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Properties

Label 348.2.m.a
Level $348$
Weight $2$
Character orbit 348.m
Analytic conductor $2.779$
Analytic rank $0$
Dimension $12$
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] = [348,2,Mod(25,348)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(348, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("348.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 348.m (of order \(7\), degree \(6\), 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: \(2.77879399034\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 8x^{10} - 8x^{9} + 29x^{8} - 64x^{7} + 99x^{6} - 105x^{5} + 168x^{4} - 63x^{3} + 98x + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{3} + ( - \beta_{11} + \beta_{10} + \beta_{6} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{11} - \beta_{9} + \beta_{5} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{3} + ( - \beta_{11} + \beta_{10} + \beta_{6} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{10} + \beta_{9} - \beta_{7} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} - 5 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} - 5 q^{5} + 2 q^{7} - 2 q^{9} - q^{13} - 5 q^{15} - 6 q^{17} - 6 q^{19} + 2 q^{21} + 12 q^{23} + 11 q^{25} - 2 q^{27} + 7 q^{29} + 20 q^{31} - 7 q^{35} + 22 q^{37} - q^{39} - 22 q^{41} + 5 q^{43} - 5 q^{45} + 9 q^{47} - 10 q^{49} - 13 q^{51} - 14 q^{53} + q^{55} - 6 q^{57} + 16 q^{59} + 2 q^{61} - 5 q^{63} - 55 q^{65} - 4 q^{67} - 16 q^{69} - 2 q^{71} - 26 q^{73} - 10 q^{75} - 4 q^{77} + 31 q^{79} - 2 q^{81} + 63 q^{85} + 20 q^{89} + 43 q^{91} + 20 q^{93} - 16 q^{95} - 22 q^{97} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 8x^{10} - 8x^{9} + 29x^{8} - 64x^{7} + 99x^{6} - 105x^{5} + 168x^{4} - 63x^{3} + 98x + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1595951032 \nu^{11} - 587324374 \nu^{10} - 16158118968 \nu^{9} - 4055561547 \nu^{8} + \cdots + 251737356066 ) / 837659674411 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2610124571 \nu^{11} + 7296039540 \nu^{10} - 26306398975 \nu^{9} + 59927714514 \nu^{8} + \cdots - 10057141920 ) / 837659674411 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4685914969 \nu^{11} + 5425402407 \nu^{10} - 39046717946 \nu^{9} + 50073374230 \nu^{8} + \cdots - 127896103979 ) / 837659674411 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7403524548 \nu^{11} - 11201568368 \nu^{10} + 69435967753 \nu^{9} - 92857273752 \nu^{8} + \cdots + 779572357683 ) / 837659674411 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7433539586 \nu^{11} - 11230849931 \nu^{10} + 67418265180 \nu^{9} - 88153181886 \nu^{8} + \cdots + 999126076361 ) / 837659674411 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8483958789 \nu^{11} + 15633173776 \nu^{10} - 72675795314 \nu^{9} + 121529030990 \nu^{8} + \cdots - 490668806831 ) / 837659674411 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8536056875 \nu^{11} - 3618899984 \nu^{10} + 62873060845 \nu^{9} - 27611878805 \nu^{8} + \cdots + 420307332585 ) / 837659674411 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8577700665 \nu^{11} - 17113757540 \nu^{10} + 72240505304 \nu^{9} - 131494666165 \nu^{8} + \cdots + 758652434337 ) / 837659674411 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 9616814992 \nu^{11} + 7840339219 \nu^{10} - 84241434983 \nu^{9} + 61155026610 \nu^{8} + \cdots - 920924475793 ) / 837659674411 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10216676392 \nu^{11} - 11297434509 \nu^{10} + 85954850371 \nu^{9} - 103101785274 \nu^{8} + \cdots + 263915343678 ) / 837659674411 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{6} + \beta_{5} + 2\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{11} + \beta_{9} + \beta_{7} - 4\beta_{4} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{11} - 7\beta_{10} + 5\beta_{9} - 7\beta_{8} - 2\beta_{6} - 13\beta_{5} - 6\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{11} - \beta_{10} - 14 \beta_{9} - 20 \beta_{7} - 8 \beta_{6} + 20 \beta_{4} - 2 \beta_{3} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 31 \beta_{11} + 19 \beta_{10} - 41 \beta_{9} + 43 \beta_{8} + 3 \beta_{6} + 68 \beta_{5} + 3 \beta_{4} + \cdots - 41 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 25 \beta_{11} + 56 \beta_{10} + 59 \beta_{9} + 13 \beta_{8} + 111 \beta_{7} + 111 \beta_{6} + \cdots - 89 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 38 \beta_{10} + 236 \beta_{9} - 140 \beta_{8} + 38 \beta_{7} - 236 \beta_{5} - 140 \beta_{4} + \cdots + 154 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 216 \beta_{11} - 648 \beta_{10} - 376 \beta_{8} - 376 \beta_{7} - 648 \beta_{6} - 577 \beta_{5} + \cdots + 577 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 800 \beta_{11} - 1520 \beta_{9} + 336 \beta_{8} - 953 \beta_{7} - 336 \beta_{6} + 616 \beta_{5} + \cdots + 953 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2182 \beta_{11} + 3885 \beta_{10} - 2182 \beta_{9} + 3885 \beta_{8} + 952 \beta_{7} + 2473 \beta_{6} + \cdots - 3807 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/348\mathbb{Z}\right)^\times\).

\(n\) \(175\) \(205\) \(233\)
\(\chi(n)\) \(1\) \(-\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
−1.33530 − 1.67442i
0.711812 + 0.892584i
0.559182 + 2.44994i
−0.336661 − 1.47501i
−0.451844 − 0.217597i
1.35281 + 0.651480i
−1.33530 + 1.67442i
0.711812 − 0.892584i
0.559182 − 2.44994i
−0.336661 + 1.47501i
−0.451844 + 0.217597i
1.35281 − 0.651480i
0 −0.900969 + 0.433884i 0 −0.0755949 + 0.331203i 0 2.09943 − 1.01103i 0 0.623490 − 0.781831i 0
25.2 0 −0.900969 + 0.433884i 0 0.655012 − 2.86979i 0 −2.50040 + 1.20413i 0 0.623490 − 0.781831i 0
49.1 0 0.623490 + 0.781831i 0 −3.38757 + 1.63137i 0 0.313167 + 0.392699i 0 −0.222521 + 0.974928i 0
49.2 0 0.623490 + 0.781831i 0 0.239624 − 0.115397i 0 0.810323 + 1.01611i 0 −0.222521 + 0.974928i 0
169.1 0 −0.222521 − 0.974928i 0 −0.590165 − 0.740043i 0 −0.584872 − 2.56249i 0 −0.900969 + 0.433884i 0
169.2 0 −0.222521 − 0.974928i 0 0.658697 + 0.825979i 0 0.862351 + 3.77821i 0 −0.900969 + 0.433884i 0
181.1 0 −0.900969 − 0.433884i 0 −0.0755949 − 0.331203i 0 2.09943 + 1.01103i 0 0.623490 + 0.781831i 0
181.2 0 −0.900969 − 0.433884i 0 0.655012 + 2.86979i 0 −2.50040 − 1.20413i 0 0.623490 + 0.781831i 0
277.1 0 0.623490 − 0.781831i 0 −3.38757 − 1.63137i 0 0.313167 − 0.392699i 0 −0.222521 − 0.974928i 0
277.2 0 0.623490 − 0.781831i 0 0.239624 + 0.115397i 0 0.810323 − 1.01611i 0 −0.222521 − 0.974928i 0
313.1 0 −0.222521 + 0.974928i 0 −0.590165 + 0.740043i 0 −0.584872 + 2.56249i 0 −0.900969 − 0.433884i 0
313.2 0 −0.222521 + 0.974928i 0 0.658697 − 0.825979i 0 0.862351 − 3.77821i 0 −0.900969 − 0.433884i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 348.2.m.a ✓ 12
3.b odd 2 1 1044.2.u.d 12
29.d even 7 1 inner 348.2.m.a ✓ 12
87.j odd 14 1 1044.2.u.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.2.m.a ✓ 12 1.a even 1 1 trivial
348.2.m.a ✓ 12 29.d even 7 1 inner
1044.2.u.d 12 3.b odd 2 1
1044.2.u.d 12 87.j odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 5 T_{5}^{11} + 12 T_{5}^{10} + 37 T_{5}^{9} + 113 T_{5}^{8} - 28 T_{5}^{7} + 85 T_{5}^{6} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(348, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 5 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} - 2 T^{11} + \cdots + 1849 \) Copy content Toggle raw display
$11$ \( T^{12} + 29 T^{10} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{12} + T^{11} + \cdots + 187489 \) Copy content Toggle raw display
$17$ \( (T^{6} + 3 T^{5} + \cdots + 1289)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} + \cdots + 67289209 \) Copy content Toggle raw display
$23$ \( (T^{6} - 6 T^{5} + 22 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 594823321 \) Copy content Toggle raw display
$31$ \( T^{12} - 20 T^{11} + \cdots + 2778889 \) Copy content Toggle raw display
$37$ \( T^{12} - 22 T^{11} + \cdots + 2181529 \) Copy content Toggle raw display
$41$ \( (T^{6} + 11 T^{5} + \cdots + 937)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 159997201 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 3994619209 \) Copy content Toggle raw display
$53$ \( T^{12} + 14 T^{11} + \cdots + 3523129 \) Copy content Toggle raw display
$59$ \( (T^{6} - 8 T^{5} + \cdots + 101459)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} - 2 T^{11} + \cdots + 528529 \) Copy content Toggle raw display
$67$ \( T^{12} + 4 T^{11} + \cdots + 312481 \) Copy content Toggle raw display
$71$ \( T^{12} + 2 T^{11} + \cdots + 6889 \) Copy content Toggle raw display
$73$ \( T^{12} + 26 T^{11} + \cdots + 4206601 \) Copy content Toggle raw display
$79$ \( T^{12} - 31 T^{11} + \cdots + 42915601 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 1287673449049 \) Copy content Toggle raw display
$89$ \( T^{12} - 20 T^{11} + \cdots + 1481089 \) Copy content Toggle raw display
$97$ \( T^{12} + 22 T^{11} + \cdots + 40666129 \) Copy content Toggle raw display
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