Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [121,4,Mod(3,121)] 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("121.3"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.c (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,10] 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: no
Analytic conductor: \(7.13923111069\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.1827904000000.7
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 26x^{6} + 676x^{4} + 17576x^{2} + 456976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 81.2
Root \(-1.57568 + 4.84946i\) of defining polynomial
Character \(\chi\) \(=\) 121.81
Dual form 121.4.c.e.3.2

$q$-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+(4.12519 + 2.99713i) q^{2} +(-1.54508 + 4.75528i) q^{3} +(5.56231 + 17.1190i) q^{4} +(-4.04508 + 2.93893i) q^{5} +(-20.6260 + 14.9856i) q^{6} +(-6.30273 - 19.3978i) q^{7} +(-15.7568 + 48.4946i) q^{8} +(1.61803 + 1.17557i) q^{9} -25.4951 q^{10} -90.0000 q^{12} +(49.5023 + 35.9655i) q^{13} +(32.1378 - 98.9099i) q^{14} +(-7.72542 - 23.7764i) q^{15} +(-93.8460 + 68.1831i) q^{16} +(-16.5008 + 11.9885i) q^{17} +(3.15137 + 9.69891i) q^{18} +(31.5137 - 96.9891i) q^{19} +(-72.8115 - 52.9007i) q^{20} +101.980 q^{21} +35.0000 q^{23} +(-206.260 - 149.856i) q^{24} +(-30.9017 + 95.1057i) q^{25} +(96.4133 + 296.730i) q^{26} +(-117.307 + 85.2289i) q^{27} +(297.014 - 215.793i) q^{28} +(63.0273 + 193.978i) q^{29} +(39.3921 - 121.236i) q^{30} +(-12.1353 - 8.81678i) q^{31} -183.565 q^{32} -104.000 q^{34} +(82.5039 + 59.9426i) q^{35} +(-11.1246 + 34.2380i) q^{36} +(-81.8895 - 252.030i) q^{37} +(420.689 - 305.648i) q^{38} +(-247.512 + 179.828i) q^{39} +(-78.7842 - 242.473i) q^{40} +(31.5137 - 96.9891i) q^{41} +(420.689 + 305.648i) q^{42} +448.714 q^{43} -10.0000 q^{45} +(144.382 + 104.899i) q^{46} +(117.426 - 361.401i) q^{47} +(-179.230 - 551.613i) q^{48} +(-59.0582 + 42.9083i) q^{49} +(-412.519 + 299.713i) q^{50} +(-31.5137 - 96.9891i) q^{51} +(-340.348 + 1047.48i) q^{52} +(-412.599 - 299.770i) q^{53} -739.358 q^{54} +1040.00 q^{56} +(412.519 + 299.713i) q^{57} +(-321.378 + 989.099i) q^{58} +(6.48936 + 19.9722i) q^{59} +(364.058 - 264.503i) q^{60} +(165.008 - 119.885i) q^{61} +(-23.6353 - 72.7418i) q^{62} +(12.6055 - 38.7956i) q^{63} +(-6.47214 - 4.70228i) q^{64} -305.941 q^{65} +585.000 q^{67} +(-297.014 - 215.793i) q^{68} +(-54.0780 + 166.435i) q^{69} +(160.689 + 494.549i) q^{70} +(-253.222 + 183.977i) q^{71} +(-82.5039 + 59.9426i) q^{72} +(-144.963 - 446.150i) q^{73} +(417.556 - 1285.11i) q^{74} +(-404.508 - 293.893i) q^{75} +1835.65 q^{76} -1560.00 q^{78} +(-495.023 - 359.655i) q^{79} +(179.230 - 551.613i) q^{80} +(-207.350 - 638.159i) q^{81} +(420.689 - 305.648i) q^{82} +(528.025 - 383.632i) q^{83} +(567.246 + 1745.80i) q^{84} +(31.5137 - 96.9891i) q^{85} +(1851.03 + 1344.85i) q^{86} -1019.80 q^{87} -185.000 q^{89} +(-41.2519 - 29.9713i) q^{90} +(385.653 - 1186.92i) q^{91} +(194.681 + 599.166i) q^{92} +(60.6763 - 44.0839i) q^{93} +(1567.57 - 1138.91i) q^{94} +(157.568 + 484.946i) q^{95} +(283.623 - 872.902i) q^{96} +(-635.078 - 461.411i) q^{97} -372.228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{3} - 36 q^{4} - 10 q^{5} + 4 q^{9} - 720 q^{12} - 208 q^{14} + 50 q^{15} - 232 q^{16} - 180 q^{20} + 280 q^{23} + 200 q^{25} - 624 q^{26} - 290 q^{27} - 30 q^{31} - 832 q^{34} + 72 q^{36} + 530 q^{37}+ \cdots - 1570 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.12519 + 2.99713i 1.45848 + 1.05964i 0.983757 + 0.179508i \(0.0574505\pi\)
0.474720 + 0.880137i \(0.342549\pi\)
\(3\) −1.54508 + 4.75528i −0.297352 + 0.915155i 0.685070 + 0.728478i \(0.259772\pi\)
−0.982421 + 0.186677i \(0.940228\pi\)
\(4\) 5.56231 + 17.1190i 0.695288 + 2.13988i
\(5\) −4.04508 + 2.93893i −0.361803 + 0.262866i −0.753804 − 0.657099i \(-0.771783\pi\)
0.392000 + 0.919965i \(0.371783\pi\)
\(6\) −20.6260 + 14.9856i −1.40342 + 1.01964i
\(7\) −6.30273 − 19.3978i −0.340316 − 1.04738i −0.964044 − 0.265743i \(-0.914383\pi\)
0.623728 − 0.781641i \(-0.285617\pi\)
\(8\) −15.7568 + 48.4946i −0.696360 + 2.14318i
\(9\) 1.61803 + 1.17557i 0.0599272 + 0.0435396i
\(10\) −25.4951 −0.806226
\(11\) 0 0
\(12\) −90.0000 −2.16506
\(13\) 49.5023 + 35.9655i 1.05611 + 0.767311i 0.973365 − 0.229259i \(-0.0736304\pi\)
0.0827480 + 0.996571i \(0.473630\pi\)
\(14\) 32.1378 − 98.9099i 0.613513 − 1.88820i
\(15\) −7.72542 − 23.7764i −0.132980 − 0.409270i
\(16\) −93.8460 + 68.1831i −1.46634 + 1.06536i
\(17\) −16.5008 + 11.9885i −0.235413 + 0.171038i −0.699237 − 0.714890i \(-0.746477\pi\)
0.463824 + 0.885927i \(0.346477\pi\)
\(18\) 3.15137 + 9.69891i 0.0412658 + 0.127003i
\(19\) 31.5137 − 96.9891i 0.380512 − 1.17110i −0.559172 − 0.829052i \(-0.688881\pi\)
0.939684 − 0.342044i \(-0.111119\pi\)
\(20\) −72.8115 − 52.9007i −0.814058 − 0.591448i
\(21\) 101.980 1.05971
\(22\) 0 0
\(23\) 35.0000 0.317305 0.158652 − 0.987335i \(-0.449285\pi\)
0.158652 + 0.987335i \(0.449285\pi\)
\(24\) −206.260 − 149.856i −1.75427 − 1.27455i
\(25\) −30.9017 + 95.1057i −0.247214 + 0.760845i
\(26\) 96.4133 + 296.730i 0.727239 + 2.23821i
\(27\) −117.307 + 85.2289i −0.836142 + 0.607493i
\(28\) 297.014 − 215.793i 2.00466 − 1.45647i
\(29\) 63.0273 + 193.978i 0.403582 + 1.24210i 0.922073 + 0.387015i \(0.126494\pi\)
−0.518491 + 0.855083i \(0.673506\pi\)
\(30\) 39.3921 − 121.236i 0.239733 − 0.737821i
\(31\) −12.1353 − 8.81678i −0.0703083 − 0.0510819i 0.552076 − 0.833794i \(-0.313836\pi\)
−0.622384 + 0.782712i \(0.713836\pi\)
\(32\) −183.565 −1.01406
\(33\) 0 0
\(34\) −104.000 −0.524584
\(35\) 82.5039 + 59.9426i 0.398449 + 0.289490i
\(36\) −11.1246 + 34.2380i −0.0515028 + 0.158509i
\(37\) −81.8895 − 252.030i −0.363853 − 1.11982i −0.950696 − 0.310123i \(-0.899630\pi\)
0.586844 − 0.809700i \(-0.300370\pi\)
\(38\) 420.689 − 305.648i 1.79591 − 1.30481i
\(39\) −247.512 + 179.828i −1.01625 + 0.738346i
\(40\) −78.7842 − 242.473i −0.311422 − 0.958458i
\(41\) 31.5137 − 96.9891i 0.120039 − 0.369443i −0.872925 − 0.487854i \(-0.837780\pi\)
0.992965 + 0.118411i \(0.0377800\pi\)
\(42\) 420.689 + 305.648i 1.54556 + 1.12292i
\(43\) 448.714 1.59135 0.795677 − 0.605721i \(-0.207115\pi\)
0.795677 + 0.605721i \(0.207115\pi\)
\(44\) 0 0
\(45\) −10.0000 −0.0331269
\(46\) 144.382 + 104.899i 0.462781 + 0.336230i
\(47\) 117.426 − 361.401i 0.364434 − 1.12161i −0.585900 − 0.810383i \(-0.699259\pi\)
0.950335 − 0.311230i \(-0.100741\pi\)
\(48\) −179.230 − 551.613i −0.538950 − 1.65872i
\(49\) −59.0582 + 42.9083i −0.172181 + 0.125097i
\(50\) −412.519 + 299.713i −1.16678 + 0.847716i
\(51\) −31.5137 − 96.9891i −0.0865254 − 0.266298i
\(52\) −340.348 + 1047.48i −0.907649 + 2.79346i
\(53\) −412.599 − 299.770i −1.06934 − 0.776918i −0.0935429 − 0.995615i \(-0.529819\pi\)
−0.975793 + 0.218697i \(0.929819\pi\)
\(54\) −739.358 −1.86322
\(55\) 0 0
\(56\) 1040.00 2.48171
\(57\) 412.519 + 299.713i 0.958588 + 0.696455i
\(58\) −321.378 + 989.099i −0.727568 + 2.23922i
\(59\) 6.48936 + 19.9722i 0.0143194 + 0.0440705i 0.957961 − 0.286899i \(-0.0926244\pi\)
−0.943642 + 0.330969i \(0.892624\pi\)
\(60\) 364.058 − 264.503i 0.783327 − 0.569121i
\(61\) 165.008 − 119.885i 0.346346 − 0.251635i −0.400989 − 0.916083i \(-0.631333\pi\)
0.747334 + 0.664448i \(0.231333\pi\)
\(62\) −23.6353 − 72.7418i −0.0484142 − 0.149004i
\(63\) 12.6055 − 38.7956i 0.0252086 − 0.0775840i
\(64\) −6.47214 − 4.70228i −0.0126409 − 0.00918414i
\(65\) −305.941 −0.583805
\(66\) 0 0
\(67\) 585.000 1.06670 0.533352 − 0.845894i \(-0.320932\pi\)
0.533352 + 0.845894i \(0.320932\pi\)
\(68\) −297.014 − 215.793i −0.529680 − 0.384835i
\(69\) −54.0780 + 166.435i −0.0943511 + 0.290383i
\(70\) 160.689 + 494.549i 0.274371 + 0.844428i
\(71\) −253.222 + 183.977i −0.423267 + 0.307522i −0.778951 − 0.627085i \(-0.784248\pi\)
0.355684 + 0.934606i \(0.384248\pi\)
\(72\) −82.5039 + 59.9426i −0.135044 + 0.0981153i
\(73\) −144.963 − 446.150i −0.232420 − 0.715314i −0.997453 − 0.0713235i \(-0.977278\pi\)
0.765034 − 0.643990i \(-0.222722\pi\)
\(74\) 417.556 − 1285.11i 0.655945 − 2.01879i
\(75\) −404.508 − 293.893i −0.622782 − 0.452477i
\(76\) 1835.65 2.77057
\(77\) 0 0
\(78\) −1560.00 −2.26455
\(79\) −495.023 − 359.655i −0.704993 − 0.512208i 0.176561 − 0.984290i \(-0.443503\pi\)
−0.881555 + 0.472082i \(0.843503\pi\)
\(80\) 179.230 − 551.613i 0.250481 − 0.770902i
\(81\) −207.350 − 638.159i −0.284431 − 0.875389i
\(82\) 420.689 − 305.648i 0.566553 − 0.411625i
\(83\) 528.025 − 383.632i 0.698292 − 0.507339i −0.181083 − 0.983468i \(-0.557960\pi\)
0.879376 + 0.476129i \(0.157960\pi\)
\(84\) 567.246 + 1745.80i 0.736805 + 2.26765i
\(85\) 31.5137 − 96.9891i 0.0402134 − 0.123764i
\(86\) 1851.03 + 1344.85i 2.32095 + 1.68627i
\(87\) −1019.80 −1.25672
\(88\) 0 0
\(89\) −185.000 −0.220337 −0.110168 − 0.993913i \(-0.535139\pi\)
−0.110168 + 0.993913i \(0.535139\pi\)
\(90\) −41.2519 − 29.9713i −0.0483148 − 0.0351028i
\(91\) 385.653 − 1186.92i 0.444258 − 1.36728i
\(92\) 194.681 + 599.166i 0.220618 + 0.678993i
\(93\) 60.6763 − 44.0839i 0.0676542 − 0.0491536i
\(94\) 1567.57 − 1138.91i 1.72003 − 1.24968i
\(95\) 157.568 + 484.946i 0.170170 + 0.523730i
\(96\) 283.623 − 872.902i 0.301533 − 0.928023i
\(97\) −635.078 − 461.411i −0.664767 − 0.482982i 0.203502 − 0.979074i \(-0.434768\pi\)
−0.868270 + 0.496093i \(0.834768\pi\)
\(98\) −372.228 −0.383681
\(99\) 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 121.4.c.e.81.2 8
11.2 odd 10 inner 121.4.c.e.27.2 8
11.3 even 5 inner 121.4.c.e.3.2 8
11.4 even 5 inner 121.4.c.e.9.1 8
11.5 even 5 121.4.a.d.1.1 ✓ 2
11.6 odd 10 121.4.a.d.1.2 yes 2
11.7 odd 10 inner 121.4.c.e.9.2 8
11.8 odd 10 inner 121.4.c.e.3.1 8
11.9 even 5 inner 121.4.c.e.27.1 8
11.10 odd 2 inner 121.4.c.e.81.1 8
33.5 odd 10 1089.4.a.r.1.2 2
33.17 even 10 1089.4.a.r.1.1 2
44.27 odd 10 1936.4.a.ba.1.2 2
44.39 even 10 1936.4.a.ba.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.a.d.1.1 ✓ 2 11.5 even 5
121.4.a.d.1.2 yes 2 11.6 odd 10
121.4.c.e.3.1 8 11.8 odd 10 inner
121.4.c.e.3.2 8 11.3 even 5 inner
121.4.c.e.9.1 8 11.4 even 5 inner
121.4.c.e.9.2 8 11.7 odd 10 inner
121.4.c.e.27.1 8 11.9 even 5 inner
121.4.c.e.27.2 8 11.2 odd 10 inner
121.4.c.e.81.1 8 11.10 odd 2 inner
121.4.c.e.81.2 8 1.1 even 1 trivial
1089.4.a.r.1.1 2 33.17 even 10
1089.4.a.r.1.2 2 33.5 odd 10
1936.4.a.ba.1.1 2 44.39 even 10
1936.4.a.ba.1.2 2 44.27 odd 10