LMFDB - Embedded newform 3330.2.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.23544108.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 20x^{3} + 39x^{2} + 9x - 36 $ x^5 - x^4 - 20x^3 + 39x^2 + 9*x - 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.892002$ of defining polynomial
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.647226 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.647226 q^{7} -1.00000 q^{8} +1.00000 q^{10} -3.68293 q^{11} -3.22455 q^{13} -0.647226 q^{14} +1.00000 q^{16} -1.35277 q^{17} +6.30301 q^{19} -1.00000 q^{20} +3.68293 q^{22} -5.22455 q^{23} +1.00000 q^{25} +3.22455 q^{26} +0.647226 q^{28} +3.87178 q^{29} -5.98594 q^{31} -1.00000 q^{32} +1.35277 q^{34} -0.647226 q^{35} +1.00000 q^{37} -6.30301 q^{38} +1.00000 q^{40} +0.242382 q^{41} +1.75762 q^{43} -3.68293 q^{44} +5.22455 q^{46} +6.44432 q^{47} -6.58110 q^{49} -1.00000 q^{50} -3.22455 q^{52} -5.46694 q^{53} +3.68293 q^{55} -0.647226 q^{56} -3.87178 q^{58} +7.36586 q^{59} +14.4755 q^{61} +5.98594 q^{62} +1.00000 q^{64} +3.22455 q^{65} +6.07141 q^{67} -1.35277 q^{68} +0.647226 q^{70} -8.22832 q^{71} +8.08702 q^{73} -1.00000 q^{74} +6.30301 q^{76} -2.38369 q^{77} +10.2283 q^{79} -1.00000 q^{80} -0.242382 q^{82} +2.14131 q^{83} +1.35277 q^{85} -1.75762 q^{86} +3.68293 q^{88} +5.81346 q^{89} -2.08702 q^{91} -5.22455 q^{92} -6.44432 q^{94} -6.30301 q^{95} -4.20194 q^{97} +6.58110 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8	$ 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8} + 5 q^{10} - 5 q^{11} + 5 q^{13} - 3 q^{14} + 5 q^{16} - 7 q^{17} - q^{19} - 5 q^{20} + 5 q^{22} - 5 q^{23} + 5 q^{25} - 5 q^{26} + 3 q^{28} - 2 q^{29} + 16 q^{31} - 5 q^{32} + 7 q^{34} - 3 q^{35} + 5 q^{37} + q^{38} + 5 q^{40} - 2 q^{41} + 12 q^{43} - 5 q^{44} + 5 q^{46} - 6 q^{47} + 16 q^{49} - 5 q^{50} + 5 q^{52} - 3 q^{53} + 5 q^{55} - 3 q^{56} + 2 q^{58} + 10 q^{59} + 16 q^{61} - 16 q^{62} + 5 q^{64} - 5 q^{65} + 4 q^{67} - 7 q^{68} + 3 q^{70} + 8 q^{71} - 3 q^{73} - 5 q^{74} - q^{76} - 3 q^{77} + 2 q^{79} - 5 q^{80} + 2 q^{82} + 5 q^{83} + 7 q^{85} - 12 q^{86} + 5 q^{88} + 7 q^{89} + 33 q^{91} - 5 q^{92} + 6 q^{94} + q^{95} + 14 q^{97} - 16 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8 + 5 * q^10 - 5 * q^11 + 5 * q^13 - 3 * q^14 + 5 * q^16 - 7 * q^17 - q^19 - 5 * q^20 + 5 * q^22 - 5 * q^23 + 5 * q^25 - 5 * q^26 + 3 * q^28 - 2 * q^29 + 16 * q^31 - 5 * q^32 + 7 * q^34 - 3 * q^35 + 5 * q^37 + q^38 + 5 * q^40 - 2 * q^41 + 12 * q^43 - 5 * q^44 + 5 * q^46 - 6 * q^47 + 16 * q^49 - 5 * q^50 + 5 * q^52 - 3 * q^53 + 5 * q^55 - 3 * q^56 + 2 * q^58 + 10 * q^59 + 16 * q^61 - 16 * q^62 + 5 * q^64 - 5 * q^65 + 4 * q^67 - 7 * q^68 + 3 * q^70 + 8 * q^71 - 3 * q^73 - 5 * q^74 - q^76 - 3 * q^77 + 2 * q^79 - 5 * q^80 + 2 * q^82 + 5 * q^83 + 7 * q^85 - 12 * q^86 + 5 * q^88 + 7 * q^89 + 33 * q^91 - 5 * q^92 + 6 * q^94 + q^95 + 14 * q^97 - 16 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.647226	0.244629	0.122314	−	0.992491i	$-0.460968\pi$
$7$	0.647226	0.244629	0.122314	+	0.992491i	$0.460968\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−3.68293	−1.11045	−0.555223	−	0.831702i	$-0.687367\pi$
$11$	−3.68293	−1.11045	−0.555223	+	0.831702i	$0.687367\pi$
$12$	0	0
$13$	−3.22455	−0.894330	−0.447165	−	0.894451i	$-0.647566\pi$
$13$	−3.22455	−0.894330	−0.447165	+	0.894451i	$0.647566\pi$
$14$	−0.647226	−0.172979
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.35277	−0.328096	−0.164048	−	0.986452i	$-0.552455\pi$
$17$	−1.35277	−0.328096	−0.164048	+	0.986452i	$0.552455\pi$
$18$	0	0
$19$	6.30301	1.44601	0.723005	−	0.690843i	$-0.242760\pi$
$19$	6.30301	1.44601	0.723005	+	0.690843i	$0.242760\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	3.68293	0.785204
$23$	−5.22455	−1.08939	−0.544697	−	0.838633i	$-0.683355\pi$
$23$	−5.22455	−1.08939	−0.544697	+	0.838633i	$0.683355\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.22455	0.632387
$27$	0	0
$28$	0.647226	0.122314
$29$	3.87178	0.718972	0.359486	−	0.933151i	$-0.382952\pi$
$29$	3.87178	0.718972	0.359486	+	0.933151i	$0.382952\pi$
$30$	0	0
$31$	−5.98594	−1.07511	−0.537553	−	0.843230i	$-0.680651\pi$
$31$	−5.98594	−1.07511	−0.537553	+	0.843230i	$0.680651\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.35277	0.231999
$35$	−0.647226	−0.109401
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−6.30301	−1.02248
$39$	0	0
$40$	1.00000	0.158114
$41$	0.242382	0.0378537	0.0189269	−	0.999821i	$-0.493975\pi$
$41$	0.242382	0.0378537	0.0189269	+	0.999821i	$0.493975\pi$
$42$	0	0
$43$	1.75762	0.268034	0.134017	−	0.990979i	$-0.457212\pi$
$43$	1.75762	0.268034	0.134017	+	0.990979i	$0.457212\pi$
$44$	−3.68293	−0.555223
$45$	0	0
$46$	5.22455	0.770318
$47$	6.44432	0.940001	0.470000	−	0.882666i	$-0.344254\pi$
$47$	6.44432	0.940001	0.470000	+	0.882666i	$0.344254\pi$
$48$	0	0
$49$	−6.58110	−0.940157
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.22455	−0.447165
$53$	−5.46694	−0.750941	−0.375471	−	0.926834i	$-0.622519\pi$
$53$	−5.46694	−0.750941	−0.375471	+	0.926834i	$0.622519\pi$
$54$	0	0
$55$	3.68293	0.496606
$56$	−0.647226	−0.0864893
$57$	0	0
$58$	−3.87178	−0.508390
$59$	7.36586	0.958954	0.479477	−	0.877555i	$-0.340826\pi$
$59$	7.36586	0.958954	0.479477	+	0.877555i	$0.340826\pi$
$60$	0	0
$61$	14.4755	1.85340	0.926699	−	0.375805i	$-0.122634\pi$
$61$	14.4755	1.85340	0.926699	+	0.375805i	$0.122634\pi$
$62$	5.98594	0.760215
$63$	0	0
$64$	1.00000	0.125000
$65$	3.22455	0.399957
$66$	0	0
$67$	6.07141	0.741741	0.370870	−	0.928685i	$-0.379059\pi$
$67$	6.07141	0.741741	0.370870	+	0.928685i	$0.379059\pi$
$68$	−1.35277	−0.164048
$69$	0	0
$70$	0.647226	0.0773583
$71$	−8.22832	−0.976522	−0.488261	−	0.872698i	$-0.662369\pi$
$71$	−8.22832	−0.976522	−0.488261	+	0.872698i	$0.662369\pi$
$72$	0	0
$73$	8.08702	0.946514	0.473257	−	0.880924i	$-0.343078\pi$
$73$	8.08702	0.946514	0.473257	+	0.880924i	$0.343078\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	6.30301	0.723005
$77$	−2.38369	−0.271647
$78$	0	0
$79$	10.2283	1.15078	0.575388	−	0.817880i	$-0.304851\pi$
$79$	10.2283	1.15078	0.575388	+	0.817880i	$0.304851\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−0.242382	−0.0267666
$83$	2.14131	0.235039	0.117520	−	0.993071i	$-0.462506\pi$
$83$	2.14131	0.235039	0.117520	+	0.993071i	$0.462506\pi$
$84$	0	0
$85$	1.35277	0.146729
$86$	−1.75762	−0.189529
$87$	0	0
$88$	3.68293	0.392602
$89$	5.81346	0.616225	0.308113	−	0.951350i	$-0.400303\pi$
$89$	5.81346	0.616225	0.308113	+	0.951350i	$0.400303\pi$
$90$	0	0
$91$	−2.08702	−0.218779
$92$	−5.22455	−0.544697
$93$	0	0
$94$	−6.44432	−0.664681
$95$	−6.30301	−0.646675
$96$	0	0
$97$	−4.20194	−0.426642	−0.213321	−	0.976982i	$-0.568428\pi$
$97$	−4.20194	−0.426642	−0.213321	+	0.976982i	$0.568428\pi$
$98$	6.58110	0.664791
$99$	0	0
$100$	1.00000	0.100000
$101$	17.3782	1.72919	0.864597	−	0.502465i	$-0.167574\pi$
$101$	17.3782	1.72919	0.864597	+	0.502465i	$0.167574\pi$
$102$	0	0
$103$	8.44432	0.832044	0.416022	−	0.909355i	$-0.363424\pi$
$103$	8.44432	0.832044	0.416022	+	0.909355i	$0.363424\pi$
$104$	3.22455	0.316194
$105$	0	0
$106$	5.46694	0.530996
$107$	−12.9681	−1.25367	−0.626837	−	0.779150i	$-0.715651\pi$
$107$	−12.9681	−1.25367	−0.626837	+	0.779150i	$0.715651\pi$
$108$	0	0
$109$	−9.91126	−0.949326	−0.474663	−	0.880168i	$-0.657430\pi$
$109$	−9.91126	−0.949326	−0.474663	+	0.880168i	$0.657430\pi$
$110$	−3.68293	−0.351154
$111$	0	0
$112$	0.647226	0.0611571
$113$	−11.2376	−1.05715	−0.528574	−	0.848887i	$-0.677273\pi$
$113$	−11.2376	−1.05715	−0.528574	+	0.848887i	$0.677273\pi$
$114$	0	0
$115$	5.22455	0.487192
$116$	3.87178	0.359486
$117$	0	0
$118$	−7.36586	−0.678083
$119$	−0.875551	−0.0802616
$120$	0	0
$121$	2.56398	0.233089
$122$	−14.4755	−1.31055
$123$	0	0
$124$	−5.98594	−0.537553
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	19.9020	1.76602	0.883008	−	0.469358i	$-0.155515\pi$
$127$	19.9020	1.76602	0.883008	+	0.469358i	$0.155515\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−3.22455	−0.282812
$131$	−1.95477	−0.170789	−0.0853944	−	0.996347i	$-0.527215\pi$
$131$	−1.95477	−0.170789	−0.0853944	+	0.996347i	$0.527215\pi$
$132$	0	0
$133$	4.07948	0.353735
$134$	−6.07141	−0.524490
$135$	0	0
$136$	1.35277	0.115999
$137$	−1.78400	−0.152418	−0.0762089	−	0.997092i	$-0.524282\pi$
$137$	−1.78400	−0.152418	−0.0762089	+	0.997092i	$0.524282\pi$
$138$	0	0
$139$	−1.00932	−0.0856092	−0.0428046	−	0.999083i	$-0.513629\pi$
$139$	−1.00932	−0.0856092	−0.0428046	+	0.999083i	$0.513629\pi$
$140$	−0.647226	−0.0547006
$141$	0	0
$142$	8.22832	0.690506
$143$	11.8758	0.993105
$144$	0	0
$145$	−3.87178	−0.321534
$146$	−8.08702	−0.669286
$147$	0	0
$148$	1.00000	0.0821995
$149$	20.4614	1.67627	0.838133	−	0.545466i	$-0.183647\pi$
$149$	20.4614	1.67627	0.838133	+	0.545466i	$0.183647\pi$
$150$	0	0
$151$	−13.1964	−1.07391	−0.536955	−	0.843611i	$-0.680426\pi$
$151$	−13.1964	−1.07391	−0.536955	+	0.843611i	$0.680426\pi$
$152$	−6.30301	−0.511242
$153$	0	0
$154$	2.38369	0.192083
$155$	5.98594	0.480802
$156$	0	0
$157$	−8.24948	−0.658380	−0.329190	−	0.944264i	$-0.606776\pi$
$157$	−8.24948	−0.658380	−0.329190	+	0.944264i	$0.606776\pi$
$158$	−10.2283	−0.813722
$159$	0	0
$160$	1.00000	0.0790569
$161$	−3.38147	−0.266497
$162$	0	0
$163$	−14.7785	−1.15754	−0.578771	−	0.815490i	$-0.696467\pi$
$163$	−14.7785	−1.15754	−0.578771	+	0.815490i	$0.696467\pi$
$164$	0.242382	0.0189269
$165$	0	0
$166$	−2.14131	−0.166198
$167$	5.22455	0.404288	0.202144	−	0.979356i	$-0.435209\pi$
$167$	5.22455	0.404288	0.202144	+	0.979356i	$0.435209\pi$
$168$	0	0
$169$	−2.60225	−0.200173
$170$	−1.35277	−0.103753
$171$	0	0
$172$	1.75762	0.134017
$173$	20.9541	1.59311	0.796554	−	0.604568i	$-0.206654\pi$
$173$	20.9541	1.59311	0.796554	+	0.604568i	$0.206654\pi$
$174$	0	0
$175$	0.647226	0.0489257
$176$	−3.68293	−0.277611
$177$	0	0
$178$	−5.81346	−0.435737
$179$	9.77922	0.730933	0.365466	−	0.930824i	$-0.380909\pi$
$179$	9.77922	0.730933	0.365466	+	0.930824i	$0.380909\pi$
$180$	0	0
$181$	3.92859	0.292010	0.146005	−	0.989284i	$-0.453358\pi$
$181$	3.92859	0.292010	0.146005	+	0.989284i	$0.453358\pi$
$182$	2.08702	0.154700
$183$	0	0
$184$	5.22455	0.385159
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	4.98217	0.364332
$188$	6.44432	0.470000
$189$	0	0
$190$	6.30301	0.457269
$191$	6.86801	0.496952	0.248476	−	0.968638i	$-0.420070\pi$
$191$	6.86801	0.496952	0.248476	+	0.968638i	$0.420070\pi$
$192$	0	0
$193$	23.5537	1.69544	0.847718	−	0.530448i	$-0.177976\pi$
$193$	23.5537	1.69544	0.847718	+	0.530448i	$0.177976\pi$
$194$	4.20194	0.301682
$195$	0	0
$196$	−6.58110	−0.470078
$197$	−15.2960	−1.08979	−0.544896	−	0.838504i	$-0.683431\pi$
$197$	−15.2960	−1.08979	−0.544896	+	0.838504i	$0.683431\pi$
$198$	0	0
$199$	−3.78879	−0.268580	−0.134290	−	0.990942i	$-0.542875\pi$
$199$	−3.78879	−0.268580	−0.134290	+	0.990942i	$0.542875\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−17.3782	−1.22273
$203$	2.50592	0.175881
$204$	0	0
$205$	−0.242382	−0.0169287
$206$	−8.44432	−0.588344
$207$	0	0
$208$	−3.22455	−0.223583
$209$	−23.2136	−1.60572
$210$	0	0
$211$	14.4778	0.996693	0.498347	−	0.866978i	$-0.333941\pi$
$211$	14.4778	0.996693	0.498347	+	0.866978i	$0.333941\pi$
$212$	−5.46694	−0.375471
$213$	0	0
$214$	12.9681	0.886482
$215$	−1.75762	−0.119869
$216$	0	0
$217$	−3.87426	−0.263002
$218$	9.91126	0.671275
$219$	0	0
$220$	3.68293	0.248303
$221$	4.36209	0.293426
$222$	0	0
$223$	9.06209	0.606843	0.303421	−	0.952857i	$-0.401871\pi$
$223$	9.06209	0.606843	0.303421	+	0.952857i	$0.401871\pi$
$224$	−0.647226	−0.0432446
$225$	0	0
$226$	11.2376	0.747517
$227$	−7.94319	−0.527208	−0.263604	−	0.964631i	$-0.584911\pi$
$227$	−7.94319	−0.527208	−0.263604	+	0.964631i	$0.584911\pi$
$228$	0	0
$229$	23.0099	1.52054	0.760268	−	0.649609i	$-0.225067\pi$
$229$	23.0099	1.52054	0.760268	+	0.649609i	$0.225067\pi$
$230$	−5.22455	−0.344497
$231$	0	0
$232$	−3.87178	−0.254195
$233$	−11.3707	−0.744916	−0.372458	−	0.928049i	$-0.621485\pi$
$233$	−11.3707	−0.744916	−0.372458	+	0.928049i	$0.621485\pi$
$234$	0	0
$235$	−6.44432	−0.420381
$236$	7.36586	0.479477
$237$	0	0
$238$	0.875551	0.0567535
$239$	16.1453	1.04436	0.522178	−	0.852837i	$-0.325120\pi$
$239$	16.1453	1.04436	0.522178	+	0.852837i	$0.325120\pi$
$240$	0	0
$241$	29.4871	1.89943	0.949716	−	0.313112i	$-0.101372\pi$
$241$	29.4871	1.89943	0.949716	+	0.313112i	$0.101372\pi$
$242$	−2.56398	−0.164819
$243$	0	0
$244$	14.4755	0.926699
$245$	6.58110	0.420451
$246$	0	0
$247$	−20.3244	−1.29321
$248$	5.98594	0.380108
$249$	0	0
$250$	1.00000	0.0632456
$251$	7.85063	0.495527	0.247764	−	0.968821i	$-0.420304\pi$
$251$	7.85063	0.495527	0.247764	+	0.968821i	$0.420304\pi$
$252$	0	0
$253$	19.2417	1.20971
$254$	−19.9020	−1.24876
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.1846	1.38384	0.691919	−	0.721975i	$-0.256766\pi$
$257$	22.1846	1.38384	0.691919	+	0.721975i	$0.256766\pi$
$258$	0	0
$259$	0.647226	0.0402167
$260$	3.22455	0.199978
$261$	0	0
$262$	1.95477	0.120766
$263$	17.3395	1.06920	0.534599	−	0.845106i	$-0.320463\pi$
$263$	17.3395	1.06920	0.534599	+	0.845106i	$0.320463\pi$
$264$	0	0
$265$	5.46694	0.335831
$266$	−4.07948	−0.250129
$267$	0	0
$268$	6.07141	0.370870
$269$	−12.8235	−0.781864	−0.390932	−	0.920419i	$-0.627847\pi$
$269$	−12.8235	−0.781864	−0.390932	+	0.920419i	$0.627847\pi$
$270$	0	0
$271$	20.8886	1.26889	0.634447	−	0.772966i	$-0.281228\pi$
$271$	20.8886	1.26889	0.634447	+	0.772966i	$0.281228\pi$
$272$	−1.35277	−0.0820239
$273$	0	0
$274$	1.78400	0.107776
$275$	−3.68293	−0.222089
$276$	0	0
$277$	−5.17932	−0.311195	−0.155598	−	0.987821i	$-0.549730\pi$
$277$	−5.17932	−0.311195	−0.155598	+	0.987821i	$0.549730\pi$
$278$	1.00932	0.0605349
$279$	0	0
$280$	0.647226	0.0386792
$281$	2.99623	0.178740	0.0893700	−	0.995998i	$-0.471515\pi$
$281$	2.99623	0.178740	0.0893700	+	0.995998i	$0.471515\pi$
$282$	0	0
$283$	−11.8306	−0.703255	−0.351627	−	0.936140i	$-0.614372\pi$
$283$	−11.8306	−0.703255	−0.351627	+	0.936140i	$0.614372\pi$
$284$	−8.22832	−0.488261
$285$	0	0
$286$	−11.8758	−0.702231
$287$	0.156876	0.00926010
$288$	0	0
$289$	−15.1700	−0.892353
$290$	3.87178	0.227359
$291$	0	0
$292$	8.08702	0.473257
$293$	−8.55018	−0.499507	−0.249753	−	0.968309i	$-0.580350\pi$
$293$	−8.55018	−0.499507	−0.249753	+	0.968309i	$0.580350\pi$
$294$	0	0
$295$	−7.36586	−0.428857
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	−20.4614	−1.18530
$299$	16.8469	0.974279
$300$	0	0
$301$	1.13758	0.0655688
$302$	13.1964	0.759370
$303$	0	0
$304$	6.30301	0.361502
$305$	−14.4755	−0.828864
$306$	0	0
$307$	−11.3659	−0.648684	−0.324342	−	0.945940i	$-0.605143\pi$
$307$	−11.3659	−0.648684	−0.324342	+	0.945940i	$0.605143\pi$
$308$	−2.38369	−0.135823
$309$	0	0
$310$	−5.98594	−0.339979
$311$	17.3231	0.982306	0.491153	−	0.871073i	$-0.336576\pi$
$311$	17.3231	0.982306	0.491153	+	0.871073i	$0.336576\pi$
$312$	0	0
$313$	13.4777	0.761806	0.380903	−	0.924615i	$-0.375613\pi$
$313$	13.4777	0.761806	0.380903	+	0.924615i	$0.375613\pi$
$314$	8.24948	0.465545
$315$	0	0
$316$	10.2283	0.575388
$317$	4.96505	0.278865	0.139432	−	0.990232i	$-0.455472\pi$
$317$	4.96505	0.278865	0.139432	+	0.990232i	$0.455472\pi$
$318$	0	0
$319$	−14.2595	−0.798379
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	3.38147	0.188442
$323$	−8.52655	−0.474430
$324$	0	0
$325$	−3.22455	−0.178866
$326$	14.7785	0.818506
$327$	0	0
$328$	−0.242382	−0.0133833
$329$	4.17093	0.229951
$330$	0	0
$331$	6.97432	0.383343	0.191672	−	0.981459i	$-0.438609\pi$
$331$	6.97432	0.383343	0.191672	+	0.981459i	$0.438609\pi$
$332$	2.14131	0.117520
$333$	0	0
$334$	−5.22455	−0.285875
$335$	−6.07141	−0.331717
$336$	0	0
$337$	−2.46916	−0.134504	−0.0672518	−	0.997736i	$-0.521423\pi$
$337$	−2.46916	−0.134504	−0.0672518	+	0.997736i	$0.521423\pi$
$338$	2.60225	0.141544
$339$	0	0
$340$	1.35277	0.0733645
$341$	22.0458	1.19385
$342$	0	0
$343$	−8.79004	−0.474618
$344$	−1.75762	−0.0947644
$345$	0	0
$346$	−20.9541	−1.12650
$347$	5.03801	0.270455	0.135227	−	0.990815i	$-0.456824\pi$
$347$	5.03801	0.270455	0.135227	+	0.990815i	$0.456824\pi$
$348$	0	0
$349$	−8.64168	−0.462578	−0.231289	−	0.972885i	$-0.574294\pi$
$349$	−8.64168	−0.462578	−0.231289	+	0.972885i	$0.574294\pi$
$350$	−0.647226	−0.0345957
$351$	0	0
$352$	3.68293	0.196301
$353$	−6.98120	−0.371572	−0.185786	−	0.982590i	$-0.559483\pi$
$353$	−6.98120	−0.371572	−0.185786	+	0.982590i	$0.559483\pi$
$354$	0	0
$355$	8.22832	0.436714
$356$	5.81346	0.308113
$357$	0	0
$358$	−9.77922	−0.516848
$359$	34.4753	1.81954	0.909768	−	0.415117i	$-0.136259\pi$
$359$	34.4753	1.81954	0.909768	+	0.415117i	$0.136259\pi$
$360$	0	0
$361$	20.7280	1.09094
$362$	−3.92859	−0.206482
$363$	0	0
$364$	−2.08702	−0.109389
$365$	−8.08702	−0.423294
$366$	0	0
$367$	−26.1515	−1.36510	−0.682548	−	0.730841i	$-0.739128\pi$
$367$	−26.1515	−1.36510	−0.682548	+	0.730841i	$0.739128\pi$
$368$	−5.22455	−0.272349
$369$	0	0
$370$	1.00000	0.0519875
$371$	−3.53835	−0.183702
$372$	0	0
$373$	−26.7488	−1.38500	−0.692501	−	0.721417i	$-0.743491\pi$
$373$	−26.7488	−1.38500	−0.692501	+	0.721417i	$0.743491\pi$
$374$	−4.98217	−0.257622
$375$	0	0
$376$	−6.44432	−0.332340
$377$	−12.4848	−0.642998
$378$	0	0
$379$	6.74130	0.346277	0.173139	−	0.984897i	$-0.444609\pi$
$379$	6.74130	0.346277	0.173139	+	0.984897i	$0.444609\pi$
$380$	−6.30301	−0.323338
$381$	0	0
$382$	−6.86801	−0.351398
$383$	28.9354	1.47853	0.739264	−	0.673415i	$-0.235173\pi$
$383$	28.9354	1.47853	0.739264	+	0.673415i	$0.235173\pi$
$384$	0	0
$385$	2.38369	0.121484
$386$	−23.5537	−1.19885
$387$	0	0
$388$	−4.20194	−0.213321
$389$	28.0458	1.42198	0.710990	−	0.703203i	$-0.248247\pi$
$389$	28.0458	1.42198	0.710990	+	0.703203i	$0.248247\pi$
$390$	0	0
$391$	7.06764	0.357426
$392$	6.58110	0.332396
$393$	0	0
$394$	15.2960	0.770599
$395$	−10.2283	−0.514643
$396$	0	0
$397$	−29.0551	−1.45824	−0.729118	−	0.684388i	$-0.760069\pi$
$397$	−29.0551	−1.45824	−0.729118	+	0.684388i	$0.760069\pi$
$398$	3.78879	0.189915
$399$	0	0
$400$	1.00000	0.0500000
$401$	−11.3131	−0.564948	−0.282474	−	0.959275i	$-0.591155\pi$
$401$	−11.3131	−0.564948	−0.282474	+	0.959275i	$0.591155\pi$
$402$	0	0
$403$	19.3020	0.961501
$404$	17.3782	0.864597
$405$	0	0
$406$	−2.50592	−0.124367
$407$	−3.68293	−0.182556
$408$	0	0
$409$	8.29973	0.410395	0.205198	−	0.978721i	$-0.434216\pi$
$409$	8.29973	0.410395	0.205198	+	0.978721i	$0.434216\pi$
$410$	0.242382	0.0119704
$411$	0	0
$412$	8.44432	0.416022
$413$	4.76738	0.234587
$414$	0	0
$415$	−2.14131	−0.105113
$416$	3.22455	0.158097
$417$	0	0
$418$	23.2136	1.13541
$419$	−13.3083	−0.650153	−0.325076	−	0.945688i	$-0.605390\pi$
$419$	−13.3083	−0.650153	−0.325076	+	0.945688i	$0.605390\pi$
$420$	0	0
$421$	15.7840	0.769265	0.384633	−	0.923070i	$-0.374328\pi$
$421$	15.7840	0.769265	0.384633	+	0.923070i	$0.374328\pi$
$422$	−14.4778	−0.704768
$423$	0	0
$424$	5.46694	0.265498
$425$	−1.35277	−0.0656192
$426$	0	0
$427$	9.36892	0.453394
$428$	−12.9681	−0.626837
$429$	0	0
$430$	1.75762	0.0847599
$431$	27.7551	1.33692	0.668459	−	0.743749i	$-0.266954\pi$
$431$	27.7551	1.33692	0.668459	+	0.743749i	$0.266954\pi$
$432$	0	0
$433$	−12.5190	−0.601625	−0.300813	−	0.953683i	$-0.597258\pi$
$433$	−12.5190	−0.601625	−0.300813	+	0.953683i	$0.597258\pi$
$434$	3.87426	0.185970
$435$	0	0
$436$	−9.91126	−0.474663
$437$	−32.9304	−1.57528
$438$	0	0
$439$	−33.4946	−1.59861	−0.799306	−	0.600925i	$-0.794799\pi$
$439$	−33.4946	−1.59861	−0.799306	+	0.600925i	$0.794799\pi$
$440$	−3.68293	−0.175577
$441$	0	0
$442$	−4.36209	−0.207484
$443$	20.7131	0.984109	0.492054	−	0.870565i	$-0.336246\pi$
$443$	20.7131	0.984109	0.492054	+	0.870565i	$0.336246\pi$
$444$	0	0
$445$	−5.81346	−0.275584
$446$	−9.06209	−0.429102
$447$	0	0
$448$	0.647226	0.0305786
$449$	9.08325	0.428665	0.214332	−	0.976761i	$-0.431242\pi$
$449$	9.08325	0.428665	0.214332	+	0.976761i	$0.431242\pi$
$450$	0	0
$451$	−0.892676	−0.0420345
$452$	−11.2376	−0.528574
$453$	0	0
$454$	7.94319	0.372792
$455$	2.08702	0.0978408
$456$	0	0
$457$	−10.8532	−0.507691	−0.253845	−	0.967245i	$-0.581695\pi$
$457$	−10.8532	−0.507691	−0.253845	+	0.967245i	$0.581695\pi$
$458$	−23.0099	−1.07518
$459$	0	0
$460$	5.22455	0.243596
$461$	−34.3455	−1.59963	−0.799816	−	0.600246i	$-0.795070\pi$
$461$	−34.3455	−1.59963	−0.799816	+	0.600246i	$0.795070\pi$
$462$	0	0
$463$	18.4087	0.855523	0.427762	−	0.903892i	$-0.359302\pi$
$463$	18.4087	0.855523	0.427762	+	0.903892i	$0.359302\pi$
$464$	3.87178	0.179743
$465$	0	0
$466$	11.3707	0.526735
$467$	−28.2570	−1.30758	−0.653789	−	0.756677i	$-0.726822\pi$
$467$	−28.2570	−1.30758	−0.653789	+	0.756677i	$0.726822\pi$
$468$	0	0
$469$	3.92958	0.181451
$470$	6.44432	0.297254
$471$	0	0
$472$	−7.36586	−0.339041
$473$	−6.47319	−0.297637
$474$	0	0
$475$	6.30301	0.289202
$476$	−0.875551	−0.0401308
$477$	0	0
$478$	−16.1453	−0.738471
$479$	24.5642	1.12237	0.561184	−	0.827691i	$-0.310346\pi$
$479$	24.5642	1.12237	0.561184	+	0.827691i	$0.310346\pi$
$480$	0	0
$481$	−3.22455	−0.147027
$482$	−29.4871	−1.34310
$483$	0	0
$484$	2.56398	0.116545
$485$	4.20194	0.190800
$486$	0	0
$487$	9.22128	0.417856	0.208928	−	0.977931i	$-0.433003\pi$
$487$	9.22128	0.417856	0.208928	+	0.977931i	$0.433003\pi$
$488$	−14.4755	−0.655275
$489$	0	0
$490$	−6.58110	−0.297304
$491$	27.2415	1.22939	0.614696	−	0.788764i	$-0.289279\pi$
$491$	27.2415	1.22939	0.614696	+	0.788764i	$0.289279\pi$
$492$	0	0
$493$	−5.23764	−0.235892
$494$	20.3244	0.914438
$495$	0	0
$496$	−5.98594	−0.268777
$497$	−5.32559	−0.238885
$498$	0	0
$499$	15.3083	0.685293	0.342647	−	0.939464i	$-0.388677\pi$
$499$	15.3083	0.685293	0.342647	+	0.939464i	$0.388677\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−7.85063	−0.350391
$503$	39.1979	1.74775	0.873875	−	0.486151i	$-0.161599\pi$
$503$	39.1979	1.74775	0.873875	+	0.486151i	$0.161599\pi$
$504$	0	0
$505$	−17.3782	−0.773319
$506$	−19.2417	−0.855397
$507$	0	0
$508$	19.9020	0.883008
$509$	−23.6951	−1.05026	−0.525132	−	0.851021i	$-0.675984\pi$
$509$	−23.6951	−1.05026	−0.525132	+	0.851021i	$0.675984\pi$
$510$	0	0
$511$	5.23413	0.231544
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−22.1846	−0.978521
$515$	−8.44432	−0.372101
$516$	0	0
$517$	−23.7340	−1.04382
$518$	−0.647226	−0.0284375
$519$	0	0
$520$	−3.22455	−0.141406
$521$	39.0672	1.71157	0.855784	−	0.517334i	$-0.173075\pi$
$521$	39.0672	1.71157	0.855784	+	0.517334i	$0.173075\pi$
$522$	0	0
$523$	−30.9867	−1.35496	−0.677478	−	0.735543i	$-0.736927\pi$
$523$	−30.9867	−1.35496	−0.677478	+	0.735543i	$0.736927\pi$
$524$	−1.95477	−0.0853944
$525$	0	0
$526$	−17.3395	−0.756037
$527$	8.09763	0.352738
$528$	0	0
$529$	4.29596	0.186781
$530$	−5.46694	−0.237469
$531$	0	0
$532$	4.07948	0.176868
$533$	−0.781574	−0.0338537
$534$	0	0
$535$	12.9681	0.560660
$536$	−6.07141	−0.262245
$537$	0	0
$538$	12.8235	0.552862
$539$	24.2377	1.04399
$540$	0	0
$541$	18.7069	0.804272	0.402136	−	0.915580i	$-0.368268\pi$
$541$	18.7069	0.804272	0.402136	+	0.915580i	$0.368268\pi$
$542$	−20.8886	−0.897244
$543$	0	0
$544$	1.35277	0.0579997
$545$	9.91126	0.424552
$546$	0	0
$547$	3.79186	0.162128	0.0810641	−	0.996709i	$-0.474168\pi$
$547$	3.79186	0.162128	0.0810641	+	0.996709i	$0.474168\pi$
$548$	−1.78400	−0.0762089
$549$	0	0
$550$	3.68293	0.157041
$551$	24.4039	1.03964
$552$	0	0
$553$	6.62004	0.281513
$554$	5.17932	0.220048
$555$	0	0
$556$	−1.00932	−0.0428046
$557$	−25.2588	−1.07025	−0.535125	−	0.844773i	$-0.679735\pi$
$557$	−25.2588	−1.07025	−0.535125	+	0.844773i	$0.679735\pi$
$558$	0	0
$559$	−5.66753	−0.239711
$560$	−0.647226	−0.0273503
$561$	0	0
$562$	−2.99623	−0.126388
$563$	13.3774	0.563792	0.281896	−	0.959445i	$-0.409037\pi$
$563$	13.3774	0.563792	0.281896	+	0.959445i	$0.409037\pi$
$564$	0	0
$565$	11.2376	0.472771
$566$	11.8306	0.497276
$567$	0	0
$568$	8.22832	0.345253
$569$	26.1303	1.09544	0.547720	−	0.836662i	$-0.315496\pi$
$569$	26.1303	1.09544	0.547720	+	0.836662i	$0.315496\pi$
$570$	0	0
$571$	−4.60048	−0.192524	−0.0962621	−	0.995356i	$-0.530689\pi$
$571$	−4.60048	−0.192524	−0.0962621	+	0.995356i	$0.530689\pi$
$572$	11.8758	0.496553
$573$	0	0
$574$	−0.156876	−0.00654788
$575$	−5.22455	−0.217879
$576$	0	0
$577$	21.7107	0.903826	0.451913	−	0.892062i	$-0.350742\pi$
$577$	21.7107	0.903826	0.451913	+	0.892062i	$0.350742\pi$
$578$	15.1700	0.630989
$579$	0	0
$580$	−3.87178	−0.160767
$581$	1.38591	0.0574973
$582$	0	0
$583$	20.1343	0.833879
$584$	−8.08702	−0.334643
$585$	0	0
$586$	8.55018	0.353205
$587$	42.5925	1.75798	0.878990	−	0.476839i	$-0.158218\pi$
$587$	42.5925	1.75798	0.878990	+	0.476839i	$0.158218\pi$
$588$	0	0
$589$	−37.7295	−1.55462
$590$	7.36586	0.303248
$591$	0	0
$592$	1.00000	0.0410997
$593$	−19.7915	−0.812741	−0.406371	−	0.913708i	$-0.633206\pi$
$593$	−19.7915	−0.812741	−0.406371	+	0.913708i	$0.633206\pi$
$594$	0	0
$595$	0.875551	0.0358941
$596$	20.4614	0.838133
$597$	0	0
$598$	−16.8469	−0.688919
$599$	10.0714	0.411507	0.205753	−	0.978604i	$-0.434036\pi$
$599$	10.0714	0.411507	0.205753	+	0.978604i	$0.434036\pi$
$600$	0	0
$601$	−11.1677	−0.455542	−0.227771	−	0.973715i	$-0.573144\pi$
$601$	−11.1677	−0.455542	−0.227771	+	0.973715i	$0.573144\pi$
$602$	−1.13758	−0.0463642
$603$	0	0
$604$	−13.1964	−0.536955
$605$	−2.56398	−0.104241
$606$	0	0
$607$	10.8653	0.441009	0.220505	−	0.975386i	$-0.429230\pi$
$607$	10.8653	0.441009	0.220505	+	0.975386i	$0.429230\pi$
$608$	−6.30301	−0.255621
$609$	0	0
$610$	14.4755	0.586096
$611$	−20.7801	−0.840671
$612$	0	0
$613$	22.8057	0.921112	0.460556	−	0.887631i	$-0.347650\pi$
$613$	22.8057	0.921112	0.460556	+	0.887631i	$0.347650\pi$
$614$	11.3659	0.458689
$615$	0	0
$616$	2.38369	0.0960416
$617$	−23.6860	−0.953562	−0.476781	−	0.879022i	$-0.658197\pi$
$617$	−23.6860	−0.953562	−0.476781	+	0.879022i	$0.658197\pi$
$618$	0	0
$619$	−46.7962	−1.88090	−0.940448	−	0.339936i	$-0.889595\pi$
$619$	−46.7962	−1.88090	−0.940448	+	0.339936i	$0.889595\pi$
$620$	5.98594	0.240401
$621$	0	0
$622$	−17.3231	−0.694595
$623$	3.76262	0.150746
$624$	0	0
$625$	1.00000	0.0400000
$626$	−13.4777	−0.538678
$627$	0	0
$628$	−8.24948	−0.329190
$629$	−1.35277	−0.0539386
$630$	0	0
$631$	−28.1459	−1.12047	−0.560235	−	0.828334i	$-0.689289\pi$
$631$	−28.1459	−1.12047	−0.560235	+	0.828334i	$0.689289\pi$
$632$	−10.2283	−0.406861
$633$	0	0
$634$	−4.96505	−0.197187
$635$	−19.9020	−0.789786
$636$	0	0
$637$	21.2211	0.840811
$638$	14.2595	0.564539
$639$	0	0
$640$	1.00000	0.0395285
$641$	1.48254	0.0585569	0.0292785	−	0.999571i	$-0.490679\pi$
$641$	1.48254	0.0585569	0.0292785	+	0.999571i	$0.490679\pi$
$642$	0	0
$643$	−35.3845	−1.39543	−0.697715	−	0.716376i	$-0.745800\pi$
$643$	−35.3845	−1.39543	−0.697715	+	0.716376i	$0.745800\pi$
$644$	−3.38147	−0.133249
$645$	0	0
$646$	8.52655	0.335473
$647$	−8.03717	−0.315974	−0.157987	−	0.987441i	$-0.550500\pi$
$647$	−8.03717	−0.315974	−0.157987	+	0.987441i	$0.550500\pi$
$648$	0	0
$649$	−27.1280	−1.06487
$650$	3.22455	0.126477
$651$	0	0
$652$	−14.7785	−0.578771
$653$	14.4135	0.564042	0.282021	−	0.959408i	$-0.408995\pi$
$653$	14.4135	0.564042	0.282021	+	0.959408i	$0.408995\pi$
$654$	0	0
$655$	1.95477	0.0763791
$656$	0.242382	0.00946343
$657$	0	0
$658$	−4.17093	−0.162600
$659$	−13.4995	−0.525864	−0.262932	−	0.964814i	$-0.584689\pi$
$659$	−13.4995	−0.525864	−0.262932	+	0.964814i	$0.584689\pi$
$660$	0	0
$661$	−33.6005	−1.30691	−0.653454	−	0.756966i	$-0.726681\pi$
$661$	−33.6005	−1.30691	−0.653454	+	0.756966i	$0.726681\pi$
$662$	−6.97432	−0.271064
$663$	0	0
$664$	−2.14131	−0.0830989
$665$	−4.07948	−0.158195
$666$	0	0
$667$	−20.2283	−0.783244
$668$	5.22455	0.202144
$669$	0	0
$670$	6.07141	0.234559
$671$	−53.3122	−2.05810
$672$	0	0
$673$	13.8396	0.533479	0.266739	−	0.963769i	$-0.414054\pi$
$673$	13.8396	0.533479	0.266739	+	0.963769i	$0.414054\pi$
$674$	2.46916	0.0951084
$675$	0	0
$676$	−2.60225	−0.100087
$677$	−42.1987	−1.62183	−0.810914	−	0.585166i	$-0.801030\pi$
$677$	−42.1987	−1.62183	−0.810914	+	0.585166i	$0.801030\pi$
$678$	0	0
$679$	−2.71960	−0.104369
$680$	−1.35277	−0.0518765
$681$	0	0
$682$	−22.0458	−0.844178
$683$	−35.4303	−1.35570	−0.677852	−	0.735199i	$-0.737089\pi$
$683$	−35.4303	−1.35570	−0.677852	+	0.735199i	$0.737089\pi$
$684$	0	0
$685$	1.78400	0.0681633
$686$	8.79004	0.335605
$687$	0	0
$688$	1.75762	0.0670086
$689$	17.6284	0.671590
$690$	0	0
$691$	−10.8740	−0.413668	−0.206834	−	0.978376i	$-0.566316\pi$
$691$	−10.8740	−0.413668	−0.206834	+	0.978376i	$0.566316\pi$
$692$	20.9541	0.796554
$693$	0	0
$694$	−5.03801	−0.191240
$695$	1.00932	0.0382856
$696$	0	0
$697$	−0.327888	−0.0124196
$698$	8.64168	0.327092
$699$	0	0
$700$	0.647226	0.0244629
$701$	−8.30629	−0.313724	−0.156862	−	0.987621i	$-0.550138\pi$
$701$	−8.30629	−0.313724	−0.156862	+	0.987621i	$0.550138\pi$
$702$	0	0
$703$	6.30301	0.237723
$704$	−3.68293	−0.138806
$705$	0	0
$706$	6.98120	0.262741
$707$	11.2476	0.423010
$708$	0	0
$709$	−10.5550	−0.396400	−0.198200	−	0.980162i	$-0.563510\pi$
$709$	−10.5550	−0.396400	−0.198200	+	0.980162i	$0.563510\pi$
$710$	−8.22832	−0.308804
$711$	0	0
$712$	−5.81346	−0.217869
$713$	31.2739	1.17122
$714$	0	0
$715$	−11.8758	−0.444130
$716$	9.77922	0.365466
$717$	0	0
$718$	−34.4753	−1.28661
$719$	24.4400	0.911460	0.455730	−	0.890118i	$-0.349378\pi$
$719$	24.4400	0.911460	0.455730	+	0.890118i	$0.349378\pi$
$720$	0	0
$721$	5.46539	0.203542
$722$	−20.7280	−0.771415
$723$	0	0
$724$	3.92859	0.146005
$725$	3.87178	0.143794
$726$	0	0
$727$	12.8748	0.477500	0.238750	−	0.971081i	$-0.423262\pi$
$727$	12.8748	0.477500	0.238750	+	0.971081i	$0.423262\pi$
$728$	2.08702	0.0773500
$729$	0	0
$730$	8.08702	0.299314
$731$	−2.37766	−0.0879409
$732$	0	0
$733$	−26.5230	−0.979651	−0.489825	−	0.871821i	$-0.662939\pi$
$733$	−26.5230	−0.979651	−0.489825	+	0.871821i	$0.662939\pi$
$734$	26.1515	0.965268
$735$	0	0
$736$	5.22455	0.192580
$737$	−22.3606	−0.823663
$738$	0	0
$739$	−20.8599	−0.767345	−0.383673	−	0.923469i	$-0.625341\pi$
$739$	−20.8599	−0.767345	−0.383673	+	0.923469i	$0.625341\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	3.53835	0.129897
$743$	−12.8794	−0.472498	−0.236249	−	0.971693i	$-0.575918\pi$
$743$	−12.8794	−0.472498	−0.236249	+	0.971693i	$0.575918\pi$
$744$	0	0
$745$	−20.4614	−0.749649
$746$	26.7488	0.979345
$747$	0	0
$748$	4.98217	0.182166
$749$	−8.39331	−0.306685
$750$	0	0
$751$	−4.20367	−0.153394	−0.0766970	−	0.997054i	$-0.524437\pi$
$751$	−4.20367	−0.153394	−0.0766970	+	0.997054i	$0.524437\pi$
$752$	6.44432	0.235000
$753$	0	0
$754$	12.4848	0.454668
$755$	13.1964	0.480268
$756$	0	0
$757$	−8.46623	−0.307710	−0.153855	−	0.988093i	$-0.549169\pi$
$757$	−8.46623	−0.307710	−0.153855	+	0.988093i	$0.549169\pi$
$758$	−6.74130	−0.244855
$759$	0	0
$760$	6.30301	0.228634
$761$	−10.3993	−0.376974	−0.188487	−	0.982076i	$-0.560358\pi$
$761$	−10.3993	−0.376974	−0.188487	+	0.982076i	$0.560358\pi$
$762$	0	0
$763$	−6.41483	−0.232232
$764$	6.86801	0.248476
$765$	0	0
$766$	−28.9354	−1.04548
$767$	−23.7516	−0.857621
$768$	0	0
$769$	3.18083	0.114704	0.0573519	−	0.998354i	$-0.481734\pi$
$769$	3.18083	0.114704	0.0573519	+	0.998354i	$0.481734\pi$
$770$	−2.38369	−0.0859022
$771$	0	0
$772$	23.5537	0.847718
$773$	35.7481	1.28577	0.642885	−	0.765962i	$-0.277737\pi$
$773$	35.7481	1.28577	0.642885	+	0.765962i	$0.277737\pi$
$774$	0	0
$775$	−5.98594	−0.215021
$776$	4.20194	0.150841
$777$	0	0
$778$	−28.0458	−1.00549
$779$	1.52774	0.0547368
$780$	0	0
$781$	30.3044	1.08437
$782$	−7.06764	−0.252738
$783$	0	0
$784$	−6.58110	−0.235039
$785$	8.24948	0.294437
$786$	0	0
$787$	−24.1474	−0.860763	−0.430382	−	0.902647i	$-0.641621\pi$
$787$	−24.1474	−0.860763	−0.430382	+	0.902647i	$0.641621\pi$
$788$	−15.2960	−0.544896
$789$	0	0
$790$	10.2283	0.363907
$791$	−7.27330	−0.258609
$792$	0	0
$793$	−46.6770	−1.65755
$794$	29.0551	1.03113
$795$	0	0
$796$	−3.78879	−0.134290
$797$	−28.4753	−1.00865	−0.504323	−	0.863515i	$-0.668258\pi$
$797$	−28.4753	−1.00865	−0.504323	+	0.863515i	$0.668258\pi$
$798$	0	0
$799$	−8.71771	−0.308410
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	11.3131	0.399479
$803$	−29.7839	−1.05105
$804$	0	0
$805$	3.38147	0.119181
$806$	−19.3020	−0.679884
$807$	0	0
$808$	−17.3782	−0.611363
$809$	−25.9825	−0.913494	−0.456747	−	0.889597i	$-0.650986\pi$
$809$	−25.9825	−0.913494	−0.456747	+	0.889597i	$0.650986\pi$
$810$	0	0
$811$	28.2360	0.991499	0.495749	−	0.868466i	$-0.334893\pi$
$811$	28.2360	0.991499	0.495749	+	0.868466i	$0.334893\pi$
$812$	2.50592	0.0879405
$813$	0	0
$814$	3.68293	0.129087
$815$	14.7785	0.517668
$816$	0	0
$817$	11.0783	0.387580
$818$	−8.29973	−0.290193
$819$	0	0
$820$	−0.242382	−0.00846435
$821$	−31.8087	−1.11013	−0.555065	−	0.831807i	$-0.687307\pi$
$821$	−31.8087	−1.11013	−0.555065	+	0.831807i	$0.687307\pi$
$822$	0	0
$823$	−10.3169	−0.359623	−0.179812	−	0.983701i	$-0.557549\pi$
$823$	−10.3169	−0.359623	−0.179812	+	0.983701i	$0.557549\pi$
$824$	−8.44432	−0.294172
$825$	0	0
$826$	−4.76738	−0.165878
$827$	−22.6487	−0.787574	−0.393787	−	0.919202i	$-0.628835\pi$
$827$	−22.6487	−0.787574	−0.393787	+	0.919202i	$0.628835\pi$
$828$	0	0
$829$	−39.8071	−1.38256	−0.691279	−	0.722588i	$-0.742952\pi$
$829$	−39.8071	−1.38256	−0.691279	+	0.722588i	$0.742952\pi$
$830$	2.14131	0.0743259
$831$	0	0
$832$	−3.22455	−0.111791
$833$	8.90274	0.308462
$834$	0	0
$835$	−5.22455	−0.180803
$836$	−23.2136	−0.802858
$837$	0	0
$838$	13.3083	0.459727
$839$	−3.37544	−0.116533	−0.0582665	−	0.998301i	$-0.518557\pi$
$839$	−3.37544	−0.116533	−0.0582665	+	0.998301i	$0.518557\pi$
$840$	0	0
$841$	−14.0093	−0.483080
$842$	−15.7840	−0.543953
$843$	0	0
$844$	14.4778	0.498347
$845$	2.60225	0.0895202
$846$	0	0
$847$	1.65948	0.0570203
$848$	−5.46694	−0.187735
$849$	0	0
$850$	1.35277	0.0463998
$851$	−5.22455	−0.179095
$852$	0	0
$853$	33.2417	1.13817	0.569087	−	0.822278i	$-0.307297\pi$
$853$	33.2417	1.13817	0.569087	+	0.822278i	$0.307297\pi$
$854$	−9.36892	−0.320598
$855$	0	0
$856$	12.9681	0.443241
$857$	−23.0465	−0.787253	−0.393626	−	0.919270i	$-0.628780\pi$
$857$	−23.0465	−0.787253	−0.393626	+	0.919270i	$0.628780\pi$
$858$	0	0
$859$	−40.6123	−1.38567	−0.692837	−	0.721094i	$-0.743640\pi$
$859$	−40.6123	−1.38567	−0.692837	+	0.721094i	$0.743640\pi$
$860$	−1.75762	−0.0599343
$861$	0	0
$862$	−27.7551	−0.945344
$863$	30.4021	1.03490	0.517451	−	0.855713i	$-0.326881\pi$
$863$	30.4021	1.03490	0.517451	+	0.855713i	$0.326881\pi$
$864$	0	0
$865$	−20.9541	−0.712459
$866$	12.5190	0.425413
$867$	0	0
$868$	−3.87426	−0.131501
$869$	−37.6702	−1.27787
$870$	0	0
$871$	−19.5776	−0.663361
$872$	9.91126	0.335637
$873$	0	0
$874$	32.9304	1.11389
$875$	−0.647226	−0.0218802
$876$	0	0
$877$	−48.9721	−1.65367	−0.826836	−	0.562443i	$-0.809861\pi$
$877$	−48.9721	−1.65367	−0.826836	+	0.562443i	$0.809861\pi$
$878$	33.4946	1.13039
$879$	0	0
$880$	3.68293	0.124152
$881$	−9.31809	−0.313934	−0.156967	−	0.987604i	$-0.550172\pi$
$881$	−9.31809	−0.313934	−0.156967	+	0.987604i	$0.550172\pi$
$882$	0	0
$883$	19.5555	0.658094	0.329047	−	0.944314i	$-0.393273\pi$
$883$	19.5555	0.658094	0.329047	+	0.944314i	$0.393273\pi$
$884$	4.36209	0.146713
$885$	0	0
$886$	−20.7131	−0.695870
$887$	9.95507	0.334259	0.167129	−	0.985935i	$-0.446550\pi$
$887$	9.95507	0.334259	0.167129	+	0.985935i	$0.446550\pi$
$888$	0	0
$889$	12.8811	0.432018
$890$	5.81346	0.194868
$891$	0	0
$892$	9.06209	0.303421
$893$	40.6186	1.35925
$894$	0	0
$895$	−9.77922	−0.326883
$896$	−0.647226	−0.0216223
$897$	0	0
$898$	−9.08325	−0.303112
$899$	−23.1763	−0.772971
$900$	0	0
$901$	7.39553	0.246381
$902$	0.892676	0.0297229
$903$	0	0
$904$	11.2376	0.373758
$905$	−3.92859	−0.130591
$906$	0	0
$907$	50.2797	1.66951	0.834755	−	0.550622i	$-0.185609\pi$
$907$	50.2797	1.66951	0.834755	+	0.550622i	$0.185609\pi$
$908$	−7.94319	−0.263604
$909$	0	0
$910$	−2.08702	−0.0691839
$911$	−23.0456	−0.763533	−0.381767	−	0.924259i	$-0.624684\pi$
$911$	−23.0456	−0.763533	−0.381767	+	0.924259i	$0.624684\pi$
$912$	0	0
$913$	−7.88629	−0.260998
$914$	10.8532	0.358992
$915$	0	0
$916$	23.0099	0.760268
$917$	−1.26518	−0.0417798
$918$	0	0
$919$	−19.3252	−0.637481	−0.318741	−	0.947842i	$-0.603260\pi$
$919$	−19.3252	−0.637481	−0.318741	+	0.947842i	$0.603260\pi$
$920$	−5.22455	−0.172248
$921$	0	0
$922$	34.3455	1.13111
$923$	26.5327	0.873334
$924$	0	0
$925$	1.00000	0.0328798
$926$	−18.4087	−0.604946
$927$	0	0
$928$	−3.87178	−0.127097
$929$	20.3013	0.666064	0.333032	−	0.942916i	$-0.391928\pi$
$929$	20.3013	0.666064	0.333032	+	0.942916i	$0.391928\pi$
$930$	0	0
$931$	−41.4807	−1.35948
$932$	−11.3707	−0.372458
$933$	0	0
$934$	28.2570	0.924598
$935$	−4.98217	−0.162934
$936$	0	0
$937$	5.08786	0.166213	0.0831066	−	0.996541i	$-0.473516\pi$
$937$	5.08786	0.166213	0.0831066	+	0.996541i	$0.473516\pi$
$938$	−3.92958	−0.128305
$939$	0	0
$940$	−6.44432	−0.210191
$941$	28.8079	0.939111	0.469556	−	0.882903i	$-0.344414\pi$
$941$	28.8079	0.939111	0.469556	+	0.882903i	$0.344414\pi$
$942$	0	0
$943$	−1.26634	−0.0412376
$944$	7.36586	0.239738
$945$	0	0
$946$	6.47319	0.210461
$947$	−35.4303	−1.15133	−0.575665	−	0.817686i	$-0.695257\pi$
$947$	−35.4303	−1.15133	−0.575665	+	0.817686i	$0.695257\pi$
$948$	0	0
$949$	−26.0770	−0.846496
$950$	−6.30301	−0.204497
$951$	0	0
$952$	0.875551	0.0283768
$953$	−14.8501	−0.481043	−0.240521	−	0.970644i	$-0.577318\pi$
$953$	−14.8501	−0.481043	−0.240521	+	0.970644i	$0.577318\pi$
$954$	0	0
$955$	−6.86801	−0.222244
$956$	16.1453	0.522178
$957$	0	0
$958$	−24.5642	−0.793635
$959$	−1.15465	−0.0372858
$960$	0	0
$961$	4.83151	0.155855
$962$	3.22455	0.103964
$963$	0	0
$964$	29.4871	0.949716
$965$	−23.5537	−0.758222
$966$	0	0
$967$	0.593694	0.0190919	0.00954595	−	0.999954i	$-0.496961\pi$
$967$	0.593694	0.0190919	0.00954595	+	0.999954i	$0.496961\pi$
$968$	−2.56398	−0.0824095
$969$	0	0
$970$	−4.20194	−0.134916
$971$	−31.8152	−1.02100	−0.510499	−	0.859878i	$-0.670539\pi$
$971$	−31.8152	−1.02100	−0.510499	+	0.859878i	$0.670539\pi$
$972$	0	0
$973$	−0.653257	−0.0209425
$974$	−9.22128	−0.295469
$975$	0	0
$976$	14.4755	0.463349
$977$	7.05606	0.225743	0.112872	−	0.993610i	$-0.463995\pi$
$977$	7.05606	0.225743	0.112872	+	0.993610i	$0.463995\pi$
$978$	0	0
$979$	−21.4106	−0.684285
$980$	6.58110	0.210225
$981$	0	0
$982$	−27.2415	−0.869311
$983$	−4.73957	−0.151169	−0.0755844	−	0.997139i	$-0.524082\pi$
$983$	−4.73957	−0.151169	−0.0755844	+	0.997139i	$0.524082\pi$
$984$	0	0
$985$	15.2960	0.487370
$986$	5.23764	0.166801
$987$	0	0
$988$	−20.3244	−0.646605
$989$	−9.18277	−0.291995
$990$	0	0
$991$	30.2404	0.960619	0.480310	−	0.877099i	$-0.340524\pi$
$991$	30.2404	0.960619	0.480310	+	0.877099i	$0.340524\pi$
$992$	5.98594	0.190054
$993$	0	0
$994$	5.32559	0.168917
$995$	3.78879	0.120113
$996$	0	0
$997$	56.4939	1.78918	0.894590	−	0.446888i	$-0.147468\pi$
$997$	56.4939	1.78918	0.894590	+	0.446888i	$0.147468\pi$
$998$	−15.3083	−0.484576
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bk.1.3	✓	5
3.2	odd	2		3330.2.a.bl.1.3	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3330.2.a.bk.1.3	✓	5	1.1	even	1	trivial
3330.2.a.bl.1.3	yes	5	3.2	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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.647226 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.647226 q^{7} -1.00000 q^{8} +1.00000 q^{10} -3.68293 q^{11} -3.22455 q^{13} -0.647226 q^{14} +1.00000 q^{16} -1.35277 q^{17} +6.30301 q^{19} -1.00000 q^{20} +3.68293 q^{22} -5.22455 q^{23} +1.00000 q^{25} +3.22455 q^{26} +0.647226 q^{28} +3.87178 q^{29} -5.98594 q^{31} -1.00000 q^{32} +1.35277 q^{34} -0.647226 q^{35} +1.00000 q^{37} -6.30301 q^{38} +1.00000 q^{40} +0.242382 q^{41} +1.75762 q^{43} -3.68293 q^{44} +5.22455 q^{46} +6.44432 q^{47} -6.58110 q^{49} -1.00000 q^{50} -3.22455 q^{52} -5.46694 q^{53} +3.68293 q^{55} -0.647226 q^{56} -3.87178 q^{58} +7.36586 q^{59} +14.4755 q^{61} +5.98594 q^{62} +1.00000 q^{64} +3.22455 q^{65} +6.07141 q^{67} -1.35277 q^{68} +0.647226 q^{70} -8.22832 q^{71} +8.08702 q^{73} -1.00000 q^{74} +6.30301 q^{76} -2.38369 q^{77} +10.2283 q^{79} -1.00000 q^{80} -0.242382 q^{82} +2.14131 q^{83} +1.35277 q^{85} -1.75762 q^{86} +3.68293 q^{88} +5.81346 q^{89} -2.08702 q^{91} -5.22455 q^{92} -6.44432 q^{94} -6.30301 q^{95} -4.20194 q^{97} +6.58110 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8	\( 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8} + 5 q^{10} - 5 q^{11} + 5 q^{13} - 3 q^{14} + 5 q^{16} - 7 q^{17} - q^{19} - 5 q^{20} + 5 q^{22} - 5 q^{23} + 5 q^{25} - 5 q^{26} + 3 q^{28} - 2 q^{29} + 16 q^{31} - 5 q^{32} + 7 q^{34} - 3 q^{35} + 5 q^{37} + q^{38} + 5 q^{40} - 2 q^{41} + 12 q^{43} - 5 q^{44} + 5 q^{46} - 6 q^{47} + 16 q^{49} - 5 q^{50} + 5 q^{52} - 3 q^{53} + 5 q^{55} - 3 q^{56} + 2 q^{58} + 10 q^{59} + 16 q^{61} - 16 q^{62} + 5 q^{64} - 5 q^{65} + 4 q^{67} - 7 q^{68} + 3 q^{70} + 8 q^{71} - 3 q^{73} - 5 q^{74} - q^{76} - 3 q^{77} + 2 q^{79} - 5 q^{80} + 2 q^{82} + 5 q^{83} + 7 q^{85} - 12 q^{86} + 5 q^{88} + 7 q^{89} + 33 q^{91} - 5 q^{92} + 6 q^{94} + q^{95} + 14 q^{97} - 16 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8 + 5 * q^10 - 5 * q^11 + 5 * q^13 - 3 * q^14 + 5 * q^16 - 7 * q^17 - q^19 - 5 * q^20 + 5 * q^22 - 5 * q^23 + 5 * q^25 - 5 * q^26 + 3 * q^28 - 2 * q^29 + 16 * q^31 - 5 * q^32 + 7 * q^34 - 3 * q^35 + 5 * q^37 + q^38 + 5 * q^40 - 2 * q^41 + 12 * q^43 - 5 * q^44 + 5 * q^46 - 6 * q^47 + 16 * q^49 - 5 * q^50 + 5 * q^52 - 3 * q^53 + 5 * q^55 - 3 * q^56 + 2 * q^58 + 10 * q^59 + 16 * q^61 - 16 * q^62 + 5 * q^64 - 5 * q^65 + 4 * q^67 - 7 * q^68 + 3 * q^70 + 8 * q^71 - 3 * q^73 - 5 * q^74 - q^76 - 3 * q^77 + 2 * q^79 - 5 * q^80 + 2 * q^82 + 5 * q^83 + 7 * q^85 - 12 * q^86 + 5 * q^88 + 7 * q^89 + 33 * q^91 - 5 * q^92 + 6 * q^94 + q^95 + 14 * q^97 - 16 * q^98

Introduction

L-functions

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