LMFDB - Embedded newform 4012.2.a.i.1.7

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:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 8 x^{17} - 4 x^{16} + 178 x^{15} - 265 x^{14} - 1405 x^{13} + 3503 x^{12} + 4295 x^{11} + \cdots - 16 $ x^18 - 8x^17 - 4x^16 + 178x^15 - 265x^14 - 1405x^13 + 3503x^12 + 4295x^11 - 17374x^10 - 893x^9 + 38112x^8 - 18700x^7 - 32137x^6 + 26381x^5 + 3964x^4 - 5788x^3 - 108x^2 + 232*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.267006$ 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)

$f(q)$	$=$	$q-0.267006 q^{3} -2.18930 q^{5} +1.68463 q^{7} -2.92871 q^{9} +O(q^{10})$	$q-0.267006 q^{3} -2.18930 q^{5} +1.68463 q^{7} -2.92871 q^{9} +0.464539 q^{11} -0.483728 q^{13} +0.584555 q^{15} -1.00000 q^{17} +5.36037 q^{19} -0.449807 q^{21} -0.258026 q^{23} -0.206984 q^{25} +1.58300 q^{27} +6.75658 q^{29} -10.3430 q^{31} -0.124035 q^{33} -3.68816 q^{35} +4.64695 q^{37} +0.129158 q^{39} -3.11085 q^{41} -8.86431 q^{43} +6.41181 q^{45} -1.83810 q^{47} -4.16202 q^{49} +0.267006 q^{51} +7.12843 q^{53} -1.01701 q^{55} -1.43125 q^{57} -1.00000 q^{59} -0.701224 q^{61} -4.93379 q^{63} +1.05902 q^{65} -4.17534 q^{67} +0.0688946 q^{69} +7.71230 q^{71} +8.25999 q^{73} +0.0552661 q^{75} +0.782577 q^{77} +15.0902 q^{79} +8.36345 q^{81} +3.65833 q^{83} +2.18930 q^{85} -1.80405 q^{87} +4.99257 q^{89} -0.814904 q^{91} +2.76165 q^{93} -11.7354 q^{95} +12.9650 q^{97} -1.36050 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * 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
$3$	−0.267006	−0.154156	−0.0770780	−	0.997025i	$-0.524559\pi$
$3$	−0.267006	−0.154156	−0.0770780	+	0.997025i	$0.524559\pi$
$4$	0	0
$5$	−2.18930	−0.979083	−0.489541	−	0.871980i	$-0.662836\pi$
$5$	−2.18930	−0.979083	−0.489541	+	0.871980i	$0.662836\pi$
$6$	0	0
$7$	1.68463	0.636731	0.318365	−	0.947968i	$-0.396866\pi$
$7$	1.68463	0.636731	0.318365	+	0.947968i	$0.396866\pi$
$8$	0	0
$9$	−2.92871	−0.976236
$10$	0	0
$11$	0.464539	0.140064	0.0700319	−	0.997545i	$-0.477690\pi$
$11$	0.464539	0.140064	0.0700319	+	0.997545i	$0.477690\pi$
$12$	0	0
$13$	−0.483728	−0.134162	−0.0670810	−	0.997748i	$-0.521369\pi$
$13$	−0.483728	−0.134162	−0.0670810	+	0.997748i	$0.521369\pi$
$14$	0	0
$15$	0.584555	0.150932
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.36037	1.22975	0.614877	−	0.788623i	$-0.289206\pi$
$19$	5.36037	1.22975	0.614877	+	0.788623i	$0.289206\pi$
$20$	0	0
$21$	−0.449807	−0.0981559
$22$	0	0
$23$	−0.258026	−0.0538022	−0.0269011	−	0.999638i	$-0.508564\pi$
$23$	−0.258026	−0.0538022	−0.0269011	+	0.999638i	$0.508564\pi$
$24$	0	0
$25$	−0.206984	−0.0413969
$26$	0	0
$27$	1.58300	0.304649
$28$	0	0
$29$	6.75658	1.25467	0.627333	−	0.778751i	$-0.284147\pi$
$29$	6.75658	1.25467	0.627333	+	0.778751i	$0.284147\pi$
$30$	0	0
$31$	−10.3430	−1.85766	−0.928832	−	0.370501i	$-0.879186\pi$
$31$	−10.3430	−1.85766	−0.928832	+	0.370501i	$0.879186\pi$
$32$	0	0
$33$	−0.124035	−0.0215917
$34$	0	0
$35$	−3.68816	−0.623412
$36$	0	0
$37$	4.64695	0.763954	0.381977	−	0.924172i	$-0.375243\pi$
$37$	4.64695	0.763954	0.381977	+	0.924172i	$0.375243\pi$
$38$	0	0
$39$	0.129158	0.0206819
$40$	0	0
$41$	−3.11085	−0.485833	−0.242916	−	0.970047i	$-0.578104\pi$
$41$	−3.11085	−0.485833	−0.242916	+	0.970047i	$0.578104\pi$
$42$	0	0
$43$	−8.86431	−1.35179	−0.675897	−	0.736996i	$-0.736244\pi$
$43$	−8.86431	−1.35179	−0.675897	+	0.736996i	$0.736244\pi$
$44$	0	0
$45$	6.41181	0.955816
$46$	0	0
$47$	−1.83810	−0.268114	−0.134057	−	0.990974i	$-0.542801\pi$
$47$	−1.83810	−0.268114	−0.134057	+	0.990974i	$0.542801\pi$
$48$	0	0
$49$	−4.16202	−0.594574
$50$	0	0
$51$	0.267006	0.0373883
$52$	0	0
$53$	7.12843	0.979165	0.489583	−	0.871957i	$-0.337149\pi$
$53$	7.12843	0.979165	0.489583	+	0.871957i	$0.337149\pi$
$54$	0	0
$55$	−1.01701	−0.137134
$56$	0	0
$57$	−1.43125	−0.189574
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−0.701224	−0.0897826	−0.0448913	−	0.998992i	$-0.514294\pi$
$61$	−0.701224	−0.0897826	−0.0448913	+	0.998992i	$0.514294\pi$
$62$	0	0
$63$	−4.93379	−0.621600
$64$	0	0
$65$	1.05902	0.131356
$66$	0	0
$67$	−4.17534	−0.510099	−0.255050	−	0.966928i	$-0.582092\pi$
$67$	−4.17534	−0.510099	−0.255050	+	0.966928i	$0.582092\pi$
$68$	0	0
$69$	0.0688946	0.00829393
$70$	0	0
$71$	7.71230	0.915281	0.457641	−	0.889137i	$-0.348695\pi$
$71$	7.71230	0.915281	0.457641	+	0.889137i	$0.348695\pi$
$72$	0	0
$73$	8.25999	0.966759	0.483379	−	0.875411i	$-0.339409\pi$
$73$	8.25999	0.966759	0.483379	+	0.875411i	$0.339409\pi$
$74$	0	0
$75$	0.0552661	0.00638158
$76$	0	0
$77$	0.782577	0.0891829
$78$	0	0
$79$	15.0902	1.69778	0.848892	−	0.528567i	$-0.177270\pi$
$79$	15.0902	1.69778	0.848892	+	0.528567i	$0.177270\pi$
$80$	0	0
$81$	8.36345	0.929272
$82$	0	0
$83$	3.65833	0.401554	0.200777	−	0.979637i	$-0.435653\pi$
$83$	3.65833	0.401554	0.200777	+	0.979637i	$0.435653\pi$
$84$	0	0
$85$	2.18930	0.237462
$86$	0	0
$87$	−1.80405	−0.193414
$88$	0	0
$89$	4.99257	0.529211	0.264606	−	0.964357i	$-0.414758\pi$
$89$	4.99257	0.529211	0.264606	+	0.964357i	$0.414758\pi$
$90$	0	0
$91$	−0.814904	−0.0854251
$92$	0	0
$93$	2.76165	0.286370
$94$	0	0
$95$	−11.7354	−1.20403
$96$	0	0
$97$	12.9650	1.31640	0.658198	−	0.752845i	$-0.271319\pi$
$97$	12.9650	1.31640	0.658198	+	0.752845i	$0.271319\pi$
$98$	0	0
$99$	−1.36050	−0.136735
$100$	0	0
$101$	13.2214	1.31558	0.657790	−	0.753201i	$-0.271491\pi$
$101$	13.2214	1.31558	0.657790	+	0.753201i	$0.271491\pi$
$102$	0	0
$103$	5.04379	0.496980	0.248490	−	0.968635i	$-0.420066\pi$
$103$	5.04379	0.496980	0.248490	+	0.968635i	$0.420066\pi$
$104$	0	0
$105$	0.984760	0.0961028
$106$	0	0
$107$	−10.7365	−1.03794	−0.518969	−	0.854793i	$-0.673684\pi$
$107$	−10.7365	−1.03794	−0.518969	+	0.854793i	$0.673684\pi$
$108$	0	0
$109$	−11.6269	−1.11365	−0.556826	−	0.830629i	$-0.687981\pi$
$109$	−11.6269	−1.11365	−0.556826	+	0.830629i	$0.687981\pi$
$110$	0	0
$111$	−1.24076	−0.117768
$112$	0	0
$113$	2.33945	0.220077	0.110039	−	0.993927i	$-0.464903\pi$
$113$	2.33945	0.220077	0.110039	+	0.993927i	$0.464903\pi$
$114$	0	0
$115$	0.564896	0.0526768
$116$	0	0
$117$	1.41670	0.130974
$118$	0	0
$119$	−1.68463	−0.154430
$120$	0	0
$121$	−10.7842	−0.980382
$122$	0	0
$123$	0.830616	0.0748941
$124$	0	0
$125$	11.3996	1.01961
$126$	0	0
$127$	5.62279	0.498942	0.249471	−	0.968382i	$-0.419743\pi$
$127$	5.62279	0.498942	0.249471	+	0.968382i	$0.419743\pi$
$128$	0	0
$129$	2.36682	0.208387
$130$	0	0
$131$	12.1667	1.06301	0.531504	−	0.847056i	$-0.321627\pi$
$131$	12.1667	1.06301	0.531504	+	0.847056i	$0.321627\pi$
$132$	0	0
$133$	9.03025	0.783022
$134$	0	0
$135$	−3.46566	−0.298276
$136$	0	0
$137$	14.4225	1.23220	0.616100	−	0.787668i	$-0.288712\pi$
$137$	14.4225	1.23220	0.616100	+	0.787668i	$0.288712\pi$
$138$	0	0
$139$	16.3683	1.38834	0.694169	−	0.719812i	$-0.255772\pi$
$139$	16.3683	1.38834	0.694169	+	0.719812i	$0.255772\pi$
$140$	0	0
$141$	0.490783	0.0413314
$142$	0	0
$143$	−0.224710	−0.0187912
$144$	0	0
$145$	−14.7921	−1.22842
$146$	0	0
$147$	1.11128	0.0916571
$148$	0	0
$149$	12.6821	1.03896	0.519480	−	0.854482i	$-0.326126\pi$
$149$	12.6821	1.03896	0.519480	+	0.854482i	$0.326126\pi$
$150$	0	0
$151$	−13.8796	−1.12951	−0.564753	−	0.825260i	$-0.691029\pi$
$151$	−13.8796	−1.12951	−0.564753	+	0.825260i	$0.691029\pi$
$152$	0	0
$153$	2.92871	0.236772
$154$	0	0
$155$	22.6440	1.81881
$156$	0	0
$157$	11.3898	0.909003	0.454501	−	0.890746i	$-0.349817\pi$
$157$	11.3898	0.909003	0.454501	+	0.890746i	$0.349817\pi$
$158$	0	0
$159$	−1.90333	−0.150944
$160$	0	0
$161$	−0.434679	−0.0342575
$162$	0	0
$163$	−0.241579	−0.0189219	−0.00946096	−	0.999955i	$-0.503012\pi$
$163$	−0.241579	−0.0189219	−0.00946096	+	0.999955i	$0.503012\pi$
$164$	0	0
$165$	0.271549	0.0211400
$166$	0	0
$167$	−1.50344	−0.116339	−0.0581697	−	0.998307i	$-0.518526\pi$
$167$	−1.50344	−0.116339	−0.0581697	+	0.998307i	$0.518526\pi$
$168$	0	0
$169$	−12.7660	−0.982001
$170$	0	0
$171$	−15.6990	−1.20053
$172$	0	0
$173$	−5.81044	−0.441759	−0.220880	−	0.975301i	$-0.570893\pi$
$173$	−5.81044	−0.441759	−0.220880	+	0.975301i	$0.570893\pi$
$174$	0	0
$175$	−0.348692	−0.0263587
$176$	0	0
$177$	0.267006	0.0200694
$178$	0	0
$179$	14.1495	1.05759	0.528793	−	0.848751i	$-0.322645\pi$
$179$	14.1495	1.05759	0.528793	+	0.848751i	$0.322645\pi$
$180$	0	0
$181$	−23.6747	−1.75972	−0.879862	−	0.475230i	$-0.842365\pi$
$181$	−23.6747	−1.75972	−0.879862	+	0.475230i	$0.842365\pi$
$182$	0	0
$183$	0.187231	0.0138405
$184$	0	0
$185$	−10.1736	−0.747974
$186$	0	0
$187$	−0.464539	−0.0339704
$188$	0	0
$189$	2.66677	0.193979
$190$	0	0
$191$	19.0309	1.37703	0.688515	−	0.725222i	$-0.258263\pi$
$191$	19.0309	1.37703	0.688515	+	0.725222i	$0.258263\pi$
$192$	0	0
$193$	16.1951	1.16575	0.582875	−	0.812562i	$-0.301928\pi$
$193$	16.1951	1.16575	0.582875	+	0.812562i	$0.301928\pi$
$194$	0	0
$195$	−0.282766	−0.0202493
$196$	0	0
$197$	−12.3512	−0.879984	−0.439992	−	0.898002i	$-0.645019\pi$
$197$	−12.3512	−0.879984	−0.439992	+	0.898002i	$0.645019\pi$
$198$	0	0
$199$	−2.90452	−0.205896	−0.102948	−	0.994687i	$-0.532828\pi$
$199$	−2.90452	−0.205896	−0.102948	+	0.994687i	$0.532828\pi$
$200$	0	0
$201$	1.11484	0.0786349
$202$	0	0
$203$	11.3823	0.798884
$204$	0	0
$205$	6.81057	0.475671
$206$	0	0
$207$	0.755683	0.0525236
$208$	0	0
$209$	2.49010	0.172244
$210$	0	0
$211$	0.205512	0.0141480	0.00707401	−	0.999975i	$-0.497748\pi$
$211$	0.205512	0.0141480	0.00707401	+	0.999975i	$0.497748\pi$
$212$	0	0
$213$	−2.05923	−0.141096
$214$	0	0
$215$	19.4066	1.32352
$216$	0	0
$217$	−17.4242	−1.18283
$218$	0	0
$219$	−2.20547	−0.149032
$220$	0	0
$221$	0.483728	0.0325391
$222$	0	0
$223$	−1.73850	−0.116419	−0.0582093	−	0.998304i	$-0.518539\pi$
$223$	−1.73850	−0.116419	−0.0582093	+	0.998304i	$0.518539\pi$
$224$	0	0
$225$	0.606197	0.0404131
$226$	0	0
$227$	22.9369	1.52238	0.761188	−	0.648531i	$-0.224616\pi$
$227$	22.9369	1.52238	0.761188	+	0.648531i	$0.224616\pi$
$228$	0	0
$229$	5.07228	0.335186	0.167593	−	0.985856i	$-0.446401\pi$
$229$	5.07228	0.335186	0.167593	+	0.985856i	$0.446401\pi$
$230$	0	0
$231$	−0.208953	−0.0137481
$232$	0	0
$233$	−12.1864	−0.798355	−0.399178	−	0.916874i	$-0.630704\pi$
$233$	−12.1864	−0.798355	−0.399178	+	0.916874i	$0.630704\pi$
$234$	0	0
$235$	4.02414	0.262506
$236$	0	0
$237$	−4.02918	−0.261724
$238$	0	0
$239$	24.1466	1.56192	0.780958	−	0.624583i	$-0.214731\pi$
$239$	24.1466	1.56192	0.780958	+	0.624583i	$0.214731\pi$
$240$	0	0
$241$	8.94611	0.576270	0.288135	−	0.957590i	$-0.406965\pi$
$241$	8.94611	0.576270	0.288135	+	0.957590i	$0.406965\pi$
$242$	0	0
$243$	−6.98210	−0.447902
$244$	0	0
$245$	9.11189	0.582137
$246$	0	0
$247$	−2.59296	−0.164986
$248$	0	0
$249$	−0.976797	−0.0619020
$250$	0	0
$251$	20.8416	1.31551	0.657755	−	0.753232i	$-0.271506\pi$
$251$	20.8416	1.31551	0.657755	+	0.753232i	$0.271506\pi$
$252$	0	0
$253$	−0.119863	−0.00753573
$254$	0	0
$255$	−0.584555	−0.0366063
$256$	0	0
$257$	12.9884	0.810196	0.405098	−	0.914273i	$-0.367237\pi$
$257$	12.9884	0.810196	0.405098	+	0.914273i	$0.367237\pi$
$258$	0	0
$259$	7.82840	0.486433
$260$	0	0
$261$	−19.7880	−1.22485
$262$	0	0
$263$	−4.93178	−0.304106	−0.152053	−	0.988372i	$-0.548589\pi$
$263$	−4.93178	−0.304106	−0.152053	+	0.988372i	$0.548589\pi$
$264$	0	0
$265$	−15.6062	−0.958684
$266$	0	0
$267$	−1.33305	−0.0815811
$268$	0	0
$269$	−13.5087	−0.823638	−0.411819	−	0.911266i	$-0.635106\pi$
$269$	−13.5087	−0.823638	−0.411819	+	0.911266i	$0.635106\pi$
$270$	0	0
$271$	−16.4229	−0.997620	−0.498810	−	0.866711i	$-0.666230\pi$
$271$	−16.4229	−0.997620	−0.498810	+	0.866711i	$0.666230\pi$
$272$	0	0
$273$	0.217584	0.0131688
$274$	0	0
$275$	−0.0961523	−0.00579820
$276$	0	0
$277$	1.04622	0.0628612	0.0314306	−	0.999506i	$-0.489994\pi$
$277$	1.04622	0.0628612	0.0314306	+	0.999506i	$0.489994\pi$
$278$	0	0
$279$	30.2917	1.81352
$280$	0	0
$281$	18.1047	1.08004	0.540019	−	0.841653i	$-0.318417\pi$
$281$	18.1047	1.08004	0.540019	+	0.841653i	$0.318417\pi$
$282$	0	0
$283$	25.2697	1.50213	0.751064	−	0.660229i	$-0.229541\pi$
$283$	25.2697	1.50213	0.751064	+	0.660229i	$0.229541\pi$
$284$	0	0
$285$	3.13343	0.185609
$286$	0	0
$287$	−5.24063	−0.309345
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−3.46173	−0.202930
$292$	0	0
$293$	−3.11211	−0.181811	−0.0909057	−	0.995860i	$-0.528976\pi$
$293$	−3.11211	−0.181811	−0.0909057	+	0.995860i	$0.528976\pi$
$294$	0	0
$295$	2.18930	0.127466
$296$	0	0
$297$	0.735365	0.0426702
$298$	0	0
$299$	0.124815	0.00721821
$300$	0	0
$301$	−14.9331	−0.860729
$302$	0	0
$303$	−3.53020	−0.202805
$304$	0	0
$305$	1.53519	0.0879046
$306$	0	0
$307$	−3.13677	−0.179025	−0.0895124	−	0.995986i	$-0.528531\pi$
$307$	−3.13677	−0.179025	−0.0895124	+	0.995986i	$0.528531\pi$
$308$	0	0
$309$	−1.34672	−0.0766124
$310$	0	0
$311$	8.55659	0.485200	0.242600	−	0.970126i	$-0.422000\pi$
$311$	8.55659	0.485200	0.242600	+	0.970126i	$0.422000\pi$
$312$	0	0
$313$	17.3702	0.981821	0.490910	−	0.871210i	$-0.336664\pi$
$313$	17.3702	0.981821	0.490910	+	0.871210i	$0.336664\pi$
$314$	0	0
$315$	10.8015	0.608597
$316$	0	0
$317$	−29.5379	−1.65901	−0.829507	−	0.558497i	$-0.811378\pi$
$317$	−29.5379	−1.65901	−0.829507	+	0.558497i	$0.811378\pi$
$318$	0	0
$319$	3.13869	0.175733
$320$	0	0
$321$	2.86671	0.160004
$322$	0	0
$323$	−5.36037	−0.298259
$324$	0	0
$325$	0.100124	0.00555389
$326$	0	0
$327$	3.10445	0.171676
$328$	0	0
$329$	−3.09652	−0.170717
$330$	0	0
$331$	−1.79098	−0.0984411	−0.0492205	−	0.998788i	$-0.515674\pi$
$331$	−1.79098	−0.0984411	−0.0492205	+	0.998788i	$0.515674\pi$
$332$	0	0
$333$	−13.6096	−0.745799
$334$	0	0
$335$	9.14106	0.499429
$336$	0	0
$337$	1.49953	0.0816844	0.0408422	−	0.999166i	$-0.486996\pi$
$337$	1.49953	0.0816844	0.0408422	+	0.999166i	$0.486996\pi$
$338$	0	0
$339$	−0.624648	−0.0339262
$340$	0	0
$341$	−4.80474	−0.260191
$342$	0	0
$343$	−18.8039	−1.01531
$344$	0	0
$345$	−0.150831	−0.00812044
$346$	0	0
$347$	−23.8148	−1.27845	−0.639223	−	0.769022i	$-0.720744\pi$
$347$	−23.8148	−1.27845	−0.639223	+	0.769022i	$0.720744\pi$
$348$	0	0
$349$	9.58873	0.513273	0.256636	−	0.966508i	$-0.417386\pi$
$349$	9.58873	0.513273	0.256636	+	0.966508i	$0.417386\pi$
$350$	0	0
$351$	−0.765742	−0.0408723
$352$	0	0
$353$	33.4399	1.77983	0.889914	−	0.456128i	$-0.150764\pi$
$353$	33.4399	1.77983	0.889914	+	0.456128i	$0.150764\pi$
$354$	0	0
$355$	−16.8845	−0.896136
$356$	0	0
$357$	0.449807	0.0238063
$358$	0	0
$359$	22.6029	1.19294	0.596468	−	0.802637i	$-0.296570\pi$
$359$	22.6029	1.19294	0.596468	+	0.802637i	$0.296570\pi$
$360$	0	0
$361$	9.73356	0.512293
$362$	0	0
$363$	2.87945	0.151132
$364$	0	0
$365$	−18.0836	−0.946537
$366$	0	0
$367$	15.4288	0.805375	0.402688	−	0.915337i	$-0.368076\pi$
$367$	15.4288	0.805375	0.402688	+	0.915337i	$0.368076\pi$
$368$	0	0
$369$	9.11077	0.474288
$370$	0	0
$371$	12.0088	0.623465
$372$	0	0
$373$	4.36783	0.226157	0.113079	−	0.993586i	$-0.463929\pi$
$373$	4.36783	0.226157	0.113079	+	0.993586i	$0.463929\pi$
$374$	0	0
$375$	−3.04377	−0.157180
$376$	0	0
$377$	−3.26835	−0.168328
$378$	0	0
$379$	−19.6346	−1.00856	−0.504280	−	0.863540i	$-0.668242\pi$
$379$	−19.6346	−1.00856	−0.504280	+	0.863540i	$0.668242\pi$
$380$	0	0
$381$	−1.50132	−0.0769149
$382$	0	0
$383$	24.4688	1.25030	0.625149	−	0.780506i	$-0.285038\pi$
$383$	24.4688	1.25030	0.625149	+	0.780506i	$0.285038\pi$
$384$	0	0
$385$	−1.71329	−0.0873174
$386$	0	0
$387$	25.9610	1.31967
$388$	0	0
$389$	−24.3775	−1.23599	−0.617995	−	0.786182i	$-0.712055\pi$
$389$	−24.3775	−1.23599	−0.617995	+	0.786182i	$0.712055\pi$
$390$	0	0
$391$	0.258026	0.0130489
$392$	0	0
$393$	−3.24858	−0.163869
$394$	0	0
$395$	−33.0370	−1.66227
$396$	0	0
$397$	3.88957	0.195212	0.0976060	−	0.995225i	$-0.468882\pi$
$397$	3.88957	0.195212	0.0976060	+	0.995225i	$0.468882\pi$
$398$	0	0
$399$	−2.41113	−0.120708
$400$	0	0
$401$	24.0322	1.20011	0.600055	−	0.799959i	$-0.295146\pi$
$401$	24.0322	1.20011	0.600055	+	0.799959i	$0.295146\pi$
$402$	0	0
$403$	5.00322	0.249228
$404$	0	0
$405$	−18.3101	−0.909835
$406$	0	0
$407$	2.15869	0.107002
$408$	0	0
$409$	−3.25464	−0.160931	−0.0804657	−	0.996757i	$-0.525641\pi$
$409$	−3.25464	−0.160931	−0.0804657	+	0.996757i	$0.525641\pi$
$410$	0	0
$411$	−3.85091	−0.189951
$412$	0	0
$413$	−1.68463	−0.0828953
$414$	0	0
$415$	−8.00917	−0.393155
$416$	0	0
$417$	−4.37043	−0.214021
$418$	0	0
$419$	−15.3207	−0.748463	−0.374232	−	0.927335i	$-0.622093\pi$
$419$	−15.3207	−0.748463	−0.374232	+	0.927335i	$0.622093\pi$
$420$	0	0
$421$	34.4060	1.67684	0.838422	−	0.545022i	$-0.183479\pi$
$421$	34.4060	1.67684	0.838422	+	0.545022i	$0.183479\pi$
$422$	0	0
$423$	5.38325	0.261743
$424$	0	0
$425$	0.206984	0.0100402
$426$	0	0
$427$	−1.18130	−0.0571673
$428$	0	0
$429$	0.0599991	0.00289678
$430$	0	0
$431$	2.31071	0.111303	0.0556515	−	0.998450i	$-0.482276\pi$
$431$	2.31071	0.111303	0.0556515	+	0.998450i	$0.482276\pi$
$432$	0	0
$433$	−36.7259	−1.76493	−0.882467	−	0.470374i	$-0.844119\pi$
$433$	−36.7259	−1.76493	−0.882467	+	0.470374i	$0.844119\pi$
$434$	0	0
$435$	3.94959	0.189369
$436$	0	0
$437$	−1.38312	−0.0661634
$438$	0	0
$439$	9.88353	0.471715	0.235857	−	0.971788i	$-0.424210\pi$
$439$	9.88353	0.471715	0.235857	+	0.971788i	$0.424210\pi$
$440$	0	0
$441$	12.1893	0.580444
$442$	0	0
$443$	−13.1968	−0.626998	−0.313499	−	0.949589i	$-0.601501\pi$
$443$	−13.1968	−0.626998	−0.313499	+	0.949589i	$0.601501\pi$
$444$	0	0
$445$	−10.9302	−0.518142
$446$	0	0
$447$	−3.38621	−0.160162
$448$	0	0
$449$	16.0987	0.759746	0.379873	−	0.925039i	$-0.375968\pi$
$449$	16.0987	0.759746	0.379873	+	0.925039i	$0.375968\pi$
$450$	0	0
$451$	−1.44511	−0.0680476
$452$	0	0
$453$	3.70594	0.174120
$454$	0	0
$455$	1.78407	0.0836383
$456$	0	0
$457$	−1.41621	−0.0662474	−0.0331237	−	0.999451i	$-0.510546\pi$
$457$	−1.41621	−0.0662474	−0.0331237	+	0.999451i	$0.510546\pi$
$458$	0	0
$459$	−1.58300	−0.0738882
$460$	0	0
$461$	−29.8488	−1.39020	−0.695099	−	0.718914i	$-0.744640\pi$
$461$	−29.8488	−1.39020	−0.695099	+	0.718914i	$0.744640\pi$
$462$	0	0
$463$	28.1632	1.30885	0.654427	−	0.756125i	$-0.272910\pi$
$463$	28.1632	1.30885	0.654427	+	0.756125i	$0.272910\pi$
$464$	0	0
$465$	−6.04608	−0.280380
$466$	0	0
$467$	−9.69505	−0.448634	−0.224317	−	0.974516i	$-0.572015\pi$
$467$	−9.69505	−0.448634	−0.224317	+	0.974516i	$0.572015\pi$
$468$	0	0
$469$	−7.03391	−0.324796
$470$	0	0
$471$	−3.04114	−0.140128
$472$	0	0
$473$	−4.11782	−0.189337
$474$	0	0
$475$	−1.10951	−0.0509079
$476$	0	0
$477$	−20.8771	−0.955896
$478$	0	0
$479$	−6.63563	−0.303190	−0.151595	−	0.988443i	$-0.548441\pi$
$479$	−6.63563	−0.303190	−0.151595	+	0.988443i	$0.548441\pi$
$480$	0	0
$481$	−2.24786	−0.102494
$482$	0	0
$483$	0.116062	0.00528100
$484$	0	0
$485$	−28.3842	−1.28886
$486$	0	0
$487$	−30.0800	−1.36305	−0.681527	−	0.731793i	$-0.738684\pi$
$487$	−30.0800	−1.36305	−0.681527	+	0.731793i	$0.738684\pi$
$488$	0	0
$489$	0.0645030	0.00291693
$490$	0	0
$491$	12.1583	0.548695	0.274348	−	0.961631i	$-0.411538\pi$
$491$	12.1583	0.548695	0.274348	+	0.961631i	$0.411538\pi$
$492$	0	0
$493$	−6.75658	−0.304301
$494$	0	0
$495$	2.97853	0.133875
$496$	0	0
$497$	12.9924	0.582788
$498$	0	0
$499$	2.05381	0.0919410	0.0459705	−	0.998943i	$-0.485362\pi$
$499$	2.05381	0.0919410	0.0459705	+	0.998943i	$0.485362\pi$
$500$	0	0
$501$	0.401427	0.0179344
$502$	0	0
$503$	9.37588	0.418050	0.209025	−	0.977910i	$-0.432971\pi$
$503$	9.37588	0.418050	0.209025	+	0.977910i	$0.432971\pi$
$504$	0	0
$505$	−28.9456	−1.28806
$506$	0	0
$507$	3.40860	0.151381
$508$	0	0
$509$	−31.6921	−1.40473	−0.702363	−	0.711819i	$-0.747872\pi$
$509$	−31.6921	−1.40473	−0.702363	+	0.711819i	$0.747872\pi$
$510$	0	0
$511$	13.9150	0.615565
$512$	0	0
$513$	8.48547	0.374643
$514$	0	0
$515$	−11.0424	−0.486584
$516$	0	0
$517$	−0.853868	−0.0375531
$518$	0	0
$519$	1.55142	0.0680999
$520$	0	0
$521$	2.17615	0.0953388	0.0476694	−	0.998863i	$-0.484821\pi$
$521$	2.17615	0.0953388	0.0476694	+	0.998863i	$0.484821\pi$
$522$	0	0
$523$	−23.8591	−1.04328	−0.521642	−	0.853164i	$-0.674680\pi$
$523$	−23.8591	−1.04328	−0.521642	+	0.853164i	$0.674680\pi$
$524$	0	0
$525$	0.0931030	0.00406335
$526$	0	0
$527$	10.3430	0.450550
$528$	0	0
$529$	−22.9334	−0.997105
$530$	0	0
$531$	2.92871	0.127095
$532$	0	0
$533$	1.50481	0.0651803
$534$	0	0
$535$	23.5054	1.01623
$536$	0	0
$537$	−3.77801	−0.163033
$538$	0	0
$539$	−1.93342	−0.0832782
$540$	0	0
$541$	−12.2401	−0.526243	−0.263121	−	0.964763i	$-0.584752\pi$
$541$	−12.2401	−0.526243	−0.263121	+	0.964763i	$0.584752\pi$
$542$	0	0
$543$	6.32128	0.271272
$544$	0	0
$545$	25.4547	1.09036
$546$	0	0
$547$	0.426540	0.0182375	0.00911877	−	0.999958i	$-0.497097\pi$
$547$	0.426540	0.0182375	0.00911877	+	0.999958i	$0.497097\pi$
$548$	0	0
$549$	2.05368	0.0876490
$550$	0	0
$551$	36.2178	1.54293
$552$	0	0
$553$	25.4215	1.08103
$554$	0	0
$555$	2.71640	0.115305
$556$	0	0
$557$	41.2567	1.74810	0.874052	−	0.485832i	$-0.161483\pi$
$557$	41.2567	1.74810	0.874052	+	0.485832i	$0.161483\pi$
$558$	0	0
$559$	4.28792	0.181360
$560$	0	0
$561$	0.124035	0.00523675
$562$	0	0
$563$	10.4493	0.440385	0.220193	−	0.975456i	$-0.429331\pi$
$563$	10.4493	0.440385	0.220193	+	0.975456i	$0.429331\pi$
$564$	0	0
$565$	−5.12175	−0.215474
$566$	0	0
$567$	14.0893	0.591696
$568$	0	0
$569$	−15.5590	−0.652265	−0.326133	−	0.945324i	$-0.605746\pi$
$569$	−15.5590	−0.652265	−0.326133	+	0.945324i	$0.605746\pi$
$570$	0	0
$571$	−35.7683	−1.49686	−0.748428	−	0.663216i	$-0.769191\pi$
$571$	−35.7683	−1.49686	−0.748428	+	0.663216i	$0.769191\pi$
$572$	0	0
$573$	−5.08138	−0.212278
$574$	0	0
$575$	0.0534074	0.00222724
$576$	0	0
$577$	−13.1126	−0.545886	−0.272943	−	0.962030i	$-0.587997\pi$
$577$	−13.1126	−0.545886	−0.272943	+	0.962030i	$0.587997\pi$
$578$	0	0
$579$	−4.32420	−0.179707
$580$	0	0
$581$	6.16294	0.255682
$582$	0	0
$583$	3.31143	0.137146
$584$	0	0
$585$	−3.10157	−0.128234
$586$	0	0
$587$	−28.0322	−1.15701	−0.578507	−	0.815677i	$-0.696364\pi$
$587$	−28.0322	−1.15701	−0.578507	+	0.815677i	$0.696364\pi$
$588$	0	0
$589$	−55.4425	−2.28447
$590$	0	0
$591$	3.29783	0.135655
$592$	0	0
$593$	−5.29773	−0.217552	−0.108776	−	0.994066i	$-0.534693\pi$
$593$	−5.29773	−0.217552	−0.108776	+	0.994066i	$0.534693\pi$
$594$	0	0
$595$	3.68816	0.151200
$596$	0	0
$597$	0.775526	0.0317402
$598$	0	0
$599$	−41.3056	−1.68770	−0.843850	−	0.536579i	$-0.819716\pi$
$599$	−41.3056	−1.68770	−0.843850	+	0.536579i	$0.819716\pi$
$600$	0	0
$601$	23.2254	0.947383	0.473692	−	0.880691i	$-0.342921\pi$
$601$	23.2254	0.947383	0.473692	+	0.880691i	$0.342921\pi$
$602$	0	0
$603$	12.2284	0.497977
$604$	0	0
$605$	23.6098	0.959875
$606$	0	0
$607$	7.19746	0.292136	0.146068	−	0.989275i	$-0.453338\pi$
$607$	7.19746	0.292136	0.146068	+	0.989275i	$0.453338\pi$
$608$	0	0
$609$	−3.03916	−0.123153
$610$	0	0
$611$	0.889140	0.0359707
$612$	0	0
$613$	−37.1607	−1.50091	−0.750453	−	0.660924i	$-0.770164\pi$
$613$	−37.1607	−1.50091	−0.750453	+	0.660924i	$0.770164\pi$
$614$	0	0
$615$	−1.81846	−0.0733275
$616$	0	0
$617$	−41.1280	−1.65575	−0.827876	−	0.560911i	$-0.810451\pi$
$617$	−41.1280	−1.65575	−0.827876	+	0.560911i	$0.810451\pi$
$618$	0	0
$619$	−29.5796	−1.18891	−0.594453	−	0.804131i	$-0.702631\pi$
$619$	−29.5796	−1.18891	−0.594453	+	0.804131i	$0.702631\pi$
$620$	0	0
$621$	−0.408456	−0.0163908
$622$	0	0
$623$	8.41064	0.336965
$624$	0	0
$625$	−23.9222	−0.956889
$626$	0	0
$627$	−0.664872	−0.0265524
$628$	0	0
$629$	−4.64695	−0.185286
$630$	0	0
$631$	−11.4133	−0.454358	−0.227179	−	0.973853i	$-0.572950\pi$
$631$	−11.4133	−0.454358	−0.227179	+	0.973853i	$0.572950\pi$
$632$	0	0
$633$	−0.0548729	−0.00218100
$634$	0	0
$635$	−12.3099	−0.488505
$636$	0	0
$637$	2.01328	0.0797692
$638$	0	0
$639$	−22.5871	−0.893530
$640$	0	0
$641$	10.4620	0.413223	0.206611	−	0.978423i	$-0.433756\pi$
$641$	10.4620	0.413223	0.206611	+	0.978423i	$0.433756\pi$
$642$	0	0
$643$	41.6596	1.64290	0.821448	−	0.570284i	$-0.193167\pi$
$643$	41.6596	1.64290	0.821448	+	0.570284i	$0.193167\pi$
$644$	0	0
$645$	−5.18168	−0.204028
$646$	0	0
$647$	−1.56291	−0.0614443	−0.0307221	−	0.999528i	$-0.509781\pi$
$647$	−1.56291	−0.0614443	−0.0307221	+	0.999528i	$0.509781\pi$
$648$	0	0
$649$	−0.464539	−0.0182347
$650$	0	0
$651$	4.65237	0.182341
$652$	0	0
$653$	−0.382330	−0.0149617	−0.00748086	−	0.999972i	$-0.502381\pi$
$653$	−0.382330	−0.0149617	−0.00748086	+	0.999972i	$0.502381\pi$
$654$	0	0
$655$	−26.6364	−1.04077
$656$	0	0
$657$	−24.1911	−0.943785
$658$	0	0
$659$	27.8793	1.08602	0.543011	−	0.839725i	$-0.317284\pi$
$659$	27.8793	1.08602	0.543011	+	0.839725i	$0.317284\pi$
$660$	0	0
$661$	−6.81592	−0.265109	−0.132554	−	0.991176i	$-0.542318\pi$
$661$	−6.81592	−0.265109	−0.132554	+	0.991176i	$0.542318\pi$
$662$	0	0
$663$	−0.129158	−0.00501609
$664$	0	0
$665$	−19.7699	−0.766643
$666$	0	0
$667$	−1.74337	−0.0675037
$668$	0	0
$669$	0.464190	0.0179466
$670$	0	0
$671$	−0.325746	−0.0125753
$672$	0	0
$673$	12.3897	0.477590	0.238795	−	0.971070i	$-0.423248\pi$
$673$	12.3897	0.477590	0.238795	+	0.971070i	$0.423248\pi$
$674$	0	0
$675$	−0.327657	−0.0126115
$676$	0	0
$677$	30.6337	1.17735	0.588674	−	0.808370i	$-0.299650\pi$
$677$	30.6337	1.17735	0.588674	+	0.808370i	$0.299650\pi$
$678$	0	0
$679$	21.8412	0.838190
$680$	0	0
$681$	−6.12430	−0.234684
$682$	0	0
$683$	4.92154	0.188318	0.0941588	−	0.995557i	$-0.469984\pi$
$683$	4.92154	0.188318	0.0941588	+	0.995557i	$0.469984\pi$
$684$	0	0
$685$	−31.5752	−1.20643
$686$	0	0
$687$	−1.35433	−0.0516709
$688$	0	0
$689$	−3.44822	−0.131367
$690$	0	0
$691$	27.2073	1.03501	0.517507	−	0.855679i	$-0.326860\pi$
$691$	27.2073	1.03501	0.517507	+	0.855679i	$0.326860\pi$
$692$	0	0
$693$	−2.29194	−0.0870635
$694$	0	0
$695$	−35.8350	−1.35930
$696$	0	0
$697$	3.11085	0.117832
$698$	0	0
$699$	3.25383	0.123071
$700$	0	0
$701$	45.0000	1.69963	0.849814	−	0.527083i	$-0.176714\pi$
$701$	45.0000	1.69963	0.849814	+	0.527083i	$0.176714\pi$
$702$	0	0
$703$	24.9094	0.939475
$704$	0	0
$705$	−1.07447	−0.0404669
$706$	0	0
$707$	22.2732	0.837671
$708$	0	0
$709$	−32.0474	−1.20356	−0.601782	−	0.798660i	$-0.705542\pi$
$709$	−32.0474	−1.20356	−0.601782	+	0.798660i	$0.705542\pi$
$710$	0	0
$711$	−44.1949	−1.65744
$712$	0	0
$713$	2.66877	0.0999464
$714$	0	0
$715$	0.491958	0.0183982
$716$	0	0
$717$	−6.44730	−0.240779
$718$	0	0
$719$	−6.73146	−0.251041	−0.125521	−	0.992091i	$-0.540060\pi$
$719$	−6.73146	−0.251041	−0.125521	+	0.992091i	$0.540060\pi$
$720$	0	0
$721$	8.49693	0.316442
$722$	0	0
$723$	−2.38867	−0.0888355
$724$	0	0
$725$	−1.39851	−0.0519392
$726$	0	0
$727$	29.5969	1.09769	0.548844	−	0.835925i	$-0.315068\pi$
$727$	29.5969	1.09769	0.548844	+	0.835925i	$0.315068\pi$
$728$	0	0
$729$	−23.2261	−0.860226
$730$	0	0
$731$	8.86431	0.327858
$732$	0	0
$733$	39.7440	1.46798	0.733988	−	0.679162i	$-0.237657\pi$
$733$	39.7440	1.46798	0.733988	+	0.679162i	$0.237657\pi$
$734$	0	0
$735$	−2.43293	−0.0897399
$736$	0	0
$737$	−1.93961	−0.0714464
$738$	0	0
$739$	15.1667	0.557916	0.278958	−	0.960303i	$-0.410011\pi$
$739$	15.1667	0.557916	0.278958	+	0.960303i	$0.410011\pi$
$740$	0	0
$741$	0.692337	0.0254336
$742$	0	0
$743$	4.92547	0.180698	0.0903490	−	0.995910i	$-0.471202\pi$
$743$	4.92547	0.180698	0.0903490	+	0.995910i	$0.471202\pi$
$744$	0	0
$745$	−27.7649	−1.01723
$746$	0	0
$747$	−10.7142	−0.392012
$748$	0	0
$749$	−18.0871	−0.660887
$750$	0	0
$751$	52.0614	1.89975	0.949873	−	0.312635i	$-0.101212\pi$
$751$	52.0614	1.89975	0.949873	+	0.312635i	$0.101212\pi$
$752$	0	0
$753$	−5.56484	−0.202794
$754$	0	0
$755$	30.3866	1.10588
$756$	0	0
$757$	13.0936	0.475896	0.237948	−	0.971278i	$-0.423525\pi$
$757$	13.0936	0.475896	0.237948	+	0.971278i	$0.423525\pi$
$758$	0	0
$759$	0.0320042	0.00116168
$760$	0	0
$761$	12.2540	0.444208	0.222104	−	0.975023i	$-0.428708\pi$
$761$	12.2540	0.444208	0.222104	+	0.975023i	$0.428708\pi$
$762$	0	0
$763$	−19.5870	−0.709097
$764$	0	0
$765$	−6.41181	−0.231819
$766$	0	0
$767$	0.483728	0.0174664
$768$	0	0
$769$	−9.80748	−0.353667	−0.176833	−	0.984241i	$-0.556585\pi$
$769$	−9.80748	−0.353667	−0.176833	+	0.984241i	$0.556585\pi$
$770$	0	0
$771$	−3.46799	−0.124897
$772$	0	0
$773$	−48.3398	−1.73866	−0.869331	−	0.494231i	$-0.835450\pi$
$773$	−48.3398	−1.73866	−0.869331	+	0.494231i	$0.835450\pi$
$774$	0	0
$775$	2.14085	0.0769015
$776$	0	0
$777$	−2.09023	−0.0749866
$778$	0	0
$779$	−16.6753	−0.597455
$780$	0	0
$781$	3.58266	0.128198
$782$	0	0
$783$	10.6957	0.382232
$784$	0	0
$785$	−24.9356	−0.889989
$786$	0	0
$787$	−15.9033	−0.566890	−0.283445	−	0.958989i	$-0.591477\pi$
$787$	−15.9033	−0.566890	−0.283445	+	0.958989i	$0.591477\pi$
$788$	0	0
$789$	1.31681	0.0468798
$790$	0	0
$791$	3.94111	0.140130
$792$	0	0
$793$	0.339202	0.0120454
$794$	0	0
$795$	4.16696	0.147787
$796$	0	0
$797$	−24.2114	−0.857612	−0.428806	−	0.903397i	$-0.641066\pi$
$797$	−24.2114	−0.857612	−0.428806	+	0.903397i	$0.641066\pi$
$798$	0	0
$799$	1.83810	0.0650272
$800$	0	0
$801$	−14.6218	−0.516635
$802$	0	0
$803$	3.83709	0.135408
$804$	0	0
$805$	0.951641	0.0335409
$806$	0	0
$807$	3.60690	0.126969
$808$	0	0
$809$	−29.1794	−1.02589	−0.512946	−	0.858421i	$-0.671446\pi$
$809$	−29.1794	−1.02589	−0.512946	+	0.858421i	$0.671446\pi$
$810$	0	0
$811$	−16.6441	−0.584453	−0.292226	−	0.956349i	$-0.594396\pi$
$811$	−16.6441	−0.584453	−0.292226	+	0.956349i	$0.594396\pi$
$812$	0	0
$813$	4.38501	0.153789
$814$	0	0
$815$	0.528888	0.0185261
$816$	0	0
$817$	−47.5160	−1.66237
$818$	0	0
$819$	2.38661	0.0833951
$820$	0	0
$821$	46.1392	1.61027	0.805134	−	0.593093i	$-0.202093\pi$
$821$	46.1392	1.61027	0.805134	+	0.593093i	$0.202093\pi$
$822$	0	0
$823$	47.3867	1.65180	0.825898	−	0.563819i	$-0.190668\pi$
$823$	47.3867	1.65180	0.825898	+	0.563819i	$0.190668\pi$
$824$	0	0
$825$	0.0256732	0.000893828 0
$826$	0	0
$827$	54.6582	1.90065	0.950325	−	0.311259i	$-0.100751\pi$
$827$	54.6582	1.90065	0.950325	+	0.311259i	$0.100751\pi$
$828$	0	0
$829$	9.30933	0.323326	0.161663	−	0.986846i	$-0.448314\pi$
$829$	9.30933	0.323326	0.161663	+	0.986846i	$0.448314\pi$
$830$	0	0
$831$	−0.279347	−0.00969043
$832$	0	0
$833$	4.16202	0.144205
$834$	0	0
$835$	3.29147	0.113906
$836$	0	0
$837$	−16.3730	−0.565935
$838$	0	0
$839$	9.68083	0.334219	0.167110	−	0.985938i	$-0.446557\pi$
$839$	9.68083	0.334219	0.167110	+	0.985938i	$0.446557\pi$
$840$	0	0
$841$	16.6514	0.574185
$842$	0	0
$843$	−4.83408	−0.166494
$844$	0	0
$845$	27.9486	0.961460
$846$	0	0
$847$	−18.1674	−0.624240
$848$	0	0
$849$	−6.74717	−0.231562
$850$	0	0
$851$	−1.19904	−0.0411024
$852$	0	0
$853$	−42.2461	−1.44648	−0.723239	−	0.690597i	$-0.757348\pi$
$853$	−42.2461	−1.44648	−0.723239	+	0.690597i	$0.757348\pi$
$854$	0	0
$855$	34.3697	1.17542
$856$	0	0
$857$	17.0940	0.583920	0.291960	−	0.956431i	$-0.405693\pi$
$857$	17.0940	0.583920	0.291960	+	0.956431i	$0.405693\pi$
$858$	0	0
$859$	−16.7987	−0.573163	−0.286581	−	0.958056i	$-0.592519\pi$
$859$	−16.7987	−0.573163	−0.286581	+	0.958056i	$0.592519\pi$
$860$	0	0
$861$	1.39928	0.0476874
$862$	0	0
$863$	16.8201	0.572562	0.286281	−	0.958146i	$-0.407581\pi$
$863$	16.8201	0.572562	0.286281	+	0.958146i	$0.407581\pi$
$864$	0	0
$865$	12.7208	0.432519
$866$	0	0
$867$	−0.267006	−0.00906800
$868$	0	0
$869$	7.01000	0.237798
$870$	0	0
$871$	2.01973	0.0684360
$872$	0	0
$873$	−37.9707	−1.28511
$874$	0	0
$875$	19.2042	0.649220
$876$	0	0
$877$	−28.5158	−0.962911	−0.481455	−	0.876471i	$-0.659892\pi$
$877$	−28.5158	−0.962911	−0.481455	+	0.876471i	$0.659892\pi$
$878$	0	0
$879$	0.830952	0.0280273
$880$	0	0
$881$	−34.9953	−1.17902	−0.589510	−	0.807761i	$-0.700679\pi$
$881$	−34.9953	−1.17902	−0.589510	+	0.807761i	$0.700679\pi$
$882$	0	0
$883$	−23.4697	−0.789818	−0.394909	−	0.918720i	$-0.629224\pi$
$883$	−23.4697	−0.789818	−0.394909	+	0.918720i	$0.629224\pi$
$884$	0	0
$885$	−0.584555	−0.0196496
$886$	0	0
$887$	−42.0057	−1.41041	−0.705206	−	0.709002i	$-0.749146\pi$
$887$	−42.0057	−1.41041	−0.705206	+	0.709002i	$0.749146\pi$
$888$	0	0
$889$	9.47232	0.317691
$890$	0	0
$891$	3.88515	0.130157
$892$	0	0
$893$	−9.85288	−0.329714
$894$	0	0
$895$	−30.9775	−1.03546
$896$	0	0
$897$	−0.0333262	−0.00111273
$898$	0	0
$899$	−69.8835	−2.33075
$900$	0	0
$901$	−7.12843	−0.237482
$902$	0	0
$903$	3.98723	0.132687
$904$	0	0
$905$	51.8308	1.72291
$906$	0	0
$907$	−6.16274	−0.204631	−0.102315	−	0.994752i	$-0.532625\pi$
$907$	−6.16274	−0.204631	−0.102315	+	0.994752i	$0.532625\pi$
$908$	0	0
$909$	−38.7217	−1.28432
$910$	0	0
$911$	27.4174	0.908379	0.454189	−	0.890905i	$-0.349929\pi$
$911$	27.4174	0.908379	0.454189	+	0.890905i	$0.349929\pi$
$912$	0	0
$913$	1.69944	0.0562432
$914$	0	0
$915$	−0.409904	−0.0135510
$916$	0	0
$917$	20.4964	0.676850
$918$	0	0
$919$	37.0111	1.22088	0.610442	−	0.792061i	$-0.290992\pi$
$919$	37.0111	1.22088	0.610442	+	0.792061i	$0.290992\pi$
$920$	0	0
$921$	0.837536	0.0275977
$922$	0	0
$923$	−3.73065	−0.122796
$924$	0	0
$925$	−0.961847	−0.0316253
$926$	0	0
$927$	−14.7718	−0.485169
$928$	0	0
$929$	4.69458	0.154024	0.0770121	−	0.997030i	$-0.475462\pi$
$929$	4.69458	0.154024	0.0770121	+	0.997030i	$0.475462\pi$
$930$	0	0
$931$	−22.3099	−0.731179
$932$	0	0
$933$	−2.28466	−0.0747965
$934$	0	0
$935$	1.01701	0.0332599
$936$	0	0
$937$	39.2038	1.28073	0.640366	−	0.768070i	$-0.278783\pi$
$937$	39.2038	1.28073	0.640366	+	0.768070i	$0.278783\pi$
$938$	0	0
$939$	−4.63794	−0.151354
$940$	0	0
$941$	−48.9736	−1.59649	−0.798247	−	0.602330i	$-0.794239\pi$
$941$	−48.9736	−1.59649	−0.798247	+	0.602330i	$0.794239\pi$
$942$	0	0
$943$	0.802680	0.0261389
$944$	0	0
$945$	−5.83836	−0.189922
$946$	0	0
$947$	−21.0263	−0.683263	−0.341631	−	0.939834i	$-0.610979\pi$
$947$	−21.0263	−0.683263	−0.341631	+	0.939834i	$0.610979\pi$
$948$	0	0
$949$	−3.99559	−0.129702
$950$	0	0
$951$	7.88680	0.255747
$952$	0	0
$953$	12.1924	0.394952	0.197476	−	0.980308i	$-0.436726\pi$
$953$	12.1924	0.394952	0.197476	+	0.980308i	$0.436726\pi$
$954$	0	0
$955$	−41.6644	−1.34823
$956$	0	0
$957$	−0.838050	−0.0270903
$958$	0	0
$959$	24.2967	0.784580
$960$	0	0
$961$	75.9784	2.45091
$962$	0	0
$963$	31.4441	1.01327
$964$	0	0
$965$	−35.4559	−1.14137
$966$	0	0
$967$	−30.8980	−0.993614	−0.496807	−	0.867861i	$-0.665494\pi$
$967$	−30.8980	−0.993614	−0.496807	+	0.867861i	$0.665494\pi$
$968$	0	0
$969$	1.43125	0.0459784
$970$	0	0
$971$	6.12775	0.196649	0.0983245	−	0.995154i	$-0.468652\pi$
$971$	6.12775	0.196649	0.0983245	+	0.995154i	$0.468652\pi$
$972$	0	0
$973$	27.5745	0.883998
$974$	0	0
$975$	−0.0267338	−0.000856166 0
$976$	0	0
$977$	3.57730	0.114448	0.0572239	−	0.998361i	$-0.481775\pi$
$977$	3.57730	0.114448	0.0572239	+	0.998361i	$0.481775\pi$
$978$	0	0
$979$	2.31924	0.0741233
$980$	0	0
$981$	34.0517	1.08719
$982$	0	0
$983$	−55.8465	−1.78123	−0.890613	−	0.454762i	$-0.849724\pi$
$983$	−55.8465	−1.78123	−0.890613	+	0.454762i	$0.849724\pi$
$984$	0	0
$985$	27.0403	0.861577
$986$	0	0
$987$	0.826789	0.0263170
$988$	0	0
$989$	2.28722	0.0727295
$990$	0	0
$991$	−27.0577	−0.859517	−0.429759	−	0.902944i	$-0.641401\pi$
$991$	−27.0577	−0.859517	−0.429759	+	0.902944i	$0.641401\pi$
$992$	0	0
$993$	0.478202	0.0151753
$994$	0	0
$995$	6.35886	0.201590
$996$	0	0
$997$	−44.5801	−1.41187	−0.705934	−	0.708278i	$-0.749472\pi$
$997$	−44.5801	−1.41187	−0.705934	+	0.708278i	$0.749472\pi$
$998$	0	0
$999$	7.35613	0.232738

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.i.1.7	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.i.1.7	✓	18	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)

\(f(q)\)	\(=\)	\(q-0.267006 q^{3} -2.18930 q^{5} +1.68463 q^{7} -2.92871 q^{9} +O(q^{10})\)	\(q-0.267006 q^{3} -2.18930 q^{5} +1.68463 q^{7} -2.92871 q^{9} +0.464539 q^{11} -0.483728 q^{13} +0.584555 q^{15} -1.00000 q^{17} +5.36037 q^{19} -0.449807 q^{21} -0.258026 q^{23} -0.206984 q^{25} +1.58300 q^{27} +6.75658 q^{29} -10.3430 q^{31} -0.124035 q^{33} -3.68816 q^{35} +4.64695 q^{37} +0.129158 q^{39} -3.11085 q^{41} -8.86431 q^{43} +6.41181 q^{45} -1.83810 q^{47} -4.16202 q^{49} +0.267006 q^{51} +7.12843 q^{53} -1.01701 q^{55} -1.43125 q^{57} -1.00000 q^{59} -0.701224 q^{61} -4.93379 q^{63} +1.05902 q^{65} -4.17534 q^{67} +0.0688946 q^{69} +7.71230 q^{71} +8.25999 q^{73} +0.0552661 q^{75} +0.782577 q^{77} +15.0902 q^{79} +8.36345 q^{81} +3.65833 q^{83} +2.18930 q^{85} -1.80405 q^{87} +4.99257 q^{89} -0.814904 q^{91} +2.76165 q^{93} -11.7354 q^{95} +12.9650 q^{97} -1.36050 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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