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Properties

Label 1148.2.n.c
Level $1148$
Weight $2$
Character orbit 1148.n
Analytic conductor $9.167$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1148,2,Mod(57,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.n (of order \(5\), 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: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 12 x^{14} - 19 x^{13} + 49 x^{12} - 91 x^{11} + 269 x^{10} - 367 x^{9} + 1058 x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{5} + \cdots - \beta_1) q^{3}+ \cdots + ( - \beta_{6} + \beta_{5} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{5} + \cdots - \beta_1) q^{3}+ \cdots + ( - \beta_{15} - \beta_{12} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{3} - 13 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{3} - 13 q^{5} + 4 q^{7} + 2 q^{9} - q^{11} - 6 q^{13} - q^{17} + 15 q^{19} + 2 q^{21} + 27 q^{23} - 3 q^{25} + 28 q^{27} - q^{29} - 14 q^{31} - 13 q^{33} - 12 q^{35} - 16 q^{37} + 10 q^{39} + 26 q^{41} + 5 q^{43} - 9 q^{45} - 14 q^{47} - 4 q^{49} + 4 q^{51} - 20 q^{53} + 10 q^{55} - 13 q^{57} - 47 q^{61} + 3 q^{63} - 29 q^{65} - 27 q^{67} + 15 q^{69} - 11 q^{71} + 70 q^{73} + 14 q^{75} + q^{77} + 30 q^{79} - 72 q^{81} - 78 q^{83} + 72 q^{85} + 21 q^{87} + 17 q^{89} - 24 q^{91} - 7 q^{93} + 27 q^{95} - 17 q^{97} + 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 12 x^{14} - 19 x^{13} + 49 x^{12} - 91 x^{11} + 269 x^{10} - 367 x^{9} + 1058 x^{8} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 48\!\cdots\!55 \nu^{15} + \cdots - 37\!\cdots\!44 ) / 10\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 44\!\cdots\!38 \nu^{15} + \cdots + 33\!\cdots\!63 ) / 10\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 69\!\cdots\!41 \nu^{15} + \cdots - 25\!\cdots\!27 ) / 10\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 77\!\cdots\!92 \nu^{15} + \cdots + 21\!\cdots\!85 ) / 10\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 99\!\cdots\!98 \nu^{15} + \cdots + 36\!\cdots\!92 ) / 10\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21\!\cdots\!85 \nu^{15} + \cdots - 23\!\cdots\!80 ) / 10\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22\!\cdots\!37 \nu^{15} + \cdots + 17\!\cdots\!50 ) / 10\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 36\!\cdots\!92 \nu^{15} + \cdots + 10\!\cdots\!02 ) / 10\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 50\!\cdots\!68 \nu^{15} + \cdots - 54\!\cdots\!53 ) / 10\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 68\!\cdots\!49 \nu^{15} + \cdots - 52\!\cdots\!52 ) / 10\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 84\!\cdots\!78 \nu^{15} + \cdots - 24\!\cdots\!08 ) / 10\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 86\!\cdots\!58 \nu^{15} + \cdots + 40\!\cdots\!61 ) / 10\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10\!\cdots\!70 \nu^{15} + \cdots + 31\!\cdots\!99 ) / 10\!\cdots\!27 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!18 \nu^{15} + \cdots - 52\!\cdots\!23 ) / 10\!\cdots\!27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - 3\beta_{9} + \beta_{6} - \beta_{5} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{11} + \beta_{9} + \beta_{8} - 2\beta_{7} + 5\beta_{6} - \beta_{5} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{13} - 7 \beta_{11} + \beta_{9} - 13 \beta_{8} - \beta_{7} - \beta_{6} + \cdots - 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{15} - 9 \beta_{14} - 11 \beta_{8} + 11 \beta_{7} - 30 \beta_{6} + 30 \beta_{5} + 7 \beta_{4} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 46 \beta_{15} - 34 \beta_{14} - 9 \beta_{12} + 46 \beta_{11} - 3 \beta_{10} - 18 \beta_{9} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16 \beta_{14} - 3 \beta_{13} - 43 \beta_{12} + 68 \beta_{11} - 3 \beta_{10} - 67 \beta_{9} + 94 \beta_{8} + \cdots - 94 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 311 \beta_{15} + 114 \beta_{14} + 24 \beta_{13} - 114 \beta_{11} + 65 \beta_{10} + 348 \beta_{9} + \cdots - 538 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 495 \beta_{15} + 221 \beta_{13} + 270 \beta_{12} - 665 \beta_{11} + 270 \beta_{10} + 741 \beta_{9} + \cdots - 758 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 977 \beta_{15} - 977 \beta_{14} + 446 \beta_{13} + 446 \beta_{12} - 1679 \beta_{8} + 1679 \beta_{7} + \cdots + 3735 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5118 \beta_{15} - 1533 \beta_{14} - 531 \beta_{12} + 5118 \beta_{11} - 1778 \beta_{10} - 5653 \beta_{9} + \cdots + 5653 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7638 \beta_{14} - 3054 \beta_{13} - 3340 \beta_{12} + 7910 \beta_{11} - 3054 \beta_{10} - 13005 \beta_{9} + \cdots - 13708 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 38808 \beta_{15} + 26062 \beta_{14} - 7352 \beta_{13} - 26062 \beta_{11} + 4856 \beta_{10} + \cdots - 68642 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 61942 \beta_{15} + 5394 \beta_{13} + 26600 \beta_{12} - 112945 \beta_{11} + 26600 \beta_{10} + \cdots - 118294 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 190503 \beta_{15} - 190503 \beta_{14} + 40736 \beta_{13} + 40736 \beta_{12} - 317343 \beta_{8} + \cdots + 511184 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
1.75205 + 1.27294i
1.25138 + 0.909177i
−0.0169945 − 0.0123472i
−1.67741 − 1.21871i
1.75205 − 1.27294i
1.25138 − 0.909177i
−0.0169945 + 0.0123472i
−1.67741 + 1.21871i
−0.567301 + 1.74597i
−0.406200 + 1.25015i
0.321891 − 0.990677i
0.842594 − 2.59324i
−0.567301 − 1.74597i
−0.406200 − 1.25015i
0.321891 + 0.990677i
0.842594 + 2.59324i
0 −2.16565 0 −0.636054 − 0.462120i 0 −0.309017 − 0.951057i 0 1.69003 0
57.2 0 −1.54678 0 −2.70780 − 1.96733i 0 −0.309017 − 0.951057i 0 −0.607457 0
57.3 0 0.0210063 0 0.567750 + 0.412494i 0 −0.309017 − 0.951057i 0 −2.99956 0
57.4 0 2.07339 0 −2.15094 − 1.56275i 0 −0.309017 − 0.951057i 0 1.29895 0
141.1 0 −2.16565 0 −0.636054 + 0.462120i 0 −0.309017 + 0.951057i 0 1.69003 0
141.2 0 −1.54678 0 −2.70780 + 1.96733i 0 −0.309017 + 0.951057i 0 −0.607457 0
141.3 0 0.0210063 0 0.567750 − 0.412494i 0 −0.309017 + 0.951057i 0 −2.99956 0
141.4 0 2.07339 0 −2.15094 + 1.56275i 0 −0.309017 + 0.951057i 0 1.29895 0
365.1 0 −1.83583 0 −0.827871 + 2.54793i 0 0.809017 + 0.587785i 0 0.370254 0
365.2 0 −1.31449 0 0.558990 − 1.72039i 0 0.809017 + 0.587785i 0 −1.27211 0
365.3 0 1.04166 0 −0.980821 + 3.01866i 0 0.809017 + 0.587785i 0 −1.91494 0
365.4 0 2.72669 0 −0.323246 + 0.994850i 0 0.809017 + 0.587785i 0 4.43484 0
953.1 0 −1.83583 0 −0.827871 − 2.54793i 0 0.809017 − 0.587785i 0 0.370254 0
953.2 0 −1.31449 0 0.558990 + 1.72039i 0 0.809017 − 0.587785i 0 −1.27211 0
953.3 0 1.04166 0 −0.980821 − 3.01866i 0 0.809017 − 0.587785i 0 −1.91494 0
953.4 0 2.72669 0 −0.323246 − 0.994850i 0 0.809017 − 0.587785i 0 4.43484 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.n.c ✓ 16
41.d even 5 1 inner 1148.2.n.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.n.c ✓ 16 1.a even 1 1 trivial
1148.2.n.c ✓ 16 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + T_{3}^{7} - 12T_{3}^{6} - 15T_{3}^{5} + 40T_{3}^{4} + 57T_{3}^{3} - 30T_{3}^{2} - 47T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1148, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + T^{7} - 12 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + 13 T^{15} + \cdots + 6241 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} + T^{15} + \cdots + 1739761 \) Copy content Toggle raw display
$13$ \( T^{16} + 6 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} + T^{15} + \cdots + 151321 \) Copy content Toggle raw display
$19$ \( T^{16} - 15 T^{15} + \cdots + 524176 \) Copy content Toggle raw display
$23$ \( T^{16} - 27 T^{15} + \cdots + 57121 \) Copy content Toggle raw display
$29$ \( T^{16} + T^{15} + \cdots + 9339136 \) Copy content Toggle raw display
$31$ \( T^{16} + 14 T^{15} + \cdots + 516961 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 121773779521 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 7984925229121 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 2781276656656 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 59136998761 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 19447975936 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2792638712161 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 594007496208361 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 651140560711936 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 53983463854336 \) Copy content Toggle raw display
$73$ \( (T^{8} - 35 T^{7} + \cdots - 108821)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 15 T^{7} + \cdots - 2357429)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 39 T^{7} + \cdots - 176521)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 431729157721 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 19920234577681 \) Copy content Toggle raw display
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