Properties

Label 1573.4.a.b
Level $1573$
Weight $4$
Character orbit 1573.a
Self dual yes
Analytic conductor $92.810$
Analytic rank $0$
Dimension $2$
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] = [1573,4,Mod(1,1573)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1573, 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("1573.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1573 = 11^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1573.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: \(92.8100044390\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + ( - 3 \beta + 4) q^{3} + (\beta - 4) q^{4} + (\beta - 2) q^{5} + ( - \beta + 12) q^{6} + ( - 11 \beta + 10) q^{7} + (11 \beta - 4) q^{8} + ( - 15 \beta + 25) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + ( - 3 \beta + 4) q^{3} + (\beta - 4) q^{4} + (\beta - 2) q^{5} + ( - \beta + 12) q^{6} + ( - 11 \beta + 10) q^{7} + (11 \beta - 4) q^{8} + ( - 15 \beta + 25) q^{9} + (\beta - 4) q^{10} + (13 \beta - 28) q^{12} + 13 q^{13} + (\beta + 44) q^{14} + (7 \beta - 20) q^{15} + ( - 15 \beta - 12) q^{16} + (17 \beta - 18) q^{17} + ( - 10 \beta + 60) q^{18} + (32 \beta + 26) q^{19} + ( - 5 \beta + 12) q^{20} + ( - 41 \beta + 172) q^{21} + ( - 12 \beta + 104) q^{23} + (23 \beta - 148) q^{24} + ( - 3 \beta - 117) q^{25} - 13 \beta q^{26} + ( - 9 \beta + 172) q^{27} + (43 \beta - 84) q^{28} + ( - 96 \beta + 70) q^{29} + (13 \beta - 28) q^{30} + ( - 34 \beta - 26) q^{31} + ( - 61 \beta + 92) q^{32} + (\beta - 68) q^{34} + (21 \beta - 64) q^{35} + (70 \beta - 160) q^{36} + (5 \beta + 102) q^{37} + ( - 58 \beta - 128) q^{38} + ( - 39 \beta + 52) q^{39} + ( - 15 \beta + 52) q^{40} + ( - 22 \beta + 126) q^{41} + ( - 131 \beta + 164) q^{42} + ( - 143 \beta - 72) q^{43} + (40 \beta - 110) q^{45} + ( - 92 \beta + 48) q^{46} + ( - 121 \beta + 278) q^{47} + (21 \beta + 132) q^{48} + ( - 99 \beta + 241) q^{49} + (120 \beta + 12) q^{50} + (71 \beta - 276) q^{51} + (13 \beta - 52) q^{52} + (30 \beta - 74) q^{53} + ( - 163 \beta + 36) q^{54} + (33 \beta - 524) q^{56} + ( - 46 \beta - 280) q^{57} + (26 \beta + 384) q^{58} + (124 \beta - 246) q^{59} + ( - 41 \beta + 108) q^{60} + (190 \beta + 434) q^{61} + (60 \beta + 136) q^{62} + ( - 260 \beta + 910) q^{63} + (89 \beta + 340) q^{64} + (13 \beta - 26) q^{65} + ( - 232 \beta + 150) q^{67} + ( - 69 \beta + 140) q^{68} + ( - 324 \beta + 560) q^{69} + (43 \beta - 84) q^{70} + ( - 231 \beta + 50) q^{71} + (170 \beta - 760) q^{72} + ( - 260 \beta - 98) q^{73} + ( - 107 \beta - 20) q^{74} + (348 \beta - 432) q^{75} + ( - 70 \beta + 24) q^{76} + ( - 13 \beta + 156) q^{78} + ( - 40 \beta + 524) q^{79} + (3 \beta - 36) q^{80} + ( - 120 \beta + 121) q^{81} + ( - 104 \beta + 88) q^{82} + (182 \beta - 1070) q^{83} + (295 \beta - 852) q^{84} + ( - 35 \beta + 104) q^{85} + (215 \beta + 572) q^{86} + ( - 306 \beta + 1432) q^{87} + ( - 388 \beta - 166) q^{89} + (70 \beta - 160) q^{90} + ( - 143 \beta + 130) q^{91} + (140 \beta - 464) q^{92} + (44 \beta + 304) q^{93} + ( - 157 \beta + 484) q^{94} + ( - 6 \beta + 76) q^{95} + ( - 337 \beta + 1100) q^{96} + (508 \beta - 718) q^{97} + ( - 142 \beta + 396) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 5 q^{3} - 7 q^{4} - 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 5 q^{3} - 7 q^{4} - 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9} - 7 q^{10} - 43 q^{12} + 26 q^{13} + 89 q^{14} - 33 q^{15} - 39 q^{16} - 19 q^{17} + 110 q^{18} + 84 q^{19} + 19 q^{20} + 303 q^{21} + 196 q^{23} - 273 q^{24} - 237 q^{25} - 13 q^{26} + 335 q^{27} - 125 q^{28} + 44 q^{29} - 43 q^{30} - 86 q^{31} + 123 q^{32} - 135 q^{34} - 107 q^{35} - 250 q^{36} + 209 q^{37} - 314 q^{38} + 65 q^{39} + 89 q^{40} + 230 q^{41} + 197 q^{42} - 287 q^{43} - 180 q^{45} + 4 q^{46} + 435 q^{47} + 285 q^{48} + 383 q^{49} + 144 q^{50} - 481 q^{51} - 91 q^{52} - 118 q^{53} - 91 q^{54} - 1015 q^{56} - 606 q^{57} + 794 q^{58} - 368 q^{59} + 175 q^{60} + 1058 q^{61} + 332 q^{62} + 1560 q^{63} + 769 q^{64} - 39 q^{65} + 68 q^{67} + 211 q^{68} + 796 q^{69} - 125 q^{70} - 131 q^{71} - 1350 q^{72} - 456 q^{73} - 147 q^{74} - 516 q^{75} - 22 q^{76} + 299 q^{78} + 1008 q^{79} - 69 q^{80} + 122 q^{81} + 72 q^{82} - 1958 q^{83} - 1409 q^{84} + 173 q^{85} + 1359 q^{86} + 2558 q^{87} - 720 q^{89} - 250 q^{90} + 117 q^{91} - 788 q^{92} + 652 q^{93} + 811 q^{94} + 146 q^{95} + 1863 q^{96} - 928 q^{97} + 650 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
2.56155
−1.56155
−2.56155 −3.68466 −1.43845 0.561553 9.43845 −18.1771 24.1771 −13.4233 −1.43845
1.2 1.56155 8.68466 −5.56155 −3.56155 13.5616 27.1771 −21.1771 48.4233 −5.56155
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( -1 \)
\(13\) \( -1 \)

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 1573.4.a.b 2
11.b odd 2 1 13.4.a.b 2
33.d even 2 1 117.4.a.d 2
44.c even 2 1 208.4.a.h 2
55.d odd 2 1 325.4.a.f 2
55.e even 4 2 325.4.b.e 4
77.b even 2 1 637.4.a.b 2
88.b odd 2 1 832.4.a.s 2
88.g even 2 1 832.4.a.z 2
132.d odd 2 1 1872.4.a.bb 2
143.d odd 2 1 169.4.a.g 2
143.g even 4 2 169.4.b.f 4
143.i odd 6 2 169.4.c.j 4
143.k odd 6 2 169.4.c.g 4
143.o even 12 4 169.4.e.f 8
429.e even 2 1 1521.4.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 11.b odd 2 1
117.4.a.d 2 33.d even 2 1
169.4.a.g 2 143.d odd 2 1
169.4.b.f 4 143.g even 4 2
169.4.c.g 4 143.k odd 6 2
169.4.c.j 4 143.i odd 6 2
169.4.e.f 8 143.o even 12 4
208.4.a.h 2 44.c even 2 1
325.4.a.f 2 55.d odd 2 1
325.4.b.e 4 55.e even 4 2
637.4.a.b 2 77.b even 2 1
832.4.a.s 2 88.b odd 2 1
832.4.a.z 2 88.g even 2 1
1521.4.a.r 2 429.e even 2 1
1573.4.a.b 2 1.a even 1 1 trivial
1872.4.a.bb 2 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1573))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} - 9T - 494 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 19T - 1138 \) Copy content Toggle raw display
$19$ \( T^{2} - 84T - 2588 \) Copy content Toggle raw display
$23$ \( T^{2} - 196T + 8992 \) Copy content Toggle raw display
$29$ \( T^{2} - 44T - 38684 \) Copy content Toggle raw display
$31$ \( T^{2} + 86T - 3064 \) Copy content Toggle raw display
$37$ \( T^{2} - 209T + 10814 \) Copy content Toggle raw display
$41$ \( T^{2} - 230T + 11168 \) Copy content Toggle raw display
$43$ \( T^{2} + 287T - 66316 \) Copy content Toggle raw display
$47$ \( T^{2} - 435T - 14918 \) Copy content Toggle raw display
$53$ \( T^{2} + 118T - 344 \) Copy content Toggle raw display
$59$ \( T^{2} + 368T - 31492 \) Copy content Toggle raw display
$61$ \( T^{2} - 1058 T + 126416 \) Copy content Toggle raw display
$67$ \( T^{2} - 68T - 227596 \) Copy content Toggle raw display
$71$ \( T^{2} + 131T - 222494 \) Copy content Toggle raw display
$73$ \( T^{2} + 456T - 235316 \) Copy content Toggle raw display
$79$ \( T^{2} - 1008 T + 247216 \) Copy content Toggle raw display
$83$ \( T^{2} + 1958 T + 817664 \) Copy content Toggle raw display
$89$ \( T^{2} + 720T - 510212 \) Copy content Toggle raw display
$97$ \( T^{2} + 928T - 881476 \) Copy content Toggle raw display
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