Properties

Label 1694.4.a.i
Level $1694$
Weight $4$
Character orbit 1694.a
Self dual yes
Analytic conductor $99.949$
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] = [1694,4,Mod(1,1694)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1694, 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("1694.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1694.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: \(99.9492355497\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{133}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 33 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{133})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + ( - \beta - 2) q^{3} + 4 q^{4} - q^{5} + (2 \beta + 4) q^{6} - 7 q^{7} - 8 q^{8} + (5 \beta + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + ( - \beta - 2) q^{3} + 4 q^{4} - q^{5} + (2 \beta + 4) q^{6} - 7 q^{7} - 8 q^{8} + (5 \beta + 10) q^{9} + 2 q^{10} + ( - 4 \beta - 8) q^{12} + ( - 2 \beta + 10) q^{13} + 14 q^{14} + (\beta + 2) q^{15} + 16 q^{16} + (7 \beta + 37) q^{17} + ( - 10 \beta - 20) q^{18} + ( - 12 \beta + 9) q^{19} - 4 q^{20} + (7 \beta + 14) q^{21} + ( - 11 \beta - 21) q^{23} + (8 \beta + 16) q^{24} - 124 q^{25} + (4 \beta - 20) q^{26} + (2 \beta - 131) q^{27} - 28 q^{28} + ( - 9 \beta - 23) q^{29} + ( - 2 \beta - 4) q^{30} + (12 \beta + 49) q^{31} - 32 q^{32} + ( - 14 \beta - 74) q^{34} + 7 q^{35} + (20 \beta + 40) q^{36} + ( - 12 \beta - 8) q^{37} + (24 \beta - 18) q^{38} + ( - 4 \beta + 46) q^{39} + 8 q^{40} + ( - 46 \beta + 83) q^{41} + ( - 14 \beta - 28) q^{42} + ( - 3 \beta + 244) q^{43} + ( - 5 \beta - 10) q^{45} + (22 \beta + 42) q^{46} + (13 \beta + 307) q^{47} + ( - 16 \beta - 32) q^{48} + 49 q^{49} + 248 q^{50} + ( - 58 \beta - 305) q^{51} + ( - 8 \beta + 40) q^{52} + (5 \beta - 200) q^{53} + ( - 4 \beta + 262) q^{54} + 56 q^{56} + (27 \beta + 378) q^{57} + (18 \beta + 46) q^{58} + (27 \beta + 39) q^{59} + (4 \beta + 8) q^{60} + (33 \beta - 182) q^{61} + ( - 24 \beta - 98) q^{62} + ( - 35 \beta - 70) q^{63} + 64 q^{64} + (2 \beta - 10) q^{65} + (129 \beta + 165) q^{67} + (28 \beta + 148) q^{68} + (54 \beta + 405) q^{69} - 14 q^{70} + ( - 19 \beta - 766) q^{71} + ( - 40 \beta - 80) q^{72} + ( - 110 \beta - 225) q^{73} + (24 \beta + 16) q^{74} + (124 \beta + 248) q^{75} + ( - 48 \beta + 36) q^{76} + (8 \beta - 92) q^{78} + (149 \beta + 346) q^{79} - 16 q^{80} + ( - 10 \beta - 74) q^{81} + (92 \beta - 166) q^{82} + (68 \beta - 521) q^{83} + (28 \beta + 56) q^{84} + ( - 7 \beta - 37) q^{85} + (6 \beta - 488) q^{86} + (50 \beta + 343) q^{87} + (113 \beta - 434) q^{89} + (10 \beta + 20) q^{90} + (14 \beta - 70) q^{91} + ( - 44 \beta - 84) q^{92} + ( - 85 \beta - 494) q^{93} + ( - 26 \beta - 614) q^{94} + (12 \beta - 9) q^{95} + (32 \beta + 64) q^{96} + ( - 12 \beta - 131) q^{97} - 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 5 q^{3} + 8 q^{4} - 2 q^{5} + 10 q^{6} - 14 q^{7} - 16 q^{8} + 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} - 5 q^{3} + 8 q^{4} - 2 q^{5} + 10 q^{6} - 14 q^{7} - 16 q^{8} + 25 q^{9} + 4 q^{10} - 20 q^{12} + 18 q^{13} + 28 q^{14} + 5 q^{15} + 32 q^{16} + 81 q^{17} - 50 q^{18} + 6 q^{19} - 8 q^{20} + 35 q^{21} - 53 q^{23} + 40 q^{24} - 248 q^{25} - 36 q^{26} - 260 q^{27} - 56 q^{28} - 55 q^{29} - 10 q^{30} + 110 q^{31} - 64 q^{32} - 162 q^{34} + 14 q^{35} + 100 q^{36} - 28 q^{37} - 12 q^{38} + 88 q^{39} + 16 q^{40} + 120 q^{41} - 70 q^{42} + 485 q^{43} - 25 q^{45} + 106 q^{46} + 627 q^{47} - 80 q^{48} + 98 q^{49} + 496 q^{50} - 668 q^{51} + 72 q^{52} - 395 q^{53} + 520 q^{54} + 112 q^{56} + 783 q^{57} + 110 q^{58} + 105 q^{59} + 20 q^{60} - 331 q^{61} - 220 q^{62} - 175 q^{63} + 128 q^{64} - 18 q^{65} + 459 q^{67} + 324 q^{68} + 864 q^{69} - 28 q^{70} - 1551 q^{71} - 200 q^{72} - 560 q^{73} + 56 q^{74} + 620 q^{75} + 24 q^{76} - 176 q^{78} + 841 q^{79} - 32 q^{80} - 158 q^{81} - 240 q^{82} - 974 q^{83} + 140 q^{84} - 81 q^{85} - 970 q^{86} + 736 q^{87} - 755 q^{89} + 50 q^{90} - 126 q^{91} - 212 q^{92} - 1073 q^{93} - 1254 q^{94} - 6 q^{95} + 160 q^{96} - 274 q^{97} - 196 q^{98}+O(q^{100}) \) 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
6.26628
−5.26628
−2.00000 −8.26628 4.00000 −1.00000 16.5326 −7.00000 −8.00000 41.3314 2.00000
1.2 −2.00000 3.26628 4.00000 −1.00000 −6.53256 −7.00000 −8.00000 −16.3314 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)

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 1694.4.a.i 2
11.b odd 2 1 1694.4.a.m yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1694.4.a.i 2 1.a even 1 1 trivial
1694.4.a.m yes 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1694))\):

\( T_{3}^{2} + 5T_{3} - 27 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{13}^{2} - 18T_{13} - 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 5T - 27 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 18T - 52 \) Copy content Toggle raw display
$17$ \( T^{2} - 81T + 11 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T - 4779 \) Copy content Toggle raw display
$23$ \( T^{2} + 53T - 3321 \) Copy content Toggle raw display
$29$ \( T^{2} + 55T - 1937 \) Copy content Toggle raw display
$31$ \( T^{2} - 110T - 1763 \) Copy content Toggle raw display
$37$ \( T^{2} + 28T - 4592 \) Copy content Toggle raw display
$41$ \( T^{2} - 120T - 66757 \) Copy content Toggle raw display
$43$ \( T^{2} - 485T + 58507 \) Copy content Toggle raw display
$47$ \( T^{2} - 627T + 92663 \) Copy content Toggle raw display
$53$ \( T^{2} + 395T + 38175 \) Copy content Toggle raw display
$59$ \( T^{2} - 105T - 21483 \) Copy content Toggle raw display
$61$ \( T^{2} + 331T - 8819 \) Copy content Toggle raw display
$67$ \( T^{2} - 459T - 500643 \) Copy content Toggle raw display
$71$ \( T^{2} + 1551 T + 589397 \) Copy content Toggle raw display
$73$ \( T^{2} + 560T - 323925 \) Copy content Toggle raw display
$79$ \( T^{2} - 841T - 561363 \) Copy content Toggle raw display
$83$ \( T^{2} + 974T + 83421 \) Copy content Toggle raw display
$89$ \( T^{2} + 755T - 282063 \) Copy content Toggle raw display
$97$ \( T^{2} + 274T + 13981 \) Copy content Toggle raw display
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