Properties

Label 980.4.i.e
Level $980$
Weight $4$
Character orbit 980.i
Analytic conductor $57.822$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 980.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(57.8218718056\)
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 20)
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.

\(f(q)\) \(=\) \( q + (4 \zeta_{6} - 4) q^{3} - 5 \zeta_{6} q^{5} + 11 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (4 \zeta_{6} - 4) q^{3} - 5 \zeta_{6} q^{5} + 11 \zeta_{6} q^{9} + ( - 60 \zeta_{6} + 60) q^{11} + 86 q^{13} + 20 q^{15} + (18 \zeta_{6} - 18) q^{17} - 44 \zeta_{6} q^{19} - 48 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} - 152 q^{27} - 186 q^{29} + (176 \zeta_{6} - 176) q^{31} + 240 \zeta_{6} q^{33} - 254 \zeta_{6} q^{37} + (344 \zeta_{6} - 344) q^{39} + 186 q^{41} - 100 q^{43} + ( - 55 \zeta_{6} + 55) q^{45} - 168 \zeta_{6} q^{47} - 72 \zeta_{6} q^{51} + ( - 498 \zeta_{6} + 498) q^{53} - 300 q^{55} + 176 q^{57} + ( - 252 \zeta_{6} + 252) q^{59} + 58 \zeta_{6} q^{61} - 430 \zeta_{6} q^{65} + ( - 1036 \zeta_{6} + 1036) q^{67} + 192 q^{69} + 168 q^{71} + (506 \zeta_{6} - 506) q^{73} - 100 \zeta_{6} q^{75} - 272 \zeta_{6} q^{79} + ( - 311 \zeta_{6} + 311) q^{81} + 948 q^{83} + 90 q^{85} + ( - 744 \zeta_{6} + 744) q^{87} + 1014 \zeta_{6} q^{89} - 704 \zeta_{6} q^{93} + (220 \zeta_{6} - 220) q^{95} - 766 q^{97} + 660 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} - 5 q^{5} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} - 5 q^{5} + 11 q^{9} + 60 q^{11} + 172 q^{13} + 40 q^{15} - 18 q^{17} - 44 q^{19} - 48 q^{23} - 25 q^{25} - 304 q^{27} - 372 q^{29} - 176 q^{31} + 240 q^{33} - 254 q^{37} - 344 q^{39} + 372 q^{41} - 200 q^{43} + 55 q^{45} - 168 q^{47} - 72 q^{51} + 498 q^{53} - 600 q^{55} + 352 q^{57} + 252 q^{59} + 58 q^{61} - 430 q^{65} + 1036 q^{67} + 384 q^{69} + 336 q^{71} - 506 q^{73} - 100 q^{75} - 272 q^{79} + 311 q^{81} + 1896 q^{83} + 180 q^{85} + 744 q^{87} + 1014 q^{89} - 704 q^{93} - 220 q^{95} - 1532 q^{97} + 1320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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
0 −2.00000 + 3.46410i 0 −2.50000 4.33013i 0 0 0 5.50000 + 9.52628i 0
961.1 0 −2.00000 3.46410i 0 −2.50000 + 4.33013i 0 0 0 5.50000 9.52628i 0
\(n\): 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 980.4.i.e 2
7.b odd 2 1 980.4.i.n 2
7.c even 3 1 20.4.a.a 1
7.c even 3 1 inner 980.4.i.e 2
7.d odd 6 1 980.4.a.c 1
7.d odd 6 1 980.4.i.n 2
21.h odd 6 1 180.4.a.a 1
28.g odd 6 1 80.4.a.c 1
35.j even 6 1 100.4.a.a 1
35.l odd 12 2 100.4.c.a 2
56.k odd 6 1 320.4.a.k 1
56.p even 6 1 320.4.a.d 1
63.g even 3 1 1620.4.i.d 2
63.h even 3 1 1620.4.i.d 2
63.j odd 6 1 1620.4.i.j 2
63.n odd 6 1 1620.4.i.j 2
77.h odd 6 1 2420.4.a.d 1
84.n even 6 1 720.4.a.k 1
105.o odd 6 1 900.4.a.m 1
105.x even 12 2 900.4.d.k 2
112.u odd 12 2 1280.4.d.c 2
112.w even 12 2 1280.4.d.n 2
140.p odd 6 1 400.4.a.o 1
140.w even 12 2 400.4.c.j 2
280.bf even 6 1 1600.4.a.bl 1
280.bi odd 6 1 1600.4.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 7.c even 3 1
80.4.a.c 1 28.g odd 6 1
100.4.a.a 1 35.j even 6 1
100.4.c.a 2 35.l odd 12 2
180.4.a.a 1 21.h odd 6 1
320.4.a.d 1 56.p even 6 1
320.4.a.k 1 56.k odd 6 1
400.4.a.o 1 140.p odd 6 1
400.4.c.j 2 140.w even 12 2
720.4.a.k 1 84.n even 6 1
900.4.a.m 1 105.o odd 6 1
900.4.d.k 2 105.x even 12 2
980.4.a.c 1 7.d odd 6 1
980.4.i.e 2 1.a even 1 1 trivial
980.4.i.e 2 7.c even 3 1 inner
980.4.i.n 2 7.b odd 2 1
980.4.i.n 2 7.d odd 6 1
1280.4.d.c 2 112.u odd 12 2
1280.4.d.n 2 112.w even 12 2
1600.4.a.p 1 280.bi odd 6 1
1600.4.a.bl 1 280.bf even 6 1
1620.4.i.d 2 63.g even 3 1
1620.4.i.d 2 63.h even 3 1
1620.4.i.j 2 63.j odd 6 1
1620.4.i.j 2 63.n odd 6 1
2420.4.a.d 1 77.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(980, [\chi])\):

\( T_{3}^{2} + 4T_{3} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 60T_{11} + 3600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 60T + 3600 \) Copy content Toggle raw display
$13$ \( (T - 86)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
$19$ \( T^{2} + 44T + 1936 \) Copy content Toggle raw display
$23$ \( T^{2} + 48T + 2304 \) Copy content Toggle raw display
$29$ \( (T + 186)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 176T + 30976 \) Copy content Toggle raw display
$37$ \( T^{2} + 254T + 64516 \) Copy content Toggle raw display
$41$ \( (T - 186)^{2} \) Copy content Toggle raw display
$43$ \( (T + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 168T + 28224 \) Copy content Toggle raw display
$53$ \( T^{2} - 498T + 248004 \) Copy content Toggle raw display
$59$ \( T^{2} - 252T + 63504 \) Copy content Toggle raw display
$61$ \( T^{2} - 58T + 3364 \) Copy content Toggle raw display
$67$ \( T^{2} - 1036 T + 1073296 \) Copy content Toggle raw display
$71$ \( (T - 168)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 506T + 256036 \) Copy content Toggle raw display
$79$ \( T^{2} + 272T + 73984 \) Copy content Toggle raw display
$83$ \( (T - 948)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1014 T + 1028196 \) Copy content Toggle raw display
$97$ \( (T + 766)^{2} \) Copy content Toggle raw display
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