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Properties

Label 465.2.ba.a.4.5
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 4.5
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.5

$q$-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+(-2.28150 + 0.741303i) q^{2} +(0.951057 + 0.309017i) q^{3} +(3.03766 - 2.20699i) q^{4} +(1.23533 + 1.86386i) q^{5} -2.39891 q^{6} +(-2.71113 - 3.73155i) q^{7} +(-2.47429 + 3.40556i) q^{8} +(0.809017 + 0.587785i) q^{9} +(-4.20008 - 3.33663i) q^{10} +(0.981656 - 0.713215i) q^{11} +(3.57099 - 1.16028i) q^{12} +(4.41533 + 1.43463i) q^{13} +(8.95165 + 6.50376i) q^{14} +(0.598905 + 2.15437i) q^{15} +(0.799953 - 2.46200i) q^{16} +(2.60818 - 3.58985i) q^{17} +(-2.28150 - 0.741303i) q^{18} +(-2.17912 - 6.70665i) q^{19} +(7.86603 + 2.93541i) q^{20} +(-1.42533 - 4.38670i) q^{21} +(-1.71094 + 2.35490i) q^{22} +(0.516415 - 0.710785i) q^{23} +(-3.40556 + 2.47429i) q^{24} +(-1.94792 + 4.60495i) q^{25} -11.1371 q^{26} +(0.587785 + 0.809017i) q^{27} +(-16.4710 - 5.35176i) q^{28} +(1.85230 + 5.70078i) q^{29} +(-2.96344 - 4.47122i) q^{30} +(2.04875 - 5.17712i) q^{31} -2.20896i q^{32} +(1.15401 - 0.374959i) q^{33} +(-3.28938 + 10.1237i) q^{34} +(3.60594 - 9.66286i) q^{35} +3.75476 q^{36} -2.72468i q^{37} +(9.94332 + 13.6858i) q^{38} +(3.75590 + 2.72882i) q^{39} +(-9.40404 - 0.404721i) q^{40} +(0.915375 + 2.81723i) q^{41} +(6.50376 + 8.95165i) q^{42} +(8.46042 - 2.74896i) q^{43} +(1.40788 - 4.33301i) q^{44} +(-0.0961444 + 2.23400i) q^{45} +(-0.651293 + 2.00447i) q^{46} +(2.03214 + 0.660282i) q^{47} +(1.52160 - 2.09430i) q^{48} +(-4.41113 + 13.5761i) q^{49} +(1.03051 - 11.9502i) q^{50} +(3.58985 - 2.60818i) q^{51} +(16.5785 - 5.38668i) q^{52} +(1.04290 - 1.43542i) q^{53} +(-1.94076 - 1.41004i) q^{54} +(2.54200 + 0.948610i) q^{55} +19.4162 q^{56} -7.05178i q^{57} +(-8.45202 - 11.6332i) q^{58} +(-2.08128 + 6.40551i) q^{59} +(6.57395 + 5.22248i) q^{60} +12.0376 q^{61} +(-0.836398 + 13.3303i) q^{62} -4.61245i q^{63} +(3.23741 + 9.96373i) q^{64} +(2.78045 + 10.0018i) q^{65} +(-2.35490 + 1.71094i) q^{66} +2.62546i q^{67} -16.6610i q^{68} +(0.710785 - 0.516415i) q^{69} +(-1.06382 + 24.7189i) q^{70} +(1.96329 + 1.42641i) q^{71} +(-4.00348 + 1.30081i) q^{72} +(4.47451 + 6.15864i) q^{73} +(2.01981 + 6.21634i) q^{74} +(-3.27559 + 3.77763i) q^{75} +(-21.4210 - 15.5632i) q^{76} +(-5.32280 - 1.72948i) q^{77} +(-10.5920 - 3.44154i) q^{78} +(-3.02338 - 2.19661i) q^{79} +(5.57702 - 1.55039i) q^{80} +(0.309017 + 0.951057i) q^{81} +(-4.17685 - 5.74894i) q^{82} +(-10.7150 + 3.48153i) q^{83} +(-14.0111 - 10.1797i) q^{84} +(9.91292 + 0.426621i) q^{85} +(-17.2646 + 12.5435i) q^{86} +5.99416i q^{87} +5.10779i q^{88} +(3.71151 - 2.69657i) q^{89} +(-1.43672 - 5.16814i) q^{90} +(-6.61715 - 20.3655i) q^{91} -3.29885i q^{92} +(3.54830 - 4.29064i) q^{93} -5.12579 q^{94} +(9.80829 - 12.3465i) q^{95} +(0.682606 - 2.10084i) q^{96} +(-4.25259 - 5.85319i) q^{97} -34.2438i q^{98} +1.21339 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.28150 + 0.741303i −1.61326 + 0.524181i −0.970339 − 0.241750i \(-0.922279\pi\)
−0.642923 + 0.765930i \(0.722279\pi\)
\(3\) 0.951057 + 0.309017i 0.549093 + 0.178411i
\(4\) 3.03766 − 2.20699i 1.51883 − 1.10350i
\(5\) 1.23533 + 1.86386i 0.552456 + 0.833542i
\(6\) −2.39891 −0.979350
\(7\) −2.71113 − 3.73155i −1.02471 − 1.41039i −0.908847 − 0.417130i \(-0.863036\pi\)
−0.115865 − 0.993265i \(-0.536964\pi\)
\(8\) −2.47429 + 3.40556i −0.874792 + 1.20405i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) −4.20008 − 3.33663i −1.32818 − 1.05513i
\(11\) 0.981656 − 0.713215i 0.295980 − 0.215042i −0.429877 − 0.902887i \(-0.641443\pi\)
0.725858 + 0.687845i \(0.241443\pi\)
\(12\) 3.57099 − 1.16028i 1.03086 − 0.334945i
\(13\) 4.41533 + 1.43463i 1.22459 + 0.397894i 0.848752 − 0.528790i \(-0.177354\pi\)
0.375840 + 0.926685i \(0.377354\pi\)
\(14\) 8.95165 + 6.50376i 2.39243 + 1.73820i
\(15\) 0.598905 + 2.15437i 0.154637 + 0.556256i
\(16\) 0.799953 − 2.46200i 0.199988 − 0.615500i
\(17\) 2.60818 − 3.58985i 0.632576 − 0.870666i −0.365617 − 0.930766i \(-0.619142\pi\)
0.998192 + 0.0600996i \(0.0191418\pi\)
\(18\) −2.28150 − 0.741303i −0.537754 − 0.174727i
\(19\) −2.17912 − 6.70665i −0.499925 − 1.53861i −0.809139 − 0.587618i \(-0.800066\pi\)
0.309214 − 0.950992i \(-0.399934\pi\)
\(20\) 7.86603 + 2.93541i 1.75890 + 0.656377i
\(21\) −1.42533 − 4.38670i −0.311032 − 0.957257i
\(22\) −1.71094 + 2.35490i −0.364773 + 0.502067i
\(23\) 0.516415 − 0.710785i 0.107680 − 0.148209i −0.751776 − 0.659419i \(-0.770802\pi\)
0.859456 + 0.511210i \(0.170802\pi\)
\(24\) −3.40556 + 2.47429i −0.695158 + 0.505062i
\(25\) −1.94792 + 4.60495i −0.389584 + 0.920991i
\(26\) −11.1371 −2.18416
\(27\) 0.587785 + 0.809017i 0.113119 + 0.155695i
\(28\) −16.4710 − 5.35176i −3.11273 − 1.01139i
\(29\) 1.85230 + 5.70078i 0.343963 + 1.05861i 0.962137 + 0.272567i \(0.0878728\pi\)
−0.618174 + 0.786041i \(0.712127\pi\)
\(30\) −2.96344 − 4.47122i −0.541048 − 0.816329i
\(31\) 2.04875 − 5.17712i 0.367966 − 0.929839i
\(32\) − 2.20896i − 0.390492i
\(33\) 1.15401 − 0.374959i 0.200887 − 0.0652720i
\(34\) −3.28938 + 10.1237i −0.564124 + 1.73620i
\(35\) 3.60594 − 9.66286i 0.609515 − 1.63332i
\(36\) 3.75476 0.625793
\(37\) − 2.72468i − 0.447934i −0.974597 − 0.223967i \(-0.928099\pi\)
0.974597 − 0.223967i \(-0.0719008\pi\)
\(38\) 9.94332 + 13.6858i 1.61302 + 2.22013i
\(39\) 3.75590 + 2.72882i 0.601426 + 0.436962i
\(40\) −9.40404 − 0.404721i −1.48691 − 0.0639920i
\(41\) 0.915375 + 2.81723i 0.142958 + 0.439978i 0.996743 − 0.0806474i \(-0.0256988\pi\)
−0.853785 + 0.520626i \(0.825699\pi\)
\(42\) 6.50376 + 8.95165i 1.00355 + 1.38127i
\(43\) 8.46042 − 2.74896i 1.29020 − 0.419212i 0.418039 − 0.908429i \(-0.362717\pi\)
0.872162 + 0.489217i \(0.162717\pi\)
\(44\) 1.40788 − 4.33301i 0.212246 − 0.653226i
\(45\) −0.0961444 + 2.23400i −0.0143324 + 0.333025i
\(46\) −0.651293 + 2.00447i −0.0960279 + 0.295544i
\(47\) 2.03214 + 0.660282i 0.296418 + 0.0963121i 0.453451 − 0.891281i \(-0.350193\pi\)
−0.157033 + 0.987593i \(0.550193\pi\)
\(48\) 1.52160 − 2.09430i 0.219624 − 0.302287i
\(49\) −4.41113 + 13.5761i −0.630162 + 1.93944i
\(50\) 1.03051 − 11.9502i 0.145736 − 1.69001i
\(51\) 3.58985 − 2.60818i 0.502679 − 0.365218i
\(52\) 16.5785 − 5.38668i 2.29902 − 0.746998i
\(53\) 1.04290 − 1.43542i 0.143253 − 0.197171i −0.731361 − 0.681990i \(-0.761115\pi\)
0.874614 + 0.484820i \(0.161115\pi\)
\(54\) −1.94076 − 1.41004i −0.264104 − 0.191883i
\(55\) 2.54200 + 0.948610i 0.342763 + 0.127911i
\(56\) 19.4162 2.59459
\(57\) − 7.05178i − 0.934032i
\(58\) −8.45202 − 11.6332i −1.10980 − 1.52751i
\(59\) −2.08128 + 6.40551i −0.270959 + 0.833926i 0.719301 + 0.694698i \(0.244462\pi\)
−0.990260 + 0.139228i \(0.955538\pi\)
\(60\) 6.57395 + 5.22248i 0.848694 + 0.674219i
\(61\) 12.0376 1.54126 0.770631 − 0.637282i \(-0.219941\pi\)
0.770631 + 0.637282i \(0.219941\pi\)
\(62\) −0.836398 + 13.3303i −0.106223 + 1.69296i
\(63\) − 4.61245i − 0.581115i
\(64\) 3.23741 + 9.96373i 0.404677 + 1.24547i
\(65\) 2.78045 + 10.0018i 0.344872 + 1.24057i
\(66\) −2.35490 + 1.71094i −0.289868 + 0.210602i
\(67\) 2.62546i 0.320751i 0.987056 + 0.160375i \(0.0512705\pi\)
−0.987056 + 0.160375i \(0.948730\pi\)
\(68\) − 16.6610i − 2.02044i
\(69\) 0.710785 − 0.516415i 0.0855685 − 0.0621691i
\(70\) −1.06382 + 24.7189i −0.127151 + 2.95447i
\(71\) 1.96329 + 1.42641i 0.233000 + 0.169284i 0.698159 − 0.715943i \(-0.254003\pi\)
−0.465159 + 0.885227i \(0.654003\pi\)
\(72\) −4.00348 + 1.30081i −0.471815 + 0.153302i
\(73\) 4.47451 + 6.15864i 0.523702 + 0.720814i 0.986154 − 0.165830i \(-0.0530303\pi\)
−0.462452 + 0.886644i \(0.653030\pi\)
\(74\) 2.01981 + 6.21634i 0.234798 + 0.722635i
\(75\) −3.27559 + 3.77763i −0.378233 + 0.436203i
\(76\) −21.4210 − 15.5632i −2.45715 − 1.78523i
\(77\) −5.32280 − 1.72948i −0.606589 − 0.197093i
\(78\) −10.5920 − 3.44154i −1.19930 − 0.389678i
\(79\) −3.02338 − 2.19661i −0.340157 − 0.247138i 0.404571 − 0.914507i \(-0.367421\pi\)
−0.744728 + 0.667368i \(0.767421\pi\)
\(80\) 5.57702 − 1.55039i 0.623530 − 0.173339i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) −4.17685 − 5.74894i −0.461256 − 0.634865i
\(83\) −10.7150 + 3.48153i −1.17613 + 0.382147i −0.830927 − 0.556382i \(-0.812189\pi\)
−0.345201 + 0.938529i \(0.612189\pi\)
\(84\) −14.0111 − 10.1797i −1.52873 − 1.11069i
\(85\) 9.91292 + 0.426621i 1.07521 + 0.0462736i
\(86\) −17.2646 + 12.5435i −1.86169 + 1.35260i
\(87\) 5.99416i 0.642641i
\(88\) 5.10779i 0.544492i
\(89\) 3.71151 − 2.69657i 0.393419 − 0.285836i −0.373436 − 0.927656i \(-0.621820\pi\)
0.766855 + 0.641820i \(0.221820\pi\)
\(90\) −1.43672 − 5.16814i −0.151443 − 0.544769i
\(91\) −6.61715 − 20.3655i −0.693666 − 2.13489i
\(92\) − 3.29885i − 0.343929i
\(93\) 3.54830 − 4.29064i 0.367941 − 0.444919i
\(94\) −5.12579 −0.528685
\(95\) 9.80829 − 12.3465i 1.00631 − 1.26672i
\(96\) 0.682606 − 2.10084i 0.0696681 − 0.214416i
\(97\) −4.25259 − 5.85319i −0.431786 − 0.594302i 0.536576 − 0.843852i \(-0.319717\pi\)
−0.968362 + 0.249550i \(0.919717\pi\)
\(98\) − 34.2438i − 3.45914i
\(99\) 1.21339 0.121951
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.4.5 ✓ 128
5.4 even 2 inner 465.2.ba.a.4.28 yes 128
31.8 even 5 inner 465.2.ba.a.349.28 yes 128
155.39 even 10 inner 465.2.ba.a.349.5 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.5 ✓ 128 1.1 even 1 trivial
465.2.ba.a.4.28 yes 128 5.4 even 2 inner
465.2.ba.a.349.5 yes 128 155.39 even 10 inner
465.2.ba.a.349.28 yes 128 31.8 even 5 inner