LMFDB - Embedded newform 2583.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2583,2,Mod(1,2583)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2583, 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("2583.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2583 = 3^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2583.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:	$20.6253588421$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.633117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 4x^{2} + 6x - 3 $ x^5 - x^4 - 6x^3 + 4x^2 + 6*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 287)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.08727$ of defining polynomial
Character	$\chi$	$=$	2583.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.08727 q^{2} -0.817843 q^{4} -0.209668 q^{5} +1.00000 q^{7} +3.06376 q^{8} +O(q^{10})$	\(q-1.08727 q^{2} -0.817843 q^{4} -0.209668 q^{5} +1.00000 q^{7} +3.06376 q^{8} +0.227965 q^{10} -6.03819 q^{11} +3.67193 q^{13} -1.08727 q^{14} -1.69545 q^{16} +5.37138 q^{17} -3.54285 q^{19} +0.171475 q^{20} +6.56515 q^{22} +1.30362 q^{23} -4.95604 q^{25} -3.99238 q^{26} -0.817843 q^{28} +8.00307 q^{29} -0.384208 q^{31} -4.28411 q^{32} -5.84014 q^{34} -0.209668 q^{35} -3.68876 q^{37} +3.85204 q^{38} -0.642370 q^{40} +1.00000 q^{41} +0.824527 q^{43} +4.93829 q^{44} -1.41739 q^{46} -5.11625 q^{47} +1.00000 q^{49} +5.38855 q^{50} -3.00307 q^{52} -1.53217 q^{53} +1.26601 q^{55} +3.06376 q^{56} -8.70150 q^{58} -10.2669 q^{59} +9.36070 q^{61} +0.417738 q^{62} +8.04887 q^{64} -0.769885 q^{65} +11.3638 q^{67} -4.39294 q^{68} +0.227965 q^{70} +14.9494 q^{71} +7.77203 q^{73} +4.01068 q^{74} +2.89750 q^{76} -6.03819 q^{77} -6.04703 q^{79} +0.355480 q^{80} -1.08727 q^{82} -14.1871 q^{83} -1.12620 q^{85} -0.896484 q^{86} -18.4996 q^{88} -0.520905 q^{89} +3.67193 q^{91} -1.06616 q^{92} +5.56275 q^{94} +0.742821 q^{95} -3.65270 q^{97} -1.08727 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 3 q^{4} + 5 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 5 * q + q^2 + 3 * q^4 + 5 * q^5 + 5 * q^7 + 3 * q^8	$ 5 q + q^{2} + 3 q^{4} + 5 q^{5} + 5 q^{7} + 3 q^{8} - 2 q^{11} + 5 q^{13} + q^{14} - q^{16} - 13 q^{17} + 23 q^{20} + q^{22} - 2 q^{23} + 22 q^{25} + 3 q^{28} + 5 q^{29} + 17 q^{31} + 12 q^{32} - 8 q^{34} + 5 q^{35} - 7 q^{37} + 3 q^{38} + 7 q^{40} + 5 q^{41} + q^{43} + 47 q^{44} - 24 q^{46} - 9 q^{47} + 5 q^{49} - 2 q^{50} + 20 q^{52} - 5 q^{53} + 33 q^{55} + 3 q^{56} - 27 q^{58} - 7 q^{59} + 22 q^{61} + 28 q^{62} - 3 q^{64} + 31 q^{65} - 3 q^{67} - 17 q^{68} + 24 q^{71} + 40 q^{73} + 5 q^{74} - 19 q^{76} - 2 q^{77} - 42 q^{79} - 24 q^{80} + q^{82} + 12 q^{83} - 23 q^{85} - 16 q^{86} + 26 q^{88} - 8 q^{89} + 5 q^{91} - 12 q^{92} - 23 q^{94} + 17 q^{95} + 16 q^{97} + q^{98}+O(q^{100}) $ 5 * q + q^2 + 3 * q^4 + 5 * q^5 + 5 * q^7 + 3 * q^8 - 2 * q^11 + 5 * q^13 + q^14 - q^16 - 13 * q^17 + 23 * q^20 + q^22 - 2 * q^23 + 22 * q^25 + 3 * q^28 + 5 * q^29 + 17 * q^31 + 12 * q^32 - 8 * q^34 + 5 * q^35 - 7 * q^37 + 3 * q^38 + 7 * q^40 + 5 * q^41 + q^43 + 47 * q^44 - 24 * q^46 - 9 * q^47 + 5 * q^49 - 2 * q^50 + 20 * q^52 - 5 * q^53 + 33 * q^55 + 3 * q^56 - 27 * q^58 - 7 * q^59 + 22 * q^61 + 28 * q^62 - 3 * q^64 + 31 * q^65 - 3 * q^67 - 17 * q^68 + 24 * q^71 + 40 * q^73 + 5 * q^74 - 19 * q^76 - 2 * q^77 - 42 * q^79 - 24 * q^80 + q^82 + 12 * q^83 - 23 * q^85 - 16 * q^86 + 26 * q^88 - 8 * q^89 + 5 * q^91 - 12 * q^92 - 23 * q^94 + 17 * q^95 + 16 * q^97 + 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 $

$2$	−1.08727	−0.768816	−0.384408	−	0.923163i	$-0.625594\pi$
$2$	−1.08727	−0.768816	−0.384408	+	0.923163i	$0.625594\pi$
$3$	0	0
$3$	0	0
$4$	−0.817843	−0.408922
$5$	−0.209668	−0.0937662	−0.0468831	−	0.998900i	$-0.514929\pi$
$5$	−0.209668	−0.0937662	−0.0468831	+	0.998900i	$0.514929\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.06376	1.08320
$9$	0	0
$10$	0.227965	0.0720889
$11$	−6.03819	−1.82058	−0.910292	−	0.413967i	$-0.864143\pi$
$11$	−6.03819	−1.82058	−0.910292	+	0.413967i	$0.864143\pi$
$12$	0	0
$13$	3.67193	1.01841	0.509205	−	0.860645i	$-0.329939\pi$
$13$	3.67193	1.01841	0.509205	+	0.860645i	$0.329939\pi$
$14$	−1.08727	−0.290585
$15$	0	0
$16$	−1.69545	−0.423862
$17$	5.37138	1.30275	0.651375	−	0.758756i	$-0.274192\pi$
$17$	5.37138	1.30275	0.651375	+	0.758756i	$0.274192\pi$
$18$	0	0
$19$	−3.54285	−0.812786	−0.406393	−	0.913698i	$-0.633214\pi$
$19$	−3.54285	−0.812786	−0.406393	+	0.913698i	$0.633214\pi$
$20$	0.171475	0.0383430
$21$	0	0
$22$	6.56515	1.39969
$23$	1.30362	0.271824	0.135912	−	0.990721i	$-0.456604\pi$
$23$	1.30362	0.271824	0.135912	+	0.990721i	$0.456604\pi$
$24$	0	0
$25$	−4.95604	−0.991208
$26$	−3.99238	−0.782971
$27$	0	0
$28$	−0.817843	−0.154558
$29$	8.00307	1.48613	0.743066	−	0.669218i	$-0.233371\pi$
$29$	8.00307	1.48613	0.743066	+	0.669218i	$0.233371\pi$
$30$	0	0
$31$	−0.384208	−0.0690058	−0.0345029	−	0.999405i	$-0.510985\pi$
$31$	−0.384208	−0.0690058	−0.0345029	+	0.999405i	$0.510985\pi$
$32$	−4.28411	−0.757330
$33$	0	0
$34$	−5.84014	−1.00158
$35$	−0.209668	−0.0354403
$36$	0	0
$37$	−3.68876	−0.606429	−0.303214	−	0.952922i	$-0.598060\pi$
$37$	−3.68876	−0.606429	−0.303214	+	0.952922i	$0.598060\pi$
$38$	3.85204	0.624883
$39$	0	0
$40$	−0.642370	−0.101568
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	0.824527	0.125739	0.0628696	−	0.998022i	$-0.479975\pi$
$43$	0.824527	0.125739	0.0628696	+	0.998022i	$0.479975\pi$
$44$	4.93829	0.744476
$45$	0	0
$46$	−1.41739	−0.208983
$47$	−5.11625	−0.746282	−0.373141	−	0.927775i	$-0.621719\pi$
$47$	−5.11625	−0.746282	−0.373141	+	0.927775i	$0.621719\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.38855	0.762057
$51$	0	0
$52$	−3.00307	−0.416450
$53$	−1.53217	−0.210460	−0.105230	−	0.994448i	$-0.533558\pi$
$53$	−1.53217	−0.210460	−0.105230	+	0.994448i	$0.533558\pi$
$54$	0	0
$55$	1.26601	0.170709
$56$	3.06376	0.409412
$57$	0	0
$58$	−8.70150	−1.14256
$59$	−10.2669	−1.33664	−0.668320	−	0.743874i	$-0.732986\pi$
$59$	−10.2669	−1.33664	−0.668320	+	0.743874i	$0.732986\pi$
$60$	0	0
$61$	9.36070	1.19851	0.599257	−	0.800557i	$-0.295463\pi$
$61$	9.36070	1.19851	0.599257	+	0.800557i	$0.295463\pi$
$62$	0.417738	0.0530528
$63$	0	0
$64$	8.04887	1.00611
$65$	−0.769885	−0.0954925
$66$	0	0
$67$	11.3638	1.38830	0.694152	−	0.719828i	$-0.255779\pi$
$67$	11.3638	1.38830	0.694152	+	0.719828i	$0.255779\pi$
$68$	−4.39294	−0.532723
$69$	0	0
$70$	0.227965	0.0272471
$71$	14.9494	1.77416	0.887081	−	0.461614i	$-0.152729\pi$
$71$	14.9494	1.77416	0.887081	+	0.461614i	$0.152729\pi$
$72$	0	0
$73$	7.77203	0.909648	0.454824	−	0.890581i	$-0.349702\pi$
$73$	7.77203	0.909648	0.454824	+	0.890581i	$0.349702\pi$
$74$	4.01068	0.466232
$75$	0	0
$76$	2.89750	0.332366
$77$	−6.03819	−0.688116
$78$	0	0
$79$	−6.04703	−0.680344	−0.340172	−	0.940363i	$-0.610485\pi$
$79$	−6.04703	−0.680344	−0.340172	+	0.940363i	$0.610485\pi$
$80$	0.355480	0.0397439
$81$	0	0
$82$	−1.08727	−0.120069
$83$	−14.1871	−1.55723	−0.778617	−	0.627500i	$-0.784078\pi$
$83$	−14.1871	−1.55723	−0.778617	+	0.627500i	$0.784078\pi$
$84$	0	0
$85$	−1.12620	−0.122154
$86$	−0.896484	−0.0966703
$87$	0	0
$88$	−18.4996	−1.97206
$89$	−0.520905	−0.0552159	−0.0276079	−	0.999619i	$-0.508789\pi$
$89$	−0.520905	−0.0552159	−0.0276079	+	0.999619i	$0.508789\pi$
$90$	0	0
$91$	3.67193	0.384923
$92$	−1.06616	−0.111155
$93$	0	0
$94$	5.56275	0.573754
$95$	0.742821	0.0762118
$96$	0	0
$97$	−3.65270	−0.370876	−0.185438	−	0.982656i	$-0.559370\pi$
$97$	−3.65270	−0.370876	−0.185438	+	0.982656i	$0.559370\pi$
$98$	−1.08727	−0.109831
$99$	0	0
$100$	4.05326	0.405326
$101$	2.45465	0.244247	0.122123	−	0.992515i	$-0.461030\pi$
$101$	2.45465	0.244247	0.122123	+	0.992515i	$0.461030\pi$
$102$	0	0
$103$	−10.2479	−1.00975	−0.504876	−	0.863192i	$-0.668462\pi$
$103$	−10.2479	−1.00975	−0.504876	+	0.863192i	$0.668462\pi$
$104$	11.2499	1.10314
$105$	0	0
$106$	1.66588	0.161805
$107$	5.33318	0.515578	0.257789	−	0.966201i	$-0.417006\pi$
$107$	5.33318	0.515578	0.257789	+	0.966201i	$0.417006\pi$
$108$	0	0
$109$	8.82638	0.845413	0.422707	−	0.906267i	$-0.361080\pi$
$109$	8.82638	0.845413	0.422707	+	0.906267i	$0.361080\pi$
$110$	−1.37650	−0.131244
$111$	0	0
$112$	−1.69545	−0.160205
$113$	18.2045	1.71253	0.856267	−	0.516534i	$-0.172778\pi$
$113$	18.2045	1.71253	0.856267	+	0.516534i	$0.172778\pi$
$114$	0	0
$115$	−0.273327	−0.0254879
$116$	−6.54525	−0.607711
$117$	0	0
$118$	11.1629	1.02763
$119$	5.37138	0.492393
$120$	0	0
$121$	25.4598	2.31452
$122$	−10.1776	−0.921437
$123$	0	0
$124$	0.314222	0.0282180
$125$	2.08746	0.186708
$126$	0	0
$127$	−13.1751	−1.16910	−0.584551	−	0.811357i	$-0.698729\pi$
$127$	−13.1751	−1.16910	−0.584551	+	0.811357i	$0.698729\pi$
$128$	−0.183089	−0.0161829
$129$	0	0
$130$	0.837073	0.0734162
$131$	−13.0506	−1.14023	−0.570117	−	0.821563i	$-0.693102\pi$
$131$	−13.0506	−1.14023	−0.570117	+	0.821563i	$0.693102\pi$
$132$	0	0
$133$	−3.54285	−0.307204
$134$	−12.3555	−1.06735
$135$	0	0
$136$	16.4566	1.41114
$137$	−12.7699	−1.09100	−0.545502	−	0.838109i	$-0.683661\pi$
$137$	−12.7699	−1.09100	−0.545502	+	0.838109i	$0.683661\pi$
$138$	0	0
$139$	13.9759	1.18542	0.592712	−	0.805415i	$-0.298057\pi$
$139$	13.9759	1.18542	0.592712	+	0.805415i	$0.298057\pi$
$140$	0.171475	0.0144923
$141$	0	0
$142$	−16.2540	−1.36400
$143$	−22.1718	−1.85410
$144$	0	0
$145$	−1.67798	−0.139349
$146$	−8.45030	−0.699352
$147$	0	0
$148$	3.01683	0.247982
$149$	2.02136	0.165597	0.0827983	−	0.996566i	$-0.473614\pi$
$149$	2.02136	0.165597	0.0827983	+	0.996566i	$0.473614\pi$
$150$	0	0
$151$	11.7648	0.957406	0.478703	−	0.877977i	$-0.341107\pi$
$151$	11.7648	0.957406	0.478703	+	0.877977i	$0.341107\pi$
$152$	−10.8544	−0.880411
$153$	0	0
$154$	6.56515	0.529035
$155$	0.0805560	0.00647041
$156$	0	0
$157$	18.8859	1.50726	0.753631	−	0.657297i	$-0.228300\pi$
$157$	18.8859	1.50726	0.753631	+	0.657297i	$0.228300\pi$
$158$	6.57475	0.523059
$159$	0	0
$160$	0.898238	0.0710119
$161$	1.30362	0.102740
$162$	0	0
$163$	16.3263	1.27877	0.639387	−	0.768885i	$-0.279188\pi$
$163$	16.3263	1.27877	0.639387	+	0.768885i	$0.279188\pi$
$164$	−0.817843	−0.0638628
$165$	0	0
$166$	15.4252	1.19723
$167$	22.5846	1.74765	0.873824	−	0.486242i	$-0.161633\pi$
$167$	22.5846	1.74765	0.873824	+	0.486242i	$0.161633\pi$
$168$	0	0
$169$	0.483092	0.0371609
$170$	1.22449	0.0939139
$171$	0	0
$172$	−0.674334	−0.0514175
$173$	22.2230	1.68958	0.844790	−	0.535097i	$-0.179725\pi$
$173$	22.2230	1.68958	0.844790	+	0.535097i	$0.179725\pi$
$174$	0	0
$175$	−4.95604	−0.374641
$176$	10.2374	0.771675
$177$	0	0
$178$	0.566365	0.0424508
$179$	10.0095	0.748142	0.374071	−	0.927400i	$-0.377962\pi$
$179$	10.0095	0.748142	0.374071	+	0.927400i	$0.377962\pi$
$180$	0	0
$181$	10.8101	0.803511	0.401755	−	0.915747i	$-0.368400\pi$
$181$	10.8101	0.803511	0.401755	+	0.915747i	$0.368400\pi$
$182$	−3.99238	−0.295935
$183$	0	0
$184$	3.99398	0.294440
$185$	0.773414	0.0568625
$186$	0	0
$187$	−32.4334	−2.37177
$188$	4.18429	0.305171
$189$	0	0
$190$	−0.807647	−0.0585929
$191$	19.7693	1.43046	0.715229	−	0.698890i	$-0.246322\pi$
$191$	19.7693	1.43046	0.715229	+	0.698890i	$0.246322\pi$
$192$	0	0
$193$	18.0955	1.30254	0.651270	−	0.758846i	$-0.274237\pi$
$193$	18.0955	1.30254	0.651270	+	0.758846i	$0.274237\pi$
$194$	3.97148	0.285135
$195$	0	0
$196$	−0.817843	−0.0584174
$197$	−4.47538	−0.318858	−0.159429	−	0.987209i	$-0.550965\pi$
$197$	−4.47538	−0.318858	−0.159429	+	0.987209i	$0.550965\pi$
$198$	0	0
$199$	10.1956	0.722744	0.361372	−	0.932422i	$-0.382308\pi$
$199$	10.1956	0.722744	0.361372	+	0.932422i	$0.382308\pi$
$200$	−15.1841	−1.07368
$201$	0	0
$202$	−2.66887	−0.187781
$203$	8.00307	0.561705
$204$	0	0
$205$	−0.209668	−0.0146438
$206$	11.1422	0.776313
$207$	0	0
$208$	−6.22556	−0.431665
$209$	21.3924	1.47974
$210$	0	0
$211$	−14.0477	−0.967081	−0.483540	−	0.875322i	$-0.660649\pi$
$211$	−14.0477	−0.967081	−0.483540	+	0.875322i	$0.660649\pi$
$212$	1.25308	0.0860616
$213$	0	0
$214$	−5.79861	−0.396385
$215$	−0.172877	−0.0117901
$216$	0	0
$217$	−0.384208	−0.0260818
$218$	−9.59666	−0.649968
$219$	0	0
$220$	−1.03540	−0.0698066
$221$	19.7233	1.32674
$222$	0	0
$223$	−0.701496	−0.0469756	−0.0234878	−	0.999724i	$-0.507477\pi$
$223$	−0.701496	−0.0469756	−0.0234878	+	0.999724i	$0.507477\pi$
$224$	−4.28411	−0.286244
$225$	0	0
$226$	−19.7932	−1.31662
$227$	−9.36869	−0.621822	−0.310911	−	0.950439i	$-0.600634\pi$
$227$	−9.36869	−0.621822	−0.310911	+	0.950439i	$0.600634\pi$
$228$	0	0
$229$	12.4012	0.819496	0.409748	−	0.912199i	$-0.365617\pi$
$229$	12.4012	0.819496	0.409748	+	0.912199i	$0.365617\pi$
$230$	0.297180	0.0195955
$231$	0	0
$232$	24.5195	1.60978
$233$	−9.61739	−0.630056	−0.315028	−	0.949082i	$-0.602014\pi$
$233$	−9.61739	−0.630056	−0.315028	+	0.949082i	$0.602014\pi$
$234$	0	0
$235$	1.07271	0.0699760
$236$	8.39674	0.546581
$237$	0	0
$238$	−5.84014	−0.378560
$239$	16.5603	1.07120	0.535600	−	0.844472i	$-0.320086\pi$
$239$	16.5603	1.07120	0.535600	+	0.844472i	$0.320086\pi$
$240$	0	0
$241$	9.29684	0.598862	0.299431	−	0.954118i	$-0.403203\pi$
$241$	9.29684	0.598862	0.299431	+	0.954118i	$0.403203\pi$
$242$	−27.6816	−1.77944
$243$	0	0
$244$	−7.65558	−0.490098
$245$	−0.209668	−0.0133952
$246$	0	0
$247$	−13.0091	−0.827750
$248$	−1.17712	−0.0747472
$249$	0	0
$250$	−2.26963	−0.143544
$251$	−2.89503	−0.182733	−0.0913664	−	0.995817i	$-0.529123\pi$
$251$	−2.89503	−0.182733	−0.0913664	+	0.995817i	$0.529123\pi$
$252$	0	0
$253$	−7.87152	−0.494878
$254$	14.3249	0.898824
$255$	0	0
$256$	−15.8987	−0.993668
$257$	−5.41470	−0.337760	−0.168880	−	0.985637i	$-0.554015\pi$
$257$	−5.41470	−0.337760	−0.168880	+	0.985637i	$0.554015\pi$
$258$	0	0
$259$	−3.68876	−0.229209
$260$	0.629645	0.0390489
$261$	0	0
$262$	14.1895	0.876631
$263$	26.7339	1.64848	0.824242	−	0.566238i	$-0.191602\pi$
$263$	26.7339	1.64848	0.824242	+	0.566238i	$0.191602\pi$
$264$	0	0
$265$	0.321246	0.0197340
$266$	3.85204	0.236184
$267$	0	0
$268$	−9.29377	−0.567708
$269$	14.6488	0.893151	0.446576	−	0.894746i	$-0.352643\pi$
$269$	14.6488	0.893151	0.446576	+	0.894746i	$0.352643\pi$
$270$	0	0
$271$	22.8241	1.38646	0.693232	−	0.720714i	$-0.256186\pi$
$271$	22.8241	1.38646	0.693232	+	0.720714i	$0.256186\pi$
$272$	−9.10688	−0.552186
$273$	0	0
$274$	13.8843	0.838782
$275$	29.9255	1.80458
$276$	0	0
$277$	−8.31190	−0.499414	−0.249707	−	0.968321i	$-0.580334\pi$
$277$	−8.31190	−0.499414	−0.249707	+	0.968321i	$0.580334\pi$
$278$	−15.1956	−0.911373
$279$	0	0
$280$	−0.642370	−0.0383890
$281$	−17.3137	−1.03285	−0.516423	−	0.856333i	$-0.672737\pi$
$281$	−17.3137	−1.03285	−0.516423	+	0.856333i	$0.672737\pi$
$282$	0	0
$283$	−14.7818	−0.878686	−0.439343	−	0.898319i	$-0.644789\pi$
$283$	−14.7818	−0.878686	−0.439343	+	0.898319i	$0.644789\pi$
$284$	−12.2262	−0.725493
$285$	0	0
$286$	24.1068	1.42546
$287$	1.00000	0.0590281
$288$	0	0
$289$	11.8517	0.697158
$290$	1.82442	0.107134
$291$	0	0
$292$	−6.35631	−0.371975
$293$	−4.99638	−0.291892	−0.145946	−	0.989293i	$-0.546623\pi$
$293$	−4.99638	−0.291892	−0.145946	+	0.989293i	$0.546623\pi$
$294$	0	0
$295$	2.15264	0.125332
$296$	−11.3015	−0.656885
$297$	0	0
$298$	−2.19777	−0.127313
$299$	4.78681	0.276828
$300$	0	0
$301$	0.824527	0.0475250
$302$	−12.7915	−0.736069
$303$	0	0
$304$	6.00671	0.344509
$305$	−1.96263	−0.112380
$306$	0	0
$307$	−7.85295	−0.448192	−0.224096	−	0.974567i	$-0.571943\pi$
$307$	−7.85295	−0.448192	−0.224096	+	0.974567i	$0.571943\pi$
$308$	4.93829	0.281385
$309$	0	0
$310$	−0.0875861	−0.00497456
$311$	27.0379	1.53318	0.766589	−	0.642138i	$-0.221952\pi$
$311$	27.0379	1.53318	0.766589	+	0.642138i	$0.221952\pi$
$312$	0	0
$313$	−0.168393	−0.00951815	−0.00475907	−	0.999989i	$-0.501515\pi$
$313$	−0.168393	−0.00951815	−0.00475907	+	0.999989i	$0.501515\pi$
$314$	−20.5341	−1.15881
$315$	0	0
$316$	4.94552	0.278207
$317$	−5.33690	−0.299750	−0.149875	−	0.988705i	$-0.547887\pi$
$317$	−5.33690	−0.299750	−0.149875	+	0.988705i	$0.547887\pi$
$318$	0	0
$319$	−48.3240	−2.70563
$320$	−1.68759	−0.0943390
$321$	0	0
$322$	−1.41739	−0.0789880
$323$	−19.0300	−1.05886
$324$	0	0
$325$	−18.1982	−1.00946
$326$	−17.7511	−0.983142
$327$	0	0
$328$	3.06376	0.169168
$329$	−5.11625	−0.282068
$330$	0	0
$331$	−8.09405	−0.444889	−0.222445	−	0.974945i	$-0.571404\pi$
$331$	−8.09405	−0.444889	−0.222445	+	0.974945i	$0.571404\pi$
$332$	11.6028	0.636786
$333$	0	0
$334$	−24.5556	−1.34362
$335$	−2.38261	−0.130176
$336$	0	0
$337$	20.1540	1.09786	0.548929	−	0.835869i	$-0.315036\pi$
$337$	20.1540	1.09786	0.548929	+	0.835869i	$0.315036\pi$
$338$	−0.525252	−0.0285699
$339$	0	0
$340$	0.921058	0.0499514
$341$	2.31992	0.125631
$342$	0	0
$343$	1.00000	0.0539949
$344$	2.52615	0.136201
$345$	0	0
$346$	−24.1624	−1.29898
$347$	−18.2807	−0.981359	−0.490679	−	0.871340i	$-0.663251\pi$
$347$	−18.2807	−0.981359	−0.490679	+	0.871340i	$0.663251\pi$
$348$	0	0
$349$	9.74567	0.521674	0.260837	−	0.965383i	$-0.416001\pi$
$349$	9.74567	0.521674	0.260837	+	0.965383i	$0.416001\pi$
$350$	5.38855	0.288030
$351$	0	0
$352$	25.8683	1.37878
$353$	−26.2143	−1.39525	−0.697624	−	0.716464i	$-0.745759\pi$
$353$	−26.2143	−1.39525	−0.697624	+	0.716464i	$0.745759\pi$
$354$	0	0
$355$	−3.13439	−0.166356
$356$	0.426019	0.0225790
$357$	0	0
$358$	−10.8830	−0.575184
$359$	−0.473108	−0.0249697	−0.0124849	−	0.999922i	$-0.503974\pi$
$359$	−0.473108	−0.0249697	−0.0124849	+	0.999922i	$0.503974\pi$
$360$	0	0
$361$	−6.44820	−0.339379
$362$	−11.7535	−0.617752
$363$	0	0
$364$	−3.00307	−0.157403
$365$	−1.62954	−0.0852942
$366$	0	0
$367$	10.2670	0.535932	0.267966	−	0.963428i	$-0.413648\pi$
$367$	10.2670	0.535932	0.267966	+	0.963428i	$0.413648\pi$
$368$	−2.21022	−0.115216
$369$	0	0
$370$	−0.840910	−0.0437168
$371$	−1.53217	−0.0795463
$372$	0	0
$373$	20.1653	1.04412	0.522061	−	0.852908i	$-0.325163\pi$
$373$	20.1653	1.04412	0.522061	+	0.852908i	$0.325163\pi$
$374$	35.2639	1.82345
$375$	0	0
$376$	−15.6749	−0.808374
$377$	29.3867	1.51349
$378$	0	0
$379$	−26.8293	−1.37813	−0.689063	−	0.724701i	$-0.741978\pi$
$379$	−26.8293	−1.37813	−0.689063	+	0.724701i	$0.741978\pi$
$380$	−0.607511	−0.0311647
$381$	0	0
$382$	−21.4946	−1.09976
$383$	24.1679	1.23492	0.617461	−	0.786602i	$-0.288161\pi$
$383$	24.1679	1.23492	0.617461	+	0.786602i	$0.288161\pi$
$384$	0	0
$385$	1.26601	0.0645220
$386$	−19.6746	−1.00141
$387$	0	0
$388$	2.98734	0.151659
$389$	−15.9595	−0.809180	−0.404590	−	0.914498i	$-0.632586\pi$
$389$	−15.9595	−0.809180	−0.404590	+	0.914498i	$0.632586\pi$
$390$	0	0
$391$	7.00224	0.354119
$392$	3.06376	0.154743
$393$	0	0
$394$	4.86595	0.245143
$395$	1.26786	0.0637932
$396$	0	0
$397$	−8.73399	−0.438346	−0.219173	−	0.975686i	$-0.570336\pi$
$397$	−8.73399	−0.438346	−0.219173	+	0.975686i	$0.570336\pi$
$398$	−11.0853	−0.555657
$399$	0	0
$400$	8.40270	0.420135
$401$	−1.77238	−0.0885086	−0.0442543	−	0.999020i	$-0.514091\pi$
$401$	−1.77238	−0.0885086	−0.0442543	+	0.999020i	$0.514091\pi$
$402$	0	0
$403$	−1.41079	−0.0702763
$404$	−2.00752	−0.0998778
$405$	0	0
$406$	−8.70150	−0.431848
$407$	22.2735	1.10405
$408$	0	0
$409$	−27.0895	−1.33949	−0.669746	−	0.742590i	$-0.733597\pi$
$409$	−27.0895	−1.33949	−0.669746	+	0.742590i	$0.733597\pi$
$410$	0.227965	0.0112584
$411$	0	0
$412$	8.38114	0.412909
$413$	−10.2669	−0.505203
$414$	0	0
$415$	2.97457	0.146016
$416$	−15.7310	−0.771273
$417$	0	0
$418$	−23.2593	−1.13765
$419$	18.3386	0.895898	0.447949	−	0.894059i	$-0.352155\pi$
$419$	18.3386	0.895898	0.447949	+	0.894059i	$0.352155\pi$
$420$	0	0
$421$	16.7202	0.814894	0.407447	−	0.913229i	$-0.366419\pi$
$421$	16.7202	0.814894	0.407447	+	0.913229i	$0.366419\pi$
$422$	15.2736	0.743507
$423$	0	0
$424$	−4.69420	−0.227970
$425$	−26.6208	−1.29130
$426$	0	0
$427$	9.36070	0.452996
$428$	−4.36171	−0.210831
$429$	0	0
$430$	0.187964	0.00906441
$431$	−0.204738	−0.00986186	−0.00493093	−	0.999988i	$-0.501570\pi$
$431$	−0.204738	−0.00986186	−0.00493093	+	0.999988i	$0.501570\pi$
$432$	0	0
$433$	31.9997	1.53781	0.768903	−	0.639366i	$-0.220803\pi$
$433$	31.9997	1.53781	0.768903	+	0.639366i	$0.220803\pi$
$434$	0.417738	0.0200521
$435$	0	0
$436$	−7.21859	−0.345708
$437$	−4.61854	−0.220935
$438$	0	0
$439$	24.9052	1.18866	0.594330	−	0.804221i	$-0.297417\pi$
$439$	24.9052	1.18866	0.594330	+	0.804221i	$0.297417\pi$
$440$	3.87876	0.184912
$441$	0	0
$442$	−21.4446	−1.02002
$443$	−19.0300	−0.904142	−0.452071	−	0.891982i	$-0.649315\pi$
$443$	−19.0300	−0.904142	−0.452071	+	0.891982i	$0.649315\pi$
$444$	0	0
$445$	0.109217	0.00517738
$446$	0.762716	0.0361156
$447$	0	0
$448$	8.04887	0.380274
$449$	5.95010	0.280802	0.140401	−	0.990095i	$-0.455161\pi$
$449$	5.95010	0.280802	0.140401	+	0.990095i	$0.455161\pi$
$450$	0	0
$451$	−6.03819	−0.284327
$452$	−14.8884	−0.700292
$453$	0	0
$454$	10.1863	0.478067
$455$	−0.769885	−0.0360928
$456$	0	0
$457$	−31.8410	−1.48946	−0.744729	−	0.667367i	$-0.767421\pi$
$457$	−31.8410	−1.48946	−0.744729	+	0.667367i	$0.767421\pi$
$458$	−13.4835	−0.630042
$459$	0	0
$460$	0.223539	0.0104225
$461$	−4.99638	−0.232705	−0.116352	−	0.993208i	$-0.537120\pi$
$461$	−4.99638	−0.232705	−0.116352	+	0.993208i	$0.537120\pi$
$462$	0	0
$463$	17.6038	0.818119	0.409060	−	0.912508i	$-0.365857\pi$
$463$	17.6038	0.818119	0.409060	+	0.912508i	$0.365857\pi$
$464$	−13.5688	−0.629914
$465$	0	0
$466$	10.4567	0.484397
$467$	42.9444	1.98723	0.993614	−	0.112833i	$-0.0359925\pi$
$467$	42.9444	1.98723	0.993614	+	0.112833i	$0.0359925\pi$
$468$	0	0
$469$	11.3638	0.524730
$470$	−1.16633	−0.0537987
$471$	0	0
$472$	−31.4554	−1.44785
$473$	−4.97865	−0.228919
$474$	0	0
$475$	17.5585	0.805640
$476$	−4.39294	−0.201350
$477$	0	0
$478$	−18.0056	−0.823556
$479$	−2.85249	−0.130334	−0.0651669	−	0.997874i	$-0.520758\pi$
$479$	−2.85249	−0.130334	−0.0651669	+	0.997874i	$0.520758\pi$
$480$	0	0
$481$	−13.5449	−0.617594
$482$	−10.1082	−0.460415
$483$	0	0
$484$	−20.8221	−0.946459
$485$	0.765853	0.0347756
$486$	0	0
$487$	−6.23225	−0.282410	−0.141205	−	0.989980i	$-0.545098\pi$
$487$	−6.23225	−0.282410	−0.141205	+	0.989980i	$0.545098\pi$
$488$	28.6789	1.29823
$489$	0	0
$490$	0.227965	0.0102984
$491$	2.13846	0.0965071	0.0482536	−	0.998835i	$-0.484634\pi$
$491$	2.13846	0.0965071	0.0482536	+	0.998835i	$0.484634\pi$
$492$	0	0
$493$	42.9875	1.93606
$494$	14.1444	0.636388
$495$	0	0
$496$	0.651404	0.0292489
$497$	14.9494	0.670570
$498$	0	0
$499$	25.1110	1.12412	0.562060	−	0.827096i	$-0.310009\pi$
$499$	25.1110	1.12412	0.562060	+	0.827096i	$0.310009\pi$
$500$	−1.70721	−0.0763489
$501$	0	0
$502$	3.14768	0.140488
$503$	−37.1641	−1.65707	−0.828534	−	0.559939i	$-0.810824\pi$
$503$	−37.1641	−1.65707	−0.828534	+	0.559939i	$0.810824\pi$
$504$	0	0
$505$	−0.514660	−0.0229021
$506$	8.55847	0.380470
$507$	0	0
$508$	10.7752	0.478071
$509$	2.36319	0.104747	0.0523734	−	0.998628i	$-0.483321\pi$
$509$	2.36319	0.104747	0.0523734	+	0.998628i	$0.483321\pi$
$510$	0	0
$511$	7.77203	0.343815
$512$	17.6523	0.780131
$513$	0	0
$514$	5.88725	0.259675
$515$	2.14864	0.0946805
$516$	0	0
$517$	30.8929	1.35867
$518$	4.01068	0.176219
$519$	0	0
$520$	−2.35874	−0.103438
$521$	2.27368	0.0996116	0.0498058	−	0.998759i	$-0.484140\pi$
$521$	2.27368	0.0996116	0.0498058	+	0.998759i	$0.484140\pi$
$522$	0	0
$523$	−35.7432	−1.56294	−0.781471	−	0.623941i	$-0.785531\pi$
$523$	−35.7432	−1.56294	−0.781471	+	0.623941i	$0.785531\pi$
$524$	10.6733	0.466266
$525$	0	0
$526$	−29.0670	−1.26738
$527$	−2.06373	−0.0898974
$528$	0	0
$529$	−21.3006	−0.926112
$530$	−0.349282	−0.0151718
$531$	0	0
$532$	2.89750	0.125622
$533$	3.67193	0.159049
$534$	0	0
$535$	−1.11820	−0.0483438
$536$	34.8158	1.50381
$537$	0	0
$538$	−15.9272	−0.686669
$539$	−6.03819	−0.260083
$540$	0	0
$541$	−22.5224	−0.968314	−0.484157	−	0.874981i	$-0.660874\pi$
$541$	−22.5224	−0.968314	−0.484157	+	0.874981i	$0.660874\pi$
$542$	−24.8159	−1.06594
$543$	0	0
$544$	−23.0116	−0.986612
$545$	−1.85060	−0.0792712
$546$	0	0
$547$	3.63915	0.155599	0.0777994	−	0.996969i	$-0.475211\pi$
$547$	3.63915	0.155599	0.0777994	+	0.996969i	$0.475211\pi$
$548$	10.4438	0.446135
$549$	0	0
$550$	−32.5371	−1.38739
$551$	−28.3537	−1.20791
$552$	0	0
$553$	−6.04703	−0.257146
$554$	9.03729	0.383957
$555$	0	0
$556$	−11.4301	−0.484745
$557$	−4.80008	−0.203386	−0.101693	−	0.994816i	$-0.532426\pi$
$557$	−4.80008	−0.203386	−0.101693	+	0.994816i	$0.532426\pi$
$558$	0	0
$559$	3.02761	0.128054
$560$	0.355480	0.0150218
$561$	0	0
$562$	18.8246	0.794069
$563$	−31.7115	−1.33648	−0.668241	−	0.743945i	$-0.732952\pi$
$563$	−31.7115	−1.33648	−0.668241	+	0.743945i	$0.732952\pi$
$564$	0	0
$565$	−3.81689	−0.160578
$566$	16.0718	0.675548
$567$	0	0
$568$	45.8012	1.92178
$569$	35.4970	1.48811	0.744055	−	0.668118i	$-0.232900\pi$
$569$	35.4970	1.48811	0.744055	+	0.668118i	$0.232900\pi$
$570$	0	0
$571$	−38.7912	−1.62336	−0.811681	−	0.584100i	$-0.801447\pi$
$571$	−38.7912	−1.62336	−0.811681	+	0.584100i	$0.801447\pi$
$572$	18.1331	0.758182
$573$	0	0
$574$	−1.08727	−0.0453818
$575$	−6.46080	−0.269434
$576$	0	0
$577$	−0.335828	−0.0139807	−0.00699035	−	0.999976i	$-0.502225\pi$
$577$	−0.335828	−0.0139807	−0.00699035	+	0.999976i	$0.502225\pi$
$578$	−12.8860	−0.535987
$579$	0	0
$580$	1.37233	0.0569828
$581$	−14.1871	−0.588579
$582$	0	0
$583$	9.25154	0.383160
$584$	23.8116	0.985332
$585$	0	0
$586$	5.43242	0.224411
$587$	−8.19386	−0.338197	−0.169098	−	0.985599i	$-0.554086\pi$
$587$	−8.19386	−0.338197	−0.169098	+	0.985599i	$0.554086\pi$
$588$	0	0
$589$	1.36119	0.0560870
$590$	−2.34050	−0.0963570
$591$	0	0
$592$	6.25410	0.257042
$593$	−30.1222	−1.23697	−0.618486	−	0.785796i	$-0.712254\pi$
$593$	−30.1222	−1.23697	−0.618486	+	0.785796i	$0.712254\pi$
$594$	0	0
$595$	−1.12620	−0.0461698
$596$	−1.65316	−0.0677160
$597$	0	0
$598$	−5.20456	−0.212830
$599$	39.6072	1.61831	0.809153	−	0.587599i	$-0.199927\pi$
$599$	39.6072	1.61831	0.809153	+	0.587599i	$0.199927\pi$
$600$	0	0
$601$	−35.0329	−1.42902	−0.714511	−	0.699624i	$-0.753351\pi$
$601$	−35.0329	−1.42902	−0.714511	+	0.699624i	$0.753351\pi$
$602$	−0.896484	−0.0365380
$603$	0	0
$604$	−9.62176	−0.391504
$605$	−5.33809	−0.217024
$606$	0	0
$607$	13.4657	0.546555	0.273278	−	0.961935i	$-0.411892\pi$
$607$	13.4657	0.546555	0.273278	+	0.961935i	$0.411892\pi$
$608$	15.1780	0.615547
$609$	0	0
$610$	2.13391	0.0863996
$611$	−18.7865	−0.760022
$612$	0	0
$613$	−6.49820	−0.262460	−0.131230	−	0.991352i	$-0.541893\pi$
$613$	−6.49820	−0.262460	−0.131230	+	0.991352i	$0.541893\pi$
$614$	8.53828	0.344577
$615$	0	0
$616$	−18.4996	−0.745368
$617$	−6.63045	−0.266932	−0.133466	−	0.991053i	$-0.542611\pi$
$617$	−6.63045	−0.266932	−0.133466	+	0.991053i	$0.542611\pi$
$618$	0	0
$619$	−4.50756	−0.181174	−0.0905871	−	0.995889i	$-0.528874\pi$
$619$	−4.50756	−0.181174	−0.0905871	+	0.995889i	$0.528874\pi$
$620$	−0.0658822	−0.00264589
$621$	0	0
$622$	−29.3975	−1.17873
$623$	−0.520905	−0.0208696
$624$	0	0
$625$	24.3425	0.973701
$626$	0.183089	0.00731771
$627$	0	0
$628$	−15.4457	−0.616352
$629$	−19.8137	−0.790025
$630$	0	0
$631$	−24.2677	−0.966081	−0.483040	−	0.875598i	$-0.660468\pi$
$631$	−24.2677	−0.966081	−0.483040	+	0.875598i	$0.660468\pi$
$632$	−18.5266	−0.736949
$633$	0	0
$634$	5.80265	0.230453
$635$	2.76239	0.109622
$636$	0	0
$637$	3.67193	0.145487
$638$	52.5413	2.08013
$639$	0	0
$640$	0.0383878	0.00151741
$641$	−22.2964	−0.880653	−0.440327	−	0.897838i	$-0.645137\pi$
$641$	−22.2964	−0.880653	−0.440327	+	0.897838i	$0.645137\pi$
$642$	0	0
$643$	−16.2405	−0.640464	−0.320232	−	0.947339i	$-0.603761\pi$
$643$	−16.2405	−0.640464	−0.320232	+	0.947339i	$0.603761\pi$
$644$	−1.06616	−0.0420125
$645$	0	0
$646$	20.6907	0.814067
$647$	−13.5814	−0.533939	−0.266969	−	0.963705i	$-0.586022\pi$
$647$	−13.5814	−0.533939	−0.266969	+	0.963705i	$0.586022\pi$
$648$	0	0
$649$	61.9937	2.43347
$650$	19.7864	0.776087
$651$	0	0
$652$	−13.3524	−0.522918
$653$	47.1222	1.84403	0.922016	−	0.387151i	$-0.126541\pi$
$653$	47.1222	1.84403	0.922016	+	0.387151i	$0.126541\pi$
$654$	0	0
$655$	2.73628	0.106915
$656$	−1.69545	−0.0661960
$657$	0	0
$658$	5.56275	0.216858
$659$	−5.27483	−0.205478	−0.102739	−	0.994708i	$-0.532761\pi$
$659$	−5.27483	−0.205478	−0.102739	+	0.994708i	$0.532761\pi$
$660$	0	0
$661$	−47.8044	−1.85938	−0.929689	−	0.368346i	$-0.879924\pi$
$661$	−47.8044	−1.85938	−0.929689	+	0.368346i	$0.879924\pi$
$662$	8.80042	0.342038
$663$	0	0
$664$	−43.4657	−1.68680
$665$	0.742821	0.0288054
$666$	0	0
$667$	10.4330	0.403966
$668$	−18.4707	−0.714651
$669$	0	0
$670$	2.59054	0.100081
$671$	−56.5217	−2.18200
$672$	0	0
$673$	−5.11960	−0.197346	−0.0986730	−	0.995120i	$-0.531460\pi$
$673$	−5.11960	−0.197346	−0.0986730	+	0.995120i	$0.531460\pi$
$674$	−21.9128	−0.844051
$675$	0	0
$676$	−0.395094	−0.0151959
$677$	−30.7985	−1.18368	−0.591842	−	0.806054i	$-0.701599\pi$
$677$	−30.7985	−1.18368	−0.591842	+	0.806054i	$0.701599\pi$
$678$	0	0
$679$	−3.65270	−0.140178
$680$	−3.45041	−0.132317
$681$	0	0
$682$	−2.52238	−0.0965870
$683$	18.8534	0.721407	0.360703	−	0.932681i	$-0.382537\pi$
$683$	18.8534	0.721407	0.360703	+	0.932681i	$0.382537\pi$
$684$	0	0
$685$	2.67743	0.102299
$686$	−1.08727	−0.0415122
$687$	0	0
$688$	−1.39794	−0.0532960
$689$	−5.62603	−0.214335
$690$	0	0
$691$	−6.22638	−0.236863	−0.118431	−	0.992962i	$-0.537787\pi$
$691$	−6.22638	−0.236863	−0.118431	+	0.992962i	$0.537787\pi$
$692$	−18.1749	−0.690906
$693$	0	0
$694$	19.8761	0.754485
$695$	−2.93030	−0.111153
$696$	0	0
$697$	5.37138	0.203455
$698$	−10.5962	−0.401071
$699$	0	0
$700$	4.05326	0.153199
$701$	−40.6743	−1.53625	−0.768124	−	0.640301i	$-0.778810\pi$
$701$	−40.6743	−1.53625	−0.768124	+	0.640301i	$0.778810\pi$
$702$	0	0
$703$	13.0687	0.492897
$704$	−48.6006	−1.83171
$705$	0	0
$706$	28.5021	1.07269
$707$	2.45465	0.0923166
$708$	0	0
$709$	−47.5660	−1.78638	−0.893189	−	0.449681i	$-0.851538\pi$
$709$	−47.5660	−1.78638	−0.893189	+	0.449681i	$0.851538\pi$
$710$	3.40793	0.127897
$711$	0	0
$712$	−1.59593	−0.0598099
$713$	−0.500862	−0.0187574
$714$	0	0
$715$	4.64871	0.173852
$716$	−8.18617	−0.305932
$717$	0	0
$718$	0.514397	0.0191971
$719$	0.700457	0.0261226	0.0130613	−	0.999915i	$-0.495842\pi$
$719$	0.700457	0.0261226	0.0130613	+	0.999915i	$0.495842\pi$
$720$	0	0
$721$	−10.2479	−0.381650
$722$	7.01094	0.260920
$723$	0	0
$724$	−8.84099	−0.328573
$725$	−39.6635	−1.47307
$726$	0	0
$727$	38.6447	1.43325	0.716625	−	0.697458i	$-0.245686\pi$
$727$	38.6447	1.43325	0.716625	+	0.697458i	$0.245686\pi$
$728$	11.2499	0.416949
$729$	0	0
$730$	1.77175	0.0655756
$731$	4.42885	0.163807
$732$	0	0
$733$	−31.3572	−1.15820	−0.579101	−	0.815255i	$-0.696596\pi$
$733$	−31.3572	−1.15820	−0.579101	+	0.815255i	$0.696596\pi$
$734$	−11.1630	−0.412033
$735$	0	0
$736$	−5.58485	−0.205860
$737$	−68.6166	−2.52752
$738$	0	0
$739$	51.5448	1.89611	0.948054	−	0.318110i	$-0.103048\pi$
$739$	51.5448	1.89611	0.948054	+	0.318110i	$0.103048\pi$
$740$	−0.632531	−0.0232523
$741$	0	0
$742$	1.66588	0.0611565
$743$	15.7347	0.577251	0.288626	−	0.957442i	$-0.406802\pi$
$743$	15.7347	0.577251	0.288626	+	0.957442i	$0.406802\pi$
$744$	0	0
$745$	−0.423814	−0.0155274
$746$	−21.9252	−0.802737
$747$	0	0
$748$	26.5254	0.969866
$749$	5.33318	0.194870
$750$	0	0
$751$	−12.1375	−0.442904	−0.221452	−	0.975171i	$-0.571080\pi$
$751$	−12.1375	−0.442904	−0.221452	+	0.975171i	$0.571080\pi$
$752$	8.67433	0.316320
$753$	0	0
$754$	−31.9513	−1.16360
$755$	−2.46670	−0.0897723
$756$	0	0
$757$	−17.9846	−0.653660	−0.326830	−	0.945083i	$-0.605981\pi$
$757$	−17.9846	−0.653660	−0.326830	+	0.945083i	$0.605981\pi$
$758$	29.1707	1.05953
$759$	0	0
$760$	2.27582	0.0825528
$761$	−23.2298	−0.842079	−0.421039	−	0.907042i	$-0.638335\pi$
$761$	−23.2298	−0.842079	−0.421039	+	0.907042i	$0.638335\pi$
$762$	0	0
$763$	8.82638	0.319536
$764$	−16.1682	−0.584945
$765$	0	0
$766$	−26.2770	−0.949428
$767$	−37.6995	−1.36125
$768$	0	0
$769$	−34.9703	−1.26106	−0.630530	−	0.776165i	$-0.717162\pi$
$769$	−34.9703	−1.26106	−0.630530	+	0.776165i	$0.717162\pi$
$770$	−1.37650	−0.0496055
$771$	0	0
$772$	−14.7992	−0.532636
$773$	26.7805	0.963228	0.481614	−	0.876384i	$-0.340051\pi$
$773$	26.7805	0.963228	0.481614	+	0.876384i	$0.340051\pi$
$774$	0	0
$775$	1.90415	0.0683991
$776$	−11.1910	−0.401733
$777$	0	0
$778$	17.3523	0.622111
$779$	−3.54285	−0.126936
$780$	0	0
$781$	−90.2671	−3.23001
$782$	−7.61333	−0.272252
$783$	0	0
$784$	−1.69545	−0.0605516
$785$	−3.95977	−0.141330
$786$	0	0
$787$	0.177103	0.00631303	0.00315652	−	0.999995i	$-0.498995\pi$
$787$	0.177103	0.00631303	0.00315652	+	0.999995i	$0.498995\pi$
$788$	3.66016	0.130388
$789$	0	0
$790$	−1.37851	−0.0490452
$791$	18.2045	0.647277
$792$	0	0
$793$	34.3718	1.22058
$794$	9.49621	0.337008
$795$	0	0
$796$	−8.33837	−0.295546
$797$	15.4863	0.548555	0.274277	−	0.961651i	$-0.411561\pi$
$797$	15.4863	0.548555	0.274277	+	0.961651i	$0.411561\pi$
$798$	0	0
$799$	−27.4813	−0.972219
$800$	21.2322	0.750672
$801$	0	0
$802$	1.92706	0.0680469
$803$	−46.9290	−1.65609
$804$	0	0
$805$	−0.273327	−0.00963352
$806$	1.53391	0.0540296
$807$	0	0
$808$	7.52045	0.264569
$809$	12.8695	0.452467	0.226234	−	0.974073i	$-0.427359\pi$
$809$	12.8695	0.452467	0.226234	+	0.974073i	$0.427359\pi$
$810$	0	0
$811$	24.8951	0.874185	0.437092	−	0.899417i	$-0.356008\pi$
$811$	24.8951	0.874185	0.437092	+	0.899417i	$0.356008\pi$
$812$	−6.54525	−0.229693
$813$	0	0
$814$	−24.2173	−0.848815
$815$	−3.42309	−0.119906
$816$	0	0
$817$	−2.92118	−0.102199
$818$	29.4537	1.02982
$819$	0	0
$820$	0.171475	0.00598817
$821$	21.8002	0.760834	0.380417	−	0.924815i	$-0.375780\pi$
$821$	21.8002	0.760834	0.380417	+	0.924815i	$0.375780\pi$
$822$	0	0
$823$	−1.76071	−0.0613744	−0.0306872	−	0.999529i	$-0.509770\pi$
$823$	−1.76071	−0.0613744	−0.0306872	+	0.999529i	$0.509770\pi$
$824$	−31.3970	−1.09376
$825$	0	0
$826$	11.1629	0.388408
$827$	−9.93134	−0.345347	−0.172673	−	0.984979i	$-0.555240\pi$
$827$	−9.93134	−0.345347	−0.172673	+	0.984979i	$0.555240\pi$
$828$	0	0
$829$	4.85768	0.168714	0.0843571	−	0.996436i	$-0.473116\pi$
$829$	4.85768	0.168714	0.0843571	+	0.996436i	$0.473116\pi$
$830$	−3.23416	−0.112259
$831$	0	0
$832$	29.5549	1.02463
$833$	5.37138	0.186107
$834$	0	0
$835$	−4.73526	−0.163870
$836$	−17.4956	−0.605100
$837$	0	0
$838$	−19.9390	−0.688781
$839$	−15.9065	−0.549152	−0.274576	−	0.961565i	$-0.588537\pi$
$839$	−15.9065	−0.549152	−0.274576	+	0.961565i	$0.588537\pi$
$840$	0	0
$841$	35.0491	1.20859
$842$	−18.1794	−0.626504
$843$	0	0
$844$	11.4888	0.395460
$845$	−0.101289	−0.00348444
$846$	0	0
$847$	25.4598	0.874808
$848$	2.59771	0.0892058
$849$	0	0
$850$	28.9440	0.992770
$851$	−4.80875	−0.164842
$852$	0	0
$853$	1.08814	0.0372572	0.0186286	−	0.999826i	$-0.494070\pi$
$853$	1.08814	0.0372572	0.0186286	+	0.999826i	$0.494070\pi$
$854$	−10.1776	−0.348271
$855$	0	0
$856$	16.3396	0.558475
$857$	1.53174	0.0523232	0.0261616	−	0.999658i	$-0.491672\pi$
$857$	1.53174	0.0523232	0.0261616	+	0.999658i	$0.491672\pi$
$858$	0	0
$859$	10.9042	0.372047	0.186024	−	0.982545i	$-0.440440\pi$
$859$	10.9042	0.372047	0.186024	+	0.982545i	$0.440440\pi$
$860$	0.141386	0.00482122
$861$	0	0
$862$	0.222605	0.00758196
$863$	54.7487	1.86367	0.931834	−	0.362884i	$-0.118208\pi$
$863$	54.7487	1.86367	0.931834	+	0.362884i	$0.118208\pi$
$864$	0	0
$865$	−4.65943	−0.158426
$866$	−34.7923	−1.18229
$867$	0	0
$868$	0.314222	0.0106654
$869$	36.5131	1.23862
$870$	0	0
$871$	41.7270	1.41386
$872$	27.0419	0.915753
$873$	0	0
$874$	5.02160	0.169858
$875$	2.08746	0.0705690
$876$	0	0
$877$	−0.0143469	−0.000484459 0	−0.000242229	−	1.00000i	$-0.500077\pi$
$877$	−0.0143469	−0.000484459 0	−0.000242229		1.00000i	$0.500077\pi$
$878$	−27.0787	−0.913862
$879$	0	0
$880$	−2.14646	−0.0723570
$881$	56.3960	1.90003	0.950014	−	0.312206i	$-0.101068\pi$
$881$	56.3960	1.90003	0.950014	+	0.312206i	$0.101068\pi$
$882$	0	0
$883$	−26.5606	−0.893835	−0.446917	−	0.894575i	$-0.647478\pi$
$883$	−26.5606	−0.893835	−0.446917	+	0.894575i	$0.647478\pi$
$884$	−16.1306	−0.542531
$885$	0	0
$886$	20.6907	0.695119
$887$	29.3990	0.987121	0.493560	−	0.869712i	$-0.335695\pi$
$887$	29.3990	0.987121	0.493560	+	0.869712i	$0.335695\pi$
$888$	0	0
$889$	−13.1751	−0.441879
$890$	−0.118748	−0.00398045
$891$	0	0
$892$	0.573714	0.0192094
$893$	18.1261	0.606567
$894$	0	0
$895$	−2.09866	−0.0701504
$896$	−0.183089	−0.00611657
$897$	0	0
$898$	−6.46936	−0.215885
$899$	−3.07484	−0.102552
$900$	0	0
$901$	−8.22986	−0.274177
$902$	6.56515	0.218595
$903$	0	0
$904$	55.7741	1.85502
$905$	−2.26653	−0.0753421
$906$	0	0
$907$	−12.9258	−0.429196	−0.214598	−	0.976702i	$-0.568844\pi$
$907$	−12.9258	−0.429196	−0.214598	+	0.976702i	$0.568844\pi$
$908$	7.66212	0.254276
$909$	0	0
$910$	0.837073	0.0277487
$911$	−33.8152	−1.12035	−0.560174	−	0.828375i	$-0.689266\pi$
$911$	−33.8152	−1.12035	−0.560174	+	0.828375i	$0.689266\pi$
$912$	0	0
$913$	85.6643	2.83507
$914$	34.6198	1.14512
$915$	0	0
$916$	−10.1423	−0.335110
$917$	−13.0506	−0.430968
$918$	0	0
$919$	−2.96555	−0.0978246	−0.0489123	−	0.998803i	$-0.515575\pi$
$919$	−2.96555	−0.0978246	−0.0489123	+	0.998803i	$0.515575\pi$
$920$	−0.837408	−0.0276085
$921$	0	0
$922$	5.43242	0.178907
$923$	54.8930	1.80683
$924$	0	0
$925$	18.2817	0.601097
$926$	−19.1401	−0.628983
$927$	0	0
$928$	−34.2860	−1.12549
$929$	24.4362	0.801726	0.400863	−	0.916138i	$-0.368710\pi$
$929$	24.4362	0.801726	0.400863	+	0.916138i	$0.368710\pi$
$930$	0	0
$931$	−3.54285	−0.116112
$932$	7.86552	0.257644
$933$	0	0
$934$	−46.6921	−1.52781
$935$	6.80023	0.222391
$936$	0	0
$937$	35.9240	1.17359	0.586793	−	0.809737i	$-0.300390\pi$
$937$	35.9240	1.17359	0.586793	+	0.809737i	$0.300390\pi$
$938$	−12.3555	−0.403421
$939$	0	0
$940$	−0.877310	−0.0286147
$941$	42.9108	1.39885	0.699426	−	0.714705i	$-0.253439\pi$
$941$	42.9108	1.39885	0.699426	+	0.714705i	$0.253439\pi$
$942$	0	0
$943$	1.30362	0.0424518
$944$	17.4070	0.566550
$945$	0	0
$946$	5.41314	0.175996
$947$	−44.6039	−1.44943	−0.724716	−	0.689048i	$-0.758029\pi$
$947$	−44.6039	−1.44943	−0.724716	+	0.689048i	$0.758029\pi$
$948$	0	0
$949$	28.5384	0.926395
$950$	−19.0909	−0.619389
$951$	0	0
$952$	16.4566	0.533361
$953$	12.8800	0.417224	0.208612	−	0.977999i	$-0.433105\pi$
$953$	12.8800	0.417224	0.208612	+	0.977999i	$0.433105\pi$
$954$	0	0
$955$	−4.14499	−0.134129
$956$	−13.5438	−0.438037
$957$	0	0
$958$	3.10143	0.100203
$959$	−12.7699	−0.412361
$960$	0	0
$961$	−30.8524	−0.995238
$962$	14.7270	0.474816
$963$	0	0
$964$	−7.60336	−0.244888
$965$	−3.79403	−0.122134
$966$	0	0
$967$	49.0909	1.57866	0.789329	−	0.613970i	$-0.210429\pi$
$967$	49.0909	1.57866	0.789329	+	0.613970i	$0.210429\pi$
$968$	78.0025	2.50710
$969$	0	0
$970$	−0.832689	−0.0267360
$971$	5.24060	0.168179	0.0840895	−	0.996458i	$-0.473202\pi$
$971$	5.24060	0.168179	0.0840895	+	0.996458i	$0.473202\pi$
$972$	0	0
$973$	13.9759	0.448048
$974$	6.77614	0.217121
$975$	0	0
$976$	−15.8706	−0.508004
$977$	−44.0772	−1.41015	−0.705077	−	0.709130i	$-0.749088\pi$
$977$	−44.0772	−1.41015	−0.705077	+	0.709130i	$0.749088\pi$
$978$	0	0
$979$	3.14533	0.100525
$980$	0.171475	0.00547757
$981$	0	0
$982$	−2.32508	−0.0741963
$983$	54.1938	1.72851	0.864256	−	0.503052i	$-0.167789\pi$
$983$	54.1938	1.72851	0.864256	+	0.503052i	$0.167789\pi$
$984$	0	0
$985$	0.938341	0.0298980
$986$	−46.7390	−1.48847
$987$	0	0
$988$	10.6394	0.338485
$989$	1.07487	0.0341789
$990$	0	0
$991$	−0.908400	−0.0288563	−0.0144281	−	0.999896i	$-0.504593\pi$
$991$	−0.908400	−0.0288563	−0.0144281	+	0.999896i	$0.504593\pi$
$992$	1.64599	0.0522602
$993$	0	0
$994$	−16.2540	−0.515545
$995$	−2.13768	−0.0677689
$996$	0	0
$997$	36.8530	1.16715	0.583573	−	0.812061i	$-0.301654\pi$
$997$	36.8530	1.16715	0.583573	+	0.812061i	$0.301654\pi$
$998$	−27.3024	−0.864242
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2583.2.a.r.1.2		5
3.2	odd	2		287.2.a.e.1.4	✓	5
12.11	even	2		4592.2.a.bb.1.2		5
15.14	odd	2		7175.2.a.n.1.2		5
21.20	even	2		2009.2.a.n.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.a.e.1.4	✓	5	3.2	odd	2
2009.2.a.n.1.4		5	21.20	even	2
2583.2.a.r.1.2		5	1.1	even	1	trivial
4592.2.a.bb.1.2		5	12.11	even	2
7175.2.a.n.1.2		5	15.14	odd	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:	\( N \)	\(=\)	\( 2583 = 3^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2583.a (trivial)

\(f(q)\)	\(=\)	\(q-1.08727 q^{2} -0.817843 q^{4} -0.209668 q^{5} +1.00000 q^{7} +3.06376 q^{8} +O(q^{10})\)	\(q-1.08727 q^{2} -0.817843 q^{4} -0.209668 q^{5} +1.00000 q^{7} +3.06376 q^{8} +0.227965 q^{10} -6.03819 q^{11} +3.67193 q^{13} -1.08727 q^{14} -1.69545 q^{16} +5.37138 q^{17} -3.54285 q^{19} +0.171475 q^{20} +6.56515 q^{22} +1.30362 q^{23} -4.95604 q^{25} -3.99238 q^{26} -0.817843 q^{28} +8.00307 q^{29} -0.384208 q^{31} -4.28411 q^{32} -5.84014 q^{34} -0.209668 q^{35} -3.68876 q^{37} +3.85204 q^{38} -0.642370 q^{40} +1.00000 q^{41} +0.824527 q^{43} +4.93829 q^{44} -1.41739 q^{46} -5.11625 q^{47} +1.00000 q^{49} +5.38855 q^{50} -3.00307 q^{52} -1.53217 q^{53} +1.26601 q^{55} +3.06376 q^{56} -8.70150 q^{58} -10.2669 q^{59} +9.36070 q^{61} +0.417738 q^{62} +8.04887 q^{64} -0.769885 q^{65} +11.3638 q^{67} -4.39294 q^{68} +0.227965 q^{70} +14.9494 q^{71} +7.77203 q^{73} +4.01068 q^{74} +2.89750 q^{76} -6.03819 q^{77} -6.04703 q^{79} +0.355480 q^{80} -1.08727 q^{82} -14.1871 q^{83} -1.12620 q^{85} -0.896484 q^{86} -18.4996 q^{88} -0.520905 q^{89} +3.67193 q^{91} -1.06616 q^{92} +5.56275 q^{94} +0.742821 q^{95} -3.65270 q^{97} -1.08727 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 3 q^{4} + 5 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 5 * q + q^2 + 3 * q^4 + 5 * q^5 + 5 * q^7 + 3 * q^8	\( 5 q + q^{2} + 3 q^{4} + 5 q^{5} + 5 q^{7} + 3 q^{8} - 2 q^{11} + 5 q^{13} + q^{14} - q^{16} - 13 q^{17} + 23 q^{20} + q^{22} - 2 q^{23} + 22 q^{25} + 3 q^{28} + 5 q^{29} + 17 q^{31} + 12 q^{32} - 8 q^{34} + 5 q^{35} - 7 q^{37} + 3 q^{38} + 7 q^{40} + 5 q^{41} + q^{43} + 47 q^{44} - 24 q^{46} - 9 q^{47} + 5 q^{49} - 2 q^{50} + 20 q^{52} - 5 q^{53} + 33 q^{55} + 3 q^{56} - 27 q^{58} - 7 q^{59} + 22 q^{61} + 28 q^{62} - 3 q^{64} + 31 q^{65} - 3 q^{67} - 17 q^{68} + 24 q^{71} + 40 q^{73} + 5 q^{74} - 19 q^{76} - 2 q^{77} - 42 q^{79} - 24 q^{80} + q^{82} + 12 q^{83} - 23 q^{85} - 16 q^{86} + 26 q^{88} - 8 q^{89} + 5 q^{91} - 12 q^{92} - 23 q^{94} + 17 q^{95} + 16 q^{97} + q^{98}+O(q^{100}) \) 5 * q + q^2 + 3 * q^4 + 5 * q^5 + 5 * q^7 + 3 * q^8 - 2 * q^11 + 5 * q^13 + q^14 - q^16 - 13 * q^17 + 23 * q^20 + q^22 - 2 * q^23 + 22 * q^25 + 3 * q^28 + 5 * q^29 + 17 * q^31 + 12 * q^32 - 8 * q^34 + 5 * q^35 - 7 * q^37 + 3 * q^38 + 7 * q^40 + 5 * q^41 + q^43 + 47 * q^44 - 24 * q^46 - 9 * q^47 + 5 * q^49 - 2 * q^50 + 20 * q^52 - 5 * q^53 + 33 * q^55 + 3 * q^56 - 27 * q^58 - 7 * q^59 + 22 * q^61 + 28 * q^62 - 3 * q^64 + 31 * q^65 - 3 * q^67 - 17 * q^68 + 24 * q^71 + 40 * q^73 + 5 * q^74 - 19 * q^76 - 2 * q^77 - 42 * q^79 - 24 * q^80 + q^82 + 12 * q^83 - 23 * q^85 - 16 * q^86 + 26 * q^88 - 8 * q^89 + 5 * q^91 - 12 * q^92 - 23 * q^94 + 17 * q^95 + 16 * q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more