LMFDB - Embedded newform 4012.2.a.g.1.8

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			1.8
Root			$-0.692763$ 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)

$f(q)$	$=$	$q+0.692763 q^{3} -3.58411 q^{5} +3.81064 q^{7} -2.52008 q^{9} +O(q^{10})$	$q+0.692763 q^{3} -3.58411 q^{5} +3.81064 q^{7} -2.52008 q^{9} -3.20083 q^{11} +3.92337 q^{13} -2.48294 q^{15} -1.00000 q^{17} +0.300487 q^{19} +2.63987 q^{21} -1.11886 q^{23} +7.84587 q^{25} -3.82410 q^{27} +7.45432 q^{29} +0.368034 q^{31} -2.21742 q^{33} -13.6578 q^{35} -5.36832 q^{37} +2.71796 q^{39} +0.470342 q^{41} +8.69365 q^{43} +9.03225 q^{45} -8.28528 q^{47} +7.52094 q^{49} -0.692763 q^{51} -14.2904 q^{53} +11.4721 q^{55} +0.208166 q^{57} +1.00000 q^{59} -2.58276 q^{61} -9.60311 q^{63} -14.0618 q^{65} -6.13324 q^{67} -0.775108 q^{69} -5.69456 q^{71} +0.352353 q^{73} +5.43533 q^{75} -12.1972 q^{77} +0.179372 q^{79} +4.91104 q^{81} +11.2386 q^{83} +3.58411 q^{85} +5.16408 q^{87} -16.2719 q^{89} +14.9505 q^{91} +0.254960 q^{93} -1.07698 q^{95} -13.6598 q^{97} +8.06635 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.692763	0.399967	0.199983	−	0.979799i	$-0.435911\pi$
$3$	0.692763	0.399967	0.199983	+	0.979799i	$0.435911\pi$
$4$	0	0
$5$	−3.58411	−1.60286	−0.801432	−	0.598086i	$-0.795928\pi$
$5$	−3.58411	−1.60286	−0.801432	+	0.598086i	$0.795928\pi$
$6$	0	0
$7$	3.81064	1.44028	0.720142	−	0.693826i	$-0.244076\pi$
$7$	3.81064	1.44028	0.720142	+	0.693826i	$0.244076\pi$
$8$	0	0
$9$	−2.52008	−0.840027
$10$	0	0
$11$	−3.20083	−0.965087	−0.482543	−	0.875872i	$-0.660287\pi$
$11$	−3.20083	−0.965087	−0.482543	+	0.875872i	$0.660287\pi$
$12$	0	0
$13$	3.92337	1.08815	0.544073	−	0.839038i	$-0.316881\pi$
$13$	3.92337	1.08815	0.544073	+	0.839038i	$0.316881\pi$
$14$	0	0
$15$	−2.48294	−0.641092
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	0.300487	0.0689364	0.0344682	−	0.999406i	$-0.489026\pi$
$19$	0.300487	0.0689364	0.0344682	+	0.999406i	$0.489026\pi$
$20$	0	0
$21$	2.63987	0.576066
$22$	0	0
$23$	−1.11886	−0.233299	−0.116650	−	0.993173i	$-0.537215\pi$
$23$	−1.11886	−0.233299	−0.116650	+	0.993173i	$0.537215\pi$
$24$	0	0
$25$	7.84587	1.56917
$26$	0	0
$27$	−3.82410	−0.735949
$28$	0	0
$29$	7.45432	1.38423	0.692117	−	0.721786i	$-0.256678\pi$
$29$	7.45432	1.38423	0.692117	+	0.721786i	$0.256678\pi$
$30$	0	0
$31$	0.368034	0.0661009	0.0330504	−	0.999454i	$-0.489478\pi$
$31$	0.368034	0.0661009	0.0330504	+	0.999454i	$0.489478\pi$
$32$	0	0
$33$	−2.21742	−0.386002
$34$	0	0
$35$	−13.6578	−2.30858
$36$	0	0
$37$	−5.36832	−0.882547	−0.441273	−	0.897373i	$-0.645473\pi$
$37$	−5.36832	−0.882547	−0.441273	+	0.897373i	$0.645473\pi$
$38$	0	0
$39$	2.71796	0.435222
$40$	0	0
$41$	0.470342	0.0734552	0.0367276	−	0.999325i	$-0.488307\pi$
$41$	0.470342	0.0734552	0.0367276	+	0.999325i	$0.488307\pi$
$42$	0	0
$43$	8.69365	1.32577	0.662884	−	0.748722i	$-0.269332\pi$
$43$	8.69365	1.32577	0.662884	+	0.748722i	$0.269332\pi$
$44$	0	0
$45$	9.03225	1.34645
$46$	0	0
$47$	−8.28528	−1.20853	−0.604266	−	0.796783i	$-0.706534\pi$
$47$	−8.28528	−1.20853	−0.604266	+	0.796783i	$0.706534\pi$
$48$	0	0
$49$	7.52094	1.07442
$50$	0	0
$51$	−0.692763	−0.0970062
$52$	0	0
$53$	−14.2904	−1.96294	−0.981469	−	0.191621i	$-0.938625\pi$
$53$	−14.2904	−1.96294	−0.981469	+	0.191621i	$0.938625\pi$
$54$	0	0
$55$	11.4721	1.54690
$56$	0	0
$57$	0.208166	0.0275723
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−2.58276	−0.330689	−0.165344	−	0.986236i	$-0.552874\pi$
$61$	−2.58276	−0.330689	−0.165344	+	0.986236i	$0.552874\pi$
$62$	0	0
$63$	−9.60311	−1.20988
$64$	0	0
$65$	−14.0618	−1.74415
$66$	0	0
$67$	−6.13324	−0.749294	−0.374647	−	0.927167i	$-0.622236\pi$
$67$	−6.13324	−0.749294	−0.374647	+	0.927167i	$0.622236\pi$
$68$	0	0
$69$	−0.775108	−0.0933120
$70$	0	0
$71$	−5.69456	−0.675820	−0.337910	−	0.941178i	$-0.609720\pi$
$71$	−5.69456	−0.675820	−0.337910	+	0.941178i	$0.609720\pi$
$72$	0	0
$73$	0.352353	0.0412398	0.0206199	−	0.999787i	$-0.493436\pi$
$73$	0.352353	0.0412398	0.0206199	+	0.999787i	$0.493436\pi$
$74$	0	0
$75$	5.43533	0.627617
$76$	0	0
$77$	−12.1972	−1.39000
$78$	0	0
$79$	0.179372	0.0201809	0.0100905	−	0.999949i	$-0.496788\pi$
$79$	0.179372	0.0201809	0.0100905	+	0.999949i	$0.496788\pi$
$80$	0	0
$81$	4.91104	0.545671
$82$	0	0
$83$	11.2386	1.23360	0.616798	−	0.787121i	$-0.288429\pi$
$83$	11.2386	1.23360	0.616798	+	0.787121i	$0.288429\pi$
$84$	0	0
$85$	3.58411	0.388752
$86$	0	0
$87$	5.16408	0.553647
$88$	0	0
$89$	−16.2719	−1.72482	−0.862409	−	0.506211i	$-0.831046\pi$
$89$	−16.2719	−1.72482	−0.862409	+	0.506211i	$0.831046\pi$
$90$	0	0
$91$	14.9505	1.56724
$92$	0	0
$93$	0.254960	0.0264382
$94$	0	0
$95$	−1.07698	−0.110496
$96$	0	0
$97$	−13.6598	−1.38694	−0.693472	−	0.720484i	$-0.743920\pi$
$97$	−13.6598	−1.38694	−0.693472	+	0.720484i	$0.743920\pi$
$98$	0	0
$99$	8.06635	0.810698
$100$	0	0
$101$	−8.66079	−0.861781	−0.430890	−	0.902404i	$-0.641800\pi$
$101$	−8.66079	−0.861781	−0.430890	+	0.902404i	$0.641800\pi$
$102$	0	0
$103$	−14.1870	−1.39789	−0.698943	−	0.715177i	$-0.746346\pi$
$103$	−14.1870	−1.39789	−0.698943	+	0.715177i	$0.746346\pi$
$104$	0	0
$105$	−9.46158	−0.923356
$106$	0	0
$107$	8.38089	0.810211	0.405105	−	0.914270i	$-0.367235\pi$
$107$	8.38089	0.810211	0.405105	+	0.914270i	$0.367235\pi$
$108$	0	0
$109$	−8.81795	−0.844606	−0.422303	−	0.906455i	$-0.638778\pi$
$109$	−8.81795	−0.844606	−0.422303	+	0.906455i	$0.638778\pi$
$110$	0	0
$111$	−3.71897	−0.352989
$112$	0	0
$113$	0.435370	0.0409562	0.0204781	−	0.999790i	$-0.493481\pi$
$113$	0.435370	0.0409562	0.0204781	+	0.999790i	$0.493481\pi$
$114$	0	0
$115$	4.01014	0.373947
$116$	0	0
$117$	−9.88721	−0.914072
$118$	0	0
$119$	−3.81064	−0.349320
$120$	0	0
$121$	−0.754687	−0.0686079
$122$	0	0
$123$	0.325836	0.0293796
$124$	0	0
$125$	−10.1999	−0.912309
$126$	0	0
$127$	−9.07527	−0.805300	−0.402650	−	0.915354i	$-0.631911\pi$
$127$	−9.07527	−0.805300	−0.402650	+	0.915354i	$0.631911\pi$
$128$	0	0
$129$	6.02263	0.530263
$130$	0	0
$131$	14.0044	1.22357	0.611787	−	0.791023i	$-0.290451\pi$
$131$	14.0044	1.22357	0.611787	+	0.791023i	$0.290451\pi$
$132$	0	0
$133$	1.14505	0.0992881
$134$	0	0
$135$	13.7060	1.17963
$136$	0	0
$137$	−9.00200	−0.769093	−0.384546	−	0.923106i	$-0.625642\pi$
$137$	−9.00200	−0.769093	−0.384546	+	0.923106i	$0.625642\pi$
$138$	0	0
$139$	−12.9014	−1.09428	−0.547141	−	0.837040i	$-0.684284\pi$
$139$	−12.9014	−1.09428	−0.547141	+	0.837040i	$0.684284\pi$
$140$	0	0
$141$	−5.73973	−0.483373
$142$	0	0
$143$	−12.5580	−1.05016
$144$	0	0
$145$	−26.7171	−2.21874
$146$	0	0
$147$	5.21023	0.429732
$148$	0	0
$149$	−3.80897	−0.312043	−0.156022	−	0.987754i	$-0.549867\pi$
$149$	−3.80897	−0.312043	−0.156022	+	0.987754i	$0.549867\pi$
$150$	0	0
$151$	12.1450	0.988348	0.494174	−	0.869363i	$-0.335471\pi$
$151$	12.1450	0.988348	0.494174	+	0.869363i	$0.335471\pi$
$152$	0	0
$153$	2.52008	0.203736
$154$	0	0
$155$	−1.31908	−0.105951
$156$	0	0
$157$	18.5153	1.47768	0.738840	−	0.673881i	$-0.235374\pi$
$157$	18.5153	1.47768	0.738840	+	0.673881i	$0.235374\pi$
$158$	0	0
$159$	−9.89986	−0.785110
$160$	0	0
$161$	−4.26359	−0.336018
$162$	0	0
$163$	−6.75710	−0.529257	−0.264629	−	0.964350i	$-0.585249\pi$
$163$	−6.75710	−0.529257	−0.264629	+	0.964350i	$0.585249\pi$
$164$	0	0
$165$	7.94747	0.618710
$166$	0	0
$167$	−10.4916	−0.811861	−0.405930	−	0.913904i	$-0.633052\pi$
$167$	−10.4916	−0.811861	−0.405930	+	0.913904i	$0.633052\pi$
$168$	0	0
$169$	2.39283	0.184064
$170$	0	0
$171$	−0.757251	−0.0579084
$172$	0	0
$173$	19.7242	1.49961	0.749803	−	0.661662i	$-0.230148\pi$
$173$	19.7242	1.49961	0.749803	+	0.661662i	$0.230148\pi$
$174$	0	0
$175$	29.8978	2.26006
$176$	0	0
$177$	0.692763	0.0520712
$178$	0	0
$179$	4.76708	0.356308	0.178154	−	0.984003i	$-0.442987\pi$
$179$	4.76708	0.356308	0.178154	+	0.984003i	$0.442987\pi$
$180$	0	0
$181$	−13.0646	−0.971084	−0.485542	−	0.874213i	$-0.661378\pi$
$181$	−13.0646	−0.971084	−0.485542	+	0.874213i	$0.661378\pi$
$182$	0	0
$183$	−1.78924	−0.132265
$184$	0	0
$185$	19.2407	1.41460
$186$	0	0
$187$	3.20083	0.234068
$188$	0	0
$189$	−14.5723	−1.05998
$190$	0	0
$191$	−5.78852	−0.418842	−0.209421	−	0.977826i	$-0.567158\pi$
$191$	−5.78852	−0.418842	−0.209421	+	0.977826i	$0.567158\pi$
$192$	0	0
$193$	7.30748	0.526004	0.263002	−	0.964795i	$-0.415287\pi$
$193$	7.30748	0.526004	0.263002	+	0.964795i	$0.415287\pi$
$194$	0	0
$195$	−9.74149	−0.697603
$196$	0	0
$197$	−13.6653	−0.973611	−0.486805	−	0.873511i	$-0.661838\pi$
$197$	−13.6653	−0.973611	−0.486805	+	0.873511i	$0.661838\pi$
$198$	0	0
$199$	20.1075	1.42538	0.712692	−	0.701477i	$-0.247476\pi$
$199$	20.1075	1.42538	0.712692	+	0.701477i	$0.247476\pi$
$200$	0	0
$201$	−4.24888	−0.299693
$202$	0	0
$203$	28.4057	1.99369
$204$	0	0
$205$	−1.68576	−0.117739
$206$	0	0
$207$	2.81963	0.195978
$208$	0	0
$209$	−0.961808	−0.0665296
$210$	0	0
$211$	−6.34017	−0.436475	−0.218238	−	0.975896i	$-0.570031\pi$
$211$	−6.34017	−0.436475	−0.218238	+	0.975896i	$0.570031\pi$
$212$	0	0
$213$	−3.94498	−0.270305
$214$	0	0
$215$	−31.1590	−2.12503
$216$	0	0
$217$	1.40244	0.0952041
$218$	0	0
$219$	0.244097	0.0164945
$220$	0	0
$221$	−3.92337	−0.263914
$222$	0	0
$223$	−24.9219	−1.66890	−0.834448	−	0.551087i	$-0.814213\pi$
$223$	−24.9219	−1.66890	−0.834448	+	0.551087i	$0.814213\pi$
$224$	0	0
$225$	−19.7722	−1.31815
$226$	0	0
$227$	−1.19663	−0.0794232	−0.0397116	−	0.999211i	$-0.512644\pi$
$227$	−1.19663	−0.0794232	−0.0397116	+	0.999211i	$0.512644\pi$
$228$	0	0
$229$	−12.2340	−0.808443	−0.404222	−	0.914661i	$-0.632458\pi$
$229$	−12.2340	−0.808443	−0.404222	+	0.914661i	$0.632458\pi$
$230$	0	0
$231$	−8.44976	−0.555954
$232$	0	0
$233$	−26.1054	−1.71022	−0.855112	−	0.518443i	$-0.826512\pi$
$233$	−26.1054	−1.71022	−0.855112	+	0.518443i	$0.826512\pi$
$234$	0	0
$235$	29.6954	1.93711
$236$	0	0
$237$	0.124262	0.00807169
$238$	0	0
$239$	−25.8474	−1.67193	−0.835964	−	0.548784i	$-0.815091\pi$
$239$	−25.8474	−1.67193	−0.835964	+	0.548784i	$0.815091\pi$
$240$	0	0
$241$	−15.0375	−0.968650	−0.484325	−	0.874888i	$-0.660935\pi$
$241$	−15.0375	−0.968650	−0.484325	+	0.874888i	$0.660935\pi$
$242$	0	0
$243$	14.8745	0.954200
$244$	0	0
$245$	−26.9559	−1.72215
$246$	0	0
$247$	1.17892	0.0750130
$248$	0	0
$249$	7.78568	0.493397
$250$	0	0
$251$	−20.8418	−1.31552	−0.657762	−	0.753226i	$-0.728497\pi$
$251$	−20.8418	−1.31552	−0.657762	+	0.753226i	$0.728497\pi$
$252$	0	0
$253$	3.58130	0.225154
$254$	0	0
$255$	2.48294	0.155488
$256$	0	0
$257$	−13.4010	−0.835934	−0.417967	−	0.908462i	$-0.637257\pi$
$257$	−13.4010	−0.835934	−0.417967	+	0.908462i	$0.637257\pi$
$258$	0	0
$259$	−20.4567	−1.27112
$260$	0	0
$261$	−18.7855	−1.16279
$262$	0	0
$263$	−8.41728	−0.519032	−0.259516	−	0.965739i	$-0.583563\pi$
$263$	−8.41728	−0.519032	−0.259516	+	0.965739i	$0.583563\pi$
$264$	0	0
$265$	51.2184	3.14632
$266$	0	0
$267$	−11.2726	−0.689870
$268$	0	0
$269$	−4.89470	−0.298435	−0.149218	−	0.988804i	$-0.547676\pi$
$269$	−4.89470	−0.298435	−0.149218	+	0.988804i	$0.547676\pi$
$270$	0	0
$271$	14.9323	0.907072	0.453536	−	0.891238i	$-0.350162\pi$
$271$	14.9323	0.907072	0.453536	+	0.891238i	$0.350162\pi$
$272$	0	0
$273$	10.3572	0.626844
$274$	0	0
$275$	−25.1133	−1.51439
$276$	0	0
$277$	6.76980	0.406758	0.203379	−	0.979100i	$-0.434808\pi$
$277$	6.76980	0.406758	0.203379	+	0.979100i	$0.434808\pi$
$278$	0	0
$279$	−0.927476	−0.0555265
$280$	0	0
$281$	12.2942	0.733411	0.366705	−	0.930337i	$-0.380486\pi$
$281$	12.2942	0.733411	0.366705	+	0.930337i	$0.380486\pi$
$282$	0	0
$283$	3.09484	0.183969	0.0919846	−	0.995760i	$-0.470679\pi$
$283$	3.09484	0.183969	0.0919846	+	0.995760i	$0.470679\pi$
$284$	0	0
$285$	−0.746091	−0.0441946
$286$	0	0
$287$	1.79230	0.105796
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−9.46300	−0.554731
$292$	0	0
$293$	18.8018	1.09841	0.549207	−	0.835686i	$-0.314930\pi$
$293$	18.8018	1.09841	0.549207	+	0.835686i	$0.314930\pi$
$294$	0	0
$295$	−3.58411	−0.208675
$296$	0	0
$297$	12.2403	0.710255
$298$	0	0
$299$	−4.38972	−0.253864
$300$	0	0
$301$	33.1283	1.90948
$302$	0	0
$303$	−5.99987	−0.344684
$304$	0	0
$305$	9.25692	0.530049
$306$	0	0
$307$	−2.82586	−0.161280	−0.0806402	−	0.996743i	$-0.525696\pi$
$307$	−2.82586	−0.161280	−0.0806402	+	0.996743i	$0.525696\pi$
$308$	0	0
$309$	−9.82822	−0.559108
$310$	0	0
$311$	22.9631	1.30212	0.651058	−	0.759028i	$-0.274325\pi$
$311$	22.9631	1.30212	0.651058	+	0.759028i	$0.274325\pi$
$312$	0	0
$313$	−17.5078	−0.989599	−0.494799	−	0.869007i	$-0.664758\pi$
$313$	−17.5078	−0.989599	−0.494799	+	0.869007i	$0.664758\pi$
$314$	0	0
$315$	34.4186	1.93927
$316$	0	0
$317$	−4.53591	−0.254762	−0.127381	−	0.991854i	$-0.540657\pi$
$317$	−4.53591	−0.254762	−0.127381	+	0.991854i	$0.540657\pi$
$318$	0	0
$319$	−23.8600	−1.33591
$320$	0	0
$321$	5.80596	0.324057
$322$	0	0
$323$	−0.300487	−0.0167195
$324$	0	0
$325$	30.7823	1.70749
$326$	0	0
$327$	−6.10874	−0.337814
$328$	0	0
$329$	−31.5722	−1.74063
$330$	0	0
$331$	−35.8706	−1.97163	−0.985813	−	0.167849i	$-0.946318\pi$
$331$	−35.8706	−1.97163	−0.985813	+	0.167849i	$0.946318\pi$
$332$	0	0
$333$	13.5286	0.741363
$334$	0	0
$335$	21.9822	1.20102
$336$	0	0
$337$	20.6893	1.12702	0.563509	−	0.826110i	$-0.309451\pi$
$337$	20.6893	1.12702	0.563509	+	0.826110i	$0.309451\pi$
$338$	0	0
$339$	0.301608	0.0163811
$340$	0	0
$341$	−1.17801	−0.0637931
$342$	0	0
$343$	1.98513	0.107187
$344$	0	0
$345$	2.77807	0.149566
$346$	0	0
$347$	7.48174	0.401641	0.200820	−	0.979628i	$-0.435639\pi$
$347$	7.48174	0.401641	0.200820	+	0.979628i	$0.435639\pi$
$348$	0	0
$349$	13.7804	0.737647	0.368823	−	0.929500i	$-0.379761\pi$
$349$	13.7804	0.737647	0.368823	+	0.929500i	$0.379761\pi$
$350$	0	0
$351$	−15.0034	−0.800821
$352$	0	0
$353$	−35.9004	−1.91079	−0.955394	−	0.295336i	$-0.904569\pi$
$353$	−35.9004	−1.91079	−0.955394	+	0.295336i	$0.904569\pi$
$354$	0	0
$355$	20.4099	1.08325
$356$	0	0
$357$	−2.63987	−0.139717
$358$	0	0
$359$	24.4040	1.28800	0.643998	−	0.765027i	$-0.277275\pi$
$359$	24.4040	1.28800	0.643998	+	0.765027i	$0.277275\pi$
$360$	0	0
$361$	−18.9097	−0.995248
$362$	0	0
$363$	−0.522819	−0.0274409
$364$	0	0
$365$	−1.26287	−0.0661018
$366$	0	0
$367$	21.6956	1.13250	0.566251	−	0.824233i	$-0.308393\pi$
$367$	21.6956	1.13250	0.566251	+	0.824233i	$0.308393\pi$
$368$	0	0
$369$	−1.18530	−0.0617043
$370$	0	0
$371$	−54.4555	−2.82719
$372$	0	0
$373$	14.9830	0.775789	0.387894	−	0.921704i	$-0.373203\pi$
$373$	14.9830	0.775789	0.387894	+	0.921704i	$0.373203\pi$
$374$	0	0
$375$	−7.06613	−0.364893
$376$	0	0
$377$	29.2461	1.50625
$378$	0	0
$379$	11.1479	0.572631	0.286315	−	0.958135i	$-0.407570\pi$
$379$	11.1479	0.572631	0.286315	+	0.958135i	$0.407570\pi$
$380$	0	0
$381$	−6.28701	−0.322093
$382$	0	0
$383$	−3.15935	−0.161435	−0.0807177	−	0.996737i	$-0.525721\pi$
$383$	−3.15935	−0.161435	−0.0807177	+	0.996737i	$0.525721\pi$
$384$	0	0
$385$	43.7161	2.22798
$386$	0	0
$387$	−21.9087	−1.11368
$388$	0	0
$389$	13.2268	0.670627	0.335313	−	0.942107i	$-0.391158\pi$
$389$	13.2268	0.670627	0.335313	+	0.942107i	$0.391158\pi$
$390$	0	0
$391$	1.11886	0.0565834
$392$	0	0
$393$	9.70175	0.489389
$394$	0	0
$395$	−0.642889	−0.0323473
$396$	0	0
$397$	19.7146	0.989446	0.494723	−	0.869051i	$-0.335270\pi$
$397$	19.7146	0.989446	0.494723	+	0.869051i	$0.335270\pi$
$398$	0	0
$399$	0.793245	0.0397119
$400$	0	0
$401$	7.86494	0.392756	0.196378	−	0.980528i	$-0.437082\pi$
$401$	7.86494	0.392756	0.196378	+	0.980528i	$0.437082\pi$
$402$	0	0
$403$	1.44393	0.0719275
$404$	0	0
$405$	−17.6017	−0.874637
$406$	0	0
$407$	17.1831	0.851734
$408$	0	0
$409$	1.23480	0.0610570	0.0305285	−	0.999534i	$-0.490281\pi$
$409$	1.23480	0.0610570	0.0305285	+	0.999534i	$0.490281\pi$
$410$	0	0
$411$	−6.23625	−0.307611
$412$	0	0
$413$	3.81064	0.187509
$414$	0	0
$415$	−40.2804	−1.97729
$416$	0	0
$417$	−8.93761	−0.437676
$418$	0	0
$419$	−23.6080	−1.15333	−0.576664	−	0.816982i	$-0.695646\pi$
$419$	−23.6080	−1.15333	−0.576664	+	0.816982i	$0.695646\pi$
$420$	0	0
$421$	−13.6336	−0.664463	−0.332231	−	0.943198i	$-0.607801\pi$
$421$	−13.6336	−0.664463	−0.332231	+	0.943198i	$0.607801\pi$
$422$	0	0
$423$	20.8796	1.01520
$424$	0	0
$425$	−7.84587	−0.380581
$426$	0	0
$427$	−9.84197	−0.476286
$428$	0	0
$429$	−8.69974	−0.420027
$430$	0	0
$431$	9.74132	0.469223	0.234611	−	0.972089i	$-0.424618\pi$
$431$	9.74132	0.469223	0.234611	+	0.972089i	$0.424618\pi$
$432$	0	0
$433$	26.3878	1.26812	0.634060	−	0.773284i	$-0.281387\pi$
$433$	26.3878	1.26812	0.634060	+	0.773284i	$0.281387\pi$
$434$	0	0
$435$	−18.5086	−0.887421
$436$	0	0
$437$	−0.336204	−0.0160828
$438$	0	0
$439$	25.7372	1.22837	0.614184	−	0.789163i	$-0.289485\pi$
$439$	25.7372	1.22837	0.614184	+	0.789163i	$0.289485\pi$
$440$	0	0
$441$	−18.9534	−0.902542
$442$	0	0
$443$	−0.988469	−0.0469636	−0.0234818	−	0.999724i	$-0.507475\pi$
$443$	−0.988469	−0.0469636	−0.0234818	+	0.999724i	$0.507475\pi$
$444$	0	0
$445$	58.3204	2.76465
$446$	0	0
$447$	−2.63871	−0.124807
$448$	0	0
$449$	2.09037	0.0986507	0.0493253	−	0.998783i	$-0.484293\pi$
$449$	2.09037	0.0986507	0.0493253	+	0.998783i	$0.484293\pi$
$450$	0	0
$451$	−1.50549	−0.0708906
$452$	0	0
$453$	8.41362	0.395306
$454$	0	0
$455$	−53.5844	−2.51208
$456$	0	0
$457$	10.9053	0.510129	0.255065	−	0.966924i	$-0.417903\pi$
$457$	10.9053	0.510129	0.255065	+	0.966924i	$0.417903\pi$
$458$	0	0
$459$	3.82410	0.178494
$460$	0	0
$461$	−13.5249	−0.629917	−0.314959	−	0.949105i	$-0.601991\pi$
$461$	−13.5249	−0.629917	−0.314959	+	0.949105i	$0.601991\pi$
$462$	0	0
$463$	−26.0581	−1.21102	−0.605510	−	0.795837i	$-0.707031\pi$
$463$	−26.0581	−1.21102	−0.605510	+	0.795837i	$0.707031\pi$
$464$	0	0
$465$	−0.913807	−0.0423768
$466$	0	0
$467$	4.03898	0.186902	0.0934509	−	0.995624i	$-0.470210\pi$
$467$	4.03898	0.186902	0.0934509	+	0.995624i	$0.470210\pi$
$468$	0	0
$469$	−23.3715	−1.07920
$470$	0	0
$471$	12.8267	0.591022
$472$	0	0
$473$	−27.8269	−1.27948
$474$	0	0
$475$	2.35758	0.108173
$476$	0	0
$477$	36.0130	1.64892
$478$	0	0
$479$	−2.89988	−0.132499	−0.0662494	−	0.997803i	$-0.521103\pi$
$479$	−2.89988	−0.132499	−0.0662494	+	0.997803i	$0.521103\pi$
$480$	0	0
$481$	−21.0619	−0.960341
$482$	0	0
$483$	−2.95365	−0.134396
$484$	0	0
$485$	48.9583	2.22308
$486$	0	0
$487$	17.2211	0.780362	0.390181	−	0.920738i	$-0.372412\pi$
$487$	17.2211	0.780362	0.390181	+	0.920738i	$0.372412\pi$
$488$	0	0
$489$	−4.68107	−0.211685
$490$	0	0
$491$	36.3473	1.64033	0.820165	−	0.572127i	$-0.193881\pi$
$491$	36.3473	1.64033	0.820165	+	0.572127i	$0.193881\pi$
$492$	0	0
$493$	−7.45432	−0.335726
$494$	0	0
$495$	−28.9107	−1.29944
$496$	0	0
$497$	−21.6999	−0.973373
$498$	0	0
$499$	−9.02762	−0.404132	−0.202066	−	0.979372i	$-0.564765\pi$
$499$	−9.02762	−0.404132	−0.202066	+	0.979372i	$0.564765\pi$
$500$	0	0
$501$	−7.26816	−0.324717
$502$	0	0
$503$	7.88686	0.351658	0.175829	−	0.984421i	$-0.443739\pi$
$503$	7.88686	0.351658	0.175829	+	0.984421i	$0.443739\pi$
$504$	0	0
$505$	31.0413	1.38132
$506$	0	0
$507$	1.65766	0.0736194
$508$	0	0
$509$	38.5727	1.70970	0.854852	−	0.518873i	$-0.173648\pi$
$509$	38.5727	1.70970	0.854852	+	0.518873i	$0.173648\pi$
$510$	0	0
$511$	1.34269	0.0593971
$512$	0	0
$513$	−1.14909	−0.0507337
$514$	0	0
$515$	50.8478	2.24062
$516$	0	0
$517$	26.5198	1.16634
$518$	0	0
$519$	13.6642	0.599792
$520$	0	0
$521$	−29.7094	−1.30159	−0.650797	−	0.759252i	$-0.725565\pi$
$521$	−29.7094	−1.30159	−0.650797	+	0.759252i	$0.725565\pi$
$522$	0	0
$523$	−28.4939	−1.24595	−0.622975	−	0.782242i	$-0.714076\pi$
$523$	−28.4939	−1.24595	−0.622975	+	0.782242i	$0.714076\pi$
$524$	0	0
$525$	20.7120	0.903948
$526$	0	0
$527$	−0.368034	−0.0160318
$528$	0	0
$529$	−21.7481	−0.945571
$530$	0	0
$531$	−2.52008	−0.109362
$532$	0	0
$533$	1.84533	0.0799300
$534$	0	0
$535$	−30.0381	−1.29866
$536$	0	0
$537$	3.30246	0.142511
$538$	0	0
$539$	−24.0733	−1.03691
$540$	0	0
$541$	−43.5547	−1.87256	−0.936282	−	0.351250i	$-0.885757\pi$
$541$	−43.5547	−1.87256	−0.936282	+	0.351250i	$0.885757\pi$
$542$	0	0
$543$	−9.05067	−0.388401
$544$	0	0
$545$	31.6045	1.35379
$546$	0	0
$547$	19.7898	0.846149	0.423075	−	0.906095i	$-0.360951\pi$
$547$	19.7898	0.846149	0.423075	+	0.906095i	$0.360951\pi$
$548$	0	0
$549$	6.50877	0.277788
$550$	0	0
$551$	2.23993	0.0954241
$552$	0	0
$553$	0.683521	0.0290663
$554$	0	0
$555$	13.3292	0.565794
$556$	0	0
$557$	39.6368	1.67947	0.839734	−	0.542999i	$-0.182711\pi$
$557$	39.6368	1.67947	0.839734	+	0.542999i	$0.182711\pi$
$558$	0	0
$559$	34.1084	1.44263
$560$	0	0
$561$	2.21742	0.0936193
$562$	0	0
$563$	11.4248	0.481496	0.240748	−	0.970588i	$-0.422607\pi$
$563$	11.4248	0.481496	0.240748	+	0.970588i	$0.422607\pi$
$564$	0	0
$565$	−1.56042	−0.0656472
$566$	0	0
$567$	18.7142	0.785922
$568$	0	0
$569$	45.1045	1.89088	0.945440	−	0.325797i	$-0.105633\pi$
$569$	45.1045	1.89088	0.945440	+	0.325797i	$0.105633\pi$
$570$	0	0
$571$	2.36082	0.0987974	0.0493987	−	0.998779i	$-0.484269\pi$
$571$	2.36082	0.0987974	0.0493987	+	0.998779i	$0.484269\pi$
$572$	0	0
$573$	−4.01007	−0.167523
$574$	0	0
$575$	−8.77847	−0.366088
$576$	0	0
$577$	−0.880750	−0.0366661	−0.0183330	−	0.999832i	$-0.505836\pi$
$577$	−0.880750	−0.0366661	−0.0183330	+	0.999832i	$0.505836\pi$
$578$	0	0
$579$	5.06235	0.210384
$580$	0	0
$581$	42.8262	1.77673
$582$	0	0
$583$	45.7412	1.89440
$584$	0	0
$585$	35.4369	1.46513
$586$	0	0
$587$	−22.6217	−0.933699	−0.466849	−	0.884337i	$-0.654611\pi$
$587$	−22.6217	−0.933699	−0.466849	+	0.884337i	$0.654611\pi$
$588$	0	0
$589$	0.110589	0.00455676
$590$	0	0
$591$	−9.46679	−0.389412
$592$	0	0
$593$	22.7829	0.935580	0.467790	−	0.883840i	$-0.345050\pi$
$593$	22.7829	0.935580	0.467790	+	0.883840i	$0.345050\pi$
$594$	0	0
$595$	13.6578	0.559913
$596$	0	0
$597$	13.9297	0.570106
$598$	0	0
$599$	−44.6684	−1.82510	−0.912550	−	0.408966i	$-0.865890\pi$
$599$	−44.6684	−1.82510	−0.912550	+	0.408966i	$0.865890\pi$
$600$	0	0
$601$	−20.7217	−0.845257	−0.422628	−	0.906303i	$-0.638892\pi$
$601$	−20.7217	−0.845257	−0.422628	+	0.906303i	$0.638892\pi$
$602$	0	0
$603$	15.4562	0.629427
$604$	0	0
$605$	2.70488	0.109969
$606$	0	0
$607$	−34.9290	−1.41772	−0.708862	−	0.705347i	$-0.750791\pi$
$607$	−34.9290	−1.41772	−0.708862	+	0.705347i	$0.750791\pi$
$608$	0	0
$609$	19.6784	0.797410
$610$	0	0
$611$	−32.5062	−1.31506
$612$	0	0
$613$	−2.34421	−0.0946819	−0.0473410	−	0.998879i	$-0.515075\pi$
$613$	−2.34421	−0.0946819	−0.0473410	+	0.998879i	$0.515075\pi$
$614$	0	0
$615$	−1.16783	−0.0470915
$616$	0	0
$617$	45.1366	1.81713	0.908566	−	0.417741i	$-0.137178\pi$
$617$	45.1366	1.81713	0.908566	+	0.417741i	$0.137178\pi$
$618$	0	0
$619$	38.4049	1.54362	0.771811	−	0.635852i	$-0.219351\pi$
$619$	38.4049	1.54362	0.771811	+	0.635852i	$0.219351\pi$
$620$	0	0
$621$	4.27866	0.171697
$622$	0	0
$623$	−62.0063	−2.48423
$624$	0	0
$625$	−2.67166	−0.106866
$626$	0	0
$627$	−0.666304	−0.0266096
$628$	0	0
$629$	5.36832	0.214049
$630$	0	0
$631$	6.61255	0.263241	0.131621	−	0.991300i	$-0.457982\pi$
$631$	6.61255	0.263241	0.131621	+	0.991300i	$0.457982\pi$
$632$	0	0
$633$	−4.39223	−0.174575
$634$	0	0
$635$	32.5268	1.29079
$636$	0	0
$637$	29.5074	1.16913
$638$	0	0
$639$	14.3507	0.567707
$640$	0	0
$641$	−1.06934	−0.0422365	−0.0211182	−	0.999777i	$-0.506723\pi$
$641$	−1.06934	−0.0422365	−0.0211182	+	0.999777i	$0.506723\pi$
$642$	0	0
$643$	40.6006	1.60113	0.800566	−	0.599245i	$-0.204532\pi$
$643$	40.6006	1.60113	0.800566	+	0.599245i	$0.204532\pi$
$644$	0	0
$645$	−21.5858	−0.849940
$646$	0	0
$647$	1.55716	0.0612183	0.0306091	−	0.999531i	$-0.490255\pi$
$647$	1.55716	0.0612183	0.0306091	+	0.999531i	$0.490255\pi$
$648$	0	0
$649$	−3.20083	−0.125644
$650$	0	0
$651$	0.971561	0.0380785
$652$	0	0
$653$	−13.1250	−0.513620	−0.256810	−	0.966462i	$-0.582671\pi$
$653$	−13.1250	−0.513620	−0.256810	+	0.966462i	$0.582671\pi$
$654$	0	0
$655$	−50.1935	−1.96122
$656$	0	0
$657$	−0.887958	−0.0346425
$658$	0	0
$659$	−8.17999	−0.318647	−0.159324	−	0.987226i	$-0.550931\pi$
$659$	−8.17999	−0.318647	−0.159324	+	0.987226i	$0.550931\pi$
$660$	0	0
$661$	−14.7286	−0.572878	−0.286439	−	0.958099i	$-0.592472\pi$
$661$	−14.7286	−0.572878	−0.286439	+	0.958099i	$0.592472\pi$
$662$	0	0
$663$	−2.71796	−0.105557
$664$	0	0
$665$	−4.10398	−0.159145
$666$	0	0
$667$	−8.34038	−0.322941
$668$	0	0
$669$	−17.2650	−0.667503
$670$	0	0
$671$	8.26699	0.319143
$672$	0	0
$673$	13.5275	0.521445	0.260723	−	0.965414i	$-0.416039\pi$
$673$	13.5275	0.521445	0.260723	+	0.965414i	$0.416039\pi$
$674$	0	0
$675$	−30.0034	−1.15483
$676$	0	0
$677$	25.7809	0.990839	0.495420	−	0.868654i	$-0.335014\pi$
$677$	25.7809	0.990839	0.495420	+	0.868654i	$0.335014\pi$
$678$	0	0
$679$	−52.0525	−1.99759
$680$	0	0
$681$	−0.828981	−0.0317666
$682$	0	0
$683$	−24.9326	−0.954020	−0.477010	−	0.878898i	$-0.658279\pi$
$683$	−24.9326	−0.954020	−0.477010	+	0.878898i	$0.658279\pi$
$684$	0	0
$685$	32.2642	1.23275
$686$	0	0
$687$	−8.47524	−0.323350
$688$	0	0
$689$	−56.0665	−2.13596
$690$	0	0
$691$	−16.3247	−0.621022	−0.310511	−	0.950570i	$-0.600500\pi$
$691$	−16.3247	−0.621022	−0.310511	+	0.950570i	$0.600500\pi$
$692$	0	0
$693$	30.7379	1.16764
$694$	0	0
$695$	46.2401	1.75399
$696$	0	0
$697$	−0.470342	−0.0178155
$698$	0	0
$699$	−18.0849	−0.684033
$700$	0	0
$701$	−5.69946	−0.215266	−0.107633	−	0.994191i	$-0.534327\pi$
$701$	−5.69946	−0.215266	−0.107633	+	0.994191i	$0.534327\pi$
$702$	0	0
$703$	−1.61311	−0.0608396
$704$	0	0
$705$	20.5719	0.774781
$706$	0	0
$707$	−33.0031	−1.24121
$708$	0	0
$709$	12.3714	0.464619	0.232309	−	0.972642i	$-0.425372\pi$
$709$	12.3714	0.464619	0.232309	+	0.972642i	$0.425372\pi$
$710$	0	0
$711$	−0.452032	−0.0169525
$712$	0	0
$713$	−0.411781	−0.0154213
$714$	0	0
$715$	45.0094	1.68326
$716$	0	0
$717$	−17.9061	−0.668716
$718$	0	0
$719$	−52.0627	−1.94161	−0.970805	−	0.239869i	$-0.922895\pi$
$719$	−52.0627	−1.94161	−0.970805	+	0.239869i	$0.922895\pi$
$720$	0	0
$721$	−54.0615	−2.01336
$722$	0	0
$723$	−10.4174	−0.387428
$724$	0	0
$725$	58.4857	2.17210
$726$	0	0
$727$	−33.8084	−1.25389	−0.626943	−	0.779065i	$-0.715694\pi$
$727$	−33.8084	−1.25389	−0.626943	+	0.779065i	$0.715694\pi$
$728$	0	0
$729$	−4.42863	−0.164023
$730$	0	0
$731$	−8.69365	−0.321546
$732$	0	0
$733$	7.37380	0.272358	0.136179	−	0.990684i	$-0.456518\pi$
$733$	7.37380	0.272358	0.136179	+	0.990684i	$0.456518\pi$
$734$	0	0
$735$	−18.6741	−0.688803
$736$	0	0
$737$	19.6314	0.723134
$738$	0	0
$739$	0.110613	0.00406896	0.00203448	−	0.999998i	$-0.499352\pi$
$739$	0.110613	0.00406896	0.00203448	+	0.999998i	$0.499352\pi$
$740$	0	0
$741$	0.816713	0.0300027
$742$	0	0
$743$	22.2344	0.815702	0.407851	−	0.913049i	$-0.366278\pi$
$743$	22.2344	0.815702	0.407851	+	0.913049i	$0.366278\pi$
$744$	0	0
$745$	13.6518	0.500163
$746$	0	0
$747$	−28.3222	−1.03625
$748$	0	0
$749$	31.9365	1.16693
$750$	0	0
$751$	28.1473	1.02711	0.513554	−	0.858057i	$-0.328328\pi$
$751$	28.1473	1.02711	0.513554	+	0.858057i	$0.328328\pi$
$752$	0	0
$753$	−14.4384	−0.526165
$754$	0	0
$755$	−43.5291	−1.58419
$756$	0	0
$757$	11.2648	0.409426	0.204713	−	0.978822i	$-0.434374\pi$
$757$	11.2648	0.409426	0.204713	+	0.978822i	$0.434374\pi$
$758$	0	0
$759$	2.48099	0.0900542
$760$	0	0
$761$	−23.5576	−0.853962	−0.426981	−	0.904261i	$-0.640423\pi$
$761$	−23.5576	−0.853962	−0.426981	+	0.904261i	$0.640423\pi$
$762$	0	0
$763$	−33.6020	−1.21647
$764$	0	0
$765$	−9.03225	−0.326562
$766$	0	0
$767$	3.92337	0.141665
$768$	0	0
$769$	−12.5349	−0.452021	−0.226011	−	0.974125i	$-0.572568\pi$
$769$	−12.5349	−0.452021	−0.226011	+	0.974125i	$0.572568\pi$
$770$	0	0
$771$	−9.28374	−0.334346
$772$	0	0
$773$	54.3872	1.95617	0.978085	−	0.208205i	$-0.0667620\pi$
$773$	54.3872	1.95617	0.978085	+	0.208205i	$0.0667620\pi$
$774$	0	0
$775$	2.88755	0.103724
$776$	0	0
$777$	−14.1717	−0.508405
$778$	0	0
$779$	0.141332	0.00506374
$780$	0	0
$781$	18.2273	0.652225
$782$	0	0
$783$	−28.5061	−1.01873
$784$	0	0
$785$	−66.3608	−2.36852
$786$	0	0
$787$	−9.50206	−0.338712	−0.169356	−	0.985555i	$-0.554169\pi$
$787$	−9.50206	−0.338712	−0.169356	+	0.985555i	$0.554169\pi$
$788$	0	0
$789$	−5.83118	−0.207595
$790$	0	0
$791$	1.65904	0.0589886
$792$	0	0
$793$	−10.1331	−0.359838
$794$	0	0
$795$	35.4822	1.25842
$796$	0	0
$797$	42.6128	1.50942	0.754711	−	0.656058i	$-0.227777\pi$
$797$	42.6128	1.50942	0.754711	+	0.656058i	$0.227777\pi$
$798$	0	0
$799$	8.28528	0.293112
$800$	0	0
$801$	41.0065	1.44889
$802$	0	0
$803$	−1.12782	−0.0398000
$804$	0	0
$805$	15.2812	0.538591
$806$	0	0
$807$	−3.39087	−0.119364
$808$	0	0
$809$	−22.8115	−0.802011	−0.401006	−	0.916076i	$-0.631339\pi$
$809$	−22.8115	−0.802011	−0.401006	+	0.916076i	$0.631339\pi$
$810$	0	0
$811$	−2.69462	−0.0946209	−0.0473105	−	0.998880i	$-0.515065\pi$
$811$	−2.69462	−0.0946209	−0.0473105	+	0.998880i	$0.515065\pi$
$812$	0	0
$813$	10.3445	0.362799
$814$	0	0
$815$	24.2182	0.848327
$816$	0	0
$817$	2.61233	0.0913938
$818$	0	0
$819$	−37.6765	−1.31652
$820$	0	0
$821$	5.74454	0.200486	0.100243	−	0.994963i	$-0.468038\pi$
$821$	5.74454	0.200486	0.100243	+	0.994963i	$0.468038\pi$
$822$	0	0
$823$	17.8613	0.622605	0.311302	−	0.950311i	$-0.399235\pi$
$823$	17.8613	0.622605	0.311302	+	0.950311i	$0.399235\pi$
$824$	0	0
$825$	−17.3976	−0.605705
$826$	0	0
$827$	32.3309	1.12425	0.562127	−	0.827051i	$-0.309983\pi$
$827$	32.3309	1.12425	0.562127	+	0.827051i	$0.309983\pi$
$828$	0	0
$829$	0.458180	0.0159132	0.00795662	−	0.999968i	$-0.497467\pi$
$829$	0.458180	0.0159132	0.00795662	+	0.999968i	$0.497467\pi$
$830$	0	0
$831$	4.68986	0.162689
$832$	0	0
$833$	−7.52094	−0.260585
$834$	0	0
$835$	37.6029	1.30130
$836$	0	0
$837$	−1.40740	−0.0486469
$838$	0	0
$839$	−14.6189	−0.504701	−0.252350	−	0.967636i	$-0.581204\pi$
$839$	−14.6189	−0.504701	−0.252350	+	0.967636i	$0.581204\pi$
$840$	0	0
$841$	26.5670	0.916102
$842$	0	0
$843$	8.51697	0.293340
$844$	0	0
$845$	−8.57617	−0.295029
$846$	0	0
$847$	−2.87584	−0.0988149
$848$	0	0
$849$	2.14399	0.0735816
$850$	0	0
$851$	6.00643	0.205898
$852$	0	0
$853$	−11.5233	−0.394549	−0.197274	−	0.980348i	$-0.563209\pi$
$853$	−11.5233	−0.394549	−0.197274	+	0.980348i	$0.563209\pi$
$854$	0	0
$855$	2.71407	0.0928194
$856$	0	0
$857$	49.2779	1.68330	0.841651	−	0.540022i	$-0.181584\pi$
$857$	49.2779	1.68330	0.841651	+	0.540022i	$0.181584\pi$
$858$	0	0
$859$	14.3279	0.488863	0.244431	−	0.969667i	$-0.421399\pi$
$859$	14.3279	0.488863	0.244431	+	0.969667i	$0.421399\pi$
$860$	0	0
$861$	1.24164	0.0423150
$862$	0	0
$863$	−15.8495	−0.539524	−0.269762	−	0.962927i	$-0.586945\pi$
$863$	−15.8495	−0.539524	−0.269762	+	0.962927i	$0.586945\pi$
$864$	0	0
$865$	−70.6939	−2.40366
$866$	0	0
$867$	0.692763	0.0235275
$868$	0	0
$869$	−0.574139	−0.0194763
$870$	0	0
$871$	−24.0630	−0.815342
$872$	0	0
$873$	34.4238	1.16507
$874$	0	0
$875$	−38.8682	−1.31399
$876$	0	0
$877$	−50.8841	−1.71823	−0.859116	−	0.511780i	$-0.828986\pi$
$877$	−50.8841	−1.71823	−0.859116	+	0.511780i	$0.828986\pi$
$878$	0	0
$879$	13.0252	0.439329
$880$	0	0
$881$	−39.7076	−1.33778	−0.668892	−	0.743359i	$-0.733231\pi$
$881$	−39.7076	−1.33778	−0.668892	+	0.743359i	$0.733231\pi$
$882$	0	0
$883$	−19.0582	−0.641358	−0.320679	−	0.947188i	$-0.603911\pi$
$883$	−19.0582	−0.641358	−0.320679	+	0.947188i	$0.603911\pi$
$884$	0	0
$885$	−2.48294	−0.0834631
$886$	0	0
$887$	27.3854	0.919512	0.459756	−	0.888045i	$-0.347937\pi$
$887$	27.3854	0.919512	0.459756	+	0.888045i	$0.347937\pi$
$888$	0	0
$889$	−34.5826	−1.15986
$890$	0	0
$891$	−15.7194	−0.526620
$892$	0	0
$893$	−2.48962	−0.0833119
$894$	0	0
$895$	−17.0858	−0.571114
$896$	0	0
$897$	−3.04103	−0.101537
$898$	0	0
$899$	2.74345	0.0914991
$900$	0	0
$901$	14.2904	0.476082
$902$	0	0
$903$	22.9501	0.763730
$904$	0	0
$905$	46.8250	1.55652
$906$	0	0
$907$	17.9406	0.595709	0.297855	−	0.954611i	$-0.403729\pi$
$907$	17.9406	0.595709	0.297855	+	0.954611i	$0.403729\pi$
$908$	0	0
$909$	21.8259	0.723919
$910$	0	0
$911$	31.3761	1.03954	0.519769	−	0.854307i	$-0.326018\pi$
$911$	31.3761	1.03954	0.519769	+	0.854307i	$0.326018\pi$
$912$	0	0
$913$	−35.9728	−1.19053
$914$	0	0
$915$	6.41285	0.212002
$916$	0	0
$917$	53.3658	1.76229
$918$	0	0
$919$	−16.9055	−0.557660	−0.278830	−	0.960340i	$-0.589947\pi$
$919$	−16.9055	−0.557660	−0.278830	+	0.960340i	$0.589947\pi$
$920$	0	0
$921$	−1.95765	−0.0645068
$922$	0	0
$923$	−22.3419	−0.735391
$924$	0	0
$925$	−42.1192	−1.38487
$926$	0	0
$927$	35.7524	1.17426
$928$	0	0
$929$	21.8038	0.715358	0.357679	−	0.933845i	$-0.383568\pi$
$929$	21.8038	0.715358	0.357679	+	0.933845i	$0.383568\pi$
$930$	0	0
$931$	2.25995	0.0740667
$932$	0	0
$933$	15.9080	0.520803
$934$	0	0
$935$	−11.4721	−0.375179
$936$	0	0
$937$	−34.3037	−1.12065	−0.560326	−	0.828272i	$-0.689324\pi$
$937$	−34.3037	−1.12065	−0.560326	+	0.828272i	$0.689324\pi$
$938$	0	0
$939$	−12.1287	−0.395807
$940$	0	0
$941$	12.7394	0.415293	0.207647	−	0.978204i	$-0.433420\pi$
$941$	12.7394	0.415293	0.207647	+	0.978204i	$0.433420\pi$
$942$	0	0
$943$	−0.526250	−0.0171370
$944$	0	0
$945$	52.2287	1.69900
$946$	0	0
$947$	−44.6263	−1.45016	−0.725081	−	0.688664i	$-0.758197\pi$
$947$	−44.6263	−1.45016	−0.725081	+	0.688664i	$0.758197\pi$
$948$	0	0
$949$	1.38241	0.0448750
$950$	0	0
$951$	−3.14231	−0.101896
$952$	0	0
$953$	15.3491	0.497206	0.248603	−	0.968605i	$-0.420028\pi$
$953$	15.3491	0.497206	0.248603	+	0.968605i	$0.420028\pi$
$954$	0	0
$955$	20.7467	0.671348
$956$	0	0
$957$	−16.5293	−0.534317
$958$	0	0
$959$	−34.3033	−1.10771
$960$	0	0
$961$	−30.8646	−0.995631
$962$	0	0
$963$	−21.1205	−0.680599
$964$	0	0
$965$	−26.1908	−0.843113
$966$	0	0
$967$	−0.432866	−0.0139200	−0.00696002	−	0.999976i	$-0.502215\pi$
$967$	−0.432866	−0.0139200	−0.00696002	+	0.999976i	$0.502215\pi$
$968$	0	0
$969$	−0.208166	−0.00668726
$970$	0	0
$971$	−31.1896	−1.00092	−0.500462	−	0.865759i	$-0.666836\pi$
$971$	−31.1896	−1.00092	−0.500462	+	0.865759i	$0.666836\pi$
$972$	0	0
$973$	−49.1625	−1.57608
$974$	0	0
$975$	21.3248	0.682940
$976$	0	0
$977$	−15.5044	−0.496029	−0.248015	−	0.968756i	$-0.579778\pi$
$977$	−15.5044	−0.496029	−0.248015	+	0.968756i	$0.579778\pi$
$978$	0	0
$979$	52.0836	1.66460
$980$	0	0
$981$	22.2219	0.709492
$982$	0	0
$983$	−6.89845	−0.220026	−0.110013	−	0.993930i	$-0.535089\pi$
$983$	−6.89845	−0.220026	−0.110013	+	0.993930i	$0.535089\pi$
$984$	0	0
$985$	48.9779	1.56057
$986$	0	0
$987$	−21.8720	−0.696194
$988$	0	0
$989$	−9.72702	−0.309301
$990$	0	0
$991$	−1.14244	−0.0362907	−0.0181453	−	0.999835i	$-0.505776\pi$
$991$	−1.14244	−0.0362907	−0.0181453	+	0.999835i	$0.505776\pi$
$992$	0	0
$993$	−24.8498	−0.788584
$994$	0	0
$995$	−72.0676	−2.28470
$996$	0	0
$997$	−2.05148	−0.0649710	−0.0324855	−	0.999472i	$-0.510342\pi$
$997$	−2.05148	−0.0649710	−0.0324855	+	0.999472i	$0.510342\pi$
$998$	0	0
$999$	20.5290	0.649510

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.g.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.g.1.8	✓	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)

\(f(q)\)	\(=\)	\(q+0.692763 q^{3} -3.58411 q^{5} +3.81064 q^{7} -2.52008 q^{9} +O(q^{10})\)	\(q+0.692763 q^{3} -3.58411 q^{5} +3.81064 q^{7} -2.52008 q^{9} -3.20083 q^{11} +3.92337 q^{13} -2.48294 q^{15} -1.00000 q^{17} +0.300487 q^{19} +2.63987 q^{21} -1.11886 q^{23} +7.84587 q^{25} -3.82410 q^{27} +7.45432 q^{29} +0.368034 q^{31} -2.21742 q^{33} -13.6578 q^{35} -5.36832 q^{37} +2.71796 q^{39} +0.470342 q^{41} +8.69365 q^{43} +9.03225 q^{45} -8.28528 q^{47} +7.52094 q^{49} -0.692763 q^{51} -14.2904 q^{53} +11.4721 q^{55} +0.208166 q^{57} +1.00000 q^{59} -2.58276 q^{61} -9.60311 q^{63} -14.0618 q^{65} -6.13324 q^{67} -0.775108 q^{69} -5.69456 q^{71} +0.352353 q^{73} +5.43533 q^{75} -12.1972 q^{77} +0.179372 q^{79} +4.91104 q^{81} +11.2386 q^{83} +3.58411 q^{85} +5.16408 q^{87} -16.2719 q^{89} +14.9505 q^{91} +0.254960 q^{93} -1.07698 q^{95} -13.6598 q^{97} +8.06635 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 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

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