LMFDB - Embedded newform 9702.2.a.cy.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9702.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$77.4708600410$
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:	no (minimal twist has level 3234)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	9702.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} -2.00000 q^{10} +1.00000 q^{11} -5.41421 q^{13} +1.00000 q^{16} +2.00000 q^{17} +2.24264 q^{19} +2.00000 q^{20} -1.00000 q^{22} +4.82843 q^{23} -1.00000 q^{25} +5.41421 q^{26} -9.65685 q^{29} -6.24264 q^{31} -1.00000 q^{32} -2.00000 q^{34} +0.828427 q^{37} -2.24264 q^{38} -2.00000 q^{40} +11.6569 q^{41} -4.82843 q^{43} +1.00000 q^{44} -4.82843 q^{46} -7.89949 q^{47} +1.00000 q^{50} -5.41421 q^{52} +10.4853 q^{53} +2.00000 q^{55} +9.65685 q^{58} +6.82843 q^{59} -2.58579 q^{61} +6.24264 q^{62} +1.00000 q^{64} -10.8284 q^{65} -1.17157 q^{67} +2.00000 q^{68} -5.65685 q^{71} -11.6569 q^{73} -0.828427 q^{74} +2.24264 q^{76} -16.4853 q^{79} +2.00000 q^{80} -11.6569 q^{82} +5.07107 q^{83} +4.00000 q^{85} +4.82843 q^{86} -1.00000 q^{88} +3.75736 q^{89} +4.82843 q^{92} +7.89949 q^{94} +4.48528 q^{95} +9.89949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8} - 4 q^{10} + 2 q^{11} - 8 q^{13} + 2 q^{16} + 4 q^{17} - 4 q^{19} + 4 q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{25} + 8 q^{26} - 8 q^{29} - 4 q^{31} - 2 q^{32} - 4 q^{34} - 4 q^{37} + 4 q^{38} - 4 q^{40} + 12 q^{41} - 4 q^{43} + 2 q^{44} - 4 q^{46} + 4 q^{47} + 2 q^{50} - 8 q^{52} + 4 q^{53} + 4 q^{55} + 8 q^{58} + 8 q^{59} - 8 q^{61} + 4 q^{62} + 2 q^{64} - 16 q^{65} - 8 q^{67} + 4 q^{68} - 12 q^{73} + 4 q^{74} - 4 q^{76} - 16 q^{79} + 4 q^{80} - 12 q^{82} - 4 q^{83} + 8 q^{85} + 4 q^{86} - 2 q^{88} + 16 q^{89} + 4 q^{92} - 4 q^{94} - 8 q^{95}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 2 * q^8 - 4 * q^10 + 2 * q^11 - 8 * q^13 + 2 * q^16 + 4 * q^17 - 4 * q^19 + 4 * q^20 - 2 * q^22 + 4 * q^23 - 2 * q^25 + 8 * q^26 - 8 * q^29 - 4 * q^31 - 2 * q^32 - 4 * q^34 - 4 * q^37 + 4 * q^38 - 4 * q^40 + 12 * q^41 - 4 * q^43 + 2 * q^44 - 4 * q^46 + 4 * q^47 + 2 * q^50 - 8 * q^52 + 4 * q^53 + 4 * q^55 + 8 * q^58 + 8 * q^59 - 8 * q^61 + 4 * q^62 + 2 * q^64 - 16 * q^65 - 8 * q^67 + 4 * q^68 - 12 * q^73 + 4 * q^74 - 4 * q^76 - 16 * q^79 + 4 * q^80 - 12 * q^82 - 4 * q^83 + 8 * q^85 + 4 * q^86 - 2 * q^88 + 16 * q^89 + 4 * q^92 - 4 * q^94 - 8 * q^95

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.00000	−0.632456
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−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
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	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	$-0.332083\pi$
$23$	4.82843	1.00680	0.503398	+	0.864054i	$0.332083\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	5.41421	1.06181
$27$	0	0
$28$	0	0
$29$	−9.65685	−1.79323	−0.896616	−	0.442808i	$-0.853982\pi$
$29$	−9.65685	−1.79323	−0.896616	+	0.442808i	$0.853982\pi$
$30$	0	0
$31$	−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
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	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
$39$	0	0
$40$	−2.00000	−0.316228
$41$	11.6569	1.82049	0.910247	−	0.414065i	$-0.135891\pi$
$41$	11.6569	1.82049	0.910247	+	0.414065i	$0.135891\pi$
$42$	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
$45$	0	0
$46$	−4.82843	−0.711913
$47$	−7.89949	−1.15226	−0.576130	−	0.817358i	$-0.695438\pi$
$47$	−7.89949	−1.15226	−0.576130	+	0.817358i	$0.695438\pi$
$48$	0	0
$49$	0	0
$50$	1.00000	0.141421
$51$	0	0
$52$	−5.41421	−0.750816
$53$	10.4853	1.44026	0.720132	−	0.693837i	$-0.244081\pi$
$53$	10.4853	1.44026	0.720132	+	0.693837i	$0.244081\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	0	0
$58$	9.65685	1.26801
$59$	6.82843	0.888985	0.444493	−	0.895782i	$-0.353384\pi$
$59$	6.82843	0.888985	0.444493	+	0.895782i	$0.353384\pi$
$60$	0	0
$61$	−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
$66$	0	0
$67$	−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
$69$	0	0
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	−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
$75$	0	0
$76$	2.24264	0.257249
$77$	0	0
$78$	0	0
$79$	−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
$81$	0	0
$82$	−11.6569	−1.28728
$83$	5.07107	0.556622	0.278311	−	0.960491i	$-0.410225\pi$
$83$	5.07107	0.556622	0.278311	+	0.960491i	$0.410225\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	4.82843	0.520663
$87$	0	0
$88$	−1.00000	−0.106600
$89$	3.75736	0.398279	0.199140	−	0.979971i	$-0.436185\pi$
$89$	3.75736	0.398279	0.199140	+	0.979971i	$0.436185\pi$
$90$	0	0
$91$	0	0
$92$	4.82843	0.503398
$93$	0	0
$94$	7.89949	0.814771
$95$	4.48528	0.460180
$96$	0	0
$97$	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
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−15.0711	−1.49963	−0.749814	−	0.661649i	$-0.769857\pi$
$101$	−15.0711	−1.49963	−0.749814	+	0.661649i	$0.769857\pi$
$102$	0	0
$103$	−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	$-0.712902\pi$
$107$	−12.8284	−1.24017	−0.620085	+	0.784534i	$0.712902\pi$
$108$	0	0
$109$	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	0
$112$	0	0
$113$	−5.65685	−0.532152	−0.266076	−	0.963952i	$-0.585727\pi$
$113$	−5.65685	−0.532152	−0.266076	+	0.963952i	$0.585727\pi$
$114$	0	0
$115$	9.65685	0.900506
$116$	−9.65685	−0.896616
$117$	0	0
$118$	−6.82843	−0.628608
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.58579	0.234106
$123$	0	0
$124$	−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
$129$	0	0
$130$	10.8284	0.949716
$131$	17.5563	1.53391	0.766953	−	0.641704i	$-0.221772\pi$
$131$	17.5563	1.53391	0.766953	+	0.641704i	$0.221772\pi$
$132$	0	0
$133$	0	0
$134$	1.17157	0.101208
$135$	0	0
$136$	−2.00000	−0.171499
$137$	5.31371	0.453981	0.226990	−	0.973897i	$-0.427111\pi$
$137$	5.31371	0.453981	0.226990	+	0.973897i	$0.427111\pi$
$138$	0	0
$139$	−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
$141$	0	0
$142$	5.65685	0.474713
$143$	−5.41421	−0.452759
$144$	0	0
$145$	−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	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	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
$153$	0	0
$154$	0	0
$155$	−12.4853	−1.00284
$156$	0	0
$157$	−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
$159$	0	0
$160$	−2.00000	−0.158114
$161$	0	0
$162$	0	0
$163$	−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
$165$	0	0
$166$	−5.07107	−0.393591
$167$	−5.17157	−0.400188	−0.200094	−	0.979777i	$-0.564125\pi$
$167$	−5.17157	−0.400188	−0.200094	+	0.979777i	$0.564125\pi$
$168$	0	0
$169$	16.3137	1.25490
$170$	−4.00000	−0.306786
$171$	0	0
$172$	−4.82843	−0.368164
$173$	−5.89949	−0.448530	−0.224265	−	0.974528i	$-0.571998\pi$
$173$	−5.89949	−0.448530	−0.224265	+	0.974528i	$0.571998\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	−3.75736	−0.281626
$179$	20.4853	1.53114	0.765571	−	0.643352i	$-0.222457\pi$
$179$	20.4853	1.53114	0.765571	+	0.643352i	$0.222457\pi$
$180$	0	0
$181$	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
$183$	0	0
$184$	−4.82843	−0.355956
$185$	1.65685	0.121814
$186$	0	0
$187$	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	$-0.789225\pi$
$191$	−21.7990	−1.57732	−0.788660	+	0.614830i	$0.789225\pi$
$192$	0	0
$193$	−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
$195$	0	0
$196$	0	0
$197$	−15.6569	−1.11550	−0.557752	−	0.830007i	$-0.688336\pi$
$197$	−15.6569	−1.11550	−0.557752	+	0.830007i	$0.688336\pi$
$198$	0	0
$199$	−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
$201$	0	0
$202$	15.0711	1.06040
$203$	0	0
$204$	0	0
$205$	23.3137	1.62830
$206$	6.24264	0.434945
$207$	0	0
$208$	−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
$213$	0	0
$214$	12.8284	0.876933
$215$	−9.65685	−0.658592
$216$	0	0
$217$	0	0
$218$	−4.34315	−0.294155
$219$	0	0
$220$	2.00000	0.134840
$221$	−10.8284	−0.728399
$222$	0	0
$223$	−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
$225$	0	0
$226$	5.65685	0.376288
$227$	−11.8995	−0.789797	−0.394899	−	0.918725i	$-0.629220\pi$
$227$	−11.8995	−0.789797	−0.394899	+	0.918725i	$0.629220\pi$
$228$	0	0
$229$	−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	$-0.419304\pi$
$233$	7.65685	0.501617	0.250809	+	0.968037i	$0.419304\pi$
$234$	0	0
$235$	−15.7990	−1.03061
$236$	6.82843	0.444493
$237$	0	0
$238$	0	0
$239$	−0.686292	−0.0443925	−0.0221963	−	0.999754i	$-0.507066\pi$
$239$	−0.686292	−0.0443925	−0.0221963	+	0.999754i	$0.507066\pi$
$240$	0	0
$241$	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
$243$	0	0
$244$	−2.58579	−0.165538
$245$	0	0
$246$	0	0
$247$	−12.1421	−0.772586
$248$	6.24264	0.396408
$249$	0	0
$250$	12.0000	0.758947
$251$	8.48528	0.535586	0.267793	−	0.963476i	$-0.413706\pi$
$251$	8.48528	0.535586	0.267793	+	0.963476i	$0.413706\pi$
$252$	0	0
$253$	4.82843	0.303561
$254$	−14.1421	−0.887357
$255$	0	0
$256$	1.00000	0.0625000
$257$	−29.2132	−1.82227	−0.911135	−	0.412108i	$-0.864792\pi$
$257$	−29.2132	−1.82227	−0.911135	+	0.412108i	$0.864792\pi$
$258$	0	0
$259$	0	0
$260$	−10.8284	−0.671551
$261$	0	0
$262$	−17.5563	−1.08463
$263$	−9.17157	−0.565543	−0.282772	−	0.959187i	$-0.591254\pi$
$263$	−9.17157	−0.565543	−0.282772	+	0.959187i	$0.591254\pi$
$264$	0	0
$265$	20.9706	1.28821
$266$	0	0
$267$	0	0
$268$	−1.17157	−0.0715652
$269$	5.31371	0.323983	0.161991	−	0.986792i	$-0.448208\pi$
$269$	5.31371	0.323983	0.161991	+	0.986792i	$0.448208\pi$
$270$	0	0
$271$	−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
$276$	0	0
$277$	−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
$279$	0	0
$280$	0	0
$281$	1.31371	0.0783693	0.0391846	−	0.999232i	$-0.487524\pi$
$281$	1.31371	0.0783693	0.0391846	+	0.999232i	$0.487524\pi$
$282$	0	0
$283$	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
$285$	0	0
$286$	5.41421	0.320149
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	19.3137	1.13414
$291$	0	0
$292$	−11.6569	−0.682166
$293$	−4.92893	−0.287951	−0.143976	−	0.989581i	$-0.545989\pi$
$293$	−4.92893	−0.287951	−0.143976	+	0.989581i	$0.545989\pi$
$294$	0	0
$295$	13.6569	0.795133
$296$	−0.828427	−0.0481513
$297$	0	0
$298$	4.00000	0.231714
$299$	−26.1421	−1.51184
$300$	0	0
$301$	0	0
$302$	−4.00000	−0.230174
$303$	0	0
$304$	2.24264	0.128624
$305$	−5.17157	−0.296123
$306$	0	0
$307$	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
$309$	0	0
$310$	12.4853	0.709116
$311$	8.10051	0.459338	0.229669	−	0.973269i	$-0.426236\pi$
$311$	8.10051	0.459338	0.229669	+	0.973269i	$0.426236\pi$
$312$	0	0
$313$	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	$-0.617311\pi$
$317$	−12.8284	−0.720516	−0.360258	+	0.932853i	$0.617311\pi$
$318$	0	0
$319$	−9.65685	−0.540680
$320$	2.00000	0.111803
$321$	0	0
$322$	0	0
$323$	4.48528	0.249568
$324$	0	0
$325$	5.41421	0.300327
$326$	2.34315	0.129775
$327$	0	0
$328$	−11.6569	−0.643642
$329$	0	0
$330$	0	0
$331$	−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	0
$334$	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
$339$	0	0
$340$	4.00000	0.216930
$341$	−6.24264	−0.338058
$342$	0	0
$343$	0	0
$344$	4.82843	0.260331
$345$	0	0
$346$	5.89949	0.317159
$347$	−19.4558	−1.04444	−0.522222	−	0.852809i	$-0.674897\pi$
$347$	−19.4558	−1.04444	−0.522222	+	0.852809i	$0.674897\pi$
$348$	0	0
$349$	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
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	21.2132	1.12906	0.564532	−	0.825411i	$-0.309057\pi$
$353$	21.2132	1.12906	0.564532	+	0.825411i	$0.309057\pi$
$354$	0	0
$355$	−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	$-0.543577\pi$
$359$	−5.17157	−0.272945	−0.136473	+	0.990644i	$0.543577\pi$
$360$	0	0
$361$	−13.9706	−0.735293
$362$	−9.31371	−0.489518
$363$	0	0
$364$	0	0
$365$	−23.3137	−1.22030
$366$	0	0
$367$	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
$369$	0	0
$370$	−1.65685	−0.0861358
$371$	0	0
$372$	0	0
$373$	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
$375$	0	0
$376$	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
$381$	0	0
$382$	21.7990	1.11533
$383$	−22.7279	−1.16134	−0.580671	−	0.814138i	$-0.697210\pi$
$383$	−22.7279	−1.16134	−0.580671	+	0.814138i	$0.697210\pi$
$384$	0	0
$385$	0	0
$386$	14.9706	0.761982
$387$	0	0
$388$	9.89949	0.502571
$389$	34.2843	1.73828	0.869141	−	0.494565i	$-0.164673\pi$
$389$	34.2843	1.73828	0.869141	+	0.494565i	$0.164673\pi$
$390$	0	0
$391$	9.65685	0.488368
$392$	0	0
$393$	0	0
$394$	15.6569	0.788781
$395$	−32.9706	−1.65893
$396$	0	0
$397$	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	$-0.585974\pi$
$401$	−10.6863	−0.533648	−0.266824	+	0.963745i	$0.585974\pi$
$402$	0	0
$403$	33.7990	1.68365
$404$	−15.0711	−0.749814
$405$	0	0
$406$	0	0
$407$	0.828427	0.0410636
$408$	0	0
$409$	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
$411$	0	0
$412$	−6.24264	−0.307553
$413$	0	0
$414$	0	0
$415$	10.1421	0.497858
$416$	5.41421	0.265454
$417$	0	0
$418$	−2.24264	−0.109691
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	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
$423$	0	0
$424$	−10.4853	−0.509210
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0	0
$428$	−12.8284	−0.620085
$429$	0	0
$430$	9.65685	0.465695
$431$	−2.14214	−0.103183	−0.0515915	−	0.998668i	$-0.516429\pi$
$431$	−2.14214	−0.103183	−0.0515915	+	0.998668i	$0.516429\pi$
$432$	0	0
$433$	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
$435$	0	0
$436$	4.34315	0.207999
$437$	10.8284	0.517994
$438$	0	0
$439$	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	$-0.595852\pi$
$443$	−12.4853	−0.593194	−0.296597	+	0.955003i	$0.595852\pi$
$444$	0	0
$445$	7.51472	0.356232
$446$	6.24264	0.295598
$447$	0	0
$448$	0	0
$449$	−16.9706	−0.800890	−0.400445	−	0.916321i	$-0.631145\pi$
$449$	−16.9706	−0.800890	−0.400445	+	0.916321i	$0.631145\pi$
$450$	0	0
$451$	11.6569	0.548900
$452$	−5.65685	−0.266076
$453$	0	0
$454$	11.8995	0.558471
$455$	0	0
$456$	0	0
$457$	−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
$459$	0	0
$460$	9.65685	0.450253
$461$	−30.5858	−1.42452	−0.712261	−	0.701915i	$-0.752329\pi$
$461$	−30.5858	−1.42452	−0.712261	+	0.701915i	$0.752329\pi$
$462$	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
$465$	0	0
$466$	−7.65685	−0.354697
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	0	0
$470$	15.7990	0.728753
$471$	0	0
$472$	−6.82843	−0.314304
$473$	−4.82843	−0.222011
$474$	0	0
$475$	−2.24264	−0.102899
$476$	0	0
$477$	0	0
$478$	0.686292	0.0313902
$479$	7.31371	0.334172	0.167086	−	0.985942i	$-0.446564\pi$
$479$	7.31371	0.334172	0.167086	+	0.985942i	$0.446564\pi$
$480$	0	0
$481$	−4.48528	−0.204511
$482$	−7.65685	−0.348760
$483$	0	0
$484$	1.00000	0.0454545
$485$	19.7990	0.899026
$486$	0	0
$487$	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
$489$	0	0
$490$	0	0
$491$	42.6274	1.92375	0.961874	−	0.273492i	$-0.0881788\pi$
$491$	42.6274	1.92375	0.961874	+	0.273492i	$0.0881788\pi$
$492$	0	0
$493$	−19.3137	−0.869846
$494$	12.1421	0.546301
$495$	0	0
$496$	−6.24264	−0.280303
$497$	0	0
$498$	0	0
$499$	−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
$501$	0	0
$502$	−8.48528	−0.378717
$503$	−8.48528	−0.378340	−0.189170	−	0.981944i	$-0.560580\pi$
$503$	−8.48528	−0.378340	−0.189170	+	0.981944i	$0.560580\pi$
$504$	0	0
$505$	−30.1421	−1.34131
$506$	−4.82843	−0.214650
$507$	0	0
$508$	14.1421	0.627456
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	29.2132	1.28854
$515$	−12.4853	−0.550167
$516$	0	0
$517$	−7.89949	−0.347419
$518$	0	0
$519$	0	0
$520$	10.8284	0.474858
$521$	21.2132	0.929367	0.464684	−	0.885477i	$-0.346168\pi$
$521$	21.2132	0.929367	0.464684	+	0.885477i	$0.346168\pi$
$522$	0	0
$523$	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
$528$	0	0
$529$	0.313708	0.0136395
$530$	−20.9706	−0.910903
$531$	0	0
$532$	0	0
$533$	−63.1127	−2.73371
$534$	0	0
$535$	−25.6569	−1.10924
$536$	1.17157	0.0506042
$537$	0	0
$538$	−5.31371	−0.229090
$539$	0	0
$540$	0	0
$541$	−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
$543$	0	0
$544$	−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
$549$	0	0
$550$	1.00000	0.0426401
$551$	−21.6569	−0.922613
$552$	0	0
$553$	0	0
$554$	16.0000	0.679775
$555$	0	0
$556$	−0.100505	−0.00426236
$557$	−19.6569	−0.832888	−0.416444	−	0.909161i	$-0.636724\pi$
$557$	−19.6569	−0.832888	−0.416444	+	0.909161i	$0.636724\pi$
$558$	0	0
$559$	26.1421	1.10569
$560$	0	0
$561$	0	0
$562$	−1.31371	−0.0554154
$563$	−21.0711	−0.888040	−0.444020	−	0.896017i	$-0.646448\pi$
$563$	−21.0711	−0.888040	−0.444020	+	0.896017i	$0.646448\pi$
$564$	0	0
$565$	−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	$-0.428090\pi$
$569$	10.6863	0.447993	0.223996	+	0.974590i	$0.428090\pi$
$570$	0	0
$571$	−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
$573$	0	0
$574$	0	0
$575$	−4.82843	−0.201359
$576$	0	0
$577$	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
$579$	0	0
$580$	−19.3137	−0.801958
$581$	0	0
$582$	0	0
$583$	10.4853	0.434256
$584$	11.6569	0.482364
$585$	0	0
$586$	4.92893	0.203612
$587$	11.7990	0.486996	0.243498	−	0.969901i	$-0.421705\pi$
$587$	11.7990	0.486996	0.243498	+	0.969901i	$0.421705\pi$
$588$	0	0
$589$	−14.0000	−0.576860
$590$	−13.6569	−0.562244
$591$	0	0
$592$	0.828427	0.0340481
$593$	−37.3137	−1.53229	−0.766145	−	0.642668i	$-0.777828\pi$
$593$	−37.3137	−1.53229	−0.766145	+	0.642668i	$0.777828\pi$
$594$	0	0
$595$	0	0
$596$	−4.00000	−0.163846
$597$	0	0
$598$	26.1421	1.06903
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	−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
$603$	0	0
$604$	4.00000	0.162758
$605$	2.00000	0.0813116
$606$	0	0
$607$	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
$612$	0	0
$613$	−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
$615$	0	0
$616$	0	0
$617$	8.00000	0.322068	0.161034	−	0.986949i	$-0.448517\pi$
$617$	8.00000	0.322068	0.161034	+	0.986949i	$0.448517\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	−8.10051	−0.324801
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−8.24264	−0.329442
$627$	0	0
$628$	−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
$633$	0	0
$634$	12.8284	0.509482
$635$	28.2843	1.12243
$636$	0	0
$637$	0	0
$638$	9.65685	0.382319
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	35.3137	1.39481	0.697404	−	0.716678i	$-0.254338\pi$
$641$	35.3137	1.39481	0.697404	+	0.716678i	$0.254338\pi$
$642$	0	0
$643$	−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
$645$	0	0
$646$	−4.48528	−0.176471
$647$	1.07107	0.0421080	0.0210540	−	0.999778i	$-0.493298\pi$
$647$	1.07107	0.0421080	0.0210540	+	0.999778i	$0.493298\pi$
$648$	0	0
$649$	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	$-0.643614\pi$
$653$	−22.2843	−0.872051	−0.436025	+	0.899934i	$0.643614\pi$
$654$	0	0
$655$	35.1127	1.37197
$656$	11.6569	0.455124
$657$	0	0
$658$	0	0
$659$	49.1127	1.91316	0.956580	−	0.291471i	$-0.0941448\pi$
$659$	49.1127	1.91316	0.956580	+	0.291471i	$0.0941448\pi$
$660$	0	0
$661$	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
$663$	0	0
$664$	−5.07107	−0.196796
$665$	0	0
$666$	0	0
$667$	−46.6274	−1.80542
$668$	−5.17157	−0.200094
$669$	0	0
$670$	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
$675$	0	0
$676$	16.3137	0.627450
$677$	−36.0416	−1.38519	−0.692596	−	0.721326i	$-0.743533\pi$
$677$	−36.0416	−1.38519	−0.692596	+	0.721326i	$0.743533\pi$
$678$	0	0
$679$	0	0
$680$	−4.00000	−0.153393
$681$	0	0
$682$	6.24264	0.239043
$683$	−28.7696	−1.10084	−0.550418	−	0.834889i	$-0.685532\pi$
$683$	−28.7696	−1.10084	−0.550418	+	0.834889i	$0.685532\pi$
$684$	0	0
$685$	10.6274	0.406053
$686$	0	0
$687$	0	0
$688$	−4.82843	−0.184082
$689$	−56.7696	−2.16275
$690$	0	0
$691$	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
$696$	0	0
$697$	23.3137	0.883070
$698$	−17.4142	−0.659138
$699$	0	0
$700$	0	0
$701$	−9.31371	−0.351774	−0.175887	−	0.984410i	$-0.556279\pi$
$701$	−9.31371	−0.351774	−0.175887	+	0.984410i	$0.556279\pi$
$702$	0	0
$703$	1.85786	0.0700707
$704$	1.00000	0.0376889
$705$	0	0
$706$	−21.2132	−0.798369
$707$	0	0
$708$	0	0
$709$	−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
$711$	0	0
$712$	−3.75736	−0.140813
$713$	−30.1421	−1.12883
$714$	0	0
$715$	−10.8284	−0.404960
$716$	20.4853	0.765571
$717$	0	0
$718$	5.17157	0.193001
$719$	−44.1838	−1.64778	−0.823888	−	0.566752i	$-0.808200\pi$
$719$	−44.1838	−1.64778	−0.823888	+	0.566752i	$0.808200\pi$
$720$	0	0
$721$	0	0
$722$	13.9706	0.519931
$723$	0	0
$724$	9.31371	0.346141
$725$	9.65685	0.358647
$726$	0	0
$727$	−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
$729$	0	0
$730$	23.3137	0.862879
$731$	−9.65685	−0.357172
$732$	0	0
$733$	−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
$738$	0	0
$739$	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
$741$	0	0
$742$	0	0
$743$	18.6274	0.683374	0.341687	−	0.939814i	$-0.389002\pi$
$743$	18.6274	0.683374	0.341687	+	0.939814i	$0.389002\pi$
$744$	0	0
$745$	−8.00000	−0.293097
$746$	−35.9411	−1.31590
$747$	0	0
$748$	2.00000	0.0731272
$749$	0	0
$750$	0	0
$751$	−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
$753$	0	0
$754$	−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
$759$	0	0
$760$	−4.48528	−0.162698
$761$	−1.51472	−0.0549085	−0.0274543	−	0.999623i	$-0.508740\pi$
$761$	−1.51472	−0.0549085	−0.0274543	+	0.999623i	$0.508740\pi$
$762$	0	0
$763$	0	0
$764$	−21.7990	−0.788660
$765$	0	0
$766$	22.7279	0.821193
$767$	−36.9706	−1.33493
$768$	0	0
$769$	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
$771$	0	0
$772$	−14.9706	−0.538802
$773$	5.51472	0.198351	0.0991753	−	0.995070i	$-0.468380\pi$
$773$	5.51472	0.198351	0.0991753	+	0.995070i	$0.468380\pi$
$774$	0	0
$775$	6.24264	0.224242
$776$	−9.89949	−0.355371
$777$	0	0
$778$	−34.2843	−1.22915
$779$	26.1421	0.936639
$780$	0	0
$781$	−5.65685	−0.202418
$782$	−9.65685	−0.345328
$783$	0	0
$784$	0	0
$785$	−25.6569	−0.915732
$786$	0	0
$787$	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
$789$	0	0
$790$	32.9706	1.17304
$791$	0	0
$792$	0	0
$793$	14.0000	0.497155
$794$	−11.1716	−0.396464
$795$	0	0
$796$	−15.2132	−0.539218
$797$	17.5147	0.620403	0.310202	−	0.950671i	$-0.399603\pi$
$797$	17.5147	0.620403	0.310202	+	0.950671i	$0.399603\pi$
$798$	0	0
$799$	−15.7990	−0.558928
$800$	1.00000	0.0353553
$801$	0	0
$802$	10.6863	0.377346
$803$	−11.6569	−0.411361
$804$	0	0
$805$	0	0
$806$	−33.7990	−1.19052
$807$	0	0
$808$	15.0711	0.530198
$809$	−26.7696	−0.941167	−0.470584	−	0.882355i	$-0.655957\pi$
$809$	−26.7696	−0.941167	−0.470584	+	0.882355i	$0.655957\pi$
$810$	0	0
$811$	−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
$813$	0	0
$814$	−0.828427	−0.0290364
$815$	−4.68629	−0.164154
$816$	0	0
$817$	−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	$-0.809180\pi$
$821$	−47.3137	−1.65126	−0.825630	+	0.564212i	$0.809180\pi$
$822$	0	0
$823$	−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
$825$	0	0
$826$	0	0
$827$	−39.3137	−1.36707	−0.683536	−	0.729917i	$-0.739559\pi$
$827$	−39.3137	−1.36707	−0.683536	+	0.729917i	$0.739559\pi$
$828$	0	0
$829$	−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
$831$	0	0
$832$	−5.41421	−0.187704
$833$	0	0
$834$	0	0
$835$	−10.3431	−0.357939
$836$	2.24264	0.0775634
$837$	0	0
$838$	−16.0000	−0.552711
$839$	−17.0711	−0.589359	−0.294679	−	0.955596i	$-0.595213\pi$
$839$	−17.0711	−0.589359	−0.294679	+	0.955596i	$0.595213\pi$
$840$	0	0
$841$	64.2548	2.21568
$842$	27.6569	0.953118
$843$	0	0
$844$	−23.1716	−0.797598
$845$	32.6274	1.12242
$846$	0	0
$847$	0	0
$848$	10.4853	0.360066
$849$	0	0
$850$	2.00000	0.0685994
$851$	4.00000	0.137118
$852$	0	0
$853$	−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
$855$	0	0
$856$	12.8284	0.438467
$857$	0.142136	0.00485526	0.00242763	−	0.999997i	$-0.499227\pi$
$857$	0.142136	0.00485526	0.00242763	+	0.999997i	$0.499227\pi$
$858$	0	0
$859$	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	$-0.270613\pi$
$863$	38.7696	1.31973	0.659865	+	0.751384i	$0.270613\pi$
$864$	0	0
$865$	−11.7990	−0.401178
$866$	−34.3848	−1.16844
$867$	0	0
$868$	0	0
$869$	−16.4853	−0.559225
$870$	0	0
$871$	6.34315	0.214929
$872$	−4.34315	−0.147077
$873$	0	0
$874$	−10.8284	−0.366277
$875$	0	0
$876$	0	0
$877$	−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
$879$	0	0
$880$	2.00000	0.0674200
$881$	37.8995	1.27687	0.638433	−	0.769677i	$-0.279583\pi$
$881$	37.8995	1.27687	0.638433	+	0.769677i	$0.279583\pi$
$882$	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
$885$	0	0
$886$	12.4853	0.419451
$887$	4.28427	0.143852	0.0719259	−	0.997410i	$-0.477085\pi$
$887$	4.28427	0.143852	0.0719259	+	0.997410i	$0.477085\pi$
$888$	0	0
$889$	0	0
$890$	−7.51472	−0.251894
$891$	0	0
$892$	−6.24264	−0.209019
$893$	−17.7157	−0.592834
$894$	0	0
$895$	40.9706	1.36949
$896$	0	0
$897$	0	0
$898$	16.9706	0.566315
$899$	60.2843	2.01059
$900$	0	0
$901$	20.9706	0.698631
$902$	−11.6569	−0.388131
$903$	0	0
$904$	5.65685	0.188144
$905$	18.6274	0.619196
$906$	0	0
$907$	−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
$909$	0	0
$910$	0	0
$911$	11.4558	0.379549	0.189775	−	0.981828i	$-0.439224\pi$
$911$	11.4558	0.379549	0.189775	+	0.981828i	$0.439224\pi$
$912$	0	0
$913$	5.07107	0.167828
$914$	29.3137	0.969611
$915$	0	0
$916$	−14.0000	−0.462573
$917$	0	0
$918$	0	0
$919$	−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
$921$	0	0
$922$	30.5858	1.00729
$923$	30.6274	1.00811
$924$	0	0
$925$	−0.828427	−0.0272385
$926$	3.31371	0.108895
$927$	0	0
$928$	9.65685	0.317002
$929$	10.7868	0.353903	0.176952	−	0.984220i	$-0.443376\pi$
$929$	10.7868	0.353903	0.176952	+	0.984220i	$0.443376\pi$
$930$	0	0
$931$	0	0
$932$	7.65685	0.250809
$933$	0	0
$934$	−12.0000	−0.392652
$935$	4.00000	0.130814
$936$	0	0
$937$	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
$939$	0	0
$940$	−15.7990	−0.515306
$941$	27.5563	0.898311	0.449156	−	0.893454i	$-0.351725\pi$
$941$	27.5563	0.898311	0.449156	+	0.893454i	$0.351725\pi$
$942$	0	0
$943$	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	$-0.174094\pi$
$947$	52.5685	1.70825	0.854124	+	0.520069i	$0.174094\pi$
$948$	0	0
$949$	63.1127	2.04872
$950$	2.24264	0.0727609
$951$	0	0
$952$	0	0
$953$	−9.51472	−0.308212	−0.154106	−	0.988054i	$-0.549250\pi$
$953$	−9.51472	−0.308212	−0.154106	+	0.988054i	$0.549250\pi$
$954$	0	0
$955$	−43.5980	−1.41080
$956$	−0.686292	−0.0221963
$957$	0	0
$958$	−7.31371	−0.236295
$959$	0	0
$960$	0	0
$961$	7.97056	0.257115
$962$	4.48528	0.144611
$963$	0	0
$964$	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
$969$	0	0
$970$	−19.7990	−0.635707
$971$	4.68629	0.150390	0.0751951	−	0.997169i	$-0.476042\pi$
$971$	4.68629	0.150390	0.0751951	+	0.997169i	$0.476042\pi$
$972$	0	0
$973$	0	0
$974$	−0.142136	−0.00455432
$975$	0	0
$976$	−2.58579	−0.0827690
$977$	−26.6274	−0.851887	−0.425943	−	0.904750i	$-0.640058\pi$
$977$	−26.6274	−0.851887	−0.425943	+	0.904750i	$0.640058\pi$
$978$	0	0
$979$	3.75736	0.120086
$980$	0	0
$981$	0	0
$982$	−42.6274	−1.36030
$983$	0.870058	0.0277505	0.0138753	−	0.999904i	$-0.495583\pi$
$983$	0.870058	0.0277505	0.0138753	+	0.999904i	$0.495583\pi$
$984$	0	0
$985$	−31.3137	−0.997738
$986$	19.3137	0.615074
$987$	0	0
$988$	−12.1421	−0.386293
$989$	−23.3137	−0.741333
$990$	0	0
$991$	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
$993$	0	0
$994$	0	0
$995$	−30.4264	−0.964582
$996$	0	0
$997$	−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	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9702.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} -2.00000 q^{10} +1.00000 q^{11} -5.41421 q^{13} +1.00000 q^{16} +2.00000 q^{17} +2.24264 q^{19} +2.00000 q^{20} -1.00000 q^{22} +4.82843 q^{23} -1.00000 q^{25} +5.41421 q^{26} -9.65685 q^{29} -6.24264 q^{31} -1.00000 q^{32} -2.00000 q^{34} +0.828427 q^{37} -2.24264 q^{38} -2.00000 q^{40} +11.6569 q^{41} -4.82843 q^{43} +1.00000 q^{44} -4.82843 q^{46} -7.89949 q^{47} +1.00000 q^{50} -5.41421 q^{52} +10.4853 q^{53} +2.00000 q^{55} +9.65685 q^{58} +6.82843 q^{59} -2.58579 q^{61} +6.24264 q^{62} +1.00000 q^{64} -10.8284 q^{65} -1.17157 q^{67} +2.00000 q^{68} -5.65685 q^{71} -11.6569 q^{73} -0.828427 q^{74} +2.24264 q^{76} -16.4853 q^{79} +2.00000 q^{80} -11.6569 q^{82} +5.07107 q^{83} +4.00000 q^{85} +4.82843 q^{86} -1.00000 q^{88} +3.75736 q^{89} +4.82843 q^{92} +7.89949 q^{94} +4.48528 q^{95} +9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8} - 4 q^{10} + 2 q^{11} - 8 q^{13} + 2 q^{16} + 4 q^{17} - 4 q^{19} + 4 q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{25} + 8 q^{26} - 8 q^{29} - 4 q^{31} - 2 q^{32} - 4 q^{34} - 4 q^{37} + 4 q^{38} - 4 q^{40} + 12 q^{41} - 4 q^{43} + 2 q^{44} - 4 q^{46} + 4 q^{47} + 2 q^{50} - 8 q^{52} + 4 q^{53} + 4 q^{55} + 8 q^{58} + 8 q^{59} - 8 q^{61} + 4 q^{62} + 2 q^{64} - 16 q^{65} - 8 q^{67} + 4 q^{68} - 12 q^{73} + 4 q^{74} - 4 q^{76} - 16 q^{79} + 4 q^{80} - 12 q^{82} - 4 q^{83} + 8 q^{85} + 4 q^{86} - 2 q^{88} + 16 q^{89} + 4 q^{92} - 4 q^{94} - 8 q^{95}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^5 - 2 * q^8 - 4 * q^10 + 2 * q^11 - 8 * q^13 + 2 * q^16 + 4 * q^17 - 4 * q^19 + 4 * q^20 - 2 * q^22 + 4 * q^23 - 2 * q^25 + 8 * q^26 - 8 * q^29 - 4 * q^31 - 2 * q^32 - 4 * q^34 - 4 * q^37 + 4 * q^38 - 4 * q^40 + 12 * q^41 - 4 * q^43 + 2 * q^44 - 4 * q^46 + 4 * q^47 + 2 * q^50 - 8 * q^52 + 4 * q^53 + 4 * q^55 + 8 * q^58 + 8 * q^59 - 8 * q^61 + 4 * q^62 + 2 * q^64 - 16 * q^65 - 8 * q^67 + 4 * q^68 - 12 * q^73 + 4 * q^74 - 4 * q^76 - 16 * q^79 + 4 * q^80 - 12 * q^82 - 4 * q^83 + 8 * q^85 + 4 * q^86 - 2 * q^88 + 16 * q^89 + 4 * q^92 - 4 * q^94 - 8 * q^95

Introduction

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