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: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 4 \nu - 25 \)\()/20\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 9 \nu + 5 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{3} + 2 \nu^{2} + 8 \nu - 25 \)\()/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} - 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 14 \beta_{1} + 13\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(8 \beta_{3} - 4 \beta_{2} - 4 \beta_{1} + 19\)\()/3\)

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 \(S_{6}^{\mathrm{new}}(49, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 9 T + 11 T^{2} + 54 T^{3} + 1284 T^{4} + 1728 T^{5} + 11264 T^{6} + 294912 T^{7} + 1048576 T^{8} \)
$3$ \( 1 + 6 T + 54 T^{2} - 3024 T^{3} - 67473 T^{4} - 734832 T^{5} + 3188646 T^{6} + 86093442 T^{7} + 3486784401 T^{8} \)
$5$ \( 1 + 18 T - 4582 T^{2} - 24192 T^{3} + 13290711 T^{4} - 75600000 T^{5} - 44746093750 T^{6} + 549316406250 T^{7} + 95367431640625 T^{8} \)
$7$ 1
$11$ \( 1 + 396 T + 14618 T^{2} - 71241984 T^{3} - 30972527013 T^{4} - 11473592765184 T^{5} + 379153272817418 T^{6} + 1654190275088597796 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \)
$13$ \( ( 1 - 350 T + 546978 T^{2} - 129952550 T^{3} + 137858491849 T^{4} )^{2} \)
$17$ \( 1 - 1800 T - 327406 T^{2} - 1309845600 T^{3} + 6110054988387 T^{4} - 1859793444079200 T^{5} - 660048498970405294 T^{6} - \)\(51\!\cdots\!00\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} \)
$19$ \( 1 + 3266 T + 3052486 T^{2} + 8694327152 T^{3} + 25434097510255 T^{4} + 21528014766740048 T^{5} + 18714993917009943286 T^{6} + \)\(49\!\cdots\!34\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} \)
$23$ \( 1 + 2088 T - 5005486 T^{2} - 7323568128 T^{3} + 18220213963059 T^{4} - 47136996455675904 T^{5} - \)\(20\!\cdots\!14\)\( T^{6} + \)\(55\!\cdots\!16\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \)
$29$ \( ( 1 - 6696 T + 51326470 T^{2} - 137342653704 T^{3} + 420707233300201 T^{4} )^{2} \)
$31$ \( 1 + 20 T - 53102702 T^{2} - 83104000 T^{3} + 2000299703381203 T^{4} - 2379196964704000 T^{5} - \)\(43\!\cdots\!02\)\( T^{6} + \)\(46\!\cdots\!20\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \)
$37$ \( 1 + 6232 T - 105404246 T^{2} + 34613500192 T^{3} + 13304021988037483 T^{4} + 2400237068933539744 T^{5} - \)\(50\!\cdots\!54\)\( T^{6} + \)\(20\!\cdots\!76\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} \)
$41$ \( ( 1 - 6048 T + 223864366 T^{2} - 700698303648 T^{3} + 13422659310152401 T^{4} )^{2} \)
$43$ \( ( 1 + 3020 T - 30383466 T^{2} + 443965497860 T^{3} + 21611482313284249 T^{4} )^{2} \)
$47$ \( 1 - 11700 T - 155845582 T^{2} + 1941666854400 T^{3} + 1699950724818675 T^{4} + \)\(44\!\cdots\!00\)\( T^{5} - \)\(81\!\cdots\!18\)\( T^{6} - \)\(14\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} \)
$53$ \( 1 + 9468 T - 768542206 T^{2} + 206347902192 T^{3} + 524106110008949019 T^{4} + 86293762686699220656 T^{5} - \)\(13\!\cdots\!94\)\( T^{6} + \)\(69\!\cdots\!76\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} \)
$59$ \( 1 + 43938 T + 77947910 T^{2} + 18574848201168 T^{3} + 1540814198098501599 T^{4} + \)\(13\!\cdots\!32\)\( T^{5} + \)\(39\!\cdots\!10\)\( T^{6} + \)\(16\!\cdots\!62\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \)
$61$ \( 1 + 64754 T + 1784759098 T^{2} + 46566467351264 T^{3} + 1545208847125876807 T^{4} + \)\(39\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!98\)\( T^{6} + \)\(39\!\cdots\!54\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} \)
$67$ \( 1 + 24784 T - 2185712534 T^{2} + 2471187261184 T^{3} + 5187623828957302459 T^{4} + \)\(33\!\cdots\!88\)\( T^{5} - \)\(39\!\cdots\!66\)\( T^{6} + \)\(60\!\cdots\!12\)\( T^{7} + \)\(33\!\cdots\!01\)\( T^{8} \)
$71$ \( ( 1 - 97416 T + 5729557966 T^{2} - 175760806457016 T^{3} + 3255243551009881201 T^{4} )^{2} \)
$73$ \( 1 - 17452 T - 3524050070 T^{2} + 5541373211024 T^{3} + 9729323261552877955 T^{4} + \)\(11\!\cdots\!32\)\( T^{5} - \)\(15\!\cdots\!30\)\( T^{6} - \)\(15\!\cdots\!64\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} \)
$79$ \( 1 + 51256 T - 1018388318 T^{2} - 128578082161664 T^{3} - 4262099135233950749 T^{4} - \)\(39\!\cdots\!36\)\( T^{5} - \)\(96\!\cdots\!18\)\( T^{6} + \)\(14\!\cdots\!44\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} \)
$83$ \( ( 1 + 117558 T + 7798161502 T^{2} + 463065739909794 T^{3} + 15516041187205853449 T^{4} )^{2} \)
$89$ \( 1 - 84276 T + 1186746746 T^{2} + 442653071637168 T^{3} - 35846714358645397725 T^{4} + \)\(24\!\cdots\!32\)\( T^{5} + \)\(37\!\cdots\!46\)\( T^{6} - \)\(14\!\cdots\!24\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} \)
$97$ \( ( 1 + 20776 T + 16174049358 T^{2} + 178410581179432 T^{3} + 73742412689492826049 T^{4} )^{2} \)
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