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Properties

Label 425.4.a.n
Level $425$
Weight $4$
Character orbit 425.a
Self dual yes
Analytic conductor $25.076$
Analytic rank $1$
Dimension $12$
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] = [425,4,Mod(1,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.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: \(25.0758117524\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 64 x^{10} + 222 x^{9} + 1546 x^{8} - 4344 x^{7} - 17308 x^{6} + 36842 x^{5} + \cdots - 23104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: no (minimal twist has level 85)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{2} + \beta_1 + 4) q^{4} + ( - \beta_{10} + \beta_{7} + \cdots + \beta_1) q^{6}+ \cdots + (\beta_{9} + \beta_{6} - 2 \beta_{4} + \cdots + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{4} - 1) q^{3} + (\beta_{2} + \beta_1 + 4) q^{4} + ( - \beta_{10} + \beta_{7} + \cdots + \beta_1) q^{6}+ \cdots + ( - 3 \beta_{11} + 53 \beta_{10} + \cdots + 121) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} - 12 q^{3} + 48 q^{4} + 8 q^{6} - 70 q^{7} - 102 q^{8} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} - 12 q^{3} + 48 q^{4} + 8 q^{6} - 70 q^{7} - 102 q^{8} + 90 q^{9} + 30 q^{11} - 120 q^{12} - 112 q^{13} + 38 q^{14} + 120 q^{16} - 204 q^{17} - 166 q^{18} - 50 q^{19} - 64 q^{21} - 400 q^{22} - 262 q^{23} - 102 q^{24} - 50 q^{26} - 678 q^{27} - 428 q^{28} + 232 q^{29} + 18 q^{31} - 880 q^{32} - 168 q^{33} + 68 q^{34} + 302 q^{36} - 852 q^{37} + 200 q^{38} + 132 q^{39} + 408 q^{41} - 1794 q^{42} - 876 q^{43} + 52 q^{44} + 104 q^{46} - 1132 q^{47} + 494 q^{48} - 12 q^{49} + 204 q^{51} - 2790 q^{52} - 492 q^{53} - 150 q^{54} + 536 q^{56} - 1842 q^{57} - 866 q^{58} - 134 q^{59} + 712 q^{61} - 1648 q^{62} - 1246 q^{63} - 12 q^{64} - 2068 q^{66} - 2680 q^{67} - 816 q^{68} - 900 q^{69} + 98 q^{71} - 3992 q^{72} - 894 q^{73} - 1236 q^{74} - 1594 q^{76} - 1164 q^{77} - 40 q^{78} - 1134 q^{79} - 608 q^{81} - 5406 q^{82} - 2272 q^{83} + 560 q^{84} - 1932 q^{86} - 2320 q^{87} - 3238 q^{88} - 1622 q^{89} - 120 q^{91} - 2226 q^{92} - 5092 q^{93} + 1982 q^{94} - 1548 q^{96} - 2706 q^{97} + 2582 q^{98} + 614 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 64 x^{10} + 222 x^{9} + 1546 x^{8} - 4344 x^{7} - 17308 x^{6} + 36842 x^{5} + \cdots - 23104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 129 \nu^{11} - 819 \nu^{10} - 5843 \nu^{9} + 39819 \nu^{8} + 80501 \nu^{7} - 628003 \nu^{6} + \cdots - 518848 ) / 32000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 347 \nu^{11} + 2449 \nu^{10} + 14361 \nu^{9} - 118937 \nu^{8} - 155631 \nu^{7} + 1888161 \nu^{6} + \cdots + 1269056 ) / 80000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2197 \nu^{11} - 16479 \nu^{10} - 86191 \nu^{9} + 800087 \nu^{8} + 761401 \nu^{7} - 12679791 \nu^{6} + \cdots - 15494336 ) / 160000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2641 \nu^{11} - 18787 \nu^{10} - 108723 \nu^{9} + 914811 \nu^{8} + 1156853 \nu^{7} - 14589523 \nu^{6} + \cdots - 7337408 ) / 160000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 541 \nu^{11} - 4039 \nu^{10} - 21255 \nu^{9} + 196831 \nu^{8} + 185121 \nu^{7} - 3137175 \nu^{6} + \cdots - 4790720 ) / 32000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 923 \nu^{11} - 6741 \nu^{10} - 36749 \nu^{9} + 327733 \nu^{8} + 338979 \nu^{7} - 5206949 \nu^{6} + \cdots - 7373504 ) / 40000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5297 \nu^{11} + 38179 \nu^{10} + 215491 \nu^{9} - 1860187 \nu^{8} - 2173701 \nu^{7} + \cdots + 33907136 ) / 160000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5521 \nu^{11} - 40787 \nu^{10} - 218803 \nu^{9} + 1983691 \nu^{8} + 1998453 \nu^{7} + \cdots - 42525888 ) / 160000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1125 \nu^{11} - 8159 \nu^{10} - 45119 \nu^{9} + 395415 \nu^{8} + 432681 \nu^{7} - 6246159 \nu^{6} + \cdots - 10221504 ) / 32000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 2\beta_{2} + 19\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - \beta_{9} + 2\beta_{8} - 3\beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} + 24\beta_{2} + 36\beta _1 + 232 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{11} + 5 \beta_{10} + 32 \beta_{9} + 27 \beta_{7} + 31 \beta_{6} - \beta_{5} + 40 \beta_{4} + \cdots + 332 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 35 \beta_{11} + 11 \beta_{10} - 38 \beta_{9} + 70 \beta_{8} - 123 \beta_{7} - 31 \beta_{6} + \cdots + 5180 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 51 \beta_{11} + 249 \beta_{10} + 839 \beta_{9} + 14 \beta_{8} + 560 \beta_{7} + 822 \beta_{6} + \cdots + 10374 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1044 \beta_{11} + 723 \beta_{10} - 1053 \beta_{9} + 2012 \beta_{8} - 3976 \beta_{7} - 694 \beta_{6} + \cdots + 123788 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2026 \beta_{11} + 9170 \beta_{10} + 20812 \beta_{9} + 1210 \beta_{8} + 9536 \beta_{7} + 20958 \beta_{6} + \cdots + 297986 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 29852 \beta_{11} + 31540 \beta_{10} - 25564 \beta_{9} + 55464 \beta_{8} - 119352 \beta_{7} + \cdots + 3066256 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 71780 \beta_{11} + 301968 \beta_{10} + 507113 \beta_{9} + 61736 \beta_{8} + 108297 \beta_{7} + \cdots + 8327972 \) 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
5.31297
4.85361
4.66854
2.51233
2.19303
0.611517
0.205607
−1.57180
−3.25846
−3.27583
−3.41186
−4.83965
−5.31297 6.01624 20.2276 0 −31.9641 −7.69179 −64.9650 9.19510 0
1.2 −4.85361 −5.07761 15.5576 0 24.6448 8.94364 −36.6815 −1.21788 0
1.3 −4.66854 −8.45783 13.7952 0 39.4857 −23.9706 −27.0552 44.5348 0
1.4 −2.51233 5.17633 −1.68821 0 −13.0046 8.44437 24.3400 −0.205570 0
1.5 −2.19303 −2.90774 −3.19061 0 6.37677 −28.8467 24.5414 −18.5450 0
1.6 −0.611517 1.94298 −7.62605 0 −1.18817 16.0577 9.55560 −23.2248 0
1.7 −0.205607 −8.50848 −7.95773 0 1.74941 −6.23269 3.28103 45.3942 0
1.8 1.57180 7.40915 −5.52946 0 11.6457 −19.5218 −21.2656 27.8955 0
1.9 3.25846 −0.620969 2.61757 0 −2.02340 18.9715 −17.5384 −26.6144 0
1.10 3.27583 4.65510 2.73108 0 15.2493 −34.5107 −17.2601 −5.33005 0
1.11 3.41186 −9.31520 3.64079 0 −31.7822 10.5093 −14.8730 59.7730 0
1.12 4.83965 −2.31197 15.4222 0 −11.1891 −12.1522 35.9208 −21.6548 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(17\) \(1\)

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 425.4.a.n 12
5.b even 2 1 425.4.a.o 12
5.c odd 4 2 85.4.b.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.4.b.a ✓ 24 5.c odd 4 2
425.4.a.n 12 1.a even 1 1 trivial
425.4.a.o 12 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(425))\):

\( T_{2}^{12} + 4 T_{2}^{11} - 64 T_{2}^{10} - 222 T_{2}^{9} + 1546 T_{2}^{8} + 4344 T_{2}^{7} + \cdots - 23104 \) Copy content Toggle raw display
\( T_{3}^{12} + 12 T_{3}^{11} - 135 T_{3}^{10} - 1754 T_{3}^{9} + 6644 T_{3}^{8} + 92452 T_{3}^{7} + \cdots - 29654000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{11} + \cdots - 23104 \) Copy content Toggle raw display
$3$ \( T^{12} + 12 T^{11} + \cdots - 29654000 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots - 65621007414784 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( (T + 17)^{12} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 42\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 93\!\cdots\!40 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 40\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 80\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 48\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 95\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 72\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 21\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 84\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
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