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Properties

Label 228.2.v.a.119.11
Level $228$
Weight $2$
Character 228.119
Analytic conductor $1.821$
Analytic rank $0$
Dimension $216$
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] = [228,2,Mod(23,228)] 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("228.23"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(228, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 228.v (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.82058916609\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(36\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 119.11
Character \(\chi\) \(=\) 228.119
Dual form 228.2.v.a.23.11

$q$-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+(-0.709055 - 1.22362i) q^{2} +(1.68017 + 0.420760i) q^{3} +(-0.994483 + 1.73522i) q^{4} +(-2.09742 + 2.49961i) q^{5} +(-0.676481 - 2.35422i) q^{6} +(-2.63219 + 1.51969i) q^{7} +(2.82839 - 0.0135021i) q^{8} +(2.64592 + 1.41389i) q^{9} +(4.54576 + 0.794084i) q^{10} +(-0.326657 + 0.565787i) q^{11} +(-2.40101 + 2.49703i) q^{12} +(0.487847 + 2.76672i) q^{13} +(3.72589 + 2.14324i) q^{14} +(-4.57576 + 3.31726i) q^{15} +(-2.02201 - 3.45130i) q^{16} +(1.31571 + 3.61488i) q^{17} +(-0.146038 - 4.24013i) q^{18} +(-1.88049 - 3.93240i) q^{19} +(-2.25154 - 6.12533i) q^{20} +(-5.06194 + 1.44582i) q^{21} +(0.923925 - 0.00147019i) q^{22} +(5.40830 - 4.53810i) q^{23} +(4.75786 + 1.16739i) q^{24} +(-0.980636 - 5.56146i) q^{25} +(3.03949 - 2.55869i) q^{26} +(3.85068 + 3.48887i) q^{27} +(-0.0193455 - 6.07874i) q^{28} +(-0.771141 + 2.11869i) q^{29} +(7.30352 + 3.24687i) q^{30} +(-2.94846 + 1.70229i) q^{31} +(-2.78936 + 4.92133i) q^{32} +(-0.786899 + 0.813172i) q^{33} +(3.49033 - 4.17308i) q^{34} +(1.72217 - 9.76689i) q^{35} +(-5.08475 + 3.18518i) q^{36} +2.89778 q^{37} +(-3.47838 + 5.08929i) q^{38} +(-0.344459 + 4.85381i) q^{39} +(-5.89860 + 7.09821i) q^{40} +(10.3840 + 1.83098i) q^{41} +(5.35832 + 5.16871i) q^{42} +(7.58372 - 9.03792i) q^{43} +(-0.656912 - 1.12949i) q^{44} +(-9.08381 + 3.64825i) q^{45} +(-9.38768 - 3.39993i) q^{46} +(-1.83801 - 0.668980i) q^{47} +(-1.94514 - 6.64954i) q^{48} +(1.11894 - 1.93805i) q^{49} +(-6.10978 + 5.14331i) q^{50} +(0.689615 + 6.62721i) q^{51} +(-5.28603 - 1.90493i) q^{52} +(-5.81935 - 6.93523i) q^{53} +(1.53871 - 7.18557i) q^{54} +(-0.729109 - 2.00321i) q^{55} +(-7.42434 + 4.33383i) q^{56} +(-1.50495 - 7.39832i) q^{57} +(3.13925 - 0.558687i) q^{58} +(-8.69888 + 3.16613i) q^{59} +(-1.20567 - 11.2389i) q^{60} +(-4.97808 + 4.17711i) q^{61} +(4.17357 + 2.40077i) q^{62} +(-9.11324 + 0.299360i) q^{63} +(7.99964 - 0.0763783i) q^{64} +(-7.93894 - 4.58355i) q^{65} +(1.55297 + 0.386280i) q^{66} +(-2.95870 + 8.12896i) q^{67} +(-7.58108 - 1.31189i) q^{68} +(10.9963 - 5.34917i) q^{69} +(-13.1721 + 4.81799i) q^{70} +(4.41322 + 3.70313i) q^{71} +(7.50280 + 3.96332i) q^{72} +(0.0225042 - 0.127627i) q^{73} +(-2.05468 - 3.54577i) q^{74} +(0.692408 - 9.75680i) q^{75} +(8.69371 + 0.647625i) q^{76} -1.98567i q^{77} +(6.18345 - 3.02013i) q^{78} +(13.9125 + 2.45315i) q^{79} +(12.8679 + 2.18461i) q^{80} +(5.00181 + 7.48210i) q^{81} +(-5.12240 - 14.0043i) q^{82} +(1.37681 + 2.38470i) q^{83} +(2.52519 - 10.2214i) q^{84} +(-11.7954 - 4.29318i) q^{85} +(-16.4362 - 2.87119i) q^{86} +(-2.18711 + 3.23529i) q^{87} +(-0.916276 + 1.60468i) q^{88} +(1.45226 - 0.256073i) q^{89} +(10.9050 + 8.52831i) q^{90} +(-5.48866 - 6.54113i) q^{91} +(2.49616 + 13.8977i) q^{92} +(-5.67016 + 1.61954i) q^{93} +(0.484672 + 2.72336i) q^{94} +(13.7737 + 3.54740i) q^{95} +(-6.75729 + 7.09500i) q^{96} +(3.69157 - 1.34362i) q^{97} +(-3.16482 + 0.00503599i) q^{98} +(-1.66427 + 1.03517i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 216 q - 18 q^{4} - 6 q^{6} - 18 q^{9} - 18 q^{10} - 3 q^{12} - 36 q^{13} - 6 q^{16} - 12 q^{18} - 30 q^{21} - 18 q^{24} - 24 q^{25} - 36 q^{28} + 12 q^{33} + 30 q^{34} - 66 q^{36} - 48 q^{37} - 42 q^{40}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/228\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\) \(115\)
\(\chi(n)\) \(-1\) \(e\left(\frac{8}{9}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.709055 − 1.22362i −0.501377 − 0.865229i
\(3\) 1.68017 + 0.420760i 0.970045 + 0.242926i
\(4\) −0.994483 + 1.73522i −0.497241 + 0.867612i
\(5\) −2.09742 + 2.49961i −0.937997 + 1.11786i 0.0548534 + 0.998494i \(0.482531\pi\)
−0.992850 + 0.119367i \(0.961914\pi\)
\(6\) −0.676481 − 2.35422i −0.276172 − 0.961108i
\(7\) −2.63219 + 1.51969i −0.994873 + 0.574390i −0.906727 − 0.421718i \(-0.861427\pi\)
−0.0881455 + 0.996108i \(0.528094\pi\)
\(8\) 2.82839 − 0.0135021i 0.999989 − 0.00477370i
\(9\) 2.64592 + 1.41389i 0.881974 + 0.471298i
\(10\) 4.54576 + 0.794084i 1.43750 + 0.251111i
\(11\) −0.326657 + 0.565787i −0.0984908 + 0.170591i −0.911060 − 0.412273i \(-0.864735\pi\)
0.812569 + 0.582865i \(0.198068\pi\)
\(12\) −2.40101 + 2.49703i −0.693112 + 0.720830i
\(13\) 0.487847 + 2.76672i 0.135304 + 0.767349i 0.974647 + 0.223747i \(0.0718288\pi\)
−0.839343 + 0.543602i \(0.817060\pi\)
\(14\) 3.72589 + 2.14324i 0.995786 + 0.572806i
\(15\) −4.57576 + 3.31726i −1.18146 + 0.856512i
\(16\) −2.02201 − 3.45130i −0.505502 − 0.862825i
\(17\) 1.31571 + 3.61488i 0.319107 + 0.876738i 0.990730 + 0.135845i \(0.0433750\pi\)
−0.671624 + 0.740893i \(0.734403\pi\)
\(18\) −0.146038 − 4.24013i −0.0344216 − 0.999407i
\(19\) −1.88049 − 3.93240i −0.431415 − 0.902154i
\(20\) −2.25154 − 6.12533i −0.503459 − 1.36966i
\(21\) −5.06194 + 1.44582i −1.10461 + 0.315504i
\(22\) 0.923925 − 0.00147019i 0.196981 − 0.000313445i
\(23\) 5.40830 − 4.53810i 1.12771 − 0.946259i 0.128740 − 0.991678i \(-0.458907\pi\)
0.998968 + 0.0454194i \(0.0144624\pi\)
\(24\) 4.75786 + 1.16739i 0.971193 + 0.238292i
\(25\) −0.980636 − 5.56146i −0.196127 − 1.11229i
\(26\) 3.03949 − 2.55869i 0.596094 − 0.501801i
\(27\) 3.85068 + 3.48887i 0.741064 + 0.671434i
\(28\) −0.0193455 − 6.07874i −0.00365596 − 1.14877i
\(29\) −0.771141 + 2.11869i −0.143197 + 0.393431i −0.990470 − 0.137726i \(-0.956021\pi\)
0.847273 + 0.531158i \(0.178243\pi\)
\(30\) 7.30352 + 3.24687i 1.33343 + 0.592794i
\(31\) −2.94846 + 1.70229i −0.529559 + 0.305741i −0.740837 − 0.671685i \(-0.765571\pi\)
0.211278 + 0.977426i \(0.432237\pi\)
\(32\) −2.78936 + 4.92133i −0.493094 + 0.869976i
\(33\) −0.786899 + 0.813172i −0.136982 + 0.141555i
\(34\) 3.49033 − 4.17308i 0.598586 − 0.715677i
\(35\) 1.72217 − 9.76689i 0.291099 − 1.65091i
\(36\) −5.08475 + 3.18518i −0.847458 + 0.530863i
\(37\) 2.89778 0.476392 0.238196 − 0.971217i \(-0.423444\pi\)
0.238196 + 0.971217i \(0.423444\pi\)
\(38\) −3.47838 + 5.08929i −0.564268 + 0.825592i
\(39\) −0.344459 + 4.85381i −0.0551576 + 0.777232i
\(40\) −5.89860 + 7.09821i −0.932650 + 1.12233i
\(41\) 10.3840 + 1.83098i 1.62171 + 0.285950i 0.909401 − 0.415920i \(-0.136541\pi\)
0.712304 + 0.701871i \(0.247652\pi\)
\(42\) 5.35832 + 5.16871i 0.826807 + 0.797550i
\(43\) 7.58372 − 9.03792i 1.15651 − 1.37827i 0.243713 − 0.969847i \(-0.421634\pi\)
0.912793 − 0.408423i \(-0.133921\pi\)
\(44\) −0.656912 − 1.12949i −0.0990332 − 0.170277i
\(45\) −9.08381 + 3.64825i −1.35413 + 0.543849i
\(46\) −9.38768 − 3.39993i −1.38414 − 0.501292i
\(47\) −1.83801 − 0.668980i −0.268101 − 0.0975808i 0.204472 − 0.978872i \(-0.434452\pi\)
−0.472573 + 0.881292i \(0.656675\pi\)
\(48\) −1.94514 − 6.64954i −0.280757 − 0.959779i
\(49\) 1.11894 − 1.93805i 0.159848 − 0.276865i
\(50\) −6.10978 + 5.14331i −0.864054 + 0.727373i
\(51\) 0.689615 + 6.62721i 0.0965654 + 0.927994i
\(52\) −5.28603 − 1.90493i −0.733040 − 0.264166i
\(53\) −5.81935 − 6.93523i −0.799349 − 0.952628i 0.200283 − 0.979738i \(-0.435814\pi\)
−0.999632 + 0.0271105i \(0.991369\pi\)
\(54\) 1.53871 − 7.18557i 0.209391 − 0.977832i
\(55\) −0.729109 − 2.00321i −0.0983131 − 0.270113i
\(56\) −7.42434 + 4.33383i −0.992119 + 0.579133i
\(57\) −1.50495 − 7.39832i −0.199335 − 0.979931i
\(58\) 3.13925 − 0.558687i 0.412204 − 0.0733592i
\(59\) −8.69888 + 3.16613i −1.13250 + 0.412195i −0.839198 − 0.543825i \(-0.816975\pi\)
−0.293299 + 0.956021i \(0.594753\pi\)
\(60\) −1.20567 − 11.2389i −0.155651 − 1.45094i
\(61\) −4.97808 + 4.17711i −0.637378 + 0.534824i −0.903212 − 0.429195i \(-0.858797\pi\)
0.265834 + 0.964019i \(0.414353\pi\)
\(62\) 4.17357 + 2.40077i 0.530044 + 0.304898i
\(63\) −9.11324 + 0.299360i −1.14816 + 0.0377159i
\(64\) 7.99964 − 0.0763783i 0.999954 − 0.00954729i
\(65\) −7.93894 − 4.58355i −0.984704 − 0.568519i
\(66\) 1.55297 + 0.386280i 0.191157 + 0.0475478i
\(67\) −2.95870 + 8.12896i −0.361463 + 0.993110i 0.617050 + 0.786924i \(0.288328\pi\)
−0.978513 + 0.206187i \(0.933895\pi\)
\(68\) −7.58108 − 1.31189i −0.919342 − 0.159090i
\(69\) 10.9963 − 5.34917i 1.32380 − 0.643964i
\(70\) −13.1721 + 4.81799i −1.57436 + 0.575859i
\(71\) 4.41322 + 3.70313i 0.523753 + 0.439481i 0.865938 − 0.500152i \(-0.166722\pi\)
−0.342185 + 0.939633i \(0.611167\pi\)
\(72\) 7.50280 + 3.96332i 0.884214 + 0.467082i
\(73\) 0.0225042 − 0.127627i 0.00263391 − 0.0149377i −0.983462 − 0.181112i \(-0.942030\pi\)
0.986096 + 0.166174i \(0.0531415\pi\)
\(74\) −2.05468 − 3.54577i −0.238852 − 0.412188i
\(75\) 0.692408 − 9.75680i 0.0799523 − 1.12662i
\(76\) 8.69371 + 0.647625i 0.997237 + 0.0742877i
\(77\) − 1.98567i − 0.226289i
\(78\) 6.18345 − 3.02013i 0.700138 − 0.341963i
\(79\) 13.9125 + 2.45315i 1.56528 + 0.276001i 0.888041 − 0.459765i \(-0.152066\pi\)
0.677237 + 0.735765i \(0.263177\pi\)
\(80\) 12.8679 + 2.18461i 1.43868 + 0.244246i
\(81\) 5.00181 + 7.48210i 0.555757 + 0.831345i
\(82\) −5.12240 − 14.0043i −0.565674 − 1.54651i
\(83\) 1.37681 + 2.38470i 0.151124 + 0.261755i 0.931641 − 0.363380i \(-0.118377\pi\)
−0.780517 + 0.625135i \(0.785044\pi\)
\(84\) 2.52519 − 10.2214i 0.275520 − 1.11525i
\(85\) −11.7954 − 4.29318i −1.27939 − 0.465661i
\(86\) −16.4362 − 2.87119i −1.77237 − 0.309609i
\(87\) −2.18711 + 3.23529i −0.234482 + 0.346860i
\(88\) −0.916276 + 1.60468i −0.0976754 + 0.171059i
\(89\) 1.45226 − 0.256073i 0.153939 − 0.0271437i −0.0961473 − 0.995367i \(-0.530652\pi\)
0.250087 + 0.968223i \(0.419541\pi\)
\(90\) 10.9050 + 8.52831i 1.14949 + 0.898962i
\(91\) −5.48866 − 6.54113i −0.575368 − 0.685697i
\(92\) 2.49616 + 13.8977i 0.260243 + 1.44893i
\(93\) −5.67016 + 1.61954i −0.587968 + 0.167939i
\(94\) 0.484672 + 2.72336i 0.0499901 + 0.280893i
\(95\) 13.7737 + 3.54740i 1.41315 + 0.363956i
\(96\) −6.75729 + 7.09500i −0.689663 + 0.724131i
\(97\) 3.69157 − 1.34362i 0.374822 − 0.136424i −0.147738 − 0.989027i \(-0.547199\pi\)
0.522560 + 0.852603i \(0.324977\pi\)
\(98\) −3.16482 + 0.00503599i −0.319695 + 0.000508712i
\(99\) −1.66427 + 1.03517i −0.167266 + 0.104038i
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 228.2.v.a.119.11 yes 216
3.2 odd 2 inner 228.2.v.a.119.26 yes 216
4.3 odd 2 inner 228.2.v.a.119.18 yes 216
12.11 even 2 inner 228.2.v.a.119.19 yes 216
19.4 even 9 inner 228.2.v.a.23.19 yes 216
57.23 odd 18 inner 228.2.v.a.23.18 yes 216
76.23 odd 18 inner 228.2.v.a.23.26 yes 216
228.23 even 18 inner 228.2.v.a.23.11 ✓ 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
228.2.v.a.23.11 ✓ 216 228.23 even 18 inner
228.2.v.a.23.18 yes 216 57.23 odd 18 inner
228.2.v.a.23.19 yes 216 19.4 even 9 inner
228.2.v.a.23.26 yes 216 76.23 odd 18 inner
228.2.v.a.119.11 yes 216 1.1 even 1 trivial
228.2.v.a.119.18 yes 216 4.3 odd 2 inner
228.2.v.a.119.19 yes 216 12.11 even 2 inner
228.2.v.a.119.26 yes 216 3.2 odd 2 inner