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Properties

Label 152.2.c.b
Level $152$
Weight $2$
Character orbit 152.c
Analytic conductor $1.214$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [152,2,Mod(77,152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("152.77"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 152.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.21372611072\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 10 x^{10} + 24 x^{9} - 40 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + \beta_{4} q^{3} + \beta_{2} q^{4} + ( - \beta_{8} + \beta_{6} + \beta_1) q^{5} + (\beta_{15} - \beta_{14} + \cdots - \beta_1) q^{6} + ( - \beta_{15} + \beta_{8} + \cdots - \beta_{5}) q^{7}+ \cdots + ( - 2 \beta_{15} - \beta_{12} + \cdots + 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{4} + 6 q^{6} - 8 q^{7} - 12 q^{8} - 24 q^{9} - 8 q^{10} + 4 q^{12} + 4 q^{14} + 2 q^{16} - 8 q^{17} + 20 q^{18} + 8 q^{20} + 20 q^{22} + 6 q^{24} - 24 q^{25} - 10 q^{26} - 14 q^{28} + 4 q^{30}+ \cdots - 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 10 x^{10} + 24 x^{9} - 40 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} - 3 \nu^{13} + 4 \nu^{12} - 4 \nu^{11} - 4 \nu^{10} + 10 \nu^{9} - 24 \nu^{8} + \cdots + 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - \nu^{14} + \nu^{13} - \nu^{12} + 8 \nu^{10} - 6 \nu^{9} + 14 \nu^{8} - 16 \nu^{7} + \cdots - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2 \nu^{14} + \nu^{13} + 4 \nu^{12} - 3 \nu^{11} + 10 \nu^{9} + 34 \nu^{7} + 36 \nu^{5} - 32 \nu^{4} + \cdots + 320 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{15} + 4 \nu^{14} + 11 \nu^{13} - 2 \nu^{12} - 4 \nu^{11} + 4 \nu^{10} + 14 \nu^{9} + 76 \nu^{8} + \cdots + 128 ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{15} - 12 \nu^{14} - \nu^{13} - 10 \nu^{12} + 8 \nu^{11} + 20 \nu^{10} - 42 \nu^{9} + \cdots - 1152 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{15} - 8 \nu^{14} - 11 \nu^{13} - 10 \nu^{12} + 12 \nu^{11} - 4 \nu^{10} - 14 \nu^{9} + \cdots - 1152 ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5 \nu^{15} - 8 \nu^{14} + 5 \nu^{13} - 18 \nu^{12} + 6 \nu^{11} + 28 \nu^{10} - 38 \nu^{9} + \cdots - 1152 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{15} - 16 \nu^{14} - 11 \nu^{13} - 18 \nu^{12} + 12 \nu^{11} + 12 \nu^{10} - 62 \nu^{9} + \cdots - 1920 ) / 256 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9 \nu^{15} - 28 \nu^{14} + 11 \nu^{13} - 58 \nu^{12} + 16 \nu^{11} + 60 \nu^{10} - 98 \nu^{9} + \cdots - 3712 ) / 256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5 \nu^{15} + 5 \nu^{14} + 7 \nu^{13} + \nu^{12} - 10 \nu^{11} + 12 \nu^{10} + 6 \nu^{9} + 74 \nu^{8} + \cdots + 384 ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25 \nu^{15} - 12 \nu^{14} + 35 \nu^{13} - 50 \nu^{12} - 8 \nu^{11} + 100 \nu^{10} - 66 \nu^{9} + \cdots - 2176 ) / 256 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 29 \nu^{15} - 40 \nu^{14} + 31 \nu^{13} - 94 \nu^{12} + 148 \nu^{10} - 170 \nu^{9} + 500 \nu^{8} + \cdots - 5760 ) / 256 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15 \nu^{15} + 14 \nu^{14} - 19 \nu^{13} + 40 \nu^{12} + 6 \nu^{11} - 60 \nu^{10} + 66 \nu^{9} + \cdots + 2176 ) / 128 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 9 \nu^{15} + 19 \nu^{14} - 9 \nu^{13} + 43 \nu^{12} - 8 \nu^{11} - 52 \nu^{10} + 74 \nu^{9} + \cdots + 2816 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21 \nu^{15} + 22 \nu^{14} - 27 \nu^{13} + 64 \nu^{12} - 100 \nu^{10} + 98 \nu^{9} - 368 \nu^{8} + \cdots + 3584 ) / 128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} - \beta_{5} + \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + \beta_{13} + \beta_{12} + \beta_{10} + 2\beta_{9} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - \beta_{13} - \beta_{12} + \beta_{10} + 2\beta_{9} + 2\beta_{6} + \beta_{4} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{15} + 2 \beta_{14} - \beta_{13} + \beta_{12} + 2 \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3 \beta_{15} + 2 \beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + \cdots - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{15} - 2 \beta_{14} - \beta_{13} - \beta_{12} + \beta_{10} - 2 \beta_{9} - 4 \beta_{8} + \cdots - 4 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - \beta_{15} + 2 \beta_{14} - 5 \beta_{13} - 5 \beta_{12} + 6 \beta_{11} - 3 \beta_{10} - 2 \beta_{9} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3 \beta_{15} + 10 \beta_{14} + 3 \beta_{13} + 11 \beta_{12} - 4 \beta_{11} + \beta_{10} + \cdots + 4 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 5 \beta_{15} - 2 \beta_{14} + 7 \beta_{13} + 7 \beta_{12} - 10 \beta_{11} - 7 \beta_{10} - 2 \beta_{9} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( \beta_{15} - 6 \beta_{14} + 7 \beta_{13} - 5 \beta_{12} + 20 \beta_{11} + \beta_{10} - 18 \beta_{9} + \cdots + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7 \beta_{15} - 10 \beta_{14} + 23 \beta_{13} - \beta_{12} - 22 \beta_{11} + 9 \beta_{10} + 30 \beta_{9} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 41 \beta_{15} - 14 \beta_{14} + 11 \beta_{13} + 47 \beta_{12} - 24 \beta_{11} + 13 \beta_{10} + \cdots - 36 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - \beta_{15} - 2 \beta_{14} - 5 \beta_{13} - 37 \beta_{12} + 38 \beta_{11} - 11 \beta_{10} + 22 \beta_{9} + \cdots - 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 39 \beta_{15} - 14 \beta_{14} - 17 \beta_{13} - 77 \beta_{12} + 12 \beta_{11} + 33 \beta_{10} + \cdots + 44 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 33 \beta_{15} - 82 \beta_{14} + 15 \beta_{13} + 23 \beta_{12} - 86 \beta_{11} - 95 \beta_{10} + \cdots - 40 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).

\(n\) \(39\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1
−0.182898 − 1.40234i
−0.182898 + 1.40234i
0.340606 + 1.37258i
0.340606 − 1.37258i
1.14052 + 0.836196i
1.14052 − 0.836196i
−1.40771 − 0.135487i
−1.40771 + 0.135487i
1.33852 − 0.456455i
1.33852 + 0.456455i
1.12629 − 0.855255i
1.12629 + 0.855255i
−0.889165 + 1.09972i
−0.889165 − 1.09972i
−0.466170 + 1.33517i
−0.466170 − 1.33517i
−1.40234 − 0.182898i − 0.840428i 1.93310 + 0.512969i 4.04855i −0.153712 + 1.17856i −3.59283 −2.61703 − 1.07291i 2.29368 0.740472 − 5.67744i
77.2 −1.40234 + 0.182898i 0.840428i 1.93310 − 0.512969i − 4.04855i −0.153712 − 1.17856i −3.59283 −2.61703 + 1.07291i 2.29368 0.740472 + 5.67744i
77.3 −1.37258 − 0.340606i − 2.95163i 1.76798 + 0.935021i − 2.13486i −1.00534 + 4.05136i 3.29464 −2.10822 − 1.88558i −5.71210 −0.727145 + 2.93027i
77.4 −1.37258 + 0.340606i 2.95163i 1.76798 − 0.935021i 2.13486i −1.00534 − 4.05136i 3.29464 −2.10822 + 1.88558i −5.71210 −0.727145 − 2.93027i
77.5 −0.836196 − 1.14052i 3.13611i −0.601554 + 1.90739i − 0.594041i 3.57679 − 2.62240i −3.48756 2.67842 − 0.908869i −6.83520 −0.677513 + 0.496734i
77.6 −0.836196 + 1.14052i − 3.13611i −0.601554 − 1.90739i 0.594041i 3.57679 + 2.62240i −3.48756 2.67842 + 0.908869i −6.83520 −0.677513 − 0.496734i
77.7 −0.135487 − 1.40771i − 1.70663i −1.96329 + 0.381452i − 1.66222i −2.40244 + 0.231226i −1.99556 0.802973 + 2.71205i 0.0874066 −2.33992 + 0.225209i
77.8 −0.135487 + 1.40771i 1.70663i −1.96329 − 0.381452i 1.66222i −2.40244 − 0.231226i −1.99556 0.802973 − 2.71205i 0.0874066 −2.33992 − 0.225209i
77.9 0.456455 − 1.33852i 2.09554i −1.58330 − 1.22195i − 3.36827i 2.80493 + 0.956520i 4.47116 −2.35832 + 1.56152i −1.39129 −4.50852 − 1.53746i
77.10 0.456455 + 1.33852i − 2.09554i −1.58330 + 1.22195i 3.36827i 2.80493 − 0.956520i 4.47116 −2.35832 − 1.56152i −1.39129 −4.50852 + 1.53746i
77.11 0.855255 − 1.12629i − 1.91886i −0.537077 − 1.92654i 1.51356i −2.16120 − 1.64111i 0.580162 −2.62919 − 1.04278i −0.682013 1.70471 + 1.29448i
77.12 0.855255 + 1.12629i 1.91886i −0.537077 + 1.92654i − 1.51356i −2.16120 + 1.64111i 0.580162 −2.62919 + 1.04278i −0.682013 1.70471 − 1.29448i
77.13 1.09972 − 0.889165i 2.32921i 0.418773 − 1.95567i 3.13887i 2.07105 + 2.56148i −0.535658 −1.27838 − 2.52304i −2.42523 2.79097 + 3.45188i
77.14 1.09972 + 0.889165i − 2.32921i 0.418773 + 1.95567i − 3.13887i 2.07105 − 2.56148i −0.535658 −1.27838 + 2.52304i −2.42523 2.79097 − 3.45188i
77.15 1.33517 − 0.466170i 0.579017i 1.56537 − 1.24484i − 2.10882i 0.269921 + 0.773088i −2.73436 1.50973 − 2.39180i 2.66474 −0.983067 − 2.81563i
77.16 1.33517 + 0.466170i − 0.579017i 1.56537 + 1.24484i 2.10882i 0.269921 − 0.773088i −2.73436 1.50973 + 2.39180i 2.66474 −0.983067 + 2.81563i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.2.c.b ✓ 16
3.b odd 2 1 1368.2.g.b 16
4.b odd 2 1 608.2.c.b 16
8.b even 2 1 inner 152.2.c.b ✓ 16
8.d odd 2 1 608.2.c.b 16
12.b even 2 1 5472.2.g.b 16
16.e even 4 1 4864.2.a.bo 8
16.e even 4 1 4864.2.a.bq 8
16.f odd 4 1 4864.2.a.bn 8
16.f odd 4 1 4864.2.a.bp 8
24.f even 2 1 5472.2.g.b 16
24.h odd 2 1 1368.2.g.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.c.b ✓ 16 1.a even 1 1 trivial
152.2.c.b ✓ 16 8.b even 2 1 inner
608.2.c.b 16 4.b odd 2 1
608.2.c.b 16 8.d odd 2 1
1368.2.g.b 16 3.b odd 2 1
1368.2.g.b 16 24.h odd 2 1
4864.2.a.bn 8 16.f odd 4 1
4864.2.a.bo 8 16.e even 4 1
4864.2.a.bp 8 16.f odd 4 1
4864.2.a.bq 8 16.e even 4 1
5472.2.g.b 16 12.b even 2 1
5472.2.g.b 16 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 36T_{3}^{14} + 526T_{3}^{12} + 4028T_{3}^{10} + 17401T_{3}^{8} + 42248T_{3}^{6} + 53472T_{3}^{4} + 29248T_{3}^{2} + 5184 \) acting on \(S_{2}^{\mathrm{new}}(152, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} + 36 T^{14} + \cdots + 5184 \) Copy content Toggle raw display
$5$ \( T^{16} + 52 T^{14} + \cdots + 82944 \) Copy content Toggle raw display
$7$ \( (T^{8} + 4 T^{7} + \cdots - 313)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 84 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( T^{16} + 112 T^{14} + \cdots + 262144 \) Copy content Toggle raw display
$17$ \( (T^{8} + 4 T^{7} + \cdots + 9409)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$23$ \( (T^{8} - 124 T^{6} + \cdots + 155824)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 248 T^{14} + \cdots + 8620096 \) Copy content Toggle raw display
$31$ \( (T^{8} - 8 T^{7} + \cdots + 7552)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 144 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( (T^{8} - 8 T^{7} + \cdots - 3072)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 204 T^{14} + \cdots + 1401856 \) Copy content Toggle raw display
$47$ \( (T^{8} - 12 T^{7} + \cdots + 34432)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 132199142464 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2722334976 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 10615398064384 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 4865341504 \) Copy content Toggle raw display
$71$ \( (T^{8} - 24 T^{7} + \cdots - 9153024)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 316 T^{6} + \cdots + 6637913)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 24 T^{7} + \cdots - 70912)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 22968008704 \) Copy content Toggle raw display
$89$ \( (T^{8} + 8 T^{7} + \cdots - 400128)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 16 T^{7} + \cdots + 8192)^{2} \) Copy content Toggle raw display
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