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Properties

Label 343.2.c.d
Level $343$
Weight $2$
Character orbit 343.c
Analytic conductor $2.739$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [343,2,Mod(18,343)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(343, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("343.18");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 343 = 7^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 343.c (of order \(3\), 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: \(2.73886878933\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 10 x^{10} - 8 x^{9} + 46 x^{8} - 31 x^{7} + 136 x^{6} - 30 x^{5} + 204 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{9} - \beta_{8} - \beta_1 + 1) q^{2} + ( - \beta_{11} + \beta_{9} - \beta_{7}) q^{3} + (\beta_{10} + \beta_{9} - \beta_{8} + \cdots - \beta_1) q^{4}+ \cdots + (\beta_{11} - 2 \beta_{9} + \beta_{8} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} - \beta_{8} - \beta_1 + 1) q^{2} + ( - \beta_{11} + \beta_{9} - \beta_{7}) q^{3} + (\beta_{10} + \beta_{9} - \beta_{8} + \cdots - \beta_1) q^{4}+ \cdots + ( - 2 \beta_{6} + 4 \beta_{5} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 5 q^{3} - 4 q^{4} - 11 q^{5} + 2 q^{6} - 12 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} - 5 q^{3} - 4 q^{4} - 11 q^{5} + 2 q^{6} - 12 q^{8} - 9 q^{9} + 2 q^{10} + q^{11} - 7 q^{12} + 14 q^{13} - 10 q^{15} + 8 q^{16} - 26 q^{17} + 18 q^{18} + 3 q^{19} + 28 q^{20} - 12 q^{22} + 9 q^{23} + 9 q^{24} - 15 q^{25} + 16 q^{27} - 4 q^{29} - 13 q^{30} + 2 q^{31} - q^{32} - 18 q^{33} - 26 q^{34} + 46 q^{36} + 2 q^{37} - 5 q^{38} - 7 q^{39} + 24 q^{40} + 56 q^{41} + 10 q^{43} + 12 q^{44} - 3 q^{45} + 12 q^{46} - 18 q^{47} - 26 q^{48} + 10 q^{50} - 13 q^{51} + 14 q^{52} + q^{53} + 32 q^{54} - 44 q^{55} - 52 q^{57} - 24 q^{58} - 5 q^{59} - 21 q^{60} + 17 q^{61} - 12 q^{62} - 36 q^{64} + 14 q^{65} + 58 q^{66} + 22 q^{67} - 7 q^{68} - 40 q^{69} - 4 q^{71} + 18 q^{72} - 12 q^{73} - 23 q^{74} + 27 q^{75} - 98 q^{76} + 42 q^{78} + 2 q^{79} + 16 q^{80} + 30 q^{81} + 35 q^{82} + 14 q^{83} + 74 q^{85} - 3 q^{86} + 25 q^{87} - 14 q^{88} - 39 q^{89} - 94 q^{90} + 22 q^{92} + 22 q^{93} + 16 q^{94} + 11 q^{95} + 28 q^{96} + 70 q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 10 x^{10} - 8 x^{9} + 46 x^{8} - 31 x^{7} + 136 x^{6} - 30 x^{5} + 204 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 34450 \nu^{11} - 551090 \nu^{10} + 808161 \nu^{9} - 3213230 \nu^{8} - 1378705 \nu^{7} + \cdots - 9920995 ) / 22005019 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2075464 \nu^{11} + 1892258 \nu^{10} + 12077486 \nu^{9} + 30669924 \nu^{8} + 94116098 \nu^{7} + \cdots + 79425037 ) / 902205779 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3967722 \nu^{11} - 10569412 \nu^{10} + 35196150 \nu^{9} - 32025170 \nu^{8} + 108297837 \nu^{7} + \cdots + 1722911057 ) / 902205779 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 299258 \nu^{11} - 73180 \nu^{10} + 2036734 \nu^{9} + 2211086 \nu^{8} + 11823469 \nu^{7} + \cdots + 27765986 ) / 22005019 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 21948604 \nu^{11} + 18771113 \nu^{10} - 178954837 \nu^{9} - 76514782 \nu^{8} + \cdots - 2185561345 ) / 902205779 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23535544 \nu^{11} + 29753220 \nu^{10} - 202666791 \nu^{9} + 23468550 \nu^{8} + \cdots - 388448031 ) / 902205779 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 76791069 \nu^{11} - 165406608 \nu^{10} + 792074718 \nu^{9} - 721695157 \nu^{8} + 3602964799 \nu^{7} + \cdots - 72256160 ) / 902205779 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 79425037 \nu^{11} + 160925538 \nu^{10} - 792358112 \nu^{9} + 647477782 \nu^{8} + \cdots - 833917341 ) / 902205779 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 100963813 \nu^{11} + 200115224 \nu^{10} - 981169262 \nu^{9} + 727116141 \nu^{8} + \cdots + 80558016 ) / 902205779 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 179933696 \nu^{11} - 365282844 \nu^{10} + 1831755987 \nu^{9} - 1493796121 \nu^{8} + \cdots + 2031334719 ) / 902205779 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + 2\beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + 4\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{10} - 8\beta_{9} - 3\beta_{8} - 4\beta_{7} + 4\beta_{2} - 6\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} + 7\beta_{10} - 9\beta_{9} - 6\beta_{7} + 7\beta_{5} - 7\beta_{4} - 18\beta_{3} + 7\beta_{2} - 18\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{6} + 16\beta_{5} - 24\beta_{4} - 33\beta_{3} + 8\beta_{2} + 37 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10\beta_{11} - 41\beta_{10} + 60\beta_{9} - 3\beta_{8} + 29\beta_{7} + 10\beta_{6} - 29\beta_{2} + 86\beta _1 + 60 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 22 \beta_{11} - 118 \beta_{10} + 185 \beta_{9} + 13 \beta_{8} + 66 \beta_{7} - 105 \beta_{5} + \cdots + 175 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -74\beta_{6} - 267\beta_{5} + 228\beta_{4} + 426\beta_{3} - 97\beta_{2} - 359 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 171 \beta_{11} + 596 \beta_{10} - 965 \beta_{9} + 47 \beta_{8} - 278 \beta_{7} - 171 \beta_{6} + \cdots - 965 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 489 \beta_{11} + 1241 \beta_{10} - 2050 \beta_{9} + 340 \beta_{8} - 574 \beta_{7} + 1581 \beta_{5} + \cdots - 2159 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/343\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 - \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
0.454446 + 0.787123i
1.05033 + 1.81922i
1.16596 + 2.01950i
−0.855415 − 1.48162i
−0.772849 − 1.33861i
−0.0424677 − 0.0735563i
0.454446 − 0.787123i
1.05033 − 1.81922i
1.16596 − 2.01950i
−0.855415 + 1.48162i
−0.772849 + 1.33861i
−0.0424677 + 0.0735563i
−0.855415 − 1.48162i −0.520274 + 0.901140i −0.463468 + 0.802751i −2.02113 − 3.50070i 1.78020 0 −1.83583 0.958631 + 1.66040i −3.45781 + 5.98910i
18.2 −0.772849 − 1.33861i −1.18625 + 2.05464i −0.194591 + 0.337041i −0.157702 − 0.273148i 3.66716 0 −2.48984 −1.31437 − 2.27655i −0.243760 + 0.422205i
18.3 −0.0424677 − 0.0735563i 0.700014 − 1.21246i 0.996393 − 1.72580i −1.64706 − 2.85279i −0.118912 0 −0.339129 0.519961 + 0.900599i −0.139894 + 0.242303i
18.4 0.454446 + 0.787123i −1.10322 + 1.91083i 0.586958 − 1.01664i 0.922099 + 1.59712i −2.00541 0 2.88475 −0.934172 − 1.61803i −0.838088 + 1.45161i
18.5 1.05033 + 1.81922i 1.08722 − 1.88311i −1.20638 + 2.08951i −1.61978 − 2.80554i 4.56774 0 −0.867058 −0.864080 − 1.49663i 3.40259 − 5.89347i
18.6 1.16596 + 2.01950i −1.47749 + 2.55909i −1.71891 + 2.97725i −0.976432 − 1.69123i −6.89078 0 −3.35289 −2.86597 − 4.96401i 2.27696 − 3.94381i
324.1 −0.855415 + 1.48162i −0.520274 − 0.901140i −0.463468 − 0.802751i −2.02113 + 3.50070i 1.78020 0 −1.83583 0.958631 − 1.66040i −3.45781 − 5.98910i
324.2 −0.772849 + 1.33861i −1.18625 − 2.05464i −0.194591 − 0.337041i −0.157702 + 0.273148i 3.66716 0 −2.48984 −1.31437 + 2.27655i −0.243760 − 0.422205i
324.3 −0.0424677 + 0.0735563i 0.700014 + 1.21246i 0.996393 + 1.72580i −1.64706 + 2.85279i −0.118912 0 −0.339129 0.519961 − 0.900599i −0.139894 − 0.242303i
324.4 0.454446 − 0.787123i −1.10322 − 1.91083i 0.586958 + 1.01664i 0.922099 − 1.59712i −2.00541 0 2.88475 −0.934172 + 1.61803i −0.838088 − 1.45161i
324.5 1.05033 − 1.81922i 1.08722 + 1.88311i −1.20638 − 2.08951i −1.61978 + 2.80554i 4.56774 0 −0.867058 −0.864080 + 1.49663i 3.40259 + 5.89347i
324.6 1.16596 − 2.01950i −1.47749 − 2.55909i −1.71891 − 2.97725i −0.976432 + 1.69123i −6.89078 0 −3.35289 −2.86597 + 4.96401i 2.27696 + 3.94381i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 343.2.c.d 12
7.b odd 2 1 343.2.c.e 12
7.c even 3 1 343.2.a.d yes 6
7.c even 3 1 inner 343.2.c.d 12
7.d odd 6 1 343.2.a.c ✓ 6
7.d odd 6 1 343.2.c.e 12
21.g even 6 1 3087.2.a.k 6
21.h odd 6 1 3087.2.a.j 6
28.f even 6 1 5488.2.a.p 6
28.g odd 6 1 5488.2.a.h 6
35.i odd 6 1 8575.2.a.o 6
35.j even 6 1 8575.2.a.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
343.2.a.c ✓ 6 7.d odd 6 1
343.2.a.d yes 6 7.c even 3 1
343.2.c.d 12 1.a even 1 1 trivial
343.2.c.d 12 7.c even 3 1 inner
343.2.c.e 12 7.b odd 2 1
343.2.c.e 12 7.d odd 6 1
3087.2.a.j 6 21.h odd 6 1
3087.2.a.k 6 21.g even 6 1
5488.2.a.h 6 28.g odd 6 1
5488.2.a.p 6 28.f even 6 1
8575.2.a.n 6 35.j even 6 1
8575.2.a.o 6 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(343, [\chi])\):

\( T_{2}^{12} - 2 T_{2}^{11} + 10 T_{2}^{10} - 8 T_{2}^{9} + 46 T_{2}^{8} - 31 T_{2}^{7} + 136 T_{2}^{6} + \cdots + 1 \) Copy content Toggle raw display
\( T_{3}^{12} + 5 T_{3}^{11} + 26 T_{3}^{10} + 63 T_{3}^{9} + 199 T_{3}^{8} + 363 T_{3}^{7} + 981 T_{3}^{6} + \cdots + 2401 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 5 T^{11} + \cdots + 2401 \) Copy content Toggle raw display
$5$ \( T^{12} + 11 T^{11} + \cdots + 2401 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - T^{11} + \cdots + 57121 \) Copy content Toggle raw display
$13$ \( (T^{6} - 7 T^{5} + \cdots + 343)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 26 T^{11} + \cdots + 12103441 \) Copy content Toggle raw display
$19$ \( T^{12} - 3 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$23$ \( T^{12} - 9 T^{11} + \cdots + 12769 \) Copy content Toggle raw display
$29$ \( (T^{6} + 2 T^{5} + \cdots - 1777)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 2 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$37$ \( T^{12} - 2 T^{11} + \cdots + 1194649 \) Copy content Toggle raw display
$41$ \( (T^{6} - 28 T^{5} + \cdots + 9947)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 5 T^{5} - 24 T^{4} + \cdots - 83)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 18 T^{11} + \cdots + 16540489 \) Copy content Toggle raw display
$53$ \( T^{12} - T^{11} + \cdots + 1681 \) Copy content Toggle raw display
$59$ \( T^{12} + 5 T^{11} + \cdots + 22591009 \) Copy content Toggle raw display
$61$ \( T^{12} - 17 T^{11} + \cdots + 4439449 \) Copy content Toggle raw display
$67$ \( T^{12} - 22 T^{11} + \cdots + 3463321 \) Copy content Toggle raw display
$71$ \( (T^{6} + 2 T^{5} + \cdots - 1189)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 12 T^{11} + \cdots + 30658369 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 625950361 \) Copy content Toggle raw display
$83$ \( (T^{6} - 7 T^{5} + \cdots - 431837)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 39 T^{11} + \cdots + 4439449 \) Copy content Toggle raw display
$97$ \( (T^{6} - 35 T^{5} + \cdots + 218491)^{2} \) Copy content Toggle raw display
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