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Properties

Label 465.2.ba.a.4.10
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 4.10
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.10

$q$-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+(-1.28040 + 0.416026i) q^{2} +(0.951057 + 0.309017i) q^{3} +(-0.151696 + 0.110214i) q^{4} +(-0.613177 + 2.15035i) q^{5} -1.34629 q^{6} +(1.52844 + 2.10372i) q^{7} +(1.73104 - 2.38257i) q^{8} +(0.809017 + 0.587785i) q^{9} +(-0.109493 - 3.00840i) q^{10} +(-3.16735 + 2.30121i) q^{11} +(-0.178330 + 0.0579429i) q^{12} +(2.83095 + 0.919832i) q^{13} +(-2.83221 - 2.05772i) q^{14} +(-1.24766 + 1.85562i) q^{15} +(-1.10932 + 3.41413i) q^{16} +(1.56893 - 2.15945i) q^{17} +(-1.28040 - 0.416026i) q^{18} +(-0.563971 - 1.73572i) q^{19} +(-0.143982 - 0.393781i) q^{20} +(0.803549 + 2.47307i) q^{21} +(3.09810 - 4.26416i) q^{22} +(-4.88417 + 6.72248i) q^{23} +(2.38257 - 1.73104i) q^{24} +(-4.24803 - 2.63709i) q^{25} -4.00742 q^{26} +(0.587785 + 0.809017i) q^{27} +(-0.463718 - 0.150671i) q^{28} +(-2.77802 - 8.54987i) q^{29} +(0.825513 - 2.89499i) q^{30} +(-1.46919 + 5.37043i) q^{31} +1.05709i q^{32} +(-3.72344 + 1.20982i) q^{33} +(-1.11047 + 3.41766i) q^{34} +(-5.46094 + 1.99673i) q^{35} -0.187507 q^{36} +5.72831i q^{37} +(1.44421 + 1.98779i) q^{38} +(2.40815 + 1.74962i) q^{39} +(4.06193 + 5.18327i) q^{40} +(0.491899 + 1.51391i) q^{41} +(-2.05772 - 2.83221i) q^{42} +(10.2142 - 3.31881i) q^{43} +(0.226850 - 0.698172i) q^{44} +(-1.76002 + 1.37925i) q^{45} +(3.45695 - 10.6394i) q^{46} +(-8.32690 - 2.70557i) q^{47} +(-2.11005 + 2.90423i) q^{48} +(0.0736180 - 0.226573i) q^{49} +(6.53626 + 1.60923i) q^{50} +(2.15945 - 1.56893i) q^{51} +(-0.530824 + 0.172475i) q^{52} +(-7.05060 + 9.70432i) q^{53} +(-1.08917 - 0.791329i) q^{54} +(-3.00627 - 8.22196i) q^{55} +7.65804 q^{56} -1.82505i q^{57} +(7.11394 + 9.79149i) q^{58} +(-1.85104 + 5.69693i) q^{59} +(-0.0152498 - 0.419001i) q^{60} +2.35019 q^{61} +(-0.353092 - 7.48750i) q^{62} +2.60034i q^{63} +(-2.65841 - 8.18175i) q^{64} +(-3.71384 + 5.52352i) q^{65} +(4.26416 - 3.09810i) q^{66} +4.19166i q^{67} +0.500498i q^{68} +(-6.72248 + 4.88417i) q^{69} +(6.16148 - 4.82851i) q^{70} +(5.85346 + 4.25279i) q^{71} +(2.80088 - 0.910060i) q^{72} +(-4.06354 - 5.59298i) q^{73} +(-2.38313 - 7.33451i) q^{74} +(-3.22521 - 3.82074i) q^{75} +(0.276853 + 0.201146i) q^{76} +(-9.68221 - 3.14594i) q^{77} +(-3.81128 - 1.23836i) q^{78} +(-4.27434 - 3.10549i) q^{79} +(-6.66137 - 4.47889i) q^{80} +(0.309017 + 0.951057i) q^{81} +(-1.25965 - 1.73376i) q^{82} +(8.66789 - 2.81637i) q^{83} +(-0.394462 - 0.286594i) q^{84} +(3.68154 + 4.69788i) q^{85} +(-11.6976 + 8.49878i) q^{86} -8.98986i q^{87} +11.5299i q^{88} +(11.3821 - 8.26960i) q^{89} +(1.67971 - 2.49821i) q^{90} +(2.39188 + 7.36144i) q^{91} -1.55808i q^{92} +(-3.05684 + 4.65357i) q^{93} +11.7873 q^{94} +(4.07823 - 0.148430i) q^{95} +(-0.326658 + 1.00535i) q^{96} +(2.79492 + 3.84687i) q^{97} +0.320730i q^{98} -3.91506 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.28040 + 0.416026i −0.905377 + 0.294175i −0.724455 − 0.689323i \(-0.757908\pi\)
−0.180922 + 0.983497i \(0.557908\pi\)
\(3\) 0.951057 + 0.309017i 0.549093 + 0.178411i
\(4\) −0.151696 + 0.110214i −0.0758482 + 0.0551069i
\(5\) −0.613177 + 2.15035i −0.274221 + 0.961667i
\(6\) −1.34629 −0.549620
\(7\) 1.52844 + 2.10372i 0.577697 + 0.795131i 0.993440 − 0.114351i \(-0.0364787\pi\)
−0.415744 + 0.909482i \(0.636479\pi\)
\(8\) 1.73104 − 2.38257i 0.612014 − 0.842365i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) −0.109493 − 3.00840i −0.0346247 − 0.951340i
\(11\) −3.16735 + 2.30121i −0.954991 + 0.693842i −0.951982 − 0.306154i \(-0.900958\pi\)
−0.00300931 + 0.999995i \(0.500958\pi\)
\(12\) −0.178330 + 0.0579429i −0.0514794 + 0.0167267i
\(13\) 2.83095 + 0.919832i 0.785165 + 0.255116i 0.674044 − 0.738691i \(-0.264556\pi\)
0.111121 + 0.993807i \(0.464556\pi\)
\(14\) −2.83221 − 2.05772i −0.756941 − 0.549950i
\(15\) −1.24766 + 1.85562i −0.322145 + 0.479120i
\(16\) −1.10932 + 3.41413i −0.277329 + 0.853532i
\(17\) 1.56893 − 2.15945i 0.380521 − 0.523743i −0.575201 − 0.818012i \(-0.695076\pi\)
0.955723 + 0.294269i \(0.0950762\pi\)
\(18\) −1.28040 − 0.416026i −0.301792 − 0.0980583i
\(19\) −0.563971 − 1.73572i −0.129384 − 0.398202i 0.865291 − 0.501271i \(-0.167134\pi\)
−0.994674 + 0.103068i \(0.967134\pi\)
\(20\) −0.143982 − 0.393781i −0.0321953 − 0.0880522i
\(21\) 0.803549 + 2.47307i 0.175349 + 0.539668i
\(22\) 3.09810 − 4.26416i 0.660517 − 0.909123i
\(23\) −4.88417 + 6.72248i −1.01842 + 1.40173i −0.105114 + 0.994460i \(0.533521\pi\)
−0.913306 + 0.407274i \(0.866479\pi\)
\(24\) 2.38257 − 1.73104i 0.486340 − 0.353346i
\(25\) −4.24803 − 2.63709i −0.849606 − 0.527419i
\(26\) −4.00742 −0.785919
\(27\) 0.587785 + 0.809017i 0.113119 + 0.155695i
\(28\) −0.463718 − 0.150671i −0.0876345 − 0.0284742i
\(29\) −2.77802 − 8.54987i −0.515866 − 1.58767i −0.781702 − 0.623653i \(-0.785648\pi\)
0.265836 − 0.964018i \(-0.414352\pi\)
\(30\) 0.825513 − 2.89499i 0.150717 − 0.528551i
\(31\) −1.46919 + 5.37043i −0.263874 + 0.964557i
\(32\) 1.05709i 0.186869i
\(33\) −3.72344 + 1.20982i −0.648168 + 0.210603i
\(34\) −1.11047 + 3.41766i −0.190443 + 0.586125i
\(35\) −5.46094 + 1.99673i −0.923068 + 0.337510i
\(36\) −0.187507 −0.0312512
\(37\) 5.72831i 0.941729i 0.882206 + 0.470864i \(0.156058\pi\)
−0.882206 + 0.470864i \(0.843942\pi\)
\(38\) 1.44421 + 1.98779i 0.234282 + 0.322462i
\(39\) 2.40815 + 1.74962i 0.385613 + 0.280164i
\(40\) 4.06193 + 5.18327i 0.642247 + 0.819548i
\(41\) 0.491899 + 1.51391i 0.0768217 + 0.236433i 0.982092 − 0.188404i \(-0.0603315\pi\)
−0.905270 + 0.424837i \(0.860331\pi\)
\(42\) −2.05772 − 2.83221i −0.317514 − 0.437020i
\(43\) 10.2142 − 3.31881i 1.55766 − 0.506113i 0.601476 − 0.798891i \(-0.294579\pi\)
0.956181 + 0.292777i \(0.0945795\pi\)
\(44\) 0.226850 − 0.698172i 0.0341989 − 0.105253i
\(45\) −1.76002 + 1.37925i −0.262368 + 0.205607i
\(46\) 3.45695 − 10.6394i 0.509699 − 1.56869i
\(47\) −8.32690 − 2.70557i −1.21460 − 0.394649i −0.369490 − 0.929235i \(-0.620467\pi\)
−0.845114 + 0.534586i \(0.820467\pi\)
\(48\) −2.11005 + 2.90423i −0.304559 + 0.419190i
\(49\) 0.0736180 − 0.226573i 0.0105169 − 0.0323676i
\(50\) 6.53626 + 1.60923i 0.924367 + 0.227580i
\(51\) 2.15945 − 1.56893i 0.302383 − 0.219694i
\(52\) −0.530824 + 0.172475i −0.0736120 + 0.0239180i
\(53\) −7.05060 + 9.70432i −0.968474 + 1.33299i −0.0256602 + 0.999671i \(0.508169\pi\)
−0.942814 + 0.333320i \(0.891831\pi\)
\(54\) −1.08917 − 0.791329i −0.148217 − 0.107686i
\(55\) −3.00627 − 8.22196i −0.405366 − 1.10865i
\(56\) 7.65804 1.02335
\(57\) − 1.82505i − 0.241733i
\(58\) 7.11394 + 9.79149i 0.934106 + 1.28569i
\(59\) −1.85104 + 5.69693i −0.240985 + 0.741676i 0.755286 + 0.655396i \(0.227498\pi\)
−0.996271 + 0.0862806i \(0.972502\pi\)
\(60\) −0.0152498 − 0.419001i −0.00196874 − 0.0540928i
\(61\) 2.35019 0.300912 0.150456 − 0.988617i \(-0.451926\pi\)
0.150456 + 0.988617i \(0.451926\pi\)
\(62\) −0.353092 − 7.48750i −0.0448427 − 0.950913i
\(63\) 2.60034i 0.327612i
\(64\) −2.65841 − 8.18175i −0.332301 − 1.02272i
\(65\) −3.71384 + 5.52352i −0.460645 + 0.685109i
\(66\) 4.26416 − 3.09810i 0.524882 − 0.381349i
\(67\) 4.19166i 0.512093i 0.966664 + 0.256046i \(0.0824200\pi\)
−0.966664 + 0.256046i \(0.917580\pi\)
\(68\) 0.500498i 0.0606943i
\(69\) −6.72248 + 4.88417i −0.809292 + 0.587985i
\(70\) 6.16148 − 4.82851i 0.736437 − 0.577117i
\(71\) 5.85346 + 4.25279i 0.694678 + 0.504713i 0.878195 − 0.478304i \(-0.158748\pi\)
−0.183517 + 0.983017i \(0.558748\pi\)
\(72\) 2.80088 − 0.910060i 0.330086 − 0.107252i
\(73\) −4.06354 − 5.59298i −0.475601 − 0.654609i 0.502051 − 0.864838i \(-0.332579\pi\)
−0.977652 + 0.210229i \(0.932579\pi\)
\(74\) −2.38313 − 7.33451i −0.277033 − 0.852619i
\(75\) −3.22521 − 3.82074i −0.372415 − 0.441181i
\(76\) 0.276853 + 0.201146i 0.0317572 + 0.0230730i
\(77\) −9.68221 − 3.14594i −1.10339 − 0.358513i
\(78\) −3.81128 − 1.23836i −0.431542 − 0.140217i
\(79\) −4.27434 − 3.10549i −0.480901 − 0.349395i 0.320773 − 0.947156i \(-0.396057\pi\)
−0.801674 + 0.597761i \(0.796057\pi\)
\(80\) −6.66137 − 4.47889i −0.744764 − 0.500755i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) −1.25965 − 1.73376i −0.139105 − 0.191462i
\(83\) 8.66789 − 2.81637i 0.951424 − 0.309136i 0.208130 − 0.978101i \(-0.433262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(84\) −0.394462 − 0.286594i −0.0430394 − 0.0312699i
\(85\) 3.68154 + 4.69788i 0.399319 + 0.509556i
\(86\) −11.6976 + 8.49878i −1.26138 + 0.916447i
\(87\) − 8.98986i − 0.963815i
\(88\) 11.5299i 1.22909i
\(89\) 11.3821 − 8.26960i 1.20650 − 0.876576i 0.211594 − 0.977358i \(-0.432134\pi\)
0.994909 + 0.100782i \(0.0321345\pi\)
\(90\) 1.67971 − 2.49821i 0.177057 − 0.263334i
\(91\) 2.39188 + 7.36144i 0.250737 + 0.771688i
\(92\) − 1.55808i − 0.162441i
\(93\) −3.05684 + 4.65357i −0.316979 + 0.482553i
\(94\) 11.7873 1.21577
\(95\) 4.07823 − 0.148430i 0.418418 − 0.0152286i
\(96\) −0.326658 + 1.00535i −0.0333394 + 0.102608i
\(97\) 2.79492 + 3.84687i 0.283781 + 0.390591i 0.926982 − 0.375107i \(-0.122394\pi\)
−0.643201 + 0.765698i \(0.722394\pi\)
\(98\) 0.320730i 0.0323986i
\(99\) −3.91506 −0.393478
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.4.10 ✓ 128
5.4 even 2 inner 465.2.ba.a.4.23 yes 128
31.8 even 5 inner 465.2.ba.a.349.23 yes 128
155.39 even 10 inner 465.2.ba.a.349.10 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.10 ✓ 128 1.1 even 1 trivial
465.2.ba.a.4.23 yes 128 5.4 even 2 inner
465.2.ba.a.349.10 yes 128 155.39 even 10 inner
465.2.ba.a.349.23 yes 128 31.8 even 5 inner