Properties

Label 49.6.c.d
Level $49$
Weight $6$
Character orbit 49.c
Analytic conductor $7.859$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 7)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2} + 5 \beta_1) q^{2} + (6 \beta_{3} - 6 \beta_1 - 6) q^{3} + (9 \beta_{3} - 7 \beta_1 - 7) q^{4} + (10 \beta_{3} - 10 \beta_{2} + 4 \beta_1) q^{5} + ( - 30 \beta_{2} + 114) q^{6} + ( - 11 \beta_{2} + 1) q^{8} + ( - 36 \beta_{3} + 36 \beta_{2} + 297 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2} + 5 \beta_1) q^{2} + (6 \beta_{3} - 6 \beta_1 - 6) q^{3} + (9 \beta_{3} - 7 \beta_1 - 7) q^{4} + (10 \beta_{3} - 10 \beta_{2} + 4 \beta_1) q^{5} + ( - 30 \beta_{2} + 114) q^{6} + ( - 11 \beta_{2} + 1) q^{8} + ( - 36 \beta_{3} + 36 \beta_{2} + 297 \beta_1) q^{9} + ( - 36 \beta_{3} + 120 \beta_1 + 120) q^{10} + ( - 124 \beta_{3} - 136 \beta_1 - 136) q^{11} + ( - 42 \beta_{3} + 42 \beta_{2} + 798 \beta_1) q^{12} + (126 \beta_{2} + 112) q^{13} + ( - 24 \beta_{2} - 816) q^{15} + (243 \beta_{3} - 243 \beta_{2} - 65 \beta_1) q^{16} + (76 \beta_{3} + 862 \beta_1 + 862) q^{17} + (441 \beta_{3} - 1989 \beta_1 - 1989) q^{18} + ( - 18 \beta_{3} + 18 \beta_{2} + 1642 \beta_1) q^{19} + ( - 56 \beta_{2} - 1232) q^{20} + (360 \beta_{2} - 1056) q^{22} + ( - 568 \beta_{3} + 568 \beta_{2} + 1328 \beta_1) q^{23} + (6 \beta_{3} - 930 \beta_1 - 930) q^{24} + ( - 180 \beta_{3} + 1709 \beta_1 + 1709) q^{25} + (392 \beta_{3} - 392 \beta_{2} - 1204 \beta_1) q^{26} + ( - 324 \beta_{2} + 3348) q^{27} + ( - 252 \beta_{2} + 3474) q^{29} + (720 \beta_{3} - 720 \beta_{2} - 3744 \beta_1) q^{30} + ( - 540 \beta_{3} + 260 \beta_1 + 260) q^{31} + ( - 1389 \beta_{3} + 3759 \beta_1 + 3759) q^{32} + ( - 816 \beta_{3} + 816 \beta_{2} - 9600 \beta_1) q^{33} + (558 \beta_{2} - 3246) q^{34} + ( - 2601 \beta_{2} + 6615) q^{36} + ( - 540 \beta_{3} + 540 \beta_{2} + 3386 \beta_1) q^{37} + (1714 \beta_{3} - 8462 \beta_1 - 8462) q^{38} + (672 \beta_{3} + 9912 \beta_1 + 9912) q^{39} + ( - 144 \beta_{3} + 144 \beta_{2} - 1536 \beta_1) q^{40} + ( - 1092 \beta_{2} + 3570) q^{41} + (4788 \beta_{2} - 3904) q^{43} + ( - 1472 \beta_{3} + 1472 \beta_{2} - 14672 \beta_1) q^{44} + ( - 2466 \beta_{3} + 3852 \beta_1 + 3852) q^{45} + (3600 \beta_{3} - 14592 \beta_1 - 14592) q^{46} + (3748 \beta_{3} - 3748 \beta_{2} - 7724 \beta_1) q^{47} + (390 \beta_{2} - 20802) q^{48} + (2429 \beta_{2} - 11065) q^{50} + (5172 \beta_{3} - 5172 \beta_{2} + 1212 \beta_1) q^{51} + (1260 \beta_{3} + 15092 \beta_1 + 15092) q^{52} + ( - 208 \beta_{3} - 4630 \beta_1 - 4630) q^{53} + ( - 4644 \beta_{3} + 4644 \beta_{2} + 21276 \beta_1) q^{54} + (3096 \beta_{2} + 17904) q^{55} + ( - 9852 \beta_{2} + 11364) q^{57} + ( - 4482 \beta_{3} + 4482 \beta_{2} + 20898 \beta_1) q^{58} + (2050 \beta_{3} - 22994 \beta_1 - 22994) q^{59} + ( - 7392 \beta_{3} + 2688 \beta_1 + 2688) q^{60} + ( - 4806 \beta_{3} + 4806 \beta_{2} + 34780 \beta_1) q^{61} + (2420 \beta_{2} - 8860) q^{62} + (1539 \beta_{2} - 36161) q^{64} + (2884 \beta_{3} - 2884 \beta_{2} + 18088 \beta_1) q^{65} + ( - 6336 \beta_{3} + 36576 \beta_1 + 36576) q^{66} + ( - 1944 \beta_{3} - 11420 \beta_1 - 11420) q^{67} + (7910 \beta_{3} - 7910 \beta_{2} + 3542 \beta_1) q^{68} + ( - 7968 \beta_{2} + 55680) q^{69} + (4200 \beta_{2} + 46608) q^{71} + ( - 2907 \beta_{3} + 2907 \beta_{2} + 5841 \beta_1) q^{72} + (5256 \beta_{3} + 6098 \beta_1 + 6098) q^{73} + (5546 \beta_{3} - 24490 \beta_1 - 24490) q^{74} + (10254 \beta_{3} - 10254 \beta_{2} - 25374 \beta_1) q^{75} + ( - 14742 \beta_{2} + 13762) q^{76} + (7224 \beta_{2} - 40152) q^{78} + ( - 14904 \beta_{3} + 14904 \beta_{2} + 33080 \beta_1) q^{79} + ( - 2752 \beta_{3} - 33760 \beta_1 - 33760) q^{80} + (11340 \beta_{3} + 24867 \beta_1 + 24867) q^{81} + ( - 7938 \beta_{3} + 7938 \beta_{2} + 33138 \beta_1) q^{82} + (15750 \beta_{2} - 66654) q^{83} + ( - 9684 \beta_{2} - 14088) q^{85} + (23056 \beta_{3} - 23056 \beta_{2} - 86552 \beta_1) q^{86} + (20844 \beta_{3} - 42012 \beta_1 - 42012) q^{87} + (2736 \beta_{3} + 18960 \beta_1 + 18960) q^{88} + ( - 22208 \beta_{3} + 22208 \beta_{2} - 31034 \beta_1) q^{89} + (13716 \beta_{2} - 53784) q^{90} + ( - 10816 \beta_{2} + 80864) q^{92} + (1560 \beta_{3} - 1560 \beta_{2} - 46920 \beta_1) q^{93} + ( - 22716 \beta_{3} + 91092 \beta_1 + 91092) q^{94} + ( - 16168 \beta_{3} - 4048 \beta_1 - 4048) q^{95} + (22554 \beta_{3} - 22554 \beta_{2} - 139230 \beta_1) q^{96} + (8820 \beta_{2} - 14798) q^{97} + (27468 \beta_{2} - 22104) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 9 q^{2} - 6 q^{3} - 5 q^{4} - 18 q^{5} + 396 q^{6} - 18 q^{8} - 558 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 9 q^{2} - 6 q^{3} - 5 q^{4} - 18 q^{5} + 396 q^{6} - 18 q^{8} - 558 q^{9} + 204 q^{10} - 396 q^{11} - 1554 q^{12} + 700 q^{13} - 3312 q^{15} - 113 q^{16} + 1800 q^{17} - 3537 q^{18} - 3266 q^{19} - 5040 q^{20} - 3504 q^{22} - 2088 q^{23} - 1854 q^{24} + 3238 q^{25} + 2016 q^{26} + 12744 q^{27} + 13392 q^{29} + 6768 q^{30} - 20 q^{31} + 6129 q^{32} + 20016 q^{33} - 11868 q^{34} + 21258 q^{36} - 6232 q^{37} - 15210 q^{38} + 20496 q^{39} + 3216 q^{40} + 12096 q^{41} - 6040 q^{43} + 30816 q^{44} + 5238 q^{45} - 25584 q^{46} + 11700 q^{47} - 82428 q^{48} - 39402 q^{50} - 7596 q^{51} + 31444 q^{52} - 9468 q^{53} - 37908 q^{54} + 77808 q^{55} + 25752 q^{57} - 37314 q^{58} - 43938 q^{59} - 2016 q^{60} - 64754 q^{61} - 30600 q^{62} - 141566 q^{64} - 39060 q^{65} + 66816 q^{66} - 24784 q^{67} - 14994 q^{68} + 206784 q^{69} + 194832 q^{71} - 8775 q^{72} + 17452 q^{73} - 43434 q^{74} + 40494 q^{75} + 25564 q^{76} - 146160 q^{78} - 51256 q^{79} - 70272 q^{80} + 61074 q^{81} - 58338 q^{82} - 235116 q^{83} - 75720 q^{85} + 150048 q^{86} - 63180 q^{87} + 40656 q^{88} + 84276 q^{89} - 187704 q^{90} + 301824 q^{92} + 92280 q^{93} + 159468 q^{94} - 24264 q^{95} + 255906 q^{96} - 41552 q^{97} - 33480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{3} + 2\nu^{2} + 8\nu - 25 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 14\beta _1 + 13 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{3} - 4\beta_{2} - 4\beta _1 + 19 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
−1.63746 + 1.52274i
2.13746 0.656712i
−1.63746 1.52274i
2.13746 + 0.656712i
−4.13746 7.16629i −12.8248 + 22.2131i −18.2371 + 31.5876i 14.3746 + 24.8975i 212.248 0 37.0241 −207.449 359.311i 118.949 206.025i
18.2 −0.362541 0.627940i 9.82475 17.0170i 15.7371 27.2575i −23.3746 40.4860i −14.2475 0 −46.0241 −71.5515 123.931i −16.9485 + 29.3557i
30.1 −4.13746 + 7.16629i −12.8248 22.2131i −18.2371 31.5876i 14.3746 24.8975i 212.248 0 37.0241 −207.449 + 359.311i 118.949 + 206.025i
30.2 −0.362541 + 0.627940i 9.82475 + 17.0170i 15.7371 + 27.2575i −23.3746 + 40.4860i −14.2475 0 −46.0241 −71.5515 + 123.931i −16.9485 29.3557i
\(n\): 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 49.6.c.d 4
7.b odd 2 1 49.6.c.e 4
7.c even 3 1 49.6.a.f 2
7.c even 3 1 inner 49.6.c.d 4
7.d odd 6 1 7.6.a.b 2
7.d odd 6 1 49.6.c.e 4
21.g even 6 1 63.6.a.f 2
21.h odd 6 1 441.6.a.l 2
28.f even 6 1 112.6.a.h 2
28.g odd 6 1 784.6.a.v 2
35.i odd 6 1 175.6.a.c 2
35.k even 12 2 175.6.b.c 4
56.j odd 6 1 448.6.a.w 2
56.m even 6 1 448.6.a.u 2
77.i even 6 1 847.6.a.c 2
84.j odd 6 1 1008.6.a.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 7.d odd 6 1
49.6.a.f 2 7.c even 3 1
49.6.c.d 4 1.a even 1 1 trivial
49.6.c.d 4 7.c even 3 1 inner
49.6.c.e 4 7.b odd 2 1
49.6.c.e 4 7.d odd 6 1
63.6.a.f 2 21.g even 6 1
112.6.a.h 2 28.f even 6 1
175.6.a.c 2 35.i odd 6 1
175.6.b.c 4 35.k even 12 2
441.6.a.l 2 21.h odd 6 1
448.6.a.u 2 56.m even 6 1
448.6.a.w 2 56.j odd 6 1
784.6.a.v 2 28.g odd 6 1
847.6.a.c 2 77.i even 6 1
1008.6.a.bq 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(49, [\chi])\):

\( T_{2}^{4} + 9T_{2}^{3} + 75T_{2}^{2} + 54T_{2} + 36 \) Copy content Toggle raw display
\( T_{3}^{4} + 6T_{3}^{3} + 540T_{3}^{2} - 3024T_{3} + 254016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9 T^{3} + 75 T^{2} + 54 T + 36 \) Copy content Toggle raw display
$3$ \( T^{4} + 6 T^{3} + 540 T^{2} + \cdots + 254016 \) Copy content Toggle raw display
$5$ \( T^{4} + 18 T^{3} + 1668 T^{2} + \cdots + 1806336 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 396 T^{3} + \cdots + 32365449216 \) Copy content Toggle raw display
$13$ \( (T^{2} - 350 T - 195608)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 1800 T^{3} + \cdots + 529535646864 \) Copy content Toggle raw display
$19$ \( T^{4} + 3266 T^{3} + \cdots + 7086627333184 \) Copy content Toggle raw display
$23$ \( T^{4} + 2088 T^{3} + \cdots + 12302247591936 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6696 T + 10304172)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 20 T^{3} + \cdots + 17265687040000 \) Copy content Toggle raw display
$37$ \( T^{4} + 6232 T^{3} + \cdots + 30848648872336 \) Copy content Toggle raw display
$41$ \( (T^{2} - 6048 T - 7848036)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 3020 T - 324400352)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 11700 T^{3} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 474989071531536 \) Copy content Toggle raw display
$59$ \( T^{4} + 43938 T^{3} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{4} + 64754 T^{3} + \cdots + 51\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{4} + 24784 T^{3} + \cdots + 99\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{2} - 97416 T + 2121099264)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 17452 T^{3} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{4} + 51256 T^{3} + \cdots + 62\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( (T^{2} + 117558 T - 79919784)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 84276 T^{3} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{2} + 20776 T - 1000631156)^{2} \) Copy content Toggle raw display
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