Properties

Label 245.4.a.g
Level $245$
Weight $4$
Character orbit 245.a
Self dual yes
Analytic conductor $14.455$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4554679514\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 3) q^{2} + ( - 3 \beta - 1) q^{3} + ( - 6 \beta + 3) q^{4} + 5 q^{5} + (8 \beta - 3) q^{6} + (13 \beta + 3) q^{8} + (6 \beta - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 3) q^{2} + ( - 3 \beta - 1) q^{3} + ( - 6 \beta + 3) q^{4} + 5 q^{5} + (8 \beta - 3) q^{6} + (13 \beta + 3) q^{8} + (6 \beta - 8) q^{9} + (5 \beta - 15) q^{10} + (10 \beta + 14) q^{11} + ( - 3 \beta + 33) q^{12} + (10 \beta - 18) q^{13} + ( - 15 \beta - 5) q^{15} + (12 \beta - 7) q^{16} + (54 \beta - 38) q^{17} + ( - 26 \beta + 36) q^{18} + (26 \beta + 80) q^{19} + ( - 30 \beta + 15) q^{20} + ( - 16 \beta - 22) q^{22} + ( - 117 \beta - 11) q^{23} + ( - 22 \beta - 81) q^{24} + 25 q^{25} + ( - 48 \beta + 74) q^{26} + (99 \beta - 1) q^{27} + ( - 60 \beta - 125) q^{29} + (40 \beta - 15) q^{30} + ( - 100 \beta - 66) q^{31} + ( - 147 \beta + 21) q^{32} + ( - 52 \beta - 74) q^{33} + ( - 200 \beta + 222) q^{34} + (66 \beta - 96) q^{36} + (136 \beta - 208) q^{37} + (2 \beta - 188) q^{38} + (44 \beta - 42) q^{39} + (65 \beta + 15) q^{40} + (30 \beta - 53) q^{41} + (5 \beta - 333) q^{43} + ( - 54 \beta - 78) q^{44} + (30 \beta - 40) q^{45} + (340 \beta - 201) q^{46} + ( - 64 \beta - 98) q^{47} + (9 \beta - 65) q^{48} + (25 \beta - 75) q^{50} + (60 \beta - 286) q^{51} + (138 \beta - 174) q^{52} + (142 \beta - 476) q^{53} + ( - 298 \beta + 201) q^{54} + (50 \beta + 70) q^{55} + ( - 266 \beta - 236) q^{57} + (55 \beta + 255) q^{58} + ( - 266 \beta + 420) q^{59} + ( - 15 \beta + 165) q^{60} + (570 \beta + 49) q^{61} + (234 \beta - 2) q^{62} + (366 \beta - 301) q^{64} + (50 \beta - 90) q^{65} + (82 \beta + 118) q^{66} + (69 \beta - 643) q^{67} + (390 \beta - 762) q^{68} + (150 \beta + 713) q^{69} + (350 \beta + 532) q^{71} + ( - 86 \beta + 132) q^{72} + ( - 118 \beta - 86) q^{73} + ( - 616 \beta + 896) q^{74} + ( - 75 \beta - 25) q^{75} + ( - 402 \beta - 72) q^{76} + ( - 174 \beta + 214) q^{78} + ( - 214 \beta - 620) q^{79} + (60 \beta - 35) q^{80} + ( - 258 \beta - 377) q^{81} + ( - 143 \beta + 219) q^{82} + ( - 5 \beta - 953) q^{83} + (270 \beta - 190) q^{85} + ( - 348 \beta + 1009) q^{86} + (435 \beta + 485) q^{87} + (212 \beta + 302) q^{88} + ( - 324 \beta + 325) q^{89} + ( - 130 \beta + 180) q^{90} + ( - 285 \beta + 1371) q^{92} + (298 \beta + 666) q^{93} + (94 \beta + 166) q^{94} + (130 \beta + 400) q^{95} + (84 \beta + 861) q^{96} + ( - 500 \beta - 314) q^{97} + (4 \beta + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 10 q^{5} - 6 q^{6} + 6 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 10 q^{5} - 6 q^{6} + 6 q^{8} - 16 q^{9} - 30 q^{10} + 28 q^{11} + 66 q^{12} - 36 q^{13} - 10 q^{15} - 14 q^{16} - 76 q^{17} + 72 q^{18} + 160 q^{19} + 30 q^{20} - 44 q^{22} - 22 q^{23} - 162 q^{24} + 50 q^{25} + 148 q^{26} - 2 q^{27} - 250 q^{29} - 30 q^{30} - 132 q^{31} + 42 q^{32} - 148 q^{33} + 444 q^{34} - 192 q^{36} - 416 q^{37} - 376 q^{38} - 84 q^{39} + 30 q^{40} - 106 q^{41} - 666 q^{43} - 156 q^{44} - 80 q^{45} - 402 q^{46} - 196 q^{47} - 130 q^{48} - 150 q^{50} - 572 q^{51} - 348 q^{52} - 952 q^{53} + 402 q^{54} + 140 q^{55} - 472 q^{57} + 510 q^{58} + 840 q^{59} + 330 q^{60} + 98 q^{61} - 4 q^{62} - 602 q^{64} - 180 q^{65} + 236 q^{66} - 1286 q^{67} - 1524 q^{68} + 1426 q^{69} + 1064 q^{71} + 264 q^{72} - 172 q^{73} + 1792 q^{74} - 50 q^{75} - 144 q^{76} + 428 q^{78} - 1240 q^{79} - 70 q^{80} - 754 q^{81} + 438 q^{82} - 1906 q^{83} - 380 q^{85} + 2018 q^{86} + 970 q^{87} + 604 q^{88} + 650 q^{89} + 360 q^{90} + 2742 q^{92} + 1332 q^{93} + 332 q^{94} + 800 q^{95} + 1722 q^{96} - 628 q^{97} + 16 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−4.41421 3.24264 11.4853 5.00000 −14.3137 0 −15.3848 −16.4853 −22.0711
1.2 −1.58579 −5.24264 −5.48528 5.00000 8.31371 0 21.3848 0.485281 −7.92893
\(n\): e.g. 2-40 or 990-1000
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 245.4.a.g 2
3.b odd 2 1 2205.4.a.bf 2
5.b even 2 1 1225.4.a.x 2
7.b odd 2 1 245.4.a.h 2
7.c even 3 2 35.4.e.b 4
7.d odd 6 2 245.4.e.l 4
21.c even 2 1 2205.4.a.bg 2
21.h odd 6 2 315.4.j.c 4
28.g odd 6 2 560.4.q.i 4
35.c odd 2 1 1225.4.a.v 2
35.j even 6 2 175.4.e.c 4
35.l odd 12 4 175.4.k.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.b 4 7.c even 3 2
175.4.e.c 4 35.j even 6 2
175.4.k.c 8 35.l odd 12 4
245.4.a.g 2 1.a even 1 1 trivial
245.4.a.h 2 7.b odd 2 1
245.4.e.l 4 7.d odd 6 2
315.4.j.c 4 21.h odd 6 2
560.4.q.i 4 28.g odd 6 2
1225.4.a.v 2 35.c odd 2 1
1225.4.a.x 2 5.b even 2 1
2205.4.a.bf 2 3.b odd 2 1
2205.4.a.bg 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{2} + 6T_{2} + 7 \) Copy content Toggle raw display
\( T_{3}^{2} + 2T_{3} - 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 6T + 7 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 17 \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 28T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 36T + 124 \) Copy content Toggle raw display
$17$ \( T^{2} + 76T - 4388 \) Copy content Toggle raw display
$19$ \( T^{2} - 160T + 5048 \) Copy content Toggle raw display
$23$ \( T^{2} + 22T - 27257 \) Copy content Toggle raw display
$29$ \( T^{2} + 250T + 8425 \) Copy content Toggle raw display
$31$ \( T^{2} + 132T - 15644 \) Copy content Toggle raw display
$37$ \( T^{2} + 416T + 6272 \) Copy content Toggle raw display
$41$ \( T^{2} + 106T + 1009 \) Copy content Toggle raw display
$43$ \( T^{2} + 666T + 110839 \) Copy content Toggle raw display
$47$ \( T^{2} + 196T + 1412 \) Copy content Toggle raw display
$53$ \( T^{2} + 952T + 186248 \) Copy content Toggle raw display
$59$ \( T^{2} - 840T + 34888 \) Copy content Toggle raw display
$61$ \( T^{2} - 98T - 647399 \) Copy content Toggle raw display
$67$ \( T^{2} + 1286 T + 403927 \) Copy content Toggle raw display
$71$ \( T^{2} - 1064T + 38024 \) Copy content Toggle raw display
$73$ \( T^{2} + 172T - 20452 \) Copy content Toggle raw display
$79$ \( T^{2} + 1240 T + 292808 \) Copy content Toggle raw display
$83$ \( T^{2} + 1906 T + 908159 \) Copy content Toggle raw display
$89$ \( T^{2} - 650T - 104327 \) Copy content Toggle raw display
$97$ \( T^{2} + 628T - 401404 \) Copy content Toggle raw display
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