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Properties

Label 575.4.a.r
Level $575$
Weight $4$
Character orbit 575.a
Self dual yes
Analytic conductor $33.926$
Analytic rank $0$
Dimension $17$
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] = [575,4,Mod(1,575)]
 
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([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("575.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 575.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: \(33.9260982533\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 4 x^{16} - 96 x^{15} + 368 x^{14} + 3705 x^{13} - 13440 x^{12} - 73933 x^{11} + 248806 x^{10} + \cdots - 2150912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 115)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} + 1) q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{8} + 2 \beta_1) q^{6} + ( - \beta_{3} - \beta_1 + 4) q^{7} + (\beta_{8} + \beta_{6} - \beta_{4} + \cdots + 1) q^{8}+ \cdots + (\beta_{9} + \beta_{4} + \beta_1 + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{4} + 1) q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{8} + 2 \beta_1) q^{6} + ( - \beta_{3} - \beta_1 + 4) q^{7} + (\beta_{8} + \beta_{6} - \beta_{4} + \cdots + 1) q^{8}+ \cdots + ( - 9 \beta_{16} + 7 \beta_{14} + \cdots + 189) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 4 q^{2} + 12 q^{3} + 72 q^{4} + 12 q^{6} + 72 q^{7} + 48 q^{8} + 155 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 4 q^{2} + 12 q^{3} + 72 q^{4} + 12 q^{6} + 72 q^{7} + 48 q^{8} + 155 q^{9} - 4 q^{11} + 342 q^{12} + 208 q^{13} - 118 q^{14} + 220 q^{16} + 268 q^{17} + 180 q^{18} - 72 q^{19} - 16 q^{21} + 318 q^{22} + 391 q^{23} - 54 q^{24} - 72 q^{26} + 408 q^{27} + 928 q^{28} - 28 q^{29} - 40 q^{31} + 268 q^{32} + 1382 q^{33} - 132 q^{34} + 932 q^{36} + 586 q^{37} + 596 q^{38} + 600 q^{39} + 134 q^{41} - 104 q^{42} + 1264 q^{43} + 806 q^{44} + 92 q^{46} + 184 q^{47} + 2928 q^{48} + 737 q^{49} - 552 q^{51} + 1558 q^{52} + 1102 q^{53} - 782 q^{54} - 1150 q^{56} + 1188 q^{57} + 3882 q^{58} + 354 q^{59} + 550 q^{61} - 1114 q^{62} + 3102 q^{63} - 50 q^{64} - 708 q^{66} + 898 q^{67} + 3774 q^{68} + 276 q^{69} + 680 q^{71} + 102 q^{72} + 4980 q^{73} - 794 q^{74} + 54 q^{76} + 584 q^{77} + 4108 q^{78} - 1984 q^{79} + 2457 q^{81} + 1798 q^{82} + 2634 q^{83} + 974 q^{84} - 3074 q^{86} - 472 q^{87} + 7584 q^{88} - 598 q^{89} - 272 q^{91} + 1656 q^{92} + 2384 q^{93} + 1170 q^{94} + 1480 q^{96} + 6266 q^{97} + 3486 q^{98} + 1908 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 4 x^{16} - 96 x^{15} + 368 x^{14} + 3705 x^{13} - 13440 x^{12} - 73933 x^{11} + 248806 x^{10} + \cdots - 2150912 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!12 \nu^{16} + \cdots + 20\!\cdots\!24 ) / 64\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 68\!\cdots\!35 \nu^{16} + \cdots - 23\!\cdots\!36 ) / 64\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\!\cdots\!33 \nu^{16} + \cdots + 11\!\cdots\!88 ) / 36\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 38\!\cdots\!21 \nu^{16} + \cdots + 61\!\cdots\!64 ) / 64\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\!\cdots\!47 \nu^{16} + \cdots - 86\!\cdots\!68 ) / 25\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 44\!\cdots\!56 \nu^{16} + \cdots - 14\!\cdots\!20 ) / 64\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22\!\cdots\!41 \nu^{16} + \cdots - 53\!\cdots\!16 ) / 25\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 49\!\cdots\!41 \nu^{16} + \cdots - 30\!\cdots\!16 ) / 51\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25\!\cdots\!53 \nu^{16} + \cdots - 10\!\cdots\!28 ) / 25\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 56\!\cdots\!57 \nu^{16} + \cdots + 24\!\cdots\!84 ) / 51\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 22\!\cdots\!49 \nu^{16} + \cdots + 11\!\cdots\!00 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 74\!\cdots\!43 \nu^{16} + \cdots - 92\!\cdots\!20 ) / 51\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 26\!\cdots\!65 \nu^{16} + \cdots + 73\!\cdots\!84 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 12\!\cdots\!62 \nu^{16} + \cdots + 58\!\cdots\!72 ) / 64\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{6} - \beta_{4} + \beta_{2} + 19\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} + \beta_{12} - \beta_{11} - \beta_{8} - 2 \beta_{7} + \beta_{6} + 5 \beta_{4} - \beta_{3} + \cdots + 237 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - \beta_{12} - \beta_{9} + 32 \beta_{8} - \beta_{7} + 34 \beta_{6} + \cdots + 45 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 38 \beta_{15} + \beta_{14} + 3 \beta_{13} + 26 \beta_{12} - 44 \beta_{11} + 6 \beta_{10} + \cdots + 5371 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9 \beta_{16} + 26 \beta_{15} + 97 \beta_{14} - 94 \beta_{13} - 35 \beta_{12} - 2 \beta_{11} + \cdots + 1795 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 22 \beta_{16} - 1146 \beta_{15} + 72 \beta_{14} + 182 \beta_{13} + 512 \beta_{12} - 1450 \beta_{11} + \cdots + 129162 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 488 \beta_{16} + 364 \beta_{15} + 3446 \beta_{14} - 3150 \beta_{13} - 968 \beta_{12} - 126 \beta_{11} + \cdots + 63299 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1138 \beta_{16} - 32455 \beta_{15} + 3532 \beta_{14} + 7430 \beta_{13} + 8713 \beta_{12} + \cdots + 3193631 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 17614 \beta_{16} - 1447 \beta_{15} + 109178 \beta_{14} - 93096 \beta_{13} - 26527 \beta_{12} + \cdots + 2049655 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 37206 \beta_{16} - 900080 \beta_{15} + 144639 \beta_{14} + 255299 \beta_{13} + 122818 \beta_{12} + \cdots + 80100557 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 536693 \beta_{16} - 337774 \beta_{15} + 3272957 \beta_{14} - 2592662 \beta_{13} - 765247 \beta_{12} + \cdots + 62885833 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 947998 \beta_{16} - 24760562 \beta_{15} + 5318234 \beta_{14} + 7996180 \beta_{13} + 955856 \beta_{12} + \cdots + 2025594478 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 15000142 \beta_{16} - 16523832 \beta_{15} + 95121210 \beta_{14} - 69937272 \beta_{13} + \cdots + 1863902263 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 19147324 \beta_{16} - 678427997 \beta_{15} + 182102576 \beta_{14} + 237025708 \beta_{13} + \cdots + 51495820397 \) 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
−5.04242
−4.93784
−4.24824
−3.17951
−3.00148
−1.98882
−1.45307
−0.407036
0.231361
0.752118
2.51792
2.81381
3.13266
3.60232
4.94561
5.07225
5.19037
−5.04242 0.0620551 17.4260 0 −0.312908 30.6525 −47.5296 −26.9961 0
1.2 −4.93784 9.92307 16.3822 0 −48.9985 14.6304 −41.3902 71.4673 0
1.3 −4.24824 2.36809 10.0475 0 −10.0602 −12.8082 −8.69841 −21.3922 0
1.4 −3.17951 −4.30546 2.10929 0 13.6892 0.115569 18.7296 −8.46305 0
1.5 −3.00148 −1.85196 1.00888 0 5.55862 4.63746 20.9837 −23.5702 0
1.6 −1.98882 −9.68598 −4.04459 0 19.2637 25.5696 23.9545 66.8182 0
1.7 −1.45307 7.24162 −5.88858 0 −10.5226 −0.356556 20.1811 25.4411 0
1.8 −0.407036 −3.62432 −7.83432 0 1.47523 −11.1245 6.44514 −13.8643 0
1.9 0.231361 2.18914 −7.94647 0 0.506482 −29.8467 −3.68940 −22.2077 0
1.10 0.752118 6.31519 −7.43432 0 4.74977 25.4579 −11.6084 12.8816 0
1.11 2.51792 −6.03797 −1.66008 0 −15.2031 24.0921 −24.3233 9.45712 0
1.12 2.81381 −0.460867 −0.0824876 0 −1.29679 −16.1192 −22.7426 −26.7876 0
1.13 3.13266 −8.19196 1.81356 0 −25.6626 −13.4923 −19.3800 40.1082 0
1.14 3.60232 8.84330 4.97671 0 31.8564 13.0981 −10.8909 51.2040 0
1.15 4.94561 5.42743 16.4591 0 26.8419 28.7748 41.8353 2.45696 0
1.16 5.07225 −3.81839 17.7277 0 −19.3678 14.6908 49.3412 −12.4199 0
1.17 5.19037 7.60701 18.9399 0 39.4832 −25.9715 56.7822 30.8666 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(23\) \(-1\)

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 575.4.a.r 17
5.b even 2 1 575.4.a.q 17
5.c odd 4 2 115.4.b.a ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.b.a ✓ 34 5.c odd 4 2
575.4.a.q 17 5.b even 2 1
575.4.a.r 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(575))\):

\( T_{2}^{17} - 4 T_{2}^{16} - 96 T_{2}^{15} + 368 T_{2}^{14} + 3705 T_{2}^{13} - 13440 T_{2}^{12} + \cdots - 2150912 \) Copy content Toggle raw display
\( T_{3}^{17} - 12 T_{3}^{16} - 235 T_{3}^{15} + 3044 T_{3}^{14} + 20159 T_{3}^{13} - 292528 T_{3}^{12} + \cdots - 1298656000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} - 4 T^{16} + \cdots - 2150912 \) Copy content Toggle raw display
$3$ \( T^{17} + \cdots - 1298656000 \) Copy content Toggle raw display
$5$ \( T^{17} \) Copy content Toggle raw display
$7$ \( T^{17} + \cdots + 17\!\cdots\!88 \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots + 66\!\cdots\!40 \) Copy content Toggle raw display
$13$ \( T^{17} + \cdots + 92\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 57\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T - 23)^{17} \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 75\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots + 10\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 15\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 12\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 40\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 59\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 33\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
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