LMFDB - Embedded newform 3087.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3087,2,Mod(1,3087)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3087, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3087.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3087 = 3^{2} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3087.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$24.6498191040$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 9x^{7} + 35x^{6} + 40x^{5} - 92x^{4} - 88x^{3} + 52x^{2} + 32x - 8 $ x^9 - 4x^8 - 9x^7 + 35x^6 + 40x^5 - 92x^4 - 88x^3 + 52x^2 + 32x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1029)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.49320$ of defining polynomial
Character	$\chi$	$=$	3087.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49320 q^{2} +4.21606 q^{4} +0.321703 q^{5} -5.52510 q^{8} +O(q^{10})$	$q-2.49320 q^{2} +4.21606 q^{4} +0.321703 q^{5} -5.52510 q^{8} -0.802071 q^{10} -2.79918 q^{11} -5.56549 q^{13} +5.34306 q^{16} -2.70782 q^{17} +5.07317 q^{19} +1.35632 q^{20} +6.97894 q^{22} +0.447454 q^{23} -4.89651 q^{25} +13.8759 q^{26} -7.86092 q^{29} -9.68879 q^{31} -2.27115 q^{32} +6.75114 q^{34} -2.85547 q^{37} -12.6485 q^{38} -1.77744 q^{40} -1.82027 q^{41} +4.04141 q^{43} -11.8015 q^{44} -1.11559 q^{46} +11.7632 q^{47} +12.2080 q^{50} -23.4645 q^{52} +13.6813 q^{53} -0.900506 q^{55} +19.5989 q^{58} +2.08051 q^{59} +8.56239 q^{61} +24.1561 q^{62} -5.02369 q^{64} -1.79043 q^{65} -9.07110 q^{67} -11.4163 q^{68} +5.92679 q^{71} +10.5771 q^{73} +7.11927 q^{74} +21.3888 q^{76} +7.26034 q^{79} +1.71888 q^{80} +4.53831 q^{82} +2.64249 q^{83} -0.871113 q^{85} -10.0761 q^{86} +15.4658 q^{88} -5.04974 q^{89} +1.88650 q^{92} -29.3279 q^{94} +1.63206 q^{95} +18.0482 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + 17 q^{4} + q^{5} - 12 q^{8}+O(q^{10}) $ 9 * q - 5 * q^2 + 17 * q^4 + q^5 - 12 * q^8	$ 9 q - 5 q^{2} + 17 q^{4} + q^{5} - 12 q^{8} - q^{10} - 21 q^{11} + 11 q^{13} + 29 q^{16} + 6 q^{17} - 2 q^{19} + 20 q^{20} - 9 q^{23} + 10 q^{25} + 23 q^{26} - 12 q^{29} - 3 q^{31} - 24 q^{32} + 12 q^{34} + 8 q^{37} + 15 q^{38} + 7 q^{40} + 17 q^{41} + 16 q^{43} - 46 q^{44} + 13 q^{46} + 7 q^{47} - 3 q^{50} + 26 q^{52} + 8 q^{53} - 6 q^{55} - 22 q^{58} + 30 q^{59} + 7 q^{61} + 43 q^{62} + 42 q^{64} + 15 q^{65} + 33 q^{67} + 66 q^{68} - 30 q^{71} + 48 q^{73} + 8 q^{74} - 34 q^{76} + 7 q^{79} + 98 q^{80} - 10 q^{82} + 11 q^{83} - 18 q^{85} + 3 q^{86} - 47 q^{88} + 25 q^{89} + 21 q^{92} + 11 q^{94} + 13 q^{95} + 50 q^{97}+O(q^{100}) $ 9 * q - 5 * q^2 + 17 * q^4 + q^5 - 12 * q^8 - q^10 - 21 * q^11 + 11 * q^13 + 29 * q^16 + 6 * q^17 - 2 * q^19 + 20 * q^20 - 9 * q^23 + 10 * q^25 + 23 * q^26 - 12 * q^29 - 3 * q^31 - 24 * q^32 + 12 * q^34 + 8 * q^37 + 15 * q^38 + 7 * q^40 + 17 * q^41 + 16 * q^43 - 46 * q^44 + 13 * q^46 + 7 * q^47 - 3 * q^50 + 26 * q^52 + 8 * q^53 - 6 * q^55 - 22 * q^58 + 30 * q^59 + 7 * q^61 + 43 * q^62 + 42 * q^64 + 15 * q^65 + 33 * q^67 + 66 * q^68 - 30 * q^71 + 48 * q^73 + 8 * q^74 - 34 * q^76 + 7 * q^79 + 98 * q^80 - 10 * q^82 + 11 * q^83 - 18 * q^85 + 3 * q^86 - 47 * q^88 + 25 * q^89 + 21 * q^92 + 11 * q^94 + 13 * q^95 + 50 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.49320	−1.76296	−0.881481	−	0.472220i	$-0.843453\pi$
$2$	−2.49320	−1.76296	−0.881481	+	0.472220i	$0.843453\pi$
$3$	0	0
$3$	0	0
$4$	4.21606	2.10803
$5$	0.321703	0.143870	0.0719350	−	0.997409i	$-0.477083\pi$
$5$	0.321703	0.143870	0.0719350	+	0.997409i	$0.477083\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−5.52510	−1.95342
$9$	0	0
$10$	−0.802071	−0.253637
$11$	−2.79918	−0.843986	−0.421993	−	0.906599i	$-0.638669\pi$
$11$	−2.79918	−0.843986	−0.421993	+	0.906599i	$0.638669\pi$
$12$	0	0
$13$	−5.56549	−1.54359	−0.771795	−	0.635872i	$-0.780641\pi$
$13$	−5.56549	−1.54359	−0.771795	+	0.635872i	$0.780641\pi$
$14$	0	0
$15$	0	0
$16$	5.34306	1.33577
$17$	−2.70782	−0.656742	−0.328371	−	0.944549i	$-0.606500\pi$
$17$	−2.70782	−0.656742	−0.328371	+	0.944549i	$0.606500\pi$
$18$	0	0
$19$	5.07317	1.16387	0.581933	−	0.813237i	$-0.302297\pi$
$19$	5.07317	1.16387	0.581933	+	0.813237i	$0.302297\pi$
$20$	1.35632	0.303282
$21$	0	0
$22$	6.97894	1.48791
$23$	0.447454	0.0933007	0.0466503	−	0.998911i	$-0.485145\pi$
$23$	0.447454	0.0933007	0.0466503	+	0.998911i	$0.485145\pi$
$24$	0	0
$25$	−4.89651	−0.979301
$26$	13.8759	2.72129
$27$	0	0
$28$	0	0
$29$	−7.86092	−1.45974	−0.729868	−	0.683588i	$-0.760418\pi$
$29$	−7.86092	−1.45974	−0.729868	+	0.683588i	$0.760418\pi$
$30$	0	0
$31$	−9.68879	−1.74016	−0.870079	−	0.492912i	$-0.835933\pi$
$31$	−9.68879	−1.74016	−0.870079	+	0.492912i	$0.835933\pi$
$32$	−2.27115	−0.401486
$33$	0	0
$34$	6.75114	1.15781
$35$	0	0
$36$	0	0
$37$	−2.85547	−0.469437	−0.234718	−	0.972063i	$-0.575417\pi$
$37$	−2.85547	−0.469437	−0.234718	+	0.972063i	$0.575417\pi$
$38$	−12.6485	−2.05185
$39$	0	0
$40$	−1.77744	−0.281038
$41$	−1.82027	−0.284279	−0.142139	−	0.989847i	$-0.545398\pi$
$41$	−1.82027	−0.284279	−0.142139	+	0.989847i	$0.545398\pi$
$42$	0	0
$43$	4.04141	0.616310	0.308155	−	0.951336i	$-0.400288\pi$
$43$	4.04141	0.616310	0.308155	+	0.951336i	$0.400288\pi$
$44$	−11.8015	−1.77915
$45$	0	0
$46$	−1.11559	−0.164485
$47$	11.7632	1.71583	0.857916	−	0.513790i	$-0.171759\pi$
$47$	11.7632	1.71583	0.857916	+	0.513790i	$0.171759\pi$
$48$	0	0
$49$	0	0
$50$	12.2080	1.72647
$51$	0	0
$52$	−23.4645	−3.25393
$53$	13.6813	1.87927	0.939635	−	0.342178i	$-0.111165\pi$
$53$	13.6813	1.87927	0.939635	+	0.342178i	$0.111165\pi$
$54$	0	0
$55$	−0.900506	−0.121424
$56$	0	0
$57$	0	0
$58$	19.5989	2.57346
$59$	2.08051	0.270859	0.135430	−	0.990787i	$-0.456759\pi$
$59$	2.08051	0.270859	0.135430	+	0.990787i	$0.456759\pi$
$60$	0	0
$61$	8.56239	1.09630	0.548151	−	0.836379i	$-0.315332\pi$
$61$	8.56239	1.09630	0.548151	+	0.836379i	$0.315332\pi$
$62$	24.1561	3.06783
$63$	0	0
$64$	−5.02369	−0.627961
$65$	−1.79043	−0.222076
$66$	0	0
$67$	−9.07110	−1.10821	−0.554106	−	0.832446i	$-0.686940\pi$
$67$	−9.07110	−1.10821	−0.554106	+	0.832446i	$0.686940\pi$
$68$	−11.4163	−1.38443
$69$	0	0
$70$	0	0
$71$	5.92679	0.703381	0.351691	−	0.936116i	$-0.385607\pi$
$71$	5.92679	0.703381	0.351691	+	0.936116i	$0.385607\pi$
$72$	0	0
$73$	10.5771	1.23796	0.618980	−	0.785407i	$-0.287546\pi$
$73$	10.5771	1.23796	0.618980	+	0.785407i	$0.287546\pi$
$74$	7.11927	0.827599
$75$	0	0
$76$	21.3888	2.45347
$77$	0	0
$78$	0	0
$79$	7.26034	0.816852	0.408426	−	0.912791i	$-0.366078\pi$
$79$	7.26034	0.816852	0.408426	+	0.912791i	$0.366078\pi$
$80$	1.71888	0.192176
$81$	0	0
$82$	4.53831	0.501172
$83$	2.64249	0.290051	0.145025	−	0.989428i	$-0.453674\pi$
$83$	2.64249	0.290051	0.145025	+	0.989428i	$0.453674\pi$
$84$	0	0
$85$	−0.871113	−0.0944854
$86$	−10.0761	−1.08653
$87$	0	0
$88$	15.4658	1.64866
$89$	−5.04974	−0.535272	−0.267636	−	0.963520i	$-0.586242\pi$
$89$	−5.04974	−0.535272	−0.267636	+	0.963520i	$0.586242\pi$
$90$	0	0
$91$	0	0
$92$	1.88650	0.196681
$93$	0	0
$94$	−29.3279	−3.02494
$95$	1.63206	0.167445
$96$	0	0
$97$	18.0482	1.83252	0.916261	−	0.400583i	$-0.131192\pi$
$97$	18.0482	1.83252	0.916261	+	0.400583i	$0.131192\pi$
$98$	0	0
$99$	0	0
$100$	−20.6440	−2.06440
$101$	3.69369	0.367536	0.183768	−	0.982970i	$-0.441170\pi$
$101$	3.69369	0.367536	0.183768	+	0.982970i	$0.441170\pi$
$102$	0	0
$103$	8.30749	0.818561	0.409281	−	0.912408i	$-0.365780\pi$
$103$	8.30749	0.818561	0.409281	+	0.912408i	$0.365780\pi$
$104$	30.7499	3.01527
$105$	0	0
$106$	−34.1102	−3.31308
$107$	15.6783	1.51568	0.757838	−	0.652443i	$-0.226256\pi$
$107$	15.6783	1.51568	0.757838	+	0.652443i	$0.226256\pi$
$108$	0	0
$109$	−12.0845	−1.15748	−0.578742	−	0.815511i	$-0.696456\pi$
$109$	−12.0845	−1.15748	−0.578742	+	0.815511i	$0.696456\pi$
$110$	2.24514	0.214066
$111$	0	0
$112$	0	0
$113$	−1.34994	−0.126991	−0.0634957	−	0.997982i	$-0.520225\pi$
$113$	−1.34994	−0.126991	−0.0634957	+	0.997982i	$0.520225\pi$
$114$	0	0
$115$	0.143947	0.0134232
$116$	−33.1421	−3.07717
$117$	0	0
$118$	−5.18714	−0.477515
$119$	0	0
$120$	0	0
$121$	−3.16456	−0.287688
$122$	−21.3478	−1.93274
$123$	0	0
$124$	−40.8486	−3.66831
$125$	−3.18374	−0.284762
$126$	0	0
$127$	3.38788	0.300626	0.150313	−	0.988638i	$-0.451972\pi$
$127$	3.38788	0.300626	0.150313	+	0.988638i	$0.451972\pi$
$128$	17.0674	1.50856
$129$	0	0
$130$	4.46392	0.391511
$131$	−17.6198	−1.53945	−0.769724	−	0.638377i	$-0.779606\pi$
$131$	−17.6198	−1.53945	−0.769724	+	0.638377i	$0.779606\pi$
$132$	0	0
$133$	0	0
$134$	22.6161	1.95373
$135$	0	0
$136$	14.9609	1.28289
$137$	−2.03493	−0.173856	−0.0869278	−	0.996215i	$-0.527705\pi$
$137$	−2.03493	−0.173856	−0.0869278	+	0.996215i	$0.527705\pi$
$138$	0	0
$139$	−6.62329	−0.561780	−0.280890	−	0.959740i	$-0.590630\pi$
$139$	−6.62329	−0.561780	−0.280890	+	0.959740i	$0.590630\pi$
$140$	0	0
$141$	0	0
$142$	−14.7767	−1.24003
$143$	15.5788	1.30277
$144$	0	0
$145$	−2.52888	−0.210012
$146$	−26.3709	−2.18247
$147$	0	0
$148$	−12.0389	−0.989587
$149$	−14.5880	−1.19510	−0.597548	−	0.801833i	$-0.703858\pi$
$149$	−14.5880	−1.19510	−0.597548	+	0.801833i	$0.703858\pi$
$150$	0	0
$151$	15.8677	1.29129	0.645647	−	0.763636i	$-0.276588\pi$
$151$	15.8677	1.29129	0.645647	+	0.763636i	$0.276588\pi$
$152$	−28.0298	−2.27352
$153$	0	0
$154$	0	0
$155$	−3.11691	−0.250356
$156$	0	0
$157$	−12.0625	−0.962694	−0.481347	−	0.876530i	$-0.659852\pi$
$157$	−12.0625	−0.962694	−0.481347	+	0.876530i	$0.659852\pi$
$158$	−18.1015	−1.44008
$159$	0	0
$160$	−0.730635	−0.0577618
$161$	0	0
$162$	0	0
$163$	10.3794	0.812976	0.406488	−	0.913656i	$-0.366753\pi$
$163$	10.3794	0.812976	0.406488	+	0.913656i	$0.366753\pi$
$164$	−7.67438	−0.599268
$165$	0	0
$166$	−6.58826	−0.511348
$167$	7.08681	0.548394	0.274197	−	0.961674i	$-0.411588\pi$
$167$	7.08681	0.548394	0.274197	+	0.961674i	$0.411588\pi$
$168$	0	0
$169$	17.9747	1.38267
$170$	2.17186	0.166574
$171$	0	0
$172$	17.0389	1.29920
$173$	19.7308	1.50011	0.750053	−	0.661378i	$-0.230028\pi$
$173$	19.7308	1.50011	0.750053	+	0.661378i	$0.230028\pi$
$174$	0	0
$175$	0	0
$176$	−14.9562	−1.12737
$177$	0	0
$178$	12.5900	0.943663
$179$	14.3001	1.06884	0.534419	−	0.845220i	$-0.320530\pi$
$179$	14.3001	1.06884	0.534419	+	0.845220i	$0.320530\pi$
$180$	0	0
$181$	−8.85665	−0.658309	−0.329155	−	0.944276i	$-0.606764\pi$
$181$	−8.85665	−0.658309	−0.329155	+	0.944276i	$0.606764\pi$
$182$	0	0
$183$	0	0
$184$	−2.47223	−0.182255
$185$	−0.918614	−0.0675378
$186$	0	0
$187$	7.57968	0.554281
$188$	49.5942	3.61703
$189$	0	0
$190$	−4.06905	−0.295200
$191$	−12.1201	−0.876977	−0.438489	−	0.898737i	$-0.644486\pi$
$191$	−12.1201	−0.876977	−0.438489	+	0.898737i	$0.644486\pi$
$192$	0	0
$193$	−3.71228	−0.267216	−0.133608	−	0.991034i	$-0.542656\pi$
$193$	−3.71228	−0.267216	−0.133608	+	0.991034i	$0.542656\pi$
$194$	−44.9979	−3.23066
$195$	0	0
$196$	0	0
$197$	−18.1223	−1.29116	−0.645580	−	0.763693i	$-0.723384\pi$
$197$	−18.1223	−1.29116	−0.645580	+	0.763693i	$0.723384\pi$
$198$	0	0
$199$	20.4261	1.44797	0.723983	−	0.689818i	$-0.242310\pi$
$199$	20.4261	1.44797	0.723983	+	0.689818i	$0.242310\pi$
$200$	27.0537	1.91298
$201$	0	0
$202$	−9.20913	−0.647952
$203$	0	0
$204$	0	0
$205$	−0.585587	−0.0408992
$206$	−20.7123	−1.44309
$207$	0	0
$208$	−29.7368	−2.06187
$209$	−14.2008	−0.982287
$210$	0	0
$211$	9.56837	0.658714	0.329357	−	0.944205i	$-0.393168\pi$
$211$	9.56837	0.658714	0.329357	+	0.944205i	$0.393168\pi$
$212$	57.6812	3.96156
$213$	0	0
$214$	−39.0891	−2.67208
$215$	1.30013	0.0886685
$216$	0	0
$217$	0	0
$218$	30.1291	2.04060
$219$	0	0
$220$	−3.79659	−0.255966
$221$	15.0703	1.01374
$222$	0	0
$223$	−7.84864	−0.525584	−0.262792	−	0.964853i	$-0.584643\pi$
$223$	−7.84864	−0.525584	−0.262792	+	0.964853i	$0.584643\pi$
$224$	0	0
$225$	0	0
$226$	3.36567	0.223881
$227$	−7.31314	−0.485390	−0.242695	−	0.970103i	$-0.578031\pi$
$227$	−7.31314	−0.485390	−0.242695	+	0.970103i	$0.578031\pi$
$228$	0	0
$229$	21.1416	1.39708	0.698539	−	0.715572i	$-0.253834\pi$
$229$	21.1416	1.39708	0.698539	+	0.715572i	$0.253834\pi$
$230$	−0.358890	−0.0236645
$231$	0	0
$232$	43.4323	2.85147
$233$	−7.21086	−0.472399	−0.236200	−	0.971705i	$-0.575902\pi$
$233$	−7.21086	−0.472399	−0.236200	+	0.971705i	$0.575902\pi$
$234$	0	0
$235$	3.78424	0.246857
$236$	8.77156	0.570980
$237$	0	0
$238$	0	0
$239$	−20.0648	−1.29788	−0.648941	−	0.760838i	$-0.724788\pi$
$239$	−20.0648	−1.29788	−0.648941	+	0.760838i	$0.724788\pi$
$240$	0	0
$241$	1.92807	0.124198	0.0620989	−	0.998070i	$-0.480221\pi$
$241$	1.92807	0.124198	0.0620989	+	0.998070i	$0.480221\pi$
$242$	7.88990	0.507182
$243$	0	0
$244$	36.0996	2.31104
$245$	0	0
$246$	0	0
$247$	−28.2347	−1.79653
$248$	53.5315	3.39925
$249$	0	0
$250$	7.93770	0.502024
$251$	−4.06687	−0.256699	−0.128349	−	0.991729i	$-0.540968\pi$
$251$	−4.06687	−0.256699	−0.128349	+	0.991729i	$0.540968\pi$
$252$	0	0
$253$	−1.25251	−0.0787445
$254$	−8.44669	−0.529992
$255$	0	0
$256$	−32.5051	−2.03157
$257$	8.23720	0.513822	0.256911	−	0.966435i	$-0.417295\pi$
$257$	8.23720	0.513822	0.256911	+	0.966435i	$0.417295\pi$
$258$	0	0
$259$	0	0
$260$	−7.54858	−0.468143
$261$	0	0
$262$	43.9297	2.71399
$263$	−22.4345	−1.38337	−0.691684	−	0.722200i	$-0.743131\pi$
$263$	−22.4345	−1.38337	−0.691684	+	0.722200i	$0.743131\pi$
$264$	0	0
$265$	4.40131	0.270370
$266$	0	0
$267$	0	0
$268$	−38.2443	−2.33614
$269$	25.8402	1.57550	0.787752	−	0.615992i	$-0.211245\pi$
$269$	25.8402	1.57550	0.787752	+	0.615992i	$0.211245\pi$
$270$	0	0
$271$	−4.06202	−0.246750	−0.123375	−	0.992360i	$-0.539372\pi$
$271$	−4.06202	−0.246750	−0.123375	+	0.992360i	$0.539372\pi$
$272$	−14.4680	−0.877254
$273$	0	0
$274$	5.07349	0.306500
$275$	13.7062	0.826517
$276$	0	0
$277$	−13.4915	−0.810626	−0.405313	−	0.914178i	$-0.632837\pi$
$277$	−13.4915	−0.810626	−0.405313	+	0.914178i	$0.632837\pi$
$278$	16.5132	0.990397
$279$	0	0
$280$	0	0
$281$	8.08133	0.482092	0.241046	−	0.970514i	$-0.422510\pi$
$281$	8.08133	0.482092	0.241046	+	0.970514i	$0.422510\pi$
$282$	0	0
$283$	6.68339	0.397286	0.198643	−	0.980072i	$-0.436347\pi$
$283$	6.68339	0.397286	0.198643	+	0.980072i	$0.436347\pi$
$284$	24.9877	1.48275
$285$	0	0
$286$	−38.8412	−2.29673
$287$	0	0
$288$	0	0
$289$	−9.66773	−0.568690
$290$	6.30501	0.370243
$291$	0	0
$292$	44.5938	2.60966
$293$	−0.707884	−0.0413550	−0.0206775	−	0.999786i	$-0.506582\pi$
$293$	−0.707884	−0.0413550	−0.0206775	+	0.999786i	$0.506582\pi$
$294$	0	0
$295$	0.669306	0.0389685
$296$	15.7768	0.917005
$297$	0	0
$298$	36.3709	2.10691
$299$	−2.49030	−0.144018
$300$	0	0
$301$	0	0
$302$	−39.5613	−2.27650
$303$	0	0
$304$	27.1063	1.55465
$305$	2.75455	0.157725
$306$	0	0
$307$	14.7634	0.842591	0.421295	−	0.906923i	$-0.361576\pi$
$307$	14.7634	0.842591	0.421295	+	0.906923i	$0.361576\pi$
$308$	0	0
$309$	0	0
$310$	7.77110	0.441369
$311$	−5.62858	−0.319167	−0.159584	−	0.987184i	$-0.551015\pi$
$311$	−5.62858	−0.319167	−0.159584	+	0.987184i	$0.551015\pi$
$312$	0	0
$313$	14.0864	0.796208	0.398104	−	0.917340i	$-0.369668\pi$
$313$	14.0864	0.796208	0.398104	+	0.917340i	$0.369668\pi$
$314$	30.0743	1.69719
$315$	0	0
$316$	30.6100	1.72195
$317$	14.4108	0.809392	0.404696	−	0.914451i	$-0.367377\pi$
$317$	14.4108	0.809392	0.404696	+	0.914451i	$0.367377\pi$
$318$	0	0
$319$	22.0042	1.23200
$320$	−1.61613	−0.0903447
$321$	0	0
$322$	0	0
$323$	−13.7372	−0.764360
$324$	0	0
$325$	27.2515	1.51164
$326$	−25.8779	−1.43324
$327$	0	0
$328$	10.0572	0.555315
$329$	0	0
$330$	0	0
$331$	22.4716	1.23515	0.617575	−	0.786512i	$-0.288115\pi$
$331$	22.4716	1.23515	0.617575	+	0.786512i	$0.288115\pi$
$332$	11.1409	0.611436
$333$	0	0
$334$	−17.6689	−0.966797
$335$	−2.91820	−0.159438
$336$	0	0
$337$	−11.5746	−0.630508	−0.315254	−	0.949007i	$-0.602090\pi$
$337$	−11.5746	−0.630508	−0.315254	+	0.949007i	$0.602090\pi$
$338$	−44.8145	−2.43759
$339$	0	0
$340$	−3.67267	−0.199178
$341$	27.1207	1.46867
$342$	0	0
$343$	0	0
$344$	−22.3292	−1.20391
$345$	0	0
$346$	−49.1929	−2.64463
$347$	−15.6206	−0.838559	−0.419280	−	0.907857i	$-0.637717\pi$
$347$	−15.6206	−0.838559	−0.419280	+	0.907857i	$0.637717\pi$
$348$	0	0
$349$	13.9055	0.744346	0.372173	−	0.928163i	$-0.378613\pi$
$349$	13.9055	0.744346	0.372173	+	0.928163i	$0.378613\pi$
$350$	0	0
$351$	0	0
$352$	6.35737	0.338849
$353$	32.6502	1.73779	0.868897	−	0.494992i	$-0.164829\pi$
$353$	32.6502	1.73779	0.868897	+	0.494992i	$0.164829\pi$
$354$	0	0
$355$	1.90667	0.101195
$356$	−21.2900	−1.12837
$357$	0	0
$358$	−35.6530	−1.88432
$359$	10.3492	0.546211	0.273106	−	0.961984i	$-0.411949\pi$
$359$	10.3492	0.546211	0.273106	+	0.961984i	$0.411949\pi$
$360$	0	0
$361$	6.73710	0.354584
$362$	22.0814	1.16057
$363$	0	0
$364$	0	0
$365$	3.40269	0.178105
$366$	0	0
$367$	−8.72685	−0.455538	−0.227769	−	0.973715i	$-0.573143\pi$
$367$	−8.72685	−0.455538	−0.227769	+	0.973715i	$0.573143\pi$
$368$	2.39078	0.124628
$369$	0	0
$370$	2.29029	0.119067
$371$	0	0
$372$	0	0
$373$	20.4922	1.06105	0.530524	−	0.847670i	$-0.321995\pi$
$373$	20.4922	1.06105	0.530524	+	0.847670i	$0.321995\pi$
$374$	−18.8977	−0.977176
$375$	0	0
$376$	−64.9925	−3.35173
$377$	43.7499	2.25323
$378$	0	0
$379$	1.08103	0.0555288	0.0277644	−	0.999614i	$-0.491161\pi$
$379$	1.08103	0.0555288	0.0277644	+	0.999614i	$0.491161\pi$
$380$	6.88085	0.352980
$381$	0	0
$382$	30.2178	1.54608
$383$	29.3865	1.50158	0.750789	−	0.660542i	$-0.229673\pi$
$383$	29.3865	1.50158	0.750789	+	0.660542i	$0.229673\pi$
$384$	0	0
$385$	0	0
$386$	9.25547	0.471091
$387$	0	0
$388$	76.0925	3.86301
$389$	−12.6031	−0.639002	−0.319501	−	0.947586i	$-0.603515\pi$
$389$	−12.6031	−0.639002	−0.319501	+	0.947586i	$0.603515\pi$
$390$	0	0
$391$	−1.21162	−0.0612745
$392$	0	0
$393$	0	0
$394$	45.1825	2.27626
$395$	2.33567	0.117520
$396$	0	0
$397$	32.7937	1.64587	0.822933	−	0.568138i	$-0.192336\pi$
$397$	32.7937	1.64587	0.822933	+	0.568138i	$0.192336\pi$
$398$	−50.9263	−2.55271
$399$	0	0
$400$	−26.1623	−1.30812
$401$	−6.87388	−0.343265	−0.171633	−	0.985161i	$-0.554904\pi$
$401$	−6.87388	−0.343265	−0.171633	+	0.985161i	$0.554904\pi$
$402$	0	0
$403$	53.9229	2.68609
$404$	15.5728	0.774778
$405$	0	0
$406$	0	0
$407$	7.99299	0.396198
$408$	0	0
$409$	11.0865	0.548190	0.274095	−	0.961703i	$-0.411622\pi$
$409$	11.0865	0.548190	0.274095	+	0.961703i	$0.411622\pi$
$410$	1.45999	0.0721036
$411$	0	0
$412$	35.0249	1.72555
$413$	0	0
$414$	0	0
$415$	0.850096	0.0417296
$416$	12.6401	0.619730
$417$	0	0
$418$	35.4054	1.73173
$419$	29.8452	1.45803	0.729017	−	0.684496i	$-0.239978\pi$
$419$	29.8452	1.45803	0.729017	+	0.684496i	$0.239978\pi$
$420$	0	0
$421$	24.9163	1.21435	0.607173	−	0.794570i	$-0.292304\pi$
$421$	24.9163	1.21435	0.607173	+	0.794570i	$0.292304\pi$
$422$	−23.8559	−1.16129
$423$	0	0
$424$	−75.5905	−3.67100
$425$	13.2588	0.643148
$426$	0	0
$427$	0	0
$428$	66.1006	3.19509
$429$	0	0
$430$	−3.24150	−0.156319
$431$	−36.1330	−1.74047	−0.870233	−	0.492641i	$-0.836032\pi$
$431$	−36.1330	−1.74047	−0.870233	+	0.492641i	$0.836032\pi$
$432$	0	0
$433$	8.81259	0.423506	0.211753	−	0.977323i	$-0.432083\pi$
$433$	8.81259	0.423506	0.211753	+	0.977323i	$0.432083\pi$
$434$	0	0
$435$	0	0
$436$	−50.9489	−2.44001
$437$	2.27001	0.108590
$438$	0	0
$439$	−0.608293	−0.0290322	−0.0145161	−	0.999895i	$-0.504621\pi$
$439$	−0.608293	−0.0290322	−0.0145161	+	0.999895i	$0.504621\pi$
$440$	4.97538	0.237192
$441$	0	0
$442$	−37.5734	−1.78718
$443$	−11.7012	−0.555943	−0.277972	−	0.960589i	$-0.589662\pi$
$443$	−11.7012	−0.555943	−0.277972	+	0.960589i	$0.589662\pi$
$444$	0	0
$445$	−1.62452	−0.0770095
$446$	19.5683	0.926584
$447$	0	0
$448$	0	0
$449$	18.7621	0.885438	0.442719	−	0.896661i	$-0.354014\pi$
$449$	18.7621	0.885438	0.442719	+	0.896661i	$0.354014\pi$
$450$	0	0
$451$	5.09528	0.239927
$452$	−5.69142	−0.267702
$453$	0	0
$454$	18.2331	0.855724
$455$	0	0
$456$	0	0
$457$	−9.66815	−0.452257	−0.226129	−	0.974097i	$-0.572607\pi$
$457$	−9.66815	−0.452257	−0.226129	+	0.974097i	$0.572607\pi$
$458$	−52.7104	−2.46299
$459$	0	0
$460$	0.606891	0.0282964
$461$	−42.6767	−1.98765	−0.993826	−	0.110950i	$-0.964611\pi$
$461$	−42.6767	−1.98765	−0.993826	+	0.110950i	$0.964611\pi$
$462$	0	0
$463$	−20.0996	−0.934108	−0.467054	−	0.884229i	$-0.654685\pi$
$463$	−20.0996	−0.934108	−0.467054	+	0.884229i	$0.654685\pi$
$464$	−42.0014	−1.94987
$465$	0	0
$466$	17.9781	0.832822
$467$	−4.45200	−0.206014	−0.103007	−	0.994681i	$-0.532846\pi$
$467$	−4.45200	−0.206014	−0.103007	+	0.994681i	$0.532846\pi$
$468$	0	0
$469$	0	0
$470$	−9.43488	−0.435199
$471$	0	0
$472$	−11.4950	−0.529101
$473$	−11.3127	−0.520157
$474$	0	0
$475$	−24.8408	−1.13978
$476$	0	0
$477$	0	0
$478$	50.0256	2.28812
$479$	3.47135	0.158610	0.0793051	−	0.996850i	$-0.474730\pi$
$479$	3.47135	0.158610	0.0793051	+	0.996850i	$0.474730\pi$
$480$	0	0
$481$	15.8921	0.724617
$482$	−4.80707	−0.218956
$483$	0	0
$484$	−13.3420	−0.606455
$485$	5.80617	0.263645
$486$	0	0
$487$	−26.8456	−1.21649	−0.608244	−	0.793750i	$-0.708126\pi$
$487$	−26.8456	−1.21649	−0.608244	+	0.793750i	$0.708126\pi$
$488$	−47.3080	−2.14153
$489$	0	0
$490$	0	0
$491$	6.34289	0.286250	0.143125	−	0.989705i	$-0.454285\pi$
$491$	6.34289	0.286250	0.143125	+	0.989705i	$0.454285\pi$
$492$	0	0
$493$	21.2859	0.958670
$494$	70.3949	3.16721
$495$	0	0
$496$	−51.7678	−2.32444
$497$	0	0
$498$	0	0
$499$	10.4115	0.466084	0.233042	−	0.972467i	$-0.425132\pi$
$499$	10.4115	0.466084	0.233042	+	0.972467i	$0.425132\pi$
$500$	−13.4228	−0.600287
$501$	0	0
$502$	10.1395	0.452550
$503$	−2.38620	−0.106395	−0.0531977	−	0.998584i	$-0.516941\pi$
$503$	−2.38620	−0.106395	−0.0531977	+	0.998584i	$0.516941\pi$
$504$	0	0
$505$	1.18827	0.0528774
$506$	3.12276	0.138823
$507$	0	0
$508$	14.2835	0.633729
$509$	24.9393	1.10541	0.552707	−	0.833376i	$-0.313595\pi$
$509$	24.9393	1.10541	0.552707	+	0.833376i	$0.313595\pi$
$510$	0	0
$511$	0	0
$512$	46.9070	2.07302
$513$	0	0
$514$	−20.5370	−0.905849
$515$	2.67254	0.117766
$516$	0	0
$517$	−32.9272	−1.44814
$518$	0	0
$519$	0	0
$520$	9.89232	0.433807
$521$	36.3021	1.59042	0.795212	−	0.606331i	$-0.207359\pi$
$521$	36.3021	1.59042	0.795212	+	0.606331i	$0.207359\pi$
$522$	0	0
$523$	−6.82016	−0.298225	−0.149112	−	0.988820i	$-0.547642\pi$
$523$	−6.82016	−0.298225	−0.149112	+	0.988820i	$0.547642\pi$
$524$	−74.2862	−3.24521
$525$	0	0
$526$	55.9337	2.43882
$527$	26.2355	1.14284
$528$	0	0
$529$	−22.7998	−0.991295
$530$	−10.9734	−0.476652
$531$	0	0
$532$	0	0
$533$	10.1307	0.438809
$534$	0	0
$535$	5.04374	0.218060
$536$	50.1187	2.16480
$537$	0	0
$538$	−64.4249	−2.77755
$539$	0	0
$540$	0	0
$541$	2.13112	0.0916240	0.0458120	−	0.998950i	$-0.485412\pi$
$541$	2.13112	0.0916240	0.0458120	+	0.998950i	$0.485412\pi$
$542$	10.1274	0.435011
$543$	0	0
$544$	6.14986	0.263673
$545$	−3.88761	−0.166527
$546$	0	0
$547$	−21.5377	−0.920886	−0.460443	−	0.887689i	$-0.652309\pi$
$547$	−21.5377	−0.920886	−0.460443	+	0.887689i	$0.652309\pi$
$548$	−8.57938	−0.366493
$549$	0	0
$550$	−34.1724	−1.45712
$551$	−39.8798	−1.69894
$552$	0	0
$553$	0	0
$554$	33.6371	1.42910
$555$	0	0
$556$	−27.9242	−1.18425
$557$	−13.4744	−0.570929	−0.285464	−	0.958389i	$-0.592148\pi$
$557$	−13.4744	−0.570929	−0.285464	+	0.958389i	$0.592148\pi$
$558$	0	0
$559$	−22.4925	−0.951329
$560$	0	0
$561$	0	0
$562$	−20.1484	−0.849909
$563$	39.3103	1.65673	0.828366	−	0.560187i	$-0.189271\pi$
$563$	39.3103	1.65673	0.828366	+	0.560187i	$0.189271\pi$
$564$	0	0
$565$	−0.434279	−0.0182703
$566$	−16.6630	−0.700400
$567$	0	0
$568$	−32.7461	−1.37400
$569$	−7.91413	−0.331778	−0.165889	−	0.986144i	$-0.553049\pi$
$569$	−7.91413	−0.331778	−0.165889	+	0.986144i	$0.553049\pi$
$570$	0	0
$571$	7.50693	0.314155	0.157078	−	0.987586i	$-0.449793\pi$
$571$	7.50693	0.314155	0.157078	+	0.987586i	$0.449793\pi$
$572$	65.6813	2.74628
$573$	0	0
$574$	0	0
$575$	−2.19096	−0.0913695
$576$	0	0
$577$	23.8747	0.993919	0.496959	−	0.867774i	$-0.334450\pi$
$577$	23.8747	0.993919	0.496959	+	0.867774i	$0.334450\pi$
$578$	24.1036	1.00258
$579$	0	0
$580$	−10.6619	−0.442712
$581$	0	0
$582$	0	0
$583$	−38.2965	−1.58608
$584$	−58.4397	−2.41825
$585$	0	0
$586$	1.76490	0.0729073
$587$	−26.2630	−1.08399	−0.541994	−	0.840382i	$-0.682331\pi$
$587$	−26.2630	−1.08399	−0.541994	+	0.840382i	$0.682331\pi$
$588$	0	0
$589$	−49.1529	−2.02531
$590$	−1.66872	−0.0687000
$591$	0	0
$592$	−15.2570	−0.627057
$593$	−18.3674	−0.754257	−0.377129	−	0.926161i	$-0.623089\pi$
$593$	−18.3674	−0.754257	−0.377129	+	0.926161i	$0.623089\pi$
$594$	0	0
$595$	0	0
$596$	−61.5039	−2.51930
$597$	0	0
$598$	6.20883	0.253898
$599$	−9.26321	−0.378484	−0.189242	−	0.981930i	$-0.560603\pi$
$599$	−9.26321	−0.378484	−0.189242	+	0.981930i	$0.560603\pi$
$600$	0	0
$601$	5.98916	0.244303	0.122152	−	0.992511i	$-0.461021\pi$
$601$	5.98916	0.244303	0.122152	+	0.992511i	$0.461021\pi$
$602$	0	0
$603$	0	0
$604$	66.8991	2.72209
$605$	−1.01805	−0.0413896
$606$	0	0
$607$	0.437136	0.0177428	0.00887140	−	0.999961i	$-0.497176\pi$
$607$	0.437136	0.0177428	0.00887140	+	0.999961i	$0.497176\pi$
$608$	−11.5219	−0.467276
$609$	0	0
$610$	−6.86764	−0.278063
$611$	−65.4677	−2.64854
$612$	0	0
$613$	14.0310	0.566705	0.283352	−	0.959016i	$-0.408553\pi$
$613$	14.0310	0.566705	0.283352	+	0.959016i	$0.408553\pi$
$614$	−36.8081	−1.48545
$615$	0	0
$616$	0	0
$617$	30.0019	1.20783	0.603916	−	0.797048i	$-0.293606\pi$
$617$	30.0019	1.20783	0.603916	+	0.797048i	$0.293606\pi$
$618$	0	0
$619$	−23.9868	−0.964111	−0.482055	−	0.876141i	$-0.660110\pi$
$619$	−23.9868	−0.964111	−0.482055	+	0.876141i	$0.660110\pi$
$620$	−13.1411	−0.527759
$621$	0	0
$622$	14.0332	0.562679
$623$	0	0
$624$	0	0
$625$	23.4583	0.938333
$626$	−35.1201	−1.40368
$627$	0	0
$628$	−50.8563	−2.02939
$629$	7.73210	0.308299
$630$	0	0
$631$	34.3769	1.36852	0.684262	−	0.729236i	$-0.260124\pi$
$631$	34.3769	1.36852	0.684262	+	0.729236i	$0.260124\pi$
$632$	−40.1141	−1.59565
$633$	0	0
$634$	−35.9291	−1.42693
$635$	1.08989	0.0432511
$636$	0	0
$637$	0	0
$638$	−54.8609	−2.17196
$639$	0	0
$640$	5.49062	0.217036
$641$	31.0085	1.22476	0.612380	−	0.790563i	$-0.290212\pi$
$641$	31.0085	1.22476	0.612380	+	0.790563i	$0.290212\pi$
$642$	0	0
$643$	30.8964	1.21844	0.609218	−	0.793002i	$-0.291483\pi$
$643$	30.8964	1.21844	0.609218	+	0.793002i	$0.291483\pi$
$644$	0	0
$645$	0	0
$646$	34.2497	1.34754
$647$	16.4402	0.646332	0.323166	−	0.946342i	$-0.395253\pi$
$647$	16.4402	0.646332	0.323166	+	0.946342i	$0.395253\pi$
$648$	0	0
$649$	−5.82373	−0.228602
$650$	−67.9434	−2.66496
$651$	0	0
$652$	43.7601	1.71378
$653$	3.99989	0.156528	0.0782639	−	0.996933i	$-0.475062\pi$
$653$	3.99989	0.156528	0.0782639	+	0.996933i	$0.475062\pi$
$654$	0	0
$655$	−5.66834	−0.221480
$656$	−9.72583	−0.379730
$657$	0	0
$658$	0	0
$659$	−22.7887	−0.887721	−0.443860	−	0.896096i	$-0.646391\pi$
$659$	−22.7887	−0.887721	−0.443860	+	0.896096i	$0.646391\pi$
$660$	0	0
$661$	−41.5358	−1.61556	−0.807779	−	0.589486i	$-0.799330\pi$
$661$	−41.5358	−1.61556	−0.807779	+	0.589486i	$0.799330\pi$
$662$	−56.0262	−2.17752
$663$	0	0
$664$	−14.6000	−0.566590
$665$	0	0
$666$	0	0
$667$	−3.51740	−0.136194
$668$	29.8784	1.15603
$669$	0	0
$670$	7.27566	0.281083
$671$	−23.9677	−0.925264
$672$	0	0
$673$	15.6207	0.602132	0.301066	−	0.953603i	$-0.402657\pi$
$673$	15.6207	0.602132	0.301066	+	0.953603i	$0.402657\pi$
$674$	28.8578	1.11156
$675$	0	0
$676$	75.7824	2.91471
$677$	−34.9379	−1.34277	−0.671386	−	0.741108i	$-0.734301\pi$
$677$	−34.9379	−1.34277	−0.671386	+	0.741108i	$0.734301\pi$
$678$	0	0
$679$	0	0
$680$	4.81298	0.184569
$681$	0	0
$682$	−67.6175	−2.58921
$683$	−48.8185	−1.86799	−0.933994	−	0.357288i	$-0.883701\pi$
$683$	−48.8185	−1.86799	−0.933994	+	0.357288i	$0.883701\pi$
$684$	0	0
$685$	−0.654642	−0.0250126
$686$	0	0
$687$	0	0
$688$	21.5935	0.823246
$689$	−76.1431	−2.90082
$690$	0	0
$691$	−19.0263	−0.723796	−0.361898	−	0.932218i	$-0.617871\pi$
$691$	−19.0263	−0.723796	−0.361898	+	0.932218i	$0.617871\pi$
$692$	83.1864	3.16227
$693$	0	0
$694$	38.9454	1.47835
$695$	−2.13073	−0.0808233
$696$	0	0
$697$	4.92896	0.186698
$698$	−34.6693	−1.31225
$699$	0	0
$700$	0	0
$701$	−23.1062	−0.872709	−0.436355	−	0.899775i	$-0.643731\pi$
$701$	−23.1062	−0.872709	−0.436355	+	0.899775i	$0.643731\pi$
$702$	0	0
$703$	−14.4863	−0.546361
$704$	14.0622	0.529990
$705$	0	0
$706$	−81.4036	−3.06366
$707$	0	0
$708$	0	0
$709$	−15.7405	−0.591149	−0.295574	−	0.955320i	$-0.595511\pi$
$709$	−15.7405	−0.591149	−0.295574	+	0.955320i	$0.595511\pi$
$710$	−4.75371	−0.178404
$711$	0	0
$712$	27.9003	1.04561
$713$	−4.33529	−0.162358
$714$	0	0
$715$	5.01176	0.187429
$716$	60.2901	2.25315
$717$	0	0
$718$	−25.8027	−0.962949
$719$	−41.3954	−1.54379	−0.771895	−	0.635751i	$-0.780691\pi$
$719$	−41.3954	−1.54379	−0.771895	+	0.635751i	$0.780691\pi$
$720$	0	0
$721$	0	0
$722$	−16.7970	−0.625118
$723$	0	0
$724$	−37.3402	−1.38774
$725$	38.4910	1.42952
$726$	0	0
$727$	43.9132	1.62865	0.814325	−	0.580410i	$-0.197108\pi$
$727$	43.9132	1.62865	0.814325	+	0.580410i	$0.197108\pi$
$728$	0	0
$729$	0	0
$730$	−8.48361	−0.313992
$731$	−10.9434	−0.404757
$732$	0	0
$733$	−40.4747	−1.49497	−0.747484	−	0.664280i	$-0.768738\pi$
$733$	−40.4747	−1.49497	−0.747484	+	0.664280i	$0.768738\pi$
$734$	21.7578	0.803096
$735$	0	0
$736$	−1.01624	−0.0374590
$737$	25.3917	0.935315
$738$	0	0
$739$	35.3552	1.30056	0.650282	−	0.759693i	$-0.274651\pi$
$739$	35.3552	1.30056	0.650282	+	0.759693i	$0.274651\pi$
$740$	−3.87293	−0.142372
$741$	0	0
$742$	0	0
$743$	14.4855	0.531422	0.265711	−	0.964053i	$-0.414393\pi$
$743$	14.4855	0.531422	0.265711	+	0.964053i	$0.414393\pi$
$744$	0	0
$745$	−4.69300	−0.171938
$746$	−51.0913	−1.87059
$747$	0	0
$748$	31.9564	1.16844
$749$	0	0
$750$	0	0
$751$	−12.9009	−0.470760	−0.235380	−	0.971903i	$-0.575634\pi$
$751$	−12.9009	−0.470760	−0.235380	+	0.971903i	$0.575634\pi$
$752$	62.8513	2.29195
$753$	0	0
$754$	−109.077	−3.97236
$755$	5.10468	0.185778
$756$	0	0
$757$	−17.2065	−0.625381	−0.312690	−	0.949855i	$-0.601230\pi$
$757$	−17.2065	−0.625381	−0.312690	+	0.949855i	$0.601230\pi$
$758$	−2.69523	−0.0978950
$759$	0	0
$760$	−9.01726	−0.327090
$761$	31.2123	1.13145	0.565723	−	0.824595i	$-0.308597\pi$
$761$	31.2123	1.13145	0.565723	+	0.824595i	$0.308597\pi$
$762$	0	0
$763$	0	0
$764$	−51.0990	−1.84870
$765$	0	0
$766$	−73.2664	−2.64722
$767$	−11.5791	−0.418096
$768$	0	0
$769$	5.41379	0.195226	0.0976132	−	0.995224i	$-0.468879\pi$
$769$	5.41379	0.195226	0.0976132	+	0.995224i	$0.468879\pi$
$770$	0	0
$771$	0	0
$772$	−15.6512	−0.563299
$773$	3.14970	0.113287	0.0566434	−	0.998394i	$-0.481960\pi$
$773$	3.14970	0.113287	0.0566434	+	0.998394i	$0.481960\pi$
$774$	0	0
$775$	47.4412	1.70414
$776$	−99.7183	−3.57968
$777$	0	0
$778$	31.4221	1.12654
$779$	−9.23456	−0.330862
$780$	0	0
$781$	−16.5902	−0.593644
$782$	3.02083	0.108025
$783$	0	0
$784$	0	0
$785$	−3.88055	−0.138503
$786$	0	0
$787$	22.2648	0.793653	0.396827	−	0.917894i	$-0.370111\pi$
$787$	22.2648	0.793653	0.396827	+	0.917894i	$0.370111\pi$
$788$	−76.4047	−2.72180
$789$	0	0
$790$	−5.82330	−0.207184
$791$	0	0
$792$	0	0
$793$	−47.6539	−1.69224
$794$	−81.7613	−2.90160
$795$	0	0
$796$	86.1176	3.05236
$797$	5.62326	0.199186	0.0995930	−	0.995028i	$-0.468246\pi$
$797$	5.62326	0.199186	0.0995930	+	0.995028i	$0.468246\pi$
$798$	0	0
$799$	−31.8525	−1.12686
$800$	11.1207	0.393176
$801$	0	0
$802$	17.1380	0.605163
$803$	−29.6073	−1.04482
$804$	0	0
$805$	0	0
$806$	−134.441	−4.73547
$807$	0	0
$808$	−20.4080	−0.717951
$809$	41.0142	1.44198	0.720992	−	0.692944i	$-0.243686\pi$
$809$	41.0142	1.44198	0.720992	+	0.692944i	$0.243686\pi$
$810$	0	0
$811$	−19.4988	−0.684696	−0.342348	−	0.939573i	$-0.611222\pi$
$811$	−19.4988	−0.684696	−0.342348	+	0.939573i	$0.611222\pi$
$812$	0	0
$813$	0	0
$814$	−19.9282	−0.698482
$815$	3.33908	0.116963
$816$	0	0
$817$	20.5028	0.717302
$818$	−27.6408	−0.966438
$819$	0	0
$820$	−2.46887	−0.0862167
$821$	33.5020	1.16923	0.584615	−	0.811311i	$-0.301246\pi$
$821$	33.5020	1.16923	0.584615	+	0.811311i	$0.301246\pi$
$822$	0	0
$823$	14.6004	0.508939	0.254470	−	0.967081i	$-0.418099\pi$
$823$	14.6004	0.508939	0.254470	+	0.967081i	$0.418099\pi$
$824$	−45.8997	−1.59899
$825$	0	0
$826$	0	0
$827$	46.8791	1.63015	0.815073	−	0.579358i	$-0.196697\pi$
$827$	46.8791	1.63015	0.815073	+	0.579358i	$0.196697\pi$
$828$	0	0
$829$	−38.9407	−1.35247	−0.676234	−	0.736687i	$-0.736389\pi$
$829$	−38.9407	−1.35247	−0.676234	+	0.736687i	$0.736389\pi$
$830$	−2.11946	−0.0735676
$831$	0	0
$832$	27.9593	0.969314
$833$	0	0
$834$	0	0
$835$	2.27985	0.0788974
$836$	−59.8713	−2.07069
$837$	0	0
$838$	−74.4102	−2.57046
$839$	29.0514	1.00296	0.501482	−	0.865168i	$-0.332788\pi$
$839$	29.0514	1.00296	0.501482	+	0.865168i	$0.332788\pi$
$840$	0	0
$841$	32.7940	1.13083
$842$	−62.1214	−2.14084
$843$	0	0
$844$	40.3409	1.38859
$845$	5.78250	0.198924
$846$	0	0
$847$	0	0
$848$	73.1000	2.51026
$849$	0	0
$850$	−33.0570	−1.13385
$851$	−1.27769	−0.0437988
$852$	0	0
$853$	10.3380	0.353967	0.176983	−	0.984214i	$-0.443366\pi$
$853$	10.3380	0.353967	0.176983	+	0.984214i	$0.443366\pi$
$854$	0	0
$855$	0	0
$856$	−86.6239	−2.96075
$857$	−13.0950	−0.447318	−0.223659	−	0.974667i	$-0.571800\pi$
$857$	−13.0950	−0.447318	−0.223659	+	0.974667i	$0.571800\pi$
$858$	0	0
$859$	−41.8040	−1.42633	−0.713166	−	0.700995i	$-0.752740\pi$
$859$	−41.8040	−1.42633	−0.713166	+	0.700995i	$0.752740\pi$
$860$	5.48145	0.186916
$861$	0	0
$862$	90.0869	3.06837
$863$	1.96226	0.0667962	0.0333981	−	0.999442i	$-0.489367\pi$
$863$	1.96226	0.0667962	0.0333981	+	0.999442i	$0.489367\pi$
$864$	0	0
$865$	6.34746	0.215820
$866$	−21.9716	−0.746625
$867$	0	0
$868$	0	0
$869$	−20.3230	−0.689411
$870$	0	0
$871$	50.4851	1.71062
$872$	66.7679	2.26105
$873$	0	0
$874$	−5.65961	−0.191439
$875$	0	0
$876$	0	0
$877$	40.5082	1.36787	0.683933	−	0.729545i	$-0.260268\pi$
$877$	40.5082	1.36787	0.683933	+	0.729545i	$0.260268\pi$
$878$	1.51660	0.0511827
$879$	0	0
$880$	−4.81146	−0.162194
$881$	16.1167	0.542986	0.271493	−	0.962440i	$-0.412483\pi$
$881$	16.1167	0.542986	0.271493	+	0.962440i	$0.412483\pi$
$882$	0	0
$883$	−15.1670	−0.510409	−0.255205	−	0.966887i	$-0.582143\pi$
$883$	−15.1670	−0.510409	−0.255205	+	0.966887i	$0.582143\pi$
$884$	63.5374	2.13700
$885$	0	0
$886$	29.1736	0.980106
$887$	−11.2847	−0.378902	−0.189451	−	0.981890i	$-0.560671\pi$
$887$	−11.2847	−0.378902	−0.189451	+	0.981890i	$0.560671\pi$
$888$	0	0
$889$	0	0
$890$	4.05025	0.135765
$891$	0	0
$892$	−33.0904	−1.10795
$893$	59.6765	1.99700
$894$	0	0
$895$	4.60038	0.153774
$896$	0	0
$897$	0	0
$898$	−46.7777	−1.56099
$899$	76.1628	2.54017
$900$	0	0
$901$	−37.0464	−1.23420
$902$	−12.7036	−0.422982
$903$	0	0
$904$	7.45854	0.248067
$905$	−2.84921	−0.0947109
$906$	0	0
$907$	−10.3421	−0.343403	−0.171701	−	0.985149i	$-0.554926\pi$
$907$	−10.3421	−0.343403	−0.171701	+	0.985149i	$0.554926\pi$
$908$	−30.8327	−1.02322
$909$	0	0
$910$	0	0
$911$	39.3514	1.30377	0.651885	−	0.758318i	$-0.273979\pi$
$911$	39.3514	1.30377	0.651885	+	0.758318i	$0.273979\pi$
$912$	0	0
$913$	−7.39681	−0.244799
$914$	24.1047	0.797312
$915$	0	0
$916$	89.1344	2.94508
$917$	0	0
$918$	0	0
$919$	24.7410	0.816129	0.408065	−	0.912953i	$-0.366204\pi$
$919$	24.7410	0.816129	0.408065	+	0.912953i	$0.366204\pi$
$920$	−0.795323	−0.0262210
$921$	0	0
$922$	106.402	3.50415
$923$	−32.9855	−1.08573
$924$	0	0
$925$	13.9818	0.459720
$926$	50.1124	1.64680
$927$	0	0
$928$	17.8533	0.586064
$929$	31.7963	1.04320	0.521602	−	0.853189i	$-0.325335\pi$
$929$	31.7963	1.04320	0.521602	+	0.853189i	$0.325335\pi$
$930$	0	0
$931$	0	0
$932$	−30.4015	−0.995833
$933$	0	0
$934$	11.0997	0.363194
$935$	2.43840	0.0797444
$936$	0	0
$937$	−50.3893	−1.64615	−0.823073	−	0.567935i	$-0.807742\pi$
$937$	−50.3893	−1.64615	−0.823073	+	0.567935i	$0.807742\pi$
$938$	0	0
$939$	0	0
$940$	15.9546	0.520381
$941$	16.5774	0.540407	0.270204	−	0.962803i	$-0.412909\pi$
$941$	16.5774	0.540407	0.270204	+	0.962803i	$0.412909\pi$
$942$	0	0
$943$	−0.814489	−0.0265234
$944$	11.1163	0.361805
$945$	0	0
$946$	28.2048	0.917016
$947$	−13.2088	−0.429227	−0.214613	−	0.976699i	$-0.568849\pi$
$947$	−13.2088	−0.429227	−0.214613	+	0.976699i	$0.568849\pi$
$948$	0	0
$949$	−58.8669	−1.91090
$950$	61.9333	2.00938
$951$	0	0
$952$	0	0
$953$	54.8490	1.77673	0.888366	−	0.459136i	$-0.151841\pi$
$953$	54.8490	1.77673	0.888366	+	0.459136i	$0.151841\pi$
$954$	0	0
$955$	−3.89906	−0.126171
$956$	−84.5944	−2.73598
$957$	0	0
$958$	−8.65479	−0.279624
$959$	0	0
$960$	0	0
$961$	62.8727	2.02815
$962$	−39.6222	−1.27747
$963$	0	0
$964$	8.12886	0.261813
$965$	−1.19425	−0.0384443
$966$	0	0
$967$	−44.1932	−1.42116	−0.710578	−	0.703618i	$-0.751567\pi$
$967$	−44.1932	−1.42116	−0.710578	+	0.703618i	$0.751567\pi$
$968$	17.4845	0.561974
$969$	0	0
$970$	−14.4760	−0.464795
$971$	−41.3069	−1.32560	−0.662801	−	0.748795i	$-0.730633\pi$
$971$	−41.3069	−1.32560	−0.662801	+	0.748795i	$0.730633\pi$
$972$	0	0
$973$	0	0
$974$	66.9315	2.14462
$975$	0	0
$976$	45.7494	1.46440
$977$	42.4623	1.35849	0.679244	−	0.733912i	$-0.262308\pi$
$977$	42.4623	1.35849	0.679244	+	0.733912i	$0.262308\pi$
$978$	0	0
$979$	14.1352	0.451762
$980$	0	0
$981$	0	0
$982$	−15.8141	−0.504648
$983$	−12.4536	−0.397210	−0.198605	−	0.980080i	$-0.563641\pi$
$983$	−12.4536	−0.397210	−0.198605	+	0.980080i	$0.563641\pi$
$984$	0	0
$985$	−5.82999	−0.185759
$986$	−53.0701	−1.69010
$987$	0	0
$988$	−119.039	−3.78714
$989$	1.80835	0.0575021
$990$	0	0
$991$	−45.6152	−1.44902	−0.724508	−	0.689266i	$-0.757933\pi$
$991$	−45.6152	−1.44902	−0.724508	+	0.689266i	$0.757933\pi$
$992$	22.0047	0.698650
$993$	0	0
$994$	0	0
$995$	6.57112	0.208319
$996$	0	0
$997$	−8.69533	−0.275384	−0.137692	−	0.990475i	$-0.543968\pi$
$997$	−8.69533	−0.275384	−0.137692	+	0.990475i	$0.543968\pi$
$998$	−25.9580	−0.821687
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3087.2.a.m.1.3		9
3.2	odd	2		1029.2.a.h.1.7	yes	9
7.6	odd	2		3087.2.a.l.1.3		9
21.2	odd	6		1029.2.e.e.361.3		18
21.5	even	6		1029.2.e.f.361.3		18
21.11	odd	6		1029.2.e.e.667.3		18
21.17	even	6		1029.2.e.f.667.3		18
21.20	even	2		1029.2.a.g.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1029.2.a.g.1.7	✓	9	21.20	even	2
1029.2.a.h.1.7	yes	9	3.2	odd	2
1029.2.e.e.361.3		18	21.2	odd	6
1029.2.e.e.667.3		18	21.11	odd	6
1029.2.e.f.361.3		18	21.5	even	6
1029.2.e.f.667.3		18	21.17	even	6
3087.2.a.l.1.3		9	7.6	odd	2
3087.2.a.m.1.3		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3087 = 3^{2} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3087.a (trivial)

\(f(q)\)	\(=\)	\(q-2.49320 q^{2} +4.21606 q^{4} +0.321703 q^{5} -5.52510 q^{8} +O(q^{10})\)	\(q-2.49320 q^{2} +4.21606 q^{4} +0.321703 q^{5} -5.52510 q^{8} -0.802071 q^{10} -2.79918 q^{11} -5.56549 q^{13} +5.34306 q^{16} -2.70782 q^{17} +5.07317 q^{19} +1.35632 q^{20} +6.97894 q^{22} +0.447454 q^{23} -4.89651 q^{25} +13.8759 q^{26} -7.86092 q^{29} -9.68879 q^{31} -2.27115 q^{32} +6.75114 q^{34} -2.85547 q^{37} -12.6485 q^{38} -1.77744 q^{40} -1.82027 q^{41} +4.04141 q^{43} -11.8015 q^{44} -1.11559 q^{46} +11.7632 q^{47} +12.2080 q^{50} -23.4645 q^{52} +13.6813 q^{53} -0.900506 q^{55} +19.5989 q^{58} +2.08051 q^{59} +8.56239 q^{61} +24.1561 q^{62} -5.02369 q^{64} -1.79043 q^{65} -9.07110 q^{67} -11.4163 q^{68} +5.92679 q^{71} +10.5771 q^{73} +7.11927 q^{74} +21.3888 q^{76} +7.26034 q^{79} +1.71888 q^{80} +4.53831 q^{82} +2.64249 q^{83} -0.871113 q^{85} -10.0761 q^{86} +15.4658 q^{88} -5.04974 q^{89} +1.88650 q^{92} -29.3279 q^{94} +1.63206 q^{95} +18.0482 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + 17 q^{4} + q^{5} - 12 q^{8}+O(q^{10}) \) 9 * q - 5 * q^2 + 17 * q^4 + q^5 - 12 * q^8	\( 9 q - 5 q^{2} + 17 q^{4} + q^{5} - 12 q^{8} - q^{10} - 21 q^{11} + 11 q^{13} + 29 q^{16} + 6 q^{17} - 2 q^{19} + 20 q^{20} - 9 q^{23} + 10 q^{25} + 23 q^{26} - 12 q^{29} - 3 q^{31} - 24 q^{32} + 12 q^{34} + 8 q^{37} + 15 q^{38} + 7 q^{40} + 17 q^{41} + 16 q^{43} - 46 q^{44} + 13 q^{46} + 7 q^{47} - 3 q^{50} + 26 q^{52} + 8 q^{53} - 6 q^{55} - 22 q^{58} + 30 q^{59} + 7 q^{61} + 43 q^{62} + 42 q^{64} + 15 q^{65} + 33 q^{67} + 66 q^{68} - 30 q^{71} + 48 q^{73} + 8 q^{74} - 34 q^{76} + 7 q^{79} + 98 q^{80} - 10 q^{82} + 11 q^{83} - 18 q^{85} + 3 q^{86} - 47 q^{88} + 25 q^{89} + 21 q^{92} + 11 q^{94} + 13 q^{95} + 50 q^{97}+O(q^{100}) \) 9 * q - 5 * q^2 + 17 * q^4 + q^5 - 12 * q^8 - q^10 - 21 * q^11 + 11 * q^13 + 29 * q^16 + 6 * q^17 - 2 * q^19 + 20 * q^20 - 9 * q^23 + 10 * q^25 + 23 * q^26 - 12 * q^29 - 3 * q^31 - 24 * q^32 + 12 * q^34 + 8 * q^37 + 15 * q^38 + 7 * q^40 + 17 * q^41 + 16 * q^43 - 46 * q^44 + 13 * q^46 + 7 * q^47 - 3 * q^50 + 26 * q^52 + 8 * q^53 - 6 * q^55 - 22 * q^58 + 30 * q^59 + 7 * q^61 + 43 * q^62 + 42 * q^64 + 15 * q^65 + 33 * q^67 + 66 * q^68 - 30 * q^71 + 48 * q^73 + 8 * q^74 - 34 * q^76 + 7 * q^79 + 98 * q^80 - 10 * q^82 + 11 * q^83 - 18 * q^85 + 3 * q^86 - 47 * q^88 + 25 * q^89 + 21 * q^92 + 11 * q^94 + 13 * q^95 + 50 * q^97

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