LMFDB - Embedded newform 6040.2.a.o.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 19 x^{13} + 119 x^{12} + 106 x^{11} - 1063 x^{10} - 48 x^{9} + 4510 x^{8} + \cdots + 272 $ x^15 - 5x^14 - 19x^13 + 119x^12 + 106x^11 - 1063x^10 - 48x^9 + 4510x^8 - 1130x^7 - 9512x^6 + 2839x^5 + 9383x^4 - 1781x^3 - 3692x^2 - 92x + 272
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			1.6
Root			$-0.532237$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.532237 q^{3} +1.00000 q^{5} -2.74903 q^{7} -2.71672 q^{9} +O(q^{10})$	$q-0.532237 q^{3} +1.00000 q^{5} -2.74903 q^{7} -2.71672 q^{9} +2.41342 q^{11} -3.48336 q^{13} -0.532237 q^{15} -4.76610 q^{17} -4.99078 q^{19} +1.46314 q^{21} -5.81301 q^{23} +1.00000 q^{25} +3.04265 q^{27} -2.34054 q^{29} -2.40404 q^{31} -1.28451 q^{33} -2.74903 q^{35} -1.48188 q^{37} +1.85397 q^{39} +4.60153 q^{41} +10.5086 q^{43} -2.71672 q^{45} +4.23084 q^{47} +0.557158 q^{49} +2.53669 q^{51} +4.02381 q^{53} +2.41342 q^{55} +2.65628 q^{57} -3.58878 q^{59} +6.68215 q^{61} +7.46835 q^{63} -3.48336 q^{65} -8.84968 q^{67} +3.09390 q^{69} +14.5781 q^{71} +3.84460 q^{73} -0.532237 q^{75} -6.63455 q^{77} +3.63207 q^{79} +6.53076 q^{81} +1.55612 q^{83} -4.76610 q^{85} +1.24572 q^{87} +11.4073 q^{89} +9.57585 q^{91} +1.27952 q^{93} -4.99078 q^{95} -6.57647 q^{97} -6.55659 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * 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
$3$	−0.532237	−0.307287	−0.153644	−	0.988126i	$-0.549101\pi$
$3$	−0.532237	−0.307287	−0.153644	+	0.988126i	$0.549101\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.74903	−1.03904	−0.519518	−	0.854460i	$-0.673888\pi$
$7$	−2.74903	−1.03904	−0.519518	+	0.854460i	$0.673888\pi$
$8$	0	0
$9$	−2.71672	−0.905574
$10$	0	0
$11$	2.41342	0.727672	0.363836	−	0.931463i	$-0.381467\pi$
$11$	2.41342	0.727672	0.363836	+	0.931463i	$0.381467\pi$
$12$	0	0
$13$	−3.48336	−0.966109	−0.483055	−	0.875590i	$-0.660473\pi$
$13$	−3.48336	−0.966109	−0.483055	+	0.875590i	$0.660473\pi$
$14$	0	0
$15$	−0.532237	−0.137423
$16$	0	0
$17$	−4.76610	−1.15595	−0.577974	−	0.816055i	$-0.696157\pi$
$17$	−4.76610	−1.15595	−0.577974	+	0.816055i	$0.696157\pi$
$18$	0	0
$19$	−4.99078	−1.14496	−0.572482	−	0.819917i	$-0.694019\pi$
$19$	−4.99078	−1.14496	−0.572482	+	0.819917i	$0.694019\pi$
$20$	0	0
$21$	1.46314	0.319282
$22$	0	0
$23$	−5.81301	−1.21210	−0.606048	−	0.795428i	$-0.707246\pi$
$23$	−5.81301	−1.21210	−0.606048	+	0.795428i	$0.707246\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.04265	0.585559
$28$	0	0
$29$	−2.34054	−0.434627	−0.217314	−	0.976102i	$-0.569729\pi$
$29$	−2.34054	−0.434627	−0.217314	+	0.976102i	$0.569729\pi$
$30$	0	0
$31$	−2.40404	−0.431779	−0.215889	−	0.976418i	$-0.569265\pi$
$31$	−2.40404	−0.431779	−0.215889	+	0.976418i	$0.569265\pi$
$32$	0	0
$33$	−1.28451	−0.223605
$34$	0	0
$35$	−2.74903	−0.464671
$36$	0	0
$37$	−1.48188	−0.243620	−0.121810	−	0.992553i	$-0.538870\pi$
$37$	−1.48188	−0.243620	−0.121810	+	0.992553i	$0.538870\pi$
$38$	0	0
$39$	1.85397	0.296873
$40$	0	0
$41$	4.60153	0.718637	0.359319	−	0.933215i	$-0.383009\pi$
$41$	4.60153	0.718637	0.359319	+	0.933215i	$0.383009\pi$
$42$	0	0
$43$	10.5086	1.60254	0.801272	−	0.598301i	$-0.204157\pi$
$43$	10.5086	1.60254	0.801272	+	0.598301i	$0.204157\pi$
$44$	0	0
$45$	−2.71672	−0.404985
$46$	0	0
$47$	4.23084	0.617132	0.308566	−	0.951203i	$-0.400151\pi$
$47$	4.23084	0.617132	0.308566	+	0.951203i	$0.400151\pi$
$48$	0	0
$49$	0.557158	0.0795940
$50$	0	0
$51$	2.53669	0.355208
$52$	0	0
$53$	4.02381	0.552713	0.276356	−	0.961055i	$-0.410873\pi$
$53$	4.02381	0.552713	0.276356	+	0.961055i	$0.410873\pi$
$54$	0	0
$55$	2.41342	0.325425
$56$	0	0
$57$	2.65628	0.351833
$58$	0	0
$59$	−3.58878	−0.467219	−0.233610	−	0.972330i	$-0.575054\pi$
$59$	−3.58878	−0.467219	−0.233610	+	0.972330i	$0.575054\pi$
$60$	0	0
$61$	6.68215	0.855561	0.427781	−	0.903883i	$-0.359296\pi$
$61$	6.68215	0.855561	0.427781	+	0.903883i	$0.359296\pi$
$62$	0	0
$63$	7.46835	0.940924
$64$	0	0
$65$	−3.48336	−0.432057
$66$	0	0
$67$	−8.84968	−1.08116	−0.540580	−	0.841293i	$-0.681795\pi$
$67$	−8.84968	−1.08116	−0.540580	+	0.841293i	$0.681795\pi$
$68$	0	0
$69$	3.09390	0.372462
$70$	0	0
$71$	14.5781	1.73010	0.865051	−	0.501684i	$-0.167286\pi$
$71$	14.5781	1.73010	0.865051	+	0.501684i	$0.167286\pi$
$72$	0	0
$73$	3.84460	0.449976	0.224988	−	0.974362i	$-0.427766\pi$
$73$	3.84460	0.449976	0.224988	+	0.974362i	$0.427766\pi$
$74$	0	0
$75$	−0.532237	−0.0614575
$76$	0	0
$77$	−6.63455	−0.756077
$78$	0	0
$79$	3.63207	0.408640	0.204320	−	0.978904i	$-0.434502\pi$
$79$	3.63207	0.408640	0.204320	+	0.978904i	$0.434502\pi$
$80$	0	0
$81$	6.53076	0.725640
$82$	0	0
$83$	1.55612	0.170806	0.0854032	−	0.996346i	$-0.472782\pi$
$83$	1.55612	0.170806	0.0854032	+	0.996346i	$0.472782\pi$
$84$	0	0
$85$	−4.76610	−0.516956
$86$	0	0
$87$	1.24572	0.133555
$88$	0	0
$89$	11.4073	1.20918	0.604588	−	0.796538i	$-0.293338\pi$
$89$	11.4073	1.20918	0.604588	+	0.796538i	$0.293338\pi$
$90$	0	0
$91$	9.57585	1.00382
$92$	0	0
$93$	1.27952	0.132680
$94$	0	0
$95$	−4.99078	−0.512043
$96$	0	0
$97$	−6.57647	−0.667739	−0.333869	−	0.942619i	$-0.608354\pi$
$97$	−6.57647	−0.667739	−0.333869	+	0.942619i	$0.608354\pi$
$98$	0	0
$99$	−6.55659	−0.658962
$100$	0	0
$101$	−1.09895	−0.109349	−0.0546746	−	0.998504i	$-0.517412\pi$
$101$	−1.09895	−0.109349	−0.0546746	+	0.998504i	$0.517412\pi$
$102$	0	0
$103$	9.73927	0.959639	0.479819	−	0.877367i	$-0.340702\pi$
$103$	9.73927	0.959639	0.479819	+	0.877367i	$0.340702\pi$
$104$	0	0
$105$	1.46314	0.142787
$106$	0	0
$107$	−7.15541	−0.691740	−0.345870	−	0.938282i	$-0.612416\pi$
$107$	−7.15541	−0.691740	−0.345870	+	0.938282i	$0.612416\pi$
$108$	0	0
$109$	−0.834476	−0.0799283	−0.0399641	−	0.999201i	$-0.512724\pi$
$109$	−0.834476	−0.0799283	−0.0399641	+	0.999201i	$0.512724\pi$
$110$	0	0
$111$	0.788713	0.0748614
$112$	0	0
$113$	−13.3128	−1.25236	−0.626181	−	0.779678i	$-0.715383\pi$
$113$	−13.3128	−1.25236	−0.626181	+	0.779678i	$0.715383\pi$
$114$	0	0
$115$	−5.81301	−0.542066
$116$	0	0
$117$	9.46331	0.874884
$118$	0	0
$119$	13.1021	1.20107
$120$	0	0
$121$	−5.17542	−0.470493
$122$	0	0
$123$	−2.44910	−0.220828
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	20.5827	1.82642	0.913210	−	0.407489i	$-0.133596\pi$
$127$	20.5827	1.82642	0.913210	+	0.407489i	$0.133596\pi$
$128$	0	0
$129$	−5.59306	−0.492441
$130$	0	0
$131$	−7.44131	−0.650151	−0.325075	−	0.945688i	$-0.605390\pi$
$131$	−7.44131	−0.650151	−0.325075	+	0.945688i	$0.605390\pi$
$132$	0	0
$133$	13.7198	1.18966
$134$	0	0
$135$	3.04265	0.261870
$136$	0	0
$137$	−2.02856	−0.173311	−0.0866556	−	0.996238i	$-0.527618\pi$
$137$	−2.02856	−0.173311	−0.0866556	+	0.996238i	$0.527618\pi$
$138$	0	0
$139$	17.4292	1.47832	0.739162	−	0.673528i	$-0.235222\pi$
$139$	17.4292	1.47832	0.739162	+	0.673528i	$0.235222\pi$
$140$	0	0
$141$	−2.25181	−0.189637
$142$	0	0
$143$	−8.40679	−0.703011
$144$	0	0
$145$	−2.34054	−0.194371
$146$	0	0
$147$	−0.296540	−0.0244582
$148$	0	0
$149$	3.21777	0.263610	0.131805	−	0.991276i	$-0.457923\pi$
$149$	3.21777	0.263610	0.131805	+	0.991276i	$0.457923\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	12.9482	1.04680
$154$	0	0
$155$	−2.40404	−0.193097
$156$	0	0
$157$	−22.6794	−1.81001	−0.905005	−	0.425401i	$-0.860133\pi$
$157$	−22.6794	−1.81001	−0.905005	+	0.425401i	$0.860133\pi$
$158$	0	0
$159$	−2.14162	−0.169842
$160$	0	0
$161$	15.9801	1.25941
$162$	0	0
$163$	−4.50542	−0.352891	−0.176446	−	0.984310i	$-0.556460\pi$
$163$	−4.50542	−0.352891	−0.176446	+	0.984310i	$0.556460\pi$
$164$	0	0
$165$	−1.28451	−0.0999990
$166$	0	0
$167$	4.14781	0.320967	0.160483	−	0.987039i	$-0.448695\pi$
$167$	4.14781	0.320967	0.160483	+	0.987039i	$0.448695\pi$
$168$	0	0
$169$	−0.866231	−0.0666331
$170$	0	0
$171$	13.5586	1.03685
$172$	0	0
$173$	−19.8577	−1.50975	−0.754877	−	0.655867i	$-0.772303\pi$
$173$	−19.8577	−1.50975	−0.754877	+	0.655867i	$0.772303\pi$
$174$	0	0
$175$	−2.74903	−0.207807
$176$	0	0
$177$	1.91008	0.143571
$178$	0	0
$179$	5.89988	0.440978	0.220489	−	0.975389i	$-0.429235\pi$
$179$	5.89988	0.440978	0.220489	+	0.975389i	$0.429235\pi$
$180$	0	0
$181$	13.1647	0.978521	0.489261	−	0.872138i	$-0.337267\pi$
$181$	13.1647	0.978521	0.489261	+	0.872138i	$0.337267\pi$
$182$	0	0
$183$	−3.55649	−0.262903
$184$	0	0
$185$	−1.48188	−0.108950
$186$	0	0
$187$	−11.5026	−0.841152
$188$	0	0
$189$	−8.36434	−0.608416
$190$	0	0
$191$	20.3467	1.47223	0.736117	−	0.676855i	$-0.236657\pi$
$191$	20.3467	1.47223	0.736117	+	0.676855i	$0.236657\pi$
$192$	0	0
$193$	3.41180	0.245587	0.122793	−	0.992432i	$-0.460815\pi$
$193$	3.41180	0.245587	0.122793	+	0.992432i	$0.460815\pi$
$194$	0	0
$195$	1.85397	0.132766
$196$	0	0
$197$	−12.8973	−0.918894	−0.459447	−	0.888205i	$-0.651952\pi$
$197$	−12.8973	−0.918894	−0.459447	+	0.888205i	$0.651952\pi$
$198$	0	0
$199$	1.57571	0.111699	0.0558496	−	0.998439i	$-0.482213\pi$
$199$	1.57571	0.111699	0.0558496	+	0.998439i	$0.482213\pi$
$200$	0	0
$201$	4.71013	0.332227
$202$	0	0
$203$	6.43421	0.451593
$204$	0	0
$205$	4.60153	0.321384
$206$	0	0
$207$	15.7923	1.09764
$208$	0	0
$209$	−12.0448	−0.833159
$210$	0	0
$211$	24.4746	1.68490	0.842451	−	0.538773i	$-0.181112\pi$
$211$	24.4746	1.68490	0.842451	+	0.538773i	$0.181112\pi$
$212$	0	0
$213$	−7.75901	−0.531638
$214$	0	0
$215$	10.5086	0.716679
$216$	0	0
$217$	6.60878	0.448633
$218$	0	0
$219$	−2.04624	−0.138272
$220$	0	0
$221$	16.6020	1.11677
$222$	0	0
$223$	11.0256	0.738331	0.369166	−	0.929364i	$-0.379644\pi$
$223$	11.0256	0.738331	0.369166	+	0.929364i	$0.379644\pi$
$224$	0	0
$225$	−2.71672	−0.181115
$226$	0	0
$227$	6.30852	0.418711	0.209356	−	0.977840i	$-0.432863\pi$
$227$	6.30852	0.418711	0.209356	+	0.977840i	$0.432863\pi$
$228$	0	0
$229$	−6.63215	−0.438265	−0.219132	−	0.975695i	$-0.570323\pi$
$229$	−6.63215	−0.438265	−0.219132	+	0.975695i	$0.570323\pi$
$230$	0	0
$231$	3.53116	0.232333
$232$	0	0
$233$	11.1606	0.731157	0.365578	−	0.930781i	$-0.380871\pi$
$233$	11.1606	0.731157	0.365578	+	0.930781i	$0.380871\pi$
$234$	0	0
$235$	4.23084	0.275990
$236$	0	0
$237$	−1.93312	−0.125570
$238$	0	0
$239$	−12.6206	−0.816360	−0.408180	−	0.912902i	$-0.633836\pi$
$239$	−12.6206	−0.816360	−0.408180	+	0.912902i	$0.633836\pi$
$240$	0	0
$241$	−18.1200	−1.16721	−0.583605	−	0.812038i	$-0.698358\pi$
$241$	−18.1200	−1.16721	−0.583605	+	0.812038i	$0.698358\pi$
$242$	0	0
$243$	−12.6039	−0.808539
$244$	0	0
$245$	0.557158	0.0355955
$246$	0	0
$247$	17.3847	1.10616
$248$	0	0
$249$	−0.828225	−0.0524866
$250$	0	0
$251$	14.9719	0.945020	0.472510	−	0.881325i	$-0.343348\pi$
$251$	14.9719	0.945020	0.472510	+	0.881325i	$0.343348\pi$
$252$	0	0
$253$	−14.0292	−0.882009
$254$	0	0
$255$	2.53669	0.158854
$256$	0	0
$257$	−12.1950	−0.760705	−0.380353	−	0.924842i	$-0.624197\pi$
$257$	−12.1950	−0.760705	−0.380353	+	0.924842i	$0.624197\pi$
$258$	0	0
$259$	4.07374	0.253130
$260$	0	0
$261$	6.35860	0.393587
$262$	0	0
$263$	−16.6885	−1.02906	−0.514528	−	0.857474i	$-0.672033\pi$
$263$	−16.6885	−1.02906	−0.514528	+	0.857474i	$0.672033\pi$
$264$	0	0
$265$	4.02381	0.247181
$266$	0	0
$267$	−6.07141	−0.371565
$268$	0	0
$269$	−5.98492	−0.364907	−0.182453	−	0.983214i	$-0.558404\pi$
$269$	−5.98492	−0.364907	−0.182453	+	0.983214i	$0.558404\pi$
$270$	0	0
$271$	30.4567	1.85011	0.925055	−	0.379833i	$-0.124018\pi$
$271$	30.4567	1.85011	0.925055	+	0.379833i	$0.124018\pi$
$272$	0	0
$273$	−5.09662	−0.308462
$274$	0	0
$275$	2.41342	0.145534
$276$	0	0
$277$	−0.844334	−0.0507311	−0.0253656	−	0.999678i	$-0.508075\pi$
$277$	−0.844334	−0.0507311	−0.0253656	+	0.999678i	$0.508075\pi$
$278$	0	0
$279$	6.53112	0.391008
$280$	0	0
$281$	−5.98472	−0.357019	−0.178509	−	0.983938i	$-0.557127\pi$
$281$	−5.98472	−0.357019	−0.178509	+	0.983938i	$0.557127\pi$
$282$	0	0
$283$	−0.874595	−0.0519893	−0.0259946	−	0.999662i	$-0.508275\pi$
$283$	−0.874595	−0.0519893	−0.0259946	+	0.999662i	$0.508275\pi$
$284$	0	0
$285$	2.65628	0.157344
$286$	0	0
$287$	−12.6497	−0.746690
$288$	0	0
$289$	5.71568	0.336217
$290$	0	0
$291$	3.50024	0.205188
$292$	0	0
$293$	−11.7192	−0.684643	−0.342322	−	0.939583i	$-0.611213\pi$
$293$	−11.7192	−0.684643	−0.342322	+	0.939583i	$0.611213\pi$
$294$	0	0
$295$	−3.58878	−0.208947
$296$	0	0
$297$	7.34319	0.426095
$298$	0	0
$299$	20.2488	1.17102
$300$	0	0
$301$	−28.8884	−1.66510
$302$	0	0
$303$	0.584900	0.0336016
$304$	0	0
$305$	6.68215	0.382619
$306$	0	0
$307$	9.90015	0.565031	0.282516	−	0.959263i	$-0.408831\pi$
$307$	9.90015	0.565031	0.282516	+	0.959263i	$0.408831\pi$
$308$	0	0
$309$	−5.18360	−0.294885
$310$	0	0
$311$	−21.2202	−1.20329	−0.601645	−	0.798764i	$-0.705488\pi$
$311$	−21.2202	−1.20329	−0.601645	+	0.798764i	$0.705488\pi$
$312$	0	0
$313$	6.46042	0.365164	0.182582	−	0.983191i	$-0.441554\pi$
$313$	6.46042	0.365164	0.182582	+	0.983191i	$0.441554\pi$
$314$	0	0
$315$	7.46835	0.420794
$316$	0	0
$317$	34.7269	1.95045	0.975227	−	0.221205i	$-0.0709989\pi$
$317$	34.7269	1.95045	0.975227	+	0.221205i	$0.0709989\pi$
$318$	0	0
$319$	−5.64870	−0.316266
$320$	0	0
$321$	3.80838	0.212563
$322$	0	0
$323$	23.7866	1.32352
$324$	0	0
$325$	−3.48336	−0.193222
$326$	0	0
$327$	0.444139	0.0245609
$328$	0	0
$329$	−11.6307	−0.641222
$330$	0	0
$331$	−4.38909	−0.241246	−0.120623	−	0.992698i	$-0.538489\pi$
$331$	−4.38909	−0.241246	−0.120623	+	0.992698i	$0.538489\pi$
$332$	0	0
$333$	4.02587	0.220616
$334$	0	0
$335$	−8.84968	−0.483509
$336$	0	0
$337$	−0.0114884	−0.000625811 0	−0.000312906	−	1.00000i	$-0.500100\pi$
$337$	−0.0114884	−0.000625811 0	−0.000312906		1.00000i	$0.500100\pi$
$338$	0	0
$339$	7.08556	0.384835
$340$	0	0
$341$	−5.80195	−0.314193
$342$	0	0
$343$	17.7116	0.956334
$344$	0	0
$345$	3.09390	0.166570
$346$	0	0
$347$	2.06953	0.111098	0.0555492	−	0.998456i	$-0.482309\pi$
$347$	2.06953	0.111098	0.0555492	+	0.998456i	$0.482309\pi$
$348$	0	0
$349$	24.6771	1.32093	0.660466	−	0.750856i	$-0.270359\pi$
$349$	24.6771	1.32093	0.660466	+	0.750856i	$0.270359\pi$
$350$	0	0
$351$	−10.5986	−0.565714
$352$	0	0
$353$	−4.82989	−0.257069	−0.128535	−	0.991705i	$-0.541027\pi$
$353$	−4.82989	−0.257069	−0.128535	+	0.991705i	$0.541027\pi$
$354$	0	0
$355$	14.5781	0.773725
$356$	0	0
$357$	−6.97345	−0.369074
$358$	0	0
$359$	2.43848	0.128698	0.0643491	−	0.997927i	$-0.479503\pi$
$359$	2.43848	0.128698	0.0643491	+	0.997927i	$0.479503\pi$
$360$	0	0
$361$	5.90790	0.310942
$362$	0	0
$363$	2.75455	0.144576
$364$	0	0
$365$	3.84460	0.201235
$366$	0	0
$367$	12.1822	0.635903	0.317952	−	0.948107i	$-0.397005\pi$
$367$	12.1822	0.635903	0.317952	+	0.948107i	$0.397005\pi$
$368$	0	0
$369$	−12.5011	−0.650780
$370$	0	0
$371$	−11.0616	−0.574288
$372$	0	0
$373$	2.86033	0.148102	0.0740512	−	0.997254i	$-0.476407\pi$
$373$	2.86033	0.148102	0.0740512	+	0.997254i	$0.476407\pi$
$374$	0	0
$375$	−0.532237	−0.0274846
$376$	0	0
$377$	8.15293	0.419897
$378$	0	0
$379$	7.38464	0.379323	0.189662	−	0.981850i	$-0.439261\pi$
$379$	7.38464	0.379323	0.189662	+	0.981850i	$0.439261\pi$
$380$	0	0
$381$	−10.9549	−0.561236
$382$	0	0
$383$	16.7497	0.855872	0.427936	−	0.903809i	$-0.359241\pi$
$383$	16.7497	0.855872	0.427936	+	0.903809i	$0.359241\pi$
$384$	0	0
$385$	−6.63455	−0.338128
$386$	0	0
$387$	−28.5489	−1.45122
$388$	0	0
$389$	2.66152	0.134944	0.0674722	−	0.997721i	$-0.478507\pi$
$389$	2.66152	0.134944	0.0674722	+	0.997721i	$0.478507\pi$
$390$	0	0
$391$	27.7054	1.40112
$392$	0	0
$393$	3.96054	0.199783
$394$	0	0
$395$	3.63207	0.182749
$396$	0	0
$397$	25.3099	1.27027	0.635134	−	0.772402i	$-0.280945\pi$
$397$	25.3099	1.27027	0.635134	+	0.772402i	$0.280945\pi$
$398$	0	0
$399$	−7.30219	−0.365567
$400$	0	0
$401$	−8.07787	−0.403390	−0.201695	−	0.979448i	$-0.564645\pi$
$401$	−8.07787	−0.403390	−0.201695	+	0.979448i	$0.564645\pi$
$402$	0	0
$403$	8.37413	0.417145
$404$	0	0
$405$	6.53076	0.324516
$406$	0	0
$407$	−3.57640	−0.177276
$408$	0	0
$409$	2.67373	0.132207	0.0661037	−	0.997813i	$-0.478943\pi$
$409$	2.67373	0.132207	0.0661037	+	0.997813i	$0.478943\pi$
$410$	0	0
$411$	1.07967	0.0532564
$412$	0	0
$413$	9.86566	0.485457
$414$	0	0
$415$	1.55612	0.0763869
$416$	0	0
$417$	−9.27646	−0.454270
$418$	0	0
$419$	−7.63434	−0.372962	−0.186481	−	0.982459i	$-0.559708\pi$
$419$	−7.63434	−0.372962	−0.186481	+	0.982459i	$0.559708\pi$
$420$	0	0
$421$	−8.17513	−0.398432	−0.199216	−	0.979956i	$-0.563839\pi$
$421$	−8.17513	−0.398432	−0.199216	+	0.979956i	$0.563839\pi$
$422$	0	0
$423$	−11.4940	−0.558859
$424$	0	0
$425$	−4.76610	−0.231190
$426$	0	0
$427$	−18.3694	−0.888958
$428$	0	0
$429$	4.47441	0.216026
$430$	0	0
$431$	−24.5539	−1.18272	−0.591361	−	0.806407i	$-0.701409\pi$
$431$	−24.5539	−1.18272	−0.591361	+	0.806407i	$0.701409\pi$
$432$	0	0
$433$	−9.66127	−0.464291	−0.232145	−	0.972681i	$-0.574575\pi$
$433$	−9.66127	−0.464291	−0.232145	+	0.972681i	$0.574575\pi$
$434$	0	0
$435$	1.24572	0.0597278
$436$	0	0
$437$	29.0115	1.38781
$438$	0	0
$439$	−18.3319	−0.874935	−0.437468	−	0.899234i	$-0.644125\pi$
$439$	−18.3319	−0.874935	−0.437468	+	0.899234i	$0.644125\pi$
$440$	0	0
$441$	−1.51364	−0.0720783
$442$	0	0
$443$	−16.1921	−0.769308	−0.384654	−	0.923061i	$-0.625679\pi$
$443$	−16.1921	−0.769308	−0.384654	+	0.923061i	$0.625679\pi$
$444$	0	0
$445$	11.4073	0.540760
$446$	0	0
$447$	−1.71262	−0.0810040
$448$	0	0
$449$	−0.701372	−0.0330998	−0.0165499	−	0.999863i	$-0.505268\pi$
$449$	−0.701372	−0.0330998	−0.0165499	+	0.999863i	$0.505268\pi$
$450$	0	0
$451$	11.1054	0.522933
$452$	0	0
$453$	0.532237	0.0250067
$454$	0	0
$455$	9.57585	0.448923
$456$	0	0
$457$	14.8406	0.694214	0.347107	−	0.937826i	$-0.387164\pi$
$457$	14.8406	0.694214	0.347107	+	0.937826i	$0.387164\pi$
$458$	0	0
$459$	−14.5016	−0.676876
$460$	0	0
$461$	−15.0630	−0.701554	−0.350777	−	0.936459i	$-0.614083\pi$
$461$	−15.0630	−0.701554	−0.350777	+	0.936459i	$0.614083\pi$
$462$	0	0
$463$	−7.93992	−0.368999	−0.184500	−	0.982833i	$-0.559066\pi$
$463$	−7.93992	−0.368999	−0.184500	+	0.982833i	$0.559066\pi$
$464$	0	0
$465$	1.27952	0.0593363
$466$	0	0
$467$	−18.6204	−0.861652	−0.430826	−	0.902435i	$-0.641778\pi$
$467$	−18.6204	−0.861652	−0.430826	+	0.902435i	$0.641778\pi$
$468$	0	0
$469$	24.3280	1.12336
$470$	0	0
$471$	12.0708	0.556193
$472$	0	0
$473$	25.3616	1.16613
$474$	0	0
$475$	−4.99078	−0.228993
$476$	0	0
$477$	−10.9316	−0.500523
$478$	0	0
$479$	−10.1466	−0.463608	−0.231804	−	0.972763i	$-0.574463\pi$
$479$	−10.1466	−0.463608	−0.231804	+	0.972763i	$0.574463\pi$
$480$	0	0
$481$	5.16192	0.235364
$482$	0	0
$483$	−8.50522	−0.387001
$484$	0	0
$485$	−6.57647	−0.298622
$486$	0	0
$487$	−1.26731	−0.0574274	−0.0287137	−	0.999588i	$-0.509141\pi$
$487$	−1.26731	−0.0574274	−0.0287137	+	0.999588i	$0.509141\pi$
$488$	0	0
$489$	2.39795	0.108439
$490$	0	0
$491$	1.33602	0.0602939	0.0301469	−	0.999545i	$-0.490402\pi$
$491$	1.33602	0.0602939	0.0301469	+	0.999545i	$0.490402\pi$
$492$	0	0
$493$	11.1552	0.502407
$494$	0	0
$495$	−6.55659	−0.294697
$496$	0	0
$497$	−40.0756	−1.79764
$498$	0	0
$499$	14.1541	0.633623	0.316811	−	0.948489i	$-0.397388\pi$
$499$	14.1541	0.633623	0.316811	+	0.948489i	$0.397388\pi$
$500$	0	0
$501$	−2.20762	−0.0986291
$502$	0	0
$503$	11.5909	0.516813	0.258406	−	0.966036i	$-0.416803\pi$
$503$	11.5909	0.516813	0.258406	+	0.966036i	$0.416803\pi$
$504$	0	0
$505$	−1.09895	−0.0489024
$506$	0	0
$507$	0.461040	0.0204755
$508$	0	0
$509$	10.3761	0.459910	0.229955	−	0.973201i	$-0.426142\pi$
$509$	10.3761	0.459910	0.229955	+	0.973201i	$0.426142\pi$
$510$	0	0
$511$	−10.5689	−0.467541
$512$	0	0
$513$	−15.1852	−0.670444
$514$	0	0
$515$	9.73927	0.429164
$516$	0	0
$517$	10.2108	0.449070
$518$	0	0
$519$	10.5690	0.463928
$520$	0	0
$521$	25.7826	1.12956	0.564778	−	0.825243i	$-0.308962\pi$
$521$	25.7826	1.12956	0.564778	+	0.825243i	$0.308962\pi$
$522$	0	0
$523$	−15.4266	−0.674559	−0.337280	−	0.941405i	$-0.609507\pi$
$523$	−15.4266	−0.674559	−0.337280	+	0.941405i	$0.609507\pi$
$524$	0	0
$525$	1.46314	0.0638565
$526$	0	0
$527$	11.4579	0.499114
$528$	0	0
$529$	10.7911	0.469176
$530$	0	0
$531$	9.74972	0.423102
$532$	0	0
$533$	−16.0288	−0.694282
$534$	0	0
$535$	−7.15541	−0.309355
$536$	0	0
$537$	−3.14014	−0.135507
$538$	0	0
$539$	1.34465	0.0579184
$540$	0	0
$541$	14.5722	0.626507	0.313253	−	0.949670i	$-0.398581\pi$
$541$	14.5722	0.626507	0.313253	+	0.949670i	$0.398581\pi$
$542$	0	0
$543$	−7.00672	−0.300687
$544$	0	0
$545$	−0.834476	−0.0357450
$546$	0	0
$547$	−37.7020	−1.61202	−0.806009	−	0.591903i	$-0.798377\pi$
$547$	−37.7020	−1.61202	−0.806009	+	0.591903i	$0.798377\pi$
$548$	0	0
$549$	−18.1535	−0.774774
$550$	0	0
$551$	11.6811	0.497632
$552$	0	0
$553$	−9.98466	−0.424591
$554$	0	0
$555$	0.788713	0.0334790
$556$	0	0
$557$	−41.4018	−1.75425	−0.877125	−	0.480261i	$-0.840542\pi$
$557$	−41.4018	−1.75425	−0.877125	+	0.480261i	$0.840542\pi$
$558$	0	0
$559$	−36.6051	−1.54823
$560$	0	0
$561$	6.12210	0.258475
$562$	0	0
$563$	−41.5395	−1.75068	−0.875341	−	0.483507i	$-0.839363\pi$
$563$	−41.5395	−1.75068	−0.875341	+	0.483507i	$0.839363\pi$
$564$	0	0
$565$	−13.3128	−0.560073
$566$	0	0
$567$	−17.9532	−0.753965
$568$	0	0
$569$	40.2083	1.68562	0.842809	−	0.538213i	$-0.180900\pi$
$569$	40.2083	1.68562	0.842809	+	0.538213i	$0.180900\pi$
$570$	0	0
$571$	33.2482	1.39140	0.695698	−	0.718335i	$-0.255095\pi$
$571$	33.2482	1.39140	0.695698	+	0.718335i	$0.255095\pi$
$572$	0	0
$573$	−10.8293	−0.452399
$574$	0	0
$575$	−5.81301	−0.242419
$576$	0	0
$577$	−20.4547	−0.851538	−0.425769	−	0.904832i	$-0.639996\pi$
$577$	−20.4547	−0.851538	−0.425769	+	0.904832i	$0.639996\pi$
$578$	0	0
$579$	−1.81589	−0.0754657
$580$	0	0
$581$	−4.27782	−0.177474
$582$	0	0
$583$	9.71113	0.402194
$584$	0	0
$585$	9.46331	0.391260
$586$	0	0
$587$	25.1157	1.03664	0.518319	−	0.855187i	$-0.326558\pi$
$587$	25.1157	1.03664	0.518319	+	0.855187i	$0.326558\pi$
$588$	0	0
$589$	11.9980	0.494371
$590$	0	0
$591$	6.86442	0.282364
$592$	0	0
$593$	17.5269	0.719745	0.359873	−	0.933001i	$-0.382820\pi$
$593$	17.5269	0.719745	0.359873	+	0.933001i	$0.382820\pi$
$594$	0	0
$595$	13.1021	0.537135
$596$	0	0
$597$	−0.838652	−0.0343237
$598$	0	0
$599$	32.1769	1.31471	0.657356	−	0.753580i	$-0.271675\pi$
$599$	32.1769	1.31471	0.657356	+	0.753580i	$0.271675\pi$
$600$	0	0
$601$	−36.1845	−1.47600	−0.737998	−	0.674803i	$-0.764229\pi$
$601$	−36.1845	−1.47600	−0.737998	+	0.674803i	$0.764229\pi$
$602$	0	0
$603$	24.0421	0.979071
$604$	0	0
$605$	−5.17542	−0.210411
$606$	0	0
$607$	20.7057	0.840418	0.420209	−	0.907427i	$-0.361957\pi$
$607$	20.7057	0.840418	0.420209	+	0.907427i	$0.361957\pi$
$608$	0	0
$609$	−3.42453	−0.138769
$610$	0	0
$611$	−14.7375	−0.596217
$612$	0	0
$613$	34.1374	1.37880	0.689399	−	0.724382i	$-0.257875\pi$
$613$	34.1374	1.37880	0.689399	+	0.724382i	$0.257875\pi$
$614$	0	0
$615$	−2.44910	−0.0987574
$616$	0	0
$617$	7.43079	0.299152	0.149576	−	0.988750i	$-0.452209\pi$
$617$	7.43079	0.299152	0.149576	+	0.988750i	$0.452209\pi$
$618$	0	0
$619$	4.76603	0.191563	0.0957814	−	0.995402i	$-0.469465\pi$
$619$	4.76603	0.191563	0.0957814	+	0.995402i	$0.469465\pi$
$620$	0	0
$621$	−17.6870	−0.709754
$622$	0	0
$623$	−31.3591	−1.25638
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.41071	0.256019
$628$	0	0
$629$	7.06280	0.281612
$630$	0	0
$631$	37.9455	1.51059	0.755293	−	0.655387i	$-0.227495\pi$
$631$	37.9455	1.51059	0.755293	+	0.655387i	$0.227495\pi$
$632$	0	0
$633$	−13.0263	−0.517749
$634$	0	0
$635$	20.5827	0.816800
$636$	0	0
$637$	−1.94078	−0.0768965
$638$	0	0
$639$	−39.6047	−1.56674
$640$	0	0
$641$	44.1334	1.74317	0.871583	−	0.490248i	$-0.163094\pi$
$641$	44.1334	1.74317	0.871583	+	0.490248i	$0.163094\pi$
$642$	0	0
$643$	−6.03660	−0.238060	−0.119030	−	0.992891i	$-0.537979\pi$
$643$	−6.03660	−0.238060	−0.119030	+	0.992891i	$0.537979\pi$
$644$	0	0
$645$	−5.59306	−0.220226
$646$	0	0
$647$	−5.22362	−0.205362	−0.102681	−	0.994714i	$-0.532742\pi$
$647$	−5.22362	−0.205362	−0.102681	+	0.994714i	$0.532742\pi$
$648$	0	0
$649$	−8.66122	−0.339983
$650$	0	0
$651$	−3.51744	−0.137859
$652$	0	0
$653$	33.5817	1.31416	0.657078	−	0.753823i	$-0.271792\pi$
$653$	33.5817	1.31416	0.657078	+	0.753823i	$0.271792\pi$
$654$	0	0
$655$	−7.44131	−0.290756
$656$	0	0
$657$	−10.4447	−0.407487
$658$	0	0
$659$	18.5530	0.722721	0.361361	−	0.932426i	$-0.382312\pi$
$659$	18.5530	0.722721	0.361361	+	0.932426i	$0.382312\pi$
$660$	0	0
$661$	−47.5870	−1.85092	−0.925460	−	0.378845i	$-0.876321\pi$
$661$	−47.5870	−1.85092	−0.925460	+	0.378845i	$0.876321\pi$
$662$	0	0
$663$	−8.83621	−0.343170
$664$	0	0
$665$	13.7198	0.532031
$666$	0	0
$667$	13.6056	0.526810
$668$	0	0
$669$	−5.86826	−0.226880
$670$	0	0
$671$	16.1268	0.622568
$672$	0	0
$673$	33.9558	1.30890	0.654451	−	0.756105i	$-0.272900\pi$
$673$	33.9558	1.30890	0.654451	+	0.756105i	$0.272900\pi$
$674$	0	0
$675$	3.04265	0.117112
$676$	0	0
$677$	30.9718	1.19034	0.595171	−	0.803599i	$-0.297084\pi$
$677$	30.9718	1.19034	0.595171	+	0.803599i	$0.297084\pi$
$678$	0	0
$679$	18.0789	0.693804
$680$	0	0
$681$	−3.35763	−0.128665
$682$	0	0
$683$	22.6317	0.865979	0.432989	−	0.901399i	$-0.357459\pi$
$683$	22.6317	0.865979	0.432989	+	0.901399i	$0.357459\pi$
$684$	0	0
$685$	−2.02856	−0.0775072
$686$	0	0
$687$	3.52988	0.134673
$688$	0	0
$689$	−14.0164	−0.533981
$690$	0	0
$691$	6.42163	0.244290	0.122145	−	0.992512i	$-0.461023\pi$
$691$	6.42163	0.244290	0.122145	+	0.992512i	$0.461023\pi$
$692$	0	0
$693$	18.0242	0.684684
$694$	0	0
$695$	17.4292	0.661126
$696$	0	0
$697$	−21.9313	−0.830708
$698$	0	0
$699$	−5.94010	−0.224675
$700$	0	0
$701$	22.6541	0.855634	0.427817	−	0.903865i	$-0.359283\pi$
$701$	22.6541	0.855634	0.427817	+	0.903865i	$0.359283\pi$
$702$	0	0
$703$	7.39575	0.278936
$704$	0	0
$705$	−2.25181	−0.0848082
$706$	0	0
$707$	3.02103	0.113618
$708$	0	0
$709$	−4.00006	−0.150225	−0.0751126	−	0.997175i	$-0.523932\pi$
$709$	−4.00006	−0.150225	−0.0751126	+	0.997175i	$0.523932\pi$
$710$	0	0
$711$	−9.86733	−0.370054
$712$	0	0
$713$	13.9747	0.523357
$714$	0	0
$715$	−8.40679	−0.314396
$716$	0	0
$717$	6.71716	0.250857
$718$	0	0
$719$	35.1109	1.30942	0.654708	−	0.755881i	$-0.272791\pi$
$719$	35.1109	1.30942	0.654708	+	0.755881i	$0.272791\pi$
$720$	0	0
$721$	−26.7735	−0.997098
$722$	0	0
$723$	9.64413	0.358669
$724$	0	0
$725$	−2.34054	−0.0869254
$726$	0	0
$727$	−9.06488	−0.336198	−0.168099	−	0.985770i	$-0.553763\pi$
$727$	−9.06488	−0.336198	−0.168099	+	0.985770i	$0.553763\pi$
$728$	0	0
$729$	−12.8840	−0.477186
$730$	0	0
$731$	−50.0849	−1.85246
$732$	0	0
$733$	2.33960	0.0864151	0.0432076	−	0.999066i	$-0.486242\pi$
$733$	2.33960	0.0864151	0.0432076	+	0.999066i	$0.486242\pi$
$734$	0	0
$735$	−0.296540	−0.0109381
$736$	0	0
$737$	−21.3580	−0.786730
$738$	0	0
$739$	4.82795	0.177599	0.0887995	−	0.996050i	$-0.471697\pi$
$739$	4.82795	0.177599	0.0887995	+	0.996050i	$0.471697\pi$
$740$	0	0
$741$	−9.25277	−0.339909
$742$	0	0
$743$	30.8326	1.13114	0.565570	−	0.824700i	$-0.308656\pi$
$743$	30.8326	1.13114	0.565570	+	0.824700i	$0.308656\pi$
$744$	0	0
$745$	3.21777	0.117890
$746$	0	0
$747$	−4.22755	−0.154678
$748$	0	0
$749$	19.6704	0.718742
$750$	0	0
$751$	23.8134	0.868963	0.434481	−	0.900681i	$-0.356932\pi$
$751$	23.8134	0.868963	0.434481	+	0.900681i	$0.356932\pi$
$752$	0	0
$753$	−7.96863	−0.290393
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	18.4373	0.670115	0.335057	−	0.942198i	$-0.391244\pi$
$757$	18.4373	0.670115	0.335057	+	0.942198i	$0.391244\pi$
$758$	0	0
$759$	7.46687	0.271030
$760$	0	0
$761$	32.1633	1.16592	0.582960	−	0.812501i	$-0.301894\pi$
$761$	32.1633	1.16592	0.582960	+	0.812501i	$0.301894\pi$
$762$	0	0
$763$	2.29400	0.0830483
$764$	0	0
$765$	12.9482	0.468142
$766$	0	0
$767$	12.5010	0.451385
$768$	0	0
$769$	−9.96423	−0.359319	−0.179660	−	0.983729i	$-0.557500\pi$
$769$	−9.96423	−0.359319	−0.179660	+	0.983729i	$0.557500\pi$
$770$	0	0
$771$	6.49065	0.233755
$772$	0	0
$773$	21.3273	0.767089	0.383545	−	0.923522i	$-0.374703\pi$
$773$	21.3273	0.767089	0.383545	+	0.923522i	$0.374703\pi$
$774$	0	0
$775$	−2.40404	−0.0863557
$776$	0	0
$777$	−2.16820	−0.0777836
$778$	0	0
$779$	−22.9652	−0.822814
$780$	0	0
$781$	35.1830	1.25895
$782$	0	0
$783$	−7.12145	−0.254500
$784$	0	0
$785$	−22.6794	−0.809461
$786$	0	0
$787$	−14.6535	−0.522340	−0.261170	−	0.965293i	$-0.584108\pi$
$787$	−14.6535	−0.522340	−0.261170	+	0.965293i	$0.584108\pi$
$788$	0	0
$789$	8.88223	0.316216
$790$	0	0
$791$	36.5972	1.30125
$792$	0	0
$793$	−23.2763	−0.826566
$794$	0	0
$795$	−2.14162	−0.0759555
$796$	0	0
$797$	−41.6980	−1.47702	−0.738510	−	0.674242i	$-0.764470\pi$
$797$	−41.6980	−1.47702	−0.738510	+	0.674242i	$0.764470\pi$
$798$	0	0
$799$	−20.1646	−0.713373
$800$	0	0
$801$	−30.9906	−1.09500
$802$	0	0
$803$	9.27862	0.327435
$804$	0	0
$805$	15.9801	0.563225
$806$	0	0
$807$	3.18540	0.112131
$808$	0	0
$809$	21.6048	0.759583	0.379792	−	0.925072i	$-0.375996\pi$
$809$	21.6048	0.759583	0.379792	+	0.925072i	$0.375996\pi$
$810$	0	0
$811$	33.1949	1.16563	0.582815	−	0.812605i	$-0.301951\pi$
$811$	33.1949	1.16563	0.582815	+	0.812605i	$0.301951\pi$
$812$	0	0
$813$	−16.2102	−0.568516
$814$	0	0
$815$	−4.50542	−0.157818
$816$	0	0
$817$	−52.4460	−1.83485
$818$	0	0
$819$	−26.0149	−0.909035
$820$	0	0
$821$	27.1966	0.949168	0.474584	−	0.880210i	$-0.342599\pi$
$821$	27.1966	0.949168	0.474584	+	0.880210i	$0.342599\pi$
$822$	0	0
$823$	−5.94200	−0.207125	−0.103563	−	0.994623i	$-0.533024\pi$
$823$	−5.94200	−0.207125	−0.103563	+	0.994623i	$0.533024\pi$
$824$	0	0
$825$	−1.28451	−0.0447209
$826$	0	0
$827$	−20.8365	−0.724555	−0.362277	−	0.932070i	$-0.618001\pi$
$827$	−20.8365	−0.724555	−0.362277	+	0.932070i	$0.618001\pi$
$828$	0	0
$829$	40.3147	1.40019	0.700094	−	0.714051i	$-0.253141\pi$
$829$	40.3147	1.40019	0.700094	+	0.714051i	$0.253141\pi$
$830$	0	0
$831$	0.449386	0.0155890
$832$	0	0
$833$	−2.65547	−0.0920066
$834$	0	0
$835$	4.14781	0.143541
$836$	0	0
$837$	−7.31467	−0.252832
$838$	0	0
$839$	−41.2940	−1.42563	−0.712813	−	0.701354i	$-0.752579\pi$
$839$	−41.2940	−1.42563	−0.712813	+	0.701354i	$0.752579\pi$
$840$	0	0
$841$	−23.5219	−0.811099
$842$	0	0
$843$	3.18529	0.109707
$844$	0	0
$845$	−0.866231	−0.0297992
$846$	0	0
$847$	14.2274	0.488858
$848$	0	0
$849$	0.465492	0.0159756
$850$	0	0
$851$	8.61420	0.295291
$852$	0	0
$853$	−25.6187	−0.877167	−0.438584	−	0.898690i	$-0.644520\pi$
$853$	−25.6187	−0.877167	−0.438584	+	0.898690i	$0.644520\pi$
$854$	0	0
$855$	13.5586	0.463693
$856$	0	0
$857$	−54.5231	−1.86248	−0.931238	−	0.364412i	$-0.881270\pi$
$857$	−54.5231	−1.86248	−0.931238	+	0.364412i	$0.881270\pi$
$858$	0	0
$859$	4.32187	0.147460	0.0737302	−	0.997278i	$-0.476510\pi$
$859$	4.32187	0.147460	0.0737302	+	0.997278i	$0.476510\pi$
$860$	0	0
$861$	6.73266	0.229448
$862$	0	0
$863$	−13.9542	−0.475005	−0.237502	−	0.971387i	$-0.576329\pi$
$863$	−13.9542	−0.475005	−0.237502	+	0.971387i	$0.576329\pi$
$864$	0	0
$865$	−19.8577	−0.675182
$866$	0	0
$867$	−3.04210	−0.103315
$868$	0	0
$869$	8.76570	0.297356
$870$	0	0
$871$	30.8266	1.04452
$872$	0	0
$873$	17.8664	0.604687
$874$	0	0
$875$	−2.74903	−0.0929341
$876$	0	0
$877$	−3.68902	−0.124569	−0.0622847	−	0.998058i	$-0.519839\pi$
$877$	−3.68902	−0.124569	−0.0622847	+	0.998058i	$0.519839\pi$
$878$	0	0
$879$	6.23740	0.210382
$880$	0	0
$881$	−3.29324	−0.110952	−0.0554760	−	0.998460i	$-0.517668\pi$
$881$	−3.29324	−0.110952	−0.0554760	+	0.998460i	$0.517668\pi$
$882$	0	0
$883$	15.5399	0.522960	0.261480	−	0.965209i	$-0.415789\pi$
$883$	15.5399	0.522960	0.261480	+	0.965209i	$0.415789\pi$
$884$	0	0
$885$	1.91008	0.0642067
$886$	0	0
$887$	41.8224	1.40426	0.702129	−	0.712050i	$-0.252233\pi$
$887$	41.8224	1.40426	0.702129	+	0.712050i	$0.252233\pi$
$888$	0	0
$889$	−56.5825	−1.89771
$890$	0	0
$891$	15.7614	0.528028
$892$	0	0
$893$	−21.1152	−0.706594
$894$	0	0
$895$	5.89988	0.197211
$896$	0	0
$897$	−10.7772	−0.359839
$898$	0	0
$899$	5.62675	0.187663
$900$	0	0
$901$	−19.1779	−0.638908
$902$	0	0
$903$	15.3755	0.511664
$904$	0	0
$905$	13.1647	0.437608
$906$	0	0
$907$	1.51795	0.0504029	0.0252014	−	0.999682i	$-0.491977\pi$
$907$	1.51795	0.0504029	0.0252014	+	0.999682i	$0.491977\pi$
$908$	0	0
$909$	2.98553	0.0990238
$910$	0	0
$911$	−23.6634	−0.784003	−0.392002	−	0.919965i	$-0.628217\pi$
$911$	−23.6634	−0.784003	−0.392002	+	0.919965i	$0.628217\pi$
$912$	0	0
$913$	3.75557	0.124291
$914$	0	0
$915$	−3.55649	−0.117574
$916$	0	0
$917$	20.4564	0.675529
$918$	0	0
$919$	−17.5882	−0.580181	−0.290090	−	0.956999i	$-0.593685\pi$
$919$	−17.5882	−0.580181	−0.290090	+	0.956999i	$0.593685\pi$
$920$	0	0
$921$	−5.26923	−0.173627
$922$	0	0
$923$	−50.7807	−1.67147
$924$	0	0
$925$	−1.48188	−0.0487240
$926$	0	0
$927$	−26.4589	−0.869024
$928$	0	0
$929$	−2.11692	−0.0694539	−0.0347269	−	0.999397i	$-0.511056\pi$
$929$	−2.11692	−0.0694539	−0.0347269	+	0.999397i	$0.511056\pi$
$930$	0	0
$931$	−2.78066	−0.0911323
$932$	0	0
$933$	11.2942	0.369756
$934$	0	0
$935$	−11.5026	−0.376175
$936$	0	0
$937$	−53.1378	−1.73594	−0.867969	−	0.496619i	$-0.834575\pi$
$937$	−53.1378	−1.73594	−0.867969	+	0.496619i	$0.834575\pi$
$938$	0	0
$939$	−3.43848	−0.112210
$940$	0	0
$941$	0.128505	0.00418914	0.00209457	−	0.999998i	$-0.499333\pi$
$941$	0.128505	0.00418914	0.00209457	+	0.999998i	$0.499333\pi$
$942$	0	0
$943$	−26.7487	−0.871058
$944$	0	0
$945$	−8.36434	−0.272092
$946$	0	0
$947$	49.7544	1.61680	0.808401	−	0.588632i	$-0.200333\pi$
$947$	49.7544	1.61680	0.808401	+	0.588632i	$0.200333\pi$
$948$	0	0
$949$	−13.3921	−0.434726
$950$	0	0
$951$	−18.4829	−0.599350
$952$	0	0
$953$	12.0562	0.390538	0.195269	−	0.980750i	$-0.437442\pi$
$953$	12.0562	0.390538	0.195269	+	0.980750i	$0.437442\pi$
$954$	0	0
$955$	20.3467	0.658403
$956$	0	0
$957$	3.00645	0.0971846
$958$	0	0
$959$	5.57656	0.180077
$960$	0	0
$961$	−25.2206	−0.813567
$962$	0	0
$963$	19.4393	0.626422
$964$	0	0
$965$	3.41180	0.109830
$966$	0	0
$967$	−14.9654	−0.481255	−0.240628	−	0.970617i	$-0.577353\pi$
$967$	−14.9654	−0.481255	−0.240628	+	0.970617i	$0.577353\pi$
$968$	0	0
$969$	−12.6601	−0.406701
$970$	0	0
$971$	14.0477	0.450813	0.225406	−	0.974265i	$-0.427629\pi$
$971$	14.0477	0.450813	0.225406	+	0.974265i	$0.427629\pi$
$972$	0	0
$973$	−47.9133	−1.53603
$974$	0	0
$975$	1.85397	0.0593746
$976$	0	0
$977$	−45.5235	−1.45643	−0.728213	−	0.685351i	$-0.759649\pi$
$977$	−45.5235	−1.45643	−0.728213	+	0.685351i	$0.759649\pi$
$978$	0	0
$979$	27.5307	0.879884
$980$	0	0
$981$	2.26704	0.0723810
$982$	0	0
$983$	11.8178	0.376928	0.188464	−	0.982080i	$-0.439649\pi$
$983$	11.8178	0.376928	0.188464	+	0.982080i	$0.439649\pi$
$984$	0	0
$985$	−12.8973	−0.410942
$986$	0	0
$987$	6.19030	0.197039
$988$	0	0
$989$	−61.0864	−1.94244
$990$	0	0
$991$	−0.922393	−0.0293008	−0.0146504	−	0.999893i	$-0.504664\pi$
$991$	−0.922393	−0.0293008	−0.0146504	+	0.999893i	$0.504664\pi$
$992$	0	0
$993$	2.33604	0.0741319
$994$	0	0
$995$	1.57571	0.0499534
$996$	0	0
$997$	25.0980	0.794862	0.397431	−	0.917632i	$-0.369902\pi$
$997$	25.0980	0.794862	0.397431	+	0.917632i	$0.369902\pi$
$998$	0	0
$999$	−4.50886	−0.142654

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.o.1.6	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.o.1.6	✓	15	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-0.532237 q^{3} +1.00000 q^{5} -2.74903 q^{7} -2.71672 q^{9} +O(q^{10})\)	\(q-0.532237 q^{3} +1.00000 q^{5} -2.74903 q^{7} -2.71672 q^{9} +2.41342 q^{11} -3.48336 q^{13} -0.532237 q^{15} -4.76610 q^{17} -4.99078 q^{19} +1.46314 q^{21} -5.81301 q^{23} +1.00000 q^{25} +3.04265 q^{27} -2.34054 q^{29} -2.40404 q^{31} -1.28451 q^{33} -2.74903 q^{35} -1.48188 q^{37} +1.85397 q^{39} +4.60153 q^{41} +10.5086 q^{43} -2.71672 q^{45} +4.23084 q^{47} +0.557158 q^{49} +2.53669 q^{51} +4.02381 q^{53} +2.41342 q^{55} +2.65628 q^{57} -3.58878 q^{59} +6.68215 q^{61} +7.46835 q^{63} -3.48336 q^{65} -8.84968 q^{67} +3.09390 q^{69} +14.5781 q^{71} +3.84460 q^{73} -0.532237 q^{75} -6.63455 q^{77} +3.63207 q^{79} +6.53076 q^{81} +1.55612 q^{83} -4.76610 q^{85} +1.24572 q^{87} +11.4073 q^{89} +9.57585 q^{91} +1.27952 q^{93} -4.99078 q^{95} -6.57647 q^{97} -6.55659 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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