Properties

Label 1150.4.a.v
Level $1150$
Weight $4$
Character orbit 1150.a
Self dual yes
Analytic conductor $67.852$
Analytic rank $0$
Dimension $5$
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] = [1150,4,Mod(1,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, 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("1150.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.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: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 107x^{3} - 3x^{2} + 2151x - 2916 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta_1 + 1) q^{3} + 4 q^{4} + ( - 2 \beta_1 + 2) q^{6} + ( - \beta_{3} - 1) q^{7} + 8 q^{8} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + ( - \beta_1 + 1) q^{3} + 4 q^{4} + ( - 2 \beta_1 + 2) q^{6} + ( - \beta_{3} - 1) q^{7} + 8 q^{8} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 16) q^{9} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 5) q^{11} + ( - 4 \beta_1 + 4) q^{12} + (\beta_{4} - \beta_1 - 12) q^{13} + ( - 2 \beta_{3} - 2) q^{14} + 16 q^{16} + (\beta_{4} + \beta_{2} - \beta_1 + 46) q^{17} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 32) q^{18} + (\beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 15) q^{19} + ( - 4 \beta_{2} - 10 \beta_1 - 16) q^{21} + (2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 10) q^{22} + 23 q^{23} + ( - 8 \beta_1 + 8) q^{24} + (2 \beta_{4} - 2 \beta_1 - 24) q^{26} + ( - 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 14 \beta_1 + 73) q^{27} + ( - 4 \beta_{3} - 4) q^{28} + ( - 6 \beta_{4} + 3 \beta_{2} + 7 \beta_1 + 5) q^{29} + ( - 6 \beta_{4} - \beta_{3} + \beta_{2} - 13 \beta_1 + 42) q^{31} + 32 q^{32} + ( - 5 \beta_{4} + 4 \beta_{3} + 7 \beta_{2} - 3 \beta_1 + 76) q^{33} + (2 \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 92) q^{34} + ( - 4 \beta_{3} + 4 \beta_{2} - 8 \beta_1 + 64) q^{36} + ( - 4 \beta_{4} + 2 \beta_{3} - 14 \beta_1 + 44) q^{37} + (2 \beta_{4} - 6 \beta_{3} - 6 \beta_{2} + 6 \beta_1 + 30) q^{38} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 5 \beta_1 + 39) q^{39} + (2 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} - 23 \beta_1 + 24) q^{41} + ( - 8 \beta_{2} - 20 \beta_1 - 32) q^{42} + (2 \beta_{4} + 5 \beta_{3} - 8 \beta_{2} + 18 \beta_1 + 25) q^{43} + (4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 20) q^{44} + 46 q^{46} + ( - \beta_{4} + 3 \beta_{3} - 3 \beta_1 + 75) q^{47} + ( - 16 \beta_1 + 16) q^{48} + (10 \beta_{4} + 5 \beta_{3} - 4 \beta_{2} + 18 \beta_1 + 42) q^{49} + ( - 5 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} - 65 \beta_1 + 112) q^{51} + (4 \beta_{4} - 4 \beta_1 - 48) q^{52} + (4 \beta_{4} + 4 \beta_{3} - 12 \beta_{2} + 48 \beta_1 + 252) q^{53} + ( - 6 \beta_{4} + 6 \beta_{3} + 6 \beta_{2} - 28 \beta_1 + 146) q^{54} + ( - 8 \beta_{3} - 8) q^{56} + (7 \beta_{4} - 12 \beta_{3} - 33 \beta_{2} - 15 \beta_1 - 192) q^{57} + ( - 12 \beta_{4} + 6 \beta_{2} + 14 \beta_1 + 10) q^{58} + (7 \beta_{4} + \beta_{3} - 11 \beta_{2} - \beta_1 - 30) q^{59} + (6 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 24 \beta_1 + 316) q^{61} + ( - 12 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 26 \beta_1 + 84) q^{62} + (12 \beta_{4} - 3 \beta_{3} - 10 \beta_{2} + 54 \beta_1 + 371) q^{63} + 64 q^{64} + ( - 10 \beta_{4} + 8 \beta_{3} + 14 \beta_{2} - 6 \beta_1 + 152) q^{66} + (13 \beta_{4} + 4 \beta_{3} + 23 \beta_{2} + 47 \beta_1 - 92) q^{67} + (4 \beta_{4} + 4 \beta_{2} - 4 \beta_1 + 184) q^{68} + ( - 23 \beta_1 + 23) q^{69} + (11 \beta_{4} - \beta_{3} + 2 \beta_{2} - 55 \beta_1 - 5) q^{71} + ( - 8 \beta_{3} + 8 \beta_{2} - 16 \beta_1 + 128) q^{72} + ( - 15 \beta_{4} + 8 \beta_{3} + 18 \beta_{2} + 60 \beta_1 + 429) q^{73} + ( - 8 \beta_{4} + 4 \beta_{3} - 28 \beta_1 + 88) q^{74} + (4 \beta_{4} - 12 \beta_{3} - 12 \beta_{2} + 12 \beta_1 + 60) q^{76} + ( - 18 \beta_{4} - 9 \beta_{3} + 4 \beta_{2} + 30 \beta_1 - 289) q^{77} + ( - 4 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 10 \beta_1 + 78) q^{78} + ( - 4 \beta_{4} + 19 \beta_{3} - 6 \beta_{2} - 48 \beta_1 + 131) q^{79} + ( - 3 \beta_{4} + 28 \beta_{3} + 23 \beta_{2} - 18 \beta_1 + 292) q^{81} + (4 \beta_{4} - 6 \beta_{3} - 12 \beta_{2} - 46 \beta_1 + 48) q^{82} + ( - 9 \beta_{4} + 21 \beta_{3} + 3 \beta_{2} + 41 \beta_1 + 409) q^{83} + ( - 16 \beta_{2} - 40 \beta_1 - 64) q^{84} + (4 \beta_{4} + 10 \beta_{3} - 16 \beta_{2} + 36 \beta_1 + 50) q^{86} + (3 \beta_{4} + 22 \beta_{3} + 26 \beta_{2} + 2 \beta_1 - 298) q^{87} + (8 \beta_{4} + 8 \beta_{3} + 8 \beta_{2} - 8 \beta_1 - 40) q^{88} + ( - \beta_{4} - 10 \beta_{3} + 3 \beta_{2} - 17 \beta_1 - 338) q^{89} + ( - 10 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} - 8 \beta_1 + 69) q^{91} + 92 q^{92} + (9 \beta_{4} - 8 \beta_{3} + 32 \beta_{2} - 42 \beta_1 + 534) q^{93} + ( - 2 \beta_{4} + 6 \beta_{3} - 6 \beta_1 + 150) q^{94} + ( - 32 \beta_1 + 32) q^{96} + (18 \beta_{4} + 6 \beta_{3} + 8 \beta_{2} + 60 \beta_1 - 252) q^{97} + (20 \beta_{4} + 10 \beta_{3} - 8 \beta_{2} + 36 \beta_1 + 84) q^{98} + ( - 38 \beta_{4} + 5 \beta_{3} + 42 \beta_{2} - 62 \beta_1 + 457) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 10 q^{2} + 5 q^{3} + 20 q^{4} + 10 q^{6} - 3 q^{7} + 40 q^{8} + 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 10 q^{2} + 5 q^{3} + 20 q^{4} + 10 q^{6} - 3 q^{7} + 40 q^{8} + 84 q^{9} - 26 q^{11} + 20 q^{12} - 61 q^{13} - 6 q^{14} + 80 q^{16} + 231 q^{17} + 168 q^{18} + 74 q^{19} - 88 q^{21} - 52 q^{22} + 115 q^{23} + 40 q^{24} - 122 q^{26} + 368 q^{27} - 12 q^{28} + 37 q^{29} + 220 q^{31} + 160 q^{32} + 391 q^{33} + 462 q^{34} + 336 q^{36} + 220 q^{37} + 148 q^{38} + 195 q^{39} + 112 q^{41} - 176 q^{42} + 97 q^{43} - 104 q^{44} + 230 q^{46} + 370 q^{47} + 80 q^{48} + 182 q^{49} + 563 q^{51} - 244 q^{52} + 1224 q^{53} + 736 q^{54} - 24 q^{56} - 1009 q^{57} + 74 q^{58} - 181 q^{59} + 1574 q^{61} + 440 q^{62} + 1829 q^{63} + 320 q^{64} + 782 q^{66} - 435 q^{67} + 924 q^{68} + 115 q^{69} - 30 q^{71} + 672 q^{72} + 2180 q^{73} + 440 q^{74} + 296 q^{76} - 1401 q^{77} + 390 q^{78} + 609 q^{79} + 1453 q^{81} + 224 q^{82} + 2018 q^{83} - 352 q^{84} + 194 q^{86} - 1485 q^{87} - 208 q^{88} - 1663 q^{89} + 361 q^{91} + 460 q^{92} + 2741 q^{93} + 740 q^{94} + 160 q^{96} - 1274 q^{97} + 364 q^{98} + 2397 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 107x^{3} - 3x^{2} + 2151x - 2916 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 3\nu^{3} - 53\nu^{2} + 111\nu - 567 ) / 45 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 3\nu^{3} - 98\nu^{2} + 111\nu + 1323 ) / 45 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{4} + 9\nu^{3} - 196\nu^{2} - 753\nu + 2646 ) / 45 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{4} - 6\beta_{3} + 65\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{4} - 71\beta_{3} + 98\beta_{2} + 84\beta _1 + 2793 \) 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
9.27140
3.88892
1.53483
−6.21006
−8.48510
2.00000 −8.27140 4.00000 0 −16.5428 22.8621 8.00000 41.4161 0
1.2 2.00000 −2.88892 4.00000 0 −5.77785 −8.21839 8.00000 −18.6541 0
1.3 2.00000 −0.534830 4.00000 0 −1.06966 −28.9380 8.00000 −26.7140 0
1.4 2.00000 7.21006 4.00000 0 14.4201 19.8879 8.00000 24.9850 0
1.5 2.00000 9.48510 4.00000 0 18.9702 −8.59355 8.00000 62.9670 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(23\) \(-1\)

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 1150.4.a.v yes 5
5.b even 2 1 1150.4.a.q 5
5.c odd 4 2 1150.4.b.r 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1150.4.a.q 5 5.b even 2 1
1150.4.a.v yes 5 1.a even 1 1 trivial
1150.4.b.r 10 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1150))\):

\( T_{3}^{5} - 5T_{3}^{4} - 97T_{3}^{3} + 314T_{3}^{2} + 1829T_{3} + 874 \) Copy content Toggle raw display
\( T_{7}^{5} + 3T_{7}^{4} - 944T_{7}^{3} - 972T_{7}^{2} + 165944T_{7} + 929248 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 5 T^{4} - 97 T^{3} + 314 T^{2} + \cdots + 874 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 3 T^{4} - 944 T^{3} + \cdots + 929248 \) Copy content Toggle raw display
$11$ \( T^{5} + 26 T^{4} - 4237 T^{3} + \cdots + 3242920 \) Copy content Toggle raw display
$13$ \( T^{5} + 61 T^{4} - 445 T^{3} + \cdots + 166878 \) Copy content Toggle raw display
$17$ \( T^{5} - 231 T^{4} + \cdots + 414548320 \) Copy content Toggle raw display
$19$ \( T^{5} - 74 T^{4} + \cdots - 112577688 \) Copy content Toggle raw display
$23$ \( (T - 23)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 37 T^{4} + \cdots - 20220755440 \) Copy content Toggle raw display
$31$ \( T^{5} - 220 T^{4} + \cdots - 51046245360 \) Copy content Toggle raw display
$37$ \( T^{5} - 220 T^{4} + \cdots - 5132681056 \) Copy content Toggle raw display
$41$ \( T^{5} - 112 T^{4} + \cdots + 4982446167 \) Copy content Toggle raw display
$43$ \( T^{5} - 97 T^{4} + \cdots - 39661228800 \) Copy content Toggle raw display
$47$ \( T^{5} - 370 T^{4} + \cdots + 1683347184 \) Copy content Toggle raw display
$53$ \( T^{5} - 1224 T^{4} + \cdots + 18447971014656 \) Copy content Toggle raw display
$59$ \( T^{5} + 181 T^{4} + \cdots + 1123269020040 \) Copy content Toggle raw display
$61$ \( T^{5} - 1574 T^{4} + \cdots + 2333556256768 \) Copy content Toggle raw display
$67$ \( T^{5} + 435 T^{4} + \cdots - 35946409847360 \) Copy content Toggle raw display
$71$ \( T^{5} + 30 T^{4} + \cdots - 5642844385736 \) Copy content Toggle raw display
$73$ \( T^{5} - 2180 T^{4} + \cdots + 95962864505643 \) Copy content Toggle raw display
$79$ \( T^{5} - 609 T^{4} + \cdots + 12948718884224 \) Copy content Toggle raw display
$83$ \( T^{5} - 2018 T^{4} + \cdots + 31156649817688 \) Copy content Toggle raw display
$89$ \( T^{5} + 1663 T^{4} + \cdots + 1214254682896 \) Copy content Toggle raw display
$97$ \( T^{5} + 1274 T^{4} + \cdots + 79142464495296 \) Copy content Toggle raw display
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