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Properties

Label 348.3.d.b
Level $348$
Weight $3$
Character orbit 348.d
Analytic conductor $9.482$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [348,3,Mod(233,348)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("348.233");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 348.d (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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 3 x^{12} + 38 x^{11} - 130 x^{10} + 222 x^{9} + 1110 x^{8} - 4356 x^{7} + \cdots + 4782969 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{11} q^{5} + (\beta_{4} + 2) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{11} q^{5} + (\beta_{4} + 2) q^{7} + \beta_{2} q^{9} + ( - \beta_{9} - \beta_{8}) q^{11} + ( - \beta_{6} + \beta_1 + 2) q^{13} + (\beta_{13} + \beta_{12} + \beta_{11} + \cdots - 2) q^{15}+ \cdots + (6 \beta_{13} + 6 \beta_{12} + \cdots - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{3} + 26 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{3} + 26 q^{7} - 2 q^{9} + 34 q^{13} - 26 q^{15} - 96 q^{19} + 12 q^{21} - 70 q^{25} + 124 q^{27} - 84 q^{31} - 18 q^{33} + 8 q^{37} - 86 q^{39} + 152 q^{43} - 112 q^{45} + 88 q^{49} + 46 q^{51} + 120 q^{55} + 84 q^{57} - 28 q^{61} + 22 q^{63} - 318 q^{67} - 192 q^{69} + 68 q^{73} + 68 q^{75} + 632 q^{79} + 202 q^{81} - 324 q^{85} - 226 q^{91} - 464 q^{93} - 104 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 3 x^{12} + 38 x^{11} - 130 x^{10} + 222 x^{9} + 1110 x^{8} - 4356 x^{7} + \cdots + 4782969 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1300 \nu^{13} - 7181 \nu^{12} + 230142 \nu^{11} - 1189468 \nu^{10} - 4329736 \nu^{9} + \cdots - 51653407995 ) / 9017490888 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1667 \nu^{13} + 6647 \nu^{12} - 54651 \nu^{11} + 51655 \nu^{10} + 42769 \nu^{9} + \cdots - 1729309014 ) / 1127186361 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15011 \nu^{13} - 15653 \nu^{12} - 43998 \nu^{11} - 1050964 \nu^{10} + 3650342 \nu^{9} + \cdots - 10586836161 ) / 9017490888 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6892 \nu^{13} + 64643 \nu^{12} - 49818 \nu^{11} + 326542 \nu^{10} + 283420 \nu^{9} + \cdots - 17527455621 ) / 4508745444 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17948 \nu^{13} - 118957 \nu^{12} - 304266 \nu^{11} + 702004 \nu^{10} - 2013200 \nu^{9} + \cdots + 5044969413 ) / 9017490888 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2677 \nu^{13} + 10772 \nu^{12} - 15060 \nu^{11} - 144116 \nu^{10} + 372130 \nu^{9} + \cdots + 1593260118 ) / 1288212984 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21931 \nu^{13} + 120766 \nu^{12} + 47172 \nu^{11} - 515812 \nu^{10} + 1131326 \nu^{9} + \cdots + 60378074892 ) / 9017490888 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2615 \nu^{13} - 2513 \nu^{12} + 44982 \nu^{11} - 259732 \nu^{10} + 85082 \nu^{9} + \cdots + 1527892875 ) / 1001943432 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 24941 \nu^{13} + 3046 \nu^{12} + 259500 \nu^{11} - 1251724 \nu^{10} + 1778462 \nu^{9} + \cdots + 21030183252 ) / 9017490888 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - \nu^{13} + 2 \nu^{12} - 3 \nu^{11} - 38 \nu^{10} + 130 \nu^{9} - 222 \nu^{8} - 1110 \nu^{7} + \cdots + 1062882 ) / 177147 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5864 \nu^{13} + 6203 \nu^{12} - 1530 \nu^{11} + 166600 \nu^{10} + 46552 \nu^{9} + \cdots - 2483778087 ) / 1001943432 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + 3\beta_{11} - 2\beta_{10} + \beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{3} + \beta_{2} - \beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{13} + \beta_{12} + 3 \beta_{11} - 5 \beta_{10} - 4 \beta_{9} + 7 \beta_{8} - 2 \beta_{7} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{13} - 7 \beta_{12} + 24 \beta_{11} + 5 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} + 5 \beta_{7} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 24 \beta_{13} + 17 \beta_{12} - 27 \beta_{11} + 20 \beta_{10} - 17 \beta_{9} + 74 \beta_{8} + 26 \beta_{7} + \cdots - 269 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{13} - 48 \beta_{12} + 105 \beta_{11} - 9 \beta_{10} + 86 \beta_{9} - 15 \beta_{8} + \cdots + 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 153 \beta_{13} - 252 \beta_{12} - 60 \beta_{11} - 303 \beta_{10} + 135 \beta_{9} + 585 \beta_{8} + \cdots - 627 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 135 \beta_{13} - 1277 \beta_{12} + 1254 \beta_{11} + 400 \beta_{10} + 46 \beta_{9} - 293 \beta_{8} + \cdots + 2976 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 906 \beta_{13} + 718 \beta_{12} - 1860 \beta_{11} + 136 \beta_{10} + 26 \beta_{9} - 4460 \beta_{8} + \cdots - 10891 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8634 \beta_{13} + 6965 \beta_{12} - 2856 \beta_{11} + 2000 \beta_{10} - 1224 \beta_{9} - 11332 \beta_{8} + \cdots + 10742 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6546 \beta_{13} + 230 \beta_{12} - 28296 \beta_{11} - 2872 \beta_{10} + 21964 \beta_{9} + \cdots + 28489 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 80898 \beta_{13} - 60882 \beta_{12} - 44562 \beta_{11} + 49884 \beta_{10} + 49778 \beta_{9} + \cdots - 283286 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
233.1
2.80278 + 1.06978i
2.80278 − 1.06978i
2.29972 + 1.92647i
2.29972 − 1.92647i
1.29474 + 2.70622i
1.29474 − 2.70622i
0.438514 + 2.96778i
0.438514 − 2.96778i
−0.193522 + 2.99375i
−0.193522 − 2.99375i
−2.69418 + 1.31961i
−2.69418 − 1.31961i
−2.94805 + 0.555893i
−2.94805 − 0.555893i
0 −2.80278 − 1.06978i 0 − 7.37221i 0 9.09982 0 6.71116 + 5.99670i 0
233.2 0 −2.80278 + 1.06978i 0 7.37221i 0 9.09982 0 6.71116 − 5.99670i 0
233.3 0 −2.29972 − 1.92647i 0 − 1.00409i 0 −9.27576 0 1.57741 + 8.86069i 0
233.4 0 −2.29972 + 1.92647i 0 1.00409i 0 −9.27576 0 1.57741 − 8.86069i 0
233.5 0 −1.29474 − 2.70622i 0 6.62602i 0 0.771037 0 −5.64729 + 7.00772i 0
233.6 0 −1.29474 + 2.70622i 0 − 6.62602i 0 0.771037 0 −5.64729 − 7.00772i 0
233.7 0 −0.438514 − 2.96778i 0 2.92821i 0 8.87217 0 −8.61541 + 2.60282i 0
233.8 0 −0.438514 + 2.96778i 0 − 2.92821i 0 8.87217 0 −8.61541 − 2.60282i 0
233.9 0 0.193522 − 2.99375i 0 − 7.16184i 0 −2.66409 0 −8.92510 − 1.15871i 0
233.10 0 0.193522 + 2.99375i 0 7.16184i 0 −2.66409 0 −8.92510 + 1.15871i 0
233.11 0 2.69418 − 1.31961i 0 − 3.78993i 0 10.5999 0 5.51726 − 7.11054i 0
233.12 0 2.69418 + 1.31961i 0 3.78993i 0 10.5999 0 5.51726 + 7.11054i 0
233.13 0 2.94805 − 0.555893i 0 − 6.04221i 0 −4.40312 0 8.38197 − 3.27760i 0
233.14 0 2.94805 + 0.555893i 0 6.04221i 0 −4.40312 0 8.38197 + 3.27760i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 233.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b 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.d.b ✓ 14
3.b odd 2 1 inner 348.3.d.b ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.3.d.b ✓ 14 1.a even 1 1 trivial
348.3.d.b ✓ 14 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 210 T_{5}^{12} + 17487 T_{5}^{10} + 729392 T_{5}^{8} + 15799023 T_{5}^{6} + \cdots + 554824404 \) acting on \(S_{3}^{\mathrm{new}}(348, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 2 T^{13} + \cdots + 4782969 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 554824404 \) Copy content Toggle raw display
$7$ \( (T^{7} - 13 T^{6} + \cdots + 71796)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 3945417984 \) Copy content Toggle raw display
$13$ \( (T^{7} - 17 T^{6} + \cdots + 7449408)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{7} + 48 T^{6} + \cdots + 18349776)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} + 29)^{7} \) Copy content Toggle raw display
$31$ \( (T^{7} + 42 T^{6} + \cdots - 7747824960)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} - 4 T^{6} + \cdots - 23670028992)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 61\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{7} - 76 T^{6} + \cdots - 688063566)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 43\!\cdots\!69 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{7} + 14 T^{6} + \cdots - 229904441376)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} + 159 T^{6} + \cdots + 257023570176)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 84\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{7} - 34 T^{6} + \cdots + 460907176320)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 15905395974168)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 56\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 18575417228544)^{2} \) Copy content Toggle raw display
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