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Properties

Label 180.2.h.a.179.1
Level $180$
Weight $2$
Character 180.179
Analytic conductor $1.437$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

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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] = [180,2,Mod(179,180)] 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("180.179"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 180.h (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: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 179.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 180.179
Dual form 180.2.h.a.179.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} +2.00000 q^{4} +(0.707107 - 2.12132i) q^{5} -2.82843 q^{8} +(-1.00000 + 3.00000i) q^{10} -6.00000i q^{13} +4.00000 q^{16} +7.07107 q^{17} +(1.41421 - 4.24264i) q^{20} +(-4.00000 - 3.00000i) q^{25} +8.48528i q^{26} -4.24264i q^{29} -5.65685 q^{32} -10.0000 q^{34} +12.0000i q^{37} +(-2.00000 + 6.00000i) q^{40} +12.7279i q^{41} -7.00000 q^{49} +(5.65685 + 4.24264i) q^{50} -12.0000i q^{52} -7.07107 q^{53} +6.00000i q^{58} +10.0000 q^{61} +8.00000 q^{64} +(-12.7279 - 4.24264i) q^{65} +14.1421 q^{68} +6.00000i q^{73} -16.9706i q^{74} +(2.82843 - 8.48528i) q^{80} -18.0000i q^{82} +(5.00000 - 15.0000i) q^{85} +4.24264i q^{89} +18.0000i q^{97} +9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} - 4 q^{10} + 16 q^{16} - 16 q^{25} - 40 q^{34} - 8 q^{40} - 28 q^{49} + 40 q^{61} + 32 q^{64} + 20 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −1.00000
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 0.707107 − 2.12132i 0.316228 − 0.948683i
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −2.82843 −1.00000
\(9\) 0 0
\(10\) −1.00000 + 3.00000i −0.316228 + 0.948683i
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 6.00000i − 1.66410i −0.554700 − 0.832050i \(-0.687167\pi\)
0.554700 − 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 7.07107 1.71499 0.857493 − 0.514496i \(-0.172021\pi\)
0.857493 + 0.514496i \(0.172021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.41421 − 4.24264i 0.316228 − 0.948683i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −4.00000 − 3.00000i −0.800000 − 0.600000i
\(26\) 8.48528i 1.66410i
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.24264i − 0.787839i −0.919145 − 0.393919i \(-0.871119\pi\)
0.919145 − 0.393919i \(-0.128881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −5.65685 −1.00000
\(33\) 0 0
\(34\) −10.0000 −1.71499
\(35\) 0 0
\(36\) 0 0
\(37\) 12.0000i 1.97279i 0.164399 + 0.986394i \(0.447432\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −2.00000 + 6.00000i −0.316228 + 0.948683i
\(41\) 12.7279i 1.98777i 0.110432 + 0.993884i \(0.464777\pi\)
−0.110432 + 0.993884i \(0.535223\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 5.65685 + 4.24264i 0.800000 + 0.600000i
\(51\) 0 0
\(52\) − 12.0000i − 1.66410i
\(53\) −7.07107 −0.971286 −0.485643 − 0.874157i \(-0.661414\pi\)
−0.485643 + 0.874157i \(0.661414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 6.00000i 0.787839i
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) −12.7279 − 4.24264i −1.57870 − 0.526235i
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 14.1421 1.71499
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) − 16.9706i − 1.97279i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 2.82843 − 8.48528i 0.316228 − 0.948683i
\(81\) 0 0
\(82\) − 18.0000i − 1.98777i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 5.00000 − 15.0000i 0.542326 − 1.62698i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.24264i 0.449719i 0.974391 + 0.224860i \(0.0721923\pi\)
−0.974391 + 0.224860i \(0.927808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.0000i 1.82762i 0.406138 + 0.913812i \(0.366875\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 9.89949 1.00000
\(99\) 0 0
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 180.2.h.a.179.1 ✓ 4
3.2 odd 2 inner 180.2.h.a.179.4 yes 4
4.3 odd 2 CM 180.2.h.a.179.1 ✓ 4
5.2 odd 4 900.2.e.a.251.1 2
5.3 odd 4 900.2.e.c.251.2 2
5.4 even 2 inner 180.2.h.a.179.3 yes 4
8.3 odd 2 2880.2.o.a.2879.2 4
8.5 even 2 2880.2.o.a.2879.2 4
12.11 even 2 inner 180.2.h.a.179.4 yes 4
15.2 even 4 900.2.e.a.251.2 2
15.8 even 4 900.2.e.c.251.1 2
15.14 odd 2 inner 180.2.h.a.179.2 yes 4
20.3 even 4 900.2.e.c.251.2 2
20.7 even 4 900.2.e.a.251.1 2
20.19 odd 2 inner 180.2.h.a.179.3 yes 4
24.5 odd 2 2880.2.o.a.2879.3 4
24.11 even 2 2880.2.o.a.2879.3 4
40.19 odd 2 2880.2.o.a.2879.4 4
40.29 even 2 2880.2.o.a.2879.4 4
60.23 odd 4 900.2.e.c.251.1 2
60.47 odd 4 900.2.e.a.251.2 2
60.59 even 2 inner 180.2.h.a.179.2 yes 4
120.29 odd 2 2880.2.o.a.2879.1 4
120.59 even 2 2880.2.o.a.2879.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.2.h.a.179.1 ✓ 4 1.1 even 1 trivial
180.2.h.a.179.1 ✓ 4 4.3 odd 2 CM
180.2.h.a.179.2 yes 4 15.14 odd 2 inner
180.2.h.a.179.2 yes 4 60.59 even 2 inner
180.2.h.a.179.3 yes 4 5.4 even 2 inner
180.2.h.a.179.3 yes 4 20.19 odd 2 inner
180.2.h.a.179.4 yes 4 3.2 odd 2 inner
180.2.h.a.179.4 yes 4 12.11 even 2 inner
900.2.e.a.251.1 2 5.2 odd 4
900.2.e.a.251.1 2 20.7 even 4
900.2.e.a.251.2 2 15.2 even 4
900.2.e.a.251.2 2 60.47 odd 4
900.2.e.c.251.1 2 15.8 even 4
900.2.e.c.251.1 2 60.23 odd 4
900.2.e.c.251.2 2 5.3 odd 4
900.2.e.c.251.2 2 20.3 even 4
2880.2.o.a.2879.1 4 120.29 odd 2
2880.2.o.a.2879.1 4 120.59 even 2
2880.2.o.a.2879.2 4 8.3 odd 2
2880.2.o.a.2879.2 4 8.5 even 2
2880.2.o.a.2879.3 4 24.5 odd 2
2880.2.o.a.2879.3 4 24.11 even 2
2880.2.o.a.2879.4 4 40.19 odd 2
2880.2.o.a.2879.4 4 40.29 even 2