↑

Properties

Label 625.4.a.a
Level $625$
Weight $4$
Character orbit 625.a
Self dual yes
Analytic conductor $36.876$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [625,4,Mod(1,625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 625.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: \(36.8761937536\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3 x^{9} - 58 x^{8} + 147 x^{7} + 1162 x^{6} - 2253 x^{5} - 9638 x^{4} + 10157 x^{3} + \cdots - 10044 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{9} q^{3} + ( - \beta_{6} + \beta_{4} + \beta_{2} + \cdots + 5) q^{4}+ \cdots + ( - \beta_{7} + \beta_{5} - \beta_{3} + \cdots + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{9} q^{3} + ( - \beta_{6} + \beta_{4} + \beta_{2} + \cdots + 5) q^{4}+ \cdots + ( - 21 \beta_{9} + 31 \beta_{8} + \cdots - 263) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} - q^{3} + 45 q^{4} - 5 q^{6} - 38 q^{7} - 60 q^{8} + 75 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{2} - q^{3} + 45 q^{4} - 5 q^{6} - 38 q^{7} - 60 q^{8} + 75 q^{9} - 100 q^{11} + 52 q^{12} - 46 q^{13} - 10 q^{14} + 125 q^{16} - 263 q^{17} - 461 q^{18} + 75 q^{19} + 10 q^{21} - 136 q^{22} - 306 q^{23} - 590 q^{24} + 685 q^{26} + 50 q^{27} - 779 q^{28} + 430 q^{29} - 330 q^{31} - 243 q^{32} - 122 q^{33} + 560 q^{34} + 350 q^{36} - 338 q^{37} - 110 q^{38} - 780 q^{39} + 325 q^{41} - 2651 q^{42} - 1431 q^{43} - 1925 q^{44} - 1640 q^{46} - 1538 q^{47} + 2599 q^{48} + 370 q^{49} + 1040 q^{51} + 97 q^{52} - 1676 q^{53} + 1835 q^{54} - 1185 q^{56} + 2375 q^{57} - 750 q^{58} + 1625 q^{59} + 570 q^{61} - 2236 q^{62} - 1456 q^{63} + 440 q^{64} + 2025 q^{66} - 1393 q^{67} - 3424 q^{68} - 3430 q^{69} - 1500 q^{71} - 4980 q^{72} + 1379 q^{73} + 265 q^{74} + 4415 q^{76} + 254 q^{77} - 4382 q^{78} + 1950 q^{79} - 2270 q^{81} + 3259 q^{82} - 1936 q^{83} + 4950 q^{84} + 675 q^{86} - 3580 q^{87} + 2625 q^{88} - 235 q^{89} + 4330 q^{91} - 6543 q^{92} - 6162 q^{93} - 5115 q^{94} - 8570 q^{96} - 2783 q^{97} + 1506 q^{98} - 2540 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3 x^{9} - 58 x^{8} + 147 x^{7} + 1162 x^{6} - 2253 x^{5} - 9638 x^{4} + 10157 x^{3} + \cdots - 10044 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17 \nu^{9} + 18 \nu^{8} + 908 \nu^{7} - 567 \nu^{6} - 14417 \nu^{5} + 1788 \nu^{4} + 58498 \nu^{3} + \cdots - 37692 ) / 46080 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11 \nu^{9} + 26 \nu^{8} - 644 \nu^{7} - 1779 \nu^{6} + 12171 \nu^{5} + 35116 \nu^{4} - 76214 \nu^{3} + \cdots + 56916 ) / 15360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11 \nu^{9} + 14 \nu^{8} + 604 \nu^{7} - 341 \nu^{6} - 10811 \nu^{5} - 2676 \nu^{4} + \cdots - 297396 ) / 15360 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19 \nu^{9} + 114 \nu^{8} - 1356 \nu^{7} - 6211 \nu^{6} + 30179 \nu^{5} + 102724 \nu^{4} + \cdots - 63276 ) / 15360 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{9} + 6 \nu^{8} + 272 \nu^{7} - 159 \nu^{6} - 4685 \nu^{5} - 624 \nu^{4} + 26302 \nu^{3} + \cdots - 33084 ) / 4608 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25 \nu^{9} + 222 \nu^{8} - 1708 \nu^{7} - 12177 \nu^{6} + 35353 \nu^{5} + 205668 \nu^{4} + \cdots + 364572 ) / 9216 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 21 \nu^{9} - 14 \nu^{8} - 1224 \nu^{7} + 471 \nu^{6} + 23021 \nu^{5} - 3304 \nu^{4} - 153974 \nu^{3} + \cdots - 21924 ) / 7680 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 51 \nu^{9} - 94 \nu^{8} - 2684 \nu^{7} + 3821 \nu^{6} + 43811 \nu^{5} - 43564 \nu^{4} - 225054 \nu^{3} + \cdots - 18924 ) / 15360 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{4} + \beta_{2} + \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} - 2\beta_{3} + 4\beta_{2} + 21\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{9} - 2 \beta_{8} + \beta_{7} - 33 \beta_{6} + \beta_{5} + 24 \beta_{4} - 11 \beta_{3} + \cdots + 265 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 44 \beta_{9} - 36 \beta_{8} + 3 \beta_{7} - 60 \beta_{6} + 33 \beta_{5} + 11 \beta_{4} - 85 \beta_{3} + \cdots + 272 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 131 \beta_{9} - 99 \beta_{8} + 47 \beta_{7} - 928 \beta_{6} + 60 \beta_{5} + 588 \beta_{4} - 527 \beta_{3} + \cdots + 6215 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1578 \beta_{9} - 1170 \beta_{8} + 178 \beta_{7} - 2376 \beta_{6} + 928 \beta_{5} + 688 \beta_{4} + \cdots + 10700 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5604 \beta_{9} - 3772 \beta_{8} + 1756 \beta_{7} - 25641 \beta_{6} + 2376 \beta_{5} + 15081 \beta_{4} + \cdots + 158029 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 52185 \beta_{9} - 36305 \beta_{8} + 7360 \beta_{7} - 82425 \beta_{6} + 25641 \beta_{5} + 29592 \beta_{4} + \cdots + 389974 \) 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
5.52344
4.16573
3.58474
3.38804
0.555063
−0.651433
−1.99429
−2.34171
−4.50680
−4.72277
−5.52344 7.37811 22.5084 0 −40.7525 2.21537 −80.1364 27.4365 0
1.2 −4.16573 −8.66031 9.35329 0 36.0765 −35.7200 −5.63742 48.0009 0
1.3 −3.58474 4.41194 4.85037 0 −15.8157 15.1009 11.2906 −7.53481 0
1.4 −3.38804 −6.23768 3.47880 0 21.1335 −7.12987 15.3180 11.9086 0
1.5 −0.555063 4.40775 −7.69191 0 −2.44658 −3.08215 8.70999 −7.57178 0
1.6 0.651433 −4.01551 −7.57564 0 −2.61584 13.3099 −10.1465 −10.8757 0
1.7 1.99429 8.64253 −4.02279 0 17.2358 −16.2723 −23.9770 47.6933 0
1.8 2.34171 −6.38955 −2.51639 0 −14.9625 29.6291 −24.6263 13.8263 0
1.9 4.50680 1.45941 12.3112 0 6.57726 −29.4521 19.4299 −24.8701 0
1.10 4.72277 −1.99668 14.3046 0 −9.42989 −6.59875 29.7752 −23.0133 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

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 625.4.a.a ✓ 10
5.b even 2 1 625.4.a.b yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.4.a.a ✓ 10 1.a even 1 1 trivial
625.4.a.b yes 10 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 3 T_{2}^{9} - 58 T_{2}^{8} - 147 T_{2}^{7} + 1162 T_{2}^{6} + 2253 T_{2}^{5} - 9638 T_{2}^{4} + \cdots - 10044 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(625))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 3 T^{9} + \cdots - 10044 \) Copy content Toggle raw display
$3$ \( T^{10} + T^{9} + \cdots - 5008241 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 32750537636 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 24138891905904 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 38\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 10\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 36\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 12\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 92\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 22\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 37\!\cdots\!75 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 36\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 66\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 40\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 68\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 96\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 84\!\cdots\!89 \) Copy content Toggle raw display
show more
show less