Properties

Label 245.4.e.l
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: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{2} + ( - \beta_{2} - 3 \beta_1 - 1) q^{3} + ( - 3 \beta_{2} + 6 \beta_1 - 3) q^{4} - 5 \beta_{2} q^{5} + (8 \beta_{3} + 3) q^{6} + ( - 13 \beta_{3} + 3) q^{8} + (6 \beta_{3} - 8 \beta_{2} + 6 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{2} + ( - \beta_{2} - 3 \beta_1 - 1) q^{3} + ( - 3 \beta_{2} + 6 \beta_1 - 3) q^{4} - 5 \beta_{2} q^{5} + (8 \beta_{3} + 3) q^{6} + ( - 13 \beta_{3} + 3) q^{8} + (6 \beta_{3} - 8 \beta_{2} + 6 \beta_1) q^{9} + ( - 15 \beta_{2} + 5 \beta_1 - 15) q^{10} + ( - 14 \beta_{2} - 10 \beta_1 - 14) q^{11} + (3 \beta_{3} - 33 \beta_{2} + 3 \beta_1) q^{12} + (10 \beta_{3} + 18) q^{13} + (15 \beta_{3} - 5) q^{15} + (12 \beta_{3} - 7 \beta_{2} + 12 \beta_1) q^{16} + ( - 38 \beta_{2} + 54 \beta_1 - 38) q^{17} + ( - 36 \beta_{2} + 26 \beta_1 - 36) q^{18} + ( - 26 \beta_{3} - 80 \beta_{2} - 26 \beta_1) q^{19} + ( - 30 \beta_{3} - 15) q^{20} + (16 \beta_{3} - 22) q^{22} + ( - 117 \beta_{3} - 11 \beta_{2} - 117 \beta_1) q^{23} + ( - 81 \beta_{2} - 22 \beta_1 - 81) q^{24} + ( - 25 \beta_{2} - 25) q^{25} + (48 \beta_{3} - 74 \beta_{2} + 48 \beta_1) q^{26} + (99 \beta_{3} + 1) q^{27} + (60 \beta_{3} - 125) q^{29} + (40 \beta_{3} - 15 \beta_{2} + 40 \beta_1) q^{30} + ( - 66 \beta_{2} - 100 \beta_1 - 66) q^{31} + ( - 21 \beta_{2} + 147 \beta_1 - 21) q^{32} + (52 \beta_{3} + 74 \beta_{2} + 52 \beta_1) q^{33} + ( - 200 \beta_{3} - 222) q^{34} + ( - 66 \beta_{3} - 96) q^{36} + (136 \beta_{3} - 208 \beta_{2} + 136 \beta_1) q^{37} + ( - 188 \beta_{2} + 2 \beta_1 - 188) q^{38} + (42 \beta_{2} - 44 \beta_1 + 42) q^{39} + ( - 65 \beta_{3} - 15 \beta_{2} - 65 \beta_1) q^{40} + (30 \beta_{3} + 53) q^{41} + ( - 5 \beta_{3} - 333) q^{43} + ( - 54 \beta_{3} - 78 \beta_{2} - 54 \beta_1) q^{44} + ( - 40 \beta_{2} + 30 \beta_1 - 40) q^{45} + (201 \beta_{2} - 340 \beta_1 + 201) q^{46} + (64 \beta_{3} + 98 \beta_{2} + 64 \beta_1) q^{47} + (9 \beta_{3} + 65) q^{48} + ( - 25 \beta_{3} - 75) q^{50} + (60 \beta_{3} - 286 \beta_{2} + 60 \beta_1) q^{51} + ( - 174 \beta_{2} + 138 \beta_1 - 174) q^{52} + (476 \beta_{2} - 142 \beta_1 + 476) q^{53} + (298 \beta_{3} - 201 \beta_{2} + 298 \beta_1) q^{54} + (50 \beta_{3} - 70) q^{55} + (266 \beta_{3} - 236) q^{57} + (55 \beta_{3} + 255 \beta_{2} + 55 \beta_1) q^{58} + (420 \beta_{2} - 266 \beta_1 + 420) q^{59} + ( - 165 \beta_{2} + 15 \beta_1 - 165) q^{60} + ( - 570 \beta_{3} - 49 \beta_{2} - 570 \beta_1) q^{61} + (234 \beta_{3} + 2) q^{62} + ( - 366 \beta_{3} - 301) q^{64} + (50 \beta_{3} - 90 \beta_{2} + 50 \beta_1) q^{65} + (118 \beta_{2} + 82 \beta_1 + 118) q^{66} + (643 \beta_{2} - 69 \beta_1 + 643) q^{67} + ( - 390 \beta_{3} + 762 \beta_{2} - 390 \beta_1) q^{68} + (150 \beta_{3} - 713) q^{69} + ( - 350 \beta_{3} + 532) q^{71} + ( - 86 \beta_{3} + 132 \beta_{2} - 86 \beta_1) q^{72} + ( - 86 \beta_{2} - 118 \beta_1 - 86) q^{73} + ( - 896 \beta_{2} + 616 \beta_1 - 896) q^{74} + (75 \beta_{3} + 25 \beta_{2} + 75 \beta_1) q^{75} + ( - 402 \beta_{3} + 72) q^{76} + (174 \beta_{3} + 214) q^{78} + ( - 214 \beta_{3} - 620 \beta_{2} - 214 \beta_1) q^{79} + ( - 35 \beta_{2} + 60 \beta_1 - 35) q^{80} + (377 \beta_{2} + 258 \beta_1 + 377) q^{81} + (143 \beta_{3} - 219 \beta_{2} + 143 \beta_1) q^{82} + ( - 5 \beta_{3} + 953) q^{83} + ( - 270 \beta_{3} - 190) q^{85} + ( - 348 \beta_{3} + 1009 \beta_{2} - 348 \beta_1) q^{86} + (485 \beta_{2} + 435 \beta_1 + 485) q^{87} + ( - 302 \beta_{2} - 212 \beta_1 - 302) q^{88} + (324 \beta_{3} - 325 \beta_{2} + 324 \beta_1) q^{89} + ( - 130 \beta_{3} - 180) q^{90} + (285 \beta_{3} + 1371) q^{92} + (298 \beta_{3} + 666 \beta_{2} + 298 \beta_1) q^{93} + (166 \beta_{2} + 94 \beta_1 + 166) q^{94} + ( - 400 \beta_{2} - 130 \beta_1 - 400) q^{95} + ( - 84 \beta_{3} - 861 \beta_{2} - 84 \beta_1) q^{96} + ( - 500 \beta_{3} + 314) q^{97} + ( - 4 \beta_{3} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} - 2 q^{3} - 6 q^{4} + 10 q^{5} + 12 q^{6} + 12 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{2} - 2 q^{3} - 6 q^{4} + 10 q^{5} + 12 q^{6} + 12 q^{8} + 16 q^{9} - 30 q^{10} - 28 q^{11} + 66 q^{12} + 72 q^{13} - 20 q^{15} + 14 q^{16} - 76 q^{17} - 72 q^{18} + 160 q^{19} - 60 q^{20} - 88 q^{22} + 22 q^{23} - 162 q^{24} - 50 q^{25} + 148 q^{26} + 4 q^{27} - 500 q^{29} + 30 q^{30} - 132 q^{31} - 42 q^{32} - 148 q^{33} - 888 q^{34} - 384 q^{36} + 416 q^{37} - 376 q^{38} + 84 q^{39} + 30 q^{40} + 212 q^{41} - 1332 q^{43} + 156 q^{44} - 80 q^{45} + 402 q^{46} - 196 q^{47} + 260 q^{48} - 300 q^{50} + 572 q^{51} - 348 q^{52} + 952 q^{53} + 402 q^{54} - 280 q^{55} - 944 q^{57} - 510 q^{58} + 840 q^{59} - 330 q^{60} + 98 q^{61} + 8 q^{62} - 1204 q^{64} + 180 q^{65} + 236 q^{66} + 1286 q^{67} - 1524 q^{68} - 2852 q^{69} + 2128 q^{71} - 264 q^{72} - 172 q^{73} - 1792 q^{74} - 50 q^{75} + 288 q^{76} + 856 q^{78} + 1240 q^{79} - 70 q^{80} + 754 q^{81} + 438 q^{82} + 3812 q^{83} - 760 q^{85} - 2018 q^{86} + 970 q^{87} - 604 q^{88} + 650 q^{89} - 720 q^{90} + 5484 q^{92} - 1332 q^{93} + 332 q^{94} - 800 q^{95} + 1722 q^{96} + 1256 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

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

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

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
0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 1.22474i
0.792893 + 1.37333i −2.62132 + 4.54026i 2.74264 4.75039i 2.50000 + 4.33013i −8.31371 0 21.3848 −0.242641 0.420266i −3.96447 + 6.86666i
116.2 2.20711 + 3.82282i 1.62132 2.80821i −5.74264 + 9.94655i 2.50000 + 4.33013i 14.3137 0 −15.3848 8.24264 + 14.2767i −11.0355 + 19.1141i
226.1 0.792893 1.37333i −2.62132 4.54026i 2.74264 + 4.75039i 2.50000 4.33013i −8.31371 0 21.3848 −0.242641 + 0.420266i −3.96447 6.86666i
226.2 2.20711 3.82282i 1.62132 + 2.80821i −5.74264 9.94655i 2.50000 4.33013i 14.3137 0 −15.3848 8.24264 14.2767i −11.0355 19.1141i
\(n\): 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.l 4
7.b odd 2 1 35.4.e.b 4
7.c even 3 1 245.4.a.h 2
7.c even 3 1 inner 245.4.e.l 4
7.d odd 6 1 35.4.e.b 4
7.d odd 6 1 245.4.a.g 2
21.c even 2 1 315.4.j.c 4
21.g even 6 1 315.4.j.c 4
21.g even 6 1 2205.4.a.bf 2
21.h odd 6 1 2205.4.a.bg 2
28.d even 2 1 560.4.q.i 4
28.f even 6 1 560.4.q.i 4
35.c odd 2 1 175.4.e.c 4
35.f even 4 2 175.4.k.c 8
35.i odd 6 1 175.4.e.c 4
35.i odd 6 1 1225.4.a.x 2
35.j even 6 1 1225.4.a.v 2
35.k even 12 2 175.4.k.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.b 4 7.b odd 2 1
35.4.e.b 4 7.d odd 6 1
175.4.e.c 4 35.c odd 2 1
175.4.e.c 4 35.i odd 6 1
175.4.k.c 8 35.f even 4 2
175.4.k.c 8 35.k even 12 2
245.4.a.g 2 7.d odd 6 1
245.4.a.h 2 7.c even 3 1
245.4.e.l 4 1.a even 1 1 trivial
245.4.e.l 4 7.c even 3 1 inner
315.4.j.c 4 21.c even 2 1
315.4.j.c 4 21.g even 6 1
560.4.q.i 4 28.d even 2 1
560.4.q.i 4 28.f even 6 1
1225.4.a.v 2 35.j even 6 1
1225.4.a.x 2 35.i odd 6 1
2205.4.a.bf 2 21.g even 6 1
2205.4.a.bg 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 \(S_{4}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{4} - 6T_{2}^{3} + 29T_{2}^{2} - 42T_{2} + 49 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{3} + 21T_{3}^{2} - 34T_{3} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 6 T^{3} + 29 T^{2} - 42 T + 49 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289 \) 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
$11$ \( T^{4} + 28 T^{3} + 788 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} - 36 T + 124)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 76 T^{3} + 10164 T^{2} + \cdots + 19254544 \) Copy content Toggle raw display
$19$ \( T^{4} - 160 T^{3} + \cdots + 25482304 \) Copy content Toggle raw display
$23$ \( T^{4} - 22 T^{3} + \cdots + 742944049 \) Copy content Toggle raw display
$29$ \( (T^{2} + 250 T + 8425)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 132 T^{3} + \cdots + 244734736 \) Copy content Toggle raw display
$37$ \( T^{4} - 416 T^{3} + \cdots + 39337984 \) Copy content Toggle raw display
$41$ \( (T^{2} - 106 T + 1009)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 666 T + 110839)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 196 T^{3} + 37004 T^{2} + \cdots + 1993744 \) Copy content Toggle raw display
$53$ \( T^{4} - 952 T^{3} + \cdots + 34688317504 \) Copy content Toggle raw display
$59$ \( T^{4} - 840 T^{3} + \cdots + 1217172544 \) Copy content Toggle raw display
$61$ \( T^{4} - 98 T^{3} + \cdots + 419125465201 \) Copy content Toggle raw display
$67$ \( T^{4} - 1286 T^{3} + \cdots + 163157021329 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1064 T + 38024)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 172 T^{3} + \cdots + 418284304 \) Copy content Toggle raw display
$79$ \( T^{4} - 1240 T^{3} + \cdots + 85736524864 \) Copy content Toggle raw display
$83$ \( (T^{2} - 1906 T + 908159)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 650 T^{3} + \cdots + 10884122929 \) Copy content Toggle raw display
$97$ \( (T^{2} - 628 T - 401404)^{2} \) Copy content Toggle raw display
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