Properties

Label 49.6.a.f
Level $49$
Weight $6$
Character orbit 49.a
Self dual yes
Analytic conductor $7.859$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 5) q^{2} + ( - 6 \beta + 6) q^{3} + ( - 9 \beta + 7) q^{4} + (10 \beta + 4) q^{5} + ( - 30 \beta + 114) q^{6} + ( - 11 \beta + 1) q^{8} + ( - 36 \beta + 297) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 5) q^{2} + ( - 6 \beta + 6) q^{3} + ( - 9 \beta + 7) q^{4} + (10 \beta + 4) q^{5} + ( - 30 \beta + 114) q^{6} + ( - 11 \beta + 1) q^{8} + ( - 36 \beta + 297) q^{9} + (36 \beta - 120) q^{10} + (124 \beta + 136) q^{11} + ( - 42 \beta + 798) q^{12} + (126 \beta + 112) q^{13} + ( - 24 \beta - 816) q^{15} + (243 \beta - 65) q^{16} + ( - 76 \beta - 862) q^{17} + ( - 441 \beta + 1989) q^{18} + ( - 18 \beta + 1642) q^{19} + ( - 56 \beta - 1232) q^{20} + (360 \beta - 1056) q^{22} + ( - 568 \beta + 1328) q^{23} + ( - 6 \beta + 930) q^{24} + (180 \beta - 1709) q^{25} + (392 \beta - 1204) q^{26} + ( - 324 \beta + 3348) q^{27} + ( - 252 \beta + 3474) q^{29} + (720 \beta - 3744) q^{30} + (540 \beta - 260) q^{31} + (1389 \beta - 3759) q^{32} + ( - 816 \beta - 9600) q^{33} + (558 \beta - 3246) q^{34} + ( - 2601 \beta + 6615) q^{36} + ( - 540 \beta + 3386) q^{37} + ( - 1714 \beta + 8462) q^{38} + ( - 672 \beta - 9912) q^{39} + ( - 144 \beta - 1536) q^{40} + ( - 1092 \beta + 3570) q^{41} + (4788 \beta - 3904) q^{43} + ( - 1472 \beta - 14672) q^{44} + (2466 \beta - 3852) q^{45} + ( - 3600 \beta + 14592) q^{46} + (3748 \beta - 7724) q^{47} + (390 \beta - 20802) q^{48} + (2429 \beta - 11065) q^{50} + (5172 \beta + 1212) q^{51} + ( - 1260 \beta - 15092) q^{52} + (208 \beta + 4630) q^{53} + ( - 4644 \beta + 21276) q^{54} + (3096 \beta + 17904) q^{55} + ( - 9852 \beta + 11364) q^{57} + ( - 4482 \beta + 20898) q^{58} + ( - 2050 \beta + 22994) q^{59} + (7392 \beta - 2688) q^{60} + ( - 4806 \beta + 34780) q^{61} + (2420 \beta - 8860) q^{62} + (1539 \beta - 36161) q^{64} + (2884 \beta + 18088) q^{65} + (6336 \beta - 36576) q^{66} + (1944 \beta + 11420) q^{67} + (7910 \beta + 3542) q^{68} + ( - 7968 \beta + 55680) q^{69} + (4200 \beta + 46608) q^{71} + ( - 2907 \beta + 5841) q^{72} + ( - 5256 \beta - 6098) q^{73} + ( - 5546 \beta + 24490) q^{74} + (10254 \beta - 25374) q^{75} + ( - 14742 \beta + 13762) q^{76} + (7224 \beta - 40152) q^{78} + ( - 14904 \beta + 33080) q^{79} + (2752 \beta + 33760) q^{80} + ( - 11340 \beta - 24867) q^{81} + ( - 7938 \beta + 33138) q^{82} + (15750 \beta - 66654) q^{83} + ( - 9684 \beta - 14088) q^{85} + (23056 \beta - 86552) q^{86} + ( - 20844 \beta + 42012) q^{87} + ( - 2736 \beta - 18960) q^{88} + ( - 22208 \beta - 31034) q^{89} + (13716 \beta - 53784) q^{90} + ( - 10816 \beta + 80864) q^{92} + (1560 \beta - 46920) q^{93} + (22716 \beta - 91092) q^{94} + (16168 \beta + 4048) q^{95} + (22554 \beta - 139230) q^{96} + (8820 \beta - 14798) q^{97} + (27468 \beta - 22104) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{2} + 6 q^{3} + 5 q^{4} + 18 q^{5} + 198 q^{6} - 9 q^{8} + 558 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 9 q^{2} + 6 q^{3} + 5 q^{4} + 18 q^{5} + 198 q^{6} - 9 q^{8} + 558 q^{9} - 204 q^{10} + 396 q^{11} + 1554 q^{12} + 350 q^{13} - 1656 q^{15} + 113 q^{16} - 1800 q^{17} + 3537 q^{18} + 3266 q^{19} - 2520 q^{20} - 1752 q^{22} + 2088 q^{23} + 1854 q^{24} - 3238 q^{25} - 2016 q^{26} + 6372 q^{27} + 6696 q^{29} - 6768 q^{30} + 20 q^{31} - 6129 q^{32} - 20016 q^{33} - 5934 q^{34} + 10629 q^{36} + 6232 q^{37} + 15210 q^{38} - 20496 q^{39} - 3216 q^{40} + 6048 q^{41} - 3020 q^{43} - 30816 q^{44} - 5238 q^{45} + 25584 q^{46} - 11700 q^{47} - 41214 q^{48} - 19701 q^{50} + 7596 q^{51} - 31444 q^{52} + 9468 q^{53} + 37908 q^{54} + 38904 q^{55} + 12876 q^{57} + 37314 q^{58} + 43938 q^{59} + 2016 q^{60} + 64754 q^{61} - 15300 q^{62} - 70783 q^{64} + 39060 q^{65} - 66816 q^{66} + 24784 q^{67} + 14994 q^{68} + 103392 q^{69} + 97416 q^{71} + 8775 q^{72} - 17452 q^{73} + 43434 q^{74} - 40494 q^{75} + 12782 q^{76} - 73080 q^{78} + 51256 q^{79} + 70272 q^{80} - 61074 q^{81} + 58338 q^{82} - 117558 q^{83} - 37860 q^{85} - 150048 q^{86} + 63180 q^{87} - 40656 q^{88} - 84276 q^{89} - 93852 q^{90} + 150912 q^{92} - 92280 q^{93} - 159468 q^{94} + 24264 q^{95} - 255906 q^{96} - 20776 q^{97} - 16740 q^{99}+O(q^{100}) \) 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
4.27492
−3.27492
0.725083 −19.6495 −31.4743 46.7492 −14.2475 0 −46.0241 143.103 33.8970
1.2 8.27492 25.6495 36.4743 −28.7492 212.248 0 37.0241 414.897 −237.897
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 49.6.a.f 2
3.b odd 2 1 441.6.a.l 2
4.b odd 2 1 784.6.a.v 2
7.b odd 2 1 7.6.a.b 2
7.c even 3 2 49.6.c.d 4
7.d odd 6 2 49.6.c.e 4
21.c even 2 1 63.6.a.f 2
28.d even 2 1 112.6.a.h 2
35.c odd 2 1 175.6.a.c 2
35.f even 4 2 175.6.b.c 4
56.e even 2 1 448.6.a.u 2
56.h odd 2 1 448.6.a.w 2
77.b even 2 1 847.6.a.c 2
84.h odd 2 1 1008.6.a.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 7.b odd 2 1
49.6.a.f 2 1.a even 1 1 trivial
49.6.c.d 4 7.c even 3 2
49.6.c.e 4 7.d odd 6 2
63.6.a.f 2 21.c even 2 1
112.6.a.h 2 28.d even 2 1
175.6.a.c 2 35.c odd 2 1
175.6.b.c 4 35.f even 4 2
441.6.a.l 2 3.b odd 2 1
448.6.a.u 2 56.e even 2 1
448.6.a.w 2 56.h odd 2 1
784.6.a.v 2 4.b odd 2 1
847.6.a.c 2 77.b even 2 1
1008.6.a.bq 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{2} - 9T_{2} + 6 \) Copy content Toggle raw display
\( T_{3}^{2} - 6T_{3} - 504 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 9T + 6 \) Copy content Toggle raw display
$3$ \( T^{2} - 6T - 504 \) Copy content Toggle raw display
$5$ \( T^{2} - 18T - 1344 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 396T - 179904 \) Copy content Toggle raw display
$13$ \( T^{2} - 350T - 195608 \) Copy content Toggle raw display
$17$ \( T^{2} + 1800 T + 727692 \) Copy content Toggle raw display
$19$ \( T^{2} - 3266 T + 2662072 \) Copy content Toggle raw display
$23$ \( T^{2} - 2088 T - 3507456 \) Copy content Toggle raw display
$29$ \( T^{2} - 6696 T + 10304172 \) Copy content Toggle raw display
$31$ \( T^{2} - 20T - 4155200 \) Copy content Toggle raw display
$37$ \( T^{2} - 6232 T + 5554156 \) Copy content Toggle raw display
$41$ \( T^{2} - 6048 T - 7848036 \) Copy content Toggle raw display
$43$ \( T^{2} + 3020 T - 324400352 \) Copy content Toggle raw display
$47$ \( T^{2} + 11700 T - 165954432 \) Copy content Toggle raw display
$53$ \( T^{2} - 9468 T + 21794244 \) Copy content Toggle raw display
$59$ \( T^{2} - 43938 T + 422751336 \) Copy content Toggle raw display
$61$ \( T^{2} - 64754 T + 719128816 \) Copy content Toggle raw display
$67$ \( T^{2} - 24784 T + 99708976 \) Copy content Toggle raw display
$71$ \( T^{2} - 97416 T + 2121099264 \) Copy content Toggle raw display
$73$ \( T^{2} + 17452 T - 317520812 \) Copy content Toggle raw display
$79$ \( T^{2} - 51256 T - 2508546944 \) Copy content Toggle raw display
$83$ \( T^{2} + 117558 T - 79919784 \) Copy content Toggle raw display
$89$ \( T^{2} + 84276 T - 5252421468 \) Copy content Toggle raw display
$97$ \( T^{2} + 20776 T - 1000631156 \) Copy content Toggle raw display
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