$
$942$	0	0
$943$	−38.2462	−1.24547
$944$	12.6847	0.412850
$945$	0	0
$946$	−28.4924	−0.926369
$947$	−32.1922	−1.04611	−0.523054	−	0.852300i	$-0.675207\pi$
$947$	−32.1922	−1.04611	−0.523054	+	0.852300i	$0.675207\pi$
$948$	0	0
$949$	6.05398	0.196520
$950$	−1.56155	−0.0506635
$951$	0	0
$952$	−3.12311	−0.101220
$953$	−30.8769	−1.00020	−0.500100	−	0.865967i	$-0.666704\pi$
$953$	−30.8769	−1.00020	−0.500100	+	0.865967i	$0.666704\pi$
$954$	0	0
$955$	0	0
$956$	−1.36932	−0.0442869
$957$	0	0
$958$	34.4384	1.11266
$959$	31.2311	1.00850
$960$	0	0
$961$	−25.0540	−0.808193
$962$	−0.492423	−0.0158763
$963$	0	0
$964$	−4.24621	−0.136761
$965$	7.75379	0.249603
$966$	0	0
$967$	52.4924	1.68804	0.844021	−	0.536310i	$-0.180182\pi$
$967$	52.4924	1.68804	0.844021	+	0.536310i	$0.180182\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	10.6847	0.343064
$971$	−3.31534	−0.106394	−0.0531972	−	0.998584i	$-0.516941\pi$
$971$	−3.31534	−0.106394	−0.0531972	+	0.998584i	$0.516941\pi$
$972$	0	0
$973$	51.5076	1.65126
$974$	15.6155	0.500354
$975$	0	0
$976$	11.5616	0.370076
$977$	18.8769	0.603925	0.301963	−	0.953320i	$-0.402358\pi$
$977$	18.8769	0.603925	0.301963	+	0.953320i	$0.402358\pi$
$978$	0	0
$979$	−10.7386	−0.343208
$980$	−2.75379	−0.0879666
$981$	0	0
$982$	−1.56155	−0.0498312
$983$	28.8769	0.921030	0.460515	−	0.887652i	$-0.347665\pi$
$983$	28.8769	0.921030	0.460515	+	0.887652i	$0.347665\pi$
$984$	0	0
$985$	−11.3693	−0.362257
$986$	−6.68466	−0.212883
$987$	0	0
$988$	0.684658	0.0217819
$989$	−22.2462	−0.707388
$990$	0	0
$991$	−27.4233	−0.871130	−0.435565	−	0.900157i	$-0.643451\pi$
$991$	−27.4233	−0.871130	−0.435565	+	0.900157i	$0.643451\pi$
$992$	2.43845	0.0774208
$993$	0	0
$994$	52.1080	1.65276
$995$	−10.4384	−0.330921
$996$	0	0
$997$	28.2462	0.894566	0.447283	−	0.894392i	$-0.352392\pi$
$997$	28.2462	0.894566	0.447283	+	0.894392i	$0.352392\pi$
$998$	0.876894	0.0277576
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1530.2.a.r.1.1		2
3.2	odd	2		170.2.a.f.1.2	✓	2
5.4	even	2		7650.2.a.de.1.2		2
12.11	even	2		1360.2.a.m.1.1		2
15.2	even	4		850.2.c.i.749.3		4
15.8	even	4		850.2.c.i.749.2		4
15.14	odd	2		850.2.a.n.1.1		2
21.20	even	2		8330.2.a.bq.1.1		2
24.5	odd	2		5440.2.a.bj.1.1		2
24.11	even	2		5440.2.a.bd.1.2		2
51.38	odd	4		2890.2.b.i.2311.2		4
51.47	odd	4		2890.2.b.i.2311.3		4
51.50	odd	2		2890.2.a.u.1.1		2
60.59	even	2		6800.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.a.f.1.2	✓	2	3.2	odd	2
850.2.a.n.1.1		2	15.14	odd	2
850.2.c.i.749.2		4	15.8	even	4
850.2.c.i.749.3		4	15.2	even	4
1360.2.a.m.1.1		2	12.11	even	2
1530.2.a.r.1.1		2	1.1	even	1	trivial
2890.2.a.u.1.1		2	51.50	odd	2
2890.2.b.i.2311.2		4	51.38	odd	4
2890.2.b.i.2311.3		4	51.47	odd	4
5440.2.a.bd.1.2		2	24.11	even	2
5440.2.a.bj.1.1		2	24.5	odd	2
6800.2.a.be.1.2		2	60.59	even	2
7650.2.a.de.1.2		2	5.4	even	2
8330.2.a.bq.1.1		2	21.20	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:	\( N \)	\(=\)	\( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1530.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.12311 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.12311 q^{7} -1.00000 q^{8} +1.00000 q^{10} +4.00000 q^{11} +0.438447 q^{13} +3.12311 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.56155 q^{19} -1.00000 q^{20} -4.00000 q^{22} -3.12311 q^{23} +1.00000 q^{25} -0.438447 q^{26} -3.12311 q^{28} -6.68466 q^{29} -2.43845 q^{31} -1.00000 q^{32} +1.00000 q^{34} +3.12311 q^{35} +1.12311 q^{37} -1.56155 q^{38} +1.00000 q^{40} +12.2462 q^{41} +7.12311 q^{43} +4.00000 q^{44} +3.12311 q^{46} -2.43845 q^{47} +2.75379 q^{49} -1.00000 q^{50} +0.438447 q^{52} -3.56155 q^{53} -4.00000 q^{55} +3.12311 q^{56} +6.68466 q^{58} +12.6847 q^{59} +11.5616 q^{61} +2.43845 q^{62} +1.00000 q^{64} -0.438447 q^{65} -0.876894 q^{67} -1.00000 q^{68} -3.12311 q^{70} +16.6847 q^{71} +13.8078 q^{73} -1.12311 q^{74} +1.56155 q^{76} -12.4924 q^{77} -1.00000 q^{80} -12.2462 q^{82} -10.2462 q^{83} +1.00000 q^{85} -7.12311 q^{86} -4.00000 q^{88} -2.68466 q^{89} -1.36932 q^{91} -3.12311 q^{92} +2.43845 q^{94} -1.56155 q^{95} +10.6847 q^{97} -2.75379 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} + 8 q^{11} + 5 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} - q^{19} - 2 q^{20} - 8 q^{22} + 2 q^{23} + 2 q^{25} - 5 q^{26} + 2 q^{28} - q^{29} - 9 q^{31} - 2 q^{32} + 2 q^{34} - 2 q^{35} - 6 q^{37} + q^{38} + 2 q^{40} + 8 q^{41} + 6 q^{43} + 8 q^{44} - 2 q^{46} - 9 q^{47} + 22 q^{49} - 2 q^{50} + 5 q^{52} - 3 q^{53} - 8 q^{55} - 2 q^{56} + q^{58} + 13 q^{59} + 19 q^{61} + 9 q^{62} + 2 q^{64} - 5 q^{65} - 10 q^{67} - 2 q^{68} + 2 q^{70} + 21 q^{71} + 7 q^{73} + 6 q^{74} - q^{76} + 8 q^{77} - 2 q^{80} - 8 q^{82} - 4 q^{83} + 2 q^{85} - 6 q^{86} - 8 q^{88} + 7 q^{89} + 22 q^{91} + 2 q^{92} + 9 q^{94} + q^{95} + 9 q^{97} - 22 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 2 * q^10 + 8 * q^11 + 5 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 - q^19 - 2 * q^20 - 8 * q^22 + 2 * q^23 + 2 * q^25 - 5 * q^26 + 2 * q^28 - q^29 - 9 * q^31 - 2 * q^32 + 2 * q^34 - 2 * q^35 - 6 * q^37 + q^38 + 2 * q^40 + 8 * q^41 + 6 * q^43 + 8 * q^44 - 2 * q^46 - 9 * q^47 + 22 * q^49 - 2 * q^50 + 5 * q^52 - 3 * q^53 - 8 * q^55 - 2 * q^56 + q^58 + 13 * q^59 + 19 * q^61 + 9 * q^62 + 2 * q^64 - 5 * q^65 - 10 * q^67 - 2 * q^68 + 2 * q^70 + 21 * q^71 + 7 * q^73 + 6 * q^74 - q^76 + 8 * q^77 - 2 * q^80 - 8 * q^82 - 4 * q^83 + 2 * q^85 - 6 * q^86 - 8 * q^88 + 7 * q^89 + 22 * q^91 + 2 * q^92 + 9 * q^94 + q^95 + 9 * q^97 - 22 * q^98

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