Properties

Label 352.2.m.f
Level $352$
Weight $2$
Character orbit 352.m
Analytic conductor $2.811$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [352,2,Mod(97,352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(352, 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("352.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.m (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: \(2.81073415115\)
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: \( x^{12} - 2 x^{11} + 11 x^{10} - 11 x^{9} + 39 x^{8} - 43 x^{7} + 99 x^{6} + 36 x^{5} + 431 x^{4} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{5}) q^{3} + \beta_{9} q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{6} + \cdots + 1) q^{7}+ \cdots + (\beta_{11} + \beta_{8} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{5}) q^{3} + \beta_{9} q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{6} + \cdots + 1) q^{7}+ \cdots + (\beta_{10} - \beta_{9} - 12 \beta_{8} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{7} - q^{9} - 11 q^{11} - 2 q^{13} + 4 q^{15} + 12 q^{17} + 5 q^{19} + 24 q^{21} - 12 q^{23} + 13 q^{25} + 3 q^{27} + 16 q^{31} - 7 q^{33} - 28 q^{35} - 4 q^{37} + 46 q^{39} - 4 q^{41} - 22 q^{43} + 28 q^{45} - 24 q^{47} - 5 q^{49} + 17 q^{51} - 14 q^{53} - 46 q^{55} - 37 q^{57} + 31 q^{59} - 14 q^{61} + 58 q^{63} - 52 q^{65} - 62 q^{67} - 18 q^{69} + 6 q^{71} - 8 q^{73} + 53 q^{75} - 46 q^{77} - 4 q^{79} - 22 q^{81} + 41 q^{83} - 36 q^{85} - 76 q^{87} - 2 q^{89} - 22 q^{91} + 8 q^{93} + 16 q^{95} + 3 q^{97} - 65 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 11 x^{10} - 11 x^{9} + 39 x^{8} - 43 x^{7} + 99 x^{6} + 36 x^{5} + 431 x^{4} + \cdots + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 12925652272 \nu^{11} - 689411783229 \nu^{10} + 1158393636547 \nu^{9} + \cdots - 68957262745775 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 171694051509 \nu^{11} - 3538073194043 \nu^{10} + 6710004699384 \nu^{9} + \cdots - 872744346462385 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1093010781921 \nu^{11} - 1619787283007 \nu^{10} + 10879795136331 \nu^{9} + \cdots - 79252706903350 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1093010781921 \nu^{11} + 1619787283007 \nu^{10} - 10879795136331 \nu^{9} + \cdots + 79252706903350 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1185075482262 \nu^{11} - 2450405323424 \nu^{10} + 12479616226722 \nu^{9} + \cdots - 68337530631950 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2733501225278 \nu^{11} - 4281926968294 \nu^{10} + 27618108154634 \nu^{9} + \cdots + 2642224095290 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2758290509831 \nu^{11} - 5503655367390 \nu^{10} + 29651783824912 \nu^{9} + \cdots - 366378456407095 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3170108276134 \nu^{11} + 5247205770347 \nu^{10} - 33251403754467 \nu^{9} + \cdots + 203977836612425 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6759751764174 \nu^{11} - 11030588267832 \nu^{10} + 70986300515617 \nu^{9} + \cdots + 7410594838995 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6923728370080 \nu^{11} + 13776035137694 \nu^{10} - 76812432137873 \nu^{9} + \cdots + 13\!\cdots\!10 ) / 206669284189945 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7100336146293 \nu^{11} + 10864290641274 \nu^{10} - 70862768140539 \nu^{9} + \cdots - 7554307496840 ) / 206669284189945 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 3\beta_{8} - \beta_{7} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{9} + \beta_{6} - 5\beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{11} - 2\beta_{10} + 9\beta_{8} + 14\beta_{7} + 9\beta_{6} + 2\beta_{5} - 2\beta_{4} - 2\beta_{3} - 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{10} - 11\beta_{9} + 15\beta_{7} - 15\beta_{6} + 11\beta_{5} + 11\beta_{4} - 20\beta_{2} + 19\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -50\beta_{11} - 22\beta_{9} - 74\beta_{8} - 151\beta_{6} - 24\beta_{5} + 30\beta_{4} + 30\beta_{3} + 24\beta _1 + 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 74 \beta_{11} + 102 \beta_{10} + 74 \beta_{9} - 64 \beta_{8} - 152 \beta_{7} - 201 \beta_{5} + \cdots - 152 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 198 \beta_{11} + 198 \beta_{10} + 377 \beta_{9} + 485 \beta_{8} - 485 \beta_{7} + 1076 \beta_{6} + \cdots - 1076 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 893 \beta_{11} - 603 \beta_{10} + 1373 \beta_{8} + 700 \beta_{7} + 1373 \beta_{6} + 1453 \beta_{5} + \cdots - 664 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2949 \beta_{10} - 2949 \beta_{9} + 4713 \beta_{7} - 4713 \beta_{6} + 1976 \beta_{5} + 1976 \beta_{4} + \cdots + 8097 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 7597 \beta_{11} - 4925 \beta_{9} - 11843 \beta_{8} - 18471 \beta_{6} - 6395 \beta_{5} + 11046 \beta_{4} + \cdots + 11843 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/352\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(287\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{6} - \beta_{7} + \beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.885530 + 2.72538i
0.289142 + 0.889888i
−0.674672 2.07643i
0.885530 2.72538i
0.289142 0.889888i
−0.674672 + 2.07643i
1.66582 1.21029i
0.198931 0.144532i
−1.36475 + 0.991547i
1.66582 + 1.21029i
0.198931 + 0.144532i
−1.36475 0.991547i
0 −1.50933 + 1.09659i 0 −0.686361 2.11240i 0 0.787585 + 0.572214i 0 0.148512 0.457073i 0
97.2 0 0.0520331 0.0378043i 0 0.714760 + 2.19980i 0 −1.31923 0.958478i 0 −0.925773 + 2.84924i 0
97.3 0 2.57533 1.87109i 0 −0.0283989 0.0874029i 0 3.14968 + 2.28838i 0 2.20431 6.78417i 0
225.1 0 −1.50933 1.09659i 0 −0.686361 + 2.11240i 0 0.787585 0.572214i 0 0.148512 + 0.457073i 0
225.2 0 0.0520331 + 0.0378043i 0 0.714760 2.19980i 0 −1.31923 + 0.958478i 0 −0.925773 2.84924i 0
225.3 0 2.57533 + 1.87109i 0 −0.0283989 + 0.0874029i 0 3.14968 2.28838i 0 2.20431 + 6.78417i 0
257.1 0 −0.945302 + 2.90934i 0 −2.43185 + 1.76684i 0 0.483582 + 1.48831i 0 −5.14361 3.73705i 0
257.2 0 −0.385002 + 1.18491i 0 3.03889 2.20788i 0 −1.04575 3.21850i 0 1.17126 + 0.850967i 0
257.3 0 0.212270 0.653300i 0 −0.607036 + 0.441038i 0 0.944137 + 2.90576i 0 2.04531 + 1.48600i 0
289.1 0 −0.945302 2.90934i 0 −2.43185 1.76684i 0 0.483582 1.48831i 0 −5.14361 + 3.73705i 0
289.2 0 −0.385002 1.18491i 0 3.03889 + 2.20788i 0 −1.04575 + 3.21850i 0 1.17126 0.850967i 0
289.3 0 0.212270 + 0.653300i 0 −0.607036 0.441038i 0 0.944137 2.90576i 0 2.04531 1.48600i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.3
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 352.2.m.f yes 12
4.b odd 2 1 352.2.m.e 12
8.b even 2 1 704.2.m.n 12
8.d odd 2 1 704.2.m.m 12
11.c even 5 1 inner 352.2.m.f yes 12
11.c even 5 1 3872.2.a.bn 6
11.d odd 10 1 3872.2.a.bo 6
44.g even 10 1 3872.2.a.bp 6
44.h odd 10 1 352.2.m.e 12
44.h odd 10 1 3872.2.a.bq 6
88.k even 10 1 7744.2.a.dt 6
88.l odd 10 1 704.2.m.m 12
88.l odd 10 1 7744.2.a.du 6
88.o even 10 1 704.2.m.n 12
88.o even 10 1 7744.2.a.dv 6
88.p odd 10 1 7744.2.a.dw 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.m.e 12 4.b odd 2 1
352.2.m.e 12 44.h odd 10 1
352.2.m.f yes 12 1.a even 1 1 trivial
352.2.m.f yes 12 11.c even 5 1 inner
704.2.m.m 12 8.d odd 2 1
704.2.m.m 12 88.l odd 10 1
704.2.m.n 12 8.b even 2 1
704.2.m.n 12 88.o even 10 1
3872.2.a.bn 6 11.c even 5 1
3872.2.a.bo 6 11.d odd 10 1
3872.2.a.bp 6 44.g even 10 1
3872.2.a.bq 6 44.h odd 10 1
7744.2.a.dt 6 88.k even 10 1
7744.2.a.du 6 88.l odd 10 1
7744.2.a.dv 6 88.o even 10 1
7744.2.a.dw 6 88.p odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 5 T_{3}^{10} - 11 T_{3}^{9} + 45 T_{3}^{8} + 175 T_{3}^{7} + 451 T_{3}^{6} + 360 T_{3}^{5} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(352, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 5 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} + T^{10} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} + \cdots + 10000 \) Copy content Toggle raw display
$11$ \( T^{12} + 11 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} + 2 T^{11} + \cdots + 3225616 \) Copy content Toggle raw display
$17$ \( T^{12} - 12 T^{11} + \cdots + 841 \) Copy content Toggle raw display
$19$ \( T^{12} - 5 T^{11} + \cdots + 7612081 \) Copy content Toggle raw display
$23$ \( (T^{6} + 6 T^{5} + \cdots + 320)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 9 T^{10} + \cdots + 6990736 \) Copy content Toggle raw display
$31$ \( T^{12} - 16 T^{11} + \cdots + 1948816 \) Copy content Toggle raw display
$37$ \( T^{12} + 4 T^{11} + \cdots + 3225616 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 839782441 \) Copy content Toggle raw display
$43$ \( (T^{6} + 11 T^{5} + \cdots - 81776)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 24 T^{11} + \cdots + 18524416 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 36796446976 \) Copy content Toggle raw display
$59$ \( T^{12} - 31 T^{11} + \cdots + 2825761 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 253446400 \) Copy content Toggle raw display
$67$ \( (T^{6} + 31 T^{5} + \cdots - 36304)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1025280400 \) Copy content Toggle raw display
$73$ \( T^{12} + 8 T^{11} + \cdots + 35700625 \) Copy content Toggle raw display
$79$ \( T^{12} + 4 T^{11} + \cdots + 54700816 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 40080440401 \) Copy content Toggle raw display
$89$ \( (T^{6} + T^{5} - 265 T^{4} + \cdots + 7600)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 3 T^{11} + \cdots + 121 \) Copy content Toggle raw display
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