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Properties

Label 465.2.ba.a.4.12
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 4.12
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.12

$q$-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+(-0.932426 + 0.302963i) q^{2} +(-0.951057 - 0.309017i) q^{3} +(-0.840403 + 0.610589i) q^{4} +(0.949797 + 2.02432i) q^{5} +0.980410 q^{6} +(-2.34180 - 3.22321i) q^{7} +(1.75117 - 2.41028i) q^{8} +(0.809017 + 0.587785i) q^{9} +(-1.49891 - 1.59978i) q^{10} +(2.64147 - 1.91914i) q^{11} +(0.987953 - 0.321005i) q^{12} +(2.34396 + 0.761598i) q^{13} +(3.16007 + 2.29593i) q^{14} +(-0.277761 - 2.21875i) q^{15} +(-0.260598 + 0.802039i) q^{16} +(-2.03112 + 2.79560i) q^{17} +(-0.932426 - 0.302963i) q^{18} +(2.35427 + 7.24570i) q^{19} +(-2.03424 - 1.12131i) q^{20} +(1.23116 + 3.78912i) q^{21} +(-1.88154 + 2.58972i) q^{22} +(0.477322 - 0.656977i) q^{23} +(-2.41028 + 1.75117i) q^{24} +(-3.19577 + 3.84539i) q^{25} -2.41630 q^{26} +(-0.587785 - 0.809017i) q^{27} +(3.93612 + 1.27892i) q^{28} +(-1.71085 - 5.26544i) q^{29} +(0.931191 + 1.98467i) q^{30} +(3.49772 + 4.33197i) q^{31} +5.13174i q^{32} +(-3.10523 + 1.00895i) q^{33} +(1.04690 - 3.22204i) q^{34} +(4.30059 - 7.80197i) q^{35} -1.03880 q^{36} +11.5362i q^{37} +(-4.39037 - 6.04282i) q^{38} +(-1.99389 - 1.44865i) q^{39} +(6.54244 + 1.25566i) q^{40} +(0.833310 + 2.56466i) q^{41} +(-2.29593 - 3.16007i) q^{42} +(5.68230 - 1.84629i) q^{43} +(-1.04809 + 3.22570i) q^{44} +(-0.421465 + 2.19599i) q^{45} +(-0.246027 + 0.757193i) q^{46} +(11.5023 + 3.73733i) q^{47} +(0.495687 - 0.682255i) q^{48} +(-2.74195 + 8.43887i) q^{49} +(1.81480 - 4.55375i) q^{50} +(2.79560 - 2.03112i) q^{51} +(-2.43489 + 0.791145i) q^{52} +(3.59081 - 4.94233i) q^{53} +(0.793169 + 0.576271i) q^{54} +(6.39381 + 3.52439i) q^{55} -11.8697 q^{56} -7.61858i q^{57} +(3.19047 + 4.39131i) q^{58} +(0.429893 - 1.32307i) q^{59} +(1.58817 + 1.69505i) q^{60} -1.54710 q^{61} +(-4.57379 - 2.97956i) q^{62} -3.98411i q^{63} +(-2.07593 - 6.38904i) q^{64} +(0.684565 + 5.46829i) q^{65} +(2.58972 - 1.88154i) q^{66} -6.32180i q^{67} -3.58961i q^{68} +(-0.656977 + 0.477322i) q^{69} +(-1.64627 + 8.57768i) q^{70} +(5.31526 + 3.86177i) q^{71} +(2.83345 - 0.920644i) q^{72} +(5.16346 + 7.10689i) q^{73} +(-3.49506 - 10.7567i) q^{74} +(4.22765 - 2.66964i) q^{75} +(-6.40268 - 4.65182i) q^{76} +(-12.3716 - 4.01977i) q^{77} +(2.29804 + 0.746679i) q^{78} +(-10.5975 - 7.69952i) q^{79} +(-1.87110 + 0.234239i) q^{80} +(0.309017 + 0.951057i) q^{81} +(-1.55400 - 2.13890i) q^{82} +(-2.31849 + 0.753324i) q^{83} +(-3.34826 - 2.43265i) q^{84} +(-7.58834 - 1.45639i) q^{85} +(-4.73896 + 3.44306i) q^{86} +5.53641i q^{87} -9.72740i q^{88} +(0.897328 - 0.651947i) q^{89} +(-0.272319 - 2.17529i) q^{90} +(-3.03429 - 9.33859i) q^{91} +0.843572i q^{92} +(-1.98787 - 5.20080i) q^{93} -11.8573 q^{94} +(-12.4316 + 11.6478i) q^{95} +(1.58579 - 4.88057i) q^{96} +(8.37310 + 11.5246i) q^{97} -8.69933i q^{98} +3.26503 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.932426 + 0.302963i −0.659325 + 0.214228i −0.619521 − 0.784980i \(-0.712673\pi\)
−0.0398035 + 0.999208i \(0.512673\pi\)
\(3\) −0.951057 − 0.309017i −0.549093 − 0.178411i
\(4\) −0.840403 + 0.610589i −0.420202 + 0.305294i
\(5\) 0.949797 + 2.02432i 0.424762 + 0.905305i
\(6\) 0.980410 0.400251
\(7\) −2.34180 − 3.22321i −0.885118 − 1.21826i −0.974977 − 0.222306i \(-0.928642\pi\)
0.0898589 − 0.995955i \(-0.471358\pi\)
\(8\) 1.75117 − 2.41028i 0.619132 − 0.852162i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) −1.49891 − 1.59978i −0.473997 − 0.505894i
\(11\) 2.64147 − 1.91914i 0.796432 − 0.578642i −0.113433 − 0.993546i \(-0.536185\pi\)
0.909865 + 0.414904i \(0.136185\pi\)
\(12\) 0.987953 − 0.321005i 0.285197 − 0.0926663i
\(13\) 2.34396 + 0.761598i 0.650097 + 0.211229i 0.615457 − 0.788170i \(-0.288971\pi\)
0.0346402 + 0.999400i \(0.488971\pi\)
\(14\) 3.16007 + 2.29593i 0.844565 + 0.613612i
\(15\) −0.277761 − 2.21875i −0.0717175 − 0.572879i
\(16\) −0.260598 + 0.802039i −0.0651496 + 0.200510i
\(17\) −2.03112 + 2.79560i −0.492619 + 0.678032i −0.980868 − 0.194673i \(-0.937636\pi\)
0.488250 + 0.872704i \(0.337636\pi\)
\(18\) −0.932426 − 0.302963i −0.219775 − 0.0714092i
\(19\) 2.35427 + 7.24570i 0.540107 + 1.66228i 0.732348 + 0.680930i \(0.238424\pi\)
−0.192241 + 0.981348i \(0.561576\pi\)
\(20\) −2.03424 − 1.12131i −0.454870 − 0.250733i
\(21\) 1.23116 + 3.78912i 0.268661 + 0.826853i
\(22\) −1.88154 + 2.58972i −0.401146 + 0.552130i
\(23\) 0.477322 − 0.656977i 0.0995284 − 0.136989i −0.756348 − 0.654169i \(-0.773018\pi\)
0.855876 + 0.517180i \(0.173018\pi\)
\(24\) −2.41028 + 1.75117i −0.491996 + 0.357456i
\(25\) −3.19577 + 3.84539i −0.639154 + 0.769079i
\(26\) −2.41630 −0.473876
\(27\) −0.587785 − 0.809017i −0.113119 − 0.155695i
\(28\) 3.93612 + 1.27892i 0.743856 + 0.241693i
\(29\) −1.71085 − 5.26544i −0.317696 − 0.977768i −0.974630 − 0.223820i \(-0.928147\pi\)
0.656934 − 0.753948i \(-0.271853\pi\)
\(30\) 0.931191 + 1.98467i 0.170011 + 0.362349i
\(31\) 3.49772 + 4.33197i 0.628208 + 0.778045i
\(32\) 5.13174i 0.907172i
\(33\) −3.10523 + 1.00895i −0.540551 + 0.175636i
\(34\) 1.04690 − 3.22204i 0.179543 − 0.552575i
\(35\) 4.30059 − 7.80197i 0.726932 − 1.31877i
\(36\) −1.03880 −0.173133
\(37\) 11.5362i 1.89655i 0.317456 + 0.948273i \(0.397171\pi\)
−0.317456 + 0.948273i \(0.602829\pi\)
\(38\) −4.39037 − 6.04282i −0.712211 − 0.980275i
\(39\) −1.99389 − 1.44865i −0.319278 − 0.231969i
\(40\) 6.54244 + 1.25566i 1.03445 + 0.198537i
\(41\) 0.833310 + 2.56466i 0.130141 + 0.400533i 0.994803 − 0.101822i \(-0.0324671\pi\)
−0.864662 + 0.502355i \(0.832467\pi\)
\(42\) −2.29593 − 3.16007i −0.354269 − 0.487610i
\(43\) 5.68230 − 1.84629i 0.866542 − 0.281557i 0.158184 − 0.987410i \(-0.449436\pi\)
0.708358 + 0.705853i \(0.249436\pi\)
\(44\) −1.04809 + 3.22570i −0.158006 + 0.486292i
\(45\) −0.421465 + 2.19599i −0.0628283 + 0.327359i
\(46\) −0.246027 + 0.757193i −0.0362747 + 0.111642i
\(47\) 11.5023 + 3.73733i 1.67779 + 0.545146i 0.984480 − 0.175496i \(-0.0561529\pi\)
0.693307 + 0.720642i \(0.256153\pi\)
\(48\) 0.495687 − 0.682255i 0.0715463 − 0.0984750i
\(49\) −2.74195 + 8.43887i −0.391708 + 1.20555i
\(50\) 1.81480 − 4.55375i 0.256652 − 0.643997i
\(51\) 2.79560 − 2.03112i 0.391462 − 0.284414i
\(52\) −2.43489 + 0.791145i −0.337659 + 0.109712i
\(53\) 3.59081 − 4.94233i 0.493236 − 0.678881i −0.487745 − 0.872986i \(-0.662180\pi\)
0.980981 + 0.194105i \(0.0621804\pi\)
\(54\) 0.793169 + 0.576271i 0.107937 + 0.0784205i
\(55\) 6.39381 + 3.52439i 0.862141 + 0.475229i
\(56\) −11.8697 −1.58616
\(57\) − 7.61858i − 1.00911i
\(58\) 3.19047 + 4.39131i 0.418930 + 0.576607i
\(59\) 0.429893 − 1.32307i 0.0559673 − 0.172250i −0.919165 − 0.393872i \(-0.871135\pi\)
0.975133 + 0.221623i \(0.0711353\pi\)
\(60\) 1.58817 + 1.69505i 0.205032 + 0.218830i
\(61\) −1.54710 −0.198086 −0.0990428 − 0.995083i \(-0.531578\pi\)
−0.0990428 + 0.995083i \(0.531578\pi\)
\(62\) −4.57379 − 2.97956i −0.580872 − 0.378405i
\(63\) − 3.98411i − 0.501951i
\(64\) −2.07593 − 6.38904i −0.259491 − 0.798631i
\(65\) 0.684565 + 5.46829i 0.0849098 + 0.678258i
\(66\) 2.58972 − 1.88154i 0.318773 − 0.231602i
\(67\) − 6.32180i − 0.772331i −0.922429 − 0.386166i \(-0.873799\pi\)
0.922429 − 0.386166i \(-0.126201\pi\)
\(68\) − 3.58961i − 0.435304i
\(69\) −0.656977 + 0.477322i −0.0790907 + 0.0574628i
\(70\) −1.64627 + 8.57768i −0.196767 + 1.02523i
\(71\) 5.31526 + 3.86177i 0.630806 + 0.458307i 0.856679 − 0.515849i \(-0.172524\pi\)
−0.225874 + 0.974157i \(0.572524\pi\)
\(72\) 2.83345 − 0.920644i 0.333925 − 0.108499i
\(73\) 5.16346 + 7.10689i 0.604337 + 0.831798i 0.996097 − 0.0882692i \(-0.0281336\pi\)
−0.391760 + 0.920068i \(0.628134\pi\)
\(74\) −3.49506 − 10.7567i −0.406292 − 1.25044i
\(75\) 4.22765 − 2.66964i 0.488167 − 0.308263i
\(76\) −6.40268 − 4.65182i −0.734438 − 0.533600i
\(77\) −12.3716 − 4.01977i −1.40987 − 0.458095i
\(78\) 2.29804 + 0.746679i 0.260202 + 0.0845447i
\(79\) −10.5975 − 7.69952i −1.19231 − 0.866264i −0.198804 − 0.980039i \(-0.563706\pi\)
−0.993507 + 0.113775i \(0.963706\pi\)
\(80\) −1.87110 + 0.234239i −0.209196 + 0.0261888i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) −1.55400 − 2.13890i −0.171610 − 0.236202i
\(83\) −2.31849 + 0.753324i −0.254488 + 0.0826881i −0.433483 − 0.901162i \(-0.642715\pi\)
0.178995 + 0.983850i \(0.442715\pi\)
\(84\) −3.34826 − 2.43265i −0.365325 − 0.265424i
\(85\) −7.58834 − 1.45639i −0.823071 − 0.157968i
\(86\) −4.73896 + 3.44306i −0.511015 + 0.371274i
\(87\) 5.53641i 0.593566i
\(88\) − 9.72740i − 1.03694i
\(89\) 0.897328 − 0.651947i 0.0951166 − 0.0691062i −0.539210 − 0.842171i \(-0.681277\pi\)
0.634327 + 0.773065i \(0.281277\pi\)
\(90\) −0.272319 − 2.17529i −0.0287050 − 0.229295i
\(91\) −3.03429 − 9.33859i −0.318080 − 0.978951i
\(92\) 0.843572i 0.0879485i
\(93\) −1.98787 − 5.20080i −0.206133 − 0.539298i
\(94\) −11.8573 −1.22299
\(95\) −12.4316 + 11.6478i −1.27545 + 1.19503i
\(96\) 1.58579 − 4.88057i 0.161850 − 0.498122i
\(97\) 8.37310 + 11.5246i 0.850160 + 1.17014i 0.983828 + 0.179118i \(0.0573245\pi\)
−0.133668 + 0.991026i \(0.542676\pi\)
\(98\) − 8.69933i − 0.878765i
\(99\) 3.26503 0.328148
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.4.12 ✓ 128
5.4 even 2 inner 465.2.ba.a.4.21 yes 128
31.8 even 5 inner 465.2.ba.a.349.21 yes 128
155.39 even 10 inner 465.2.ba.a.349.12 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.12 ✓ 128 1.1 even 1 trivial
465.2.ba.a.4.21 yes 128 5.4 even 2 inner
465.2.ba.a.349.12 yes 128 155.39 even 10 inner
465.2.ba.a.349.21 yes 128 31.8 even 5 inner