LMFDB - Embedded newform 1160.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1160,2,Mod(1,1160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1160, 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("1160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1160 = 2^{3} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1160.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:	$9.26264663447$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.6083172.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 13x^{3} + 10x^{2} + 40x - 28 $ x^5 - x^4 - 13x^3 + 10x^2 + 40*x - 28
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.36347$ of defining polynomial
Character	$\chi$	$=$	1160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.36347 q^{3} +1.00000 q^{5} +2.16993 q^{7} +2.58601 q^{9} +1.75593 q^{11} +2.60754 q^{13} +2.36347 q^{15} -5.49204 q^{17} +5.90603 q^{19} +5.12856 q^{21} -7.29849 q^{23} +1.00000 q^{25} -0.978465 q^{27} -1.00000 q^{29} -0.169927 q^{31} +4.15010 q^{33} +2.16993 q^{35} +3.75593 q^{37} +6.16285 q^{39} -10.3889 q^{41} +4.55702 q^{43} +2.58601 q^{45} +1.17909 q^{47} -2.29142 q^{49} -12.9803 q^{51} +11.8248 q^{53} +1.75593 q^{55} +13.9587 q^{57} -4.11941 q^{59} +1.00537 q^{61} +5.61144 q^{63} +2.60754 q^{65} -8.48288 q^{67} -17.2498 q^{69} +6.10703 q^{71} -1.72903 q^{73} +2.36347 q^{75} +3.81025 q^{77} -2.02544 q^{79} -10.0706 q^{81} +1.86296 q^{83} -5.49204 q^{85} -2.36347 q^{87} +16.5970 q^{89} +5.65817 q^{91} -0.401618 q^{93} +5.90603 q^{95} +0.464516 q^{97} +4.54085 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 5 q^{5} - q^{7} + 12 q^{9} - 4 q^{11} + 13 q^{13} - q^{15} + 3 q^{17} + 8 q^{21} - 7 q^{23} + 5 q^{25} - 4 q^{27} - 5 q^{29} + 11 q^{31} + 4 q^{33} - q^{35} + 6 q^{37} + 21 q^{39} + 16 q^{41}+ \cdots + 10 q^{99}+O(q^{100}) $ 5 * q - q^3 + 5 * q^5 - q^7 + 12 * q^9 - 4 * q^11 + 13 * q^13 - q^15 + 3 * q^17 + 8 * q^21 - 7 * q^23 + 5 * q^25 - 4 * q^27 - 5 * q^29 + 11 * q^31 + 4 * q^33 - q^35 + 6 * q^37 + 21 * q^39 + 16 * q^41 + 9 * q^43 + 12 * q^45 + 2 * q^47 + 16 * q^49 + 2 * q^51 + 7 * q^53 - 4 * q^55 + 2 * q^57 + 5 * q^59 + 7 * q^61 - 28 * q^63 + 13 * q^65 - 4 * q^67 + 23 * q^69 + 12 * q^71 + 7 * q^73 - q^75 + 26 * q^77 + 45 * q^79 - 7 * q^81 - 22 * q^83 + 3 * q^85 + q^87 + 24 * q^89 - 20 * q^91 - 10 * q^93 + 17 * q^97 + 10 * 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
$3$	2.36347	1.36455	0.682276	−	0.731095i	$-0.260990\pi$
$3$	2.36347	1.36455	0.682276	+	0.731095i	$0.260990\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.16993	0.820155	0.410078	−	0.912051i	$-0.365502\pi$
$7$	2.16993	0.820155	0.410078	+	0.912051i	$0.365502\pi$
$8$	0	0
$9$	2.58601	0.862002
$10$	0	0
$11$	1.75593	0.529434	0.264717	−	0.964326i	$-0.414722\pi$
$11$	1.75593	0.529434	0.264717	+	0.964326i	$0.414722\pi$
$12$	0	0
$13$	2.60754	0.723202	0.361601	−	0.932333i	$-0.382230\pi$
$13$	2.60754	0.723202	0.361601	+	0.932333i	$0.382230\pi$
$14$	0	0
$15$	2.36347	0.610246
$16$	0	0
$17$	−5.49204	−1.33201	−0.666007	−	0.745945i	$-0.731998\pi$
$17$	−5.49204	−1.33201	−0.666007	+	0.745945i	$0.731998\pi$
$18$	0	0
$19$	5.90603	1.35494	0.677468	−	0.735552i	$-0.263077\pi$
$19$	5.90603	1.35494	0.677468	+	0.735552i	$0.263077\pi$
$20$	0	0
$21$	5.12856	1.11914
$22$	0	0
$23$	−7.29849	−1.52184	−0.760920	−	0.648845i	$-0.775252\pi$
$23$	−7.29849	−1.52184	−0.760920	+	0.648845i	$0.775252\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.978465	−0.188306
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−0.169927	−0.0305198	−0.0152599	−	0.999884i	$-0.504858\pi$
$31$	−0.169927	−0.0305198	−0.0152599	+	0.999884i	$0.504858\pi$
$32$	0	0
$33$	4.15010	0.722440
$34$	0	0
$35$	2.16993	0.366785
$36$	0	0
$37$	3.75593	0.617472	0.308736	−	0.951148i	$-0.400094\pi$
$37$	3.75593	0.617472	0.308736	+	0.951148i	$0.400094\pi$
$38$	0	0
$39$	6.16285	0.986846
$40$	0	0
$41$	−10.3889	−1.62248	−0.811238	−	0.584717i	$-0.801206\pi$
$41$	−10.3889	−1.62248	−0.811238	+	0.584717i	$0.801206\pi$
$42$	0	0
$43$	4.55702	0.694939	0.347469	−	0.937691i	$-0.387041\pi$
$43$	4.55702	0.694939	0.347469	+	0.937691i	$0.387041\pi$
$44$	0	0
$45$	2.58601	0.385499
$46$	0	0
$47$	1.17909	0.171987	0.0859937	−	0.996296i	$-0.472594\pi$
$47$	1.17909	0.171987	0.0859937	+	0.996296i	$0.472594\pi$
$48$	0	0
$49$	−2.29142	−0.327345
$50$	0	0
$51$	−12.9803	−1.81760
$52$	0	0
$53$	11.8248	1.62426	0.812132	−	0.583474i	$-0.198307\pi$
$53$	11.8248	1.62426	0.812132	+	0.583474i	$0.198307\pi$
$54$	0	0
$55$	1.75593	0.236770
$56$	0	0
$57$	13.9587	1.84888
$58$	0	0
$59$	−4.11941	−0.536301	−0.268150	−	0.963377i	$-0.586412\pi$
$59$	−4.11941	−0.536301	−0.268150	+	0.963377i	$0.586412\pi$
$60$	0	0
$61$	1.00537	0.128724	0.0643620	−	0.997927i	$-0.479499\pi$
$61$	1.00537	0.128724	0.0643620	+	0.997927i	$0.479499\pi$
$62$	0	0
$63$	5.61144	0.706975
$64$	0	0
$65$	2.60754	0.323426
$66$	0	0
$67$	−8.48288	−1.03635	−0.518174	−	0.855275i	$-0.673388\pi$
$67$	−8.48288	−1.03635	−0.518174	+	0.855275i	$0.673388\pi$
$68$	0	0
$69$	−17.2498	−2.07663
$70$	0	0
$71$	6.10703	0.724771	0.362386	−	0.932028i	$-0.381962\pi$
$71$	6.10703	0.724771	0.362386	+	0.932028i	$0.381962\pi$
$72$	0	0
$73$	−1.72903	−0.202368	−0.101184	−	0.994868i	$-0.532263\pi$
$73$	−1.72903	−0.202368	−0.101184	+	0.994868i	$0.532263\pi$
$74$	0	0
$75$	2.36347	0.272910
$76$	0	0
$77$	3.81025	0.434218
$78$	0	0
$79$	−2.02544	−0.227880	−0.113940	−	0.993488i	$-0.536347\pi$
$79$	−2.02544	−0.227880	−0.113940	+	0.993488i	$0.536347\pi$
$80$	0	0
$81$	−10.0706	−1.11895
$82$	0	0
$83$	1.86296	0.204487	0.102243	−	0.994759i	$-0.467398\pi$
$83$	1.86296	0.204487	0.102243	+	0.994759i	$0.467398\pi$
$84$	0	0
$85$	−5.49204	−0.595695
$86$	0	0
$87$	−2.36347	−0.253391
$88$	0	0
$89$	16.5970	1.75928	0.879638	−	0.475643i	$-0.157785\pi$
$89$	16.5970	1.75928	0.879638	+	0.475643i	$0.157785\pi$
$90$	0	0
$91$	5.65817	0.593138
$92$	0	0
$93$	−0.401618	−0.0416458
$94$	0	0
$95$	5.90603	0.605946
$96$	0	0
$97$	0.464516	0.0471644	0.0235822	−	0.999722i	$-0.492493\pi$
$97$	0.464516	0.0471644	0.0235822	+	0.999722i	$0.492493\pi$
$98$	0	0
$99$	4.54085	0.456373
$100$	0	0
$101$	−12.5663	−1.25039	−0.625196	−	0.780468i	$-0.714981\pi$
$101$	−12.5663	−1.25039	−0.625196	+	0.780468i	$0.714981\pi$
$102$	0	0
$103$	−12.4829	−1.22997	−0.614987	−	0.788537i	$-0.710839\pi$
$103$	−12.4829	−1.22997	−0.614987	+	0.788537i	$0.710839\pi$
$104$	0	0
$105$	5.12856	0.500497
$106$	0	0
$107$	−8.29312	−0.801727	−0.400863	−	0.916138i	$-0.631290\pi$
$107$	−8.29312	−0.801727	−0.400863	+	0.916138i	$0.631290\pi$
$108$	0	0
$109$	18.3889	1.76134	0.880669	−	0.473732i	$-0.157093\pi$
$109$	18.3889	1.76134	0.880669	+	0.473732i	$0.157093\pi$
$110$	0	0
$111$	8.87705	0.842572
$112$	0	0
$113$	−0.0236192	−0.00222190	−0.00111095	−	0.999999i	$-0.500354\pi$
$113$	−0.0236192	−0.00222190	−0.00111095	+	0.999999i	$0.500354\pi$
$114$	0	0
$115$	−7.29849	−0.680588
$116$	0	0
$117$	6.74311	0.623401
$118$	0	0
$119$	−11.9173	−1.09246
$120$	0	0
$121$	−7.91670	−0.719700
$122$	0	0
$123$	−24.5539	−2.21395
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−15.3127	−1.35878	−0.679391	−	0.733777i	$-0.737756\pi$
$127$	−15.3127	−1.35878	−0.679391	+	0.733777i	$0.737756\pi$
$128$	0	0
$129$	10.7704	0.948280
$130$	0	0
$131$	−20.1018	−1.75630	−0.878150	−	0.478385i	$-0.841222\pi$
$131$	−20.1018	−1.75630	−0.878150	+	0.478385i	$0.841222\pi$
$132$	0	0
$133$	12.8157	1.11126
$134$	0	0
$135$	−0.978465	−0.0842128
$136$	0	0
$137$	10.8576	0.927627	0.463813	−	0.885933i	$-0.346481\pi$
$137$	10.8576	0.927627	0.463813	+	0.885933i	$0.346481\pi$
$138$	0	0
$139$	8.03289	0.681341	0.340670	−	0.940183i	$-0.389346\pi$
$139$	8.03289	0.681341	0.340670	+	0.940183i	$0.389346\pi$
$140$	0	0
$141$	2.78674	0.234686
$142$	0	0
$143$	4.57867	0.382887
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−5.41570	−0.446680
$148$	0	0
$149$	20.8830	1.71080	0.855402	−	0.517964i	$-0.173310\pi$
$149$	20.8830	1.71080	0.855402	+	0.517964i	$0.173310\pi$
$150$	0	0
$151$	−3.32211	−0.270350	−0.135175	−	0.990822i	$-0.543160\pi$
$151$	−3.32211	−0.270350	−0.135175	+	0.990822i	$0.543160\pi$
$152$	0	0
$153$	−14.2024	−1.14820
$154$	0	0
$155$	−0.169927	−0.0136489
$156$	0	0
$157$	0.820914	0.0655161	0.0327581	−	0.999463i	$-0.489571\pi$
$157$	0.820914	0.0655161	0.0327581	+	0.999463i	$0.489571\pi$
$158$	0	0
$159$	27.9476	2.21639
$160$	0	0
$161$	−15.8372	−1.24815
$162$	0	0
$163$	3.03497	0.237717	0.118859	−	0.992911i	$-0.462076\pi$
$163$	3.03497	0.237717	0.118859	+	0.992911i	$0.462076\pi$
$164$	0	0
$165$	4.15010	0.323085
$166$	0	0
$167$	−7.53511	−0.583084	−0.291542	−	0.956558i	$-0.594168\pi$
$167$	−7.53511	−0.583084	−0.291542	+	0.956558i	$0.594168\pi$
$168$	0	0
$169$	−6.20073	−0.476979
$170$	0	0
$171$	15.2730	1.16796
$172$	0	0
$173$	−0.177378	−0.0134858	−0.00674290	−	0.999977i	$-0.502146\pi$
$173$	−0.177378	−0.0134858	−0.00674290	+	0.999977i	$0.502146\pi$
$174$	0	0
$175$	2.16993	0.164031
$176$	0	0
$177$	−9.73610	−0.731810
$178$	0	0
$179$	−23.4415	−1.75210	−0.876051	−	0.482219i	$-0.839831\pi$
$179$	−23.4415	−1.75210	−0.876051	+	0.482219i	$0.839831\pi$
$180$	0	0
$181$	21.7128	1.61390	0.806951	−	0.590618i	$-0.201116\pi$
$181$	21.7128	1.61390	0.806951	+	0.590618i	$0.201116\pi$
$182$	0	0
$183$	2.37616	0.175651
$184$	0	0
$185$	3.75593	0.276142
$186$	0	0
$187$	−9.64365	−0.705213
$188$	0	0
$189$	−2.12320	−0.154440
$190$	0	0
$191$	14.5630	1.05374	0.526871	−	0.849945i	$-0.323365\pi$
$191$	14.5630	1.05374	0.526871	+	0.849945i	$0.323365\pi$
$192$	0	0
$193$	11.2843	0.812265	0.406132	−	0.913814i	$-0.366877\pi$
$193$	11.2843	0.812265	0.406132	+	0.913814i	$0.366877\pi$
$194$	0	0
$195$	6.16285	0.441331
$196$	0	0
$197$	−8.60375	−0.612992	−0.306496	−	0.951872i	$-0.599157\pi$
$197$	−8.60375	−0.612992	−0.306496	+	0.951872i	$0.599157\pi$
$198$	0	0
$199$	0.290797	0.0206141	0.0103070	−	0.999947i	$-0.496719\pi$
$199$	0.290797	0.0206141	0.0103070	+	0.999947i	$0.496719\pi$
$200$	0	0
$201$	−20.0491	−1.41415
$202$	0	0
$203$	−2.16993	−0.152299
$204$	0	0
$205$	−10.3889	−0.725593
$206$	0	0
$207$	−18.8739	−1.31183
$208$	0	0
$209$	10.3706	0.717349
$210$	0	0
$211$	−10.0561	−0.692293	−0.346146	−	0.938181i	$-0.612510\pi$
$211$	−10.0561	−0.692293	−0.346146	+	0.938181i	$0.612510\pi$
$212$	0	0
$213$	14.4338	0.988988
$214$	0	0
$215$	4.55702	0.310786
$216$	0	0
$217$	−0.368729	−0.0250310
$218$	0	0
$219$	−4.08652	−0.276141
$220$	0	0
$221$	−14.3207	−0.963315
$222$	0	0
$223$	−9.64251	−0.645710	−0.322855	−	0.946448i	$-0.604643\pi$
$223$	−9.64251	−0.645710	−0.322855	+	0.946448i	$0.604643\pi$
$224$	0	0
$225$	2.58601	0.172400
$226$	0	0
$227$	7.22814	0.479749	0.239874	−	0.970804i	$-0.422894\pi$
$227$	7.22814	0.479749	0.239874	+	0.970804i	$0.422894\pi$
$228$	0	0
$229$	20.5027	1.35486	0.677428	−	0.735589i	$-0.263094\pi$
$229$	20.5027	1.35486	0.677428	+	0.735589i	$0.263094\pi$
$230$	0	0
$231$	9.00541	0.592513
$232$	0	0
$233$	−0.344022	−0.0225376	−0.0112688	−	0.999937i	$-0.503587\pi$
$233$	−0.344022	−0.0225376	−0.0112688	+	0.999937i	$0.503587\pi$
$234$	0	0
$235$	1.17909	0.0769151
$236$	0	0
$237$	−4.78707	−0.310953
$238$	0	0
$239$	5.30379	0.343074	0.171537	−	0.985178i	$-0.445127\pi$
$239$	5.30379	0.343074	0.171537	+	0.985178i	$0.445127\pi$
$240$	0	0
$241$	−26.9492	−1.73595	−0.867976	−	0.496607i	$-0.834579\pi$
$241$	−26.9492	−1.73595	−0.867976	+	0.496607i	$0.834579\pi$
$242$	0	0
$243$	−20.8662	−1.33857
$244$	0	0
$245$	−2.29142	−0.146393
$246$	0	0
$247$	15.4002	0.979892
$248$	0	0
$249$	4.40306	0.279033
$250$	0	0
$251$	−11.7270	−0.740202	−0.370101	−	0.928991i	$-0.620677\pi$
$251$	−11.7270	−0.740202	−0.370101	+	0.928991i	$0.620677\pi$
$252$	0	0
$253$	−12.8157	−0.805714
$254$	0	0
$255$	−12.9803	−0.812857
$256$	0	0
$257$	−23.0911	−1.44038	−0.720192	−	0.693775i	$-0.755946\pi$
$257$	−23.0911	−1.44038	−0.720192	+	0.693775i	$0.755946\pi$
$258$	0	0
$259$	8.15010	0.506423
$260$	0	0
$261$	−2.58601	−0.160070
$262$	0	0
$263$	−26.9190	−1.65990	−0.829949	−	0.557839i	$-0.811631\pi$
$263$	−26.9190	−1.65990	−0.829949	+	0.557839i	$0.811631\pi$
$264$	0	0
$265$	11.8248	0.726393
$266$	0	0
$267$	39.2265	2.40062
$268$	0	0
$269$	9.45604	0.576545	0.288273	−	0.957548i	$-0.406919\pi$
$269$	9.45604	0.576545	0.288273	+	0.957548i	$0.406919\pi$
$270$	0	0
$271$	22.6820	1.37784	0.688918	−	0.724840i	$-0.258086\pi$
$271$	22.6820	1.37784	0.688918	+	0.724840i	$0.258086\pi$
$272$	0	0
$273$	13.3729	0.809367
$274$	0	0
$275$	1.75593	0.105887
$276$	0	0
$277$	1.76301	0.105929	0.0529644	−	0.998596i	$-0.483133\pi$
$277$	1.76301	0.105929	0.0529644	+	0.998596i	$0.483133\pi$
$278$	0	0
$279$	−0.439432	−0.0263081
$280$	0	0
$281$	−10.6959	−0.638062	−0.319031	−	0.947744i	$-0.603357\pi$
$281$	−10.6959	−0.638062	−0.319031	+	0.947744i	$0.603357\pi$
$282$	0	0
$283$	8.34218	0.495891	0.247946	−	0.968774i	$-0.420245\pi$
$283$	8.34218	0.495891	0.247946	+	0.968774i	$0.420245\pi$
$284$	0	0
$285$	13.9587	0.826845
$286$	0	0
$287$	−22.5432	−1.33068
$288$	0	0
$289$	13.1625	0.774263
$290$	0	0
$291$	1.09787	0.0643583
$292$	0	0
$293$	5.86296	0.342518	0.171259	−	0.985226i	$-0.445217\pi$
$293$	5.86296	0.342518	0.171259	+	0.985226i	$0.445217\pi$
$294$	0	0
$295$	−4.11941	−0.239841
$296$	0	0
$297$	−1.71812	−0.0996953
$298$	0	0
$299$	−19.0311	−1.10060
$300$	0	0
$301$	9.88840	0.569958
$302$	0	0
$303$	−29.7001	−1.70623
$304$	0	0
$305$	1.00537	0.0575671
$306$	0	0
$307$	−11.3127	−0.645649	−0.322825	−	0.946459i	$-0.604632\pi$
$307$	−11.3127	−0.645649	−0.322825	+	0.946459i	$0.604632\pi$
$308$	0	0
$309$	−29.5029	−1.67836
$310$	0	0
$311$	−14.5012	−0.822290	−0.411145	−	0.911570i	$-0.634871\pi$
$311$	−14.5012	−0.822290	−0.411145	+	0.911570i	$0.634871\pi$
$312$	0	0
$313$	−28.1827	−1.59298	−0.796489	−	0.604653i	$-0.793312\pi$
$313$	−28.1827	−1.59298	−0.796489	+	0.604653i	$0.793312\pi$
$314$	0	0
$315$	5.61144	0.316169
$316$	0	0
$317$	−1.01067	−0.0567648	−0.0283824	−	0.999597i	$-0.509036\pi$
$317$	−1.01067	−0.0567648	−0.0283824	+	0.999597i	$0.509036\pi$
$318$	0	0
$319$	−1.75593	−0.0983133
$320$	0	0
$321$	−19.6006	−1.09400
$322$	0	0
$323$	−32.4361	−1.80480
$324$	0	0
$325$	2.60754	0.144640
$326$	0	0
$327$	43.4617	2.40344
$328$	0	0
$329$	2.55853	0.141056
$330$	0	0
$331$	8.81675	0.484612	0.242306	−	0.970200i	$-0.422096\pi$
$331$	8.81675	0.484612	0.242306	+	0.970200i	$0.422096\pi$
$332$	0	0
$333$	9.71286	0.532262
$334$	0	0
$335$	−8.48288	−0.463469
$336$	0	0
$337$	10.9110	0.594361	0.297181	−	0.954821i	$-0.403954\pi$
$337$	10.9110	0.594361	0.297181	+	0.954821i	$0.403954\pi$
$338$	0	0
$339$	−0.0558233	−0.00303190
$340$	0	0
$341$	−0.298380	−0.0161582
$342$	0	0
$343$	−20.1617	−1.08863
$344$	0	0
$345$	−17.2498	−0.928697
$346$	0	0
$347$	−3.21874	−0.172791	−0.0863955	−	0.996261i	$-0.527535\pi$
$347$	−3.21874	−0.172791	−0.0863955	+	0.996261i	$0.527535\pi$
$348$	0	0
$349$	0.0430703	0.00230550	0.00115275	−	0.999999i	$-0.499633\pi$
$349$	0.0430703	0.00230550	0.00115275	+	0.999999i	$0.499633\pi$
$350$	0	0
$351$	−2.55139	−0.136183
$352$	0	0
$353$	−8.95977	−0.476881	−0.238440	−	0.971157i	$-0.576636\pi$
$353$	−8.95977	−0.476881	−0.238440	+	0.971157i	$0.576636\pi$
$354$	0	0
$355$	6.10703	0.324127
$356$	0	0
$357$	−28.1663	−1.49072
$358$	0	0
$359$	28.8347	1.52184	0.760918	−	0.648849i	$-0.224749\pi$
$359$	28.8347	1.52184	0.760918	+	0.648849i	$0.224749\pi$
$360$	0	0
$361$	15.8812	0.835853
$362$	0	0
$363$	−18.7109	−0.982068
$364$	0	0
$365$	−1.72903	−0.0905016
$366$	0	0
$367$	−0.452139	−0.0236015	−0.0118007	−	0.999930i	$-0.503756\pi$
$367$	−0.452139	−0.0236015	−0.0118007	+	0.999930i	$0.503756\pi$
$368$	0	0
$369$	−26.8658	−1.39858
$370$	0	0
$371$	25.6590	1.33215
$372$	0	0
$373$	−23.0375	−1.19284	−0.596420	−	0.802673i	$-0.703411\pi$
$373$	−23.0375	−1.19284	−0.596420	+	0.802673i	$0.703411\pi$
$374$	0	0
$375$	2.36347	0.122049
$376$	0	0
$377$	−2.60754	−0.134295
$378$	0	0
$379$	−18.2477	−0.937322	−0.468661	−	0.883378i	$-0.655263\pi$
$379$	−18.2477	−0.937322	−0.468661	+	0.883378i	$0.655263\pi$
$380$	0	0
$381$	−36.1911	−1.85413
$382$	0	0
$383$	−7.47190	−0.381796	−0.190898	−	0.981610i	$-0.561140\pi$
$383$	−7.47190	−0.381796	−0.190898	+	0.981610i	$0.561140\pi$
$384$	0	0
$385$	3.81025	0.194188
$386$	0	0
$387$	11.7845	0.599039
$388$	0	0
$389$	−24.6708	−1.25086	−0.625429	−	0.780281i	$-0.715076\pi$
$389$	−24.6708	−1.25086	−0.625429	+	0.780281i	$0.715076\pi$
$390$	0	0
$391$	40.0836	2.02711
$392$	0	0
$393$	−47.5100	−2.39656
$394$	0	0
$395$	−2.02544	−0.101911
$396$	0	0
$397$	−5.52985	−0.277535	−0.138768	−	0.990325i	$-0.544314\pi$
$397$	−5.52985	−0.277535	−0.138768	+	0.990325i	$0.544314\pi$
$398$	0	0
$399$	30.2895	1.51637
$400$	0	0
$401$	22.9594	1.14654	0.573268	−	0.819368i	$-0.305675\pi$
$401$	22.9594	1.14654	0.573268	+	0.819368i	$0.305675\pi$
$402$	0	0
$403$	−0.443092	−0.0220720
$404$	0	0
$405$	−10.0706	−0.500412
$406$	0	0
$407$	6.59516	0.326910
$408$	0	0
$409$	28.5698	1.41268	0.706342	−	0.707871i	$-0.250344\pi$
$409$	28.5698	1.41268	0.706342	+	0.707871i	$0.250344\pi$
$410$	0	0
$411$	25.6616	1.26580
$412$	0	0
$413$	−8.93881	−0.439850
$414$	0	0
$415$	1.86296	0.0914492
$416$	0	0
$417$	18.9855	0.929725
$418$	0	0
$419$	−4.94595	−0.241626	−0.120813	−	0.992675i	$-0.538550\pi$
$419$	−4.94595	−0.241626	−0.120813	+	0.992675i	$0.538550\pi$
$420$	0	0
$421$	−1.83500	−0.0894324	−0.0447162	−	0.999000i	$-0.514238\pi$
$421$	−1.83500	−0.0894324	−0.0447162	+	0.999000i	$0.514238\pi$
$422$	0	0
$423$	3.04912	0.148253
$424$	0	0
$425$	−5.49204	−0.266403
$426$	0	0
$427$	2.18157	0.105574
$428$	0	0
$429$	10.8216	0.522470
$430$	0	0
$431$	21.8948	1.05463	0.527317	−	0.849668i	$-0.323198\pi$
$431$	21.8948	1.05463	0.527317	+	0.849668i	$0.323198\pi$
$432$	0	0
$433$	−8.08164	−0.388379	−0.194189	−	0.980964i	$-0.562208\pi$
$433$	−8.08164	−0.388379	−0.194189	+	0.980964i	$0.562208\pi$
$434$	0	0
$435$	−2.36347	−0.113320
$436$	0	0
$437$	−43.1051	−2.06200
$438$	0	0
$439$	37.5320	1.79130	0.895652	−	0.444755i	$-0.146709\pi$
$439$	37.5320	1.79130	0.895652	+	0.444755i	$0.146709\pi$
$440$	0	0
$441$	−5.92562	−0.282172
$442$	0	0
$443$	4.25826	0.202316	0.101158	−	0.994870i	$-0.467745\pi$
$443$	4.25826	0.202316	0.101158	+	0.994870i	$0.467745\pi$
$444$	0	0
$445$	16.5970	0.786772
$446$	0	0
$447$	49.3565	2.33448
$448$	0	0
$449$	4.53302	0.213927	0.106963	−	0.994263i	$-0.465887\pi$
$449$	4.53302	0.213927	0.106963	+	0.994263i	$0.465887\pi$
$450$	0	0
$451$	−18.2422	−0.858993
$452$	0	0
$453$	−7.85172	−0.368906
$454$	0	0
$455$	5.65817	0.265259
$456$	0	0
$457$	−1.76301	−0.0824700	−0.0412350	−	0.999149i	$-0.513129\pi$
$457$	−1.76301	−0.0824700	−0.0412350	+	0.999149i	$0.513129\pi$
$458$	0	0
$459$	5.37377	0.250826
$460$	0	0
$461$	32.3148	1.50505	0.752524	−	0.658564i	$-0.228836\pi$
$461$	32.3148	1.50505	0.752524	+	0.658564i	$0.228836\pi$
$462$	0	0
$463$	20.3894	0.947577	0.473788	−	0.880639i	$-0.342886\pi$
$463$	20.3894	0.947577	0.473788	+	0.880639i	$0.342886\pi$
$464$	0	0
$465$	−0.401618	−0.0186246
$466$	0	0
$467$	−31.6392	−1.46409	−0.732044	−	0.681257i	$-0.761433\pi$
$467$	−31.6392	−1.46409	−0.732044	+	0.681257i	$0.761433\pi$
$468$	0	0
$469$	−18.4072	−0.849967
$470$	0	0
$471$	1.94021	0.0894001
$472$	0	0
$473$	8.00182	0.367924
$474$	0	0
$475$	5.90603	0.270987
$476$	0	0
$477$	30.5790	1.40012
$478$	0	0
$479$	8.02741	0.366782	0.183391	−	0.983040i	$-0.441293\pi$
$479$	8.02741	0.366782	0.183391	+	0.983040i	$0.441293\pi$
$480$	0	0
$481$	9.79375	0.446556
$482$	0	0
$483$	−37.4308	−1.70316
$484$	0	0
$485$	0.464516	0.0210926
$486$	0	0
$487$	6.05911	0.274564	0.137282	−	0.990532i	$-0.456163\pi$
$487$	6.05911	0.274564	0.137282	+	0.990532i	$0.456163\pi$
$488$	0	0
$489$	7.17308	0.324378
$490$	0	0
$491$	15.1106	0.681932	0.340966	−	0.940076i	$-0.389246\pi$
$491$	15.1106	0.681932	0.340966	+	0.940076i	$0.389246\pi$
$492$	0	0
$493$	5.49204	0.247349
$494$	0	0
$495$	4.54085	0.204096
$496$	0	0
$497$	13.2518	0.594425
$498$	0	0
$499$	−43.2431	−1.93583	−0.967913	−	0.251286i	$-0.919146\pi$
$499$	−43.2431	−1.93583	−0.967913	+	0.251286i	$0.919146\pi$
$500$	0	0
$501$	−17.8090	−0.795649
$502$	0	0
$503$	26.4309	1.17850	0.589248	−	0.807952i	$-0.299424\pi$
$503$	26.4309	1.17850	0.589248	+	0.807952i	$0.299424\pi$
$504$	0	0
$505$	−12.5663	−0.559193
$506$	0	0
$507$	−14.6553	−0.650863
$508$	0	0
$509$	−25.4126	−1.12640	−0.563198	−	0.826322i	$-0.690429\pi$
$509$	−25.4126	−1.12640	−0.563198	+	0.826322i	$0.690429\pi$
$510$	0	0
$511$	−3.75187	−0.165973
$512$	0	0
$513$	−5.77884	−0.255142
$514$	0	0
$515$	−12.4829	−0.550061
$516$	0	0
$517$	2.07039	0.0910559
$518$	0	0
$519$	−0.419228	−0.0184021
$520$	0	0
$521$	37.8715	1.65918	0.829591	−	0.558372i	$-0.188574\pi$
$521$	37.8715	1.65918	0.829591	+	0.558372i	$0.188574\pi$
$522$	0	0
$523$	6.19341	0.270819	0.135410	−	0.990790i	$-0.456765\pi$
$523$	6.19341	0.270819	0.135410	+	0.990790i	$0.456765\pi$
$524$	0	0
$525$	5.12856	0.223829
$526$	0	0
$527$	0.933246	0.0406528
$528$	0	0
$529$	30.2680	1.31600
$530$	0	0
$531$	−10.6528	−0.462292
$532$	0	0
$533$	−27.0895	−1.17338
$534$	0	0
$535$	−8.29312	−0.358543
$536$	0	0
$537$	−55.4034	−2.39083
$538$	0	0
$539$	−4.02357	−0.173308
$540$	0	0
$541$	−26.0329	−1.11924	−0.559621	−	0.828749i	$-0.689053\pi$
$541$	−26.0329	−1.11924	−0.559621	+	0.828749i	$0.689053\pi$
$542$	0	0
$543$	51.3177	2.20225
$544$	0	0
$545$	18.3889	0.787694
$546$	0	0
$547$	36.4847	1.55997	0.779987	−	0.625796i	$-0.215226\pi$
$547$	36.4847	1.55997	0.779987	+	0.625796i	$0.215226\pi$
$548$	0	0
$549$	2.59988	0.110960
$550$	0	0
$551$	−5.90603	−0.251605
$552$	0	0
$553$	−4.39505	−0.186897
$554$	0	0
$555$	8.87705	0.376810
$556$	0	0
$557$	42.0173	1.78033	0.890164	−	0.455640i	$-0.150590\pi$
$557$	42.0173	1.78033	0.890164	+	0.455640i	$0.150590\pi$
$558$	0	0
$559$	11.8826	0.502581
$560$	0	0
$561$	−22.7925	−0.962300
$562$	0	0
$563$	10.8636	0.457845	0.228923	−	0.973445i	$-0.426480\pi$
$563$	10.8636	0.457845	0.228923	+	0.973445i	$0.426480\pi$
$564$	0	0
$565$	−0.0236192	−0.000993666 0
$566$	0	0
$567$	−21.8524	−0.917717
$568$	0	0
$569$	16.2696	0.682055	0.341028	−	0.940053i	$-0.389225\pi$
$569$	16.2696	0.682055	0.341028	+	0.940053i	$0.389225\pi$
$570$	0	0
$571$	29.2502	1.22408	0.612041	−	0.790826i	$-0.290349\pi$
$571$	29.2502	1.22408	0.612041	+	0.790826i	$0.290349\pi$
$572$	0	0
$573$	34.4193	1.43789
$574$	0	0
$575$	−7.29849	−0.304368
$576$	0	0
$577$	39.6878	1.65222	0.826112	−	0.563506i	$-0.190548\pi$
$577$	39.6878	1.65222	0.826112	+	0.563506i	$0.190548\pi$
$578$	0	0
$579$	26.6702	1.10838
$580$	0	0
$581$	4.04249	0.167711
$582$	0	0
$583$	20.7636	0.859940
$584$	0	0
$585$	6.74311	0.278793
$586$	0	0
$587$	41.6739	1.72007	0.860033	−	0.510238i	$-0.170443\pi$
$587$	41.6739	1.72007	0.860033	+	0.510238i	$0.170443\pi$
$588$	0	0
$589$	−1.00359	−0.0413524
$590$	0	0
$591$	−20.3347	−0.836459
$592$	0	0
$593$	47.8719	1.96586	0.982931	−	0.183976i	$-0.0588969\pi$
$593$	47.8719	1.96586	0.982931	+	0.183976i	$0.0588969\pi$
$594$	0	0
$595$	−11.9173	−0.488563
$596$	0	0
$597$	0.687291	0.0281290
$598$	0	0
$599$	41.8248	1.70891	0.854457	−	0.519521i	$-0.173890\pi$
$599$	41.8248	1.70891	0.854457	+	0.519521i	$0.173890\pi$
$600$	0	0
$601$	−10.4180	−0.424958	−0.212479	−	0.977166i	$-0.568154\pi$
$601$	−10.4180	−0.424958	−0.212479	+	0.977166i	$0.568154\pi$
$602$	0	0
$603$	−21.9368	−0.893334
$604$	0	0
$605$	−7.91670	−0.321860
$606$	0	0
$607$	26.4309	1.07280	0.536398	−	0.843965i	$-0.319784\pi$
$607$	26.4309	1.07280	0.536398	+	0.843965i	$0.319784\pi$
$608$	0	0
$609$	−5.12856	−0.207820
$610$	0	0
$611$	3.07451	0.124382
$612$	0	0
$613$	26.9696	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$613$	26.9696	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$614$	0	0
$615$	−24.5539	−0.990109
$616$	0	0
$617$	−17.3067	−0.696740	−0.348370	−	0.937357i	$-0.613265\pi$
$617$	−17.3067	−0.696740	−0.348370	+	0.937357i	$0.613265\pi$
$618$	0	0
$619$	32.9250	1.32337	0.661684	−	0.749783i	$-0.269842\pi$
$619$	32.9250	1.32337	0.661684	+	0.749783i	$0.269842\pi$
$620$	0	0
$621$	7.14132	0.286571
$622$	0	0
$623$	36.0142	1.44288
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	24.5106	0.978860
$628$	0	0
$629$	−20.6277	−0.822481
$630$	0	0
$631$	17.8551	0.710802	0.355401	−	0.934714i	$-0.384344\pi$
$631$	17.8551	0.710802	0.355401	+	0.934714i	$0.384344\pi$
$632$	0	0
$633$	−23.7674	−0.944669
$634$	0	0
$635$	−15.3127	−0.607665
$636$	0	0
$637$	−5.97496	−0.236737
$638$	0	0
$639$	15.7928	0.624754
$640$	0	0
$641$	10.1749	0.401883	0.200941	−	0.979603i	$-0.435600\pi$
$641$	10.1749	0.401883	0.200941	+	0.979603i	$0.435600\pi$
$642$	0	0
$643$	28.7293	1.13297	0.566486	−	0.824071i	$-0.308303\pi$
$643$	28.7293	1.13297	0.566486	+	0.824071i	$0.308303\pi$
$644$	0	0
$645$	10.7704	0.424084
$646$	0	0
$647$	−44.7235	−1.75826	−0.879131	−	0.476580i	$-0.841876\pi$
$647$	−44.7235	−1.75826	−0.879131	+	0.476580i	$0.841876\pi$
$648$	0	0
$649$	−7.23340	−0.283936
$650$	0	0
$651$	−0.871482	−0.0341561
$652$	0	0
$653$	10.6401	0.416378	0.208189	−	0.978089i	$-0.433243\pi$
$653$	10.6401	0.416378	0.208189	+	0.978089i	$0.433243\pi$
$654$	0	0
$655$	−20.1018	−0.785441
$656$	0	0
$657$	−4.47128	−0.174441
$658$	0	0
$659$	7.08878	0.276140	0.138070	−	0.990423i	$-0.455910\pi$
$659$	7.08878	0.276140	0.138070	+	0.990423i	$0.455910\pi$
$660$	0	0
$661$	−16.1804	−0.629344	−0.314672	−	0.949200i	$-0.601895\pi$
$661$	−16.1804	−0.629344	−0.314672	+	0.949200i	$0.601895\pi$
$662$	0	0
$663$	−33.8466	−1.31449
$664$	0	0
$665$	12.8157	0.496970
$666$	0	0
$667$	7.29849	0.282599
$668$	0	0
$669$	−22.7898	−0.881105
$670$	0	0
$671$	1.76536	0.0681508
$672$	0	0
$673$	16.2677	0.627075	0.313538	−	0.949576i	$-0.398486\pi$
$673$	16.2677	0.627075	0.313538	+	0.949576i	$0.398486\pi$
$674$	0	0
$675$	−0.978465	−0.0376611
$676$	0	0
$677$	−26.3795	−1.01385	−0.506923	−	0.861991i	$-0.669217\pi$
$677$	−26.3795	−1.01385	−0.506923	+	0.861991i	$0.669217\pi$
$678$	0	0
$679$	1.00797	0.0386822
$680$	0	0
$681$	17.0835	0.654642
$682$	0	0
$683$	−40.4356	−1.54723	−0.773613	−	0.633658i	$-0.781553\pi$
$683$	−40.4356	−1.54723	−0.773613	+	0.633658i	$0.781553\pi$
$684$	0	0
$685$	10.8576	0.414847
$686$	0	0
$687$	48.4576	1.84877
$688$	0	0
$689$	30.8337	1.17467
$690$	0	0
$691$	−25.1960	−0.958500	−0.479250	−	0.877678i	$-0.659091\pi$
$691$	−25.1960	−0.958500	−0.479250	+	0.877678i	$0.659091\pi$
$692$	0	0
$693$	9.85332	0.374297
$694$	0	0
$695$	8.03289	0.304705
$696$	0	0
$697$	57.0563	2.16116
$698$	0	0
$699$	−0.813087	−0.0307538
$700$	0	0
$701$	12.0823	0.456341	0.228171	−	0.973621i	$-0.426726\pi$
$701$	12.0823	0.456341	0.228171	+	0.973621i	$0.426726\pi$
$702$	0	0
$703$	22.1827	0.836635
$704$	0	0
$705$	2.78674	0.104955
$706$	0	0
$707$	−27.2679	−1.02552
$708$	0	0
$709$	−4.14012	−0.155485	−0.0777427	−	0.996973i	$-0.524771\pi$
$709$	−4.14012	−0.155485	−0.0777427	+	0.996973i	$0.524771\pi$
$710$	0	0
$711$	−5.23779	−0.196433
$712$	0	0
$713$	1.24021	0.0464463
$714$	0	0
$715$	4.57867	0.171232
$716$	0	0
$717$	12.5354	0.468142
$718$	0	0
$719$	−32.0138	−1.19391	−0.596956	−	0.802274i	$-0.703624\pi$
$719$	−32.0138	−1.19391	−0.596956	+	0.802274i	$0.703624\pi$
$720$	0	0
$721$	−27.0869	−1.00877
$722$	0	0
$723$	−63.6937	−2.36880
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−29.4823	−1.09344	−0.546719	−	0.837316i	$-0.684123\pi$
$727$	−29.4823	−1.09344	−0.546719	+	0.837316i	$0.684123\pi$
$728$	0	0
$729$	−19.1049	−0.707588
$730$	0	0
$731$	−25.0273	−0.925669
$732$	0	0
$733$	9.57873	0.353798	0.176899	−	0.984229i	$-0.443393\pi$
$733$	9.57873	0.353798	0.176899	+	0.984229i	$0.443393\pi$
$734$	0	0
$735$	−5.41570	−0.199761
$736$	0	0
$737$	−14.8954	−0.548678
$738$	0	0
$739$	31.5043	1.15890	0.579451	−	0.815007i	$-0.303267\pi$
$739$	31.5043	1.15890	0.579451	+	0.815007i	$0.303267\pi$
$740$	0	0
$741$	36.3980	1.33711
$742$	0	0
$743$	−13.2845	−0.487362	−0.243681	−	0.969855i	$-0.578355\pi$
$743$	−13.2845	−0.487362	−0.243681	+	0.969855i	$0.578355\pi$
$744$	0	0
$745$	20.8830	0.765095
$746$	0	0
$747$	4.81763	0.176268
$748$	0	0
$749$	−17.9955	−0.657540
$750$	0	0
$751$	1.19323	0.0435418	0.0217709	−	0.999763i	$-0.493070\pi$
$751$	1.19323	0.0435418	0.0217709	+	0.999763i	$0.493070\pi$
$752$	0	0
$753$	−27.7165	−1.01004
$754$	0	0
$755$	−3.32211	−0.120904
$756$	0	0
$757$	−39.5804	−1.43857	−0.719287	−	0.694713i	$-0.755532\pi$
$757$	−39.5804	−1.43857	−0.719287	+	0.694713i	$0.755532\pi$
$758$	0	0
$759$	−30.2895	−1.09944
$760$	0	0
$761$	4.79093	0.173671	0.0868356	−	0.996223i	$-0.472325\pi$
$761$	4.79093	0.173671	0.0868356	+	0.996223i	$0.472325\pi$
$762$	0	0
$763$	39.9026	1.44457
$764$	0	0
$765$	−14.2024	−0.513490
$766$	0	0
$767$	−10.7415	−0.387854
$768$	0	0
$769$	13.3712	0.482177	0.241088	−	0.970503i	$-0.422496\pi$
$769$	13.3712	0.482177	0.241088	+	0.970503i	$0.422496\pi$
$770$	0	0
$771$	−54.5752	−1.96548
$772$	0	0
$773$	51.9115	1.86713	0.933563	−	0.358413i	$-0.116682\pi$
$773$	51.9115	1.86713	0.933563	+	0.358413i	$0.116682\pi$
$774$	0	0
$775$	−0.169927	−0.00610396
$776$	0	0
$777$	19.2625	0.691040
$778$	0	0
$779$	−61.3572	−2.19835
$780$	0	0
$781$	10.7235	0.383718
$782$	0	0
$783$	0.978465	0.0349675
$784$	0	0
$785$	0.820914	0.0292997
$786$	0	0
$787$	16.3073	0.581292	0.290646	−	0.956831i	$-0.406130\pi$
$787$	16.3073	0.581292	0.290646	+	0.956831i	$0.406130\pi$
$788$	0	0
$789$	−63.6224	−2.26502
$790$	0	0
$791$	−0.0512519	−0.00182231
$792$	0	0
$793$	2.62154	0.0930934
$794$	0	0
$795$	27.9476	0.991201
$796$	0	0
$797$	−7.24349	−0.256578	−0.128289	−	0.991737i	$-0.540948\pi$
$797$	−7.24349	−0.256578	−0.128289	+	0.991737i	$0.540948\pi$
$798$	0	0
$799$	−6.47558	−0.229090
$800$	0	0
$801$	42.9199	1.51650
$802$	0	0
$803$	−3.03606	−0.107140
$804$	0	0
$805$	−15.8372	−0.558188
$806$	0	0
$807$	22.3491	0.786726
$808$	0	0
$809$	50.1813	1.76428	0.882141	−	0.470986i	$-0.156102\pi$
$809$	50.1813	1.76428	0.882141	+	0.470986i	$0.156102\pi$
$810$	0	0
$811$	46.3184	1.62646	0.813230	−	0.581943i	$-0.197707\pi$
$811$	46.3184	1.62646	0.813230	+	0.581943i	$0.197707\pi$
$812$	0	0
$813$	53.6084	1.88013
$814$	0	0
$815$	3.03497	0.106310
$816$	0	0
$817$	26.9139	0.941598
$818$	0	0
$819$	14.6321	0.511286
$820$	0	0
$821$	3.80865	0.132923	0.0664614	−	0.997789i	$-0.478829\pi$
$821$	3.80865	0.132923	0.0664614	+	0.997789i	$0.478829\pi$
$822$	0	0
$823$	−26.8321	−0.935309	−0.467655	−	0.883911i	$-0.654901\pi$
$823$	−26.8321	−0.935309	−0.467655	+	0.883911i	$0.654901\pi$
$824$	0	0
$825$	4.15010	0.144488
$826$	0	0
$827$	−30.5168	−1.06117	−0.530587	−	0.847630i	$-0.678029\pi$
$827$	−30.5168	−1.06117	−0.530587	+	0.847630i	$0.678029\pi$
$828$	0	0
$829$	31.2716	1.08611	0.543054	−	0.839698i	$-0.317268\pi$
$829$	31.2716	1.08611	0.543054	+	0.839698i	$0.317268\pi$
$830$	0	0
$831$	4.16682	0.144545
$832$	0	0
$833$	12.5845	0.436029
$834$	0	0
$835$	−7.53511	−0.260763
$836$	0	0
$837$	0.166268	0.00574705
$838$	0	0
$839$	−51.3879	−1.77411	−0.887053	−	0.461667i	$-0.847251\pi$
$839$	−51.3879	−1.77411	−0.887053	+	0.461667i	$0.847251\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−25.2794	−0.870669
$844$	0	0
$845$	−6.20073	−0.213312
$846$	0	0
$847$	−17.1787	−0.590266
$848$	0	0
$849$	19.7165	0.676669
$850$	0	0
$851$	−27.4126	−0.939693
$852$	0	0
$853$	−2.98175	−0.102093	−0.0510466	−	0.998696i	$-0.516256\pi$
$853$	−2.98175	−0.102093	−0.0510466	+	0.998696i	$0.516256\pi$
$854$	0	0
$855$	15.2730	0.522327
$856$	0	0
$857$	−42.5514	−1.45353	−0.726765	−	0.686887i	$-0.758977\pi$
$857$	−42.5514	−1.45353	−0.726765	+	0.686887i	$0.758977\pi$
$858$	0	0
$859$	52.3972	1.78777	0.893885	−	0.448296i	$-0.147969\pi$
$859$	52.3972	1.78777	0.893885	+	0.448296i	$0.147969\pi$
$860$	0	0
$861$	−53.2802	−1.81578
$862$	0	0
$863$	−34.4446	−1.17251	−0.586255	−	0.810127i	$-0.699398\pi$
$863$	−34.4446	−1.17251	−0.586255	+	0.810127i	$0.699398\pi$
$864$	0	0
$865$	−0.177378	−0.00603103
$866$	0	0
$867$	31.1092	1.05652
$868$	0	0
$869$	−3.55653	−0.120647
$870$	0	0
$871$	−22.1195	−0.749489
$872$	0	0
$873$	1.20124	0.0406558
$874$	0	0
$875$	2.16993	0.0733569
$876$	0	0
$877$	−28.6880	−0.968723	−0.484362	−	0.874868i	$-0.660948\pi$
$877$	−28.6880	−0.968723	−0.484362	+	0.874868i	$0.660948\pi$
$878$	0	0
$879$	13.8570	0.467383
$880$	0	0
$881$	27.0686	0.911965	0.455982	−	0.889989i	$-0.349288\pi$
$881$	27.0686	0.911965	0.455982	+	0.889989i	$0.349288\pi$
$882$	0	0
$883$	−9.08146	−0.305615	−0.152808	−	0.988256i	$-0.548832\pi$
$883$	−9.08146	−0.305615	−0.152808	+	0.988256i	$0.548832\pi$
$884$	0	0
$885$	−9.73610	−0.327276
$886$	0	0
$887$	−50.6083	−1.69926	−0.849630	−	0.527378i	$-0.823175\pi$
$887$	−50.6083	−1.69926	−0.849630	+	0.527378i	$0.823175\pi$
$888$	0	0
$889$	−33.2274	−1.11441
$890$	0	0
$891$	−17.6833	−0.592412
$892$	0	0
$893$	6.96372	0.233032
$894$	0	0
$895$	−23.4415	−0.783563
$896$	0	0
$897$	−44.9795	−1.50182
$898$	0	0
$899$	0.169927	0.00566738
$900$	0	0
$901$	−64.9423	−2.16354
$902$	0	0
$903$	23.3710	0.777737
$904$	0	0
$905$	21.7128	0.721759
$906$	0	0
$907$	14.2527	0.473252	0.236626	−	0.971601i	$-0.423958\pi$
$907$	14.2527	0.473252	0.236626	+	0.971601i	$0.423958\pi$
$908$	0	0
$909$	−32.4965	−1.07784
$910$	0	0
$911$	−5.33340	−0.176703	−0.0883517	−	0.996089i	$-0.528160\pi$
$911$	−5.33340	−0.176703	−0.0883517	+	0.996089i	$0.528160\pi$
$912$	0	0
$913$	3.27124	0.108262
$914$	0	0
$915$	2.37616	0.0785534
$916$	0	0
$917$	−43.6194	−1.44044
$918$	0	0
$919$	−4.80076	−0.158362	−0.0791812	−	0.996860i	$-0.525231\pi$
$919$	−4.80076	−0.158362	−0.0791812	+	0.996860i	$0.525231\pi$
$920$	0	0
$921$	−26.7372	−0.881022
$922$	0	0
$923$	15.9243	0.524156
$924$	0	0
$925$	3.75593	0.123494
$926$	0	0
$927$	−32.2808	−1.06024
$928$	0	0
$929$	35.1469	1.15313	0.576567	−	0.817050i	$-0.304392\pi$
$929$	35.1469	1.15313	0.576567	+	0.817050i	$0.304392\pi$
$930$	0	0
$931$	−13.5332	−0.443532
$932$	0	0
$933$	−34.2733	−1.12206
$934$	0	0
$935$	−9.64365	−0.315381
$936$	0	0
$937$	−33.4126	−1.09154	−0.545772	−	0.837934i	$-0.683763\pi$
$937$	−33.4126	−1.09154	−0.545772	+	0.837934i	$0.683763\pi$
$938$	0	0
$939$	−66.6090	−2.17370
$940$	0	0
$941$	44.5207	1.45133	0.725667	−	0.688047i	$-0.241532\pi$
$941$	44.5207	1.45133	0.725667	+	0.688047i	$0.241532\pi$
$942$	0	0
$943$	75.8234	2.46915
$944$	0	0
$945$	−2.12320	−0.0690676
$946$	0	0
$947$	−0.670769	−0.0217971	−0.0108985	−	0.999941i	$-0.503469\pi$
$947$	−0.670769	−0.0217971	−0.0108985	+	0.999941i	$0.503469\pi$
$948$	0	0
$949$	−4.50852	−0.146353
$950$	0	0
$951$	−2.38869	−0.0774585
$952$	0	0
$953$	59.0039	1.91132	0.955661	−	0.294469i	$-0.0951427\pi$
$953$	59.0039	1.91132	0.955661	+	0.294469i	$0.0951427\pi$
$954$	0	0
$955$	14.5630	0.471248
$956$	0	0
$957$	−4.15010	−0.134154
$958$	0	0
$959$	23.5602	0.760798
$960$	0	0
$961$	−30.9711	−0.999069
$962$	0	0
$963$	−21.4461	−0.691090
$964$	0	0
$965$	11.2843	0.363256
$966$	0	0
$967$	−53.6065	−1.72387	−0.861935	−	0.507019i	$-0.830748\pi$
$967$	−53.6065	−1.72387	−0.861935	+	0.507019i	$0.830748\pi$
$968$	0	0
$969$	−76.6620	−2.46274
$970$	0	0
$971$	12.4238	0.398700	0.199350	−	0.979928i	$-0.436117\pi$
$971$	12.4238	0.398700	0.199350	+	0.979928i	$0.436117\pi$
$972$	0	0
$973$	17.4308	0.558805
$974$	0	0
$975$	6.16285	0.197369
$976$	0	0
$977$	−9.39410	−0.300544	−0.150272	−	0.988645i	$-0.548015\pi$
$977$	−9.39410	−0.300544	−0.150272	+	0.988645i	$0.548015\pi$
$978$	0	0
$979$	29.1432	0.931420
$980$	0	0
$981$	47.5538	1.51828
$982$	0	0
$983$	−21.4859	−0.685292	−0.342646	−	0.939465i	$-0.611323\pi$
$983$	−21.4859	−0.685292	−0.342646	+	0.939465i	$0.611323\pi$
$984$	0	0
$985$	−8.60375	−0.274138
$986$	0	0
$987$	6.04702	0.192479
$988$	0	0
$989$	−33.2594	−1.05759
$990$	0	0
$991$	24.8912	0.790695	0.395347	−	0.918532i	$-0.370624\pi$
$991$	24.8912	0.790695	0.395347	+	0.918532i	$0.370624\pi$
$992$	0	0
$993$	20.8381	0.661278
$994$	0	0
$995$	0.290797	0.00921889
$996$	0	0
$997$	−34.9298	−1.10624	−0.553118	−	0.833103i	$-0.686562\pi$
$997$	−34.9298	−1.10624	−0.553118	+	0.833103i	$0.686562\pi$
$998$	0	0
$999$	−3.67505	−0.116273

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.2.a.i.1.4	✓	5
4.3	odd	2		2320.2.a.w.1.2		5
5.4	even	2		5800.2.a.v.1.2		5
8.3	odd	2		9280.2.a.ch.1.4		5
8.5	even	2		9280.2.a.cj.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.i.1.4	✓	5	1.1	even	1	trivial
2320.2.a.w.1.2		5	4.3	odd	2
5800.2.a.v.1.2		5	5.4	even	2
9280.2.a.ch.1.4		5	8.3	odd	2
9280.2.a.cj.1.2		5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1160.a (trivial)

\(f(q)\)	\(=\)	\(q+2.36347 q^{3} +1.00000 q^{5} +2.16993 q^{7} +2.58601 q^{9} +1.75593 q^{11} +2.60754 q^{13} +2.36347 q^{15} -5.49204 q^{17} +5.90603 q^{19} +5.12856 q^{21} -7.29849 q^{23} +1.00000 q^{25} -0.978465 q^{27} -1.00000 q^{29} -0.169927 q^{31} +4.15010 q^{33} +2.16993 q^{35} +3.75593 q^{37} +6.16285 q^{39} -10.3889 q^{41} +4.55702 q^{43} +2.58601 q^{45} +1.17909 q^{47} -2.29142 q^{49} -12.9803 q^{51} +11.8248 q^{53} +1.75593 q^{55} +13.9587 q^{57} -4.11941 q^{59} +1.00537 q^{61} +5.61144 q^{63} +2.60754 q^{65} -8.48288 q^{67} -17.2498 q^{69} +6.10703 q^{71} -1.72903 q^{73} +2.36347 q^{75} +3.81025 q^{77} -2.02544 q^{79} -10.0706 q^{81} +1.86296 q^{83} -5.49204 q^{85} -2.36347 q^{87} +16.5970 q^{89} +5.65817 q^{91} -0.401618 q^{93} +5.90603 q^{95} +0.464516 q^{97} +4.54085 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 5 q^{5} - q^{7} + 12 q^{9} - 4 q^{11} + 13 q^{13} - q^{15} + 3 q^{17} + 8 q^{21} - 7 q^{23} + 5 q^{25} - 4 q^{27} - 5 q^{29} + 11 q^{31} + 4 q^{33} - q^{35} + 6 q^{37} + 21 q^{39} + 16 q^{41}+ \cdots + 10 q^{99}+O(q^{100}) \) 5 * q - q^3 + 5 * q^5 - q^7 + 12 * q^9 - 4 * q^11 + 13 * q^13 - q^15 + 3 * q^17 + 8 * q^21 - 7 * q^23 + 5 * q^25 - 4 * q^27 - 5 * q^29 + 11 * q^31 + 4 * q^33 - q^35 + 6 * q^37 + 21 * q^39 + 16 * q^41 + 9 * q^43 + 12 * q^45 + 2 * q^47 + 16 * q^49 + 2 * q^51 + 7 * q^53 - 4 * q^55 + 2 * q^57 + 5 * q^59 + 7 * q^61 - 28 * q^63 + 13 * q^65 - 4 * q^67 + 23 * q^69 + 12 * q^71 + 7 * q^73 - q^75 + 26 * q^77 + 45 * q^79 - 7 * q^81 - 22 * q^83 + 3 * q^85 + q^87 + 24 * q^89 - 20 * q^91 - 10 * q^93 + 17 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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