Properties

Label 114.3.f.a
Level $114$
Weight $3$
Character orbit 114.f
Analytic conductor $3.106$
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] = [114,3,Mod(31,114)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(114, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("114.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 114.f (of order \(6\), degree \(2\), 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: \(3.10627501371\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 92x^{6} + 2680x^{4} - 23592x^{2} + 79524 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( - \beta_{4} - 1) q^{3} + 2 \beta_{4} q^{4} + ( - \beta_{5} + 2 \beta_{4} + \beta_1 - 2) q^{5} + ( - 2 \beta_{6} + \beta_{5}) q^{6} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} + \beta_1) q^{7} + (2 \beta_{6} - 2 \beta_{5}) q^{8} + 3 \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + ( - \beta_{4} - 1) q^{3} + 2 \beta_{4} q^{4} + ( - \beta_{5} + 2 \beta_{4} + \beta_1 - 2) q^{5} + ( - 2 \beta_{6} + \beta_{5}) q^{6} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} + \beta_1) q^{7} + (2 \beta_{6} - 2 \beta_{5}) q^{8} + 3 \beta_{4} q^{9} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 2) q^{10} + (2 \beta_{6} + \beta_{5} + \beta_{2} - \beta_1 + 8) q^{11} + ( - 4 \beta_{4} + 2) q^{12} + ( - 3 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{13}+ \cdots + (9 \beta_{6} - 6 \beta_{5} + 24 \beta_{4} + 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} + 8 q^{4} - 8 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{3} + 8 q^{4} - 8 q^{5} + 12 q^{9} - 12 q^{10} + 64 q^{11} - 24 q^{13} + 12 q^{14} + 24 q^{15} - 16 q^{16} - 20 q^{17} - 32 q^{20} + 36 q^{22} - 68 q^{23} - 16 q^{25} + 24 q^{29} + 24 q^{30} - 96 q^{33} + 24 q^{34} + 68 q^{35} - 24 q^{36} - 36 q^{38} + 48 q^{39} - 24 q^{40} - 132 q^{41} - 12 q^{42} - 56 q^{43} + 64 q^{44} - 48 q^{45} + 52 q^{47} + 48 q^{48} + 264 q^{49} + 60 q^{51} - 48 q^{52} + 108 q^{53} - 176 q^{55} + 24 q^{57} - 144 q^{58} + 72 q^{59} + 48 q^{60} - 96 q^{61} - 36 q^{62} - 64 q^{64} - 36 q^{66} - 72 q^{67} - 80 q^{68} + 372 q^{70} - 84 q^{71} - 44 q^{73} - 168 q^{74} - 48 q^{76} - 112 q^{77} - 144 q^{79} - 32 q^{80} - 36 q^{81} + 264 q^{82} + 232 q^{83} - 8 q^{85} + 228 q^{86} - 48 q^{87} - 36 q^{90} + 744 q^{91} + 136 q^{92} - 404 q^{95} + 492 q^{97} + 48 q^{98} + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 92x^{6} + 2680x^{4} - 23592x^{2} + 79524 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{4} + 46\nu^{2} - 28\nu - 282 ) / 56 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} + 46\nu^{2} + 28\nu - 282 ) / 56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14\nu^{6} - 1123\nu^{4} + 27326\nu^{2} + 4788\nu - 141294 ) / 9576 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 92\nu^{5} + 2398\nu^{3} - 10620\nu + 7896 ) / 15792 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 91\nu^{7} - 141\nu^{6} - 7385\nu^{5} + 9588\nu^{4} + 176764\nu^{3} - 99546\nu^{2} - 774942\nu - 1280844 ) / 1350216 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 91 \nu^{7} - 141 \nu^{6} + 7385 \nu^{5} + 9588 \nu^{4} - 176764 \nu^{3} - 99546 \nu^{2} + 774942 \nu - 1280844 ) / 1350216 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 253 \nu^{7} - 1974 \nu^{6} - 26096 \nu^{5} + 134232 \nu^{4} + 918022 \nu^{3} - 2743860 \nu^{2} - 10177908 \nu + 13123152 ) / 2700432 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{6} + 7\beta_{5} + \beta_{3} - \beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 14\beta_{7} - 10\beta_{6} + 10\beta_{5} - 42\beta_{4} + 7\beta_{3} + 29\beta_{2} - 36\beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 322\beta_{6} + 322\beta_{5} + 46\beta_{3} - 74\beta_{2} - 28\beta _1 + 776 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 588\beta_{7} - 1104\beta_{6} + 1104\beta_{5} - 3220\beta_{4} + 294\beta_{3} + 1024\beta_{2} - 1318\beta _1 + 1610 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12166\beta_{6} + 12166\beta_{5} + 2422\beta_{3} - 4326\beta_{2} - 1904\beta _1 + 27446 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 20524 \beta_{7} - 77588 \beta_{6} + 77588 \beta_{5} - 179732 \beta_{4} + 10262 \beta_{3} + 35286 \beta_{2} - 45548 \beta _1 + 89866 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
2.52063 0.707107i
−2.52063 0.707107i
6.37546 + 0.707107i
−6.37546 + 0.707107i
2.52063 + 0.707107i
−2.52063 + 0.707107i
6.37546 0.707107i
−6.37546 0.707107i
−1.22474 + 0.707107i −1.50000 + 0.866025i 1.00000 1.73205i −1.64794 2.85432i 1.22474 2.12132i 7.47014 2.82843i 1.50000 2.59808i 4.03662 + 2.33054i
31.2 −1.22474 + 0.707107i −1.50000 + 0.866025i 1.00000 1.73205i 0.872687 + 1.51154i 1.22474 2.12132i −9.91963 2.82843i 1.50000 2.59808i −2.13764 1.23417i
31.3 1.22474 0.707107i −1.50000 + 0.866025i 1.00000 1.73205i −4.80010 8.31402i −1.22474 + 2.12132i −8.01641 2.82843i 1.50000 2.59808i −11.7578 6.78837i
31.4 1.22474 0.707107i −1.50000 + 0.866025i 1.00000 1.73205i 1.57536 + 2.72860i −1.22474 + 2.12132i 10.4659 2.82843i 1.50000 2.59808i 3.85882 + 2.22789i
103.1 −1.22474 0.707107i −1.50000 0.866025i 1.00000 + 1.73205i −1.64794 + 2.85432i 1.22474 + 2.12132i 7.47014 2.82843i 1.50000 + 2.59808i 4.03662 2.33054i
103.2 −1.22474 0.707107i −1.50000 0.866025i 1.00000 + 1.73205i 0.872687 1.51154i 1.22474 + 2.12132i −9.91963 2.82843i 1.50000 + 2.59808i −2.13764 + 1.23417i
103.3 1.22474 + 0.707107i −1.50000 0.866025i 1.00000 + 1.73205i −4.80010 + 8.31402i −1.22474 2.12132i −8.01641 2.82843i 1.50000 + 2.59808i −11.7578 + 6.78837i
103.4 1.22474 + 0.707107i −1.50000 0.866025i 1.00000 + 1.73205i 1.57536 2.72860i −1.22474 2.12132i 10.4659 2.82843i 1.50000 + 2.59808i 3.85882 2.22789i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.3.f.a 8
3.b odd 2 1 342.3.m.c 8
4.b odd 2 1 912.3.be.h 8
19.d odd 6 1 inner 114.3.f.a 8
57.f even 6 1 342.3.m.c 8
76.f even 6 1 912.3.be.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.3.f.a 8 1.a even 1 1 trivial
114.3.f.a 8 19.d odd 6 1 inner
342.3.m.c 8 3.b odd 2 1
342.3.m.c 8 57.f even 6 1
912.3.be.h 8 4.b odd 2 1
912.3.be.h 8 76.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 8T_{5}^{7} + 90T_{5}^{6} - 40T_{5}^{5} + 1174T_{5}^{4} - 600T_{5}^{3} + 11580T_{5}^{2} - 14616T_{5} + 30276 \) acting on \(S_{3}^{\mathrm{new}}(114, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 8 T^{7} + 90 T^{6} + \cdots + 30276 \) Copy content Toggle raw display
$7$ \( (T^{4} - 164 T^{2} - 24 T + 6217)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 32 T^{3} + 310 T^{2} - 1116 T + 1182)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 24 T^{7} - 138 T^{6} + \cdots + 21316689 \) Copy content Toggle raw display
$17$ \( T^{8} + 20 T^{7} + 396 T^{6} + \cdots + 14400 \) Copy content Toggle raw display
$19$ \( T^{8} - 1024 T^{6} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( T^{8} + 68 T^{7} + \cdots + 27238861764 \) Copy content Toggle raw display
$29$ \( T^{8} - 24 T^{7} + \cdots + 366828880896 \) Copy content Toggle raw display
$31$ \( T^{8} + 3864 T^{6} + \cdots + 586340901441 \) Copy content Toggle raw display
$37$ \( T^{8} + 4956 T^{6} + \cdots + 129865816161 \) Copy content Toggle raw display
$41$ \( T^{8} + 132 T^{7} + \cdots + 50792988686400 \) Copy content Toggle raw display
$43$ \( T^{8} + 56 T^{7} + \cdots + 7809080161 \) Copy content Toggle raw display
$47$ \( T^{8} - 52 T^{7} + \cdots + 101750792256 \) Copy content Toggle raw display
$53$ \( T^{8} - 108 T^{7} + \cdots + 5979932252100 \) Copy content Toggle raw display
$59$ \( T^{8} - 72 T^{7} + \cdots + 198640836 \) Copy content Toggle raw display
$61$ \( T^{8} + 96 T^{7} + \cdots + 49371772515025 \) Copy content Toggle raw display
$67$ \( T^{8} + 72 T^{7} + \cdots + 284883725025 \) Copy content Toggle raw display
$71$ \( T^{8} + 84 T^{7} + \cdots + 3674889000000 \) Copy content Toggle raw display
$73$ \( T^{8} + 44 T^{7} + \cdots + 8048063521 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 195890003437041 \) Copy content Toggle raw display
$83$ \( (T^{4} - 116 T^{3} - 11396 T^{2} + \cdots - 30693384)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 232835310959364 \) Copy content Toggle raw display
$97$ \( T^{8} - 492 T^{7} + \cdots + 34707690000 \) Copy content Toggle raw display
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