Properties

Label 4368.2.h.r
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: \( x^{10} + 20x^{8} + 138x^{6} + 364x^{4} + 249x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - q^{3} - \beta_{4} q^{5} - \beta_{7} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - \beta_{4} q^{5} - \beta_{7} q^{7} + q^{9} + ( - \beta_{9} + \beta_{3}) q^{11} + ( - \beta_{6} - \beta_{4} - \beta_{3}) q^{13} + \beta_{4} q^{15} + (\beta_{5} + \beta_1 + 1) q^{17} + (\beta_{8} + \beta_{7} + \beta_{6} + \beta_{3}) q^{19} + \beta_{7} q^{21} + (2 \beta_{8} - 2 \beta_{6} + 2 \beta_{5} + \beta_{2} + 1) q^{23} + \beta_{2} q^{25} - q^{27} + ( - \beta_{8} + \beta_{6} - 2 \beta_{5} + \beta_{2} - 5) q^{29} + ( - \beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3}) q^{31} + (\beta_{9} - \beta_{3}) q^{33} - \beta_{5} q^{35} + ( - \beta_{9} + \beta_{7} - \beta_{4}) q^{37} + (\beta_{6} + \beta_{4} + \beta_{3}) q^{39} + ( - 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3}) q^{41} + ( - \beta_{8} + \beta_{6} - \beta_{2} + 2 \beta_1 - 3) q^{43} - \beta_{4} q^{45} + (3 \beta_{9} + \beta_{8} + 5 \beta_{7} + \beta_{6} + 2 \beta_{4}) q^{47} - q^{49} + ( - \beta_{5} - \beta_1 - 1) q^{51} + ( - \beta_{8} + \beta_{6} + \beta_{5} - \beta_{2} - \beta_1) q^{53} + (3 \beta_{8} - 3 \beta_{6} + \beta_{5} + 2 \beta_{2} - \beta_1 + 3) q^{55} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{3}) q^{57} + (2 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} + \beta_{4} + 3 \beta_{3}) q^{59} + (\beta_{8} - \beta_{6} + \beta_{5} + 2 \beta_{2} + \beta_1 + 3) q^{61} - \beta_{7} q^{63} + (\beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 - 4) q^{65} + ( - 2 \beta_{9} + 2 \beta_{8} + 6 \beta_{7} + 2 \beta_{6}) q^{67} + ( - 2 \beta_{8} + 2 \beta_{6} - 2 \beta_{5} - \beta_{2} - 1) q^{69} + ( - 5 \beta_{7} + \beta_{4} + \beta_{3}) q^{71} + (3 \beta_{7} - 2 \beta_{4} + \beta_{3}) q^{73} - \beta_{2} q^{75} + (\beta_{2} + \beta_1) q^{77} + ( - \beta_{5} + \beta_{2} - 3 \beta_1) q^{79} + q^{81} + ( - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 4 \beta_{4}) q^{83} + ( - \beta_{9} - 2 \beta_{8} - 7 \beta_{7} - 2 \beta_{6} - 3 \beta_{4}) q^{85} + (\beta_{8} - \beta_{6} + 2 \beta_{5} - \beta_{2} + 5) q^{87} + (\beta_{9} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} - 4 \beta_{4}) q^{89} + ( - \beta_{8} - \beta_{5} - \beta_{2}) q^{91} + (\beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3}) q^{93} + (\beta_{8} - \beta_{6} + 3 \beta_{2} + 2 \beta_1 - 3) q^{95} + (\beta_{9} + 4 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + 3 \beta_{4} + 3 \beta_{3}) q^{97} + ( - \beta_{9} + \beta_{3}) 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 \( 10 q - 10 q^{3} + 10 q^{9} - 4 q^{13} + 10 q^{17} - 8 q^{23} - 2 q^{25} - 10 q^{27} - 44 q^{29} + 4 q^{39} - 20 q^{43} - 10 q^{49} - 10 q^{51} + 10 q^{53} + 2 q^{55} + 18 q^{61} - 30 q^{65} + 8 q^{69} + 2 q^{75} - 2 q^{77} - 2 q^{79} + 10 q^{81} + 44 q^{87} + 6 q^{91} - 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 20x^{8} + 138x^{6} + 364x^{4} + 249x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + 11\nu^{6} + 27\nu^{4} - 11\nu^{2} - 12 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 17\nu^{7} + 87\nu^{5} + 139\nu^{3} + 84\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{9} + 17\nu^{7} + 105\nu^{5} + 283\nu^{3} + 246\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} + 11\nu^{6} + 15\nu^{4} - 95\nu^{2} - 48 ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{8} + 14\nu^{6} + 57\nu^{4} + 58\nu^{2} + 6\nu + 12 ) / 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{9} - 17\nu^{7} - 96\nu^{5} - 193\nu^{3} - 75\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{8} - 14\nu^{6} - 57\nu^{4} - 58\nu^{2} + 6\nu - 12 ) / 6 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} + 26\nu^{7} + 213\nu^{5} + 598\nu^{3} + 264\nu ) / 18 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{8} + 2\beta_{7} - 5\beta_{6} + 2\beta_{4} + 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{5} + \beta_{2} - 7\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 31\beta_{8} - 16\beta_{7} + 31\beta_{6} - 12\beta_{4} - 20\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{8} + \beta_{6} + 10\beta_{5} - 14\beta_{2} + 47\beta _1 - 166 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4\beta_{9} - 199\beta_{8} + 122\beta_{7} - 199\beta_{6} + 66\beta_{4} + 170\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 11\beta_{8} - 11\beta_{6} - 83\beta_{5} + 139\beta_{2} - 317\beta _1 + 1119 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -68\beta_{9} + 1297\beta_{8} - 960\beta_{7} + 1297\beta_{6} - 356\beta_{4} - 1356\beta_{3} ) / 2 \) 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
2.28392i
2.56826i
0.444296i
0.854441i
2.69449i
2.69449i
0.854441i
0.444296i
2.56826i
2.28392i
0 −1.00000 0 2.89917i 0 1.00000i 0 1.00000 0
337.2 0 −1.00000 0 2.71697i 0 1.00000i 0 1.00000 0
337.3 0 −1.00000 0 2.39548i 0 1.00000i 0 1.00000 0
337.4 0 −1.00000 0 2.11295i 0 1.00000i 0 1.00000 0
337.5 0 −1.00000 0 0.100328i 0 1.00000i 0 1.00000 0
337.6 0 −1.00000 0 0.100328i 0 1.00000i 0 1.00000 0
337.7 0 −1.00000 0 2.11295i 0 1.00000i 0 1.00000 0
337.8 0 −1.00000 0 2.39548i 0 1.00000i 0 1.00000 0
337.9 0 −1.00000 0 2.71697i 0 1.00000i 0 1.00000 0
337.10 0 −1.00000 0 2.89917i 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.r 10
4.b odd 2 1 2184.2.h.f 10
13.b even 2 1 inner 4368.2.h.r 10
52.b odd 2 1 2184.2.h.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2184.2.h.f 10 4.b odd 2 1
2184.2.h.f 10 52.b odd 2 1
4368.2.h.r 10 1.a even 1 1 trivial
4368.2.h.r 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 \(S_{2}^{\mathrm{new}}(4368, [\chi])\):

\( T_{5}^{10} + 26T_{5}^{8} + 249T_{5}^{6} + 1040T_{5}^{4} + 1600T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{10} + 73T_{11}^{8} + 1976T_{11}^{6} + 24192T_{11}^{4} + 128528T_{11}^{2} + 215296 \) Copy content Toggle raw display
\( T_{17}^{5} - 5T_{17}^{4} - 26T_{17}^{3} + 104T_{17}^{2} + 96T_{17} - 384 \) 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
$5$ \( T^{10} + 26 T^{8} + 249 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{10} + 73 T^{8} + 1976 T^{6} + \cdots + 215296 \) Copy content Toggle raw display
$13$ \( T^{10} + 4 T^{9} + 33 T^{8} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( (T^{5} - 5 T^{4} - 26 T^{3} + 104 T^{2} + \cdots - 384)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 74 T^{8} + 1401 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( (T^{5} + 4 T^{4} - 87 T^{3} - 462 T^{2} + \cdots + 5272)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 22 T^{4} + 103 T^{3} - 538 T^{2} + \cdots - 216)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 161 T^{8} + 8256 T^{6} + \cdots + 4194304 \) Copy content Toggle raw display
$37$ \( T^{10} + 73 T^{8} + 1248 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$41$ \( T^{10} + 324 T^{8} + \cdots + 707134464 \) Copy content Toggle raw display
$43$ \( (T^{5} + 10 T^{4} - 75 T^{3} - 480 T^{2} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 493 T^{8} + \cdots + 259081216 \) Copy content Toggle raw display
$53$ \( (T^{5} - 5 T^{4} - 88 T^{3} - 72 T^{2} + \cdots + 2032)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 360 T^{8} + \cdots + 131974144 \) Copy content Toggle raw display
$61$ \( (T^{5} - 9 T^{4} - 58 T^{3} + 372 T^{2} + \cdots - 1312)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 676 T^{8} + \cdots + 17314349056 \) Copy content Toggle raw display
$71$ \( T^{10} + 204 T^{8} + 12944 T^{6} + \cdots + 9216 \) Copy content Toggle raw display
$73$ \( T^{10} + 186 T^{8} + 12809 T^{6} + \cdots + 2027776 \) Copy content Toggle raw display
$79$ \( (T^{5} + T^{4} - 256 T^{3} - 680 T^{2} + \cdots + 35136)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 713 T^{8} + \cdots + 1802511936 \) Copy content Toggle raw display
$89$ \( T^{10} + 473 T^{8} + \cdots + 43243776 \) Copy content Toggle raw display
$97$ \( T^{10} + 757 T^{8} + \cdots + 1109689344 \) Copy content Toggle raw display
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