Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [176,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 176.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.9797644852\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{15}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.87298\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q-43.4758 q^{3} -389.919 q^{5} +1249.17 q^{7} -296.855 q^{9} -1331.00 q^{11} -3840.41 q^{13} +16952.1 q^{15} +24162.7 q^{17} -5458.47 q^{19} -54308.6 q^{21} +63973.2 q^{23} +73912.1 q^{25} +107988. q^{27} +178351. q^{29} +185129. q^{31} +57866.3 q^{33} -487075. q^{35} -409817. q^{37} +166965. q^{39} -675273. q^{41} -38903.5 q^{43} +115749. q^{45} -949397. q^{47} +736881. q^{49} -1.05049e6 q^{51} +294003. q^{53} +518983. q^{55} +237311. q^{57} +87803.7 q^{59} -2.78573e6 q^{61} -370822. q^{63} +1.49745e6 q^{65} -2.95364e6 q^{67} -2.78129e6 q^{69} +4.08380e6 q^{71} +1.95525e6 q^{73} -3.21339e6 q^{75} -1.66264e6 q^{77} +608158. q^{79} -4.04562e6 q^{81} -214613. q^{83} -9.42151e6 q^{85} -7.75394e6 q^{87} -8.30366e6 q^{89} -4.79732e6 q^{91} -8.04863e6 q^{93} +2.12836e6 q^{95} -1.38909e6 q^{97} +395114. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 470 q^{5} + 1228 q^{7} - 36 q^{9} - 2662 q^{11} + 344 q^{13} + 12990 q^{15} - 8468 q^{17} + 35280 q^{19} - 55356 q^{21} + 61486 q^{23} + 2200 q^{25} + 12690 q^{27} + 179040 q^{29} + 57166 q^{31}+ \cdots + 47916 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −43.4758 −0.929658 −0.464829 − 0.885400i \(-0.653884\pi\)
−0.464829 + 0.885400i \(0.653884\pi\)
\(4\) 0 0
\(5\) −389.919 −1.39502 −0.697509 − 0.716576i \(-0.745708\pi\)
−0.697509 + 0.716576i \(0.745708\pi\)
\(6\) 0 0
\(7\) 1249.17 1.37651 0.688253 − 0.725471i \(-0.258378\pi\)
0.688253 + 0.725471i \(0.258378\pi\)
\(8\) 0 0
\(9\) −296.855 −0.135736
\(10\) 0 0
\(11\) −1331.00 −0.301511
\(12\) 0 0
\(13\) −3840.41 −0.484815 −0.242407 − 0.970175i \(-0.577937\pi\)
−0.242407 + 0.970175i \(0.577937\pi\)
\(14\) 0 0
\(15\) 16952.1 1.29689
\(16\) 0 0
\(17\) 24162.7 1.19282 0.596409 − 0.802680i \(-0.296593\pi\)
0.596409 + 0.802680i \(0.296593\pi\)
\(18\) 0 0
\(19\) −5458.47 −0.182572 −0.0912859 − 0.995825i \(-0.529098\pi\)
−0.0912859 + 0.995825i \(0.529098\pi\)
\(20\) 0 0
\(21\) −54308.6 −1.27968
\(22\) 0 0
\(23\) 63973.2 1.09635 0.548177 − 0.836362i \(-0.315322\pi\)
0.548177 + 0.836362i \(0.315322\pi\)
\(24\) 0 0
\(25\) 73912.1 0.946075
\(26\) 0 0
\(27\) 107988. 1.05585
\(28\) 0 0
\(29\) 178351. 1.35794 0.678972 − 0.734164i \(-0.262426\pi\)
0.678972 + 0.734164i \(0.262426\pi\)
\(30\) 0 0
\(31\) 185129. 1.11611 0.558057 − 0.829803i \(-0.311547\pi\)
0.558057 + 0.829803i \(0.311547\pi\)
\(32\) 0 0
\(33\) 57866.3 0.280302
\(34\) 0 0
\(35\) −487075. −1.92025
\(36\) 0 0
\(37\) −409817. −1.33010 −0.665050 − 0.746799i \(-0.731590\pi\)
−0.665050 + 0.746799i \(0.731590\pi\)
\(38\) 0 0
\(39\) 166965. 0.450712
\(40\) 0 0
\(41\) −675273. −1.53016 −0.765078 − 0.643938i \(-0.777300\pi\)
−0.765078 + 0.643938i \(0.777300\pi\)
\(42\) 0 0
\(43\) −38903.5 −0.0746189 −0.0373094 − 0.999304i \(-0.511879\pi\)
−0.0373094 + 0.999304i \(0.511879\pi\)
\(44\) 0 0
\(45\) 115749. 0.189354
\(46\) 0 0
\(47\) −949397. −1.33385 −0.666923 − 0.745127i \(-0.732389\pi\)
−0.666923 + 0.745127i \(0.732389\pi\)
\(48\) 0 0
\(49\) 736881. 0.894769
\(50\) 0 0
\(51\) −1.05049e6 −1.10891
\(52\) 0 0
\(53\) 294003. 0.271260 0.135630 − 0.990760i \(-0.456694\pi\)
0.135630 + 0.990760i \(0.456694\pi\)
\(54\) 0 0
\(55\) 518983. 0.420614
\(56\) 0 0
\(57\) 237311. 0.169729
\(58\) 0 0
\(59\) 87803.7 0.0556584 0.0278292 − 0.999613i \(-0.491141\pi\)
0.0278292 + 0.999613i \(0.491141\pi\)
\(60\) 0 0
\(61\) −2.78573e6 −1.57139 −0.785696 − 0.618613i \(-0.787695\pi\)
−0.785696 + 0.618613i \(0.787695\pi\)
\(62\) 0 0
\(63\) −370822. −0.186842
\(64\) 0 0
\(65\) 1.49745e6 0.676325
\(66\) 0 0
\(67\) −2.95364e6 −1.19976 −0.599882 − 0.800089i \(-0.704786\pi\)
−0.599882 + 0.800089i \(0.704786\pi\)
\(68\) 0 0
\(69\) −2.78129e6 −1.01923
\(70\) 0 0
\(71\) 4.08380e6 1.35413 0.677065 − 0.735923i \(-0.263251\pi\)
0.677065 + 0.735923i \(0.263251\pi\)
\(72\) 0 0
\(73\) 1.95525e6 0.588263 0.294132 − 0.955765i \(-0.404970\pi\)
0.294132 + 0.955765i \(0.404970\pi\)
\(74\) 0 0
\(75\) −3.21339e6 −0.879526
\(76\) 0 0
\(77\) −1.66264e6 −0.415032
\(78\) 0 0
\(79\) 608158. 0.138778 0.0693891 − 0.997590i \(-0.477895\pi\)
0.0693891 + 0.997590i \(0.477895\pi\)
\(80\) 0 0
\(81\) −4.04562e6 −0.845840
\(82\) 0 0
\(83\) −214613. −0.0411986 −0.0205993 − 0.999788i \(-0.506557\pi\)
−0.0205993 + 0.999788i \(0.506557\pi\)
\(84\) 0 0
\(85\) −9.42151e6 −1.66400
\(86\) 0 0
\(87\) −7.75394e6 −1.26242
\(88\) 0 0
\(89\) −8.30366e6 −1.24855 −0.624273 − 0.781206i \(-0.714605\pi\)
−0.624273 + 0.781206i \(0.714605\pi\)
\(90\) 0 0
\(91\) −4.79732e6 −0.667351
\(92\) 0 0
\(93\) −8.04863e6 −1.03760
\(94\) 0 0
\(95\) 2.12836e6 0.254691
\(96\) 0 0
\(97\) −1.38909e6 −0.154536 −0.0772678 − 0.997010i \(-0.524620\pi\)
−0.0772678 + 0.997010i \(0.524620\pi\)
\(98\) 0 0
\(99\) 395114. 0.0409260
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 176.8.a.d.1.1 2
4.3 odd 2 11.8.a.a.1.1 ✓ 2
12.11 even 2 99.8.a.c.1.2 2
20.19 odd 2 275.8.a.a.1.2 2
28.27 even 2 539.8.a.a.1.1 2
44.43 even 2 121.8.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.8.a.a.1.1 ✓ 2 4.3 odd 2
99.8.a.c.1.2 2 12.11 even 2
121.8.a.b.1.2 2 44.43 even 2
176.8.a.d.1.1 2 1.1 even 1 trivial
275.8.a.a.1.2 2 20.19 odd 2
539.8.a.a.1.1 2 28.27 even 2