Properties

Label 176.4.a.a.1.1
Level $176$
Weight $4$
Character 176.1
Self dual yes
Analytic conductor $10.384$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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] = [176,4,Mod(1,176)] 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("176.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 176.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-7,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.3843361610\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 176.1

$q$-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)
 
\(f(q)\) \(=\) \(q-7.00000 q^{3} +9.00000 q^{5} -2.00000 q^{7} +22.0000 q^{9} +11.0000 q^{11} -63.0000 q^{15} -38.0000 q^{17} -44.0000 q^{19} +14.0000 q^{21} -175.000 q^{23} -44.0000 q^{25} +35.0000 q^{27} -264.000 q^{29} -159.000 q^{31} -77.0000 q^{33} -18.0000 q^{35} -173.000 q^{37} -220.000 q^{41} +542.000 q^{43} +198.000 q^{45} +264.000 q^{47} -339.000 q^{49} +266.000 q^{51} +682.000 q^{53} +99.0000 q^{55} +308.000 q^{57} -421.000 q^{59} +308.000 q^{61} -44.0000 q^{63} -177.000 q^{67} +1225.00 q^{69} -365.000 q^{71} -528.000 q^{73} +308.000 q^{75} -22.0000 q^{77} -686.000 q^{79} -839.000 q^{81} -698.000 q^{83} -342.000 q^{85} +1848.00 q^{87} +967.000 q^{89} +1113.00 q^{93} -396.000 q^{95} -1127.00 q^{97} +242.000 q^{99} +O(q^{100})\)

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
\(3\) −7.00000 −1.34715 −0.673575 − 0.739119i \(-0.735242\pi\)
−0.673575 + 0.739119i \(0.735242\pi\)
\(4\) 0 0
\(5\) 9.00000 0.804984 0.402492 − 0.915423i \(-0.368144\pi\)
0.402492 + 0.915423i \(0.368144\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.107990 −0.0539949 − 0.998541i \(-0.517195\pi\)
−0.0539949 + 0.998541i \(0.517195\pi\)
\(8\) 0 0
\(9\) 22.0000 0.814815
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −63.0000 −1.08444
\(16\) 0 0
\(17\) −38.0000 −0.542138 −0.271069 − 0.962560i \(-0.587377\pi\)
−0.271069 + 0.962560i \(0.587377\pi\)
\(18\) 0 0
\(19\) −44.0000 −0.531279 −0.265639 − 0.964072i \(-0.585583\pi\)
−0.265639 + 0.964072i \(0.585583\pi\)
\(20\) 0 0
\(21\) 14.0000 0.145479
\(22\) 0 0
\(23\) −175.000 −1.58652 −0.793261 − 0.608881i \(-0.791619\pi\)
−0.793261 + 0.608881i \(0.791619\pi\)
\(24\) 0 0
\(25\) −44.0000 −0.352000
\(26\) 0 0
\(27\) 35.0000 0.249472
\(28\) 0 0
\(29\) −264.000 −1.69047 −0.845234 − 0.534396i \(-0.820539\pi\)
−0.845234 + 0.534396i \(0.820539\pi\)
\(30\) 0 0
\(31\) −159.000 −0.921201 −0.460601 − 0.887607i \(-0.652366\pi\)
−0.460601 + 0.887607i \(0.652366\pi\)
\(32\) 0 0
\(33\) −77.0000 −0.406181
\(34\) 0 0
\(35\) −18.0000 −0.0869302
\(36\) 0 0
\(37\) −173.000 −0.768676 −0.384338 − 0.923192i \(-0.625570\pi\)
−0.384338 + 0.923192i \(0.625570\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −220.000 −0.838006 −0.419003 − 0.907985i \(-0.637620\pi\)
−0.419003 + 0.907985i \(0.637620\pi\)
\(42\) 0 0
\(43\) 542.000 1.92219 0.961096 − 0.276216i \(-0.0890805\pi\)
0.961096 + 0.276216i \(0.0890805\pi\)
\(44\) 0 0
\(45\) 198.000 0.655913
\(46\) 0 0
\(47\) 264.000 0.819327 0.409663 − 0.912237i \(-0.365646\pi\)
0.409663 + 0.912237i \(0.365646\pi\)
\(48\) 0 0
\(49\) −339.000 −0.988338
\(50\) 0 0
\(51\) 266.000 0.730342
\(52\) 0 0
\(53\) 682.000 1.76755 0.883773 − 0.467916i \(-0.154995\pi\)
0.883773 + 0.467916i \(0.154995\pi\)
\(54\) 0 0
\(55\) 99.0000 0.242712
\(56\) 0 0
\(57\) 308.000 0.715712
\(58\) 0 0
\(59\) −421.000 −0.928975 −0.464488 − 0.885580i \(-0.653761\pi\)
−0.464488 + 0.885580i \(0.653761\pi\)
\(60\) 0 0
\(61\) 308.000 0.646481 0.323241 − 0.946317i \(-0.395228\pi\)
0.323241 + 0.946317i \(0.395228\pi\)
\(62\) 0 0
\(63\) −44.0000 −0.0879917
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −177.000 −0.322746 −0.161373 − 0.986893i \(-0.551592\pi\)
−0.161373 + 0.986893i \(0.551592\pi\)
\(68\) 0 0
\(69\) 1225.00 2.13729
\(70\) 0 0
\(71\) −365.000 −0.610106 −0.305053 − 0.952335i \(-0.598674\pi\)
−0.305053 + 0.952335i \(0.598674\pi\)
\(72\) 0 0
\(73\) −528.000 −0.846544 −0.423272 − 0.906003i \(-0.639119\pi\)
−0.423272 + 0.906003i \(0.639119\pi\)
\(74\) 0 0
\(75\) 308.000 0.474197
\(76\) 0 0
\(77\) −22.0000 −0.0325602
\(78\) 0 0
\(79\) −686.000 −0.976975 −0.488488 − 0.872571i \(-0.662451\pi\)
−0.488488 + 0.872571i \(0.662451\pi\)
\(80\) 0 0
\(81\) −839.000 −1.15089
\(82\) 0 0
\(83\) −698.000 −0.923078 −0.461539 − 0.887120i \(-0.652703\pi\)
−0.461539 + 0.887120i \(0.652703\pi\)
\(84\) 0 0
\(85\) −342.000 −0.436413
\(86\) 0 0
\(87\) 1848.00 2.27731
\(88\) 0 0
\(89\) 967.000 1.15171 0.575853 − 0.817553i \(-0.304670\pi\)
0.575853 + 0.817553i \(0.304670\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1113.00 1.24100
\(94\) 0 0
\(95\) −396.000 −0.427671
\(96\) 0 0
\(97\) −1127.00 −1.17969 −0.589843 − 0.807518i \(-0.700810\pi\)
−0.589843 + 0.807518i \(0.700810\pi\)
\(98\) 0 0
\(99\) 242.000 0.245676
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 176.4.a.a.1.1 1
3.2 odd 2 1584.4.a.g.1.1 1
4.3 odd 2 88.4.a.b.1.1 ✓ 1
8.3 odd 2 704.4.a.a.1.1 1
8.5 even 2 704.4.a.k.1.1 1
11.10 odd 2 1936.4.a.b.1.1 1
12.11 even 2 792.4.a.b.1.1 1
20.19 odd 2 2200.4.a.a.1.1 1
44.43 even 2 968.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
88.4.a.b.1.1 ✓ 1 4.3 odd 2
176.4.a.a.1.1 1 1.1 even 1 trivial
704.4.a.a.1.1 1 8.3 odd 2
704.4.a.k.1.1 1 8.5 even 2
792.4.a.b.1.1 1 12.11 even 2
968.4.a.e.1.1 1 44.43 even 2
1584.4.a.g.1.1 1 3.2 odd 2
1936.4.a.b.1.1 1 11.10 odd 2
2200.4.a.a.1.1 1 20.19 odd 2