Properties

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

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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: \(\Q(\sqrt{-3}, \sqrt{-13})\)
Defining polynomial: \( x^{4} - 13x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_1 + 2) q^{2} + \beta_{2} q^{3} + 28 \beta_1 q^{4} + (3 \beta_{3} - 3 \beta_{2}) q^{5} + 2 \beta_{3} q^{6} + 120 q^{8} + (381 \beta_1 - 381) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_1 + 2) q^{2} + \beta_{2} q^{3} + 28 \beta_1 q^{4} + (3 \beta_{3} - 3 \beta_{2}) q^{5} + 2 \beta_{3} q^{6} + 120 q^{8} + (381 \beta_1 - 381) q^{9} - 6 \beta_{2} q^{10} + 284 \beta_1 q^{11} + ( - 28 \beta_{3} + 28 \beta_{2}) q^{12} - 21 \beta_{3} q^{13} + 1872 q^{15} + (656 \beta_1 - 656) q^{16} - 6 \beta_{2} q^{17} + 762 \beta_1 q^{18} + (87 \beta_{3} - 87 \beta_{2}) q^{19} + 84 \beta_{3} q^{20} + 568 q^{22} + (1496 \beta_1 - 1496) q^{23} + 120 \beta_{2} q^{24} - 2491 \beta_1 q^{25} + ( - 42 \beta_{3} + 42 \beta_{2}) q^{26} - 138 \beta_{3} q^{27} - 4366 q^{29} + ( - 3744 \beta_1 + 3744) q^{30} - 258 \beta_{2} q^{31} + 5152 \beta_1 q^{32} + ( - 284 \beta_{3} + 284 \beta_{2}) q^{33} - 12 \beta_{3} q^{34} - 10668 q^{36} + ( - 12630 \beta_1 + 12630) q^{37} - 174 \beta_{2} q^{38} - 13104 \beta_1 q^{39} + (360 \beta_{3} - 360 \beta_{2}) q^{40} + 378 \beta_{3} q^{41} - 1356 q^{43} + (7952 \beta_1 - 7952) q^{44} + 1143 \beta_{2} q^{45} + 2992 \beta_1 q^{46} + (402 \beta_{3} - 402 \beta_{2}) q^{47} - 656 \beta_{3} q^{48} - 4982 q^{50} + ( - 3744 \beta_1 + 3744) q^{51} - 588 \beta_{2} q^{52} - 14150 \beta_1 q^{53} + ( - 276 \beta_{3} + 276 \beta_{2}) q^{54} + 852 \beta_{3} q^{55} + 54288 q^{57} + (8732 \beta_1 - 8732) q^{58} - 1497 \beta_{2} q^{59} + 52416 \beta_1 q^{60} + ( - 1425 \beta_{3} + 1425 \beta_{2}) q^{61} - 516 \beta_{3} q^{62} - 10688 q^{64} + (39312 \beta_1 - 39312) q^{65} + 568 \beta_{2} q^{66} + 3644 \beta_1 q^{67} + (168 \beta_{3} - 168 \beta_{2}) q^{68} - 1496 \beta_{3} q^{69} + 35632 q^{71} + (45720 \beta_1 - 45720) q^{72} + 1632 \beta_{2} q^{73} - 25260 \beta_1 q^{74} + (2491 \beta_{3} - 2491 \beta_{2}) q^{75} + 2436 \beta_{3} q^{76} - 26208 q^{78} + ( - 54616 \beta_1 + 54616) q^{79} + 1968 \beta_{2} q^{80} + 6471 \beta_1 q^{81} + (756 \beta_{3} - 756 \beta_{2}) q^{82} + 21 \beta_{3} q^{83} - 11232 q^{85} + (2712 \beta_1 - 2712) q^{86} - 4366 \beta_{2} q^{87} + 34080 \beta_1 q^{88} + ( - 816 \beta_{3} + 816 \beta_{2}) q^{89} + 2286 \beta_{3} q^{90} - 41888 q^{92} + ( - 160992 \beta_1 + 160992) q^{93} - 804 \beta_{2} q^{94} - 162864 \beta_1 q^{95} + ( - 5152 \beta_{3} + 5152 \beta_{2}) q^{96} - 7350 \beta_{3} q^{97} - 108204 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 56 q^{4} + 480 q^{8} - 762 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 56 q^{4} + 480 q^{8} - 762 q^{9} + 568 q^{11} + 7488 q^{15} - 1312 q^{16} + 1524 q^{18} + 2272 q^{22} - 2992 q^{23} - 4982 q^{25} - 17464 q^{29} + 7488 q^{30} + 10304 q^{32} - 42672 q^{36} + 25260 q^{37} - 26208 q^{39} - 5424 q^{43} - 15904 q^{44} + 5984 q^{46} - 19928 q^{50} + 7488 q^{51} - 28300 q^{53} + 217152 q^{57} - 17464 q^{58} + 104832 q^{60} - 42752 q^{64} - 78624 q^{65} + 7288 q^{67} + 142528 q^{71} - 91440 q^{72} - 50520 q^{74} - 104832 q^{78} + 109232 q^{79} + 12942 q^{81} - 44928 q^{85} - 5424 q^{86} + 68160 q^{88} - 167552 q^{92} + 321984 q^{93} - 325728 q^{95} - 432816 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 13x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 13 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{3} + 52\nu ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{3} + 104\nu ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 13\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{3} + 26\beta_{2} ) / 12 \) Copy content Toggle raw display

Character values

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

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

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
−3.12250 + 1.80278i
3.12250 1.80278i
−3.12250 1.80278i
3.12250 + 1.80278i
1.00000 + 1.73205i −12.4900 + 21.6333i 14.0000 24.2487i −37.4700 64.8999i −49.9600 0 120.000 −190.500 329.956i 74.9400 129.800i
18.2 1.00000 + 1.73205i 12.4900 21.6333i 14.0000 24.2487i 37.4700 + 64.8999i 49.9600 0 120.000 −190.500 329.956i −74.9400 + 129.800i
30.1 1.00000 1.73205i −12.4900 21.6333i 14.0000 + 24.2487i −37.4700 + 64.8999i −49.9600 0 120.000 −190.500 + 329.956i 74.9400 + 129.800i
30.2 1.00000 1.73205i 12.4900 + 21.6333i 14.0000 + 24.2487i 37.4700 64.8999i 49.9600 0 120.000 −190.500 + 329.956i −74.9400 129.800i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.6.c.g 4
7.b odd 2 1 inner 49.6.c.g 4
7.c even 3 1 49.6.a.c 2
7.c even 3 1 inner 49.6.c.g 4
7.d odd 6 1 49.6.a.c 2
7.d odd 6 1 inner 49.6.c.g 4
21.g even 6 1 441.6.a.u 2
21.h odd 6 1 441.6.a.u 2
28.f even 6 1 784.6.a.z 2
28.g odd 6 1 784.6.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.c 2 7.c even 3 1
49.6.a.c 2 7.d odd 6 1
49.6.c.g 4 1.a even 1 1 trivial
49.6.c.g 4 7.b odd 2 1 inner
49.6.c.g 4 7.c even 3 1 inner
49.6.c.g 4 7.d odd 6 1 inner
441.6.a.u 2 21.g even 6 1
441.6.a.u 2 21.h odd 6 1
784.6.a.z 2 28.f even 6 1
784.6.a.z 2 28.g odd 6 1

Hecke kernels

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

\( T_{2}^{2} - 2T_{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{4} + 624T_{3}^{2} + 389376 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 624 T^{2} + 389376 \) Copy content Toggle raw display
$5$ \( T^{4} + 5616 T^{2} + \cdots + 31539456 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 284 T + 80656)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 275184)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 22464 T^{2} + \cdots + 504631296 \) Copy content Toggle raw display
$19$ \( T^{4} + 4723056 T^{2} + \cdots + 22307257979136 \) Copy content Toggle raw display
$23$ \( (T^{2} + 1496 T + 2238016)^{2} \) Copy content Toggle raw display
$29$ \( (T + 4366)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 41535936 T^{2} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( (T^{2} - 12630 T + 159516900)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 89159616)^{2} \) Copy content Toggle raw display
$43$ \( (T + 1356)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 100840896 T^{2} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{2} + 14150 T + 200222500)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 1398389616 T^{2} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{4} + 1267110000 T^{2} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{2} - 3644 T + 13278736)^{2} \) Copy content Toggle raw display
$71$ \( (T - 35632)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 1661976576 T^{2} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{2} - 54616 T + 2982907456)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 275184)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 415494144 T^{2} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{2} - 33710040000)^{2} \) Copy content Toggle raw display
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