LMFDB - Embedded newform 4012.2.a.g.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$12$
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			1.9
Root			$-0.864716$ 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)

$f(q)$	$=$	$q+0.864716 q^{3} -1.40497 q^{5} -2.45790 q^{7} -2.25227 q^{9} +O(q^{10})$	$q+0.864716 q^{3} -1.40497 q^{5} -2.45790 q^{7} -2.25227 q^{9} +1.26992 q^{11} +3.79248 q^{13} -1.21490 q^{15} -1.00000 q^{17} +6.52341 q^{19} -2.12539 q^{21} +1.32346 q^{23} -3.02605 q^{25} -4.54172 q^{27} -7.10007 q^{29} +6.91840 q^{31} +1.09812 q^{33} +3.45329 q^{35} +6.38301 q^{37} +3.27942 q^{39} -7.21762 q^{41} +9.21157 q^{43} +3.16437 q^{45} -11.3544 q^{47} -0.958714 q^{49} -0.864716 q^{51} -12.1634 q^{53} -1.78421 q^{55} +5.64089 q^{57} +1.00000 q^{59} -2.31997 q^{61} +5.53585 q^{63} -5.32834 q^{65} -14.7359 q^{67} +1.14442 q^{69} -2.56458 q^{71} +4.85616 q^{73} -2.61667 q^{75} -3.12134 q^{77} +9.22521 q^{79} +2.82950 q^{81} -12.6201 q^{83} +1.40497 q^{85} -6.13954 q^{87} +10.0483 q^{89} -9.32155 q^{91} +5.98245 q^{93} -9.16521 q^{95} -6.67250 q^{97} -2.86020 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.864716	0.499244	0.249622	−	0.968343i	$-0.419694\pi$
$3$	0.864716	0.499244	0.249622	+	0.968343i	$0.419694\pi$
$4$	0	0
$5$	−1.40497	−0.628323	−0.314162	−	0.949370i	$-0.601723\pi$
$5$	−1.40497	−0.628323	−0.314162	+	0.949370i	$0.601723\pi$
$6$	0	0
$7$	−2.45790	−0.929000	−0.464500	−	0.885573i	$-0.653766\pi$
$7$	−2.45790	−0.929000	−0.464500	+	0.885573i	$0.653766\pi$
$8$	0	0
$9$	−2.25227	−0.750755
$10$	0	0
$11$	1.26992	0.382896	0.191448	−	0.981503i	$-0.438682\pi$
$11$	1.26992	0.382896	0.191448	+	0.981503i	$0.438682\pi$
$12$	0	0
$13$	3.79248	1.05185	0.525923	−	0.850532i	$-0.323720\pi$
$13$	3.79248	1.05185	0.525923	+	0.850532i	$0.323720\pi$
$14$	0	0
$15$	−1.21490	−0.313687
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	6.52341	1.49657	0.748286	−	0.663376i	$-0.230877\pi$
$19$	6.52341	1.49657	0.748286	+	0.663376i	$0.230877\pi$
$20$	0	0
$21$	−2.12539	−0.463798
$22$	0	0
$23$	1.32346	0.275961	0.137981	−	0.990435i	$-0.455939\pi$
$23$	1.32346	0.275961	0.137981	+	0.990435i	$0.455939\pi$
$24$	0	0
$25$	−3.02605	−0.605210
$26$	0	0
$27$	−4.54172	−0.874054
$28$	0	0
$29$	−7.10007	−1.31845	−0.659225	−	0.751946i	$-0.729115\pi$
$29$	−7.10007	−1.31845	−0.659225	+	0.751946i	$0.729115\pi$
$30$	0	0
$31$	6.91840	1.24258	0.621291	−	0.783580i	$-0.286609\pi$
$31$	6.91840	1.24258	0.621291	+	0.783580i	$0.286609\pi$
$32$	0	0
$33$	1.09812	0.191158
$34$	0	0
$35$	3.45329	0.583712
$36$	0	0
$37$	6.38301	1.04936	0.524681	−	0.851299i	$-0.324185\pi$
$37$	6.38301	1.04936	0.524681	+	0.851299i	$0.324185\pi$
$38$	0	0
$39$	3.27942	0.525127
$40$	0	0
$41$	−7.21762	−1.12720	−0.563601	−	0.826047i	$-0.690585\pi$
$41$	−7.21762	−1.12720	−0.563601	+	0.826047i	$0.690585\pi$
$42$	0	0
$43$	9.21157	1.40475	0.702376	−	0.711807i	$-0.252123\pi$
$43$	9.21157	1.40475	0.702376	+	0.711807i	$0.252123\pi$
$44$	0	0
$45$	3.16437	0.471717
$46$	0	0
$47$	−11.3544	−1.65621	−0.828103	−	0.560576i	$-0.810580\pi$
$47$	−11.3544	−1.65621	−0.828103	+	0.560576i	$0.810580\pi$
$48$	0	0
$49$	−0.958714	−0.136959
$50$	0	0
$51$	−0.864716	−0.121084
$52$	0	0
$53$	−12.1634	−1.67077	−0.835385	−	0.549665i	$-0.814755\pi$
$53$	−12.1634	−1.67077	−0.835385	+	0.549665i	$0.814755\pi$
$54$	0	0
$55$	−1.78421	−0.240582
$56$	0	0
$57$	5.64089	0.747155
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−2.31997	−0.297042	−0.148521	−	0.988909i	$-0.547451\pi$
$61$	−2.31997	−0.297042	−0.148521	+	0.988909i	$0.547451\pi$
$62$	0	0
$63$	5.53585	0.697452
$64$	0	0
$65$	−5.32834	−0.660899
$66$	0	0
$67$	−14.7359	−1.80027	−0.900137	−	0.435607i	$-0.856534\pi$
$67$	−14.7359	−1.80027	−0.900137	+	0.435607i	$0.856534\pi$
$68$	0	0
$69$	1.14442	0.137772
$70$	0	0
$71$	−2.56458	−0.304359	−0.152180	−	0.988353i	$-0.548629\pi$
$71$	−2.56458	−0.304359	−0.152180	+	0.988353i	$0.548629\pi$
$72$	0	0
$73$	4.85616	0.568370	0.284185	−	0.958769i	$-0.408277\pi$
$73$	4.85616	0.568370	0.284185	+	0.958769i	$0.408277\pi$
$74$	0	0
$75$	−2.61667	−0.302147
$76$	0	0
$77$	−3.12134	−0.355710
$78$	0	0
$79$	9.22521	1.03792	0.518958	−	0.854799i	$-0.326320\pi$
$79$	9.22521	1.03792	0.518958	+	0.854799i	$0.326320\pi$
$80$	0	0
$81$	2.82950	0.314389
$82$	0	0
$83$	−12.6201	−1.38524	−0.692619	−	0.721304i	$-0.743543\pi$
$83$	−12.6201	−1.38524	−0.692619	+	0.721304i	$0.743543\pi$
$84$	0	0
$85$	1.40497	0.152391
$86$	0	0
$87$	−6.13954	−0.658228
$88$	0	0
$89$	10.0483	1.06512	0.532558	−	0.846393i	$-0.321231\pi$
$89$	10.0483	1.06512	0.532558	+	0.846393i	$0.321231\pi$
$90$	0	0
$91$	−9.32155	−0.977164
$92$	0	0
$93$	5.98245	0.620351
$94$	0	0
$95$	−9.16521	−0.940331
$96$	0	0
$97$	−6.67250	−0.677490	−0.338745	−	0.940878i	$-0.610002\pi$
$97$	−6.67250	−0.677490	−0.338745	+	0.940878i	$0.610002\pi$
$98$	0	0
$99$	−2.86020	−0.287461
$100$	0	0
$101$	13.4085	1.33420	0.667100	−	0.744969i	$-0.267536\pi$
$101$	13.4085	1.33420	0.667100	+	0.744969i	$0.267536\pi$
$102$	0	0
$103$	10.7335	1.05760	0.528800	−	0.848746i	$-0.322642\pi$
$103$	10.7335	1.05760	0.528800	+	0.848746i	$0.322642\pi$
$104$	0	0
$105$	2.98611	0.291415
$106$	0	0
$107$	−15.7214	−1.51985	−0.759924	−	0.650012i	$-0.774764\pi$
$107$	−15.7214	−1.51985	−0.759924	+	0.650012i	$0.774764\pi$
$108$	0	0
$109$	−10.8611	−1.04030	−0.520152	−	0.854074i	$-0.674125\pi$
$109$	−10.8611	−1.04030	−0.520152	+	0.854074i	$0.674125\pi$
$110$	0	0
$111$	5.51949	0.523887
$112$	0	0
$113$	−12.1816	−1.14595	−0.572975	−	0.819573i	$-0.694211\pi$
$113$	−12.1816	−1.14595	−0.572975	+	0.819573i	$0.694211\pi$
$114$	0	0
$115$	−1.85943	−0.173393
$116$	0	0
$117$	−8.54168	−0.789678
$118$	0	0
$119$	2.45790	0.225316
$120$	0	0
$121$	−9.38730	−0.853391
$122$	0	0
$123$	−6.24119	−0.562749
$124$	0	0
$125$	11.2764	1.00859
$126$	0	0
$127$	−17.4744	−1.55060	−0.775302	−	0.631591i	$-0.782402\pi$
$127$	−17.4744	−1.55060	−0.775302	+	0.631591i	$0.782402\pi$
$128$	0	0
$129$	7.96539	0.701314
$130$	0	0
$131$	−7.99832	−0.698816	−0.349408	−	0.936971i	$-0.613617\pi$
$131$	−7.99832	−0.698816	−0.349408	+	0.936971i	$0.613617\pi$
$132$	0	0
$133$	−16.0339	−1.39032
$134$	0	0
$135$	6.38099	0.549189
$136$	0	0
$137$	3.95226	0.337665	0.168832	−	0.985645i	$-0.446000\pi$
$137$	3.95226	0.337665	0.168832	+	0.985645i	$0.446000\pi$
$138$	0	0
$139$	−10.9735	−0.930756	−0.465378	−	0.885112i	$-0.654082\pi$
$139$	−10.9735	−0.930756	−0.465378	+	0.885112i	$0.654082\pi$
$140$	0	0
$141$	−9.81831	−0.826851
$142$	0	0
$143$	4.81615	0.402747
$144$	0	0
$145$	9.97541	0.828413
$146$	0	0
$147$	−0.829015	−0.0683760
$148$	0	0
$149$	−15.9686	−1.30820	−0.654100	−	0.756408i	$-0.726953\pi$
$149$	−15.9686	−1.30820	−0.654100	+	0.756408i	$0.726953\pi$
$150$	0	0
$151$	−22.3823	−1.82145	−0.910724	−	0.413015i	$-0.864476\pi$
$151$	−22.3823	−1.82145	−0.910724	+	0.413015i	$0.864476\pi$
$152$	0	0
$153$	2.25227	0.182085
$154$	0	0
$155$	−9.72017	−0.780743
$156$	0	0
$157$	−19.8758	−1.58626	−0.793129	−	0.609054i	$-0.791549\pi$
$157$	−19.8758	−1.58626	−0.793129	+	0.609054i	$0.791549\pi$
$158$	0	0
$159$	−10.5179	−0.834122
$160$	0	0
$161$	−3.25295	−0.256368
$162$	0	0
$163$	−2.77046	−0.216999	−0.108500	−	0.994096i	$-0.534605\pi$
$163$	−2.77046	−0.216999	−0.108500	+	0.994096i	$0.534605\pi$
$164$	0	0
$165$	−1.54283	−0.120109
$166$	0	0
$167$	24.7982	1.91895	0.959473	−	0.281801i	$-0.0909319\pi$
$167$	24.7982	1.91895	0.959473	+	0.281801i	$0.0909319\pi$
$168$	0	0
$169$	1.38291	0.106378
$170$	0	0
$171$	−14.6924	−1.12356
$172$	0	0
$173$	−10.8989	−0.828630	−0.414315	−	0.910133i	$-0.635979\pi$
$173$	−10.8989	−0.828630	−0.414315	+	0.910133i	$0.635979\pi$
$174$	0	0
$175$	7.43773	0.562240
$176$	0	0
$177$	0.864716	0.0649960
$178$	0	0
$179$	−1.69364	−0.126589	−0.0632943	−	0.997995i	$-0.520161\pi$
$179$	−1.69364	−0.126589	−0.0632943	+	0.997995i	$0.520161\pi$
$180$	0	0
$181$	1.29745	0.0964390	0.0482195	−	0.998837i	$-0.484645\pi$
$181$	1.29745	0.0964390	0.0482195	+	0.998837i	$0.484645\pi$
$182$	0	0
$183$	−2.00612	−0.148296
$184$	0	0
$185$	−8.96797	−0.659338
$186$	0	0
$187$	−1.26992	−0.0928659
$188$	0	0
$189$	11.1631	0.811996
$190$	0	0
$191$	−0.465653	−0.0336935	−0.0168467	−	0.999858i	$-0.505363\pi$
$191$	−0.465653	−0.0336935	−0.0168467	+	0.999858i	$0.505363\pi$
$192$	0	0
$193$	−13.1778	−0.948560	−0.474280	−	0.880374i	$-0.657292\pi$
$193$	−13.1778	−0.948560	−0.474280	+	0.880374i	$0.657292\pi$
$194$	0	0
$195$	−4.60750	−0.329950
$196$	0	0
$197$	21.3903	1.52400	0.761998	−	0.647580i	$-0.224219\pi$
$197$	21.3903	1.52400	0.761998	+	0.647580i	$0.224219\pi$
$198$	0	0
$199$	9.46123	0.670689	0.335344	−	0.942096i	$-0.391147\pi$
$199$	9.46123	0.670689	0.335344	+	0.942096i	$0.391147\pi$
$200$	0	0
$201$	−12.7423	−0.898776
$202$	0	0
$203$	17.4513	1.22484
$204$	0	0
$205$	10.1406	0.708248
$206$	0	0
$207$	−2.98080	−0.207180
$208$	0	0
$209$	8.28421	0.573031
$210$	0	0
$211$	−5.28272	−0.363678	−0.181839	−	0.983328i	$-0.558205\pi$
$211$	−5.28272	−0.363678	−0.181839	+	0.983328i	$0.558205\pi$
$212$	0	0
$213$	−2.21763	−0.151949
$214$	0	0
$215$	−12.9420	−0.882638
$216$	0	0
$217$	−17.0048	−1.15436
$218$	0	0
$219$	4.19920	0.283756
$220$	0	0
$221$	−3.79248	−0.255110
$222$	0	0
$223$	22.4443	1.50298	0.751492	−	0.659742i	$-0.229335\pi$
$223$	22.4443	1.50298	0.751492	+	0.659742i	$0.229335\pi$
$224$	0	0
$225$	6.81547	0.454365
$226$	0	0
$227$	−1.16577	−0.0773751	−0.0386875	−	0.999251i	$-0.512318\pi$
$227$	−1.16577	−0.0773751	−0.0386875	+	0.999251i	$0.512318\pi$
$228$	0	0
$229$	−4.41708	−0.291889	−0.145944	−	0.989293i	$-0.546622\pi$
$229$	−4.41708	−0.291889	−0.145944	+	0.989293i	$0.546622\pi$
$230$	0	0
$231$	−2.69908	−0.177586
$232$	0	0
$233$	22.7157	1.48815	0.744076	−	0.668095i	$-0.232890\pi$
$233$	22.7157	1.48815	0.744076	+	0.668095i	$0.232890\pi$
$234$	0	0
$235$	15.9526	1.04063
$236$	0	0
$237$	7.97718	0.518174
$238$	0	0
$239$	−11.9563	−0.773388	−0.386694	−	0.922208i	$-0.626383\pi$
$239$	−11.9563	−0.773388	−0.386694	+	0.922208i	$0.626383\pi$
$240$	0	0
$241$	12.8737	0.829268	0.414634	−	0.909988i	$-0.363910\pi$
$241$	12.8737	0.829268	0.414634	+	0.909988i	$0.363910\pi$
$242$	0	0
$243$	16.0719	1.03101
$244$	0	0
$245$	1.34697	0.0860546
$246$	0	0
$247$	24.7399	1.57416
$248$	0	0
$249$	−10.9128	−0.691572
$250$	0	0
$251$	−13.6277	−0.860175	−0.430088	−	0.902787i	$-0.641517\pi$
$251$	−13.6277	−0.860175	−0.430088	+	0.902787i	$0.641517\pi$
$252$	0	0
$253$	1.68070	0.105664
$254$	0	0
$255$	1.21490	0.0760802
$256$	0	0
$257$	−19.6606	−1.22640	−0.613199	−	0.789929i	$-0.710117\pi$
$257$	−19.6606	−1.22640	−0.613199	+	0.789929i	$0.710117\pi$
$258$	0	0
$259$	−15.6888	−0.974856
$260$	0	0
$261$	15.9912	0.989833
$262$	0	0
$263$	5.48341	0.338121	0.169061	−	0.985606i	$-0.445927\pi$
$263$	5.48341	0.338121	0.169061	+	0.985606i	$0.445927\pi$
$264$	0	0
$265$	17.0892	1.04978
$266$	0	0
$267$	8.68892	0.531753
$268$	0	0
$269$	10.1449	0.618546	0.309273	−	0.950973i	$-0.399914\pi$
$269$	10.1449	0.618546	0.309273	+	0.950973i	$0.399914\pi$
$270$	0	0
$271$	−3.54207	−0.215165	−0.107583	−	0.994196i	$-0.534311\pi$
$271$	−3.54207	−0.215165	−0.107583	+	0.994196i	$0.534311\pi$
$272$	0	0
$273$	−8.06049	−0.487843
$274$	0	0
$275$	−3.84285	−0.231732
$276$	0	0
$277$	19.2177	1.15468	0.577339	−	0.816504i	$-0.304091\pi$
$277$	19.2177	1.15468	0.577339	+	0.816504i	$0.304091\pi$
$278$	0	0
$279$	−15.5821	−0.932875
$280$	0	0
$281$	9.71710	0.579674	0.289837	−	0.957076i	$-0.406399\pi$
$281$	9.71710	0.579674	0.289837	+	0.957076i	$0.406399\pi$
$282$	0	0
$283$	5.43775	0.323241	0.161620	−	0.986853i	$-0.448328\pi$
$283$	5.43775	0.323241	0.161620	+	0.986853i	$0.448328\pi$
$284$	0	0
$285$	−7.92531	−0.469455
$286$	0	0
$287$	17.7402	1.04717
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−5.76982	−0.338233
$292$	0	0
$293$	−12.9884	−0.758793	−0.379396	−	0.925234i	$-0.623868\pi$
$293$	−12.9884	−0.758793	−0.379396	+	0.925234i	$0.623868\pi$
$294$	0	0
$295$	−1.40497	−0.0818007
$296$	0	0
$297$	−5.76763	−0.334672
$298$	0	0
$299$	5.01922	0.290269
$300$	0	0
$301$	−22.6411	−1.30501
$302$	0	0
$303$	11.5946	0.666091
$304$	0	0
$305$	3.25950	0.186638
$306$	0	0
$307$	−13.8146	−0.788438	−0.394219	−	0.919016i	$-0.628985\pi$
$307$	−13.8146	−0.788438	−0.394219	+	0.919016i	$0.628985\pi$
$308$	0	0
$309$	9.28140	0.528000
$310$	0	0
$311$	−18.0508	−1.02357	−0.511784	−	0.859114i	$-0.671015\pi$
$311$	−18.0508	−1.02357	−0.511784	+	0.859114i	$0.671015\pi$
$312$	0	0
$313$	−23.8897	−1.35033	−0.675164	−	0.737668i	$-0.735927\pi$
$313$	−23.8897	−1.35033	−0.675164	+	0.737668i	$0.735927\pi$
$314$	0	0
$315$	−7.77773	−0.438225
$316$	0	0
$317$	8.54912	0.480166	0.240083	−	0.970752i	$-0.422825\pi$
$317$	8.54912	0.480166	0.240083	+	0.970752i	$0.422825\pi$
$318$	0	0
$319$	−9.01653	−0.504829
$320$	0	0
$321$	−13.5946	−0.758775
$322$	0	0
$323$	−6.52341	−0.362972
$324$	0	0
$325$	−11.4762	−0.636587
$326$	0	0
$327$	−9.39176	−0.519366
$328$	0	0
$329$	27.9080	1.53862
$330$	0	0
$331$	11.8675	0.652298	0.326149	−	0.945318i	$-0.394249\pi$
$331$	11.8675	0.652298	0.326149	+	0.945318i	$0.394249\pi$
$332$	0	0
$333$	−14.3762	−0.787814
$334$	0	0
$335$	20.7035	1.13115
$336$	0	0
$337$	−6.29403	−0.342858	−0.171429	−	0.985197i	$-0.554838\pi$
$337$	−6.29403	−0.342858	−0.171429	+	0.985197i	$0.554838\pi$
$338$	0	0
$339$	−10.5336	−0.572109
$340$	0	0
$341$	8.78582	0.475779
$342$	0	0
$343$	19.5617	1.05623
$344$	0	0
$345$	−1.60788	−0.0865654
$346$	0	0
$347$	3.19978	0.171773	0.0858866	−	0.996305i	$-0.472628\pi$
$347$	3.19978	0.171773	0.0858866	+	0.996305i	$0.472628\pi$
$348$	0	0
$349$	26.2383	1.40450	0.702252	−	0.711929i	$-0.252178\pi$
$349$	26.2383	1.40450	0.702252	+	0.711929i	$0.252178\pi$
$350$	0	0
$351$	−17.2244	−0.919370
$352$	0	0
$353$	11.1625	0.594119	0.297059	−	0.954859i	$-0.403994\pi$
$353$	11.1625	0.594119	0.297059	+	0.954859i	$0.403994\pi$
$354$	0	0
$355$	3.60316	0.191236
$356$	0	0
$357$	2.12539	0.112487
$358$	0	0
$359$	−31.3590	−1.65506	−0.827532	−	0.561418i	$-0.810256\pi$
$359$	−31.3590	−1.65506	−0.827532	+	0.561418i	$0.810256\pi$
$360$	0	0
$361$	23.5548	1.23973
$362$	0	0
$363$	−8.11735	−0.426050
$364$	0	0
$365$	−6.82278	−0.357120
$366$	0	0
$367$	−8.85347	−0.462147	−0.231074	−	0.972936i	$-0.574224\pi$
$367$	−8.85347	−0.462147	−0.231074	+	0.972936i	$0.574224\pi$
$368$	0	0
$369$	16.2560	0.846253
$370$	0	0
$371$	29.8964	1.55215
$372$	0	0
$373$	−31.6779	−1.64022	−0.820111	−	0.572205i	$-0.806088\pi$
$373$	−31.6779	−1.64022	−0.820111	+	0.572205i	$0.806088\pi$
$374$	0	0
$375$	9.75087	0.503533
$376$	0	0
$377$	−26.9269	−1.38680
$378$	0	0
$379$	28.2687	1.45206	0.726032	−	0.687661i	$-0.241362\pi$
$379$	28.2687	1.45206	0.726032	+	0.687661i	$0.241362\pi$
$380$	0	0
$381$	−15.1104	−0.774129
$382$	0	0
$383$	15.9554	0.815285	0.407643	−	0.913142i	$-0.366351\pi$
$383$	15.9554	0.815285	0.407643	+	0.913142i	$0.366351\pi$
$384$	0	0
$385$	4.38541	0.223501
$386$	0	0
$387$	−20.7469	−1.05462
$388$	0	0
$389$	12.9543	0.656810	0.328405	−	0.944537i	$-0.393489\pi$
$389$	12.9543	0.656810	0.328405	+	0.944537i	$0.393489\pi$
$390$	0	0
$391$	−1.32346	−0.0669305
$392$	0	0
$393$	−6.91627	−0.348880
$394$	0	0
$395$	−12.9612	−0.652147
$396$	0	0
$397$	9.98957	0.501362	0.250681	−	0.968070i	$-0.419345\pi$
$397$	9.98957	0.501362	0.250681	+	0.968070i	$0.419345\pi$
$398$	0	0
$399$	−13.8648	−0.694107
$400$	0	0
$401$	8.07820	0.403406	0.201703	−	0.979447i	$-0.435352\pi$
$401$	8.07820	0.403406	0.201703	+	0.979447i	$0.435352\pi$
$402$	0	0
$403$	26.2379	1.30700
$404$	0	0
$405$	−3.97538	−0.197538
$406$	0	0
$407$	8.10593	0.401796
$408$	0	0
$409$	−5.07272	−0.250830	−0.125415	−	0.992104i	$-0.540026\pi$
$409$	−5.07272	−0.250830	−0.125415	+	0.992104i	$0.540026\pi$
$410$	0	0
$411$	3.41759	0.168577
$412$	0	0
$413$	−2.45790	−0.120945
$414$	0	0
$415$	17.7309	0.870377
$416$	0	0
$417$	−9.48892	−0.464674
$418$	0	0
$419$	−2.11139	−0.103148	−0.0515742	−	0.998669i	$-0.516424\pi$
$419$	−2.11139	−0.103148	−0.0515742	+	0.998669i	$0.516424\pi$
$420$	0	0
$421$	17.8083	0.867922	0.433961	−	0.900932i	$-0.357116\pi$
$421$	17.8083	0.867922	0.433961	+	0.900932i	$0.357116\pi$
$422$	0	0
$423$	25.5731	1.24341
$424$	0	0
$425$	3.02605	0.146785
$426$	0	0
$427$	5.70226	0.275952
$428$	0	0
$429$	4.16461	0.201069
$430$	0	0
$431$	−18.9057	−0.910656	−0.455328	−	0.890324i	$-0.650478\pi$
$431$	−18.9057	−0.910656	−0.455328	+	0.890324i	$0.650478\pi$
$432$	0	0
$433$	4.41653	0.212245	0.106122	−	0.994353i	$-0.466156\pi$
$433$	4.41653	0.212245	0.106122	+	0.994353i	$0.466156\pi$
$434$	0	0
$435$	8.62590	0.413580
$436$	0	0
$437$	8.63350	0.412996
$438$	0	0
$439$	24.8955	1.18820	0.594099	−	0.804392i	$-0.297509\pi$
$439$	24.8955	1.18820	0.594099	+	0.804392i	$0.297509\pi$
$440$	0	0
$441$	2.15928	0.102823
$442$	0	0
$443$	−20.6050	−0.978975	−0.489487	−	0.872010i	$-0.662816\pi$
$443$	−20.6050	−0.978975	−0.489487	+	0.872010i	$0.662816\pi$
$444$	0	0
$445$	−14.1176	−0.669238
$446$	0	0
$447$	−13.8083	−0.653111
$448$	0	0
$449$	19.5452	0.922396	0.461198	−	0.887297i	$-0.347420\pi$
$449$	19.5452	0.922396	0.461198	+	0.887297i	$0.347420\pi$
$450$	0	0
$451$	−9.16581	−0.431601
$452$	0	0
$453$	−19.3544	−0.909347
$454$	0	0
$455$	13.0965	0.613975
$456$	0	0
$457$	3.66545	0.171462	0.0857312	−	0.996318i	$-0.472677\pi$
$457$	3.66545	0.171462	0.0857312	+	0.996318i	$0.472677\pi$
$458$	0	0
$459$	4.54172	0.211989
$460$	0	0
$461$	−24.5705	−1.14436	−0.572181	−	0.820127i	$-0.693903\pi$
$461$	−24.5705	−1.14436	−0.572181	+	0.820127i	$0.693903\pi$
$462$	0	0
$463$	25.5385	1.18688	0.593439	−	0.804879i	$-0.297770\pi$
$463$	25.5385	1.18688	0.593439	+	0.804879i	$0.297770\pi$
$464$	0	0
$465$	−8.40519	−0.389781
$466$	0	0
$467$	−19.2767	−0.892020	−0.446010	−	0.895028i	$-0.647155\pi$
$467$	−19.2767	−0.892020	−0.446010	+	0.895028i	$0.647155\pi$
$468$	0	0
$469$	36.2194	1.67245
$470$	0	0
$471$	−17.1869	−0.791930
$472$	0	0
$473$	11.6980	0.537873
$474$	0	0
$475$	−19.7401	−0.905740
$476$	0	0
$477$	27.3952	1.25434
$478$	0	0
$479$	7.38871	0.337599	0.168799	−	0.985650i	$-0.446011\pi$
$479$	7.38871	0.337599	0.168799	+	0.985650i	$0.446011\pi$
$480$	0	0
$481$	24.2075	1.10377
$482$	0	0
$483$	−2.81288	−0.127990
$484$	0	0
$485$	9.37469	0.425683
$486$	0	0
$487$	−31.1498	−1.41153	−0.705767	−	0.708444i	$-0.749397\pi$
$487$	−31.1498	−1.41153	−0.705767	+	0.708444i	$0.749397\pi$
$488$	0	0
$489$	−2.39566	−0.108336
$490$	0	0
$491$	−12.5165	−0.564860	−0.282430	−	0.959288i	$-0.591140\pi$
$491$	−12.5165	−0.564860	−0.282430	+	0.959288i	$0.591140\pi$
$492$	0	0
$493$	7.10007	0.319771
$494$	0	0
$495$	4.01851	0.180618
$496$	0	0
$497$	6.30348	0.282750
$498$	0	0
$499$	0.205806	0.00921315	0.00460657	−	0.999989i	$-0.498534\pi$
$499$	0.205806	0.00921315	0.00460657	+	0.999989i	$0.498534\pi$
$500$	0	0
$501$	21.4434	0.958022
$502$	0	0
$503$	−10.5155	−0.468863	−0.234431	−	0.972133i	$-0.575323\pi$
$503$	−10.5155	−0.468863	−0.234431	+	0.972133i	$0.575323\pi$
$504$	0	0
$505$	−18.8386	−0.838308
$506$	0	0
$507$	1.19583	0.0531086
$508$	0	0
$509$	−13.6417	−0.604658	−0.302329	−	0.953204i	$-0.597764\pi$
$509$	−13.6417	−0.604658	−0.302329	+	0.953204i	$0.597764\pi$
$510$	0	0
$511$	−11.9360	−0.528016
$512$	0	0
$513$	−29.6275	−1.30808
$514$	0	0
$515$	−15.0802	−0.664515
$516$	0	0
$517$	−14.4192	−0.634154
$518$	0	0
$519$	−9.42448	−0.413689
$520$	0	0
$521$	8.08894	0.354383	0.177191	−	0.984176i	$-0.443299\pi$
$521$	8.08894	0.354383	0.177191	+	0.984176i	$0.443299\pi$
$522$	0	0
$523$	9.32178	0.407613	0.203807	−	0.979011i	$-0.434669\pi$
$523$	9.32178	0.407613	0.203807	+	0.979011i	$0.434669\pi$
$524$	0	0
$525$	6.43153	0.280695
$526$	0	0
$527$	−6.91840	−0.301370
$528$	0	0
$529$	−21.2484	−0.923845
$530$	0	0
$531$	−2.25227	−0.0977400
$532$	0	0
$533$	−27.3727	−1.18564
$534$	0	0
$535$	22.0882	0.954956
$536$	0	0
$537$	−1.46452	−0.0631986
$538$	0	0
$539$	−1.21749	−0.0524411
$540$	0	0
$541$	−0.687310	−0.0295498	−0.0147749	−	0.999891i	$-0.504703\pi$
$541$	−0.687310	−0.0295498	−0.0147749	+	0.999891i	$0.504703\pi$
$542$	0	0
$543$	1.12193	0.0481466
$544$	0	0
$545$	15.2596	0.653647
$546$	0	0
$547$	−26.5431	−1.13490	−0.567450	−	0.823408i	$-0.692070\pi$
$547$	−26.5431	−1.13490	−0.567450	+	0.823408i	$0.692070\pi$
$548$	0	0
$549$	5.22519	0.223006
$550$	0	0
$551$	−46.3166	−1.97315
$552$	0	0
$553$	−22.6747	−0.964225
$554$	0	0
$555$	−7.75474	−0.329171
$556$	0	0
$557$	9.41792	0.399050	0.199525	−	0.979893i	$-0.436060\pi$
$557$	9.41792	0.399050	0.199525	+	0.979893i	$0.436060\pi$
$558$	0	0
$559$	34.9347	1.47758
$560$	0	0
$561$	−1.09812	−0.0463627
$562$	0	0
$563$	−25.8884	−1.09107	−0.545533	−	0.838090i	$-0.683673\pi$
$563$	−25.8884	−1.09107	−0.545533	+	0.838090i	$0.683673\pi$
$564$	0	0
$565$	17.1149	0.720027
$566$	0	0
$567$	−6.95464	−0.292067
$568$	0	0
$569$	−28.6163	−1.19966	−0.599830	−	0.800128i	$-0.704765\pi$
$569$	−28.6163	−1.19966	−0.599830	+	0.800128i	$0.704765\pi$
$570$	0	0
$571$	−32.4796	−1.35923	−0.679614	−	0.733570i	$-0.737853\pi$
$571$	−32.4796	−1.35923	−0.679614	+	0.733570i	$0.737853\pi$
$572$	0	0
$573$	−0.402658	−0.0168213
$574$	0	0
$575$	−4.00487	−0.167015
$576$	0	0
$577$	−16.5525	−0.689091	−0.344546	−	0.938770i	$-0.611967\pi$
$577$	−16.5525	−0.689091	−0.344546	+	0.938770i	$0.611967\pi$
$578$	0	0
$579$	−11.3951	−0.473563
$580$	0	0
$581$	31.0190	1.28689
$582$	0	0
$583$	−15.4466	−0.639731
$584$	0	0
$585$	12.0008	0.496173
$586$	0	0
$587$	21.7234	0.896622	0.448311	−	0.893878i	$-0.352026\pi$
$587$	21.7234	0.896622	0.448311	+	0.893878i	$0.352026\pi$
$588$	0	0
$589$	45.1315	1.85961
$590$	0	0
$591$	18.4965	0.760846
$592$	0	0
$593$	21.4114	0.879260	0.439630	−	0.898179i	$-0.355109\pi$
$593$	21.4114	0.879260	0.439630	+	0.898179i	$0.355109\pi$
$594$	0	0
$595$	−3.45329	−0.141571
$596$	0	0
$597$	8.18128	0.334837
$598$	0	0
$599$	26.7962	1.09486	0.547431	−	0.836851i	$-0.315606\pi$
$599$	26.7962	1.09486	0.547431	+	0.836851i	$0.315606\pi$
$600$	0	0
$601$	22.5975	0.921772	0.460886	−	0.887459i	$-0.347532\pi$
$601$	22.5975	0.921772	0.460886	+	0.887459i	$0.347532\pi$
$602$	0	0
$603$	33.1891	1.35157
$604$	0	0
$605$	13.1889	0.536205
$606$	0	0
$607$	−29.8772	−1.21268	−0.606340	−	0.795206i	$-0.707363\pi$
$607$	−29.8772	−1.21268	−0.606340	+	0.795206i	$0.707363\pi$
$608$	0	0
$609$	15.0904	0.611494
$610$	0	0
$611$	−43.0613	−1.74207
$612$	0	0
$613$	47.4094	1.91485	0.957423	−	0.288688i	$-0.0932190\pi$
$613$	47.4094	1.91485	0.957423	+	0.288688i	$0.0932190\pi$
$614$	0	0
$615$	8.76871	0.353588
$616$	0	0
$617$	−30.4721	−1.22676	−0.613380	−	0.789788i	$-0.710190\pi$
$617$	−30.4721	−1.22676	−0.613380	+	0.789788i	$0.710190\pi$
$618$	0	0
$619$	−12.9767	−0.521577	−0.260788	−	0.965396i	$-0.583982\pi$
$619$	−12.9767	−0.521577	−0.260788	+	0.965396i	$0.583982\pi$
$620$	0	0
$621$	−6.01081	−0.241205
$622$	0	0
$623$	−24.6977	−0.989493
$624$	0	0
$625$	−0.712781	−0.0285112
$626$	0	0
$627$	7.16349	0.286082
$628$	0	0
$629$	−6.38301	−0.254507
$630$	0	0
$631$	−6.03775	−0.240359	−0.120179	−	0.992752i	$-0.538347\pi$
$631$	−6.03775	−0.240359	−0.120179	+	0.992752i	$0.538347\pi$
$632$	0	0
$633$	−4.56805	−0.181564
$634$	0	0
$635$	24.5511	0.974280
$636$	0	0
$637$	−3.63590	−0.144060
$638$	0	0
$639$	5.77611	0.228499
$640$	0	0
$641$	16.3185	0.644540	0.322270	−	0.946648i	$-0.395554\pi$
$641$	16.3185	0.644540	0.322270	+	0.946648i	$0.395554\pi$
$642$	0	0
$643$	−10.4448	−0.411904	−0.205952	−	0.978562i	$-0.566029\pi$
$643$	−10.4448	−0.411904	−0.205952	+	0.978562i	$0.566029\pi$
$644$	0	0
$645$	−11.1912	−0.440652
$646$	0	0
$647$	22.7049	0.892620	0.446310	−	0.894878i	$-0.352738\pi$
$647$	22.7049	0.892620	0.446310	+	0.894878i	$0.352738\pi$
$648$	0	0
$649$	1.26992	0.0498488
$650$	0	0
$651$	−14.7043	−0.576306
$652$	0	0
$653$	−11.2960	−0.442045	−0.221023	−	0.975269i	$-0.570939\pi$
$653$	−11.2960	−0.442045	−0.221023	+	0.975269i	$0.570939\pi$
$654$	0	0
$655$	11.2374	0.439083
$656$	0	0
$657$	−10.9374	−0.426707
$658$	0	0
$659$	−29.6163	−1.15369	−0.576843	−	0.816855i	$-0.695716\pi$
$659$	−29.6163	−1.15369	−0.576843	+	0.816855i	$0.695716\pi$
$660$	0	0
$661$	5.20393	0.202409	0.101205	−	0.994866i	$-0.467730\pi$
$661$	5.20393	0.202409	0.101205	+	0.994866i	$0.467730\pi$
$662$	0	0
$663$	−3.27942	−0.127362
$664$	0	0
$665$	22.5272	0.873567
$666$	0	0
$667$	−9.39669	−0.363841
$668$	0	0
$669$	19.4080	0.750356
$670$	0	0
$671$	−2.94618	−0.113736
$672$	0	0
$673$	−7.50186	−0.289175	−0.144588	−	0.989492i	$-0.546186\pi$
$673$	−7.50186	−0.289175	−0.144588	+	0.989492i	$0.546186\pi$
$674$	0	0
$675$	13.7435	0.528986
$676$	0	0
$677$	5.95163	0.228740	0.114370	−	0.993438i	$-0.463515\pi$
$677$	5.95163	0.228740	0.114370	+	0.993438i	$0.463515\pi$
$678$	0	0
$679$	16.4004	0.629388
$680$	0	0
$681$	−1.00806	−0.0386290
$682$	0	0
$683$	21.0962	0.807222	0.403611	−	0.914931i	$-0.367755\pi$
$683$	21.0962	0.807222	0.403611	+	0.914931i	$0.367755\pi$
$684$	0	0
$685$	−5.55283	−0.212163
$686$	0	0
$687$	−3.81952	−0.145724
$688$	0	0
$689$	−46.1294	−1.75739
$690$	0	0
$691$	23.8962	0.909053	0.454527	−	0.890733i	$-0.349808\pi$
$691$	23.8962	0.909053	0.454527	+	0.890733i	$0.349808\pi$
$692$	0	0
$693$	7.03010	0.267051
$694$	0	0
$695$	15.4174	0.584816
$696$	0	0
$697$	7.21762	0.273387
$698$	0	0
$699$	19.6426	0.742951
$700$	0	0
$701$	−26.5315	−1.00208	−0.501041	−	0.865424i	$-0.667049\pi$
$701$	−26.5315	−1.00208	−0.501041	+	0.865424i	$0.667049\pi$
$702$	0	0
$703$	41.6390	1.57044
$704$	0	0
$705$	13.7945	0.519530
$706$	0	0
$707$	−32.9569	−1.23947
$708$	0	0
$709$	27.9766	1.05068	0.525341	−	0.850892i	$-0.323938\pi$
$709$	27.9766	1.05068	0.525341	+	0.850892i	$0.323938\pi$
$710$	0	0
$711$	−20.7776	−0.779222
$712$	0	0
$713$	9.15626	0.342905
$714$	0	0
$715$	−6.76657	−0.253055
$716$	0	0
$717$	−10.3388	−0.386110
$718$	0	0
$719$	7.10331	0.264909	0.132454	−	0.991189i	$-0.457714\pi$
$719$	7.10331	0.264909	0.132454	+	0.991189i	$0.457714\pi$
$720$	0	0
$721$	−26.3818	−0.982510
$722$	0	0
$723$	11.1321	0.414007
$724$	0	0
$725$	21.4852	0.797939
$726$	0	0
$727$	17.9008	0.663903	0.331951	−	0.943296i	$-0.392293\pi$
$727$	17.9008	0.663903	0.331951	+	0.943296i	$0.392293\pi$
$728$	0	0
$729$	5.40910	0.200337
$730$	0	0
$731$	−9.21157	−0.340702
$732$	0	0
$733$	47.3404	1.74856	0.874278	−	0.485425i	$-0.161335\pi$
$733$	47.3404	1.74856	0.874278	+	0.485425i	$0.161335\pi$
$734$	0	0
$735$	1.16474	0.0429622
$736$	0	0
$737$	−18.7134	−0.689317
$738$	0	0
$739$	−12.2291	−0.449855	−0.224928	−	0.974375i	$-0.572215\pi$
$739$	−12.2291	−0.449855	−0.224928	+	0.974375i	$0.572215\pi$
$740$	0	0
$741$	21.3930	0.785891
$742$	0	0
$743$	−54.1701	−1.98731	−0.993654	−	0.112478i	$-0.964121\pi$
$743$	−54.1701	−1.98731	−0.993654	+	0.112478i	$0.964121\pi$
$744$	0	0
$745$	22.4355	0.821973
$746$	0	0
$747$	28.4239	1.03997
$748$	0	0
$749$	38.6418	1.41194
$750$	0	0
$751$	21.3321	0.778419	0.389209	−	0.921149i	$-0.372748\pi$
$751$	21.3321	0.778419	0.389209	+	0.921149i	$0.372748\pi$
$752$	0	0
$753$	−11.7841	−0.429437
$754$	0	0
$755$	31.4466	1.14446
$756$	0	0
$757$	−20.5162	−0.745676	−0.372838	−	0.927897i	$-0.621615\pi$
$757$	−20.5162	−0.745676	−0.372838	+	0.927897i	$0.621615\pi$
$758$	0	0
$759$	1.45333	0.0527524
$760$	0	0
$761$	35.0095	1.26909	0.634547	−	0.772884i	$-0.281187\pi$
$761$	35.0095	1.26909	0.634547	+	0.772884i	$0.281187\pi$
$762$	0	0
$763$	26.6955	0.966443
$764$	0	0
$765$	−3.16437	−0.114408
$766$	0	0
$767$	3.79248	0.136939
$768$	0	0
$769$	−47.7151	−1.72065	−0.860325	−	0.509745i	$-0.829740\pi$
$769$	−47.7151	−1.72065	−0.860325	+	0.509745i	$0.829740\pi$
$770$	0	0
$771$	−17.0009	−0.612271
$772$	0	0
$773$	13.0274	0.468564	0.234282	−	0.972169i	$-0.424726\pi$
$773$	13.0274	0.468564	0.234282	+	0.972169i	$0.424726\pi$
$774$	0	0
$775$	−20.9354	−0.752022
$776$	0	0
$777$	−13.5664	−0.486691
$778$	0	0
$779$	−47.0834	−1.68694
$780$	0	0
$781$	−3.25681	−0.116538
$782$	0	0
$783$	32.2465	1.15240
$784$	0	0
$785$	27.9249	0.996683
$786$	0	0
$787$	−50.7483	−1.80898	−0.904491	−	0.426493i	$-0.859749\pi$
$787$	−50.7483	−1.80898	−0.904491	+	0.426493i	$0.859749\pi$
$788$	0	0
$789$	4.74159	0.168805
$790$	0	0
$791$	29.9412	1.06459
$792$	0	0
$793$	−8.79845	−0.312442
$794$	0	0
$795$	14.7773	0.524098
$796$	0	0
$797$	30.9689	1.09698	0.548488	−	0.836159i	$-0.315204\pi$
$797$	30.9689	1.09698	0.548488	+	0.836159i	$0.315204\pi$
$798$	0	0
$799$	11.3544	0.401689
$800$	0	0
$801$	−22.6314	−0.799642
$802$	0	0
$803$	6.16694	0.217627
$804$	0	0
$805$	4.57031	0.161082
$806$	0	0
$807$	8.77247	0.308805
$808$	0	0
$809$	44.2100	1.55434	0.777171	−	0.629289i	$-0.216654\pi$
$809$	44.2100	1.55434	0.777171	+	0.629289i	$0.216654\pi$
$810$	0	0
$811$	43.4611	1.52612	0.763062	−	0.646325i	$-0.223695\pi$
$811$	43.4611	1.52612	0.763062	+	0.646325i	$0.223695\pi$
$812$	0	0
$813$	−3.06288	−0.107420
$814$	0	0
$815$	3.89243	0.136346
$816$	0	0
$817$	60.0908	2.10231
$818$	0	0
$819$	20.9946	0.733611
$820$	0	0
$821$	10.9757	0.383053	0.191527	−	0.981487i	$-0.438656\pi$
$821$	10.9757	0.383053	0.191527	+	0.981487i	$0.438656\pi$
$822$	0	0
$823$	−49.9430	−1.74090	−0.870452	−	0.492254i	$-0.836173\pi$
$823$	−49.9430	−1.74090	−0.870452	+	0.492254i	$0.836173\pi$
$824$	0	0
$825$	−3.32297	−0.115691
$826$	0	0
$827$	3.81677	0.132722	0.0663610	−	0.997796i	$-0.478861\pi$
$827$	3.81677	0.132722	0.0663610	+	0.997796i	$0.478861\pi$
$828$	0	0
$829$	28.8265	1.00119	0.500594	−	0.865682i	$-0.333115\pi$
$829$	28.8265	1.00119	0.500594	+	0.865682i	$0.333115\pi$
$830$	0	0
$831$	16.6178	0.576466
$832$	0	0
$833$	0.958714	0.0332175
$834$	0	0
$835$	−34.8409	−1.20572
$836$	0	0
$837$	−31.4214	−1.08608
$838$	0	0
$839$	28.2710	0.976023	0.488011	−	0.872837i	$-0.337722\pi$
$839$	28.2710	0.976023	0.488011	+	0.872837i	$0.337722\pi$
$840$	0	0
$841$	21.4110	0.738309
$842$	0	0
$843$	8.40253	0.289399
$844$	0	0
$845$	−1.94296	−0.0668398
$846$	0	0
$847$	23.0731	0.792800
$848$	0	0
$849$	4.70211	0.161376
$850$	0	0
$851$	8.44770	0.289583
$852$	0	0
$853$	55.3536	1.89527	0.947635	−	0.319354i	$-0.103466\pi$
$853$	55.3536	1.89527	0.947635	+	0.319354i	$0.103466\pi$
$854$	0	0
$855$	20.6425	0.705959
$856$	0	0
$857$	24.0285	0.820797	0.410399	−	0.911906i	$-0.365390\pi$
$857$	24.0285	0.820797	0.410399	+	0.911906i	$0.365390\pi$
$858$	0	0
$859$	−43.9380	−1.49914	−0.749572	−	0.661923i	$-0.769741\pi$
$859$	−43.9380	−1.49914	−0.749572	+	0.661923i	$0.769741\pi$
$860$	0	0
$861$	15.3402	0.522794
$862$	0	0
$863$	−39.5113	−1.34498	−0.672490	−	0.740106i	$-0.734775\pi$
$863$	−39.5113	−1.34498	−0.672490	+	0.740106i	$0.734775\pi$
$864$	0	0
$865$	15.3127	0.520648
$866$	0	0
$867$	0.864716	0.0293673
$868$	0	0
$869$	11.7153	0.397414
$870$	0	0
$871$	−55.8855	−1.89361
$872$	0	0
$873$	15.0282	0.508629
$874$	0	0
$875$	−27.7163	−0.936981
$876$	0	0
$877$	−56.9846	−1.92423	−0.962117	−	0.272638i	$-0.912104\pi$
$877$	−56.9846	−1.92423	−0.962117	+	0.272638i	$0.912104\pi$
$878$	0	0
$879$	−11.2313	−0.378823
$880$	0	0
$881$	2.94417	0.0991915	0.0495958	−	0.998769i	$-0.484207\pi$
$881$	2.94417	0.0991915	0.0495958	+	0.998769i	$0.484207\pi$
$882$	0	0
$883$	−53.9659	−1.81610	−0.908049	−	0.418864i	$-0.862428\pi$
$883$	−53.9659	−1.81610	−0.908049	+	0.418864i	$0.862428\pi$
$884$	0	0
$885$	−1.21490	−0.0408385
$886$	0	0
$887$	34.3259	1.15255	0.576276	−	0.817255i	$-0.304505\pi$
$887$	34.3259	1.15255	0.576276	+	0.817255i	$0.304505\pi$
$888$	0	0
$889$	42.9504	1.44051
$890$	0	0
$891$	3.59325	0.120378
$892$	0	0
$893$	−74.0692	−2.47863
$894$	0	0
$895$	2.37952	0.0795385
$896$	0	0
$897$	4.34020	0.144915
$898$	0	0
$899$	−49.1211	−1.63828
$900$	0	0
$901$	12.1634	0.405221
$902$	0	0
$903$	−19.5782	−0.651520
$904$	0	0
$905$	−1.82289	−0.0605949
$906$	0	0
$907$	49.9132	1.65734	0.828671	−	0.559736i	$-0.189097\pi$
$907$	49.9132	1.65734	0.828671	+	0.559736i	$0.189097\pi$
$908$	0	0
$909$	−30.1996	−1.00166
$910$	0	0
$911$	13.2761	0.439857	0.219929	−	0.975516i	$-0.429418\pi$
$911$	13.2761	0.439857	0.219929	+	0.975516i	$0.429418\pi$
$912$	0	0
$913$	−16.0266	−0.530402
$914$	0	0
$915$	2.81854	0.0931781
$916$	0	0
$917$	19.6591	0.649200
$918$	0	0
$919$	−3.56725	−0.117673	−0.0588363	−	0.998268i	$-0.518739\pi$
$919$	−3.56725	−0.117673	−0.0588363	+	0.998268i	$0.518739\pi$
$920$	0	0
$921$	−11.9457	−0.393623
$922$	0	0
$923$	−9.72610	−0.320139
$924$	0	0
$925$	−19.3153	−0.635084
$926$	0	0
$927$	−24.1746	−0.793999
$928$	0	0
$929$	−35.8713	−1.17690	−0.588450	−	0.808534i	$-0.700262\pi$
$929$	−35.8713	−1.17690	−0.588450	+	0.808534i	$0.700262\pi$
$930$	0	0
$931$	−6.25408	−0.204969
$932$	0	0
$933$	−15.6088	−0.511010
$934$	0	0
$935$	1.78421	0.0583498
$936$	0	0
$937$	−2.19755	−0.0717907	−0.0358953	−	0.999356i	$-0.511428\pi$
$937$	−2.19755	−0.0717907	−0.0358953	+	0.999356i	$0.511428\pi$
$938$	0	0
$939$	−20.6578	−0.674143
$940$	0	0
$941$	−24.1909	−0.788599	−0.394300	−	0.918982i	$-0.629013\pi$
$941$	−24.1909	−0.788599	−0.394300	+	0.918982i	$0.629013\pi$
$942$	0	0
$943$	−9.55226	−0.311064
$944$	0	0
$945$	−15.6839	−0.510196
$946$	0	0
$947$	12.3571	0.401552	0.200776	−	0.979637i	$-0.435654\pi$
$947$	12.3571	0.401552	0.200776	+	0.979637i	$0.435654\pi$
$948$	0	0
$949$	18.4169	0.597838
$950$	0	0
$951$	7.39256	0.239720
$952$	0	0
$953$	54.8634	1.77720	0.888600	−	0.458682i	$-0.151678\pi$
$953$	54.8634	1.77720	0.888600	+	0.458682i	$0.151678\pi$
$954$	0	0
$955$	0.654230	0.0211704
$956$	0	0
$957$	−7.79674	−0.252033
$958$	0	0
$959$	−9.71428	−0.313691
$960$	0	0
$961$	16.8642	0.544008
$962$	0	0
$963$	35.4088	1.14103
$964$	0	0
$965$	18.5145	0.596003
$966$	0	0
$967$	5.44333	0.175046	0.0875228	−	0.996163i	$-0.472105\pi$
$967$	5.44333	0.175046	0.0875228	+	0.996163i	$0.472105\pi$
$968$	0	0
$969$	−5.64089	−0.181212
$970$	0	0
$971$	−19.3852	−0.622102	−0.311051	−	0.950393i	$-0.600681\pi$
$971$	−19.3852	−0.622102	−0.311051	+	0.950393i	$0.600681\pi$
$972$	0	0
$973$	26.9717	0.864672
$974$	0	0
$975$	−9.92368	−0.317812
$976$	0	0
$977$	−30.0326	−0.960827	−0.480413	−	0.877042i	$-0.659513\pi$
$977$	−30.0326	−0.960827	−0.480413	+	0.877042i	$0.659513\pi$
$978$	0	0
$979$	12.7605	0.407829
$980$	0	0
$981$	24.4621	0.781014
$982$	0	0
$983$	23.9831	0.764943	0.382471	−	0.923967i	$-0.375073\pi$
$983$	23.9831	0.764943	0.382471	+	0.923967i	$0.375073\pi$
$984$	0	0
$985$	−30.0528	−0.957562
$986$	0	0
$987$	24.1325	0.768144
$988$	0	0
$989$	12.1912	0.387657
$990$	0	0
$991$	44.0186	1.39830	0.699148	−	0.714977i	$-0.253563\pi$
$991$	44.0186	1.39830	0.699148	+	0.714977i	$0.253563\pi$
$992$	0	0
$993$	10.2620	0.325656
$994$	0	0
$995$	−13.2928	−0.421409
$996$	0	0
$997$	57.1800	1.81091	0.905454	−	0.424445i	$-0.139531\pi$
$997$	57.1800	1.81091	0.905454	+	0.424445i	$0.139531\pi$
$998$	0	0
$999$	−28.9899	−0.917198

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.g.1.9	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.g.1.9	✓	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)

\(f(q)\)	\(=\)	\(q+0.864716 q^{3} -1.40497 q^{5} -2.45790 q^{7} -2.25227 q^{9} +O(q^{10})\)	\(q+0.864716 q^{3} -1.40497 q^{5} -2.45790 q^{7} -2.25227 q^{9} +1.26992 q^{11} +3.79248 q^{13} -1.21490 q^{15} -1.00000 q^{17} +6.52341 q^{19} -2.12539 q^{21} +1.32346 q^{23} -3.02605 q^{25} -4.54172 q^{27} -7.10007 q^{29} +6.91840 q^{31} +1.09812 q^{33} +3.45329 q^{35} +6.38301 q^{37} +3.27942 q^{39} -7.21762 q^{41} +9.21157 q^{43} +3.16437 q^{45} -11.3544 q^{47} -0.958714 q^{49} -0.864716 q^{51} -12.1634 q^{53} -1.78421 q^{55} +5.64089 q^{57} +1.00000 q^{59} -2.31997 q^{61} +5.53585 q^{63} -5.32834 q^{65} -14.7359 q^{67} +1.14442 q^{69} -2.56458 q^{71} +4.85616 q^{73} -2.61667 q^{75} -3.12134 q^{77} +9.22521 q^{79} +2.82950 q^{81} -12.6201 q^{83} +1.40497 q^{85} -6.13954 q^{87} +10.0483 q^{89} -9.32155 q^{91} +5.98245 q^{93} -9.16521 q^{95} -6.67250 q^{97} -2.86020 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 3 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 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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