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Properties

Label 4650.2.a.bz.1.2
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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] = [4650,2,Mod(1,4650)] 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("4650.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,-2,2,0,2,1,-2,2,0,5,-2,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.53113\) of defining polynomial
Character \(\chi\) \(=\) 4650.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.53113 q^{7} -1.00000 q^{8} +1.00000 q^{9} +6.53113 q^{11} -1.00000 q^{12} -6.00000 q^{13} -4.53113 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +4.53113 q^{19} -4.53113 q^{21} -6.53113 q^{22} -6.53113 q^{23} +1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} +4.53113 q^{28} +1.00000 q^{31} -1.00000 q^{32} -6.53113 q^{33} -4.00000 q^{34} +1.00000 q^{36} +7.06226 q^{37} -4.53113 q^{38} +6.00000 q^{39} +7.06226 q^{41} +4.53113 q^{42} +8.53113 q^{43} +6.53113 q^{44} +6.53113 q^{46} +5.06226 q^{47} -1.00000 q^{48} +13.5311 q^{49} -4.00000 q^{51} -6.00000 q^{52} +2.53113 q^{53} +1.00000 q^{54} -4.53113 q^{56} -4.53113 q^{57} -9.06226 q^{59} +5.06226 q^{61} -1.00000 q^{62} +4.53113 q^{63} +1.00000 q^{64} +6.53113 q^{66} -5.06226 q^{67} +4.00000 q^{68} +6.53113 q^{69} -12.5311 q^{71} -1.00000 q^{72} +8.53113 q^{73} -7.06226 q^{74} +4.53113 q^{76} +29.5934 q^{77} -6.00000 q^{78} -8.53113 q^{79} +1.00000 q^{81} -7.06226 q^{82} +8.00000 q^{83} -4.53113 q^{84} -8.53113 q^{86} -6.53113 q^{88} -6.53113 q^{89} -27.1868 q^{91} -6.53113 q^{92} -1.00000 q^{93} -5.06226 q^{94} +1.00000 q^{96} -16.1245 q^{97} -13.5311 q^{98} +6.53113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9} + 5 q^{11} - 2 q^{12} - 12 q^{13} - q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{18} + q^{19} - q^{21} - 5 q^{22} - 5 q^{23} + 2 q^{24}+ \cdots + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 4.53113 1.71261 0.856303 − 0.516474i \(-0.172756\pi\)
0.856303 + 0.516474i \(0.172756\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.53113 1.96921 0.984605 − 0.174796i \(-0.0559265\pi\)
0.984605 + 0.174796i \(0.0559265\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −4.53113 −1.21100
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.53113 1.03951 0.519756 − 0.854315i \(-0.326023\pi\)
0.519756 + 0.854315i \(0.326023\pi\)
\(20\) 0 0
\(21\) −4.53113 −0.988773
\(22\) −6.53113 −1.39244
\(23\) −6.53113 −1.36183 −0.680917 − 0.732360i \(-0.738419\pi\)
−0.680917 + 0.732360i \(0.738419\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) 4.53113 0.856303
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) −6.53113 −1.13692
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 7.06226 1.16103 0.580514 − 0.814250i \(-0.302852\pi\)
0.580514 + 0.814250i \(0.302852\pi\)
\(38\) −4.53113 −0.735046
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 7.06226 1.10294 0.551470 − 0.834195i \(-0.314067\pi\)
0.551470 + 0.834195i \(0.314067\pi\)
\(42\) 4.53113 0.699168
\(43\) 8.53113 1.30098 0.650492 − 0.759513i \(-0.274563\pi\)
0.650492 + 0.759513i \(0.274563\pi\)
\(44\) 6.53113 0.984605
\(45\) 0 0
\(46\) 6.53113 0.962962
\(47\) 5.06226 0.738406 0.369203 − 0.929349i \(-0.379631\pi\)
0.369203 + 0.929349i \(0.379631\pi\)
\(48\) −1.00000 −0.144338
\(49\) 13.5311 1.93302
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) −6.00000 −0.832050
\(53\) 2.53113 0.347677 0.173839 − 0.984774i \(-0.444383\pi\)
0.173839 + 0.984774i \(0.444383\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −4.53113 −0.605498
\(57\) −4.53113 −0.600163
\(58\) 0 0
\(59\) −9.06226 −1.17981 −0.589903 − 0.807474i \(-0.700834\pi\)
−0.589903 + 0.807474i \(0.700834\pi\)
\(60\) 0 0
\(61\) 5.06226 0.648156 0.324078 − 0.946030i \(-0.394946\pi\)
0.324078 + 0.946030i \(0.394946\pi\)
\(62\) −1.00000 −0.127000
\(63\) 4.53113 0.570869
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.53113 0.803926
\(67\) −5.06226 −0.618453 −0.309227 − 0.950988i \(-0.600070\pi\)
−0.309227 + 0.950988i \(0.600070\pi\)
\(68\) 4.00000 0.485071
\(69\) 6.53113 0.786256
\(70\) 0 0
\(71\) −12.5311 −1.48717 −0.743586 − 0.668641i \(-0.766876\pi\)
−0.743586 + 0.668641i \(0.766876\pi\)
\(72\) −1.00000 −0.117851
\(73\) 8.53113 0.998493 0.499247 − 0.866460i \(-0.333610\pi\)
0.499247 + 0.866460i \(0.333610\pi\)
\(74\) −7.06226 −0.820971
\(75\) 0 0
\(76\) 4.53113 0.519756
\(77\) 29.5934 3.37248
\(78\) −6.00000 −0.679366
\(79\) −8.53113 −0.959827 −0.479913 − 0.877316i \(-0.659332\pi\)
−0.479913 + 0.877316i \(0.659332\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −7.06226 −0.779896
\(83\) 8.00000 0.878114 0.439057 − 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −4.53113 −0.494387
\(85\) 0 0
\(86\) −8.53113 −0.919935
\(87\) 0 0
\(88\) −6.53113 −0.696221
\(89\) −6.53113 −0.692298 −0.346149 − 0.938180i \(-0.612511\pi\)
−0.346149 + 0.938180i \(0.612511\pi\)
\(90\) 0 0
\(91\) −27.1868 −2.84995
\(92\) −6.53113 −0.680917
\(93\) −1.00000 −0.103695
\(94\) −5.06226 −0.522132
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −16.1245 −1.63720 −0.818598 − 0.574367i \(-0.805248\pi\)
−0.818598 + 0.574367i \(0.805248\pi\)
\(98\) −13.5311 −1.36685
\(99\) 6.53113 0.656403
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 4650.2.a.bz.1.2 2
5.2 odd 4 4650.2.d.bg.3349.2 4
5.3 odd 4 4650.2.d.bg.3349.3 4
5.4 even 2 930.2.a.q.1.1 ✓ 2
15.14 odd 2 2790.2.a.bf.1.1 2
20.19 odd 2 7440.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.q.1.1 ✓ 2 5.4 even 2
2790.2.a.bf.1.1 2 15.14 odd 2
4650.2.a.bz.1.2 2 1.1 even 1 trivial
4650.2.d.bg.3349.2 4 5.2 odd 4
4650.2.d.bg.3349.3 4 5.3 odd 4
7440.2.a.bd.1.2 2 20.19 odd 2