LMFDB - Embedded newform 2793.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2793,2,Mod(1,2793)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2793, 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("2793.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2793 = 3 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2793.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:	$22.3022172845$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2793.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} +2.82843 q^{5} -0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} +2.82843 q^{5} -0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} +1.17157 q^{10} +0.828427 q^{11} +1.82843 q^{12} -0.828427 q^{13} -2.82843 q^{15} +3.00000 q^{16} -6.82843 q^{17} +0.414214 q^{18} +1.00000 q^{19} -5.17157 q^{20} +0.343146 q^{22} +0.828427 q^{23} +1.58579 q^{24} +3.00000 q^{25} -0.343146 q^{26} -1.00000 q^{27} -8.48528 q^{29} -1.17157 q^{30} +4.41421 q^{32} -0.828427 q^{33} -2.82843 q^{34} -1.82843 q^{36} -6.00000 q^{37} +0.414214 q^{38} +0.828427 q^{39} -4.48528 q^{40} +5.65685 q^{41} -1.51472 q^{44} +2.82843 q^{45} +0.343146 q^{46} +3.17157 q^{47} -3.00000 q^{48} +1.24264 q^{50} +6.82843 q^{51} +1.51472 q^{52} +5.17157 q^{53} -0.414214 q^{54} +2.34315 q^{55} -1.00000 q^{57} -3.51472 q^{58} -5.65685 q^{59} +5.17157 q^{60} +4.00000 q^{61} -4.17157 q^{64} -2.34315 q^{65} -0.343146 q^{66} -7.17157 q^{67} +12.4853 q^{68} -0.828427 q^{69} -15.6569 q^{71} -1.58579 q^{72} +8.00000 q^{73} -2.48528 q^{74} -3.00000 q^{75} -1.82843 q^{76} +0.343146 q^{78} -14.4853 q^{79} +8.48528 q^{80} +1.00000 q^{81} +2.34315 q^{82} +0.828427 q^{83} -19.3137 q^{85} +8.48528 q^{87} -1.31371 q^{88} +6.34315 q^{89} +1.17157 q^{90} -1.51472 q^{92} +1.31371 q^{94} +2.82843 q^{95} -4.41421 q^{96} +2.48528 q^{97} +0.828427 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 8 q^{10} - 4 q^{11} - 2 q^{12} + 4 q^{13} + 6 q^{16} - 8 q^{17} - 2 q^{18} + 2 q^{19} - 16 q^{20} + 12 q^{22} - 4 q^{23} + 6 q^{24} + 6 q^{25} - 12 q^{26} - 2 q^{27} - 8 q^{30} + 6 q^{32} + 4 q^{33} + 2 q^{36} - 12 q^{37} - 2 q^{38} - 4 q^{39} + 8 q^{40} - 20 q^{44} + 12 q^{46} + 12 q^{47} - 6 q^{48} - 6 q^{50} + 8 q^{51} + 20 q^{52} + 16 q^{53} + 2 q^{54} + 16 q^{55} - 2 q^{57} - 24 q^{58} + 16 q^{60} + 8 q^{61} - 14 q^{64} - 16 q^{65} - 12 q^{66} - 20 q^{67} + 8 q^{68} + 4 q^{69} - 20 q^{71} - 6 q^{72} + 16 q^{73} + 12 q^{74} - 6 q^{75} + 2 q^{76} + 12 q^{78} - 12 q^{79} + 2 q^{81} + 16 q^{82} - 4 q^{83} - 16 q^{85} + 20 q^{88} + 24 q^{89} + 8 q^{90} - 20 q^{92} - 20 q^{94} - 6 q^{96} - 12 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9 + 8 * q^10 - 4 * q^11 - 2 * q^12 + 4 * q^13 + 6 * q^16 - 8 * q^17 - 2 * q^18 + 2 * q^19 - 16 * q^20 + 12 * q^22 - 4 * q^23 + 6 * q^24 + 6 * q^25 - 12 * q^26 - 2 * q^27 - 8 * q^30 + 6 * q^32 + 4 * q^33 + 2 * q^36 - 12 * q^37 - 2 * q^38 - 4 * q^39 + 8 * q^40 - 20 * q^44 + 12 * q^46 + 12 * q^47 - 6 * q^48 - 6 * q^50 + 8 * q^51 + 20 * q^52 + 16 * q^53 + 2 * q^54 + 16 * q^55 - 2 * q^57 - 24 * q^58 + 16 * q^60 + 8 * q^61 - 14 * q^64 - 16 * q^65 - 12 * q^66 - 20 * q^67 + 8 * q^68 + 4 * q^69 - 20 * q^71 - 6 * q^72 + 16 * q^73 + 12 * q^74 - 6 * q^75 + 2 * q^76 + 12 * q^78 - 12 * q^79 + 2 * q^81 + 16 * q^82 - 4 * q^83 - 16 * q^85 + 20 * q^88 + 24 * q^89 + 8 * q^90 - 20 * q^92 - 20 * q^94 - 6 * q^96 - 12 * q^97 - 4 * q^99

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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.82843	−0.914214
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	−0.414214	−0.169102
$7$	0	0
$7$	0	0
$8$	−1.58579	−0.560660
$9$	1.00000	0.333333
$10$	1.17157	0.370484
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	1.82843	0.527821
$13$	−0.828427	−0.229764	−0.114882	−	0.993379i	$-0.536649\pi$
$13$	−0.828427	−0.229764	−0.114882	+	0.993379i	$0.536649\pi$
$14$	0	0
$15$	−2.82843	−0.730297
$16$	3.00000	0.750000
$17$	−6.82843	−1.65614	−0.828068	−	0.560627i	$-0.810560\pi$
$17$	−6.82843	−1.65614	−0.828068	+	0.560627i	$0.810560\pi$
$18$	0.414214	0.0976311
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−5.17157	−1.15640
$21$	0	0
$22$	0.343146	0.0731589
$23$	0.828427	0.172739	0.0863695	−	0.996263i	$-0.472473\pi$
$23$	0.828427	0.172739	0.0863695	+	0.996263i	$0.472473\pi$
$24$	1.58579	0.323697
$25$	3.00000	0.600000
$26$	−0.343146	−0.0672964
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.48528	−1.57568	−0.787839	−	0.615882i	$-0.788800\pi$
$29$	−8.48528	−1.57568	−0.787839	+	0.615882i	$0.788800\pi$
$30$	−1.17157	−0.213899
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	4.41421	0.780330
$33$	−0.828427	−0.144211
$34$	−2.82843	−0.485071
$35$	0	0
$36$	−1.82843	−0.304738
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0.414214	0.0671943
$39$	0.828427	0.132655
$40$	−4.48528	−0.709185
$41$	5.65685	0.883452	0.441726	−	0.897150i	$-0.354366\pi$
$41$	5.65685	0.883452	0.441726	+	0.897150i	$0.354366\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−1.51472	−0.228352
$45$	2.82843	0.421637
$46$	0.343146	0.0505941
$47$	3.17157	0.462621	0.231311	−	0.972880i	$-0.425699\pi$
$47$	3.17157	0.462621	0.231311	+	0.972880i	$0.425699\pi$
$48$	−3.00000	−0.433013
$49$	0	0
$50$	1.24264	0.175736
$51$	6.82843	0.956171
$52$	1.51472	0.210054
$53$	5.17157	0.710370	0.355185	−	0.934796i	$-0.384418\pi$
$53$	5.17157	0.710370	0.355185	+	0.934796i	$0.384418\pi$
$54$	−0.414214	−0.0563673
$55$	2.34315	0.315950
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−3.51472	−0.461505
$59$	−5.65685	−0.736460	−0.368230	−	0.929735i	$-0.620036\pi$
$59$	−5.65685	−0.736460	−0.368230	+	0.929735i	$0.620036\pi$
$60$	5.17157	0.667647
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	0	0
$64$	−4.17157	−0.521447
$65$	−2.34315	−0.290631
$66$	−0.343146	−0.0422383
$67$	−7.17157	−0.876147	−0.438074	−	0.898939i	$-0.644339\pi$
$67$	−7.17157	−0.876147	−0.438074	+	0.898939i	$0.644339\pi$
$68$	12.4853	1.51406
$69$	−0.828427	−0.0997309
$70$	0	0
$71$	−15.6569	−1.85813	−0.929063	−	0.369921i	$-0.879385\pi$
$71$	−15.6569	−1.85813	−0.929063	+	0.369921i	$0.879385\pi$
$72$	−1.58579	−0.186887
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	−2.48528	−0.288908
$75$	−3.00000	−0.346410
$76$	−1.82843	−0.209735
$77$	0	0
$78$	0.343146	0.0388536
$79$	−14.4853	−1.62972	−0.814861	−	0.579657i	$-0.803187\pi$
$79$	−14.4853	−1.62972	−0.814861	+	0.579657i	$0.803187\pi$
$80$	8.48528	0.948683
$81$	1.00000	0.111111
$82$	2.34315	0.258757
$83$	0.828427	0.0909317	0.0454658	−	0.998966i	$-0.485523\pi$
$83$	0.828427	0.0909317	0.0454658	+	0.998966i	$0.485523\pi$
$84$	0	0
$85$	−19.3137	−2.09487
$86$	0	0
$87$	8.48528	0.909718
$88$	−1.31371	−0.140042
$89$	6.34315	0.672372	0.336186	−	0.941796i	$-0.390863\pi$
$89$	6.34315	0.672372	0.336186	+	0.941796i	$0.390863\pi$
$90$	1.17157	0.123495
$91$	0	0
$92$	−1.51472	−0.157920
$93$	0	0
$94$	1.31371	0.135499
$95$	2.82843	0.290191
$96$	−4.41421	−0.450524
$97$	2.48528	0.252342	0.126171	−	0.992009i	$-0.459731\pi$
$97$	2.48528	0.252342	0.126171	+	0.992009i	$0.459731\pi$
$98$	0	0
$99$	0.828427	0.0832601
$100$	−5.48528	−0.548528
$101$	−8.48528	−0.844317	−0.422159	−	0.906522i	$-0.638727\pi$
$101$	−8.48528	−0.844317	−0.422159	+	0.906522i	$0.638727\pi$
$102$	2.82843	0.280056
$103$	−16.9706	−1.67216	−0.836080	−	0.548608i	$-0.815158\pi$
$103$	−16.9706	−1.67216	−0.836080	+	0.548608i	$0.815158\pi$
$104$	1.31371	0.128820
$105$	0	0
$106$	2.14214	0.208063
$107$	3.65685	0.353521	0.176761	−	0.984254i	$-0.443438\pi$
$107$	3.65685	0.353521	0.176761	+	0.984254i	$0.443438\pi$
$108$	1.82843	0.175940
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0.970563	0.0925395
$111$	6.00000	0.569495
$112$	0	0
$113$	9.17157	0.862789	0.431394	−	0.902163i	$-0.358022\pi$
$113$	9.17157	0.862789	0.431394	+	0.902163i	$0.358022\pi$
$114$	−0.414214	−0.0387947
$115$	2.34315	0.218499
$116$	15.5147	1.44051
$117$	−0.828427	−0.0765881
$118$	−2.34315	−0.215704
$119$	0	0
$120$	4.48528	0.409448
$121$	−10.3137	−0.937610
$122$	1.65685	0.150005
$123$	−5.65685	−0.510061
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−12.8284	−1.13834	−0.569169	−	0.822220i	$-0.692735\pi$
$127$	−12.8284	−1.13834	−0.569169	+	0.822220i	$0.692735\pi$
$128$	−10.5563	−0.933058
$129$	0	0
$130$	−0.970563	−0.0851240
$131$	2.48528	0.217140	0.108570	−	0.994089i	$-0.465373\pi$
$131$	2.48528	0.217140	0.108570	+	0.994089i	$0.465373\pi$
$132$	1.51472	0.131839
$133$	0	0
$134$	−2.97056	−0.256618
$135$	−2.82843	−0.243432
$136$	10.8284	0.928530
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−0.343146	−0.0292105
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	−3.17157	−0.267095
$142$	−6.48528	−0.544233
$143$	−0.686292	−0.0573906
$144$	3.00000	0.250000
$145$	−24.0000	−1.99309
$146$	3.31371	0.274244
$147$	0	0
$148$	10.9706	0.901775
$149$	4.34315	0.355804	0.177902	−	0.984048i	$-0.443069\pi$
$149$	4.34315	0.355804	0.177902	+	0.984048i	$0.443069\pi$
$150$	−1.24264	−0.101461
$151$	0.828427	0.0674164	0.0337082	−	0.999432i	$-0.489268\pi$
$151$	0.828427	0.0674164	0.0337082	+	0.999432i	$0.489268\pi$
$152$	−1.58579	−0.128624
$153$	−6.82843	−0.552046
$154$	0	0
$155$	0	0
$156$	−1.51472	−0.121275
$157$	−16.9706	−1.35440	−0.677199	−	0.735800i	$-0.736806\pi$
$157$	−16.9706	−1.35440	−0.677199	+	0.735800i	$0.736806\pi$
$158$	−6.00000	−0.477334
$159$	−5.17157	−0.410132
$160$	12.4853	0.987048
$161$	0	0
$162$	0.414214	0.0325437
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−10.3431	−0.807664
$165$	−2.34315	−0.182414
$166$	0.343146	0.0266333
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	−8.00000	−0.613572
$171$	1.00000	0.0764719
$172$	0	0
$173$	−6.34315	−0.482260	−0.241130	−	0.970493i	$-0.577518\pi$
$173$	−6.34315	−0.482260	−0.241130	+	0.970493i	$0.577518\pi$
$174$	3.51472	0.266450
$175$	0	0
$176$	2.48528	0.187335
$177$	5.65685	0.425195
$178$	2.62742	0.196933
$179$	4.34315	0.324622	0.162311	−	0.986740i	$-0.448105\pi$
$179$	4.34315	0.324622	0.162311	+	0.986740i	$0.448105\pi$
$180$	−5.17157	−0.385466
$181$	18.4853	1.37400	0.687000	−	0.726657i	$-0.258927\pi$
$181$	18.4853	1.37400	0.687000	+	0.726657i	$0.258927\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	−1.31371	−0.0968479
$185$	−16.9706	−1.24770
$186$	0	0
$187$	−5.65685	−0.413670
$188$	−5.79899	−0.422935
$189$	0	0
$190$	1.17157	0.0849948
$191$	−14.4853	−1.04812	−0.524059	−	0.851682i	$-0.675583\pi$
$191$	−14.4853	−1.04812	−0.524059	+	0.851682i	$0.675583\pi$
$192$	4.17157	0.301057
$193$	19.6569	1.41493	0.707466	−	0.706748i	$-0.249838\pi$
$193$	19.6569	1.41493	0.707466	+	0.706748i	$0.249838\pi$
$194$	1.02944	0.0739093
$195$	2.34315	0.167796
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0.343146	0.0243863
$199$	13.6569	0.968109	0.484054	−	0.875038i	$-0.339164\pi$
$199$	13.6569	0.968109	0.484054	+	0.875038i	$0.339164\pi$
$200$	−4.75736	−0.336396
$201$	7.17157	0.505844
$202$	−3.51472	−0.247295
$203$	0	0
$204$	−12.4853	−0.874145
$205$	16.0000	1.11749
$206$	−7.02944	−0.489764
$207$	0.828427	0.0575797
$208$	−2.48528	−0.172323
$209$	0.828427	0.0573035
$210$	0	0
$211$	16.1421	1.11127	0.555635	−	0.831426i	$-0.312475\pi$
$211$	16.1421	1.11127	0.555635	+	0.831426i	$0.312475\pi$
$212$	−9.45584	−0.649430
$213$	15.6569	1.07279
$214$	1.51472	0.103544
$215$	0	0
$216$	1.58579	0.107899
$217$	0	0
$218$	−2.48528	−0.168324
$219$	−8.00000	−0.540590
$220$	−4.28427	−0.288846
$221$	5.65685	0.380521
$222$	2.48528	0.166801
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	3.79899	0.252705
$227$	−6.34315	−0.421009	−0.210505	−	0.977593i	$-0.567511\pi$
$227$	−6.34315	−0.421009	−0.210505	+	0.977593i	$0.567511\pi$
$228$	1.82843	0.121091
$229$	28.0000	1.85029	0.925146	−	0.379611i	$-0.123942\pi$
$229$	28.0000	1.85029	0.925146	+	0.379611i	$0.123942\pi$
$230$	0.970563	0.0639970
$231$	0	0
$232$	13.4558	0.883419
$233$	10.9706	0.718705	0.359353	−	0.933202i	$-0.382997\pi$
$233$	10.9706	0.718705	0.359353	+	0.933202i	$0.382997\pi$
$234$	−0.343146	−0.0224321
$235$	8.97056	0.585175
$236$	10.3431	0.673281
$237$	14.4853	0.940920
$238$	0	0
$239$	−2.48528	−0.160759	−0.0803797	−	0.996764i	$-0.525613\pi$
$239$	−2.48528	−0.160759	−0.0803797	+	0.996764i	$0.525613\pi$
$240$	−8.48528	−0.547723
$241$	−18.4853	−1.19074	−0.595371	−	0.803451i	$-0.702995\pi$
$241$	−18.4853	−1.19074	−0.595371	+	0.803451i	$0.702995\pi$
$242$	−4.27208	−0.274620
$243$	−1.00000	−0.0641500
$244$	−7.31371	−0.468212
$245$	0	0
$246$	−2.34315	−0.149394
$247$	−0.828427	−0.0527116
$248$	0	0
$249$	−0.828427	−0.0524994
$250$	−2.34315	−0.148194
$251$	−8.82843	−0.557245	−0.278623	−	0.960401i	$-0.589878\pi$
$251$	−8.82843	−0.557245	−0.278623	+	0.960401i	$0.589878\pi$
$252$	0	0
$253$	0.686292	0.0431468
$254$	−5.31371	−0.333412
$255$	19.3137	1.20947
$256$	3.97056	0.248160
$257$	−0.686292	−0.0428097	−0.0214048	−	0.999771i	$-0.506814\pi$
$257$	−0.686292	−0.0428097	−0.0214048	+	0.999771i	$0.506814\pi$
$258$	0	0
$259$	0	0
$260$	4.28427	0.265699
$261$	−8.48528	−0.525226
$262$	1.02944	0.0635988
$263$	−20.8284	−1.28434	−0.642168	−	0.766564i	$-0.721965\pi$
$263$	−20.8284	−1.28434	−0.642168	+	0.766564i	$0.721965\pi$
$264$	1.31371	0.0808532
$265$	14.6274	0.898555
$266$	0	0
$267$	−6.34315	−0.388194
$268$	13.1127	0.800986
$269$	−2.34315	−0.142864	−0.0714321	−	0.997445i	$-0.522757\pi$
$269$	−2.34315	−0.142864	−0.0714321	+	0.997445i	$0.522757\pi$
$270$	−1.17157	−0.0712997
$271$	9.65685	0.586612	0.293306	−	0.956019i	$-0.405245\pi$
$271$	9.65685	0.586612	0.293306	+	0.956019i	$0.405245\pi$
$272$	−20.4853	−1.24210
$273$	0	0
$274$	−7.45584	−0.450424
$275$	2.48528	0.149868
$276$	1.51472	0.0911753
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	4.97056	0.298115
$279$	0	0
$280$	0	0
$281$	11.7990	0.703869	0.351934	−	0.936025i	$-0.385524\pi$
$281$	11.7990	0.703869	0.351934	+	0.936025i	$0.385524\pi$
$282$	−1.31371	−0.0782302
$283$	−16.9706	−1.00880	−0.504398	−	0.863472i	$-0.668285\pi$
$283$	−16.9706	−1.00880	−0.504398	+	0.863472i	$0.668285\pi$
$284$	28.6274	1.69872
$285$	−2.82843	−0.167542
$286$	−0.284271	−0.0168093
$287$	0	0
$288$	4.41421	0.260110
$289$	29.6274	1.74279
$290$	−9.94113	−0.583763
$291$	−2.48528	−0.145690
$292$	−14.6274	−0.856005
$293$	−5.65685	−0.330477	−0.165238	−	0.986254i	$-0.552839\pi$
$293$	−5.65685	−0.330477	−0.165238	+	0.986254i	$0.552839\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	9.51472	0.553032
$297$	−0.828427	−0.0480702
$298$	1.79899	0.104213
$299$	−0.686292	−0.0396893
$300$	5.48528	0.316693
$301$	0	0
$302$	0.343146	0.0197458
$303$	8.48528	0.487467
$304$	3.00000	0.172062
$305$	11.3137	0.647821
$306$	−2.82843	−0.161690
$307$	−14.3431	−0.818607	−0.409303	−	0.912398i	$-0.634228\pi$
$307$	−14.3431	−0.818607	−0.409303	+	0.912398i	$0.634228\pi$
$308$	0	0
$309$	16.9706	0.965422
$310$	0	0
$311$	−20.8284	−1.18107	−0.590536	−	0.807011i	$-0.701084\pi$
$311$	−20.8284	−1.18107	−0.590536	+	0.807011i	$0.701084\pi$
$312$	−1.31371	−0.0743741
$313$	26.6274	1.50507	0.752535	−	0.658552i	$-0.228831\pi$
$313$	26.6274	1.50507	0.752535	+	0.658552i	$0.228831\pi$
$314$	−7.02944	−0.396694
$315$	0	0
$316$	26.4853	1.48991
$317$	−1.85786	−0.104348	−0.0521740	−	0.998638i	$-0.516615\pi$
$317$	−1.85786	−0.104348	−0.0521740	+	0.998638i	$0.516615\pi$
$318$	−2.14214	−0.120125
$319$	−7.02944	−0.393573
$320$	−11.7990	−0.659584
$321$	−3.65685	−0.204106
$322$	0	0
$323$	−6.82843	−0.379944
$324$	−1.82843	−0.101579
$325$	−2.48528	−0.137859
$326$	−6.62742	−0.367059
$327$	6.00000	0.331801
$328$	−8.97056	−0.495316
$329$	0	0
$330$	−0.970563	−0.0534277
$331$	−4.82843	−0.265394	−0.132697	−	0.991157i	$-0.542364\pi$
$331$	−4.82843	−0.265394	−0.132697	+	0.991157i	$0.542364\pi$
$332$	−1.51472	−0.0831310
$333$	−6.00000	−0.328798
$334$	−2.34315	−0.128211
$335$	−20.2843	−1.10825
$336$	0	0
$337$	−30.9706	−1.68707	−0.843537	−	0.537071i	$-0.819531\pi$
$337$	−30.9706	−1.68707	−0.843537	+	0.537071i	$0.819531\pi$
$338$	−5.10051	−0.277431
$339$	−9.17157	−0.498131
$340$	35.3137	1.91515
$341$	0	0
$342$	0.414214	0.0223981
$343$	0	0
$344$	0	0
$345$	−2.34315	−0.126151
$346$	−2.62742	−0.141251
$347$	13.7990	0.740769	0.370384	−	0.928879i	$-0.379226\pi$
$347$	13.7990	0.740769	0.370384	+	0.928879i	$0.379226\pi$
$348$	−15.5147	−0.831676
$349$	32.2843	1.72814	0.864069	−	0.503374i	$-0.167908\pi$
$349$	32.2843	1.72814	0.864069	+	0.503374i	$0.167908\pi$
$350$	0	0
$351$	0.828427	0.0442182
$352$	3.65685	0.194911
$353$	1.17157	0.0623565	0.0311783	−	0.999514i	$-0.490074\pi$
$353$	1.17157	0.0623565	0.0311783	+	0.999514i	$0.490074\pi$
$354$	2.34315	0.124537
$355$	−44.2843	−2.35037
$356$	−11.5980	−0.614692
$357$	0	0
$358$	1.79899	0.0950796
$359$	−20.1421	−1.06306	−0.531531	−	0.847039i	$-0.678383\pi$
$359$	−20.1421	−1.06306	−0.531531	+	0.847039i	$0.678383\pi$
$360$	−4.48528	−0.236395
$361$	1.00000	0.0526316
$362$	7.65685	0.402435
$363$	10.3137	0.541329
$364$	0	0
$365$	22.6274	1.18437
$366$	−1.65685	−0.0866052
$367$	24.2843	1.26763	0.633814	−	0.773485i	$-0.281488\pi$
$367$	24.2843	1.26763	0.633814	+	0.773485i	$0.281488\pi$
$368$	2.48528	0.129554
$369$	5.65685	0.294484
$370$	−7.02944	−0.365443
$371$	0	0
$372$	0	0
$373$	26.9706	1.39648	0.698241	−	0.715862i	$-0.253966\pi$
$373$	26.9706	1.39648	0.698241	+	0.715862i	$0.253966\pi$
$374$	−2.34315	−0.121161
$375$	5.65685	0.292119
$376$	−5.02944	−0.259373
$377$	7.02944	0.362034
$378$	0	0
$379$	−12.8284	−0.658952	−0.329476	−	0.944164i	$-0.606872\pi$
$379$	−12.8284	−0.658952	−0.329476	+	0.944164i	$0.606872\pi$
$380$	−5.17157	−0.265296
$381$	12.8284	0.657220
$382$	−6.00000	−0.306987
$383$	22.3431	1.14168	0.570841	−	0.821061i	$-0.306617\pi$
$383$	22.3431	1.14168	0.570841	+	0.821061i	$0.306617\pi$
$384$	10.5563	0.538701
$385$	0	0
$386$	8.14214	0.414424
$387$	0	0
$388$	−4.54416	−0.230695
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0.970563	0.0491464
$391$	−5.65685	−0.286079
$392$	0	0
$393$	−2.48528	−0.125366
$394$	−7.45584	−0.375620
$395$	−40.9706	−2.06145
$396$	−1.51472	−0.0761175
$397$	1.65685	0.0831551	0.0415776	−	0.999135i	$-0.486762\pi$
$397$	1.65685	0.0831551	0.0415776	+	0.999135i	$0.486762\pi$
$398$	5.65685	0.283552
$399$	0	0
$400$	9.00000	0.450000
$401$	14.1421	0.706225	0.353112	−	0.935581i	$-0.385123\pi$
$401$	14.1421	0.706225	0.353112	+	0.935581i	$0.385123\pi$
$402$	2.97056	0.148158
$403$	0	0
$404$	15.5147	0.771886
$405$	2.82843	0.140546
$406$	0	0
$407$	−4.97056	−0.246382
$408$	−10.8284	−0.536087
$409$	23.4558	1.15982	0.579908	−	0.814682i	$-0.303088\pi$
$409$	23.4558	1.15982	0.579908	+	0.814682i	$0.303088\pi$
$410$	6.62742	0.327305
$411$	18.0000	0.887875
$412$	31.0294	1.52871
$413$	0	0
$414$	0.343146	0.0168647
$415$	2.34315	0.115021
$416$	−3.65685	−0.179292
$417$	−12.0000	−0.587643
$418$	0.343146	0.0167838
$419$	−33.1127	−1.61766	−0.808831	−	0.588042i	$-0.799899\pi$
$419$	−33.1127	−1.61766	−0.808831	+	0.588042i	$0.799899\pi$
$420$	0	0
$421$	−14.9706	−0.729621	−0.364810	−	0.931082i	$-0.618866\pi$
$421$	−14.9706	−0.729621	−0.364810	+	0.931082i	$0.618866\pi$
$422$	6.68629	0.325484
$423$	3.17157	0.154207
$424$	−8.20101	−0.398276
$425$	−20.4853	−0.993682
$426$	6.48528	0.314213
$427$	0	0
$428$	−6.68629	−0.323194
$429$	0.686292	0.0331345
$430$	0	0
$431$	−34.9706	−1.68447	−0.842236	−	0.539108i	$-0.818761\pi$
$431$	−34.9706	−1.68447	−0.842236	+	0.539108i	$0.818761\pi$
$432$	−3.00000	−0.144338
$433$	18.4853	0.888346	0.444173	−	0.895941i	$-0.353498\pi$
$433$	18.4853	0.888346	0.444173	+	0.895941i	$0.353498\pi$
$434$	0	0
$435$	24.0000	1.15071
$436$	10.9706	0.525395
$437$	0.828427	0.0396290
$438$	−3.31371	−0.158335
$439$	27.3137	1.30361	0.651806	−	0.758386i	$-0.274012\pi$
$439$	27.3137	1.30361	0.651806	+	0.758386i	$0.274012\pi$
$440$	−3.71573	−0.177140
$441$	0	0
$442$	2.34315	0.111452
$443$	40.1421	1.90721	0.953605	−	0.301060i	$-0.0973404\pi$
$443$	40.1421	1.90721	0.953605	+	0.301060i	$0.0973404\pi$
$444$	−10.9706	−0.520640
$445$	17.9411	0.850491
$446$	−9.94113	−0.470726
$447$	−4.34315	−0.205424
$448$	0	0
$449$	2.82843	0.133482	0.0667409	−	0.997770i	$-0.478740\pi$
$449$	2.82843	0.133482	0.0667409	+	0.997770i	$0.478740\pi$
$450$	1.24264	0.0585786
$451$	4.68629	0.220669
$452$	−16.7696	−0.788773
$453$	−0.828427	−0.0389229
$454$	−2.62742	−0.123311
$455$	0	0
$456$	1.58579	0.0742613
$457$	37.3137	1.74546	0.872731	−	0.488202i	$-0.162347\pi$
$457$	37.3137	1.74546	0.872731	+	0.488202i	$0.162347\pi$
$458$	11.5980	0.541938
$459$	6.82843	0.318724
$460$	−4.28427	−0.199755
$461$	−29.1716	−1.35866	−0.679328	−	0.733835i	$-0.737729\pi$
$461$	−29.1716	−1.35866	−0.679328	+	0.733835i	$0.737729\pi$
$462$	0	0
$463$	19.3137	0.897584	0.448792	−	0.893636i	$-0.351854\pi$
$463$	19.3137	0.897584	0.448792	+	0.893636i	$0.351854\pi$
$464$	−25.4558	−1.18176
$465$	0	0
$466$	4.54416	0.210504
$467$	23.1716	1.07225	0.536126	−	0.844138i	$-0.319887\pi$
$467$	23.1716	1.07225	0.536126	+	0.844138i	$0.319887\pi$
$468$	1.51472	0.0700179
$469$	0	0
$470$	3.71573	0.171394
$471$	16.9706	0.781962
$472$	8.97056	0.412904
$473$	0	0
$474$	6.00000	0.275589
$475$	3.00000	0.137649
$476$	0	0
$477$	5.17157	0.236790
$478$	−1.02944	−0.0470854
$479$	38.7696	1.77143	0.885713	−	0.464233i	$-0.153670\pi$
$479$	38.7696	1.77143	0.885713	+	0.464233i	$0.153670\pi$
$480$	−12.4853	−0.569873
$481$	4.97056	0.226638
$482$	−7.65685	−0.348760
$483$	0	0
$484$	18.8579	0.857176
$485$	7.02944	0.319190
$486$	−0.414214	−0.0187891
$487$	−13.5147	−0.612410	−0.306205	−	0.951966i	$-0.599059\pi$
$487$	−13.5147	−0.612410	−0.306205	+	0.951966i	$0.599059\pi$
$488$	−6.34315	−0.287141
$489$	16.0000	0.723545
$490$	0	0
$491$	41.1127	1.85539	0.927695	−	0.373339i	$-0.121787\pi$
$491$	41.1127	1.85539	0.927695	+	0.373339i	$0.121787\pi$
$492$	10.3431	0.466305
$493$	57.9411	2.60954
$494$	−0.343146	−0.0154389
$495$	2.34315	0.105317
$496$	0	0
$497$	0	0
$498$	−0.343146	−0.0153767
$499$	−8.97056	−0.401578	−0.200789	−	0.979635i	$-0.564350\pi$
$499$	−8.97056	−0.401578	−0.200789	+	0.979635i	$0.564350\pi$
$500$	10.3431	0.462560
$501$	5.65685	0.252730
$502$	−3.65685	−0.163213
$503$	14.4853	0.645867	0.322933	−	0.946422i	$-0.395331\pi$
$503$	14.4853	0.645867	0.322933	+	0.946422i	$0.395331\pi$
$504$	0	0
$505$	−24.0000	−1.06799
$506$	0.284271	0.0126374
$507$	12.3137	0.546871
$508$	23.4558	1.04068
$509$	−17.6569	−0.782626	−0.391313	−	0.920258i	$-0.627979\pi$
$509$	−17.6569	−0.782626	−0.391313	+	0.920258i	$0.627979\pi$
$510$	8.00000	0.354246
$511$	0	0
$512$	22.7574	1.00574
$513$	−1.00000	−0.0441511
$514$	−0.284271	−0.0125387
$515$	−48.0000	−2.11513
$516$	0	0
$517$	2.62742	0.115554
$518$	0	0
$519$	6.34315	0.278433
$520$	3.71573	0.162945
$521$	39.5980	1.73482	0.867409	−	0.497595i	$-0.165783\pi$
$521$	39.5980	1.73482	0.867409	+	0.497595i	$0.165783\pi$
$522$	−3.51472	−0.153835
$523$	−3.02944	−0.132468	−0.0662340	−	0.997804i	$-0.521098\pi$
$523$	−3.02944	−0.132468	−0.0662340	+	0.997804i	$0.521098\pi$
$524$	−4.54416	−0.198512
$525$	0	0
$526$	−8.62742	−0.376173
$527$	0	0
$528$	−2.48528	−0.108158
$529$	−22.3137	−0.970161
$530$	6.05887	0.263181
$531$	−5.65685	−0.245487
$532$	0	0
$533$	−4.68629	−0.202986
$534$	−2.62742	−0.113699
$535$	10.3431	0.447173
$536$	11.3726	0.491221
$537$	−4.34315	−0.187421
$538$	−0.970563	−0.0418439
$539$	0	0
$540$	5.17157	0.222549
$541$	−1.31371	−0.0564807	−0.0282404	−	0.999601i	$-0.508990\pi$
$541$	−1.31371	−0.0564807	−0.0282404	+	0.999601i	$0.508990\pi$
$542$	4.00000	0.171815
$543$	−18.4853	−0.793279
$544$	−30.1421	−1.29233
$545$	−16.9706	−0.726939
$546$	0	0
$547$	21.5147	0.919903	0.459951	−	0.887944i	$-0.347867\pi$
$547$	21.5147	0.919903	0.459951	+	0.887944i	$0.347867\pi$
$548$	32.9117	1.40592
$549$	4.00000	0.170716
$550$	1.02944	0.0438954
$551$	−8.48528	−0.361485
$552$	1.31371	0.0559151
$553$	0	0
$554$	−2.48528	−0.105589
$555$	16.9706	0.720360
$556$	−21.9411	−0.930511
$557$	−29.3137	−1.24206	−0.621031	−	0.783786i	$-0.713286\pi$
$557$	−29.3137	−1.24206	−0.621031	+	0.783786i	$0.713286\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	5.65685	0.238833
$562$	4.88730	0.206158
$563$	−39.3137	−1.65688	−0.828438	−	0.560081i	$-0.810770\pi$
$563$	−39.3137	−1.65688	−0.828438	+	0.560081i	$0.810770\pi$
$564$	5.79899	0.244182
$565$	25.9411	1.09135
$566$	−7.02944	−0.295469
$567$	0	0
$568$	24.8284	1.04178
$569$	25.1716	1.05525	0.527624	−	0.849478i	$-0.323083\pi$
$569$	25.1716	1.05525	0.527624	+	0.849478i	$0.323083\pi$
$570$	−1.17157	−0.0490718
$571$	−18.6274	−0.779533	−0.389767	−	0.920914i	$-0.627444\pi$
$571$	−18.6274	−0.779533	−0.389767	+	0.920914i	$0.627444\pi$
$572$	1.25483	0.0524672
$573$	14.4853	0.605131
$574$	0	0
$575$	2.48528	0.103643
$576$	−4.17157	−0.173816
$577$	31.3137	1.30361	0.651803	−	0.758388i	$-0.274013\pi$
$577$	31.3137	1.30361	0.651803	+	0.758388i	$0.274013\pi$
$578$	12.2721	0.510451
$579$	−19.6569	−0.816911
$580$	43.8823	1.82211
$581$	0	0
$582$	−1.02944	−0.0426715
$583$	4.28427	0.177436
$584$	−12.6863	−0.524962
$585$	−2.34315	−0.0968772
$586$	−2.34315	−0.0967945
$587$	−42.7696	−1.76529	−0.882644	−	0.470042i	$-0.844239\pi$
$587$	−42.7696	−1.76529	−0.882644	+	0.470042i	$0.844239\pi$
$588$	0	0
$589$	0	0
$590$	−6.62742	−0.272846
$591$	18.0000	0.740421
$592$	−18.0000	−0.739795
$593$	27.5147	1.12989	0.564947	−	0.825127i	$-0.308897\pi$
$593$	27.5147	1.12989	0.564947	+	0.825127i	$0.308897\pi$
$594$	−0.343146	−0.0140794
$595$	0	0
$596$	−7.94113	−0.325281
$597$	−13.6569	−0.558938
$598$	−0.284271	−0.0116247
$599$	22.2843	0.910511	0.455255	−	0.890361i	$-0.349548\pi$
$599$	22.2843	0.910511	0.455255	+	0.890361i	$0.349548\pi$
$600$	4.75736	0.194218
$601$	34.4853	1.40668	0.703342	−	0.710852i	$-0.251690\pi$
$601$	34.4853	1.40668	0.703342	+	0.710852i	$0.251690\pi$
$602$	0	0
$603$	−7.17157	−0.292049
$604$	−1.51472	−0.0616330
$605$	−29.1716	−1.18599
$606$	3.51472	0.142776
$607$	−0.970563	−0.0393939	−0.0196970	−	0.999806i	$-0.506270\pi$
$607$	−0.970563	−0.0393939	−0.0196970	+	0.999806i	$0.506270\pi$
$608$	4.41421	0.179020
$609$	0	0
$610$	4.68629	0.189742
$611$	−2.62742	−0.106294
$612$	12.4853	0.504688
$613$	−27.9411	−1.12853	−0.564266	−	0.825593i	$-0.690841\pi$
$613$	−27.9411	−1.12853	−0.564266	+	0.825593i	$0.690841\pi$
$614$	−5.94113	−0.239764
$615$	−16.0000	−0.645182
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	7.02944	0.282765
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	−0.828427	−0.0332436
$622$	−8.62742	−0.345928
$623$	0	0
$624$	2.48528	0.0994909
$625$	−31.0000	−1.24000
$626$	11.0294	0.440825
$627$	−0.828427	−0.0330842
$628$	31.0294	1.23821
$629$	40.9706	1.63360
$630$	0	0
$631$	42.6274	1.69697	0.848485	−	0.529219i	$-0.177515\pi$
$631$	42.6274	1.69697	0.848485	+	0.529219i	$0.177515\pi$
$632$	22.9706	0.913720
$633$	−16.1421	−0.641592
$634$	−0.769553	−0.0305628
$635$	−36.2843	−1.43990
$636$	9.45584	0.374949
$637$	0	0
$638$	−2.91169	−0.115275
$639$	−15.6569	−0.619376
$640$	−29.8579	−1.18024
$641$	−7.79899	−0.308042	−0.154021	−	0.988068i	$-0.549222\pi$
$641$	−7.79899	−0.308042	−0.154021	+	0.988068i	$0.549222\pi$
$642$	−1.51472	−0.0597812
$643$	4.97056	0.196020	0.0980099	−	0.995185i	$-0.468752\pi$
$643$	4.97056	0.196020	0.0980099	+	0.995185i	$0.468752\pi$
$644$	0	0
$645$	0	0
$646$	−2.82843	−0.111283
$647$	−12.8284	−0.504338	−0.252169	−	0.967683i	$-0.581144\pi$
$647$	−12.8284	−0.504338	−0.252169	+	0.967683i	$0.581144\pi$
$648$	−1.58579	−0.0622956
$649$	−4.68629	−0.183953
$650$	−1.02944	−0.0403779
$651$	0	0
$652$	29.2548	1.14571
$653$	−44.6274	−1.74641	−0.873203	−	0.487357i	$-0.837961\pi$
$653$	−44.6274	−1.74641	−0.873203	+	0.487357i	$0.837961\pi$
$654$	2.48528	0.0971822
$655$	7.02944	0.274663
$656$	16.9706	0.662589
$657$	8.00000	0.312110
$658$	0	0
$659$	−26.2843	−1.02389	−0.511945	−	0.859018i	$-0.671075\pi$
$659$	−26.2843	−1.02389	−0.511945	+	0.859018i	$0.671075\pi$
$660$	4.28427	0.166765
$661$	−2.48528	−0.0966662	−0.0483331	−	0.998831i	$-0.515391\pi$
$661$	−2.48528	−0.0966662	−0.0483331	+	0.998831i	$0.515391\pi$
$662$	−2.00000	−0.0777322
$663$	−5.65685	−0.219694
$664$	−1.31371	−0.0509818
$665$	0	0
$666$	−2.48528	−0.0963027
$667$	−7.02944	−0.272181
$668$	10.3431	0.400188
$669$	24.0000	0.927894
$670$	−8.40202	−0.324598
$671$	3.31371	0.127924
$672$	0	0
$673$	10.9706	0.422884	0.211442	−	0.977391i	$-0.432184\pi$
$673$	10.9706	0.422884	0.211442	+	0.977391i	$0.432184\pi$
$674$	−12.8284	−0.494133
$675$	−3.00000	−0.115470
$676$	22.5147	0.865951
$677$	−0.686292	−0.0263763	−0.0131882	−	0.999913i	$-0.504198\pi$
$677$	−0.686292	−0.0263763	−0.0131882	+	0.999913i	$0.504198\pi$
$678$	−3.79899	−0.145899
$679$	0	0
$680$	30.6274	1.17451
$681$	6.34315	0.243070
$682$	0	0
$683$	−24.6274	−0.942342	−0.471171	−	0.882042i	$-0.656169\pi$
$683$	−24.6274	−0.942342	−0.471171	+	0.882042i	$0.656169\pi$
$684$	−1.82843	−0.0699117
$685$	−50.9117	−1.94524
$686$	0	0
$687$	−28.0000	−1.06827
$688$	0	0
$689$	−4.28427	−0.163218
$690$	−0.970563	−0.0369487
$691$	16.9706	0.645591	0.322795	−	0.946469i	$-0.395377\pi$
$691$	16.9706	0.645591	0.322795	+	0.946469i	$0.395377\pi$
$692$	11.5980	0.440889
$693$	0	0
$694$	5.71573	0.216966
$695$	33.9411	1.28746
$696$	−13.4558	−0.510042
$697$	−38.6274	−1.46312
$698$	13.3726	0.506160
$699$	−10.9706	−0.414945
$700$	0	0
$701$	29.3137	1.10716	0.553582	−	0.832795i	$-0.313261\pi$
$701$	29.3137	1.10716	0.553582	+	0.832795i	$0.313261\pi$
$702$	0.343146	0.0129512
$703$	−6.00000	−0.226294
$704$	−3.45584	−0.130247
$705$	−8.97056	−0.337851
$706$	0.485281	0.0182638
$707$	0	0
$708$	−10.3431	−0.388719
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	−18.3431	−0.688406
$711$	−14.4853	−0.543240
$712$	−10.0589	−0.376972
$713$	0	0
$714$	0	0
$715$	−1.94113	−0.0725940
$716$	−7.94113	−0.296774
$717$	2.48528	0.0928145
$718$	−8.34315	−0.311363
$719$	−33.7990	−1.26049	−0.630245	−	0.776396i	$-0.717045\pi$
$719$	−33.7990	−1.26049	−0.630245	+	0.776396i	$0.717045\pi$
$720$	8.48528	0.316228
$721$	0	0
$722$	0.414214	0.0154154
$723$	18.4853	0.687475
$724$	−33.7990	−1.25613
$725$	−25.4558	−0.945406
$726$	4.27208	0.158552
$727$	−20.9706	−0.777755	−0.388878	−	0.921289i	$-0.627137\pi$
$727$	−20.9706	−0.777755	−0.388878	+	0.921289i	$0.627137\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	9.37258	0.346895
$731$	0	0
$732$	7.31371	0.270322
$733$	9.65685	0.356684	0.178342	−	0.983969i	$-0.442927\pi$
$733$	9.65685	0.356684	0.178342	+	0.983969i	$0.442927\pi$
$734$	10.0589	0.371280
$735$	0	0
$736$	3.65685	0.134793
$737$	−5.94113	−0.218844
$738$	2.34315	0.0862524
$739$	9.65685	0.355233	0.177617	−	0.984100i	$-0.443161\pi$
$739$	9.65685	0.355233	0.177617	+	0.984100i	$0.443161\pi$
$740$	31.0294	1.14066
$741$	0.828427	0.0304330
$742$	0	0
$743$	5.31371	0.194941	0.0974705	−	0.995238i	$-0.468925\pi$
$743$	5.31371	0.194941	0.0974705	+	0.995238i	$0.468925\pi$
$744$	0	0
$745$	12.2843	0.450061
$746$	11.1716	0.409020
$747$	0.828427	0.0303106
$748$	10.3431	0.378183
$749$	0	0
$750$	2.34315	0.0855596
$751$	−12.1421	−0.443073	−0.221536	−	0.975152i	$-0.571107\pi$
$751$	−12.1421	−0.443073	−0.221536	+	0.975152i	$0.571107\pi$
$752$	9.51472	0.346966
$753$	8.82843	0.321726
$754$	2.91169	0.106037
$755$	2.34315	0.0852758
$756$	0	0
$757$	19.9411	0.724773	0.362386	−	0.932028i	$-0.381962\pi$
$757$	19.9411	0.724773	0.362386	+	0.932028i	$0.381962\pi$
$758$	−5.31371	−0.193003
$759$	−0.686292	−0.0249108
$760$	−4.48528	−0.162698
$761$	13.4558	0.487774	0.243887	−	0.969804i	$-0.421577\pi$
$761$	13.4558	0.487774	0.243887	+	0.969804i	$0.421577\pi$
$762$	5.31371	0.192495
$763$	0	0
$764$	26.4853	0.958204
$765$	−19.3137	−0.698289
$766$	9.25483	0.334391
$767$	4.68629	0.169212
$768$	−3.97056	−0.143275
$769$	−12.0000	−0.432731	−0.216366	−	0.976312i	$-0.569420\pi$
$769$	−12.0000	−0.432731	−0.216366	+	0.976312i	$0.569420\pi$
$770$	0	0
$771$	0.686292	0.0247162
$772$	−35.9411	−1.29355
$773$	−24.2843	−0.873445	−0.436722	−	0.899596i	$-0.643861\pi$
$773$	−24.2843	−0.873445	−0.436722	+	0.899596i	$0.643861\pi$
$774$	0	0
$775$	0	0
$776$	−3.94113	−0.141478
$777$	0	0
$778$	7.45584	0.267305
$779$	5.65685	0.202678
$780$	−4.28427	−0.153402
$781$	−12.9706	−0.464123
$782$	−2.34315	−0.0837907
$783$	8.48528	0.303239
$784$	0	0
$785$	−48.0000	−1.71319
$786$	−1.02944	−0.0367188
$787$	−12.9706	−0.462351	−0.231175	−	0.972912i	$-0.574257\pi$
$787$	−12.9706	−0.462351	−0.231175	+	0.972912i	$0.574257\pi$
$788$	32.9117	1.17243
$789$	20.8284	0.741512
$790$	−16.9706	−0.603786
$791$	0	0
$792$	−1.31371	−0.0466806
$793$	−3.31371	−0.117673
$794$	0.686292	0.0243556
$795$	−14.6274	−0.518781
$796$	−24.9706	−0.885058
$797$	−40.9706	−1.45125	−0.725626	−	0.688089i	$-0.758450\pi$
$797$	−40.9706	−1.45125	−0.725626	+	0.688089i	$0.758450\pi$
$798$	0	0
$799$	−21.6569	−0.766164
$800$	13.2426	0.468198
$801$	6.34315	0.224124
$802$	5.85786	0.206848
$803$	6.62742	0.233876
$804$	−13.1127	−0.462449
$805$	0	0
$806$	0	0
$807$	2.34315	0.0824826
$808$	13.4558	0.473375
$809$	−12.3431	−0.433962	−0.216981	−	0.976176i	$-0.569621\pi$
$809$	−12.3431	−0.433962	−0.216981	+	0.976176i	$0.569621\pi$
$810$	1.17157	0.0411649
$811$	−16.6863	−0.585935	−0.292967	−	0.956122i	$-0.594643\pi$
$811$	−16.6863	−0.585935	−0.292967	+	0.956122i	$0.594643\pi$
$812$	0	0
$813$	−9.65685	−0.338681
$814$	−2.05887	−0.0721635
$815$	−45.2548	−1.58521
$816$	20.4853	0.717128
$817$	0	0
$818$	9.71573	0.339702
$819$	0	0
$820$	−29.2548	−1.02162
$821$	2.68629	0.0937522	0.0468761	−	0.998901i	$-0.485073\pi$
$821$	2.68629	0.0937522	0.0468761	+	0.998901i	$0.485073\pi$
$822$	7.45584	0.260052
$823$	−32.9706	−1.14928	−0.574641	−	0.818406i	$-0.694858\pi$
$823$	−32.9706	−1.14928	−0.574641	+	0.818406i	$0.694858\pi$
$824$	26.9117	0.937513
$825$	−2.48528	−0.0865264
$826$	0	0
$827$	−30.2843	−1.05309	−0.526544	−	0.850148i	$-0.676512\pi$
$827$	−30.2843	−1.05309	−0.526544	+	0.850148i	$0.676512\pi$
$828$	−1.51472	−0.0526401
$829$	5.51472	0.191534	0.0957670	−	0.995404i	$-0.469470\pi$
$829$	5.51472	0.191534	0.0957670	+	0.995404i	$0.469470\pi$
$830$	0.970563	0.0336887
$831$	6.00000	0.208138
$832$	3.45584	0.119810
$833$	0	0
$834$	−4.97056	−0.172117
$835$	−16.0000	−0.553703
$836$	−1.51472	−0.0523876
$837$	0	0
$838$	−13.7157	−0.473802
$839$	15.3137	0.528688	0.264344	−	0.964428i	$-0.414845\pi$
$839$	15.3137	0.528688	0.264344	+	0.964428i	$0.414845\pi$
$840$	0	0
$841$	43.0000	1.48276
$842$	−6.20101	−0.213701
$843$	−11.7990	−0.406379
$844$	−29.5147	−1.01594
$845$	−34.8284	−1.19813
$846$	1.31371	0.0451662
$847$	0	0
$848$	15.5147	0.532778
$849$	16.9706	0.582428
$850$	−8.48528	−0.291043
$851$	−4.97056	−0.170389
$852$	−28.6274	−0.980759
$853$	28.6863	0.982200	0.491100	−	0.871103i	$-0.336595\pi$
$853$	28.6863	0.982200	0.491100	+	0.871103i	$0.336595\pi$
$854$	0	0
$855$	2.82843	0.0967302
$856$	−5.79899	−0.198205
$857$	−51.5980	−1.76255	−0.881277	−	0.472601i	$-0.843315\pi$
$857$	−51.5980	−1.76255	−0.881277	+	0.472601i	$0.843315\pi$
$858$	0.284271	0.00970486
$859$	49.9411	1.70397	0.851985	−	0.523567i	$-0.175399\pi$
$859$	49.9411	1.70397	0.851985	+	0.523567i	$0.175399\pi$
$860$	0	0
$861$	0	0
$862$	−14.4853	−0.493371
$863$	4.62742	0.157519	0.0787596	−	0.996894i	$-0.474904\pi$
$863$	4.62742	0.157519	0.0787596	+	0.996894i	$0.474904\pi$
$864$	−4.41421	−0.150175
$865$	−17.9411	−0.610017
$866$	7.65685	0.260190
$867$	−29.6274	−1.00620
$868$	0	0
$869$	−12.0000	−0.407072
$870$	9.94113	0.337036
$871$	5.94113	0.201307
$872$	9.51472	0.322209
$873$	2.48528	0.0841140
$874$	0.343146	0.0116071
$875$	0	0
$876$	14.6274	0.494215
$877$	46.9706	1.58608	0.793042	−	0.609167i	$-0.208496\pi$
$877$	46.9706	1.58608	0.793042	+	0.609167i	$0.208496\pi$
$878$	11.3137	0.381819
$879$	5.65685	0.190801
$880$	7.02944	0.236962
$881$	−3.51472	−0.118414	−0.0592069	−	0.998246i	$-0.518857\pi$
$881$	−3.51472	−0.118414	−0.0592069	+	0.998246i	$0.518857\pi$
$882$	0	0
$883$	23.5980	0.794135	0.397068	−	0.917789i	$-0.370028\pi$
$883$	23.5980	0.794135	0.397068	+	0.917789i	$0.370028\pi$
$884$	−10.3431	−0.347878
$885$	16.0000	0.537834
$886$	16.6274	0.558609
$887$	−43.3137	−1.45433	−0.727166	−	0.686462i	$-0.759163\pi$
$887$	−43.3137	−1.45433	−0.727166	+	0.686462i	$0.759163\pi$
$888$	−9.51472	−0.319293
$889$	0	0
$890$	7.43146	0.249103
$891$	0.828427	0.0277534
$892$	43.8823	1.46929
$893$	3.17157	0.106133
$894$	−1.79899	−0.0601672
$895$	12.2843	0.410618
$896$	0	0
$897$	0.686292	0.0229146
$898$	1.17157	0.0390959
$899$	0	0
$900$	−5.48528	−0.182843
$901$	−35.3137	−1.17647
$902$	1.94113	0.0646324
$903$	0	0
$904$	−14.5442	−0.483731
$905$	52.2843	1.73799
$906$	−0.343146	−0.0114003
$907$	−17.5147	−0.581567	−0.290783	−	0.956789i	$-0.593916\pi$
$907$	−17.5147	−0.581567	−0.290783	+	0.956789i	$0.593916\pi$
$908$	11.5980	0.384892
$909$	−8.48528	−0.281439
$910$	0	0
$911$	−10.2843	−0.340733	−0.170367	−	0.985381i	$-0.554495\pi$
$911$	−10.2843	−0.340733	−0.170367	+	0.985381i	$0.554495\pi$
$912$	−3.00000	−0.0993399
$913$	0.686292	0.0227129
$914$	15.4558	0.511234
$915$	−11.3137	−0.374020
$916$	−51.1960	−1.69156
$917$	0	0
$918$	2.82843	0.0933520
$919$	−30.6274	−1.01031	−0.505153	−	0.863030i	$-0.668564\pi$
$919$	−30.6274	−1.01031	−0.505153	+	0.863030i	$0.668564\pi$
$920$	−3.71573	−0.122504
$921$	14.3431	0.472623
$922$	−12.0833	−0.397941
$923$	12.9706	0.426931
$924$	0	0
$925$	−18.0000	−0.591836
$926$	8.00000	0.262896
$927$	−16.9706	−0.557386
$928$	−37.4558	−1.22955
$929$	−34.1421	−1.12017	−0.560084	−	0.828436i	$-0.689231\pi$
$929$	−34.1421	−1.12017	−0.560084	+	0.828436i	$0.689231\pi$
$930$	0	0
$931$	0	0
$932$	−20.0589	−0.657050
$933$	20.8284	0.681892
$934$	9.59798	0.314055
$935$	−16.0000	−0.523256
$936$	1.31371	0.0429399
$937$	−14.3431	−0.468570	−0.234285	−	0.972168i	$-0.575275\pi$
$937$	−14.3431	−0.468570	−0.234285	+	0.972168i	$0.575275\pi$
$938$	0	0
$939$	−26.6274	−0.868953
$940$	−16.4020	−0.534975
$941$	−16.2843	−0.530852	−0.265426	−	0.964131i	$-0.585513\pi$
$941$	−16.2843	−0.530852	−0.265426	+	0.964131i	$0.585513\pi$
$942$	7.02944	0.229031
$943$	4.68629	0.152607
$944$	−16.9706	−0.552345
$945$	0	0
$946$	0	0
$947$	33.7990	1.09832	0.549160	−	0.835717i	$-0.314948\pi$
$947$	33.7990	1.09832	0.549160	+	0.835717i	$0.314948\pi$
$948$	−26.4853	−0.860202
$949$	−6.62742	−0.215135
$950$	1.24264	0.0403166
$951$	1.85786	0.0602454
$952$	0	0
$953$	36.7696	1.19108	0.595541	−	0.803325i	$-0.296938\pi$
$953$	36.7696	1.19108	0.595541	+	0.803325i	$0.296938\pi$
$954$	2.14214	0.0693542
$955$	−40.9706	−1.32578
$956$	4.54416	0.146969
$957$	7.02944	0.227229
$958$	16.0589	0.518839
$959$	0	0
$960$	11.7990	0.380811
$961$	−31.0000	−1.00000
$962$	2.05887	0.0663808
$963$	3.65685	0.117840
$964$	33.7990	1.08859
$965$	55.5980	1.78976
$966$	0	0
$967$	−16.6863	−0.536595	−0.268297	−	0.963336i	$-0.586461\pi$
$967$	−16.6863	−0.536595	−0.268297	+	0.963336i	$0.586461\pi$
$968$	16.3553	0.525681
$969$	6.82843	0.219361
$970$	2.91169	0.0934887
$971$	−41.6569	−1.33683	−0.668416	−	0.743788i	$-0.733027\pi$
$971$	−41.6569	−1.33683	−0.668416	+	0.743788i	$0.733027\pi$
$972$	1.82843	0.0586468
$973$	0	0
$974$	−5.59798	−0.179371
$975$	2.48528	0.0795927
$976$	12.0000	0.384111
$977$	28.7696	0.920420	0.460210	−	0.887810i	$-0.347774\pi$
$977$	28.7696	0.920420	0.460210	+	0.887810i	$0.347774\pi$
$978$	6.62742	0.211921
$979$	5.25483	0.167945
$980$	0	0
$981$	−6.00000	−0.191565
$982$	17.0294	0.543431
$983$	28.2843	0.902128	0.451064	−	0.892492i	$-0.351045\pi$
$983$	28.2843	0.902128	0.451064	+	0.892492i	$0.351045\pi$
$984$	8.97056	0.285971
$985$	−50.9117	−1.62218
$986$	24.0000	0.764316
$987$	0	0
$988$	1.51472	0.0481896
$989$	0	0
$990$	0.970563	0.0308465
$991$	26.7696	0.850363	0.425181	−	0.905108i	$-0.360210\pi$
$991$	26.7696	0.850363	0.425181	+	0.905108i	$0.360210\pi$
$992$	0	0
$993$	4.82843	0.153226
$994$	0	0
$995$	38.6274	1.22457
$996$	1.51472	0.0479957
$997$	−40.9706	−1.29755	−0.648775	−	0.760980i	$-0.724719\pi$
$997$	−40.9706	−1.29755	−0.648775	+	0.760980i	$0.724719\pi$
$998$	−3.71573	−0.117619
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2793.2.a.m.1.2	✓	2
3.2	odd	2		8379.2.a.bk.1.1		2
7.6	odd	2		2793.2.a.p.1.2	yes	2
21.20	even	2		8379.2.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2793.2.a.m.1.2	✓	2	1.1	even	1	trivial
2793.2.a.p.1.2	yes	2	7.6	odd	2
8379.2.a.bj.1.1		2	21.20	even	2
8379.2.a.bk.1.1		2	3.2	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:	\( N \)	\(=\)	\( 2793 = 3 \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2793.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} +2.82843 q^{5} -0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} +2.82843 q^{5} -0.414214 q^{6} -1.58579 q^{8} +1.00000 q^{9} +1.17157 q^{10} +0.828427 q^{11} +1.82843 q^{12} -0.828427 q^{13} -2.82843 q^{15} +3.00000 q^{16} -6.82843 q^{17} +0.414214 q^{18} +1.00000 q^{19} -5.17157 q^{20} +0.343146 q^{22} +0.828427 q^{23} +1.58579 q^{24} +3.00000 q^{25} -0.343146 q^{26} -1.00000 q^{27} -8.48528 q^{29} -1.17157 q^{30} +4.41421 q^{32} -0.828427 q^{33} -2.82843 q^{34} -1.82843 q^{36} -6.00000 q^{37} +0.414214 q^{38} +0.828427 q^{39} -4.48528 q^{40} +5.65685 q^{41} -1.51472 q^{44} +2.82843 q^{45} +0.343146 q^{46} +3.17157 q^{47} -3.00000 q^{48} +1.24264 q^{50} +6.82843 q^{51} +1.51472 q^{52} +5.17157 q^{53} -0.414214 q^{54} +2.34315 q^{55} -1.00000 q^{57} -3.51472 q^{58} -5.65685 q^{59} +5.17157 q^{60} +4.00000 q^{61} -4.17157 q^{64} -2.34315 q^{65} -0.343146 q^{66} -7.17157 q^{67} +12.4853 q^{68} -0.828427 q^{69} -15.6569 q^{71} -1.58579 q^{72} +8.00000 q^{73} -2.48528 q^{74} -3.00000 q^{75} -1.82843 q^{76} +0.343146 q^{78} -14.4853 q^{79} +8.48528 q^{80} +1.00000 q^{81} +2.34315 q^{82} +0.828427 q^{83} -19.3137 q^{85} +8.48528 q^{87} -1.31371 q^{88} +6.34315 q^{89} +1.17157 q^{90} -1.51472 q^{92} +1.31371 q^{94} +2.82843 q^{95} -4.41421 q^{96} +2.48528 q^{97} +0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 8 q^{10} - 4 q^{11} - 2 q^{12} + 4 q^{13} + 6 q^{16} - 8 q^{17} - 2 q^{18} + 2 q^{19} - 16 q^{20} + 12 q^{22} - 4 q^{23} + 6 q^{24} + 6 q^{25} - 12 q^{26} - 2 q^{27} - 8 q^{30} + 6 q^{32} + 4 q^{33} + 2 q^{36} - 12 q^{37} - 2 q^{38} - 4 q^{39} + 8 q^{40} - 20 q^{44} + 12 q^{46} + 12 q^{47} - 6 q^{48} - 6 q^{50} + 8 q^{51} + 20 q^{52} + 16 q^{53} + 2 q^{54} + 16 q^{55} - 2 q^{57} - 24 q^{58} + 16 q^{60} + 8 q^{61} - 14 q^{64} - 16 q^{65} - 12 q^{66} - 20 q^{67} + 8 q^{68} + 4 q^{69} - 20 q^{71} - 6 q^{72} + 16 q^{73} + 12 q^{74} - 6 q^{75} + 2 q^{76} + 12 q^{78} - 12 q^{79} + 2 q^{81} + 16 q^{82} - 4 q^{83} - 16 q^{85} + 20 q^{88} + 24 q^{89} + 8 q^{90} - 20 q^{92} - 20 q^{94} - 6 q^{96} - 12 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 2 * q^9 + 8 * q^10 - 4 * q^11 - 2 * q^12 + 4 * q^13 + 6 * q^16 - 8 * q^17 - 2 * q^18 + 2 * q^19 - 16 * q^20 + 12 * q^22 - 4 * q^23 + 6 * q^24 + 6 * q^25 - 12 * q^26 - 2 * q^27 - 8 * q^30 + 6 * q^32 + 4 * q^33 + 2 * q^36 - 12 * q^37 - 2 * q^38 - 4 * q^39 + 8 * q^40 - 20 * q^44 + 12 * q^46 + 12 * q^47 - 6 * q^48 - 6 * q^50 + 8 * q^51 + 20 * q^52 + 16 * q^53 + 2 * q^54 + 16 * q^55 - 2 * q^57 - 24 * q^58 + 16 * q^60 + 8 * q^61 - 14 * q^64 - 16 * q^65 - 12 * q^66 - 20 * q^67 + 8 * q^68 + 4 * q^69 - 20 * q^71 - 6 * q^72 + 16 * q^73 + 12 * q^74 - 6 * q^75 + 2 * q^76 + 12 * q^78 - 12 * q^79 + 2 * q^81 + 16 * q^82 - 4 * q^83 - 16 * q^85 + 20 * q^88 + 24 * q^89 + 8 * q^90 - 20 * q^92 - 20 * q^94 - 6 * q^96 - 12 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more