LMFDB - Embedded newform 3380.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, 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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.756.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 2 $ x^3 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.339877$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.339877 q^{3} +1.00000 q^{5} -3.88448 q^{7} -2.88448 q^{9} +O(q^{10})$	$q-0.339877 q^{3} +1.00000 q^{5} -3.88448 q^{7} -2.88448 q^{9} -1.54461 q^{11} -0.339877 q^{15} +2.86485 q^{19} +1.32025 q^{21} -5.42909 q^{23} +1.00000 q^{25} +2.00000 q^{27} -5.20473 q^{29} +6.22436 q^{31} +0.524976 q^{33} -3.88448 q^{35} +8.56424 q^{37} -9.08921 q^{41} -0.980369 q^{43} -2.88448 q^{45} +6.52498 q^{47} +8.08921 q^{49} +6.44872 q^{53} -1.54461 q^{55} -0.973697 q^{57} -4.45539 q^{59} +9.65345 q^{61} +11.2047 q^{63} +6.97370 q^{67} +1.84522 q^{69} -12.6731 q^{71} +3.43576 q^{73} -0.339877 q^{75} +6.00000 q^{77} -13.1285 q^{79} +7.97370 q^{81} +8.56424 q^{83} +1.76897 q^{87} +17.1285 q^{89} -2.11552 q^{93} +2.86485 q^{95} +13.7690 q^{97} +4.45539 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^5 + 3 * q^9	$ 3 q + 3 q^{5} + 3 q^{9} + 6 q^{11} + 6 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} - 6 q^{29} + 6 q^{31} - 6 q^{33} + 12 q^{37} - 6 q^{41} - 6 q^{43} + 3 q^{45} + 12 q^{47} + 3 q^{49} - 6 q^{53} + 6 q^{55} + 30 q^{57} - 24 q^{59} - 6 q^{61} + 24 q^{63} - 12 q^{67} + 24 q^{73} + 18 q^{77} - 12 q^{79} - 9 q^{81} + 12 q^{83} - 18 q^{87} + 24 q^{89} - 18 q^{93} + 18 q^{97} + 24 q^{99}+O(q^{100}) $ 3 * q + 3 * q^5 + 3 * q^9 + 6 * q^11 + 6 * q^21 + 6 * q^23 + 3 * q^25 + 6 * q^27 - 6 * q^29 + 6 * q^31 - 6 * q^33 + 12 * q^37 - 6 * q^41 - 6 * q^43 + 3 * q^45 + 12 * q^47 + 3 * q^49 - 6 * q^53 + 6 * q^55 + 30 * q^57 - 24 * q^59 - 6 * q^61 + 24 * q^63 - 12 * q^67 + 24 * q^73 + 18 * q^77 - 12 * q^79 - 9 * q^81 + 12 * q^83 - 18 * q^87 + 24 * q^89 - 18 * q^93 + 18 * q^97 + 24 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−0.339877	−0.196228	−0.0981140	−	0.995175i	$-0.531281\pi$
$3$	−0.339877	−0.196228	−0.0981140	+	0.995175i	$0.531281\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.88448	−1.46820	−0.734098	−	0.679043i	$-0.762395\pi$
$7$	−3.88448	−1.46820	−0.734098	+	0.679043i	$0.762395\pi$
$8$	0	0
$9$	−2.88448	−0.961495
$10$	0	0
$11$	−1.54461	−0.465716	−0.232858	−	0.972511i	$-0.574808\pi$
$11$	−1.54461	−0.465716	−0.232858	+	0.972511i	$0.574808\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.339877	−0.0877558
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	2.86485	0.657242	0.328621	−	0.944462i	$-0.393416\pi$
$19$	2.86485	0.657242	0.328621	+	0.944462i	$0.393416\pi$
$20$	0	0
$21$	1.32025	0.288101
$22$	0	0
$23$	−5.42909	−1.13204	−0.566022	−	0.824390i	$-0.691518\pi$
$23$	−5.42909	−1.13204	−0.566022	+	0.824390i	$0.691518\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.00000	0.384900
$28$	0	0
$29$	−5.20473	−0.966494	−0.483247	−	0.875484i	$-0.660543\pi$
$29$	−5.20473	−0.966494	−0.483247	+	0.875484i	$0.660543\pi$
$30$	0	0
$31$	6.22436	1.11793	0.558964	−	0.829192i	$-0.311199\pi$
$31$	6.22436	1.11793	0.558964	+	0.829192i	$0.311199\pi$
$32$	0	0
$33$	0.524976	0.0913866
$34$	0	0
$35$	−3.88448	−0.656598
$36$	0	0
$37$	8.56424	1.40795	0.703976	−	0.710224i	$-0.251406\pi$
$37$	8.56424	1.40795	0.703976	+	0.710224i	$0.251406\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.08921	−1.41950	−0.709748	−	0.704455i	$-0.751191\pi$
$41$	−9.08921	−1.41950	−0.709748	+	0.704455i	$0.751191\pi$
$42$	0	0
$43$	−0.980369	−0.149505	−0.0747525	−	0.997202i	$-0.523817\pi$
$43$	−0.980369	−0.149505	−0.0747525	+	0.997202i	$0.523817\pi$
$44$	0	0
$45$	−2.88448	−0.429993
$46$	0	0
$47$	6.52498	0.951766	0.475883	−	0.879509i	$-0.342129\pi$
$47$	6.52498	0.951766	0.475883	+	0.879509i	$0.342129\pi$
$48$	0	0
$49$	8.08921	1.15560
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.44872	0.885800	0.442900	−	0.896571i	$-0.353950\pi$
$53$	6.44872	0.885800	0.442900	+	0.896571i	$0.353950\pi$
$54$	0	0
$55$	−1.54461	−0.208275
$56$	0	0
$57$	−0.973697	−0.128969
$58$	0	0
$59$	−4.45539	−0.580043	−0.290021	−	0.957020i	$-0.593662\pi$
$59$	−4.45539	−0.580043	−0.290021	+	0.957020i	$0.593662\pi$
$60$	0	0
$61$	9.65345	1.23600	0.617999	−	0.786179i	$-0.287944\pi$
$61$	9.65345	1.23600	0.617999	+	0.786179i	$0.287944\pi$
$62$	0	0
$63$	11.2047	1.41166
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.97370	0.851973	0.425986	−	0.904730i	$-0.359927\pi$
$67$	6.97370	0.851973	0.425986	+	0.904730i	$0.359927\pi$
$68$	0	0
$69$	1.84522	0.222139
$70$	0	0
$71$	−12.6731	−1.50402	−0.752009	−	0.659153i	$-0.770915\pi$
$71$	−12.6731	−1.50402	−0.752009	+	0.659153i	$0.770915\pi$
$72$	0	0
$73$	3.43576	0.402126	0.201063	−	0.979578i	$-0.435560\pi$
$73$	3.43576	0.402126	0.201063	+	0.979578i	$0.435560\pi$
$74$	0	0
$75$	−0.339877	−0.0392456
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	−13.1285	−1.47707	−0.738534	−	0.674216i	$-0.764482\pi$
$79$	−13.1285	−1.47707	−0.738534	+	0.674216i	$0.764482\pi$
$80$	0	0
$81$	7.97370	0.885966
$82$	0	0
$83$	8.56424	0.940047	0.470024	−	0.882654i	$-0.344245\pi$
$83$	8.56424	0.940047	0.470024	+	0.882654i	$0.344245\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.76897	0.189653
$88$	0	0
$89$	17.1285	1.81561	0.907807	−	0.419387i	$-0.137755\pi$
$89$	17.1285	1.81561	0.907807	+	0.419387i	$0.137755\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.11552	−0.219369
$94$	0	0
$95$	2.86485	0.293928
$96$	0	0
$97$	13.7690	1.39803	0.699013	−	0.715109i	$-0.253623\pi$
$97$	13.7690	1.39803	0.699013	+	0.715109i	$0.253623\pi$
$98$	0	0
$99$	4.45539	0.447784
$100$	0	0
$101$	6.44872	0.641672	0.320836	−	0.947135i	$-0.396036\pi$
$101$	6.44872	0.641672	0.320836	+	0.947135i	$0.396036\pi$
$102$	0	0
$103$	−12.1088	−1.19312	−0.596560	−	0.802569i	$-0.703466\pi$
$103$	−12.1088	−1.19312	−0.596560	+	0.802569i	$0.703466\pi$
$104$	0	0
$105$	1.32025	0.128843
$106$	0	0
$107$	17.4291	1.68493	0.842467	−	0.538748i	$-0.181103\pi$
$107$	17.4291	1.68493	0.842467	+	0.538748i	$0.181103\pi$
$108$	0	0
$109$	16.4095	1.57174	0.785871	−	0.618391i	$-0.212215\pi$
$109$	16.4095	1.57174	0.785871	+	0.618391i	$0.212215\pi$
$110$	0	0
$111$	−2.91079	−0.276280
$112$	0	0
$113$	1.59054	0.149625	0.0748127	−	0.997198i	$-0.476164\pi$
$113$	1.59054	0.149625	0.0748127	+	0.997198i	$0.476164\pi$
$114$	0	0
$115$	−5.42909	−0.506265
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.61419	−0.783108
$122$	0	0
$123$	3.08921	0.278545
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.108844	0.00965837	0.00482918	−	0.999988i	$-0.498463\pi$
$127$	0.108844	0.00965837	0.00482918	+	0.999988i	$0.498463\pi$
$128$	0	0
$129$	0.333205	0.0293371
$130$	0	0
$131$	−10.4095	−0.909479	−0.454739	−	0.890625i	$-0.650268\pi$
$131$	−10.4095	−0.909479	−0.454739	+	0.890625i	$0.650268\pi$
$132$	0	0
$133$	−11.1285	−0.964961
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	−9.08921	−0.776544	−0.388272	−	0.921545i	$-0.626928\pi$
$137$	−9.08921	−0.776544	−0.388272	+	0.921545i	$0.626928\pi$
$138$	0	0
$139$	9.04995	0.767607	0.383803	−	0.923415i	$-0.374614\pi$
$139$	9.04995	0.767607	0.383803	+	0.923415i	$0.374614\pi$
$140$	0	0
$141$	−2.21769	−0.186763
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.20473	−0.432229
$146$	0	0
$147$	−2.74934	−0.226761
$148$	0	0
$149$	11.1285	0.911680	0.455840	−	0.890062i	$-0.349339\pi$
$149$	11.1285	0.911680	0.455840	+	0.890062i	$0.349339\pi$
$150$	0	0
$151$	9.13515	0.743408	0.371704	−	0.928351i	$-0.378774\pi$
$151$	9.13515	0.743408	0.371704	+	0.928351i	$0.378774\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.22436	0.499953
$156$	0	0
$157$	4.91079	0.391923	0.195962	−	0.980612i	$-0.437217\pi$
$157$	4.91079	0.391923	0.195962	+	0.980612i	$0.437217\pi$
$158$	0	0
$159$	−2.19177	−0.173819
$160$	0	0
$161$	21.0892	1.66206
$162$	0	0
$163$	−16.9344	−1.32641	−0.663204	−	0.748439i	$-0.730804\pi$
$163$	−16.9344	−1.32641	−0.663204	+	0.748439i	$0.730804\pi$
$164$	0	0
$165$	0.524976	0.0408693
$166$	0	0
$167$	21.6142	1.67256	0.836278	−	0.548306i	$-0.184727\pi$
$167$	21.6142	1.67256	0.836278	+	0.548306i	$0.184727\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−8.26362	−0.631935
$172$	0	0
$173$	8.03926	0.611214	0.305607	−	0.952158i	$-0.401141\pi$
$173$	8.03926	0.611214	0.305607	+	0.952158i	$0.401141\pi$
$174$	0	0
$175$	−3.88448	−0.293639
$176$	0	0
$177$	1.51429	0.113821
$178$	0	0
$179$	−10.8582	−0.811579	−0.405789	−	0.913967i	$-0.633003\pi$
$179$	−10.8582	−0.811579	−0.405789	+	0.913967i	$0.633003\pi$
$180$	0	0
$181$	−10.5642	−0.785234	−0.392617	−	0.919702i	$-0.628430\pi$
$181$	−10.5642	−0.785234	−0.392617	+	0.919702i	$0.628430\pi$
$182$	0	0
$183$	−3.28098	−0.242537
$184$	0	0
$185$	8.56424	0.629655
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−7.76897	−0.565109
$190$	0	0
$191$	−8.81892	−0.638115	−0.319057	−	0.947735i	$-0.603366\pi$
$191$	−8.81892	−0.638115	−0.319057	+	0.947735i	$0.603366\pi$
$192$	0	0
$193$	0.270294	0.0194562	0.00972809	−	0.999953i	$-0.496903\pi$
$193$	0.270294	0.0194562	0.00972809	+	0.999953i	$0.496903\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.96074	−0.282191	−0.141095	−	0.989996i	$-0.545062\pi$
$197$	−3.96074	−0.282191	−0.141095	+	0.989996i	$0.545062\pi$
$198$	0	0
$199$	−6.03926	−0.428112	−0.214056	−	0.976821i	$-0.568667\pi$
$199$	−6.03926	−0.428112	−0.214056	+	0.976821i	$0.568667\pi$
$200$	0	0
$201$	−2.37020	−0.167181
$202$	0	0
$203$	20.2177	1.41900
$204$	0	0
$205$	−9.08921	−0.634818
$206$	0	0
$207$	15.6601	1.08845
$208$	0	0
$209$	−4.42507	−0.306089
$210$	0	0
$211$	−16.2177	−1.11647	−0.558236	−	0.829682i	$-0.688522\pi$
$211$	−16.2177	−1.11647	−0.558236	+	0.829682i	$0.688522\pi$
$212$	0	0
$213$	4.30729	0.295130
$214$	0	0
$215$	−0.980369	−0.0668606
$216$	0	0
$217$	−24.1784	−1.64134
$218$	0	0
$219$	−1.16774	−0.0789083
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−10.0629	−0.673862	−0.336931	−	0.941529i	$-0.609389\pi$
$223$	−10.0629	−0.673862	−0.336931	+	0.941529i	$0.609389\pi$
$224$	0	0
$225$	−2.88448	−0.192299
$226$	0	0
$227$	−3.43576	−0.228040	−0.114020	−	0.993478i	$-0.536373\pi$
$227$	−3.43576	−0.228040	−0.114020	+	0.993478i	$0.536373\pi$
$228$	0	0
$229$	22.8582	1.51051	0.755256	−	0.655430i	$-0.227513\pi$
$229$	22.8582	1.51051	0.755256	+	0.655430i	$0.227513\pi$
$230$	0	0
$231$	−2.03926	−0.134174
$232$	0	0
$233$	4.40946	0.288873	0.144437	−	0.989514i	$-0.453863\pi$
$233$	4.40946	0.288873	0.144437	+	0.989514i	$0.453863\pi$
$234$	0	0
$235$	6.52498	0.425643
$236$	0	0
$237$	4.46207	0.289842
$238$	0	0
$239$	−22.8122	−1.47560	−0.737801	−	0.675018i	$-0.764136\pi$
$239$	−22.8122	−1.47560	−0.737801	+	0.675018i	$0.764136\pi$
$240$	0	0
$241$	10.6798	0.687943	0.343972	−	0.938980i	$-0.388228\pi$
$241$	10.6798	0.687943	0.343972	+	0.938980i	$0.388228\pi$
$242$	0	0
$243$	−8.71008	−0.558752
$244$	0	0
$245$	8.08921	0.516801
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−2.91079	−0.184464
$250$	0	0
$251$	21.2676	1.34240	0.671201	−	0.741276i	$-0.265779\pi$
$251$	21.2676	1.34240	0.671201	+	0.741276i	$0.265779\pi$
$252$	0	0
$253$	8.38581	0.527211
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.8974	−1.17879	−0.589395	−	0.807845i	$-0.700634\pi$
$257$	−18.8974	−1.17879	−0.589395	+	0.807845i	$0.700634\pi$
$258$	0	0
$259$	−33.2676	−2.06715
$260$	0	0
$261$	15.0130	0.929279
$262$	0	0
$263$	−13.0196	−0.802825	−0.401412	−	0.915897i	$-0.631481\pi$
$263$	−13.0196	−0.802825	−0.401412	+	0.915897i	$0.631481\pi$
$264$	0	0
$265$	6.44872	0.396142
$266$	0	0
$267$	−5.82157	−0.356274
$268$	0	0
$269$	14.3702	0.876166	0.438083	−	0.898934i	$-0.355658\pi$
$269$	14.3702	0.876166	0.438083	+	0.898934i	$0.355658\pi$
$270$	0	0
$271$	13.7230	0.833615	0.416807	−	0.908995i	$-0.363149\pi$
$271$	13.7230	0.833615	0.416807	+	0.908995i	$0.363149\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.54461	−0.0931433
$276$	0	0
$277$	19.2676	1.15768	0.578840	−	0.815441i	$-0.303506\pi$
$277$	19.2676	1.15768	0.578840	+	0.815441i	$0.303506\pi$
$278$	0	0
$279$	−17.9541	−1.07488
$280$	0	0
$281$	−3.96074	−0.236278	−0.118139	−	0.992997i	$-0.537693\pi$
$281$	−3.96074	−0.236278	−0.118139	+	0.992997i	$0.537693\pi$
$282$	0	0
$283$	21.1981	1.26009	0.630047	−	0.776557i	$-0.283036\pi$
$283$	21.1981	1.26009	0.630047	+	0.776557i	$0.283036\pi$
$284$	0	0
$285$	−0.973697	−0.0576769
$286$	0	0
$287$	35.3069	2.08410
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−4.67975	−0.274332
$292$	0	0
$293$	13.6927	0.799937	0.399968	−	0.916529i	$-0.369021\pi$
$293$	13.6927	0.799937	0.399968	+	0.916529i	$0.369021\pi$
$294$	0	0
$295$	−4.45539	−0.259403
$296$	0	0
$297$	−3.08921	−0.179254
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3.80823	0.219503
$302$	0	0
$303$	−2.19177	−0.125914
$304$	0	0
$305$	9.65345	0.552755
$306$	0	0
$307$	12.7953	0.730265	0.365132	−	0.930956i	$-0.381024\pi$
$307$	12.7953	0.730265	0.365132	+	0.930956i	$0.381024\pi$
$308$	0	0
$309$	4.11552	0.234124
$310$	0	0
$311$	22.8582	1.29617	0.648084	−	0.761569i	$-0.275570\pi$
$311$	22.8582	1.29617	0.648084	+	0.761569i	$0.275570\pi$
$312$	0	0
$313$	−27.2284	−1.53904	−0.769520	−	0.638623i	$-0.779504\pi$
$313$	−27.2284	−1.53904	−0.769520	+	0.638623i	$0.779504\pi$
$314$	0	0
$315$	11.2047	0.631315
$316$	0	0
$317$	27.7926	1.56099	0.780494	−	0.625163i	$-0.214967\pi$
$317$	27.7926	1.56099	0.780494	+	0.625163i	$0.214967\pi$
$318$	0	0
$319$	8.03926	0.450112
$320$	0	0
$321$	−5.92375	−0.330631
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.57720	−0.308420
$328$	0	0
$329$	−25.3462	−1.39738
$330$	0	0
$331$	19.7230	1.08408	0.542038	−	0.840354i	$-0.317653\pi$
$331$	19.7230	1.08408	0.542038	+	0.840354i	$0.317653\pi$
$332$	0	0
$333$	−24.7034	−1.35374
$334$	0	0
$335$	6.97370	0.381014
$336$	0	0
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	0	0
$339$	−0.540588	−0.0293607
$340$	0	0
$341$	−9.61419	−0.520638
$342$	0	0
$343$	−4.23103	−0.228454
$344$	0	0
$345$	1.84522	0.0993434
$346$	0	0
$347$	−2.61017	−0.140121	−0.0700607	−	0.997543i	$-0.522319\pi$
$347$	−2.61017	−0.140121	−0.0700607	+	0.997543i	$0.522319\pi$
$348$	0	0
$349$	3.08921	0.165362	0.0826809	−	0.996576i	$-0.473652\pi$
$349$	3.08921	0.165362	0.0826809	+	0.996576i	$0.473652\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.9211	1.11352	0.556759	−	0.830674i	$-0.312045\pi$
$353$	20.9211	1.11352	0.556759	+	0.830674i	$0.312045\pi$
$354$	0	0
$355$	−12.6731	−0.672617
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.8515	−1.62828	−0.814140	−	0.580668i	$-0.802791\pi$
$359$	−30.8515	−1.62828	−0.814140	+	0.580668i	$0.802791\pi$
$360$	0	0
$361$	−10.7926	−0.568032
$362$	0	0
$363$	2.92776	0.153668
$364$	0	0
$365$	3.43576	0.179836
$366$	0	0
$367$	21.0196	1.09722	0.548608	−	0.836080i	$-0.315158\pi$
$367$	21.0196	1.09722	0.548608	+	0.836080i	$0.315158\pi$
$368$	0	0
$369$	26.2177	1.36484
$370$	0	0
$371$	−25.0500	−1.30053
$372$	0	0
$373$	29.3462	1.51949	0.759743	−	0.650223i	$-0.225325\pi$
$373$	29.3462	1.51949	0.759743	+	0.650223i	$0.225325\pi$
$374$	0	0
$375$	−0.339877	−0.0175512
$376$	0	0
$377$	0	0
$378$	0	0
$379$	11.3528	0.583156	0.291578	−	0.956547i	$-0.405820\pi$
$379$	11.3528	0.583156	0.291578	+	0.956547i	$0.405820\pi$
$380$	0	0
$381$	−0.0369937	−0.00189524
$382$	0	0
$383$	−20.5642	−1.05078	−0.525392	−	0.850860i	$-0.676081\pi$
$383$	−20.5642	−1.05078	−0.525392	+	0.850860i	$0.676081\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	2.82786	0.143748
$388$	0	0
$389$	−26.8189	−1.35977	−0.679887	−	0.733317i	$-0.737971\pi$
$389$	−26.8189	−1.35977	−0.679887	+	0.733317i	$0.737971\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	3.53793	0.178465
$394$	0	0
$395$	−13.1285	−0.660565
$396$	0	0
$397$	36.1022	1.81192	0.905958	−	0.423367i	$-0.139152\pi$
$397$	36.1022	1.81192	0.905958	+	0.423367i	$0.139152\pi$
$398$	0	0
$399$	3.78231	0.189352
$400$	0	0
$401$	4.07852	0.203672	0.101836	−	0.994801i	$-0.467528\pi$
$401$	4.07852	0.203672	0.101836	+	0.994801i	$0.467528\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	7.97370	0.396216
$406$	0	0
$407$	−13.2284	−0.655706
$408$	0	0
$409$	−0.627148	−0.0310105	−0.0155052	−	0.999880i	$-0.504936\pi$
$409$	−0.627148	−0.0310105	−0.0155052	+	0.999880i	$0.504936\pi$
$410$	0	0
$411$	3.08921	0.152380
$412$	0	0
$413$	17.3069	0.851617
$414$	0	0
$415$	8.56424	0.420402
$416$	0	0
$417$	−3.07587	−0.150626
$418$	0	0
$419$	−11.5513	−0.564317	−0.282158	−	0.959368i	$-0.591050\pi$
$419$	−11.5513	−0.564317	−0.282158	+	0.959368i	$0.591050\pi$
$420$	0	0
$421$	−35.4853	−1.72945	−0.864725	−	0.502246i	$-0.832507\pi$
$421$	−35.4853	−1.72945	−0.864725	+	0.502246i	$0.832507\pi$
$422$	0	0
$423$	−18.8212	−0.915117
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−37.4987	−1.81469
$428$	0	0
$429$	0	0
$430$	0	0
$431$	33.7623	1.62627	0.813136	−	0.582073i	$-0.197758\pi$
$431$	33.7623	1.62627	0.813136	+	0.582073i	$0.197758\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	1.76897	0.0848155
$436$	0	0
$437$	−15.5535	−0.744027
$438$	0	0
$439$	19.0892	0.911078	0.455539	−	0.890216i	$-0.349446\pi$
$439$	19.0892	0.911078	0.455539	+	0.890216i	$0.349446\pi$
$440$	0	0
$441$	−23.3332	−1.11110
$442$	0	0
$443$	−10.2873	−0.488763	−0.244382	−	0.969679i	$-0.578585\pi$
$443$	−10.2873	−0.488763	−0.244382	+	0.969679i	$0.578585\pi$
$444$	0	0
$445$	17.1285	0.811968
$446$	0	0
$447$	−3.78231	−0.178897
$448$	0	0
$449$	−2.21769	−0.104659	−0.0523296	−	0.998630i	$-0.516665\pi$
$449$	−2.21769	−0.104659	−0.0523296	+	0.998630i	$0.516665\pi$
$450$	0	0
$451$	14.0393	0.661083
$452$	0	0
$453$	−3.10483	−0.145877
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17.3069	−0.809583	−0.404791	−	0.914409i	$-0.632656\pi$
$457$	−17.3069	−0.809583	−0.404791	+	0.914409i	$0.632656\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16.1392	0.751676	0.375838	−	0.926685i	$-0.377355\pi$
$461$	16.1392	0.751676	0.375838	+	0.926685i	$0.377355\pi$
$462$	0	0
$463$	6.52498	0.303241	0.151621	−	0.988439i	$-0.451551\pi$
$463$	6.52498	0.303241	0.151621	+	0.988439i	$0.451551\pi$
$464$	0	0
$465$	−2.11552	−0.0981047
$466$	0	0
$467$	35.4291	1.63946	0.819731	−	0.572748i	$-0.194123\pi$
$467$	35.4291	1.63946	0.819731	+	0.572748i	$0.194123\pi$
$468$	0	0
$469$	−27.0892	−1.25086
$470$	0	0
$471$	−1.66906	−0.0769064
$472$	0	0
$473$	1.51429	0.0696269
$474$	0	0
$475$	2.86485	0.131448
$476$	0	0
$477$	−18.6012	−0.851692
$478$	0	0
$479$	−26.7730	−1.22329	−0.611644	−	0.791133i	$-0.709492\pi$
$479$	−26.7730	−1.22329	−0.611644	+	0.791133i	$0.709492\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−7.16774	−0.326143
$484$	0	0
$485$	13.7690	0.625217
$486$	0	0
$487$	−18.3725	−0.832536	−0.416268	−	0.909242i	$-0.636662\pi$
$487$	−18.3725	−0.832536	−0.416268	+	0.909242i	$0.636662\pi$
$488$	0	0
$489$	5.75562	0.260278
$490$	0	0
$491$	8.81892	0.397992	0.198996	−	0.980000i	$-0.436232\pi$
$491$	8.81892	0.397992	0.198996	+	0.980000i	$0.436232\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4.45539	0.200255
$496$	0	0
$497$	49.2284	2.20819
$498$	0	0
$499$	−23.7756	−1.06434	−0.532172	−	0.846636i	$-0.678624\pi$
$499$	−23.7756	−1.06434	−0.532172	+	0.846636i	$0.678624\pi$
$500$	0	0
$501$	−7.34616	−0.328202
$502$	0	0
$503$	39.1455	1.74541	0.872705	−	0.488248i	$-0.162364\pi$
$503$	39.1455	1.74541	0.872705	+	0.488248i	$0.162364\pi$
$504$	0	0
$505$	6.44872	0.286964
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−44.5745	−1.97573	−0.987866	−	0.155309i	$-0.950363\pi$
$509$	−44.5745	−1.97573	−0.987866	+	0.155309i	$0.950363\pi$
$510$	0	0
$511$	−13.3462	−0.590400
$512$	0	0
$513$	5.72971	0.252973
$514$	0	0
$515$	−12.1088	−0.533579
$516$	0	0
$517$	−10.0785	−0.443253
$518$	0	0
$519$	−2.73236	−0.119937
$520$	0	0
$521$	−17.2047	−0.753753	−0.376876	−	0.926264i	$-0.623002\pi$
$521$	−17.2047	−0.753753	−0.376876	+	0.926264i	$0.623002\pi$
$522$	0	0
$523$	3.19806	0.139841	0.0699207	−	0.997553i	$-0.477725\pi$
$523$	3.19806	0.139841	0.0699207	+	0.997553i	$0.477725\pi$
$524$	0	0
$525$	1.32025	0.0576203
$526$	0	0
$527$	0	0
$528$	0	0
$529$	6.47502	0.281523
$530$	0	0
$531$	12.8515	0.557708
$532$	0	0
$533$	0	0
$534$	0	0
$535$	17.4291	0.753525
$536$	0	0
$537$	3.69044	0.159254
$538$	0	0
$539$	−12.4947	−0.538183
$540$	0	0
$541$	1.65118	0.0709899	0.0354950	−	0.999370i	$-0.488699\pi$
$541$	1.65118	0.0709899	0.0354950	+	0.999370i	$0.488699\pi$
$542$	0	0
$543$	3.59054	0.154085
$544$	0	0
$545$	16.4095	0.702904
$546$	0	0
$547$	38.3265	1.63872	0.819362	−	0.573276i	$-0.194328\pi$
$547$	38.3265	1.63872	0.819362	+	0.573276i	$0.194328\pi$
$548$	0	0
$549$	−27.8452	−1.18841
$550$	0	0
$551$	−14.9108	−0.635221
$552$	0	0
$553$	50.9973	2.16863
$554$	0	0
$555$	−2.91079	−0.123556
$556$	0	0
$557$	−31.8711	−1.35042	−0.675212	−	0.737624i	$-0.735948\pi$
$557$	−31.8711	−1.35042	−0.675212	+	0.737624i	$0.735948\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.87781	−0.247720	−0.123860	−	0.992300i	$-0.539527\pi$
$563$	−5.87781	−0.247720	−0.123860	+	0.992300i	$0.539527\pi$
$564$	0	0
$565$	1.59054	0.0669145
$566$	0	0
$567$	−30.9737	−1.30077
$568$	0	0
$569$	16.0629	0.673392	0.336696	−	0.941613i	$-0.390690\pi$
$569$	16.0629	0.673392	0.336696	+	0.941613i	$0.390690\pi$
$570$	0	0
$571$	−5.04995	−0.211334	−0.105667	−	0.994402i	$-0.533698\pi$
$571$	−5.04995	−0.211334	−0.105667	+	0.994402i	$0.533698\pi$
$572$	0	0
$573$	2.99735	0.125216
$574$	0	0
$575$	−5.42909	−0.226409
$576$	0	0
$577$	−17.3832	−0.723670	−0.361835	−	0.932242i	$-0.617850\pi$
$577$	−17.3832	−0.723670	−0.361835	+	0.932242i	$0.617850\pi$
$578$	0	0
$579$	−0.0918667	−0.00381785
$580$	0	0
$581$	−33.2676	−1.38017
$582$	0	0
$583$	−9.96074	−0.412532
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.57493	−0.312651	−0.156325	−	0.987706i	$-0.549965\pi$
$587$	−7.57493	−0.312651	−0.156325	+	0.987706i	$0.549965\pi$
$588$	0	0
$589$	17.8319	0.734750
$590$	0	0
$591$	1.34616	0.0553738
$592$	0	0
$593$	−16.9500	−0.696055	−0.348028	−	0.937484i	$-0.613148\pi$
$593$	−16.9500	−0.696055	−0.348028	+	0.937484i	$0.613148\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.05261	0.0840075
$598$	0	0
$599$	−43.6771	−1.78460	−0.892299	−	0.451445i	$-0.850909\pi$
$599$	−43.6771	−1.78460	−0.892299	+	0.451445i	$0.850909\pi$
$600$	0	0
$601$	2.07852	0.0847847	0.0423924	−	0.999101i	$-0.486502\pi$
$601$	2.07852	0.0847847	0.0423924	+	0.999101i	$0.486502\pi$
$602$	0	0
$603$	−20.1155	−0.819167
$604$	0	0
$605$	−8.61419	−0.350217
$606$	0	0
$607$	−34.3265	−1.39327	−0.696635	−	0.717425i	$-0.745320\pi$
$607$	−34.3265	−1.39327	−0.696635	+	0.717425i	$0.745320\pi$
$608$	0	0
$609$	−6.87153	−0.278448
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−18.3569	−0.741426	−0.370713	−	0.928747i	$-0.620887\pi$
$613$	−18.3569	−0.741426	−0.370713	+	0.928747i	$0.620887\pi$
$614$	0	0
$615$	3.08921	0.124569
$616$	0	0
$617$	−37.5246	−1.51068	−0.755342	−	0.655331i	$-0.772529\pi$
$617$	−37.5246	−1.51068	−0.755342	+	0.655331i	$0.772529\pi$
$618$	0	0
$619$	12.8256	0.515504	0.257752	−	0.966211i	$-0.417018\pi$
$619$	12.8256	0.515504	0.257752	+	0.966211i	$0.417018\pi$
$620$	0	0
$621$	−10.8582	−0.435724
$622$	0	0
$623$	−66.5353	−2.66568
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.50398	0.0600632
$628$	0	0
$629$	0	0
$630$	0	0
$631$	12.6472	0.503476	0.251738	−	0.967795i	$-0.418998\pi$
$631$	12.6472	0.503476	0.251738	+	0.967795i	$0.418998\pi$
$632$	0	0
$633$	5.51202	0.219083
$634$	0	0
$635$	0.108844	0.00431935
$636$	0	0
$637$	0	0
$638$	0	0
$639$	36.5553	1.44611
$640$	0	0
$641$	−5.10256	−0.201539	−0.100769	−	0.994910i	$-0.532130\pi$
$641$	−5.10256	−0.201539	−0.100769	+	0.994910i	$0.532130\pi$
$642$	0	0
$643$	0.346549	0.0136666	0.00683328	−	0.999977i	$-0.497825\pi$
$643$	0.346549	0.0136666	0.00683328	+	0.999977i	$0.497825\pi$
$644$	0	0
$645$	0.333205	0.0131199
$646$	0	0
$647$	−44.2480	−1.73957	−0.869784	−	0.493432i	$-0.835742\pi$
$647$	−44.2480	−1.73957	−0.869784	+	0.493432i	$0.835742\pi$
$648$	0	0
$649$	6.88183	0.270135
$650$	0	0
$651$	8.21769	0.322077
$652$	0	0
$653$	−14.0393	−0.549399	−0.274699	−	0.961530i	$-0.588578\pi$
$653$	−14.0393	−0.549399	−0.274699	+	0.961530i	$0.588578\pi$
$654$	0	0
$655$	−10.4095	−0.406731
$656$	0	0
$657$	−9.91040	−0.386642
$658$	0	0
$659$	12.4487	0.484933	0.242467	−	0.970160i	$-0.422043\pi$
$659$	12.4487	0.484933	0.242467	+	0.970160i	$0.422043\pi$
$660$	0	0
$661$	−11.8216	−0.459806	−0.229903	−	0.973214i	$-0.573841\pi$
$661$	−11.8216	−0.459806	−0.229903	+	0.973214i	$0.573841\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−11.1285	−0.431544
$666$	0	0
$667$	28.2569	1.09411
$668$	0	0
$669$	3.42015	0.132231
$670$	0	0
$671$	−14.9108	−0.575625
$672$	0	0
$673$	−31.3069	−1.20679	−0.603396	−	0.797442i	$-0.706186\pi$
$673$	−31.3069	−1.20679	−0.603396	+	0.797442i	$0.706186\pi$
$674$	0	0
$675$	2.00000	0.0769800
$676$	0	0
$677$	28.8582	1.10911	0.554555	−	0.832147i	$-0.312889\pi$
$677$	28.8582	1.10911	0.554555	+	0.832147i	$0.312889\pi$
$678$	0	0
$679$	−53.4853	−2.05258
$680$	0	0
$681$	1.16774	0.0447478
$682$	0	0
$683$	−20.5642	−0.786869	−0.393434	−	0.919353i	$-0.628713\pi$
$683$	−20.5642	−0.786869	−0.393434	+	0.919353i	$0.628713\pi$
$684$	0	0
$685$	−9.08921	−0.347281
$686$	0	0
$687$	−7.76897	−0.296405
$688$	0	0
$689$	0	0
$690$	0	0
$691$	6.67308	0.253856	0.126928	−	0.991912i	$-0.459488\pi$
$691$	6.67308	0.253856	0.126928	+	0.991912i	$0.459488\pi$
$692$	0	0
$693$	−17.3069	−0.657435
$694$	0	0
$695$	9.04995	0.343284
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−1.49867	−0.0566850
$700$	0	0
$701$	−9.62980	−0.363713	−0.181856	−	0.983325i	$-0.558211\pi$
$701$	−9.62980	−0.363713	−0.181856	+	0.983325i	$0.558211\pi$
$702$	0	0
$703$	24.5353	0.925366
$704$	0	0
$705$	−2.21769	−0.0835230
$706$	0	0
$707$	−25.0500	−0.942100
$708$	0	0
$709$	33.8082	1.26969	0.634847	−	0.772638i	$-0.281063\pi$
$709$	33.8082	1.26969	0.634847	+	0.772638i	$0.281063\pi$
$710$	0	0
$711$	37.8689	1.42019
$712$	0	0
$713$	−33.7926	−1.26554
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.75336	0.289554
$718$	0	0
$719$	36.2043	1.35019	0.675097	−	0.737729i	$-0.264102\pi$
$719$	36.2043	1.35019	0.675097	+	0.737729i	$0.264102\pi$
$720$	0	0
$721$	47.0366	1.75173
$722$	0	0
$723$	−3.62980	−0.134994
$724$	0	0
$725$	−5.20473	−0.193299
$726$	0	0
$727$	46.2480	1.71524	0.857622	−	0.514281i	$-0.171941\pi$
$727$	46.2480	1.71524	0.857622	+	0.514281i	$0.171941\pi$
$728$	0	0
$729$	−20.9607	−0.776324
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−20.0393	−0.740167	−0.370084	−	0.928998i	$-0.620671\pi$
$733$	−20.0393	−0.740167	−0.370084	+	0.928998i	$0.620671\pi$
$734$	0	0
$735$	−2.74934	−0.101411
$736$	0	0
$737$	−10.7716	−0.396778
$738$	0	0
$739$	−52.6338	−1.93617	−0.968083	−	0.250629i	$-0.919363\pi$
$739$	−52.6338	−1.93617	−0.968083	+	0.250629i	$0.919363\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	47.9497	1.75910	0.879551	−	0.475804i	$-0.157843\pi$
$743$	47.9497	1.75910	0.879551	+	0.475804i	$0.157843\pi$
$744$	0	0
$745$	11.1285	0.407716
$746$	0	0
$747$	−24.7034	−0.903850
$748$	0	0
$749$	−67.7030	−2.47381
$750$	0	0
$751$	−23.0892	−0.842537	−0.421269	−	0.906936i	$-0.638415\pi$
$751$	−23.0892	−0.842537	−0.421269	+	0.906936i	$0.638415\pi$
$752$	0	0
$753$	−7.22838	−0.263417
$754$	0	0
$755$	9.13515	0.332462
$756$	0	0
$757$	44.3176	1.61075	0.805375	−	0.592765i	$-0.201964\pi$
$757$	44.3176	1.61075	0.805375	+	0.592765i	$0.201964\pi$
$758$	0	0
$759$	−2.85014	−0.103454
$760$	0	0
$761$	41.4853	1.50384	0.751921	−	0.659253i	$-0.229127\pi$
$761$	41.4853	1.50384	0.751921	+	0.659253i	$0.229127\pi$
$762$	0	0
$763$	−63.7423	−2.30763
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	28.5879	1.03091	0.515453	−	0.856918i	$-0.327624\pi$
$769$	28.5879	1.03091	0.515453	+	0.856918i	$0.327624\pi$
$770$	0	0
$771$	6.42280	0.231312
$772$	0	0
$773$	−15.4358	−0.555186	−0.277593	−	0.960699i	$-0.589537\pi$
$773$	−15.4358	−0.555186	−0.277593	+	0.960699i	$0.589537\pi$
$774$	0	0
$775$	6.22436	0.223586
$776$	0	0
$777$	11.3069	0.405633
$778$	0	0
$779$	−26.0393	−0.932953
$780$	0	0
$781$	19.5749	0.700446
$782$	0	0
$783$	−10.4095	−0.372004
$784$	0	0
$785$	4.91079	0.175273
$786$	0	0
$787$	−5.38316	−0.191889	−0.0959444	−	0.995387i	$-0.530587\pi$
$787$	−5.38316	−0.191889	−0.0959444	+	0.995387i	$0.530587\pi$
$788$	0	0
$789$	4.42507	0.157537
$790$	0	0
$791$	−6.17843	−0.219680
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−2.19177	−0.0777341
$796$	0	0
$797$	−35.7556	−1.26653	−0.633265	−	0.773935i	$-0.718286\pi$
$797$	−35.7556	−1.26653	−0.633265	+	0.773935i	$0.718286\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−49.4068	−1.74570
$802$	0	0
$803$	−5.30690	−0.187277
$804$	0	0
$805$	21.0892	0.743297
$806$	0	0
$807$	−4.88410	−0.171928
$808$	0	0
$809$	−35.7712	−1.25765	−0.628825	−	0.777547i	$-0.716464\pi$
$809$	−35.7712	−1.25765	−0.628825	+	0.777547i	$0.716464\pi$
$810$	0	0
$811$	−30.2244	−1.06132	−0.530660	−	0.847585i	$-0.678056\pi$
$811$	−30.2244	−1.06132	−0.530660	+	0.847585i	$0.678056\pi$
$812$	0	0
$813$	−4.66414	−0.163579
$814$	0	0
$815$	−16.9344	−0.593187
$816$	0	0
$817$	−2.80861	−0.0982610
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.2284	0.880477	0.440238	−	0.897881i	$-0.354894\pi$
$821$	25.2284	0.880477	0.440238	+	0.897881i	$0.354894\pi$
$822$	0	0
$823$	−30.9804	−1.07991	−0.539954	−	0.841695i	$-0.681558\pi$
$823$	−30.9804	−1.07991	−0.539954	+	0.841695i	$0.681558\pi$
$824$	0	0
$825$	0.524976	0.0182773
$826$	0	0
$827$	6.82119	0.237196	0.118598	−	0.992942i	$-0.462160\pi$
$827$	6.82119	0.237196	0.118598	+	0.992942i	$0.462160\pi$
$828$	0	0
$829$	3.39650	0.117965	0.0589827	−	0.998259i	$-0.481214\pi$
$829$	3.39650	0.117965	0.0589827	+	0.998259i	$0.481214\pi$
$830$	0	0
$831$	−6.54863	−0.227169
$832$	0	0
$833$	0	0
$834$	0	0
$835$	21.6142	0.747990
$836$	0	0
$837$	12.4487	0.430291
$838$	0	0
$839$	9.76230	0.337032	0.168516	−	0.985699i	$-0.446103\pi$
$839$	9.76230	0.337032	0.168516	+	0.985699i	$0.446103\pi$
$840$	0	0
$841$	−1.91079	−0.0658892
$842$	0	0
$843$	1.34616	0.0463643
$844$	0	0
$845$	0	0
$846$	0	0
$847$	33.4617	1.14976
$848$	0	0
$849$	−7.20473	−0.247266
$850$	0	0
$851$	−46.4960	−1.59386
$852$	0	0
$853$	−18.2806	−0.625916	−0.312958	−	0.949767i	$-0.601320\pi$
$853$	−18.2806	−0.625916	−0.312958	+	0.949767i	$0.601320\pi$
$854$	0	0
$855$	−8.26362	−0.282610
$856$	0	0
$857$	11.6691	0.398608	0.199304	−	0.979938i	$-0.436132\pi$
$857$	11.6691	0.398608	0.199304	+	0.979938i	$0.436132\pi$
$858$	0	0
$859$	−3.92148	−0.133799	−0.0668995	−	0.997760i	$-0.521311\pi$
$859$	−3.92148	−0.133799	−0.0668995	+	0.997760i	$0.521311\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	0	0
$863$	5.83188	0.198519	0.0992597	−	0.995062i	$-0.468353\pi$
$863$	5.83188	0.198519	0.0992597	+	0.995062i	$0.468353\pi$
$864$	0	0
$865$	8.03926	0.273343
$866$	0	0
$867$	5.77791	0.196228
$868$	0	0
$869$	20.2783	0.687895
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−39.7164	−1.34420
$874$	0	0
$875$	−3.88448	−0.131320
$876$	0	0
$877$	57.5379	1.94292	0.971459	−	0.237208i	$-0.0762321\pi$
$877$	57.5379	1.94292	0.971459	+	0.237208i	$0.0762321\pi$
$878$	0	0
$879$	−4.65384	−0.156970
$880$	0	0
$881$	55.4483	1.86810	0.934051	−	0.357140i	$-0.116248\pi$
$881$	55.4483	1.86810	0.934051	+	0.357140i	$0.116248\pi$
$882$	0	0
$883$	−1.75199	−0.0589591	−0.0294796	−	0.999565i	$-0.509385\pi$
$883$	−1.75199	−0.0589591	−0.0294796	+	0.999565i	$0.509385\pi$
$884$	0	0
$885$	1.51429	0.0509021
$886$	0	0
$887$	36.3265	1.21973	0.609863	−	0.792507i	$-0.291225\pi$
$887$	36.3265	1.21973	0.609863	+	0.792507i	$0.291225\pi$
$888$	0	0
$889$	−0.422804	−0.0141804
$890$	0	0
$891$	−12.3162	−0.412609
$892$	0	0
$893$	18.6931	0.625541
$894$	0	0
$895$	−10.8582	−0.362949
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−32.3961	−1.08047
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−1.29433	−0.0430726
$904$	0	0
$905$	−10.5642	−0.351167
$906$	0	0
$907$	10.0696	0.334355	0.167178	−	0.985927i	$-0.446535\pi$
$907$	10.0696	0.334355	0.167178	+	0.985927i	$0.446535\pi$
$908$	0	0
$909$	−18.6012	−0.616964
$910$	0	0
$911$	−39.8341	−1.31976	−0.659882	−	0.751369i	$-0.729394\pi$
$911$	−39.8341	−1.31976	−0.659882	+	0.751369i	$0.729394\pi$
$912$	0	0
$913$	−13.2284	−0.437795
$914$	0	0
$915$	−3.28098	−0.108466
$916$	0	0
$917$	40.4354	1.33529
$918$	0	0
$919$	11.8608	0.391253	0.195626	−	0.980678i	$-0.437326\pi$
$919$	11.8608	0.391253	0.195626	+	0.980678i	$0.437326\pi$
$920$	0	0
$921$	−4.34882	−0.143298
$922$	0	0
$923$	0	0
$924$	0	0
$925$	8.56424	0.281590
$926$	0	0
$927$	34.9278	1.14718
$928$	0	0
$929$	7.34616	0.241020	0.120510	−	0.992712i	$-0.461547\pi$
$929$	7.34616	0.241020	0.120510	+	0.992712i	$0.461547\pi$
$930$	0	0
$931$	23.1744	0.759511
$932$	0	0
$933$	−7.76897	−0.254345
$934$	0	0
$935$	0	0
$936$	0	0
$937$	45.3069	1.48011	0.740056	−	0.672545i	$-0.234799\pi$
$937$	45.3069	1.48011	0.740056	+	0.672545i	$0.234799\pi$
$938$	0	0
$939$	9.25430	0.302003
$940$	0	0
$941$	5.88222	0.191755	0.0958774	−	0.995393i	$-0.469434\pi$
$941$	5.88222	0.191755	0.0958774	+	0.995393i	$0.469434\pi$
$942$	0	0
$943$	49.3462	1.60693
$944$	0	0
$945$	−7.76897	−0.252725
$946$	0	0
$947$	14.7427	0.479072	0.239536	−	0.970887i	$-0.423005\pi$
$947$	14.7427	0.479072	0.239536	+	0.970887i	$0.423005\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−9.44607	−0.306310
$952$	0	0
$953$	37.2284	1.20595	0.602973	−	0.797762i	$-0.293983\pi$
$953$	37.2284	1.20595	0.602973	+	0.797762i	$0.293983\pi$
$954$	0	0
$955$	−8.81892	−0.285374
$956$	0	0
$957$	−2.73236	−0.0883246
$958$	0	0
$959$	35.3069	1.14012
$960$	0	0
$961$	7.74266	0.249763
$962$	0	0
$963$	−50.2739	−1.62005
$964$	0	0
$965$	0.270294	0.00870107
$966$	0	0
$967$	−31.8711	−1.02491	−0.512453	−	0.858715i	$-0.671263\pi$
$967$	−31.8711	−1.02491	−0.512453	+	0.858715i	$0.671263\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.6531	1.17625	0.588126	−	0.808769i	$-0.299866\pi$
$971$	36.6531	1.17625	0.588126	+	0.808769i	$0.299866\pi$
$972$	0	0
$973$	−35.1544	−1.12700
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.61419	0.307585	0.153793	−	0.988103i	$-0.450851\pi$
$977$	9.61419	0.307585	0.153793	+	0.988103i	$0.450851\pi$
$978$	0	0
$979$	−26.4568	−0.845562
$980$	0	0
$981$	−47.3328	−1.51122
$982$	0	0
$983$	4.78193	0.152520	0.0762599	−	0.997088i	$-0.475702\pi$
$983$	4.78193	0.152520	0.0762599	+	0.997088i	$0.475702\pi$
$984$	0	0
$985$	−3.96074	−0.126200
$986$	0	0
$987$	8.61458	0.274205
$988$	0	0
$989$	5.32251	0.169246
$990$	0	0
$991$	51.8814	1.64807	0.824034	−	0.566540i	$-0.191718\pi$
$991$	51.8814	1.64807	0.824034	+	0.566540i	$0.191718\pi$
$992$	0	0
$993$	−6.70340	−0.212726
$994$	0	0
$995$	−6.03926	−0.191457
$996$	0	0
$997$	12.3176	0.390102	0.195051	−	0.980793i	$-0.437513\pi$
$997$	12.3176	0.390102	0.195051	+	0.980793i	$0.437513\pi$
$998$	0	0
$999$	17.1285	0.541921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.n.1.2		3
13.5	odd	4		260.2.f.a.181.3	✓	6
13.8	odd	4		260.2.f.a.181.4	yes	6
13.12	even	2		3380.2.a.m.1.2		3
39.5	even	4		2340.2.c.d.181.4		6
39.8	even	4		2340.2.c.d.181.3		6
52.31	even	4		1040.2.k.c.961.3		6
52.47	even	4		1040.2.k.c.961.4		6
65.8	even	4		1300.2.d.c.649.3		6
65.18	even	4		1300.2.d.d.649.3		6
65.34	odd	4		1300.2.f.e.701.3		6
65.44	odd	4		1300.2.f.e.701.4		6
65.47	even	4		1300.2.d.d.649.4		6
65.57	even	4		1300.2.d.c.649.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.f.a.181.3	✓	6	13.5	odd	4
260.2.f.a.181.4	yes	6	13.8	odd	4
1040.2.k.c.961.3		6	52.31	even	4
1040.2.k.c.961.4		6	52.47	even	4
1300.2.d.c.649.3		6	65.8	even	4
1300.2.d.c.649.4		6	65.57	even	4
1300.2.d.d.649.3		6	65.18	even	4
1300.2.d.d.649.4		6	65.47	even	4
1300.2.f.e.701.3		6	65.34	odd	4
1300.2.f.e.701.4		6	65.44	odd	4
2340.2.c.d.181.3		6	39.8	even	4
2340.2.c.d.181.4		6	39.5	even	4
3380.2.a.m.1.2		3	13.12	even	2
3380.2.a.n.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q-0.339877 q^{3} +1.00000 q^{5} -3.88448 q^{7} -2.88448 q^{9} +O(q^{10})\)	\(q-0.339877 q^{3} +1.00000 q^{5} -3.88448 q^{7} -2.88448 q^{9} -1.54461 q^{11} -0.339877 q^{15} +2.86485 q^{19} +1.32025 q^{21} -5.42909 q^{23} +1.00000 q^{25} +2.00000 q^{27} -5.20473 q^{29} +6.22436 q^{31} +0.524976 q^{33} -3.88448 q^{35} +8.56424 q^{37} -9.08921 q^{41} -0.980369 q^{43} -2.88448 q^{45} +6.52498 q^{47} +8.08921 q^{49} +6.44872 q^{53} -1.54461 q^{55} -0.973697 q^{57} -4.45539 q^{59} +9.65345 q^{61} +11.2047 q^{63} +6.97370 q^{67} +1.84522 q^{69} -12.6731 q^{71} +3.43576 q^{73} -0.339877 q^{75} +6.00000 q^{77} -13.1285 q^{79} +7.97370 q^{81} +8.56424 q^{83} +1.76897 q^{87} +17.1285 q^{89} -2.11552 q^{93} +2.86485 q^{95} +13.7690 q^{97} +4.45539 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^5 + 3 * q^9	\( 3 q + 3 q^{5} + 3 q^{9} + 6 q^{11} + 6 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} - 6 q^{29} + 6 q^{31} - 6 q^{33} + 12 q^{37} - 6 q^{41} - 6 q^{43} + 3 q^{45} + 12 q^{47} + 3 q^{49} - 6 q^{53} + 6 q^{55} + 30 q^{57} - 24 q^{59} - 6 q^{61} + 24 q^{63} - 12 q^{67} + 24 q^{73} + 18 q^{77} - 12 q^{79} - 9 q^{81} + 12 q^{83} - 18 q^{87} + 24 q^{89} - 18 q^{93} + 18 q^{97} + 24 q^{99}+O(q^{100}) \) 3 * q + 3 * q^5 + 3 * q^9 + 6 * q^11 + 6 * q^21 + 6 * q^23 + 3 * q^25 + 6 * q^27 - 6 * q^29 + 6 * q^31 - 6 * q^33 + 12 * q^37 - 6 * q^41 - 6 * q^43 + 3 * q^45 + 12 * q^47 + 3 * q^49 - 6 * q^53 + 6 * q^55 + 30 * q^57 - 24 * q^59 - 6 * q^61 + 24 * q^63 - 12 * q^67 + 24 * q^73 + 18 * q^77 - 12 * q^79 - 9 * q^81 + 12 * q^83 - 18 * q^87 + 24 * q^89 - 18 * q^93 + 18 * q^97 + 24 * q^99

Introduction

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