Properties

Label 575.4.b.g
Level $575$
Weight $4$
Character orbit 575.b
Analytic conductor $33.926$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 36x^{6} + 244x^{4} + 153x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 23)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{3}) q^{2} + ( - \beta_{7} + \beta_{6} - \beta_1) q^{3} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{2} - 5) q^{4} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_{2} - 5) q^{6} + (4 \beta_{7} - 4 \beta_{6} - 2 \beta_{3} - 4 \beta_1) q^{7} + (\beta_{7} - 15 \beta_{6} - 5 \beta_{3} - 4 \beta_1) q^{8} + (2 \beta_{5} + \beta_{4} - 4 \beta_{2} + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{3}) q^{2} + ( - \beta_{7} + \beta_{6} - \beta_1) q^{3} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{2} - 5) q^{4} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_{2} - 5) q^{6} + (4 \beta_{7} - 4 \beta_{6} - 2 \beta_{3} - 4 \beta_1) q^{7} + (\beta_{7} - 15 \beta_{6} - 5 \beta_{3} - 4 \beta_1) q^{8} + (2 \beta_{5} + \beta_{4} - 4 \beta_{2} + 7) q^{9} + ( - 2 \beta_{5} + 4 \beta_{4} - 8 \beta_{2} - 4) q^{11} + (\beta_{7} + 16 \beta_{6} - 6 \beta_{3} - 8 \beta_1) q^{12} + (11 \beta_{7} + 31 \beta_{6} + 10 \beta_{3} + 11 \beta_1) q^{13} + (18 \beta_{5} + 24 \beta_{4} + 14 \beta_{2} + 46) q^{14} + (19 \beta_{5} + 2 \beta_{4} + 10 \beta_{2} + 30) q^{16} + ( - 6 \beta_{7} - 32 \beta_{6} + 10 \beta_{3} - 2 \beta_1) q^{17} + (5 \beta_{7} + 6 \beta_{6} + 20 \beta_{3} + 5 \beta_1) q^{18} + (24 \beta_{5} + 24 \beta_{4} - 8 \beta_{2} - 22) q^{19} + ( - 6 \beta_{5} + 8 \beta_{4} - 20 \beta_{2} + 30) q^{21} + ( - 8 \beta_{7} - 68 \beta_{6} + 12 \beta_{3} - 10 \beta_1) q^{22} - 23 \beta_{6} q^{23} + (3 \beta_{5} - 11 \beta_{2} + 43) q^{24} + ( - 61 \beta_{5} + 13 \beta_{4} + 32 \beta_{2} - 75) q^{26} + ( - 29 \beta_{7} + 25 \beta_{6} + 4 \beta_{3} - 9 \beta_1) q^{27} + ( - 18 \beta_{7} + 34 \beta_{6} + 62 \beta_{3} - 2 \beta_1) q^{28} + ( - 26 \beta_{5} + 23 \beta_{4} - 12 \beta_{2} - 30) q^{29} + ( - 12 \beta_{5} - 9 \beta_{4} - 28 \beta_{2} - 66) q^{31} + (30 \beta_{7} + 122 \beta_{6} + 10 \beta_{3} + 23 \beta_1) q^{32} + ( - 6 \beta_{7} - 82 \beta_{6} + 6 \beta_{3} + 54 \beta_1) q^{33} + ( - 2 \beta_{5} - 42 \beta_{4} - 6 \beta_{2} - 160) q^{34} + ( - 50 \beta_{5} - 17 \beta_{4} - 2 \beta_{2} - 179) q^{36} + ( - 54 \beta_{7} - 46 \beta_{6} + 10 \beta_{3} + 30 \beta_1) q^{37} + ( - 16 \beta_{7} + 174 \beta_{6} + 66 \beta_{3} + 48 \beta_1) q^{38} + ( - 32 \beta_{5} + 11 \beta_{4} + 24 \beta_{2} + 66) q^{39} + (18 \beta_{5} + 7 \beta_{4} + 28 \beta_{2} - 10) q^{41} + ( - 16 \beta_{7} - 210 \beta_{6} + 66 \beta_{3} - 26 \beta_1) q^{42} + ( - 14 \beta_{7} - 24 \beta_{6} - 28 \beta_{3} - 70 \beta_1) q^{43} + (18 \beta_{5} - 14 \beta_{4} - 66 \beta_{2} - 228) q^{44} + 23 \beta_{5} q^{46} + (13 \beta_{7} + 119 \beta_{6} + 36 \beta_{3} - 71 \beta_1) q^{47} + (25 \beta_{7} + 105 \beta_{6} + 23 \beta_{3} - 55 \beta_1) q^{48} + (32 \beta_{5} + 8 \beta_{4} + 100 \beta_{2} - 241) q^{49} + (2 \beta_{5} - 56 \beta_{4} - 24 \beta_{2} - 82) q^{51} + ( - 105 \beta_{7} - 558 \beta_{6} - 168 \beta_{3} - 108 \beta_1) q^{52} + (26 \beta_{7} - 118 \beta_{6} + 120 \beta_{3} - 14 \beta_1) q^{53} + ( - 57 \beta_{5} - 115 \beta_{4} - 74 \beta_{2} - 197) q^{54} + ( - 92 \beta_{5} - 2 \beta_{4} + 122 \beta_{2} - 528) q^{56} + (70 \beta_{7} - 270 \beta_{6} + 72 \beta_{3} + 158 \beta_1) q^{57} + ( - 109 \beta_{7} - 519 \beta_{6} - 35 \beta_{3} - 101 \beta_1) q^{58} + (28 \beta_{5} + 60 \beta_{4} + 24 \beta_{2} + 296) q^{59} + ( - 36 \beta_{5} + 26 \beta_{4} + 40 \beta_{2} + 184) q^{61} + (31 \beta_{7} - 95 \beta_{6} - 43 \beta_{3} - 27 \beta_1) q^{62} + (52 \beta_{7} - 248 \beta_{6} - 44 \beta_{3} - 20 \beta_1) q^{63} + (7 \beta_{5} + 93 \beta_{4} + 157 \beta_{2} + 260) q^{64} + (4 \beta_{5} - 90 \beta_{4} - 66 \beta_{2} - 108) q^{66} + (84 \beta_{7} - 110 \beta_{6} + 110 \beta_{3} + 92 \beta_1) q^{67} + (80 \beta_{7} + 158 \beta_{6} - 114 \beta_{3} + 20 \beta_1) q^{68} + ( - 23 \beta_{4} + 46) q^{69} + ( - 68 \beta_{5} + 111 \beta_{4} - 32 \beta_{2} - 178) q^{71} + ( - 7 \beta_{7} - 358 \beta_{6} - 132 \beta_{3} - 93 \beta_1) q^{72} + (\beta_{7} + 281 \beta_{6} + 66 \beta_{3} - 127 \beta_1) q^{73} + ( - 68 \beta_{5} - 266 \beta_{4} - 182 \beta_{2} - 400) q^{74} + ( - 244 \beta_{5} - 52 \beta_{4} - 94 \beta_{2} - 1114) q^{76} + (208 \beta_{7} - 84 \beta_{6} + 36 \beta_{3} + 132 \beta_1) q^{77} + ( - 121 \beta_{7} - 543 \beta_{6} - 35 \beta_{3} - 107 \beta_1) q^{78} + ( - 132 \beta_{5} - 50 \beta_{4} + 48 \beta_{2} + 180) q^{79} + (108 \beta_{5} + 24 \beta_{4} - 132 \beta_{2} - 251) q^{81} + ( - 13 \beta_{7} + 269 \beta_{6} - 23 \beta_{3} + 47 \beta_1) q^{82} + ( - 6 \beta_{7} + 96 \beta_{6} + 22 \beta_{3} + 146 \beta_1) q^{83} + ( - 26 \beta_{5} - 106 \beta_{4} - 116 \beta_{2} - 698) q^{84} + (164 \beta_{5} + 70 \beta_{4} + 294) q^{86} + ( - 71 \beta_{7} - 517 \beta_{6} + 20 \beta_{3} + 133 \beta_1) q^{87} + (80 \beta_{7} - 98 \beta_{6} + 22 \beta_{3} - 12 \beta_1) q^{88} + (160 \beta_{5} + 82 \beta_{4} - 526) q^{89} + (350 \beta_{5} - 56 \beta_{4} - 56 \beta_{2} + 234) q^{91} + (46 \beta_{7} + 138 \beta_{6} + 46 \beta_{3} + 69 \beta_1) q^{92} + (39 \beta_{7} - 19 \beta_{6} - 30 \beta_{3} + 167 \beta_1) q^{93} + ( - 143 \beta_{5} + 51 \beta_{4} + 146 \beta_{2} - 403) q^{94} + ( - 70 \beta_{5} + 109 \beta_{4} + 65 \beta_{2} + 170) q^{96} + (150 \beta_{7} - 388 \beta_{6} + 282 \beta_{3} - 38 \beta_1) q^{97} + ( - 60 \beta_{7} + 833 \beta_{6} - 369 \beta_{3} + 88 \beta_1) q^{98} + ( - 48 \beta_{5} - 46 \beta_{4} + 48 \beta_{2} + 386) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 40 q^{4} - 34 q^{6} + 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{4} - 34 q^{6} + 66 q^{9} + 16 q^{11} + 288 q^{14} + 128 q^{16} - 192 q^{19} + 360 q^{21} + 376 q^{24} - 458 q^{26} - 42 q^{29} - 386 q^{31} - 1332 q^{34} - 1258 q^{36} + 582 q^{39} - 250 q^{41} - 1660 q^{44} - 92 q^{46} - 2440 q^{49} - 680 q^{51} - 1282 q^{54} - 4348 q^{56} + 2280 q^{59} + 1508 q^{61} + 1610 q^{64} - 796 q^{66} + 322 q^{69} - 802 q^{71} - 2732 q^{74} - 7664 q^{76} + 1676 q^{79} - 1864 q^{81} - 5228 q^{84} + 1836 q^{86} - 4684 q^{89} + 584 q^{91} - 3134 q^{94} + 1598 q^{96} + 2996 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 36x^{6} + 244x^{4} + 153x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{6} + 77\nu^{4} + 580\nu^{2} - 53 ) / 201 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 72\nu^{5} + 1496\nu^{3} + 7377\nu ) / 804 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\nu^{6} + 467\nu^{4} + 3167\nu^{2} + 1699 ) / 201 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -22\nu^{6} - 780\nu^{4} - 4973\nu^{2} - 1494 ) / 201 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -37\nu^{7} - 1324\nu^{5} - 8720\nu^{3} - 3341\nu ) / 804 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -63\nu^{7} - 2258\nu^{5} - 15054\nu^{3} - 8347\nu ) / 402 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} - 2\beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{7} + 17\beta_{6} - \beta_{3} - 24\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -18\beta_{5} - 42\beta_{4} + 75\beta_{2} + 241 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 174\beta_{7} - 591\beta_{6} + 57\beta_{3} + 634\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 403\beta_{5} + 1037\beta_{4} - 2207\beta_{2} - 6352 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5048\beta_{7} + 17120\beta_{6} - 1804\beta_{3} - 17121\beta_1 \) Copy content Toggle raw display

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
0.743529i
0.362907i
2.83969i
5.22031i
5.22031i
2.83969i
0.362907i
0.743529i
5.07751i 1.55870i −17.7811 0 −7.91434 24.3381i 49.6639i 24.5704 0
24.2 4.24143i 4.41777i −9.98977 0 −18.7377 27.0572i 8.43948i 7.48328 0
24.3 2.86845i 3.43737i −0.228032 0 9.85995 32.7301i 22.2935i 15.1845 0
24.4 0.0323756i 6.42170i 7.99895 0 −0.207906 14.0109i 0.517976i −14.2382 0
24.5 0.0323756i 6.42170i 7.99895 0 −0.207906 14.0109i 0.517976i −14.2382 0
24.6 2.86845i 3.43737i −0.228032 0 9.85995 32.7301i 22.2935i 15.1845 0
24.7 4.24143i 4.41777i −9.98977 0 −18.7377 27.0572i 8.43948i 7.48328 0
24.8 5.07751i 1.55870i −17.7811 0 −7.91434 24.3381i 49.6639i 24.5704 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.8
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.g 8
5.b even 2 1 inner 575.4.b.g 8
5.c odd 4 1 23.4.a.b 4
5.c odd 4 1 575.4.a.i 4
15.e even 4 1 207.4.a.e 4
20.e even 4 1 368.4.a.l 4
35.f even 4 1 1127.4.a.c 4
40.i odd 4 1 1472.4.a.y 4
40.k even 4 1 1472.4.a.bf 4
115.e even 4 1 529.4.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.4.a.b 4 5.c odd 4 1
207.4.a.e 4 15.e even 4 1
368.4.a.l 4 20.e even 4 1
529.4.a.g 4 115.e even 4 1
575.4.a.i 4 5.c odd 4 1
575.4.b.g 8 1.a even 1 1 trivial
575.4.b.g 8 5.b even 2 1 inner
1127.4.a.c 4 35.f even 4 1
1472.4.a.y 4 40.i odd 4 1
1472.4.a.bf 4 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(575, [\chi])\):

\( T_{2}^{8} + 52T_{2}^{6} + 824T_{2}^{4} + 3817T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{3}^{8} + 75T_{3}^{6} + 1699T_{3}^{4} + 13209T_{3}^{2} + 23104 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 52 T^{6} + 824 T^{4} + 3817 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{8} + 75 T^{6} + 1699 T^{4} + \cdots + 23104 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 2592 T^{6} + \cdots + 91194336256 \) Copy content Toggle raw display
$11$ \( (T^{4} - 8 T^{3} - 2488 T^{2} + \cdots - 81440)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 11263 T^{6} + \cdots + 1749424184964 \) Copy content Toggle raw display
$17$ \( T^{8} + 11620 T^{6} + \cdots + 731503878400 \) Copy content Toggle raw display
$19$ \( (T^{4} + 96 T^{3} - 21208 T^{2} + \cdots + 66996944)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 529)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 21 T^{3} - 54527 T^{2} + \cdots + 325399050)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 193 T^{3} - 685 T^{2} + \cdots - 58104720)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 298924 T^{6} + \cdots + 57\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{4} + 125 T^{3} - 13663 T^{2} + \cdots + 29467114)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 160868 T^{6} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{8} + 405027 T^{6} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + 720124 T^{6} + \cdots + 58\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{4} - 1140 T^{3} + \cdots + 1146071296)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 754 T^{3} + 135708 T^{2} + \cdots - 621762112)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 759280 T^{6} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{4} + 401 T^{3} - 687701 T^{2} + \cdots - 5581505296)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 1425327 T^{6} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{4} - 838 T^{3} + \cdots - 61908677856)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 787364 T^{6} + \cdots + 49\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{4} + 2342 T^{3} + \cdots - 213195182848)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 4689428 T^{6} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
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