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Properties

Label 230.2.l.a.33.6
Level $230$
Weight $2$
Character 230.33
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 33.6
Character \(\chi\) \(=\) 230.33
Dual form 230.2.l.a.7.6

$q$-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+(-0.997452 - 0.0713392i) q^{2} +(0.445581 + 2.04831i) q^{3} +(0.989821 + 0.142315i) q^{4} +(0.313060 + 2.21404i) q^{5} +(-0.298322 - 2.07487i) q^{6} +(0.390085 + 0.714388i) q^{7} +(-0.977147 - 0.212565i) q^{8} +(-1.26811 + 0.579129i) q^{9} +(-0.154315 - 2.23074i) q^{10} +(0.227504 + 0.197133i) q^{11} +(0.149542 + 2.09087i) q^{12} +(-2.16625 - 1.18286i) q^{13} +(-0.338128 - 0.740396i) q^{14} +(-4.39554 + 1.62778i) q^{15} +(0.959493 + 0.281733i) q^{16} +(-2.33418 + 3.11810i) q^{17} +(1.30620 - 0.487187i) q^{18} +(0.734740 - 5.11023i) q^{19} +(-0.00521751 + 2.23606i) q^{20} +(-1.28947 + 1.11733i) q^{21} +(-0.212861 - 0.212861i) q^{22} +(2.45971 + 4.11702i) q^{23} -2.09621i q^{24} +(-4.80399 + 1.38626i) q^{25} +(2.07635 + 1.33439i) q^{26} +(2.01736 + 2.69487i) q^{27} +(0.284447 + 0.762632i) q^{28} +(2.53852 - 0.364985i) q^{29} +(4.50047 - 1.31006i) q^{30} +(-1.50555 + 0.967558i) q^{31} +(-0.936950 - 0.349464i) q^{32} +(-0.302418 + 0.553837i) q^{33} +(2.55068 - 2.94364i) q^{34} +(-1.45957 + 1.08731i) q^{35} +(-1.33763 + 0.392762i) q^{36} +(-0.257255 + 0.689729i) q^{37} +(-1.09743 + 5.04480i) q^{38} +(1.45762 - 4.96421i) q^{39} +(0.164723 - 2.22999i) q^{40} +(-0.0867320 + 0.189917i) q^{41} +(1.36589 - 1.02250i) q^{42} +(10.0465 - 2.18548i) 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} +(-0.149542 + 2.09087i) q^{48} +(3.42630 - 5.33143i) q^{49} +(4.89064 - 1.04001i) q^{50} +(-7.42690 - 3.39175i) q^{51} +(-1.97587 - 1.47912i) q^{52} +(-8.05563 + 4.39871i) q^{53} +(-1.81997 - 2.83192i) q^{54} +(-0.365240 + 0.565419i) q^{55} +(-0.229317 - 0.780981i) q^{56} +(10.7947 - 0.772052i) q^{57} +(-2.55809 + 0.182959i) q^{58} +(-0.263189 - 0.896340i) q^{59} +(-4.58246 + 0.985661i) q^{60} +(-2.69749 - 4.19738i) q^{61} +(1.57074 - 0.857688i) q^{62} +(-0.908396 - 0.680016i) q^{63} +(0.909632 + 0.415415i) q^{64} +(1.94075 - 5.16649i) q^{65} +(0.341158 - 0.530852i) q^{66} +(0.705970 - 9.87075i) 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} +(1.36224 - 0.296336i) q^{72} +(3.89022 - 2.91218i) q^{73} +(0.305805 - 0.669619i) q^{74} +(-4.98005 - 9.22234i) q^{75} +(1.45452 - 4.95365i) q^{76} +(-0.0520837 + 0.239425i) q^{77} +(-1.80805 + 4.84758i) q^{78} +(11.3120 - 3.32150i) q^{79} +(-0.323389 + 2.21256i) q^{80} +(-7.35984 + 8.49371i) q^{81} +(0.100060 - 0.183245i) q^{82} +(1.05349 + 0.392931i) q^{83} +(-1.43536 + 0.922448i) q^{84} +(-7.63436 - 4.19183i) q^{85} +(-10.1768 + 1.46321i) q^{86} +(1.87872 + 5.03704i) q^{87} +(-0.180401 - 0.240988i) q^{88} +(11.4942 + 7.38685i) q^{89} +(1.48757 + 2.73946i) q^{90} -2.00896i q^{91} +(1.84876 + 4.42516i) q^{92} +(-2.65270 - 2.65270i) q^{93} +(-8.90933 + 7.71997i) q^{94} +(11.5443 + 0.0269369i) q^{95} +(0.298322 - 2.07487i) q^{96} +(-11.4609 + 4.27471i) q^{97} +(-3.79791 + 5.07342i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.997452 − 0.0713392i −0.705305 − 0.0504444i
\(3\) 0.445581 + 2.04831i 0.257257 + 1.18259i 0.906566 + 0.422065i \(0.138695\pi\)
−0.649309 + 0.760525i \(0.724942\pi\)
\(4\) 0.989821 + 0.142315i 0.494911 + 0.0711574i
\(5\) 0.313060 + 2.21404i 0.140005 + 0.990151i
\(6\) −0.298322 − 2.07487i −0.121789 − 0.847064i
\(7\) 0.390085 + 0.714388i 0.147438 + 0.270013i 0.940953 − 0.338539i \(-0.109933\pi\)
−0.793514 + 0.608552i \(0.791751\pi\)
\(8\) −0.977147 − 0.212565i −0.345474 − 0.0751532i
\(9\) −1.26811 + 0.579129i −0.422705 + 0.193043i
\(10\) −0.154315 − 2.23074i −0.0487986 − 0.705421i
\(11\) 0.227504 + 0.197133i 0.0685951 + 0.0594380i 0.688484 − 0.725252i \(-0.258277\pi\)
−0.619888 + 0.784690i \(0.712822\pi\)
\(12\) 0.149542 + 2.09087i 0.0431690 + 0.603582i
\(13\) −2.16625 − 1.18286i −0.600811 − 0.328068i 0.149897 − 0.988702i \(-0.452106\pi\)
−0.750708 + 0.660634i \(0.770288\pi\)
\(14\) −0.338128 − 0.740396i −0.0903684 − 0.197879i
\(15\) −4.39554 + 1.62778i −1.13492 + 0.420291i
\(16\) 0.959493 + 0.281733i 0.239873 + 0.0704331i
\(17\) −2.33418 + 3.11810i −0.566123 + 0.756251i −0.988673 − 0.150088i \(-0.952044\pi\)
0.422550 + 0.906340i \(0.361135\pi\)
\(18\) 1.30620 − 0.487187i 0.307874 − 0.114831i
\(19\) 0.734740 − 5.11023i 0.168561 − 1.17237i −0.713300 − 0.700859i \(-0.752800\pi\)
0.881861 − 0.471509i \(-0.156291\pi\)
\(20\) −0.00521751 + 2.23606i −0.00116667 + 0.499999i
\(21\) −1.28947 + 1.11733i −0.281385 + 0.243822i
\(22\) −0.212861 − 0.212861i −0.0453821 − 0.0453821i
\(23\) 2.45971 + 4.11702i 0.512885 + 0.858457i
\(24\) − 2.09621i − 0.427887i
\(25\) −4.80399 + 1.38626i −0.960797 + 0.277252i
\(26\) 2.07635 + 1.33439i 0.407206 + 0.261695i
\(27\) 2.01736 + 2.69487i 0.388240 + 0.518628i
\(28\) 0.284447 + 0.762632i 0.0537554 + 0.144124i
\(29\) 2.53852 − 0.364985i 0.471392 − 0.0677760i 0.0974748 − 0.995238i \(-0.468923\pi\)
0.373917 + 0.927462i \(0.378014\pi\)
\(30\) 4.50047 − 1.31006i 0.821670 − 0.239183i
\(31\) −1.50555 + 0.967558i −0.270405 + 0.173778i −0.668815 − 0.743429i \(-0.733198\pi\)
0.398411 + 0.917207i \(0.369562\pi\)
\(32\) −0.936950 − 0.349464i −0.165631 − 0.0617771i
\(33\) −0.302418 + 0.553837i −0.0526442 + 0.0964106i
\(34\) 2.55068 − 2.94364i 0.437438 − 0.504830i
\(35\) −1.45957 + 1.08731i −0.246712 + 0.183789i
\(36\) −1.33763 + 0.392762i −0.222938 + 0.0654604i
\(37\) −0.257255 + 0.689729i −0.0422925 + 0.113391i −0.956379 − 0.292128i \(-0.905637\pi\)
0.914087 + 0.405519i \(0.132909\pi\)
\(38\) −1.09743 + 5.04480i −0.178026 + 0.818374i
\(39\) 1.45762 − 4.96421i 0.233407 − 0.794910i
\(40\) 0.164723 − 2.22999i 0.0260450 − 0.352593i
\(41\) −0.0867320 + 0.189917i −0.0135453 + 0.0296600i −0.916284 − 0.400530i \(-0.868826\pi\)
0.902738 + 0.430190i \(0.141553\pi\)
\(42\) 1.36589 − 1.02250i 0.210762 − 0.157774i
\(43\) 10.0465 − 2.18548i 1.53208 − 0.333283i 0.634075 − 0.773272i \(-0.281381\pi\)
0.898003 + 0.439988i \(0.145017\pi\)
\(44\) 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 \(-0.0794002\pi\)
\(48\) −0.149542 + 2.09087i −0.0215845 + 0.301791i
\(49\) 3.42630 − 5.33143i 0.489472 − 0.761633i
\(50\) 4.89064 − 1.04001i 0.691641 − 0.147080i
\(51\) −7.42690 − 3.39175i −1.03997 − 0.474940i
\(52\) −1.97587 − 1.47912i −0.274003 − 0.205116i
\(53\) −8.05563 + 4.39871i −1.10653 + 0.604209i −0.925301 − 0.379233i \(-0.876188\pi\)
−0.181225 + 0.983442i \(0.558006\pi\)
\(54\) −1.81997 − 2.83192i −0.247666 − 0.385376i
\(55\) −0.365240 + 0.565419i −0.0492489 + 0.0762411i
\(56\) −0.229317 − 0.780981i −0.0306437 − 0.104363i
\(57\) 10.7947 − 0.772052i 1.42979 − 0.102261i
\(58\) −2.55809 + 0.182959i −0.335894 + 0.0240236i
\(59\) −0.263189 − 0.896340i −0.0342643 − 0.116693i 0.940585 − 0.339558i \(-0.110277\pi\)
−0.974849 + 0.222864i \(0.928459\pi\)
\(60\) −4.58246 + 0.985661i −0.591593 + 0.127248i
\(61\) −2.69749 − 4.19738i −0.345379 − 0.537420i 0.624495 − 0.781029i \(-0.285305\pi\)
−0.969873 + 0.243609i \(0.921669\pi\)
\(62\) 1.57074 − 0.857688i 0.199484 − 0.108926i
\(63\) −0.908396 − 0.680016i −0.114447 − 0.0856740i
\(64\) 0.909632 + 0.415415i 0.113704 + 0.0519269i
\(65\) 1.94075 − 5.16649i 0.240720 − 0.640825i
\(66\) 0.341158 − 0.530852i 0.0419936 − 0.0653433i
\(67\) 0.705970 − 9.87075i 0.0862479 − 1.20590i −0.751458 − 0.659781i \(-0.770649\pi\)
0.837706 − 0.546122i \(-0.183897\pi\)
\(68\) −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 0.975709 − 0.219069i \(-0.0703020\pi\)
−0.355697 + 0.934601i \(0.615757\pi\)
\(72\) 1.36224 − 0.296336i 0.160541 − 0.0349236i
\(73\) 3.89022 − 2.91218i 0.455316 − 0.340845i −0.346876 − 0.937911i \(-0.612758\pi\)
0.802192 + 0.597066i \(0.203667\pi\)
\(74\) 0.305805 − 0.669619i 0.0355491 − 0.0778416i
\(75\) −4.98005 − 9.22234i −0.575047 − 1.06490i
\(76\) 1.45452 − 4.95365i 0.166845 − 0.568223i
\(77\) −0.0520837 + 0.239425i −0.00593549 + 0.0272850i
\(78\) −1.80805 + 4.84758i −0.204722 + 0.548880i
\(79\) 11.3120 − 3.32150i 1.27270 − 0.373697i 0.425491 − 0.904963i \(-0.360101\pi\)
0.847206 + 0.531265i \(0.178283\pi\)
\(80\) −0.323389 + 2.21256i −0.0361560 + 0.247372i
\(81\) −7.35984 + 8.49371i −0.817760 + 0.943745i
\(82\) 0.100060 − 0.183245i 0.0110497 − 0.0202361i
\(83\) 1.05349 + 0.392931i 0.115635 + 0.0431298i 0.406616 − 0.913599i \(-0.366709\pi\)
−0.290981 + 0.956729i \(0.593981\pi\)
\(84\) −1.43536 + 0.922448i −0.156610 + 0.100647i
\(85\) −7.63436 − 4.19183i −0.828063 − 0.454668i
\(86\) −10.1768 + 1.46321i −1.09740 + 0.157782i
\(87\) 1.87872 + 5.03704i 0.201420 + 0.540028i
\(88\) −0.180401 − 0.240988i −0.0192308 − 0.0256894i
\(89\) 11.4942 + 7.38685i 1.21838 + 0.783004i 0.982041 − 0.188667i \(-0.0604167\pi\)
0.236337 + 0.971671i \(0.424053\pi\)
\(90\) 1.48757 + 2.73946i 0.156804 + 0.288765i
\(91\) − 2.00896i − 0.210597i
\(92\) 1.84876 + 4.42516i 0.192747 + 0.461355i
\(93\) −2.65270 − 2.65270i −0.275072 − 0.275072i
\(94\) −8.90933 + 7.71997i −0.918927 + 0.796255i
\(95\) 11.5443 + 0.0269369i 1.18442 + 0.00276367i
\(96\) 0.298322 − 2.07487i 0.0304473 − 0.211766i
\(97\) −11.4609 + 4.27471i −1.16368 + 0.434031i −0.855756 − 0.517380i \(-0.826907\pi\)
−0.307926 + 0.951410i \(0.599635\pi\)
\(98\) −3.79791 + 5.07342i −0.383647 + 0.512492i
\(99\) −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.33.6 yes 240
5.2 odd 4 inner 230.2.l.a.217.12 yes 240
23.7 odd 22 inner 230.2.l.a.53.12 yes 240
115.7 even 44 inner 230.2.l.a.7.6 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.l.a.7.6 ✓ 240 115.7 even 44 inner
230.2.l.a.33.6 yes 240 1.1 even 1 trivial
230.2.l.a.53.12 yes 240 23.7 odd 22 inner
230.2.l.a.217.12 yes 240 5.2 odd 4 inner