LMFDB - Embedded newform 8040.2.a.t.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8040,2,Mod(1,8040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8040.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:	$64.1997232251$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 3x^{4} + 43x^{3} - 6x^{2} - 29x + 6 $ x^7 - x^6 - 15x^5 + 3x^4 + 43x^3 - 6x^2 - 29*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.05266$ of defining polynomial
Character	$\chi$	$=$	8040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +0.670274 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +0.670274 q^{7} +1.00000 q^{9} +3.78339 q^{11} +4.11429 q^{13} -1.00000 q^{15} +1.83923 q^{17} +5.73969 q^{19} +0.670274 q^{21} +7.13944 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.82187 q^{29} -5.12208 q^{31} +3.78339 q^{33} -0.670274 q^{35} +2.09121 q^{37} +4.11429 q^{39} +7.12208 q^{41} -3.02515 q^{43} -1.00000 q^{45} -5.47041 q^{47} -6.55073 q^{49} +1.83923 q^{51} +6.86073 q^{53} -3.78339 q^{55} +5.73969 q^{57} +1.63478 q^{59} +5.50889 q^{61} +0.670274 q^{63} -4.11429 q^{65} +1.00000 q^{67} +7.13944 q^{69} -12.4491 q^{71} -10.4317 q^{73} +1.00000 q^{75} +2.53591 q^{77} +7.01717 q^{79} +1.00000 q^{81} -5.16000 q^{83} -1.83923 q^{85} -1.82187 q^{87} +5.63153 q^{89} +2.75770 q^{91} -5.12208 q^{93} -5.73969 q^{95} -9.80937 q^{97} +3.78339 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9	$ 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9} - q^{13} - 7 q^{15} - 2 q^{17} + 9 q^{19} + 10 q^{21} + 2 q^{23} + 7 q^{25} + 7 q^{27} - q^{29} + 9 q^{31} - 10 q^{35} + 23 q^{37} - q^{39} + 5 q^{41} - 3 q^{43} - 7 q^{45} + 11 q^{47} + 13 q^{49} - 2 q^{51} + 13 q^{53} + 9 q^{57} + q^{59} + 4 q^{61} + 10 q^{63} + q^{65} + 7 q^{67} + 2 q^{69} + q^{71} + 14 q^{73} + 7 q^{75} + 18 q^{77} + 25 q^{79} + 7 q^{81} - 29 q^{83} + 2 q^{85} - q^{87} + 7 q^{89} + 27 q^{91} + 9 q^{93} - 9 q^{95} + 38 q^{97}+O(q^{100}) $ 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9 - q^13 - 7 * q^15 - 2 * q^17 + 9 * q^19 + 10 * q^21 + 2 * q^23 + 7 * q^25 + 7 * q^27 - q^29 + 9 * q^31 - 10 * q^35 + 23 * q^37 - q^39 + 5 * q^41 - 3 * q^43 - 7 * q^45 + 11 * q^47 + 13 * q^49 - 2 * q^51 + 13 * q^53 + 9 * q^57 + q^59 + 4 * q^61 + 10 * q^63 + q^65 + 7 * q^67 + 2 * q^69 + q^71 + 14 * q^73 + 7 * q^75 + 18 * q^77 + 25 * q^79 + 7 * q^81 - 29 * q^83 + 2 * q^85 - q^87 + 7 * q^89 + 27 * q^91 + 9 * q^93 - 9 * q^95 + 38 * q^97

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.670274	0.253340	0.126670	−	0.991945i	$-0.459571\pi$
$7$	0.670274	0.253340	0.126670	+	0.991945i	$0.459571\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.78339	1.14073	0.570367	−	0.821390i	$-0.306801\pi$
$11$	3.78339	1.14073	0.570367	+	0.821390i	$0.306801\pi$
$12$	0	0
$13$	4.11429	1.14110	0.570550	−	0.821263i	$-0.306730\pi$
$13$	4.11429	1.14110	0.570550	+	0.821263i	$0.306730\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.83923	0.446080	0.223040	−	0.974809i	$-0.428402\pi$
$17$	1.83923	0.446080	0.223040	+	0.974809i	$0.428402\pi$
$18$	0	0
$19$	5.73969	1.31677	0.658387	−	0.752679i	$-0.271239\pi$
$19$	5.73969	1.31677	0.658387	+	0.752679i	$0.271239\pi$
$20$	0	0
$21$	0.670274	0.146266
$22$	0	0
$23$	7.13944	1.48868	0.744338	−	0.667803i	$-0.232765\pi$
$23$	7.13944	1.48868	0.744338	+	0.667803i	$0.232765\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.82187	−0.338313	−0.169156	−	0.985589i	$-0.554104\pi$
$29$	−1.82187	−0.338313	−0.169156	+	0.985589i	$0.554104\pi$
$30$	0	0
$31$	−5.12208	−0.919952	−0.459976	−	0.887931i	$-0.652142\pi$
$31$	−5.12208	−0.919952	−0.459976	+	0.887931i	$0.652142\pi$
$32$	0	0
$33$	3.78339	0.658603
$34$	0	0
$35$	−0.670274	−0.113297
$36$	0	0
$37$	2.09121	0.343793	0.171896	−	0.985115i	$-0.445011\pi$
$37$	2.09121	0.343793	0.171896	+	0.985115i	$0.445011\pi$
$38$	0	0
$39$	4.11429	0.658814
$40$	0	0
$41$	7.12208	1.11228	0.556141	−	0.831088i	$-0.312282\pi$
$41$	7.12208	1.11228	0.556141	+	0.831088i	$0.312282\pi$
$42$	0	0
$43$	−3.02515	−0.461331	−0.230665	−	0.973033i	$-0.574090\pi$
$43$	−3.02515	−0.461331	−0.230665	+	0.973033i	$0.574090\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−5.47041	−0.797941	−0.398971	−	0.916964i	$-0.630632\pi$
$47$	−5.47041	−0.797941	−0.398971	+	0.916964i	$0.630632\pi$
$48$	0	0
$49$	−6.55073	−0.935819
$50$	0	0
$51$	1.83923	0.257544
$52$	0	0
$53$	6.86073	0.942393	0.471197	−	0.882028i	$-0.343822\pi$
$53$	6.86073	0.942393	0.471197	+	0.882028i	$0.343822\pi$
$54$	0	0
$55$	−3.78339	−0.510152
$56$	0	0
$57$	5.73969	0.760240
$58$	0	0
$59$	1.63478	0.212830	0.106415	−	0.994322i	$-0.466063\pi$
$59$	1.63478	0.212830	0.106415	+	0.994322i	$0.466063\pi$
$60$	0	0
$61$	5.50889	0.705341	0.352671	−	0.935748i	$-0.385274\pi$
$61$	5.50889	0.705341	0.352671	+	0.935748i	$0.385274\pi$
$62$	0	0
$63$	0.670274	0.0844466
$64$	0	0
$65$	−4.11429	−0.510315
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	7.13944	0.859488
$70$	0	0
$71$	−12.4491	−1.47744	−0.738718	−	0.674015i	$-0.764568\pi$
$71$	−12.4491	−1.47744	−0.738718	+	0.674015i	$0.764568\pi$
$72$	0	0
$73$	−10.4317	−1.22093	−0.610467	−	0.792042i	$-0.709018\pi$
$73$	−10.4317	−1.22093	−0.610467	+	0.792042i	$0.709018\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.53591	0.288993
$78$	0	0
$79$	7.01717	0.789493	0.394746	−	0.918790i	$-0.370832\pi$
$79$	7.01717	0.789493	0.394746	+	0.918790i	$0.370832\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.16000	−0.566383	−0.283192	−	0.959063i	$-0.591393\pi$
$83$	−5.16000	−0.566383	−0.283192	+	0.959063i	$0.591393\pi$
$84$	0	0
$85$	−1.83923	−0.199493
$86$	0	0
$87$	−1.82187	−0.195325
$88$	0	0
$89$	5.63153	0.596941	0.298470	−	0.954419i	$-0.403524\pi$
$89$	5.63153	0.596941	0.298470	+	0.954419i	$0.403524\pi$
$90$	0	0
$91$	2.75770	0.289086
$92$	0	0
$93$	−5.12208	−0.531135
$94$	0	0
$95$	−5.73969	−0.588879
$96$	0	0
$97$	−9.80937	−0.995991	−0.497995	−	0.867180i	$-0.665930\pi$
$97$	−9.80937	−0.995991	−0.497995	+	0.867180i	$0.665930\pi$
$98$	0	0
$99$	3.78339	0.380245
$100$	0	0
$101$	3.47407	0.345683	0.172842	−	0.984950i	$-0.444705\pi$
$101$	3.47407	0.345683	0.172842	+	0.984950i	$0.444705\pi$
$102$	0	0
$103$	16.1717	1.59345	0.796723	−	0.604345i	$-0.206565\pi$
$103$	16.1717	1.59345	0.796723	+	0.604345i	$0.206565\pi$
$104$	0	0
$105$	−0.670274	−0.0654121
$106$	0	0
$107$	−17.9034	−1.73079	−0.865393	−	0.501093i	$-0.832931\pi$
$107$	−17.9034	−1.73079	−0.865393	+	0.501093i	$0.832931\pi$
$108$	0	0
$109$	17.3694	1.66369	0.831844	−	0.555009i	$-0.187285\pi$
$109$	17.3694	1.66369	0.831844	+	0.555009i	$0.187285\pi$
$110$	0	0
$111$	2.09121	0.198489
$112$	0	0
$113$	−19.3577	−1.82102	−0.910508	−	0.413491i	$-0.864309\pi$
$113$	−19.3577	−1.82102	−0.910508	+	0.413491i	$0.864309\pi$
$114$	0	0
$115$	−7.13944	−0.665756
$116$	0	0
$117$	4.11429	0.380366
$118$	0	0
$119$	1.23279	0.113010
$120$	0	0
$121$	3.31401	0.301274
$122$	0	0
$123$	7.12208	0.642176
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.6572	1.12315	0.561573	−	0.827427i	$-0.310196\pi$
$127$	12.6572	1.12315	0.561573	+	0.827427i	$0.310196\pi$
$128$	0	0
$129$	−3.02515	−0.266349
$130$	0	0
$131$	−13.7384	−1.20033	−0.600166	−	0.799876i	$-0.704899\pi$
$131$	−13.7384	−1.20033	−0.600166	+	0.799876i	$0.704899\pi$
$132$	0	0
$133$	3.84716	0.333591
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−10.2004	−0.871476	−0.435738	−	0.900074i	$-0.643513\pi$
$137$	−10.2004	−0.871476	−0.435738	+	0.900074i	$0.643513\pi$
$138$	0	0
$139$	−12.0665	−1.02347	−0.511733	−	0.859144i	$-0.670996\pi$
$139$	−12.0665	−1.02347	−0.511733	+	0.859144i	$0.670996\pi$
$140$	0	0
$141$	−5.47041	−0.460691
$142$	0	0
$143$	15.5660	1.30169
$144$	0	0
$145$	1.82187	0.151298
$146$	0	0
$147$	−6.55073	−0.540295
$148$	0	0
$149$	−1.31188	−0.107474	−0.0537368	−	0.998555i	$-0.517113\pi$
$149$	−1.31188	−0.107474	−0.0537368	+	0.998555i	$0.517113\pi$
$150$	0	0
$151$	3.61324	0.294041	0.147021	−	0.989133i	$-0.453032\pi$
$151$	3.61324	0.294041	0.147021	+	0.989133i	$0.453032\pi$
$152$	0	0
$153$	1.83923	0.148693
$154$	0	0
$155$	5.12208	0.411415
$156$	0	0
$157$	5.72452	0.456867	0.228433	−	0.973560i	$-0.426640\pi$
$157$	5.72452	0.456867	0.228433	+	0.973560i	$0.426640\pi$
$158$	0	0
$159$	6.86073	0.544091
$160$	0	0
$161$	4.78538	0.377141
$162$	0	0
$163$	14.8151	1.16041	0.580204	−	0.814471i	$-0.302973\pi$
$163$	14.8151	1.16041	0.580204	+	0.814471i	$0.302973\pi$
$164$	0	0
$165$	−3.78339	−0.294536
$166$	0	0
$167$	−3.97042	−0.307240	−0.153620	−	0.988130i	$-0.549093\pi$
$167$	−3.97042	−0.307240	−0.153620	+	0.988130i	$0.549093\pi$
$168$	0	0
$169$	3.92740	0.302107
$170$	0	0
$171$	5.73969	0.438925
$172$	0	0
$173$	24.2043	1.84022	0.920109	−	0.391662i	$-0.128100\pi$
$173$	24.2043	1.84022	0.920109	+	0.391662i	$0.128100\pi$
$174$	0	0
$175$	0.670274	0.0506680
$176$	0	0
$177$	1.63478	0.122877
$178$	0	0
$179$	−9.26827	−0.692743	−0.346371	−	0.938097i	$-0.612586\pi$
$179$	−9.26827	−0.692743	−0.346371	+	0.938097i	$0.612586\pi$
$180$	0	0
$181$	−24.7538	−1.83994	−0.919969	−	0.391991i	$-0.871786\pi$
$181$	−24.7538	−1.83994	−0.919969	+	0.391991i	$0.871786\pi$
$182$	0	0
$183$	5.50889	0.407229
$184$	0	0
$185$	−2.09121	−0.153749
$186$	0	0
$187$	6.95853	0.508858
$188$	0	0
$189$	0.670274	0.0487553
$190$	0	0
$191$	1.65532	0.119775	0.0598875	−	0.998205i	$-0.480926\pi$
$191$	1.65532	0.119775	0.0598875	+	0.998205i	$0.480926\pi$
$192$	0	0
$193$	−13.2421	−0.953191	−0.476595	−	0.879123i	$-0.658129\pi$
$193$	−13.2421	−0.953191	−0.476595	+	0.879123i	$0.658129\pi$
$194$	0	0
$195$	−4.11429	−0.294631
$196$	0	0
$197$	−3.51881	−0.250705	−0.125352	−	0.992112i	$-0.540006\pi$
$197$	−3.51881	−0.250705	−0.125352	+	0.992112i	$0.540006\pi$
$198$	0	0
$199$	13.9365	0.987933	0.493966	−	0.869481i	$-0.335547\pi$
$199$	13.9365	0.987933	0.493966	+	0.869481i	$0.335547\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	−1.22115	−0.0857081
$204$	0	0
$205$	−7.12208	−0.497427
$206$	0	0
$207$	7.13944	0.496225
$208$	0	0
$209$	21.7154	1.50209
$210$	0	0
$211$	24.8657	1.71183	0.855914	−	0.517118i	$-0.172995\pi$
$211$	24.8657	1.71183	0.855914	+	0.517118i	$0.172995\pi$
$212$	0	0
$213$	−12.4491	−0.852998
$214$	0	0
$215$	3.02515	0.206313
$216$	0	0
$217$	−3.43320	−0.233060
$218$	0	0
$219$	−10.4317	−0.704906
$220$	0	0
$221$	7.56715	0.509021
$222$	0	0
$223$	−9.60254	−0.643033	−0.321517	−	0.946904i	$-0.604193\pi$
$223$	−9.60254	−0.643033	−0.321517	+	0.946904i	$0.604193\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	13.6064	0.903089	0.451545	−	0.892249i	$-0.350873\pi$
$227$	13.6064	0.903089	0.451545	+	0.892249i	$0.350873\pi$
$228$	0	0
$229$	3.78622	0.250200	0.125100	−	0.992144i	$-0.460075\pi$
$229$	3.78622	0.250200	0.125100	+	0.992144i	$0.460075\pi$
$230$	0	0
$231$	2.53591	0.166850
$232$	0	0
$233$	13.4793	0.883059	0.441530	−	0.897247i	$-0.354436\pi$
$233$	13.4793	0.883059	0.441530	+	0.897247i	$0.354436\pi$
$234$	0	0
$235$	5.47041	0.356850
$236$	0	0
$237$	7.01717	0.455814
$238$	0	0
$239$	−7.81123	−0.505266	−0.252633	−	0.967562i	$-0.581297\pi$
$239$	−7.81123	−0.505266	−0.252633	+	0.967562i	$0.581297\pi$
$240$	0	0
$241$	13.8308	0.890923	0.445462	−	0.895301i	$-0.353040\pi$
$241$	13.8308	0.890923	0.445462	+	0.895301i	$0.353040\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.55073	0.418511
$246$	0	0
$247$	23.6147	1.50257
$248$	0	0
$249$	−5.16000	−0.327001
$250$	0	0
$251$	−13.7731	−0.869352	−0.434676	−	0.900587i	$-0.643137\pi$
$251$	−13.7731	−0.869352	−0.434676	+	0.900587i	$0.643137\pi$
$252$	0	0
$253$	27.0113	1.69818
$254$	0	0
$255$	−1.83923	−0.115177
$256$	0	0
$257$	20.1620	1.25767	0.628837	−	0.777538i	$-0.283531\pi$
$257$	20.1620	1.25767	0.628837	+	0.777538i	$0.283531\pi$
$258$	0	0
$259$	1.40168	0.0870964
$260$	0	0
$261$	−1.82187	−0.112771
$262$	0	0
$263$	0.295696	0.0182334	0.00911671	−	0.999958i	$-0.497098\pi$
$263$	0.295696	0.0182334	0.00911671	+	0.999958i	$0.497098\pi$
$264$	0	0
$265$	−6.86073	−0.421451
$266$	0	0
$267$	5.63153	0.344644
$268$	0	0
$269$	−19.4531	−1.18608	−0.593038	−	0.805175i	$-0.702071\pi$
$269$	−19.4531	−1.18608	−0.593038	+	0.805175i	$0.702071\pi$
$270$	0	0
$271$	−10.8772	−0.660743	−0.330372	−	0.943851i	$-0.607174\pi$
$271$	−10.8772	−0.660743	−0.330372	+	0.943851i	$0.607174\pi$
$272$	0	0
$273$	2.75770	0.166904
$274$	0	0
$275$	3.78339	0.228147
$276$	0	0
$277$	25.1608	1.51177	0.755883	−	0.654707i	$-0.227208\pi$
$277$	25.1608	1.51177	0.755883	+	0.654707i	$0.227208\pi$
$278$	0	0
$279$	−5.12208	−0.306651
$280$	0	0
$281$	8.91436	0.531786	0.265893	−	0.964003i	$-0.414333\pi$
$281$	8.91436	0.531786	0.265893	+	0.964003i	$0.414333\pi$
$282$	0	0
$283$	−19.4711	−1.15744	−0.578720	−	0.815526i	$-0.696447\pi$
$283$	−19.4711	−1.15744	−0.578720	+	0.815526i	$0.696447\pi$
$284$	0	0
$285$	−5.73969	−0.339990
$286$	0	0
$287$	4.77374	0.281785
$288$	0	0
$289$	−13.6172	−0.801013
$290$	0	0
$291$	−9.80937	−0.575035
$292$	0	0
$293$	−10.3591	−0.605185	−0.302592	−	0.953120i	$-0.597852\pi$
$293$	−10.3591	−0.605185	−0.302592	+	0.953120i	$0.597852\pi$
$294$	0	0
$295$	−1.63478	−0.0951803
$296$	0	0
$297$	3.78339	0.219534
$298$	0	0
$299$	29.3737	1.69873
$300$	0	0
$301$	−2.02768	−0.116873
$302$	0	0
$303$	3.47407	0.199580
$304$	0	0
$305$	−5.50889	−0.315438
$306$	0	0
$307$	25.7029	1.46694	0.733470	−	0.679721i	$-0.237899\pi$
$307$	25.7029	1.46694	0.733470	+	0.679721i	$0.237899\pi$
$308$	0	0
$309$	16.1717	0.919976
$310$	0	0
$311$	−2.07146	−0.117462	−0.0587308	−	0.998274i	$-0.518705\pi$
$311$	−2.07146	−0.117462	−0.0587308	+	0.998274i	$0.518705\pi$
$312$	0	0
$313$	11.9262	0.674110	0.337055	−	0.941485i	$-0.390569\pi$
$313$	11.9262	0.674110	0.337055	+	0.941485i	$0.390569\pi$
$314$	0	0
$315$	−0.670274	−0.0377657
$316$	0	0
$317$	−32.0644	−1.80092	−0.900458	−	0.434943i	$-0.856769\pi$
$317$	−32.0644	−1.80092	−0.900458	+	0.434943i	$0.856769\pi$
$318$	0	0
$319$	−6.89284	−0.385925
$320$	0	0
$321$	−17.9034	−0.999270
$322$	0	0
$323$	10.5566	0.587386
$324$	0	0
$325$	4.11429	0.228220
$326$	0	0
$327$	17.3694	0.960531
$328$	0	0
$329$	−3.66667	−0.202150
$330$	0	0
$331$	10.3465	0.568697	0.284349	−	0.958721i	$-0.408223\pi$
$331$	10.3465	0.568697	0.284349	+	0.958721i	$0.408223\pi$
$332$	0	0
$333$	2.09121	0.114598
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	3.03793	0.165486	0.0827432	−	0.996571i	$-0.473632\pi$
$337$	3.03793	0.165486	0.0827432	+	0.996571i	$0.473632\pi$
$338$	0	0
$339$	−19.3577	−1.05136
$340$	0	0
$341$	−19.3788	−1.04942
$342$	0	0
$343$	−9.08271	−0.490420
$344$	0	0
$345$	−7.13944	−0.384375
$346$	0	0
$347$	−14.1956	−0.762058	−0.381029	−	0.924563i	$-0.624430\pi$
$347$	−14.1956	−0.762058	−0.381029	+	0.924563i	$0.624430\pi$
$348$	0	0
$349$	29.5421	1.58135	0.790677	−	0.612234i	$-0.209729\pi$
$349$	29.5421	1.58135	0.790677	+	0.612234i	$0.209729\pi$
$350$	0	0
$351$	4.11429	0.219605
$352$	0	0
$353$	32.2119	1.71447	0.857234	−	0.514927i	$-0.172181\pi$
$353$	32.2119	1.71447	0.857234	+	0.514927i	$0.172181\pi$
$354$	0	0
$355$	12.4491	0.660729
$356$	0	0
$357$	1.23279	0.0652462
$358$	0	0
$359$	30.8548	1.62845	0.814226	−	0.580548i	$-0.197162\pi$
$359$	30.8548	1.62845	0.814226	+	0.580548i	$0.197162\pi$
$360$	0	0
$361$	13.9440	0.733894
$362$	0	0
$363$	3.31401	0.173941
$364$	0	0
$365$	10.4317	0.546018
$366$	0	0
$367$	−30.8580	−1.61077	−0.805386	−	0.592750i	$-0.798042\pi$
$367$	−30.8580	−1.61077	−0.805386	+	0.592750i	$0.798042\pi$
$368$	0	0
$369$	7.12208	0.370760
$370$	0	0
$371$	4.59857	0.238746
$372$	0	0
$373$	15.4483	0.799881	0.399940	−	0.916541i	$-0.369031\pi$
$373$	15.4483	0.799881	0.399940	+	0.916541i	$0.369031\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−7.49570	−0.386048
$378$	0	0
$379$	18.9680	0.974320	0.487160	−	0.873313i	$-0.338033\pi$
$379$	18.9680	0.974320	0.487160	+	0.873313i	$0.338033\pi$
$380$	0	0
$381$	12.6572	0.648449
$382$	0	0
$383$	−10.5529	−0.539230	−0.269615	−	0.962968i	$-0.586896\pi$
$383$	−10.5529	−0.539230	−0.269615	+	0.962968i	$0.586896\pi$
$384$	0	0
$385$	−2.53591	−0.129242
$386$	0	0
$387$	−3.02515	−0.153777
$388$	0	0
$389$	−5.90885	−0.299591	−0.149795	−	0.988717i	$-0.547861\pi$
$389$	−5.90885	−0.299591	−0.149795	+	0.988717i	$0.547861\pi$
$390$	0	0
$391$	13.1311	0.664068
$392$	0	0
$393$	−13.7384	−0.693012
$394$	0	0
$395$	−7.01717	−0.353072
$396$	0	0
$397$	19.2533	0.966295	0.483148	−	0.875539i	$-0.339493\pi$
$397$	19.2533	0.966295	0.483148	+	0.875539i	$0.339493\pi$
$398$	0	0
$399$	3.84716	0.192599
$400$	0	0
$401$	4.05547	0.202520	0.101260	−	0.994860i	$-0.467713\pi$
$401$	4.05547	0.202520	0.101260	+	0.994860i	$0.467713\pi$
$402$	0	0
$403$	−21.0737	−1.04976
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	7.91185	0.392176
$408$	0	0
$409$	1.26712	0.0626550	0.0313275	−	0.999509i	$-0.490027\pi$
$409$	1.26712	0.0626550	0.0313275	+	0.999509i	$0.490027\pi$
$410$	0	0
$411$	−10.2004	−0.503147
$412$	0	0
$413$	1.09575	0.0539182
$414$	0	0
$415$	5.16000	0.253294
$416$	0	0
$417$	−12.0665	−0.590899
$418$	0	0
$419$	31.0552	1.51714	0.758572	−	0.651589i	$-0.225897\pi$
$419$	31.0552	1.51714	0.758572	+	0.651589i	$0.225897\pi$
$420$	0	0
$421$	−22.9337	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$421$	−22.9337	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$422$	0	0
$423$	−5.47041	−0.265980
$424$	0	0
$425$	1.83923	0.0892160
$426$	0	0
$427$	3.69247	0.178691
$428$	0	0
$429$	15.5660	0.751531
$430$	0	0
$431$	−10.8073	−0.520568	−0.260284	−	0.965532i	$-0.583816\pi$
$431$	−10.8073	−0.520568	−0.260284	+	0.965532i	$0.583816\pi$
$432$	0	0
$433$	−39.1113	−1.87957	−0.939785	−	0.341766i	$-0.888975\pi$
$433$	−39.1113	−1.87957	−0.939785	+	0.341766i	$0.888975\pi$
$434$	0	0
$435$	1.82187	0.0873519
$436$	0	0
$437$	40.9781	1.96025
$438$	0	0
$439$	−3.92985	−0.187561	−0.0937807	−	0.995593i	$-0.529895\pi$
$439$	−3.92985	−0.187561	−0.0937807	+	0.995593i	$0.529895\pi$
$440$	0	0
$441$	−6.55073	−0.311940
$442$	0	0
$443$	−21.6585	−1.02903	−0.514513	−	0.857483i	$-0.672027\pi$
$443$	−21.6585	−1.02903	−0.514513	+	0.857483i	$0.672027\pi$
$444$	0	0
$445$	−5.63153	−0.266960
$446$	0	0
$447$	−1.31188	−0.0620499
$448$	0	0
$449$	−17.5401	−0.827770	−0.413885	−	0.910329i	$-0.635828\pi$
$449$	−17.5401	−0.827770	−0.413885	+	0.910329i	$0.635828\pi$
$450$	0	0
$451$	26.9456	1.26882
$452$	0	0
$453$	3.61324	0.169765
$454$	0	0
$455$	−2.75770	−0.129283
$456$	0	0
$457$	−12.1088	−0.566424	−0.283212	−	0.959057i	$-0.591400\pi$
$457$	−12.1088	−0.566424	−0.283212	+	0.959057i	$0.591400\pi$
$458$	0	0
$459$	1.83923	0.0858481
$460$	0	0
$461$	−26.6861	−1.24290	−0.621448	−	0.783455i	$-0.713455\pi$
$461$	−26.6861	−1.24290	−0.621448	+	0.783455i	$0.713455\pi$
$462$	0	0
$463$	24.7128	1.14850	0.574250	−	0.818680i	$-0.305294\pi$
$463$	24.7128	1.14850	0.574250	+	0.818680i	$0.305294\pi$
$464$	0	0
$465$	5.12208	0.237531
$466$	0	0
$467$	0.926018	0.0428510	0.0214255	−	0.999770i	$-0.493180\pi$
$467$	0.926018	0.0428510	0.0214255	+	0.999770i	$0.493180\pi$
$468$	0	0
$469$	0.670274	0.0309504
$470$	0	0
$471$	5.72452	0.263772
$472$	0	0
$473$	−11.4453	−0.526256
$474$	0	0
$475$	5.73969	0.263355
$476$	0	0
$477$	6.86073	0.314131
$478$	0	0
$479$	31.8360	1.45462	0.727312	−	0.686307i	$-0.240769\pi$
$479$	31.8360	1.45462	0.727312	+	0.686307i	$0.240769\pi$
$480$	0	0
$481$	8.60384	0.392302
$482$	0	0
$483$	4.78538	0.217742
$484$	0	0
$485$	9.80937	0.445421
$486$	0	0
$487$	25.2782	1.14547	0.572733	−	0.819742i	$-0.305883\pi$
$487$	25.2782	1.14547	0.572733	+	0.819742i	$0.305883\pi$
$488$	0	0
$489$	14.8151	0.669962
$490$	0	0
$491$	−15.9319	−0.718997	−0.359499	−	0.933146i	$-0.617052\pi$
$491$	−15.9319	−0.718997	−0.359499	+	0.933146i	$0.617052\pi$
$492$	0	0
$493$	−3.35084	−0.150914
$494$	0	0
$495$	−3.78339	−0.170051
$496$	0	0
$497$	−8.34430	−0.374293
$498$	0	0
$499$	−5.67237	−0.253930	−0.126965	−	0.991907i	$-0.540524\pi$
$499$	−5.67237	−0.253930	−0.126965	+	0.991907i	$0.540524\pi$
$500$	0	0
$501$	−3.97042	−0.177385
$502$	0	0
$503$	−15.2206	−0.678654	−0.339327	−	0.940668i	$-0.610199\pi$
$503$	−15.2206	−0.678654	−0.339327	+	0.940668i	$0.610199\pi$
$504$	0	0
$505$	−3.47407	−0.154594
$506$	0	0
$507$	3.92740	0.174422
$508$	0	0
$509$	−9.30717	−0.412533	−0.206266	−	0.978496i	$-0.566131\pi$
$509$	−9.30717	−0.412533	−0.206266	+	0.978496i	$0.566131\pi$
$510$	0	0
$511$	−6.99207	−0.309311
$512$	0	0
$513$	5.73969	0.253413
$514$	0	0
$515$	−16.1717	−0.712610
$516$	0	0
$517$	−20.6967	−0.910238
$518$	0	0
$519$	24.2043	1.06245
$520$	0	0
$521$	−11.3936	−0.499162	−0.249581	−	0.968354i	$-0.580293\pi$
$521$	−11.3936	−0.499162	−0.249581	+	0.968354i	$0.580293\pi$
$522$	0	0
$523$	21.3201	0.932261	0.466130	−	0.884716i	$-0.345648\pi$
$523$	21.3201	0.932261	0.466130	+	0.884716i	$0.345648\pi$
$524$	0	0
$525$	0.670274	0.0292532
$526$	0	0
$527$	−9.42070	−0.410372
$528$	0	0
$529$	27.9716	1.21616
$530$	0	0
$531$	1.63478	0.0709432
$532$	0	0
$533$	29.3023	1.26922
$534$	0	0
$535$	17.9034	0.774031
$536$	0	0
$537$	−9.26827	−0.399955
$538$	0	0
$539$	−24.7840	−1.06752
$540$	0	0
$541$	−29.2004	−1.25542	−0.627712	−	0.778446i	$-0.716008\pi$
$541$	−29.2004	−1.25542	−0.627712	+	0.778446i	$0.716008\pi$
$542$	0	0
$543$	−24.7538	−1.06229
$544$	0	0
$545$	−17.3694	−0.744024
$546$	0	0
$547$	34.0743	1.45691	0.728455	−	0.685093i	$-0.240239\pi$
$547$	34.0743	1.45691	0.728455	+	0.685093i	$0.240239\pi$
$548$	0	0
$549$	5.50889	0.235114
$550$	0	0
$551$	−10.4570	−0.445481
$552$	0	0
$553$	4.70342	0.200010
$554$	0	0
$555$	−2.09121	−0.0887669
$556$	0	0
$557$	37.7922	1.60131	0.800654	−	0.599127i	$-0.204486\pi$
$557$	37.7922	1.60131	0.800654	+	0.599127i	$0.204486\pi$
$558$	0	0
$559$	−12.4463	−0.526424
$560$	0	0
$561$	6.95853	0.293789
$562$	0	0
$563$	26.1466	1.10195	0.550974	−	0.834522i	$-0.314256\pi$
$563$	26.1466	1.10195	0.550974	+	0.834522i	$0.314256\pi$
$564$	0	0
$565$	19.3577	0.814383
$566$	0	0
$567$	0.670274	0.0281489
$568$	0	0
$569$	−37.3200	−1.56454	−0.782268	−	0.622942i	$-0.785937\pi$
$569$	−37.3200	−1.56454	−0.782268	+	0.622942i	$0.785937\pi$
$570$	0	0
$571$	1.81060	0.0757713	0.0378856	−	0.999282i	$-0.487938\pi$
$571$	1.81060	0.0757713	0.0378856	+	0.999282i	$0.487938\pi$
$572$	0	0
$573$	1.65532	0.0691521
$574$	0	0
$575$	7.13944	0.297735
$576$	0	0
$577$	−18.7019	−0.778568	−0.389284	−	0.921118i	$-0.627278\pi$
$577$	−18.7019	−0.778568	−0.389284	+	0.921118i	$0.627278\pi$
$578$	0	0
$579$	−13.2421	−0.550325
$580$	0	0
$581$	−3.45861	−0.143487
$582$	0	0
$583$	25.9568	1.07502
$584$	0	0
$585$	−4.11429	−0.170105
$586$	0	0
$587$	−1.91146	−0.0788944	−0.0394472	−	0.999222i	$-0.512560\pi$
$587$	−1.91146	−0.0788944	−0.0394472	+	0.999222i	$0.512560\pi$
$588$	0	0
$589$	−29.3991	−1.21137
$590$	0	0
$591$	−3.51881	−0.144744
$592$	0	0
$593$	1.36025	0.0558588	0.0279294	−	0.999610i	$-0.491109\pi$
$593$	1.36025	0.0558588	0.0279294	+	0.999610i	$0.491109\pi$
$594$	0	0
$595$	−1.23279	−0.0505395
$596$	0	0
$597$	13.9365	0.570383
$598$	0	0
$599$	30.3993	1.24208	0.621040	−	0.783779i	$-0.286710\pi$
$599$	30.3993	1.24208	0.621040	+	0.783779i	$0.286710\pi$
$600$	0	0
$601$	−25.6037	−1.04439	−0.522197	−	0.852825i	$-0.674888\pi$
$601$	−25.6037	−1.04439	−0.522197	+	0.852825i	$0.674888\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	−3.31401	−0.134734
$606$	0	0
$607$	1.07196	0.0435094	0.0217547	−	0.999763i	$-0.493075\pi$
$607$	1.07196	0.0435094	0.0217547	+	0.999763i	$0.493075\pi$
$608$	0	0
$609$	−1.22115	−0.0494836
$610$	0	0
$611$	−22.5069	−0.910530
$612$	0	0
$613$	19.7022	0.795765	0.397883	−	0.917436i	$-0.369745\pi$
$613$	19.7022	0.795765	0.397883	+	0.917436i	$0.369745\pi$
$614$	0	0
$615$	−7.12208	−0.287190
$616$	0	0
$617$	−9.84791	−0.396462	−0.198231	−	0.980155i	$-0.563520\pi$
$617$	−9.84791	−0.396462	−0.198231	+	0.980155i	$0.563520\pi$
$618$	0	0
$619$	5.39967	0.217031	0.108516	−	0.994095i	$-0.465390\pi$
$619$	5.39967	0.217031	0.108516	+	0.994095i	$0.465390\pi$
$620$	0	0
$621$	7.13944	0.286496
$622$	0	0
$623$	3.77467	0.151229
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	21.7154	0.867232
$628$	0	0
$629$	3.84622	0.153359
$630$	0	0
$631$	45.8600	1.82566	0.912829	−	0.408343i	$-0.133893\pi$
$631$	45.8600	1.82566	0.912829	+	0.408343i	$0.133893\pi$
$632$	0	0
$633$	24.8657	0.988324
$634$	0	0
$635$	−12.6572	−0.502286
$636$	0	0
$637$	−26.9516	−1.06786
$638$	0	0
$639$	−12.4491	−0.492478
$640$	0	0
$641$	−33.9181	−1.33969	−0.669843	−	0.742503i	$-0.733639\pi$
$641$	−33.9181	−1.33969	−0.669843	+	0.742503i	$0.733639\pi$
$642$	0	0
$643$	−2.51361	−0.0991272	−0.0495636	−	0.998771i	$-0.515783\pi$
$643$	−2.51361	−0.0991272	−0.0495636	+	0.998771i	$0.515783\pi$
$644$	0	0
$645$	3.02515	0.119115
$646$	0	0
$647$	27.6877	1.08852	0.544259	−	0.838917i	$-0.316811\pi$
$647$	27.6877	1.08852	0.544259	+	0.838917i	$0.316811\pi$
$648$	0	0
$649$	6.18499	0.242782
$650$	0	0
$651$	−3.43320	−0.134558
$652$	0	0
$653$	−39.6926	−1.55329	−0.776645	−	0.629938i	$-0.783080\pi$
$653$	−39.6926	−1.55329	−0.776645	+	0.629938i	$0.783080\pi$
$654$	0	0
$655$	13.7384	0.536805
$656$	0	0
$657$	−10.4317	−0.406978
$658$	0	0
$659$	−16.4133	−0.639370	−0.319685	−	0.947524i	$-0.603577\pi$
$659$	−16.4133	−0.639370	−0.319685	+	0.947524i	$0.603577\pi$
$660$	0	0
$661$	3.56972	0.138846	0.0694230	−	0.997587i	$-0.477884\pi$
$661$	3.56972	0.138846	0.0694230	+	0.997587i	$0.477884\pi$
$662$	0	0
$663$	7.56715	0.293884
$664$	0	0
$665$	−3.84716	−0.149187
$666$	0	0
$667$	−13.0071	−0.503638
$668$	0	0
$669$	−9.60254	−0.371256
$670$	0	0
$671$	20.8423	0.804607
$672$	0	0
$673$	−6.07992	−0.234364	−0.117182	−	0.993110i	$-0.537386\pi$
$673$	−6.07992	−0.234364	−0.117182	+	0.993110i	$0.537386\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−32.9303	−1.26561	−0.632807	−	0.774310i	$-0.718097\pi$
$677$	−32.9303	−1.26561	−0.632807	+	0.774310i	$0.718097\pi$
$678$	0	0
$679$	−6.57497	−0.252324
$680$	0	0
$681$	13.6064	0.521399
$682$	0	0
$683$	21.7093	0.830685	0.415342	−	0.909665i	$-0.363662\pi$
$683$	21.7093	0.830685	0.415342	+	0.909665i	$0.363662\pi$
$684$	0	0
$685$	10.2004	0.389736
$686$	0	0
$687$	3.78622	0.144453
$688$	0	0
$689$	28.2270	1.07536
$690$	0	0
$691$	48.7214	1.85345	0.926724	−	0.375742i	$-0.122612\pi$
$691$	48.7214	1.85345	0.926724	+	0.375742i	$0.122612\pi$
$692$	0	0
$693$	2.53591	0.0963311
$694$	0	0
$695$	12.0665	0.457708
$696$	0	0
$697$	13.0992	0.496166
$698$	0	0
$699$	13.4793	0.509835
$700$	0	0
$701$	−11.2299	−0.424146	−0.212073	−	0.977254i	$-0.568021\pi$
$701$	−11.2299	−0.424146	−0.212073	+	0.977254i	$0.568021\pi$
$702$	0	0
$703$	12.0029	0.452697
$704$	0	0
$705$	5.47041	0.206027
$706$	0	0
$707$	2.32858	0.0875753
$708$	0	0
$709$	−6.42425	−0.241268	−0.120634	−	0.992697i	$-0.538493\pi$
$709$	−6.42425	−0.241268	−0.120634	+	0.992697i	$0.538493\pi$
$710$	0	0
$711$	7.01717	0.263164
$712$	0	0
$713$	−36.5688	−1.36951
$714$	0	0
$715$	−15.5660	−0.582134
$716$	0	0
$717$	−7.81123	−0.291716
$718$	0	0
$719$	−14.1488	−0.527662	−0.263831	−	0.964569i	$-0.584986\pi$
$719$	−14.1488	−0.527662	−0.263831	+	0.964569i	$0.584986\pi$
$720$	0	0
$721$	10.8395	0.403683
$722$	0	0
$723$	13.8308	0.514375
$724$	0	0
$725$	−1.82187	−0.0676625
$726$	0	0
$727$	0.0186600	0.000692059 0	0.000346030	−	1.00000i	$-0.499890\pi$
$727$	0.0186600	0.000692059 0	0.000346030		1.00000i	$0.499890\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.56396	−0.205790
$732$	0	0
$733$	−38.2015	−1.41101	−0.705503	−	0.708707i	$-0.749279\pi$
$733$	−38.2015	−1.41101	−0.705503	+	0.708707i	$0.749279\pi$
$734$	0	0
$735$	6.55073	0.241627
$736$	0	0
$737$	3.78339	0.139363
$738$	0	0
$739$	−17.1980	−0.632640	−0.316320	−	0.948653i	$-0.602447\pi$
$739$	−17.1980	−0.632640	−0.316320	+	0.948653i	$0.602447\pi$
$740$	0	0
$741$	23.6147	0.867509
$742$	0	0
$743$	−33.1297	−1.21541	−0.607705	−	0.794163i	$-0.707910\pi$
$743$	−33.1297	−1.21541	−0.607705	+	0.794163i	$0.707910\pi$
$744$	0	0
$745$	1.31188	0.0480636
$746$	0	0
$747$	−5.16000	−0.188794
$748$	0	0
$749$	−12.0002	−0.438477
$750$	0	0
$751$	−31.0531	−1.13314	−0.566572	−	0.824012i	$-0.691731\pi$
$751$	−31.0531	−1.13314	−0.566572	+	0.824012i	$0.691731\pi$
$752$	0	0
$753$	−13.7731	−0.501920
$754$	0	0
$755$	−3.61324	−0.131499
$756$	0	0
$757$	−23.1862	−0.842716	−0.421358	−	0.906894i	$-0.638446\pi$
$757$	−23.1862	−0.842716	−0.421358	+	0.906894i	$0.638446\pi$
$758$	0	0
$759$	27.0113	0.980447
$760$	0	0
$761$	−2.58898	−0.0938503	−0.0469251	−	0.998898i	$-0.514942\pi$
$761$	−2.58898	−0.0938503	−0.0469251	+	0.998898i	$0.514942\pi$
$762$	0	0
$763$	11.6423	0.421479
$764$	0	0
$765$	−1.83923	−0.0664976
$766$	0	0
$767$	6.72594	0.242860
$768$	0	0
$769$	−23.4029	−0.843929	−0.421964	−	0.906612i	$-0.638659\pi$
$769$	−23.4029	−0.843929	−0.421964	+	0.906612i	$0.638659\pi$
$770$	0	0
$771$	20.1620	0.726118
$772$	0	0
$773$	−30.9422	−1.11292	−0.556458	−	0.830876i	$-0.687840\pi$
$773$	−30.9422	−1.11292	−0.556458	+	0.830876i	$0.687840\pi$
$774$	0	0
$775$	−5.12208	−0.183990
$776$	0	0
$777$	1.40168	0.0502851
$778$	0	0
$779$	40.8785	1.46462
$780$	0	0
$781$	−47.0997	−1.68536
$782$	0	0
$783$	−1.82187	−0.0651083
$784$	0	0
$785$	−5.72452	−0.204317
$786$	0	0
$787$	−44.1093	−1.57233	−0.786163	−	0.618019i	$-0.787935\pi$
$787$	−44.1093	−1.57233	−0.786163	+	0.618019i	$0.787935\pi$
$788$	0	0
$789$	0.295696	0.0105271
$790$	0	0
$791$	−12.9749	−0.461336
$792$	0	0
$793$	22.6652	0.804864
$794$	0	0
$795$	−6.86073	−0.243325
$796$	0	0
$797$	−30.2678	−1.07214	−0.536070	−	0.844173i	$-0.680092\pi$
$797$	−30.2678	−1.07214	−0.536070	+	0.844173i	$0.680092\pi$
$798$	0	0
$799$	−10.0614	−0.355945
$800$	0	0
$801$	5.63153	0.198980
$802$	0	0
$803$	−39.4670	−1.39276
$804$	0	0
$805$	−4.78538	−0.168663
$806$	0	0
$807$	−19.4531	−0.684781
$808$	0	0
$809$	−20.5517	−0.722561	−0.361280	−	0.932457i	$-0.617660\pi$
$809$	−20.5517	−0.722561	−0.361280	+	0.932457i	$0.617660\pi$
$810$	0	0
$811$	−23.4426	−0.823182	−0.411591	−	0.911369i	$-0.635027\pi$
$811$	−23.4426	−0.823182	−0.411591	+	0.911369i	$0.635027\pi$
$812$	0	0
$813$	−10.8772	−0.381480
$814$	0	0
$815$	−14.8151	−0.518951
$816$	0	0
$817$	−17.3634	−0.607469
$818$	0	0
$819$	2.75770	0.0963620
$820$	0	0
$821$	−27.1275	−0.946755	−0.473377	−	0.880860i	$-0.656965\pi$
$821$	−27.1275	−0.946755	−0.473377	+	0.880860i	$0.656965\pi$
$822$	0	0
$823$	−13.6446	−0.475622	−0.237811	−	0.971311i	$-0.576430\pi$
$823$	−13.6446	−0.475622	−0.237811	+	0.971311i	$0.576430\pi$
$824$	0	0
$825$	3.78339	0.131721
$826$	0	0
$827$	−37.5205	−1.30472	−0.652358	−	0.757911i	$-0.726220\pi$
$827$	−37.5205	−1.30472	−0.652358	+	0.757911i	$0.726220\pi$
$828$	0	0
$829$	−9.84587	−0.341961	−0.170981	−	0.985274i	$-0.554694\pi$
$829$	−9.84587	−0.341961	−0.170981	+	0.985274i	$0.554694\pi$
$830$	0	0
$831$	25.1608	0.872818
$832$	0	0
$833$	−12.0483	−0.417450
$834$	0	0
$835$	3.97042	0.137402
$836$	0	0
$837$	−5.12208	−0.177045
$838$	0	0
$839$	0.762629	0.0263289	0.0131644	−	0.999913i	$-0.495810\pi$
$839$	0.762629	0.0263289	0.0131644	+	0.999913i	$0.495810\pi$
$840$	0	0
$841$	−25.6808	−0.885545
$842$	0	0
$843$	8.91436	0.307027
$844$	0	0
$845$	−3.92740	−0.135107
$846$	0	0
$847$	2.22130	0.0763247
$848$	0	0
$849$	−19.4711	−0.668248
$850$	0	0
$851$	14.9301	0.511796
$852$	0	0
$853$	42.8841	1.46832	0.734162	−	0.678974i	$-0.237575\pi$
$853$	42.8841	1.46832	0.734162	+	0.678974i	$0.237575\pi$
$854$	0	0
$855$	−5.73969	−0.196293
$856$	0	0
$857$	35.6280	1.21703	0.608514	−	0.793543i	$-0.291766\pi$
$857$	35.6280	1.21703	0.608514	+	0.793543i	$0.291766\pi$
$858$	0	0
$859$	7.95039	0.271264	0.135632	−	0.990759i	$-0.456694\pi$
$859$	7.95039	0.271264	0.135632	+	0.990759i	$0.456694\pi$
$860$	0	0
$861$	4.77374	0.162689
$862$	0	0
$863$	54.3602	1.85044	0.925221	−	0.379430i	$-0.123880\pi$
$863$	54.3602	1.85044	0.925221	+	0.379430i	$0.123880\pi$
$864$	0	0
$865$	−24.2043	−0.822971
$866$	0	0
$867$	−13.6172	−0.462465
$868$	0	0
$869$	26.5486	0.900601
$870$	0	0
$871$	4.11429	0.139407
$872$	0	0
$873$	−9.80937	−0.331997
$874$	0	0
$875$	−0.670274	−0.0226594
$876$	0	0
$877$	10.3772	0.350413	0.175207	−	0.984532i	$-0.443941\pi$
$877$	10.3772	0.350413	0.175207	+	0.984532i	$0.443941\pi$
$878$	0	0
$879$	−10.3591	−0.349404
$880$	0	0
$881$	25.8167	0.869786	0.434893	−	0.900482i	$-0.356786\pi$
$881$	25.8167	0.869786	0.434893	+	0.900482i	$0.356786\pi$
$882$	0	0
$883$	1.52466	0.0513088	0.0256544	−	0.999671i	$-0.491833\pi$
$883$	1.52466	0.0513088	0.0256544	+	0.999671i	$0.491833\pi$
$884$	0	0
$885$	−1.63478	−0.0549524
$886$	0	0
$887$	13.2881	0.446172	0.223086	−	0.974799i	$-0.428387\pi$
$887$	13.2881	0.446172	0.223086	+	0.974799i	$0.428387\pi$
$888$	0	0
$889$	8.48380	0.284538
$890$	0	0
$891$	3.78339	0.126748
$892$	0	0
$893$	−31.3984	−1.05071
$894$	0	0
$895$	9.26827	0.309804
$896$	0	0
$897$	29.3737	0.980761
$898$	0	0
$899$	9.33175	0.311231
$900$	0	0
$901$	12.6185	0.420383
$902$	0	0
$903$	−2.02768	−0.0674769
$904$	0	0
$905$	24.7538	0.822845
$906$	0	0
$907$	−38.0633	−1.26387	−0.631936	−	0.775021i	$-0.717739\pi$
$907$	−38.0633	−1.26387	−0.631936	+	0.775021i	$0.717739\pi$
$908$	0	0
$909$	3.47407	0.115228
$910$	0	0
$911$	−16.9805	−0.562589	−0.281295	−	0.959621i	$-0.590764\pi$
$911$	−16.9805	−0.562589	−0.281295	+	0.959621i	$0.590764\pi$
$912$	0	0
$913$	−19.5223	−0.646093
$914$	0	0
$915$	−5.50889	−0.182118
$916$	0	0
$917$	−9.20851	−0.304092
$918$	0	0
$919$	19.4705	0.642272	0.321136	−	0.947033i	$-0.395935\pi$
$919$	19.4705	0.642272	0.321136	+	0.947033i	$0.395935\pi$
$920$	0	0
$921$	25.7029	0.846939
$922$	0	0
$923$	−51.2192	−1.68590
$924$	0	0
$925$	2.09121	0.0687585
$926$	0	0
$927$	16.1717	0.531148
$928$	0	0
$929$	−43.5080	−1.42745	−0.713725	−	0.700426i	$-0.752993\pi$
$929$	−43.5080	−1.42745	−0.713725	+	0.700426i	$0.752993\pi$
$930$	0	0
$931$	−37.5991	−1.23226
$932$	0	0
$933$	−2.07146	−0.0678165
$934$	0	0
$935$	−6.95853	−0.227568
$936$	0	0
$937$	16.6446	0.543756	0.271878	−	0.962332i	$-0.412355\pi$
$937$	16.6446	0.543756	0.271878	+	0.962332i	$0.412355\pi$
$938$	0	0
$939$	11.9262	0.389198
$940$	0	0
$941$	5.64480	0.184015	0.0920077	−	0.995758i	$-0.470672\pi$
$941$	5.64480	0.184015	0.0920077	+	0.995758i	$0.470672\pi$
$942$	0	0
$943$	50.8476	1.65583
$944$	0	0
$945$	−0.670274	−0.0218040
$946$	0	0
$947$	−44.4840	−1.44553	−0.722767	−	0.691092i	$-0.757130\pi$
$947$	−44.4840	−1.44553	−0.722767	+	0.691092i	$0.757130\pi$
$948$	0	0
$949$	−42.9189	−1.39321
$950$	0	0
$951$	−32.0644	−1.03976
$952$	0	0
$953$	−10.4655	−0.339012	−0.169506	−	0.985529i	$-0.554217\pi$
$953$	−10.4655	−0.339012	−0.169506	+	0.985529i	$0.554217\pi$
$954$	0	0
$955$	−1.65532	−0.0535650
$956$	0	0
$957$	−6.89284	−0.222814
$958$	0	0
$959$	−6.83704	−0.220779
$960$	0	0
$961$	−4.76434	−0.153688
$962$	0	0
$963$	−17.9034	−0.576929
$964$	0	0
$965$	13.2421	0.426280
$966$	0	0
$967$	47.4772	1.52676	0.763382	−	0.645948i	$-0.223538\pi$
$967$	47.4772	1.52676	0.763382	+	0.645948i	$0.223538\pi$
$968$	0	0
$969$	10.5566	0.339128
$970$	0	0
$971$	38.5501	1.23713	0.618567	−	0.785732i	$-0.287714\pi$
$971$	38.5501	1.23713	0.618567	+	0.785732i	$0.287714\pi$
$972$	0	0
$973$	−8.08786	−0.259285
$974$	0	0
$975$	4.11429	0.131763
$976$	0	0
$977$	18.3124	0.585866	0.292933	−	0.956133i	$-0.405369\pi$
$977$	18.3124	0.585866	0.292933	+	0.956133i	$0.405369\pi$
$978$	0	0
$979$	21.3063	0.680951
$980$	0	0
$981$	17.3694	0.554563
$982$	0	0
$983$	16.1671	0.515650	0.257825	−	0.966192i	$-0.416994\pi$
$983$	16.1671	0.515650	0.257825	+	0.966192i	$0.416994\pi$
$984$	0	0
$985$	3.51881	0.112118
$986$	0	0
$987$	−3.66667	−0.116711
$988$	0	0
$989$	−21.5979	−0.686772
$990$	0	0
$991$	57.2685	1.81919	0.909597	−	0.415492i	$-0.136391\pi$
$991$	57.2685	1.81919	0.909597	+	0.415492i	$0.136391\pi$
$992$	0	0
$993$	10.3465	0.328337
$994$	0	0
$995$	−13.9365	−0.441817
$996$	0	0
$997$	43.2429	1.36952	0.684758	−	0.728771i	$-0.259908\pi$
$997$	43.2429	1.36952	0.684758	+	0.728771i	$0.259908\pi$
$998$	0	0
$999$	2.09121	0.0661629

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8040.2.a.t.1.4	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8040.2.a.t.1.4	✓	7	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:	\( N \)	\(=\)	\( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8040.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +0.670274 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +0.670274 q^{7} +1.00000 q^{9} +3.78339 q^{11} +4.11429 q^{13} -1.00000 q^{15} +1.83923 q^{17} +5.73969 q^{19} +0.670274 q^{21} +7.13944 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.82187 q^{29} -5.12208 q^{31} +3.78339 q^{33} -0.670274 q^{35} +2.09121 q^{37} +4.11429 q^{39} +7.12208 q^{41} -3.02515 q^{43} -1.00000 q^{45} -5.47041 q^{47} -6.55073 q^{49} +1.83923 q^{51} +6.86073 q^{53} -3.78339 q^{55} +5.73969 q^{57} +1.63478 q^{59} +5.50889 q^{61} +0.670274 q^{63} -4.11429 q^{65} +1.00000 q^{67} +7.13944 q^{69} -12.4491 q^{71} -10.4317 q^{73} +1.00000 q^{75} +2.53591 q^{77} +7.01717 q^{79} +1.00000 q^{81} -5.16000 q^{83} -1.83923 q^{85} -1.82187 q^{87} +5.63153 q^{89} +2.75770 q^{91} -5.12208 q^{93} -5.73969 q^{95} -9.80937 q^{97} +3.78339 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9	\( 7 q + 7 q^{3} - 7 q^{5} + 10 q^{7} + 7 q^{9} - q^{13} - 7 q^{15} - 2 q^{17} + 9 q^{19} + 10 q^{21} + 2 q^{23} + 7 q^{25} + 7 q^{27} - q^{29} + 9 q^{31} - 10 q^{35} + 23 q^{37} - q^{39} + 5 q^{41} - 3 q^{43} - 7 q^{45} + 11 q^{47} + 13 q^{49} - 2 q^{51} + 13 q^{53} + 9 q^{57} + q^{59} + 4 q^{61} + 10 q^{63} + q^{65} + 7 q^{67} + 2 q^{69} + q^{71} + 14 q^{73} + 7 q^{75} + 18 q^{77} + 25 q^{79} + 7 q^{81} - 29 q^{83} + 2 q^{85} - q^{87} + 7 q^{89} + 27 q^{91} + 9 q^{93} - 9 q^{95} + 38 q^{97}+O(q^{100}) \) 7 * q + 7 * q^3 - 7 * q^5 + 10 * q^7 + 7 * q^9 - q^13 - 7 * q^15 - 2 * q^17 + 9 * q^19 + 10 * q^21 + 2 * q^23 + 7 * q^25 + 7 * q^27 - q^29 + 9 * q^31 - 10 * q^35 + 23 * q^37 - q^39 + 5 * q^41 - 3 * q^43 - 7 * q^45 + 11 * q^47 + 13 * q^49 - 2 * q^51 + 13 * q^53 + 9 * q^57 + q^59 + 4 * q^61 + 10 * q^63 + q^65 + 7 * q^67 + 2 * q^69 + q^71 + 14 * q^73 + 7 * q^75 + 18 * q^77 + 25 * q^79 + 7 * q^81 - 29 * q^83 + 2 * q^85 - q^87 + 7 * q^89 + 27 * q^91 + 9 * q^93 - 9 * q^95 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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