Properties

Label 315.2.d.e
Level 315
Weight 2
Character orbit 315.d
Analytic conductor 2.515
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -1 + \beta_{3} + \beta_{5} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + \beta_{4} q^{7} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -1 + \beta_{3} + \beta_{5} ) q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + \beta_{4} q^{7} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{8} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{10} -2 q^{11} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{13} + \beta_{2} q^{14} + ( 3 - 4 \beta_{2} ) q^{16} + ( 2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{17} + ( -2 + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{19} + ( 4 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{20} + 2 \beta_{1} q^{22} + ( -\beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{23} + ( -1 - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{25} + ( -4 - 4 \beta_{2} + \beta_{3} + \beta_{5} ) q^{26} + ( \beta_{3} - \beta_{4} - \beta_{5} ) q^{28} + ( 2 + 2 \beta_{3} + 2 \beta_{5} ) q^{29} + ( 2 + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{31} + ( -\beta_{1} - 2 \beta_{3} + 8 \beta_{4} + 2 \beta_{5} ) q^{32} + ( 4 + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{34} + ( \beta_{2} - \beta_{5} ) q^{35} -4 \beta_{1} q^{37} + ( \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{38} + ( -3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{40} + ( 4 \beta_{2} - \beta_{3} - \beta_{5} ) q^{41} + ( 4 \beta_{1} - 4 \beta_{3} + 4 \beta_{5} ) q^{43} + ( 2 - 2 \beta_{3} - 2 \beta_{5} ) q^{44} + ( -2 + \beta_{3} + \beta_{5} ) q^{46} + ( 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{47} - q^{49} + ( 2 + \beta_{1} - 3 \beta_{3} + 6 \beta_{4} + \beta_{5} ) q^{50} + ( 2 \beta_{1} - \beta_{3} + 6 \beta_{4} + \beta_{5} ) q^{52} + ( -2 \beta_{1} - \beta_{3} - 6 \beta_{4} + \beta_{5} ) q^{53} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{55} + ( 2 - \beta_{2} - \beta_{3} - \beta_{5} ) q^{56} + ( 2 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{58} + ( 4 + 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{5} ) q^{59} + ( -2 + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{61} + ( 3 \beta_{3} - 8 \beta_{4} - 3 \beta_{5} ) q^{62} + ( -1 + 4 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} ) q^{64} + ( -6 - 4 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{65} + ( -2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{67} + ( -2 \beta_{1} - \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{68} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{70} -2 q^{71} + ( 2 \beta_{1} - \beta_{3} - 6 \beta_{4} + \beta_{5} ) q^{73} + ( -12 + 4 \beta_{3} + 4 \beta_{5} ) q^{74} + ( -2 - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{5} ) q^{76} -2 \beta_{4} q^{77} + ( -4 + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{79} + ( -4 + \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} ) q^{80} + ( -2 \beta_{1} + 3 \beta_{3} - 10 \beta_{4} - 3 \beta_{5} ) q^{82} + ( -4 \beta_{1} - 4 \beta_{4} ) q^{83} + ( 6 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{85} + ( 4 + 8 \beta_{2} ) q^{86} + ( -2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{88} + ( 4 - \beta_{3} - \beta_{5} ) q^{89} + ( 2 + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{91} + ( 4 \beta_{1} - \beta_{3} - 6 \beta_{4} + \beta_{5} ) q^{92} + ( 4 - 2 \beta_{3} - 2 \beta_{5} ) q^{94} + ( -2 - 3 \beta_{3} - 6 \beta_{4} - \beta_{5} ) q^{95} + ( 2 \beta_{1} + 3 \beta_{3} + 10 \beta_{4} - 3 \beta_{5} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 10q^{4} - 2q^{5} + O(q^{10}) \) \( 6q - 10q^{4} - 2q^{5} - 12q^{10} - 12q^{11} - 2q^{14} + 26q^{16} - 12q^{19} + 30q^{20} - 2q^{25} - 20q^{26} + 4q^{29} + 4q^{31} + 24q^{34} + 4q^{40} - 4q^{41} + 20q^{44} - 16q^{46} - 6q^{49} + 16q^{50} + 4q^{55} + 18q^{56} + 32q^{59} - 12q^{61} - 26q^{64} - 32q^{65} + 10q^{70} - 12q^{71} - 88q^{74} + 4q^{76} - 24q^{79} - 46q^{80} + 32q^{85} + 8q^{86} + 28q^{89} + 12q^{91} + 32q^{94} - 4q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{5} + \nu^{4} + 11 \nu^{3} - 26 \nu^{2} + 6 \nu - 1 \)\()/23\)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{5} + 9 \nu^{4} - 16 \nu^{3} - 4 \nu^{2} + 8 \nu - 9 \)\()/23\)
\(\beta_{3}\)\(=\)\((\)\( 6 \nu^{5} - 2 \nu^{4} + \nu^{3} + 6 \nu^{2} + 80 \nu + 2 \)\()/23\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13 \)\()/23\)
\(\beta_{5}\)\(=\)\((\)\( -16 \nu^{5} + 36 \nu^{4} - 41 \nu^{3} - 16 \nu^{2} - 60 \nu + 56 \)\()/23\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 4 \beta_{4} - \beta_{3} + 2 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 2 \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} + 2 \beta_{3} - 5 \beta_{2} - 7\)
\(\nu^{5}\)\(=\)\(-9 \beta_{4} + 5 \beta_{3} - 8 \beta_{2} - 8 \beta_{1} - 9\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
−0.854638 + 0.854638i
1.45161 + 1.45161i
0.403032 0.403032i
0.403032 + 0.403032i
1.45161 1.45161i
−0.854638 0.854638i
2.70928i 0 −5.34017 −2.17009 0.539189i 0 1.00000i 9.04945i 0 −1.46081 + 5.87936i
64.2 1.90321i 0 −1.62222 −0.311108 2.21432i 0 1.00000i 0.719004i 0 −4.21432 + 0.592104i
64.3 0.193937i 0 1.96239 1.48119 1.67513i 0 1.00000i 0.768452i 0 −0.324869 0.287258i
64.4 0.193937i 0 1.96239 1.48119 + 1.67513i 0 1.00000i 0.768452i 0 −0.324869 + 0.287258i
64.5 1.90321i 0 −1.62222 −0.311108 + 2.21432i 0 1.00000i 0.719004i 0 −4.21432 0.592104i
64.6 2.70928i 0 −5.34017 −2.17009 + 0.539189i 0 1.00000i 9.04945i 0 −1.46081 5.87936i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.6
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 315.2.d.e 6
3.b odd 2 1 105.2.d.b 6
4.b odd 2 1 5040.2.t.v 6
5.b even 2 1 inner 315.2.d.e 6
5.c odd 4 1 1575.2.a.w 3
5.c odd 4 1 1575.2.a.x 3
7.b odd 2 1 2205.2.d.l 6
12.b even 2 1 1680.2.t.k 6
15.d odd 2 1 105.2.d.b 6
15.e even 4 1 525.2.a.j 3
15.e even 4 1 525.2.a.k 3
20.d odd 2 1 5040.2.t.v 6
21.c even 2 1 735.2.d.b 6
21.g even 6 2 735.2.q.f 12
21.h odd 6 2 735.2.q.e 12
35.c odd 2 1 2205.2.d.l 6
60.h even 2 1 1680.2.t.k 6
60.l odd 4 1 8400.2.a.dg 3
60.l odd 4 1 8400.2.a.dj 3
105.g even 2 1 735.2.d.b 6
105.k odd 4 1 3675.2.a.bi 3
105.k odd 4 1 3675.2.a.bj 3
105.o odd 6 2 735.2.q.e 12
105.p even 6 2 735.2.q.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.b 6 3.b odd 2 1
105.2.d.b 6 15.d odd 2 1
315.2.d.e 6 1.a even 1 1 trivial
315.2.d.e 6 5.b even 2 1 inner
525.2.a.j 3 15.e even 4 1
525.2.a.k 3 15.e even 4 1
735.2.d.b 6 21.c even 2 1
735.2.d.b 6 105.g even 2 1
735.2.q.e 12 21.h odd 6 2
735.2.q.e 12 105.o odd 6 2
735.2.q.f 12 21.g even 6 2
735.2.q.f 12 105.p even 6 2
1575.2.a.w 3 5.c odd 4 1
1575.2.a.x 3 5.c odd 4 1
1680.2.t.k 6 12.b even 2 1
1680.2.t.k 6 60.h even 2 1
2205.2.d.l 6 7.b odd 2 1
2205.2.d.l 6 35.c odd 2 1
3675.2.a.bi 3 105.k odd 4 1
3675.2.a.bj 3 105.k odd 4 1
5040.2.t.v 6 4.b odd 2 1
5040.2.t.v 6 20.d odd 2 1
8400.2.a.dg 3 60.l odd 4 1
8400.2.a.dj 3 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{6} + 11 T_{2}^{4} + 27 T_{2}^{2} + 1 \)
\( T_{11} + 2 \)
\( T_{29}^{3} - 2 T_{29}^{2} - 52 T_{29} + 40 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} - T^{4} - 3 T^{6} - 4 T^{8} - 16 T^{10} + 64 T^{12} \)
$3$ \( \)
$5$ \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 15 T^{4} + 50 T^{5} + 125 T^{6} \)
$7$ \( ( 1 + T^{2} )^{3} \)
$11$ \( ( 1 + 2 T + 11 T^{2} )^{6} \)
$13$ \( 1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 60671 T^{8} - 971074 T^{10} + 4826809 T^{12} \)
$17$ \( 1 - 70 T^{2} + 2415 T^{4} - 51220 T^{6} + 697935 T^{8} - 5846470 T^{10} + 24137569 T^{12} \)
$19$ \( ( 1 + 6 T + 53 T^{2} + 188 T^{3} + 1007 T^{4} + 2166 T^{5} + 6859 T^{6} )^{2} \)
$23$ \( 1 - 106 T^{2} + 5183 T^{4} - 150348 T^{6} + 2741807 T^{8} - 29663146 T^{10} + 148035889 T^{12} \)
$29$ \( ( 1 - 2 T + 35 T^{2} - 76 T^{3} + 1015 T^{4} - 1682 T^{5} + 24389 T^{6} )^{2} \)
$31$ \( ( 1 - 2 T + 41 T^{2} + 60 T^{3} + 1271 T^{4} - 1922 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( 1 - 46 T^{2} + 1399 T^{4} - 74788 T^{6} + 1915231 T^{8} - 86211406 T^{10} + 2565726409 T^{12} \)
$41$ \( ( 1 + 2 T + 63 T^{2} - 36 T^{3} + 2583 T^{4} + 3362 T^{5} + 68921 T^{6} )^{2} \)
$43$ \( 1 + 46 T^{2} + 2839 T^{4} + 118948 T^{6} + 5249311 T^{8} + 157264846 T^{10} + 6321363049 T^{12} \)
$47$ \( 1 - 154 T^{2} + 12143 T^{4} - 652332 T^{6} + 26823887 T^{8} - 751470874 T^{10} + 10779215329 T^{12} \)
$53$ \( 1 - 146 T^{2} + 14103 T^{4} - 884828 T^{6} + 39615327 T^{8} - 1152010226 T^{10} + 22164361129 T^{12} \)
$59$ \( ( 1 - 16 T + 113 T^{2} - 608 T^{3} + 6667 T^{4} - 55696 T^{5} + 205379 T^{6} )^{2} \)
$61$ \( ( 1 + 6 T + 131 T^{2} + 484 T^{3} + 7991 T^{4} + 22326 T^{5} + 226981 T^{6} )^{2} \)
$67$ \( 1 - 274 T^{2} + 36103 T^{4} - 2962972 T^{6} + 162066367 T^{8} - 5521407154 T^{10} + 90458382169 T^{12} \)
$71$ \( ( 1 + 2 T + 71 T^{2} )^{6} \)
$73$ \( 1 - 298 T^{2} + 43775 T^{4} - 3982284 T^{6} + 233276975 T^{8} - 8462675818 T^{10} + 151334226289 T^{12} \)
$79$ \( ( 1 + 12 T + 221 T^{2} + 1576 T^{3} + 17459 T^{4} + 74892 T^{5} + 493039 T^{6} )^{2} \)
$83$ \( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 329177087 T^{8} - 14522246226 T^{10} + 326940373369 T^{12} \)
$89$ \( ( 1 - 14 T + 319 T^{2} - 2532 T^{3} + 28391 T^{4} - 110894 T^{5} + 704969 T^{6} )^{2} \)
$97$ \( 1 - 26 T^{2} + 8719 T^{4} + 446932 T^{6} + 82037071 T^{8} - 2301761306 T^{10} + 832972004929 T^{12} \)
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