LMFDB - Embedded newform 9680.2.a.cy.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.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:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.45753625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 $ x^6 - x^5 - 13x^4 + 11x^3 + 41x^2 - 30x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.444728$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.719585 q^{3} -1.00000 q^{5} +4.18418 q^{7} -2.48220 q^{9} +6.05056 q^{13} +0.719585 q^{15} -6.52180 q^{17} +0.739455 q^{19} -3.01088 q^{21} +5.13093 q^{23} +1.00000 q^{25} +3.94491 q^{27} -2.08036 q^{29} +1.12698 q^{31} -4.18418 q^{35} +7.11676 q^{37} -4.35390 q^{39} +4.02170 q^{41} +6.29891 q^{43} +2.48220 q^{45} -9.24147 q^{47} +10.5074 q^{49} +4.69299 q^{51} +7.77015 q^{53} -0.532101 q^{57} -6.16416 q^{59} -14.6869 q^{61} -10.3860 q^{63} -6.05056 q^{65} +5.53268 q^{67} -3.69214 q^{69} -8.87252 q^{71} +8.09351 q^{73} -0.719585 q^{75} -0.570482 q^{79} +4.60789 q^{81} -15.2224 q^{83} +6.52180 q^{85} +1.49700 q^{87} -1.33867 q^{89} +25.3167 q^{91} -0.810957 q^{93} -0.739455 q^{95} +9.96643 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} + 6 q^{13} + 2 q^{15} - 11 q^{17} - 11 q^{19} - 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} + 6 q^{29} - q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} + 4 q^{41} + 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 10 * q^9 + 6 * q^13 + 2 * q^15 - 11 * q^17 - 11 * q^19 - 2 * q^21 - 18 * q^23 + 6 * q^25 + q^27 + 6 * q^29 - q^31 - 6 * q^35 + 4 * q^37 + 27 * q^39 + 4 * q^41 + 3 * q^43 - 10 * q^45 - 14 * q^47 + 8 * q^49 + 31 * q^51 + 14 * q^53 + 5 * q^57 - 2 * q^59 + 4 * q^61 - 16 * q^63 - 6 * q^65 - 11 * q^67 + 8 * q^69 - 7 * q^71 + 9 * q^73 - 2 * q^75 + 36 * q^79 + 30 * q^81 - 45 * q^83 + 11 * q^85 + 25 * q^87 + q^89 + 8 * q^91 + 55 * q^93 + 11 * q^95 + 20 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	−0.719585	−0.415453	−0.207726	−	0.978187i	$-0.566606\pi$
$3$	−0.719585	−0.415453	−0.207726	+	0.978187i	$0.566606\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.18418	1.58147	0.790736	−	0.612157i	$-0.209698\pi$
$7$	4.18418	1.58147	0.790736	+	0.612157i	$0.209698\pi$
$8$	0	0
$9$	−2.48220	−0.827399
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	6.05056	1.67812	0.839062	−	0.544035i	$-0.183104\pi$
$13$	6.05056	1.67812	0.839062	+	0.544035i	$0.183104\pi$
$14$	0	0
$15$	0.719585	0.185796
$16$	0	0
$17$	−6.52180	−1.58177	−0.790885	−	0.611965i	$-0.790379\pi$
$17$	−6.52180	−1.58177	−0.790885	+	0.611965i	$0.790379\pi$
$18$	0	0
$19$	0.739455	0.169642	0.0848212	−	0.996396i	$-0.472968\pi$
$19$	0.739455	0.169642	0.0848212	+	0.996396i	$0.472968\pi$
$20$	0	0
$21$	−3.01088	−0.657027
$22$	0	0
$23$	5.13093	1.06987	0.534936	−	0.844893i	$-0.320336\pi$
$23$	5.13093	1.06987	0.534936	+	0.844893i	$0.320336\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.94491	0.759198
$28$	0	0
$29$	−2.08036	−0.386313	−0.193157	−	0.981168i	$-0.561873\pi$
$29$	−2.08036	−0.386313	−0.193157	+	0.981168i	$0.561873\pi$
$30$	0	0
$31$	1.12698	0.202411	0.101206	−	0.994866i	$-0.467730\pi$
$31$	1.12698	0.202411	0.101206	+	0.994866i	$0.467730\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.18418	−0.707256
$36$	0	0
$37$	7.11676	1.16999	0.584994	−	0.811037i	$-0.301097\pi$
$37$	7.11676	1.16999	0.584994	+	0.811037i	$0.301097\pi$
$38$	0	0
$39$	−4.35390	−0.697182
$40$	0	0
$41$	4.02170	0.628085	0.314042	−	0.949409i	$-0.398317\pi$
$41$	4.02170	0.628085	0.314042	+	0.949409i	$0.398317\pi$
$42$	0	0
$43$	6.29891	0.960575	0.480288	−	0.877111i	$-0.340532\pi$
$43$	6.29891	0.960575	0.480288	+	0.877111i	$0.340532\pi$
$44$	0	0
$45$	2.48220	0.370024
$46$	0	0
$47$	−9.24147	−1.34801	−0.674003	−	0.738728i	$-0.735427\pi$
$47$	−9.24147	−1.34801	−0.674003	+	0.738728i	$0.735427\pi$
$48$	0	0
$49$	10.5074	1.50106
$50$	0	0
$51$	4.69299	0.657151
$52$	0	0
$53$	7.77015	1.06731	0.533656	−	0.845702i	$-0.320818\pi$
$53$	7.77015	1.06731	0.533656	+	0.845702i	$0.320818\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.532101	−0.0704785
$58$	0	0
$59$	−6.16416	−0.802505	−0.401253	−	0.915967i	$-0.631425\pi$
$59$	−6.16416	−0.802505	−0.401253	+	0.915967i	$0.631425\pi$
$60$	0	0
$61$	−14.6869	−1.88046	−0.940230	−	0.340541i	$-0.889390\pi$
$61$	−14.6869	−1.88046	−0.940230	+	0.340541i	$0.889390\pi$
$62$	0	0
$63$	−10.3860	−1.30851
$64$	0	0
$65$	−6.05056	−0.750480
$66$	0	0
$67$	5.53268	0.675924	0.337962	−	0.941160i	$-0.390262\pi$
$67$	5.53268	0.675924	0.337962	+	0.941160i	$0.390262\pi$
$68$	0	0
$69$	−3.69214	−0.444481
$70$	0	0
$71$	−8.87252	−1.05297	−0.526487	−	0.850183i	$-0.676491\pi$
$71$	−8.87252	−1.05297	−0.526487	+	0.850183i	$0.676491\pi$
$72$	0	0
$73$	8.09351	0.947274	0.473637	−	0.880720i	$-0.342941\pi$
$73$	8.09351	0.947274	0.473637	+	0.880720i	$0.342941\pi$
$74$	0	0
$75$	−0.719585	−0.0830906
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.570482	−0.0641842	−0.0320921	−	0.999485i	$-0.510217\pi$
$79$	−0.570482	−0.0641842	−0.0320921	+	0.999485i	$0.510217\pi$
$80$	0	0
$81$	4.60789	0.511988
$82$	0	0
$83$	−15.2224	−1.67088	−0.835439	−	0.549584i	$-0.814786\pi$
$83$	−15.2224	−1.67088	−0.835439	+	0.549584i	$0.814786\pi$
$84$	0	0
$85$	6.52180	0.707389
$86$	0	0
$87$	1.49700	0.160495
$88$	0	0
$89$	−1.33867	−0.141899	−0.0709495	−	0.997480i	$-0.522603\pi$
$89$	−1.33867	−0.141899	−0.0709495	+	0.997480i	$0.522603\pi$
$90$	0	0
$91$	25.3167	2.65391
$92$	0	0
$93$	−0.810957	−0.0840923
$94$	0	0
$95$	−0.739455	−0.0758664
$96$	0	0
$97$	9.96643	1.01194	0.505969	−	0.862552i	$-0.331135\pi$
$97$	9.96643	1.01194	0.505969	+	0.862552i	$0.331135\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.8701	−1.18112	−0.590562	−	0.806992i	$-0.701094\pi$
$101$	−11.8701	−1.18112	−0.590562	+	0.806992i	$0.701094\pi$
$102$	0	0
$103$	10.3935	1.02410	0.512050	−	0.858955i	$-0.328886\pi$
$103$	10.3935	1.02410	0.512050	+	0.858955i	$0.328886\pi$
$104$	0	0
$105$	3.01088	0.293832
$106$	0	0
$107$	8.34807	0.807039	0.403519	−	0.914971i	$-0.367787\pi$
$107$	8.34807	0.807039	0.403519	+	0.914971i	$0.367787\pi$
$108$	0	0
$109$	6.25046	0.598686	0.299343	−	0.954146i	$-0.403233\pi$
$109$	6.25046	0.598686	0.299343	+	0.954146i	$0.403233\pi$
$110$	0	0
$111$	−5.12112	−0.486075
$112$	0	0
$113$	16.4937	1.55160	0.775798	−	0.630982i	$-0.217348\pi$
$113$	16.4937	1.55160	0.775798	+	0.630982i	$0.217348\pi$
$114$	0	0
$115$	−5.13093	−0.478461
$116$	0	0
$117$	−15.0187	−1.38848
$118$	0	0
$119$	−27.2884	−2.50152
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−2.89396	−0.260940
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.5082	1.46487	0.732434	−	0.680838i	$-0.238384\pi$
$127$	16.5082	1.46487	0.732434	+	0.680838i	$0.238384\pi$
$128$	0	0
$129$	−4.53261	−0.399074
$130$	0	0
$131$	12.1354	1.06027	0.530137	−	0.847912i	$-0.322141\pi$
$131$	12.1354	1.06027	0.530137	+	0.847912i	$0.322141\pi$
$132$	0	0
$133$	3.09401	0.268285
$134$	0	0
$135$	−3.94491	−0.339524
$136$	0	0
$137$	6.18352	0.528294	0.264147	−	0.964482i	$-0.414910\pi$
$137$	6.18352	0.528294	0.264147	+	0.964482i	$0.414910\pi$
$138$	0	0
$139$	−5.20245	−0.441266	−0.220633	−	0.975357i	$-0.570812\pi$
$139$	−5.20245	−0.441266	−0.220633	+	0.975357i	$0.570812\pi$
$140$	0	0
$141$	6.65003	0.560033
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.08036	0.172765
$146$	0	0
$147$	−7.56096	−0.623618
$148$	0	0
$149$	11.2912	0.925013	0.462506	−	0.886616i	$-0.346950\pi$
$149$	11.2912	0.925013	0.462506	+	0.886616i	$0.346950\pi$
$150$	0	0
$151$	−0.156619	−0.0127455	−0.00637275	−	0.999980i	$-0.502029\pi$
$151$	−0.156619	−0.0127455	−0.00637275	+	0.999980i	$0.502029\pi$
$152$	0	0
$153$	16.1884	1.30875
$154$	0	0
$155$	−1.12698	−0.0905210
$156$	0	0
$157$	−13.6990	−1.09330	−0.546650	−	0.837361i	$-0.684097\pi$
$157$	−13.6990	−1.09330	−0.546650	+	0.837361i	$0.684097\pi$
$158$	0	0
$159$	−5.59129	−0.443418
$160$	0	0
$161$	21.4687	1.69197
$162$	0	0
$163$	−5.85320	−0.458458	−0.229229	−	0.973372i	$-0.573621\pi$
$163$	−5.85320	−0.458458	−0.229229	+	0.973372i	$0.573621\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.2892	−0.950965	−0.475483	−	0.879725i	$-0.657727\pi$
$167$	−12.2892	−0.950965	−0.475483	+	0.879725i	$0.657727\pi$
$168$	0	0
$169$	23.6093	1.81610
$170$	0	0
$171$	−1.83547	−0.140362
$172$	0	0
$173$	19.0157	1.44574	0.722868	−	0.690987i	$-0.242824\pi$
$173$	19.0157	1.44574	0.722868	+	0.690987i	$0.242824\pi$
$174$	0	0
$175$	4.18418	0.316294
$176$	0	0
$177$	4.43564	0.333403
$178$	0	0
$179$	0.166428	0.0124394	0.00621969	−	0.999981i	$-0.498020\pi$
$179$	0.166428	0.0124394	0.00621969	+	0.999981i	$0.498020\pi$
$180$	0	0
$181$	4.08469	0.303613	0.151806	−	0.988410i	$-0.451491\pi$
$181$	4.08469	0.303613	0.151806	+	0.988410i	$0.451491\pi$
$182$	0	0
$183$	10.5684	0.781242
$184$	0	0
$185$	−7.11676	−0.523235
$186$	0	0
$187$	0	0
$188$	0	0
$189$	16.5062	1.20065
$190$	0	0
$191$	−14.7337	−1.06609	−0.533047	−	0.846086i	$-0.678953\pi$
$191$	−14.7337	−1.06609	−0.533047	+	0.846086i	$0.678953\pi$
$192$	0	0
$193$	−0.314976	−0.0226725	−0.0113362	−	0.999936i	$-0.503609\pi$
$193$	−0.314976	−0.0226725	−0.0113362	+	0.999936i	$0.503609\pi$
$194$	0	0
$195$	4.35390	0.311789
$196$	0	0
$197$	−1.41533	−0.100838	−0.0504191	−	0.998728i	$-0.516056\pi$
$197$	−1.41533	−0.100838	−0.0504191	+	0.998728i	$0.516056\pi$
$198$	0	0
$199$	−22.3049	−1.58115	−0.790575	−	0.612365i	$-0.790218\pi$
$199$	−22.3049	−1.58115	−0.790575	+	0.612365i	$0.790218\pi$
$200$	0	0
$201$	−3.98124	−0.280815
$202$	0	0
$203$	−8.70461	−0.610944
$204$	0	0
$205$	−4.02170	−0.280888
$206$	0	0
$207$	−12.7360	−0.885211
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−24.3488	−1.67624	−0.838120	−	0.545486i	$-0.816345\pi$
$211$	−24.3488	−1.67624	−0.838120	+	0.545486i	$0.816345\pi$
$212$	0	0
$213$	6.38453	0.437461
$214$	0	0
$215$	−6.29891	−0.429582
$216$	0	0
$217$	4.71548	0.320108
$218$	0	0
$219$	−5.82397	−0.393547
$220$	0	0
$221$	−39.4606	−2.65441
$222$	0	0
$223$	−6.22145	−0.416619	−0.208310	−	0.978063i	$-0.566796\pi$
$223$	−6.22145	−0.416619	−0.208310	+	0.978063i	$0.566796\pi$
$224$	0	0
$225$	−2.48220	−0.165480
$226$	0	0
$227$	−16.0658	−1.06632	−0.533162	−	0.846013i	$-0.678996\pi$
$227$	−16.0658	−1.06632	−0.533162	+	0.846013i	$0.678996\pi$
$228$	0	0
$229$	−15.6832	−1.03638	−0.518189	−	0.855266i	$-0.673394\pi$
$229$	−15.6832	−1.03638	−0.518189	+	0.855266i	$0.673394\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0236	0.918718	0.459359	−	0.888251i	$-0.348079\pi$
$233$	14.0236	0.918718	0.459359	+	0.888251i	$0.348079\pi$
$234$	0	0
$235$	9.24147	0.602847
$236$	0	0
$237$	0.410510	0.0266655
$238$	0	0
$239$	10.8094	0.699200	0.349600	−	0.936899i	$-0.386317\pi$
$239$	10.8094	0.699200	0.349600	+	0.936899i	$0.386317\pi$
$240$	0	0
$241$	−19.7912	−1.27486	−0.637431	−	0.770507i	$-0.720003\pi$
$241$	−19.7912	−1.27486	−0.637431	+	0.770507i	$0.720003\pi$
$242$	0	0
$243$	−15.1505	−0.971905
$244$	0	0
$245$	−10.5074	−0.671292
$246$	0	0
$247$	4.47412	0.284681
$248$	0	0
$249$	10.9538	0.694171
$250$	0	0
$251$	−3.82947	−0.241714	−0.120857	−	0.992670i	$-0.538564\pi$
$251$	−3.82947	−0.241714	−0.120857	+	0.992670i	$0.538564\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−4.69299	−0.293887
$256$	0	0
$257$	19.0304	1.18709	0.593543	−	0.804802i	$-0.297729\pi$
$257$	19.0304	1.18709	0.593543	+	0.804802i	$0.297729\pi$
$258$	0	0
$259$	29.7778	1.85030
$260$	0	0
$261$	5.16387	0.319635
$262$	0	0
$263$	26.6567	1.64372	0.821862	−	0.569686i	$-0.192935\pi$
$263$	26.6567	1.64372	0.821862	+	0.569686i	$0.192935\pi$
$264$	0	0
$265$	−7.77015	−0.477317
$266$	0	0
$267$	0.963290	0.0589524
$268$	0	0
$269$	0.454544	0.0277141	0.0138570	−	0.999904i	$-0.495589\pi$
$269$	0.454544	0.0277141	0.0138570	+	0.999904i	$0.495589\pi$
$270$	0	0
$271$	16.2608	0.987771	0.493885	−	0.869527i	$-0.335576\pi$
$271$	16.2608	0.987771	0.493885	+	0.869527i	$0.335576\pi$
$272$	0	0
$273$	−18.2175	−1.10257
$274$	0	0
$275$	0	0
$276$	0	0
$277$	11.4338	0.686989	0.343495	−	0.939155i	$-0.388389\pi$
$277$	11.4338	0.686989	0.343495	+	0.939155i	$0.388389\pi$
$278$	0	0
$279$	−2.79738	−0.167475
$280$	0	0
$281$	5.69217	0.339566	0.169783	−	0.985481i	$-0.445693\pi$
$281$	5.69217	0.339566	0.169783	+	0.985481i	$0.445693\pi$
$282$	0	0
$283$	−0.580914	−0.0345318	−0.0172659	−	0.999851i	$-0.505496\pi$
$283$	−0.580914	−0.0345318	−0.0172659	+	0.999851i	$0.505496\pi$
$284$	0	0
$285$	0.532101	0.0315189
$286$	0	0
$287$	16.8275	0.993298
$288$	0	0
$289$	25.5339	1.50199
$290$	0	0
$291$	−7.17170	−0.420413
$292$	0	0
$293$	1.17741	0.0687850	0.0343925	−	0.999408i	$-0.489050\pi$
$293$	1.17741	0.0687850	0.0343925	+	0.999408i	$0.489050\pi$
$294$	0	0
$295$	6.16416	0.358891
$296$	0	0
$297$	0	0
$298$	0	0
$299$	31.0450	1.79538
$300$	0	0
$301$	26.3558	1.51912
$302$	0	0
$303$	8.54158	0.490701
$304$	0	0
$305$	14.6869	0.840967
$306$	0	0
$307$	−19.1615	−1.09360	−0.546802	−	0.837262i	$-0.684155\pi$
$307$	−19.1615	−1.09360	−0.546802	+	0.837262i	$0.684155\pi$
$308$	0	0
$309$	−7.47900	−0.425466
$310$	0	0
$311$	−18.6728	−1.05884	−0.529419	−	0.848361i	$-0.677590\pi$
$311$	−18.6728	−1.05884	−0.529419	+	0.848361i	$0.677590\pi$
$312$	0	0
$313$	−3.58974	−0.202904	−0.101452	−	0.994840i	$-0.532349\pi$
$313$	−3.58974	−0.202904	−0.101452	+	0.994840i	$0.532349\pi$
$314$	0	0
$315$	10.3860	0.585183
$316$	0	0
$317$	2.50027	0.140429	0.0702146	−	0.997532i	$-0.477632\pi$
$317$	2.50027	0.140429	0.0702146	+	0.997532i	$0.477632\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−6.00715	−0.335287
$322$	0	0
$323$	−4.82258	−0.268335
$324$	0	0
$325$	6.05056	0.335625
$326$	0	0
$327$	−4.49774	−0.248726
$328$	0	0
$329$	−38.6680	−2.13184
$330$	0	0
$331$	6.58629	0.362015	0.181008	−	0.983482i	$-0.442064\pi$
$331$	6.58629	0.362015	0.181008	+	0.983482i	$0.442064\pi$
$332$	0	0
$333$	−17.6652	−0.968047
$334$	0	0
$335$	−5.53268	−0.302283
$336$	0	0
$337$	15.2151	0.828821	0.414410	−	0.910090i	$-0.363988\pi$
$337$	15.2151	0.828821	0.414410	+	0.910090i	$0.363988\pi$
$338$	0	0
$339$	−11.8686	−0.644615
$340$	0	0
$341$	0	0
$342$	0	0
$343$	14.6755	0.792405
$344$	0	0
$345$	3.69214	0.198778
$346$	0	0
$347$	7.66135	0.411283	0.205641	−	0.978627i	$-0.434072\pi$
$347$	7.66135	0.411283	0.205641	+	0.978627i	$0.434072\pi$
$348$	0	0
$349$	8.99401	0.481438	0.240719	−	0.970595i	$-0.422617\pi$
$349$	8.99401	0.481438	0.240719	+	0.970595i	$0.422617\pi$
$350$	0	0
$351$	23.8689	1.27403
$352$	0	0
$353$	10.3310	0.549862	0.274931	−	0.961464i	$-0.411345\pi$
$353$	10.3310	0.549862	0.274931	+	0.961464i	$0.411345\pi$
$354$	0	0
$355$	8.87252	0.470904
$356$	0	0
$357$	19.6363	1.03927
$358$	0	0
$359$	16.2627	0.858310	0.429155	−	0.903231i	$-0.358811\pi$
$359$	16.2627	0.858310	0.429155	+	0.903231i	$0.358811\pi$
$360$	0	0
$361$	−18.4532	−0.971221
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.09351	−0.423634
$366$	0	0
$367$	34.5427	1.80311	0.901557	−	0.432660i	$-0.142425\pi$
$367$	34.5427	1.80311	0.901557	+	0.432660i	$0.142425\pi$
$368$	0	0
$369$	−9.98266	−0.519676
$370$	0	0
$371$	32.5117	1.68792
$372$	0	0
$373$	−31.2168	−1.61634	−0.808172	−	0.588946i	$-0.799543\pi$
$373$	−31.2168	−1.61634	−0.808172	+	0.588946i	$0.799543\pi$
$374$	0	0
$375$	0.719585	0.0371592
$376$	0	0
$377$	−12.5874	−0.648282
$378$	0	0
$379$	33.4663	1.71905	0.859525	−	0.511094i	$-0.170759\pi$
$379$	33.4663	1.71905	0.859525	+	0.511094i	$0.170759\pi$
$380$	0	0
$381$	−11.8791	−0.608584
$382$	0	0
$383$	0.134288	0.00686178	0.00343089	−	0.999994i	$-0.498908\pi$
$383$	0.134288	0.00686178	0.00343089	+	0.999994i	$0.498908\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−15.6351	−0.794779
$388$	0	0
$389$	−8.53807	−0.432897	−0.216449	−	0.976294i	$-0.569447\pi$
$389$	−8.53807	−0.432897	−0.216449	+	0.976294i	$0.569447\pi$
$390$	0	0
$391$	−33.4629	−1.69229
$392$	0	0
$393$	−8.73245	−0.440494
$394$	0	0
$395$	0.570482	0.0287040
$396$	0	0
$397$	38.0250	1.90842	0.954211	−	0.299135i	$-0.0966982\pi$
$397$	38.0250	1.90842	0.954211	+	0.299135i	$0.0966982\pi$
$398$	0	0
$399$	−2.22641	−0.111460
$400$	0	0
$401$	−11.7343	−0.585982	−0.292991	−	0.956115i	$-0.594651\pi$
$401$	−11.7343	−0.585982	−0.292991	+	0.956115i	$0.594651\pi$
$402$	0	0
$403$	6.81885	0.339671
$404$	0	0
$405$	−4.60789	−0.228968
$406$	0	0
$407$	0	0
$408$	0	0
$409$	28.9837	1.43315	0.716575	−	0.697510i	$-0.245709\pi$
$409$	28.9837	1.43315	0.716575	+	0.697510i	$0.245709\pi$
$410$	0	0
$411$	−4.44957	−0.219481
$412$	0	0
$413$	−25.7920	−1.26914
$414$	0	0
$415$	15.2224	0.747239
$416$	0	0
$417$	3.74361	0.183325
$418$	0	0
$419$	−12.5318	−0.612221	−0.306110	−	0.951996i	$-0.599028\pi$
$419$	−12.5318	−0.612221	−0.306110	+	0.951996i	$0.599028\pi$
$420$	0	0
$421$	−11.4281	−0.556973	−0.278486	−	0.960440i	$-0.589833\pi$
$421$	−11.4281	−0.556973	−0.278486	+	0.960440i	$0.589833\pi$
$422$	0	0
$423$	22.9391	1.11534
$424$	0	0
$425$	−6.52180	−0.316354
$426$	0	0
$427$	−61.4525	−2.97389
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.78747	0.182436	0.0912181	−	0.995831i	$-0.470924\pi$
$431$	3.78747	0.182436	0.0912181	+	0.995831i	$0.470924\pi$
$432$	0	0
$433$	25.6848	1.23433	0.617166	−	0.786833i	$-0.288281\pi$
$433$	25.6848	1.23433	0.617166	+	0.786833i	$0.288281\pi$
$434$	0	0
$435$	−1.49700	−0.0717756
$436$	0	0
$437$	3.79409	0.181496
$438$	0	0
$439$	29.1218	1.38991	0.694955	−	0.719054i	$-0.255424\pi$
$439$	29.1218	1.38991	0.694955	+	0.719054i	$0.255424\pi$
$440$	0	0
$441$	−26.0814	−1.24197
$442$	0	0
$443$	−12.1115	−0.575437	−0.287718	−	0.957715i	$-0.592897\pi$
$443$	−12.1115	−0.575437	−0.287718	+	0.957715i	$0.592897\pi$
$444$	0	0
$445$	1.33867	0.0634592
$446$	0	0
$447$	−8.12500	−0.384299
$448$	0	0
$449$	29.6886	1.40109	0.700546	−	0.713607i	$-0.252940\pi$
$449$	29.6886	1.40109	0.700546	+	0.713607i	$0.252940\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0.112701	0.00529516
$454$	0	0
$455$	−25.3167	−1.18686
$456$	0	0
$457$	−0.751497	−0.0351535	−0.0175768	−	0.999846i	$-0.505595\pi$
$457$	−0.751497	−0.0351535	−0.0175768	+	0.999846i	$0.505595\pi$
$458$	0	0
$459$	−25.7279	−1.20088
$460$	0	0
$461$	9.62920	0.448477	0.224238	−	0.974534i	$-0.428011\pi$
$461$	9.62920	0.448477	0.224238	+	0.974534i	$0.428011\pi$
$462$	0	0
$463$	3.33661	0.155065	0.0775327	−	0.996990i	$-0.475296\pi$
$463$	3.33661	0.155065	0.0775327	+	0.996990i	$0.475296\pi$
$464$	0	0
$465$	0.810957	0.0376072
$466$	0	0
$467$	−1.69675	−0.0785161	−0.0392581	−	0.999229i	$-0.512499\pi$
$467$	−1.69675	−0.0785161	−0.0392581	+	0.999229i	$0.512499\pi$
$468$	0	0
$469$	23.1497	1.06896
$470$	0	0
$471$	9.85761	0.454214
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0.739455	0.0339285
$476$	0	0
$477$	−19.2870	−0.883093
$478$	0	0
$479$	22.1315	1.01121	0.505607	−	0.862764i	$-0.331269\pi$
$479$	22.1315	1.01121	0.505607	+	0.862764i	$0.331269\pi$
$480$	0	0
$481$	43.0604	1.96339
$482$	0	0
$483$	−15.4486	−0.702935
$484$	0	0
$485$	−9.96643	−0.452552
$486$	0	0
$487$	19.9414	0.903631	0.451815	−	0.892111i	$-0.350777\pi$
$487$	19.9414	0.903631	0.451815	+	0.892111i	$0.350777\pi$
$488$	0	0
$489$	4.21188	0.190468
$490$	0	0
$491$	26.2451	1.18443	0.592213	−	0.805782i	$-0.298254\pi$
$491$	26.2451	1.18443	0.592213	+	0.805782i	$0.298254\pi$
$492$	0	0
$493$	13.5677	0.611059
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−37.1242	−1.66525
$498$	0	0
$499$	8.34480	0.373565	0.186782	−	0.982401i	$-0.440194\pi$
$499$	8.34480	0.373565	0.186782	+	0.982401i	$0.440194\pi$
$500$	0	0
$501$	8.84311	0.395081
$502$	0	0
$503$	30.3824	1.35469	0.677343	−	0.735667i	$-0.263131\pi$
$503$	30.3824	1.35469	0.677343	+	0.735667i	$0.263131\pi$
$504$	0	0
$505$	11.8701	0.528214
$506$	0	0
$507$	−16.9889	−0.754505
$508$	0	0
$509$	17.2197	0.763248	0.381624	−	0.924318i	$-0.375365\pi$
$509$	17.2197	0.763248	0.381624	+	0.924318i	$0.375365\pi$
$510$	0	0
$511$	33.8647	1.49809
$512$	0	0
$513$	2.91708	0.128792
$514$	0	0
$515$	−10.3935	−0.457992
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−13.6834	−0.600635
$520$	0	0
$521$	−21.4975	−0.941822	−0.470911	−	0.882181i	$-0.656075\pi$
$521$	−21.4975	−0.941822	−0.470911	+	0.882181i	$0.656075\pi$
$522$	0	0
$523$	−0.0139271	−0.000608991 0	−0.000304496	−	1.00000i	$-0.500097\pi$
$523$	−0.0139271	−0.000608991 0	−0.000304496		1.00000i	$0.500097\pi$
$524$	0	0
$525$	−3.01088	−0.131405
$526$	0	0
$527$	−7.34993	−0.320168
$528$	0	0
$529$	3.32640	0.144626
$530$	0	0
$531$	15.3007	0.663992
$532$	0	0
$533$	24.3336	1.05400
$534$	0	0
$535$	−8.34807	−0.360919
$536$	0	0
$537$	−0.119759	−0.00516798
$538$	0	0
$539$	0	0
$540$	0	0
$541$	42.7268	1.83697	0.918483	−	0.395459i	$-0.129415\pi$
$541$	42.7268	1.83697	0.918483	+	0.395459i	$0.129415\pi$
$542$	0	0
$543$	−2.93929	−0.126137
$544$	0	0
$545$	−6.25046	−0.267740
$546$	0	0
$547$	17.8319	0.762438	0.381219	−	0.924485i	$-0.375504\pi$
$547$	17.8319	0.762438	0.381219	+	0.924485i	$0.375504\pi$
$548$	0	0
$549$	36.4557	1.55589
$550$	0	0
$551$	−1.53833	−0.0655352
$552$	0	0
$553$	−2.38700	−0.101506
$554$	0	0
$555$	5.12112	0.217379
$556$	0	0
$557$	17.8786	0.757541	0.378771	−	0.925491i	$-0.376347\pi$
$557$	17.8786	0.757541	0.378771	+	0.925491i	$0.376347\pi$
$558$	0	0
$559$	38.1120	1.61196
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.6011	−0.446781	−0.223391	−	0.974729i	$-0.571713\pi$
$563$	−10.6011	−0.446781	−0.223391	+	0.974729i	$0.571713\pi$
$564$	0	0
$565$	−16.4937	−0.693894
$566$	0	0
$567$	19.2803	0.809695
$568$	0	0
$569$	22.8718	0.958836	0.479418	−	0.877587i	$-0.340848\pi$
$569$	22.8718	0.958836	0.479418	+	0.877587i	$0.340848\pi$
$570$	0	0
$571$	−20.5139	−0.858480	−0.429240	−	0.903191i	$-0.641218\pi$
$571$	−20.5139	−0.858480	−0.429240	+	0.903191i	$0.641218\pi$
$572$	0	0
$573$	10.6022	0.442912
$574$	0	0
$575$	5.13093	0.213974
$576$	0	0
$577$	24.2174	1.00819	0.504093	−	0.863650i	$-0.331827\pi$
$577$	24.2174	1.00819	0.504093	+	0.863650i	$0.331827\pi$
$578$	0	0
$579$	0.226652	0.00941935
$580$	0	0
$581$	−63.6934	−2.64245
$582$	0	0
$583$	0	0
$584$	0	0
$585$	15.0187	0.620946
$586$	0	0
$587$	31.9904	1.32039	0.660194	−	0.751095i	$-0.270474\pi$
$587$	31.9904	1.32039	0.660194	+	0.751095i	$0.270474\pi$
$588$	0	0
$589$	0.833349	0.0343375
$590$	0	0
$591$	1.01845	0.0418935
$592$	0	0
$593$	42.5206	1.74611	0.873056	−	0.487620i	$-0.162135\pi$
$593$	42.5206	1.74611	0.873056	+	0.487620i	$0.162135\pi$
$594$	0	0
$595$	27.2884	1.11872
$596$	0	0
$597$	16.0503	0.656894
$598$	0	0
$599$	4.29039	0.175301	0.0876503	−	0.996151i	$-0.472064\pi$
$599$	4.29039	0.175301	0.0876503	+	0.996151i	$0.472064\pi$
$600$	0	0
$601$	−19.9972	−0.815704	−0.407852	−	0.913048i	$-0.633722\pi$
$601$	−19.9972	−0.815704	−0.407852	+	0.913048i	$0.633722\pi$
$602$	0	0
$603$	−13.7332	−0.559259
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23.2114	−0.942121	−0.471060	−	0.882101i	$-0.656129\pi$
$607$	−23.2114	−0.942121	−0.471060	+	0.882101i	$0.656129\pi$
$608$	0	0
$609$	6.26371	0.253818
$610$	0	0
$611$	−55.9161	−2.26212
$612$	0	0
$613$	5.94916	0.240284	0.120142	−	0.992757i	$-0.461665\pi$
$613$	5.94916	0.240284	0.120142	+	0.992757i	$0.461665\pi$
$614$	0	0
$615$	2.89396	0.116696
$616$	0	0
$617$	39.7576	1.60058	0.800291	−	0.599612i	$-0.204678\pi$
$617$	39.7576	1.60058	0.800291	+	0.599612i	$0.204678\pi$
$618$	0	0
$619$	11.8287	0.475436	0.237718	−	0.971334i	$-0.423601\pi$
$619$	11.8287	0.475436	0.237718	+	0.971334i	$0.423601\pi$
$620$	0	0
$621$	20.2410	0.812245
$622$	0	0
$623$	−5.60125	−0.224410
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−46.4141	−1.85065
$630$	0	0
$631$	−14.8112	−0.589623	−0.294812	−	0.955555i	$-0.595257\pi$
$631$	−14.8112	−0.589623	−0.294812	+	0.955555i	$0.595257\pi$
$632$	0	0
$633$	17.5210	0.696398
$634$	0	0
$635$	−16.5082	−0.655109
$636$	0	0
$637$	63.5756	2.51896
$638$	0	0
$639$	22.0233	0.871230
$640$	0	0
$641$	−8.51035	−0.336139	−0.168069	−	0.985775i	$-0.553753\pi$
$641$	−8.51035	−0.336139	−0.168069	+	0.985775i	$0.553753\pi$
$642$	0	0
$643$	−34.4340	−1.35794	−0.678971	−	0.734165i	$-0.737574\pi$
$643$	−34.4340	−1.35794	−0.678971	+	0.734165i	$0.737574\pi$
$644$	0	0
$645$	4.53261	0.178471
$646$	0	0
$647$	−5.32491	−0.209344	−0.104672	−	0.994507i	$-0.533379\pi$
$647$	−5.32491	−0.209344	−0.104672	+	0.994507i	$0.533379\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−3.39319	−0.132990
$652$	0	0
$653$	28.2099	1.10394	0.551970	−	0.833864i	$-0.313876\pi$
$653$	28.2099	1.10394	0.551970	+	0.833864i	$0.313876\pi$
$654$	0	0
$655$	−12.1354	−0.474169
$656$	0	0
$657$	−20.0897	−0.783773
$658$	0	0
$659$	−19.9228	−0.776082	−0.388041	−	0.921642i	$-0.626848\pi$
$659$	−19.9228	−0.776082	−0.388041	+	0.921642i	$0.626848\pi$
$660$	0	0
$661$	−16.6144	−0.646225	−0.323112	−	0.946361i	$-0.604729\pi$
$661$	−16.6144	−0.646225	−0.323112	+	0.946361i	$0.604729\pi$
$662$	0	0
$663$	28.3953	1.10278
$664$	0	0
$665$	−3.09401	−0.119981
$666$	0	0
$667$	−10.6742	−0.413306
$668$	0	0
$669$	4.47687	0.173086
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−32.7246	−1.26144	−0.630720	−	0.776010i	$-0.717240\pi$
$673$	−32.7246	−1.26144	−0.630720	+	0.776010i	$0.717240\pi$
$674$	0	0
$675$	3.94491	0.151840
$676$	0	0
$677$	−41.0308	−1.57694	−0.788472	−	0.615071i	$-0.789127\pi$
$677$	−41.0308	−1.57694	−0.788472	+	0.615071i	$0.789127\pi$
$678$	0	0
$679$	41.7014	1.60035
$680$	0	0
$681$	11.5607	0.443007
$682$	0	0
$683$	−5.71534	−0.218691	−0.109346	−	0.994004i	$-0.534876\pi$
$683$	−5.71534	−0.218691	−0.109346	+	0.994004i	$0.534876\pi$
$684$	0	0
$685$	−6.18352	−0.236260
$686$	0	0
$687$	11.2854	0.430566
$688$	0	0
$689$	47.0138	1.79108
$690$	0	0
$691$	38.8709	1.47872	0.739358	−	0.673312i	$-0.235129\pi$
$691$	38.8709	1.47872	0.739358	+	0.673312i	$0.235129\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.20245	0.197340
$696$	0	0
$697$	−26.2288	−0.993485
$698$	0	0
$699$	−10.0912	−0.381684
$700$	0	0
$701$	−3.37005	−0.127285	−0.0636426	−	0.997973i	$-0.520272\pi$
$701$	−3.37005	−0.127285	−0.0636426	+	0.997973i	$0.520272\pi$
$702$	0	0
$703$	5.26252	0.198480
$704$	0	0
$705$	−6.65003	−0.250455
$706$	0	0
$707$	−49.6668	−1.86791
$708$	0	0
$709$	−22.6143	−0.849300	−0.424650	−	0.905358i	$-0.639603\pi$
$709$	−22.6143	−0.849300	−0.424650	+	0.905358i	$0.639603\pi$
$710$	0	0
$711$	1.41605	0.0531059
$712$	0	0
$713$	5.78244	0.216554
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7.77827	−0.290485
$718$	0	0
$719$	18.7071	0.697655	0.348828	−	0.937187i	$-0.386580\pi$
$719$	18.7071	0.697655	0.348828	+	0.937187i	$0.386580\pi$
$720$	0	0
$721$	43.4883	1.61959
$722$	0	0
$723$	14.2415	0.529645
$724$	0	0
$725$	−2.08036	−0.0772627
$726$	0	0
$727$	2.98028	0.110532	0.0552662	−	0.998472i	$-0.482399\pi$
$727$	2.98028	0.110532	0.0552662	+	0.998472i	$0.482399\pi$
$728$	0	0
$729$	−2.92160	−0.108207
$730$	0	0
$731$	−41.0803	−1.51941
$732$	0	0
$733$	20.1444	0.744050	0.372025	−	0.928223i	$-0.378664\pi$
$733$	20.1444	0.744050	0.372025	+	0.928223i	$0.378664\pi$
$734$	0	0
$735$	7.56096	0.278890
$736$	0	0
$737$	0	0
$738$	0	0
$739$	23.2196	0.854146	0.427073	−	0.904217i	$-0.359545\pi$
$739$	23.2196	0.854146	0.427073	+	0.904217i	$0.359545\pi$
$740$	0	0
$741$	−3.21951	−0.118272
$742$	0	0
$743$	31.4412	1.15346	0.576732	−	0.816933i	$-0.304328\pi$
$743$	31.4412	1.15346	0.576732	+	0.816933i	$0.304328\pi$
$744$	0	0
$745$	−11.2912	−0.413678
$746$	0	0
$747$	37.7850	1.38248
$748$	0	0
$749$	34.9299	1.27631
$750$	0	0
$751$	−40.5769	−1.48067	−0.740337	−	0.672236i	$-0.765334\pi$
$751$	−40.5769	−1.48067	−0.740337	+	0.672236i	$0.765334\pi$
$752$	0	0
$753$	2.75563	0.100421
$754$	0	0
$755$	0.156619	0.00569996
$756$	0	0
$757$	−22.5716	−0.820378	−0.410189	−	0.912000i	$-0.634537\pi$
$757$	−22.5716	−0.820378	−0.410189	+	0.912000i	$0.634537\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.9901	−0.797139	−0.398570	−	0.917138i	$-0.630493\pi$
$761$	−21.9901	−0.797139	−0.398570	+	0.917138i	$0.630493\pi$
$762$	0	0
$763$	26.1531	0.946805
$764$	0	0
$765$	−16.1884	−0.585293
$766$	0	0
$767$	−37.2966	−1.34670
$768$	0	0
$769$	41.1834	1.48511	0.742556	−	0.669784i	$-0.233613\pi$
$769$	41.1834	1.48511	0.742556	+	0.669784i	$0.233613\pi$
$770$	0	0
$771$	−13.6940	−0.493178
$772$	0	0
$773$	−10.8352	−0.389716	−0.194858	−	0.980831i	$-0.562425\pi$
$773$	−10.8352	−0.389716	−0.194858	+	0.980831i	$0.562425\pi$
$774$	0	0
$775$	1.12698	0.0404822
$776$	0	0
$777$	−21.4277	−0.768714
$778$	0	0
$779$	2.97387	0.106550
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−8.20684	−0.293288
$784$	0	0
$785$	13.6990	0.488938
$786$	0	0
$787$	−31.3348	−1.11696	−0.558482	−	0.829517i	$-0.688616\pi$
$787$	−31.3348	−1.11696	−0.558482	+	0.829517i	$0.688616\pi$
$788$	0	0
$789$	−19.1818	−0.682890
$790$	0	0
$791$	69.0126	2.45381
$792$	0	0
$793$	−88.8638	−3.15565
$794$	0	0
$795$	5.59129	0.198303
$796$	0	0
$797$	−9.52077	−0.337243	−0.168621	−	0.985681i	$-0.553932\pi$
$797$	−9.52077	−0.337243	−0.168621	+	0.985681i	$0.553932\pi$
$798$	0	0
$799$	60.2710	2.13224
$800$	0	0
$801$	3.32285	0.117407
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−21.4687	−0.756673
$806$	0	0
$807$	−0.327084	−0.0115139
$808$	0	0
$809$	29.0902	1.02276	0.511379	−	0.859355i	$-0.329135\pi$
$809$	29.0902	1.02276	0.511379	+	0.859355i	$0.329135\pi$
$810$	0	0
$811$	−43.6205	−1.53172	−0.765862	−	0.643005i	$-0.777687\pi$
$811$	−43.6205	−1.53172	−0.765862	+	0.643005i	$0.777687\pi$
$812$	0	0
$813$	−11.7010	−0.410372
$814$	0	0
$815$	5.85320	0.205029
$816$	0	0
$817$	4.65776	0.162954
$818$	0	0
$819$	−62.8410	−2.19584
$820$	0	0
$821$	−26.6375	−0.929656	−0.464828	−	0.885401i	$-0.653884\pi$
$821$	−26.6375	−0.929656	−0.464828	+	0.885401i	$0.653884\pi$
$822$	0	0
$823$	−19.9584	−0.695706	−0.347853	−	0.937549i	$-0.613089\pi$
$823$	−19.9584	−0.695706	−0.347853	+	0.937549i	$0.613089\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.6819	0.927819	0.463910	−	0.885883i	$-0.346446\pi$
$827$	26.6819	0.927819	0.463910	+	0.885883i	$0.346446\pi$
$828$	0	0
$829$	−50.7077	−1.76115	−0.880576	−	0.473905i	$-0.842844\pi$
$829$	−50.7077	−1.76115	−0.880576	+	0.473905i	$0.842844\pi$
$830$	0	0
$831$	−8.22758	−0.285412
$832$	0	0
$833$	−68.5271	−2.37432
$834$	0	0
$835$	12.2892	0.425285
$836$	0	0
$837$	4.44583	0.153670
$838$	0	0
$839$	2.69881	0.0931733	0.0465867	−	0.998914i	$-0.485166\pi$
$839$	2.69881	0.0931733	0.0465867	+	0.998914i	$0.485166\pi$
$840$	0	0
$841$	−24.6721	−0.850762
$842$	0	0
$843$	−4.09600	−0.141074
$844$	0	0
$845$	−23.6093	−0.812186
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0.418017	0.0143463
$850$	0	0
$851$	36.5156	1.25174
$852$	0	0
$853$	17.6391	0.603951	0.301976	−	0.953316i	$-0.402354\pi$
$853$	17.6391	0.603951	0.301976	+	0.953316i	$0.402354\pi$
$854$	0	0
$855$	1.83547	0.0627718
$856$	0	0
$857$	8.72254	0.297956	0.148978	−	0.988840i	$-0.452402\pi$
$857$	8.72254	0.297956	0.148978	+	0.988840i	$0.452402\pi$
$858$	0	0
$859$	34.2322	1.16799	0.583993	−	0.811759i	$-0.301490\pi$
$859$	34.2322	1.16799	0.583993	+	0.811759i	$0.301490\pi$
$860$	0	0
$861$	−12.1089	−0.412669
$862$	0	0
$863$	40.8286	1.38982	0.694911	−	0.719096i	$-0.255444\pi$
$863$	40.8286	1.38982	0.694911	+	0.719096i	$0.255444\pi$
$864$	0	0
$865$	−19.0157	−0.646552
$866$	0	0
$867$	−18.3738	−0.624008
$868$	0	0
$869$	0	0
$870$	0	0
$871$	33.4758	1.13429
$872$	0	0
$873$	−24.7387	−0.837276
$874$	0	0
$875$	−4.18418	−0.141451
$876$	0	0
$877$	4.01395	0.135541	0.0677707	−	0.997701i	$-0.478411\pi$
$877$	4.01395	0.135541	0.0677707	+	0.997701i	$0.478411\pi$
$878$	0	0
$879$	−0.847247	−0.0285769
$880$	0	0
$881$	21.1713	0.713278	0.356639	−	0.934242i	$-0.383923\pi$
$881$	21.1713	0.713278	0.356639	+	0.934242i	$0.383923\pi$
$882$	0	0
$883$	−6.66990	−0.224460	−0.112230	−	0.993682i	$-0.535799\pi$
$883$	−6.66990	−0.224460	−0.112230	+	0.993682i	$0.535799\pi$
$884$	0	0
$885$	−4.43564	−0.149102
$886$	0	0
$887$	2.53548	0.0851332	0.0425666	−	0.999094i	$-0.486447\pi$
$887$	2.53548	0.0851332	0.0425666	+	0.999094i	$0.486447\pi$
$888$	0	0
$889$	69.0735	2.31665
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−6.83365	−0.228679
$894$	0	0
$895$	−0.166428	−0.00556306
$896$	0	0
$897$	−22.3395	−0.745895
$898$	0	0
$899$	−2.34452	−0.0781942
$900$	0	0
$901$	−50.6754	−1.68824
$902$	0	0
$903$	−18.9653	−0.631124
$904$	0	0
$905$	−4.08469	−0.135780
$906$	0	0
$907$	36.3808	1.20801	0.604003	−	0.796982i	$-0.293572\pi$
$907$	36.3808	1.20801	0.604003	+	0.796982i	$0.293572\pi$
$908$	0	0
$909$	29.4640	0.977260
$910$	0	0
$911$	58.3264	1.93244	0.966220	−	0.257719i	$-0.0829709\pi$
$911$	58.3264	1.93244	0.966220	+	0.257719i	$0.0829709\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−10.5684	−0.349382
$916$	0	0
$917$	50.7767	1.67679
$918$	0	0
$919$	29.7560	0.981560	0.490780	−	0.871283i	$-0.336712\pi$
$919$	29.7560	0.981560	0.490780	+	0.871283i	$0.336712\pi$
$920$	0	0
$921$	13.7883	0.454341
$922$	0	0
$923$	−53.6837	−1.76702
$924$	0	0
$925$	7.11676	0.233998
$926$	0	0
$927$	−25.7987	−0.847340
$928$	0	0
$929$	−26.0033	−0.853140	−0.426570	−	0.904455i	$-0.640278\pi$
$929$	−26.0033	−0.853140	−0.426570	+	0.904455i	$0.640278\pi$
$930$	0	0
$931$	7.76973	0.254643
$932$	0	0
$933$	13.4367	0.439897
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6.46058	0.211058	0.105529	−	0.994416i	$-0.466346\pi$
$937$	6.46058	0.211058	0.105529	+	0.994416i	$0.466346\pi$
$938$	0	0
$939$	2.58312	0.0842970
$940$	0	0
$941$	29.8484	0.973030	0.486515	−	0.873672i	$-0.338268\pi$
$941$	29.8484	0.973030	0.486515	+	0.873672i	$0.338268\pi$
$942$	0	0
$943$	20.6351	0.671970
$944$	0	0
$945$	−16.5062	−0.536947
$946$	0	0
$947$	−36.1069	−1.17332	−0.586658	−	0.809834i	$-0.699557\pi$
$947$	−36.1069	−1.17332	−0.586658	+	0.809834i	$0.699557\pi$
$948$	0	0
$949$	48.9703	1.58964
$950$	0	0
$951$	−1.79916	−0.0583417
$952$	0	0
$953$	34.5190	1.11818	0.559091	−	0.829107i	$-0.311150\pi$
$953$	34.5190	1.11818	0.559091	+	0.829107i	$0.311150\pi$
$954$	0	0
$955$	14.7337	0.476772
$956$	0	0
$957$	0	0
$958$	0	0
$959$	25.8730	0.835482
$960$	0	0
$961$	−29.7299	−0.959030
$962$	0	0
$963$	−20.7216	−0.667743
$964$	0	0
$965$	0.314976	0.0101394
$966$	0	0
$967$	−28.4086	−0.913559	−0.456780	−	0.889580i	$-0.650997\pi$
$967$	−28.4086	−0.913559	−0.456780	+	0.889580i	$0.650997\pi$
$968$	0	0
$969$	3.47026	0.111481
$970$	0	0
$971$	30.9718	0.993932	0.496966	−	0.867770i	$-0.334447\pi$
$971$	30.9718	0.993932	0.496966	+	0.867770i	$0.334447\pi$
$972$	0	0
$973$	−21.7680	−0.697850
$974$	0	0
$975$	−4.35390	−0.139436
$976$	0	0
$977$	−38.7235	−1.23887	−0.619437	−	0.785046i	$-0.712639\pi$
$977$	−38.7235	−1.23887	−0.619437	+	0.785046i	$0.712639\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−15.5149	−0.495352
$982$	0	0
$983$	−11.1197	−0.354663	−0.177331	−	0.984151i	$-0.556746\pi$
$983$	−11.1197	−0.354663	−0.177331	+	0.984151i	$0.556746\pi$
$984$	0	0
$985$	1.41533	0.0450962
$986$	0	0
$987$	27.8249	0.885677
$988$	0	0
$989$	32.3193	1.02769
$990$	0	0
$991$	11.5682	0.367475	0.183737	−	0.982975i	$-0.441180\pi$
$991$	11.5682	0.367475	0.183737	+	0.982975i	$0.441180\pi$
$992$	0	0
$993$	−4.73940	−0.150400
$994$	0	0
$995$	22.3049	0.707112
$996$	0	0
$997$	−28.1683	−0.892097	−0.446049	−	0.895009i	$-0.647169\pi$
$997$	−28.1683	−0.892097	−0.446049	+	0.895009i	$0.647169\pi$
$998$	0	0
$999$	28.0750	0.888253

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.cy.1.4		6
4.3	odd	2		4840.2.a.be.1.3		6
11.7	odd	10		880.2.bo.j.401.2		12
11.8	odd	10		880.2.bo.j.801.2		12
11.10	odd	2		9680.2.a.cx.1.4		6
44.7	even	10		440.2.y.b.401.2	yes	12
44.19	even	10		440.2.y.b.361.2	✓	12
44.43	even	2		4840.2.a.bf.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.b.361.2	✓	12	44.19	even	10
440.2.y.b.401.2	yes	12	44.7	even	10
880.2.bo.j.401.2		12	11.7	odd	10
880.2.bo.j.801.2		12	11.8	odd	10
4840.2.a.be.1.3		6	4.3	odd	2
4840.2.a.bf.1.3		6	44.43	even	2
9680.2.a.cx.1.4		6	11.10	odd	2
9680.2.a.cy.1.4		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q-0.719585 q^{3} -1.00000 q^{5} +4.18418 q^{7} -2.48220 q^{9} +6.05056 q^{13} +0.719585 q^{15} -6.52180 q^{17} +0.739455 q^{19} -3.01088 q^{21} +5.13093 q^{23} +1.00000 q^{25} +3.94491 q^{27} -2.08036 q^{29} +1.12698 q^{31} -4.18418 q^{35} +7.11676 q^{37} -4.35390 q^{39} +4.02170 q^{41} +6.29891 q^{43} +2.48220 q^{45} -9.24147 q^{47} +10.5074 q^{49} +4.69299 q^{51} +7.77015 q^{53} -0.532101 q^{57} -6.16416 q^{59} -14.6869 q^{61} -10.3860 q^{63} -6.05056 q^{65} +5.53268 q^{67} -3.69214 q^{69} -8.87252 q^{71} +8.09351 q^{73} -0.719585 q^{75} -0.570482 q^{79} +4.60789 q^{81} -15.2224 q^{83} +6.52180 q^{85} +1.49700 q^{87} -1.33867 q^{89} +25.3167 q^{91} -0.810957 q^{93} -0.739455 q^{95} +9.96643 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} + 6 q^{13} + 2 q^{15} - 11 q^{17} - 11 q^{19} - 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} + 6 q^{29} - q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} + 4 q^{41} + 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 10 * q^9 + 6 * q^13 + 2 * q^15 - 11 * q^17 - 11 * q^19 - 2 * q^21 - 18 * q^23 + 6 * q^25 + q^27 + 6 * q^29 - q^31 - 6 * q^35 + 4 * q^37 + 27 * q^39 + 4 * q^41 + 3 * q^43 - 10 * q^45 - 14 * q^47 + 8 * q^49 + 31 * q^51 + 14 * q^53 + 5 * q^57 - 2 * q^59 + 4 * q^61 - 16 * q^63 - 6 * q^65 - 11 * q^67 + 8 * q^69 - 7 * q^71 + 9 * q^73 - 2 * q^75 + 36 * q^79 + 30 * q^81 - 45 * q^83 + 11 * q^85 + 25 * q^87 + q^89 + 8 * q^91 + 55 * q^93 + 11 * q^95 + 20 * q^97

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