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Properties

Label 121.4.c.e.3.2
Level $121$
Weight $4$
Character 121.3
Analytic conductor $7.139$
Analytic rank $0$
Dimension $8$
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] = [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 3.2
Root \(-1.57568 - 4.84946i\) of defining polynomial
Character \(\chi\) \(=\) 121.3
Dual form 121.4.c.e.81.2

$q$-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 \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\)

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 0.474720 − 0.880137i \(-0.342549\pi\)
0.983757 − 0.179508i \(-0.0574505\pi\)
\(3\) −1.54508 − 4.75528i −0.297352 − 0.915155i −0.982421 − 0.186677i \(-0.940228\pi\)
0.685070 − 0.728478i \(-0.259772\pi\)
\(4\) 5.56231 − 17.1190i 0.695288 − 2.13988i
\(5\) −4.04508 − 2.93893i −0.361803 − 0.262866i 0.392000 − 0.919965i \(-0.371783\pi\)
−0.753804 + 0.657099i \(0.771783\pi\)
\(6\) −20.6260 − 14.9856i −1.40342 − 1.01964i
\(7\) −6.30273 + 19.3978i −0.340316 + 1.04738i 0.623728 + 0.781641i \(0.285617\pi\)
−0.964044 + 0.265743i \(0.914383\pi\)
\(8\) −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 0.0827480 − 0.996571i \(-0.473630\pi\)
0.973365 + 0.229259i \(0.0736304\pi\)
\(14\) 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 0.463824 − 0.885927i \(-0.346477\pi\)
−0.699237 + 0.714890i \(0.746477\pi\)
\(18\) 3.15137 − 9.69891i 0.0412658 − 0.127003i
\(19\) 31.5137 + 96.9891i 0.380512 + 1.17110i 0.939684 + 0.342044i \(0.111119\pi\)
−0.559172 + 0.829052i \(0.688881\pi\)
\(20\) −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 −0.518491 − 0.855083i \(-0.673506\pi\)
0.922073 − 0.387015i \(-0.126494\pi\)
\(30\) 39.3921 + 121.236i 0.239733 + 0.737821i
\(31\) −12.1353 + 8.81678i −0.0703083 + 0.0510819i −0.622384 − 0.782712i \(-0.713836\pi\)
0.552076 + 0.833794i \(0.313836\pi\)
\(32\) −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 0.586844 + 0.809700i \(0.300370\pi\)
−0.950696 + 0.310123i \(0.899630\pi\)
\(38\) 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 0.992965 − 0.118411i \(-0.0377800\pi\)
−0.872925 + 0.487854i \(0.837780\pi\)
\(42\) 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 0.950335 + 0.311230i \(0.100741\pi\)
−0.585900 + 0.810383i \(0.699259\pi\)
\(48\) −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 −0.975793 − 0.218697i \(-0.929819\pi\)
−0.0935429 + 0.995615i \(0.529819\pi\)
\(54\) −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 −0.943642 − 0.330969i \(-0.892624\pi\)
0.957961 + 0.286899i \(0.0926244\pi\)
\(60\) 364.058 + 264.503i 0.783327 + 0.569121i
\(61\) 165.008 + 119.885i 0.346346 + 0.251635i 0.747334 − 0.664448i \(-0.231333\pi\)
−0.400989 + 0.916083i \(0.631333\pi\)
\(62\) −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 0.355684 − 0.934606i \(-0.384248\pi\)
−0.778951 + 0.627085i \(0.784248\pi\)
\(72\) −82.5039 − 59.9426i −0.135044 − 0.0981153i
\(73\) −144.963 + 446.150i −0.232420 + 0.715314i 0.765034 + 0.643990i \(0.222722\pi\)
−0.997453 + 0.0713235i \(0.977278\pi\)
\(74\) 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 −0.881555 − 0.472082i \(-0.843503\pi\)
0.176561 + 0.984290i \(0.443503\pi\)
\(80\) 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 0.879376 − 0.476129i \(-0.157960\pi\)
−0.181083 + 0.983468i \(0.557960\pi\)
\(84\) 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 −0.868270 − 0.496093i \(-0.834768\pi\)
0.203502 + 0.979074i \(0.434768\pi\)
\(98\) −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.3.2 8
11.2 odd 10 121.4.a.d.1.2 yes 2
11.3 even 5 inner 121.4.c.e.27.1 8
11.4 even 5 inner 121.4.c.e.81.2 8
11.5 even 5 inner 121.4.c.e.9.1 8
11.6 odd 10 inner 121.4.c.e.9.2 8
11.7 odd 10 inner 121.4.c.e.81.1 8
11.8 odd 10 inner 121.4.c.e.27.2 8
11.9 even 5 121.4.a.d.1.1 ✓ 2
11.10 odd 2 inner 121.4.c.e.3.1 8
33.2 even 10 1089.4.a.r.1.1 2
33.20 odd 10 1089.4.a.r.1.2 2
44.31 odd 10 1936.4.a.ba.1.2 2
44.35 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 11.9 even 5
121.4.a.d.1.2 yes 2 11.2 odd 10
121.4.c.e.3.1 8 11.10 odd 2 inner
121.4.c.e.3.2 8 1.1 even 1 trivial
121.4.c.e.9.1 8 11.5 even 5 inner
121.4.c.e.9.2 8 11.6 odd 10 inner
121.4.c.e.27.1 8 11.3 even 5 inner
121.4.c.e.27.2 8 11.8 odd 10 inner
121.4.c.e.81.1 8 11.7 odd 10 inner
121.4.c.e.81.2 8 11.4 even 5 inner
1089.4.a.r.1.1 2 33.2 even 10
1089.4.a.r.1.2 2 33.20 odd 10
1936.4.a.ba.1.1 2 44.35 even 10
1936.4.a.ba.1.2 2 44.31 odd 10