Properties

Label 4368.2.h.s
Level $4368$
Weight $2$
Character orbit 4368.h
Analytic conductor $34.879$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \( x^{10} + 21x^{8} + 124x^{6} + 212x^{4} + 116x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 2184)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( - \beta_{6} - \beta_1) q^{5} - \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + ( - \beta_{6} - \beta_1) q^{5} - \beta_1 q^{7} + q^{9} + ( - \beta_{6} + \beta_{4} + \beta_{3} - \beta_1) q^{11} + ( - \beta_{5} + \beta_{3}) q^{13} + (\beta_{6} + \beta_1) q^{15} + ( - \beta_{5} + \beta_{2} - 2) q^{17} + ( - \beta_{9} + 2 \beta_1) q^{19} + \beta_1 q^{21} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{23} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - 1) q^{25} - q^{27} + ( - \beta_{5} + \beta_{2}) q^{29} + (\beta_{6} - \beta_{4} - \beta_{3} + \beta_1) q^{33} + ( - \beta_{2} - 1) q^{35} + ( - \beta_{8} - 2 \beta_{6}) q^{37} + (\beta_{5} - \beta_{3}) q^{39} + (\beta_{8} + \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - \beta_1) q^{41} + (\beta_{7} - 2 \beta_{2}) q^{43} + ( - \beta_{6} - \beta_1) q^{45} + (\beta_{8} - \beta_{6} + \beta_{4} + \beta_{3} - \beta_1) q^{47} - q^{49} + (\beta_{5} - \beta_{2} + 2) q^{51} + (\beta_{7} - \beta_{5} - \beta_{2}) q^{53} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} - 2) q^{55} + (\beta_{9} - 2 \beta_1) q^{57} + (\beta_{9} + 3 \beta_{6} - \beta_{4} - \beta_{3} - \beta_1) q^{59} + (\beta_{7} - \beta_{4} + \beta_{3}) q^{61} - \beta_1 q^{63} + ( - \beta_{9} + \beta_{8} + 3 \beta_{6} - \beta_{4} - \beta_{3} + \beta_1 + 2) q^{65} + (\beta_{9} + \beta_{8} + 4 \beta_{6}) q^{67} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 2) q^{69} + (\beta_{9} + 2 \beta_{8} + 3 \beta_{6} - \beta_{4} - \beta_{3} + \beta_1) q^{71} + (\beta_{9} - \beta_{4} - \beta_{3} + 2 \beta_1) q^{73} + (\beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} + 1) q^{75} + ( - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{77} + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} - 4) q^{79} + q^{81} + ( - \beta_{9} - 2 \beta_{8} - 5 \beta_{6} + \beta_{4} + \beta_{3} - \beta_1) q^{83} + ( - \beta_{9} + 2 \beta_{6}) q^{85} + (\beta_{5} - \beta_{2}) q^{87} + (\beta_{8} + 5 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - \beta_1) q^{89} + (\beta_{8} + \beta_{6} - \beta_{4}) q^{91} + ( - \beta_{7} + 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 4) q^{95} + (2 \beta_{8} - \beta_{4} - \beta_{3} - 4 \beta_1) q^{97} + ( - \beta_{6} + \beta_{4} + \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{3} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{3} + 10 q^{9} - 22 q^{17} + 18 q^{23} - 8 q^{25} - 10 q^{27} - 2 q^{29} - 10 q^{35} + 2 q^{43} - 10 q^{49} + 22 q^{51} - 18 q^{55} + 6 q^{61} + 20 q^{65} - 18 q^{69} + 8 q^{75} - 6 q^{77} - 40 q^{79} + 10 q^{81} + 2 q^{87} + 2 q^{91} + 38 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 21x^{8} + 124x^{6} + 212x^{4} + 116x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{9} - 89\nu^{7} - 296\nu^{5} + 632\nu^{3} + 1004\nu ) / 392 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{8} - 89\nu^{6} - 296\nu^{4} + 632\nu^{2} + 808 ) / 196 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{8} - 81\nu^{6} - 423\nu^{4} - 396\nu^{2} + 98\nu - 20 ) / 98 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{8} + 81\nu^{6} + 423\nu^{4} + 396\nu^{2} + 98\nu + 20 ) / 98 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\nu^{8} + 243\nu^{6} + 1318\nu^{4} + 1727\nu^{2} + 501 ) / 49 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27\nu^{9} + 559\nu^{7} + 3186\nu^{5} + 4878\nu^{3} + 2340\nu ) / 196 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -65\nu^{8} - 1353\nu^{6} - 7768\nu^{4} - 11580\nu^{2} - 3020 ) / 196 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 65\nu^{9} + 1353\nu^{7} + 7768\nu^{5} + 11580\nu^{3} + 2628\nu ) / 392 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -289\nu^{9} - 5889\nu^{7} - 32240\nu^{5} - 42380\nu^{3} - 11588\nu ) / 392 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{5} + 2\beta_{4} - 2\beta_{3} + 3\beta_{2} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} - \beta_{8} + 6\beta_{6} - 10\beta_{4} - 10\beta_{3} - 6\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11\beta_{7} - 9\beta_{5} - 28\beta_{4} + 28\beta_{3} - 33\beta_{2} + 70 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -19\beta_{9} + 3\beta_{8} - 94\beta_{6} + 114\beta_{4} + 114\beta_{3} + 122\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 117\beta_{7} + 79\beta_{5} + 358\beta_{4} - 358\beta_{3} + 383\beta_{2} - 730 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 279\beta_{9} + 45\beta_{8} + 1274\beta_{6} - 1350\beta_{4} - 1350\beta_{3} - 1782\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1305\beta_{7} - 747\beta_{5} - 4462\beta_{4} + 4462\beta_{3} - 4563\beta_{2} + 8162 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -3715\beta_{9} - 1105\beta_{8} - 16354\beta_{6} + 16218\beta_{4} + 16218\beta_{3} + 23582\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times\).

\(n\) \(1093\) \(1249\) \(1457\) \(2017\) \(3823\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
0.455234i
0.812474i
1.21919i
2.54035i
3.49183i
3.49183i
2.54035i
1.21919i
0.812474i
0.455234i
0 −1.00000 0 4.39335i 0 1.00000i 0 1.00000 0
337.2 0 −1.00000 0 2.46162i 0 1.00000i 0 1.00000 0
337.3 0 −1.00000 0 1.64044i 0 1.00000i 0 1.00000 0
337.4 0 −1.00000 0 0.787293i 0 1.00000i 0 1.00000 0
337.5 0 −1.00000 0 0.572766i 0 1.00000i 0 1.00000 0
337.6 0 −1.00000 0 0.572766i 0 1.00000i 0 1.00000 0
337.7 0 −1.00000 0 0.787293i 0 1.00000i 0 1.00000 0
337.8 0 −1.00000 0 1.64044i 0 1.00000i 0 1.00000 0
337.9 0 −1.00000 0 2.46162i 0 1.00000i 0 1.00000 0
337.10 0 −1.00000 0 4.39335i 0 1.00000i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4368.2.h.s 10
4.b odd 2 1 2184.2.h.g 10
13.b even 2 1 inner 4368.2.h.s 10
52.b odd 2 1 2184.2.h.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2184.2.h.g 10 4.b odd 2 1
2184.2.h.g 10 52.b odd 2 1
4368.2.h.s 10 1.a even 1 1 trivial
4368.2.h.s 10 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4368, [\chi])\):

\( T_{5}^{10} + 29T_{5}^{8} + 212T_{5}^{6} + 496T_{5}^{4} + 336T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{10} + 73T_{11}^{8} + 1576T_{11}^{6} + 11200T_{11}^{4} + 12944T_{11}^{2} + 4096 \) Copy content Toggle raw display
\( T_{17}^{5} + 11T_{17}^{4} + 18T_{17}^{3} - 96T_{17}^{2} - 128T_{17} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 29 T^{8} + 212 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{10} + 73 T^{8} + 1576 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{10} - 23 T^{8} + 64 T^{7} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( (T^{5} + 11 T^{4} + 18 T^{3} - 96 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 177 T^{8} + \cdots + 39337984 \) Copy content Toggle raw display
$23$ \( (T^{5} - 9 T^{4} - 8 T^{3} + 96 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + T^{4} - 30 T^{3} - 20 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} \) Copy content Toggle raw display
$37$ \( T^{10} + 109 T^{8} + 3284 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( T^{10} + 256 T^{8} + 19072 T^{6} + \cdots + 6553600 \) Copy content Toggle raw display
$43$ \( (T^{5} - T^{4} - 148 T^{3} + 160 T^{2} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 196 T^{8} + \cdots + 16777216 \) Copy content Toggle raw display
$53$ \( (T^{5} - 160 T^{3} + 304 T^{2} + \cdots - 9920)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 316 T^{8} + 29984 T^{6} + \cdots + 2408704 \) Copy content Toggle raw display
$61$ \( (T^{5} - 3 T^{4} - 110 T^{3} + 316 T^{2} + \cdots - 4192)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 420 T^{8} + 53632 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$71$ \( T^{10} + 316 T^{8} + \cdots + 203689984 \) Copy content Toggle raw display
$73$ \( T^{10} + 277 T^{8} + \cdots + 50922496 \) Copy content Toggle raw display
$79$ \( (T^{5} + 20 T^{4} - 8 T^{3} - 1168 T^{2} + \cdots + 5120)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 492 T^{8} + \cdots + 1774768384 \) Copy content Toggle raw display
$89$ \( T^{10} + 448 T^{8} + \cdots + 10240000 \) Copy content Toggle raw display
$97$ \( T^{10} + 328 T^{8} + \cdots + 125440000 \) Copy content Toggle raw display
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