Properties

Label 720.2.z.d
Level $720$
Weight $2$
Character orbit 720.z
Analytic conductor $5.749$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.z (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - i + 1) q^{2} - 2 i q^{4} + ( - i + 2) q^{5} + ( - 3 i - 3) q^{7} + ( - 2 i - 2) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i + 1) q^{2} - 2 i q^{4} + ( - i + 2) q^{5} + ( - 3 i - 3) q^{7} + ( - 2 i - 2) q^{8} + ( - 3 i + 1) q^{10} + ( - i + 1) q^{11} + 2 i q^{13} - 6 q^{14} - 4 q^{16} + ( - i - 1) q^{17} + (3 i - 3) q^{19} + ( - 4 i - 2) q^{20} - 2 i q^{22} + ( - i + 1) q^{23} + ( - 4 i + 3) q^{25} + (2 i + 2) q^{26} + (6 i - 6) q^{28} + (7 i + 7) q^{29} - 2 i q^{31} + (4 i - 4) q^{32} - 2 q^{34} + ( - 3 i - 9) q^{35} - 6 i q^{37} + 6 i q^{38} + ( - 2 i - 6) q^{40} - 4 i q^{41} - 4 i q^{43} + ( - 2 i - 2) q^{44} - 2 i q^{46} + ( - 7 i + 7) q^{47} + 11 i q^{49} + ( - 7 i - 1) q^{50} + 4 q^{52} + 8 q^{53} + ( - 3 i + 1) q^{55} + 12 i q^{56} + 14 q^{58} + ( - 3 i - 3) q^{59} + (i - 1) q^{61} + ( - 2 i - 2) q^{62} + 8 i q^{64} + (4 i + 2) q^{65} + 4 i q^{67} + (2 i - 2) q^{68} + (6 i - 12) q^{70} + (3 i + 3) q^{73} + ( - 6 i - 6) q^{74} + (6 i + 6) q^{76} - 6 q^{77} + 8 q^{79} + (4 i - 8) q^{80} + ( - 4 i - 4) q^{82} + 2 q^{83} + ( - i - 3) q^{85} + ( - 4 i - 4) q^{86} - 4 q^{88} + 6 q^{89} + ( - 6 i + 6) q^{91} + ( - 2 i - 2) q^{92} - 14 i q^{94} + (9 i - 3) q^{95} + ( - 11 i - 11) q^{97} + (11 i + 11) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{5} - 6 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 4 q^{5} - 6 q^{7} - 4 q^{8} + 2 q^{10} + 2 q^{11} - 12 q^{14} - 8 q^{16} - 2 q^{17} - 6 q^{19} - 4 q^{20} + 2 q^{23} + 6 q^{25} + 4 q^{26} - 12 q^{28} + 14 q^{29} - 8 q^{32} - 4 q^{34} - 18 q^{35} - 12 q^{40} - 4 q^{44} + 14 q^{47} - 2 q^{50} + 8 q^{52} + 16 q^{53} + 2 q^{55} + 28 q^{58} - 6 q^{59} - 2 q^{61} - 4 q^{62} + 4 q^{65} - 4 q^{68} - 24 q^{70} + 6 q^{73} - 12 q^{74} + 12 q^{76} - 12 q^{77} + 16 q^{79} - 16 q^{80} - 8 q^{82} + 4 q^{83} - 6 q^{85} - 8 q^{86} - 8 q^{88} + 12 q^{89} + 12 q^{91} - 4 q^{92} - 6 q^{95} - 22 q^{97} + 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(i\) \(-1\) \(i\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
1.00000i
1.00000i
1.00000 + 1.00000i 0 2.00000i 2.00000 + 1.00000i 0 −3.00000 + 3.00000i −2.00000 + 2.00000i 0 1.00000 + 3.00000i
667.1 1.00000 1.00000i 0 2.00000i 2.00000 1.00000i 0 −3.00000 3.00000i −2.00000 2.00000i 0 1.00000 3.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.z.d 2
3.b odd 2 1 80.2.s.a yes 2
5.c odd 4 1 720.2.bd.a 2
12.b even 2 1 320.2.s.a 2
15.d odd 2 1 400.2.s.a 2
15.e even 4 1 80.2.j.a 2
15.e even 4 1 400.2.j.a 2
16.f odd 4 1 720.2.bd.a 2
24.f even 2 1 640.2.s.a 2
24.h odd 2 1 640.2.s.b 2
48.i odd 4 1 320.2.j.a 2
48.i odd 4 1 640.2.j.b 2
48.k even 4 1 80.2.j.a 2
48.k even 4 1 640.2.j.a 2
60.h even 2 1 1600.2.s.a 2
60.l odd 4 1 320.2.j.a 2
60.l odd 4 1 1600.2.j.a 2
80.s even 4 1 inner 720.2.z.d 2
120.q odd 4 1 640.2.j.b 2
120.w even 4 1 640.2.j.a 2
240.t even 4 1 400.2.j.a 2
240.z odd 4 1 80.2.s.a yes 2
240.bb even 4 1 320.2.s.a 2
240.bd odd 4 1 400.2.s.a 2
240.bd odd 4 1 640.2.s.b 2
240.bf even 4 1 640.2.s.a 2
240.bf even 4 1 1600.2.s.a 2
240.bm odd 4 1 1600.2.j.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.a 2 15.e even 4 1
80.2.j.a 2 48.k even 4 1
80.2.s.a yes 2 3.b odd 2 1
80.2.s.a yes 2 240.z odd 4 1
320.2.j.a 2 48.i odd 4 1
320.2.j.a 2 60.l odd 4 1
320.2.s.a 2 12.b even 2 1
320.2.s.a 2 240.bb even 4 1
400.2.j.a 2 15.e even 4 1
400.2.j.a 2 240.t even 4 1
400.2.s.a 2 15.d odd 2 1
400.2.s.a 2 240.bd odd 4 1
640.2.j.a 2 48.k even 4 1
640.2.j.a 2 120.w even 4 1
640.2.j.b 2 48.i odd 4 1
640.2.j.b 2 120.q odd 4 1
640.2.s.a 2 24.f even 2 1
640.2.s.a 2 240.bf even 4 1
640.2.s.b 2 24.h odd 2 1
640.2.s.b 2 240.bd odd 4 1
720.2.z.d 2 1.a even 1 1 trivial
720.2.z.d 2 80.s even 4 1 inner
720.2.bd.a 2 5.c odd 4 1
720.2.bd.a 2 16.f odd 4 1
1600.2.j.a 2 60.l odd 4 1
1600.2.j.a 2 240.bm odd 4 1
1600.2.s.a 2 60.h even 2 1
1600.2.s.a 2 240.bf even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{2} + 6T_{7} + 18 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$31$ \( T^{2} + 4 \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( T^{2} + 16 \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$53$ \( (T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( (T - 2)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 22T + 242 \) Copy content Toggle raw display
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