Properties

Label 1666.4.a.c
Level $1666$
Weight $4$
Character orbit 1666.a
Self dual yes
Analytic conductor $98.297$
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] = [1666,4,Mod(1,1666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1666, 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("1666.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1666 = 2 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1666.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: \(98.2971820696\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{93}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 23 \) 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 238)
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 \(\beta = \frac{1}{2}(1 + \sqrt{93})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + ( - \beta + 3) q^{3} + 4 q^{4} + ( - \beta - 4) q^{5} + (2 \beta - 6) q^{6} - 8 q^{8} + ( - 5 \beta + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + ( - \beta + 3) q^{3} + 4 q^{4} + ( - \beta - 4) q^{5} + (2 \beta - 6) q^{6} - 8 q^{8} + ( - 5 \beta + 5) q^{9} + (2 \beta + 8) q^{10} + ( - 8 \beta + 2) q^{11} + ( - 4 \beta + 12) q^{12} + (8 \beta + 10) q^{13} + (2 \beta + 11) q^{15} + 16 q^{16} - 17 q^{17} + (10 \beta - 10) q^{18} + (4 \beta - 8) q^{19} + ( - 4 \beta - 16) q^{20} + (16 \beta - 4) q^{22} + ( - 2 \beta + 26) q^{23} + (8 \beta - 24) q^{24} + (9 \beta - 86) q^{25} + ( - 16 \beta - 20) q^{26} + (12 \beta + 49) q^{27} + (28 \beta - 118) q^{29} + ( - 4 \beta - 22) q^{30} + (27 \beta - 108) q^{31} - 32 q^{32} + ( - 18 \beta + 190) q^{33} + 34 q^{34} + ( - 20 \beta + 20) q^{36} + (32 \beta - 176) q^{37} + ( - 8 \beta + 16) q^{38} + (6 \beta - 154) q^{39} + (8 \beta + 32) q^{40} + (7 \beta + 225) q^{41} + (69 \beta - 69) q^{43} + ( - 32 \beta + 8) q^{44} + (20 \beta + 95) q^{45} + (4 \beta - 52) q^{46} + ( - 6 \beta + 390) q^{47} + ( - 16 \beta + 48) q^{48} + ( - 18 \beta + 172) q^{50} + (17 \beta - 51) q^{51} + (32 \beta + 40) q^{52} + ( - 15 \beta - 358) q^{53} + ( - 24 \beta - 98) q^{54} + (38 \beta + 176) q^{55} + (16 \beta - 116) q^{57} + ( - 56 \beta + 236) q^{58} - 148 q^{59} + (8 \beta + 44) q^{60} + (69 \beta + 135) q^{61} + ( - 54 \beta + 216) q^{62} + 64 q^{64} + ( - 50 \beta - 224) q^{65} + (36 \beta - 380) q^{66} + ( - 49 \beta + 258) q^{67} - 68 q^{68} + ( - 30 \beta + 124) q^{69} + (16 \beta + 100) q^{71} + (40 \beta - 40) q^{72} + (21 \beta - 421) q^{73} + ( - 64 \beta + 352) q^{74} + (104 \beta - 465) q^{75} + (16 \beta - 32) q^{76} + ( - 12 \beta + 308) q^{78} + ( - 78 \beta + 54) q^{79} + ( - 16 \beta - 64) q^{80} + (110 \beta - 264) q^{81} + ( - 14 \beta - 450) q^{82} + ( - 158 \beta + 258) q^{83} + (17 \beta + 68) q^{85} + ( - 138 \beta + 138) q^{86} + (174 \beta - 998) q^{87} + (64 \beta - 16) q^{88} + ( - 38 \beta + 478) q^{89} + ( - 40 \beta - 190) q^{90} + ( - 8 \beta + 104) q^{92} + (162 \beta - 945) q^{93} + (12 \beta - 780) q^{94} + ( - 12 \beta - 60) q^{95} + (32 \beta - 96) q^{96} + ( - 109 \beta + 512) q^{97} + ( - 10 \beta + 930) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 5 q^{3} + 8 q^{4} - 9 q^{5} - 10 q^{6} - 16 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 5 q^{3} + 8 q^{4} - 9 q^{5} - 10 q^{6} - 16 q^{8} + 5 q^{9} + 18 q^{10} - 4 q^{11} + 20 q^{12} + 28 q^{13} + 24 q^{15} + 32 q^{16} - 34 q^{17} - 10 q^{18} - 12 q^{19} - 36 q^{20} + 8 q^{22} + 50 q^{23} - 40 q^{24} - 163 q^{25} - 56 q^{26} + 110 q^{27} - 208 q^{29} - 48 q^{30} - 189 q^{31} - 64 q^{32} + 362 q^{33} + 68 q^{34} + 20 q^{36} - 320 q^{37} + 24 q^{38} - 302 q^{39} + 72 q^{40} + 457 q^{41} - 69 q^{43} - 16 q^{44} + 210 q^{45} - 100 q^{46} + 774 q^{47} + 80 q^{48} + 326 q^{50} - 85 q^{51} + 112 q^{52} - 731 q^{53} - 220 q^{54} + 390 q^{55} - 216 q^{57} + 416 q^{58} - 296 q^{59} + 96 q^{60} + 339 q^{61} + 378 q^{62} + 128 q^{64} - 498 q^{65} - 724 q^{66} + 467 q^{67} - 136 q^{68} + 218 q^{69} + 216 q^{71} - 40 q^{72} - 821 q^{73} + 640 q^{74} - 826 q^{75} - 48 q^{76} + 604 q^{78} + 30 q^{79} - 144 q^{80} - 418 q^{81} - 914 q^{82} + 358 q^{83} + 153 q^{85} + 138 q^{86} - 1822 q^{87} + 32 q^{88} + 918 q^{89} - 420 q^{90} + 200 q^{92} - 1728 q^{93} - 1548 q^{94} - 132 q^{95} - 160 q^{96} + 915 q^{97} + 1850 q^{99}+O(q^{100}) \) 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
5.32183
−4.32183
−2.00000 −2.32183 4.00000 −9.32183 4.64365 0 −8.00000 −21.6091 18.6437
1.2 −2.00000 7.32183 4.00000 0.321825 −14.6437 0 −8.00000 26.6091 −0.643651
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(17\) \(1\)

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 1666.4.a.c 2
7.b odd 2 1 238.4.a.a 2
21.c even 2 1 2142.4.a.l 2
28.d even 2 1 1904.4.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
238.4.a.a 2 7.b odd 2 1
1666.4.a.c 2 1.a even 1 1 trivial
1904.4.a.b 2 28.d even 2 1
2142.4.a.l 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1666))\):

\( T_{3}^{2} - 5T_{3} - 17 \) Copy content Toggle raw display
\( T_{5}^{2} + 9T_{5} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 5T - 17 \) Copy content Toggle raw display
$5$ \( T^{2} + 9T - 3 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 1484 \) Copy content Toggle raw display
$13$ \( T^{2} - 28T - 1292 \) Copy content Toggle raw display
$17$ \( (T + 17)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 12T - 336 \) Copy content Toggle raw display
$23$ \( T^{2} - 50T + 532 \) Copy content Toggle raw display
$29$ \( T^{2} + 208T - 7412 \) Copy content Toggle raw display
$31$ \( T^{2} + 189T - 8019 \) Copy content Toggle raw display
$37$ \( T^{2} + 320T + 1792 \) Copy content Toggle raw display
$41$ \( T^{2} - 457T + 51073 \) Copy content Toggle raw display
$43$ \( T^{2} + 69T - 109503 \) Copy content Toggle raw display
$47$ \( T^{2} - 774T + 148932 \) Copy content Toggle raw display
$53$ \( T^{2} + 731T + 128359 \) Copy content Toggle raw display
$59$ \( (T + 148)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 339T - 81963 \) Copy content Toggle raw display
$67$ \( T^{2} - 467T - 1301 \) Copy content Toggle raw display
$71$ \( T^{2} - 216T + 5712 \) Copy content Toggle raw display
$73$ \( T^{2} + 821T + 158257 \) Copy content Toggle raw display
$79$ \( T^{2} - 30T - 141228 \) Copy content Toggle raw display
$83$ \( T^{2} - 358T - 548372 \) Copy content Toggle raw display
$89$ \( T^{2} - 918T + 177108 \) Copy content Toggle raw display
$97$ \( T^{2} - 915T - 66927 \) Copy content Toggle raw display
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