LMFDB - Embedded newform 9702.2.a.cy.1.2

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			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} -2.00000 q^{10} +1.00000 q^{11} -2.58579 q^{13} +1.00000 q^{16} +2.00000 q^{17} -6.24264 q^{19} +2.00000 q^{20} -1.00000 q^{22} -0.828427 q^{23} -1.00000 q^{25} +2.58579 q^{26} +1.65685 q^{29} +2.24264 q^{31} -1.00000 q^{32} -2.00000 q^{34} -4.82843 q^{37} +6.24264 q^{38} -2.00000 q^{40} +0.343146 q^{41} +0.828427 q^{43} +1.00000 q^{44} +0.828427 q^{46} +11.8995 q^{47} +1.00000 q^{50} -2.58579 q^{52} -6.48528 q^{53} +2.00000 q^{55} -1.65685 q^{58} +1.17157 q^{59} -5.41421 q^{61} -2.24264 q^{62} +1.00000 q^{64} -5.17157 q^{65} -6.82843 q^{67} +2.00000 q^{68} +5.65685 q^{71} -0.343146 q^{73} +4.82843 q^{74} -6.24264 q^{76} +0.485281 q^{79} +2.00000 q^{80} -0.343146 q^{82} -9.07107 q^{83} +4.00000 q^{85} -0.828427 q^{86} -1.00000 q^{88} +12.2426 q^{89} -0.828427 q^{92} -11.8995 q^{94} -12.4853 q^{95} -9.89949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$13$	−2.58579	−0.717168	−0.358584	−	0.933497i	$-0.616740\pi$
$13$	−2.58579	−0.717168	−0.358584	+	0.933497i	$0.616740\pi$
$14$	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
$19$	−6.24264	−1.43216	−0.716080	−	0.698018i	$-0.754065\pi$
$19$	−6.24264	−1.43216	−0.716080	+	0.698018i	$0.754065\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−0.828427	−0.172739	−0.0863695	−	0.996263i	$-0.527527\pi$
$23$	−0.828427	−0.172739	−0.0863695	+	0.996263i	$0.527527\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	2.58579	0.507114
$27$	0	0
$28$	0	0
$29$	1.65685	0.307670	0.153835	−	0.988097i	$-0.450838\pi$
$29$	1.65685	0.307670	0.153835	+	0.988097i	$0.450838\pi$
$30$	0	0
$31$	2.24264	0.402790	0.201395	−	0.979510i	$-0.435452\pi$
$31$	2.24264	0.402790	0.201395	+	0.979510i	$0.435452\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	−4.82843	−0.793789	−0.396894	−	0.917864i	$-0.629912\pi$
$37$	−4.82843	−0.793789	−0.396894	+	0.917864i	$0.629912\pi$
$38$	6.24264	1.01269
$39$	0	0
$40$	−2.00000	−0.316228
$41$	0.343146	0.0535904	0.0267952	−	0.999641i	$-0.491470\pi$
$41$	0.343146	0.0535904	0.0267952	+	0.999641i	$0.491470\pi$
$42$	0	0
$43$	0.828427	0.126334	0.0631670	−	0.998003i	$-0.479880\pi$
$43$	0.828427	0.126334	0.0631670	+	0.998003i	$0.479880\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	0.828427	0.122145
$47$	11.8995	1.73572	0.867860	−	0.496809i	$-0.165495\pi$
$47$	11.8995	1.73572	0.867860	+	0.496809i	$0.165495\pi$
$48$	0	0
$49$	0	0
$50$	1.00000	0.141421
$51$	0	0
$52$	−2.58579	−0.358584
$53$	−6.48528	−0.890822	−0.445411	−	0.895326i	$-0.646942\pi$
$53$	−6.48528	−0.890822	−0.445411	+	0.895326i	$0.646942\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	0	0
$58$	−1.65685	−0.217556
$59$	1.17157	0.152526	0.0762629	−	0.997088i	$-0.475701\pi$
$59$	1.17157	0.152526	0.0762629	+	0.997088i	$0.475701\pi$
$60$	0	0
$61$	−5.41421	−0.693219	−0.346610	−	0.938010i	$-0.612667\pi$
$61$	−5.41421	−0.693219	−0.346610	+	0.938010i	$0.612667\pi$
$62$	−2.24264	−0.284816
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.17157	−0.641455
$66$	0	0
$67$	−6.82843	−0.834225	−0.417113	−	0.908855i	$-0.636958\pi$
$67$	−6.82843	−0.834225	−0.417113	+	0.908855i	$0.636958\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	−0.343146	−0.0401622	−0.0200811	−	0.999798i	$-0.506392\pi$
$73$	−0.343146	−0.0401622	−0.0200811	+	0.999798i	$0.506392\pi$
$74$	4.82843	0.561293
$75$	0	0
$76$	−6.24264	−0.716080
$77$	0	0
$78$	0	0
$79$	0.485281	0.0545984	0.0272992	−	0.999627i	$-0.491309\pi$
$79$	0.485281	0.0545984	0.0272992	+	0.999627i	$0.491309\pi$
$80$	2.00000	0.223607
$81$	0	0
$82$	−0.343146	−0.0378941
$83$	−9.07107	−0.995679	−0.497840	−	0.867269i	$-0.665873\pi$
$83$	−9.07107	−0.995679	−0.497840	+	0.867269i	$0.665873\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	−0.828427	−0.0893316
$87$	0	0
$88$	−1.00000	−0.106600
$89$	12.2426	1.29772	0.648859	−	0.760909i	$-0.275247\pi$
$89$	12.2426	1.29772	0.648859	+	0.760909i	$0.275247\pi$
$90$	0	0
$91$	0	0
$92$	−0.828427	−0.0863695
$93$	0	0
$94$	−11.8995	−1.22734
$95$	−12.4853	−1.28096
$96$	0	0
$97$	−9.89949	−1.00514	−0.502571	−	0.864536i	$-0.667612\pi$
$97$	−9.89949	−1.00514	−0.502571	+	0.864536i	$0.667612\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−0.928932	−0.0924322	−0.0462161	−	0.998931i	$-0.514716\pi$
$101$	−0.928932	−0.0924322	−0.0462161	+	0.998931i	$0.514716\pi$
$102$	0	0
$103$	2.24264	0.220974	0.110487	−	0.993878i	$-0.464759\pi$
$103$	2.24264	0.220974	0.110487	+	0.993878i	$0.464759\pi$
$104$	2.58579	0.253557
$105$	0	0
$106$	6.48528	0.629906
$107$	−7.17157	−0.693302	−0.346651	−	0.937994i	$-0.612681\pi$
$107$	−7.17157	−0.693302	−0.346651	+	0.937994i	$0.612681\pi$
$108$	0	0
$109$	15.6569	1.49965	0.749827	−	0.661634i	$-0.230137\pi$
$109$	15.6569	1.49965	0.749827	+	0.661634i	$0.230137\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	0	0
$113$	5.65685	0.532152	0.266076	−	0.963952i	$-0.414273\pi$
$113$	5.65685	0.532152	0.266076	+	0.963952i	$0.414273\pi$
$114$	0	0
$115$	−1.65685	−0.154502
$116$	1.65685	0.153835
$117$	0	0
$118$	−1.17157	−0.107852
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	5.41421	0.490180
$123$	0	0
$124$	2.24264	0.201395
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−14.1421	−1.25491	−0.627456	−	0.778652i	$-0.715904\pi$
$127$	−14.1421	−1.25491	−0.627456	+	0.778652i	$0.715904\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	5.17157	0.453577
$131$	−13.5563	−1.18442	−0.592212	−	0.805782i	$-0.701745\pi$
$131$	−13.5563	−1.18442	−0.592212	+	0.805782i	$0.701745\pi$
$132$	0	0
$133$	0	0
$134$	6.82843	0.589886
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−17.3137	−1.47921	−0.739605	−	0.673041i	$-0.764988\pi$
$137$	−17.3137	−1.47921	−0.739605	+	0.673041i	$0.764988\pi$
$138$	0	0
$139$	−19.8995	−1.68785	−0.843927	−	0.536459i	$-0.819762\pi$
$139$	−19.8995	−1.68785	−0.843927	+	0.536459i	$0.819762\pi$
$140$	0	0
$141$	0	0
$142$	−5.65685	−0.474713
$143$	−2.58579	−0.216234
$144$	0	0
$145$	3.31371	0.275189
$146$	0.343146	0.0283989
$147$	0	0
$148$	−4.82843	−0.396894
$149$	−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	$0.447961\pi$
$152$	6.24264	0.506345
$153$	0	0
$154$	0	0
$155$	4.48528	0.360266
$156$	0	0
$157$	−7.17157	−0.572354	−0.286177	−	0.958177i	$-0.592385\pi$
$157$	−7.17157	−0.572354	−0.286177	+	0.958177i	$0.592385\pi$
$158$	−0.485281	−0.0386069
$159$	0	0
$160$	−2.00000	−0.158114
$161$	0	0
$162$	0	0
$163$	−13.6569	−1.06969	−0.534844	−	0.844951i	$-0.679630\pi$
$163$	−13.6569	−1.06969	−0.534844	+	0.844951i	$0.679630\pi$
$164$	0.343146	0.0267952
$165$	0	0
$166$	9.07107	0.704051
$167$	−10.8284	−0.837929	−0.418964	−	0.908003i	$-0.637607\pi$
$167$	−10.8284	−0.837929	−0.418964	+	0.908003i	$0.637607\pi$
$168$	0	0
$169$	−6.31371	−0.485670
$170$	−4.00000	−0.306786
$171$	0	0
$172$	0.828427	0.0631670
$173$	13.8995	1.05676	0.528380	−	0.849008i	$-0.322800\pi$
$173$	13.8995	1.05676	0.528380	+	0.849008i	$0.322800\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	−12.2426	−0.917625
$179$	3.51472	0.262702	0.131351	−	0.991336i	$-0.458068\pi$
$179$	3.51472	0.262702	0.131351	+	0.991336i	$0.458068\pi$
$180$	0	0
$181$	−13.3137	−0.989600	−0.494800	−	0.869007i	$-0.664759\pi$
$181$	−13.3137	−0.989600	−0.494800	+	0.869007i	$0.664759\pi$
$182$	0	0
$183$	0	0
$184$	0.828427	0.0610725
$185$	−9.65685	−0.709986
$186$	0	0
$187$	2.00000	0.146254
$188$	11.8995	0.867860
$189$	0	0
$190$	12.4853	0.905778
$191$	17.7990	1.28789	0.643945	−	0.765072i	$-0.277297\pi$
$191$	17.7990	1.28789	0.643945	+	0.765072i	$0.277297\pi$
$192$	0	0
$193$	18.9706	1.36553	0.682765	−	0.730638i	$-0.260777\pi$
$193$	18.9706	1.36553	0.682765	+	0.730638i	$0.260777\pi$
$194$	9.89949	0.710742
$195$	0	0
$196$	0	0
$197$	−4.34315	−0.309436	−0.154718	−	0.987959i	$-0.549447\pi$
$197$	−4.34315	−0.309436	−0.154718	+	0.987959i	$0.549447\pi$
$198$	0	0
$199$	27.2132	1.92909	0.964546	−	0.263913i	$-0.0850133\pi$
$199$	27.2132	1.92909	0.964546	+	0.263913i	$0.0850133\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	0.928932	0.0653594
$203$	0	0
$204$	0	0
$205$	0.686292	0.0479327
$206$	−2.24264	−0.156252
$207$	0	0
$208$	−2.58579	−0.179292
$209$	−6.24264	−0.431812
$210$	0	0
$211$	−28.8284	−1.98463	−0.992315	−	0.123734i	$-0.960513\pi$
$211$	−28.8284	−1.98463	−0.992315	+	0.123734i	$0.960513\pi$
$212$	−6.48528	−0.445411
$213$	0	0
$214$	7.17157	0.490239
$215$	1.65685	0.112997
$216$	0	0
$217$	0	0
$218$	−15.6569	−1.06042
$219$	0	0
$220$	2.00000	0.134840
$221$	−5.17157	−0.347878
$222$	0	0
$223$	2.24264	0.150178	0.0750892	−	0.997177i	$-0.476076\pi$
$223$	2.24264	0.150178	0.0750892	+	0.997177i	$0.476076\pi$
$224$	0	0
$225$	0	0
$226$	−5.65685	−0.376288
$227$	7.89949	0.524308	0.262154	−	0.965026i	$-0.415567\pi$
$227$	7.89949	0.524308	0.262154	+	0.965026i	$0.415567\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	1.65685	0.109250
$231$	0	0
$232$	−1.65685	−0.108778
$233$	−3.65685	−0.239568	−0.119784	−	0.992800i	$-0.538220\pi$
$233$	−3.65685	−0.239568	−0.119784	+	0.992800i	$0.538220\pi$
$234$	0	0
$235$	23.7990	1.55247
$236$	1.17157	0.0762629
$237$	0	0
$238$	0	0
$239$	−23.3137	−1.50804	−0.754019	−	0.656852i	$-0.771887\pi$
$239$	−23.3137	−1.50804	−0.754019	+	0.656852i	$0.771887\pi$
$240$	0	0
$241$	−3.65685	−0.235559	−0.117779	−	0.993040i	$-0.537578\pi$
$241$	−3.65685	−0.235559	−0.117779	+	0.993040i	$0.537578\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−5.41421	−0.346610
$245$	0	0
$246$	0	0
$247$	16.1421	1.02710
$248$	−2.24264	−0.142408
$249$	0	0
$250$	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
$253$	−0.828427	−0.0520828
$254$	14.1421	0.887357
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.2132	0.824217	0.412108	−	0.911135i	$-0.364792\pi$
$257$	13.2132	0.824217	0.412108	+	0.911135i	$0.364792\pi$
$258$	0	0
$259$	0	0
$260$	−5.17157	−0.320727
$261$	0	0
$262$	13.5563	0.837514
$263$	−14.8284	−0.914360	−0.457180	−	0.889374i	$-0.651140\pi$
$263$	−14.8284	−0.914360	−0.457180	+	0.889374i	$0.651140\pi$
$264$	0	0
$265$	−12.9706	−0.796775
$266$	0	0
$267$	0	0
$268$	−6.82843	−0.417113
$269$	−17.3137	−1.05564	−0.527818	−	0.849358i	$-0.676990\pi$
$269$	−17.3137	−1.05564	−0.527818	+	0.849358i	$0.676990\pi$
$270$	0	0
$271$	6.14214	0.373108	0.186554	−	0.982445i	$-0.440268\pi$
$271$	6.14214	0.373108	0.186554	+	0.982445i	$0.440268\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	17.3137	1.04596
$275$	−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	$0.659608\pi$
$278$	19.8995	1.19349
$279$	0	0
$280$	0	0
$281$	−21.3137	−1.27147	−0.635735	−	0.771908i	$-0.719303\pi$
$281$	−21.3137	−1.27147	−0.635735	+	0.771908i	$0.719303\pi$
$282$	0	0
$283$	−6.24264	−0.371086	−0.185543	−	0.982636i	$-0.559404\pi$
$283$	−6.24264	−0.371086	−0.185543	+	0.982636i	$0.559404\pi$
$284$	5.65685	0.335673
$285$	0	0
$286$	2.58579	0.152901
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−3.31371	−0.194588
$291$	0	0
$292$	−0.343146	−0.0200811
$293$	−19.0711	−1.11414	−0.557072	−	0.830464i	$-0.688075\pi$
$293$	−19.0711	−1.11414	−0.557072	+	0.830464i	$0.688075\pi$
$294$	0	0
$295$	2.34315	0.136423
$296$	4.82843	0.280647
$297$	0	0
$298$	4.00000	0.231714
$299$	2.14214	0.123883
$300$	0	0
$301$	0	0
$302$	−4.00000	−0.230174
$303$	0	0
$304$	−6.24264	−0.358040
$305$	−10.8284	−0.620034
$306$	0	0
$307$	10.9289	0.623747	0.311874	−	0.950124i	$-0.399043\pi$
$307$	10.9289	0.623747	0.311874	+	0.950124i	$0.399043\pi$
$308$	0	0
$309$	0	0
$310$	−4.48528	−0.254747
$311$	27.8995	1.58204	0.791018	−	0.611793i	$-0.209552\pi$
$311$	27.8995	1.58204	0.791018	+	0.611793i	$0.209552\pi$
$312$	0	0
$313$	−0.242641	−0.0137149	−0.00685743	−	0.999976i	$-0.502183\pi$
$313$	−0.242641	−0.0137149	−0.00685743	+	0.999976i	$0.502183\pi$
$314$	7.17157	0.404715
$315$	0	0
$316$	0.485281	0.0272992
$317$	−7.17157	−0.402796	−0.201398	−	0.979510i	$-0.564548\pi$
$317$	−7.17157	−0.402796	−0.201398	+	0.979510i	$0.564548\pi$
$318$	0	0
$319$	1.65685	0.0927660
$320$	2.00000	0.111803
$321$	0	0
$322$	0	0
$323$	−12.4853	−0.694700
$324$	0	0
$325$	2.58579	0.143434
$326$	13.6569	0.756383
$327$	0	0
$328$	−0.343146	−0.0189471
$329$	0	0
$330$	0	0
$331$	11.7990	0.648531	0.324266	−	0.945966i	$-0.394883\pi$
$331$	11.7990	0.648531	0.324266	+	0.945966i	$0.394883\pi$
$332$	−9.07107	−0.497840
$333$	0	0
$334$	10.8284	0.592505
$335$	−13.6569	−0.746154
$336$	0	0
$337$	7.17157	0.390660	0.195330	−	0.980738i	$-0.437422\pi$
$337$	7.17157	0.390660	0.195330	+	0.980738i	$0.437422\pi$
$338$	6.31371	0.343420
$339$	0	0
$340$	4.00000	0.216930
$341$	2.24264	0.121446
$342$	0	0
$343$	0	0
$344$	−0.828427	−0.0446658
$345$	0	0
$346$	−13.8995	−0.747241
$347$	31.4558	1.68864	0.844319	−	0.535841i	$-0.180005\pi$
$347$	31.4558	1.68864	0.844319	+	0.535841i	$0.180005\pi$
$348$	0	0
$349$	14.5858	0.780759	0.390380	−	0.920654i	$-0.372344\pi$
$349$	14.5858	0.780759	0.390380	+	0.920654i	$0.372344\pi$
$350$	0	0
$351$	0	0
$352$	−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	$0.690943\pi$
$354$	0	0
$355$	11.3137	0.600469
$356$	12.2426	0.648859
$357$	0	0
$358$	−3.51472	−0.185759
$359$	−10.8284	−0.571503	−0.285751	−	0.958304i	$-0.592243\pi$
$359$	−10.8284	−0.571503	−0.285751	+	0.958304i	$0.592243\pi$
$360$	0	0
$361$	19.9706	1.05108
$362$	13.3137	0.699753
$363$	0	0
$364$	0	0
$365$	−0.686292	−0.0359221
$366$	0	0
$367$	35.4142	1.84861	0.924303	−	0.381658i	$-0.124647\pi$
$367$	35.4142	1.84861	0.924303	+	0.381658i	$0.124647\pi$
$368$	−0.828427	−0.0431847
$369$	0	0
$370$	9.65685	0.502036
$371$	0	0
$372$	0	0
$373$	−31.9411	−1.65385	−0.826924	−	0.562313i	$-0.809912\pi$
$373$	−31.9411	−1.65385	−0.826924	+	0.562313i	$0.809912\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	−11.8995	−0.613670
$377$	−4.28427	−0.220651
$378$	0	0
$379$	−24.2843	−1.24740	−0.623700	−	0.781664i	$-0.714371\pi$
$379$	−24.2843	−1.24740	−0.623700	+	0.781664i	$0.714371\pi$
$380$	−12.4853	−0.640481
$381$	0	0
$382$	−17.7990	−0.910676
$383$	2.72792	0.139390	0.0696952	−	0.997568i	$-0.477797\pi$
$383$	2.72792	0.139390	0.0696952	+	0.997568i	$0.477797\pi$
$384$	0	0
$385$	0	0
$386$	−18.9706	−0.965576
$387$	0	0
$388$	−9.89949	−0.502571
$389$	−22.2843	−1.12986	−0.564929	−	0.825140i	$-0.691096\pi$
$389$	−22.2843	−1.12986	−0.564929	+	0.825140i	$0.691096\pi$
$390$	0	0
$391$	−1.65685	−0.0837907
$392$	0	0
$393$	0	0
$394$	4.34315	0.218805
$395$	0.970563	0.0488343
$396$	0	0
$397$	16.8284	0.844595	0.422297	−	0.906457i	$-0.361224\pi$
$397$	16.8284	0.844595	0.422297	+	0.906457i	$0.361224\pi$
$398$	−27.2132	−1.36407
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−33.3137	−1.66361	−0.831804	−	0.555070i	$-0.812691\pi$
$401$	−33.3137	−1.66361	−0.831804	+	0.555070i	$0.812691\pi$
$402$	0	0
$403$	−5.79899	−0.288868
$404$	−0.928932	−0.0462161
$405$	0	0
$406$	0	0
$407$	−4.82843	−0.239336
$408$	0	0
$409$	18.4853	0.914038	0.457019	−	0.889457i	$-0.348917\pi$
$409$	18.4853	0.914038	0.457019	+	0.889457i	$0.348917\pi$
$410$	−0.686292	−0.0338935
$411$	0	0
$412$	2.24264	0.110487
$413$	0	0
$414$	0	0
$415$	−18.1421	−0.890562
$416$	2.58579	0.126779
$417$	0	0
$418$	6.24264	0.305338
$419$	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
$421$	−16.3431	−0.796516	−0.398258	−	0.917273i	$-0.630385\pi$
$421$	−16.3431	−0.796516	−0.398258	+	0.917273i	$0.630385\pi$
$422$	28.8284	1.40335
$423$	0	0
$424$	6.48528	0.314953
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0	0
$428$	−7.17157	−0.346651
$429$	0	0
$430$	−1.65685	−0.0799006
$431$	26.1421	1.25922	0.629611	−	0.776910i	$-0.283214\pi$
$431$	26.1421	1.25922	0.629611	+	0.776910i	$0.283214\pi$
$432$	0	0
$433$	−2.38478	−0.114605	−0.0573025	−	0.998357i	$-0.518250\pi$
$433$	−2.38478	−0.114605	−0.0573025	+	0.998357i	$0.518250\pi$
$434$	0	0
$435$	0	0
$436$	15.6569	0.749827
$437$	5.17157	0.247390
$438$	0	0
$439$	34.8284	1.66227	0.831135	−	0.556071i	$-0.187692\pi$
$439$	34.8284	1.66227	0.831135	+	0.556071i	$0.187692\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	5.17157	0.245987
$443$	4.48528	0.213102	0.106551	−	0.994307i	$-0.466019\pi$
$443$	4.48528	0.213102	0.106551	+	0.994307i	$0.466019\pi$
$444$	0	0
$445$	24.4853	1.16071
$446$	−2.24264	−0.106192
$447$	0	0
$448$	0	0
$449$	16.9706	0.800890	0.400445	−	0.916321i	$-0.368855\pi$
$449$	16.9706	0.800890	0.400445	+	0.916321i	$0.368855\pi$
$450$	0	0
$451$	0.343146	0.0161581
$452$	5.65685	0.266076
$453$	0	0
$454$	−7.89949	−0.370742
$455$	0	0
$456$	0	0
$457$	−6.68629	−0.312772	−0.156386	−	0.987696i	$-0.549984\pi$
$457$	−6.68629	−0.312772	−0.156386	+	0.987696i	$0.549984\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	−1.65685	−0.0772512
$461$	−33.4142	−1.55626	−0.778128	−	0.628106i	$-0.783830\pi$
$461$	−33.4142	−1.55626	−0.778128	+	0.628106i	$0.783830\pi$
$462$	0	0
$463$	19.3137	0.897584	0.448792	−	0.893636i	$-0.351854\pi$
$463$	19.3137	0.897584	0.448792	+	0.893636i	$0.351854\pi$
$464$	1.65685	0.0769175
$465$	0	0
$466$	3.65685	0.169401
$467$	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
$470$	−23.7990	−1.09777
$471$	0	0
$472$	−1.17157	−0.0539260
$473$	0.828427	0.0380911
$474$	0	0
$475$	6.24264	0.286432
$476$	0	0
$477$	0	0
$478$	23.3137	1.06634
$479$	−15.3137	−0.699701	−0.349851	−	0.936806i	$-0.613768\pi$
$479$	−15.3137	−0.699701	−0.349851	+	0.936806i	$0.613768\pi$
$480$	0	0
$481$	12.4853	0.569280
$482$	3.65685	0.166565
$483$	0	0
$484$	1.00000	0.0454545
$485$	−19.7990	−0.899026
$486$	0	0
$487$	−28.1421	−1.27524	−0.637621	−	0.770350i	$-0.720081\pi$
$487$	−28.1421	−1.27524	−0.637621	+	0.770350i	$0.720081\pi$
$488$	5.41421	0.245090
$489$	0	0
$490$	0	0
$491$	−2.62742	−0.118574	−0.0592868	−	0.998241i	$-0.518883\pi$
$491$	−2.62742	−0.118574	−0.0592868	+	0.998241i	$0.518883\pi$
$492$	0	0
$493$	3.31371	0.149242
$494$	−16.1421	−0.726269
$495$	0	0
$496$	2.24264	0.100698
$497$	0	0
$498$	0	0
$499$	22.1421	0.991218	0.495609	−	0.868546i	$-0.334945\pi$
$499$	22.1421	0.991218	0.495609	+	0.868546i	$0.334945\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	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
$505$	−1.85786	−0.0826739
$506$	0.828427	0.0368281
$507$	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
$514$	−13.2132	−0.582809
$515$	4.48528	0.197645
$516$	0	0
$517$	11.8995	0.523339
$518$	0	0
$519$	0	0
$520$	5.17157	0.226788
$521$	−21.2132	−0.929367	−0.464684	−	0.885477i	$-0.653832\pi$
$521$	−21.2132	−0.929367	−0.464684	+	0.885477i	$0.653832\pi$
$522$	0	0
$523$	−5.07107	−0.221742	−0.110871	−	0.993835i	$-0.535364\pi$
$523$	−5.07107	−0.221742	−0.110871	+	0.993835i	$0.535364\pi$
$524$	−13.5563	−0.592212
$525$	0	0
$526$	14.8284	0.646550
$527$	4.48528	0.195382
$528$	0	0
$529$	−22.3137	−0.970161
$530$	12.9706	0.563405
$531$	0	0
$532$	0	0
$533$	−0.887302	−0.0384333
$534$	0	0
$535$	−14.3431	−0.620108
$536$	6.82843	0.294943
$537$	0	0
$538$	17.3137	0.746447
$539$	0	0
$540$	0	0
$541$	−22.6863	−0.975360	−0.487680	−	0.873023i	$-0.662157\pi$
$541$	−22.6863	−0.975360	−0.487680	+	0.873023i	$0.662157\pi$
$542$	−6.14214	−0.263827
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	31.3137	1.34133
$546$	0	0
$547$	2.62742	0.112340	0.0561701	−	0.998421i	$-0.482111\pi$
$547$	2.62742	0.112340	0.0561701	+	0.998421i	$0.482111\pi$
$548$	−17.3137	−0.739605
$549$	0	0
$550$	1.00000	0.0426401
$551$	−10.3431	−0.440633
$552$	0	0
$553$	0	0
$554$	16.0000	0.679775
$555$	0	0
$556$	−19.8995	−0.843927
$557$	−8.34315	−0.353510	−0.176755	−	0.984255i	$-0.556560\pi$
$557$	−8.34315	−0.353510	−0.176755	+	0.984255i	$0.556560\pi$
$558$	0	0
$559$	−2.14214	−0.0906027
$560$	0	0
$561$	0	0
$562$	21.3137	0.899065
$563$	−6.92893	−0.292020	−0.146010	−	0.989283i	$-0.546643\pi$
$563$	−6.92893	−0.292020	−0.146010	+	0.989283i	$0.546643\pi$
$564$	0	0
$565$	11.3137	0.475971
$566$	6.24264	0.262398
$567$	0	0
$568$	−5.65685	−0.237356
$569$	33.3137	1.39658	0.698292	−	0.715813i	$-0.253944\pi$
$569$	33.3137	1.39658	0.698292	+	0.715813i	$0.253944\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−2.58579	−0.108117
$573$	0	0
$574$	0	0
$575$	0.828427	0.0345478
$576$	0	0
$577$	35.5563	1.48023	0.740115	−	0.672480i	$-0.234771\pi$
$577$	35.5563	1.48023	0.740115	+	0.672480i	$0.234771\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	3.31371	0.137594
$581$	0	0
$582$	0	0
$583$	−6.48528	−0.268593
$584$	0.343146	0.0141995
$585$	0	0
$586$	19.0711	0.787819
$587$	−27.7990	−1.14739	−0.573694	−	0.819070i	$-0.694490\pi$
$587$	−27.7990	−1.14739	−0.573694	+	0.819070i	$0.694490\pi$
$588$	0	0
$589$	−14.0000	−0.576860
$590$	−2.34315	−0.0964658
$591$	0	0
$592$	−4.82843	−0.198447
$593$	−14.6863	−0.603094	−0.301547	−	0.953451i	$-0.597503\pi$
$593$	−14.6863	−0.603094	−0.301547	+	0.953451i	$0.597503\pi$
$594$	0	0
$595$	0	0
$596$	−4.00000	−0.163846
$597$	0	0
$598$	−2.14214	−0.0875984
$599$	−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
$601$	−20.8284	−0.849609	−0.424805	−	0.905285i	$-0.639657\pi$
$601$	−20.8284	−0.849609	−0.424805	+	0.905285i	$0.639657\pi$
$602$	0	0
$603$	0	0
$604$	4.00000	0.162758
$605$	2.00000	0.0813116
$606$	0	0
$607$	12.6863	0.514921	0.257460	−	0.966289i	$-0.417114\pi$
$607$	12.6863	0.514921	0.257460	+	0.966289i	$0.417114\pi$
$608$	6.24264	0.253173
$609$	0	0
$610$	10.8284	0.438430
$611$	−30.7696	−1.24480
$612$	0	0
$613$	−41.6569	−1.68250	−0.841252	−	0.540643i	$-0.818181\pi$
$613$	−41.6569	−1.68250	−0.841252	+	0.540643i	$0.818181\pi$
$614$	−10.9289	−0.441056
$615$	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
$619$	10.3431	0.415726	0.207863	−	0.978158i	$-0.433349\pi$
$619$	10.3431	0.415726	0.207863	+	0.978158i	$0.433349\pi$
$620$	4.48528	0.180133
$621$	0	0
$622$	−27.8995	−1.11867
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0.242641	0.00969787
$627$	0	0
$628$	−7.17157	−0.286177
$629$	−9.65685	−0.385044
$630$	0	0
$631$	7.85786	0.312817	0.156408	−	0.987692i	$-0.450008\pi$
$631$	7.85786	0.312817	0.156408	+	0.987692i	$0.450008\pi$
$632$	−0.485281	−0.0193035
$633$	0	0
$634$	7.17157	0.284820
$635$	−28.2843	−1.12243
$636$	0	0
$637$	0	0
$638$	−1.65685	−0.0655955
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	12.6863	0.501078	0.250539	−	0.968106i	$-0.419392\pi$
$641$	12.6863	0.501078	0.250539	+	0.968106i	$0.419392\pi$
$642$	0	0
$643$	13.4558	0.530647	0.265323	−	0.964159i	$-0.414521\pi$
$643$	13.4558	0.530647	0.265323	+	0.964159i	$0.414521\pi$
$644$	0	0
$645$	0	0
$646$	12.4853	0.491227
$647$	−13.0711	−0.513877	−0.256938	−	0.966428i	$-0.582714\pi$
$647$	−13.0711	−0.513877	−0.256938	+	0.966428i	$0.582714\pi$
$648$	0	0
$649$	1.17157	0.0459883
$650$	−2.58579	−0.101423
$651$	0	0
$652$	−13.6569	−0.534844
$653$	34.2843	1.34165	0.670824	−	0.741617i	$-0.265941\pi$
$653$	34.2843	1.34165	0.670824	+	0.741617i	$0.265941\pi$
$654$	0	0
$655$	−27.1127	−1.05938
$656$	0.343146	0.0133976
$657$	0	0
$658$	0	0
$659$	−13.1127	−0.510798	−0.255399	−	0.966836i	$-0.582207\pi$
$659$	−13.1127	−0.510798	−0.255399	+	0.966836i	$0.582207\pi$
$660$	0	0
$661$	−27.9411	−1.08678	−0.543392	−	0.839479i	$-0.682860\pi$
$661$	−27.9411	−1.08678	−0.543392	+	0.839479i	$0.682860\pi$
$662$	−11.7990	−0.458581
$663$	0	0
$664$	9.07107	0.352026
$665$	0	0
$666$	0	0
$667$	−1.37258	−0.0531466
$668$	−10.8284	−0.418964
$669$	0	0
$670$	13.6569	0.527610
$671$	−5.41421	−0.209013
$672$	0	0
$673$	−34.4853	−1.32931	−0.664655	−	0.747150i	$-0.731421\pi$
$673$	−34.4853	−1.32931	−0.664655	+	0.747150i	$0.731421\pi$
$674$	−7.17157	−0.276239
$675$	0	0
$676$	−6.31371	−0.242835
$677$	12.0416	0.462797	0.231399	−	0.972859i	$-0.425670\pi$
$677$	12.0416	0.462797	0.231399	+	0.972859i	$0.425670\pi$
$678$	0	0
$679$	0	0
$680$	−4.00000	−0.153393
$681$	0	0
$682$	−2.24264	−0.0858752
$683$	44.7696	1.71306	0.856530	−	0.516098i	$-0.172616\pi$
$683$	44.7696	1.71306	0.856530	+	0.516098i	$0.172616\pi$
$684$	0	0
$685$	−34.6274	−1.32305
$686$	0	0
$687$	0	0
$688$	0.828427	0.0315835
$689$	16.7696	0.638869
$690$	0	0
$691$	−14.1421	−0.537992	−0.268996	−	0.963141i	$-0.586692\pi$
$691$	−14.1421	−0.537992	−0.268996	+	0.963141i	$0.586692\pi$
$692$	13.8995	0.528380
$693$	0	0
$694$	−31.4558	−1.19405
$695$	−39.7990	−1.50966
$696$	0	0
$697$	0.686292	0.0259951
$698$	−14.5858	−0.552080
$699$	0	0
$700$	0	0
$701$	13.3137	0.502852	0.251426	−	0.967877i	$-0.419101\pi$
$701$	13.3137	0.502852	0.251426	+	0.967877i	$0.419101\pi$
$702$	0	0
$703$	30.1421	1.13683
$704$	1.00000	0.0376889
$705$	0	0
$706$	21.2132	0.798369
$707$	0	0
$708$	0	0
$709$	38.7696	1.45602	0.728011	−	0.685566i	$-0.240445\pi$
$709$	38.7696	1.45602	0.728011	+	0.685566i	$0.240445\pi$
$710$	−11.3137	−0.424596
$711$	0	0
$712$	−12.2426	−0.458812
$713$	−1.85786	−0.0695776
$714$	0	0
$715$	−5.17157	−0.193406
$716$	3.51472	0.131351
$717$	0	0
$718$	10.8284	0.404113
$719$	32.1838	1.20025	0.600126	−	0.799906i	$-0.295117\pi$
$719$	32.1838	1.20025	0.600126	+	0.799906i	$0.295117\pi$
$720$	0	0
$721$	0	0
$722$	−19.9706	−0.743227
$723$	0	0
$724$	−13.3137	−0.494800
$725$	−1.65685	−0.0615340
$726$	0	0
$727$	11.2132	0.415875	0.207937	−	0.978142i	$-0.433325\pi$
$727$	11.2132	0.415875	0.207937	+	0.978142i	$0.433325\pi$
$728$	0	0
$729$	0	0
$730$	0.686292	0.0254008
$731$	1.65685	0.0612810
$732$	0	0
$733$	11.5563	0.426843	0.213422	−	0.976960i	$-0.431539\pi$
$733$	11.5563	0.426843	0.213422	+	0.976960i	$0.431539\pi$
$734$	−35.4142	−1.30716
$735$	0	0
$736$	0.828427	0.0305362
$737$	−6.82843	−0.251528
$738$	0	0
$739$	−42.6274	−1.56807	−0.784037	−	0.620714i	$-0.786843\pi$
$739$	−42.6274	−1.56807	−0.784037	+	0.620714i	$0.786843\pi$
$740$	−9.65685	−0.354993
$741$	0	0
$742$	0	0
$743$	−26.6274	−0.976865	−0.488433	−	0.872602i	$-0.662431\pi$
$743$	−26.6274	−0.976865	−0.488433	+	0.872602i	$0.662431\pi$
$744$	0	0
$745$	−8.00000	−0.293097
$746$	31.9411	1.16945
$747$	0	0
$748$	2.00000	0.0731272
$749$	0	0
$750$	0	0
$751$	16.1421	0.589035	0.294517	−	0.955646i	$-0.404841\pi$
$751$	16.1421	0.589035	0.294517	+	0.955646i	$0.404841\pi$
$752$	11.8995	0.433930
$753$	0	0
$754$	4.28427	0.156024
$755$	8.00000	0.291150
$756$	0	0
$757$	−19.8579	−0.721746	−0.360873	−	0.932615i	$-0.617521\pi$
$757$	−19.8579	−0.721746	−0.360873	+	0.932615i	$0.617521\pi$
$758$	24.2843	0.882044
$759$	0	0
$760$	12.4853	0.452889
$761$	−18.4853	−0.670091	−0.335045	−	0.942202i	$-0.608752\pi$
$761$	−18.4853	−0.670091	−0.335045	+	0.942202i	$0.608752\pi$
$762$	0	0
$763$	0	0
$764$	17.7990	0.643945
$765$	0	0
$766$	−2.72792	−0.0985638
$767$	−3.02944	−0.109387
$768$	0	0
$769$	44.1421	1.59181	0.795903	−	0.605424i	$-0.206996\pi$
$769$	44.1421	1.59181	0.795903	+	0.605424i	$0.206996\pi$
$770$	0	0
$771$	0	0
$772$	18.9706	0.682765
$773$	22.4853	0.808739	0.404370	−	0.914596i	$-0.367491\pi$
$773$	22.4853	0.808739	0.404370	+	0.914596i	$0.367491\pi$
$774$	0	0
$775$	−2.24264	−0.0805580
$776$	9.89949	0.355371
$777$	0	0
$778$	22.2843	0.798930
$779$	−2.14214	−0.0767500
$780$	0	0
$781$	5.65685	0.202418
$782$	1.65685	0.0592490
$783$	0	0
$784$	0	0
$785$	−14.3431	−0.511929
$786$	0	0
$787$	0.585786	0.0208810	0.0104405	−	0.999945i	$-0.496677\pi$
$787$	0.585786	0.0208810	0.0104405	+	0.999945i	$0.496677\pi$
$788$	−4.34315	−0.154718
$789$	0	0
$790$	−0.970563	−0.0345311
$791$	0	0
$792$	0	0
$793$	14.0000	0.497155
$794$	−16.8284	−0.597219
$795$	0	0
$796$	27.2132	0.964546
$797$	34.4853	1.22153	0.610766	−	0.791811i	$-0.290862\pi$
$797$	34.4853	1.22153	0.610766	+	0.791811i	$0.290862\pi$
$798$	0	0
$799$	23.7990	0.841948
$800$	1.00000	0.0353553
$801$	0	0
$802$	33.3137	1.17635
$803$	−0.343146	−0.0121094
$804$	0	0
$805$	0	0
$806$	5.79899	0.204261
$807$	0	0
$808$	0.928932	0.0326797
$809$	46.7696	1.64433	0.822165	−	0.569249i	$-0.192766\pi$
$809$	46.7696	1.64433	0.822165	+	0.569249i	$0.192766\pi$
$810$	0	0
$811$	0.384776	0.0135113	0.00675566	−	0.999977i	$-0.497850\pi$
$811$	0.384776	0.0135113	0.00675566	+	0.999977i	$0.497850\pi$
$812$	0	0
$813$	0	0
$814$	4.82843	0.169236
$815$	−27.3137	−0.956757
$816$	0	0
$817$	−5.17157	−0.180930
$818$	−18.4853	−0.646323
$819$	0	0
$820$	0.686292	0.0239663
$821$	−24.6863	−0.861558	−0.430779	−	0.902458i	$-0.641761\pi$
$821$	−24.6863	−0.861558	−0.430779	+	0.902458i	$0.641761\pi$
$822$	0	0
$823$	0.142136	0.00495454	0.00247727	−	0.999997i	$-0.499211\pi$
$823$	0.142136	0.00495454	0.00247727	+	0.999997i	$0.499211\pi$
$824$	−2.24264	−0.0781261
$825$	0	0
$826$	0	0
$827$	−16.6863	−0.580239	−0.290120	−	0.956990i	$-0.593695\pi$
$827$	−16.6863	−0.580239	−0.290120	+	0.956990i	$0.593695\pi$
$828$	0	0
$829$	6.28427	0.218262	0.109131	−	0.994027i	$-0.465193\pi$
$829$	6.28427	0.218262	0.109131	+	0.994027i	$0.465193\pi$
$830$	18.1421	0.629723
$831$	0	0
$832$	−2.58579	−0.0896460
$833$	0	0
$834$	0	0
$835$	−21.6569	−0.749466
$836$	−6.24264	−0.215906
$837$	0	0
$838$	−16.0000	−0.552711
$839$	−2.92893	−0.101118	−0.0505590	−	0.998721i	$-0.516100\pi$
$839$	−2.92893	−0.101118	−0.0505590	+	0.998721i	$0.516100\pi$
$840$	0	0
$841$	−26.2548	−0.905339
$842$	16.3431	0.563222
$843$	0	0
$844$	−28.8284	−0.992315
$845$	−12.6274	−0.434396
$846$	0	0
$847$	0	0
$848$	−6.48528	−0.222705
$849$	0	0
$850$	2.00000	0.0685994
$851$	4.00000	0.137118
$852$	0	0
$853$	−4.44365	−0.152148	−0.0760739	−	0.997102i	$-0.524238\pi$
$853$	−4.44365	−0.152148	−0.0760739	+	0.997102i	$0.524238\pi$
$854$	0	0
$855$	0	0
$856$	7.17157	0.245119
$857$	−28.1421	−0.961317	−0.480659	−	0.876908i	$-0.659602\pi$
$857$	−28.1421	−0.961317	−0.480659	+	0.876908i	$0.659602\pi$
$858$	0	0
$859$	−54.4264	−1.85701	−0.928503	−	0.371326i	$-0.878903\pi$
$859$	−54.4264	−1.85701	−0.928503	+	0.371326i	$0.878903\pi$
$860$	1.65685	0.0564983
$861$	0	0
$862$	−26.1421	−0.890405
$863$	−34.7696	−1.18357	−0.591785	−	0.806096i	$-0.701576\pi$
$863$	−34.7696	−1.18357	−0.591785	+	0.806096i	$0.701576\pi$
$864$	0	0
$865$	27.7990	0.945194
$866$	2.38478	0.0810380
$867$	0	0
$868$	0	0
$869$	0.485281	0.0164620
$870$	0	0
$871$	17.6569	0.598280
$872$	−15.6569	−0.530208
$873$	0	0
$874$	−5.17157	−0.174931
$875$	0	0
$876$	0	0
$877$	−29.3137	−0.989854	−0.494927	−	0.868935i	$-0.664805\pi$
$877$	−29.3137	−0.989854	−0.494927	+	0.868935i	$0.664805\pi$
$878$	−34.8284	−1.17540
$879$	0	0
$880$	2.00000	0.0674200
$881$	18.1005	0.609822	0.304911	−	0.952381i	$-0.401373\pi$
$881$	18.1005	0.609822	0.304911	+	0.952381i	$0.401373\pi$
$882$	0	0
$883$	34.6274	1.16531	0.582653	−	0.812721i	$-0.302015\pi$
$883$	34.6274	1.16531	0.582653	+	0.812721i	$0.302015\pi$
$884$	−5.17157	−0.173939
$885$	0	0
$886$	−4.48528	−0.150686
$887$	−52.2843	−1.75553	−0.877767	−	0.479088i	$-0.840968\pi$
$887$	−52.2843	−1.75553	−0.877767	+	0.479088i	$0.840968\pi$
$888$	0	0
$889$	0	0
$890$	−24.4853	−0.820748
$891$	0	0
$892$	2.24264	0.0750892
$893$	−74.2843	−2.48583
$894$	0	0
$895$	7.02944	0.234968
$896$	0	0
$897$	0	0
$898$	−16.9706	−0.566315
$899$	3.71573	0.123926
$900$	0	0
$901$	−12.9706	−0.432112
$902$	−0.343146	−0.0114255
$903$	0	0
$904$	−5.65685	−0.188144
$905$	−26.6274	−0.885125
$906$	0	0
$907$	−47.7990	−1.58714	−0.793570	−	0.608479i	$-0.791780\pi$
$907$	−47.7990	−1.58714	−0.793570	+	0.608479i	$0.791780\pi$
$908$	7.89949	0.262154
$909$	0	0
$910$	0	0
$911$	−39.4558	−1.30723	−0.653615	−	0.756827i	$-0.726749\pi$
$911$	−39.4558	−1.30723	−0.653615	+	0.756827i	$0.726749\pi$
$912$	0	0
$913$	−9.07107	−0.300209
$914$	6.68629	0.221163
$915$	0	0
$916$	−14.0000	−0.462573
$917$	0	0
$918$	0	0
$919$	32.9706	1.08760	0.543799	−	0.839215i	$-0.316985\pi$
$919$	32.9706	1.08760	0.543799	+	0.839215i	$0.316985\pi$
$920$	1.65685	0.0546249
$921$	0	0
$922$	33.4142	1.10044
$923$	−14.6274	−0.481467
$924$	0	0
$925$	4.82843	0.158758
$926$	−19.3137	−0.634688
$927$	0	0
$928$	−1.65685	−0.0543889
$929$	53.2132	1.74587	0.872934	−	0.487838i	$-0.162214\pi$
$929$	53.2132	1.74587	0.872934	+	0.487838i	$0.162214\pi$
$930$	0	0
$931$	0	0
$932$	−3.65685	−0.119784
$933$	0	0
$934$	−12.0000	−0.392652
$935$	4.00000	0.130814
$936$	0	0
$937$	−44.9117	−1.46720	−0.733600	−	0.679581i	$-0.762162\pi$
$937$	−44.9117	−1.46720	−0.733600	+	0.679581i	$0.762162\pi$
$938$	0	0
$939$	0	0
$940$	23.7990	0.776237
$941$	−3.55635	−0.115934	−0.0579668	−	0.998319i	$-0.518462\pi$
$941$	−3.55635	−0.115934	−0.0579668	+	0.998319i	$0.518462\pi$
$942$	0	0
$943$	−0.284271	−0.00925715
$944$	1.17157	0.0381314
$945$	0	0
$946$	−0.828427	−0.0269345
$947$	−60.5685	−1.96821	−0.984107	−	0.177579i	$-0.943174\pi$
$947$	−60.5685	−1.96821	−0.984107	+	0.177579i	$0.943174\pi$
$948$	0	0
$949$	0.887302	0.0288030
$950$	−6.24264	−0.202538
$951$	0	0
$952$	0	0
$953$	−26.4853	−0.857942	−0.428971	−	0.903318i	$-0.641124\pi$
$953$	−26.4853	−0.857942	−0.428971	+	0.903318i	$0.641124\pi$
$954$	0	0
$955$	35.5980	1.15192
$956$	−23.3137	−0.754019
$957$	0	0
$958$	15.3137	0.494763
$959$	0	0
$960$	0	0
$961$	−25.9706	−0.837760
$962$	−12.4853	−0.402542
$963$	0	0
$964$	−3.65685	−0.117779
$965$	37.9411	1.22137
$966$	0	0
$967$	57.9411	1.86326	0.931630	−	0.363407i	$-0.118387\pi$
$967$	57.9411	1.86326	0.931630	+	0.363407i	$0.118387\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	19.7990	0.635707
$971$	27.3137	0.876539	0.438269	−	0.898844i	$-0.355592\pi$
$971$	27.3137	0.876539	0.438269	+	0.898844i	$0.355592\pi$
$972$	0	0
$973$	0	0
$974$	28.1421	0.901732
$975$	0	0
$976$	−5.41421	−0.173305
$977$	18.6274	0.595944	0.297972	−	0.954575i	$-0.403690\pi$
$977$	18.6274	0.595944	0.297972	+	0.954575i	$0.403690\pi$
$978$	0	0
$979$	12.2426	0.391276
$980$	0	0
$981$	0	0
$982$	2.62742	0.0838442
$983$	−52.8701	−1.68629	−0.843146	−	0.537684i	$-0.819299\pi$
$983$	−52.8701	−1.68629	−0.843146	+	0.537684i	$0.819299\pi$
$984$	0	0
$985$	−8.68629	−0.276768
$986$	−3.31371	−0.105530
$987$	0	0
$988$	16.1421	0.513550
$989$	−0.686292	−0.0218228
$990$	0	0
$991$	−50.9117	−1.61726	−0.808632	−	0.588315i	$-0.799791\pi$
$991$	−50.9117	−1.61726	−0.808632	+	0.588315i	$0.799791\pi$
$992$	−2.24264	−0.0712039
$993$	0	0
$994$	0	0
$995$	54.4264	1.72543
$996$	0	0
$997$	−33.4142	−1.05824	−0.529119	−	0.848547i	$-0.677478\pi$
$997$	−33.4142	−1.05824	−0.529119	+	0.848547i	$0.677478\pi$
$998$	−22.1421	−0.700897
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{8} -2.00000 q^{10} +1.00000 q^{11} -2.58579 q^{13} +1.00000 q^{16} +2.00000 q^{17} -6.24264 q^{19} +2.00000 q^{20} -1.00000 q^{22} -0.828427 q^{23} -1.00000 q^{25} +2.58579 q^{26} +1.65685 q^{29} +2.24264 q^{31} -1.00000 q^{32} -2.00000 q^{34} -4.82843 q^{37} +6.24264 q^{38} -2.00000 q^{40} +0.343146 q^{41} +0.828427 q^{43} +1.00000 q^{44} +0.828427 q^{46} +11.8995 q^{47} +1.00000 q^{50} -2.58579 q^{52} -6.48528 q^{53} +2.00000 q^{55} -1.65685 q^{58} +1.17157 q^{59} -5.41421 q^{61} -2.24264 q^{62} +1.00000 q^{64} -5.17157 q^{65} -6.82843 q^{67} +2.00000 q^{68} +5.65685 q^{71} -0.343146 q^{73} +4.82843 q^{74} -6.24264 q^{76} +0.485281 q^{79} +2.00000 q^{80} -0.343146 q^{82} -9.07107 q^{83} +4.00000 q^{85} -0.828427 q^{86} -1.00000 q^{88} +12.2426 q^{89} -0.828427 q^{92} -11.8995 q^{94} -12.4853 q^{95} -9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 2 q^{8}+O(q^{10}) \) 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

L-functions

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