Properties

Label 49.4.a.e
Level $49$
Weight $4$
Character orbit 49.a
Self dual yes
Analytic conductor $2.891$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.89109359028\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{65})\)
Defining polynomial: \( x^{4} - 2x^{3} - 35x^{2} + 36x + 194 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + \beta_{2} q^{3} + (\beta_1 + 9) q^{4} - \beta_{3} q^{5} + (2 \beta_{3} - 3 \beta_{2}) q^{6} + (\beta_1 + 17) q^{8} + ( - 6 \beta_1 + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + \beta_{2} q^{3} + (\beta_1 + 9) q^{4} - \beta_{3} q^{5} + (2 \beta_{3} - 3 \beta_{2}) q^{6} + (\beta_1 + 17) q^{8} + ( - 6 \beta_1 + 7) q^{9} + ( - 4 \beta_{3} - 2 \beta_{2}) q^{10} + ( - 6 \beta_1 + 22) q^{11} + (2 \beta_{3} + 5 \beta_{2}) q^{12} + (\beta_{3} + 6 \beta_{2}) q^{13} + ( - 8 \beta_1 - 20) q^{15} + (9 \beta_1 - 39) q^{16} + (9 \beta_{3} - \beta_{2}) q^{17} + (7 \beta_1 - 89) q^{18} + ( - 4 \beta_{3} - 11 \beta_{2}) q^{19} + ( - 12 \beta_{3} - 2 \beta_{2}) q^{20} + (22 \beta_1 - 74) q^{22} + (8 \beta_1 + 92) q^{23} + (2 \beta_{3} + 13 \beta_{2}) q^{24} + (22 \beta_1 - 21) q^{25} + (16 \beta_{3} - 16 \beta_{2}) q^{26} + ( - 12 \beta_{3} + 4 \beta_{2}) q^{27} + ( - 14 \beta_1 + 58) q^{29} + ( - 20 \beta_1 - 148) q^{30} + (4 \beta_{3} - 38 \beta_{2}) q^{31} + ( - 47 \beta_1 - 31) q^{32} + ( - 12 \beta_{3} + 46 \beta_{2}) q^{33} + (34 \beta_{3} + 21 \beta_{2}) q^{34} + ( - 41 \beta_1 - 33) q^{36} + ( - 6 \beta_1 + 50) q^{37} + ( - 38 \beta_{3} + 25 \beta_{2}) q^{38} + ( - 28 \beta_1 + 224) q^{39} + ( - 20 \beta_{3} - 2 \beta_{2}) q^{40} + (3 \beta_{3} - 31 \beta_{2}) q^{41} + (70 \beta_1 + 170) q^{43} + ( - 26 \beta_1 + 102) q^{44} + (11 \beta_{3} + 12 \beta_{2}) q^{45} + (92 \beta_1 + 220) q^{46} + (32 \beta_{3} - 26 \beta_{2}) q^{47} + (18 \beta_{3} - 75 \beta_{2}) q^{48} + ( - 21 \beta_1 + 331) q^{50} + (78 \beta_1 + 146) q^{51} + (24 \beta_{3} + 32 \beta_{2}) q^{52} + (36 \beta_1 + 22) q^{53} + ( - 40 \beta_{3} - 36 \beta_{2}) q^{54} + ( - 4 \beta_{3} + 12 \beta_{2}) q^{55} + (34 \beta_1 - 454) q^{57} + (58 \beta_1 - 166) q^{58} + ( - 20 \beta_{3} + 49 \beta_{2}) q^{59} + ( - 84 \beta_1 - 308) q^{60} + ( - 27 \beta_{3} + 68 \beta_{2}) q^{61} + ( - 60 \beta_{3} + 122 \beta_{2}) q^{62} + ( - 103 \beta_1 - 471) q^{64} + ( - 70 \beta_1 - 224) q^{65} + (44 \beta_{3} - 162 \beta_{2}) q^{66} + ( - 76 \beta_1 - 524) q^{67} + (106 \beta_{3} + 13 \beta_{2}) q^{68} + (16 \beta_{3} + 60 \beta_{2}) q^{69} + ( - 28 \beta_1 + 548) q^{71} + ( - 89 \beta_1 + 23) q^{72} + ( - 25 \beta_{3} - 37 \beta_{2}) q^{73} + (50 \beta_1 - 46) q^{74} + (44 \beta_{3} - 109 \beta_{2}) q^{75} + ( - 70 \beta_{3} - 63 \beta_{2}) q^{76} + (224 \beta_1 - 224) q^{78} + ( - 188 \beta_1 - 356) q^{79} + (12 \beta_{3} - 18 \beta_{2}) q^{80} + (42 \beta_1 - 293) q^{81} + ( - 50 \beta_{3} + 99 \beta_{2}) q^{82} + ( - 8 \beta_{3} + \beta_{2}) q^{83} + ( - 190 \beta_1 - 916) q^{85} + (170 \beta_1 + 1290) q^{86} + ( - 28 \beta_{3} + 114 \beta_{2}) q^{87} + ( - 74 \beta_1 + 278) q^{88} + ( - 75 \beta_{3} - 157 \beta_{2}) q^{89} + (68 \beta_{3} - 14 \beta_{2}) q^{90} + (156 \beta_1 + 956) q^{92} + (260 \beta_1 - 1212) q^{93} + (76 \beta_{3} + 142 \beta_{2}) q^{94} + (176 \beta_1 + 636) q^{95} + ( - 94 \beta_{3} + 157 \beta_{2}) q^{96} + (91 \beta_{3} - 189 \beta_{2}) q^{97} + ( - 210 \beta_1 + 730) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 34 q^{4} + 66 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 34 q^{4} + 66 q^{8} + 40 q^{9} + 100 q^{11} - 64 q^{15} - 174 q^{16} - 370 q^{18} - 340 q^{22} + 352 q^{23} - 128 q^{25} + 260 q^{29} - 552 q^{30} - 30 q^{32} - 50 q^{36} + 212 q^{37} + 952 q^{39} + 540 q^{43} + 460 q^{44} + 696 q^{46} + 1366 q^{50} + 428 q^{51} + 16 q^{53} - 1884 q^{57} - 780 q^{58} - 1064 q^{60} - 1678 q^{64} - 756 q^{65} - 1944 q^{67} + 2248 q^{71} + 270 q^{72} - 284 q^{74} - 1344 q^{78} - 1048 q^{79} - 1256 q^{81} - 3284 q^{85} + 4820 q^{86} + 1260 q^{88} + 3512 q^{92} - 5368 q^{93} + 2192 q^{95} + 3340 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 35x^{2} + 36x + 194 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} + 100\nu - 79 ) / 57 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 11\nu^{2} + 12\nu - 182 ) / 19 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{3} + 12\nu^{2} - 265\nu - 392 ) / 57 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + 7\beta _1 + 7 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{3} + 10\beta_{2} + 7\beta _1 + 133 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{3} - 5\beta_{2} + 23\beta _1 + 39 \) 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
−2.11692
−4.94534
3.11692
5.94534
−3.53113 −7.82220 4.46887 −2.07730 27.6212 0 12.4689 34.1868 7.33521
1.2 −3.53113 7.82220 4.46887 2.07730 −27.6212 0 12.4689 34.1868 −7.33521
1.3 4.53113 −3.57956 12.5311 13.4791 −16.2194 0 20.5311 −14.1868 61.0753
1.4 4.53113 3.57956 12.5311 −13.4791 16.2194 0 20.5311 −14.1868 −61.0753
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.4.a.e 4
3.b odd 2 1 441.4.a.u 4
4.b odd 2 1 784.4.a.bf 4
5.b even 2 1 1225.4.a.bb 4
7.b odd 2 1 inner 49.4.a.e 4
7.c even 3 2 49.4.c.e 8
7.d odd 6 2 49.4.c.e 8
21.c even 2 1 441.4.a.u 4
21.g even 6 2 441.4.e.y 8
21.h odd 6 2 441.4.e.y 8
28.d even 2 1 784.4.a.bf 4
35.c odd 2 1 1225.4.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.e 4 1.a even 1 1 trivial
49.4.a.e 4 7.b odd 2 1 inner
49.4.c.e 8 7.c even 3 2
49.4.c.e 8 7.d odd 6 2
441.4.a.u 4 3.b odd 2 1
441.4.a.u 4 21.c even 2 1
441.4.e.y 8 21.g even 6 2
441.4.e.y 8 21.h odd 6 2
784.4.a.bf 4 4.b odd 2 1
784.4.a.bf 4 28.d even 2 1
1225.4.a.bb 4 5.b even 2 1
1225.4.a.bb 4 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{2} - T_{2} - 16 \) Copy content Toggle raw display
\( T_{3}^{4} - 74T_{3}^{2} + 784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T - 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 74T^{2} + 784 \) Copy content Toggle raw display
$5$ \( T^{4} - 186T^{2} + 784 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 50 T + 40)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 3234 T^{2} + \cdots + 2458624 \) Copy content Toggle raw display
$17$ \( T^{4} - 14564 T^{2} + \cdots + 9746884 \) Copy content Toggle raw display
$19$ \( T^{4} - 14746 T^{2} + \cdots + 52591504 \) Copy content Toggle raw display
$23$ \( (T^{2} - 176 T + 6704)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 130 T + 1040)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 100104 T^{2} + \cdots + 629407744 \) Copy content Toggle raw display
$37$ \( (T^{2} - 106 T + 2224)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 66836 T^{2} + \cdots + 307721764 \) Copy content Toggle raw display
$43$ \( (T^{2} - 270 T - 61400)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 187240 T^{2} + \cdots + 8332038400 \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T - 21044)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 189354 T^{2} + \cdots + 1600960144 \) Copy content Toggle raw display
$61$ \( T^{4} - 360266 T^{2} + \cdots + 5022273424 \) Copy content Toggle raw display
$67$ \( (T^{2} + 972 T + 142336)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1124 T + 303104)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 276756 T^{2} + \cdots + 12428236324 \) Copy content Toggle raw display
$79$ \( (T^{2} + 524 T - 505696)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 11466 T^{2} + \cdots + 6492304 \) Copy content Toggle raw display
$89$ \( T^{4} - 3623876 T^{2} + \cdots + 2844693770884 \) Copy content Toggle raw display
$97$ \( T^{4} - 3082884 T^{2} + \cdots + 841222821124 \) Copy content Toggle raw display
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