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Properties

Label 182.2.n.b
Level $182$
Weight $2$
Character orbit 182.n
Analytic conductor $1.453$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [182,2,Mod(25,182)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(182, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("182.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 182 = 2 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 182.n (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: \(1.45327731679\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} + (\beta_{9} - \beta_{7} + 1) q^{3} + ( - \beta_{7} + 1) q^{4} + (\beta_{11} + \beta_{8} - \beta_1) q^{5} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_{2}) q^{6}+ \cdots + (2 \beta_{9} - \beta_{5} + \beta_{4} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} + (\beta_{9} - \beta_{7} + 1) q^{3} + ( - \beta_{7} + 1) q^{4} + (\beta_{11} + \beta_{8} - \beta_1) q^{5} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_{2}) q^{6}+ \cdots + (2 \beta_{11} + 2 \beta_{10} + \cdots + 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} + 6 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{3} + 6 q^{4} + 6 q^{9} + 4 q^{10} - 4 q^{12} - 8 q^{13} - 6 q^{16} - 8 q^{17} - 8 q^{22} - 18 q^{25} - 2 q^{26} + 4 q^{27} + 32 q^{29} + 2 q^{30} - 28 q^{35} + 12 q^{36} - 16 q^{38} - 12 q^{39} - 4 q^{40} + 28 q^{42} - 40 q^{43} - 8 q^{48} - 18 q^{51} - 4 q^{52} + 24 q^{53} - 24 q^{55} - 4 q^{61} - 20 q^{62} - 12 q^{64} + 8 q^{65} - 12 q^{66} + 8 q^{68} + 56 q^{69} - 12 q^{74} + 12 q^{75} + 56 q^{77} + 16 q^{78} - 16 q^{79} + 2 q^{81} + 12 q^{82} + 20 q^{87} - 4 q^{88} + 36 q^{90} - 28 q^{91} + 8 q^{94} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 3\beta_{8} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{5} + 9\beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{6} - 14\beta_{2} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/182\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(157\)
\(\chi(n)\) \(-1\) \(-\beta_{7}\)

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
−1.07992 − 0.623490i
1.56052 + 0.900969i
0.385418 + 0.222521i
1.07992 + 0.623490i
−1.56052 − 0.900969i
−0.385418 − 0.222521i
−1.07992 + 0.623490i
1.56052 − 0.900969i
0.385418 − 0.222521i
1.07992 − 0.623490i
−1.56052 + 0.900969i
−0.385418 + 0.222521i
−0.866025 + 0.500000i −0.400969 + 0.694498i 0.500000 − 0.866025i −0.480608 + 0.277479i − 0.801938i −1.46533 + 2.20291i 1.00000i 1.17845 + 2.04113i 0.277479 − 0.480608i
25.2 −0.866025 + 0.500000i 0.277479 − 0.480608i 0.500000 − 0.866025i −1.94594 + 1.12349i 0.554958i 2.64044 + 0.167563i 1.00000i 1.34601 + 2.33136i 1.12349 − 1.94594i
25.3 −0.866025 + 0.500000i 1.12349 − 1.94594i 0.500000 − 0.866025i 0.694498 − 0.400969i 2.24698i −1.17511 − 2.37047i 1.00000i −1.02446 − 1.77441i −0.400969 + 0.694498i
25.4 0.866025 − 0.500000i −0.400969 + 0.694498i 0.500000 − 0.866025i 0.480608 − 0.277479i 0.801938i 1.46533 − 2.20291i − 1.00000i 1.17845 + 2.04113i 0.277479 − 0.480608i
25.5 0.866025 − 0.500000i 0.277479 − 0.480608i 0.500000 − 0.866025i 1.94594 − 1.12349i − 0.554958i −2.64044 − 0.167563i − 1.00000i 1.34601 + 2.33136i 1.12349 − 1.94594i
25.6 0.866025 − 0.500000i 1.12349 − 1.94594i 0.500000 − 0.866025i −0.694498 + 0.400969i − 2.24698i 1.17511 + 2.37047i − 1.00000i −1.02446 − 1.77441i −0.400969 + 0.694498i
51.1 −0.866025 − 0.500000i −0.400969 − 0.694498i 0.500000 + 0.866025i −0.480608 − 0.277479i 0.801938i −1.46533 − 2.20291i − 1.00000i 1.17845 − 2.04113i 0.277479 + 0.480608i
51.2 −0.866025 − 0.500000i 0.277479 + 0.480608i 0.500000 + 0.866025i −1.94594 − 1.12349i − 0.554958i 2.64044 − 0.167563i − 1.00000i 1.34601 − 2.33136i 1.12349 + 1.94594i
51.3 −0.866025 − 0.500000i 1.12349 + 1.94594i 0.500000 + 0.866025i 0.694498 + 0.400969i − 2.24698i −1.17511 + 2.37047i − 1.00000i −1.02446 + 1.77441i −0.400969 − 0.694498i
51.4 0.866025 + 0.500000i −0.400969 − 0.694498i 0.500000 + 0.866025i 0.480608 + 0.277479i − 0.801938i 1.46533 + 2.20291i 1.00000i 1.17845 − 2.04113i 0.277479 + 0.480608i
51.5 0.866025 + 0.500000i 0.277479 + 0.480608i 0.500000 + 0.866025i 1.94594 + 1.12349i 0.554958i −2.64044 + 0.167563i 1.00000i 1.34601 − 2.33136i 1.12349 + 1.94594i
51.6 0.866025 + 0.500000i 1.12349 + 1.94594i 0.500000 + 0.866025i −0.694498 − 0.400969i 2.24698i 1.17511 − 2.37047i 1.00000i −1.02446 + 1.77441i −0.400969 − 0.694498i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 182.2.n.b ✓ 12
3.b odd 2 1 1638.2.dm.d 12
7.b odd 2 1 1274.2.n.l 12
7.c even 3 1 inner 182.2.n.b ✓ 12
7.c even 3 1 1274.2.d.k 6
7.d odd 6 1 1274.2.d.m 6
7.d odd 6 1 1274.2.n.l 12
13.b even 2 1 inner 182.2.n.b ✓ 12
21.h odd 6 1 1638.2.dm.d 12
39.d odd 2 1 1638.2.dm.d 12
91.b odd 2 1 1274.2.n.l 12
91.r even 6 1 inner 182.2.n.b ✓ 12
91.r even 6 1 1274.2.d.k 6
91.s odd 6 1 1274.2.d.m 6
91.s odd 6 1 1274.2.n.l 12
273.w odd 6 1 1638.2.dm.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.n.b ✓ 12 1.a even 1 1 trivial
182.2.n.b ✓ 12 7.c even 3 1 inner
182.2.n.b ✓ 12 13.b even 2 1 inner
182.2.n.b ✓ 12 91.r even 6 1 inner
1274.2.d.k 6 7.c even 3 1
1274.2.d.k 6 91.r even 6 1
1274.2.d.m 6 7.d odd 6 1
1274.2.d.m 6 91.s odd 6 1
1274.2.n.l 12 7.b odd 2 1
1274.2.n.l 12 7.d odd 6 1
1274.2.n.l 12 91.b odd 2 1
1274.2.n.l 12 91.s odd 6 1
1638.2.dm.d 12 3.b odd 2 1
1638.2.dm.d 12 21.h odd 6 1
1638.2.dm.d 12 39.d odd 2 1
1638.2.dm.d 12 273.w odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 2T_{3}^{5} + 5T_{3}^{4} + 3T_{3}^{2} - T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(182, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{6} - 2 T^{5} + 5 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} - 637 T^{6} + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} - 20 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( (T^{6} + 4 T^{5} + \cdots + 2197)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} + 27 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 40 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( (T^{6} + 28 T^{4} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 8 T^{2} + \cdots + 344)^{4} \) Copy content Toggle raw display
$31$ \( T^{12} - 83 T^{10} + \cdots + 2825761 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 45165175441 \) Copy content Toggle raw display
$41$ \( (T^{6} + 68 T^{4} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 10 T^{2} + \cdots - 349)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} - 10 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( (T^{6} - 12 T^{5} + \cdots + 46656)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 1065552449536 \) Copy content Toggle raw display
$61$ \( (T^{6} + 2 T^{5} + \cdots + 1032256)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 116985856 \) Copy content Toggle raw display
$71$ \( (T^{6} + 66 T^{4} + \cdots + 5041)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 2897022976 \) Copy content Toggle raw display
$79$ \( (T^{6} + 8 T^{5} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 244 T^{4} + \cdots + 322624)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 104086245376 \) Copy content Toggle raw display
$97$ \( (T^{6} + 404 T^{4} + \cdots + 118336)^{2} \) Copy content Toggle raw display
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