Properties

Label 950.4.a.e
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: \(\Q(\sqrt{73}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 18 \) 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 38)
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 \(\beta = \frac{1}{2}(1 + \sqrt{73})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + ( - \beta - 4) q^{3} + 4 q^{4} + (2 \beta + 8) q^{6} + (4 \beta + 7) q^{7} - 8 q^{8} + (9 \beta + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + ( - \beta - 4) q^{3} + 4 q^{4} + (2 \beta + 8) q^{6} + (4 \beta + 7) q^{7} - 8 q^{8} + (9 \beta + 7) q^{9} + (\beta - 9) q^{11} + ( - 4 \beta - 16) q^{12} + ( - 13 \beta - 2) q^{13} + ( - 8 \beta - 14) q^{14} + 16 q^{16} + (2 \beta + 39) q^{17} + ( - 18 \beta - 14) q^{18} + 19 q^{19} + ( - 27 \beta - 100) q^{21} + ( - 2 \beta + 18) q^{22} + ( - 13 \beta - 30) q^{23} + (8 \beta + 32) q^{24} + (26 \beta + 4) q^{26} + ( - 25 \beta - 82) q^{27} + (16 \beta + 28) q^{28} + ( - 21 \beta + 12) q^{29} + ( - 44 \beta + 128) q^{31} - 32 q^{32} + (4 \beta + 18) q^{33} + ( - 4 \beta - 78) q^{34} + (36 \beta + 28) q^{36} + (28 \beta - 110) q^{37} - 38 q^{38} + (67 \beta + 242) q^{39} + (10 \beta - 30) q^{41} + (54 \beta + 200) q^{42} + ( - 7 \beta - 335) q^{43} + (4 \beta - 36) q^{44} + (26 \beta + 60) q^{46} + (71 \beta + 159) q^{47} + ( - 16 \beta - 64) q^{48} + (72 \beta - 6) q^{49} + ( - 49 \beta - 192) q^{51} + ( - 52 \beta - 8) q^{52} + ( - 17 \beta + 618) q^{53} + (50 \beta + 164) q^{54} + ( - 32 \beta - 56) q^{56} + ( - 19 \beta - 76) q^{57} + (42 \beta - 24) q^{58} + (25 \beta - 156) q^{59} + (111 \beta + 101) q^{61} + (88 \beta - 256) q^{62} + (127 \beta + 697) q^{63} + 64 q^{64} + ( - 8 \beta - 36) q^{66} + (77 \beta - 650) q^{67} + (8 \beta + 156) q^{68} + (95 \beta + 354) q^{69} + (116 \beta + 42) q^{71} + ( - 72 \beta - 56) q^{72} + (184 \beta - 281) q^{73} + ( - 56 \beta + 220) q^{74} + 76 q^{76} + ( - 25 \beta + 9) q^{77} + ( - 134 \beta - 484) q^{78} + ( - 58 \beta + 704) q^{79} + ( - 36 \beta + 589) q^{81} + ( - 20 \beta + 60) q^{82} + ( - 194 \beta + 432) q^{83} + ( - 108 \beta - 400) q^{84} + (14 \beta + 670) q^{86} + (93 \beta + 330) q^{87} + ( - 8 \beta + 72) q^{88} + ( - 188 \beta - 24) q^{89} + ( - 151 \beta - 950) q^{91} + ( - 52 \beta - 120) q^{92} + (92 \beta + 280) q^{93} + ( - 142 \beta - 318) q^{94} + (32 \beta + 128) q^{96} + ( - 102 \beta - 596) q^{97} + ( - 144 \beta + 12) q^{98} + ( - 65 \beta + 99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 9 q^{3} + 8 q^{4} + 18 q^{6} + 18 q^{7} - 16 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} - 9 q^{3} + 8 q^{4} + 18 q^{6} + 18 q^{7} - 16 q^{8} + 23 q^{9} - 17 q^{11} - 36 q^{12} - 17 q^{13} - 36 q^{14} + 32 q^{16} + 80 q^{17} - 46 q^{18} + 38 q^{19} - 227 q^{21} + 34 q^{22} - 73 q^{23} + 72 q^{24} + 34 q^{26} - 189 q^{27} + 72 q^{28} + 3 q^{29} + 212 q^{31} - 64 q^{32} + 40 q^{33} - 160 q^{34} + 92 q^{36} - 192 q^{37} - 76 q^{38} + 551 q^{39} - 50 q^{41} + 454 q^{42} - 677 q^{43} - 68 q^{44} + 146 q^{46} + 389 q^{47} - 144 q^{48} + 60 q^{49} - 433 q^{51} - 68 q^{52} + 1219 q^{53} + 378 q^{54} - 144 q^{56} - 171 q^{57} - 6 q^{58} - 287 q^{59} + 313 q^{61} - 424 q^{62} + 1521 q^{63} + 128 q^{64} - 80 q^{66} - 1223 q^{67} + 320 q^{68} + 803 q^{69} + 200 q^{71} - 184 q^{72} - 378 q^{73} + 384 q^{74} + 152 q^{76} - 7 q^{77} - 1102 q^{78} + 1350 q^{79} + 1142 q^{81} + 100 q^{82} + 670 q^{83} - 908 q^{84} + 1354 q^{86} + 753 q^{87} + 136 q^{88} - 236 q^{89} - 2051 q^{91} - 292 q^{92} + 652 q^{93} - 778 q^{94} + 288 q^{96} - 1294 q^{97} - 120 q^{98} + 133 q^{99}+O(q^{100}) \) 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
4.77200
−3.77200
−2.00000 −8.77200 4.00000 0 17.5440 26.0880 −8.00000 49.9480 0
1.2 −2.00000 −0.227998 4.00000 0 0.455996 −8.08801 −8.00000 −26.9480 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.e 2
5.b even 2 1 38.4.a.c 2
5.c odd 4 2 950.4.b.i 4
15.d odd 2 1 342.4.a.h 2
20.d odd 2 1 304.4.a.c 2
35.c odd 2 1 1862.4.a.e 2
40.e odd 2 1 1216.4.a.p 2
40.f even 2 1 1216.4.a.g 2
95.d odd 2 1 722.4.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.c 2 5.b even 2 1
304.4.a.c 2 20.d odd 2 1
342.4.a.h 2 15.d odd 2 1
722.4.a.f 2 95.d odd 2 1
950.4.a.e 2 1.a even 1 1 trivial
950.4.b.i 4 5.c odd 4 2
1216.4.a.g 2 40.f even 2 1
1216.4.a.p 2 40.e odd 2 1
1862.4.a.e 2 35.c odd 2 1

Hecke kernels

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

\( T_{3}^{2} + 9T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 18T_{7} - 211 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 18T - 211 \) Copy content Toggle raw display
$11$ \( T^{2} + 17T + 54 \) Copy content Toggle raw display
$13$ \( T^{2} + 17T - 3012 \) Copy content Toggle raw display
$17$ \( T^{2} - 80T + 1527 \) Copy content Toggle raw display
$19$ \( (T - 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 73T - 1752 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 8046 \) Copy content Toggle raw display
$31$ \( T^{2} - 212T - 24096 \) Copy content Toggle raw display
$37$ \( T^{2} + 192T - 5092 \) Copy content Toggle raw display
$41$ \( T^{2} + 50T - 1200 \) Copy content Toggle raw display
$43$ \( T^{2} + 677T + 113688 \) Copy content Toggle raw display
$47$ \( T^{2} - 389T - 54168 \) Copy content Toggle raw display
$53$ \( T^{2} - 1219 T + 366216 \) Copy content Toggle raw display
$59$ \( T^{2} + 287T + 9186 \) Copy content Toggle raw display
$61$ \( T^{2} - 313T - 200366 \) Copy content Toggle raw display
$67$ \( T^{2} + 1223 T + 265728 \) Copy content Toggle raw display
$71$ \( T^{2} - 200T - 235572 \) Copy content Toggle raw display
$73$ \( T^{2} + 378T - 582151 \) Copy content Toggle raw display
$79$ \( T^{2} - 1350 T + 394232 \) Copy content Toggle raw display
$83$ \( T^{2} - 670T - 574632 \) Copy content Toggle raw display
$89$ \( T^{2} + 236T - 631104 \) Copy content Toggle raw display
$97$ \( T^{2} + 1294 T + 228736 \) Copy content Toggle raw display
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