LMFDB - Embedded newform 1755.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1755,2,Mod(1,1755)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1755, 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("1755.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1755 = 3^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1755.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:	$14.0137455547$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.8957.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + x + 1 $ x^4 - x^3 - 5*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			$0.542055$ of defining polynomial
Character	$\chi$	$=$	1755.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.84483 q^{2} +1.40340 q^{4} +1.00000 q^{5} +0.457945 q^{7} +1.10062 q^{8} +O(q^{10})$	\(q-1.84483 q^{2} +1.40340 q^{4} +1.00000 q^{5} +0.457945 q^{7} +1.10062 q^{8} -1.84483 q^{10} +3.16412 q^{11} -1.00000 q^{13} -0.844831 q^{14} -4.83727 q^{16} +4.99244 q^{17} +1.38689 q^{19} +1.40340 q^{20} -5.83727 q^{22} +1.89938 q^{23} +1.00000 q^{25} +1.84483 q^{26} +0.642680 q^{28} +5.02957 q^{29} -3.57162 q^{31} +6.72269 q^{32} -9.21020 q^{34} +0.457945 q^{35} +0.377934 q^{37} -2.55857 q^{38} +1.10062 q^{40} -6.66058 q^{41} +6.71513 q^{43} +4.44053 q^{44} -3.50403 q^{46} +8.03852 q^{47} -6.79029 q^{49} -1.84483 q^{50} -1.40340 q^{52} -3.78272 q^{53} +3.16412 q^{55} +0.504026 q^{56} -9.27870 q^{58} +2.34081 q^{59} -4.26475 q^{61} +6.58904 q^{62} -2.72769 q^{64} -1.00000 q^{65} -7.15656 q^{67} +7.00639 q^{68} -0.844831 q^{70} -1.25718 q^{71} +4.56752 q^{73} -0.697224 q^{74} +1.94636 q^{76} +1.44899 q^{77} -1.62617 q^{79} -4.83727 q^{80} +12.2877 q^{82} -6.95094 q^{83} +4.99244 q^{85} -12.3883 q^{86} +3.48251 q^{88} +7.94546 q^{89} -0.457945 q^{91} +2.66558 q^{92} -14.8297 q^{94} +1.38689 q^{95} +5.71028 q^{97} +12.5269 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 3 q^{4} + 4 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10}) $ 4 * q + q^2 + 3 * q^4 + 4 * q^5 + 3 * q^7 + 9 * q^8	$ 4 q + q^{2} + 3 q^{4} + 4 q^{5} + 3 q^{7} + 9 q^{8} + q^{10} + 4 q^{11} - 4 q^{13} + 5 q^{14} + 13 q^{16} - 4 q^{17} - 4 q^{19} + 3 q^{20} + 9 q^{22} + 3 q^{23} + 4 q^{25} - q^{26} + 6 q^{28} + 14 q^{29} - 7 q^{31} + 24 q^{32} - 29 q^{34} + 3 q^{35} + 9 q^{37} - 16 q^{38} + 9 q^{40} + 4 q^{41} + 33 q^{44} - 16 q^{46} + 9 q^{47} - 15 q^{49} + q^{50} - 3 q^{52} + 21 q^{53} + 4 q^{55} + 4 q^{56} + q^{58} - q^{59} - 13 q^{61} + 5 q^{62} + 9 q^{64} - 4 q^{65} + 4 q^{67} - 30 q^{68} + 5 q^{70} + 23 q^{71} + 7 q^{73} - 10 q^{74} - 17 q^{76} + 24 q^{77} - 3 q^{79} + 13 q^{80} - 6 q^{82} + 13 q^{83} - 4 q^{85} + q^{86} + 26 q^{88} + 28 q^{89} - 3 q^{91} - 37 q^{92} - 3 q^{94} - 4 q^{95} + 17 q^{97} + q^{98}+O(q^{100}) $ 4 * q + q^2 + 3 * q^4 + 4 * q^5 + 3 * q^7 + 9 * q^8 + q^10 + 4 * q^11 - 4 * q^13 + 5 * q^14 + 13 * q^16 - 4 * q^17 - 4 * q^19 + 3 * q^20 + 9 * q^22 + 3 * q^23 + 4 * q^25 - q^26 + 6 * q^28 + 14 * q^29 - 7 * q^31 + 24 * q^32 - 29 * q^34 + 3 * q^35 + 9 * q^37 - 16 * q^38 + 9 * q^40 + 4 * q^41 + 33 * q^44 - 16 * q^46 + 9 * q^47 - 15 * q^49 + q^50 - 3 * q^52 + 21 * q^53 + 4 * q^55 + 4 * q^56 + q^58 - q^59 - 13 * q^61 + 5 * q^62 + 9 * q^64 - 4 * q^65 + 4 * q^67 - 30 * q^68 + 5 * q^70 + 23 * q^71 + 7 * q^73 - 10 * q^74 - 17 * q^76 + 24 * q^77 - 3 * q^79 + 13 * q^80 - 6 * q^82 + 13 * q^83 - 4 * q^85 + q^86 + 26 * q^88 + 28 * q^89 - 3 * q^91 - 37 * q^92 - 3 * q^94 - 4 * q^95 + 17 * q^97 + 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 $

$2$	−1.84483	−1.30449	−0.652246	−	0.758007i	$-0.726173\pi$
$2$	−1.84483	−1.30449	−0.652246	+	0.758007i	$0.726173\pi$
$3$	0	0
$3$	0	0
$4$	1.40340	0.701700
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.457945	0.173087	0.0865434	−	0.996248i	$-0.472418\pi$
$7$	0.457945	0.173087	0.0865434	+	0.996248i	$0.472418\pi$
$8$	1.10062	0.389130
$9$	0	0
$10$	−1.84483	−0.583387
$11$	3.16412	0.954018	0.477009	−	0.878898i	$-0.341721\pi$
$11$	3.16412	0.954018	0.477009	+	0.878898i	$0.341721\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.844831	−0.225791
$15$	0	0
$16$	−4.83727	−1.20932
$17$	4.99244	1.21084	0.605422	−	0.795905i	$-0.293004\pi$
$17$	4.99244	1.21084	0.605422	+	0.795905i	$0.293004\pi$
$18$	0	0
$19$	1.38689	0.318173	0.159087	−	0.987265i	$-0.449145\pi$
$19$	1.38689	0.318173	0.159087	+	0.987265i	$0.449145\pi$
$20$	1.40340	0.313810
$21$	0	0
$22$	−5.83727	−1.24451
$23$	1.89938	0.396047	0.198024	−	0.980197i	$-0.436548\pi$
$23$	1.89938	0.396047	0.198024	+	0.980197i	$0.436548\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.84483	0.361801
$27$	0	0
$28$	0.642680	0.121455
$29$	5.02957	0.933967	0.466983	−	0.884266i	$-0.345341\pi$
$29$	5.02957	0.933967	0.466983	+	0.884266i	$0.345341\pi$
$30$	0	0
$31$	−3.57162	−0.641482	−0.320741	−	0.947167i	$-0.603932\pi$
$31$	−3.57162	−0.641482	−0.320741	+	0.947167i	$0.603932\pi$
$32$	6.72269	1.18842
$33$	0	0
$34$	−9.21020	−1.57954
$35$	0.457945	0.0774068
$36$	0	0
$37$	0.377934	0.0621320	0.0310660	−	0.999517i	$-0.490110\pi$
$37$	0.377934	0.0621320	0.0310660	+	0.999517i	$0.490110\pi$
$38$	−2.55857	−0.415055
$39$	0	0
$40$	1.10062	0.174024
$41$	−6.66058	−1.04021	−0.520104	−	0.854103i	$-0.674107\pi$
$41$	−6.66058	−1.04021	−0.520104	+	0.854103i	$0.674107\pi$
$42$	0	0
$43$	6.71513	1.02405	0.512024	−	0.858971i	$-0.328896\pi$
$43$	6.71513	1.02405	0.512024	+	0.858971i	$0.328896\pi$
$44$	4.44053	0.669435
$45$	0	0
$46$	−3.50403	−0.516640
$47$	8.03852	1.17254	0.586269	−	0.810116i	$-0.300596\pi$
$47$	8.03852	1.17254	0.586269	+	0.810116i	$0.300596\pi$
$48$	0	0
$49$	−6.79029	−0.970041
$50$	−1.84483	−0.260898
$51$	0	0
$52$	−1.40340	−0.194617
$53$	−3.78272	−0.519597	−0.259799	−	0.965663i	$-0.583656\pi$
$53$	−3.78272	−0.519597	−0.259799	+	0.965663i	$0.583656\pi$
$54$	0	0
$55$	3.16412	0.426650
$56$	0.504026	0.0673532
$57$	0	0
$58$	−9.27870	−1.21835
$59$	2.34081	0.304747	0.152373	−	0.988323i	$-0.451308\pi$
$59$	2.34081	0.304747	0.152373	+	0.988323i	$0.451308\pi$
$60$	0	0
$61$	−4.26475	−0.546045	−0.273022	−	0.962008i	$-0.588023\pi$
$61$	−4.26475	−0.546045	−0.273022	+	0.962008i	$0.588023\pi$
$62$	6.58904	0.836808
$63$	0	0
$64$	−2.72769	−0.340961
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−7.15656	−0.874313	−0.437156	−	0.899386i	$-0.644014\pi$
$67$	−7.15656	−0.874313	−0.437156	+	0.899386i	$0.644014\pi$
$68$	7.00639	0.849650
$69$	0	0
$70$	−0.844831	−0.100977
$71$	−1.25718	−0.149200	−0.0746001	−	0.997214i	$-0.523768\pi$
$71$	−1.25718	−0.149200	−0.0746001	+	0.997214i	$0.523768\pi$
$72$	0	0
$73$	4.56752	0.534588	0.267294	−	0.963615i	$-0.413871\pi$
$73$	4.56752	0.534588	0.267294	+	0.963615i	$0.413871\pi$
$74$	−0.697224	−0.0810507
$75$	0	0
$76$	1.94636	0.223262
$77$	1.44899	0.165128
$78$	0	0
$79$	−1.62617	−0.182958	−0.0914790	−	0.995807i	$-0.529159\pi$
$79$	−1.62617	−0.182958	−0.0914790	+	0.995807i	$0.529159\pi$
$80$	−4.83727	−0.540823
$81$	0	0
$82$	12.2877	1.35694
$83$	−6.95094	−0.762965	−0.381483	−	0.924376i	$-0.624586\pi$
$83$	−6.95094	−0.762965	−0.381483	+	0.924376i	$0.624586\pi$
$84$	0	0
$85$	4.99244	0.541506
$86$	−12.3883	−1.33586
$87$	0	0
$88$	3.48251	0.371237
$89$	7.94546	0.842217	0.421108	−	0.907010i	$-0.361641\pi$
$89$	7.94546	0.842217	0.421108	+	0.907010i	$0.361641\pi$
$90$	0	0
$91$	−0.457945	−0.0480057
$92$	2.66558	0.277906
$93$	0	0
$94$	−14.8297	−1.52957
$95$	1.38689	0.142291
$96$	0	0
$97$	5.71028	0.579791	0.289895	−	0.957058i	$-0.406380\pi$
$97$	5.71028	0.579791	0.289895	+	0.957058i	$0.406380\pi$
$98$	12.5269	1.26541
$99$	0	0
$100$	1.40340	0.140340
$101$	−5.05913	−0.503402	−0.251701	−	0.967805i	$-0.580990\pi$
$101$	−5.05913	−0.503402	−0.251701	+	0.967805i	$0.580990\pi$
$102$	0	0
$103$	−9.41325	−0.927515	−0.463758	−	0.885962i	$-0.653499\pi$
$103$	−9.41325	−0.927515	−0.463758	+	0.885962i	$0.653499\pi$
$104$	−1.10062	−0.107925
$105$	0	0
$106$	6.97849	0.677810
$107$	6.55052	0.633263	0.316631	−	0.948549i	$-0.397448\pi$
$107$	6.55052	0.633263	0.316631	+	0.948549i	$0.397448\pi$
$108$	0	0
$109$	13.6365	1.30614	0.653070	−	0.757297i	$-0.273481\pi$
$109$	13.6365	1.30614	0.653070	+	0.757297i	$0.273481\pi$
$110$	−5.83727	−0.556562
$111$	0	0
$112$	−2.21520	−0.209317
$113$	18.3413	1.72540	0.862702	−	0.505713i	$-0.168770\pi$
$113$	18.3413	1.72540	0.862702	+	0.505713i	$0.168770\pi$
$114$	0	0
$115$	1.89938	0.177118
$116$	7.05850	0.655365
$117$	0	0
$118$	−4.31839	−0.397540
$119$	2.28626	0.209581
$120$	0	0
$121$	−0.988338	−0.0898489
$122$	7.86773	0.712311
$123$	0	0
$124$	−5.01242	−0.450128
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.3422	1.36140	0.680700	−	0.732562i	$-0.261676\pi$
$127$	15.3422	1.36140	0.680700	+	0.732562i	$0.261676\pi$
$128$	−8.41325	−0.743634
$129$	0	0
$130$	1.84483	0.161802
$131$	−14.2421	−1.24433	−0.622167	−	0.782884i	$-0.713748\pi$
$131$	−14.2421	−1.24433	−0.622167	+	0.782884i	$0.713748\pi$
$132$	0	0
$133$	0.635117	0.0550717
$134$	13.2026	1.14053
$135$	0	0
$136$	5.49480	0.471175
$137$	−9.23657	−0.789133	−0.394567	−	0.918867i	$-0.629105\pi$
$137$	−9.23657	−0.789133	−0.394567	+	0.918867i	$0.629105\pi$
$138$	0	0
$139$	−2.96287	−0.251307	−0.125654	−	0.992074i	$-0.540103\pi$
$139$	−2.96287	−0.251307	−0.125654	+	0.992074i	$0.540103\pi$
$140$	0.642680	0.0543164
$141$	0	0
$142$	2.31929	0.194631
$143$	−3.16412	−0.264597
$144$	0	0
$145$	5.02957	0.417683
$146$	−8.42630	−0.697366
$147$	0	0
$148$	0.530393	0.0435980
$149$	15.2746	1.25134	0.625672	−	0.780086i	$-0.284825\pi$
$149$	15.2746	1.25134	0.625672	+	0.780086i	$0.284825\pi$
$150$	0	0
$151$	−10.2111	−0.830968	−0.415484	−	0.909601i	$-0.636388\pi$
$151$	−10.2111	−0.830968	−0.415484	+	0.909601i	$0.636388\pi$
$152$	1.52644	0.123811
$153$	0	0
$154$	−2.67315	−0.215408
$155$	−3.57162	−0.286880
$156$	0	0
$157$	13.1575	1.05008	0.525040	−	0.851078i	$-0.324050\pi$
$157$	13.1575	1.05008	0.525040	+	0.851078i	$0.324050\pi$
$158$	3.00000	0.238667
$159$	0	0
$160$	6.72269	0.531475
$161$	0.869809	0.0685506
$162$	0	0
$163$	6.89591	0.540129	0.270065	−	0.962842i	$-0.412955\pi$
$163$	6.89591	0.540129	0.270065	+	0.962842i	$0.412955\pi$
$164$	−9.34747	−0.729915
$165$	0	0
$166$	12.8233	0.995282
$167$	12.4715	0.965072	0.482536	−	0.875876i	$-0.339716\pi$
$167$	12.4715	0.965072	0.482536	+	0.875876i	$0.339716\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−9.21020	−0.706390
$171$	0	0
$172$	9.42401	0.718574
$173$	14.2666	1.08467	0.542336	−	0.840162i	$-0.317540\pi$
$173$	14.2666	1.08467	0.542336	+	0.840162i	$0.317540\pi$
$174$	0	0
$175$	0.457945	0.0346174
$176$	−15.3057	−1.15371
$177$	0	0
$178$	−14.6580	−1.09867
$179$	7.24018	0.541156	0.270578	−	0.962698i	$-0.412785\pi$
$179$	7.24018	0.541156	0.270578	+	0.962698i	$0.412785\pi$
$180$	0	0
$181$	2.76621	0.205611	0.102805	−	0.994701i	$-0.467218\pi$
$181$	2.76621	0.205611	0.102805	+	0.994701i	$0.467218\pi$
$182$	0.844831	0.0626230
$183$	0	0
$184$	2.09050	0.154114
$185$	0.377934	0.0277863
$186$	0	0
$187$	15.7967	1.15517
$188$	11.2813	0.822770
$189$	0	0
$190$	−2.55857	−0.185618
$191$	−12.4298	−0.899386	−0.449693	−	0.893183i	$-0.648467\pi$
$191$	−12.4298	−0.899386	−0.449693	+	0.893183i	$0.648467\pi$
$192$	0	0
$193$	19.3633	1.39380	0.696900	−	0.717168i	$-0.254562\pi$
$193$	19.3633	1.39380	0.696900	+	0.717168i	$0.254562\pi$
$194$	−10.5345	−0.756332
$195$	0	0
$196$	−9.52949	−0.680678
$197$	11.3213	0.806610	0.403305	−	0.915066i	$-0.367861\pi$
$197$	11.3213	0.806610	0.403305	+	0.915066i	$0.367861\pi$
$198$	0	0
$199$	20.7591	1.47158	0.735788	−	0.677212i	$-0.236812\pi$
$199$	20.7591	1.47158	0.735788	+	0.677212i	$0.236812\pi$
$200$	1.10062	0.0778259
$201$	0	0
$202$	9.33324	0.656685
$203$	2.30326	0.161657
$204$	0	0
$205$	−6.66058	−0.465195
$206$	17.3659	1.20994
$207$	0	0
$208$	4.83727	0.335404
$209$	4.38827	0.303543
$210$	0	0
$211$	−12.4845	−0.859471	−0.429735	−	0.902955i	$-0.641393\pi$
$211$	−12.4845	−0.859471	−0.429735	+	0.902955i	$0.641393\pi$
$212$	−5.30868	−0.364601
$213$	0	0
$214$	−12.0846	−0.826086
$215$	6.71513	0.457968
$216$	0	0
$217$	−1.63561	−0.111032
$218$	−25.1570	−1.70385
$219$	0	0
$220$	4.44053	0.299380
$221$	−4.99244	−0.335828
$222$	0	0
$223$	25.5923	1.71379	0.856894	−	0.515493i	$-0.172391\pi$
$223$	25.5923	1.71379	0.856894	+	0.515493i	$0.172391\pi$
$224$	3.07862	0.205699
$225$	0	0
$226$	−33.8366	−2.25078
$227$	20.9398	1.38982	0.694911	−	0.719096i	$-0.255444\pi$
$227$	20.9398	1.38982	0.694911	+	0.719096i	$0.255444\pi$
$228$	0	0
$229$	−27.6702	−1.82850	−0.914248	−	0.405155i	$-0.867218\pi$
$229$	−27.6702	−1.82850	−0.914248	+	0.405155i	$0.867218\pi$
$230$	−3.50403	−0.231049
$231$	0	0
$232$	5.53567	0.363434
$233$	−7.35871	−0.482085	−0.241043	−	0.970515i	$-0.577489\pi$
$233$	−7.35871	−0.482085	−0.241043	+	0.970515i	$0.577489\pi$
$234$	0	0
$235$	8.03852	0.524375
$236$	3.28509	0.213841
$237$	0	0
$238$	−4.21776	−0.273397
$239$	23.6702	1.53110	0.765548	−	0.643379i	$-0.222468\pi$
$239$	23.6702	1.53110	0.765548	+	0.643379i	$0.222468\pi$
$240$	0	0
$241$	18.0248	1.16108	0.580539	−	0.814232i	$-0.302842\pi$
$241$	18.0248	1.16108	0.580539	+	0.814232i	$0.302842\pi$
$242$	1.82332	0.117207
$243$	0	0
$244$	−5.98515	−0.383160
$245$	−6.79029	−0.433815
$246$	0	0
$247$	−1.38689	−0.0882454
$248$	−3.93102	−0.249620
$249$	0	0
$250$	−1.84483	−0.116677
$251$	−22.8767	−1.44396	−0.721982	−	0.691912i	$-0.756769\pi$
$251$	−22.8767	−1.44396	−0.721982	+	0.691912i	$0.756769\pi$
$252$	0	0
$253$	6.00985	0.377836
$254$	−28.3038	−1.77594
$255$	0	0
$256$	20.9764	1.31103
$257$	−12.4353	−0.775690	−0.387845	−	0.921725i	$-0.626780\pi$
$257$	−12.4353	−0.775690	−0.387845	+	0.921725i	$0.626780\pi$
$258$	0	0
$259$	0.173073	0.0107542
$260$	−1.40340	−0.0870352
$261$	0	0
$262$	26.2742	1.62322
$263$	−10.7752	−0.664425	−0.332212	−	0.943205i	$-0.607795\pi$
$263$	−10.7752	−0.664425	−0.332212	+	0.943205i	$0.607795\pi$
$264$	0	0
$265$	−3.78272	−0.232371
$266$	−1.17168	−0.0718405
$267$	0	0
$268$	−10.0435	−0.613506
$269$	1.19715	0.0729916	0.0364958	−	0.999334i	$-0.488380\pi$
$269$	1.19715	0.0729916	0.0364958	+	0.999334i	$0.488380\pi$
$270$	0	0
$271$	−2.89479	−0.175846	−0.0879229	−	0.996127i	$-0.528023\pi$
$271$	−2.89479	−0.175846	−0.0879229	+	0.996127i	$0.528023\pi$
$272$	−24.1498	−1.46429
$273$	0	0
$274$	17.0399	1.02942
$275$	3.16412	0.190804
$276$	0	0
$277$	13.9840	0.840216	0.420108	−	0.907474i	$-0.361992\pi$
$277$	13.9840	0.840216	0.420108	+	0.907474i	$0.361992\pi$
$278$	5.46600	0.327829
$279$	0	0
$280$	0.504026	0.0301213
$281$	9.89279	0.590154	0.295077	−	0.955473i	$-0.404655\pi$
$281$	9.89279	0.590154	0.295077	+	0.955473i	$0.404655\pi$
$282$	0	0
$283$	7.07218	0.420398	0.210199	−	0.977659i	$-0.432589\pi$
$283$	7.07218	0.420398	0.210199	+	0.977659i	$0.432589\pi$
$284$	−1.76433	−0.104694
$285$	0	0
$286$	5.83727	0.345165
$287$	−3.05018	−0.180046
$288$	0	0
$289$	7.92443	0.466143
$290$	−9.27870	−0.544864
$291$	0	0
$292$	6.41006	0.375121
$293$	24.2331	1.41571	0.707857	−	0.706356i	$-0.249662\pi$
$293$	24.2331	1.41571	0.707857	+	0.706356i	$0.249662\pi$
$294$	0	0
$295$	2.34081	0.136287
$296$	0.415964	0.0241774
$297$	0	0
$298$	−28.1790	−1.63237
$299$	−1.89938	−0.109844
$300$	0	0
$301$	3.07516	0.177249
$302$	18.8378	1.08399
$303$	0	0
$304$	−6.70874	−0.384773
$305$	−4.26475	−0.244199
$306$	0	0
$307$	−31.6308	−1.80527	−0.902633	−	0.430411i	$-0.858369\pi$
$307$	−31.6308	−1.80527	−0.902633	+	0.430411i	$0.858369\pi$
$308$	2.03352	0.115870
$309$	0	0
$310$	6.58904	0.374232
$311$	−6.09667	−0.345711	−0.172855	−	0.984947i	$-0.555299\pi$
$311$	−6.09667	−0.345711	−0.172855	+	0.984947i	$0.555299\pi$
$312$	0	0
$313$	−17.5918	−0.994348	−0.497174	−	0.867651i	$-0.665629\pi$
$313$	−17.5918	−0.994348	−0.497174	+	0.867651i	$0.665629\pi$
$314$	−24.2733	−1.36982
$315$	0	0
$316$	−2.28216	−0.128382
$317$	6.96558	0.391226	0.195613	−	0.980681i	$-0.437330\pi$
$317$	6.96558	0.391226	0.195613	+	0.980681i	$0.437330\pi$
$318$	0	0
$319$	15.9142	0.891022
$320$	−2.72769	−0.152483
$321$	0	0
$322$	−1.60465	−0.0894237
$323$	6.92394	0.385258
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−12.7218	−0.704595
$327$	0	0
$328$	−7.33080	−0.404776
$329$	3.68120	0.202951
$330$	0	0
$331$	28.5978	1.57188	0.785938	−	0.618305i	$-0.212180\pi$
$331$	28.5978	1.57188	0.785938	+	0.618305i	$0.212180\pi$
$332$	−9.75496	−0.535373
$333$	0	0
$334$	−23.0078	−1.25893
$335$	−7.15656	−0.391005
$336$	0	0
$337$	−13.3248	−0.725847	−0.362923	−	0.931819i	$-0.618221\pi$
$337$	−13.3248	−0.725847	−0.362923	+	0.931819i	$0.618221\pi$
$338$	−1.84483	−0.100346
$339$	0	0
$340$	7.00639	0.379975
$341$	−11.3010	−0.611986
$342$	0	0
$343$	−6.31519	−0.340988
$344$	7.39084	0.398487
$345$	0	0
$346$	−26.3195	−1.41495
$347$	−0.894110	−0.0479983	−0.0239992	−	0.999712i	$-0.507640\pi$
$347$	−0.894110	−0.0479983	−0.0239992	+	0.999712i	$0.507640\pi$
$348$	0	0
$349$	−34.5534	−1.84960	−0.924800	−	0.380454i	$-0.875768\pi$
$349$	−34.5534	−1.84960	−0.924800	+	0.380454i	$0.875768\pi$
$350$	−0.844831	−0.0451581
$351$	0	0
$352$	21.2714	1.13377
$353$	−28.2971	−1.50610	−0.753051	−	0.657962i	$-0.771419\pi$
$353$	−28.2971	−1.50610	−0.753051	+	0.657962i	$0.771419\pi$
$354$	0	0
$355$	−1.25718	−0.0667244
$356$	11.1507	0.590984
$357$	0	0
$358$	−13.3569	−0.705934
$359$	8.28467	0.437248	0.218624	−	0.975809i	$-0.429843\pi$
$359$	8.28467	0.437248	0.218624	+	0.975809i	$0.429843\pi$
$360$	0	0
$361$	−17.0765	−0.898766
$362$	−5.10319	−0.268218
$363$	0	0
$364$	−0.642680	−0.0336856
$365$	4.56752	0.239075
$366$	0	0
$367$	−18.8088	−0.981813	−0.490906	−	0.871212i	$-0.663334\pi$
$367$	−18.8088	−0.981813	−0.490906	+	0.871212i	$0.663334\pi$
$368$	−9.18779	−0.478946
$369$	0	0
$370$	−0.697224	−0.0362470
$371$	−1.73228	−0.0899354
$372$	0	0
$373$	−32.1260	−1.66342	−0.831712	−	0.555208i	$-0.812639\pi$
$373$	−32.1260	−1.66342	−0.831712	+	0.555208i	$0.812639\pi$
$374$	−29.1422	−1.50691
$375$	0	0
$376$	8.84739	0.456269
$377$	−5.02957	−0.259036
$378$	0	0
$379$	−19.6441	−1.00905	−0.504524	−	0.863398i	$-0.668332\pi$
$379$	−19.6441	−1.00905	−0.504524	+	0.863398i	$0.668332\pi$
$380$	1.94636	0.0998460
$381$	0	0
$382$	22.9308	1.17324
$383$	−11.5515	−0.590254	−0.295127	−	0.955458i	$-0.595362\pi$
$383$	−11.5515	−0.590254	−0.295127	+	0.955458i	$0.595362\pi$
$384$	0	0
$385$	1.44899	0.0738475
$386$	−35.7220	−1.81820
$387$	0	0
$388$	8.01380	0.406839
$389$	−7.62782	−0.386746	−0.193373	−	0.981125i	$-0.561943\pi$
$389$	−7.62782	−0.386746	−0.193373	+	0.981125i	$0.561943\pi$
$390$	0	0
$391$	9.48251	0.479551
$392$	−7.47356	−0.377472
$393$	0	0
$394$	−20.8859	−1.05222
$395$	−1.62617	−0.0818213
$396$	0	0
$397$	32.2737	1.61977	0.809885	−	0.586588i	$-0.199529\pi$
$397$	32.2737	1.61977	0.809885	+	0.586588i	$0.199529\pi$
$398$	−38.2971	−1.91966
$399$	0	0
$400$	−4.83727	−0.241863
$401$	16.1023	0.804109	0.402055	−	0.915616i	$-0.368296\pi$
$401$	16.1023	0.804109	0.402055	+	0.915616i	$0.368296\pi$
$402$	0	0
$403$	3.57162	0.177915
$404$	−7.09999	−0.353238
$405$	0	0
$406$	−4.24913	−0.210881
$407$	1.19583	0.0592750
$408$	0	0
$409$	9.35483	0.462567	0.231283	−	0.972886i	$-0.425708\pi$
$409$	9.35483	0.462567	0.231283	+	0.972886i	$0.425708\pi$
$410$	12.2877	0.606844
$411$	0	0
$412$	−13.2106	−0.650838
$413$	1.07196	0.0527477
$414$	0	0
$415$	−6.95094	−0.341208
$416$	−6.72269	−0.329607
$417$	0	0
$418$	−8.09562	−0.395970
$419$	−13.8749	−0.677832	−0.338916	−	0.940817i	$-0.610060\pi$
$419$	−13.8749	−0.677832	−0.338916	+	0.940817i	$0.610060\pi$
$420$	0	0
$421$	−0.143019	−0.00697030	−0.00348515	−	0.999994i	$-0.501109\pi$
$421$	−0.143019	−0.00697030	−0.00348515	+	0.999994i	$0.501109\pi$
$422$	23.0319	1.12117
$423$	0	0
$424$	−4.16336	−0.202191
$425$	4.99244	0.242169
$426$	0	0
$427$	−1.95302	−0.0945132
$428$	9.19300	0.444361
$429$	0	0
$430$	−12.3883	−0.597416
$431$	4.66058	0.224492	0.112246	−	0.993680i	$-0.464195\pi$
$431$	4.66058	0.224492	0.112246	+	0.993680i	$0.464195\pi$
$432$	0	0
$433$	−1.93492	−0.0929862	−0.0464931	−	0.998919i	$-0.514805\pi$
$433$	−1.93492	−0.0929862	−0.0464931	+	0.998919i	$0.514805\pi$
$434$	3.01742	0.144841
$435$	0	0
$436$	19.1375	0.916519
$437$	2.63422	0.126012
$438$	0	0
$439$	−5.27509	−0.251766	−0.125883	−	0.992045i	$-0.540176\pi$
$439$	−5.27509	−0.251766	−0.125883	+	0.992045i	$0.540176\pi$
$440$	3.48251	0.166022
$441$	0	0
$442$	9.21020	0.438085
$443$	−29.5061	−1.40187	−0.700937	−	0.713223i	$-0.747235\pi$
$443$	−29.5061	−1.40187	−0.700937	+	0.713223i	$0.747235\pi$
$444$	0	0
$445$	7.94546	0.376651
$446$	−47.2135	−2.23562
$447$	0	0
$448$	−1.24913	−0.0590159
$449$	23.0019	1.08553	0.542763	−	0.839886i	$-0.317378\pi$
$449$	23.0019	1.08553	0.542763	+	0.839886i	$0.317378\pi$
$450$	0	0
$451$	−21.0749	−0.992378
$452$	25.7402	1.21072
$453$	0	0
$454$	−38.6303	−1.81301
$455$	−0.457945	−0.0214688
$456$	0	0
$457$	−7.17217	−0.335500	−0.167750	−	0.985830i	$-0.553650\pi$
$457$	−7.17217	−0.335500	−0.167750	+	0.985830i	$0.553650\pi$
$458$	51.0468	2.38526
$459$	0	0
$460$	2.66558	0.124283
$461$	15.9796	0.744245	0.372122	−	0.928184i	$-0.378630\pi$
$461$	15.9796	0.744245	0.372122	+	0.928184i	$0.378630\pi$
$462$	0	0
$463$	−25.0741	−1.16529	−0.582645	−	0.812727i	$-0.697982\pi$
$463$	−25.0741	−1.16529	−0.582645	+	0.812727i	$0.697982\pi$
$464$	−24.3294	−1.12946
$465$	0	0
$466$	13.5756	0.628876
$467$	12.4694	0.577015	0.288508	−	0.957478i	$-0.406841\pi$
$467$	12.4694	0.577015	0.288508	+	0.957478i	$0.406841\pi$
$468$	0	0
$469$	−3.27731	−0.151332
$470$	−14.8297	−0.684043
$471$	0	0
$472$	2.57635	0.118586
$473$	21.2475	0.976960
$474$	0	0
$475$	1.38689	0.0636347
$476$	3.20854	0.147063
$477$	0	0
$478$	−43.6675	−1.99730
$479$	−7.70257	−0.351939	−0.175970	−	0.984396i	$-0.556306\pi$
$479$	−7.70257	−0.351939	−0.175970	+	0.984396i	$0.556306\pi$
$480$	0	0
$481$	−0.377934	−0.0172323
$482$	−33.2527	−1.51462
$483$	0	0
$484$	−1.38703	−0.0630470
$485$	5.71028	0.259290
$486$	0	0
$487$	7.30368	0.330961	0.165481	−	0.986213i	$-0.447082\pi$
$487$	7.30368	0.330961	0.165481	+	0.986213i	$0.447082\pi$
$488$	−4.69389	−0.212482
$489$	0	0
$490$	12.5269	0.565909
$491$	−21.4898	−0.969821	−0.484911	−	0.874564i	$-0.661148\pi$
$491$	−21.4898	−0.969821	−0.484911	+	0.874564i	$0.661148\pi$
$492$	0	0
$493$	25.1098	1.13089
$494$	2.55857	0.115115
$495$	0	0
$496$	17.2769	0.775755
$497$	−0.575721	−0.0258246
$498$	0	0
$499$	−39.4165	−1.76453	−0.882263	−	0.470757i	$-0.843981\pi$
$499$	−39.4165	−1.76453	−0.882263	+	0.470757i	$0.843981\pi$
$500$	1.40340	0.0627620
$501$	0	0
$502$	42.2036	1.88364
$503$	−33.1944	−1.48007	−0.740033	−	0.672570i	$-0.765190\pi$
$503$	−33.1944	−1.48007	−0.740033	+	0.672570i	$0.765190\pi$
$504$	0	0
$505$	−5.05913	−0.225128
$506$	−11.0872	−0.492884
$507$	0	0
$508$	21.5312	0.955294
$509$	22.3608	0.991124	0.495562	−	0.868572i	$-0.334962\pi$
$509$	22.3608	0.991124	0.495562	+	0.868572i	$0.334962\pi$
$510$	0	0
$511$	2.09167	0.0925302
$512$	−21.8714	−0.966589
$513$	0	0
$514$	22.9409	1.01188
$515$	−9.41325	−0.414797
$516$	0	0
$517$	25.4348	1.11862
$518$	−0.319290	−0.0140288
$519$	0	0
$520$	−1.10062	−0.0482656
$521$	16.6004	0.727275	0.363637	−	0.931541i	$-0.381535\pi$
$521$	16.6004	0.727275	0.363637	+	0.931541i	$0.381535\pi$
$522$	0	0
$523$	−39.9457	−1.74671	−0.873353	−	0.487088i	$-0.838059\pi$
$523$	−39.9457	−1.74671	−0.873353	+	0.487088i	$0.838059\pi$
$524$	−19.9873	−0.873150
$525$	0	0
$526$	19.8783	0.866737
$527$	−17.8311	−0.776735
$528$	0	0
$529$	−19.3924	−0.843147
$530$	6.97849	0.303126
$531$	0	0
$532$	0.891324	0.0386438
$533$	6.66058	0.288502
$534$	0	0
$535$	6.55052	0.283204
$536$	−7.87669	−0.340221
$537$	0	0
$538$	−2.20854	−0.0952169
$539$	−21.4853	−0.925437
$540$	0	0
$541$	−28.7251	−1.23499	−0.617495	−	0.786575i	$-0.711852\pi$
$541$	−28.7251	−1.23499	−0.617495	+	0.786575i	$0.711852\pi$
$542$	5.34039	0.229390
$543$	0	0
$544$	33.5626	1.43899
$545$	13.6365	0.584124
$546$	0	0
$547$	−7.65100	−0.327133	−0.163566	−	0.986532i	$-0.552300\pi$
$547$	−7.65100	−0.327133	−0.163566	+	0.986532i	$0.552300\pi$
$548$	−12.9626	−0.553735
$549$	0	0
$550$	−5.83727	−0.248902
$551$	6.97543	0.297163
$552$	0	0
$553$	−0.744694	−0.0316676
$554$	−25.7981	−1.09605
$555$	0	0
$556$	−4.15810	−0.176343
$557$	23.8829	1.01195	0.505975	−	0.862548i	$-0.331133\pi$
$557$	23.8829	1.01195	0.505975	+	0.862548i	$0.331133\pi$
$558$	0	0
$559$	−6.71513	−0.284020
$560$	−2.21520	−0.0936094
$561$	0	0
$562$	−18.2505	−0.769852
$563$	17.9693	0.757317	0.378659	−	0.925536i	$-0.376385\pi$
$563$	17.9693	0.757317	0.378659	+	0.925536i	$0.376385\pi$
$564$	0	0
$565$	18.3413	0.771624
$566$	−13.0470	−0.548406
$567$	0	0
$568$	−1.38369	−0.0580582
$569$	19.9787	0.837551	0.418775	−	0.908090i	$-0.362459\pi$
$569$	19.9787	0.837551	0.418775	+	0.908090i	$0.362459\pi$
$570$	0	0
$571$	−0.867322	−0.0362963	−0.0181482	−	0.999835i	$-0.505777\pi$
$571$	−0.867322	−0.0362963	−0.0181482	+	0.999835i	$0.505777\pi$
$572$	−4.44053	−0.185668
$573$	0	0
$574$	5.62707	0.234869
$575$	1.89938	0.0792094
$576$	0	0
$577$	20.6677	0.860406	0.430203	−	0.902732i	$-0.358442\pi$
$577$	20.6677	0.860406	0.430203	+	0.902732i	$0.358442\pi$
$578$	−14.6192	−0.608080
$579$	0	0
$580$	7.05850	0.293088
$581$	−3.18315	−0.132059
$582$	0	0
$583$	−11.9690	−0.495705
$584$	5.02713	0.208024
$585$	0	0
$586$	−44.7060	−1.84679
$587$	8.13365	0.335712	0.167856	−	0.985812i	$-0.446316\pi$
$587$	8.13365	0.335712	0.167856	+	0.985812i	$0.446316\pi$
$588$	0	0
$589$	−4.95343	−0.204103
$590$	−4.31839	−0.177785
$591$	0	0
$592$	−1.82817	−0.0751373
$593$	−38.8439	−1.59513	−0.797564	−	0.603235i	$-0.793878\pi$
$593$	−38.8439	−1.59513	−0.797564	+	0.603235i	$0.793878\pi$
$594$	0	0
$595$	2.28626	0.0937276
$596$	21.4364	0.878068
$597$	0	0
$598$	3.50403	0.143290
$599$	5.88384	0.240407	0.120204	−	0.992749i	$-0.461645\pi$
$599$	5.88384	0.240407	0.120204	+	0.992749i	$0.461645\pi$
$600$	0	0
$601$	−29.1509	−1.18909	−0.594544	−	0.804063i	$-0.702668\pi$
$601$	−29.1509	−1.18909	−0.594544	+	0.804063i	$0.702668\pi$
$602$	−5.67315	−0.231220
$603$	0	0
$604$	−14.3303	−0.583090
$605$	−0.988338	−0.0401816
$606$	0	0
$607$	19.3464	0.785245	0.392622	−	0.919700i	$-0.371568\pi$
$607$	19.3464	0.785245	0.392622	+	0.919700i	$0.371568\pi$
$608$	9.32361	0.378122
$609$	0	0
$610$	7.86773	0.318555
$611$	−8.03852	−0.325204
$612$	0	0
$613$	−14.0156	−0.566085	−0.283042	−	0.959107i	$-0.591344\pi$
$613$	−14.0156	−0.566085	−0.283042	+	0.959107i	$0.591344\pi$
$614$	58.3535	2.35496
$615$	0	0
$616$	1.59480	0.0642562
$617$	−8.34817	−0.336085	−0.168042	−	0.985780i	$-0.553745\pi$
$617$	−8.34817	−0.336085	−0.168042	+	0.985780i	$0.553745\pi$
$618$	0	0
$619$	3.79390	0.152490	0.0762448	−	0.997089i	$-0.475707\pi$
$619$	3.79390	0.152490	0.0762448	+	0.997089i	$0.475707\pi$
$620$	−5.01242	−0.201303
$621$	0	0
$622$	11.2473	0.450977
$623$	3.63858	0.145777
$624$	0	0
$625$	1.00000	0.0400000
$626$	32.4539	1.29712
$627$	0	0
$628$	18.4652	0.736841
$629$	1.88681	0.0752321
$630$	0	0
$631$	−27.4007	−1.09080	−0.545402	−	0.838175i	$-0.683623\pi$
$631$	−27.4007	−1.09080	−0.545402	+	0.838175i	$0.683623\pi$
$632$	−1.78980	−0.0711944
$633$	0	0
$634$	−12.8503	−0.510351
$635$	15.3422	0.608836
$636$	0	0
$637$	6.79029	0.269041
$638$	−29.3589	−1.16233
$639$	0	0
$640$	−8.41325	−0.332563
$641$	10.2354	0.404274	0.202137	−	0.979357i	$-0.435211\pi$
$641$	10.2354	0.404274	0.202137	+	0.979357i	$0.435211\pi$
$642$	0	0
$643$	−0.319290	−0.0125916	−0.00629579	−	0.999980i	$-0.502004\pi$
$643$	−0.319290	−0.0125916	−0.00629579	+	0.999980i	$0.502004\pi$
$644$	1.22069	0.0481019
$645$	0	0
$646$	−12.7735	−0.502567
$647$	−1.56383	−0.0614807	−0.0307403	−	0.999527i	$-0.509786\pi$
$647$	−1.56383	−0.0614807	−0.0307403	+	0.999527i	$0.509786\pi$
$648$	0	0
$649$	7.40659	0.290734
$650$	1.84483	0.0723602
$651$	0	0
$652$	9.67773	0.379009
$653$	34.1020	1.33451	0.667257	−	0.744827i	$-0.267468\pi$
$653$	34.1020	1.33451	0.667257	+	0.744827i	$0.267468\pi$
$654$	0	0
$655$	−14.2421	−0.556483
$656$	32.2190	1.25794
$657$	0	0
$658$	−6.79119	−0.264748
$659$	33.0473	1.28734	0.643669	−	0.765304i	$-0.277411\pi$
$659$	33.0473	1.28734	0.643669	+	0.765304i	$0.277411\pi$
$660$	0	0
$661$	31.0509	1.20774	0.603870	−	0.797083i	$-0.293625\pi$
$661$	31.0509	1.20774	0.603870	+	0.797083i	$0.293625\pi$
$662$	−52.7581	−2.05050
$663$	0	0
$664$	−7.65038	−0.296892
$665$	0.635117	0.0246288
$666$	0	0
$667$	9.55303	0.369895
$668$	17.5025	0.677192
$669$	0	0
$670$	13.2026	0.510062
$671$	−13.4942	−0.520937
$672$	0	0
$673$	17.2287	0.664119	0.332060	−	0.943258i	$-0.392256\pi$
$673$	17.2287	0.664119	0.332060	+	0.943258i	$0.392256\pi$
$674$	24.5820	0.946862
$675$	0	0
$676$	1.40340	0.0539769
$677$	−17.5327	−0.673836	−0.336918	−	0.941534i	$-0.609384\pi$
$677$	−17.5327	−0.673836	−0.336918	+	0.941534i	$0.609384\pi$
$678$	0	0
$679$	2.61499	0.100354
$680$	5.49480	0.210716
$681$	0	0
$682$	20.8485	0.798331
$683$	12.5834	0.481489	0.240744	−	0.970589i	$-0.422608\pi$
$683$	12.5834	0.481489	0.240744	+	0.970589i	$0.422608\pi$
$684$	0	0
$685$	−9.23657	−0.352911
$686$	11.6505	0.444817
$687$	0	0
$688$	−32.4829	−1.23840
$689$	3.78272	0.144110
$690$	0	0
$691$	18.1836	0.691737	0.345869	−	0.938283i	$-0.387584\pi$
$691$	18.1836	0.691737	0.345869	+	0.938283i	$0.387584\pi$
$692$	20.0218	0.761114
$693$	0	0
$694$	1.64948	0.0626135
$695$	−2.96287	−0.112388
$696$	0	0
$697$	−33.2525	−1.25953
$698$	63.7451	2.41279
$699$	0	0
$700$	0.642680	0.0242910
$701$	35.7143	1.34891	0.674455	−	0.738316i	$-0.264379\pi$
$701$	35.7143	1.34891	0.674455	+	0.738316i	$0.264379\pi$
$702$	0	0
$703$	0.524151	0.0197687
$704$	−8.63075	−0.325283
$705$	0	0
$706$	52.2033	1.96470
$707$	−2.31680	−0.0871324
$708$	0	0
$709$	48.0648	1.80511	0.902555	−	0.430574i	$-0.141689\pi$
$709$	48.0648	1.80511	0.902555	+	0.430574i	$0.141689\pi$
$710$	2.31929	0.0870414
$711$	0	0
$712$	8.74497	0.327731
$713$	−6.78385	−0.254057
$714$	0	0
$715$	−3.16412	−0.118331
$716$	10.1609	0.379730
$717$	0	0
$718$	−15.2838	−0.570387
$719$	−36.6376	−1.36635	−0.683177	−	0.730253i	$-0.739402\pi$
$719$	−36.6376	−1.36635	−0.683177	+	0.730253i	$0.739402\pi$
$720$	0	0
$721$	−4.31075	−0.160541
$722$	31.5033	1.17243
$723$	0	0
$724$	3.88210	0.144277
$725$	5.02957	0.186793
$726$	0	0
$727$	−0.399714	−0.0148246	−0.00741228	−	0.999973i	$-0.502359\pi$
$727$	−0.399714	−0.0148246	−0.00741228	+	0.999973i	$0.502359\pi$
$728$	−0.504026	−0.0186804
$729$	0	0
$730$	−8.42630	−0.311872
$731$	33.5249	1.23996
$732$	0	0
$733$	32.8135	1.21200	0.605998	−	0.795466i	$-0.292774\pi$
$733$	32.8135	1.21200	0.605998	+	0.795466i	$0.292774\pi$
$734$	34.6991	1.28077
$735$	0	0
$736$	12.7689	0.470668
$737$	−22.6442	−0.834110
$738$	0	0
$739$	−38.8765	−1.43009	−0.715047	−	0.699076i	$-0.753595\pi$
$739$	−38.8765	−1.43009	−0.715047	+	0.699076i	$0.753595\pi$
$740$	0.530393	0.0194976
$741$	0	0
$742$	3.19576	0.117320
$743$	−40.8736	−1.49951	−0.749753	−	0.661718i	$-0.769828\pi$
$743$	−40.8736	−1.49951	−0.749753	+	0.661718i	$0.769828\pi$
$744$	0	0
$745$	15.2746	0.559618
$746$	59.2671	2.16992
$747$	0	0
$748$	22.1691	0.810581
$749$	2.99978	0.109609
$750$	0	0
$751$	13.8051	0.503757	0.251878	−	0.967759i	$-0.418952\pi$
$751$	13.8051	0.503757	0.251878	+	0.967759i	$0.418952\pi$
$752$	−38.8845	−1.41797
$753$	0	0
$754$	9.27870	0.337910
$755$	−10.2111	−0.371620
$756$	0	0
$757$	−14.9954	−0.545018	−0.272509	−	0.962153i	$-0.587854\pi$
$757$	−14.9954	−0.545018	−0.272509	+	0.962153i	$0.587854\pi$
$758$	36.2400	1.31630
$759$	0	0
$760$	1.52644	0.0553698
$761$	16.0621	0.582251	0.291125	−	0.956685i	$-0.405970\pi$
$761$	16.0621	0.582251	0.291125	+	0.956685i	$0.405970\pi$
$762$	0	0
$763$	6.24477	0.226076
$764$	−17.4439	−0.631100
$765$	0	0
$766$	21.3106	0.769982
$767$	−2.34081	−0.0845216
$768$	0	0
$769$	−34.6736	−1.25036	−0.625182	−	0.780479i	$-0.714975\pi$
$769$	−34.6736	−1.25036	−0.625182	+	0.780479i	$0.714975\pi$
$770$	−2.67315	−0.0963335
$771$	0	0
$772$	27.1745	0.978030
$773$	−17.5155	−0.629988	−0.314994	−	0.949094i	$-0.602003\pi$
$773$	−17.5155	−0.629988	−0.314994	+	0.949094i	$0.602003\pi$
$774$	0	0
$775$	−3.57162	−0.128296
$776$	6.28487	0.225614
$777$	0	0
$778$	14.0720	0.504507
$779$	−9.23747	−0.330967
$780$	0	0
$781$	−3.97788	−0.142340
$782$	−17.4936	−0.625571
$783$	0	0
$784$	32.8464	1.17309
$785$	13.1575	0.469610
$786$	0	0
$787$	−19.6242	−0.699527	−0.349764	−	0.936838i	$-0.613738\pi$
$787$	−19.6242	−0.699527	−0.349764	+	0.936838i	$0.613738\pi$
$788$	15.8883	0.565999
$789$	0	0
$790$	3.00000	0.106735
$791$	8.39930	0.298645
$792$	0	0
$793$	4.26475	0.151446
$794$	−59.5395	−2.11298
$795$	0	0
$796$	29.1334	1.03261
$797$	4.46843	0.158280	0.0791400	−	0.996864i	$-0.474783\pi$
$797$	4.46843	0.158280	0.0791400	+	0.996864i	$0.474783\pi$
$798$	0	0
$799$	40.1318	1.41976
$800$	6.72269	0.237683
$801$	0	0
$802$	−29.7060	−1.04895
$803$	14.4522	0.510007
$804$	0	0
$805$	0.869809	0.0306567
$806$	−6.58904	−0.232089
$807$	0	0
$808$	−5.56821	−0.195889
$809$	3.25572	0.114465	0.0572325	−	0.998361i	$-0.481772\pi$
$809$	3.25572	0.114465	0.0572325	+	0.998361i	$0.481772\pi$
$810$	0	0
$811$	−3.87432	−0.136046	−0.0680229	−	0.997684i	$-0.521669\pi$
$811$	−3.87432	−0.136046	−0.0680229	+	0.997684i	$0.521669\pi$
$812$	3.23240	0.113435
$813$	0	0
$814$	−2.20610	−0.0773238
$815$	6.89591	0.241553
$816$	0	0
$817$	9.31312	0.325825
$818$	−17.2581	−0.603415
$819$	0	0
$820$	−9.34747	−0.326428
$821$	47.3865	1.65380	0.826900	−	0.562350i	$-0.190103\pi$
$821$	47.3865	1.65380	0.826900	+	0.562350i	$0.190103\pi$
$822$	0	0
$823$	54.3879	1.89584	0.947921	−	0.318506i	$-0.103181\pi$
$823$	54.3879	1.89584	0.947921	+	0.318506i	$0.103181\pi$
$824$	−10.3605	−0.360924
$825$	0	0
$826$	−1.97758	−0.0688090
$827$	−37.2003	−1.29358	−0.646791	−	0.762667i	$-0.723890\pi$
$827$	−37.2003	−1.29358	−0.646791	+	0.762667i	$0.723890\pi$
$828$	0	0
$829$	−2.75711	−0.0957584	−0.0478792	−	0.998853i	$-0.515246\pi$
$829$	−2.75711	−0.0957584	−0.0478792	+	0.998853i	$0.515246\pi$
$830$	12.8233	0.445104
$831$	0	0
$832$	2.72769	0.0945657
$833$	−33.9001	−1.17457
$834$	0	0
$835$	12.4715	0.431594
$836$	6.15851	0.212996
$837$	0	0
$838$	25.5968	0.884227
$839$	10.9514	0.378083	0.189041	−	0.981969i	$-0.439462\pi$
$839$	10.9514	0.378083	0.189041	+	0.981969i	$0.439462\pi$
$840$	0	0
$841$	−3.70347	−0.127706
$842$	0.263845	0.00909271
$843$	0	0
$844$	−17.5208	−0.603091
$845$	1.00000	0.0344010
$846$	0	0
$847$	−0.452604	−0.0155517
$848$	18.2980	0.628358
$849$	0	0
$850$	−9.21020	−0.315907
$851$	0.717838	0.0246072
$852$	0	0
$853$	−28.0239	−0.959519	−0.479760	−	0.877400i	$-0.659276\pi$
$853$	−28.0239	−0.959519	−0.479760	+	0.877400i	$0.659276\pi$
$854$	3.60299	0.123292
$855$	0	0
$856$	7.20966	0.246421
$857$	−41.6469	−1.42263	−0.711316	−	0.702872i	$-0.751900\pi$
$857$	−41.6469	−1.42263	−0.711316	+	0.702872i	$0.751900\pi$
$858$	0	0
$859$	28.2643	0.964365	0.482182	−	0.876071i	$-0.339844\pi$
$859$	28.2643	0.964365	0.482182	+	0.876071i	$0.339844\pi$
$860$	9.42401	0.321356
$861$	0	0
$862$	−8.59799	−0.292849
$863$	1.83215	0.0623671	0.0311836	−	0.999514i	$-0.490072\pi$
$863$	1.83215	0.0623671	0.0311836	+	0.999514i	$0.490072\pi$
$864$	0	0
$865$	14.2666	0.485080
$866$	3.56960	0.121300
$867$	0	0
$868$	−2.29541	−0.0779113
$869$	−5.14538	−0.174545
$870$	0	0
$871$	7.15656	0.242491
$872$	15.0087	0.508258
$873$	0	0
$874$	−4.85968	−0.164381
$875$	0.457945	0.0154814
$876$	0	0
$877$	−34.7092	−1.17205	−0.586023	−	0.810295i	$-0.699307\pi$
$877$	−34.7092	−1.17205	−0.586023	+	0.810295i	$0.699307\pi$
$878$	9.73164	0.328427
$879$	0	0
$880$	−15.3057	−0.515955
$881$	−28.7675	−0.969200	−0.484600	−	0.874736i	$-0.661035\pi$
$881$	−28.7675	−0.969200	−0.484600	+	0.874736i	$0.661035\pi$
$882$	0	0
$883$	28.2568	0.950916	0.475458	−	0.879738i	$-0.342282\pi$
$883$	28.2568	0.950916	0.475458	+	0.879738i	$0.342282\pi$
$884$	−7.00639	−0.235650
$885$	0	0
$886$	54.4337	1.82873
$887$	−26.4940	−0.889581	−0.444790	−	0.895635i	$-0.646722\pi$
$887$	−26.4940	−0.889581	−0.444790	+	0.895635i	$0.646722\pi$
$888$	0	0
$889$	7.02588	0.235640
$890$	−14.6580	−0.491338
$891$	0	0
$892$	35.9163	1.20257
$893$	11.1485	0.373071
$894$	0	0
$895$	7.24018	0.242013
$896$	−3.85281	−0.128713
$897$	0	0
$898$	−42.4346	−1.41606
$899$	−17.9637	−0.599123
$900$	0	0
$901$	−18.8850	−0.629151
$902$	38.8796	1.29455
$903$	0	0
$904$	20.1869	0.671406
$905$	2.76621	0.0919519
$906$	0	0
$907$	−13.9846	−0.464351	−0.232176	−	0.972674i	$-0.574584\pi$
$907$	−13.9846	−0.464351	−0.232176	+	0.972674i	$0.574584\pi$
$908$	29.3869	0.975238
$909$	0	0
$910$	0.844831	0.0280059
$911$	47.9383	1.58827	0.794134	−	0.607743i	$-0.207925\pi$
$911$	47.9383	1.58827	0.794134	+	0.607743i	$0.207925\pi$
$912$	0	0
$913$	−21.9936	−0.727883
$914$	13.2314	0.437657
$915$	0	0
$916$	−38.8323	−1.28306
$917$	−6.52208	−0.215378
$918$	0	0
$919$	−19.1925	−0.633103	−0.316551	−	0.948575i	$-0.602525\pi$
$919$	−19.1925	−0.633103	−0.316551	+	0.948575i	$0.602525\pi$
$920$	2.09050	0.0689217
$921$	0	0
$922$	−29.4797	−0.970861
$923$	1.25718	0.0413807
$924$	0	0
$925$	0.377934	0.0124264
$926$	46.2574	1.52011
$927$	0	0
$928$	33.8122	1.10994
$929$	37.9924	1.24649	0.623245	−	0.782027i	$-0.285814\pi$
$929$	37.9924	1.24649	0.623245	+	0.782027i	$0.285814\pi$
$930$	0	0
$931$	−9.41735	−0.308641
$932$	−10.3272	−0.338279
$933$	0	0
$934$	−23.0040	−0.752712
$935$	15.7967	0.516607
$936$	0	0
$937$	31.4652	1.02792	0.513962	−	0.857813i	$-0.328177\pi$
$937$	31.4652	1.02792	0.513962	+	0.857813i	$0.328177\pi$
$938$	6.04608	0.197412
$939$	0	0
$940$	11.2813	0.367954
$941$	27.6637	0.901811	0.450906	−	0.892572i	$-0.351101\pi$
$941$	27.6637	0.901811	0.450906	+	0.892572i	$0.351101\pi$
$942$	0	0
$943$	−12.6509	−0.411972
$944$	−11.3231	−0.368536
$945$	0	0
$946$	−39.1980	−1.27444
$947$	−7.93936	−0.257994	−0.128997	−	0.991645i	$-0.541176\pi$
$947$	−7.93936	−0.257994	−0.128997	+	0.991645i	$0.541176\pi$
$948$	0	0
$949$	−4.56752	−0.148268
$950$	−2.55857	−0.0830110
$951$	0	0
$952$	2.51632	0.0815543
$953$	−54.0271	−1.75011	−0.875054	−	0.484025i	$-0.839174\pi$
$953$	−54.0271	−1.75011	−0.875054	+	0.484025i	$0.839174\pi$
$954$	0	0
$955$	−12.4298	−0.402218
$956$	33.2187	1.07437
$957$	0	0
$958$	14.2099	0.459102
$959$	−4.22984	−0.136589
$960$	0	0
$961$	−18.2435	−0.588501
$962$	0.697224	0.0224794
$963$	0	0
$964$	25.2960	0.814729
$965$	19.3633	0.623327
$966$	0	0
$967$	−48.1793	−1.54934	−0.774670	−	0.632365i	$-0.782084\pi$
$967$	−48.1793	−1.54934	−0.774670	+	0.632365i	$0.782084\pi$
$968$	−1.08779	−0.0349629
$969$	0	0
$970$	−10.5345	−0.338242
$971$	−19.4430	−0.623956	−0.311978	−	0.950089i	$-0.600992\pi$
$971$	−19.4430	−0.623956	−0.311978	+	0.950089i	$0.600992\pi$
$972$	0	0
$973$	−1.35683	−0.0434980
$974$	−13.4740	−0.431736
$975$	0	0
$976$	20.6297	0.660341
$977$	42.2264	1.35094	0.675472	−	0.737386i	$-0.263940\pi$
$977$	42.2264	1.35094	0.675472	+	0.737386i	$0.263940\pi$
$978$	0	0
$979$	25.1404	0.803490
$980$	−9.52949	−0.304408
$981$	0	0
$982$	39.6450	1.26512
$983$	−34.7147	−1.10723	−0.553614	−	0.832774i	$-0.686752\pi$
$983$	−34.7147	−1.10723	−0.553614	+	0.832774i	$0.686752\pi$
$984$	0	0
$985$	11.3213	0.360727
$986$	−46.3233	−1.47523
$987$	0	0
$988$	−1.94636	−0.0619218
$989$	12.7545	0.405571
$990$	0	0
$991$	9.06398	0.287927	0.143963	−	0.989583i	$-0.454015\pi$
$991$	9.06398	0.287927	0.143963	+	0.989583i	$0.454015\pi$
$992$	−24.0109	−0.762347
$993$	0	0
$994$	1.06211	0.0336880
$995$	20.7591	0.658109
$996$	0	0
$997$	−32.6185	−1.03304	−0.516518	−	0.856276i	$-0.672772\pi$
$997$	−32.6185	−1.03304	−0.516518	+	0.856276i	$0.672772\pi$
$998$	72.7168	2.30181
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.r.1.1	yes	4
3.2	odd	2		1755.2.a.p.1.4	✓	4
5.4	even	2		8775.2.a.bi.1.4		4
15.14	odd	2		8775.2.a.bq.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.p.1.4	✓	4	3.2	odd	2
1755.2.a.r.1.1	yes	4	1.1	even	1	trivial
8775.2.a.bi.1.4		4	5.4	even	2
8775.2.a.bq.1.1		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1755.a (trivial)

\(f(q)\)	\(=\)	\(q-1.84483 q^{2} +1.40340 q^{4} +1.00000 q^{5} +0.457945 q^{7} +1.10062 q^{8} +O(q^{10})\)	\(q-1.84483 q^{2} +1.40340 q^{4} +1.00000 q^{5} +0.457945 q^{7} +1.10062 q^{8} -1.84483 q^{10} +3.16412 q^{11} -1.00000 q^{13} -0.844831 q^{14} -4.83727 q^{16} +4.99244 q^{17} +1.38689 q^{19} +1.40340 q^{20} -5.83727 q^{22} +1.89938 q^{23} +1.00000 q^{25} +1.84483 q^{26} +0.642680 q^{28} +5.02957 q^{29} -3.57162 q^{31} +6.72269 q^{32} -9.21020 q^{34} +0.457945 q^{35} +0.377934 q^{37} -2.55857 q^{38} +1.10062 q^{40} -6.66058 q^{41} +6.71513 q^{43} +4.44053 q^{44} -3.50403 q^{46} +8.03852 q^{47} -6.79029 q^{49} -1.84483 q^{50} -1.40340 q^{52} -3.78272 q^{53} +3.16412 q^{55} +0.504026 q^{56} -9.27870 q^{58} +2.34081 q^{59} -4.26475 q^{61} +6.58904 q^{62} -2.72769 q^{64} -1.00000 q^{65} -7.15656 q^{67} +7.00639 q^{68} -0.844831 q^{70} -1.25718 q^{71} +4.56752 q^{73} -0.697224 q^{74} +1.94636 q^{76} +1.44899 q^{77} -1.62617 q^{79} -4.83727 q^{80} +12.2877 q^{82} -6.95094 q^{83} +4.99244 q^{85} -12.3883 q^{86} +3.48251 q^{88} +7.94546 q^{89} -0.457945 q^{91} +2.66558 q^{92} -14.8297 q^{94} +1.38689 q^{95} +5.71028 q^{97} +12.5269 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 3 q^{4} + 4 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10}) \) 4 * q + q^2 + 3 * q^4 + 4 * q^5 + 3 * q^7 + 9 * q^8	\( 4 q + q^{2} + 3 q^{4} + 4 q^{5} + 3 q^{7} + 9 q^{8} + q^{10} + 4 q^{11} - 4 q^{13} + 5 q^{14} + 13 q^{16} - 4 q^{17} - 4 q^{19} + 3 q^{20} + 9 q^{22} + 3 q^{23} + 4 q^{25} - q^{26} + 6 q^{28} + 14 q^{29} - 7 q^{31} + 24 q^{32} - 29 q^{34} + 3 q^{35} + 9 q^{37} - 16 q^{38} + 9 q^{40} + 4 q^{41} + 33 q^{44} - 16 q^{46} + 9 q^{47} - 15 q^{49} + q^{50} - 3 q^{52} + 21 q^{53} + 4 q^{55} + 4 q^{56} + q^{58} - q^{59} - 13 q^{61} + 5 q^{62} + 9 q^{64} - 4 q^{65} + 4 q^{67} - 30 q^{68} + 5 q^{70} + 23 q^{71} + 7 q^{73} - 10 q^{74} - 17 q^{76} + 24 q^{77} - 3 q^{79} + 13 q^{80} - 6 q^{82} + 13 q^{83} - 4 q^{85} + q^{86} + 26 q^{88} + 28 q^{89} - 3 q^{91} - 37 q^{92} - 3 q^{94} - 4 q^{95} + 17 q^{97} + q^{98}+O(q^{100}) \) 4 * q + q^2 + 3 * q^4 + 4 * q^5 + 3 * q^7 + 9 * q^8 + q^10 + 4 * q^11 - 4 * q^13 + 5 * q^14 + 13 * q^16 - 4 * q^17 - 4 * q^19 + 3 * q^20 + 9 * q^22 + 3 * q^23 + 4 * q^25 - q^26 + 6 * q^28 + 14 * q^29 - 7 * q^31 + 24 * q^32 - 29 * q^34 + 3 * q^35 + 9 * q^37 - 16 * q^38 + 9 * q^40 + 4 * q^41 + 33 * q^44 - 16 * q^46 + 9 * q^47 - 15 * q^49 + q^50 - 3 * q^52 + 21 * q^53 + 4 * q^55 + 4 * q^56 + q^58 - q^59 - 13 * q^61 + 5 * q^62 + 9 * q^64 - 4 * q^65 + 4 * q^67 - 30 * q^68 + 5 * q^70 + 23 * q^71 + 7 * q^73 - 10 * q^74 - 17 * q^76 + 24 * q^77 - 3 * q^79 + 13 * q^80 - 6 * q^82 + 13 * q^83 - 4 * q^85 + q^86 + 26 * q^88 + 28 * q^89 - 3 * q^91 - 37 * q^92 - 3 * q^94 - 4 * q^95 + 17 * q^97 + q^98

Introduction

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