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Properties

Label 150.3.b.b
Level $150$
Weight $3$
Character orbit 150.b
Analytic conductor $4.087$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,3,Mod(149,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.149");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 150.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(4.08720396540\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{3} q^{3} + 2 q^{4} + ( - \beta_{4} - 1) q^{6} + (\beta_{7} - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{3} q^{3} + 2 q^{4} + ( - \beta_{4} - 1) q^{6} + (\beta_{7} - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + 2) q^{9} - 3 \beta_{5} q^{11} - 2 \beta_{3} q^{12} + 5 \beta_1 q^{13} + ( - 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4}) q^{14} + 4 q^{16} + ( - \beta_{7} - 4 \beta_{3} + 14 \beta_{2} + 2 \beta_1) q^{17} + (\beta_{7} + 2 \beta_{3} + \beta_{2} - 5 \beta_1) q^{18} + ( - 2 \beta_{6} + \beta_{5} + 4 \beta_{4} - 8) q^{19} + (\beta_{6} + 10 \beta_{5} + 3 \beta_{4} - 13) q^{21} + 6 \beta_1 q^{22} + ( - \beta_{7} - 4 \beta_{3} - \beta_{2} + 2 \beta_1) q^{23} + ( - 2 \beta_{4} - 2) q^{24} - 5 \beta_{5} q^{26} + (\beta_{7} - \beta_{3} - 17 \beta_{2} + 4 \beta_1) q^{27} + (2 \beta_{7} - 4 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{28} + (4 \beta_{6} + 4 \beta_{5} + 4 \beta_{4}) q^{29} + 8 q^{31} + 4 \beta_{2} q^{32} + ( - 3 \beta_{7} + 6 \beta_{2} - 3 \beta_1) q^{33} + (2 \beta_{6} - \beta_{5} - 4 \beta_{4} + 24) q^{34} + ( - 2 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 4) q^{36} + ( - 4 \beta_{7} + 8 \beta_{3} - 4 \beta_{2} - 15 \beta_1) q^{37} + (2 \beta_{7} + 8 \beta_{3} - 12 \beta_{2} - 4 \beta_1) q^{38} + (5 \beta_{6} + 5 \beta_{5} + 10) q^{39} + (2 \beta_{6} - 10 \beta_{5} + 2 \beta_{4}) q^{41} + ( - \beta_{7} + 6 \beta_{3} - 16 \beta_{2} - 19 \beta_1) q^{42} + ( - 3 \beta_{7} + 6 \beta_{3} - 3 \beta_{2} - 10 \beta_1) q^{43} - 6 \beta_{5} q^{44} + (2 \beta_{6} - \beta_{5} - 4 \beta_{4} - 6) q^{46} + (5 \beta_{7} + 20 \beta_{3} + 5 \beta_{2} - 10 \beta_1) q^{47} - 4 \beta_{3} q^{48} + (4 \beta_{6} - 2 \beta_{5} - 8 \beta_{4} - 45) q^{49} + ( - 3 \beta_{6} + 6 \beta_{5} - 12 \beta_{4} + 18) q^{51} + 10 \beta_1 q^{52} + (\beta_{7} + 4 \beta_{3} - 14 \beta_{2} - 2 \beta_1) q^{53} + ( - 2 \beta_{6} - 5 \beta_{5} - \beta_{4} - 35) q^{54} + ( - 4 \beta_{6} - 6 \beta_{5} - 4 \beta_{4}) q^{56} + ( - 3 \beta_{7} + 8 \beta_{3} - 30 \beta_{2} + 15 \beta_1) q^{57} + ( - 4 \beta_{7} + 8 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{58} + ( - 4 \beta_{6} - 22 \beta_{5} - 4 \beta_{4}) q^{59} + ( - 4 \beta_{6} + 2 \beta_{5} + 8 \beta_{4} - 16) q^{61} + 8 \beta_{2} q^{62} + (7 \beta_{7} + 8 \beta_{3} - 11 \beta_{2} + 28 \beta_1) q^{63} + 8 q^{64} + (6 \beta_{6} + 6 \beta_{5} + 12) q^{66} + (3 \beta_{7} - 6 \beta_{3} + 3 \beta_{2} - 38 \beta_1) q^{67} + ( - 2 \beta_{7} - 8 \beta_{3} + 28 \beta_{2} + 4 \beta_1) q^{68} + ( - 3 \beta_{6} + 6 \beta_{5} + 3 \beta_{4} + 33) q^{69} + ( - 4 \beta_{6} - 19 \beta_{5} - 4 \beta_{4}) q^{71} + (2 \beta_{7} + 4 \beta_{3} + 2 \beta_{2} - 10 \beta_1) q^{72} + (4 \beta_{7} - 8 \beta_{3} + 4 \beta_{2} - 21 \beta_1) q^{73} + (8 \beta_{6} + 19 \beta_{5} + 8 \beta_{4}) q^{74} + ( - 4 \beta_{6} + 2 \beta_{5} + 8 \beta_{4} - 16) q^{76} + ( - 6 \beta_{7} - 24 \beta_{3} + 12 \beta_1) q^{77} + ( - 5 \beta_{7} + 10 \beta_{2} - 5 \beta_1) q^{78} + (2 \beta_{6} - \beta_{5} - 4 \beta_{4} + 28) q^{79} + (4 \beta_{6} + 10 \beta_{5} + 20 \beta_{4} + 7) q^{81} + ( - 2 \beta_{7} + 4 \beta_{3} - 2 \beta_{2} + 22 \beta_1) q^{82} + ( - \beta_{7} - 4 \beta_{3} + 11 \beta_{2} + 2 \beta_1) q^{83} + (2 \beta_{6} + 20 \beta_{5} + 6 \beta_{4} - 26) q^{84} + (6 \beta_{6} + 13 \beta_{5} + 6 \beta_{4}) q^{86} + ( - 12 \beta_{3} + 36 \beta_{2} + 36 \beta_1) q^{87} + 12 \beta_1 q^{88} + ( - 4 \beta_{6} + 8 \beta_{5} - 4 \beta_{4}) q^{89} + (10 \beta_{6} - 5 \beta_{5} - 20 \beta_{4} - 20) q^{91} + ( - 2 \beta_{7} - 8 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{92} - 8 \beta_{3} q^{93} + ( - 10 \beta_{6} + 5 \beta_{5} + 20 \beta_{4} + 30) q^{94} + ( - 4 \beta_{4} - 4) q^{96} + (4 \beta_{7} - 8 \beta_{3} + 4 \beta_{2} + 41 \beta_1) q^{97} + ( - 4 \beta_{7} - 16 \beta_{3} - 37 \beta_{2} + 8 \beta_1) q^{98} + ( - 6 \beta_{6} - 15 \beta_{5} - 12 \beta_{4} + 48) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} - 8 q^{6} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} - 8 q^{6} + 16 q^{9} + 32 q^{16} - 64 q^{19} - 104 q^{21} - 16 q^{24} + 64 q^{31} + 192 q^{34} + 32 q^{36} + 80 q^{39} - 48 q^{46} - 360 q^{49} + 144 q^{51} - 280 q^{54} - 128 q^{61} + 64 q^{64} + 96 q^{66} + 264 q^{69} - 128 q^{76} + 224 q^{79} + 56 q^{81} - 208 q^{84} - 160 q^{91} + 240 q^{94} - 32 q^{96} + 384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{6} - 16\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} + \nu^{5} - 13\nu^{3} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - \nu^{6} - 2\nu^{4} - 21\nu^{3} - 8\nu^{2} - 3\nu - 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 3\nu^{6} - 2\nu^{5} + 16\nu^{3} - 18\nu^{2} - 16\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{7} + 2\nu^{5} + 26\nu^{3} + 10\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\nu^{7} - 2\nu^{6} - 14\nu^{3} - 12\nu^{2} + 2\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{7} + 2\nu^{5} - 4\nu^{4} + 58\nu^{3} + 22\nu - 14 ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - 2\beta_{5} - 2\beta_{4} - 2\beta_{3} - 2\beta_{2} + \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} + 2\beta_{5} + 2\beta_{4} - 9\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} - 5\beta_{5} + 4\beta_{4} - 4\beta_{3} + 14\beta_{2} + 2\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} - 4\beta_{3} + 2\beta_{2} + 2\beta _1 - 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} - 5\beta_{6} + 19\beta_{5} + 10\beta_{4} + 10\beta_{3} + 28\beta_{2} - 5\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -8\beta_{6} - 8\beta_{5} - 8\beta_{4} + 27\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} + 13\beta_{6} + 37\beta_{5} - 26\beta_{4} + 26\beta_{3} - 100\beta_{2} - 13\beta_1 ) / 12 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.14412 − 1.14412i
1.14412 + 1.14412i
−0.437016 − 0.437016i
−0.437016 + 0.437016i
−1.14412 − 1.14412i
−1.14412 + 1.14412i
0.437016 − 0.437016i
0.437016 + 0.437016i
−1.41421 −1.52896 − 2.58114i 2.00000 0 2.16228 + 3.65028i − 7.48683i −2.82843 −4.32456 + 7.89292i 0
149.2 −1.41421 −1.52896 + 2.58114i 2.00000 0 2.16228 − 3.65028i 7.48683i −2.82843 −4.32456 − 7.89292i 0
149.3 −1.41421 2.94317 − 0.581139i 2.00000 0 −4.16228 + 0.821854i − 11.4868i −2.82843 8.32456 − 3.42079i 0
149.4 −1.41421 2.94317 + 0.581139i 2.00000 0 −4.16228 − 0.821854i 11.4868i −2.82843 8.32456 + 3.42079i 0
149.5 1.41421 −2.94317 − 0.581139i 2.00000 0 −4.16228 − 0.821854i − 11.4868i 2.82843 8.32456 + 3.42079i 0
149.6 1.41421 −2.94317 + 0.581139i 2.00000 0 −4.16228 + 0.821854i 11.4868i 2.82843 8.32456 − 3.42079i 0
149.7 1.41421 1.52896 − 2.58114i 2.00000 0 2.16228 − 3.65028i − 7.48683i 2.82843 −4.32456 − 7.89292i 0
149.8 1.41421 1.52896 + 2.58114i 2.00000 0 2.16228 + 3.65028i 7.48683i 2.82843 −4.32456 + 7.89292i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.3.b.b 8
3.b odd 2 1 inner 150.3.b.b 8
4.b odd 2 1 1200.3.c.k 8
5.b even 2 1 inner 150.3.b.b 8
5.c odd 4 1 30.3.d.a ✓ 4
5.c odd 4 1 150.3.d.c 4
12.b even 2 1 1200.3.c.k 8
15.d odd 2 1 inner 150.3.b.b 8
15.e even 4 1 30.3.d.a ✓ 4
15.e even 4 1 150.3.d.c 4
20.d odd 2 1 1200.3.c.k 8
20.e even 4 1 240.3.l.c 4
20.e even 4 1 1200.3.l.u 4
40.i odd 4 1 960.3.l.e 4
40.k even 4 1 960.3.l.f 4
45.k odd 12 2 810.3.h.a 8
45.l even 12 2 810.3.h.a 8
60.h even 2 1 1200.3.c.k 8
60.l odd 4 1 240.3.l.c 4
60.l odd 4 1 1200.3.l.u 4
120.q odd 4 1 960.3.l.f 4
120.w even 4 1 960.3.l.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.d.a ✓ 4 5.c odd 4 1
30.3.d.a ✓ 4 15.e even 4 1
150.3.b.b 8 1.a even 1 1 trivial
150.3.b.b 8 3.b odd 2 1 inner
150.3.b.b 8 5.b even 2 1 inner
150.3.b.b 8 15.d odd 2 1 inner
150.3.d.c 4 5.c odd 4 1
150.3.d.c 4 15.e even 4 1
240.3.l.c 4 20.e even 4 1
240.3.l.c 4 60.l odd 4 1
810.3.h.a 8 45.k odd 12 2
810.3.h.a 8 45.l even 12 2
960.3.l.e 4 40.i odd 4 1
960.3.l.e 4 120.w even 4 1
960.3.l.f 4 40.k even 4 1
960.3.l.f 4 120.q odd 4 1
1200.3.c.k 8 4.b odd 2 1
1200.3.c.k 8 12.b even 2 1
1200.3.c.k 8 20.d odd 2 1
1200.3.c.k 8 60.h even 2 1
1200.3.l.u 4 20.e even 4 1
1200.3.l.u 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 188T_{7}^{2} + 7396 \) acting on \(S_{3}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 8 T^{6} + 18 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 188 T^{2} + 7396)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 936 T^{2} + 11664)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16 T - 296)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 396 T^{2} + 26244)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 720)^{4} \) Copy content Toggle raw display
$31$ \( (T - 8)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3848 T^{2} + 913936)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 2664 T^{2} + 944784)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2012 T^{2} + 376996)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 9900 T^{2} + 16402500)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 936 T^{2} + 11664)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 6624 T^{2} + 3504384)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 32 T - 1184)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 15068 T^{2} + 34975396)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 5040 T^{2} + 1166400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 7880 T^{2} + 1123600)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 56 T + 424)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 684 T^{2} + 324)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 3744 T^{2} + 186624)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 13832 T^{2} + 16289296)^{2} \) Copy content Toggle raw display
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