LMFDB - Embedded newform 4664.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4664,2,Mod(1,4664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4664, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4664.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4664 = 2^{3} \cdot 11 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4664.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:	$37.2422275027$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5x^{10} - 7x^{9} + 53x^{8} + 13x^{7} - 189x^{6} - 16x^{5} + 260x^{4} + 32x^{3} - 118x^{2} - 8x + 16 $ x^11 - 5x^10 - 7x^9 + 53x^8 + 13x^7 - 189x^6 - 16x^5 + 260x^4 + 32x^3 - 118x^2 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.67321$ of defining polynomial
Character	$\chi$	$=$	4664.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67321 q^{3} +1.73724 q^{5} +1.26953 q^{7} +4.14608 q^{9} +O(q^{10})$	$q-2.67321 q^{3} +1.73724 q^{5} +1.26953 q^{7} +4.14608 q^{9} +1.00000 q^{11} +0.398033 q^{13} -4.64402 q^{15} +4.77699 q^{17} -3.52835 q^{19} -3.39371 q^{21} -6.70473 q^{23} -1.98199 q^{25} -3.06371 q^{27} +2.17464 q^{29} -2.14337 q^{31} -2.67321 q^{33} +2.20547 q^{35} -9.51345 q^{37} -1.06403 q^{39} -4.25616 q^{41} +3.23929 q^{43} +7.20274 q^{45} -7.48638 q^{47} -5.38831 q^{49} -12.7699 q^{51} +1.00000 q^{53} +1.73724 q^{55} +9.43205 q^{57} -5.82416 q^{59} +13.7392 q^{61} +5.26355 q^{63} +0.691480 q^{65} -6.18103 q^{67} +17.9232 q^{69} +3.77261 q^{71} +15.7705 q^{73} +5.29828 q^{75} +1.26953 q^{77} -5.77213 q^{79} -4.24828 q^{81} -7.77309 q^{83} +8.29879 q^{85} -5.81327 q^{87} -8.96613 q^{89} +0.505313 q^{91} +5.72968 q^{93} -6.12961 q^{95} +2.58558 q^{97} +4.14608 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 6 q^{3} + 3 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 11 * q - 6 * q^3 + 3 * q^5 - 5 * q^7 + 7 * q^9	$ 11 q - 6 q^{3} + 3 q^{5} - 5 q^{7} + 7 q^{9} + 11 q^{11} - 13 q^{13} - 8 q^{15} - 7 q^{17} - 21 q^{19} + 6 q^{21} + 11 q^{23} + 4 q^{25} - 6 q^{27} - 5 q^{31} - 6 q^{33} - 25 q^{35} - 4 q^{37} - 19 q^{39} - 11 q^{41} + 16 q^{45} - 17 q^{47} - 2 q^{49} - 18 q^{51} + 11 q^{53} + 3 q^{55} - 5 q^{57} - 19 q^{59} - 2 q^{61} - 36 q^{63} - 13 q^{65} + 25 q^{67} + 3 q^{69} - 30 q^{71} + 5 q^{73} - 5 q^{75} - 5 q^{77} - 23 q^{79} - 9 q^{81} - 19 q^{83} + 2 q^{85} - 7 q^{87} + 6 q^{89} - 20 q^{91} + 43 q^{93} - 50 q^{95} - 35 q^{97} + 7 q^{99}+O(q^{100}) $ 11 * q - 6 * q^3 + 3 * q^5 - 5 * q^7 + 7 * q^9 + 11 * q^11 - 13 * q^13 - 8 * q^15 - 7 * q^17 - 21 * q^19 + 6 * q^21 + 11 * q^23 + 4 * q^25 - 6 * q^27 - 5 * q^31 - 6 * q^33 - 25 * q^35 - 4 * q^37 - 19 * q^39 - 11 * q^41 + 16 * q^45 - 17 * q^47 - 2 * q^49 - 18 * q^51 + 11 * q^53 + 3 * q^55 - 5 * q^57 - 19 * q^59 - 2 * q^61 - 36 * q^63 - 13 * q^65 + 25 * q^67 + 3 * q^69 - 30 * q^71 + 5 * q^73 - 5 * q^75 - 5 * q^77 - 23 * q^79 - 9 * q^81 - 19 * q^83 + 2 * q^85 - 7 * q^87 + 6 * q^89 - 20 * q^91 + 43 * q^93 - 50 * q^95 - 35 * q^97 + 7 * 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
$3$	−2.67321	−1.54338	−0.771691	−	0.635998i	$-0.780589\pi$
$3$	−2.67321	−1.54338	−0.771691	+	0.635998i	$0.780589\pi$
$4$	0	0
$5$	1.73724	0.776919	0.388459	−	0.921466i	$-0.373007\pi$
$5$	1.73724	0.776919	0.388459	+	0.921466i	$0.373007\pi$
$6$	0	0
$7$	1.26953	0.479835	0.239918	−	0.970793i	$-0.422880\pi$
$7$	1.26953	0.479835	0.239918	+	0.970793i	$0.422880\pi$
$8$	0	0
$9$	4.14608	1.38203
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.398033	0.110395	0.0551973	−	0.998475i	$-0.482421\pi$
$13$	0.398033	0.110395	0.0551973	+	0.998475i	$0.482421\pi$
$14$	0	0
$15$	−4.64402	−1.19908
$16$	0	0
$17$	4.77699	1.15859	0.579295	−	0.815118i	$-0.303328\pi$
$17$	4.77699	1.15859	0.579295	+	0.815118i	$0.303328\pi$
$18$	0	0
$19$	−3.52835	−0.809460	−0.404730	−	0.914436i	$-0.632634\pi$
$19$	−3.52835	−0.809460	−0.404730	+	0.914436i	$0.632634\pi$
$20$	0	0
$21$	−3.39371	−0.740569
$22$	0	0
$23$	−6.70473	−1.39803	−0.699017	−	0.715105i	$-0.746379\pi$
$23$	−6.70473	−1.39803	−0.699017	+	0.715105i	$0.746379\pi$
$24$	0	0
$25$	−1.98199	−0.396397
$26$	0	0
$27$	−3.06371	−0.589611
$28$	0	0
$29$	2.17464	0.403820	0.201910	−	0.979404i	$-0.435285\pi$
$29$	2.17464	0.403820	0.201910	+	0.979404i	$0.435285\pi$
$30$	0	0
$31$	−2.14337	−0.384960	−0.192480	−	0.981301i	$-0.561653\pi$
$31$	−2.14337	−0.384960	−0.192480	+	0.981301i	$0.561653\pi$
$32$	0	0
$33$	−2.67321	−0.465347
$34$	0	0
$35$	2.20547	0.372793
$36$	0	0
$37$	−9.51345	−1.56400	−0.782001	−	0.623277i	$-0.785801\pi$
$37$	−9.51345	−1.56400	−0.782001	+	0.623277i	$0.785801\pi$
$38$	0	0
$39$	−1.06403	−0.170381
$40$	0	0
$41$	−4.25616	−0.664700	−0.332350	−	0.943156i	$-0.607842\pi$
$41$	−4.25616	−0.664700	−0.332350	+	0.943156i	$0.607842\pi$
$42$	0	0
$43$	3.23929	0.493988	0.246994	−	0.969017i	$-0.420557\pi$
$43$	3.23929	0.493988	0.246994	+	0.969017i	$0.420557\pi$
$44$	0	0
$45$	7.20274	1.07372
$46$	0	0
$47$	−7.48638	−1.09200	−0.546001	−	0.837785i	$-0.683850\pi$
$47$	−7.48638	−1.09200	−0.546001	+	0.837785i	$0.683850\pi$
$48$	0	0
$49$	−5.38831	−0.769758
$50$	0	0
$51$	−12.7699	−1.78815
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	1.73724	0.234250
$56$	0	0
$57$	9.43205	1.24931
$58$	0	0
$59$	−5.82416	−0.758240	−0.379120	−	0.925347i	$-0.623773\pi$
$59$	−5.82416	−0.758240	−0.379120	+	0.925347i	$0.623773\pi$
$60$	0	0
$61$	13.7392	1.75912	0.879560	−	0.475788i	$-0.157837\pi$
$61$	13.7392	1.75912	0.879560	+	0.475788i	$0.157837\pi$
$62$	0	0
$63$	5.26355	0.663145
$64$	0	0
$65$	0.691480	0.0857676
$66$	0	0
$67$	−6.18103	−0.755133	−0.377567	−	0.925982i	$-0.623239\pi$
$67$	−6.18103	−0.755133	−0.377567	+	0.925982i	$0.623239\pi$
$68$	0	0
$69$	17.9232	2.15770
$70$	0	0
$71$	3.77261	0.447726	0.223863	−	0.974621i	$-0.428133\pi$
$71$	3.77261	0.447726	0.223863	+	0.974621i	$0.428133\pi$
$72$	0	0
$73$	15.7705	1.84580	0.922900	−	0.385040i	$-0.125812\pi$
$73$	15.7705	1.84580	0.922900	+	0.385040i	$0.125812\pi$
$74$	0	0
$75$	5.29828	0.611792
$76$	0	0
$77$	1.26953	0.144676
$78$	0	0
$79$	−5.77213	−0.649415	−0.324708	−	0.945814i	$-0.605266\pi$
$79$	−5.77213	−0.649415	−0.324708	+	0.945814i	$0.605266\pi$
$80$	0	0
$81$	−4.24828	−0.472031
$82$	0	0
$83$	−7.77309	−0.853208	−0.426604	−	0.904439i	$-0.640290\pi$
$83$	−7.77309	−0.853208	−0.426604	+	0.904439i	$0.640290\pi$
$84$	0	0
$85$	8.29879	0.900130
$86$	0	0
$87$	−5.81327	−0.623248
$88$	0	0
$89$	−8.96613	−0.950408	−0.475204	−	0.879876i	$-0.657626\pi$
$89$	−8.96613	−0.950408	−0.475204	+	0.879876i	$0.657626\pi$
$90$	0	0
$91$	0.505313	0.0529712
$92$	0	0
$93$	5.72968	0.594140
$94$	0	0
$95$	−6.12961	−0.628884
$96$	0	0
$97$	2.58558	0.262526	0.131263	−	0.991348i	$-0.458097\pi$
$97$	2.58558	0.262526	0.131263	+	0.991348i	$0.458097\pi$
$98$	0	0
$99$	4.14608	0.416696
$100$	0	0
$101$	−4.69969	−0.467636	−0.233818	−	0.972280i	$-0.575122\pi$
$101$	−4.69969	−0.467636	−0.233818	+	0.972280i	$0.575122\pi$
$102$	0	0
$103$	−9.03478	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$103$	−9.03478	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$104$	0	0
$105$	−5.89570	−0.575362
$106$	0	0
$107$	12.0557	1.16547	0.582734	−	0.812663i	$-0.301983\pi$
$107$	12.0557	1.16547	0.582734	+	0.812663i	$0.301983\pi$
$108$	0	0
$109$	6.09746	0.584031	0.292016	−	0.956414i	$-0.405674\pi$
$109$	6.09746	0.584031	0.292016	+	0.956414i	$0.405674\pi$
$110$	0	0
$111$	25.4315	2.41385
$112$	0	0
$113$	−11.6584	−1.09673	−0.548366	−	0.836239i	$-0.684750\pi$
$113$	−11.6584	−1.09673	−0.548366	+	0.836239i	$0.684750\pi$
$114$	0	0
$115$	−11.6478	−1.08616
$116$	0	0
$117$	1.65028	0.152568
$118$	0	0
$119$	6.06450	0.555932
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	11.3776	1.02589
$124$	0	0
$125$	−12.1294	−1.08489
$126$	0	0
$127$	17.1604	1.52274	0.761371	−	0.648316i	$-0.224526\pi$
$127$	17.1604	1.52274	0.761371	+	0.648316i	$0.224526\pi$
$128$	0	0
$129$	−8.65933	−0.762411
$130$	0	0
$131$	−4.05268	−0.354084	−0.177042	−	0.984203i	$-0.556653\pi$
$131$	−4.05268	−0.354084	−0.177042	+	0.984203i	$0.556653\pi$
$132$	0	0
$133$	−4.47933	−0.388407
$134$	0	0
$135$	−5.32240	−0.458080
$136$	0	0
$137$	−0.754517	−0.0644628	−0.0322314	−	0.999480i	$-0.510261\pi$
$137$	−0.754517	−0.0644628	−0.0322314	+	0.999480i	$0.510261\pi$
$138$	0	0
$139$	−1.31391	−0.111445	−0.0557223	−	0.998446i	$-0.517746\pi$
$139$	−1.31391	−0.111445	−0.0557223	+	0.998446i	$0.517746\pi$
$140$	0	0
$141$	20.0127	1.68537
$142$	0	0
$143$	0.398033	0.0332852
$144$	0	0
$145$	3.77787	0.313735
$146$	0	0
$147$	14.4041	1.18803
$148$	0	0
$149$	4.39693	0.360211	0.180105	−	0.983647i	$-0.442356\pi$
$149$	4.39693	0.360211	0.180105	+	0.983647i	$0.442356\pi$
$150$	0	0
$151$	−10.8954	−0.886657	−0.443328	−	0.896359i	$-0.646202\pi$
$151$	−10.8954	−0.886657	−0.443328	+	0.896359i	$0.646202\pi$
$152$	0	0
$153$	19.8058	1.60120
$154$	0	0
$155$	−3.72355	−0.299083
$156$	0	0
$157$	5.24954	0.418959	0.209479	−	0.977813i	$-0.432823\pi$
$157$	5.24954	0.418959	0.209479	+	0.977813i	$0.432823\pi$
$158$	0	0
$159$	−2.67321	−0.212000
$160$	0	0
$161$	−8.51183	−0.670826
$162$	0	0
$163$	1.78939	0.140156	0.0700778	−	0.997542i	$-0.477675\pi$
$163$	1.78939	0.140156	0.0700778	+	0.997542i	$0.477675\pi$
$164$	0	0
$165$	−4.64402	−0.361537
$166$	0	0
$167$	5.02713	0.389011	0.194506	−	0.980901i	$-0.437690\pi$
$167$	5.02713	0.389011	0.194506	+	0.980901i	$0.437690\pi$
$168$	0	0
$169$	−12.8416	−0.987813
$170$	0	0
$171$	−14.6288	−1.11869
$172$	0	0
$173$	10.0385	0.763214	0.381607	−	0.924325i	$-0.375371\pi$
$173$	10.0385	0.763214	0.381607	+	0.924325i	$0.375371\pi$
$174$	0	0
$175$	−2.51618	−0.190206
$176$	0	0
$177$	15.5692	1.17025
$178$	0	0
$179$	4.89368	0.365771	0.182886	−	0.983134i	$-0.441456\pi$
$179$	4.89368	0.365771	0.182886	+	0.983134i	$0.441456\pi$
$180$	0	0
$181$	14.4546	1.07440	0.537201	−	0.843454i	$-0.319482\pi$
$181$	14.4546	1.07440	0.537201	+	0.843454i	$0.319482\pi$
$182$	0	0
$183$	−36.7277	−2.71499
$184$	0	0
$185$	−16.5272	−1.21510
$186$	0	0
$187$	4.77699	0.349328
$188$	0	0
$189$	−3.88945	−0.282916
$190$	0	0
$191$	−21.4129	−1.54938	−0.774691	−	0.632341i	$-0.782094\pi$
$191$	−21.4129	−1.54938	−0.774691	+	0.632341i	$0.782094\pi$
$192$	0	0
$193$	−11.7984	−0.849267	−0.424633	−	0.905365i	$-0.639597\pi$
$193$	−11.7984	−0.849267	−0.424633	+	0.905365i	$0.639597\pi$
$194$	0	0
$195$	−1.84848	−0.132372
$196$	0	0
$197$	−7.60767	−0.542024	−0.271012	−	0.962576i	$-0.587358\pi$
$197$	−7.60767	−0.542024	−0.271012	+	0.962576i	$0.587358\pi$
$198$	0	0
$199$	9.18665	0.651225	0.325612	−	0.945503i	$-0.394430\pi$
$199$	9.18665	0.651225	0.325612	+	0.945503i	$0.394430\pi$
$200$	0	0
$201$	16.5232	1.16546
$202$	0	0
$203$	2.76075	0.193767
$204$	0	0
$205$	−7.39398	−0.516418
$206$	0	0
$207$	−27.7983	−1.93212
$208$	0	0
$209$	−3.52835	−0.244061
$210$	0	0
$211$	−20.0701	−1.38168	−0.690842	−	0.723006i	$-0.742760\pi$
$211$	−20.0701	−1.38168	−0.690842	+	0.723006i	$0.742760\pi$
$212$	0	0
$213$	−10.0850	−0.691012
$214$	0	0
$215$	5.62744	0.383788
$216$	0	0
$217$	−2.72106	−0.184718
$218$	0	0
$219$	−42.1580	−2.84877
$220$	0	0
$221$	1.90140	0.127902
$222$	0	0
$223$	−24.0433	−1.61006	−0.805028	−	0.593237i	$-0.797850\pi$
$223$	−24.0433	−1.61006	−0.805028	+	0.593237i	$0.797850\pi$
$224$	0	0
$225$	−8.21747	−0.547831
$226$	0	0
$227$	0.0119927	0.000795980 0	0.000397990	−	1.00000i	$-0.499873\pi$
$227$	0.0119927	0.000795980 0	0.000397990		1.00000i	$0.499873\pi$
$228$	0	0
$229$	−2.71451	−0.179380	−0.0896900	−	0.995970i	$-0.528588\pi$
$229$	−2.71451	−0.179380	−0.0896900	+	0.995970i	$0.528588\pi$
$230$	0	0
$231$	−3.39371	−0.223290
$232$	0	0
$233$	20.7294	1.35803	0.679014	−	0.734126i	$-0.262408\pi$
$233$	20.7294	1.35803	0.679014	+	0.734126i	$0.262408\pi$
$234$	0	0
$235$	−13.0057	−0.848396
$236$	0	0
$237$	15.4301	1.00230
$238$	0	0
$239$	11.5613	0.747840	0.373920	−	0.927461i	$-0.378013\pi$
$239$	11.5613	0.747840	0.373920	+	0.927461i	$0.378013\pi$
$240$	0	0
$241$	−8.85779	−0.570580	−0.285290	−	0.958441i	$-0.592090\pi$
$241$	−8.85779	−0.570580	−0.285290	+	0.958441i	$0.592090\pi$
$242$	0	0
$243$	20.5477	1.31813
$244$	0	0
$245$	−9.36080	−0.598039
$246$	0	0
$247$	−1.40440	−0.0893600
$248$	0	0
$249$	20.7791	1.31682
$250$	0	0
$251$	17.5949	1.11058	0.555289	−	0.831657i	$-0.312608\pi$
$251$	17.5949	1.11058	0.555289	+	0.831657i	$0.312608\pi$
$252$	0	0
$253$	−6.70473	−0.421523
$254$	0	0
$255$	−22.1844	−1.38924
$256$	0	0
$257$	−15.3768	−0.959180	−0.479590	−	0.877493i	$-0.659215\pi$
$257$	−15.3768	−0.959180	−0.479590	+	0.877493i	$0.659215\pi$
$258$	0	0
$259$	−12.0776	−0.750464
$260$	0	0
$261$	9.01621	0.558089
$262$	0	0
$263$	−13.1100	−0.808394	−0.404197	−	0.914672i	$-0.632449\pi$
$263$	−13.1100	−0.808394	−0.404197	+	0.914672i	$0.632449\pi$
$264$	0	0
$265$	1.73724	0.106718
$266$	0	0
$267$	23.9684	1.46684
$268$	0	0
$269$	−24.6234	−1.50132	−0.750658	−	0.660691i	$-0.770263\pi$
$269$	−24.6234	−1.50132	−0.750658	+	0.660691i	$0.770263\pi$
$270$	0	0
$271$	−12.2927	−0.746727	−0.373363	−	0.927685i	$-0.621796\pi$
$271$	−12.2927	−0.746727	−0.373363	+	0.927685i	$0.621796\pi$
$272$	0	0
$273$	−1.35081	−0.0817548
$274$	0	0
$275$	−1.98199	−0.119518
$276$	0	0
$277$	9.98689	0.600054	0.300027	−	0.953931i	$-0.403004\pi$
$277$	9.98689	0.600054	0.300027	+	0.953931i	$0.403004\pi$
$278$	0	0
$279$	−8.88657	−0.532025
$280$	0	0
$281$	−16.7355	−0.998358	−0.499179	−	0.866499i	$-0.666365\pi$
$281$	−16.7355	−0.998358	−0.499179	+	0.866499i	$0.666365\pi$
$282$	0	0
$283$	5.66890	0.336981	0.168491	−	0.985703i	$-0.446111\pi$
$283$	5.66890	0.336981	0.168491	+	0.985703i	$0.446111\pi$
$284$	0	0
$285$	16.3858	0.970608
$286$	0	0
$287$	−5.40330	−0.318947
$288$	0	0
$289$	5.81960	0.342330
$290$	0	0
$291$	−6.91180	−0.405177
$292$	0	0
$293$	11.9879	0.700338	0.350169	−	0.936687i	$-0.386124\pi$
$293$	11.9879	0.700338	0.350169	+	0.936687i	$0.386124\pi$
$294$	0	0
$295$	−10.1180	−0.589091
$296$	0	0
$297$	−3.06371	−0.177774
$298$	0	0
$299$	−2.66871	−0.154335
$300$	0	0
$301$	4.11237	0.237033
$302$	0	0
$303$	12.5633	0.721741
$304$	0	0
$305$	23.8683	1.36669
$306$	0	0
$307$	−28.8686	−1.64762	−0.823808	−	0.566869i	$-0.808154\pi$
$307$	−28.8686	−1.64762	−0.823808	+	0.566869i	$0.808154\pi$
$308$	0	0
$309$	24.1519	1.37395
$310$	0	0
$311$	−26.9373	−1.52748	−0.763738	−	0.645527i	$-0.776638\pi$
$311$	−26.9373	−1.52748	−0.763738	+	0.645527i	$0.776638\pi$
$312$	0	0
$313$	−10.3310	−0.583945	−0.291973	−	0.956427i	$-0.594312\pi$
$313$	−10.3310	−0.583945	−0.291973	+	0.956427i	$0.594312\pi$
$314$	0	0
$315$	9.14406	0.515209
$316$	0	0
$317$	−15.8411	−0.889725	−0.444862	−	0.895599i	$-0.646747\pi$
$317$	−15.8411	−0.889725	−0.444862	+	0.895599i	$0.646747\pi$
$318$	0	0
$319$	2.17464	0.121756
$320$	0	0
$321$	−32.2274	−1.79876
$322$	0	0
$323$	−16.8549	−0.937832
$324$	0	0
$325$	−0.788897	−0.0437601
$326$	0	0
$327$	−16.2998	−0.901383
$328$	0	0
$329$	−9.50415	−0.523981
$330$	0	0
$331$	−21.1804	−1.16418	−0.582089	−	0.813125i	$-0.697764\pi$
$331$	−21.1804	−1.16418	−0.582089	+	0.813125i	$0.697764\pi$
$332$	0	0
$333$	−39.4435	−2.16149
$334$	0	0
$335$	−10.7380	−0.586677
$336$	0	0
$337$	5.40409	0.294380	0.147190	−	0.989108i	$-0.452977\pi$
$337$	5.40409	0.294380	0.147190	+	0.989108i	$0.452977\pi$
$338$	0	0
$339$	31.1654	1.69267
$340$	0	0
$341$	−2.14337	−0.116070
$342$	0	0
$343$	−15.7273	−0.849192
$344$	0	0
$345$	31.1369	1.67636
$346$	0	0
$347$	6.73673	0.361646	0.180823	−	0.983516i	$-0.442124\pi$
$347$	6.73673	0.361646	0.180823	+	0.983516i	$0.442124\pi$
$348$	0	0
$349$	−18.4243	−0.986233	−0.493116	−	0.869963i	$-0.664142\pi$
$349$	−18.4243	−0.986233	−0.493116	+	0.869963i	$0.664142\pi$
$350$	0	0
$351$	−1.21946	−0.0650898
$352$	0	0
$353$	−17.3090	−0.921265	−0.460632	−	0.887591i	$-0.652377\pi$
$353$	−17.3090	−0.921265	−0.460632	+	0.887591i	$0.652377\pi$
$354$	0	0
$355$	6.55394	0.347847
$356$	0	0
$357$	−16.2117	−0.858015
$358$	0	0
$359$	−16.8101	−0.887204	−0.443602	−	0.896224i	$-0.646300\pi$
$359$	−16.8101	−0.887204	−0.443602	+	0.896224i	$0.646300\pi$
$360$	0	0
$361$	−6.55072	−0.344775
$362$	0	0
$363$	−2.67321	−0.140307
$364$	0	0
$365$	27.3972	1.43404
$366$	0	0
$367$	−6.92922	−0.361702	−0.180851	−	0.983510i	$-0.557885\pi$
$367$	−6.92922	−0.361702	−0.180851	+	0.983510i	$0.557885\pi$
$368$	0	0
$369$	−17.6464	−0.918633
$370$	0	0
$371$	1.26953	0.0659105
$372$	0	0
$373$	−36.3288	−1.88103	−0.940516	−	0.339750i	$-0.889657\pi$
$373$	−36.3288	−1.88103	−0.940516	+	0.339750i	$0.889657\pi$
$374$	0	0
$375$	32.4245	1.67439
$376$	0	0
$377$	0.865577	0.0445795
$378$	0	0
$379$	34.8546	1.79036	0.895180	−	0.445706i	$-0.147047\pi$
$379$	34.8546	1.79036	0.895180	+	0.445706i	$0.147047\pi$
$380$	0	0
$381$	−45.8735	−2.35017
$382$	0	0
$383$	22.5160	1.15051	0.575257	−	0.817973i	$-0.304902\pi$
$383$	22.5160	1.15051	0.575257	+	0.817973i	$0.304902\pi$
$384$	0	0
$385$	2.20547	0.112401
$386$	0	0
$387$	13.4304	0.682704
$388$	0	0
$389$	31.9920	1.62206	0.811029	−	0.585005i	$-0.198908\pi$
$389$	31.9920	1.62206	0.811029	+	0.585005i	$0.198908\pi$
$390$	0	0
$391$	−32.0284	−1.61975
$392$	0	0
$393$	10.8337	0.546486
$394$	0	0
$395$	−10.0276	−0.504543
$396$	0	0
$397$	8.36157	0.419655	0.209828	−	0.977738i	$-0.432710\pi$
$397$	8.36157	0.419655	0.209828	+	0.977738i	$0.432710\pi$
$398$	0	0
$399$	11.9742	0.599461
$400$	0	0
$401$	32.5373	1.62484	0.812418	−	0.583075i	$-0.198151\pi$
$401$	32.5373	1.62484	0.812418	+	0.583075i	$0.198151\pi$
$402$	0	0
$403$	−0.853132	−0.0424975
$404$	0	0
$405$	−7.38029	−0.366730
$406$	0	0
$407$	−9.51345	−0.471564
$408$	0	0
$409$	−7.57108	−0.374366	−0.187183	−	0.982325i	$-0.559936\pi$
$409$	−7.57108	−0.374366	−0.187183	+	0.982325i	$0.559936\pi$
$410$	0	0
$411$	2.01699	0.0994906
$412$	0	0
$413$	−7.39391	−0.363831
$414$	0	0
$415$	−13.5038	−0.662873
$416$	0	0
$417$	3.51237	0.172001
$418$	0	0
$419$	34.5429	1.68753	0.843765	−	0.536713i	$-0.180334\pi$
$419$	34.5429	1.68753	0.843765	+	0.536713i	$0.180334\pi$
$420$	0	0
$421$	−29.3104	−1.42850	−0.714252	−	0.699889i	$-0.753233\pi$
$421$	−29.3104	−1.42850	−0.714252	+	0.699889i	$0.753233\pi$
$422$	0	0
$423$	−31.0391	−1.50917
$424$	0	0
$425$	−9.46793	−0.459262
$426$	0	0
$427$	17.4422	0.844088
$428$	0	0
$429$	−1.06403	−0.0513718
$430$	0	0
$431$	−21.9914	−1.05929	−0.529645	−	0.848219i	$-0.677675\pi$
$431$	−21.9914	−1.05929	−0.529645	+	0.848219i	$0.677675\pi$
$432$	0	0
$433$	−19.3483	−0.929818	−0.464909	−	0.885359i	$-0.653913\pi$
$433$	−19.3483	−0.929818	−0.464909	+	0.885359i	$0.653913\pi$
$434$	0	0
$435$	−10.0991	−0.484213
$436$	0	0
$437$	23.6567	1.13165
$438$	0	0
$439$	13.3902	0.639081	0.319540	−	0.947573i	$-0.396472\pi$
$439$	13.3902	0.639081	0.319540	+	0.947573i	$0.396472\pi$
$440$	0	0
$441$	−22.3403	−1.06383
$442$	0	0
$443$	−20.9114	−0.993529	−0.496765	−	0.867885i	$-0.665479\pi$
$443$	−20.9114	−0.993529	−0.496765	+	0.867885i	$0.665479\pi$
$444$	0	0
$445$	−15.5764	−0.738390
$446$	0	0
$447$	−11.7539	−0.555942
$448$	0	0
$449$	−26.1728	−1.23517	−0.617584	−	0.786505i	$-0.711889\pi$
$449$	−26.1728	−1.23517	−0.617584	+	0.786505i	$0.711889\pi$
$450$	0	0
$451$	−4.25616	−0.200415
$452$	0	0
$453$	29.1258	1.36845
$454$	0	0
$455$	0.877852	0.0411543
$456$	0	0
$457$	26.8079	1.25402	0.627011	−	0.779011i	$-0.284278\pi$
$457$	26.8079	1.25402	0.627011	+	0.779011i	$0.284278\pi$
$458$	0	0
$459$	−14.6353	−0.683117
$460$	0	0
$461$	36.8196	1.71486	0.857430	−	0.514601i	$-0.172060\pi$
$461$	36.8196	1.71486	0.857430	+	0.514601i	$0.172060\pi$
$462$	0	0
$463$	−1.30540	−0.0606669	−0.0303334	−	0.999540i	$-0.509657\pi$
$463$	−1.30540	−0.0606669	−0.0303334	+	0.999540i	$0.509657\pi$
$464$	0	0
$465$	9.95385	0.461599
$466$	0	0
$467$	14.9044	0.689695	0.344847	−	0.938659i	$-0.387931\pi$
$467$	14.9044	0.689695	0.344847	+	0.938659i	$0.387931\pi$
$468$	0	0
$469$	−7.84697	−0.362340
$470$	0	0
$471$	−14.0331	−0.646613
$472$	0	0
$473$	3.23929	0.148943
$474$	0	0
$475$	6.99315	0.320868
$476$	0	0
$477$	4.14608	0.189836
$478$	0	0
$479$	2.22872	0.101833	0.0509164	−	0.998703i	$-0.483786\pi$
$479$	2.22872	0.101833	0.0509164	+	0.998703i	$0.483786\pi$
$480$	0	0
$481$	−3.78667	−0.172657
$482$	0	0
$483$	22.7539	1.03534
$484$	0	0
$485$	4.49178	0.203961
$486$	0	0
$487$	28.5319	1.29290	0.646451	−	0.762955i	$-0.276252\pi$
$487$	28.5319	1.29290	0.646451	+	0.762955i	$0.276252\pi$
$488$	0	0
$489$	−4.78341	−0.216313
$490$	0	0
$491$	−28.7181	−1.29603	−0.648015	−	0.761628i	$-0.724400\pi$
$491$	−28.7181	−1.29603	−0.648015	+	0.761628i	$0.724400\pi$
$492$	0	0
$493$	10.3882	0.467861
$494$	0	0
$495$	7.20274	0.323739
$496$	0	0
$497$	4.78942	0.214835
$498$	0	0
$499$	−8.35781	−0.374147	−0.187073	−	0.982346i	$-0.559900\pi$
$499$	−8.35781	−0.374147	−0.187073	+	0.982346i	$0.559900\pi$
$500$	0	0
$501$	−13.4386	−0.600392
$502$	0	0
$503$	−9.32874	−0.415948	−0.207974	−	0.978134i	$-0.566687\pi$
$503$	−9.32874	−0.415948	−0.207974	+	0.978134i	$0.566687\pi$
$504$	0	0
$505$	−8.16450	−0.363315
$506$	0	0
$507$	34.3283	1.52457
$508$	0	0
$509$	−25.7342	−1.14065	−0.570325	−	0.821419i	$-0.693183\pi$
$509$	−25.7342	−1.14065	−0.570325	+	0.821419i	$0.693183\pi$
$510$	0	0
$511$	20.0211	0.885680
$512$	0	0
$513$	10.8098	0.477266
$514$	0	0
$515$	−15.6956	−0.691631
$516$	0	0
$517$	−7.48638	−0.329251
$518$	0	0
$519$	−26.8351	−1.17793
$520$	0	0
$521$	−8.00760	−0.350819	−0.175410	−	0.984496i	$-0.556125\pi$
$521$	−8.00760	−0.350819	−0.175410	+	0.984496i	$0.556125\pi$
$522$	0	0
$523$	4.56720	0.199710	0.0998549	−	0.995002i	$-0.468162\pi$
$523$	4.56720	0.199710	0.0998549	+	0.995002i	$0.468162\pi$
$524$	0	0
$525$	6.72630	0.293560
$526$	0	0
$527$	−10.2388	−0.446011
$528$	0	0
$529$	21.9535	0.954498
$530$	0	0
$531$	−24.1474	−1.04791
$532$	0	0
$533$	−1.69409	−0.0733793
$534$	0	0
$535$	20.9437	0.905473
$536$	0	0
$537$	−13.0819	−0.564524
$538$	0	0
$539$	−5.38831	−0.232091
$540$	0	0
$541$	−9.64486	−0.414665	−0.207332	−	0.978271i	$-0.566478\pi$
$541$	−9.64486	−0.414665	−0.207332	+	0.978271i	$0.566478\pi$
$542$	0	0
$543$	−38.6402	−1.65821
$544$	0	0
$545$	10.5928	0.453745
$546$	0	0
$547$	3.30206	0.141186	0.0705928	−	0.997505i	$-0.477511\pi$
$547$	3.30206	0.141186	0.0705928	+	0.997505i	$0.477511\pi$
$548$	0	0
$549$	56.9636	2.43115
$550$	0	0
$551$	−7.67289	−0.326876
$552$	0	0
$553$	−7.32786	−0.311612
$554$	0	0
$555$	44.1807	1.87537
$556$	0	0
$557$	−23.9121	−1.01319	−0.506593	−	0.862185i	$-0.669095\pi$
$557$	−23.9121	−1.01319	−0.506593	+	0.862185i	$0.669095\pi$
$558$	0	0
$559$	1.28935	0.0545336
$560$	0	0
$561$	−12.7699	−0.539146
$562$	0	0
$563$	28.6414	1.20709	0.603546	−	0.797328i	$-0.293754\pi$
$563$	28.6414	1.20709	0.603546	+	0.797328i	$0.293754\pi$
$564$	0	0
$565$	−20.2535	−0.852071
$566$	0	0
$567$	−5.39330	−0.226497
$568$	0	0
$569$	14.4668	0.606480	0.303240	−	0.952914i	$-0.401932\pi$
$569$	14.4668	0.606480	0.303240	+	0.952914i	$0.401932\pi$
$570$	0	0
$571$	3.55242	0.148664	0.0743322	−	0.997234i	$-0.476317\pi$
$571$	3.55242	0.148664	0.0743322	+	0.997234i	$0.476317\pi$
$572$	0	0
$573$	57.2412	2.39129
$574$	0	0
$575$	13.2887	0.554177
$576$	0	0
$577$	21.3025	0.886836	0.443418	−	0.896315i	$-0.353766\pi$
$577$	21.3025	0.886836	0.443418	+	0.896315i	$0.353766\pi$
$578$	0	0
$579$	31.5396	1.31074
$580$	0	0
$581$	−9.86814	−0.409399
$582$	0	0
$583$	1.00000	0.0414158
$584$	0	0
$585$	2.86693	0.118533
$586$	0	0
$587$	−15.1277	−0.624389	−0.312194	−	0.950018i	$-0.601064\pi$
$587$	−15.1277	−0.624389	−0.312194	+	0.950018i	$0.601064\pi$
$588$	0	0
$589$	7.56256	0.311610
$590$	0	0
$591$	20.3369	0.836549
$592$	0	0
$593$	9.26680	0.380542	0.190271	−	0.981732i	$-0.439063\pi$
$593$	9.26680	0.380542	0.190271	+	0.981732i	$0.439063\pi$
$594$	0	0
$595$	10.5355	0.431914
$596$	0	0
$597$	−24.5579	−1.00509
$598$	0	0
$599$	20.5336	0.838978	0.419489	−	0.907760i	$-0.362209\pi$
$599$	20.5336	0.838978	0.419489	+	0.907760i	$0.362209\pi$
$600$	0	0
$601$	−21.0842	−0.860043	−0.430021	−	0.902819i	$-0.641494\pi$
$601$	−21.0842	−0.860043	−0.430021	+	0.902819i	$0.641494\pi$
$602$	0	0
$603$	−25.6270	−1.04361
$604$	0	0
$605$	1.73724	0.0706290
$606$	0	0
$607$	23.0661	0.936226	0.468113	−	0.883669i	$-0.344934\pi$
$607$	23.0661	0.936226	0.468113	+	0.883669i	$0.344934\pi$
$608$	0	0
$609$	−7.38009	−0.299056
$610$	0	0
$611$	−2.97983	−0.120551
$612$	0	0
$613$	−11.3548	−0.458615	−0.229307	−	0.973354i	$-0.573646\pi$
$613$	−11.3548	−0.458615	−0.229307	+	0.973354i	$0.573646\pi$
$614$	0	0
$615$	19.7657	0.797030
$616$	0	0
$617$	29.6305	1.19288	0.596440	−	0.802658i	$-0.296581\pi$
$617$	29.6305	1.19288	0.596440	+	0.802658i	$0.296581\pi$
$618$	0	0
$619$	−43.1853	−1.73576	−0.867882	−	0.496770i	$-0.834519\pi$
$619$	−43.1853	−1.73576	−0.867882	+	0.496770i	$0.834519\pi$
$620$	0	0
$621$	20.5413	0.824296
$622$	0	0
$623$	−11.3827	−0.456040
$624$	0	0
$625$	−11.1618	−0.446472
$626$	0	0
$627$	9.43205	0.376680
$628$	0	0
$629$	−45.4456	−1.81204
$630$	0	0
$631$	26.2444	1.04477	0.522387	−	0.852709i	$-0.325042\pi$
$631$	26.2444	1.04477	0.522387	+	0.852709i	$0.325042\pi$
$632$	0	0
$633$	53.6517	2.13246
$634$	0	0
$635$	29.8119	1.18305
$636$	0	0
$637$	−2.14472	−0.0849771
$638$	0	0
$639$	15.6415	0.618769
$640$	0	0
$641$	−10.2426	−0.404558	−0.202279	−	0.979328i	$-0.564835\pi$
$641$	−10.2426	−0.404558	−0.202279	+	0.979328i	$0.564835\pi$
$642$	0	0
$643$	−6.86569	−0.270756	−0.135378	−	0.990794i	$-0.543225\pi$
$643$	−6.86569	−0.270756	−0.135378	+	0.990794i	$0.543225\pi$
$644$	0	0
$645$	−15.0434	−0.592332
$646$	0	0
$647$	24.6607	0.969510	0.484755	−	0.874650i	$-0.338909\pi$
$647$	24.6607	0.969510	0.484755	+	0.874650i	$0.338909\pi$
$648$	0	0
$649$	−5.82416	−0.228618
$650$	0	0
$651$	7.27397	0.285090
$652$	0	0
$653$	17.5784	0.687894	0.343947	−	0.938989i	$-0.388236\pi$
$653$	17.5784	0.687894	0.343947	+	0.938989i	$0.388236\pi$
$654$	0	0
$655$	−7.04048	−0.275094
$656$	0	0
$657$	65.3858	2.55094
$658$	0	0
$659$	36.4055	1.41816	0.709079	−	0.705129i	$-0.249111\pi$
$659$	36.4055	1.41816	0.709079	+	0.705129i	$0.249111\pi$
$660$	0	0
$661$	−5.68153	−0.220986	−0.110493	−	0.993877i	$-0.535243\pi$
$661$	−5.68153	−0.220986	−0.110493	+	0.993877i	$0.535243\pi$
$662$	0	0
$663$	−5.08285	−0.197401
$664$	0	0
$665$	−7.78169	−0.301761
$666$	0	0
$667$	−14.5804	−0.564554
$668$	0	0
$669$	64.2728	2.48493
$670$	0	0
$671$	13.7392	0.530395
$672$	0	0
$673$	−30.7201	−1.18417	−0.592087	−	0.805874i	$-0.701696\pi$
$673$	−30.7201	−1.18417	−0.592087	+	0.805874i	$0.701696\pi$
$674$	0	0
$675$	6.07223	0.233720
$676$	0	0
$677$	−40.9519	−1.57391	−0.786954	−	0.617011i	$-0.788343\pi$
$677$	−40.9519	−1.57391	−0.786954	+	0.617011i	$0.788343\pi$
$678$	0	0
$679$	3.28246	0.125969
$680$	0	0
$681$	−0.0320589	−0.00122850
$682$	0	0
$683$	28.2313	1.08024	0.540121	−	0.841587i	$-0.318378\pi$
$683$	28.2313	1.08024	0.540121	+	0.841587i	$0.318378\pi$
$684$	0	0
$685$	−1.31078	−0.0500823
$686$	0	0
$687$	7.25647	0.276852
$688$	0	0
$689$	0.398033	0.0151639
$690$	0	0
$691$	−9.16825	−0.348777	−0.174388	−	0.984677i	$-0.555795\pi$
$691$	−9.16825	−0.348777	−0.174388	+	0.984677i	$0.555795\pi$
$692$	0	0
$693$	5.26355	0.199946
$694$	0	0
$695$	−2.28258	−0.0865833
$696$	0	0
$697$	−20.3316	−0.770115
$698$	0	0
$699$	−55.4141	−2.09595
$700$	0	0
$701$	9.38718	0.354549	0.177274	−	0.984161i	$-0.443272\pi$
$701$	9.38718	0.354549	0.177274	+	0.984161i	$0.443272\pi$
$702$	0	0
$703$	33.5668	1.26600
$704$	0	0
$705$	34.7669	1.30940
$706$	0	0
$707$	−5.96637	−0.224388
$708$	0	0
$709$	35.9038	1.34840	0.674198	−	0.738551i	$-0.264489\pi$
$709$	35.9038	1.34840	0.674198	+	0.738551i	$0.264489\pi$
$710$	0	0
$711$	−23.9317	−0.897508
$712$	0	0
$713$	14.3707	0.538187
$714$	0	0
$715$	0.691480	0.0258599
$716$	0	0
$717$	−30.9059	−1.15420
$718$	0	0
$719$	−28.0520	−1.04616	−0.523081	−	0.852283i	$-0.675217\pi$
$719$	−28.0520	−1.04616	−0.523081	+	0.852283i	$0.675217\pi$
$720$	0	0
$721$	−11.4699	−0.427161
$722$	0	0
$723$	23.6788	0.880623
$724$	0	0
$725$	−4.31010	−0.160073
$726$	0	0
$727$	28.9900	1.07518	0.537591	−	0.843206i	$-0.319335\pi$
$727$	28.9900	1.07518	0.537591	+	0.843206i	$0.319335\pi$
$728$	0	0
$729$	−42.1835	−1.56235
$730$	0	0
$731$	15.4741	0.572329
$732$	0	0
$733$	22.6981	0.838372	0.419186	−	0.907900i	$-0.362316\pi$
$733$	22.6981	0.838372	0.419186	+	0.907900i	$0.362316\pi$
$734$	0	0
$735$	25.0234	0.923003
$736$	0	0
$737$	−6.18103	−0.227681
$738$	0	0
$739$	29.8047	1.09638	0.548192	−	0.836353i	$-0.315316\pi$
$739$	29.8047	1.09638	0.548192	+	0.836353i	$0.315316\pi$
$740$	0	0
$741$	3.75427	0.137916
$742$	0	0
$743$	7.27201	0.266784	0.133392	−	0.991063i	$-0.457413\pi$
$743$	7.27201	0.266784	0.133392	+	0.991063i	$0.457413\pi$
$744$	0	0
$745$	7.63854	0.279854
$746$	0	0
$747$	−32.2278	−1.17915
$748$	0	0
$749$	15.3050	0.559232
$750$	0	0
$751$	−54.3101	−1.98180	−0.990902	−	0.134586i	$-0.957030\pi$
$751$	−54.3101	−1.98180	−0.990902	+	0.134586i	$0.957030\pi$
$752$	0	0
$753$	−47.0349	−1.71405
$754$	0	0
$755$	−18.9280	−0.688860
$756$	0	0
$757$	−28.0071	−1.01794	−0.508968	−	0.860785i	$-0.669973\pi$
$757$	−28.0071	−1.01794	−0.508968	+	0.860785i	$0.669973\pi$
$758$	0	0
$759$	17.9232	0.650571
$760$	0	0
$761$	−17.2546	−0.625479	−0.312739	−	0.949839i	$-0.601247\pi$
$761$	−17.2546	−0.625479	−0.312739	+	0.949839i	$0.601247\pi$
$762$	0	0
$763$	7.74088	0.280239
$764$	0	0
$765$	34.4074	1.24400
$766$	0	0
$767$	−2.31821	−0.0837056
$768$	0	0
$769$	−6.81143	−0.245626	−0.122813	−	0.992430i	$-0.539192\pi$
$769$	−6.81143	−0.245626	−0.122813	+	0.992430i	$0.539192\pi$
$770$	0	0
$771$	41.1056	1.48038
$772$	0	0
$773$	10.7190	0.385534	0.192767	−	0.981245i	$-0.438254\pi$
$773$	10.7190	0.385534	0.192767	+	0.981245i	$0.438254\pi$
$774$	0	0
$775$	4.24813	0.152597
$776$	0	0
$777$	32.2859	1.15825
$778$	0	0
$779$	15.0172	0.538048
$780$	0	0
$781$	3.77261	0.134995
$782$	0	0
$783$	−6.66245	−0.238097
$784$	0	0
$785$	9.11972	0.325497
$786$	0	0
$787$	13.8701	0.494415	0.247208	−	0.968963i	$-0.420487\pi$
$787$	13.8701	0.494415	0.247208	+	0.968963i	$0.420487\pi$
$788$	0	0
$789$	35.0457	1.24766
$790$	0	0
$791$	−14.8006	−0.526250
$792$	0	0
$793$	5.46864	0.194197
$794$	0	0
$795$	−4.64402	−0.164707
$796$	0	0
$797$	−40.3757	−1.43018	−0.715090	−	0.699032i	$-0.753614\pi$
$797$	−40.3757	−1.43018	−0.715090	+	0.699032i	$0.753614\pi$
$798$	0	0
$799$	−35.7624	−1.26518
$800$	0	0
$801$	−37.1743	−1.31349
$802$	0	0
$803$	15.7705	0.556530
$804$	0	0
$805$	−14.7871	−0.521177
$806$	0	0
$807$	65.8237	2.31710
$808$	0	0
$809$	14.0404	0.493634	0.246817	−	0.969062i	$-0.420615\pi$
$809$	14.0404	0.493634	0.246817	+	0.969062i	$0.420615\pi$
$810$	0	0
$811$	6.25161	0.219524	0.109762	−	0.993958i	$-0.464991\pi$
$811$	6.25161	0.219524	0.109762	+	0.993958i	$0.464991\pi$
$812$	0	0
$813$	32.8610	1.15248
$814$	0	0
$815$	3.10860	0.108889
$816$	0	0
$817$	−11.4294	−0.399863
$818$	0	0
$819$	2.09507	0.0732076
$820$	0	0
$821$	1.93119	0.0673992	0.0336996	−	0.999432i	$-0.489271\pi$
$821$	1.93119	0.0673992	0.0336996	+	0.999432i	$0.489271\pi$
$822$	0	0
$823$	−44.5804	−1.55397	−0.776987	−	0.629517i	$-0.783253\pi$
$823$	−44.5804	−1.55397	−0.776987	+	0.629517i	$0.783253\pi$
$824$	0	0
$825$	5.29828	0.184462
$826$	0	0
$827$	38.1538	1.32674	0.663369	−	0.748293i	$-0.269126\pi$
$827$	38.1538	1.32674	0.663369	+	0.748293i	$0.269126\pi$
$828$	0	0
$829$	24.2059	0.840706	0.420353	−	0.907361i	$-0.361906\pi$
$829$	24.2059	0.840706	0.420353	+	0.907361i	$0.361906\pi$
$830$	0	0
$831$	−26.6971	−0.926112
$832$	0	0
$833$	−25.7399	−0.891834
$834$	0	0
$835$	8.73335	0.302230
$836$	0	0
$837$	6.56665	0.226977
$838$	0	0
$839$	16.4633	0.568378	0.284189	−	0.958768i	$-0.408276\pi$
$839$	16.4633	0.568378	0.284189	+	0.958768i	$0.408276\pi$
$840$	0	0
$841$	−24.2710	−0.836930
$842$	0	0
$843$	44.7377	1.54085
$844$	0	0
$845$	−22.3089	−0.767450
$846$	0	0
$847$	1.26953	0.0436214
$848$	0	0
$849$	−15.1542	−0.520090
$850$	0	0
$851$	63.7852	2.18653
$852$	0	0
$853$	−41.5128	−1.42137	−0.710685	−	0.703510i	$-0.751615\pi$
$853$	−41.5128	−1.42137	−0.710685	+	0.703510i	$0.751615\pi$
$854$	0	0
$855$	−25.4138	−0.869134
$856$	0	0
$857$	41.0525	1.40233	0.701163	−	0.713001i	$-0.252664\pi$
$857$	41.0525	1.40233	0.701163	+	0.713001i	$0.252664\pi$
$858$	0	0
$859$	4.05057	0.138204	0.0691019	−	0.997610i	$-0.477987\pi$
$859$	4.05057	0.138204	0.0691019	+	0.997610i	$0.477987\pi$
$860$	0	0
$861$	14.4442	0.492256
$862$	0	0
$863$	−31.1361	−1.05988	−0.529942	−	0.848034i	$-0.677786\pi$
$863$	−31.1361	−1.05988	−0.529942	+	0.848034i	$0.677786\pi$
$864$	0	0
$865$	17.4393	0.592955
$866$	0	0
$867$	−15.5570	−0.528345
$868$	0	0
$869$	−5.77213	−0.195806
$870$	0	0
$871$	−2.46026	−0.0833626
$872$	0	0
$873$	10.7200	0.362817
$874$	0	0
$875$	−15.3986	−0.520567
$876$	0	0
$877$	25.8780	0.873838	0.436919	−	0.899501i	$-0.356070\pi$
$877$	25.8780	0.873838	0.436919	+	0.899501i	$0.356070\pi$
$878$	0	0
$879$	−32.0461	−1.08089
$880$	0	0
$881$	−33.9348	−1.14329	−0.571646	−	0.820501i	$-0.693695\pi$
$881$	−33.9348	−1.14329	−0.571646	+	0.820501i	$0.693695\pi$
$882$	0	0
$883$	3.23734	0.108945	0.0544726	−	0.998515i	$-0.482652\pi$
$883$	3.23734	0.108945	0.0544726	+	0.998515i	$0.482652\pi$
$884$	0	0
$885$	27.0475	0.909192
$886$	0	0
$887$	−40.4404	−1.35786	−0.678928	−	0.734205i	$-0.737555\pi$
$887$	−40.4404	−1.35786	−0.678928	+	0.734205i	$0.737555\pi$
$888$	0	0
$889$	21.7856	0.730666
$890$	0	0
$891$	−4.24828	−0.142323
$892$	0	0
$893$	26.4146	0.883931
$894$	0	0
$895$	8.50152	0.284174
$896$	0	0
$897$	7.13403	0.238198
$898$	0	0
$899$	−4.66104	−0.155455
$900$	0	0
$901$	4.77699	0.159145
$902$	0	0
$903$	−10.9932	−0.365832
$904$	0	0
$905$	25.1111	0.834723
$906$	0	0
$907$	32.4412	1.07719	0.538596	−	0.842564i	$-0.318955\pi$
$907$	32.4412	1.07719	0.538596	+	0.842564i	$0.318955\pi$
$908$	0	0
$909$	−19.4853	−0.646285
$910$	0	0
$911$	−4.34869	−0.144079	−0.0720393	−	0.997402i	$-0.522951\pi$
$911$	−4.34869	−0.144079	−0.0720393	+	0.997402i	$0.522951\pi$
$912$	0	0
$913$	−7.77309	−0.257252
$914$	0	0
$915$	−63.8050	−2.10933
$916$	0	0
$917$	−5.14497	−0.169902
$918$	0	0
$919$	37.9292	1.25117	0.625585	−	0.780156i	$-0.284860\pi$
$919$	37.9292	1.25117	0.625585	+	0.780156i	$0.284860\pi$
$920$	0	0
$921$	77.1718	2.54290
$922$	0	0
$923$	1.50162	0.0494265
$924$	0	0
$925$	18.8555	0.619967
$926$	0	0
$927$	−37.4589	−1.23031
$928$	0	0
$929$	−56.0018	−1.83736	−0.918680	−	0.395002i	$-0.870744\pi$
$929$	−56.0018	−1.83736	−0.918680	+	0.395002i	$0.870744\pi$
$930$	0	0
$931$	19.0119	0.623088
$932$	0	0
$933$	72.0092	2.35748
$934$	0	0
$935$	8.29879	0.271399
$936$	0	0
$937$	31.8832	1.04158	0.520789	−	0.853685i	$-0.325638\pi$
$937$	31.8832	1.04158	0.520789	+	0.853685i	$0.325638\pi$
$938$	0	0
$939$	27.6171	0.901250
$940$	0	0
$941$	34.5865	1.12749	0.563744	−	0.825949i	$-0.309360\pi$
$941$	34.5865	1.12749	0.563744	+	0.825949i	$0.309360\pi$
$942$	0	0
$943$	28.5364	0.929273
$944$	0	0
$945$	−6.75693	−0.219803
$946$	0	0
$947$	54.9644	1.78610	0.893051	−	0.449955i	$-0.148560\pi$
$947$	54.9644	1.78610	0.893051	+	0.449955i	$0.148560\pi$
$948$	0	0
$949$	6.27719	0.203766
$950$	0	0
$951$	42.3467	1.37318
$952$	0	0
$953$	51.4910	1.66796	0.833978	−	0.551797i	$-0.186058\pi$
$953$	51.4910	1.66796	0.833978	+	0.551797i	$0.186058\pi$
$954$	0	0
$955$	−37.1994	−1.20374
$956$	0	0
$957$	−5.81327	−0.187916
$958$	0	0
$959$	−0.957879	−0.0309315
$960$	0	0
$961$	−26.4060	−0.851806
$962$	0	0
$963$	49.9838	1.61071
$964$	0	0
$965$	−20.4967	−0.659811
$966$	0	0
$967$	48.0974	1.54671	0.773354	−	0.633975i	$-0.218578\pi$
$967$	48.0974	1.54671	0.773354	+	0.633975i	$0.218578\pi$
$968$	0	0
$969$	45.0568	1.44743
$970$	0	0
$971$	13.2119	0.423989	0.211995	−	0.977271i	$-0.432004\pi$
$971$	13.2119	0.423989	0.211995	+	0.977271i	$0.432004\pi$
$972$	0	0
$973$	−1.66804	−0.0534750
$974$	0	0
$975$	2.10889	0.0675386
$976$	0	0
$977$	33.1717	1.06126	0.530629	−	0.847604i	$-0.321956\pi$
$977$	33.1717	1.06126	0.530629	+	0.847604i	$0.321956\pi$
$978$	0	0
$979$	−8.96613	−0.286559
$980$	0	0
$981$	25.2806	0.807146
$982$	0	0
$983$	48.5466	1.54840	0.774199	−	0.632943i	$-0.218153\pi$
$983$	48.5466	1.54840	0.774199	+	0.632943i	$0.218153\pi$
$984$	0	0
$985$	−13.2164	−0.421108
$986$	0	0
$987$	25.4066	0.808702
$988$	0	0
$989$	−21.7186	−0.690611
$990$	0	0
$991$	−20.9986	−0.667042	−0.333521	−	0.942743i	$-0.608237\pi$
$991$	−20.9986	−0.667042	−0.333521	+	0.942743i	$0.608237\pi$
$992$	0	0
$993$	56.6196	1.79677
$994$	0	0
$995$	15.9594	0.505948
$996$	0	0
$997$	9.03842	0.286250	0.143125	−	0.989705i	$-0.454285\pi$
$997$	9.03842	0.286250	0.143125	+	0.989705i	$0.454285\pi$
$998$	0	0
$999$	29.1464	0.922153

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4664.2.a.k.1.2	✓	11
4.3	odd	2		9328.2.a.bm.1.10		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4664.2.a.k.1.2	✓	11	1.1	even	1	trivial
9328.2.a.bm.1.10		11	4.3	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:	\( N \)	\(=\)	\( 4664 = 2^{3} \cdot 11 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4664.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67321 q^{3} +1.73724 q^{5} +1.26953 q^{7} +4.14608 q^{9} +O(q^{10})\)	\(q-2.67321 q^{3} +1.73724 q^{5} +1.26953 q^{7} +4.14608 q^{9} +1.00000 q^{11} +0.398033 q^{13} -4.64402 q^{15} +4.77699 q^{17} -3.52835 q^{19} -3.39371 q^{21} -6.70473 q^{23} -1.98199 q^{25} -3.06371 q^{27} +2.17464 q^{29} -2.14337 q^{31} -2.67321 q^{33} +2.20547 q^{35} -9.51345 q^{37} -1.06403 q^{39} -4.25616 q^{41} +3.23929 q^{43} +7.20274 q^{45} -7.48638 q^{47} -5.38831 q^{49} -12.7699 q^{51} +1.00000 q^{53} +1.73724 q^{55} +9.43205 q^{57} -5.82416 q^{59} +13.7392 q^{61} +5.26355 q^{63} +0.691480 q^{65} -6.18103 q^{67} +17.9232 q^{69} +3.77261 q^{71} +15.7705 q^{73} +5.29828 q^{75} +1.26953 q^{77} -5.77213 q^{79} -4.24828 q^{81} -7.77309 q^{83} +8.29879 q^{85} -5.81327 q^{87} -8.96613 q^{89} +0.505313 q^{91} +5.72968 q^{93} -6.12961 q^{95} +2.58558 q^{97} +4.14608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 6 q^{3} + 3 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 11 * q - 6 * q^3 + 3 * q^5 - 5 * q^7 + 7 * q^9	\( 11 q - 6 q^{3} + 3 q^{5} - 5 q^{7} + 7 q^{9} + 11 q^{11} - 13 q^{13} - 8 q^{15} - 7 q^{17} - 21 q^{19} + 6 q^{21} + 11 q^{23} + 4 q^{25} - 6 q^{27} - 5 q^{31} - 6 q^{33} - 25 q^{35} - 4 q^{37} - 19 q^{39} - 11 q^{41} + 16 q^{45} - 17 q^{47} - 2 q^{49} - 18 q^{51} + 11 q^{53} + 3 q^{55} - 5 q^{57} - 19 q^{59} - 2 q^{61} - 36 q^{63} - 13 q^{65} + 25 q^{67} + 3 q^{69} - 30 q^{71} + 5 q^{73} - 5 q^{75} - 5 q^{77} - 23 q^{79} - 9 q^{81} - 19 q^{83} + 2 q^{85} - 7 q^{87} + 6 q^{89} - 20 q^{91} + 43 q^{93} - 50 q^{95} - 35 q^{97} + 7 q^{99}+O(q^{100}) \) 11 * q - 6 * q^3 + 3 * q^5 - 5 * q^7 + 7 * q^9 + 11 * q^11 - 13 * q^13 - 8 * q^15 - 7 * q^17 - 21 * q^19 + 6 * q^21 + 11 * q^23 + 4 * q^25 - 6 * q^27 - 5 * q^31 - 6 * q^33 - 25 * q^35 - 4 * q^37 - 19 * q^39 - 11 * q^41 + 16 * q^45 - 17 * q^47 - 2 * q^49 - 18 * q^51 + 11 * q^53 + 3 * q^55 - 5 * q^57 - 19 * q^59 - 2 * q^61 - 36 * q^63 - 13 * q^65 + 25 * q^67 + 3 * q^69 - 30 * q^71 + 5 * q^73 - 5 * q^75 - 5 * q^77 - 23 * q^79 - 9 * q^81 - 19 * q^83 + 2 * q^85 - 7 * q^87 + 6 * q^89 - 20 * q^91 + 43 * q^93 - 50 * q^95 - 35 * q^97 + 7 * q^99

Introduction

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