LMFDB - Embedded newform 3234.2.a.bd.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3234,2,Mod(1,3234)] 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("3234.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3234, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3234.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,2,2,-4,2,0,2,2,-4,-2,2,-8,0,-4,2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.8236200137$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3234.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -5.41421 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +2.24264 q^{19} -2.00000 q^{20} -1.00000 q^{22} -4.82843 q^{23} +1.00000 q^{24} -1.00000 q^{25} -5.41421 q^{26} +1.00000 q^{27} +9.65685 q^{29} -2.00000 q^{30} -6.24264 q^{31} +1.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +0.828427 q^{37} +2.24264 q^{38} -5.41421 q^{39} -2.00000 q^{40} -11.6569 q^{41} -4.82843 q^{43} -1.00000 q^{44} -2.00000 q^{45} -4.82843 q^{46} +7.89949 q^{47} +1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} -5.41421 q^{52} -10.4853 q^{53} +1.00000 q^{54} +2.00000 q^{55} +2.24264 q^{57} +9.65685 q^{58} -6.82843 q^{59} -2.00000 q^{60} -2.58579 q^{61} -6.24264 q^{62} +1.00000 q^{64} +10.8284 q^{65} -1.00000 q^{66} -1.17157 q^{67} -2.00000 q^{68} -4.82843 q^{69} +5.65685 q^{71} +1.00000 q^{72} -11.6569 q^{73} +0.828427 q^{74} -1.00000 q^{75} +2.24264 q^{76} -5.41421 q^{78} -16.4853 q^{79} -2.00000 q^{80} +1.00000 q^{81} -11.6569 q^{82} -5.07107 q^{83} +4.00000 q^{85} -4.82843 q^{86} +9.65685 q^{87} -1.00000 q^{88} -3.75736 q^{89} -2.00000 q^{90} -4.82843 q^{92} -6.24264 q^{93} +7.89949 q^{94} -4.48528 q^{95} +1.00000 q^{96} +9.89949 q^{97} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{10} - 2 q^{11} + 2 q^{12} - 8 q^{13} - 4 q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 4 q^{19} - 4 q^{20} - 2 q^{22} - 4 q^{23}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 4 * q^10 - 2 * q^11 + 2 * q^12 - 8 * q^13 - 4 * q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 - 4 * q^19 - 4 * q^20 - 2 * q^22 - 4 * q^23 + 2 * q^24 - 2 * q^25 - 8 * q^26 + 2 * q^27 + 8 * q^29 - 4 * q^30 - 4 * q^31 + 2 * q^32 - 2 * q^33 - 4 * q^34 + 2 * q^36 - 4 * q^37 - 4 * q^38 - 8 * q^39 - 4 * q^40 - 12 * q^41 - 4 * q^43 - 2 * q^44 - 4 * q^45 - 4 * q^46 - 4 * q^47 + 2 * q^48 - 2 * q^50 - 4 * q^51 - 8 * q^52 - 4 * q^53 + 2 * q^54 + 4 * q^55 - 4 * q^57 + 8 * q^58 - 8 * q^59 - 4 * q^60 - 8 * q^61 - 4 * q^62 + 2 * q^64 + 16 * q^65 - 2 * q^66 - 8 * q^67 - 4 * q^68 - 4 * q^69 + 2 * q^72 - 12 * q^73 - 4 * q^74 - 2 * q^75 - 4 * q^76 - 8 * q^78 - 16 * q^79 - 4 * q^80 + 2 * q^81 - 12 * q^82 + 4 * q^83 + 8 * q^85 - 4 * q^86 + 8 * q^87 - 2 * q^88 - 16 * q^89 - 4 * q^90 - 4 * q^92 - 4 * q^93 - 4 * q^94 + 8 * q^95 + 2 * q^96 - 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	−5.41421	−1.50163	−0.750816	−	0.660511i	$-0.770340\pi$
$13$	−5.41421	−1.50163	−0.750816	+	0.660511i	$0.770340\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	2.24264	0.514497	0.257249	−	0.966345i	$-0.417184\pi$
$19$	2.24264	0.514497	0.257249	+	0.966345i	$0.417184\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−4.82843	−1.00680	−0.503398	−	0.864054i	$-0.667917\pi$
$23$	−4.82843	−1.00680	−0.503398	+	0.864054i	$0.667917\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	−5.41421	−1.06181
$27$	1.00000	0.192450
$28$	0	0
$29$	9.65685	1.79323	0.896616	−	0.442808i	$-0.146018\pi$
$29$	9.65685	1.79323	0.896616	+	0.442808i	$0.146018\pi$
$30$	−2.00000	−0.365148
$31$	−6.24264	−1.12121	−0.560606	−	0.828083i	$-0.689432\pi$
$31$	−6.24264	−1.12121	−0.560606	+	0.828083i	$0.689432\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	0.828427	0.136193	0.0680963	−	0.997679i	$-0.478307\pi$
$37$	0.828427	0.136193	0.0680963	+	0.997679i	$0.478307\pi$
$38$	2.24264	0.363804
$39$	−5.41421	−0.866968
$40$	−2.00000	−0.316228
$41$	−11.6569	−1.82049	−0.910247	−	0.414065i	$-0.864109\pi$
$41$	−11.6569	−1.82049	−0.910247	+	0.414065i	$0.864109\pi$
$42$	0	0
$43$	−4.82843	−0.736328	−0.368164	−	0.929761i	$-0.620014\pi$
$43$	−4.82843	−0.736328	−0.368164	+	0.929761i	$0.620014\pi$
$44$	−1.00000	−0.150756
$45$	−2.00000	−0.298142
$46$	−4.82843	−0.711913
$47$	7.89949	1.15226	0.576130	−	0.817358i	$-0.304562\pi$
$47$	7.89949	1.15226	0.576130	+	0.817358i	$0.304562\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	−5.41421	−0.750816
$53$	−10.4853	−1.44026	−0.720132	−	0.693837i	$-0.755919\pi$
$53$	−10.4853	−1.44026	−0.720132	+	0.693837i	$0.755919\pi$
$54$	1.00000	0.136083
$55$	2.00000	0.269680
$56$	0	0
$57$	2.24264	0.297045
$58$	9.65685	1.26801
$59$	−6.82843	−0.888985	−0.444493	−	0.895782i	$-0.646616\pi$
$59$	−6.82843	−0.888985	−0.444493	+	0.895782i	$0.646616\pi$
$60$	−2.00000	−0.258199
$61$	−2.58579	−0.331076	−0.165538	−	0.986203i	$-0.552936\pi$
$61$	−2.58579	−0.331076	−0.165538	+	0.986203i	$0.552936\pi$
$62$	−6.24264	−0.792816
$63$	0	0
$64$	1.00000	0.125000
$65$	10.8284	1.34310
$66$	−1.00000	−0.123091
$67$	−1.17157	−0.143130	−0.0715652	−	0.997436i	$-0.522799\pi$
$67$	−1.17157	−0.143130	−0.0715652	+	0.997436i	$0.522799\pi$
$68$	−2.00000	−0.242536
$69$	−4.82843	−0.581274
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	1.00000	0.117851
$73$	−11.6569	−1.36433	−0.682166	−	0.731198i	$-0.738962\pi$
$73$	−11.6569	−1.36433	−0.682166	+	0.731198i	$0.738962\pi$
$74$	0.828427	0.0963027
$75$	−1.00000	−0.115470
$76$	2.24264	0.257249
$77$	0	0
$78$	−5.41421	−0.613039
$79$	−16.4853	−1.85474	−0.927370	−	0.374147i	$-0.877936\pi$
$79$	−16.4853	−1.85474	−0.927370	+	0.374147i	$0.877936\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	−11.6569	−1.28728
$83$	−5.07107	−0.556622	−0.278311	−	0.960491i	$-0.589775\pi$
$83$	−5.07107	−0.556622	−0.278311	+	0.960491i	$0.589775\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	−4.82843	−0.520663
$87$	9.65685	1.03532
$88$	−1.00000	−0.106600
$89$	−3.75736	−0.398279	−0.199140	−	0.979971i	$-0.563815\pi$
$89$	−3.75736	−0.398279	−0.199140	+	0.979971i	$0.563815\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	−4.82843	−0.503398
$93$	−6.24264	−0.647332
$94$	7.89949	0.814771
$95$	−4.48528	−0.460180
$96$	1.00000	0.102062
$97$	9.89949	1.00514	0.502571	−	0.864536i	$-0.332388\pi$
$97$	9.89949	1.00514	0.502571	+	0.864536i	$0.332388\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	−1.00000	−0.100000
$101$	15.0711	1.49963	0.749814	−	0.661649i	$-0.230143\pi$
$101$	15.0711	1.49963	0.749814	+	0.661649i	$0.230143\pi$
$102$	−2.00000	−0.198030
$103$	−6.24264	−0.615106	−0.307553	−	0.951531i	$-0.599510\pi$
$103$	−6.24264	−0.615106	−0.307553	+	0.951531i	$0.599510\pi$
$104$	−5.41421	−0.530907
$105$	0	0
$106$	−10.4853	−1.01842
$107$	12.8284	1.24017	0.620085	−	0.784534i	$-0.287098\pi$
$107$	12.8284	1.24017	0.620085	+	0.784534i	$0.287098\pi$
$108$	1.00000	0.0962250
$109$	4.34315	0.415998	0.207999	−	0.978129i	$-0.433305\pi$
$109$	4.34315	0.415998	0.207999	+	0.978129i	$0.433305\pi$
$110$	2.00000	0.190693
$111$	0.828427	0.0786308
$112$	0	0
$113$	5.65685	0.532152	0.266076	−	0.963952i	$-0.414273\pi$
$113$	5.65685	0.532152	0.266076	+	0.963952i	$0.414273\pi$
$114$	2.24264	0.210043
$115$	9.65685	0.900506
$116$	9.65685	0.896616
$117$	−5.41421	−0.500544
$118$	−6.82843	−0.628608
$119$	0	0
$120$	−2.00000	−0.182574
$121$	1.00000	0.0909091
$122$	−2.58579	−0.234106
$123$	−11.6569	−1.05106
$124$	−6.24264	−0.560606
$125$	12.0000	1.07331
$126$	0	0
$127$	14.1421	1.25491	0.627456	−	0.778652i	$-0.284096\pi$
$127$	14.1421	1.25491	0.627456	+	0.778652i	$0.284096\pi$
$128$	1.00000	0.0883883
$129$	−4.82843	−0.425119
$130$	10.8284	0.949716
$131$	−17.5563	−1.53391	−0.766953	−	0.641704i	$-0.778228\pi$
$131$	−17.5563	−1.53391	−0.766953	+	0.641704i	$0.778228\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	−1.17157	−0.101208
$135$	−2.00000	−0.172133
$136$	−2.00000	−0.171499
$137$	−5.31371	−0.453981	−0.226990	−	0.973897i	$-0.572889\pi$
$137$	−5.31371	−0.453981	−0.226990	+	0.973897i	$0.572889\pi$
$138$	−4.82843	−0.411023
$139$	−0.100505	−0.00852473	−0.00426236	−	0.999991i	$-0.501357\pi$
$139$	−0.100505	−0.00852473	−0.00426236	+	0.999991i	$0.501357\pi$
$140$	0	0
$141$	7.89949	0.665257
$142$	5.65685	0.474713
$143$	5.41421	0.452759
$144$	1.00000	0.0833333
$145$	−19.3137	−1.60392
$146$	−11.6569	−0.964728
$147$	0	0
$148$	0.828427	0.0680963
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	−1.00000	−0.0816497
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	2.24264	0.181902
$153$	−2.00000	−0.161690
$154$	0	0
$155$	12.4853	1.00284
$156$	−5.41421	−0.433484
$157$	−12.8284	−1.02382	−0.511910	−	0.859039i	$-0.671062\pi$
$157$	−12.8284	−1.02382	−0.511910	+	0.859039i	$0.671062\pi$
$158$	−16.4853	−1.31150
$159$	−10.4853	−0.831537
$160$	−2.00000	−0.158114
$161$	0	0
$162$	1.00000	0.0785674
$163$	−2.34315	−0.183529	−0.0917647	−	0.995781i	$-0.529251\pi$
$163$	−2.34315	−0.183529	−0.0917647	+	0.995781i	$0.529251\pi$
$164$	−11.6569	−0.910247
$165$	2.00000	0.155700
$166$	−5.07107	−0.393591
$167$	5.17157	0.400188	0.200094	−	0.979777i	$-0.435875\pi$
$167$	5.17157	0.400188	0.200094	+	0.979777i	$0.435875\pi$
$168$	0	0
$169$	16.3137	1.25490
$170$	4.00000	0.306786
$171$	2.24264	0.171499
$172$	−4.82843	−0.368164
$173$	5.89949	0.448530	0.224265	−	0.974528i	$-0.428002\pi$
$173$	5.89949	0.448530	0.224265	+	0.974528i	$0.428002\pi$
$174$	9.65685	0.732084
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−6.82843	−0.513256
$178$	−3.75736	−0.281626
$179$	−20.4853	−1.53114	−0.765571	−	0.643352i	$-0.777543\pi$
$179$	−20.4853	−1.53114	−0.765571	+	0.643352i	$0.777543\pi$
$180$	−2.00000	−0.149071
$181$	9.31371	0.692283	0.346141	−	0.938182i	$-0.387492\pi$
$181$	9.31371	0.692283	0.346141	+	0.938182i	$0.387492\pi$
$182$	0	0
$183$	−2.58579	−0.191147
$184$	−4.82843	−0.355956
$185$	−1.65685	−0.121814
$186$	−6.24264	−0.457733
$187$	2.00000	0.146254
$188$	7.89949	0.576130
$189$	0	0
$190$	−4.48528	−0.325397
$191$	21.7990	1.57732	0.788660	−	0.614830i	$-0.210775\pi$
$191$	21.7990	1.57732	0.788660	+	0.614830i	$0.210775\pi$
$192$	1.00000	0.0721688
$193$	−14.9706	−1.07760	−0.538802	−	0.842432i	$-0.681123\pi$
$193$	−14.9706	−1.07760	−0.538802	+	0.842432i	$0.681123\pi$
$194$	9.89949	0.710742
$195$	10.8284	0.775440
$196$	0	0
$197$	15.6569	1.11550	0.557752	−	0.830007i	$-0.311664\pi$
$197$	15.6569	1.11550	0.557752	+	0.830007i	$0.311664\pi$
$198$	−1.00000	−0.0710669
$199$	−15.2132	−1.07844	−0.539218	−	0.842166i	$-0.681280\pi$
$199$	−15.2132	−1.07844	−0.539218	+	0.842166i	$0.681280\pi$
$200$	−1.00000	−0.0707107
$201$	−1.17157	−0.0826364
$202$	15.0711	1.06040
$203$	0	0
$204$	−2.00000	−0.140028
$205$	23.3137	1.62830
$206$	−6.24264	−0.434945
$207$	−4.82843	−0.335599
$208$	−5.41421	−0.375408
$209$	−2.24264	−0.155127
$210$	0	0
$211$	−23.1716	−1.59520	−0.797598	−	0.603189i	$-0.793897\pi$
$211$	−23.1716	−1.59520	−0.797598	+	0.603189i	$0.793897\pi$
$212$	−10.4853	−0.720132
$213$	5.65685	0.387601
$214$	12.8284	0.876933
$215$	9.65685	0.658592
$216$	1.00000	0.0680414
$217$	0	0
$218$	4.34315	0.294155
$219$	−11.6569	−0.787697
$220$	2.00000	0.134840
$221$	10.8284	0.728399
$222$	0.828427	0.0556004
$223$	−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
$225$	−1.00000	−0.0666667
$226$	5.65685	0.376288
$227$	11.8995	0.789797	0.394899	−	0.918725i	$-0.370780\pi$
$227$	11.8995	0.789797	0.394899	+	0.918725i	$0.370780\pi$
$228$	2.24264	0.148523
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	9.65685	0.636754
$231$	0	0
$232$	9.65685	0.634004
$233$	−7.65685	−0.501617	−0.250809	−	0.968037i	$-0.580696\pi$
$233$	−7.65685	−0.501617	−0.250809	+	0.968037i	$0.580696\pi$
$234$	−5.41421	−0.353938
$235$	−15.7990	−1.03061
$236$	−6.82843	−0.444493
$237$	−16.4853	−1.07083
$238$	0	0
$239$	0.686292	0.0443925	0.0221963	−	0.999754i	$-0.492934\pi$
$239$	0.686292	0.0443925	0.0221963	+	0.999754i	$0.492934\pi$
$240$	−2.00000	−0.129099
$241$	7.65685	0.493221	0.246611	−	0.969115i	$-0.420683\pi$
$241$	7.65685	0.493221	0.246611	+	0.969115i	$0.420683\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	−2.58579	−0.165538
$245$	0	0
$246$	−11.6569	−0.743214
$247$	−12.1421	−0.772586
$248$	−6.24264	−0.396408
$249$	−5.07107	−0.321366
$250$	12.0000	0.758947
$251$	−8.48528	−0.535586	−0.267793	−	0.963476i	$-0.586294\pi$
$251$	−8.48528	−0.535586	−0.267793	+	0.963476i	$0.586294\pi$
$252$	0	0
$253$	4.82843	0.303561
$254$	14.1421	0.887357
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	29.2132	1.82227	0.911135	−	0.412108i	$-0.135208\pi$
$257$	29.2132	1.82227	0.911135	+	0.412108i	$0.135208\pi$
$258$	−4.82843	−0.300605
$259$	0	0
$260$	10.8284	0.671551
$261$	9.65685	0.597744
$262$	−17.5563	−1.08463
$263$	9.17157	0.565543	0.282772	−	0.959187i	$-0.408746\pi$
$263$	9.17157	0.565543	0.282772	+	0.959187i	$0.408746\pi$
$264$	−1.00000	−0.0615457
$265$	20.9706	1.28821
$266$	0	0
$267$	−3.75736	−0.229947
$268$	−1.17157	−0.0715652
$269$	−5.31371	−0.323983	−0.161991	−	0.986792i	$-0.551792\pi$
$269$	−5.31371	−0.323983	−0.161991	+	0.986792i	$0.551792\pi$
$270$	−2.00000	−0.121716
$271$	−22.1421	−1.34504	−0.672519	−	0.740079i	$-0.734788\pi$
$271$	−22.1421	−1.34504	−0.672519	+	0.740079i	$0.734788\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−5.31371	−0.321013
$275$	1.00000	0.0603023
$276$	−4.82843	−0.290637
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	−0.100505	−0.00602789
$279$	−6.24264	−0.373737
$280$	0	0
$281$	−1.31371	−0.0783693	−0.0391846	−	0.999232i	$-0.512476\pi$
$281$	−1.31371	−0.0783693	−0.0391846	+	0.999232i	$0.512476\pi$
$282$	7.89949	0.470408
$283$	2.24264	0.133311	0.0666556	−	0.997776i	$-0.478767\pi$
$283$	2.24264	0.133311	0.0666556	+	0.997776i	$0.478767\pi$
$284$	5.65685	0.335673
$285$	−4.48528	−0.265685
$286$	5.41421	0.320149
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	−19.3137	−1.13414
$291$	9.89949	0.580319
$292$	−11.6569	−0.682166
$293$	4.92893	0.287951	0.143976	−	0.989581i	$-0.454011\pi$
$293$	4.92893	0.287951	0.143976	+	0.989581i	$0.454011\pi$
$294$	0	0
$295$	13.6569	0.795133
$296$	0.828427	0.0481513
$297$	−1.00000	−0.0580259
$298$	4.00000	0.231714
$299$	26.1421	1.51184
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	4.00000	0.230174
$303$	15.0711	0.865810
$304$	2.24264	0.128624
$305$	5.17157	0.296123
$306$	−2.00000	−0.114332
$307$	25.0711	1.43088	0.715441	−	0.698673i	$-0.246226\pi$
$307$	25.0711	1.43088	0.715441	+	0.698673i	$0.246226\pi$
$308$	0	0
$309$	−6.24264	−0.355131
$310$	12.4853	0.709116
$311$	−8.10051	−0.459338	−0.229669	−	0.973269i	$-0.573764\pi$
$311$	−8.10051	−0.459338	−0.229669	+	0.973269i	$0.573764\pi$
$312$	−5.41421	−0.306519
$313$	8.24264	0.465902	0.232951	−	0.972489i	$-0.425162\pi$
$313$	8.24264	0.465902	0.232951	+	0.972489i	$0.425162\pi$
$314$	−12.8284	−0.723950
$315$	0	0
$316$	−16.4853	−0.927370
$317$	12.8284	0.720516	0.360258	−	0.932853i	$-0.382689\pi$
$317$	12.8284	0.720516	0.360258	+	0.932853i	$0.382689\pi$
$318$	−10.4853	−0.587985
$319$	−9.65685	−0.540680
$320$	−2.00000	−0.111803
$321$	12.8284	0.716013
$322$	0	0
$323$	−4.48528	−0.249568
$324$	1.00000	0.0555556
$325$	5.41421	0.300327
$326$	−2.34315	−0.129775
$327$	4.34315	0.240177
$328$	−11.6569	−0.643642
$329$	0	0
$330$	2.00000	0.110096
$331$	−27.7990	−1.52797	−0.763985	−	0.645234i	$-0.776760\pi$
$331$	−27.7990	−1.52797	−0.763985	+	0.645234i	$0.776760\pi$
$332$	−5.07107	−0.278311
$333$	0.828427	0.0453975
$334$	5.17157	0.282976
$335$	2.34315	0.128020
$336$	0	0
$337$	12.8284	0.698809	0.349404	−	0.936972i	$-0.386384\pi$
$337$	12.8284	0.698809	0.349404	+	0.936972i	$0.386384\pi$
$338$	16.3137	0.887349
$339$	5.65685	0.307238
$340$	4.00000	0.216930
$341$	6.24264	0.338058
$342$	2.24264	0.121268
$343$	0	0
$344$	−4.82843	−0.260331
$345$	9.65685	0.519908
$346$	5.89949	0.317159
$347$	19.4558	1.04444	0.522222	−	0.852809i	$-0.325103\pi$
$347$	19.4558	1.04444	0.522222	+	0.852809i	$0.325103\pi$
$348$	9.65685	0.517662
$349$	17.4142	0.932161	0.466081	−	0.884742i	$-0.345666\pi$
$349$	17.4142	0.932161	0.466081	+	0.884742i	$0.345666\pi$
$350$	0	0
$351$	−5.41421	−0.288989
$352$	−1.00000	−0.0533002
$353$	−21.2132	−1.12906	−0.564532	−	0.825411i	$-0.690943\pi$
$353$	−21.2132	−1.12906	−0.564532	+	0.825411i	$0.690943\pi$
$354$	−6.82843	−0.362927
$355$	−11.3137	−0.600469
$356$	−3.75736	−0.199140
$357$	0	0
$358$	−20.4853	−1.08268
$359$	5.17157	0.272945	0.136473	−	0.990644i	$-0.456423\pi$
$359$	5.17157	0.272945	0.136473	+	0.990644i	$0.456423\pi$
$360$	−2.00000	−0.105409
$361$	−13.9706	−0.735293
$362$	9.31371	0.489518
$363$	1.00000	0.0524864
$364$	0	0
$365$	23.3137	1.22030
$366$	−2.58579	−0.135161
$367$	32.5858	1.70096	0.850482	−	0.526004i	$-0.176310\pi$
$367$	32.5858	1.70096	0.850482	+	0.526004i	$0.176310\pi$
$368$	−4.82843	−0.251699
$369$	−11.6569	−0.606832
$370$	−1.65685	−0.0861358
$371$	0	0
$372$	−6.24264	−0.323666
$373$	35.9411	1.86096	0.930480	−	0.366342i	$-0.119390\pi$
$373$	35.9411	1.86096	0.930480	+	0.366342i	$0.119390\pi$
$374$	2.00000	0.103418
$375$	12.0000	0.619677
$376$	7.89949	0.407385
$377$	−52.2843	−2.69278
$378$	0	0
$379$	32.2843	1.65833	0.829166	−	0.559003i	$-0.188816\pi$
$379$	32.2843	1.65833	0.829166	+	0.559003i	$0.188816\pi$
$380$	−4.48528	−0.230090
$381$	14.1421	0.724524
$382$	21.7990	1.11533
$383$	22.7279	1.16134	0.580671	−	0.814138i	$-0.302790\pi$
$383$	22.7279	1.16134	0.580671	+	0.814138i	$0.302790\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−14.9706	−0.761982
$387$	−4.82843	−0.245443
$388$	9.89949	0.502571
$389$	−34.2843	−1.73828	−0.869141	−	0.494565i	$-0.835327\pi$
$389$	−34.2843	−1.73828	−0.869141	+	0.494565i	$0.835327\pi$
$390$	10.8284	0.548319
$391$	9.65685	0.488368
$392$	0	0
$393$	−17.5563	−0.885601
$394$	15.6569	0.788781
$395$	32.9706	1.65893
$396$	−1.00000	−0.0502519
$397$	11.1716	0.560685	0.280343	−	0.959900i	$-0.409552\pi$
$397$	11.1716	0.560685	0.280343	+	0.959900i	$0.409552\pi$
$398$	−15.2132	−0.762569
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	10.6863	0.533648	0.266824	−	0.963745i	$-0.414026\pi$
$401$	10.6863	0.533648	0.266824	+	0.963745i	$0.414026\pi$
$402$	−1.17157	−0.0584327
$403$	33.7990	1.68365
$404$	15.0711	0.749814
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−0.828427	−0.0410636
$408$	−2.00000	−0.0990148
$409$	1.51472	0.0748980	0.0374490	−	0.999299i	$-0.488077\pi$
$409$	1.51472	0.0748980	0.0374490	+	0.999299i	$0.488077\pi$
$410$	23.3137	1.15138
$411$	−5.31371	−0.262106
$412$	−6.24264	−0.307553
$413$	0	0
$414$	−4.82843	−0.237304
$415$	10.1421	0.497858
$416$	−5.41421	−0.265454
$417$	−0.100505	−0.00492175
$418$	−2.24264	−0.109691
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	−27.6569	−1.34791	−0.673956	−	0.738771i	$-0.735406\pi$
$421$	−27.6569	−1.34791	−0.673956	+	0.738771i	$0.735406\pi$
$422$	−23.1716	−1.12797
$423$	7.89949	0.384087
$424$	−10.4853	−0.509210
$425$	2.00000	0.0970143
$426$	5.65685	0.274075
$427$	0	0
$428$	12.8284	0.620085
$429$	5.41421	0.261401
$430$	9.65685	0.465695
$431$	2.14214	0.103183	0.0515915	−	0.998668i	$-0.483571\pi$
$431$	2.14214	0.103183	0.0515915	+	0.998668i	$0.483571\pi$
$432$	1.00000	0.0481125
$433$	34.3848	1.65243	0.826213	−	0.563357i	$-0.190491\pi$
$433$	34.3848	1.65243	0.826213	+	0.563357i	$0.190491\pi$
$434$	0	0
$435$	−19.3137	−0.926021
$436$	4.34315	0.207999
$437$	−10.8284	−0.517994
$438$	−11.6569	−0.556986
$439$	29.1716	1.39228	0.696142	−	0.717904i	$-0.254899\pi$
$439$	29.1716	1.39228	0.696142	+	0.717904i	$0.254899\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	10.8284	0.515056
$443$	12.4853	0.593194	0.296597	−	0.955003i	$-0.404148\pi$
$443$	12.4853	0.593194	0.296597	+	0.955003i	$0.404148\pi$
$444$	0.828427	0.0393154
$445$	7.51472	0.356232
$446$	−6.24264	−0.295598
$447$	4.00000	0.189194
$448$	0	0
$449$	16.9706	0.800890	0.400445	−	0.916321i	$-0.368855\pi$
$449$	16.9706	0.800890	0.400445	+	0.916321i	$0.368855\pi$
$450$	−1.00000	−0.0471405
$451$	11.6569	0.548900
$452$	5.65685	0.266076
$453$	4.00000	0.187936
$454$	11.8995	0.558471
$455$	0	0
$456$	2.24264	0.105021
$457$	−29.3137	−1.37124	−0.685619	−	0.727961i	$-0.740468\pi$
$457$	−29.3137	−1.37124	−0.685619	+	0.727961i	$0.740468\pi$
$458$	−14.0000	−0.654177
$459$	−2.00000	−0.0933520
$460$	9.65685	0.450253
$461$	30.5858	1.42452	0.712261	−	0.701915i	$-0.247671\pi$
$461$	30.5858	1.42452	0.712261	+	0.701915i	$0.247671\pi$
$462$	0	0
$463$	−3.31371	−0.154001	−0.0770005	−	0.997031i	$-0.524534\pi$
$463$	−3.31371	−0.154001	−0.0770005	+	0.997031i	$0.524534\pi$
$464$	9.65685	0.448308
$465$	12.4853	0.578991
$466$	−7.65685	−0.354697
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−5.41421	−0.250272
$469$	0	0
$470$	−15.7990	−0.728753
$471$	−12.8284	−0.591103
$472$	−6.82843	−0.314304
$473$	4.82843	0.222011
$474$	−16.4853	−0.757194
$475$	−2.24264	−0.102899
$476$	0	0
$477$	−10.4853	−0.480088
$478$	0.686292	0.0313902
$479$	−7.31371	−0.334172	−0.167086	−	0.985942i	$-0.553436\pi$
$479$	−7.31371	−0.334172	−0.167086	+	0.985942i	$0.553436\pi$
$480$	−2.00000	−0.0912871
$481$	−4.48528	−0.204511
$482$	7.65685	0.348760
$483$	0	0
$484$	1.00000	0.0454545
$485$	−19.7990	−0.899026
$486$	1.00000	0.0453609
$487$	0.142136	0.00644078	0.00322039	−	0.999995i	$-0.498975\pi$
$487$	0.142136	0.00644078	0.00322039	+	0.999995i	$0.498975\pi$
$488$	−2.58579	−0.117053
$489$	−2.34315	−0.105961
$490$	0	0
$491$	−42.6274	−1.92375	−0.961874	−	0.273492i	$-0.911821\pi$
$491$	−42.6274	−1.92375	−0.961874	+	0.273492i	$0.911821\pi$
$492$	−11.6569	−0.525532
$493$	−19.3137	−0.869846
$494$	−12.1421	−0.546301
$495$	2.00000	0.0898933
$496$	−6.24264	−0.280303
$497$	0	0
$498$	−5.07107	−0.227240
$499$	−6.14214	−0.274960	−0.137480	−	0.990505i	$-0.543900\pi$
$499$	−6.14214	−0.274960	−0.137480	+	0.990505i	$0.543900\pi$
$500$	12.0000	0.536656
$501$	5.17157	0.231049
$502$	−8.48528	−0.378717
$503$	8.48528	0.378340	0.189170	−	0.981944i	$-0.439420\pi$
$503$	8.48528	0.378340	0.189170	+	0.981944i	$0.439420\pi$
$504$	0	0
$505$	−30.1421	−1.34131
$506$	4.82843	0.214650
$507$	16.3137	0.724517
$508$	14.1421	0.627456
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	4.00000	0.177123
$511$	0	0
$512$	1.00000	0.0441942
$513$	2.24264	0.0990150
$514$	29.2132	1.28854
$515$	12.4853	0.550167
$516$	−4.82843	−0.212560
$517$	−7.89949	−0.347419
$518$	0	0
$519$	5.89949	0.258959
$520$	10.8284	0.474858
$521$	−21.2132	−0.929367	−0.464684	−	0.885477i	$-0.653832\pi$
$521$	−21.2132	−0.929367	−0.464684	+	0.885477i	$0.653832\pi$
$522$	9.65685	0.422669
$523$	9.07107	0.396650	0.198325	−	0.980136i	$-0.436450\pi$
$523$	9.07107	0.396650	0.198325	+	0.980136i	$0.436450\pi$
$524$	−17.5563	−0.766953
$525$	0	0
$526$	9.17157	0.399900
$527$	12.4853	0.543867
$528$	−1.00000	−0.0435194
$529$	0.313708	0.0136395
$530$	20.9706	0.910903
$531$	−6.82843	−0.296328
$532$	0	0
$533$	63.1127	2.73371
$534$	−3.75736	−0.162597
$535$	−25.6569	−1.10924
$536$	−1.17157	−0.0506042
$537$	−20.4853	−0.884005
$538$	−5.31371	−0.229090
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	−45.3137	−1.94819	−0.974094	−	0.226142i	$-0.927389\pi$
$541$	−45.3137	−1.94819	−0.974094	+	0.226142i	$0.927389\pi$
$542$	−22.1421	−0.951086
$543$	9.31371	0.399689
$544$	−2.00000	−0.0857493
$545$	−8.68629	−0.372080
$546$	0	0
$547$	−42.6274	−1.82262	−0.911308	−	0.411724i	$-0.864927\pi$
$547$	−42.6274	−1.82262	−0.911308	+	0.411724i	$0.864927\pi$
$548$	−5.31371	−0.226990
$549$	−2.58579	−0.110359
$550$	1.00000	0.0426401
$551$	21.6569	0.922613
$552$	−4.82843	−0.205512
$553$	0	0
$554$	−16.0000	−0.679775
$555$	−1.65685	−0.0703295
$556$	−0.100505	−0.00426236
$557$	19.6569	0.832888	0.416444	−	0.909161i	$-0.363276\pi$
$557$	19.6569	0.832888	0.416444	+	0.909161i	$0.363276\pi$
$558$	−6.24264	−0.264272
$559$	26.1421	1.10569
$560$	0	0
$561$	2.00000	0.0844401
$562$	−1.31371	−0.0554154
$563$	21.0711	0.888040	0.444020	−	0.896017i	$-0.353552\pi$
$563$	21.0711	0.888040	0.444020	+	0.896017i	$0.353552\pi$
$564$	7.89949	0.332629
$565$	−11.3137	−0.475971
$566$	2.24264	0.0942652
$567$	0	0
$568$	5.65685	0.237356
$569$	−10.6863	−0.447993	−0.223996	−	0.974590i	$-0.571910\pi$
$569$	−10.6863	−0.447993	−0.223996	+	0.974590i	$0.571910\pi$
$570$	−4.48528	−0.187868
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	5.41421	0.226380
$573$	21.7990	0.910666
$574$	0	0
$575$	4.82843	0.201359
$576$	1.00000	0.0416667
$577$	4.44365	0.184992	0.0924958	−	0.995713i	$-0.470516\pi$
$577$	4.44365	0.184992	0.0924958	+	0.995713i	$0.470516\pi$
$578$	−13.0000	−0.540729
$579$	−14.9706	−0.622155
$580$	−19.3137	−0.801958
$581$	0	0
$582$	9.89949	0.410347
$583$	10.4853	0.434256
$584$	−11.6569	−0.482364
$585$	10.8284	0.447700
$586$	4.92893	0.203612
$587$	−11.7990	−0.486996	−0.243498	−	0.969901i	$-0.578295\pi$
$587$	−11.7990	−0.486996	−0.243498	+	0.969901i	$0.578295\pi$
$588$	0	0
$589$	−14.0000	−0.576860
$590$	13.6569	0.562244
$591$	15.6569	0.644037
$592$	0.828427	0.0340481
$593$	37.3137	1.53229	0.766145	−	0.642668i	$-0.222172\pi$
$593$	37.3137	1.53229	0.766145	+	0.642668i	$0.222172\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	4.00000	0.163846
$597$	−15.2132	−0.622635
$598$	26.1421	1.06903
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	−1.00000	−0.0408248
$601$	−15.1716	−0.618861	−0.309431	−	0.950922i	$-0.600138\pi$
$601$	−15.1716	−0.618861	−0.309431	+	0.950922i	$0.600138\pi$
$602$	0	0
$603$	−1.17157	−0.0477101
$604$	4.00000	0.162758
$605$	−2.00000	−0.0813116
$606$	15.0711	0.612220
$607$	35.3137	1.43334	0.716670	−	0.697413i	$-0.245666\pi$
$607$	35.3137	1.43334	0.716670	+	0.697413i	$0.245666\pi$
$608$	2.24264	0.0909511
$609$	0	0
$610$	5.17157	0.209391
$611$	−42.7696	−1.73027
$612$	−2.00000	−0.0808452
$613$	−30.3431	−1.22555	−0.612774	−	0.790258i	$-0.709946\pi$
$613$	−30.3431	−1.22555	−0.612774	+	0.790258i	$0.709946\pi$
$614$	25.0711	1.01179
$615$	23.3137	0.940099
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	−6.24264	−0.251116
$619$	21.6569	0.870462	0.435231	−	0.900319i	$-0.356667\pi$
$619$	21.6569	0.870462	0.435231	+	0.900319i	$0.356667\pi$
$620$	12.4853	0.501421
$621$	−4.82843	−0.193758
$622$	−8.10051	−0.324801
$623$	0	0
$624$	−5.41421	−0.216742
$625$	−19.0000	−0.760000
$626$	8.24264	0.329442
$627$	−2.24264	−0.0895624
$628$	−12.8284	−0.511910
$629$	−1.65685	−0.0660631
$630$	0	0
$631$	36.1421	1.43880	0.719398	−	0.694598i	$-0.244418\pi$
$631$	36.1421	1.43880	0.719398	+	0.694598i	$0.244418\pi$
$632$	−16.4853	−0.655749
$633$	−23.1716	−0.920987
$634$	12.8284	0.509482
$635$	−28.2843	−1.12243
$636$	−10.4853	−0.415768
$637$	0	0
$638$	−9.65685	−0.382319
$639$	5.65685	0.223782
$640$	−2.00000	−0.0790569
$641$	−35.3137	−1.39481	−0.697404	−	0.716678i	$-0.745662\pi$
$641$	−35.3137	−1.39481	−0.697404	+	0.716678i	$0.745662\pi$
$642$	12.8284	0.506298
$643$	−37.4558	−1.47711	−0.738557	−	0.674191i	$-0.764493\pi$
$643$	−37.4558	−1.47711	−0.738557	+	0.674191i	$0.764493\pi$
$644$	0	0
$645$	9.65685	0.380238
$646$	−4.48528	−0.176471
$647$	−1.07107	−0.0421080	−0.0210540	−	0.999778i	$-0.506702\pi$
$647$	−1.07107	−0.0421080	−0.0210540	+	0.999778i	$0.506702\pi$
$648$	1.00000	0.0392837
$649$	6.82843	0.268039
$650$	5.41421	0.212363
$651$	0	0
$652$	−2.34315	−0.0917647
$653$	22.2843	0.872051	0.436025	−	0.899934i	$-0.356386\pi$
$653$	22.2843	0.872051	0.436025	+	0.899934i	$0.356386\pi$
$654$	4.34315	0.169830
$655$	35.1127	1.37197
$656$	−11.6569	−0.455124
$657$	−11.6569	−0.454777
$658$	0	0
$659$	−49.1127	−1.91316	−0.956580	−	0.291471i	$-0.905855\pi$
$659$	−49.1127	−1.91316	−0.956580	+	0.291471i	$0.905855\pi$
$660$	2.00000	0.0778499
$661$	39.9411	1.55353	0.776765	−	0.629791i	$-0.216859\pi$
$661$	39.9411	1.55353	0.776765	+	0.629791i	$0.216859\pi$
$662$	−27.7990	−1.08044
$663$	10.8284	0.420541
$664$	−5.07107	−0.196796
$665$	0	0
$666$	0.828427	0.0321009
$667$	−46.6274	−1.80542
$668$	5.17157	0.200094
$669$	−6.24264	−0.241354
$670$	2.34315	0.0905236
$671$	2.58579	0.0998232
$672$	0	0
$673$	−17.5147	−0.675143	−0.337571	−	0.941300i	$-0.609605\pi$
$673$	−17.5147	−0.675143	−0.337571	+	0.941300i	$0.609605\pi$
$674$	12.8284	0.494133
$675$	−1.00000	−0.0384900
$676$	16.3137	0.627450
$677$	36.0416	1.38519	0.692596	−	0.721326i	$-0.256467\pi$
$677$	36.0416	1.38519	0.692596	+	0.721326i	$0.256467\pi$
$678$	5.65685	0.217250
$679$	0	0
$680$	4.00000	0.153393
$681$	11.8995	0.455990
$682$	6.24264	0.239043
$683$	28.7696	1.10084	0.550418	−	0.834889i	$-0.314468\pi$
$683$	28.7696	1.10084	0.550418	+	0.834889i	$0.314468\pi$
$684$	2.24264	0.0857495
$685$	10.6274	0.406053
$686$	0	0
$687$	−14.0000	−0.534133
$688$	−4.82843	−0.184082
$689$	56.7696	2.16275
$690$	9.65685	0.367630
$691$	14.1421	0.537992	0.268996	−	0.963141i	$-0.413308\pi$
$691$	14.1421	0.537992	0.268996	+	0.963141i	$0.413308\pi$
$692$	5.89949	0.224265
$693$	0	0
$694$	19.4558	0.738534
$695$	0.201010	0.00762475
$696$	9.65685	0.366042
$697$	23.3137	0.883070
$698$	17.4142	0.659138
$699$	−7.65685	−0.289609
$700$	0	0
$701$	9.31371	0.351774	0.175887	−	0.984410i	$-0.443721\pi$
$701$	9.31371	0.351774	0.175887	+	0.984410i	$0.443721\pi$
$702$	−5.41421	−0.204346
$703$	1.85786	0.0700707
$704$	−1.00000	−0.0376889
$705$	−15.7990	−0.595024
$706$	−21.2132	−0.798369
$707$	0	0
$708$	−6.82843	−0.256628
$709$	−34.7696	−1.30580	−0.652899	−	0.757445i	$-0.726447\pi$
$709$	−34.7696	−1.30580	−0.652899	+	0.757445i	$0.726447\pi$
$710$	−11.3137	−0.424596
$711$	−16.4853	−0.618246
$712$	−3.75736	−0.140813
$713$	30.1421	1.12883
$714$	0	0
$715$	−10.8284	−0.404960
$716$	−20.4853	−0.765571
$717$	0.686292	0.0256300
$718$	5.17157	0.193001
$719$	44.1838	1.64778	0.823888	−	0.566752i	$-0.191800\pi$
$719$	44.1838	1.64778	0.823888	+	0.566752i	$0.191800\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	−13.9706	−0.519931
$723$	7.65685	0.284761
$724$	9.31371	0.346141
$725$	−9.65685	−0.358647
$726$	1.00000	0.0371135
$727$	−31.2132	−1.15763	−0.578817	−	0.815458i	$-0.696485\pi$
$727$	−31.2132	−1.15763	−0.578817	+	0.815458i	$0.696485\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	23.3137	0.862879
$731$	9.65685	0.357172
$732$	−2.58579	−0.0955734
$733$	−19.5563	−0.722330	−0.361165	−	0.932502i	$-0.617621\pi$
$733$	−19.5563	−0.722330	−0.361165	+	0.932502i	$0.617621\pi$
$734$	32.5858	1.20276
$735$	0	0
$736$	−4.82843	−0.177978
$737$	1.17157	0.0431554
$738$	−11.6569	−0.429095
$739$	2.62742	0.0966511	0.0483255	−	0.998832i	$-0.484612\pi$
$739$	2.62742	0.0966511	0.0483255	+	0.998832i	$0.484612\pi$
$740$	−1.65685	−0.0609072
$741$	−12.1421	−0.446052
$742$	0	0
$743$	−18.6274	−0.683374	−0.341687	−	0.939814i	$-0.610998\pi$
$743$	−18.6274	−0.683374	−0.341687	+	0.939814i	$0.610998\pi$
$744$	−6.24264	−0.228866
$745$	−8.00000	−0.293097
$746$	35.9411	1.31590
$747$	−5.07107	−0.185541
$748$	2.00000	0.0731272
$749$	0	0
$750$	12.0000	0.438178
$751$	−12.1421	−0.443073	−0.221536	−	0.975152i	$-0.571107\pi$
$751$	−12.1421	−0.443073	−0.221536	+	0.975152i	$0.571107\pi$
$752$	7.89949	0.288065
$753$	−8.48528	−0.309221
$754$	−52.2843	−1.90408
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−48.1421	−1.74976	−0.874878	−	0.484344i	$-0.839058\pi$
$757$	−48.1421	−1.74976	−0.874878	+	0.484344i	$0.839058\pi$
$758$	32.2843	1.17262
$759$	4.82843	0.175261
$760$	−4.48528	−0.162698
$761$	1.51472	0.0549085	0.0274543	−	0.999623i	$-0.491260\pi$
$761$	1.51472	0.0549085	0.0274543	+	0.999623i	$0.491260\pi$
$762$	14.1421	0.512316
$763$	0	0
$764$	21.7990	0.788660
$765$	4.00000	0.144620
$766$	22.7279	0.821193
$767$	36.9706	1.33493
$768$	1.00000	0.0360844
$769$	15.8579	0.571849	0.285925	−	0.958252i	$-0.407699\pi$
$769$	15.8579	0.571849	0.285925	+	0.958252i	$0.407699\pi$
$770$	0	0
$771$	29.2132	1.05209
$772$	−14.9706	−0.538802
$773$	−5.51472	−0.198351	−0.0991753	−	0.995070i	$-0.531620\pi$
$773$	−5.51472	−0.198351	−0.0991753	+	0.995070i	$0.531620\pi$
$774$	−4.82843	−0.173554
$775$	6.24264	0.224242
$776$	9.89949	0.355371
$777$	0	0
$778$	−34.2843	−1.22915
$779$	−26.1421	−0.936639
$780$	10.8284	0.387720
$781$	−5.65685	−0.202418
$782$	9.65685	0.345328
$783$	9.65685	0.345108
$784$	0	0
$785$	25.6569	0.915732
$786$	−17.5563	−0.626214
$787$	3.41421	0.121704	0.0608518	−	0.998147i	$-0.480618\pi$
$787$	3.41421	0.121704	0.0608518	+	0.998147i	$0.480618\pi$
$788$	15.6569	0.557752
$789$	9.17157	0.326517
$790$	32.9706	1.17304
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	14.0000	0.497155
$794$	11.1716	0.396464
$795$	20.9706	0.743749
$796$	−15.2132	−0.539218
$797$	−17.5147	−0.620403	−0.310202	−	0.950671i	$-0.600397\pi$
$797$	−17.5147	−0.620403	−0.310202	+	0.950671i	$0.600397\pi$
$798$	0	0
$799$	−15.7990	−0.558928
$800$	−1.00000	−0.0353553
$801$	−3.75736	−0.132760
$802$	10.6863	0.377346
$803$	11.6569	0.411361
$804$	−1.17157	−0.0413182
$805$	0	0
$806$	33.7990	1.19052
$807$	−5.31371	−0.187051
$808$	15.0711	0.530198
$809$	26.7696	0.941167	0.470584	−	0.882355i	$-0.344043\pi$
$809$	26.7696	0.941167	0.470584	+	0.882355i	$0.344043\pi$
$810$	−2.00000	−0.0702728
$811$	−36.3848	−1.27764	−0.638821	−	0.769355i	$-0.720578\pi$
$811$	−36.3848	−1.27764	−0.638821	+	0.769355i	$0.720578\pi$
$812$	0	0
$813$	−22.1421	−0.776559
$814$	−0.828427	−0.0290364
$815$	4.68629	0.164154
$816$	−2.00000	−0.0700140
$817$	−10.8284	−0.378839
$818$	1.51472	0.0529609
$819$	0	0
$820$	23.3137	0.814150
$821$	47.3137	1.65126	0.825630	−	0.564212i	$-0.190820\pi$
$821$	47.3137	1.65126	0.825630	+	0.564212i	$0.190820\pi$
$822$	−5.31371	−0.185337
$823$	−28.1421	−0.980973	−0.490487	−	0.871449i	$-0.663181\pi$
$823$	−28.1421	−0.980973	−0.490487	+	0.871449i	$0.663181\pi$
$824$	−6.24264	−0.217473
$825$	1.00000	0.0348155
$826$	0	0
$827$	39.3137	1.36707	0.683536	−	0.729917i	$-0.260441\pi$
$827$	39.3137	1.36707	0.683536	+	0.729917i	$0.260441\pi$
$828$	−4.82843	−0.167799
$829$	−50.2843	−1.74644	−0.873222	−	0.487322i	$-0.837974\pi$
$829$	−50.2843	−1.74644	−0.873222	+	0.487322i	$0.837974\pi$
$830$	10.1421	0.352039
$831$	−16.0000	−0.555034
$832$	−5.41421	−0.187704
$833$	0	0
$834$	−0.100505	−0.00348021
$835$	−10.3431	−0.357939
$836$	−2.24264	−0.0775634
$837$	−6.24264	−0.215777
$838$	−16.0000	−0.552711
$839$	17.0711	0.589359	0.294679	−	0.955596i	$-0.404787\pi$
$839$	17.0711	0.589359	0.294679	+	0.955596i	$0.404787\pi$
$840$	0	0
$841$	64.2548	2.21568
$842$	−27.6569	−0.953118
$843$	−1.31371	−0.0452465
$844$	−23.1716	−0.797598
$845$	−32.6274	−1.12242
$846$	7.89949	0.271590
$847$	0	0
$848$	−10.4853	−0.360066
$849$	2.24264	0.0769672
$850$	2.00000	0.0685994
$851$	−4.00000	−0.137118
$852$	5.65685	0.193801
$853$	−35.5563	−1.21743	−0.608713	−	0.793390i	$-0.708314\pi$
$853$	−35.5563	−1.21743	−0.608713	+	0.793390i	$0.708314\pi$
$854$	0	0
$855$	−4.48528	−0.153393
$856$	12.8284	0.438467
$857$	−0.142136	−0.00485526	−0.00242763	−	0.999997i	$-0.500773\pi$
$857$	−0.142136	−0.00485526	−0.00242763	+	0.999997i	$0.500773\pi$
$858$	5.41421	0.184838
$859$	30.4264	1.03814	0.519068	−	0.854733i	$-0.326279\pi$
$859$	30.4264	1.03814	0.519068	+	0.854733i	$0.326279\pi$
$860$	9.65685	0.329296
$861$	0	0
$862$	2.14214	0.0729614
$863$	−38.7696	−1.31973	−0.659865	−	0.751384i	$-0.729387\pi$
$863$	−38.7696	−1.31973	−0.659865	+	0.751384i	$0.729387\pi$
$864$	1.00000	0.0340207
$865$	−11.7990	−0.401178
$866$	34.3848	1.16844
$867$	−13.0000	−0.441503
$868$	0	0
$869$	16.4853	0.559225
$870$	−19.3137	−0.654796
$871$	6.34315	0.214929
$872$	4.34315	0.147077
$873$	9.89949	0.335047
$874$	−10.8284	−0.366277
$875$	0	0
$876$	−11.6569	−0.393849
$877$	−6.68629	−0.225780	−0.112890	−	0.993607i	$-0.536011\pi$
$877$	−6.68629	−0.225780	−0.112890	+	0.993607i	$0.536011\pi$
$878$	29.1716	0.984493
$879$	4.92893	0.166249
$880$	2.00000	0.0674200
$881$	−37.8995	−1.27687	−0.638433	−	0.769677i	$-0.720417\pi$
$881$	−37.8995	−1.27687	−0.638433	+	0.769677i	$0.720417\pi$
$882$	0	0
$883$	−10.6274	−0.357641	−0.178821	−	0.983882i	$-0.557228\pi$
$883$	−10.6274	−0.357641	−0.178821	+	0.983882i	$0.557228\pi$
$884$	10.8284	0.364199
$885$	13.6569	0.459070
$886$	12.4853	0.419451
$887$	−4.28427	−0.143852	−0.0719259	−	0.997410i	$-0.522915\pi$
$887$	−4.28427	−0.143852	−0.0719259	+	0.997410i	$0.522915\pi$
$888$	0.828427	0.0278002
$889$	0	0
$890$	7.51472	0.251894
$891$	−1.00000	−0.0335013
$892$	−6.24264	−0.209019
$893$	17.7157	0.592834
$894$	4.00000	0.133780
$895$	40.9706	1.36949
$896$	0	0
$897$	26.1421	0.872861
$898$	16.9706	0.566315
$899$	−60.2843	−2.01059
$900$	−1.00000	−0.0333333
$901$	20.9706	0.698631
$902$	11.6569	0.388131
$903$	0	0
$904$	5.65685	0.188144
$905$	−18.6274	−0.619196
$906$	4.00000	0.132891
$907$	−8.20101	−0.272310	−0.136155	−	0.990688i	$-0.543475\pi$
$907$	−8.20101	−0.272310	−0.136155	+	0.990688i	$0.543475\pi$
$908$	11.8995	0.394899
$909$	15.0711	0.499876
$910$	0	0
$911$	−11.4558	−0.379549	−0.189775	−	0.981828i	$-0.560776\pi$
$911$	−11.4558	−0.379549	−0.189775	+	0.981828i	$0.560776\pi$
$912$	2.24264	0.0742613
$913$	5.07107	0.167828
$914$	−29.3137	−0.969611
$915$	5.17157	0.170967
$916$	−14.0000	−0.462573
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	−0.970563	−0.0320159	−0.0160080	−	0.999872i	$-0.505096\pi$
$919$	−0.970563	−0.0320159	−0.0160080	+	0.999872i	$0.505096\pi$
$920$	9.65685	0.318377
$921$	25.0711	0.826120
$922$	30.5858	1.00729
$923$	−30.6274	−1.00811
$924$	0	0
$925$	−0.828427	−0.0272385
$926$	−3.31371	−0.108895
$927$	−6.24264	−0.205035
$928$	9.65685	0.317002
$929$	−10.7868	−0.353903	−0.176952	−	0.984220i	$-0.556624\pi$
$929$	−10.7868	−0.353903	−0.176952	+	0.984220i	$0.556624\pi$
$930$	12.4853	0.409409
$931$	0	0
$932$	−7.65685	−0.250809
$933$	−8.10051	−0.265199
$934$	−12.0000	−0.392652
$935$	−4.00000	−0.130814
$936$	−5.41421	−0.176969
$937$	56.9117	1.85922	0.929612	−	0.368540i	$-0.120142\pi$
$937$	56.9117	1.85922	0.929612	+	0.368540i	$0.120142\pi$
$938$	0	0
$939$	8.24264	0.268988
$940$	−15.7990	−0.515306
$941$	−27.5563	−0.898311	−0.449156	−	0.893454i	$-0.648275\pi$
$941$	−27.5563	−0.898311	−0.449156	+	0.893454i	$0.648275\pi$
$942$	−12.8284	−0.417973
$943$	56.2843	1.83287
$944$	−6.82843	−0.222246
$945$	0	0
$946$	4.82843	0.156986
$947$	−52.5685	−1.70825	−0.854124	−	0.520069i	$-0.825906\pi$
$947$	−52.5685	−1.70825	−0.854124	+	0.520069i	$0.825906\pi$
$948$	−16.4853	−0.535417
$949$	63.1127	2.04872
$950$	−2.24264	−0.0727609
$951$	12.8284	0.415990
$952$	0	0
$953$	9.51472	0.308212	0.154106	−	0.988054i	$-0.450750\pi$
$953$	9.51472	0.308212	0.154106	+	0.988054i	$0.450750\pi$
$954$	−10.4853	−0.339474
$955$	−43.5980	−1.41080
$956$	0.686292	0.0221963
$957$	−9.65685	−0.312162
$958$	−7.31371	−0.236295
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	7.97056	0.257115
$962$	−4.48528	−0.144611
$963$	12.8284	0.413390
$964$	7.65685	0.246611
$965$	29.9411	0.963839
$966$	0	0
$967$	−9.94113	−0.319685	−0.159843	−	0.987143i	$-0.551099\pi$
$967$	−9.94113	−0.319685	−0.159843	+	0.987143i	$0.551099\pi$
$968$	1.00000	0.0321412
$969$	−4.48528	−0.144088
$970$	−19.7990	−0.635707
$971$	−4.68629	−0.150390	−0.0751951	−	0.997169i	$-0.523958\pi$
$971$	−4.68629	−0.150390	−0.0751951	+	0.997169i	$0.523958\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	0.142136	0.00455432
$975$	5.41421	0.173394
$976$	−2.58579	−0.0827690
$977$	26.6274	0.851887	0.425943	−	0.904750i	$-0.359942\pi$
$977$	26.6274	0.851887	0.425943	+	0.904750i	$0.359942\pi$
$978$	−2.34315	−0.0749255
$979$	3.75736	0.120086
$980$	0	0
$981$	4.34315	0.138666
$982$	−42.6274	−1.36030
$983$	−0.870058	−0.0277505	−0.0138753	−	0.999904i	$-0.504417\pi$
$983$	−0.870058	−0.0277505	−0.0138753	+	0.999904i	$0.504417\pi$
$984$	−11.6569	−0.371607
$985$	−31.3137	−0.997738
$986$	−19.3137	−0.615074
$987$	0	0
$988$	−12.1421	−0.386293
$989$	23.3137	0.741333
$990$	2.00000	0.0635642
$991$	50.9117	1.61726	0.808632	−	0.588315i	$-0.200209\pi$
$991$	50.9117	1.61726	0.808632	+	0.588315i	$0.200209\pi$
$992$	−6.24264	−0.198204
$993$	−27.7990	−0.882174
$994$	0	0
$995$	30.4264	0.964582
$996$	−5.07107	−0.160683
$997$	−30.5858	−0.968662	−0.484331	−	0.874885i	$-0.660937\pi$
$997$	−30.5858	−0.968662	−0.484331	+	0.874885i	$0.660937\pi$
$998$	−6.14214	−0.194426
$999$	0.828427	0.0262103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.bd.1.1	yes	2
3.2	odd	2		9702.2.a.cy.1.1		2
7.6	odd	2		3234.2.a.bc.1.2	✓	2
21.20	even	2		9702.2.a.ci.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.bc.1.2	✓	2	7.6	odd	2
3234.2.a.bd.1.1	yes	2	1.1	even	1	trivial
9702.2.a.ci.1.2		2	21.20	even	2
9702.2.a.cy.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -5.41421 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +2.24264 q^{19} -2.00000 q^{20} -1.00000 q^{22} -4.82843 q^{23} +1.00000 q^{24} -1.00000 q^{25} -5.41421 q^{26} +1.00000 q^{27} +9.65685 q^{29} -2.00000 q^{30} -6.24264 q^{31} +1.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +0.828427 q^{37} +2.24264 q^{38} -5.41421 q^{39} -2.00000 q^{40} -11.6569 q^{41} -4.82843 q^{43} -1.00000 q^{44} -2.00000 q^{45} -4.82843 q^{46} +7.89949 q^{47} +1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} -5.41421 q^{52} -10.4853 q^{53} +1.00000 q^{54} +2.00000 q^{55} +2.24264 q^{57} +9.65685 q^{58} -6.82843 q^{59} -2.00000 q^{60} -2.58579 q^{61} -6.24264 q^{62} +1.00000 q^{64} +10.8284 q^{65} -1.00000 q^{66} -1.17157 q^{67} -2.00000 q^{68} -4.82843 q^{69} +5.65685 q^{71} +1.00000 q^{72} -11.6569 q^{73} +0.828427 q^{74} -1.00000 q^{75} +2.24264 q^{76} -5.41421 q^{78} -16.4853 q^{79} -2.00000 q^{80} +1.00000 q^{81} -11.6569 q^{82} -5.07107 q^{83} +4.00000 q^{85} -4.82843 q^{86} +9.65685 q^{87} -1.00000 q^{88} -3.75736 q^{89} -2.00000 q^{90} -4.82843 q^{92} -6.24264 q^{93} +7.89949 q^{94} -4.48528 q^{95} +1.00000 q^{96} +9.89949 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{10} - 2 q^{11} + 2 q^{12} - 8 q^{13} - 4 q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 4 q^{19} - 4 q^{20} - 2 q^{22} - 4 q^{23}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 - 4 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 - 4 * q^10 - 2 * q^11 + 2 * q^12 - 8 * q^13 - 4 * q^15 + 2 * q^16 - 4 * q^17 + 2 * q^18 - 4 * q^19 - 4 * q^20 - 2 * q^22 - 4 * q^23 + 2 * q^24 - 2 * q^25 - 8 * q^26 + 2 * q^27 + 8 * q^29 - 4 * q^30 - 4 * q^31 + 2 * q^32 - 2 * q^33 - 4 * q^34 + 2 * q^36 - 4 * q^37 - 4 * q^38 - 8 * q^39 - 4 * q^40 - 12 * q^41 - 4 * q^43 - 2 * q^44 - 4 * q^45 - 4 * q^46 - 4 * q^47 + 2 * q^48 - 2 * q^50 - 4 * q^51 - 8 * q^52 - 4 * q^53 + 2 * q^54 + 4 * q^55 - 4 * q^57 + 8 * q^58 - 8 * q^59 - 4 * q^60 - 8 * q^61 - 4 * q^62 + 2 * q^64 + 16 * q^65 - 2 * q^66 - 8 * q^67 - 4 * q^68 - 4 * q^69 + 2 * q^72 - 12 * q^73 - 4 * q^74 - 2 * q^75 - 4 * q^76 - 8 * q^78 - 16 * q^79 - 4 * q^80 + 2 * q^81 - 12 * q^82 + 4 * q^83 + 8 * q^85 - 4 * q^86 + 8 * q^87 - 2 * q^88 - 16 * q^89 - 4 * q^90 - 4 * q^92 - 4 * q^93 - 4 * q^94 + 8 * q^95 + 2 * q^96 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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