LMFDB - Embedded newform 2790.2.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2790,2,Mod(1,2790)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2790, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2790.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2790.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:	$22.2782621639$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 310)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	2790.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.07838 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.07838 q^{7} -1.00000 q^{8} +1.00000 q^{10} -6.04945 q^{11} -0.290725 q^{13} -1.07838 q^{14} +1.00000 q^{16} -1.07838 q^{17} +5.26180 q^{19} -1.00000 q^{20} +6.04945 q^{22} +4.34017 q^{23} +1.00000 q^{25} +0.290725 q^{26} +1.07838 q^{28} +9.31124 q^{29} +1.00000 q^{31} -1.00000 q^{32} +1.07838 q^{34} -1.07838 q^{35} -2.44748 q^{37} -5.26180 q^{38} +1.00000 q^{40} -5.60197 q^{41} -7.86603 q^{43} -6.04945 q^{44} -4.34017 q^{46} +1.75872 q^{47} -5.83710 q^{49} -1.00000 q^{50} -0.290725 q^{52} +6.44748 q^{53} +6.04945 q^{55} -1.07838 q^{56} -9.31124 q^{58} -11.4186 q^{59} -12.0494 q^{61} -1.00000 q^{62} +1.00000 q^{64} +0.290725 q^{65} -13.9421 q^{67} -1.07838 q^{68} +1.07838 q^{70} +4.68035 q^{71} -8.18342 q^{73} +2.44748 q^{74} +5.26180 q^{76} -6.52359 q^{77} +14.2557 q^{79} -1.00000 q^{80} +5.60197 q^{82} -3.55252 q^{83} +1.07838 q^{85} +7.86603 q^{86} +6.04945 q^{88} -3.84324 q^{89} -0.313511 q^{91} +4.34017 q^{92} -1.75872 q^{94} -5.26180 q^{95} +2.49693 q^{97} +5.83710 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{8} + 3 q^{10} - 8 q^{13} + 3 q^{16} + 8 q^{19} - 3 q^{20} + 2 q^{23} + 3 q^{25} + 8 q^{26} + 2 q^{29} + 3 q^{31} - 3 q^{32} - 8 q^{37} - 8 q^{38} + 3 q^{40} + 2 q^{41} - 10 q^{43} - 2 q^{46} - 20 q^{47} + 11 q^{49} - 3 q^{50} - 8 q^{52} + 20 q^{53} - 2 q^{58} - 20 q^{59} - 18 q^{61} - 3 q^{62} + 3 q^{64} + 8 q^{65} - 12 q^{67} - 8 q^{71} - 20 q^{73} + 8 q^{74} + 8 q^{76} - 4 q^{77} - 3 q^{80} - 2 q^{82} - 10 q^{83} + 10 q^{86} - 18 q^{89} + 12 q^{91} + 2 q^{92} + 20 q^{94} - 8 q^{95} - 10 q^{97} - 11 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^8 + 3 * q^10 - 8 * q^13 + 3 * q^16 + 8 * q^19 - 3 * q^20 + 2 * q^23 + 3 * q^25 + 8 * q^26 + 2 * q^29 + 3 * q^31 - 3 * q^32 - 8 * q^37 - 8 * q^38 + 3 * q^40 + 2 * q^41 - 10 * q^43 - 2 * q^46 - 20 * q^47 + 11 * q^49 - 3 * q^50 - 8 * q^52 + 20 * q^53 - 2 * q^58 - 20 * q^59 - 18 * q^61 - 3 * q^62 + 3 * q^64 + 8 * q^65 - 12 * q^67 - 8 * q^71 - 20 * q^73 + 8 * q^74 + 8 * q^76 - 4 * q^77 - 3 * q^80 - 2 * q^82 - 10 * q^83 + 10 * q^86 - 18 * q^89 + 12 * q^91 + 2 * q^92 + 20 * q^94 - 8 * q^95 - 10 * q^97 - 11 * q^98

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 $

$2$	−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
$7$	1.07838	0.407588	0.203794	−	0.979014i	$-0.434673\pi$
$7$	1.07838	0.407588	0.203794	+	0.979014i	$0.434673\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−6.04945	−1.82398	−0.911989	−	0.410215i	$-0.865454\pi$
$11$	−6.04945	−1.82398	−0.911989	+	0.410215i	$0.865454\pi$
$12$	0	0
$13$	−0.290725	−0.0806325	−0.0403163	−	0.999187i	$-0.512837\pi$
$13$	−0.290725	−0.0806325	−0.0403163	+	0.999187i	$0.512837\pi$
$14$	−1.07838	−0.288209
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.07838	−0.261545	−0.130773	−	0.991412i	$-0.541746\pi$
$17$	−1.07838	−0.261545	−0.130773	+	0.991412i	$0.541746\pi$
$18$	0	0
$19$	5.26180	1.20714	0.603569	−	0.797311i	$-0.293745\pi$
$19$	5.26180	1.20714	0.603569	+	0.797311i	$0.293745\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	6.04945	1.28975
$23$	4.34017	0.904989	0.452494	−	0.891767i	$-0.350534\pi$
$23$	4.34017	0.904989	0.452494	+	0.891767i	$0.350534\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.290725	0.0570158
$27$	0	0
$28$	1.07838	0.203794
$29$	9.31124	1.72905	0.864527	−	0.502586i	$-0.167618\pi$
$29$	9.31124	1.72905	0.864527	+	0.502586i	$0.167618\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.07838	0.184940
$35$	−1.07838	−0.182279
$36$	0	0
$37$	−2.44748	−0.402363	−0.201182	−	0.979554i	$-0.564478\pi$
$37$	−2.44748	−0.402363	−0.201182	+	0.979554i	$0.564478\pi$
$38$	−5.26180	−0.853576
$39$	0	0
$40$	1.00000	0.158114
$41$	−5.60197	−0.874880	−0.437440	−	0.899247i	$-0.644115\pi$
$41$	−5.60197	−0.874880	−0.437440	+	0.899247i	$0.644115\pi$
$42$	0	0
$43$	−7.86603	−1.19956	−0.599779	−	0.800166i	$-0.704745\pi$
$43$	−7.86603	−1.19956	−0.599779	+	0.800166i	$0.704745\pi$
$44$	−6.04945	−0.911989
$45$	0	0
$46$	−4.34017	−0.639924
$47$	1.75872	0.256536	0.128268	−	0.991740i	$-0.459058\pi$
$47$	1.75872	0.256536	0.128268	+	0.991740i	$0.459058\pi$
$48$	0	0
$49$	−5.83710	−0.833872
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−0.290725	−0.0403163
$53$	6.44748	0.885630	0.442815	−	0.896613i	$-0.353980\pi$
$53$	6.44748	0.885630	0.442815	+	0.896613i	$0.353980\pi$
$54$	0	0
$55$	6.04945	0.815707
$56$	−1.07838	−0.144104
$57$	0	0
$58$	−9.31124	−1.22263
$59$	−11.4186	−1.48657	−0.743284	−	0.668976i	$-0.766733\pi$
$59$	−11.4186	−1.48657	−0.743284	+	0.668976i	$0.766733\pi$
$60$	0	0
$61$	−12.0494	−1.54277	−0.771387	−	0.636366i	$-0.780437\pi$
$61$	−12.0494	−1.54277	−0.771387	+	0.636366i	$0.780437\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	0.290725	0.0360600
$66$	0	0
$67$	−13.9421	−1.70330	−0.851652	−	0.524108i	$-0.824399\pi$
$67$	−13.9421	−1.70330	−0.851652	+	0.524108i	$0.824399\pi$
$68$	−1.07838	−0.130773
$69$	0	0
$70$	1.07838	0.128891
$71$	4.68035	0.555455	0.277727	−	0.960660i	$-0.410419\pi$
$71$	4.68035	0.555455	0.277727	+	0.960660i	$0.410419\pi$
$72$	0	0
$73$	−8.18342	−0.957797	−0.478898	−	0.877870i	$-0.658964\pi$
$73$	−8.18342	−0.957797	−0.478898	+	0.877870i	$0.658964\pi$
$74$	2.44748	0.284514
$75$	0	0
$76$	5.26180	0.603569
$77$	−6.52359	−0.743432
$78$	0	0
$79$	14.2557	1.60389	0.801943	−	0.597400i	$-0.203800\pi$
$79$	14.2557	1.60389	0.801943	+	0.597400i	$0.203800\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	5.60197	0.618634
$83$	−3.55252	−0.389940	−0.194970	−	0.980809i	$-0.562461\pi$
$83$	−3.55252	−0.389940	−0.194970	+	0.980809i	$0.562461\pi$
$84$	0	0
$85$	1.07838	0.116966
$86$	7.86603	0.848216
$87$	0	0
$88$	6.04945	0.644873
$89$	−3.84324	−0.407383	−0.203692	−	0.979035i	$-0.565294\pi$
$89$	−3.84324	−0.407383	−0.203692	+	0.979035i	$0.565294\pi$
$90$	0	0
$91$	−0.313511	−0.0328649
$92$	4.34017	0.452494
$93$	0	0
$94$	−1.75872	−0.181398
$95$	−5.26180	−0.539849
$96$	0	0
$97$	2.49693	0.253525	0.126762	−	0.991933i	$-0.459541\pi$
$97$	2.49693	0.253525	0.126762	+	0.991933i	$0.459541\pi$
$98$	5.83710	0.589636
$99$	0	0
$100$	1.00000	0.100000
$101$	−4.83710	−0.481310	−0.240655	−	0.970611i	$-0.577362\pi$
$101$	−4.83710	−0.481310	−0.240655	+	0.970611i	$0.577362\pi$
$102$	0	0
$103$	6.83710	0.673680	0.336840	−	0.941562i	$-0.390642\pi$
$103$	6.83710	0.673680	0.336840	+	0.941562i	$0.390642\pi$
$104$	0.290725	0.0285079
$105$	0	0
$106$	−6.44748	−0.626235
$107$	2.52359	0.243965	0.121982	−	0.992532i	$-0.461075\pi$
$107$	2.52359	0.243965	0.121982	+	0.992532i	$0.461075\pi$
$108$	0	0
$109$	−1.41855	−0.135873	−0.0679363	−	0.997690i	$-0.521641\pi$
$109$	−1.41855	−0.135873	−0.0679363	+	0.997690i	$0.521641\pi$
$110$	−6.04945	−0.576792
$111$	0	0
$112$	1.07838	0.101897
$113$	−15.1773	−1.42776	−0.713879	−	0.700269i	$-0.753063\pi$
$113$	−15.1773	−1.42776	−0.713879	+	0.700269i	$0.753063\pi$
$114$	0	0
$115$	−4.34017	−0.404723
$116$	9.31124	0.864527
$117$	0	0
$118$	11.4186	1.05116
$119$	−1.16290	−0.106603
$120$	0	0
$121$	25.5958	2.32689
$122$	12.0494	1.09091
$123$	0	0
$124$	1.00000	0.0898027
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.23513	−0.109600	−0.0548002	−	0.998497i	$-0.517452\pi$
$127$	−1.23513	−0.109600	−0.0548002	+	0.998497i	$0.517452\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−0.290725	−0.0254982
$131$	−0.581449	−0.0508015	−0.0254007	−	0.999677i	$-0.508086\pi$
$131$	−0.581449	−0.0508015	−0.0254007	+	0.999677i	$0.508086\pi$
$132$	0	0
$133$	5.67420	0.492016
$134$	13.9421	1.20442
$135$	0	0
$136$	1.07838	0.0924701
$137$	−3.91548	−0.334522	−0.167261	−	0.985913i	$-0.553492\pi$
$137$	−3.91548	−0.334522	−0.167261	+	0.985913i	$0.553492\pi$
$138$	0	0
$139$	−15.9916	−1.35639	−0.678194	−	0.734882i	$-0.737237\pi$
$139$	−15.9916	−1.35639	−0.678194	+	0.734882i	$0.737237\pi$
$140$	−1.07838	−0.0911396
$141$	0	0
$142$	−4.68035	−0.392766
$143$	1.75872	0.147072
$144$	0	0
$145$	−9.31124	−0.773257
$146$	8.18342	0.677264
$147$	0	0
$148$	−2.44748	−0.201182
$149$	−1.90110	−0.155744	−0.0778722	−	0.996963i	$-0.524813\pi$
$149$	−1.90110	−0.155744	−0.0778722	+	0.996963i	$0.524813\pi$
$150$	0	0
$151$	−9.94214	−0.809080	−0.404540	−	0.914520i	$-0.632568\pi$
$151$	−9.94214	−0.809080	−0.404540	+	0.914520i	$0.632568\pi$
$152$	−5.26180	−0.426788
$153$	0	0
$154$	6.52359	0.525686
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−14.3135	−1.14234	−0.571171	−	0.820831i	$-0.693511\pi$
$157$	−14.3135	−1.14234	−0.571171	+	0.820831i	$0.693511\pi$
$158$	−14.2557	−1.13412
$159$	0	0
$160$	1.00000	0.0790569
$161$	4.68035	0.368863
$162$	0	0
$163$	−16.4124	−1.28552	−0.642759	−	0.766068i	$-0.722211\pi$
$163$	−16.4124	−1.28552	−0.642759	+	0.766068i	$0.722211\pi$
$164$	−5.60197	−0.437440
$165$	0	0
$166$	3.55252	0.275729
$167$	−19.1773	−1.48398	−0.741991	−	0.670410i	$-0.766118\pi$
$167$	−19.1773	−1.48398	−0.741991	+	0.670410i	$0.766118\pi$
$168$	0	0
$169$	−12.9155	−0.993498
$170$	−1.07838	−0.0827078
$171$	0	0
$172$	−7.86603	−0.599779
$173$	5.51745	0.419484	0.209742	−	0.977757i	$-0.432738\pi$
$173$	5.51745	0.419484	0.209742	+	0.977757i	$0.432738\pi$
$174$	0	0
$175$	1.07838	0.0815177
$176$	−6.04945	−0.455994
$177$	0	0
$178$	3.84324	0.288063
$179$	6.36296	0.475590	0.237795	−	0.971315i	$-0.423575\pi$
$179$	6.36296	0.475590	0.237795	+	0.971315i	$0.423575\pi$
$180$	0	0
$181$	0.264063	0.0196276	0.00981381	−	0.999952i	$-0.496876\pi$
$181$	0.264063	0.0196276	0.00981381	+	0.999952i	$0.496876\pi$
$182$	0.313511	0.0232390
$183$	0	0
$184$	−4.34017	−0.319962
$185$	2.44748	0.179942
$186$	0	0
$187$	6.52359	0.477052
$188$	1.75872	0.128268
$189$	0	0
$190$	5.26180	0.381731
$191$	−2.65368	−0.192014	−0.0960069	−	0.995381i	$-0.530607\pi$
$191$	−2.65368	−0.192014	−0.0960069	+	0.995381i	$0.530607\pi$
$192$	0	0
$193$	0.156755	0.0112835	0.00564175	−	0.999984i	$-0.498204\pi$
$193$	0.156755	0.0112835	0.00564175	+	0.999984i	$0.498204\pi$
$194$	−2.49693	−0.179269
$195$	0	0
$196$	−5.83710	−0.416936
$197$	−3.70928	−0.264275	−0.132137	−	0.991231i	$-0.542184\pi$
$197$	−3.70928	−0.264275	−0.132137	+	0.991231i	$0.542184\pi$
$198$	0	0
$199$	6.15676	0.436441	0.218220	−	0.975900i	$-0.429975\pi$
$199$	6.15676	0.436441	0.218220	+	0.975900i	$0.429975\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	4.83710	0.340337
$203$	10.0410	0.704743
$204$	0	0
$205$	5.60197	0.391258
$206$	−6.83710	−0.476363
$207$	0	0
$208$	−0.290725	−0.0201581
$209$	−31.8310	−2.20179
$210$	0	0
$211$	25.3607	1.74590	0.872951	−	0.487808i	$-0.162203\pi$
$211$	25.3607	1.74590	0.872951	+	0.487808i	$0.162203\pi$
$212$	6.44748	0.442815
$213$	0	0
$214$	−2.52359	−0.172509
$215$	7.86603	0.536459
$216$	0	0
$217$	1.07838	0.0732051
$218$	1.41855	0.0960764
$219$	0	0
$220$	6.04945	0.407854
$221$	0.313511	0.0210890
$222$	0	0
$223$	−24.7526	−1.65756	−0.828778	−	0.559578i	$-0.810963\pi$
$223$	−24.7526	−1.65756	−0.828778	+	0.559578i	$0.810963\pi$
$224$	−1.07838	−0.0720521
$225$	0	0
$226$	15.1773	1.00958
$227$	−11.7321	−0.778684	−0.389342	−	0.921093i	$-0.627298\pi$
$227$	−11.7321	−0.778684	−0.389342	+	0.921093i	$0.627298\pi$
$228$	0	0
$229$	6.88655	0.455076	0.227538	−	0.973769i	$-0.426932\pi$
$229$	6.88655	0.455076	0.227538	+	0.973769i	$0.426932\pi$
$230$	4.34017	0.286183
$231$	0	0
$232$	−9.31124	−0.611313
$233$	−25.7009	−1.68372	−0.841860	−	0.539696i	$-0.818539\pi$
$233$	−25.7009	−1.68372	−0.841860	+	0.539696i	$0.818539\pi$
$234$	0	0
$235$	−1.75872	−0.114726
$236$	−11.4186	−0.743284
$237$	0	0
$238$	1.16290	0.0753795
$239$	23.7321	1.53510	0.767550	−	0.640989i	$-0.221476\pi$
$239$	23.7321	1.53510	0.767550	+	0.640989i	$0.221476\pi$
$240$	0	0
$241$	−6.09890	−0.392864	−0.196432	−	0.980517i	$-0.562936\pi$
$241$	−6.09890	−0.392864	−0.196432	+	0.980517i	$0.562936\pi$
$242$	−25.5958	−1.64536
$243$	0	0
$244$	−12.0494	−0.771387
$245$	5.83710	0.372919
$246$	0	0
$247$	−1.52973	−0.0973346
$248$	−1.00000	−0.0635001
$249$	0	0
$250$	1.00000	0.0632456
$251$	15.8927	1.00314	0.501569	−	0.865118i	$-0.332756\pi$
$251$	15.8927	1.00314	0.501569	+	0.865118i	$0.332756\pi$
$252$	0	0
$253$	−26.2557	−1.65068
$254$	1.23513	0.0774992
$255$	0	0
$256$	1.00000	0.0625000
$257$	−1.31965	−0.0823178	−0.0411589	−	0.999153i	$-0.513105\pi$
$257$	−1.31965	−0.0823178	−0.0411589	+	0.999153i	$0.513105\pi$
$258$	0	0
$259$	−2.63931	−0.163999
$260$	0.290725	0.0180300
$261$	0	0
$262$	0.581449	0.0359221
$263$	−26.5958	−1.63997	−0.819984	−	0.572386i	$-0.806018\pi$
$263$	−26.5958	−1.63997	−0.819984	+	0.572386i	$0.806018\pi$
$264$	0	0
$265$	−6.44748	−0.396066
$266$	−5.67420	−0.347908
$267$	0	0
$268$	−13.9421	−0.851652
$269$	26.8865	1.63930	0.819651	−	0.572863i	$-0.194167\pi$
$269$	26.8865	1.63930	0.819651	+	0.572863i	$0.194167\pi$
$270$	0	0
$271$	17.2618	1.04858	0.524290	−	0.851540i	$-0.324331\pi$
$271$	17.2618	1.04858	0.524290	+	0.851540i	$0.324331\pi$
$272$	−1.07838	−0.0653863
$273$	0	0
$274$	3.91548	0.236543
$275$	−6.04945	−0.364795
$276$	0	0
$277$	−22.4475	−1.34874	−0.674369	−	0.738394i	$-0.735584\pi$
$277$	−22.4475	−1.34874	−0.674369	+	0.738394i	$0.735584\pi$
$278$	15.9916	0.959112
$279$	0	0
$280$	1.07838	0.0644454
$281$	23.6742	1.41228	0.706142	−	0.708070i	$-0.250434\pi$
$281$	23.6742	1.41228	0.706142	+	0.708070i	$0.250434\pi$
$282$	0	0
$283$	−26.9360	−1.60118	−0.800589	−	0.599213i	$-0.795480\pi$
$283$	−26.9360	−1.60118	−0.800589	+	0.599213i	$0.795480\pi$
$284$	4.68035	0.277727
$285$	0	0
$286$	−1.75872	−0.103996
$287$	−6.04104	−0.356591
$288$	0	0
$289$	−15.8371	−0.931594
$290$	9.31124	0.546775
$291$	0	0
$292$	−8.18342	−0.478898
$293$	−22.0989	−1.29103	−0.645516	−	0.763747i	$-0.723357\pi$
$293$	−22.0989	−1.29103	−0.645516	+	0.763747i	$0.723357\pi$
$294$	0	0
$295$	11.4186	0.664814
$296$	2.44748	0.142257
$297$	0	0
$298$	1.90110	0.110128
$299$	−1.26180	−0.0729715
$300$	0	0
$301$	−8.48255	−0.488926
$302$	9.94214	0.572106
$303$	0	0
$304$	5.26180	0.301785
$305$	12.0494	0.689949
$306$	0	0
$307$	16.9939	0.969891	0.484945	−	0.874544i	$-0.338839\pi$
$307$	16.9939	0.969891	0.484945	+	0.874544i	$0.338839\pi$
$308$	−6.52359	−0.371716
$309$	0	0
$310$	1.00000	0.0567962
$311$	20.8781	1.18389	0.591945	−	0.805978i	$-0.298360\pi$
$311$	20.8781	1.18389	0.591945	+	0.805978i	$0.298360\pi$
$312$	0	0
$313$	22.7526	1.28605	0.643026	−	0.765844i	$-0.277679\pi$
$313$	22.7526	1.28605	0.643026	+	0.765844i	$0.277679\pi$
$314$	14.3135	0.807758
$315$	0	0
$316$	14.2557	0.801943
$317$	−6.09890	−0.342548	−0.171274	−	0.985223i	$-0.554788\pi$
$317$	−6.09890	−0.342548	−0.171274	+	0.985223i	$0.554788\pi$
$318$	0	0
$319$	−56.3279	−3.15376
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−4.68035	−0.260825
$323$	−5.67420	−0.315721
$324$	0	0
$325$	−0.290725	−0.0161265
$326$	16.4124	0.908999
$327$	0	0
$328$	5.60197	0.309317
$329$	1.89657	0.104561
$330$	0	0
$331$	25.5669	1.40528	0.702642	−	0.711544i	$-0.252004\pi$
$331$	25.5669	1.40528	0.702642	+	0.711544i	$0.252004\pi$
$332$	−3.55252	−0.194970
$333$	0	0
$334$	19.1773	1.04933
$335$	13.9421	0.761741
$336$	0	0
$337$	−4.18342	−0.227885	−0.113943	−	0.993487i	$-0.536348\pi$
$337$	−4.18342	−0.227885	−0.113943	+	0.993487i	$0.536348\pi$
$338$	12.9155	0.702509
$339$	0	0
$340$	1.07838	0.0584832
$341$	−6.04945	−0.327596
$342$	0	0
$343$	−13.8432	−0.747465
$344$	7.86603	0.424108
$345$	0	0
$346$	−5.51745	−0.296620
$347$	1.02893	0.0552358	0.0276179	−	0.999619i	$-0.491208\pi$
$347$	1.02893	0.0552358	0.0276179	+	0.999619i	$0.491208\pi$
$348$	0	0
$349$	−27.5753	−1.47607	−0.738036	−	0.674761i	$-0.764247\pi$
$349$	−27.5753	−1.47607	−0.738036	+	0.674761i	$0.764247\pi$
$350$	−1.07838	−0.0576417
$351$	0	0
$352$	6.04945	0.322437
$353$	20.0144	1.06526	0.532629	−	0.846349i	$-0.321204\pi$
$353$	20.0144	1.06526	0.532629	+	0.846349i	$0.321204\pi$
$354$	0	0
$355$	−4.68035	−0.248407
$356$	−3.84324	−0.203692
$357$	0	0
$358$	−6.36296	−0.336293
$359$	−8.86376	−0.467812	−0.233906	−	0.972259i	$-0.575151\pi$
$359$	−8.86376	−0.467812	−0.233906	+	0.972259i	$0.575151\pi$
$360$	0	0
$361$	8.68649	0.457184
$362$	−0.264063	−0.0138788
$363$	0	0
$364$	−0.313511	−0.0164324
$365$	8.18342	0.428340
$366$	0	0
$367$	29.7009	1.55037	0.775186	−	0.631733i	$-0.217656\pi$
$367$	29.7009	1.55037	0.775186	+	0.631733i	$0.217656\pi$
$368$	4.34017	0.226247
$369$	0	0
$370$	−2.44748	−0.127238
$371$	6.95282	0.360972
$372$	0	0
$373$	34.9770	1.81104	0.905521	−	0.424301i	$-0.139480\pi$
$373$	34.9770	1.81104	0.905521	+	0.424301i	$0.139480\pi$
$374$	−6.52359	−0.337327
$375$	0	0
$376$	−1.75872	−0.0906992
$377$	−2.70701	−0.139418
$378$	0	0
$379$	8.09890	0.416012	0.208006	−	0.978128i	$-0.433303\pi$
$379$	8.09890	0.416012	0.208006	+	0.978128i	$0.433303\pi$
$380$	−5.26180	−0.269924
$381$	0	0
$382$	2.65368	0.135774
$383$	−37.6020	−1.92137	−0.960685	−	0.277639i	$-0.910448\pi$
$383$	−37.6020	−1.92137	−0.960685	+	0.277639i	$0.910448\pi$
$384$	0	0
$385$	6.52359	0.332473
$386$	−0.156755	−0.00797864
$387$	0	0
$388$	2.49693	0.126762
$389$	−27.9337	−1.41630	−0.708148	−	0.706064i	$-0.750469\pi$
$389$	−27.9337	−1.41630	−0.708148	+	0.706064i	$0.750469\pi$
$390$	0	0
$391$	−4.68035	−0.236695
$392$	5.83710	0.294818
$393$	0	0
$394$	3.70928	0.186871
$395$	−14.2557	−0.717280
$396$	0	0
$397$	−20.6225	−1.03501	−0.517506	−	0.855679i	$-0.673140\pi$
$397$	−20.6225	−1.03501	−0.517506	+	0.855679i	$0.673140\pi$
$398$	−6.15676	−0.308610
$399$	0	0
$400$	1.00000	0.0500000
$401$	−16.2557	−0.811769	−0.405884	−	0.913924i	$-0.633036\pi$
$401$	−16.2557	−0.811769	−0.405884	+	0.913924i	$0.633036\pi$
$402$	0	0
$403$	−0.290725	−0.0144820
$404$	−4.83710	−0.240655
$405$	0	0
$406$	−10.0410	−0.498328
$407$	14.8059	0.733901
$408$	0	0
$409$	4.73820	0.234289	0.117145	−	0.993115i	$-0.462626\pi$
$409$	4.73820	0.234289	0.117145	+	0.993115i	$0.462626\pi$
$410$	−5.60197	−0.276661
$411$	0	0
$412$	6.83710	0.336840
$413$	−12.3135	−0.605908
$414$	0	0
$415$	3.55252	0.174386
$416$	0.290725	0.0142539
$417$	0	0
$418$	31.8310	1.55690
$419$	−4.21461	−0.205897	−0.102949	−	0.994687i	$-0.532828\pi$
$419$	−4.21461	−0.205897	−0.102949	+	0.994687i	$0.532828\pi$
$420$	0	0
$421$	−7.62863	−0.371797	−0.185898	−	0.982569i	$-0.559520\pi$
$421$	−7.62863	−0.371797	−0.185898	+	0.982569i	$0.559520\pi$
$422$	−25.3607	−1.23454
$423$	0	0
$424$	−6.44748	−0.313117
$425$	−1.07838	−0.0523090
$426$	0	0
$427$	−12.9939	−0.628817
$428$	2.52359	0.121982
$429$	0	0
$430$	−7.86603	−0.379334
$431$	38.1711	1.83864	0.919319	−	0.393512i	$-0.128740\pi$
$431$	38.1711	1.83864	0.919319	+	0.393512i	$0.128740\pi$
$432$	0	0
$433$	13.7587	0.661202	0.330601	−	0.943771i	$-0.392749\pi$
$433$	13.7587	0.661202	0.330601	+	0.943771i	$0.392749\pi$
$434$	−1.07838	−0.0517638
$435$	0	0
$436$	−1.41855	−0.0679363
$437$	22.8371	1.09245
$438$	0	0
$439$	−1.65983	−0.0792192	−0.0396096	−	0.999215i	$-0.512611\pi$
$439$	−1.65983	−0.0792192	−0.0396096	+	0.999215i	$0.512611\pi$
$440$	−6.04945	−0.288396
$441$	0	0
$442$	−0.313511	−0.0149122
$443$	−17.8432	−0.847758	−0.423879	−	0.905719i	$-0.639332\pi$
$443$	−17.8432	−0.847758	−0.423879	+	0.905719i	$0.639332\pi$
$444$	0	0
$445$	3.84324	0.182187
$446$	24.7526	1.17207
$447$	0	0
$448$	1.07838	0.0509486
$449$	−15.2618	−0.720249	−0.360124	−	0.932904i	$-0.617266\pi$
$449$	−15.2618	−0.720249	−0.360124	+	0.932904i	$0.617266\pi$
$450$	0	0
$451$	33.8888	1.59576
$452$	−15.1773	−0.713879
$453$	0	0
$454$	11.7321	0.550613
$455$	0.313511	0.0146976
$456$	0	0
$457$	16.2290	0.759160	0.379580	−	0.925159i	$-0.376068\pi$
$457$	16.2290	0.759160	0.379580	+	0.925159i	$0.376068\pi$
$458$	−6.88655	−0.321787
$459$	0	0
$460$	−4.34017	−0.202362
$461$	−8.99773	−0.419066	−0.209533	−	0.977802i	$-0.567194\pi$
$461$	−8.99773	−0.419066	−0.209533	+	0.977802i	$0.567194\pi$
$462$	0	0
$463$	4.65368	0.216275	0.108138	−	0.994136i	$-0.465511\pi$
$463$	4.65368	0.216275	0.108138	+	0.994136i	$0.465511\pi$
$464$	9.31124	0.432264
$465$	0	0
$466$	25.7009	1.19057
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	−15.0349	−0.694247
$470$	1.75872	0.0811239
$471$	0	0
$472$	11.4186	0.525581
$473$	47.5851	2.18797
$474$	0	0
$475$	5.26180	0.241428
$476$	−1.16290	−0.0533014
$477$	0	0
$478$	−23.7321	−1.08548
$479$	15.6332	0.714298	0.357149	−	0.934048i	$-0.383749\pi$
$479$	15.6332	0.714298	0.357149	+	0.934048i	$0.383749\pi$
$480$	0	0
$481$	0.711543	0.0324436
$482$	6.09890	0.277797
$483$	0	0
$484$	25.5958	1.16345
$485$	−2.49693	−0.113380
$486$	0	0
$487$	4.43907	0.201153	0.100577	−	0.994929i	$-0.467931\pi$
$487$	4.43907	0.201153	0.100577	+	0.994929i	$0.467931\pi$
$488$	12.0494	0.545453
$489$	0	0
$490$	−5.83710	−0.263693
$491$	−31.2411	−1.40989	−0.704946	−	0.709261i	$-0.749029\pi$
$491$	−31.2411	−1.40989	−0.704946	+	0.709261i	$0.749029\pi$
$492$	0	0
$493$	−10.0410	−0.452226
$494$	1.52973	0.0688260
$495$	0	0
$496$	1.00000	0.0449013
$497$	5.04718	0.226397
$498$	0	0
$499$	−24.9854	−1.11850	−0.559251	−	0.828998i	$-0.688911\pi$
$499$	−24.9854	−1.11850	−0.559251	+	0.828998i	$0.688911\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−15.8927	−0.709326
$503$	−14.0410	−0.626059	−0.313029	−	0.949743i	$-0.601344\pi$
$503$	−14.0410	−0.626059	−0.313029	+	0.949743i	$0.601344\pi$
$504$	0	0
$505$	4.83710	0.215248
$506$	26.2557	1.16721
$507$	0	0
$508$	−1.23513	−0.0548002
$509$	−24.9444	−1.10564	−0.552821	−	0.833300i	$-0.686449\pi$
$509$	−24.9444	−1.10564	−0.552821	+	0.833300i	$0.686449\pi$
$510$	0	0
$511$	−8.82482	−0.390387
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	1.31965	0.0582074
$515$	−6.83710	−0.301279
$516$	0	0
$517$	−10.6393	−0.467916
$518$	2.63931	0.115965
$519$	0	0
$520$	−0.290725	−0.0127491
$521$	−11.4764	−0.502791	−0.251395	−	0.967884i	$-0.580889\pi$
$521$	−11.4764	−0.502791	−0.251395	+	0.967884i	$0.580889\pi$
$522$	0	0
$523$	40.7975	1.78395	0.891975	−	0.452085i	$-0.149320\pi$
$523$	40.7975	1.78395	0.891975	+	0.452085i	$0.149320\pi$
$524$	−0.581449	−0.0254007
$525$	0	0
$526$	26.5958	1.15963
$527$	−1.07838	−0.0469749
$528$	0	0
$529$	−4.16290	−0.180996
$530$	6.44748	0.280061
$531$	0	0
$532$	5.67420	0.246008
$533$	1.62863	0.0705438
$534$	0	0
$535$	−2.52359	−0.109104
$536$	13.9421	0.602209
$537$	0	0
$538$	−26.8865	−1.15916
$539$	35.3112	1.52096
$540$	0	0
$541$	−13.3028	−0.571933	−0.285967	−	0.958240i	$-0.592315\pi$
$541$	−13.3028	−0.571933	−0.285967	+	0.958240i	$0.592315\pi$
$542$	−17.2618	−0.741458
$543$	0	0
$544$	1.07838	0.0462351
$545$	1.41855	0.0607640
$546$	0	0
$547$	−32.4124	−1.38585	−0.692927	−	0.721008i	$-0.743679\pi$
$547$	−32.4124	−1.38585	−0.692927	+	0.721008i	$0.743679\pi$
$548$	−3.91548	−0.167261
$549$	0	0
$550$	6.04945	0.257949
$551$	48.9939	2.08721
$552$	0	0
$553$	15.3730	0.653726
$554$	22.4475	0.953702
$555$	0	0
$556$	−15.9916	−0.678194
$557$	8.60424	0.364573	0.182287	−	0.983245i	$-0.441650\pi$
$557$	8.60424	0.364573	0.182287	+	0.983245i	$0.441650\pi$
$558$	0	0
$559$	2.28685	0.0967234
$560$	−1.07838	−0.0455698
$561$	0	0
$562$	−23.6742	−0.998636
$563$	−34.4534	−1.45204	−0.726020	−	0.687674i	$-0.758632\pi$
$563$	−34.4534	−1.45204	−0.726020	+	0.687674i	$0.758632\pi$
$564$	0	0
$565$	15.1773	0.638513
$566$	26.9360	1.13220
$567$	0	0
$568$	−4.68035	−0.196383
$569$	21.9832	0.921583	0.460791	−	0.887508i	$-0.347566\pi$
$569$	21.9832	0.921583	0.460791	+	0.887508i	$0.347566\pi$
$570$	0	0
$571$	−3.57918	−0.149784	−0.0748920	−	0.997192i	$-0.523861\pi$
$571$	−3.57918	−0.149784	−0.0748920	+	0.997192i	$0.523861\pi$
$572$	1.75872	0.0735359
$573$	0	0
$574$	6.04104	0.252148
$575$	4.34017	0.180998
$576$	0	0
$577$	1.18956	0.0495221	0.0247610	−	0.999693i	$-0.492118\pi$
$577$	1.18956	0.0495221	0.0247610	+	0.999693i	$0.492118\pi$
$578$	15.8371	0.658737
$579$	0	0
$580$	−9.31124	−0.386628
$581$	−3.83096	−0.158935
$582$	0	0
$583$	−39.0037	−1.61537
$584$	8.18342	0.338632
$585$	0	0
$586$	22.0989	0.912897
$587$	29.1278	1.20223	0.601117	−	0.799161i	$-0.294723\pi$
$587$	29.1278	1.20223	0.601117	+	0.799161i	$0.294723\pi$
$588$	0	0
$589$	5.26180	0.216808
$590$	−11.4186	−0.470094
$591$	0	0
$592$	−2.44748	−0.100591
$593$	−39.0616	−1.60407	−0.802033	−	0.597280i	$-0.796248\pi$
$593$	−39.0616	−1.60407	−0.802033	+	0.597280i	$0.796248\pi$
$594$	0	0
$595$	1.16290	0.0476742
$596$	−1.90110	−0.0778722
$597$	0	0
$598$	1.26180	0.0515986
$599$	4.97948	0.203456	0.101728	−	0.994812i	$-0.467563\pi$
$599$	4.97948	0.203456	0.101728	+	0.994812i	$0.467563\pi$
$600$	0	0
$601$	36.4079	1.48511	0.742554	−	0.669786i	$-0.233614\pi$
$601$	36.4079	1.48511	0.742554	+	0.669786i	$0.233614\pi$
$602$	8.48255	0.345723
$603$	0	0
$604$	−9.94214	−0.404540
$605$	−25.5958	−1.04062
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	−5.26180	−0.213394
$609$	0	0
$610$	−12.0494	−0.487868
$611$	−0.511304	−0.0206852
$612$	0	0
$613$	41.1689	1.66279	0.831397	−	0.555678i	$-0.187541\pi$
$613$	41.1689	1.66279	0.831397	+	0.555678i	$0.187541\pi$
$614$	−16.9939	−0.685816
$615$	0	0
$616$	6.52359	0.262843
$617$	0.952819	0.0383591	0.0191795	−	0.999816i	$-0.493895\pi$
$617$	0.952819	0.0383591	0.0191795	+	0.999816i	$0.493895\pi$
$618$	0	0
$619$	−3.52586	−0.141716	−0.0708581	−	0.997486i	$-0.522574\pi$
$619$	−3.52586	−0.141716	−0.0708581	+	0.997486i	$0.522574\pi$
$620$	−1.00000	−0.0401610
$621$	0	0
$622$	−20.8781	−0.837137
$623$	−4.14447	−0.166045
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.7526	−0.909376
$627$	0	0
$628$	−14.3135	−0.571171
$629$	2.63931	0.105236
$630$	0	0
$631$	47.5630	1.89345	0.946727	−	0.322037i	$-0.104368\pi$
$631$	47.5630	1.89345	0.946727	+	0.322037i	$0.104368\pi$
$632$	−14.2557	−0.567059
$633$	0	0
$634$	6.09890	0.242218
$635$	1.23513	0.0490148
$636$	0	0
$637$	1.69699	0.0672372
$638$	56.3279	2.23004
$639$	0	0
$640$	1.00000	0.0395285
$641$	−38.6102	−1.52501	−0.762506	−	0.646982i	$-0.776031\pi$
$641$	−38.6102	−1.52501	−0.762506	+	0.646982i	$0.776031\pi$
$642$	0	0
$643$	7.91935	0.312309	0.156154	−	0.987733i	$-0.450090\pi$
$643$	7.91935	0.312309	0.156154	+	0.987733i	$0.450090\pi$
$644$	4.68035	0.184431
$645$	0	0
$646$	5.67420	0.223249
$647$	29.4863	1.15922	0.579612	−	0.814893i	$-0.303204\pi$
$647$	29.4863	1.15922	0.579612	+	0.814893i	$0.303204\pi$
$648$	0	0
$649$	69.0759	2.71147
$650$	0.290725	0.0114032
$651$	0	0
$652$	−16.4124	−0.642759
$653$	34.9939	1.36942	0.684708	−	0.728818i	$-0.259930\pi$
$653$	34.9939	1.36942	0.684708	+	0.728818i	$0.259930\pi$
$654$	0	0
$655$	0.581449	0.0227191
$656$	−5.60197	−0.218720
$657$	0	0
$658$	−1.89657	−0.0739359
$659$	−13.9955	−0.545186	−0.272593	−	0.962129i	$-0.587881\pi$
$659$	−13.9955	−0.545186	−0.272593	+	0.962129i	$0.587881\pi$
$660$	0	0
$661$	2.63477	0.102481	0.0512404	−	0.998686i	$-0.483683\pi$
$661$	2.63477	0.102481	0.0512404	+	0.998686i	$0.483683\pi$
$662$	−25.5669	−0.993686
$663$	0	0
$664$	3.55252	0.137865
$665$	−5.67420	−0.220036
$666$	0	0
$667$	40.4124	1.56477
$668$	−19.1773	−0.741991
$669$	0	0
$670$	−13.9421	−0.538632
$671$	72.8925	2.81398
$672$	0	0
$673$	−36.2823	−1.39858	−0.699290	−	0.714838i	$-0.746500\pi$
$673$	−36.2823	−1.39858	−0.699290	+	0.714838i	$0.746500\pi$
$674$	4.18342	0.161139
$675$	0	0
$676$	−12.9155	−0.496749
$677$	32.4885	1.24864	0.624318	−	0.781171i	$-0.285377\pi$
$677$	32.4885	1.24864	0.624318	+	0.781171i	$0.285377\pi$
$678$	0	0
$679$	2.69263	0.103334
$680$	−1.07838	−0.0413539
$681$	0	0
$682$	6.04945	0.231645
$683$	26.5236	1.01490	0.507448	−	0.861682i	$-0.330589\pi$
$683$	26.5236	1.01490	0.507448	+	0.861682i	$0.330589\pi$
$684$	0	0
$685$	3.91548	0.149603
$686$	13.8432	0.528538
$687$	0	0
$688$	−7.86603	−0.299890
$689$	−1.87444	−0.0714105
$690$	0	0
$691$	−27.0349	−1.02846	−0.514228	−	0.857654i	$-0.671921\pi$
$691$	−27.0349	−1.02846	−0.514228	+	0.857654i	$0.671921\pi$
$692$	5.51745	0.209742
$693$	0	0
$694$	−1.02893	−0.0390576
$695$	15.9916	0.606596
$696$	0	0
$697$	6.04104	0.228821
$698$	27.5753	1.04374
$699$	0	0
$700$	1.07838	0.0407588
$701$	1.80221	0.0680684	0.0340342	−	0.999421i	$-0.489164\pi$
$701$	1.80221	0.0680684	0.0340342	+	0.999421i	$0.489164\pi$
$702$	0	0
$703$	−12.8781	−0.485708
$704$	−6.04945	−0.227997
$705$	0	0
$706$	−20.0144	−0.753251
$707$	−5.21622	−0.196176
$708$	0	0
$709$	−20.7298	−0.778524	−0.389262	−	0.921127i	$-0.627270\pi$
$709$	−20.7298	−0.778524	−0.389262	+	0.921127i	$0.627270\pi$
$710$	4.68035	0.175650
$711$	0	0
$712$	3.84324	0.144032
$713$	4.34017	0.162541
$714$	0	0
$715$	−1.75872	−0.0657725
$716$	6.36296	0.237795
$717$	0	0
$718$	8.86376	0.330793
$719$	−28.9939	−1.08129	−0.540644	−	0.841251i	$-0.681819\pi$
$719$	−28.9939	−1.08129	−0.540644	+	0.841251i	$0.681819\pi$
$720$	0	0
$721$	7.37298	0.274584
$722$	−8.68649	−0.323278
$723$	0	0
$724$	0.264063	0.00981381
$725$	9.31124	0.345811
$726$	0	0
$727$	16.5113	0.612370	0.306185	−	0.951972i	$-0.400947\pi$
$727$	16.5113	0.612370	0.306185	+	0.951972i	$0.400947\pi$
$728$	0.313511	0.0116195
$729$	0	0
$730$	−8.18342	−0.302882
$731$	8.48255	0.313739
$732$	0	0
$733$	14.8950	0.550158	0.275079	−	0.961422i	$-0.411296\pi$
$733$	14.8950	0.550158	0.275079	+	0.961422i	$0.411296\pi$
$734$	−29.7009	−1.09628
$735$	0	0
$736$	−4.34017	−0.159981
$737$	84.3423	3.10679
$738$	0	0
$739$	−3.67808	−0.135300	−0.0676502	−	0.997709i	$-0.521550\pi$
$739$	−3.67808	−0.135300	−0.0676502	+	0.997709i	$0.521550\pi$
$740$	2.44748	0.0899712
$741$	0	0
$742$	−6.95282	−0.255246
$743$	33.7464	1.23804	0.619018	−	0.785377i	$-0.287531\pi$
$743$	33.7464	1.23804	0.619018	+	0.785377i	$0.287531\pi$
$744$	0	0
$745$	1.90110	0.0696510
$746$	−34.9770	−1.28060
$747$	0	0
$748$	6.52359	0.238526
$749$	2.72138	0.0994372
$750$	0	0
$751$	−18.6537	−0.680683	−0.340341	−	0.940302i	$-0.610543\pi$
$751$	−18.6537	−0.680683	−0.340341	+	0.940302i	$0.610543\pi$
$752$	1.75872	0.0641341
$753$	0	0
$754$	2.70701	0.0985834
$755$	9.94214	0.361832
$756$	0	0
$757$	−13.8660	−0.503969	−0.251985	−	0.967731i	$-0.581083\pi$
$757$	−13.8660	−0.503969	−0.251985	+	0.967731i	$0.581083\pi$
$758$	−8.09890	−0.294165
$759$	0	0
$760$	5.26180	0.190865
$761$	−12.3714	−0.448462	−0.224231	−	0.974536i	$-0.571987\pi$
$761$	−12.3714	−0.448462	−0.224231	+	0.974536i	$0.571987\pi$
$762$	0	0
$763$	−1.52973	−0.0553801
$764$	−2.65368	−0.0960069
$765$	0	0
$766$	37.6020	1.35861
$767$	3.31965	0.119866
$768$	0	0
$769$	4.60811	0.166173	0.0830864	−	0.996542i	$-0.473522\pi$
$769$	4.60811	0.166173	0.0830864	+	0.996542i	$0.473522\pi$
$770$	−6.52359	−0.235094
$771$	0	0
$772$	0.156755	0.00564175
$773$	−11.6104	−0.417596	−0.208798	−	0.977959i	$-0.566955\pi$
$773$	−11.6104	−0.417596	−0.208798	+	0.977959i	$0.566955\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	−2.49693	−0.0896345
$777$	0	0
$778$	27.9337	1.00147
$779$	−29.4764	−1.05610
$780$	0	0
$781$	−28.3135	−1.01314
$782$	4.68035	0.167369
$783$	0	0
$784$	−5.83710	−0.208468
$785$	14.3135	0.510871
$786$	0	0
$787$	22.7031	0.809279	0.404640	−	0.914476i	$-0.367397\pi$
$787$	22.7031	0.809279	0.404640	+	0.914476i	$0.367397\pi$
$788$	−3.70928	−0.132137
$789$	0	0
$790$	14.2557	0.507193
$791$	−16.3668	−0.581938
$792$	0	0
$793$	3.50307	0.124398
$794$	20.6225	0.731865
$795$	0	0
$796$	6.15676	0.218220
$797$	−30.9132	−1.09500	−0.547501	−	0.836805i	$-0.684421\pi$
$797$	−30.9132	−1.09500	−0.547501	+	0.836805i	$0.684421\pi$
$798$	0	0
$799$	−1.89657	−0.0670958
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	16.2557	0.574007
$803$	49.5052	1.74700
$804$	0	0
$805$	−4.68035	−0.164961
$806$	0.290725	0.0102403
$807$	0	0
$808$	4.83710	0.170169
$809$	47.0928	1.65569	0.827847	−	0.560955i	$-0.189566\pi$
$809$	47.0928	1.65569	0.827847	+	0.560955i	$0.189566\pi$
$810$	0	0
$811$	30.1399	1.05836	0.529178	−	0.848511i	$-0.322500\pi$
$811$	30.1399	1.05836	0.529178	+	0.848511i	$0.322500\pi$
$812$	10.0410	0.352371
$813$	0	0
$814$	−14.8059	−0.518947
$815$	16.4124	0.574902
$816$	0	0
$817$	−41.3894	−1.44803
$818$	−4.73820	−0.165667
$819$	0	0
$820$	5.60197	0.195629
$821$	33.5090	1.16947	0.584737	−	0.811223i	$-0.301198\pi$
$821$	33.5090	1.16947	0.584737	+	0.811223i	$0.301198\pi$
$822$	0	0
$823$	36.0722	1.25740	0.628700	−	0.777648i	$-0.283588\pi$
$823$	36.0722	1.25740	0.628700	+	0.777648i	$0.283588\pi$
$824$	−6.83710	−0.238182
$825$	0	0
$826$	12.3135	0.428442
$827$	28.2329	0.981753	0.490876	−	0.871229i	$-0.336677\pi$
$827$	28.2329	0.981753	0.490876	+	0.871229i	$0.336677\pi$
$828$	0	0
$829$	26.2518	0.911762	0.455881	−	0.890041i	$-0.349324\pi$
$829$	26.2518	0.911762	0.455881	+	0.890041i	$0.349324\pi$
$830$	−3.55252	−0.123310
$831$	0	0
$832$	−0.290725	−0.0100791
$833$	6.29460	0.218095
$834$	0	0
$835$	19.1773	0.663657
$836$	−31.8310	−1.10090
$837$	0	0
$838$	4.21461	0.145591
$839$	5.54411	0.191404	0.0957020	−	0.995410i	$-0.469490\pi$
$839$	5.54411	0.191404	0.0957020	+	0.995410i	$0.469490\pi$
$840$	0	0
$841$	57.6993	1.98963
$842$	7.62863	0.262900
$843$	0	0
$844$	25.3607	0.872951
$845$	12.9155	0.444306
$846$	0	0
$847$	27.6020	0.948415
$848$	6.44748	0.221407
$849$	0	0
$850$	1.07838	0.0369881
$851$	−10.6225	−0.364134
$852$	0	0
$853$	−15.4764	−0.529902	−0.264951	−	0.964262i	$-0.585356\pi$
$853$	−15.4764	−0.529902	−0.264951	+	0.964262i	$0.585356\pi$
$854$	12.9939	0.444641
$855$	0	0
$856$	−2.52359	−0.0862545
$857$	−46.0288	−1.57231	−0.786156	−	0.618028i	$-0.787932\pi$
$857$	−46.0288	−1.57231	−0.786156	+	0.618028i	$0.787932\pi$
$858$	0	0
$859$	−6.99773	−0.238760	−0.119380	−	0.992849i	$-0.538091\pi$
$859$	−6.99773	−0.238760	−0.119380	+	0.992849i	$0.538091\pi$
$860$	7.86603	0.268229
$861$	0	0
$862$	−38.1711	−1.30011
$863$	−4.24128	−0.144375	−0.0721874	−	0.997391i	$-0.522998\pi$
$863$	−4.24128	−0.144375	−0.0721874	+	0.997391i	$0.522998\pi$
$864$	0	0
$865$	−5.51745	−0.187599
$866$	−13.7587	−0.467540
$867$	0	0
$868$	1.07838	0.0366025
$869$	−86.2388	−2.92545
$870$	0	0
$871$	4.05332	0.137342
$872$	1.41855	0.0480382
$873$	0	0
$874$	−22.8371	−0.772476
$875$	−1.07838	−0.0364558
$876$	0	0
$877$	−36.2557	−1.22427	−0.612133	−	0.790755i	$-0.709688\pi$
$877$	−36.2557	−1.22427	−0.612133	+	0.790755i	$0.709688\pi$
$878$	1.65983	0.0560164
$879$	0	0
$880$	6.04945	0.203927
$881$	41.6619	1.40363	0.701813	−	0.712361i	$-0.252374\pi$
$881$	41.6619	1.40363	0.701813	+	0.712361i	$0.252374\pi$
$882$	0	0
$883$	4.85989	0.163548	0.0817741	−	0.996651i	$-0.473941\pi$
$883$	4.85989	0.163548	0.0817741	+	0.996651i	$0.473941\pi$
$884$	0.313511	0.0105445
$885$	0	0
$886$	17.8432	0.599456
$887$	44.1133	1.48118	0.740589	−	0.671958i	$-0.234546\pi$
$887$	44.1133	1.48118	0.740589	+	0.671958i	$0.234546\pi$
$888$	0	0
$889$	−1.33194	−0.0446718
$890$	−3.84324	−0.128826
$891$	0	0
$892$	−24.7526	−0.828778
$893$	9.25404	0.309675
$894$	0	0
$895$	−6.36296	−0.212690
$896$	−1.07838	−0.0360261
$897$	0	0
$898$	15.2618	0.509293
$899$	9.31124	0.310547
$900$	0	0
$901$	−6.95282	−0.231632
$902$	−33.8888	−1.12837
$903$	0	0
$904$	15.1773	0.504789
$905$	−0.264063	−0.00877774
$906$	0	0
$907$	−34.0944	−1.13208	−0.566042	−	0.824376i	$-0.691526\pi$
$907$	−34.0944	−1.13208	−0.566042	+	0.824376i	$0.691526\pi$
$908$	−11.7321	−0.389342
$909$	0	0
$910$	−0.313511	−0.0103928
$911$	11.3197	0.375037	0.187518	−	0.982261i	$-0.439956\pi$
$911$	11.3197	0.375037	0.187518	+	0.982261i	$0.439956\pi$
$912$	0	0
$913$	21.4908	0.711241
$914$	−16.2290	−0.536807
$915$	0	0
$916$	6.88655	0.227538
$917$	−0.627022	−0.0207061
$918$	0	0
$919$	47.2039	1.55711	0.778557	−	0.627574i	$-0.215952\pi$
$919$	47.2039	1.55711	0.778557	+	0.627574i	$0.215952\pi$
$920$	4.34017	0.143091
$921$	0	0
$922$	8.99773	0.296325
$923$	−1.36069	−0.0447877
$924$	0	0
$925$	−2.44748	−0.0804727
$926$	−4.65368	−0.152930
$927$	0	0
$928$	−9.31124	−0.305657
$929$	−28.1399	−0.923241	−0.461621	−	0.887077i	$-0.652732\pi$
$929$	−28.1399	−0.923241	−0.461621	+	0.887077i	$0.652732\pi$
$930$	0	0
$931$	−30.7136	−1.00660
$932$	−25.7009	−0.841860
$933$	0	0
$934$	8.00000	0.261768
$935$	−6.52359	−0.213344
$936$	0	0
$937$	−50.0288	−1.63437	−0.817184	−	0.576377i	$-0.804466\pi$
$937$	−50.0288	−1.63437	−0.817184	+	0.576377i	$0.804466\pi$
$938$	15.0349	0.490907
$939$	0	0
$940$	−1.75872	−0.0573632
$941$	−42.7708	−1.39429	−0.697145	−	0.716931i	$-0.745546\pi$
$941$	−42.7708	−1.39429	−0.697145	+	0.716931i	$0.745546\pi$
$942$	0	0
$943$	−24.3135	−0.791757
$944$	−11.4186	−0.371642
$945$	0	0
$946$	−47.5851	−1.54713
$947$	−52.8964	−1.71890	−0.859451	−	0.511218i	$-0.829194\pi$
$947$	−52.8964	−1.71890	−0.859451	+	0.511218i	$0.829194\pi$
$948$	0	0
$949$	2.37912	0.0772295
$950$	−5.26180	−0.170715
$951$	0	0
$952$	1.16290	0.0376898
$953$	4.71154	0.152622	0.0763109	−	0.997084i	$-0.475686\pi$
$953$	4.71154	0.152622	0.0763109	+	0.997084i	$0.475686\pi$
$954$	0	0
$955$	2.65368	0.0858712
$956$	23.7321	0.767550
$957$	0	0
$958$	−15.6332	−0.505085
$959$	−4.22237	−0.136347
$960$	0	0
$961$	1.00000	0.0322581
$962$	−0.711543	−0.0229411
$963$	0	0
$964$	−6.09890	−0.196432
$965$	−0.156755	−0.00504614
$966$	0	0
$967$	25.4863	0.819583	0.409791	−	0.912179i	$-0.365602\pi$
$967$	25.4863	0.819583	0.409791	+	0.912179i	$0.365602\pi$
$968$	−25.5958	−0.822681
$969$	0	0
$970$	2.49693	0.0801715
$971$	16.2511	0.521523	0.260762	−	0.965403i	$-0.416026\pi$
$971$	16.2511	0.521523	0.260762	+	0.965403i	$0.416026\pi$
$972$	0	0
$973$	−17.2450	−0.552848
$974$	−4.43907	−0.142237
$975$	0	0
$976$	−12.0494	−0.385693
$977$	10.5113	0.336286	0.168143	−	0.985763i	$-0.446223\pi$
$977$	10.5113	0.336286	0.168143	+	0.985763i	$0.446223\pi$
$978$	0	0
$979$	23.2495	0.743058
$980$	5.83710	0.186459
$981$	0	0
$982$	31.2411	0.996944
$983$	17.5486	0.559715	0.279857	−	0.960042i	$-0.409713\pi$
$983$	17.5486	0.559715	0.279857	+	0.960042i	$0.409713\pi$
$984$	0	0
$985$	3.70928	0.118187
$986$	10.0410	0.319772
$987$	0	0
$988$	−1.52973	−0.0486673
$989$	−34.1399	−1.08559
$990$	0	0
$991$	−4.19779	−0.133347	−0.0666736	−	0.997775i	$-0.521239\pi$
$991$	−4.19779	−0.133347	−0.0666736	+	0.997775i	$0.521239\pi$
$992$	−1.00000	−0.0317500
$993$	0	0
$994$	−5.04718	−0.160087
$995$	−6.15676	−0.195182
$996$	0	0
$997$	−40.4001	−1.27948	−0.639742	−	0.768589i	$-0.720959\pi$
$997$	−40.4001	−1.27948	−0.639742	+	0.768589i	$0.720959\pi$
$998$	24.9854	0.790900
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.bi.1.2		3
3.2	odd	2		310.2.a.e.1.1	✓	3
12.11	even	2		2480.2.a.u.1.3		3
15.2	even	4		1550.2.b.j.249.6		6
15.8	even	4		1550.2.b.j.249.1		6
15.14	odd	2		1550.2.a.k.1.3		3
24.5	odd	2		9920.2.a.bw.1.3		3
24.11	even	2		9920.2.a.bx.1.1		3
93.92	even	2		9610.2.a.u.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.a.e.1.1	✓	3	3.2	odd	2
1550.2.a.k.1.3		3	15.14	odd	2
1550.2.b.j.249.1		6	15.8	even	4
1550.2.b.j.249.6		6	15.2	even	4
2480.2.a.u.1.3		3	12.11	even	2
2790.2.a.bi.1.2		3	1.1	even	1	trivial
9610.2.a.u.1.3		3	93.92	even	2
9920.2.a.bw.1.3		3	24.5	odd	2
9920.2.a.bx.1.1		3	24.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.07838 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.07838 q^{7} -1.00000 q^{8} +1.00000 q^{10} -6.04945 q^{11} -0.290725 q^{13} -1.07838 q^{14} +1.00000 q^{16} -1.07838 q^{17} +5.26180 q^{19} -1.00000 q^{20} +6.04945 q^{22} +4.34017 q^{23} +1.00000 q^{25} +0.290725 q^{26} +1.07838 q^{28} +9.31124 q^{29} +1.00000 q^{31} -1.00000 q^{32} +1.07838 q^{34} -1.07838 q^{35} -2.44748 q^{37} -5.26180 q^{38} +1.00000 q^{40} -5.60197 q^{41} -7.86603 q^{43} -6.04945 q^{44} -4.34017 q^{46} +1.75872 q^{47} -5.83710 q^{49} -1.00000 q^{50} -0.290725 q^{52} +6.44748 q^{53} +6.04945 q^{55} -1.07838 q^{56} -9.31124 q^{58} -11.4186 q^{59} -12.0494 q^{61} -1.00000 q^{62} +1.00000 q^{64} +0.290725 q^{65} -13.9421 q^{67} -1.07838 q^{68} +1.07838 q^{70} +4.68035 q^{71} -8.18342 q^{73} +2.44748 q^{74} +5.26180 q^{76} -6.52359 q^{77} +14.2557 q^{79} -1.00000 q^{80} +5.60197 q^{82} -3.55252 q^{83} +1.07838 q^{85} +7.86603 q^{86} +6.04945 q^{88} -3.84324 q^{89} -0.313511 q^{91} +4.34017 q^{92} -1.75872 q^{94} -5.26180 q^{95} +2.49693 q^{97} +5.83710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{8} + 3 q^{10} - 8 q^{13} + 3 q^{16} + 8 q^{19} - 3 q^{20} + 2 q^{23} + 3 q^{25} + 8 q^{26} + 2 q^{29} + 3 q^{31} - 3 q^{32} - 8 q^{37} - 8 q^{38} + 3 q^{40} + 2 q^{41} - 10 q^{43} - 2 q^{46} - 20 q^{47} + 11 q^{49} - 3 q^{50} - 8 q^{52} + 20 q^{53} - 2 q^{58} - 20 q^{59} - 18 q^{61} - 3 q^{62} + 3 q^{64} + 8 q^{65} - 12 q^{67} - 8 q^{71} - 20 q^{73} + 8 q^{74} + 8 q^{76} - 4 q^{77} - 3 q^{80} - 2 q^{82} - 10 q^{83} + 10 q^{86} - 18 q^{89} + 12 q^{91} + 2 q^{92} + 20 q^{94} - 8 q^{95} - 10 q^{97} - 11 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^5 - 3 * q^8 + 3 * q^10 - 8 * q^13 + 3 * q^16 + 8 * q^19 - 3 * q^20 + 2 * q^23 + 3 * q^25 + 8 * q^26 + 2 * q^29 + 3 * q^31 - 3 * q^32 - 8 * q^37 - 8 * q^38 + 3 * q^40 + 2 * q^41 - 10 * q^43 - 2 * q^46 - 20 * q^47 + 11 * q^49 - 3 * q^50 - 8 * q^52 + 20 * q^53 - 2 * q^58 - 20 * q^59 - 18 * q^61 - 3 * q^62 + 3 * q^64 + 8 * q^65 - 12 * q^67 - 8 * q^71 - 20 * q^73 + 8 * q^74 + 8 * q^76 - 4 * q^77 - 3 * q^80 - 2 * q^82 - 10 * q^83 + 10 * q^86 - 18 * q^89 + 12 * q^91 + 2 * q^92 + 20 * q^94 - 8 * q^95 - 10 * q^97 - 11 * q^98

Introduction

L-functions

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