LMFDB - Embedded newform 538.2.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [538,2,Mod(1,538)] 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("538.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(538, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	538.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,-7,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.29595162874$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 10x^{5} + 7x^{4} + 27x^{3} - 15x^{2} - 20x + 12 $ x^7 - x^6 - 10x^5 + 7x^4 + 27x^3 - 15x^2 - 20*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.870832$ of defining polynomial
Character	$\chi$	$=$	538.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -2.79016 q^{3} +1.00000 q^{4} +2.36234 q^{5} +2.79016 q^{6} -3.88429 q^{7} -1.00000 q^{8} +4.78499 q^{9} -2.36234 q^{10} +2.24531 q^{11} -2.79016 q^{12} -0.152297 q^{13} +3.88429 q^{14} -6.59131 q^{15} +1.00000 q^{16} -0.664994 q^{17} -4.78499 q^{18} +1.76305 q^{19} +2.36234 q^{20} +10.8378 q^{21} -2.24531 q^{22} +1.84606 q^{23} +2.79016 q^{24} +0.580659 q^{25} +0.152297 q^{26} -4.98041 q^{27} -3.88429 q^{28} -5.78775 q^{29} +6.59131 q^{30} -8.74754 q^{31} -1.00000 q^{32} -6.26478 q^{33} +0.664994 q^{34} -9.17602 q^{35} +4.78499 q^{36} +1.17330 q^{37} -1.76305 q^{38} +0.424932 q^{39} -2.36234 q^{40} -2.82985 q^{41} -10.8378 q^{42} -10.8695 q^{43} +2.24531 q^{44} +11.3038 q^{45} -1.84606 q^{46} -5.51807 q^{47} -2.79016 q^{48} +8.08771 q^{49} -0.580659 q^{50} +1.85544 q^{51} -0.152297 q^{52} -10.2524 q^{53} +4.98041 q^{54} +5.30420 q^{55} +3.88429 q^{56} -4.91920 q^{57} +5.78775 q^{58} -10.1961 q^{59} -6.59131 q^{60} +9.64218 q^{61} +8.74754 q^{62} -18.5863 q^{63} +1.00000 q^{64} -0.359777 q^{65} +6.26478 q^{66} -5.11712 q^{67} -0.664994 q^{68} -5.15081 q^{69} +9.17602 q^{70} -6.15492 q^{71} -4.78499 q^{72} -5.99070 q^{73} -1.17330 q^{74} -1.62013 q^{75} +1.76305 q^{76} -8.72145 q^{77} -0.424932 q^{78} +1.58845 q^{79} +2.36234 q^{80} -0.458837 q^{81} +2.82985 q^{82} +14.5243 q^{83} +10.8378 q^{84} -1.57094 q^{85} +10.8695 q^{86} +16.1487 q^{87} -2.24531 q^{88} -5.61030 q^{89} -11.3038 q^{90} +0.591565 q^{91} +1.84606 q^{92} +24.4070 q^{93} +5.51807 q^{94} +4.16494 q^{95} +2.79016 q^{96} +9.43184 q^{97} -8.08771 q^{98} +10.7438 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} - 4 q^{3} + 7 q^{4} - 6 q^{5} + 4 q^{6} - 3 q^{7} - 7 q^{8} + 7 q^{9} + 6 q^{10} - 12 q^{11} - 4 q^{12} + 3 q^{13} + 3 q^{14} - 6 q^{15} + 7 q^{16} - 8 q^{17} - 7 q^{18} - 7 q^{19} - 6 q^{20}+ \cdots - 9 q^{99}+O(q^{100}) $ 7 * q - 7 * q^2 - 4 * q^3 + 7 * q^4 - 6 * q^5 + 4 * q^6 - 3 * q^7 - 7 * q^8 + 7 * q^9 + 6 * q^10 - 12 * q^11 - 4 * q^12 + 3 * q^13 + 3 * q^14 - 6 * q^15 + 7 * q^16 - 8 * q^17 - 7 * q^18 - 7 * q^19 - 6 * q^20 - 5 * q^21 + 12 * q^22 - 22 * q^23 + 4 * q^24 + 9 * q^25 - 3 * q^26 - 22 * q^27 - 3 * q^28 - 7 * q^29 + 6 * q^30 - 3 * q^31 - 7 * q^32 - 19 * q^33 + 8 * q^34 - 25 * q^35 + 7 * q^36 - 13 * q^37 + 7 * q^38 + q^39 + 6 * q^40 - 11 * q^41 + 5 * q^42 - 14 * q^43 - 12 * q^44 - 17 * q^45 + 22 * q^46 - 23 * q^47 - 4 * q^48 + 6 * q^49 - 9 * q^50 - 21 * q^51 + 3 * q^52 - 21 * q^53 + 22 * q^54 + 15 * q^55 + 3 * q^56 - 14 * q^57 + 7 * q^58 - 20 * q^59 - 6 * q^60 + 13 * q^61 + 3 * q^62 - 12 * q^63 + 7 * q^64 - 29 * q^65 + 19 * q^66 - 30 * q^67 - 8 * q^68 + 13 * q^69 + 25 * q^70 - 23 * q^71 - 7 * q^72 + 9 * q^73 + 13 * q^74 + 3 * q^75 - 7 * q^76 + 8 * q^77 - q^78 + 11 * q^79 - 6 * q^80 + 43 * q^81 + 11 * q^82 - 3 * q^83 - 5 * q^84 + 21 * q^85 + 14 * q^86 + 28 * q^87 + 12 * q^88 + 16 * q^89 + 17 * q^90 - 10 * q^91 - 22 * q^92 - 13 * q^93 + 23 * q^94 + 4 * q^96 + 21 * q^97 - 6 * q^98 - 9 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.79016	−1.61090	−0.805450	−	0.592664i	$-0.798076\pi$
$3$	−2.79016	−1.61090	−0.805450	+	0.592664i	$0.798076\pi$
$4$	1.00000	0.500000
$5$	2.36234	1.05647	0.528236	−	0.849098i	$-0.322854\pi$
$5$	2.36234	1.05647	0.528236	+	0.849098i	$0.322854\pi$
$6$	2.79016	1.13908
$7$	−3.88429	−1.46812	−0.734062	−	0.679083i	$-0.762378\pi$
$7$	−3.88429	−1.46812	−0.734062	+	0.679083i	$0.762378\pi$
$8$	−1.00000	−0.353553
$9$	4.78499	1.59500
$10$	−2.36234	−0.747038
$11$	2.24531	0.676987	0.338494	−	0.940969i	$-0.390083\pi$
$11$	2.24531	0.676987	0.338494	+	0.940969i	$0.390083\pi$
$12$	−2.79016	−0.805450
$13$	−0.152297	−0.0422395	−0.0211198	−	0.999777i	$-0.506723\pi$
$13$	−0.152297	−0.0422395	−0.0211198	+	0.999777i	$0.506723\pi$
$14$	3.88429	1.03812
$15$	−6.59131	−1.70187
$16$	1.00000	0.250000
$17$	−0.664994	−0.161285	−0.0806424	−	0.996743i	$-0.525697\pi$
$17$	−0.664994	−0.161285	−0.0806424	+	0.996743i	$0.525697\pi$
$18$	−4.78499	−1.12783
$19$	1.76305	0.404473	0.202236	−	0.979337i	$-0.435179\pi$
$19$	1.76305	0.404473	0.202236	+	0.979337i	$0.435179\pi$
$20$	2.36234	0.528236
$21$	10.8378	2.36500
$22$	−2.24531	−0.478702
$23$	1.84606	0.384931	0.192465	−	0.981304i	$-0.438352\pi$
$23$	1.84606	0.384931	0.192465	+	0.981304i	$0.438352\pi$
$24$	2.79016	0.569539
$25$	0.580659	0.116132
$26$	0.152297	0.0298678
$27$	−4.98041	−0.958480
$28$	−3.88429	−0.734062
$29$	−5.78775	−1.07476	−0.537379	−	0.843341i	$-0.680585\pi$
$29$	−5.78775	−1.07476	−0.537379	+	0.843341i	$0.680585\pi$
$30$	6.59131	1.20340
$31$	−8.74754	−1.57110	−0.785552	−	0.618795i	$-0.787621\pi$
$31$	−8.74754	−1.57110	−0.785552	+	0.618795i	$0.787621\pi$
$32$	−1.00000	−0.176777
$33$	−6.26478	−1.09056
$34$	0.664994	0.114046
$35$	−9.17602	−1.55103
$36$	4.78499	0.797498
$37$	1.17330	0.192889	0.0964444	−	0.995338i	$-0.469253\pi$
$37$	1.17330	0.192889	0.0964444	+	0.995338i	$0.469253\pi$
$38$	−1.76305	−0.286005
$39$	0.424932	0.0680436
$40$	−2.36234	−0.373519
$41$	−2.82985	−0.441948	−0.220974	−	0.975280i	$-0.570924\pi$
$41$	−2.82985	−0.441948	−0.220974	+	0.975280i	$0.570924\pi$
$42$	−10.8378	−1.67231
$43$	−10.8695	−1.65759	−0.828794	−	0.559554i	$-0.810972\pi$
$43$	−10.8695	−1.65759	−0.828794	+	0.559554i	$0.810972\pi$
$44$	2.24531	0.338494
$45$	11.3038	1.68507
$46$	−1.84606	−0.272187
$47$	−5.51807	−0.804893	−0.402447	−	0.915443i	$-0.631840\pi$
$47$	−5.51807	−0.804893	−0.402447	+	0.915443i	$0.631840\pi$
$48$	−2.79016	−0.402725
$49$	8.08771	1.15539
$50$	−0.580659	−0.0821176
$51$	1.85544	0.259813
$52$	−0.152297	−0.0211198
$53$	−10.2524	−1.40828	−0.704139	−	0.710062i	$-0.748667\pi$
$53$	−10.2524	−1.40828	−0.704139	+	0.710062i	$0.748667\pi$
$54$	4.98041	0.677748
$55$	5.30420	0.715218
$56$	3.88429	0.519060
$57$	−4.91920	−0.651565
$58$	5.78775	0.759968
$59$	−10.1961	−1.32742	−0.663711	−	0.747989i	$-0.731019\pi$
$59$	−10.1961	−1.32742	−0.663711	+	0.747989i	$0.731019\pi$
$60$	−6.59131	−0.850935
$61$	9.64218	1.23455	0.617277	−	0.786746i	$-0.288236\pi$
$61$	9.64218	1.23455	0.617277	+	0.786746i	$0.288236\pi$
$62$	8.74754	1.11094
$63$	−18.5863	−2.34165
$64$	1.00000	0.125000
$65$	−0.359777	−0.0446248
$66$	6.26478	0.771141
$67$	−5.11712	−0.625155	−0.312578	−	0.949892i	$-0.601192\pi$
$67$	−5.11712	−0.625155	−0.312578	+	0.949892i	$0.601192\pi$
$68$	−0.664994	−0.0806424
$69$	−5.15081	−0.620085
$70$	9.17602	1.09674
$71$	−6.15492	−0.730454	−0.365227	−	0.930918i	$-0.619009\pi$
$71$	−6.15492	−0.730454	−0.365227	+	0.930918i	$0.619009\pi$
$72$	−4.78499	−0.563917
$73$	−5.99070	−0.701158	−0.350579	−	0.936533i	$-0.614015\pi$
$73$	−5.99070	−0.701158	−0.350579	+	0.936533i	$0.614015\pi$
$74$	−1.17330	−0.136393
$75$	−1.62013	−0.187077
$76$	1.76305	0.202236
$77$	−8.72145	−0.993901
$78$	−0.424932	−0.0481141
$79$	1.58845	0.178714	0.0893572	−	0.996000i	$-0.471519\pi$
$79$	1.58845	0.178714	0.0893572	+	0.996000i	$0.471519\pi$
$80$	2.36234	0.264118
$81$	−0.458837	−0.0509819
$82$	2.82985	0.312505
$83$	14.5243	1.59425	0.797125	−	0.603814i	$-0.206353\pi$
$83$	14.5243	1.59425	0.797125	+	0.603814i	$0.206353\pi$
$84$	10.8378	1.18250
$85$	−1.57094	−0.170393
$86$	10.8695	1.17209
$87$	16.1487	1.73133
$88$	−2.24531	−0.239351
$89$	−5.61030	−0.594691	−0.297345	−	0.954770i	$-0.596101\pi$
$89$	−5.61030	−0.594691	−0.297345	+	0.954770i	$0.596101\pi$
$90$	−11.3038	−1.19152
$91$	0.591565	0.0620128
$92$	1.84606	0.192465
$93$	24.4070	2.53089
$94$	5.51807	0.569145
$95$	4.16494	0.427314
$96$	2.79016	0.284769
$97$	9.43184	0.957658	0.478829	−	0.877908i	$-0.341061\pi$
$97$	9.43184	0.957658	0.478829	+	0.877908i	$0.341061\pi$
$98$	−8.08771	−0.816983
$99$	10.7438	1.07979
$100$	0.580659	0.0580659
$101$	18.9376	1.88436	0.942180	−	0.335106i	$-0.108772\pi$
$101$	18.9376	1.88436	0.942180	+	0.335106i	$0.108772\pi$
$102$	−1.85544	−0.183716
$103$	11.3673	1.12005	0.560026	−	0.828475i	$-0.310791\pi$
$103$	11.3673	1.12005	0.560026	+	0.828475i	$0.310791\pi$
$104$	0.152297	0.0149339
$105$	25.6026	2.49855
$106$	10.2524	0.995804
$107$	−14.2088	−1.37361	−0.686806	−	0.726841i	$-0.740988\pi$
$107$	−14.2088	−1.37361	−0.686806	+	0.726841i	$0.740988\pi$
$108$	−4.98041	−0.479240
$109$	13.2500	1.26912	0.634560	−	0.772873i	$-0.281181\pi$
$109$	13.2500	1.26912	0.634560	+	0.772873i	$0.281181\pi$
$110$	−5.30420	−0.505735
$111$	−3.27368	−0.310724
$112$	−3.88429	−0.367031
$113$	−17.5143	−1.64761	−0.823803	−	0.566876i	$-0.808152\pi$
$113$	−17.5143	−1.64761	−0.823803	+	0.566876i	$0.808152\pi$
$114$	4.91920	0.460726
$115$	4.36103	0.406668
$116$	−5.78775	−0.537379
$117$	−0.728738	−0.0673719
$118$	10.1961	0.938629
$119$	2.58303	0.236786
$120$	6.59131	0.601702
$121$	−5.95857	−0.541688
$122$	−9.64218	−0.872962
$123$	7.89573	0.711934
$124$	−8.74754	−0.785552
$125$	−10.4400	−0.933781
$126$	18.5863	1.65580
$127$	14.4012	1.27790	0.638949	−	0.769249i	$-0.279370\pi$
$127$	14.4012	1.27790	0.638949	+	0.769249i	$0.279370\pi$
$128$	−1.00000	−0.0883883
$129$	30.3277	2.67021
$130$	0.359777	0.0315545
$131$	−0.204126	−0.0178346	−0.00891730	−	0.999960i	$-0.502839\pi$
$131$	−0.204126	−0.0178346	−0.00891730	+	0.999960i	$0.502839\pi$
$132$	−6.26478	−0.545279
$133$	−6.84822	−0.593816
$134$	5.11712	0.442052
$135$	−11.7654	−1.01261
$136$	0.664994	0.0570228
$137$	−9.93816	−0.849074	−0.424537	−	0.905411i	$-0.639563\pi$
$137$	−9.93816	−0.849074	−0.424537	+	0.905411i	$0.639563\pi$
$138$	5.15081	0.438466
$139$	−6.11879	−0.518989	−0.259495	−	0.965745i	$-0.583556\pi$
$139$	−6.11879	−0.518989	−0.259495	+	0.965745i	$0.583556\pi$
$140$	−9.17602	−0.775515
$141$	15.3963	1.29660
$142$	6.15492	0.516509
$143$	−0.341954	−0.0285956
$144$	4.78499	0.398749
$145$	−13.6726	−1.13545
$146$	5.99070	0.495794
$147$	−22.5660	−1.86121
$148$	1.17330	0.0964444
$149$	−13.1330	−1.07590	−0.537949	−	0.842977i	$-0.680801\pi$
$149$	−13.1330	−1.07590	−0.537949	+	0.842977i	$0.680801\pi$
$150$	1.62013	0.132283
$151$	6.57750	0.535270	0.267635	−	0.963520i	$-0.413758\pi$
$151$	6.57750	0.535270	0.267635	+	0.963520i	$0.413758\pi$
$152$	−1.76305	−0.143003
$153$	−3.18199	−0.257249
$154$	8.72145	0.702794
$155$	−20.6647	−1.65983
$156$	0.424932	0.0340218
$157$	4.45872	0.355845	0.177922	−	0.984045i	$-0.443062\pi$
$157$	4.45872	0.355845	0.177922	+	0.984045i	$0.443062\pi$
$158$	−1.58845	−0.126370
$159$	28.6059	2.26860
$160$	−2.36234	−0.186760
$161$	−7.17065	−0.565126
$162$	0.458837	0.0360496
$163$	8.57083	0.671319	0.335660	−	0.941983i	$-0.391041\pi$
$163$	8.57083	0.671319	0.335660	+	0.941983i	$0.391041\pi$
$164$	−2.82985	−0.220974
$165$	−14.7996	−1.15214
$166$	−14.5243	−1.12731
$167$	1.45022	0.112221	0.0561105	−	0.998425i	$-0.482130\pi$
$167$	1.45022	0.112221	0.0561105	+	0.998425i	$0.482130\pi$
$168$	−10.8378	−0.836154
$169$	−12.9768	−0.998216
$170$	1.57094	0.120486
$171$	8.43620	0.645132
$172$	−10.8695	−0.828794
$173$	−13.8967	−1.05655	−0.528274	−	0.849074i	$-0.677161\pi$
$173$	−13.8967	−1.05655	−0.528274	+	0.849074i	$0.677161\pi$
$174$	−16.1487	−1.22423
$175$	−2.25545	−0.170496
$176$	2.24531	0.169247
$177$	28.4488	2.13834
$178$	5.61030	0.420510
$179$	−1.93736	−0.144805	−0.0724025	−	0.997375i	$-0.523067\pi$
$179$	−1.93736	−0.144805	−0.0724025	+	0.997375i	$0.523067\pi$
$180$	11.3038	0.842534
$181$	−21.8586	−1.62473	−0.812367	−	0.583146i	$-0.801822\pi$
$181$	−21.8586	−1.62473	−0.812367	+	0.583146i	$0.801822\pi$
$182$	−0.591565	−0.0438497
$183$	−26.9032	−1.98874
$184$	−1.84606	−0.136094
$185$	2.77173	0.203781
$186$	−24.4070	−1.78961
$187$	−1.49312	−0.109188
$188$	−5.51807	−0.402447
$189$	19.3454	1.40717
$190$	−4.16494	−0.302156
$191$	20.6013	1.49066	0.745329	−	0.666697i	$-0.232293\pi$
$191$	20.6013	1.49066	0.745329	+	0.666697i	$0.232293\pi$
$192$	−2.79016	−0.201362
$193$	15.9686	1.14945	0.574724	−	0.818347i	$-0.305110\pi$
$193$	15.9686	1.14945	0.574724	+	0.818347i	$0.305110\pi$
$194$	−9.43184	−0.677166
$195$	1.00383	0.0718861
$196$	8.08771	0.577694
$197$	−21.2617	−1.51483	−0.757417	−	0.652932i	$-0.773539\pi$
$197$	−21.2617	−1.51483	−0.757417	+	0.652932i	$0.773539\pi$
$198$	−10.7438	−0.763529
$199$	−15.4590	−1.09586	−0.547930	−	0.836524i	$-0.684584\pi$
$199$	−15.4590	−1.09586	−0.547930	+	0.836524i	$0.684584\pi$
$200$	−0.580659	−0.0410588
$201$	14.2776	1.00706
$202$	−18.9376	−1.33244
$203$	22.4813	1.57788
$204$	1.85544	0.129907
$205$	−6.68507	−0.466906
$206$	−11.3673	−0.791996
$207$	8.83340	0.613963
$208$	−0.152297	−0.0105599
$209$	3.95861	0.273823
$210$	−25.6026	−1.76675
$211$	6.05713	0.416990	0.208495	−	0.978023i	$-0.433143\pi$
$211$	6.05713	0.416990	0.208495	+	0.978023i	$0.433143\pi$
$212$	−10.2524	−0.704139
$213$	17.1732	1.17669
$214$	14.2088	0.971291
$215$	−25.6775	−1.75119
$216$	4.98041	0.338874
$217$	33.9780	2.30658
$218$	−13.2500	−0.897404
$219$	16.7150	1.12950
$220$	5.30420	0.357609
$221$	0.101276	0.00681259
$222$	3.27368	0.219715
$223$	23.7788	1.59234	0.796172	−	0.605070i	$-0.206855\pi$
$223$	23.7788	1.59234	0.796172	+	0.605070i	$0.206855\pi$
$224$	3.88429	0.259530
$225$	2.77845	0.185230
$226$	17.5143	1.16503
$227$	−2.91310	−0.193349	−0.0966746	−	0.995316i	$-0.530821\pi$
$227$	−2.91310	−0.193349	−0.0966746	+	0.995316i	$0.530821\pi$
$228$	−4.91920	−0.325782
$229$	−23.4623	−1.55043	−0.775216	−	0.631696i	$-0.782359\pi$
$229$	−23.4623	−1.55043	−0.775216	+	0.631696i	$0.782359\pi$
$230$	−4.36103	−0.287558
$231$	24.3342	1.60108
$232$	5.78775	0.379984
$233$	27.7100	1.81535	0.907673	−	0.419679i	$-0.137857\pi$
$233$	27.7100	1.81535	0.907673	+	0.419679i	$0.137857\pi$
$234$	0.728738	0.0476391
$235$	−13.0356	−0.850347
$236$	−10.1961	−0.663711
$237$	−4.43202	−0.287891
$238$	−2.58303	−0.167433
$239$	10.2652	0.664000	0.332000	−	0.943279i	$-0.392277\pi$
$239$	10.2652	0.664000	0.332000	+	0.943279i	$0.392277\pi$
$240$	−6.59131	−0.425467
$241$	12.6345	0.813860	0.406930	−	0.913459i	$-0.366599\pi$
$241$	12.6345	0.813860	0.406930	+	0.913459i	$0.366599\pi$
$242$	5.95857	0.383031
$243$	16.2215	1.04061
$244$	9.64218	0.617277
$245$	19.1059	1.22063
$246$	−7.89573	−0.503414
$247$	−0.268507	−0.0170847
$248$	8.74754	0.555469
$249$	−40.5252	−2.56818
$250$	10.4400	0.660283
$251$	−27.4666	−1.73368	−0.866839	−	0.498589i	$-0.833852\pi$
$251$	−27.4666	−1.73368	−0.866839	+	0.498589i	$0.833852\pi$
$252$	−18.5863	−1.17083
$253$	4.14499	0.260593
$254$	−14.4012	−0.903610
$255$	4.38318	0.274486
$256$	1.00000	0.0625000
$257$	1.72998	0.107913	0.0539565	−	0.998543i	$-0.482817\pi$
$257$	1.72998	0.107913	0.0539565	+	0.998543i	$0.482817\pi$
$258$	−30.3277	−1.88812
$259$	−4.55743	−0.283185
$260$	−0.359777	−0.0223124
$261$	−27.6943	−1.71423
$262$	0.204126	0.0126110
$263$	22.4198	1.38246	0.691231	−	0.722633i	$-0.257069\pi$
$263$	22.4198	1.38246	0.691231	+	0.722633i	$0.257069\pi$
$264$	6.26478	0.385571
$265$	−24.2197	−1.48781
$266$	6.84822	0.419891
$267$	15.6536	0.957987
$268$	−5.11712	−0.312578
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	11.7654	0.716021
$271$	−5.05568	−0.307111	−0.153555	−	0.988140i	$-0.549072\pi$
$271$	−5.05568	−0.307111	−0.153555	+	0.988140i	$0.549072\pi$
$272$	−0.664994	−0.0403212
$273$	−1.65056	−0.0998964
$274$	9.93816	0.600386
$275$	1.30376	0.0786198
$276$	−5.15081	−0.310042
$277$	−17.0935	−1.02705	−0.513525	−	0.858074i	$-0.671661\pi$
$277$	−17.0935	−1.02705	−0.513525	+	0.858074i	$0.671661\pi$
$278$	6.11879	0.366981
$279$	−41.8569	−2.50591
$280$	9.17602	0.548372
$281$	24.4306	1.45741	0.728704	−	0.684829i	$-0.240123\pi$
$281$	24.4306	1.45741	0.728704	+	0.684829i	$0.240123\pi$
$282$	−15.3963	−0.916836
$283$	12.5843	0.748057	0.374028	−	0.927417i	$-0.377976\pi$
$283$	12.5843	0.748057	0.374028	+	0.927417i	$0.377976\pi$
$284$	−6.15492	−0.365227
$285$	−11.6208	−0.688359
$286$	0.341954	0.0202202
$287$	10.9920	0.648835
$288$	−4.78499	−0.281958
$289$	−16.5578	−0.973987
$290$	13.6726	0.802885
$291$	−26.3163	−1.54269
$292$	−5.99070	−0.350579
$293$	−5.03934	−0.294401	−0.147201	−	0.989107i	$-0.547026\pi$
$293$	−5.03934	−0.294401	−0.147201	+	0.989107i	$0.547026\pi$
$294$	22.5660	1.31608
$295$	−24.0867	−1.40238
$296$	−1.17330	−0.0681965
$297$	−11.1826	−0.648879
$298$	13.1330	0.760775
$299$	−0.281149	−0.0162593
$300$	−1.62013	−0.0935384
$301$	42.2204	2.43354
$302$	−6.57750	−0.378493
$303$	−52.8389	−3.03552
$304$	1.76305	0.101118
$305$	22.7781	1.30427
$306$	3.18199	0.181902
$307$	−19.6450	−1.12120	−0.560601	−	0.828086i	$-0.689430\pi$
$307$	−19.6450	−1.12120	−0.560601	+	0.828086i	$0.689430\pi$
$308$	−8.72145	−0.496951
$309$	−31.7165	−1.80429
$310$	20.6647	1.17367
$311$	26.7504	1.51687	0.758437	−	0.651746i	$-0.225963\pi$
$311$	26.7504	1.51687	0.758437	+	0.651746i	$0.225963\pi$
$312$	−0.424932	−0.0240570
$313$	28.2351	1.59594	0.797971	−	0.602696i	$-0.205907\pi$
$313$	28.2351	1.59594	0.797971	+	0.602696i	$0.205907\pi$
$314$	−4.45872	−0.251620
$315$	−43.9072	−2.47389
$316$	1.58845	0.0893572
$317$	−34.9342	−1.96210	−0.981049	−	0.193761i	$-0.937931\pi$
$317$	−34.9342	−1.96210	−0.981049	+	0.193761i	$0.937931\pi$
$318$	−28.6059	−1.60414
$319$	−12.9953	−0.727597
$320$	2.36234	0.132059
$321$	39.6447	2.21275
$322$	7.17065	0.399605
$323$	−1.17242	−0.0652352
$324$	−0.458837	−0.0254910
$325$	−0.0884325	−0.00490535
$326$	−8.57083	−0.474694
$327$	−36.9696	−2.04443
$328$	2.82985	0.156252
$329$	21.4338	1.18168
$330$	14.7996	0.814689
$331$	−6.94889	−0.381946	−0.190973	−	0.981595i	$-0.561164\pi$
$331$	−6.94889	−0.381946	−0.190973	+	0.981595i	$0.561164\pi$
$332$	14.5243	0.797125
$333$	5.61421	0.307657
$334$	−1.45022	−0.0793523
$335$	−12.0884	−0.660459
$336$	10.8378	0.591250
$337$	−10.2113	−0.556246	−0.278123	−	0.960546i	$-0.589712\pi$
$337$	−10.2113	−0.556246	−0.278123	+	0.960546i	$0.589712\pi$
$338$	12.9768	0.705845
$339$	48.8677	2.65413
$340$	−1.57094	−0.0851964
$341$	−19.6410	−1.06362
$342$	−8.43620	−0.456178
$343$	−4.22500	−0.228129
$344$	10.8695	0.586046
$345$	−12.1680	−0.655102
$346$	13.8967	0.747092
$347$	1.35144	0.0725492	0.0362746	−	0.999342i	$-0.488451\pi$
$347$	1.35144	0.0725492	0.0362746	+	0.999342i	$0.488451\pi$
$348$	16.1487	0.865663
$349$	18.8717	1.01018	0.505089	−	0.863067i	$-0.331460\pi$
$349$	18.8717	1.01018	0.505089	+	0.863067i	$0.331460\pi$
$350$	2.25545	0.120559
$351$	0.758500	0.0404857
$352$	−2.24531	−0.119676
$353$	23.1371	1.23146	0.615732	−	0.787956i	$-0.288860\pi$
$353$	23.1371	1.23146	0.615732	+	0.787956i	$0.288860\pi$
$354$	−28.4488	−1.51204
$355$	−14.5400	−0.771704
$356$	−5.61030	−0.297345
$357$	−7.20707	−0.381438
$358$	1.93736	0.102393
$359$	−3.97007	−0.209532	−0.104766	−	0.994497i	$-0.533409\pi$
$359$	−3.97007	−0.209532	−0.104766	+	0.994497i	$0.533409\pi$
$360$	−11.3038	−0.595762
$361$	−15.8916	−0.836402
$362$	21.8586	1.14886
$363$	16.6254	0.872605
$364$	0.591565	0.0310064
$365$	−14.1521	−0.740754
$366$	26.9032	1.40625
$367$	36.6793	1.91464	0.957322	−	0.289025i	$-0.0933311\pi$
$367$	36.6793	1.91464	0.957322	+	0.289025i	$0.0933311\pi$
$368$	1.84606	0.0962327
$369$	−13.5408	−0.704906
$370$	−2.77173	−0.144095
$371$	39.8234	2.06753
$372$	24.4070	1.26545
$373$	−5.11979	−0.265092	−0.132546	−	0.991177i	$-0.542315\pi$
$373$	−5.11979	−0.265092	−0.132546	+	0.991177i	$0.542315\pi$
$374$	1.49312	0.0772074
$375$	29.1292	1.50423
$376$	5.51807	0.284573
$377$	0.881454	0.0453972
$378$	−19.3454	−0.995018
$379$	−14.3927	−0.739301	−0.369650	−	0.929171i	$-0.620523\pi$
$379$	−14.3927	−0.739301	−0.369650	+	0.929171i	$0.620523\pi$
$380$	4.16494	0.213657
$381$	−40.1816	−2.05856
$382$	−20.6013	−1.05405
$383$	−7.15093	−0.365395	−0.182698	−	0.983169i	$-0.558483\pi$
$383$	−7.15093	−0.365395	−0.182698	+	0.983169i	$0.558483\pi$
$384$	2.79016	0.142385
$385$	−20.6030	−1.05003
$386$	−15.9686	−0.812782
$387$	−52.0106	−2.64385
$388$	9.43184	0.478829
$389$	1.63839	0.0830694	0.0415347	−	0.999137i	$-0.486775\pi$
$389$	1.63839	0.0830694	0.0415347	+	0.999137i	$0.486775\pi$
$390$	−1.00383	−0.0508312
$391$	−1.22762	−0.0620835
$392$	−8.08771	−0.408491
$393$	0.569545	0.0287297
$394$	21.2617	1.07115
$395$	3.75246	0.188807
$396$	10.7438	0.539896
$397$	9.16066	0.459760	0.229880	−	0.973219i	$-0.426167\pi$
$397$	9.16066	0.459760	0.229880	+	0.973219i	$0.426167\pi$
$398$	15.4590	0.774890
$399$	19.1076	0.956578
$400$	0.580659	0.0290330
$401$	−23.5962	−1.17834	−0.589170	−	0.808009i	$-0.700545\pi$
$401$	−23.5962	−1.17834	−0.589170	+	0.808009i	$0.700545\pi$
$402$	−14.2776	−0.712101
$403$	1.33222	0.0663627
$404$	18.9376	0.942180
$405$	−1.08393	−0.0538609
$406$	−22.4813	−1.11573
$407$	2.63442	0.130583
$408$	−1.85544	−0.0918579
$409$	−2.71755	−0.134374	−0.0671870	−	0.997740i	$-0.521402\pi$
$409$	−2.71755	−0.134374	−0.0671870	+	0.997740i	$0.521402\pi$
$410$	6.68507	0.330152
$411$	27.7291	1.36777
$412$	11.3673	0.560026
$413$	39.6047	1.94882
$414$	−8.83340	−0.434138
$415$	34.3114	1.68428
$416$	0.152297	0.00746696
$417$	17.0724	0.836039
$418$	−3.95861	−0.193622
$419$	−27.9763	−1.36673	−0.683366	−	0.730076i	$-0.739484\pi$
$419$	−27.9763	−1.36673	−0.683366	+	0.730076i	$0.739484\pi$
$420$	25.6026	1.24928
$421$	20.3928	0.993883	0.496941	−	0.867784i	$-0.334456\pi$
$421$	20.3928	0.993883	0.496941	+	0.867784i	$0.334456\pi$
$422$	−6.05713	−0.294856
$423$	−26.4039	−1.28380
$424$	10.2524	0.497902
$425$	−0.386135	−0.0187303
$426$	−17.1732	−0.832044
$427$	−37.4530	−1.81248
$428$	−14.2088	−0.686806
$429$	0.954106	0.0460647
$430$	25.6775	1.23828
$431$	−9.56842	−0.460894	−0.230447	−	0.973085i	$-0.574019\pi$
$431$	−9.56842	−0.460894	−0.230447	+	0.973085i	$0.574019\pi$
$432$	−4.98041	−0.239620
$433$	−25.7870	−1.23924	−0.619622	−	0.784901i	$-0.712714\pi$
$433$	−25.7870	−1.23924	−0.619622	+	0.784901i	$0.712714\pi$
$434$	−33.9780	−1.63100
$435$	38.1488	1.82910
$436$	13.2500	0.634560
$437$	3.25471	0.155694
$438$	−16.7150	−0.798674
$439$	10.5835	0.505124	0.252562	−	0.967581i	$-0.418727\pi$
$439$	10.5835	0.505124	0.252562	+	0.967581i	$0.418727\pi$
$440$	−5.30420	−0.252868
$441$	38.6996	1.84284
$442$	−0.101276	−0.00481723
$443$	−7.91213	−0.375917	−0.187958	−	0.982177i	$-0.560187\pi$
$443$	−7.91213	−0.375917	−0.187958	+	0.982177i	$0.560187\pi$
$444$	−3.27368	−0.155362
$445$	−13.2534	−0.628274
$446$	−23.7788	−1.12596
$447$	36.6432	1.73316
$448$	−3.88429	−0.183515
$449$	−6.03733	−0.284919	−0.142460	−	0.989801i	$-0.545501\pi$
$449$	−6.03733	−0.284919	−0.142460	+	0.989801i	$0.545501\pi$
$450$	−2.77845	−0.130977
$451$	−6.35390	−0.299193
$452$	−17.5143	−0.823803
$453$	−18.3523	−0.862266
$454$	2.91310	0.136718
$455$	1.39748	0.0655148
$456$	4.91920	0.230363
$457$	−4.72498	−0.221025	−0.110513	−	0.993875i	$-0.535249\pi$
$457$	−4.72498	−0.221025	−0.110513	+	0.993875i	$0.535249\pi$
$458$	23.4623	1.09632
$459$	3.31194	0.154588
$460$	4.36103	0.203334
$461$	25.3520	1.18076	0.590381	−	0.807125i	$-0.298978\pi$
$461$	25.3520	1.18076	0.590381	+	0.807125i	$0.298978\pi$
$462$	−24.3342	−1.13213
$463$	−35.4324	−1.64668	−0.823341	−	0.567547i	$-0.807893\pi$
$463$	−35.4324	−1.64668	−0.823341	+	0.567547i	$0.807893\pi$
$464$	−5.78775	−0.268689
$465$	57.6578	2.67381
$466$	−27.7100	−1.28364
$467$	20.1572	0.932765	0.466383	−	0.884583i	$-0.345557\pi$
$467$	20.1572	0.932765	0.466383	+	0.884583i	$0.345557\pi$
$468$	−0.728738	−0.0336859
$469$	19.8764	0.917806
$470$	13.0356	0.601286
$471$	−12.4405	−0.573230
$472$	10.1961	0.469314
$473$	−24.4055	−1.12217
$474$	4.43202	0.203570
$475$	1.02373	0.0469722
$476$	2.58303	0.118393
$477$	−49.0578	−2.24620
$478$	−10.2652	−0.469519
$479$	−21.2364	−0.970314	−0.485157	−	0.874427i	$-0.661238\pi$
$479$	−21.2364	−0.970314	−0.485157	+	0.874427i	$0.661238\pi$
$480$	6.59131	0.300851
$481$	−0.178689	−0.00814753
$482$	−12.6345	−0.575486
$483$	20.0072	0.910361
$484$	−5.95857	−0.270844
$485$	22.2812	1.01174
$486$	−16.2215	−0.735820
$487$	1.46670	0.0664624	0.0332312	−	0.999448i	$-0.489420\pi$
$487$	1.46670	0.0664624	0.0332312	+	0.999448i	$0.489420\pi$
$488$	−9.64218	−0.436481
$489$	−23.9140	−1.08143
$490$	−19.1059	−0.863119
$491$	14.3389	0.647105	0.323553	−	0.946210i	$-0.395123\pi$
$491$	14.3389	0.647105	0.323553	+	0.946210i	$0.395123\pi$
$492$	7.89573	0.355967
$493$	3.84882	0.173342
$494$	0.268507	0.0120807
$495$	25.3805	1.14077
$496$	−8.74754	−0.392776
$497$	23.9075	1.07240
$498$	40.5252	1.81598
$499$	−13.7751	−0.616659	−0.308329	−	0.951280i	$-0.599770\pi$
$499$	−13.7751	−0.616659	−0.308329	+	0.951280i	$0.599770\pi$
$500$	−10.4400	−0.466891
$501$	−4.04633	−0.180777
$502$	27.4666	1.22590
$503$	−16.4029	−0.731370	−0.365685	−	0.930739i	$-0.619165\pi$
$503$	−16.4029	−0.731370	−0.365685	+	0.930739i	$0.619165\pi$
$504$	18.5863	0.827899
$505$	44.7371	1.99077
$506$	−4.14499	−0.184267
$507$	36.2074	1.60803
$508$	14.4012	0.638949
$509$	14.6232	0.648160	0.324080	−	0.946030i	$-0.394945\pi$
$509$	14.6232	0.648160	0.324080	+	0.946030i	$0.394945\pi$
$510$	−4.38318	−0.194091
$511$	23.2696	1.02939
$512$	−1.00000	−0.0441942
$513$	−8.78073	−0.387679
$514$	−1.72998	−0.0763060
$515$	26.8534	1.18330
$516$	30.3277	1.33510
$517$	−12.3898	−0.544902
$518$	4.55743	0.200242
$519$	38.7741	1.70199
$520$	0.359777	0.0157773
$521$	13.6453	0.597811	0.298905	−	0.954283i	$-0.403378\pi$
$521$	13.6453	0.597811	0.298905	+	0.954283i	$0.403378\pi$
$522$	27.6943	1.21215
$523$	2.36730	0.103515	0.0517574	−	0.998660i	$-0.483518\pi$
$523$	2.36730	0.103515	0.0517574	+	0.998660i	$0.483518\pi$
$524$	−0.204126	−0.00891730
$525$	6.29307	0.274652
$526$	−22.4198	−0.977549
$527$	5.81706	0.253395
$528$	−6.26478	−0.272640
$529$	−19.5921	−0.851828
$530$	24.2197	1.05204
$531$	−48.7883	−2.11723
$532$	−6.84822	−0.296908
$533$	0.430977	0.0186677
$534$	−15.6536	−0.677399
$535$	−33.5659	−1.45118
$536$	5.11712	0.221026
$537$	5.40554	0.233266
$538$	1.00000	0.0431131
$539$	18.1595	0.782183
$540$	−11.7654	−0.506303
$541$	8.38943	0.360690	0.180345	−	0.983603i	$-0.442279\pi$
$541$	8.38943	0.360690	0.180345	+	0.983603i	$0.442279\pi$
$542$	5.05568	0.217160
$543$	60.9889	2.61728
$544$	0.664994	0.0285114
$545$	31.3011	1.34079
$546$	1.65056	0.0706374
$547$	4.07641	0.174295	0.0871473	−	0.996195i	$-0.472225\pi$
$547$	4.07641	0.174295	0.0871473	+	0.996195i	$0.472225\pi$
$548$	−9.93816	−0.424537
$549$	46.1377	1.96911
$550$	−1.30376	−0.0555926
$551$	−10.2041	−0.434710
$552$	5.15081	0.219233
$553$	−6.16999	−0.262375
$554$	17.0935	0.726235
$555$	−7.73356	−0.328271
$556$	−6.11879	−0.259495
$557$	−17.9458	−0.760387	−0.380193	−	0.924907i	$-0.624143\pi$
$557$	−17.9458	−0.760387	−0.380193	+	0.924907i	$0.624143\pi$
$558$	41.8569	1.77194
$559$	1.65539	0.0700157
$560$	−9.17602	−0.387758
$561$	4.16604	0.175890
$562$	−24.4306	−1.03054
$563$	36.1794	1.52478	0.762391	−	0.647117i	$-0.224025\pi$
$563$	36.1794	1.52478	0.762391	+	0.647117i	$0.224025\pi$
$564$	15.3963	0.648301
$565$	−41.3748	−1.74065
$566$	−12.5843	−0.528956
$567$	1.78226	0.0748477
$568$	6.15492	0.258255
$569$	20.7037	0.867945	0.433973	−	0.900926i	$-0.357111\pi$
$569$	20.7037	0.867945	0.433973	+	0.900926i	$0.357111\pi$
$570$	11.6208	0.486744
$571$	11.8564	0.496175	0.248087	−	0.968738i	$-0.420198\pi$
$571$	11.8564	0.496175	0.248087	+	0.968738i	$0.420198\pi$
$572$	−0.341954	−0.0142978
$573$	−57.4809	−2.40130
$574$	−10.9920	−0.458796
$575$	1.07193	0.0447027
$576$	4.78499	0.199375
$577$	34.0176	1.41617	0.708086	−	0.706127i	$-0.249559\pi$
$577$	34.0176	1.41617	0.708086	+	0.706127i	$0.249559\pi$
$578$	16.5578	0.688713
$579$	−44.5550	−1.85164
$580$	−13.6726	−0.567725
$581$	−56.4167	−2.34056
$582$	26.3163	1.09085
$583$	−23.0199	−0.953387
$584$	5.99070	0.247897
$585$	−1.72153	−0.0711765
$586$	5.03934	0.208173
$587$	20.2066	0.834015	0.417008	−	0.908903i	$-0.363079\pi$
$587$	20.2066	0.834015	0.417008	+	0.908903i	$0.363079\pi$
$588$	−22.5660	−0.930607
$589$	−15.4224	−0.635469
$590$	24.0867	0.991634
$591$	59.3235	2.44024
$592$	1.17330	0.0482222
$593$	−27.6602	−1.13587	−0.567934	−	0.823074i	$-0.692257\pi$
$593$	−27.6602	−1.13587	−0.567934	+	0.823074i	$0.692257\pi$
$594$	11.1826	0.458827
$595$	6.10200	0.250158
$596$	−13.1330	−0.537949
$597$	43.1331	1.76532
$598$	0.281149	0.0114971
$599$	−11.2617	−0.460141	−0.230070	−	0.973174i	$-0.573896\pi$
$599$	−11.2617	−0.460141	−0.230070	+	0.973174i	$0.573896\pi$
$600$	1.62013	0.0661416
$601$	−3.94629	−0.160972	−0.0804862	−	0.996756i	$-0.525647\pi$
$601$	−3.94629	−0.160972	−0.0804862	+	0.996756i	$0.525647\pi$
$602$	−42.2204	−1.72078
$603$	−24.4854	−0.997121
$604$	6.57750	0.267635
$605$	−14.0762	−0.572278
$606$	52.8389	2.14643
$607$	−35.3920	−1.43652	−0.718258	−	0.695777i	$-0.755060\pi$
$607$	−35.3920	−1.43652	−0.718258	+	0.695777i	$0.755060\pi$
$608$	−1.76305	−0.0715013
$609$	−62.7264	−2.54180
$610$	−22.7781	−0.922259
$611$	0.840384	0.0339983
$612$	−3.18199	−0.128624
$613$	23.7537	0.959401	0.479701	−	0.877432i	$-0.340745\pi$
$613$	23.7537	0.959401	0.479701	+	0.877432i	$0.340745\pi$
$614$	19.6450	0.792809
$615$	18.6524	0.752138
$616$	8.72145	0.351397
$617$	−5.90000	−0.237525	−0.118762	−	0.992923i	$-0.537893\pi$
$617$	−5.90000	−0.237525	−0.118762	+	0.992923i	$0.537893\pi$
$618$	31.7165	1.27583
$619$	−20.7014	−0.832059	−0.416029	−	0.909351i	$-0.636579\pi$
$619$	−20.7014	−0.832059	−0.416029	+	0.909351i	$0.636579\pi$
$620$	−20.6647	−0.829913
$621$	−9.19415	−0.368949
$622$	−26.7504	−1.07259
$623$	21.7920	0.873080
$624$	0.424932	0.0170109
$625$	−27.5661	−1.10265
$626$	−28.2351	−1.12850
$627$	−11.0452	−0.441101
$628$	4.45872	0.177922
$629$	−0.780235	−0.0311100
$630$	43.9072	1.74930
$631$	5.36235	0.213472	0.106736	−	0.994287i	$-0.465960\pi$
$631$	5.36235	0.213472	0.106736	+	0.994287i	$0.465960\pi$
$632$	−1.58845	−0.0631851
$633$	−16.9004	−0.671729
$634$	34.9342	1.38741
$635$	34.0205	1.35006
$636$	28.6059	1.13430
$637$	−1.23173	−0.0488030
$638$	12.9953	0.514489
$639$	−29.4512	−1.16507
$640$	−2.36234	−0.0933798
$641$	−26.7918	−1.05821	−0.529107	−	0.848555i	$-0.677473\pi$
$641$	−26.7918	−1.05821	−0.529107	+	0.848555i	$0.677473\pi$
$642$	−39.6447	−1.56465
$643$	8.07707	0.318529	0.159264	−	0.987236i	$-0.449088\pi$
$643$	8.07707	0.318529	0.159264	+	0.987236i	$0.449088\pi$
$644$	−7.17065	−0.282563
$645$	71.6445	2.82100
$646$	1.17242	0.0461283
$647$	−13.4690	−0.529523	−0.264761	−	0.964314i	$-0.585293\pi$
$647$	−13.4690	−0.529523	−0.264761	+	0.964314i	$0.585293\pi$
$648$	0.458837	0.0180248
$649$	−22.8935	−0.898647
$650$	0.0884325	0.00346861
$651$	−94.8040	−3.71566
$652$	8.57083	0.335660
$653$	−15.6835	−0.613744	−0.306872	−	0.951751i	$-0.599282\pi$
$653$	−15.6835	−0.613744	−0.306872	+	0.951751i	$0.599282\pi$
$654$	36.9696	1.44563
$655$	−0.482216	−0.0188417
$656$	−2.82985	−0.110487
$657$	−28.6654	−1.11835
$658$	−21.4338	−0.835576
$659$	−13.5401	−0.527447	−0.263724	−	0.964598i	$-0.584951\pi$
$659$	−13.5401	−0.527447	−0.263724	+	0.964598i	$0.584951\pi$
$660$	−14.7996	−0.576072
$661$	50.0074	1.94506	0.972531	−	0.232775i	$-0.0747806\pi$
$661$	50.0074	1.94506	0.972531	+	0.232775i	$0.0747806\pi$
$662$	6.94889	0.270076
$663$	−0.282577	−0.0109744
$664$	−14.5243	−0.563653
$665$	−16.1778	−0.627349
$666$	−5.61421	−0.217546
$667$	−10.6845	−0.413707
$668$	1.45022	0.0561105
$669$	−66.3466	−2.56511
$670$	12.0884	0.467015
$671$	21.6497	0.835778
$672$	−10.8378	−0.418077
$673$	16.4521	0.634180	0.317090	−	0.948395i	$-0.397294\pi$
$673$	16.4521	0.634180	0.317090	+	0.948395i	$0.397294\pi$
$674$	10.2113	0.393325
$675$	−2.89192	−0.111310
$676$	−12.9768	−0.499108
$677$	42.6665	1.63981	0.819903	−	0.572503i	$-0.194027\pi$
$677$	42.6665	1.63981	0.819903	+	0.572503i	$0.194027\pi$
$678$	−48.8677	−1.87675
$679$	−36.6360	−1.40596
$680$	1.57094	0.0602429
$681$	8.12801	0.311466
$682$	19.6410	0.752091
$683$	−43.3205	−1.65761	−0.828807	−	0.559534i	$-0.810980\pi$
$683$	−43.3205	−1.65761	−0.828807	+	0.559534i	$0.810980\pi$
$684$	8.43620	0.322566
$685$	−23.4773	−0.897023
$686$	4.22500	0.161311
$687$	65.4636	2.49759
$688$	−10.8695	−0.414397
$689$	1.56141	0.0594850
$690$	12.1680	0.463227
$691$	19.8389	0.754707	0.377354	−	0.926069i	$-0.376834\pi$
$691$	19.8389	0.754707	0.377354	+	0.926069i	$0.376834\pi$
$692$	−13.8967	−0.528274
$693$	−41.7321	−1.58527
$694$	−1.35144	−0.0513000
$695$	−14.4547	−0.548297
$696$	−16.1487	−0.612116
$697$	1.88183	0.0712795
$698$	−18.8717	−0.714303
$699$	−77.3154	−2.92434
$700$	−2.25545	−0.0852480
$701$	−0.479263	−0.0181015	−0.00905077	−	0.999959i	$-0.502881\pi$
$701$	−0.479263	−0.0181015	−0.00905077	+	0.999959i	$0.502881\pi$
$702$	−0.758500	−0.0286277
$703$	2.06859	0.0780182
$704$	2.24531	0.0846234
$705$	36.3713	1.36982
$706$	−23.1371	−0.870776
$707$	−73.5591	−2.76647
$708$	28.4488	1.06917
$709$	37.2475	1.39886	0.699429	−	0.714702i	$-0.253438\pi$
$709$	37.2475	1.39886	0.699429	+	0.714702i	$0.253438\pi$
$710$	14.5400	0.545677
$711$	7.60071	0.285049
$712$	5.61030	0.210255
$713$	−16.1485	−0.604767
$714$	7.20707	0.269718
$715$	−0.807812	−0.0302104
$716$	−1.93736	−0.0724025
$717$	−28.6415	−1.06964
$718$	3.97007	0.148162
$719$	−20.0777	−0.748772	−0.374386	−	0.927273i	$-0.622147\pi$
$719$	−20.0777	−0.748772	−0.374386	+	0.927273i	$0.622147\pi$
$720$	11.3038	0.421267
$721$	−44.1538	−1.64437
$722$	15.8916	0.591425
$723$	−35.2523	−1.31105
$724$	−21.8586	−0.812367
$725$	−3.36071	−0.124814
$726$	−16.6254	−0.617025
$727$	36.5795	1.35666	0.678329	−	0.734758i	$-0.262704\pi$
$727$	36.5795	1.35666	0.678329	+	0.734758i	$0.262704\pi$
$728$	−0.591565	−0.0219248
$729$	−43.8839	−1.62533
$730$	14.1521	0.523792
$731$	7.22817	0.267344
$732$	−26.9032	−0.994371
$733$	18.6190	0.687710	0.343855	−	0.939023i	$-0.388267\pi$
$733$	18.6190	0.687710	0.343855	+	0.939023i	$0.388267\pi$
$734$	−36.6793	−1.35386
$735$	−53.3086	−1.96632
$736$	−1.84606	−0.0680468
$737$	−11.4895	−0.423222
$738$	13.5408	0.498444
$739$	28.8894	1.06271	0.531356	−	0.847149i	$-0.321683\pi$
$739$	28.8894	1.06271	0.531356	+	0.847149i	$0.321683\pi$
$740$	2.77173	0.101891
$741$	0.749179	0.0275218
$742$	−39.8234	−1.46196
$743$	−11.5934	−0.425320	−0.212660	−	0.977126i	$-0.568213\pi$
$743$	−11.5934	−0.425320	−0.212660	+	0.977126i	$0.568213\pi$
$744$	−24.4070	−0.894805
$745$	−31.0247	−1.13666
$746$	5.11979	0.187449
$747$	69.4987	2.54282
$748$	−1.49312	−0.0545939
$749$	55.1909	2.01663
$750$	−29.1292	−1.06365
$751$	54.6438	1.99398	0.996990	−	0.0775296i	$-0.0247032\pi$
$751$	54.6438	1.99398	0.996990	+	0.0775296i	$0.0247032\pi$
$752$	−5.51807	−0.201223
$753$	76.6362	2.79278
$754$	−0.881454	−0.0321007
$755$	15.5383	0.565497
$756$	19.3454	0.703584
$757$	37.1233	1.34927	0.674634	−	0.738153i	$-0.264302\pi$
$757$	37.1233	1.34927	0.674634	+	0.738153i	$0.264302\pi$
$758$	14.3927	0.522765
$759$	−11.5652	−0.419790
$760$	−4.16494	−0.151078
$761$	38.3748	1.39109	0.695543	−	0.718485i	$-0.255164\pi$
$761$	38.3748	1.39109	0.695543	+	0.718485i	$0.255164\pi$
$762$	40.1816	1.45562
$763$	−51.4669	−1.86323
$764$	20.6013	0.745329
$765$	−7.51695	−0.271776
$766$	7.15093	0.258373
$767$	1.55283	0.0560696
$768$	−2.79016	−0.100681
$769$	30.1412	1.08692	0.543460	−	0.839435i	$-0.317114\pi$
$769$	30.1412	1.08692	0.543460	+	0.839435i	$0.317114\pi$
$770$	20.6030	0.742482
$771$	−4.82691	−0.173837
$772$	15.9686	0.574724
$773$	14.7449	0.530336	0.265168	−	0.964202i	$-0.414573\pi$
$773$	14.7449	0.530336	0.265168	+	0.964202i	$0.414573\pi$
$774$	52.0106	1.86948
$775$	−5.07934	−0.182455
$776$	−9.43184	−0.338583
$777$	12.7159	0.456182
$778$	−1.63839	−0.0587389
$779$	−4.98918	−0.178756
$780$	1.00383	0.0359431
$781$	−13.8197	−0.494508
$782$	1.22762	0.0438996
$783$	28.8253	1.03013
$784$	8.08771	0.288847
$785$	10.5330	0.375940
$786$	−0.569545	−0.0203150
$787$	−44.0842	−1.57143	−0.785715	−	0.618588i	$-0.787705\pi$
$787$	−44.0842	−1.57143	−0.785715	+	0.618588i	$0.787705\pi$
$788$	−21.2617	−0.757417
$789$	−62.5548	−2.22701
$790$	−3.75246	−0.133506
$791$	68.0306	2.41889
$792$	−10.7438	−0.381764
$793$	−1.46847	−0.0521470
$794$	−9.16066	−0.325100
$795$	67.5769	2.39671
$796$	−15.4590	−0.547930
$797$	38.7280	1.37181	0.685907	−	0.727689i	$-0.259405\pi$
$797$	38.7280	1.37181	0.685907	+	0.727689i	$0.259405\pi$
$798$	−19.1076	−0.676402
$799$	3.66948	0.129817
$800$	−0.580659	−0.0205294
$801$	−26.8452	−0.948530
$802$	23.5962	0.833212
$803$	−13.4510	−0.474675
$804$	14.2776	0.503531
$805$	−16.9395	−0.597040
$806$	−1.33222	−0.0469255
$807$	2.79016	0.0982183
$808$	−18.9376	−0.666222
$809$	12.7907	0.449697	0.224848	−	0.974394i	$-0.427811\pi$
$809$	12.7907	0.449697	0.224848	+	0.974394i	$0.427811\pi$
$810$	1.08393	0.0380854
$811$	−6.26400	−0.219959	−0.109979	−	0.993934i	$-0.535079\pi$
$811$	−6.26400	−0.219959	−0.109979	+	0.993934i	$0.535079\pi$
$812$	22.4813	0.788938
$813$	14.1061	0.494724
$814$	−2.63442	−0.0923363
$815$	20.2472	0.709230
$816$	1.85544	0.0649534
$817$	−19.1636	−0.670449
$818$	2.71755	0.0950168
$819$	2.83063	0.0989103
$820$	−6.68507	−0.233453
$821$	−31.8437	−1.11135	−0.555676	−	0.831399i	$-0.687540\pi$
$821$	−31.8437	−1.11135	−0.555676	+	0.831399i	$0.687540\pi$
$822$	−27.7291	−0.967162
$823$	23.6039	0.822781	0.411391	−	0.911459i	$-0.365043\pi$
$823$	23.6039	0.822781	0.411391	+	0.911459i	$0.365043\pi$
$824$	−11.3673	−0.395998
$825$	−3.63770	−0.126649
$826$	−39.6047	−1.37802
$827$	−23.1364	−0.804532	−0.402266	−	0.915523i	$-0.631777\pi$
$827$	−23.1364	−0.804532	−0.402266	+	0.915523i	$0.631777\pi$
$828$	8.83340	0.306982
$829$	5.12273	0.177920	0.0889598	−	0.996035i	$-0.471646\pi$
$829$	5.12273	0.177920	0.0889598	+	0.996035i	$0.471646\pi$
$830$	−34.3114	−1.19097
$831$	47.6937	1.65448
$832$	−0.152297	−0.00527994
$833$	−5.37828	−0.186346
$834$	−17.0724	−0.591169
$835$	3.42591	0.118558
$836$	3.95861	0.136911
$837$	43.5663	1.50587
$838$	27.9763	0.966425
$839$	−5.94481	−0.205238	−0.102619	−	0.994721i	$-0.532722\pi$
$839$	−5.94481	−0.205238	−0.102619	+	0.994721i	$0.532722\pi$
$840$	−25.6026	−0.883373
$841$	4.49799	0.155103
$842$	−20.3928	−0.702781
$843$	−68.1653	−2.34774
$844$	6.05713	0.208495
$845$	−30.6557	−1.05459
$846$	26.4039	0.907785
$847$	23.1448	0.795265
$848$	−10.2524	−0.352070
$849$	−35.1121	−1.20504
$850$	0.386135	0.0132443
$851$	2.16598	0.0742488
$852$	17.1732	0.588344
$853$	−38.0526	−1.30290	−0.651448	−	0.758693i	$-0.725838\pi$
$853$	−38.0526	−1.30290	−0.651448	+	0.758693i	$0.725838\pi$
$854$	37.4530	1.28162
$855$	19.9292	0.681564
$856$	14.2088	0.485645
$857$	−23.5758	−0.805335	−0.402667	−	0.915346i	$-0.631917\pi$
$857$	−23.5758	−0.805335	−0.402667	+	0.915346i	$0.631917\pi$
$858$	−0.954106	−0.0325726
$859$	−9.42527	−0.321586	−0.160793	−	0.986988i	$-0.551405\pi$
$859$	−9.42527	−0.321586	−0.160793	+	0.986988i	$0.551405\pi$
$860$	−25.6775	−0.875597
$861$	−30.6693	−1.04521
$862$	9.56842	0.325902
$863$	−28.1396	−0.957884	−0.478942	−	0.877846i	$-0.658980\pi$
$863$	−28.1396	−0.957884	−0.478942	+	0.877846i	$0.658980\pi$
$864$	4.98041	0.169437
$865$	−32.8288	−1.11621
$866$	25.7870	0.876278
$867$	46.1989	1.56900
$868$	33.9780	1.15329
$869$	3.56656	0.120987
$870$	−38.1488	−1.29337
$871$	0.779320	0.0264063
$872$	−13.2500	−0.448702
$873$	45.1313	1.52746
$874$	−3.25471	−0.110092
$875$	40.5520	1.37091
$876$	16.7150	0.564748
$877$	30.7237	1.03747	0.518733	−	0.854936i	$-0.326404\pi$
$877$	30.7237	1.03747	0.518733	+	0.854936i	$0.326404\pi$
$878$	−10.5835	−0.357176
$879$	14.0606	0.474251
$880$	5.30420	0.178804
$881$	−26.6012	−0.896217	−0.448108	−	0.893979i	$-0.647902\pi$
$881$	−26.6012	−0.896217	−0.448108	+	0.893979i	$0.647902\pi$
$882$	−38.6996	−1.30308
$883$	−39.0409	−1.31383	−0.656915	−	0.753965i	$-0.728139\pi$
$883$	−39.0409	−1.31383	−0.656915	+	0.753965i	$0.728139\pi$
$884$	0.101276	0.00340629
$885$	67.2058	2.25910
$886$	7.91213	0.265813
$887$	−39.9648	−1.34189	−0.670943	−	0.741509i	$-0.734111\pi$
$887$	−39.9648	−1.34189	−0.670943	+	0.741509i	$0.734111\pi$
$888$	3.27368	0.109858
$889$	−55.9383	−1.87611
$890$	13.2534	0.444257
$891$	−1.03023	−0.0345141
$892$	23.7788	0.796172
$893$	−9.72866	−0.325557
$894$	−36.6432	−1.22553
$895$	−4.57670	−0.152982
$896$	3.88429	0.129765
$897$	0.784452	0.0261921
$898$	6.03733	0.201468
$899$	50.6285	1.68856
$900$	2.77845	0.0926150
$901$	6.81780	0.227134
$902$	6.35390	0.211562
$903$	−117.802	−3.92019
$904$	17.5143	0.582517
$905$	−51.6374	−1.71649
$906$	18.3523	0.609714
$907$	−41.5798	−1.38064	−0.690318	−	0.723506i	$-0.742529\pi$
$907$	−41.5798	−1.38064	−0.690318	+	0.723506i	$0.742529\pi$
$908$	−2.91310	−0.0966746
$909$	90.6162	3.00555
$910$	−1.39748	−0.0463259
$911$	−0.891781	−0.0295460	−0.0147730	−	0.999891i	$-0.504703\pi$
$911$	−0.891781	−0.0295460	−0.0147730	+	0.999891i	$0.504703\pi$
$912$	−4.91920	−0.162891
$913$	32.6116	1.07929
$914$	4.72498	0.156289
$915$	−63.5546	−2.10105
$916$	−23.4623	−0.775216
$917$	0.792886	0.0261834
$918$	−3.31194	−0.109310
$919$	−9.62868	−0.317621	−0.158810	−	0.987309i	$-0.550766\pi$
$919$	−9.62868	−0.317621	−0.158810	+	0.987309i	$0.550766\pi$
$920$	−4.36103	−0.143779
$921$	54.8128	1.80614
$922$	−25.3520	−0.834924
$923$	0.937373	0.0308540
$924$	24.3342	0.800538
$925$	0.681286	0.0224005
$926$	35.4324	1.16438
$927$	54.3923	1.78648
$928$	5.78775	0.189992
$929$	0.396563	0.0130108	0.00650540	−	0.999979i	$-0.497929\pi$
$929$	0.396563	0.0130108	0.00650540	+	0.999979i	$0.497929\pi$
$930$	−57.6578	−1.89067
$931$	14.2591	0.467323
$932$	27.7100	0.907673
$933$	−74.6378	−2.44353
$934$	−20.1572	−0.659565
$935$	−3.52726	−0.115354
$936$	0.728738	0.0238196
$937$	15.4786	0.505664	0.252832	−	0.967510i	$-0.418638\pi$
$937$	15.4786	0.505664	0.252832	+	0.967510i	$0.418638\pi$
$938$	−19.8764	−0.648987
$939$	−78.7804	−2.57090
$940$	−13.0356	−0.425173
$941$	−29.4435	−0.959832	−0.479916	−	0.877315i	$-0.659333\pi$
$941$	−29.4435	−0.959832	−0.479916	+	0.877315i	$0.659333\pi$
$942$	12.4405	0.405335
$943$	−5.22408	−0.170120
$944$	−10.1961	−0.331855
$945$	45.7003	1.48663
$946$	24.4055	0.793491
$947$	1.41962	0.0461316	0.0230658	−	0.999734i	$-0.492657\pi$
$947$	1.41962	0.0461316	0.0230658	+	0.999734i	$0.492657\pi$
$948$	−4.43202	−0.143945
$949$	0.912363	0.0296166
$950$	−1.02373	−0.0332143
$951$	97.4719	3.16074
$952$	−2.58303	−0.0837165
$953$	21.4840	0.695936	0.347968	−	0.937506i	$-0.386872\pi$
$953$	21.4840	0.695936	0.347968	+	0.937506i	$0.386872\pi$
$954$	49.0578	1.58830
$955$	48.6673	1.57484
$956$	10.2652	0.332000
$957$	36.2590	1.17209
$958$	21.2364	0.686116
$959$	38.6027	1.24655
$960$	−6.59131	−0.212734
$961$	45.5194	1.46837
$962$	0.178689	0.00576117
$963$	−67.9888	−2.19091
$964$	12.6345	0.406930
$965$	37.7234	1.21436
$966$	−20.0072	−0.643723
$967$	40.8566	1.31386	0.656929	−	0.753952i	$-0.271855\pi$
$967$	40.8566	1.31386	0.656929	+	0.753952i	$0.271855\pi$
$968$	5.95857	0.191516
$969$	3.27124	0.105087
$970$	−22.2812	−0.715407
$971$	32.3471	1.03807	0.519034	−	0.854754i	$-0.326292\pi$
$971$	32.3471	1.03807	0.519034	+	0.854754i	$0.326292\pi$
$972$	16.2215	0.520303
$973$	23.7672	0.761941
$974$	−1.46670	−0.0469960
$975$	0.246741	0.00790203
$976$	9.64218	0.308639
$977$	−33.6227	−1.07569	−0.537843	−	0.843045i	$-0.680761\pi$
$977$	−33.6227	−1.07569	−0.537843	+	0.843045i	$0.680761\pi$
$978$	23.9140	0.764685
$979$	−12.5969	−0.402598
$980$	19.1059	0.610317
$981$	63.4012	2.02424
$982$	−14.3389	−0.457572
$983$	47.1311	1.50325	0.751624	−	0.659592i	$-0.229271\pi$
$983$	47.1311	1.50325	0.751624	+	0.659592i	$0.229271\pi$
$984$	−7.89573	−0.251707
$985$	−50.2274	−1.60038
$986$	−3.84882	−0.122571
$987$	−59.8037	−1.90357
$988$	−0.268507	−0.00854236
$989$	−20.0658	−0.638057
$990$	−25.3805	−0.806646
$991$	59.6574	1.89508	0.947541	−	0.319635i	$-0.103560\pi$
$991$	59.6574	1.89508	0.947541	+	0.319635i	$0.103560\pi$
$992$	8.74754	0.277735
$993$	19.3885	0.615276
$994$	−23.9075	−0.758299
$995$	−36.5195	−1.15774
$996$	−40.5252	−1.28409
$997$	−51.0492	−1.61674	−0.808371	−	0.588673i	$-0.799651\pi$
$997$	−51.0492	−1.61674	−0.808371	+	0.588673i	$0.799651\pi$
$998$	13.7751	0.436044
$999$	−5.84350	−0.184880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.2.a.d.1.2	✓	7
3.2	odd	2		4842.2.a.o.1.2		7
4.3	odd	2		4304.2.a.i.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.d.1.2	✓	7	1.1	even	1	trivial
4304.2.a.i.1.6		7	4.3	odd	2
4842.2.a.o.1.2		7	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.79016 q^{3} +1.00000 q^{4} +2.36234 q^{5} +2.79016 q^{6} -3.88429 q^{7} -1.00000 q^{8} +4.78499 q^{9} -2.36234 q^{10} +2.24531 q^{11} -2.79016 q^{12} -0.152297 q^{13} +3.88429 q^{14} -6.59131 q^{15} +1.00000 q^{16} -0.664994 q^{17} -4.78499 q^{18} +1.76305 q^{19} +2.36234 q^{20} +10.8378 q^{21} -2.24531 q^{22} +1.84606 q^{23} +2.79016 q^{24} +0.580659 q^{25} +0.152297 q^{26} -4.98041 q^{27} -3.88429 q^{28} -5.78775 q^{29} +6.59131 q^{30} -8.74754 q^{31} -1.00000 q^{32} -6.26478 q^{33} +0.664994 q^{34} -9.17602 q^{35} +4.78499 q^{36} +1.17330 q^{37} -1.76305 q^{38} +0.424932 q^{39} -2.36234 q^{40} -2.82985 q^{41} -10.8378 q^{42} -10.8695 q^{43} +2.24531 q^{44} +11.3038 q^{45} -1.84606 q^{46} -5.51807 q^{47} -2.79016 q^{48} +8.08771 q^{49} -0.580659 q^{50} +1.85544 q^{51} -0.152297 q^{52} -10.2524 q^{53} +4.98041 q^{54} +5.30420 q^{55} +3.88429 q^{56} -4.91920 q^{57} +5.78775 q^{58} -10.1961 q^{59} -6.59131 q^{60} +9.64218 q^{61} +8.74754 q^{62} -18.5863 q^{63} +1.00000 q^{64} -0.359777 q^{65} +6.26478 q^{66} -5.11712 q^{67} -0.664994 q^{68} -5.15081 q^{69} +9.17602 q^{70} -6.15492 q^{71} -4.78499 q^{72} -5.99070 q^{73} -1.17330 q^{74} -1.62013 q^{75} +1.76305 q^{76} -8.72145 q^{77} -0.424932 q^{78} +1.58845 q^{79} +2.36234 q^{80} -0.458837 q^{81} +2.82985 q^{82} +14.5243 q^{83} +10.8378 q^{84} -1.57094 q^{85} +10.8695 q^{86} +16.1487 q^{87} -2.24531 q^{88} -5.61030 q^{89} -11.3038 q^{90} +0.591565 q^{91} +1.84606 q^{92} +24.4070 q^{93} +5.51807 q^{94} +4.16494 q^{95} +2.79016 q^{96} +9.43184 q^{97} -8.08771 q^{98} +10.7438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} - 4 q^{3} + 7 q^{4} - 6 q^{5} + 4 q^{6} - 3 q^{7} - 7 q^{8} + 7 q^{9} + 6 q^{10} - 12 q^{11} - 4 q^{12} + 3 q^{13} + 3 q^{14} - 6 q^{15} + 7 q^{16} - 8 q^{17} - 7 q^{18} - 7 q^{19} - 6 q^{20}+ \cdots - 9 q^{99}+O(q^{100}) \) 7 * q - 7 * q^2 - 4 * q^3 + 7 * q^4 - 6 * q^5 + 4 * q^6 - 3 * q^7 - 7 * q^8 + 7 * q^9 + 6 * q^10 - 12 * q^11 - 4 * q^12 + 3 * q^13 + 3 * q^14 - 6 * q^15 + 7 * q^16 - 8 * q^17 - 7 * q^18 - 7 * q^19 - 6 * q^20 - 5 * q^21 + 12 * q^22 - 22 * q^23 + 4 * q^24 + 9 * q^25 - 3 * q^26 - 22 * q^27 - 3 * q^28 - 7 * q^29 + 6 * q^30 - 3 * q^31 - 7 * q^32 - 19 * q^33 + 8 * q^34 - 25 * q^35 + 7 * q^36 - 13 * q^37 + 7 * q^38 + q^39 + 6 * q^40 - 11 * q^41 + 5 * q^42 - 14 * q^43 - 12 * q^44 - 17 * q^45 + 22 * q^46 - 23 * q^47 - 4 * q^48 + 6 * q^49 - 9 * q^50 - 21 * q^51 + 3 * q^52 - 21 * q^53 + 22 * q^54 + 15 * q^55 + 3 * q^56 - 14 * q^57 + 7 * q^58 - 20 * q^59 - 6 * q^60 + 13 * q^61 + 3 * q^62 - 12 * q^63 + 7 * q^64 - 29 * q^65 + 19 * q^66 - 30 * q^67 - 8 * q^68 + 13 * q^69 + 25 * q^70 - 23 * q^71 - 7 * q^72 + 9 * q^73 + 13 * q^74 + 3 * q^75 - 7 * q^76 + 8 * q^77 - q^78 + 11 * q^79 - 6 * q^80 + 43 * q^81 + 11 * q^82 - 3 * q^83 - 5 * q^84 + 21 * q^85 + 14 * q^86 + 28 * q^87 + 12 * q^88 + 16 * q^89 + 17 * q^90 - 10 * q^91 - 22 * q^92 - 13 * q^93 + 23 * q^94 + 4 * q^96 + 21 * q^97 - 6 * q^98 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more