Properties

Label 3450.2.d.z
Level $3450$
Weight $2$
Character orbit 3450.d
Analytic conductor $27.548$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{57})\)
Defining polynomial: \(x^{4} + 29 x^{2} + 196\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + \beta_{2} q^{3} - q^{4} + q^{6} -3 \beta_{2} q^{7} + \beta_{2} q^{8} - q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + \beta_{2} q^{3} - q^{4} + q^{6} -3 \beta_{2} q^{7} + \beta_{2} q^{8} - q^{9} + ( 3 - \beta_{3} ) q^{11} -\beta_{2} q^{12} + \beta_{1} q^{13} -3 q^{14} + q^{16} + ( \beta_{1} - \beta_{2} ) q^{17} + \beta_{2} q^{18} + ( 1 - \beta_{3} ) q^{19} + 3 q^{21} + ( \beta_{1} - 2 \beta_{2} ) q^{22} -\beta_{2} q^{23} - q^{24} + ( -1 + \beta_{3} ) q^{26} -\beta_{2} q^{27} + 3 \beta_{2} q^{28} + ( 1 + 2 \beta_{3} ) q^{29} + 6 q^{31} -\beta_{2} q^{32} + ( -\beta_{1} + 2 \beta_{2} ) q^{33} + ( -2 + \beta_{3} ) q^{34} + q^{36} + ( -3 \beta_{1} - \beta_{2} ) q^{37} + \beta_{1} q^{38} + ( 1 - \beta_{3} ) q^{39} + ( -7 + \beta_{3} ) q^{41} -3 \beta_{2} q^{42} + ( \beta_{1} + 4 \beta_{2} ) q^{43} + ( -3 + \beta_{3} ) q^{44} - q^{46} + ( -\beta_{1} - 5 \beta_{2} ) q^{47} + \beta_{2} q^{48} -2 q^{49} + ( 2 - \beta_{3} ) q^{51} -\beta_{1} q^{52} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{53} - q^{54} + 3 q^{56} -\beta_{1} q^{57} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{58} + ( -4 - 2 \beta_{3} ) q^{59} + ( -8 + 2 \beta_{3} ) q^{61} -6 \beta_{2} q^{62} + 3 \beta_{2} q^{63} - q^{64} + ( 3 - \beta_{3} ) q^{66} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -\beta_{1} + \beta_{2} ) q^{68} + q^{69} + ( 4 - 3 \beta_{3} ) q^{71} -\beta_{2} q^{72} + ( -2 \beta_{1} - 9 \beta_{2} ) q^{73} + ( 2 - 3 \beta_{3} ) q^{74} + ( -1 + \beta_{3} ) q^{76} + ( 3 \beta_{1} - 6 \beta_{2} ) q^{77} + \beta_{1} q^{78} + ( -1 + 3 \beta_{3} ) q^{79} + q^{81} + ( -\beta_{1} + 6 \beta_{2} ) q^{82} + ( 2 \beta_{1} - 9 \beta_{2} ) q^{83} -3 q^{84} + ( 3 + \beta_{3} ) q^{86} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{87} + ( -\beta_{1} + 2 \beta_{2} ) q^{88} + ( -4 - \beta_{3} ) q^{89} + ( -3 + 3 \beta_{3} ) q^{91} + \beta_{2} q^{92} + 6 \beta_{2} q^{93} + ( -4 - \beta_{3} ) q^{94} + q^{96} -14 \beta_{2} q^{97} + 2 \beta_{2} q^{98} + ( -3 + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + 10q^{11} - 12q^{14} + 4q^{16} + 2q^{19} + 12q^{21} - 4q^{24} - 2q^{26} + 8q^{29} + 24q^{31} - 6q^{34} + 4q^{36} + 2q^{39} - 26q^{41} - 10q^{44} - 4q^{46} - 8q^{49} + 6q^{51} - 4q^{54} + 12q^{56} - 20q^{59} - 28q^{61} - 4q^{64} + 10q^{66} + 4q^{69} + 10q^{71} + 2q^{74} - 2q^{76} + 2q^{79} + 4q^{81} - 12q^{84} + 14q^{86} - 18q^{89} - 6q^{91} - 18q^{94} + 4q^{96} - 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 29 x^{2} + 196\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 15 \nu \)\()/14\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 15 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 15\)
\(\nu^{3}\)\(=\)\(14 \beta_{2} - 15 \beta_{1}\)

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
3.27492i
4.27492i
3.27492i
4.27492i
1.00000i 1.00000i −1.00000 0 1.00000 3.00000i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 1.00000 3.00000i 1.00000i −1.00000 0
2899.3 1.00000i 1.00000i −1.00000 0 1.00000 3.00000i 1.00000i −1.00000 0
2899.4 1.00000i 1.00000i −1.00000 0 1.00000 3.00000i 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.z 4
5.b even 2 1 inner 3450.2.d.z 4
5.c odd 4 1 3450.2.a.bd 2
5.c odd 4 1 3450.2.a.bm yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.bd 2 5.c odd 4 1
3450.2.a.bm yes 2 5.c odd 4 1
3450.2.d.z 4 1.a even 1 1 trivial
3450.2.d.z 4 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 \(S_{2}^{\mathrm{new}}(3450, [\chi])\):

\( T_{7}^{2} + 9 \)
\( T_{11}^{2} - 5 T_{11} - 8 \)
\( T_{13}^{4} + 29 T_{13}^{2} + 196 \)
\( T_{17}^{4} + 33 T_{17}^{2} + 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 9 + T^{2} )^{2} \)
$11$ \( ( -8 - 5 T + T^{2} )^{2} \)
$13$ \( 196 + 29 T^{2} + T^{4} \)
$17$ \( 144 + 33 T^{2} + T^{4} \)
$19$ \( ( -14 - T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( -53 - 4 T + T^{2} )^{2} \)
$31$ \( ( -6 + T )^{4} \)
$37$ \( 16384 + 257 T^{2} + T^{4} \)
$41$ \( ( 28 + 13 T + T^{2} )^{2} \)
$43$ \( 4 + 53 T^{2} + T^{4} \)
$47$ \( 36 + 69 T^{2} + T^{4} \)
$53$ \( 2304 + 132 T^{2} + T^{4} \)
$59$ \( ( -32 + 10 T + T^{2} )^{2} \)
$61$ \( ( -8 + 14 T + T^{2} )^{2} \)
$67$ \( 2304 + 132 T^{2} + T^{4} \)
$71$ \( ( -122 - 5 T + T^{2} )^{2} \)
$73$ \( 49 + 242 T^{2} + T^{4} \)
$79$ \( ( -128 - T + T^{2} )^{2} \)
$83$ \( 1849 + 314 T^{2} + T^{4} \)
$89$ \( ( 6 + 9 T + T^{2} )^{2} \)
$97$ \( ( 196 + T^{2} )^{2} \)
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