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Properties

Label 575.4.b.k
Level $575$
Weight $4$
Character orbit 575.b
Analytic conductor $33.926$
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] = [575,4,Mod(24,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.24");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.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: \(33.9260982533\)
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: \( x^{16} + 102 x^{14} + 4025 x^{12} + 79249 x^{10} + 832798 x^{8} + 4596761 x^{6} + 12272424 x^{4} + \cdots + 5760000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 115)
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.

\(f(q)\) \(=\) \( q + (\beta_{9} + \beta_1) q^{2} + \beta_{6} q^{3} + (\beta_{4} - 5) q^{4} + (\beta_{12} + \beta_{10} + \beta_{7} + \cdots + 1) q^{6}+ \cdots + (\beta_{12} + \beta_{10} + \beta_{8} + \cdots - 20) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} + \beta_1) q^{2} + \beta_{6} q^{3} + (\beta_{4} - 5) q^{4} + (\beta_{12} + \beta_{10} + \beta_{7} + \cdots + 1) q^{6}+ \cdots + (40 \beta_{12} - 27 \beta_{10} + \cdots - 184) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 84 q^{4} - 316 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 84 q^{4} - 316 q^{9} + 82 q^{11} - 322 q^{14} + 196 q^{16} - 354 q^{19} + 584 q^{21} - 354 q^{24} + 792 q^{26} - 450 q^{29} - 72 q^{31} + 2102 q^{34} - 792 q^{36} + 2154 q^{39} + 1240 q^{41} + 282 q^{44} + 276 q^{46} - 1762 q^{49} + 1914 q^{51} + 5898 q^{54} + 2062 q^{56} + 1352 q^{59} + 2894 q^{61} + 5766 q^{64} - 238 q^{66} + 2792 q^{71} + 1928 q^{74} + 4542 q^{76} - 1416 q^{79} + 8632 q^{81} + 7940 q^{84} + 7140 q^{86} - 2720 q^{89} + 5386 q^{91} + 6414 q^{94} + 9912 q^{96} - 2638 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 102 x^{14} + 4025 x^{12} + 79249 x^{10} + 832798 x^{8} + 4596761 x^{6} + 12272424 x^{4} + \cdots + 5760000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 62769 \nu^{15} + 563950738 \nu^{13} + 51758819825 \nu^{11} + 1766106717981 \nu^{9} + \cdots + 951202090787200 \nu ) / 14991564864000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 141169 \nu^{14} + 14067473 \nu^{12} + 536170050 \nu^{10} + 10010725561 \nu^{8} + \cdots + 533657491200 ) / 37478912160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 141169 \nu^{14} + 14067473 \nu^{12} + 536170050 \nu^{10} + 10010725561 \nu^{8} + \cdots + 758530964160 ) / 18739456080 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6282689 \nu^{14} + 656570778 \nu^{12} + 26409149625 \nu^{10} + 520375102061 \nu^{8} + \cdots + 37783897094400 ) / 499718828800 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 874957 \nu^{15} - 82180858 \nu^{13} - 2897440573 \nu^{11} - 49674841513 \nu^{9} + \cdots - 12525583909376 \nu ) / 599662594560 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23189833 \nu^{14} - 2385557066 \nu^{12} - 94345593825 \nu^{10} - 1826036116117 \nu^{8} + \cdots - 21907643155200 ) / 1499156486400 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7881277 \nu^{14} - 819260154 \nu^{12} - 33001934925 \nu^{10} - 660533726873 \nu^{8} + \cdots - 52618804793600 ) / 499718828800 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9264887 \nu^{15} - 930901574 \nu^{13} - 35884422875 \nu^{11} - 680616024863 \nu^{9} + \cdots - 43331904120400 \nu ) / 3747891216000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21443417 \nu^{14} - 2128995234 \nu^{12} - 80985778425 \nu^{10} - 1519072131333 \nu^{8} + \cdots - 109657444192000 ) / 499718828800 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 91316057 \nu^{15} - 9123442114 \nu^{13} - 351514664025 \nu^{11} + \cdots - 686930647315200 \nu ) / 14991564864000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 87184903 \nu^{14} + 8438114606 \nu^{12} + 308905490775 \nu^{10} + 5474867662747 \nu^{8} + \cdots + 225470085312000 ) / 1499156486400 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 315487263 \nu^{15} - 30965976926 \nu^{13} - 1158193311775 \nu^{11} + \cdots - 11\!\cdots\!00 \nu ) / 14991564864000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 24265436 \nu^{15} + 2441017897 \nu^{13} + 94249017375 \nu^{11} + 1791579935564 \nu^{9} + \cdots + 118528220689200 \nu ) / 936972804000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 530114417 \nu^{15} + 54053210834 \nu^{13} + 2127574165025 \nu^{11} + \cdots + 33\!\cdots\!00 \nu ) / 14991564864000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 2\beta_{3} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + 3\beta_{14} - \beta_{11} + 19\beta_{9} + \beta_{6} + \beta_{2} - 23\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{12} + 6\beta_{10} - 9\beta_{8} + 5\beta_{7} - 2\beta_{5} - 30\beta_{4} + 87\beta_{3} + 268 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 45\beta_{15} - 133\beta_{14} + 14\beta_{13} + 36\beta_{11} - 911\beta_{9} - 79\beta_{6} - 54\beta_{2} + 652\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -128\beta_{12} - 353\beta_{10} + 548\beta_{8} - 266\beta_{7} + 133\beta_{5} + 951\beta_{4} - 3189\beta_{3} - 7538 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1852 \beta_{15} + 5030 \beta_{14} - 860 \beta_{13} - 1258 \beta_{11} + 34352 \beta_{9} + \cdots - 20593 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6058 \beta_{12} + 15786 \beta_{10} - 25120 \beta_{8} + 11344 \beta_{7} - 6168 \beta_{5} - 32181 \beta_{4} + \cdots + 238458 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 73087 \beta_{15} - 183427 \beta_{14} + 39290 \beta_{13} + 45249 \beta_{11} - 1225695 \beta_{9} + \cdots + 690215 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 255060 \beta_{12} - 640896 \beta_{10} + 1034649 \beta_{8} - 448993 \beta_{7} + 251422 \beta_{5} + \cdots - 8026794 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2803191 \beta_{15} + 6632341 \beta_{14} - 1612388 \beta_{13} - 1655172 \beta_{11} + \cdots - 23912428 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 10121142 \beta_{12} + 24858207 \beta_{10} - 40461468 \beta_{8} + 17157366 \beta_{7} - 9690217 \beta_{5} + \cdots + 279251100 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 105525306 \beta_{15} - 239672806 \beta_{14} + 62899374 \beta_{13} + 60851394 \beta_{11} + \cdots + 844359561 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 388418204 \beta_{12} - 940539900 \beta_{10} + 1538242336 \beta_{8} - 643235900 \beta_{7} + \cdots - 9892729928 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3925097025 \beta_{15} + 8673534011 \beta_{14} - 2387668032 \beta_{13} - 2236359977 \beta_{11} + \cdots - 30154848263 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
− 6.03341i
− 3.81977i
− 3.57019i
− 3.05234i
− 4.85207i
− 1.16661i
− 1.76920i
0.954235i
− 0.954235i
1.76920i
1.16661i
4.85207i
3.05234i
3.57019i
3.81977i
6.03341i
− 5.03341i 4.21852i −17.3352 0 21.2335 − 19.9088i 46.9881i 9.20409 0
24.2 − 4.81977i 1.39521i −15.2302 0 6.72460 − 13.5587i 34.8481i 25.0534 0
24.3 − 4.57019i 4.96142i −12.8867 0 22.6746 18.4259i 22.3330i 2.38431 0
24.4 − 4.05234i − 8.94300i −8.42146 0 −36.2401 − 32.7645i 1.70790i −52.9772 0
24.5 − 3.85207i − 8.45088i −6.83843 0 −32.5534 16.9190i − 4.47444i −44.4174 0
24.6 − 2.16661i 8.59146i 3.30580 0 18.6143 − 9.86011i − 24.4953i −46.8131 0
24.7 − 0.769202i 0.0181662i 7.40833 0 0.0139735 10.0918i − 11.8521i 26.9997 0
24.8 − 0.0457645i − 10.2193i 7.99791 0 −0.467681 33.8594i − 0.732137i −77.4338 0
24.9 0.0457645i 10.2193i 7.99791 0 −0.467681 − 33.8594i 0.732137i −77.4338 0
24.10 0.769202i − 0.0181662i 7.40833 0 0.0139735 − 10.0918i 11.8521i 26.9997 0
24.11 2.16661i − 8.59146i 3.30580 0 18.6143 9.86011i 24.4953i −46.8131 0
24.12 3.85207i 8.45088i −6.83843 0 −32.5534 − 16.9190i 4.47444i −44.4174 0
24.13 4.05234i 8.94300i −8.42146 0 −36.2401 32.7645i − 1.70790i −52.9772 0
24.14 4.57019i − 4.96142i −12.8867 0 22.6746 − 18.4259i − 22.3330i 2.38431 0
24.15 4.81977i − 1.39521i −15.2302 0 6.72460 13.5587i − 34.8481i 25.0534 0
24.16 5.03341i − 4.21852i −17.3352 0 21.2335 19.9088i − 46.9881i 9.20409 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.16
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 575.4.b.k 16
5.b even 2 1 inner 575.4.b.k 16
5.c odd 4 1 115.4.a.f ✓ 8
5.c odd 4 1 575.4.a.n 8
15.e even 4 1 1035.4.a.r 8
20.e even 4 1 1840.4.a.v 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.f ✓ 8 5.c odd 4 1
575.4.a.n 8 5.c odd 4 1
575.4.b.k 16 1.a even 1 1 trivial
575.4.b.k 16 5.b even 2 1 inner
1035.4.a.r 8 15.e even 4 1
1840.4.a.v 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 \(S_{4}^{\mathrm{new}}(575, [\chi])\):

\( T_{2}^{16} + 106 T_{2}^{14} + 4553 T_{2}^{12} + 100849 T_{2}^{10} + 1205510 T_{2}^{8} + 7313941 T_{2}^{6} + \cdots + 17424 \) Copy content Toggle raw display
\( T_{3}^{16} + 374 T_{3}^{14} + 55549 T_{3}^{12} + 4149965 T_{3}^{10} + 162277030 T_{3}^{8} + \cdots + 12390400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 106 T^{14} + \cdots + 17424 \) Copy content Toggle raw display
$3$ \( T^{16} + 374 T^{14} + \cdots + 12390400 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 86\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( (T^{8} - 41 T^{7} + \cdots - 11345758080)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 54772847726720)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 529)^{8} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 18\!\cdots\!22)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 94\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 30\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 72\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 13\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
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