Properties

Label 49.6.c.b
Level $49$
Weight $6$
Character orbit 49.c
Analytic conductor $7.859$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 10 \zeta_{6} q^{2} + (14 \zeta_{6} - 14) q^{3} + (68 \zeta_{6} - 68) q^{4} - 56 \zeta_{6} q^{5} - 140 q^{6} - 360 q^{8} + 47 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 10 \zeta_{6} q^{2} + (14 \zeta_{6} - 14) q^{3} + (68 \zeta_{6} - 68) q^{4} - 56 \zeta_{6} q^{5} - 140 q^{6} - 360 q^{8} + 47 \zeta_{6} q^{9} + ( - 560 \zeta_{6} + 560) q^{10} + (232 \zeta_{6} - 232) q^{11} - 952 \zeta_{6} q^{12} + 140 q^{13} + 784 q^{15} - 1424 \zeta_{6} q^{16} + (1722 \zeta_{6} - 1722) q^{17} + (470 \zeta_{6} - 470) q^{18} - 98 \zeta_{6} q^{19} + 3808 q^{20} - 2320 q^{22} - 1824 \zeta_{6} q^{23} + ( - 5040 \zeta_{6} + 5040) q^{24} + (11 \zeta_{6} - 11) q^{25} + 1400 \zeta_{6} q^{26} - 4060 q^{27} + 3418 q^{29} + 7840 \zeta_{6} q^{30} + (7644 \zeta_{6} - 7644) q^{31} + ( - 2720 \zeta_{6} + 2720) q^{32} - 3248 \zeta_{6} q^{33} - 17220 q^{34} - 3196 q^{36} + 10398 \zeta_{6} q^{37} + ( - 980 \zeta_{6} + 980) q^{38} + (1960 \zeta_{6} - 1960) q^{39} + 20160 \zeta_{6} q^{40} + 17962 q^{41} + 10880 q^{43} - 15776 \zeta_{6} q^{44} + ( - 2632 \zeta_{6} + 2632) q^{45} + ( - 18240 \zeta_{6} + 18240) q^{46} + 9324 \zeta_{6} q^{47} + 19936 q^{48} - 110 q^{50} - 24108 \zeta_{6} q^{51} + (9520 \zeta_{6} - 9520) q^{52} + (2262 \zeta_{6} - 2262) q^{53} - 40600 \zeta_{6} q^{54} + 12992 q^{55} + 1372 q^{57} + 34180 \zeta_{6} q^{58} + (2730 \zeta_{6} - 2730) q^{59} + (53312 \zeta_{6} - 53312) q^{60} + 25648 \zeta_{6} q^{61} - 76440 q^{62} - 18368 q^{64} - 7840 \zeta_{6} q^{65} + ( - 32480 \zeta_{6} + 32480) q^{66} + ( - 48404 \zeta_{6} + 48404) q^{67} - 117096 \zeta_{6} q^{68} + 25536 q^{69} - 58560 q^{71} - 16920 \zeta_{6} q^{72} + ( - 68082 \zeta_{6} + 68082) q^{73} + (103980 \zeta_{6} - 103980) q^{74} - 154 \zeta_{6} q^{75} + 6664 q^{76} - 19600 q^{78} - 31784 \zeta_{6} q^{79} + (79744 \zeta_{6} - 79744) q^{80} + ( - 45419 \zeta_{6} + 45419) q^{81} + 179620 \zeta_{6} q^{82} + 20538 q^{83} + 96432 q^{85} + 108800 \zeta_{6} q^{86} + (47852 \zeta_{6} - 47852) q^{87} + ( - 83520 \zeta_{6} + 83520) q^{88} - 50582 \zeta_{6} q^{89} + 26320 q^{90} + 124032 q^{92} - 107016 \zeta_{6} q^{93} + (93240 \zeta_{6} - 93240) q^{94} + (5488 \zeta_{6} - 5488) q^{95} + 38080 \zeta_{6} q^{96} + 58506 q^{97} - 10904 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{2} - 14 q^{3} - 68 q^{4} - 56 q^{5} - 280 q^{6} - 720 q^{8} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{2} - 14 q^{3} - 68 q^{4} - 56 q^{5} - 280 q^{6} - 720 q^{8} + 47 q^{9} + 560 q^{10} - 232 q^{11} - 952 q^{12} + 280 q^{13} + 1568 q^{15} - 1424 q^{16} - 1722 q^{17} - 470 q^{18} - 98 q^{19} + 7616 q^{20} - 4640 q^{22} - 1824 q^{23} + 5040 q^{24} - 11 q^{25} + 1400 q^{26} - 8120 q^{27} + 6836 q^{29} + 7840 q^{30} - 7644 q^{31} + 2720 q^{32} - 3248 q^{33} - 34440 q^{34} - 6392 q^{36} + 10398 q^{37} + 980 q^{38} - 1960 q^{39} + 20160 q^{40} + 35924 q^{41} + 21760 q^{43} - 15776 q^{44} + 2632 q^{45} + 18240 q^{46} + 9324 q^{47} + 39872 q^{48} - 220 q^{50} - 24108 q^{51} - 9520 q^{52} - 2262 q^{53} - 40600 q^{54} + 25984 q^{55} + 2744 q^{57} + 34180 q^{58} - 2730 q^{59} - 53312 q^{60} + 25648 q^{61} - 152880 q^{62} - 36736 q^{64} - 7840 q^{65} + 32480 q^{66} + 48404 q^{67} - 117096 q^{68} + 51072 q^{69} - 117120 q^{71} - 16920 q^{72} + 68082 q^{73} - 103980 q^{74} - 154 q^{75} + 13328 q^{76} - 39200 q^{78} - 31784 q^{79} - 79744 q^{80} + 45419 q^{81} + 179620 q^{82} + 41076 q^{83} + 192864 q^{85} + 108800 q^{86} - 47852 q^{87} + 83520 q^{88} - 50582 q^{89} + 52640 q^{90} + 248064 q^{92} - 107016 q^{93} - 93240 q^{94} - 5488 q^{95} + 38080 q^{96} + 117012 q^{97} - 21808 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
5.00000 + 8.66025i −7.00000 + 12.1244i −34.0000 + 58.8897i −28.0000 48.4974i −140.000 0 −360.000 23.5000 + 40.7032i 280.000 484.974i
30.1 5.00000 8.66025i −7.00000 12.1244i −34.0000 58.8897i −28.0000 + 48.4974i −140.000 0 −360.000 23.5000 40.7032i 280.000 + 484.974i
\(n\): 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.b 2
7.b odd 2 1 49.6.c.c 2
7.c even 3 1 49.6.a.a 1
7.c even 3 1 inner 49.6.c.b 2
7.d odd 6 1 7.6.a.a 1
7.d odd 6 1 49.6.c.c 2
21.g even 6 1 63.6.a.e 1
21.h odd 6 1 441.6.a.k 1
28.f even 6 1 112.6.a.g 1
28.g odd 6 1 784.6.a.c 1
35.i odd 6 1 175.6.a.b 1
35.k even 12 2 175.6.b.a 2
56.j odd 6 1 448.6.a.m 1
56.m even 6 1 448.6.a.c 1
77.i even 6 1 847.6.a.b 1
84.j odd 6 1 1008.6.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 7.d odd 6 1
49.6.a.a 1 7.c even 3 1
49.6.c.b 2 1.a even 1 1 trivial
49.6.c.b 2 7.c even 3 1 inner
49.6.c.c 2 7.b odd 2 1
49.6.c.c 2 7.d odd 6 1
63.6.a.e 1 21.g even 6 1
112.6.a.g 1 28.f even 6 1
175.6.a.b 1 35.i odd 6 1
175.6.b.a 2 35.k even 12 2
441.6.a.k 1 21.h odd 6 1
448.6.a.c 1 56.m even 6 1
448.6.a.m 1 56.j odd 6 1
784.6.a.c 1 28.g odd 6 1
847.6.a.b 1 77.i even 6 1
1008.6.a.y 1 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 \(S_{6}^{\mathrm{new}}(49, [\chi])\):

\( T_{2}^{2} - 10T_{2} + 100 \) Copy content Toggle raw display
\( T_{3}^{2} + 14T_{3} + 196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$3$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$5$ \( T^{2} + 56T + 3136 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 232T + 53824 \) Copy content Toggle raw display
$13$ \( (T - 140)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 1722 T + 2965284 \) Copy content Toggle raw display
$19$ \( T^{2} + 98T + 9604 \) Copy content Toggle raw display
$23$ \( T^{2} + 1824 T + 3326976 \) Copy content Toggle raw display
$29$ \( (T - 3418)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 7644 T + 58430736 \) Copy content Toggle raw display
$37$ \( T^{2} - 10398 T + 108118404 \) Copy content Toggle raw display
$41$ \( (T - 17962)^{2} \) Copy content Toggle raw display
$43$ \( (T - 10880)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 9324 T + 86936976 \) Copy content Toggle raw display
$53$ \( T^{2} + 2262 T + 5116644 \) Copy content Toggle raw display
$59$ \( T^{2} + 2730 T + 7452900 \) Copy content Toggle raw display
$61$ \( T^{2} - 25648 T + 657819904 \) Copy content Toggle raw display
$67$ \( T^{2} - 48404 T + 2342947216 \) Copy content Toggle raw display
$71$ \( (T + 58560)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 68082 T + 4635158724 \) Copy content Toggle raw display
$79$ \( T^{2} + 31784 T + 1010222656 \) Copy content Toggle raw display
$83$ \( (T - 20538)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 50582 T + 2558538724 \) Copy content Toggle raw display
$97$ \( (T - 58506)^{2} \) Copy content Toggle raw display
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