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Properties

Label 625.2.b.d
Level $625$
Weight $2$
Character orbit 625.b
Analytic conductor $4.991$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [625,2,Mod(624,625)]
 
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([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.624");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 25x^{14} + 236x^{12} + 1080x^{10} + 2606x^{8} + 3285x^{6} + 1901x^{4} + 310x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} - \beta_{10} q^{3} + ( - \beta_{12} - 1) q^{4} + (\beta_{15} - \beta_{9} + \beta_{5} - 1) q^{6} + \beta_{14} q^{7} + ( - \beta_{14} - \beta_{10} + \cdots + \beta_{4}) q^{8}+ \cdots + (\beta_{15} - \beta_{11} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} - \beta_{10} q^{3} + ( - \beta_{12} - 1) q^{4} + (\beta_{15} - \beta_{9} + \beta_{5} - 1) q^{6} + \beta_{14} q^{7} + ( - \beta_{14} - \beta_{10} + \cdots + \beta_{4}) q^{8}+ \cdots + (\beta_{12} - \beta_{11} + 4 \beta_{9} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 22 q^{4} - 8 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 22 q^{4} - 8 q^{6} - 18 q^{9} + 2 q^{11} + 16 q^{14} + 26 q^{16} + 20 q^{19} - 28 q^{21} - 10 q^{24} + 22 q^{26} - 20 q^{29} - 18 q^{31} - 14 q^{34} + 6 q^{36} - 16 q^{39} - 8 q^{41} + 36 q^{44} - 18 q^{46} + 8 q^{49} - 28 q^{51} + 40 q^{54} + 10 q^{59} + 12 q^{61} + 18 q^{64} - 36 q^{66} - 6 q^{69} - 18 q^{71} + 36 q^{74} - 20 q^{76} + 40 q^{79} + 16 q^{81} + 56 q^{84} - 48 q^{86} + 10 q^{89} + 12 q^{91} - 94 q^{94} + 142 q^{96} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 25x^{14} + 236x^{12} + 1080x^{10} + 2606x^{8} + 3285x^{6} + 1901x^{4} + 310x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 34 \nu^{15} + 883 \nu^{13} + 8770 \nu^{11} + 42745 \nu^{9} + 110333 \nu^{7} + 149651 \nu^{5} + \cdots + 17966 \nu ) / 453 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 661 \nu^{14} - 15359 \nu^{12} - 128832 \nu^{10} - 485041 \nu^{8} - 854544 \nu^{6} - 621968 \nu^{4} + \cdots + 498 ) / 453 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1486 \nu^{14} - 34613 \nu^{12} - 291537 \nu^{10} - 1105729 \nu^{8} - 1974048 \nu^{6} - 1477448 \nu^{4} + \cdots - 162 ) / 453 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3817 \nu^{15} - 88351 \nu^{13} - 736504 \nu^{11} - 2744200 \nu^{9} - 4750454 \nu^{7} + \cdots + 46867 \nu ) / 453 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6214 \nu^{14} - 144487 \nu^{12} - 1213906 \nu^{10} - 4588756 \nu^{8} - 8169938 \nu^{6} + \cdots - 1619 ) / 453 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6150 \nu^{15} + 142967 \nu^{13} + 1200853 \nu^{11} + 4539312 \nu^{9} + 8091233 \nu^{7} + \cdots + 12470 \nu ) / 453 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8777 \nu^{15} + 203886 \nu^{13} + 1710286 \nu^{11} + 6448364 \nu^{9} + 11431970 \nu^{7} + \cdots - 8773 \nu ) / 453 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2996 \nu^{15} - 69645 \nu^{13} - 584930 \nu^{11} - 2210445 \nu^{9} - 3936595 \nu^{7} + \cdots - 4541 \nu ) / 151 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -88\nu^{14} - 2045\nu^{12} - 17166\nu^{10} - 64804\nu^{8} - 115179\nu^{6} - 86177\nu^{4} - 15427\nu^{2} - 48 ) / 3 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17909 \nu^{15} - 416241 \nu^{13} - 3494785 \nu^{11} - 13198196 \nu^{9} - 23471276 \nu^{7} + \cdots - 15656 \nu ) / 453 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 27952 \nu^{14} - 649666 \nu^{12} - 5454730 \nu^{10} - 20600785 \nu^{8} - 36638918 \nu^{6} + \cdots - 19100 ) / 453 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10510 \nu^{14} + 244252 \nu^{12} + 2050470 \nu^{10} + 7741962 \nu^{8} + 13763486 \nu^{6} + 10303192 \nu^{4} + \cdots + 5690 ) / 151 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28756 \nu^{15} + 668308 \nu^{13} + 5610526 \nu^{11} + 21183307 \nu^{9} + 37650611 \nu^{7} + \cdots + 4160 \nu ) / 453 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 31374 \nu^{15} + 729202 \nu^{13} + 6122585 \nu^{11} + 23123826 \nu^{9} + 41130787 \nu^{7} + \cdots + 25975 \nu ) / 453 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 41934 \nu^{14} - 974566 \nu^{12} - 8181653 \nu^{10} - 30892959 \nu^{8} - 54923452 \nu^{6} + \cdots - 25363 ) / 453 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{14} - 2\beta_{10} - 3\beta_{8} - \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{15} - \beta_{12} - 2\beta_{11} - 6\beta_{9} + 4\beta_{5} - 4\beta_{3} + 2\beta_{2} - 20 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 18\beta_{14} - 3\beta_{13} + 18\beta_{10} + 10\beta_{8} - \beta_{7} - 10\beta_{6} - \beta_{4} + 7\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -27\beta_{15} + 5\beta_{12} + 20\beta_{11} + 68\beta_{9} - 35\beta_{5} + 39\beta_{3} - 20\beta_{2} + 142 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -147\beta_{14} + 25\beta_{13} - 162\beta_{10} - 43\beta_{8} + 105\beta_{6} + 10\beta_{4} - 66\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 281\beta_{15} - 41\beta_{12} - 202\beta_{11} - 667\beta_{9} + 294\beta_{5} - 337\beta_{3} + 162\beta_{2} - 1159 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1184 \beta_{14} - 187 \beta_{13} + 1409 \beta_{10} + 223 \beta_{8} + 116 \beta_{7} - 955 \beta_{6} + \cdots + 654 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2734 \beta_{15} + 430 \beta_{12} + 1985 \beta_{11} + 6366 \beta_{9} - 2460 \beta_{5} + 2868 \beta_{3} + \cdots + 9914 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9577 \beta_{14} + 1375 \beta_{13} - 12177 \beta_{10} - 1238 \beta_{8} - 2050 \beta_{7} + \cdots - 6376 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 25940 \beta_{15} - 4591 \beta_{12} - 19097 \beta_{11} - 60038 \beta_{9} + 20719 \beta_{5} - 24515 \beta_{3} + \cdots - 86383 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 78270 \beta_{14} - 10116 \beta_{13} + 105635 \beta_{10} + 6591 \beta_{8} + 26288 \beta_{7} + \cdots + 61061 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 243046 \beta_{15} + 47415 \beta_{12} + 181050 \beta_{11} + 561524 \beta_{9} - 176185 \beta_{5} + \cdots + 760001 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 647502 \beta_{14} + 74785 \beta_{13} - 922312 \beta_{10} - 28878 \beta_{8} - 295450 \beta_{7} + \cdots - 577121 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2259634 \beta_{15} - 474016 \beta_{12} - 1698757 \beta_{11} - 5218064 \beta_{9} + 1513019 \beta_{5} + \cdots - 6730167 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 5422786 \beta_{14} - 556355 \beta_{13} + 8105816 \beta_{10} + 45549 \beta_{8} + 3095890 \beta_{7} + \cdots + 5403208 \beta_1 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
624.1
− 1.66501i
− 1.47435i
− 1.32610i
− 3.01367i
− 2.68341i
− 0.0573749i
0.499011i
− 1.32675i
1.32675i
− 0.499011i
0.0573749i
2.68341i
3.01367i
1.32610i
1.47435i
1.66501i
− 2.66501i 0.759083i −5.10229 0 2.02296 2.04213i 8.26764i 2.42379 0
624.2 − 2.47435i 2.11675i −4.12242 0 5.23759 0.973070i 5.25163i −1.48063 0
624.3 − 2.32610i − 2.30231i −3.41075 0 −5.35542 − 3.59425i 3.28154i −2.30065 0
624.4 − 2.01367i − 3.02566i −2.05487 0 −6.09269 0.369971i 0.110485i −6.15465 0
624.5 − 1.68341i − 0.710340i −0.833870 0 −1.19579 4.59110i − 1.96307i 2.49542 0
624.6 − 1.05737i − 0.687404i 0.881958 0 −0.726843 − 1.01199i − 3.04731i 2.52748 0
624.7 − 0.500989i 3.09326i 1.74901 0 1.54969 − 0.0237879i − 1.87821i −6.56824 0
624.8 − 0.326751i 1.71538i 1.89323 0 0.560502 3.42409i − 1.27212i 0.0574791 0
624.9 0.326751i − 1.71538i 1.89323 0 0.560502 − 3.42409i 1.27212i 0.0574791 0
624.10 0.500989i − 3.09326i 1.74901 0 1.54969 0.0237879i 1.87821i −6.56824 0
624.11 1.05737i 0.687404i 0.881958 0 −0.726843 1.01199i 3.04731i 2.52748 0
624.12 1.68341i 0.710340i −0.833870 0 −1.19579 − 4.59110i 1.96307i 2.49542 0
624.13 2.01367i 3.02566i −2.05487 0 −6.09269 − 0.369971i − 0.110485i −6.15465 0
624.14 2.32610i 2.30231i −3.41075 0 −5.35542 3.59425i − 3.28154i −2.30065 0
624.15 2.47435i − 2.11675i −4.12242 0 5.23759 − 0.973070i − 5.25163i −1.48063 0
624.16 2.66501i − 0.759083i −5.10229 0 2.02296 − 2.04213i − 8.26764i 2.42379 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 624.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.b.d 16
5.b even 2 1 inner 625.2.b.d 16
5.c odd 4 1 625.2.a.e ✓ 8
5.c odd 4 1 625.2.a.g yes 8
15.e even 4 1 5625.2.a.s 8
15.e even 4 1 5625.2.a.be 8
20.e even 4 1 10000.2.a.be 8
20.e even 4 1 10000.2.a.bn 8
25.d even 5 2 625.2.e.j 32
25.d even 5 2 625.2.e.k 32
25.e even 10 2 625.2.e.j 32
25.e even 10 2 625.2.e.k 32
25.f odd 20 2 625.2.d.m 16
25.f odd 20 2 625.2.d.n 16
25.f odd 20 2 625.2.d.p 16
25.f odd 20 2 625.2.d.q 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.e ✓ 8 5.c odd 4 1
625.2.a.g yes 8 5.c odd 4 1
625.2.b.d 16 1.a even 1 1 trivial
625.2.b.d 16 5.b even 2 1 inner
625.2.d.m 16 25.f odd 20 2
625.2.d.n 16 25.f odd 20 2
625.2.d.p 16 25.f odd 20 2
625.2.d.q 16 25.f odd 20 2
625.2.e.j 32 25.d even 5 2
625.2.e.j 32 25.e even 10 2
625.2.e.k 32 25.d even 5 2
625.2.e.k 32 25.e even 10 2
5625.2.a.s 8 15.e even 4 1
5625.2.a.be 8 15.e even 4 1
10000.2.a.be 8 20.e even 4 1
10000.2.a.bn 8 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\):

\( T_{2}^{16} + 27T_{2}^{14} + 293T_{2}^{12} + 1629T_{2}^{10} + 4885T_{2}^{8} + 7584T_{2}^{6} + 5283T_{2}^{4} + 1242T_{2}^{2} + 81 \) Copy content Toggle raw display
\( T_{3}^{16} + 33T_{3}^{14} + 428T_{3}^{12} + 2781T_{3}^{10} + 9585T_{3}^{8} + 17091T_{3}^{6} + 14543T_{3}^{4} + 5718T_{3}^{2} + 841 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 27 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} + 33 T^{14} + \cdots + 841 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 52 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{8} - T^{7} - 58 T^{6} + \cdots + 2421)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 78 T^{14} + \cdots + 130321 \) Copy content Toggle raw display
$17$ \( T^{16} + 107 T^{14} + \cdots + 2595321 \) Copy content Toggle raw display
$19$ \( (T^{8} - 10 T^{7} + \cdots + 10525)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 2124195921 \) Copy content Toggle raw display
$29$ \( (T^{8} + 10 T^{7} + \cdots - 60975)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 9 T^{7} + \cdots + 24001)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 282 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$41$ \( (T^{8} + 4 T^{7} + \cdots - 487629)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 198 T^{14} + \cdots + 3798601 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 3244555521 \) Copy content Toggle raw display
$53$ \( T^{16} + 378 T^{14} + \cdots + 3606201 \) Copy content Toggle raw display
$59$ \( (T^{8} - 5 T^{7} + \cdots - 225)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 6 T^{7} + \cdots - 103529)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 3049560197401 \) Copy content Toggle raw display
$71$ \( (T^{8} + 9 T^{7} + \cdots - 16749)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 56212142281 \) Copy content Toggle raw display
$79$ \( (T^{8} - 20 T^{7} + \cdots + 249525)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 150377514648201 \) Copy content Toggle raw display
$89$ \( (T^{8} - 5 T^{7} + \cdots + 1849275)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 945602601241 \) Copy content Toggle raw display
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