Properties

Label 93.4.a.d
Level $93$
Weight $4$
Character orbit 93.a
Self dual yes
Analytic conductor $5.487$
Analytic rank $1$
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] = [93,4,Mod(1,93)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(93, 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("93.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 93.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: \(5.48717763053\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} - 3 q^{3} + (\beta + 2) q^{4} + (3 \beta - 7) q^{5} + 3 \beta q^{6} + ( - \beta + 15) q^{7} + (5 \beta - 10) q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - 3 q^{3} + (\beta + 2) q^{4} + (3 \beta - 7) q^{5} + 3 \beta q^{6} + ( - \beta + 15) q^{7} + (5 \beta - 10) q^{8} + 9 q^{9} + (4 \beta - 30) q^{10} - 68 q^{11} + ( - 3 \beta - 6) q^{12} + (16 \beta - 18) q^{13} + ( - 14 \beta + 10) q^{14} + ( - 9 \beta + 21) q^{15} + ( - 3 \beta - 66) q^{16} + ( - 22 \beta - 4) q^{17} - 9 \beta q^{18} + ( - 17 \beta + 67) q^{19} + (2 \beta + 16) q^{20} + (3 \beta - 45) q^{21} + 68 \beta q^{22} + ( - 42 \beta - 50) q^{23} + ( - 15 \beta + 30) q^{24} + ( - 33 \beta + 14) q^{25} + (2 \beta - 160) q^{26} - 27 q^{27} + (12 \beta + 20) q^{28} + (26 \beta + 24) q^{29} + ( - 12 \beta + 90) q^{30} + 31 q^{31} + (29 \beta + 110) q^{32} + 204 q^{33} + (26 \beta + 220) q^{34} + (49 \beta - 135) q^{35} + (9 \beta + 18) q^{36} + ( - 6 \beta - 80) q^{37} + ( - 50 \beta + 170) q^{38} + ( - 48 \beta + 54) q^{39} + ( - 50 \beta + 220) q^{40} + (105 \beta - 133) q^{41} + (42 \beta - 30) q^{42} + (8 \beta - 140) q^{43} + ( - 68 \beta - 136) q^{44} + (27 \beta - 63) q^{45} + (92 \beta + 420) q^{46} + (8 \beta - 344) q^{47} + (9 \beta + 198) q^{48} + ( - 29 \beta - 108) q^{49} + (19 \beta + 330) q^{50} + (66 \beta + 12) q^{51} + (30 \beta + 124) q^{52} + ( - 134 \beta - 128) q^{53} + 27 \beta q^{54} + ( - 204 \beta + 476) q^{55} + (80 \beta - 200) q^{56} + (51 \beta - 201) q^{57} + ( - 50 \beta - 260) q^{58} + (223 \beta - 293) q^{59} + ( - 6 \beta - 48) q^{60} + ( - 26 \beta - 332) q^{61} - 31 \beta q^{62} + ( - 9 \beta + 135) q^{63} + ( - 115 \beta + 238) q^{64} + ( - 118 \beta + 606) q^{65} - 204 \beta q^{66} + ( - 168 \beta + 212) q^{67} + ( - 70 \beta - 228) q^{68} + (126 \beta + 150) q^{69} + (86 \beta - 490) q^{70} + ( - 161 \beta + 383) q^{71} + (45 \beta - 90) q^{72} + (58 \beta + 260) q^{73} + (86 \beta + 60) q^{74} + (99 \beta - 42) q^{75} + (16 \beta - 36) q^{76} + (68 \beta - 1020) q^{77} + ( - 6 \beta + 480) q^{78} + (30 \beta - 530) q^{79} + ( - 186 \beta + 372) q^{80} + 81 q^{81} + (28 \beta - 1050) q^{82} + (62 \beta - 1054) q^{83} + ( - 36 \beta - 60) q^{84} + (76 \beta - 632) q^{85} + (132 \beta - 80) q^{86} + ( - 78 \beta - 72) q^{87} + ( - 340 \beta + 680) q^{88} + (192 \beta + 298) q^{89} + (36 \beta - 270) q^{90} + (242 \beta - 430) q^{91} + ( - 176 \beta - 520) q^{92} - 93 q^{93} + (336 \beta - 80) q^{94} + (269 \beta - 979) q^{95} + ( - 87 \beta - 330) q^{96} + ( - 179 \beta + 903) q^{97} + (137 \beta + 290) q^{98} - 612 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 6 q^{3} + 5 q^{4} - 11 q^{5} + 3 q^{6} + 29 q^{7} - 15 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 6 q^{3} + 5 q^{4} - 11 q^{5} + 3 q^{6} + 29 q^{7} - 15 q^{8} + 18 q^{9} - 56 q^{10} - 136 q^{11} - 15 q^{12} - 20 q^{13} + 6 q^{14} + 33 q^{15} - 135 q^{16} - 30 q^{17} - 9 q^{18} + 117 q^{19} + 34 q^{20} - 87 q^{21} + 68 q^{22} - 142 q^{23} + 45 q^{24} - 5 q^{25} - 318 q^{26} - 54 q^{27} + 52 q^{28} + 74 q^{29} + 168 q^{30} + 62 q^{31} + 249 q^{32} + 408 q^{33} + 466 q^{34} - 221 q^{35} + 45 q^{36} - 166 q^{37} + 290 q^{38} + 60 q^{39} + 390 q^{40} - 161 q^{41} - 18 q^{42} - 272 q^{43} - 340 q^{44} - 99 q^{45} + 932 q^{46} - 680 q^{47} + 405 q^{48} - 245 q^{49} + 679 q^{50} + 90 q^{51} + 278 q^{52} - 390 q^{53} + 27 q^{54} + 748 q^{55} - 320 q^{56} - 351 q^{57} - 570 q^{58} - 363 q^{59} - 102 q^{60} - 690 q^{61} - 31 q^{62} + 261 q^{63} + 361 q^{64} + 1094 q^{65} - 204 q^{66} + 256 q^{67} - 526 q^{68} + 426 q^{69} - 894 q^{70} + 605 q^{71} - 135 q^{72} + 578 q^{73} + 206 q^{74} + 15 q^{75} - 56 q^{76} - 1972 q^{77} + 954 q^{78} - 1030 q^{79} + 558 q^{80} + 162 q^{81} - 2072 q^{82} - 2046 q^{83} - 156 q^{84} - 1188 q^{85} - 28 q^{86} - 222 q^{87} + 1020 q^{88} + 788 q^{89} - 504 q^{90} - 618 q^{91} - 1216 q^{92} - 186 q^{93} + 176 q^{94} - 1689 q^{95} - 747 q^{96} + 1627 q^{97} + 717 q^{98} - 1224 q^{99}+O(q^{100}) \) 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
3.70156
−2.70156
−3.70156 −3.00000 5.70156 4.10469 11.1047 11.2984 8.50781 9.00000 −15.1938
1.2 2.70156 −3.00000 −0.701562 −15.1047 −8.10469 17.7016 −23.5078 9.00000 −40.8062
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(31\) \(-1\)

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 93.4.a.d 2
3.b odd 2 1 279.4.a.b 2
4.b odd 2 1 1488.4.a.l 2
5.b even 2 1 2325.4.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.4.a.d 2 1.a even 1 1 trivial
279.4.a.b 2 3.b odd 2 1
1488.4.a.l 2 4.b odd 2 1
2325.4.a.k 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 10 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(93))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 10 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 11T - 62 \) Copy content Toggle raw display
$7$ \( T^{2} - 29T + 200 \) Copy content Toggle raw display
$11$ \( (T + 68)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 20T - 2524 \) Copy content Toggle raw display
$17$ \( T^{2} + 30T - 4736 \) Copy content Toggle raw display
$19$ \( T^{2} - 117T + 460 \) Copy content Toggle raw display
$23$ \( T^{2} + 142T - 13040 \) Copy content Toggle raw display
$29$ \( T^{2} - 74T - 5560 \) Copy content Toggle raw display
$31$ \( (T - 31)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 166T + 6520 \) Copy content Toggle raw display
$41$ \( T^{2} + 161T - 106526 \) Copy content Toggle raw display
$43$ \( T^{2} + 272T + 17840 \) Copy content Toggle raw display
$47$ \( T^{2} + 680T + 114944 \) Copy content Toggle raw display
$53$ \( T^{2} + 390T - 146024 \) Copy content Toggle raw display
$59$ \( T^{2} + 363T - 476780 \) Copy content Toggle raw display
$61$ \( T^{2} + 690T + 112096 \) Copy content Toggle raw display
$67$ \( T^{2} - 256T - 272912 \) Copy content Toggle raw display
$71$ \( T^{2} - 605T - 174184 \) Copy content Toggle raw display
$73$ \( T^{2} - 578T + 49040 \) Copy content Toggle raw display
$79$ \( T^{2} + 1030 T + 256000 \) Copy content Toggle raw display
$83$ \( T^{2} + 2046 T + 1007128 \) Copy content Toggle raw display
$89$ \( T^{2} - 788T - 222620 \) Copy content Toggle raw display
$97$ \( T^{2} - 1627 T + 333362 \) Copy content Toggle raw display
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