LMFDB - Embedded newform 790.6.a.e.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [790,6,Mod(1,790)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("790.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(790, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 790 = 2 \cdot 5 \cdot 79 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	790.a (trivial)

Newform invariants

comment:select newform

sage:traces = [17,-68,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$126.703217652$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 5 x^{16} - 2808 x^{15} + 13814 x^{14} + 3138420 x^{13} - 13599532 x^{12} + \cdots + 28\!\cdots\!40 $ x^17 - 5x^16 - 2808x^15 + 13814x^14 + 3138420x^13 - 13599532x^12 - 1794235241x^11 + 5842269565x^10 + 561598097426x^9 - 1037253761968x^8 - 93966714227253x^7 + 31556306807371x^6 + 7318090515571747x^5 + 6375769962622993x^4 - 170883861722576126x^3 - 55033009686120922x^2 + 422953390225755138x + 281395624894688340
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.87160$ of defining polynomial
Character	$\chi$	$=$	790.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} -2.87160 q^{3} +16.0000 q^{4} +25.0000 q^{5} +11.4864 q^{6} -101.453 q^{7} -64.0000 q^{8} -234.754 q^{9} -100.000 q^{10} +444.280 q^{11} -45.9456 q^{12} -702.437 q^{13} +405.812 q^{14} -71.7899 q^{15} +256.000 q^{16} +794.711 q^{17} +939.016 q^{18} -2194.30 q^{19} +400.000 q^{20} +291.332 q^{21} -1777.12 q^{22} -2707.30 q^{23} +183.782 q^{24} +625.000 q^{25} +2809.75 q^{26} +1371.92 q^{27} -1623.25 q^{28} -5361.62 q^{29} +287.160 q^{30} +8632.57 q^{31} -1024.00 q^{32} -1275.79 q^{33} -3178.85 q^{34} -2536.32 q^{35} -3756.06 q^{36} -12353.9 q^{37} +8777.18 q^{38} +2017.12 q^{39} -1600.00 q^{40} -948.749 q^{41} -1165.33 q^{42} +5053.83 q^{43} +7108.49 q^{44} -5868.85 q^{45} +10829.2 q^{46} +573.732 q^{47} -735.129 q^{48} -6514.29 q^{49} -2500.00 q^{50} -2282.09 q^{51} -11239.0 q^{52} -5801.59 q^{53} -5487.67 q^{54} +11107.0 q^{55} +6492.99 q^{56} +6301.14 q^{57} +21446.5 q^{58} -11221.7 q^{59} -1148.64 q^{60} +48396.5 q^{61} -34530.3 q^{62} +23816.5 q^{63} +4096.00 q^{64} -17560.9 q^{65} +5103.18 q^{66} +5341.47 q^{67} +12715.4 q^{68} +7774.27 q^{69} +10145.3 q^{70} +3051.21 q^{71} +15024.3 q^{72} +1279.40 q^{73} +49415.8 q^{74} -1794.75 q^{75} -35108.7 q^{76} -45073.6 q^{77} -8068.46 q^{78} -6241.00 q^{79} +6400.00 q^{80} +53105.6 q^{81} +3795.00 q^{82} +70537.8 q^{83} +4661.31 q^{84} +19867.8 q^{85} -20215.3 q^{86} +15396.4 q^{87} -28434.0 q^{88} -29243.4 q^{89} +23475.4 q^{90} +71264.3 q^{91} -43316.8 q^{92} -24789.3 q^{93} -2294.93 q^{94} -54857.4 q^{95} +2940.52 q^{96} +18133.0 q^{97} +26057.2 q^{98} -104297. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 68 q^{2} - 22 q^{3} + 272 q^{4} + 425 q^{5} + 88 q^{6} - 4 q^{7} - 1088 q^{8} + 1537 q^{9} - 1700 q^{10} + 426 q^{11} - 352 q^{12} + 750 q^{13} + 16 q^{14} - 550 q^{15} + 4352 q^{16} + 2074 q^{17}+ \cdots + 126104 q^{99}+O(q^{100}) $ 17 * q - 68 * q^2 - 22 * q^3 + 272 * q^4 + 425 * q^5 + 88 * q^6 - 4 * q^7 - 1088 * q^8 + 1537 * q^9 - 1700 * q^10 + 426 * q^11 - 352 * q^12 + 750 * q^13 + 16 * q^14 - 550 * q^15 + 4352 * q^16 + 2074 * q^17 - 6148 * q^18 + 1546 * q^19 + 6800 * q^20 + 4478 * q^21 - 1704 * q^22 - 6366 * q^23 + 1408 * q^24 + 10625 * q^25 - 3000 * q^26 - 7066 * q^27 - 64 * q^28 + 13612 * q^29 + 2200 * q^30 + 4578 * q^31 - 17408 * q^32 + 1274 * q^33 - 8296 * q^34 - 100 * q^35 + 24592 * q^36 + 6908 * q^37 - 6184 * q^38 + 49306 * q^39 - 27200 * q^40 + 39394 * q^41 - 17912 * q^42 + 12302 * q^43 + 6816 * q^44 + 38425 * q^45 + 25464 * q^46 - 3814 * q^47 - 5632 * q^48 + 76123 * q^49 - 42500 * q^50 + 26042 * q^51 + 12000 * q^52 + 103642 * q^53 + 28264 * q^54 + 10650 * q^55 + 256 * q^56 + 35532 * q^57 - 54448 * q^58 + 88660 * q^59 - 8800 * q^60 + 75624 * q^61 - 18312 * q^62 + 80298 * q^63 + 69632 * q^64 + 18750 * q^65 - 5096 * q^66 - 82964 * q^67 + 33184 * q^68 + 77392 * q^69 + 400 * q^70 + 96816 * q^71 - 98368 * q^72 + 36940 * q^73 - 27632 * q^74 - 13750 * q^75 + 24736 * q^76 + 53744 * q^77 - 197224 * q^78 - 106097 * q^79 + 108800 * q^80 + 130149 * q^81 - 157576 * q^82 + 11764 * q^83 + 71648 * q^84 + 51850 * q^85 - 49208 * q^86 - 232276 * q^87 - 27264 * q^88 + 195846 * q^89 - 153700 * q^90 - 217334 * q^91 - 101856 * q^92 + 53668 * q^93 + 15256 * q^94 + 38650 * q^95 + 22528 * q^96 - 261292 * q^97 - 304492 * q^98 + 126104 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	−2.87160	−0.184213	−0.0921065	−	0.995749i	$-0.529360\pi$
$3$	−2.87160	−0.184213	−0.0921065	+	0.995749i	$0.529360\pi$
$4$	16.0000	0.500000
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	11.4864	0.130258
$7$	−101.453	−0.782564	−0.391282	−	0.920271i	$-0.627968\pi$
$7$	−101.453	−0.782564	−0.391282	+	0.920271i	$0.627968\pi$
$8$	−64.0000	−0.353553
$9$	−234.754	−0.966066
$10$	−100.000	−0.316228
$11$	444.280	1.10707	0.553536	−	0.832826i	$-0.313278\pi$
$11$	444.280	1.10707	0.553536	+	0.832826i	$0.313278\pi$
$12$	−45.9456	−0.0921065
$13$	−702.437	−1.15279	−0.576393	−	0.817173i	$-0.695540\pi$
$13$	−702.437	−1.15279	−0.576393	+	0.817173i	$0.695540\pi$
$14$	405.812	0.553356
$15$	−71.7899	−0.0823826
$16$	256.000	0.250000
$17$	794.711	0.666941	0.333470	−	0.942761i	$-0.391780\pi$
$17$	794.711	0.666941	0.333470	+	0.942761i	$0.391780\pi$
$18$	939.016	0.683111
$19$	−2194.30	−1.39448	−0.697238	−	0.716839i	$-0.745588\pi$
$19$	−2194.30	−1.39448	−0.697238	+	0.716839i	$0.745588\pi$
$20$	400.000	0.223607
$21$	291.332	0.144158
$22$	−1777.12	−0.782817
$23$	−2707.30	−1.06713	−0.533564	−	0.845760i	$-0.679148\pi$
$23$	−2707.30	−1.06713	−0.533564	+	0.845760i	$0.679148\pi$
$24$	183.782	0.0651292
$25$	625.000	0.200000
$26$	2809.75	0.815143
$27$	1371.92	0.362175
$28$	−1623.25	−0.391282
$29$	−5361.62	−1.18386	−0.591931	−	0.805988i	$-0.701634\pi$
$29$	−5361.62	−1.18386	−0.591931	+	0.805988i	$0.701634\pi$
$30$	287.160	0.0582533
$31$	8632.57	1.61338	0.806689	−	0.590977i	$-0.201258\pi$
$31$	8632.57	1.61338	0.806689	+	0.590977i	$0.201258\pi$
$32$	−1024.00	−0.176777
$33$	−1275.79	−0.203937
$34$	−3178.85	−0.471598
$35$	−2536.32	−0.349973
$36$	−3756.06	−0.483033
$37$	−12353.9	−1.48355	−0.741773	−	0.670651i	$-0.766015\pi$
$37$	−12353.9	−1.48355	−0.741773	+	0.670651i	$0.766015\pi$
$38$	8777.18	0.986044
$39$	2017.12	0.212358
$40$	−1600.00	−0.158114
$41$	−948.749	−0.0881438	−0.0440719	−	0.999028i	$-0.514033\pi$
$41$	−948.749	−0.0881438	−0.0440719	+	0.999028i	$0.514033\pi$
$42$	−1165.33	−0.101935
$43$	5053.83	0.416821	0.208410	−	0.978041i	$-0.433171\pi$
$43$	5053.83	0.416821	0.208410	+	0.978041i	$0.433171\pi$
$44$	7108.49	0.553536
$45$	−5868.85	−0.432038
$46$	10829.2	0.754573
$47$	573.732	0.0378848	0.0189424	−	0.999821i	$-0.493970\pi$
$47$	573.732	0.0378848	0.0189424	+	0.999821i	$0.493970\pi$
$48$	−735.129	−0.0460533
$49$	−6514.29	−0.387594
$50$	−2500.00	−0.141421
$51$	−2282.09	−0.122859
$52$	−11239.0	−0.576393
$53$	−5801.59	−0.283699	−0.141849	−	0.989888i	$-0.545305\pi$
$53$	−5801.59	−0.283699	−0.141849	+	0.989888i	$0.545305\pi$
$54$	−5487.67	−0.256096
$55$	11107.0	0.495097
$56$	6492.99	0.276678
$57$	6301.14	0.256881
$58$	21446.5	0.837117
$59$	−11221.7	−0.419690	−0.209845	−	0.977735i	$-0.567296\pi$
$59$	−11221.7	−0.419690	−0.209845	+	0.977735i	$0.567296\pi$
$60$	−1148.64	−0.0411913
$61$	48396.5	1.66529	0.832644	−	0.553809i	$-0.186826\pi$
$61$	48396.5	1.66529	0.832644	+	0.553809i	$0.186826\pi$
$62$	−34530.3	−1.14083
$63$	23816.5	0.756008
$64$	4096.00	0.125000
$65$	−17560.9	−0.515542
$66$	5103.18	0.144205
$67$	5341.47	0.145370	0.0726848	−	0.997355i	$-0.476843\pi$
$67$	5341.47	0.145370	0.0726848	+	0.997355i	$0.476843\pi$
$68$	12715.4	0.333470
$69$	7774.27	0.196579
$70$	10145.3	0.247468
$71$	3051.21	0.0718334	0.0359167	−	0.999355i	$-0.488565\pi$
$71$	3051.21	0.0718334	0.0359167	+	0.999355i	$0.488565\pi$
$72$	15024.3	0.341556
$73$	1279.40	0.0280996	0.0140498	−	0.999901i	$-0.495528\pi$
$73$	1279.40	0.0280996	0.0140498	+	0.999901i	$0.495528\pi$
$74$	49415.8	1.04903
$75$	−1794.75	−0.0368426
$76$	−35108.7	−0.697238
$77$	−45073.6	−0.866354
$78$	−8068.46	−0.150160
$79$	−6241.00	−0.112509
$79$	−6241.00	−0.112509
$80$	6400.00	0.111803
$81$	53105.6	0.899348
$82$	3795.00	0.0623271
$83$	70537.8	1.12390	0.561948	−	0.827172i	$-0.310052\pi$
$83$	70537.8	1.12390	0.561948	+	0.827172i	$0.310052\pi$
$84$	4661.31	0.0720792
$85$	19867.8	0.298265
$86$	−20215.3	−0.294737
$87$	15396.4	0.218083
$88$	−28434.0	−0.391409
$89$	−29243.4	−0.391339	−0.195669	−	0.980670i	$-0.562688\pi$
$89$	−29243.4	−0.391339	−0.195669	+	0.980670i	$0.562688\pi$
$90$	23475.4	0.305497
$91$	71264.3	0.902129
$92$	−43316.8	−0.533564
$93$	−24789.3	−0.297205
$94$	−2294.93	−0.0267886
$95$	−54857.4	−0.623629
$96$	2940.52	0.0325646
$97$	18133.0	0.195678	0.0978388	−	0.995202i	$-0.468807\pi$
$97$	18133.0	0.195678	0.0978388	+	0.995202i	$0.468807\pi$
$98$	26057.2	0.274070
$99$	−104297.	−1.06950

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	790.6.a.e.1.8	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
790.6.a.e.1.8	✓	17	1.1	even	1	trivial

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 790 = 2 \cdot 5 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	790.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} -2.87160 q^{3} +16.0000 q^{4} +25.0000 q^{5} +11.4864 q^{6} -101.453 q^{7} -64.0000 q^{8} -234.754 q^{9} -100.000 q^{10} +444.280 q^{11} -45.9456 q^{12} -702.437 q^{13} +405.812 q^{14} -71.7899 q^{15} +256.000 q^{16} +794.711 q^{17} +939.016 q^{18} -2194.30 q^{19} +400.000 q^{20} +291.332 q^{21} -1777.12 q^{22} -2707.30 q^{23} +183.782 q^{24} +625.000 q^{25} +2809.75 q^{26} +1371.92 q^{27} -1623.25 q^{28} -5361.62 q^{29} +287.160 q^{30} +8632.57 q^{31} -1024.00 q^{32} -1275.79 q^{33} -3178.85 q^{34} -2536.32 q^{35} -3756.06 q^{36} -12353.9 q^{37} +8777.18 q^{38} +2017.12 q^{39} -1600.00 q^{40} -948.749 q^{41} -1165.33 q^{42} +5053.83 q^{43} +7108.49 q^{44} -5868.85 q^{45} +10829.2 q^{46} +573.732 q^{47} -735.129 q^{48} -6514.29 q^{49} -2500.00 q^{50} -2282.09 q^{51} -11239.0 q^{52} -5801.59 q^{53} -5487.67 q^{54} +11107.0 q^{55} +6492.99 q^{56} +6301.14 q^{57} +21446.5 q^{58} -11221.7 q^{59} -1148.64 q^{60} +48396.5 q^{61} -34530.3 q^{62} +23816.5 q^{63} +4096.00 q^{64} -17560.9 q^{65} +5103.18 q^{66} +5341.47 q^{67} +12715.4 q^{68} +7774.27 q^{69} +10145.3 q^{70} +3051.21 q^{71} +15024.3 q^{72} +1279.40 q^{73} +49415.8 q^{74} -1794.75 q^{75} -35108.7 q^{76} -45073.6 q^{77} -8068.46 q^{78} -6241.00 q^{79} +6400.00 q^{80} +53105.6 q^{81} +3795.00 q^{82} +70537.8 q^{83} +4661.31 q^{84} +19867.8 q^{85} -20215.3 q^{86} +15396.4 q^{87} -28434.0 q^{88} -29243.4 q^{89} +23475.4 q^{90} +71264.3 q^{91} -43316.8 q^{92} -24789.3 q^{93} -2294.93 q^{94} -54857.4 q^{95} +2940.52 q^{96} +18133.0 q^{97} +26057.2 q^{98} -104297. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 68 q^{2} - 22 q^{3} + 272 q^{4} + 425 q^{5} + 88 q^{6} - 4 q^{7} - 1088 q^{8} + 1537 q^{9} - 1700 q^{10} + 426 q^{11} - 352 q^{12} + 750 q^{13} + 16 q^{14} - 550 q^{15} + 4352 q^{16} + 2074 q^{17}+ \cdots + 126104 q^{99}+O(q^{100}) \) 17 * q - 68 * q^2 - 22 * q^3 + 272 * q^4 + 425 * q^5 + 88 * q^6 - 4 * q^7 - 1088 * q^8 + 1537 * q^9 - 1700 * q^10 + 426 * q^11 - 352 * q^12 + 750 * q^13 + 16 * q^14 - 550 * q^15 + 4352 * q^16 + 2074 * q^17 - 6148 * q^18 + 1546 * q^19 + 6800 * q^20 + 4478 * q^21 - 1704 * q^22 - 6366 * q^23 + 1408 * q^24 + 10625 * q^25 - 3000 * q^26 - 7066 * q^27 - 64 * q^28 + 13612 * q^29 + 2200 * q^30 + 4578 * q^31 - 17408 * q^32 + 1274 * q^33 - 8296 * q^34 - 100 * q^35 + 24592 * q^36 + 6908 * q^37 - 6184 * q^38 + 49306 * q^39 - 27200 * q^40 + 39394 * q^41 - 17912 * q^42 + 12302 * q^43 + 6816 * q^44 + 38425 * q^45 + 25464 * q^46 - 3814 * q^47 - 5632 * q^48 + 76123 * q^49 - 42500 * q^50 + 26042 * q^51 + 12000 * q^52 + 103642 * q^53 + 28264 * q^54 + 10650 * q^55 + 256 * q^56 + 35532 * q^57 - 54448 * q^58 + 88660 * q^59 - 8800 * q^60 + 75624 * q^61 - 18312 * q^62 + 80298 * q^63 + 69632 * q^64 + 18750 * q^65 - 5096 * q^66 - 82964 * q^67 + 33184 * q^68 + 77392 * q^69 + 400 * q^70 + 96816 * q^71 - 98368 * q^72 + 36940 * q^73 - 27632 * q^74 - 13750 * q^75 + 24736 * q^76 + 53744 * q^77 - 197224 * q^78 - 106097 * q^79 + 108800 * q^80 + 130149 * q^81 - 157576 * q^82 + 11764 * q^83 + 71648 * q^84 + 51850 * q^85 - 49208 * q^86 - 232276 * q^87 - 27264 * q^88 + 195846 * q^89 - 153700 * q^90 - 217334 * q^91 - 101856 * q^92 + 53668 * q^93 + 15256 * q^94 + 38650 * q^95 + 22528 * q^96 - 261292 * q^97 - 304492 * q^98 + 126104 * q^99

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