LMFDB - Embedded newform 1530.2.a.r.1.2

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			1.2
Root			$2.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +5.12311 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +5.12311 q^{7} -1.00000 q^{8} +1.00000 q^{10} +4.00000 q^{11} +4.56155 q^{13} -5.12311 q^{14} +1.00000 q^{16} -1.00000 q^{17} -2.56155 q^{19} -1.00000 q^{20} -4.00000 q^{22} +5.12311 q^{23} +1.00000 q^{25} -4.56155 q^{26} +5.12311 q^{28} +5.68466 q^{29} -6.56155 q^{31} -1.00000 q^{32} +1.00000 q^{34} -5.12311 q^{35} -7.12311 q^{37} +2.56155 q^{38} +1.00000 q^{40} -4.24621 q^{41} -1.12311 q^{43} +4.00000 q^{44} -5.12311 q^{46} -6.56155 q^{47} +19.2462 q^{49} -1.00000 q^{50} +4.56155 q^{52} +0.561553 q^{53} -4.00000 q^{55} -5.12311 q^{56} -5.68466 q^{58} +0.315342 q^{59} +7.43845 q^{61} +6.56155 q^{62} +1.00000 q^{64} -4.56155 q^{65} -9.12311 q^{67} -1.00000 q^{68} +5.12311 q^{70} +4.31534 q^{71} -6.80776 q^{73} +7.12311 q^{74} -2.56155 q^{76} +20.4924 q^{77} -1.00000 q^{80} +4.24621 q^{82} +6.24621 q^{83} +1.00000 q^{85} +1.12311 q^{86} -4.00000 q^{88} +9.68466 q^{89} +23.3693 q^{91} +5.12311 q^{92} +6.56155 q^{94} +2.56155 q^{95} -1.68466 q^{97} -19.2462 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	5.12311	1.93635	0.968176	−	0.250270i	$-0.0805195\pi$
$7$	5.12311	1.93635	0.968176	+	0.250270i	$0.0805195\pi$
$8$	−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
$13$	4.56155	1.26515	0.632574	−	0.774500i	$-0.281999\pi$
$13$	4.56155	1.26515	0.632574	+	0.774500i	$0.281999\pi$
$14$	−5.12311	−1.36921
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−2.56155	−0.587661	−0.293830	−	0.955858i	$-0.594930\pi$
$19$	−2.56155	−0.587661	−0.293830	+	0.955858i	$0.594930\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−4.00000	−0.852803
$23$	5.12311	1.06824	0.534121	−	0.845408i	$-0.320643\pi$
$23$	5.12311	1.06824	0.534121	+	0.845408i	$0.320643\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.56155	−0.894594
$27$	0	0
$28$	5.12311	0.968176
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	−6.56155	−1.17849	−0.589245	−	0.807955i	$-0.700575\pi$
$31$	−6.56155	−1.17849	−0.589245	+	0.807955i	$0.700575\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	−5.12311	−0.865963
$36$	0	0
$37$	−7.12311	−1.17103	−0.585516	−	0.810661i	$-0.699108\pi$
$37$	−7.12311	−1.17103	−0.585516	+	0.810661i	$0.699108\pi$
$38$	2.56155	0.415539
$39$	0	0
$40$	1.00000	0.158114
$41$	−4.24621	−0.663147	−0.331573	−	0.943429i	$-0.607579\pi$
$41$	−4.24621	−0.663147	−0.331573	+	0.943429i	$0.607579\pi$
$42$	0	0
$43$	−1.12311	−0.171272	−0.0856360	−	0.996326i	$-0.527292\pi$
$43$	−1.12311	−0.171272	−0.0856360	+	0.996326i	$0.527292\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−5.12311	−0.755361
$47$	−6.56155	−0.957101	−0.478550	−	0.878060i	$-0.658838\pi$
$47$	−6.56155	−0.957101	−0.478550	+	0.878060i	$0.658838\pi$
$48$	0	0
$49$	19.2462	2.74946
$50$	−1.00000	−0.141421
$51$	0	0
$52$	4.56155	0.632574
$53$	0.561553	0.0771352	0.0385676	−	0.999256i	$-0.487721\pi$
$53$	0.561553	0.0771352	0.0385676	+	0.999256i	$0.487721\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	−5.12311	−0.684604
$57$	0	0
$58$	−5.68466	−0.746432
$59$	0.315342	0.0410540	0.0205270	−	0.999789i	$-0.493466\pi$
$59$	0.315342	0.0410540	0.0205270	+	0.999789i	$0.493466\pi$
$60$	0	0
$61$	7.43845	0.952396	0.476198	−	0.879338i	$-0.342015\pi$
$61$	7.43845	0.952396	0.476198	+	0.879338i	$0.342015\pi$
$62$	6.56155	0.833318
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.56155	−0.565791
$66$	0	0
$67$	−9.12311	−1.11456	−0.557282	−	0.830323i	$-0.688156\pi$
$67$	−9.12311	−1.11456	−0.557282	+	0.830323i	$0.688156\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	5.12311	0.612328
$71$	4.31534	0.512137	0.256068	−	0.966659i	$-0.417573\pi$
$71$	4.31534	0.512137	0.256068	+	0.966659i	$0.417573\pi$
$72$	0	0
$73$	−6.80776	−0.796789	−0.398394	−	0.917214i	$-0.630432\pi$
$73$	−6.80776	−0.796789	−0.398394	+	0.917214i	$0.630432\pi$
$74$	7.12311	0.828044
$75$	0	0
$76$	−2.56155	−0.293830
$77$	20.4924	2.33533
$78$	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
$82$	4.24621	0.468916
$83$	6.24621	0.685611	0.342805	−	0.939406i	$-0.388623\pi$
$83$	6.24621	0.685611	0.342805	+	0.939406i	$0.388623\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	1.12311	0.121108
$87$	0	0
$88$	−4.00000	−0.426401
$89$	9.68466	1.02657	0.513286	−	0.858218i	$-0.328428\pi$
$89$	9.68466	1.02657	0.513286	+	0.858218i	$0.328428\pi$
$90$	0	0
$91$	23.3693	2.44977
$92$	5.12311	0.534121
$93$	0	0
$94$	6.56155	0.676772
$95$	2.56155	0.262810
$96$	0	0
$97$	−1.68466	−0.171051	−0.0855256	−	0.996336i	$-0.527257\pi$
$97$	−1.68466	−0.171051	−0.0855256	+	0.996336i	$0.527257\pi$
$98$	−19.2462	−1.94416
$99$	0	0
$100$	1.00000	0.100000
$101$	14.4924	1.44205	0.721025	−	0.692909i	$-0.243671\pi$
$101$	14.4924	1.44205	0.721025	+	0.692909i	$0.243671\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−4.56155	−0.447297
$105$	0	0
$106$	−0.561553	−0.0545428
$107$	−16.4924	−1.59438	−0.797191	−	0.603727i	$-0.793682\pi$
$107$	−16.4924	−1.59438	−0.797191	+	0.603727i	$0.793682\pi$
$108$	0	0
$109$	−8.56155	−0.820048	−0.410024	−	0.912075i	$-0.634480\pi$
$109$	−8.56155	−0.820048	−0.410024	+	0.912075i	$0.634480\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	5.12311	0.484088
$113$	−2.80776	−0.264132	−0.132066	−	0.991241i	$-0.542161\pi$
$113$	−2.80776	−0.264132	−0.132066	+	0.991241i	$0.542161\pi$
$114$	0	0
$115$	−5.12311	−0.477732
$116$	5.68466	0.527807
$117$	0	0
$118$	−0.315342	−0.0290295
$119$	−5.12311	−0.469634
$120$	0	0
$121$	5.00000	0.454545
$122$	−7.43845	−0.673445
$123$	0	0
$124$	−6.56155	−0.589245
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.43845	0.837527	0.418763	−	0.908095i	$-0.362464\pi$
$127$	9.43845	0.837527	0.418763	+	0.908095i	$0.362464\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	4.56155	0.400075
$131$	−1.12311	−0.0981262	−0.0490631	−	0.998796i	$-0.515624\pi$
$131$	−1.12311	−0.0981262	−0.0490631	+	0.998796i	$0.515624\pi$
$132$	0	0
$133$	−13.1231	−1.13792
$134$	9.12311	0.788116
$135$	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	$-0.253435\pi$
$139$	16.4924	1.39887	0.699435	+	0.714697i	$0.253435\pi$
$140$	−5.12311	−0.432981
$141$	0	0
$142$	−4.31534	−0.362135
$143$	18.2462	1.52582
$144$	0	0
$145$	−5.68466	−0.472085
$146$	6.80776	0.563415
$147$	0	0
$148$	−7.12311	−0.585516
$149$	−3.12311	−0.255855	−0.127927	−	0.991784i	$-0.540832\pi$
$149$	−3.12311	−0.255855	−0.127927	+	0.991784i	$0.540832\pi$
$150$	0	0
$151$	−15.3693	−1.25074	−0.625369	−	0.780329i	$-0.715051\pi$
$151$	−15.3693	−1.25074	−0.625369	+	0.780329i	$0.715051\pi$
$152$	2.56155	0.207769
$153$	0	0
$154$	−20.4924	−1.65133
$155$	6.56155	0.527037
$156$	0	0
$157$	18.4924	1.47586	0.737928	−	0.674879i	$-0.235804\pi$
$157$	18.4924	1.47586	0.737928	+	0.674879i	$0.235804\pi$
$158$	0	0
$159$	0	0
$160$	1.00000	0.0790569
$161$	26.2462	2.06849
$162$	0	0
$163$	14.2462	1.11585	0.557925	−	0.829892i	$-0.311598\pi$
$163$	14.2462	1.11585	0.557925	+	0.829892i	$0.311598\pi$
$164$	−4.24621	−0.331573
$165$	0	0
$166$	−6.24621	−0.484800
$167$	−5.12311	−0.396438	−0.198219	−	0.980158i	$-0.563516\pi$
$167$	−5.12311	−0.396438	−0.198219	+	0.980158i	$0.563516\pi$
$168$	0	0
$169$	7.80776	0.600597
$170$	−1.00000	−0.0766965
$171$	0	0
$172$	−1.12311	−0.0856360
$173$	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
$175$	5.12311	0.387270
$176$	4.00000	0.301511
$177$	0	0
$178$	−9.68466	−0.725896
$179$	−8.49242	−0.634753	−0.317377	−	0.948300i	$-0.602802\pi$
$179$	−8.49242	−0.634753	−0.317377	+	0.948300i	$0.602802\pi$
$180$	0	0
$181$	16.2462	1.20757	0.603786	−	0.797147i	$-0.293658\pi$
$181$	16.2462	1.20757	0.603786	+	0.797147i	$0.293658\pi$
$182$	−23.3693	−1.73225
$183$	0	0
$184$	−5.12311	−0.377680
$185$	7.12311	0.523701
$186$	0	0
$187$	−4.00000	−0.292509
$188$	−6.56155	−0.478550
$189$	0	0
$190$	−2.56155	−0.185835
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−24.2462	−1.74528	−0.872640	−	0.488364i	$-0.837594\pi$
$193$	−24.2462	−1.74528	−0.872640	+	0.488364i	$0.837594\pi$
$194$	1.68466	0.120951
$195$	0	0
$196$	19.2462	1.37473
$197$	−13.3693	−0.952524	−0.476262	−	0.879303i	$-0.658009\pi$
$197$	−13.3693	−0.952524	−0.476262	+	0.879303i	$0.658009\pi$
$198$	0	0
$199$	14.5616	1.03224	0.516121	−	0.856516i	$-0.327376\pi$
$199$	14.5616	1.03224	0.516121	+	0.856516i	$0.327376\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−14.4924	−1.01968
$203$	29.1231	2.04404
$204$	0	0
$205$	4.24621	0.296568
$206$	−8.00000	−0.557386
$207$	0	0
$208$	4.56155	0.316287
$209$	−10.2462	−0.708745
$210$	0	0
$211$	−3.36932	−0.231953	−0.115977	−	0.993252i	$-0.537000\pi$
$211$	−3.36932	−0.231953	−0.115977	+	0.993252i	$0.537000\pi$
$212$	0.561553	0.0385676
$213$	0	0
$214$	16.4924	1.12740
$215$	1.12311	0.0765952
$216$	0	0
$217$	−33.6155	−2.28197
$218$	8.56155	0.579862
$219$	0	0
$220$	−4.00000	−0.269680
$221$	−4.56155	−0.306843
$222$	0	0
$223$	3.68466	0.246743	0.123371	−	0.992361i	$-0.460629\pi$
$223$	3.68466	0.246743	0.123371	+	0.992361i	$0.460629\pi$
$224$	−5.12311	−0.342302
$225$	0	0
$226$	2.80776	0.186770
$227$	−16.3153	−1.08289	−0.541444	−	0.840737i	$-0.682122\pi$
$227$	−16.3153	−1.08289	−0.541444	+	0.840737i	$0.682122\pi$
$228$	0	0
$229$	−14.4924	−0.957686	−0.478843	−	0.877900i	$-0.658944\pi$
$229$	−14.4924	−0.957686	−0.478843	+	0.877900i	$0.658944\pi$
$230$	5.12311	0.337808
$231$	0	0
$232$	−5.68466	−0.373216
$233$	−6.31534	−0.413732	−0.206866	−	0.978369i	$-0.566326\pi$
$233$	−6.31534	−0.413732	−0.206866	+	0.978369i	$0.566326\pi$
$234$	0	0
$235$	6.56155	0.428029
$236$	0.315342	0.0205270
$237$	0	0
$238$	5.12311	0.332082
$239$	23.3693	1.51164	0.755818	−	0.654782i	$-0.227240\pi$
$239$	23.3693	1.51164	0.755818	+	0.654782i	$0.227240\pi$
$240$	0	0
$241$	12.2462	0.788848	0.394424	−	0.918929i	$-0.370944\pi$
$241$	12.2462	0.788848	0.394424	+	0.918929i	$0.370944\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	7.43845	0.476198
$245$	−19.2462	−1.22960
$246$	0	0
$247$	−11.6847	−0.743477
$248$	6.56155	0.416659
$249$	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
$253$	20.4924	1.28835
$254$	−9.43845	−0.592221
$255$	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
$259$	−36.4924	−2.26753
$260$	−4.56155	−0.282895
$261$	0	0
$262$	1.12311	0.0693857
$263$	−24.8078	−1.52971	−0.764856	−	0.644201i	$-0.777190\pi$
$263$	−24.8078	−1.52971	−0.764856	+	0.644201i	$0.777190\pi$
$264$	0	0
$265$	−0.561553	−0.0344959
$266$	13.1231	0.804629
$267$	0	0
$268$	−9.12311	−0.557282
$269$	2.80776	0.171192	0.0855962	−	0.996330i	$-0.472721\pi$
$269$	2.80776	0.171192	0.0855962	+	0.996330i	$0.472721\pi$
$270$	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
$277$	−15.7538	−0.946553	−0.473277	−	0.880914i	$-0.656929\pi$
$277$	−15.7538	−0.946553	−0.473277	+	0.880914i	$0.656929\pi$
$278$	−16.4924	−0.989150
$279$	0	0
$280$	5.12311	0.306164
$281$	−16.5616	−0.987979	−0.493990	−	0.869468i	$-0.664462\pi$
$281$	−16.5616	−0.987979	−0.493990	+	0.869468i	$0.664462\pi$
$282$	0	0
$283$	8.31534	0.494296	0.247148	−	0.968978i	$-0.420507\pi$
$283$	8.31534	0.494296	0.247148	+	0.968978i	$0.420507\pi$
$284$	4.31534	0.256068
$285$	0	0
$286$	−18.2462	−1.07892
$287$	−21.7538	−1.28409
$288$	0	0
$289$	1.00000	0.0588235
$290$	5.68466	0.333815
$291$	0	0
$292$	−6.80776	−0.398394
$293$	16.5616	0.967536	0.483768	−	0.875196i	$-0.339268\pi$
$293$	16.5616	0.967536	0.483768	+	0.875196i	$0.339268\pi$
$294$	0	0
$295$	−0.315342	−0.0183599
$296$	7.12311	0.414022
$297$	0	0
$298$	3.12311	0.180917
$299$	23.3693	1.35148
$300$	0	0
$301$	−5.75379	−0.331643
$302$	15.3693	0.884405
$303$	0	0
$304$	−2.56155	−0.146915
$305$	−7.43845	−0.425924
$306$	0	0
$307$	−17.7538	−1.01326	−0.506631	−	0.862163i	$-0.669109\pi$
$307$	−17.7538	−1.01326	−0.506631	+	0.862163i	$0.669109\pi$
$308$	20.4924	1.16766
$309$	0	0
$310$	−6.56155	−0.372671
$311$	−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	$0.554238\pi$
$314$	−18.4924	−1.04359
$315$	0	0
$316$	0	0
$317$	−31.6155	−1.77570	−0.887852	−	0.460128i	$-0.847803\pi$
$317$	−31.6155	−1.77570	−0.887852	+	0.460128i	$0.847803\pi$
$318$	0	0
$319$	22.7386	1.27312
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−26.2462	−1.46264
$323$	2.56155	0.142529
$324$	0	0
$325$	4.56155	0.253029
$326$	−14.2462	−0.789025
$327$	0	0
$328$	4.24621	0.234458
$329$	−33.6155	−1.85328
$330$	0	0
$331$	23.0540	1.26716	0.633581	−	0.773677i	$-0.281585\pi$
$331$	23.0540	1.26716	0.633581	+	0.773677i	$0.281585\pi$
$332$	6.24621	0.342805
$333$	0	0
$334$	5.12311	0.280324
$335$	9.12311	0.498449
$336$	0	0
$337$	−10.3153	−0.561912	−0.280956	−	0.959721i	$-0.590652\pi$
$337$	−10.3153	−0.561912	−0.280956	+	0.959721i	$0.590652\pi$
$338$	−7.80776	−0.424686
$339$	0	0
$340$	1.00000	0.0542326
$341$	−26.2462	−1.42131
$342$	0	0
$343$	62.7386	3.38757
$344$	1.12311	0.0605538
$345$	0	0
$346$	−18.0000	−0.967686
$347$	10.5616	0.566974	0.283487	−	0.958976i	$-0.408509\pi$
$347$	10.5616	0.566974	0.283487	+	0.958976i	$0.408509\pi$
$348$	0	0
$349$	−15.1231	−0.809521	−0.404761	−	0.914423i	$-0.632645\pi$
$349$	−15.1231	−0.809521	−0.404761	+	0.914423i	$0.632645\pi$
$350$	−5.12311	−0.273842
$351$	0	0
$352$	−4.00000	−0.213201
$353$	−22.4924	−1.19715	−0.598575	−	0.801066i	$-0.704266\pi$
$353$	−22.4924	−1.19715	−0.598575	+	0.801066i	$0.704266\pi$
$354$	0	0
$355$	−4.31534	−0.229035
$356$	9.68466	0.513286
$357$	0	0
$358$	8.49242	0.448838
$359$	10.8769	0.574061	0.287030	−	0.957922i	$-0.407332\pi$
$359$	10.8769	0.574061	0.287030	+	0.957922i	$0.407332\pi$
$360$	0	0
$361$	−12.4384	−0.654655
$362$	−16.2462	−0.853882
$363$	0	0
$364$	23.3693	1.22489
$365$	6.80776	0.356335
$366$	0	0
$367$	−33.6155	−1.75472	−0.877358	−	0.479836i	$-0.840696\pi$
$367$	−33.6155	−1.75472	−0.877358	+	0.479836i	$0.840696\pi$
$368$	5.12311	0.267060
$369$	0	0
$370$	−7.12311	−0.370313
$371$	2.87689	0.149361
$372$	0	0
$373$	−24.7386	−1.28092	−0.640459	−	0.767992i	$-0.721256\pi$
$373$	−24.7386	−1.28092	−0.640459	+	0.767992i	$0.721256\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	6.56155	0.338386
$377$	25.9309	1.33551
$378$	0	0
$379$	−14.2462	−0.731779	−0.365889	−	0.930658i	$-0.619235\pi$
$379$	−14.2462	−0.731779	−0.365889	+	0.930658i	$0.619235\pi$
$380$	2.56155	0.131405
$381$	0	0
$382$	0	0
$383$	16.8078	0.858837	0.429418	−	0.903106i	$-0.358719\pi$
$383$	16.8078	0.858837	0.429418	+	0.903106i	$0.358719\pi$
$384$	0	0
$385$	−20.4924	−1.04439
$386$	24.2462	1.23410
$387$	0	0
$388$	−1.68466	−0.0855256
$389$	−13.3693	−0.677851	−0.338926	−	0.940813i	$-0.610064\pi$
$389$	−13.3693	−0.677851	−0.338926	+	0.940813i	$0.610064\pi$
$390$	0	0
$391$	−5.12311	−0.259087
$392$	−19.2462	−0.972080
$393$	0	0
$394$	13.3693	0.673536
$395$	0	0
$396$	0	0
$397$	21.3693	1.07250	0.536248	−	0.844061i	$-0.319841\pi$
$397$	21.3693	1.07250	0.536248	+	0.844061i	$0.319841\pi$
$398$	−14.5616	−0.729905
$399$	0	0
$400$	1.00000	0.0500000
$401$	−10.6307	−0.530871	−0.265435	−	0.964129i	$-0.585516\pi$
$401$	−10.6307	−0.530871	−0.265435	+	0.964129i	$0.585516\pi$
$402$	0	0
$403$	−29.9309	−1.49096
$404$	14.4924	0.721025
$405$	0	0
$406$	−29.1231	−1.44536
$407$	−28.4924	−1.41232
$408$	0	0
$409$	−2.31534	−0.114486	−0.0572431	−	0.998360i	$-0.518231\pi$
$409$	−2.31534	−0.114486	−0.0572431	+	0.998360i	$0.518231\pi$
$410$	−4.24621	−0.209705
$411$	0	0
$412$	8.00000	0.394132
$413$	1.61553	0.0794949
$414$	0	0
$415$	−6.24621	−0.306614
$416$	−4.56155	−0.223649
$417$	0	0
$418$	10.2462	0.501159
$419$	25.1231	1.22734	0.613672	−	0.789561i	$-0.289692\pi$
$419$	25.1231	1.22734	0.613672	+	0.789561i	$0.289692\pi$
$420$	0	0
$421$	7.61553	0.371158	0.185579	−	0.982629i	$-0.440584\pi$
$421$	7.61553	0.371158	0.185579	+	0.982629i	$0.440584\pi$
$422$	3.36932	0.164016
$423$	0	0
$424$	−0.561553	−0.0272714
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	38.1080	1.84417
$428$	−16.4924	−0.797191
$429$	0	0
$430$	−1.12311	−0.0541610
$431$	20.4924	0.987085	0.493543	−	0.869722i	$-0.335702\pi$
$431$	20.4924	0.987085	0.493543	+	0.869722i	$0.335702\pi$
$432$	0	0
$433$	−2.49242	−0.119778	−0.0598891	−	0.998205i	$-0.519075\pi$
$433$	−2.49242	−0.119778	−0.0598891	+	0.998205i	$0.519075\pi$
$434$	33.6155	1.61360
$435$	0	0
$436$	−8.56155	−0.410024
$437$	−13.1231	−0.627763
$438$	0	0
$439$	−32.9848	−1.57428	−0.787140	−	0.616774i	$-0.788439\pi$
$439$	−32.9848	−1.57428	−0.787140	+	0.616774i	$0.788439\pi$
$440$	4.00000	0.190693
$441$	0	0
$442$	4.56155	0.216971
$443$	−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
$445$	−9.68466	−0.459097
$446$	−3.68466	−0.174474
$447$	0	0
$448$	5.12311	0.242044
$449$	16.8769	0.796470	0.398235	−	0.917283i	$-0.369623\pi$
$449$	16.8769	0.796470	0.398235	+	0.917283i	$0.369623\pi$
$450$	0	0
$451$	−16.9848	−0.799785
$452$	−2.80776	−0.132066
$453$	0	0
$454$	16.3153	0.765717
$455$	−23.3693	−1.09557
$456$	0	0
$457$	−16.2462	−0.759966	−0.379983	−	0.924994i	$-0.624070\pi$
$457$	−16.2462	−0.759966	−0.379983	+	0.924994i	$0.624070\pi$
$458$	14.4924	0.677186
$459$	0	0
$460$	−5.12311	−0.238866
$461$	−2.49242	−0.116084	−0.0580418	−	0.998314i	$-0.518486\pi$
$461$	−2.49242	−0.116084	−0.0580418	+	0.998314i	$0.518486\pi$
$462$	0	0
$463$	−13.9309	−0.647422	−0.323711	−	0.946156i	$-0.604931\pi$
$463$	−13.9309	−0.647422	−0.323711	+	0.946156i	$0.604931\pi$
$464$	5.68466	0.263904
$465$	0	0
$466$	6.31534	0.292553
$467$	32.4924	1.50357	0.751785	−	0.659408i	$-0.229193\pi$
$467$	32.4924	1.50357	0.751785	+	0.659408i	$0.229193\pi$
$468$	0	0
$469$	−46.7386	−2.15819
$470$	−6.56155	−0.302662
$471$	0	0
$472$	−0.315342	−0.0145148
$473$	−4.49242	−0.206562
$474$	0	0
$475$	−2.56155	−0.117532
$476$	−5.12311	−0.234817
$477$	0	0
$478$	−23.3693	−1.06889
$479$	−38.5616	−1.76192	−0.880961	−	0.473189i	$-0.843103\pi$
$479$	−38.5616	−1.76192	−0.880961	+	0.473189i	$0.843103\pi$
$480$	0	0
$481$	−32.4924	−1.48153
$482$	−12.2462	−0.557800
$483$	0	0
$484$	5.00000	0.227273
$485$	1.68466	0.0764964
$486$	0	0
$487$	25.6155	1.16075	0.580375	−	0.814349i	$-0.302906\pi$
$487$	25.6155	1.16075	0.580375	+	0.814349i	$0.302906\pi$
$488$	−7.43845	−0.336723
$489$	0	0
$490$	19.2462	0.869455
$491$	−2.56155	−0.115601	−0.0578006	−	0.998328i	$-0.518409\pi$
$491$	−2.56155	−0.115601	−0.0578006	+	0.998328i	$0.518409\pi$
$492$	0	0
$493$	−5.68466	−0.256024
$494$	11.6847	0.525718
$495$	0	0
$496$	−6.56155	−0.294622
$497$	22.1080	0.991677
$498$	0	0
$499$	−9.12311	−0.408406	−0.204203	−	0.978929i	$-0.565460\pi$
$499$	−9.12311	−0.408406	−0.204203	+	0.978929i	$0.565460\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−4.00000	−0.178529
$503$	−22.7386	−1.01387	−0.506933	−	0.861986i	$-0.669221\pi$
$503$	−22.7386	−1.01387	−0.506933	+	0.861986i	$0.669221\pi$
$504$	0	0
$505$	−14.4924	−0.644904
$506$	−20.4924	−0.910999
$507$	0	0
$508$	9.43845	0.418763
$509$	−16.8769	−0.748055	−0.374028	−	0.927418i	$-0.622023\pi$
$509$	−16.8769	−0.748055	−0.374028	+	0.927418i	$0.622023\pi$
$510$	0	0
$511$	−34.8769	−1.54286
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	2.00000	0.0882162
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−26.2462	−1.15431
$518$	36.4924	1.60338
$519$	0	0
$520$	4.56155	0.200037
$521$	−43.6155	−1.91083	−0.955415	−	0.295265i	$-0.904592\pi$
$521$	−43.6155	−1.91083	−0.955415	+	0.295265i	$0.904592\pi$
$522$	0	0
$523$	39.8617	1.74303	0.871516	−	0.490367i	$-0.163137\pi$
$523$	39.8617	1.74303	0.871516	+	0.490367i	$0.163137\pi$
$524$	−1.12311	−0.0490631
$525$	0	0
$526$	24.8078	1.08167
$527$	6.56155	0.285826
$528$	0	0
$529$	3.24621	0.141140
$530$	0.561553	0.0243923
$531$	0	0
$532$	−13.1231	−0.568959
$533$	−19.3693	−0.838978
$534$	0	0
$535$	16.4924	0.713030
$536$	9.12311	0.394058
$537$	0	0
$538$	−2.80776	−0.121051
$539$	76.9848	3.31597
$540$	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
$545$	8.56155	0.366737
$546$	0	0
$547$	29.4384	1.25870	0.629349	−	0.777123i	$-0.283322\pi$
$547$	29.4384	1.25870	0.629349	+	0.777123i	$0.283322\pi$
$548$	−10.0000	−0.427179
$549$	0	0
$550$	−4.00000	−0.170561
$551$	−14.5616	−0.620343
$552$	0	0
$553$	0	0
$554$	15.7538	0.669314
$555$	0	0
$556$	16.4924	0.699435
$557$	−36.5616	−1.54916	−0.774581	−	0.632474i	$-0.782039\pi$
$557$	−36.5616	−1.54916	−0.774581	+	0.632474i	$0.782039\pi$
$558$	0	0
$559$	−5.12311	−0.216684
$560$	−5.12311	−0.216491
$561$	0	0
$562$	16.5616	0.698607
$563$	4.63068	0.195160	0.0975800	−	0.995228i	$-0.468890\pi$
$563$	4.63068	0.195160	0.0975800	+	0.995228i	$0.468890\pi$
$564$	0	0
$565$	2.80776	0.118124
$566$	−8.31534	−0.349520
$567$	0	0
$568$	−4.31534	−0.181068
$569$	3.93087	0.164791	0.0823953	−	0.996600i	$-0.473743\pi$
$569$	3.93087	0.164791	0.0823953	+	0.996600i	$0.473743\pi$
$570$	0	0
$571$	25.1231	1.05137	0.525685	−	0.850680i	$-0.323809\pi$
$571$	25.1231	1.05137	0.525685	+	0.850680i	$0.323809\pi$
$572$	18.2462	0.762912
$573$	0	0
$574$	21.7538	0.907986
$575$	5.12311	0.213648
$576$	0	0
$577$	−5.36932	−0.223528	−0.111764	−	0.993735i	$-0.535650\pi$
$577$	−5.36932	−0.223528	−0.111764	+	0.993735i	$0.535650\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	−5.68466	−0.236043
$581$	32.0000	1.32758
$582$	0	0
$583$	2.24621	0.0930286
$584$	6.80776	0.281707
$585$	0	0
$586$	−16.5616	−0.684151
$587$	6.87689	0.283840	0.141920	−	0.989878i	$-0.454672\pi$
$587$	6.87689	0.283840	0.141920	+	0.989878i	$0.454672\pi$
$588$	0	0
$589$	16.8078	0.692552
$590$	0.315342	0.0129824
$591$	0	0
$592$	−7.12311	−0.292758
$593$	8.24621	0.338631	0.169316	−	0.985562i	$-0.445844\pi$
$593$	8.24621	0.338631	0.169316	+	0.985562i	$0.445844\pi$
$594$	0	0
$595$	5.12311	0.210027
$596$	−3.12311	−0.127927
$597$	0	0
$598$	−23.3693	−0.955642
$599$	9.61553	0.392880	0.196440	−	0.980516i	$-0.437062\pi$
$599$	9.61553	0.392880	0.196440	+	0.980516i	$0.437062\pi$
$600$	0	0
$601$	−7.61553	−0.310644	−0.155322	−	0.987864i	$-0.549641\pi$
$601$	−7.61553	−0.310644	−0.155322	+	0.987864i	$0.549641\pi$
$602$	5.75379	0.234507
$603$	0	0
$604$	−15.3693	−0.625369
$605$	−5.00000	−0.203279
$606$	0	0
$607$	−43.8617	−1.78029	−0.890147	−	0.455674i	$-0.849398\pi$
$607$	−43.8617	−1.78029	−0.890147	+	0.455674i	$0.849398\pi$
$608$	2.56155	0.103885
$609$	0	0
$610$	7.43845	0.301174
$611$	−29.9309	−1.21087
$612$	0	0
$613$	−23.9309	−0.966559	−0.483279	−	0.875466i	$-0.660554\pi$
$613$	−23.9309	−0.966559	−0.483279	+	0.875466i	$0.660554\pi$
$614$	17.7538	0.716485
$615$	0	0
$616$	−20.4924	−0.825663
$617$	−6.31534	−0.254246	−0.127123	−	0.991887i	$-0.540574\pi$
$617$	−6.31534	−0.254246	−0.127123	+	0.991887i	$0.540574\pi$
$618$	0	0
$619$	9.12311	0.366689	0.183344	−	0.983049i	$-0.441308\pi$
$619$	9.12311	0.366689	0.183344	+	0.983049i	$0.441308\pi$
$620$	6.56155	0.263518
$621$	0	0
$622$	8.00000	0.320771
$623$	49.6155	1.98780
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	0	0
$628$	18.4924	0.737928
$629$	7.12311	0.284017
$630$	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
$634$	31.6155	1.25561
$635$	−9.43845	−0.374553
$636$	0	0
$637$	87.7926	3.47847
$638$	−22.7386	−0.900231
$639$	0	0
$640$	1.00000	0.0395285
$641$	24.2462	0.957668	0.478834	−	0.877906i	$-0.341060\pi$
$641$	24.2462	0.957668	0.478834	+	0.877906i	$0.341060\pi$
$642$	0	0
$643$	40.4924	1.59687	0.798433	−	0.602084i	$-0.205663\pi$
$643$	40.4924	1.59687	0.798433	+	0.602084i	$0.205663\pi$
$644$	26.2462	1.03425
$645$	0	0
$646$	−2.56155	−0.100783
$647$	−11.6847	−0.459371	−0.229686	−	0.973265i	$-0.573770\pi$
$647$	−11.6847	−0.459371	−0.229686	+	0.973265i	$0.573770\pi$
$648$	0	0
$649$	1.26137	0.0495130
$650$	−4.56155	−0.178919
$651$	0	0
$652$	14.2462	0.557925
$653$	38.4924	1.50632	0.753162	−	0.657835i	$-0.228527\pi$
$653$	38.4924	1.50632	0.753162	+	0.657835i	$0.228527\pi$
$654$	0	0
$655$	1.12311	0.0438834
$656$	−4.24621	−0.165787
$657$	0	0
$658$	33.6155	1.31047
$659$	41.9309	1.63339	0.816697	−	0.577066i	$-0.195803\pi$
$659$	41.9309	1.63339	0.816697	+	0.577066i	$0.195803\pi$
$660$	0	0
$661$	17.8617	0.694741	0.347371	−	0.937728i	$-0.387075\pi$
$661$	17.8617	0.694741	0.347371	+	0.937728i	$0.387075\pi$
$662$	−23.0540	−0.896018
$663$	0	0
$664$	−6.24621	−0.242400
$665$	13.1231	0.508892
$666$	0	0
$667$	29.1231	1.12765
$668$	−5.12311	−0.198219
$669$	0	0
$670$	−9.12311	−0.352456
$671$	29.7538	1.14863
$672$	0	0
$673$	20.4233	0.787260	0.393630	−	0.919269i	$-0.371219\pi$
$673$	20.4233	0.787260	0.393630	+	0.919269i	$0.371219\pi$
$674$	10.3153	0.397332
$675$	0	0
$676$	7.80776	0.300299
$677$	40.7386	1.56571	0.782856	−	0.622202i	$-0.213762\pi$
$677$	40.7386	1.56571	0.782856	+	0.622202i	$0.213762\pi$
$678$	0	0
$679$	−8.63068	−0.331215
$680$	−1.00000	−0.0383482
$681$	0	0
$682$	26.2462	1.00502
$683$	41.3002	1.58031	0.790154	−	0.612909i	$-0.210001\pi$
$683$	41.3002	1.58031	0.790154	+	0.612909i	$0.210001\pi$
$684$	0	0
$685$	10.0000	0.382080
$686$	−62.7386	−2.39537
$687$	0	0
$688$	−1.12311	−0.0428180
$689$	2.56155	0.0975874
$690$	0	0
$691$	9.75379	0.371052	0.185526	−	0.982639i	$-0.440601\pi$
$691$	9.75379	0.371052	0.185526	+	0.982639i	$0.440601\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−10.5616	−0.400911
$695$	−16.4924	−0.625593
$696$	0	0
$697$	4.24621	0.160837
$698$	15.1231	0.572418
$699$	0	0
$700$	5.12311	0.193635
$701$	−46.3542	−1.75077	−0.875386	−	0.483424i	$-0.839393\pi$
$701$	−46.3542	−1.75077	−0.875386	+	0.483424i	$0.839393\pi$
$702$	0	0
$703$	18.2462	0.688169
$704$	4.00000	0.150756
$705$	0	0
$706$	22.4924	0.846513
$707$	74.2462	2.79232
$708$	0	0
$709$	37.5464	1.41008	0.705042	−	0.709165i	$-0.250928\pi$
$709$	37.5464	1.41008	0.705042	+	0.709165i	$0.250928\pi$
$710$	4.31534	0.161952
$711$	0	0
$712$	−9.68466	−0.362948
$713$	−33.6155	−1.25891
$714$	0	0
$715$	−18.2462	−0.682370
$716$	−8.49242	−0.317377
$717$	0	0
$718$	−10.8769	−0.405922
$719$	−15.1922	−0.566575	−0.283287	−	0.959035i	$-0.591425\pi$
$719$	−15.1922	−0.566575	−0.283287	+	0.959035i	$0.591425\pi$
$720$	0	0
$721$	40.9848	1.52636
$722$	12.4384	0.462911
$723$	0	0
$724$	16.2462	0.603786
$725$	5.68466	0.211123
$726$	0	0
$727$	46.5616	1.72687	0.863436	−	0.504458i	$-0.168308\pi$
$727$	46.5616	1.72687	0.863436	+	0.504458i	$0.168308\pi$
$728$	−23.3693	−0.866125
$729$	0	0
$730$	−6.80776	−0.251967
$731$	1.12311	0.0415396
$732$	0	0
$733$	−22.4924	−0.830777	−0.415388	−	0.909644i	$-0.636354\pi$
$733$	−22.4924	−0.830777	−0.415388	+	0.909644i	$0.636354\pi$
$734$	33.6155	1.24077
$735$	0	0
$736$	−5.12311	−0.188840
$737$	−36.4924	−1.34422
$738$	0	0
$739$	−20.1771	−0.742226	−0.371113	−	0.928588i	$-0.621024\pi$
$739$	−20.1771	−0.742226	−0.371113	+	0.928588i	$0.621024\pi$
$740$	7.12311	0.261851
$741$	0	0
$742$	−2.87689	−0.105614
$743$	21.1231	0.774932	0.387466	−	0.921884i	$-0.373351\pi$
$743$	21.1231	0.774932	0.387466	+	0.921884i	$0.373351\pi$
$744$	0	0
$745$	3.12311	0.114422
$746$	24.7386	0.905746
$747$	0	0
$748$	−4.00000	−0.146254
$749$	−84.4924	−3.08729
$750$	0	0
$751$	−24.1771	−0.882234	−0.441117	−	0.897450i	$-0.645418\pi$
$751$	−24.1771	−0.882234	−0.441117	+	0.897450i	$0.645418\pi$
$752$	−6.56155	−0.239275
$753$	0	0
$754$	−25.9309	−0.944347
$755$	15.3693	0.559347
$756$	0	0
$757$	47.7926	1.73705	0.868526	−	0.495644i	$-0.165068\pi$
$757$	47.7926	1.73705	0.868526	+	0.495644i	$0.165068\pi$
$758$	14.2462	0.517446
$759$	0	0
$760$	−2.56155	−0.0929173
$761$	20.7386	0.751775	0.375887	−	0.926665i	$-0.377338\pi$
$761$	20.7386	0.751775	0.375887	+	0.926665i	$0.377338\pi$
$762$	0	0
$763$	−43.8617	−1.58790
$764$	0	0
$765$	0	0
$766$	−16.8078	−0.607289
$767$	1.43845	0.0519393
$768$	0	0
$769$	11.4384	0.412481	0.206240	−	0.978501i	$-0.433877\pi$
$769$	11.4384	0.412481	0.206240	+	0.978501i	$0.433877\pi$
$770$	20.4924	0.738496
$771$	0	0
$772$	−24.2462	−0.872640
$773$	−20.7386	−0.745917	−0.372958	−	0.927848i	$-0.621657\pi$
$773$	−20.7386	−0.745917	−0.372958	+	0.927848i	$0.621657\pi$
$774$	0	0
$775$	−6.56155	−0.235698
$776$	1.68466	0.0604757
$777$	0	0
$778$	13.3693	0.479313
$779$	10.8769	0.389705
$780$	0	0
$781$	17.2614	0.617660
$782$	5.12311	0.183202
$783$	0	0
$784$	19.2462	0.687365
$785$	−18.4924	−0.660023
$786$	0	0
$787$	−27.5464	−0.981923	−0.490962	−	0.871181i	$-0.663354\pi$
$787$	−27.5464	−0.981923	−0.490962	+	0.871181i	$0.663354\pi$
$788$	−13.3693	−0.476262
$789$	0	0
$790$	0	0
$791$	−14.3845	−0.511453
$792$	0	0
$793$	33.9309	1.20492
$794$	−21.3693	−0.758369
$795$	0	0
$796$	14.5616	0.516121
$797$	−18.4924	−0.655035	−0.327518	−	0.944845i	$-0.606212\pi$
$797$	−18.4924	−0.655035	−0.327518	+	0.944845i	$0.606212\pi$
$798$	0	0
$799$	6.56155	0.232131
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	10.6307	0.375382
$803$	−27.2311	−0.960963
$804$	0	0
$805$	−26.2462	−0.925057
$806$	29.9309	1.05427
$807$	0	0
$808$	−14.4924	−0.509842
$809$	−18.9848	−0.667472	−0.333736	−	0.942667i	$-0.608309\pi$
$809$	−18.9848	−0.667472	−0.333736	+	0.942667i	$0.608309\pi$
$810$	0	0
$811$	3.36932	0.118313	0.0591564	−	0.998249i	$-0.481159\pi$
$811$	3.36932	0.118313	0.0591564	+	0.998249i	$0.481159\pi$
$812$	29.1231	1.02202
$813$	0	0
$814$	28.4924	0.998659
$815$	−14.2462	−0.499023
$816$	0	0
$817$	2.87689	0.100650
$818$	2.31534	0.0809540
$819$	0	0
$820$	4.24621	0.148284
$821$	15.3002	0.533980	0.266990	−	0.963699i	$-0.413971\pi$
$821$	15.3002	0.533980	0.266990	+	0.963699i	$0.413971\pi$
$822$	0	0
$823$	0.630683	0.0219842	0.0109921	−	0.999940i	$-0.496501\pi$
$823$	0.630683	0.0219842	0.0109921	+	0.999940i	$0.496501\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−1.61553	−0.0562114
$827$	−0.492423	−0.0171232	−0.00856160	−	0.999963i	$-0.502725\pi$
$827$	−0.492423	−0.0171232	−0.00856160	+	0.999963i	$0.502725\pi$
$828$	0	0
$829$	−9.36932	−0.325410	−0.162705	−	0.986675i	$-0.552022\pi$
$829$	−9.36932	−0.325410	−0.162705	+	0.986675i	$0.552022\pi$
$830$	6.24621	0.216809
$831$	0	0
$832$	4.56155	0.158143
$833$	−19.2462	−0.666842
$834$	0	0
$835$	5.12311	0.177292
$836$	−10.2462	−0.354373
$837$	0	0
$838$	−25.1231	−0.867863
$839$	−35.0540	−1.21020	−0.605099	−	0.796150i	$-0.706866\pi$
$839$	−35.0540	−1.21020	−0.605099	+	0.796150i	$0.706866\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	−7.61553	−0.262448
$843$	0	0
$844$	−3.36932	−0.115977
$845$	−7.80776	−0.268595
$846$	0	0
$847$	25.6155	0.880160
$848$	0.561553	0.0192838
$849$	0	0
$850$	1.00000	0.0342997
$851$	−36.4924	−1.25094
$852$	0	0
$853$	48.2462	1.65192	0.825959	−	0.563730i	$-0.190634\pi$
$853$	48.2462	1.65192	0.825959	+	0.563730i	$0.190634\pi$
$854$	−38.1080	−1.30403
$855$	0	0
$856$	16.4924	0.563699
$857$	−1.82292	−0.0622697	−0.0311349	−	0.999515i	$-0.509912\pi$
$857$	−1.82292	−0.0622697	−0.0311349	+	0.999515i	$0.509912\pi$
$858$	0	0
$859$	−35.5464	−1.21283	−0.606414	−	0.795149i	$-0.707392\pi$
$859$	−35.5464	−1.21283	−0.606414	+	0.795149i	$0.707392\pi$
$860$	1.12311	0.0382976
$861$	0	0
$862$	−20.4924	−0.697975
$863$	−11.5076	−0.391722	−0.195861	−	0.980632i	$-0.562750\pi$
$863$	−11.5076	−0.391722	−0.195861	+	0.980632i	$0.562750\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	2.49242	0.0846960
$867$	0	0
$868$	−33.6155	−1.14099
$869$	0	0
$870$	0	0
$871$	−41.6155	−1.41009
$872$	8.56155	0.289931
$873$	0	0
$874$	13.1231	0.443896
$875$	−5.12311	−0.173193
$876$	0	0
$877$	−51.6155	−1.74293	−0.871466	−	0.490455i	$-0.836830\pi$
$877$	−51.6155	−1.74293	−0.871466	+	0.490455i	$0.836830\pi$
$878$	32.9848	1.11318
$879$	0	0
$880$	−4.00000	−0.134840
$881$	28.7386	0.968229	0.484115	−	0.875005i	$-0.339142\pi$
$881$	28.7386	0.968229	0.484115	+	0.875005i	$0.339142\pi$
$882$	0	0
$883$	27.3693	0.921051	0.460525	−	0.887647i	$-0.347661\pi$
$883$	27.3693	0.921051	0.460525	+	0.887647i	$0.347661\pi$
$884$	−4.56155	−0.153422
$885$	0	0
$886$	28.0000	0.940678
$887$	−50.6004	−1.69899	−0.849497	−	0.527593i	$-0.823095\pi$
$887$	−50.6004	−1.69899	−0.849497	+	0.527593i	$0.823095\pi$
$888$	0	0
$889$	48.3542	1.62175
$890$	9.68466	0.324630
$891$	0	0
$892$	3.68466	0.123371
$893$	16.8078	0.562450
$894$	0	0
$895$	8.49242	0.283870
$896$	−5.12311	−0.171151
$897$	0	0
$898$	−16.8769	−0.563189
$899$	−37.3002	−1.24403
$900$	0	0
$901$	−0.561553	−0.0187080
$902$	16.9848	0.565533
$903$	0	0
$904$	2.80776	0.0933848
$905$	−16.2462	−0.540042
$906$	0	0
$907$	18.5616	0.616326	0.308163	−	0.951334i	$-0.400286\pi$
$907$	18.5616	0.616326	0.308163	+	0.951334i	$0.400286\pi$
$908$	−16.3153	−0.541444
$909$	0	0
$910$	23.3693	0.774685
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	24.9848	0.826878
$914$	16.2462	0.537377
$915$	0	0
$916$	−14.4924	−0.478843
$917$	−5.75379	−0.190007
$918$	0	0
$919$	−21.1231	−0.696787	−0.348393	−	0.937348i	$-0.613273\pi$
$919$	−21.1231	−0.696787	−0.348393	+	0.937348i	$0.613273\pi$
$920$	5.12311	0.168904
$921$	0	0
$922$	2.49242	0.0820836
$923$	19.6847	0.647928
$924$	0	0
$925$	−7.12311	−0.234206
$926$	13.9309	0.457797
$927$	0	0
$928$	−5.68466	−0.186608
$929$	59.4773	1.95139	0.975693	−	0.219142i	$-0.0703259\pi$
$929$	59.4773	1.95139	0.975693	+	0.219142i	$0.0703259\pi$
$930$	0	0
$931$	−49.3002	−1.61575
$932$	−6.31534	−0.206866
$933$	0	0
$934$	−32.4924	−1.06318
$935$	4.00000	0.130814
$936$	0	0
$937$	−44.1080	−1.44094	−0.720472	−	0.693484i	$-0.756075\pi$
$937$	−44.1080	−1.44094	−0.720472	+	0.693484i	$0.756075\pi$
$938$	46.7386	1.52607
$939$	0	0
$940$	6.56155	0.214014
$941$	27.4384	0.894468	0.447234	−	0.894417i	$-0.352409\pi$
$941$	27.4384	0.894468	0.447234	+	0.894417i	$0.352409\pi$
$942$	0	0
$943$	−21.7538	−0.708401
$944$	0.315342	0.0102635
$945$	0	0
$946$	4.49242	0.146061
$947$	−52.8078	−1.71602	−0.858011	−	0.513632i	$-0.828300\pi$
$947$	−52.8078	−1.71602	−0.858011	+	0.513632i	$0.828300\pi$
$948$	0	0
$949$	−31.0540	−1.00805
$950$	2.56155	0.0831077
$951$	0	0
$952$	5.12311	0.166041
$953$	−39.1231	−1.26732	−0.633661	−	0.773611i	$-0.718449\pi$
$953$	−39.1231	−1.26732	−0.633661	+	0.773611i	$0.718449\pi$
$954$	0	0
$955$	0	0
$956$	23.3693	0.755818
$957$	0	0
$958$	38.5616	1.24587
$959$	−51.2311	−1.65434
$960$	0	0
$961$	12.0540	0.388838
$962$	32.4924	1.04760
$963$	0	0
$964$	12.2462	0.394424
$965$	24.2462	0.780513
$966$	0	0
$967$	19.5076	0.627321	0.313661	−	0.949535i	$-0.398445\pi$
$967$	19.5076	0.627321	0.313661	+	0.949535i	$0.398445\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	−1.68466	−0.0540911
$971$	−15.6847	−0.503345	−0.251672	−	0.967813i	$-0.580981\pi$
$971$	−15.6847	−0.503345	−0.251672	+	0.967813i	$0.580981\pi$
$972$	0	0
$973$	84.4924	2.70870
$974$	−25.6155	−0.820774
$975$	0	0
$976$	7.43845	0.238099
$977$	27.1231	0.867745	0.433873	−	0.900974i	$-0.357147\pi$
$977$	27.1231	0.867745	0.433873	+	0.900974i	$0.357147\pi$
$978$	0	0
$979$	38.7386	1.23809
$980$	−19.2462	−0.614798
$981$	0	0
$982$	2.56155	0.0817424
$983$	37.1231	1.18404	0.592022	−	0.805922i	$-0.298330\pi$
$983$	37.1231	1.18404	0.592022	+	0.805922i	$0.298330\pi$
$984$	0	0
$985$	13.3693	0.425982
$986$	5.68466	0.181036
$987$	0	0
$988$	−11.6847	−0.371739
$989$	−5.75379	−0.182960
$990$	0	0
$991$	34.4233	1.09349	0.546746	−	0.837299i	$-0.315866\pi$
$991$	34.4233	1.09349	0.546746	+	0.837299i	$0.315866\pi$
$992$	6.56155	0.208330
$993$	0	0
$994$	−22.1080	−0.701222
$995$	−14.5616	−0.461632
$996$	0	0
$997$	11.7538	0.372246	0.186123	−	0.982526i	$-0.440408\pi$
$997$	11.7538	0.372246	0.186123	+	0.982526i	$0.440408\pi$
$998$	9.12311	0.288787
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1530.2.a.r.1.2		2
3.2	odd	2		170.2.a.f.1.1	✓	2
5.4	even	2		7650.2.a.de.1.1		2
12.11	even	2		1360.2.a.m.1.2		2
15.2	even	4		850.2.c.i.749.4		4
15.8	even	4		850.2.c.i.749.1		4
15.14	odd	2		850.2.a.n.1.2		2
21.20	even	2		8330.2.a.bq.1.2		2
24.5	odd	2		5440.2.a.bj.1.2		2
24.11	even	2		5440.2.a.bd.1.1		2
51.38	odd	4		2890.2.b.i.2311.4		4
51.47	odd	4		2890.2.b.i.2311.1		4
51.50	odd	2		2890.2.a.u.1.2		2
60.59	even	2		6800.2.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.a.f.1.1	✓	2	3.2	odd	2
850.2.a.n.1.2		2	15.14	odd	2
850.2.c.i.749.1		4	15.8	even	4
850.2.c.i.749.4		4	15.2	even	4
1360.2.a.m.1.2		2	12.11	even	2
1530.2.a.r.1.2		2	1.1	even	1	trivial
2890.2.a.u.1.2		2	51.50	odd	2
2890.2.b.i.2311.1		4	51.47	odd	4
2890.2.b.i.2311.4		4	51.38	odd	4
5440.2.a.bd.1.1		2	24.11	even	2
5440.2.a.bj.1.2		2	24.5	odd	2
6800.2.a.be.1.1		2	60.59	even	2
7650.2.a.de.1.1		2	5.4	even	2
8330.2.a.bq.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +5.12311 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +5.12311 q^{7} -1.00000 q^{8} +1.00000 q^{10} +4.00000 q^{11} +4.56155 q^{13} -5.12311 q^{14} +1.00000 q^{16} -1.00000 q^{17} -2.56155 q^{19} -1.00000 q^{20} -4.00000 q^{22} +5.12311 q^{23} +1.00000 q^{25} -4.56155 q^{26} +5.12311 q^{28} +5.68466 q^{29} -6.56155 q^{31} -1.00000 q^{32} +1.00000 q^{34} -5.12311 q^{35} -7.12311 q^{37} +2.56155 q^{38} +1.00000 q^{40} -4.24621 q^{41} -1.12311 q^{43} +4.00000 q^{44} -5.12311 q^{46} -6.56155 q^{47} +19.2462 q^{49} -1.00000 q^{50} +4.56155 q^{52} +0.561553 q^{53} -4.00000 q^{55} -5.12311 q^{56} -5.68466 q^{58} +0.315342 q^{59} +7.43845 q^{61} +6.56155 q^{62} +1.00000 q^{64} -4.56155 q^{65} -9.12311 q^{67} -1.00000 q^{68} +5.12311 q^{70} +4.31534 q^{71} -6.80776 q^{73} +7.12311 q^{74} -2.56155 q^{76} +20.4924 q^{77} -1.00000 q^{80} +4.24621 q^{82} +6.24621 q^{83} +1.00000 q^{85} +1.12311 q^{86} -4.00000 q^{88} +9.68466 q^{89} +23.3693 q^{91} +5.12311 q^{92} +6.56155 q^{94} +2.56155 q^{95} -1.68466 q^{97} -19.2462 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 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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