Properties

Label 315.4.a.g
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: \(\Q(\sqrt{41}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{2} + ( 3 + 3 \beta ) q^{4} + 5 q^{5} + 7 q^{7} + ( -25 - \beta ) q^{8} +O(q^{10})\) \( q + ( -1 - \beta ) q^{2} + ( 3 + 3 \beta ) q^{4} + 5 q^{5} + 7 q^{7} + ( -25 - \beta ) q^{8} + ( -5 - 5 \beta ) q^{10} + ( -32 + 2 \beta ) q^{11} + ( 2 - 10 \beta ) q^{13} + ( -7 - 7 \beta ) q^{14} + ( 11 + 3 \beta ) q^{16} + ( -26 + 12 \beta ) q^{17} + ( -60 - 2 \beta ) q^{19} + ( 15 + 15 \beta ) q^{20} + ( 12 + 28 \beta ) q^{22} + ( -32 + 48 \beta ) q^{23} + 25 q^{25} + ( 98 + 18 \beta ) q^{26} + ( 21 + 21 \beta ) q^{28} + ( -170 - 12 \beta ) q^{29} + ( 52 - 38 \beta ) q^{31} + ( 159 - 9 \beta ) q^{32} + ( -94 + 2 \beta ) q^{34} + 35 q^{35} + ( -134 + 80 \beta ) q^{37} + ( 80 + 64 \beta ) q^{38} + ( -125 - 5 \beta ) q^{40} + ( -62 + 108 \beta ) q^{41} + ( -248 + 100 \beta ) q^{43} + ( -36 - 84 \beta ) q^{44} + ( -448 - 64 \beta ) q^{46} + ( 164 - 140 \beta ) q^{47} + 49 q^{49} + ( -25 - 25 \beta ) q^{50} + ( -294 - 54 \beta ) q^{52} + ( -462 - 58 \beta ) q^{53} + ( -160 + 10 \beta ) q^{55} + ( -175 - 7 \beta ) q^{56} + ( 290 + 194 \beta ) q^{58} + ( -220 - 76 \beta ) q^{59} + ( -398 - 84 \beta ) q^{61} + ( 328 + 24 \beta ) q^{62} + ( -157 - 165 \beta ) q^{64} + ( 10 - 50 \beta ) q^{65} + ( -64 - 228 \beta ) q^{67} + ( 282 - 6 \beta ) q^{68} + ( -35 - 35 \beta ) q^{70} + ( -112 - 86 \beta ) q^{71} + ( 182 - 38 \beta ) q^{73} + ( -666 - 26 \beta ) q^{74} + ( -240 - 192 \beta ) q^{76} + ( -224 + 14 \beta ) q^{77} + ( 920 - 8 \beta ) q^{79} + ( 55 + 15 \beta ) q^{80} + ( -1018 - 154 \beta ) q^{82} + ( 228 + 224 \beta ) q^{83} + ( -130 + 60 \beta ) q^{85} + ( -752 + 48 \beta ) q^{86} + ( 780 - 20 \beta ) q^{88} + ( -230 - 336 \beta ) q^{89} + ( 14 - 70 \beta ) q^{91} + ( 1344 + 192 \beta ) q^{92} + ( 1236 + 116 \beta ) q^{94} + ( -300 - 10 \beta ) q^{95} + ( -474 + 278 \beta ) q^{97} + ( -49 - 49 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} + 9q^{4} + 10q^{5} + 14q^{7} - 51q^{8} + O(q^{10}) \) \( 2q - 3q^{2} + 9q^{4} + 10q^{5} + 14q^{7} - 51q^{8} - 15q^{10} - 62q^{11} - 6q^{13} - 21q^{14} + 25q^{16} - 40q^{17} - 122q^{19} + 45q^{20} + 52q^{22} - 16q^{23} + 50q^{25} + 214q^{26} + 63q^{28} - 352q^{29} + 66q^{31} + 309q^{32} - 186q^{34} + 70q^{35} - 188q^{37} + 224q^{38} - 255q^{40} - 16q^{41} - 396q^{43} - 156q^{44} - 960q^{46} + 188q^{47} + 98q^{49} - 75q^{50} - 642q^{52} - 982q^{53} - 310q^{55} - 357q^{56} + 774q^{58} - 516q^{59} - 880q^{61} + 680q^{62} - 479q^{64} - 30q^{65} - 356q^{67} + 558q^{68} - 105q^{70} - 310q^{71} + 326q^{73} - 1358q^{74} - 672q^{76} - 434q^{77} + 1832q^{79} + 125q^{80} - 2190q^{82} + 680q^{83} - 200q^{85} - 1456q^{86} + 1540q^{88} - 796q^{89} - 42q^{91} + 2880q^{92} + 2588q^{94} - 610q^{95} - 670q^{97} - 147q^{98} + O(q^{100}) \)

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
3.70156
−2.70156
−4.70156 0 14.1047 5.00000 0 7.00000 −28.7016 0 −23.5078
1.2 1.70156 0 −5.10469 5.00000 0 7.00000 −22.2984 0 8.50781
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.g 2
3.b odd 2 1 105.4.a.g 2
5.b even 2 1 1575.4.a.y 2
7.b odd 2 1 2205.4.a.v 2
12.b even 2 1 1680.4.a.y 2
15.d odd 2 1 525.4.a.i 2
15.e even 4 2 525.4.d.j 4
21.c even 2 1 735.4.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.g 2 3.b odd 2 1
315.4.a.g 2 1.a even 1 1 trivial
525.4.a.i 2 15.d odd 2 1
525.4.d.j 4 15.e even 4 2
735.4.a.q 2 21.c even 2 1
1575.4.a.y 2 5.b even 2 1
1680.4.a.y 2 12.b even 2 1
2205.4.a.v 2 7.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 T_{2} - 8 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 8 T^{2} + 24 T^{3} + 64 T^{4} \)
$3$ 1
$5$ \( ( 1 - 5 T )^{2} \)
$7$ \( ( 1 - 7 T )^{2} \)
$11$ \( 1 + 62 T + 3582 T^{2} + 82522 T^{3} + 1771561 T^{4} \)
$13$ \( 1 + 6 T + 3378 T^{2} + 13182 T^{3} + 4826809 T^{4} \)
$17$ \( 1 + 40 T + 8750 T^{2} + 196520 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 122 T + 17398 T^{2} + 836798 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 16 T + 782 T^{2} + 194672 T^{3} + 148035889 T^{4} \)
$29$ \( 1 + 352 T + 78278 T^{2} + 8584928 T^{3} + 594823321 T^{4} \)
$31$ \( 1 - 66 T + 45870 T^{2} - 1966206 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 188 T + 44542 T^{2} + 9522764 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 + 16 T + 18350 T^{2} + 1102736 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 396 T + 95718 T^{2} + 31484772 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 188 T + 15582 T^{2} - 19518724 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 982 T + 504354 T^{2} + 146197214 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 516 T + 418118 T^{2} + 105975564 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 880 T + 575238 T^{2} + 199743280 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 356 T + 100374 T^{2} + 107071628 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 + 310 T + 664038 T^{2} + 110952410 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 - 326 T + 789802 T^{2} - 126819542 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 1832 T + 1824478 T^{2} - 903247448 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 680 T + 744870 T^{2} - 388815160 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 + 796 T + 411158 T^{2} + 561155324 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 + 670 T + 1145410 T^{2} + 611490910 T^{3} + 832972004929 T^{4} \)
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