LMFDB - Embedded newform 6534.2.a.bp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6534,2,Mod(1,6534)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6534, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6534.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6534 = 2 \cdot 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6534.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:	$52.1742526807$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.54138$ of defining polynomial
Character	$\chi$	$=$	6534.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.54138 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +3.54138 q^{5} -1.00000 q^{7} -1.00000 q^{8} -3.54138 q^{10} -0.458619 q^{13} +1.00000 q^{14} +1.00000 q^{16} -7.08276 q^{17} -4.54138 q^{19} +3.54138 q^{20} -0.541381 q^{23} +7.54138 q^{25} +0.458619 q^{26} -1.00000 q^{28} -3.00000 q^{29} +2.54138 q^{31} -1.00000 q^{32} +7.08276 q^{34} -3.54138 q^{35} -2.08276 q^{37} +4.54138 q^{38} -3.54138 q^{40} +9.00000 q^{41} +6.08276 q^{43} +0.541381 q^{46} +10.6241 q^{47} -6.00000 q^{49} -7.54138 q^{50} -0.458619 q^{52} -10.0828 q^{53} +1.00000 q^{56} +3.00000 q^{58} -7.62414 q^{59} +6.08276 q^{61} -2.54138 q^{62} +1.00000 q^{64} -1.62414 q^{65} -15.1655 q^{67} -7.08276 q^{68} +3.54138 q^{70} +6.00000 q^{71} +8.54138 q^{73} +2.08276 q^{74} -4.54138 q^{76} -14.6241 q^{79} +3.54138 q^{80} -9.00000 q^{82} +5.45862 q^{83} -25.0828 q^{85} -6.08276 q^{86} -6.54138 q^{89} +0.458619 q^{91} -0.541381 q^{92} -10.6241 q^{94} -16.0828 q^{95} -18.1655 q^{97} +6.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + q^5 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} - 2 q^{8} - q^{10} - 7 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} - 3 q^{19} + q^{20} + 5 q^{23} + 9 q^{25} + 7 q^{26} - 2 q^{28} - 6 q^{29} - q^{31} - 2 q^{32} + 2 q^{34} - q^{35} + 8 q^{37} + 3 q^{38} - q^{40} + 18 q^{41} - 5 q^{46} + 3 q^{47} - 12 q^{49} - 9 q^{50} - 7 q^{52} - 8 q^{53} + 2 q^{56} + 6 q^{58} + 3 q^{59} + q^{62} + 2 q^{64} + 15 q^{65} - 6 q^{67} - 2 q^{68} + q^{70} + 12 q^{71} + 11 q^{73} - 8 q^{74} - 3 q^{76} - 11 q^{79} + q^{80} - 18 q^{82} + 17 q^{83} - 38 q^{85} - 7 q^{89} + 7 q^{91} + 5 q^{92} - 3 q^{94} - 20 q^{95} - 12 q^{97} + 12 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + q^5 - 2 * q^7 - 2 * q^8 - q^10 - 7 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 - 3 * q^19 + q^20 + 5 * q^23 + 9 * q^25 + 7 * q^26 - 2 * q^28 - 6 * q^29 - q^31 - 2 * q^32 + 2 * q^34 - q^35 + 8 * q^37 + 3 * q^38 - q^40 + 18 * q^41 - 5 * q^46 + 3 * q^47 - 12 * q^49 - 9 * q^50 - 7 * q^52 - 8 * q^53 + 2 * q^56 + 6 * q^58 + 3 * q^59 + q^62 + 2 * q^64 + 15 * q^65 - 6 * q^67 - 2 * q^68 + q^70 + 12 * q^71 + 11 * q^73 - 8 * q^74 - 3 * q^76 - 11 * q^79 + q^80 - 18 * q^82 + 17 * q^83 - 38 * q^85 - 7 * q^89 + 7 * q^91 + 5 * q^92 - 3 * q^94 - 20 * q^95 - 12 * q^97 + 12 * q^98

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
$5$	3.54138	1.58375	0.791877	−	0.610681i	$-0.209104\pi$
$5$	3.54138	1.58375	0.791877	+	0.610681i	$0.209104\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.54138	−1.11988
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.458619	−0.127198	−0.0635990	−	0.997976i	$-0.520258\pi$
$13$	−0.458619	−0.127198	−0.0635990	+	0.997976i	$0.520258\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.08276	−1.71782	−0.858911	−	0.512125i	$-0.828859\pi$
$17$	−7.08276	−1.71782	−0.858911	+	0.512125i	$0.828859\pi$
$18$	0	0
$19$	−4.54138	−1.04186	−0.520932	−	0.853598i	$-0.674416\pi$
$19$	−4.54138	−1.04186	−0.520932	+	0.853598i	$0.674416\pi$
$20$	3.54138	0.791877
$21$	0	0
$22$	0	0
$23$	−0.541381	−0.112886	−0.0564429	−	0.998406i	$-0.517976\pi$
$23$	−0.541381	−0.112886	−0.0564429	+	0.998406i	$0.517976\pi$
$24$	0	0
$25$	7.54138	1.50828
$26$	0.458619	0.0899425
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	2.54138	0.456446	0.228223	−	0.973609i	$-0.426709\pi$
$31$	2.54138	0.456446	0.228223	+	0.973609i	$0.426709\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.08276	1.21468
$35$	−3.54138	−0.598603
$36$	0	0
$37$	−2.08276	−0.342404	−0.171202	−	0.985236i	$-0.554765\pi$
$37$	−2.08276	−0.342404	−0.171202	+	0.985236i	$0.554765\pi$
$38$	4.54138	0.736709
$39$	0	0
$40$	−3.54138	−0.559942
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	6.08276	0.927613	0.463806	−	0.885937i	$-0.346483\pi$
$43$	6.08276	0.927613	0.463806	+	0.885937i	$0.346483\pi$
$44$	0	0
$45$	0	0
$46$	0.541381	0.0798223
$47$	10.6241	1.54969	0.774845	−	0.632151i	$-0.217828\pi$
$47$	10.6241	1.54969	0.774845	+	0.632151i	$0.217828\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	−7.54138	−1.06651
$51$	0	0
$52$	−0.458619	−0.0635990
$53$	−10.0828	−1.38497	−0.692487	−	0.721430i	$-0.743485\pi$
$53$	−10.0828	−1.38497	−0.692487	+	0.721430i	$0.743485\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	3.00000	0.393919
$59$	−7.62414	−0.992579	−0.496289	−	0.868157i	$-0.665305\pi$
$59$	−7.62414	−0.992579	−0.496289	+	0.868157i	$0.665305\pi$
$60$	0	0
$61$	6.08276	0.778818	0.389409	−	0.921065i	$-0.372679\pi$
$61$	6.08276	0.778818	0.389409	+	0.921065i	$0.372679\pi$
$62$	−2.54138	−0.322756
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.62414	−0.201450
$66$	0	0
$67$	−15.1655	−1.85276	−0.926382	−	0.376585i	$-0.877098\pi$
$67$	−15.1655	−1.85276	−0.926382	+	0.376585i	$0.877098\pi$
$68$	−7.08276	−0.858911
$69$	0	0
$70$	3.54138	0.423276
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	8.54138	0.999693	0.499847	−	0.866114i	$-0.333390\pi$
$73$	8.54138	0.999693	0.499847	+	0.866114i	$0.333390\pi$
$74$	2.08276	0.242116
$75$	0	0
$76$	−4.54138	−0.520932
$77$	0	0
$78$	0	0
$79$	−14.6241	−1.64534	−0.822672	−	0.568516i	$-0.807518\pi$
$79$	−14.6241	−1.64534	−0.822672	+	0.568516i	$0.807518\pi$
$80$	3.54138	0.395938
$81$	0	0
$82$	−9.00000	−0.993884
$83$	5.45862	0.599161	0.299581	−	0.954071i	$-0.403153\pi$
$83$	5.45862	0.599161	0.299581	+	0.954071i	$0.403153\pi$
$84$	0	0
$85$	−25.0828	−2.72061
$86$	−6.08276	−0.655921
$87$	0	0
$88$	0	0
$89$	−6.54138	−0.693385	−0.346693	−	0.937979i	$-0.612695\pi$
$89$	−6.54138	−0.693385	−0.346693	+	0.937979i	$0.612695\pi$
$90$	0	0
$91$	0.458619	0.0480763
$92$	−0.541381	−0.0564429
$93$	0	0
$94$	−10.6241	−1.09580
$95$	−16.0828	−1.65006
$96$	0	0
$97$	−18.1655	−1.84443	−0.922215	−	0.386678i	$-0.873623\pi$
$97$	−18.1655	−1.84443	−0.922215	+	0.386678i	$0.873623\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	7.54138	0.754138
$101$	−19.0828	−1.89881	−0.949403	−	0.314061i	$-0.898310\pi$
$101$	−19.0828	−1.89881	−0.949403	+	0.314061i	$0.898310\pi$
$102$	0	0
$103$	13.1655	1.29724	0.648619	−	0.761113i	$-0.275347\pi$
$103$	13.1655	1.29724	0.648619	+	0.761113i	$0.275347\pi$
$104$	0.458619	0.0449713
$105$	0	0
$106$	10.0828	0.979324
$107$	13.0828	1.26476	0.632379	−	0.774659i	$-0.282078\pi$
$107$	13.0828	1.26476	0.632379	+	0.774659i	$0.282078\pi$
$108$	0	0
$109$	−9.45862	−0.905971	−0.452986	−	0.891518i	$-0.649641\pi$
$109$	−9.45862	−0.905971	−0.452986	+	0.891518i	$0.649641\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	15.5414	1.46201	0.731005	−	0.682372i	$-0.239052\pi$
$113$	15.5414	1.46201	0.731005	+	0.682372i	$0.239052\pi$
$114$	0	0
$115$	−1.91724	−0.178783
$116$	−3.00000	−0.278543
$117$	0	0
$118$	7.62414	0.701859
$119$	7.08276	0.649276
$120$	0	0
$121$	0	0
$122$	−6.08276	−0.550707
$123$	0	0
$124$	2.54138	0.228223
$125$	9.00000	0.804984
$126$	0	0
$127$	−9.45862	−0.839317	−0.419658	−	0.907682i	$-0.637850\pi$
$127$	−9.45862	−0.839317	−0.419658	+	0.907682i	$0.637850\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.62414	0.142447
$131$	4.08276	0.356713	0.178356	−	0.983966i	$-0.442922\pi$
$131$	4.08276	0.356713	0.178356	+	0.983966i	$0.442922\pi$
$132$	0	0
$133$	4.54138	0.393788
$134$	15.1655	1.31010
$135$	0	0
$136$	7.08276	0.607342
$137$	−11.4586	−0.978976	−0.489488	−	0.872010i	$-0.662816\pi$
$137$	−11.4586	−0.978976	−0.489488	+	0.872010i	$0.662816\pi$
$138$	0	0
$139$	−16.2483	−1.37816	−0.689081	−	0.724684i	$-0.741986\pi$
$139$	−16.2483	−1.37816	−0.689081	+	0.724684i	$0.741986\pi$
$140$	−3.54138	−0.299301
$141$	0	0
$142$	−6.00000	−0.503509
$143$	0	0
$144$	0	0
$145$	−10.6241	−0.882287
$146$	−8.54138	−0.706890
$147$	0	0
$148$	−2.08276	−0.171202
$149$	0.541381	0.0443517	0.0221758	−	0.999754i	$-0.492941\pi$
$149$	0.541381	0.0443517	0.0221758	+	0.999754i	$0.492941\pi$
$150$	0	0
$151$	−11.9172	−0.969811	−0.484906	−	0.874567i	$-0.661146\pi$
$151$	−11.9172	−0.969811	−0.484906	+	0.874567i	$0.661146\pi$
$152$	4.54138	0.368355
$153$	0	0
$154$	0	0
$155$	9.00000	0.722897
$156$	0	0
$157$	−16.2483	−1.29675	−0.648377	−	0.761319i	$-0.724552\pi$
$157$	−16.2483	−1.29675	−0.648377	+	0.761319i	$0.724552\pi$
$158$	14.6241	1.16343
$159$	0	0
$160$	−3.54138	−0.279971
$161$	0.541381	0.0426668
$162$	0	0
$163$	−19.5414	−1.53060	−0.765300	−	0.643674i	$-0.777409\pi$
$163$	−19.5414	−1.53060	−0.765300	+	0.643674i	$0.777409\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	−5.45862	−0.423671
$167$	−1.37586	−0.106467	−0.0532335	−	0.998582i	$-0.516953\pi$
$167$	−1.37586	−0.106467	−0.0532335	+	0.998582i	$0.516953\pi$
$168$	0	0
$169$	−12.7897	−0.983821
$170$	25.0828	1.92376
$171$	0	0
$172$	6.08276	0.463806
$173$	−10.0828	−0.766578	−0.383289	−	0.923628i	$-0.625209\pi$
$173$	−10.0828	−0.766578	−0.383289	+	0.923628i	$0.625209\pi$
$174$	0	0
$175$	−7.54138	−0.570075
$176$	0	0
$177$	0	0
$178$	6.54138	0.490297
$179$	−14.7069	−1.09925	−0.549623	−	0.835413i	$-0.685229\pi$
$179$	−14.7069	−1.09925	−0.549623	+	0.835413i	$0.685229\pi$
$180$	0	0
$181$	−10.5414	−0.783535	−0.391767	−	0.920064i	$-0.628136\pi$
$181$	−10.5414	−0.783535	−0.391767	+	0.920064i	$0.628136\pi$
$182$	−0.458619	−0.0339951
$183$	0	0
$184$	0.541381	0.0399112
$185$	−7.37586	−0.542284
$186$	0	0
$187$	0	0
$188$	10.6241	0.774845
$189$	0	0
$190$	16.0828	1.16677
$191$	16.6241	1.20288	0.601440	−	0.798918i	$-0.294594\pi$
$191$	16.6241	1.20288	0.601440	+	0.798918i	$0.294594\pi$
$192$	0	0
$193$	−7.54138	−0.542841	−0.271420	−	0.962461i	$-0.587493\pi$
$193$	−7.54138	−0.542841	−0.271420	+	0.962461i	$0.587493\pi$
$194$	18.1655	1.30421
$195$	0	0
$196$	−6.00000	−0.428571
$197$	13.0828	0.932108	0.466054	−	0.884756i	$-0.345675\pi$
$197$	13.0828	0.932108	0.466054	+	0.884756i	$0.345675\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	−7.54138	−0.533256
$201$	0	0
$202$	19.0828	1.34266
$203$	3.00000	0.210559
$204$	0	0
$205$	31.8724	2.22607
$206$	−13.1655	−0.917286
$207$	0	0
$208$	−0.458619	−0.0317995
$209$	0	0
$210$	0	0
$211$	−7.00000	−0.481900	−0.240950	−	0.970538i	$-0.577459\pi$
$211$	−7.00000	−0.481900	−0.240950	+	0.970538i	$0.577459\pi$
$212$	−10.0828	−0.692487
$213$	0	0
$214$	−13.0828	−0.894319
$215$	21.5414	1.46911
$216$	0	0
$217$	−2.54138	−0.172520
$218$	9.45862	0.640618
$219$	0	0
$220$	0	0
$221$	3.24829	0.218503
$222$	0	0
$223$	23.5414	1.57645	0.788224	−	0.615389i	$-0.211001\pi$
$223$	23.5414	1.57645	0.788224	+	0.615389i	$0.211001\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−15.5414	−1.03380
$227$	−7.37586	−0.489553	−0.244776	−	0.969580i	$-0.578715\pi$
$227$	−7.37586	−0.489553	−0.244776	+	0.969580i	$0.578715\pi$
$228$	0	0
$229$	−11.3759	−0.751738	−0.375869	−	0.926673i	$-0.622656\pi$
$229$	−11.3759	−0.751738	−0.375869	+	0.926673i	$0.622656\pi$
$230$	1.91724	0.126419
$231$	0	0
$232$	3.00000	0.196960
$233$	11.4586	0.750679	0.375340	−	0.926887i	$-0.377526\pi$
$233$	11.4586	0.750679	0.375340	+	0.926887i	$0.377526\pi$
$234$	0	0
$235$	37.6241	2.45433
$236$	−7.62414	−0.496289
$237$	0	0
$238$	−7.08276	−0.459107
$239$	−4.08276	−0.264092	−0.132046	−	0.991244i	$-0.542155\pi$
$239$	−4.08276	−0.264092	−0.132046	+	0.991244i	$0.542155\pi$
$240$	0	0
$241$	−6.16553	−0.397156	−0.198578	−	0.980085i	$-0.563632\pi$
$241$	−6.16553	−0.397156	−0.198578	+	0.980085i	$0.563632\pi$
$242$	0	0
$243$	0	0
$244$	6.08276	0.389409
$245$	−21.2483	−1.35750
$246$	0	0
$247$	2.08276	0.132523
$248$	−2.54138	−0.161378
$249$	0	0
$250$	−9.00000	−0.569210
$251$	−14.4586	−0.912620	−0.456310	−	0.889821i	$-0.650829\pi$
$251$	−14.4586	−0.912620	−0.456310	+	0.889821i	$0.650829\pi$
$252$	0	0
$253$	0	0
$254$	9.45862	0.593487
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.16553	0.322217	0.161108	−	0.986937i	$-0.448493\pi$
$257$	5.16553	0.322217	0.161108	+	0.986937i	$0.448493\pi$
$258$	0	0
$259$	2.08276	0.129417
$260$	−1.62414	−0.100725
$261$	0	0
$262$	−4.08276	−0.252234
$263$	15.2483	0.940250	0.470125	−	0.882600i	$-0.344209\pi$
$263$	15.2483	0.940250	0.470125	+	0.882600i	$0.344209\pi$
$264$	0	0
$265$	−35.7069	−2.19346
$266$	−4.54138	−0.278450
$267$	0	0
$268$	−15.1655	−0.926382
$269$	−14.7069	−0.896696	−0.448348	−	0.893859i	$-0.647987\pi$
$269$	−14.7069	−0.896696	−0.448348	+	0.893859i	$0.647987\pi$
$270$	0	0
$271$	−8.62414	−0.523879	−0.261940	−	0.965084i	$-0.584362\pi$
$271$	−8.62414	−0.523879	−0.261940	+	0.965084i	$0.584362\pi$
$272$	−7.08276	−0.429456
$273$	0	0
$274$	11.4586	0.692240
$275$	0	0
$276$	0	0
$277$	14.2483	0.856097	0.428048	−	0.903756i	$-0.359201\pi$
$277$	14.2483	0.856097	0.428048	+	0.903756i	$0.359201\pi$
$278$	16.2483	0.974508
$279$	0	0
$280$	3.54138	0.211638
$281$	15.7897	0.941933	0.470966	−	0.882151i	$-0.343905\pi$
$281$	15.7897	0.941933	0.470966	+	0.882151i	$0.343905\pi$
$282$	0	0
$283$	−33.1655	−1.97149	−0.985743	−	0.168258i	$-0.946186\pi$
$283$	−33.1655	−1.97149	−0.985743	+	0.168258i	$0.946186\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	−9.00000	−0.531253
$288$	0	0
$289$	33.1655	1.95091
$290$	10.6241	0.623871
$291$	0	0
$292$	8.54138	0.499847
$293$	−18.5414	−1.08320	−0.541600	−	0.840637i	$-0.682181\pi$
$293$	−18.5414	−1.08320	−0.541600	+	0.840637i	$0.682181\pi$
$294$	0	0
$295$	−27.0000	−1.57200
$296$	2.08276	0.121058
$297$	0	0
$298$	−0.541381	−0.0313614
$299$	0.248288	0.0143588
$300$	0	0
$301$	−6.08276	−0.350605
$302$	11.9172	0.685760
$303$	0	0
$304$	−4.54138	−0.260466
$305$	21.5414	1.23346
$306$	0	0
$307$	−1.29309	−0.0738008	−0.0369004	−	0.999319i	$-0.511748\pi$
$307$	−1.29309	−0.0738008	−0.0369004	+	0.999319i	$0.511748\pi$
$308$	0	0
$309$	0	0
$310$	−9.00000	−0.511166
$311$	10.0828	0.571741	0.285871	−	0.958268i	$-0.407717\pi$
$311$	10.0828	0.571741	0.285871	+	0.958268i	$0.407717\pi$
$312$	0	0
$313$	−2.62414	−0.148325	−0.0741627	−	0.997246i	$-0.523628\pi$
$313$	−2.62414	−0.148325	−0.0741627	+	0.997246i	$0.523628\pi$
$314$	16.2483	0.916944
$315$	0	0
$316$	−14.6241	−0.822672
$317$	−9.24829	−0.519436	−0.259718	−	0.965685i	$-0.583630\pi$
$317$	−9.24829	−0.519436	−0.259718	+	0.965685i	$0.583630\pi$
$318$	0	0
$319$	0	0
$320$	3.54138	0.197969
$321$	0	0
$322$	−0.541381	−0.0301700
$323$	32.1655	1.78974
$324$	0	0
$325$	−3.45862	−0.191850
$326$	19.5414	1.08230
$327$	0	0
$328$	−9.00000	−0.496942
$329$	−10.6241	−0.585728
$330$	0	0
$331$	8.24829	0.453367	0.226683	−	0.973968i	$-0.427212\pi$
$331$	8.24829	0.453367	0.226683	+	0.973968i	$0.427212\pi$
$332$	5.45862	0.299581
$333$	0	0
$334$	1.37586	0.0752835
$335$	−53.7069	−2.93432
$336$	0	0
$337$	0.917237	0.0499651	0.0249826	−	0.999688i	$-0.492047\pi$
$337$	0.917237	0.0499651	0.0249826	+	0.999688i	$0.492047\pi$
$338$	12.7897	0.695666
$339$	0	0
$340$	−25.0828	−1.36030
$341$	0	0
$342$	0	0
$343$	13.0000	0.701934
$344$	−6.08276	−0.327961
$345$	0	0
$346$	10.0828	0.542053
$347$	36.7897	1.97497	0.987486	−	0.157704i	$-0.0504091\pi$
$347$	36.7897	1.97497	0.987486	+	0.157704i	$0.0504091\pi$
$348$	0	0
$349$	−24.7069	−1.32253	−0.661265	−	0.750152i	$-0.729980\pi$
$349$	−24.7069	−1.32253	−0.661265	+	0.750152i	$0.729980\pi$
$350$	7.54138	0.403104
$351$	0	0
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	21.2483	1.12774
$356$	−6.54138	−0.346693
$357$	0	0
$358$	14.7069	0.777284
$359$	24.2483	1.27978	0.639888	−	0.768468i	$-0.278981\pi$
$359$	24.2483	1.27978	0.639888	+	0.768468i	$0.278981\pi$
$360$	0	0
$361$	1.62414	0.0854813
$362$	10.5414	0.554043
$363$	0	0
$364$	0.458619	0.0240382
$365$	30.2483	1.58327
$366$	0	0
$367$	36.6241	1.91176	0.955882	−	0.293750i	$-0.0949032\pi$
$367$	36.6241	1.91176	0.955882	+	0.293750i	$0.0949032\pi$
$368$	−0.541381	−0.0282214
$369$	0	0
$370$	7.37586	0.383453
$371$	10.0828	0.523471
$372$	0	0
$373$	−19.5414	−1.01181	−0.505907	−	0.862588i	$-0.668842\pi$
$373$	−19.5414	−1.01181	−0.505907	+	0.862588i	$0.668842\pi$
$374$	0	0
$375$	0	0
$376$	−10.6241	−0.547898
$377$	1.37586	0.0708602
$378$	0	0
$379$	3.62414	0.186160	0.0930799	−	0.995659i	$-0.470329\pi$
$379$	3.62414	0.186160	0.0930799	+	0.995659i	$0.470329\pi$
$380$	−16.0828	−0.825028
$381$	0	0
$382$	−16.6241	−0.850565
$383$	1.91724	0.0979663	0.0489831	−	0.998800i	$-0.484402\pi$
$383$	1.91724	0.0979663	0.0489831	+	0.998800i	$0.484402\pi$
$384$	0	0
$385$	0	0
$386$	7.54138	0.383846
$387$	0	0
$388$	−18.1655	−0.922215
$389$	−36.2483	−1.83786	−0.918931	−	0.394419i	$-0.870946\pi$
$389$	−36.2483	−1.83786	−0.918931	+	0.394419i	$0.870946\pi$
$390$	0	0
$391$	3.83447	0.193918
$392$	6.00000	0.303046
$393$	0	0
$394$	−13.0828	−0.659100
$395$	−51.7897	−2.60582
$396$	0	0
$397$	27.0828	1.35924	0.679622	−	0.733562i	$-0.262144\pi$
$397$	27.0828	1.35924	0.679622	+	0.733562i	$0.262144\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	7.54138	0.377069
$401$	−9.54138	−0.476474	−0.238237	−	0.971207i	$-0.576569\pi$
$401$	−9.54138	−0.476474	−0.238237	+	0.971207i	$0.576569\pi$
$402$	0	0
$403$	−1.16553	−0.0580589
$404$	−19.0828	−0.949403
$405$	0	0
$406$	−3.00000	−0.148888
$407$	0	0
$408$	0	0
$409$	25.7069	1.27112	0.635562	−	0.772050i	$-0.280768\pi$
$409$	25.7069	1.27112	0.635562	+	0.772050i	$0.280768\pi$
$410$	−31.8724	−1.57407
$411$	0	0
$412$	13.1655	0.648619
$413$	7.62414	0.375160
$414$	0	0
$415$	19.3311	0.948924
$416$	0.458619	0.0224856
$417$	0	0
$418$	0	0
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	7.00000	0.340755
$423$	0	0
$424$	10.0828	0.489662
$425$	−53.4138	−2.59095
$426$	0	0
$427$	−6.08276	−0.294366
$428$	13.0828	0.632379
$429$	0	0
$430$	−21.5414	−1.03882
$431$	−3.24829	−0.156465	−0.0782323	−	0.996935i	$-0.524928\pi$
$431$	−3.24829	−0.156465	−0.0782323	+	0.996935i	$0.524928\pi$
$432$	0	0
$433$	0.375856	0.0180625	0.00903125	−	0.999959i	$-0.497125\pi$
$433$	0.375856	0.0180625	0.00903125	+	0.999959i	$0.497125\pi$
$434$	2.54138	0.121990
$435$	0	0
$436$	−9.45862	−0.452986
$437$	2.45862	0.117612
$438$	0	0
$439$	15.3311	0.731711	0.365856	−	0.930672i	$-0.380776\pi$
$439$	15.3311	0.731711	0.365856	+	0.930672i	$0.380776\pi$
$440$	0	0
$441$	0	0
$442$	−3.24829	−0.154505
$443$	1.08276	0.0514436	0.0257218	−	0.999669i	$-0.491812\pi$
$443$	1.08276	0.0514436	0.0257218	+	0.999669i	$0.491812\pi$
$444$	0	0
$445$	−23.1655	−1.09815
$446$	−23.5414	−1.11472
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−12.8345	−0.605696	−0.302848	−	0.953039i	$-0.597938\pi$
$449$	−12.8345	−0.605696	−0.302848	+	0.953039i	$0.597938\pi$
$450$	0	0
$451$	0	0
$452$	15.5414	0.731005
$453$	0	0
$454$	7.37586	0.346166
$455$	1.62414	0.0761410
$456$	0	0
$457$	−24.1655	−1.13042	−0.565208	−	0.824949i	$-0.691204\pi$
$457$	−24.1655	−1.13042	−0.565208	+	0.824949i	$0.691204\pi$
$458$	11.3759	0.531559
$459$	0	0
$460$	−1.91724	−0.0893917
$461$	17.1655	0.799478	0.399739	−	0.916629i	$-0.369101\pi$
$461$	17.1655	0.799478	0.399739	+	0.916629i	$0.369101\pi$
$462$	0	0
$463$	2.78967	0.129647	0.0648235	−	0.997897i	$-0.479352\pi$
$463$	2.78967	0.129647	0.0648235	+	0.997897i	$0.479352\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−11.4586	−0.530810
$467$	10.6241	0.491627	0.245813	−	0.969317i	$-0.420945\pi$
$467$	10.6241	0.491627	0.245813	+	0.969317i	$0.420945\pi$
$468$	0	0
$469$	15.1655	0.700279
$470$	−37.6241	−1.73547
$471$	0	0
$472$	7.62414	0.350930
$473$	0	0
$474$	0	0
$475$	−34.2483	−1.57142
$476$	7.08276	0.324638
$477$	0	0
$478$	4.08276	0.186741
$479$	−19.8724	−0.907995	−0.453997	−	0.891003i	$-0.650002\pi$
$479$	−19.8724	−0.907995	−0.453997	+	0.891003i	$0.650002\pi$
$480$	0	0
$481$	0.955194	0.0435531
$482$	6.16553	0.280832
$483$	0	0
$484$	0	0
$485$	−64.3311	−2.92112
$486$	0	0
$487$	11.0000	0.498458	0.249229	−	0.968445i	$-0.419823\pi$
$487$	11.0000	0.498458	0.249229	+	0.968445i	$0.419823\pi$
$488$	−6.08276	−0.275354
$489$	0	0
$490$	21.2483	0.959900
$491$	7.91724	0.357300	0.178650	−	0.983913i	$-0.442827\pi$
$491$	7.91724	0.357300	0.178650	+	0.983913i	$0.442827\pi$
$492$	0	0
$493$	21.2483	0.956975
$494$	−2.08276	−0.0937079
$495$	0	0
$496$	2.54138	0.114111
$497$	−6.00000	−0.269137
$498$	0	0
$499$	−3.75171	−0.167950	−0.0839749	−	0.996468i	$-0.526762\pi$
$499$	−3.75171	−0.167950	−0.0839749	+	0.996468i	$0.526762\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	14.4586	0.645320
$503$	30.5414	1.36177	0.680886	−	0.732389i	$-0.261595\pi$
$503$	30.5414	1.36177	0.680886	+	0.732389i	$0.261595\pi$
$504$	0	0
$505$	−67.5793	−3.00724
$506$	0	0
$507$	0	0
$508$	−9.45862	−0.419658
$509$	20.4138	0.904826	0.452413	−	0.891808i	$-0.350563\pi$
$509$	20.4138	0.904826	0.452413	+	0.891808i	$0.350563\pi$
$510$	0	0
$511$	−8.54138	−0.377848
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−5.16553	−0.227842
$515$	46.6241	2.05451
$516$	0	0
$517$	0	0
$518$	−2.08276	−0.0915113
$519$	0	0
$520$	1.62414	0.0712234
$521$	−7.37586	−0.323142	−0.161571	−	0.986861i	$-0.551656\pi$
$521$	−7.37586	−0.323142	−0.161571	+	0.986861i	$0.551656\pi$
$522$	0	0
$523$	−26.6241	−1.16419	−0.582096	−	0.813120i	$-0.697767\pi$
$523$	−26.6241	−1.16419	−0.582096	+	0.813120i	$0.697767\pi$
$524$	4.08276	0.178356
$525$	0	0
$526$	−15.2483	−0.664857
$527$	−18.0000	−0.784092
$528$	0	0
$529$	−22.7069	−0.987257
$530$	35.7069	1.55101
$531$	0	0
$532$	4.54138	0.196894
$533$	−4.12757	−0.178785
$534$	0	0
$535$	46.3311	2.00307
$536$	15.1655	0.655051
$537$	0	0
$538$	14.7069	0.634060
$539$	0	0
$540$	0	0
$541$	21.3311	0.917093	0.458547	−	0.888670i	$-0.348370\pi$
$541$	21.3311	0.917093	0.458547	+	0.888670i	$0.348370\pi$
$542$	8.62414	0.370439
$543$	0	0
$544$	7.08276	0.303671
$545$	−33.4966	−1.43484
$546$	0	0
$547$	39.8724	1.70482	0.852411	−	0.522872i	$-0.175140\pi$
$547$	39.8724	1.70482	0.852411	+	0.522872i	$0.175140\pi$
$548$	−11.4586	−0.489488
$549$	0	0
$550$	0	0
$551$	13.6241	0.580408
$552$	0	0
$553$	14.6241	0.621882
$554$	−14.2483	−0.605352
$555$	0	0
$556$	−16.2483	−0.689081
$557$	13.8724	0.587794	0.293897	−	0.955837i	$-0.405048\pi$
$557$	13.8724	0.587794	0.293897	+	0.955837i	$0.405048\pi$
$558$	0	0
$559$	−2.78967	−0.117990
$560$	−3.54138	−0.149651
$561$	0	0
$562$	−15.7897	−0.666047
$563$	−17.4138	−0.733905	−0.366952	−	0.930240i	$-0.619599\pi$
$563$	−17.4138	−0.733905	−0.366952	+	0.930240i	$0.619599\pi$
$564$	0	0
$565$	55.0380	2.31546
$566$	33.1655	1.39405
$567$	0	0
$568$	−6.00000	−0.251754
$569$	−29.1655	−1.22268	−0.611341	−	0.791367i	$-0.709370\pi$
$569$	−29.1655	−1.22268	−0.611341	+	0.791367i	$0.709370\pi$
$570$	0	0
$571$	12.0828	0.505648	0.252824	−	0.967512i	$-0.418641\pi$
$571$	12.0828	0.505648	0.252824	+	0.967512i	$0.418641\pi$
$572$	0	0
$573$	0	0
$574$	9.00000	0.375653
$575$	−4.08276	−0.170263
$576$	0	0
$577$	−37.2483	−1.55067	−0.775333	−	0.631552i	$-0.782418\pi$
$577$	−37.2483	−1.55067	−0.775333	+	0.631552i	$0.782418\pi$
$578$	−33.1655	−1.37950
$579$	0	0
$580$	−10.6241	−0.441144
$581$	−5.45862	−0.226462
$582$	0	0
$583$	0	0
$584$	−8.54138	−0.353445
$585$	0	0
$586$	18.5414	0.765937
$587$	−46.6241	−1.92438	−0.962192	−	0.272371i	$-0.912192\pi$
$587$	−46.6241	−1.92438	−0.962192	+	0.272371i	$0.912192\pi$
$588$	0	0
$589$	−11.5414	−0.475554
$590$	27.0000	1.11157
$591$	0	0
$592$	−2.08276	−0.0856010
$593$	−13.3759	−0.549281	−0.274640	−	0.961547i	$-0.588559\pi$
$593$	−13.3759	−0.549281	−0.274640	+	0.961547i	$0.588559\pi$
$594$	0	0
$595$	25.0828	1.02829
$596$	0.541381	0.0221758
$597$	0	0
$598$	−0.248288	−0.0101532
$599$	4.62414	0.188937	0.0944687	−	0.995528i	$-0.469885\pi$
$599$	4.62414	0.188937	0.0944687	+	0.995528i	$0.469885\pi$
$600$	0	0
$601$	−36.4138	−1.48535	−0.742675	−	0.669652i	$-0.766443\pi$
$601$	−36.4138	−1.48535	−0.742675	+	0.669652i	$0.766443\pi$
$602$	6.08276	0.247915
$603$	0	0
$604$	−11.9172	−0.484906
$605$	0	0
$606$	0	0
$607$	−30.4138	−1.23446	−0.617229	−	0.786783i	$-0.711745\pi$
$607$	−30.4138	−1.23446	−0.617229	+	0.786783i	$0.711745\pi$
$608$	4.54138	0.184177
$609$	0	0
$610$	−21.5414	−0.872185
$611$	−4.87243	−0.197117
$612$	0	0
$613$	−14.3311	−0.578826	−0.289413	−	0.957204i	$-0.593460\pi$
$613$	−14.3311	−0.578826	−0.289413	+	0.957204i	$0.593460\pi$
$614$	1.29309	0.0521850
$615$	0	0
$616$	0	0
$617$	30.5414	1.22955	0.614775	−	0.788703i	$-0.289247\pi$
$617$	30.5414	1.22955	0.614775	+	0.788703i	$0.289247\pi$
$618$	0	0
$619$	−2.62414	−0.105473	−0.0527366	−	0.998608i	$-0.516794\pi$
$619$	−2.62414	−0.105473	−0.0527366	+	0.998608i	$0.516794\pi$
$620$	9.00000	0.361449
$621$	0	0
$622$	−10.0828	−0.404282
$623$	6.54138	0.262075
$624$	0	0
$625$	−5.83447	−0.233379
$626$	2.62414	0.104882
$627$	0	0
$628$	−16.2483	−0.648377
$629$	14.7517	0.588189
$630$	0	0
$631$	9.08276	0.361579	0.180790	−	0.983522i	$-0.442135\pi$
$631$	9.08276	0.361579	0.180790	+	0.983522i	$0.442135\pi$
$632$	14.6241	0.581717
$633$	0	0
$634$	9.24829	0.367296
$635$	−33.4966	−1.32927
$636$	0	0
$637$	2.75171	0.109027
$638$	0	0
$639$	0	0
$640$	−3.54138	−0.139985
$641$	4.62414	0.182643	0.0913213	−	0.995821i	$-0.470891\pi$
$641$	4.62414	0.182643	0.0913213	+	0.995821i	$0.470891\pi$
$642$	0	0
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0.541381	0.0213334
$645$	0	0
$646$	−32.1655	−1.26554
$647$	−19.3311	−0.759982	−0.379991	−	0.924990i	$-0.624073\pi$
$647$	−19.3311	−0.759982	−0.379991	+	0.924990i	$0.624073\pi$
$648$	0	0
$649$	0	0
$650$	3.45862	0.135658
$651$	0	0
$652$	−19.5414	−0.765300
$653$	−27.2483	−1.06631	−0.533154	−	0.846018i	$-0.678993\pi$
$653$	−27.2483	−1.06631	−0.533154	+	0.846018i	$0.678993\pi$
$654$	0	0
$655$	14.4586	0.564945
$656$	9.00000	0.351391
$657$	0	0
$658$	10.6241	0.414172
$659$	−31.8724	−1.24157	−0.620787	−	0.783979i	$-0.713187\pi$
$659$	−31.8724	−1.24157	−0.620787	+	0.783979i	$0.713187\pi$
$660$	0	0
$661$	16.4138	0.638423	0.319212	−	0.947683i	$-0.396582\pi$
$661$	16.4138	0.638423	0.319212	+	0.947683i	$0.396582\pi$
$662$	−8.24829	−0.320579
$663$	0	0
$664$	−5.45862	−0.211835
$665$	16.0828	0.623663
$666$	0	0
$667$	1.62414	0.0628871
$668$	−1.37586	−0.0532335
$669$	0	0
$670$	53.7069	2.07488
$671$	0	0
$672$	0	0
$673$	−4.00000	−0.154189	−0.0770943	−	0.997024i	$-0.524564\pi$
$673$	−4.00000	−0.154189	−0.0770943	+	0.997024i	$0.524564\pi$
$674$	−0.917237	−0.0353307
$675$	0	0
$676$	−12.7897	−0.491910
$677$	−15.5414	−0.597304	−0.298652	−	0.954362i	$-0.596537\pi$
$677$	−15.5414	−0.597304	−0.298652	+	0.954362i	$0.596537\pi$
$678$	0	0
$679$	18.1655	0.697129
$680$	25.0828	0.961880
$681$	0	0
$682$	0	0
$683$	−22.3759	−0.856188	−0.428094	−	0.903734i	$-0.640815\pi$
$683$	−22.3759	−0.856188	−0.428094	+	0.903734i	$0.640815\pi$
$684$	0	0
$685$	−40.5793	−1.55046
$686$	−13.0000	−0.496342
$687$	0	0
$688$	6.08276	0.231903
$689$	4.62414	0.176166
$690$	0	0
$691$	−3.45862	−0.131572	−0.0657860	−	0.997834i	$-0.520955\pi$
$691$	−3.45862	−0.131572	−0.0657860	+	0.997834i	$0.520955\pi$
$692$	−10.0828	−0.383289
$693$	0	0
$694$	−36.7897	−1.39652
$695$	−57.5414	−2.18267
$696$	0	0
$697$	−63.7449	−2.41451
$698$	24.7069	0.935170
$699$	0	0
$700$	−7.54138	−0.285037
$701$	46.8724	1.77035	0.885174	−	0.465261i	$-0.154039\pi$
$701$	46.8724	1.77035	0.885174	+	0.465261i	$0.154039\pi$
$702$	0	0
$703$	9.45862	0.356739
$704$	0	0
$705$	0	0
$706$	−30.0000	−1.12906
$707$	19.0828	0.717681
$708$	0	0
$709$	31.1655	1.17045	0.585223	−	0.810872i	$-0.301007\pi$
$709$	31.1655	1.17045	0.585223	+	0.810872i	$0.301007\pi$
$710$	−21.2483	−0.797434
$711$	0	0
$712$	6.54138	0.245149
$713$	−1.37586	−0.0515262
$714$	0	0
$715$	0	0
$716$	−14.7069	−0.549623
$717$	0	0
$718$	−24.2483	−0.904938
$719$	−45.4966	−1.69674	−0.848368	−	0.529407i	$-0.822414\pi$
$719$	−45.4966	−1.69674	−0.848368	+	0.529407i	$0.822414\pi$
$720$	0	0
$721$	−13.1655	−0.490310
$722$	−1.62414	−0.0604444
$723$	0	0
$724$	−10.5414	−0.391767
$725$	−22.6241	−0.840240
$726$	0	0
$727$	18.9172	0.701602	0.350801	−	0.936450i	$-0.385909\pi$
$727$	18.9172	0.701602	0.350801	+	0.936450i	$0.385909\pi$
$728$	−0.458619	−0.0169975
$729$	0	0
$730$	−30.2483	−1.11954
$731$	−43.0828	−1.59347
$732$	0	0
$733$	36.8724	1.36191	0.680957	−	0.732323i	$-0.261564\pi$
$733$	36.8724	1.36191	0.680957	+	0.732323i	$0.261564\pi$
$734$	−36.6241	−1.35182
$735$	0	0
$736$	0.541381	0.0199556
$737$	0	0
$738$	0	0
$739$	−5.37586	−0.197754	−0.0988770	−	0.995100i	$-0.531525\pi$
$739$	−5.37586	−0.197754	−0.0988770	+	0.995100i	$0.531525\pi$
$740$	−7.37586	−0.271142
$741$	0	0
$742$	−10.0828	−0.370150
$743$	20.7069	0.759663	0.379831	−	0.925056i	$-0.375982\pi$
$743$	20.7069	0.759663	0.379831	+	0.925056i	$0.375982\pi$
$744$	0	0
$745$	1.91724	0.0702421
$746$	19.5414	0.715461
$747$	0	0
$748$	0	0
$749$	−13.0828	−0.478034
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	10.6241	0.387423
$753$	0	0
$754$	−1.37586	−0.0501057
$755$	−42.2035	−1.53594
$756$	0	0
$757$	26.2483	0.954010	0.477005	−	0.878901i	$-0.341722\pi$
$757$	26.2483	0.954010	0.477005	+	0.878901i	$0.341722\pi$
$758$	−3.62414	−0.131635
$759$	0	0
$760$	16.0828	0.583383
$761$	36.2483	1.31400	0.657000	−	0.753891i	$-0.271825\pi$
$761$	36.2483	1.31400	0.657000	+	0.753891i	$0.271825\pi$
$762$	0	0
$763$	9.45862	0.342425
$764$	16.6241	0.601440
$765$	0	0
$766$	−1.91724	−0.0692726
$767$	3.49658	0.126254
$768$	0	0
$769$	−40.7897	−1.47091	−0.735457	−	0.677572i	$-0.763032\pi$
$769$	−40.7897	−1.47091	−0.735457	+	0.677572i	$0.763032\pi$
$770$	0	0
$771$	0	0
$772$	−7.54138	−0.271420
$773$	−11.7517	−0.422680	−0.211340	−	0.977413i	$-0.567783\pi$
$773$	−11.7517	−0.422680	−0.211340	+	0.977413i	$0.567783\pi$
$774$	0	0
$775$	19.1655	0.688446
$776$	18.1655	0.652104
$777$	0	0
$778$	36.2483	1.29956
$779$	−40.8724	−1.46441
$780$	0	0
$781$	0	0
$782$	−3.83447	−0.137121
$783$	0	0
$784$	−6.00000	−0.214286
$785$	−57.5414	−2.05374
$786$	0	0
$787$	33.3311	1.18812	0.594062	−	0.804419i	$-0.297523\pi$
$787$	33.3311	1.18812	0.594062	+	0.804419i	$0.297523\pi$
$788$	13.0828	0.466054
$789$	0	0
$790$	51.7897	1.84259
$791$	−15.5414	−0.552588
$792$	0	0
$793$	−2.78967	−0.0990640
$794$	−27.0828	−0.961131
$795$	0	0
$796$	8.00000	0.283552
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	−75.2483	−2.66209
$800$	−7.54138	−0.266628
$801$	0	0
$802$	9.54138	0.336918
$803$	0	0
$804$	0	0
$805$	1.91724	0.0675737
$806$	1.16553	0.0410539
$807$	0	0
$808$	19.0828	0.671329
$809$	6.83447	0.240287	0.120144	−	0.992757i	$-0.461664\pi$
$809$	6.83447	0.240287	0.120144	+	0.992757i	$0.461664\pi$
$810$	0	0
$811$	12.6241	0.443294	0.221647	−	0.975127i	$-0.428857\pi$
$811$	12.6241	0.443294	0.221647	+	0.975127i	$0.428857\pi$
$812$	3.00000	0.105279
$813$	0	0
$814$	0	0
$815$	−69.2035	−2.42409
$816$	0	0
$817$	−27.6241	−0.966446
$818$	−25.7069	−0.898821
$819$	0	0
$820$	31.8724	1.11303
$821$	−18.2931	−0.638433	−0.319217	−	0.947682i	$-0.603420\pi$
$821$	−18.2931	−0.638433	−0.319217	+	0.947682i	$0.603420\pi$
$822$	0	0
$823$	−42.1207	−1.46824	−0.734118	−	0.679022i	$-0.762404\pi$
$823$	−42.1207	−1.46824	−0.734118	+	0.679022i	$0.762404\pi$
$824$	−13.1655	−0.458643
$825$	0	0
$826$	−7.62414	−0.265278
$827$	44.1655	1.53579	0.767893	−	0.640578i	$-0.221305\pi$
$827$	44.1655	1.53579	0.767893	+	0.640578i	$0.221305\pi$
$828$	0	0
$829$	7.70691	0.267672	0.133836	−	0.991003i	$-0.457270\pi$
$829$	7.70691	0.267672	0.133836	+	0.991003i	$0.457270\pi$
$830$	−19.3311	−0.670991
$831$	0	0
$832$	−0.458619	−0.0158997
$833$	42.4966	1.47242
$834$	0	0
$835$	−4.87243	−0.168617
$836$	0	0
$837$	0	0
$838$	21.0000	0.725433
$839$	43.6241	1.50607	0.753036	−	0.657979i	$-0.228588\pi$
$839$	43.6241	1.50607	0.753036	+	0.657979i	$0.228588\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	1.00000	0.0344623
$843$	0	0
$844$	−7.00000	−0.240950
$845$	−45.2931	−1.55813
$846$	0	0
$847$	0	0
$848$	−10.0828	−0.346243
$849$	0	0
$850$	53.4138	1.83208
$851$	1.12757	0.0386526
$852$	0	0
$853$	−17.3311	−0.593404	−0.296702	−	0.954970i	$-0.595887\pi$
$853$	−17.3311	−0.593404	−0.296702	+	0.954970i	$0.595887\pi$
$854$	6.08276	0.208148
$855$	0	0
$856$	−13.0828	−0.447160
$857$	9.54138	0.325927	0.162964	−	0.986632i	$-0.447895\pi$
$857$	9.54138	0.325927	0.162964	+	0.986632i	$0.447895\pi$
$858$	0	0
$859$	48.5793	1.65751	0.828753	−	0.559615i	$-0.189051\pi$
$859$	48.5793	1.65751	0.828753	+	0.559615i	$0.189051\pi$
$860$	21.5414	0.734555
$861$	0	0
$862$	3.24829	0.110637
$863$	−7.08276	−0.241100	−0.120550	−	0.992707i	$-0.538466\pi$
$863$	−7.08276	−0.241100	−0.120550	+	0.992707i	$0.538466\pi$
$864$	0	0
$865$	−35.7069	−1.21407
$866$	−0.375856	−0.0127721
$867$	0	0
$868$	−2.54138	−0.0862601
$869$	0	0
$870$	0	0
$871$	6.95519	0.235668
$872$	9.45862	0.320309
$873$	0	0
$874$	−2.45862	−0.0831640
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−50.5793	−1.70794	−0.853971	−	0.520320i	$-0.825813\pi$
$877$	−50.5793	−1.70794	−0.853971	+	0.520320i	$0.825813\pi$
$878$	−15.3311	−0.517398
$879$	0	0
$880$	0	0
$881$	−0.541381	−0.0182396	−0.00911980	−	0.999958i	$-0.502903\pi$
$881$	−0.541381	−0.0182396	−0.00911980	+	0.999958i	$0.502903\pi$
$882$	0	0
$883$	30.5793	1.02908	0.514538	−	0.857467i	$-0.327963\pi$
$883$	30.5793	1.02908	0.514538	+	0.857467i	$0.327963\pi$
$884$	3.24829	0.109252
$885$	0	0
$886$	−1.08276	−0.0363761
$887$	−0.248288	−0.00833668	−0.00416834	−	0.999991i	$-0.501327\pi$
$887$	−0.248288	−0.00833668	−0.00416834	+	0.999991i	$0.501327\pi$
$888$	0	0
$889$	9.45862	0.317232
$890$	23.1655	0.776510
$891$	0	0
$892$	23.5414	0.788224
$893$	−48.2483	−1.61457
$894$	0	0
$895$	−52.0828	−1.74093
$896$	1.00000	0.0334077
$897$	0	0
$898$	12.8345	0.428292
$899$	−7.62414	−0.254279
$900$	0	0
$901$	71.4138	2.37914
$902$	0	0
$903$	0	0
$904$	−15.5414	−0.516899
$905$	−37.3311	−1.24093
$906$	0	0
$907$	−25.5414	−0.848088	−0.424044	−	0.905642i	$-0.639390\pi$
$907$	−25.5414	−0.848088	−0.424044	+	0.905642i	$0.639390\pi$
$908$	−7.37586	−0.244776
$909$	0	0
$910$	−1.62414	−0.0538398
$911$	54.2483	1.79733	0.898663	−	0.438640i	$-0.144540\pi$
$911$	54.2483	1.79733	0.898663	+	0.438640i	$0.144540\pi$
$912$	0	0
$913$	0	0
$914$	24.1655	0.799325
$915$	0	0
$916$	−11.3759	−0.375869
$917$	−4.08276	−0.134825
$918$	0	0
$919$	−17.3759	−0.573177	−0.286588	−	0.958054i	$-0.592521\pi$
$919$	−17.3759	−0.573177	−0.286588	+	0.958054i	$0.592521\pi$
$920$	1.91724	0.0632094
$921$	0	0
$922$	−17.1655	−0.565316
$923$	−2.75171	−0.0905737
$924$	0	0
$925$	−15.7069	−0.516440
$926$	−2.78967	−0.0916742
$927$	0	0
$928$	3.00000	0.0984798
$929$	−28.6241	−0.939128	−0.469564	−	0.882899i	$-0.655589\pi$
$929$	−28.6241	−0.939128	−0.469564	+	0.882899i	$0.655589\pi$
$930$	0	0
$931$	27.2483	0.893027
$932$	11.4586	0.375340
$933$	0	0
$934$	−10.6241	−0.347633
$935$	0	0
$936$	0	0
$937$	0.0827625	0.00270373	0.00135187	−	0.999999i	$-0.499570\pi$
$937$	0.0827625	0.00270373	0.00135187	+	0.999999i	$0.499570\pi$
$938$	−15.1655	−0.495172
$939$	0	0
$940$	37.6241	1.22716
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	−4.87243	−0.158668
$944$	−7.62414	−0.248145
$945$	0	0
$946$	0	0
$947$	30.2483	0.982937	0.491469	−	0.870895i	$-0.336460\pi$
$947$	30.2483	0.982937	0.491469	+	0.870895i	$0.336460\pi$
$948$	0	0
$949$	−3.91724	−0.127159
$950$	34.2483	1.11116
$951$	0	0
$952$	−7.08276	−0.229554
$953$	5.75171	0.186316	0.0931581	−	0.995651i	$-0.470304\pi$
$953$	5.75171	0.186316	0.0931581	+	0.995651i	$0.470304\pi$
$954$	0	0
$955$	58.8724	1.90507
$956$	−4.08276	−0.132046
$957$	0	0
$958$	19.8724	0.642049
$959$	11.4586	0.370018
$960$	0	0
$961$	−24.5414	−0.791657
$962$	−0.955194	−0.0307967
$963$	0	0
$964$	−6.16553	−0.198578
$965$	−26.7069	−0.859726
$966$	0	0
$967$	34.9552	1.12408	0.562042	−	0.827109i	$-0.310016\pi$
$967$	34.9552	1.12408	0.562042	+	0.827109i	$0.310016\pi$
$968$	0	0
$969$	0	0
$970$	64.3311	2.06555
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	0	0
$973$	16.2483	0.520896
$974$	−11.0000	−0.352463
$975$	0	0
$976$	6.08276	0.194704
$977$	2.16553	0.0692813	0.0346406	−	0.999400i	$-0.488971\pi$
$977$	2.16553	0.0692813	0.0346406	+	0.999400i	$0.488971\pi$
$978$	0	0
$979$	0	0
$980$	−21.2483	−0.678752
$981$	0	0
$982$	−7.91724	−0.252649
$983$	33.4966	1.06837	0.534187	−	0.845366i	$-0.320618\pi$
$983$	33.4966	1.06837	0.534187	+	0.845366i	$0.320618\pi$
$984$	0	0
$985$	46.3311	1.47623
$986$	−21.2483	−0.676683
$987$	0	0
$988$	2.08276	0.0662615
$989$	−3.29309	−0.104714
$990$	0	0
$991$	−7.00000	−0.222362	−0.111181	−	0.993800i	$-0.535463\pi$
$991$	−7.00000	−0.222362	−0.111181	+	0.993800i	$0.535463\pi$
$992$	−2.54138	−0.0806889
$993$	0	0
$994$	6.00000	0.190308
$995$	28.3311	0.898155
$996$	0	0
$997$	−7.54138	−0.238838	−0.119419	−	0.992844i	$-0.538103\pi$
$997$	−7.54138	−0.238838	−0.119419	+	0.992844i	$0.538103\pi$
$998$	3.75171	0.118758
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6534.2.a.bp.1.2	yes	2
3.2	odd	2		6534.2.a.cc.1.1	yes	2
11.10	odd	2		6534.2.a.cl.1.2	yes	2
33.32	even	2		6534.2.a.bk.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6534.2.a.bk.1.1	✓	2	33.32	even	2
6534.2.a.bp.1.2	yes	2	1.1	even	1	trivial
6534.2.a.cc.1.1	yes	2	3.2	odd	2
6534.2.a.cl.1.2	yes	2	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6534 = 2 \cdot 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6534.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.54138 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.54138 q^{5} -1.00000 q^{7} -1.00000 q^{8} -3.54138 q^{10} -0.458619 q^{13} +1.00000 q^{14} +1.00000 q^{16} -7.08276 q^{17} -4.54138 q^{19} +3.54138 q^{20} -0.541381 q^{23} +7.54138 q^{25} +0.458619 q^{26} -1.00000 q^{28} -3.00000 q^{29} +2.54138 q^{31} -1.00000 q^{32} +7.08276 q^{34} -3.54138 q^{35} -2.08276 q^{37} +4.54138 q^{38} -3.54138 q^{40} +9.00000 q^{41} +6.08276 q^{43} +0.541381 q^{46} +10.6241 q^{47} -6.00000 q^{49} -7.54138 q^{50} -0.458619 q^{52} -10.0828 q^{53} +1.00000 q^{56} +3.00000 q^{58} -7.62414 q^{59} +6.08276 q^{61} -2.54138 q^{62} +1.00000 q^{64} -1.62414 q^{65} -15.1655 q^{67} -7.08276 q^{68} +3.54138 q^{70} +6.00000 q^{71} +8.54138 q^{73} +2.08276 q^{74} -4.54138 q^{76} -14.6241 q^{79} +3.54138 q^{80} -9.00000 q^{82} +5.45862 q^{83} -25.0828 q^{85} -6.08276 q^{86} -6.54138 q^{89} +0.458619 q^{91} -0.541381 q^{92} -10.6241 q^{94} -16.0828 q^{95} -18.1655 q^{97} +6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} - 2 q^{8} - q^{10} - 7 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} - 3 q^{19} + q^{20} + 5 q^{23} + 9 q^{25} + 7 q^{26} - 2 q^{28} - 6 q^{29} - q^{31} - 2 q^{32} + 2 q^{34} - q^{35} + 8 q^{37} + 3 q^{38} - q^{40} + 18 q^{41} - 5 q^{46} + 3 q^{47} - 12 q^{49} - 9 q^{50} - 7 q^{52} - 8 q^{53} + 2 q^{56} + 6 q^{58} + 3 q^{59} + q^{62} + 2 q^{64} + 15 q^{65} - 6 q^{67} - 2 q^{68} + q^{70} + 12 q^{71} + 11 q^{73} - 8 q^{74} - 3 q^{76} - 11 q^{79} + q^{80} - 18 q^{82} + 17 q^{83} - 38 q^{85} - 7 q^{89} + 7 q^{91} + 5 q^{92} - 3 q^{94} - 20 q^{95} - 12 q^{97} + 12 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + q^5 - 2 * q^7 - 2 * q^8 - q^10 - 7 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 - 3 * q^19 + q^20 + 5 * q^23 + 9 * q^25 + 7 * q^26 - 2 * q^28 - 6 * q^29 - q^31 - 2 * q^32 + 2 * q^34 - q^35 + 8 * q^37 + 3 * q^38 - q^40 + 18 * q^41 - 5 * q^46 + 3 * q^47 - 12 * q^49 - 9 * q^50 - 7 * q^52 - 8 * q^53 + 2 * q^56 + 6 * q^58 + 3 * q^59 + q^62 + 2 * q^64 + 15 * q^65 - 6 * q^67 - 2 * q^68 + q^70 + 12 * q^71 + 11 * q^73 - 8 * q^74 - 3 * q^76 - 11 * q^79 + q^80 - 18 * q^82 + 17 * q^83 - 38 * q^85 - 7 * q^89 + 7 * q^91 + 5 * q^92 - 3 * q^94 - 20 * q^95 - 12 * q^97 + 12 * q^98

Introduction

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