LMFDB - Embedded newform 1881.2.a.p.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1881,2,Mod(1,1881)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [7,1,0,15,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 $ x^7 - x^6 - 14x^5 + 10x^4 + 59x^3 - 27x^2 - 66*x + 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.19313$ 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)

$f(q)$	$=$	\(q+1.19313 q^{2} -0.576442 q^{4} -1.08235 q^{5} -1.30958 q^{7} -3.07403 q^{8} -1.29138 q^{10} +1.00000 q^{11} +2.53997 q^{13} -1.56249 q^{14} -2.51483 q^{16} +5.43892 q^{17} +1.00000 q^{19} +0.623909 q^{20} +1.19313 q^{22} -3.87095 q^{23} -3.82853 q^{25} +3.03052 q^{26} +0.754894 q^{28} +2.41412 q^{29} +3.03647 q^{31} +3.14754 q^{32} +6.48934 q^{34} +1.41741 q^{35} +6.85067 q^{37} +1.19313 q^{38} +3.32716 q^{40} +6.11344 q^{41} -2.95329 q^{43} -0.576442 q^{44} -4.61854 q^{46} +12.0923 q^{47} -5.28501 q^{49} -4.56793 q^{50} -1.46415 q^{52} +0.992927 q^{53} -1.08235 q^{55} +4.02567 q^{56} +2.88036 q^{58} +14.2251 q^{59} -5.82518 q^{61} +3.62290 q^{62} +8.78508 q^{64} -2.74913 q^{65} +8.79718 q^{67} -3.13522 q^{68} +1.69116 q^{70} +2.44975 q^{71} +5.84063 q^{73} +8.17374 q^{74} -0.576442 q^{76} -1.30958 q^{77} +17.0397 q^{79} +2.72192 q^{80} +7.29413 q^{82} +7.01303 q^{83} -5.88679 q^{85} -3.52366 q^{86} -3.07403 q^{88} +9.13103 q^{89} -3.32629 q^{91} +2.23138 q^{92} +14.4277 q^{94} -1.08235 q^{95} -14.7950 q^{97} -6.30570 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 15 q^{4} - 2 q^{5} + 10 q^{7} + 9 q^{8} - 6 q^{10} + 7 q^{11} - 4 q^{13} - 6 q^{14} + 27 q^{16} - 2 q^{17} + 7 q^{19} + 4 q^{20} + q^{22} - 10 q^{23} + 9 q^{25} + 8 q^{26} + 26 q^{28} + 18 q^{29}+ \cdots - 19 q^{98}+O(q^{100}) $ 7 * q + q^2 + 15 * q^4 - 2 * q^5 + 10 * q^7 + 9 * q^8 - 6 * q^10 + 7 * q^11 - 4 * q^13 - 6 * q^14 + 27 * q^16 - 2 * q^17 + 7 * q^19 + 4 * q^20 + q^22 - 10 * q^23 + 9 * q^25 + 8 * q^26 + 26 * q^28 + 18 * q^29 + 24 * q^31 + 49 * q^32 - 6 * q^34 - 8 * q^35 + q^38 - 2 * q^40 + 12 * q^41 + 2 * q^43 + 15 * q^44 - 4 * q^46 - 8 * q^47 + 17 * q^49 + 33 * q^50 - 60 * q^52 - 2 * q^53 - 2 * q^55 - 26 * q^56 - 8 * q^58 + 10 * q^59 + 14 * q^61 - 14 * q^62 + 55 * q^64 + 14 * q^65 + 8 * q^67 + 18 * q^68 - 66 * q^70 - 10 * q^71 - 6 * q^73 - 26 * q^74 + 15 * q^76 + 10 * q^77 + 52 * q^79 + 12 * q^80 + 24 * q^82 + 10 * q^83 - 12 * q^85 - 8 * q^86 + 9 * q^88 + 12 * q^91 + 24 * q^94 - 2 * q^95 - 24 * q^97 - 19 * 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 $

$2$	1.19313	0.843670	0.421835	−	0.906673i	$-0.361386\pi$
$2$	1.19313	0.843670	0.421835	+	0.906673i	$0.361386\pi$
$3$	0	0
$3$	0	0
$4$	−0.576442	−0.288221
$5$	−1.08235	−0.484040	−0.242020	−	0.970271i	$-0.577810\pi$
$5$	−1.08235	−0.484040	−0.242020	+	0.970271i	$0.577810\pi$
$6$	0	0
$7$	−1.30958	−0.494973	−0.247487	−	0.968891i	$-0.579605\pi$
$7$	−1.30958	−0.494973	−0.247487	+	0.968891i	$0.579605\pi$
$8$	−3.07403	−1.08683
$9$	0	0
$10$	−1.29138	−0.408370
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.53997	0.704462	0.352231	−	0.935913i	$-0.385423\pi$
$13$	2.53997	0.704462	0.352231	+	0.935913i	$0.385423\pi$
$14$	−1.56249	−0.417594
$15$	0	0
$16$	−2.51483	−0.628708
$17$	5.43892	1.31913	0.659566	−	0.751646i	$-0.270740\pi$
$17$	5.43892	1.31913	0.659566	+	0.751646i	$0.270740\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.623909	0.139510
$21$	0	0
$22$	1.19313	0.254376
$23$	−3.87095	−0.807148	−0.403574	−	0.914947i	$-0.632232\pi$
$23$	−3.87095	−0.807148	−0.403574	+	0.914947i	$0.632232\pi$
$24$	0	0
$25$	−3.82853	−0.765706
$26$	3.03052	0.594334
$27$	0	0
$28$	0.754894	0.142662
$29$	2.41412	0.448290	0.224145	−	0.974556i	$-0.428041\pi$
$29$	2.41412	0.448290	0.224145	+	0.974556i	$0.428041\pi$
$30$	0	0
$31$	3.03647	0.545366	0.272683	−	0.962104i	$-0.412089\pi$
$31$	3.03647	0.545366	0.272683	+	0.962104i	$0.412089\pi$
$32$	3.14754	0.556412
$33$	0	0
$34$	6.48934	1.11291
$35$	1.41741	0.239587
$36$	0	0
$37$	6.85067	1.12624	0.563122	−	0.826374i	$-0.309600\pi$
$37$	6.85067	1.12624	0.563122	+	0.826374i	$0.309600\pi$
$38$	1.19313	0.193551
$39$	0	0
$40$	3.32716	0.526070
$41$	6.11344	0.954759	0.477380	−	0.878697i	$-0.341587\pi$
$41$	6.11344	0.954759	0.477380	+	0.878697i	$0.341587\pi$
$42$	0	0
$43$	−2.95329	−0.450373	−0.225186	−	0.974316i	$-0.572299\pi$
$43$	−2.95329	−0.450373	−0.225186	+	0.974316i	$0.572299\pi$
$44$	−0.576442	−0.0869019
$45$	0	0
$46$	−4.61854	−0.680966
$47$	12.0923	1.76385	0.881925	−	0.471391i	$-0.156248\pi$
$47$	12.0923	1.76385	0.881925	+	0.471391i	$0.156248\pi$
$48$	0	0
$49$	−5.28501	−0.755002
$50$	−4.56793	−0.646003
$51$	0	0
$52$	−1.46415	−0.203041
$53$	0.992927	0.136389	0.0681945	−	0.997672i	$-0.478276\pi$
$53$	0.992927	0.136389	0.0681945	+	0.997672i	$0.478276\pi$
$54$	0	0
$55$	−1.08235	−0.145943
$56$	4.02567	0.537953
$57$	0	0
$58$	2.88036	0.378209
$59$	14.2251	1.85195	0.925977	−	0.377580i	$-0.123244\pi$
$59$	14.2251	1.85195	0.925977	+	0.377580i	$0.123244\pi$
$60$	0	0
$61$	−5.82518	−0.745838	−0.372919	−	0.927864i	$-0.621643\pi$
$61$	−5.82518	−0.745838	−0.372919	+	0.927864i	$0.621643\pi$
$62$	3.62290	0.460109
$63$	0	0
$64$	8.78508	1.09814
$65$	−2.74913	−0.340988
$66$	0	0
$67$	8.79718	1.07475	0.537373	−	0.843345i	$-0.319417\pi$
$67$	8.79718	1.07475	0.537373	+	0.843345i	$0.319417\pi$
$68$	−3.13522	−0.380202
$69$	0	0
$70$	1.69116	0.202132
$71$	2.44975	0.290732	0.145366	−	0.989378i	$-0.453564\pi$
$71$	2.44975	0.290732	0.145366	+	0.989378i	$0.453564\pi$
$72$	0	0
$73$	5.84063	0.683594	0.341797	−	0.939774i	$-0.388965\pi$
$73$	5.84063	0.683594	0.341797	+	0.939774i	$0.388965\pi$
$74$	8.17374	0.950178
$75$	0	0
$76$	−0.576442	−0.0661224
$77$	−1.30958	−0.149240
$78$	0	0
$79$	17.0397	1.91711	0.958557	−	0.284902i	$-0.0919610\pi$
$79$	17.0397	1.91711	0.958557	+	0.284902i	$0.0919610\pi$
$80$	2.72192	0.304319
$81$	0	0
$82$	7.29413	0.805502
$83$	7.01303	0.769780	0.384890	−	0.922963i	$-0.374239\pi$
$83$	7.01303	0.769780	0.384890	+	0.922963i	$0.374239\pi$
$84$	0	0
$85$	−5.88679	−0.638512
$86$	−3.52366	−0.379966
$87$	0	0
$88$	−3.07403	−0.327693
$89$	9.13103	0.967887	0.483944	−	0.875099i	$-0.339204\pi$
$89$	9.13103	0.967887	0.483944	+	0.875099i	$0.339204\pi$
$90$	0	0
$91$	−3.32629	−0.348690
$92$	2.23138	0.232637
$93$	0	0
$94$	14.4277	1.48811
$95$	−1.08235	−0.111046
$96$	0	0
$97$	−14.7950	−1.50220	−0.751100	−	0.660188i	$-0.770476\pi$
$97$	−14.7950	−1.50220	−0.751100	+	0.660188i	$0.770476\pi$
$98$	−6.30570	−0.636972
$99$	0	0
$100$	2.20692	0.220692
$101$	−19.0516	−1.89570	−0.947852	−	0.318711i	$-0.896750\pi$
$101$	−19.0516	−1.89570	−0.947852	+	0.318711i	$0.896750\pi$
$102$	0	0
$103$	10.4722	1.03186	0.515928	−	0.856632i	$-0.327447\pi$
$103$	10.4722	1.03186	0.515928	+	0.856632i	$0.327447\pi$
$104$	−7.80795	−0.765633
$105$	0	0
$106$	1.18469	0.115067
$107$	−14.4916	−1.40095	−0.700477	−	0.713675i	$-0.747029\pi$
$107$	−14.4916	−1.40095	−0.700477	+	0.713675i	$0.747029\pi$
$108$	0	0
$109$	−6.39919	−0.612931	−0.306465	−	0.951882i	$-0.599146\pi$
$109$	−6.39919	−0.612931	−0.306465	+	0.951882i	$0.599146\pi$
$110$	−1.29138	−0.123128
$111$	0	0
$112$	3.29336	0.311193
$113$	2.91702	0.274410	0.137205	−	0.990543i	$-0.456188\pi$
$113$	2.91702	0.274410	0.137205	+	0.990543i	$0.456188\pi$
$114$	0	0
$115$	4.18970	0.390692
$116$	−1.39160	−0.129207
$117$	0	0
$118$	16.9724	1.56244
$119$	−7.12268	−0.652935
$120$	0	0
$121$	1.00000	0.0909091
$122$	−6.95020	−0.629241
$123$	0	0
$124$	−1.75035	−0.157186
$125$	9.55552	0.854671
$126$	0	0
$127$	−12.1995	−1.08253	−0.541267	−	0.840851i	$-0.682055\pi$
$127$	−12.1995	−1.08253	−0.541267	+	0.840851i	$0.682055\pi$
$128$	4.18666	0.370052
$129$	0	0
$130$	−3.28007	−0.287681
$131$	−18.7468	−1.63792	−0.818959	−	0.573852i	$-0.805449\pi$
$131$	−18.7468	−1.63792	−0.818959	+	0.573852i	$0.805449\pi$
$132$	0	0
$133$	−1.30958	−0.113555
$134$	10.4962	0.906731
$135$	0	0
$136$	−16.7194	−1.43368
$137$	−16.8291	−1.43781	−0.718904	−	0.695109i	$-0.755356\pi$
$137$	−16.8291	−1.43781	−0.718904	+	0.695109i	$0.755356\pi$
$138$	0	0
$139$	12.5906	1.06792	0.533959	−	0.845511i	$-0.320704\pi$
$139$	12.5906	1.06792	0.533959	+	0.845511i	$0.320704\pi$
$140$	−0.817056	−0.0690539
$141$	0	0
$142$	2.92287	0.245282
$143$	2.53997	0.212403
$144$	0	0
$145$	−2.61291	−0.216990
$146$	6.96862	0.576727
$147$	0	0
$148$	−3.94902	−0.324607
$149$	−6.22689	−0.510126	−0.255063	−	0.966924i	$-0.582096\pi$
$149$	−6.22689	−0.510126	−0.255063	+	0.966924i	$0.582096\pi$
$150$	0	0
$151$	6.14114	0.499759	0.249879	−	0.968277i	$-0.419609\pi$
$151$	6.14114	0.499759	0.249879	+	0.968277i	$0.419609\pi$
$152$	−3.07403	−0.249337
$153$	0	0
$154$	−1.56249	−0.125909
$155$	−3.28651	−0.263979
$156$	0	0
$157$	19.7489	1.57613	0.788065	−	0.615592i	$-0.211083\pi$
$157$	19.7489	1.57613	0.788065	+	0.615592i	$0.211083\pi$
$158$	20.3305	1.61741
$159$	0	0
$160$	−3.40672	−0.269325
$161$	5.06930	0.399517
$162$	0	0
$163$	−5.70383	−0.446759	−0.223379	−	0.974732i	$-0.571709\pi$
$163$	−5.70383	−0.446759	−0.223379	+	0.974732i	$0.571709\pi$
$164$	−3.52405	−0.275182
$165$	0	0
$166$	8.36745	0.649440
$167$	10.1612	0.786294	0.393147	−	0.919476i	$-0.371386\pi$
$167$	10.1612	0.786294	0.393147	+	0.919476i	$0.371386\pi$
$168$	0	0
$169$	−6.54853	−0.503733
$170$	−7.02371	−0.538694
$171$	0	0
$172$	1.70240	0.129807
$173$	−2.22385	−0.169076	−0.0845382	−	0.996420i	$-0.526942\pi$
$173$	−2.22385	−0.169076	−0.0845382	+	0.996420i	$0.526942\pi$
$174$	0	0
$175$	5.01375	0.379004
$176$	−2.51483	−0.189562
$177$	0	0
$178$	10.8945	0.816577
$179$	−12.0997	−0.904376	−0.452188	−	0.891923i	$-0.649356\pi$
$179$	−12.0997	−0.904376	−0.452188	+	0.891923i	$0.649356\pi$
$180$	0	0
$181$	−0.359088	−0.0266908	−0.0133454	−	0.999911i	$-0.504248\pi$
$181$	−0.359088	−0.0266908	−0.0133454	+	0.999911i	$0.504248\pi$
$182$	−3.96869	−0.294179
$183$	0	0
$184$	11.8994	0.877235
$185$	−7.41479	−0.545147
$186$	0	0
$187$	5.43892	0.397733
$188$	−6.97053	−0.508378
$189$	0	0
$190$	−1.29138	−0.0936864
$191$	−21.3246	−1.54299	−0.771496	−	0.636235i	$-0.780491\pi$
$191$	−21.3246	−1.54299	−0.771496	+	0.636235i	$0.780491\pi$
$192$	0	0
$193$	−14.5451	−1.04698	−0.523491	−	0.852031i	$-0.675370\pi$
$193$	−14.5451	−1.04698	−0.523491	+	0.852031i	$0.675370\pi$
$194$	−17.6523	−1.26736
$195$	0	0
$196$	3.04650	0.217607
$197$	4.71318	0.335800	0.167900	−	0.985804i	$-0.446301\pi$
$197$	4.71318	0.335800	0.167900	+	0.985804i	$0.446301\pi$
$198$	0	0
$199$	−1.77204	−0.125616	−0.0628081	−	0.998026i	$-0.520006\pi$
$199$	−1.77204	−0.125616	−0.0628081	+	0.998026i	$0.520006\pi$
$200$	11.7690	0.832194
$201$	0	0
$202$	−22.7310	−1.59935
$203$	−3.16147	−0.221892
$204$	0	0
$205$	−6.61686	−0.462141
$206$	12.4947	0.870546
$207$	0	0
$208$	−6.38761	−0.442901
$209$	1.00000	0.0691714
$210$	0	0
$211$	17.8069	1.22588	0.612940	−	0.790129i	$-0.289987\pi$
$211$	17.8069	1.22588	0.612940	+	0.790129i	$0.289987\pi$
$212$	−0.572365	−0.0393102
$213$	0	0
$214$	−17.2903	−1.18194
$215$	3.19648	0.217998
$216$	0	0
$217$	−3.97648	−0.269941
$218$	−7.63506	−0.517111
$219$	0	0
$220$	0.623909	0.0420640
$221$	13.8147	0.929279
$222$	0	0
$223$	−9.69081	−0.648945	−0.324472	−	0.945895i	$-0.605187\pi$
$223$	−9.69081	−0.648945	−0.324472	+	0.945895i	$0.605187\pi$
$224$	−4.12194	−0.275409
$225$	0	0
$226$	3.48038	0.231511
$227$	6.02204	0.399697	0.199848	−	0.979827i	$-0.435955\pi$
$227$	6.02204	0.399697	0.199848	+	0.979827i	$0.435955\pi$
$228$	0	0
$229$	21.7503	1.43730	0.718649	−	0.695373i	$-0.244761\pi$
$229$	21.7503	1.43730	0.718649	+	0.695373i	$0.244761\pi$
$230$	4.99885	0.329615
$231$	0	0
$232$	−7.42107	−0.487217
$233$	15.6866	1.02766	0.513831	−	0.857891i	$-0.328226\pi$
$233$	15.6866	1.02766	0.513831	+	0.857891i	$0.328226\pi$
$234$	0	0
$235$	−13.0881	−0.853773
$236$	−8.19996	−0.533772
$237$	0	0
$238$	−8.49828	−0.550862
$239$	−21.8841	−1.41556	−0.707782	−	0.706430i	$-0.750304\pi$
$239$	−21.8841	−1.41556	−0.707782	+	0.706430i	$0.750304\pi$
$240$	0	0
$241$	5.98510	0.385534	0.192767	−	0.981245i	$-0.438254\pi$
$241$	5.98510	0.385534	0.192767	+	0.981245i	$0.438254\pi$
$242$	1.19313	0.0766973
$243$	0	0
$244$	3.35788	0.214966
$245$	5.72021	0.365451
$246$	0	0
$247$	2.53997	0.161615
$248$	−9.33419	−0.592722
$249$	0	0
$250$	11.4010	0.721061
$251$	6.63439	0.418759	0.209379	−	0.977834i	$-0.432856\pi$
$251$	6.63439	0.418759	0.209379	+	0.977834i	$0.432856\pi$
$252$	0	0
$253$	−3.87095	−0.243364
$254$	−14.5556	−0.913302
$255$	0	0
$256$	−12.5749	−0.785933
$257$	9.31591	0.581111	0.290555	−	0.956858i	$-0.406160\pi$
$257$	9.31591	0.581111	0.290555	+	0.956858i	$0.406160\pi$
$258$	0	0
$259$	−8.97148	−0.557460
$260$	1.58471	0.0982798
$261$	0	0
$262$	−22.3674	−1.38186
$263$	9.25004	0.570382	0.285191	−	0.958471i	$-0.407943\pi$
$263$	9.25004	0.570382	0.285191	+	0.958471i	$0.407943\pi$
$264$	0	0
$265$	−1.07469	−0.0660177
$266$	−1.56249	−0.0958026
$267$	0	0
$268$	−5.07106	−0.309764
$269$	17.0076	1.03697	0.518486	−	0.855086i	$-0.326496\pi$
$269$	17.0076	1.03697	0.518486	+	0.855086i	$0.326496\pi$
$270$	0	0
$271$	7.43527	0.451660	0.225830	−	0.974167i	$-0.427491\pi$
$271$	7.43527	0.451660	0.225830	+	0.974167i	$0.427491\pi$
$272$	−13.6780	−0.829349
$273$	0	0
$274$	−20.0793	−1.21304
$275$	−3.82853	−0.230869
$276$	0	0
$277$	2.31338	0.138998	0.0694989	−	0.997582i	$-0.477860\pi$
$277$	2.31338	0.138998	0.0694989	+	0.997582i	$0.477860\pi$
$278$	15.0222	0.900970
$279$	0	0
$280$	−4.35717	−0.260391
$281$	5.20248	0.310354	0.155177	−	0.987887i	$-0.450405\pi$
$281$	5.20248	0.310354	0.155177	+	0.987887i	$0.450405\pi$
$282$	0	0
$283$	26.7125	1.58789	0.793947	−	0.607987i	$-0.208023\pi$
$283$	26.7125	1.58789	0.793947	+	0.607987i	$0.208023\pi$
$284$	−1.41214	−0.0837952
$285$	0	0
$286$	3.03052	0.179198
$287$	−8.00602	−0.472580
$288$	0	0
$289$	12.5819	0.740110
$290$	−3.11754	−0.183068
$291$	0	0
$292$	−3.36678	−0.197026
$293$	0.937617	0.0547762	0.0273881	−	0.999625i	$-0.491281\pi$
$293$	0.937617	0.0547762	0.0273881	+	0.999625i	$0.491281\pi$
$294$	0	0
$295$	−15.3965	−0.896419
$296$	−21.0592	−1.22404
$297$	0	0
$298$	−7.42948	−0.430378
$299$	−9.83210	−0.568605
$300$	0	0
$301$	3.86756	0.222922
$302$	7.32717	0.421631
$303$	0	0
$304$	−2.51483	−0.144235
$305$	6.30486	0.361015
$306$	0	0
$307$	−4.83167	−0.275758	−0.137879	−	0.990449i	$-0.544029\pi$
$307$	−4.83167	−0.275758	−0.137879	+	0.990449i	$0.544029\pi$
$308$	0.754894	0.0430141
$309$	0	0
$310$	−3.92123	−0.222711
$311$	14.3916	0.816071	0.408036	−	0.912966i	$-0.366214\pi$
$311$	14.3916	0.816071	0.408036	+	0.912966i	$0.366214\pi$
$312$	0	0
$313$	−29.3478	−1.65884	−0.829418	−	0.558628i	$-0.811328\pi$
$313$	−29.3478	−1.65884	−0.829418	+	0.558628i	$0.811328\pi$
$314$	23.5629	1.32973
$315$	0	0
$316$	−9.82238	−0.552552
$317$	16.4027	0.921267	0.460633	−	0.887590i	$-0.347622\pi$
$317$	16.4027	0.921267	0.460633	+	0.887590i	$0.347622\pi$
$318$	0	0
$319$	2.41412	0.135165
$320$	−9.50849	−0.531541
$321$	0	0
$322$	6.04833	0.337060
$323$	5.43892	0.302630
$324$	0	0
$325$	−9.72436	−0.539411
$326$	−6.80541	−0.376917
$327$	0	0
$328$	−18.7929	−1.03766
$329$	−15.8358	−0.873058
$330$	0	0
$331$	14.1995	0.780477	0.390239	−	0.920714i	$-0.372392\pi$
$331$	14.1995	0.780477	0.390239	+	0.920714i	$0.372392\pi$
$332$	−4.04260	−0.221867
$333$	0	0
$334$	12.1236	0.663372
$335$	−9.52159	−0.520220
$336$	0	0
$337$	−22.1449	−1.20631	−0.603154	−	0.797624i	$-0.706090\pi$
$337$	−22.1449	−1.20631	−0.603154	+	0.797624i	$0.706090\pi$
$338$	−7.81325	−0.424985
$339$	0	0
$340$	3.39339	0.184033
$341$	3.03647	0.164434
$342$	0	0
$343$	16.0882	0.868679
$344$	9.07850	0.489480
$345$	0	0
$346$	−2.65334	−0.142645
$347$	19.1475	1.02789	0.513946	−	0.857823i	$-0.328183\pi$
$347$	19.1475	1.02789	0.513946	+	0.857823i	$0.328183\pi$
$348$	0	0
$349$	−16.0709	−0.860255	−0.430127	−	0.902768i	$-0.641531\pi$
$349$	−16.0709	−0.860255	−0.430127	+	0.902768i	$0.641531\pi$
$350$	5.98205	0.319754
$351$	0	0
$352$	3.14754	0.167764
$353$	−10.9142	−0.580902	−0.290451	−	0.956890i	$-0.593805\pi$
$353$	−10.9142	−0.580902	−0.290451	+	0.956890i	$0.593805\pi$
$354$	0	0
$355$	−2.65148	−0.140726
$356$	−5.26351	−0.278965
$357$	0	0
$358$	−14.4365	−0.762995
$359$	35.4105	1.86889	0.934446	−	0.356104i	$-0.115895\pi$
$359$	35.4105	1.86889	0.934446	+	0.356104i	$0.115895\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−0.428438	−0.0225182
$363$	0	0
$364$	1.91741	0.100500
$365$	−6.32158	−0.330886
$366$	0	0
$367$	−23.3384	−1.21825	−0.609127	−	0.793073i	$-0.708480\pi$
$367$	−23.3384	−1.21825	−0.609127	+	0.793073i	$0.708480\pi$
$368$	9.73477	0.507460
$369$	0	0
$370$	−8.84681	−0.459924
$371$	−1.30031	−0.0675089
$372$	0	0
$373$	−17.1972	−0.890440	−0.445220	−	0.895421i	$-0.646874\pi$
$373$	−17.1972	−0.890440	−0.445220	+	0.895421i	$0.646874\pi$
$374$	6.48934	0.335556
$375$	0	0
$376$	−37.1722	−1.91701
$377$	6.13180	0.315804
$378$	0	0
$379$	25.2353	1.29625	0.648125	−	0.761534i	$-0.275553\pi$
$379$	25.2353	1.29625	0.648125	+	0.761534i	$0.275553\pi$
$380$	0.623909	0.0320059
$381$	0	0
$382$	−25.4430	−1.30178
$383$	−29.6247	−1.51375	−0.756877	−	0.653558i	$-0.773276\pi$
$383$	−29.6247	−1.51375	−0.756877	+	0.653558i	$0.773276\pi$
$384$	0	0
$385$	1.41741	0.0722381
$386$	−17.3542	−0.883307
$387$	0	0
$388$	8.52843	0.432966
$389$	−7.70975	−0.390900	−0.195450	−	0.980714i	$-0.562617\pi$
$389$	−7.70975	−0.390900	−0.195450	+	0.980714i	$0.562617\pi$
$390$	0	0
$391$	−21.0538	−1.06474
$392$	16.2463	0.820561
$393$	0	0
$394$	5.62344	0.283305
$395$	−18.4428	−0.927959
$396$	0	0
$397$	8.26973	0.415046	0.207523	−	0.978230i	$-0.433460\pi$
$397$	8.26973	0.415046	0.207523	+	0.978230i	$0.433460\pi$
$398$	−2.11427	−0.105979
$399$	0	0
$400$	9.62810	0.481405
$401$	−1.98297	−0.0990247	−0.0495123	−	0.998774i	$-0.515767\pi$
$401$	−1.98297	−0.0990247	−0.0495123	+	0.998774i	$0.515767\pi$
$402$	0	0
$403$	7.71255	0.384189
$404$	10.9821	0.546382
$405$	0	0
$406$	−3.77204	−0.187203
$407$	6.85067	0.339575
$408$	0	0
$409$	25.8761	1.27949	0.639746	−	0.768586i	$-0.279040\pi$
$409$	25.8761	1.27949	0.639746	+	0.768586i	$0.279040\pi$
$410$	−7.89477	−0.389895
$411$	0	0
$412$	−6.03661	−0.297402
$413$	−18.6289	−0.916667
$414$	0	0
$415$	−7.59052	−0.372604
$416$	7.99467	0.391971
$417$	0	0
$418$	1.19313	0.0583579
$419$	−36.2262	−1.76977	−0.884883	−	0.465813i	$-0.845762\pi$
$419$	−36.2262	−1.76977	−0.884883	+	0.465813i	$0.845762\pi$
$420$	0	0
$421$	−5.91248	−0.288156	−0.144078	−	0.989566i	$-0.546022\pi$
$421$	−5.91248	−0.288156	−0.144078	+	0.989566i	$0.546022\pi$
$422$	21.2460	1.03424
$423$	0	0
$424$	−3.05229	−0.148232
$425$	−20.8231	−1.01007
$426$	0	0
$427$	7.62852	0.369170
$428$	8.35356	0.403784
$429$	0	0
$430$	3.81382	0.183919
$431$	11.1886	0.538935	0.269468	−	0.963009i	$-0.413152\pi$
$431$	11.1886	0.538935	0.269468	+	0.963009i	$0.413152\pi$
$432$	0	0
$433$	13.3903	0.643497	0.321748	−	0.946825i	$-0.395729\pi$
$433$	13.3903	0.643497	0.321748	+	0.946825i	$0.395729\pi$
$434$	−4.74446	−0.227741
$435$	0	0
$436$	3.68876	0.176660
$437$	−3.87095	−0.185172
$438$	0	0
$439$	23.6338	1.12798	0.563989	−	0.825782i	$-0.309266\pi$
$439$	23.6338	1.12798	0.563989	+	0.825782i	$0.309266\pi$
$440$	3.32716	0.158616
$441$	0	0
$442$	16.4828	0.784005
$443$	25.6914	1.22063	0.610317	−	0.792158i	$-0.291042\pi$
$443$	25.6914	1.22063	0.610317	+	0.792158i	$0.291042\pi$
$444$	0	0
$445$	−9.88293	−0.468496
$446$	−11.5624	−0.547495
$447$	0	0
$448$	−11.5047	−0.543547
$449$	9.72223	0.458820	0.229410	−	0.973330i	$-0.426320\pi$
$449$	9.72223	0.458820	0.229410	+	0.973330i	$0.426320\pi$
$450$	0	0
$451$	6.11344	0.287871
$452$	−1.68149	−0.0790907
$453$	0	0
$454$	7.18507	0.337212
$455$	3.60019	0.168780
$456$	0	0
$457$	−2.04841	−0.0958207	−0.0479103	−	0.998852i	$-0.515256\pi$
$457$	−2.04841	−0.0958207	−0.0479103	+	0.998852i	$0.515256\pi$
$458$	25.9509	1.21261
$459$	0	0
$460$	−2.41512	−0.112606
$461$	22.8905	1.06612	0.533059	−	0.846078i	$-0.321042\pi$
$461$	22.8905	1.06612	0.533059	+	0.846078i	$0.321042\pi$
$462$	0	0
$463$	34.9486	1.62420	0.812098	−	0.583520i	$-0.198325\pi$
$463$	34.9486	1.62420	0.812098	+	0.583520i	$0.198325\pi$
$464$	−6.07110	−0.281844
$465$	0	0
$466$	18.7161	0.867008
$467$	−13.7353	−0.635595	−0.317797	−	0.948159i	$-0.602943\pi$
$467$	−13.7353	−0.635595	−0.317797	+	0.948159i	$0.602943\pi$
$468$	0	0
$469$	−11.5206	−0.531971
$470$	−15.6158	−0.720302
$471$	0	0
$472$	−43.7284	−2.01276
$473$	−2.95329	−0.135792
$474$	0	0
$475$	−3.82853	−0.175665
$476$	4.10581	0.188190
$477$	0	0
$478$	−26.1106	−1.19427
$479$	−3.13630	−0.143301	−0.0716506	−	0.997430i	$-0.522827\pi$
$479$	−3.13630	−0.143301	−0.0716506	+	0.997430i	$0.522827\pi$
$480$	0	0
$481$	17.4005	0.793396
$482$	7.14100	0.325264
$483$	0	0
$484$	−0.576442	−0.0262019
$485$	16.0133	0.727124
$486$	0	0
$487$	21.2263	0.961854	0.480927	−	0.876761i	$-0.340300\pi$
$487$	21.2263	0.961854	0.480927	+	0.876761i	$0.340300\pi$
$488$	17.9068	0.810602
$489$	0	0
$490$	6.82495	0.308320
$491$	−28.5840	−1.28998	−0.644988	−	0.764193i	$-0.723138\pi$
$491$	−28.5840	−1.28998	−0.644988	+	0.764193i	$0.723138\pi$
$492$	0	0
$493$	13.1302	0.591354
$494$	3.03052	0.136349
$495$	0	0
$496$	−7.63620	−0.342876
$497$	−3.20814	−0.143905
$498$	0	0
$499$	−20.1316	−0.901212	−0.450606	−	0.892723i	$-0.648792\pi$
$499$	−20.1316	−0.901212	−0.450606	+	0.892723i	$0.648792\pi$
$500$	−5.50820	−0.246334
$501$	0	0
$502$	7.91568	0.353294
$503$	6.37170	0.284100	0.142050	−	0.989859i	$-0.454631\pi$
$503$	6.37170	0.284100	0.142050	+	0.989859i	$0.454631\pi$
$504$	0	0
$505$	20.6204	0.917596
$506$	−4.61854	−0.205319
$507$	0	0
$508$	7.03233	0.312009
$509$	9.83799	0.436061	0.218031	−	0.975942i	$-0.430037\pi$
$509$	9.83799	0.436061	0.218031	+	0.975942i	$0.430037\pi$
$510$	0	0
$511$	−7.64874	−0.338360
$512$	−23.3769	−1.03312
$513$	0	0
$514$	11.1151	0.490266
$515$	−11.3345	−0.499459
$516$	0	0
$517$	12.0923	0.531820
$518$	−10.7041	−0.470313
$519$	0	0
$520$	8.45090	0.370597
$521$	−21.1840	−0.928088	−0.464044	−	0.885812i	$-0.653602\pi$
$521$	−21.1840	−0.928088	−0.464044	+	0.885812i	$0.653602\pi$
$522$	0	0
$523$	22.1326	0.967790	0.483895	−	0.875126i	$-0.339222\pi$
$523$	22.1326	0.967790	0.483895	+	0.875126i	$0.339222\pi$
$524$	10.8065	0.472083
$525$	0	0
$526$	11.0365	0.481214
$527$	16.5151	0.719410
$528$	0	0
$529$	−8.01578	−0.348512
$530$	−1.28224	−0.0556971
$531$	0	0
$532$	0.754894	0.0327288
$533$	15.5280	0.672592
$534$	0	0
$535$	15.6849	0.678117
$536$	−27.0428	−1.16807
$537$	0	0
$538$	20.2923	0.874862
$539$	−5.28501	−0.227642
$540$	0	0
$541$	−1.83412	−0.0788551	−0.0394275	−	0.999222i	$-0.512553\pi$
$541$	−1.83412	−0.0788551	−0.0394275	+	0.999222i	$0.512553\pi$
$542$	8.87124	0.381052
$543$	0	0
$544$	17.1192	0.733981
$545$	6.92613	0.296683
$546$	0	0
$547$	−20.2583	−0.866181	−0.433091	−	0.901350i	$-0.642577\pi$
$547$	−20.2583	−0.866181	−0.433091	+	0.901350i	$0.642577\pi$
$548$	9.70101	0.414406
$549$	0	0
$550$	−4.56793	−0.194777
$551$	2.41412	0.102845
$552$	0	0
$553$	−22.3147	−0.948920
$554$	2.76017	0.117268
$555$	0	0
$556$	−7.25773	−0.307796
$557$	−15.1944	−0.643805	−0.321903	−	0.946773i	$-0.604322\pi$
$557$	−15.1944	−0.643805	−0.321903	+	0.946773i	$0.604322\pi$
$558$	0	0
$559$	−7.50128	−0.317270
$560$	−3.56455	−0.150630
$561$	0	0
$562$	6.20723	0.261836
$563$	22.4513	0.946210	0.473105	−	0.881006i	$-0.343133\pi$
$563$	22.4513	0.946210	0.473105	+	0.881006i	$0.343133\pi$
$564$	0	0
$565$	−3.15722	−0.132825
$566$	31.8715	1.33966
$567$	0	0
$568$	−7.53062	−0.315978
$569$	7.54832	0.316442	0.158221	−	0.987404i	$-0.449424\pi$
$569$	7.54832	0.316442	0.158221	+	0.987404i	$0.449424\pi$
$570$	0	0
$571$	−0.101602	−0.00425192	−0.00212596	−	0.999998i	$-0.500677\pi$
$571$	−0.101602	−0.00425192	−0.00212596	+	0.999998i	$0.500677\pi$
$572$	−1.46415	−0.0612191
$573$	0	0
$574$	−9.55221	−0.398702
$575$	14.8200	0.618038
$576$	0	0
$577$	−27.8194	−1.15814	−0.579068	−	0.815279i	$-0.696584\pi$
$577$	−27.8194	−1.15814	−0.579068	+	0.815279i	$0.696584\pi$
$578$	15.0118	0.624409
$579$	0	0
$580$	1.50619	0.0625412
$581$	−9.18409	−0.381020
$582$	0	0
$583$	0.992927	0.0411228
$584$	−17.9543	−0.742952
$585$	0	0
$586$	1.11870	0.0462130
$587$	−15.7385	−0.649596	−0.324798	−	0.945783i	$-0.605296\pi$
$587$	−15.7385	−0.649596	−0.324798	+	0.945783i	$0.605296\pi$
$588$	0	0
$589$	3.03647	0.125115
$590$	−18.3700	−0.756282
$591$	0	0
$592$	−17.2283	−0.708078
$593$	−33.1616	−1.36178	−0.680892	−	0.732384i	$-0.738408\pi$
$593$	−33.1616	−1.36178	−0.680892	+	0.732384i	$0.738408\pi$
$594$	0	0
$595$	7.70920	0.316046
$596$	3.58944	0.147029
$597$	0	0
$598$	−11.7310	−0.479715
$599$	5.80115	0.237029	0.118514	−	0.992952i	$-0.462187\pi$
$599$	5.80115	0.237029	0.118514	+	0.992952i	$0.462187\pi$
$600$	0	0
$601$	−21.7148	−0.885766	−0.442883	−	0.896580i	$-0.646044\pi$
$601$	−21.7148	−0.885766	−0.442883	+	0.896580i	$0.646044\pi$
$602$	4.61450	0.188073
$603$	0	0
$604$	−3.54001	−0.144041
$605$	−1.08235	−0.0440036
$606$	0	0
$607$	−30.3992	−1.23386	−0.616932	−	0.787016i	$-0.711625\pi$
$607$	−30.3992	−1.23386	−0.616932	+	0.787016i	$0.711625\pi$
$608$	3.14754	0.127650
$609$	0	0
$610$	7.52251	0.304578
$611$	30.7142	1.24256
$612$	0	0
$613$	8.70645	0.351650	0.175825	−	0.984421i	$-0.443741\pi$
$613$	8.70645	0.351650	0.175825	+	0.984421i	$0.443741\pi$
$614$	−5.76481	−0.232649
$615$	0	0
$616$	4.02567	0.162199
$617$	14.9292	0.601026	0.300513	−	0.953778i	$-0.402842\pi$
$617$	14.9292	0.601026	0.300513	+	0.953778i	$0.402842\pi$
$618$	0	0
$619$	−16.5120	−0.663673	−0.331836	−	0.943337i	$-0.607668\pi$
$619$	−16.5120	−0.663673	−0.331836	+	0.943337i	$0.607668\pi$
$620$	1.89448	0.0760842
$621$	0	0
$622$	17.1710	0.688495
$623$	−11.9578	−0.479078
$624$	0	0
$625$	8.80027	0.352011
$626$	−35.0157	−1.39951
$627$	0	0
$628$	−11.3841	−0.454274
$629$	37.2603	1.48566
$630$	0	0
$631$	33.1047	1.31788	0.658938	−	0.752197i	$-0.271006\pi$
$631$	33.1047	1.31788	0.658938	+	0.752197i	$0.271006\pi$
$632$	−52.3805	−2.08358
$633$	0	0
$634$	19.5705	0.777245
$635$	13.2041	0.523989
$636$	0	0
$637$	−13.4238	−0.531870
$638$	2.88036	0.114034
$639$	0	0
$640$	−4.53142	−0.179120
$641$	−9.80138	−0.387131	−0.193566	−	0.981087i	$-0.562005\pi$
$641$	−9.80138	−0.387131	−0.193566	+	0.981087i	$0.562005\pi$
$642$	0	0
$643$	−14.1775	−0.559106	−0.279553	−	0.960130i	$-0.590186\pi$
$643$	−14.1775	−0.559106	−0.279553	+	0.960130i	$0.590186\pi$
$644$	−2.92216	−0.115149
$645$	0	0
$646$	6.48934	0.255320
$647$	−2.89498	−0.113813	−0.0569067	−	0.998379i	$-0.518124\pi$
$647$	−2.89498	−0.113813	−0.0569067	+	0.998379i	$0.518124\pi$
$648$	0	0
$649$	14.2251	0.558385
$650$	−11.6024	−0.455085
$651$	0	0
$652$	3.28793	0.128765
$653$	−14.5415	−0.569054	−0.284527	−	0.958668i	$-0.591836\pi$
$653$	−14.5415	−0.569054	−0.284527	+	0.958668i	$0.591836\pi$
$654$	0	0
$655$	20.2906	0.792817
$656$	−15.3743	−0.600265
$657$	0	0
$658$	−18.8942	−0.736573
$659$	−23.8625	−0.929550	−0.464775	−	0.885429i	$-0.653865\pi$
$659$	−23.8625	−0.929550	−0.464775	+	0.885429i	$0.653865\pi$
$660$	0	0
$661$	−5.72037	−0.222497	−0.111248	−	0.993793i	$-0.535485\pi$
$661$	−5.72037	−0.222497	−0.111248	+	0.993793i	$0.535485\pi$
$662$	16.9419	0.658465
$663$	0	0
$664$	−21.5582	−0.836622
$665$	1.41741	0.0549649
$666$	0	0
$667$	−9.34492	−0.361837
$668$	−5.85732	−0.226626
$669$	0	0
$670$	−11.3605	−0.438894
$671$	−5.82518	−0.224879
$672$	0	0
$673$	17.0169	0.655954	0.327977	−	0.944686i	$-0.393633\pi$
$673$	17.0169	0.655954	0.327977	+	0.944686i	$0.393633\pi$
$674$	−26.4217	−1.01773
$675$	0	0
$676$	3.77485	0.145186
$677$	−30.2237	−1.16159	−0.580796	−	0.814049i	$-0.697259\pi$
$677$	−30.2237	−1.16159	−0.580796	+	0.814049i	$0.697259\pi$
$678$	0	0
$679$	19.3751	0.743549
$680$	18.0962	0.693956
$681$	0	0
$682$	3.62290	0.138728
$683$	30.7218	1.17554	0.587768	−	0.809029i	$-0.300007\pi$
$683$	30.7218	1.17554	0.587768	+	0.809029i	$0.300007\pi$
$684$	0	0
$685$	18.2149	0.695956
$686$	19.1952	0.732878
$687$	0	0
$688$	7.42703	0.283153
$689$	2.52201	0.0960809
$690$	0	0
$691$	−29.7093	−1.13020	−0.565098	−	0.825024i	$-0.691162\pi$
$691$	−29.7093	−1.13020	−0.565098	+	0.825024i	$0.691162\pi$
$692$	1.28192	0.0487314
$693$	0	0
$694$	22.8454	0.867201
$695$	−13.6273	−0.516914
$696$	0	0
$697$	33.2505	1.25945
$698$	−19.1747	−0.725771
$699$	0	0
$700$	−2.89013	−0.109237
$701$	−9.54914	−0.360666	−0.180333	−	0.983606i	$-0.557718\pi$
$701$	−9.54914	−0.360666	−0.180333	+	0.983606i	$0.557718\pi$
$702$	0	0
$703$	6.85067	0.258378
$704$	8.78508	0.331100
$705$	0	0
$706$	−13.0220	−0.490090
$707$	24.9495	0.938322
$708$	0	0
$709$	42.6564	1.60200	0.800998	−	0.598667i	$-0.204303\pi$
$709$	42.6564	1.60200	0.800998	+	0.598667i	$0.204303\pi$
$710$	−3.16356	−0.118726
$711$	0	0
$712$	−28.0691	−1.05193
$713$	−11.7540	−0.440191
$714$	0	0
$715$	−2.74913	−0.102812
$716$	6.97479	0.260660
$717$	0	0
$718$	42.2493	1.57673
$719$	16.7679	0.625335	0.312668	−	0.949863i	$-0.398777\pi$
$719$	16.7679	0.625335	0.312668	+	0.949863i	$0.398777\pi$
$720$	0	0
$721$	−13.7141	−0.510741
$722$	1.19313	0.0444037
$723$	0	0
$724$	0.206993	0.00769285
$725$	−9.24252	−0.343259
$726$	0	0
$727$	21.4858	0.796864	0.398432	−	0.917198i	$-0.369554\pi$
$727$	21.4858	0.796864	0.398432	+	0.917198i	$0.369554\pi$
$728$	10.2251	0.378968
$729$	0	0
$730$	−7.54246	−0.279159
$731$	−16.0627	−0.594101
$732$	0	0
$733$	−34.1734	−1.26222	−0.631111	−	0.775692i	$-0.717401\pi$
$733$	−34.1734	−1.26222	−0.631111	+	0.775692i	$0.717401\pi$
$734$	−27.8457	−1.02780
$735$	0	0
$736$	−12.1840	−0.449106
$737$	8.79718	0.324048
$738$	0	0
$739$	−11.5412	−0.424548	−0.212274	−	0.977210i	$-0.568087\pi$
$739$	−11.5412	−0.424548	−0.212274	+	0.977210i	$0.568087\pi$
$740$	4.27420	0.157123
$741$	0	0
$742$	−1.55144	−0.0569552
$743$	−24.7316	−0.907313	−0.453657	−	0.891177i	$-0.649881\pi$
$743$	−24.7316	−0.907313	−0.453657	+	0.891177i	$0.649881\pi$
$744$	0	0
$745$	6.73964	0.246921
$746$	−20.5185	−0.751237
$747$	0	0
$748$	−3.13522	−0.114635
$749$	18.9778	0.693435
$750$	0	0
$751$	46.2026	1.68596	0.842978	−	0.537949i	$-0.180801\pi$
$751$	46.2026	1.68596	0.842978	+	0.537949i	$0.180801\pi$
$752$	−30.4102	−1.10895
$753$	0	0
$754$	7.31603	0.266434
$755$	−6.64683	−0.241903
$756$	0	0
$757$	−43.9301	−1.59667	−0.798333	−	0.602216i	$-0.794285\pi$
$757$	−43.9301	−1.59667	−0.798333	+	0.602216i	$0.794285\pi$
$758$	30.1090	1.09361
$759$	0	0
$760$	3.32716	0.120689
$761$	32.1299	1.16471	0.582355	−	0.812935i	$-0.302131\pi$
$761$	32.1299	1.16471	0.582355	+	0.812935i	$0.302131\pi$
$762$	0	0
$763$	8.38022	0.303384
$764$	12.2924	0.444723
$765$	0	0
$766$	−35.3461	−1.27711
$767$	36.1315	1.30463
$768$	0	0
$769$	27.7514	1.00074	0.500371	−	0.865811i	$-0.333197\pi$
$769$	27.7514	1.00074	0.500371	+	0.865811i	$0.333197\pi$
$770$	1.69116	0.0609451
$771$	0	0
$772$	8.38442	0.301762
$773$	39.5999	1.42431	0.712154	−	0.702023i	$-0.247720\pi$
$773$	39.5999	1.42431	0.712154	+	0.702023i	$0.247720\pi$
$774$	0	0
$775$	−11.6252	−0.417590
$776$	45.4801	1.63264
$777$	0	0
$778$	−9.19873	−0.329790
$779$	6.11344	0.219037
$780$	0	0
$781$	2.44975	0.0876591
$782$	−25.1199	−0.898285
$783$	0	0
$784$	13.2909	0.474675
$785$	−21.3751	−0.762910
$786$	0	0
$787$	−26.9286	−0.959902	−0.479951	−	0.877295i	$-0.659346\pi$
$787$	−26.9286	−0.959902	−0.479951	+	0.877295i	$0.659346\pi$
$788$	−2.71688	−0.0967847
$789$	0	0
$790$	−22.0047	−0.782891
$791$	−3.82006	−0.135826
$792$	0	0
$793$	−14.7958	−0.525415
$794$	9.86685	0.350162
$795$	0	0
$796$	1.02148	0.0362052
$797$	−18.3883	−0.651348	−0.325674	−	0.945482i	$-0.605591\pi$
$797$	−18.3883	−0.651348	−0.325674	+	0.945482i	$0.605591\pi$
$798$	0	0
$799$	65.7693	2.32675
$800$	−12.0504	−0.426048
$801$	0	0
$802$	−2.36594	−0.0835442
$803$	5.84063	0.206111
$804$	0	0
$805$	−5.48673	−0.193382
$806$	9.20207	0.324129
$807$	0	0
$808$	58.5651	2.06031
$809$	−47.5038	−1.67014	−0.835072	−	0.550141i	$-0.814574\pi$
$809$	−47.5038	−1.67014	−0.835072	+	0.550141i	$0.814574\pi$
$810$	0	0
$811$	−39.8438	−1.39910	−0.699552	−	0.714582i	$-0.746617\pi$
$811$	−39.8438	−1.39910	−0.699552	+	0.714582i	$0.746617\pi$
$812$	1.82240	0.0639539
$813$	0	0
$814$	8.17374	0.286489
$815$	6.17352	0.216249
$816$	0	0
$817$	−2.95329	−0.103323
$818$	30.8736	1.07947
$819$	0	0
$820$	3.81423	0.133199
$821$	39.8464	1.39065	0.695325	−	0.718695i	$-0.255260\pi$
$821$	39.8464	1.39065	0.695325	+	0.718695i	$0.255260\pi$
$822$	0	0
$823$	−24.5258	−0.854917	−0.427459	−	0.904035i	$-0.640591\pi$
$823$	−24.5258	−0.854917	−0.427459	+	0.904035i	$0.640591\pi$
$824$	−32.1918	−1.12146
$825$	0	0
$826$	−22.2267	−0.773365
$827$	38.5836	1.34168	0.670841	−	0.741601i	$-0.265933\pi$
$827$	38.5836	1.34168	0.670841	+	0.741601i	$0.265933\pi$
$828$	0	0
$829$	−5.07772	−0.176357	−0.0881783	−	0.996105i	$-0.528105\pi$
$829$	−5.07772	−0.176357	−0.0881783	+	0.996105i	$0.528105\pi$
$830$	−9.05647	−0.314355
$831$	0	0
$832$	22.3139	0.773595
$833$	−28.7448	−0.995947
$834$	0	0
$835$	−10.9979	−0.380597
$836$	−0.576442	−0.0199367
$837$	0	0
$838$	−43.2226	−1.49310
$839$	44.8328	1.54780	0.773900	−	0.633308i	$-0.218303\pi$
$839$	44.8328	1.54780	0.773900	+	0.633308i	$0.218303\pi$
$840$	0	0
$841$	−23.1720	−0.799036
$842$	−7.05435	−0.243109
$843$	0	0
$844$	−10.2647	−0.353325
$845$	7.08777	0.243827
$846$	0	0
$847$	−1.30958	−0.0449976
$848$	−2.49704	−0.0857488
$849$	0	0
$850$	−24.8446	−0.852163
$851$	−26.5186	−0.909045
$852$	0	0
$853$	−23.4081	−0.801477	−0.400738	−	0.916193i	$-0.631246\pi$
$853$	−23.4081	−0.801477	−0.400738	+	0.916193i	$0.631246\pi$
$854$	9.10181	0.311457
$855$	0	0
$856$	44.5476	1.52260
$857$	12.1302	0.414359	0.207180	−	0.978303i	$-0.433572\pi$
$857$	12.1302	0.414359	0.207180	+	0.978303i	$0.433572\pi$
$858$	0	0
$859$	−4.25984	−0.145344	−0.0726720	−	0.997356i	$-0.523153\pi$
$859$	−4.25984	−0.145344	−0.0726720	+	0.997356i	$0.523153\pi$
$860$	−1.84259	−0.0628317
$861$	0	0
$862$	13.3494	0.454684
$863$	−19.0125	−0.647194	−0.323597	−	0.946195i	$-0.604892\pi$
$863$	−19.0125	−0.647194	−0.323597	+	0.946195i	$0.604892\pi$
$864$	0	0
$865$	2.40698	0.0818397
$866$	15.9764	0.542899
$867$	0	0
$868$	2.29221	0.0778028
$869$	17.0397	0.578031
$870$	0	0
$871$	22.3446	0.757118
$872$	19.6713	0.666154
$873$	0	0
$874$	−4.61854	−0.156224
$875$	−12.5137	−0.423039
$876$	0	0
$877$	−2.17273	−0.0733678	−0.0366839	−	0.999327i	$-0.511679\pi$
$877$	−2.17273	−0.0733678	−0.0366839	+	0.999327i	$0.511679\pi$
$878$	28.1981	0.951641
$879$	0	0
$880$	2.72192	0.0917558
$881$	−21.5283	−0.725305	−0.362653	−	0.931924i	$-0.618129\pi$
$881$	−21.5283	−0.725305	−0.362653	+	0.931924i	$0.618129\pi$
$882$	0	0
$883$	−5.14254	−0.173060	−0.0865302	−	0.996249i	$-0.527578\pi$
$883$	−5.14254	−0.173060	−0.0865302	+	0.996249i	$0.527578\pi$
$884$	−7.96339	−0.267838
$885$	0	0
$886$	30.6531	1.02981
$887$	42.8806	1.43979	0.719895	−	0.694083i	$-0.244190\pi$
$887$	42.8806	1.43979	0.719895	+	0.694083i	$0.244190\pi$
$888$	0	0
$889$	15.9762	0.535825
$890$	−11.7916	−0.395256
$891$	0	0
$892$	5.58619	0.187039
$893$	12.0923	0.404655
$894$	0	0
$895$	13.0961	0.437754
$896$	−5.48275	−0.183166
$897$	0	0
$898$	11.5999	0.387093
$899$	7.33039	0.244482
$900$	0	0
$901$	5.40045	0.179915
$902$	7.29413	0.242868
$903$	0	0
$904$	−8.96700	−0.298238
$905$	0.388657	0.0129194
$906$	0	0
$907$	−28.3695	−0.941992	−0.470996	−	0.882135i	$-0.656105\pi$
$907$	−28.3695	−0.941992	−0.470996	+	0.882135i	$0.656105\pi$
$908$	−3.47135	−0.115201
$909$	0	0
$910$	4.29550	0.142394
$911$	−40.0237	−1.32604	−0.663022	−	0.748600i	$-0.730727\pi$
$911$	−40.0237	−1.32604	−0.663022	+	0.748600i	$0.730727\pi$
$912$	0	0
$913$	7.01303	0.232097
$914$	−2.44402	−0.0808410
$915$	0	0
$916$	−12.5378	−0.414259
$917$	24.5504	0.810726
$918$	0	0
$919$	51.0340	1.68345	0.841727	−	0.539903i	$-0.181539\pi$
$919$	51.0340	1.68345	0.841727	+	0.539903i	$0.181539\pi$
$920$	−12.8793	−0.424617
$921$	0	0
$922$	27.3113	0.899451
$923$	6.22231	0.204810
$924$	0	0
$925$	−26.2280	−0.862371
$926$	41.6981	1.37029
$927$	0	0
$928$	7.59853	0.249434
$929$	−30.7910	−1.01022	−0.505110	−	0.863055i	$-0.668548\pi$
$929$	−30.7910	−1.01022	−0.505110	+	0.863055i	$0.668548\pi$
$930$	0	0
$931$	−5.28501	−0.173209
$932$	−9.04241	−0.296194
$933$	0	0
$934$	−16.3880	−0.536232
$935$	−5.88679	−0.192519
$936$	0	0
$937$	25.9052	0.846287	0.423143	−	0.906063i	$-0.360927\pi$
$937$	25.9052	0.846287	0.423143	+	0.906063i	$0.360927\pi$
$938$	−13.7455	−0.448808
$939$	0	0
$940$	7.54452	0.246075
$941$	31.0254	1.01140	0.505699	−	0.862710i	$-0.331235\pi$
$941$	31.0254	1.01140	0.505699	+	0.862710i	$0.331235\pi$
$942$	0	0
$943$	−23.6648	−0.770632
$944$	−35.7738	−1.16434
$945$	0	0
$946$	−3.52366	−0.114564
$947$	−32.3742	−1.05202	−0.526009	−	0.850479i	$-0.676312\pi$
$947$	−32.3742	−1.05202	−0.526009	+	0.850479i	$0.676312\pi$
$948$	0	0
$949$	14.8350	0.481566
$950$	−4.56793	−0.148203
$951$	0	0
$952$	21.8953	0.709632
$953$	−3.88933	−0.125988	−0.0629938	−	0.998014i	$-0.520065\pi$
$953$	−3.88933	−0.125988	−0.0629938	+	0.998014i	$0.520065\pi$
$954$	0	0
$955$	23.0805	0.746869
$956$	12.6149	0.407996
$957$	0	0
$958$	−3.74201	−0.120899
$959$	22.0390	0.711676
$960$	0	0
$961$	−21.7799	−0.702576
$962$	20.7611	0.669364
$963$	0	0
$964$	−3.45006	−0.111119
$965$	15.7429	0.506780
$966$	0	0
$967$	15.6679	0.503846	0.251923	−	0.967747i	$-0.418937\pi$
$967$	15.6679	0.503846	0.251923	+	0.967747i	$0.418937\pi$
$968$	−3.07403	−0.0988030
$969$	0	0
$970$	19.1059	0.613453
$971$	−25.6584	−0.823417	−0.411708	−	0.911316i	$-0.635068\pi$
$971$	−25.6584	−0.823417	−0.411708	+	0.911316i	$0.635068\pi$
$972$	0	0
$973$	−16.4883	−0.528590
$974$	25.3257	0.811487
$975$	0	0
$976$	14.6493	0.468914
$977$	−44.6708	−1.42914	−0.714572	−	0.699562i	$-0.753379\pi$
$977$	−44.6708	−1.42914	−0.714572	+	0.699562i	$0.753379\pi$
$978$	0	0
$979$	9.13103	0.291829
$980$	−3.29737	−0.105331
$981$	0	0
$982$	−34.1044	−1.08831
$983$	−48.0898	−1.53383	−0.766913	−	0.641751i	$-0.778208\pi$
$983$	−48.0898	−1.53383	−0.766913	+	0.641751i	$0.778208\pi$
$984$	0	0
$985$	−5.10129	−0.162541
$986$	15.6660	0.498908
$987$	0	0
$988$	−1.46415	−0.0465807
$989$	11.4320	0.363517
$990$	0	0
$991$	−57.3200	−1.82083	−0.910415	−	0.413697i	$-0.864237\pi$
$991$	−57.3200	−1.82083	−0.910415	+	0.413697i	$0.864237\pi$
$992$	9.55740	0.303448
$993$	0	0
$994$	−3.82773	−0.121408
$995$	1.91795	0.0608032
$996$	0	0
$997$	14.1271	0.447411	0.223705	−	0.974657i	$-0.428185\pi$
$997$	14.1271	0.447411	0.223705	+	0.974657i	$0.428185\pi$
$998$	−24.0196	−0.760326
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.p.1.5		7
3.2	odd	2		209.2.a.d.1.3	✓	7
12.11	even	2		3344.2.a.ba.1.1		7
15.14	odd	2		5225.2.a.n.1.5		7
33.32	even	2		2299.2.a.q.1.5		7
57.56	even	2		3971.2.a.i.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.d.1.3	✓	7	3.2	odd	2
1881.2.a.p.1.5		7	1.1	even	1	trivial
2299.2.a.q.1.5		7	33.32	even	2
3344.2.a.ba.1.1		7	12.11	even	2
3971.2.a.i.1.5		7	57.56	even	2
5225.2.a.n.1.5		7	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.19313 q^{2} -0.576442 q^{4} -1.08235 q^{5} -1.30958 q^{7} -3.07403 q^{8} -1.29138 q^{10} +1.00000 q^{11} +2.53997 q^{13} -1.56249 q^{14} -2.51483 q^{16} +5.43892 q^{17} +1.00000 q^{19} +0.623909 q^{20} +1.19313 q^{22} -3.87095 q^{23} -3.82853 q^{25} +3.03052 q^{26} +0.754894 q^{28} +2.41412 q^{29} +3.03647 q^{31} +3.14754 q^{32} +6.48934 q^{34} +1.41741 q^{35} +6.85067 q^{37} +1.19313 q^{38} +3.32716 q^{40} +6.11344 q^{41} -2.95329 q^{43} -0.576442 q^{44} -4.61854 q^{46} +12.0923 q^{47} -5.28501 q^{49} -4.56793 q^{50} -1.46415 q^{52} +0.992927 q^{53} -1.08235 q^{55} +4.02567 q^{56} +2.88036 q^{58} +14.2251 q^{59} -5.82518 q^{61} +3.62290 q^{62} +8.78508 q^{64} -2.74913 q^{65} +8.79718 q^{67} -3.13522 q^{68} +1.69116 q^{70} +2.44975 q^{71} +5.84063 q^{73} +8.17374 q^{74} -0.576442 q^{76} -1.30958 q^{77} +17.0397 q^{79} +2.72192 q^{80} +7.29413 q^{82} +7.01303 q^{83} -5.88679 q^{85} -3.52366 q^{86} -3.07403 q^{88} +9.13103 q^{89} -3.32629 q^{91} +2.23138 q^{92} +14.4277 q^{94} -1.08235 q^{95} -14.7950 q^{97} -6.30570 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 15 q^{4} - 2 q^{5} + 10 q^{7} + 9 q^{8} - 6 q^{10} + 7 q^{11} - 4 q^{13} - 6 q^{14} + 27 q^{16} - 2 q^{17} + 7 q^{19} + 4 q^{20} + q^{22} - 10 q^{23} + 9 q^{25} + 8 q^{26} + 26 q^{28} + 18 q^{29}+ \cdots - 19 q^{98}+O(q^{100}) \) 7 * q + q^2 + 15 * q^4 - 2 * q^5 + 10 * q^7 + 9 * q^8 - 6 * q^10 + 7 * q^11 - 4 * q^13 - 6 * q^14 + 27 * q^16 - 2 * q^17 + 7 * q^19 + 4 * q^20 + q^22 - 10 * q^23 + 9 * q^25 + 8 * q^26 + 26 * q^28 + 18 * q^29 + 24 * q^31 + 49 * q^32 - 6 * q^34 - 8 * q^35 + q^38 - 2 * q^40 + 12 * q^41 + 2 * q^43 + 15 * q^44 - 4 * q^46 - 8 * q^47 + 17 * q^49 + 33 * q^50 - 60 * q^52 - 2 * q^53 - 2 * q^55 - 26 * q^56 - 8 * q^58 + 10 * q^59 + 14 * q^61 - 14 * q^62 + 55 * q^64 + 14 * q^65 + 8 * q^67 + 18 * q^68 - 66 * q^70 - 10 * q^71 - 6 * q^73 - 26 * q^74 + 15 * q^76 + 10 * q^77 + 52 * q^79 + 12 * q^80 + 24 * q^82 + 10 * q^83 - 12 * q^85 - 8 * q^86 + 9 * q^88 + 12 * q^91 + 24 * q^94 - 2 * q^95 - 24 * q^97 - 19 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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