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Properties

Label 425.4.a.l
Level $425$
Weight $4$
Character orbit 425.a
Self dual yes
Analytic conductor $25.076$
Analytic rank $0$
Dimension $10$
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: \(0\)
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: \( x^{10} - x^{9} - 70 x^{8} + 34 x^{7} + 1721 x^{6} + 15 x^{5} - 17204 x^{4} - 6512 x^{3} + 56640 x^{2} + \cdots - 29184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{6} + 1) q^{3} + (\beta_{2} + \beta_1 + 6) q^{4} + ( - \beta_{8} - \beta_1) q^{6} + ( - \beta_{6} - \beta_{3} + 2) q^{7} + ( - \beta_{9} - \beta_{8} - \beta_{3} + \cdots - 9) q^{8}+ \cdots + (\beta_{9} + \beta_{8} + \beta_{5} + \cdots + 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{6} + 1) q^{3} + (\beta_{2} + \beta_1 + 6) q^{4} + ( - \beta_{8} - \beta_1) q^{6} + ( - \beta_{6} - \beta_{3} + 2) q^{7} + ( - \beta_{9} - \beta_{8} - \beta_{3} + \cdots - 9) q^{8}+ \cdots + (44 \beta_{9} + 4 \beta_{8} + \cdots + 746) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 6 q^{3} + 61 q^{4} + q^{6} + 16 q^{7} - 93 q^{8} + 128 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} + 6 q^{3} + 61 q^{4} + q^{6} + 16 q^{7} - 93 q^{8} + 128 q^{9} + 68 q^{11} + 39 q^{12} - 58 q^{13} + 67 q^{14} + 317 q^{16} + 170 q^{17} + 134 q^{18} + 226 q^{19} + 450 q^{21} - 64 q^{22} + 272 q^{23} + 137 q^{24} + 581 q^{26} - 234 q^{27} + 309 q^{28} - 88 q^{29} + 654 q^{31} - 805 q^{32} + 696 q^{33} - 17 q^{34} + 1050 q^{36} - 568 q^{37} + 318 q^{38} + 736 q^{39} + 364 q^{41} - 1663 q^{42} + 882 q^{43} - 160 q^{44} + 2132 q^{46} - 672 q^{47} + 2127 q^{48} + 904 q^{49} + 102 q^{51} - 3145 q^{52} + 494 q^{53} - 849 q^{54} + 2363 q^{56} - 708 q^{57} + 2952 q^{58} + 202 q^{59} + 1276 q^{61} - 2569 q^{62} + 1458 q^{63} + 1797 q^{64} + 1968 q^{66} - 664 q^{67} + 1037 q^{68} - 308 q^{69} + 1724 q^{71} - 2594 q^{72} + 24 q^{73} - 452 q^{74} + 4122 q^{76} + 932 q^{77} + 3523 q^{78} + 3354 q^{79} + 3174 q^{81} + 994 q^{82} - 2708 q^{83} + 3979 q^{84} + 2348 q^{86} + 1194 q^{87} + 2336 q^{88} + 264 q^{89} + 3532 q^{91} - 2764 q^{92} - 358 q^{93} + 1254 q^{94} + 1417 q^{96} + 1868 q^{97} - 2366 q^{98} + 7106 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 70 x^{8} + 34 x^{7} + 1721 x^{6} + 15 x^{5} - 17204 x^{4} - 6512 x^{3} + 56640 x^{2} + \cdots - 29184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21 \nu^{9} + 6 \nu^{8} + 1789 \nu^{7} + 1508 \nu^{6} - 55835 \nu^{5} - 72298 \nu^{4} + \cdots - 392736 ) / 70496 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 129 \nu^{9} - 981 \nu^{8} - 5010 \nu^{7} + 53050 \nu^{6} + 33937 \nu^{5} - 886181 \nu^{4} + \cdots - 1384192 ) / 281984 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 85 \nu^{9} - 339 \nu^{8} - 5248 \nu^{7} + 20542 \nu^{6} + 111233 \nu^{5} - 409283 \nu^{4} + \cdots - 3012736 ) / 140992 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 383 \nu^{9} + 1683 \nu^{8} + 22662 \nu^{7} - 88102 \nu^{6} - 444495 \nu^{5} + 1380307 \nu^{4} + \cdots + 3300864 ) / 281984 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 641 \nu^{9} - 3645 \nu^{8} - 32682 \nu^{7} + 194202 \nu^{6} + 477121 \nu^{5} - 3258413 \nu^{4} + \cdots - 16502656 ) / 281984 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 325 \nu^{9} + 1037 \nu^{8} + 18770 \nu^{7} - 53662 \nu^{6} - 346513 \nu^{5} + 848801 \nu^{4} + \cdots + 2794368 ) / 70496 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 346 \nu^{9} - 1043 \nu^{8} - 20559 \nu^{7} + 52154 \nu^{6} + 402348 \nu^{5} - 776503 \nu^{4} + \cdots - 2049152 ) / 70496 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{3} + \beta_{2} + 22\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - \beta_{8} - 2\beta_{7} - 4\beta_{6} - 2\beta_{5} + 2\beta_{4} + \beta_{3} + 28\beta_{2} + 34\beta _1 + 305 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 36 \beta_{9} + 36 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} + 6 \beta_{5} + 10 \beta_{4} + 30 \beta_{3} + \cdots + 360 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 34 \beta_{9} - 34 \beta_{8} - 92 \beta_{7} - 168 \beta_{6} - 52 \beta_{5} + 92 \beta_{4} + \cdots + 7500 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1079 \beta_{9} + 1055 \beta_{8} - 116 \beta_{7} + 256 \beta_{6} + 252 \beta_{5} + 588 \beta_{4} + \cdots + 11687 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 787 \beta_{9} - 939 \beta_{8} - 3334 \beta_{7} - 5140 \beta_{6} - 982 \beta_{5} + 3406 \beta_{4} + \cdots + 195751 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 31162 \beta_{9} + 29074 \beta_{8} - 5238 \beta_{7} + 11412 \beta_{6} + 8386 \beta_{5} + 24198 \beta_{4} + \cdots + 362534 \) 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
5.55245
4.90944
4.25398
2.06883
0.605497
−1.02999
−2.86867
−3.16532
−4.15022
−5.17600
−5.55245 −0.854197 22.8297 0 4.74288 −22.1474 −82.3411 −26.2703 0
1.2 −4.90944 8.54295 16.1026 0 −41.9411 29.3547 −39.7790 45.9820 0
1.3 −4.25398 −8.37256 10.0964 0 35.6167 −5.95662 −8.91795 43.0997 0
1.4 −2.06883 0.643144 −3.71993 0 −1.33056 −2.16624 24.2466 −26.5864 0
1.5 −0.605497 4.96930 −7.63337 0 −3.00890 29.7343 9.46596 −2.30608 0
1.6 1.02999 0.400571 −6.93913 0 0.412584 −33.7017 −15.3871 −26.8395 0
1.7 2.86867 9.10998 0.229276 0 26.1335 −4.02694 −22.2917 55.9918 0
1.8 3.16532 −9.73823 2.01923 0 −30.8246 −10.5373 −18.9310 67.8331 0
1.9 4.15022 −4.36306 9.22436 0 −18.1077 26.8391 5.08135 −7.96367 0
1.10 5.17600 5.66210 18.7909 0 29.3070 8.60812 55.8539 5.05942 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
\(5\) \(-1\)
\(17\) \(-1\)

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.l ✓ 10
5.b even 2 1 425.4.a.m yes 10
5.c odd 4 2 425.4.b.k 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.4.a.l ✓ 10 1.a even 1 1 trivial
425.4.a.m yes 10 5.b even 2 1
425.4.b.k 20 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(425))\):

\( T_{2}^{10} + T_{2}^{9} - 70 T_{2}^{8} - 34 T_{2}^{7} + 1721 T_{2}^{6} - 15 T_{2}^{5} - 17204 T_{2}^{4} + \cdots - 29184 \) Copy content Toggle raw display
\( T_{3}^{10} - 6 T_{3}^{9} - 181 T_{3}^{8} + 1128 T_{3}^{7} + 9422 T_{3}^{6} - 61450 T_{3}^{5} + \cdots + 171425 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + \cdots - 29184 \) Copy content Toggle raw display
$3$ \( T^{10} - 6 T^{9} + \cdots + 171425 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 82413303397 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots - 17834351412480 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 17\!\cdots\!79 \) Copy content Toggle raw display
$17$ \( (T - 17)^{10} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 18\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 60\!\cdots\!27 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 97\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 67\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 12\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 19\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 49\!\cdots\!09 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 20\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 41\!\cdots\!03 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 18\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 13\!\cdots\!55 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 48\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 84\!\cdots\!00 \) Copy content Toggle raw display
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