Properties

Label 1320.2.bw.f
Level $1320$
Weight $2$
Character orbit 1320.bw
Analytic conductor $10.540$
Analytic rank $0$
Dimension $12$
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] = [1320,2,Mod(361,1320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1320, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1320.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1320.bw (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: \(10.5402530668\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 7 x^{10} + 15 x^{9} + 51 x^{8} + 175 x^{7} + 1103 x^{6} + 2884 x^{5} + 5561 x^{4} + 5670 x^{3} + 2840 x^{2} - 125 x + 25 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + (\beta_{8} - \beta_{7} + \beta_{6} - 1) q^{5} + (\beta_{7} - \beta_{2}) q^{7} - \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + (\beta_{8} - \beta_{7} + \beta_{6} - 1) q^{5} + (\beta_{7} - \beta_{2}) q^{7} - \beta_{8} q^{9} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{3} - \beta_{2} - 1) q^{11} + ( - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} - \beta_1) q^{13} - \beta_{7} q^{15} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} + 1) q^{17} + (\beta_{10} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{19} + ( - \beta_{4} - 1) q^{21} + (\beta_{3} + \beta_{2} - \beta_1) q^{23} - \beta_{6} q^{25} + ( - \beta_{8} + \beta_{7} - \beta_{6} + 1) q^{27} + ( - \beta_{10} + 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2}) q^{29} + ( - 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{7} - 2 \beta_{5} + \beta_{4} + \cdots + 2 \beta_1) q^{31}+ \cdots + (\beta_{8} - \beta_{7} + \beta_{6} + \beta_{3} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - 3 q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{3} - 3 q^{5} - 3 q^{9} - 4 q^{11} + 3 q^{13} + 3 q^{15} + 12 q^{17} - 4 q^{19} - 10 q^{21} - 12 q^{23} - 3 q^{25} + 3 q^{27} + 16 q^{29} + 3 q^{31} + 4 q^{33} - 5 q^{35} - 19 q^{37} - 3 q^{39} + 22 q^{41} + 52 q^{43} + 12 q^{45} + 25 q^{47} - 11 q^{49} + 8 q^{51} - q^{53} - 4 q^{55} + 4 q^{57} - 9 q^{59} + 21 q^{61} - 12 q^{65} - 18 q^{67} - 8 q^{69} + 17 q^{71} + 37 q^{73} + 3 q^{75} - 13 q^{77} - 18 q^{79} - 3 q^{81} + 19 q^{83} - 8 q^{85} + 14 q^{87} + 2 q^{89} - 30 q^{91} - 3 q^{93} - 4 q^{95} + q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 7 x^{10} + 15 x^{9} + 51 x^{8} + 175 x^{7} + 1103 x^{6} + 2884 x^{5} + 5561 x^{4} + 5670 x^{3} + 2840 x^{2} - 125 x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 136390865097927 \nu^{11} + 914034935057519 \nu^{10} + \cdots + 73\!\cdots\!75 ) / 11\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 301209731965466 \nu^{11} + 144579968687973 \nu^{10} - 506029326150353 \nu^{9} + \cdots - 77\!\cdots\!25 ) / 11\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 718085572404367 \nu^{11} - 695454595132424 \nu^{10} + \cdots - 52\!\cdots\!80 ) / 11\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 795868267650904 \nu^{11} + \cdots - 11\!\cdots\!25 ) / 11\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!97 \nu^{11} + \cdots + 33\!\cdots\!75 ) / 11\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\!\cdots\!23 \nu^{11} + \cdots + 81\!\cdots\!30 ) / 11\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 29\!\cdots\!91 \nu^{11} + \cdots - 10\!\cdots\!00 ) / 11\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 30\!\cdots\!13 \nu^{11} + \cdots - 16\!\cdots\!30 ) / 11\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 392954540830935 \nu^{11} + 582859541666266 \nu^{10} + \cdots - 19\!\cdots\!35 ) / 60\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 79\!\cdots\!34 \nu^{11} + \cdots + 29\!\cdots\!25 ) / 11\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!94 \nu^{11} + \cdots - 36\!\cdots\!75 ) / 11\!\cdots\!10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( -\beta_{5} + \beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{8} + 4\beta_{7} + \beta_{6} - 2\beta_{5} + 2\beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} - \beta_{9} + 9\beta_{8} + 8\beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} - 10\beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9 \beta_{11} - 9 \beta_{10} + \beta_{9} + 47 \beta_{8} - 38 \beta_{7} + 65 \beta_{6} + 9 \beta_{5} - 30 \beta_{4} - 10 \beta_{3} - 30 \beta _1 - 65 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20 \beta_{11} + 20 \beta_{9} - 133 \beta_{7} + 153 \beta_{6} - 146 \beta_{4} - 10 \beta_{3} + 30 \beta_{2} - 30 \beta _1 - 154 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20 \beta_{11} + 96 \beta_{10} + 96 \beta_{9} - 550 \beta_{8} - 296 \beta_{7} - 55 \beta_{5} - 360 \beta_{4} - 20 \beta_{3} + 360 \beta_{2} + 55 \beta _1 - 316 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 75 \beta_{11} + 299 \beta_{10} - 2243 \beta_{8} - 367 \beta_{7} - 2243 \beta_{6} + 621 \beta_{5} - 621 \beta_{4} + 2056 \beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 397 \beta_{10} - 1136 \beta_{9} - 5311 \beta_{8} - 12359 \beta_{6} + 5734 \beta_{5} + 397 \beta_{3} + 5734 \beta_{2} + 99 \beta _1 + 5311 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 298 \beta_{11} - 298 \beta_{10} - 4201 \beta_{9} - 7707 \beta_{8} + 8005 \beta_{7} - 39938 \beta_{6} + 18417 \beta_{5} + 11243 \beta_{4} + 3903 \beta_{3} + 11243 \beta _1 + 39938 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7340 \beta_{11} - 7340 \beta_{9} + 79131 \beta_{7} - 86471 \beta_{6} + 92084 \beta_{4} + 21556 \beta_{3} - 80308 \beta_{2} + 80308 \beta _1 + 180801 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 58752 \beta_{11} - 4436 \beta_{10} - 4436 \beta_{9} + 152940 \beta_{8} + 406440 \beta_{7} - 243170 \beta_{5} + 433501 \beta_{4} + 58752 \beta_{3} - 433501 \beta_{2} + 243170 \beta _1 + 465192 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1320\mathbb{Z}\right)^\times\).

\(n\) \(661\) \(881\) \(991\) \(1057\) \(1201\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.550743 + 1.69501i
0.0276441 0.0850798i
1.02310 3.14877i
−0.550743 1.69501i
0.0276441 + 0.0850798i
1.02310 + 3.14877i
−1.70426 1.23822i
−0.936904 0.680700i
3.14116 + 2.28219i
−1.70426 + 1.23822i
−0.936904 + 0.680700i
3.14116 2.28219i
0 0.809017 0.587785i 0 0.309017 + 0.951057i 0 −2.25088 1.63536i 0 0.309017 0.951057i 0
361.2 0 0.809017 0.587785i 0 0.309017 + 0.951057i 0 −0.736644 0.535203i 0 0.309017 0.951057i 0
361.3 0 0.809017 0.587785i 0 0.309017 + 0.951057i 0 1.86949 + 1.35826i 0 0.309017 0.951057i 0
841.1 0 0.809017 + 0.587785i 0 0.309017 0.951057i 0 −2.25088 + 1.63536i 0 0.309017 + 0.951057i 0
841.2 0 0.809017 + 0.587785i 0 0.309017 0.951057i 0 −0.736644 + 0.535203i 0 0.309017 + 0.951057i 0
841.3 0 0.809017 + 0.587785i 0 0.309017 0.951057i 0 1.86949 1.35826i 0 0.309017 + 0.951057i 0
961.1 0 −0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0 −0.341952 1.05242i 0 −0.809017 0.587785i 0
961.2 0 −0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0 −0.0488484 0.150340i 0 −0.809017 0.587785i 0
961.3 0 −0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0 1.50883 + 4.64371i 0 −0.809017 0.587785i 0
1081.1 0 −0.309017 0.951057i 0 −0.809017 0.587785i 0 −0.341952 + 1.05242i 0 −0.809017 + 0.587785i 0
1081.2 0 −0.309017 0.951057i 0 −0.809017 0.587785i 0 −0.0488484 + 0.150340i 0 −0.809017 + 0.587785i 0
1081.3 0 −0.309017 0.951057i 0 −0.809017 0.587785i 0 1.50883 4.64371i 0 −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1320.2.bw.f 12
11.c even 5 1 inner 1320.2.bw.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.bw.f 12 1.a even 1 1 trivial
1320.2.bw.f 12 11.c even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 16 T_{7}^{10} + 60 T_{7}^{9} + 76 T_{7}^{8} - 190 T_{7}^{7} + 421 T_{7}^{6} + 1370 T_{7}^{5} + 2551 T_{7}^{4} + 2360 T_{7}^{3} + 1265 T_{7}^{2} + 150 T_{7} + 25 \) acting on \(S_{2}^{\mathrm{new}}(1320, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} + 16 T^{10} + 60 T^{9} + 76 T^{8} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + 32 T^{10} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} - 3 T^{11} + 33 T^{10} + \cdots + 83521 \) Copy content Toggle raw display
$17$ \( T^{12} - 12 T^{11} + 125 T^{10} + \cdots + 20736 \) Copy content Toggle raw display
$19$ \( T^{12} + 4 T^{11} + 7 T^{10} + \cdots + 249001 \) Copy content Toggle raw display
$23$ \( (T^{6} + 6 T^{5} - 41 T^{4} - 275 T^{3} + \cdots + 3275)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 16 T^{11} + 203 T^{10} + \cdots + 250000 \) Copy content Toggle raw display
$31$ \( T^{12} - 3 T^{11} - T^{10} + \cdots + 156250000 \) Copy content Toggle raw display
$37$ \( T^{12} + 19 T^{11} + 213 T^{10} + \cdots + 198025 \) Copy content Toggle raw display
$41$ \( T^{12} - 22 T^{11} + 278 T^{10} + \cdots + 45495025 \) Copy content Toggle raw display
$43$ \( (T^{6} - 26 T^{5} + 238 T^{4} - 857 T^{3} + \cdots + 20)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 25 T^{11} + 359 T^{10} + \cdots + 10208025 \) Copy content Toggle raw display
$53$ \( T^{12} + T^{11} + 122 T^{10} + \cdots + 10773402025 \) Copy content Toggle raw display
$59$ \( T^{12} + 9 T^{11} + 45 T^{10} + \cdots + 128881 \) Copy content Toggle raw display
$61$ \( T^{12} - 21 T^{11} + 138 T^{10} + \cdots + 32400 \) Copy content Toggle raw display
$67$ \( (T^{6} + 9 T^{5} + 2 T^{4} - 88 T^{3} + \cdots + 44)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 17 T^{11} + 299 T^{10} + \cdots + 24010000 \) Copy content Toggle raw display
$73$ \( T^{12} - 37 T^{11} + \cdots + 356605456 \) Copy content Toggle raw display
$79$ \( T^{12} + 18 T^{11} + \cdots + 365758848400 \) Copy content Toggle raw display
$83$ \( T^{12} - 19 T^{11} + \cdots + 237285894400 \) Copy content Toggle raw display
$89$ \( (T^{6} - T^{5} - 174 T^{4} + 259 T^{3} + \cdots - 1375)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - T^{11} - 58 T^{10} + \cdots + 6990736 \) Copy content Toggle raw display
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