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Properties

Label 78.5.l.b
Level $78$
Weight $5$
Character orbit 78.l
Analytic conductor $8.063$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [78,5,Mod(7,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("78.7");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 78.l (of order \(12\), degree \(4\), minimal)

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: no
Analytic conductor: \(8.06285712054\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.1579585536.10
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2}\cdot 13^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{2} - 2 \beta_1 + 2) q^{2} + (6 \beta_{3} + 3 \beta_{2}) q^{3} - 8 \beta_{3} q^{4} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots + 3) q^{5}+ \cdots - 27 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{2} - 2 \beta_1 + 2) q^{2} + (6 \beta_{3} + 3 \beta_{2}) q^{3} - 8 \beta_{3} q^{4} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots + 3) q^{5}+ \cdots + (27 \beta_{7} + 162 \beta_{6} + \cdots + 567) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 24 q^{5} - 72 q^{6} + 22 q^{7} + 128 q^{8} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 24 q^{5} - 72 q^{6} + 22 q^{7} + 128 q^{8} - 108 q^{9} - 240 q^{10} - 102 q^{11} + 114 q^{13} + 512 q^{14} + 288 q^{15} + 256 q^{16} + 258 q^{17} - 432 q^{18} - 1238 q^{19} - 576 q^{20} + 252 q^{21} + 312 q^{22} + 210 q^{23} + 576 q^{24} - 320 q^{26} + 176 q^{28} - 912 q^{29} + 432 q^{30} - 1024 q^{31} - 512 q^{32} - 630 q^{33} + 2520 q^{34} - 558 q^{35} + 38 q^{37} - 558 q^{39} - 768 q^{40} - 2202 q^{41} + 504 q^{42} + 6162 q^{43} + 1248 q^{44} - 1296 q^{45} - 1464 q^{46} + 10068 q^{47} - 8268 q^{49} + 2576 q^{50} - 160 q^{52} + 5328 q^{53} + 1944 q^{54} - 3138 q^{55} - 1344 q^{56} - 5472 q^{57} - 1608 q^{58} - 924 q^{59} + 2880 q^{60} - 9426 q^{61} - 11760 q^{62} + 2862 q^{63} - 2634 q^{65} + 576 q^{66} - 5570 q^{67} + 5040 q^{68} + 8478 q^{69} - 2232 q^{70} - 7074 q^{71} - 1728 q^{72} + 46072 q^{73} + 7216 q^{74} + 4788 q^{75} + 9904 q^{76} - 3672 q^{78} - 4752 q^{79} - 4608 q^{80} - 2916 q^{81} - 11784 q^{82} - 42840 q^{83} - 5616 q^{84} - 17958 q^{85} + 6072 q^{86} - 5148 q^{87} - 768 q^{88} + 22416 q^{89} - 8768 q^{91} - 15072 q^{92} + 13428 q^{93} + 20136 q^{94} - 10374 q^{95} + 22472 q^{97} - 13024 q^{98} + 4212 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 46\nu^{6} + 150\nu^{5} - 27\nu^{4} - 720\nu^{3} + 1768\nu^{2} - 2792\nu + 2816 ) / 1870 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3006 \nu^{7} - 12986 \nu^{6} + 9580 \nu^{5} + 30103 \nu^{4} - 124150 \nu^{3} + 454478 \nu^{2} + \cdots + 77946 ) / 753610 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4304 \nu^{7} - 6309 \nu^{6} + 4190 \nu^{5} + 34327 \nu^{4} - 28340 \nu^{3} + 236062 \nu^{2} + \cdots + 1083324 ) / 753610 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9336 \nu^{7} - 70416 \nu^{6} + 360680 \nu^{5} - 692457 \nu^{4} - 1212900 \nu^{3} + 6256578 \nu^{2} + \cdots + 10950566 ) / 753610 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1553 \nu^{7} + 12533 \nu^{6} - 49690 \nu^{5} + 56891 \nu^{4} + 166330 \nu^{3} - 746674 \nu^{2} + \cdots - 1708498 ) / 57970 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 27252 \nu^{7} + 122242 \nu^{6} + 240315 \nu^{5} - 1191231 \nu^{4} + 872820 \nu^{3} + \cdots + 11892628 ) / 753610 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2932\nu^{7} - 8647\nu^{6} - 2455\nu^{5} + 62956\nu^{4} - 113880\nu^{3} + 99246\nu^{2} + 97956\nu - 324918 ) / 24310 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} + 4\beta_{5} + 4\beta_{4} + 23\beta_{3} + 20\beta _1 - 1 ) / 39 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 9\beta_{6} + 9\beta_{5} - 2\beta_{4} + 8\beta_{3} + 161\beta_{2} + 45\beta _1 + 43 ) / 39 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} - 3\beta_{6} + 4\beta_{5} - 3\beta_{4} + 103\beta_{3} + 67\beta_{2} + 46\beta _1 - 109 ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{6} + 2\beta_{5} - 6\beta_{4} + 18\beta_{3} + 38\beta_{2} + 34\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -34\beta_{7} - 4\beta_{6} + 34\beta_{5} - 38\beta_{4} + 1556\beta_{3} - 1038\beta_{2} + 1106\beta _1 - 2632 ) / 39 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 82\beta_{7} + 166\beta_{6} - 82\beta_{4} - 296\beta_{3} - 296\beta_{2} + 124 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 272 \beta_{7} - 272 \beta_{6} - 502 \beta_{5} + 502 \beta_{4} + 4622 \beta_{3} - 11738 \beta_{2} + \cdots - 11466 ) / 39 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(67\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
2.22833 + 1.32913i
−2.59436 + 0.0368949i
0.252411 + 1.79004i
1.11361 − 1.42401i
0.252411 − 1.79004i
1.11361 + 1.42401i
2.22833 − 1.32913i
−2.59436 − 0.0368949i
2.73205 − 0.732051i −2.59808 + 4.50000i 6.92820 − 4.00000i −24.4898 − 24.4898i −3.80385 + 14.1962i −23.1946 − 6.21497i 16.0000 − 16.0000i −13.5000 − 23.3827i −84.8353 − 48.9797i
7.2 2.73205 − 0.732051i −2.59808 + 4.50000i 6.92820 − 4.00000i 1.16932 + 1.16932i −3.80385 + 14.1962i 50.3452 + 13.4900i 16.0000 − 16.0000i −13.5000 − 23.3827i 4.05066 + 2.33865i
19.1 −0.732051 − 2.73205i 2.59808 − 4.50000i −6.92820 + 4.00000i −6.96519 + 6.96519i −14.1962 − 3.80385i −22.3923 + 83.5690i 16.0000 + 16.0000i −13.5000 − 23.3827i 24.1281 + 13.9304i
19.2 −0.732051 − 2.73205i 2.59808 − 4.50000i −6.92820 + 4.00000i 18.2857 − 18.2857i −14.1962 − 3.80385i 6.24162 − 23.2940i 16.0000 + 16.0000i −13.5000 − 23.3827i −63.3435 − 36.5714i
37.1 −0.732051 + 2.73205i 2.59808 + 4.50000i −6.92820 − 4.00000i −6.96519 − 6.96519i −14.1962 + 3.80385i −22.3923 − 83.5690i 16.0000 − 16.0000i −13.5000 + 23.3827i 24.1281 − 13.9304i
37.2 −0.732051 + 2.73205i 2.59808 + 4.50000i −6.92820 − 4.00000i 18.2857 + 18.2857i −14.1962 + 3.80385i 6.24162 + 23.2940i 16.0000 − 16.0000i −13.5000 + 23.3827i −63.3435 + 36.5714i
67.1 2.73205 + 0.732051i −2.59808 − 4.50000i 6.92820 + 4.00000i −24.4898 + 24.4898i −3.80385 − 14.1962i −23.1946 + 6.21497i 16.0000 + 16.0000i −13.5000 + 23.3827i −84.8353 + 48.9797i
67.2 2.73205 + 0.732051i −2.59808 − 4.50000i 6.92820 + 4.00000i 1.16932 − 1.16932i −3.80385 − 14.1962i 50.3452 − 13.4900i 16.0000 + 16.0000i −13.5000 + 23.3827i 4.05066 − 2.33865i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.5.l.b ✓ 8
3.b odd 2 1 234.5.bb.b 8
13.f odd 12 1 inner 78.5.l.b ✓ 8
39.k even 12 1 234.5.bb.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.5.l.b ✓ 8 1.a even 1 1 trivial
78.5.l.b ✓ 8 13.f odd 12 1 inner
234.5.bb.b 8 3.b odd 2 1
234.5.bb.b 8 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 24 T_{5}^{7} + 288 T_{5}^{6} - 9576 T_{5}^{5} + 676422 T_{5}^{4} + 8540424 T_{5}^{3} + \cdots + 212838921 \) acting on \(S_{5}^{\mathrm{new}}(78, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 27 T^{2} + 729)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 24 T^{7} + \cdots + 212838921 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 6818981697124 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 66\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 36\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 35\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 79\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 13\!\cdots\!69 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 108105777079947)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 76\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 18\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 14\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 39\!\cdots\!96 \) Copy content Toggle raw display
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