LMFDB - Embedded newform 4012.2.a.g.1.11

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 13 x^{10} + 60 x^{9} + 48 x^{8} - 289 x^{7} - 89 x^{6} + 602 x^{5} + 161 x^{4} + \cdots + 82 $ x^12 - 4x^11 - 13x^10 + 60x^9 + 48x^8 - 289x^7 - 89x^6 + 602x^5 + 161x^4 - 555x^3 - 197x^2 + 181*x + 82
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-1.70453$ 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)

$f(q)$	$=$	$q+1.70453 q^{3} -0.637713 q^{5} +1.14740 q^{7} -0.0945840 q^{9} +O(q^{10})$	$q+1.70453 q^{3} -0.637713 q^{5} +1.14740 q^{7} -0.0945840 q^{9} +0.485761 q^{11} -4.68138 q^{13} -1.08700 q^{15} -1.00000 q^{17} +1.90066 q^{19} +1.95578 q^{21} -0.312411 q^{23} -4.59332 q^{25} -5.27481 q^{27} -3.80295 q^{29} -1.49914 q^{31} +0.827993 q^{33} -0.731715 q^{35} +3.52029 q^{37} -7.97955 q^{39} -5.10566 q^{41} -2.74040 q^{43} +0.0603175 q^{45} +5.63321 q^{47} -5.68346 q^{49} -1.70453 q^{51} -5.67494 q^{53} -0.309776 q^{55} +3.23974 q^{57} +1.00000 q^{59} -11.8658 q^{61} -0.108526 q^{63} +2.98538 q^{65} +7.64584 q^{67} -0.532514 q^{69} +1.03706 q^{71} +10.9506 q^{73} -7.82945 q^{75} +0.557364 q^{77} -0.162183 q^{79} -8.70730 q^{81} -1.63200 q^{83} +0.637713 q^{85} -6.48224 q^{87} -18.8243 q^{89} -5.37144 q^{91} -2.55533 q^{93} -1.21208 q^{95} -14.5538 q^{97} -0.0459452 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9	$ 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9} - 5 q^{11} + 9 q^{13} - 16 q^{15} - 12 q^{17} - 15 q^{19} + 5 q^{21} - 22 q^{23} + 5 q^{25} - 16 q^{27} - 17 q^{29} + 5 q^{31} - 33 q^{33} - 9 q^{35} - 2 q^{37} - 24 q^{39} - 18 q^{41} - 16 q^{43} + 30 q^{45} - 60 q^{47} + 10 q^{49} + 4 q^{51} - 24 q^{53} - q^{55} + 41 q^{57} + 12 q^{59} - 17 q^{61} - 23 q^{63} + 2 q^{65} - 22 q^{67} + 6 q^{69} - 10 q^{71} + 4 q^{73} - 62 q^{75} - 6 q^{77} + 48 q^{79} + 4 q^{81} - 20 q^{83} - 3 q^{85} + 13 q^{87} - 29 q^{89} - 28 q^{91} - 31 q^{93} - 48 q^{95} - 26 q^{97} - 3 q^{99}+O(q^{100}) $ 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9 - 5 * q^11 + 9 * q^13 - 16 * q^15 - 12 * q^17 - 15 * q^19 + 5 * q^21 - 22 * q^23 + 5 * q^25 - 16 * q^27 - 17 * q^29 + 5 * q^31 - 33 * q^33 - 9 * q^35 - 2 * q^37 - 24 * q^39 - 18 * q^41 - 16 * q^43 + 30 * q^45 - 60 * q^47 + 10 * q^49 + 4 * q^51 - 24 * q^53 - q^55 + 41 * q^57 + 12 * q^59 - 17 * q^61 - 23 * q^63 + 2 * q^65 - 22 * q^67 + 6 * q^69 - 10 * q^71 + 4 * q^73 - 62 * q^75 - 6 * q^77 + 48 * q^79 + 4 * q^81 - 20 * q^83 - 3 * q^85 + 13 * q^87 - 29 * q^89 - 28 * q^91 - 31 * q^93 - 48 * q^95 - 26 * q^97 - 3 * 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
$3$	1.70453	0.984110	0.492055	−	0.870564i	$-0.336246\pi$
$3$	1.70453	0.984110	0.492055	+	0.870564i	$0.336246\pi$
$4$	0	0
$5$	−0.637713	−0.285194	−0.142597	−	0.989781i	$-0.545545\pi$
$5$	−0.637713	−0.285194	−0.142597	+	0.989781i	$0.545545\pi$
$6$	0	0
$7$	1.14740	0.433678	0.216839	−	0.976207i	$-0.430425\pi$
$7$	1.14740	0.433678	0.216839	+	0.976207i	$0.430425\pi$
$8$	0	0
$9$	−0.0945840	−0.0315280
$10$	0	0
$11$	0.485761	0.146462	0.0732312	−	0.997315i	$-0.476669\pi$
$11$	0.485761	0.146462	0.0732312	+	0.997315i	$0.476669\pi$
$12$	0	0
$13$	−4.68138	−1.29838	−0.649191	−	0.760625i	$-0.724892\pi$
$13$	−4.68138	−1.29838	−0.649191	+	0.760625i	$0.724892\pi$
$14$	0	0
$15$	−1.08700	−0.280662
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	1.90066	0.436042	0.218021	−	0.975944i	$-0.430040\pi$
$19$	1.90066	0.436042	0.218021	+	0.975944i	$0.430040\pi$
$20$	0	0
$21$	1.95578	0.426787
$22$	0	0
$23$	−0.312411	−0.0651422	−0.0325711	−	0.999469i	$-0.510370\pi$
$23$	−0.312411	−0.0651422	−0.0325711	+	0.999469i	$0.510370\pi$
$24$	0	0
$25$	−4.59332	−0.918664
$26$	0	0
$27$	−5.27481	−1.01514
$28$	0	0
$29$	−3.80295	−0.706190	−0.353095	−	0.935587i	$-0.614871\pi$
$29$	−3.80295	−0.706190	−0.353095	+	0.935587i	$0.614871\pi$
$30$	0	0
$31$	−1.49914	−0.269254	−0.134627	−	0.990896i	$-0.542984\pi$
$31$	−1.49914	−0.269254	−0.134627	+	0.990896i	$0.542984\pi$
$32$	0	0
$33$	0.827993	0.144135
$34$	0	0
$35$	−0.731715	−0.123682
$36$	0	0
$37$	3.52029	0.578732	0.289366	−	0.957218i	$-0.406555\pi$
$37$	3.52029	0.578732	0.289366	+	0.957218i	$0.406555\pi$
$38$	0	0
$39$	−7.97955	−1.27775
$40$	0	0
$41$	−5.10566	−0.797371	−0.398685	−	0.917088i	$-0.630533\pi$
$41$	−5.10566	−0.797371	−0.398685	+	0.917088i	$0.630533\pi$
$42$	0	0
$43$	−2.74040	−0.417908	−0.208954	−	0.977926i	$-0.567006\pi$
$43$	−2.74040	−0.417908	−0.208954	+	0.977926i	$0.567006\pi$
$44$	0	0
$45$	0.0603175	0.00899159
$46$	0	0
$47$	5.63321	0.821689	0.410844	−	0.911706i	$-0.365234\pi$
$47$	5.63321	0.821689	0.410844	+	0.911706i	$0.365234\pi$
$48$	0	0
$49$	−5.68346	−0.811923
$50$	0	0
$51$	−1.70453	−0.238682
$52$	0	0
$53$	−5.67494	−0.779513	−0.389756	−	0.920918i	$-0.627441\pi$
$53$	−5.67494	−0.779513	−0.389756	+	0.920918i	$0.627441\pi$
$54$	0	0
$55$	−0.309776	−0.0417702
$56$	0	0
$57$	3.23974	0.429113
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−11.8658	−1.51926	−0.759628	−	0.650358i	$-0.774619\pi$
$61$	−11.8658	−1.51926	−0.759628	+	0.650358i	$0.774619\pi$
$62$	0	0
$63$	−0.108526	−0.0136730
$64$	0	0
$65$	2.98538	0.370291
$66$	0	0
$67$	7.64584	0.934088	0.467044	−	0.884234i	$-0.345319\pi$
$67$	7.64584	0.934088	0.467044	+	0.884234i	$0.345319\pi$
$68$	0	0
$69$	−0.532514	−0.0641071
$70$	0	0
$71$	1.03706	0.123077	0.0615385	−	0.998105i	$-0.480399\pi$
$71$	1.03706	0.123077	0.0615385	+	0.998105i	$0.480399\pi$
$72$	0	0
$73$	10.9506	1.28167	0.640833	−	0.767680i	$-0.278589\pi$
$73$	10.9506	1.28167	0.640833	+	0.767680i	$0.278589\pi$
$74$	0	0
$75$	−7.82945	−0.904067
$76$	0	0
$77$	0.557364	0.0635175
$78$	0	0
$79$	−0.162183	−0.0182471	−0.00912353	−	0.999958i	$-0.502904\pi$
$79$	−0.162183	−0.0182471	−0.00912353	+	0.999958i	$0.502904\pi$
$80$	0	0
$81$	−8.70730	−0.967478
$82$	0	0
$83$	−1.63200	−0.179135	−0.0895676	−	0.995981i	$-0.528548\pi$
$83$	−1.63200	−0.179135	−0.0895676	+	0.995981i	$0.528548\pi$
$84$	0	0
$85$	0.637713	0.0691697
$86$	0	0
$87$	−6.48224	−0.694969
$88$	0	0
$89$	−18.8243	−1.99537	−0.997686	−	0.0679960i	$-0.978339\pi$
$89$	−18.8243	−1.99537	−0.997686	+	0.0679960i	$0.978339\pi$
$90$	0	0
$91$	−5.37144	−0.563080
$92$	0	0
$93$	−2.55533	−0.264976
$94$	0	0
$95$	−1.21208	−0.124357
$96$	0	0
$97$	−14.5538	−1.47771	−0.738857	−	0.673862i	$-0.764634\pi$
$97$	−14.5538	−1.47771	−0.738857	+	0.673862i	$0.764634\pi$
$98$	0	0
$99$	−0.0459452	−0.00461766
$100$	0	0
$101$	−9.44284	−0.939598	−0.469799	−	0.882774i	$-0.655674\pi$
$101$	−9.44284	−0.939598	−0.469799	+	0.882774i	$0.655674\pi$
$102$	0	0
$103$	3.38169	0.333208	0.166604	−	0.986024i	$-0.446720\pi$
$103$	3.38169	0.333208	0.166604	+	0.986024i	$0.446720\pi$
$104$	0	0
$105$	−1.24723	−0.121717
$106$	0	0
$107$	16.0081	1.54756	0.773782	−	0.633452i	$-0.218363\pi$
$107$	16.0081	1.54756	0.773782	+	0.633452i	$0.218363\pi$
$108$	0	0
$109$	4.29824	0.411697	0.205849	−	0.978584i	$-0.434005\pi$
$109$	4.29824	0.411697	0.205849	+	0.978584i	$0.434005\pi$
$110$	0	0
$111$	6.00044	0.569536
$112$	0	0
$113$	−1.77885	−0.167340	−0.0836699	−	0.996494i	$-0.526664\pi$
$113$	−1.77885	−0.167340	−0.0836699	+	0.996494i	$0.526664\pi$
$114$	0	0
$115$	0.199229	0.0185782
$116$	0	0
$117$	0.442784	0.0409354
$118$	0	0
$119$	−1.14740	−0.105182
$120$	0	0
$121$	−10.7640	−0.978549
$122$	0	0
$123$	−8.70275	−0.784700
$124$	0	0
$125$	6.11779	0.547192
$126$	0	0
$127$	14.0428	1.24609	0.623047	−	0.782185i	$-0.285895\pi$
$127$	14.0428	1.24609	0.623047	+	0.782185i	$0.285895\pi$
$128$	0	0
$129$	−4.67109	−0.411267
$130$	0	0
$131$	−8.95537	−0.782435	−0.391217	−	0.920298i	$-0.627946\pi$
$131$	−8.95537	−0.782435	−0.391217	+	0.920298i	$0.627946\pi$
$132$	0	0
$133$	2.18083	0.189102
$134$	0	0
$135$	3.36381	0.289511
$136$	0	0
$137$	3.70810	0.316804	0.158402	−	0.987375i	$-0.449366\pi$
$137$	3.70810	0.316804	0.158402	+	0.987375i	$0.449366\pi$
$138$	0	0
$139$	−14.6235	−1.24035	−0.620173	−	0.784465i	$-0.712938\pi$
$139$	−14.6235	−1.24035	−0.620173	+	0.784465i	$0.712938\pi$
$140$	0	0
$141$	9.60197	0.808632
$142$	0	0
$143$	−2.27403	−0.190164
$144$	0	0
$145$	2.42519	0.201401
$146$	0	0
$147$	−9.68762	−0.799022
$148$	0	0
$149$	18.1980	1.49084	0.745419	−	0.666596i	$-0.232250\pi$
$149$	18.1980	1.49084	0.745419	+	0.666596i	$0.232250\pi$
$150$	0	0
$151$	5.68115	0.462325	0.231163	−	0.972915i	$-0.425747\pi$
$151$	5.68115	0.462325	0.231163	+	0.972915i	$0.425747\pi$
$152$	0	0
$153$	0.0945840	0.00764666
$154$	0	0
$155$	0.956025	0.0767897
$156$	0	0
$157$	−21.0170	−1.67734	−0.838670	−	0.544639i	$-0.816667\pi$
$157$	−21.0170	−1.67734	−0.838670	+	0.544639i	$0.816667\pi$
$158$	0	0
$159$	−9.67309	−0.767126
$160$	0	0
$161$	−0.358462	−0.0282507
$162$	0	0
$163$	12.1591	0.952372	0.476186	−	0.879345i	$-0.342019\pi$
$163$	12.1591	0.952372	0.476186	+	0.879345i	$0.342019\pi$
$164$	0	0
$165$	−0.528022	−0.0411065
$166$	0	0
$167$	−7.38644	−0.571580	−0.285790	−	0.958292i	$-0.592256\pi$
$167$	−7.38644	−0.571580	−0.285790	+	0.958292i	$0.592256\pi$
$168$	0	0
$169$	8.91535	0.685796
$170$	0	0
$171$	−0.179772	−0.0137475
$172$	0	0
$173$	−10.6279	−0.808022	−0.404011	−	0.914754i	$-0.632384\pi$
$173$	−10.6279	−0.808022	−0.404011	+	0.914754i	$0.632384\pi$
$174$	0	0
$175$	−5.27040	−0.398404
$176$	0	0
$177$	1.70453	0.128120
$178$	0	0
$179$	−9.34917	−0.698790	−0.349395	−	0.936976i	$-0.613613\pi$
$179$	−9.34917	−0.698790	−0.349395	+	0.936976i	$0.613613\pi$
$180$	0	0
$181$	9.31537	0.692406	0.346203	−	0.938160i	$-0.387471\pi$
$181$	9.31537	0.692406	0.346203	+	0.938160i	$0.387471\pi$
$182$	0	0
$183$	−20.2255	−1.49512
$184$	0	0
$185$	−2.24494	−0.165051
$186$	0	0
$187$	−0.485761	−0.0355223
$188$	0	0
$189$	−6.05233	−0.440242
$190$	0	0
$191$	3.95791	0.286384	0.143192	−	0.989695i	$-0.454263\pi$
$191$	3.95791	0.286384	0.143192	+	0.989695i	$0.454263\pi$
$192$	0	0
$193$	−9.55163	−0.687541	−0.343771	−	0.939054i	$-0.611704\pi$
$193$	−9.55163	−0.687541	−0.343771	+	0.939054i	$0.611704\pi$
$194$	0	0
$195$	5.08866	0.364407
$196$	0	0
$197$	−0.701314	−0.0499666	−0.0249833	−	0.999688i	$-0.507953\pi$
$197$	−0.701314	−0.0499666	−0.0249833	+	0.999688i	$0.507953\pi$
$198$	0	0
$199$	21.9001	1.55246	0.776229	−	0.630451i	$-0.217130\pi$
$199$	21.9001	1.55246	0.776229	+	0.630451i	$0.217130\pi$
$200$	0	0
$201$	13.0326	0.919245
$202$	0	0
$203$	−4.36352	−0.306259
$204$	0	0
$205$	3.25595	0.227405
$206$	0	0
$207$	0.0295491	0.00205380
$208$	0	0
$209$	0.923268	0.0638638
$210$	0	0
$211$	−1.40422	−0.0966703	−0.0483351	−	0.998831i	$-0.515392\pi$
$211$	−1.40422	−0.0966703	−0.0483351	+	0.998831i	$0.515392\pi$
$212$	0	0
$213$	1.76771	0.121121
$214$	0	0
$215$	1.74759	0.119185
$216$	0	0
$217$	−1.72012	−0.116770
$218$	0	0
$219$	18.6655	1.26130
$220$	0	0
$221$	4.68138	0.314904
$222$	0	0
$223$	7.04980	0.472090	0.236045	−	0.971742i	$-0.424149\pi$
$223$	7.04980	0.472090	0.236045	+	0.971742i	$0.424149\pi$
$224$	0	0
$225$	0.434455	0.0289636
$226$	0	0
$227$	2.36642	0.157065	0.0785324	−	0.996912i	$-0.474977\pi$
$227$	2.36642	0.157065	0.0785324	+	0.996912i	$0.474977\pi$
$228$	0	0
$229$	−1.83183	−0.121051	−0.0605255	−	0.998167i	$-0.519278\pi$
$229$	−1.83183	−0.121051	−0.0605255	+	0.998167i	$0.519278\pi$
$230$	0	0
$231$	0.950042	0.0625082
$232$	0	0
$233$	−21.2098	−1.38950	−0.694750	−	0.719251i	$-0.744485\pi$
$233$	−21.2098	−1.38950	−0.694750	+	0.719251i	$0.744485\pi$
$234$	0	0
$235$	−3.59238	−0.234341
$236$	0	0
$237$	−0.276446	−0.0179571
$238$	0	0
$239$	11.1137	0.718882	0.359441	−	0.933168i	$-0.382967\pi$
$239$	11.1137	0.718882	0.359441	+	0.933168i	$0.382967\pi$
$240$	0	0
$241$	6.29182	0.405292	0.202646	−	0.979252i	$-0.435046\pi$
$241$	6.29182	0.405292	0.202646	+	0.979252i	$0.435046\pi$
$242$	0	0
$243$	0.982575	0.0630322
$244$	0	0
$245$	3.62442	0.231556
$246$	0	0
$247$	−8.89774	−0.566150
$248$	0	0
$249$	−2.78179	−0.176289
$250$	0	0
$251$	2.30846	0.145708	0.0728542	−	0.997343i	$-0.476789\pi$
$251$	2.30846	0.145708	0.0728542	+	0.997343i	$0.476789\pi$
$252$	0	0
$253$	−0.151757	−0.00954089
$254$	0	0
$255$	1.08700	0.0680706
$256$	0	0
$257$	19.6249	1.22417	0.612084	−	0.790793i	$-0.290331\pi$
$257$	19.6249	1.22417	0.612084	+	0.790793i	$0.290331\pi$
$258$	0	0
$259$	4.03920	0.250983
$260$	0	0
$261$	0.359698	0.0222648
$262$	0	0
$263$	−27.1353	−1.67323	−0.836617	−	0.547789i	$-0.815470\pi$
$263$	−27.1353	−1.67323	−0.836617	+	0.547789i	$0.815470\pi$
$264$	0	0
$265$	3.61898	0.222312
$266$	0	0
$267$	−32.0865	−1.96366
$268$	0	0
$269$	−4.18261	−0.255018	−0.127509	−	0.991837i	$-0.540698\pi$
$269$	−4.18261	−0.255018	−0.127509	+	0.991837i	$0.540698\pi$
$270$	0	0
$271$	−18.8254	−1.14356	−0.571781	−	0.820406i	$-0.693747\pi$
$271$	−18.8254	−1.14356	−0.571781	+	0.820406i	$0.693747\pi$
$272$	0	0
$273$	−9.15577	−0.554132
$274$	0	0
$275$	−2.23126	−0.134550
$276$	0	0
$277$	−1.20874	−0.0726263	−0.0363132	−	0.999340i	$-0.511561\pi$
$277$	−1.20874	−0.0726263	−0.0363132	+	0.999340i	$0.511561\pi$
$278$	0	0
$279$	0.141795	0.00848905
$280$	0	0
$281$	−14.8339	−0.884915	−0.442458	−	0.896789i	$-0.645893\pi$
$281$	−14.8339	−0.884915	−0.442458	+	0.896789i	$0.645893\pi$
$282$	0	0
$283$	−0.0173913	−0.00103381	−0.000516903	−	1.00000i	$-0.500165\pi$
$283$	−0.0173913	−0.00103381	−0.000516903		1.00000i	$0.500165\pi$
$284$	0	0
$285$	−2.06602	−0.122381
$286$	0	0
$287$	−5.85826	−0.345802
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−24.8073	−1.45423
$292$	0	0
$293$	−0.862325	−0.0503775	−0.0251888	−	0.999683i	$-0.508019\pi$
$293$	−0.862325	−0.0503775	−0.0251888	+	0.999683i	$0.508019\pi$
$294$	0	0
$295$	−0.637713	−0.0371291
$296$	0	0
$297$	−2.56229	−0.148679
$298$	0	0
$299$	1.46252	0.0845795
$300$	0	0
$301$	−3.14435	−0.181237
$302$	0	0
$303$	−16.0956	−0.924667
$304$	0	0
$305$	7.56696	0.433283
$306$	0	0
$307$	21.4961	1.22685	0.613424	−	0.789754i	$-0.289792\pi$
$307$	21.4961	1.22685	0.613424	+	0.789754i	$0.289792\pi$
$308$	0	0
$309$	5.76419	0.327913
$310$	0	0
$311$	−8.09850	−0.459224	−0.229612	−	0.973282i	$-0.573746\pi$
$311$	−8.09850	−0.459224	−0.229612	+	0.973282i	$0.573746\pi$
$312$	0	0
$313$	24.6870	1.39539	0.697696	−	0.716394i	$-0.254209\pi$
$313$	24.6870	1.39539	0.697696	+	0.716394i	$0.254209\pi$
$314$	0	0
$315$	0.0692085	0.00389946
$316$	0	0
$317$	−7.51323	−0.421985	−0.210993	−	0.977488i	$-0.567670\pi$
$317$	−7.51323	−0.421985	−0.210993	+	0.977488i	$0.567670\pi$
$318$	0	0
$319$	−1.84732	−0.103430
$320$	0	0
$321$	27.2863	1.52297
$322$	0	0
$323$	−1.90066	−0.105756
$324$	0	0
$325$	21.5031	1.19278
$326$	0	0
$327$	7.32648	0.405155
$328$	0	0
$329$	6.46357	0.356348
$330$	0	0
$331$	−1.10110	−0.0605218	−0.0302609	−	0.999542i	$-0.509634\pi$
$331$	−1.10110	−0.0605218	−0.0302609	+	0.999542i	$0.509634\pi$
$332$	0	0
$333$	−0.332963	−0.0182463
$334$	0	0
$335$	−4.87585	−0.266396
$336$	0	0
$337$	−23.1140	−1.25910	−0.629549	−	0.776961i	$-0.716760\pi$
$337$	−23.1140	−1.25910	−0.629549	+	0.776961i	$0.716760\pi$
$338$	0	0
$339$	−3.03209	−0.164681
$340$	0	0
$341$	−0.728226	−0.0394356
$342$	0	0
$343$	−14.5531	−0.785791
$344$	0	0
$345$	0.339591	0.0182830
$346$	0	0
$347$	−14.3881	−0.772392	−0.386196	−	0.922417i	$-0.626211\pi$
$347$	−14.3881	−0.772392	−0.386196	+	0.922417i	$0.626211\pi$
$348$	0	0
$349$	17.3754	0.930083	0.465041	−	0.885289i	$-0.346039\pi$
$349$	17.3754	0.930083	0.465041	+	0.885289i	$0.346039\pi$
$350$	0	0
$351$	24.6934	1.31804
$352$	0	0
$353$	−16.2975	−0.867428	−0.433714	−	0.901051i	$-0.642797\pi$
$353$	−16.2975	−0.867428	−0.433714	+	0.901051i	$0.642797\pi$
$354$	0	0
$355$	−0.661350	−0.0351008
$356$	0	0
$357$	−1.95578	−0.103511
$358$	0	0
$359$	29.0062	1.53089	0.765444	−	0.643502i	$-0.222519\pi$
$359$	29.0062	1.53089	0.765444	+	0.643502i	$0.222519\pi$
$360$	0	0
$361$	−15.3875	−0.809867
$362$	0	0
$363$	−18.3476	−0.962999
$364$	0	0
$365$	−6.98332	−0.365524
$366$	0	0
$367$	−17.6968	−0.923768	−0.461884	−	0.886940i	$-0.652826\pi$
$367$	−17.6968	−0.923768	−0.461884	+	0.886940i	$0.652826\pi$
$368$	0	0
$369$	0.482914	0.0251395
$370$	0	0
$371$	−6.51145	−0.338058
$372$	0	0
$373$	12.8060	0.663068	0.331534	−	0.943443i	$-0.392434\pi$
$373$	12.8060	0.663068	0.331534	+	0.943443i	$0.392434\pi$
$374$	0	0
$375$	10.4279	0.538497
$376$	0	0
$377$	17.8031	0.916905
$378$	0	0
$379$	−10.0149	−0.514431	−0.257215	−	0.966354i	$-0.582805\pi$
$379$	−10.0149	−0.514431	−0.257215	+	0.966354i	$0.582805\pi$
$380$	0	0
$381$	23.9363	1.22629
$382$	0	0
$383$	24.0024	1.22646	0.613232	−	0.789903i	$-0.289869\pi$
$383$	24.0024	1.22646	0.613232	+	0.789903i	$0.289869\pi$
$384$	0	0
$385$	−0.355438	−0.0181148
$386$	0	0
$387$	0.259198	0.0131758
$388$	0	0
$389$	−6.31841	−0.320356	−0.160178	−	0.987088i	$-0.551207\pi$
$389$	−6.31841	−0.320356	−0.160178	+	0.987088i	$0.551207\pi$
$390$	0	0
$391$	0.312411	0.0157993
$392$	0	0
$393$	−15.2647	−0.770002
$394$	0	0
$395$	0.103427	0.00520395
$396$	0	0
$397$	−17.1736	−0.861917	−0.430958	−	0.902372i	$-0.641824\pi$
$397$	−17.1736	−0.861917	−0.430958	+	0.902372i	$0.641824\pi$
$398$	0	0
$399$	3.71729	0.186097
$400$	0	0
$401$	−11.1027	−0.554441	−0.277220	−	0.960806i	$-0.589413\pi$
$401$	−11.1027	−0.554441	−0.277220	+	0.960806i	$0.589413\pi$
$402$	0	0
$403$	7.01807	0.349595
$404$	0	0
$405$	5.55276	0.275919
$406$	0	0
$407$	1.71002	0.0847625
$408$	0	0
$409$	29.4879	1.45808	0.729040	−	0.684471i	$-0.239967\pi$
$409$	29.4879	1.45808	0.729040	+	0.684471i	$0.239967\pi$
$410$	0	0
$411$	6.32055	0.311770
$412$	0	0
$413$	1.14740	0.0564601
$414$	0	0
$415$	1.04075	0.0510883
$416$	0	0
$417$	−24.9261	−1.22064
$418$	0	0
$419$	−3.42964	−0.167549	−0.0837746	−	0.996485i	$-0.526698\pi$
$419$	−3.42964	−0.167549	−0.0837746	+	0.996485i	$0.526698\pi$
$420$	0	0
$421$	20.5576	1.00191	0.500957	−	0.865472i	$-0.332981\pi$
$421$	20.5576	1.00191	0.500957	+	0.865472i	$0.332981\pi$
$422$	0	0
$423$	−0.532812	−0.0259062
$424$	0	0
$425$	4.59332	0.222809
$426$	0	0
$427$	−13.6148	−0.658868
$428$	0	0
$429$	−3.87615	−0.187142
$430$	0	0
$431$	3.27470	0.157737	0.0788684	−	0.996885i	$-0.474869\pi$
$431$	3.27470	0.157737	0.0788684	+	0.996885i	$0.474869\pi$
$432$	0	0
$433$	−1.38688	−0.0666490	−0.0333245	−	0.999445i	$-0.510609\pi$
$433$	−1.38688	−0.0666490	−0.0333245	+	0.999445i	$0.510609\pi$
$434$	0	0
$435$	4.13381	0.198201
$436$	0	0
$437$	−0.593789	−0.0284048
$438$	0	0
$439$	12.2628	0.585271	0.292636	−	0.956224i	$-0.405468\pi$
$439$	12.2628	0.585271	0.292636	+	0.956224i	$0.405468\pi$
$440$	0	0
$441$	0.537565	0.0255983
$442$	0	0
$443$	39.6510	1.88388	0.941938	−	0.335786i	$-0.109002\pi$
$443$	39.6510	1.88388	0.941938	+	0.335786i	$0.109002\pi$
$444$	0	0
$445$	12.0045	0.569068
$446$	0	0
$447$	31.0190	1.46715
$448$	0	0
$449$	−27.3128	−1.28897	−0.644486	−	0.764616i	$-0.722929\pi$
$449$	−27.3128	−1.28897	−0.644486	+	0.764616i	$0.722929\pi$
$450$	0	0
$451$	−2.48013	−0.116785
$452$	0	0
$453$	9.68367	0.454979
$454$	0	0
$455$	3.42544	0.160587
$456$	0	0
$457$	27.2849	1.27633	0.638167	−	0.769898i	$-0.279693\pi$
$457$	27.2849	1.27633	0.638167	+	0.769898i	$0.279693\pi$
$458$	0	0
$459$	5.27481	0.246207
$460$	0	0
$461$	15.5523	0.724341	0.362170	−	0.932112i	$-0.382036\pi$
$461$	15.5523	0.724341	0.362170	+	0.932112i	$0.382036\pi$
$462$	0	0
$463$	31.5039	1.46411	0.732055	−	0.681245i	$-0.238561\pi$
$463$	31.5039	1.46411	0.732055	+	0.681245i	$0.238561\pi$
$464$	0	0
$465$	1.62957	0.0755695
$466$	0	0
$467$	12.5065	0.578732	0.289366	−	0.957218i	$-0.406555\pi$
$467$	12.5065	0.578732	0.289366	+	0.957218i	$0.406555\pi$
$468$	0	0
$469$	8.77287	0.405093
$470$	0	0
$471$	−35.8241	−1.65069
$472$	0	0
$473$	−1.33118	−0.0612078
$474$	0	0
$475$	−8.73036	−0.400577
$476$	0	0
$477$	0.536758	0.0245765
$478$	0	0
$479$	−0.596960	−0.0272758	−0.0136379	−	0.999907i	$-0.504341\pi$
$479$	−0.596960	−0.0272758	−0.0136379	+	0.999907i	$0.504341\pi$
$480$	0	0
$481$	−16.4798	−0.751416
$482$	0	0
$483$	−0.611008	−0.0278018
$484$	0	0
$485$	9.28115	0.421435
$486$	0	0
$487$	14.3507	0.650292	0.325146	−	0.945664i	$-0.394587\pi$
$487$	14.3507	0.650292	0.325146	+	0.945664i	$0.394587\pi$
$488$	0	0
$489$	20.7255	0.937239
$490$	0	0
$491$	13.1501	0.593457	0.296729	−	0.954962i	$-0.404104\pi$
$491$	13.1501	0.593457	0.296729	+	0.954962i	$0.404104\pi$
$492$	0	0
$493$	3.80295	0.171276
$494$	0	0
$495$	0.0292999	0.00131693
$496$	0	0
$497$	1.18993	0.0533758
$498$	0	0
$499$	14.2142	0.636316	0.318158	−	0.948038i	$-0.396936\pi$
$499$	14.2142	0.636316	0.318158	+	0.948038i	$0.396936\pi$
$500$	0	0
$501$	−12.5904	−0.562497
$502$	0	0
$503$	−21.0178	−0.937138	−0.468569	−	0.883427i	$-0.655230\pi$
$503$	−21.0178	−0.937138	−0.468569	+	0.883427i	$0.655230\pi$
$504$	0	0
$505$	6.02182	0.267968
$506$	0	0
$507$	15.1965	0.674899
$508$	0	0
$509$	22.9540	1.01742	0.508709	−	0.860938i	$-0.330123\pi$
$509$	22.9540	1.01742	0.508709	+	0.860938i	$0.330123\pi$
$510$	0	0
$511$	12.5647	0.555831
$512$	0	0
$513$	−10.0256	−0.442643
$514$	0	0
$515$	−2.15655	−0.0950290
$516$	0	0
$517$	2.73639	0.120346
$518$	0	0
$519$	−18.1155	−0.795182
$520$	0	0
$521$	−39.4341	−1.72764	−0.863819	−	0.503802i	$-0.831934\pi$
$521$	−39.4341	−1.72764	−0.863819	+	0.503802i	$0.831934\pi$
$522$	0	0
$523$	−19.2025	−0.839665	−0.419832	−	0.907602i	$-0.637911\pi$
$523$	−19.2025	−0.839665	−0.419832	+	0.907602i	$0.637911\pi$
$524$	0	0
$525$	−8.98354	−0.392074
$526$	0	0
$527$	1.49914	0.0653038
$528$	0	0
$529$	−22.9024	−0.995756
$530$	0	0
$531$	−0.0945840	−0.00410459
$532$	0	0
$533$	23.9016	1.03529
$534$	0	0
$535$	−10.2086	−0.441356
$536$	0	0
$537$	−15.9359	−0.687686
$538$	0	0
$539$	−2.76080	−0.118916
$540$	0	0
$541$	36.1214	1.55298	0.776489	−	0.630130i	$-0.216999\pi$
$541$	36.1214	1.55298	0.776489	+	0.630130i	$0.216999\pi$
$542$	0	0
$543$	15.8783	0.681403
$544$	0	0
$545$	−2.74105	−0.117414
$546$	0	0
$547$	−24.7174	−1.05684	−0.528420	−	0.848983i	$-0.677215\pi$
$547$	−24.7174	−1.05684	−0.528420	+	0.848983i	$0.677215\pi$
$548$	0	0
$549$	1.12231	0.0478991
$550$	0	0
$551$	−7.22813	−0.307929
$552$	0	0
$553$	−0.186090	−0.00791335
$554$	0	0
$555$	−3.82656	−0.162428
$556$	0	0
$557$	−8.46580	−0.358707	−0.179354	−	0.983785i	$-0.557401\pi$
$557$	−8.46580	−0.358707	−0.179354	+	0.983785i	$0.557401\pi$
$558$	0	0
$559$	12.8289	0.542604
$560$	0	0
$561$	−0.827993	−0.0349579
$562$	0	0
$563$	−39.9654	−1.68434	−0.842171	−	0.539210i	$-0.818723\pi$
$563$	−39.9654	−1.68434	−0.842171	+	0.539210i	$0.818723\pi$
$564$	0	0
$565$	1.13439	0.0477243
$566$	0	0
$567$	−9.99079	−0.419574
$568$	0	0
$569$	−24.3309	−1.02001	−0.510003	−	0.860173i	$-0.670356\pi$
$569$	−24.3309	−1.02001	−0.510003	+	0.860173i	$0.670356\pi$
$570$	0	0
$571$	37.2309	1.55807	0.779033	−	0.626983i	$-0.215711\pi$
$571$	37.2309	1.55807	0.779033	+	0.626983i	$0.215711\pi$
$572$	0	0
$573$	6.74636	0.281833
$574$	0	0
$575$	1.43500	0.0598438
$576$	0	0
$577$	22.8250	0.950217	0.475108	−	0.879927i	$-0.342409\pi$
$577$	22.8250	0.950217	0.475108	+	0.879927i	$0.342409\pi$
$578$	0	0
$579$	−16.2810	−0.676616
$580$	0	0
$581$	−1.87256	−0.0776870
$582$	0	0
$583$	−2.75666	−0.114169
$584$	0	0
$585$	−0.282369	−0.0116745
$586$	0	0
$587$	−37.2429	−1.53718	−0.768590	−	0.639742i	$-0.779041\pi$
$587$	−37.2429	−1.53718	−0.768590	+	0.639742i	$0.779041\pi$
$588$	0	0
$589$	−2.84937	−0.117406
$590$	0	0
$591$	−1.19541	−0.0491726
$592$	0	0
$593$	19.8471	0.815023	0.407512	−	0.913200i	$-0.366397\pi$
$593$	19.8471	0.815023	0.407512	+	0.913200i	$0.366397\pi$
$594$	0	0
$595$	0.731715	0.0299974
$596$	0	0
$597$	37.3293	1.52779
$598$	0	0
$599$	−37.2999	−1.52403	−0.762016	−	0.647559i	$-0.775790\pi$
$599$	−37.2999	−1.52403	−0.762016	+	0.647559i	$0.775790\pi$
$600$	0	0
$601$	5.86276	0.239147	0.119574	−	0.992825i	$-0.461847\pi$
$601$	5.86276	0.239147	0.119574	+	0.992825i	$0.461847\pi$
$602$	0	0
$603$	−0.723174	−0.0294499
$604$	0	0
$605$	6.86437	0.279076
$606$	0	0
$607$	−1.20210	−0.0487918	−0.0243959	−	0.999702i	$-0.507766\pi$
$607$	−1.20210	−0.0487918	−0.0243959	+	0.999702i	$0.507766\pi$
$608$	0	0
$609$	−7.43774	−0.301393
$610$	0	0
$611$	−26.3712	−1.06687
$612$	0	0
$613$	−7.18529	−0.290211	−0.145106	−	0.989416i	$-0.546352\pi$
$613$	−7.18529	−0.290211	−0.145106	+	0.989416i	$0.546352\pi$
$614$	0	0
$615$	5.54986	0.223792
$616$	0	0
$617$	−12.9495	−0.521327	−0.260664	−	0.965430i	$-0.583941\pi$
$617$	−12.9495	−0.521327	−0.260664	+	0.965430i	$0.583941\pi$
$618$	0	0
$619$	−40.3802	−1.62302	−0.811509	−	0.584340i	$-0.801353\pi$
$619$	−40.3802	−1.62302	−0.811509	+	0.584340i	$0.801353\pi$
$620$	0	0
$621$	1.64791	0.0661283
$622$	0	0
$623$	−21.5991	−0.865348
$624$	0	0
$625$	19.0652	0.762609
$626$	0	0
$627$	1.57374	0.0628490
$628$	0	0
$629$	−3.52029	−0.140363
$630$	0	0
$631$	3.09133	0.123064	0.0615320	−	0.998105i	$-0.480401\pi$
$631$	3.09133	0.123064	0.0615320	+	0.998105i	$0.480401\pi$
$632$	0	0
$633$	−2.39353	−0.0951342
$634$	0	0
$635$	−8.95526	−0.355378
$636$	0	0
$637$	26.6065	1.05419
$638$	0	0
$639$	−0.0980897	−0.00388037
$640$	0	0
$641$	−17.0742	−0.674390	−0.337195	−	0.941435i	$-0.609478\pi$
$641$	−17.0742	−0.674390	−0.337195	+	0.941435i	$0.609478\pi$
$642$	0	0
$643$	16.5473	0.652562	0.326281	−	0.945273i	$-0.394204\pi$
$643$	16.5473	0.652562	0.326281	+	0.945273i	$0.394204\pi$
$644$	0	0
$645$	2.97882	0.117291
$646$	0	0
$647$	−39.2654	−1.54368	−0.771841	−	0.635816i	$-0.780664\pi$
$647$	−39.2654	−1.54368	−0.771841	+	0.635816i	$0.780664\pi$
$648$	0	0
$649$	0.485761	0.0190678
$650$	0	0
$651$	−2.93200	−0.114914
$652$	0	0
$653$	−13.1985	−0.516497	−0.258249	−	0.966078i	$-0.583145\pi$
$653$	−13.1985	−0.516497	−0.258249	+	0.966078i	$0.583145\pi$
$654$	0	0
$655$	5.71096	0.223146
$656$	0	0
$657$	−1.03575	−0.0404084
$658$	0	0
$659$	−16.0782	−0.626318	−0.313159	−	0.949701i	$-0.601387\pi$
$659$	−16.0782	−0.626318	−0.313159	+	0.949701i	$0.601387\pi$
$660$	0	0
$661$	7.55488	0.293851	0.146925	−	0.989148i	$-0.453062\pi$
$661$	7.55488	0.293851	0.146925	+	0.989148i	$0.453062\pi$
$662$	0	0
$663$	7.97955	0.309900
$664$	0	0
$665$	−1.39074	−0.0539307
$666$	0	0
$667$	1.18808	0.0460028
$668$	0	0
$669$	12.0166	0.464588
$670$	0	0
$671$	−5.76393	−0.222514
$672$	0	0
$673$	−25.0573	−0.965887	−0.482943	−	0.875652i	$-0.660432\pi$
$673$	−25.0573	−0.965887	−0.482943	+	0.875652i	$0.660432\pi$
$674$	0	0
$675$	24.2289	0.932570
$676$	0	0
$677$	−8.42370	−0.323749	−0.161875	−	0.986811i	$-0.551754\pi$
$677$	−8.42370	−0.323749	−0.161875	+	0.986811i	$0.551754\pi$
$678$	0	0
$679$	−16.6991	−0.640852
$680$	0	0
$681$	4.03363	0.154569
$682$	0	0
$683$	46.6844	1.78633	0.893165	−	0.449729i	$-0.148480\pi$
$683$	46.6844	1.78633	0.893165	+	0.449729i	$0.148480\pi$
$684$	0	0
$685$	−2.36470	−0.0903506
$686$	0	0
$687$	−3.12241	−0.119127
$688$	0	0
$689$	26.5666	1.01211
$690$	0	0
$691$	−16.7165	−0.635926	−0.317963	−	0.948103i	$-0.602999\pi$
$691$	−16.7165	−0.635926	−0.317963	+	0.948103i	$0.602999\pi$
$692$	0	0
$693$	−0.0527177	−0.00200258
$694$	0	0
$695$	9.32557	0.353739
$696$	0	0
$697$	5.10566	0.193391
$698$	0	0
$699$	−36.1527	−1.36742
$700$	0	0
$701$	9.32012	0.352016	0.176008	−	0.984389i	$-0.443682\pi$
$701$	9.32012	0.352016	0.176008	+	0.984389i	$0.443682\pi$
$702$	0	0
$703$	6.69089	0.252352
$704$	0	0
$705$	−6.12330	−0.230617
$706$	0	0
$707$	−10.8348	−0.407483
$708$	0	0
$709$	39.6498	1.48908	0.744540	−	0.667578i	$-0.232669\pi$
$709$	39.6498	1.48908	0.744540	+	0.667578i	$0.232669\pi$
$710$	0	0
$711$	0.0153400	0.000575293 0
$712$	0	0
$713$	0.468350	0.0175398
$714$	0	0
$715$	1.45018	0.0542337
$716$	0	0
$717$	18.9435	0.707459
$718$	0	0
$719$	44.6919	1.66673	0.833363	−	0.552726i	$-0.186412\pi$
$719$	44.6919	1.66673	0.833363	+	0.552726i	$0.186412\pi$
$720$	0	0
$721$	3.88017	0.144505
$722$	0	0
$723$	10.7246	0.398852
$724$	0	0
$725$	17.4682	0.648752
$726$	0	0
$727$	−26.6270	−0.987539	−0.493770	−	0.869593i	$-0.664381\pi$
$727$	−26.6270	−0.987539	−0.493770	+	0.869593i	$0.664381\pi$
$728$	0	0
$729$	27.7967	1.02951
$730$	0	0
$731$	2.74040	0.101357
$732$	0	0
$733$	−29.6174	−1.09394	−0.546971	−	0.837152i	$-0.684219\pi$
$733$	−29.6174	−1.09394	−0.546971	+	0.837152i	$0.684219\pi$
$734$	0	0
$735$	6.17793	0.227876
$736$	0	0
$737$	3.71405	0.136809
$738$	0	0
$739$	1.08773	0.0400126	0.0200063	−	0.999800i	$-0.493631\pi$
$739$	1.08773	0.0400126	0.0200063	+	0.999800i	$0.493631\pi$
$740$	0	0
$741$	−15.1664	−0.557153
$742$	0	0
$743$	40.9563	1.50254	0.751271	−	0.659994i	$-0.229441\pi$
$743$	40.9563	1.50254	0.751271	+	0.659994i	$0.229441\pi$
$744$	0	0
$745$	−11.6051	−0.425178
$746$	0	0
$747$	0.154361	0.00564777
$748$	0	0
$749$	18.3678	0.671144
$750$	0	0
$751$	−16.1748	−0.590228	−0.295114	−	0.955462i	$-0.595358\pi$
$751$	−16.1748	−0.590228	−0.295114	+	0.955462i	$0.595358\pi$
$752$	0	0
$753$	3.93483	0.143393
$754$	0	0
$755$	−3.62294	−0.131852
$756$	0	0
$757$	40.3359	1.46603	0.733016	−	0.680211i	$-0.238112\pi$
$757$	40.3359	1.46603	0.733016	+	0.680211i	$0.238112\pi$
$758$	0	0
$759$	−0.258674	−0.00938928
$760$	0	0
$761$	−32.9761	−1.19538	−0.597692	−	0.801726i	$-0.703915\pi$
$761$	−32.9761	−1.19538	−0.597692	+	0.801726i	$0.703915\pi$
$762$	0	0
$763$	4.93182	0.178544
$764$	0	0
$765$	−0.0603175	−0.00218078
$766$	0	0
$767$	−4.68138	−0.169035
$768$	0	0
$769$	9.18564	0.331243	0.165621	−	0.986189i	$-0.447037\pi$
$769$	9.18564	0.331243	0.165621	+	0.986189i	$0.447037\pi$
$770$	0	0
$771$	33.4512	1.20472
$772$	0	0
$773$	−45.0972	−1.62203	−0.811017	−	0.585023i	$-0.801085\pi$
$773$	−45.0972	−1.62203	−0.811017	+	0.585023i	$0.801085\pi$
$774$	0	0
$775$	6.88605	0.247354
$776$	0	0
$777$	6.88492	0.246995
$778$	0	0
$779$	−9.70415	−0.347687
$780$	0	0
$781$	0.503765	0.0180261
$782$	0	0
$783$	20.0598	0.716880
$784$	0	0
$785$	13.4028	0.478368
$786$	0	0
$787$	27.7593	0.989513	0.494756	−	0.869032i	$-0.335257\pi$
$787$	27.7593	0.989513	0.494756	+	0.869032i	$0.335257\pi$
$788$	0	0
$789$	−46.2528	−1.64665
$790$	0	0
$791$	−2.04106	−0.0725716
$792$	0	0
$793$	55.5482	1.97258
$794$	0	0
$795$	6.16866	0.218780
$796$	0	0
$797$	−4.96118	−0.175734	−0.0878670	−	0.996132i	$-0.528005\pi$
$797$	−4.96118	−0.175734	−0.0878670	+	0.996132i	$0.528005\pi$
$798$	0	0
$799$	−5.63321	−0.199289
$800$	0	0
$801$	1.78048	0.0629100
$802$	0	0
$803$	5.31935	0.187716
$804$	0	0
$805$	0.228596	0.00805694
$806$	0	0
$807$	−7.12937	−0.250966
$808$	0	0
$809$	22.3202	0.784737	0.392369	−	0.919808i	$-0.371656\pi$
$809$	22.3202	0.784737	0.392369	+	0.919808i	$0.371656\pi$
$810$	0	0
$811$	27.0718	0.950618	0.475309	−	0.879819i	$-0.342336\pi$
$811$	27.0718	0.950618	0.475309	+	0.879819i	$0.342336\pi$
$812$	0	0
$813$	−32.0884	−1.12539
$814$	0	0
$815$	−7.75400	−0.271611
$816$	0	0
$817$	−5.20859	−0.182225
$818$	0	0
$819$	0.508052	0.0177528
$820$	0	0
$821$	15.5750	0.543571	0.271785	−	0.962358i	$-0.412386\pi$
$821$	15.5750	0.543571	0.271785	+	0.962358i	$0.412386\pi$
$822$	0	0
$823$	−29.6838	−1.03471	−0.517357	−	0.855770i	$-0.673084\pi$
$823$	−29.6838	−1.03471	−0.517357	+	0.855770i	$0.673084\pi$
$824$	0	0
$825$	−3.80324	−0.132412
$826$	0	0
$827$	−46.9658	−1.63316	−0.816581	−	0.577231i	$-0.804133\pi$
$827$	−46.9658	−1.63316	−0.816581	+	0.577231i	$0.804133\pi$
$828$	0	0
$829$	5.63810	0.195819	0.0979097	−	0.995195i	$-0.468784\pi$
$829$	5.63810	0.195819	0.0979097	+	0.995195i	$0.468784\pi$
$830$	0	0
$831$	−2.06034	−0.0714723
$832$	0	0
$833$	5.68346	0.196920
$834$	0	0
$835$	4.71043	0.163011
$836$	0	0
$837$	7.90770	0.273330
$838$	0	0
$839$	26.4978	0.914807	0.457403	−	0.889259i	$-0.348780\pi$
$839$	26.4978	0.914807	0.457403	+	0.889259i	$0.348780\pi$
$840$	0	0
$841$	−14.5376	−0.501295
$842$	0	0
$843$	−25.2848	−0.870854
$844$	0	0
$845$	−5.68544	−0.195585
$846$	0	0
$847$	−12.3507	−0.424375
$848$	0	0
$849$	−0.0296440	−0.00101738
$850$	0	0
$851$	−1.09978	−0.0376999
$852$	0	0
$853$	−7.77672	−0.266270	−0.133135	−	0.991098i	$-0.542504\pi$
$853$	−7.77672	−0.266270	−0.133135	+	0.991098i	$0.542504\pi$
$854$	0	0
$855$	0.114643	0.00392072
$856$	0	0
$857$	9.93663	0.339429	0.169714	−	0.985493i	$-0.445715\pi$
$857$	9.93663	0.339429	0.169714	+	0.985493i	$0.445715\pi$
$858$	0	0
$859$	29.9040	1.02031	0.510156	−	0.860082i	$-0.329588\pi$
$859$	29.9040	1.02031	0.510156	+	0.860082i	$0.329588\pi$
$860$	0	0
$861$	−9.98557	−0.340307
$862$	0	0
$863$	−36.0132	−1.22590	−0.612952	−	0.790120i	$-0.710018\pi$
$863$	−36.0132	−1.22590	−0.612952	+	0.790120i	$0.710018\pi$
$864$	0	0
$865$	6.77753	0.230443
$866$	0	0
$867$	1.70453	0.0578888
$868$	0	0
$869$	−0.0787824	−0.00267251
$870$	0	0
$871$	−35.7931	−1.21280
$872$	0	0
$873$	1.37656	0.0465893
$874$	0	0
$875$	7.01958	0.237305
$876$	0	0
$877$	45.4356	1.53425	0.767125	−	0.641498i	$-0.221687\pi$
$877$	45.4356	1.53425	0.767125	+	0.641498i	$0.221687\pi$
$878$	0	0
$879$	−1.46986	−0.0495770
$880$	0	0
$881$	26.6125	0.896599	0.448299	−	0.893883i	$-0.352030\pi$
$881$	26.6125	0.896599	0.448299	+	0.893883i	$0.352030\pi$
$882$	0	0
$883$	21.5010	0.723567	0.361784	−	0.932262i	$-0.382168\pi$
$883$	21.5010	0.723567	0.361784	+	0.932262i	$0.382168\pi$
$884$	0	0
$885$	−1.08700	−0.0365391
$886$	0	0
$887$	6.25498	0.210022	0.105011	−	0.994471i	$-0.466512\pi$
$887$	6.25498	0.210022	0.105011	+	0.994471i	$0.466512\pi$
$888$	0	0
$889$	16.1127	0.540403
$890$	0	0
$891$	−4.22967	−0.141699
$892$	0	0
$893$	10.7068	0.358291
$894$	0	0
$895$	5.96209	0.199291
$896$	0	0
$897$	2.49290	0.0832355
$898$	0	0
$899$	5.70117	0.190145
$900$	0	0
$901$	5.67494	0.189060
$902$	0	0
$903$	−5.35963	−0.178357
$904$	0	0
$905$	−5.94053	−0.197470
$906$	0	0
$907$	−22.7009	−0.753770	−0.376885	−	0.926260i	$-0.623005\pi$
$907$	−22.7009	−0.753770	−0.376885	+	0.926260i	$0.623005\pi$
$908$	0	0
$909$	0.893141	0.0296236
$910$	0	0
$911$	−29.5418	−0.978765	−0.489382	−	0.872069i	$-0.662778\pi$
$911$	−29.5418	−0.978765	−0.489382	+	0.872069i	$0.662778\pi$
$912$	0	0
$913$	−0.792761	−0.0262366
$914$	0	0
$915$	12.8981	0.426398
$916$	0	0
$917$	−10.2754	−0.339325
$918$	0	0
$919$	10.0345	0.331006	0.165503	−	0.986209i	$-0.447075\pi$
$919$	10.0345	0.331006	0.165503	+	0.986209i	$0.447075\pi$
$920$	0	0
$921$	36.6407	1.20735
$922$	0	0
$923$	−4.85490	−0.159801
$924$	0	0
$925$	−16.1698	−0.531661
$926$	0	0
$927$	−0.319854	−0.0105054
$928$	0	0
$929$	−7.13078	−0.233953	−0.116977	−	0.993135i	$-0.537320\pi$
$929$	−7.13078	−0.233953	−0.116977	+	0.993135i	$0.537320\pi$
$930$	0	0
$931$	−10.8024	−0.354033
$932$	0	0
$933$	−13.8041	−0.451927
$934$	0	0
$935$	0.309776	0.0101308
$936$	0	0
$937$	0.648603	0.0211889	0.0105945	−	0.999944i	$-0.496628\pi$
$937$	0.648603	0.0211889	0.0105945	+	0.999944i	$0.496628\pi$
$938$	0	0
$939$	42.0797	1.37322
$940$	0	0
$941$	33.7146	1.09907	0.549533	−	0.835472i	$-0.314806\pi$
$941$	33.7146	1.09907	0.549533	+	0.835472i	$0.314806\pi$
$942$	0	0
$943$	1.59507	0.0519425
$944$	0	0
$945$	3.85965	0.125555
$946$	0	0
$947$	−39.2779	−1.27636	−0.638181	−	0.769887i	$-0.720313\pi$
$947$	−39.2779	−1.27636	−0.638181	+	0.769887i	$0.720313\pi$
$948$	0	0
$949$	−51.2638	−1.66409
$950$	0	0
$951$	−12.8065	−0.415280
$952$	0	0
$953$	44.6513	1.44640	0.723199	−	0.690640i	$-0.242671\pi$
$953$	44.6513	1.44640	0.723199	+	0.690640i	$0.242671\pi$
$954$	0	0
$955$	−2.52401	−0.0816750
$956$	0	0
$957$	−3.14882	−0.101787
$958$	0	0
$959$	4.25468	0.137391
$960$	0	0
$961$	−28.7526	−0.927502
$962$	0	0
$963$	−1.51411	−0.0487916
$964$	0	0
$965$	6.09120	0.196083
$966$	0	0
$967$	56.9750	1.83219	0.916096	−	0.400959i	$-0.131323\pi$
$967$	56.9750	1.83219	0.916096	+	0.400959i	$0.131323\pi$
$968$	0	0
$969$	−3.23974	−0.104075
$970$	0	0
$971$	−32.1333	−1.03121	−0.515604	−	0.856827i	$-0.672432\pi$
$971$	−32.1333	−1.03121	−0.515604	+	0.856827i	$0.672432\pi$
$972$	0	0
$973$	−16.7790	−0.537911
$974$	0	0
$975$	36.6526	1.17382
$976$	0	0
$977$	53.2357	1.70316	0.851580	−	0.524224i	$-0.175645\pi$
$977$	53.2357	1.70316	0.851580	+	0.524224i	$0.175645\pi$
$978$	0	0
$979$	−9.14410	−0.292247
$980$	0	0
$981$	−0.406545	−0.0129800
$982$	0	0
$983$	37.1645	1.18536	0.592681	−	0.805437i	$-0.298069\pi$
$983$	37.1645	1.18536	0.592681	+	0.805437i	$0.298069\pi$
$984$	0	0
$985$	0.447237	0.0142502
$986$	0	0
$987$	11.0173	0.350686
$988$	0	0
$989$	0.856132	0.0272234
$990$	0	0
$991$	50.1445	1.59289	0.796446	−	0.604710i	$-0.206711\pi$
$991$	50.1445	1.59289	0.796446	+	0.604710i	$0.206711\pi$
$992$	0	0
$993$	−1.87685	−0.0595601
$994$	0	0
$995$	−13.9660	−0.442752
$996$	0	0
$997$	23.4874	0.743852	0.371926	−	0.928262i	$-0.378697\pi$
$997$	23.4874	0.743852	0.371926	+	0.928262i	$0.378697\pi$
$998$	0	0
$999$	−18.5689	−0.587493

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.g.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.g.1.11	✓	12	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+1.70453 q^{3} -0.637713 q^{5} +1.14740 q^{7} -0.0945840 q^{9} +O(q^{10})\)	\(q+1.70453 q^{3} -0.637713 q^{5} +1.14740 q^{7} -0.0945840 q^{9} +0.485761 q^{11} -4.68138 q^{13} -1.08700 q^{15} -1.00000 q^{17} +1.90066 q^{19} +1.95578 q^{21} -0.312411 q^{23} -4.59332 q^{25} -5.27481 q^{27} -3.80295 q^{29} -1.49914 q^{31} +0.827993 q^{33} -0.731715 q^{35} +3.52029 q^{37} -7.97955 q^{39} -5.10566 q^{41} -2.74040 q^{43} +0.0603175 q^{45} +5.63321 q^{47} -5.68346 q^{49} -1.70453 q^{51} -5.67494 q^{53} -0.309776 q^{55} +3.23974 q^{57} +1.00000 q^{59} -11.8658 q^{61} -0.108526 q^{63} +2.98538 q^{65} +7.64584 q^{67} -0.532514 q^{69} +1.03706 q^{71} +10.9506 q^{73} -7.82945 q^{75} +0.557364 q^{77} -0.162183 q^{79} -8.70730 q^{81} -1.63200 q^{83} +0.637713 q^{85} -6.48224 q^{87} -18.8243 q^{89} -5.37144 q^{91} -2.55533 q^{93} -1.21208 q^{95} -14.5538 q^{97} -0.0459452 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9} - 5 q^{11} + 9 q^{13} - 16 q^{15} - 12 q^{17} - 15 q^{19} + 5 q^{21} - 22 q^{23} + 5 q^{25} - 16 q^{27} - 17 q^{29} + 5 q^{31} - 33 q^{33} - 9 q^{35} - 2 q^{37} - 24 q^{39} - 18 q^{41} - 16 q^{43} + 30 q^{45} - 60 q^{47} + 10 q^{49} + 4 q^{51} - 24 q^{53} - q^{55} + 41 q^{57} + 12 q^{59} - 17 q^{61} - 23 q^{63} + 2 q^{65} - 22 q^{67} + 6 q^{69} - 10 q^{71} + 4 q^{73} - 62 q^{75} - 6 q^{77} + 48 q^{79} + 4 q^{81} - 20 q^{83} - 3 q^{85} + 13 q^{87} - 29 q^{89} - 28 q^{91} - 31 q^{93} - 48 q^{95} - 26 q^{97} - 3 q^{99}+O(q^{100}) \) 12 * q - 4 * q^3 + 3 * q^5 + 2 * q^7 + 6 * q^9 - 5 * q^11 + 9 * q^13 - 16 * q^15 - 12 * q^17 - 15 * q^19 + 5 * q^21 - 22 * q^23 + 5 * q^25 - 16 * q^27 - 17 * q^29 + 5 * q^31 - 33 * q^33 - 9 * q^35 - 2 * q^37 - 24 * q^39 - 18 * q^41 - 16 * q^43 + 30 * q^45 - 60 * q^47 + 10 * q^49 + 4 * q^51 - 24 * q^53 - q^55 + 41 * q^57 + 12 * q^59 - 17 * q^61 - 23 * q^63 + 2 * q^65 - 22 * q^67 + 6 * q^69 - 10 * q^71 + 4 * q^73 - 62 * q^75 - 6 * q^77 + 48 * q^79 + 4 * q^81 - 20 * q^83 - 3 * q^85 + 13 * q^87 - 29 * q^89 - 28 * q^91 - 31 * q^93 - 48 * q^95 - 26 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more