LMFDB - Embedded newform 790.6.a.d.1.7

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 = [15,60,-26] 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:	$1$
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} - 4 x^{14} - 2364 x^{13} + 6706 x^{12} + 2211698 x^{11} - 3792234 x^{10} - 1046418217 x^{9} + \cdots - 64\!\cdots\!80 $ x^15 - 4x^14 - 2364x^13 + 6706x^12 + 2211698x^11 - 3792234x^10 - 1046418217x^9 + 627531604x^8 + 264767532093x^7 + 124409909828x^6 - 34668973086029x^5 - 45509869153326x^4 + 2061334836635454x^3 + 3210313317981444x^2 - 42787781024724504x - 64913202418067280
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-6.74656$ 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)

$f(q)$	$=$	\(q+4.00000 q^{2} -8.74656 q^{3} +16.0000 q^{4} -25.0000 q^{5} -34.9863 q^{6} -48.9370 q^{7} +64.0000 q^{8} -166.498 q^{9} -100.000 q^{10} -367.183 q^{11} -139.945 q^{12} +627.041 q^{13} -195.748 q^{14} +218.664 q^{15} +256.000 q^{16} +664.445 q^{17} -665.991 q^{18} +1872.03 q^{19} -400.000 q^{20} +428.031 q^{21} -1468.73 q^{22} +2185.31 q^{23} -559.780 q^{24} +625.000 q^{25} +2508.17 q^{26} +3581.70 q^{27} -782.993 q^{28} -5478.52 q^{29} +874.656 q^{30} +484.869 q^{31} +1024.00 q^{32} +3211.59 q^{33} +2657.78 q^{34} +1223.43 q^{35} -2663.96 q^{36} +10965.8 q^{37} +7488.10 q^{38} -5484.46 q^{39} -1600.00 q^{40} -17591.7 q^{41} +1712.12 q^{42} +18924.8 q^{43} -5874.92 q^{44} +4162.44 q^{45} +8741.23 q^{46} +1886.44 q^{47} -2239.12 q^{48} -14412.2 q^{49} +2500.00 q^{50} -5811.61 q^{51} +10032.7 q^{52} -40423.0 q^{53} +14326.8 q^{54} +9179.57 q^{55} -3131.97 q^{56} -16373.8 q^{57} -21914.1 q^{58} -10373.7 q^{59} +3498.63 q^{60} +12856.2 q^{61} +1939.48 q^{62} +8147.90 q^{63} +4096.00 q^{64} -15676.0 q^{65} +12846.4 q^{66} -36439.8 q^{67} +10631.1 q^{68} -19113.9 q^{69} +4893.70 q^{70} -39211.1 q^{71} -10655.8 q^{72} -6988.15 q^{73} +43863.0 q^{74} -5466.60 q^{75} +29952.4 q^{76} +17968.8 q^{77} -21937.8 q^{78} +6241.00 q^{79} -6400.00 q^{80} +9131.39 q^{81} -70366.6 q^{82} -58486.9 q^{83} +6848.49 q^{84} -16611.1 q^{85} +75699.1 q^{86} +47918.2 q^{87} -23499.7 q^{88} -106541. q^{89} +16649.8 q^{90} -30685.5 q^{91} +34964.9 q^{92} -4240.94 q^{93} +7545.76 q^{94} -46800.6 q^{95} -8956.48 q^{96} +144441. q^{97} -57648.7 q^{98} +61135.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 60 q^{2} - 26 q^{3} + 240 q^{4} - 375 q^{5} - 104 q^{6} - 156 q^{7} + 960 q^{8} + 1143 q^{9} - 1500 q^{10} - 1198 q^{11} - 416 q^{12} - 266 q^{13} - 624 q^{14} + 650 q^{15} + 3840 q^{16} - 3256 q^{17}+ \cdots - 534352 q^{99}+O(q^{100}) $ 15 * q + 60 * q^2 - 26 * q^3 + 240 * q^4 - 375 * q^5 - 104 * q^6 - 156 * q^7 + 960 * q^8 + 1143 * q^9 - 1500 * q^10 - 1198 * q^11 - 416 * q^12 - 266 * q^13 - 624 * q^14 + 650 * q^15 + 3840 * q^16 - 3256 * q^17 + 4572 * q^18 + 1042 * q^19 - 6000 * q^20 + 4026 * q^21 - 4792 * q^22 - 1280 * q^23 - 1664 * q^24 + 9375 * q^25 - 1064 * q^26 - 7586 * q^27 - 2496 * q^28 - 12880 * q^29 + 2600 * q^30 + 5606 * q^31 + 15360 * q^32 - 1366 * q^33 - 13024 * q^34 + 3900 * q^35 + 18288 * q^36 - 12154 * q^37 + 4168 * q^38 - 41782 * q^39 - 24000 * q^40 + 15618 * q^41 + 16104 * q^42 - 30294 * q^43 - 19168 * q^44 - 28575 * q^45 - 5120 * q^46 - 2846 * q^47 - 6656 * q^48 - 21927 * q^49 + 37500 * q^50 - 7010 * q^51 - 4256 * q^52 - 40344 * q^53 - 30344 * q^54 + 29950 * q^55 - 9984 * q^56 - 43704 * q^57 - 51520 * q^58 - 43120 * q^59 + 10400 * q^60 - 54312 * q^61 + 22424 * q^62 - 213090 * q^63 + 61440 * q^64 + 6650 * q^65 - 5464 * q^66 - 149106 * q^67 - 52096 * q^68 - 68632 * q^69 + 15600 * q^70 - 132408 * q^71 + 73152 * q^72 - 178412 * q^73 - 48616 * q^74 - 16250 * q^75 + 16672 * q^76 - 24688 * q^77 - 167128 * q^78 + 93615 * q^79 - 96000 * q^80 - 182805 * q^81 + 62472 * q^82 - 208266 * q^83 + 64416 * q^84 + 81400 * q^85 - 121176 * q^86 - 116392 * q^87 - 76672 * q^88 - 44658 * q^89 - 114300 * q^90 + 14718 * q^91 - 20480 * q^92 - 275232 * q^93 - 11384 * q^94 - 26050 * q^95 - 26624 * q^96 - 196772 * q^97 - 87708 * q^98 - 534352 * 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
$3$	−8.74656	−0.561092	−0.280546	−	0.959841i	$-0.590516\pi$
$3$	−8.74656	−0.561092	−0.280546	+	0.959841i	$0.590516\pi$
$4$	16.0000	0.500000
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	−34.9863	−0.396752
$7$	−48.9370	−0.377479	−0.188739	−	0.982027i	$-0.560440\pi$
$7$	−48.9370	−0.377479	−0.188739	+	0.982027i	$0.560440\pi$
$8$	64.0000	0.353553
$9$	−166.498	−0.685175
$10$	−100.000	−0.316228
$11$	−367.183	−0.914957	−0.457478	−	0.889221i	$-0.651247\pi$
$11$	−367.183	−0.914957	−0.457478	+	0.889221i	$0.651247\pi$
$12$	−139.945	−0.280546
$13$	627.041	1.02905	0.514526	−	0.857474i	$-0.327968\pi$
$13$	627.041	1.02905	0.514526	+	0.857474i	$0.327968\pi$
$14$	−195.748	−0.266918
$15$	218.664	0.250928
$16$	256.000	0.250000
$17$	664.445	0.557618	0.278809	−	0.960347i	$-0.410060\pi$
$17$	664.445	0.557618	0.278809	+	0.960347i	$0.410060\pi$
$18$	−665.991	−0.484492
$19$	1872.03	1.18967	0.594837	−	0.803847i	$-0.297217\pi$
$19$	1872.03	1.18967	0.594837	+	0.803847i	$0.297217\pi$
$20$	−400.000	−0.223607
$21$	428.031	0.211800
$22$	−1468.73	−0.646972
$23$	2185.31	0.861376	0.430688	−	0.902501i	$-0.358271\pi$
$23$	2185.31	0.861376	0.430688	+	0.902501i	$0.358271\pi$
$24$	−559.780	−0.198376
$25$	625.000	0.200000
$26$	2508.17	0.727650
$27$	3581.70	0.945539
$28$	−782.993	−0.188739
$29$	−5478.52	−1.20967	−0.604837	−	0.796350i	$-0.706762\pi$
$29$	−5478.52	−1.20967	−0.604837	+	0.796350i	$0.706762\pi$
$30$	874.656	0.177433
$31$	484.869	0.0906192	0.0453096	−	0.998973i	$-0.485573\pi$
$31$	484.869	0.0906192	0.0453096	+	0.998973i	$0.485573\pi$
$32$	1024.00	0.176777
$33$	3211.59	0.513375
$34$	2657.78	0.394295
$35$	1223.43	0.168814
$36$	−2663.96	−0.342588
$37$	10965.8	1.31684	0.658422	−	0.752649i	$-0.271224\pi$
$37$	10965.8	1.31684	0.658422	+	0.752649i	$0.271224\pi$
$38$	7488.10	0.841226
$39$	−5484.46	−0.577394
$40$	−1600.00	−0.158114
$41$	−17591.7	−1.63436	−0.817179	−	0.576384i	$-0.804463\pi$
$41$	−17591.7	−1.63436	−0.817179	+	0.576384i	$0.804463\pi$
$42$	1712.12	0.149766
$43$	18924.8	1.56084	0.780422	−	0.625253i	$-0.215004\pi$
$43$	18924.8	1.56084	0.780422	+	0.625253i	$0.215004\pi$
$44$	−5874.92	−0.457478
$45$	4162.44	0.306420
$46$	8741.23	0.609085
$47$	1886.44	0.124566	0.0622828	−	0.998059i	$-0.480162\pi$
$47$	1886.44	0.124566	0.0622828	+	0.998059i	$0.480162\pi$
$48$	−2239.12	−0.140273
$49$	−14412.2	−0.857510
$50$	2500.00	0.141421
$51$	−5811.61	−0.312875
$52$	10032.7	0.514526
$53$	−40423.0	−1.97669	−0.988346	−	0.152224i	$-0.951356\pi$
$53$	−40423.0	−1.97669	−0.988346	+	0.152224i	$0.951356\pi$
$54$	14326.8	0.668597
$55$	9179.57	0.409181
$56$	−3131.97	−0.133459
$57$	−16373.8	−0.667517
$58$	−21914.1	−0.855368
$59$	−10373.7	−0.387973	−0.193987	−	0.981004i	$-0.562142\pi$
$59$	−10373.7	−0.387973	−0.193987	+	0.981004i	$0.562142\pi$
$60$	3498.63	0.125464
$61$	12856.2	0.442374	0.221187	−	0.975231i	$-0.429007\pi$
$61$	12856.2	0.442374	0.221187	+	0.975231i	$0.429007\pi$
$62$	1939.48	0.0640774
$63$	8147.90	0.258639
$64$	4096.00	0.125000
$65$	−15676.0	−0.460206
$66$	12846.4	0.363011
$67$	−36439.8	−0.991720	−0.495860	−	0.868403i	$-0.665147\pi$
$67$	−36439.8	−0.991720	−0.495860	+	0.868403i	$0.665147\pi$
$68$	10631.1	0.278809
$69$	−19113.9	−0.483312
$70$	4893.70	0.119369
$71$	−39211.1	−0.923130	−0.461565	−	0.887106i	$-0.652712\pi$
$71$	−39211.1	−0.923130	−0.461565	+	0.887106i	$0.652712\pi$
$72$	−10655.8	−0.242246
$73$	−6988.15	−0.153481	−0.0767405	−	0.997051i	$-0.524451\pi$
$73$	−6988.15	−0.153481	−0.0767405	+	0.997051i	$0.524451\pi$
$74$	43863.0	0.931150
$75$	−5466.60	−0.112218
$76$	29952.4	0.594837
$77$	17968.8	0.345377
$78$	−21937.8	−0.408279
$79$	6241.00	0.112509
$79$	6241.00	0.112509
$80$	−6400.00	−0.111803
$81$	9131.39	0.154641
$82$	−70366.6	−1.15567
$83$	−58486.9	−0.931888	−0.465944	−	0.884814i	$-0.654285\pi$
$83$	−58486.9	−0.931888	−0.465944	+	0.884814i	$0.654285\pi$
$84$	6848.49	0.105900
$85$	−16611.1	−0.249374
$86$	75699.1	1.10368
$87$	47918.2	0.678738
$88$	−23499.7	−0.323486
$89$	−106541.	−1.42575	−0.712874	−	0.701292i	$-0.752607\pi$
$89$	−106541.	−1.42575	−0.712874	+	0.701292i	$0.752607\pi$
$90$	16649.8	0.216672
$91$	−30685.5	−0.388446
$92$	34964.9	0.430688
$93$	−4240.94	−0.0508457
$94$	7545.76	0.0880812
$95$	−46800.6	−0.532038
$96$	−8956.48	−0.0991880
$97$	144441.	1.55870	0.779348	−	0.626591i	$-0.215550\pi$
$97$	144441.	1.55870	0.779348	+	0.626591i	$0.215550\pi$
$98$	−57648.7	−0.606351
$99$	61135.1	0.626906

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.d.1.7	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
790.6.a.d.1.7	✓	15	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)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} -8.74656 q^{3} +16.0000 q^{4} -25.0000 q^{5} -34.9863 q^{6} -48.9370 q^{7} +64.0000 q^{8} -166.498 q^{9} -100.000 q^{10} -367.183 q^{11} -139.945 q^{12} +627.041 q^{13} -195.748 q^{14} +218.664 q^{15} +256.000 q^{16} +664.445 q^{17} -665.991 q^{18} +1872.03 q^{19} -400.000 q^{20} +428.031 q^{21} -1468.73 q^{22} +2185.31 q^{23} -559.780 q^{24} +625.000 q^{25} +2508.17 q^{26} +3581.70 q^{27} -782.993 q^{28} -5478.52 q^{29} +874.656 q^{30} +484.869 q^{31} +1024.00 q^{32} +3211.59 q^{33} +2657.78 q^{34} +1223.43 q^{35} -2663.96 q^{36} +10965.8 q^{37} +7488.10 q^{38} -5484.46 q^{39} -1600.00 q^{40} -17591.7 q^{41} +1712.12 q^{42} +18924.8 q^{43} -5874.92 q^{44} +4162.44 q^{45} +8741.23 q^{46} +1886.44 q^{47} -2239.12 q^{48} -14412.2 q^{49} +2500.00 q^{50} -5811.61 q^{51} +10032.7 q^{52} -40423.0 q^{53} +14326.8 q^{54} +9179.57 q^{55} -3131.97 q^{56} -16373.8 q^{57} -21914.1 q^{58} -10373.7 q^{59} +3498.63 q^{60} +12856.2 q^{61} +1939.48 q^{62} +8147.90 q^{63} +4096.00 q^{64} -15676.0 q^{65} +12846.4 q^{66} -36439.8 q^{67} +10631.1 q^{68} -19113.9 q^{69} +4893.70 q^{70} -39211.1 q^{71} -10655.8 q^{72} -6988.15 q^{73} +43863.0 q^{74} -5466.60 q^{75} +29952.4 q^{76} +17968.8 q^{77} -21937.8 q^{78} +6241.00 q^{79} -6400.00 q^{80} +9131.39 q^{81} -70366.6 q^{82} -58486.9 q^{83} +6848.49 q^{84} -16611.1 q^{85} +75699.1 q^{86} +47918.2 q^{87} -23499.7 q^{88} -106541. q^{89} +16649.8 q^{90} -30685.5 q^{91} +34964.9 q^{92} -4240.94 q^{93} +7545.76 q^{94} -46800.6 q^{95} -8956.48 q^{96} +144441. q^{97} -57648.7 q^{98} +61135.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 60 q^{2} - 26 q^{3} + 240 q^{4} - 375 q^{5} - 104 q^{6} - 156 q^{7} + 960 q^{8} + 1143 q^{9} - 1500 q^{10} - 1198 q^{11} - 416 q^{12} - 266 q^{13} - 624 q^{14} + 650 q^{15} + 3840 q^{16} - 3256 q^{17}+ \cdots - 534352 q^{99}+O(q^{100}) \) 15 * q + 60 * q^2 - 26 * q^3 + 240 * q^4 - 375 * q^5 - 104 * q^6 - 156 * q^7 + 960 * q^8 + 1143 * q^9 - 1500 * q^10 - 1198 * q^11 - 416 * q^12 - 266 * q^13 - 624 * q^14 + 650 * q^15 + 3840 * q^16 - 3256 * q^17 + 4572 * q^18 + 1042 * q^19 - 6000 * q^20 + 4026 * q^21 - 4792 * q^22 - 1280 * q^23 - 1664 * q^24 + 9375 * q^25 - 1064 * q^26 - 7586 * q^27 - 2496 * q^28 - 12880 * q^29 + 2600 * q^30 + 5606 * q^31 + 15360 * q^32 - 1366 * q^33 - 13024 * q^34 + 3900 * q^35 + 18288 * q^36 - 12154 * q^37 + 4168 * q^38 - 41782 * q^39 - 24000 * q^40 + 15618 * q^41 + 16104 * q^42 - 30294 * q^43 - 19168 * q^44 - 28575 * q^45 - 5120 * q^46 - 2846 * q^47 - 6656 * q^48 - 21927 * q^49 + 37500 * q^50 - 7010 * q^51 - 4256 * q^52 - 40344 * q^53 - 30344 * q^54 + 29950 * q^55 - 9984 * q^56 - 43704 * q^57 - 51520 * q^58 - 43120 * q^59 + 10400 * q^60 - 54312 * q^61 + 22424 * q^62 - 213090 * q^63 + 61440 * q^64 + 6650 * q^65 - 5464 * q^66 - 149106 * q^67 - 52096 * q^68 - 68632 * q^69 + 15600 * q^70 - 132408 * q^71 + 73152 * q^72 - 178412 * q^73 - 48616 * q^74 - 16250 * q^75 + 16672 * q^76 - 24688 * q^77 - 167128 * q^78 + 93615 * q^79 - 96000 * q^80 - 182805 * q^81 + 62472 * q^82 - 208266 * q^83 + 64416 * q^84 + 81400 * q^85 - 121176 * q^86 - 116392 * q^87 - 76672 * q^88 - 44658 * q^89 - 114300 * q^90 + 14718 * q^91 - 20480 * q^92 - 275232 * q^93 - 11384 * q^94 - 26050 * q^95 - 26624 * q^96 - 196772 * q^97 - 87708 * q^98 - 534352 * q^99

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