Properties

Label 575.2.p.b
Level $575$
Weight $2$
Character orbit 575.p
Analytic conductor $4.591$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,2,Mod(49,575)]
 
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([11, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.p (of order \(22\), degree \(10\), 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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{44})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{22}]$

$q$-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 primitive root of unity \(\zeta_{44}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{44}^{11} + \zeta_{44}^{7} - \zeta_{44}^{5} - \zeta_{44}) q^{2} + ( - \zeta_{44}^{19} + \zeta_{44}^{17} + \zeta_{44}^{13} - \zeta_{44}^{11}) q^{3} + (2 \zeta_{44}^{18} - 2 \zeta_{44}^{16} + \zeta_{44}^{14} - 2 \zeta_{44}^{12} + \zeta_{44}^{10} - 2 \zeta_{44}^{8} + 2 \zeta_{44}^{6} + \cdots - 1) q^{4}+ \cdots + ( - \zeta_{44}^{16} + 2 \zeta_{44}^{14} - \zeta_{44}^{12} + 2 \zeta_{44}^{10} - \zeta_{44}^{8} + 2 \zeta_{44}^{6} - \zeta_{44}^{4} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{44}^{11} + \zeta_{44}^{7} - \zeta_{44}^{5} - \zeta_{44}) q^{2} + ( - \zeta_{44}^{19} + \zeta_{44}^{17} + \zeta_{44}^{13} - \zeta_{44}^{11}) q^{3} + (2 \zeta_{44}^{18} - 2 \zeta_{44}^{16} + \zeta_{44}^{14} - 2 \zeta_{44}^{12} + \zeta_{44}^{10} - 2 \zeta_{44}^{8} + 2 \zeta_{44}^{6} + \cdots - 1) q^{4}+ \cdots + ( - 3 \zeta_{44}^{18} - \zeta_{44}^{16} + 2 \zeta_{44}^{14} - 2 \zeta_{44}^{12} + \zeta_{44}^{10} + 3 \zeta_{44}^{8} + 3 \zeta_{44}^{4} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 6 q^{4} + 12 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 6 q^{4} + 12 q^{6} + 4 q^{9} + 14 q^{11} - 18 q^{14} + 2 q^{16} - 4 q^{19} - 4 q^{21} + 76 q^{24} + 24 q^{26} - 28 q^{29} + 20 q^{31} - 58 q^{34} + 54 q^{36} - 2 q^{39} + 14 q^{41} + 68 q^{44} - 58 q^{46} + 36 q^{49} + 14 q^{51} + 12 q^{54} - 4 q^{56} + 42 q^{59} + 6 q^{61} - 48 q^{64} + 4 q^{66} - 52 q^{69} - 28 q^{71} - 20 q^{74} - 32 q^{76} + 30 q^{79} - 88 q^{81} + 34 q^{84} - 22 q^{86} - 50 q^{89} - 8 q^{91} - 34 q^{94} - 102 q^{96} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(51\) \(277\)
\(\chi(n)\) \(-\zeta_{44}^{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.281733 + 0.959493i
−0.281733 0.959493i
0.540641 + 0.841254i
−0.540641 0.841254i
0.989821 + 0.142315i
−0.989821 0.142315i
0.909632 0.415415i
−0.909632 + 0.415415i
−0.755750 0.654861i
0.755750 + 0.654861i
−0.755750 + 0.654861i
0.755750 0.654861i
0.281733 0.959493i
−0.281733 + 0.959493i
0.909632 + 0.415415i
−0.909632 0.415415i
0.989821 0.142315i
−0.989821 + 0.142315i
0.540641 0.841254i
−0.540641 + 0.841254i
−2.18119 + 0.313607i −2.28621 1.04408i 2.74024 0.804606i 0 5.31408 + 1.56036i 1.28282 + 1.99611i −1.71568 + 0.783524i 2.17208 + 2.50672i 0
49.2 2.18119 0.313607i 2.28621 + 1.04408i 2.74024 0.804606i 0 5.31408 + 1.56036i −1.28282 1.99611i 1.71568 0.783524i 2.17208 + 2.50672i 0
124.1 −0.0666238 0.226900i −0.361922 + 0.313607i 1.63546 1.05105i 0 0.0952700 + 0.0612263i −2.31611 1.05773i −0.704881 0.610783i −0.394306 + 2.74246i 0
124.2 0.0666238 + 0.226900i 0.361922 0.313607i 1.63546 1.05105i 0 0.0952700 + 0.0612263i 2.31611 + 1.05773i 0.704881 + 0.610783i −0.394306 + 2.74246i 0
174.1 −1.20493 + 1.04408i −0.127850 + 0.198939i 0.0771283 0.536439i 0 −0.0536570 0.373193i 0.256896 0.874908i −1.25679 1.95561i 1.22301 + 2.67803i 0
174.2 1.20493 1.04408i 0.127850 0.198939i 0.0771283 0.536439i 0 −0.0536570 0.373193i −0.256896 + 0.874908i 1.25679 + 1.95561i 1.22301 + 2.67803i 0
324.1 −1.35881 + 2.11435i 1.57812 0.226900i −1.79329 3.92676i 0 −1.66463 + 3.64502i 0.928595 + 0.804632i 5.76379 + 0.828708i −0.439490 + 0.129046i 0
324.2 1.35881 2.11435i −1.57812 + 0.226900i −1.79329 3.92676i 0 −1.66463 + 3.64502i −0.928595 0.804632i −5.76379 0.828708i −0.439490 + 0.129046i 0
349.1 −0.435615 + 0.198939i 0.620830 + 2.11435i −1.15954 + 1.33818i 0 −0.691070 0.797537i −3.36329 + 0.483568i 0.508735 1.73259i −1.56130 + 1.00339i 0
349.2 0.435615 0.198939i −0.620830 2.11435i −1.15954 + 1.33818i 0 −0.691070 0.797537i 3.36329 0.483568i −0.508735 + 1.73259i −1.56130 + 1.00339i 0
374.1 −0.435615 0.198939i 0.620830 2.11435i −1.15954 1.33818i 0 −0.691070 + 0.797537i −3.36329 0.483568i 0.508735 + 1.73259i −1.56130 1.00339i 0
374.2 0.435615 + 0.198939i −0.620830 + 2.11435i −1.15954 1.33818i 0 −0.691070 + 0.797537i 3.36329 + 0.483568i −0.508735 1.73259i −1.56130 1.00339i 0
399.1 −2.18119 0.313607i −2.28621 + 1.04408i 2.74024 + 0.804606i 0 5.31408 1.56036i 1.28282 1.99611i −1.71568 0.783524i 2.17208 2.50672i 0
399.2 2.18119 + 0.313607i 2.28621 1.04408i 2.74024 + 0.804606i 0 5.31408 1.56036i −1.28282 + 1.99611i 1.71568 + 0.783524i 2.17208 2.50672i 0
449.1 −1.35881 2.11435i 1.57812 + 0.226900i −1.79329 + 3.92676i 0 −1.66463 3.64502i 0.928595 0.804632i 5.76379 0.828708i −0.439490 0.129046i 0
449.2 1.35881 + 2.11435i −1.57812 0.226900i −1.79329 + 3.92676i 0 −1.66463 3.64502i −0.928595 + 0.804632i −5.76379 + 0.828708i −0.439490 0.129046i 0
499.1 −1.20493 1.04408i −0.127850 0.198939i 0.0771283 + 0.536439i 0 −0.0536570 + 0.373193i 0.256896 + 0.874908i −1.25679 + 1.95561i 1.22301 2.67803i 0
499.2 1.20493 + 1.04408i 0.127850 + 0.198939i 0.0771283 + 0.536439i 0 −0.0536570 + 0.373193i −0.256896 0.874908i 1.25679 1.95561i 1.22301 2.67803i 0
524.1 −0.0666238 + 0.226900i −0.361922 0.313607i 1.63546 + 1.05105i 0 0.0952700 0.0612263i −2.31611 + 1.05773i −0.704881 + 0.610783i −0.394306 2.74246i 0
524.2 0.0666238 0.226900i 0.361922 + 0.313607i 1.63546 + 1.05105i 0 0.0952700 0.0612263i 2.31611 1.05773i 0.704881 0.610783i −0.394306 2.74246i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.c even 11 1 inner
115.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.p.b 20
5.b even 2 1 inner 575.2.p.b 20
5.c odd 4 1 23.2.c.a 10
5.c odd 4 1 575.2.k.b 10
15.e even 4 1 207.2.i.c 10
20.e even 4 1 368.2.m.c 10
23.c even 11 1 inner 575.2.p.b 20
115.e even 4 1 529.2.c.a 10
115.j even 22 1 inner 575.2.p.b 20
115.k odd 44 1 23.2.c.a 10
115.k odd 44 1 529.2.a.i 5
115.k odd 44 2 529.2.c.b 10
115.k odd 44 2 529.2.c.d 10
115.k odd 44 2 529.2.c.g 10
115.k odd 44 2 529.2.c.i 10
115.k odd 44 1 575.2.k.b 10
115.l even 44 1 529.2.a.j 5
115.l even 44 1 529.2.c.a 10
115.l even 44 2 529.2.c.c 10
115.l even 44 2 529.2.c.e 10
115.l even 44 2 529.2.c.f 10
115.l even 44 2 529.2.c.h 10
345.u odd 44 1 4761.2.a.bn 5
345.x even 44 1 207.2.i.c 10
345.x even 44 1 4761.2.a.bo 5
460.v odd 44 1 8464.2.a.bt 5
460.w even 44 1 368.2.m.c 10
460.w even 44 1 8464.2.a.bs 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.c.a 10 5.c odd 4 1
23.2.c.a 10 115.k odd 44 1
207.2.i.c 10 15.e even 4 1
207.2.i.c 10 345.x even 44 1
368.2.m.c 10 20.e even 4 1
368.2.m.c 10 460.w even 44 1
529.2.a.i 5 115.k odd 44 1
529.2.a.j 5 115.l even 44 1
529.2.c.a 10 115.e even 4 1
529.2.c.a 10 115.l even 44 1
529.2.c.b 10 115.k odd 44 2
529.2.c.c 10 115.l even 44 2
529.2.c.d 10 115.k odd 44 2
529.2.c.e 10 115.l even 44 2
529.2.c.f 10 115.l even 44 2
529.2.c.g 10 115.k odd 44 2
529.2.c.h 10 115.l even 44 2
529.2.c.i 10 115.k odd 44 2
575.2.k.b 10 5.c odd 4 1
575.2.k.b 10 115.k odd 44 1
575.2.p.b 20 1.a even 1 1 trivial
575.2.p.b 20 5.b even 2 1 inner
575.2.p.b 20 23.c even 11 1 inner
575.2.p.b 20 115.j even 22 1 inner
4761.2.a.bn 5 345.u odd 44 1
4761.2.a.bo 5 345.x even 44 1
8464.2.a.bs 5 460.w even 44 1
8464.2.a.bt 5 460.v odd 44 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 5 T_{2}^{18} + 25 T_{2}^{16} - 290 T_{2}^{14} + 1274 T_{2}^{12} - 2542 T_{2}^{10} + 6583 T_{2}^{8} - 1312 T_{2}^{6} + 158 T_{2}^{4} + 24 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 5 T^{18} + 25 T^{16} - 290 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{20} - 5 T^{18} + 3 T^{16} + 150 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 25 T^{18} + 229 T^{16} + \cdots + 279841 \) Copy content Toggle raw display
$11$ \( (T^{10} - 7 T^{9} + 16 T^{8} + 31 T^{7} + \cdots + 529)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} - 9 T^{18} - 95 T^{16} + 1207 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{20} - 12 T^{18} + 1288 T^{16} + \cdots + 279841 \) Copy content Toggle raw display
$19$ \( (T^{10} + 2 T^{9} - 40 T^{8} + 184 T^{7} + \cdots + 541696)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + 164 T^{18} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( (T^{10} + 14 T^{9} + 86 T^{8} + \cdots + 4932841)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 10 T^{9} + 56 T^{8} - 164 T^{7} + \cdots + 17161)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + 13 T^{18} + 1742 T^{16} + \cdots + 279841 \) Copy content Toggle raw display
$41$ \( (T^{10} - 7 T^{9} + 16 T^{8} + 9 T^{7} + \cdots + 1849)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} - 11 T^{18} + \cdots + 4097152081 \) Copy content Toggle raw display
$47$ \( (T^{10} + 91 T^{8} + 1559 T^{6} + 8367 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} - 115 T^{18} + \cdots + 26\!\cdots\!81 \) Copy content Toggle raw display
$59$ \( (T^{10} - 21 T^{9} + 265 T^{8} - 1649 T^{7} + \cdots + 4489)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - 3 T^{9} + 218 T^{8} + 281 T^{7} + \cdots + 529)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} - 155 T^{18} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( (T^{10} + 14 T^{9} + 130 T^{8} + 566 T^{7} + \cdots + 529)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} - 141 T^{18} + \cdots + 964483090561 \) Copy content Toggle raw display
$79$ \( (T^{10} - 15 T^{9} + 115 T^{8} + \cdots + 517426009)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} - 60 T^{18} + 28218 T^{16} + \cdots + 279841 \) Copy content Toggle raw display
$89$ \( (T^{10} + 25 T^{9} + 262 T^{8} + \cdots + 78310985281)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + 296 T^{18} + \cdots + 5639109851761 \) Copy content Toggle raw display
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