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Properties

Label 1680.2.ba.c
Level $1680$
Weight $2$
Character orbit 1680.ba
Analytic conductor $13.415$
Analytic rank $0$
Dimension $16$
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] = [1680,2,Mod(911,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.911");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.ba (of order \(2\), degree \(1\), 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: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 640 x^{12} - 1656 x^{11} + 3522 x^{10} - 6148 x^{9} + \cdots + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{5} q^{5} + \beta_{5} q^{7} + ( - \beta_{9} - \beta_{5} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{5} q^{5} + \beta_{5} q^{7} + ( - \beta_{9} - \beta_{5} - 1) q^{9} + (\beta_{7} - 1) q^{11} + (\beta_{10} + \beta_{7} + \beta_{3}) q^{13} + \beta_{3} q^{15} + ( - \beta_{15} + \beta_{6} + \cdots + \beta_{3}) q^{17}+ \cdots + (\beta_{15} + 2 \beta_{14} + \beta_{11} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 10 q^{9} - 12 q^{11} + 2 q^{15} + 2 q^{21} - 8 q^{23} - 16 q^{25} + 12 q^{27} + 12 q^{33} - 16 q^{35} - 8 q^{37} + 14 q^{39} + 8 q^{45} - 16 q^{47} - 16 q^{49} + 34 q^{51} + 12 q^{57} - 32 q^{59} + 8 q^{61} + 8 q^{63} - 20 q^{69} + 32 q^{71} + 48 q^{73} - 22 q^{81} - 72 q^{83} - 12 q^{85} + 44 q^{87} + 28 q^{93} - 8 q^{95} - 8 q^{97} + 50 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 640 x^{12} - 1656 x^{11} + 3522 x^{10} - 6148 x^{9} + \cdots + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 681 \nu^{15} + 4590 \nu^{14} - 26391 \nu^{13} + 96763 \nu^{12} - 294044 \nu^{11} + 684451 \nu^{10} + \cdots + 1629 ) / 92 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 47 \nu^{14} + 329 \nu^{13} - 1894 \nu^{12} + 7087 \nu^{11} - 21663 \nu^{10} + 51192 \nu^{9} + \cdots - 833 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 509 \nu^{15} - 3806 \nu^{14} + 22161 \nu^{13} - 86211 \nu^{12} + 269326 \nu^{11} - 659527 \nu^{10} + \cdots - 653 ) / 46 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 509 \nu^{15} + 3829 \nu^{14} - 22322 \nu^{13} + 87131 \nu^{12} - 272753 \nu^{11} + 669900 \nu^{10} + \cdots + 952 ) / 46 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 962 \nu^{15} + 7215 \nu^{14} - 42419 \nu^{13} + 166296 \nu^{12} - 526483 \nu^{11} + 1307141 \nu^{10} + \cdots + 17399 ) / 92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 681 \nu^{15} - 5625 \nu^{14} + 33636 \nu^{13} - 138485 \nu^{12} + 450191 \nu^{11} - 1161954 \nu^{10} + \cdots - 19730 ) / 92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 29 \nu^{14} - 203 \nu^{13} + 1171 \nu^{12} - 4387 \nu^{11} + 13446 \nu^{10} - 31854 \nu^{9} + 61933 \nu^{8} + \cdots + 617 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2911 \nu^{15} + 19843 \nu^{14} - 114408 \nu^{13} + 422755 \nu^{12} - 1291517 \nu^{11} + 3031194 \nu^{10} + \cdots + 15334 ) / 92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3458 \nu^{15} + 25705 \nu^{14} - 149877 \nu^{13} + 582072 \nu^{12} - 1821137 \nu^{11} + 4461875 \nu^{10} + \cdots + 20609 ) / 92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1018 \nu^{15} + 689 \nu^{14} + 4139 \nu^{13} - 107120 \nu^{12} + 508607 \nu^{11} - 1890757 \nu^{10} + \cdots - 144491 ) / 92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3458 \nu^{15} + 26165 \nu^{14} - 153097 \nu^{13} + 600564 \nu^{12} - 1890229 \nu^{11} + 4672371 \nu^{10} + \cdots + 28521 ) / 92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2911 \nu^{15} + 23822 \nu^{14} - 142261 \nu^{13} + 583341 \nu^{12} - 1892944 \nu^{11} + 4873241 \nu^{10} + \cdots + 93879 ) / 92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1120 \nu^{15} - 8400 \nu^{14} + 49192 \nu^{13} - 192348 \nu^{12} + 605768 \nu^{11} - 1496176 \nu^{10} + \cdots - 13986 ) / 23 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3574 \nu^{15} - 26805 \nu^{14} + 157241 \nu^{13} - 615524 \nu^{12} + 1942913 \nu^{11} - 4809651 \nu^{10} + \cdots - 54537 ) / 46 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12983 \nu^{15} + 96855 \nu^{14} - 566580 \nu^{13} + 2208639 \nu^{12} - 6942865 \nu^{11} + \cdots + 148682 ) / 92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{14} - \beta_{13} + \beta_{6} + \beta_{4} - \beta_{3} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{15} - \beta_{14} - \beta_{13} + 3\beta_{6} - \beta_{4} - 3\beta_{3} - 2\beta_{2} + 2\beta _1 - 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{15} + \beta_{14} + 6 \beta_{13} + \beta_{12} + 2 \beta_{11} + 2 \beta_{9} + \beta_{8} + \cdots - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5 \beta_{15} + 3 \beta_{14} + 13 \beta_{13} + 6 \beta_{11} + 2 \beta_{9} + 4 \beta_{8} + 2 \beta_{7} + \cdots + 34 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4 \beta_{15} + 9 \beta_{14} - 30 \beta_{13} - 5 \beta_{12} - 7 \beta_{11} - 17 \beta_{9} + 5 \beta_{8} + \cdots + 102 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 25 \beta_{15} + 19 \beta_{14} - 123 \beta_{13} + 12 \beta_{12} - 55 \beta_{11} - 2 \beta_{10} + \cdots - 126 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 79 \beta_{14} + 116 \beta_{13} + 63 \beta_{12} - 9 \beta_{11} - 7 \beta_{10} + 89 \beta_{9} + \cdots - 810 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 129 \beta_{15} - 397 \beta_{14} + 1069 \beta_{13} - 94 \beta_{12} + 392 \beta_{11} + 16 \beta_{10} + \cdots + 126 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 92 \beta_{15} + 182 \beta_{14} + 127 \beta_{13} - 715 \beta_{12} + 443 \beta_{11} + 114 \beta_{10} + \cdots + 5989 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 709 \beta_{15} + 4004 \beta_{14} - 8310 \beta_{13} + 160 \beta_{12} - 2465 \beta_{11} - 2 \beta_{10} + \cdots + 4785 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1110 \beta_{15} + 3431 \beta_{14} - 9275 \beta_{13} + 6480 \beta_{12} - 5551 \beta_{11} + \cdots - 40389 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 3905 \beta_{15} - 30372 \beta_{14} + 56324 \beta_{13} + 5342 \beta_{12} + 13398 \beta_{11} + \cdots - 75449 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 10859 \beta_{15} - 63476 \beta_{14} + 129646 \beta_{13} - 48602 \beta_{12} + 54363 \beta_{11} + \cdots + 235879 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 18998 \beta_{15} + 179947 \beta_{14} - 313757 \beta_{13} - 94164 \beta_{12} - 51911 \beta_{11} + \cdots + 812031 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 96027 \beta_{15} + 711656 \beta_{14} - 1335995 \beta_{13} + 297869 \beta_{12} - 466799 \beta_{11} + \cdots - 1028652 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
911.1
0.500000 − 0.492896i
0.500000 + 0.492896i
0.500000 + 2.75709i
0.500000 − 2.75709i
0.500000 + 0.601654i
0.500000 − 0.601654i
0.500000 − 1.50326i
0.500000 + 1.50326i
0.500000 + 0.617921i
0.500000 − 0.617921i
0.500000 + 1.62829i
0.500000 − 1.62829i
0.500000 − 2.12118i
0.500000 + 2.12118i
0.500000 + 0.0342497i
0.500000 − 0.0342497i
0 −1.43348 − 0.972180i 0 − 1.00000i 0 − 1.00000i 0 1.10973 + 2.78720i 0
911.2 0 −1.43348 + 0.972180i 0 1.00000i 0 1.00000i 0 1.10973 − 2.78720i 0
911.3 0 −1.29877 − 1.14595i 0 1.00000i 0 1.00000i 0 0.373584 + 2.97665i 0
911.4 0 −1.29877 + 1.14595i 0 − 1.00000i 0 − 1.00000i 0 0.373584 − 2.97665i 0
911.5 0 −0.715763 − 1.57724i 0 − 1.00000i 0 − 1.00000i 0 −1.97537 + 2.25786i 0
911.6 0 −0.715763 + 1.57724i 0 1.00000i 0 1.00000i 0 −1.97537 − 2.25786i 0
911.7 0 −0.486448 − 1.66234i 0 1.00000i 0 1.00000i 0 −2.52674 + 1.61728i 0
911.8 0 −0.486448 + 1.66234i 0 − 1.00000i 0 − 1.00000i 0 −2.52674 − 1.61728i 0
911.9 0 0.234465 − 1.71611i 0 − 1.00000i 0 − 1.00000i 0 −2.89005 − 0.804736i 0
911.10 0 0.234465 + 1.71611i 0 1.00000i 0 1.00000i 0 −2.89005 + 0.804736i 0
911.11 0 0.819437 − 1.52595i 0 1.00000i 0 1.00000i 0 −1.65704 − 2.50084i 0
911.12 0 0.819437 + 1.52595i 0 − 1.00000i 0 − 1.00000i 0 −1.65704 + 2.50084i 0
911.13 0 1.18129 − 1.26671i 0 1.00000i 0 1.00000i 0 −0.209100 − 2.99270i 0
911.14 0 1.18129 + 1.26671i 0 − 1.00000i 0 − 1.00000i 0 −0.209100 + 2.99270i 0
911.15 0 1.69926 − 0.335422i 0 − 1.00000i 0 − 1.00000i 0 2.77498 − 1.13994i 0
911.16 0 1.69926 + 0.335422i 0 1.00000i 0 1.00000i 0 2.77498 + 1.13994i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 911.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.ba.c ✓ 16
3.b odd 2 1 1680.2.ba.d yes 16
4.b odd 2 1 1680.2.ba.d yes 16
12.b even 2 1 inner 1680.2.ba.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1680.2.ba.c ✓ 16 1.a even 1 1 trivial
1680.2.ba.c ✓ 16 12.b even 2 1 inner
1680.2.ba.d yes 16 3.b odd 2 1
1680.2.ba.d yes 16 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} + 6T_{11}^{7} - 23T_{11}^{6} - 168T_{11}^{5} + 8T_{11}^{4} + 1008T_{11}^{3} + 848T_{11}^{2} - 576T_{11} + 64 \) acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 5 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$11$ \( (T^{8} + 6 T^{7} - 23 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 43 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 122 T^{14} + \cdots + 495616 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 349241344 \) Copy content Toggle raw display
$23$ \( (T^{8} + 4 T^{7} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 725979136 \) Copy content Toggle raw display
$31$ \( T^{16} + 264 T^{14} + \cdots + 262144 \) Copy content Toggle raw display
$37$ \( (T^{8} + 4 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 70068207616 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 349241344 \) Copy content Toggle raw display
$47$ \( (T^{8} + 8 T^{7} + \cdots - 135872)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 401124622336 \) Copy content Toggle raw display
$59$ \( (T^{8} + 16 T^{7} + \cdots + 859648)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 4 T^{7} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1396965376 \) Copy content Toggle raw display
$71$ \( (T^{8} - 16 T^{7} + \cdots - 14624768)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 24 T^{7} + \cdots + 37888)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 83371800825856 \) Copy content Toggle raw display
$83$ \( (T^{8} + 36 T^{7} + \cdots - 773888)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 8115479117824 \) Copy content Toggle raw display
$97$ \( (T^{8} + 4 T^{7} + \cdots - 7629248)^{2} \) Copy content Toggle raw display
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