Properties

Label 950.4.a.i
Level $950$
Weight $4$
Character orbit 950.a
Self dual yes
Analytic conductor $56.052$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{313}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 78 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{313})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} - \beta q^{3} + 4 q^{4} - 2 \beta q^{6} + ( - \beta + 14) q^{7} + 8 q^{8} + (\beta + 51) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - \beta q^{3} + 4 q^{4} - 2 \beta q^{6} + ( - \beta + 14) q^{7} + 8 q^{8} + (\beta + 51) q^{9} + (4 \beta - 28) q^{11} - 4 \beta q^{12} + ( - 3 \beta + 10) q^{13} + ( - 2 \beta + 28) q^{14} + 16 q^{16} + ( - 3 \beta - 36) q^{17} + (2 \beta + 102) q^{18} + 19 q^{19} + ( - 13 \beta + 78) q^{21} + (8 \beta - 56) q^{22} + (7 \beta + 98) q^{23} - 8 \beta q^{24} + ( - 6 \beta + 20) q^{26} + ( - 25 \beta - 78) q^{27} + ( - 4 \beta + 56) q^{28} + ( - 9 \beta + 96) q^{29} + ( - 26 \beta + 4) q^{31} + 32 q^{32} + (24 \beta - 312) q^{33} + ( - 6 \beta - 72) q^{34} + (4 \beta + 204) q^{36} + ( - 22 \beta - 28) q^{37} + 38 q^{38} + ( - 7 \beta + 234) q^{39} + 390 q^{41} + ( - 26 \beta + 156) q^{42} + ( - 2 \beta - 168) q^{43} + (16 \beta - 112) q^{44} + (14 \beta + 196) q^{46} - 24 \beta q^{47} - 16 \beta q^{48} + ( - 27 \beta - 69) q^{49} + (39 \beta + 234) q^{51} + ( - 12 \beta + 40) q^{52} + (53 \beta - 26) q^{53} + ( - 50 \beta - 156) q^{54} + ( - 8 \beta + 112) q^{56} - 19 \beta q^{57} + ( - 18 \beta + 192) q^{58} + (45 \beta - 366) q^{59} + (14 \beta - 618) q^{61} + ( - 52 \beta + 8) q^{62} + ( - 38 \beta + 636) q^{63} + 64 q^{64} + (48 \beta - 624) q^{66} + (67 \beta - 12) q^{67} + ( - 12 \beta - 144) q^{68} + ( - 105 \beta - 546) q^{69} + ( - 36 \beta + 768) q^{71} + (8 \beta + 408) q^{72} + (39 \beta + 472) q^{73} + ( - 44 \beta - 56) q^{74} + 76 q^{76} + (80 \beta - 704) q^{77} + ( - 14 \beta + 468) q^{78} + ( - 2 \beta + 376) q^{79} + (76 \beta + 573) q^{81} + 780 q^{82} + (86 \beta - 548) q^{83} + ( - 52 \beta + 312) q^{84} + ( - 4 \beta - 336) q^{86} + ( - 87 \beta + 702) q^{87} + (32 \beta - 224) q^{88} + (88 \beta + 38) q^{89} + ( - 49 \beta + 374) q^{91} + (28 \beta + 392) q^{92} + (22 \beta + 2028) q^{93} - 48 \beta q^{94} - 32 \beta q^{96} + (98 \beta + 428) q^{97} + ( - 54 \beta - 138) q^{98} + (180 \beta - 1116) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - q^{3} + 8 q^{4} - 2 q^{6} + 27 q^{7} + 16 q^{8} + 103 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - q^{3} + 8 q^{4} - 2 q^{6} + 27 q^{7} + 16 q^{8} + 103 q^{9} - 52 q^{11} - 4 q^{12} + 17 q^{13} + 54 q^{14} + 32 q^{16} - 75 q^{17} + 206 q^{18} + 38 q^{19} + 143 q^{21} - 104 q^{22} + 203 q^{23} - 8 q^{24} + 34 q^{26} - 181 q^{27} + 108 q^{28} + 183 q^{29} - 18 q^{31} + 64 q^{32} - 600 q^{33} - 150 q^{34} + 412 q^{36} - 78 q^{37} + 76 q^{38} + 461 q^{39} + 780 q^{41} + 286 q^{42} - 338 q^{43} - 208 q^{44} + 406 q^{46} - 24 q^{47} - 16 q^{48} - 165 q^{49} + 507 q^{51} + 68 q^{52} + q^{53} - 362 q^{54} + 216 q^{56} - 19 q^{57} + 366 q^{58} - 687 q^{59} - 1222 q^{61} - 36 q^{62} + 1234 q^{63} + 128 q^{64} - 1200 q^{66} + 43 q^{67} - 300 q^{68} - 1197 q^{69} + 1500 q^{71} + 824 q^{72} + 983 q^{73} - 156 q^{74} + 152 q^{76} - 1328 q^{77} + 922 q^{78} + 750 q^{79} + 1222 q^{81} + 1560 q^{82} - 1010 q^{83} + 572 q^{84} - 676 q^{86} + 1317 q^{87} - 416 q^{88} + 164 q^{89} + 699 q^{91} + 812 q^{92} + 4078 q^{93} - 48 q^{94} - 32 q^{96} + 954 q^{97} - 330 q^{98} - 2052 q^{99}+O(q^{100}) \) 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
9.34590
−8.34590
2.00000 −9.34590 4.00000 0 −18.6918 4.65410 8.00000 60.3459 0
1.2 2.00000 8.34590 4.00000 0 16.6918 22.3459 8.00000 42.6541 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.i 2
5.b even 2 1 190.4.a.d 2
5.c odd 4 2 950.4.b.f 4
15.d odd 2 1 1710.4.a.p 2
20.d odd 2 1 1520.4.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.4.a.d 2 5.b even 2 1
950.4.a.i 2 1.a even 1 1 trivial
950.4.b.f 4 5.c odd 4 2
1520.4.a.k 2 20.d odd 2 1
1710.4.a.p 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(950))\):

\( T_{3}^{2} + T_{3} - 78 \) Copy content Toggle raw display
\( T_{7}^{2} - 27T_{7} + 104 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 78 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 27T + 104 \) Copy content Toggle raw display
$11$ \( T^{2} + 52T - 576 \) Copy content Toggle raw display
$13$ \( T^{2} - 17T - 632 \) Copy content Toggle raw display
$17$ \( T^{2} + 75T + 702 \) Copy content Toggle raw display
$19$ \( (T - 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 203T + 6468 \) Copy content Toggle raw display
$29$ \( T^{2} - 183T + 2034 \) Copy content Toggle raw display
$31$ \( T^{2} + 18T - 52816 \) Copy content Toggle raw display
$37$ \( T^{2} + 78T - 36352 \) Copy content Toggle raw display
$41$ \( (T - 390)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 338T + 28248 \) Copy content Toggle raw display
$47$ \( T^{2} + 24T - 44928 \) Copy content Toggle raw display
$53$ \( T^{2} - T - 219804 \) Copy content Toggle raw display
$59$ \( T^{2} + 687T - 40464 \) Copy content Toggle raw display
$61$ \( T^{2} + 1222 T + 357984 \) Copy content Toggle raw display
$67$ \( T^{2} - 43T - 350802 \) Copy content Toggle raw display
$71$ \( T^{2} - 1500 T + 461088 \) Copy content Toggle raw display
$73$ \( T^{2} - 983T + 122554 \) Copy content Toggle raw display
$79$ \( T^{2} - 750T + 140312 \) Copy content Toggle raw display
$83$ \( T^{2} + 1010 T - 323712 \) Copy content Toggle raw display
$89$ \( T^{2} - 164T - 599244 \) Copy content Toggle raw display
$97$ \( T^{2} - 954T - 523984 \) Copy content Toggle raw display
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