Properties

Label 3450.2.d.l
Level $3450$
Weight $2$
Character orbit 3450.d
Analytic conductor $27.548$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.5483886973\)
Analytic rank: \(1\)
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: yes
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.

\(f(q)\) \(=\) \( q + i q^{2} -i q^{3} - q^{4} + q^{6} + 5 i q^{7} -i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} -i q^{3} - q^{4} + q^{6} + 5 i q^{7} -i q^{8} - q^{9} -3 q^{11} + i q^{12} -5 i q^{13} -5 q^{14} + q^{16} + 6 i q^{17} -i q^{18} + q^{19} + 5 q^{21} -3 i q^{22} -i q^{23} - q^{24} + 5 q^{26} + i q^{27} -5 i q^{28} + 5 q^{29} -8 q^{31} + i q^{32} + 3 i q^{33} -6 q^{34} + q^{36} + 4 i q^{37} + i q^{38} -5 q^{39} -7 q^{41} + 5 i q^{42} + 7 i q^{43} + 3 q^{44} + q^{46} -6 i q^{47} -i q^{48} -18 q^{49} + 6 q^{51} + 5 i q^{52} -8 i q^{53} - q^{54} + 5 q^{56} -i q^{57} + 5 i q^{58} + 10 q^{59} -12 q^{61} -8 i q^{62} -5 i q^{63} - q^{64} -3 q^{66} -12 i q^{67} -6 i q^{68} - q^{69} + 10 q^{71} + i q^{72} -15 i q^{73} -4 q^{74} - q^{76} -15 i q^{77} -5 i q^{78} + 5 q^{79} + q^{81} -7 i q^{82} + 9 i q^{83} -5 q^{84} -7 q^{86} -5 i q^{87} + 3 i q^{88} -14 q^{89} + 25 q^{91} + i q^{92} + 8 i q^{93} + 6 q^{94} + q^{96} + 16 i q^{97} -18 i q^{98} + 3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} - 6q^{11} - 10q^{14} + 2q^{16} + 2q^{19} + 10q^{21} - 2q^{24} + 10q^{26} + 10q^{29} - 16q^{31} - 12q^{34} + 2q^{36} - 10q^{39} - 14q^{41} + 6q^{44} + 2q^{46} - 36q^{49} + 12q^{51} - 2q^{54} + 10q^{56} + 20q^{59} - 24q^{61} - 2q^{64} - 6q^{66} - 2q^{69} + 20q^{71} - 8q^{74} - 2q^{76} + 10q^{79} + 2q^{81} - 10q^{84} - 14q^{86} - 28q^{89} + 50q^{91} + 12q^{94} + 2q^{96} + 6q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2899.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 5.00000i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 1.00000 5.00000i 1.00000i −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
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 3450.2.d.l 2
5.b even 2 1 inner 3450.2.d.l 2
5.c odd 4 1 3450.2.a.a 1
5.c odd 4 1 3450.2.a.bb yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.a 1 5.c odd 4 1
3450.2.a.bb yes 1 5.c odd 4 1
3450.2.d.l 2 1.a even 1 1 trivial
3450.2.d.l 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3450, [\chi])\):

\( T_{7}^{2} + 25 \)
\( T_{11} + 3 \)
\( T_{13}^{2} + 25 \)
\( T_{17}^{2} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 25 + T^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( 25 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( 1 + T^{2} \)
$29$ \( ( -5 + T )^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( 16 + T^{2} \)
$41$ \( ( 7 + T )^{2} \)
$43$ \( 49 + T^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( 64 + T^{2} \)
$59$ \( ( -10 + T )^{2} \)
$61$ \( ( 12 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( ( -10 + T )^{2} \)
$73$ \( 225 + T^{2} \)
$79$ \( ( -5 + T )^{2} \)
$83$ \( 81 + T^{2} \)
$89$ \( ( 14 + T )^{2} \)
$97$ \( 256 + T^{2} \)
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