Properties

Label 49.6.c.c
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.c 2
7.b odd 2 1 49.6.c.b 2
7.c even 3 1 7.6.a.a 1
7.c even 3 1 inner 49.6.c.c 2
7.d odd 6 1 49.6.a.a 1
7.d odd 6 1 49.6.c.b 2
21.g even 6 1 441.6.a.k 1
21.h odd 6 1 63.6.a.e 1
28.f even 6 1 784.6.a.c 1
28.g odd 6 1 112.6.a.g 1
35.j even 6 1 175.6.a.b 1
35.l odd 12 2 175.6.b.a 2
56.k odd 6 1 448.6.a.c 1
56.p even 6 1 448.6.a.m 1
77.h odd 6 1 847.6.a.b 1
84.n even 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.c even 3 1
49.6.a.a 1 7.d odd 6 1
49.6.c.b 2 7.b odd 2 1
49.6.c.b 2 7.d odd 6 1
49.6.c.c 2 1.a even 1 1 trivial
49.6.c.c 2 7.c even 3 1 inner
63.6.a.e 1 21.h odd 6 1
112.6.a.g 1 28.g odd 6 1
175.6.a.b 1 35.j even 6 1
175.6.b.a 2 35.l odd 12 2
441.6.a.k 1 21.g even 6 1
448.6.a.c 1 56.k odd 6 1
448.6.a.m 1 56.p even 6 1
784.6.a.c 1 28.f even 6 1
847.6.a.b 1 77.h odd 6 1
1008.6.a.y 1 84.n even 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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