Properties

Label 58.2.d.b
Level $58$
Weight $2$
Character orbit 58.d
Analytic conductor $0.463$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [58,2,Mod(7,58)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(58, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("58.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 58.d (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: \(0.463132331723\)
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: \( x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{11} q^{3} + \beta_{6} q^{4} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots + ( - \beta_{10} + \beta_{8} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{11} q^{3} + \beta_{6} q^{4} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{11} + \beta_{10} + 2 \beta_{9} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 3 q^{3} - 2 q^{4} + 4 q^{6} + q^{7} - 2 q^{8} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - 3 q^{3} - 2 q^{4} + 4 q^{6} + q^{7} - 2 q^{8} - 11 q^{9} - 7 q^{10} - 2 q^{11} + 4 q^{12} + q^{13} + q^{14} - 9 q^{15} - 2 q^{16} - 12 q^{17} - 11 q^{18} - 6 q^{19} + 7 q^{20} - 13 q^{21} - 2 q^{22} + 35 q^{23} + 4 q^{24} - 6 q^{25} + 8 q^{26} + 39 q^{27} - 6 q^{28} - 14 q^{29} + 54 q^{30} - 8 q^{31} - 2 q^{32} + 33 q^{33} - 5 q^{34} - 18 q^{35} - 4 q^{36} + 31 q^{37} + q^{38} - 22 q^{39} + 7 q^{40} - 30 q^{41} - 13 q^{42} - 5 q^{43} + 5 q^{44} - 20 q^{45} - 28 q^{46} - 33 q^{47} - 3 q^{48} + 37 q^{49} + q^{50} - 15 q^{51} - 6 q^{52} + 13 q^{53} - 24 q^{54} + 36 q^{55} + q^{56} - 38 q^{57} + 38 q^{59} - 9 q^{60} - 5 q^{61} - 8 q^{62} + 4 q^{63} - 2 q^{64} - 20 q^{65} - 30 q^{66} - 7 q^{67} - 19 q^{68} + 20 q^{69} - 18 q^{70} + 3 q^{71} + 3 q^{72} - 22 q^{73} + 31 q^{74} - 2 q^{75} + q^{76} + 10 q^{77} + 41 q^{78} - 60 q^{79} - 7 q^{80} + 88 q^{81} + 26 q^{82} - 39 q^{83} + 43 q^{84} + 38 q^{85} + 2 q^{86} + 9 q^{87} - 2 q^{88} + 39 q^{89} - 34 q^{90} + 61 q^{91} + 35 q^{92} - 54 q^{93} + 16 q^{94} + 55 q^{95} - 3 q^{96} - 19 q^{97} + 37 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 97 \nu^{10} - 291 \nu^{9} + 1067 \nu^{8} - 233 \nu^{7} - 1945 \nu^{6} + 1185 \nu^{5} + \cdots + 293952 ) / 443456 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 53 \nu^{11} - 435 \nu^{10} + 1469 \nu^{9} - 3473 \nu^{8} + 443 \nu^{7} + 7123 \nu^{6} + \cdots - 838912 ) / 1773824 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 30 \nu^{11} + 40 \nu^{10} - 79 \nu^{9} + 1137 \nu^{8} - 307 \nu^{7} - 8945 \nu^{6} + 25423 \nu^{5} + \cdots + 324672 ) / 443456 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 69 \nu^{11} + 195 \nu^{10} - 749 \nu^{9} + 177 \nu^{8} + 1317 \nu^{7} - 1587 \nu^{6} + \cdots - 54272 ) / 443456 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 325 \nu^{11} - 250 \nu^{10} + 750 \nu^{9} + 12256 \nu^{8} - 28698 \nu^{7} + 30394 \nu^{6} + \cdots + 1298688 ) / 1773824 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 25 \nu^{11} + 76 \nu^{10} - 269 \nu^{9} + 57 \nu^{8} + 495 \nu^{7} - 247 \nu^{6} - 12733 \nu^{5} + \cdots + 1024 ) / 110864 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1961 \nu^{11} - 9535 \nu^{10} + 37297 \nu^{9} - 72085 \nu^{8} + 46567 \nu^{7} + 31327 \nu^{6} + \cdots - 3739648 ) / 7095296 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 130 \nu^{11} - 469 \nu^{10} + 1407 \nu^{9} - 157 \nu^{8} - 9995 \nu^{7} + 19513 \nu^{6} + \cdots - 122880 ) / 443456 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 725 \nu^{11} - 3475 \nu^{10} + 15181 \nu^{9} - 27073 \nu^{8} + 19019 \nu^{7} + 54723 \nu^{6} + \cdots - 1331200 ) / 1773824 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 913 \nu^{11} - 2951 \nu^{10} + 13609 \nu^{9} - 14093 \nu^{8} + 9327 \nu^{7} + 30183 \nu^{6} + \cdots + 2008064 ) / 1773824 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{7} + 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{9} + 4\beta_{8} + \beta_{7} + 4\beta_{6} + 6\beta_{5} + 4\beta_{3} + 4\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 14 \beta_{11} + 14 \beta_{10} - \beta_{9} - 12 \beta_{7} + 4 \beta_{6} + 12 \beta_{5} + 4 \beta_{4} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18 \beta_{10} + 17 \beta_{9} - 56 \beta_{8} - 49 \beta_{7} - 52 \beta_{6} - 17 \beta_{5} + \cdots + 18 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 175 \beta_{11} - 153 \beta_{10} + 43 \beta_{9} - 72 \beta_{8} + 43 \beta_{7} - 140 \beta_{6} - 153 \beta_{5} + \cdots - 72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 247 \beta_{11} - 468 \beta_{10} - 234 \beta_{9} + 612 \beta_{8} + 468 \beta_{7} + 440 \beta_{6} + \cdots - 88 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1782 \beta_{11} + 1142 \beta_{10} - 1142 \beta_{9} + 1872 \beta_{8} + 2808 \beta_{6} + 1782 \beta_{5} + \cdots + 884 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5771 \beta_{11} + 7849 \beta_{10} - 4568 \beta_{8} - 5771 \beta_{7} + 2666 \beta_{5} + 4568 \beta_{4} + \cdots + 2560 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 12417 \beta_{11} + 12417 \beta_{9} - 31396 \beta_{8} - 8331 \beta_{7} - 31396 \beta_{6} - 16862 \beta_{5} + \cdots - 8312 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 89754 \beta_{11} - 89754 \beta_{10} + 20729 \beta_{9} + 58342 \beta_{7} - 49668 \beta_{6} - 58342 \beta_{5} + \cdots - 49668 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
2.06920 0.996473i
−1.56920 + 0.755686i
0.760453 + 3.33176i
−0.260453 1.14112i
2.06920 + 0.996473i
−1.56920 0.755686i
1.52179 + 1.90827i
−1.02179 1.28129i
1.52179 1.90827i
−1.02179 + 1.28129i
0.760453 3.33176i
−0.260453 + 1.14112i
−0.222521 0.974928i −2.06920 0.996473i −0.900969 + 0.433884i −0.788529 3.45477i −0.511050 + 2.23905i 3.72857 + 1.79558i 0.623490 + 0.781831i 1.41815 + 1.77830i −3.19269 + 1.53752i
7.2 −0.222521 0.974928i 1.56920 + 0.755686i −0.900969 + 0.433884i 0.110081 + 0.482295i 0.387560 1.69801i −2.82760 1.36170i 0.623490 + 0.781831i 0.0208506 + 0.0261458i 0.445708 0.214642i
23.1 0.623490 0.781831i −0.760453 + 3.33176i −0.222521 0.974928i 1.00725 1.26305i 2.13074 + 2.67187i 0.338433 1.48277i −0.900969 0.433884i −7.81944 3.76565i −0.359484 1.57500i
23.2 0.623490 0.781831i 0.260453 1.14112i −0.222521 0.974928i −1.85326 + 2.32392i −0.729773 0.915107i −0.115912 + 0.507846i −0.900969 0.433884i 1.46859 + 0.707235i 0.661422 + 2.89788i
25.1 −0.222521 + 0.974928i −2.06920 + 0.996473i −0.900969 0.433884i −0.788529 + 3.45477i −0.511050 2.23905i 3.72857 1.79558i 0.623490 0.781831i 1.41815 1.77830i −3.19269 1.53752i
25.2 −0.222521 + 0.974928i 1.56920 0.755686i −0.900969 0.433884i 0.110081 0.482295i 0.387560 + 1.69801i −2.82760 + 1.36170i 0.623490 0.781831i 0.0208506 0.0261458i 0.445708 + 0.214642i
45.1 −0.900969 0.433884i −1.52179 + 1.90827i 0.623490 + 0.781831i 2.60002 + 1.25211i 2.19905 1.05901i −1.89765 + 2.37957i −0.222521 0.974928i −0.658071 2.88320i −1.79927 2.25622i
45.2 −0.900969 0.433884i 1.02179 1.28129i 0.623490 + 0.781831i −1.07557 0.517965i −1.47653 + 0.711061i 1.27416 1.59774i −0.222521 0.974928i 0.0699247 + 0.306360i 0.744314 + 0.933340i
49.1 −0.900969 + 0.433884i −1.52179 1.90827i 0.623490 0.781831i 2.60002 1.25211i 2.19905 + 1.05901i −1.89765 2.37957i −0.222521 + 0.974928i −0.658071 + 2.88320i −1.79927 + 2.25622i
49.2 −0.900969 + 0.433884i 1.02179 + 1.28129i 0.623490 0.781831i −1.07557 + 0.517965i −1.47653 0.711061i 1.27416 + 1.59774i −0.222521 + 0.974928i 0.0699247 0.306360i 0.744314 0.933340i
53.1 0.623490 + 0.781831i −0.760453 3.33176i −0.222521 + 0.974928i 1.00725 + 1.26305i 2.13074 2.67187i 0.338433 + 1.48277i −0.900969 + 0.433884i −7.81944 + 3.76565i −0.359484 + 1.57500i
53.2 0.623490 + 0.781831i 0.260453 + 1.14112i −0.222521 + 0.974928i −1.85326 2.32392i −0.729773 + 0.915107i −0.115912 0.507846i −0.900969 + 0.433884i 1.46859 0.707235i 0.661422 2.89788i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.2
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 58.2.d.b 12
3.b odd 2 1 522.2.k.h 12
4.b odd 2 1 464.2.u.h 12
29.d even 7 1 inner 58.2.d.b 12
29.d even 7 1 1682.2.a.t 6
29.e even 14 1 1682.2.a.q 6
29.f odd 28 2 1682.2.b.i 12
87.j odd 14 1 522.2.k.h 12
116.j odd 14 1 464.2.u.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.d.b 12 1.a even 1 1 trivial
58.2.d.b 12 29.d even 7 1 inner
464.2.u.h 12 4.b odd 2 1
464.2.u.h 12 116.j odd 14 1
522.2.k.h 12 3.b odd 2 1
522.2.k.h 12 87.j odd 14 1
1682.2.a.q 6 29.e even 14 1
1682.2.a.t 6 29.d even 7 1
1682.2.b.i 12 29.f odd 28 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 3 T_{3}^{11} + 13 T_{3}^{10} + 9 T_{3}^{9} - 5 T_{3}^{8} - 35 T_{3}^{7} + 197 T_{3}^{6} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(58, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 3 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$5$ \( T^{12} + 8 T^{10} + \cdots + 841 \) Copy content Toggle raw display
$7$ \( T^{12} - T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( T^{12} + 2 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{12} - T^{11} + \cdots + 49 \) Copy content Toggle raw display
$17$ \( (T^{6} + 6 T^{5} + \cdots - 1259)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} + \cdots + 12845056 \) Copy content Toggle raw display
$23$ \( T^{12} - 35 T^{11} + \cdots + 7573504 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 594823321 \) Copy content Toggle raw display
$31$ \( T^{12} + 8 T^{11} + \cdots + 3444736 \) Copy content Toggle raw display
$37$ \( T^{12} - 31 T^{11} + \cdots + 68442529 \) Copy content Toggle raw display
$41$ \( (T^{6} + 15 T^{5} + \cdots - 9653)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 5 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$47$ \( T^{12} + 33 T^{11} + \cdots + 7573504 \) Copy content Toggle raw display
$53$ \( T^{12} - 13 T^{11} + \cdots + 35724529 \) Copy content Toggle raw display
$59$ \( (T^{6} - 19 T^{5} + \cdots - 3584)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 5 T^{11} + \cdots + 97160449 \) Copy content Toggle raw display
$67$ \( T^{12} + 7 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 3720024064 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 490724067289 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 113844158464 \) Copy content Toggle raw display
$83$ \( T^{12} + 39 T^{11} + \cdots + 200704 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 126405049 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 36316162624 \) Copy content Toggle raw display
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