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Properties

Label 574.2.h.j
Level $574$
Weight $2$
Character orbit 574.h
Analytic conductor $4.583$
Analytic rank $0$
Dimension $12$
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] = [574,2,Mod(57,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.h (of order \(5\), degree \(4\), 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: \(4.58341307602\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 53x^{10} + 1039x^{8} + 9262x^{6} + 35771x^{4} + 40565x^{2} + 605 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + \beta_{6} q^{4} + ( - \beta_{10} - \beta_{6} + \beta_{2} - 1) q^{5} + ( - \beta_{6} + \beta_1 - 1) q^{6} - \beta_{2} q^{7} + (\beta_{6} - \beta_{2} - \beta_1 + 1) q^{8} + ( - \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - \beta_1 + 1) q^{3} + \beta_{6} q^{4} + ( - \beta_{10} - \beta_{6} + \beta_{2} - 1) q^{5} + ( - \beta_{6} + \beta_1 - 1) q^{6} - \beta_{2} q^{7} + (\beta_{6} - \beta_{2} - \beta_1 + 1) q^{8} + ( - \beta_{2} - \beta_1 - 1) q^{9} + (\beta_{11} - \beta_{6} + \beta_{2}) q^{10} + ( - \beta_{11} - \beta_{9} + \beta_{8} + \cdots + 1) q^{11}+ \cdots + (\beta_{11} + \beta_{9} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} + 6 q^{3} - 3 q^{4} - 4 q^{5} - 6 q^{6} - 3 q^{7} + 3 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{2} + 6 q^{3} - 3 q^{4} - 4 q^{5} - 6 q^{6} - 3 q^{7} + 3 q^{8} - 18 q^{9} + 4 q^{10} + 11 q^{11} - 9 q^{12} - q^{13} - 12 q^{14} - 2 q^{15} - 3 q^{16} - 5 q^{17} - 12 q^{18} + 2 q^{19} + 6 q^{20} + 6 q^{21} + 4 q^{22} - 17 q^{23} + 9 q^{24} - 11 q^{25} + q^{26} - 12 q^{27} - 3 q^{28} - 9 q^{29} + 2 q^{30} - 11 q^{31} - 12 q^{32} + 18 q^{33} + 5 q^{34} + 6 q^{35} - 3 q^{36} + 9 q^{37} - 12 q^{38} + 2 q^{39} + 4 q^{40} - 10 q^{41} - 6 q^{42} + 34 q^{43} - 14 q^{44} + 6 q^{45} + 2 q^{46} - 5 q^{47} + 6 q^{48} - 3 q^{49} - 64 q^{50} - 15 q^{51} - q^{52} + 31 q^{53} + 27 q^{54} + 64 q^{55} + 3 q^{56} + 6 q^{57} + 9 q^{58} - 17 q^{59} - 2 q^{60} + 5 q^{61} - 9 q^{62} + 12 q^{63} - 3 q^{64} - 31 q^{65} + 7 q^{66} + 16 q^{67} + 20 q^{68} + 19 q^{69} + 4 q^{70} - 13 q^{71} + 3 q^{72} - 9 q^{74} + 32 q^{75} + 2 q^{76} + 11 q^{77} + 3 q^{78} - 28 q^{79} + 6 q^{80} - 12 q^{81} + 10 q^{82} + 26 q^{83} + 6 q^{84} - 12 q^{85} + 26 q^{86} - 12 q^{87} + 4 q^{88} - 23 q^{89} - 6 q^{90} + 4 q^{91} - 2 q^{92} - 13 q^{93} + 9 q^{95} - 6 q^{96} - 21 q^{97} + 3 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 53x^{10} + 1039x^{8} + 9262x^{6} + 35771x^{4} + 40565x^{2} + 605 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 79 \nu^{11} + 2343 \nu^{10} - 4492 \nu^{9} + 99264 \nu^{8} - 299751 \nu^{7} + 1366101 \nu^{6} + \cdots - 1412037 ) / 4817472 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 79 \nu^{11} + 2343 \nu^{10} + 4492 \nu^{9} + 99264 \nu^{8} + 299751 \nu^{7} + 1366101 \nu^{6} + \cdots - 1412037 ) / 4817472 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 97 \nu^{11} + 3168 \nu^{10} - 9962 \nu^{9} + 141284 \nu^{8} - 531299 \nu^{7} + 2102980 \nu^{6} + \cdots + 11569228 ) / 4817472 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 72\nu^{10} + 3211\nu^{8} + 47795\nu^{6} + 267756\nu^{4} + 433205\nu^{2} + 262937 ) / 54744 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 764 \nu^{11} + 2343 \nu^{10} + 41042 \nu^{9} + 99264 \nu^{8} + 805082 \nu^{7} + 1366101 \nu^{6} + \cdots - 3820773 ) / 4817472 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 764 \nu^{11} + 2343 \nu^{10} + 41042 \nu^{9} + 99264 \nu^{8} + 805082 \nu^{7} + 1366101 \nu^{6} + \cdots - 3820773 ) / 4817472 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 213 \nu^{11} - 50 \nu^{10} + 9024 \nu^{9} - 1026 \nu^{8} + 124191 \nu^{7} + 8564 \nu^{6} + \cdots + 42020 ) / 437952 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 213 \nu^{11} - 50 \nu^{10} - 9024 \nu^{9} - 1026 \nu^{8} - 124191 \nu^{7} + 8564 \nu^{6} + \cdots + 42020 ) / 437952 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 213 \nu^{11} - 789 \nu^{10} - 9024 \nu^{9} - 34712 \nu^{8} - 124191 \nu^{7} - 506551 \nu^{6} + \cdots - 4345 ) / 437952 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11269 \nu^{11} - 13057 \nu^{10} + 542642 \nu^{9} - 568018 \nu^{8} + 9378405 \nu^{7} + \cdots - 37798783 ) / 9634944 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11269 \nu^{11} - 13057 \nu^{10} - 542642 \nu^{9} - 568018 \nu^{8} - 9378405 \nu^{7} + \cdots - 37798783 ) / 9634944 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{4} + \beta_{2} + \beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} - 2\beta_{10} - 4\beta_{8} + 4\beta_{7} + 16\beta_{6} - 10\beta_{5} + 2\beta_{2} - 8\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{11} + 2 \beta_{10} - 42 \beta_{9} + 14 \beta_{8} - 28 \beta_{7} + 21 \beta_{6} - 21 \beta_{5} + \cdots + 130 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 54 \beta_{11} + 54 \beta_{10} + 101 \beta_{8} - 101 \beta_{7} - 305 \beta_{6} + 103 \beta_{5} + \cdots - 101 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 96 \beta_{11} - 96 \beta_{10} + 848 \beta_{9} - 208 \beta_{8} + 640 \beta_{7} - 424 \beta_{6} + \cdots - 2119 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1184 \beta_{11} - 1184 \beta_{10} - 2172 \beta_{8} + 2172 \beta_{7} + 5751 \beta_{6} - 1071 \beta_{5} + \cdots + 2340 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2664 \beta_{11} + 2664 \beta_{10} - 17078 \beta_{9} + 3559 \beta_{8} - 13519 \beta_{7} + 8539 \beta_{6} + \cdots + 37443 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 24374 \beta_{11} + 24374 \beta_{10} + 44612 \beta_{8} - 44612 \beta_{7} - 108072 \beta_{6} + \cdots - 48525 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 62518 \beta_{11} - 62518 \beta_{10} + 342870 \beta_{9} - 66694 \beta_{8} + 276176 \beta_{7} + \cdots - 696170 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 489834 \beta_{11} - 489834 \beta_{10} - 900051 \beta_{8} + 900051 \beta_{7} + 2046755 \beta_{6} + \cdots + 969055 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/574\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(493\)
\(\chi(n)\) \(\beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
− 3.15836i
4.45688i
− 0.122943i
3.15836i
− 4.45688i
0.122943i
1.35269i
− 2.97838i
3.52781i
− 1.35269i
2.97838i
− 3.52781i
−0.309017 + 0.951057i 1.61803 −0.809017 − 0.587785i −3.07461 − 2.23383i −0.500000 + 1.53884i 0.309017 + 0.951057i 0.809017 − 0.587785i −0.381966 3.07461 − 2.23383i
57.2 −0.309017 + 0.951057i 1.61803 −0.809017 − 0.587785i −1.21526 − 0.882939i −0.500000 + 1.53884i 0.309017 + 0.951057i 0.809017 − 0.587785i −0.381966 1.21526 − 0.882939i
57.3 −0.309017 + 0.951057i 1.61803 −0.809017 − 0.587785i 3.28987 + 2.39023i −0.500000 + 1.53884i 0.309017 + 0.951057i 0.809017 − 0.587785i −0.381966 −3.28987 + 2.39023i
141.1 −0.309017 − 0.951057i 1.61803 −0.809017 + 0.587785i −3.07461 + 2.23383i −0.500000 − 1.53884i 0.309017 − 0.951057i 0.809017 + 0.587785i −0.381966 3.07461 + 2.23383i
141.2 −0.309017 − 0.951057i 1.61803 −0.809017 + 0.587785i −1.21526 + 0.882939i −0.500000 − 1.53884i 0.309017 − 0.951057i 0.809017 + 0.587785i −0.381966 1.21526 + 0.882939i
141.3 −0.309017 − 0.951057i 1.61803 −0.809017 + 0.587785i 3.28987 − 2.39023i −0.500000 − 1.53884i 0.309017 − 0.951057i 0.809017 + 0.587785i −0.381966 −3.28987 − 2.39023i
365.1 0.809017 − 0.587785i −0.618034 0.309017 − 0.951057i −1.11217 + 3.42290i −0.500000 + 0.363271i −0.809017 − 0.587785i −0.309017 − 0.951057i −2.61803 1.11217 + 3.42290i
365.2 0.809017 − 0.587785i −0.618034 0.309017 − 0.951057i −0.810952 + 2.49586i −0.500000 + 0.363271i −0.809017 − 0.587785i −0.309017 − 0.951057i −2.61803 0.810952 + 2.49586i
365.3 0.809017 − 0.587785i −0.618034 0.309017 − 0.951057i 0.923120 − 2.84107i −0.500000 + 0.363271i −0.809017 − 0.587785i −0.309017 − 0.951057i −2.61803 −0.923120 − 2.84107i
379.1 0.809017 + 0.587785i −0.618034 0.309017 + 0.951057i −1.11217 − 3.42290i −0.500000 − 0.363271i −0.809017 + 0.587785i −0.309017 + 0.951057i −2.61803 1.11217 − 3.42290i
379.2 0.809017 + 0.587785i −0.618034 0.309017 + 0.951057i −0.810952 − 2.49586i −0.500000 − 0.363271i −0.809017 + 0.587785i −0.309017 + 0.951057i −2.61803 0.810952 − 2.49586i
379.3 0.809017 + 0.587785i −0.618034 0.309017 + 0.951057i 0.923120 + 2.84107i −0.500000 − 0.363271i −0.809017 + 0.587785i −0.309017 + 0.951057i −2.61803 −0.923120 + 2.84107i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.h.j ✓ 12
41.d even 5 1 inner 574.2.h.j ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.h.j ✓ 12 1.a even 1 1 trivial
574.2.h.j ✓ 12 41.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\):

\( T_{3}^{2} - T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{12} + 4 T_{5}^{11} + 21 T_{5}^{10} + 44 T_{5}^{9} + 276 T_{5}^{8} + 1020 T_{5}^{7} + \cdots + 429025 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} - T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 4 T^{11} + \cdots + 429025 \) Copy content Toggle raw display
$7$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} - 11 T^{11} + \cdots + 35940025 \) Copy content Toggle raw display
$13$ \( T^{12} + T^{11} + \cdots + 281961 \) Copy content Toggle raw display
$17$ \( T^{12} + 5 T^{11} + \cdots + 19971961 \) Copy content Toggle raw display
$19$ \( T^{12} - 2 T^{11} + \cdots + 81558961 \) Copy content Toggle raw display
$23$ \( T^{12} + 17 T^{11} + \cdots + 16810000 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 3685582681 \) Copy content Toggle raw display
$31$ \( T^{12} + 11 T^{11} + \cdots + 9801 \) Copy content Toggle raw display
$37$ \( T^{12} - 9 T^{11} + \cdots + 16128256 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 4750104241 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 885360025 \) Copy content Toggle raw display
$47$ \( T^{12} + 5 T^{11} + \cdots + 45373696 \) Copy content Toggle raw display
$53$ \( T^{12} - 31 T^{11} + \cdots + 61716736 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 11010095041 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 61488624961 \) Copy content Toggle raw display
$67$ \( T^{12} - 16 T^{11} + \cdots + 1324801 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 332880025 \) Copy content Toggle raw display
$73$ \( (T^{6} - 194 T^{4} + \cdots - 7744)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 14 T^{5} + \cdots + 2209)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 13 T^{5} + \cdots + 3231)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 23 T^{11} + \cdots + 44289025 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 864066025 \) Copy content Toggle raw display
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