LMFDB - Embedded newform 2790.2.a.bd.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
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			$-2.37228$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.37228 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.37228 q^{7} -1.00000 q^{8} +1.00000 q^{10} -6.37228 q^{11} -2.00000 q^{13} +2.37228 q^{14} +1.00000 q^{16} -6.74456 q^{17} -6.37228 q^{19} -1.00000 q^{20} +6.37228 q^{22} +2.37228 q^{23} +1.00000 q^{25} +2.00000 q^{26} -2.37228 q^{28} -2.74456 q^{29} -1.00000 q^{31} -1.00000 q^{32} +6.74456 q^{34} +2.37228 q^{35} +10.7446 q^{37} +6.37228 q^{38} +1.00000 q^{40} +10.7446 q^{41} +6.37228 q^{43} -6.37228 q^{44} -2.37228 q^{46} -4.74456 q^{47} -1.37228 q^{49} -1.00000 q^{50} -2.00000 q^{52} +4.37228 q^{53} +6.37228 q^{55} +2.37228 q^{56} +2.74456 q^{58} +8.74456 q^{59} -11.4891 q^{61} +1.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -0.744563 q^{67} -6.74456 q^{68} -2.37228 q^{70} +2.37228 q^{71} +9.11684 q^{73} -10.7446 q^{74} -6.37228 q^{76} +15.1168 q^{77} -10.3723 q^{79} -1.00000 q^{80} -10.7446 q^{82} +12.0000 q^{83} +6.74456 q^{85} -6.37228 q^{86} +6.37228 q^{88} -4.37228 q^{89} +4.74456 q^{91} +2.37228 q^{92} +4.74456 q^{94} +6.37228 q^{95} +2.00000 q^{97} +1.37228 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 7 q^{11} - 4 q^{13} - q^{14} + 2 q^{16} - 2 q^{17} - 7 q^{19} - 2 q^{20} + 7 q^{22} - q^{23} + 2 q^{25} + 4 q^{26} + q^{28} + 6 q^{29} - 2 q^{31} - 2 q^{32} + 2 q^{34} - q^{35} + 10 q^{37} + 7 q^{38} + 2 q^{40} + 10 q^{41} + 7 q^{43} - 7 q^{44} + q^{46} + 2 q^{47} + 3 q^{49} - 2 q^{50} - 4 q^{52} + 3 q^{53} + 7 q^{55} - q^{56} - 6 q^{58} + 6 q^{59} + 2 q^{62} + 2 q^{64} + 4 q^{65} + 10 q^{67} - 2 q^{68} + q^{70} - q^{71} + q^{73} - 10 q^{74} - 7 q^{76} + 13 q^{77} - 15 q^{79} - 2 q^{80} - 10 q^{82} + 24 q^{83} + 2 q^{85} - 7 q^{86} + 7 q^{88} - 3 q^{89} - 2 q^{91} - q^{92} - 2 q^{94} + 7 q^{95} + 4 q^{97} - 3 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8 + 2 * q^10 - 7 * q^11 - 4 * q^13 - q^14 + 2 * q^16 - 2 * q^17 - 7 * q^19 - 2 * q^20 + 7 * q^22 - q^23 + 2 * q^25 + 4 * q^26 + q^28 + 6 * q^29 - 2 * q^31 - 2 * q^32 + 2 * q^34 - q^35 + 10 * q^37 + 7 * q^38 + 2 * q^40 + 10 * q^41 + 7 * q^43 - 7 * q^44 + q^46 + 2 * q^47 + 3 * q^49 - 2 * q^50 - 4 * q^52 + 3 * q^53 + 7 * q^55 - q^56 - 6 * q^58 + 6 * q^59 + 2 * q^62 + 2 * q^64 + 4 * q^65 + 10 * q^67 - 2 * q^68 + q^70 - q^71 + q^73 - 10 * q^74 - 7 * q^76 + 13 * q^77 - 15 * q^79 - 2 * q^80 - 10 * q^82 + 24 * q^83 + 2 * q^85 - 7 * q^86 + 7 * q^88 - 3 * q^89 - 2 * q^91 - q^92 - 2 * q^94 + 7 * q^95 + 4 * q^97 - 3 * 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
$7$	−2.37228	−0.896638	−0.448319	−	0.893874i	$-0.647977\pi$
$7$	−2.37228	−0.896638	−0.448319	+	0.893874i	$0.647977\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−6.37228	−1.92132	−0.960658	−	0.277736i	$-0.910416\pi$
$11$	−6.37228	−1.92132	−0.960658	+	0.277736i	$0.910416\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	2.37228	0.634019
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.74456	−1.63580	−0.817898	−	0.575363i	$-0.804861\pi$
$17$	−6.74456	−1.63580	−0.817898	+	0.575363i	$0.804861\pi$
$18$	0	0
$19$	−6.37228	−1.46190	−0.730951	−	0.682430i	$-0.760923\pi$
$19$	−6.37228	−1.46190	−0.730951	+	0.682430i	$0.760923\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	6.37228	1.35857
$23$	2.37228	0.494655	0.247327	−	0.968932i	$-0.420448\pi$
$23$	2.37228	0.494655	0.247327	+	0.968932i	$0.420448\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	−2.37228	−0.448319
$29$	−2.74456	−0.509652	−0.254826	−	0.966987i	$-0.582018\pi$
$29$	−2.74456	−0.509652	−0.254826	+	0.966987i	$0.582018\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.74456	1.15668
$35$	2.37228	0.400989
$36$	0	0
$37$	10.7446	1.76640	0.883198	−	0.469001i	$-0.155386\pi$
$37$	10.7446	1.76640	0.883198	+	0.469001i	$0.155386\pi$
$38$	6.37228	1.03372
$39$	0	0
$40$	1.00000	0.158114
$41$	10.7446	1.67802	0.839009	−	0.544117i	$-0.183135\pi$
$41$	10.7446	1.67802	0.839009	+	0.544117i	$0.183135\pi$
$42$	0	0
$43$	6.37228	0.971764	0.485882	−	0.874024i	$-0.338499\pi$
$43$	6.37228	0.971764	0.485882	+	0.874024i	$0.338499\pi$
$44$	−6.37228	−0.960658
$45$	0	0
$46$	−2.37228	−0.349774
$47$	−4.74456	−0.692066	−0.346033	−	0.938222i	$-0.612471\pi$
$47$	−4.74456	−0.692066	−0.346033	+	0.938222i	$0.612471\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	4.37228	0.600579	0.300290	−	0.953848i	$-0.402917\pi$
$53$	4.37228	0.600579	0.300290	+	0.953848i	$0.402917\pi$
$54$	0	0
$55$	6.37228	0.859238
$56$	2.37228	0.317009
$57$	0	0
$58$	2.74456	0.360379
$59$	8.74456	1.13845	0.569223	−	0.822183i	$-0.307244\pi$
$59$	8.74456	1.13845	0.569223	+	0.822183i	$0.307244\pi$
$60$	0	0
$61$	−11.4891	−1.47103	−0.735516	−	0.677507i	$-0.763060\pi$
$61$	−11.4891	−1.47103	−0.735516	+	0.677507i	$0.763060\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	−0.744563	−0.0909628	−0.0454814	−	0.998965i	$-0.514482\pi$
$67$	−0.744563	−0.0909628	−0.0454814	+	0.998965i	$0.514482\pi$
$68$	−6.74456	−0.817898
$69$	0	0
$70$	−2.37228	−0.283542
$71$	2.37228	0.281538	0.140769	−	0.990042i	$-0.455043\pi$
$71$	2.37228	0.281538	0.140769	+	0.990042i	$0.455043\pi$
$72$	0	0
$73$	9.11684	1.06705	0.533523	−	0.845786i	$-0.320868\pi$
$73$	9.11684	1.06705	0.533523	+	0.845786i	$0.320868\pi$
$74$	−10.7446	−1.24903
$75$	0	0
$76$	−6.37228	−0.730951
$77$	15.1168	1.72272
$78$	0	0
$79$	−10.3723	−1.16697	−0.583486	−	0.812123i	$-0.698312\pi$
$79$	−10.3723	−1.16697	−0.583486	+	0.812123i	$0.698312\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−10.7446	−1.18654
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	6.74456	0.731551
$86$	−6.37228	−0.687141
$87$	0	0
$88$	6.37228	0.679287
$89$	−4.37228	−0.463461	−0.231730	−	0.972780i	$-0.574439\pi$
$89$	−4.37228	−0.463461	−0.231730	+	0.972780i	$0.574439\pi$
$90$	0	0
$91$	4.74456	0.497365
$92$	2.37228	0.247327
$93$	0	0
$94$	4.74456	0.489364
$95$	6.37228	0.653782
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	1.37228	0.138621
$99$	0	0
$100$	1.00000	0.100000
$101$	9.11684	0.907160	0.453580	−	0.891216i	$-0.350147\pi$
$101$	9.11684	0.907160	0.453580	+	0.891216i	$0.350147\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−4.37228	−0.424674
$107$	−6.37228	−0.616032	−0.308016	−	0.951381i	$-0.599665\pi$
$107$	−6.37228	−0.616032	−0.308016	+	0.951381i	$0.599665\pi$
$108$	0	0
$109$	−6.74456	−0.646012	−0.323006	−	0.946397i	$-0.604693\pi$
$109$	−6.74456	−0.646012	−0.323006	+	0.946397i	$0.604693\pi$
$110$	−6.37228	−0.607573
$111$	0	0
$112$	−2.37228	−0.224160
$113$	8.37228	0.787598	0.393799	−	0.919197i	$-0.371161\pi$
$113$	8.37228	0.787598	0.393799	+	0.919197i	$0.371161\pi$
$114$	0	0
$115$	−2.37228	−0.221216
$116$	−2.74456	−0.254826
$117$	0	0
$118$	−8.74456	−0.805002
$119$	16.0000	1.46672
$120$	0	0
$121$	29.6060	2.69145
$122$	11.4891	1.04018
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.48913	−0.842024	−0.421012	−	0.907055i	$-0.638325\pi$
$127$	−9.48913	−0.842024	−0.421012	+	0.907055i	$0.638325\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.00000	−0.175412
$131$	−18.2337	−1.59308	−0.796542	−	0.604583i	$-0.793340\pi$
$131$	−18.2337	−1.59308	−0.796542	+	0.604583i	$0.793340\pi$
$132$	0	0
$133$	15.1168	1.31080
$134$	0.744563	0.0643204
$135$	0	0
$136$	6.74456	0.578341
$137$	−19.4891	−1.66507	−0.832534	−	0.553974i	$-0.813111\pi$
$137$	−19.4891	−1.66507	−0.832534	+	0.553974i	$0.813111\pi$
$138$	0	0
$139$	0.744563	0.0631530	0.0315765	−	0.999501i	$-0.489947\pi$
$139$	0.744563	0.0631530	0.0315765	+	0.999501i	$0.489947\pi$
$140$	2.37228	0.200494
$141$	0	0
$142$	−2.37228	−0.199077
$143$	12.7446	1.06575
$144$	0	0
$145$	2.74456	0.227924
$146$	−9.11684	−0.754515
$147$	0	0
$148$	10.7446	0.883198
$149$	−5.11684	−0.419188	−0.209594	−	0.977788i	$-0.567214\pi$
$149$	−5.11684	−0.419188	−0.209594	+	0.977788i	$0.567214\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	6.37228	0.516860
$153$	0	0
$154$	−15.1168	−1.21815
$155$	1.00000	0.0803219
$156$	0	0
$157$	3.62772	0.289523	0.144762	−	0.989467i	$-0.453758\pi$
$157$	3.62772	0.289523	0.144762	+	0.989467i	$0.453758\pi$
$158$	10.3723	0.825174
$159$	0	0
$160$	1.00000	0.0790569
$161$	−5.62772	−0.443526
$162$	0	0
$163$	−10.2337	−0.801564	−0.400782	−	0.916173i	$-0.631262\pi$
$163$	−10.2337	−0.801564	−0.400782	+	0.916173i	$0.631262\pi$
$164$	10.7446	0.839009
$165$	0	0
$166$	−12.0000	−0.931381
$167$	18.3723	1.42169	0.710845	−	0.703349i	$-0.248313\pi$
$167$	18.3723	1.42169	0.710845	+	0.703349i	$0.248313\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−6.74456	−0.517284
$171$	0	0
$172$	6.37228	0.485882
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	−2.37228	−0.179328
$176$	−6.37228	−0.480329
$177$	0	0
$178$	4.37228	0.327716
$179$	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
$181$	−13.8614	−1.03031	−0.515155	−	0.857097i	$-0.672266\pi$
$181$	−13.8614	−1.03031	−0.515155	+	0.857097i	$0.672266\pi$
$182$	−4.74456	−0.351690
$183$	0	0
$184$	−2.37228	−0.174887
$185$	−10.7446	−0.789956
$186$	0	0
$187$	42.9783	3.14288
$188$	−4.74456	−0.346033
$189$	0	0
$190$	−6.37228	−0.462294
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	−7.48913	−0.539079	−0.269540	−	0.962989i	$-0.586871\pi$
$193$	−7.48913	−0.539079	−0.269540	+	0.962989i	$0.586871\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−1.37228	−0.0980201
$197$	3.48913	0.248590	0.124295	−	0.992245i	$-0.460333\pi$
$197$	3.48913	0.248590	0.124295	+	0.992245i	$0.460333\pi$
$198$	0	0
$199$	−18.3723	−1.30238	−0.651188	−	0.758916i	$-0.725729\pi$
$199$	−18.3723	−1.30238	−0.651188	+	0.758916i	$0.725729\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−9.11684	−0.641459
$203$	6.51087	0.456974
$204$	0	0
$205$	−10.7446	−0.750433
$206$	8.00000	0.557386
$207$	0	0
$208$	−2.00000	−0.138675
$209$	40.6060	2.80877
$210$	0	0
$211$	−6.37228	−0.438686	−0.219343	−	0.975648i	$-0.570391\pi$
$211$	−6.37228	−0.438686	−0.219343	+	0.975648i	$0.570391\pi$
$212$	4.37228	0.300290
$213$	0	0
$214$	6.37228	0.435600
$215$	−6.37228	−0.434586
$216$	0	0
$217$	2.37228	0.161041
$218$	6.74456	0.456799
$219$	0	0
$220$	6.37228	0.429619
$221$	13.4891	0.907377
$222$	0	0
$223$	20.7446	1.38916	0.694579	−	0.719416i	$-0.255591\pi$
$223$	20.7446	1.38916	0.694579	+	0.719416i	$0.255591\pi$
$224$	2.37228	0.158505
$225$	0	0
$226$	−8.37228	−0.556916
$227$	11.1168	0.737851	0.368925	−	0.929459i	$-0.379726\pi$
$227$	11.1168	0.737851	0.368925	+	0.929459i	$0.379726\pi$
$228$	0	0
$229$	21.1168	1.39544	0.697720	−	0.716370i	$-0.254198\pi$
$229$	21.1168	1.39544	0.697720	+	0.716370i	$0.254198\pi$
$230$	2.37228	0.156424
$231$	0	0
$232$	2.74456	0.180189
$233$	−13.8614	−0.908091	−0.454045	−	0.890979i	$-0.650020\pi$
$233$	−13.8614	−0.908091	−0.454045	+	0.890979i	$0.650020\pi$
$234$	0	0
$235$	4.74456	0.309501
$236$	8.74456	0.569223
$237$	0	0
$238$	−16.0000	−1.03713
$239$	6.51087	0.421153	0.210577	−	0.977577i	$-0.432466\pi$
$239$	6.51087	0.421153	0.210577	+	0.977577i	$0.432466\pi$
$240$	0	0
$241$	27.4891	1.77073	0.885365	−	0.464896i	$-0.153908\pi$
$241$	27.4891	1.77073	0.885365	+	0.464896i	$0.153908\pi$
$242$	−29.6060	−1.90314
$243$	0	0
$244$	−11.4891	−0.735516
$245$	1.37228	0.0876718
$246$	0	0
$247$	12.7446	0.810917
$248$	1.00000	0.0635001
$249$	0	0
$250$	1.00000	0.0632456
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	−15.1168	−0.950388
$254$	9.48913	0.595401
$255$	0	0
$256$	1.00000	0.0625000
$257$	−7.62772	−0.475804	−0.237902	−	0.971289i	$-0.576460\pi$
$257$	−7.62772	−0.475804	−0.237902	+	0.971289i	$0.576460\pi$
$258$	0	0
$259$	−25.4891	−1.58382
$260$	2.00000	0.124035
$261$	0	0
$262$	18.2337	1.12648
$263$	−26.9783	−1.66355	−0.831775	−	0.555113i	$-0.812675\pi$
$263$	−26.9783	−1.66355	−0.831775	+	0.555113i	$0.812675\pi$
$264$	0	0
$265$	−4.37228	−0.268587
$266$	−15.1168	−0.926873
$267$	0	0
$268$	−0.744563	−0.0454814
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−31.1168	−1.89021	−0.945107	−	0.326762i	$-0.894043\pi$
$271$	−31.1168	−1.89021	−0.945107	+	0.326762i	$0.894043\pi$
$272$	−6.74456	−0.408949
$273$	0	0
$274$	19.4891	1.17738
$275$	−6.37228	−0.384263
$276$	0	0
$277$	26.7446	1.60693	0.803463	−	0.595355i	$-0.202989\pi$
$277$	26.7446	1.60693	0.803463	+	0.595355i	$0.202989\pi$
$278$	−0.744563	−0.0446559
$279$	0	0
$280$	−2.37228	−0.141771
$281$	−5.25544	−0.313513	−0.156757	−	0.987637i	$-0.550104\pi$
$281$	−5.25544	−0.313513	−0.156757	+	0.987637i	$0.550104\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	2.37228	0.140769
$285$	0	0
$286$	−12.7446	−0.753602
$287$	−25.4891	−1.50458
$288$	0	0
$289$	28.4891	1.67583
$290$	−2.74456	−0.161166
$291$	0	0
$292$	9.11684	0.533523
$293$	−15.4891	−0.904884	−0.452442	−	0.891794i	$-0.649447\pi$
$293$	−15.4891	−0.904884	−0.452442	+	0.891794i	$0.649447\pi$
$294$	0	0
$295$	−8.74456	−0.509128
$296$	−10.7446	−0.624515
$297$	0	0
$298$	5.11684	0.296411
$299$	−4.74456	−0.274385
$300$	0	0
$301$	−15.1168	−0.871320
$302$	8.00000	0.460348
$303$	0	0
$304$	−6.37228	−0.365475
$305$	11.4891	0.657865
$306$	0	0
$307$	−14.9783	−0.854854	−0.427427	−	0.904050i	$-0.640580\pi$
$307$	−14.9783	−0.854854	−0.427427	+	0.904050i	$0.640580\pi$
$308$	15.1168	0.861362
$309$	0	0
$310$	−1.00000	−0.0567962
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−3.62772	−0.204724
$315$	0	0
$316$	−10.3723	−0.583486
$317$	13.2554	0.744500	0.372250	−	0.928133i	$-0.378586\pi$
$317$	13.2554	0.744500	0.372250	+	0.928133i	$0.378586\pi$
$318$	0	0
$319$	17.4891	0.979203
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	5.62772	0.313621
$323$	42.9783	2.39137
$324$	0	0
$325$	−2.00000	−0.110940
$326$	10.2337	0.566792
$327$	0	0
$328$	−10.7446	−0.593269
$329$	11.2554	0.620532
$330$	0	0
$331$	21.4891	1.18115	0.590575	−	0.806983i	$-0.298901\pi$
$331$	21.4891	1.18115	0.590575	+	0.806983i	$0.298901\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−18.3723	−1.00529
$335$	0.744563	0.0406798
$336$	0	0
$337$	−16.9783	−0.924864	−0.462432	−	0.886655i	$-0.653023\pi$
$337$	−16.9783	−0.924864	−0.462432	+	0.886655i	$0.653023\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	6.74456	0.365775
$341$	6.37228	0.345078
$342$	0	0
$343$	19.8614	1.07242
$344$	−6.37228	−0.343570
$345$	0	0
$346$	−18.0000	−0.967686
$347$	32.4674	1.74294	0.871470	−	0.490449i	$-0.163167\pi$
$347$	32.4674	1.74294	0.871470	+	0.490449i	$0.163167\pi$
$348$	0	0
$349$	18.7446	1.00337	0.501687	−	0.865049i	$-0.332713\pi$
$349$	18.7446	1.00337	0.501687	+	0.865049i	$0.332713\pi$
$350$	2.37228	0.126804
$351$	0	0
$352$	6.37228	0.339644
$353$	7.48913	0.398606	0.199303	−	0.979938i	$-0.436132\pi$
$353$	7.48913	0.398606	0.199303	+	0.979938i	$0.436132\pi$
$354$	0	0
$355$	−2.37228	−0.125908
$356$	−4.37228	−0.231730
$357$	0	0
$358$	−12.0000	−0.634220
$359$	2.37228	0.125204	0.0626021	−	0.998039i	$-0.480060\pi$
$359$	2.37228	0.125204	0.0626021	+	0.998039i	$0.480060\pi$
$360$	0	0
$361$	21.6060	1.13716
$362$	13.8614	0.728539
$363$	0	0
$364$	4.74456	0.248683
$365$	−9.11684	−0.477197
$366$	0	0
$367$	14.2337	0.742992	0.371496	−	0.928434i	$-0.378845\pi$
$367$	14.2337	0.742992	0.371496	+	0.928434i	$0.378845\pi$
$368$	2.37228	0.123664
$369$	0	0
$370$	10.7446	0.558583
$371$	−10.3723	−0.538502
$372$	0	0
$373$	−13.8614	−0.717716	−0.358858	−	0.933392i	$-0.616834\pi$
$373$	−13.8614	−0.717716	−0.358858	+	0.933392i	$0.616834\pi$
$374$	−42.9783	−2.22235
$375$	0	0
$376$	4.74456	0.244682
$377$	5.48913	0.282704
$378$	0	0
$379$	−0.138593	−0.00711906	−0.00355953	−	0.999994i	$-0.501133\pi$
$379$	−0.138593	−0.00711906	−0.00355953	+	0.999994i	$0.501133\pi$
$380$	6.37228	0.326891
$381$	0	0
$382$	16.0000	0.818631
$383$	9.48913	0.484872	0.242436	−	0.970167i	$-0.422054\pi$
$383$	9.48913	0.484872	0.242436	+	0.970167i	$0.422054\pi$
$384$	0	0
$385$	−15.1168	−0.770426
$386$	7.48913	0.381186
$387$	0	0
$388$	2.00000	0.101535
$389$	28.9783	1.46926	0.734628	−	0.678470i	$-0.237357\pi$
$389$	28.9783	1.46926	0.734628	+	0.678470i	$0.237357\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	1.37228	0.0693107
$393$	0	0
$394$	−3.48913	−0.175780
$395$	10.3723	0.521886
$396$	0	0
$397$	38.6060	1.93758	0.968789	−	0.247887i	$-0.0797361\pi$
$397$	38.6060	1.93758	0.968789	+	0.247887i	$0.0797361\pi$
$398$	18.3723	0.920919
$399$	0	0
$400$	1.00000	0.0500000
$401$	−28.3723	−1.41684	−0.708422	−	0.705789i	$-0.750593\pi$
$401$	−28.3723	−1.41684	−0.708422	+	0.705789i	$0.750593\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	9.11684	0.453580
$405$	0	0
$406$	−6.51087	−0.323129
$407$	−68.4674	−3.39380
$408$	0	0
$409$	24.2337	1.19828	0.599139	−	0.800645i	$-0.295510\pi$
$409$	24.2337	1.19828	0.599139	+	0.800645i	$0.295510\pi$
$410$	10.7446	0.530636
$411$	0	0
$412$	−8.00000	−0.394132
$413$	−20.7446	−1.02077
$414$	0	0
$415$	−12.0000	−0.589057
$416$	2.00000	0.0980581
$417$	0	0
$418$	−40.6060	−1.98610
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−3.48913	−0.170050	−0.0850248	−	0.996379i	$-0.527097\pi$
$421$	−3.48913	−0.170050	−0.0850248	+	0.996379i	$0.527097\pi$
$422$	6.37228	0.310198
$423$	0	0
$424$	−4.37228	−0.212337
$425$	−6.74456	−0.327159
$426$	0	0
$427$	27.2554	1.31898
$428$	−6.37228	−0.308016
$429$	0	0
$430$	6.37228	0.307299
$431$	−18.9783	−0.914150	−0.457075	−	0.889428i	$-0.651103\pi$
$431$	−18.9783	−0.914150	−0.457075	+	0.889428i	$0.651103\pi$
$432$	0	0
$433$	18.8832	0.907467	0.453733	−	0.891138i	$-0.350092\pi$
$433$	18.8832	0.907467	0.453733	+	0.891138i	$0.350092\pi$
$434$	−2.37228	−0.113873
$435$	0	0
$436$	−6.74456	−0.323006
$437$	−15.1168	−0.723137
$438$	0	0
$439$	3.25544	0.155374	0.0776868	−	0.996978i	$-0.475247\pi$
$439$	3.25544	0.155374	0.0776868	+	0.996978i	$0.475247\pi$
$440$	−6.37228	−0.303787
$441$	0	0
$442$	−13.4891	−0.641612
$443$	17.3505	0.824349	0.412174	−	0.911105i	$-0.364769\pi$
$443$	17.3505	0.824349	0.412174	+	0.911105i	$0.364769\pi$
$444$	0	0
$445$	4.37228	0.207266
$446$	−20.7446	−0.982284
$447$	0	0
$448$	−2.37228	−0.112080
$449$	−36.9783	−1.74511	−0.872556	−	0.488515i	$-0.837539\pi$
$449$	−36.9783	−1.74511	−0.872556	+	0.488515i	$0.837539\pi$
$450$	0	0
$451$	−68.4674	−3.22400
$452$	8.37228	0.393799
$453$	0	0
$454$	−11.1168	−0.521739
$455$	−4.74456	−0.222429
$456$	0	0
$457$	−34.4674	−1.61232	−0.806158	−	0.591700i	$-0.798457\pi$
$457$	−34.4674	−1.61232	−0.806158	+	0.591700i	$0.798457\pi$
$458$	−21.1168	−0.986725
$459$	0	0
$460$	−2.37228	−0.110608
$461$	−37.7228	−1.75693	−0.878463	−	0.477810i	$-0.841431\pi$
$461$	−37.7228	−1.75693	−0.878463	+	0.477810i	$0.841431\pi$
$462$	0	0
$463$	25.4891	1.18458	0.592290	−	0.805725i	$-0.298224\pi$
$463$	25.4891	1.18458	0.592290	+	0.805725i	$0.298224\pi$
$464$	−2.74456	−0.127413
$465$	0	0
$466$	13.8614	0.642117
$467$	−29.4891	−1.36459	−0.682297	−	0.731075i	$-0.739019\pi$
$467$	−29.4891	−1.36459	−0.682297	+	0.731075i	$0.739019\pi$
$468$	0	0
$469$	1.76631	0.0815607
$470$	−4.74456	−0.218850
$471$	0	0
$472$	−8.74456	−0.402501
$473$	−40.6060	−1.86706
$474$	0	0
$475$	−6.37228	−0.292380
$476$	16.0000	0.733359
$477$	0	0
$478$	−6.51087	−0.297800
$479$	19.8614	0.907491	0.453745	−	0.891131i	$-0.350088\pi$
$479$	19.8614	0.907491	0.453745	+	0.891131i	$0.350088\pi$
$480$	0	0
$481$	−21.4891	−0.979820
$482$	−27.4891	−1.25210
$483$	0	0
$484$	29.6060	1.34573
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−14.5109	−0.657550	−0.328775	−	0.944408i	$-0.606636\pi$
$487$	−14.5109	−0.657550	−0.328775	+	0.944408i	$0.606636\pi$
$488$	11.4891	0.520088
$489$	0	0
$490$	−1.37228	−0.0619934
$491$	12.6060	0.568899	0.284450	−	0.958691i	$-0.408189\pi$
$491$	12.6060	0.568899	0.284450	+	0.958691i	$0.408189\pi$
$492$	0	0
$493$	18.5109	0.833688
$494$	−12.7446	−0.573405
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	−5.62772	−0.252438
$498$	0	0
$499$	10.5109	0.470531	0.235266	−	0.971931i	$-0.424404\pi$
$499$	10.5109	0.470531	0.235266	+	0.971931i	$0.424404\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−4.00000	−0.178529
$503$	17.4891	0.779802	0.389901	−	0.920857i	$-0.372509\pi$
$503$	17.4891	0.779802	0.389901	+	0.920857i	$0.372509\pi$
$504$	0	0
$505$	−9.11684	−0.405694
$506$	15.1168	0.672026
$507$	0	0
$508$	−9.48913	−0.421012
$509$	38.7446	1.71732	0.858661	−	0.512543i	$-0.171297\pi$
$509$	38.7446	1.71732	0.858661	+	0.512543i	$0.171297\pi$
$510$	0	0
$511$	−21.6277	−0.956754
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	7.62772	0.336444
$515$	8.00000	0.352522
$516$	0	0
$517$	30.2337	1.32968
$518$	25.4891	1.11993
$519$	0	0
$520$	−2.00000	−0.0877058
$521$	8.97825	0.393344	0.196672	−	0.980469i	$-0.436987\pi$
$521$	8.97825	0.393344	0.196672	+	0.980469i	$0.436987\pi$
$522$	0	0
$523$	15.8614	0.693571	0.346785	−	0.937944i	$-0.387273\pi$
$523$	15.8614	0.693571	0.346785	+	0.937944i	$0.387273\pi$
$524$	−18.2337	−0.796542
$525$	0	0
$526$	26.9783	1.17631
$527$	6.74456	0.293798
$528$	0	0
$529$	−17.3723	−0.755317
$530$	4.37228	0.189920
$531$	0	0
$532$	15.1168	0.655398
$533$	−21.4891	−0.930797
$534$	0	0
$535$	6.37228	0.275498
$536$	0.744563	0.0321602
$537$	0	0
$538$	−2.00000	−0.0862261
$539$	8.74456	0.376655
$540$	0	0
$541$	7.48913	0.321983	0.160991	−	0.986956i	$-0.448531\pi$
$541$	7.48913	0.321983	0.160991	+	0.986956i	$0.448531\pi$
$542$	31.1168	1.33658
$543$	0	0
$544$	6.74456	0.289171
$545$	6.74456	0.288905
$546$	0	0
$547$	8.74456	0.373890	0.186945	−	0.982370i	$-0.440141\pi$
$547$	8.74456	0.373890	0.186945	+	0.982370i	$0.440141\pi$
$548$	−19.4891	−0.832534
$549$	0	0
$550$	6.37228	0.271715
$551$	17.4891	0.745062
$552$	0	0
$553$	24.6060	1.04635
$554$	−26.7446	−1.13627
$555$	0	0
$556$	0.744563	0.0315765
$557$	−29.1168	−1.23372	−0.616860	−	0.787073i	$-0.711596\pi$
$557$	−29.1168	−1.23372	−0.616860	+	0.787073i	$0.711596\pi$
$558$	0	0
$559$	−12.7446	−0.539038
$560$	2.37228	0.100247
$561$	0	0
$562$	5.25544	0.221687
$563$	−38.9783	−1.64274	−0.821369	−	0.570398i	$-0.806789\pi$
$563$	−38.9783	−1.64274	−0.821369	+	0.570398i	$0.806789\pi$
$564$	0	0
$565$	−8.37228	−0.352225
$566$	−12.0000	−0.504398
$567$	0	0
$568$	−2.37228	−0.0995387
$569$	−4.37228	−0.183296	−0.0916478	−	0.995791i	$-0.529213\pi$
$569$	−4.37228	−0.183296	−0.0916478	+	0.995791i	$0.529213\pi$
$570$	0	0
$571$	−15.2554	−0.638420	−0.319210	−	0.947684i	$-0.603418\pi$
$571$	−15.2554	−0.638420	−0.319210	+	0.947684i	$0.603418\pi$
$572$	12.7446	0.532877
$573$	0	0
$574$	25.4891	1.06390
$575$	2.37228	0.0989310
$576$	0	0
$577$	32.2337	1.34191	0.670953	−	0.741500i	$-0.265885\pi$
$577$	32.2337	1.34191	0.670953	+	0.741500i	$0.265885\pi$
$578$	−28.4891	−1.18499
$579$	0	0
$580$	2.74456	0.113962
$581$	−28.4674	−1.18103
$582$	0	0
$583$	−27.8614	−1.15390
$584$	−9.11684	−0.377258
$585$	0	0
$586$	15.4891	0.639850
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	6.37228	0.262565
$590$	8.74456	0.360008
$591$	0	0
$592$	10.7446	0.441599
$593$	0.978251	0.0401719	0.0200860	−	0.999798i	$-0.493606\pi$
$593$	0.978251	0.0401719	0.0200860	+	0.999798i	$0.493606\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	−5.11684	−0.209594
$597$	0	0
$598$	4.74456	0.194020
$599$	7.11684	0.290786	0.145393	−	0.989374i	$-0.453555\pi$
$599$	7.11684	0.290786	0.145393	+	0.989374i	$0.453555\pi$
$600$	0	0
$601$	35.4891	1.44763	0.723816	−	0.689993i	$-0.242387\pi$
$601$	35.4891	1.44763	0.723816	+	0.689993i	$0.242387\pi$
$602$	15.1168	0.616117
$603$	0	0
$604$	−8.00000	−0.325515
$605$	−29.6060	−1.20365
$606$	0	0
$607$	−10.3723	−0.420998	−0.210499	−	0.977594i	$-0.567509\pi$
$607$	−10.3723	−0.420998	−0.210499	+	0.977594i	$0.567509\pi$
$608$	6.37228	0.258430
$609$	0	0
$610$	−11.4891	−0.465181
$611$	9.48913	0.383889
$612$	0	0
$613$	12.5109	0.505309	0.252655	−	0.967557i	$-0.418696\pi$
$613$	12.5109	0.505309	0.252655	+	0.967557i	$0.418696\pi$
$614$	14.9783	0.604473
$615$	0	0
$616$	−15.1168	−0.609075
$617$	−25.1168	−1.01117	−0.505583	−	0.862778i	$-0.668723\pi$
$617$	−25.1168	−1.01117	−0.505583	+	0.862778i	$0.668723\pi$
$618$	0	0
$619$	−18.2337	−0.732874	−0.366437	−	0.930443i	$-0.619422\pi$
$619$	−18.2337	−0.732874	−0.366437	+	0.930443i	$0.619422\pi$
$620$	1.00000	0.0401610
$621$	0	0
$622$	8.00000	0.320771
$623$	10.3723	0.415557
$624$	0	0
$625$	1.00000	0.0400000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	3.62772	0.144762
$629$	−72.4674	−2.88946
$630$	0	0
$631$	5.35053	0.213001	0.106501	−	0.994313i	$-0.466035\pi$
$631$	5.35053	0.213001	0.106501	+	0.994313i	$0.466035\pi$
$632$	10.3723	0.412587
$633$	0	0
$634$	−13.2554	−0.526441
$635$	9.48913	0.376564
$636$	0	0
$637$	2.74456	0.108744
$638$	−17.4891	−0.692401
$639$	0	0
$640$	1.00000	0.0395285
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	38.0951	1.50232	0.751162	−	0.660118i	$-0.229494\pi$
$643$	38.0951	1.50232	0.751162	+	0.660118i	$0.229494\pi$
$644$	−5.62772	−0.221763
$645$	0	0
$646$	−42.9783	−1.69096
$647$	35.5842	1.39896	0.699480	−	0.714652i	$-0.253415\pi$
$647$	35.5842	1.39896	0.699480	+	0.714652i	$0.253415\pi$
$648$	0	0
$649$	−55.7228	−2.18731
$650$	2.00000	0.0784465
$651$	0	0
$652$	−10.2337	−0.400782
$653$	4.97825	0.194814	0.0974070	−	0.995245i	$-0.468945\pi$
$653$	4.97825	0.194814	0.0974070	+	0.995245i	$0.468945\pi$
$654$	0	0
$655$	18.2337	0.712449
$656$	10.7446	0.419505
$657$	0	0
$658$	−11.2554	−0.438783
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	40.9783	1.59387	0.796935	−	0.604066i	$-0.206454\pi$
$661$	40.9783	1.59387	0.796935	+	0.604066i	$0.206454\pi$
$662$	−21.4891	−0.835199
$663$	0	0
$664$	−12.0000	−0.465690
$665$	−15.1168	−0.586206
$666$	0	0
$667$	−6.51087	−0.252102
$668$	18.3723	0.710845
$669$	0	0
$670$	−0.744563	−0.0287650
$671$	73.2119	2.82632
$672$	0	0
$673$	−23.4891	−0.905439	−0.452720	−	0.891653i	$-0.649546\pi$
$673$	−23.4891	−0.905439	−0.452720	+	0.891653i	$0.649546\pi$
$674$	16.9783	0.653978
$675$	0	0
$676$	−9.00000	−0.346154
$677$	1.39403	0.0535770	0.0267885	−	0.999641i	$-0.491472\pi$
$677$	1.39403	0.0535770	0.0267885	+	0.999641i	$0.491472\pi$
$678$	0	0
$679$	−4.74456	−0.182080
$680$	−6.74456	−0.258642
$681$	0	0
$682$	−6.37228	−0.244007
$683$	−15.8614	−0.606920	−0.303460	−	0.952844i	$-0.598142\pi$
$683$	−15.8614	−0.606920	−0.303460	+	0.952844i	$0.598142\pi$
$684$	0	0
$685$	19.4891	0.744641
$686$	−19.8614	−0.758312
$687$	0	0
$688$	6.37228	0.242941
$689$	−8.74456	−0.333141
$690$	0	0
$691$	−47.8614	−1.82073	−0.910367	−	0.413802i	$-0.864201\pi$
$691$	−47.8614	−1.82073	−0.910367	+	0.413802i	$0.864201\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−32.4674	−1.23244
$695$	−0.744563	−0.0282429
$696$	0	0
$697$	−72.4674	−2.74490
$698$	−18.7446	−0.709492
$699$	0	0
$700$	−2.37228	−0.0896638
$701$	−5.39403	−0.203730	−0.101865	−	0.994798i	$-0.532481\pi$
$701$	−5.39403	−0.203730	−0.101865	+	0.994798i	$0.532481\pi$
$702$	0	0
$703$	−68.4674	−2.58230
$704$	−6.37228	−0.240164
$705$	0	0
$706$	−7.48913	−0.281857
$707$	−21.6277	−0.813394
$708$	0	0
$709$	22.8832	0.859395	0.429697	−	0.902973i	$-0.358620\pi$
$709$	22.8832	0.859395	0.429697	+	0.902973i	$0.358620\pi$
$710$	2.37228	0.0890301
$711$	0	0
$712$	4.37228	0.163858
$713$	−2.37228	−0.0888426
$714$	0	0
$715$	−12.7446	−0.476620
$716$	12.0000	0.448461
$717$	0	0
$718$	−2.37228	−0.0885328
$719$	−27.2554	−1.01646	−0.508228	−	0.861222i	$-0.669699\pi$
$719$	−27.2554	−1.01646	−0.508228	+	0.861222i	$0.669699\pi$
$720$	0	0
$721$	18.9783	0.706787
$722$	−21.6060	−0.804091
$723$	0	0
$724$	−13.8614	−0.515155
$725$	−2.74456	−0.101930
$726$	0	0
$727$	13.6277	0.505424	0.252712	−	0.967542i	$-0.418677\pi$
$727$	13.6277	0.505424	0.252712	+	0.967542i	$0.418677\pi$
$728$	−4.74456	−0.175845
$729$	0	0
$730$	9.11684	0.337430
$731$	−42.9783	−1.58961
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−14.2337	−0.525375
$735$	0	0
$736$	−2.37228	−0.0874434
$737$	4.74456	0.174768
$738$	0	0
$739$	22.9783	0.845269	0.422634	−	0.906300i	$-0.361105\pi$
$739$	22.9783	0.845269	0.422634	+	0.906300i	$0.361105\pi$
$740$	−10.7446	−0.394978
$741$	0	0
$742$	10.3723	0.380778
$743$	29.6277	1.08694	0.543468	−	0.839430i	$-0.317111\pi$
$743$	29.6277	1.08694	0.543468	+	0.839430i	$0.317111\pi$
$744$	0	0
$745$	5.11684	0.187467
$746$	13.8614	0.507502
$747$	0	0
$748$	42.9783	1.57144
$749$	15.1168	0.552357
$750$	0	0
$751$	−27.2554	−0.994565	−0.497283	−	0.867589i	$-0.665669\pi$
$751$	−27.2554	−0.994565	−0.497283	+	0.867589i	$0.665669\pi$
$752$	−4.74456	−0.173016
$753$	0	0
$754$	−5.48913	−0.199902
$755$	8.00000	0.291150
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	0.138593	0.00503394
$759$	0	0
$760$	−6.37228	−0.231147
$761$	18.1386	0.657523	0.328762	−	0.944413i	$-0.393369\pi$
$761$	18.1386	0.657523	0.328762	+	0.944413i	$0.393369\pi$
$762$	0	0
$763$	16.0000	0.579239
$764$	−16.0000	−0.578860
$765$	0	0
$766$	−9.48913	−0.342856
$767$	−17.4891	−0.631496
$768$	0	0
$769$	−3.62772	−0.130819	−0.0654094	−	0.997859i	$-0.520835\pi$
$769$	−3.62772	−0.130819	−0.0654094	+	0.997859i	$0.520835\pi$
$770$	15.1168	0.544773
$771$	0	0
$772$	−7.48913	−0.269540
$773$	18.6060	0.669210	0.334605	−	0.942358i	$-0.391397\pi$
$773$	18.6060	0.669210	0.334605	+	0.942358i	$0.391397\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	−28.9783	−1.03892
$779$	−68.4674	−2.45310
$780$	0	0
$781$	−15.1168	−0.540923
$782$	16.0000	0.572159
$783$	0	0
$784$	−1.37228	−0.0490100
$785$	−3.62772	−0.129479
$786$	0	0
$787$	−27.1168	−0.966611	−0.483306	−	0.875452i	$-0.660564\pi$
$787$	−27.1168	−0.966611	−0.483306	+	0.875452i	$0.660564\pi$
$788$	3.48913	0.124295
$789$	0	0
$790$	−10.3723	−0.369029
$791$	−19.8614	−0.706190
$792$	0	0
$793$	22.9783	0.815982
$794$	−38.6060	−1.37007
$795$	0	0
$796$	−18.3723	−0.651188
$797$	43.4891	1.54046	0.770232	−	0.637764i	$-0.220140\pi$
$797$	43.4891	1.54046	0.770232	+	0.637764i	$0.220140\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	28.3723	1.00186
$803$	−58.0951	−2.05013
$804$	0	0
$805$	5.62772	0.198351
$806$	−2.00000	−0.0704470
$807$	0	0
$808$	−9.11684	−0.320729
$809$	30.6060	1.07605	0.538024	−	0.842929i	$-0.319171\pi$
$809$	30.6060	1.07605	0.538024	+	0.842929i	$0.319171\pi$
$810$	0	0
$811$	−46.3723	−1.62835	−0.814176	−	0.580619i	$-0.802811\pi$
$811$	−46.3723	−1.62835	−0.814176	+	0.580619i	$0.802811\pi$
$812$	6.51087	0.228487
$813$	0	0
$814$	68.4674	2.39978
$815$	10.2337	0.358470
$816$	0	0
$817$	−40.6060	−1.42062
$818$	−24.2337	−0.847311
$819$	0	0
$820$	−10.7446	−0.375216
$821$	−26.7446	−0.933392	−0.466696	−	0.884418i	$-0.654556\pi$
$821$	−26.7446	−0.933392	−0.466696	+	0.884418i	$0.654556\pi$
$822$	0	0
$823$	−28.7446	−1.00197	−0.500986	−	0.865455i	$-0.667029\pi$
$823$	−28.7446	−1.00197	−0.500986	+	0.865455i	$0.667029\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	20.7446	0.721796
$827$	−5.48913	−0.190876	−0.0954378	−	0.995435i	$-0.530425\pi$
$827$	−5.48913	−0.190876	−0.0954378	+	0.995435i	$0.530425\pi$
$828$	0	0
$829$	32.0951	1.11471	0.557354	−	0.830275i	$-0.311816\pi$
$829$	32.0951	1.11471	0.557354	+	0.830275i	$0.311816\pi$
$830$	12.0000	0.416526
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	9.25544	0.320682
$834$	0	0
$835$	−18.3723	−0.635799
$836$	40.6060	1.40439
$837$	0	0
$838$	−12.0000	−0.414533
$839$	11.8614	0.409501	0.204751	−	0.978814i	$-0.434362\pi$
$839$	11.8614	0.409501	0.204751	+	0.978814i	$0.434362\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	3.48913	0.120243
$843$	0	0
$844$	−6.37228	−0.219343
$845$	9.00000	0.309609
$846$	0	0
$847$	−70.2337	−2.41326
$848$	4.37228	0.150145
$849$	0	0
$850$	6.74456	0.231337
$851$	25.4891	0.873756
$852$	0	0
$853$	24.0951	0.825000	0.412500	−	0.910958i	$-0.364656\pi$
$853$	24.0951	0.825000	0.412500	+	0.910958i	$0.364656\pi$
$854$	−27.2554	−0.932662
$855$	0	0
$856$	6.37228	0.217800
$857$	15.4891	0.529098	0.264549	−	0.964372i	$-0.414777\pi$
$857$	15.4891	0.529098	0.264549	+	0.964372i	$0.414777\pi$
$858$	0	0
$859$	5.48913	0.187287	0.0936433	−	0.995606i	$-0.470149\pi$
$859$	5.48913	0.187287	0.0936433	+	0.995606i	$0.470149\pi$
$860$	−6.37228	−0.217293
$861$	0	0
$862$	18.9783	0.646402
$863$	−45.3505	−1.54375	−0.771875	−	0.635774i	$-0.780681\pi$
$863$	−45.3505	−1.54375	−0.771875	+	0.635774i	$0.780681\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	−18.8832	−0.641676
$867$	0	0
$868$	2.37228	0.0805205
$869$	66.0951	2.24212
$870$	0	0
$871$	1.48913	0.0504571
$872$	6.74456	0.228400
$873$	0	0
$874$	15.1168	0.511335
$875$	2.37228	0.0801977
$876$	0	0
$877$	0.978251	0.0330332	0.0165166	−	0.999864i	$-0.494742\pi$
$877$	0.978251	0.0330332	0.0165166	+	0.999864i	$0.494742\pi$
$878$	−3.25544	−0.109866
$879$	0	0
$880$	6.37228	0.214810
$881$	48.9783	1.65012	0.825060	−	0.565046i	$-0.191141\pi$
$881$	48.9783	1.65012	0.825060	+	0.565046i	$0.191141\pi$
$882$	0	0
$883$	16.1386	0.543107	0.271553	−	0.962423i	$-0.412463\pi$
$883$	16.1386	0.543107	0.271553	+	0.962423i	$0.412463\pi$
$884$	13.4891	0.453688
$885$	0	0
$886$	−17.3505	−0.582903
$887$	19.2554	0.646534	0.323267	−	0.946308i	$-0.395219\pi$
$887$	19.2554	0.646534	0.323267	+	0.946308i	$0.395219\pi$
$888$	0	0
$889$	22.5109	0.754991
$890$	−4.37228	−0.146559
$891$	0	0
$892$	20.7446	0.694579
$893$	30.2337	1.01173
$894$	0	0
$895$	−12.0000	−0.401116
$896$	2.37228	0.0792524
$897$	0	0
$898$	36.9783	1.23398
$899$	2.74456	0.0915363
$900$	0	0
$901$	−29.4891	−0.982425
$902$	68.4674	2.27971
$903$	0	0
$904$	−8.37228	−0.278458
$905$	13.8614	0.460769
$906$	0	0
$907$	5.48913	0.182263	0.0911317	−	0.995839i	$-0.470952\pi$
$907$	5.48913	0.182263	0.0911317	+	0.995839i	$0.470952\pi$
$908$	11.1168	0.368925
$909$	0	0
$910$	4.74456	0.157281
$911$	25.4891	0.844492	0.422246	−	0.906481i	$-0.361242\pi$
$911$	25.4891	0.844492	0.422246	+	0.906481i	$0.361242\pi$
$912$	0	0
$913$	−76.4674	−2.53070
$914$	34.4674	1.14008
$915$	0	0
$916$	21.1168	0.697720
$917$	43.2554	1.42842
$918$	0	0
$919$	22.2337	0.733422	0.366711	−	0.930335i	$-0.380484\pi$
$919$	22.2337	0.733422	0.366711	+	0.930335i	$0.380484\pi$
$920$	2.37228	0.0782118
$921$	0	0
$922$	37.7228	1.24233
$923$	−4.74456	−0.156169
$924$	0	0
$925$	10.7446	0.353279
$926$	−25.4891	−0.837625
$927$	0	0
$928$	2.74456	0.0900947
$929$	−1.11684	−0.0366425	−0.0183212	−	0.999832i	$-0.505832\pi$
$929$	−1.11684	−0.0366425	−0.0183212	+	0.999832i	$0.505832\pi$
$930$	0	0
$931$	8.74456	0.286591
$932$	−13.8614	−0.454045
$933$	0	0
$934$	29.4891	0.964914
$935$	−42.9783	−1.40554
$936$	0	0
$937$	−42.7446	−1.39640	−0.698202	−	0.715901i	$-0.746016\pi$
$937$	−42.7446	−1.39640	−0.698202	+	0.715901i	$0.746016\pi$
$938$	−1.76631	−0.0576721
$939$	0	0
$940$	4.74456	0.154751
$941$	−37.7228	−1.22973	−0.614864	−	0.788633i	$-0.710789\pi$
$941$	−37.7228	−1.22973	−0.614864	+	0.788633i	$0.710789\pi$
$942$	0	0
$943$	25.4891	0.830040
$944$	8.74456	0.284611
$945$	0	0
$946$	40.6060	1.32021
$947$	−8.74456	−0.284160	−0.142080	−	0.989855i	$-0.545379\pi$
$947$	−8.74456	−0.284160	−0.142080	+	0.989855i	$0.545379\pi$
$948$	0	0
$949$	−18.2337	−0.591891
$950$	6.37228	0.206744
$951$	0	0
$952$	−16.0000	−0.518563
$953$	45.7228	1.48111	0.740554	−	0.671997i	$-0.234563\pi$
$953$	45.7228	1.48111	0.740554	+	0.671997i	$0.234563\pi$
$954$	0	0
$955$	16.0000	0.517748
$956$	6.51087	0.210577
$957$	0	0
$958$	−19.8614	−0.641693
$959$	46.2337	1.49296
$960$	0	0
$961$	1.00000	0.0322581
$962$	21.4891	0.692837
$963$	0	0
$964$	27.4891	0.885365
$965$	7.48913	0.241083
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−29.6060	−0.951572
$969$	0	0
$970$	2.00000	0.0642161
$971$	−54.7011	−1.75544	−0.877720	−	0.479173i	$-0.840937\pi$
$971$	−54.7011	−1.75544	−0.877720	+	0.479173i	$0.840937\pi$
$972$	0	0
$973$	−1.76631	−0.0566254
$974$	14.5109	0.464958
$975$	0	0
$976$	−11.4891	−0.367758
$977$	46.0000	1.47167	0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000	1.47167	0.735835	+	0.677161i	$0.236790\pi$
$978$	0	0
$979$	27.8614	0.890454
$980$	1.37228	0.0438359
$981$	0	0
$982$	−12.6060	−0.402273
$983$	−42.9783	−1.37079	−0.685397	−	0.728170i	$-0.740371\pi$
$983$	−42.9783	−1.37079	−0.685397	+	0.728170i	$0.740371\pi$
$984$	0	0
$985$	−3.48913	−0.111173
$986$	−18.5109	−0.589506
$987$	0	0
$988$	12.7446	0.405459
$989$	15.1168	0.480688
$990$	0	0
$991$	−7.39403	−0.234879	−0.117440	−	0.993080i	$-0.537469\pi$
$991$	−7.39403	−0.234879	−0.117440	+	0.993080i	$0.537469\pi$
$992$	1.00000	0.0317500
$993$	0	0
$994$	5.62772	0.178500
$995$	18.3723	0.582440
$996$	0	0
$997$	6.00000	0.190022	0.0950110	−	0.995476i	$-0.469711\pi$
$997$	6.00000	0.190022	0.0950110	+	0.995476i	$0.469711\pi$
$998$	−10.5109	−0.332716
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.r.1.1	✓	2	3.2	odd	2
2790.2.a.bd.1.1		2	1.1	even	1	trivial
4650.2.a.by.1.2		2	15.14	odd	2
4650.2.d.bh.3349.2		4	15.8	even	4
4650.2.d.bh.3349.3		4	15.2	even	4
7440.2.a.bg.1.2		2	12.11	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.37228 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.37228 q^{7} -1.00000 q^{8} +1.00000 q^{10} -6.37228 q^{11} -2.00000 q^{13} +2.37228 q^{14} +1.00000 q^{16} -6.74456 q^{17} -6.37228 q^{19} -1.00000 q^{20} +6.37228 q^{22} +2.37228 q^{23} +1.00000 q^{25} +2.00000 q^{26} -2.37228 q^{28} -2.74456 q^{29} -1.00000 q^{31} -1.00000 q^{32} +6.74456 q^{34} +2.37228 q^{35} +10.7446 q^{37} +6.37228 q^{38} +1.00000 q^{40} +10.7446 q^{41} +6.37228 q^{43} -6.37228 q^{44} -2.37228 q^{46} -4.74456 q^{47} -1.37228 q^{49} -1.00000 q^{50} -2.00000 q^{52} +4.37228 q^{53} +6.37228 q^{55} +2.37228 q^{56} +2.74456 q^{58} +8.74456 q^{59} -11.4891 q^{61} +1.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -0.744563 q^{67} -6.74456 q^{68} -2.37228 q^{70} +2.37228 q^{71} +9.11684 q^{73} -10.7446 q^{74} -6.37228 q^{76} +15.1168 q^{77} -10.3723 q^{79} -1.00000 q^{80} -10.7446 q^{82} +12.0000 q^{83} +6.74456 q^{85} -6.37228 q^{86} +6.37228 q^{88} -4.37228 q^{89} +4.74456 q^{91} +2.37228 q^{92} +4.74456 q^{94} +6.37228 q^{95} +2.00000 q^{97} +1.37228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - 7 q^{11} - 4 q^{13} - q^{14} + 2 q^{16} - 2 q^{17} - 7 q^{19} - 2 q^{20} + 7 q^{22} - q^{23} + 2 q^{25} + 4 q^{26} + q^{28} + 6 q^{29} - 2 q^{31} - 2 q^{32} + 2 q^{34} - q^{35} + 10 q^{37} + 7 q^{38} + 2 q^{40} + 10 q^{41} + 7 q^{43} - 7 q^{44} + q^{46} + 2 q^{47} + 3 q^{49} - 2 q^{50} - 4 q^{52} + 3 q^{53} + 7 q^{55} - q^{56} - 6 q^{58} + 6 q^{59} + 2 q^{62} + 2 q^{64} + 4 q^{65} + 10 q^{67} - 2 q^{68} + q^{70} - q^{71} + q^{73} - 10 q^{74} - 7 q^{76} + 13 q^{77} - 15 q^{79} - 2 q^{80} - 10 q^{82} + 24 q^{83} + 2 q^{85} - 7 q^{86} + 7 q^{88} - 3 q^{89} - 2 q^{91} - q^{92} - 2 q^{94} + 7 q^{95} + 4 q^{97} - 3 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + q^7 - 2 * q^8 + 2 * q^10 - 7 * q^11 - 4 * q^13 - q^14 + 2 * q^16 - 2 * q^17 - 7 * q^19 - 2 * q^20 + 7 * q^22 - q^23 + 2 * q^25 + 4 * q^26 + q^28 + 6 * q^29 - 2 * q^31 - 2 * q^32 + 2 * q^34 - q^35 + 10 * q^37 + 7 * q^38 + 2 * q^40 + 10 * q^41 + 7 * q^43 - 7 * q^44 + q^46 + 2 * q^47 + 3 * q^49 - 2 * q^50 - 4 * q^52 + 3 * q^53 + 7 * q^55 - q^56 - 6 * q^58 + 6 * q^59 + 2 * q^62 + 2 * q^64 + 4 * q^65 + 10 * q^67 - 2 * q^68 + q^70 - q^71 + q^73 - 10 * q^74 - 7 * q^76 + 13 * q^77 - 15 * q^79 - 2 * q^80 - 10 * q^82 + 24 * q^83 + 2 * q^85 - 7 * q^86 + 7 * q^88 - 3 * q^89 - 2 * q^91 - q^92 - 2 * q^94 + 7 * q^95 + 4 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more