Properties

Label 1575.4.a.q
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( 1 - 2 \beta ) q^{4} + 7 q^{7} + ( -9 - 5 \beta ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( 1 - 2 \beta ) q^{4} + 7 q^{7} + ( -9 - 5 \beta ) q^{8} + ( 8 + 20 \beta ) q^{11} + ( 38 - 2 \beta ) q^{13} + ( -7 + 7 \beta ) q^{14} + ( -39 + 12 \beta ) q^{16} + ( -62 + 2 \beta ) q^{17} + ( -48 + 16 \beta ) q^{19} + ( 152 - 12 \beta ) q^{22} + ( -8 + 34 \beta ) q^{23} + ( -54 + 40 \beta ) q^{26} + ( 7 - 14 \beta ) q^{28} + ( -94 - 54 \beta ) q^{29} + ( -60 - 18 \beta ) q^{31} + ( 207 - 11 \beta ) q^{32} + ( 78 - 64 \beta ) q^{34} + ( 66 - 66 \beta ) q^{37} + ( 176 - 64 \beta ) q^{38} + ( -50 + 80 \beta ) q^{41} + ( 268 + 62 \beta ) q^{43} + ( -312 + 4 \beta ) q^{44} + ( 280 - 42 \beta ) q^{46} + ( -464 + 42 \beta ) q^{47} + 49 q^{49} + ( 70 - 78 \beta ) q^{52} + ( 442 - 64 \beta ) q^{53} + ( -63 - 35 \beta ) q^{56} + ( -338 - 40 \beta ) q^{58} + ( -52 - 204 \beta ) q^{59} + ( -234 + 42 \beta ) q^{61} + ( -84 - 42 \beta ) q^{62} + ( 17 + 122 \beta ) q^{64} + ( 844 + 38 \beta ) q^{67} + ( -94 + 126 \beta ) q^{68} + ( 68 - 150 \beta ) q^{71} + ( -254 + 322 \beta ) q^{73} + ( -594 + 132 \beta ) q^{74} + ( -304 + 112 \beta ) q^{76} + ( 56 + 140 \beta ) q^{77} + ( -216 + 232 \beta ) q^{79} + ( 690 - 130 \beta ) q^{82} + ( -292 + 84 \beta ) q^{83} + ( 228 + 206 \beta ) q^{86} + ( -872 - 220 \beta ) q^{88} + ( 702 + 112 \beta ) q^{89} + ( 266 - 14 \beta ) q^{91} + ( -552 + 50 \beta ) q^{92} + ( 800 - 506 \beta ) q^{94} + ( 594 + 46 \beta ) q^{97} + ( -49 + 49 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 14 q^{7} - 18 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} + 2 q^{4} + 14 q^{7} - 18 q^{8} + 16 q^{11} + 76 q^{13} - 14 q^{14} - 78 q^{16} - 124 q^{17} - 96 q^{19} + 304 q^{22} - 16 q^{23} - 108 q^{26} + 14 q^{28} - 188 q^{29} - 120 q^{31} + 414 q^{32} + 156 q^{34} + 132 q^{37} + 352 q^{38} - 100 q^{41} + 536 q^{43} - 624 q^{44} + 560 q^{46} - 928 q^{47} + 98 q^{49} + 140 q^{52} + 884 q^{53} - 126 q^{56} - 676 q^{58} - 104 q^{59} - 468 q^{61} - 168 q^{62} + 34 q^{64} + 1688 q^{67} - 188 q^{68} + 136 q^{71} - 508 q^{73} - 1188 q^{74} - 608 q^{76} + 112 q^{77} - 432 q^{79} + 1380 q^{82} - 584 q^{83} + 456 q^{86} - 1744 q^{88} + 1404 q^{89} + 532 q^{91} - 1104 q^{92} + 1600 q^{94} + 1188 q^{97} - 98 q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−3.82843 0 6.65685 0 0 7.00000 5.14214 0 0
1.2 1.82843 0 −4.65685 0 0 7.00000 −23.1421 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.4.a.q 2
3.b odd 2 1 525.4.a.l 2
5.b even 2 1 315.4.a.k 2
15.d odd 2 1 105.4.a.e 2
15.e even 4 2 525.4.d.l 4
35.c odd 2 1 2205.4.a.bb 2
60.h even 2 1 1680.4.a.bo 2
105.g even 2 1 735.4.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.e 2 15.d odd 2 1
315.4.a.k 2 5.b even 2 1
525.4.a.l 2 3.b odd 2 1
525.4.d.l 4 15.e even 4 2
735.4.a.o 2 105.g even 2 1
1575.4.a.q 2 1.a even 1 1 trivial
1680.4.a.bo 2 60.h even 2 1
2205.4.a.bb 2 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} + 2 T_{2} - 7 \)
\( T_{11}^{2} - 16 T_{11} - 3136 \)
\( T_{13}^{2} - 76 T_{13} + 1412 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -7 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( -3136 - 16 T + T^{2} \)
$13$ \( 1412 - 76 T + T^{2} \)
$17$ \( 3812 + 124 T + T^{2} \)
$19$ \( 256 + 96 T + T^{2} \)
$23$ \( -9184 + 16 T + T^{2} \)
$29$ \( -14492 + 188 T + T^{2} \)
$31$ \( 1008 + 120 T + T^{2} \)
$37$ \( -30492 - 132 T + T^{2} \)
$41$ \( -48700 + 100 T + T^{2} \)
$43$ \( 41072 - 536 T + T^{2} \)
$47$ \( 201184 + 928 T + T^{2} \)
$53$ \( 162596 - 884 T + T^{2} \)
$59$ \( -330224 + 104 T + T^{2} \)
$61$ \( 40644 + 468 T + T^{2} \)
$67$ \( 700784 - 1688 T + T^{2} \)
$71$ \( -175376 - 136 T + T^{2} \)
$73$ \( -764956 + 508 T + T^{2} \)
$79$ \( -383936 + 432 T + T^{2} \)
$83$ \( 28816 + 584 T + T^{2} \)
$89$ \( 392452 - 1404 T + T^{2} \)
$97$ \( 335908 - 1188 T + T^{2} \)
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