LMFDB - Embedded newform 2790.2.a.be.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.56155 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.56155 q^{11} +2.00000 q^{13} +1.56155 q^{14} +1.00000 q^{16} -5.12311 q^{17} +4.68466 q^{19} -1.00000 q^{20} -1.56155 q^{22} -5.56155 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.56155 q^{28} +1.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} +5.12311 q^{34} +1.56155 q^{35} +5.12311 q^{37} -4.68466 q^{38} +1.00000 q^{40} +1.12311 q^{41} -7.80776 q^{43} +1.56155 q^{44} +5.56155 q^{46} +3.12311 q^{47} -4.56155 q^{49} -1.00000 q^{50} +2.00000 q^{52} -11.5616 q^{53} -1.56155 q^{55} +1.56155 q^{56} -1.12311 q^{58} +4.87689 q^{59} +6.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +9.36932 q^{67} -5.12311 q^{68} -1.56155 q^{70} -4.68466 q^{71} +9.80776 q^{73} -5.12311 q^{74} +4.68466 q^{76} -2.43845 q^{77} -16.6847 q^{79} -1.00000 q^{80} -1.12311 q^{82} +2.24621 q^{83} +5.12311 q^{85} +7.80776 q^{86} -1.56155 q^{88} +1.31534 q^{89} -3.12311 q^{91} -5.56155 q^{92} -3.12311 q^{94} -4.68466 q^{95} -6.00000 q^{97} +4.56155 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} + 4 q^{13} - q^{14} + 2 q^{16} - 2 q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 7 q^{23} + 2 q^{25} - 4 q^{26} + q^{28} - 6 q^{29} + 2 q^{31} - 2 q^{32} + 2 q^{34} - q^{35} + 2 q^{37} + 3 q^{38} + 2 q^{40} - 6 q^{41} + 5 q^{43} - q^{44} + 7 q^{46} - 2 q^{47} - 5 q^{49} - 2 q^{50} + 4 q^{52} - 19 q^{53} + q^{55} - q^{56} + 6 q^{58} + 18 q^{59} + 12 q^{61} - 2 q^{62} + 2 q^{64} - 4 q^{65} - 6 q^{67} - 2 q^{68} + q^{70} + 3 q^{71} - q^{73} - 2 q^{74} - 3 q^{76} - 9 q^{77} - 21 q^{79} - 2 q^{80} + 6 q^{82} - 12 q^{83} + 2 q^{85} - 5 q^{86} + q^{88} + 15 q^{89} + 2 q^{91} - 7 q^{92} + 2 q^{94} + 3 q^{95} - 12 q^{97} + 5 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8 + 2 * q^10 - q^11 + 4 * q^13 - q^14 + 2 * q^16 - 2 * q^17 - 3 * q^19 - 2 * q^20 + q^22 - 7 * q^23 + 2 * q^25 - 4 * q^26 + q^28 - 6 * q^29 + 2 * q^31 - 2 * q^32 + 2 * q^34 - q^35 + 2 * q^37 + 3 * q^38 + 2 * q^40 - 6 * q^41 + 5 * q^43 - q^44 + 7 * q^46 - 2 * q^47 - 5 * q^49 - 2 * q^50 + 4 * q^52 - 19 * q^53 + q^55 - q^56 + 6 * q^58 + 18 * q^59 + 12 * q^61 - 2 * q^62 + 2 * q^64 - 4 * q^65 - 6 * q^67 - 2 * q^68 + q^70 + 3 * q^71 - q^73 - 2 * q^74 - 3 * q^76 - 9 * q^77 - 21 * q^79 - 2 * q^80 + 6 * q^82 - 12 * q^83 + 2 * q^85 - 5 * q^86 + q^88 + 15 * q^89 + 2 * q^91 - 7 * q^92 + 2 * q^94 + 3 * q^95 - 12 * q^97 + 5 * 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
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	1.56155	0.470826	0.235413	−	0.971895i	$-0.424356\pi$
$11$	1.56155	0.470826	0.235413	+	0.971895i	$0.424356\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.56155	0.417343
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.12311	−1.24254	−0.621268	−	0.783598i	$-0.713382\pi$
$17$	−5.12311	−1.24254	−0.621268	+	0.783598i	$0.713382\pi$
$18$	0	0
$19$	4.68466	1.07473	0.537367	−	0.843348i	$-0.319419\pi$
$19$	4.68466	1.07473	0.537367	+	0.843348i	$0.319419\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.56155	−0.332924
$23$	−5.56155	−1.15966	−0.579832	−	0.814736i	$-0.696882\pi$
$23$	−5.56155	−1.15966	−0.579832	+	0.814736i	$0.696882\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	−1.56155	−0.295106
$29$	1.12311	0.208555	0.104278	−	0.994548i	$-0.466747\pi$
$29$	1.12311	0.208555	0.104278	+	0.994548i	$0.466747\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.12311	0.878605
$35$	1.56155	0.263951
$36$	0	0
$37$	5.12311	0.842233	0.421117	−	0.907006i	$-0.361638\pi$
$37$	5.12311	0.842233	0.421117	+	0.907006i	$0.361638\pi$
$38$	−4.68466	−0.759952
$39$	0	0
$40$	1.00000	0.158114
$41$	1.12311	0.175400	0.0876998	−	0.996147i	$-0.472048\pi$
$41$	1.12311	0.175400	0.0876998	+	0.996147i	$0.472048\pi$
$42$	0	0
$43$	−7.80776	−1.19067	−0.595336	−	0.803477i	$-0.702981\pi$
$43$	−7.80776	−1.19067	−0.595336	+	0.803477i	$0.702981\pi$
$44$	1.56155	0.235413
$45$	0	0
$46$	5.56155	0.820006
$47$	3.12311	0.455552	0.227776	−	0.973714i	$-0.426855\pi$
$47$	3.12311	0.455552	0.227776	+	0.973714i	$0.426855\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	−1.00000	−0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	−11.5616	−1.58810	−0.794051	−	0.607852i	$-0.792032\pi$
$53$	−11.5616	−1.58810	−0.794051	+	0.607852i	$0.792032\pi$
$54$	0	0
$55$	−1.56155	−0.210560
$56$	1.56155	0.208671
$57$	0	0
$58$	−1.12311	−0.147471
$59$	4.87689	0.634918	0.317459	−	0.948272i	$-0.397170\pi$
$59$	4.87689	0.634918	0.317459	+	0.948272i	$0.397170\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	9.36932	1.14464	0.572322	−	0.820029i	$-0.306043\pi$
$67$	9.36932	1.14464	0.572322	+	0.820029i	$0.306043\pi$
$68$	−5.12311	−0.621268
$69$	0	0
$70$	−1.56155	−0.186641
$71$	−4.68466	−0.555967	−0.277983	−	0.960586i	$-0.589666\pi$
$71$	−4.68466	−0.555967	−0.277983	+	0.960586i	$0.589666\pi$
$72$	0	0
$73$	9.80776	1.14791	0.573956	−	0.818886i	$-0.305408\pi$
$73$	9.80776	1.14791	0.573956	+	0.818886i	$0.305408\pi$
$74$	−5.12311	−0.595549
$75$	0	0
$76$	4.68466	0.537367
$77$	−2.43845	−0.277887
$78$	0	0
$79$	−16.6847	−1.87717	−0.938585	−	0.345047i	$-0.887863\pi$
$79$	−16.6847	−1.87717	−0.938585	+	0.345047i	$0.887863\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−1.12311	−0.124026
$83$	2.24621	0.246554	0.123277	−	0.992372i	$-0.460660\pi$
$83$	2.24621	0.246554	0.123277	+	0.992372i	$0.460660\pi$
$84$	0	0
$85$	5.12311	0.555679
$86$	7.80776	0.841933
$87$	0	0
$88$	−1.56155	−0.166462
$89$	1.31534	0.139426	0.0697130	−	0.997567i	$-0.477792\pi$
$89$	1.31534	0.139426	0.0697130	+	0.997567i	$0.477792\pi$
$90$	0	0
$91$	−3.12311	−0.327390
$92$	−5.56155	−0.579832
$93$	0	0
$94$	−3.12311	−0.322124
$95$	−4.68466	−0.480636
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	4.56155	0.460786
$99$	0	0
$100$	1.00000	0.100000
$101$	−3.56155	−0.354388	−0.177194	−	0.984176i	$-0.556702\pi$
$101$	−3.56155	−0.354388	−0.177194	+	0.984176i	$0.556702\pi$
$102$	0	0
$103$	−18.2462	−1.79785	−0.898926	−	0.438100i	$-0.855652\pi$
$103$	−18.2462	−1.79785	−0.898926	+	0.438100i	$0.855652\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	11.5616	1.12296
$107$	−1.56155	−0.150961	−0.0754805	−	0.997147i	$-0.524049\pi$
$107$	−1.56155	−0.150961	−0.0754805	+	0.997147i	$0.524049\pi$
$108$	0	0
$109$	−19.3693	−1.85524	−0.927622	−	0.373520i	$-0.878151\pi$
$109$	−19.3693	−1.85524	−0.927622	+	0.373520i	$0.878151\pi$
$110$	1.56155	0.148888
$111$	0	0
$112$	−1.56155	−0.147553
$113$	−10.6847	−1.00513	−0.502564	−	0.864540i	$-0.667610\pi$
$113$	−10.6847	−1.00513	−0.502564	+	0.864540i	$0.667610\pi$
$114$	0	0
$115$	5.56155	0.518617
$116$	1.12311	0.104278
$117$	0	0
$118$	−4.87689	−0.448955
$119$	8.00000	0.733359
$120$	0	0
$121$	−8.56155	−0.778323
$122$	−6.00000	−0.543214
$123$	0	0
$124$	1.00000	0.0898027
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.00000	0.175412
$131$	−9.36932	−0.818601	−0.409301	−	0.912400i	$-0.634227\pi$
$131$	−9.36932	−0.818601	−0.409301	+	0.912400i	$0.634227\pi$
$132$	0	0
$133$	−7.31534	−0.634321
$134$	−9.36932	−0.809386
$135$	0	0
$136$	5.12311	0.439303
$137$	−3.75379	−0.320708	−0.160354	−	0.987060i	$-0.551264\pi$
$137$	−3.75379	−0.320708	−0.160354	+	0.987060i	$0.551264\pi$
$138$	0	0
$139$	−8.87689	−0.752928	−0.376464	−	0.926431i	$-0.622860\pi$
$139$	−8.87689	−0.752928	−0.376464	+	0.926431i	$0.622860\pi$
$140$	1.56155	0.131975
$141$	0	0
$142$	4.68466	0.393128
$143$	3.12311	0.261167
$144$	0	0
$145$	−1.12311	−0.0932688
$146$	−9.80776	−0.811696
$147$	0	0
$148$	5.12311	0.421117
$149$	20.0540	1.64289	0.821443	−	0.570291i	$-0.193170\pi$
$149$	20.0540	1.64289	0.821443	+	0.570291i	$0.193170\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−4.68466	−0.379976
$153$	0	0
$154$	2.43845	0.196496
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−20.0540	−1.60048	−0.800241	−	0.599679i	$-0.795295\pi$
$157$	−20.0540	−1.60048	−0.800241	+	0.599679i	$0.795295\pi$
$158$	16.6847	1.32736
$159$	0	0
$160$	1.00000	0.0790569
$161$	8.68466	0.684447
$162$	0	0
$163$	9.36932	0.733862	0.366931	−	0.930248i	$-0.380409\pi$
$163$	9.36932	0.733862	0.366931	+	0.930248i	$0.380409\pi$
$164$	1.12311	0.0876998
$165$	0	0
$166$	−2.24621	−0.174340
$167$	2.43845	0.188693	0.0943464	−	0.995539i	$-0.469924\pi$
$167$	2.43845	0.188693	0.0943464	+	0.995539i	$0.469924\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−5.12311	−0.392924
$171$	0	0
$172$	−7.80776	−0.595336
$173$	8.24621	0.626948	0.313474	−	0.949597i	$-0.398507\pi$
$173$	8.24621	0.626948	0.313474	+	0.949597i	$0.398507\pi$
$174$	0	0
$175$	−1.56155	−0.118042
$176$	1.56155	0.117706
$177$	0	0
$178$	−1.31534	−0.0985890
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	24.0540	1.78792	0.893959	−	0.448149i	$-0.147917\pi$
$181$	24.0540	1.78792	0.893959	+	0.448149i	$0.147917\pi$
$182$	3.12311	0.231500
$183$	0	0
$184$	5.56155	0.410003
$185$	−5.12311	−0.376658
$186$	0	0
$187$	−8.00000	−0.585018
$188$	3.12311	0.227776
$189$	0	0
$190$	4.68466	0.339861
$191$	−2.24621	−0.162530	−0.0812651	−	0.996693i	$-0.525896\pi$
$191$	−2.24621	−0.162530	−0.0812651	+	0.996693i	$0.525896\pi$
$192$	0	0
$193$	−18.4924	−1.33111	−0.665557	−	0.746347i	$-0.731806\pi$
$193$	−18.4924	−1.33111	−0.665557	+	0.746347i	$0.731806\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	−4.56155	−0.325825
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	3.80776	0.269925	0.134963	−	0.990851i	$-0.456909\pi$
$199$	3.80776	0.269925	0.134963	+	0.990851i	$0.456909\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	3.56155	0.250590
$203$	−1.75379	−0.123092
$204$	0	0
$205$	−1.12311	−0.0784411
$206$	18.2462	1.27127
$207$	0	0
$208$	2.00000	0.138675
$209$	7.31534	0.506013
$210$	0	0
$211$	−11.3153	−0.778980	−0.389490	−	0.921031i	$-0.627349\pi$
$211$	−11.3153	−0.778980	−0.389490	+	0.921031i	$0.627349\pi$
$212$	−11.5616	−0.794051
$213$	0	0
$214$	1.56155	0.106746
$215$	7.80776	0.532485
$216$	0	0
$217$	−1.56155	−0.106005
$218$	19.3693	1.31186
$219$	0	0
$220$	−1.56155	−0.105280
$221$	−10.2462	−0.689235
$222$	0	0
$223$	−12.8769	−0.862301	−0.431150	−	0.902280i	$-0.641892\pi$
$223$	−12.8769	−0.862301	−0.431150	+	0.902280i	$0.641892\pi$
$224$	1.56155	0.104336
$225$	0	0
$226$	10.6847	0.710733
$227$	4.68466	0.310932	0.155466	−	0.987841i	$-0.450312\pi$
$227$	4.68466	0.310932	0.155466	+	0.987841i	$0.450312\pi$
$228$	0	0
$229$	−12.4384	−0.821956	−0.410978	−	0.911645i	$-0.634813\pi$
$229$	−12.4384	−0.821956	−0.410978	+	0.911645i	$0.634813\pi$
$230$	−5.56155	−0.366718
$231$	0	0
$232$	−1.12311	−0.0737355
$233$	−20.0540	−1.31378	−0.656890	−	0.753987i	$-0.728128\pi$
$233$	−20.0540	−1.31378	−0.656890	+	0.753987i	$0.728128\pi$
$234$	0	0
$235$	−3.12311	−0.203729
$236$	4.87689	0.317459
$237$	0	0
$238$	−8.00000	−0.518563
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−4.24621	−0.273523	−0.136761	−	0.990604i	$-0.543669\pi$
$241$	−4.24621	−0.273523	−0.136761	+	0.990604i	$0.543669\pi$
$242$	8.56155	0.550357
$243$	0	0
$244$	6.00000	0.384111
$245$	4.56155	0.291427
$246$	0	0
$247$	9.36932	0.596155
$248$	−1.00000	−0.0635001
$249$	0	0
$250$	1.00000	0.0632456
$251$	16.4924	1.04099	0.520496	−	0.853864i	$-0.325747\pi$
$251$	16.4924	1.04099	0.520496	+	0.853864i	$0.325747\pi$
$252$	0	0
$253$	−8.68466	−0.546000
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.6847	−0.666491	−0.333245	−	0.942840i	$-0.608144\pi$
$257$	−10.6847	−0.666491	−0.333245	+	0.942840i	$0.608144\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	−2.00000	−0.124035
$261$	0	0
$262$	9.36932	0.578838
$263$	−12.4924	−0.770316	−0.385158	−	0.922851i	$-0.625853\pi$
$263$	−12.4924	−0.770316	−0.385158	+	0.922851i	$0.625853\pi$
$264$	0	0
$265$	11.5616	0.710221
$266$	7.31534	0.448532
$267$	0	0
$268$	9.36932	0.572322
$269$	−16.2462	−0.990549	−0.495274	−	0.868737i	$-0.664933\pi$
$269$	−16.2462	−0.990549	−0.495274	+	0.868737i	$0.664933\pi$
$270$	0	0
$271$	−10.0540	−0.610736	−0.305368	−	0.952234i	$-0.598779\pi$
$271$	−10.0540	−0.610736	−0.305368	+	0.952234i	$0.598779\pi$
$272$	−5.12311	−0.310634
$273$	0	0
$274$	3.75379	0.226775
$275$	1.56155	0.0941652
$276$	0	0
$277$	19.3693	1.16379	0.581895	−	0.813264i	$-0.302312\pi$
$277$	19.3693	1.16379	0.581895	+	0.813264i	$0.302312\pi$
$278$	8.87689	0.532401
$279$	0	0
$280$	−1.56155	−0.0933206
$281$	−13.1231	−0.782859	−0.391429	−	0.920208i	$-0.628019\pi$
$281$	−13.1231	−0.782859	−0.391429	+	0.920208i	$0.628019\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−4.68466	−0.277983
$285$	0	0
$286$	−3.12311	−0.184673
$287$	−1.75379	−0.103523
$288$	0	0
$289$	9.24621	0.543895
$290$	1.12311	0.0659510
$291$	0	0
$292$	9.80776	0.573956
$293$	6.49242	0.379291	0.189646	−	0.981853i	$-0.439266\pi$
$293$	6.49242	0.379291	0.189646	+	0.981853i	$0.439266\pi$
$294$	0	0
$295$	−4.87689	−0.283944
$296$	−5.12311	−0.297774
$297$	0	0
$298$	−20.0540	−1.16170
$299$	−11.1231	−0.643266
$300$	0	0
$301$	12.1922	0.702749
$302$	0	0
$303$	0	0
$304$	4.68466	0.268684
$305$	−6.00000	−0.343559
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	−2.43845	−0.138943
$309$	0	0
$310$	1.00000	0.0567962
$311$	18.2462	1.03465	0.517324	−	0.855790i	$-0.326928\pi$
$311$	18.2462	1.03465	0.517324	+	0.855790i	$0.326928\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	20.0540	1.13171
$315$	0	0
$316$	−16.6847	−0.938585
$317$	−25.1231	−1.41105	−0.705527	−	0.708683i	$-0.749290\pi$
$317$	−25.1231	−1.41105	−0.705527	+	0.708683i	$0.749290\pi$
$318$	0	0
$319$	1.75379	0.0981933
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−8.68466	−0.483977
$323$	−24.0000	−1.33540
$324$	0	0
$325$	2.00000	0.110940
$326$	−9.36932	−0.518918
$327$	0	0
$328$	−1.12311	−0.0620131
$329$	−4.87689	−0.268872
$330$	0	0
$331$	10.2462	0.563183	0.281591	−	0.959534i	$-0.409138\pi$
$331$	10.2462	0.563183	0.281591	+	0.959534i	$0.409138\pi$
$332$	2.24621	0.123277
$333$	0	0
$334$	−2.43845	−0.133426
$335$	−9.36932	−0.511900
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	5.12311	0.277839
$341$	1.56155	0.0845628
$342$	0	0
$343$	18.0540	0.974823
$344$	7.80776	0.420966
$345$	0	0
$346$	−8.24621	−0.443319
$347$	2.24621	0.120583	0.0602915	−	0.998181i	$-0.480797\pi$
$347$	2.24621	0.120583	0.0602915	+	0.998181i	$0.480797\pi$
$348$	0	0
$349$	−19.3693	−1.03682	−0.518408	−	0.855133i	$-0.673475\pi$
$349$	−19.3693	−1.03682	−0.518408	+	0.855133i	$0.673475\pi$
$350$	1.56155	0.0834685
$351$	0	0
$352$	−1.56155	−0.0832310
$353$	−16.2462	−0.864699	−0.432349	−	0.901706i	$-0.642315\pi$
$353$	−16.2462	−0.864699	−0.432349	+	0.901706i	$0.642315\pi$
$354$	0	0
$355$	4.68466	0.248636
$356$	1.31534	0.0697130
$357$	0	0
$358$	−12.0000	−0.634220
$359$	19.3153	1.01942	0.509712	−	0.860345i	$-0.329752\pi$
$359$	19.3153	1.01942	0.509712	+	0.860345i	$0.329752\pi$
$360$	0	0
$361$	2.94602	0.155054
$362$	−24.0540	−1.26425
$363$	0	0
$364$	−3.12311	−0.163695
$365$	−9.80776	−0.513362
$366$	0	0
$367$	−9.36932	−0.489074	−0.244537	−	0.969640i	$-0.578636\pi$
$367$	−9.36932	−0.489074	−0.244537	+	0.969640i	$0.578636\pi$
$368$	−5.56155	−0.289916
$369$	0	0
$370$	5.12311	0.266338
$371$	18.0540	0.937316
$372$	0	0
$373$	6.68466	0.346118	0.173059	−	0.984911i	$-0.444635\pi$
$373$	6.68466	0.346118	0.173059	+	0.984911i	$0.444635\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	−3.12311	−0.161062
$377$	2.24621	0.115686
$378$	0	0
$379$	30.0540	1.54377	0.771885	−	0.635763i	$-0.219314\pi$
$379$	30.0540	1.54377	0.771885	+	0.635763i	$0.219314\pi$
$380$	−4.68466	−0.240318
$381$	0	0
$382$	2.24621	0.114926
$383$	−6.24621	−0.319166	−0.159583	−	0.987184i	$-0.551015\pi$
$383$	−6.24621	−0.319166	−0.159583	+	0.987184i	$0.551015\pi$
$384$	0	0
$385$	2.43845	0.124275
$386$	18.4924	0.941240
$387$	0	0
$388$	−6.00000	−0.304604
$389$	−24.2462	−1.22933	−0.614666	−	0.788788i	$-0.710709\pi$
$389$	−24.2462	−1.22933	−0.614666	+	0.788788i	$0.710709\pi$
$390$	0	0
$391$	28.4924	1.44092
$392$	4.56155	0.230393
$393$	0	0
$394$	6.00000	0.302276
$395$	16.6847	0.839496
$396$	0	0
$397$	−28.0540	−1.40799	−0.703994	−	0.710206i	$-0.748602\pi$
$397$	−28.0540	−1.40799	−0.703994	+	0.710206i	$0.748602\pi$
$398$	−3.80776	−0.190866
$399$	0	0
$400$	1.00000	0.0500000
$401$	1.31534	0.0656850	0.0328425	−	0.999461i	$-0.489544\pi$
$401$	1.31534	0.0656850	0.0328425	+	0.999461i	$0.489544\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	−3.56155	−0.177194
$405$	0	0
$406$	1.75379	0.0870391
$407$	8.00000	0.396545
$408$	0	0
$409$	−12.6307	−0.624547	−0.312274	−	0.949992i	$-0.601091\pi$
$409$	−12.6307	−0.624547	−0.312274	+	0.949992i	$0.601091\pi$
$410$	1.12311	0.0554662
$411$	0	0
$412$	−18.2462	−0.898926
$413$	−7.61553	−0.374736
$414$	0	0
$415$	−2.24621	−0.110262
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	−7.31534	−0.357805
$419$	−22.2462	−1.08680	−0.543399	−	0.839474i	$-0.682863\pi$
$419$	−22.2462	−1.08680	−0.543399	+	0.839474i	$0.682863\pi$
$420$	0	0
$421$	18.4924	0.901266	0.450633	−	0.892709i	$-0.351198\pi$
$421$	18.4924	0.901266	0.450633	+	0.892709i	$0.351198\pi$
$422$	11.3153	0.550822
$423$	0	0
$424$	11.5616	0.561479
$425$	−5.12311	−0.248507
$426$	0	0
$427$	−9.36932	−0.453413
$428$	−1.56155	−0.0754805
$429$	0	0
$430$	−7.80776	−0.376524
$431$	13.7538	0.662497	0.331248	−	0.943544i	$-0.392530\pi$
$431$	13.7538	0.662497	0.331248	+	0.943544i	$0.392530\pi$
$432$	0	0
$433$	−18.3002	−0.879451	−0.439725	−	0.898132i	$-0.644924\pi$
$433$	−18.3002	−0.879451	−0.439725	+	0.898132i	$0.644924\pi$
$434$	1.56155	0.0749569
$435$	0	0
$436$	−19.3693	−0.927622
$437$	−26.0540	−1.24633
$438$	0	0
$439$	−31.6155	−1.50893	−0.754463	−	0.656342i	$-0.772103\pi$
$439$	−31.6155	−1.50893	−0.754463	+	0.656342i	$0.772103\pi$
$440$	1.56155	0.0744441
$441$	0	0
$442$	10.2462	0.487363
$443$	14.0540	0.667725	0.333862	−	0.942622i	$-0.391648\pi$
$443$	14.0540	0.667725	0.333862	+	0.942622i	$0.391648\pi$
$444$	0	0
$445$	−1.31534	−0.0623532
$446$	12.8769	0.609739
$447$	0	0
$448$	−1.56155	−0.0737764
$449$	22.4924	1.06148	0.530742	−	0.847534i	$-0.321913\pi$
$449$	22.4924	1.06148	0.530742	+	0.847534i	$0.321913\pi$
$450$	0	0
$451$	1.75379	0.0825827
$452$	−10.6847	−0.502564
$453$	0	0
$454$	−4.68466	−0.219862
$455$	3.12311	0.146413
$456$	0	0
$457$	24.7386	1.15722	0.578612	−	0.815603i	$-0.303594\pi$
$457$	24.7386	1.15722	0.578612	+	0.815603i	$0.303594\pi$
$458$	12.4384	0.581210
$459$	0	0
$460$	5.56155	0.259309
$461$	15.3693	0.715820	0.357910	−	0.933756i	$-0.383489\pi$
$461$	15.3693	0.715820	0.357910	+	0.933756i	$0.383489\pi$
$462$	0	0
$463$	−18.7386	−0.870858	−0.435429	−	0.900223i	$-0.643403\pi$
$463$	−18.7386	−0.870858	−0.435429	+	0.900223i	$0.643403\pi$
$464$	1.12311	0.0521389
$465$	0	0
$466$	20.0540	0.928982
$467$	26.2462	1.21453	0.607265	−	0.794499i	$-0.292267\pi$
$467$	26.2462	1.21453	0.607265	+	0.794499i	$0.292267\pi$
$468$	0	0
$469$	−14.6307	−0.675582
$470$	3.12311	0.144058
$471$	0	0
$472$	−4.87689	−0.224477
$473$	−12.1922	−0.560600
$474$	0	0
$475$	4.68466	0.214947
$476$	8.00000	0.366679
$477$	0	0
$478$	24.0000	1.09773
$479$	−6.43845	−0.294180	−0.147090	−	0.989123i	$-0.546991\pi$
$479$	−6.43845	−0.294180	−0.147090	+	0.989123i	$0.546991\pi$
$480$	0	0
$481$	10.2462	0.467187
$482$	4.24621	0.193410
$483$	0	0
$484$	−8.56155	−0.389161
$485$	6.00000	0.272446
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	−4.56155	−0.206070
$491$	−2.93087	−0.132268	−0.0661341	−	0.997811i	$-0.521067\pi$
$491$	−2.93087	−0.132268	−0.0661341	+	0.997811i	$0.521067\pi$
$492$	0	0
$493$	−5.75379	−0.259138
$494$	−9.36932	−0.421545
$495$	0	0
$496$	1.00000	0.0449013
$497$	7.31534	0.328138
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−16.4924	−0.736093
$503$	−9.75379	−0.434900	−0.217450	−	0.976071i	$-0.569774\pi$
$503$	−9.75379	−0.434900	−0.217450	+	0.976071i	$0.569774\pi$
$504$	0	0
$505$	3.56155	0.158487
$506$	8.68466	0.386080
$507$	0	0
$508$	0	0
$509$	21.6155	0.958091	0.479046	−	0.877790i	$-0.340983\pi$
$509$	21.6155	0.958091	0.479046	+	0.877790i	$0.340983\pi$
$510$	0	0
$511$	−15.3153	−0.677511
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	10.6847	0.471280
$515$	18.2462	0.804024
$516$	0	0
$517$	4.87689	0.214486
$518$	8.00000	0.351500
$519$	0	0
$520$	2.00000	0.0877058
$521$	15.7538	0.690186	0.345093	−	0.938568i	$-0.387847\pi$
$521$	15.7538	0.690186	0.345093	+	0.938568i	$0.387847\pi$
$522$	0	0
$523$	39.4233	1.72386	0.861930	−	0.507027i	$-0.169256\pi$
$523$	39.4233	1.72386	0.861930	+	0.507027i	$0.169256\pi$
$524$	−9.36932	−0.409301
$525$	0	0
$526$	12.4924	0.544696
$527$	−5.12311	−0.223166
$528$	0	0
$529$	7.93087	0.344820
$530$	−11.5616	−0.502202
$531$	0	0
$532$	−7.31534	−0.317160
$533$	2.24621	0.0972942
$534$	0	0
$535$	1.56155	0.0675118
$536$	−9.36932	−0.404693
$537$	0	0
$538$	16.2462	0.700424
$539$	−7.12311	−0.306814
$540$	0	0
$541$	36.2462	1.55835	0.779173	−	0.626809i	$-0.215639\pi$
$541$	36.2462	1.55835	0.779173	+	0.626809i	$0.215639\pi$
$542$	10.0540	0.431855
$543$	0	0
$544$	5.12311	0.219651
$545$	19.3693	0.829690
$546$	0	0
$547$	35.1231	1.50176	0.750878	−	0.660441i	$-0.229631\pi$
$547$	35.1231	1.50176	0.750878	+	0.660441i	$0.229631\pi$
$548$	−3.75379	−0.160354
$549$	0	0
$550$	−1.56155	−0.0665848
$551$	5.26137	0.224142
$552$	0	0
$553$	26.0540	1.10793
$554$	−19.3693	−0.822923
$555$	0	0
$556$	−8.87689	−0.376464
$557$	6.19224	0.262373	0.131187	−	0.991358i	$-0.458121\pi$
$557$	6.19224	0.262373	0.131187	+	0.991358i	$0.458121\pi$
$558$	0	0
$559$	−15.6155	−0.660466
$560$	1.56155	0.0659877
$561$	0	0
$562$	13.1231	0.553565
$563$	16.4924	0.695073	0.347536	−	0.937667i	$-0.387018\pi$
$563$	16.4924	0.695073	0.347536	+	0.937667i	$0.387018\pi$
$564$	0	0
$565$	10.6847	0.449507
$566$	0	0
$567$	0	0
$568$	4.68466	0.196564
$569$	−10.1922	−0.427281	−0.213640	−	0.976912i	$-0.568532\pi$
$569$	−10.1922	−0.427281	−0.213640	+	0.976912i	$0.568532\pi$
$570$	0	0
$571$	32.8769	1.37586	0.687928	−	0.725779i	$-0.258521\pi$
$571$	32.8769	1.37586	0.687928	+	0.725779i	$0.258521\pi$
$572$	3.12311	0.130584
$573$	0	0
$574$	1.75379	0.0732017
$575$	−5.56155	−0.231933
$576$	0	0
$577$	6.87689	0.286289	0.143144	−	0.989702i	$-0.454279\pi$
$577$	6.87689	0.286289	0.143144	+	0.989702i	$0.454279\pi$
$578$	−9.24621	−0.384592
$579$	0	0
$580$	−1.12311	−0.0466344
$581$	−3.50758	−0.145519
$582$	0	0
$583$	−18.0540	−0.747719
$584$	−9.80776	−0.405848
$585$	0	0
$586$	−6.49242	−0.268200
$587$	−32.4924	−1.34111	−0.670553	−	0.741862i	$-0.733943\pi$
$587$	−32.4924	−1.34111	−0.670553	+	0.741862i	$0.733943\pi$
$588$	0	0
$589$	4.68466	0.193028
$590$	4.87689	0.200779
$591$	0	0
$592$	5.12311	0.210558
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	−8.00000	−0.327968
$596$	20.0540	0.821443
$597$	0	0
$598$	11.1231	0.454858
$599$	23.4233	0.957050	0.478525	−	0.878074i	$-0.341172\pi$
$599$	23.4233	0.957050	0.478525	+	0.878074i	$0.341172\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−12.1922	−0.496918
$603$	0	0
$604$	0	0
$605$	8.56155	0.348077
$606$	0	0
$607$	−30.0540	−1.21985	−0.609927	−	0.792458i	$-0.708801\pi$
$607$	−30.0540	−1.21985	−0.609927	+	0.792458i	$0.708801\pi$
$608$	−4.68466	−0.189988
$609$	0	0
$610$	6.00000	0.242933
$611$	6.24621	0.252695
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	0	0
$616$	2.43845	0.0982478
$617$	19.5616	0.787518	0.393759	−	0.919214i	$-0.371174\pi$
$617$	19.5616	0.787518	0.393759	+	0.919214i	$0.371174\pi$
$618$	0	0
$619$	−13.3693	−0.537358	−0.268679	−	0.963230i	$-0.586587\pi$
$619$	−13.3693	−0.537358	−0.268679	+	0.963230i	$0.586587\pi$
$620$	−1.00000	−0.0401610
$621$	0	0
$622$	−18.2462	−0.731606
$623$	−2.05398	−0.0822908
$624$	0	0
$625$	1.00000	0.0400000
$626$	10.0000	0.399680
$627$	0	0
$628$	−20.0540	−0.800241
$629$	−26.2462	−1.04650
$630$	0	0
$631$	−24.3002	−0.967375	−0.483688	−	0.875241i	$-0.660703\pi$
$631$	−24.3002	−0.967375	−0.483688	+	0.875241i	$0.660703\pi$
$632$	16.6847	0.663680
$633$	0	0
$634$	25.1231	0.997766
$635$	0	0
$636$	0	0
$637$	−9.12311	−0.361471
$638$	−1.75379	−0.0694332
$639$	0	0
$640$	1.00000	0.0395285
$641$	38.4924	1.52036	0.760180	−	0.649713i	$-0.225111\pi$
$641$	38.4924	1.52036	0.760180	+	0.649713i	$0.225111\pi$
$642$	0	0
$643$	30.0540	1.18521	0.592607	−	0.805492i	$-0.298099\pi$
$643$	30.0540	1.18521	0.592607	+	0.805492i	$0.298099\pi$
$644$	8.68466	0.342223
$645$	0	0
$646$	24.0000	0.944267
$647$	17.0691	0.671057	0.335528	−	0.942030i	$-0.391085\pi$
$647$	17.0691	0.671057	0.335528	+	0.942030i	$0.391085\pi$
$648$	0	0
$649$	7.61553	0.298936
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	9.36932	0.366931
$653$	−23.7538	−0.929558	−0.464779	−	0.885427i	$-0.653866\pi$
$653$	−23.7538	−0.929558	−0.464779	+	0.885427i	$0.653866\pi$
$654$	0	0
$655$	9.36932	0.366090
$656$	1.12311	0.0438499
$657$	0	0
$658$	4.87689	0.190121
$659$	−18.7386	−0.729954	−0.364977	−	0.931017i	$-0.618923\pi$
$659$	−18.7386	−0.729954	−0.364977	+	0.931017i	$0.618923\pi$
$660$	0	0
$661$	−6.49242	−0.252526	−0.126263	−	0.991997i	$-0.540298\pi$
$661$	−6.49242	−0.252526	−0.126263	+	0.991997i	$0.540298\pi$
$662$	−10.2462	−0.398230
$663$	0	0
$664$	−2.24621	−0.0871699
$665$	7.31534	0.283677
$666$	0	0
$667$	−6.24621	−0.241854
$668$	2.43845	0.0943464
$669$	0	0
$670$	9.36932	0.361968
$671$	9.36932	0.361698
$672$	0	0
$673$	28.2462	1.08881	0.544406	−	0.838822i	$-0.316755\pi$
$673$	28.2462	1.08881	0.544406	+	0.838822i	$0.316755\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	29.4233	1.13083	0.565414	−	0.824807i	$-0.308716\pi$
$677$	29.4233	1.13083	0.565414	+	0.824807i	$0.308716\pi$
$678$	0	0
$679$	9.36932	0.359561
$680$	−5.12311	−0.196462
$681$	0	0
$682$	−1.56155	−0.0597949
$683$	−19.3153	−0.739081	−0.369541	−	0.929215i	$-0.620485\pi$
$683$	−19.3153	−0.739081	−0.369541	+	0.929215i	$0.620485\pi$
$684$	0	0
$685$	3.75379	0.143425
$686$	−18.0540	−0.689304
$687$	0	0
$688$	−7.80776	−0.297668
$689$	−23.1231	−0.880920
$690$	0	0
$691$	−22.0540	−0.838973	−0.419486	−	0.907762i	$-0.637790\pi$
$691$	−22.0540	−0.838973	−0.419486	+	0.907762i	$0.637790\pi$
$692$	8.24621	0.313474
$693$	0	0
$694$	−2.24621	−0.0852650
$695$	8.87689	0.336720
$696$	0	0
$697$	−5.75379	−0.217940
$698$	19.3693	0.733139
$699$	0	0
$700$	−1.56155	−0.0590211
$701$	−25.8078	−0.974746	−0.487373	−	0.873194i	$-0.662045\pi$
$701$	−25.8078	−0.974746	−0.487373	+	0.873194i	$0.662045\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	1.56155	0.0588532
$705$	0	0
$706$	16.2462	0.611434
$707$	5.56155	0.209164
$708$	0	0
$709$	−20.0540	−0.753143	−0.376571	−	0.926388i	$-0.622897\pi$
$709$	−20.0540	−0.753143	−0.376571	+	0.926388i	$0.622897\pi$
$710$	−4.68466	−0.175812
$711$	0	0
$712$	−1.31534	−0.0492945
$713$	−5.56155	−0.208282
$714$	0	0
$715$	−3.12311	−0.116798
$716$	12.0000	0.448461
$717$	0	0
$718$	−19.3153	−0.720842
$719$	43.1231	1.60822	0.804110	−	0.594480i	$-0.202642\pi$
$719$	43.1231	1.60822	0.804110	+	0.594480i	$0.202642\pi$
$720$	0	0
$721$	28.4924	1.06111
$722$	−2.94602	−0.109640
$723$	0	0
$724$	24.0540	0.893959
$725$	1.12311	0.0417111
$726$	0	0
$727$	−14.0540	−0.521233	−0.260617	−	0.965442i	$-0.583926\pi$
$727$	−14.0540	−0.521233	−0.260617	+	0.965442i	$0.583926\pi$
$728$	3.12311	0.115750
$729$	0	0
$730$	9.80776	0.363002
$731$	40.0000	1.47945
$732$	0	0
$733$	26.4924	0.978520	0.489260	−	0.872138i	$-0.337267\pi$
$733$	26.4924	0.978520	0.489260	+	0.872138i	$0.337267\pi$
$734$	9.36932	0.345828
$735$	0	0
$736$	5.56155	0.205002
$737$	14.6307	0.538928
$738$	0	0
$739$	−12.9848	−0.477655	−0.238828	−	0.971062i	$-0.576763\pi$
$739$	−12.9848	−0.477655	−0.238828	+	0.971062i	$0.576763\pi$
$740$	−5.12311	−0.188329
$741$	0	0
$742$	−18.0540	−0.662782
$743$	−11.4233	−0.419080	−0.209540	−	0.977800i	$-0.567197\pi$
$743$	−11.4233	−0.419080	−0.209540	+	0.977800i	$0.567197\pi$
$744$	0	0
$745$	−20.0540	−0.734721
$746$	−6.68466	−0.244743
$747$	0	0
$748$	−8.00000	−0.292509
$749$	2.43845	0.0890989
$750$	0	0
$751$	36.8769	1.34566	0.672828	−	0.739798i	$-0.265079\pi$
$751$	36.8769	1.34566	0.672828	+	0.739798i	$0.265079\pi$
$752$	3.12311	0.113888
$753$	0	0
$754$	−2.24621	−0.0818022
$755$	0	0
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	−30.0540	−1.09161
$759$	0	0
$760$	4.68466	0.169930
$761$	−41.4233	−1.50159	−0.750797	−	0.660533i	$-0.770330\pi$
$761$	−41.4233	−1.50159	−0.750797	+	0.660533i	$0.770330\pi$
$762$	0	0
$763$	30.2462	1.09499
$764$	−2.24621	−0.0812651
$765$	0	0
$766$	6.24621	0.225685
$767$	9.75379	0.352189
$768$	0	0
$769$	−16.4384	−0.592786	−0.296393	−	0.955066i	$-0.595784\pi$
$769$	−16.4384	−0.592786	−0.296393	+	0.955066i	$0.595784\pi$
$770$	−2.43845	−0.0878755
$771$	0	0
$772$	−18.4924	−0.665557
$773$	15.5616	0.559710	0.279855	−	0.960042i	$-0.409714\pi$
$773$	15.5616	0.559710	0.279855	+	0.960042i	$0.409714\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	6.00000	0.215387
$777$	0	0
$778$	24.2462	0.869269
$779$	5.26137	0.188508
$780$	0	0
$781$	−7.31534	−0.261764
$782$	−28.4924	−1.01889
$783$	0	0
$784$	−4.56155	−0.162913
$785$	20.0540	0.715757
$786$	0	0
$787$	10.9309	0.389643	0.194822	−	0.980839i	$-0.437587\pi$
$787$	10.9309	0.389643	0.194822	+	0.980839i	$0.437587\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−16.6847	−0.593614
$791$	16.6847	0.593238
$792$	0	0
$793$	12.0000	0.426132
$794$	28.0540	0.995598
$795$	0	0
$796$	3.80776	0.134963
$797$	−38.9848	−1.38091	−0.690457	−	0.723373i	$-0.742591\pi$
$797$	−38.9848	−1.38091	−0.690457	+	0.723373i	$0.742591\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−1.31534	−0.0464463
$803$	15.3153	0.540467
$804$	0	0
$805$	−8.68466	−0.306094
$806$	−2.00000	−0.0704470
$807$	0	0
$808$	3.56155	0.125295
$809$	25.3153	0.890040	0.445020	−	0.895521i	$-0.353197\pi$
$809$	25.3153	0.890040	0.445020	+	0.895521i	$0.353197\pi$
$810$	0	0
$811$	53.6695	1.88459	0.942296	−	0.334782i	$-0.108663\pi$
$811$	53.6695	1.88459	0.942296	+	0.334782i	$0.108663\pi$
$812$	−1.75379	−0.0615459
$813$	0	0
$814$	−8.00000	−0.280400
$815$	−9.36932	−0.328193
$816$	0	0
$817$	−36.5767	−1.27966
$818$	12.6307	0.441621
$819$	0	0
$820$	−1.12311	−0.0392205
$821$	21.6155	0.754387	0.377194	−	0.926134i	$-0.376889\pi$
$821$	21.6155	0.754387	0.377194	+	0.926134i	$0.376889\pi$
$822$	0	0
$823$	15.6155	0.544323	0.272162	−	0.962252i	$-0.412261\pi$
$823$	15.6155	0.544323	0.272162	+	0.962252i	$0.412261\pi$
$824$	18.2462	0.635637
$825$	0	0
$826$	7.61553	0.264978
$827$	−18.2462	−0.634483	−0.317241	−	0.948345i	$-0.602757\pi$
$827$	−18.2462	−0.634483	−0.317241	+	0.948345i	$0.602757\pi$
$828$	0	0
$829$	−37.4233	−1.29976	−0.649882	−	0.760035i	$-0.725182\pi$
$829$	−37.4233	−1.29976	−0.649882	+	0.760035i	$0.725182\pi$
$830$	2.24621	0.0779671
$831$	0	0
$832$	2.00000	0.0693375
$833$	23.3693	0.809699
$834$	0	0
$835$	−2.43845	−0.0843859
$836$	7.31534	0.253006
$837$	0	0
$838$	22.2462	0.768483
$839$	−34.9309	−1.20595	−0.602974	−	0.797761i	$-0.706018\pi$
$839$	−34.9309	−1.20595	−0.602974	+	0.797761i	$0.706018\pi$
$840$	0	0
$841$	−27.7386	−0.956505
$842$	−18.4924	−0.637291
$843$	0	0
$844$	−11.3153	−0.389490
$845$	9.00000	0.309609
$846$	0	0
$847$	13.3693	0.459375
$848$	−11.5616	−0.397025
$849$	0	0
$850$	5.12311	0.175721
$851$	−28.4924	−0.976708
$852$	0	0
$853$	−38.7926	−1.32823	−0.664117	−	0.747629i	$-0.731192\pi$
$853$	−38.7926	−1.32823	−0.664117	+	0.747629i	$0.731192\pi$
$854$	9.36932	0.320611
$855$	0	0
$856$	1.56155	0.0533728
$857$	53.2311	1.81834	0.909169	−	0.416427i	$-0.136718\pi$
$857$	53.2311	1.81834	0.909169	+	0.416427i	$0.136718\pi$
$858$	0	0
$859$	−22.7386	−0.775832	−0.387916	−	0.921695i	$-0.626805\pi$
$859$	−22.7386	−0.775832	−0.387916	+	0.921695i	$0.626805\pi$
$860$	7.80776	0.266243
$861$	0	0
$862$	−13.7538	−0.468456
$863$	30.5464	1.03981	0.519906	−	0.854224i	$-0.325967\pi$
$863$	30.5464	1.03981	0.519906	+	0.854224i	$0.325967\pi$
$864$	0	0
$865$	−8.24621	−0.280380
$866$	18.3002	0.621866
$867$	0	0
$868$	−1.56155	−0.0530026
$869$	−26.0540	−0.883821
$870$	0	0
$871$	18.7386	0.634934
$872$	19.3693	0.655928
$873$	0	0
$874$	26.0540	0.881289
$875$	1.56155	0.0527901
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	31.6155	1.06697
$879$	0	0
$880$	−1.56155	−0.0526399
$881$	−42.4924	−1.43161	−0.715803	−	0.698302i	$-0.753939\pi$
$881$	−42.4924	−1.43161	−0.715803	+	0.698302i	$0.753939\pi$
$882$	0	0
$883$	−31.4233	−1.05748	−0.528739	−	0.848784i	$-0.677335\pi$
$883$	−31.4233	−1.05748	−0.528739	+	0.848784i	$0.677335\pi$
$884$	−10.2462	−0.344617
$885$	0	0
$886$	−14.0540	−0.472153
$887$	57.3693	1.92627	0.963137	−	0.269013i	$-0.0866974\pi$
$887$	57.3693	1.92627	0.963137	+	0.269013i	$0.0866974\pi$
$888$	0	0
$889$	0	0
$890$	1.31534	0.0440903
$891$	0	0
$892$	−12.8769	−0.431150
$893$	14.6307	0.489597
$894$	0	0
$895$	−12.0000	−0.401116
$896$	1.56155	0.0521678
$897$	0	0
$898$	−22.4924	−0.750582
$899$	1.12311	0.0374577
$900$	0	0
$901$	59.2311	1.97327
$902$	−1.75379	−0.0583948
$903$	0	0
$904$	10.6847	0.355366
$905$	−24.0540	−0.799581
$906$	0	0
$907$	24.0000	0.796907	0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000	0.796907	0.398453	+	0.917189i	$0.369547\pi$
$908$	4.68466	0.155466
$909$	0	0
$910$	−3.12311	−0.103530
$911$	44.4924	1.47410	0.737050	−	0.675838i	$-0.236218\pi$
$911$	44.4924	1.47410	0.737050	+	0.675838i	$0.236218\pi$
$912$	0	0
$913$	3.50758	0.116084
$914$	−24.7386	−0.818281
$915$	0	0
$916$	−12.4384	−0.410978
$917$	14.6307	0.483148
$918$	0	0
$919$	39.6155	1.30680	0.653398	−	0.757015i	$-0.273343\pi$
$919$	39.6155	1.30680	0.653398	+	0.757015i	$0.273343\pi$
$920$	−5.56155	−0.183359
$921$	0	0
$922$	−15.3693	−0.506161
$923$	−9.36932	−0.308395
$924$	0	0
$925$	5.12311	0.168447
$926$	18.7386	0.615790
$927$	0	0
$928$	−1.12311	−0.0368677
$929$	−11.5616	−0.379322	−0.189661	−	0.981850i	$-0.560739\pi$
$929$	−11.5616	−0.379322	−0.189661	+	0.981850i	$0.560739\pi$
$930$	0	0
$931$	−21.3693	−0.700351
$932$	−20.0540	−0.656890
$933$	0	0
$934$	−26.2462	−0.858802
$935$	8.00000	0.261628
$936$	0	0
$937$	16.6307	0.543301	0.271650	−	0.962396i	$-0.412431\pi$
$937$	16.6307	0.543301	0.271650	+	0.962396i	$0.412431\pi$
$938$	14.6307	0.477709
$939$	0	0
$940$	−3.12311	−0.101864
$941$	−27.3693	−0.892214	−0.446107	−	0.894980i	$-0.647190\pi$
$941$	−27.3693	−0.892214	−0.446107	+	0.894980i	$0.647190\pi$
$942$	0	0
$943$	−6.24621	−0.203405
$944$	4.87689	0.158729
$945$	0	0
$946$	12.1922	0.396404
$947$	0.876894	0.0284952	0.0142476	−	0.999898i	$-0.495465\pi$
$947$	0.876894	0.0284952	0.0142476	+	0.999898i	$0.495465\pi$
$948$	0	0
$949$	19.6155	0.636747
$950$	−4.68466	−0.151990
$951$	0	0
$952$	−8.00000	−0.259281
$953$	58.1080	1.88230	0.941151	−	0.337988i	$-0.109746\pi$
$953$	58.1080	1.88230	0.941151	+	0.337988i	$0.109746\pi$
$954$	0	0
$955$	2.24621	0.0726857
$956$	−24.0000	−0.776215
$957$	0	0
$958$	6.43845	0.208017
$959$	5.86174	0.189285
$960$	0	0
$961$	1.00000	0.0322581
$962$	−10.2462	−0.330351
$963$	0	0
$964$	−4.24621	−0.136761
$965$	18.4924	0.595292
$966$	0	0
$967$	−25.7538	−0.828186	−0.414093	−	0.910235i	$-0.635901\pi$
$967$	−25.7538	−0.828186	−0.414093	+	0.910235i	$0.635901\pi$
$968$	8.56155	0.275179
$969$	0	0
$970$	−6.00000	−0.192648
$971$	28.8769	0.926704	0.463352	−	0.886174i	$-0.346647\pi$
$971$	28.8769	0.926704	0.463352	+	0.886174i	$0.346647\pi$
$972$	0	0
$973$	13.8617	0.444387
$974$	−24.0000	−0.769010
$975$	0	0
$976$	6.00000	0.192055
$977$	22.9848	0.735350	0.367675	−	0.929954i	$-0.380154\pi$
$977$	22.9848	0.735350	0.367675	+	0.929954i	$0.380154\pi$
$978$	0	0
$979$	2.05398	0.0656453
$980$	4.56155	0.145713
$981$	0	0
$982$	2.93087	0.0935278
$983$	24.9848	0.796893	0.398446	−	0.917192i	$-0.369549\pi$
$983$	24.9848	0.796893	0.398446	+	0.917192i	$0.369549\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	5.75379	0.183238
$987$	0	0
$988$	9.36932	0.298078
$989$	43.4233	1.38078
$990$	0	0
$991$	23.3153	0.740636	0.370318	−	0.928905i	$-0.379249\pi$
$991$	23.3153	0.740636	0.370318	+	0.928905i	$0.379249\pi$
$992$	−1.00000	−0.0317500
$993$	0	0
$994$	−7.31534	−0.232029
$995$	−3.80776	−0.120714
$996$	0	0
$997$	−62.4924	−1.97915	−0.989577	−	0.144002i	$-0.954003\pi$
$997$	−62.4924	−1.97915	−0.989577	+	0.144002i	$0.954003\pi$
$998$	−20.0000	−0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.be.1.1		2
3.2	odd	2		930.2.a.p.1.1	✓	2
12.11	even	2		7440.2.a.bl.1.2		2
15.2	even	4		4650.2.d.bd.3349.3		4
15.8	even	4		4650.2.d.bd.3349.2		4
15.14	odd	2		4650.2.a.ce.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.p.1.1	✓	2	3.2	odd	2
2790.2.a.be.1.1		2	1.1	even	1	trivial
4650.2.a.ce.1.2		2	15.14	odd	2
4650.2.d.bd.3349.2		4	15.8	even	4
4650.2.d.bd.3349.3		4	15.2	even	4
7440.2.a.bl.1.2		2	12.11	even	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.56155 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.56155 q^{11} +2.00000 q^{13} +1.56155 q^{14} +1.00000 q^{16} -5.12311 q^{17} +4.68466 q^{19} -1.00000 q^{20} -1.56155 q^{22} -5.56155 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.56155 q^{28} +1.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} +5.12311 q^{34} +1.56155 q^{35} +5.12311 q^{37} -4.68466 q^{38} +1.00000 q^{40} +1.12311 q^{41} -7.80776 q^{43} +1.56155 q^{44} +5.56155 q^{46} +3.12311 q^{47} -4.56155 q^{49} -1.00000 q^{50} +2.00000 q^{52} -11.5616 q^{53} -1.56155 q^{55} +1.56155 q^{56} -1.12311 q^{58} +4.87689 q^{59} +6.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +9.36932 q^{67} -5.12311 q^{68} -1.56155 q^{70} -4.68466 q^{71} +9.80776 q^{73} -5.12311 q^{74} +4.68466 q^{76} -2.43845 q^{77} -16.6847 q^{79} -1.00000 q^{80} -1.12311 q^{82} +2.24621 q^{83} +5.12311 q^{85} +7.80776 q^{86} -1.56155 q^{88} +1.31534 q^{89} -3.12311 q^{91} -5.56155 q^{92} -3.12311 q^{94} -4.68466 q^{95} -6.00000 q^{97} +4.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} + 4 q^{13} - q^{14} + 2 q^{16} - 2 q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 7 q^{23} + 2 q^{25} - 4 q^{26} + q^{28} - 6 q^{29} + 2 q^{31} - 2 q^{32} + 2 q^{34} - q^{35} + 2 q^{37} + 3 q^{38} + 2 q^{40} - 6 q^{41} + 5 q^{43} - q^{44} + 7 q^{46} - 2 q^{47} - 5 q^{49} - 2 q^{50} + 4 q^{52} - 19 q^{53} + q^{55} - q^{56} + 6 q^{58} + 18 q^{59} + 12 q^{61} - 2 q^{62} + 2 q^{64} - 4 q^{65} - 6 q^{67} - 2 q^{68} + q^{70} + 3 q^{71} - q^{73} - 2 q^{74} - 3 q^{76} - 9 q^{77} - 21 q^{79} - 2 q^{80} + 6 q^{82} - 12 q^{83} + 2 q^{85} - 5 q^{86} + q^{88} + 15 q^{89} + 2 q^{91} - 7 q^{92} + 2 q^{94} + 3 q^{95} - 12 q^{97} + 5 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8 + 2 * q^10 - q^11 + 4 * q^13 - q^14 + 2 * q^16 - 2 * q^17 - 3 * q^19 - 2 * q^20 + q^22 - 7 * q^23 + 2 * q^25 - 4 * q^26 + q^28 - 6 * q^29 + 2 * q^31 - 2 * q^32 + 2 * q^34 - q^35 + 2 * q^37 + 3 * q^38 + 2 * q^40 - 6 * q^41 + 5 * q^43 - q^44 + 7 * q^46 - 2 * q^47 - 5 * q^49 - 2 * q^50 + 4 * q^52 - 19 * q^53 + q^55 - q^56 + 6 * q^58 + 18 * q^59 + 12 * q^61 - 2 * q^62 + 2 * q^64 - 4 * q^65 - 6 * q^67 - 2 * q^68 + q^70 + 3 * q^71 - q^73 - 2 * q^74 - 3 * q^76 - 9 * q^77 - 21 * q^79 - 2 * q^80 + 6 * q^82 - 12 * q^83 + 2 * q^85 - 5 * q^86 + q^88 + 15 * q^89 + 2 * q^91 - 7 * q^92 + 2 * q^94 + 3 * q^95 - 12 * q^97 + 5 * q^98

Introduction

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