LMFDB - Embedded newform 1521.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.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:	$12.1452461474$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{5} +1.73205 q^{8} -3.00000 q^{10} -5.00000 q^{16} -3.00000 q^{17} -3.46410 q^{19} +1.73205 q^{20} -6.00000 q^{23} -2.00000 q^{25} -3.00000 q^{29} +3.46410 q^{31} +5.19615 q^{32} +5.19615 q^{34} +8.66025 q^{37} +6.00000 q^{38} +3.00000 q^{40} -5.19615 q^{41} -8.00000 q^{43} +10.3923 q^{46} -3.46410 q^{47} -7.00000 q^{49} +3.46410 q^{50} +3.00000 q^{53} +5.19615 q^{58} +6.92820 q^{59} +1.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} -3.46410 q^{67} -3.00000 q^{68} -3.46410 q^{71} -1.73205 q^{73} -15.0000 q^{74} -3.46410 q^{76} +4.00000 q^{79} -8.66025 q^{80} +9.00000 q^{82} -13.8564 q^{83} -5.19615 q^{85} +13.8564 q^{86} +6.92820 q^{89} -6.00000 q^{92} +6.00000 q^{94} -6.00000 q^{95} +6.92820 q^{97} +12.1244 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 6 q^{10} - 10 q^{16} - 6 q^{17} - 12 q^{23} - 4 q^{25} - 6 q^{29} + 12 q^{38} + 6 q^{40} - 16 q^{43} - 14 q^{49} + 6 q^{53} + 2 q^{61} - 12 q^{62} + 2 q^{64} - 6 q^{68} - 30 q^{74} + 8 q^{79}+ \cdots - 12 q^{95}+O(q^{100}) $ 2 * q + 2 * q^4 - 6 * q^10 - 10 * q^16 - 6 * q^17 - 12 * q^23 - 4 * q^25 - 6 * q^29 + 12 * q^38 + 6 * q^40 - 16 * q^43 - 14 * q^49 + 6 * q^53 + 2 * q^61 - 12 * q^62 + 2 * q^64 - 6 * q^68 - 30 * q^74 + 8 * q^79 + 18 * q^82 - 12 * q^92 + 12 * q^94 - 12 * q^95

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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.73205	0.612372
$9$	0	0
$10$	−3.00000	−0.948683
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	1.73205	0.387298
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−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
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	5.19615	0.891133
$35$	0	0
$36$	0	0
$37$	8.66025	1.42374	0.711868	−	0.702313i	$-0.247849\pi$
$37$	8.66025	1.42374	0.711868	+	0.702313i	$0.247849\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	3.00000	0.474342
$41$	−5.19615	−0.811503	−0.405751	−	0.913984i	$-0.632990\pi$
$41$	−5.19615	−0.811503	−0.405751	+	0.913984i	$0.632990\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	0	0
$46$	10.3923	1.53226
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	3.46410	0.489898
$51$	0	0
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	5.19615	0.682288
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−3.46410	−0.423207	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	−3.46410	−0.423207	−0.211604	+	0.977356i	$0.567869\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	0	0
$71$	−3.46410	−0.411113	−0.205557	−	0.978645i	$-0.565900\pi$
$71$	−3.46410	−0.411113	−0.205557	+	0.978645i	$0.565900\pi$
$72$	0	0
$73$	−1.73205	−0.202721	−0.101361	−	0.994850i	$-0.532320\pi$
$73$	−1.73205	−0.202721	−0.101361	+	0.994850i	$0.532320\pi$
$74$	−15.0000	−1.74371
$75$	0	0
$76$	−3.46410	−0.397360
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−8.66025	−0.968246
$81$	0	0
$82$	9.00000	0.993884
$83$	−13.8564	−1.52094	−0.760469	−	0.649374i	$-0.775031\pi$
$83$	−13.8564	−1.52094	−0.760469	+	0.649374i	$0.775031\pi$
$84$	0	0
$85$	−5.19615	−0.563602
$86$	13.8564	1.49417
$87$	0	0
$88$	0	0
$89$	6.92820	0.734388	0.367194	−	0.930144i	$-0.380318\pi$
$89$	6.92820	0.734388	0.367194	+	0.930144i	$0.380318\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	0	0
$94$	6.00000	0.618853
$95$	−6.00000	−0.615587
$96$	0	0
$97$	6.92820	0.703452	0.351726	−	0.936103i	$-0.385595\pi$
$97$	6.92820	0.703452	0.351726	+	0.936103i	$0.385595\pi$
$98$	12.1244	1.22474
$99$	0	0
$100$	−2.00000	−0.200000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	0	0
$106$	−5.19615	−0.504695
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−13.8564	−1.32720	−0.663602	−	0.748086i	$-0.730973\pi$
$109$	−13.8564	−1.32720	−0.663602	+	0.748086i	$0.730973\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	0	0
$115$	−10.3923	−0.969087
$116$	−3.00000	−0.278543
$117$	0	0
$118$	−12.0000	−1.10469
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−1.73205	−0.156813
$123$	0	0
$124$	3.46410	0.311086
$125$	−12.1244	−1.08444
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	0	0
$134$	6.00000	0.518321
$135$	0	0
$136$	−5.19615	−0.445566
$137$	−15.5885	−1.33181	−0.665906	−	0.746036i	$-0.731955\pi$
$137$	−15.5885	−1.33181	−0.665906	+	0.746036i	$0.731955\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	6.00000	0.503509
$143$	0	0
$144$	0	0
$145$	−5.19615	−0.431517
$146$	3.00000	0.248282
$147$	0	0
$148$	8.66025	0.711868
$149$	19.0526	1.56085	0.780423	−	0.625252i	$-0.215004\pi$
$149$	19.0526	1.56085	0.780423	+	0.625252i	$0.215004\pi$
$150$	0	0
$151$	−17.3205	−1.40952	−0.704761	−	0.709444i	$-0.748946\pi$
$151$	−17.3205	−1.40952	−0.704761	+	0.709444i	$0.748946\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	−6.92820	−0.551178
$159$	0	0
$160$	9.00000	0.711512
$161$	0	0
$162$	0	0
$163$	−20.7846	−1.62798	−0.813988	−	0.580881i	$-0.802708\pi$
$163$	−20.7846	−1.62798	−0.813988	+	0.580881i	$0.802708\pi$
$164$	−5.19615	−0.405751
$165$	0	0
$166$	24.0000	1.86276
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	0	0
$169$	0	0
$170$	9.00000	0.690268
$171$	0	0
$172$	−8.00000	−0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−12.0000	−0.899438
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−11.0000	−0.817624	−0.408812	−	0.912619i	$-0.634057\pi$
$181$	−11.0000	−0.817624	−0.408812	+	0.912619i	$0.634057\pi$
$182$	0	0
$183$	0	0
$184$	−10.3923	−0.766131
$185$	15.0000	1.10282
$186$	0	0
$187$	0	0
$188$	−3.46410	−0.252646
$189$	0	0
$190$	10.3923	0.753937
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	−5.19615	−0.374027	−0.187014	−	0.982357i	$-0.559881\pi$
$193$	−5.19615	−0.374027	−0.187014	+	0.982357i	$0.559881\pi$
$194$	−12.0000	−0.861550
$195$	0	0
$196$	−7.00000	−0.500000
$197$	13.8564	0.987228	0.493614	−	0.869681i	$-0.335676\pi$
$197$	13.8564	0.987228	0.493614	+	0.869681i	$0.335676\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	−3.46410	−0.244949
$201$	0	0
$202$	5.19615	0.365600
$203$	0	0
$204$	0	0
$205$	−9.00000	−0.628587
$206$	−17.3205	−1.20678
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	3.00000	0.206041
$213$	0	0
$214$	10.3923	0.710403
$215$	−13.8564	−0.944999
$216$	0	0
$217$	0	0
$218$	24.0000	1.62549
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	10.3923	0.695920	0.347960	−	0.937509i	$-0.386874\pi$
$223$	10.3923	0.695920	0.347960	+	0.937509i	$0.386874\pi$
$224$	0	0
$225$	0	0
$226$	−25.9808	−1.72821
$227$	−24.2487	−1.60944	−0.804722	−	0.593652i	$-0.797686\pi$
$227$	−24.2487	−1.60944	−0.804722	+	0.593652i	$0.797686\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	18.0000	1.18688
$231$	0	0
$232$	−5.19615	−0.341144
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	6.92820	0.450988
$237$	0	0
$238$	0	0
$239$	20.7846	1.34444	0.672222	−	0.740349i	$-0.265340\pi$
$239$	20.7846	1.34444	0.672222	+	0.740349i	$0.265340\pi$
$240$	0	0
$241$	1.73205	0.111571	0.0557856	−	0.998443i	$-0.482234\pi$
$241$	1.73205	0.111571	0.0557856	+	0.998443i	$0.482234\pi$
$242$	19.0526	1.22474
$243$	0	0
$244$	1.00000	0.0640184
$245$	−12.1244	−0.774597
$246$	0	0
$247$	0	0
$248$	6.00000	0.381000
$249$	0	0
$250$	21.0000	1.32816
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	0	0
$254$	−3.46410	−0.217357
$255$	0	0
$256$	19.0000	1.18750
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	31.1769	1.92612
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	5.19615	0.319197
$266$	0	0
$267$	0	0
$268$	−3.46410	−0.211604
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	20.7846	1.26258	0.631288	−	0.775549i	$-0.282527\pi$
$271$	20.7846	1.26258	0.631288	+	0.775549i	$0.282527\pi$
$272$	15.0000	0.909509
$273$	0	0
$274$	27.0000	1.63113
$275$	0	0
$276$	0	0
$277$	7.00000	0.420589	0.210295	−	0.977638i	$-0.432558\pi$
$277$	7.00000	0.420589	0.210295	+	0.977638i	$0.432558\pi$
$278$	6.92820	0.415526
$279$	0	0
$280$	0	0
$281$	−22.5167	−1.34323	−0.671616	−	0.740900i	$-0.734399\pi$
$281$	−22.5167	−1.34323	−0.671616	+	0.740900i	$0.734399\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−3.46410	−0.205557
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	9.00000	0.528498
$291$	0	0
$292$	−1.73205	−0.101361
$293$	5.19615	0.303562	0.151781	−	0.988414i	$-0.451499\pi$
$293$	5.19615	0.303562	0.151781	+	0.988414i	$0.451499\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	15.0000	0.871857
$297$	0	0
$298$	−33.0000	−1.91164
$299$	0	0
$300$	0	0
$301$	0	0
$302$	30.0000	1.72631
$303$	0	0
$304$	17.3205	0.993399
$305$	1.73205	0.0991769
$306$	0	0
$307$	17.3205	0.988534	0.494267	−	0.869310i	$-0.335437\pi$
$307$	17.3205	0.988534	0.494267	+	0.869310i	$0.335437\pi$
$308$	0	0
$309$	0	0
$310$	−10.3923	−0.590243
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	22.5167	1.27069
$315$	0	0
$316$	4.00000	0.225018
$317$	5.19615	0.291845	0.145922	−	0.989296i	$-0.453385\pi$
$317$	5.19615	0.291845	0.145922	+	0.989296i	$0.453385\pi$
$318$	0	0
$319$	0	0
$320$	1.73205	0.0968246
$321$	0	0
$322$	0	0
$323$	10.3923	0.578243
$324$	0	0
$325$	0	0
$326$	36.0000	1.99386
$327$	0	0
$328$	−9.00000	−0.496942
$329$	0	0
$330$	0	0
$331$	−27.7128	−1.52323	−0.761617	−	0.648027i	$-0.775594\pi$
$331$	−27.7128	−1.52323	−0.761617	+	0.648027i	$0.775594\pi$
$332$	−13.8564	−0.760469
$333$	0	0
$334$	−24.0000	−1.31322
$335$	−6.00000	−0.327815
$336$	0	0
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	0	0
$339$	0	0
$340$	−5.19615	−0.281801
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−13.8564	−0.747087
$345$	0	0
$346$	−10.3923	−0.558694
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	13.8564	0.741716	0.370858	−	0.928689i	$-0.379064\pi$
$349$	13.8564	0.741716	0.370858	+	0.928689i	$0.379064\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.9090	1.75157	0.875784	−	0.482704i	$-0.160345\pi$
$353$	32.9090	1.75157	0.875784	+	0.482704i	$0.160345\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	6.92820	0.367194
$357$	0	0
$358$	0	0
$359$	−6.92820	−0.365657	−0.182828	−	0.983145i	$-0.558525\pi$
$359$	−6.92820	−0.365657	−0.182828	+	0.983145i	$0.558525\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	19.0526	1.00138
$363$	0	0
$364$	0	0
$365$	−3.00000	−0.157027
$366$	0	0
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	30.0000	1.56386
$369$	0	0
$370$	−25.9808	−1.35068
$371$	0	0
$372$	0	0
$373$	19.0000	0.983783	0.491891	−	0.870657i	$-0.336306\pi$
$373$	19.0000	0.983783	0.491891	+	0.870657i	$0.336306\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	0	0
$378$	0	0
$379$	24.2487	1.24557	0.622786	−	0.782392i	$-0.286001\pi$
$379$	24.2487	1.24557	0.622786	+	0.782392i	$0.286001\pi$
$380$	−6.00000	−0.307794
$381$	0	0
$382$	31.1769	1.59515
$383$	20.7846	1.06204	0.531022	−	0.847358i	$-0.321808\pi$
$383$	20.7846	1.06204	0.531022	+	0.847358i	$0.321808\pi$
$384$	0	0
$385$	0	0
$386$	9.00000	0.458088
$387$	0	0
$388$	6.92820	0.351726
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	−12.1244	−0.612372
$393$	0	0
$394$	−24.0000	−1.20910
$395$	6.92820	0.348596
$396$	0	0
$397$	13.8564	0.695433	0.347717	−	0.937600i	$-0.386957\pi$
$397$	13.8564	0.695433	0.347717	+	0.937600i	$0.386957\pi$
$398$	−3.46410	−0.173640
$399$	0	0
$400$	10.0000	0.500000
$401$	1.73205	0.0864945	0.0432472	−	0.999064i	$-0.486230\pi$
$401$	1.73205	0.0864945	0.0432472	+	0.999064i	$0.486230\pi$
$402$	0	0
$403$	0	0
$404$	−3.00000	−0.149256
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	15.5885	0.770800	0.385400	−	0.922750i	$-0.374064\pi$
$409$	15.5885	0.770800	0.385400	+	0.922750i	$0.374064\pi$
$410$	15.5885	0.769859
$411$	0	0
$412$	10.0000	0.492665
$413$	0	0
$414$	0	0
$415$	−24.0000	−1.17811
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	15.5885	0.759735	0.379867	−	0.925041i	$-0.375970\pi$
$421$	15.5885	0.759735	0.379867	+	0.925041i	$0.375970\pi$
$422$	−17.3205	−0.843149
$423$	0	0
$424$	5.19615	0.252347
$425$	6.00000	0.291043
$426$	0	0
$427$	0	0
$428$	−6.00000	−0.290021
$429$	0	0
$430$	24.0000	1.15738
$431$	−6.92820	−0.333720	−0.166860	−	0.985981i	$-0.553363\pi$
$431$	−6.92820	−0.333720	−0.166860	+	0.985981i	$0.553363\pi$
$432$	0	0
$433$	17.0000	0.816968	0.408484	−	0.912766i	$-0.366058\pi$
$433$	17.0000	0.816968	0.408484	+	0.912766i	$0.366058\pi$
$434$	0	0
$435$	0	0
$436$	−13.8564	−0.663602
$437$	20.7846	0.994263
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	−18.0000	−0.852325
$447$	0	0
$448$	0	0
$449$	−6.92820	−0.326962	−0.163481	−	0.986546i	$-0.552272\pi$
$449$	−6.92820	−0.326962	−0.163481	+	0.986546i	$0.552272\pi$
$450$	0	0
$451$	0	0
$452$	15.0000	0.705541
$453$	0	0
$454$	42.0000	1.97116
$455$	0	0
$456$	0	0
$457$	−1.73205	−0.0810219	−0.0405110	−	0.999179i	$-0.512899\pi$
$457$	−1.73205	−0.0810219	−0.0405110	+	0.999179i	$0.512899\pi$
$458$	0	0
$459$	0	0
$460$	−10.3923	−0.484544
$461$	−22.5167	−1.04871	−0.524353	−	0.851501i	$-0.675693\pi$
$461$	−22.5167	−1.04871	−0.524353	+	0.851501i	$0.675693\pi$
$462$	0	0
$463$	−13.8564	−0.643962	−0.321981	−	0.946746i	$-0.604349\pi$
$463$	−13.8564	−0.643962	−0.321981	+	0.946746i	$0.604349\pi$
$464$	15.0000	0.696358
$465$	0	0
$466$	−10.3923	−0.481414
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	0	0
$470$	10.3923	0.479361
$471$	0	0
$472$	12.0000	0.552345
$473$	0	0
$474$	0	0
$475$	6.92820	0.317888
$476$	0	0
$477$	0	0
$478$	−36.0000	−1.64660
$479$	−24.2487	−1.10795	−0.553976	−	0.832533i	$-0.686890\pi$
$479$	−24.2487	−1.10795	−0.553976	+	0.832533i	$0.686890\pi$
$480$	0	0
$481$	0	0
$482$	−3.00000	−0.136646
$483$	0	0
$484$	−11.0000	−0.500000
$485$	12.0000	0.544892
$486$	0	0
$487$	−6.92820	−0.313947	−0.156973	−	0.987603i	$-0.550174\pi$
$487$	−6.92820	−0.313947	−0.156973	+	0.987603i	$0.550174\pi$
$488$	1.73205	0.0784063
$489$	0	0
$490$	21.0000	0.948683
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	0	0
$495$	0	0
$496$	−17.3205	−0.777714
$497$	0	0
$498$	0	0
$499$	31.1769	1.39567	0.697835	−	0.716258i	$-0.254147\pi$
$499$	31.1769	1.39567	0.697835	+	0.716258i	$0.254147\pi$
$500$	−12.1244	−0.542218
$501$	0	0
$502$	31.1769	1.39149
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	−5.19615	−0.231226
$506$	0	0
$507$	0	0
$508$	2.00000	0.0887357
$509$	19.0526	0.844490	0.422245	−	0.906482i	$-0.361242\pi$
$509$	19.0526	0.844490	0.422245	+	0.906482i	$0.361242\pi$
$510$	0	0
$511$	0	0
$512$	−8.66025	−0.382733
$513$	0	0
$514$	−5.19615	−0.229192
$515$	17.3205	0.763233
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9.00000	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	20.7846	0.906252
$527$	−10.3923	−0.452696
$528$	0	0
$529$	13.0000	0.565217
$530$	−9.00000	−0.390935
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−10.3923	−0.449299
$536$	−6.00000	−0.259161
$537$	0	0
$538$	−10.3923	−0.448044
$539$	0	0
$540$	0	0
$541$	−29.4449	−1.26593	−0.632967	−	0.774179i	$-0.718163\pi$
$541$	−29.4449	−1.26593	−0.632967	+	0.774179i	$0.718163\pi$
$542$	−36.0000	−1.54633
$543$	0	0
$544$	−15.5885	−0.668350
$545$	−24.0000	−1.02805
$546$	0	0
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	−15.5885	−0.665906
$549$	0	0
$550$	0	0
$551$	10.3923	0.442727
$552$	0	0
$553$	0	0
$554$	−12.1244	−0.515115
$555$	0	0
$556$	−4.00000	−0.169638
$557$	15.5885	0.660504	0.330252	−	0.943893i	$-0.392866\pi$
$557$	15.5885	0.660504	0.330252	+	0.943893i	$0.392866\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	39.0000	1.64512
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	25.9808	1.09302
$566$	6.92820	0.291214
$567$	0	0
$568$	−6.00000	−0.251754
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	−19.0526	−0.793168	−0.396584	−	0.917998i	$-0.629805\pi$
$577$	−19.0526	−0.793168	−0.396584	+	0.917998i	$0.629805\pi$
$578$	13.8564	0.576351
$579$	0	0
$580$	−5.19615	−0.215758
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−3.00000	−0.124141
$585$	0	0
$586$	−9.00000	−0.371787
$587$	−20.7846	−0.857873	−0.428936	−	0.903335i	$-0.641112\pi$
$587$	−20.7846	−0.857873	−0.428936	+	0.903335i	$0.641112\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	−20.7846	−0.855689
$591$	0	0
$592$	−43.3013	−1.77967
$593$	−25.9808	−1.06690	−0.533451	−	0.845831i	$-0.679105\pi$
$593$	−25.9808	−1.06690	−0.533451	+	0.845831i	$0.679105\pi$
$594$	0	0
$595$	0	0
$596$	19.0526	0.780423
$597$	0	0
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	25.0000	1.01977	0.509886	−	0.860242i	$-0.329688\pi$
$601$	25.0000	1.01977	0.509886	+	0.860242i	$0.329688\pi$
$602$	0	0
$603$	0	0
$604$	−17.3205	−0.704761
$605$	−19.0526	−0.774597
$606$	0	0
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	−18.0000	−0.729996
$609$	0	0
$610$	−3.00000	−0.121466
$611$	0	0
$612$	0	0
$613$	12.1244	0.489698	0.244849	−	0.969561i	$-0.421262\pi$
$613$	12.1244	0.489698	0.244849	+	0.969561i	$0.421262\pi$
$614$	−30.0000	−1.21070
$615$	0	0
$616$	0	0
$617$	22.5167	0.906487	0.453243	−	0.891387i	$-0.350267\pi$
$617$	22.5167	0.906487	0.453243	+	0.891387i	$0.350267\pi$
$618$	0	0
$619$	20.7846	0.835404	0.417702	−	0.908584i	$-0.362836\pi$
$619$	20.7846	0.835404	0.417702	+	0.908584i	$0.362836\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	51.9615	2.08347
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	−17.3205	−0.692267
$627$	0	0
$628$	−13.0000	−0.518756
$629$	−25.9808	−1.03592
$630$	0	0
$631$	48.4974	1.93065	0.965326	−	0.261048i	$-0.0840679\pi$
$631$	48.4974	1.93065	0.965326	+	0.261048i	$0.0840679\pi$
$632$	6.92820	0.275589
$633$	0	0
$634$	−9.00000	−0.357436
$635$	3.46410	0.137469
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	−21.0000	−0.830098
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	13.8564	0.546443	0.273222	−	0.961951i	$-0.411911\pi$
$643$	13.8564	0.546443	0.273222	+	0.961951i	$0.411911\pi$
$644$	0	0
$645$	0	0
$646$	−18.0000	−0.708201
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−20.7846	−0.813988
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	−31.1769	−1.21818
$656$	25.9808	1.01438
$657$	0	0
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−46.7654	−1.81896	−0.909481	−	0.415745i	$-0.863521\pi$
$661$	−46.7654	−1.81896	−0.909481	+	0.415745i	$0.863521\pi$
$662$	48.0000	1.86557
$663$	0	0
$664$	−24.0000	−0.931381
$665$	0	0
$666$	0	0
$667$	18.0000	0.696963
$668$	13.8564	0.536120
$669$	0	0
$670$	10.3923	0.401490
$671$	0	0
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	−39.8372	−1.53447
$675$	0	0
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	0	0
$680$	−9.00000	−0.345134
$681$	0	0
$682$	0	0
$683$	24.2487	0.927851	0.463926	−	0.885874i	$-0.346441\pi$
$683$	24.2487	0.927851	0.463926	+	0.885874i	$0.346441\pi$
$684$	0	0
$685$	−27.0000	−1.03162
$686$	0	0
$687$	0	0
$688$	40.0000	1.52499
$689$	0	0
$690$	0	0
$691$	−13.8564	−0.527123	−0.263561	−	0.964643i	$-0.584897\pi$
$691$	−13.8564	−0.527123	−0.263561	+	0.964643i	$0.584897\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−51.9615	−1.97243
$695$	−6.92820	−0.262802
$696$	0	0
$697$	15.5885	0.590455
$698$	−24.0000	−0.908413
$699$	0	0
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−30.0000	−1.13147
$704$	0	0
$705$	0	0
$706$	−57.0000	−2.14522
$707$	0	0
$708$	0	0
$709$	5.19615	0.195146	0.0975728	−	0.995228i	$-0.468892\pi$
$709$	5.19615	0.195146	0.0975728	+	0.995228i	$0.468892\pi$
$710$	10.3923	0.390016
$711$	0	0
$712$	12.0000	0.449719
$713$	−20.7846	−0.778390
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	12.0000	0.447836
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	0	0
$722$	12.1244	0.451222
$723$	0	0
$724$	−11.0000	−0.408812
$725$	6.00000	0.222834
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	0	0
$730$	5.19615	0.192318
$731$	24.0000	0.887672
$732$	0	0
$733$	12.1244	0.447823	0.223912	−	0.974609i	$-0.428117\pi$
$733$	12.1244	0.447823	0.223912	+	0.974609i	$0.428117\pi$
$734$	38.1051	1.40649
$735$	0	0
$736$	−31.1769	−1.14920
$737$	0	0
$738$	0	0
$739$	−20.7846	−0.764574	−0.382287	−	0.924044i	$-0.624863\pi$
$739$	−20.7846	−0.764574	−0.382287	+	0.924044i	$0.624863\pi$
$740$	15.0000	0.551411
$741$	0	0
$742$	0	0
$743$	−34.6410	−1.27086	−0.635428	−	0.772160i	$-0.719176\pi$
$743$	−34.6410	−1.27086	−0.635428	+	0.772160i	$0.719176\pi$
$744$	0	0
$745$	33.0000	1.20903
$746$	−32.9090	−1.20488
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	17.3205	0.631614
$753$	0	0
$754$	0	0
$755$	−30.0000	−1.09181
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	−42.0000	−1.52551
$759$	0	0
$760$	−10.3923	−0.376969
$761$	34.6410	1.25574	0.627868	−	0.778320i	$-0.283928\pi$
$761$	34.6410	1.25574	0.627868	+	0.778320i	$0.283928\pi$
$762$	0	0
$763$	0	0
$764$	−18.0000	−0.651217
$765$	0	0
$766$	−36.0000	−1.30073
$767$	0	0
$768$	0	0
$769$	−6.92820	−0.249837	−0.124919	−	0.992167i	$-0.539867\pi$
$769$	−6.92820	−0.249837	−0.124919	+	0.992167i	$0.539867\pi$
$770$	0	0
$771$	0	0
$772$	−5.19615	−0.187014
$773$	34.6410	1.24595	0.622975	−	0.782241i	$-0.285924\pi$
$773$	34.6410	1.24595	0.622975	+	0.782241i	$0.285924\pi$
$774$	0	0
$775$	−6.92820	−0.248868
$776$	12.0000	0.430775
$777$	0	0
$778$	15.5885	0.558873
$779$	18.0000	0.644917
$780$	0	0
$781$	0	0
$782$	−31.1769	−1.11488
$783$	0	0
$784$	35.0000	1.25000
$785$	−22.5167	−0.803654
$786$	0	0
$787$	−38.1051	−1.35830	−0.679150	−	0.733999i	$-0.737652\pi$
$787$	−38.1051	−1.35830	−0.679150	+	0.733999i	$0.737652\pi$
$788$	13.8564	0.493614
$789$	0	0
$790$	−12.0000	−0.426941
$791$	0	0
$792$	0	0
$793$	0	0
$794$	−24.0000	−0.851728
$795$	0	0
$796$	2.00000	0.0708881
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	10.3923	0.367653
$800$	−10.3923	−0.367423
$801$	0	0
$802$	−3.00000	−0.105934
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−5.19615	−0.182800
$809$	−33.0000	−1.16022	−0.580109	−	0.814539i	$-0.696990\pi$
$809$	−33.0000	−1.16022	−0.580109	+	0.814539i	$0.696990\pi$
$810$	0	0
$811$	−38.1051	−1.33805	−0.669026	−	0.743239i	$-0.733288\pi$
$811$	−38.1051	−1.33805	−0.669026	+	0.743239i	$0.733288\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−36.0000	−1.26102
$816$	0	0
$817$	27.7128	0.969549
$818$	−27.0000	−0.944033
$819$	0	0
$820$	−9.00000	−0.314294
$821$	−41.5692	−1.45078	−0.725388	−	0.688340i	$-0.758340\pi$
$821$	−41.5692	−1.45078	−0.725388	+	0.688340i	$0.758340\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	17.3205	0.603388
$825$	0	0
$826$	0	0
$827$	20.7846	0.722752	0.361376	−	0.932420i	$-0.382307\pi$
$827$	20.7846	0.722752	0.361376	+	0.932420i	$0.382307\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	41.5692	1.44289
$831$	0	0
$832$	0	0
$833$	21.0000	0.727607
$834$	0	0
$835$	24.0000	0.830554
$836$	0	0
$837$	0	0
$838$	31.1769	1.07699
$839$	45.0333	1.55472	0.777361	−	0.629054i	$-0.216558\pi$
$839$	45.0333	1.55472	0.777361	+	0.629054i	$0.216558\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−27.0000	−0.930481
$843$	0	0
$844$	10.0000	0.344214
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−15.0000	−0.515102
$849$	0	0
$850$	−10.3923	−0.356453
$851$	−51.9615	−1.78122
$852$	0	0
$853$	−25.9808	−0.889564	−0.444782	−	0.895639i	$-0.646719\pi$
$853$	−25.9808	−0.889564	−0.444782	+	0.895639i	$0.646719\pi$
$854$	0	0
$855$	0	0
$856$	−10.3923	−0.355202
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	−13.8564	−0.472500
$861$	0	0
$862$	12.0000	0.408722
$863$	27.7128	0.943355	0.471678	−	0.881771i	$-0.343649\pi$
$863$	27.7128	0.943355	0.471678	+	0.881771i	$0.343649\pi$
$864$	0	0
$865$	10.3923	0.353349
$866$	−29.4449	−1.00058
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−24.0000	−0.812743
$873$	0	0
$874$	−36.0000	−1.21772
$875$	0	0
$876$	0	0
$877$	12.1244	0.409410	0.204705	−	0.978824i	$-0.434376\pi$
$877$	12.1244	0.409410	0.204705	+	0.978824i	$0.434376\pi$
$878$	−48.4974	−1.63671
$879$	0	0
$880$	0	0
$881$	27.0000	0.909653	0.454827	−	0.890580i	$-0.349701\pi$
$881$	27.0000	0.909653	0.454827	+	0.890580i	$0.349701\pi$
$882$	0	0
$883$	−10.0000	−0.336527	−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000	−0.336527	−0.168263	+	0.985742i	$0.553816\pi$
$884$	0	0
$885$	0	0
$886$	−20.7846	−0.698273
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	0	0
$890$	−20.7846	−0.696702
$891$	0	0
$892$	10.3923	0.347960
$893$	12.0000	0.401565
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	12.0000	0.400445
$899$	−10.3923	−0.346603
$900$	0	0
$901$	−9.00000	−0.299833
$902$	0	0
$903$	0	0
$904$	25.9808	0.864107
$905$	−19.0526	−0.633328
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−24.2487	−0.804722
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	3.00000	0.0992312
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	22.0000	0.725713	0.362857	−	0.931845i	$-0.381802\pi$
$919$	22.0000	0.725713	0.362857	+	0.931845i	$0.381802\pi$
$920$	−18.0000	−0.593442
$921$	0	0
$922$	39.0000	1.28440
$923$	0	0
$924$	0	0
$925$	−17.3205	−0.569495
$926$	24.0000	0.788689
$927$	0	0
$928$	−15.5885	−0.511716
$929$	46.7654	1.53432	0.767161	−	0.641455i	$-0.221669\pi$
$929$	46.7654	1.53432	0.767161	+	0.641455i	$0.221669\pi$
$930$	0	0
$931$	24.2487	0.794719
$932$	6.00000	0.196537
$933$	0	0
$934$	−20.7846	−0.680093
$935$	0	0
$936$	0	0
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	0	0
$939$	0	0
$940$	−6.00000	−0.195698
$941$	−20.7846	−0.677559	−0.338779	−	0.940866i	$-0.610014\pi$
$941$	−20.7846	−0.677559	−0.338779	+	0.940866i	$0.610014\pi$
$942$	0	0
$943$	31.1769	1.01526
$944$	−34.6410	−1.12747
$945$	0	0
$946$	0	0
$947$	17.3205	0.562841	0.281420	−	0.959585i	$-0.409194\pi$
$947$	17.3205	0.562841	0.281420	+	0.959585i	$0.409194\pi$
$948$	0	0
$949$	0	0
$950$	−12.0000	−0.389331
$951$	0	0
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	−31.1769	−1.00886
$956$	20.7846	0.672222
$957$	0	0
$958$	42.0000	1.35696
$959$	0	0
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	0	0
$964$	1.73205	0.0557856
$965$	−9.00000	−0.289720
$966$	0	0
$967$	58.8897	1.89377	0.946883	−	0.321578i	$-0.104213\pi$
$967$	58.8897	1.89377	0.946883	+	0.321578i	$0.104213\pi$
$968$	−19.0526	−0.612372
$969$	0	0
$970$	−20.7846	−0.667354
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	0	0
$973$	0	0
$974$	12.0000	0.384505
$975$	0	0
$976$	−5.00000	−0.160046
$977$	43.3013	1.38533	0.692665	−	0.721259i	$-0.256436\pi$
$977$	43.3013	1.38533	0.692665	+	0.721259i	$0.256436\pi$
$978$	0	0
$979$	0	0
$980$	−12.1244	−0.387298
$981$	0	0
$982$	−20.7846	−0.663264
$983$	51.9615	1.65732	0.828658	−	0.559756i	$-0.189105\pi$
$983$	51.9615	1.65732	0.828658	+	0.559756i	$0.189105\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	−15.5885	−0.496438
$987$	0	0
$988$	0	0
$989$	48.0000	1.52631
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	18.0000	0.571501
$993$	0	0
$994$	0	0
$995$	3.46410	0.109819
$996$	0	0
$997$	17.0000	0.538395	0.269198	−	0.963085i	$-0.413241\pi$
$997$	17.0000	0.538395	0.269198	+	0.963085i	$0.413241\pi$
$998$	−54.0000	−1.70934
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.k.1.1		2
3.2	odd	2		169.2.a.a.1.2		2
12.11	even	2		2704.2.a.o.1.1		2
13.5	odd	4		1521.2.b.a.1351.2		2
13.6	odd	12		117.2.q.c.10.1		2
13.8	odd	4		1521.2.b.a.1351.1		2
13.11	odd	12		117.2.q.c.82.1		2
13.12	even	2	inner	1521.2.a.k.1.2		2
15.14	odd	2		4225.2.a.v.1.1		2
21.20	even	2		8281.2.a.q.1.2		2
39.2	even	12		169.2.e.a.147.1		2
39.5	even	4		169.2.b.a.168.1		2
39.8	even	4		169.2.b.a.168.2		2
39.11	even	12		13.2.e.a.4.1	✓	2
39.17	odd	6		169.2.c.a.146.2		4
39.20	even	12		169.2.e.a.23.1		2
39.23	odd	6		169.2.c.a.22.2		4
39.29	odd	6		169.2.c.a.22.1		4
39.32	even	12		13.2.e.a.10.1	yes	2
39.35	odd	6		169.2.c.a.146.1		4
39.38	odd	2		169.2.a.a.1.1		2
52.11	even	12		1872.2.by.d.433.1		2
52.19	even	12		1872.2.by.d.1297.1		2
156.11	odd	12		208.2.w.b.17.1		2
156.47	odd	4		2704.2.f.b.337.1		2
156.71	odd	12		208.2.w.b.49.1		2
156.83	odd	4		2704.2.f.b.337.2		2
156.155	even	2		2704.2.a.o.1.2		2
195.32	odd	12		325.2.m.a.49.1		4
195.89	even	12		325.2.n.a.251.1		2
195.128	odd	12		325.2.m.a.199.1		4
195.149	even	12		325.2.n.a.101.1		2
195.167	odd	12		325.2.m.a.199.2		4
195.188	odd	12		325.2.m.a.49.2		4
195.194	odd	2		4225.2.a.v.1.2		2
273.11	even	12		637.2.u.c.30.1		2
273.32	even	12		637.2.k.a.569.1		2
273.89	odd	12		637.2.k.c.459.1		2
273.110	odd	12		637.2.u.b.361.1		2
273.128	even	12		637.2.k.a.459.1		2
273.149	even	12		637.2.u.c.361.1		2
273.167	odd	12		637.2.q.a.589.1		2
273.188	odd	12		637.2.q.a.491.1		2
273.206	odd	12		637.2.u.b.30.1		2
273.227	odd	12		637.2.k.c.569.1		2
273.272	even	2		8281.2.a.q.1.1		2
312.11	odd	12		832.2.w.a.641.1		2
312.149	even	12		832.2.w.d.257.1		2
312.227	odd	12		832.2.w.a.257.1		2
312.245	even	12		832.2.w.d.641.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.2.e.a.4.1	✓	2	39.11	even	12
13.2.e.a.10.1	yes	2	39.32	even	12
117.2.q.c.10.1		2	13.6	odd	12
117.2.q.c.82.1		2	13.11	odd	12
169.2.a.a.1.1		2	39.38	odd	2
169.2.a.a.1.2		2	3.2	odd	2
169.2.b.a.168.1		2	39.5	even	4
169.2.b.a.168.2		2	39.8	even	4
169.2.c.a.22.1		4	39.29	odd	6
169.2.c.a.22.2		4	39.23	odd	6
169.2.c.a.146.1		4	39.35	odd	6
169.2.c.a.146.2		4	39.17	odd	6
169.2.e.a.23.1		2	39.20	even	12
169.2.e.a.147.1		2	39.2	even	12
208.2.w.b.17.1		2	156.11	odd	12
208.2.w.b.49.1		2	156.71	odd	12
325.2.m.a.49.1		4	195.32	odd	12
325.2.m.a.49.2		4	195.188	odd	12
325.2.m.a.199.1		4	195.128	odd	12
325.2.m.a.199.2		4	195.167	odd	12
325.2.n.a.101.1		2	195.149	even	12
325.2.n.a.251.1		2	195.89	even	12
637.2.k.a.459.1		2	273.128	even	12
637.2.k.a.569.1		2	273.32	even	12
637.2.k.c.459.1		2	273.89	odd	12
637.2.k.c.569.1		2	273.227	odd	12
637.2.q.a.491.1		2	273.188	odd	12
637.2.q.a.589.1		2	273.167	odd	12
637.2.u.b.30.1		2	273.206	odd	12
637.2.u.b.361.1		2	273.110	odd	12
637.2.u.c.30.1		2	273.11	even	12
637.2.u.c.361.1		2	273.149	even	12
832.2.w.a.257.1		2	312.227	odd	12
832.2.w.a.641.1		2	312.11	odd	12
832.2.w.d.257.1		2	312.149	even	12
832.2.w.d.641.1		2	312.245	even	12
1521.2.a.k.1.1		2	1.1	even	1	trivial
1521.2.a.k.1.2		2	13.12	even	2	inner
1521.2.b.a.1351.1		2	13.8	odd	4
1521.2.b.a.1351.2		2	13.5	odd	4
1872.2.by.d.433.1		2	52.11	even	12
1872.2.by.d.1297.1		2	52.19	even	12
2704.2.a.o.1.1		2	12.11	even	2
2704.2.a.o.1.2		2	156.155	even	2
2704.2.f.b.337.1		2	156.47	odd	4
2704.2.f.b.337.2		2	156.83	odd	4
4225.2.a.v.1.1		2	15.14	odd	2
4225.2.a.v.1.2		2	195.194	odd	2
8281.2.a.q.1.1		2	273.272	even	2
8281.2.a.q.1.2		2	21.20	even	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{5} +1.73205 q^{8} -3.00000 q^{10} -5.00000 q^{16} -3.00000 q^{17} -3.46410 q^{19} +1.73205 q^{20} -6.00000 q^{23} -2.00000 q^{25} -3.00000 q^{29} +3.46410 q^{31} +5.19615 q^{32} +5.19615 q^{34} +8.66025 q^{37} +6.00000 q^{38} +3.00000 q^{40} -5.19615 q^{41} -8.00000 q^{43} +10.3923 q^{46} -3.46410 q^{47} -7.00000 q^{49} +3.46410 q^{50} +3.00000 q^{53} +5.19615 q^{58} +6.92820 q^{59} +1.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} -3.46410 q^{67} -3.00000 q^{68} -3.46410 q^{71} -1.73205 q^{73} -15.0000 q^{74} -3.46410 q^{76} +4.00000 q^{79} -8.66025 q^{80} +9.00000 q^{82} -13.8564 q^{83} -5.19615 q^{85} +13.8564 q^{86} +6.92820 q^{89} -6.00000 q^{92} +6.00000 q^{94} -6.00000 q^{95} +6.92820 q^{97} +12.1244 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 6 q^{10} - 10 q^{16} - 6 q^{17} - 12 q^{23} - 4 q^{25} - 6 q^{29} + 12 q^{38} + 6 q^{40} - 16 q^{43} - 14 q^{49} + 6 q^{53} + 2 q^{61} - 12 q^{62} + 2 q^{64} - 6 q^{68} - 30 q^{74} + 8 q^{79}+ \cdots - 12 q^{95}+O(q^{100}) \) 2 * q + 2 * q^4 - 6 * q^10 - 10 * q^16 - 6 * q^17 - 12 * q^23 - 4 * q^25 - 6 * q^29 + 12 * q^38 + 6 * q^40 - 16 * q^43 - 14 * q^49 + 6 * q^53 + 2 * q^61 - 12 * q^62 + 2 * q^64 - 6 * q^68 - 30 * q^74 + 8 * q^79 + 18 * q^82 - 12 * q^92 + 12 * q^94 - 12 * q^95

Introduction

L-functions

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