↑

Properties

Label 462.2.y.a.37.3
Level $462$
Weight $2$
Character 462.37
Analytic conductor $3.689$
Analytic rank $0$
Dimension $24$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [462,2,Mod(25,462)] 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("462.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(462, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([0, 20, 24])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.y (of order \(15\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label 37.3
Character \(\chi\) \(=\) 462.37
Dual form 462.2.y.a.25.3

$q$-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+(-0.913545 + 0.406737i) q^{2} +(0.978148 - 0.207912i) q^{3} +(0.669131 - 0.743145i) q^{4} +(0.0177181 + 0.168577i) q^{5} +(-0.809017 + 0.587785i) q^{6} +(2.46190 - 0.969054i) q^{7} +(-0.309017 + 0.951057i) q^{8} +(0.913545 - 0.406737i) q^{9} +(-0.0847527 - 0.146796i) q^{10} +(0.324907 - 3.30067i) q^{11} +(0.500000 - 0.866025i) q^{12} +(0.621990 + 0.451902i) q^{13} +(-1.85490 + 1.88662i) q^{14} +(0.0523800 + 0.161209i) q^{15} +(-0.104528 - 0.994522i) q^{16} +(-3.72774 - 1.65970i) q^{17} +(-0.669131 + 0.743145i) q^{18} +(2.42727 + 2.69576i) q^{19} +(0.137133 + 0.0996327i) q^{20} +(2.20662 - 1.45974i) q^{21} +(1.04569 + 3.14747i) q^{22} +(0.717830 - 1.24332i) q^{23} +(-0.104528 + 0.994522i) q^{24} +(4.86263 - 1.03358i) q^{25} +(-0.752021 - 0.159847i) q^{26} +(0.809017 - 0.587785i) q^{27} +(0.927182 - 2.47797i) q^{28} +(1.02582 + 3.15714i) q^{29} +(-0.113421 - 0.125967i) q^{30} +(0.541632 - 5.15328i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-0.368442 - 3.29610i) q^{33} +4.08052 q^{34} +(0.206980 + 0.397849i) q^{35} +(0.309017 - 0.951057i) q^{36} +(-3.94124 - 0.837737i) q^{37} +(-3.31389 - 1.47544i) q^{38} +(0.702353 + 0.312708i) q^{39} +(-0.165801 - 0.0352421i) q^{40} +(-1.29807 + 3.99504i) q^{41} +(-1.42212 + 2.23105i) q^{42} +3.56184 q^{43} +(-2.23547 - 2.45003i) q^{44} +(0.0847527 + 0.146796i) q^{45} +(-0.150067 + 1.42780i) q^{46} +(5.60119 + 6.22075i) q^{47} +(-0.309017 - 0.951057i) q^{48} +(5.12187 - 4.77142i) q^{49} +(-4.02184 + 2.92204i) q^{50} +(-3.99135 - 0.848388i) q^{51} +(0.752021 - 0.159847i) q^{52} +(0.491146 - 4.67294i) q^{53} +(-0.500000 + 0.866025i) q^{54} +(0.562173 - 0.00371003i) q^{55} +(0.160858 + 2.64086i) q^{56} +(2.93471 + 2.13219i) q^{57} +(-2.22125 - 2.46695i) q^{58} +(2.22714 - 2.47349i) q^{59} +(0.154851 + 0.0689440i) q^{60} +(0.627005 + 5.96555i) q^{61} +(1.60122 + 4.92806i) q^{62} +(1.85490 - 1.88662i) q^{63} +(-0.809017 - 0.587785i) q^{64} +(-0.0651597 + 0.112860i) q^{65} +(1.67723 + 2.86128i) q^{66} +(2.70447 + 4.68428i) q^{67} +(-3.72774 + 1.65970i) q^{68} +(0.443644 - 1.36539i) q^{69} +(-0.350906 - 0.279266i) q^{70} +(-2.83494 + 2.05970i) q^{71} +(0.104528 + 0.994522i) q^{72} +(1.41042 - 1.56642i) q^{73} +(3.94124 - 0.837737i) q^{74} +(4.54148 - 2.02200i) q^{75} +3.62750 q^{76} +(-2.39864 - 8.44076i) q^{77} -0.768822 q^{78} +(-13.0264 + 5.79971i) q^{79} +(0.165801 - 0.0352421i) q^{80} +(0.669131 - 0.743145i) q^{81} +(-0.439085 - 4.17762i) q^{82} +(-13.2534 + 9.62913i) q^{83} +(0.391723 - 2.61659i) q^{84} +(0.213738 - 0.657818i) q^{85} +(-3.25390 + 1.44873i) q^{86} +(1.65981 + 2.87487i) q^{87} +(3.03872 + 1.32897i) q^{88} +(-0.842129 + 1.45861i) q^{89} +(-0.137133 - 0.0996327i) q^{90} +(1.96919 + 0.509794i) q^{91} +(-0.443644 - 1.36539i) q^{92} +(-0.541632 - 5.15328i) q^{93} +(-7.64715 - 3.40473i) q^{94} +(-0.411435 + 0.456945i) q^{95} +(0.669131 + 0.743145i) q^{96} +(-12.4725 - 9.06181i) q^{97} +(-2.73835 + 6.44216i) q^{98} +(-1.04569 - 3.14747i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 5 q^{5} - 6 q^{6} + 11 q^{7} + 6 q^{8} + 3 q^{9} + 10 q^{10} + 6 q^{11} + 12 q^{12} - 2 q^{13} - 5 q^{14} + 3 q^{16} - 2 q^{17} - 3 q^{18} + 7 q^{19} - 10 q^{20} - 4 q^{21}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.913545 + 0.406737i −0.645974 + 0.287606i
\(3\) 0.978148 − 0.207912i 0.564734 − 0.120038i
\(4\) 0.669131 − 0.743145i 0.334565 − 0.371572i
\(5\) 0.0177181 + 0.168577i 0.00792379 + 0.0753898i 0.997769 − 0.0667584i \(-0.0212657\pi\)
−0.989845 + 0.142148i \(0.954599\pi\)
\(6\) −0.809017 + 0.587785i −0.330280 + 0.239962i
\(7\) 2.46190 − 0.969054i 0.930509 − 0.366268i
\(8\) −0.309017 + 0.951057i −0.109254 + 0.336249i
\(9\) 0.913545 − 0.406737i 0.304515 − 0.135579i
\(10\) −0.0847527 − 0.146796i −0.0268011 − 0.0464209i
\(11\) 0.324907 − 3.30067i 0.0979630 − 0.995190i
\(12\) 0.500000 − 0.866025i 0.144338 − 0.250000i
\(13\) 0.621990 + 0.451902i 0.172509 + 0.125335i 0.670689 − 0.741738i \(-0.265998\pi\)
−0.498181 + 0.867073i \(0.665998\pi\)
\(14\) −1.85490 + 1.88662i −0.495744 + 0.504220i
\(15\) 0.0523800 + 0.161209i 0.0135245 + 0.0416240i
\(16\) −0.104528 − 0.994522i −0.0261321 − 0.248630i
\(17\) −3.72774 − 1.65970i −0.904110 − 0.402536i −0.0986066 − 0.995126i \(-0.531439\pi\)
−0.805504 + 0.592591i \(0.798105\pi\)
\(18\) −0.669131 + 0.743145i −0.157716 + 0.175161i
\(19\) 2.42727 + 2.69576i 0.556854 + 0.618449i 0.954182 − 0.299228i \(-0.0967290\pi\)
−0.397328 + 0.917677i \(0.630062\pi\)
\(20\) 0.137133 + 0.0996327i 0.0306638 + 0.0222786i
\(21\) 2.20662 − 1.45974i 0.481524 − 0.318540i
\(22\) 1.04569 + 3.14747i 0.222941 + 0.671042i
\(23\) 0.717830 − 1.24332i 0.149678 − 0.259250i −0.781430 − 0.623992i \(-0.785510\pi\)
0.931108 + 0.364742i \(0.118843\pi\)
\(24\) −0.104528 + 0.994522i −0.0213368 + 0.203006i
\(25\) 4.86263 − 1.03358i 0.972527 − 0.206717i
\(26\) −0.752021 − 0.159847i −0.147483 − 0.0313486i
\(27\) 0.809017 − 0.587785i 0.155695 − 0.113119i
\(28\) 0.927182 − 2.47797i 0.175221 − 0.468292i
\(29\) 1.02582 + 3.15714i 0.190489 + 0.586266i 1.00000 0.000839712i \(-0.000267289\pi\)
−0.809510 + 0.587106i \(0.800267\pi\)
\(30\) −0.113421 − 0.125967i −0.0207078 − 0.0229983i
\(31\) 0.541632 − 5.15328i 0.0972799 − 0.925557i −0.831650 − 0.555301i \(-0.812603\pi\)
0.928930 − 0.370256i \(-0.120730\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) −0.368442 − 3.29610i −0.0641375 − 0.573777i
\(34\) 4.08052 0.699804
\(35\) 0.206980 + 0.397849i 0.0349860 + 0.0672487i
\(36\) 0.309017 − 0.951057i 0.0515028 − 0.158509i
\(37\) −3.94124 − 0.837737i −0.647937 − 0.137723i −0.127791 − 0.991801i \(-0.540789\pi\)
−0.520145 + 0.854078i \(0.674122\pi\)
\(38\) −3.31389 − 1.47544i −0.537583 − 0.239347i
\(39\) 0.702353 + 0.312708i 0.112467 + 0.0500733i
\(40\) −0.165801 − 0.0352421i −0.0262155 − 0.00557227i
\(41\) −1.29807 + 3.99504i −0.202724 + 0.623920i 0.797075 + 0.603880i \(0.206379\pi\)
−0.999799 + 0.0200399i \(0.993621\pi\)
\(42\) −1.42212 + 2.23105i −0.219438 + 0.344258i
\(43\) 3.56184 0.543175 0.271587 − 0.962414i \(-0.412451\pi\)
0.271587 + 0.962414i \(0.412451\pi\)
\(44\) −2.23547 − 2.45003i −0.337010 − 0.369356i
\(45\) 0.0847527 + 0.146796i 0.0126342 + 0.0218830i
\(46\) −0.150067 + 1.42780i −0.0221262 + 0.210517i
\(47\) 5.60119 + 6.22075i 0.817018 + 0.907390i 0.997088 − 0.0762636i \(-0.0242990\pi\)
−0.180070 + 0.983654i \(0.557632\pi\)
\(48\) −0.309017 − 0.951057i −0.0446028 − 0.137273i
\(49\) 5.12187 − 4.77142i 0.731695 − 0.681632i
\(50\) −4.02184 + 2.92204i −0.568774 + 0.413239i
\(51\) −3.99135 − 0.848388i −0.558901 − 0.118798i
\(52\) 0.752021 − 0.159847i 0.104287 − 0.0221668i
\(53\) 0.491146 − 4.67294i 0.0674641 − 0.641878i −0.907582 − 0.419875i \(-0.862074\pi\)
0.975046 − 0.222003i \(-0.0712594\pi\)
\(54\) −0.500000 + 0.866025i −0.0680414 + 0.117851i
\(55\) 0.562173 − 0.00371003i 0.0758034 − 0.000500260i
\(56\) 0.160858 + 2.64086i 0.0214955 + 0.352899i
\(57\) 2.93471 + 2.13219i 0.388712 + 0.282416i
\(58\) −2.22125 − 2.46695i −0.291665 − 0.323927i
\(59\) 2.22714 − 2.47349i 0.289948 − 0.322020i −0.580519 − 0.814247i \(-0.697150\pi\)
0.870467 + 0.492227i \(0.163817\pi\)
\(60\) 0.154851 + 0.0689440i 0.0199912 + 0.00890064i
\(61\) 0.627005 + 5.96555i 0.0802798 + 0.763811i 0.958411 + 0.285391i \(0.0921236\pi\)
−0.878131 + 0.478420i \(0.841210\pi\)
\(62\) 1.60122 + 4.92806i 0.203356 + 0.625864i
\(63\) 1.85490 − 1.88662i 0.233696 − 0.237692i
\(64\) −0.809017 − 0.587785i −0.101127 − 0.0734732i
\(65\) −0.0651597 + 0.112860i −0.00808206 + 0.0139985i
\(66\) 1.67723 + 2.86128i 0.206453 + 0.352199i
\(67\) 2.70447 + 4.68428i 0.330404 + 0.572276i 0.982591 − 0.185782i \(-0.0594817\pi\)
−0.652187 + 0.758058i \(0.726148\pi\)
\(68\) −3.72774 + 1.65970i −0.452055 + 0.201268i
\(69\) 0.443644 − 1.36539i 0.0534084 − 0.164374i
\(70\) −0.350906 − 0.279266i −0.0419412 − 0.0333787i
\(71\) −2.83494 + 2.05970i −0.336445 + 0.244442i −0.743160 − 0.669113i \(-0.766674\pi\)
0.406715 + 0.913555i \(0.366674\pi\)
\(72\) 0.104528 + 0.994522i 0.0123188 + 0.117206i
\(73\) 1.41042 − 1.56642i 0.165077 − 0.183336i −0.654931 − 0.755689i \(-0.727302\pi\)
0.820008 + 0.572352i \(0.193969\pi\)
\(74\) 3.94124 − 0.837737i 0.458160 − 0.0973850i
\(75\) 4.54148 − 2.02200i 0.524405 − 0.233480i
\(76\) 3.62750 0.416103
\(77\) −2.39864 − 8.44076i −0.273351 − 0.961914i
\(78\) −0.768822 −0.0870519
\(79\) −13.0264 + 5.79971i −1.46558 + 0.652519i −0.975668 − 0.219253i \(-0.929638\pi\)
−0.489913 + 0.871771i \(0.662971\pi\)
\(80\) 0.165801 − 0.0352421i 0.0185371 − 0.00394019i
\(81\) 0.669131 − 0.743145i 0.0743478 − 0.0825716i
\(82\) −0.439085 − 4.17762i −0.0484889 − 0.461341i
\(83\) −13.2534 + 9.62913i −1.45475 + 1.05693i −0.470053 + 0.882638i \(0.655765\pi\)
−0.984693 + 0.174296i \(0.944235\pi\)
\(84\) 0.391723 − 2.61659i 0.0427404 − 0.285494i
\(85\) 0.213738 − 0.657818i 0.0231831 − 0.0713503i
\(86\) −3.25390 + 1.44873i −0.350877 + 0.156220i
\(87\) 1.65981 + 2.87487i 0.177950 + 0.308218i
\(88\) 3.03872 + 1.32897i 0.323929 + 0.141669i
\(89\) −0.842129 + 1.45861i −0.0892655 + 0.154612i −0.907201 − 0.420698i \(-0.861785\pi\)
0.817935 + 0.575310i \(0.195119\pi\)
\(90\) −0.137133 − 0.0996327i −0.0144551 − 0.0105022i
\(91\) 1.96919 + 0.509794i 0.206427 + 0.0534409i
\(92\) −0.443644 − 1.36539i −0.0462530 − 0.142352i
\(93\) −0.541632 − 5.15328i −0.0561646 − 0.534370i
\(94\) −7.64715 − 3.40473i −0.788744 − 0.351171i
\(95\) −0.411435 + 0.456945i −0.0422124 + 0.0468816i
\(96\) 0.669131 + 0.743145i 0.0682929 + 0.0758469i
\(97\) −12.4725 − 9.06181i −1.26639 − 0.920087i −0.267338 − 0.963603i \(-0.586144\pi\)
−0.999053 + 0.0435157i \(0.986144\pi\)
\(98\) −2.73835 + 6.44216i −0.276615 + 0.650757i
\(99\) −1.04569 − 3.14747i −0.105096 − 0.316332i
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 462.2.y.a.37.3 yes 24
7.4 even 3 inner 462.2.y.a.235.1 yes 24
11.3 even 5 inner 462.2.y.a.289.1 yes 24
77.25 even 15 inner 462.2.y.a.25.3 ✓ 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.y.a.25.3 ✓ 24 77.25 even 15 inner
462.2.y.a.37.3 yes 24 1.1 even 1 trivial
462.2.y.a.235.1 yes 24 7.4 even 3 inner
462.2.y.a.289.1 yes 24 11.3 even 5 inner