LMFDB - Embedded newform 1881.2.a.r.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1881,2,Mod(1,1881)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1881, 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("1881.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1881 = 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1881.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:	$15.0198606202$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 9x^{5} + 7x^{4} + 22x^{3} - 12x^{2} - 9x - 1 $ x^7 - x^6 - 9x^5 + 7x^4 + 22x^3 - 12x^2 - 9*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			1.7
Root			$-0.327599$ of defining polynomial
Character	$\chi$	$=$	1881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.44300 q^{2} +3.96824 q^{4} +0.800323 q^{5} +0.588123 q^{7} +4.80840 q^{8} +1.95519 q^{10} -1.00000 q^{11} +2.31077 q^{13} +1.43678 q^{14} +3.81044 q^{16} +6.57145 q^{17} +1.00000 q^{19} +3.17587 q^{20} -2.44300 q^{22} -4.10043 q^{23} -4.35948 q^{25} +5.64521 q^{26} +2.33381 q^{28} +4.73715 q^{29} +4.02301 q^{31} -0.307914 q^{32} +16.0540 q^{34} +0.470688 q^{35} -5.91366 q^{37} +2.44300 q^{38} +3.84827 q^{40} +11.1741 q^{41} +7.80457 q^{43} -3.96824 q^{44} -10.0174 q^{46} -2.56595 q^{47} -6.65411 q^{49} -10.6502 q^{50} +9.16969 q^{52} -11.4718 q^{53} -0.800323 q^{55} +2.82793 q^{56} +11.5728 q^{58} +2.46799 q^{59} -8.36670 q^{61} +9.82821 q^{62} -8.37311 q^{64} +1.84936 q^{65} -2.82416 q^{67} +26.0771 q^{68} +1.14989 q^{70} +4.05247 q^{71} -8.44391 q^{73} -14.4470 q^{74} +3.96824 q^{76} -0.588123 q^{77} +12.6320 q^{79} +3.04958 q^{80} +27.2983 q^{82} -0.0137315 q^{83} +5.25928 q^{85} +19.0666 q^{86} -4.80840 q^{88} -10.7746 q^{89} +1.35902 q^{91} -16.2715 q^{92} -6.26860 q^{94} +0.800323 q^{95} -7.37915 q^{97} -16.2560 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 6 q^{4} + 8 q^{5} - 2 q^{7} + 6 q^{8} + 6 q^{10} - 7 q^{11} - 7 q^{13} + 6 q^{14} + 5 q^{17} + 7 q^{19} + 20 q^{20} - 2 q^{22} + 22 q^{23} + 3 q^{25} + 10 q^{26} + 2 q^{28} + 18 q^{29} + 14 q^{32}+ \cdots - 62 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 + 6 * q^4 + 8 * q^5 - 2 * q^7 + 6 * q^8 + 6 * q^10 - 7 * q^11 - 7 * q^13 + 6 * q^14 + 5 * q^17 + 7 * q^19 + 20 * q^20 - 2 * q^22 + 22 * q^23 + 3 * q^25 + 10 * q^26 + 2 * q^28 + 18 * q^29 + 14 * q^32 + 6 * q^34 + 8 * q^35 + 6 * q^37 + 2 * q^38 + 28 * q^40 + 6 * q^41 + 8 * q^43 - 6 * q^44 + 8 * q^46 + 14 * q^47 + 5 * q^49 + 24 * q^50 - 12 * q^52 - q^53 - 8 * q^55 + 20 * q^56 - 2 * q^58 + 23 * q^59 - 28 * q^61 + 2 * q^62 - 26 * q^64 + 10 * q^65 + 14 * q^67 + 12 * q^68 - 18 * q^70 + 31 * q^71 - 12 * q^73 + 20 * q^74 + 6 * q^76 + 2 * q^77 - 17 * q^79 - 6 * q^80 + 12 * q^82 + 5 * q^83 - 12 * q^85 + 38 * q^86 - 6 * q^88 - 3 * q^89 - 12 * q^91 - 60 * q^94 + 8 * q^95 + 12 * q^97 - 62 * q^98

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 $

$2$	2.44300	1.72746	0.863730	−	0.503955i	$-0.168122\pi$
$2$	2.44300	1.72746	0.863730	+	0.503955i	$0.168122\pi$
$3$	0	0
$3$	0	0
$4$	3.96824	1.98412
$5$	0.800323	0.357915	0.178958	−	0.983857i	$-0.442728\pi$
$5$	0.800323	0.357915	0.178958	+	0.983857i	$0.442728\pi$
$6$	0	0
$7$	0.588123	0.222290	0.111145	−	0.993804i	$-0.464548\pi$
$7$	0.588123	0.222290	0.111145	+	0.993804i	$0.464548\pi$
$8$	4.80840	1.70003
$9$	0	0
$10$	1.95519	0.618284
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.31077	0.640892	0.320446	−	0.947267i	$-0.396167\pi$
$13$	2.31077	0.640892	0.320446	+	0.947267i	$0.396167\pi$
$14$	1.43678	0.383997
$15$	0	0
$16$	3.81044	0.952609
$17$	6.57145	1.59381	0.796905	−	0.604104i	$-0.206469\pi$
$17$	6.57145	1.59381	0.796905	+	0.604104i	$0.206469\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.17587	0.710146
$21$	0	0
$22$	−2.44300	−0.520849
$23$	−4.10043	−0.855000	−0.427500	−	0.904015i	$-0.640606\pi$
$23$	−4.10043	−0.855000	−0.427500	+	0.904015i	$0.640606\pi$
$24$	0	0
$25$	−4.35948	−0.871897
$26$	5.64521	1.10712
$27$	0	0
$28$	2.33381	0.441049
$29$	4.73715	0.879666	0.439833	−	0.898079i	$-0.355038\pi$
$29$	4.73715	0.879666	0.439833	+	0.898079i	$0.355038\pi$
$30$	0	0
$31$	4.02301	0.722554	0.361277	−	0.932458i	$-0.382341\pi$
$31$	4.02301	0.722554	0.361277	+	0.932458i	$0.382341\pi$
$32$	−0.307914	−0.0544320
$33$	0	0
$34$	16.0540	2.75324
$35$	0.470688	0.0795608
$36$	0	0
$37$	−5.91366	−0.972199	−0.486099	−	0.873903i	$-0.661581\pi$
$37$	−5.91366	−0.972199	−0.486099	+	0.873903i	$0.661581\pi$
$38$	2.44300	0.396307
$39$	0	0
$40$	3.84827	0.608465
$41$	11.1741	1.74510	0.872551	−	0.488524i	$-0.162464\pi$
$41$	11.1741	1.74510	0.872551	+	0.488524i	$0.162464\pi$
$42$	0	0
$43$	7.80457	1.19019	0.595093	−	0.803657i	$-0.297115\pi$
$43$	7.80457	1.19019	0.595093	+	0.803657i	$0.297115\pi$
$44$	−3.96824	−0.598234
$45$	0	0
$46$	−10.0174	−1.47698
$47$	−2.56595	−0.374282	−0.187141	−	0.982333i	$-0.559922\pi$
$47$	−2.56595	−0.374282	−0.187141	+	0.982333i	$0.559922\pi$
$48$	0	0
$49$	−6.65411	−0.950587
$50$	−10.6502	−1.50617
$51$	0	0
$52$	9.16969	1.27161
$53$	−11.4718	−1.57578	−0.787889	−	0.615818i	$-0.788826\pi$
$53$	−11.4718	−1.57578	−0.787889	+	0.615818i	$0.788826\pi$
$54$	0	0
$55$	−0.800323	−0.107915
$56$	2.82793	0.377898
$57$	0	0
$58$	11.5728	1.51959
$59$	2.46799	0.321304	0.160652	−	0.987011i	$-0.448640\pi$
$59$	2.46799	0.321304	0.160652	+	0.987011i	$0.448640\pi$
$60$	0	0
$61$	−8.36670	−1.07125	−0.535623	−	0.844457i	$-0.679923\pi$
$61$	−8.36670	−1.07125	−0.535623	+	0.844457i	$0.679923\pi$
$62$	9.82821	1.24818
$63$	0	0
$64$	−8.37311	−1.04664
$65$	1.84936	0.229385
$66$	0	0
$67$	−2.82416	−0.345025	−0.172513	−	0.985007i	$-0.555189\pi$
$67$	−2.82416	−0.345025	−0.172513	+	0.985007i	$0.555189\pi$
$68$	26.0771	3.16231
$69$	0	0
$70$	1.14989	0.137438
$71$	4.05247	0.480939	0.240470	−	0.970657i	$-0.422699\pi$
$71$	4.05247	0.480939	0.240470	+	0.970657i	$0.422699\pi$
$72$	0	0
$73$	−8.44391	−0.988285	−0.494143	−	0.869381i	$-0.664518\pi$
$73$	−8.44391	−0.988285	−0.494143	+	0.869381i	$0.664518\pi$
$74$	−14.4470	−1.67944
$75$	0	0
$76$	3.96824	0.455188
$77$	−0.588123	−0.0670229
$78$	0	0
$79$	12.6320	1.42121	0.710603	−	0.703593i	$-0.248422\pi$
$79$	12.6320	1.42121	0.710603	+	0.703593i	$0.248422\pi$
$80$	3.04958	0.340953
$81$	0	0
$82$	27.2983	3.01459
$83$	−0.0137315	−0.00150723	−0.000753615	−	1.00000i	$-0.500240\pi$
$83$	−0.0137315	−0.00150723	−0.000753615		1.00000i	$0.500240\pi$
$84$	0	0
$85$	5.25928	0.570449
$86$	19.0666	2.05600
$87$	0	0
$88$	−4.80840	−0.512577
$89$	−10.7746	−1.14211	−0.571054	−	0.820912i	$-0.693465\pi$
$89$	−10.7746	−1.14211	−0.571054	+	0.820912i	$0.693465\pi$
$90$	0	0
$91$	1.35902	0.142464
$92$	−16.2715	−1.69642
$93$	0	0
$94$	−6.26860	−0.646557
$95$	0.800323	0.0821114
$96$	0	0
$97$	−7.37915	−0.749239	−0.374619	−	0.927179i	$-0.622227\pi$
$97$	−7.37915	−0.749239	−0.374619	+	0.927179i	$0.622227\pi$
$98$	−16.2560	−1.64210
$99$	0	0
$100$	−17.2995	−1.72995
$101$	−0.938512	−0.0933854	−0.0466927	−	0.998909i	$-0.514868\pi$
$101$	−0.938512	−0.0933854	−0.0466927	+	0.998909i	$0.514868\pi$
$102$	0	0
$103$	10.0710	0.992323	0.496161	−	0.868230i	$-0.334742\pi$
$103$	10.0710	0.992323	0.496161	+	0.868230i	$0.334742\pi$
$104$	11.1111	1.08953
$105$	0	0
$106$	−28.0257	−2.72209
$107$	15.6571	1.51363	0.756817	−	0.653627i	$-0.226754\pi$
$107$	15.6571	1.51363	0.756817	+	0.653627i	$0.226754\pi$
$108$	0	0
$109$	−7.55580	−0.723714	−0.361857	−	0.932234i	$-0.617857\pi$
$109$	−7.55580	−0.723714	−0.361857	+	0.932234i	$0.617857\pi$
$110$	−1.95519	−0.186420
$111$	0	0
$112$	2.24101	0.211755
$113$	−16.3998	−1.54276	−0.771379	−	0.636375i	$-0.780433\pi$
$113$	−16.3998	−1.54276	−0.771379	+	0.636375i	$0.780433\pi$
$114$	0	0
$115$	−3.28167	−0.306017
$116$	18.7981	1.74536
$117$	0	0
$118$	6.02928	0.555040
$119$	3.86482	0.354288
$120$	0	0
$121$	1.00000	0.0909091
$122$	−20.4398	−1.85054
$123$	0	0
$124$	15.9643	1.43363
$125$	−7.49061	−0.669980
$126$	0	0
$127$	−12.5780	−1.11611	−0.558057	−	0.829802i	$-0.688453\pi$
$127$	−12.5780	−1.11611	−0.558057	+	0.829802i	$0.688453\pi$
$128$	−19.8397	−1.75359
$129$	0	0
$130$	4.51799	0.396254
$131$	−4.21358	−0.368142	−0.184071	−	0.982913i	$-0.558928\pi$
$131$	−4.21358	−0.368142	−0.184071	+	0.982913i	$0.558928\pi$
$132$	0	0
$133$	0.588123	0.0509967
$134$	−6.89940	−0.596018
$135$	0	0
$136$	31.5982	2.70952
$137$	−6.74711	−0.576444	−0.288222	−	0.957564i	$-0.593064\pi$
$137$	−6.74711	−0.576444	−0.288222	+	0.957564i	$0.593064\pi$
$138$	0	0
$139$	−18.3905	−1.55986	−0.779930	−	0.625867i	$-0.784745\pi$
$139$	−18.3905	−1.55986	−0.779930	+	0.625867i	$0.784745\pi$
$140$	1.86780	0.157858
$141$	0	0
$142$	9.90016	0.830803
$143$	−2.31077	−0.193236
$144$	0	0
$145$	3.79125	0.314846
$146$	−20.6285	−1.70722
$147$	0	0
$148$	−23.4668	−1.92896
$149$	22.6306	1.85397	0.926984	−	0.375101i	$-0.122392\pi$
$149$	22.6306	1.85397	0.926984	+	0.375101i	$0.122392\pi$
$150$	0	0
$151$	11.3444	0.923197	0.461599	−	0.887089i	$-0.347276\pi$
$151$	11.3444	0.923197	0.461599	+	0.887089i	$0.347276\pi$
$152$	4.80840	0.390013
$153$	0	0
$154$	−1.43678	−0.115779
$155$	3.21971	0.258613
$156$	0	0
$157$	3.14378	0.250901	0.125451	−	0.992100i	$-0.459962\pi$
$157$	3.14378	0.250901	0.125451	+	0.992100i	$0.459962\pi$
$158$	30.8599	2.45508
$159$	0	0
$160$	−0.246430	−0.0194820
$161$	−2.41156	−0.190058
$162$	0	0
$163$	−17.9653	−1.40715	−0.703576	−	0.710620i	$-0.748415\pi$
$163$	−17.9653	−1.40715	−0.703576	+	0.710620i	$0.748415\pi$
$164$	44.3415	3.46249
$165$	0	0
$166$	−0.0335461	−0.00260368
$167$	−7.19830	−0.557021	−0.278511	−	0.960433i	$-0.589841\pi$
$167$	−7.19830	−0.557021	−0.278511	+	0.960433i	$0.589841\pi$
$168$	0	0
$169$	−7.66034	−0.589257
$170$	12.8484	0.985428
$171$	0	0
$172$	30.9704	2.36147
$173$	−4.80689	−0.365461	−0.182731	−	0.983163i	$-0.558494\pi$
$173$	−4.80689	−0.365461	−0.182731	+	0.983163i	$0.558494\pi$
$174$	0	0
$175$	−2.56391	−0.193814
$176$	−3.81044	−0.287222
$177$	0	0
$178$	−26.3224	−1.97295
$179$	15.4138	1.15208	0.576040	−	0.817422i	$-0.304597\pi$
$179$	15.4138	1.15208	0.576040	+	0.817422i	$0.304597\pi$
$180$	0	0
$181$	1.03075	0.0766149	0.0383074	−	0.999266i	$-0.487803\pi$
$181$	1.03075	0.0766149	0.0383074	+	0.999266i	$0.487803\pi$
$182$	3.32008	0.246100
$183$	0	0
$184$	−19.7165	−1.45352
$185$	−4.73283	−0.347965
$186$	0	0
$187$	−6.57145	−0.480552
$188$	−10.1823	−0.742619
$189$	0	0
$190$	1.95519	0.141844
$191$	21.9471	1.58803	0.794017	−	0.607895i	$-0.207986\pi$
$191$	21.9471	1.58803	0.794017	+	0.607895i	$0.207986\pi$
$192$	0	0
$193$	−14.3468	−1.03271	−0.516354	−	0.856375i	$-0.672711\pi$
$193$	−14.3468	−1.03271	−0.516354	+	0.856375i	$0.672711\pi$
$194$	−18.0272	−1.29428
$195$	0	0
$196$	−26.4051	−1.88608
$197$	7.47661	0.532687	0.266343	−	0.963878i	$-0.414185\pi$
$197$	7.47661	0.532687	0.266343	+	0.963878i	$0.414185\pi$
$198$	0	0
$199$	−12.9940	−0.921119	−0.460559	−	0.887629i	$-0.652351\pi$
$199$	−12.9940	−0.921119	−0.460559	+	0.887629i	$0.652351\pi$
$200$	−20.9621	−1.48225
$201$	0	0
$202$	−2.29278	−0.161320
$203$	2.78603	0.195541
$204$	0	0
$205$	8.94288	0.624598
$206$	24.6034	1.71420
$207$	0	0
$208$	8.80504	0.610520
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−27.9511	−1.92423	−0.962117	−	0.272636i	$-0.912105\pi$
$211$	−27.9511	−1.92423	−0.962117	+	0.272636i	$0.912105\pi$
$212$	−45.5230	−3.12653
$213$	0	0
$214$	38.2504	2.61474
$215$	6.24618	0.425986
$216$	0	0
$217$	2.36603	0.160616
$218$	−18.4588	−1.25019
$219$	0	0
$220$	−3.17587	−0.214117
$221$	15.1851	1.02146
$222$	0	0
$223$	11.3216	0.758150	0.379075	−	0.925366i	$-0.376242\pi$
$223$	11.3216	0.758150	0.379075	+	0.925366i	$0.376242\pi$
$224$	−0.181091	−0.0120997
$225$	0	0
$226$	−40.0646	−2.66505
$227$	−1.50898	−0.100154	−0.0500772	−	0.998745i	$-0.515947\pi$
$227$	−1.50898	−0.100154	−0.0500772	+	0.998745i	$0.515947\pi$
$228$	0	0
$229$	−21.2793	−1.40618	−0.703089	−	0.711102i	$-0.748196\pi$
$229$	−21.2793	−1.40618	−0.703089	+	0.711102i	$0.748196\pi$
$230$	−8.01711	−0.528633
$231$	0	0
$232$	22.7781	1.49546
$233$	25.8325	1.69234	0.846172	−	0.532910i	$-0.178902\pi$
$233$	25.8325	1.69234	0.846172	+	0.532910i	$0.178902\pi$
$234$	0	0
$235$	−2.05358	−0.133961
$236$	9.79355	0.637506
$237$	0	0
$238$	9.44175	0.612018
$239$	22.5612	1.45936	0.729682	−	0.683787i	$-0.239668\pi$
$239$	22.5612	1.45936	0.729682	+	0.683787i	$0.239668\pi$
$240$	0	0
$241$	1.23059	0.0792691	0.0396346	−	0.999214i	$-0.487381\pi$
$241$	1.23059	0.0792691	0.0396346	+	0.999214i	$0.487381\pi$
$242$	2.44300	0.157042
$243$	0	0
$244$	−33.2011	−2.12548
$245$	−5.32544	−0.340230
$246$	0	0
$247$	2.31077	0.147031
$248$	19.3443	1.22836
$249$	0	0
$250$	−18.2995	−1.15736
$251$	3.67960	0.232254	0.116127	−	0.993234i	$-0.462952\pi$
$251$	3.67960	0.232254	0.116127	+	0.993234i	$0.462952\pi$
$252$	0	0
$253$	4.10043	0.257792
$254$	−30.7280	−1.92804
$255$	0	0
$256$	−31.7220	−1.98263
$257$	9.00012	0.561412	0.280706	−	0.959794i	$-0.409431\pi$
$257$	9.00012	0.561412	0.280706	+	0.959794i	$0.409431\pi$
$258$	0	0
$259$	−3.47796	−0.216110
$260$	7.33871	0.455127
$261$	0	0
$262$	−10.2938	−0.635950
$263$	−0.317206	−0.0195598	−0.00977989	−	0.999952i	$-0.503113\pi$
$263$	−0.317206	−0.0195598	−0.00977989	+	0.999952i	$0.503113\pi$
$264$	0	0
$265$	−9.18117	−0.563994
$266$	1.43678	0.0880949
$267$	0	0
$268$	−11.2069	−0.684572
$269$	−17.9973	−1.09731	−0.548656	−	0.836048i	$-0.684860\pi$
$269$	−17.9973	−1.09731	−0.548656	+	0.836048i	$0.684860\pi$
$270$	0	0
$271$	−0.262808	−0.0159644	−0.00798222	−	0.999968i	$-0.502541\pi$
$271$	−0.262808	−0.0159644	−0.00798222	+	0.999968i	$0.502541\pi$
$272$	25.0401	1.51828
$273$	0	0
$274$	−16.4832	−0.995785
$275$	4.35948	0.262887
$276$	0	0
$277$	−6.20953	−0.373094	−0.186547	−	0.982446i	$-0.559730\pi$
$277$	−6.20953	−0.373094	−0.186547	+	0.982446i	$0.559730\pi$
$278$	−44.9279	−2.69460
$279$	0	0
$280$	2.26326	0.135256
$281$	−7.29039	−0.434908	−0.217454	−	0.976071i	$-0.569775\pi$
$281$	−7.29039	−0.434908	−0.217454	+	0.976071i	$0.569775\pi$
$282$	0	0
$283$	−15.9557	−0.948468	−0.474234	−	0.880399i	$-0.657275\pi$
$283$	−15.9557	−0.948468	−0.474234	+	0.880399i	$0.657275\pi$
$284$	16.0811	0.954241
$285$	0	0
$286$	−5.64521	−0.333808
$287$	6.57175	0.387918
$288$	0	0
$289$	26.1839	1.54023
$290$	9.26201	0.543884
$291$	0	0
$292$	−33.5075	−1.96088
$293$	2.82613	0.165105	0.0825523	−	0.996587i	$-0.473693\pi$
$293$	2.82613	0.165105	0.0825523	+	0.996587i	$0.473693\pi$
$294$	0	0
$295$	1.97518	0.115000
$296$	−28.4352	−1.65276
$297$	0	0
$298$	55.2864	3.20266
$299$	−9.47516	−0.547963
$300$	0	0
$301$	4.59005	0.264566
$302$	27.7144	1.59479
$303$	0	0
$304$	3.81044	0.218544
$305$	−6.69606	−0.383415
$306$	0	0
$307$	16.7858	0.958016	0.479008	−	0.877811i	$-0.340997\pi$
$307$	16.7858	0.958016	0.479008	+	0.877811i	$0.340997\pi$
$308$	−2.33381	−0.132981
$309$	0	0
$310$	7.86574	0.446744
$311$	14.6719	0.831970	0.415985	−	0.909372i	$-0.363437\pi$
$311$	14.6719	0.831970	0.415985	+	0.909372i	$0.363437\pi$
$312$	0	0
$313$	27.4425	1.55114	0.775572	−	0.631259i	$-0.217462\pi$
$313$	27.4425	1.55114	0.775572	+	0.631259i	$0.217462\pi$
$314$	7.68026	0.433422
$315$	0	0
$316$	50.1266	2.81984
$317$	3.15940	0.177450	0.0887249	−	0.996056i	$-0.471721\pi$
$317$	3.15940	0.177450	0.0887249	+	0.996056i	$0.471721\pi$
$318$	0	0
$319$	−4.73715	−0.265229
$320$	−6.70119	−0.374608
$321$	0	0
$322$	−5.89144	−0.328317
$323$	6.57145	0.365645
$324$	0	0
$325$	−10.0738	−0.558792
$326$	−43.8892	−2.43080
$327$	0	0
$328$	53.7295	2.96672
$329$	−1.50909	−0.0831989
$330$	0	0
$331$	9.26332	0.509158	0.254579	−	0.967052i	$-0.418063\pi$
$331$	9.26332	0.509158	0.254579	+	0.967052i	$0.418063\pi$
$332$	−0.0544899	−0.00299052
$333$	0	0
$334$	−17.5854	−0.962232
$335$	−2.26024	−0.123490
$336$	0	0
$337$	−13.9986	−0.762555	−0.381277	−	0.924461i	$-0.624516\pi$
$337$	−13.9986	−0.762555	−0.381277	+	0.924461i	$0.624516\pi$
$338$	−18.7142	−1.01792
$339$	0	0
$340$	20.8701	1.13184
$341$	−4.02301	−0.217858
$342$	0	0
$343$	−8.03030	−0.433595
$344$	37.5275	2.02335
$345$	0	0
$346$	−11.7432	−0.631320
$347$	30.9325	1.66054	0.830272	−	0.557358i	$-0.188185\pi$
$347$	30.9325	1.66054	0.830272	+	0.557358i	$0.188185\pi$
$348$	0	0
$349$	−17.3337	−0.927852	−0.463926	−	0.885874i	$-0.653560\pi$
$349$	−17.3337	−0.927852	−0.463926	+	0.885874i	$0.653560\pi$
$350$	−6.26363	−0.334805
$351$	0	0
$352$	0.307914	0.0164119
$353$	−7.09753	−0.377763	−0.188882	−	0.982000i	$-0.560486\pi$
$353$	−7.09753	−0.377763	−0.188882	+	0.982000i	$0.560486\pi$
$354$	0	0
$355$	3.24328	0.172135
$356$	−42.7563	−2.26608
$357$	0	0
$358$	37.6558	1.99017
$359$	−25.6409	−1.35328	−0.676638	−	0.736316i	$-0.736564\pi$
$359$	−25.6409	−1.35328	−0.676638	+	0.736316i	$0.736564\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.51811	0.132349
$363$	0	0
$364$	5.39290	0.282665
$365$	−6.75785	−0.353722
$366$	0	0
$367$	8.81606	0.460195	0.230097	−	0.973168i	$-0.426096\pi$
$367$	8.81606	0.460195	0.230097	+	0.973168i	$0.426096\pi$
$368$	−15.6244	−0.814481
$369$	0	0
$370$	−11.5623	−0.601095
$371$	−6.74685	−0.350279
$372$	0	0
$373$	14.5020	0.750887	0.375443	−	0.926845i	$-0.377490\pi$
$373$	14.5020	0.750887	0.375443	+	0.926845i	$0.377490\pi$
$374$	−16.0540	−0.830134
$375$	0	0
$376$	−12.3381	−0.636289
$377$	10.9465	0.563771
$378$	0	0
$379$	10.5223	0.540495	0.270248	−	0.962791i	$-0.412894\pi$
$379$	10.5223	0.540495	0.270248	+	0.962791i	$0.412894\pi$
$380$	3.17587	0.162919
$381$	0	0
$382$	53.6166	2.74327
$383$	−27.5133	−1.40586	−0.702932	−	0.711257i	$-0.748126\pi$
$383$	−27.5133	−1.40586	−0.702932	+	0.711257i	$0.748126\pi$
$384$	0	0
$385$	−0.470688	−0.0239885
$386$	−35.0493	−1.78396
$387$	0	0
$388$	−29.2822	−1.48658
$389$	−6.36132	−0.322532	−0.161266	−	0.986911i	$-0.551558\pi$
$389$	−6.36132	−0.322532	−0.161266	+	0.986911i	$0.551558\pi$
$390$	0	0
$391$	−26.9458	−1.36271
$392$	−31.9956	−1.61602
$393$	0	0
$394$	18.2653	0.920195
$395$	10.1096	0.508671
$396$	0	0
$397$	18.5939	0.933203	0.466601	−	0.884468i	$-0.345478\pi$
$397$	18.5939	0.933203	0.466601	+	0.884468i	$0.345478\pi$
$398$	−31.7443	−1.59120
$399$	0	0
$400$	−16.6115	−0.830577
$401$	17.7160	0.884695	0.442348	−	0.896844i	$-0.354146\pi$
$401$	17.7160	0.884695	0.442348	+	0.896844i	$0.354146\pi$
$402$	0	0
$403$	9.29626	0.463079
$404$	−3.72424	−0.185288
$405$	0	0
$406$	6.80626	0.337789
$407$	5.91366	0.293129
$408$	0	0
$409$	−22.6211	−1.11854	−0.559271	−	0.828985i	$-0.688919\pi$
$409$	−22.6211	−1.11854	−0.559271	+	0.828985i	$0.688919\pi$
$410$	21.8474	1.07897
$411$	0	0
$412$	39.9640	1.96889
$413$	1.45148	0.0714226
$414$	0	0
$415$	−0.0109896	−0.000539460 0
$416$	−0.711518	−0.0348850
$417$	0	0
$418$	−2.44300	−0.119491
$419$	−19.1301	−0.934569	−0.467284	−	0.884107i	$-0.654768\pi$
$419$	−19.1301	−0.934569	−0.467284	+	0.884107i	$0.654768\pi$
$420$	0	0
$421$	16.6696	0.812428	0.406214	−	0.913778i	$-0.366849\pi$
$421$	16.6696	0.812428	0.406214	+	0.913778i	$0.366849\pi$
$422$	−68.2845	−3.32404
$423$	0	0
$424$	−55.1612	−2.67886
$425$	−28.6481	−1.38964
$426$	0	0
$427$	−4.92065	−0.238127
$428$	62.1313	3.00323
$429$	0	0
$430$	15.2594	0.735873
$431$	−19.6828	−0.948087	−0.474044	−	0.880501i	$-0.657206\pi$
$431$	−19.6828	−0.948087	−0.474044	+	0.880501i	$0.657206\pi$
$432$	0	0
$433$	−19.6530	−0.944462	−0.472231	−	0.881475i	$-0.656551\pi$
$433$	−19.6530	−0.944462	−0.472231	+	0.881475i	$0.656551\pi$
$434$	5.78020	0.277458
$435$	0	0
$436$	−29.9832	−1.43593
$437$	−4.10043	−0.196150
$438$	0	0
$439$	−6.05821	−0.289143	−0.144571	−	0.989494i	$-0.546180\pi$
$439$	−6.05821	−0.289143	−0.144571	+	0.989494i	$0.546180\pi$
$440$	−3.84827	−0.183459
$441$	0	0
$442$	37.0972	1.76453
$443$	−7.01333	−0.333214	−0.166607	−	0.986023i	$-0.553281\pi$
$443$	−7.01333	−0.333214	−0.166607	+	0.986023i	$0.553281\pi$
$444$	0	0
$445$	−8.62318	−0.408778
$446$	27.6586	1.30967
$447$	0	0
$448$	−4.92442	−0.232657
$449$	16.7703	0.791439	0.395719	−	0.918372i	$-0.370495\pi$
$449$	16.7703	0.791439	0.395719	+	0.918372i	$0.370495\pi$
$450$	0	0
$451$	−11.1741	−0.526168
$452$	−65.0781	−3.06102
$453$	0	0
$454$	−3.68643	−0.173013
$455$	1.08765	0.0509899
$456$	0	0
$457$	31.9584	1.49495	0.747476	−	0.664289i	$-0.231265\pi$
$457$	31.9584	1.49495	0.747476	+	0.664289i	$0.231265\pi$
$458$	−51.9854	−2.42912
$459$	0	0
$460$	−13.0225	−0.607175
$461$	13.3466	0.621612	0.310806	−	0.950473i	$-0.399401\pi$
$461$	13.3466	0.621612	0.310806	+	0.950473i	$0.399401\pi$
$462$	0	0
$463$	−27.6018	−1.28277	−0.641383	−	0.767221i	$-0.721639\pi$
$463$	−27.6018	−1.28277	−0.641383	+	0.767221i	$0.721639\pi$
$464$	18.0506	0.837978
$465$	0	0
$466$	63.1087	2.92346
$467$	23.2142	1.07422	0.537112	−	0.843511i	$-0.319515\pi$
$467$	23.2142	1.07422	0.537112	+	0.843511i	$0.319515\pi$
$468$	0	0
$469$	−1.66095	−0.0766956
$470$	−5.01690	−0.231412
$471$	0	0
$472$	11.8671	0.546226
$473$	−7.80457	−0.358855
$474$	0	0
$475$	−4.35948	−0.200027
$476$	15.3365	0.702949
$477$	0	0
$478$	55.1170	2.52099
$479$	−4.07012	−0.185969	−0.0929843	−	0.995668i	$-0.529641\pi$
$479$	−4.07012	−0.185969	−0.0929843	+	0.995668i	$0.529641\pi$
$480$	0	0
$481$	−13.6651	−0.623075
$482$	3.00632	0.136934
$483$	0	0
$484$	3.96824	0.180374
$485$	−5.90570	−0.268164
$486$	0	0
$487$	−7.50704	−0.340177	−0.170088	−	0.985429i	$-0.554405\pi$
$487$	−7.50704	−0.340177	−0.170088	+	0.985429i	$0.554405\pi$
$488$	−40.2305	−1.82115
$489$	0	0
$490$	−13.0100	−0.587733
$491$	−16.2778	−0.734606	−0.367303	−	0.930101i	$-0.619719\pi$
$491$	−16.2778	−0.734606	−0.367303	+	0.930101i	$0.619719\pi$
$492$	0	0
$493$	31.1299	1.40202
$494$	5.64521	0.253990
$495$	0	0
$496$	15.3294	0.688312
$497$	2.38335	0.106908
$498$	0	0
$499$	41.0613	1.83816	0.919079	−	0.394073i	$-0.128934\pi$
$499$	41.0613	1.83816	0.919079	+	0.394073i	$0.128934\pi$
$500$	−29.7245	−1.32932
$501$	0	0
$502$	8.98926	0.401210
$503$	29.1287	1.29878	0.649392	−	0.760454i	$-0.275024\pi$
$503$	29.1287	1.29878	0.649392	+	0.760454i	$0.275024\pi$
$504$	0	0
$505$	−0.751112	−0.0334241
$506$	10.0174	0.445326
$507$	0	0
$508$	−49.9124	−2.21450
$509$	−14.3959	−0.638087	−0.319044	−	0.947740i	$-0.603362\pi$
$509$	−14.3959	−0.638087	−0.319044	+	0.947740i	$0.603362\pi$
$510$	0	0
$511$	−4.96606	−0.219686
$512$	−37.8175	−1.67131
$513$	0	0
$514$	21.9873	0.969817
$515$	8.06003	0.355167
$516$	0	0
$517$	2.56595	0.112850
$518$	−8.49664	−0.373321
$519$	0	0
$520$	8.89247	0.389961
$521$	20.3390	0.891067	0.445534	−	0.895265i	$-0.353014\pi$
$521$	20.3390	0.891067	0.445534	+	0.895265i	$0.353014\pi$
$522$	0	0
$523$	45.0421	1.96955	0.984776	−	0.173829i	$-0.0556140\pi$
$523$	45.0421	1.96955	0.984776	+	0.173829i	$0.0556140\pi$
$524$	−16.7205	−0.730437
$525$	0	0
$526$	−0.774934	−0.0337887
$527$	26.4370	1.15161
$528$	0	0
$529$	−6.18643	−0.268975
$530$	−22.4296	−0.974278
$531$	0	0
$532$	2.33381	0.101184
$533$	25.8208	1.11842
$534$	0	0
$535$	12.5308	0.541752
$536$	−13.5797	−0.586552
$537$	0	0
$538$	−43.9673	−1.89556
$539$	6.65411	0.286613
$540$	0	0
$541$	10.4800	0.450569	0.225284	−	0.974293i	$-0.427669\pi$
$541$	10.4800	0.450569	0.225284	+	0.974293i	$0.427669\pi$
$542$	−0.642039	−0.0275779
$543$	0	0
$544$	−2.02344	−0.0867542
$545$	−6.04708	−0.259028
$546$	0	0
$547$	−9.59366	−0.410195	−0.205098	−	0.978742i	$-0.565751\pi$
$547$	−9.59366	−0.410195	−0.205098	+	0.978742i	$0.565751\pi$
$548$	−26.7741	−1.14373
$549$	0	0
$550$	10.6502	0.454126
$551$	4.73715	0.201809
$552$	0	0
$553$	7.42915	0.315920
$554$	−15.1699	−0.644506
$555$	0	0
$556$	−72.9778	−3.09495
$557$	10.5593	0.447413	0.223706	−	0.974657i	$-0.428184\pi$
$557$	10.5593	0.447413	0.223706	+	0.974657i	$0.428184\pi$
$558$	0	0
$559$	18.0346	0.762781
$560$	1.79353	0.0757904
$561$	0	0
$562$	−17.8104	−0.751287
$563$	−36.1211	−1.52232	−0.761162	−	0.648562i	$-0.775371\pi$
$563$	−36.1211	−1.52232	−0.761162	+	0.648562i	$0.775371\pi$
$564$	0	0
$565$	−13.1251	−0.552177
$566$	−38.9797	−1.63844
$567$	0	0
$568$	19.4859	0.817609
$569$	2.74039	0.114883	0.0574416	−	0.998349i	$-0.481706\pi$
$569$	2.74039	0.114883	0.0574416	+	0.998349i	$0.481706\pi$
$570$	0	0
$571$	4.82157	0.201776	0.100888	−	0.994898i	$-0.467832\pi$
$571$	4.82157	0.201776	0.100888	+	0.994898i	$0.467832\pi$
$572$	−9.16969	−0.383404
$573$	0	0
$574$	16.0548	0.670113
$575$	17.8758	0.745472
$576$	0	0
$577$	−27.2712	−1.13531	−0.567657	−	0.823265i	$-0.692150\pi$
$577$	−27.2712	−1.13531	−0.567657	+	0.823265i	$0.692150\pi$
$578$	63.9673	2.66069
$579$	0	0
$580$	15.0446	0.624692
$581$	−0.00807582	−0.000335041 0
$582$	0	0
$583$	11.4718	0.475115
$584$	−40.6017	−1.68011
$585$	0	0
$586$	6.90424	0.285212
$587$	21.7232	0.896613	0.448307	−	0.893880i	$-0.352027\pi$
$587$	21.7232	0.896613	0.448307	+	0.893880i	$0.352027\pi$
$588$	0	0
$589$	4.02301	0.165765
$590$	4.82537	0.198657
$591$	0	0
$592$	−22.5336	−0.926126
$593$	−19.5478	−0.802730	−0.401365	−	0.915918i	$-0.631464\pi$
$593$	−19.5478	−0.802730	−0.401365	+	0.915918i	$0.631464\pi$
$594$	0	0
$595$	3.09310	0.126805
$596$	89.8034	3.67849
$597$	0	0
$598$	−23.1478	−0.946584
$599$	5.21220	0.212965	0.106482	−	0.994315i	$-0.466041\pi$
$599$	5.21220	0.212965	0.106482	+	0.994315i	$0.466041\pi$
$600$	0	0
$601$	41.4095	1.68913	0.844564	−	0.535454i	$-0.179860\pi$
$601$	41.4095	1.68913	0.844564	+	0.535454i	$0.179860\pi$
$602$	11.2135	0.457027
$603$	0	0
$604$	45.0174	1.83173
$605$	0.800323	0.0325377
$606$	0	0
$607$	26.5388	1.07718	0.538589	−	0.842568i	$-0.318957\pi$
$607$	26.5388	1.07718	0.538589	+	0.842568i	$0.318957\pi$
$608$	−0.307914	−0.0124875
$609$	0	0
$610$	−16.3585	−0.662335
$611$	−5.92931	−0.239874
$612$	0	0
$613$	−17.4161	−0.703428	−0.351714	−	0.936107i	$-0.614401\pi$
$613$	−17.4161	−0.703428	−0.351714	+	0.936107i	$0.614401\pi$
$614$	41.0076	1.65493
$615$	0	0
$616$	−2.82793	−0.113941
$617$	20.2862	0.816694	0.408347	−	0.912827i	$-0.366105\pi$
$617$	20.2862	0.816694	0.408347	+	0.912827i	$0.366105\pi$
$618$	0	0
$619$	14.6703	0.589648	0.294824	−	0.955551i	$-0.404739\pi$
$619$	14.6703	0.589648	0.294824	+	0.955551i	$0.404739\pi$
$620$	12.7766	0.513119
$621$	0	0
$622$	35.8435	1.43719
$623$	−6.33681	−0.253879
$624$	0	0
$625$	15.8025	0.632101
$626$	67.0420	2.67954
$627$	0	0
$628$	12.4753	0.497818
$629$	−38.8613	−1.54950
$630$	0	0
$631$	−6.86206	−0.273174	−0.136587	−	0.990628i	$-0.543613\pi$
$631$	−6.86206	−0.273174	−0.136587	+	0.990628i	$0.543613\pi$
$632$	60.7395	2.41609
$633$	0	0
$634$	7.71842	0.306538
$635$	−10.0664	−0.399474
$636$	0	0
$637$	−15.3761	−0.609224
$638$	−11.5728	−0.458173
$639$	0	0
$640$	−15.8781	−0.627638
$641$	14.2086	0.561206	0.280603	−	0.959824i	$-0.409466\pi$
$641$	14.2086	0.561206	0.280603	+	0.959824i	$0.409466\pi$
$642$	0	0
$643$	28.5456	1.12573	0.562864	−	0.826549i	$-0.309699\pi$
$643$	28.5456	1.12573	0.562864	+	0.826549i	$0.309699\pi$
$644$	−9.56965	−0.377097
$645$	0	0
$646$	16.0540	0.631638
$647$	16.0786	0.632116	0.316058	−	0.948740i	$-0.397641\pi$
$647$	16.0786	0.632116	0.316058	+	0.948740i	$0.397641\pi$
$648$	0	0
$649$	−2.46799	−0.0968769
$650$	−24.6102	−0.965291
$651$	0	0
$652$	−71.2907	−2.79196
$653$	−31.3855	−1.22821	−0.614106	−	0.789224i	$-0.710483\pi$
$653$	−31.3855	−1.22821	−0.614106	+	0.789224i	$0.710483\pi$
$654$	0	0
$655$	−3.37222	−0.131764
$656$	42.5782	1.66240
$657$	0	0
$658$	−3.68671	−0.143723
$659$	1.37686	0.0536349	0.0268174	−	0.999640i	$-0.491463\pi$
$659$	1.37686	0.0536349	0.0268174	+	0.999640i	$0.491463\pi$
$660$	0	0
$661$	−20.2558	−0.787860	−0.393930	−	0.919141i	$-0.628885\pi$
$661$	−20.2558	−0.787860	−0.393930	+	0.919141i	$0.628885\pi$
$662$	22.6303	0.879551
$663$	0	0
$664$	−0.0660266	−0.00256233
$665$	0.470688	0.0182525
$666$	0	0
$667$	−19.4244	−0.752115
$668$	−28.5646	−1.10520
$669$	0	0
$670$	−5.52175	−0.213324
$671$	8.36670	0.322993
$672$	0	0
$673$	−10.4733	−0.403715	−0.201858	−	0.979415i	$-0.564698\pi$
$673$	−10.4733	−0.403715	−0.201858	+	0.979415i	$0.564698\pi$
$674$	−34.1987	−1.31728
$675$	0	0
$676$	−30.3981	−1.16916
$677$	10.1457	0.389933	0.194966	−	0.980810i	$-0.437540\pi$
$677$	10.1457	0.389933	0.194966	+	0.980810i	$0.437540\pi$
$678$	0	0
$679$	−4.33985	−0.166548
$680$	25.2887	0.969778
$681$	0	0
$682$	−9.82821	−0.376342
$683$	12.9495	0.495499	0.247750	−	0.968824i	$-0.420309\pi$
$683$	12.9495	0.495499	0.247750	+	0.968824i	$0.420309\pi$
$684$	0	0
$685$	−5.39986	−0.206318
$686$	−19.6180	−0.749019
$687$	0	0
$688$	29.7388	1.13378
$689$	−26.5088	−1.00990
$690$	0	0
$691$	32.7663	1.24649	0.623245	−	0.782027i	$-0.285814\pi$
$691$	32.7663	1.24649	0.623245	+	0.782027i	$0.285814\pi$
$692$	−19.0749	−0.725119
$693$	0	0
$694$	75.5681	2.86852
$695$	−14.7183	−0.558298
$696$	0	0
$697$	73.4300	2.78136
$698$	−42.3462	−1.60283
$699$	0	0
$700$	−10.1742	−0.384549
$701$	−0.0542369	−0.00204850	−0.00102425	−	0.999999i	$-0.500326\pi$
$701$	−0.0542369	−0.00204850	−0.00102425	+	0.999999i	$0.500326\pi$
$702$	0	0
$703$	−5.91366	−0.223038
$704$	8.37311	0.315573
$705$	0	0
$706$	−17.3392	−0.652571
$707$	−0.551960	−0.0207586
$708$	0	0
$709$	9.61093	0.360946	0.180473	−	0.983580i	$-0.442237\pi$
$709$	9.61093	0.360946	0.180473	+	0.983580i	$0.442237\pi$
$710$	7.92333	0.297357
$711$	0	0
$712$	−51.8087	−1.94161
$713$	−16.4961	−0.617784
$714$	0	0
$715$	−1.84936	−0.0691622
$716$	61.1655	2.28586
$717$	0	0
$718$	−62.6407	−2.33773
$719$	22.3065	0.831892	0.415946	−	0.909389i	$-0.363450\pi$
$719$	22.3065	0.831892	0.415946	+	0.909389i	$0.363450\pi$
$720$	0	0
$721$	5.92298	0.220583
$722$	2.44300	0.0909190
$723$	0	0
$724$	4.09025	0.152013
$725$	−20.6515	−0.766978
$726$	0	0
$727$	36.6445	1.35907	0.679534	−	0.733644i	$-0.262182\pi$
$727$	36.6445	1.35907	0.679534	+	0.733644i	$0.262182\pi$
$728$	6.53470	0.242192
$729$	0	0
$730$	−16.5094	−0.611041
$731$	51.2874	1.89693
$732$	0	0
$733$	31.2843	1.15551	0.577755	−	0.816210i	$-0.303929\pi$
$733$	31.2843	1.15551	0.577755	+	0.816210i	$0.303929\pi$
$734$	21.5376	0.794968
$735$	0	0
$736$	1.26258	0.0465393
$737$	2.82416	0.104029
$738$	0	0
$739$	41.0054	1.50841	0.754205	−	0.656639i	$-0.228023\pi$
$739$	41.0054	1.50841	0.754205	+	0.656639i	$0.228023\pi$
$740$	−18.7810	−0.690403
$741$	0	0
$742$	−16.4825	−0.605093
$743$	39.7923	1.45984	0.729918	−	0.683535i	$-0.239558\pi$
$743$	39.7923	1.45984	0.729918	+	0.683535i	$0.239558\pi$
$744$	0	0
$745$	18.1117	0.663563
$746$	35.4284	1.29713
$747$	0	0
$748$	−26.0771	−0.953472
$749$	9.20833	0.336465
$750$	0	0
$751$	3.25214	0.118672	0.0593361	−	0.998238i	$-0.481102\pi$
$751$	3.25214	0.118672	0.0593361	+	0.998238i	$0.481102\pi$
$752$	−9.77737	−0.356544
$753$	0	0
$754$	26.7422	0.973893
$755$	9.07921	0.330426
$756$	0	0
$757$	26.5536	0.965108	0.482554	−	0.875866i	$-0.339709\pi$
$757$	26.5536	0.965108	0.482554	+	0.875866i	$0.339709\pi$
$758$	25.7060	0.933684
$759$	0	0
$760$	3.84827	0.139591
$761$	3.21425	0.116516	0.0582582	−	0.998302i	$-0.481445\pi$
$761$	3.21425	0.116516	0.0582582	+	0.998302i	$0.481445\pi$
$762$	0	0
$763$	−4.44374	−0.160874
$764$	87.0912	3.15085
$765$	0	0
$766$	−67.2149	−2.42857
$767$	5.70295	0.205921
$768$	0	0
$769$	31.9235	1.15119	0.575596	−	0.817734i	$-0.304770\pi$
$769$	31.9235	1.15119	0.575596	+	0.817734i	$0.304770\pi$
$770$	−1.14989	−0.0414392
$771$	0	0
$772$	−56.9317	−2.04902
$773$	2.02574	0.0728609	0.0364305	−	0.999336i	$-0.488401\pi$
$773$	2.02574	0.0728609	0.0364305	+	0.999336i	$0.488401\pi$
$774$	0	0
$775$	−17.5383	−0.629993
$776$	−35.4819	−1.27373
$777$	0	0
$778$	−15.5407	−0.557161
$779$	11.1741	0.400354
$780$	0	0
$781$	−4.05247	−0.145009
$782$	−65.8285	−2.35402
$783$	0	0
$784$	−25.3551	−0.905538
$785$	2.51604	0.0898013
$786$	0	0
$787$	−48.7384	−1.73734	−0.868668	−	0.495395i	$-0.835023\pi$
$787$	−48.7384	−1.73734	−0.868668	+	0.495395i	$0.835023\pi$
$788$	29.6690	1.05691
$789$	0	0
$790$	24.6978	0.878710
$791$	−9.64507	−0.342939
$792$	0	0
$793$	−19.3335	−0.686554
$794$	45.4249	1.61207
$795$	0	0
$796$	−51.5632	−1.82761
$797$	−8.62944	−0.305670	−0.152835	−	0.988252i	$-0.548840\pi$
$797$	−8.62944	−0.305670	−0.152835	+	0.988252i	$0.548840\pi$
$798$	0	0
$799$	−16.8620	−0.596534
$800$	1.34234	0.0474590
$801$	0	0
$802$	43.2802	1.52828
$803$	8.44391	0.297979
$804$	0	0
$805$	−1.93003	−0.0680245
$806$	22.7107	0.799951
$807$	0	0
$808$	−4.51274	−0.158758
$809$	46.5536	1.63674	0.818369	−	0.574693i	$-0.194879\pi$
$809$	46.5536	1.63674	0.818369	+	0.574693i	$0.194879\pi$
$810$	0	0
$811$	−16.4402	−0.577292	−0.288646	−	0.957436i	$-0.593205\pi$
$811$	−16.4402	−0.577292	−0.288646	+	0.957436i	$0.593205\pi$
$812$	11.0556	0.387976
$813$	0	0
$814$	14.4470	0.506369
$815$	−14.3781	−0.503641
$816$	0	0
$817$	7.80457	0.273047
$818$	−55.2633	−1.93224
$819$	0	0
$820$	35.4875	1.23928
$821$	7.61708	0.265838	0.132919	−	0.991127i	$-0.457565\pi$
$821$	7.61708	0.265838	0.132919	+	0.991127i	$0.457565\pi$
$822$	0	0
$823$	6.33183	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$823$	6.33183	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$824$	48.4253	1.68698
$825$	0	0
$826$	3.54596	0.123380
$827$	−1.70709	−0.0593615	−0.0296807	−	0.999559i	$-0.509449\pi$
$827$	−1.70709	−0.0593615	−0.0296807	+	0.999559i	$0.509449\pi$
$828$	0	0
$829$	−53.8836	−1.87145	−0.935727	−	0.352724i	$-0.885255\pi$
$829$	−53.8836	−1.87145	−0.935727	+	0.352724i	$0.885255\pi$
$830$	−0.0268477	−0.000931896 0
$831$	0	0
$832$	−19.3483	−0.670782
$833$	−43.7271	−1.51506
$834$	0	0
$835$	−5.76096	−0.199366
$836$	−3.96824	−0.137244
$837$	0	0
$838$	−46.7349	−1.61443
$839$	40.6382	1.40299	0.701494	−	0.712676i	$-0.252517\pi$
$839$	40.6382	1.40299	0.701494	+	0.712676i	$0.252517\pi$
$840$	0	0
$841$	−6.55942	−0.226187
$842$	40.7238	1.40344
$843$	0	0
$844$	−110.917	−3.81791
$845$	−6.13074	−0.210904
$846$	0	0
$847$	0.588123	0.0202082
$848$	−43.7127	−1.50110
$849$	0	0
$850$	−69.9873	−2.40054
$851$	24.2486	0.831230
$852$	0	0
$853$	44.3438	1.51830	0.759151	−	0.650915i	$-0.225614\pi$
$853$	44.3438	1.51830	0.759151	+	0.650915i	$0.225614\pi$
$854$	−12.0211	−0.411355
$855$	0	0
$856$	75.2858	2.57322
$857$	32.4832	1.10961	0.554803	−	0.831982i	$-0.312794\pi$
$857$	32.4832	1.10961	0.554803	+	0.831982i	$0.312794\pi$
$858$	0	0
$859$	−4.04454	−0.137998	−0.0689990	−	0.997617i	$-0.521981\pi$
$859$	−4.04454	−0.137998	−0.0689990	+	0.997617i	$0.521981\pi$
$860$	24.7863	0.845206
$861$	0	0
$862$	−48.0850	−1.63778
$863$	−29.4268	−1.00170	−0.500850	−	0.865534i	$-0.666979\pi$
$863$	−29.4268	−1.00170	−0.500850	+	0.865534i	$0.666979\pi$
$864$	0	0
$865$	−3.84707	−0.130804
$866$	−48.0122	−1.63152
$867$	0	0
$868$	9.38896	0.318682
$869$	−12.6320	−0.428510
$870$	0	0
$871$	−6.52597	−0.221124
$872$	−36.3313	−1.23033
$873$	0	0
$874$	−10.0174	−0.338842
$875$	−4.40540	−0.148930
$876$	0	0
$877$	31.1047	1.05033	0.525166	−	0.851000i	$-0.324003\pi$
$877$	31.1047	1.05033	0.525166	+	0.851000i	$0.324003\pi$
$878$	−14.8002	−0.499483
$879$	0	0
$880$	−3.04958	−0.102801
$881$	−25.1336	−0.846774	−0.423387	−	0.905949i	$-0.639159\pi$
$881$	−25.1336	−0.846774	−0.423387	+	0.905949i	$0.639159\pi$
$882$	0	0
$883$	−38.8941	−1.30889	−0.654446	−	0.756109i	$-0.727098\pi$
$883$	−38.8941	−1.30889	−0.654446	+	0.756109i	$0.727098\pi$
$884$	60.2581	2.02670
$885$	0	0
$886$	−17.1336	−0.575613
$887$	−38.1500	−1.28095	−0.640476	−	0.767978i	$-0.721263\pi$
$887$	−38.1500	−1.28095	−0.640476	+	0.767978i	$0.721263\pi$
$888$	0	0
$889$	−7.39740	−0.248101
$890$	−21.0664	−0.706147
$891$	0	0
$892$	44.9268	1.50426
$893$	−2.56595	−0.0858661
$894$	0	0
$895$	12.3360	0.412347
$896$	−11.6682	−0.389806
$897$	0	0
$898$	40.9698	1.36718
$899$	19.0576	0.635607
$900$	0	0
$901$	−75.3865	−2.51149
$902$	−27.2983	−0.908934
$903$	0	0
$904$	−78.8566	−2.62273
$905$	0.824931	0.0274216
$906$	0	0
$907$	−2.68659	−0.0892067	−0.0446034	−	0.999005i	$-0.514202\pi$
$907$	−2.68659	−0.0892067	−0.0446034	+	0.999005i	$0.514202\pi$
$908$	−5.98798	−0.198718
$909$	0	0
$910$	2.65713	0.0880831
$911$	38.5099	1.27589	0.637945	−	0.770082i	$-0.279785\pi$
$911$	38.5099	1.27589	0.637945	+	0.770082i	$0.279785\pi$
$912$	0	0
$913$	0.0137315	0.000454447 0
$914$	78.0744	2.58247
$915$	0	0
$916$	−84.4414	−2.79002
$917$	−2.47810	−0.0818341
$918$	0	0
$919$	3.56656	0.117650	0.0588250	−	0.998268i	$-0.481265\pi$
$919$	3.56656	0.117650	0.0588250	+	0.998268i	$0.481265\pi$
$920$	−15.7796	−0.520238
$921$	0	0
$922$	32.6056	1.07381
$923$	9.36432	0.308230
$924$	0	0
$925$	25.7805	0.847657
$926$	−67.4312	−2.21593
$927$	0	0
$928$	−1.45863	−0.0478820
$929$	27.3654	0.897829	0.448914	−	0.893575i	$-0.351811\pi$
$929$	27.3654	0.897829	0.448914	+	0.893575i	$0.351811\pi$
$930$	0	0
$931$	−6.65411	−0.218080
$932$	102.510	3.35781
$933$	0	0
$934$	56.7122	1.85568
$935$	−5.25928	−0.171997
$936$	0	0
$937$	−47.8748	−1.56400	−0.782001	−	0.623277i	$-0.785801\pi$
$937$	−47.8748	−1.56400	−0.782001	+	0.623277i	$0.785801\pi$
$938$	−4.05770	−0.132489
$939$	0	0
$940$	−8.14911	−0.265795
$941$	7.37140	0.240301	0.120150	−	0.992756i	$-0.461662\pi$
$941$	7.37140	0.240301	0.120150	+	0.992756i	$0.461662\pi$
$942$	0	0
$943$	−45.8187	−1.49206
$944$	9.40410	0.306077
$945$	0	0
$946$	−19.0666	−0.619907
$947$	−56.4061	−1.83295	−0.916475	−	0.400091i	$-0.868978\pi$
$947$	−56.4061	−1.83295	−0.916475	+	0.400091i	$0.868978\pi$
$948$	0	0
$949$	−19.5119	−0.633384
$950$	−10.6502	−0.345538
$951$	0	0
$952$	18.5836	0.602298
$953$	−37.0454	−1.20002	−0.600009	−	0.799993i	$-0.704836\pi$
$953$	−37.0454	−1.20002	−0.600009	+	0.799993i	$0.704836\pi$
$954$	0	0
$955$	17.5647	0.568382
$956$	89.5283	2.89555
$957$	0	0
$958$	−9.94330	−0.321253
$959$	−3.96813	−0.128138
$960$	0	0
$961$	−14.8154	−0.477915
$962$	−33.3838	−1.07634
$963$	0	0
$964$	4.88327	0.157279
$965$	−11.4821	−0.369622
$966$	0	0
$967$	54.7029	1.75913	0.879563	−	0.475783i	$-0.157835\pi$
$967$	54.7029	1.75913	0.879563	+	0.475783i	$0.157835\pi$
$968$	4.80840	0.154548
$969$	0	0
$970$	−14.4276	−0.463243
$971$	−50.3142	−1.61466	−0.807330	−	0.590101i	$-0.799088\pi$
$971$	−50.3142	−1.61466	−0.807330	+	0.590101i	$0.799088\pi$
$972$	0	0
$973$	−10.8159	−0.346741
$974$	−18.3397	−0.587641
$975$	0	0
$976$	−31.8808	−1.02048
$977$	−51.7607	−1.65597	−0.827985	−	0.560750i	$-0.810513\pi$
$977$	−51.7607	−1.65597	−0.827985	+	0.560750i	$0.810513\pi$
$978$	0	0
$979$	10.7746	0.344359
$980$	−21.1326	−0.675056
$981$	0	0
$982$	−39.7666	−1.26900
$983$	25.8291	0.823821	0.411910	−	0.911224i	$-0.364862\pi$
$983$	25.8291	0.823821	0.411910	+	0.911224i	$0.364862\pi$
$984$	0	0
$985$	5.98370	0.190657
$986$	76.0503	2.42194
$987$	0	0
$988$	9.16969	0.291727
$989$	−32.0021	−1.01761
$990$	0	0
$991$	−54.6641	−1.73646	−0.868232	−	0.496159i	$-0.834743\pi$
$991$	−54.6641	−1.73646	−0.868232	+	0.496159i	$0.834743\pi$
$992$	−1.23874	−0.0393300
$993$	0	0
$994$	5.82252	0.184679
$995$	−10.3994	−0.329682
$996$	0	0
$997$	−4.18723	−0.132611	−0.0663055	−	0.997799i	$-0.521121\pi$
$997$	−4.18723	−0.132611	−0.0663055	+	0.997799i	$0.521121\pi$
$998$	100.313	3.17534
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.r.1.7	yes	7
3.2	odd	2		1881.2.a.n.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1881.2.a.n.1.1	✓	7	3.2	odd	2
1881.2.a.r.1.7	yes	7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1881.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44300 q^{2} +3.96824 q^{4} +0.800323 q^{5} +0.588123 q^{7} +4.80840 q^{8} +1.95519 q^{10} -1.00000 q^{11} +2.31077 q^{13} +1.43678 q^{14} +3.81044 q^{16} +6.57145 q^{17} +1.00000 q^{19} +3.17587 q^{20} -2.44300 q^{22} -4.10043 q^{23} -4.35948 q^{25} +5.64521 q^{26} +2.33381 q^{28} +4.73715 q^{29} +4.02301 q^{31} -0.307914 q^{32} +16.0540 q^{34} +0.470688 q^{35} -5.91366 q^{37} +2.44300 q^{38} +3.84827 q^{40} +11.1741 q^{41} +7.80457 q^{43} -3.96824 q^{44} -10.0174 q^{46} -2.56595 q^{47} -6.65411 q^{49} -10.6502 q^{50} +9.16969 q^{52} -11.4718 q^{53} -0.800323 q^{55} +2.82793 q^{56} +11.5728 q^{58} +2.46799 q^{59} -8.36670 q^{61} +9.82821 q^{62} -8.37311 q^{64} +1.84936 q^{65} -2.82416 q^{67} +26.0771 q^{68} +1.14989 q^{70} +4.05247 q^{71} -8.44391 q^{73} -14.4470 q^{74} +3.96824 q^{76} -0.588123 q^{77} +12.6320 q^{79} +3.04958 q^{80} +27.2983 q^{82} -0.0137315 q^{83} +5.25928 q^{85} +19.0666 q^{86} -4.80840 q^{88} -10.7746 q^{89} +1.35902 q^{91} -16.2715 q^{92} -6.26860 q^{94} +0.800323 q^{95} -7.37915 q^{97} -16.2560 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 6 q^{4} + 8 q^{5} - 2 q^{7} + 6 q^{8} + 6 q^{10} - 7 q^{11} - 7 q^{13} + 6 q^{14} + 5 q^{17} + 7 q^{19} + 20 q^{20} - 2 q^{22} + 22 q^{23} + 3 q^{25} + 10 q^{26} + 2 q^{28} + 18 q^{29} + 14 q^{32}+ \cdots - 62 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 + 6 * q^4 + 8 * q^5 - 2 * q^7 + 6 * q^8 + 6 * q^10 - 7 * q^11 - 7 * q^13 + 6 * q^14 + 5 * q^17 + 7 * q^19 + 20 * q^20 - 2 * q^22 + 22 * q^23 + 3 * q^25 + 10 * q^26 + 2 * q^28 + 18 * q^29 + 14 * q^32 + 6 * q^34 + 8 * q^35 + 6 * q^37 + 2 * q^38 + 28 * q^40 + 6 * q^41 + 8 * q^43 - 6 * q^44 + 8 * q^46 + 14 * q^47 + 5 * q^49 + 24 * q^50 - 12 * q^52 - q^53 - 8 * q^55 + 20 * q^56 - 2 * q^58 + 23 * q^59 - 28 * q^61 + 2 * q^62 - 26 * q^64 + 10 * q^65 + 14 * q^67 + 12 * q^68 - 18 * q^70 + 31 * q^71 - 12 * q^73 + 20 * q^74 + 6 * q^76 + 2 * q^77 - 17 * q^79 - 6 * q^80 + 12 * q^82 + 5 * q^83 - 12 * q^85 + 38 * q^86 - 6 * q^88 - 3 * q^89 - 12 * q^91 - 60 * q^94 + 8 * q^95 + 12 * q^97 - 62 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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