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Properties

Label 116.2.g.b
Level $116$
Weight $2$
Character orbit 116.g
Analytic conductor $0.926$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [116,2,Mod(25,116)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(116, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("116.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 116 = 2^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 116.g (of order \(7\), degree \(6\), 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: \(0.926264663447\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} + 12 x^{10} - 16 x^{9} + 22 x^{8} + 28 x^{7} + 71 x^{6} + 154 x^{5} + 442 x^{4} + \cdots + 841 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{8} - \beta_1) q^{3} + (\beta_{6} + \beta_{5} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots - 2 \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} - \beta_1) q^{3} + (\beta_{6} + \beta_{5} + \cdots - \beta_{2}) q^{5}+ \cdots + (\beta_{10} + \beta_{9} + 6 \beta_{8} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{3} - q^{5} - 7 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{3} - q^{5} - 7 q^{7} + q^{9} + 3 q^{11} - 21 q^{13} + 29 q^{15} - 10 q^{17} + 17 q^{19} + 3 q^{21} - 19 q^{23} - 3 q^{25} + 9 q^{27} - 5 q^{29} - 27 q^{31} - 47 q^{33} + 27 q^{35} - 3 q^{37} - 37 q^{39} - 2 q^{41} - 9 q^{43} + 22 q^{45} + 19 q^{47} + 17 q^{49} + 2 q^{51} + 22 q^{53} + 49 q^{55} + 30 q^{57} + 36 q^{59} - 33 q^{61} + 9 q^{63} + 38 q^{65} - 3 q^{67} + 47 q^{69} - 35 q^{71} + 34 q^{73} - 44 q^{75} - 27 q^{77} - 19 q^{79} - 39 q^{81} - q^{83} - 34 q^{85} - 25 q^{87} - 11 q^{89} - 23 q^{91} + 7 q^{93} - 105 q^{95} + 38 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} + 12 x^{10} - 16 x^{9} + 22 x^{8} + 28 x^{7} + 71 x^{6} + 154 x^{5} + 442 x^{4} + \cdots + 841 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2666113593 \nu^{11} - 103378288493 \nu^{10} + 730364594638 \nu^{9} + \cdots - 87085829035390 ) / 87597662955238 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 76569689033 \nu^{11} - 329670261352 \nu^{10} + 1009877030172 \nu^{9} + \cdots + 235137749968678 ) / 175195325910476 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 80877989929 \nu^{11} + 560470291448 \nu^{10} - 1931233965956 \nu^{9} + \cdots + 92982388988572 ) / 175195325910476 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 154305296456 \nu^{11} - 821175970637 \nu^{10} + 2060526694663 \nu^{9} + \cdots + 132826473739629 ) / 175195325910476 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 182774232211 \nu^{11} + 793118206506 \nu^{10} - 1236048179004 \nu^{9} + \cdots - 277769000014142 ) / 175195325910476 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 103550331790 \nu^{11} - 520417772543 \nu^{10} + 1139225692987 \nu^{9} + \cdots + 108496393085623 ) / 87597662955238 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 230492168800 \nu^{11} + 1461546568214 \nu^{10} - 4597897267111 \nu^{9} + \cdots + 9060333114751 ) / 175195325910476 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 232650030836 \nu^{11} + 1290368348160 \nu^{10} - 3776871348421 \nu^{9} + \cdots - 303156139498967 ) / 175195325910476 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 309085724214 \nu^{11} - 1831991241511 \nu^{10} + 4952928996914 \nu^{9} + \cdots + 193843913960800 ) / 175195325910476 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 513040509027 \nu^{11} + 2862030349339 \nu^{10} - 7361090329451 \nu^{9} + \cdots - 456441142019925 ) / 175195325910476 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{10} - \beta_{8} - \beta_{6} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 6 \beta_{11} - 6 \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{6} - 10 \beta_{5} - 4 \beta_{4} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 13 \beta_{11} - 11 \beta_{10} + 13 \beta_{9} + 8 \beta_{8} + 13 \beta_{7} - 26 \beta_{5} - 26 \beta_{4} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 29 \beta_{11} - 16 \beta_{10} + 50 \beta_{9} + 34 \beta_{8} + 63 \beta_{7} + 14 \beta_{6} - 63 \beta_{5} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 49 \beta_{11} + 115 \beta_{9} + 153 \beta_{8} + 229 \beta_{7} + 76 \beta_{6} - 125 \beta_{5} + \cdots - 49 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 192 \beta_{10} + 192 \beta_{9} + 536 \beta_{8} + 536 \beta_{7} + 262 \beta_{6} - 262 \beta_{4} + \cdots - 147 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 606 \beta_{11} + 1214 \beta_{10} + 1498 \beta_{8} + 920 \beta_{7} + 920 \beta_{6} + 1396 \beta_{5} + \cdots - 790 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3829 \beta_{11} + 5015 \beta_{10} - 2134 \beta_{9} + 2729 \beta_{8} + 2729 \beta_{6} + 8366 \beta_{5} + \cdots - 3032 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 16110 \beta_{11} + 16110 \beta_{10} - 12910 \beta_{9} - 10231 \beta_{7} + 5253 \beta_{6} + 34746 \beta_{5} + \cdots - 10231 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 52909 \beta_{11} + 41607 \beta_{10} - 52909 \beta_{9} - 28208 \beta_{8} - 61701 \beta_{7} + 114610 \beta_{5} + \cdots - 28208 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\) \(59\) \(89\)
\(\chi(n)\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
2.94643 − 1.41893i
−1.32294 + 0.637094i
1.30120 − 1.63166i
−0.523722 + 0.656727i
1.30120 + 1.63166i
−0.523722 − 0.656727i
0.465490 + 2.03945i
−0.366459 − 1.60556i
2.94643 + 1.41893i
−1.32294 − 0.637094i
0.465490 − 2.03945i
−0.366459 + 1.60556i
0 −2.04546 + 0.985042i 0 −0.299629 + 1.31276i 0 −3.84740 + 1.85281i 0 1.34313 − 1.68423i 0
25.2 0 2.22391 − 1.07098i 0 −0.768902 + 3.36878i 0 0.421971 − 0.203210i 0 1.92831 − 2.41802i 0
45.1 0 −1.92469 + 2.41349i 0 −3.16565 − 1.52449i 0 −0.677712 + 0.849824i 0 −1.45292 − 6.36565i 0
45.2 0 −0.0997674 + 0.125104i 0 1.58623 + 0.763888i 0 1.14721 − 1.43856i 0 0.661865 + 2.89982i 0
49.1 0 −1.92469 − 2.41349i 0 −3.16565 + 1.52449i 0 −0.677712 − 0.849824i 0 −1.45292 + 6.36565i 0
49.2 0 −0.0997674 − 0.125104i 0 1.58623 − 0.763888i 0 1.14721 + 1.43856i 0 0.661865 − 2.89982i 0
53.1 0 −0.242969 − 1.06452i 0 2.52737 + 3.16923i 0 −0.688011 − 3.01437i 0 1.62874 − 0.784360i 0
53.2 0 0.588980 + 2.58049i 0 −0.379425 − 0.475783i 0 0.143938 + 0.630635i 0 −3.60913 + 1.73806i 0
65.1 0 −2.04546 − 0.985042i 0 −0.299629 − 1.31276i 0 −3.84740 − 1.85281i 0 1.34313 + 1.68423i 0
65.2 0 2.22391 + 1.07098i 0 −0.768902 − 3.36878i 0 0.421971 + 0.203210i 0 1.92831 + 2.41802i 0
81.1 0 −0.242969 + 1.06452i 0 2.52737 − 3.16923i 0 −0.688011 + 3.01437i 0 1.62874 + 0.784360i 0
81.2 0 0.588980 − 2.58049i 0 −0.379425 + 0.475783i 0 0.143938 − 0.630635i 0 −3.60913 − 1.73806i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 116.2.g.b ✓ 12
3.b odd 2 1 1044.2.u.c 12
4.b odd 2 1 464.2.u.g 12
29.d even 7 1 inner 116.2.g.b ✓ 12
29.d even 7 1 3364.2.a.m 6
29.e even 14 1 3364.2.a.p 6
29.f odd 28 2 3364.2.c.j 12
87.j odd 14 1 1044.2.u.c 12
116.j odd 14 1 464.2.u.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.2.g.b ✓ 12 1.a even 1 1 trivial
116.2.g.b ✓ 12 29.d even 7 1 inner
464.2.u.g 12 4.b odd 2 1
464.2.u.g 12 116.j odd 14 1
1044.2.u.c 12 3.b odd 2 1
1044.2.u.c 12 87.j odd 14 1
3364.2.a.m 6 29.d even 7 1
3364.2.a.p 6 29.e even 14 1
3364.2.c.j 12 29.f odd 28 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 3 T_{3}^{11} + 7 T_{3}^{10} + T_{3}^{9} + 17 T_{3}^{8} - 21 T_{3}^{7} - 55 T_{3}^{6} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(116, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 3 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{12} + T^{11} + \cdots + 5041 \) Copy content Toggle raw display
$7$ \( T^{12} + 7 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{12} - 3 T^{11} + \cdots + 46656 \) Copy content Toggle raw display
$13$ \( T^{12} + 21 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( (T^{6} + 5 T^{5} + \cdots + 344)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 285745216 \) Copy content Toggle raw display
$23$ \( T^{12} + 19 T^{11} + \cdots + 1032256 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 594823321 \) Copy content Toggle raw display
$31$ \( T^{12} + 27 T^{11} + \cdots + 53824 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 161620369 \) Copy content Toggle raw display
$41$ \( (T^{6} + T^{5} - 142 T^{4} + \cdots - 1624)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 9 T^{11} + \cdots + 11235904 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 16533845056 \) Copy content Toggle raw display
$53$ \( (T^{6} - 11 T^{5} + \cdots + 841)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 18 T^{5} + \cdots + 19264)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 2812499089 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 5027377216 \) Copy content Toggle raw display
$71$ \( T^{12} + 35 T^{11} + \cdots + 96983104 \) Copy content Toggle raw display
$73$ \( T^{12} - 34 T^{11} + \cdots + 37442161 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 397313387584 \) Copy content Toggle raw display
$83$ \( T^{12} + T^{11} + \cdots + 13601344 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 876692881 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1819798302001 \) Copy content Toggle raw display
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