Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [121,8,Mod(1,121)] 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("121.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 121.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(37.7985880836\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.87298\) of defining polynomial
Character \(\chi\) \(=\) 121.1

$q$-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)
 
\(f(q)\) \(=\) \(q+11.7460 q^{2} +43.4758 q^{3} +9.96773 q^{4} -389.919 q^{5} +510.665 q^{6} +1249.17 q^{7} -1386.40 q^{8} -296.855 q^{9} -4579.98 q^{10} +433.355 q^{12} +3840.41 q^{13} +14672.7 q^{14} -16952.1 q^{15} -17560.5 q^{16} -24162.7 q^{17} -3486.85 q^{18} -5458.47 q^{19} -3886.61 q^{20} +54308.6 q^{21} -63973.2 q^{23} -60275.0 q^{24} +73912.1 q^{25} +45109.3 q^{26} -107988. q^{27} +12451.4 q^{28} -178351. q^{29} -199118. q^{30} -185129. q^{31} -28805.6 q^{32} -283814. q^{34} -487075. q^{35} -2958.97 q^{36} -409817. q^{37} -64115.0 q^{38} +166965. q^{39} +540585. q^{40} +675273. q^{41} +637907. q^{42} -38903.5 q^{43} +115749. q^{45} -751427. q^{46} +949397. q^{47} -763457. q^{48} +736881. q^{49} +868169. q^{50} -1.05049e6 q^{51} +38280.2 q^{52} +294003. q^{53} -1.26842e6 q^{54} -1.73185e6 q^{56} -237311. q^{57} -2.09490e6 q^{58} -87803.7 q^{59} -168974. q^{60} +2.78573e6 q^{61} -2.17452e6 q^{62} -370822. q^{63} +1.90940e6 q^{64} -1.49745e6 q^{65} +2.95364e6 q^{67} -240847. q^{68} -2.78129e6 q^{69} -5.72117e6 q^{70} -4.08380e6 q^{71} +411560. q^{72} -1.95525e6 q^{73} -4.81370e6 q^{74} +3.21339e6 q^{75} -54408.6 q^{76} +1.96116e6 q^{78} +608158. q^{79} +6.84718e6 q^{80} -4.04562e6 q^{81} +7.93173e6 q^{82} -214613. q^{83} +541334. q^{84} +9.42151e6 q^{85} -456959. q^{86} -7.75394e6 q^{87} -8.30366e6 q^{89} +1.35959e6 q^{90} +4.79732e6 q^{91} -637668. q^{92} -8.04863e6 q^{93} +1.11516e7 q^{94} +2.12836e6 q^{95} -1.25235e6 q^{96} -1.38909e6 q^{97} +8.65538e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} - 6 q^{3} - 104 q^{4} - 470 q^{5} + 696 q^{6} + 1228 q^{7} - 480 q^{8} - 36 q^{9} - 4280 q^{10} + 6072 q^{12} - 344 q^{13} + 14752 q^{14} - 12990 q^{15} - 6368 q^{16} + 8468 q^{17} - 4464 q^{18}+ \cdots + 11738664 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 11.7460 1.03821 0.519103 − 0.854712i \(-0.326266\pi\)
0.519103 + 0.854712i \(0.326266\pi\)
\(3\) 43.4758 0.929658 0.464829 − 0.885400i \(-0.346116\pi\)
0.464829 + 0.885400i \(0.346116\pi\)
\(4\) 9.96773 0.0778729
\(5\) −389.919 −1.39502 −0.697509 − 0.716576i \(-0.745708\pi\)
−0.697509 + 0.716576i \(0.745708\pi\)
\(6\) 510.665 0.965177
\(7\) 1249.17 1.37651 0.688253 − 0.725471i \(-0.258378\pi\)
0.688253 + 0.725471i \(0.258378\pi\)
\(8\) −1386.40 −0.957358
\(9\) −296.855 −0.135736
\(10\) −4579.98 −1.44832
\(11\) 0 0
\(12\) 433.355 0.0723952
\(13\) 3840.41 0.484815 0.242407 − 0.970175i \(-0.422063\pi\)
0.242407 + 0.970175i \(0.422063\pi\)
\(14\) 14672.7 1.42910
\(15\) −16952.1 −1.29689
\(16\) −17560.5 −1.07181
\(17\) −24162.7 −1.19282 −0.596409 − 0.802680i \(-0.703407\pi\)
−0.596409 + 0.802680i \(0.703407\pi\)
\(18\) −3486.85 −0.140922
\(19\) −5458.47 −0.182572 −0.0912859 − 0.995825i \(-0.529098\pi\)
−0.0912859 + 0.995825i \(0.529098\pi\)
\(20\) −3886.61 −0.108634
\(21\) 54308.6 1.27968
\(22\) 0 0
\(23\) −63973.2 −1.09635 −0.548177 − 0.836362i \(-0.684678\pi\)
−0.548177 + 0.836362i \(0.684678\pi\)
\(24\) −60275.0 −0.890016
\(25\) 73912.1 0.946075
\(26\) 45109.3 0.503338
\(27\) −107988. −1.05585
\(28\) 12451.4 0.107193
\(29\) −178351. −1.35794 −0.678972 − 0.734164i \(-0.737574\pi\)
−0.678972 + 0.734164i \(0.737574\pi\)
\(30\) −199118. −1.34644
\(31\) −185129. −1.11611 −0.558057 − 0.829803i \(-0.688453\pi\)
−0.558057 + 0.829803i \(0.688453\pi\)
\(32\) −28805.6 −0.155400
\(33\) 0 0
\(34\) −283814. −1.23839
\(35\) −487075. −1.92025
\(36\) −2958.97 −0.0105702
\(37\) −409817. −1.33010 −0.665050 − 0.746799i \(-0.731590\pi\)
−0.665050 + 0.746799i \(0.731590\pi\)
\(38\) −64115.0 −0.189547
\(39\) 166965. 0.450712
\(40\) 540585. 1.33553
\(41\) 675273. 1.53016 0.765078 − 0.643938i \(-0.222700\pi\)
0.765078 + 0.643938i \(0.222700\pi\)
\(42\) 637907. 1.32857
\(43\) −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
\(46\) −751427. −1.13824
\(47\) 949397. 1.33385 0.666923 − 0.745127i \(-0.267611\pi\)
0.666923 + 0.745127i \(0.267611\pi\)
\(48\) −763457. −0.996416
\(49\) 736881. 0.894769
\(50\) 868169. 0.982221
\(51\) −1.05049e6 −1.10891
\(52\) 38280.2 0.0377539
\(53\) 294003. 0.271260 0.135630 − 0.990760i \(-0.456694\pi\)
0.135630 + 0.990760i \(0.456694\pi\)
\(54\) −1.26842e6 −1.09619
\(55\) 0 0
\(56\) −1.73185e6 −1.31781
\(57\) −237311. −0.169729
\(58\) −2.09490e6 −1.40983
\(59\) −87803.7 −0.0556584 −0.0278292 − 0.999613i \(-0.508859\pi\)
−0.0278292 + 0.999613i \(0.508859\pi\)
\(60\) −168974. −0.100993
\(61\) 2.78573e6 1.57139 0.785696 − 0.618613i \(-0.212305\pi\)
0.785696 + 0.618613i \(0.212305\pi\)
\(62\) −2.17452e6 −1.15876
\(63\) −370822. −0.186842
\(64\) 1.90940e6 0.910471
\(65\) −1.49745e6 −0.676325
\(66\) 0 0
\(67\) 2.95364e6 1.19976 0.599882 − 0.800089i \(-0.295214\pi\)
0.599882 + 0.800089i \(0.295214\pi\)
\(68\) −240847. −0.0928883
\(69\) −2.78129e6 −1.01923
\(70\) −5.72117e6 −1.99362
\(71\) −4.08380e6 −1.35413 −0.677065 − 0.735923i \(-0.736749\pi\)
−0.677065 + 0.735923i \(0.736749\pi\)
\(72\) 411560. 0.129948
\(73\) −1.95525e6 −0.588263 −0.294132 − 0.955765i \(-0.595030\pi\)
−0.294132 + 0.955765i \(0.595030\pi\)
\(74\) −4.81370e6 −1.38092
\(75\) 3.21339e6 0.879526
\(76\) −54408.6 −0.0142174
\(77\) 0 0
\(78\) 1.96116e6 0.467932
\(79\) 608158. 0.138778 0.0693891 − 0.997590i \(-0.477895\pi\)
0.0693891 + 0.997590i \(0.477895\pi\)
\(80\) 6.84718e6 1.49519
\(81\) −4.04562e6 −0.845840
\(82\) 7.93173e6 1.58862
\(83\) −214613. −0.0411986 −0.0205993 − 0.999788i \(-0.506557\pi\)
−0.0205993 + 0.999788i \(0.506557\pi\)
\(84\) 541334. 0.0996524
\(85\) 9.42151e6 1.66400
\(86\) −456959. −0.0774698
\(87\) −7.75394e6 −1.26242
\(88\) 0 0
\(89\) −8.30366e6 −1.24855 −0.624273 − 0.781206i \(-0.714605\pi\)
−0.624273 + 0.781206i \(0.714605\pi\)
\(90\) 1.35959e6 0.196589
\(91\) 4.79732e6 0.667351
\(92\) −637668. −0.0853763
\(93\) −8.04863e6 −1.03760
\(94\) 1.11516e7 1.38481
\(95\) 2.12836e6 0.254691
\(96\) −1.25235e6 −0.144469
\(97\) −1.38909e6 −0.154536 −0.0772678 − 0.997010i \(-0.524620\pi\)
−0.0772678 + 0.997010i \(0.524620\pi\)
\(98\) 8.65538e6 0.928955
\(99\) 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 121.8.a.b.1.2 2
11.10 odd 2 11.8.a.a.1.1 ✓ 2
33.32 even 2 99.8.a.c.1.2 2
44.43 even 2 176.8.a.d.1.1 2
55.54 odd 2 275.8.a.a.1.2 2
77.76 even 2 539.8.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.8.a.a.1.1 ✓ 2 11.10 odd 2
99.8.a.c.1.2 2 33.32 even 2
121.8.a.b.1.2 2 1.1 even 1 trivial
176.8.a.d.1.1 2 44.43 even 2
275.8.a.a.1.2 2 55.54 odd 2
539.8.a.a.1.1 2 77.76 even 2