Properties

Label 1275.2.d.b
Level $1275$
Weight $2$
Character orbit 1275.d
Analytic conductor $10.181$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1275.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.1809262577\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{3} + q^{4} -i q^{6} -4 q^{7} + 3 i q^{8} + q^{9} +O(q^{10})\) \( q + i q^{2} - q^{3} + q^{4} -i q^{6} -4 q^{7} + 3 i q^{8} + q^{9} -4 i q^{11} - q^{12} -2 i q^{13} -4 i q^{14} - q^{16} + ( 4 + i ) q^{17} + i q^{18} + 4 q^{19} + 4 q^{21} + 4 q^{22} + 4 q^{23} -3 i q^{24} + 2 q^{26} - q^{27} -4 q^{28} + 4 i q^{31} + 5 i q^{32} + 4 i q^{33} + ( -1 + 4 i ) q^{34} + q^{36} + 8 q^{37} + 4 i q^{38} + 2 i q^{39} + 8 i q^{41} + 4 i q^{42} + 4 i q^{43} -4 i q^{44} + 4 i q^{46} -8 i q^{47} + q^{48} + 9 q^{49} + ( -4 - i ) q^{51} -2 i q^{52} -6 i q^{53} -i q^{54} -12 i q^{56} -4 q^{57} + 12 q^{59} + 8 i q^{61} -4 q^{62} -4 q^{63} -7 q^{64} -4 q^{66} + 12 i q^{67} + ( 4 + i ) q^{68} -4 q^{69} -12 i q^{71} + 3 i q^{72} + 8 i q^{74} + 4 q^{76} + 16 i q^{77} -2 q^{78} -4 i q^{79} + q^{81} -8 q^{82} -12 i q^{83} + 4 q^{84} -4 q^{86} + 12 q^{88} + 10 q^{89} + 8 i q^{91} + 4 q^{92} -4 i q^{93} + 8 q^{94} -5 i q^{96} + 16 q^{97} + 9 i q^{98} -4 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{4} - 8q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{4} - 8q^{7} + 2q^{9} - 2q^{12} - 2q^{16} + 8q^{17} + 8q^{19} + 8q^{21} + 8q^{22} + 8q^{23} + 4q^{26} - 2q^{27} - 8q^{28} - 2q^{34} + 2q^{36} + 16q^{37} + 2q^{48} + 18q^{49} - 8q^{51} - 8q^{57} + 24q^{59} - 8q^{62} - 8q^{63} - 14q^{64} - 8q^{66} + 8q^{68} - 8q^{69} + 8q^{76} - 4q^{78} + 2q^{81} - 16q^{82} + 8q^{84} - 8q^{86} + 24q^{88} + 20q^{89} + 8q^{92} + 16q^{94} + 32q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1275\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(751\) \(851\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1
1.00000i
1.00000i
1.00000i −1.00000 1.00000 0 1.00000i −4.00000 3.00000i 1.00000 0
424.2 1.00000i −1.00000 1.00000 0 1.00000i −4.00000 3.00000i 1.00000 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
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1275.2.d.b 2
5.b even 2 1 1275.2.d.d 2
5.c odd 4 1 51.2.d.b 2
5.c odd 4 1 1275.2.g.a 2
15.e even 4 1 153.2.d.a 2
17.b even 2 1 1275.2.d.d 2
20.e even 4 1 816.2.c.c 2
40.i odd 4 1 3264.2.c.e 2
40.k even 4 1 3264.2.c.d 2
60.l odd 4 1 2448.2.c.j 2
85.c even 2 1 inner 1275.2.d.b 2
85.f odd 4 1 867.2.a.b 1
85.g odd 4 1 51.2.d.b 2
85.g odd 4 1 1275.2.g.a 2
85.i odd 4 1 867.2.a.a 1
85.k odd 8 2 867.2.e.d 4
85.n odd 8 2 867.2.e.d 4
85.o even 16 4 867.2.h.d 8
85.r even 16 4 867.2.h.d 8
255.k even 4 1 2601.2.a.j 1
255.o even 4 1 153.2.d.a 2
255.r even 4 1 2601.2.a.i 1
340.r even 4 1 816.2.c.c 2
680.u even 4 1 3264.2.c.d 2
680.bi odd 4 1 3264.2.c.e 2
1020.x odd 4 1 2448.2.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 5.c odd 4 1
51.2.d.b 2 85.g odd 4 1
153.2.d.a 2 15.e even 4 1
153.2.d.a 2 255.o even 4 1
816.2.c.c 2 20.e even 4 1
816.2.c.c 2 340.r even 4 1
867.2.a.a 1 85.i odd 4 1
867.2.a.b 1 85.f odd 4 1
867.2.e.d 4 85.k odd 8 2
867.2.e.d 4 85.n odd 8 2
867.2.h.d 8 85.o even 16 4
867.2.h.d 8 85.r even 16 4
1275.2.d.b 2 1.a even 1 1 trivial
1275.2.d.b 2 85.c even 2 1 inner
1275.2.d.d 2 5.b even 2 1
1275.2.d.d 2 17.b even 2 1
1275.2.g.a 2 5.c odd 4 1
1275.2.g.a 2 85.g odd 4 1
2448.2.c.j 2 60.l odd 4 1
2448.2.c.j 2 1020.x odd 4 1
2601.2.a.i 1 255.r even 4 1
2601.2.a.j 1 255.k even 4 1
3264.2.c.d 2 40.k even 4 1
3264.2.c.d 2 680.u even 4 1
3264.2.c.e 2 40.i odd 4 1
3264.2.c.e 2 680.bi odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1275, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 4 + T )^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 17 - 8 T + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( 16 + T^{2} \)
$37$ \( ( -8 + T )^{2} \)
$41$ \( 64 + T^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 64 + T^{2} \)
$53$ \( 36 + T^{2} \)
$59$ \( ( -12 + T )^{2} \)
$61$ \( 64 + T^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( 144 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( 16 + T^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( ( -16 + T )^{2} \)
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