Properties

Label 1450.2.b.b
Level $1450$
Weight $2$
Character orbit 1450.b
Analytic conductor $11.578$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.5783082931\)
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 58)
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.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} - 2 i q^{7} - i q^{8} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + i q^{3} - q^{4} - q^{6} - 2 i q^{7} - i q^{8} + 2 q^{9} - 3 q^{11} - i q^{12} + i q^{13} + 2 q^{14} + q^{16} + 8 i q^{17} + 2 i q^{18} + 2 q^{21} - 3 i q^{22} - 4 i q^{23} + q^{24} - q^{26} + 5 i q^{27} + 2 i q^{28} + q^{29} - 3 q^{31} + i q^{32} - 3 i q^{33} - 8 q^{34} - 2 q^{36} + 8 i q^{37} - q^{39} + 2 q^{41} + 2 i q^{42} + 11 i q^{43} + 3 q^{44} + 4 q^{46} + 13 i q^{47} + i q^{48} + 3 q^{49} - 8 q^{51} - i q^{52} + 11 i q^{53} - 5 q^{54} - 2 q^{56} + i q^{58} - 8 q^{61} - 3 i q^{62} - 4 i q^{63} - q^{64} + 3 q^{66} - 12 i q^{67} - 8 i q^{68} + 4 q^{69} + 2 q^{71} - 2 i q^{72} - 4 i q^{73} - 8 q^{74} + 6 i q^{77} - i q^{78} - 15 q^{79} + q^{81} + 2 i q^{82} - 4 i q^{83} - 2 q^{84} - 11 q^{86} + i q^{87} + 3 i q^{88} + 10 q^{89} + 2 q^{91} + 4 i q^{92} - 3 i q^{93} - 13 q^{94} - q^{96} - 2 i q^{97} + 3 i q^{98} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} - 6 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{21} + 2 q^{24} - 2 q^{26} + 2 q^{29} - 6 q^{31} - 16 q^{34} - 4 q^{36} - 2 q^{39} + 4 q^{41} + 6 q^{44} + 8 q^{46} + 6 q^{49} - 16 q^{51} - 10 q^{54} - 4 q^{56} - 16 q^{61} - 2 q^{64} + 6 q^{66} + 8 q^{69} + 4 q^{71} - 16 q^{74} - 30 q^{79} + 2 q^{81} - 4 q^{84} - 22 q^{86} + 20 q^{89} + 4 q^{91} - 26 q^{94} - 2 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 2.00000i 1.00000i 2.00000 0
349.2 1.00000i 1.00000i −1.00000 0 −1.00000 2.00000i 1.00000i 2.00000 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 1450.2.b.b 2
5.b even 2 1 inner 1450.2.b.b 2
5.c odd 4 1 58.2.a.b 1
5.c odd 4 1 1450.2.a.c 1
15.e even 4 1 522.2.a.b 1
20.e even 4 1 464.2.a.e 1
35.f even 4 1 2842.2.a.e 1
40.i odd 4 1 1856.2.a.k 1
40.k even 4 1 1856.2.a.f 1
55.e even 4 1 7018.2.a.a 1
60.l odd 4 1 4176.2.a.n 1
65.h odd 4 1 9802.2.a.a 1
145.e even 4 1 1682.2.b.a 2
145.h odd 4 1 1682.2.a.d 1
145.j even 4 1 1682.2.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.a.b 1 5.c odd 4 1
464.2.a.e 1 20.e even 4 1
522.2.a.b 1 15.e even 4 1
1450.2.a.c 1 5.c odd 4 1
1450.2.b.b 2 1.a even 1 1 trivial
1450.2.b.b 2 5.b even 2 1 inner
1682.2.a.d 1 145.h odd 4 1
1682.2.b.a 2 145.e even 4 1
1682.2.b.a 2 145.j even 4 1
1856.2.a.f 1 40.k even 4 1
1856.2.a.k 1 40.i odd 4 1
2842.2.a.e 1 35.f even 4 1
4176.2.a.n 1 60.l odd 4 1
7018.2.a.a 1 55.e even 4 1
9802.2.a.a 1 65.h odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1450, [\chi])\):

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 64 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 121 \) Copy content Toggle raw display
$47$ \( T^{2} + 169 \) Copy content Toggle raw display
$53$ \( T^{2} + 121 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T - 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T + 15)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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