Properties

Label 275.2.b.d
Level $275$
Weight $2$
Character orbit 275.b
Analytic conductor $2.196$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 55)
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.

\(f(q)\) \(=\) \( q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{2} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{4} + ( -4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{6} -2 \zeta_{8}^{2} q^{7} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{8} -5 q^{9} +O(q^{10})\) \( q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{2} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{4} + ( -4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{6} -2 \zeta_{8}^{2} q^{7} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{8} -5 q^{9} + q^{11} + ( -2 \zeta_{8} - 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{12} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{13} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{14} + 3 q^{16} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{17} + ( -5 \zeta_{8} - 5 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{18} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{21} + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{22} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{23} + ( 4 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{24} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{26} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{27} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{28} + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{29} + ( \zeta_{8} - 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{32} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{33} + ( -8 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{34} + ( 5 + 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{36} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{37} + ( 8 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{39} + 6 q^{41} + ( 4 \zeta_{8} + 8 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{42} + 6 \zeta_{8}^{2} q^{43} + ( -1 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{44} + ( -4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{46} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{47} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{48} + 3 q^{49} + ( -8 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{51} + ( -6 \zeta_{8} + 4 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{52} + ( -4 \zeta_{8} - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{53} + ( 8 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{54} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{56} + ( 2 \zeta_{8} + 6 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{58} + ( 4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{59} + ( 2 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{61} + 10 \zeta_{8}^{2} q^{63} + ( 7 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{64} + ( -4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{66} + ( 6 \zeta_{8} + 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{67} + ( -10 \zeta_{8} - 12 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{68} -8 q^{69} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{71} + ( 5 \zeta_{8} + 15 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{72} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{73} + ( 10 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{74} -2 \zeta_{8}^{2} q^{77} -8 \zeta_{8}^{2} q^{78} -4 q^{79} + q^{81} + ( 6 \zeta_{8} + 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{82} + 6 \zeta_{8}^{2} q^{83} + ( -16 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{84} + ( -6 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{86} + ( -4 \zeta_{8} + 16 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{87} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{88} + ( 2 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{89} + ( 8 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{91} + ( -2 \zeta_{8} - 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{92} + ( -4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{94} + ( -4 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{96} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{97} + ( 3 \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{98} -5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 16 q^{6} - 20 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{4} - 16 q^{6} - 20 q^{9} + 4 q^{11} + 8 q^{14} + 12 q^{16} + 16 q^{24} - 8 q^{29} - 32 q^{34} + 20 q^{36} + 32 q^{39} + 24 q^{41} - 4 q^{44} - 16 q^{46} + 12 q^{49} - 32 q^{51} + 32 q^{54} - 24 q^{56} + 16 q^{59} + 8 q^{61} + 28 q^{64} - 16 q^{66} - 32 q^{69} + 40 q^{74} - 16 q^{79} + 4 q^{81} - 64 q^{84} - 24 q^{86} + 8 q^{89} + 32 q^{91} - 16 q^{94} - 16 q^{96} - 20 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
2.41421i 2.82843i −3.82843 0 −6.82843 2.00000i 4.41421i −5.00000 0
199.2 0.414214i 2.82843i 1.82843 0 −1.17157 2.00000i 1.58579i −5.00000 0
199.3 0.414214i 2.82843i 1.82843 0 −1.17157 2.00000i 1.58579i −5.00000 0
199.4 2.41421i 2.82843i −3.82843 0 −6.82843 2.00000i 4.41421i −5.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.b.d 4
3.b odd 2 1 2475.2.c.l 4
4.b odd 2 1 4400.2.b.q 4
5.b even 2 1 inner 275.2.b.d 4
5.c odd 4 1 55.2.a.b 2
5.c odd 4 1 275.2.a.c 2
15.d odd 2 1 2475.2.c.l 4
15.e even 4 1 495.2.a.b 2
15.e even 4 1 2475.2.a.x 2
20.d odd 2 1 4400.2.b.q 4
20.e even 4 1 880.2.a.m 2
20.e even 4 1 4400.2.a.bn 2
35.f even 4 1 2695.2.a.f 2
40.i odd 4 1 3520.2.a.bn 2
40.k even 4 1 3520.2.a.bo 2
55.e even 4 1 605.2.a.d 2
55.e even 4 1 3025.2.a.o 2
55.k odd 20 4 605.2.g.f 8
55.l even 20 4 605.2.g.l 8
60.l odd 4 1 7920.2.a.ch 2
65.h odd 4 1 9295.2.a.g 2
165.l odd 4 1 5445.2.a.y 2
220.i odd 4 1 9680.2.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 5.c odd 4 1
275.2.a.c 2 5.c odd 4 1
275.2.b.d 4 1.a even 1 1 trivial
275.2.b.d 4 5.b even 2 1 inner
495.2.a.b 2 15.e even 4 1
605.2.a.d 2 55.e even 4 1
605.2.g.f 8 55.k odd 20 4
605.2.g.l 8 55.l even 20 4
880.2.a.m 2 20.e even 4 1
2475.2.a.x 2 15.e even 4 1
2475.2.c.l 4 3.b odd 2 1
2475.2.c.l 4 15.d odd 2 1
2695.2.a.f 2 35.f even 4 1
3025.2.a.o 2 55.e even 4 1
3520.2.a.bn 2 40.i odd 4 1
3520.2.a.bo 2 40.k even 4 1
4400.2.a.bn 2 20.e even 4 1
4400.2.b.q 4 4.b odd 2 1
4400.2.b.q 4 20.d odd 2 1
5445.2.a.y 2 165.l odd 4 1
7920.2.a.ch 2 60.l odd 4 1
9295.2.a.g 2 65.h odd 4 1
9680.2.a.bn 2 220.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 6 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T^{2} + T^{4} \)
$3$ \( ( 8 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 4 + T^{2} )^{2} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( 64 + 48 T^{2} + T^{4} \)
$17$ \( 64 + 48 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 8 + T^{2} )^{2} \)
$29$ \( ( -28 + 4 T + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( 784 + 72 T^{2} + T^{4} \)
$41$ \( ( -6 + T )^{4} \)
$43$ \( ( 36 + T^{2} )^{2} \)
$47$ \( ( 8 + T^{2} )^{2} \)
$53$ \( 16 + 136 T^{2} + T^{4} \)
$59$ \( ( -16 - 8 T + T^{2} )^{2} \)
$61$ \( ( -124 - 4 T + T^{2} )^{2} \)
$67$ \( 3136 + 176 T^{2} + T^{4} \)
$71$ \( ( -128 + T^{2} )^{2} \)
$73$ \( 64 + 48 T^{2} + T^{4} \)
$79$ \( ( 4 + T )^{4} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( ( -124 - 4 T + T^{2} )^{2} \)
$97$ \( 784 + 72 T^{2} + T^{4} \)
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