Properties

Label 270.3.b.c
Level $270$
Weight $3$
Character orbit 270.b
Analytic conductor $7.357$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 270.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.35696713773\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{8}^{3} + \zeta_{8}) q^{2} + 2 q^{4} + 5 \zeta_{8} q^{5} + 5 \zeta_{8}^{2} q^{7} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{8}^{3} + \zeta_{8}) q^{2} + 2 q^{4} + 5 \zeta_{8} q^{5} + 5 \zeta_{8}^{2} q^{7} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{8} + (5 \zeta_{8}^{2} + 5) q^{10} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{11} + 9 \zeta_{8}^{2} q^{13} + (5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{14} + 4 q^{16} + (8 \zeta_{8}^{3} - 8 \zeta_{8}) q^{17} + 21 q^{19} + 10 \zeta_{8} q^{20} - 2 \zeta_{8}^{2} q^{22} + (\zeta_{8}^{3} - \zeta_{8}) q^{23} + 25 \zeta_{8}^{2} q^{25} + (9 \zeta_{8}^{3} + 9 \zeta_{8}) q^{26} + 10 \zeta_{8}^{2} q^{28} + ( - 27 \zeta_{8}^{3} - 27 \zeta_{8}) q^{29} + 40 q^{31} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8}) q^{32} - 16 q^{34} + 25 \zeta_{8}^{3} q^{35} - 25 \zeta_{8}^{2} q^{37} + ( - 21 \zeta_{8}^{3} + 21 \zeta_{8}) q^{38} + (10 \zeta_{8}^{2} + 10) q^{40} + ( - 37 \zeta_{8}^{3} - 37 \zeta_{8}) q^{41} + 64 \zeta_{8}^{2} q^{43} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{44} - 2 q^{46} + (16 \zeta_{8}^{3} - 16 \zeta_{8}) q^{47} + 24 q^{49} + (25 \zeta_{8}^{3} + 25 \zeta_{8}) q^{50} + 18 \zeta_{8}^{2} q^{52} + (51 \zeta_{8}^{3} - 51 \zeta_{8}) q^{53} + ( - 5 \zeta_{8}^{2} + 5) q^{55} + (10 \zeta_{8}^{3} + 10 \zeta_{8}) q^{56} - 54 \zeta_{8}^{2} q^{58} + ( - 64 \zeta_{8}^{3} - 64 \zeta_{8}) q^{59} - 97 q^{61} + ( - 40 \zeta_{8}^{3} + 40 \zeta_{8}) q^{62} + 8 q^{64} + 45 \zeta_{8}^{3} q^{65} - 131 \zeta_{8}^{2} q^{67} + (16 \zeta_{8}^{3} - 16 \zeta_{8}) q^{68} + (25 \zeta_{8}^{2} - 25) q^{70} + (63 \zeta_{8}^{3} + 63 \zeta_{8}) q^{71} - 17 \zeta_{8}^{2} q^{73} + ( - 25 \zeta_{8}^{3} - 25 \zeta_{8}) q^{74} + 42 q^{76} + ( - 5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{77} - 117 q^{79} + 20 \zeta_{8} q^{80} - 74 \zeta_{8}^{2} q^{82} + ( - 41 \zeta_{8}^{3} + 41 \zeta_{8}) q^{83} + ( - 40 \zeta_{8}^{2} - 40) q^{85} + (64 \zeta_{8}^{3} + 64 \zeta_{8}) q^{86} - 4 \zeta_{8}^{2} q^{88} + ( - 104 \zeta_{8}^{3} - 104 \zeta_{8}) q^{89} - 45 q^{91} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{92} - 32 q^{94} + 105 \zeta_{8} q^{95} - 41 \zeta_{8}^{2} q^{97} + ( - 24 \zeta_{8}^{3} + 24 \zeta_{8}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 20 q^{10} + 16 q^{16} + 84 q^{19} + 160 q^{31} - 64 q^{34} + 40 q^{40} - 8 q^{46} + 96 q^{49} + 20 q^{55} - 388 q^{61} + 32 q^{64} - 100 q^{70} + 168 q^{76} - 468 q^{79} - 160 q^{85} - 180 q^{91} - 128 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−1.41421 0 2.00000 −3.53553 3.53553i 0 5.00000i −2.82843 0 5.00000 + 5.00000i
269.2 −1.41421 0 2.00000 −3.53553 + 3.53553i 0 5.00000i −2.82843 0 5.00000 5.00000i
269.3 1.41421 0 2.00000 3.53553 3.53553i 0 5.00000i 2.82843 0 5.00000 5.00000i
269.4 1.41421 0 2.00000 3.53553 + 3.53553i 0 5.00000i 2.82843 0 5.00000 + 5.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.3.b.c 4
3.b odd 2 1 inner 270.3.b.c 4
4.b odd 2 1 2160.3.c.i 4
5.b even 2 1 inner 270.3.b.c 4
5.c odd 4 1 1350.3.d.f 2
5.c odd 4 1 1350.3.d.g 2
9.c even 3 2 810.3.j.e 8
9.d odd 6 2 810.3.j.e 8
12.b even 2 1 2160.3.c.i 4
15.d odd 2 1 inner 270.3.b.c 4
15.e even 4 1 1350.3.d.f 2
15.e even 4 1 1350.3.d.g 2
20.d odd 2 1 2160.3.c.i 4
45.h odd 6 2 810.3.j.e 8
45.j even 6 2 810.3.j.e 8
60.h even 2 1 2160.3.c.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.c 4 1.a even 1 1 trivial
270.3.b.c 4 3.b odd 2 1 inner
270.3.b.c 4 5.b even 2 1 inner
270.3.b.c 4 15.d odd 2 1 inner
810.3.j.e 8 9.c even 3 2
810.3.j.e 8 9.d odd 6 2
810.3.j.e 8 45.h odd 6 2
810.3.j.e 8 45.j even 6 2
1350.3.d.f 2 5.c odd 4 1
1350.3.d.f 2 15.e even 4 1
1350.3.d.g 2 5.c odd 4 1
1350.3.d.g 2 15.e even 4 1
2160.3.c.i 4 4.b odd 2 1
2160.3.c.i 4 12.b even 2 1
2160.3.c.i 4 20.d odd 2 1
2160.3.c.i 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(270, [\chi])\):

\( T_{7}^{2} + 25 \) Copy content Toggle raw display
\( T_{17}^{2} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$19$ \( (T - 21)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1458)^{2} \) Copy content Toggle raw display
$31$ \( (T - 40)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 625)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2738)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4096)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 512)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 5202)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 8192)^{2} \) Copy content Toggle raw display
$61$ \( (T + 97)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 17161)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 7938)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 289)^{2} \) Copy content Toggle raw display
$79$ \( (T + 117)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 3362)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 21632)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1681)^{2} \) Copy content Toggle raw display
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