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Properties

Label 78.3.l.c
Level $78$
Weight $3$
Character orbit 78.l
Analytic conductor $2.125$
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] = [78,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("78.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
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: \(2.12534606201\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 82x^{5} + 5053x^{4} - 6736x^{3} + 6728x^{2} + 275384x + 5635876 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{4}) q^{2} + ( - \beta_{5} + 2 \beta_{2}) q^{3} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{4} + ( - \beta_{6} - \beta_{3} + \beta_{2}) q^{5} + (\beta_{5} - \beta_{4} + \beta_{2} + 1) q^{6} + ( - \beta_{7} - \beta_{5} - \beta_1 + 1) q^{7} + (2 \beta_{2} + 2) q^{8} + ( - 3 \beta_{4} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{4}) q^{2} + ( - \beta_{5} + 2 \beta_{2}) q^{3} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{4} + ( - \beta_{6} - \beta_{3} + \beta_{2}) q^{5} + (\beta_{5} - \beta_{4} + \beta_{2} + 1) q^{6} + ( - \beta_{7} - \beta_{5} - \beta_1 + 1) q^{7} + (2 \beta_{2} + 2) q^{8} + ( - 3 \beta_{4} - 3) q^{9} + (\beta_{7} - \beta_{6} + 2 \beta_{5}) q^{10} + ( - 2 \beta_{7} + 2 \beta_{5} + \cdots + 2) q^{11}+ \cdots + ( - 6 \beta_{5} - 6 \beta_{4} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 6 q^{5} + 12 q^{6} + 10 q^{7} + 16 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 6 q^{5} + 12 q^{6} + 10 q^{7} + 16 q^{8} - 12 q^{9} - 6 q^{10} + 24 q^{11} + 4 q^{14} - 12 q^{15} + 16 q^{16} - 84 q^{17} - 24 q^{18} + 10 q^{19} - 12 q^{20} + 18 q^{21} - 12 q^{22} - 12 q^{23} - 24 q^{24} + 26 q^{26} + 20 q^{28} + 36 q^{29} - 18 q^{30} - 94 q^{31} - 16 q^{32} + 60 q^{34} - 204 q^{35} + 140 q^{37} + 66 q^{39} - 24 q^{40} + 72 q^{41} + 18 q^{42} - 222 q^{43} - 24 q^{44} - 84 q^{46} + 300 q^{47} + 42 q^{49} - 62 q^{50} + 44 q^{52} + 84 q^{53} - 36 q^{54} + 396 q^{55} + 36 q^{56} + 24 q^{57} - 66 q^{58} - 60 q^{59} - 12 q^{60} - 90 q^{61} + 198 q^{62} - 24 q^{63} - 108 q^{65} + 72 q^{66} + 304 q^{67} + 60 q^{68} - 216 q^{69} - 408 q^{70} - 192 q^{71} - 24 q^{72} + 16 q^{73} - 46 q^{74} + 312 q^{75} - 20 q^{76} + 114 q^{78} - 96 q^{79} - 24 q^{80} - 36 q^{81} + 114 q^{82} - 12 q^{84} - 390 q^{85} + 168 q^{86} + 30 q^{87} + 72 q^{88} + 354 q^{89} - 218 q^{91} - 288 q^{92} - 42 q^{93} + 300 q^{94} - 576 q^{95} - 460 q^{97} + 58 q^{98} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 82x^{5} + 5053x^{4} - 6736x^{3} + 6728x^{2} + 275384x + 5635876 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4081829 \nu^{7} + 125878448 \nu^{6} - 175318185 \nu^{5} - 167602931 \nu^{4} + \cdots - 561922481148 ) / 28390156409030 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 900565 \nu^{7} + 114752803 \nu^{6} + 496632472 \nu^{5} + 5297934372 \nu^{4} + \cdots + 15143464458814 ) / 813197403460 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 887 \nu^{7} - 2974 \nu^{6} - 42132 \nu^{5} - 34750 \nu^{4} - 2587559 \nu^{3} + \cdots - 276718188 ) / 320442520 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5758773899 \nu^{7} - 1375000768 \nu^{6} - 272272114360 \nu^{5} - 695422749614 \nu^{4} + \cdots - 24\!\cdots\!12 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4272345 \nu^{7} - 119540717 \nu^{6} - 491845918 \nu^{5} - 5101492538 \nu^{4} + \cdots - 13671329236226 ) / 813197403460 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} - 17\nu^{6} + 16\nu^{5} + 798\nu^{4} - 6888\nu^{3} - 47339\nu^{2} - 13670\nu + 2105738 ) / 134980 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} - 6\beta_{4} + 49\beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{7} + 51\beta_{6} - 46\beta_{5} + 46\beta_{4} + 51\beta_{3} + 43\beta_{2} - 2\beta _1 - 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 95\beta_{7} - 5\beta_{6} + 502\beta_{5} + 90\beta_{3} - 251\beta_{2} + 5\beta _1 - 2607 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 161\beta_{6} - 4400\beta_{5} - 4400\beta_{4} - 161\beta_{3} + 545\beta_{2} - 2702\beta _1 - 3694 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -6941\beta_{7} - 6235\beta_{6} + 31990\beta_{4} + 706\beta_{3} - 133847\beta_{2} - 6235\beta _1 + 15289 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 19520 \beta_{7} - 147023 \beta_{6} + 323522 \beta_{5} - 323522 \beta_{4} - 147023 \beta_{3} + \cdots + 89488 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(53\) \(67\)
\(\chi(n)\) \(1\) \(\beta_{2} - \beta_{5}\)

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
−5.39181 + 5.39181i
5.02578 − 5.02578i
−4.04651 − 4.04651i
5.41254 + 5.41254i
−4.04651 + 4.04651i
5.41254 − 5.41254i
−5.39181 − 5.39181i
5.02578 + 5.02578i
1.36603 − 0.366025i 0.866025 − 1.50000i 1.73205 − 1.00000i −6.39181 − 6.39181i 0.633975 − 2.36603i 9.23137 + 2.47354i 2.00000 − 2.00000i −1.50000 − 2.59808i −11.0709 − 6.39181i
7.2 1.36603 − 0.366025i 0.866025 − 1.50000i 1.73205 − 1.00000i 4.02578 + 4.02578i 0.633975 − 2.36603i −4.99932 − 1.33956i 2.00000 − 2.00000i −1.50000 − 2.59808i 6.97286 + 4.02578i
19.1 −0.366025 − 1.36603i −0.866025 + 1.50000i −1.73205 + 1.00000i −5.04651 + 5.04651i 2.36603 + 0.633975i −1.34715 + 5.02764i 2.00000 + 2.00000i −1.50000 − 2.59808i 8.74082 + 5.04651i
19.2 −0.366025 − 1.36603i −0.866025 + 1.50000i −1.73205 + 1.00000i 4.41254 − 4.41254i 2.36603 + 0.633975i 2.11510 − 7.89367i 2.00000 + 2.00000i −1.50000 − 2.59808i −7.64274 − 4.41254i
37.1 −0.366025 + 1.36603i −0.866025 − 1.50000i −1.73205 − 1.00000i −5.04651 − 5.04651i 2.36603 − 0.633975i −1.34715 − 5.02764i 2.00000 − 2.00000i −1.50000 + 2.59808i 8.74082 − 5.04651i
37.2 −0.366025 + 1.36603i −0.866025 − 1.50000i −1.73205 − 1.00000i 4.41254 + 4.41254i 2.36603 − 0.633975i 2.11510 + 7.89367i 2.00000 − 2.00000i −1.50000 + 2.59808i −7.64274 + 4.41254i
67.1 1.36603 + 0.366025i 0.866025 + 1.50000i 1.73205 + 1.00000i −6.39181 + 6.39181i 0.633975 + 2.36603i 9.23137 − 2.47354i 2.00000 + 2.00000i −1.50000 + 2.59808i −11.0709 + 6.39181i
67.2 1.36603 + 0.366025i 0.866025 + 1.50000i 1.73205 + 1.00000i 4.02578 − 4.02578i 0.633975 + 2.36603i −4.99932 + 1.33956i 2.00000 + 2.00000i −1.50000 + 2.59808i 6.97286 − 4.02578i
\(n\): 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 78.3.l.c ✓ 8
3.b odd 2 1 234.3.bb.d 8
13.c even 3 1 1014.3.f.h 8
13.e even 6 1 1014.3.f.j 8
13.f odd 12 1 inner 78.3.l.c ✓ 8
13.f odd 12 1 1014.3.f.h 8
13.f odd 12 1 1014.3.f.j 8
39.k even 12 1 234.3.bb.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.3.l.c ✓ 8 1.a even 1 1 trivial
78.3.l.c ✓ 8 13.f odd 12 1 inner
234.3.bb.d 8 3.b odd 2 1
234.3.bb.d 8 39.k even 12 1
1014.3.f.h 8 13.c even 3 1
1014.3.f.h 8 13.f odd 12 1
1014.3.f.j 8 13.e even 6 1
1014.3.f.j 8 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 6T_{5}^{7} + 18T_{5}^{6} - 282T_{5}^{5} + 4065T_{5}^{4} + 11916T_{5}^{3} + 38088T_{5}^{2} - 632592T_{5} + 5253264 \) acting on \(S_{3}^{\mathrm{new}}(78, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} + \cdots + 5253264 \) Copy content Toggle raw display
$7$ \( T^{8} - 10 T^{7} + \cdots + 4426816 \) Copy content Toggle raw display
$11$ \( T^{8} - 24 T^{7} + \cdots + 973440000 \) Copy content Toggle raw display
$13$ \( T^{8} - 50 T^{6} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 4567597056 \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{7} + \cdots + 1763584 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 17831863296 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 48599084304 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 33544655104 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 10744151716 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 370617958656 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 1819196698176 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 20494380893184 \) Copy content Toggle raw display
$53$ \( (T^{4} - 42 T^{3} + \cdots - 109128)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 15611728564224 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 295174493893449 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 42600893548096 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 1623606027264 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 261324417601 \) Copy content Toggle raw display
$79$ \( (T^{4} + 48 T^{3} + \cdots - 8209344)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13865554427904 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
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