LMFDB - Embedded newform 2583.2.a.p.1.1

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:	$1$
Dimension:	$5$
Coefficient field:	5.5.626512.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 4x^{2} + 6x - 2 $ x^5 - 2x^4 - 6x^3 + 4x^2 + 6x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 861)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56695$ 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)

$f(q)$	$=$	$q-2.56695 q^{2} +4.58924 q^{4} -1.31287 q^{5} -1.00000 q^{7} -6.64645 q^{8} +O(q^{10})$	\(q-2.56695 q^{2} +4.58924 q^{4} -1.31287 q^{5} -1.00000 q^{7} -6.64645 q^{8} +3.37008 q^{10} +4.88262 q^{11} -1.82542 q^{13} +2.56695 q^{14} +7.88262 q^{16} -1.68713 q^{17} +1.82262 q^{19} -6.02509 q^{20} -12.5335 q^{22} -3.03931 q^{23} -3.27636 q^{25} +4.68575 q^{26} -4.58924 q^{28} +2.73736 q^{29} +2.17738 q^{31} -6.94142 q^{32} +4.33077 q^{34} +1.31287 q^{35} -3.96349 q^{37} -4.67857 q^{38} +8.72594 q^{40} +1.00000 q^{41} -7.49135 q^{43} +22.4075 q^{44} +7.80176 q^{46} +2.68015 q^{47} +1.00000 q^{49} +8.41027 q^{50} -8.37727 q^{52} +5.35306 q^{53} -6.41027 q^{55} +6.64645 q^{56} -7.02667 q^{58} -4.82103 q^{59} +12.7554 q^{61} -5.58924 q^{62} +2.05304 q^{64} +2.39654 q^{65} +4.06001 q^{67} -7.74263 q^{68} -3.37008 q^{70} -11.2580 q^{71} -4.33077 q^{73} +10.1741 q^{74} +8.36442 q^{76} -4.88262 q^{77} +6.58226 q^{79} -10.3489 q^{80} -2.56695 q^{82} -13.4974 q^{83} +2.21498 q^{85} +19.2299 q^{86} -32.4521 q^{88} +8.01653 q^{89} +1.82542 q^{91} -13.9481 q^{92} -6.87982 q^{94} -2.39286 q^{95} -11.3241 q^{97} -2.56695 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{2} + 7 q^{4} - 3 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) $ 5 * q - 3 * q^2 + 7 * q^4 - 3 * q^5 - 5 * q^7 - 3 * q^8	$ 5 q - 3 q^{2} + 7 q^{4} - 3 q^{5} - 5 q^{7} - 3 q^{8} - q^{10} - 4 q^{11} + 5 q^{13} + 3 q^{14} + 11 q^{16} - 12 q^{17} + 10 q^{19} - 9 q^{20} - 6 q^{22} - 9 q^{23} - 4 q^{25} - 13 q^{26} - 7 q^{28} - 7 q^{29} + 10 q^{31} - 9 q^{32} + 10 q^{34} + 3 q^{35} - 11 q^{37} - 7 q^{40} + 5 q^{41} - 2 q^{43} - 9 q^{46} + 7 q^{47} + 5 q^{49} + 10 q^{50} - 11 q^{52} + 9 q^{53} + 3 q^{56} - 31 q^{58} - 8 q^{59} - 6 q^{61} - 12 q^{62} - 29 q^{64} + 13 q^{65} - 9 q^{67} - 12 q^{68} + q^{70} - 4 q^{71} - 10 q^{73} + 17 q^{74} + 36 q^{76} + 4 q^{77} + 7 q^{79} - 9 q^{80} - 3 q^{82} - 50 q^{83} - 12 q^{85} + 8 q^{86} - 38 q^{88} - 8 q^{89} - 5 q^{91} + 17 q^{92} - 21 q^{94} - 36 q^{95} - 11 q^{97} - 3 q^{98}+O(q^{100}) $ 5 * q - 3 * q^2 + 7 * q^4 - 3 * q^5 - 5 * q^7 - 3 * q^8 - q^10 - 4 * q^11 + 5 * q^13 + 3 * q^14 + 11 * q^16 - 12 * q^17 + 10 * q^19 - 9 * q^20 - 6 * q^22 - 9 * q^23 - 4 * q^25 - 13 * q^26 - 7 * q^28 - 7 * q^29 + 10 * q^31 - 9 * q^32 + 10 * q^34 + 3 * q^35 - 11 * q^37 - 7 * q^40 + 5 * q^41 - 2 * q^43 - 9 * q^46 + 7 * q^47 + 5 * q^49 + 10 * q^50 - 11 * q^52 + 9 * q^53 + 3 * q^56 - 31 * q^58 - 8 * q^59 - 6 * q^61 - 12 * q^62 - 29 * q^64 + 13 * q^65 - 9 * q^67 - 12 * q^68 + q^70 - 4 * q^71 - 10 * q^73 + 17 * q^74 + 36 * q^76 + 4 * q^77 + 7 * q^79 - 9 * q^80 - 3 * q^82 - 50 * q^83 - 12 * q^85 + 8 * q^86 - 38 * q^88 - 8 * q^89 - 5 * q^91 + 17 * q^92 - 21 * q^94 - 36 * q^95 - 11 * 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 $

$2$	−2.56695	−1.81511	−0.907554	−	0.419935i	$-0.862053\pi$
$2$	−2.56695	−1.81511	−0.907554	+	0.419935i	$0.862053\pi$
$3$	0	0
$3$	0	0
$4$	4.58924	2.29462
$5$	−1.31287	−0.587135	−0.293567	−	0.955938i	$-0.594842\pi$
$5$	−1.31287	−0.587135	−0.293567	+	0.955938i	$0.594842\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−6.64645	−2.34987
$9$	0	0
$10$	3.37008	1.06571
$11$	4.88262	1.47217	0.736083	−	0.676891i	$-0.236673\pi$
$11$	4.88262	1.47217	0.736083	+	0.676891i	$0.236673\pi$
$12$	0	0
$13$	−1.82542	−0.506279	−0.253140	−	0.967430i	$-0.581463\pi$
$13$	−1.82542	−0.506279	−0.253140	+	0.967430i	$0.581463\pi$
$14$	2.56695	0.686047
$15$	0	0
$16$	7.88262	1.97066
$17$	−1.68713	−0.409188	−0.204594	−	0.978847i	$-0.565587\pi$
$17$	−1.68713	−0.409188	−0.204594	+	0.978847i	$0.565587\pi$
$18$	0	0
$19$	1.82262	0.418137	0.209068	−	0.977901i	$-0.432957\pi$
$19$	1.82262	0.418137	0.209068	+	0.977901i	$0.432957\pi$
$20$	−6.02509	−1.34725
$21$	0	0
$22$	−12.5335	−2.67214
$23$	−3.03931	−0.633740	−0.316870	−	0.948469i	$-0.602632\pi$
$23$	−3.03931	−0.633740	−0.316870	+	0.948469i	$0.602632\pi$
$24$	0	0
$25$	−3.27636	−0.655273
$26$	4.68575	0.918952
$27$	0	0
$28$	−4.58924	−0.867284
$29$	2.73736	0.508315	0.254158	−	0.967163i	$-0.418202\pi$
$29$	2.73736	0.508315	0.254158	+	0.967163i	$0.418202\pi$
$30$	0	0
$31$	2.17738	0.391070	0.195535	−	0.980697i	$-0.437356\pi$
$31$	2.17738	0.391070	0.195535	+	0.980697i	$0.437356\pi$
$32$	−6.94142	−1.22708
$33$	0	0
$34$	4.33077	0.742721
$35$	1.31287	0.221916
$36$	0	0
$37$	−3.96349	−0.651594	−0.325797	−	0.945440i	$-0.605633\pi$
$37$	−3.96349	−0.651594	−0.325797	+	0.945440i	$0.605633\pi$
$38$	−4.67857	−0.758964
$39$	0	0
$40$	8.72594	1.37969
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−7.49135	−1.14242	−0.571210	−	0.820804i	$-0.693526\pi$
$43$	−7.49135	−1.14242	−0.571210	+	0.820804i	$0.693526\pi$
$44$	22.4075	3.37806
$45$	0	0
$46$	7.80176	1.15031
$47$	2.68015	0.390941	0.195470	−	0.980710i	$-0.437377\pi$
$47$	2.68015	0.390941	0.195470	+	0.980710i	$0.437377\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	8.41027	1.18939
$51$	0	0
$52$	−8.37727	−1.16172
$53$	5.35306	0.735299	0.367650	−	0.929964i	$-0.380163\pi$
$53$	5.35306	0.735299	0.367650	+	0.929964i	$0.380163\pi$
$54$	0	0
$55$	−6.41027	−0.864360
$56$	6.64645	0.888169
$57$	0	0
$58$	−7.02667	−0.922647
$59$	−4.82103	−0.627644	−0.313822	−	0.949482i	$-0.601610\pi$
$59$	−4.82103	−0.627644	−0.313822	+	0.949482i	$0.601610\pi$
$60$	0	0
$61$	12.7554	1.63316	0.816582	−	0.577230i	$-0.195866\pi$
$61$	12.7554	1.63316	0.816582	+	0.577230i	$0.195866\pi$
$62$	−5.58924	−0.709834
$63$	0	0
$64$	2.05304	0.256629
$65$	2.39654	0.297254
$66$	0	0
$67$	4.06001	0.496009	0.248004	−	0.968759i	$-0.420225\pi$
$67$	4.06001	0.496009	0.248004	+	0.968759i	$0.420225\pi$
$68$	−7.74263	−0.938931
$69$	0	0
$70$	−3.37008	−0.402802
$71$	−11.2580	−1.33608	−0.668038	−	0.744128i	$-0.732865\pi$
$71$	−11.2580	−1.33608	−0.668038	+	0.744128i	$0.732865\pi$
$72$	0	0
$73$	−4.33077	−0.506879	−0.253439	−	0.967351i	$-0.581562\pi$
$73$	−4.33077	−0.506879	−0.253439	+	0.967351i	$0.581562\pi$
$74$	10.1741	1.18271
$75$	0	0
$76$	8.36442	0.959465
$77$	−4.88262	−0.556427
$78$	0	0
$79$	6.58226	0.740563	0.370281	−	0.928920i	$-0.379261\pi$
$79$	6.58226	0.740563	0.370281	+	0.928920i	$0.379261\pi$
$80$	−10.3489	−1.15704
$81$	0	0
$82$	−2.56695	−0.283472
$83$	−13.4974	−1.48154	−0.740768	−	0.671760i	$-0.765538\pi$
$83$	−13.4974	−1.48154	−0.740768	+	0.671760i	$0.765538\pi$
$84$	0	0
$85$	2.21498	0.240249
$86$	19.2299	2.07362
$87$	0	0
$88$	−32.4521	−3.45940
$89$	8.01653	0.849750	0.424875	−	0.905252i	$-0.360318\pi$
$89$	8.01653	0.849750	0.424875	+	0.905252i	$0.360318\pi$
$90$	0	0
$91$	1.82542	0.191356
$92$	−13.9481	−1.45419
$93$	0	0
$94$	−6.87982	−0.709600
$95$	−2.39286	−0.245503
$96$	0	0
$97$	−11.3241	−1.14979	−0.574896	−	0.818227i	$-0.694957\pi$
$97$	−11.3241	−1.14979	−0.574896	+	0.818227i	$0.694957\pi$
$98$	−2.56695	−0.259301
$99$	0	0
$100$	−15.0360	−1.50360
$101$	−4.01093	−0.399102	−0.199551	−	0.979887i	$-0.563948\pi$
$101$	−4.01093	−0.399102	−0.199551	+	0.979887i	$0.563948\pi$
$102$	0	0
$103$	10.5595	1.04046	0.520228	−	0.854027i	$-0.325847\pi$
$103$	10.5595	1.04046	0.520228	+	0.854027i	$0.325847\pi$
$104$	12.1325	1.18969
$105$	0	0
$106$	−13.7410	−1.33465
$107$	−4.86193	−0.470020	−0.235010	−	0.971993i	$-0.575512\pi$
$107$	−4.86193	−0.470020	−0.235010	+	0.971993i	$0.575512\pi$
$108$	0	0
$109$	−17.5419	−1.68021	−0.840106	−	0.542423i	$-0.817507\pi$
$109$	−17.5419	−1.68021	−0.840106	+	0.542423i	$0.817507\pi$
$110$	16.4548	1.56891
$111$	0	0
$112$	−7.88262	−0.744838
$113$	−0.890970	−0.0838154	−0.0419077	−	0.999121i	$-0.513344\pi$
$113$	−0.890970	−0.0838154	−0.0419077	+	0.999121i	$0.513344\pi$
$114$	0	0
$115$	3.99023	0.372091
$116$	12.5624	1.16639
$117$	0	0
$118$	12.3753	1.13924
$119$	1.68713	0.154659
$120$	0	0
$121$	12.8400	1.16727
$122$	−32.7425	−2.96437
$123$	0	0
$124$	9.99253	0.897356
$125$	10.8658	0.971868
$126$	0	0
$127$	−1.34061	−0.118960	−0.0594798	−	0.998230i	$-0.518944\pi$
$127$	−1.34061	−0.118960	−0.0594798	+	0.998230i	$0.518944\pi$
$128$	8.61280	0.761271
$129$	0	0
$130$	−6.15180	−0.539549
$131$	−9.55026	−0.834411	−0.417205	−	0.908812i	$-0.636990\pi$
$131$	−9.55026	−0.834411	−0.417205	+	0.908812i	$0.636990\pi$
$132$	0	0
$133$	−1.82262	−0.158041
$134$	−10.4218	−0.900310
$135$	0	0
$136$	11.2134	0.961541
$137$	16.5363	1.41279	0.706394	−	0.707819i	$-0.250321\pi$
$137$	16.5363	1.41279	0.706394	+	0.707819i	$0.250321\pi$
$138$	0	0
$139$	17.4941	1.48383	0.741917	−	0.670492i	$-0.233917\pi$
$139$	17.4941	1.48383	0.741917	+	0.670492i	$0.233917\pi$
$140$	6.02509	0.509213
$141$	0	0
$142$	28.8987	2.42512
$143$	−8.91282	−0.745328
$144$	0	0
$145$	−3.59381	−0.298450
$146$	11.1169	0.920039
$147$	0	0
$148$	−18.1894	−1.49516
$149$	0.260929	0.0213761	0.0106881	−	0.999943i	$-0.496598\pi$
$149$	0.260929	0.0213761	0.0106881	+	0.999943i	$0.496598\pi$
$150$	0	0
$151$	−17.9296	−1.45909	−0.729544	−	0.683934i	$-0.760267\pi$
$151$	−17.9296	−1.45909	−0.729544	+	0.683934i	$0.760267\pi$
$152$	−12.1139	−0.982569
$153$	0	0
$154$	12.5335	1.00997
$155$	−2.85863	−0.229611
$156$	0	0
$157$	22.1072	1.76435	0.882174	−	0.470923i	$-0.156079\pi$
$157$	22.1072	1.76435	0.882174	+	0.470923i	$0.156079\pi$
$158$	−16.8964	−1.34420
$159$	0	0
$160$	9.11320	0.720462
$161$	3.03931	0.239531
$162$	0	0
$163$	−3.11513	−0.243996	−0.121998	−	0.992530i	$-0.538930\pi$
$163$	−3.11513	−0.243996	−0.121998	+	0.992530i	$0.538930\pi$
$164$	4.58924	0.358359
$165$	0	0
$166$	34.6473	2.68915
$167$	19.6015	1.51681	0.758406	−	0.651783i	$-0.225979\pi$
$167$	19.6015	1.51681	0.758406	+	0.651783i	$0.225979\pi$
$168$	0	0
$169$	−9.66786	−0.743681
$170$	−5.68575	−0.436077
$171$	0	0
$172$	−34.3796	−2.62142
$173$	3.01970	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$173$	3.01970	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$174$	0	0
$175$	3.27636	0.247670
$176$	38.4879	2.90113
$177$	0	0
$178$	−20.5780	−1.54239
$179$	−22.2192	−1.66074	−0.830371	−	0.557211i	$-0.811871\pi$
$179$	−22.2192	−1.66074	−0.830371	+	0.557211i	$0.811871\pi$
$180$	0	0
$181$	−6.88350	−0.511647	−0.255823	−	0.966724i	$-0.582347\pi$
$181$	−6.88350	−0.511647	−0.255823	+	0.966724i	$0.582347\pi$
$182$	−4.68575	−0.347331
$183$	0	0
$184$	20.2006	1.48921
$185$	5.20356	0.382573
$186$	0	0
$187$	−8.23761	−0.602393
$188$	12.2999	0.897060
$189$	0	0
$190$	6.14236	0.445614
$191$	−2.81366	−0.203589	−0.101795	−	0.994805i	$-0.532458\pi$
$191$	−2.81366	−0.203589	−0.101795	+	0.994805i	$0.532458\pi$
$192$	0	0
$193$	19.3710	1.39436	0.697178	−	0.716898i	$-0.254439\pi$
$193$	19.3710	1.39436	0.697178	+	0.716898i	$0.254439\pi$
$194$	29.0685	2.08700
$195$	0	0
$196$	4.58924	0.327803
$197$	−11.2985	−0.804984	−0.402492	−	0.915423i	$-0.631856\pi$
$197$	−11.2985	−0.804984	−0.402492	+	0.915423i	$0.631856\pi$
$198$	0	0
$199$	−15.4956	−1.09845	−0.549226	−	0.835674i	$-0.685077\pi$
$199$	−15.4956	−1.09845	−0.549226	+	0.835674i	$0.685077\pi$
$200$	21.7762	1.53981
$201$	0	0
$202$	10.2959	0.724414
$203$	−2.73736	−0.192125
$204$	0	0
$205$	−1.31287	−0.0916950
$206$	−27.1057	−1.88854
$207$	0	0
$208$	−14.3891	−0.997703
$209$	8.89915	0.615567
$210$	0	0
$211$	−22.5367	−1.55149	−0.775745	−	0.631046i	$-0.782626\pi$
$211$	−22.5367	−1.55149	−0.775745	+	0.631046i	$0.782626\pi$
$212$	24.5665	1.68723
$213$	0	0
$214$	12.4803	0.853137
$215$	9.83519	0.670754
$216$	0	0
$217$	−2.17738	−0.147810
$218$	45.0292	3.04977
$219$	0	0
$220$	−29.4182	−1.98338
$221$	3.07971	0.207164
$222$	0	0
$223$	4.61752	0.309212	0.154606	−	0.987976i	$-0.450589\pi$
$223$	4.61752	0.309212	0.154606	+	0.987976i	$0.450589\pi$
$224$	6.94142	0.463793
$225$	0	0
$226$	2.28708	0.152134
$227$	−17.9390	−1.19065	−0.595327	−	0.803484i	$-0.702977\pi$
$227$	−17.9390	−1.19065	−0.595327	+	0.803484i	$0.702977\pi$
$228$	0	0
$229$	−18.1453	−1.19907	−0.599536	−	0.800348i	$-0.704648\pi$
$229$	−18.1453	−1.19907	−0.599536	+	0.800348i	$0.704648\pi$
$230$	−10.2427	−0.675385
$231$	0	0
$232$	−18.1937	−1.19448
$233$	−25.5795	−1.67577	−0.837885	−	0.545847i	$-0.816208\pi$
$233$	−25.5795	−1.67577	−0.837885	+	0.545847i	$0.816208\pi$
$234$	0	0
$235$	−3.51870	−0.229535
$236$	−22.1248	−1.44020
$237$	0	0
$238$	−4.33077	−0.280722
$239$	−12.3963	−0.801851	−0.400926	−	0.916111i	$-0.631311\pi$
$239$	−12.3963	−0.801851	−0.400926	+	0.916111i	$0.631311\pi$
$240$	0	0
$241$	13.1898	0.849630	0.424815	−	0.905280i	$-0.360339\pi$
$241$	13.1898	0.849630	0.424815	+	0.905280i	$0.360339\pi$
$242$	−32.9597	−2.11873
$243$	0	0
$244$	58.5376	3.74749
$245$	−1.31287	−0.0838764
$246$	0	0
$247$	−3.32703	−0.211694
$248$	−14.4719	−0.918964
$249$	0	0
$250$	−27.8920	−1.76405
$251$	2.11974	0.133797	0.0668985	−	0.997760i	$-0.478690\pi$
$251$	2.11974	0.133797	0.0668985	+	0.997760i	$0.478690\pi$
$252$	0	0
$253$	−14.8398	−0.932970
$254$	3.44127	0.215925
$255$	0	0
$256$	−26.2147	−1.63842
$257$	−20.9577	−1.30730	−0.653652	−	0.756796i	$-0.726764\pi$
$257$	−20.9577	−1.30730	−0.653652	+	0.756796i	$0.726764\pi$
$258$	0	0
$259$	3.96349	0.246279
$260$	10.9983	0.682085
$261$	0	0
$262$	24.5151	1.51455
$263$	−11.3323	−0.698777	−0.349389	−	0.936978i	$-0.613611\pi$
$263$	−11.3323	−0.698777	−0.349389	+	0.936978i	$0.613611\pi$
$264$	0	0
$265$	−7.02789	−0.431720
$266$	4.67857	0.286861
$267$	0	0
$268$	18.6323	1.13815
$269$	−15.6834	−0.956234	−0.478117	−	0.878296i	$-0.658680\pi$
$269$	−15.6834	−0.956234	−0.478117	+	0.878296i	$0.658680\pi$
$270$	0	0
$271$	−17.6977	−1.07506	−0.537529	−	0.843245i	$-0.680642\pi$
$271$	−17.6977	−1.07506	−0.537529	+	0.843245i	$0.680642\pi$
$272$	−13.2990	−0.806370
$273$	0	0
$274$	−42.4478	−2.56436
$275$	−15.9973	−0.964671
$276$	0	0
$277$	7.95465	0.477949	0.238974	−	0.971026i	$-0.423189\pi$
$277$	7.95465	0.477949	0.238974	+	0.971026i	$0.423189\pi$
$278$	−44.9066	−2.69332
$279$	0	0
$280$	−8.72594	−0.521475
$281$	1.53517	0.0915803	0.0457902	−	0.998951i	$-0.485419\pi$
$281$	1.53517	0.0915803	0.0457902	+	0.998951i	$0.485419\pi$
$282$	0	0
$283$	−23.8827	−1.41968	−0.709840	−	0.704363i	$-0.751233\pi$
$283$	−23.8827	−1.41968	−0.709840	+	0.704363i	$0.751233\pi$
$284$	−51.6655	−3.06578
$285$	0	0
$286$	22.8788	1.35285
$287$	−1.00000	−0.0590281
$288$	0	0
$289$	−14.1536	−0.832565
$290$	9.22513	0.541718
$291$	0	0
$292$	−19.8749	−1.16309
$293$	−21.4580	−1.25359	−0.626794	−	0.779185i	$-0.715633\pi$
$293$	−21.4580	−1.25359	−0.626794	+	0.779185i	$0.715633\pi$
$294$	0	0
$295$	6.32940	0.368512
$296$	26.3431	1.53116
$297$	0	0
$298$	−0.669792	−0.0388000
$299$	5.54800	0.320849
$300$	0	0
$301$	7.49135	0.431794
$302$	46.0243	2.64840
$303$	0	0
$304$	14.3670	0.824004
$305$	−16.7462	−0.958887
$306$	0	0
$307$	−5.76942	−0.329278	−0.164639	−	0.986354i	$-0.552646\pi$
$307$	−5.76942	−0.329278	−0.164639	+	0.986354i	$0.552646\pi$
$308$	−22.4075	−1.27679
$309$	0	0
$310$	7.33796	0.416768
$311$	4.30612	0.244177	0.122089	−	0.992519i	$-0.461041\pi$
$311$	4.30612	0.244177	0.122089	+	0.992519i	$0.461041\pi$
$312$	0	0
$313$	−17.7494	−1.00325	−0.501627	−	0.865084i	$-0.667265\pi$
$313$	−17.7494	−1.00325	−0.501627	+	0.865084i	$0.667265\pi$
$314$	−56.7482	−3.20248
$315$	0	0
$316$	30.2076	1.69931
$317$	−25.2987	−1.42092	−0.710460	−	0.703738i	$-0.751513\pi$
$317$	−25.2987	−1.42092	−0.710460	+	0.703738i	$0.751513\pi$
$318$	0	0
$319$	13.3655	0.748325
$320$	−2.69537	−0.150676
$321$	0	0
$322$	−7.80176	−0.434775
$323$	−3.07498	−0.171097
$324$	0	0
$325$	5.98073	0.331751
$326$	7.99638	0.442878
$327$	0	0
$328$	−6.64645	−0.366989
$329$	−2.68015	−0.147762
$330$	0	0
$331$	24.3413	1.33792	0.668958	−	0.743300i	$-0.266740\pi$
$331$	24.3413	1.33792	0.668958	+	0.743300i	$0.266740\pi$
$332$	−61.9430	−3.39956
$333$	0	0
$334$	−50.3162	−2.75318
$335$	−5.33028	−0.291224
$336$	0	0
$337$	−20.8089	−1.13353	−0.566766	−	0.823879i	$-0.691806\pi$
$337$	−20.8089	−1.13353	−0.566766	+	0.823879i	$0.691806\pi$
$338$	24.8169	1.34986
$339$	0	0
$340$	10.1651	0.551279
$341$	10.6313	0.575720
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	49.7908	2.68454
$345$	0	0
$346$	−7.75142	−0.416719
$347$	−22.2830	−1.19622	−0.598108	−	0.801416i	$-0.704081\pi$
$347$	−22.2830	−1.19622	−0.598108	+	0.801416i	$0.704081\pi$
$348$	0	0
$349$	20.8218	1.11457	0.557283	−	0.830323i	$-0.311844\pi$
$349$	20.8218	1.11457	0.557283	+	0.830323i	$0.311844\pi$
$350$	−8.41027	−0.449548
$351$	0	0
$352$	−33.8923	−1.80647
$353$	6.17551	0.328689	0.164345	−	0.986403i	$-0.447449\pi$
$353$	6.17551	0.328689	0.164345	+	0.986403i	$0.447449\pi$
$354$	0	0
$355$	14.7803	0.784456
$356$	36.7897	1.94985
$357$	0	0
$358$	57.0356	3.01443
$359$	10.6861	0.563990	0.281995	−	0.959416i	$-0.409004\pi$
$359$	10.6861	0.563990	0.281995	+	0.959416i	$0.409004\pi$
$360$	0	0
$361$	−15.6781	−0.825162
$362$	17.6696	0.928694
$363$	0	0
$364$	8.37727	0.439088
$365$	5.68575	0.297606
$366$	0	0
$367$	15.0205	0.784063	0.392031	−	0.919952i	$-0.371772\pi$
$367$	15.0205	0.784063	0.392031	+	0.919952i	$0.371772\pi$
$368$	−23.9577	−1.24888
$369$	0	0
$370$	−13.3573	−0.694412
$371$	−5.35306	−0.277917
$372$	0	0
$373$	−18.3738	−0.951357	−0.475679	−	0.879619i	$-0.657797\pi$
$373$	−18.3738	−0.951357	−0.475679	+	0.879619i	$0.657797\pi$
$374$	21.1455	1.09341
$375$	0	0
$376$	−17.8135	−0.918661
$377$	−4.99683	−0.257350
$378$	0	0
$379$	18.6617	0.958588	0.479294	−	0.877654i	$-0.340893\pi$
$379$	18.6617	0.958588	0.479294	+	0.877654i	$0.340893\pi$
$380$	−10.9814	−0.563335
$381$	0	0
$382$	7.22252	0.369536
$383$	36.3994	1.85992	0.929962	−	0.367655i	$-0.119839\pi$
$383$	36.3994	1.85992	0.929962	+	0.367655i	$0.119839\pi$
$384$	0	0
$385$	6.41027	0.326697
$386$	−49.7244	−2.53091
$387$	0	0
$388$	−51.9691	−2.63833
$389$	−19.5262	−0.990016	−0.495008	−	0.868888i	$-0.664835\pi$
$389$	−19.5262	−0.990016	−0.495008	+	0.868888i	$0.664835\pi$
$390$	0	0
$391$	5.12770	0.259319
$392$	−6.64645	−0.335696
$393$	0	0
$394$	29.0027	1.46113
$395$	−8.64168	−0.434810
$396$	0	0
$397$	−17.0369	−0.855057	−0.427528	−	0.904002i	$-0.640616\pi$
$397$	−17.0369	−0.855057	−0.427528	+	0.904002i	$0.640616\pi$
$398$	39.7764	1.99381
$399$	0	0
$400$	−25.8263	−1.29132
$401$	4.68642	0.234028	0.117014	−	0.993130i	$-0.462668\pi$
$401$	4.68642	0.234028	0.117014	+	0.993130i	$0.462668\pi$
$402$	0	0
$403$	−3.97463	−0.197991
$404$	−18.4071	−0.915787
$405$	0	0
$406$	7.02667	0.348728
$407$	−19.3522	−0.959255
$408$	0	0
$409$	26.3484	1.30285	0.651423	−	0.758714i	$-0.274172\pi$
$409$	26.3484	1.30285	0.651423	+	0.758714i	$0.274172\pi$
$410$	3.37008	0.166436
$411$	0	0
$412$	48.4600	2.38745
$413$	4.82103	0.237227
$414$	0	0
$415$	17.7204	0.869862
$416$	12.6710	0.621246
$417$	0	0
$418$	−22.8437	−1.11732
$419$	11.5944	0.566425	0.283213	−	0.959057i	$-0.408600\pi$
$419$	11.5944	0.566425	0.283213	+	0.959057i	$0.408600\pi$
$420$	0	0
$421$	−37.9490	−1.84952	−0.924761	−	0.380549i	$-0.875735\pi$
$421$	−37.9490	−1.84952	−0.924761	+	0.380549i	$0.875735\pi$
$422$	57.8506	2.81612
$423$	0	0
$424$	−35.5788	−1.72786
$425$	5.52764	0.268130
$426$	0	0
$427$	−12.7554	−0.617278
$428$	−22.3125	−1.07852
$429$	0	0
$430$	−25.2464	−1.21749
$431$	−39.1044	−1.88359	−0.941797	−	0.336183i	$-0.890864\pi$
$431$	−39.1044	−1.88359	−0.941797	+	0.336183i	$0.890864\pi$
$432$	0	0
$433$	−24.7981	−1.19172	−0.595861	−	0.803088i	$-0.703189\pi$
$433$	−24.7981	−1.19172	−0.595861	+	0.803088i	$0.703189\pi$
$434$	5.58924	0.268292
$435$	0	0
$436$	−80.5040	−3.85544
$437$	−5.53949	−0.264990
$438$	0	0
$439$	8.41774	0.401757	0.200878	−	0.979616i	$-0.435620\pi$
$439$	8.41774	0.401757	0.200878	+	0.979616i	$0.435620\pi$
$440$	42.6055	2.03114
$441$	0	0
$442$	−7.90546	−0.376024
$443$	30.7906	1.46291	0.731453	−	0.681892i	$-0.238842\pi$
$443$	30.7906	1.46291	0.731453	+	0.681892i	$0.238842\pi$
$444$	0	0
$445$	−10.5247	−0.498918
$446$	−11.8530	−0.561254
$447$	0	0
$448$	−2.05304	−0.0969968
$449$	2.60695	0.123030	0.0615148	−	0.998106i	$-0.480407\pi$
$449$	2.60695	0.123030	0.0615148	+	0.998106i	$0.480407\pi$
$450$	0	0
$451$	4.88262	0.229914
$452$	−4.08887	−0.192324
$453$	0	0
$454$	46.0485	2.16116
$455$	−2.39654	−0.112352
$456$	0	0
$457$	−26.4155	−1.23566	−0.617832	−	0.786310i	$-0.711989\pi$
$457$	−26.4155	−1.23566	−0.617832	+	0.786310i	$0.711989\pi$
$458$	46.5780	2.17645
$459$	0	0
$460$	18.3121	0.853806
$461$	13.7760	0.641610	0.320805	−	0.947145i	$-0.396047\pi$
$461$	13.7760	0.641610	0.320805	+	0.947145i	$0.396047\pi$
$462$	0	0
$463$	−24.7308	−1.14934	−0.574668	−	0.818387i	$-0.694869\pi$
$463$	−24.7308	−1.14934	−0.574668	+	0.818387i	$0.694869\pi$
$464$	21.5776	1.00171
$465$	0	0
$466$	65.6614	3.04170
$467$	−9.06714	−0.419577	−0.209789	−	0.977747i	$-0.567278\pi$
$467$	−9.06714	−0.419577	−0.209789	+	0.977747i	$0.567278\pi$
$468$	0	0
$469$	−4.06001	−0.187474
$470$	9.03234	0.416631
$471$	0	0
$472$	32.0427	1.47488
$473$	−36.5774	−1.68183
$474$	0	0
$475$	−5.97155	−0.273994
$476$	7.74263	0.354883
$477$	0	0
$478$	31.8207	1.45545
$479$	−5.89718	−0.269449	−0.134725	−	0.990883i	$-0.543015\pi$
$479$	−5.89718	−0.269449	−0.134725	+	0.990883i	$0.543015\pi$
$480$	0	0
$481$	7.23502	0.329889
$482$	−33.8576	−1.54217
$483$	0	0
$484$	58.9259	2.67845
$485$	14.8672	0.675083
$486$	0	0
$487$	24.7646	1.12219	0.561095	−	0.827751i	$-0.310380\pi$
$487$	24.7646	1.12219	0.561095	+	0.827751i	$0.310380\pi$
$488$	−84.7782	−3.83773
$489$	0	0
$490$	3.37008	0.152245
$491$	−3.02036	−0.136307	−0.0681535	−	0.997675i	$-0.521711\pi$
$491$	−3.02036	−0.136307	−0.0681535	+	0.997675i	$0.521711\pi$
$492$	0	0
$493$	−4.61828	−0.207997
$494$	8.54033	0.384248
$495$	0	0
$496$	17.1635	0.770664
$497$	11.2580	0.504989
$498$	0	0
$499$	31.7421	1.42097	0.710486	−	0.703711i	$-0.248475\pi$
$499$	31.7421	1.42097	0.710486	+	0.703711i	$0.248475\pi$
$500$	49.8658	2.23007
$501$	0	0
$502$	−5.44127	−0.242856
$503$	21.7388	0.969285	0.484642	−	0.874712i	$-0.338950\pi$
$503$	21.7388	0.969285	0.484642	+	0.874712i	$0.338950\pi$
$504$	0	0
$505$	5.26584	0.234327
$506$	38.0930	1.69344
$507$	0	0
$508$	−6.15236	−0.272967
$509$	28.9296	1.28228	0.641140	−	0.767424i	$-0.278462\pi$
$509$	28.9296	1.28228	0.641140	+	0.767424i	$0.278462\pi$
$510$	0	0
$511$	4.33077	0.191582
$512$	50.0663	2.21264
$513$	0	0
$514$	53.7973	2.37290
$515$	−13.8633	−0.610888
$516$	0	0
$517$	13.0862	0.575530
$518$	−10.1741	−0.447024
$519$	0	0
$520$	−15.9285	−0.698510
$521$	−34.9796	−1.53248	−0.766241	−	0.642553i	$-0.777875\pi$
$521$	−34.9796	−1.53248	−0.766241	+	0.642553i	$0.777875\pi$
$522$	0	0
$523$	−39.6515	−1.73384	−0.866920	−	0.498446i	$-0.833904\pi$
$523$	−39.6515	−1.73384	−0.866920	+	0.498446i	$0.833904\pi$
$524$	−43.8284	−1.91465
$525$	0	0
$526$	29.0894	1.26836
$527$	−3.67352	−0.160021
$528$	0	0
$529$	−13.7626	−0.598374
$530$	18.0402	0.783618
$531$	0	0
$532$	−8.36442	−0.362644
$533$	−1.82542	−0.0790676
$534$	0	0
$535$	6.38309	0.275965
$536$	−26.9846	−1.16556
$537$	0	0
$538$	40.2585	1.73567
$539$	4.88262	0.210310
$540$	0	0
$541$	8.54367	0.367321	0.183661	−	0.982990i	$-0.441205\pi$
$541$	8.54367	0.367321	0.183661	+	0.982990i	$0.441205\pi$
$542$	45.4291	1.95135
$543$	0	0
$544$	11.7111	0.502107
$545$	23.0303	0.986510
$546$	0	0
$547$	10.7543	0.459819	0.229910	−	0.973212i	$-0.426157\pi$
$547$	10.7543	0.459819	0.229910	+	0.973212i	$0.426157\pi$
$548$	75.8888	3.24181
$549$	0	0
$550$	41.0642	1.75098
$551$	4.98916	0.212545
$552$	0	0
$553$	−6.58226	−0.279906
$554$	−20.4192	−0.867528
$555$	0	0
$556$	80.2848	3.40483
$557$	26.1476	1.10791	0.553955	−	0.832547i	$-0.313118\pi$
$557$	26.1476	1.10791	0.553955	+	0.832547i	$0.313118\pi$
$558$	0	0
$559$	13.6748	0.578384
$560$	10.3489	0.437320
$561$	0	0
$562$	−3.94070	−0.166228
$563$	−12.8165	−0.540152	−0.270076	−	0.962839i	$-0.587049\pi$
$563$	−12.8165	−0.540152	−0.270076	+	0.962839i	$0.587049\pi$
$564$	0	0
$565$	1.16973	0.0492109
$566$	61.3058	2.57687
$567$	0	0
$568$	74.8255	3.13961
$569$	−22.9583	−0.962462	−0.481231	−	0.876594i	$-0.659810\pi$
$569$	−22.9583	−0.962462	−0.481231	+	0.876594i	$0.659810\pi$
$570$	0	0
$571$	−39.7733	−1.66446	−0.832230	−	0.554430i	$-0.812936\pi$
$571$	−39.7733	−1.66446	−0.832230	+	0.554430i	$0.812936\pi$
$572$	−40.9031	−1.71024
$573$	0	0
$574$	2.56695	0.107142
$575$	9.95788	0.415272
$576$	0	0
$577$	19.2880	0.802970	0.401485	−	0.915866i	$-0.368494\pi$
$577$	19.2880	0.802970	0.401485	+	0.915866i	$0.368494\pi$
$578$	36.3316	1.51120
$579$	0	0
$580$	−16.4928	−0.684828
$581$	13.4974	0.559968
$582$	0	0
$583$	26.1370	1.08248
$584$	28.7842	1.19110
$585$	0	0
$586$	55.0816	2.27540
$587$	−21.7390	−0.897264	−0.448632	−	0.893717i	$-0.648089\pi$
$587$	−21.7390	−0.897264	−0.448632	+	0.893717i	$0.648089\pi$
$588$	0	0
$589$	3.96854	0.163521
$590$	−16.2473	−0.668889
$591$	0	0
$592$	−31.2427	−1.28407
$593$	−21.3404	−0.876347	−0.438173	−	0.898890i	$-0.644374\pi$
$593$	−21.3404	−0.876347	−0.438173	+	0.898890i	$0.644374\pi$
$594$	0	0
$595$	−2.21498	−0.0908055
$596$	1.19746	0.0490501
$597$	0	0
$598$	−14.2415	−0.582376
$599$	13.7030	0.559891	0.279945	−	0.960016i	$-0.409684\pi$
$599$	13.7030	0.559891	0.279945	+	0.960016i	$0.409684\pi$
$600$	0	0
$601$	−15.6838	−0.639757	−0.319878	−	0.947459i	$-0.603642\pi$
$601$	−15.6838	−0.639757	−0.319878	+	0.947459i	$0.603642\pi$
$602$	−19.2299	−0.783753
$603$	0	0
$604$	−82.2830	−3.34805
$605$	−16.8573	−0.685347
$606$	0	0
$607$	30.8537	1.25231	0.626157	−	0.779697i	$-0.284627\pi$
$607$	30.8537	1.25231	0.626157	+	0.779697i	$0.284627\pi$
$608$	−12.6515	−0.513088
$609$	0	0
$610$	42.9868	1.74048
$611$	−4.89240	−0.197925
$612$	0	0
$613$	13.2957	0.537008	0.268504	−	0.963279i	$-0.413471\pi$
$613$	13.2957	0.537008	0.268504	+	0.963279i	$0.413471\pi$
$614$	14.8098	0.597676
$615$	0	0
$616$	32.4521	1.30753
$617$	10.0553	0.404813	0.202406	−	0.979302i	$-0.435124\pi$
$617$	10.0553	0.404813	0.202406	+	0.979302i	$0.435124\pi$
$618$	0	0
$619$	36.1745	1.45398	0.726988	−	0.686650i	$-0.240919\pi$
$619$	36.1745	1.45398	0.726988	+	0.686650i	$0.240919\pi$
$620$	−13.1189	−0.526869
$621$	0	0
$622$	−11.0536	−0.443208
$623$	−8.01653	−0.321175
$624$	0	0
$625$	2.11638	0.0846553
$626$	45.5618	1.82102
$627$	0	0
$628$	101.455	4.04851
$629$	6.68691	0.266625
$630$	0	0
$631$	−40.8063	−1.62447	−0.812236	−	0.583329i	$-0.801750\pi$
$631$	−40.8063	−1.62447	−0.812236	+	0.583329i	$0.801750\pi$
$632$	−43.7487	−1.74023
$633$	0	0
$634$	64.9406	2.57912
$635$	1.76005	0.0698453
$636$	0	0
$637$	−1.82542	−0.0723256
$638$	−34.3086	−1.35829
$639$	0	0
$640$	−11.3075	−0.446969
$641$	34.5505	1.36466	0.682331	−	0.731044i	$-0.260966\pi$
$641$	34.5505	1.36466	0.682331	+	0.731044i	$0.260966\pi$
$642$	0	0
$643$	19.8893	0.784358	0.392179	−	0.919889i	$-0.371721\pi$
$643$	19.8893	0.784358	0.392179	+	0.919889i	$0.371721\pi$
$644$	13.9481	0.549633
$645$	0	0
$646$	7.89334	0.310559
$647$	37.5903	1.47783	0.738914	−	0.673799i	$-0.235339\pi$
$647$	37.5903	1.47783	0.738914	+	0.673799i	$0.235339\pi$
$648$	0	0
$649$	−23.5393	−0.923997
$650$	−15.3522	−0.602164
$651$	0	0
$652$	−14.2961	−0.559877
$653$	27.9003	1.09182	0.545911	−	0.837843i	$-0.316184\pi$
$653$	27.9003	1.09182	0.545911	+	0.837843i	$0.316184\pi$
$654$	0	0
$655$	12.5383	0.489911
$656$	7.88262	0.307765
$657$	0	0
$658$	6.87982	0.268203
$659$	−41.4881	−1.61615	−0.808073	−	0.589082i	$-0.799489\pi$
$659$	−41.4881	−1.61615	−0.808073	+	0.589082i	$0.799489\pi$
$660$	0	0
$661$	−13.1749	−0.512444	−0.256222	−	0.966618i	$-0.582478\pi$
$661$	−13.1749	−0.512444	−0.256222	+	0.966618i	$0.582478\pi$
$662$	−62.4828	−2.42846
$663$	0	0
$664$	89.7100	3.48142
$665$	2.39286	0.0927913
$666$	0	0
$667$	−8.31969	−0.322140
$668$	89.9561	3.48050
$669$	0	0
$670$	13.6826	0.528603
$671$	62.2799	2.40429
$672$	0	0
$673$	29.2106	1.12599	0.562993	−	0.826462i	$-0.309650\pi$
$673$	29.2106	1.12599	0.562993	+	0.826462i	$0.309650\pi$
$674$	53.4154	2.05748
$675$	0	0
$676$	−44.3681	−1.70646
$677$	33.3295	1.28096	0.640479	−	0.767976i	$-0.278736\pi$
$677$	33.3295	1.28096	0.640479	+	0.767976i	$0.278736\pi$
$678$	0	0
$679$	11.3241	0.434580
$680$	−14.7218	−0.564554
$681$	0	0
$682$	−27.2901	−1.04499
$683$	−30.6771	−1.17383	−0.586913	−	0.809650i	$-0.699657\pi$
$683$	−30.6771	−1.17383	−0.586913	+	0.809650i	$0.699657\pi$
$684$	0	0
$685$	−21.7100	−0.829497
$686$	2.56695	0.0980066
$687$	0	0
$688$	−59.0515	−2.25132
$689$	−9.77156	−0.372267
$690$	0	0
$691$	1.87148	0.0711946	0.0355973	−	0.999366i	$-0.488667\pi$
$691$	1.87148	0.0711946	0.0355973	+	0.999366i	$0.488667\pi$
$692$	13.8581	0.526807
$693$	0	0
$694$	57.1994	2.17126
$695$	−22.9676	−0.871211
$696$	0	0
$697$	−1.68713	−0.0639045
$698$	−53.4486	−2.02306
$699$	0	0
$700$	15.0360	0.568308
$701$	−16.8636	−0.636930	−0.318465	−	0.947935i	$-0.603167\pi$
$701$	−16.8636	−0.636930	−0.318465	+	0.947935i	$0.603167\pi$
$702$	0	0
$703$	−7.22392	−0.272455
$704$	10.0242	0.377801
$705$	0	0
$706$	−15.8522	−0.596607
$707$	4.01093	0.150846
$708$	0	0
$709$	24.6942	0.927410	0.463705	−	0.885990i	$-0.346520\pi$
$709$	24.6942	0.927410	0.463705	+	0.885990i	$0.346520\pi$
$710$	−37.9403	−1.42387
$711$	0	0
$712$	−53.2814	−1.99680
$713$	−6.61774	−0.247836
$714$	0	0
$715$	11.7014	0.437608
$716$	−101.969	−3.81077
$717$	0	0
$718$	−27.4307	−1.02370
$719$	−15.1916	−0.566550	−0.283275	−	0.959039i	$-0.591421\pi$
$719$	−15.1916	−0.566550	−0.283275	+	0.959039i	$0.591421\pi$
$720$	0	0
$721$	−10.5595	−0.393256
$722$	40.2448	1.49776
$723$	0	0
$724$	−31.5900	−1.17403
$725$	−8.96859	−0.333085
$726$	0	0
$727$	−18.9276	−0.701986	−0.350993	−	0.936378i	$-0.614156\pi$
$727$	−18.9276	−0.701986	−0.350993	+	0.936378i	$0.614156\pi$
$728$	−12.1325	−0.449661
$729$	0	0
$730$	−14.5951	−0.540187
$731$	12.6389	0.467465
$732$	0	0
$733$	−18.0703	−0.667441	−0.333720	−	0.942672i	$-0.608304\pi$
$733$	−18.0703	−0.667441	−0.333720	+	0.942672i	$0.608304\pi$
$734$	−38.5568	−1.42316
$735$	0	0
$736$	21.0971	0.777650
$737$	19.8235	0.730208
$738$	0	0
$739$	11.7429	0.431970	0.215985	−	0.976397i	$-0.430704\pi$
$739$	11.7429	0.431970	0.215985	+	0.976397i	$0.430704\pi$
$740$	23.8804	0.877860
$741$	0	0
$742$	13.7410	0.504449
$743$	1.00208	0.0367629	0.0183815	−	0.999831i	$-0.494149\pi$
$743$	1.00208	0.0367629	0.0183815	+	0.999831i	$0.494149\pi$
$744$	0	0
$745$	−0.342566	−0.0125507
$746$	47.1645	1.72682
$747$	0	0
$748$	−37.8043	−1.38226
$749$	4.86193	0.177651
$750$	0	0
$751$	−3.28392	−0.119832	−0.0599160	−	0.998203i	$-0.519083\pi$
$751$	−3.28392	−0.119832	−0.0599160	+	0.998203i	$0.519083\pi$
$752$	21.1266	0.770410
$753$	0	0
$754$	12.8266	0.467117
$755$	23.5392	0.856681
$756$	0	0
$757$	31.3116	1.13804	0.569020	−	0.822324i	$-0.307323\pi$
$757$	31.3116	1.13804	0.569020	+	0.822324i	$0.307323\pi$
$758$	−47.9037	−1.73994
$759$	0	0
$760$	15.9040	0.576900
$761$	−28.8300	−1.04509	−0.522544	−	0.852613i	$-0.675017\pi$
$761$	−28.8300	−1.04509	−0.522544	+	0.852613i	$0.675017\pi$
$762$	0	0
$763$	17.5419	0.635060
$764$	−12.9125	−0.467159
$765$	0	0
$766$	−93.4356	−3.37596
$767$	8.80038	0.317763
$768$	0	0
$769$	−20.4272	−0.736624	−0.368312	−	0.929702i	$-0.620064\pi$
$769$	−20.4272	−0.736624	−0.368312	+	0.929702i	$0.620064\pi$
$770$	−16.4548	−0.592991
$771$	0	0
$772$	88.8982	3.19952
$773$	−4.79951	−0.172626	−0.0863132	−	0.996268i	$-0.527509\pi$
$773$	−4.79951	−0.172626	−0.0863132	+	0.996268i	$0.527509\pi$
$774$	0	0
$775$	−7.13390	−0.256257
$776$	75.2652	2.70186
$777$	0	0
$778$	50.1227	1.79699
$779$	1.82262	0.0653020
$780$	0	0
$781$	−54.9684	−1.96693
$782$	−13.1626	−0.470692
$783$	0	0
$784$	7.88262	0.281522
$785$	−29.0240	−1.03591
$786$	0	0
$787$	−16.2114	−0.577875	−0.288938	−	0.957348i	$-0.593302\pi$
$787$	−16.2114	−0.577875	−0.288938	+	0.957348i	$0.593302\pi$
$788$	−51.8515	−1.84713
$789$	0	0
$790$	22.1828	0.789227
$791$	0.890970	0.0316792
$792$	0	0
$793$	−23.2839	−0.826837
$794$	43.7329	1.55202
$795$	0	0
$796$	−71.1129	−2.52053
$797$	36.6223	1.29723	0.648613	−	0.761118i	$-0.275349\pi$
$797$	36.6223	1.29723	0.648613	+	0.761118i	$0.275349\pi$
$798$	0	0
$799$	−4.52176	−0.159968
$800$	22.7426	0.804073
$801$	0	0
$802$	−12.0298	−0.424787
$803$	−21.1455	−0.746210
$804$	0	0
$805$	−3.99023	−0.140637
$806$	10.2027	0.359374
$807$	0	0
$808$	26.6584	0.937839
$809$	−7.01611	−0.246673	−0.123337	−	0.992365i	$-0.539360\pi$
$809$	−7.01611	−0.246673	−0.123337	+	0.992365i	$0.539360\pi$
$810$	0	0
$811$	−34.8787	−1.22476	−0.612379	−	0.790564i	$-0.709787\pi$
$811$	−34.8787	−1.22476	−0.612379	+	0.790564i	$0.709787\pi$
$812$	−12.5624	−0.440854
$813$	0	0
$814$	49.6762	1.74115
$815$	4.08977	0.143258
$816$	0	0
$817$	−13.6539	−0.477688
$818$	−67.6352	−2.36481
$819$	0	0
$820$	−6.02509	−0.210405
$821$	−6.28647	−0.219399	−0.109700	−	0.993965i	$-0.534989\pi$
$821$	−6.28647	−0.219399	−0.109700	+	0.993965i	$0.534989\pi$
$822$	0	0
$823$	16.6048	0.578806	0.289403	−	0.957207i	$-0.406543\pi$
$823$	16.6048	0.578806	0.289403	+	0.957207i	$0.406543\pi$
$824$	−70.1830	−2.44494
$825$	0	0
$826$	−12.3753	−0.430593
$827$	54.3533	1.89005	0.945025	−	0.326997i	$-0.106037\pi$
$827$	54.3533	1.89005	0.945025	+	0.326997i	$0.106037\pi$
$828$	0	0
$829$	−33.3938	−1.15981	−0.579907	−	0.814683i	$-0.696911\pi$
$829$	−33.3938	−1.15981	−0.579907	+	0.814683i	$0.696911\pi$
$830$	−45.4875	−1.57889
$831$	0	0
$832$	−3.74764	−0.129926
$833$	−1.68713	−0.0584555
$834$	0	0
$835$	−25.7343	−0.890573
$836$	40.8403	1.41249
$837$	0	0
$838$	−29.7624	−1.02812
$839$	−0.866263	−0.0299067	−0.0149534	−	0.999888i	$-0.504760\pi$
$839$	−0.866263	−0.0299067	−0.0149534	+	0.999888i	$0.504760\pi$
$840$	0	0
$841$	−21.5068	−0.741616
$842$	97.4132	3.35708
$843$	0	0
$844$	−103.426	−3.56008
$845$	12.6927	0.436641
$846$	0	0
$847$	−12.8400	−0.441188
$848$	42.1961	1.44902
$849$	0	0
$850$	−14.1892	−0.486685
$851$	12.0463	0.412941
$852$	0	0
$853$	−38.9886	−1.33494	−0.667472	−	0.744635i	$-0.732623\pi$
$853$	−38.9886	−1.33494	−0.667472	+	0.744635i	$0.732623\pi$
$854$	32.7425	1.12043
$855$	0	0
$856$	32.3145	1.10449
$857$	−3.55581	−0.121464	−0.0607320	−	0.998154i	$-0.519343\pi$
$857$	−3.55581	−0.121464	−0.0607320	+	0.998154i	$0.519343\pi$
$858$	0	0
$859$	19.5753	0.667901	0.333951	−	0.942591i	$-0.391618\pi$
$859$	19.5753	0.667901	0.333951	+	0.942591i	$0.391618\pi$
$860$	45.1360	1.53913
$861$	0	0
$862$	100.379	3.41893
$863$	30.9582	1.05383	0.526915	−	0.849918i	$-0.323349\pi$
$863$	30.9582	1.05383	0.526915	+	0.849918i	$0.323349\pi$
$864$	0	0
$865$	−3.96448	−0.134797
$866$	63.6555	2.16310
$867$	0	0
$868$	−9.99253	−0.339169
$869$	32.1387	1.09023
$870$	0	0
$871$	−7.41120	−0.251119
$872$	116.591	3.94828
$873$	0	0
$874$	14.2196	0.480985
$875$	−10.8658	−0.367332
$876$	0	0
$877$	−9.25642	−0.312567	−0.156284	−	0.987712i	$-0.549951\pi$
$877$	−9.25642	−0.312567	−0.156284	+	0.987712i	$0.549951\pi$
$878$	−21.6079	−0.729232
$879$	0	0
$880$	−50.5297	−1.70336
$881$	23.6835	0.797917	0.398959	−	0.916969i	$-0.369372\pi$
$881$	23.6835	0.797917	0.398959	+	0.916969i	$0.369372\pi$
$882$	0	0
$883$	−44.4147	−1.49467	−0.747337	−	0.664445i	$-0.768668\pi$
$883$	−44.4147	−1.49467	−0.747337	+	0.664445i	$0.768668\pi$
$884$	14.1335	0.475362
$885$	0	0
$886$	−79.0380	−2.65533
$887$	−6.18660	−0.207726	−0.103863	−	0.994592i	$-0.533120\pi$
$887$	−6.18660	−0.207726	−0.103863	+	0.994592i	$0.533120\pi$
$888$	0	0
$889$	1.34061	0.0449625
$890$	27.0163	0.905590
$891$	0	0
$892$	21.1909	0.709524
$893$	4.88489	0.163467
$894$	0	0
$895$	29.1710	0.975079
$896$	−8.61280	−0.287733
$897$	0	0
$898$	−6.69192	−0.223312
$899$	5.96029	0.198787
$900$	0	0
$901$	−9.03129	−0.300876
$902$	−12.5335	−0.417318
$903$	0	0
$904$	5.92178	0.196956
$905$	9.03716	0.300405
$906$	0	0
$907$	32.1863	1.06873	0.534365	−	0.845254i	$-0.320551\pi$
$907$	32.1863	1.06873	0.534365	+	0.845254i	$0.320551\pi$
$908$	−82.3263	−2.73209
$909$	0	0
$910$	6.15180	0.203930
$911$	55.0672	1.82446	0.912228	−	0.409682i	$-0.134360\pi$
$911$	55.0672	1.82446	0.912228	+	0.409682i	$0.134360\pi$
$912$	0	0
$913$	−65.9029	−2.18107
$914$	67.8072	2.24286
$915$	0	0
$916$	−83.2729	−2.75141
$917$	9.55026	0.315378
$918$	0	0
$919$	13.1533	0.433887	0.216943	−	0.976184i	$-0.430391\pi$
$919$	13.1533	0.433887	0.216943	+	0.976184i	$0.430391\pi$
$920$	−26.5208	−0.874366
$921$	0	0
$922$	−35.3622	−1.16459
$923$	20.5505	0.676427
$924$	0	0
$925$	12.9858	0.426972
$926$	63.4826	2.08617
$927$	0	0
$928$	−19.0012	−0.623744
$929$	57.3506	1.88161	0.940806	−	0.338946i	$-0.110070\pi$
$929$	57.3506	1.88161	0.940806	+	0.338946i	$0.110070\pi$
$930$	0	0
$931$	1.82262	0.0597338
$932$	−117.390	−3.84525
$933$	0	0
$934$	23.2749	0.761578
$935$	10.8149	0.353686
$936$	0	0
$937$	−5.23621	−0.171060	−0.0855298	−	0.996336i	$-0.527258\pi$
$937$	−5.23621	−0.171060	−0.0855298	+	0.996336i	$0.527258\pi$
$938$	10.4218	0.340285
$939$	0	0
$940$	−16.1482	−0.526695
$941$	10.0115	0.326367	0.163183	−	0.986596i	$-0.447824\pi$
$941$	10.0115	0.326367	0.163183	+	0.986596i	$0.447824\pi$
$942$	0	0
$943$	−3.03931	−0.0989735
$944$	−38.0024	−1.23687
$945$	0	0
$946$	93.8925	3.05271
$947$	27.3885	0.890006	0.445003	−	0.895529i	$-0.353203\pi$
$947$	27.3885	0.890006	0.445003	+	0.895529i	$0.353203\pi$
$948$	0	0
$949$	7.90546	0.256622
$950$	15.3287	0.497328
$951$	0	0
$952$	−11.2134	−0.363428
$953$	2.54292	0.0823733	0.0411867	−	0.999151i	$-0.486886\pi$
$953$	2.54292	0.0823733	0.0411867	+	0.999151i	$0.486886\pi$
$954$	0	0
$955$	3.69397	0.119534
$956$	−56.8897	−1.83994
$957$	0	0
$958$	15.1378	0.489079
$959$	−16.5363	−0.533984
$960$	0	0
$961$	−26.2590	−0.847065
$962$	−18.5719	−0.598784
$963$	0	0
$964$	60.5311	1.94958
$965$	−25.4317	−0.818675
$966$	0	0
$967$	9.44817	0.303833	0.151916	−	0.988393i	$-0.451456\pi$
$967$	9.44817	0.303833	0.151916	+	0.988393i	$0.451456\pi$
$968$	−85.3405	−2.74295
$969$	0	0
$970$	−38.1633	−1.22535
$971$	−0.540148	−0.0173342	−0.00866708	−	0.999962i	$-0.502759\pi$
$971$	−0.540148	−0.0173342	−0.00866708	+	0.999962i	$0.502759\pi$
$972$	0	0
$973$	−17.4941	−0.560837
$974$	−63.5695	−2.03690
$975$	0	0
$976$	100.546	3.21840
$977$	51.4138	1.64487	0.822437	−	0.568856i	$-0.192614\pi$
$977$	51.4138	1.64487	0.822437	+	0.568856i	$0.192614\pi$
$978$	0	0
$979$	39.1417	1.25097
$980$	−6.02509	−0.192464
$981$	0	0
$982$	7.75312	0.247412
$983$	5.25359	0.167563	0.0837817	−	0.996484i	$-0.473300\pi$
$983$	5.25359	0.167563	0.0837817	+	0.996484i	$0.473300\pi$
$984$	0	0
$985$	14.8335	0.472634
$986$	11.8549	0.377537
$987$	0	0
$988$	−15.2685	−0.485757
$989$	22.7685	0.723997
$990$	0	0
$991$	23.1189	0.734397	0.367198	−	0.930143i	$-0.380317\pi$
$991$	23.1189	0.734397	0.367198	+	0.930143i	$0.380317\pi$
$992$	−15.1141	−0.479874
$993$	0	0
$994$	−28.8987	−0.916610
$995$	20.3437	0.644939
$996$	0	0
$997$	4.35145	0.137812	0.0689059	−	0.997623i	$-0.478049\pi$
$997$	4.35145	0.137812	0.0689059	+	0.997623i	$0.478049\pi$
$998$	−81.4805	−2.57922
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2583.2.a.p.1.1		5
3.2	odd	2		861.2.a.l.1.5	✓	5
21.20	even	2		6027.2.a.w.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.a.l.1.5	✓	5	3.2	odd	2
2583.2.a.p.1.1		5	1.1	even	1	trivial
6027.2.a.w.1.5		5	21.20	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:	\( N \)	\(=\)	\( 2583 = 3^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2583.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56695 q^{2} +4.58924 q^{4} -1.31287 q^{5} -1.00000 q^{7} -6.64645 q^{8} +O(q^{10})\)	\(q-2.56695 q^{2} +4.58924 q^{4} -1.31287 q^{5} -1.00000 q^{7} -6.64645 q^{8} +3.37008 q^{10} +4.88262 q^{11} -1.82542 q^{13} +2.56695 q^{14} +7.88262 q^{16} -1.68713 q^{17} +1.82262 q^{19} -6.02509 q^{20} -12.5335 q^{22} -3.03931 q^{23} -3.27636 q^{25} +4.68575 q^{26} -4.58924 q^{28} +2.73736 q^{29} +2.17738 q^{31} -6.94142 q^{32} +4.33077 q^{34} +1.31287 q^{35} -3.96349 q^{37} -4.67857 q^{38} +8.72594 q^{40} +1.00000 q^{41} -7.49135 q^{43} +22.4075 q^{44} +7.80176 q^{46} +2.68015 q^{47} +1.00000 q^{49} +8.41027 q^{50} -8.37727 q^{52} +5.35306 q^{53} -6.41027 q^{55} +6.64645 q^{56} -7.02667 q^{58} -4.82103 q^{59} +12.7554 q^{61} -5.58924 q^{62} +2.05304 q^{64} +2.39654 q^{65} +4.06001 q^{67} -7.74263 q^{68} -3.37008 q^{70} -11.2580 q^{71} -4.33077 q^{73} +10.1741 q^{74} +8.36442 q^{76} -4.88262 q^{77} +6.58226 q^{79} -10.3489 q^{80} -2.56695 q^{82} -13.4974 q^{83} +2.21498 q^{85} +19.2299 q^{86} -32.4521 q^{88} +8.01653 q^{89} +1.82542 q^{91} -13.9481 q^{92} -6.87982 q^{94} -2.39286 q^{95} -11.3241 q^{97} -2.56695 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{2} + 7 q^{4} - 3 q^{5} - 5 q^{7} - 3 q^{8}+O(q^{10}) \) 5 * q - 3 * q^2 + 7 * q^4 - 3 * q^5 - 5 * q^7 - 3 * q^8	\( 5 q - 3 q^{2} + 7 q^{4} - 3 q^{5} - 5 q^{7} - 3 q^{8} - q^{10} - 4 q^{11} + 5 q^{13} + 3 q^{14} + 11 q^{16} - 12 q^{17} + 10 q^{19} - 9 q^{20} - 6 q^{22} - 9 q^{23} - 4 q^{25} - 13 q^{26} - 7 q^{28} - 7 q^{29} + 10 q^{31} - 9 q^{32} + 10 q^{34} + 3 q^{35} - 11 q^{37} - 7 q^{40} + 5 q^{41} - 2 q^{43} - 9 q^{46} + 7 q^{47} + 5 q^{49} + 10 q^{50} - 11 q^{52} + 9 q^{53} + 3 q^{56} - 31 q^{58} - 8 q^{59} - 6 q^{61} - 12 q^{62} - 29 q^{64} + 13 q^{65} - 9 q^{67} - 12 q^{68} + q^{70} - 4 q^{71} - 10 q^{73} + 17 q^{74} + 36 q^{76} + 4 q^{77} + 7 q^{79} - 9 q^{80} - 3 q^{82} - 50 q^{83} - 12 q^{85} + 8 q^{86} - 38 q^{88} - 8 q^{89} - 5 q^{91} + 17 q^{92} - 21 q^{94} - 36 q^{95} - 11 q^{97} - 3 q^{98}+O(q^{100}) \) 5 * q - 3 * q^2 + 7 * q^4 - 3 * q^5 - 5 * q^7 - 3 * q^8 - q^10 - 4 * q^11 + 5 * q^13 + 3 * q^14 + 11 * q^16 - 12 * q^17 + 10 * q^19 - 9 * q^20 - 6 * q^22 - 9 * q^23 - 4 * q^25 - 13 * q^26 - 7 * q^28 - 7 * q^29 + 10 * q^31 - 9 * q^32 + 10 * q^34 + 3 * q^35 - 11 * q^37 - 7 * q^40 + 5 * q^41 - 2 * q^43 - 9 * q^46 + 7 * q^47 + 5 * q^49 + 10 * q^50 - 11 * q^52 + 9 * q^53 + 3 * q^56 - 31 * q^58 - 8 * q^59 - 6 * q^61 - 12 * q^62 - 29 * q^64 + 13 * q^65 - 9 * q^67 - 12 * q^68 + q^70 - 4 * q^71 - 10 * q^73 + 17 * q^74 + 36 * q^76 + 4 * q^77 + 7 * q^79 - 9 * q^80 - 3 * q^82 - 50 * q^83 - 12 * q^85 + 8 * q^86 - 38 * q^88 - 8 * q^89 - 5 * q^91 + 17 * q^92 - 21 * q^94 - 36 * q^95 - 11 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more