LMFDB - Embedded newform 931.2.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [931,2,Mod(1,931)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("931.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(931, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 931 = 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	931.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.43407242818$
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:	no (minimal twist has level 133)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	931.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.414214 q^{2} -1.41421 q^{3} -1.82843 q^{4} +1.00000 q^{5} -0.585786 q^{6} -1.58579 q^{8} -1.00000 q^{9} +0.414214 q^{10} +2.41421 q^{11} +2.58579 q^{12} +6.24264 q^{13} -1.41421 q^{15} +3.00000 q^{16} -4.00000 q^{17} -0.414214 q^{18} -1.00000 q^{19} -1.82843 q^{20} +1.00000 q^{22} -8.41421 q^{23} +2.24264 q^{24} -4.00000 q^{25} +2.58579 q^{26} +5.65685 q^{27} -9.41421 q^{29} -0.585786 q^{30} +2.24264 q^{31} +4.41421 q^{32} -3.41421 q^{33} -1.65685 q^{34} +1.82843 q^{36} -5.07107 q^{37} -0.414214 q^{38} -8.82843 q^{39} -1.58579 q^{40} -8.82843 q^{41} -6.07107 q^{43} -4.41421 q^{44} -1.00000 q^{45} -3.48528 q^{46} -1.58579 q^{47} -4.24264 q^{48} -1.65685 q^{50} +5.65685 q^{51} -11.4142 q^{52} +4.24264 q^{53} +2.34315 q^{54} +2.41421 q^{55} +1.41421 q^{57} -3.89949 q^{58} +1.17157 q^{59} +2.58579 q^{60} +6.17157 q^{61} +0.928932 q^{62} -4.17157 q^{64} +6.24264 q^{65} -1.41421 q^{66} +6.82843 q^{67} +7.31371 q^{68} +11.8995 q^{69} -5.75736 q^{71} +1.58579 q^{72} -13.1421 q^{73} -2.10051 q^{74} +5.65685 q^{75} +1.82843 q^{76} -3.65685 q^{78} -1.41421 q^{79} +3.00000 q^{80} -5.00000 q^{81} -3.65685 q^{82} +0.757359 q^{83} -4.00000 q^{85} -2.51472 q^{86} +13.3137 q^{87} -3.82843 q^{88} -8.58579 q^{89} -0.414214 q^{90} +15.3848 q^{92} -3.17157 q^{93} -0.656854 q^{94} -1.00000 q^{95} -6.24264 q^{96} -8.82843 q^{97} -2.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{6} - 6 q^{8} - 2 q^{9} - 2 q^{10} + 2 q^{11} + 8 q^{12} + 4 q^{13} + 6 q^{16} - 8 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{20} + 2 q^{22} - 14 q^{23} - 4 q^{24} - 8 q^{25}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 4 * q^6 - 6 * q^8 - 2 * q^9 - 2 * q^10 + 2 * q^11 + 8 * q^12 + 4 * q^13 + 6 * q^16 - 8 * q^17 + 2 * q^18 - 2 * q^19 + 2 * q^20 + 2 * q^22 - 14 * q^23 - 4 * q^24 - 8 * q^25 + 8 * q^26 - 16 * q^29 - 4 * q^30 - 4 * q^31 + 6 * q^32 - 4 * q^33 + 8 * q^34 - 2 * q^36 + 4 * q^37 + 2 * q^38 - 12 * q^39 - 6 * q^40 - 12 * q^41 + 2 * q^43 - 6 * q^44 - 2 * q^45 + 10 * q^46 - 6 * q^47 + 8 * q^50 - 20 * q^52 + 16 * q^54 + 2 * q^55 + 12 * q^58 + 8 * q^59 + 8 * q^60 + 18 * q^61 + 16 * q^62 - 14 * q^64 + 4 * q^65 + 8 * q^67 - 8 * q^68 + 4 * q^69 - 20 * q^71 + 6 * q^72 + 2 * q^73 - 24 * q^74 - 2 * q^76 + 4 * q^78 + 6 * q^80 - 10 * q^81 + 4 * q^82 + 10 * q^83 - 8 * q^85 - 22 * q^86 + 4 * q^87 - 2 * q^88 - 20 * q^89 + 2 * q^90 - 6 * q^92 - 12 * q^93 + 10 * q^94 - 2 * q^95 - 4 * q^96 - 12 * q^97 - 2 * q^99

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	$0.453216\pi$
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	−1.82843	−0.914214
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−0.585786	−0.239146
$7$	0	0
$7$	0	0
$8$	−1.58579	−0.560660
$9$	−1.00000	−0.333333
$10$	0.414214	0.130986
$11$	2.41421	0.727913	0.363956	−	0.931416i	$-0.381426\pi$
$11$	2.41421	0.727913	0.363956	+	0.931416i	$0.381426\pi$
$12$	2.58579	0.746452
$13$	6.24264	1.73140	0.865699	−	0.500566i	$-0.166875\pi$
$13$	6.24264	1.73140	0.865699	+	0.500566i	$0.166875\pi$
$14$	0	0
$15$	−1.41421	−0.365148
$16$	3.00000	0.750000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−0.414214	−0.0976311
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.82843	−0.408849
$21$	0	0
$22$	1.00000	0.213201
$23$	−8.41421	−1.75448	−0.877242	−	0.480048i	$-0.840619\pi$
$23$	−8.41421	−1.75448	−0.877242	+	0.480048i	$0.840619\pi$
$24$	2.24264	0.457777
$25$	−4.00000	−0.800000
$26$	2.58579	0.507114
$27$	5.65685	1.08866
$28$	0	0
$29$	−9.41421	−1.74818	−0.874088	−	0.485768i	$-0.838540\pi$
$29$	−9.41421	−1.74818	−0.874088	+	0.485768i	$0.838540\pi$
$30$	−0.585786	−0.106949
$31$	2.24264	0.402790	0.201395	−	0.979510i	$-0.435452\pi$
$31$	2.24264	0.402790	0.201395	+	0.979510i	$0.435452\pi$
$32$	4.41421	0.780330
$33$	−3.41421	−0.594338
$34$	−1.65685	−0.284148
$35$	0	0
$36$	1.82843	0.304738
$37$	−5.07107	−0.833678	−0.416839	−	0.908980i	$-0.636862\pi$
$37$	−5.07107	−0.833678	−0.416839	+	0.908980i	$0.636862\pi$
$38$	−0.414214	−0.0671943
$39$	−8.82843	−1.41368
$40$	−1.58579	−0.250735
$41$	−8.82843	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$41$	−8.82843	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$42$	0	0
$43$	−6.07107	−0.925829	−0.462915	−	0.886403i	$-0.653196\pi$
$43$	−6.07107	−0.925829	−0.462915	+	0.886403i	$0.653196\pi$
$44$	−4.41421	−0.665468
$45$	−1.00000	−0.149071
$46$	−3.48528	−0.513877
$47$	−1.58579	−0.231311	−0.115655	−	0.993289i	$-0.536897\pi$
$47$	−1.58579	−0.231311	−0.115655	+	0.993289i	$0.536897\pi$
$48$	−4.24264	−0.612372
$49$	0	0
$50$	−1.65685	−0.234315
$51$	5.65685	0.792118
$52$	−11.4142	−1.58287
$53$	4.24264	0.582772	0.291386	−	0.956606i	$-0.405884\pi$
$53$	4.24264	0.582772	0.291386	+	0.956606i	$0.405884\pi$
$54$	2.34315	0.318862
$55$	2.41421	0.325532
$56$	0	0
$57$	1.41421	0.187317
$58$	−3.89949	−0.512029
$59$	1.17157	0.152526	0.0762629	−	0.997088i	$-0.475701\pi$
$59$	1.17157	0.152526	0.0762629	+	0.997088i	$0.475701\pi$
$60$	2.58579	0.333824
$61$	6.17157	0.790189	0.395094	−	0.918640i	$-0.370712\pi$
$61$	6.17157	0.790189	0.395094	+	0.918640i	$0.370712\pi$
$62$	0.928932	0.117975
$63$	0	0
$64$	−4.17157	−0.521447
$65$	6.24264	0.774304
$66$	−1.41421	−0.174078
$67$	6.82843	0.834225	0.417113	−	0.908855i	$-0.363042\pi$
$67$	6.82843	0.834225	0.417113	+	0.908855i	$0.363042\pi$
$68$	7.31371	0.886917
$69$	11.8995	1.43253
$70$	0	0
$71$	−5.75736	−0.683273	−0.341636	−	0.939832i	$-0.610981\pi$
$71$	−5.75736	−0.683273	−0.341636	+	0.939832i	$0.610981\pi$
$72$	1.58579	0.186887
$73$	−13.1421	−1.53817	−0.769085	−	0.639146i	$-0.779288\pi$
$73$	−13.1421	−1.53817	−0.769085	+	0.639146i	$0.779288\pi$
$74$	−2.10051	−0.244179
$75$	5.65685	0.653197
$76$	1.82843	0.209735
$77$	0	0
$78$	−3.65685	−0.414057
$79$	−1.41421	−0.159111	−0.0795557	−	0.996830i	$-0.525350\pi$
$79$	−1.41421	−0.159111	−0.0795557	+	0.996830i	$0.525350\pi$
$80$	3.00000	0.335410
$81$	−5.00000	−0.555556
$82$	−3.65685	−0.403832
$83$	0.757359	0.0831310	0.0415655	−	0.999136i	$-0.486765\pi$
$83$	0.757359	0.0831310	0.0415655	+	0.999136i	$0.486765\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	−2.51472	−0.271169
$87$	13.3137	1.42738
$88$	−3.82843	−0.408112
$89$	−8.58579	−0.910092	−0.455046	−	0.890468i	$-0.650377\pi$
$89$	−8.58579	−0.910092	−0.455046	+	0.890468i	$0.650377\pi$
$90$	−0.414214	−0.0436619
$91$	0	0
$92$	15.3848	1.60397
$93$	−3.17157	−0.328877
$94$	−0.656854	−0.0677493
$95$	−1.00000	−0.102598
$96$	−6.24264	−0.637137
$97$	−8.82843	−0.896391	−0.448195	−	0.893936i	$-0.647933\pi$
$97$	−8.82843	−0.896391	−0.448195	+	0.893936i	$0.647933\pi$
$98$	0	0
$99$	−2.41421	−0.242638
$100$	7.31371	0.731371
$101$	−4.31371	−0.429230	−0.214615	−	0.976699i	$-0.568850\pi$
$101$	−4.31371	−0.429230	−0.214615	+	0.976699i	$0.568850\pi$
$102$	2.34315	0.232006
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	−9.89949	−0.970725
$105$	0	0
$106$	1.75736	0.170690
$107$	5.89949	0.570326	0.285163	−	0.958479i	$-0.407952\pi$
$107$	5.89949	0.570326	0.285163	+	0.958479i	$0.407952\pi$
$108$	−10.3431	−0.995270
$109$	1.65685	0.158698	0.0793489	−	0.996847i	$-0.474716\pi$
$109$	1.65685	0.158698	0.0793489	+	0.996847i	$0.474716\pi$
$110$	1.00000	0.0953463
$111$	7.17157	0.680696
$112$	0	0
$113$	8.24264	0.775402	0.387701	−	0.921785i	$-0.373269\pi$
$113$	8.24264	0.775402	0.387701	+	0.921785i	$0.373269\pi$
$114$	0.585786	0.0548639
$115$	−8.41421	−0.784629
$116$	17.2132	1.59821
$117$	−6.24264	−0.577132
$118$	0.485281	0.0446738
$119$	0	0
$120$	2.24264	0.204724
$121$	−5.17157	−0.470143
$122$	2.55635	0.231441
$123$	12.4853	1.12576
$124$	−4.10051	−0.368236
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−4.24264	−0.376473	−0.188237	−	0.982124i	$-0.560277\pi$
$127$	−4.24264	−0.376473	−0.188237	+	0.982124i	$0.560277\pi$
$128$	−10.5563	−0.933058
$129$	8.58579	0.755936
$130$	2.58579	0.226788
$131$	20.4853	1.78981	0.894904	−	0.446259i	$-0.147244\pi$
$131$	20.4853	1.78981	0.894904	+	0.446259i	$0.147244\pi$
$132$	6.24264	0.543352
$133$	0	0
$134$	2.82843	0.244339
$135$	5.65685	0.486864
$136$	6.34315	0.543920
$137$	−14.1716	−1.21076	−0.605380	−	0.795937i	$-0.706979\pi$
$137$	−14.1716	−1.21076	−0.605380	+	0.795937i	$0.706979\pi$
$138$	4.92893	0.419579
$139$	8.07107	0.684579	0.342290	−	0.939595i	$-0.388798\pi$
$139$	8.07107	0.684579	0.342290	+	0.939595i	$0.388798\pi$
$140$	0	0
$141$	2.24264	0.188864
$142$	−2.38478	−0.200126
$143$	15.0711	1.26031
$144$	−3.00000	−0.250000
$145$	−9.41421	−0.781808
$146$	−5.44365	−0.450520
$147$	0	0
$148$	9.27208	0.762160
$149$	−5.82843	−0.477483	−0.238742	−	0.971083i	$-0.576735\pi$
$149$	−5.82843	−0.477483	−0.238742	+	0.971083i	$0.576735\pi$
$150$	2.34315	0.191317
$151$	5.31371	0.432423	0.216212	−	0.976346i	$-0.430630\pi$
$151$	5.31371	0.432423	0.216212	+	0.976346i	$0.430630\pi$
$152$	1.58579	0.128624
$153$	4.00000	0.323381
$154$	0	0
$155$	2.24264	0.180133
$156$	16.1421	1.29241
$157$	11.8284	0.944011	0.472006	−	0.881596i	$-0.343530\pi$
$157$	11.8284	0.944011	0.472006	+	0.881596i	$0.343530\pi$
$158$	−0.585786	−0.0466027
$159$	−6.00000	−0.475831
$160$	4.41421	0.348974
$161$	0	0
$162$	−2.07107	−0.162718
$163$	18.0711	1.41544	0.707718	−	0.706495i	$-0.249725\pi$
$163$	18.0711	1.41544	0.707718	+	0.706495i	$0.249725\pi$
$164$	16.1421	1.26049
$165$	−3.41421	−0.265796
$166$	0.313708	0.0243485
$167$	−1.07107	−0.0828817	−0.0414409	−	0.999141i	$-0.513195\pi$
$167$	−1.07107	−0.0828817	−0.0414409	+	0.999141i	$0.513195\pi$
$168$	0	0
$169$	25.9706	1.99774
$170$	−1.65685	−0.127075
$171$	1.00000	0.0764719
$172$	11.1005	0.846406
$173$	15.3137	1.16428	0.582140	−	0.813089i	$-0.302216\pi$
$173$	15.3137	1.16428	0.582140	+	0.813089i	$0.302216\pi$
$174$	5.51472	0.418070
$175$	0	0
$176$	7.24264	0.545935
$177$	−1.65685	−0.124537
$178$	−3.55635	−0.266560
$179$	−21.7990	−1.62933	−0.814667	−	0.579930i	$-0.803080\pi$
$179$	−21.7990	−1.62933	−0.814667	+	0.579930i	$0.803080\pi$
$180$	1.82843	0.136283
$181$	−9.89949	−0.735824	−0.367912	−	0.929861i	$-0.619927\pi$
$181$	−9.89949	−0.735824	−0.367912	+	0.929861i	$0.619927\pi$
$182$	0	0
$183$	−8.72792	−0.645187
$184$	13.3431	0.983670
$185$	−5.07107	−0.372832
$186$	−1.31371	−0.0963258
$187$	−9.65685	−0.706179
$188$	2.89949	0.211467
$189$	0	0
$190$	−0.414214	−0.0300502
$191$	5.58579	0.404173	0.202087	−	0.979368i	$-0.435228\pi$
$191$	5.58579	0.404173	0.202087	+	0.979368i	$0.435228\pi$
$192$	5.89949	0.425759
$193$	−11.8995	−0.856544	−0.428272	−	0.903650i	$-0.640878\pi$
$193$	−11.8995	−0.856544	−0.428272	+	0.903650i	$0.640878\pi$
$194$	−3.65685	−0.262547
$195$	−8.82843	−0.632217
$196$	0	0
$197$	21.4853	1.53076	0.765381	−	0.643577i	$-0.222550\pi$
$197$	21.4853	1.53076	0.765381	+	0.643577i	$0.222550\pi$
$198$	−1.00000	−0.0710669
$199$	1.92893	0.136738	0.0683692	−	0.997660i	$-0.478220\pi$
$199$	1.92893	0.136738	0.0683692	+	0.997660i	$0.478220\pi$
$200$	6.34315	0.448528
$201$	−9.65685	−0.681142
$202$	−1.78680	−0.125719
$203$	0	0
$204$	−10.3431	−0.724165
$205$	−8.82843	−0.616604
$206$	−2.48528	−0.173158
$207$	8.41421	0.584828
$208$	18.7279	1.29855
$209$	−2.41421	−0.166995
$210$	0	0
$211$	−6.82843	−0.470088	−0.235044	−	0.971985i	$-0.575523\pi$
$211$	−6.82843	−0.470088	−0.235044	+	0.971985i	$0.575523\pi$
$212$	−7.75736	−0.532778
$213$	8.14214	0.557890
$214$	2.44365	0.167045
$215$	−6.07107	−0.414043
$216$	−8.97056	−0.610369
$217$	0	0
$218$	0.686292	0.0464815
$219$	18.5858	1.25591
$220$	−4.41421	−0.297606
$221$	−24.9706	−1.67970
$222$	2.97056	0.199371
$223$	−6.24264	−0.418038	−0.209019	−	0.977912i	$-0.567027\pi$
$223$	−6.24264	−0.418038	−0.209019	+	0.977912i	$0.567027\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	3.41421	0.227110
$227$	−11.7574	−0.780363	−0.390182	−	0.920738i	$-0.627588\pi$
$227$	−11.7574	−0.780363	−0.390182	+	0.920738i	$0.627588\pi$
$228$	−2.58579	−0.171248
$229$	−4.48528	−0.296396	−0.148198	−	0.988958i	$-0.547347\pi$
$229$	−4.48528	−0.296396	−0.148198	+	0.988958i	$0.547347\pi$
$230$	−3.48528	−0.229813
$231$	0	0
$232$	14.9289	0.980132
$233$	−16.4853	−1.07999	−0.539993	−	0.841669i	$-0.681573\pi$
$233$	−16.4853	−1.07999	−0.539993	+	0.841669i	$0.681573\pi$
$234$	−2.58579	−0.169038
$235$	−1.58579	−0.103445
$236$	−2.14214	−0.139441
$237$	2.00000	0.129914
$238$	0	0
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	−4.24264	−0.273861
$241$	22.0416	1.41983	0.709913	−	0.704289i	$-0.248734\pi$
$241$	22.0416	1.41983	0.709913	+	0.704289i	$0.248734\pi$
$242$	−2.14214	−0.137702
$243$	−9.89949	−0.635053
$244$	−11.2843	−0.722401
$245$	0	0
$246$	5.17157	0.329727
$247$	−6.24264	−0.397210
$248$	−3.55635	−0.225828
$249$	−1.07107	−0.0678762
$250$	−3.72792	−0.235774
$251$	11.7279	0.740260	0.370130	−	0.928980i	$-0.379313\pi$
$251$	11.7279	0.740260	0.370130	+	0.928980i	$0.379313\pi$
$252$	0	0
$253$	−20.3137	−1.27711
$254$	−1.75736	−0.110267
$255$	5.65685	0.354246
$256$	3.97056	0.248160
$257$	−13.3137	−0.830486	−0.415243	−	0.909710i	$-0.636304\pi$
$257$	−13.3137	−0.830486	−0.415243	+	0.909710i	$0.636304\pi$
$258$	3.55635	0.221409
$259$	0	0
$260$	−11.4142	−0.707879
$261$	9.41421	0.582725
$262$	8.48528	0.524222
$263$	−0.485281	−0.0299237	−0.0149619	−	0.999888i	$-0.504763\pi$
$263$	−0.485281	−0.0299237	−0.0149619	+	0.999888i	$0.504763\pi$
$264$	5.41421	0.333222
$265$	4.24264	0.260623
$266$	0	0
$267$	12.1421	0.743087
$268$	−12.4853	−0.762660
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	2.34315	0.142599
$271$	−6.75736	−0.410480	−0.205240	−	0.978712i	$-0.565798\pi$
$271$	−6.75736	−0.410480	−0.205240	+	0.978712i	$0.565798\pi$
$272$	−12.0000	−0.727607
$273$	0	0
$274$	−5.87006	−0.354623
$275$	−9.65685	−0.582330
$276$	−21.7574	−1.30964
$277$	28.4558	1.70975	0.854873	−	0.518837i	$-0.173635\pi$
$277$	28.4558	1.70975	0.854873	+	0.518837i	$0.173635\pi$
$278$	3.34315	0.200509
$279$	−2.24264	−0.134263
$280$	0	0
$281$	−19.7574	−1.17863	−0.589313	−	0.807905i	$-0.700601\pi$
$281$	−19.7574	−1.17863	−0.589313	+	0.807905i	$0.700601\pi$
$282$	0.928932	0.0553171
$283$	15.9289	0.946877	0.473438	−	0.880827i	$-0.343013\pi$
$283$	15.9289	0.946877	0.473438	+	0.880827i	$0.343013\pi$
$284$	10.5269	0.624657
$285$	1.41421	0.0837708
$286$	6.24264	0.369135
$287$	0	0
$288$	−4.41421	−0.260110
$289$	−1.00000	−0.0588235
$290$	−3.89949	−0.228986
$291$	12.4853	0.731900
$292$	24.0294	1.40622
$293$	8.82843	0.515762	0.257881	−	0.966177i	$-0.416976\pi$
$293$	8.82843	0.515762	0.257881	+	0.966177i	$0.416976\pi$
$294$	0	0
$295$	1.17157	0.0682116
$296$	8.04163	0.467410
$297$	13.6569	0.792451
$298$	−2.41421	−0.139852
$299$	−52.5269	−3.03771
$300$	−10.3431	−0.597162
$301$	0	0
$302$	2.20101	0.126654
$303$	6.10051	0.350465
$304$	−3.00000	−0.172062
$305$	6.17157	0.353383
$306$	1.65685	0.0947161
$307$	6.68629	0.381607	0.190803	−	0.981628i	$-0.438891\pi$
$307$	6.68629	0.381607	0.190803	+	0.981628i	$0.438891\pi$
$308$	0	0
$309$	8.48528	0.482711
$310$	0.928932	0.0527598
$311$	15.7990	0.895879	0.447939	−	0.894064i	$-0.352158\pi$
$311$	15.7990	0.895879	0.447939	+	0.894064i	$0.352158\pi$
$312$	14.0000	0.792594
$313$	4.65685	0.263221	0.131610	−	0.991302i	$-0.457985\pi$
$313$	4.65685	0.263221	0.131610	+	0.991302i	$0.457985\pi$
$314$	4.89949	0.276494
$315$	0	0
$316$	2.58579	0.145462
$317$	−11.7574	−0.660359	−0.330180	−	0.943918i	$-0.607109\pi$
$317$	−11.7574	−0.660359	−0.330180	+	0.943918i	$0.607109\pi$
$318$	−2.48528	−0.139368
$319$	−22.7279	−1.27252
$320$	−4.17157	−0.233198
$321$	−8.34315	−0.465669
$322$	0	0
$323$	4.00000	0.222566
$324$	9.14214	0.507896
$325$	−24.9706	−1.38512
$326$	7.48528	0.414571
$327$	−2.34315	−0.129576
$328$	14.0000	0.773021
$329$	0	0
$330$	−1.41421	−0.0778499
$331$	19.4558	1.06939	0.534695	−	0.845045i	$-0.320427\pi$
$331$	19.4558	1.06939	0.534695	+	0.845045i	$0.320427\pi$
$332$	−1.38478	−0.0759995
$333$	5.07107	0.277893
$334$	−0.443651	−0.0242755
$335$	6.82843	0.373077
$336$	0	0
$337$	−11.5563	−0.629514	−0.314757	−	0.949172i	$-0.601923\pi$
$337$	−11.5563	−0.629514	−0.314757	+	0.949172i	$0.601923\pi$
$338$	10.7574	0.585123
$339$	−11.6569	−0.633113
$340$	7.31371	0.396642
$341$	5.41421	0.293196
$342$	0.414214	0.0223981
$343$	0	0
$344$	9.62742	0.519076
$345$	11.8995	0.640647
$346$	6.34315	0.341010
$347$	−21.0416	−1.12957	−0.564787	−	0.825237i	$-0.691042\pi$
$347$	−21.0416	−1.12957	−0.564787	+	0.825237i	$0.691042\pi$
$348$	−24.3431	−1.30493
$349$	−12.4853	−0.668322	−0.334161	−	0.942516i	$-0.608453\pi$
$349$	−12.4853	−0.668322	−0.334161	+	0.942516i	$0.608453\pi$
$350$	0	0
$351$	35.3137	1.88491
$352$	10.6569	0.568012
$353$	−16.4853	−0.877423	−0.438711	−	0.898628i	$-0.644565\pi$
$353$	−16.4853	−0.877423	−0.438711	+	0.898628i	$0.644565\pi$
$354$	−0.686292	−0.0364760
$355$	−5.75736	−0.305569
$356$	15.6985	0.832018
$357$	0	0
$358$	−9.02944	−0.477221
$359$	−3.24264	−0.171140	−0.0855700	−	0.996332i	$-0.527271\pi$
$359$	−3.24264	−0.171140	−0.0855700	+	0.996332i	$0.527271\pi$
$360$	1.58579	0.0835783
$361$	1.00000	0.0526316
$362$	−4.10051	−0.215518
$363$	7.31371	0.383870
$364$	0	0
$365$	−13.1421	−0.687891
$366$	−3.61522	−0.188971
$367$	−35.6569	−1.86127	−0.930636	−	0.365945i	$-0.880746\pi$
$367$	−35.6569	−1.86127	−0.930636	+	0.365945i	$0.880746\pi$
$368$	−25.2426	−1.31586
$369$	8.82843	0.459590
$370$	−2.10051	−0.109200
$371$	0	0
$372$	5.79899	0.300664
$373$	−11.3137	−0.585802	−0.292901	−	0.956143i	$-0.594621\pi$
$373$	−11.3137	−0.585802	−0.292901	+	0.956143i	$0.594621\pi$
$374$	−4.00000	−0.206835
$375$	12.7279	0.657267
$376$	2.51472	0.129687
$377$	−58.7696	−3.02679
$378$	0	0
$379$	−4.24264	−0.217930	−0.108965	−	0.994046i	$-0.534754\pi$
$379$	−4.24264	−0.217930	−0.108965	+	0.994046i	$0.534754\pi$
$380$	1.82843	0.0937963
$381$	6.00000	0.307389
$382$	2.31371	0.118380
$383$	28.2426	1.44313	0.721566	−	0.692346i	$-0.243423\pi$
$383$	28.2426	1.44313	0.721566	+	0.692346i	$0.243423\pi$
$384$	14.9289	0.761839
$385$	0	0
$386$	−4.92893	−0.250876
$387$	6.07107	0.308610
$388$	16.1421	0.819493
$389$	−28.4853	−1.44426	−0.722131	−	0.691757i	$-0.756837\pi$
$389$	−28.4853	−1.44426	−0.722131	+	0.691757i	$0.756837\pi$
$390$	−3.65685	−0.185172
$391$	33.6569	1.70210
$392$	0	0
$393$	−28.9706	−1.46137
$394$	8.89949	0.448350
$395$	−1.41421	−0.0711568
$396$	4.41421	0.221823
$397$	36.6274	1.83828	0.919139	−	0.393934i	$-0.128886\pi$
$397$	36.6274	1.83828	0.919139	+	0.393934i	$0.128886\pi$
$398$	0.798990	0.0400497
$399$	0	0
$400$	−12.0000	−0.600000
$401$	−16.4853	−0.823236	−0.411618	−	0.911357i	$-0.635036\pi$
$401$	−16.4853	−0.823236	−0.411618	+	0.911357i	$0.635036\pi$
$402$	−4.00000	−0.199502
$403$	14.0000	0.697390
$404$	7.88730	0.392408
$405$	−5.00000	−0.248452
$406$	0	0
$407$	−12.2426	−0.606845
$408$	−8.97056	−0.444109
$409$	−25.3137	−1.25168	−0.625841	−	0.779951i	$-0.715244\pi$
$409$	−25.3137	−1.25168	−0.625841	+	0.779951i	$0.715244\pi$
$410$	−3.65685	−0.180599
$411$	20.0416	0.988581
$412$	10.9706	0.540481
$413$	0	0
$414$	3.48528	0.171292
$415$	0.757359	0.0371773
$416$	27.5563	1.35106
$417$	−11.4142	−0.558956
$418$	−1.00000	−0.0489116
$419$	−8.07107	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$419$	−8.07107	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$420$	0	0
$421$	5.21320	0.254076	0.127038	−	0.991898i	$-0.459453\pi$
$421$	5.21320	0.254076	0.127038	+	0.991898i	$0.459453\pi$
$422$	−2.82843	−0.137686
$423$	1.58579	0.0771036
$424$	−6.72792	−0.326737
$425$	16.0000	0.776114
$426$	3.37258	0.163402
$427$	0	0
$428$	−10.7868	−0.521399
$429$	−21.3137	−1.02904
$430$	−2.51472	−0.121271
$431$	38.0416	1.83240	0.916200	−	0.400720i	$-0.131240\pi$
$431$	38.0416	1.83240	0.916200	+	0.400720i	$0.131240\pi$
$432$	16.9706	0.816497
$433$	28.0000	1.34559	0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000	1.34559	0.672797	+	0.739827i	$0.265093\pi$
$434$	0	0
$435$	13.3137	0.638343
$436$	−3.02944	−0.145084
$437$	8.41421	0.402506
$438$	7.69848	0.367848
$439$	−7.51472	−0.358658	−0.179329	−	0.983789i	$-0.557393\pi$
$439$	−7.51472	−0.358658	−0.179329	+	0.983789i	$0.557393\pi$
$440$	−3.82843	−0.182513
$441$	0	0
$442$	−10.3431	−0.491973
$443$	6.34315	0.301372	0.150686	−	0.988582i	$-0.451852\pi$
$443$	6.34315	0.301372	0.150686	+	0.988582i	$0.451852\pi$
$444$	−13.1127	−0.622301
$445$	−8.58579	−0.407005
$446$	−2.58579	−0.122441
$447$	8.24264	0.389864
$448$	0	0
$449$	−1.65685	−0.0781918	−0.0390959	−	0.999235i	$-0.512448\pi$
$449$	−1.65685	−0.0781918	−0.0390959	+	0.999235i	$0.512448\pi$
$450$	1.65685	0.0781049
$451$	−21.3137	−1.00362
$452$	−15.0711	−0.708883
$453$	−7.51472	−0.353072
$454$	−4.87006	−0.228563
$455$	0	0
$456$	−2.24264	−0.105021
$457$	−2.17157	−0.101582	−0.0507909	−	0.998709i	$-0.516174\pi$
$457$	−2.17157	−0.101582	−0.0507909	+	0.998709i	$0.516174\pi$
$458$	−1.85786	−0.0868123
$459$	−22.6274	−1.05616
$460$	15.3848	0.717319
$461$	−3.97056	−0.184928	−0.0924638	−	0.995716i	$-0.529474\pi$
$461$	−3.97056	−0.184928	−0.0924638	+	0.995716i	$0.529474\pi$
$462$	0	0
$463$	10.0711	0.468042	0.234021	−	0.972232i	$-0.424812\pi$
$463$	10.0711	0.468042	0.234021	+	0.972232i	$0.424812\pi$
$464$	−28.2426	−1.31113
$465$	−3.17157	−0.147078
$466$	−6.82843	−0.316321
$467$	38.8995	1.80005	0.900027	−	0.435834i	$-0.143547\pi$
$467$	38.8995	1.80005	0.900027	+	0.435834i	$0.143547\pi$
$468$	11.4142	0.527622
$469$	0	0
$470$	−0.656854	−0.0302984
$471$	−16.7279	−0.770782
$472$	−1.85786	−0.0855151
$473$	−14.6569	−0.673923
$474$	0.828427	0.0380509
$475$	4.00000	0.183533
$476$	0	0
$477$	−4.24264	−0.194257
$478$	0.828427	0.0378914
$479$	11.3848	0.520184	0.260092	−	0.965584i	$-0.416247\pi$
$479$	11.3848	0.520184	0.260092	+	0.965584i	$0.416247\pi$
$480$	−6.24264	−0.284936
$481$	−31.6569	−1.44343
$482$	9.12994	0.415857
$483$	0	0
$484$	9.45584	0.429811
$485$	−8.82843	−0.400878
$486$	−4.10051	−0.186003
$487$	20.8284	0.943826	0.471913	−	0.881645i	$-0.343564\pi$
$487$	20.8284	0.943826	0.471913	+	0.881645i	$0.343564\pi$
$488$	−9.78680	−0.443027
$489$	−25.5563	−1.15570
$490$	0	0
$491$	−32.5563	−1.46925	−0.734624	−	0.678475i	$-0.762641\pi$
$491$	−32.5563	−1.46925	−0.734624	+	0.678475i	$0.762641\pi$
$492$	−22.8284	−1.02918
$493$	37.6569	1.69598
$494$	−2.58579	−0.116340
$495$	−2.41421	−0.108511
$496$	6.72792	0.302093
$497$	0	0
$498$	−0.443651	−0.0198805
$499$	3.44365	0.154159	0.0770795	−	0.997025i	$-0.475440\pi$
$499$	3.44365	0.154159	0.0770795	+	0.997025i	$0.475440\pi$
$500$	16.4558	0.735928
$501$	1.51472	0.0676726
$502$	4.85786	0.216817
$503$	19.0416	0.849024	0.424512	−	0.905422i	$-0.360446\pi$
$503$	19.0416	0.849024	0.424512	+	0.905422i	$0.360446\pi$
$504$	0	0
$505$	−4.31371	−0.191958
$506$	−8.41421	−0.374057
$507$	−36.7279	−1.63114
$508$	7.75736	0.344177
$509$	18.9706	0.840855	0.420428	−	0.907326i	$-0.361880\pi$
$509$	18.9706	0.840855	0.420428	+	0.907326i	$0.361880\pi$
$510$	2.34315	0.103756
$511$	0	0
$512$	22.7574	1.00574
$513$	−5.65685	−0.249756
$514$	−5.51472	−0.243244
$515$	−6.00000	−0.264392
$516$	−15.6985	−0.691087
$517$	−3.82843	−0.168374
$518$	0	0
$519$	−21.6569	−0.950630
$520$	−9.89949	−0.434122
$521$	10.4437	0.457545	0.228772	−	0.973480i	$-0.426529\pi$
$521$	10.4437	0.457545	0.228772	+	0.973480i	$0.426529\pi$
$522$	3.89949	0.170676
$523$	−23.6569	−1.03444	−0.517221	−	0.855852i	$-0.673033\pi$
$523$	−23.6569	−1.03444	−0.517221	+	0.855852i	$0.673033\pi$
$524$	−37.4558	−1.63627
$525$	0	0
$526$	−0.201010	−0.00876446
$527$	−8.97056	−0.390764
$528$	−10.2426	−0.445754
$529$	47.7990	2.07822
$530$	1.75736	0.0763348
$531$	−1.17157	−0.0508419
$532$	0	0
$533$	−55.1127	−2.38720
$534$	5.02944	0.217645
$535$	5.89949	0.255057
$536$	−10.8284	−0.467717
$537$	30.8284	1.33034
$538$	−1.65685	−0.0714321
$539$	0	0
$540$	−10.3431	−0.445098
$541$	5.14214	0.221078	0.110539	−	0.993872i	$-0.464742\pi$
$541$	5.14214	0.221078	0.110539	+	0.993872i	$0.464742\pi$
$542$	−2.79899	−0.120227
$543$	14.0000	0.600798
$544$	−17.6569	−0.757031
$545$	1.65685	0.0709718
$546$	0	0
$547$	29.5563	1.26374	0.631869	−	0.775075i	$-0.282288\pi$
$547$	29.5563	1.26374	0.631869	+	0.775075i	$0.282288\pi$
$548$	25.9117	1.10689
$549$	−6.17157	−0.263396
$550$	−4.00000	−0.170561
$551$	9.41421	0.401059
$552$	−18.8701	−0.803163
$553$	0	0
$554$	11.7868	0.500773
$555$	7.17157	0.304416
$556$	−14.7574	−0.625851
$557$	10.1716	0.430983	0.215492	−	0.976506i	$-0.430865\pi$
$557$	10.1716	0.430983	0.215492	+	0.976506i	$0.430865\pi$
$558$	−0.928932	−0.0393248
$559$	−37.8995	−1.60298
$560$	0	0
$561$	13.6569	0.576593
$562$	−8.18377	−0.345211
$563$	−13.8995	−0.585794	−0.292897	−	0.956144i	$-0.594619\pi$
$563$	−13.8995	−0.585794	−0.292897	+	0.956144i	$0.594619\pi$
$564$	−4.10051	−0.172662
$565$	8.24264	0.346770
$566$	6.59798	0.277334
$567$	0	0
$568$	9.12994	0.383084
$569$	−42.9706	−1.80142	−0.900710	−	0.434421i	$-0.856953\pi$
$569$	−42.9706	−1.80142	−0.900710	+	0.434421i	$0.856953\pi$
$570$	0.585786	0.0245359
$571$	−18.8995	−0.790919	−0.395460	−	0.918483i	$-0.629415\pi$
$571$	−18.8995	−0.790919	−0.395460	+	0.918483i	$0.629415\pi$
$572$	−27.5563	−1.15219
$573$	−7.89949	−0.330006
$574$	0	0
$575$	33.6569	1.40359
$576$	4.17157	0.173816
$577$	27.9706	1.16443	0.582215	−	0.813035i	$-0.302186\pi$
$577$	27.9706	1.16443	0.582215	+	0.813035i	$0.302186\pi$
$578$	−0.414214	−0.0172290
$579$	16.8284	0.699366
$580$	17.2132	0.714739
$581$	0	0
$582$	5.17157	0.214369
$583$	10.2426	0.424207
$584$	20.8406	0.862391
$585$	−6.24264	−0.258101
$586$	3.65685	0.151063
$587$	14.0000	0.577842	0.288921	−	0.957353i	$-0.406704\pi$
$587$	14.0000	0.577842	0.288921	+	0.957353i	$0.406704\pi$
$588$	0	0
$589$	−2.24264	−0.0924064
$590$	0.485281	0.0199787
$591$	−30.3848	−1.24986
$592$	−15.2132	−0.625259
$593$	6.17157	0.253436	0.126718	−	0.991939i	$-0.459556\pi$
$593$	6.17157	0.253436	0.126718	+	0.991939i	$0.459556\pi$
$594$	5.65685	0.232104
$595$	0	0
$596$	10.6569	0.436522
$597$	−2.72792	−0.111646
$598$	−21.7574	−0.889725
$599$	−31.6985	−1.29516	−0.647582	−	0.761995i	$-0.724220\pi$
$599$	−31.6985	−1.29516	−0.647582	+	0.761995i	$0.724220\pi$
$600$	−8.97056	−0.366222
$601$	−9.27208	−0.378216	−0.189108	−	0.981956i	$-0.560560\pi$
$601$	−9.27208	−0.378216	−0.189108	+	0.981956i	$0.560560\pi$
$602$	0	0
$603$	−6.82843	−0.278075
$604$	−9.71573	−0.395327
$605$	−5.17157	−0.210254
$606$	2.52691	0.102649
$607$	32.5269	1.32023	0.660113	−	0.751166i	$-0.270508\pi$
$607$	32.5269	1.32023	0.660113	+	0.751166i	$0.270508\pi$
$608$	−4.41421	−0.179020
$609$	0	0
$610$	2.55635	0.103504
$611$	−9.89949	−0.400491
$612$	−7.31371	−0.295639
$613$	−8.68629	−0.350836	−0.175418	−	0.984494i	$-0.556128\pi$
$613$	−8.68629	−0.350836	−0.175418	+	0.984494i	$0.556128\pi$
$614$	2.76955	0.111770
$615$	12.4853	0.503455
$616$	0	0
$617$	−42.4558	−1.70921	−0.854604	−	0.519280i	$-0.826200\pi$
$617$	−42.4558	−1.70921	−0.854604	+	0.519280i	$0.826200\pi$
$618$	3.51472	0.141383
$619$	3.44365	0.138412	0.0692060	−	0.997602i	$-0.477953\pi$
$619$	3.44365	0.138412	0.0692060	+	0.997602i	$0.477953\pi$
$620$	−4.10051	−0.164680
$621$	−47.5980	−1.91004
$622$	6.54416	0.262397
$623$	0	0
$624$	−26.4853	−1.06026
$625$	11.0000	0.440000
$626$	1.92893	0.0770956
$627$	3.41421	0.136351
$628$	−21.6274	−0.863028
$629$	20.2843	0.808787
$630$	0	0
$631$	−5.44365	−0.216708	−0.108354	−	0.994112i	$-0.534558\pi$
$631$	−5.44365	−0.216708	−0.108354	+	0.994112i	$0.534558\pi$
$632$	2.24264	0.0892075
$633$	9.65685	0.383825
$634$	−4.87006	−0.193415
$635$	−4.24264	−0.168364
$636$	10.9706	0.435011
$637$	0	0
$638$	−9.41421	−0.372712
$639$	5.75736	0.227758
$640$	−10.5563	−0.417276
$641$	3.79899	0.150051	0.0750255	−	0.997182i	$-0.476096\pi$
$641$	3.79899	0.150051	0.0750255	+	0.997182i	$0.476096\pi$
$642$	−3.45584	−0.136391
$643$	30.1421	1.18869	0.594345	−	0.804210i	$-0.297411\pi$
$643$	30.1421	1.18869	0.594345	+	0.804210i	$0.297411\pi$
$644$	0	0
$645$	8.58579	0.338065
$646$	1.65685	0.0651881
$647$	6.21320	0.244266	0.122133	−	0.992514i	$-0.461027\pi$
$647$	6.21320	0.244266	0.122133	+	0.992514i	$0.461027\pi$
$648$	7.92893	0.311478
$649$	2.82843	0.111025
$650$	−10.3431	−0.405692
$651$	0	0
$652$	−33.0416	−1.29401
$653$	−1.85786	−0.0727039	−0.0363519	−	0.999339i	$-0.511574\pi$
$653$	−1.85786	−0.0727039	−0.0363519	+	0.999339i	$0.511574\pi$
$654$	−0.970563	−0.0379520
$655$	20.4853	0.800426
$656$	−26.4853	−1.03408
$657$	13.1421	0.512724
$658$	0	0
$659$	−38.0416	−1.48189	−0.740946	−	0.671565i	$-0.765622\pi$
$659$	−38.0416	−1.48189	−0.740946	+	0.671565i	$0.765622\pi$
$660$	6.24264	0.242994
$661$	−22.7279	−0.884014	−0.442007	−	0.897012i	$-0.645733\pi$
$661$	−22.7279	−0.884014	−0.442007	+	0.897012i	$0.645733\pi$
$662$	8.05887	0.313217
$663$	35.3137	1.37147
$664$	−1.20101	−0.0466082
$665$	0	0
$666$	2.10051	0.0813929
$667$	79.2132	3.06715
$668$	1.95837	0.0757716
$669$	8.82843	0.341327
$670$	2.82843	0.109272
$671$	14.8995	0.575189
$672$	0	0
$673$	35.7990	1.37995	0.689975	−	0.723833i	$-0.257622\pi$
$673$	35.7990	1.37995	0.689975	+	0.723833i	$0.257622\pi$
$674$	−4.78680	−0.184381
$675$	−22.6274	−0.870930
$676$	−47.4853	−1.82636
$677$	0.686292	0.0263763	0.0131882	−	0.999913i	$-0.495802\pi$
$677$	0.686292	0.0263763	0.0131882	+	0.999913i	$0.495802\pi$
$678$	−4.82843	−0.185435
$679$	0	0
$680$	6.34315	0.243249
$681$	16.6274	0.637164
$682$	2.24264	0.0858752
$683$	10.4853	0.401208	0.200604	−	0.979672i	$-0.435710\pi$
$683$	10.4853	0.401208	0.200604	+	0.979672i	$0.435710\pi$
$684$	−1.82843	−0.0699117
$685$	−14.1716	−0.541468
$686$	0	0
$687$	6.34315	0.242006
$688$	−18.2132	−0.694372
$689$	26.4853	1.00901
$690$	4.92893	0.187641
$691$	−2.97056	−0.113006	−0.0565028	−	0.998402i	$-0.517995\pi$
$691$	−2.97056	−0.113006	−0.0565028	+	0.998402i	$0.517995\pi$
$692$	−28.0000	−1.06440
$693$	0	0
$694$	−8.71573	−0.330845
$695$	8.07107	0.306153
$696$	−21.1127	−0.800275
$697$	35.3137	1.33760
$698$	−5.17157	−0.195747
$699$	23.3137	0.881805
$700$	0	0
$701$	3.82843	0.144598	0.0722988	−	0.997383i	$-0.476966\pi$
$701$	3.82843	0.144598	0.0722988	+	0.997383i	$0.476966\pi$
$702$	14.6274	0.552076
$703$	5.07107	0.191259
$704$	−10.0711	−0.379568
$705$	2.24264	0.0844627
$706$	−6.82843	−0.256991
$707$	0	0
$708$	3.02944	0.113853
$709$	24.9411	0.936684	0.468342	−	0.883547i	$-0.344852\pi$
$709$	24.9411	0.936684	0.468342	+	0.883547i	$0.344852\pi$
$710$	−2.38478	−0.0894991
$711$	1.41421	0.0530372
$712$	13.6152	0.510252
$713$	−18.8701	−0.706689
$714$	0	0
$715$	15.0711	0.563626
$716$	39.8579	1.48956
$717$	−2.82843	−0.105630
$718$	−1.34315	−0.0501258
$719$	−44.0833	−1.64403	−0.822014	−	0.569467i	$-0.807150\pi$
$719$	−44.0833	−1.64403	−0.822014	+	0.569467i	$0.807150\pi$
$720$	−3.00000	−0.111803
$721$	0	0
$722$	0.414214	0.0154154
$723$	−31.1716	−1.15928
$724$	18.1005	0.672700
$725$	37.6569	1.39854
$726$	3.02944	0.112433
$727$	5.92893	0.219892	0.109946	−	0.993938i	$-0.464932\pi$
$727$	5.92893	0.219892	0.109946	+	0.993938i	$0.464932\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	−5.44365	−0.201479
$731$	24.2843	0.898186
$732$	15.9584	0.589838
$733$	−29.4558	−1.08798	−0.543988	−	0.839093i	$-0.683086\pi$
$733$	−29.4558	−1.08798	−0.543988	+	0.839093i	$0.683086\pi$
$734$	−14.7696	−0.545154
$735$	0	0
$736$	−37.1421	−1.36908
$737$	16.4853	0.607243
$738$	3.65685	0.134611
$739$	−19.6569	−0.723089	−0.361545	−	0.932355i	$-0.617750\pi$
$739$	−19.6569	−0.723089	−0.361545	+	0.932355i	$0.617750\pi$
$740$	9.27208	0.340848
$741$	8.82843	0.324320
$742$	0	0
$743$	50.8701	1.86624	0.933121	−	0.359563i	$-0.117074\pi$
$743$	50.8701	1.86624	0.933121	+	0.359563i	$0.117074\pi$
$744$	5.02944	0.184388
$745$	−5.82843	−0.213537
$746$	−4.68629	−0.171577
$747$	−0.757359	−0.0277103
$748$	17.6569	0.645599
$749$	0	0
$750$	5.27208	0.192509
$751$	−41.4558	−1.51275	−0.756373	−	0.654141i	$-0.773030\pi$
$751$	−41.4558	−1.51275	−0.756373	+	0.654141i	$0.773030\pi$
$752$	−4.75736	−0.173483
$753$	−16.5858	−0.604420
$754$	−24.3431	−0.886525
$755$	5.31371	0.193386
$756$	0	0
$757$	−5.00000	−0.181728	−0.0908640	−	0.995863i	$-0.528963\pi$
$757$	−5.00000	−0.181728	−0.0908640	+	0.995863i	$0.528963\pi$
$758$	−1.75736	−0.0638302
$759$	28.7279	1.04276
$760$	1.58579	0.0575225
$761$	28.1127	1.01908	0.509542	−	0.860446i	$-0.329815\pi$
$761$	28.1127	1.01908	0.509542	+	0.860446i	$0.329815\pi$
$762$	2.48528	0.0900322
$763$	0	0
$764$	−10.2132	−0.369501
$765$	4.00000	0.144620
$766$	11.6985	0.422683
$767$	7.31371	0.264083
$768$	−5.61522	−0.202622
$769$	41.4264	1.49387	0.746937	−	0.664895i	$-0.231524\pi$
$769$	41.4264	1.49387	0.746937	+	0.664895i	$0.231524\pi$
$770$	0	0
$771$	18.8284	0.678089
$772$	21.7574	0.783064
$773$	−25.7574	−0.926428	−0.463214	−	0.886247i	$-0.653304\pi$
$773$	−25.7574	−0.926428	−0.463214	+	0.886247i	$0.653304\pi$
$774$	2.51472	0.0903897
$775$	−8.97056	−0.322232
$776$	14.0000	0.502571
$777$	0	0
$778$	−11.7990	−0.423014
$779$	8.82843	0.316311
$780$	16.1421	0.577981
$781$	−13.8995	−0.497363
$782$	13.9411	0.498534
$783$	−53.2548	−1.90317
$784$	0	0
$785$	11.8284	0.422175
$786$	−12.0000	−0.428026
$787$	−41.0122	−1.46193	−0.730963	−	0.682417i	$-0.760929\pi$
$787$	−41.0122	−1.46193	−0.730963	+	0.682417i	$0.760929\pi$
$788$	−39.2843	−1.39944
$789$	0.686292	0.0244326
$790$	−0.585786	−0.0208413
$791$	0	0
$792$	3.82843	0.136037
$793$	38.5269	1.36813
$794$	15.1716	0.538419
$795$	−6.00000	−0.212798
$796$	−3.52691	−0.125008
$797$	−21.7574	−0.770685	−0.385343	−	0.922774i	$-0.625917\pi$
$797$	−21.7574	−0.770685	−0.385343	+	0.922774i	$0.625917\pi$
$798$	0	0
$799$	6.34315	0.224404
$800$	−17.6569	−0.624264
$801$	8.58579	0.303364
$802$	−6.82843	−0.241120
$803$	−31.7279	−1.11965
$804$	17.6569	0.622709
$805$	0	0
$806$	5.79899	0.204261
$807$	5.65685	0.199131
$808$	6.84062	0.240652
$809$	2.85786	0.100477	0.0502386	−	0.998737i	$-0.484002\pi$
$809$	2.85786	0.100477	0.0502386	+	0.998737i	$0.484002\pi$
$810$	−2.07107	−0.0727699
$811$	−51.4558	−1.80686	−0.903430	−	0.428737i	$-0.858959\pi$
$811$	−51.4558	−1.80686	−0.903430	+	0.428737i	$0.858959\pi$
$812$	0	0
$813$	9.55635	0.335156
$814$	−5.07107	−0.177741
$815$	18.0711	0.633002
$816$	16.9706	0.594089
$817$	6.07107	0.212400
$818$	−10.4853	−0.366609
$819$	0	0
$820$	16.1421	0.563708
$821$	−52.1127	−1.81875	−0.909373	−	0.415982i	$-0.863438\pi$
$821$	−52.1127	−1.81875	−0.909373	+	0.415982i	$0.863438\pi$
$822$	8.30152	0.289549
$823$	36.2132	1.26231	0.631156	−	0.775656i	$-0.282581\pi$
$823$	36.2132	1.26231	0.631156	+	0.775656i	$0.282581\pi$
$824$	9.51472	0.331461
$825$	13.6569	0.475471
$826$	0	0
$827$	−44.2843	−1.53991	−0.769957	−	0.638095i	$-0.779723\pi$
$827$	−44.2843	−1.53991	−0.769957	+	0.638095i	$0.779723\pi$
$828$	−15.3848	−0.534658
$829$	28.2843	0.982353	0.491177	−	0.871060i	$-0.336567\pi$
$829$	28.2843	0.982353	0.491177	+	0.871060i	$0.336567\pi$
$830$	0.313708	0.0108890
$831$	−40.2426	−1.39600
$832$	−26.0416	−0.902831
$833$	0	0
$834$	−4.72792	−0.163715
$835$	−1.07107	−0.0370658
$836$	4.41421	0.152669
$837$	12.6863	0.438502
$838$	−3.34315	−0.115487
$839$	19.7990	0.683537	0.341769	−	0.939784i	$-0.388974\pi$
$839$	19.7990	0.683537	0.341769	+	0.939784i	$0.388974\pi$
$840$	0	0
$841$	59.6274	2.05612
$842$	2.15938	0.0744171
$843$	27.9411	0.962343
$844$	12.4853	0.429761
$845$	25.9706	0.893415
$846$	0.656854	0.0225831
$847$	0	0
$848$	12.7279	0.437079
$849$	−22.5269	−0.773122
$850$	6.62742	0.227319
$851$	42.6690	1.46268
$852$	−14.8873	−0.510031
$853$	10.0294	0.343401	0.171701	−	0.985149i	$-0.445074\pi$
$853$	10.0294	0.343401	0.171701	+	0.985149i	$0.445074\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	−9.35534	−0.319759
$857$	20.9706	0.716341	0.358170	−	0.933656i	$-0.383401\pi$
$857$	20.9706	0.716341	0.358170	+	0.933656i	$0.383401\pi$
$858$	−8.82843	−0.301398
$859$	2.07107	0.0706639	0.0353320	−	0.999376i	$-0.488751\pi$
$859$	2.07107	0.0706639	0.0353320	+	0.999376i	$0.488751\pi$
$860$	11.1005	0.378524
$861$	0	0
$862$	15.7574	0.536698
$863$	7.41421	0.252383	0.126191	−	0.992006i	$-0.459725\pi$
$863$	7.41421	0.252383	0.126191	+	0.992006i	$0.459725\pi$
$864$	24.9706	0.849516
$865$	15.3137	0.520682
$866$	11.5980	0.394115
$867$	1.41421	0.0480292
$868$	0	0
$869$	−3.41421	−0.115819
$870$	5.51472	0.186966
$871$	42.6274	1.44437
$872$	−2.62742	−0.0889756
$873$	8.82843	0.298797
$874$	3.48528	0.117891
$875$	0	0
$876$	−33.9828	−1.14817
$877$	−8.10051	−0.273535	−0.136767	−	0.990603i	$-0.543671\pi$
$877$	−8.10051	−0.273535	−0.136767	+	0.990603i	$0.543671\pi$
$878$	−3.11270	−0.105048
$879$	−12.4853	−0.421118
$880$	7.24264	0.244149
$881$	4.28427	0.144341	0.0721704	−	0.997392i	$-0.477007\pi$
$881$	4.28427	0.144341	0.0721704	+	0.997392i	$0.477007\pi$
$882$	0	0
$883$	5.65685	0.190368	0.0951842	−	0.995460i	$-0.469656\pi$
$883$	5.65685	0.190368	0.0951842	+	0.995460i	$0.469656\pi$
$884$	45.6569	1.53561
$885$	−1.65685	−0.0556945
$886$	2.62742	0.0882698
$887$	1.31371	0.0441100	0.0220550	−	0.999757i	$-0.492979\pi$
$887$	1.31371	0.0441100	0.0220550	+	0.999757i	$0.492979\pi$
$888$	−11.3726	−0.381639
$889$	0	0
$890$	−3.55635	−0.119209
$891$	−12.0711	−0.404396
$892$	11.4142	0.382176
$893$	1.58579	0.0530663
$894$	3.41421	0.114188
$895$	−21.7990	−0.728660
$896$	0	0
$897$	74.2843	2.48028
$898$	−0.686292	−0.0229018
$899$	−21.1127	−0.704148
$900$	−7.31371	−0.243790
$901$	−16.9706	−0.565371
$902$	−8.82843	−0.293954
$903$	0	0
$904$	−13.0711	−0.434737
$905$	−9.89949	−0.329070
$906$	−3.11270	−0.103412
$907$	28.5858	0.949175	0.474588	−	0.880208i	$-0.342597\pi$
$907$	28.5858	0.949175	0.474588	+	0.880208i	$0.342597\pi$
$908$	21.4975	0.713419
$909$	4.31371	0.143077
$910$	0	0
$911$	34.7279	1.15059	0.575294	−	0.817947i	$-0.304888\pi$
$911$	34.7279	1.15059	0.575294	+	0.817947i	$0.304888\pi$
$912$	4.24264	0.140488
$913$	1.82843	0.0605121
$914$	−0.899495	−0.0297526
$915$	−8.72792	−0.288536
$916$	8.20101	0.270969
$917$	0	0
$918$	−9.37258	−0.309341
$919$	29.0416	0.957995	0.478997	−	0.877816i	$-0.341000\pi$
$919$	29.0416	0.957995	0.478997	+	0.877816i	$0.341000\pi$
$920$	13.3431	0.439910
$921$	−9.45584	−0.311581
$922$	−1.64466	−0.0541640
$923$	−35.9411	−1.18302
$924$	0	0
$925$	20.2843	0.666943
$926$	4.17157	0.137086
$927$	6.00000	0.197066
$928$	−41.5563	−1.36415
$929$	15.0000	0.492134	0.246067	−	0.969253i	$-0.420862\pi$
$929$	15.0000	0.492134	0.246067	+	0.969253i	$0.420862\pi$
$930$	−1.31371	−0.0430782
$931$	0	0
$932$	30.1421	0.987338
$933$	−22.3431	−0.731482
$934$	16.1127	0.527224
$935$	−9.65685	−0.315813
$936$	9.89949	0.323575
$937$	−56.3137	−1.83969	−0.919844	−	0.392284i	$-0.871685\pi$
$937$	−56.3137	−1.83969	−0.919844	+	0.392284i	$0.871685\pi$
$938$	0	0
$939$	−6.58579	−0.214919
$940$	2.89949	0.0945711
$941$	−44.0416	−1.43572	−0.717858	−	0.696189i	$-0.754877\pi$
$941$	−44.0416	−1.43572	−0.717858	+	0.696189i	$0.754877\pi$
$942$	−6.92893	−0.225757
$943$	74.2843	2.41903
$944$	3.51472	0.114394
$945$	0	0
$946$	−6.07107	−0.197387
$947$	−7.65685	−0.248814	−0.124407	−	0.992231i	$-0.539703\pi$
$947$	−7.65685	−0.248814	−0.124407	+	0.992231i	$0.539703\pi$
$948$	−3.65685	−0.118769
$949$	−82.0416	−2.66318
$950$	1.65685	0.0537555
$951$	16.6274	0.539181
$952$	0	0
$953$	−0.585786	−0.0189755	−0.00948774	−	0.999955i	$-0.503020\pi$
$953$	−0.585786	−0.0189755	−0.00948774	+	0.999955i	$0.503020\pi$
$954$	−1.75736	−0.0568966
$955$	5.58579	0.180752
$956$	−3.65685	−0.118271
$957$	32.1421	1.03901
$958$	4.71573	0.152358
$959$	0	0
$960$	5.89949	0.190405
$961$	−25.9706	−0.837760
$962$	−13.1127	−0.422770
$963$	−5.89949	−0.190109
$964$	−40.3015	−1.29802
$965$	−11.8995	−0.383058
$966$	0	0
$967$	20.2843	0.652298	0.326149	−	0.945318i	$-0.394249\pi$
$967$	20.2843	0.652298	0.326149	+	0.945318i	$0.394249\pi$
$968$	8.20101	0.263590
$969$	−5.65685	−0.181724
$970$	−3.65685	−0.117415
$971$	−26.8701	−0.862301	−0.431151	−	0.902280i	$-0.641892\pi$
$971$	−26.8701	−0.862301	−0.431151	+	0.902280i	$0.641892\pi$
$972$	18.1005	0.580574
$973$	0	0
$974$	8.62742	0.276440
$975$	35.3137	1.13094
$976$	18.5147	0.592642
$977$	28.7696	0.920420	0.460210	−	0.887810i	$-0.347774\pi$
$977$	28.7696	0.920420	0.460210	+	0.887810i	$0.347774\pi$
$978$	−10.5858	−0.338496
$979$	−20.7279	−0.662467
$980$	0	0
$981$	−1.65685	−0.0528993
$982$	−13.4853	−0.430333
$983$	−46.0000	−1.46717	−0.733586	−	0.679597i	$-0.762155\pi$
$983$	−46.0000	−1.46717	−0.733586	+	0.679597i	$0.762155\pi$
$984$	−19.7990	−0.631169
$985$	21.4853	0.684578
$986$	15.5980	0.496741
$987$	0	0
$988$	11.4142	0.363135
$989$	51.0833	1.62435
$990$	−1.00000	−0.0317821
$991$	−35.7990	−1.13719	−0.568596	−	0.822617i	$-0.692513\pi$
$991$	−35.7990	−1.13719	−0.568596	+	0.822617i	$0.692513\pi$
$992$	9.89949	0.314309
$993$	−27.5147	−0.873153
$994$	0	0
$995$	1.92893	0.0611513
$996$	1.95837	0.0620533
$997$	28.2843	0.895772	0.447886	−	0.894091i	$-0.352177\pi$
$997$	28.2843	0.895772	0.447886	+	0.894091i	$0.352177\pi$
$998$	1.42641	0.0451521
$999$	−28.6863	−0.907594

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	931.2.a.f.1.2		2
3.2	odd	2		8379.2.a.bi.1.1		2
7.2	even	3		133.2.f.c.39.1	✓	4
7.3	odd	6		931.2.f.i.324.1		4
7.4	even	3		133.2.f.c.58.1	yes	4
7.5	odd	6		931.2.f.i.704.1		4
7.6	odd	2		931.2.a.e.1.2		2
21.2	odd	6		1197.2.j.e.172.2		4
21.11	odd	6		1197.2.j.e.856.2		4
21.20	even	2		8379.2.a.bl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
133.2.f.c.39.1	✓	4	7.2	even	3
133.2.f.c.58.1	yes	4	7.4	even	3
931.2.a.e.1.2		2	7.6	odd	2
931.2.a.f.1.2		2	1.1	even	1	trivial
931.2.f.i.324.1		4	7.3	odd	6
931.2.f.i.704.1		4	7.5	odd	6
1197.2.j.e.172.2		4	21.2	odd	6
1197.2.j.e.856.2		4	21.11	odd	6
8379.2.a.bi.1.1		2	3.2	odd	2
8379.2.a.bl.1.1		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 931 = 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	931.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -1.41421 q^{3} -1.82843 q^{4} +1.00000 q^{5} -0.585786 q^{6} -1.58579 q^{8} -1.00000 q^{9} +0.414214 q^{10} +2.41421 q^{11} +2.58579 q^{12} +6.24264 q^{13} -1.41421 q^{15} +3.00000 q^{16} -4.00000 q^{17} -0.414214 q^{18} -1.00000 q^{19} -1.82843 q^{20} +1.00000 q^{22} -8.41421 q^{23} +2.24264 q^{24} -4.00000 q^{25} +2.58579 q^{26} +5.65685 q^{27} -9.41421 q^{29} -0.585786 q^{30} +2.24264 q^{31} +4.41421 q^{32} -3.41421 q^{33} -1.65685 q^{34} +1.82843 q^{36} -5.07107 q^{37} -0.414214 q^{38} -8.82843 q^{39} -1.58579 q^{40} -8.82843 q^{41} -6.07107 q^{43} -4.41421 q^{44} -1.00000 q^{45} -3.48528 q^{46} -1.58579 q^{47} -4.24264 q^{48} -1.65685 q^{50} +5.65685 q^{51} -11.4142 q^{52} +4.24264 q^{53} +2.34315 q^{54} +2.41421 q^{55} +1.41421 q^{57} -3.89949 q^{58} +1.17157 q^{59} +2.58579 q^{60} +6.17157 q^{61} +0.928932 q^{62} -4.17157 q^{64} +6.24264 q^{65} -1.41421 q^{66} +6.82843 q^{67} +7.31371 q^{68} +11.8995 q^{69} -5.75736 q^{71} +1.58579 q^{72} -13.1421 q^{73} -2.10051 q^{74} +5.65685 q^{75} +1.82843 q^{76} -3.65685 q^{78} -1.41421 q^{79} +3.00000 q^{80} -5.00000 q^{81} -3.65685 q^{82} +0.757359 q^{83} -4.00000 q^{85} -2.51472 q^{86} +13.3137 q^{87} -3.82843 q^{88} -8.58579 q^{89} -0.414214 q^{90} +15.3848 q^{92} -3.17157 q^{93} -0.656854 q^{94} -1.00000 q^{95} -6.24264 q^{96} -8.82843 q^{97} -2.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 4 q^{6} - 6 q^{8} - 2 q^{9} - 2 q^{10} + 2 q^{11} + 8 q^{12} + 4 q^{13} + 6 q^{16} - 8 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{20} + 2 q^{22} - 14 q^{23} - 4 q^{24} - 8 q^{25}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 4 * q^6 - 6 * q^8 - 2 * q^9 - 2 * q^10 + 2 * q^11 + 8 * q^12 + 4 * q^13 + 6 * q^16 - 8 * q^17 + 2 * q^18 - 2 * q^19 + 2 * q^20 + 2 * q^22 - 14 * q^23 - 4 * q^24 - 8 * q^25 + 8 * q^26 - 16 * q^29 - 4 * q^30 - 4 * q^31 + 6 * q^32 - 4 * q^33 + 8 * q^34 - 2 * q^36 + 4 * q^37 + 2 * q^38 - 12 * q^39 - 6 * q^40 - 12 * q^41 + 2 * q^43 - 6 * q^44 - 2 * q^45 + 10 * q^46 - 6 * q^47 + 8 * q^50 - 20 * q^52 + 16 * q^54 + 2 * q^55 + 12 * q^58 + 8 * q^59 + 8 * q^60 + 18 * q^61 + 16 * q^62 - 14 * q^64 + 4 * q^65 + 8 * q^67 - 8 * q^68 + 4 * q^69 - 20 * q^71 + 6 * q^72 + 2 * q^73 - 24 * q^74 - 2 * q^76 + 4 * q^78 + 6 * q^80 - 10 * q^81 + 4 * q^82 + 10 * q^83 - 8 * q^85 - 22 * q^86 + 4 * q^87 - 2 * q^88 - 20 * q^89 + 2 * q^90 - 6 * q^92 - 12 * q^93 + 10 * q^94 - 2 * q^95 - 4 * q^96 - 12 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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