LMFDB - Embedded newform 4012.2.a.g.1.4

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			1.4
Root			$1.85534$ 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)

$f(q)$	$=$	$q-1.85534 q^{3} -3.75753 q^{5} -3.33364 q^{7} +0.442279 q^{9} +O(q^{10})$	$q-1.85534 q^{3} -3.75753 q^{5} -3.33364 q^{7} +0.442279 q^{9} +2.31567 q^{11} -1.30252 q^{13} +6.97149 q^{15} -1.00000 q^{17} -1.32303 q^{19} +6.18503 q^{21} -2.50661 q^{23} +9.11905 q^{25} +4.74544 q^{27} +6.82690 q^{29} +7.79295 q^{31} -4.29635 q^{33} +12.5263 q^{35} +2.02635 q^{37} +2.41661 q^{39} +6.45582 q^{41} -4.87421 q^{43} -1.66188 q^{45} -7.41036 q^{47} +4.11316 q^{49} +1.85534 q^{51} -0.619651 q^{53} -8.70120 q^{55} +2.45467 q^{57} +1.00000 q^{59} -1.42317 q^{61} -1.47440 q^{63} +4.89425 q^{65} +0.724862 q^{67} +4.65060 q^{69} +8.70782 q^{71} -10.8925 q^{73} -16.9189 q^{75} -7.71961 q^{77} +14.5916 q^{79} -10.1312 q^{81} -1.49687 q^{83} +3.75753 q^{85} -12.6662 q^{87} -6.21141 q^{89} +4.34213 q^{91} -14.4586 q^{93} +4.97134 q^{95} -4.07138 q^{97} +1.02417 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.85534	−1.07118	−0.535590	−	0.844478i	$-0.679911\pi$
$3$	−1.85534	−1.07118	−0.535590	+	0.844478i	$0.679911\pi$
$4$	0	0
$5$	−3.75753	−1.68042	−0.840210	−	0.542261i	$-0.817568\pi$
$5$	−3.75753	−1.68042	−0.840210	+	0.542261i	$0.817568\pi$
$6$	0	0
$7$	−3.33364	−1.26000	−0.629999	−	0.776596i	$-0.716945\pi$
$7$	−3.33364	−1.26000	−0.629999	+	0.776596i	$0.716945\pi$
$8$	0	0
$9$	0.442279	0.147426
$10$	0	0
$11$	2.31567	0.698200	0.349100	−	0.937085i	$-0.386487\pi$
$11$	2.31567	0.698200	0.349100	+	0.937085i	$0.386487\pi$
$12$	0	0
$13$	−1.30252	−0.361253	−0.180627	−	0.983552i	$-0.557813\pi$
$13$	−1.30252	−0.361253	−0.180627	+	0.983552i	$0.557813\pi$
$14$	0	0
$15$	6.97149	1.80003
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−1.32303	−0.303524	−0.151762	−	0.988417i	$-0.548495\pi$
$19$	−1.32303	−0.303524	−0.151762	+	0.988417i	$0.548495\pi$
$20$	0	0
$21$	6.18503	1.34968
$22$	0	0
$23$	−2.50661	−0.522664	−0.261332	−	0.965249i	$-0.584162\pi$
$23$	−2.50661	−0.522664	−0.261332	+	0.965249i	$0.584162\pi$
$24$	0	0
$25$	9.11905	1.82381
$26$	0	0
$27$	4.74544	0.913260
$28$	0	0
$29$	6.82690	1.26772	0.633862	−	0.773446i	$-0.281469\pi$
$29$	6.82690	1.26772	0.633862	+	0.773446i	$0.281469\pi$
$30$	0	0
$31$	7.79295	1.39966	0.699828	−	0.714312i	$-0.253260\pi$
$31$	7.79295	1.39966	0.699828	+	0.714312i	$0.253260\pi$
$32$	0	0
$33$	−4.29635	−0.747898
$34$	0	0
$35$	12.5263	2.11733
$36$	0	0
$37$	2.02635	0.333130	0.166565	−	0.986030i	$-0.446732\pi$
$37$	2.02635	0.333130	0.166565	+	0.986030i	$0.446732\pi$
$38$	0	0
$39$	2.41661	0.386967
$40$	0	0
$41$	6.45582	1.00823	0.504115	−	0.863637i	$-0.331819\pi$
$41$	6.45582	1.00823	0.504115	+	0.863637i	$0.331819\pi$
$42$	0	0
$43$	−4.87421	−0.743310	−0.371655	−	0.928371i	$-0.621210\pi$
$43$	−4.87421	−0.743310	−0.371655	+	0.928371i	$0.621210\pi$
$44$	0	0
$45$	−1.66188	−0.247738
$46$	0	0
$47$	−7.41036	−1.08091	−0.540456	−	0.841372i	$-0.681748\pi$
$47$	−7.41036	−1.08091	−0.540456	+	0.841372i	$0.681748\pi$
$48$	0	0
$49$	4.11316	0.587595
$50$	0	0
$51$	1.85534	0.259799
$52$	0	0
$53$	−0.619651	−0.0851156	−0.0425578	−	0.999094i	$-0.513551\pi$
$53$	−0.619651	−0.0851156	−0.0425578	+	0.999094i	$0.513551\pi$
$54$	0	0
$55$	−8.70120	−1.17327
$56$	0	0
$57$	2.45467	0.325129
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−1.42317	−0.182219	−0.0911093	−	0.995841i	$-0.529041\pi$
$61$	−1.42317	−0.182219	−0.0911093	+	0.995841i	$0.529041\pi$
$62$	0	0
$63$	−1.47440	−0.185757
$64$	0	0
$65$	4.89425	0.607057
$66$	0	0
$67$	0.724862	0.0885559	0.0442780	−	0.999019i	$-0.485901\pi$
$67$	0.724862	0.0885559	0.0442780	+	0.999019i	$0.485901\pi$
$68$	0	0
$69$	4.65060	0.559867
$70$	0	0
$71$	8.70782	1.03343	0.516714	−	0.856158i	$-0.327155\pi$
$71$	8.70782	1.03343	0.516714	+	0.856158i	$0.327155\pi$
$72$	0	0
$73$	−10.8925	−1.27487	−0.637435	−	0.770504i	$-0.720005\pi$
$73$	−10.8925	−1.27487	−0.637435	+	0.770504i	$0.720005\pi$
$74$	0	0
$75$	−16.9189	−1.95363
$76$	0	0
$77$	−7.71961	−0.879731
$78$	0	0
$79$	14.5916	1.64169	0.820843	−	0.571154i	$-0.193504\pi$
$79$	14.5916	1.64169	0.820843	+	0.571154i	$0.193504\pi$
$80$	0	0
$81$	−10.1312	−1.12569
$82$	0	0
$83$	−1.49687	−0.164303	−0.0821515	−	0.996620i	$-0.526179\pi$
$83$	−1.49687	−0.164303	−0.0821515	+	0.996620i	$0.526179\pi$
$84$	0	0
$85$	3.75753	0.407562
$86$	0	0
$87$	−12.6662	−1.35796
$88$	0	0
$89$	−6.21141	−0.658408	−0.329204	−	0.944259i	$-0.606780\pi$
$89$	−6.21141	−0.658408	−0.329204	+	0.944259i	$0.606780\pi$
$90$	0	0
$91$	4.34213	0.455179
$92$	0	0
$93$	−14.4586	−1.49928
$94$	0	0
$95$	4.97134	0.510048
$96$	0	0
$97$	−4.07138	−0.413386	−0.206693	−	0.978406i	$-0.566270\pi$
$97$	−4.07138	−0.413386	−0.206693	+	0.978406i	$0.566270\pi$
$98$	0	0
$99$	1.02417	0.102933
$100$	0	0
$101$	7.13480	0.709939	0.354970	−	0.934878i	$-0.384491\pi$
$101$	7.13480	0.709939	0.354970	+	0.934878i	$0.384491\pi$
$102$	0	0
$103$	3.16352	0.311711	0.155856	−	0.987780i	$-0.450187\pi$
$103$	3.16352	0.311711	0.155856	+	0.987780i	$0.450187\pi$
$104$	0	0
$105$	−23.2405	−2.26804
$106$	0	0
$107$	−2.34450	−0.226651	−0.113326	−	0.993558i	$-0.536150\pi$
$107$	−2.34450	−0.226651	−0.113326	+	0.993558i	$0.536150\pi$
$108$	0	0
$109$	9.44291	0.904467	0.452234	−	0.891900i	$-0.350627\pi$
$109$	9.44291	0.904467	0.452234	+	0.891900i	$0.350627\pi$
$110$	0	0
$111$	−3.75956	−0.356842
$112$	0	0
$113$	4.66821	0.439149	0.219574	−	0.975596i	$-0.429533\pi$
$113$	4.66821	0.439149	0.219574	+	0.975596i	$0.429533\pi$
$114$	0	0
$115$	9.41866	0.878294
$116$	0	0
$117$	−0.576077	−0.0532583
$118$	0	0
$119$	3.33364	0.305594
$120$	0	0
$121$	−5.63768	−0.512517
$122$	0	0
$123$	−11.9777	−1.07999
$124$	0	0
$125$	−15.4775	−1.38435
$126$	0	0
$127$	12.2936	1.09088	0.545439	−	0.838151i	$-0.316363\pi$
$127$	12.2936	1.09088	0.545439	+	0.838151i	$0.316363\pi$
$128$	0	0
$129$	9.04331	0.796219
$130$	0	0
$131$	−14.2561	−1.24557	−0.622783	−	0.782395i	$-0.713998\pi$
$131$	−14.2561	−1.24557	−0.622783	+	0.782395i	$0.713998\pi$
$132$	0	0
$133$	4.41051	0.382440
$134$	0	0
$135$	−17.8311	−1.53466
$136$	0	0
$137$	6.26542	0.535291	0.267646	−	0.963517i	$-0.413754\pi$
$137$	6.26542	0.535291	0.267646	+	0.963517i	$0.413754\pi$
$138$	0	0
$139$	−3.32589	−0.282098	−0.141049	−	0.990003i	$-0.545048\pi$
$139$	−3.32589	−0.282098	−0.141049	+	0.990003i	$0.545048\pi$
$140$	0	0
$141$	13.7487	1.15785
$142$	0	0
$143$	−3.01620	−0.252227
$144$	0	0
$145$	−25.6523	−2.13031
$146$	0	0
$147$	−7.63131	−0.629420
$148$	0	0
$149$	13.0987	1.07309	0.536544	−	0.843872i	$-0.319729\pi$
$149$	13.0987	1.07309	0.536544	+	0.843872i	$0.319729\pi$
$150$	0	0
$151$	−4.44446	−0.361685	−0.180842	−	0.983512i	$-0.557882\pi$
$151$	−4.44446	−0.361685	−0.180842	+	0.983512i	$0.557882\pi$
$152$	0	0
$153$	−0.442279	−0.0357562
$154$	0	0
$155$	−29.2823	−2.35201
$156$	0	0
$157$	16.6740	1.33073	0.665366	−	0.746517i	$-0.268275\pi$
$157$	16.6740	1.33073	0.665366	+	0.746517i	$0.268275\pi$
$158$	0	0
$159$	1.14966	0.0911741
$160$	0	0
$161$	8.35613	0.658555
$162$	0	0
$163$	−10.2534	−0.803110	−0.401555	−	0.915835i	$-0.631530\pi$
$163$	−10.2534	−0.803110	−0.401555	+	0.915835i	$0.631530\pi$
$164$	0	0
$165$	16.1437	1.25678
$166$	0	0
$167$	−23.1224	−1.78926	−0.894632	−	0.446804i	$-0.852562\pi$
$167$	−23.1224	−1.78926	−0.894632	+	0.446804i	$0.852562\pi$
$168$	0	0
$169$	−11.3034	−0.869496
$170$	0	0
$171$	−0.585150	−0.0447475
$172$	0	0
$173$	−24.1860	−1.83883	−0.919413	−	0.393294i	$-0.871335\pi$
$173$	−24.1860	−1.83883	−0.919413	+	0.393294i	$0.871335\pi$
$174$	0	0
$175$	−30.3997	−2.29800
$176$	0	0
$177$	−1.85534	−0.139456
$178$	0	0
$179$	−1.42312	−0.106369	−0.0531844	−	0.998585i	$-0.516937\pi$
$179$	−1.42312	−0.106369	−0.0531844	+	0.998585i	$0.516937\pi$
$180$	0	0
$181$	−3.43803	−0.255547	−0.127773	−	0.991803i	$-0.540783\pi$
$181$	−3.43803	−0.255547	−0.127773	+	0.991803i	$0.540783\pi$
$182$	0	0
$183$	2.64047	0.195189
$184$	0	0
$185$	−7.61408	−0.559798
$186$	0	0
$187$	−2.31567	−0.169338
$188$	0	0
$189$	−15.8196	−1.15071
$190$	0	0
$191$	−23.7509	−1.71855	−0.859276	−	0.511512i	$-0.829086\pi$
$191$	−23.7509	−1.71855	−0.859276	+	0.511512i	$0.829086\pi$
$192$	0	0
$193$	3.29389	0.237099	0.118550	−	0.992948i	$-0.462176\pi$
$193$	3.29389	0.237099	0.118550	+	0.992948i	$0.462176\pi$
$194$	0	0
$195$	−9.08050	−0.650268
$196$	0	0
$197$	13.7650	0.980715	0.490358	−	0.871521i	$-0.336866\pi$
$197$	13.7650	0.980715	0.490358	+	0.871521i	$0.336866\pi$
$198$	0	0
$199$	9.49296	0.672938	0.336469	−	0.941694i	$-0.390767\pi$
$199$	9.49296	0.672938	0.336469	+	0.941694i	$0.390767\pi$
$200$	0	0
$201$	−1.34486	−0.0948593
$202$	0	0
$203$	−22.7584	−1.59733
$204$	0	0
$205$	−24.2579	−1.69425
$206$	0	0
$207$	−1.10862	−0.0770544
$208$	0	0
$209$	−3.06370	−0.211921
$210$	0	0
$211$	0.313416	0.0215765	0.0107882	−	0.999942i	$-0.496566\pi$
$211$	0.313416	0.0215765	0.0107882	+	0.999942i	$0.496566\pi$
$212$	0	0
$213$	−16.1560	−1.10699
$214$	0	0
$215$	18.3150	1.24907
$216$	0	0
$217$	−25.9789	−1.76356
$218$	0	0
$219$	20.2093	1.36562
$220$	0	0
$221$	1.30252	0.0876168
$222$	0	0
$223$	2.29570	0.153732	0.0768658	−	0.997041i	$-0.475509\pi$
$223$	2.29570	0.153732	0.0768658	+	0.997041i	$0.475509\pi$
$224$	0	0
$225$	4.03317	0.268878
$226$	0	0
$227$	8.08017	0.536300	0.268150	−	0.963377i	$-0.413588\pi$
$227$	8.08017	0.536300	0.268150	+	0.963377i	$0.413588\pi$
$228$	0	0
$229$	16.4214	1.08516	0.542580	−	0.840004i	$-0.317448\pi$
$229$	16.4214	1.08516	0.542580	+	0.840004i	$0.317448\pi$
$230$	0	0
$231$	14.3225	0.942350
$232$	0	0
$233$	6.47619	0.424269	0.212135	−	0.977240i	$-0.431958\pi$
$233$	6.47619	0.424269	0.212135	+	0.977240i	$0.431958\pi$
$234$	0	0
$235$	27.8447	1.81639
$236$	0	0
$237$	−27.0724	−1.75854
$238$	0	0
$239$	−20.9063	−1.35232	−0.676158	−	0.736757i	$-0.736356\pi$
$239$	−20.9063	−1.35232	−0.676158	+	0.736757i	$0.736356\pi$
$240$	0	0
$241$	8.60994	0.554615	0.277307	−	0.960781i	$-0.410558\pi$
$241$	8.60994	0.554615	0.277307	+	0.960781i	$0.410558\pi$
$242$	0	0
$243$	4.56054	0.292559
$244$	0	0
$245$	−15.4553	−0.987406
$246$	0	0
$247$	1.72327	0.109649
$248$	0	0
$249$	2.77720	0.175998
$250$	0	0
$251$	−25.8758	−1.63327	−0.816633	−	0.577157i	$-0.804162\pi$
$251$	−25.8758	−1.63327	−0.816633	+	0.577157i	$0.804162\pi$
$252$	0	0
$253$	−5.80447	−0.364924
$254$	0	0
$255$	−6.97149	−0.436572
$256$	0	0
$257$	11.5391	0.719787	0.359893	−	0.932993i	$-0.382813\pi$
$257$	11.5391	0.719787	0.359893	+	0.932993i	$0.382813\pi$
$258$	0	0
$259$	−6.75512	−0.419743
$260$	0	0
$261$	3.01940	0.186896
$262$	0	0
$263$	−10.4215	−0.642615	−0.321308	−	0.946975i	$-0.604122\pi$
$263$	−10.4215	−0.642615	−0.321308	+	0.946975i	$0.604122\pi$
$264$	0	0
$265$	2.32836	0.143030
$266$	0	0
$267$	11.5243	0.705274
$268$	0	0
$269$	26.8465	1.63686	0.818429	−	0.574608i	$-0.194845\pi$
$269$	26.8465	1.63686	0.818429	+	0.574608i	$0.194845\pi$
$270$	0	0
$271$	−12.2274	−0.742761	−0.371381	−	0.928481i	$-0.621116\pi$
$271$	−12.2274	−0.742761	−0.371381	+	0.928481i	$0.621116\pi$
$272$	0	0
$273$	−8.05611	−0.487578
$274$	0	0
$275$	21.1167	1.27338
$276$	0	0
$277$	29.3809	1.76533	0.882664	−	0.470004i	$-0.155748\pi$
$277$	29.3809	1.76533	0.882664	+	0.470004i	$0.155748\pi$
$278$	0	0
$279$	3.44666	0.206346
$280$	0	0
$281$	−2.83346	−0.169030	−0.0845150	−	0.996422i	$-0.526934\pi$
$281$	−2.83346	−0.169030	−0.0845150	+	0.996422i	$0.526934\pi$
$282$	0	0
$283$	−17.2973	−1.02822	−0.514109	−	0.857725i	$-0.671877\pi$
$283$	−17.2973	−1.02822	−0.514109	+	0.857725i	$0.671877\pi$
$284$	0	0
$285$	−9.22351	−0.546354
$286$	0	0
$287$	−21.5214	−1.27037
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	7.55379	0.442811
$292$	0	0
$293$	17.7358	1.03614	0.518068	−	0.855339i	$-0.326651\pi$
$293$	17.7358	1.03614	0.518068	+	0.855339i	$0.326651\pi$
$294$	0	0
$295$	−3.75753	−0.218772
$296$	0	0
$297$	10.9889	0.637638
$298$	0	0
$299$	3.26490	0.188814
$300$	0	0
$301$	16.2489	0.936570
$302$	0	0
$303$	−13.2375	−0.760473
$304$	0	0
$305$	5.34762	0.306204
$306$	0	0
$307$	−15.9930	−0.912769	−0.456385	−	0.889783i	$-0.650856\pi$
$307$	−15.9930	−0.912769	−0.456385	+	0.889783i	$0.650856\pi$
$308$	0	0
$309$	−5.86940	−0.333899
$310$	0	0
$311$	−11.1535	−0.632457	−0.316229	−	0.948683i	$-0.602417\pi$
$311$	−11.1535	−0.632457	−0.316229	+	0.948683i	$0.602417\pi$
$312$	0	0
$313$	−10.7340	−0.606721	−0.303361	−	0.952876i	$-0.598109\pi$
$313$	−10.7340	−0.606721	−0.303361	+	0.952876i	$0.598109\pi$
$314$	0	0
$315$	5.54011	0.312150
$316$	0	0
$317$	4.66865	0.262218	0.131109	−	0.991368i	$-0.458146\pi$
$317$	4.66865	0.262218	0.131109	+	0.991368i	$0.458146\pi$
$318$	0	0
$319$	15.8088	0.885125
$320$	0	0
$321$	4.34984	0.242784
$322$	0	0
$323$	1.32303	0.0736155
$324$	0	0
$325$	−11.8777	−0.658858
$326$	0	0
$327$	−17.5198	−0.968847
$328$	0	0
$329$	24.7035	1.36195
$330$	0	0
$331$	9.03028	0.496349	0.248174	−	0.968715i	$-0.420169\pi$
$331$	9.03028	0.496349	0.248174	+	0.968715i	$0.420169\pi$
$332$	0	0
$333$	0.896213	0.0491122
$334$	0	0
$335$	−2.72369	−0.148811
$336$	0	0
$337$	10.6963	0.582663	0.291331	−	0.956622i	$-0.405902\pi$
$337$	10.6963	0.582663	0.291331	+	0.956622i	$0.405902\pi$
$338$	0	0
$339$	−8.66111	−0.470407
$340$	0	0
$341$	18.0459	0.977239
$342$	0	0
$343$	9.62368	0.519630
$344$	0	0
$345$	−17.4748	−0.940811
$346$	0	0
$347$	27.6983	1.48692	0.743462	−	0.668778i	$-0.233182\pi$
$347$	27.6983	1.48692	0.743462	+	0.668778i	$0.233182\pi$
$348$	0	0
$349$	−1.72901	−0.0925519	−0.0462760	−	0.998929i	$-0.514735\pi$
$349$	−1.72901	−0.0925519	−0.0462760	+	0.998929i	$0.514735\pi$
$350$	0	0
$351$	−6.18102	−0.329918
$352$	0	0
$353$	−5.82177	−0.309862	−0.154931	−	0.987925i	$-0.549515\pi$
$353$	−5.82177	−0.309862	−0.154931	+	0.987925i	$0.549515\pi$
$354$	0	0
$355$	−32.7199	−1.73659
$356$	0	0
$357$	−6.18503	−0.327347
$358$	0	0
$359$	−0.912868	−0.0481793	−0.0240897	−	0.999710i	$-0.507669\pi$
$359$	−0.912868	−0.0481793	−0.0240897	+	0.999710i	$0.507669\pi$
$360$	0	0
$361$	−17.2496	−0.907873
$362$	0	0
$363$	10.4598	0.548998
$364$	0	0
$365$	40.9289	2.14232
$366$	0	0
$367$	3.83714	0.200297	0.100149	−	0.994972i	$-0.468068\pi$
$367$	3.83714	0.200297	0.100149	+	0.994972i	$0.468068\pi$
$368$	0	0
$369$	2.85527	0.148640
$370$	0	0
$371$	2.06569	0.107245
$372$	0	0
$373$	11.3125	0.585736	0.292868	−	0.956153i	$-0.405390\pi$
$373$	11.3125	0.585736	0.292868	+	0.956153i	$0.405390\pi$
$374$	0	0
$375$	28.7160	1.48289
$376$	0	0
$377$	−8.89216	−0.457970
$378$	0	0
$379$	−5.54089	−0.284617	−0.142308	−	0.989822i	$-0.545452\pi$
$379$	−5.54089	−0.284617	−0.142308	+	0.989822i	$0.545452\pi$
$380$	0	0
$381$	−22.8087	−1.16853
$382$	0	0
$383$	−5.46082	−0.279035	−0.139517	−	0.990220i	$-0.544555\pi$
$383$	−5.46082	−0.279035	−0.139517	+	0.990220i	$0.544555\pi$
$384$	0	0
$385$	29.0067	1.47832
$386$	0	0
$387$	−2.15576	−0.109584
$388$	0	0
$389$	−6.88390	−0.349028	−0.174514	−	0.984655i	$-0.555835\pi$
$389$	−6.88390	−0.349028	−0.174514	+	0.984655i	$0.555835\pi$
$390$	0	0
$391$	2.50661	0.126765
$392$	0	0
$393$	26.4500	1.33422
$394$	0	0
$395$	−54.8285	−2.75872
$396$	0	0
$397$	19.9273	1.00012	0.500060	−	0.865991i	$-0.333311\pi$
$397$	19.9273	1.00012	0.500060	+	0.865991i	$0.333311\pi$
$398$	0	0
$399$	−8.18300	−0.409662
$400$	0	0
$401$	−16.4500	−0.821473	−0.410736	−	0.911754i	$-0.634728\pi$
$401$	−16.4500	−0.821473	−0.410736	+	0.911754i	$0.634728\pi$
$402$	0	0
$403$	−10.1505	−0.505630
$404$	0	0
$405$	38.0684	1.89163
$406$	0	0
$407$	4.69235	0.232591
$408$	0	0
$409$	−15.2889	−0.755989	−0.377994	−	0.925808i	$-0.623386\pi$
$409$	−15.2889	−0.755989	−0.377994	+	0.925808i	$0.623386\pi$
$410$	0	0
$411$	−11.6245	−0.573393
$412$	0	0
$413$	−3.33364	−0.164038
$414$	0	0
$415$	5.62455	0.276098
$416$	0	0
$417$	6.17065	0.302178
$418$	0	0
$419$	−2.72035	−0.132898	−0.0664488	−	0.997790i	$-0.521167\pi$
$419$	−2.72035	−0.132898	−0.0664488	+	0.997790i	$0.521167\pi$
$420$	0	0
$421$	8.86840	0.432219	0.216110	−	0.976369i	$-0.430663\pi$
$421$	8.86840	0.432219	0.216110	+	0.976369i	$0.430663\pi$
$422$	0	0
$423$	−3.27745	−0.159355
$424$	0	0
$425$	−9.11905	−0.442339
$426$	0	0
$427$	4.74435	0.229595
$428$	0	0
$429$	5.59607	0.270181
$430$	0	0
$431$	−18.6159	−0.896698	−0.448349	−	0.893859i	$-0.647988\pi$
$431$	−18.6159	−0.896698	−0.448349	+	0.893859i	$0.647988\pi$
$432$	0	0
$433$	29.6155	1.42323	0.711614	−	0.702570i	$-0.247964\pi$
$433$	29.6155	1.42323	0.711614	+	0.702570i	$0.247964\pi$
$434$	0	0
$435$	47.5937	2.28194
$436$	0	0
$437$	3.31632	0.158641
$438$	0	0
$439$	3.72431	0.177752	0.0888759	−	0.996043i	$-0.471673\pi$
$439$	3.72431	0.177752	0.0888759	+	0.996043i	$0.471673\pi$
$440$	0	0
$441$	1.81917	0.0866270
$442$	0	0
$443$	5.22885	0.248430	0.124215	−	0.992255i	$-0.460359\pi$
$443$	5.22885	0.248430	0.124215	+	0.992255i	$0.460359\pi$
$444$	0	0
$445$	23.3396	1.10640
$446$	0	0
$447$	−24.3025	−1.14947
$448$	0	0
$449$	−27.0624	−1.27716	−0.638578	−	0.769557i	$-0.720477\pi$
$449$	−27.0624	−1.27716	−0.638578	+	0.769557i	$0.720477\pi$
$450$	0	0
$451$	14.9495	0.703946
$452$	0	0
$453$	8.24597	0.387430
$454$	0	0
$455$	−16.3157	−0.764891
$456$	0	0
$457$	−41.0564	−1.92054	−0.960268	−	0.279080i	$-0.909971\pi$
$457$	−41.0564	−1.92054	−0.960268	+	0.279080i	$0.909971\pi$
$458$	0	0
$459$	−4.74544	−0.221498
$460$	0	0
$461$	3.16975	0.147630	0.0738150	−	0.997272i	$-0.476483\pi$
$461$	3.16975	0.147630	0.0738150	+	0.997272i	$0.476483\pi$
$462$	0	0
$463$	−21.9669	−1.02089	−0.510445	−	0.859911i	$-0.670519\pi$
$463$	−21.9669	−1.02089	−0.510445	+	0.859911i	$0.670519\pi$
$464$	0	0
$465$	54.3285	2.51942
$466$	0	0
$467$	4.73251	0.218995	0.109497	−	0.993987i	$-0.465076\pi$
$467$	4.73251	0.218995	0.109497	+	0.993987i	$0.465076\pi$
$468$	0	0
$469$	−2.41643	−0.111580
$470$	0	0
$471$	−30.9360	−1.42545
$472$	0	0
$473$	−11.2871	−0.518979
$474$	0	0
$475$	−12.0648	−0.553571
$476$	0	0
$477$	−0.274059	−0.0125483
$478$	0	0
$479$	−8.48804	−0.387828	−0.193914	−	0.981018i	$-0.562118\pi$
$479$	−8.48804	−0.387828	−0.193914	+	0.981018i	$0.562118\pi$
$480$	0	0
$481$	−2.63936	−0.120344
$482$	0	0
$483$	−15.5034	−0.705431
$484$	0	0
$485$	15.2983	0.694662
$486$	0	0
$487$	−24.8532	−1.12620	−0.563102	−	0.826387i	$-0.690392\pi$
$487$	−24.8532	−1.12620	−0.563102	+	0.826387i	$0.690392\pi$
$488$	0	0
$489$	19.0236	0.860275
$490$	0	0
$491$	−24.2916	−1.09627	−0.548133	−	0.836391i	$-0.684661\pi$
$491$	−24.2916	−1.09627	−0.548133	+	0.836391i	$0.684661\pi$
$492$	0	0
$493$	−6.82690	−0.307468
$494$	0	0
$495$	−3.84836	−0.172971
$496$	0	0
$497$	−29.0288	−1.30212
$498$	0	0
$499$	28.7278	1.28603	0.643017	−	0.765852i	$-0.277682\pi$
$499$	28.7278	1.28603	0.643017	+	0.765852i	$0.277682\pi$
$500$	0	0
$501$	42.8998	1.91662
$502$	0	0
$503$	15.9124	0.709499	0.354750	−	0.934961i	$-0.384566\pi$
$503$	15.9124	0.709499	0.354750	+	0.934961i	$0.384566\pi$
$504$	0	0
$505$	−26.8093	−1.19300
$506$	0	0
$507$	20.9717	0.931387
$508$	0	0
$509$	19.6646	0.871618	0.435809	−	0.900039i	$-0.356462\pi$
$509$	19.6646	0.871618	0.435809	+	0.900039i	$0.356462\pi$
$510$	0	0
$511$	36.3117	1.60633
$512$	0	0
$513$	−6.27837	−0.277197
$514$	0	0
$515$	−11.8870	−0.523805
$516$	0	0
$517$	−17.1599	−0.754693
$518$	0	0
$519$	44.8732	1.96971
$520$	0	0
$521$	8.18680	0.358670	0.179335	−	0.983788i	$-0.442605\pi$
$521$	8.18680	0.358670	0.179335	+	0.983788i	$0.442605\pi$
$522$	0	0
$523$	−19.4485	−0.850423	−0.425211	−	0.905094i	$-0.639800\pi$
$523$	−19.4485	−0.850423	−0.425211	+	0.905094i	$0.639800\pi$
$524$	0	0
$525$	56.4016	2.46157
$526$	0	0
$527$	−7.79295	−0.339466
$528$	0	0
$529$	−16.7169	−0.726823
$530$	0	0
$531$	0.442279	0.0191933
$532$	0	0
$533$	−8.40882	−0.364226
$534$	0	0
$535$	8.80953	0.380869
$536$	0	0
$537$	2.64036	0.113940
$538$	0	0
$539$	9.52472	0.410259
$540$	0	0
$541$	21.6221	0.929605	0.464802	−	0.885414i	$-0.346125\pi$
$541$	21.6221	0.929605	0.464802	+	0.885414i	$0.346125\pi$
$542$	0	0
$543$	6.37871	0.273737
$544$	0	0
$545$	−35.4821	−1.51988
$546$	0	0
$547$	30.4016	1.29988	0.649938	−	0.759987i	$-0.274795\pi$
$547$	30.4016	1.29988	0.649938	+	0.759987i	$0.274795\pi$
$548$	0	0
$549$	−0.629440	−0.0268638
$550$	0	0
$551$	−9.03221	−0.384785
$552$	0	0
$553$	−48.6432	−2.06852
$554$	0	0
$555$	14.1267	0.599644
$556$	0	0
$557$	−18.4359	−0.781155	−0.390577	−	0.920570i	$-0.627725\pi$
$557$	−18.4359	−0.781155	−0.390577	+	0.920570i	$0.627725\pi$
$558$	0	0
$559$	6.34875	0.268523
$560$	0	0
$561$	4.29635	0.181392
$562$	0	0
$563$	−21.8813	−0.922188	−0.461094	−	0.887351i	$-0.652543\pi$
$563$	−21.8813	−0.922188	−0.461094	+	0.887351i	$0.652543\pi$
$564$	0	0
$565$	−17.5410	−0.737954
$566$	0	0
$567$	33.7739	1.41837
$568$	0	0
$569$	−4.89113	−0.205047	−0.102523	−	0.994731i	$-0.532692\pi$
$569$	−4.89113	−0.205047	−0.102523	+	0.994731i	$0.532692\pi$
$570$	0	0
$571$	13.9503	0.583804	0.291902	−	0.956448i	$-0.405712\pi$
$571$	13.9503	0.583804	0.291902	+	0.956448i	$0.405712\pi$
$572$	0	0
$573$	44.0659	1.84088
$574$	0	0
$575$	−22.8579	−0.953239
$576$	0	0
$577$	12.1864	0.507326	0.253663	−	0.967293i	$-0.418365\pi$
$577$	12.1864	0.507326	0.253663	+	0.967293i	$0.418365\pi$
$578$	0	0
$579$	−6.11127	−0.253976
$580$	0	0
$581$	4.99003	0.207022
$582$	0	0
$583$	−1.43491	−0.0594277
$584$	0	0
$585$	2.16463	0.0894963
$586$	0	0
$587$	0.143998	0.00594342	0.00297171	−	0.999996i	$-0.499054\pi$
$587$	0.143998	0.00594342	0.00297171	+	0.999996i	$0.499054\pi$
$588$	0	0
$589$	−10.3103	−0.424830
$590$	0	0
$591$	−25.5387	−1.05052
$592$	0	0
$593$	14.0804	0.578212	0.289106	−	0.957297i	$-0.406642\pi$
$593$	14.0804	0.578212	0.289106	+	0.957297i	$0.406642\pi$
$594$	0	0
$595$	−12.5263	−0.513527
$596$	0	0
$597$	−17.6127	−0.720838
$598$	0	0
$599$	−1.99932	−0.0816899	−0.0408450	−	0.999165i	$-0.513005\pi$
$599$	−1.99932	−0.0816899	−0.0408450	+	0.999165i	$0.513005\pi$
$600$	0	0
$601$	16.3950	0.668767	0.334383	−	0.942437i	$-0.391472\pi$
$601$	16.3950	0.668767	0.334383	+	0.942437i	$0.391472\pi$
$602$	0	0
$603$	0.320591	0.0130555
$604$	0	0
$605$	21.1838	0.861243
$606$	0	0
$607$	−19.9240	−0.808690	−0.404345	−	0.914606i	$-0.632501\pi$
$607$	−19.9240	−0.808690	−0.404345	+	0.914606i	$0.632501\pi$
$608$	0	0
$609$	42.2246	1.71103
$610$	0	0
$611$	9.65212	0.390483
$612$	0	0
$613$	−5.07961	−0.205164	−0.102582	−	0.994725i	$-0.532710\pi$
$613$	−5.07961	−0.205164	−0.102582	+	0.994725i	$0.532710\pi$
$614$	0	0
$615$	45.0067	1.81484
$616$	0	0
$617$	8.85737	0.356584	0.178292	−	0.983978i	$-0.442943\pi$
$617$	8.85737	0.356584	0.178292	+	0.983978i	$0.442943\pi$
$618$	0	0
$619$	1.30527	0.0524632	0.0262316	−	0.999656i	$-0.491649\pi$
$619$	1.30527	0.0524632	0.0262316	+	0.999656i	$0.491649\pi$
$620$	0	0
$621$	−11.8949	−0.477328
$622$	0	0
$623$	20.7066	0.829593
$624$	0	0
$625$	12.5619	0.502475
$626$	0	0
$627$	5.68420	0.227005
$628$	0	0
$629$	−2.02635	−0.0807959
$630$	0	0
$631$	42.8589	1.70619	0.853094	−	0.521758i	$-0.174723\pi$
$631$	42.8589	1.70619	0.853094	+	0.521758i	$0.174723\pi$
$632$	0	0
$633$	−0.581493	−0.0231123
$634$	0	0
$635$	−46.1935	−1.83313
$636$	0	0
$637$	−5.35747	−0.212271
$638$	0	0
$639$	3.85129	0.152355
$640$	0	0
$641$	−48.6014	−1.91964	−0.959819	−	0.280618i	$-0.909460\pi$
$641$	−48.6014	−1.91964	−0.959819	+	0.280618i	$0.909460\pi$
$642$	0	0
$643$	44.7479	1.76469	0.882343	−	0.470607i	$-0.155965\pi$
$643$	44.7479	1.76469	0.882343	+	0.470607i	$0.155965\pi$
$644$	0	0
$645$	−33.9805	−1.33798
$646$	0	0
$647$	−29.6310	−1.16491	−0.582457	−	0.812861i	$-0.697909\pi$
$647$	−29.6310	−1.16491	−0.582457	+	0.812861i	$0.697909\pi$
$648$	0	0
$649$	2.31567	0.0908979
$650$	0	0
$651$	48.1996	1.88909
$652$	0	0
$653$	−19.7112	−0.771358	−0.385679	−	0.922633i	$-0.626033\pi$
$653$	−19.7112	−0.771358	−0.385679	+	0.922633i	$0.626033\pi$
$654$	0	0
$655$	53.5679	2.09307
$656$	0	0
$657$	−4.81753	−0.187950
$658$	0	0
$659$	−20.8278	−0.811335	−0.405667	−	0.914021i	$-0.632961\pi$
$659$	−20.8278	−0.811335	−0.405667	+	0.914021i	$0.632961\pi$
$660$	0	0
$661$	21.1383	0.822186	0.411093	−	0.911593i	$-0.365147\pi$
$661$	21.1383	0.822186	0.411093	+	0.911593i	$0.365147\pi$
$662$	0	0
$663$	−2.41661	−0.0938534
$664$	0	0
$665$	−16.5727	−0.642660
$666$	0	0
$667$	−17.1124	−0.662593
$668$	0	0
$669$	−4.25930	−0.164674
$670$	0	0
$671$	−3.29560	−0.127225
$672$	0	0
$673$	23.1361	0.891832	0.445916	−	0.895075i	$-0.352878\pi$
$673$	23.1361	0.891832	0.445916	+	0.895075i	$0.352878\pi$
$674$	0	0
$675$	43.2739	1.66561
$676$	0	0
$677$	−30.5934	−1.17580	−0.587899	−	0.808934i	$-0.700045\pi$
$677$	−30.5934	−1.17580	−0.587899	+	0.808934i	$0.700045\pi$
$678$	0	0
$679$	13.5725	0.520866
$680$	0	0
$681$	−14.9914	−0.574473
$682$	0	0
$683$	41.8411	1.60100	0.800502	−	0.599330i	$-0.204566\pi$
$683$	41.8411	1.60100	0.800502	+	0.599330i	$0.204566\pi$
$684$	0	0
$685$	−23.5425	−0.899514
$686$	0	0
$687$	−30.4673	−1.16240
$688$	0	0
$689$	0.807106	0.0307483
$690$	0	0
$691$	−29.3358	−1.11599	−0.557993	−	0.829846i	$-0.688428\pi$
$691$	−29.3358	−1.11599	−0.557993	+	0.829846i	$0.688428\pi$
$692$	0	0
$693$	−3.41422	−0.129696
$694$	0	0
$695$	12.4971	0.474044
$696$	0	0
$697$	−6.45582	−0.244531
$698$	0	0
$699$	−12.0155	−0.454469
$700$	0	0
$701$	32.0370	1.21002	0.605010	−	0.796218i	$-0.293169\pi$
$701$	32.0370	1.21002	0.605010	+	0.796218i	$0.293169\pi$
$702$	0	0
$703$	−2.68093	−0.101113
$704$	0	0
$705$	−51.6613	−1.94568
$706$	0	0
$707$	−23.7849	−0.894522
$708$	0	0
$709$	42.0818	1.58042	0.790208	−	0.612839i	$-0.209973\pi$
$709$	42.0818	1.58042	0.790208	+	0.612839i	$0.209973\pi$
$710$	0	0
$711$	6.45357	0.242028
$712$	0	0
$713$	−19.5339	−0.731549
$714$	0	0
$715$	11.3335	0.423848
$716$	0	0
$717$	38.7882	1.44857
$718$	0	0
$719$	10.8796	0.405742	0.202871	−	0.979205i	$-0.434973\pi$
$719$	10.8796	0.405742	0.202871	+	0.979205i	$0.434973\pi$
$720$	0	0
$721$	−10.5460	−0.392755
$722$	0	0
$723$	−15.9743	−0.594092
$724$	0	0
$725$	62.2549	2.31209
$726$	0	0
$727$	43.9809	1.63116	0.815581	−	0.578643i	$-0.196417\pi$
$727$	43.9809	1.63116	0.815581	+	0.578643i	$0.196417\pi$
$728$	0	0
$729$	21.9323	0.812309
$730$	0	0
$731$	4.87421	0.180279
$732$	0	0
$733$	5.17663	0.191203	0.0956016	−	0.995420i	$-0.469523\pi$
$733$	5.17663	0.191203	0.0956016	+	0.995420i	$0.469523\pi$
$734$	0	0
$735$	28.6749	1.05769
$736$	0	0
$737$	1.67854	0.0618298
$738$	0	0
$739$	−14.9165	−0.548713	−0.274357	−	0.961628i	$-0.588465\pi$
$739$	−14.9165	−0.548713	−0.274357	+	0.961628i	$0.588465\pi$
$740$	0	0
$741$	−3.19725	−0.117454
$742$	0	0
$743$	−26.7072	−0.979792	−0.489896	−	0.871781i	$-0.662965\pi$
$743$	−26.7072	−0.979792	−0.489896	+	0.871781i	$0.662965\pi$
$744$	0	0
$745$	−49.2189	−1.80324
$746$	0	0
$747$	−0.662036	−0.0242226
$748$	0	0
$749$	7.81572	0.285580
$750$	0	0
$751$	−8.17483	−0.298304	−0.149152	−	0.988814i	$-0.547654\pi$
$751$	−8.17483	−0.298304	−0.149152	+	0.988814i	$0.547654\pi$
$752$	0	0
$753$	48.0084	1.74952
$754$	0	0
$755$	16.7002	0.607782
$756$	0	0
$757$	17.2068	0.625392	0.312696	−	0.949853i	$-0.398768\pi$
$757$	17.2068	0.625392	0.312696	+	0.949853i	$0.398768\pi$
$758$	0	0
$759$	10.7692	0.390899
$760$	0	0
$761$	−24.0300	−0.871086	−0.435543	−	0.900168i	$-0.643444\pi$
$761$	−24.0300	−0.871086	−0.435543	+	0.900168i	$0.643444\pi$
$762$	0	0
$763$	−31.4793	−1.13963
$764$	0	0
$765$	1.66188	0.0600854
$766$	0	0
$767$	−1.30252	−0.0470312
$768$	0	0
$769$	−5.03458	−0.181552	−0.0907758	−	0.995871i	$-0.528935\pi$
$769$	−5.03458	−0.181552	−0.0907758	+	0.995871i	$0.528935\pi$
$770$	0	0
$771$	−21.4089	−0.771021
$772$	0	0
$773$	−8.61254	−0.309772	−0.154886	−	0.987932i	$-0.549501\pi$
$773$	−8.61254	−0.309772	−0.154886	+	0.987932i	$0.549501\pi$
$774$	0	0
$775$	71.0643	2.55271
$776$	0	0
$777$	12.5330	0.449620
$778$	0	0
$779$	−8.54125	−0.306022
$780$	0	0
$781$	20.1644	0.721540
$782$	0	0
$783$	32.3966	1.15776
$784$	0	0
$785$	−62.6532	−2.23619
$786$	0	0
$787$	−6.41439	−0.228648	−0.114324	−	0.993443i	$-0.536470\pi$
$787$	−6.41439	−0.228648	−0.114324	+	0.993443i	$0.536470\pi$
$788$	0	0
$789$	19.3354	0.688357
$790$	0	0
$791$	−15.5621	−0.553326
$792$	0	0
$793$	1.85371	0.0658271
$794$	0	0
$795$	−4.31989	−0.153211
$796$	0	0
$797$	−36.6551	−1.29839	−0.649196	−	0.760621i	$-0.724894\pi$
$797$	−36.6551	−1.29839	−0.649196	+	0.760621i	$0.724894\pi$
$798$	0	0
$799$	7.41036	0.262160
$800$	0	0
$801$	−2.74718	−0.0970668
$802$	0	0
$803$	−25.2234	−0.890115
$804$	0	0
$805$	−31.3984	−1.10665
$806$	0	0
$807$	−49.8093	−1.75337
$808$	0	0
$809$	6.81501	0.239603	0.119801	−	0.992798i	$-0.461774\pi$
$809$	6.81501	0.239603	0.119801	+	0.992798i	$0.461774\pi$
$810$	0	0
$811$	−37.5929	−1.32006	−0.660032	−	0.751237i	$-0.729457\pi$
$811$	−37.5929	−1.32006	−0.660032	+	0.751237i	$0.729457\pi$
$812$	0	0
$813$	22.6860	0.795631
$814$	0	0
$815$	38.5276	1.34956
$816$	0	0
$817$	6.44874	0.225613
$818$	0	0
$819$	1.92043	0.0671054
$820$	0	0
$821$	10.3793	0.362240	0.181120	−	0.983461i	$-0.442028\pi$
$821$	10.3793	0.362240	0.181120	+	0.983461i	$0.442028\pi$
$822$	0	0
$823$	2.53711	0.0884382	0.0442191	−	0.999022i	$-0.485920\pi$
$823$	2.53711	0.0884382	0.0442191	+	0.999022i	$0.485920\pi$
$824$	0	0
$825$	−39.1786	−1.36402
$826$	0	0
$827$	20.8288	0.724289	0.362145	−	0.932122i	$-0.382045\pi$
$827$	20.8288	0.724289	0.362145	+	0.932122i	$0.382045\pi$
$828$	0	0
$829$	−53.9717	−1.87452	−0.937258	−	0.348638i	$-0.886644\pi$
$829$	−53.9717	−1.87452	−0.937258	+	0.348638i	$0.886644\pi$
$830$	0	0
$831$	−54.5115	−1.89098
$832$	0	0
$833$	−4.11316	−0.142513
$834$	0	0
$835$	86.8831	3.00671
$836$	0	0
$837$	36.9810	1.27825
$838$	0	0
$839$	−55.8041	−1.92657	−0.963286	−	0.268478i	$-0.913479\pi$
$839$	−55.8041	−1.92657	−0.963286	+	0.268478i	$0.913479\pi$
$840$	0	0
$841$	17.6066	0.607123
$842$	0	0
$843$	5.25703	0.181062
$844$	0	0
$845$	42.4731	1.46112
$846$	0	0
$847$	18.7940	0.645770
$848$	0	0
$849$	32.0923	1.10141
$850$	0	0
$851$	−5.07926	−0.174115
$852$	0	0
$853$	6.07550	0.208021	0.104011	−	0.994576i	$-0.466832\pi$
$853$	6.07550	0.208021	0.104011	+	0.994576i	$0.466832\pi$
$854$	0	0
$855$	2.19872	0.0751946
$856$	0	0
$857$	−24.8784	−0.849831	−0.424916	−	0.905233i	$-0.639696\pi$
$857$	−24.8784	−0.849831	−0.424916	+	0.905233i	$0.639696\pi$
$858$	0	0
$859$	22.1644	0.756239	0.378119	−	0.925757i	$-0.376571\pi$
$859$	22.1644	0.756239	0.378119	+	0.925757i	$0.376571\pi$
$860$	0	0
$861$	39.9294	1.36079
$862$	0	0
$863$	−53.0075	−1.80440	−0.902198	−	0.431322i	$-0.858047\pi$
$863$	−53.0075	−1.80440	−0.902198	+	0.431322i	$0.858047\pi$
$864$	0	0
$865$	90.8796	3.09000
$866$	0	0
$867$	−1.85534	−0.0630106
$868$	0	0
$869$	33.7894	1.14623
$870$	0	0
$871$	−0.944145	−0.0319911
$872$	0	0
$873$	−1.80069	−0.0609440
$874$	0	0
$875$	51.5964	1.74428
$876$	0	0
$877$	11.1445	0.376322	0.188161	−	0.982138i	$-0.439747\pi$
$877$	11.1445	0.376322	0.188161	+	0.982138i	$0.439747\pi$
$878$	0	0
$879$	−32.9059	−1.10989
$880$	0	0
$881$	38.7999	1.30720	0.653601	−	0.756839i	$-0.273257\pi$
$881$	38.7999	1.30720	0.653601	+	0.756839i	$0.273257\pi$
$882$	0	0
$883$	−27.3775	−0.921327	−0.460664	−	0.887575i	$-0.652389\pi$
$883$	−27.3775	−0.921327	−0.460664	+	0.887575i	$0.652389\pi$
$884$	0	0
$885$	6.97149	0.234344
$886$	0	0
$887$	−36.8543	−1.23745	−0.618723	−	0.785609i	$-0.712350\pi$
$887$	−36.8543	−1.23745	−0.618723	+	0.785609i	$0.712350\pi$
$888$	0	0
$889$	−40.9823	−1.37450
$890$	0	0
$891$	−23.4606	−0.785958
$892$	0	0
$893$	9.80414	0.328083
$894$	0	0
$895$	5.34741	0.178744
$896$	0	0
$897$	−6.05749	−0.202254
$898$	0	0
$899$	53.2017	1.77438
$900$	0	0
$901$	0.619651	0.0206436
$902$	0	0
$903$	−30.1472	−1.00323
$904$	0	0
$905$	12.9185	0.429426
$906$	0	0
$907$	−46.8188	−1.55459	−0.777296	−	0.629136i	$-0.783409\pi$
$907$	−46.8188	−1.55459	−0.777296	+	0.629136i	$0.783409\pi$
$908$	0	0
$909$	3.15558	0.104664
$910$	0	0
$911$	33.2475	1.10154	0.550770	−	0.834657i	$-0.314334\pi$
$911$	33.2475	1.10154	0.550770	+	0.834657i	$0.314334\pi$
$912$	0	0
$913$	−3.46626	−0.114716
$914$	0	0
$915$	−9.92164	−0.327999
$916$	0	0
$917$	47.5249	1.56941
$918$	0	0
$919$	26.7710	0.883094	0.441547	−	0.897238i	$-0.354430\pi$
$919$	26.7710	0.883094	0.441547	+	0.897238i	$0.354430\pi$
$920$	0	0
$921$	29.6724	0.977740
$922$	0	0
$923$	−11.3421	−0.373330
$924$	0	0
$925$	18.4784	0.607566
$926$	0	0
$927$	1.39916	0.0459545
$928$	0	0
$929$	−28.5662	−0.937227	−0.468613	−	0.883403i	$-0.655246\pi$
$929$	−28.5662	−0.937227	−0.468613	+	0.883403i	$0.655246\pi$
$930$	0	0
$931$	−5.44185	−0.178349
$932$	0	0
$933$	20.6935	0.677475
$934$	0	0
$935$	8.70120	0.284560
$936$	0	0
$937$	45.0107	1.47044	0.735218	−	0.677831i	$-0.237080\pi$
$937$	45.0107	1.47044	0.735218	+	0.677831i	$0.237080\pi$
$938$	0	0
$939$	19.9152	0.649907
$940$	0	0
$941$	−10.5546	−0.344069	−0.172035	−	0.985091i	$-0.555034\pi$
$941$	−10.5546	−0.344069	−0.172035	+	0.985091i	$0.555034\pi$
$942$	0	0
$943$	−16.1822	−0.526965
$944$	0	0
$945$	59.4426	1.93367
$946$	0	0
$947$	−46.5326	−1.51211	−0.756053	−	0.654511i	$-0.772875\pi$
$947$	−46.5326	−1.51211	−0.756053	+	0.654511i	$0.772875\pi$
$948$	0	0
$949$	14.1877	0.460551
$950$	0	0
$951$	−8.66193	−0.280882
$952$	0	0
$953$	−32.6950	−1.05910	−0.529548	−	0.848280i	$-0.677638\pi$
$953$	−32.6950	−1.05910	−0.529548	+	0.848280i	$0.677638\pi$
$954$	0	0
$955$	89.2447	2.88789
$956$	0	0
$957$	−29.3307	−0.948128
$958$	0	0
$959$	−20.8867	−0.674466
$960$	0	0
$961$	29.7301	0.959035
$962$	0	0
$963$	−1.03692	−0.0334144
$964$	0	0
$965$	−12.3769	−0.398426
$966$	0	0
$967$	31.5525	1.01466	0.507329	−	0.861752i	$-0.330633\pi$
$967$	31.5525	1.01466	0.507329	+	0.861752i	$0.330633\pi$
$968$	0	0
$969$	−2.45467	−0.0788554
$970$	0	0
$971$	8.18250	0.262589	0.131294	−	0.991343i	$-0.458087\pi$
$971$	8.18250	0.262589	0.131294	+	0.991343i	$0.458087\pi$
$972$	0	0
$973$	11.0873	0.355443
$974$	0	0
$975$	22.0372	0.705755
$976$	0	0
$977$	39.0246	1.24851	0.624253	−	0.781222i	$-0.285403\pi$
$977$	39.0246	1.24851	0.624253	+	0.781222i	$0.285403\pi$
$978$	0	0
$979$	−14.3836	−0.459701
$980$	0	0
$981$	4.17641	0.133342
$982$	0	0
$983$	−16.6447	−0.530884	−0.265442	−	0.964127i	$-0.585518\pi$
$983$	−16.6447	−0.530884	−0.265442	+	0.964127i	$0.585518\pi$
$984$	0	0
$985$	−51.7224	−1.64801
$986$	0	0
$987$	−45.8333	−1.45889
$988$	0	0
$989$	12.2177	0.388501
$990$	0	0
$991$	58.9404	1.87230	0.936151	−	0.351598i	$-0.114361\pi$
$991$	58.9404	1.87230	0.936151	+	0.351598i	$0.114361\pi$
$992$	0	0
$993$	−16.7542	−0.531679
$994$	0	0
$995$	−35.6701	−1.13082
$996$	0	0
$997$	20.6381	0.653616	0.326808	−	0.945091i	$-0.394027\pi$
$997$	20.6381	0.653616	0.326808	+	0.945091i	$0.394027\pi$
$998$	0	0
$999$	9.61591	0.304234

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.g.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.g.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-1.85534 q^{3} -3.75753 q^{5} -3.33364 q^{7} +0.442279 q^{9} +O(q^{10})\)	\(q-1.85534 q^{3} -3.75753 q^{5} -3.33364 q^{7} +0.442279 q^{9} +2.31567 q^{11} -1.30252 q^{13} +6.97149 q^{15} -1.00000 q^{17} -1.32303 q^{19} +6.18503 q^{21} -2.50661 q^{23} +9.11905 q^{25} +4.74544 q^{27} +6.82690 q^{29} +7.79295 q^{31} -4.29635 q^{33} +12.5263 q^{35} +2.02635 q^{37} +2.41661 q^{39} +6.45582 q^{41} -4.87421 q^{43} -1.66188 q^{45} -7.41036 q^{47} +4.11316 q^{49} +1.85534 q^{51} -0.619651 q^{53} -8.70120 q^{55} +2.45467 q^{57} +1.00000 q^{59} -1.42317 q^{61} -1.47440 q^{63} +4.89425 q^{65} +0.724862 q^{67} +4.65060 q^{69} +8.70782 q^{71} -10.8925 q^{73} -16.9189 q^{75} -7.71961 q^{77} +14.5916 q^{79} -10.1312 q^{81} -1.49687 q^{83} +3.75753 q^{85} -12.6662 q^{87} -6.21141 q^{89} +4.34213 q^{91} -14.4586 q^{93} +4.97134 q^{95} -4.07138 q^{97} +1.02417 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 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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