LMFDB - Embedded newform 4012.2.a.h.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} + \cdots - 2 $ x^15 - 5x^14 - 16x^13 + 123x^12 - 76x^11 - 630x^10 + 937x^9 + 1083x^8 - 2602x^7 - 249x^6 + 2736x^5 - 801x^4 - 900x^3 + 429x^2 - 36x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.346366$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.915888 q^{3} +1.84475 q^{5} -3.15913 q^{7} -2.16115 q^{9} +O(q^{10})$	$q+0.915888 q^{3} +1.84475 q^{5} -3.15913 q^{7} -2.16115 q^{9} +3.23665 q^{11} +0.662216 q^{13} +1.68958 q^{15} +1.00000 q^{17} +3.12338 q^{19} -2.89341 q^{21} -8.89145 q^{23} -1.59691 q^{25} -4.72703 q^{27} -7.93350 q^{29} -8.13921 q^{31} +2.96441 q^{33} -5.82779 q^{35} +8.26516 q^{37} +0.606516 q^{39} -8.91628 q^{41} -6.95317 q^{43} -3.98678 q^{45} +0.528689 q^{47} +2.98009 q^{49} +0.915888 q^{51} +13.1011 q^{53} +5.97080 q^{55} +2.86066 q^{57} -1.00000 q^{59} +2.06122 q^{61} +6.82735 q^{63} +1.22162 q^{65} -3.33917 q^{67} -8.14357 q^{69} -1.13637 q^{71} -10.2484 q^{73} -1.46259 q^{75} -10.2250 q^{77} +12.3047 q^{79} +2.15401 q^{81} +14.4855 q^{83} +1.84475 q^{85} -7.26619 q^{87} +3.07375 q^{89} -2.09203 q^{91} -7.45460 q^{93} +5.76184 q^{95} +4.23320 q^{97} -6.99488 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * 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
$3$	0.915888	0.528788	0.264394	−	0.964415i	$-0.414828\pi$
$3$	0.915888	0.528788	0.264394	+	0.964415i	$0.414828\pi$
$4$	0	0
$5$	1.84475	0.824996	0.412498	−	0.910958i	$-0.364656\pi$
$5$	1.84475	0.824996	0.412498	+	0.910958i	$0.364656\pi$
$6$	0	0
$7$	−3.15913	−1.19404	−0.597019	−	0.802227i	$-0.703648\pi$
$7$	−3.15913	−1.19404	−0.597019	+	0.802227i	$0.703648\pi$
$8$	0	0
$9$	−2.16115	−0.720383
$10$	0	0
$11$	3.23665	0.975887	0.487943	−	0.872875i	$-0.337747\pi$
$11$	3.23665	0.975887	0.487943	+	0.872875i	$0.337747\pi$
$12$	0	0
$13$	0.662216	0.183666	0.0918329	−	0.995774i	$-0.470727\pi$
$13$	0.662216	0.183666	0.0918329	+	0.995774i	$0.470727\pi$
$14$	0	0
$15$	1.68958	0.436248
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	3.12338	0.716552	0.358276	−	0.933616i	$-0.383365\pi$
$19$	3.12338	0.716552	0.358276	+	0.933616i	$0.383365\pi$
$20$	0	0
$21$	−2.89341	−0.631393
$22$	0	0
$23$	−8.89145	−1.85400	−0.926998	−	0.375067i	$-0.877620\pi$
$23$	−8.89145	−1.85400	−0.926998	+	0.375067i	$0.877620\pi$
$24$	0	0
$25$	−1.59691	−0.319381
$26$	0	0
$27$	−4.72703	−0.909718
$28$	0	0
$29$	−7.93350	−1.47321	−0.736607	−	0.676321i	$-0.763573\pi$
$29$	−7.93350	−1.47321	−0.736607	+	0.676321i	$0.763573\pi$
$30$	0	0
$31$	−8.13921	−1.46184	−0.730922	−	0.682461i	$-0.760910\pi$
$31$	−8.13921	−1.46184	−0.730922	+	0.682461i	$0.760910\pi$
$32$	0	0
$33$	2.96441	0.516037
$34$	0	0
$35$	−5.82779	−0.985077
$36$	0	0
$37$	8.26516	1.35878	0.679392	−	0.733776i	$-0.262244\pi$
$37$	8.26516	1.35878	0.679392	+	0.733776i	$0.262244\pi$
$38$	0	0
$39$	0.606516	0.0971203
$40$	0	0
$41$	−8.91628	−1.39249	−0.696244	−	0.717805i	$-0.745147\pi$
$41$	−8.91628	−1.39249	−0.696244	+	0.717805i	$0.745147\pi$
$42$	0	0
$43$	−6.95317	−1.06035	−0.530174	−	0.847889i	$-0.677873\pi$
$43$	−6.95317	−1.06035	−0.530174	+	0.847889i	$0.677873\pi$
$44$	0	0
$45$	−3.98678	−0.594313
$46$	0	0
$47$	0.528689	0.0771172	0.0385586	−	0.999256i	$-0.487723\pi$
$47$	0.528689	0.0771172	0.0385586	+	0.999256i	$0.487723\pi$
$48$	0	0
$49$	2.98009	0.425727
$50$	0	0
$51$	0.915888	0.128250
$52$	0	0
$53$	13.1011	1.79957	0.899786	−	0.436332i	$-0.143723\pi$
$53$	13.1011	1.79957	0.899786	+	0.436332i	$0.143723\pi$
$54$	0	0
$55$	5.97080	0.805103
$56$	0	0
$57$	2.86066	0.378904
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	2.06122	0.263912	0.131956	−	0.991256i	$-0.457874\pi$
$61$	2.06122	0.263912	0.131956	+	0.991256i	$0.457874\pi$
$62$	0	0
$63$	6.82735	0.860165
$64$	0	0
$65$	1.22162	0.151524
$66$	0	0
$67$	−3.33917	−0.407944	−0.203972	−	0.978977i	$-0.565385\pi$
$67$	−3.33917	−0.407944	−0.203972	+	0.978977i	$0.565385\pi$
$68$	0	0
$69$	−8.14357	−0.980371
$70$	0	0
$71$	−1.13637	−0.134862	−0.0674309	−	0.997724i	$-0.521480\pi$
$71$	−1.13637	−0.134862	−0.0674309	+	0.997724i	$0.521480\pi$
$72$	0	0
$73$	−10.2484	−1.19948	−0.599741	−	0.800194i	$-0.704730\pi$
$73$	−10.2484	−1.19948	−0.599741	+	0.800194i	$0.704730\pi$
$74$	0	0
$75$	−1.46259	−0.168885
$76$	0	0
$77$	−10.2250	−1.16525
$78$	0	0
$79$	12.3047	1.38438	0.692191	−	0.721714i	$-0.256646\pi$
$79$	12.3047	1.38438	0.692191	+	0.721714i	$0.256646\pi$
$80$	0	0
$81$	2.15401	0.239335
$82$	0	0
$83$	14.4855	1.58999	0.794995	−	0.606616i	$-0.207473\pi$
$83$	14.4855	1.58999	0.794995	+	0.606616i	$0.207473\pi$
$84$	0	0
$85$	1.84475	0.200091
$86$	0	0
$87$	−7.26619	−0.779018
$88$	0	0
$89$	3.07375	0.325817	0.162909	−	0.986641i	$-0.447912\pi$
$89$	3.07375	0.325817	0.162909	+	0.986641i	$0.447912\pi$
$90$	0	0
$91$	−2.09203	−0.219304
$92$	0	0
$93$	−7.45460	−0.773006
$94$	0	0
$95$	5.76184	0.591153
$96$	0	0
$97$	4.23320	0.429816	0.214908	−	0.976634i	$-0.431055\pi$
$97$	4.23320	0.429816	0.214908	+	0.976634i	$0.431055\pi$
$98$	0	0
$99$	−6.99488	−0.703012
$100$	0	0
$101$	−5.06678	−0.504164	−0.252082	−	0.967706i	$-0.581115\pi$
$101$	−5.06678	−0.504164	−0.252082	+	0.967706i	$0.581115\pi$
$102$	0	0
$103$	−10.1538	−1.00048	−0.500240	−	0.865887i	$-0.666755\pi$
$103$	−10.1538	−1.00048	−0.500240	+	0.865887i	$0.666755\pi$
$104$	0	0
$105$	−5.33761	−0.520897
$106$	0	0
$107$	−12.1166	−1.17135	−0.585677	−	0.810544i	$-0.699171\pi$
$107$	−12.1166	−1.17135	−0.585677	+	0.810544i	$0.699171\pi$
$108$	0	0
$109$	−10.5962	−1.01493	−0.507465	−	0.861672i	$-0.669417\pi$
$109$	−10.5962	−1.01493	−0.507465	+	0.861672i	$0.669417\pi$
$110$	0	0
$111$	7.56996	0.718508
$112$	0	0
$113$	0.622980	0.0586050	0.0293025	−	0.999571i	$-0.490671\pi$
$113$	0.622980	0.0586050	0.0293025	+	0.999571i	$0.490671\pi$
$114$	0	0
$115$	−16.4025	−1.52954
$116$	0	0
$117$	−1.43115	−0.132310
$118$	0	0
$119$	−3.15913	−0.289597
$120$	0	0
$121$	−0.524095	−0.0476450
$122$	0	0
$123$	−8.16631	−0.736332
$124$	0	0
$125$	−12.1696	−1.08848
$126$	0	0
$127$	15.8467	1.40617	0.703084	−	0.711107i	$-0.251806\pi$
$127$	15.8467	1.40617	0.703084	+	0.711107i	$0.251806\pi$
$128$	0	0
$129$	−6.36832	−0.560700
$130$	0	0
$131$	−11.6997	−1.02221	−0.511106	−	0.859518i	$-0.670764\pi$
$131$	−11.6997	−1.02221	−0.511106	+	0.859518i	$0.670764\pi$
$132$	0	0
$133$	−9.86715	−0.855590
$134$	0	0
$135$	−8.72019	−0.750514
$136$	0	0
$137$	−7.15525	−0.611314	−0.305657	−	0.952142i	$-0.598876\pi$
$137$	−7.15525	−0.611314	−0.305657	+	0.952142i	$0.598876\pi$
$138$	0	0
$139$	−22.0331	−1.86882	−0.934410	−	0.356199i	$-0.884073\pi$
$139$	−22.0331	−1.86882	−0.934410	+	0.356199i	$0.884073\pi$
$140$	0	0
$141$	0.484219	0.0407786
$142$	0	0
$143$	2.14336	0.179237
$144$	0	0
$145$	−14.6353	−1.21540
$146$	0	0
$147$	2.72943	0.225119
$148$	0	0
$149$	20.5196	1.68103	0.840517	−	0.541786i	$-0.182252\pi$
$149$	20.5196	1.68103	0.840517	+	0.541786i	$0.182252\pi$
$150$	0	0
$151$	12.8129	1.04270	0.521350	−	0.853343i	$-0.325429\pi$
$151$	12.8129	1.04270	0.521350	+	0.853343i	$0.325429\pi$
$152$	0	0
$153$	−2.16115	−0.174719
$154$	0	0
$155$	−15.0148	−1.20602
$156$	0	0
$157$	−8.73405	−0.697053	−0.348526	−	0.937299i	$-0.613318\pi$
$157$	−8.73405	−0.697053	−0.348526	+	0.937299i	$0.613318\pi$
$158$	0	0
$159$	11.9991	0.951592
$160$	0	0
$161$	28.0892	2.21374
$162$	0	0
$163$	6.52201	0.510843	0.255422	−	0.966830i	$-0.417786\pi$
$163$	6.52201	0.510843	0.255422	+	0.966830i	$0.417786\pi$
$164$	0	0
$165$	5.46859	0.425729
$166$	0	0
$167$	−17.4619	−1.35124	−0.675622	−	0.737248i	$-0.736125\pi$
$167$	−17.4619	−1.35124	−0.675622	+	0.737248i	$0.736125\pi$
$168$	0	0
$169$	−12.5615	−0.966267
$170$	0	0
$171$	−6.75008	−0.516192
$172$	0	0
$173$	1.70207	0.129406	0.0647031	−	0.997905i	$-0.479390\pi$
$173$	1.70207	0.129406	0.0647031	+	0.997905i	$0.479390\pi$
$174$	0	0
$175$	5.04483	0.381353
$176$	0	0
$177$	−0.915888	−0.0688424
$178$	0	0
$179$	−13.1415	−0.982239	−0.491120	−	0.871092i	$-0.663412\pi$
$179$	−13.1415	−0.982239	−0.491120	+	0.871092i	$0.663412\pi$
$180$	0	0
$181$	1.57990	0.117433	0.0587165	−	0.998275i	$-0.481299\pi$
$181$	1.57990	0.117433	0.0587165	+	0.998275i	$0.481299\pi$
$182$	0	0
$183$	1.88785	0.139554
$184$	0	0
$185$	15.2471	1.12099
$186$	0	0
$187$	3.23665	0.236687
$188$	0	0
$189$	14.9333	1.08624
$190$	0	0
$191$	−17.0454	−1.23336	−0.616681	−	0.787213i	$-0.711523\pi$
$191$	−17.0454	−1.23336	−0.616681	+	0.787213i	$0.711523\pi$
$192$	0	0
$193$	−8.21709	−0.591479	−0.295739	−	0.955269i	$-0.595566\pi$
$193$	−8.21709	−0.591479	−0.295739	+	0.955269i	$0.595566\pi$
$194$	0	0
$195$	1.11887	0.0801239
$196$	0	0
$197$	10.4939	0.747659	0.373829	−	0.927498i	$-0.378045\pi$
$197$	10.4939	0.747659	0.373829	+	0.927498i	$0.378045\pi$
$198$	0	0
$199$	−20.5757	−1.45857	−0.729285	−	0.684210i	$-0.760147\pi$
$199$	−20.5757	−1.45857	−0.729285	+	0.684210i	$0.760147\pi$
$200$	0	0
$201$	−3.05830	−0.215716
$202$	0	0
$203$	25.0629	1.75907
$204$	0	0
$205$	−16.4483	−1.14880
$206$	0	0
$207$	19.2158	1.33559
$208$	0	0
$209$	10.1093	0.699273
$210$	0	0
$211$	−24.1288	−1.66109	−0.830546	−	0.556950i	$-0.811972\pi$
$211$	−24.1288	−1.66109	−0.830546	+	0.556950i	$0.811972\pi$
$212$	0	0
$213$	−1.04078	−0.0713133
$214$	0	0
$215$	−12.8268	−0.874783
$216$	0	0
$217$	25.7128	1.74550
$218$	0	0
$219$	−9.38636	−0.634272
$220$	0	0
$221$	0.662216	0.0445455
$222$	0	0
$223$	3.66724	0.245577	0.122788	−	0.992433i	$-0.460816\pi$
$223$	3.66724	0.245577	0.122788	+	0.992433i	$0.460816\pi$
$224$	0	0
$225$	3.45115	0.230077
$226$	0	0
$227$	−15.9992	−1.06191	−0.530953	−	0.847401i	$-0.678166\pi$
$227$	−15.9992	−1.06191	−0.530953	+	0.847401i	$0.678166\pi$
$228$	0	0
$229$	15.8808	1.04944	0.524718	−	0.851276i	$-0.324171\pi$
$229$	15.8808	1.04944	0.524718	+	0.851276i	$0.324171\pi$
$230$	0	0
$231$	−9.36495	−0.616168
$232$	0	0
$233$	17.4989	1.14639	0.573195	−	0.819419i	$-0.305704\pi$
$233$	17.4989	1.14639	0.573195	+	0.819419i	$0.305704\pi$
$234$	0	0
$235$	0.975297	0.0636214
$236$	0	0
$237$	11.2697	0.732045
$238$	0	0
$239$	16.1933	1.04746	0.523730	−	0.851884i	$-0.324540\pi$
$239$	16.1933	1.04746	0.523730	+	0.851884i	$0.324540\pi$
$240$	0	0
$241$	−16.6706	−1.07385	−0.536923	−	0.843631i	$-0.680414\pi$
$241$	−16.6706	−1.07385	−0.536923	+	0.843631i	$0.680414\pi$
$242$	0	0
$243$	16.1539	1.03628
$244$	0	0
$245$	5.49751	0.351223
$246$	0	0
$247$	2.06835	0.131606
$248$	0	0
$249$	13.2671	0.840768
$250$	0	0
$251$	5.91556	0.373387	0.186693	−	0.982418i	$-0.440223\pi$
$251$	5.91556	0.373387	0.186693	+	0.982418i	$0.440223\pi$
$252$	0	0
$253$	−28.7785	−1.80929
$254$	0	0
$255$	1.68958	0.105806
$256$	0	0
$257$	−2.86587	−0.178768	−0.0893840	−	0.995997i	$-0.528490\pi$
$257$	−2.86587	−0.178768	−0.0893840	+	0.995997i	$0.528490\pi$
$258$	0	0
$259$	−26.1107	−1.62244
$260$	0	0
$261$	17.1455	1.06128
$262$	0	0
$263$	−12.0456	−0.742761	−0.371381	−	0.928481i	$-0.621115\pi$
$263$	−12.0456	−0.742761	−0.371381	+	0.928481i	$0.621115\pi$
$264$	0	0
$265$	24.1682	1.48464
$266$	0	0
$267$	2.81521	0.172288
$268$	0	0
$269$	23.1030	1.40862	0.704308	−	0.709894i	$-0.251257\pi$
$269$	23.1030	1.40862	0.704308	+	0.709894i	$0.251257\pi$
$270$	0	0
$271$	26.5571	1.61323	0.806614	−	0.591079i	$-0.201298\pi$
$271$	26.5571	1.61323	0.806614	+	0.591079i	$0.201298\pi$
$272$	0	0
$273$	−1.91606	−0.115965
$274$	0	0
$275$	−5.16863	−0.311680
$276$	0	0
$277$	−4.12100	−0.247607	−0.123804	−	0.992307i	$-0.539509\pi$
$277$	−4.12100	−0.247607	−0.123804	+	0.992307i	$0.539509\pi$
$278$	0	0
$279$	17.5900	1.05309
$280$	0	0
$281$	8.39833	0.501003	0.250501	−	0.968116i	$-0.419405\pi$
$281$	8.39833	0.501003	0.250501	+	0.968116i	$0.419405\pi$
$282$	0	0
$283$	−2.61306	−0.155330	−0.0776650	−	0.996980i	$-0.524746\pi$
$283$	−2.61306	−0.155330	−0.0776650	+	0.996980i	$0.524746\pi$
$284$	0	0
$285$	5.27720	0.312594
$286$	0	0
$287$	28.1677	1.66268
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	3.87713	0.227282
$292$	0	0
$293$	11.6077	0.678130	0.339065	−	0.940763i	$-0.389889\pi$
$293$	11.6077	0.678130	0.339065	+	0.940763i	$0.389889\pi$
$294$	0	0
$295$	−1.84475	−0.107405
$296$	0	0
$297$	−15.2998	−0.887782
$298$	0	0
$299$	−5.88806	−0.340516
$300$	0	0
$301$	21.9659	1.26610
$302$	0	0
$303$	−4.64060	−0.266596
$304$	0	0
$305$	3.80243	0.217727
$306$	0	0
$307$	−12.5964	−0.718914	−0.359457	−	0.933162i	$-0.617038\pi$
$307$	−12.5964	−0.718914	−0.359457	+	0.933162i	$0.617038\pi$
$308$	0	0
$309$	−9.29971	−0.529042
$310$	0	0
$311$	15.3210	0.868775	0.434387	−	0.900726i	$-0.356965\pi$
$311$	15.3210	0.868775	0.434387	+	0.900726i	$0.356965\pi$
$312$	0	0
$313$	−0.926117	−0.0523472	−0.0261736	−	0.999657i	$-0.508332\pi$
$313$	−0.926117	−0.0523472	−0.0261736	+	0.999657i	$0.508332\pi$
$314$	0	0
$315$	12.5947	0.709633
$316$	0	0
$317$	−4.50320	−0.252925	−0.126462	−	0.991971i	$-0.540362\pi$
$317$	−4.50320	−0.252925	−0.126462	+	0.991971i	$0.540362\pi$
$318$	0	0
$319$	−25.6780	−1.43769
$320$	0	0
$321$	−11.0974	−0.619398
$322$	0	0
$323$	3.12338	0.173789
$324$	0	0
$325$	−1.05750	−0.0586594
$326$	0	0
$327$	−9.70492	−0.536683
$328$	0	0
$329$	−1.67019	−0.0920808
$330$	0	0
$331$	−21.3346	−1.17265	−0.586327	−	0.810074i	$-0.699427\pi$
$331$	−21.3346	−1.17265	−0.586327	+	0.810074i	$0.699427\pi$
$332$	0	0
$333$	−17.8622	−0.978844
$334$	0	0
$335$	−6.15992	−0.336552
$336$	0	0
$337$	−17.9930	−0.980141	−0.490071	−	0.871683i	$-0.663029\pi$
$337$	−17.9930	−0.980141	−0.490071	+	0.871683i	$0.663029\pi$
$338$	0	0
$339$	0.570580	0.0309896
$340$	0	0
$341$	−26.3438	−1.42659
$342$	0	0
$343$	12.6994	0.685704
$344$	0	0
$345$	−15.0228	−0.808802
$346$	0	0
$347$	10.2855	0.552154	0.276077	−	0.961136i	$-0.410966\pi$
$347$	10.2855	0.552154	0.276077	+	0.961136i	$0.410966\pi$
$348$	0	0
$349$	31.1025	1.66488	0.832439	−	0.554117i	$-0.186944\pi$
$349$	31.1025	1.66488	0.832439	+	0.554117i	$0.186944\pi$
$350$	0	0
$351$	−3.13032	−0.167084
$352$	0	0
$353$	−11.6021	−0.617520	−0.308760	−	0.951140i	$-0.599914\pi$
$353$	−11.6021	−0.617520	−0.308760	+	0.951140i	$0.599914\pi$
$354$	0	0
$355$	−2.09631	−0.111260
$356$	0	0
$357$	−2.89341	−0.153135
$358$	0	0
$359$	−16.1530	−0.852521	−0.426260	−	0.904601i	$-0.640169\pi$
$359$	−16.1530	−0.852521	−0.426260	+	0.904601i	$0.640169\pi$
$360$	0	0
$361$	−9.24451	−0.486553
$362$	0	0
$363$	−0.480013	−0.0251941
$364$	0	0
$365$	−18.9057	−0.989568
$366$	0	0
$367$	−25.8078	−1.34716	−0.673578	−	0.739116i	$-0.735243\pi$
$367$	−25.8078	−1.34716	−0.673578	+	0.739116i	$0.735243\pi$
$368$	0	0
$369$	19.2694	1.00313
$370$	0	0
$371$	−41.3880	−2.14876
$372$	0	0
$373$	−29.3609	−1.52025	−0.760124	−	0.649778i	$-0.774862\pi$
$373$	−29.3609	−1.52025	−0.760124	+	0.649778i	$0.774862\pi$
$374$	0	0
$375$	−11.1460	−0.575578
$376$	0	0
$377$	−5.25369	−0.270579
$378$	0	0
$379$	13.2235	0.679244	0.339622	−	0.940562i	$-0.389701\pi$
$379$	13.2235	0.679244	0.339622	+	0.940562i	$0.389701\pi$
$380$	0	0
$381$	14.5138	0.743565
$382$	0	0
$383$	−9.47898	−0.484353	−0.242177	−	0.970232i	$-0.577861\pi$
$383$	−9.47898	−0.484353	−0.242177	+	0.970232i	$0.577861\pi$
$384$	0	0
$385$	−18.8625	−0.961324
$386$	0	0
$387$	15.0268	0.763857
$388$	0	0
$389$	15.7161	0.796835	0.398418	−	0.917204i	$-0.369559\pi$
$389$	15.7161	0.796835	0.398418	+	0.917204i	$0.369559\pi$
$390$	0	0
$391$	−8.89145	−0.449660
$392$	0	0
$393$	−10.7157	−0.540533
$394$	0	0
$395$	22.6990	1.14211
$396$	0	0
$397$	15.3591	0.770850	0.385425	−	0.922739i	$-0.374055\pi$
$397$	15.3591	0.770850	0.385425	+	0.922739i	$0.374055\pi$
$398$	0	0
$399$	−9.03720	−0.452426
$400$	0	0
$401$	25.2749	1.26217	0.631084	−	0.775715i	$-0.282610\pi$
$401$	25.2749	1.26217	0.631084	+	0.775715i	$0.282610\pi$
$402$	0	0
$403$	−5.38992	−0.268491
$404$	0	0
$405$	3.97361	0.197450
$406$	0	0
$407$	26.7514	1.32602
$408$	0	0
$409$	25.0728	1.23977	0.619884	−	0.784693i	$-0.287180\pi$
$409$	25.0728	1.23977	0.619884	+	0.784693i	$0.287180\pi$
$410$	0	0
$411$	−6.55341	−0.323256
$412$	0	0
$413$	3.15913	0.155451
$414$	0	0
$415$	26.7221	1.31174
$416$	0	0
$417$	−20.1798	−0.988210
$418$	0	0
$419$	29.5888	1.44551	0.722754	−	0.691105i	$-0.242876\pi$
$419$	29.5888	1.44551	0.722754	+	0.691105i	$0.242876\pi$
$420$	0	0
$421$	2.54542	0.124056	0.0620281	−	0.998074i	$-0.480243\pi$
$421$	2.54542	0.124056	0.0620281	+	0.998074i	$0.480243\pi$
$422$	0	0
$423$	−1.14257	−0.0555539
$424$	0	0
$425$	−1.59691	−0.0774613
$426$	0	0
$427$	−6.51166	−0.315121
$428$	0	0
$429$	1.96308	0.0947784
$430$	0	0
$431$	35.0466	1.68814	0.844068	−	0.536237i	$-0.180155\pi$
$431$	35.0466	1.68814	0.844068	+	0.536237i	$0.180155\pi$
$432$	0	0
$433$	31.1157	1.49532	0.747662	−	0.664080i	$-0.231176\pi$
$433$	31.1157	1.49532	0.747662	+	0.664080i	$0.231176\pi$
$434$	0	0
$435$	−13.4043	−0.642687
$436$	0	0
$437$	−27.7714	−1.32848
$438$	0	0
$439$	15.1219	0.721727	0.360864	−	0.932619i	$-0.382482\pi$
$439$	15.1219	0.721727	0.360864	+	0.932619i	$0.382482\pi$
$440$	0	0
$441$	−6.44042	−0.306687
$442$	0	0
$443$	−40.8545	−1.94105	−0.970527	−	0.240992i	$-0.922527\pi$
$443$	−40.8545	−1.94105	−0.970527	+	0.240992i	$0.922527\pi$
$444$	0	0
$445$	5.67030	0.268798
$446$	0	0
$447$	18.7937	0.888910
$448$	0	0
$449$	9.58763	0.452468	0.226234	−	0.974073i	$-0.427359\pi$
$449$	9.58763	0.452468	0.226234	+	0.974073i	$0.427359\pi$
$450$	0	0
$451$	−28.8589	−1.35891
$452$	0	0
$453$	11.7352	0.551367
$454$	0	0
$455$	−3.85926	−0.180925
$456$	0	0
$457$	11.2685	0.527117	0.263558	−	0.964643i	$-0.415104\pi$
$457$	11.2685	0.527117	0.263558	+	0.964643i	$0.415104\pi$
$458$	0	0
$459$	−4.72703	−0.220639
$460$	0	0
$461$	17.7799	0.828093	0.414046	−	0.910256i	$-0.364115\pi$
$461$	17.7799	0.828093	0.414046	+	0.910256i	$0.364115\pi$
$462$	0	0
$463$	−26.2577	−1.22030	−0.610149	−	0.792287i	$-0.708890\pi$
$463$	−26.2577	−1.22030	−0.610149	+	0.792287i	$0.708890\pi$
$464$	0	0
$465$	−13.7519	−0.637727
$466$	0	0
$467$	−19.8824	−0.920048	−0.460024	−	0.887906i	$-0.652159\pi$
$467$	−19.8824	−0.920048	−0.460024	+	0.887906i	$0.652159\pi$
$468$	0	0
$469$	10.5489	0.487101
$470$	0	0
$471$	−7.99941	−0.368593
$472$	0	0
$473$	−22.5050	−1.03478
$474$	0	0
$475$	−4.98774	−0.228853
$476$	0	0
$477$	−28.3134	−1.29638
$478$	0	0
$479$	1.34892	0.0616335	0.0308168	−	0.999525i	$-0.490189\pi$
$479$	1.34892	0.0616335	0.0308168	+	0.999525i	$0.490189\pi$
$480$	0	0
$481$	5.47332	0.249562
$482$	0	0
$483$	25.7266	1.17060
$484$	0	0
$485$	7.80918	0.354597
$486$	0	0
$487$	13.3040	0.602863	0.301431	−	0.953488i	$-0.402536\pi$
$487$	13.3040	0.602863	0.301431	+	0.953488i	$0.402536\pi$
$488$	0	0
$489$	5.97343	0.270128
$490$	0	0
$491$	−26.2329	−1.18387	−0.591937	−	0.805984i	$-0.701637\pi$
$491$	−26.2329	−1.18387	−0.591937	+	0.805984i	$0.701637\pi$
$492$	0	0
$493$	−7.93350	−0.357307
$494$	0	0
$495$	−12.9038	−0.579983
$496$	0	0
$497$	3.58992	0.161030
$498$	0	0
$499$	15.3003	0.684935	0.342468	−	0.939530i	$-0.388737\pi$
$499$	15.3003	0.684935	0.342468	+	0.939530i	$0.388737\pi$
$500$	0	0
$501$	−15.9932	−0.714522
$502$	0	0
$503$	22.9219	1.02204	0.511018	−	0.859570i	$-0.329269\pi$
$503$	22.9219	1.02204	0.511018	+	0.859570i	$0.329269\pi$
$504$	0	0
$505$	−9.34693	−0.415933
$506$	0	0
$507$	−11.5049	−0.510950
$508$	0	0
$509$	5.50940	0.244200	0.122100	−	0.992518i	$-0.461037\pi$
$509$	5.50940	0.244200	0.122100	+	0.992518i	$0.461037\pi$
$510$	0	0
$511$	32.3759	1.43223
$512$	0	0
$513$	−14.7643	−0.651860
$514$	0	0
$515$	−18.7311	−0.825392
$516$	0	0
$517$	1.71118	0.0752576
$518$	0	0
$519$	1.55891	0.0684285
$520$	0	0
$521$	−21.0960	−0.924233	−0.462117	−	0.886819i	$-0.652910\pi$
$521$	−21.0960	−0.924233	−0.462117	+	0.886819i	$0.652910\pi$
$522$	0	0
$523$	−21.8246	−0.954324	−0.477162	−	0.878815i	$-0.658334\pi$
$523$	−21.8246	−0.954324	−0.477162	+	0.878815i	$0.658334\pi$
$524$	0	0
$525$	4.62050	0.201655
$526$	0	0
$527$	−8.13921	−0.354549
$528$	0	0
$529$	56.0579	2.43730
$530$	0	0
$531$	2.16115	0.0937859
$532$	0	0
$533$	−5.90451	−0.255753
$534$	0	0
$535$	−22.3520	−0.966363
$536$	0	0
$537$	−12.0361	−0.519396
$538$	0	0
$539$	9.64551	0.415461
$540$	0	0
$541$	−8.09307	−0.347948	−0.173974	−	0.984750i	$-0.555661\pi$
$541$	−8.09307	−0.347948	−0.173974	+	0.984750i	$0.555661\pi$
$542$	0	0
$543$	1.44701	0.0620972
$544$	0	0
$545$	−19.5473	−0.837314
$546$	0	0
$547$	31.7574	1.35785	0.678924	−	0.734209i	$-0.262447\pi$
$547$	31.7574	1.35785	0.678924	+	0.734209i	$0.262447\pi$
$548$	0	0
$549$	−4.45461	−0.190118
$550$	0	0
$551$	−24.7793	−1.05563
$552$	0	0
$553$	−38.8720	−1.65300
$554$	0	0
$555$	13.9647	0.592767
$556$	0	0
$557$	−38.2911	−1.62245	−0.811223	−	0.584737i	$-0.801198\pi$
$557$	−38.2911	−1.62245	−0.811223	+	0.584737i	$0.801198\pi$
$558$	0	0
$559$	−4.60450	−0.194750
$560$	0	0
$561$	2.96441	0.125157
$562$	0	0
$563$	37.5210	1.58132	0.790660	−	0.612256i	$-0.209738\pi$
$563$	37.5210	1.58132	0.790660	+	0.612256i	$0.209738\pi$
$564$	0	0
$565$	1.14924	0.0483489
$566$	0	0
$567$	−6.80481	−0.285775
$568$	0	0
$569$	43.0682	1.80551	0.902756	−	0.430153i	$-0.141540\pi$
$569$	43.0682	1.80551	0.902756	+	0.430153i	$0.141540\pi$
$570$	0	0
$571$	−26.2991	−1.10058	−0.550291	−	0.834973i	$-0.685483\pi$
$571$	−26.2991	−1.10058	−0.550291	+	0.834973i	$0.685483\pi$
$572$	0	0
$573$	−15.6117	−0.652187
$574$	0	0
$575$	14.1988	0.592131
$576$	0	0
$577$	7.20536	0.299963	0.149981	−	0.988689i	$-0.452079\pi$
$577$	7.20536	0.299963	0.149981	+	0.988689i	$0.452079\pi$
$578$	0	0
$579$	−7.52593	−0.312767
$580$	0	0
$581$	−45.7616	−1.89851
$582$	0	0
$583$	42.4036	1.75618
$584$	0	0
$585$	−2.64011	−0.109155
$586$	0	0
$587$	−43.4097	−1.79171	−0.895855	−	0.444347i	$-0.853436\pi$
$587$	−43.4097	−1.79171	−0.895855	+	0.444347i	$0.853436\pi$
$588$	0	0
$589$	−25.4218	−1.04749
$590$	0	0
$591$	9.61122	0.395353
$592$	0	0
$593$	6.78778	0.278741	0.139370	−	0.990240i	$-0.455492\pi$
$593$	6.78778	0.278741	0.139370	+	0.990240i	$0.455492\pi$
$594$	0	0
$595$	−5.82779	−0.238916
$596$	0	0
$597$	−18.8450	−0.771275
$598$	0	0
$599$	−3.00322	−0.122708	−0.0613542	−	0.998116i	$-0.519542\pi$
$599$	−3.00322	−0.122708	−0.0613542	+	0.998116i	$0.519542\pi$
$600$	0	0
$601$	31.0452	1.26636	0.633180	−	0.774005i	$-0.281749\pi$
$601$	31.0452	1.26636	0.633180	+	0.774005i	$0.281749\pi$
$602$	0	0
$603$	7.21643	0.293876
$604$	0	0
$605$	−0.966824	−0.0393070
$606$	0	0
$607$	−19.5575	−0.793816	−0.396908	−	0.917858i	$-0.629917\pi$
$607$	−19.5575	−0.793816	−0.396908	+	0.917858i	$0.629917\pi$
$608$	0	0
$609$	22.9548	0.930177
$610$	0	0
$611$	0.350106	0.0141638
$612$	0	0
$613$	5.81296	0.234783	0.117392	−	0.993086i	$-0.462547\pi$
$613$	5.81296	0.234783	0.117392	+	0.993086i	$0.462547\pi$
$614$	0	0
$615$	−15.0648	−0.607471
$616$	0	0
$617$	9.03058	0.363557	0.181779	−	0.983339i	$-0.441815\pi$
$617$	9.03058	0.363557	0.181779	+	0.983339i	$0.441815\pi$
$618$	0	0
$619$	19.2424	0.773416	0.386708	−	0.922202i	$-0.373612\pi$
$619$	19.2424	0.773416	0.386708	+	0.922202i	$0.373612\pi$
$620$	0	0
$621$	42.0302	1.68661
$622$	0	0
$623$	−9.71038	−0.389038
$624$	0	0
$625$	−14.4654	−0.578614
$626$	0	0
$627$	9.25897	0.369768
$628$	0	0
$629$	8.26516	0.329553
$630$	0	0
$631$	43.2804	1.72296	0.861482	−	0.507788i	$-0.169537\pi$
$631$	43.2804	1.72296	0.861482	+	0.507788i	$0.169537\pi$
$632$	0	0
$633$	−22.0992	−0.878366
$634$	0	0
$635$	29.2332	1.16008
$636$	0	0
$637$	1.97346	0.0781915
$638$	0	0
$639$	2.45585	0.0971521
$640$	0	0
$641$	25.3279	1.00039	0.500195	−	0.865913i	$-0.333261\pi$
$641$	25.3279	1.00039	0.500195	+	0.865913i	$0.333261\pi$
$642$	0	0
$643$	22.3660	0.882029	0.441015	−	0.897500i	$-0.354619\pi$
$643$	22.3660	0.882029	0.441015	+	0.897500i	$0.354619\pi$
$644$	0	0
$645$	−11.7479	−0.462575
$646$	0	0
$647$	−12.6427	−0.497038	−0.248519	−	0.968627i	$-0.579944\pi$
$647$	−12.6427	−0.497038	−0.248519	+	0.968627i	$0.579944\pi$
$648$	0	0
$649$	−3.23665	−0.127050
$650$	0	0
$651$	23.5500	0.922999
$652$	0	0
$653$	−9.93570	−0.388814	−0.194407	−	0.980921i	$-0.562278\pi$
$653$	−9.93570	−0.388814	−0.194407	+	0.980921i	$0.562278\pi$
$654$	0	0
$655$	−21.5831	−0.843320
$656$	0	0
$657$	22.1483	0.864086
$658$	0	0
$659$	−9.19330	−0.358120	−0.179060	−	0.983838i	$-0.557306\pi$
$659$	−9.19330	−0.358120	−0.179060	+	0.983838i	$0.557306\pi$
$660$	0	0
$661$	1.70312	0.0662438	0.0331219	−	0.999451i	$-0.489455\pi$
$661$	1.70312	0.0662438	0.0331219	+	0.999451i	$0.489455\pi$
$662$	0	0
$663$	0.606516	0.0235551
$664$	0	0
$665$	−18.2024	−0.705859
$666$	0	0
$667$	70.5403	2.73133
$668$	0	0
$669$	3.35878	0.129858
$670$	0	0
$671$	6.67145	0.257549
$672$	0	0
$673$	6.82245	0.262986	0.131493	−	0.991317i	$-0.458023\pi$
$673$	6.82245	0.262986	0.131493	+	0.991317i	$0.458023\pi$
$674$	0	0
$675$	7.54863	0.290547
$676$	0	0
$677$	−27.5768	−1.05986	−0.529931	−	0.848041i	$-0.677782\pi$
$677$	−27.5768	−1.05986	−0.529931	+	0.848041i	$0.677782\pi$
$678$	0	0
$679$	−13.3732	−0.513217
$680$	0	0
$681$	−14.6535	−0.561523
$682$	0	0
$683$	32.6125	1.24788	0.623941	−	0.781472i	$-0.285531\pi$
$683$	32.6125	1.24788	0.623941	+	0.781472i	$0.285531\pi$
$684$	0	0
$685$	−13.1996	−0.504332
$686$	0	0
$687$	14.5451	0.554929
$688$	0	0
$689$	8.67575	0.330520
$690$	0	0
$691$	−9.21148	−0.350421	−0.175211	−	0.984531i	$-0.556061\pi$
$691$	−9.21148	−0.350421	−0.175211	+	0.984531i	$0.556061\pi$
$692$	0	0
$693$	22.0977	0.839424
$694$	0	0
$695$	−40.6455	−1.54177
$696$	0	0
$697$	−8.91628	−0.337728
$698$	0	0
$699$	16.0270	0.606198
$700$	0	0
$701$	−34.3991	−1.29924	−0.649618	−	0.760261i	$-0.725071\pi$
$701$	−34.3991	−1.29924	−0.649618	+	0.760261i	$0.725071\pi$
$702$	0	0
$703$	25.8152	0.973639
$704$	0	0
$705$	0.893263	0.0336422
$706$	0	0
$707$	16.0066	0.601991
$708$	0	0
$709$	33.7300	1.26676	0.633378	−	0.773843i	$-0.281668\pi$
$709$	33.7300	1.26676	0.633378	+	0.773843i	$0.281668\pi$
$710$	0	0
$711$	−26.5922	−0.997285
$712$	0	0
$713$	72.3694	2.71025
$714$	0	0
$715$	3.95396	0.147870
$716$	0	0
$717$	14.8313	0.553885
$718$	0	0
$719$	−15.6047	−0.581956	−0.290978	−	0.956730i	$-0.593981\pi$
$719$	−15.6047	−0.581956	−0.290978	+	0.956730i	$0.593981\pi$
$720$	0	0
$721$	32.0770	1.19461
$722$	0	0
$723$	−15.2684	−0.567837
$724$	0	0
$725$	12.6691	0.470517
$726$	0	0
$727$	−2.83578	−0.105173	−0.0525866	−	0.998616i	$-0.516747\pi$
$727$	−2.83578	−0.105173	−0.0525866	+	0.998616i	$0.516747\pi$
$728$	0	0
$729$	8.33315	0.308635
$730$	0	0
$731$	−6.95317	−0.257172
$732$	0	0
$733$	7.17556	0.265035	0.132518	−	0.991181i	$-0.457694\pi$
$733$	7.17556	0.265035	0.132518	+	0.991181i	$0.457694\pi$
$734$	0	0
$735$	5.03511	0.185723
$736$	0	0
$737$	−10.8077	−0.398107
$738$	0	0
$739$	26.4331	0.972358	0.486179	−	0.873859i	$-0.338390\pi$
$739$	26.4331	0.972358	0.486179	+	0.873859i	$0.338390\pi$
$740$	0	0
$741$	1.89438	0.0695917
$742$	0	0
$743$	−30.4434	−1.11686	−0.558431	−	0.829551i	$-0.688596\pi$
$743$	−30.4434	−1.11686	−0.558431	+	0.829551i	$0.688596\pi$
$744$	0	0
$745$	37.8535	1.38685
$746$	0	0
$747$	−31.3053	−1.14540
$748$	0	0
$749$	38.2778	1.39864
$750$	0	0
$751$	−41.2984	−1.50700	−0.753500	−	0.657448i	$-0.771636\pi$
$751$	−41.2984	−1.50700	−0.753500	+	0.657448i	$0.771636\pi$
$752$	0	0
$753$	5.41799	0.197442
$754$	0	0
$755$	23.6366	0.860223
$756$	0	0
$757$	−1.25106	−0.0454706	−0.0227353	−	0.999742i	$-0.507237\pi$
$757$	−1.25106	−0.0454706	−0.0227353	+	0.999742i	$0.507237\pi$
$758$	0	0
$759$	−26.3579	−0.956731
$760$	0	0
$761$	−9.22534	−0.334418	−0.167209	−	0.985921i	$-0.553476\pi$
$761$	−9.22534	−0.334418	−0.167209	+	0.985921i	$0.553476\pi$
$762$	0	0
$763$	33.4747	1.21187
$764$	0	0
$765$	−3.98678	−0.144142
$766$	0	0
$767$	−0.662216	−0.0239112
$768$	0	0
$769$	21.5674	0.777740	0.388870	−	0.921293i	$-0.372866\pi$
$769$	21.5674	0.777740	0.388870	+	0.921293i	$0.372866\pi$
$770$	0	0
$771$	−2.62482	−0.0945304
$772$	0	0
$773$	23.4891	0.844845	0.422423	−	0.906399i	$-0.361180\pi$
$773$	23.4891	0.844845	0.422423	+	0.906399i	$0.361180\pi$
$774$	0	0
$775$	12.9975	0.466886
$776$	0	0
$777$	−23.9145	−0.857926
$778$	0	0
$779$	−27.8489	−0.997790
$780$	0	0
$781$	−3.67802	−0.131610
$782$	0	0
$783$	37.5019	1.34021
$784$	0	0
$785$	−16.1121	−0.575066
$786$	0	0
$787$	47.4743	1.69227	0.846137	−	0.532965i	$-0.178922\pi$
$787$	47.4743	1.69227	0.846137	+	0.532965i	$0.178922\pi$
$788$	0	0
$789$	−11.0324	−0.392763
$790$	0	0
$791$	−1.96807	−0.0699766
$792$	0	0
$793$	1.36497	0.0484717
$794$	0	0
$795$	22.1354	0.785060
$796$	0	0
$797$	−22.6671	−0.802910	−0.401455	−	0.915879i	$-0.631495\pi$
$797$	−22.6671	−0.802910	−0.401455	+	0.915879i	$0.631495\pi$
$798$	0	0
$799$	0.528689	0.0187037
$800$	0	0
$801$	−6.64284	−0.234713
$802$	0	0
$803$	−33.1704	−1.17056
$804$	0	0
$805$	51.8175	1.82633
$806$	0	0
$807$	21.1598	0.744860
$808$	0	0
$809$	35.8361	1.25993	0.629965	−	0.776624i	$-0.283069\pi$
$809$	35.8361	1.25993	0.629965	+	0.776624i	$0.283069\pi$
$810$	0	0
$811$	22.7993	0.800591	0.400296	−	0.916386i	$-0.368907\pi$
$811$	22.7993	0.800591	0.400296	+	0.916386i	$0.368907\pi$
$812$	0	0
$813$	24.3233	0.853056
$814$	0	0
$815$	12.0315	0.421444
$816$	0	0
$817$	−21.7174	−0.759795
$818$	0	0
$819$	4.52118	0.157983
$820$	0	0
$821$	33.7564	1.17811	0.589053	−	0.808094i	$-0.299501\pi$
$821$	33.7564	1.17811	0.589053	+	0.808094i	$0.299501\pi$
$822$	0	0
$823$	11.2393	0.391777	0.195888	−	0.980626i	$-0.437241\pi$
$823$	11.2393	0.391777	0.195888	+	0.980626i	$0.437241\pi$
$824$	0	0
$825$	−4.73388	−0.164813
$826$	0	0
$827$	−50.5755	−1.75868	−0.879341	−	0.476192i	$-0.842017\pi$
$827$	−50.5755	−1.75868	−0.879341	+	0.476192i	$0.842017\pi$
$828$	0	0
$829$	−15.7224	−0.546063	−0.273032	−	0.962005i	$-0.588026\pi$
$829$	−15.7224	−0.546063	−0.273032	+	0.962005i	$0.588026\pi$
$830$	0	0
$831$	−3.77438	−0.130932
$832$	0	0
$833$	2.98009	0.103254
$834$	0	0
$835$	−32.2128	−1.11477
$836$	0	0
$837$	38.4743	1.32987
$838$	0	0
$839$	23.2404	0.802348	0.401174	−	0.916002i	$-0.368602\pi$
$839$	23.2404	0.802348	0.401174	+	0.916002i	$0.368602\pi$
$840$	0	0
$841$	33.9404	1.17036
$842$	0	0
$843$	7.69193	0.264924
$844$	0	0
$845$	−23.1727	−0.797167
$846$	0	0
$847$	1.65568	0.0568900
$848$	0	0
$849$	−2.39327	−0.0821367
$850$	0	0
$851$	−73.4892	−2.51918
$852$	0	0
$853$	−18.3343	−0.627755	−0.313878	−	0.949463i	$-0.601628\pi$
$853$	−18.3343	−0.627755	−0.313878	+	0.949463i	$0.601628\pi$
$854$	0	0
$855$	−12.4522	−0.425856
$856$	0	0
$857$	4.11023	0.140403	0.0702013	−	0.997533i	$-0.477636\pi$
$857$	4.11023	0.140403	0.0702013	+	0.997533i	$0.477636\pi$
$858$	0	0
$859$	12.6289	0.430894	0.215447	−	0.976516i	$-0.430879\pi$
$859$	12.6289	0.430894	0.215447	+	0.976516i	$0.430879\pi$
$860$	0	0
$861$	25.7984	0.879208
$862$	0	0
$863$	−21.7030	−0.738779	−0.369389	−	0.929275i	$-0.620433\pi$
$863$	−21.7030	−0.738779	−0.369389	+	0.929275i	$0.620433\pi$
$864$	0	0
$865$	3.13989	0.106760
$866$	0	0
$867$	0.915888	0.0311052
$868$	0	0
$869$	39.8259	1.35100
$870$	0	0
$871$	−2.21125	−0.0749253
$872$	0	0
$873$	−9.14857	−0.309632
$874$	0	0
$875$	38.4454	1.29969
$876$	0	0
$877$	−11.0161	−0.371988	−0.185994	−	0.982551i	$-0.559551\pi$
$877$	−11.0161	−0.371988	−0.185994	+	0.982551i	$0.559551\pi$
$878$	0	0
$879$	10.6314	0.358587
$880$	0	0
$881$	−40.8123	−1.37500	−0.687501	−	0.726183i	$-0.741292\pi$
$881$	−40.8123	−1.37500	−0.687501	+	0.726183i	$0.741292\pi$
$882$	0	0
$883$	−9.20501	−0.309773	−0.154887	−	0.987932i	$-0.549501\pi$
$883$	−9.20501	−0.309773	−0.154887	+	0.987932i	$0.549501\pi$
$884$	0	0
$885$	−1.68958	−0.0567947
$886$	0	0
$887$	−43.5379	−1.46186	−0.730930	−	0.682452i	$-0.760913\pi$
$887$	−43.5379	−1.46186	−0.730930	+	0.682452i	$0.760913\pi$
$888$	0	0
$889$	−50.0618	−1.67902
$890$	0	0
$891$	6.97179	0.233564
$892$	0	0
$893$	1.65129	0.0552584
$894$	0	0
$895$	−24.2427	−0.810344
$896$	0	0
$897$	−5.39281	−0.180061
$898$	0	0
$899$	64.5724	2.15361
$900$	0	0
$901$	13.1011	0.436460
$902$	0	0
$903$	20.1183	0.669497
$904$	0	0
$905$	2.91452	0.0968819
$906$	0	0
$907$	−18.0233	−0.598455	−0.299228	−	0.954182i	$-0.596729\pi$
$907$	−18.0233	−0.598455	−0.299228	+	0.954182i	$0.596729\pi$
$908$	0	0
$909$	10.9501	0.363191
$910$	0	0
$911$	35.2584	1.16816	0.584081	−	0.811695i	$-0.301455\pi$
$911$	35.2584	1.16816	0.584081	+	0.811695i	$0.301455\pi$
$912$	0	0
$913$	46.8845	1.55165
$914$	0	0
$915$	3.48260	0.115131
$916$	0	0
$917$	36.9610	1.22056
$918$	0	0
$919$	48.1478	1.58825	0.794124	−	0.607756i	$-0.207930\pi$
$919$	48.1478	1.58825	0.794124	+	0.607756i	$0.207930\pi$
$920$	0	0
$921$	−11.5369	−0.380153
$922$	0	0
$923$	−0.752520	−0.0247695
$924$	0	0
$925$	−13.1987	−0.433970
$926$	0	0
$927$	21.9438	0.720729
$928$	0	0
$929$	24.0237	0.788193	0.394096	−	0.919069i	$-0.371058\pi$
$929$	24.0237	0.788193	0.394096	+	0.919069i	$0.371058\pi$
$930$	0	0
$931$	9.30794	0.305056
$932$	0	0
$933$	14.0323	0.459398
$934$	0	0
$935$	5.97080	0.195266
$936$	0	0
$937$	38.4967	1.25763	0.628816	−	0.777554i	$-0.283540\pi$
$937$	38.4967	1.25763	0.628816	+	0.777554i	$0.283540\pi$
$938$	0	0
$939$	−0.848220	−0.0276806
$940$	0	0
$941$	−12.6082	−0.411015	−0.205508	−	0.978656i	$-0.565885\pi$
$941$	−12.6082	−0.411015	−0.205508	+	0.978656i	$0.565885\pi$
$942$	0	0
$943$	79.2787	2.58167
$944$	0	0
$945$	27.5482	0.896142
$946$	0	0
$947$	−36.4841	−1.18557	−0.592786	−	0.805360i	$-0.701972\pi$
$947$	−36.4841	−1.18557	−0.592786	+	0.805360i	$0.701972\pi$
$948$	0	0
$949$	−6.78664	−0.220304
$950$	0	0
$951$	−4.12443	−0.133744
$952$	0	0
$953$	−58.0365	−1.87999	−0.939994	−	0.341192i	$-0.889170\pi$
$953$	−58.0365	−1.87999	−0.939994	+	0.341192i	$0.889170\pi$
$954$	0	0
$955$	−31.4445	−1.01752
$956$	0	0
$957$	−23.5181	−0.760233
$958$	0	0
$959$	22.6044	0.729933
$960$	0	0
$961$	35.2467	1.13699
$962$	0	0
$963$	26.1858	0.843824
$964$	0	0
$965$	−15.1585	−0.487968
$966$	0	0
$967$	−31.8553	−1.02440	−0.512199	−	0.858867i	$-0.671169\pi$
$967$	−31.8553	−1.02440	−0.512199	+	0.858867i	$0.671169\pi$
$968$	0	0
$969$	2.86066	0.0918977
$970$	0	0
$971$	−51.0631	−1.63869	−0.819346	−	0.573299i	$-0.805663\pi$
$971$	−51.0631	−1.63869	−0.819346	+	0.573299i	$0.805663\pi$
$972$	0	0
$973$	69.6053	2.23144
$974$	0	0
$975$	−0.968549	−0.0310184
$976$	0	0
$977$	31.2803	1.00075	0.500373	−	0.865810i	$-0.333196\pi$
$977$	31.2803	1.00075	0.500373	+	0.865810i	$0.333196\pi$
$978$	0	0
$979$	9.94867	0.317961
$980$	0	0
$981$	22.8999	0.731139
$982$	0	0
$983$	17.1401	0.546684	0.273342	−	0.961917i	$-0.411871\pi$
$983$	17.1401	0.546684	0.273342	+	0.961917i	$0.411871\pi$
$984$	0	0
$985$	19.3586	0.616815
$986$	0	0
$987$	−1.52971	−0.0486912
$988$	0	0
$989$	61.8238	1.96588
$990$	0	0
$991$	−48.5412	−1.54196	−0.770980	−	0.636859i	$-0.780233\pi$
$991$	−48.5412	−1.54196	−0.770980	+	0.636859i	$0.780233\pi$
$992$	0	0
$993$	−19.5401	−0.620086
$994$	0	0
$995$	−37.9569	−1.20332
$996$	0	0
$997$	18.4598	0.584628	0.292314	−	0.956322i	$-0.405575\pi$
$997$	18.4598	0.584628	0.292314	+	0.956322i	$0.405575\pi$
$998$	0	0
$999$	−39.0697	−1.23611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.h.1.9	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.h.1.9	✓	15	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q+0.915888 q^{3} +1.84475 q^{5} -3.15913 q^{7} -2.16115 q^{9} +O(q^{10})\)	\(q+0.915888 q^{3} +1.84475 q^{5} -3.15913 q^{7} -2.16115 q^{9} +3.23665 q^{11} +0.662216 q^{13} +1.68958 q^{15} +1.00000 q^{17} +3.12338 q^{19} -2.89341 q^{21} -8.89145 q^{23} -1.59691 q^{25} -4.72703 q^{27} -7.93350 q^{29} -8.13921 q^{31} +2.96441 q^{33} -5.82779 q^{35} +8.26516 q^{37} +0.606516 q^{39} -8.91628 q^{41} -6.95317 q^{43} -3.98678 q^{45} +0.528689 q^{47} +2.98009 q^{49} +0.915888 q^{51} +13.1011 q^{53} +5.97080 q^{55} +2.86066 q^{57} -1.00000 q^{59} +2.06122 q^{61} +6.82735 q^{63} +1.22162 q^{65} -3.33917 q^{67} -8.14357 q^{69} -1.13637 q^{71} -10.2484 q^{73} -1.46259 q^{75} -10.2250 q^{77} +12.3047 q^{79} +2.15401 q^{81} +14.4855 q^{83} +1.84475 q^{85} -7.26619 q^{87} +3.07375 q^{89} -2.09203 q^{91} -7.45460 q^{93} +5.76184 q^{95} +4.23320 q^{97} -6.99488 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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