Properties

Label 490.4.e.a
Level $490$
Weight $4$
Character orbit 490.e
Analytic conductor $28.911$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 490.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.9109359028\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
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.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{3} + (4 \zeta_{6} - 4) q^{4} + 5 \zeta_{6} q^{5} + 16 q^{6} + 8 q^{8} - 37 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{6} q^{2} + (8 \zeta_{6} - 8) q^{3} + (4 \zeta_{6} - 4) q^{4} + 5 \zeta_{6} q^{5} + 16 q^{6} + 8 q^{8} - 37 \zeta_{6} q^{9} + ( - 10 \zeta_{6} + 10) q^{10} + (12 \zeta_{6} - 12) q^{11} - 32 \zeta_{6} q^{12} + 58 q^{13} - 40 q^{15} - 16 \zeta_{6} q^{16} + ( - 66 \zeta_{6} + 66) q^{17} + (74 \zeta_{6} - 74) q^{18} - 100 \zeta_{6} q^{19} - 20 q^{20} + 24 q^{22} - 132 \zeta_{6} q^{23} + (64 \zeta_{6} - 64) q^{24} + (25 \zeta_{6} - 25) q^{25} - 116 \zeta_{6} q^{26} + 80 q^{27} - 90 q^{29} + 80 \zeta_{6} q^{30} + ( - 152 \zeta_{6} + 152) q^{31} + (32 \zeta_{6} - 32) q^{32} - 96 \zeta_{6} q^{33} - 132 q^{34} + 148 q^{36} + 34 \zeta_{6} q^{37} + (200 \zeta_{6} - 200) q^{38} + (464 \zeta_{6} - 464) q^{39} + 40 \zeta_{6} q^{40} + 438 q^{41} + 32 q^{43} - 48 \zeta_{6} q^{44} + ( - 185 \zeta_{6} + 185) q^{45} + (264 \zeta_{6} - 264) q^{46} - 204 \zeta_{6} q^{47} + 128 q^{48} + 50 q^{50} + 528 \zeta_{6} q^{51} + (232 \zeta_{6} - 232) q^{52} + (222 \zeta_{6} - 222) q^{53} - 160 \zeta_{6} q^{54} - 60 q^{55} + 800 q^{57} + 180 \zeta_{6} q^{58} + ( - 420 \zeta_{6} + 420) q^{59} + ( - 160 \zeta_{6} + 160) q^{60} + 902 \zeta_{6} q^{61} - 304 q^{62} + 64 q^{64} + 290 \zeta_{6} q^{65} + (192 \zeta_{6} - 192) q^{66} + ( - 1024 \zeta_{6} + 1024) q^{67} + 264 \zeta_{6} q^{68} + 1056 q^{69} + 432 q^{71} - 296 \zeta_{6} q^{72} + ( - 362 \zeta_{6} + 362) q^{73} + ( - 68 \zeta_{6} + 68) q^{74} - 200 \zeta_{6} q^{75} + 400 q^{76} + 928 q^{78} + 160 \zeta_{6} q^{79} + ( - 80 \zeta_{6} + 80) q^{80} + ( - 359 \zeta_{6} + 359) q^{81} - 876 \zeta_{6} q^{82} - 72 q^{83} + 330 q^{85} - 64 \zeta_{6} q^{86} + ( - 720 \zeta_{6} + 720) q^{87} + (96 \zeta_{6} - 96) q^{88} + 810 \zeta_{6} q^{89} - 370 q^{90} + 528 q^{92} + 1216 \zeta_{6} q^{93} + (408 \zeta_{6} - 408) q^{94} + ( - 500 \zeta_{6} + 500) q^{95} - 256 \zeta_{6} q^{96} - 1106 q^{97} + 444 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 8 q^{3} - 4 q^{4} + 5 q^{5} + 32 q^{6} + 16 q^{8} - 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 8 q^{3} - 4 q^{4} + 5 q^{5} + 32 q^{6} + 16 q^{8} - 37 q^{9} + 10 q^{10} - 12 q^{11} - 32 q^{12} + 116 q^{13} - 80 q^{15} - 16 q^{16} + 66 q^{17} - 74 q^{18} - 100 q^{19} - 40 q^{20} + 48 q^{22} - 132 q^{23} - 64 q^{24} - 25 q^{25} - 116 q^{26} + 160 q^{27} - 180 q^{29} + 80 q^{30} + 152 q^{31} - 32 q^{32} - 96 q^{33} - 264 q^{34} + 296 q^{36} + 34 q^{37} - 200 q^{38} - 464 q^{39} + 40 q^{40} + 876 q^{41} + 64 q^{43} - 48 q^{44} + 185 q^{45} - 264 q^{46} - 204 q^{47} + 256 q^{48} + 100 q^{50} + 528 q^{51} - 232 q^{52} - 222 q^{53} - 160 q^{54} - 120 q^{55} + 1600 q^{57} + 180 q^{58} + 420 q^{59} + 160 q^{60} + 902 q^{61} - 608 q^{62} + 128 q^{64} + 290 q^{65} - 192 q^{66} + 1024 q^{67} + 264 q^{68} + 2112 q^{69} + 864 q^{71} - 296 q^{72} + 362 q^{73} + 68 q^{74} - 200 q^{75} + 800 q^{76} + 1856 q^{78} + 160 q^{79} + 80 q^{80} + 359 q^{81} - 876 q^{82} - 144 q^{83} + 660 q^{85} - 64 q^{86} + 720 q^{87} - 96 q^{88} + 810 q^{89} - 740 q^{90} + 1056 q^{92} + 1216 q^{93} - 408 q^{94} + 500 q^{95} - 256 q^{96} - 2212 q^{97} + 888 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 1.73205i −4.00000 + 6.92820i −2.00000 + 3.46410i 2.50000 + 4.33013i 16.0000 0 8.00000 −18.5000 32.0429i 5.00000 8.66025i
471.1 −1.00000 + 1.73205i −4.00000 6.92820i −2.00000 3.46410i 2.50000 4.33013i 16.0000 0 8.00000 −18.5000 + 32.0429i 5.00000 + 8.66025i
\(n\): 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 490.4.e.a 2
7.b odd 2 1 490.4.e.i 2
7.c even 3 1 490.4.a.o 1
7.c even 3 1 inner 490.4.e.a 2
7.d odd 6 1 10.4.a.a 1
7.d odd 6 1 490.4.e.i 2
21.g even 6 1 90.4.a.a 1
28.f even 6 1 80.4.a.f 1
35.i odd 6 1 50.4.a.c 1
35.j even 6 1 2450.4.a.b 1
35.k even 12 2 50.4.b.a 2
56.j odd 6 1 320.4.a.m 1
56.m even 6 1 320.4.a.b 1
63.i even 6 1 810.4.e.w 2
63.k odd 6 1 810.4.e.c 2
63.s even 6 1 810.4.e.w 2
63.t odd 6 1 810.4.e.c 2
77.i even 6 1 1210.4.a.b 1
84.j odd 6 1 720.4.a.j 1
91.s odd 6 1 1690.4.a.a 1
105.p even 6 1 450.4.a.q 1
105.w odd 12 2 450.4.c.d 2
112.v even 12 2 1280.4.d.g 2
112.x odd 12 2 1280.4.d.j 2
140.s even 6 1 400.4.a.b 1
140.x odd 12 2 400.4.c.c 2
280.ba even 6 1 1600.4.a.bx 1
280.bk odd 6 1 1600.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 7.d odd 6 1
50.4.a.c 1 35.i odd 6 1
50.4.b.a 2 35.k even 12 2
80.4.a.f 1 28.f even 6 1
90.4.a.a 1 21.g even 6 1
320.4.a.b 1 56.m even 6 1
320.4.a.m 1 56.j odd 6 1
400.4.a.b 1 140.s even 6 1
400.4.c.c 2 140.x odd 12 2
450.4.a.q 1 105.p even 6 1
450.4.c.d 2 105.w odd 12 2
490.4.a.o 1 7.c even 3 1
490.4.e.a 2 1.a even 1 1 trivial
490.4.e.a 2 7.c even 3 1 inner
490.4.e.i 2 7.b odd 2 1
490.4.e.i 2 7.d odd 6 1
720.4.a.j 1 84.j odd 6 1
810.4.e.c 2 63.k odd 6 1
810.4.e.c 2 63.t odd 6 1
810.4.e.w 2 63.i even 6 1
810.4.e.w 2 63.s even 6 1
1210.4.a.b 1 77.i even 6 1
1280.4.d.g 2 112.v even 12 2
1280.4.d.j 2 112.x odd 12 2
1600.4.a.d 1 280.bk odd 6 1
1600.4.a.bx 1 280.ba even 6 1
1690.4.a.a 1 91.s odd 6 1
2450.4.a.b 1 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{2} + 8T_{3} + 64 \) Copy content Toggle raw display
\( T_{11}^{2} + 12T_{11} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$5$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$13$ \( (T - 58)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 66T + 4356 \) Copy content Toggle raw display
$19$ \( T^{2} + 100T + 10000 \) Copy content Toggle raw display
$23$ \( T^{2} + 132T + 17424 \) Copy content Toggle raw display
$29$ \( (T + 90)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 152T + 23104 \) Copy content Toggle raw display
$37$ \( T^{2} - 34T + 1156 \) Copy content Toggle raw display
$41$ \( (T - 438)^{2} \) Copy content Toggle raw display
$43$ \( (T - 32)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 204T + 41616 \) Copy content Toggle raw display
$53$ \( T^{2} + 222T + 49284 \) Copy content Toggle raw display
$59$ \( T^{2} - 420T + 176400 \) Copy content Toggle raw display
$61$ \( T^{2} - 902T + 813604 \) Copy content Toggle raw display
$67$ \( T^{2} - 1024 T + 1048576 \) Copy content Toggle raw display
$71$ \( (T - 432)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 362T + 131044 \) Copy content Toggle raw display
$79$ \( T^{2} - 160T + 25600 \) Copy content Toggle raw display
$83$ \( (T + 72)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 810T + 656100 \) Copy content Toggle raw display
$97$ \( (T + 1106)^{2} \) Copy content Toggle raw display
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