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Properties

Label 8925.2.a.cu
Level $8925$
Weight $2$
Character orbit 8925.a
Self dual yes
Analytic conductor $71.266$
Analytic rank $1$
Dimension $14$
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] = [8925,2,Mod(1,8925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8925, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8925.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8925 = 3 \cdot 5^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8925.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: \(71.2664838040\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} - 18 x^{12} + 54 x^{11} + 124 x^{10} - 366 x^{9} - 416 x^{8} + 1164 x^{7} + 727 x^{6} + \cdots - 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 1785)
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 a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} - q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} - q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} + q^{9} + \beta_{11} q^{11} + ( - \beta_{2} - 1) q^{12} - \beta_{6} q^{13} + \beta_1 q^{14} + (\beta_{4} + \beta_{3} + \beta_{2} + \cdots + 1) q^{16}+ \cdots + \beta_{11} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3 q^{2} - 14 q^{3} + 17 q^{4} + 3 q^{6} - 14 q^{7} - 15 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 3 q^{2} - 14 q^{3} + 17 q^{4} + 3 q^{6} - 14 q^{7} - 15 q^{8} + 14 q^{9} + 4 q^{11} - 17 q^{12} + q^{13} + 3 q^{14} + 19 q^{16} + 14 q^{17} - 3 q^{18} - 6 q^{19} + 14 q^{21} - 12 q^{22} - 7 q^{23} + 15 q^{24} - 2 q^{26} - 14 q^{27} - 17 q^{28} + 20 q^{29} - 7 q^{31} - 33 q^{32} - 4 q^{33} - 3 q^{34} + 17 q^{36} - 9 q^{37} - 18 q^{38} - q^{39} + 25 q^{41} - 3 q^{42} - 28 q^{43} + 24 q^{44} - 10 q^{46} - 29 q^{47} - 19 q^{48} + 14 q^{49} - 14 q^{51} - 4 q^{52} - 26 q^{53} + 3 q^{54} + 15 q^{56} + 6 q^{57} - 2 q^{58} + 14 q^{59} - 9 q^{61} - 28 q^{62} - 14 q^{63} + 27 q^{64} + 12 q^{66} - 32 q^{67} + 17 q^{68} + 7 q^{69} + 2 q^{71} - 15 q^{72} - 18 q^{73} + 64 q^{74} - 42 q^{76} - 4 q^{77} + 2 q^{78} - 4 q^{79} + 14 q^{81} - 42 q^{82} - 51 q^{83} + 17 q^{84} + 44 q^{86} - 20 q^{87} - 32 q^{88} + 38 q^{89} - q^{91} + 16 q^{92} + 7 q^{93} + 8 q^{94} + 33 q^{96} - 14 q^{97} - 3 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} - 18 x^{12} + 54 x^{11} + 124 x^{10} - 366 x^{9} - 416 x^{8} + 1164 x^{7} + 727 x^{6} + \cdots - 40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 7\nu^{2} + 4\nu + 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23 \nu^{13} + 30 \nu^{12} - 678 \nu^{11} - 491 \nu^{10} + 7255 \nu^{9} + 2951 \nu^{8} - 35893 \nu^{7} + \cdots + 2212 ) / 274 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 62 \nu^{13} + 211 \nu^{12} + 958 \nu^{11} - 3531 \nu^{10} - 5029 \nu^{9} + 21381 \nu^{8} + \cdots + 1054 ) / 274 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 177 \nu^{13} - 609 \nu^{12} - 2704 \nu^{11} + 10118 \nu^{10} + 13904 \nu^{9} - 60330 \nu^{8} + \cdots - 1776 ) / 548 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 169 \nu^{13} + 679 \nu^{12} + 2218 \nu^{11} - 11218 \nu^{10} - 7068 \nu^{9} + 66622 \nu^{8} + \cdots + 3832 ) / 548 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 133 \nu^{13} - 446 \nu^{12} - 2086 \nu^{11} + 7537 \nu^{10} + 11241 \nu^{9} - 46301 \nu^{8} + \cdots - 3494 ) / 274 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 140 \nu^{13} + 556 \nu^{12} + 1929 \nu^{11} - 9246 \nu^{10} - 7564 \nu^{9} + 55386 \nu^{8} + \cdots + 3750 ) / 274 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 181 \nu^{13} + 985 \nu^{12} + 1714 \nu^{11} - 16418 \nu^{10} + 3228 \nu^{9} + 99106 \nu^{8} + \cdots + 11160 ) / 548 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 387 \nu^{13} + 1443 \nu^{12} + 5666 \nu^{11} - 24124 \nu^{10} - 26346 \nu^{9} + 145464 \nu^{8} + \cdots + 5072 ) / 548 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 383 \nu^{13} - 1889 \nu^{12} - 4190 \nu^{11} + 31524 \nu^{10} + 2378 \nu^{9} - 190532 \nu^{8} + \cdots - 23088 ) / 548 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 9\beta_{3} + 2\beta_{2} + 28\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} + \beta_{12} + 2 \beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{12} + 13 \beta_{11} - 13 \beta_{10} + \beta_{8} - 11 \beta_{7} - 15 \beta_{6} - 14 \beta_{5} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 11 \beta_{13} + 14 \beta_{12} + 28 \beta_{11} - 17 \beta_{10} + \beta_{9} + 14 \beta_{8} + 13 \beta_{7} + \cdots + 541 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3 \beta_{13} + 21 \beta_{12} + 120 \beta_{11} - 129 \beta_{10} + 3 \beta_{9} + 19 \beta_{8} + \cdots + 674 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 81 \beta_{13} + 147 \beta_{12} + 276 \beta_{11} - 203 \beta_{10} + 24 \beta_{9} + 143 \beta_{8} + \cdots + 3528 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 64 \beta_{13} + 282 \beta_{12} + 978 \beta_{11} - 1160 \beta_{10} + 66 \beta_{9} + 234 \beta_{8} + \cdots + 5323 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 468 \beta_{13} + 1400 \beta_{12} + 2372 \beta_{11} - 2090 \beta_{10} + 346 \beta_{9} + 1290 \beta_{8} + \cdots + 23895 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 880 \beta_{13} + 3118 \beta_{12} + 7545 \beta_{11} - 9939 \beta_{10} + 932 \beta_{9} + 2392 \beta_{8} + \cdots + 41776 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.80343
2.53468
2.27949
2.13433
1.20959
1.10180
0.679959
−0.126917
−0.739867
−1.13259
−1.15341
−1.87407
−2.27385
−2.44257
−2.80343 −1.00000 5.85923 0 2.80343 −1.00000 −10.8191 1.00000 0
1.2 −2.53468 −1.00000 4.42458 0 2.53468 −1.00000 −6.14554 1.00000 0
1.3 −2.27949 −1.00000 3.19608 0 2.27949 −1.00000 −2.72646 1.00000 0
1.4 −2.13433 −1.00000 2.55536 0 2.13433 −1.00000 −1.18533 1.00000 0
1.5 −1.20959 −1.00000 −0.536888 0 1.20959 −1.00000 3.06860 1.00000 0
1.6 −1.10180 −1.00000 −0.786043 0 1.10180 −1.00000 3.06965 1.00000 0
1.7 −0.679959 −1.00000 −1.53766 0 0.679959 −1.00000 2.40546 1.00000 0
1.8 0.126917 −1.00000 −1.98389 0 −0.126917 −1.00000 −0.505624 1.00000 0
1.9 0.739867 −1.00000 −1.45260 0 −0.739867 −1.00000 −2.55446 1.00000 0
1.10 1.13259 −1.00000 −0.717247 0 −1.13259 −1.00000 −3.07752 1.00000 0
1.11 1.15341 −1.00000 −0.669644 0 −1.15341 −1.00000 −3.07920 1.00000 0
1.12 1.87407 −1.00000 1.51216 0 −1.87407 −1.00000 −0.914256 1.00000 0
1.13 2.27385 −1.00000 3.17040 0 −2.27385 −1.00000 2.66132 1.00000 0
1.14 2.44257 −1.00000 3.96615 0 −2.44257 −1.00000 4.80245 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(7\) \( +1 \)
\(17\) \( -1 \)

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 8925.2.a.cu 14
5.b even 2 1 8925.2.a.cx 14
5.c odd 4 2 1785.2.g.g ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1785.2.g.g ✓ 28 5.c odd 4 2
8925.2.a.cu 14 1.a even 1 1 trivial
8925.2.a.cx 14 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8925))\):

\( T_{2}^{14} + 3 T_{2}^{13} - 18 T_{2}^{12} - 54 T_{2}^{11} + 124 T_{2}^{10} + 366 T_{2}^{9} - 416 T_{2}^{8} + \cdots - 40 \) Copy content Toggle raw display
\( T_{11}^{14} - 4 T_{11}^{13} - 82 T_{11}^{12} + 360 T_{11}^{11} + 2457 T_{11}^{10} - 12448 T_{11}^{9} + \cdots + 3962880 \) Copy content Toggle raw display
\( T_{13}^{14} - T_{13}^{13} - 90 T_{13}^{12} - 26 T_{13}^{11} + 2883 T_{13}^{10} + 4053 T_{13}^{9} + \cdots - 116672 \) Copy content Toggle raw display
\( T_{23}^{14} + 7 T_{23}^{13} - 176 T_{23}^{12} - 1130 T_{23}^{11} + 12713 T_{23}^{10} + 69587 T_{23}^{9} + \cdots - 1349052416 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 3 T^{13} + \cdots - 40 \) Copy content Toggle raw display
$3$ \( (T + 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( (T + 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} - 4 T^{13} + \cdots + 3962880 \) Copy content Toggle raw display
$13$ \( T^{14} - T^{13} + \cdots - 116672 \) Copy content Toggle raw display
$17$ \( (T - 1)^{14} \) Copy content Toggle raw display
$19$ \( T^{14} + 6 T^{13} + \cdots - 3276800 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 1349052416 \) Copy content Toggle raw display
$29$ \( T^{14} - 20 T^{13} + \cdots - 46704640 \) Copy content Toggle raw display
$31$ \( T^{14} + 7 T^{13} + \cdots - 256 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 910219264 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 2280095744 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 1092384256 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 6444288000 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 7055585280 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 5327749120 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 3486030848 \) Copy content Toggle raw display
$67$ \( T^{14} + 32 T^{13} + \cdots + 4476928 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 229803335680 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 12322979840 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 1015313219584 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 254517248 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 275565772800 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 1204273152 \) Copy content Toggle raw display
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