LMFDB - Embedded newform 9702.2.a.du.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:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.66908$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +0.454904 q^{5} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +0.454904 q^{5} -1.00000 q^{8} -0.454904 q^{10} -1.00000 q^{11} -0.909808 q^{13} +1.00000 q^{16} +1.00000 q^{17} -3.88325 q^{19} +0.454904 q^{20} +1.00000 q^{22} -8.33816 q^{23} -4.79306 q^{25} +0.909808 q^{26} +2.79306 q^{29} +9.58612 q^{31} -1.00000 q^{32} -1.00000 q^{34} +7.88325 q^{37} +3.88325 q^{38} -0.454904 q^{40} -1.79306 q^{41} +7.88325 q^{43} -1.00000 q^{44} +8.33816 q^{46} +0.338158 q^{47} +4.79306 q^{50} -0.909808 q^{52} -0.909808 q^{53} -0.454904 q^{55} -2.79306 q^{58} +6.97345 q^{59} +14.0410 q^{61} -9.58612 q^{62} +1.00000 q^{64} -0.413875 q^{65} -3.79306 q^{67} +1.00000 q^{68} +4.79306 q^{71} -10.6763 q^{73} -7.88325 q^{74} -3.88325 q^{76} +5.36471 q^{79} +0.454904 q^{80} +1.79306 q^{82} +1.97345 q^{83} +0.454904 q^{85} -7.88325 q^{86} +1.00000 q^{88} -11.0902 q^{89} -8.33816 q^{92} -0.338158 q^{94} -1.76651 q^{95} -14.5861 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} - 3 q^{11} + 3 q^{16} + 3 q^{17} + 3 q^{19} + 3 q^{22} - 9 q^{23} + 3 q^{25} - 9 q^{29} - 6 q^{31} - 3 q^{32} - 3 q^{34} + 9 q^{37} - 3 q^{38} + 12 q^{41} + 9 q^{43} - 3 q^{44} + 9 q^{46} - 15 q^{47} - 3 q^{50} + 9 q^{58} + 9 q^{59} + 6 q^{61} + 6 q^{62} + 3 q^{64} - 36 q^{65} + 6 q^{67} + 3 q^{68} - 3 q^{71} - 9 q^{74} + 3 q^{76} + 12 q^{79} - 12 q^{82} - 6 q^{83} - 9 q^{86} + 3 q^{88} - 36 q^{89} - 9 q^{92} + 15 q^{94} + 24 q^{95} - 9 q^{97}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 - 3 * q^11 + 3 * q^16 + 3 * q^17 + 3 * q^19 + 3 * q^22 - 9 * q^23 + 3 * q^25 - 9 * q^29 - 6 * q^31 - 3 * q^32 - 3 * q^34 + 9 * q^37 - 3 * q^38 + 12 * q^41 + 9 * q^43 - 3 * q^44 + 9 * q^46 - 15 * q^47 - 3 * q^50 + 9 * q^58 + 9 * q^59 + 6 * q^61 + 6 * q^62 + 3 * q^64 - 36 * q^65 + 6 * q^67 + 3 * q^68 - 3 * q^71 - 9 * q^74 + 3 * q^76 + 12 * q^79 - 12 * q^82 - 6 * q^83 - 9 * q^86 + 3 * q^88 - 36 * q^89 - 9 * q^92 + 15 * q^94 + 24 * q^95 - 9 * q^97

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
$5$	0.454904	0.203439	0.101720	−	0.994813i	$-0.467566\pi$
$5$	0.454904	0.203439	0.101720	+	0.994813i	$0.467566\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.454904	−0.143853
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.909808	−0.252335	−0.126168	−	0.992009i	$-0.540268\pi$
$13$	−0.909808	−0.252335	−0.126168	+	0.992009i	$0.540268\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−3.88325	−0.890880	−0.445440	−	0.895312i	$-0.646953\pi$
$19$	−3.88325	−0.890880	−0.445440	+	0.895312i	$0.646953\pi$
$20$	0.454904	0.101720
$21$	0	0
$22$	1.00000	0.213201
$23$	−8.33816	−1.73863	−0.869313	−	0.494262i	$-0.835438\pi$
$23$	−8.33816	−1.73863	−0.869313	+	0.494262i	$0.835438\pi$
$24$	0	0
$25$	−4.79306	−0.958612
$26$	0.909808	0.178428
$27$	0	0
$28$	0	0
$29$	2.79306	0.518659	0.259329	−	0.965789i	$-0.416498\pi$
$29$	2.79306	0.518659	0.259329	+	0.965789i	$0.416498\pi$
$30$	0	0
$31$	9.58612	1.72172	0.860859	−	0.508843i	$-0.169927\pi$
$31$	9.58612	1.72172	0.860859	+	0.508843i	$0.169927\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	0	0
$37$	7.88325	1.29600	0.647999	−	0.761641i	$-0.275606\pi$
$37$	7.88325	1.29600	0.647999	+	0.761641i	$0.275606\pi$
$38$	3.88325	0.629947
$39$	0	0
$40$	−0.454904	−0.0719267
$41$	−1.79306	−0.280029	−0.140015	−	0.990149i	$-0.544715\pi$
$41$	−1.79306	−0.280029	−0.140015	+	0.990149i	$0.544715\pi$
$42$	0	0
$43$	7.88325	1.20218	0.601092	−	0.799179i	$-0.294732\pi$
$43$	7.88325	1.20218	0.601092	+	0.799179i	$0.294732\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	8.33816	1.22939
$47$	0.338158	0.0493254	0.0246627	−	0.999696i	$-0.492149\pi$
$47$	0.338158	0.0493254	0.0246627	+	0.999696i	$0.492149\pi$
$48$	0	0
$49$	0	0
$50$	4.79306	0.677841
$51$	0	0
$52$	−0.909808	−0.126168
$53$	−0.909808	−0.124972	−0.0624859	−	0.998046i	$-0.519903\pi$
$53$	−0.909808	−0.124972	−0.0624859	+	0.998046i	$0.519903\pi$
$54$	0	0
$55$	−0.454904	−0.0613393
$56$	0	0
$57$	0	0
$58$	−2.79306	−0.366747
$59$	6.97345	0.907865	0.453933	−	0.891036i	$-0.350021\pi$
$59$	6.97345	0.907865	0.453933	+	0.891036i	$0.350021\pi$
$60$	0	0
$61$	14.0410	1.79777	0.898885	−	0.438185i	$-0.144379\pi$
$61$	14.0410	1.79777	0.898885	+	0.438185i	$0.144379\pi$
$62$	−9.58612	−1.21744
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.413875	−0.0513349
$66$	0	0
$67$	−3.79306	−0.463396	−0.231698	−	0.972788i	$-0.574428\pi$
$67$	−3.79306	−0.463396	−0.231698	+	0.972788i	$0.574428\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	4.79306	0.568832	0.284416	−	0.958701i	$-0.408200\pi$
$71$	4.79306	0.568832	0.284416	+	0.958701i	$0.408200\pi$
$72$	0	0
$73$	−10.6763	−1.24957	−0.624784	−	0.780798i	$-0.714813\pi$
$73$	−10.6763	−1.24957	−0.624784	+	0.780798i	$0.714813\pi$
$74$	−7.88325	−0.916410
$75$	0	0
$76$	−3.88325	−0.445440
$77$	0	0
$78$	0	0
$79$	5.36471	0.603577	0.301789	−	0.953375i	$-0.402416\pi$
$79$	5.36471	0.603577	0.301789	+	0.953375i	$0.402416\pi$
$80$	0.454904	0.0508598
$81$	0	0
$82$	1.79306	0.198011
$83$	1.97345	0.216614	0.108307	−	0.994118i	$-0.465457\pi$
$83$	1.97345	0.216614	0.108307	+	0.994118i	$0.465457\pi$
$84$	0	0
$85$	0.454904	0.0493413
$86$	−7.88325	−0.850073
$87$	0	0
$88$	1.00000	0.106600
$89$	−11.0902	−1.17556	−0.587779	−	0.809022i	$-0.699998\pi$
$89$	−11.0902	−1.17556	−0.587779	+	0.809022i	$0.699998\pi$
$90$	0	0
$91$	0	0
$92$	−8.33816	−0.869313
$93$	0	0
$94$	−0.338158	−0.0348784
$95$	−1.76651	−0.181240
$96$	0	0
$97$	−14.5861	−1.48100	−0.740498	−	0.672058i	$-0.765410\pi$
$97$	−14.5861	−1.48100	−0.740498	+	0.672058i	$0.765410\pi$
$98$	0	0
$99$	0	0
$100$	−4.79306	−0.479306
$101$	−8.06364	−0.802362	−0.401181	−	0.915999i	$-0.631400\pi$
$101$	−8.06364	−0.802362	−0.401181	+	0.915999i	$0.631400\pi$
$102$	0	0
$103$	−5.76651	−0.568191	−0.284095	−	0.958796i	$-0.591693\pi$
$103$	−5.76651	−0.568191	−0.284095	+	0.958796i	$0.591693\pi$
$104$	0.909808	0.0892140
$105$	0	0
$106$	0.909808	0.0883684
$107$	−15.6127	−1.50933	−0.754667	−	0.656108i	$-0.772202\pi$
$107$	−15.6127	−1.50933	−0.754667	+	0.656108i	$0.772202\pi$
$108$	0	0
$109$	14.0410	1.34489	0.672443	−	0.740149i	$-0.265245\pi$
$109$	14.0410	1.34489	0.672443	+	0.740149i	$0.265245\pi$
$110$	0.454904	0.0433734
$111$	0	0
$112$	0	0
$113$	−7.58612	−0.713643	−0.356821	−	0.934173i	$-0.616139\pi$
$113$	−7.58612	−0.713643	−0.356821	+	0.934173i	$0.616139\pi$
$114$	0	0
$115$	−3.79306	−0.353705
$116$	2.79306	0.259329
$117$	0	0
$118$	−6.97345	−0.641958
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−14.0410	−1.27122
$123$	0	0
$124$	9.58612	0.860859
$125$	−4.45490	−0.398459
$126$	0	0
$127$	7.42835	0.659159	0.329580	−	0.944128i	$-0.393093\pi$
$127$	7.42835	0.659159	0.329580	+	0.944128i	$0.393093\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0.413875	0.0362993
$131$	−19.4057	−1.69549	−0.847744	−	0.530406i	$-0.822039\pi$
$131$	−19.4057	−1.69549	−0.847744	+	0.530406i	$0.822039\pi$
$132$	0	0
$133$	0	0
$134$	3.79306	0.327671
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−0.909808	−0.0777302	−0.0388651	−	0.999244i	$-0.512374\pi$
$137$	−0.909808	−0.0777302	−0.0388651	+	0.999244i	$0.512374\pi$
$138$	0	0
$139$	8.37919	0.710713	0.355357	−	0.934731i	$-0.384359\pi$
$139$	8.37919	0.710713	0.355357	+	0.934731i	$0.384359\pi$
$140$	0	0
$141$	0	0
$142$	−4.79306	−0.402225
$143$	0.909808	0.0760820
$144$	0	0
$145$	1.27058	0.105516
$146$	10.6763	0.883578
$147$	0	0
$148$	7.88325	0.647999
$149$	−4.11675	−0.337257	−0.168628	−	0.985680i	$-0.553934\pi$
$149$	−4.11675	−0.337257	−0.168628	+	0.985680i	$0.553934\pi$
$150$	0	0
$151$	−6.15777	−0.501113	−0.250556	−	0.968102i	$-0.580614\pi$
$151$	−6.15777	−0.501113	−0.250556	+	0.968102i	$0.580614\pi$
$152$	3.88325	0.314973
$153$	0	0
$154$	0	0
$155$	4.36077	0.350265
$156$	0	0
$157$	17.6498	1.40860	0.704302	−	0.709900i	$-0.251260\pi$
$157$	17.6498	1.40860	0.704302	+	0.709900i	$0.251260\pi$
$158$	−5.36471	−0.426794
$159$	0	0
$160$	−0.454904	−0.0359633
$161$	0	0
$162$	0	0
$163$	−11.6127	−0.909575	−0.454788	−	0.890600i	$-0.650285\pi$
$163$	−11.6127	−0.909575	−0.454788	+	0.890600i	$0.650285\pi$
$164$	−1.79306	−0.140015
$165$	0	0
$166$	−1.97345	−0.153169
$167$	−4.85670	−0.375823	−0.187911	−	0.982186i	$-0.560172\pi$
$167$	−4.85670	−0.375823	−0.187911	+	0.982186i	$0.560172\pi$
$168$	0	0
$169$	−12.1722	−0.936327
$170$	−0.454904	−0.0348896
$171$	0	0
$172$	7.88325	0.601092
$173$	−25.5861	−1.94528	−0.972639	−	0.232324i	$-0.925367\pi$
$173$	−25.5861	−1.94528	−0.972639	+	0.232324i	$0.925367\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	11.0902	0.831245
$179$	15.7029	1.17369	0.586844	−	0.809700i	$-0.300370\pi$
$179$	15.7029	1.17369	0.586844	+	0.809700i	$0.300370\pi$
$180$	0	0
$181$	2.49593	0.185521	0.0927606	−	0.995688i	$-0.470431\pi$
$181$	2.49593	0.185521	0.0927606	+	0.995688i	$0.470431\pi$
$182$	0	0
$183$	0	0
$184$	8.33816	0.614697
$185$	3.58612	0.263657
$186$	0	0
$187$	−1.00000	−0.0731272
$188$	0.338158	0.0246627
$189$	0	0
$190$	1.76651	0.128156
$191$	−9.81962	−0.710523	−0.355261	−	0.934767i	$-0.615608\pi$
$191$	−9.81962	−0.710523	−0.355261	+	0.934767i	$0.615608\pi$
$192$	0	0
$193$	6.85670	0.493556	0.246778	−	0.969072i	$-0.420628\pi$
$193$	6.85670	0.493556	0.246778	+	0.969072i	$0.420628\pi$
$194$	14.5861	1.04722
$195$	0	0
$196$	0	0
$197$	−23.4694	−1.67212	−0.836062	−	0.548635i	$-0.815148\pi$
$197$	−23.4694	−1.67212	−0.836062	+	0.548635i	$0.815148\pi$
$198$	0	0
$199$	2.23349	0.158328	0.0791640	−	0.996862i	$-0.474775\pi$
$199$	2.23349	0.158328	0.0791640	+	0.996862i	$0.474775\pi$
$200$	4.79306	0.338921
$201$	0	0
$202$	8.06364	0.567356
$203$	0	0
$204$	0	0
$205$	−0.815671	−0.0569690
$206$	5.76651	0.401772
$207$	0	0
$208$	−0.909808	−0.0630838
$209$	3.88325	0.268610
$210$	0	0
$211$	−18.2624	−1.25724	−0.628619	−	0.777713i	$-0.716380\pi$
$211$	−18.2624	−1.25724	−0.628619	+	0.777713i	$0.716380\pi$
$212$	−0.909808	−0.0624859
$213$	0	0
$214$	15.6127	1.06726
$215$	3.58612	0.244572
$216$	0	0
$217$	0	0
$218$	−14.0410	−0.950978
$219$	0	0
$220$	−0.454904	−0.0306696
$221$	−0.909808	−0.0612003
$222$	0	0
$223$	−12.4959	−0.836790	−0.418395	−	0.908265i	$-0.637407\pi$
$223$	−12.4959	−0.836790	−0.418395	+	0.908265i	$0.637407\pi$
$224$	0	0
$225$	0	0
$226$	7.58612	0.504621
$227$	9.79306	0.649988	0.324994	−	0.945716i	$-0.394638\pi$
$227$	9.79306	0.649988	0.324994	+	0.945716i	$0.394638\pi$
$228$	0	0
$229$	−8.67632	−0.573347	−0.286674	−	0.958028i	$-0.592550\pi$
$229$	−8.67632	−0.573347	−0.286674	+	0.958028i	$0.592550\pi$
$230$	3.79306	0.250107
$231$	0	0
$232$	−2.79306	−0.183374
$233$	−16.7665	−1.09841	−0.549205	−	0.835688i	$-0.685069\pi$
$233$	−16.7665	−1.09841	−0.549205	+	0.835688i	$0.685069\pi$
$234$	0	0
$235$	0.153830	0.0100347
$236$	6.97345	0.453933
$237$	0	0
$238$	0	0
$239$	26.2624	1.69878	0.849388	−	0.527769i	$-0.176971\pi$
$239$	26.2624	1.69878	0.849388	+	0.527769i	$0.176971\pi$
$240$	0	0
$241$	−7.94689	−0.511904	−0.255952	−	0.966689i	$-0.582389\pi$
$241$	−7.94689	−0.511904	−0.255952	+	0.966689i	$0.582389\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	14.0410	0.898885
$245$	0	0
$246$	0	0
$247$	3.53302	0.224800
$248$	−9.58612	−0.608720
$249$	0	0
$250$	4.45490	0.281753
$251$	−3.93636	−0.248461	−0.124230	−	0.992253i	$-0.539646\pi$
$251$	−3.93636	−0.248461	−0.124230	+	0.992253i	$0.539646\pi$
$252$	0	0
$253$	8.33816	0.524216
$254$	−7.42835	−0.466096
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.0902	−0.941300	−0.470650	−	0.882320i	$-0.655981\pi$
$257$	−15.0902	−0.941300	−0.470650	+	0.882320i	$0.655981\pi$
$258$	0	0
$259$	0	0
$260$	−0.413875	−0.0256675
$261$	0	0
$262$	19.4057	1.19889
$263$	28.6232	1.76498	0.882491	−	0.470329i	$-0.155865\pi$
$263$	28.6232	1.76498	0.882491	+	0.470329i	$0.155865\pi$
$264$	0	0
$265$	−0.413875	−0.0254242
$266$	0	0
$267$	0	0
$268$	−3.79306	−0.231698
$269$	−22.9508	−1.39934	−0.699669	−	0.714468i	$-0.746669\pi$
$269$	−22.9508	−1.39934	−0.699669	+	0.714468i	$0.746669\pi$
$270$	0	0
$271$	−9.81962	−0.596499	−0.298250	−	0.954488i	$-0.596403\pi$
$271$	−9.81962	−0.596499	−0.298250	+	0.954488i	$0.596403\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	0.909808	0.0549635
$275$	4.79306	0.289033
$276$	0	0
$277$	10.7294	0.644669	0.322334	−	0.946626i	$-0.395532\pi$
$277$	10.7294	0.644669	0.322334	+	0.946626i	$0.395532\pi$
$278$	−8.37919	−0.502550
$279$	0	0
$280$	0	0
$281$	−2.58612	−0.154275	−0.0771376	−	0.997020i	$-0.524578\pi$
$281$	−2.58612	−0.154275	−0.0771376	+	0.997020i	$0.524578\pi$
$282$	0	0
$283$	20.0821	1.19375	0.596877	−	0.802333i	$-0.296408\pi$
$283$	20.0821	1.19375	0.596877	+	0.802333i	$0.296408\pi$
$284$	4.79306	0.284416
$285$	0	0
$286$	−0.909808	−0.0537981
$287$	0	0
$288$	0	0
$289$	−16.0000	−0.941176
$290$	−1.27058	−0.0746108
$291$	0	0
$292$	−10.6763	−0.624784
$293$	10.3792	0.606359	0.303179	−	0.952934i	$-0.401952\pi$
$293$	10.3792	0.606359	0.303179	+	0.952934i	$0.401952\pi$
$294$	0	0
$295$	3.17225	0.184695
$296$	−7.88325	−0.458205
$297$	0	0
$298$	4.11675	0.238477
$299$	7.58612	0.438717
$300$	0	0
$301$	0	0
$302$	6.15777	0.354340
$303$	0	0
$304$	−3.88325	−0.222720
$305$	6.38732	0.365737
$306$	0	0
$307$	−9.76651	−0.557404	−0.278702	−	0.960378i	$-0.589904\pi$
$307$	−9.76651	−0.557404	−0.278702	+	0.960378i	$0.589904\pi$
$308$	0	0
$309$	0	0
$310$	−4.36077	−0.247675
$311$	−6.51854	−0.369633	−0.184816	−	0.982773i	$-0.559169\pi$
$311$	−6.51854	−0.369633	−0.184816	+	0.982773i	$0.559169\pi$
$312$	0	0
$313$	−30.7931	−1.74053	−0.870263	−	0.492587i	$-0.836051\pi$
$313$	−30.7931	−1.74053	−0.870263	+	0.492587i	$0.836051\pi$
$314$	−17.6498	−0.996034
$315$	0	0
$316$	5.36471	0.301789
$317$	27.6272	1.55170	0.775848	−	0.630920i	$-0.217322\pi$
$317$	27.6272	1.55170	0.775848	+	0.630920i	$0.217322\pi$
$318$	0	0
$319$	−2.79306	−0.156381
$320$	0.454904	0.0254299
$321$	0	0
$322$	0	0
$323$	−3.88325	−0.216070
$324$	0	0
$325$	4.36077	0.241892
$326$	11.6127	0.643167
$327$	0	0
$328$	1.79306	0.0990053
$329$	0	0
$330$	0	0
$331$	−28.9653	−1.59208	−0.796039	−	0.605246i	$-0.793075\pi$
$331$	−28.9653	−1.59208	−0.796039	+	0.605246i	$0.793075\pi$
$332$	1.97345	0.108307
$333$	0	0
$334$	4.85670	0.265747
$335$	−1.72548	−0.0942730
$336$	0	0
$337$	−9.03708	−0.492281	−0.246141	−	0.969234i	$-0.579163\pi$
$337$	−9.03708	−0.492281	−0.246141	+	0.969234i	$0.579163\pi$
$338$	12.1722	0.662083
$339$	0	0
$340$	0.454904	0.0246706
$341$	−9.58612	−0.519118
$342$	0	0
$343$	0	0
$344$	−7.88325	−0.425037
$345$	0	0
$346$	25.5861	1.37552
$347$	21.6127	1.16023	0.580115	−	0.814535i	$-0.303008\pi$
$347$	21.6127	1.16023	0.580115	+	0.814535i	$0.303008\pi$
$348$	0	0
$349$	−14.2745	−0.764098	−0.382049	−	0.924142i	$-0.624781\pi$
$349$	−14.2745	−0.764098	−0.382049	+	0.924142i	$0.624781\pi$
$350$	0	0
$351$	0	0
$352$	1.00000	0.0533002
$353$	−28.4959	−1.51669	−0.758343	−	0.651856i	$-0.773991\pi$
$353$	−28.4959	−1.51669	−0.758343	+	0.651856i	$0.773991\pi$
$354$	0	0
$355$	2.18038	0.115723
$356$	−11.0902	−0.587779
$357$	0	0
$358$	−15.7029	−0.829922
$359$	13.6392	0.719851	0.359926	−	0.932981i	$-0.382802\pi$
$359$	13.6392	0.719851	0.359926	+	0.932981i	$0.382802\pi$
$360$	0	0
$361$	−3.92034	−0.206334
$362$	−2.49593	−0.131183
$363$	0	0
$364$	0	0
$365$	−4.85670	−0.254211
$366$	0	0
$367$	−22.7294	−1.18647	−0.593233	−	0.805031i	$-0.702149\pi$
$367$	−22.7294	−1.18647	−0.593233	+	0.805031i	$0.702149\pi$
$368$	−8.33816	−0.434657
$369$	0	0
$370$	−3.58612	−0.186434
$371$	0	0
$372$	0	0
$373$	10.4018	0.538585	0.269292	−	0.963058i	$-0.413210\pi$
$373$	10.4018	0.538585	0.269292	+	0.963058i	$0.413210\pi$
$374$	1.00000	0.0517088
$375$	0	0
$376$	−0.338158	−0.0174392
$377$	−2.54115	−0.130876
$378$	0	0
$379$	7.61268	0.391037	0.195519	−	0.980700i	$-0.437361\pi$
$379$	7.61268	0.391037	0.195519	+	0.980700i	$0.437361\pi$
$380$	−1.76651	−0.0906200
$381$	0	0
$382$	9.81962	0.502415
$383$	−22.1457	−1.13159	−0.565796	−	0.824545i	$-0.691431\pi$
$383$	−22.1457	−1.13159	−0.565796	+	0.824545i	$0.691431\pi$
$384$	0	0
$385$	0	0
$386$	−6.85670	−0.348997
$387$	0	0
$388$	−14.5861	−0.740498
$389$	3.54510	0.179743	0.0898717	−	0.995953i	$-0.471354\pi$
$389$	3.54510	0.179743	0.0898717	+	0.995953i	$0.471354\pi$
$390$	0	0
$391$	−8.33816	−0.421679
$392$	0	0
$393$	0	0
$394$	23.4694	1.18237
$395$	2.44043	0.122791
$396$	0	0
$397$	2.55957	0.128461	0.0642306	−	0.997935i	$-0.479541\pi$
$397$	2.55957	0.128461	0.0642306	+	0.997935i	$0.479541\pi$
$398$	−2.23349	−0.111955
$399$	0	0
$400$	−4.79306	−0.239653
$401$	−10.4959	−0.524142	−0.262071	−	0.965049i	$-0.584405\pi$
$401$	−10.4959	−0.524142	−0.262071	+	0.965049i	$0.584405\pi$
$402$	0	0
$403$	−8.72153	−0.434451
$404$	−8.06364	−0.401181
$405$	0	0
$406$	0	0
$407$	−7.88325	−0.390758
$408$	0	0
$409$	−15.8486	−0.783661	−0.391831	−	0.920037i	$-0.628158\pi$
$409$	−15.8486	−0.783661	−0.391831	+	0.920037i	$0.628158\pi$
$410$	0.815671	0.0402831
$411$	0	0
$412$	−5.76651	−0.284095
$413$	0	0
$414$	0	0
$415$	0.897729	0.0440678
$416$	0.909808	0.0446070
$417$	0	0
$418$	−3.88325	−0.189936
$419$	−9.52249	−0.465204	−0.232602	−	0.972572i	$-0.574724\pi$
$419$	−9.52249	−0.465204	−0.232602	+	0.972572i	$0.574724\pi$
$420$	0	0
$421$	−0.530621	−0.0258609	−0.0129305	−	0.999916i	$-0.504116\pi$
$421$	−0.530621	−0.0258609	−0.0129305	+	0.999916i	$0.504116\pi$
$422$	18.2624	0.889002
$423$	0	0
$424$	0.909808	0.0441842
$425$	−4.79306	−0.232498
$426$	0	0
$427$	0	0
$428$	−15.6127	−0.754667
$429$	0	0
$430$	−3.58612	−0.172938
$431$	−20.8036	−1.00207	−0.501037	−	0.865426i	$-0.667048\pi$
$431$	−20.8036	−1.00207	−0.501037	+	0.865426i	$0.667048\pi$
$432$	0	0
$433$	4.40574	0.211726	0.105863	−	0.994381i	$-0.466239\pi$
$433$	4.40574	0.211726	0.105863	+	0.994381i	$0.466239\pi$
$434$	0	0
$435$	0	0
$436$	14.0410	0.672443
$437$	32.3792	1.54891
$438$	0	0
$439$	15.1949	0.725211	0.362606	−	0.931943i	$-0.381887\pi$
$439$	15.1949	0.725211	0.362606	+	0.931943i	$0.381887\pi$
$440$	0.454904	0.0216867
$441$	0	0
$442$	0.909808	0.0432752
$443$	11.3873	0.541028	0.270514	−	0.962716i	$-0.412806\pi$
$443$	11.3873	0.541028	0.270514	+	0.962716i	$0.412806\pi$
$444$	0	0
$445$	−5.04497	−0.239155
$446$	12.4959	0.591700
$447$	0	0
$448$	0	0
$449$	24.0821	1.13650	0.568251	−	0.822855i	$-0.307620\pi$
$449$	24.0821	1.13650	0.568251	+	0.822855i	$0.307620\pi$
$450$	0	0
$451$	1.79306	0.0844320
$452$	−7.58612	−0.356821
$453$	0	0
$454$	−9.79306	−0.459611
$455$	0	0
$456$	0	0
$457$	−29.5330	−1.38150	−0.690748	−	0.723095i	$-0.742719\pi$
$457$	−29.5330	−1.38150	−0.690748	+	0.723095i	$0.742719\pi$
$458$	8.67632	0.405418
$459$	0	0
$460$	−3.79306	−0.176852
$461$	4.14569	0.193084	0.0965421	−	0.995329i	$-0.469222\pi$
$461$	4.14569	0.193084	0.0965421	+	0.995329i	$0.469222\pi$
$462$	0	0
$463$	−9.32368	−0.433308	−0.216654	−	0.976248i	$-0.569514\pi$
$463$	−9.32368	−0.433308	−0.216654	+	0.976248i	$0.569514\pi$
$464$	2.79306	0.129665
$465$	0	0
$466$	16.7665	0.776693
$467$	36.5596	1.69178	0.845888	−	0.533361i	$-0.179071\pi$
$467$	36.5596	1.69178	0.845888	+	0.533361i	$0.179071\pi$
$468$	0	0
$469$	0	0
$470$	−0.153830	−0.00709563
$471$	0	0
$472$	−6.97345	−0.320979
$473$	−7.88325	−0.362472
$474$	0	0
$475$	18.6127	0.854008
$476$	0	0
$477$	0	0
$478$	−26.2624	−1.20122
$479$	0.729425	0.0333283	0.0166641	−	0.999861i	$-0.494695\pi$
$479$	0.729425	0.0333283	0.0166641	+	0.999861i	$0.494695\pi$
$480$	0	0
$481$	−7.17225	−0.327026
$482$	7.94689	0.361971
$483$	0	0
$484$	1.00000	0.0454545
$485$	−6.63529	−0.301293
$486$	0	0
$487$	−36.8567	−1.67014	−0.835068	−	0.550146i	$-0.814572\pi$
$487$	−36.8567	−1.67014	−0.835068	+	0.550146i	$0.814572\pi$
$488$	−14.0410	−0.635608
$489$	0	0
$490$	0	0
$491$	−25.5065	−1.15109	−0.575545	−	0.817770i	$-0.695210\pi$
$491$	−25.5065	−1.15109	−0.575545	+	0.817770i	$0.695210\pi$
$492$	0	0
$493$	2.79306	0.125793
$494$	−3.53302	−0.158958
$495$	0	0
$496$	9.58612	0.430430
$497$	0	0
$498$	0	0
$499$	−4.12728	−0.184762	−0.0923811	−	0.995724i	$-0.529448\pi$
$499$	−4.12728	−0.184762	−0.0923811	+	0.995724i	$0.529448\pi$
$500$	−4.45490	−0.199229
$501$	0	0
$502$	3.93636	0.175688
$503$	−6.85670	−0.305725	−0.152863	−	0.988247i	$-0.548849\pi$
$503$	−6.85670	−0.305725	−0.152863	+	0.988247i	$0.548849\pi$
$504$	0	0
$505$	−3.66818	−0.163232
$506$	−8.33816	−0.370676
$507$	0	0
$508$	7.42835	0.329580
$509$	16.4428	0.728815	0.364408	−	0.931240i	$-0.381271\pi$
$509$	16.4428	0.728815	0.364408	+	0.931240i	$0.381271\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	15.0902	0.665600
$515$	−2.62321	−0.115592
$516$	0	0
$517$	−0.338158	−0.0148722
$518$	0	0
$519$	0	0
$520$	0.413875	0.0181496
$521$	−5.63923	−0.247059	−0.123530	−	0.992341i	$-0.539421\pi$
$521$	−5.63923	−0.247059	−0.123530	+	0.992341i	$0.539421\pi$
$522$	0	0
$523$	4.08206	0.178496	0.0892480	−	0.996009i	$-0.471554\pi$
$523$	4.08206	0.178496	0.0892480	+	0.996009i	$0.471554\pi$
$524$	−19.4057	−0.847744
$525$	0	0
$526$	−28.6232	−1.24803
$527$	9.58612	0.417578
$528$	0	0
$529$	46.5249	2.02282
$530$	0.413875	0.0179776
$531$	0	0
$532$	0	0
$533$	1.63134	0.0706613
$534$	0	0
$535$	−7.10227	−0.307058
$536$	3.79306	0.163835
$537$	0	0
$538$	22.9508	0.989481
$539$	0	0
$540$	0	0
$541$	−4.89773	−0.210570	−0.105285	−	0.994442i	$-0.533575\pi$
$541$	−4.89773	−0.210570	−0.105285	+	0.994442i	$0.533575\pi$
$542$	9.81962	0.421789
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	6.38732	0.273603
$546$	0	0
$547$	21.4694	0.917964	0.458982	−	0.888445i	$-0.348214\pi$
$547$	21.4694	0.917964	0.458982	+	0.888445i	$0.348214\pi$
$548$	−0.909808	−0.0388651
$549$	0	0
$550$	−4.79306	−0.204377
$551$	−10.8462	−0.462062
$552$	0	0
$553$	0	0
$554$	−10.7294	−0.455850
$555$	0	0
$556$	8.37919	0.355357
$557$	39.4694	1.67237	0.836186	−	0.548447i	$-0.184781\pi$
$557$	39.4694	1.67237	0.836186	+	0.548447i	$0.184781\pi$
$558$	0	0
$559$	−7.17225	−0.303354
$560$	0	0
$561$	0	0
$562$	2.58612	0.109089
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	−3.45096	−0.145183
$566$	−20.0821	−0.844112
$567$	0	0
$568$	−4.79306	−0.201112
$569$	−34.1988	−1.43369	−0.716844	−	0.697233i	$-0.754414\pi$
$569$	−34.1988	−1.43369	−0.716844	+	0.697233i	$0.754414\pi$
$570$	0	0
$571$	14.7931	0.619070	0.309535	−	0.950888i	$-0.399827\pi$
$571$	14.7931	0.619070	0.309535	+	0.950888i	$0.399827\pi$
$572$	0.909808	0.0380410
$573$	0	0
$574$	0	0
$575$	39.9653	1.66667
$576$	0	0
$577$	7.19880	0.299690	0.149845	−	0.988709i	$-0.452123\pi$
$577$	7.19880	0.299690	0.149845	+	0.988709i	$0.452123\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	1.27058	0.0527578
$581$	0	0
$582$	0	0
$583$	0.909808	0.0376804
$584$	10.6763	0.441789
$585$	0	0
$586$	−10.3792	−0.428760
$587$	−44.4959	−1.83654	−0.918272	−	0.395951i	$-0.870415\pi$
$587$	−44.4959	−1.83654	−0.918272	+	0.395951i	$0.870415\pi$
$588$	0	0
$589$	−37.2254	−1.53384
$590$	−3.17225	−0.130599
$591$	0	0
$592$	7.88325	0.324000
$593$	17.9653	0.737747	0.368873	−	0.929480i	$-0.379744\pi$
$593$	17.9653	0.737747	0.368873	+	0.929480i	$0.379744\pi$
$594$	0	0
$595$	0	0
$596$	−4.11675	−0.168628
$597$	0	0
$598$	−7.58612	−0.310220
$599$	25.8075	1.05447	0.527234	−	0.849720i	$-0.323229\pi$
$599$	25.8075	1.05447	0.527234	+	0.849720i	$0.323229\pi$
$600$	0	0
$601$	−10.4139	−0.424791	−0.212395	−	0.977184i	$-0.568126\pi$
$601$	−10.4139	−0.424791	−0.212395	+	0.977184i	$0.568126\pi$
$602$	0	0
$603$	0	0
$604$	−6.15777	−0.250556
$605$	0.454904	0.0184945
$606$	0	0
$607$	8.89773	0.361148	0.180574	−	0.983561i	$-0.442205\pi$
$607$	8.89773	0.361148	0.180574	+	0.983561i	$0.442205\pi$
$608$	3.88325	0.157487
$609$	0	0
$610$	−6.38732	−0.258615
$611$	−0.307659	−0.0124466
$612$	0	0
$613$	−23.6272	−0.954292	−0.477146	−	0.878824i	$-0.658329\pi$
$613$	−23.6272	−0.954292	−0.477146	+	0.878824i	$0.658329\pi$
$614$	9.76651	0.394144
$615$	0	0
$616$	0	0
$617$	35.8486	1.44321	0.721604	−	0.692306i	$-0.243405\pi$
$617$	35.8486	1.44321	0.721604	+	0.692306i	$0.243405\pi$
$618$	0	0
$619$	−27.9203	−1.12221	−0.561107	−	0.827744i	$-0.689624\pi$
$619$	−27.9203	−1.12221	−0.561107	+	0.827744i	$0.689624\pi$
$620$	4.36077	0.175133
$621$	0	0
$622$	6.51854	0.261370
$623$	0	0
$624$	0	0
$625$	21.9388	0.877550
$626$	30.7931	1.23074
$627$	0	0
$628$	17.6498	0.704302
$629$	7.88325	0.314326
$630$	0	0
$631$	43.6682	1.73840	0.869201	−	0.494458i	$-0.164633\pi$
$631$	43.6682	1.73840	0.869201	+	0.494458i	$0.164633\pi$
$632$	−5.36471	−0.213397
$633$	0	0
$634$	−27.6272	−1.09721
$635$	3.37919	0.134099
$636$	0	0
$637$	0	0
$638$	2.79306	0.110578
$639$	0	0
$640$	−0.454904	−0.0179817
$641$	19.7665	0.780730	0.390365	−	0.920660i	$-0.372349\pi$
$641$	19.7665	0.780730	0.390365	+	0.920660i	$0.372349\pi$
$642$	0	0
$643$	−24.7584	−0.976375	−0.488187	−	0.872739i	$-0.662342\pi$
$643$	−24.7584	−0.976375	−0.488187	+	0.872739i	$0.662342\pi$
$644$	0	0
$645$	0	0
$646$	3.88325	0.152785
$647$	31.0781	1.22181	0.610903	−	0.791705i	$-0.290806\pi$
$647$	31.0781	1.22181	0.610903	+	0.791705i	$0.290806\pi$
$648$	0	0
$649$	−6.97345	−0.273732
$650$	−4.36077	−0.171043
$651$	0	0
$652$	−11.6127	−0.454788
$653$	−22.7173	−0.888998	−0.444499	−	0.895779i	$-0.646618\pi$
$653$	−22.7173	−0.888998	−0.444499	+	0.895779i	$0.646618\pi$
$654$	0	0
$655$	−8.82775	−0.344929
$656$	−1.79306	−0.0700073
$657$	0	0
$658$	0	0
$659$	−43.6127	−1.69891	−0.849454	−	0.527662i	$-0.823069\pi$
$659$	−43.6127	−1.69891	−0.849454	+	0.527662i	$0.823069\pi$
$660$	0	0
$661$	7.93636	0.308689	0.154344	−	0.988017i	$-0.450673\pi$
$661$	7.93636	0.308689	0.154344	+	0.988017i	$0.450673\pi$
$662$	28.9653	1.12577
$663$	0	0
$664$	−1.97345	−0.0765846
$665$	0	0
$666$	0	0
$667$	−23.2890	−0.901753
$668$	−4.85670	−0.187911
$669$	0	0
$670$	1.72548	0.0666611
$671$	−14.0410	−0.542048
$672$	0	0
$673$	36.9388	1.42388	0.711942	−	0.702238i	$-0.247816\pi$
$673$	36.9388	1.42388	0.711942	+	0.702238i	$0.247816\pi$
$674$	9.03708	0.348095
$675$	0	0
$676$	−12.1722	−0.468163
$677$	−39.6498	−1.52386	−0.761932	−	0.647657i	$-0.775749\pi$
$677$	−39.6498	−1.52386	−0.761932	+	0.647657i	$0.775749\pi$
$678$	0	0
$679$	0	0
$680$	−0.454904	−0.0174448
$681$	0	0
$682$	9.58612	0.367072
$683$	45.5514	1.74298	0.871489	−	0.490416i	$-0.163155\pi$
$683$	45.5514	1.74298	0.871489	+	0.490416i	$0.163155\pi$
$684$	0	0
$685$	−0.413875	−0.0158134
$686$	0	0
$687$	0	0
$688$	7.88325	0.300546
$689$	0.827751	0.0315348
$690$	0	0
$691$	19.1988	0.730357	0.365178	−	0.930938i	$-0.381008\pi$
$691$	19.1988	0.730357	0.365178	+	0.930938i	$0.381008\pi$
$692$	−25.5861	−0.972639
$693$	0	0
$694$	−21.6127	−0.820406
$695$	3.81173	0.144587
$696$	0	0
$697$	−1.79306	−0.0679171
$698$	14.2745	0.540299
$699$	0	0
$700$	0	0
$701$	−23.2890	−0.879613	−0.439807	−	0.898093i	$-0.644953\pi$
$701$	−23.2890	−0.879613	−0.439807	+	0.898093i	$0.644953\pi$
$702$	0	0
$703$	−30.6127	−1.15458
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	28.4959	1.07246
$707$	0	0
$708$	0	0
$709$	−20.7931	−0.780900	−0.390450	−	0.920624i	$-0.627681\pi$
$709$	−20.7931	−0.780900	−0.390450	+	0.920624i	$0.627681\pi$
$710$	−2.18038	−0.0818283
$711$	0	0
$712$	11.0902	0.415623
$713$	−79.9306	−2.99343
$714$	0	0
$715$	0.413875	0.0154781
$716$	15.7029	0.586844
$717$	0	0
$718$	−13.6392	−0.509012
$719$	−5.48146	−0.204424	−0.102212	−	0.994763i	$-0.532592\pi$
$719$	−5.48146	−0.204424	−0.102212	+	0.994763i	$0.532592\pi$
$720$	0	0
$721$	0	0
$722$	3.92034	0.145900
$723$	0	0
$724$	2.49593	0.0927606
$725$	−13.3873	−0.497193
$726$	0	0
$727$	−25.5330	−0.946967	−0.473484	−	0.880803i	$-0.657004\pi$
$727$	−25.5330	−0.946967	−0.473484	+	0.880803i	$0.657004\pi$
$728$	0	0
$729$	0	0
$730$	4.85670	0.179755
$731$	7.88325	0.291573
$732$	0	0
$733$	−16.2214	−0.599152	−0.299576	−	0.954073i	$-0.596845\pi$
$733$	−16.2214	−0.599152	−0.299576	+	0.954073i	$0.596845\pi$
$734$	22.7294	0.838958
$735$	0	0
$736$	8.33816	0.307349
$737$	3.79306	0.139719
$738$	0	0
$739$	18.7584	0.690038	0.345019	−	0.938596i	$-0.387872\pi$
$739$	18.7584	0.690038	0.345019	+	0.938596i	$0.387872\pi$
$740$	3.58612	0.131829
$741$	0	0
$742$	0	0
$743$	44.6763	1.63902	0.819508	−	0.573068i	$-0.194247\pi$
$743$	44.6763	1.63902	0.819508	+	0.573068i	$0.194247\pi$
$744$	0	0
$745$	−1.87272	−0.0686113
$746$	−10.4018	−0.380837
$747$	0	0
$748$	−1.00000	−0.0365636
$749$	0	0
$750$	0	0
$751$	−2.96292	−0.108118	−0.0540592	−	0.998538i	$-0.517216\pi$
$751$	−2.96292	−0.108118	−0.0540592	+	0.998538i	$0.517216\pi$
$752$	0.338158	0.0123314
$753$	0	0
$754$	2.54115	0.0925433
$755$	−2.80120	−0.101946
$756$	0	0
$757$	26.1988	0.952212	0.476106	−	0.879388i	$-0.342048\pi$
$757$	26.1988	0.952212	0.476106	+	0.879388i	$0.342048\pi$
$758$	−7.61268	−0.276505
$759$	0	0
$760$	1.76651	0.0640780
$761$	25.7931	0.934998	0.467499	−	0.883994i	$-0.345155\pi$
$761$	25.7931	0.934998	0.467499	+	0.883994i	$0.345155\pi$
$762$	0	0
$763$	0	0
$764$	−9.81962	−0.355261
$765$	0	0
$766$	22.1457	0.800156
$767$	−6.34450	−0.229087
$768$	0	0
$769$	37.2012	1.34151	0.670755	−	0.741679i	$-0.265970\pi$
$769$	37.2012	1.34151	0.670755	+	0.741679i	$0.265970\pi$
$770$	0	0
$771$	0	0
$772$	6.85670	0.246778
$773$	−40.7705	−1.46641	−0.733206	−	0.680007i	$-0.761977\pi$
$773$	−40.7705	−1.46641	−0.733206	+	0.680007i	$0.761977\pi$
$774$	0	0
$775$	−45.9469	−1.65046
$776$	14.5861	0.523611
$777$	0	0
$778$	−3.54510	−0.127098
$779$	6.96292	0.249472
$780$	0	0
$781$	−4.79306	−0.171509
$782$	8.33816	0.298172
$783$	0	0
$784$	0	0
$785$	8.02895	0.286565
$786$	0	0
$787$	−54.5885	−1.94587	−0.972935	−	0.231078i	$-0.925775\pi$
$787$	−54.5885	−1.94587	−0.972935	+	0.231078i	$0.925775\pi$
$788$	−23.4694	−0.836062
$789$	0	0
$790$	−2.44043	−0.0868266
$791$	0	0
$792$	0	0
$793$	−12.7746	−0.453641
$794$	−2.55957	−0.0908358
$795$	0	0
$796$	2.23349	0.0791640
$797$	−16.7705	−0.594040	−0.297020	−	0.954871i	$-0.595993\pi$
$797$	−16.7705	−0.594040	−0.297020	+	0.954871i	$0.595993\pi$
$798$	0	0
$799$	0.338158	0.0119632
$800$	4.79306	0.169460
$801$	0	0
$802$	10.4959	0.370624
$803$	10.6763	0.376759
$804$	0	0
$805$	0	0
$806$	8.72153	0.307203
$807$	0	0
$808$	8.06364	0.283678
$809$	10.2069	0.358857	0.179428	−	0.983771i	$-0.442575\pi$
$809$	10.2069	0.358857	0.179428	+	0.983771i	$0.442575\pi$
$810$	0	0
$811$	28.4139	0.997746	0.498873	−	0.866675i	$-0.333747\pi$
$811$	28.4139	0.997746	0.498873	+	0.866675i	$0.333747\pi$
$812$	0	0
$813$	0	0
$814$	7.88325	0.276308
$815$	−5.28266	−0.185043
$816$	0	0
$817$	−30.6127	−1.07100
$818$	15.8486	0.554132
$819$	0	0
$820$	−0.815671	−0.0284845
$821$	3.40574	0.118861	0.0594306	−	0.998232i	$-0.481072\pi$
$821$	3.40574	0.118861	0.0594306	+	0.998232i	$0.481072\pi$
$822$	0	0
$823$	−38.0289	−1.32561	−0.662803	−	0.748794i	$-0.730633\pi$
$823$	−38.0289	−1.32561	−0.662803	+	0.748794i	$0.730633\pi$
$824$	5.76651	0.200886
$825$	0	0
$826$	0	0
$827$	12.9653	0.450848	0.225424	−	0.974261i	$-0.427623\pi$
$827$	12.9653	0.450848	0.225424	+	0.974261i	$0.427623\pi$
$828$	0	0
$829$	−51.8833	−1.80198	−0.900990	−	0.433840i	$-0.857158\pi$
$829$	−51.8833	−1.80198	−0.900990	+	0.433840i	$0.857158\pi$
$830$	−0.897729	−0.0311606
$831$	0	0
$832$	−0.909808	−0.0315419
$833$	0	0
$834$	0	0
$835$	−2.20933	−0.0764571
$836$	3.88325	0.134305
$837$	0	0
$838$	9.52249	0.328949
$839$	−50.2504	−1.73484	−0.867418	−	0.497581i	$-0.834222\pi$
$839$	−50.2504	−1.73484	−0.867418	+	0.497581i	$0.834222\pi$
$840$	0	0
$841$	−21.1988	−0.730993
$842$	0.530621	0.0182864
$843$	0	0
$844$	−18.2624	−0.628619
$845$	−5.53721	−0.190486
$846$	0	0
$847$	0	0
$848$	−0.909808	−0.0312429
$849$	0	0
$850$	4.79306	0.164401
$851$	−65.7318	−2.25326
$852$	0	0
$853$	−13.3647	−0.457599	−0.228800	−	0.973474i	$-0.573480\pi$
$853$	−13.3647	−0.457599	−0.228800	+	0.973474i	$0.573480\pi$
$854$	0	0
$855$	0	0
$856$	15.6127	0.533630
$857$	26.3526	0.900189	0.450094	−	0.892981i	$-0.351390\pi$
$857$	26.3526	0.900189	0.450094	+	0.892981i	$0.351390\pi$
$858$	0	0
$859$	−1.97345	−0.0673331	−0.0336666	−	0.999433i	$-0.510718\pi$
$859$	−1.97345	−0.0673331	−0.0336666	+	0.999433i	$0.510718\pi$
$860$	3.58612	0.122286
$861$	0	0
$862$	20.8036	0.708573
$863$	49.8075	1.69547	0.847734	−	0.530421i	$-0.177966\pi$
$863$	49.8075	1.69547	0.847734	+	0.530421i	$0.177966\pi$
$864$	0	0
$865$	−11.6392	−0.395746
$866$	−4.40574	−0.149713
$867$	0	0
$868$	0	0
$869$	−5.36471	−0.181985
$870$	0	0
$871$	3.45096	0.116931
$872$	−14.0410	−0.475489
$873$	0	0
$874$	−32.3792	−1.09524
$875$	0	0
$876$	0	0
$877$	−24.5370	−0.828554	−0.414277	−	0.910151i	$-0.635966\pi$
$877$	−24.5370	−0.828554	−0.414277	+	0.910151i	$0.635966\pi$
$878$	−15.1949	−0.512802
$879$	0	0
$880$	−0.454904	−0.0153348
$881$	11.5861	0.390346	0.195173	−	0.980769i	$-0.437473\pi$
$881$	11.5861	0.390346	0.195173	+	0.980769i	$0.437473\pi$
$882$	0	0
$883$	−44.1376	−1.48535	−0.742674	−	0.669654i	$-0.766443\pi$
$883$	−44.1376	−1.48535	−0.742674	+	0.669654i	$0.766443\pi$
$884$	−0.909808	−0.0306002
$885$	0	0
$886$	−11.3873	−0.382565
$887$	−20.0289	−0.672506	−0.336253	−	0.941772i	$-0.609160\pi$
$887$	−20.0289	−0.672506	−0.336253	+	0.941772i	$0.609160\pi$
$888$	0	0
$889$	0	0
$890$	5.04497	0.169108
$891$	0	0
$892$	−12.4959	−0.418395
$893$	−1.31315	−0.0439430
$894$	0	0
$895$	7.14330	0.238774
$896$	0	0
$897$	0	0
$898$	−24.0821	−0.803629
$899$	26.7746	0.892984
$900$	0	0
$901$	−0.909808	−0.0303101
$902$	−1.79306	−0.0597024
$903$	0	0
$904$	7.58612	0.252311
$905$	1.13541	0.0377423
$906$	0	0
$907$	−17.9203	−0.595035	−0.297518	−	0.954716i	$-0.596159\pi$
$907$	−17.9203	−0.595035	−0.297518	+	0.954716i	$0.596159\pi$
$908$	9.79306	0.324994
$909$	0	0
$910$	0	0
$911$	−13.9695	−0.462830	−0.231415	−	0.972855i	$-0.574336\pi$
$911$	−13.9695	−0.462830	−0.231415	+	0.972855i	$0.574336\pi$
$912$	0	0
$913$	−1.97345	−0.0653115
$914$	29.5330	0.976865
$915$	0	0
$916$	−8.67632	−0.286674
$917$	0	0
$918$	0	0
$919$	−42.9614	−1.41716	−0.708582	−	0.705628i	$-0.750665\pi$
$919$	−42.9614	−1.41716	−0.708582	+	0.705628i	$0.750665\pi$
$920$	3.79306	0.125054
$921$	0	0
$922$	−4.14569	−0.136531
$923$	−4.36077	−0.143536
$924$	0	0
$925$	−37.7849	−1.24236
$926$	9.32368	0.306395
$927$	0	0
$928$	−2.79306	−0.0916868
$929$	−25.6682	−0.842146	−0.421073	−	0.907027i	$-0.638346\pi$
$929$	−25.6682	−0.842146	−0.421073	+	0.907027i	$0.638346\pi$
$930$	0	0
$931$	0	0
$932$	−16.7665	−0.549205
$933$	0	0
$934$	−36.5596	−1.19627
$935$	−0.454904	−0.0148770
$936$	0	0
$937$	−26.7584	−0.874158	−0.437079	−	0.899423i	$-0.643987\pi$
$937$	−26.7584	−0.874158	−0.437079	+	0.899423i	$0.643987\pi$
$938$	0	0
$939$	0	0
$940$	0.153830	0.00501737
$941$	−15.6208	−0.509224	−0.254612	−	0.967043i	$-0.581948\pi$
$941$	−15.6208	−0.509224	−0.254612	+	0.967043i	$0.581948\pi$
$942$	0	0
$943$	14.9508	0.486866
$944$	6.97345	0.226966
$945$	0	0
$946$	7.88325	0.256307
$947$	−19.7849	−0.642924	−0.321462	−	0.946923i	$-0.604174\pi$
$947$	−19.7849	−0.642924	−0.321462	+	0.946923i	$0.604174\pi$
$948$	0	0
$949$	9.71340	0.315310
$950$	−18.6127	−0.603875
$951$	0	0
$952$	0	0
$953$	−43.5065	−1.40931	−0.704656	−	0.709549i	$-0.748899\pi$
$953$	−43.5065	−1.40931	−0.704656	+	0.709549i	$0.748899\pi$
$954$	0	0
$955$	−4.46698	−0.144548
$956$	26.2624	0.849388
$957$	0	0
$958$	−0.729425	−0.0235666
$959$	0	0
$960$	0	0
$961$	60.8938	1.96432
$962$	7.17225	0.231243
$963$	0	0
$964$	−7.94689	−0.255952
$965$	3.11914	0.100409
$966$	0	0
$967$	34.5185	1.11004	0.555021	−	0.831837i	$-0.312710\pi$
$967$	34.5185	1.11004	0.555021	+	0.831837i	$0.312710\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	6.63529	0.213046
$971$	−44.9388	−1.44215	−0.721077	−	0.692855i	$-0.756352\pi$
$971$	−44.9388	−1.44215	−0.721077	+	0.692855i	$0.756352\pi$
$972$	0	0
$973$	0	0
$974$	36.8567	1.18096
$975$	0	0
$976$	14.0410	0.449442
$977$	−32.5249	−1.04056	−0.520282	−	0.853995i	$-0.674173\pi$
$977$	−32.5249	−1.04056	−0.520282	+	0.853995i	$0.674173\pi$
$978$	0	0
$979$	11.0902	0.354444
$980$	0	0
$981$	0	0
$982$	25.5065	0.813944
$983$	−47.1949	−1.50528	−0.752641	−	0.658431i	$-0.771220\pi$
$983$	−47.1949	−1.50528	−0.752641	+	0.658431i	$0.771220\pi$
$984$	0	0
$985$	−10.6763	−0.340176
$986$	−2.79306	−0.0889492
$987$	0	0
$988$	3.53302	0.112400
$989$	−65.7318	−2.09015
$990$	0	0
$991$	12.5490	0.398633	0.199317	−	0.979935i	$-0.436128\pi$
$991$	12.5490	0.398633	0.199317	+	0.979935i	$0.436128\pi$
$992$	−9.58612	−0.304360
$993$	0	0
$994$	0	0
$995$	1.01602	0.0322101
$996$	0	0
$997$	−40.9098	−1.29563	−0.647813	−	0.761799i	$-0.724316\pi$
$997$	−40.9098	−1.29563	−0.647813	+	0.761799i	$0.724316\pi$
$998$	4.12728	0.130647
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.du.1.2		3
3.2	odd	2		3234.2.a.bg.1.2		3
7.3	odd	6		1386.2.k.w.793.2		6
7.5	odd	6		1386.2.k.w.991.2		6
7.6	odd	2		9702.2.a.dt.1.2		3
21.5	even	6		462.2.i.f.67.2	✓	6
21.17	even	6		462.2.i.f.331.2	yes	6
21.20	even	2		3234.2.a.bi.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.f.67.2	✓	6	21.5	even	6
462.2.i.f.331.2	yes	6	21.17	even	6
1386.2.k.w.793.2		6	7.3	odd	6
1386.2.k.w.991.2		6	7.5	odd	6
3234.2.a.bg.1.2		3	3.2	odd	2
3234.2.a.bi.1.2		3	21.20	even	2
9702.2.a.dt.1.2		3	7.6	odd	2
9702.2.a.du.1.2		3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.454904 q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.454904 q^{5} -1.00000 q^{8} -0.454904 q^{10} -1.00000 q^{11} -0.909808 q^{13} +1.00000 q^{16} +1.00000 q^{17} -3.88325 q^{19} +0.454904 q^{20} +1.00000 q^{22} -8.33816 q^{23} -4.79306 q^{25} +0.909808 q^{26} +2.79306 q^{29} +9.58612 q^{31} -1.00000 q^{32} -1.00000 q^{34} +7.88325 q^{37} +3.88325 q^{38} -0.454904 q^{40} -1.79306 q^{41} +7.88325 q^{43} -1.00000 q^{44} +8.33816 q^{46} +0.338158 q^{47} +4.79306 q^{50} -0.909808 q^{52} -0.909808 q^{53} -0.454904 q^{55} -2.79306 q^{58} +6.97345 q^{59} +14.0410 q^{61} -9.58612 q^{62} +1.00000 q^{64} -0.413875 q^{65} -3.79306 q^{67} +1.00000 q^{68} +4.79306 q^{71} -10.6763 q^{73} -7.88325 q^{74} -3.88325 q^{76} +5.36471 q^{79} +0.454904 q^{80} +1.79306 q^{82} +1.97345 q^{83} +0.454904 q^{85} -7.88325 q^{86} +1.00000 q^{88} -11.0902 q^{89} -8.33816 q^{92} -0.338158 q^{94} -1.76651 q^{95} -14.5861 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} - 3 q^{11} + 3 q^{16} + 3 q^{17} + 3 q^{19} + 3 q^{22} - 9 q^{23} + 3 q^{25} - 9 q^{29} - 6 q^{31} - 3 q^{32} - 3 q^{34} + 9 q^{37} - 3 q^{38} + 12 q^{41} + 9 q^{43} - 3 q^{44} + 9 q^{46} - 15 q^{47} - 3 q^{50} + 9 q^{58} + 9 q^{59} + 6 q^{61} + 6 q^{62} + 3 q^{64} - 36 q^{65} + 6 q^{67} + 3 q^{68} - 3 q^{71} - 9 q^{74} + 3 q^{76} + 12 q^{79} - 12 q^{82} - 6 q^{83} - 9 q^{86} + 3 q^{88} - 36 q^{89} - 9 q^{92} + 15 q^{94} + 24 q^{95} - 9 q^{97}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 - 3 * q^11 + 3 * q^16 + 3 * q^17 + 3 * q^19 + 3 * q^22 - 9 * q^23 + 3 * q^25 - 9 * q^29 - 6 * q^31 - 3 * q^32 - 3 * q^34 + 9 * q^37 - 3 * q^38 + 12 * q^41 + 9 * q^43 - 3 * q^44 + 9 * q^46 - 15 * q^47 - 3 * q^50 + 9 * q^58 + 9 * q^59 + 6 * q^61 + 6 * q^62 + 3 * q^64 - 36 * q^65 + 6 * q^67 + 3 * q^68 - 3 * q^71 - 9 * q^74 + 3 * q^76 + 12 * q^79 - 12 * q^82 - 6 * q^83 - 9 * q^86 + 3 * q^88 - 36 * q^89 - 9 * q^92 + 15 * q^94 + 24 * q^95 - 9 * q^97

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