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Properties

Label 79.8.a.a
Level $79$
Weight $8$
Character orbit 79.a
Self dual yes
Analytic conductor $24.678$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [79,8,Mod(1,79)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(79, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("79.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 79 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 79.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: \(24.6784170132\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} - 1802 x^{18} + 7542 x^{17} + 1343302 x^{16} - 5975914 x^{15} - 536909742 x^{14} + \cdots + 47\!\cdots\!04 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: multiple of \( 2^{23} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + ( - \beta_{4} - 3) q^{3} + (\beta_{2} + 2 \beta_1 + 54) q^{4} + ( - \beta_{8} - 50) q^{5} + (\beta_{11} + \beta_{8} + 3 \beta_{4} + \cdots - 53) q^{6}+ \cdots + (2 \beta_{15} - 2 \beta_{11} + \cdots + 319) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + ( - \beta_{4} - 3) q^{3} + (\beta_{2} + 2 \beta_1 + 54) q^{4} + ( - \beta_{8} - 50) q^{5} + (\beta_{11} + \beta_{8} + 3 \beta_{4} + \cdots - 53) q^{6}+ \cdots + ( - 173 \beta_{19} - 7926 \beta_{18} + \cdots - 693197) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 24 q^{2} - 55 q^{3} + 1088 q^{4} - 1001 q^{5} - 1047 q^{6} - 489 q^{7} - 3810 q^{8} + 6405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 24 q^{2} - 55 q^{3} + 1088 q^{4} - 1001 q^{5} - 1047 q^{6} - 489 q^{7} - 3810 q^{8} + 6405 q^{9} + 1125 q^{10} - 13546 q^{11} - 15737 q^{12} - 14687 q^{13} - 46523 q^{14} + 6191 q^{15} + 60964 q^{16} - 88244 q^{17} - 73283 q^{18} - 51768 q^{19} - 79521 q^{20} + 19797 q^{21} + 262450 q^{22} - 23542 q^{23} - 173195 q^{24} + 65813 q^{25} - 513251 q^{26} - 290491 q^{27} - 596337 q^{28} - 465390 q^{29} - 1326743 q^{30} - 851754 q^{31} - 1669114 q^{32} - 1198078 q^{33} - 974758 q^{34} - 723905 q^{35} - 1837841 q^{36} - 1168894 q^{37} - 2331450 q^{38} - 1574535 q^{39} - 2888103 q^{40} - 2709480 q^{41} - 3136289 q^{42} - 1761930 q^{43} - 4042472 q^{44} - 3330820 q^{45} - 2426730 q^{46} - 1556809 q^{47} - 670625 q^{48} + 911277 q^{49} - 1921205 q^{50} - 1814526 q^{51} - 1913059 q^{52} + 374768 q^{53} + 189233 q^{54} - 2063058 q^{55} - 8438231 q^{56} - 767060 q^{57} + 3505396 q^{58} - 2976205 q^{59} + 3192255 q^{60} - 702724 q^{61} - 629238 q^{62} - 1281150 q^{63} + 8315004 q^{64} - 5904005 q^{65} + 7195320 q^{66} - 1253156 q^{67} - 4691670 q^{68} - 7591826 q^{69} + 9711975 q^{70} - 1075617 q^{71} + 17265399 q^{72} - 4110844 q^{73} + 2427230 q^{74} - 3567682 q^{75} + 27564618 q^{76} - 4171150 q^{77} + 34780911 q^{78} + 9860780 q^{79} + 7941219 q^{80} + 9964324 q^{81} + 31468838 q^{82} + 25739262 q^{83} + 51786145 q^{84} + 20447958 q^{85} + 27708418 q^{86} + 24494826 q^{87} + 68674672 q^{88} - 17667565 q^{89} + 71356356 q^{90} + 18103209 q^{91} + 36412438 q^{92} + 3445962 q^{93} + 31859317 q^{94} + 54120 q^{95} + 26423037 q^{96} + 24731719 q^{97} + 17601201 q^{98} - 14714332 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} - 1802 x^{18} + 7542 x^{17} + 1343302 x^{16} - 5975914 x^{15} - 536909742 x^{14} + \cdots + 47\!\cdots\!04 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 181 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 24\!\cdots\!33 \nu^{19} + \cdots - 76\!\cdots\!52 ) / 79\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25\!\cdots\!57 \nu^{19} + \cdots - 18\!\cdots\!68 ) / 79\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 44\!\cdots\!33 \nu^{19} + \cdots + 12\!\cdots\!28 ) / 79\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 65\!\cdots\!99 \nu^{19} + \cdots - 21\!\cdots\!36 ) / 79\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 32\!\cdots\!93 \nu^{19} + \cdots - 44\!\cdots\!52 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\!\cdots\!33 \nu^{19} + \cdots - 51\!\cdots\!88 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17\!\cdots\!61 \nu^{19} + \cdots + 90\!\cdots\!88 ) / 79\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!23 \nu^{19} + \cdots + 35\!\cdots\!48 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 50\!\cdots\!67 \nu^{19} + \cdots + 85\!\cdots\!80 ) / 15\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 50\!\cdots\!39 \nu^{19} + \cdots - 16\!\cdots\!00 ) / 15\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\!\cdots\!31 \nu^{19} + \cdots - 83\!\cdots\!96 ) / 49\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 57\!\cdots\!93 \nu^{19} + \cdots - 60\!\cdots\!76 ) / 15\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 15\!\cdots\!13 \nu^{19} + \cdots - 43\!\cdots\!68 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 27\!\cdots\!33 \nu^{19} + \cdots - 11\!\cdots\!92 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 14\!\cdots\!13 \nu^{19} + \cdots - 21\!\cdots\!72 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 68\!\cdots\!79 \nu^{19} + \cdots - 12\!\cdots\!84 ) / 79\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 10\!\cdots\!09 \nu^{19} + \cdots + 94\!\cdots\!16 ) / 79\!\cdots\!80 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 181 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{11} - 2\beta_{8} - \beta_{5} - 4\beta_{4} + 2\beta_{2} + 308\beta _1 - 109 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{19} - 4 \beta_{18} - 12 \beta_{17} + 2 \beta_{16} - \beta_{15} - 6 \beta_{14} + 3 \beta_{13} + \cdots + 55793 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 13 \beta_{19} - 5 \beta_{18} + 17 \beta_{17} + 30 \beta_{16} + 554 \beta_{15} - 111 \beta_{14} + \cdots - 23473 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3491 \beta_{19} - 1495 \beta_{18} - 7727 \beta_{17} + 1476 \beta_{16} - 502 \beta_{15} + \cdots + 20026365 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 7338 \beta_{19} - 5674 \beta_{18} + 13442 \beta_{17} + 23388 \beta_{16} + 261639 \beta_{15} + \cdots - 892117 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2119258 \beta_{19} - 403394 \beta_{18} - 3875162 \beta_{17} + 851438 \beta_{16} - 255407 \beta_{15} + \cdots + 7683003709 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3747123 \beta_{19} - 3743819 \beta_{18} + 7968527 \beta_{17} + 13510738 \beta_{16} + \cdots + 2162910699 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1128256997 \beta_{19} - 78172241 \beta_{18} - 1803473881 \beta_{17} + 445754784 \beta_{16} + \cdots + 3059600157537 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2024168120 \beta_{19} - 1886141816 \beta_{18} + 4208973480 \beta_{17} + 7113842624 \beta_{16} + \cdots + 1496444679791 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 564103864836 \beta_{19} - 1032485060 \beta_{18} - 816902147756 \beta_{17} + 221751067578 \beta_{16} + \cdots + 12\!\cdots\!45 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1124993804297 \beta_{19} - 813196881473 \beta_{18} + 2089740311837 \beta_{17} + \cdots + 713648542403943 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 272418777042983 \beta_{19} + 10221027016885 \beta_{18} - 366263051277251 \beta_{17} + \cdots + 51\!\cdots\!25 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 618341725552886 \beta_{19} - 314060114820918 \beta_{18} + \cdots + 28\!\cdots\!07 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 12\!\cdots\!74 \beta_{19} + \cdots + 21\!\cdots\!77 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 33\!\cdots\!87 \beta_{19} + \cdots + 95\!\cdots\!11 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 60\!\cdots\!33 \beta_{19} + \cdots + 93\!\cdots\!97 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 17\!\cdots\!60 \beta_{19} + \cdots + 24\!\cdots\!19 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
21.1706
19.3373
19.0170
13.6188
13.0823
9.73216
8.14996
7.27980
5.32930
3.23257
−0.246753
−1.39809
−6.31644
−8.94077
−11.1037
−13.1053
−17.3673
−17.5964
−18.6218
−21.2534
−22.1706 36.3394 363.536 124.897 −805.667 1494.17 −5221.99 −866.450 −2769.03
1.2 −20.3373 −54.9003 285.607 −378.869 1116.53 −656.062 −3205.30 827.042 7705.19
1.3 −20.0170 −12.5255 272.681 293.482 250.724 −775.836 −2896.09 −2030.11 −5874.64
1.4 −14.6188 25.1265 85.7088 −363.578 −367.319 1010.61 618.246 −1555.66 5315.07
1.5 −14.0823 49.4083 70.3121 361.912 −695.784 −749.689 812.380 254.180 −5096.57
1.6 −10.7322 −80.0754 −12.8206 −307.245 859.383 −1129.83 1511.31 4225.07 3297.40
1.7 −9.14996 86.8126 −44.2782 −279.754 −794.332 208.209 1576.34 5349.43 2559.74
1.8 −8.27980 −29.4770 −59.4449 47.3201 244.064 −270.990 1552.01 −1318.11 −391.801
1.9 −6.32930 −59.5746 −87.9399 −496.391 377.066 1426.57 1366.75 1362.13 3141.81
1.10 −4.23257 57.4526 −110.085 −163.668 −243.172 299.643 1007.71 1113.80 692.735
1.11 −0.753247 16.8197 −127.433 316.500 −12.6694 −33.2636 192.404 −1904.10 −238.403
1.12 0.398090 −79.7604 −127.842 138.272 −31.7518 516.258 −101.848 4174.72 55.0447
1.13 5.31644 28.3797 −99.7355 255.218 150.879 −324.432 −1210.74 −1381.59 1356.85
1.14 7.94077 −19.2607 −64.9441 8.05061 −152.945 1776.21 −1532.13 −1816.02 63.9281
1.15 10.1037 69.8474 −25.9145 −333.967 705.720 −652.355 −1555.11 2691.66 −3374.31
1.16 12.1053 28.3495 18.5371 65.4780 343.178 −825.100 −1325.08 −1383.31 792.628
1.17 16.3673 −4.40920 139.888 −277.875 −72.1666 769.137 194.572 −2167.56 −4548.06
1.18 16.5964 −77.6123 147.441 194.731 −1288.08 164.135 322.648 3836.66 3231.83
1.19 17.6218 −36.9617 182.528 240.180 −651.331 −1466.06 960.878 −820.835 4232.41
1.20 20.2534 1.02136 282.198 −445.695 20.6860 −1270.31 3123.03 −2185.96 −9026.81
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(79\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 79.8.a.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
79.8.a.a ✓ 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 24 T_{2}^{19} - 1536 T_{2}^{18} - 38154 T_{2}^{17} + 948103 T_{2}^{16} + \cdots - 60\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(79))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots - 60\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 40\!\cdots\!44 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots - 68\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots - 19\!\cdots\!68 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots - 73\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 94\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 36\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots - 92\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 14\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 21\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( (T - 493039)^{20} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots - 12\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 17\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 69\!\cdots\!36 \) Copy content Toggle raw display
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