Properties

Label 245.4.e.k
Level $245$
Weight $4$
Character orbit 245.e
Analytic conductor $14.455$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{11})\)
Defining polynomial: \( x^{4} + 11x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{2} - 5 \beta_{2} q^{3} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{4} + (5 \beta_{2} + 5) q^{5} + ( - 5 \beta_{3} - 5) q^{6} + ( - 2 \beta_{3} + 18) q^{8} + (2 \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 1) q^{2} - 5 \beta_{2} q^{3} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{4} + (5 \beta_{2} + 5) q^{5} + ( - 5 \beta_{3} - 5) q^{6} + ( - 2 \beta_{3} + 18) q^{8} + (2 \beta_{2} + 2) q^{9} + (5 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{10} + (4 \beta_{3} + 33 \beta_{2} + 4 \beta_1) q^{11} + (20 \beta_{2} - 10 \beta_1 + 20) q^{12} + ( - 20 \beta_{3} - 5) q^{13} + 25 q^{15} + (36 \beta_{2} + 36) q^{16} + ( - 20 \beta_{3} - 35 \beta_{2} - 20 \beta_1) q^{17} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{18} + (70 \beta_{2} - 20 \beta_1 + 70) q^{19} + ( - 10 \beta_{3} - 20) q^{20} + (29 \beta_{3} - 11) q^{22} + (8 \beta_{2} - 28 \beta_1 + 8) q^{23} + ( - 10 \beta_{3} - 90 \beta_{2} - 10 \beta_1) q^{24} + 25 \beta_{2} q^{25} + (225 \beta_{2} - 25 \beta_1 + 225) q^{26} + 145 q^{27} + ( - 48 \beta_{3} - 129) q^{29} + ( - 25 \beta_{2} + 25 \beta_1 - 25) q^{30} + ( - 60 \beta_{3} + 10 \beta_{2} - 60 \beta_1) q^{31} + (20 \beta_{3} - 180 \beta_{2} + 20 \beta_1) q^{32} + (165 \beta_{2} + 20 \beta_1 + 165) q^{33} + ( - 15 \beta_{3} + 185) q^{34} + ( - 4 \beta_{3} - 8) q^{36} + ( - 164 \beta_{2} + 44 \beta_1 - 164) q^{37} + (90 \beta_{3} - 290 \beta_{2} + 90 \beta_1) q^{38} + ( - 100 \beta_{3} + 25 \beta_{2} - 100 \beta_1) q^{39} + (90 \beta_{2} + 10 \beta_1 + 90) q^{40} + (100 \beta_{3} - 150) q^{41} + (12 \beta_{3} - 58) q^{43} + ( - 44 \beta_{2} + 50 \beta_1 - 44) q^{44} + 10 \beta_{2} q^{45} + (36 \beta_{3} - 316 \beta_{2} + 36 \beta_1) q^{46} + ( - 15 \beta_{2} + 40 \beta_1 - 15) q^{47} + 180 q^{48} + (25 \beta_{3} + 25) q^{50} + ( - 175 \beta_{2} - 100 \beta_1 - 175) q^{51} + (90 \beta_{3} - 460 \beta_{2} + 90 \beta_1) q^{52} + (120 \beta_{3} + 270 \beta_{2} + 120 \beta_1) q^{53} + ( - 145 \beta_{2} + 145 \beta_1 - 145) q^{54} + (20 \beta_{3} - 165) q^{55} + (100 \beta_{3} + 350) q^{57} + (657 \beta_{2} - 177 \beta_1 + 657) q^{58} + (40 \beta_{3} - 190 \beta_{2} + 40 \beta_1) q^{59} + ( - 50 \beta_{3} + 100 \beta_{2} - 50 \beta_1) q^{60} + (540 \beta_{2} + 60 \beta_1 + 540) q^{61} + (70 \beta_{3} + 670) q^{62} + ( - 200 \beta_{3} - 112) q^{64} + ( - 25 \beta_{2} + 100 \beta_1 - 25) q^{65} + (145 \beta_{3} + 55 \beta_{2} + 145 \beta_1) q^{66} + (96 \beta_{3} + 234 \beta_{2} + 96 \beta_1) q^{67} + ( - 300 \beta_{2} + 10 \beta_1 - 300) q^{68} + (140 \beta_{3} + 40) q^{69} + (64 \beta_{3} - 528) q^{71} + (36 \beta_{2} + 4 \beta_1 + 36) q^{72} + ( - 200 \beta_{3} + 430 \beta_{2} - 200 \beta_1) q^{73} + ( - 208 \beta_{3} + 648 \beta_{2} - 208 \beta_1) q^{74} + (125 \beta_{2} + 125) q^{75} + ( - 220 \beta_{3} - 720) q^{76} + (125 \beta_{3} + 1125) q^{78} + ( - 79 \beta_{2} + 348 \beta_1 - 79) q^{79} + 180 \beta_{2} q^{80} - 671 \beta_{2} q^{81} + ( - 950 \beta_{2} - 50 \beta_1 - 950) q^{82} + (200 \beta_{3} - 20) q^{83} + ( - 100 \beta_{3} + 175) q^{85} + ( - 74 \beta_{2} - 46 \beta_1 - 74) q^{86} + ( - 240 \beta_{3} + 645 \beta_{2} - 240 \beta_1) q^{87} + (138 \beta_{3} + 682 \beta_{2} + 138 \beta_1) q^{88} + ( - 120 \beta_{2} - 380 \beta_1 - 120) q^{89} + (10 \beta_{3} + 10) q^{90} + ( - 128 \beta_{3} - 648) q^{92} + (50 \beta_{2} - 300 \beta_1 + 50) q^{93} + ( - 55 \beta_{3} + 455 \beta_{2} - 55 \beta_1) q^{94} + ( - 100 \beta_{3} + 350 \beta_{2} - 100 \beta_1) q^{95} + ( - 900 \beta_{2} + 100 \beta_1 - 900) q^{96} + ( - 180 \beta_{3} - 815) q^{97} + (8 \beta_{3} - 66) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 10 q^{3} - 8 q^{4} + 10 q^{5} - 20 q^{6} + 72 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 10 q^{3} - 8 q^{4} + 10 q^{5} - 20 q^{6} + 72 q^{8} + 4 q^{9} + 10 q^{10} - 66 q^{11} + 40 q^{12} - 20 q^{13} + 100 q^{15} + 72 q^{16} + 70 q^{17} + 4 q^{18} + 140 q^{19} - 80 q^{20} - 44 q^{22} + 16 q^{23} + 180 q^{24} - 50 q^{25} + 450 q^{26} + 580 q^{27} - 516 q^{29} - 50 q^{30} - 20 q^{31} + 360 q^{32} + 330 q^{33} + 740 q^{34} - 32 q^{36} - 328 q^{37} + 580 q^{38} - 50 q^{39} + 180 q^{40} - 600 q^{41} - 232 q^{43} - 88 q^{44} - 20 q^{45} + 632 q^{46} - 30 q^{47} + 720 q^{48} + 100 q^{50} - 350 q^{51} + 920 q^{52} - 540 q^{53} - 290 q^{54} - 660 q^{55} + 1400 q^{57} + 1314 q^{58} + 380 q^{59} - 200 q^{60} + 1080 q^{61} + 2680 q^{62} - 448 q^{64} - 50 q^{65} - 110 q^{66} - 468 q^{67} - 600 q^{68} + 160 q^{69} - 2112 q^{71} + 72 q^{72} - 860 q^{73} - 1296 q^{74} + 250 q^{75} - 2880 q^{76} + 4500 q^{78} - 158 q^{79} - 360 q^{80} + 1342 q^{81} - 1900 q^{82} - 80 q^{83} + 700 q^{85} - 148 q^{86} - 1290 q^{87} - 1364 q^{88} - 240 q^{89} + 40 q^{90} - 2592 q^{92} + 100 q^{93} - 910 q^{94} - 700 q^{95} - 1800 q^{96} - 3260 q^{97} - 264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 11x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 11\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 11\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−1.65831 2.87228i
1.65831 + 2.87228i
−1.65831 + 2.87228i
1.65831 2.87228i
−2.15831 3.73831i 2.50000 4.33013i −5.31662 + 9.20866i 2.50000 + 4.33013i −21.5831 0 11.3668 1.00000 + 1.73205i 10.7916 18.6915i
116.2 1.15831 + 2.00626i 2.50000 4.33013i 1.31662 2.28046i 2.50000 + 4.33013i 11.5831 0 24.6332 1.00000 + 1.73205i −5.79156 + 10.0313i
226.1 −2.15831 + 3.73831i 2.50000 + 4.33013i −5.31662 9.20866i 2.50000 4.33013i −21.5831 0 11.3668 1.00000 1.73205i 10.7916 + 18.6915i
226.2 1.15831 2.00626i 2.50000 + 4.33013i 1.31662 + 2.28046i 2.50000 4.33013i 11.5831 0 24.6332 1.00000 1.73205i −5.79156 10.0313i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.e.k 4
7.b odd 2 1 245.4.e.j 4
7.c even 3 1 245.4.a.i 2
7.c even 3 1 inner 245.4.e.k 4
7.d odd 6 1 245.4.a.j yes 2
7.d odd 6 1 245.4.e.j 4
21.g even 6 1 2205.4.a.w 2
21.h odd 6 1 2205.4.a.x 2
35.i odd 6 1 1225.4.a.p 2
35.j even 6 1 1225.4.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.i 2 7.c even 3 1
245.4.a.j yes 2 7.d odd 6 1
245.4.e.j 4 7.b odd 2 1
245.4.e.j 4 7.d odd 6 1
245.4.e.k 4 1.a even 1 1 trivial
245.4.e.k 4 7.c even 3 1 inner
1225.4.a.p 2 35.i odd 6 1
1225.4.a.q 2 35.j even 6 1
2205.4.a.w 2 21.g even 6 1
2205.4.a.x 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{4} + 2T_{2}^{3} + 14T_{2}^{2} - 20T_{2} + 100 \) Copy content Toggle raw display
\( T_{3}^{2} - 5T_{3} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 14 T^{2} - 20 T + 100 \) Copy content Toggle raw display
$3$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 66 T^{3} + 3443 T^{2} + \cdots + 833569 \) Copy content Toggle raw display
$13$ \( (T^{2} + 10 T - 4375)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 70 T^{3} + 8075 T^{2} + \cdots + 10080625 \) Copy content Toggle raw display
$19$ \( T^{4} - 140 T^{3} + 19100 T^{2} + \cdots + 250000 \) Copy content Toggle raw display
$23$ \( T^{4} - 16 T^{3} + 8816 T^{2} + \cdots + 73273600 \) Copy content Toggle raw display
$29$ \( (T^{2} + 258 T - 8703)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 20 T^{3} + \cdots + 1560250000 \) Copy content Toggle raw display
$37$ \( T^{4} + 328 T^{3} + \cdots + 31360000 \) Copy content Toggle raw display
$41$ \( (T^{2} + 300 T - 87500)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 116 T + 1780)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 30 T^{3} + \cdots + 301890625 \) Copy content Toggle raw display
$53$ \( T^{4} + 540 T^{3} + \cdots + 7310250000 \) Copy content Toggle raw display
$59$ \( T^{4} - 380 T^{3} + \cdots + 342250000 \) Copy content Toggle raw display
$61$ \( T^{4} - 1080 T^{3} + \cdots + 63504000000 \) Copy content Toggle raw display
$67$ \( T^{4} + 468 T^{3} + \cdots + 2173424400 \) Copy content Toggle raw display
$71$ \( (T^{2} + 1056 T + 233728)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 860 T^{3} + \cdots + 65076010000 \) Copy content Toggle raw display
$79$ \( T^{4} + 158 T^{3} + \cdots + 1758018765409 \) Copy content Toggle raw display
$83$ \( (T^{2} + 40 T - 439600)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 240 T^{3} + \cdots + 2477476000000 \) Copy content Toggle raw display
$97$ \( (T^{2} + 1630 T + 307825)^{2} \) Copy content Toggle raw display
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