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Properties

Label 348.3.j.a
Level $348$
Weight $3$
Character orbit 348.j
Analytic conductor $9.482$
Analytic rank $0$
Dimension $20$
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] = [348,3,Mod(133,348)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(348, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("348.133");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 348.j (of order \(4\), degree \(2\), 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: \(9.48231319974\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 8 x^{18} + 480 x^{17} + 8664 x^{16} - 4140 x^{15} + 62448 x^{14} + 2397840 x^{13} + \cdots + 1236544 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + (\beta_{12} - \beta_{11} + \cdots - \beta_1) q^{5}+ \cdots + 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + (\beta_{12} - \beta_{11} + \cdots - \beta_1) q^{5}+ \cdots + (3 \beta_{11} - 3 \beta_{9}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 36 q^{15} + 8 q^{17} + 8 q^{19} - 72 q^{23} - 124 q^{25} + 36 q^{29} + 44 q^{31} + 32 q^{37} - 48 q^{39} - 56 q^{41} + 64 q^{43} + 48 q^{45} - 56 q^{47} + 44 q^{49} + 176 q^{53} + 168 q^{55} - 80 q^{59} - 64 q^{61} - 296 q^{65} - 48 q^{69} - 172 q^{73} + 48 q^{75} + 336 q^{77} - 244 q^{79} - 180 q^{81} - 368 q^{83} + 592 q^{85} + 12 q^{87} + 176 q^{89} - 184 q^{95} + 188 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 8 x^{18} + 480 x^{17} + 8664 x^{16} - 4140 x^{15} + 62448 x^{14} + 2397840 x^{13} + \cdots + 1236544 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 17\!\cdots\!08 \nu^{19} + \cdots + 71\!\cdots\!00 ) / 63\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!79 \nu^{19} + \cdots - 27\!\cdots\!80 ) / 32\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 45\!\cdots\!21 \nu^{19} + \cdots + 10\!\cdots\!28 ) / 74\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19\!\cdots\!37 \nu^{19} + \cdots + 36\!\cdots\!28 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 45\!\cdots\!07 \nu^{19} + \cdots - 18\!\cdots\!08 ) / 15\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 87\!\cdots\!17 \nu^{19} + \cdots + 49\!\cdots\!24 ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 44\!\cdots\!47 \nu^{19} + \cdots - 51\!\cdots\!20 ) / 74\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 70\!\cdots\!01 \nu^{19} + \cdots - 29\!\cdots\!44 ) / 44\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14\!\cdots\!17 \nu^{19} + \cdots + 16\!\cdots\!16 ) / 74\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!99 \nu^{19} + \cdots - 74\!\cdots\!04 ) / 44\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 29\!\cdots\!57 \nu^{19} + \cdots - 35\!\cdots\!40 ) / 10\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16\!\cdots\!59 \nu^{19} + \cdots - 40\!\cdots\!12 ) / 44\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16\!\cdots\!31 \nu^{19} + \cdots - 56\!\cdots\!20 ) / 32\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13\!\cdots\!63 \nu^{19} + \cdots - 34\!\cdots\!32 ) / 22\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 70\!\cdots\!42 \nu^{19} + \cdots + 16\!\cdots\!80 ) / 11\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 11\!\cdots\!93 \nu^{19} + \cdots + 18\!\cdots\!88 ) / 16\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 35\!\cdots\!97 \nu^{19} + \cdots + 14\!\cdots\!92 ) / 40\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 11\!\cdots\!76 \nu^{19} + \cdots - 13\!\cdots\!40 ) / 11\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 40\!\cdots\!35 \nu^{19} + \cdots - 49\!\cdots\!88 ) / 22\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{19} - 2 \beta_{18} - \beta_{17} - \beta_{16} - \beta_{15} + \beta_{14} + 2 \beta_{13} + \cdots + 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 7 \beta_{19} - 5 \beta_{18} - 23 \beta_{15} + 2 \beta_{14} + \beta_{12} - 5 \beta_{11} - 7 \beta_{10} + \cdots - 5 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 29 \beta_{17} + 19 \beta_{16} - 29 \beta_{15} + 19 \beta_{14} - 49 \beta_{13} + 49 \beta_{12} + \cdots - 197 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 625 \beta_{19} + 665 \beta_{18} + 1997 \beta_{17} + 380 \beta_{16} - 475 \beta_{13} + 521 \beta_{11} + \cdots - 13906 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7757 \beta_{19} + 13546 \beta_{18} + 8645 \beta_{17} + 3962 \beta_{16} + 8645 \beta_{15} - 3962 \beta_{14} + \cdots - 51698 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 18478 \beta_{19} + 24473 \beta_{18} + 57319 \beta_{15} - 13011 \beta_{14} - 21638 \beta_{12} + \cdots + 24473 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 847955 \beta_{17} - 304307 \beta_{16} + 847955 \beta_{15} - 304307 \beta_{14} + 865894 \beta_{13} + \cdots + 6507651 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4861163 \beta_{19} - 7606099 \beta_{18} - 14966587 \beta_{17} - 3523810 \beta_{16} + 7014557 \beta_{13} + \cdots + 99239114 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 20163384 \beta_{19} - 39707289 \beta_{18} - 27208011 \beta_{17} - 8178093 \beta_{16} + \cdots + 168600616 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 425581841 \beta_{19} - 765156433 \beta_{18} - 1320717721 \beta_{15} + 303718486 \beta_{14} + \cdots - 765156433 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7743039391 \beta_{17} + 2018519554 \beta_{16} - 7743039391 \beta_{15} + 2018519554 \beta_{14} + \cdots - 58990108590 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 12449595258 \beta_{19} + 25253464041 \beta_{18} + 39340406783 \beta_{17} + 8551599985 \beta_{16} + \cdots - 261903346103 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 479685714001 \beta_{19} + 1098738759812 \beta_{18} + 727428902785 \beta_{17} + 167149479667 \beta_{16} + \cdots - 4424083805641 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 3289508919295 \beta_{19} + 7429910165813 \beta_{18} + 10655058422531 \beta_{15} - 2144021765498 \beta_{14} + \cdots + 7429910165813 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 22646532905753 \beta_{17} - 4613660934639 \beta_{16} + 22646532905753 \beta_{15} + \cdots + 171689759006537 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 290800813054729 \beta_{19} - 724121920929689 \beta_{18} - 969633600945749 \beta_{17} + \cdots + 64\!\cdots\!42 ) / 6 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 38\!\cdots\!81 \beta_{19} + \cdots + 37\!\cdots\!86 ) / 6 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 85\!\cdots\!22 \beta_{19} + \cdots - 23\!\cdots\!09 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 58\!\cdots\!99 \beta_{17} + \cdots - 44\!\cdots\!03 ) / 6 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
133.1
−5.45002 − 5.45002i
−0.175515 − 0.175515i
−0.580857 − 0.580857i
0.652561 + 0.652561i
6.55383 + 6.55383i
−4.21478 − 4.21478i
0.531346 + 0.531346i
6.89315 + 6.89315i
−0.581969 − 0.581969i
−1.62775 − 1.62775i
6.55383 − 6.55383i
0.652561 − 0.652561i
−0.580857 + 0.580857i
−0.175515 + 0.175515i
−5.45002 + 5.45002i
−1.62775 + 1.62775i
−0.581969 + 0.581969i
6.89315 − 6.89315i
0.531346 − 0.531346i
−4.21478 + 4.21478i
0 −1.22474 + 1.22474i 0 − 4.39478i 0 −1.89847 0 − 3.00000i 0
133.2 0 −1.22474 + 1.22474i 0 − 1.04133i 0 8.43078 0 − 3.00000i 0
133.3 0 −1.22474 + 1.22474i 0 − 0.130512i 0 −7.59506 0 − 3.00000i 0
133.4 0 −1.22474 + 1.22474i 0 7.37999i 0 −6.31238 0 − 3.00000i 0
133.5 0 −1.22474 + 1.22474i 0 9.53510i 0 7.37514 0 − 3.00000i 0
133.6 0 1.22474 − 1.22474i 0 − 7.29638i 0 10.1980 0 − 3.00000i 0
133.7 0 1.22474 − 1.22474i 0 − 6.22487i 0 −8.38632 0 − 3.00000i 0
133.8 0 1.22474 − 1.22474i 0 0.824812i 0 7.15456 0 − 3.00000i 0
133.9 0 1.22474 − 1.22474i 0 2.45552i 0 −7.62003 0 − 3.00000i 0
133.10 0 1.22474 − 1.22474i 0 6.89246i 0 −1.34618 0 − 3.00000i 0
157.1 0 −1.22474 − 1.22474i 0 − 9.53510i 0 7.37514 0 3.00000i 0
157.2 0 −1.22474 − 1.22474i 0 − 7.37999i 0 −6.31238 0 3.00000i 0
157.3 0 −1.22474 − 1.22474i 0 0.130512i 0 −7.59506 0 3.00000i 0
157.4 0 −1.22474 − 1.22474i 0 1.04133i 0 8.43078 0 3.00000i 0
157.5 0 −1.22474 − 1.22474i 0 4.39478i 0 −1.89847 0 3.00000i 0
157.6 0 1.22474 + 1.22474i 0 − 6.89246i 0 −1.34618 0 3.00000i 0
157.7 0 1.22474 + 1.22474i 0 − 2.45552i 0 −7.62003 0 3.00000i 0
157.8 0 1.22474 + 1.22474i 0 − 0.824812i 0 7.15456 0 3.00000i 0
157.9 0 1.22474 + 1.22474i 0 6.22487i 0 −8.38632 0 3.00000i 0
157.10 0 1.22474 + 1.22474i 0 7.29638i 0 10.1980 0 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 348.3.j.a ✓ 20
3.b odd 2 1 1044.3.k.b 20
29.c odd 4 1 inner 348.3.j.a ✓ 20
87.f even 4 1 1044.3.k.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.3.j.a ✓ 20 1.a even 1 1 trivial
348.3.j.a ✓ 20 29.c odd 4 1 inner
1044.3.k.b 20 3.b odd 2 1
1044.3.k.b 20 87.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(348, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 710115904 \) Copy content Toggle raw display
$7$ \( (T^{10} - 256 T^{8} + \cdots + 35521768)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 26\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 21\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( (T^{10} + 36 T^{9} + \cdots - 14365250816)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 17\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 83\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 98\!\cdots\!24)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 99\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 99\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 62\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 65\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 14\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 67\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 51\!\cdots\!64 \) Copy content Toggle raw display
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