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Properties

Label 228.2.v.a.119.12
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 119.12
Character \(\chi\) \(=\) 228.119
Dual form 228.2.v.a.23.12

$q$-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+(-0.687167 + 1.23604i) q^{2} +(0.927511 + 1.46278i) q^{3} +(-1.05560 - 1.69873i) q^{4} +(0.118249 - 0.140923i) q^{5} +(-2.44541 + 0.141271i) q^{6} +(-1.88835 + 1.09024i) q^{7} +(2.82509 - 0.137458i) q^{8} +(-1.27945 + 2.71349i) q^{9} +(0.0929307 + 0.242998i) q^{10} +(-2.97017 + 5.14448i) q^{11} +(1.50579 - 3.11971i) q^{12} +(0.572936 + 3.24928i) q^{13} +(-0.0499714 - 3.08326i) q^{14} +(0.315817 + 0.0422638i) q^{15} +(-1.77140 + 3.58638i) q^{16} +(-1.63631 - 4.49572i) q^{17} +(-2.47480 - 3.44607i) q^{18} +(4.17803 - 1.24260i) q^{19} +(-0.364215 - 0.0521140i) q^{20} +(-3.34625 - 1.75103i) q^{21} +(-4.31780 - 7.20637i) q^{22} +(4.68466 - 3.93090i) q^{23} +(2.82137 + 4.00498i) q^{24} +(0.862364 + 4.89071i) q^{25} +(-4.40995 - 1.52462i) q^{26} +(-5.15593 + 0.645244i) q^{27} +(3.84538 + 2.05695i) q^{28} +(0.421181 - 1.15719i) q^{29} +(-0.269259 + 0.361321i) q^{30} +(6.65444 - 3.84194i) q^{31} +(-3.21567 - 4.65397i) q^{32} +(-10.2801 + 0.426865i) q^{33} +(6.68132 + 1.06676i) q^{34} +(-0.0696549 + 0.395032i) q^{35} +(5.96009 - 0.690930i) q^{36} -1.63611 q^{37} +(-1.33509 + 6.01810i) q^{38} +(-4.22158 + 3.85183i) q^{39} +(0.314692 - 0.414375i) q^{40} +(1.14765 + 0.202362i) q^{41} +(4.46378 - 2.93285i) q^{42} +(-1.51151 + 1.80135i) q^{43} +(11.8744 - 0.385007i) q^{44} +(0.231101 + 0.501170i) q^{45} +(1.63962 + 8.49163i) q^{46} +(6.83260 + 2.48686i) q^{47} +(-6.88908 + 0.735242i) q^{48} +(-1.12275 + 1.94467i) q^{49} +(-6.63772 - 2.29481i) q^{50} +(5.05855 - 6.56339i) q^{51} +(4.91488 - 4.40322i) q^{52} +(1.42960 + 1.70373i) q^{53} +(2.74544 - 6.81635i) q^{54} +(0.373759 + 1.02690i) q^{55} +(-5.18489 + 3.33959i) q^{56} +(5.69282 + 4.95901i) q^{57} +(1.14091 + 1.31578i) q^{58} +(5.82579 - 2.12042i) q^{59} +(-0.261583 - 0.581103i) q^{60} +(9.65727 - 8.10341i) q^{61} +(0.176097 + 10.8652i) q^{62} +(-0.542311 - 6.51892i) q^{63} +(7.96221 - 0.776659i) q^{64} +(0.525649 + 0.303484i) q^{65} +(6.53652 - 13.0000i) q^{66} +(-3.18066 + 8.73880i) q^{67} +(-5.90974 + 7.52535i) q^{68} +(10.0951 + 3.20667i) q^{69} +(-0.440412 - 0.357549i) q^{70} +(-3.36704 - 2.82528i) q^{71} +(-3.24155 + 7.84170i) q^{72} +(1.20880 - 6.85547i) q^{73} +(1.12428 - 2.02230i) q^{74} +(-6.35418 + 5.79764i) q^{75} +(-6.52120 - 5.78567i) q^{76} -12.9528i q^{77} +(-1.86009 - 7.86490i) q^{78} +(-6.94773 - 1.22507i) q^{79} +(0.295939 + 0.673717i) q^{80} +(-5.72604 - 6.94352i) q^{81} +(-1.03876 + 1.27949i) q^{82} +(1.86888 + 3.23699i) q^{83} +(0.557775 + 7.53278i) q^{84} +(-0.827043 - 0.301019i) q^{85} +(-1.18788 - 3.10612i) q^{86} +(2.08336 - 0.457208i) q^{87} +(-7.68383 + 14.9419i) q^{88} +(1.23073 - 0.217011i) q^{89} +(-0.778273 - 0.0587368i) q^{90} +(-4.62440 - 5.51115i) q^{91} +(-11.6227 - 3.80853i) q^{92} +(11.7920 + 6.17053i) q^{93} +(-7.76901 + 6.73650i) q^{94} +(0.318935 - 0.735718i) q^{95} +(3.82516 - 9.02043i) q^{96} +(-10.6136 + 3.86302i) q^{97} +(-1.63217 - 2.72408i) q^{98} +(-10.1593 - 14.6416i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.687167 + 1.23604i −0.485900 + 0.874014i
\(3\) 0.927511 + 1.46278i 0.535499 + 0.844536i
\(4\) −1.05560 − 1.69873i −0.527802 − 0.849367i
\(5\) 0.118249 − 0.140923i 0.0528825 − 0.0630229i −0.738955 − 0.673755i \(-0.764680\pi\)
0.791837 + 0.610732i \(0.209125\pi\)
\(6\) −2.44541 + 0.141271i −0.998335 + 0.0576735i
\(7\) −1.88835 + 1.09024i −0.713730 + 0.412072i −0.812440 − 0.583044i \(-0.801861\pi\)
0.0987109 + 0.995116i \(0.468528\pi\)
\(8\) 2.82509 − 0.137458i 0.998818 − 0.0485986i
\(9\) −1.27945 + 2.71349i −0.426482 + 0.904496i
\(10\) 0.0929307 + 0.242998i 0.0293873 + 0.0768428i
\(11\) −2.97017 + 5.14448i −0.895540 + 1.55112i −0.0624045 + 0.998051i \(0.519877\pi\)
−0.833135 + 0.553069i \(0.813456\pi\)
\(12\) 1.50579 − 3.11971i 0.434684 − 0.900583i
\(13\) 0.572936 + 3.24928i 0.158904 + 0.901189i 0.955129 + 0.296189i \(0.0957158\pi\)
−0.796226 + 0.605000i \(0.793173\pi\)
\(14\) −0.0499714 − 3.08326i −0.0133554 − 0.824036i
\(15\) 0.315817 + 0.0422638i 0.0815436 + 0.0109125i
\(16\) −1.77140 + 3.58638i −0.442850 + 0.896596i
\(17\) −1.63631 − 4.49572i −0.396863 − 1.09037i −0.963804 − 0.266613i \(-0.914095\pi\)
0.566941 − 0.823759i \(-0.308127\pi\)
\(18\) −2.47480 − 3.44607i −0.583315 − 0.812246i
\(19\) 4.17803 − 1.24260i 0.958506 − 0.285072i
\(20\) −0.364215 − 0.0521140i −0.0814410 − 0.0116531i
\(21\) −3.34625 − 1.75103i −0.730211 − 0.382106i
\(22\) −4.31780 − 7.20637i −0.920558 − 1.53640i
\(23\) 4.68466 − 3.93090i 0.976820 − 0.819649i −0.00678675 − 0.999977i \(-0.502160\pi\)
0.983607 + 0.180328i \(0.0577159\pi\)
\(24\) 2.82137 + 4.00498i 0.575909 + 0.817514i
\(25\) 0.862364 + 4.89071i 0.172473 + 0.978142i
\(26\) −4.40995 − 1.52462i −0.864863 − 0.299004i
\(27\) −5.15593 + 0.645244i −0.992260 + 0.124177i
\(28\) 3.84538 + 2.05695i 0.726708 + 0.388726i
\(29\) 0.421181 − 1.15719i 0.0782113 − 0.214884i −0.894424 − 0.447219i \(-0.852414\pi\)
0.972636 + 0.232335i \(0.0746366\pi\)
\(30\) −0.269259 + 0.361321i −0.0491597 + 0.0659679i
\(31\) 6.65444 − 3.84194i 1.19517 − 0.690034i 0.235698 − 0.971826i \(-0.424262\pi\)
0.959475 + 0.281793i \(0.0909291\pi\)
\(32\) −3.21567 − 4.65397i −0.568456 − 0.822713i
\(33\) −10.2801 + 0.426865i −1.78954 + 0.0743077i
\(34\) 6.68132 + 1.06676i 1.14584 + 0.182948i
\(35\) −0.0696549 + 0.395032i −0.0117738 + 0.0667727i
\(36\) 5.96009 − 0.690930i 0.993348 − 0.115155i
\(37\) −1.63611 −0.268974 −0.134487 − 0.990915i \(-0.542939\pi\)
−0.134487 + 0.990915i \(0.542939\pi\)
\(38\) −1.33509 + 6.01810i −0.216581 + 0.976265i
\(39\) −4.22158 + 3.85183i −0.675993 + 0.616786i
\(40\) 0.314692 − 0.414375i 0.0497571 − 0.0655184i
\(41\) 1.14765 + 0.202362i 0.179233 + 0.0316036i 0.262544 − 0.964920i \(-0.415438\pi\)
−0.0833112 + 0.996524i \(0.526550\pi\)
\(42\) 4.46378 − 2.93285i 0.688776 − 0.452549i
\(43\) −1.51151 + 1.80135i −0.230503 + 0.274703i −0.868882 − 0.495019i \(-0.835161\pi\)
0.638379 + 0.769723i \(0.279605\pi\)
\(44\) 11.8744 − 0.385007i 1.79014 − 0.0580420i
\(45\) 0.231101 + 0.501170i 0.0344505 + 0.0747101i
\(46\) 1.63962 + 8.49163i 0.241748 + 1.25202i
\(47\) 6.83260 + 2.48686i 0.996637 + 0.362746i 0.788287 − 0.615308i \(-0.210968\pi\)
0.208350 + 0.978054i \(0.433191\pi\)
\(48\) −6.88908 + 0.735242i −0.994353 + 0.106123i
\(49\) −1.12275 + 1.94467i −0.160393 + 0.277810i
\(50\) −6.63772 − 2.29481i −0.938715 − 0.324536i
\(51\) 5.05855 − 6.56339i 0.708339 − 0.919058i
\(52\) 4.91488 − 4.40322i 0.681571 − 0.610617i
\(53\) 1.42960 + 1.70373i 0.196371 + 0.234026i 0.855240 − 0.518231i \(-0.173409\pi\)
−0.658869 + 0.752257i \(0.728965\pi\)
\(54\) 2.74544 − 6.81635i 0.373607 − 0.927587i
\(55\) 0.373759 + 1.02690i 0.0503977 + 0.138467i
\(56\) −5.18489 + 3.33959i −0.692860 + 0.446271i
\(57\) 5.69282 + 4.95901i 0.754033 + 0.656837i
\(58\) 1.14091 + 1.31578i 0.149809 + 0.172770i
\(59\) 5.82579 − 2.12042i 0.758454 − 0.276055i 0.0662951 − 0.997800i \(-0.478882\pi\)
0.692159 + 0.721745i \(0.256660\pi\)
\(60\) −0.261583 − 0.581103i −0.0337702 − 0.0750201i
\(61\) 9.65727 − 8.10341i 1.23649 − 1.03754i 0.238697 − 0.971094i \(-0.423280\pi\)
0.997790 − 0.0664414i \(-0.0211646\pi\)
\(62\) 0.176097 + 10.8652i 0.0223643 + 1.37989i
\(63\) −0.542311 − 6.51892i −0.0683247 − 0.821307i
\(64\) 7.96221 − 0.776659i 0.995276 − 0.0970824i
\(65\) 0.525649 + 0.303484i 0.0651987 + 0.0376425i
\(66\) 6.53652 − 13.0000i 0.804591 − 1.60019i
\(67\) −3.18066 + 8.73880i −0.388580 + 1.06761i 0.579061 + 0.815284i \(0.303419\pi\)
−0.967641 + 0.252330i \(0.918803\pi\)
\(68\) −5.90974 + 7.52535i −0.716662 + 0.912583i
\(69\) 10.0951 + 3.20667i 1.21531 + 0.386038i
\(70\) −0.440412 − 0.357549i −0.0526393 − 0.0427353i
\(71\) −3.36704 − 2.82528i −0.399594 − 0.335300i 0.420742 − 0.907180i \(-0.361770\pi\)
−0.820337 + 0.571881i \(0.806214\pi\)
\(72\) −3.24155 + 7.84170i −0.382021 + 0.924154i
\(73\) 1.20880 − 6.85547i 0.141480 − 0.802372i −0.828647 − 0.559772i \(-0.810889\pi\)
0.970126 − 0.242600i \(-0.0780002\pi\)
\(74\) 1.12428 − 2.02230i 0.130695 − 0.235087i
\(75\) −6.35418 + 5.79764i −0.733717 + 0.669454i
\(76\) −6.52120 − 5.78567i −0.748033 − 0.663662i
\(77\) − 12.9528i − 1.47611i
\(78\) −1.86009 − 7.86490i −0.210614 − 0.890524i
\(79\) −6.94773 − 1.22507i −0.781681 − 0.137831i −0.231453 − 0.972846i \(-0.574348\pi\)
−0.550228 + 0.835015i \(0.685459\pi\)
\(80\) 0.295939 + 0.673717i 0.0330870 + 0.0753239i
\(81\) −5.72604 − 6.94352i −0.636226 − 0.771502i
\(82\) −1.03876 + 1.27949i −0.114711 + 0.141296i
\(83\) 1.86888 + 3.23699i 0.205136 + 0.355306i 0.950176 − 0.311714i \(-0.100903\pi\)
−0.745040 + 0.667020i \(0.767570\pi\)
\(84\) 0.557775 + 7.53278i 0.0608582 + 0.821894i
\(85\) −0.827043 − 0.301019i −0.0897055 − 0.0326501i
\(86\) −1.18788 − 3.10612i −0.128093 − 0.334942i
\(87\) 2.08336 − 0.457208i 0.223359 − 0.0490178i
\(88\) −7.68383 + 14.9419i −0.819099 + 1.59281i
\(89\) 1.23073 − 0.217011i 0.130457 − 0.0230031i −0.108038 − 0.994147i \(-0.534457\pi\)
0.238495 + 0.971144i \(0.423346\pi\)
\(90\) −0.778273 − 0.0587368i −0.0820372 − 0.00619140i
\(91\) −4.62440 − 5.51115i −0.484769 − 0.577725i
\(92\) −11.6227 − 3.80853i −1.21175 − 0.397067i
\(93\) 11.7920 + 6.17053i 1.22277 + 0.639855i
\(94\) −7.76901 + 6.73650i −0.801312 + 0.694817i
\(95\) 0.318935 − 0.735718i 0.0327221 − 0.0754831i
\(96\) 3.82516 − 9.02043i 0.390403 − 0.920644i
\(97\) −10.6136 + 3.86302i −1.07764 + 0.392231i −0.819031 − 0.573750i \(-0.805488\pi\)
−0.258614 + 0.965981i \(0.583266\pi\)
\(98\) −1.63217 − 2.72408i −0.164874 − 0.275174i
\(99\) −10.1593 − 14.6416i −1.02105 − 1.47154i
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.119.12 yes 216
3.2 odd 2 inner 228.2.v.a.119.25 yes 216
4.3 odd 2 inner 228.2.v.a.119.33 yes 216
12.11 even 2 inner 228.2.v.a.119.4 yes 216
19.4 even 9 inner 228.2.v.a.23.4 ✓ 216
57.23 odd 18 inner 228.2.v.a.23.33 yes 216
76.23 odd 18 inner 228.2.v.a.23.25 yes 216
228.23 even 18 inner 228.2.v.a.23.12 yes 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
228.2.v.a.23.4 ✓ 216 19.4 even 9 inner
228.2.v.a.23.12 yes 216 228.23 even 18 inner
228.2.v.a.23.25 yes 216 76.23 odd 18 inner
228.2.v.a.23.33 yes 216 57.23 odd 18 inner
228.2.v.a.119.4 yes 216 12.11 even 2 inner
228.2.v.a.119.12 yes 216 1.1 even 1 trivial
228.2.v.a.119.25 yes 216 3.2 odd 2 inner
228.2.v.a.119.33 yes 216 4.3 odd 2 inner