LMFDB - Embedded newform 4640.2.a.j.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4640,2,Mod(1,4640)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4640.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4640, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4640 = 2^{5} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4640.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,-3,0,-7,0,2,0,2,0,-9,0,3,0,1,0,6,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$37.0505865379$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	4640.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.860806 q^{3} -1.00000 q^{5} +2.18421 q^{7} -2.25901 q^{9} -0.925197 q^{11} -4.86081 q^{13} -0.860806 q^{15} -0.462598 q^{17} +5.72161 q^{19} +1.88018 q^{21} -0.860806 q^{23} +1.00000 q^{25} -4.52699 q^{27} +1.00000 q^{29} +9.10941 q^{31} -0.796415 q^{33} -2.18421 q^{35} -6.36842 q^{37} -4.18421 q^{39} +1.07480 q^{41} +5.25901 q^{43} +2.25901 q^{45} -6.79641 q^{47} -2.22923 q^{49} -0.398207 q^{51} +2.98062 q^{53} +0.925197 q^{55} +4.92520 q^{57} -2.73202 q^{59} -13.2292 q^{61} -4.93416 q^{63} +4.86081 q^{65} -10.7964 q^{67} -0.740987 q^{69} +5.85039 q^{71} -13.7562 q^{73} +0.860806 q^{75} -2.02082 q^{77} -14.4328 q^{79} +2.88018 q^{81} -2.92520 q^{83} +0.462598 q^{85} +0.860806 q^{87} +17.5720 q^{89} -10.6170 q^{91} +7.84143 q^{93} -5.72161 q^{95} -8.30403 q^{97} +2.09003 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} - 7 q^{7} + 2 q^{9} + 2 q^{11} - 9 q^{13} + 3 q^{15} + q^{17} + 6 q^{19} + 20 q^{21} + 3 q^{23} + 3 q^{25} - 12 q^{27} + 3 q^{29} + 9 q^{31} + 4 q^{33} + 7 q^{35} + 8 q^{37} + q^{39}+ \cdots - 32 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 - 7 * q^7 + 2 * q^9 + 2 * q^11 - 9 * q^13 + 3 * q^15 + q^17 + 6 * q^19 + 20 * q^21 + 3 * q^23 + 3 * q^25 - 12 * q^27 + 3 * q^29 + 9 * q^31 + 4 * q^33 + 7 * q^35 + 8 * q^37 + q^39 + 8 * q^41 + 7 * q^43 - 2 * q^45 - 14 * q^47 + 26 * q^49 + 2 * q^51 - 11 * q^53 - 2 * q^55 + 10 * q^57 - q^59 - 7 * q^61 - 26 * q^63 + 9 * q^65 - 26 * q^67 - 11 * q^69 + 8 * q^71 - 7 * q^73 - 3 * q^75 - 14 * q^77 - 17 * q^79 + 23 * q^81 - 4 * q^83 - q^85 - 3 * q^87 + 32 * q^89 + 8 * q^91 - 2 * q^93 - 6 * q^95 + 3 * q^97 - 32 * q^99

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
$3$	0.860806	0.496986	0.248493	−	0.968634i	$-0.420065\pi$
$3$	0.860806	0.496986	0.248493	+	0.968634i	$0.420065\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.18421	0.825554	0.412777	−	0.910832i	$-0.364559\pi$
$7$	2.18421	0.825554	0.412777	+	0.910832i	$0.364559\pi$
$8$	0	0
$9$	−2.25901	−0.753004
$10$	0	0
$11$	−0.925197	−0.278957	−0.139479	−	0.990225i	$-0.544543\pi$
$11$	−0.925197	−0.278957	−0.139479	+	0.990225i	$0.544543\pi$
$12$	0	0
$13$	−4.86081	−1.34814	−0.674072	−	0.738665i	$-0.735456\pi$
$13$	−4.86081	−1.34814	−0.674072	+	0.738665i	$0.735456\pi$
$14$	0	0
$15$	−0.860806	−0.222259
$16$	0	0
$17$	−0.462598	−0.112197	−0.0560983	−	0.998425i	$-0.517866\pi$
$17$	−0.462598	−0.112197	−0.0560983	+	0.998425i	$0.517866\pi$
$18$	0	0
$19$	5.72161	1.31263	0.656314	−	0.754488i	$-0.272115\pi$
$19$	5.72161	1.31263	0.656314	+	0.754488i	$0.272115\pi$
$20$	0	0
$21$	1.88018	0.410289
$22$	0	0
$23$	−0.860806	−0.179490	−0.0897452	−	0.995965i	$-0.528605\pi$
$23$	−0.860806	−0.179490	−0.0897452	+	0.995965i	$0.528605\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.52699	−0.871220
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	9.10941	1.63610	0.818049	−	0.575149i	$-0.195056\pi$
$31$	9.10941	1.63610	0.818049	+	0.575149i	$0.195056\pi$
$32$	0	0
$33$	−0.796415	−0.138638
$34$	0	0
$35$	−2.18421	−0.369199
$36$	0	0
$37$	−6.36842	−1.04696	−0.523481	−	0.852037i	$-0.675367\pi$
$37$	−6.36842	−1.04696	−0.523481	+	0.852037i	$0.675367\pi$
$38$	0	0
$39$	−4.18421	−0.670010
$40$	0	0
$41$	1.07480	0.167856	0.0839280	−	0.996472i	$-0.473253\pi$
$41$	1.07480	0.167856	0.0839280	+	0.996472i	$0.473253\pi$
$42$	0	0
$43$	5.25901	0.801992	0.400996	−	0.916080i	$-0.368664\pi$
$43$	5.25901	0.801992	0.400996	+	0.916080i	$0.368664\pi$
$44$	0	0
$45$	2.25901	0.336754
$46$	0	0
$47$	−6.79641	−0.991359	−0.495679	−	0.868506i	$-0.665081\pi$
$47$	−6.79641	−0.991359	−0.495679	+	0.868506i	$0.665081\pi$
$48$	0	0
$49$	−2.22923	−0.318461
$50$	0	0
$51$	−0.398207	−0.0557602
$52$	0	0
$53$	2.98062	0.409420	0.204710	−	0.978823i	$-0.434375\pi$
$53$	2.98062	0.409420	0.204710	+	0.978823i	$0.434375\pi$
$54$	0	0
$55$	0.925197	0.124754
$56$	0	0
$57$	4.92520	0.652358
$58$	0	0
$59$	−2.73202	−0.355679	−0.177840	−	0.984059i	$-0.556911\pi$
$59$	−2.73202	−0.355679	−0.177840	+	0.984059i	$0.556911\pi$
$60$	0	0
$61$	−13.2292	−1.69383	−0.846914	−	0.531729i	$-0.821542\pi$
$61$	−13.2292	−1.69383	−0.846914	+	0.531729i	$0.821542\pi$
$62$	0	0
$63$	−4.93416	−0.621646
$64$	0	0
$65$	4.86081	0.602909
$66$	0	0
$67$	−10.7964	−1.31899	−0.659496	−	0.751708i	$-0.729230\pi$
$67$	−10.7964	−1.31899	−0.659496	+	0.751708i	$0.729230\pi$
$68$	0	0
$69$	−0.740987	−0.0892043
$70$	0	0
$71$	5.85039	0.694314	0.347157	−	0.937807i	$-0.387147\pi$
$71$	5.85039	0.694314	0.347157	+	0.937807i	$0.387147\pi$
$72$	0	0
$73$	−13.7562	−1.61004	−0.805022	−	0.593245i	$-0.797846\pi$
$73$	−13.7562	−1.61004	−0.805022	+	0.593245i	$0.797846\pi$
$74$	0	0
$75$	0.860806	0.0993973
$76$	0	0
$77$	−2.02082	−0.230294
$78$	0	0
$79$	−14.4328	−1.62382	−0.811909	−	0.583784i	$-0.801572\pi$
$79$	−14.4328	−1.62382	−0.811909	+	0.583784i	$0.801572\pi$
$80$	0	0
$81$	2.88018	0.320020
$82$	0	0
$83$	−2.92520	−0.321082	−0.160541	−	0.987029i	$-0.551324\pi$
$83$	−2.92520	−0.321082	−0.160541	+	0.987029i	$0.551324\pi$
$84$	0	0
$85$	0.462598	0.0501758
$86$	0	0
$87$	0.860806	0.0922881
$88$	0	0
$89$	17.5720	1.86263	0.931314	−	0.364216i	$-0.118663\pi$
$89$	17.5720	1.86263	0.931314	+	0.364216i	$0.118663\pi$
$90$	0	0
$91$	−10.6170	−1.11297
$92$	0	0
$93$	7.84143	0.813119
$94$	0	0
$95$	−5.72161	−0.587025
$96$	0	0
$97$	−8.30403	−0.843146	−0.421573	−	0.906794i	$-0.638522\pi$
$97$	−8.30403	−0.843146	−0.421573	+	0.906794i	$0.638522\pi$
$98$	0	0
$99$	2.09003	0.210056
$100$	0	0
$101$	−13.3580	−1.32917	−0.664586	−	0.747212i	$-0.731392\pi$
$101$	−13.3580	−1.32917	−0.664586	+	0.747212i	$0.731392\pi$
$102$	0	0
$103$	−2.79641	−0.275539	−0.137769	−	0.990464i	$-0.543993\pi$
$103$	−2.79641	−0.275539	−0.137769	+	0.990464i	$0.543993\pi$
$104$	0	0
$105$	−1.88018	−0.183487
$106$	0	0
$107$	−8.51803	−0.823469	−0.411734	−	0.911304i	$-0.635077\pi$
$107$	−8.51803	−0.823469	−0.411734	+	0.911304i	$0.635077\pi$
$108$	0	0
$109$	−12.7756	−1.22368	−0.611840	−	0.790982i	$-0.709570\pi$
$109$	−12.7756	−1.22368	−0.611840	+	0.790982i	$0.709570\pi$
$110$	0	0
$111$	−5.48197	−0.520326
$112$	0	0
$113$	6.83998	0.643451	0.321726	−	0.946833i	$-0.395737\pi$
$113$	6.83998	0.643451	0.321726	+	0.946833i	$0.395737\pi$
$114$	0	0
$115$	0.860806	0.0802706
$116$	0	0
$117$	10.9806	1.01516
$118$	0	0
$119$	−1.01041	−0.0926243
$120$	0	0
$121$	−10.1440	−0.922183
$122$	0	0
$123$	0.925197	0.0834222
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−20.0900	−1.78270	−0.891351	−	0.453314i	$-0.850242\pi$
$127$	−20.0900	−1.78270	−0.891351	+	0.453314i	$0.850242\pi$
$128$	0	0
$129$	4.52699	0.398579
$130$	0	0
$131$	3.20359	0.279899	0.139949	−	0.990159i	$-0.455306\pi$
$131$	3.20359	0.279899	0.139949	+	0.990159i	$0.455306\pi$
$132$	0	0
$133$	12.4972	1.08364
$134$	0	0
$135$	4.52699	0.389621
$136$	0	0
$137$	11.8760	1.01464	0.507319	−	0.861758i	$-0.330637\pi$
$137$	11.8760	1.01464	0.507319	+	0.861758i	$0.330637\pi$
$138$	0	0
$139$	−14.7022	−1.24703	−0.623514	−	0.781812i	$-0.714295\pi$
$139$	−14.7022	−1.24703	−0.623514	+	0.781812i	$0.714295\pi$
$140$	0	0
$141$	−5.85039	−0.492692
$142$	0	0
$143$	4.49720	0.376075
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−1.91893	−0.158271
$148$	0	0
$149$	−9.87122	−0.808682	−0.404341	−	0.914608i	$-0.632499\pi$
$149$	−9.87122	−0.808682	−0.404341	+	0.914608i	$0.632499\pi$
$150$	0	0
$151$	−1.72161	−0.140103	−0.0700514	−	0.997543i	$-0.522316\pi$
$151$	−1.72161	−0.140103	−0.0700514	+	0.997543i	$0.522316\pi$
$152$	0	0
$153$	1.04502	0.0844845
$154$	0	0
$155$	−9.10941	−0.731685
$156$	0	0
$157$	−2.55678	−0.204053	−0.102026	−	0.994782i	$-0.532533\pi$
$157$	−2.55678	−0.204053	−0.102026	+	0.994782i	$0.532533\pi$
$158$	0	0
$159$	2.56574	0.203476
$160$	0	0
$161$	−1.88018	−0.148179
$162$	0	0
$163$	−3.35319	−0.262642	−0.131321	−	0.991340i	$-0.541922\pi$
$163$	−3.35319	−0.262642	−0.131321	+	0.991340i	$0.541922\pi$
$164$	0	0
$165$	0.796415	0.0620008
$166$	0	0
$167$	−1.51658	−0.117356	−0.0586781	−	0.998277i	$-0.518689\pi$
$167$	−1.51658	−0.117356	−0.0586781	+	0.998277i	$0.518689\pi$
$168$	0	0
$169$	10.6274	0.817495
$170$	0	0
$171$	−12.9252	−0.988415
$172$	0	0
$173$	2.71120	0.206129	0.103064	−	0.994675i	$-0.467135\pi$
$173$	2.71120	0.206129	0.103064	+	0.994675i	$0.467135\pi$
$174$	0	0
$175$	2.18421	0.165111
$176$	0	0
$177$	−2.35174	−0.176768
$178$	0	0
$179$	13.3193	0.995528	0.497764	−	0.867312i	$-0.334155\pi$
$179$	13.3193	0.995528	0.497764	+	0.867312i	$0.334155\pi$
$180$	0	0
$181$	12.7756	0.949602	0.474801	−	0.880093i	$-0.342520\pi$
$181$	12.7756	0.949602	0.474801	+	0.880093i	$0.342520\pi$
$182$	0	0
$183$	−11.3878	−0.841810
$184$	0	0
$185$	6.36842	0.468216
$186$	0	0
$187$	0.427995	0.0312981
$188$	0	0
$189$	−9.88790	−0.719239
$190$	0	0
$191$	13.7770	0.996872	0.498436	−	0.866927i	$-0.333908\pi$
$191$	13.7770	0.996872	0.498436	+	0.866927i	$0.333908\pi$
$192$	0	0
$193$	12.5616	0.904203	0.452102	−	0.891966i	$-0.350674\pi$
$193$	12.5616	0.904203	0.452102	+	0.891966i	$0.350674\pi$
$194$	0	0
$195$	4.18421	0.299638
$196$	0	0
$197$	−5.01938	−0.357616	−0.178808	−	0.983884i	$-0.557224\pi$
$197$	−5.01938	−0.357616	−0.178808	+	0.983884i	$0.557224\pi$
$198$	0	0
$199$	−0.368420	−0.0261166	−0.0130583	−	0.999915i	$-0.504157\pi$
$199$	−0.368420	−0.0261166	−0.0130583	+	0.999915i	$0.504157\pi$
$200$	0	0
$201$	−9.29362	−0.655521
$202$	0	0
$203$	2.18421	0.153301
$204$	0	0
$205$	−1.07480	−0.0750675
$206$	0	0
$207$	1.94457	0.135157
$208$	0	0
$209$	−5.29362	−0.366167
$210$	0	0
$211$	16.0692	1.10625	0.553125	−	0.833098i	$-0.313435\pi$
$211$	16.0692	1.10625	0.553125	+	0.833098i	$0.313435\pi$
$212$	0	0
$213$	5.03605	0.345065
$214$	0	0
$215$	−5.25901	−0.358662
$216$	0	0
$217$	19.8969	1.35069
$218$	0	0
$219$	−11.8414	−0.800170
$220$	0	0
$221$	2.24860	0.151257
$222$	0	0
$223$	18.5824	1.24437	0.622185	−	0.782870i	$-0.286245\pi$
$223$	18.5824	1.24437	0.622185	+	0.782870i	$0.286245\pi$
$224$	0	0
$225$	−2.25901	−0.150601
$226$	0	0
$227$	−0.149606	−0.00992972	−0.00496486	−	0.999988i	$-0.501580\pi$
$227$	−0.149606	−0.00992972	−0.00496486	+	0.999988i	$0.501580\pi$
$228$	0	0
$229$	2.37257	0.156784	0.0783918	−	0.996923i	$-0.475021\pi$
$229$	2.37257	0.156784	0.0783918	+	0.996923i	$0.475021\pi$
$230$	0	0
$231$	−1.73954	−0.114453
$232$	0	0
$233$	7.29362	0.477821	0.238910	−	0.971042i	$-0.423210\pi$
$233$	7.29362	0.477821	0.238910	+	0.971042i	$0.423210\pi$
$234$	0	0
$235$	6.79641	0.443349
$236$	0	0
$237$	−12.4238	−0.807016
$238$	0	0
$239$	−25.7908	−1.66827	−0.834135	−	0.551561i	$-0.814032\pi$
$239$	−25.7908	−1.66827	−0.834135	+	0.551561i	$0.814032\pi$
$240$	0	0
$241$	−1.91478	−0.123342	−0.0616711	−	0.998097i	$-0.519643\pi$
$241$	−1.91478	−0.123342	−0.0616711	+	0.998097i	$0.519643\pi$
$242$	0	0
$243$	16.0602	1.03027
$244$	0	0
$245$	2.22923	0.142420
$246$	0	0
$247$	−27.8116	−1.76961
$248$	0	0
$249$	−2.51803	−0.159573
$250$	0	0
$251$	−8.86562	−0.559593	−0.279797	−	0.960059i	$-0.590267\pi$
$251$	−8.86562	−0.559593	−0.279797	+	0.960059i	$0.590267\pi$
$252$	0	0
$253$	0.796415	0.0500702
$254$	0	0
$255$	0.398207	0.0249367
$256$	0	0
$257$	27.1053	1.69078	0.845390	−	0.534150i	$-0.179368\pi$
$257$	27.1053	1.69078	0.845390	+	0.534150i	$0.179368\pi$
$258$	0	0
$259$	−13.9100	−0.864323
$260$	0	0
$261$	−2.25901	−0.139829
$262$	0	0
$263$	−24.3476	−1.50134	−0.750669	−	0.660679i	$-0.770268\pi$
$263$	−24.3476	−1.50134	−0.750669	+	0.660679i	$0.770268\pi$
$264$	0	0
$265$	−2.98062	−0.183098
$266$	0	0
$267$	15.1261	0.925701
$268$	0	0
$269$	−19.2203	−1.17188	−0.585940	−	0.810354i	$-0.699275\pi$
$269$	−19.2203	−1.17188	−0.585940	+	0.810354i	$0.699275\pi$
$270$	0	0
$271$	−23.4432	−1.42407	−0.712037	−	0.702142i	$-0.752227\pi$
$271$	−23.4432	−1.42407	−0.712037	+	0.702142i	$0.752227\pi$
$272$	0	0
$273$	−9.13919	−0.553129
$274$	0	0
$275$	−0.925197	−0.0557915
$276$	0	0
$277$	18.4793	1.11031	0.555156	−	0.831746i	$-0.312659\pi$
$277$	18.4793	1.11031	0.555156	+	0.831746i	$0.312659\pi$
$278$	0	0
$279$	−20.5783	−1.23199
$280$	0	0
$281$	25.2590	1.50683	0.753413	−	0.657547i	$-0.228406\pi$
$281$	25.2590	1.50683	0.753413	+	0.657547i	$0.228406\pi$
$282$	0	0
$283$	−14.3684	−0.854114	−0.427057	−	0.904225i	$-0.640450\pi$
$283$	−14.3684	−0.854114	−0.427057	+	0.904225i	$0.640450\pi$
$284$	0	0
$285$	−4.92520	−0.291743
$286$	0	0
$287$	2.34760	0.138574
$288$	0	0
$289$	−16.7860	−0.987412
$290$	0	0
$291$	−7.14816	−0.419032
$292$	0	0
$293$	−16.2188	−0.947513	−0.473757	−	0.880656i	$-0.657102\pi$
$293$	−16.2188	−0.947513	−0.473757	+	0.880656i	$0.657102\pi$
$294$	0	0
$295$	2.73202	0.159065
$296$	0	0
$297$	4.18836	0.243033
$298$	0	0
$299$	4.18421	0.241979
$300$	0	0
$301$	11.4868	0.662088
$302$	0	0
$303$	−11.4987	−0.660580
$304$	0	0
$305$	13.2292	0.757503
$306$	0	0
$307$	19.2340	1.09774	0.548872	−	0.835906i	$-0.315057\pi$
$307$	19.2340	1.09774	0.548872	+	0.835906i	$0.315057\pi$
$308$	0	0
$309$	−2.40717	−0.136939
$310$	0	0
$311$	17.3193	0.982085	0.491043	−	0.871136i	$-0.336616\pi$
$311$	17.3193	0.982085	0.491043	+	0.871136i	$0.336616\pi$
$312$	0	0
$313$	14.5568	0.822798	0.411399	−	0.911455i	$-0.365040\pi$
$313$	14.5568	0.822798	0.411399	+	0.911455i	$0.365040\pi$
$314$	0	0
$315$	4.93416	0.278008
$316$	0	0
$317$	−29.5720	−1.66093	−0.830465	−	0.557071i	$-0.811925\pi$
$317$	−29.5720	−1.66093	−0.830465	+	0.557071i	$0.811925\pi$
$318$	0	0
$319$	−0.925197	−0.0518011
$320$	0	0
$321$	−7.33237	−0.409253
$322$	0	0
$323$	−2.64681	−0.147272
$324$	0	0
$325$	−4.86081	−0.269629
$326$	0	0
$327$	−10.9973	−0.608152
$328$	0	0
$329$	−14.8448	−0.818420
$330$	0	0
$331$	13.9612	0.767380	0.383690	−	0.923462i	$-0.374653\pi$
$331$	13.9612	0.767380	0.383690	+	0.923462i	$0.374653\pi$
$332$	0	0
$333$	14.3863	0.788367
$334$	0	0
$335$	10.7964	0.589871
$336$	0	0
$337$	14.3130	0.779678	0.389839	−	0.920883i	$-0.372531\pi$
$337$	14.3130	0.779678	0.389839	+	0.920883i	$0.372531\pi$
$338$	0	0
$339$	5.88790	0.319787
$340$	0	0
$341$	−8.42799	−0.456401
$342$	0	0
$343$	−20.1586	−1.08846
$344$	0	0
$345$	0.740987	0.0398934
$346$	0	0
$347$	1.07480	0.0576985	0.0288492	−	0.999584i	$-0.490816\pi$
$347$	1.07480	0.0576985	0.0288492	+	0.999584i	$0.490816\pi$
$348$	0	0
$349$	−8.34760	−0.446837	−0.223418	−	0.974723i	$-0.571722\pi$
$349$	−8.34760	−0.446837	−0.223418	+	0.974723i	$0.571722\pi$
$350$	0	0
$351$	22.0048	1.17453
$352$	0	0
$353$	−22.6773	−1.20699	−0.603495	−	0.797367i	$-0.706225\pi$
$353$	−22.6773	−1.20699	−0.603495	+	0.797367i	$0.706225\pi$
$354$	0	0
$355$	−5.85039	−0.310507
$356$	0	0
$357$	−0.869769	−0.0460330
$358$	0	0
$359$	−30.8608	−1.62877	−0.814386	−	0.580324i	$-0.802926\pi$
$359$	−30.8608	−1.62877	−0.814386	+	0.580324i	$0.802926\pi$
$360$	0	0
$361$	13.7368	0.722992
$362$	0	0
$363$	−8.73202	−0.458312
$364$	0	0
$365$	13.7562	0.720033
$366$	0	0
$367$	−9.07480	−0.473701	−0.236850	−	0.971546i	$-0.576115\pi$
$367$	−9.07480	−0.473701	−0.236850	+	0.971546i	$0.576115\pi$
$368$	0	0
$369$	−2.42799	−0.126396
$370$	0	0
$371$	6.51031	0.337999
$372$	0	0
$373$	−10.3130	−0.533986	−0.266993	−	0.963698i	$-0.586030\pi$
$373$	−10.3130	−0.533986	−0.266993	+	0.963698i	$0.586030\pi$
$374$	0	0
$375$	−0.860806	−0.0444518
$376$	0	0
$377$	−4.86081	−0.250344
$378$	0	0
$379$	−14.5180	−0.745741	−0.372870	−	0.927883i	$-0.621626\pi$
$379$	−14.5180	−0.745741	−0.372870	+	0.927883i	$0.621626\pi$
$380$	0	0
$381$	−17.2936	−0.885979
$382$	0	0
$383$	−8.70224	−0.444664	−0.222332	−	0.974971i	$-0.571367\pi$
$383$	−8.70224	−0.444664	−0.222332	+	0.974971i	$0.571367\pi$
$384$	0	0
$385$	2.02082	0.102991
$386$	0	0
$387$	−11.8802	−0.603904
$388$	0	0
$389$	18.9944	0.963055	0.481527	−	0.876431i	$-0.340082\pi$
$389$	18.9944	0.963055	0.481527	+	0.876431i	$0.340082\pi$
$390$	0	0
$391$	0.398207	0.0201382
$392$	0	0
$393$	2.75766	0.139106
$394$	0	0
$395$	14.4328	0.726194
$396$	0	0
$397$	−28.2742	−1.41904	−0.709522	−	0.704684i	$-0.751089\pi$
$397$	−28.2742	−1.41904	−0.709522	+	0.704684i	$0.751089\pi$
$398$	0	0
$399$	10.7577	0.538557
$400$	0	0
$401$	9.48679	0.473748	0.236874	−	0.971540i	$-0.423877\pi$
$401$	9.48679	0.473748	0.236874	+	0.971540i	$0.423877\pi$
$402$	0	0
$403$	−44.2791	−2.20570
$404$	0	0
$405$	−2.88018	−0.143117
$406$	0	0
$407$	5.89204	0.292058
$408$	0	0
$409$	14.5389	0.718900	0.359450	−	0.933164i	$-0.382964\pi$
$409$	14.5389	0.718900	0.359450	+	0.933164i	$0.382964\pi$
$410$	0	0
$411$	10.2230	0.504261
$412$	0	0
$413$	−5.96731	−0.293632
$414$	0	0
$415$	2.92520	0.143592
$416$	0	0
$417$	−12.6558	−0.619756
$418$	0	0
$419$	18.4030	0.899047	0.449523	−	0.893269i	$-0.351594\pi$
$419$	18.4030	0.899047	0.449523	+	0.893269i	$0.351594\pi$
$420$	0	0
$421$	−24.3297	−1.18576	−0.592878	−	0.805292i	$-0.702008\pi$
$421$	−24.3297	−1.18576	−0.592878	+	0.805292i	$0.702008\pi$
$422$	0	0
$423$	15.3532	0.746498
$424$	0	0
$425$	−0.462598	−0.0224393
$426$	0	0
$427$	−28.8954	−1.39835
$428$	0	0
$429$	3.87122	0.186904
$430$	0	0
$431$	33.7216	1.62431	0.812156	−	0.583440i	$-0.198294\pi$
$431$	33.7216	1.62431	0.812156	+	0.583440i	$0.198294\pi$
$432$	0	0
$433$	17.1440	0.823889	0.411944	−	0.911209i	$-0.364850\pi$
$433$	17.1440	0.823889	0.411944	+	0.911209i	$0.364850\pi$
$434$	0	0
$435$	−0.860806	−0.0412725
$436$	0	0
$437$	−4.92520	−0.235604
$438$	0	0
$439$	36.1205	1.72394	0.861968	−	0.506962i	$-0.169232\pi$
$439$	36.1205	1.72394	0.861968	+	0.506962i	$0.169232\pi$
$440$	0	0
$441$	5.03585	0.239802
$442$	0	0
$443$	−32.7833	−1.55758	−0.778791	−	0.627284i	$-0.784167\pi$
$443$	−32.7833	−1.55758	−0.778791	+	0.627284i	$0.784167\pi$
$444$	0	0
$445$	−17.5720	−0.832993
$446$	0	0
$447$	−8.49720	−0.401904
$448$	0	0
$449$	−27.9612	−1.31957	−0.659786	−	0.751453i	$-0.729353\pi$
$449$	−27.9612	−1.31957	−0.659786	+	0.751453i	$0.729353\pi$
$450$	0	0
$451$	−0.994404	−0.0468247
$452$	0	0
$453$	−1.48197	−0.0696292
$454$	0	0
$455$	10.6170	0.497734
$456$	0	0
$457$	31.2936	1.46385	0.731927	−	0.681383i	$-0.238621\pi$
$457$	31.2936	1.46385	0.731927	+	0.681383i	$0.238621\pi$
$458$	0	0
$459$	2.09418	0.0977479
$460$	0	0
$461$	23.8879	1.11257	0.556285	−	0.830991i	$-0.312226\pi$
$461$	23.8879	1.11257	0.556285	+	0.830991i	$0.312226\pi$
$462$	0	0
$463$	37.0844	1.72346	0.861730	−	0.507367i	$-0.169381\pi$
$463$	37.0844	1.72346	0.861730	+	0.507367i	$0.169381\pi$
$464$	0	0
$465$	−7.84143	−0.363638
$466$	0	0
$467$	−24.2147	−1.12052	−0.560261	−	0.828316i	$-0.689299\pi$
$467$	−24.2147	−1.12052	−0.560261	+	0.828316i	$0.689299\pi$
$468$	0	0
$469$	−23.5816	−1.08890
$470$	0	0
$471$	−2.20089	−0.101412
$472$	0	0
$473$	−4.86562	−0.223722
$474$	0	0
$475$	5.72161	0.262526
$476$	0	0
$477$	−6.73327	−0.308295
$478$	0	0
$479$	4.74099	0.216621	0.108311	−	0.994117i	$-0.465456\pi$
$479$	4.74099	0.216621	0.108311	+	0.994117i	$0.465456\pi$
$480$	0	0
$481$	30.9557	1.41146
$482$	0	0
$483$	−1.61847	−0.0736430
$484$	0	0
$485$	8.30403	0.377067
$486$	0	0
$487$	9.11837	0.413193	0.206596	−	0.978426i	$-0.433761\pi$
$487$	9.11837	0.413193	0.206596	+	0.978426i	$0.433761\pi$
$488$	0	0
$489$	−2.88645	−0.130530
$490$	0	0
$491$	8.55678	0.386162	0.193081	−	0.981183i	$-0.438152\pi$
$491$	8.55678	0.386162	0.193081	+	0.981183i	$0.438152\pi$
$492$	0	0
$493$	−0.462598	−0.0208344
$494$	0	0
$495$	−2.09003	−0.0939400
$496$	0	0
$497$	12.7785	0.573194
$498$	0	0
$499$	−29.0104	−1.29868	−0.649342	−	0.760496i	$-0.724956\pi$
$499$	−29.0104	−1.29868	−0.649342	+	0.760496i	$0.724956\pi$
$500$	0	0
$501$	−1.30548	−0.0583245
$502$	0	0
$503$	25.6412	1.14329	0.571643	−	0.820503i	$-0.306306\pi$
$503$	25.6412	1.14329	0.571643	+	0.820503i	$0.306306\pi$
$504$	0	0
$505$	13.3580	0.594424
$506$	0	0
$507$	9.14816	0.406284
$508$	0	0
$509$	18.0692	0.800904	0.400452	−	0.916318i	$-0.368853\pi$
$509$	18.0692	0.800904	0.400452	+	0.916318i	$0.368853\pi$
$510$	0	0
$511$	−30.0465	−1.32918
$512$	0	0
$513$	−25.9017	−1.14359
$514$	0	0
$515$	2.79641	0.123225
$516$	0	0
$517$	6.28802	0.276547
$518$	0	0
$519$	2.33382	0.102443
$520$	0	0
$521$	36.0263	1.57834	0.789171	−	0.614174i	$-0.210511\pi$
$521$	36.0263	1.57834	0.789171	+	0.614174i	$0.210511\pi$
$522$	0	0
$523$	10.6260	0.464642	0.232321	−	0.972639i	$-0.425368\pi$
$523$	10.6260	0.464642	0.232321	+	0.972639i	$0.425368\pi$
$524$	0	0
$525$	1.88018	0.0820578
$526$	0	0
$527$	−4.21400	−0.183565
$528$	0	0
$529$	−22.2590	−0.967783
$530$	0	0
$531$	6.17168	0.267828
$532$	0	0
$533$	−5.22441	−0.226294
$534$	0	0
$535$	8.51803	0.368266
$536$	0	0
$537$	11.4653	0.494764
$538$	0	0
$539$	2.06247	0.0888370
$540$	0	0
$541$	38.1759	1.64131	0.820655	−	0.571423i	$-0.193609\pi$
$541$	38.1759	1.64131	0.820655	+	0.571423i	$0.193609\pi$
$542$	0	0
$543$	10.9973	0.471939
$544$	0	0
$545$	12.7756	0.547246
$546$	0	0
$547$	13.2549	0.566737	0.283369	−	0.959011i	$-0.408548\pi$
$547$	13.2549	0.566737	0.283369	+	0.959011i	$0.408548\pi$
$548$	0	0
$549$	29.8850	1.27546
$550$	0	0
$551$	5.72161	0.243749
$552$	0	0
$553$	−31.5243	−1.34055
$554$	0	0
$555$	5.48197	0.232697
$556$	0	0
$557$	6.15442	0.260771	0.130386	−	0.991463i	$-0.458378\pi$
$557$	6.15442	0.260771	0.130386	+	0.991463i	$0.458378\pi$
$558$	0	0
$559$	−25.5630	−1.08120
$560$	0	0
$561$	0.368420	0.0155547
$562$	0	0
$563$	−28.0561	−1.18242	−0.591212	−	0.806516i	$-0.701350\pi$
$563$	−28.0561	−1.18242	−0.591212	+	0.806516i	$0.701350\pi$
$564$	0	0
$565$	−6.83998	−0.287760
$566$	0	0
$567$	6.29092	0.264194
$568$	0	0
$569$	46.9348	1.96761	0.983805	−	0.179241i	$-0.0573641\pi$
$569$	46.9348	1.96761	0.983805	+	0.179241i	$0.0573641\pi$
$570$	0	0
$571$	1.69597	0.0709742	0.0354871	−	0.999370i	$-0.488702\pi$
$571$	1.69597	0.0709742	0.0354871	+	0.999370i	$0.488702\pi$
$572$	0	0
$573$	11.8594	0.495432
$574$	0	0
$575$	−0.860806	−0.0358981
$576$	0	0
$577$	−45.6371	−1.89990	−0.949948	−	0.312408i	$-0.898864\pi$
$577$	−45.6371	−1.89990	−0.949948	+	0.312408i	$0.898864\pi$
$578$	0	0
$579$	10.8131	0.449377
$580$	0	0
$581$	−6.38924	−0.265071
$582$	0	0
$583$	−2.75766	−0.114211
$584$	0	0
$585$	−10.9806	−0.453993
$586$	0	0
$587$	0.107958	0.00445589	0.00222794	−	0.999998i	$-0.499291\pi$
$587$	0.107958	0.00445589	0.00222794	+	0.999998i	$0.499291\pi$
$588$	0	0
$589$	52.1205	2.14759
$590$	0	0
$591$	−4.32071	−0.177730
$592$	0	0
$593$	0.577601	0.0237192	0.0118596	−	0.999930i	$-0.496225\pi$
$593$	0.577601	0.0237192	0.0118596	+	0.999930i	$0.496225\pi$
$594$	0	0
$595$	1.01041	0.0414229
$596$	0	0
$597$	−0.317138	−0.0129796
$598$	0	0
$599$	24.8102	1.01372	0.506859	−	0.862029i	$-0.330807\pi$
$599$	24.8102	1.01372	0.506859	+	0.862029i	$0.330807\pi$
$600$	0	0
$601$	−23.7216	−0.967625	−0.483812	−	0.875172i	$-0.660748\pi$
$601$	−23.7216	−0.967625	−0.483812	+	0.875172i	$0.660748\pi$
$602$	0	0
$603$	24.3892	0.993207
$604$	0	0
$605$	10.1440	0.412413
$606$	0	0
$607$	−24.0484	−0.976094	−0.488047	−	0.872817i	$-0.662290\pi$
$607$	−24.0484	−0.976094	−0.488047	+	0.872817i	$0.662290\pi$
$608$	0	0
$609$	1.88018	0.0761888
$610$	0	0
$611$	33.0361	1.33650
$612$	0	0
$613$	−9.79786	−0.395732	−0.197866	−	0.980229i	$-0.563401\pi$
$613$	−9.79786	−0.395732	−0.197866	+	0.980229i	$0.563401\pi$
$614$	0	0
$615$	−0.925197	−0.0373075
$616$	0	0
$617$	28.9419	1.16516	0.582578	−	0.812775i	$-0.302044\pi$
$617$	28.9419	1.16516	0.582578	+	0.812775i	$0.302044\pi$
$618$	0	0
$619$	−19.7604	−0.794236	−0.397118	−	0.917768i	$-0.629990\pi$
$619$	−19.7604	−0.794236	−0.397118	+	0.917768i	$0.629990\pi$
$620$	0	0
$621$	3.89686	0.156376
$622$	0	0
$623$	38.3810	1.53770
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.55678	−0.181980
$628$	0	0
$629$	2.94602	0.117466
$630$	0	0
$631$	11.6829	0.465087	0.232544	−	0.972586i	$-0.425295\pi$
$631$	11.6829	0.465087	0.232544	+	0.972586i	$0.425295\pi$
$632$	0	0
$633$	13.8325	0.549791
$634$	0	0
$635$	20.0900	0.797249
$636$	0	0
$637$	10.8358	0.429331
$638$	0	0
$639$	−13.2161	−0.522822
$640$	0	0
$641$	−34.3684	−1.35747	−0.678735	−	0.734383i	$-0.737472\pi$
$641$	−34.3684	−1.35747	−0.678735	+	0.734383i	$0.737472\pi$
$642$	0	0
$643$	10.9073	0.430141	0.215070	−	0.976599i	$-0.431002\pi$
$643$	10.9073	0.430141	0.215070	+	0.976599i	$0.431002\pi$
$644$	0	0
$645$	−4.52699	−0.178250
$646$	0	0
$647$	16.8269	0.661533	0.330766	−	0.943713i	$-0.392693\pi$
$647$	16.8269	0.661533	0.330766	+	0.943713i	$0.392693\pi$
$648$	0	0
$649$	2.52766	0.0992193
$650$	0	0
$651$	17.1273	0.671273
$652$	0	0
$653$	−40.8961	−1.60039	−0.800194	−	0.599742i	$-0.795270\pi$
$653$	−40.8961	−1.60039	−0.800194	+	0.599742i	$0.795270\pi$
$654$	0	0
$655$	−3.20359	−0.125174
$656$	0	0
$657$	31.0755	1.21237
$658$	0	0
$659$	33.9612	1.32294	0.661471	−	0.749971i	$-0.269932\pi$
$659$	33.9612	1.32294	0.661471	+	0.749971i	$0.269932\pi$
$660$	0	0
$661$	−39.4224	−1.53335	−0.766677	−	0.642033i	$-0.778091\pi$
$661$	−39.4224	−1.53335	−0.766677	+	0.642033i	$0.778091\pi$
$662$	0	0
$663$	1.93561	0.0751728
$664$	0	0
$665$	−12.4972	−0.484621
$666$	0	0
$667$	−0.860806	−0.0333305
$668$	0	0
$669$	15.9959	0.618435
$670$	0	0
$671$	12.2396	0.472506
$672$	0	0
$673$	−17.8116	−0.686588	−0.343294	−	0.939228i	$-0.611543\pi$
$673$	−17.8116	−0.686588	−0.343294	+	0.939228i	$0.611543\pi$
$674$	0	0
$675$	−4.52699	−0.174244
$676$	0	0
$677$	−3.09563	−0.118975	−0.0594873	−	0.998229i	$-0.518947\pi$
$677$	−3.09563	−0.118975	−0.0594873	+	0.998229i	$0.518947\pi$
$678$	0	0
$679$	−18.1377	−0.696063
$680$	0	0
$681$	−0.128782	−0.00493494
$682$	0	0
$683$	2.62598	0.100480	0.0502402	−	0.998737i	$-0.484001\pi$
$683$	2.62598	0.100480	0.0502402	+	0.998737i	$0.484001\pi$
$684$	0	0
$685$	−11.8760	−0.453760
$686$	0	0
$687$	2.04232	0.0779193
$688$	0	0
$689$	−14.4882	−0.551958
$690$	0	0
$691$	−27.4391	−1.04383	−0.521916	−	0.852997i	$-0.674783\pi$
$691$	−27.4391	−1.04383	−0.521916	+	0.852997i	$0.674783\pi$
$692$	0	0
$693$	4.56507	0.173413
$694$	0	0
$695$	14.7022	0.557688
$696$	0	0
$697$	−0.497202	−0.0188329
$698$	0	0
$699$	6.27839	0.237470
$700$	0	0
$701$	−7.90168	−0.298442	−0.149221	−	0.988804i	$-0.547677\pi$
$701$	−7.90168	−0.298442	−0.149221	+	0.988804i	$0.547677\pi$
$702$	0	0
$703$	−36.4376	−1.37427
$704$	0	0
$705$	5.85039	0.220339
$706$	0	0
$707$	−29.1767	−1.09730
$708$	0	0
$709$	1.39194	0.0522755	0.0261377	−	0.999658i	$-0.491679\pi$
$709$	1.39194	0.0522755	0.0261377	+	0.999658i	$0.491679\pi$
$710$	0	0
$711$	32.6039	1.22274
$712$	0	0
$713$	−7.84143	−0.293664
$714$	0	0
$715$	−4.49720	−0.168186
$716$	0	0
$717$	−22.2009	−0.829107
$718$	0	0
$719$	−19.2728	−0.718754	−0.359377	−	0.933192i	$-0.617011\pi$
$719$	−19.2728	−0.718754	−0.359377	+	0.933192i	$0.617011\pi$
$720$	0	0
$721$	−6.10796	−0.227472
$722$	0	0
$723$	−1.64826	−0.0612994
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	−7.16484	−0.265729	−0.132865	−	0.991134i	$-0.542418\pi$
$727$	−7.16484	−0.265729	−0.132865	+	0.991134i	$0.542418\pi$
$728$	0	0
$729$	5.18421	0.192008
$730$	0	0
$731$	−2.43281	−0.0899808
$732$	0	0
$733$	−25.9821	−0.959670	−0.479835	−	0.877359i	$-0.659303\pi$
$733$	−25.9821	−0.959670	−0.479835	+	0.877359i	$0.659303\pi$
$734$	0	0
$735$	1.91893	0.0707808
$736$	0	0
$737$	9.98881	0.367943
$738$	0	0
$739$	14.4072	0.529976	0.264988	−	0.964252i	$-0.414632\pi$
$739$	14.4072	0.529976	0.264988	+	0.964252i	$0.414632\pi$
$740$	0	0
$741$	−23.9404	−0.879474
$742$	0	0
$743$	−38.4972	−1.41233	−0.706163	−	0.708050i	$-0.749575\pi$
$743$	−38.4972	−1.41233	−0.706163	+	0.708050i	$0.749575\pi$
$744$	0	0
$745$	9.87122	0.361653
$746$	0	0
$747$	6.60806	0.241776
$748$	0	0
$749$	−18.6052	−0.679818
$750$	0	0
$751$	29.4737	1.07551	0.537755	−	0.843101i	$-0.319273\pi$
$751$	29.4737	1.07551	0.537755	+	0.843101i	$0.319273\pi$
$752$	0	0
$753$	−7.63158	−0.278110
$754$	0	0
$755$	1.72161	0.0626559
$756$	0	0
$757$	51.9612	1.88856	0.944282	−	0.329138i	$-0.106758\pi$
$757$	51.9612	1.88856	0.944282	+	0.329138i	$0.106758\pi$
$758$	0	0
$759$	0.685559	0.0248842
$760$	0	0
$761$	−8.40302	−0.304609	−0.152305	−	0.988334i	$-0.548670\pi$
$761$	−8.40302	−0.304609	−0.152305	+	0.988334i	$0.548670\pi$
$762$	0	0
$763$	−27.9046	−1.01021
$764$	0	0
$765$	−1.04502	−0.0377826
$766$	0	0
$767$	13.2798	0.479507
$768$	0	0
$769$	1.31444	0.0474000	0.0237000	−	0.999719i	$-0.492455\pi$
$769$	1.31444	0.0474000	0.0237000	+	0.999719i	$0.492455\pi$
$770$	0	0
$771$	23.3324	0.840295
$772$	0	0
$773$	−18.7785	−0.675415	−0.337708	−	0.941251i	$-0.609652\pi$
$773$	−18.7785	−0.675415	−0.337708	+	0.941251i	$0.609652\pi$
$774$	0	0
$775$	9.10941	0.327220
$776$	0	0
$777$	−11.9738	−0.429557
$778$	0	0
$779$	6.14961	0.220333
$780$	0	0
$781$	−5.41277	−0.193684
$782$	0	0
$783$	−4.52699	−0.161781
$784$	0	0
$785$	2.55678	0.0912553
$786$	0	0
$787$	−28.4764	−1.01507	−0.507537	−	0.861630i	$-0.669444\pi$
$787$	−28.4764	−1.01507	−0.507537	+	0.861630i	$0.669444\pi$
$788$	0	0
$789$	−20.9586	−0.746144
$790$	0	0
$791$	14.9400	0.531204
$792$	0	0
$793$	64.3047	2.28353
$794$	0	0
$795$	−2.56574	−0.0909974
$796$	0	0
$797$	−17.7604	−0.629104	−0.314552	−	0.949240i	$-0.601854\pi$
$797$	−17.7604	−0.629104	−0.314552	+	0.949240i	$0.601854\pi$
$798$	0	0
$799$	3.14401	0.111227
$800$	0	0
$801$	−39.6954	−1.40257
$802$	0	0
$803$	12.7272	0.449133
$804$	0	0
$805$	1.88018	0.0662677
$806$	0	0
$807$	−16.5449	−0.582409
$808$	0	0
$809$	−36.8269	−1.29476	−0.647382	−	0.762166i	$-0.724136\pi$
$809$	−36.8269	−1.29476	−0.647382	+	0.762166i	$0.724136\pi$
$810$	0	0
$811$	26.9598	0.946687	0.473343	−	0.880878i	$-0.343047\pi$
$811$	26.9598	0.946687	0.473343	+	0.880878i	$0.343047\pi$
$812$	0	0
$813$	−20.1801	−0.707746
$814$	0	0
$815$	3.35319	0.117457
$816$	0	0
$817$	30.0900	1.05272
$818$	0	0
$819$	23.9840	0.838069
$820$	0	0
$821$	37.4945	1.30857	0.654284	−	0.756249i	$-0.272970\pi$
$821$	37.4945	1.30857	0.654284	+	0.756249i	$0.272970\pi$
$822$	0	0
$823$	12.1496	0.423509	0.211754	−	0.977323i	$-0.432082\pi$
$823$	12.1496	0.423509	0.211754	+	0.977323i	$0.432082\pi$
$824$	0	0
$825$	−0.796415	−0.0277276
$826$	0	0
$827$	−22.7327	−0.790493	−0.395247	−	0.918575i	$-0.629341\pi$
$827$	−22.7327	−0.790493	−0.395247	+	0.918575i	$0.629341\pi$
$828$	0	0
$829$	−19.7354	−0.685438	−0.342719	−	0.939438i	$-0.611348\pi$
$829$	−19.7354	−0.685438	−0.342719	+	0.939438i	$0.611348\pi$
$830$	0	0
$831$	15.9071	0.551810
$832$	0	0
$833$	1.03124	0.0357302
$834$	0	0
$835$	1.51658	0.0524833
$836$	0	0
$837$	−41.2382	−1.42540
$838$	0	0
$839$	40.7368	1.40639	0.703196	−	0.710996i	$-0.251756\pi$
$839$	40.7368	1.40639	0.703196	+	0.710996i	$0.251756\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	21.7431	0.748872
$844$	0	0
$845$	−10.6274	−0.365595
$846$	0	0
$847$	−22.1567	−0.761312
$848$	0	0
$849$	−12.3684	−0.424483
$850$	0	0
$851$	5.48197	0.187920
$852$	0	0
$853$	41.2549	1.41254	0.706270	−	0.707943i	$-0.250377\pi$
$853$	41.2549	1.41254	0.706270	+	0.707943i	$0.250377\pi$
$854$	0	0
$855$	12.9252	0.442032
$856$	0	0
$857$	31.7216	1.08359	0.541795	−	0.840511i	$-0.317745\pi$
$857$	31.7216	1.08359	0.541795	+	0.840511i	$0.317745\pi$
$858$	0	0
$859$	−6.20089	−0.211572	−0.105786	−	0.994389i	$-0.533736\pi$
$859$	−6.20089	−0.211572	−0.105786	+	0.994389i	$0.533736\pi$
$860$	0	0
$861$	2.02082	0.0688695
$862$	0	0
$863$	2.41199	0.0821050	0.0410525	−	0.999157i	$-0.486929\pi$
$863$	2.41199	0.0821050	0.0410525	+	0.999157i	$0.486929\pi$
$864$	0	0
$865$	−2.71120	−0.0921836
$866$	0	0
$867$	−14.4495	−0.490730
$868$	0	0
$869$	13.3532	0.452976
$870$	0	0
$871$	52.4793	1.77819
$872$	0	0
$873$	18.7589	0.634893
$874$	0	0
$875$	−2.18421	−0.0738398
$876$	0	0
$877$	−26.2057	−0.884904	−0.442452	−	0.896792i	$-0.645891\pi$
$877$	−26.2057	−0.884904	−0.442452	+	0.896792i	$0.645891\pi$
$878$	0	0
$879$	−13.9612	−0.470901
$880$	0	0
$881$	28.9973	0.976944	0.488472	−	0.872580i	$-0.337554\pi$
$881$	28.9973	0.976944	0.488472	+	0.872580i	$0.337554\pi$
$882$	0	0
$883$	32.5180	1.09432	0.547159	−	0.837028i	$-0.315709\pi$
$883$	32.5180	1.09432	0.547159	+	0.837028i	$0.315709\pi$
$884$	0	0
$885$	2.35174	0.0790529
$886$	0	0
$887$	−9.98207	−0.335165	−0.167583	−	0.985858i	$-0.553596\pi$
$887$	−9.98207	−0.335165	−0.167583	+	0.985858i	$0.553596\pi$
$888$	0	0
$889$	−43.8809	−1.47172
$890$	0	0
$891$	−2.66473	−0.0892720
$892$	0	0
$893$	−38.8864	−1.30129
$894$	0	0
$895$	−13.3193	−0.445214
$896$	0	0
$897$	3.60179	0.120260
$898$	0	0
$899$	9.10941	0.303816
$900$	0	0
$901$	−1.37883	−0.0459356
$902$	0	0
$903$	9.88790	0.329049
$904$	0	0
$905$	−12.7756	−0.424675
$906$	0	0
$907$	−51.0319	−1.69449	−0.847243	−	0.531205i	$-0.821739\pi$
$907$	−51.0319	−1.69449	−0.847243	+	0.531205i	$0.821739\pi$
$908$	0	0
$909$	30.1759	1.00087
$910$	0	0
$911$	55.7264	1.84630	0.923149	−	0.384441i	$-0.125606\pi$
$911$	55.7264	1.84630	0.923149	+	0.384441i	$0.125606\pi$
$912$	0	0
$913$	2.70638	0.0895682
$914$	0	0
$915$	11.3878	0.376469
$916$	0	0
$917$	6.99730	0.231071
$918$	0	0
$919$	−28.5568	−0.942001	−0.471001	−	0.882133i	$-0.656107\pi$
$919$	−28.5568	−0.942001	−0.471001	+	0.882133i	$0.656107\pi$
$920$	0	0
$921$	16.5568	0.545564
$922$	0	0
$923$	−28.4376	−0.936036
$924$	0	0
$925$	−6.36842	−0.209392
$926$	0	0
$927$	6.31714	0.207482
$928$	0	0
$929$	−27.5470	−0.903789	−0.451894	−	0.892071i	$-0.649252\pi$
$929$	−27.5470	−0.903789	−0.451894	+	0.892071i	$0.649252\pi$
$930$	0	0
$931$	−12.7548	−0.418021
$932$	0	0
$933$	14.9085	0.488083
$934$	0	0
$935$	−0.427995	−0.0139969
$936$	0	0
$937$	−18.1884	−0.594188	−0.297094	−	0.954848i	$-0.596017\pi$
$937$	−18.1884	−0.594188	−0.297094	+	0.954848i	$0.596017\pi$
$938$	0	0
$939$	12.5306	0.408919
$940$	0	0
$941$	−5.98207	−0.195010	−0.0975050	−	0.995235i	$-0.531086\pi$
$941$	−5.98207	−0.195010	−0.0975050	+	0.995235i	$0.531086\pi$
$942$	0	0
$943$	−0.925197	−0.0301286
$944$	0	0
$945$	9.88790	0.321653
$946$	0	0
$947$	−2.02564	−0.0658245	−0.0329122	−	0.999458i	$-0.510478\pi$
$947$	−2.02564	−0.0658245	−0.0329122	+	0.999458i	$0.510478\pi$
$948$	0	0
$949$	66.8663	2.17057
$950$	0	0
$951$	−25.4558	−0.825459
$952$	0	0
$953$	−3.55118	−0.115034	−0.0575170	−	0.998345i	$-0.518318\pi$
$953$	−3.55118	−0.115034	−0.0575170	+	0.998345i	$0.518318\pi$
$954$	0	0
$955$	−13.7770	−0.445815
$956$	0	0
$957$	−0.796415	−0.0257444
$958$	0	0
$959$	25.9398	0.837638
$960$	0	0
$961$	51.9813	1.67682
$962$	0	0
$963$	19.2423	0.620076
$964$	0	0
$965$	−12.5616	−0.404372
$966$	0	0
$967$	−23.4224	−0.753214	−0.376607	−	0.926373i	$-0.622909\pi$
$967$	−23.4224	−0.753214	−0.376607	+	0.926373i	$0.622909\pi$
$968$	0	0
$969$	−2.27839	−0.0731924
$970$	0	0
$971$	38.6385	1.23997	0.619984	−	0.784614i	$-0.287139\pi$
$971$	38.6385	1.23997	0.619984	+	0.784614i	$0.287139\pi$
$972$	0	0
$973$	−32.1128	−1.02949
$974$	0	0
$975$	−4.18421	−0.134002
$976$	0	0
$977$	4.02082	0.128638	0.0643188	−	0.997929i	$-0.479513\pi$
$977$	4.02082	0.128638	0.0643188	+	0.997929i	$0.479513\pi$
$978$	0	0
$979$	−16.2576	−0.519594
$980$	0	0
$981$	28.8602	0.921436
$982$	0	0
$983$	54.1384	1.72675	0.863374	−	0.504565i	$-0.168347\pi$
$983$	54.1384	1.72675	0.863374	+	0.504565i	$0.168347\pi$
$984$	0	0
$985$	5.01938	0.159931
$986$	0	0
$987$	−12.7785	−0.406744
$988$	0	0
$989$	−4.52699	−0.143950
$990$	0	0
$991$	1.78119	0.0565812	0.0282906	−	0.999600i	$-0.490994\pi$
$991$	1.78119	0.0565812	0.0282906	+	0.999600i	$0.490994\pi$
$992$	0	0
$993$	12.0179	0.381377
$994$	0	0
$995$	0.368420	0.0116797
$996$	0	0
$997$	−3.72161	−0.117865	−0.0589323	−	0.998262i	$-0.518770\pi$
$997$	−3.72161	−0.117865	−0.0589323	+	0.998262i	$0.518770\pi$
$998$	0	0
$999$	28.8298	0.912134

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4640.2.a.j.1.3	✓	3
4.3	odd	2		4640.2.a.k.1.1	yes	3
8.3	odd	2		9280.2.a.bg.1.3		3
8.5	even	2		9280.2.a.bx.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4640.2.a.j.1.3	✓	3	1.1	even	1	trivial
4640.2.a.k.1.1	yes	3	4.3	odd	2
9280.2.a.bg.1.3		3	8.3	odd	2
9280.2.a.bx.1.1		3	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4640 = 2^{5} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4640.a (trivial)

\(f(q)\)	\(=\)	\(q+0.860806 q^{3} -1.00000 q^{5} +2.18421 q^{7} -2.25901 q^{9} -0.925197 q^{11} -4.86081 q^{13} -0.860806 q^{15} -0.462598 q^{17} +5.72161 q^{19} +1.88018 q^{21} -0.860806 q^{23} +1.00000 q^{25} -4.52699 q^{27} +1.00000 q^{29} +9.10941 q^{31} -0.796415 q^{33} -2.18421 q^{35} -6.36842 q^{37} -4.18421 q^{39} +1.07480 q^{41} +5.25901 q^{43} +2.25901 q^{45} -6.79641 q^{47} -2.22923 q^{49} -0.398207 q^{51} +2.98062 q^{53} +0.925197 q^{55} +4.92520 q^{57} -2.73202 q^{59} -13.2292 q^{61} -4.93416 q^{63} +4.86081 q^{65} -10.7964 q^{67} -0.740987 q^{69} +5.85039 q^{71} -13.7562 q^{73} +0.860806 q^{75} -2.02082 q^{77} -14.4328 q^{79} +2.88018 q^{81} -2.92520 q^{83} +0.462598 q^{85} +0.860806 q^{87} +17.5720 q^{89} -10.6170 q^{91} +7.84143 q^{93} -5.72161 q^{95} -8.30403 q^{97} +2.09003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 7 q^{7} + 2 q^{9} + 2 q^{11} - 9 q^{13} + 3 q^{15} + q^{17} + 6 q^{19} + 20 q^{21} + 3 q^{23} + 3 q^{25} - 12 q^{27} + 3 q^{29} + 9 q^{31} + 4 q^{33} + 7 q^{35} + 8 q^{37} + q^{39}+ \cdots - 32 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 - 7 * q^7 + 2 * q^9 + 2 * q^11 - 9 * q^13 + 3 * q^15 + q^17 + 6 * q^19 + 20 * q^21 + 3 * q^23 + 3 * q^25 - 12 * q^27 + 3 * q^29 + 9 * q^31 + 4 * q^33 + 7 * q^35 + 8 * q^37 + q^39 + 8 * q^41 + 7 * q^43 - 2 * q^45 - 14 * q^47 + 26 * q^49 + 2 * q^51 - 11 * q^53 - 2 * q^55 + 10 * q^57 - q^59 - 7 * q^61 - 26 * q^63 + 9 * q^65 - 26 * q^67 - 11 * q^69 + 8 * q^71 - 7 * q^73 - 3 * q^75 - 14 * q^77 - 17 * q^79 + 23 * q^81 - 4 * q^83 - q^85 - 3 * q^87 + 32 * q^89 + 8 * q^91 - 2 * q^93 - 6 * q^95 + 3 * q^97 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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