LMFDB - Embedded newform 8910.2.a.cd.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8910,2,Mod(1,8910)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8910, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8910.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8910 = 2 \cdot 3^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8910.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,7,0,7,7,0,5,7,0,7,7,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$71.1467082010$
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} - 3x^{6} - 13x^{5} + 31x^{4} + 21x^{3} - 37x^{2} - 15x + 6 $ x^7 - 3x^6 - 13x^5 + 31x^4 + 21x^3 - 37x^2 - 15x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 990)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.10401$ of defining polynomial
Character	$\chi$	$=$	8910.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.17324 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +4.48444 q^{13} +2.17324 q^{14} +1.00000 q^{16} +0.0509592 q^{17} +2.99191 q^{19} +1.00000 q^{20} +1.00000 q^{22} +6.53625 q^{23} +1.00000 q^{25} +4.48444 q^{26} +2.17324 q^{28} -1.16515 q^{29} -6.71607 q^{31} +1.00000 q^{32} +0.0509592 q^{34} +2.17324 q^{35} +8.52258 q^{37} +2.99191 q^{38} +1.00000 q^{40} -9.02069 q^{41} +4.22169 q^{43} +1.00000 q^{44} +6.53625 q^{46} +6.66762 q^{47} -2.27701 q^{49} +1.00000 q^{50} +4.48444 q^{52} -2.99527 q^{53} +1.00000 q^{55} +2.17324 q^{56} -1.16515 q^{58} -5.20217 q^{59} -0.0438644 q^{61} -6.71607 q^{62} +1.00000 q^{64} +4.48444 q^{65} -11.5425 q^{67} +0.0509592 q^{68} +2.17324 q^{70} +6.04471 q^{71} +15.8864 q^{73} +8.52258 q^{74} +2.99191 q^{76} +2.17324 q^{77} +4.69956 q^{79} +1.00000 q^{80} -9.02069 q^{82} +1.50162 q^{83} +0.0509592 q^{85} +4.22169 q^{86} +1.00000 q^{88} -16.3756 q^{89} +9.74578 q^{91} +6.53625 q^{92} +6.66762 q^{94} +2.99191 q^{95} +7.13655 q^{97} -2.27701 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + 5 q^{7} + 7 q^{8} + 7 q^{10} + 7 q^{11} + 5 q^{13} + 5 q^{14} + 7 q^{16} + 3 q^{17} + 11 q^{19} + 7 q^{20} + 7 q^{22} + 3 q^{23} + 7 q^{25} + 5 q^{26} + 5 q^{28} + 12 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) $ 7 * q + 7 * q^2 + 7 * q^4 + 7 * q^5 + 5 * q^7 + 7 * q^8 + 7 * q^10 + 7 * q^11 + 5 * q^13 + 5 * q^14 + 7 * q^16 + 3 * q^17 + 11 * q^19 + 7 * q^20 + 7 * q^22 + 3 * q^23 + 7 * q^25 + 5 * q^26 + 5 * q^28 + 12 * q^29 + 5 * q^31 + 7 * q^32 + 3 * q^34 + 5 * q^35 + 2 * q^37 + 11 * q^38 + 7 * q^40 + 6 * q^41 + 17 * q^43 + 7 * q^44 + 3 * q^46 - 3 * q^47 + 12 * q^49 + 7 * q^50 + 5 * q^52 + 3 * q^53 + 7 * q^55 + 5 * q^56 + 12 * q^58 + 18 * q^59 + 11 * q^61 + 5 * q^62 + 7 * q^64 + 5 * q^65 + 20 * q^67 + 3 * q^68 + 5 * q^70 + 12 * q^71 + 2 * q^73 + 2 * q^74 + 11 * q^76 + 5 * q^77 - 7 * q^79 + 7 * q^80 + 6 * q^82 + 3 * q^85 + 17 * q^86 + 7 * q^88 - 3 * q^89 + 7 * q^91 + 3 * q^92 - 3 * q^94 + 11 * q^95 + 23 * q^97 + 12 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.17324	0.821409	0.410705	−	0.911768i	$-0.365283\pi$
$7$	2.17324	0.821409	0.410705	+	0.911768i	$0.365283\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.48444	1.24376	0.621880	−	0.783113i	$-0.286369\pi$
$13$	4.48444	1.24376	0.621880	+	0.783113i	$0.286369\pi$
$14$	2.17324	0.580824
$15$	0	0
$16$	1.00000	0.250000
$17$	0.0509592	0.0123594	0.00617971	−	0.999981i	$-0.498033\pi$
$17$	0.0509592	0.0123594	0.00617971	+	0.999981i	$0.498033\pi$
$18$	0	0
$19$	2.99191	0.686391	0.343195	−	0.939264i	$-0.388491\pi$
$19$	2.99191	0.686391	0.343195	+	0.939264i	$0.388491\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	6.53625	1.36290	0.681451	−	0.731864i	$-0.261349\pi$
$23$	6.53625	1.36290	0.681451	+	0.731864i	$0.261349\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.48444	0.879471
$27$	0	0
$28$	2.17324	0.410705
$29$	−1.16515	−0.216364	−0.108182	−	0.994131i	$-0.534503\pi$
$29$	−1.16515	−0.216364	−0.108182	+	0.994131i	$0.534503\pi$
$30$	0	0
$31$	−6.71607	−1.20624	−0.603121	−	0.797650i	$-0.706076\pi$
$31$	−6.71607	−1.20624	−0.603121	+	0.797650i	$0.706076\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.0509592	0.00873943
$35$	2.17324	0.367345
$36$	0	0
$37$	8.52258	1.40110	0.700551	−	0.713602i	$-0.252937\pi$
$37$	8.52258	1.40110	0.700551	+	0.713602i	$0.252937\pi$
$38$	2.99191	0.485352
$39$	0	0
$40$	1.00000	0.158114
$41$	−9.02069	−1.40879	−0.704397	−	0.709806i	$-0.748783\pi$
$41$	−9.02069	−1.40879	−0.704397	+	0.709806i	$0.748783\pi$
$42$	0	0
$43$	4.22169	0.643802	0.321901	−	0.946773i	$-0.395678\pi$
$43$	4.22169	0.643802	0.321901	+	0.946773i	$0.395678\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	6.53625	0.963717
$47$	6.66762	0.972572	0.486286	−	0.873800i	$-0.338351\pi$
$47$	6.66762	0.972572	0.486286	+	0.873800i	$0.338351\pi$
$48$	0	0
$49$	−2.27701	−0.325287
$50$	1.00000	0.141421
$51$	0	0
$52$	4.48444	0.621880
$53$	−2.99527	−0.411432	−0.205716	−	0.978612i	$-0.565952\pi$
$53$	−2.99527	−0.411432	−0.205716	+	0.978612i	$0.565952\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	2.17324	0.290412
$57$	0	0
$58$	−1.16515	−0.152992
$59$	−5.20217	−0.677265	−0.338632	−	0.940919i	$-0.609964\pi$
$59$	−5.20217	−0.677265	−0.338632	+	0.940919i	$0.609964\pi$
$60$	0	0
$61$	−0.0438644	−0.00561627	−0.00280813	−	0.999996i	$-0.500894\pi$
$61$	−0.0438644	−0.00561627	−0.00280813	+	0.999996i	$0.500894\pi$
$62$	−6.71607	−0.852942
$63$	0	0
$64$	1.00000	0.125000
$65$	4.48444	0.556226
$66$	0	0
$67$	−11.5425	−1.41015	−0.705073	−	0.709135i	$-0.749086\pi$
$67$	−11.5425	−1.41015	−0.705073	+	0.709135i	$0.749086\pi$
$68$	0.0509592	0.00617971
$69$	0	0
$70$	2.17324	0.259752
$71$	6.04471	0.717376	0.358688	−	0.933458i	$-0.383224\pi$
$71$	6.04471	0.717376	0.358688	+	0.933458i	$0.383224\pi$
$72$	0	0
$73$	15.8864	1.85937	0.929683	−	0.368361i	$-0.120081\pi$
$73$	15.8864	1.85937	0.929683	+	0.368361i	$0.120081\pi$
$74$	8.52258	0.990729
$75$	0	0
$76$	2.99191	0.343195
$77$	2.17324	0.247664
$78$	0	0
$79$	4.69956	0.528741	0.264371	−	0.964421i	$-0.414836\pi$
$79$	4.69956	0.528741	0.264371	+	0.964421i	$0.414836\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−9.02069	−0.996168
$83$	1.50162	0.164824	0.0824120	−	0.996598i	$-0.473738\pi$
$83$	1.50162	0.164824	0.0824120	+	0.996598i	$0.473738\pi$
$84$	0	0
$85$	0.0509592	0.00552730
$86$	4.22169	0.455237
$87$	0	0
$88$	1.00000	0.106600
$89$	−16.3756	−1.73581	−0.867905	−	0.496730i	$-0.834534\pi$
$89$	−16.3756	−1.73581	−0.867905	+	0.496730i	$0.834534\pi$
$90$	0	0
$91$	9.74578	1.02164
$92$	6.53625	0.681451
$93$	0	0
$94$	6.66762	0.687713
$95$	2.99191	0.306963
$96$	0	0
$97$	7.13655	0.724607	0.362303	−	0.932060i	$-0.381990\pi$
$97$	7.13655	0.724607	0.362303	+	0.932060i	$0.381990\pi$
$98$	−2.27701	−0.230013
$99$	0	0
$100$	1.00000	0.100000
$101$	−17.5693	−1.74822	−0.874108	−	0.485732i	$-0.838553\pi$
$101$	−17.5693	−1.74822	−0.874108	+	0.485732i	$0.838553\pi$
$102$	0	0
$103$	−18.6707	−1.83968	−0.919840	−	0.392293i	$-0.871682\pi$
$103$	−18.6707	−1.83968	−0.919840	+	0.392293i	$0.871682\pi$
$104$	4.48444	0.439735
$105$	0	0
$106$	−2.99527	−0.290926
$107$	−8.99123	−0.869215	−0.434608	−	0.900620i	$-0.643113\pi$
$107$	−8.99123	−0.869215	−0.434608	+	0.900620i	$0.643113\pi$
$108$	0	0
$109$	5.02945	0.481734	0.240867	−	0.970558i	$-0.422568\pi$
$109$	5.02945	0.481734	0.240867	+	0.970558i	$0.422568\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	2.17324	0.205352
$113$	10.0035	0.941051	0.470525	−	0.882386i	$-0.344064\pi$
$113$	10.0035	0.941051	0.470525	+	0.882386i	$0.344064\pi$
$114$	0	0
$115$	6.53625	0.609508
$116$	−1.16515	−0.108182
$117$	0	0
$118$	−5.20217	−0.478898
$119$	0.110747	0.0101521
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.0438644	−0.00397130
$123$	0	0
$124$	−6.71607	−0.603121
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.56837	0.139170	0.0695850	−	0.997576i	$-0.477833\pi$
$127$	1.56837	0.139170	0.0695850	+	0.997576i	$0.477833\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.48444	0.393311
$131$	−6.52716	−0.570281	−0.285140	−	0.958486i	$-0.592040\pi$
$131$	−6.52716	−0.570281	−0.285140	+	0.958486i	$0.592040\pi$
$132$	0	0
$133$	6.50215	0.563808
$134$	−11.5425	−0.997123
$135$	0	0
$136$	0.0509592	0.00436972
$137$	13.9447	1.19138	0.595689	−	0.803215i	$-0.296879\pi$
$137$	13.9447	1.19138	0.595689	+	0.803215i	$0.296879\pi$
$138$	0	0
$139$	−15.0553	−1.27698	−0.638488	−	0.769632i	$-0.720440\pi$
$139$	−15.0553	−1.27698	−0.638488	+	0.769632i	$0.720440\pi$
$140$	2.17324	0.183673
$141$	0	0
$142$	6.04471	0.507261
$143$	4.48444	0.375008
$144$	0	0
$145$	−1.16515	−0.0967607
$146$	15.8864	1.31477
$147$	0	0
$148$	8.52258	0.700551
$149$	2.34416	0.192041	0.0960206	−	0.995379i	$-0.469389\pi$
$149$	2.34416	0.192041	0.0960206	+	0.995379i	$0.469389\pi$
$150$	0	0
$151$	−21.2074	−1.72583	−0.862916	−	0.505348i	$-0.831364\pi$
$151$	−21.2074	−1.72583	−0.862916	+	0.505348i	$0.831364\pi$
$152$	2.99191	0.242676
$153$	0	0
$154$	2.17324	0.175125
$155$	−6.71607	−0.539448
$156$	0	0
$157$	8.50606	0.678858	0.339429	−	0.940632i	$-0.389766\pi$
$157$	8.50606	0.678858	0.339429	+	0.940632i	$0.389766\pi$
$158$	4.69956	0.373877
$159$	0	0
$160$	1.00000	0.0790569
$161$	14.2049	1.11950
$162$	0	0
$163$	24.1125	1.88863	0.944317	−	0.329036i	$-0.106724\pi$
$163$	24.1125	1.88863	0.944317	+	0.329036i	$0.106724\pi$
$164$	−9.02069	−0.704397
$165$	0	0
$166$	1.50162	0.116548
$167$	−11.1014	−0.859050	−0.429525	−	0.903055i	$-0.641319\pi$
$167$	−11.1014	−0.859050	−0.429525	+	0.903055i	$0.641319\pi$
$168$	0	0
$169$	7.11019	0.546938
$170$	0.0509592	0.00390839
$171$	0	0
$172$	4.22169	0.321901
$173$	−8.85369	−0.673133	−0.336567	−	0.941660i	$-0.609266\pi$
$173$	−8.85369	−0.673133	−0.336567	+	0.941660i	$0.609266\pi$
$174$	0	0
$175$	2.17324	0.164282
$176$	1.00000	0.0753778
$177$	0	0
$178$	−16.3756	−1.22740
$179$	1.94083	0.145064	0.0725321	−	0.997366i	$-0.476892\pi$
$179$	1.94083	0.145064	0.0725321	+	0.997366i	$0.476892\pi$
$180$	0	0
$181$	−15.6795	−1.16545	−0.582724	−	0.812670i	$-0.698013\pi$
$181$	−15.6795	−1.16545	−0.582724	+	0.812670i	$0.698013\pi$
$182$	9.74578	0.722405
$183$	0	0
$184$	6.53625	0.481859
$185$	8.52258	0.626592
$186$	0	0
$187$	0.0509592	0.00372651
$188$	6.66762	0.486286
$189$	0	0
$190$	2.99191	0.217056
$191$	13.7569	0.995418	0.497709	−	0.867344i	$-0.334175\pi$
$191$	13.7569	0.995418	0.497709	+	0.867344i	$0.334175\pi$
$192$	0	0
$193$	−1.77539	−0.127795	−0.0638977	−	0.997956i	$-0.520353\pi$
$193$	−1.77539	−0.127795	−0.0638977	+	0.997956i	$0.520353\pi$
$194$	7.13655	0.512374
$195$	0	0
$196$	−2.27701	−0.162643
$197$	23.7919	1.69510	0.847552	−	0.530712i	$-0.178075\pi$
$197$	23.7919	1.69510	0.847552	+	0.530712i	$0.178075\pi$
$198$	0	0
$199$	2.15809	0.152983	0.0764916	−	0.997070i	$-0.475628\pi$
$199$	2.15809	0.152983	0.0764916	+	0.997070i	$0.475628\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−17.5693	−1.23617
$203$	−2.53216	−0.177723
$204$	0	0
$205$	−9.02069	−0.630032
$206$	−18.6707	−1.30085
$207$	0	0
$208$	4.48444	0.310940
$209$	2.99191	0.206955
$210$	0	0
$211$	22.1373	1.52400	0.761998	−	0.647580i	$-0.224219\pi$
$211$	22.1373	1.52400	0.761998	+	0.647580i	$0.224219\pi$
$212$	−2.99527	−0.205716
$213$	0	0
$214$	−8.99123	−0.614628
$215$	4.22169	0.287917
$216$	0	0
$217$	−14.5957	−0.990818
$218$	5.02945	0.340638
$219$	0	0
$220$	1.00000	0.0674200
$221$	0.228523	0.0153721
$222$	0	0
$223$	−8.75047	−0.585975	−0.292987	−	0.956116i	$-0.594649\pi$
$223$	−8.75047	−0.585975	−0.292987	+	0.956116i	$0.594649\pi$
$224$	2.17324	0.145206
$225$	0	0
$226$	10.0035	0.665423
$227$	−5.93425	−0.393870	−0.196935	−	0.980417i	$-0.563099\pi$
$227$	−5.93425	−0.393870	−0.196935	+	0.980417i	$0.563099\pi$
$228$	0	0
$229$	1.12065	0.0740549	0.0370275	−	0.999314i	$-0.488211\pi$
$229$	1.12065	0.0740549	0.0370275	+	0.999314i	$0.488211\pi$
$230$	6.53625	0.430987
$231$	0	0
$232$	−1.16515	−0.0764961
$233$	−18.6243	−1.22012	−0.610060	−	0.792356i	$-0.708855\pi$
$233$	−18.6243	−1.22012	−0.610060	+	0.792356i	$0.708855\pi$
$234$	0	0
$235$	6.66762	0.434948
$236$	−5.20217	−0.338632
$237$	0	0
$238$	0.110747	0.00717865
$239$	7.19057	0.465119	0.232560	−	0.972582i	$-0.425290\pi$
$239$	7.19057	0.465119	0.232560	+	0.972582i	$0.425290\pi$
$240$	0	0
$241$	−1.40835	−0.0907199	−0.0453600	−	0.998971i	$-0.514443\pi$
$241$	−1.40835	−0.0907199	−0.0453600	+	0.998971i	$0.514443\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−0.0438644	−0.00280813
$245$	−2.27701	−0.145473
$246$	0	0
$247$	13.4170	0.853705
$248$	−6.71607	−0.426471
$249$	0	0
$250$	1.00000	0.0632456
$251$	7.90649	0.499053	0.249527	−	0.968368i	$-0.419725\pi$
$251$	7.90649	0.499053	0.249527	+	0.968368i	$0.419725\pi$
$252$	0	0
$253$	6.53625	0.410930
$254$	1.56837	0.0984080
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.5444	−1.71818	−0.859088	−	0.511829i	$-0.828968\pi$
$257$	−27.5444	−1.71818	−0.859088	+	0.511829i	$0.828968\pi$
$258$	0	0
$259$	18.5216	1.15088
$260$	4.48444	0.278113
$261$	0	0
$262$	−6.52716	−0.403249
$263$	−0.311983	−0.0192377	−0.00961885	−	0.999954i	$-0.503062\pi$
$263$	−0.311983	−0.0192377	−0.00961885	+	0.999954i	$0.503062\pi$
$264$	0	0
$265$	−2.99527	−0.183998
$266$	6.50215	0.398672
$267$	0	0
$268$	−11.5425	−0.705073
$269$	16.7369	1.02047	0.510233	−	0.860036i	$-0.329559\pi$
$269$	16.7369	1.02047	0.510233	+	0.860036i	$0.329559\pi$
$270$	0	0
$271$	22.7888	1.38432	0.692161	−	0.721744i	$-0.256659\pi$
$271$	22.7888	1.38432	0.692161	+	0.721744i	$0.256659\pi$
$272$	0.0509592	0.00308986
$273$	0	0
$274$	13.9447	0.842431
$275$	1.00000	0.0603023
$276$	0	0
$277$	8.95288	0.537926	0.268963	−	0.963150i	$-0.413319\pi$
$277$	8.95288	0.537926	0.268963	+	0.963150i	$0.413319\pi$
$278$	−15.0553	−0.902958
$279$	0	0
$280$	2.17324	0.129876
$281$	−8.40687	−0.501512	−0.250756	−	0.968050i	$-0.580679\pi$
$281$	−8.40687	−0.501512	−0.250756	+	0.968050i	$0.580679\pi$
$282$	0	0
$283$	14.0918	0.837670	0.418835	−	0.908062i	$-0.362439\pi$
$283$	14.0918	0.837670	0.418835	+	0.908062i	$0.362439\pi$
$284$	6.04471	0.358688
$285$	0	0
$286$	4.48444	0.265170
$287$	−19.6042	−1.15720
$288$	0	0
$289$	−16.9974	−0.999847
$290$	−1.16515	−0.0684202
$291$	0	0
$292$	15.8864	0.929683
$293$	−14.1511	−0.826714	−0.413357	−	0.910569i	$-0.635644\pi$
$293$	−14.1511	−0.826714	−0.413357	+	0.910569i	$0.635644\pi$
$294$	0	0
$295$	−5.20217	−0.302882
$296$	8.52258	0.495365
$297$	0	0
$298$	2.34416	0.135794
$299$	29.3114	1.69512
$300$	0	0
$301$	9.17477	0.528825
$302$	−21.2074	−1.22035
$303$	0	0
$304$	2.99191	0.171598
$305$	−0.0438644	−0.00251167
$306$	0	0
$307$	0.924724	0.0527768	0.0263884	−	0.999652i	$-0.491599\pi$
$307$	0.924724	0.0527768	0.0263884	+	0.999652i	$0.491599\pi$
$308$	2.17324	0.123832
$309$	0	0
$310$	−6.71607	−0.381447
$311$	−3.62726	−0.205683	−0.102842	−	0.994698i	$-0.532793\pi$
$311$	−3.62726	−0.205683	−0.102842	+	0.994698i	$0.532793\pi$
$312$	0	0
$313$	21.8892	1.23725	0.618626	−	0.785685i	$-0.287689\pi$
$313$	21.8892	1.23725	0.618626	+	0.785685i	$0.287689\pi$
$314$	8.50606	0.480025
$315$	0	0
$316$	4.69956	0.264371
$317$	−19.4322	−1.09142	−0.545710	−	0.837974i	$-0.683740\pi$
$317$	−19.4322	−1.09142	−0.545710	+	0.837974i	$0.683740\pi$
$318$	0	0
$319$	−1.16515	−0.0652361
$320$	1.00000	0.0559017
$321$	0	0
$322$	14.2049	0.791606
$323$	0.152465	0.00848340
$324$	0	0
$325$	4.48444	0.248752
$326$	24.1125	1.33547
$327$	0	0
$328$	−9.02069	−0.498084
$329$	14.4904	0.798880
$330$	0	0
$331$	5.47362	0.300857	0.150429	−	0.988621i	$-0.451935\pi$
$331$	5.47362	0.300857	0.150429	+	0.988621i	$0.451935\pi$
$332$	1.50162	0.0824120
$333$	0	0
$334$	−11.1014	−0.607440
$335$	−11.5425	−0.630636
$336$	0	0
$337$	25.7022	1.40009	0.700043	−	0.714101i	$-0.253164\pi$
$337$	25.7022	1.40009	0.700043	+	0.714101i	$0.253164\pi$
$338$	7.11019	0.386744
$339$	0	0
$340$	0.0509592	0.00276365
$341$	−6.71607	−0.363696
$342$	0	0
$343$	−20.1612	−1.08860
$344$	4.22169	0.227618
$345$	0	0
$346$	−8.85369	−0.475977
$347$	30.8475	1.65598	0.827992	−	0.560740i	$-0.189483\pi$
$347$	30.8475	1.65598	0.827992	+	0.560740i	$0.189483\pi$
$348$	0	0
$349$	11.0961	0.593962	0.296981	−	0.954883i	$-0.404020\pi$
$349$	11.0961	0.593962	0.296981	+	0.954883i	$0.404020\pi$
$350$	2.17324	0.116165
$351$	0	0
$352$	1.00000	0.0533002
$353$	13.4324	0.714934	0.357467	−	0.933926i	$-0.383640\pi$
$353$	13.4324	0.714934	0.357467	+	0.933926i	$0.383640\pi$
$354$	0	0
$355$	6.04471	0.320820
$356$	−16.3756	−0.867905
$357$	0	0
$358$	1.94083	0.102576
$359$	−11.2447	−0.593474	−0.296737	−	0.954959i	$-0.595899\pi$
$359$	−11.2447	−0.593474	−0.296737	+	0.954959i	$0.595899\pi$
$360$	0	0
$361$	−10.0485	−0.528867
$362$	−15.6795	−0.824096
$363$	0	0
$364$	9.74578	0.510818
$365$	15.8864	0.831534
$366$	0	0
$367$	22.1691	1.15722	0.578608	−	0.815606i	$-0.303596\pi$
$367$	22.1691	1.15722	0.578608	+	0.815606i	$0.303596\pi$
$368$	6.53625	0.340725
$369$	0	0
$370$	8.52258	0.443068
$371$	−6.50945	−0.337954
$372$	0	0
$373$	−21.5923	−1.11801	−0.559003	−	0.829166i	$-0.688816\pi$
$373$	−21.5923	−1.11801	−0.559003	+	0.829166i	$0.688816\pi$
$374$	0.0509592	0.00263504
$375$	0	0
$376$	6.66762	0.343856
$377$	−5.22506	−0.269104
$378$	0	0
$379$	−11.2563	−0.578199	−0.289099	−	0.957299i	$-0.593356\pi$
$379$	−11.2563	−0.578199	−0.289099	+	0.957299i	$0.593356\pi$
$380$	2.99191	0.153482
$381$	0	0
$382$	13.7569	0.703867
$383$	0.547464	0.0279741	0.0139870	−	0.999902i	$-0.495548\pi$
$383$	0.547464	0.0279741	0.0139870	+	0.999902i	$0.495548\pi$
$384$	0	0
$385$	2.17324	0.110759
$386$	−1.77539	−0.0903650
$387$	0	0
$388$	7.13655	0.362303
$389$	−22.2724	−1.12926	−0.564628	−	0.825345i	$-0.690981\pi$
$389$	−22.2724	−1.12926	−0.564628	+	0.825345i	$0.690981\pi$
$390$	0	0
$391$	0.333082	0.0168447
$392$	−2.27701	−0.115006
$393$	0	0
$394$	23.7919	1.19862
$395$	4.69956	0.236460
$396$	0	0
$397$	−13.3297	−0.668999	−0.334500	−	0.942396i	$-0.608567\pi$
$397$	−13.3297	−0.668999	−0.334500	+	0.942396i	$0.608567\pi$
$398$	2.15809	0.108175
$399$	0	0
$400$	1.00000	0.0500000
$401$	−8.07960	−0.403476	−0.201738	−	0.979440i	$-0.564659\pi$
$401$	−8.07960	−0.403476	−0.201738	+	0.979440i	$0.564659\pi$
$402$	0	0
$403$	−30.1178	−1.50027
$404$	−17.5693	−0.874108
$405$	0	0
$406$	−2.53216	−0.125669
$407$	8.52258	0.422448
$408$	0	0
$409$	−8.83292	−0.436760	−0.218380	−	0.975864i	$-0.570077\pi$
$409$	−8.83292	−0.436760	−0.218380	+	0.975864i	$0.570077\pi$
$410$	−9.02069	−0.445500
$411$	0	0
$412$	−18.6707	−0.919840
$413$	−11.3056	−0.556312
$414$	0	0
$415$	1.50162	0.0737115
$416$	4.48444	0.219868
$417$	0	0
$418$	2.99191	0.146339
$419$	−5.17941	−0.253031	−0.126515	−	0.991965i	$-0.540379\pi$
$419$	−5.17941	−0.253031	−0.126515	+	0.991965i	$0.540379\pi$
$420$	0	0
$421$	6.64176	0.323699	0.161850	−	0.986815i	$-0.448254\pi$
$421$	6.64176	0.323699	0.161850	+	0.986815i	$0.448254\pi$
$422$	22.1373	1.07763
$423$	0	0
$424$	−2.99527	−0.145463
$425$	0.0509592	0.00247188
$426$	0	0
$427$	−0.0953282	−0.00461325
$428$	−8.99123	−0.434608
$429$	0	0
$430$	4.22169	0.203588
$431$	−29.6036	−1.42595	−0.712977	−	0.701188i	$-0.752654\pi$
$431$	−29.6036	−1.42595	−0.712977	+	0.701188i	$0.752654\pi$
$432$	0	0
$433$	26.1933	1.25877	0.629384	−	0.777094i	$-0.283307\pi$
$433$	26.1933	1.25877	0.629384	+	0.777094i	$0.283307\pi$
$434$	−14.5957	−0.700614
$435$	0	0
$436$	5.02945	0.240867
$437$	19.5559	0.935484
$438$	0	0
$439$	14.5524	0.694550	0.347275	−	0.937763i	$-0.387107\pi$
$439$	14.5524	0.694550	0.347275	+	0.937763i	$0.387107\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	0.228523	0.0108698
$443$	−19.6834	−0.935185	−0.467593	−	0.883944i	$-0.654879\pi$
$443$	−19.6834	−0.935185	−0.467593	+	0.883944i	$0.654879\pi$
$444$	0	0
$445$	−16.3756	−0.776278
$446$	−8.75047	−0.414347
$447$	0	0
$448$	2.17324	0.102676
$449$	24.0348	1.13427	0.567137	−	0.823623i	$-0.308051\pi$
$449$	24.0348	1.13427	0.567137	+	0.823623i	$0.308051\pi$
$450$	0	0
$451$	−9.02069	−0.424768
$452$	10.0035	0.470525
$453$	0	0
$454$	−5.93425	−0.278508
$455$	9.74578	0.456889
$456$	0	0
$457$	−20.3728	−0.953000	−0.476500	−	0.879174i	$-0.658095\pi$
$457$	−20.3728	−0.953000	−0.476500	+	0.879174i	$0.658095\pi$
$458$	1.12065	0.0523648
$459$	0	0
$460$	6.53625	0.304754
$461$	−0.0295634	−0.00137690	−0.000688452	−	1.00000i	$-0.500219\pi$
$461$	−0.0295634	−0.00137690	−0.000688452		1.00000i	$0.500219\pi$
$462$	0	0
$463$	−27.6808	−1.28644	−0.643218	−	0.765683i	$-0.722401\pi$
$463$	−27.6808	−1.28644	−0.643218	+	0.765683i	$0.722401\pi$
$464$	−1.16515	−0.0540909
$465$	0	0
$466$	−18.6243	−0.862755
$467$	−7.38942	−0.341941	−0.170971	−	0.985276i	$-0.554690\pi$
$467$	−7.38942	−0.341941	−0.170971	+	0.985276i	$0.554690\pi$
$468$	0	0
$469$	−25.0848	−1.15831
$470$	6.66762	0.307554
$471$	0	0
$472$	−5.20217	−0.239449
$473$	4.22169	0.194114
$474$	0	0
$475$	2.99191	0.137278
$476$	0.110747	0.00507607
$477$	0	0
$478$	7.19057	0.328889
$479$	22.4611	1.02628	0.513138	−	0.858306i	$-0.328483\pi$
$479$	22.4611	1.02628	0.513138	+	0.858306i	$0.328483\pi$
$480$	0	0
$481$	38.2190	1.74264
$482$	−1.40835	−0.0641487
$483$	0	0
$484$	1.00000	0.0454545
$485$	7.13655	0.324054
$486$	0	0
$487$	33.9578	1.53878	0.769388	−	0.638782i	$-0.220561\pi$
$487$	33.9578	1.53878	0.769388	+	0.638782i	$0.220561\pi$
$488$	−0.0438644	−0.00198565
$489$	0	0
$490$	−2.27701	−0.102865
$491$	−29.6810	−1.33948	−0.669741	−	0.742594i	$-0.733595\pi$
$491$	−29.6810	−1.33948	−0.669741	+	0.742594i	$0.733595\pi$
$492$	0	0
$493$	−0.0593753	−0.00267413
$494$	13.4170	0.603661
$495$	0	0
$496$	−6.71607	−0.301560
$497$	13.1366	0.589259
$498$	0	0
$499$	−9.82210	−0.439697	−0.219849	−	0.975534i	$-0.570556\pi$
$499$	−9.82210	−0.439697	−0.219849	+	0.975534i	$0.570556\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	7.90649	0.352884
$503$	−0.682461	−0.0304294	−0.0152147	−	0.999884i	$-0.504843\pi$
$503$	−0.682461	−0.0304294	−0.0152147	+	0.999884i	$0.504843\pi$
$504$	0	0
$505$	−17.5693	−0.781826
$506$	6.53625	0.290572
$507$	0	0
$508$	1.56837	0.0695850
$509$	−21.7766	−0.965230	−0.482615	−	0.875833i	$-0.660313\pi$
$509$	−21.7766	−0.965230	−0.482615	+	0.875833i	$0.660313\pi$
$510$	0	0
$511$	34.5251	1.52730
$512$	1.00000	0.0441942
$513$	0	0
$514$	−27.5444	−1.21493
$515$	−18.6707	−0.822730
$516$	0	0
$517$	6.66762	0.293242
$518$	18.5216	0.813794
$519$	0	0
$520$	4.48444	0.196656
$521$	38.5129	1.68728	0.843640	−	0.536910i	$-0.180408\pi$
$521$	38.5129	1.68728	0.843640	+	0.536910i	$0.180408\pi$
$522$	0	0
$523$	−2.35549	−0.102998	−0.0514991	−	0.998673i	$-0.516400\pi$
$523$	−2.35549	−0.102998	−0.0514991	+	0.998673i	$0.516400\pi$
$524$	−6.52716	−0.285140
$525$	0	0
$526$	−0.311983	−0.0136031
$527$	−0.342246	−0.0149084
$528$	0	0
$529$	19.7225	0.857501
$530$	−2.99527	−0.130106
$531$	0	0
$532$	6.50215	0.281904
$533$	−40.4527	−1.75220
$534$	0	0
$535$	−8.99123	−0.388725
$536$	−11.5425	−0.498562
$537$	0	0
$538$	16.7369	0.721578
$539$	−2.27701	−0.0980777
$540$	0	0
$541$	−25.7546	−1.10728	−0.553638	−	0.832757i	$-0.686761\pi$
$541$	−25.7546	−1.10728	−0.553638	+	0.832757i	$0.686761\pi$
$542$	22.7888	0.978863
$543$	0	0
$544$	0.0509592	0.00218486
$545$	5.02945	0.215438
$546$	0	0
$547$	36.6960	1.56901	0.784504	−	0.620124i	$-0.212918\pi$
$547$	36.6960	1.56901	0.784504	+	0.620124i	$0.212918\pi$
$548$	13.9447	0.595689
$549$	0	0
$550$	1.00000	0.0426401
$551$	−3.48603	−0.148510
$552$	0	0
$553$	10.2133	0.434313
$554$	8.95288	0.380371
$555$	0	0
$556$	−15.0553	−0.638488
$557$	4.32507	0.183259	0.0916295	−	0.995793i	$-0.470792\pi$
$557$	4.32507	0.183259	0.0916295	+	0.995793i	$0.470792\pi$
$558$	0	0
$559$	18.9319	0.800735
$560$	2.17324	0.0918363
$561$	0	0
$562$	−8.40687	−0.354622
$563$	−28.2678	−1.19135	−0.595674	−	0.803227i	$-0.703115\pi$
$563$	−28.2678	−1.19135	−0.595674	+	0.803227i	$0.703115\pi$
$564$	0	0
$565$	10.0035	0.420851
$566$	14.0918	0.592322
$567$	0	0
$568$	6.04471	0.253631
$569$	−9.81446	−0.411443	−0.205722	−	0.978611i	$-0.565954\pi$
$569$	−9.81446	−0.411443	−0.205722	+	0.978611i	$0.565954\pi$
$570$	0	0
$571$	34.7843	1.45568	0.727838	−	0.685749i	$-0.240525\pi$
$571$	34.7843	1.45568	0.727838	+	0.685749i	$0.240525\pi$
$572$	4.48444	0.187504
$573$	0	0
$574$	−19.6042	−0.818262
$575$	6.53625	0.272580
$576$	0	0
$577$	26.6306	1.10865	0.554323	−	0.832302i	$-0.312977\pi$
$577$	26.6306	1.10865	0.554323	+	0.832302i	$0.312977\pi$
$578$	−16.9974	−0.706999
$579$	0	0
$580$	−1.16515	−0.0483804
$581$	3.26338	0.135388
$582$	0	0
$583$	−2.99527	−0.124051
$584$	15.8864	0.657385
$585$	0	0
$586$	−14.1511	−0.584575
$587$	38.4009	1.58498	0.792488	−	0.609887i	$-0.208785\pi$
$587$	38.4009	1.58498	0.792488	+	0.609887i	$0.208785\pi$
$588$	0	0
$589$	−20.0939	−0.827953
$590$	−5.20217	−0.214170
$591$	0	0
$592$	8.52258	0.350276
$593$	−29.5196	−1.21223	−0.606113	−	0.795378i	$-0.707272\pi$
$593$	−29.5196	−1.21223	−0.606113	+	0.795378i	$0.707272\pi$
$594$	0	0
$595$	0.110747	0.00454018
$596$	2.34416	0.0960206
$597$	0	0
$598$	29.3114	1.19863
$599$	22.7841	0.930934	0.465467	−	0.885065i	$-0.345886\pi$
$599$	22.7841	0.930934	0.465467	+	0.885065i	$0.345886\pi$
$600$	0	0
$601$	19.7176	0.804299	0.402150	−	0.915574i	$-0.368263\pi$
$601$	19.7176	0.804299	0.402150	+	0.915574i	$0.368263\pi$
$602$	9.17477	0.373936
$603$	0	0
$604$	−21.2074	−0.862916
$605$	1.00000	0.0406558
$606$	0	0
$607$	39.2182	1.59182	0.795908	−	0.605418i	$-0.206994\pi$
$607$	39.2182	1.59182	0.795908	+	0.605418i	$0.206994\pi$
$608$	2.99191	0.121338
$609$	0	0
$610$	−0.0438644	−0.00177602
$611$	29.9005	1.20965
$612$	0	0
$613$	21.4815	0.867628	0.433814	−	0.901002i	$-0.357167\pi$
$613$	21.4815	0.867628	0.433814	+	0.901002i	$0.357167\pi$
$614$	0.924724	0.0373188
$615$	0	0
$616$	2.17324	0.0875625
$617$	3.96947	0.159805	0.0799024	−	0.996803i	$-0.474539\pi$
$617$	3.96947	0.159805	0.0799024	+	0.996803i	$0.474539\pi$
$618$	0	0
$619$	−36.3919	−1.46271	−0.731357	−	0.681995i	$-0.761113\pi$
$619$	−36.3919	−1.46271	−0.731357	+	0.681995i	$0.761113\pi$
$620$	−6.71607	−0.269724
$621$	0	0
$622$	−3.62726	−0.145440
$623$	−35.5882	−1.42581
$624$	0	0
$625$	1.00000	0.0400000
$626$	21.8892	0.874870
$627$	0	0
$628$	8.50606	0.339429
$629$	0.434304	0.0173168
$630$	0	0
$631$	−30.1410	−1.19989	−0.599946	−	0.800040i	$-0.704811\pi$
$631$	−30.1410	−1.19989	−0.599946	+	0.800040i	$0.704811\pi$
$632$	4.69956	0.186938
$633$	0	0
$634$	−19.4322	−0.771750
$635$	1.56837	0.0622387
$636$	0	0
$637$	−10.2111	−0.404579
$638$	−1.16515	−0.0461289
$639$	0	0
$640$	1.00000	0.0395285
$641$	24.2764	0.958860	0.479430	−	0.877580i	$-0.340843\pi$
$641$	24.2764	0.958860	0.479430	+	0.877580i	$0.340843\pi$
$642$	0	0
$643$	25.3102	0.998138	0.499069	−	0.866562i	$-0.333675\pi$
$643$	25.3102	0.998138	0.499069	+	0.866562i	$0.333675\pi$
$644$	14.2049	0.559750
$645$	0	0
$646$	0.152465	0.00599867
$647$	9.98234	0.392446	0.196223	−	0.980559i	$-0.437132\pi$
$647$	9.98234	0.392446	0.196223	+	0.980559i	$0.437132\pi$
$648$	0	0
$649$	−5.20217	−0.204203
$650$	4.48444	0.175894
$651$	0	0
$652$	24.1125	0.944317
$653$	−49.1351	−1.92280	−0.961402	−	0.275147i	$-0.911273\pi$
$653$	−49.1351	−1.92280	−0.961402	+	0.275147i	$0.911273\pi$
$654$	0	0
$655$	−6.52716	−0.255037
$656$	−9.02069	−0.352199
$657$	0	0
$658$	14.4904	0.564893
$659$	−33.7939	−1.31642	−0.658212	−	0.752832i	$-0.728687\pi$
$659$	−33.7939	−1.31642	−0.658212	+	0.752832i	$0.728687\pi$
$660$	0	0
$661$	−15.0390	−0.584948	−0.292474	−	0.956273i	$-0.594479\pi$
$661$	−15.0390	−0.584948	−0.292474	+	0.956273i	$0.594479\pi$
$662$	5.47362	0.212738
$663$	0	0
$664$	1.50162	0.0582741
$665$	6.50215	0.252143
$666$	0	0
$667$	−7.61573	−0.294882
$668$	−11.1014	−0.429525
$669$	0	0
$670$	−11.5425	−0.445927
$671$	−0.0438644	−0.00169337
$672$	0	0
$673$	−17.8656	−0.688667	−0.344334	−	0.938847i	$-0.611895\pi$
$673$	−17.8656	−0.688667	−0.344334	+	0.938847i	$0.611895\pi$
$674$	25.7022	0.990010
$675$	0	0
$676$	7.11019	0.273469
$677$	−26.2476	−1.00878	−0.504389	−	0.863476i	$-0.668282\pi$
$677$	−26.2476	−1.00878	−0.504389	+	0.863476i	$0.668282\pi$
$678$	0	0
$679$	15.5095	0.595199
$680$	0.0509592	0.00195420
$681$	0	0
$682$	−6.71607	−0.257172
$683$	14.6318	0.559872	0.279936	−	0.960019i	$-0.409687\pi$
$683$	14.6318	0.559872	0.279936	+	0.960019i	$0.409687\pi$
$684$	0	0
$685$	13.9447	0.532800
$686$	−20.1612	−0.769758
$687$	0	0
$688$	4.22169	0.160951
$689$	−13.4321	−0.511722
$690$	0	0
$691$	26.7461	1.01747	0.508735	−	0.860923i	$-0.330113\pi$
$691$	26.7461	1.01747	0.508735	+	0.860923i	$0.330113\pi$
$692$	−8.85369	−0.336567
$693$	0	0
$694$	30.8475	1.17096
$695$	−15.0553	−0.571081
$696$	0	0
$697$	−0.459687	−0.0174119
$698$	11.0961	0.419994
$699$	0	0
$700$	2.17324	0.0821409
$701$	0.623098	0.0235341	0.0117670	−	0.999931i	$-0.496254\pi$
$701$	0.623098	0.0235341	0.0117670	+	0.999931i	$0.496254\pi$
$702$	0	0
$703$	25.4988	0.961704
$704$	1.00000	0.0376889
$705$	0	0
$706$	13.4324	0.505535
$707$	−38.1825	−1.43600
$708$	0	0
$709$	−18.0552	−0.678077	−0.339038	−	0.940773i	$-0.610102\pi$
$709$	−18.0552	−0.678077	−0.339038	+	0.940773i	$0.610102\pi$
$710$	6.04471	0.226854
$711$	0	0
$712$	−16.3756	−0.613702
$713$	−43.8979	−1.64399
$714$	0	0
$715$	4.48444	0.167709
$716$	1.94083	0.0725321
$717$	0	0
$718$	−11.2447	−0.419650
$719$	−50.7877	−1.89406	−0.947031	−	0.321143i	$-0.895933\pi$
$719$	−50.7877	−1.89406	−0.947031	+	0.321143i	$0.895933\pi$
$720$	0	0
$721$	−40.5760	−1.51113
$722$	−10.0485	−0.373966
$723$	0	0
$724$	−15.6795	−0.582724
$725$	−1.16515	−0.0432727
$726$	0	0
$727$	4.21428	0.156299	0.0781494	−	0.996942i	$-0.475099\pi$
$727$	4.21428	0.156299	0.0781494	+	0.996942i	$0.475099\pi$
$728$	9.74578	0.361203
$729$	0	0
$730$	15.8864	0.587983
$731$	0.215134	0.00795702
$732$	0	0
$733$	−49.4129	−1.82511	−0.912553	−	0.408958i	$-0.865892\pi$
$733$	−49.4129	−1.82511	−0.912553	+	0.408958i	$0.865892\pi$
$734$	22.1691	0.818275
$735$	0	0
$736$	6.53625	0.240929
$737$	−11.5425	−0.425175
$738$	0	0
$739$	8.54493	0.314330	0.157165	−	0.987572i	$-0.449764\pi$
$739$	8.54493	0.314330	0.157165	+	0.987572i	$0.449764\pi$
$740$	8.52258	0.313296
$741$	0	0
$742$	−6.50945	−0.238969
$743$	29.7018	1.08965	0.544827	−	0.838549i	$-0.316595\pi$
$743$	29.7018	1.08965	0.544827	+	0.838549i	$0.316595\pi$
$744$	0	0
$745$	2.34416	0.0858834
$746$	−21.5923	−0.790549
$747$	0	0
$748$	0.0509592	0.00186325
$749$	−19.5401	−0.713981
$750$	0	0
$751$	29.0023	1.05831	0.529155	−	0.848525i	$-0.322509\pi$
$751$	29.0023	1.05831	0.529155	+	0.848525i	$0.322509\pi$
$752$	6.66762	0.243143
$753$	0	0
$754$	−5.22506	−0.190285
$755$	−21.2074	−0.771815
$756$	0	0
$757$	38.6709	1.40552	0.702758	−	0.711429i	$-0.251952\pi$
$757$	38.6709	1.40552	0.702758	+	0.711429i	$0.251952\pi$
$758$	−11.2563	−0.408848
$759$	0	0
$760$	2.99191	0.108528
$761$	−18.3920	−0.666710	−0.333355	−	0.942801i	$-0.608181\pi$
$761$	−18.3920	−0.666710	−0.333355	+	0.942801i	$0.608181\pi$
$762$	0	0
$763$	10.9302	0.395701
$764$	13.7569	0.497709
$765$	0	0
$766$	0.547464	0.0197807
$767$	−23.3288	−0.842355
$768$	0	0
$769$	22.6827	0.817960	0.408980	−	0.912543i	$-0.365885\pi$
$769$	22.6827	0.817960	0.408980	+	0.912543i	$0.365885\pi$
$770$	2.17324	0.0783183
$771$	0	0
$772$	−1.77539	−0.0638977
$773$	−26.7574	−0.962396	−0.481198	−	0.876612i	$-0.659798\pi$
$773$	−26.7574	−0.962396	−0.481198	+	0.876612i	$0.659798\pi$
$774$	0	0
$775$	−6.71607	−0.241248
$776$	7.13655	0.256187
$777$	0	0
$778$	−22.2724	−0.798505
$779$	−26.9891	−0.966984
$780$	0	0
$781$	6.04471	0.216297
$782$	0.333082	0.0119110
$783$	0	0
$784$	−2.27701	−0.0813217
$785$	8.50606	0.303594
$786$	0	0
$787$	22.8617	0.814932	0.407466	−	0.913220i	$-0.366412\pi$
$787$	22.8617	0.814932	0.407466	+	0.913220i	$0.366412\pi$
$788$	23.7919	0.847552
$789$	0	0
$790$	4.69956	0.167203
$791$	21.7401	0.772988
$792$	0	0
$793$	−0.196707	−0.00698529
$794$	−13.3297	−0.473054
$795$	0	0
$796$	2.15809	0.0764916
$797$	6.14102	0.217526	0.108763	−	0.994068i	$-0.465311\pi$
$797$	6.14102	0.217526	0.108763	+	0.994068i	$0.465311\pi$
$798$	0	0
$799$	0.339777	0.0120204
$800$	1.00000	0.0353553
$801$	0	0
$802$	−8.07960	−0.285301
$803$	15.8864	0.560620
$804$	0	0
$805$	14.2049	0.500656
$806$	−30.1178	−1.06085
$807$	0	0
$808$	−17.5693	−0.618087
$809$	55.8430	1.96334	0.981668	−	0.190599i	$-0.0610432\pi$
$809$	55.8430	1.96334	0.981668	+	0.190599i	$0.0610432\pi$
$810$	0	0
$811$	−23.7202	−0.832929	−0.416464	−	0.909152i	$-0.636731\pi$
$811$	−23.7202	−0.832929	−0.416464	+	0.909152i	$0.636731\pi$
$812$	−2.53216	−0.0888615
$813$	0	0
$814$	8.52258	0.298716
$815$	24.1125	0.844623
$816$	0	0
$817$	12.6309	0.441900
$818$	−8.83292	−0.308836
$819$	0	0
$820$	−9.02069	−0.315016
$821$	10.3178	0.360092	0.180046	−	0.983658i	$-0.442375\pi$
$821$	10.3178	0.360092	0.180046	+	0.983658i	$0.442375\pi$
$822$	0	0
$823$	−35.4214	−1.23471	−0.617356	−	0.786684i	$-0.711796\pi$
$823$	−35.4214	−1.23471	−0.617356	+	0.786684i	$0.711796\pi$
$824$	−18.6707	−0.650425
$825$	0	0
$826$	−11.3056	−0.393372
$827$	−40.8064	−1.41898	−0.709489	−	0.704717i	$-0.751074\pi$
$827$	−40.8064	−1.41898	−0.709489	+	0.704717i	$0.751074\pi$
$828$	0	0
$829$	16.9261	0.587868	0.293934	−	0.955826i	$-0.405035\pi$
$829$	16.9261	0.587868	0.293934	+	0.955826i	$0.405035\pi$
$830$	1.50162	0.0521219
$831$	0	0
$832$	4.48444	0.155470
$833$	−0.116035	−0.00402036
$834$	0	0
$835$	−11.1014	−0.384179
$836$	2.99191	0.103477
$837$	0	0
$838$	−5.17941	−0.178920
$839$	12.7446	0.439992	0.219996	−	0.975501i	$-0.429396\pi$
$839$	12.7446	0.439992	0.219996	+	0.975501i	$0.429396\pi$
$840$	0	0
$841$	−27.6424	−0.953187
$842$	6.64176	0.228890
$843$	0	0
$844$	22.1373	0.761998
$845$	7.11019	0.244598
$846$	0	0
$847$	2.17324	0.0746736
$848$	−2.99527	−0.102858
$849$	0	0
$850$	0.0509592	0.00174789
$851$	55.7057	1.90957
$852$	0	0
$853$	7.90817	0.270770	0.135385	−	0.990793i	$-0.456773\pi$
$853$	7.90817	0.270770	0.135385	+	0.990793i	$0.456773\pi$
$854$	−0.0953282	−0.00326206
$855$	0	0
$856$	−8.99123	−0.307314
$857$	−32.0389	−1.09443	−0.547214	−	0.836993i	$-0.684312\pi$
$857$	−32.0389	−1.09443	−0.547214	+	0.836993i	$0.684312\pi$
$858$	0	0
$859$	−7.85760	−0.268098	−0.134049	−	0.990975i	$-0.542798\pi$
$859$	−7.85760	−0.268098	−0.134049	+	0.990975i	$0.542798\pi$
$860$	4.22169	0.143959
$861$	0	0
$862$	−29.6036	−1.00830
$863$	10.0931	0.343574	0.171787	−	0.985134i	$-0.445046\pi$
$863$	10.0931	0.343574	0.171787	+	0.985134i	$0.445046\pi$
$864$	0	0
$865$	−8.85369	−0.301034
$866$	26.1933	0.890084
$867$	0	0
$868$	−14.5957	−0.495409
$869$	4.69956	0.159422
$870$	0	0
$871$	−51.7618	−1.75388
$872$	5.02945	0.170319
$873$	0	0
$874$	19.5559	0.661487
$875$	2.17324	0.0734691
$876$	0	0
$877$	0.204196	0.00689522	0.00344761	−	0.999994i	$-0.498903\pi$
$877$	0.204196	0.00689522	0.00344761	+	0.999994i	$0.498903\pi$
$878$	14.5524	0.491121
$879$	0	0
$880$	1.00000	0.0337100
$881$	−15.6635	−0.527718	−0.263859	−	0.964561i	$-0.584995\pi$
$881$	−15.6635	−0.527718	−0.263859	+	0.964561i	$0.584995\pi$
$882$	0	0
$883$	40.1215	1.35019	0.675097	−	0.737729i	$-0.264102\pi$
$883$	40.1215	1.35019	0.675097	+	0.737729i	$0.264102\pi$
$884$	0.228523	0.00768607
$885$	0	0
$886$	−19.6834	−0.661276
$887$	13.4026	0.450015	0.225008	−	0.974357i	$-0.427759\pi$
$887$	13.4026	0.450015	0.225008	+	0.974357i	$0.427759\pi$
$888$	0	0
$889$	3.40844	0.114315
$890$	−16.3756	−0.548911
$891$	0	0
$892$	−8.75047	−0.292987
$893$	19.9489	0.667565
$894$	0	0
$895$	1.94083	0.0648746
$896$	2.17324	0.0726030
$897$	0	0
$898$	24.0348	0.802053
$899$	7.82525	0.260987
$900$	0	0
$901$	−0.152636	−0.00508506
$902$	−9.02069	−0.300356
$903$	0	0
$904$	10.0035	0.332712
$905$	−15.6795	−0.521204
$906$	0	0
$907$	14.1787	0.470795	0.235398	−	0.971899i	$-0.424361\pi$
$907$	14.1787	0.470795	0.235398	+	0.971899i	$0.424361\pi$
$908$	−5.93425	−0.196935
$909$	0	0
$910$	9.74578	0.323070
$911$	−14.6298	−0.484707	−0.242353	−	0.970188i	$-0.577919\pi$
$911$	−14.6298	−0.484707	−0.242353	+	0.970188i	$0.577919\pi$
$912$	0	0
$913$	1.50162	0.0496963
$914$	−20.3728	−0.673873
$915$	0	0
$916$	1.12065	0.0370275
$917$	−14.1851	−0.468434
$918$	0	0
$919$	−1.15658	−0.0381521	−0.0190760	−	0.999818i	$-0.506072\pi$
$919$	−1.15658	−0.0381521	−0.0190760	+	0.999818i	$0.506072\pi$
$920$	6.53625	0.215494
$921$	0	0
$922$	−0.0295634	−0.000973618 0
$923$	27.1072	0.892243
$924$	0	0
$925$	8.52258	0.280221
$926$	−27.6808	−0.909647
$927$	0	0
$928$	−1.16515	−0.0382480
$929$	−2.36225	−0.0775028	−0.0387514	−	0.999249i	$-0.512338\pi$
$929$	−2.36225	−0.0775028	−0.0387514	+	0.999249i	$0.512338\pi$
$930$	0	0
$931$	−6.81260	−0.223274
$932$	−18.6243	−0.610060
$933$	0	0
$934$	−7.38942	−0.241789
$935$	0.0509592	0.00166654
$936$	0	0
$937$	0.756699	0.0247203	0.0123601	−	0.999924i	$-0.496066\pi$
$937$	0.756699	0.0247203	0.0123601	+	0.999924i	$0.496066\pi$
$938$	−25.0848	−0.819046
$939$	0	0
$940$	6.66762	0.217474
$941$	−21.2252	−0.691921	−0.345961	−	0.938249i	$-0.612447\pi$
$941$	−21.2252	−0.691921	−0.345961	+	0.938249i	$0.612447\pi$
$942$	0	0
$943$	−58.9614	−1.92005
$944$	−5.20217	−0.169316
$945$	0	0
$946$	4.22169	0.137259
$947$	−22.3925	−0.727658	−0.363829	−	0.931466i	$-0.618531\pi$
$947$	−22.3925	−0.727658	−0.363829	+	0.931466i	$0.618531\pi$
$948$	0	0
$949$	71.2417	2.31260
$950$	2.99191	0.0970703
$951$	0	0
$952$	0.110747	0.00358932
$953$	26.8894	0.871034	0.435517	−	0.900181i	$-0.356566\pi$
$953$	26.8894	0.871034	0.435517	+	0.900181i	$0.356566\pi$
$954$	0	0
$955$	13.7569	0.445164
$956$	7.19057	0.232560
$957$	0	0
$958$	22.4611	0.725686
$959$	30.3053	0.978609
$960$	0	0
$961$	14.1056	0.455019
$962$	38.2190	1.23223
$963$	0	0
$964$	−1.40835	−0.0453600
$965$	−1.77539	−0.0571519
$966$	0	0
$967$	59.4607	1.91213	0.956063	−	0.293161i	$-0.0947072\pi$
$967$	59.4607	1.91213	0.956063	+	0.293161i	$0.0947072\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	7.13655	0.229141
$971$	−56.6074	−1.81662	−0.908309	−	0.418301i	$-0.862626\pi$
$971$	−56.6074	−1.81662	−0.908309	+	0.418301i	$0.862626\pi$
$972$	0	0
$973$	−32.7189	−1.04892
$974$	33.9578	1.08808
$975$	0	0
$976$	−0.0438644	−0.00140407
$977$	−38.3417	−1.22666	−0.613330	−	0.789827i	$-0.710170\pi$
$977$	−38.3417	−1.22666	−0.613330	+	0.789827i	$0.710170\pi$
$978$	0	0
$979$	−16.3756	−0.523366
$980$	−2.27701	−0.0727364
$981$	0	0
$982$	−29.6810	−0.947157
$983$	−50.4365	−1.60867	−0.804337	−	0.594173i	$-0.797479\pi$
$983$	−50.4365	−1.60867	−0.804337	+	0.594173i	$0.797479\pi$
$984$	0	0
$985$	23.7919	0.758074
$986$	−0.0593753	−0.00189089
$987$	0	0
$988$	13.4170	0.426853
$989$	27.5940	0.877439
$990$	0	0
$991$	2.77450	0.0881348	0.0440674	−	0.999029i	$-0.485968\pi$
$991$	2.77450	0.0881348	0.0440674	+	0.999029i	$0.485968\pi$
$992$	−6.71607	−0.213235
$993$	0	0
$994$	13.1366	0.416669
$995$	2.15809	0.0684162
$996$	0	0
$997$	−51.2718	−1.62379	−0.811897	−	0.583801i	$-0.801565\pi$
$997$	−51.2718	−1.62379	−0.811897	+	0.583801i	$0.801565\pi$
$998$	−9.82210	−0.310913
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8910.2.a.cd.1.5		7
3.2	odd	2		8910.2.a.ca.1.5		7
9.2	odd	6		2970.2.i.k.1981.3		14
9.4	even	3		990.2.i.j.331.1	✓	14
9.5	odd	6		2970.2.i.k.991.3		14
9.7	even	3		990.2.i.j.661.1	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
990.2.i.j.331.1	✓	14	9.4	even	3
990.2.i.j.661.1	yes	14	9.7	even	3
2970.2.i.k.991.3		14	9.5	odd	6
2970.2.i.k.1981.3		14	9.2	odd	6
8910.2.a.ca.1.5		7	3.2	odd	2
8910.2.a.cd.1.5		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8910 = 2 \cdot 3^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8910.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.17324 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +4.48444 q^{13} +2.17324 q^{14} +1.00000 q^{16} +0.0509592 q^{17} +2.99191 q^{19} +1.00000 q^{20} +1.00000 q^{22} +6.53625 q^{23} +1.00000 q^{25} +4.48444 q^{26} +2.17324 q^{28} -1.16515 q^{29} -6.71607 q^{31} +1.00000 q^{32} +0.0509592 q^{34} +2.17324 q^{35} +8.52258 q^{37} +2.99191 q^{38} +1.00000 q^{40} -9.02069 q^{41} +4.22169 q^{43} +1.00000 q^{44} +6.53625 q^{46} +6.66762 q^{47} -2.27701 q^{49} +1.00000 q^{50} +4.48444 q^{52} -2.99527 q^{53} +1.00000 q^{55} +2.17324 q^{56} -1.16515 q^{58} -5.20217 q^{59} -0.0438644 q^{61} -6.71607 q^{62} +1.00000 q^{64} +4.48444 q^{65} -11.5425 q^{67} +0.0509592 q^{68} +2.17324 q^{70} +6.04471 q^{71} +15.8864 q^{73} +8.52258 q^{74} +2.99191 q^{76} +2.17324 q^{77} +4.69956 q^{79} +1.00000 q^{80} -9.02069 q^{82} +1.50162 q^{83} +0.0509592 q^{85} +4.22169 q^{86} +1.00000 q^{88} -16.3756 q^{89} +9.74578 q^{91} +6.53625 q^{92} +6.66762 q^{94} +2.99191 q^{95} +7.13655 q^{97} -2.27701 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + 5 q^{7} + 7 q^{8} + 7 q^{10} + 7 q^{11} + 5 q^{13} + 5 q^{14} + 7 q^{16} + 3 q^{17} + 11 q^{19} + 7 q^{20} + 7 q^{22} + 3 q^{23} + 7 q^{25} + 5 q^{26} + 5 q^{28} + 12 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) \) 7 * q + 7 * q^2 + 7 * q^4 + 7 * q^5 + 5 * q^7 + 7 * q^8 + 7 * q^10 + 7 * q^11 + 5 * q^13 + 5 * q^14 + 7 * q^16 + 3 * q^17 + 11 * q^19 + 7 * q^20 + 7 * q^22 + 3 * q^23 + 7 * q^25 + 5 * q^26 + 5 * q^28 + 12 * q^29 + 5 * q^31 + 7 * q^32 + 3 * q^34 + 5 * q^35 + 2 * q^37 + 11 * q^38 + 7 * q^40 + 6 * q^41 + 17 * q^43 + 7 * q^44 + 3 * q^46 - 3 * q^47 + 12 * q^49 + 7 * q^50 + 5 * q^52 + 3 * q^53 + 7 * q^55 + 5 * q^56 + 12 * q^58 + 18 * q^59 + 11 * q^61 + 5 * q^62 + 7 * q^64 + 5 * q^65 + 20 * q^67 + 3 * q^68 + 5 * q^70 + 12 * q^71 + 2 * q^73 + 2 * q^74 + 11 * q^76 + 5 * q^77 - 7 * q^79 + 7 * q^80 + 6 * q^82 + 3 * q^85 + 17 * q^86 + 7 * q^88 - 3 * q^89 + 7 * q^91 + 3 * q^92 - 3 * q^94 + 11 * q^95 + 23 * q^97 + 12 * q^98

Introduction

L-functions

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Database

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