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Properties

Label 1260.2.ba.b
Level $1260$
Weight $2$
Character orbit 1260.ba
Analytic conductor $10.061$
Analytic rank $0$
Dimension $16$
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] = [1260,2,Mod(433,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.433");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.ba (of order \(4\), 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: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8x^{14} + 8x^{12} - 8x^{10} + 212x^{8} + 248x^{6} + 368x^{4} + 32x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{13} + \beta_{11} + \beta_{7}) q^{5} + ( - \beta_{12} - \beta_{5} + \cdots + \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{13} + \beta_{11} + \beta_{7}) q^{5} + ( - \beta_{12} - \beta_{5} + \cdots + \beta_{3}) q^{7}+ \cdots + (3 \beta_{14} + 5 \beta_{13} + \cdots - \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{11} - 8 q^{23} + 16 q^{25} - 16 q^{35} + 16 q^{37} + 48 q^{43} + 40 q^{53} + 56 q^{65} - 48 q^{67} + 32 q^{71} + 24 q^{77} - 64 q^{85} + 32 q^{91} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8x^{14} + 8x^{12} - 8x^{10} + 212x^{8} + 248x^{6} + 368x^{4} + 32x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2329 \nu^{14} - 16984 \nu^{12} + 3874 \nu^{10} + 8131 \nu^{8} + 466620 \nu^{6} + 946682 \nu^{4} + \cdots + 235200 ) / 425450 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8550 \nu^{15} + 2406 \nu^{14} - 70680 \nu^{13} - 16486 \nu^{12} + 70230 \nu^{11} + 31586 \nu^{10} + \cdots + 1327700 ) / 2552700 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13373 \nu^{14} - 97503 \nu^{12} + 3648 \nu^{10} + 217087 \nu^{8} + 2330180 \nu^{6} + \cdots + 1398550 ) / 1276350 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1710 \nu^{15} + 936 \nu^{14} + 14136 \nu^{13} - 4334 \nu^{12} - 14046 \nu^{11} - 16316 \nu^{10} + \cdots + 1712440 ) / 510540 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8550 \nu^{15} + 5938 \nu^{14} - 70680 \nu^{13} - 84258 \nu^{12} + 70230 \nu^{11} + 349068 \nu^{10} + \cdots - 5005900 ) / 2552700 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9036 \nu^{15} - 8064 \nu^{14} - 96821 \nu^{13} + 44539 \nu^{12} + 270556 \nu^{11} + 130096 \nu^{10} + \cdots + 379600 ) / 2552700 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 39\nu^{15} - 309\nu^{13} + 314\nu^{11} - 494\nu^{9} + 8230\nu^{7} + 11542\nu^{5} + 18796\nu^{3} + 10940\nu ) / 6700 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8550 \nu^{15} - 22792 \nu^{14} + 70680 \nu^{13} + 192952 \nu^{12} - 70230 \nu^{11} + \cdots + 3365600 ) / 2552700 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15088 \nu^{15} + 8064 \nu^{14} - 159898 \nu^{13} - 44539 \nu^{12} + 462293 \nu^{11} + \cdots - 379600 ) / 2552700 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 8550 \nu^{15} + 31004 \nu^{14} + 70680 \nu^{13} - 282394 \nu^{12} - 70230 \nu^{11} + \cdots - 3689700 ) / 2552700 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16334 \nu^{15} + 8064 \nu^{14} - 101559 \nu^{13} - 44539 \nu^{12} - 102621 \nu^{11} + \cdots - 379600 ) / 2552700 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 19866 \nu^{15} + 8064 \nu^{14} - 169331 \nu^{13} - 44539 \nu^{12} + 214861 \nu^{11} + \cdots - 379600 ) / 2552700 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 28136 \nu^{15} - 8064 \nu^{14} + 226351 \nu^{13} + 44539 \nu^{12} - 242336 \nu^{11} + \cdots + 379600 ) / 2552700 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 159 \nu^{15} - 1289 \nu^{13} + 1294 \nu^{11} - 364 \nu^{9} + 31790 \nu^{7} + 37982 \nu^{5} + \cdots - 8320 \nu ) / 12700 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 52682 \nu^{15} + 8064 \nu^{14} + 417352 \nu^{13} - 44539 \nu^{12} - 375617 \nu^{11} + \cdots - 379600 ) / 2552700 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{7} - \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} + 2\beta_{10} + 2\beta_{8} - \beta_{7} - 2\beta_{5} - 2\beta_{4} + 2\beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{15} + 8\beta_{14} - 5\beta_{12} + \beta_{11} + \beta_{9} - 2\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{14} + 4\beta_{10} + 6\beta_{8} - 2\beta_{7} - 3\beta_{5} - 5\beta_{4} - 2\beta_{3} + 4\beta_{2} + 7\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11 \beta_{15} + 26 \beta_{14} - 14 \beta_{13} - 23 \beta_{12} + \beta_{11} - 5 \beta_{9} + \cdots + 7 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20 \beta_{14} + 27 \beta_{10} + 30 \beta_{8} - 20 \beta_{7} - 26 \beta_{5} - 29 \beta_{4} - 16 \beta_{3} + \cdots + 35 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{15} + 67 \beta_{14} - 41 \beta_{13} - 73 \beta_{12} - 5 \beta_{11} - 7 \beta_{9} + \cdots + 25 \beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 156 \beta_{14} + 172 \beta_{10} + 178 \beta_{8} - 156 \beta_{7} - 162 \beta_{5} - 142 \beta_{4} + \cdots + 138 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 18 \beta_{15} + 349 \beta_{14} - 155 \beta_{13} - 429 \beta_{12} - \beta_{11} - 26 \beta_{9} + \cdots + 212 \beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1017 \beta_{14} + 988 \beta_{10} + 980 \beta_{8} - 1017 \beta_{7} - 918 \beta_{5} - 670 \beta_{4} + \cdots + 522 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 277 \beta_{15} + 1406 \beta_{14} - 768 \beta_{13} - 2353 \beta_{12} + 45 \beta_{11} + \cdots + 1538 \beta_{3} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6486 \beta_{14} + 5608 \beta_{10} + 5128 \beta_{8} - 6486 \beta_{7} - 5368 \beta_{5} - 3096 \beta_{4} + \cdots + 1786 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3819 \beta_{15} + 5050 \beta_{14} - 3562 \beta_{13} - 12765 \beta_{12} + 279 \beta_{11} + \cdots + 10101 \beta_{3} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 40072 \beta_{14} + 31276 \beta_{10} + 26354 \beta_{8} - 40072 \beta_{7} - 30284 \beta_{5} - 12784 \beta_{4} + \cdots + 2188 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 32894 \beta_{15} + 13058 \beta_{14} - 13138 \beta_{13} - 67474 \beta_{12} + 2642 \beta_{11} + \cdots + 64022 \beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
0.919224 − 1.30296i
2.36188 + 0.195512i
0.522506 − 1.01508i
0.550947 + 0.483398i
−0.550947 − 0.483398i
−0.522506 + 1.01508i
−2.36188 − 0.195512i
−0.919224 + 1.30296i
0.919224 + 1.30296i
2.36188 − 0.195512i
0.522506 + 1.01508i
0.550947 − 0.483398i
−0.550947 + 0.483398i
−0.522506 − 1.01508i
−2.36188 + 0.195512i
−0.919224 − 1.30296i
0 0 0 −2.22219 + 0.248747i 0 2.16604 − 1.51930i 0 0 0
433.2 0 0 0 −2.16637 − 0.553944i 0 2.12678 + 1.57379i 0 0 0
433.3 0 0 0 −1.53758 + 1.62353i 0 −2.63522 + 0.235858i 0 0 0
433.4 0 0 0 −0.0675488 + 2.23505i 0 1.05782 + 2.42508i 0 0 0
433.5 0 0 0 0.0675488 − 2.23505i 0 −2.42508 − 1.05782i 0 0 0
433.6 0 0 0 1.53758 − 1.62353i 0 −0.235858 + 2.63522i 0 0 0
433.7 0 0 0 2.16637 + 0.553944i 0 −1.57379 − 2.12678i 0 0 0
433.8 0 0 0 2.22219 − 0.248747i 0 1.51930 − 2.16604i 0 0 0
937.1 0 0 0 −2.22219 − 0.248747i 0 2.16604 + 1.51930i 0 0 0
937.2 0 0 0 −2.16637 + 0.553944i 0 2.12678 − 1.57379i 0 0 0
937.3 0 0 0 −1.53758 − 1.62353i 0 −2.63522 − 0.235858i 0 0 0
937.4 0 0 0 −0.0675488 − 2.23505i 0 1.05782 − 2.42508i 0 0 0
937.5 0 0 0 0.0675488 + 2.23505i 0 −2.42508 + 1.05782i 0 0 0
937.6 0 0 0 1.53758 + 1.62353i 0 −0.235858 − 2.63522i 0 0 0
937.7 0 0 0 2.16637 − 0.553944i 0 −1.57379 + 2.12678i 0 0 0
937.8 0 0 0 2.22219 + 0.248747i 0 1.51930 + 2.16604i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.ba.b 16
3.b odd 2 1 420.2.x.a ✓ 16
5.c odd 4 1 inner 1260.2.ba.b 16
7.b odd 2 1 inner 1260.2.ba.b 16
12.b even 2 1 1680.2.cz.c 16
15.d odd 2 1 2100.2.x.d 16
15.e even 4 1 420.2.x.a ✓ 16
15.e even 4 1 2100.2.x.d 16
21.c even 2 1 420.2.x.a ✓ 16
35.f even 4 1 inner 1260.2.ba.b 16
60.l odd 4 1 1680.2.cz.c 16
84.h odd 2 1 1680.2.cz.c 16
105.g even 2 1 2100.2.x.d 16
105.k odd 4 1 420.2.x.a ✓ 16
105.k odd 4 1 2100.2.x.d 16
420.w even 4 1 1680.2.cz.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.x.a ✓ 16 3.b odd 2 1
420.2.x.a ✓ 16 15.e even 4 1
420.2.x.a ✓ 16 21.c even 2 1
420.2.x.a ✓ 16 105.k odd 4 1
1260.2.ba.b 16 1.a even 1 1 trivial
1260.2.ba.b 16 5.c odd 4 1 inner
1260.2.ba.b 16 7.b odd 2 1 inner
1260.2.ba.b 16 35.f even 4 1 inner
1680.2.cz.c 16 12.b even 2 1
1680.2.cz.c 16 60.l odd 4 1
1680.2.cz.c 16 84.h odd 2 1
1680.2.cz.c 16 420.w even 4 1
2100.2.x.d 16 15.d odd 2 1
2100.2.x.d 16 15.e even 4 1
2100.2.x.d 16 105.g even 2 1
2100.2.x.d 16 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 4T_{11}^{3} - 8T_{11}^{2} - 16T_{11} - 4 \) acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 8 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 32 T^{13} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} - 8 T^{2} + \cdots - 4)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 116985856 \) Copy content Toggle raw display
$17$ \( T^{16} + 2224 T^{12} + \cdots + 3748096 \) Copy content Toggle raw display
$19$ \( (T^{8} - 32 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 4 T^{7} + \cdots + 760384)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 96 T^{6} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 112 T^{6} + \cdots + 8464)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 8 T^{7} + \cdots + 7507600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 128 T^{6} + \cdots + 270400)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 24 T^{7} + \cdots + 692224)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 8704 T^{12} + \cdots + 1048576 \) Copy content Toggle raw display
$53$ \( (T^{8} - 20 T^{7} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 192 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 176 T^{6} + \cdots + 173056)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 24 T^{7} + \cdots + 127599616)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 8 T^{3} + \cdots + 5164)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 133633600000000 \) Copy content Toggle raw display
$79$ \( (T^{8} + 384 T^{6} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 655360000 \) Copy content Toggle raw display
$89$ \( (T^{8} - 384 T^{6} + \cdots + 193600)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 744753706110976 \) Copy content Toggle raw display
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