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Properties

Label 465.2.ba.a.4.11
Level $465$
Weight $2$
Character 465.4
Analytic conductor $3.713$
Analytic rank $0$
Dimension $128$
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] = [465,2,Mod(4,465)] 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("465.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 5, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.ba (of order \(10\), degree \(4\), 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 4.11
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.11

$q$-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+(-1.03892 + 0.337565i) q^{2} +(0.951057 + 0.309017i) q^{3} +(-0.652633 + 0.474165i) q^{4} +(-1.88420 + 1.20407i) q^{5} -1.09238 q^{6} +(-1.49943 - 2.06379i) q^{7} +(1.80214 - 2.48044i) q^{8} +(0.809017 + 0.587785i) q^{9} +(1.55108 - 1.88697i) q^{10} +(1.85450 - 1.34737i) q^{11} +(-0.767216 + 0.249284i) q^{12} +(-5.74222 - 1.86576i) q^{13} +(2.25445 + 1.63796i) q^{14} +(-2.16406 + 0.562892i) q^{15} +(-0.536404 + 1.65088i) q^{16} +(3.66324 - 5.04202i) q^{17} +(-1.03892 - 0.337565i) q^{18} +(1.61635 + 4.97461i) q^{19} +(0.658760 - 1.67924i) q^{20} +(-0.788299 - 2.42614i) q^{21} +(-1.47185 + 2.02582i) q^{22} +(3.93228 - 5.41232i) q^{23} +(2.48044 - 1.80214i) q^{24} +(2.10041 - 4.53743i) q^{25} +6.59552 q^{26} +(0.587785 + 0.809017i) q^{27} +(1.95716 + 0.635920i) q^{28} +(1.01508 + 3.12411i) q^{29} +(2.05827 - 1.31531i) q^{30} +(-0.0310873 - 5.56768i) q^{31} +4.23578i q^{32} +(2.18009 - 0.708355i) q^{33} +(-2.10380 + 6.47483i) q^{34} +(5.31019 + 2.08317i) q^{35} -0.806698 q^{36} +3.20681i q^{37} +(-3.35851 - 4.62259i) q^{38} +(-4.88463 - 3.54889i) q^{39} +(-0.408969 + 6.84355i) q^{40} +(-3.91280 - 12.0424i) q^{41} +(1.63796 + 2.25445i) q^{42} +(-2.84817 + 0.925426i) q^{43} +(-0.571429 + 1.75868i) q^{44} +(-2.23209 - 0.133389i) q^{45} +(-2.25831 + 6.95035i) q^{46} +(-6.26098 - 2.03432i) q^{47} +(-1.02030 + 1.40432i) q^{48} +(0.152176 - 0.468350i) q^{49} +(-0.650482 + 5.42304i) q^{50} +(5.04202 - 3.66324i) q^{51} +(4.63224 - 1.50511i) q^{52} +(4.52143 - 6.22321i) q^{53} +(-0.883757 - 0.642087i) q^{54} +(-1.87191 + 4.77167i) q^{55} -7.82131 q^{56} +5.23061i q^{57} +(-2.10918 - 2.90304i) q^{58} +(-2.46107 + 7.57440i) q^{59} +(1.14543 - 1.39348i) q^{60} -3.75935 q^{61} +(1.91175 + 5.77387i) q^{62} -2.55099i q^{63} +(-2.50266 - 7.70239i) q^{64} +(13.0660 - 3.39859i) q^{65} +(-2.02582 + 1.47185i) q^{66} -9.07884i q^{67} +5.02757i q^{68} +(5.41232 - 3.93228i) q^{69} +(-6.22006 - 0.371709i) q^{70} +(4.83228 + 3.51086i) q^{71} +(2.91593 - 0.947443i) q^{72} +(-2.29723 - 3.16186i) q^{73} +(-1.08251 - 3.33161i) q^{74} +(3.39976 - 3.66629i) q^{75} +(-3.41367 - 2.48018i) q^{76} +(-5.56139 - 1.80701i) q^{77} +(6.27271 + 2.03813i) q^{78} +(-8.72392 - 6.33830i) q^{79} +(-0.977091 - 3.75646i) q^{80} +(0.309017 + 0.951057i) q^{81} +(8.13016 + 11.1902i) q^{82} +(-0.304798 + 0.0990349i) q^{83} +(1.66486 + 1.20959i) q^{84} +(-0.831316 + 13.9110i) q^{85} +(2.64662 - 1.92288i) q^{86} +3.28488i q^{87} -7.02812i q^{88} +(7.45504 - 5.41640i) q^{89} +(2.36398 - 0.614894i) q^{90} +(4.75954 + 14.6484i) q^{91} +5.39680i q^{92} +(1.69094 - 5.30478i) q^{93} +7.19136 q^{94} +(-9.03532 - 7.42695i) q^{95} +(-1.30893 + 4.02846i) q^{96} +(-4.12257 - 5.67423i) q^{97} +0.537947i q^{98} +2.29228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 36 q^{4} - 20 q^{6} + 32 q^{9} + 2 q^{10} + 4 q^{14} - 4 q^{15} - 40 q^{16} - 4 q^{19} + 36 q^{20} + 20 q^{21} - 20 q^{24} + 12 q^{25} + 24 q^{26} - 28 q^{29} - 8 q^{30} - 12 q^{31} - 44 q^{34}+ \cdots - 20 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.03892 + 0.337565i −0.734626 + 0.238695i −0.652353 − 0.757915i \(-0.726218\pi\)
−0.0822733 + 0.996610i \(0.526218\pi\)
\(3\) 0.951057 + 0.309017i 0.549093 + 0.178411i
\(4\) −0.652633 + 0.474165i −0.326316 + 0.237083i
\(5\) −1.88420 + 1.20407i −0.842640 + 0.538478i
\(6\) −1.09238 −0.445964
\(7\) −1.49943 − 2.06379i −0.566733 − 0.780041i 0.425430 − 0.904991i \(-0.360123\pi\)
−0.992163 + 0.124950i \(0.960123\pi\)
\(8\) 1.80214 − 2.48044i 0.637154 − 0.876968i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) 1.55108 − 1.88697i 0.490493 − 0.596714i
\(11\) 1.85450 − 1.34737i 0.559152 − 0.406248i −0.271997 − 0.962298i \(-0.587684\pi\)
0.831148 + 0.556051i \(0.187684\pi\)
\(12\) −0.767216 + 0.249284i −0.221476 + 0.0719619i
\(13\) −5.74222 − 1.86576i −1.59261 − 0.517469i −0.627342 − 0.778744i \(-0.715857\pi\)
−0.965265 + 0.261275i \(0.915857\pi\)
\(14\) 2.25445 + 1.63796i 0.602528 + 0.437762i
\(15\) −2.16406 + 0.562892i −0.558758 + 0.145338i
\(16\) −0.536404 + 1.65088i −0.134101 + 0.412721i
\(17\) 3.66324 − 5.04202i 0.888466 − 1.22287i −0.0855372 − 0.996335i \(-0.527261\pi\)
0.974003 − 0.226534i \(-0.0727393\pi\)
\(18\) −1.03892 − 0.337565i −0.244875 − 0.0795648i
\(19\) 1.61635 + 4.97461i 0.370816 + 1.14125i 0.946258 + 0.323411i \(0.104830\pi\)
−0.575443 + 0.817842i \(0.695170\pi\)
\(20\) 0.658760 − 1.67924i 0.147303 − 0.375489i
\(21\) −0.788299 − 2.42614i −0.172021 − 0.529426i
\(22\) −1.47185 + 2.02582i −0.313799 + 0.431907i
\(23\) 3.93228 − 5.41232i 0.819937 − 1.12855i −0.169777 − 0.985483i \(-0.554305\pi\)
0.989714 − 0.143063i \(-0.0456953\pi\)
\(24\) 2.48044 − 1.80214i 0.506317 − 0.367861i
\(25\) 2.10041 − 4.53743i 0.420083 − 0.907486i
\(26\) 6.59552 1.29349
\(27\) 0.587785 + 0.809017i 0.113119 + 0.155695i
\(28\) 1.95716 + 0.635920i 0.369868 + 0.120178i
\(29\) 1.01508 + 3.12411i 0.188496 + 0.580132i 0.999991 − 0.00422760i \(-0.00134569\pi\)
−0.811495 + 0.584360i \(0.801346\pi\)
\(30\) 2.05827 − 1.31531i 0.375787 − 0.240142i
\(31\) −0.0310873 − 5.56768i −0.00558345 − 0.999984i
\(32\) 4.23578i 0.748787i
\(33\) 2.18009 − 0.708355i 0.379505 − 0.123309i
\(34\) −2.10380 + 6.47483i −0.360798 + 1.11042i
\(35\) 5.31019 + 2.08317i 0.897586 + 0.352120i
\(36\) −0.806698 −0.134450
\(37\) 3.20681i 0.527196i 0.964633 + 0.263598i \(0.0849092\pi\)
−0.964633 + 0.263598i \(0.915091\pi\)
\(38\) −3.35851 − 4.62259i −0.544822 − 0.749883i
\(39\) −4.88463 − 3.54889i −0.782167 − 0.568277i
\(40\) −0.408969 + 6.84355i −0.0646637 + 1.08206i
\(41\) −3.91280 − 12.0424i −0.611077 − 1.88070i −0.447837 − 0.894115i \(-0.647806\pi\)
−0.163240 − 0.986586i \(-0.552194\pi\)
\(42\) 1.63796 + 2.25445i 0.252742 + 0.347870i
\(43\) −2.84817 + 0.925426i −0.434341 + 0.141126i −0.518023 − 0.855366i \(-0.673332\pi\)
0.0836820 + 0.996493i \(0.473332\pi\)
\(44\) −0.571429 + 1.75868i −0.0861461 + 0.265131i
\(45\) −2.23209 − 0.133389i −0.332740 − 0.0198844i
\(46\) −2.25831 + 6.95035i −0.332969 + 1.02477i
\(47\) −6.26098 − 2.03432i −0.913258 − 0.296736i −0.185560 − 0.982633i \(-0.559410\pi\)
−0.727698 + 0.685897i \(0.759410\pi\)
\(48\) −1.02030 + 1.40432i −0.147268 + 0.202697i
\(49\) 0.152176 − 0.468350i 0.0217395 − 0.0669072i
\(50\) −0.650482 + 5.42304i −0.0919921 + 0.766934i
\(51\) 5.04202 − 3.66324i 0.706024 − 0.512956i
\(52\) 4.63224 − 1.50511i 0.642377 − 0.208721i
\(53\) 4.52143 − 6.22321i 0.621066 − 0.854824i −0.376364 − 0.926472i \(-0.622826\pi\)
0.997430 + 0.0716477i \(0.0228257\pi\)
\(54\) −0.883757 − 0.642087i −0.120264 − 0.0873770i
\(55\) −1.87191 + 4.77167i −0.252408 + 0.643411i
\(56\) −7.82131 −1.04517
\(57\) 5.23061i 0.692812i
\(58\) −2.10918 − 2.90304i −0.276949 − 0.381187i
\(59\) −2.46107 + 7.57440i −0.320404 + 0.986103i 0.653068 + 0.757299i \(0.273481\pi\)
−0.973473 + 0.228804i \(0.926519\pi\)
\(60\) 1.14543 − 1.39348i 0.147875 − 0.179898i
\(61\) −3.75935 −0.481335 −0.240668 − 0.970608i \(-0.577366\pi\)
−0.240668 + 0.970608i \(0.577366\pi\)
\(62\) 1.91175 + 5.77387i 0.242793 + 0.733282i
\(63\) − 2.55099i − 0.321394i
\(64\) −2.50266 − 7.70239i −0.312832 − 0.962799i
\(65\) 13.0660 − 3.39859i 1.62064 − 0.421543i
\(66\) −2.02582 + 1.47185i −0.249361 + 0.181172i
\(67\) − 9.07884i − 1.10916i −0.832132 − 0.554578i \(-0.812880\pi\)
0.832132 − 0.554578i \(-0.187120\pi\)
\(68\) 5.02757i 0.609682i
\(69\) 5.41232 − 3.93228i 0.651566 − 0.473391i
\(70\) −6.22006 − 0.371709i −0.743440 − 0.0444278i
\(71\) 4.83228 + 3.51086i 0.573486 + 0.416662i 0.836370 − 0.548165i \(-0.184674\pi\)
−0.262884 + 0.964828i \(0.584674\pi\)
\(72\) 2.91593 − 0.947443i 0.343646 − 0.111657i
\(73\) −2.29723 − 3.16186i −0.268870 − 0.370068i 0.653138 − 0.757239i \(-0.273452\pi\)
−0.922008 + 0.387171i \(0.873452\pi\)
\(74\) −1.08251 − 3.33161i −0.125839 − 0.387292i
\(75\) 3.39976 − 3.66629i 0.392570 − 0.423346i
\(76\) −3.41367 − 2.48018i −0.391575 − 0.284496i
\(77\) −5.56139 − 1.80701i −0.633779 − 0.205927i
\(78\) 6.27271 + 2.03813i 0.710245 + 0.230772i
\(79\) −8.72392 − 6.33830i −0.981518 − 0.713114i −0.0234704 − 0.999725i \(-0.507472\pi\)
−0.958047 + 0.286610i \(0.907472\pi\)
\(80\) −0.977091 − 3.75646i −0.109242 − 0.419985i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) 8.13016 + 11.1902i 0.897826 + 1.23575i
\(83\) −0.304798 + 0.0990349i −0.0334559 + 0.0108705i −0.325697 − 0.945474i \(-0.605599\pi\)
0.292241 + 0.956345i \(0.405599\pi\)
\(84\) 1.66486 + 1.20959i 0.181651 + 0.131977i
\(85\) −0.831316 + 13.9110i −0.0901689 + 1.50886i
\(86\) 2.64662 − 1.92288i 0.285393 − 0.207350i
\(87\) 3.28488i 0.352176i
\(88\) − 7.02812i − 0.749200i
\(89\) 7.45504 − 5.41640i 0.790232 − 0.574137i −0.117800 − 0.993037i \(-0.537584\pi\)
0.908032 + 0.418900i \(0.137584\pi\)
\(90\) 2.36398 − 0.614894i 0.249186 − 0.0648155i
\(91\) 4.75954 + 14.6484i 0.498935 + 1.53556i
\(92\) 5.39680i 0.562656i
\(93\) 1.69094 − 5.30478i 0.175342 − 0.550080i
\(94\) 7.19136 0.741732
\(95\) −9.03532 − 7.42695i −0.927004 − 0.761989i
\(96\) −1.30893 + 4.02846i −0.133592 + 0.411153i
\(97\) −4.12257 − 5.67423i −0.418583 − 0.576130i 0.546702 − 0.837327i \(-0.315883\pi\)
−0.965286 + 0.261197i \(0.915883\pi\)
\(98\) 0.537947i 0.0543409i
\(99\) 2.29228 0.230383
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 465.2.ba.a.4.11 ✓ 128
5.4 even 2 inner 465.2.ba.a.4.22 yes 128
31.8 even 5 inner 465.2.ba.a.349.22 yes 128
155.39 even 10 inner 465.2.ba.a.349.11 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.11 ✓ 128 1.1 even 1 trivial
465.2.ba.a.4.22 yes 128 5.4 even 2 inner
465.2.ba.a.349.11 yes 128 155.39 even 10 inner
465.2.ba.a.349.22 yes 128 31.8 even 5 inner