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Properties

Label 465.2.ba.a.4.4
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 4.4
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.4

$q$-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+(-2.30094 + 0.747620i) q^{2} +(-0.951057 - 0.309017i) q^{3} +(3.11734 - 2.26488i) q^{4} +(-1.05352 + 1.97233i) q^{5} +2.41935 q^{6} +(-0.528967 - 0.728060i) q^{7} +(-2.63543 + 3.62735i) q^{8} +(0.809017 + 0.587785i) q^{9} +(0.949523 - 5.32585i) q^{10} +(-1.34972 + 0.980628i) q^{11} +(-3.66466 + 1.19072i) q^{12} +(0.815758 + 0.265056i) q^{13} +(1.76143 + 1.27975i) q^{14} +(1.61144 - 1.55025i) q^{15} +(0.970631 - 2.98730i) q^{16} +(0.0405735 - 0.0558446i) q^{17} +(-2.30094 - 0.747620i) q^{18} +(0.722419 + 2.22338i) q^{19} +(1.18293 + 8.53454i) q^{20} +(0.278094 + 0.855886i) q^{21} +(2.37248 - 3.26544i) q^{22} +(-1.66773 + 2.29543i) q^{23} +(3.62735 - 2.63543i) q^{24} +(-2.78020 - 4.15578i) q^{25} -2.07517 q^{26} +(-0.587785 - 0.809017i) q^{27} +(-3.29794 - 1.07157i) q^{28} +(0.184455 + 0.567695i) q^{29} +(-2.54883 + 4.77176i) q^{30} +(-5.46534 + 1.06302i) q^{31} -1.36807i q^{32} +(1.58669 - 0.515547i) q^{33} +(-0.0516065 + 0.158828i) q^{34} +(1.99325 - 0.276274i) q^{35} +3.85325 q^{36} -8.05981i q^{37} +(-3.32448 - 4.57575i) q^{38} +(-0.693925 - 0.504166i) q^{39} +(-4.37788 - 9.01942i) q^{40} +(-2.74234 - 8.44006i) q^{41} +(-1.27975 - 1.76143i) q^{42} +(-2.62431 + 0.852690i) q^{43} +(-1.98653 + 6.11391i) q^{44} +(-2.01162 + 0.976409i) q^{45} +(2.12123 - 6.52847i) q^{46} +(-5.40206 - 1.75524i) q^{47} +(-1.84625 + 2.54115i) q^{48} +(1.91285 - 5.88716i) q^{49} +(9.50401 + 7.48365i) q^{50} +(-0.0558446 + 0.0405735i) q^{51} +(3.14332 - 1.02133i) q^{52} +(5.23319 - 7.20287i) q^{53} +(1.95729 + 1.42206i) q^{54} +(-0.512173 - 3.69521i) q^{55} +4.03498 q^{56} -2.33780i q^{57} +(-0.848839 - 1.16833i) q^{58} +(1.64440 - 5.06096i) q^{59} +(1.51229 - 8.48237i) q^{60} -4.53756 q^{61} +(11.7807 - 6.53195i) q^{62} -0.899932i q^{63} +(2.96406 + 9.12243i) q^{64} +(-1.38219 + 1.32971i) q^{65} +(-3.26544 + 2.37248i) q^{66} +2.55040i q^{67} -0.265981i q^{68} +(2.29543 - 1.66773i) q^{69} +(-4.37980 + 2.12589i) q^{70} +(-10.7161 - 7.78571i) q^{71} +(-4.26421 + 1.38553i) q^{72} +(-6.49858 - 8.94452i) q^{73} +(6.02567 + 18.5451i) q^{74} +(1.35992 + 4.81151i) q^{75} +(7.28771 + 5.29483i) q^{76} +(1.42791 + 0.463957i) q^{77} +(1.97360 + 0.641263i) q^{78} +(2.24115 + 1.62829i) q^{79} +(4.86937 + 5.06158i) q^{80} +(0.309017 + 0.951057i) q^{81} +(12.6199 + 17.3698i) q^{82} +(-7.32492 + 2.38001i) q^{83} +(2.80540 + 2.03824i) q^{84} +(0.0673993 + 0.138858i) q^{85} +(5.40089 - 3.92397i) q^{86} -0.596909i q^{87} -7.48028i q^{88} +(-0.0832648 + 0.0604954i) q^{89} +(3.89863 - 3.75059i) q^{90} +(-0.238532 - 0.734127i) q^{91} +10.9329i q^{92} +(5.52634 + 0.677891i) q^{93} +13.7421 q^{94} +(-5.14632 - 0.917516i) q^{95} +(-0.422756 + 1.30111i) q^{96} +(10.3192 + 14.2032i) q^{97} +14.9761i q^{98} -1.66834 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.30094 + 0.747620i −1.62701 + 0.528647i −0.973581 − 0.228342i \(-0.926670\pi\)
−0.653427 + 0.756989i \(0.726670\pi\)
\(3\) −0.951057 − 0.309017i −0.549093 − 0.178411i
\(4\) 3.11734 − 2.26488i 1.55867 − 1.13244i
\(5\) −1.05352 + 1.97233i −0.471148 + 0.882054i
\(6\) 2.41935 0.987695
\(7\) −0.528967 − 0.728060i −0.199931 − 0.275181i 0.697266 − 0.716813i \(-0.254400\pi\)
−0.897196 + 0.441632i \(0.854400\pi\)
\(8\) −2.63543 + 3.62735i −0.931764 + 1.28246i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) 0.949523 − 5.32585i 0.300266 − 1.68418i
\(11\) −1.34972 + 0.980628i −0.406956 + 0.295671i −0.772368 − 0.635175i \(-0.780928\pi\)
0.365413 + 0.930846i \(0.380928\pi\)
\(12\) −3.66466 + 1.19072i −1.05790 + 0.343731i
\(13\) 0.815758 + 0.265056i 0.226251 + 0.0735133i 0.419948 − 0.907548i \(-0.362048\pi\)
−0.193698 + 0.981061i \(0.562048\pi\)
\(14\) 1.76143 + 1.27975i 0.470762 + 0.342029i
\(15\) 1.61144 − 1.55025i 0.416072 − 0.400272i
\(16\) 0.970631 − 2.98730i 0.242658 − 0.746824i
\(17\) 0.0405735 − 0.0558446i 0.00984051 − 0.0135443i −0.804068 − 0.594537i \(-0.797335\pi\)
0.813909 + 0.580993i \(0.197335\pi\)
\(18\) −2.30094 − 0.747620i −0.542336 − 0.176216i
\(19\) 0.722419 + 2.22338i 0.165734 + 0.510077i 0.999090 − 0.0426600i \(-0.0135832\pi\)
−0.833355 + 0.552737i \(0.813583\pi\)
\(20\) 1.18293 + 8.53454i 0.264510 + 1.90838i
\(21\) 0.278094 + 0.855886i 0.0606851 + 0.186770i
\(22\) 2.37248 − 3.26544i 0.505815 − 0.696194i
\(23\) −1.66773 + 2.29543i −0.347745 + 0.478631i −0.946684 − 0.322165i \(-0.895590\pi\)
0.598938 + 0.800795i \(0.295590\pi\)
\(24\) 3.62735 − 2.63543i 0.740430 − 0.537954i
\(25\) −2.78020 − 4.15578i −0.556040 − 0.831156i
\(26\) −2.07517 −0.406974
\(27\) −0.587785 − 0.809017i −0.113119 − 0.155695i
\(28\) −3.29794 − 1.07157i −0.623252 − 0.202507i
\(29\) 0.184455 + 0.567695i 0.0342525 + 0.105418i 0.966721 − 0.255833i \(-0.0823497\pi\)
−0.932469 + 0.361251i \(0.882350\pi\)
\(30\) −2.54883 + 4.77176i −0.465350 + 0.871201i
\(31\) −5.46534 + 1.06302i −0.981605 + 0.190924i
\(32\) − 1.36807i − 0.241842i
\(33\) 1.58669 − 0.515547i 0.276207 − 0.0897452i
\(34\) −0.0516065 + 0.158828i −0.00885044 + 0.0272389i
\(35\) 1.99325 − 0.276274i 0.336921 − 0.0466989i
\(36\) 3.85325 0.642208
\(37\) − 8.05981i − 1.32502i −0.749051 − 0.662512i \(-0.769490\pi\)
0.749051 − 0.662512i \(-0.230510\pi\)
\(38\) −3.32448 − 4.57575i −0.539302 − 0.742285i
\(39\) −0.693925 − 0.504166i −0.111117 − 0.0807312i
\(40\) −4.37788 − 9.01942i −0.692204 − 1.42610i
\(41\) −2.74234 − 8.44006i −0.428282 − 1.31812i −0.899816 − 0.436269i \(-0.856300\pi\)
0.471535 − 0.881848i \(-0.343700\pi\)
\(42\) −1.27975 − 1.76143i −0.197470 − 0.271795i
\(43\) −2.62431 + 0.852690i −0.400204 + 0.130034i −0.502202 − 0.864751i \(-0.667476\pi\)
0.101998 + 0.994785i \(0.467476\pi\)
\(44\) −1.98653 + 6.11391i −0.299481 + 0.921707i
\(45\) −2.01162 + 0.976409i −0.299875 + 0.145554i
\(46\) 2.12123 − 6.52847i 0.312758 − 0.962570i
\(47\) −5.40206 − 1.75524i −0.787972 − 0.256028i −0.112732 − 0.993625i \(-0.535960\pi\)
−0.675240 + 0.737598i \(0.735960\pi\)
\(48\) −1.84625 + 2.54115i −0.266483 + 0.366783i
\(49\) 1.91285 − 5.88716i 0.273265 − 0.841022i
\(50\) 9.50401 + 7.48365i 1.34407 + 1.05835i
\(51\) −0.0558446 + 0.0405735i −0.00781981 + 0.00568142i
\(52\) 3.14332 − 1.02133i 0.435900 − 0.141632i
\(53\) 5.23319 − 7.20287i 0.718834 − 0.989390i −0.280727 − 0.959788i \(-0.590576\pi\)
0.999562 − 0.0296030i \(-0.00942429\pi\)
\(54\) 1.95729 + 1.42206i 0.266354 + 0.193518i
\(55\) −0.512173 − 3.69521i −0.0690614 − 0.498261i
\(56\) 4.03498 0.539197
\(57\) − 2.33780i − 0.309649i
\(58\) −0.848839 − 1.16833i −0.111458 − 0.153409i
\(59\) 1.64440 − 5.06096i 0.214083 − 0.658881i −0.785134 − 0.619326i \(-0.787406\pi\)
0.999217 − 0.0395548i \(-0.0125940\pi\)
\(60\) 1.51229 − 8.48237i 0.195235 − 1.09507i
\(61\) −4.53756 −0.580975 −0.290488 − 0.956879i \(-0.593817\pi\)
−0.290488 + 0.956879i \(0.593817\pi\)
\(62\) 11.7807 − 6.53195i 1.49615 − 0.829558i
\(63\) − 0.899932i − 0.113381i
\(64\) 2.96406 + 9.12243i 0.370507 + 1.14030i
\(65\) −1.38219 + 1.32971i −0.171440 + 0.164930i
\(66\) −3.26544 + 2.37248i −0.401948 + 0.292032i
\(67\) 2.55040i 0.311581i 0.987790 + 0.155790i \(0.0497924\pi\)
−0.987790 + 0.155790i \(0.950208\pi\)
\(68\) − 0.265981i − 0.0322549i
\(69\) 2.29543 − 1.66773i 0.276337 − 0.200771i
\(70\) −4.37980 + 2.12589i −0.523487 + 0.254092i
\(71\) −10.7161 − 7.78571i −1.27177 − 0.923993i −0.272496 − 0.962157i \(-0.587849\pi\)
−0.999272 + 0.0381636i \(0.987849\pi\)
\(72\) −4.26421 + 1.38553i −0.502542 + 0.163286i
\(73\) −6.49858 − 8.94452i −0.760601 − 1.04688i −0.997164 − 0.0752611i \(-0.976021\pi\)
0.236563 − 0.971616i \(-0.423979\pi\)
\(74\) 6.02567 + 18.5451i 0.700470 + 2.15582i
\(75\) 1.35992 + 4.81151i 0.157030 + 0.555585i
\(76\) 7.28771 + 5.29483i 0.835958 + 0.607359i
\(77\) 1.42791 + 0.463957i 0.162726 + 0.0528728i
\(78\) 1.97360 + 0.641263i 0.223467 + 0.0726087i
\(79\) 2.24115 + 1.62829i 0.252149 + 0.183197i 0.706679 − 0.707535i \(-0.250193\pi\)
−0.454530 + 0.890732i \(0.650193\pi\)
\(80\) 4.86937 + 5.06158i 0.544412 + 0.565902i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) 12.6199 + 17.3698i 1.39364 + 1.91818i
\(83\) −7.32492 + 2.38001i −0.804014 + 0.261240i −0.682061 − 0.731296i \(-0.738916\pi\)
−0.121954 + 0.992536i \(0.538916\pi\)
\(84\) 2.80540 + 2.03824i 0.306094 + 0.222390i
\(85\) 0.0673993 + 0.138858i 0.00731048 + 0.0150612i
\(86\) 5.40089 − 3.92397i 0.582393 − 0.423133i
\(87\) − 0.596909i − 0.0639954i
\(88\) − 7.48028i − 0.797401i
\(89\) −0.0832648 + 0.0604954i −0.00882605 + 0.00641250i −0.592190 − 0.805799i \(-0.701736\pi\)
0.583363 + 0.812211i \(0.301736\pi\)
\(90\) 3.89863 − 3.75059i 0.410952 − 0.395346i
\(91\) −0.238532 − 0.734127i −0.0250050 − 0.0769574i
\(92\) 10.9329i 1.13983i
\(93\) 5.52634 + 0.677891i 0.573055 + 0.0702940i
\(94\) 13.7421 1.41739
\(95\) −5.14632 − 0.917516i −0.528001 − 0.0941352i
\(96\) −0.422756 + 1.30111i −0.0431474 + 0.132794i
\(97\) 10.3192 + 14.2032i 1.04776 + 1.44212i 0.890735 + 0.454522i \(0.150190\pi\)
0.157024 + 0.987595i \(0.449810\pi\)
\(98\) 14.9761i 1.51281i
\(99\) −1.66834 −0.167675
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.4.4 ✓ 128
5.4 even 2 inner 465.2.ba.a.4.29 yes 128
31.8 even 5 inner 465.2.ba.a.349.29 yes 128
155.39 even 10 inner 465.2.ba.a.349.4 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.4 ✓ 128 1.1 even 1 trivial
465.2.ba.a.4.29 yes 128 5.4 even 2 inner
465.2.ba.a.349.4 yes 128 155.39 even 10 inner
465.2.ba.a.349.29 yes 128 31.8 even 5 inner