Properties

Label 315.8.a.c
Level $315$
Weight $8$
Character orbit 315.a
Self dual yes
Analytic conductor $98.401$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,8,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(98.4012830275\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 8) q^{2} + ( - 16 \beta - 20) q^{4} - 125 q^{5} - 343 q^{7} + ( - 20 \beta + 480) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 8) q^{2} + ( - 16 \beta - 20) q^{4} - 125 q^{5} - 343 q^{7} + ( - 20 \beta + 480) q^{8} + ( - 125 \beta + 1000) q^{10} + ( - 380 \beta + 3953) q^{11} + (418 \beta - 8909) q^{13} + ( - 343 \beta + 2744) q^{14} + (2688 \beta - 2160) q^{16} + (2210 \beta + 1199) q^{17} + ( - 5542 \beta - 1806) q^{19} + (2000 \beta + 2500) q^{20} + (6993 \beta - 48344) q^{22} + (10690 \beta - 6922) q^{23} + 15625 q^{25} + ( - 12253 \beta + 89664) q^{26} + (5488 \beta + 6860) q^{28} + ( - 23772 \beta + 63449) q^{29} + (22554 \beta + 126384) q^{31} + ( - 21104 \beta + 74112) q^{32} + ( - 16481 \beta + 87648) q^{34} + 42875 q^{35} + (43638 \beta - 132930) q^{37} + (42530 \beta - 229400) q^{38} + (2500 \beta - 60000) q^{40} + (37354 \beta + 55960) q^{41} + (24578 \beta + 473786) q^{43} + ( - 55648 \beta + 188460) q^{44} + ( - 92442 \beta + 525736) q^{46} + ( - 56742 \beta - 135637) q^{47} + 117649 q^{49} + (15625 \beta - 125000) q^{50} + (134184 \beta - 116092) q^{52} + ( - 65224 \beta + 633896) q^{53} + (47500 \beta - 494125) q^{55} + (6860 \beta - 164640) q^{56} + (253625 \beta - 1553560) q^{58} + (170640 \beta + 680060) q^{59} + ( - 11334 \beta - 906840) q^{61} + ( - 54048 \beta - 18696) q^{62} + ( - 101120 \beta - 1244992) q^{64} + ( - 52250 \beta + 1113625) q^{65} + ( - 506344 \beta - 1094656) q^{67} + ( - 63384 \beta - 1579820) q^{68} + (42875 \beta - 343000) q^{70} + ( - 222048 \beta + 747464) q^{71} + ( - 212396 \beta + 3584894) q^{73} + ( - 482034 \beta + 2983512) q^{74} + (139736 \beta + 3937688) q^{76} + (130340 \beta - 1355879) q^{77} + ( - 187504 \beta - 3971487) q^{79} + ( - 336000 \beta + 270000) q^{80} + ( - 242872 \beta + 1195896) q^{82} + ( - 939444 \beta + 152356) q^{83} + ( - 276250 \beta - 149875) q^{85} + (277162 \beta - 2708856) q^{86} + ( - 261460 \beta + 2231840) q^{88} + (247538 \beta + 8971764) q^{89} + ( - 143374 \beta + 3055787) q^{91} + ( - 103048 \beta - 7387320) q^{92} + (318299 \beta - 1411552) q^{94} + (692750 \beta + 225750) q^{95} + (680782 \beta + 2129037) q^{97} + (117649 \beta - 941192) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{2} - 40 q^{4} - 250 q^{5} - 686 q^{7} + 960 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{2} - 40 q^{4} - 250 q^{5} - 686 q^{7} + 960 q^{8} + 2000 q^{10} + 7906 q^{11} - 17818 q^{13} + 5488 q^{14} - 4320 q^{16} + 2398 q^{17} - 3612 q^{19} + 5000 q^{20} - 96688 q^{22} - 13844 q^{23} + 31250 q^{25} + 179328 q^{26} + 13720 q^{28} + 126898 q^{29} + 252768 q^{31} + 148224 q^{32} + 175296 q^{34} + 85750 q^{35} - 265860 q^{37} - 458800 q^{38} - 120000 q^{40} + 111920 q^{41} + 947572 q^{43} + 376920 q^{44} + 1051472 q^{46} - 271274 q^{47} + 235298 q^{49} - 250000 q^{50} - 232184 q^{52} + 1267792 q^{53} - 988250 q^{55} - 329280 q^{56} - 3107120 q^{58} + 1360120 q^{59} - 1813680 q^{61} - 37392 q^{62} - 2489984 q^{64} + 2227250 q^{65} - 2189312 q^{67} - 3159640 q^{68} - 686000 q^{70} + 1494928 q^{71} + 7169788 q^{73} + 5967024 q^{74} + 7875376 q^{76} - 2711758 q^{77} - 7942974 q^{79} + 540000 q^{80} + 2391792 q^{82} + 304712 q^{83} - 299750 q^{85} - 5417712 q^{86} + 4463680 q^{88} + 17943528 q^{89} + 6111574 q^{91} - 14774640 q^{92} - 2823104 q^{94} + 451500 q^{95} + 4258074 q^{97} - 1882384 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.31662
3.31662
−14.6332 0 86.1320 −125.000 0 −343.000 612.665 0 1829.16
1.2 −1.36675 0 −126.132 −125.000 0 −343.000 347.335 0 170.844
\(n\): 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 315.8.a.c 2
3.b odd 2 1 35.8.a.a 2
12.b even 2 1 560.8.a.i 2
15.d odd 2 1 175.8.a.b 2
15.e even 4 2 175.8.b.c 4
21.c even 2 1 245.8.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.8.a.a 2 3.b odd 2 1
175.8.a.b 2 15.d odd 2 1
175.8.b.c 4 15.e even 4 2
245.8.a.b 2 21.c even 2 1
315.8.a.c 2 1.a even 1 1 trivial
560.8.a.i 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 16T_{2} + 20 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(315))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16T + 20 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 125)^{2} \) Copy content Toggle raw display
$7$ \( (T + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 7906 T + 9272609 \) Copy content Toggle raw display
$13$ \( T^{2} + 17818 T + 71682425 \) Copy content Toggle raw display
$17$ \( T^{2} - 2398 T - 213462799 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 1348143980 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 4980234316 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 20838975695 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 6409132848 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 66117717036 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 58262616304 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 197893738100 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 123267405047 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 214640651072 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 818710818800 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 816706565136 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 10082635080448 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 1610731398080 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 10866534315332 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 14225767990465 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 38809208931248 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 77796446568160 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 15859623239687 \) Copy content Toggle raw display
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