Properties

Label 315.4.a.k
Level $315$
Weight $4$
Character orbit 315.a
Self dual yes
Analytic conductor $18.586$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.5856016518\)
Analytic rank: \(1\)
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.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - 5 q^{5} - 7 q^{7} + ( - 5 \beta + 9) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + (2 \beta + 1) q^{4} - 5 q^{5} - 7 q^{7} + ( - 5 \beta + 9) q^{8} + ( - 5 \beta - 5) q^{10} + ( - 20 \beta + 8) q^{11} + ( - 2 \beta - 38) q^{13} + ( - 7 \beta - 7) q^{14} + ( - 12 \beta - 39) q^{16} + (2 \beta + 62) q^{17} + ( - 16 \beta - 48) q^{19} + ( - 10 \beta - 5) q^{20} + ( - 12 \beta - 152) q^{22} + (34 \beta + 8) q^{23} + 25 q^{25} + ( - 40 \beta - 54) q^{26} + ( - 14 \beta - 7) q^{28} + (54 \beta - 94) q^{29} + (18 \beta - 60) q^{31} + ( - 11 \beta - 207) q^{32} + (64 \beta + 78) q^{34} + 35 q^{35} + ( - 66 \beta - 66) q^{37} + ( - 64 \beta - 176) q^{38} + (25 \beta - 45) q^{40} + ( - 80 \beta - 50) q^{41} + (62 \beta - 268) q^{43} + ( - 4 \beta - 312) q^{44} + (42 \beta + 280) q^{46} + (42 \beta + 464) q^{47} + 49 q^{49} + (25 \beta + 25) q^{50} + ( - 78 \beta - 70) q^{52} + ( - 64 \beta - 442) q^{53} + (100 \beta - 40) q^{55} + (35 \beta - 63) q^{56} + ( - 40 \beta + 338) q^{58} + (204 \beta - 52) q^{59} + ( - 42 \beta - 234) q^{61} + ( - 42 \beta + 84) q^{62} + ( - 122 \beta + 17) q^{64} + (10 \beta + 190) q^{65} + (38 \beta - 844) q^{67} + (126 \beta + 94) q^{68} + (35 \beta + 35) q^{70} + (150 \beta + 68) q^{71} + (322 \beta + 254) q^{73} + ( - 132 \beta - 594) q^{74} + ( - 112 \beta - 304) q^{76} + (140 \beta - 56) q^{77} + ( - 232 \beta - 216) q^{79} + (60 \beta + 195) q^{80} + ( - 130 \beta - 690) q^{82} + (84 \beta + 292) q^{83} + ( - 10 \beta - 310) q^{85} + ( - 206 \beta + 228) q^{86} + ( - 220 \beta + 872) q^{88} + ( - 112 \beta + 702) q^{89} + (14 \beta + 266) q^{91} + (50 \beta + 552) q^{92} + (506 \beta + 800) q^{94} + (80 \beta + 240) q^{95} + (46 \beta - 594) q^{97} + (49 \beta + 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 10 q^{5} - 14 q^{7} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 10 q^{5} - 14 q^{7} + 18 q^{8} - 10 q^{10} + 16 q^{11} - 76 q^{13} - 14 q^{14} - 78 q^{16} + 124 q^{17} - 96 q^{19} - 10 q^{20} - 304 q^{22} + 16 q^{23} + 50 q^{25} - 108 q^{26} - 14 q^{28} - 188 q^{29} - 120 q^{31} - 414 q^{32} + 156 q^{34} + 70 q^{35} - 132 q^{37} - 352 q^{38} - 90 q^{40} - 100 q^{41} - 536 q^{43} - 624 q^{44} + 560 q^{46} + 928 q^{47} + 98 q^{49} + 50 q^{50} - 140 q^{52} - 884 q^{53} - 80 q^{55} - 126 q^{56} + 676 q^{58} - 104 q^{59} - 468 q^{61} + 168 q^{62} + 34 q^{64} + 380 q^{65} - 1688 q^{67} + 188 q^{68} + 70 q^{70} + 136 q^{71} + 508 q^{73} - 1188 q^{74} - 608 q^{76} - 112 q^{77} - 432 q^{79} + 390 q^{80} - 1380 q^{82} + 584 q^{83} - 620 q^{85} + 456 q^{86} + 1744 q^{88} + 1404 q^{89} + 532 q^{91} + 1104 q^{92} + 1600 q^{94} + 480 q^{95} - 1188 q^{97} + 98 q^{98}+O(q^{100}) \) 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
−1.82843 0 −4.65685 −5.00000 0 −7.00000 23.1421 0 9.14214
1.2 3.82843 0 6.65685 −5.00000 0 −7.00000 −5.14214 0 −19.1421
\(n\): 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 315.4.a.k 2
3.b odd 2 1 105.4.a.e 2
5.b even 2 1 1575.4.a.q 2
7.b odd 2 1 2205.4.a.bb 2
12.b even 2 1 1680.4.a.bo 2
15.d odd 2 1 525.4.a.l 2
15.e even 4 2 525.4.d.l 4
21.c 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 3.b odd 2 1
315.4.a.k 2 1.a even 1 1 trivial
525.4.a.l 2 15.d odd 2 1
525.4.d.l 4 15.e even 4 2
735.4.a.o 2 21.c even 2 1
1575.4.a.q 2 5.b even 2 1
1680.4.a.bo 2 12.b even 2 1
2205.4.a.bb 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2T_{2} - 7 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\). 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
$5$ \( (T + 5)^{2} \) 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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