LMFDB - Embedded newform 1881.2.a.p.1.3

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			1.3
Root			$-1.45416$ 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)

$f(q)$	$=$	\(q-1.45416 q^{2} +0.114592 q^{4} -2.59296 q^{5} -2.00933 q^{7} +2.74169 q^{8} +3.77059 q^{10} +1.00000 q^{11} -4.45988 q^{13} +2.92189 q^{14} -4.21605 q^{16} -4.54435 q^{17} +1.00000 q^{19} -0.297132 q^{20} -1.45416 q^{22} -7.48175 q^{23} +1.72344 q^{25} +6.48540 q^{26} -0.230253 q^{28} +3.17425 q^{29} +9.34529 q^{31} +0.647446 q^{32} +6.60824 q^{34} +5.21010 q^{35} -6.84486 q^{37} -1.45416 q^{38} -7.10910 q^{40} -0.644299 q^{41} -8.07470 q^{43} +0.114592 q^{44} +10.8797 q^{46} -11.6485 q^{47} -2.96260 q^{49} -2.50616 q^{50} -0.511066 q^{52} +5.53442 q^{53} -2.59296 q^{55} -5.50896 q^{56} -4.61588 q^{58} -9.16829 q^{59} +9.45268 q^{61} -13.5896 q^{62} +7.49061 q^{64} +11.5643 q^{65} +0.113535 q^{67} -0.520746 q^{68} -7.57634 q^{70} +9.84191 q^{71} -2.38027 q^{73} +9.95355 q^{74} +0.114592 q^{76} -2.00933 q^{77} -2.01251 q^{79} +10.9321 q^{80} +0.936917 q^{82} +2.90426 q^{83} +11.7833 q^{85} +11.7419 q^{86} +2.74169 q^{88} +8.82031 q^{89} +8.96136 q^{91} -0.857347 q^{92} +16.9389 q^{94} -2.59296 q^{95} +11.0100 q^{97} +4.30811 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.45416	−1.02825	−0.514124	−	0.857716i	$-0.671883\pi$
$2$	−1.45416	−1.02825	−0.514124	+	0.857716i	$0.671883\pi$
$3$	0	0
$3$	0	0
$4$	0.114592	0.0572959
$5$	−2.59296	−1.15961	−0.579803	−	0.814756i	$-0.696871\pi$
$5$	−2.59296	−1.15961	−0.579803	+	0.814756i	$0.696871\pi$
$6$	0	0
$7$	−2.00933	−0.759454	−0.379727	−	0.925099i	$-0.623982\pi$
$7$	−2.00933	−0.759454	−0.379727	+	0.925099i	$0.623982\pi$
$8$	2.74169	0.969334
$9$	0	0
$10$	3.77059	1.19236
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.45988	−1.23695	−0.618475	−	0.785805i	$-0.712249\pi$
$13$	−4.45988	−1.23695	−0.618475	+	0.785805i	$0.712249\pi$
$14$	2.92189	0.780908
$15$	0	0
$16$	−4.21605	−1.05401
$17$	−4.54435	−1.10217	−0.551084	−	0.834450i	$-0.685786\pi$
$17$	−4.54435	−1.10217	−0.551084	+	0.834450i	$0.685786\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.297132	−0.0664407
$21$	0	0
$22$	−1.45416	−0.310029
$23$	−7.48175	−1.56005	−0.780026	−	0.625747i	$-0.784794\pi$
$23$	−7.48175	−1.56005	−0.780026	+	0.625747i	$0.784794\pi$
$24$	0	0
$25$	1.72344	0.344687
$26$	6.48540	1.27189
$27$	0	0
$28$	−0.230253	−0.0435136
$29$	3.17425	0.589443	0.294722	−	0.955583i	$-0.404773\pi$
$29$	3.17425	0.589443	0.294722	+	0.955583i	$0.404773\pi$
$30$	0	0
$31$	9.34529	1.67846	0.839232	−	0.543774i	$-0.183005\pi$
$31$	9.34529	1.67846	0.839232	+	0.543774i	$0.183005\pi$
$32$	0.647446	0.114453
$33$	0	0
$34$	6.60824	1.13330
$35$	5.21010	0.880668
$36$	0	0
$37$	−6.84486	−1.12529	−0.562644	−	0.826699i	$-0.690216\pi$
$37$	−6.84486	−1.12529	−0.562644	+	0.826699i	$0.690216\pi$
$38$	−1.45416	−0.235896
$39$	0	0
$40$	−7.10910	−1.12405
$41$	−0.644299	−0.100623	−0.0503113	−	0.998734i	$-0.516021\pi$
$41$	−0.644299	−0.100623	−0.0503113	+	0.998734i	$0.516021\pi$
$42$	0	0
$43$	−8.07470	−1.23138	−0.615690	−	0.787988i	$-0.711123\pi$
$43$	−8.07470	−1.23138	−0.615690	+	0.787988i	$0.711123\pi$
$44$	0.114592	0.0172754
$45$	0	0
$46$	10.8797	1.60412
$47$	−11.6485	−1.69911	−0.849557	−	0.527497i	$-0.823131\pi$
$47$	−11.6485	−1.69911	−0.849557	+	0.527497i	$0.823131\pi$
$48$	0	0
$49$	−2.96260	−0.423229
$50$	−2.50616	−0.354425
$51$	0	0
$52$	−0.511066	−0.0708722
$53$	5.53442	0.760211	0.380105	−	0.924943i	$-0.375888\pi$
$53$	5.53442	0.760211	0.380105	+	0.924943i	$0.375888\pi$
$54$	0	0
$55$	−2.59296	−0.349635
$56$	−5.50896	−0.736165
$57$	0	0
$58$	−4.61588	−0.606095
$59$	−9.16829	−1.19361	−0.596805	−	0.802386i	$-0.703563\pi$
$59$	−9.16829	−1.19361	−0.596805	+	0.802386i	$0.703563\pi$
$60$	0	0
$61$	9.45268	1.21029	0.605146	−	0.796115i	$-0.293115\pi$
$61$	9.45268	1.21029	0.605146	+	0.796115i	$0.293115\pi$
$62$	−13.5896	−1.72588
$63$	0	0
$64$	7.49061	0.936327
$65$	11.5643	1.43437
$66$	0	0
$67$	0.113535	0.0138705	0.00693524	−	0.999976i	$-0.497792\pi$
$67$	0.113535	0.0138705	0.00693524	+	0.999976i	$0.497792\pi$
$68$	−0.520746	−0.0631497
$69$	0	0
$70$	−7.57634	−0.905546
$71$	9.84191	1.16802	0.584010	−	0.811746i	$-0.301483\pi$
$71$	9.84191	1.16802	0.584010	+	0.811746i	$0.301483\pi$
$72$	0	0
$73$	−2.38027	−0.278590	−0.139295	−	0.990251i	$-0.544484\pi$
$73$	−2.38027	−0.278590	−0.139295	+	0.990251i	$0.544484\pi$
$74$	9.95355	1.15708
$75$	0	0
$76$	0.114592	0.0131446
$77$	−2.00933	−0.228984
$78$	0	0
$79$	−2.01251	−0.226425	−0.113213	−	0.993571i	$-0.536114\pi$
$79$	−2.01251	−0.226425	−0.113213	+	0.993571i	$0.536114\pi$
$80$	10.9321	1.22224
$81$	0	0
$82$	0.936917	0.103465
$83$	2.90426	0.318784	0.159392	−	0.987215i	$-0.449047\pi$
$83$	2.90426	0.318784	0.159392	+	0.987215i	$0.449047\pi$
$84$	0	0
$85$	11.7833	1.27808
$86$	11.7419	1.26617
$87$	0	0
$88$	2.74169	0.292265
$89$	8.82031	0.934951	0.467475	−	0.884006i	$-0.345164\pi$
$89$	8.82031	0.934951	0.467475	+	0.884006i	$0.345164\pi$
$90$	0	0
$91$	8.96136	0.939406
$92$	−0.857347	−0.0893846
$93$	0	0
$94$	16.9389	1.74711
$95$	−2.59296	−0.266032
$96$	0	0
$97$	11.0100	1.11790	0.558949	−	0.829202i	$-0.311205\pi$
$97$	11.0100	1.11790	0.558949	+	0.829202i	$0.311205\pi$
$98$	4.30811	0.435185
$99$	0	0
$100$	0.197492	0.0197492
$101$	−0.864844	−0.0860552	−0.0430276	−	0.999074i	$-0.513700\pi$
$101$	−0.864844	−0.0860552	−0.0430276	+	0.999074i	$0.513700\pi$
$102$	0	0
$103$	2.64855	0.260969	0.130485	−	0.991450i	$-0.458347\pi$
$103$	2.64855	0.260969	0.130485	+	0.991450i	$0.458347\pi$
$104$	−12.2276	−1.19902
$105$	0	0
$106$	−8.04795	−0.781686
$107$	0.180382	0.0174382	0.00871910	−	0.999962i	$-0.497225\pi$
$107$	0.180382	0.0174382	0.00871910	+	0.999962i	$0.497225\pi$
$108$	0	0
$109$	6.60778	0.632910	0.316455	−	0.948607i	$-0.397507\pi$
$109$	6.60778	0.632910	0.316455	+	0.948607i	$0.397507\pi$
$110$	3.77059	0.359511
$111$	0	0
$112$	8.47143	0.800475
$113$	−4.22599	−0.397548	−0.198774	−	0.980045i	$-0.563696\pi$
$113$	−4.22599	−0.397548	−0.198774	+	0.980045i	$0.563696\pi$
$114$	0	0
$115$	19.3999	1.80905
$116$	0.363743	0.0337727
$117$	0	0
$118$	13.3322	1.22733
$119$	9.13110	0.837046
$120$	0	0
$121$	1.00000	0.0909091
$122$	−13.7457	−1.24448
$123$	0	0
$124$	1.07089	0.0961692
$125$	8.49599	0.759905
$126$	0	0
$127$	14.0808	1.24947	0.624734	−	0.780838i	$-0.285207\pi$
$127$	14.0808	1.24947	0.624734	+	0.780838i	$0.285207\pi$
$128$	−12.1875	−1.07723
$129$	0	0
$130$	−16.8164	−1.47489
$131$	−10.1097	−0.883290	−0.441645	−	0.897190i	$-0.645605\pi$
$131$	−10.1097	−0.883290	−0.441645	+	0.897190i	$0.645605\pi$
$132$	0	0
$133$	−2.00933	−0.174231
$134$	−0.165098	−0.0142623
$135$	0	0
$136$	−12.4592	−1.06837
$137$	2.86731	0.244971	0.122485	−	0.992470i	$-0.460914\pi$
$137$	2.86731	0.244971	0.122485	+	0.992470i	$0.460914\pi$
$138$	0	0
$139$	−21.3049	−1.80706	−0.903529	−	0.428526i	$-0.859033\pi$
$139$	−21.3049	−1.80706	−0.903529	+	0.428526i	$0.859033\pi$
$140$	0.597036	0.0504587
$141$	0	0
$142$	−14.3118	−1.20102
$143$	−4.45988	−0.372954
$144$	0	0
$145$	−8.23070	−0.683522
$146$	3.46130	0.286460
$147$	0	0
$148$	−0.784366	−0.0644745
$149$	7.28860	0.597105	0.298553	−	0.954393i	$-0.403496\pi$
$149$	7.28860	0.597105	0.298553	+	0.954393i	$0.403496\pi$
$150$	0	0
$151$	22.4316	1.82546	0.912731	−	0.408562i	$-0.133969\pi$
$151$	22.4316	1.82546	0.912731	+	0.408562i	$0.133969\pi$
$152$	2.74169	0.222381
$153$	0	0
$154$	2.92189	0.235453
$155$	−24.2320	−1.94636
$156$	0	0
$157$	15.9630	1.27399	0.636994	−	0.770869i	$-0.280178\pi$
$157$	15.9630	1.27399	0.636994	+	0.770869i	$0.280178\pi$
$158$	2.92652	0.232821
$159$	0	0
$160$	−1.67880	−0.132721
$161$	15.0333	1.18479
$162$	0	0
$163$	−2.04841	−0.160444	−0.0802218	−	0.996777i	$-0.525563\pi$
$163$	−2.04841	−0.160444	−0.0802218	+	0.996777i	$0.525563\pi$
$164$	−0.0738315	−0.00576527
$165$	0	0
$166$	−4.22327	−0.327789
$167$	6.99086	0.540969	0.270485	−	0.962724i	$-0.412816\pi$
$167$	6.99086	0.540969	0.270485	+	0.962724i	$0.412816\pi$
$168$	0	0
$169$	6.89056	0.530043
$170$	−17.1349	−1.31419
$171$	0	0
$172$	−0.925296	−0.0705531
$173$	19.3508	1.47122	0.735609	−	0.677407i	$-0.236896\pi$
$173$	19.3508	1.47122	0.735609	+	0.677407i	$0.236896\pi$
$174$	0	0
$175$	−3.46295	−0.261774
$176$	−4.21605	−0.317797
$177$	0	0
$178$	−12.8262	−0.961362
$179$	22.8792	1.71007	0.855037	−	0.518567i	$-0.173534\pi$
$179$	22.8792	1.71007	0.855037	+	0.518567i	$0.173534\pi$
$180$	0	0
$181$	−1.33552	−0.0992682	−0.0496341	−	0.998767i	$-0.515806\pi$
$181$	−1.33552	−0.0992682	−0.0496341	+	0.998767i	$0.515806\pi$
$182$	−13.0313	−0.965944
$183$	0	0
$184$	−20.5126	−1.51221
$185$	17.7485	1.30489
$186$	0	0
$187$	−4.54435	−0.332316
$188$	−1.33483	−0.0973523
$189$	0	0
$190$	3.77059	0.273547
$191$	−0.690960	−0.0499961	−0.0249981	−	0.999687i	$-0.507958\pi$
$191$	−0.690960	−0.0499961	−0.0249981	+	0.999687i	$0.507958\pi$
$192$	0	0
$193$	−19.3391	−1.39206	−0.696031	−	0.718012i	$-0.745052\pi$
$193$	−19.3391	−1.39206	−0.696031	+	0.718012i	$0.745052\pi$
$194$	−16.0104	−1.14948
$195$	0	0
$196$	−0.339490	−0.0242493
$197$	12.7676	0.909653	0.454826	−	0.890580i	$-0.349701\pi$
$197$	12.7676	0.909653	0.454826	+	0.890580i	$0.349701\pi$
$198$	0	0
$199$	−1.78947	−0.126852	−0.0634261	−	0.997987i	$-0.520203\pi$
$199$	−1.78947	−0.126852	−0.0634261	+	0.997987i	$0.520203\pi$
$200$	4.72513	0.334117
$201$	0	0
$202$	1.25763	0.0884862
$203$	−6.37811	−0.447655
$204$	0	0
$205$	1.67064	0.116683
$206$	−3.85142	−0.268341
$207$	0	0
$208$	18.8031	1.30376
$209$	1.00000	0.0691714
$210$	0	0
$211$	10.5987	0.729644	0.364822	−	0.931077i	$-0.381130\pi$
$211$	10.5987	0.729644	0.364822	+	0.931077i	$0.381130\pi$
$212$	0.634199	0.0435570
$213$	0	0
$214$	−0.262305	−0.0179308
$215$	20.9374	1.42792
$216$	0	0
$217$	−18.7777	−1.27472
$218$	−9.60879	−0.650789
$219$	0	0
$220$	−0.297132	−0.0200326
$221$	20.2673	1.36333
$222$	0	0
$223$	−3.22342	−0.215856	−0.107928	−	0.994159i	$-0.534422\pi$
$223$	−3.22342	−0.215856	−0.107928	+	0.994159i	$0.534422\pi$
$224$	−1.30093	−0.0869221
$225$	0	0
$226$	6.14528	0.408778
$227$	19.1441	1.27064	0.635320	−	0.772249i	$-0.280868\pi$
$227$	19.1441	1.27064	0.635320	+	0.772249i	$0.280868\pi$
$228$	0	0
$229$	−29.4688	−1.94735	−0.973676	−	0.227935i	$-0.926803\pi$
$229$	−29.4688	−1.94735	−0.973676	+	0.227935i	$0.926803\pi$
$230$	−28.2106	−1.86015
$231$	0	0
$232$	8.70281	0.571368
$233$	−19.5192	−1.27875	−0.639374	−	0.768896i	$-0.720806\pi$
$233$	−19.5192	−1.27875	−0.639374	+	0.768896i	$0.720806\pi$
$234$	0	0
$235$	30.2042	1.97030
$236$	−1.05061	−0.0683890
$237$	0	0
$238$	−13.2781	−0.860692
$239$	−11.1392	−0.720535	−0.360267	−	0.932849i	$-0.617315\pi$
$239$	−11.1392	−0.720535	−0.360267	+	0.932849i	$0.617315\pi$
$240$	0	0
$241$	−16.7483	−1.07885	−0.539427	−	0.842033i	$-0.681359\pi$
$241$	−16.7483	−1.07885	−0.539427	+	0.842033i	$0.681359\pi$
$242$	−1.45416	−0.0934772
$243$	0	0
$244$	1.08320	0.0693448
$245$	7.68191	0.490779
$246$	0	0
$247$	−4.45988	−0.283776
$248$	25.6219	1.62699
$249$	0	0
$250$	−12.3546	−0.781371
$251$	−30.1448	−1.90272	−0.951360	−	0.308081i	$-0.900313\pi$
$251$	−30.1448	−1.90272	−0.951360	+	0.308081i	$0.900313\pi$
$252$	0	0
$253$	−7.48175	−0.470373
$254$	−20.4758	−1.28476
$255$	0	0
$256$	2.74135	0.171334
$257$	11.6015	0.723680	0.361840	−	0.932240i	$-0.382149\pi$
$257$	11.6015	0.723680	0.361840	+	0.932240i	$0.382149\pi$
$258$	0	0
$259$	13.7536	0.854605
$260$	1.32517	0.0821838
$261$	0	0
$262$	14.7012	0.908242
$263$	3.88242	0.239401	0.119700	−	0.992810i	$-0.461807\pi$
$263$	3.88242	0.239401	0.119700	+	0.992810i	$0.461807\pi$
$264$	0	0
$265$	−14.3505	−0.881545
$266$	2.92189	0.179153
$267$	0	0
$268$	0.0130102	0.000794723 0
$269$	−23.7542	−1.44832	−0.724160	−	0.689632i	$-0.757772\pi$
$269$	−23.7542	−1.44832	−0.724160	+	0.689632i	$0.757772\pi$
$270$	0	0
$271$	17.3686	1.05507	0.527533	−	0.849535i	$-0.323117\pi$
$271$	17.3686	1.05507	0.527533	+	0.849535i	$0.323117\pi$
$272$	19.1592	1.16170
$273$	0	0
$274$	−4.16954	−0.251891
$275$	1.72344	0.103927
$276$	0	0
$277$	6.47702	0.389167	0.194583	−	0.980886i	$-0.437665\pi$
$277$	6.47702	0.389167	0.194583	+	0.980886i	$0.437665\pi$
$278$	30.9808	1.85811
$279$	0	0
$280$	14.2845	0.853662
$281$	11.8438	0.706542	0.353271	−	0.935521i	$-0.385069\pi$
$281$	11.8438	0.706542	0.353271	+	0.935521i	$0.385069\pi$
$282$	0	0
$283$	11.1566	0.663192	0.331596	−	0.943421i	$-0.392413\pi$
$283$	11.1566	0.663192	0.331596	+	0.943421i	$0.392413\pi$
$284$	1.12780	0.0669228
$285$	0	0
$286$	6.48540	0.383490
$287$	1.29461	0.0764183
$288$	0	0
$289$	3.65116	0.214774
$290$	11.9688	0.702831
$291$	0	0
$292$	−0.272760	−0.0159621
$293$	25.7289	1.50310	0.751551	−	0.659675i	$-0.229306\pi$
$293$	25.7289	1.50310	0.751551	+	0.659675i	$0.229306\pi$
$294$	0	0
$295$	23.7730	1.38412
$296$	−18.7665	−1.09078
$297$	0	0
$298$	−10.5988	−0.613973
$299$	33.3677	1.92970
$300$	0	0
$301$	16.2247	0.935178
$302$	−32.6193	−1.87703
$303$	0	0
$304$	−4.21605	−0.241807
$305$	−24.5104	−1.40346
$306$	0	0
$307$	0.843227	0.0481255	0.0240627	−	0.999710i	$-0.492340\pi$
$307$	0.843227	0.0481255	0.0240627	+	0.999710i	$0.492340\pi$
$308$	−0.230253	−0.0131199
$309$	0	0
$310$	35.2372	2.00134
$311$	3.89732	0.220997	0.110498	−	0.993876i	$-0.464755\pi$
$311$	3.89732	0.220997	0.110498	+	0.993876i	$0.464755\pi$
$312$	0	0
$313$	8.00036	0.452207	0.226104	−	0.974103i	$-0.427401\pi$
$313$	8.00036	0.452207	0.226104	+	0.974103i	$0.427401\pi$
$314$	−23.2128	−1.30998
$315$	0	0
$316$	−0.230617	−0.0129732
$317$	−24.3173	−1.36579	−0.682897	−	0.730515i	$-0.739280\pi$
$317$	−24.3173	−1.36579	−0.682897	+	0.730515i	$0.739280\pi$
$318$	0	0
$319$	3.17425	0.177724
$320$	−19.4229	−1.08577
$321$	0	0
$322$	−21.8608	−1.21826
$323$	−4.54435	−0.252855
$324$	0	0
$325$	−7.68633	−0.426361
$326$	2.97872	0.164976
$327$	0	0
$328$	−1.76647	−0.0975370
$329$	23.4057	1.29040
$330$	0	0
$331$	1.00272	0.0551145	0.0275573	−	0.999620i	$-0.491227\pi$
$331$	1.00272	0.0551145	0.0275573	+	0.999620i	$0.491227\pi$
$332$	0.332805	0.0182650
$333$	0	0
$334$	−10.1659	−0.556251
$335$	−0.294391	−0.0160843
$336$	0	0
$337$	−24.3229	−1.32495	−0.662477	−	0.749082i	$-0.730495\pi$
$337$	−24.3229	−1.32495	−0.662477	+	0.749082i	$0.730495\pi$
$338$	−10.0200	−0.545016
$339$	0	0
$340$	1.35027	0.0732289
$341$	9.34529	0.506076
$342$	0	0
$343$	20.0181	1.08088
$344$	−22.1384	−1.19362
$345$	0	0
$346$	−28.1393	−1.51278
$347$	14.4536	0.775908	0.387954	−	0.921679i	$-0.373182\pi$
$347$	14.4536	0.775908	0.387954	+	0.921679i	$0.373182\pi$
$348$	0	0
$349$	34.9365	1.87011	0.935055	−	0.354503i	$-0.115350\pi$
$349$	34.9365	1.87011	0.935055	+	0.354503i	$0.115350\pi$
$350$	5.03570	0.269169
$351$	0	0
$352$	0.647446	0.0345090
$353$	−1.66429	−0.0885813	−0.0442906	−	0.999019i	$-0.514103\pi$
$353$	−1.66429	−0.0885813	−0.0442906	+	0.999019i	$0.514103\pi$
$354$	0	0
$355$	−25.5197	−1.35444
$356$	1.01074	0.0535689
$357$	0	0
$358$	−33.2701	−1.75838
$359$	−11.6721	−0.616032	−0.308016	−	0.951381i	$-0.599665\pi$
$359$	−11.6721	−0.616032	−0.308016	+	0.951381i	$0.599665\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	1.94206	0.102072
$363$	0	0
$364$	1.02690	0.0538242
$365$	6.17195	0.323054
$366$	0	0
$367$	−1.44184	−0.0752633	−0.0376317	−	0.999292i	$-0.511981\pi$
$367$	−1.44184	−0.0752633	−0.0376317	+	0.999292i	$0.511981\pi$
$368$	31.5434	1.64432
$369$	0	0
$370$	−25.8092	−1.34175
$371$	−11.1205	−0.577345
$372$	0	0
$373$	7.90630	0.409372	0.204686	−	0.978828i	$-0.434383\pi$
$373$	7.90630	0.409372	0.204686	+	0.978828i	$0.434383\pi$
$374$	6.60824	0.341704
$375$	0	0
$376$	−31.9367	−1.64701
$377$	−14.1568	−0.729111
$378$	0	0
$379$	−25.0206	−1.28522	−0.642611	−	0.766192i	$-0.722149\pi$
$379$	−25.0206	−1.28522	−0.642611	+	0.766192i	$0.722149\pi$
$380$	−0.297132	−0.0152426
$381$	0	0
$382$	1.00477	0.0514085
$383$	−18.3767	−0.939005	−0.469503	−	0.882931i	$-0.655567\pi$
$383$	−18.3767	−0.939005	−0.469503	+	0.882931i	$0.655567\pi$
$384$	0	0
$385$	5.21010	0.265531
$386$	28.1223	1.43139
$387$	0	0
$388$	1.26166	0.0640510
$389$	−16.0386	−0.813188	−0.406594	−	0.913609i	$-0.633284\pi$
$389$	−16.0386	−0.813188	−0.406594	+	0.913609i	$0.633284\pi$
$390$	0	0
$391$	33.9997	1.71944
$392$	−8.12255	−0.410251
$393$	0	0
$394$	−18.5662	−0.935349
$395$	5.21836	0.262564
$396$	0	0
$397$	−28.7655	−1.44370	−0.721849	−	0.692051i	$-0.756707\pi$
$397$	−28.7655	−1.44370	−0.721849	+	0.692051i	$0.756707\pi$
$398$	2.60218	0.130436
$399$	0	0
$400$	−7.26610	−0.363305
$401$	22.0831	1.10278	0.551389	−	0.834249i	$-0.314098\pi$
$401$	22.0831	1.10278	0.551389	+	0.834249i	$0.314098\pi$
$402$	0	0
$403$	−41.6789	−2.07617
$404$	−0.0991041	−0.00493061
$405$	0	0
$406$	9.27481	0.460301
$407$	−6.84486	−0.339287
$408$	0	0
$409$	6.78338	0.335416	0.167708	−	0.985837i	$-0.446363\pi$
$409$	6.78338	0.335416	0.167708	+	0.985837i	$0.446363\pi$
$410$	−2.42939	−0.119979
$411$	0	0
$412$	0.303502	0.0149525
$413$	18.4221	0.906492
$414$	0	0
$415$	−7.53063	−0.369664
$416$	−2.88754	−0.141573
$417$	0	0
$418$	−1.45416	−0.0711255
$419$	−18.6261	−0.909943	−0.454971	−	0.890506i	$-0.650351\pi$
$419$	−18.6261	−0.909943	−0.454971	+	0.890506i	$0.650351\pi$
$420$	0	0
$421$	−1.65361	−0.0805921	−0.0402960	−	0.999188i	$-0.512830\pi$
$421$	−1.65361	−0.0805921	−0.0402960	+	0.999188i	$0.512830\pi$
$422$	−15.4122	−0.750256
$423$	0	0
$424$	15.1737	0.736898
$425$	−7.83191	−0.379903
$426$	0	0
$427$	−18.9935	−0.919161
$428$	0.0206703	0.000999138 0
$429$	0	0
$430$	−30.4464	−1.46825
$431$	22.0877	1.06393	0.531964	−	0.846767i	$-0.321454\pi$
$431$	22.0877	1.06393	0.531964	+	0.846767i	$0.321454\pi$
$432$	0	0
$433$	−17.5885	−0.845249	−0.422625	−	0.906305i	$-0.638891\pi$
$433$	−17.5885	−0.845249	−0.422625	+	0.906305i	$0.638891\pi$
$434$	27.3059	1.31073
$435$	0	0
$436$	0.757198	0.0362632
$437$	−7.48175	−0.357900
$438$	0	0
$439$	9.46754	0.451861	0.225931	−	0.974143i	$-0.427458\pi$
$439$	9.46754	0.451861	0.225931	+	0.974143i	$0.427458\pi$
$440$	−7.10910	−0.338913
$441$	0	0
$442$	−29.4720	−1.40184
$443$	14.0458	0.667337	0.333669	−	0.942690i	$-0.391713\pi$
$443$	14.0458	0.667337	0.333669	+	0.942690i	$0.391713\pi$
$444$	0	0
$445$	−22.8707	−1.08418
$446$	4.68738	0.221954
$447$	0	0
$448$	−15.0511	−0.711097
$449$	−18.8516	−0.889660	−0.444830	−	0.895615i	$-0.646736\pi$
$449$	−18.8516	−0.889660	−0.444830	+	0.895615i	$0.646736\pi$
$450$	0	0
$451$	−0.644299	−0.0303389
$452$	−0.484264	−0.0227779
$453$	0	0
$454$	−27.8387	−1.30653
$455$	−23.2365	−1.08934
$456$	0	0
$457$	22.1917	1.03808	0.519041	−	0.854749i	$-0.326289\pi$
$457$	22.1917	1.03808	0.519041	+	0.854749i	$0.326289\pi$
$458$	42.8524	2.00236
$459$	0	0
$460$	2.22307	0.103651
$461$	−29.0705	−1.35395	−0.676973	−	0.736008i	$-0.736709\pi$
$461$	−29.0705	−1.35395	−0.676973	+	0.736008i	$0.736709\pi$
$462$	0	0
$463$	21.1214	0.981594	0.490797	−	0.871274i	$-0.336706\pi$
$463$	21.1214	0.981594	0.490797	+	0.871274i	$0.336706\pi$
$464$	−13.3828	−0.621281
$465$	0	0
$466$	28.3842	1.31487
$467$	6.37732	0.295107	0.147554	−	0.989054i	$-0.452860\pi$
$467$	6.37732	0.295107	0.147554	+	0.989054i	$0.452860\pi$
$468$	0	0
$469$	−0.228129	−0.0105340
$470$	−43.9218	−2.02596
$471$	0	0
$472$	−25.1366	−1.15701
$473$	−8.07470	−0.371275
$474$	0	0
$475$	1.72344	0.0790767
$476$	1.04635	0.0479593
$477$	0	0
$478$	16.1982	0.740889
$479$	−29.3832	−1.34255	−0.671276	−	0.741208i	$-0.734253\pi$
$479$	−29.3832	−1.34255	−0.671276	+	0.741208i	$0.734253\pi$
$480$	0	0
$481$	30.5273	1.39193
$482$	24.3548	1.10933
$483$	0	0
$484$	0.114592	0.00520872
$485$	−28.5485	−1.29632
$486$	0	0
$487$	−34.3525	−1.55666	−0.778331	−	0.627854i	$-0.783934\pi$
$487$	−34.3525	−1.55666	−0.778331	+	0.627854i	$0.783934\pi$
$488$	25.9163	1.17318
$489$	0	0
$490$	−11.1708	−0.504643
$491$	−23.0123	−1.03853	−0.519264	−	0.854614i	$-0.673794\pi$
$491$	−23.0123	−1.03853	−0.519264	+	0.854614i	$0.673794\pi$
$492$	0	0
$493$	−14.4249	−0.649666
$494$	6.48540	0.291792
$495$	0	0
$496$	−39.4002	−1.76912
$497$	−19.7756	−0.887058
$498$	0	0
$499$	−18.9873	−0.849988	−0.424994	−	0.905196i	$-0.639724\pi$
$499$	−18.9873	−0.849988	−0.424994	+	0.905196i	$0.639724\pi$
$500$	0.973572	0.0435395
$501$	0	0
$502$	43.8354	1.95647
$503$	33.3366	1.48641	0.743203	−	0.669066i	$-0.233306\pi$
$503$	33.3366	1.48641	0.743203	+	0.669066i	$0.233306\pi$
$504$	0	0
$505$	2.24251	0.0997902
$506$	10.8797	0.483661
$507$	0	0
$508$	1.61354	0.0715895
$509$	−10.1122	−0.448217	−0.224109	−	0.974564i	$-0.571947\pi$
$509$	−10.1122	−0.448217	−0.224109	+	0.974564i	$0.571947\pi$
$510$	0	0
$511$	4.78274	0.211576
$512$	20.3886	0.901056
$513$	0	0
$514$	−16.8704	−0.744124
$515$	−6.86758	−0.302622
$516$	0	0
$517$	−11.6485	−0.512302
$518$	−19.9999	−0.878747
$519$	0	0
$520$	31.7057	1.39039
$521$	−24.9735	−1.09411	−0.547055	−	0.837097i	$-0.684251\pi$
$521$	−24.9735	−1.09411	−0.547055	+	0.837097i	$0.684251\pi$
$522$	0	0
$523$	−27.5980	−1.20678	−0.603388	−	0.797448i	$-0.706183\pi$
$523$	−27.5980	−1.20678	−0.603388	+	0.797448i	$0.706183\pi$
$524$	−1.15849	−0.0506089
$525$	0	0
$526$	−5.64568	−0.246163
$527$	−42.4683	−1.84995
$528$	0	0
$529$	32.9765	1.43376
$530$	20.8680	0.906448
$531$	0	0
$532$	−0.230253	−0.00998272
$533$	2.87350	0.124465
$534$	0	0
$535$	−0.467724	−0.0202215
$536$	0.311278	0.0134451
$537$	0	0
$538$	34.5425	1.48923
$539$	−2.96260	−0.127608
$540$	0	0
$541$	8.78281	0.377603	0.188801	−	0.982015i	$-0.439540\pi$
$541$	8.78281	0.377603	0.188801	+	0.982015i	$0.439540\pi$
$542$	−25.2567	−1.08487
$543$	0	0
$544$	−2.94223	−0.126147
$545$	−17.1337	−0.733927
$546$	0	0
$547$	20.2388	0.865347	0.432674	−	0.901551i	$-0.357570\pi$
$547$	20.2388	0.865347	0.432674	+	0.901551i	$0.357570\pi$
$548$	0.328570	0.0140358
$549$	0	0
$550$	−2.50616	−0.106863
$551$	3.17425	0.135228
$552$	0	0
$553$	4.04379	0.171959
$554$	−9.41865	−0.400160
$555$	0	0
$556$	−2.44137	−0.103537
$557$	−21.6180	−0.915985	−0.457993	−	0.888956i	$-0.651431\pi$
$557$	−21.6180	−0.915985	−0.457993	+	0.888956i	$0.651431\pi$
$558$	0	0
$559$	36.0122	1.52316
$560$	−21.9661	−0.928236
$561$	0	0
$562$	−17.2228	−0.726501
$563$	2.72687	0.114924	0.0574620	−	0.998348i	$-0.481699\pi$
$563$	2.72687	0.114924	0.0574620	+	0.998348i	$0.481699\pi$
$564$	0	0
$565$	10.9578	0.460999
$566$	−16.2236	−0.681927
$567$	0	0
$568$	26.9835	1.13220
$569$	−18.8953	−0.792131	−0.396065	−	0.918222i	$-0.629625\pi$
$569$	−18.8953	−0.792131	−0.396065	+	0.918222i	$0.629625\pi$
$570$	0	0
$571$	32.7597	1.37095	0.685475	−	0.728096i	$-0.259595\pi$
$571$	32.7597	1.37095	0.685475	+	0.728096i	$0.259595\pi$
$572$	−0.511066	−0.0213688
$573$	0	0
$574$	−1.88257	−0.0785771
$575$	−12.8943	−0.537730
$576$	0	0
$577$	−39.3665	−1.63885	−0.819425	−	0.573186i	$-0.805707\pi$
$577$	−39.3665	−1.63885	−0.819425	+	0.573186i	$0.805707\pi$
$578$	−5.30938	−0.220841
$579$	0	0
$580$	−0.943171	−0.0391631
$581$	−5.83561	−0.242102
$582$	0	0
$583$	5.53442	0.229212
$584$	−6.52597	−0.270047
$585$	0	0
$586$	−37.4141	−1.54556
$587$	46.0274	1.89976	0.949878	−	0.312621i	$-0.101207\pi$
$587$	46.0274	1.89976	0.949878	+	0.312621i	$0.101207\pi$
$588$	0	0
$589$	9.34529	0.385066
$590$	−34.5699	−1.42322
$591$	0	0
$592$	28.8583	1.18607
$593$	11.6575	0.478715	0.239358	−	0.970932i	$-0.423063\pi$
$593$	11.6575	0.478715	0.239358	+	0.970932i	$0.423063\pi$
$594$	0	0
$595$	−23.6766	−0.970644
$596$	0.835214	0.0342117
$597$	0	0
$598$	−48.5221	−1.98422
$599$	36.7024	1.49962	0.749811	−	0.661653i	$-0.230145\pi$
$599$	36.7024	1.49962	0.749811	+	0.661653i	$0.230145\pi$
$600$	0	0
$601$	−22.7859	−0.929456	−0.464728	−	0.885454i	$-0.653848\pi$
$601$	−22.7859	−0.929456	−0.464728	+	0.885454i	$0.653848\pi$
$602$	−23.5934	−0.961595
$603$	0	0
$604$	2.57048	0.104592
$605$	−2.59296	−0.105419
$606$	0	0
$607$	18.4924	0.750584	0.375292	−	0.926907i	$-0.377542\pi$
$607$	18.4924	0.750584	0.375292	+	0.926907i	$0.377542\pi$
$608$	0.647446	0.0262574
$609$	0	0
$610$	35.6422	1.44311
$611$	51.9511	2.10172
$612$	0	0
$613$	21.1249	0.853229	0.426614	−	0.904434i	$-0.359706\pi$
$613$	21.1249	0.853229	0.426614	+	0.904434i	$0.359706\pi$
$614$	−1.22619	−0.0494850
$615$	0	0
$616$	−5.50896	−0.221962
$617$	−24.6204	−0.991178	−0.495589	−	0.868557i	$-0.665048\pi$
$617$	−24.6204	−0.991178	−0.495589	+	0.868557i	$0.665048\pi$
$618$	0	0
$619$	−8.89942	−0.357698	−0.178849	−	0.983877i	$-0.557237\pi$
$619$	−8.89942	−0.357698	−0.178849	+	0.983877i	$0.557237\pi$
$620$	−2.77679	−0.111518
$621$	0	0
$622$	−5.66734	−0.227240
$623$	−17.7229	−0.710052
$624$	0	0
$625$	−30.6470	−1.22588
$626$	−11.6338	−0.464982
$627$	0	0
$628$	1.82923	0.0729943
$629$	31.1055	1.24026
$630$	0	0
$631$	15.2980	0.609005	0.304503	−	0.952512i	$-0.401510\pi$
$631$	15.2980	0.609005	0.304503	+	0.952512i	$0.401510\pi$
$632$	−5.51768	−0.219482
$633$	0	0
$634$	35.3613	1.40438
$635$	−36.5109	−1.44889
$636$	0	0
$637$	13.2129	0.523513
$638$	−4.61588	−0.182744
$639$	0	0
$640$	31.6016	1.24916
$641$	28.4765	1.12475	0.562377	−	0.826881i	$-0.309887\pi$
$641$	28.4765	1.12475	0.562377	+	0.826881i	$0.309887\pi$
$642$	0	0
$643$	−12.5264	−0.493992	−0.246996	−	0.969017i	$-0.579443\pi$
$643$	−12.5264	−0.493992	−0.246996	+	0.969017i	$0.579443\pi$
$644$	1.72269	0.0678835
$645$	0	0
$646$	6.60824	0.259998
$647$	17.3701	0.682889	0.341445	−	0.939902i	$-0.389084\pi$
$647$	17.3701	0.682889	0.341445	+	0.939902i	$0.389084\pi$
$648$	0	0
$649$	−9.16829	−0.359887
$650$	11.1772	0.438405
$651$	0	0
$652$	−0.234731	−0.00919277
$653$	32.5362	1.27324	0.636621	−	0.771177i	$-0.280332\pi$
$653$	32.5362	1.27324	0.636621	+	0.771177i	$0.280332\pi$
$654$	0	0
$655$	26.2141	1.02427
$656$	2.71640	0.106058
$657$	0	0
$658$	−34.0358	−1.32685
$659$	−48.1592	−1.87602	−0.938008	−	0.346614i	$-0.887332\pi$
$659$	−48.1592	−1.87602	−0.938008	+	0.346614i	$0.887332\pi$
$660$	0	0
$661$	−6.19327	−0.240890	−0.120445	−	0.992720i	$-0.538432\pi$
$661$	−6.19327	−0.240890	−0.120445	+	0.992720i	$0.538432\pi$
$662$	−1.45812	−0.0566714
$663$	0	0
$664$	7.96259	0.309008
$665$	5.21010	0.202039
$666$	0	0
$667$	−23.7489	−0.919562
$668$	0.801096	0.0309953
$669$	0	0
$670$	0.428093	0.0165387
$671$	9.45268	0.364917
$672$	0	0
$673$	22.2416	0.857351	0.428676	−	0.903459i	$-0.358980\pi$
$673$	22.2416	0.857351	0.428676	+	0.903459i	$0.358980\pi$
$674$	35.3695	1.36238
$675$	0	0
$676$	0.789602	0.0303693
$677$	−45.2780	−1.74018	−0.870088	−	0.492897i	$-0.835938\pi$
$677$	−45.2780	−1.74018	−0.870088	+	0.492897i	$0.835938\pi$
$678$	0	0
$679$	−22.1227	−0.848992
$680$	32.3063	1.23889
$681$	0	0
$682$	−13.5896	−0.520372
$683$	36.8106	1.40852	0.704259	−	0.709943i	$-0.251279\pi$
$683$	36.8106	1.40852	0.704259	+	0.709943i	$0.251279\pi$
$684$	0	0
$685$	−7.43482	−0.284070
$686$	−29.1096	−1.11141
$687$	0	0
$688$	34.0434	1.29789
$689$	−24.6829	−0.940342
$690$	0	0
$691$	−35.0477	−1.33328	−0.666638	−	0.745382i	$-0.732267\pi$
$691$	−35.0477	−1.33328	−0.666638	+	0.745382i	$0.732267\pi$
$692$	2.21745	0.0842948
$693$	0	0
$694$	−21.0179	−0.797827
$695$	55.2428	2.09548
$696$	0	0
$697$	2.92793	0.110903
$698$	−50.8034	−1.92294
$699$	0	0
$700$	−0.396826	−0.0149986
$701$	29.2997	1.10664	0.553318	−	0.832970i	$-0.313361\pi$
$701$	29.2997	1.10664	0.553318	+	0.832970i	$0.313361\pi$
$702$	0	0
$703$	−6.84486	−0.258159
$704$	7.49061	0.282313
$705$	0	0
$706$	2.42015	0.0910836
$707$	1.73776	0.0653550
$708$	0	0
$709$	34.3627	1.29052	0.645259	−	0.763964i	$-0.276750\pi$
$709$	34.3627	1.29052	0.645259	+	0.763964i	$0.276750\pi$
$710$	37.1098	1.39271
$711$	0	0
$712$	24.1826	0.906280
$713$	−69.9191	−2.61849
$714$	0	0
$715$	11.5643	0.432480
$716$	2.62177	0.0979803
$717$	0	0
$718$	16.9732	0.633435
$719$	−23.2839	−0.868345	−0.434172	−	0.900830i	$-0.642959\pi$
$719$	−23.2839	−0.868345	−0.434172	+	0.900830i	$0.642959\pi$
$720$	0	0
$721$	−5.32180	−0.198194
$722$	−1.45416	−0.0541184
$723$	0	0
$724$	−0.153039	−0.00568767
$725$	5.47062	0.203174
$726$	0	0
$727$	−10.5331	−0.390652	−0.195326	−	0.980738i	$-0.562576\pi$
$727$	−10.5331	−0.390652	−0.195326	+	0.980738i	$0.562576\pi$
$728$	24.5693	0.910599
$729$	0	0
$730$	−8.97502	−0.332180
$731$	36.6943	1.35719
$732$	0	0
$733$	46.7251	1.72583	0.862915	−	0.505349i	$-0.168636\pi$
$733$	46.7251	1.72583	0.862915	+	0.505349i	$0.168636\pi$
$734$	2.09667	0.0773894
$735$	0	0
$736$	−4.84403	−0.178553
$737$	0.113535	0.00418211
$738$	0	0
$739$	6.14537	0.226061	0.113031	−	0.993592i	$-0.463944\pi$
$739$	6.14537	0.226061	0.113031	+	0.993592i	$0.463944\pi$
$740$	2.03383	0.0747650
$741$	0	0
$742$	16.1710	0.593655
$743$	−5.00527	−0.183625	−0.0918127	−	0.995776i	$-0.529266\pi$
$743$	−5.00527	−0.183625	−0.0918127	+	0.995776i	$0.529266\pi$
$744$	0	0
$745$	−18.8990	−0.692407
$746$	−11.4970	−0.420937
$747$	0	0
$748$	−0.520746	−0.0190404
$749$	−0.362447	−0.0132435
$750$	0	0
$751$	45.1888	1.64896	0.824482	−	0.565889i	$-0.191467\pi$
$751$	45.1888	1.64896	0.824482	+	0.565889i	$0.191467\pi$
$752$	49.1108	1.79089
$753$	0	0
$754$	20.5863	0.749708
$755$	−58.1643	−2.11682
$756$	0	0
$757$	−2.72001	−0.0988606	−0.0494303	−	0.998778i	$-0.515741\pi$
$757$	−2.72001	−0.0988606	−0.0494303	+	0.998778i	$0.515741\pi$
$758$	36.3841	1.32153
$759$	0	0
$760$	−7.10910	−0.257874
$761$	9.10142	0.329926	0.164963	−	0.986300i	$-0.447249\pi$
$761$	9.10142	0.329926	0.164963	+	0.986300i	$0.447249\pi$
$762$	0	0
$763$	−13.2772	−0.480666
$764$	−0.0791784	−0.00286457
$765$	0	0
$766$	26.7227	0.965531
$767$	40.8895	1.47644
$768$	0	0
$769$	37.2427	1.34301	0.671503	−	0.741002i	$-0.265649\pi$
$769$	37.2427	1.34301	0.671503	+	0.741002i	$0.265649\pi$
$770$	−7.57634	−0.273032
$771$	0	0
$772$	−2.21611	−0.0797595
$773$	14.7996	0.532306	0.266153	−	0.963931i	$-0.414247\pi$
$773$	14.7996	0.532306	0.266153	+	0.963931i	$0.414247\pi$
$774$	0	0
$775$	16.1060	0.578545
$776$	30.1861	1.08362
$777$	0	0
$778$	23.3227	0.836160
$779$	−0.644299	−0.0230844
$780$	0	0
$781$	9.84191	0.352171
$782$	−49.4411	−1.76801
$783$	0	0
$784$	12.4905	0.446089
$785$	−41.3915	−1.47732
$786$	0	0
$787$	−43.0866	−1.53587	−0.767936	−	0.640527i	$-0.778716\pi$
$787$	−43.0866	−1.53587	−0.767936	+	0.640527i	$0.778716\pi$
$788$	1.46306	0.0521194
$789$	0	0
$790$	−7.58834	−0.269981
$791$	8.49140	0.301919
$792$	0	0
$793$	−42.1579	−1.49707
$794$	41.8297	1.48448
$795$	0	0
$796$	−0.205059	−0.00726812
$797$	−29.2827	−1.03724	−0.518622	−	0.855003i	$-0.673555\pi$
$797$	−29.2827	−1.03724	−0.518622	+	0.855003i	$0.673555\pi$
$798$	0	0
$799$	52.9351	1.87271
$800$	1.11583	0.0394507
$801$	0	0
$802$	−32.1124	−1.13393
$803$	−2.38027	−0.0839980
$804$	0	0
$805$	−38.9807	−1.37389
$806$	60.6080	2.13482
$807$	0	0
$808$	−2.37114	−0.0834163
$809$	34.4478	1.21112	0.605560	−	0.795800i	$-0.292949\pi$
$809$	34.4478	1.21112	0.605560	+	0.795800i	$0.292949\pi$
$810$	0	0
$811$	41.0171	1.44031	0.720153	−	0.693815i	$-0.244072\pi$
$811$	41.0171	1.44031	0.720153	+	0.693815i	$0.244072\pi$
$812$	−0.730879	−0.0256488
$813$	0	0
$814$	9.95355	0.348872
$815$	5.31144	0.186052
$816$	0	0
$817$	−8.07470	−0.282498
$818$	−9.86414	−0.344891
$819$	0	0
$820$	0.191442	0.00668545
$821$	−14.6696	−0.511973	−0.255987	−	0.966680i	$-0.582400\pi$
$821$	−14.6696	−0.511973	−0.255987	+	0.966680i	$0.582400\pi$
$822$	0	0
$823$	8.16556	0.284634	0.142317	−	0.989821i	$-0.454545\pi$
$823$	8.16556	0.284634	0.142317	+	0.989821i	$0.454545\pi$
$824$	7.26151	0.252967
$825$	0	0
$826$	−26.7888	−0.932100
$827$	21.7995	0.758043	0.379021	−	0.925388i	$-0.376261\pi$
$827$	21.7995	0.758043	0.379021	+	0.925388i	$0.376261\pi$
$828$	0	0
$829$	−13.3293	−0.462944	−0.231472	−	0.972842i	$-0.574354\pi$
$829$	−13.3293	−0.462944	−0.231472	+	0.972842i	$0.574354\pi$
$830$	10.9508	0.380107
$831$	0	0
$832$	−33.4073	−1.15819
$833$	13.4631	0.466470
$834$	0	0
$835$	−18.1270	−0.627311
$836$	0.114592	0.00396324
$837$	0	0
$838$	27.0853	0.935648
$839$	−24.2904	−0.838598	−0.419299	−	0.907848i	$-0.637724\pi$
$839$	−24.2904	−0.838598	−0.419299	+	0.907848i	$0.637724\pi$
$840$	0	0
$841$	−18.9241	−0.652557
$842$	2.40462	0.0828687
$843$	0	0
$844$	1.21452	0.0418056
$845$	−17.8669	−0.614641
$846$	0	0
$847$	−2.00933	−0.0690413
$848$	−23.3334	−0.801272
$849$	0	0
$850$	11.3889	0.390635
$851$	51.2115	1.75551
$852$	0	0
$853$	−4.18601	−0.143326	−0.0716631	−	0.997429i	$-0.522831\pi$
$853$	−4.18601	−0.143326	−0.0716631	+	0.997429i	$0.522831\pi$
$854$	27.6197	0.945127
$855$	0	0
$856$	0.494552	0.0169035
$857$	14.1228	0.482425	0.241212	−	0.970472i	$-0.422455\pi$
$857$	14.1228	0.482425	0.241212	+	0.970472i	$0.422455\pi$
$858$	0	0
$859$	24.1260	0.823169	0.411584	−	0.911372i	$-0.364976\pi$
$859$	24.1260	0.823169	0.411584	+	0.911372i	$0.364976\pi$
$860$	2.39925	0.0818139
$861$	0	0
$862$	−32.1192	−1.09398
$863$	28.2661	0.962188	0.481094	−	0.876669i	$-0.340240\pi$
$863$	28.2661	0.962188	0.481094	+	0.876669i	$0.340240\pi$
$864$	0	0
$865$	−50.1759	−1.70603
$866$	25.5766	0.869127
$867$	0	0
$868$	−2.15178	−0.0730361
$869$	−2.01251	−0.0682697
$870$	0	0
$871$	−0.506352	−0.0171571
$872$	18.1165	0.613502
$873$	0	0
$874$	10.8797	0.368011
$875$	−17.0712	−0.577113
$876$	0	0
$877$	−28.4762	−0.961573	−0.480787	−	0.876838i	$-0.659649\pi$
$877$	−28.4762	−0.961573	−0.480787	+	0.876838i	$0.659649\pi$
$878$	−13.7674	−0.464626
$879$	0	0
$880$	10.9321	0.368519
$881$	−13.0669	−0.440234	−0.220117	−	0.975473i	$-0.570644\pi$
$881$	−13.0669	−0.440234	−0.220117	+	0.975473i	$0.570644\pi$
$882$	0	0
$883$	15.9805	0.537786	0.268893	−	0.963170i	$-0.413342\pi$
$883$	15.9805	0.537786	0.268893	+	0.963170i	$0.413342\pi$
$884$	2.32247	0.0781130
$885$	0	0
$886$	−20.4249	−0.686189
$887$	7.47191	0.250882	0.125441	−	0.992101i	$-0.459965\pi$
$887$	7.47191	0.250882	0.125441	+	0.992101i	$0.459965\pi$
$888$	0	0
$889$	−28.2929	−0.948914
$890$	33.2577	1.11480
$891$	0	0
$892$	−0.369377	−0.0123677
$893$	−11.6485	−0.389803
$894$	0	0
$895$	−59.3249	−1.98301
$896$	24.4886	0.818107
$897$	0	0
$898$	27.4133	0.914792
$899$	29.6643	0.989359
$900$	0	0
$901$	−25.1504	−0.837880
$902$	0.936917	0.0311959
$903$	0	0
$904$	−11.5864	−0.385357
$905$	3.46294	0.115112
$906$	0	0
$907$	48.3649	1.60593	0.802966	−	0.596025i	$-0.203254\pi$
$907$	48.3649	1.60593	0.802966	+	0.596025i	$0.203254\pi$
$908$	2.19376	0.0728025
$909$	0	0
$910$	33.7896	1.12011
$911$	−33.2104	−1.10031	−0.550154	−	0.835063i	$-0.685431\pi$
$911$	−33.2104	−1.10031	−0.550154	+	0.835063i	$0.685431\pi$
$912$	0	0
$913$	2.90426	0.0961170
$914$	−32.2703	−1.06741
$915$	0	0
$916$	−3.37688	−0.111575
$917$	20.3137	0.670818
$918$	0	0
$919$	55.5155	1.83129	0.915644	−	0.401990i	$-0.131681\pi$
$919$	55.5155	1.83129	0.915644	+	0.401990i	$0.131681\pi$
$920$	53.1884	1.75357
$921$	0	0
$922$	42.2732	1.39219
$923$	−43.8938	−1.44478
$924$	0	0
$925$	−11.7967	−0.387873
$926$	−30.7140	−1.00932
$927$	0	0
$928$	2.05516	0.0674638
$929$	19.2731	0.632330	0.316165	−	0.948704i	$-0.397605\pi$
$929$	19.2731	0.632330	0.316165	+	0.948704i	$0.397605\pi$
$930$	0	0
$931$	−2.96260	−0.0970954
$932$	−2.23675	−0.0732670
$933$	0	0
$934$	−9.27367	−0.303444
$935$	11.7833	0.385356
$936$	0	0
$937$	33.0983	1.08127	0.540637	−	0.841256i	$-0.318183\pi$
$937$	33.0983	1.08127	0.540637	+	0.841256i	$0.318183\pi$
$938$	0.331736	0.0108316
$939$	0	0
$940$	3.46115	0.112890
$941$	6.91821	0.225527	0.112764	−	0.993622i	$-0.464030\pi$
$941$	6.91821	0.225527	0.112764	+	0.993622i	$0.464030\pi$
$942$	0	0
$943$	4.82048	0.156977
$944$	38.6540	1.25808
$945$	0	0
$946$	11.7419	0.381763
$947$	−47.4763	−1.54277	−0.771386	−	0.636367i	$-0.780436\pi$
$947$	−47.4763	−1.54277	−0.771386	+	0.636367i	$0.780436\pi$
$948$	0	0
$949$	10.6157	0.344601
$950$	−2.50616	−0.0813106
$951$	0	0
$952$	25.0347	0.811378
$953$	12.5758	0.407370	0.203685	−	0.979036i	$-0.434708\pi$
$953$	12.5758	0.407370	0.203685	+	0.979036i	$0.434708\pi$
$954$	0	0
$955$	1.79163	0.0579758
$956$	−1.27646	−0.0412837
$957$	0	0
$958$	42.7280	1.38048
$959$	−5.76136	−0.186044
$960$	0	0
$961$	56.3345	1.81724
$962$	−44.3917	−1.43125
$963$	0	0
$964$	−1.91922	−0.0618139
$965$	50.1456	1.61424
$966$	0	0
$967$	−12.1925	−0.392084	−0.196042	−	0.980595i	$-0.562809\pi$
$967$	−12.1925	−0.392084	−0.196042	+	0.980595i	$0.562809\pi$
$968$	2.74169	0.0881213
$969$	0	0
$970$	41.5142	1.33294
$971$	15.2425	0.489155	0.244577	−	0.969630i	$-0.421351\pi$
$971$	15.2425	0.489155	0.244577	+	0.969630i	$0.421351\pi$
$972$	0	0
$973$	42.8085	1.37238
$974$	49.9542	1.60064
$975$	0	0
$976$	−39.8530	−1.27566
$977$	17.7548	0.568027	0.284013	−	0.958820i	$-0.408334\pi$
$977$	17.7548	0.568027	0.284013	+	0.958820i	$0.408334\pi$
$978$	0	0
$979$	8.82031	0.281898
$980$	0.880285	0.0281197
$981$	0	0
$982$	33.4636	1.06787
$983$	41.1104	1.31122	0.655609	−	0.755100i	$-0.272412\pi$
$983$	41.1104	1.31122	0.655609	+	0.755100i	$0.272412\pi$
$984$	0	0
$985$	−33.1058	−1.05484
$986$	20.9762	0.668018
$987$	0	0
$988$	−0.511066	−0.0162592
$989$	60.4129	1.92102
$990$	0	0
$991$	−18.6736	−0.593185	−0.296593	−	0.955004i	$-0.595850\pi$
$991$	−18.6736	−0.593185	−0.296593	+	0.955004i	$0.595850\pi$
$992$	6.05057	0.192106
$993$	0	0
$994$	28.7570	0.912116
$995$	4.64002	0.147099
$996$	0	0
$997$	−51.5507	−1.63263	−0.816313	−	0.577609i	$-0.803986\pi$
$997$	−51.5507	−1.63263	−0.816313	+	0.577609i	$0.803986\pi$
$998$	27.6106	0.873999
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.45416 q^{2} +0.114592 q^{4} -2.59296 q^{5} -2.00933 q^{7} +2.74169 q^{8} +3.77059 q^{10} +1.00000 q^{11} -4.45988 q^{13} +2.92189 q^{14} -4.21605 q^{16} -4.54435 q^{17} +1.00000 q^{19} -0.297132 q^{20} -1.45416 q^{22} -7.48175 q^{23} +1.72344 q^{25} +6.48540 q^{26} -0.230253 q^{28} +3.17425 q^{29} +9.34529 q^{31} +0.647446 q^{32} +6.60824 q^{34} +5.21010 q^{35} -6.84486 q^{37} -1.45416 q^{38} -7.10910 q^{40} -0.644299 q^{41} -8.07470 q^{43} +0.114592 q^{44} +10.8797 q^{46} -11.6485 q^{47} -2.96260 q^{49} -2.50616 q^{50} -0.511066 q^{52} +5.53442 q^{53} -2.59296 q^{55} -5.50896 q^{56} -4.61588 q^{58} -9.16829 q^{59} +9.45268 q^{61} -13.5896 q^{62} +7.49061 q^{64} +11.5643 q^{65} +0.113535 q^{67} -0.520746 q^{68} -7.57634 q^{70} +9.84191 q^{71} -2.38027 q^{73} +9.95355 q^{74} +0.114592 q^{76} -2.00933 q^{77} -2.01251 q^{79} +10.9321 q^{80} +0.936917 q^{82} +2.90426 q^{83} +11.7833 q^{85} +11.7419 q^{86} +2.74169 q^{88} +8.82031 q^{89} +8.96136 q^{91} -0.857347 q^{92} +16.9389 q^{94} -2.59296 q^{95} +11.0100 q^{97} +4.30811 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 15 q^{4} - 2 q^{5} + 10 q^{7} + 9 q^{8} - 6 q^{10} + 7 q^{11} - 4 q^{13} - 6 q^{14} + 27 q^{16} - 2 q^{17} + 7 q^{19} + 4 q^{20} + q^{22} - 10 q^{23} + 9 q^{25} + 8 q^{26} + 26 q^{28} + 18 q^{29}+ \cdots - 19 q^{98}+O(q^{100}) \) 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

Properties

Related objects

Downloads

Learn more