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Properties

Label 348.3.e.b
Level $348$
Weight $3$
Character orbit 348.e
Analytic conductor $9.482$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [348,3,Mod(173,348)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(348, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("348.173");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 348.e (of order \(2\), degree \(1\), 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8x^{14} + 33x^{12} - 572x^{10} + 4516x^{8} - 46332x^{6} + 216513x^{4} - 4251528x^{2} + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{3} - \beta_{5} q^{5} + (\beta_{4} + 1) q^{7} + ( - \beta_{7} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{3} - \beta_{5} q^{5} + (\beta_{4} + 1) q^{7} + ( - \beta_{7} + 1) q^{9} + ( - \beta_{6} + \beta_{3}) q^{11} + ( - \beta_{7} + \beta_{4} + \beta_{2} - 7) q^{13} + (\beta_{15} + \beta_{14} + \cdots + \beta_1) q^{15}+ \cdots + (\beta_{15} + 5 \beta_{14} + \cdots - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{7} + 16 q^{9} - 116 q^{13} - 184 q^{25} + 48 q^{33} - 164 q^{45} + 356 q^{49} - 24 q^{51} + 292 q^{57} - 480 q^{63} - 44 q^{67} - 4 q^{81} - 164 q^{87} + 60 q^{91} + 88 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8x^{14} + 33x^{12} - 572x^{10} + 4516x^{8} - 46332x^{6} + 216513x^{4} - 4251528x^{2} + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1153 \nu^{15} + 101645 \nu^{13} + 554466 \nu^{11} - 6414214 \nu^{9} + \cdots - 14528002617 \nu ) / 36503619408 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 67 \nu^{14} - 374 \nu^{12} + 7476 \nu^{10} - 85466 \nu^{8} - 105020 \nu^{6} - 2936898 \nu^{4} + \cdots - 291229668 ) / 76527504 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3245 \nu^{14} + 6845 \nu^{12} + 244866 \nu^{10} - 1318138 \nu^{8} + 20257514 \nu^{6} + \cdots - 19236569877 ) / 2703971808 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2269 \nu^{15} - 16759 \nu^{13} + 827706 \nu^{11} + 2068178 \nu^{9} - 87157438 \nu^{7} + \cdots + 19341795195 \nu ) / 24335746272 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{14} - 8\nu^{12} + 33\nu^{10} - 572\nu^{8} + 4516\nu^{6} - 46332\nu^{4} + 216513\nu^{2} - 3720087 ) / 531441 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{15} + 8\nu^{13} - 33\nu^{11} + 572\nu^{9} - 4516\nu^{7} + 46332\nu^{5} - 216513\nu^{3} + 4251528\nu ) / 4782969 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24599 \nu^{15} + 146259 \nu^{14} + 152971 \nu^{13} + 424251 \nu^{12} - 375906 \nu^{11} + \cdots - 311902191459 ) / 73007238816 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14383 \nu^{15} + 56096 \nu^{13} + 1880112 \nu^{11} - 22600390 \nu^{9} + \cdots - 5897932218 \nu ) / 36503619408 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14989 \nu^{15} - 121063 \nu^{13} + 2056458 \nu^{11} + 2543906 \nu^{9} + \cdots + 28805696523 \nu ) / 36503619408 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 24599 \nu^{15} - 310968 \nu^{14} - 152971 \nu^{13} - 929808 \nu^{12} + 375906 \nu^{11} + \cdots + 1175385933936 ) / 73007238816 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 24599 \nu^{15} + 332703 \nu^{14} + 152971 \nu^{13} - 650313 \nu^{12} - 375906 \nu^{11} + \cdots - 1060781213727 ) / 73007238816 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 38435 \nu^{15} + 69737 \nu^{13} - 1126086 \nu^{11} - 7614286 \nu^{9} - 139075390 \nu^{7} + \cdots - 64481330853 \nu ) / 73007238816 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 74657 \nu^{15} + 103637 \nu^{13} + 4998738 \nu^{11} - 14030290 \nu^{9} + \cdots - 425436058053 \nu ) / 73007238816 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{11} - \beta_{8} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{13} - \beta_{12} - 3\beta_{9} - 2\beta_{7} - 3\beta_{5} + 2\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{15} + 2\beta_{14} - 7\beta_{11} + 9\beta_{10} + 17\beta_{8} + 7\beta_{6} - 24\beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{14} - 7 \beta_{13} - \beta_{12} - 18 \beta_{9} + 12 \beta_{8} + 31 \beta_{7} - 12 \beta_{6} + \cdots + 157 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 45\beta_{15} - 84\beta_{14} + 33\beta_{11} - 9\beta_{10} + 33\beta_{8} - 159\beta_{6} - 9\beta_{3} + 254\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 27 \beta_{14} + 285 \beta_{13} + 186 \beta_{12} - 45 \beta_{9} - 27 \beta_{8} - 114 \beta_{7} + \cdots - 397 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 450 \beta_{15} - 428 \beta_{14} + 427 \beta_{11} - 1098 \beta_{10} + 2209 \beta_{8} + 392 \beta_{6} + \cdots - 624 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 846 \beta_{14} + 2278 \beta_{13} + 2002 \beta_{12} - 1968 \beta_{9} + 846 \beta_{8} - 4051 \beta_{7} + \cdots + 16849 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 6018 \beta_{15} - 824 \beta_{14} + 3568 \beta_{11} + 3222 \beta_{10} + 10411 \beta_{8} + \cdots + 13174 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5160 \beta_{14} - 10778 \beta_{13} - 12422 \beta_{12} + 8676 \beta_{9} - 5160 \beta_{8} - 41248 \beta_{7} + \cdots + 131081 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 45486 \beta_{15} + 22218 \beta_{14} - 40440 \beta_{11} + 16218 \beta_{10} + 319506 \beta_{8} + \cdots + 72823 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 138834 \beta_{14} + 125898 \beta_{13} - 100866 \beta_{12} + 50904 \beta_{9} - 138834 \beta_{8} + \cdots + 3013777 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 358470 \beta_{15} + 648641 \beta_{14} - 453859 \beta_{11} - 147006 \beta_{10} - 451807 \beta_{8} + \cdots + 2710050 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(175\) \(205\) \(233\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
173.1
2.99317 − 0.202391i
2.99317 + 0.202391i
2.62778 − 1.44734i
2.62778 + 1.44734i
1.80752 − 2.39434i
1.80752 + 2.39434i
0.931994 − 2.85156i
0.931994 + 2.85156i
−0.931994 − 2.85156i
−0.931994 + 2.85156i
−1.80752 − 2.39434i
−1.80752 + 2.39434i
−2.62778 − 1.44734i
−2.62778 + 1.44734i
−2.99317 − 0.202391i
−2.99317 + 0.202391i
0 −2.99317 − 0.202391i 0 8.22159i 0 −8.77407 0 8.91808 + 1.21158i 0
173.2 0 −2.99317 + 0.202391i 0 − 8.22159i 0 −8.77407 0 8.91808 − 1.21158i 0
173.3 0 −2.62778 − 1.44734i 0 − 1.83280i 0 5.97616 0 4.81043 + 7.60656i 0
173.4 0 −2.62778 + 1.44734i 0 1.83280i 0 5.97616 0 4.81043 − 7.60656i 0
173.5 0 −1.80752 − 2.39434i 0 − 0.122583i 0 −5.91845 0 −2.46573 + 8.65564i 0
173.6 0 −1.80752 + 2.39434i 0 0.122583i 0 −5.91845 0 −2.46573 − 8.65564i 0
173.7 0 −0.931994 − 2.85156i 0 8.66206i 0 11.7164 0 −7.26278 + 5.31527i 0
173.8 0 −0.931994 + 2.85156i 0 − 8.66206i 0 11.7164 0 −7.26278 − 5.31527i 0
173.9 0 0.931994 − 2.85156i 0 − 8.66206i 0 11.7164 0 −7.26278 − 5.31527i 0
173.10 0 0.931994 + 2.85156i 0 8.66206i 0 11.7164 0 −7.26278 + 5.31527i 0
173.11 0 1.80752 − 2.39434i 0 0.122583i 0 −5.91845 0 −2.46573 − 8.65564i 0
173.12 0 1.80752 + 2.39434i 0 − 0.122583i 0 −5.91845 0 −2.46573 + 8.65564i 0
173.13 0 2.62778 − 1.44734i 0 1.83280i 0 5.97616 0 4.81043 − 7.60656i 0
173.14 0 2.62778 + 1.44734i 0 − 1.83280i 0 5.97616 0 4.81043 + 7.60656i 0
173.15 0 2.99317 − 0.202391i 0 − 8.22159i 0 −8.77407 0 8.91808 − 1.21158i 0
173.16 0 2.99317 + 0.202391i 0 8.22159i 0 −8.77407 0 8.91808 + 1.21158i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 173.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
29.b even 2 1 inner
87.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 348.3.e.b ✓ 16
3.b odd 2 1 inner 348.3.e.b ✓ 16
29.b even 2 1 inner 348.3.e.b ✓ 16
87.d odd 2 1 inner 348.3.e.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.3.e.b ✓ 16 1.a even 1 1 trivial
348.3.e.b ✓ 16 3.b odd 2 1 inner
348.3.e.b ✓ 16 29.b even 2 1 inner
348.3.e.b ✓ 16 87.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 146T_{5}^{6} + 5553T_{5}^{4} + 17120T_{5}^{2} + 256 \) acting on \(S_{3}^{\mathrm{new}}(348, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 8 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( (T^{8} + 146 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 3 T^{3} + \cdots + 3636)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} - 645 T^{6} + \cdots + 427716)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 29 T^{3} + \cdots - 1402)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} - 681 T^{6} + \cdots + 106750224)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1780 T^{6} + \cdots + 4729037824)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2328 T^{6} + \cdots + 4698279936)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 25\!\cdots\!21 \) Copy content Toggle raw display
$31$ \( (T^{8} + 3976 T^{6} + \cdots + 293920116736)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 3368428867584)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 3644 T^{6} + \cdots + 108683627584)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 10365515080704)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 3317 T^{6} + \cdots + 942612804)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 2611817996544)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 5768 T^{6} + \cdots + 82465460224)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 30953110856704)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 11 T^{3} + \cdots + 2107448)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 22497642676224)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 17720193335296)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 63920152920064)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 387735796056064)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 82\!\cdots\!56)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 83\!\cdots\!56)^{2} \) Copy content Toggle raw display
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