Properties

Label 245.6.a.h
Level $245$
Weight $6$
Character orbit 245.a
Self dual yes
Analytic conductor $39.294$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.2940358542\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - x^{5} - 109x^{4} + 41x^{3} + 2208x^{2} - 3204x + 560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{3} - \beta_1 - 3) q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + 25 q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 5 \beta_1 - 16) q^{6} + (2 \beta_{5} + 4 \beta_{3} + 5 \beta_1 - 23) q^{8} + ( - 3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 65) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{3} - \beta_1 - 3) q^{3} + (\beta_{2} - \beta_1 + 5) q^{4} + 25 q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 5 \beta_1 - 16) q^{6} + (2 \beta_{5} + 4 \beta_{3} + 5 \beta_1 - 23) q^{8} + ( - 3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 65) q^{9} + (25 \beta_1 - 25) q^{10} + (\beta_{5} + 4 \beta_{4} - 9 \beta_{3} - 4 \beta_{2} - 32 \beta_1 - 151) q^{11} + (9 \beta_{5} - 5 \beta_{4} - 13 \beta_{3} - 10 \beta_{2} + 27 \beta_1 - 64) q^{12} + (3 \beta_{5} - 10 \beta_{4} - 7 \beta_{3} - 4 \beta_1 + 29) q^{13} + (25 \beta_{3} - 25 \beta_1 - 75) q^{15} + ( - 8 \beta_{5} + 8 \beta_{4} - 8 \beta_{3} - 11 \beta_{2} + 19 \beta_1 + 69) q^{16} + (4 \beta_{5} - 34 \beta_{3} - 20 \beta_{2} - 2 \beta_1 - 256) q^{17} + ( - 7 \beta_{5} - 5 \beta_{4} - 31 \beta_{3} - 20 \beta_{2} - 56 \beta_1 + 47) q^{18} + ( - 3 \beta_{5} + 8 \beta_{4} - 43 \beta_{3} + 8 \beta_{2} + 230 \beta_1 - 67) q^{19} + (25 \beta_{2} - 25 \beta_1 + 125) q^{20} + (15 \beta_{5} - 15 \beta_{4} + 39 \beta_{3} - 50 \beta_{2} - 308 \beta_1 - 1063) q^{22} + (53 \beta_{5} - 20 \beta_{4} - 34 \beta_{3} - 60 \beta_{2} + 49 \beta_1 - 646) q^{23} + ( - 13 \beta_{5} - 17 \beta_{4} - 73 \beta_{3} - 5 \beta_{2} - 47 \beta_1 + 1208) q^{24} + 625 q^{25} + ( - 39 \beta_{5} + 19 \beta_{4} - 119 \beta_{3} - 6 \beta_{2} + 210 \beta_1 - 495) q^{26} + (6 \beta_{5} + 60 \beta_{4} + 13 \beta_{3} + 60 \beta_{2} - 103 \beta_1 - 375) q^{27} + ( - 103 \beta_{5} + 38 \beta_{4} + 147 \beta_{3} + 112 \beta_{2} + \cdots + 240) q^{29}+ \cdots + (461 \beta_{5} + 524 \beta_{4} + 3201 \beta_{3} + 2532 \beta_{2} + \cdots - 32765) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{2} - 20 q^{3} + 31 q^{4} + 150 q^{5} - 96 q^{6} - 135 q^{8} + 378 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 5 q^{2} - 20 q^{3} + 31 q^{4} + 150 q^{5} - 96 q^{6} - 135 q^{8} + 378 q^{9} - 125 q^{10} - 924 q^{11} - 370 q^{12} + 150 q^{13} - 500 q^{15} + 435 q^{16} - 1540 q^{17} + 195 q^{18} - 92 q^{19} + 775 q^{20} - 6855 q^{22} - 3920 q^{23} + 7200 q^{24} + 3750 q^{25} - 2635 q^{26} - 2060 q^{27} + 1264 q^{29} - 2400 q^{30} + 7160 q^{31} + 9105 q^{32} - 4460 q^{33} - 2166 q^{34} - 26375 q^{36} - 14170 q^{37} + 46215 q^{38} - 15376 q^{39} - 3375 q^{40} - 4098 q^{41} - 24460 q^{43} - 27873 q^{44} + 9450 q^{45} + 6815 q^{46} - 42940 q^{47} - 11610 q^{48} - 3125 q^{50} - 42008 q^{51} + 36115 q^{52} - 2450 q^{53} - 19566 q^{54} - 23100 q^{55} - 97100 q^{57} - 36110 q^{58} + 64600 q^{59} - 9250 q^{60} - 73620 q^{61} - 111440 q^{62} - 157997 q^{64} + 3750 q^{65} + 139138 q^{66} - 142620 q^{67} - 124330 q^{68} - 17344 q^{69} - 154256 q^{71} - 117495 q^{72} + 5120 q^{73} + 2785 q^{74} - 12500 q^{75} - 7775 q^{76} - 214090 q^{78} - 222504 q^{79} + 10875 q^{80} - 43986 q^{81} - 31665 q^{82} - 179580 q^{83} - 38500 q^{85} - 207160 q^{86} + 209300 q^{87} - 45145 q^{88} - 41648 q^{89} + 4875 q^{90} - 292185 q^{92} - 198520 q^{93} + 333699 q^{94} - 2300 q^{95} - 61824 q^{96} + 73980 q^{97} - 190772 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 109x^{4} + 41x^{3} + 2208x^{2} - 3204x + 560 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 103\nu^{3} - 86\nu^{2} + 1744\nu - 704 ) / 48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 6\nu^{4} + 103\nu^{3} - 460\nu^{2} - 2380\nu + 5552 ) / 48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 115\nu^{3} + 50\nu^{2} - 2536\nu + 1736 ) / 24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + 4\beta_{3} + 3\beta_{2} + 69\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{4} + 8\beta_{3} + 91\beta_{2} + 197\beta _1 + 2468 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 206\beta_{5} + 460\beta_{3} + 395\beta_{2} + 5449\beta _1 + 6066 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.17253
−6.22733
0.203319
1.35048
4.12050
9.72556
−9.17253 −14.3258 52.1353 25.0000 131.404 0 −184.691 −37.7723 −229.313
1.2 −7.22733 15.9209 20.2343 25.0000 −115.066 0 85.0347 10.4759 −180.683
1.3 −0.796681 −10.5748 −31.3653 25.0000 8.42477 0 50.4819 −131.173 −19.9170
1.4 0.350479 21.5910 −31.8772 25.0000 7.56720 0 −22.3876 223.173 8.76197
1.5 3.12050 −27.8717 −22.2625 25.0000 −86.9736 0 −169.326 533.832 78.0125
1.6 8.72556 −4.73965 44.1354 25.0000 −41.3561 0 105.888 −220.536 218.139
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.6.a.h 6
7.b odd 2 1 245.6.a.i 6
7.d odd 6 2 35.6.e.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.e.a 12 7.d odd 6 2
245.6.a.h 6 1.a even 1 1 trivial
245.6.a.i 6 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{6} + 5T_{2}^{5} - 99T_{2}^{4} - 385T_{2}^{3} + 1682T_{2}^{2} + 900T_{2} - 504 \) Copy content Toggle raw display
\( T_{3}^{6} + 20T_{3}^{5} - 718T_{3}^{4} - 13100T_{3}^{3} + 87921T_{3}^{2} + 2078280T_{3} + 6879276 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 5 T^{5} - 99 T^{4} - 385 T^{3} + \cdots - 504 \) Copy content Toggle raw display
$3$ \( T^{6} + 20 T^{5} - 718 T^{4} + \cdots + 6879276 \) Copy content Toggle raw display
$5$ \( (T - 25)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 924 T^{5} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{6} - 150 T^{5} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{6} + 1540 T^{5} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{6} + 92 T^{5} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{6} + 3920 T^{5} + \cdots - 17\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( T^{6} - 1264 T^{5} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{6} - 7160 T^{5} + \cdots + 60\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{6} + 14170 T^{5} + \cdots + 68\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{6} + 4098 T^{5} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{6} + 24460 T^{5} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{6} + 42940 T^{5} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{6} + 2450 T^{5} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{6} - 64600 T^{5} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + 73620 T^{5} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{6} + 142620 T^{5} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{6} + 154256 T^{5} + \cdots + 99\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{6} - 5120 T^{5} + \cdots - 51\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{6} + 222504 T^{5} + \cdots + 71\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{6} + 179580 T^{5} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{6} + 41648 T^{5} + \cdots - 46\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{6} - 73980 T^{5} + \cdots + 99\!\cdots\!84 \) Copy content Toggle raw display
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