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: \( x^{2} - 2 \) Copy content Toggle raw display
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.

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

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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} + 2T_{2} - 7 \) Copy content Toggle raw display
\( T_{11}^{2} - 16T_{11} - 3136 \) Copy content Toggle raw display
\( T_{13}^{2} - 76T_{13} + 1412 \) Copy content Toggle raw display

Hecke characteristic polynomials

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