Properties

Label 950.4.a.f
Level $950$
Weight $4$
Character orbit 950.a
Self dual yes
Analytic conductor $56.052$
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] = [950,4,Mod(1,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, 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("950.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 950.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: \(56.0518145055\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{34}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 34 \) 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 190)
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 \(\beta = \sqrt{34}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + \beta q^{3} + 4 q^{4} - 2 \beta q^{6} + (2 \beta - 12) q^{7} - 8 q^{8} + 7 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + \beta q^{3} + 4 q^{4} - 2 \beta q^{6} + (2 \beta - 12) q^{7} - 8 q^{8} + 7 q^{9} + (2 \beta + 26) q^{11} + 4 \beta q^{12} + ( - 5 \beta - 22) q^{13} + ( - 4 \beta + 24) q^{14} + 16 q^{16} + ( - 2 \beta - 50) q^{17} - 14 q^{18} + 19 q^{19} + ( - 12 \beta + 68) q^{21} + ( - 4 \beta - 52) q^{22} + ( - 8 \beta - 8) q^{23} - 8 \beta q^{24} + (10 \beta + 44) q^{26} - 20 \beta q^{27} + (8 \beta - 48) q^{28} + ( - 18 \beta - 6) q^{29} + (20 \beta - 8) q^{31} - 32 q^{32} + (26 \beta + 68) q^{33} + (4 \beta + 100) q^{34} + 28 q^{36} + ( - 31 \beta - 18) q^{37} - 38 q^{38} + ( - 22 \beta - 170) q^{39} + (26 \beta + 74) q^{41} + (24 \beta - 136) q^{42} + ( - 26 \beta + 236) q^{43} + (8 \beta + 104) q^{44} + (16 \beta + 16) q^{46} + ( - 26 \beta - 188) q^{47} + 16 \beta q^{48} + ( - 48 \beta - 63) q^{49} + ( - 50 \beta - 68) q^{51} + ( - 20 \beta - 88) q^{52} + (47 \beta - 58) q^{53} + 40 \beta q^{54} + ( - 16 \beta + 96) q^{56} + 19 \beta q^{57} + (36 \beta + 12) q^{58} + ( - 82 \beta + 56) q^{59} + ( - 36 \beta - 340) q^{61} + ( - 40 \beta + 16) q^{62} + (14 \beta - 84) q^{63} + 64 q^{64} + ( - 52 \beta - 136) q^{66} + ( - 95 \beta + 104) q^{67} + ( - 8 \beta - 200) q^{68} + ( - 8 \beta - 272) q^{69} + ( - 20 \beta - 308) q^{71} - 56 q^{72} + (14 \beta + 666) q^{73} + (62 \beta + 36) q^{74} + 76 q^{76} + (28 \beta - 176) q^{77} + (44 \beta + 340) q^{78} + (94 \beta + 108) q^{79} - 869 q^{81} + ( - 52 \beta - 148) q^{82} + ( - 184 \beta + 20) q^{83} + ( - 48 \beta + 272) q^{84} + (52 \beta - 472) q^{86} + ( - 6 \beta - 612) q^{87} + ( - 16 \beta - 208) q^{88} + (26 \beta + 242) q^{89} + (16 \beta - 76) q^{91} + ( - 32 \beta - 32) q^{92} + ( - 8 \beta + 680) q^{93} + (52 \beta + 376) q^{94} - 32 \beta q^{96} + (39 \beta - 278) q^{97} + (96 \beta + 126) q^{98} + (14 \beta + 182) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 8 q^{4} - 24 q^{7} - 16 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 8 q^{4} - 24 q^{7} - 16 q^{8} + 14 q^{9} + 52 q^{11} - 44 q^{13} + 48 q^{14} + 32 q^{16} - 100 q^{17} - 28 q^{18} + 38 q^{19} + 136 q^{21} - 104 q^{22} - 16 q^{23} + 88 q^{26} - 96 q^{28} - 12 q^{29} - 16 q^{31} - 64 q^{32} + 136 q^{33} + 200 q^{34} + 56 q^{36} - 36 q^{37} - 76 q^{38} - 340 q^{39} + 148 q^{41} - 272 q^{42} + 472 q^{43} + 208 q^{44} + 32 q^{46} - 376 q^{47} - 126 q^{49} - 136 q^{51} - 176 q^{52} - 116 q^{53} + 192 q^{56} + 24 q^{58} + 112 q^{59} - 680 q^{61} + 32 q^{62} - 168 q^{63} + 128 q^{64} - 272 q^{66} + 208 q^{67} - 400 q^{68} - 544 q^{69} - 616 q^{71} - 112 q^{72} + 1332 q^{73} + 72 q^{74} + 152 q^{76} - 352 q^{77} + 680 q^{78} + 216 q^{79} - 1738 q^{81} - 296 q^{82} + 40 q^{83} + 544 q^{84} - 944 q^{86} - 1224 q^{87} - 416 q^{88} + 484 q^{89} - 152 q^{91} - 64 q^{92} + 1360 q^{93} + 752 q^{94} - 556 q^{97} + 252 q^{98} + 364 q^{99}+O(q^{100}) \) 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
−5.83095
5.83095
−2.00000 −5.83095 4.00000 0 11.6619 −23.6619 −8.00000 7.00000 0
1.2 −2.00000 5.83095 4.00000 0 −11.6619 −0.338096 −8.00000 7.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(19\) \( -1 \)

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 950.4.a.f 2
5.b even 2 1 190.4.a.f 2
5.c odd 4 2 950.4.b.h 4
15.d odd 2 1 1710.4.a.l 2
20.d odd 2 1 1520.4.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.4.a.f 2 5.b even 2 1
950.4.a.f 2 1.a even 1 1 trivial
950.4.b.h 4 5.c odd 4 2
1520.4.a.l 2 20.d odd 2 1
1710.4.a.l 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(950))\):

\( T_{3}^{2} - 34 \) Copy content Toggle raw display
\( T_{7}^{2} + 24T_{7} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 34 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 24T + 8 \) Copy content Toggle raw display
$11$ \( T^{2} - 52T + 540 \) Copy content Toggle raw display
$13$ \( T^{2} + 44T - 366 \) Copy content Toggle raw display
$17$ \( T^{2} + 100T + 2364 \) Copy content Toggle raw display
$19$ \( (T - 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16T - 2112 \) Copy content Toggle raw display
$29$ \( T^{2} + 12T - 10980 \) Copy content Toggle raw display
$31$ \( T^{2} + 16T - 13536 \) Copy content Toggle raw display
$37$ \( T^{2} + 36T - 32350 \) Copy content Toggle raw display
$41$ \( T^{2} - 148T - 17508 \) Copy content Toggle raw display
$43$ \( T^{2} - 472T + 32712 \) Copy content Toggle raw display
$47$ \( T^{2} + 376T + 12360 \) Copy content Toggle raw display
$53$ \( T^{2} + 116T - 71742 \) Copy content Toggle raw display
$59$ \( T^{2} - 112T - 225480 \) Copy content Toggle raw display
$61$ \( T^{2} + 680T + 71536 \) Copy content Toggle raw display
$67$ \( T^{2} - 208T - 296034 \) Copy content Toggle raw display
$71$ \( T^{2} + 616T + 81264 \) Copy content Toggle raw display
$73$ \( T^{2} - 1332 T + 436892 \) Copy content Toggle raw display
$79$ \( T^{2} - 216T - 288760 \) Copy content Toggle raw display
$83$ \( T^{2} - 40T - 1150704 \) Copy content Toggle raw display
$89$ \( T^{2} - 484T + 35580 \) Copy content Toggle raw display
$97$ \( T^{2} + 556T + 25570 \) Copy content Toggle raw display
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