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Properties

Label 230.2.l.a.53.12
Level $230$
Weight $2$
Character 230.53
Analytic conductor $1.837$
Analytic rank $0$
Dimension $240$
Inner twists $4$

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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] = [230,2,Mod(7,230)] 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("230.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(44)) chi = DirichletCharacter(H, H._module([11, 38])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.l (of order \(44\), degree \(20\), 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: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label 53.12
Character \(\chi\) \(=\) 230.53
Dual form 230.2.l.a.217.12

$q$-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+(0.0713392 + 0.997452i) q^{2} +(2.04831 + 0.445581i) q^{3} +(-0.989821 + 0.142315i) q^{4} +(1.87827 + 1.21330i) q^{5} +(-0.298322 + 2.07487i) q^{6} +(-0.714388 - 0.390085i) q^{7} +(-0.212565 - 0.977147i) q^{8} +(1.26811 + 0.579129i) q^{9} +(-1.07621 + 1.96004i) q^{10} +(0.227504 - 0.197133i) q^{11} +(-2.09087 - 0.149542i) q^{12} +(-1.18286 - 2.16625i) q^{13} +(0.338128 - 0.740396i) q^{14} +(3.30666 + 3.32212i) q^{15} +(0.959493 - 0.281733i) q^{16} +(-3.11810 + 2.33418i) q^{17} +(-0.487187 + 1.30620i) q^{18} +(-0.734740 - 5.11023i) q^{19} +(-2.03183 - 0.933640i) q^{20} +(-1.28947 - 1.11733i) q^{21} +(0.212861 + 0.212861i) q^{22} +(4.11702 + 2.45971i) q^{23} -2.09621i q^{24} +(2.05583 + 4.55780i) q^{25} +(2.07635 - 1.33439i) q^{26} +(-2.69487 - 2.01736i) q^{27} +(0.762632 + 0.284447i) q^{28} +(-2.53852 - 0.364985i) q^{29} +(-3.07777 + 3.53523i) q^{30} +(-1.50555 - 0.967558i) q^{31} +(0.349464 + 0.936950i) q^{32} +(0.553837 - 0.302418i) q^{33} +(-2.55068 - 2.94364i) q^{34} +(-0.868528 - 1.59945i) q^{35} +(-1.33763 - 0.392762i) q^{36} +(-0.689729 + 0.257255i) q^{37} +(5.04480 - 1.09743i) q^{38} +(-1.45762 - 4.96421i) q^{39} +(0.786312 - 2.09325i) q^{40} +(-0.0867320 - 0.189917i) q^{41} +(1.02250 - 1.36589i) q^{42} +(-2.18548 + 10.0465i) q^{43} +(-0.197133 + 0.227504i) q^{44} +(1.67921 + 2.62636i) q^{45} +(-2.15974 + 4.28200i) q^{46} +(8.33589 - 8.33589i) q^{47} +(2.09087 - 0.149542i) q^{48} +(-3.42630 - 5.33143i) q^{49} +(-4.39953 + 2.37574i) q^{50} +(-7.42690 + 3.39175i) q^{51} +(1.47912 + 1.97587i) q^{52} +(4.39871 - 8.05563i) q^{53} +(1.81997 - 2.83192i) q^{54} +(0.666496 - 0.0942409i) q^{55} +(-0.229317 + 0.780981i) q^{56} +(0.772052 - 10.7947i) q^{57} +(0.182959 - 2.55809i) q^{58} +(0.263189 - 0.896340i) q^{59} +(-3.74579 - 2.81772i) q^{60} +(-2.69749 + 4.19738i) q^{61} +(0.857688 - 1.57074i) q^{62} +(-0.680016 - 0.908396i) q^{63} +(-0.909632 + 0.415415i) q^{64} +(0.406563 - 5.50399i) q^{65} +(0.341158 + 0.530852i) q^{66} +(9.87075 - 0.705970i) q^{67} +(2.75418 - 2.75418i) q^{68} +(7.33690 + 6.87271i) q^{69} +(1.53342 - 0.980418i) q^{70} +(5.22431 - 6.02918i) q^{71} +(0.296336 - 1.36224i) q^{72} +(-2.91218 + 3.89022i) q^{73} +(-0.305805 - 0.669619i) q^{74} +(2.18009 + 10.2518i) q^{75} +(1.45452 + 4.95365i) q^{76} +(-0.239425 + 0.0520837i) q^{77} +(4.84758 - 1.80805i) q^{78} +(-11.3120 - 3.32150i) q^{79} +(2.14402 + 0.634978i) q^{80} +(-7.35984 - 8.49371i) q^{81} +(0.183245 - 0.100060i) q^{82} +(0.392931 + 1.05349i) q^{83} +(1.43536 + 0.922448i) q^{84} +(-8.68871 + 0.601055i) q^{85} +(-10.1768 - 1.46321i) q^{86} +(-5.03704 - 1.87872i) q^{87} +(-0.240988 - 0.180401i) q^{88} +(-11.4942 + 7.38685i) q^{89} +(-2.49987 + 1.86230i) q^{90} +2.00896i q^{91} +(-4.42516 - 1.84876i) q^{92} +(-2.65270 - 2.65270i) q^{93} +(8.90933 + 7.71997i) q^{94} +(4.82018 - 10.4899i) q^{95} +(0.298322 + 2.07487i) q^{96} +(-4.27471 + 11.4609i) q^{97} +(5.07342 - 3.79791i) q^{98} +(0.402667 - 0.118234i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q - 8 q^{3} - 8 q^{6} - 8 q^{12} + 16 q^{13} + 24 q^{16} - 72 q^{18} - 80 q^{23} - 8 q^{26} + 16 q^{27} - 44 q^{28} + 24 q^{31} - 44 q^{33} - 8 q^{35} - 32 q^{36} - 88 q^{37} - 24 q^{41} - 8 q^{46}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{19}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0713392 + 0.997452i 0.0504444 + 0.705305i
\(3\) 2.04831 + 0.445581i 1.18259 + 0.257257i 0.760525 − 0.649309i \(-0.224942\pi\)
0.422065 + 0.906566i \(0.361305\pi\)
\(4\) −0.989821 + 0.142315i −0.494911 + 0.0711574i
\(5\) 1.87827 + 1.21330i 0.839990 + 0.542602i
\(6\) −0.298322 + 2.07487i −0.121789 + 0.847064i
\(7\) −0.714388 − 0.390085i −0.270013 − 0.147438i 0.338539 − 0.940953i \(-0.390067\pi\)
−0.608552 + 0.793514i \(0.708249\pi\)
\(8\) −0.212565 − 0.977147i −0.0751532 − 0.345474i
\(9\) 1.26811 + 0.579129i 0.422705 + 0.193043i
\(10\) −1.07621 + 1.96004i −0.340327 + 0.619820i
\(11\) 0.227504 − 0.197133i 0.0685951 − 0.0594380i −0.619888 − 0.784690i \(-0.712822\pi\)
0.688484 + 0.725252i \(0.258277\pi\)
\(12\) −2.09087 − 0.149542i −0.603582 − 0.0431690i
\(13\) −1.18286 − 2.16625i −0.328068 − 0.600811i 0.660634 − 0.750708i \(-0.270288\pi\)
−0.988702 + 0.149897i \(0.952106\pi\)
\(14\) 0.338128 − 0.740396i 0.0903684 − 0.197879i
\(15\) 3.30666 + 3.32212i 0.853775 + 0.857769i
\(16\) 0.959493 − 0.281733i 0.239873 − 0.0704331i
\(17\) −3.11810 + 2.33418i −0.756251 + 0.566123i −0.906340 − 0.422550i \(-0.861135\pi\)
0.150088 + 0.988673i \(0.452044\pi\)
\(18\) −0.487187 + 1.30620i −0.114831 + 0.307874i
\(19\) −0.734740 − 5.11023i −0.168561 − 1.17237i −0.881861 − 0.471509i \(-0.843709\pi\)
0.713300 − 0.700859i \(-0.247200\pi\)
\(20\) −2.03183 − 0.933640i −0.454330 − 0.208768i
\(21\) −1.28947 − 1.11733i −0.281385 − 0.243822i
\(22\) 0.212861 + 0.212861i 0.0453821 + 0.0453821i
\(23\) 4.11702 + 2.45971i 0.858457 + 0.512885i
\(24\) − 2.09621i − 0.427887i
\(25\) 2.05583 + 4.55780i 0.411166 + 0.911561i
\(26\) 2.07635 − 1.33439i 0.407206 − 0.261695i
\(27\) −2.69487 − 2.01736i −0.518628 − 0.388240i
\(28\) 0.762632 + 0.284447i 0.144124 + 0.0537554i
\(29\) −2.53852 − 0.364985i −0.471392 − 0.0677760i −0.0974748 − 0.995238i \(-0.531077\pi\)
−0.373917 + 0.927462i \(0.621986\pi\)
\(30\) −3.07777 + 3.53523i −0.561920 + 0.645442i
\(31\) −1.50555 − 0.967558i −0.270405 − 0.173778i 0.398411 − 0.917207i \(-0.369562\pi\)
−0.668815 + 0.743429i \(0.733198\pi\)
\(32\) 0.349464 + 0.936950i 0.0617771 + 0.165631i
\(33\) 0.553837 − 0.302418i 0.0964106 − 0.0526442i
\(34\) −2.55068 − 2.94364i −0.437438 − 0.504830i
\(35\) −0.868528 − 1.59945i −0.146808 − 0.270357i
\(36\) −1.33763 − 0.392762i −0.222938 − 0.0654604i
\(37\) −0.689729 + 0.257255i −0.113391 + 0.0422925i −0.405519 − 0.914087i \(-0.632909\pi\)
0.292128 + 0.956379i \(0.405637\pi\)
\(38\) 5.04480 − 1.09743i 0.818374 − 0.178026i
\(39\) −1.45762 − 4.96421i −0.233407 − 0.794910i
\(40\) 0.786312 − 2.09325i 0.124327 − 0.330973i
\(41\) −0.0867320 − 0.189917i −0.0135453 − 0.0296600i 0.902738 − 0.430190i \(-0.141553\pi\)
−0.916284 + 0.400530i \(0.868826\pi\)
\(42\) 1.02250 − 1.36589i 0.157774 − 0.210762i
\(43\) −2.18548 + 10.0465i −0.333283 + 1.53208i 0.439988 + 0.898003i \(0.354983\pi\)
−0.773272 + 0.634075i \(0.781381\pi\)
\(44\) −0.197133 + 0.227504i −0.0297190 + 0.0342975i
\(45\) 1.67921 + 2.62636i 0.250322 + 0.391515i
\(46\) −2.15974 + 4.28200i −0.318436 + 0.631346i
\(47\) 8.33589 − 8.33589i 1.21591 − 1.21591i 0.246864 − 0.969050i \(-0.420600\pi\)
0.969050 − 0.246864i \(-0.0794002\pi\)
\(48\) 2.09087 − 0.149542i 0.301791 − 0.0215845i
\(49\) −3.42630 − 5.33143i −0.489472 − 0.761633i
\(50\) −4.39953 + 2.37574i −0.622187 + 0.335980i
\(51\) −7.42690 + 3.39175i −1.03997 + 0.474940i
\(52\) 1.47912 + 1.97587i 0.205116 + 0.274003i
\(53\) 4.39871 − 8.05563i 0.604209 − 1.10653i −0.379233 − 0.925301i \(-0.623812\pi\)
0.983442 − 0.181225i \(-0.0580061\pi\)
\(54\) 1.81997 − 2.83192i 0.247666 − 0.385376i
\(55\) 0.666496 − 0.0942409i 0.0898703 − 0.0127074i
\(56\) −0.229317 + 0.780981i −0.0306437 + 0.104363i
\(57\) 0.772052 − 10.7947i 0.102261 − 1.42979i
\(58\) 0.182959 − 2.55809i 0.0240236 − 0.335894i
\(59\) 0.263189 − 0.896340i 0.0342643 − 0.116693i −0.940585 − 0.339558i \(-0.889723\pi\)
0.974849 + 0.222864i \(0.0715407\pi\)
\(60\) −3.74579 − 2.81772i −0.483579 − 0.363766i
\(61\) −2.69749 + 4.19738i −0.345379 + 0.537420i −0.969873 − 0.243609i \(-0.921669\pi\)
0.624495 + 0.781029i \(0.285305\pi\)
\(62\) 0.857688 − 1.57074i 0.108926 − 0.199484i
\(63\) −0.680016 − 0.908396i −0.0856740 − 0.114447i
\(64\) −0.909632 + 0.415415i −0.113704 + 0.0519269i
\(65\) 0.406563 − 5.50399i 0.0504280 − 0.682685i
\(66\) 0.341158 + 0.530852i 0.0419936 + 0.0653433i
\(67\) 9.87075 − 0.705970i 1.20590 − 0.0862479i 0.546122 − 0.837706i \(-0.316103\pi\)
0.659781 + 0.751458i \(0.270649\pi\)
\(68\) 2.75418 − 2.75418i 0.333993 − 0.333993i
\(69\) 7.33690 + 6.87271i 0.883259 + 0.827377i
\(70\) 1.53342 − 0.980418i 0.183278 − 0.117182i
\(71\) 5.22431 − 6.02918i 0.620012 − 0.715532i −0.355697 − 0.934601i \(-0.615757\pi\)
0.975709 + 0.219069i \(0.0703020\pi\)
\(72\) 0.296336 − 1.36224i 0.0349236 − 0.160541i
\(73\) −2.91218 + 3.89022i −0.340845 + 0.455316i −0.937911 − 0.346876i \(-0.887242\pi\)
0.597066 + 0.802192i \(0.296333\pi\)
\(74\) −0.305805 − 0.669619i −0.0355491 − 0.0778416i
\(75\) 2.18009 + 10.2518i 0.251735 + 1.18378i
\(76\) 1.45452 + 4.95365i 0.166845 + 0.568223i
\(77\) −0.239425 + 0.0520837i −0.0272850 + 0.00593549i
\(78\) 4.84758 − 1.80805i 0.548880 − 0.204722i
\(79\) −11.3120 − 3.32150i −1.27270 − 0.373697i −0.425491 − 0.904963i \(-0.639899\pi\)
−0.847206 + 0.531265i \(0.821717\pi\)
\(80\) 2.14402 + 0.634978i 0.239708 + 0.0709927i
\(81\) −7.35984 − 8.49371i −0.817760 − 0.943745i
\(82\) 0.183245 − 0.100060i 0.0202361 − 0.0110497i
\(83\) 0.392931 + 1.05349i 0.0431298 + 0.115635i 0.956729 − 0.290981i \(-0.0939815\pi\)
−0.913599 + 0.406616i \(0.866709\pi\)
\(84\) 1.43536 + 0.922448i 0.156610 + 0.100647i
\(85\) −8.68871 + 0.601055i −0.942423 + 0.0651935i
\(86\) −10.1768 − 1.46321i −1.09740 − 0.157782i
\(87\) −5.03704 − 1.87872i −0.540028 − 0.201420i
\(88\) −0.240988 − 0.180401i −0.0256894 − 0.0192308i
\(89\) −11.4942 + 7.38685i −1.21838 + 0.783004i −0.982041 − 0.188667i \(-0.939583\pi\)
−0.236337 + 0.971671i \(0.575947\pi\)
\(90\) −2.49987 + 1.86230i −0.263510 + 0.196303i
\(91\) 2.00896i 0.210597i
\(92\) −4.42516 − 1.84876i −0.461355 − 0.192747i
\(93\) −2.65270 − 2.65270i −0.275072 − 0.275072i
\(94\) 8.90933 + 7.71997i 0.918927 + 0.796255i
\(95\) 4.82018 − 10.4899i 0.494540 − 1.07624i
\(96\) 0.298322 + 2.07487i 0.0304473 + 0.211766i
\(97\) −4.27471 + 11.4609i −0.434031 + 1.16368i 0.517380 + 0.855756i \(0.326907\pi\)
−0.951410 + 0.307926i \(0.900365\pi\)
\(98\) 5.07342 − 3.79791i 0.512492 − 0.383647i
\(99\) 0.402667 − 0.118234i 0.0404696 − 0.0118829i
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 230.2.l.a.53.12 yes 240
5.2 odd 4 inner 230.2.l.a.7.6 ✓ 240
23.10 odd 22 inner 230.2.l.a.33.6 yes 240
115.102 even 44 inner 230.2.l.a.217.12 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.7.6 ✓ 240 5.2 odd 4 inner
230.2.l.a.33.6 yes 240 23.10 odd 22 inner
230.2.l.a.53.12 yes 240 1.1 even 1 trivial
230.2.l.a.217.12 yes 240 115.102 even 44 inner