LMFDB - Embedded newform 9152.2.a.cg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9152,2,Mod(1,9152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9152, 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("9152.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9152 = 2^{6} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9152.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:	$73.0790879299$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.1957.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} - x + 1 $ x^4 - 4*x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 143)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.06150$ of defining polynomial
Character	$\chi$	$=$	9152.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.188279 q^{3} +4.12300 q^{5} -2.15845 q^{7} -2.96455 q^{9} +O(q^{10})$	$q-0.188279 q^{3} +4.12300 q^{5} -2.15845 q^{7} -2.96455 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.776272 q^{15} -3.09317 q^{17} -2.68783 q^{19} +0.406389 q^{21} -3.31128 q^{23} +11.9991 q^{25} +1.12300 q^{27} +3.34673 q^{29} -9.96928 q^{31} -0.188279 q^{33} -8.89927 q^{35} -2.05966 q^{37} -0.188279 q^{39} +9.49394 q^{41} +2.84717 q^{43} -12.2228 q^{45} +9.46972 q^{47} -2.34111 q^{49} +0.582377 q^{51} +5.34111 q^{53} +4.12300 q^{55} +0.506062 q^{57} +4.62255 q^{59} +0.193895 q^{61} +6.39883 q^{63} +4.12300 q^{65} -5.78662 q^{67} +0.623443 q^{69} +3.78295 q^{71} +7.47534 q^{73} -2.25918 q^{75} -2.15845 q^{77} +17.2049 q^{79} +8.68222 q^{81} -0.218111 q^{83} -12.7531 q^{85} -0.630117 q^{87} -1.52939 q^{89} -2.15845 q^{91} +1.87700 q^{93} -11.0819 q^{95} +12.1462 q^{97} -2.96455 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{7} + 2 q^{9}+O(q^{10}) $ 4 * q - 6 * q^7 + 2 * q^9	$ 4 q - 6 q^{7} + 2 q^{9} + 4 q^{11} + 4 q^{13} + 10 q^{15} + 6 q^{17} + 8 q^{19} + 2 q^{21} + 4 q^{23} + 12 q^{25} - 12 q^{27} + 10 q^{29} - 2 q^{31} - 6 q^{35} - 12 q^{37} + 8 q^{41} + 26 q^{43} - 26 q^{45} + 18 q^{47} + 6 q^{49} - 22 q^{51} + 6 q^{53} + 32 q^{57} - 16 q^{59} + 12 q^{61} - 22 q^{63} + 2 q^{67} + 4 q^{69} + 14 q^{71} + 22 q^{73} - 36 q^{75} - 6 q^{77} + 10 q^{79} + 4 q^{81} - 2 q^{83} - 48 q^{85} - 16 q^{87} + 10 q^{89} - 6 q^{91} + 24 q^{93} - 2 q^{95} + 22 q^{97} + 2 q^{99}+O(q^{100}) $ 4 * q - 6 * q^7 + 2 * q^9 + 4 * q^11 + 4 * q^13 + 10 * q^15 + 6 * q^17 + 8 * q^19 + 2 * q^21 + 4 * q^23 + 12 * q^25 - 12 * q^27 + 10 * q^29 - 2 * q^31 - 6 * q^35 - 12 * q^37 + 8 * q^41 + 26 * q^43 - 26 * q^45 + 18 * q^47 + 6 * q^49 - 22 * q^51 + 6 * q^53 + 32 * q^57 - 16 * q^59 + 12 * q^61 - 22 * q^63 + 2 * q^67 + 4 * q^69 + 14 * q^71 + 22 * q^73 - 36 * q^75 - 6 * q^77 + 10 * q^79 + 4 * q^81 - 2 * q^83 - 48 * q^85 - 16 * q^87 + 10 * q^89 - 6 * q^91 + 24 * q^93 - 2 * q^95 + 22 * q^97 + 2 * 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
$3$	−0.188279	−0.108703	−0.0543514	−	0.998522i	$-0.517309\pi$
$3$	−0.188279	−0.108703	−0.0543514	+	0.998522i	$0.517309\pi$
$4$	0	0
$5$	4.12300	1.84386	0.921930	−	0.387356i	$-0.126611\pi$
$5$	4.12300	1.84386	0.921930	+	0.387356i	$0.126611\pi$
$6$	0	0
$7$	−2.15845	−0.815816	−0.407908	−	0.913023i	$-0.633742\pi$
$7$	−2.15845	−0.815816	−0.407908	+	0.913023i	$0.633742\pi$
$8$	0	0
$9$	−2.96455	−0.988184
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.776272	−0.200433
$16$	0	0
$17$	−3.09317	−0.750203	−0.375101	−	0.926984i	$-0.622392\pi$
$17$	−3.09317	−0.750203	−0.375101	+	0.926984i	$0.622392\pi$
$18$	0	0
$19$	−2.68783	−0.616631	−0.308316	−	0.951284i	$-0.599765\pi$
$19$	−2.68783	−0.616631	−0.308316	+	0.951284i	$0.599765\pi$
$20$	0	0
$21$	0.406389	0.0886814
$22$	0	0
$23$	−3.31128	−0.690449	−0.345224	−	0.938520i	$-0.612197\pi$
$23$	−3.31128	−0.690449	−0.345224	+	0.938520i	$0.612197\pi$
$24$	0	0
$25$	11.9991	2.39982
$26$	0	0
$27$	1.12300	0.216121
$28$	0	0
$29$	3.34673	0.621471	0.310736	−	0.950496i	$-0.399425\pi$
$29$	3.34673	0.621471	0.310736	+	0.950496i	$0.399425\pi$
$30$	0	0
$31$	−9.96928	−1.79054	−0.895268	−	0.445529i	$-0.853016\pi$
$31$	−9.96928	−1.79054	−0.895268	+	0.445529i	$0.853016\pi$
$32$	0	0
$33$	−0.188279	−0.0327751
$34$	0	0
$35$	−8.89927	−1.50425
$36$	0	0
$37$	−2.05966	−0.338607	−0.169303	−	0.985564i	$-0.554152\pi$
$37$	−2.05966	−0.338607	−0.169303	+	0.985564i	$0.554152\pi$
$38$	0	0
$39$	−0.188279	−0.0301487
$40$	0	0
$41$	9.49394	1.48270	0.741352	−	0.671116i	$-0.234185\pi$
$41$	9.49394	1.48270	0.741352	+	0.671116i	$0.234185\pi$
$42$	0	0
$43$	2.84717	0.434189	0.217095	−	0.976151i	$-0.430342\pi$
$43$	2.84717	0.434189	0.217095	+	0.976151i	$0.430342\pi$
$44$	0	0
$45$	−12.2228	−1.82207
$46$	0	0
$47$	9.46972	1.38130	0.690651	−	0.723189i	$-0.257324\pi$
$47$	9.46972	1.38130	0.690651	+	0.723189i	$0.257324\pi$
$48$	0	0
$49$	−2.34111	−0.334444
$50$	0	0
$51$	0.582377	0.0815491
$52$	0	0
$53$	5.34111	0.733658	0.366829	−	0.930288i	$-0.380443\pi$
$53$	5.34111	0.733658	0.366829	+	0.930288i	$0.380443\pi$
$54$	0	0
$55$	4.12300	0.555945
$56$	0	0
$57$	0.506062	0.0670295
$58$	0	0
$59$	4.62255	0.601805	0.300903	−	0.953655i	$-0.402712\pi$
$59$	4.62255	0.601805	0.300903	+	0.953655i	$0.402712\pi$
$60$	0	0
$61$	0.193895	0.0248258	0.0124129	−	0.999923i	$-0.496049\pi$
$61$	0.193895	0.0248258	0.0124129	+	0.999923i	$0.496049\pi$
$62$	0	0
$63$	6.39883	0.806176
$64$	0	0
$65$	4.12300	0.511395
$66$	0	0
$67$	−5.78662	−0.706948	−0.353474	−	0.935444i	$-0.615000\pi$
$67$	−5.78662	−0.706948	−0.353474	+	0.935444i	$0.615000\pi$
$68$	0	0
$69$	0.623443	0.0750537
$70$	0	0
$71$	3.78295	0.448953	0.224477	−	0.974479i	$-0.427933\pi$
$71$	3.78295	0.448953	0.224477	+	0.974479i	$0.427933\pi$
$72$	0	0
$73$	7.47534	0.874922	0.437461	−	0.899237i	$-0.355878\pi$
$73$	7.47534	0.874922	0.437461	+	0.899237i	$0.355878\pi$
$74$	0	0
$75$	−2.25918	−0.260867
$76$	0	0
$77$	−2.15845	−0.245978
$78$	0	0
$79$	17.2049	1.93571	0.967853	−	0.251517i	$-0.0809294\pi$
$79$	17.2049	1.93571	0.967853	+	0.251517i	$0.0809294\pi$
$80$	0	0
$81$	8.68222	0.964691
$82$	0	0
$83$	−0.218111	−0.0239408	−0.0119704	−	0.999928i	$-0.503810\pi$
$83$	−0.218111	−0.0239408	−0.0119704	+	0.999928i	$0.503810\pi$
$84$	0	0
$85$	−12.7531	−1.38327
$86$	0	0
$87$	−0.630117	−0.0675556
$88$	0	0
$89$	−1.52939	−0.162115	−0.0810574	−	0.996709i	$-0.525830\pi$
$89$	−1.52939	−0.162115	−0.0810574	+	0.996709i	$0.525830\pi$
$90$	0	0
$91$	−2.15845	−0.226267
$92$	0	0
$93$	1.87700	0.194636
$94$	0	0
$95$	−11.0819	−1.13698
$96$	0	0
$97$	12.1462	1.23326	0.616628	−	0.787255i	$-0.288498\pi$
$97$	12.1462	1.23326	0.616628	+	0.787255i	$0.288498\pi$
$98$	0	0
$99$	−2.96455	−0.297949
$100$	0	0
$101$	13.2162	1.31506	0.657529	−	0.753429i	$-0.271602\pi$
$101$	13.2162	1.31506	0.657529	+	0.753429i	$0.271602\pi$
$102$	0	0
$103$	4.61783	0.455008	0.227504	−	0.973777i	$-0.426944\pi$
$103$	4.61783	0.455008	0.227504	+	0.973777i	$0.426944\pi$
$104$	0	0
$105$	1.67554	0.163516
$106$	0	0
$107$	10.3458	1.00017	0.500085	−	0.865976i	$-0.333302\pi$
$107$	10.3458	1.00017	0.500085	+	0.865976i	$0.333302\pi$
$108$	0	0
$109$	−6.83032	−0.654226	−0.327113	−	0.944985i	$-0.606076\pi$
$109$	−6.83032	−0.654226	−0.327113	+	0.944985i	$0.606076\pi$
$110$	0	0
$111$	0.387791	0.0368075
$112$	0	0
$113$	−7.40077	−0.696206	−0.348103	−	0.937456i	$-0.613174\pi$
$113$	−7.40077	−0.696206	−0.348103	+	0.937456i	$0.613174\pi$
$114$	0	0
$115$	−13.6524	−1.27309
$116$	0	0
$117$	−2.96455	−0.274073
$118$	0	0
$119$	6.67643	0.612028
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.78751	−0.161174
$124$	0	0
$125$	28.8573	2.58108
$126$	0	0
$127$	−8.40550	−0.745867	−0.372934	−	0.927858i	$-0.621648\pi$
$127$	−8.40550	−0.745867	−0.372934	+	0.927858i	$0.621648\pi$
$128$	0	0
$129$	−0.536061	−0.0471976
$130$	0	0
$131$	−0.630117	−0.0550536	−0.0275268	−	0.999621i	$-0.508763\pi$
$131$	−0.630117	−0.0550536	−0.0275268	+	0.999621i	$0.508763\pi$
$132$	0	0
$133$	5.80155	0.503058
$134$	0	0
$135$	4.63012	0.398497
$136$	0	0
$137$	4.81645	0.411497	0.205748	−	0.978605i	$-0.434037\pi$
$137$	4.81645	0.411497	0.205748	+	0.978605i	$0.434037\pi$
$138$	0	0
$139$	−10.5592	−0.895621	−0.447811	−	0.894128i	$-0.647796\pi$
$139$	−10.5592	−0.895621	−0.447811	+	0.894128i	$0.647796\pi$
$140$	0	0
$141$	−1.78295	−0.150151
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	13.7985	1.14591
$146$	0	0
$147$	0.440781	0.0363550
$148$	0	0
$149$	2.80049	0.229425	0.114712	−	0.993399i	$-0.463405\pi$
$149$	2.80049	0.229425	0.114712	+	0.993399i	$0.463405\pi$
$150$	0	0
$151$	8.24600	0.671050	0.335525	−	0.942031i	$-0.391086\pi$
$151$	8.24600	0.671050	0.335525	+	0.942031i	$0.391086\pi$
$152$	0	0
$153$	9.16985	0.741338
$154$	0	0
$155$	−41.1033	−3.30150
$156$	0	0
$157$	13.2918	1.06080	0.530400	−	0.847747i	$-0.322042\pi$
$157$	13.2918	1.06080	0.530400	+	0.847747i	$0.322042\pi$
$158$	0	0
$159$	−1.00562	−0.0797506
$160$	0	0
$161$	7.14721	0.563279
$162$	0	0
$163$	−2.85084	−0.223295	−0.111647	−	0.993748i	$-0.535613\pi$
$163$	−2.85084	−0.223295	−0.111647	+	0.993748i	$0.535613\pi$
$164$	0	0
$165$	−0.776272	−0.0604327
$166$	0	0
$167$	1.30093	0.100669	0.0503346	−	0.998732i	$-0.483971\pi$
$167$	1.30093	0.100669	0.0503346	+	0.998732i	$0.483971\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.96822	0.609345
$172$	0	0
$173$	−8.52850	−0.648410	−0.324205	−	0.945987i	$-0.605097\pi$
$173$	−8.52850	−0.648410	−0.324205	+	0.945987i	$0.605097\pi$
$174$	0	0
$175$	−25.8994	−1.95781
$176$	0	0
$177$	−0.870328	−0.0654179
$178$	0	0
$179$	19.3048	1.44291	0.721453	−	0.692463i	$-0.243475\pi$
$179$	19.3048	1.44291	0.721453	+	0.692463i	$0.243475\pi$
$180$	0	0
$181$	26.3402	1.95785	0.978927	−	0.204213i	$-0.0654635\pi$
$181$	26.3402	1.95785	0.978927	+	0.204213i	$0.0654635\pi$
$182$	0	0
$183$	−0.0365064	−0.00269863
$184$	0	0
$185$	−8.49199	−0.624344
$186$	0	0
$187$	−3.09317	−0.226195
$188$	0	0
$189$	−2.42393	−0.176315
$190$	0	0
$191$	1.86015	0.134596	0.0672979	−	0.997733i	$-0.478562\pi$
$191$	1.86015	0.134596	0.0672979	+	0.997733i	$0.478562\pi$
$192$	0	0
$193$	16.4239	1.18222	0.591110	−	0.806591i	$-0.298690\pi$
$193$	16.4239	1.18222	0.591110	+	0.806591i	$0.298690\pi$
$194$	0	0
$195$	−0.776272	−0.0555900
$196$	0	0
$197$	17.5929	1.25344	0.626721	−	0.779244i	$-0.284397\pi$
$197$	17.5929	1.25344	0.626721	+	0.779244i	$0.284397\pi$
$198$	0	0
$199$	−0.434274	−0.0307849	−0.0153924	−	0.999882i	$-0.504900\pi$
$199$	−0.434274	−0.0307849	−0.0153924	+	0.999882i	$0.504900\pi$
$200$	0	0
$201$	1.08950	0.0768471
$202$	0	0
$203$	−7.22373	−0.507006
$204$	0	0
$205$	39.1435	2.73390
$206$	0	0
$207$	9.81645	0.682290
$208$	0	0
$209$	−2.68783	−0.185921
$210$	0	0
$211$	4.04018	0.278137	0.139069	−	0.990283i	$-0.455589\pi$
$211$	4.04018	0.278137	0.139069	+	0.990283i	$0.455589\pi$
$212$	0	0
$213$	−0.712248	−0.0488024
$214$	0	0
$215$	11.7389	0.800585
$216$	0	0
$217$	21.5182	1.46075
$218$	0	0
$219$	−1.40745	−0.0951065
$220$	0	0
$221$	−3.09317	−0.208069
$222$	0	0
$223$	−21.4390	−1.43566	−0.717831	−	0.696218i	$-0.754865\pi$
$223$	−21.4390	−1.43566	−0.717831	+	0.696218i	$0.754865\pi$
$224$	0	0
$225$	−35.5720	−2.37147
$226$	0	0
$227$	18.8647	1.25209	0.626047	−	0.779785i	$-0.284672\pi$
$227$	18.8647	1.25209	0.626047	+	0.779785i	$0.284672\pi$
$228$	0	0
$229$	−13.3048	−0.879204	−0.439602	−	0.898193i	$-0.644880\pi$
$229$	−13.3048	−0.879204	−0.439602	+	0.898193i	$0.644880\pi$
$230$	0	0
$231$	0.406389	0.0267385
$232$	0	0
$233$	21.3160	1.39646	0.698229	−	0.715875i	$-0.253972\pi$
$233$	21.3160	1.39646	0.698229	+	0.715875i	$0.253972\pi$
$234$	0	0
$235$	39.0436	2.54693
$236$	0	0
$237$	−3.23932	−0.210417
$238$	0	0
$239$	−13.9123	−0.899909	−0.449954	−	0.893052i	$-0.648560\pi$
$239$	−13.9123	−0.899909	−0.449954	+	0.893052i	$0.648560\pi$
$240$	0	0
$241$	0.565726	0.0364416	0.0182208	−	0.999834i	$-0.494200\pi$
$241$	0.565726	0.0364416	0.0182208	+	0.999834i	$0.494200\pi$
$242$	0	0
$243$	−5.00367	−0.320986
$244$	0	0
$245$	−9.65238	−0.616668
$246$	0	0
$247$	−2.68783	−0.171023
$248$	0	0
$249$	0.0410656	0.00260243
$250$	0	0
$251$	12.4985	0.788898	0.394449	−	0.918918i	$-0.370935\pi$
$251$	12.4985	0.788898	0.394449	+	0.918918i	$0.370935\pi$
$252$	0	0
$253$	−3.31128	−0.208178
$254$	0	0
$255$	2.40114	0.150365
$256$	0	0
$257$	14.7640	0.920952	0.460476	−	0.887672i	$-0.347679\pi$
$257$	14.7640	0.920952	0.460476	+	0.887672i	$0.347679\pi$
$258$	0	0
$259$	4.44568	0.276241
$260$	0	0
$261$	−9.92154	−0.614128
$262$	0	0
$263$	15.8445	0.977014	0.488507	−	0.872560i	$-0.337542\pi$
$263$	15.8445	0.977014	0.488507	+	0.872560i	$0.337542\pi$
$264$	0	0
$265$	22.0214	1.35276
$266$	0	0
$267$	0.287951	0.0176223
$268$	0	0
$269$	2.34289	0.142848	0.0714242	−	0.997446i	$-0.477246\pi$
$269$	2.34289	0.142848	0.0714242	+	0.997446i	$0.477246\pi$
$270$	0	0
$271$	−26.0038	−1.57962	−0.789810	−	0.613351i	$-0.789821\pi$
$271$	−26.0038	−1.57962	−0.789810	+	0.613351i	$0.789821\pi$
$272$	0	0
$273$	0.406389	0.0245958
$274$	0	0
$275$	11.9991	0.723574
$276$	0	0
$277$	−28.0019	−1.68247	−0.841235	−	0.540669i	$-0.818171\pi$
$277$	−28.0019	−1.68247	−0.841235	+	0.540669i	$0.818171\pi$
$278$	0	0
$279$	29.5544	1.76938
$280$	0	0
$281$	−20.4632	−1.22073	−0.610367	−	0.792119i	$-0.708978\pi$
$281$	−20.4632	−1.22073	−0.610367	+	0.792119i	$0.708978\pi$
$282$	0	0
$283$	−9.17810	−0.545582	−0.272791	−	0.962073i	$-0.587947\pi$
$283$	−9.17810	−0.545582	−0.272791	+	0.962073i	$0.587947\pi$
$284$	0	0
$285$	2.08649	0.123593
$286$	0	0
$287$	−20.4922	−1.20961
$288$	0	0
$289$	−7.43233	−0.437196
$290$	0	0
$291$	−2.28686	−0.134058
$292$	0	0
$293$	−18.7380	−1.09468	−0.547342	−	0.836909i	$-0.684360\pi$
$293$	−18.7380	−1.09468	−0.547342	+	0.836909i	$0.684360\pi$
$294$	0	0
$295$	19.0588	1.10964
$296$	0	0
$297$	1.12300	0.0651629
$298$	0	0
$299$	−3.31128	−0.191496
$300$	0	0
$301$	−6.14546	−0.354219
$302$	0	0
$303$	−2.48832	−0.142950
$304$	0	0
$305$	0.799430	0.0457752
$306$	0	0
$307$	3.49288	0.199349	0.0996746	−	0.995020i	$-0.468220\pi$
$307$	3.49288	0.199349	0.0996746	+	0.995020i	$0.468220\pi$
$308$	0	0
$309$	−0.869438	−0.0494606
$310$	0	0
$311$	4.19951	0.238132	0.119066	−	0.992886i	$-0.462010\pi$
$311$	4.19951	0.238132	0.119066	+	0.992886i	$0.462010\pi$
$312$	0	0
$313$	−12.4678	−0.704720	−0.352360	−	0.935864i	$-0.614621\pi$
$313$	−12.4678	−0.704720	−0.352360	+	0.935864i	$0.614621\pi$
$314$	0	0
$315$	26.3823	1.48648
$316$	0	0
$317$	8.27494	0.464767	0.232383	−	0.972624i	$-0.425348\pi$
$317$	8.27494	0.464767	0.232383	+	0.972624i	$0.425348\pi$
$318$	0	0
$319$	3.34673	0.187381
$320$	0	0
$321$	−1.94790	−0.108721
$322$	0	0
$323$	8.31391	0.462599
$324$	0	0
$325$	11.9991	0.665591
$326$	0	0
$327$	1.28600	0.0711162
$328$	0	0
$329$	−20.4399	−1.12689
$330$	0	0
$331$	−23.5852	−1.29636	−0.648179	−	0.761488i	$-0.724469\pi$
$331$	−23.5852	−1.29636	−0.648179	+	0.761488i	$0.724469\pi$
$332$	0	0
$333$	6.10598	0.334606
$334$	0	0
$335$	−23.8582	−1.30351
$336$	0	0
$337$	0.664508	0.0361981	0.0180990	−	0.999836i	$-0.494239\pi$
$337$	0.664508	0.0361981	0.0180990	+	0.999836i	$0.494239\pi$
$338$	0	0
$339$	1.39341	0.0756795
$340$	0	0
$341$	−9.96928	−0.539867
$342$	0	0
$343$	20.1623	1.08866
$344$	0	0
$345$	2.57045	0.138389
$346$	0	0
$347$	−1.33916	−0.0718899	−0.0359450	−	0.999354i	$-0.511444\pi$
$347$	−1.33916	−0.0718899	−0.0359450	+	0.999354i	$0.511444\pi$
$348$	0	0
$349$	1.48448	0.0794626	0.0397313	−	0.999210i	$-0.487350\pi$
$349$	1.48448	0.0794626	0.0397313	+	0.999210i	$0.487350\pi$
$350$	0	0
$351$	1.12300	0.0599412
$352$	0	0
$353$	6.04843	0.321925	0.160963	−	0.986960i	$-0.448540\pi$
$353$	6.04843	0.321925	0.160963	+	0.986960i	$0.448540\pi$
$354$	0	0
$355$	15.5971	0.827807
$356$	0	0
$357$	−1.25703	−0.0665291
$358$	0	0
$359$	16.7101	0.881925	0.440963	−	0.897525i	$-0.354637\pi$
$359$	16.7101	0.881925	0.440963	+	0.897525i	$0.354637\pi$
$360$	0	0
$361$	−11.7756	−0.619766
$362$	0	0
$363$	−0.188279	−0.00988207
$364$	0	0
$365$	30.8208	1.61323
$366$	0	0
$367$	−18.2943	−0.954953	−0.477476	−	0.878645i	$-0.658448\pi$
$367$	−18.2943	−0.954953	−0.477476	+	0.878645i	$0.658448\pi$
$368$	0	0
$369$	−28.1453	−1.46518
$370$	0	0
$371$	−11.5285	−0.598530
$372$	0	0
$373$	−16.5748	−0.858211	−0.429106	−	0.903254i	$-0.641171\pi$
$373$	−16.5748	−0.858211	−0.429106	+	0.903254i	$0.641171\pi$
$374$	0	0
$375$	−5.43322	−0.280570
$376$	0	0
$377$	3.34673	0.172365
$378$	0	0
$379$	−26.9952	−1.38665	−0.693326	−	0.720625i	$-0.743855\pi$
$379$	−26.9952	−1.38665	−0.693326	+	0.720625i	$0.743855\pi$
$380$	0	0
$381$	1.58258	0.0810778
$382$	0	0
$383$	−9.35039	−0.477783	−0.238891	−	0.971046i	$-0.576784\pi$
$383$	−9.35039	−0.477783	−0.238891	+	0.971046i	$0.576784\pi$
$384$	0	0
$385$	−8.89927	−0.453549
$386$	0	0
$387$	−8.44058	−0.429059
$388$	0	0
$389$	−17.8258	−0.903802	−0.451901	−	0.892068i	$-0.649254\pi$
$389$	−17.8258	−0.903802	−0.451901	+	0.892068i	$0.649254\pi$
$390$	0	0
$391$	10.2423	0.517977
$392$	0	0
$393$	0.118638	0.00598447
$394$	0	0
$395$	70.9359	3.56917
$396$	0	0
$397$	−36.0101	−1.80730	−0.903649	−	0.428275i	$-0.859122\pi$
$397$	−36.0101	−1.80730	−0.903649	+	0.428275i	$0.859122\pi$
$398$	0	0
$399$	−1.09231	−0.0546838
$400$	0	0
$401$	23.4604	1.17156	0.585778	−	0.810472i	$-0.300789\pi$
$401$	23.4604	1.17156	0.585778	+	0.810472i	$0.300789\pi$
$402$	0	0
$403$	−9.96928	−0.496605
$404$	0	0
$405$	35.7968	1.77876
$406$	0	0
$407$	−2.05966	−0.102094
$408$	0	0
$409$	14.8798	0.735758	0.367879	−	0.929874i	$-0.380084\pi$
$409$	14.8798	0.735758	0.367879	+	0.929874i	$0.380084\pi$
$410$	0	0
$411$	−0.906834	−0.0447308
$412$	0	0
$413$	−9.97753	−0.490962
$414$	0	0
$415$	−0.899270	−0.0441434
$416$	0	0
$417$	1.98808	0.0973565
$418$	0	0
$419$	−27.9935	−1.36757	−0.683786	−	0.729683i	$-0.739668\pi$
$419$	−27.9935	−1.36757	−0.683786	+	0.729683i	$0.739668\pi$
$420$	0	0
$421$	13.3185	0.649103	0.324551	−	0.945868i	$-0.394787\pi$
$421$	13.3185	0.649103	0.324551	+	0.945868i	$0.394787\pi$
$422$	0	0
$423$	−28.0735	−1.36498
$424$	0	0
$425$	−37.1152	−1.80035
$426$	0	0
$427$	−0.418513	−0.0202533
$428$	0	0
$429$	−0.188279	−0.00909018
$430$	0	0
$431$	−12.2181	−0.588526	−0.294263	−	0.955725i	$-0.595074\pi$
$431$	−12.2181	−0.588526	−0.294263	+	0.955725i	$0.595074\pi$
$432$	0	0
$433$	−27.2663	−1.31034	−0.655168	−	0.755483i	$-0.727402\pi$
$433$	−27.2663	−1.31034	−0.655168	+	0.755483i	$0.727402\pi$
$434$	0	0
$435$	−2.59797	−0.124563
$436$	0	0
$437$	8.90016	0.425752
$438$	0	0
$439$	34.0428	1.62477	0.812386	−	0.583120i	$-0.198168\pi$
$439$	34.0428	1.62477	0.812386	+	0.583120i	$0.198168\pi$
$440$	0	0
$441$	6.94034	0.330492
$442$	0	0
$443$	−39.2983	−1.86712	−0.933558	−	0.358425i	$-0.883314\pi$
$443$	−39.2983	−1.86712	−0.933558	+	0.358425i	$0.883314\pi$
$444$	0	0
$445$	−6.30566	−0.298917
$446$	0	0
$447$	−0.527272	−0.0249391
$448$	0	0
$449$	14.8508	0.700854	0.350427	−	0.936590i	$-0.386036\pi$
$449$	14.8508	0.700854	0.350427	+	0.936590i	$0.386036\pi$
$450$	0	0
$451$	9.49394	0.447052
$452$	0	0
$453$	−1.55254	−0.0729449
$454$	0	0
$455$	−8.89927	−0.417204
$456$	0	0
$457$	19.7213	0.922525	0.461263	−	0.887264i	$-0.347397\pi$
$457$	19.7213	0.922525	0.461263	+	0.887264i	$0.347397\pi$
$458$	0	0
$459$	−3.47362	−0.162135
$460$	0	0
$461$	29.8161	1.38867	0.694337	−	0.719650i	$-0.255698\pi$
$461$	29.8161	1.38867	0.694337	+	0.719650i	$0.255698\pi$
$462$	0	0
$463$	16.5769	0.770395	0.385198	−	0.922834i	$-0.374133\pi$
$463$	16.5769	0.770395	0.385198	+	0.922834i	$0.374133\pi$
$464$	0	0
$465$	7.73888	0.358882
$466$	0	0
$467$	−1.67187	−0.0773651	−0.0386825	−	0.999252i	$-0.512316\pi$
$467$	−1.67187	−0.0773651	−0.0386825	+	0.999252i	$0.512316\pi$
$468$	0	0
$469$	12.4901	0.576739
$470$	0	0
$471$	−2.50256	−0.115312
$472$	0	0
$473$	2.84717	0.130913
$474$	0	0
$475$	−32.2516	−1.47981
$476$	0	0
$477$	−15.8340	−0.724989
$478$	0	0
$479$	20.0579	0.916468	0.458234	−	0.888832i	$-0.348482\pi$
$479$	20.0579	0.916468	0.458234	+	0.888832i	$0.348482\pi$
$480$	0	0
$481$	−2.05966	−0.0939126
$482$	0	0
$483$	−1.34567	−0.0612300
$484$	0	0
$485$	50.0786	2.27395
$486$	0	0
$487$	−4.74733	−0.215122	−0.107561	−	0.994198i	$-0.534304\pi$
$487$	−4.74733	−0.215122	−0.107561	+	0.994198i	$0.534304\pi$
$488$	0	0
$489$	0.536752	0.0242728
$490$	0	0
$491$	42.4522	1.91584	0.957919	−	0.287037i	$-0.0926704\pi$
$491$	42.4522	1.91584	0.957919	+	0.287037i	$0.0926704\pi$
$492$	0	0
$493$	−10.3520	−0.466230
$494$	0	0
$495$	−12.2228	−0.549376
$496$	0	0
$497$	−8.16529	−0.366263
$498$	0	0
$499$	−20.6880	−0.926122	−0.463061	−	0.886326i	$-0.653249\pi$
$499$	−20.6880	−0.926122	−0.463061	+	0.886326i	$0.653249\pi$
$500$	0	0
$501$	−0.244938	−0.0109430
$502$	0	0
$503$	−3.88526	−0.173235	−0.0866175	−	0.996242i	$-0.527606\pi$
$503$	−3.88526	−0.173235	−0.0866175	+	0.996242i	$0.527606\pi$
$504$	0	0
$505$	54.4902	2.42478
$506$	0	0
$507$	−0.188279	−0.00836175
$508$	0	0
$509$	−4.35776	−0.193154	−0.0965771	−	0.995326i	$-0.530789\pi$
$509$	−4.35776	−0.193154	−0.0965771	+	0.995326i	$0.530789\pi$
$510$	0	0
$511$	−16.1351	−0.713776
$512$	0	0
$513$	−3.01843	−0.133267
$514$	0	0
$515$	19.0393	0.838971
$516$	0	0
$517$	9.46972	0.416478
$518$	0	0
$519$	1.60573	0.0704839
$520$	0	0
$521$	−21.4185	−0.938360	−0.469180	−	0.883103i	$-0.655450\pi$
$521$	−21.4185	−0.938360	−0.469180	+	0.883103i	$0.655450\pi$
$522$	0	0
$523$	27.8835	1.21926	0.609630	−	0.792686i	$-0.291318\pi$
$523$	27.8835	1.21926	0.609630	+	0.792686i	$0.291318\pi$
$524$	0	0
$525$	4.87631	0.212820
$526$	0	0
$527$	30.8366	1.34326
$528$	0	0
$529$	−12.0354	−0.523280
$530$	0	0
$531$	−13.7038	−0.594694
$532$	0	0
$533$	9.49394	0.411228
$534$	0	0
$535$	42.6559	1.84417
$536$	0	0
$537$	−3.63468	−0.156848
$538$	0	0
$539$	−2.34111	−0.100839
$540$	0	0
$541$	−16.1874	−0.695950	−0.347975	−	0.937504i	$-0.613131\pi$
$541$	−16.1874	−0.695950	−0.347975	+	0.937504i	$0.613131\pi$
$542$	0	0
$543$	−4.95930	−0.212824
$544$	0	0
$545$	−28.1614	−1.20630
$546$	0	0
$547$	−16.5264	−0.706617	−0.353309	−	0.935507i	$-0.614943\pi$
$547$	−16.5264	−0.706617	−0.353309	+	0.935507i	$0.614943\pi$
$548$	0	0
$549$	−0.574813	−0.0245324
$550$	0	0
$551$	−8.99544	−0.383219
$552$	0	0
$553$	−37.1359	−1.57918
$554$	0	0
$555$	1.59886	0.0678679
$556$	0	0
$557$	27.0075	1.14434	0.572172	−	0.820133i	$-0.306101\pi$
$557$	27.0075	1.14434	0.572172	+	0.820133i	$0.306101\pi$
$558$	0	0
$559$	2.84717	0.120422
$560$	0	0
$561$	0.582377	0.0245880
$562$	0	0
$563$	37.7276	1.59003	0.795015	−	0.606589i	$-0.207463\pi$
$563$	37.7276	1.59003	0.795015	+	0.606589i	$0.207463\pi$
$564$	0	0
$565$	−30.5134	−1.28371
$566$	0	0
$567$	−18.7401	−0.787010
$568$	0	0
$569$	−7.39694	−0.310096	−0.155048	−	0.987907i	$-0.549553\pi$
$569$	−7.39694	−0.310096	−0.155048	+	0.987907i	$0.549553\pi$
$570$	0	0
$571$	24.3578	1.01934	0.509670	−	0.860370i	$-0.329767\pi$
$571$	24.3578	1.01934	0.509670	+	0.860370i	$0.329767\pi$
$572$	0	0
$573$	−0.350227	−0.0146309
$574$	0	0
$575$	−39.7324	−1.65695
$576$	0	0
$577$	4.42797	0.184339	0.0921693	−	0.995743i	$-0.470620\pi$
$577$	4.42797	0.184339	0.0921693	+	0.995743i	$0.470620\pi$
$578$	0	0
$579$	−3.09228	−0.128511
$580$	0	0
$581$	0.470780	0.0195313
$582$	0	0
$583$	5.34111	0.221206
$584$	0	0
$585$	−12.2228	−0.505352
$586$	0	0
$587$	9.80799	0.404819	0.202410	−	0.979301i	$-0.435123\pi$
$587$	9.80799	0.404819	0.202410	+	0.979301i	$0.435123\pi$
$588$	0	0
$589$	26.7958	1.10410
$590$	0	0
$591$	−3.31237	−0.136253
$592$	0	0
$593$	23.4213	0.961796	0.480898	−	0.876777i	$-0.340311\pi$
$593$	23.4213	0.961796	0.480898	+	0.876777i	$0.340311\pi$
$594$	0	0
$595$	27.5269	1.12849
$596$	0	0
$597$	0.0817645	0.00334640
$598$	0	0
$599$	2.61327	0.106775	0.0533876	−	0.998574i	$-0.482998\pi$
$599$	2.61327	0.106775	0.0533876	+	0.998574i	$0.482998\pi$
$600$	0	0
$601$	−27.4904	−1.12136	−0.560679	−	0.828034i	$-0.689460\pi$
$601$	−27.4904	−1.12136	−0.560679	+	0.828034i	$0.689460\pi$
$602$	0	0
$603$	17.1547	0.698594
$604$	0	0
$605$	4.12300	0.167624
$606$	0	0
$607$	−35.0458	−1.42246	−0.711232	−	0.702958i	$-0.751862\pi$
$607$	−35.0458	−1.42246	−0.711232	+	0.702958i	$0.751862\pi$
$608$	0	0
$609$	1.36007	0.0551130
$610$	0	0
$611$	9.46972	0.383104
$612$	0	0
$613$	15.8368	0.639641	0.319821	−	0.947478i	$-0.396377\pi$
$613$	15.8368	0.639641	0.319821	+	0.947478i	$0.396377\pi$
$614$	0	0
$615$	−7.36988	−0.297182
$616$	0	0
$617$	19.0168	0.765588	0.382794	−	0.923834i	$-0.374962\pi$
$617$	19.0168	0.765588	0.382794	+	0.923834i	$0.374962\pi$
$618$	0	0
$619$	−26.7417	−1.07484	−0.537419	−	0.843315i	$-0.680601\pi$
$619$	−26.7417	−1.07484	−0.537419	+	0.843315i	$0.680601\pi$
$620$	0	0
$621$	−3.71856	−0.149221
$622$	0	0
$623$	3.30110	0.132256
$624$	0	0
$625$	58.9831	2.35932
$626$	0	0
$627$	0.506062	0.0202102
$628$	0	0
$629$	6.37088	0.254024
$630$	0	0
$631$	39.8528	1.58651	0.793257	−	0.608888i	$-0.208384\pi$
$631$	39.8528	1.58651	0.793257	+	0.608888i	$0.208384\pi$
$632$	0	0
$633$	−0.760679	−0.0302343
$634$	0	0
$635$	−34.6559	−1.37528
$636$	0	0
$637$	−2.34111	−0.0927581
$638$	0	0
$639$	−11.2147	−0.443648
$640$	0	0
$641$	9.85468	0.389236	0.194618	−	0.980879i	$-0.437653\pi$
$641$	9.85468	0.389236	0.194618	+	0.980879i	$0.437653\pi$
$642$	0	0
$643$	10.3078	0.406499	0.203249	−	0.979127i	$-0.434850\pi$
$643$	10.3078	0.406499	0.203249	+	0.979127i	$0.434850\pi$
$644$	0	0
$645$	−2.21018	−0.0870257
$646$	0	0
$647$	24.4455	0.961052	0.480526	−	0.876980i	$-0.340446\pi$
$647$	24.4455	0.961052	0.480526	+	0.876980i	$0.340446\pi$
$648$	0	0
$649$	4.62255	0.181451
$650$	0	0
$651$	−4.05141	−0.158787
$652$	0	0
$653$	−26.5723	−1.03986	−0.519928	−	0.854210i	$-0.674041\pi$
$653$	−26.5723	−1.03986	−0.519928	+	0.854210i	$0.674041\pi$
$654$	0	0
$655$	−2.59797	−0.101511
$656$	0	0
$657$	−22.1610	−0.864584
$658$	0	0
$659$	22.5480	0.878345	0.439172	−	0.898403i	$-0.355272\pi$
$659$	22.5480	0.878345	0.439172	+	0.898403i	$0.355272\pi$
$660$	0	0
$661$	44.8824	1.74572	0.872861	−	0.487969i	$-0.162262\pi$
$661$	44.8824	1.74572	0.872861	+	0.487969i	$0.162262\pi$
$662$	0	0
$663$	0.582377	0.0226177
$664$	0	0
$665$	23.9198	0.927568
$666$	0	0
$667$	−11.0819	−0.429094
$668$	0	0
$669$	4.03651	0.156060
$670$	0	0
$671$	0.193895	0.00748525
$672$	0	0
$673$	−0.766928	−0.0295629	−0.0147815	−	0.999891i	$-0.504705\pi$
$673$	−0.766928	−0.0295629	−0.0147815	+	0.999891i	$0.504705\pi$
$674$	0	0
$675$	13.4750	0.518652
$676$	0	0
$677$	−22.6545	−0.870682	−0.435341	−	0.900266i	$-0.643372\pi$
$677$	−22.6545	−0.870682	−0.435341	+	0.900266i	$0.643372\pi$
$678$	0	0
$679$	−26.2168	−1.00611
$680$	0	0
$681$	−3.55182	−0.136106
$682$	0	0
$683$	6.69712	0.256258	0.128129	−	0.991757i	$-0.459103\pi$
$683$	6.69712	0.256258	0.128129	+	0.991757i	$0.459103\pi$
$684$	0	0
$685$	19.8582	0.758743
$686$	0	0
$687$	2.50500	0.0955719
$688$	0	0
$689$	5.34111	0.203480
$690$	0	0
$691$	−22.1563	−0.842865	−0.421433	−	0.906860i	$-0.638473\pi$
$691$	−22.1563	−0.842865	−0.421433	+	0.906860i	$0.638473\pi$
$692$	0	0
$693$	6.39883	0.243071
$694$	0	0
$695$	−43.5356	−1.65140
$696$	0	0
$697$	−29.3663	−1.11233
$698$	0	0
$699$	−4.01335	−0.151799
$700$	0	0
$701$	−14.7084	−0.555527	−0.277763	−	0.960650i	$-0.589593\pi$
$701$	−14.7084	−0.555527	−0.277763	+	0.960650i	$0.589593\pi$
$702$	0	0
$703$	5.53603	0.208796
$704$	0	0
$705$	−7.35109	−0.276858
$706$	0	0
$707$	−28.5264	−1.07285
$708$	0	0
$709$	−28.8284	−1.08267	−0.541336	−	0.840806i	$-0.682081\pi$
$709$	−28.8284	−1.08267	−0.541336	+	0.840806i	$0.682081\pi$
$710$	0	0
$711$	−51.0049	−1.91283
$712$	0	0
$713$	33.0110	1.23627
$714$	0	0
$715$	4.12300	0.154191
$716$	0	0
$717$	2.61938	0.0978225
$718$	0	0
$719$	8.10420	0.302236	0.151118	−	0.988516i	$-0.451713\pi$
$719$	8.10420	0.302236	0.151118	+	0.988516i	$0.451713\pi$
$720$	0	0
$721$	−9.96733	−0.371203
$722$	0	0
$723$	−0.106514	−0.00396130
$724$	0	0
$725$	40.1577	1.49142
$726$	0	0
$727$	19.1116	0.708810	0.354405	−	0.935092i	$-0.384683\pi$
$727$	19.1116	0.708810	0.354405	+	0.935092i	$0.384683\pi$
$728$	0	0
$729$	−25.1046	−0.929799
$730$	0	0
$731$	−8.80677	−0.325730
$732$	0	0
$733$	27.4827	1.01510	0.507548	−	0.861623i	$-0.330552\pi$
$733$	27.4827	1.01510	0.507548	+	0.861623i	$0.330552\pi$
$734$	0	0
$735$	1.81734	0.0670335
$736$	0	0
$737$	−5.78662	−0.213153
$738$	0	0
$739$	−3.99458	−0.146943	−0.0734715	−	0.997297i	$-0.523408\pi$
$739$	−3.99458	−0.146943	−0.0734715	+	0.997297i	$0.523408\pi$
$740$	0	0
$741$	0.506062	0.0185906
$742$	0	0
$743$	−27.9386	−1.02497	−0.512483	−	0.858697i	$-0.671274\pi$
$743$	−27.9386	−1.02497	−0.512483	+	0.858697i	$0.671274\pi$
$744$	0	0
$745$	11.5464	0.423028
$746$	0	0
$747$	0.646601	0.0236579
$748$	0	0
$749$	−22.3309	−0.815955
$750$	0	0
$751$	16.3149	0.595341	0.297670	−	0.954669i	$-0.403790\pi$
$751$	16.3149	0.595341	0.297670	+	0.954669i	$0.403790\pi$
$752$	0	0
$753$	−2.35320	−0.0857554
$754$	0	0
$755$	33.9982	1.23732
$756$	0	0
$757$	−50.8067	−1.84660	−0.923301	−	0.384077i	$-0.874520\pi$
$757$	−50.8067	−1.84660	−0.923301	+	0.384077i	$0.874520\pi$
$758$	0	0
$759$	0.623443	0.0226295
$760$	0	0
$761$	−7.06895	−0.256249	−0.128125	−	0.991758i	$-0.540896\pi$
$761$	−7.06895	−0.256249	−0.128125	+	0.991758i	$0.540896\pi$
$762$	0	0
$763$	14.7429	0.533728
$764$	0	0
$765$	37.8073	1.36692
$766$	0	0
$767$	4.62255	0.166911
$768$	0	0
$769$	−19.0465	−0.686834	−0.343417	−	0.939183i	$-0.611584\pi$
$769$	−19.0465	−0.686834	−0.343417	+	0.939183i	$0.611584\pi$
$770$	0	0
$771$	−2.77974	−0.100110
$772$	0	0
$773$	34.6902	1.24772	0.623861	−	0.781536i	$-0.285563\pi$
$773$	34.6902	1.24772	0.623861	+	0.781536i	$0.285563\pi$
$774$	0	0
$775$	−119.622	−4.29697
$776$	0	0
$777$	−0.837026	−0.0300281
$778$	0	0
$779$	−25.5181	−0.914282
$780$	0	0
$781$	3.78295	0.135364
$782$	0	0
$783$	3.75836	0.134313
$784$	0	0
$785$	54.8020	1.95597
$786$	0	0
$787$	−29.2246	−1.04174	−0.520872	−	0.853635i	$-0.674393\pi$
$787$	−29.2246	−1.04174	−0.520872	+	0.853635i	$0.674393\pi$
$788$	0	0
$789$	−2.98318	−0.106204
$790$	0	0
$791$	15.9742	0.567976
$792$	0	0
$793$	0.193895	0.00688543
$794$	0	0
$795$	−4.14616	−0.147049
$796$	0	0
$797$	3.30655	0.117124	0.0585620	−	0.998284i	$-0.481348\pi$
$797$	3.30655	0.117124	0.0585620	+	0.998284i	$0.481348\pi$
$798$	0	0
$799$	−29.2914	−1.03626
$800$	0	0
$801$	4.53395	0.160199
$802$	0	0
$803$	7.47534	0.263799
$804$	0	0
$805$	29.4679	1.03861
$806$	0	0
$807$	−0.441116	−0.0155280
$808$	0	0
$809$	−9.10142	−0.319989	−0.159994	−	0.987118i	$-0.551148\pi$
$809$	−9.10142	−0.319989	−0.159994	+	0.987118i	$0.551148\pi$
$810$	0	0
$811$	25.0849	0.880849	0.440425	−	0.897790i	$-0.354828\pi$
$811$	25.0849	0.880849	0.440425	+	0.897790i	$0.354828\pi$
$812$	0	0
$813$	4.89597	0.171709
$814$	0	0
$815$	−11.7540	−0.411725
$816$	0	0
$817$	−7.65272	−0.267735
$818$	0	0
$819$	6.39883	0.223593
$820$	0	0
$821$	−6.29231	−0.219603	−0.109802	−	0.993954i	$-0.535022\pi$
$821$	−6.29231	−0.219603	−0.109802	+	0.993954i	$0.535022\pi$
$822$	0	0
$823$	32.7466	1.14147	0.570737	−	0.821133i	$-0.306657\pi$
$823$	32.7466	1.14147	0.570737	+	0.821133i	$0.306657\pi$
$824$	0	0
$825$	−2.25918	−0.0786544
$826$	0	0
$827$	−6.16948	−0.214534	−0.107267	−	0.994230i	$-0.534210\pi$
$827$	−6.16948	−0.214534	−0.107267	+	0.994230i	$0.534210\pi$
$828$	0	0
$829$	−16.0410	−0.557127	−0.278564	−	0.960418i	$-0.589858\pi$
$829$	−16.0410	−0.557127	−0.278564	+	0.960418i	$0.589858\pi$
$830$	0	0
$831$	5.27216	0.182889
$832$	0	0
$833$	7.24144	0.250901
$834$	0	0
$835$	5.36374	0.185620
$836$	0	0
$837$	−11.1955	−0.386972
$838$	0	0
$839$	−27.1837	−0.938484	−0.469242	−	0.883070i	$-0.655473\pi$
$839$	−27.1837	−0.938484	−0.469242	+	0.883070i	$0.655473\pi$
$840$	0	0
$841$	−17.7994	−0.613773
$842$	0	0
$843$	3.85279	0.132697
$844$	0	0
$845$	4.12300	0.141835
$846$	0	0
$847$	−2.15845	−0.0741651
$848$	0	0
$849$	1.72804	0.0593062
$850$	0	0
$851$	6.82012	0.233791
$852$	0	0
$853$	−24.2748	−0.831152	−0.415576	−	0.909558i	$-0.636420\pi$
$853$	−24.2748	−0.831152	−0.415576	+	0.909558i	$0.636420\pi$
$854$	0	0
$855$	32.8530	1.12355
$856$	0	0
$857$	16.7036	0.570584	0.285292	−	0.958441i	$-0.407909\pi$
$857$	16.7036	0.570584	0.285292	+	0.958441i	$0.407909\pi$
$858$	0	0
$859$	−28.9414	−0.987468	−0.493734	−	0.869613i	$-0.664368\pi$
$859$	−28.9414	−0.987468	−0.493734	+	0.869613i	$0.664368\pi$
$860$	0	0
$861$	3.85824	0.131488
$862$	0	0
$863$	33.6601	1.14580	0.572901	−	0.819625i	$-0.305818\pi$
$863$	33.6601	1.14580	0.572901	+	0.819625i	$0.305818\pi$
$864$	0	0
$865$	−35.1630	−1.19558
$866$	0	0
$867$	1.39935	0.0475244
$868$	0	0
$869$	17.2049	0.583637
$870$	0	0
$871$	−5.78662	−0.196072
$872$	0	0
$873$	−36.0079	−1.21868
$874$	0	0
$875$	−62.2870	−2.10568
$876$	0	0
$877$	56.6214	1.91197	0.955985	−	0.293417i	$-0.0947925\pi$
$877$	56.6214	1.91197	0.955985	+	0.293417i	$0.0947925\pi$
$878$	0	0
$879$	3.52796	0.118995
$880$	0	0
$881$	35.8236	1.20693	0.603465	−	0.797390i	$-0.293786\pi$
$881$	35.8236	1.20693	0.603465	+	0.797390i	$0.293786\pi$
$882$	0	0
$883$	−29.2843	−0.985493	−0.492747	−	0.870173i	$-0.664007\pi$
$883$	−29.2843	−0.985493	−0.492747	+	0.870173i	$0.664007\pi$
$884$	0	0
$885$	−3.58836	−0.120621
$886$	0	0
$887$	36.8342	1.23677	0.618385	−	0.785875i	$-0.287787\pi$
$887$	36.8342	1.23677	0.618385	+	0.785875i	$0.287787\pi$
$888$	0	0
$889$	18.1428	0.608491
$890$	0	0
$891$	8.68222	0.290865
$892$	0	0
$893$	−25.4530	−0.851754
$894$	0	0
$895$	79.5935	2.66052
$896$	0	0
$897$	0.623443	0.0208161
$898$	0	0
$899$	−33.3644	−1.11277
$900$	0	0
$901$	−16.5209	−0.550392
$902$	0	0
$903$	1.15706	0.0385045
$904$	0	0
$905$	108.601	3.61001
$906$	0	0
$907$	0.491159	0.0163087	0.00815433	−	0.999967i	$-0.497404\pi$
$907$	0.491159	0.0163087	0.00815433	+	0.999967i	$0.497404\pi$
$908$	0	0
$909$	−39.1800	−1.29952
$910$	0	0
$911$	44.1275	1.46201	0.731005	−	0.682373i	$-0.239052\pi$
$911$	44.1275	1.46201	0.731005	+	0.682373i	$0.239052\pi$
$912$	0	0
$913$	−0.218111	−0.00721841
$914$	0	0
$915$	−0.150516	−0.00497589
$916$	0	0
$917$	1.36007	0.0449136
$918$	0	0
$919$	43.8564	1.44669	0.723345	−	0.690487i	$-0.242604\pi$
$919$	43.8564	1.44669	0.723345	+	0.690487i	$0.242604\pi$
$920$	0	0
$921$	−0.657635	−0.0216698
$922$	0	0
$923$	3.78295	0.124517
$924$	0	0
$925$	−24.7141	−0.812596
$926$	0	0
$927$	−13.6898	−0.449631
$928$	0	0
$929$	−21.9917	−0.721526	−0.360763	−	0.932658i	$-0.617484\pi$
$929$	−21.9917	−0.721526	−0.360763	+	0.932658i	$0.617484\pi$
$930$	0	0
$931$	6.29251	0.206229
$932$	0	0
$933$	−0.790679	−0.0258857
$934$	0	0
$935$	−12.7531	−0.417071
$936$	0	0
$937$	−15.9198	−0.520076	−0.260038	−	0.965598i	$-0.583735\pi$
$937$	−15.9198	−0.520076	−0.260038	+	0.965598i	$0.583735\pi$
$938$	0	0
$939$	2.34742	0.0766050
$940$	0	0
$941$	−7.79435	−0.254088	−0.127044	−	0.991897i	$-0.540549\pi$
$941$	−7.79435	−0.254088	−0.127044	+	0.991897i	$0.540549\pi$
$942$	0	0
$943$	−31.4371	−1.02373
$944$	0	0
$945$	−9.99386	−0.325100
$946$	0	0
$947$	37.4253	1.21616	0.608079	−	0.793876i	$-0.291940\pi$
$947$	37.4253	1.21616	0.608079	+	0.793876i	$0.291940\pi$
$948$	0	0
$949$	7.47534	0.242660
$950$	0	0
$951$	−1.55799	−0.0505214
$952$	0	0
$953$	40.9457	1.32636	0.663181	−	0.748459i	$-0.269206\pi$
$953$	40.9457	1.32636	0.663181	+	0.748459i	$0.269206\pi$
$954$	0	0
$955$	7.66940	0.248176
$956$	0	0
$957$	−0.630117	−0.0203688
$958$	0	0
$959$	−10.3960	−0.335706
$960$	0	0
$961$	68.3865	2.20602
$962$	0	0
$963$	−30.6708	−0.988351
$964$	0	0
$965$	67.7158	2.17985
$966$	0	0
$967$	37.0738	1.19221	0.596107	−	0.802905i	$-0.296713\pi$
$967$	37.0738	1.19221	0.596107	+	0.802905i	$0.296713\pi$
$968$	0	0
$969$	−1.56533	−0.0502857
$970$	0	0
$971$	−9.13718	−0.293226	−0.146613	−	0.989194i	$-0.546837\pi$
$971$	−9.13718	−0.293226	−0.146613	+	0.989194i	$0.546837\pi$
$972$	0	0
$973$	22.7915	0.730662
$974$	0	0
$975$	−2.25918	−0.0723515
$976$	0	0
$977$	24.8647	0.795491	0.397745	−	0.917496i	$-0.369793\pi$
$977$	24.8647	0.795491	0.397745	+	0.917496i	$0.369793\pi$
$978$	0	0
$979$	−1.52939	−0.0488794
$980$	0	0
$981$	20.2488	0.646495
$982$	0	0
$983$	22.0037	0.701808	0.350904	−	0.936411i	$-0.385874\pi$
$983$	22.0037	0.701808	0.350904	+	0.936411i	$0.385874\pi$
$984$	0	0
$985$	72.5354	2.31117
$986$	0	0
$987$	3.84840	0.122496
$988$	0	0
$989$	−9.42777	−0.299786
$990$	0	0
$991$	−44.3616	−1.40919	−0.704596	−	0.709608i	$-0.748872\pi$
$991$	−44.3616	−1.40919	−0.704596	+	0.709608i	$0.748872\pi$
$992$	0	0
$993$	4.44058	0.140918
$994$	0	0
$995$	−1.79051	−0.0567630
$996$	0	0
$997$	0.262678	0.00831910	0.00415955	−	0.999991i	$-0.498676\pi$
$997$	0.262678	0.00831910	0.00415955	+	0.999991i	$0.498676\pi$
$998$	0	0
$999$	−2.31300	−0.0731800

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9152.2.a.cg.1.2		4
4.3	odd	2		9152.2.a.ch.1.3		4
8.3	odd	2		143.2.a.b.1.4	✓	4
8.5	even	2		2288.2.a.x.1.3		4
24.11	even	2		1287.2.a.k.1.1		4
40.19	odd	2		3575.2.a.k.1.1		4
56.27	even	2		7007.2.a.n.1.4		4
88.43	even	2		1573.2.a.f.1.1		4
104.51	odd	2		1859.2.a.i.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.2.a.b.1.4	✓	4	8.3	odd	2
1287.2.a.k.1.1		4	24.11	even	2
1573.2.a.f.1.1		4	88.43	even	2
1859.2.a.i.1.1		4	104.51	odd	2
2288.2.a.x.1.3		4	8.5	even	2
3575.2.a.k.1.1		4	40.19	odd	2
7007.2.a.n.1.4		4	56.27	even	2
9152.2.a.cg.1.2		4	1.1	even	1	trivial
9152.2.a.ch.1.3		4	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:	\( N \)	\(=\)	\( 9152 = 2^{6} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9152.a (trivial)

\(f(q)\)	\(=\)	\(q-0.188279 q^{3} +4.12300 q^{5} -2.15845 q^{7} -2.96455 q^{9} +O(q^{10})\)	\(q-0.188279 q^{3} +4.12300 q^{5} -2.15845 q^{7} -2.96455 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.776272 q^{15} -3.09317 q^{17} -2.68783 q^{19} +0.406389 q^{21} -3.31128 q^{23} +11.9991 q^{25} +1.12300 q^{27} +3.34673 q^{29} -9.96928 q^{31} -0.188279 q^{33} -8.89927 q^{35} -2.05966 q^{37} -0.188279 q^{39} +9.49394 q^{41} +2.84717 q^{43} -12.2228 q^{45} +9.46972 q^{47} -2.34111 q^{49} +0.582377 q^{51} +5.34111 q^{53} +4.12300 q^{55} +0.506062 q^{57} +4.62255 q^{59} +0.193895 q^{61} +6.39883 q^{63} +4.12300 q^{65} -5.78662 q^{67} +0.623443 q^{69} +3.78295 q^{71} +7.47534 q^{73} -2.25918 q^{75} -2.15845 q^{77} +17.2049 q^{79} +8.68222 q^{81} -0.218111 q^{83} -12.7531 q^{85} -0.630117 q^{87} -1.52939 q^{89} -2.15845 q^{91} +1.87700 q^{93} -11.0819 q^{95} +12.1462 q^{97} -2.96455 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 4 * q - 6 * q^7 + 2 * q^9	\( 4 q - 6 q^{7} + 2 q^{9} + 4 q^{11} + 4 q^{13} + 10 q^{15} + 6 q^{17} + 8 q^{19} + 2 q^{21} + 4 q^{23} + 12 q^{25} - 12 q^{27} + 10 q^{29} - 2 q^{31} - 6 q^{35} - 12 q^{37} + 8 q^{41} + 26 q^{43} - 26 q^{45} + 18 q^{47} + 6 q^{49} - 22 q^{51} + 6 q^{53} + 32 q^{57} - 16 q^{59} + 12 q^{61} - 22 q^{63} + 2 q^{67} + 4 q^{69} + 14 q^{71} + 22 q^{73} - 36 q^{75} - 6 q^{77} + 10 q^{79} + 4 q^{81} - 2 q^{83} - 48 q^{85} - 16 q^{87} + 10 q^{89} - 6 q^{91} + 24 q^{93} - 2 q^{95} + 22 q^{97} + 2 q^{99}+O(q^{100}) \) 4 * q - 6 * q^7 + 2 * q^9 + 4 * q^11 + 4 * q^13 + 10 * q^15 + 6 * q^17 + 8 * q^19 + 2 * q^21 + 4 * q^23 + 12 * q^25 - 12 * q^27 + 10 * q^29 - 2 * q^31 - 6 * q^35 - 12 * q^37 + 8 * q^41 + 26 * q^43 - 26 * q^45 + 18 * q^47 + 6 * q^49 - 22 * q^51 + 6 * q^53 + 32 * q^57 - 16 * q^59 + 12 * q^61 - 22 * q^63 + 2 * q^67 + 4 * q^69 + 14 * q^71 + 22 * q^73 - 36 * q^75 - 6 * q^77 + 10 * q^79 + 4 * q^81 - 2 * q^83 - 48 * q^85 - 16 * q^87 + 10 * q^89 - 6 * q^91 + 24 * q^93 - 2 * q^95 + 22 * q^97 + 2 * q^99

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