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Properties

Label 575.2.k.c
Level $575$
Weight $2$
Character orbit 575.k
Analytic conductor $4.591$
Analytic rank $0$
Dimension $20$
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] = [575,2,Mod(26,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.k (of order \(11\), degree \(10\), 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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + 12 x^{18} - 12 x^{17} + 56 x^{16} - 155 x^{15} + 551 x^{14} - 1189 x^{13} + \cdots + 1437601 \) 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 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{17} - \beta_{16} + \beta_{13} + \cdots + 1) q^{2}+ \cdots + ( - \beta_{16} - \beta_{14} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{17} - \beta_{16} + \beta_{13} + \cdots + 1) q^{2}+ \cdots + ( - \beta_{19} - \beta_{18} - \beta_{17} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{2} - q^{3} - 4 q^{4} - 9 q^{6} + 4 q^{7} - 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{2} - q^{3} - 4 q^{4} - 9 q^{6} + 4 q^{7} - 17 q^{9} - 8 q^{11} - 2 q^{12} - 10 q^{13} + 3 q^{14} - 36 q^{16} + 9 q^{17} - 10 q^{18} + 24 q^{19} + 25 q^{21} - 6 q^{22} - 11 q^{23} - 2 q^{26} - 28 q^{27} + 8 q^{28} + 12 q^{29} + 10 q^{31} + 50 q^{32} + 49 q^{33} - 7 q^{34} + 43 q^{36} + 18 q^{37} + 29 q^{38} - 52 q^{39} + 10 q^{41} + 5 q^{42} - 35 q^{43} + 6 q^{44} - 33 q^{46} + 2 q^{47} + 4 q^{48} + 4 q^{49} + 50 q^{51} - 9 q^{52} + 5 q^{53} + 23 q^{54} - 33 q^{57} - 35 q^{58} + 41 q^{59} - 13 q^{61} + 2 q^{62} + 46 q^{63} + 16 q^{64} + 56 q^{66} + 9 q^{67} - 26 q^{68} - 45 q^{69} - 19 q^{71} + 27 q^{73} + 8 q^{74} + 4 q^{76} + 64 q^{77} + 27 q^{78} - 83 q^{79} + 12 q^{81} - 31 q^{82} - 67 q^{83} + 28 q^{84} - 73 q^{86} - 69 q^{87} - 22 q^{88} - 5 q^{89} - 84 q^{91} - 11 q^{92} - 28 q^{93} - 37 q^{94} + 14 q^{96} - 18 q^{97} - 8 q^{98} + 62 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 12 x^{18} - 12 x^{17} + 56 x^{16} - 155 x^{15} + 551 x^{14} - 1189 x^{13} + \cdots + 1437601 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35\!\cdots\!00 \nu^{19} + \cdots + 26\!\cdots\!27 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 44\!\cdots\!48 \nu^{19} + \cdots - 32\!\cdots\!68 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17\!\cdots\!49 \nu^{19} + \cdots - 13\!\cdots\!32 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 18\!\cdots\!41 \nu^{19} + \cdots + 25\!\cdots\!69 ) / 18\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23\!\cdots\!67 \nu^{19} + \cdots + 22\!\cdots\!20 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 26\!\cdots\!03 \nu^{19} + \cdots + 30\!\cdots\!94 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 29\!\cdots\!05 \nu^{19} + \cdots - 26\!\cdots\!93 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37\!\cdots\!41 \nu^{19} + \cdots + 38\!\cdots\!45 ) / 18\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 41\!\cdots\!00 \nu^{19} + \cdots - 18\!\cdots\!32 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 46\!\cdots\!76 \nu^{19} + \cdots + 60\!\cdots\!23 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 51\!\cdots\!80 \nu^{19} + \cdots - 55\!\cdots\!43 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 69\!\cdots\!92 \nu^{19} + \cdots - 14\!\cdots\!69 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 83\!\cdots\!47 \nu^{19} + \cdots + 59\!\cdots\!60 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 12\!\cdots\!05 \nu^{19} + \cdots + 34\!\cdots\!73 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 13\!\cdots\!77 \nu^{19} + \cdots - 86\!\cdots\!27 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 14\!\cdots\!04 \nu^{19} + \cdots - 11\!\cdots\!76 ) / 20\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 14\!\cdots\!12 \nu^{19} + \cdots - 92\!\cdots\!88 ) / 18\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 24\!\cdots\!47 \nu^{19} + \cdots + 17\!\cdots\!53 ) / 18\!\cdots\!49 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} - \beta_{13} + \beta_{10} + \beta_{6} - 5\beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} - \beta_{16} + \beta_{15} - \beta_{14} + 2\beta_{13} + 5\beta_{12} - 2\beta_{8} + 9\beta_{7} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{19} - 2 \beta_{18} - 11 \beta_{17} + 12 \beta_{13} + 9 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{18} + 23 \beta_{16} - 13 \beta_{15} + 44 \beta_{14} - 34 \beta_{13} - 12 \beta_{11} + \cdots + 14 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6 \beta_{19} + 3 \beta_{18} + 35 \beta_{17} + 17 \beta_{16} - 31 \beta_{15} - 160 \beta_{13} + \cdots - 127 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 220 \beta_{19} + 366 \beta_{18} + 349 \beta_{17} - 284 \beta_{16} + 241 \beta_{15} - 366 \beta_{14} + \cdots - 46 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 360 \beta_{19} - 905 \beta_{17} - 905 \beta_{16} + 639 \beta_{15} - 70 \beta_{14} + 2943 \beta_{13} + \cdots + 1155 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1265 \beta_{19} - 3303 \beta_{18} - 2567 \beta_{17} + 3834 \beta_{16} - 1265 \beta_{15} + 1877 \beta_{14} + \cdots + 2567 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4776 \beta_{19} - 1096 \beta_{18} + 9665 \beta_{17} + 12822 \beta_{16} - 4928 \beta_{15} + 4703 \beta_{14} + \cdots - 12822 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7992 \beta_{19} + 20742 \beta_{18} + 19170 \beta_{17} - 26889 \beta_{16} + 3739 \beta_{15} + \cdots - 37937 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4047 \beta_{19} + 50143 \beta_{18} - 12252 \beta_{17} - 127125 \beta_{16} + 62754 \beta_{15} + \cdots + 89371 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 165016 \beta_{19} - 209586 \beta_{18} - 359596 \beta_{17} + 104634 \beta_{16} + 165016 \beta_{14} + \cdots + 402946 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 673816 \beta_{18} + 1539092 \beta_{16} - 673816 \beta_{15} + 733465 \beta_{14} - 2212908 \beta_{13} + \cdots - 359595 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1704109 \beta_{19} + 1704109 \beta_{18} + 3743159 \beta_{17} - 508799 \beta_{15} - 612415 \beta_{14} + \cdots - 4924236 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 786830 \beta_{19} + 7923974 \beta_{18} + 2008094 \beta_{17} - 16110986 \beta_{16} + 6234098 \beta_{15} + \cdots - 7923974 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 12590309 \beta_{19} - 6939408 \beta_{18} - 29138056 \beta_{17} - 12993602 \beta_{16} + \cdots + 51826803 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 35787356 \beta_{19} - 101905026 \beta_{18} - 76928222 \beta_{17} + 149268766 \beta_{16} + \cdots + 42995757 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 138501888 \beta_{19} + 306081383 \beta_{17} + 306081383 \beta_{16} - 189600126 \beta_{15} + \cdots - 483179004 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(51\) \(277\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−0.318034 − 2.21198i
0.460349 + 3.20180i
2.10016 − 0.616663i
−1.14067 + 0.334930i
−0.932866 − 1.07658i
1.58773 + 1.83233i
−2.36430 − 1.51944i
1.52304 + 0.978800i
−1.02569 − 2.24594i
0.610271 + 1.33631i
−1.02569 + 2.24594i
0.610271 − 1.33631i
−0.318034 + 2.21198i
0.460349 − 3.20180i
−2.36430 + 1.51944i
1.52304 − 0.978800i
−0.932866 + 1.07658i
1.58773 − 1.83233i
2.10016 + 0.616663i
−1.14067 − 0.334930i
−0.273100 − 1.89945i −0.928337 + 2.03278i −1.61435 + 0.474017i 0 4.11469 + 1.20818i 0.720681 − 0.463153i −0.253098 − 0.554206i −1.30578 − 1.50695i 0
26.2 −0.273100 − 1.89945i 1.34375 − 2.94241i −1.61435 + 0.474017i 0 −5.95594 − 1.74882i −3.10213 + 1.99362i −0.253098 − 0.554206i −4.88751 − 5.64048i 0
101.1 1.61435 − 0.474017i −1.43337 − 1.65420i 0.698939 − 0.449181i 0 −3.09809 − 1.99102i 0.775475 − 1.69805i −1.28820 + 1.48666i −0.254878 + 1.77272i 0
101.2 1.61435 − 0.474017i 0.778513 + 0.898452i 0.698939 − 0.449181i 0 1.68268 + 1.08139i −1.06223 + 2.32595i −1.28820 + 1.48666i 0.225811 − 1.57055i 0
151.1 −0.186393 − 0.215109i −1.19839 − 0.770157i 0.273100 − 1.89945i 0 0.0577033 + 0.401335i −1.34020 − 0.393517i −0.938384 + 0.603063i −0.403254 − 0.883004i 0
151.2 −0.186393 − 0.215109i 2.03964 + 1.31080i 0.273100 − 1.89945i 0 −0.0982103 − 0.683068i 4.87354 + 1.43100i −0.938384 + 0.603063i 1.19570 + 2.61822i 0
301.1 −0.698939 − 0.449181i −0.399968 − 2.78184i −0.544078 − 1.19136i 0 −0.969995 + 2.12399i 0.131014 − 0.151198i −0.391340 + 2.72183i −4.70017 + 1.38010i 0
301.2 −0.698939 − 0.449181i 0.257653 + 1.79202i −0.544078 − 1.19136i 0 0.624856 − 1.36824i 0.992315 − 1.14519i −0.391340 + 2.72183i −0.266462 + 0.0782402i 0
326.1 0.544078 + 1.19136i −2.36905 + 0.695616i 0.186393 − 0.215109i 0 −2.11768 − 2.44393i −0.531987 − 3.70005i 2.87102 + 0.843008i 2.60476 − 1.67397i 0
326.2 0.544078 + 1.19136i 1.40956 − 0.413883i 0.186393 − 0.215109i 0 1.25999 + 1.45411i 0.543517 + 3.78024i 2.87102 + 0.843008i −0.708209 + 0.455138i 0
351.1 0.544078 − 1.19136i −2.36905 − 0.695616i 0.186393 + 0.215109i 0 −2.11768 + 2.44393i −0.531987 + 3.70005i 2.87102 − 0.843008i 2.60476 + 1.67397i 0
351.2 0.544078 − 1.19136i 1.40956 + 0.413883i 0.186393 + 0.215109i 0 1.25999 − 1.45411i 0.543517 − 3.78024i 2.87102 − 0.843008i −0.708209 − 0.455138i 0
376.1 −0.273100 + 1.89945i −0.928337 − 2.03278i −1.61435 − 0.474017i 0 4.11469 − 1.20818i 0.720681 + 0.463153i −0.253098 + 0.554206i −1.30578 + 1.50695i 0
376.2 −0.273100 + 1.89945i 1.34375 + 2.94241i −1.61435 − 0.474017i 0 −5.95594 + 1.74882i −3.10213 − 1.99362i −0.253098 + 0.554206i −4.88751 + 5.64048i 0
426.1 −0.698939 + 0.449181i −0.399968 + 2.78184i −0.544078 + 1.19136i 0 −0.969995 − 2.12399i 0.131014 + 0.151198i −0.391340 − 2.72183i −4.70017 − 1.38010i 0
426.2 −0.698939 + 0.449181i 0.257653 − 1.79202i −0.544078 + 1.19136i 0 0.624856 + 1.36824i 0.992315 + 1.14519i −0.391340 − 2.72183i −0.266462 − 0.0782402i 0
476.1 −0.186393 + 0.215109i −1.19839 + 0.770157i 0.273100 + 1.89945i 0 0.0577033 − 0.401335i −1.34020 + 0.393517i −0.938384 − 0.603063i −0.403254 + 0.883004i 0
476.2 −0.186393 + 0.215109i 2.03964 − 1.31080i 0.273100 + 1.89945i 0 −0.0982103 + 0.683068i 4.87354 − 1.43100i −0.938384 − 0.603063i 1.19570 − 2.61822i 0
501.1 1.61435 + 0.474017i −1.43337 + 1.65420i 0.698939 + 0.449181i 0 −3.09809 + 1.99102i 0.775475 + 1.69805i −1.28820 − 1.48666i −0.254878 − 1.77272i 0
501.2 1.61435 + 0.474017i 0.778513 − 0.898452i 0.698939 + 0.449181i 0 1.68268 − 1.08139i −1.06223 − 2.32595i −1.28820 − 1.48666i 0.225811 + 1.57055i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.k.c 20
5.b even 2 1 115.2.g.b ✓ 20
5.c odd 4 2 575.2.p.c 40
23.c even 11 1 inner 575.2.k.c 20
115.i odd 22 1 2645.2.a.u 10
115.j even 22 1 115.2.g.b ✓ 20
115.j even 22 1 2645.2.a.t 10
115.k odd 44 2 575.2.p.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.g.b ✓ 20 5.b even 2 1
115.2.g.b ✓ 20 115.j even 22 1
575.2.k.c 20 1.a even 1 1 trivial
575.2.k.c 20 23.c even 11 1 inner
575.2.p.c 40 5.c odd 4 2
575.2.p.c 40 115.k odd 44 2
2645.2.a.t 10 115.j even 22 1
2645.2.a.u 10 115.i odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 2T_{2}^{9} + 4T_{2}^{8} - 8T_{2}^{7} + 5T_{2}^{6} + T_{2}^{5} - 2T_{2}^{4} + 4T_{2}^{3} + 14T_{2}^{2} + 5T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} + T^{19} + \cdots + 1437601 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 4 T^{19} + \cdots + 214369 \) Copy content Toggle raw display
$11$ \( T^{20} + 8 T^{19} + \cdots + 4489 \) Copy content Toggle raw display
$13$ \( T^{20} + 10 T^{19} + \cdots + 279841 \) Copy content Toggle raw display
$17$ \( T^{20} - 9 T^{19} + \cdots + 776161 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 214358881 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} - 12 T^{19} + \cdots + 734449 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 61681199449 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 91426407424 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 15954973969 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 135494512124521 \) Copy content Toggle raw display
$47$ \( (T^{10} - T^{9} + \cdots - 3222967)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 162544573994209 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 167533079110849 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 36339424311961 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 135387938401 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 37\!\cdots\!49 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 13\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 22553276442841 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 63\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 298615252849 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 15\!\cdots\!89 \) Copy content Toggle raw display
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