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Properties

Label 1232.2.bn.c
Level $1232$
Weight $2$
Character orbit 1232.bn
Analytic conductor $9.838$
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] = [1232,2,Mod(241,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.bn (of order \(6\), degree \(2\), not 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: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 2x^{14} - 17x^{12} - 343x^{10} + 490x^{8} - 16807x^{6} - 40817x^{4} + 235298x^{2} + 5764801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 308)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{5} q^{5} - \beta_{9} q^{7} + \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_{5} q^{5} - \beta_{9} q^{7} + \beta_{7} q^{9} + ( - \beta_{8} + \beta_{6} + \cdots + \beta_1) q^{11}+ \cdots + ( - \beta_{15} - 2 \beta_{14} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{5} + 4 q^{9} + q^{11} - 16 q^{15} + 2 q^{23} - 6 q^{25} + 12 q^{31} + 21 q^{33} + 2 q^{37} - 18 q^{45} + 12 q^{47} - 4 q^{49} + 8 q^{53} + 30 q^{59} + 18 q^{67} + 76 q^{71} - 6 q^{75} - 25 q^{77} - 8 q^{81} - 12 q^{89} - 6 q^{91} + 10 q^{93} + 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2x^{14} - 17x^{12} - 343x^{10} + 490x^{8} - 16807x^{6} - 40817x^{4} + 235298x^{2} + 5764801 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 125 \nu^{14} - 4013 \nu^{12} + 9586 \nu^{10} - 61985 \nu^{8} + 293461 \nu^{6} - 2482634 \nu^{4} + \cdots - 68824665 ) / 31059336 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{14} + 5\nu^{12} - 298\nu^{10} + 409\nu^{8} - 2205\nu^{6} - 25774\nu^{4} - 328937\nu^{2} + 5015689 ) / 633864 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 599 \nu^{14} - 8455 \nu^{12} + 25734 \nu^{10} - 173411 \nu^{8} + 852943 \nu^{6} + \cdots - 163179163 ) / 93178008 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 641 \nu^{14} + 1919 \nu^{12} - 34662 \nu^{10} - 19061 \nu^{8} - 182231 \nu^{6} - 7246218 \nu^{4} + \cdots + 563185763 ) / 93178008 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 827 \nu^{14} + 10939 \nu^{12} - 53022 \nu^{10} + 156359 \nu^{8} - 1084027 \nu^{6} + \cdots + 497772919 ) / 93178008 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 445 \nu^{14} + 2311 \nu^{12} - 18786 \nu^{10} + 37583 \nu^{8} - 234367 \nu^{6} - 4048086 \nu^{4} + \cdots + 342476239 ) / 46589004 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 66 \nu^{14} - 260 \nu^{12} + 1181 \nu^{10} - 4998 \nu^{8} + 73500 \nu^{6} - 1248520 \nu^{4} + \cdots - 4705960 ) / 3882417 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 519 \nu^{15} - 6545 \nu^{14} - 29391 \nu^{13} - 85463 \nu^{12} + 312078 \nu^{11} + \cdots - 8858852051 ) / 1304492112 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} + 2\nu^{13} - 17\nu^{11} - 343\nu^{9} + 490\nu^{7} - 16807\nu^{5} - 40817\nu^{3} + 235298\nu ) / 823543 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3293 \nu^{15} + 6545 \nu^{14} + 34859 \nu^{13} + 85463 \nu^{12} - 563670 \nu^{11} + \cdots + 8858852051 ) / 1304492112 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 499 \nu^{15} - 214 \nu^{14} + 6863 \nu^{13} + 12410 \nu^{12} - 13182 \nu^{11} + \cdots + 830366642 ) / 93178008 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 7725 \nu^{15} - 18809 \nu^{14} + 31149 \nu^{13} - 11207 \nu^{12} - 56394 \nu^{11} + \cdots - 11507366339 ) / 1304492112 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1283 \nu^{15} - 935 \nu^{14} + 9797 \nu^{13} - 12209 \nu^{12} - 129114 \nu^{11} + \cdots - 1265550293 ) / 186356016 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2353 \nu^{15} - 2497 \nu^{13} - 52006 \nu^{11} - 91581 \nu^{9} + 170961 \nu^{7} + \cdots + 1343433931 \nu ) / 217415352 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 16595 \nu^{15} - 18809 \nu^{14} + 159331 \nu^{13} - 11207 \nu^{12} - 281238 \nu^{11} + \cdots - 11507366339 ) / 1304492112 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{15} + 3 \beta_{14} - \beta_{13} + 5 \beta_{12} + 3 \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots - 2 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{6} - 2\beta_{5} + 3\beta_{4} - 3\beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12 \beta_{15} + 31 \beta_{14} - \beta_{13} + 26 \beta_{12} - 18 \beta_{11} - 29 \beta_{10} - 30 \beta_{9} + \cdots + 12 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 10\beta_{6} + 8\beta_{5} - 5\beta_{4} - 6\beta_{3} - 8\beta_{2} + 13\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 40 \beta_{15} + 59 \beta_{14} + 34 \beta_{13} + 166 \beta_{12} - 74 \beta_{11} + 34 \beta_{10} + \cdots + 159 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 94\beta_{7} - 54\beta_{6} + 111\beta_{5} - 51\beta_{4} + 3\beta_{3} + 30\beta_{2} + 23\beta _1 + 45 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 317 \beta_{15} - 914 \beta_{14} + 993 \beta_{13} + 159 \beta_{12} + 38 \beta_{11} - 561 \beta_{10} + \cdots + 1160 ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 140\beta_{7} - 457\beta_{6} - 772\beta_{5} - 81\beta_{4} - 1137\beta_{3} + 735\beta_{2} - 119\beta _1 + 72 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1951 \beta_{15} + 1018 \beta_{14} + 1826 \beta_{13} - 3376 \beta_{12} - 1775 \beta_{11} - 2955 \beta_{10} + \cdots + 3078 ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -397\beta_{7} + 2348\beta_{6} - 734\beta_{5} - 1489\beta_{4} - 321\beta_{3} - 2332\beta_{2} + 286\beta _1 + 13704 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 23060 \beta_{15} + 54701 \beta_{14} - 25040 \beta_{13} + 60856 \beta_{12} + 54862 \beta_{11} + \cdots - 26616 ) / 7 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3223 \beta_{7} - 26283 \beta_{6} - 20721 \beta_{5} + 48984 \beta_{4} - 36327 \beta_{3} - 14250 \beta_{2} + \cdots + 47855 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 188655 \beta_{15} + 561907 \beta_{14} - 5510 \beta_{13} + 435160 \beta_{12} - 153031 \beta_{11} + \cdots + 192631 ) / 7 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 22379 \beta_{7} + 166276 \beta_{6} - 2102 \beta_{5} + 29991 \beta_{4} - 194610 \beta_{3} - 143493 \beta_{2} + \cdots - 165023 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1687873 \beta_{15} + 2154183 \beta_{14} + 491007 \beta_{13} + 1603068 \beta_{12} - 2073047 \beta_{11} + \cdots + 3282333 ) / 7 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1
−2.64407 − 0.0942693i
2.64407 + 0.0942693i
−0.978837 + 2.45802i
0.978837 − 2.45802i
−2.16092 − 1.52658i
2.16092 + 1.52658i
−0.938723 − 2.47362i
0.938723 + 2.47362i
−2.64407 + 0.0942693i
2.64407 − 0.0942693i
−0.978837 − 2.45802i
0.978837 + 2.45802i
−2.16092 + 1.52658i
2.16092 − 1.52658i
−0.938723 + 2.47362i
0.938723 − 2.47362i
0 −2.37157 − 1.36923i 0 0.316246 − 0.182585i 0 −2.64407 + 0.0942693i 0 2.24957 + 3.89637i 0
241.2 0 −2.37157 − 1.36923i 0 0.316246 − 0.182585i 0 2.64407 − 0.0942693i 0 2.24957 + 3.89637i 0
241.3 0 −0.238162 − 0.137503i 0 3.14912 − 1.81815i 0 −0.978837 − 2.45802i 0 −1.46219 − 2.53258i 0
241.4 0 −0.238162 − 0.137503i 0 3.14912 − 1.81815i 0 0.978837 + 2.45802i 0 −1.46219 − 2.53258i 0
241.5 0 0.464164 + 0.267985i 0 −1.61581 + 0.932888i 0 −2.16092 + 1.52658i 0 −1.35637 − 2.34930i 0
241.6 0 0.464164 + 0.267985i 0 −1.61581 + 0.932888i 0 2.16092 − 1.52658i 0 −1.35637 − 2.34930i 0
241.7 0 2.14557 + 1.23875i 0 −0.349557 + 0.201817i 0 −0.938723 + 2.47362i 0 1.56898 + 2.71756i 0
241.8 0 2.14557 + 1.23875i 0 −0.349557 + 0.201817i 0 0.938723 − 2.47362i 0 1.56898 + 2.71756i 0
593.1 0 −2.37157 + 1.36923i 0 0.316246 + 0.182585i 0 −2.64407 − 0.0942693i 0 2.24957 − 3.89637i 0
593.2 0 −2.37157 + 1.36923i 0 0.316246 + 0.182585i 0 2.64407 + 0.0942693i 0 2.24957 − 3.89637i 0
593.3 0 −0.238162 + 0.137503i 0 3.14912 + 1.81815i 0 −0.978837 + 2.45802i 0 −1.46219 + 2.53258i 0
593.4 0 −0.238162 + 0.137503i 0 3.14912 + 1.81815i 0 0.978837 − 2.45802i 0 −1.46219 + 2.53258i 0
593.5 0 0.464164 − 0.267985i 0 −1.61581 − 0.932888i 0 −2.16092 − 1.52658i 0 −1.35637 + 2.34930i 0
593.6 0 0.464164 − 0.267985i 0 −1.61581 − 0.932888i 0 2.16092 + 1.52658i 0 −1.35637 + 2.34930i 0
593.7 0 2.14557 − 1.23875i 0 −0.349557 − 0.201817i 0 −0.938723 − 2.47362i 0 1.56898 − 2.71756i 0
593.8 0 2.14557 − 1.23875i 0 −0.349557 − 0.201817i 0 0.938723 + 2.47362i 0 1.56898 − 2.71756i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.b odd 2 1 inner
77.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.bn.c 16
4.b odd 2 1 308.2.q.a ✓ 16
7.d odd 6 1 inner 1232.2.bn.c 16
11.b odd 2 1 inner 1232.2.bn.c 16
12.b even 2 1 2772.2.cs.a 16
28.d even 2 1 2156.2.q.c 16
28.f even 6 1 308.2.q.a ✓ 16
28.f even 6 1 2156.2.c.c 16
28.g odd 6 1 2156.2.c.c 16
28.g odd 6 1 2156.2.q.c 16
44.c even 2 1 308.2.q.a ✓ 16
77.i even 6 1 inner 1232.2.bn.c 16
84.j odd 6 1 2772.2.cs.a 16
132.d odd 2 1 2772.2.cs.a 16
308.g odd 2 1 2156.2.q.c 16
308.m odd 6 1 308.2.q.a ✓ 16
308.m odd 6 1 2156.2.c.c 16
308.n even 6 1 2156.2.c.c 16
308.n even 6 1 2156.2.q.c 16
924.y even 6 1 2772.2.cs.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.q.a ✓ 16 4.b odd 2 1
308.2.q.a ✓ 16 28.f even 6 1
308.2.q.a ✓ 16 44.c even 2 1
308.2.q.a ✓ 16 308.m odd 6 1
1232.2.bn.c 16 1.a even 1 1 trivial
1232.2.bn.c 16 7.d odd 6 1 inner
1232.2.bn.c 16 11.b odd 2 1 inner
1232.2.bn.c 16 77.i even 6 1 inner
2156.2.c.c 16 28.f even 6 1
2156.2.c.c 16 28.g odd 6 1
2156.2.c.c 16 308.m odd 6 1
2156.2.c.c 16 308.n even 6 1
2156.2.q.c 16 28.d even 2 1
2156.2.q.c 16 28.g odd 6 1
2156.2.q.c 16 308.g odd 2 1
2156.2.q.c 16 308.n even 6 1
2772.2.cs.a 16 12.b even 2 1
2772.2.cs.a 16 84.j odd 6 1
2772.2.cs.a 16 132.d odd 2 1
2772.2.cs.a 16 924.y even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 7T_{3}^{6} + 48T_{3}^{4} - 21T_{3}^{3} - 4T_{3}^{2} + 3T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1232, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 7 T^{6} + 48 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 3 T^{7} - 4 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 2 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} - 95 T^{6} + \cdots + 240100)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 13841287201 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 13841287201 \) Copy content Toggle raw display
$23$ \( (T^{8} - T^{7} + \cdots + 44521)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 195 T^{6} + \cdots + 1382976)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 6 T^{7} + \cdots + 790321)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - T^{7} + 37 T^{6} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 95 T^{6} + \cdots + 240100)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 208 T^{6} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 6 T^{7} + \cdots + 39601)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 4 T^{7} + \cdots + 866761)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 15 T^{7} + \cdots + 2337841)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 154922431942656 \) Copy content Toggle raw display
$67$ \( (T^{8} - 9 T^{7} + \cdots + 69696)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 19 T^{3} + \cdots + 16)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 700715164550625 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$83$ \( (T^{8} - 263 T^{6} + \cdots + 38416)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 6 T^{7} + \cdots + 3381921)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 191 T^{6} + \cdots + 283024)^{2} \) Copy content Toggle raw display
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