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Properties

Label 150.7.b.a
Level $150$
Weight $7$
Character orbit 150.b
Analytic conductor $34.508$
Analytic rank $0$
Dimension $4$
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] = [150,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.149");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(34.5081125430\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 6)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - 3 \beta_{3} - 21 \beta_1) q^{3} + 32 q^{4} + ( - 21 \beta_{2} - 96) q^{6} + 2 \beta_1 q^{7} + 32 \beta_{3} q^{8} + (126 \beta_{2} - 153) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - 3 \beta_{3} - 21 \beta_1) q^{3} + 32 q^{4} + ( - 21 \beta_{2} - 96) q^{6} + 2 \beta_1 q^{7} + 32 \beta_{3} q^{8} + (126 \beta_{2} - 153) q^{9} + 6 \beta_{2} q^{11} + ( - 96 \beta_{3} - 672 \beta_1) q^{12} + 2950 \beta_1 q^{13} + 2 \beta_{2} q^{14} + 1024 q^{16} + 792 \beta_{3} q^{17} + ( - 153 \beta_{3} + 4032 \beta_1) q^{18} - 5258 q^{19} + ( - 6 \beta_{2} + 42) q^{21} + 192 \beta_1 q^{22} + 1812 \beta_{3} q^{23} + ( - 672 \beta_{2} - 3072) q^{24} + 2950 \beta_{2} q^{26} + (3105 \beta_{3} - 8883 \beta_1) q^{27} + 64 \beta_1 q^{28} + 390 \beta_{2} q^{29} + 22898 q^{31} + 1024 \beta_{3} q^{32} + (126 \beta_{3} - 576 \beta_1) q^{33} + 25344 q^{34} + (4032 \beta_{2} - 4896) q^{36} + 34058 \beta_1 q^{37} - 5258 \beta_{3} q^{38} + ( - 8850 \beta_{2} + 61950) q^{39} + 2964 \beta_{2} q^{41} + (42 \beta_{3} - 192 \beta_1) q^{42} + 6406 \beta_1 q^{43} + 192 \beta_{2} q^{44} + 57984 q^{46} + 31800 \beta_{3} q^{47} + ( - 3072 \beta_{3} - 21504 \beta_1) q^{48} + 117645 q^{49} + ( - 16632 \beta_{2} - 76032) q^{51} + 94400 \beta_1 q^{52} + 34038 \beta_{3} q^{53} + ( - 8883 \beta_{2} + 99360) q^{54} + 64 \beta_{2} q^{56} + (15774 \beta_{3} + 110418 \beta_1) q^{57} + 12480 \beta_1 q^{58} - 57774 \beta_{2} q^{59} - 62566 q^{61} + 22898 \beta_{3} q^{62} + ( - 252 \beta_{3} - 306 \beta_1) q^{63} + 32768 q^{64} + ( - 576 \beta_{2} + 4032) q^{66} + 438698 \beta_1 q^{67} + 25344 \beta_{3} q^{68} + ( - 38052 \beta_{2} - 173952) q^{69} + 12060 \beta_{2} q^{71} + ( - 4896 \beta_{3} + 129024 \beta_1) q^{72} + 730510 \beta_1 q^{73} + 34058 \beta_{2} q^{74} - 168256 q^{76} - 12 \beta_{3} q^{77} + (61950 \beta_{3} - 283200 \beta_1) q^{78} - 340562 q^{79} + ( - 38556 \beta_{2} - 484623) q^{81} + 94848 \beta_1 q^{82} - 87726 \beta_{3} q^{83} + ( - 192 \beta_{2} + 1344) q^{84} + 6406 \beta_{2} q^{86} + (8190 \beta_{3} - 37440 \beta_1) q^{87} + 6144 \beta_1 q^{88} - 68364 \beta_{2} q^{89} - 5900 q^{91} + 57984 \beta_{3} q^{92} + ( - 68694 \beta_{3} - 480858 \beta_1) q^{93} + 1017600 q^{94} + ( - 21504 \beta_{2} - 98304) q^{96} - 281086 \beta_1 q^{97} + 117645 \beta_{3} q^{98} + ( - 918 \beta_{2} - 24192) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 128 q^{4} - 384 q^{6} - 612 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 128 q^{4} - 384 q^{6} - 612 q^{9} + 4096 q^{16} - 21032 q^{19} + 168 q^{21} - 12288 q^{24} + 91592 q^{31} + 101376 q^{34} - 19584 q^{36} + 247800 q^{39} + 231936 q^{46} + 470580 q^{49} - 304128 q^{51} + 397440 q^{54} - 250264 q^{61} + 131072 q^{64} + 16128 q^{66} - 695808 q^{69} - 673024 q^{76} - 1362248 q^{79} - 1938492 q^{81} + 5376 q^{84} - 23600 q^{91} + 4070400 q^{94} - 393216 q^{96} - 96768 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\zeta_{8}^{3} + 4\zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\zeta_{8}^{3} + 4\zeta_{8} \) Copy content Toggle raw display

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

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
−0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 − 0.707107i
−5.65685 16.9706 − 21.0000i 32.0000 0 −96.0000 + 118.794i 2.00000i −181.019 −153.000 − 712.764i 0
149.2 −5.65685 16.9706 + 21.0000i 32.0000 0 −96.0000 − 118.794i − 2.00000i −181.019 −153.000 + 712.764i 0
149.3 5.65685 −16.9706 − 21.0000i 32.0000 0 −96.0000 − 118.794i 2.00000i 181.019 −153.000 + 712.764i 0
149.4 5.65685 −16.9706 + 21.0000i 32.0000 0 −96.0000 + 118.794i − 2.00000i 181.019 −153.000 − 712.764i 0
\(n\): e.g. 2-40 or 990-1000
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 150.7.b.a 4
3.b odd 2 1 inner 150.7.b.a 4
5.b even 2 1 inner 150.7.b.a 4
5.c odd 4 1 6.7.b.a ✓ 2
5.c odd 4 1 150.7.d.a 2
15.d odd 2 1 inner 150.7.b.a 4
15.e even 4 1 6.7.b.a ✓ 2
15.e even 4 1 150.7.d.a 2
20.e even 4 1 48.7.e.b 2
35.f even 4 1 294.7.b.a 2
40.i odd 4 1 192.7.e.c 2
40.k even 4 1 192.7.e.f 2
45.k odd 12 2 162.7.d.b 4
45.l even 12 2 162.7.d.b 4
60.l odd 4 1 48.7.e.b 2
105.k odd 4 1 294.7.b.a 2
120.q odd 4 1 192.7.e.f 2
120.w even 4 1 192.7.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.7.b.a ✓ 2 5.c odd 4 1
6.7.b.a ✓ 2 15.e even 4 1
48.7.e.b 2 20.e even 4 1
48.7.e.b 2 60.l odd 4 1
150.7.b.a 4 1.a even 1 1 trivial
150.7.b.a 4 3.b odd 2 1 inner
150.7.b.a 4 5.b even 2 1 inner
150.7.b.a 4 15.d odd 2 1 inner
150.7.d.a 2 5.c odd 4 1
150.7.d.a 2 15.e even 4 1
162.7.d.b 4 45.k odd 12 2
162.7.d.b 4 45.l even 12 2
192.7.e.c 2 40.i odd 4 1
192.7.e.c 2 120.w even 4 1
192.7.e.f 2 40.k even 4 1
192.7.e.f 2 120.q odd 4 1
294.7.b.a 2 35.f even 4 1
294.7.b.a 2 105.k odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 306 T^{2} + 531441 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1152)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 8702500)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 20072448)^{2} \) Copy content Toggle raw display
$19$ \( (T + 5258)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 105067008)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4867200)^{2} \) Copy content Toggle raw display
$31$ \( (T - 22898)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1159947364)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 281129472)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 41036836)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 32359680000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 37074734208)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 106810722432)^{2} \) Copy content Toggle raw display
$61$ \( (T + 62566)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 192455935204)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4654195200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 533644860100)^{2} \) Copy content Toggle raw display
$79$ \( (T + 340562)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 246267234432)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 149556367872)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 79009339396)^{2} \) Copy content Toggle raw display
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