Properties

Label 315.5.e.c
Level $315$
Weight $5$
Character orbit 315.e
Analytic conductor $32.562$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(32.5615383714\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Defining polynomial: \( x^{2} + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 10 q^{4} + ( - 10 \beta - 5) q^{5} + ( - 14 \beta + 35) q^{7} + 26 \beta q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 10 q^{4} + ( - 10 \beta - 5) q^{5} + ( - 14 \beta + 35) q^{7} + 26 \beta q^{8} + ( - 5 \beta + 60) q^{10} - 89 q^{11} - 5 q^{13} + (35 \beta + 84) q^{14} + 4 q^{16} + 485 q^{17} + 90 \beta q^{19} + ( - 100 \beta - 50) q^{20} - 89 \beta q^{22} - 286 \beta q^{23} + (100 \beta - 575) q^{25} - 5 \beta q^{26} + ( - 140 \beta + 350) q^{28} - 191 q^{29} - 430 \beta q^{31} + 420 \beta q^{32} + 485 \beta q^{34} + ( - 280 \beta - 1015) q^{35} - 666 \beta q^{37} - 540 q^{38} + ( - 130 \beta + 1560) q^{40} - 1190 \beta q^{41} - 154 \beta q^{43} - 890 q^{44} + 1716 q^{46} + 2195 q^{47} + ( - 980 \beta + 49) q^{49} + ( - 575 \beta - 600) q^{50} - 50 q^{52} - 648 \beta q^{53} + (890 \beta + 445) q^{55} + (910 \beta + 2184) q^{56} - 191 \beta q^{58} + 1480 \beta q^{59} + 790 \beta q^{61} + 2580 q^{62} - 2456 q^{64} + (50 \beta + 25) q^{65} - 836 \beta q^{67} + 4850 q^{68} + ( - 1015 \beta + 1680) q^{70} - 4454 q^{71} + 8650 q^{73} + 3996 q^{74} + 900 \beta q^{76} + (1246 \beta - 3115) q^{77} + 5561 q^{79} + ( - 40 \beta - 20) q^{80} + 7140 q^{82} - 1990 q^{83} + ( - 4850 \beta - 2425) q^{85} + 924 q^{86} - 2314 \beta q^{88} + 330 \beta q^{89} + (70 \beta - 175) q^{91} - 2860 \beta q^{92} + 2195 \beta q^{94} + ( - 450 \beta + 5400) q^{95} + 9235 q^{97} + (49 \beta + 5880) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{4} - 10 q^{5} + 70 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 20 q^{4} - 10 q^{5} + 70 q^{7} + 120 q^{10} - 178 q^{11} - 10 q^{13} + 168 q^{14} + 8 q^{16} + 970 q^{17} - 100 q^{20} - 1150 q^{25} + 700 q^{28} - 382 q^{29} - 2030 q^{35} - 1080 q^{38} + 3120 q^{40} - 1780 q^{44} + 3432 q^{46} + 4390 q^{47} + 98 q^{49} - 1200 q^{50} - 100 q^{52} + 890 q^{55} + 4368 q^{56} + 5160 q^{62} - 4912 q^{64} + 50 q^{65} + 9700 q^{68} + 3360 q^{70} - 8908 q^{71} + 17300 q^{73} + 7992 q^{74} - 6230 q^{77} + 11122 q^{79} - 40 q^{80} + 14280 q^{82} - 3980 q^{83} - 4850 q^{85} + 1848 q^{86} - 350 q^{91} + 10800 q^{95} + 18470 q^{97} + 11760 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
2.44949i
2.44949i
2.44949i 0 10.0000 −5.00000 + 24.4949i 0 35.0000 + 34.2929i 63.6867i 0 60.0000 + 12.2474i
244.2 2.44949i 0 10.0000 −5.00000 24.4949i 0 35.0000 34.2929i 63.6867i 0 60.0000 12.2474i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.5.e.c 2
3.b odd 2 1 35.5.c.d yes 2
5.b even 2 1 315.5.e.d 2
7.b odd 2 1 315.5.e.d 2
12.b even 2 1 560.5.p.d 2
15.d odd 2 1 35.5.c.c 2
15.e even 4 2 175.5.d.e 4
21.c even 2 1 35.5.c.c 2
35.c odd 2 1 inner 315.5.e.c 2
60.h even 2 1 560.5.p.e 2
84.h odd 2 1 560.5.p.e 2
105.g even 2 1 35.5.c.d yes 2
105.k odd 4 2 175.5.d.e 4
420.o odd 2 1 560.5.p.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.c 2 15.d odd 2 1
35.5.c.c 2 21.c even 2 1
35.5.c.d yes 2 3.b odd 2 1
35.5.c.d yes 2 105.g even 2 1
175.5.d.e 4 15.e even 4 2
175.5.d.e 4 105.k odd 4 2
315.5.e.c 2 1.a even 1 1 trivial
315.5.e.c 2 35.c odd 2 1 inner
315.5.e.d 2 5.b even 2 1
315.5.e.d 2 7.b odd 2 1
560.5.p.d 2 12.b even 2 1
560.5.p.d 2 420.o odd 2 1
560.5.p.e 2 60.h even 2 1
560.5.p.e 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{2} + 6 \) Copy content Toggle raw display
\( T_{13} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 6 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 10T + 625 \) Copy content Toggle raw display
$7$ \( T^{2} - 70T + 2401 \) Copy content Toggle raw display
$11$ \( (T + 89)^{2} \) Copy content Toggle raw display
$13$ \( (T + 5)^{2} \) Copy content Toggle raw display
$17$ \( (T - 485)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 48600 \) Copy content Toggle raw display
$23$ \( T^{2} + 490776 \) Copy content Toggle raw display
$29$ \( (T + 191)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1109400 \) Copy content Toggle raw display
$37$ \( T^{2} + 2661336 \) Copy content Toggle raw display
$41$ \( T^{2} + 8496600 \) Copy content Toggle raw display
$43$ \( T^{2} + 142296 \) Copy content Toggle raw display
$47$ \( (T - 2195)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 2519424 \) Copy content Toggle raw display
$59$ \( T^{2} + 13142400 \) Copy content Toggle raw display
$61$ \( T^{2} + 3744600 \) Copy content Toggle raw display
$67$ \( T^{2} + 4193376 \) Copy content Toggle raw display
$71$ \( (T + 4454)^{2} \) Copy content Toggle raw display
$73$ \( (T - 8650)^{2} \) Copy content Toggle raw display
$79$ \( (T - 5561)^{2} \) Copy content Toggle raw display
$83$ \( (T + 1990)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 653400 \) Copy content Toggle raw display
$97$ \( (T - 9235)^{2} \) Copy content Toggle raw display
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