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Properties

Label 176.2.e.b.175.2
Level $176$
Weight $2$
Character 176.175
Analytic conductor $1.405$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [176,2,Mod(175,176)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("176.175"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 176.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.40536707557\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 175.2
Root \(1.68614 + 0.396143i\) of defining polynomial
Character \(\chi\) \(=\) 176.175
Dual form 176.2.e.b.175.3

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.792287i q^{3} +1.37228 q^{5} +2.37228 q^{9} -3.31662i q^{11} -1.08724i q^{15} +6.13592i q^{23} -3.11684 q^{25} -4.25639i q^{27} +9.30506i q^{31} -2.62772 q^{33} -12.1168 q^{37} +3.25544 q^{45} +6.63325i q^{47} -7.00000 q^{49} +6.00000 q^{53} -4.55134i q^{55} -14.6487i q^{59} +16.2333i q^{67} +4.86141 q^{69} -10.8896i q^{71} +2.46943i q^{75} +3.74456 q^{81} +18.8614 q^{89} +7.37228 q^{93} -0.116844 q^{97} -7.86797i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} - 2 q^{9} + 22 q^{25} - 22 q^{33} - 14 q^{37} + 36 q^{45} - 28 q^{49} + 24 q^{53} - 38 q^{69} - 8 q^{81} + 18 q^{89} + 18 q^{93} + 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/176\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(133\) \(145\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 0.792287i − 0.457427i −0.973494 − 0.228714i \(-0.926548\pi\)
0.973494 − 0.228714i \(-0.0734519\pi\)
\(4\) 0 0
\(5\) 1.37228 0.613703 0.306851 − 0.951757i \(-0.400725\pi\)
0.306851 + 0.951757i \(0.400725\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 2.37228 0.790760
\(10\) 0 0
\(11\) − 3.31662i − 1.00000i
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) − 1.08724i − 0.280724i
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.13592i 1.27943i 0.768613 + 0.639713i \(0.220947\pi\)
−0.768613 + 0.639713i \(0.779053\pi\)
\(24\) 0 0
\(25\) −3.11684 −0.623369
\(26\) 0 0
\(27\) − 4.25639i − 0.819142i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 9.30506i 1.67124i 0.549309 + 0.835619i \(0.314891\pi\)
−0.549309 + 0.835619i \(0.685109\pi\)
\(32\) 0 0
\(33\) −2.62772 −0.457427
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −12.1168 −1.99200 −0.995998 − 0.0893706i \(-0.971514\pi\)
−0.995998 + 0.0893706i \(0.971514\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 3.25544 0.485292
\(46\) 0 0
\(47\) 6.63325i 0.967559i 0.875190 + 0.483779i \(0.160736\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) − 4.55134i − 0.613703i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 14.6487i − 1.90710i −0.301239 − 0.953549i \(-0.597400\pi\)
0.301239 − 0.953549i \(-0.402600\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 16.2333i 1.98321i 0.129307 + 0.991605i \(0.458725\pi\)
−0.129307 + 0.991605i \(0.541275\pi\)
\(68\) 0 0
\(69\) 4.86141 0.585245
\(70\) 0 0
\(71\) − 10.8896i − 1.29236i −0.763184 − 0.646181i \(-0.776365\pi\)
0.763184 − 0.646181i \(-0.223635\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 2.46943i 0.285146i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 3.74456 0.416063
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.8614 1.99931 0.999653 − 0.0263586i \(-0.00839118\pi\)
0.999653 + 0.0263586i \(0.00839118\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 7.37228 0.764470
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.116844 −0.0118637 −0.00593185 − 0.999982i \(-0.501888\pi\)
−0.00593185 + 0.999982i \(0.501888\pi\)
\(98\) 0 0
\(99\) − 7.86797i − 0.790760i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 176.2.e.b.175.2 ✓ 4
3.2 odd 2 1584.2.o.e.703.2 4
4.3 odd 2 inner 176.2.e.b.175.3 yes 4
8.3 odd 2 704.2.e.c.703.2 4
8.5 even 2 704.2.e.c.703.3 4
11.10 odd 2 CM 176.2.e.b.175.2 ✓ 4
12.11 even 2 1584.2.o.e.703.1 4
16.3 odd 4 2816.2.g.c.1407.3 8
16.5 even 4 2816.2.g.c.1407.4 8
16.11 odd 4 2816.2.g.c.1407.6 8
16.13 even 4 2816.2.g.c.1407.5 8
33.32 even 2 1584.2.o.e.703.2 4
44.43 even 2 inner 176.2.e.b.175.3 yes 4
88.21 odd 2 704.2.e.c.703.3 4
88.43 even 2 704.2.e.c.703.2 4
132.131 odd 2 1584.2.o.e.703.1 4
176.21 odd 4 2816.2.g.c.1407.4 8
176.43 even 4 2816.2.g.c.1407.6 8
176.109 odd 4 2816.2.g.c.1407.5 8
176.131 even 4 2816.2.g.c.1407.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
176.2.e.b.175.2 ✓ 4 1.1 even 1 trivial
176.2.e.b.175.2 ✓ 4 11.10 odd 2 CM
176.2.e.b.175.3 yes 4 4.3 odd 2 inner
176.2.e.b.175.3 yes 4 44.43 even 2 inner
704.2.e.c.703.2 4 8.3 odd 2
704.2.e.c.703.2 4 88.43 even 2
704.2.e.c.703.3 4 8.5 even 2
704.2.e.c.703.3 4 88.21 odd 2
1584.2.o.e.703.1 4 12.11 even 2
1584.2.o.e.703.1 4 132.131 odd 2
1584.2.o.e.703.2 4 3.2 odd 2
1584.2.o.e.703.2 4 33.32 even 2
2816.2.g.c.1407.3 8 16.3 odd 4
2816.2.g.c.1407.3 8 176.131 even 4
2816.2.g.c.1407.4 8 16.5 even 4
2816.2.g.c.1407.4 8 176.21 odd 4
2816.2.g.c.1407.5 8 16.13 even 4
2816.2.g.c.1407.5 8 176.109 odd 4
2816.2.g.c.1407.6 8 16.11 odd 4
2816.2.g.c.1407.6 8 176.43 even 4