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Properties

Label 1150.4.a.v.1.4
Level $1150$
Weight $4$
Character 1150.1
Self dual yes
Analytic conductor $67.852$
Analytic rank $0$
Dimension $5$
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] = [1150,4,Mod(1,1150)] 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("1150.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,10,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 107x^{3} - 3x^{2} + 2151x - 2916 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-6.21006\) of defining polynomial
Character \(\chi\) \(=\) 1150.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +7.21006 q^{3} +4.00000 q^{4} +14.4201 q^{6} +19.8879 q^{7} +8.00000 q^{8} +24.9850 q^{9} -31.0553 q^{11} +28.8402 q^{12} +7.15563 q^{13} +39.7757 q^{14} +16.0000 q^{16} +40.8326 q^{17} +49.9699 q^{18} +144.948 q^{19} +143.393 q^{21} -62.1106 q^{22} +23.0000 q^{23} +57.6805 q^{24} +14.3113 q^{26} -14.5286 q^{27} +79.5515 q^{28} -189.113 q^{29} +41.6222 q^{31} +32.0000 q^{32} -223.910 q^{33} +81.6652 q^{34} +99.9398 q^{36} +37.3828 q^{37} +289.896 q^{38} +51.5925 q^{39} +401.324 q^{41} +286.785 q^{42} +29.2550 q^{43} -124.221 q^{44} +46.0000 q^{46} +18.0210 q^{47} +115.361 q^{48} +52.5274 q^{49} +294.405 q^{51} +28.6225 q^{52} +214.024 q^{53} -29.0572 q^{54} +159.103 q^{56} +1045.08 q^{57} -378.226 q^{58} +313.495 q^{59} +452.293 q^{61} +83.2444 q^{62} +496.898 q^{63} +64.0000 q^{64} -447.821 q^{66} -858.562 q^{67} +163.330 q^{68} +165.831 q^{69} +451.196 q^{71} +199.880 q^{72} -742.705 q^{73} +74.7656 q^{74} +579.792 q^{76} -617.623 q^{77} +103.185 q^{78} +126.369 q^{79} -779.346 q^{81} +802.649 q^{82} -473.737 q^{83} +573.571 q^{84} +58.5100 q^{86} -1363.52 q^{87} -248.442 q^{88} -109.465 q^{89} +142.310 q^{91} +92.0000 q^{92} +300.099 q^{93} +36.0420 q^{94} +230.722 q^{96} -711.495 q^{97} +105.055 q^{98} -775.915 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 10 q^{2} + 5 q^{3} + 20 q^{4} + 10 q^{6} - 3 q^{7} + 40 q^{8} + 84 q^{9} - 26 q^{11} + 20 q^{12} - 61 q^{13} - 6 q^{14} + 80 q^{16} + 231 q^{17} + 168 q^{18} + 74 q^{19} - 88 q^{21} - 52 q^{22} + 115 q^{23}+ \cdots + 2397 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 7.21006 1.38758 0.693788 − 0.720179i \(-0.255940\pi\)
0.693788 + 0.720179i \(0.255940\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 14.4201 0.981165
\(7\) 19.8879 1.07384 0.536922 − 0.843632i \(-0.319587\pi\)
0.536922 + 0.843632i \(0.319587\pi\)
\(8\) 8.00000 0.353553
\(9\) 24.9850 0.925369
\(10\) 0 0
\(11\) −31.0553 −0.851229 −0.425614 − 0.904905i \(-0.639942\pi\)
−0.425614 + 0.904905i \(0.639942\pi\)
\(12\) 28.8402 0.693788
\(13\) 7.15563 0.152663 0.0763313 − 0.997083i \(-0.475679\pi\)
0.0763313 + 0.997083i \(0.475679\pi\)
\(14\) 39.7757 0.759322
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 40.8326 0.582551 0.291275 − 0.956639i \(-0.405920\pi\)
0.291275 + 0.956639i \(0.405920\pi\)
\(18\) 49.9699 0.654335
\(19\) 144.948 1.75018 0.875089 − 0.483962i \(-0.160803\pi\)
0.875089 + 0.483962i \(0.160803\pi\)
\(20\) 0 0
\(21\) 143.393 1.49004
\(22\) −62.1106 −0.601910
\(23\) 23.0000 0.208514
\(24\) 57.6805 0.490582
\(25\) 0 0
\(26\) 14.3113 0.107949
\(27\) −14.5286 −0.103557
\(28\) 79.5515 0.536922
\(29\) −189.113 −1.21094 −0.605472 − 0.795866i \(-0.707016\pi\)
−0.605472 + 0.795866i \(0.707016\pi\)
\(30\) 0 0
\(31\) 41.6222 0.241147 0.120574 − 0.992704i \(-0.461527\pi\)
0.120574 + 0.992704i \(0.461527\pi\)
\(32\) 32.0000 0.176777
\(33\) −223.910 −1.18115
\(34\) 81.6652 0.411925
\(35\) 0 0
\(36\) 99.9398 0.462684
\(37\) 37.3828 0.166100 0.0830500 − 0.996545i \(-0.473534\pi\)
0.0830500 + 0.996545i \(0.473534\pi\)
\(38\) 289.896 1.23756
\(39\) 51.5925 0.211831
\(40\) 0 0
\(41\) 401.324 1.52869 0.764345 − 0.644807i \(-0.223062\pi\)
0.764345 + 0.644807i \(0.223062\pi\)
\(42\) 286.785 1.05362
\(43\) 29.2550 0.103752 0.0518761 − 0.998654i \(-0.483480\pi\)
0.0518761 + 0.998654i \(0.483480\pi\)
\(44\) −124.221 −0.425614
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) 18.0210 0.0559284 0.0279642 − 0.999609i \(-0.491098\pi\)
0.0279642 + 0.999609i \(0.491098\pi\)
\(48\) 115.361 0.346894
\(49\) 52.5274 0.153141
\(50\) 0 0
\(51\) 294.405 0.808333
\(52\) 28.6225 0.0763313
\(53\) 214.024 0.554689 0.277344 − 0.960771i \(-0.410546\pi\)
0.277344 + 0.960771i \(0.410546\pi\)
\(54\) −29.0572 −0.0732256
\(55\) 0 0
\(56\) 159.103 0.379661
\(57\) 1045.08 2.42851
\(58\) −378.226 −0.856267
\(59\) 313.495 0.691754 0.345877 − 0.938280i \(-0.387581\pi\)
0.345877 + 0.938280i \(0.387581\pi\)
\(60\) 0 0
\(61\) 452.293 0.949347 0.474674 − 0.880162i \(-0.342566\pi\)
0.474674 + 0.880162i \(0.342566\pi\)
\(62\) 83.2444 0.170517
\(63\) 496.898 0.993702
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −447.821 −0.835196
\(67\) −858.562 −1.56552 −0.782761 − 0.622322i \(-0.786189\pi\)
−0.782761 + 0.622322i \(0.786189\pi\)
\(68\) 163.330 0.291275
\(69\) 165.831 0.289330
\(70\) 0 0
\(71\) 451.196 0.754185 0.377093 − 0.926176i \(-0.376924\pi\)
0.377093 + 0.926176i \(0.376924\pi\)
\(72\) 199.880 0.327167
\(73\) −742.705 −1.19078 −0.595390 − 0.803437i \(-0.703003\pi\)
−0.595390 + 0.803437i \(0.703003\pi\)
\(74\) 74.7656 0.117450
\(75\) 0 0
\(76\) 579.792 0.875089
\(77\) −617.623 −0.914087
\(78\) 103.185 0.149787
\(79\) 126.369 0.179970 0.0899851 − 0.995943i \(-0.471318\pi\)
0.0899851 + 0.995943i \(0.471318\pi\)
\(80\) 0 0
\(81\) −779.346 −1.06906
\(82\) 802.649 1.08095
\(83\) −473.737 −0.626499 −0.313249 − 0.949671i \(-0.601418\pi\)
−0.313249 + 0.949671i \(0.601418\pi\)
\(84\) 573.571 0.745020
\(85\) 0 0
\(86\) 58.5100 0.0733639
\(87\) −1363.52 −1.68028
\(88\) −248.442 −0.300955
\(89\) −109.465 −0.130374 −0.0651869 − 0.997873i \(-0.520764\pi\)
−0.0651869 + 0.997873i \(0.520764\pi\)
\(90\) 0 0
\(91\) 142.310 0.163936
\(92\) 92.0000 0.104257
\(93\) 300.099 0.334610
\(94\) 36.0420 0.0395473
\(95\) 0 0
\(96\) 230.722 0.245291
\(97\) −711.495 −0.744756 −0.372378 − 0.928081i \(-0.621458\pi\)
−0.372378 + 0.928081i \(0.621458\pi\)
\(98\) 105.055 0.108287
\(99\) −775.915 −0.787701
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 1150.4.a.v.1.4 yes 5
5.2 odd 4 1150.4.b.r.599.7 10
5.3 odd 4 1150.4.b.r.599.4 10
5.4 even 2 1150.4.a.q.1.2 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.q.1.2 ✓ 5 5.4 even 2
1150.4.a.v.1.4 yes 5 1.1 even 1 trivial
1150.4.b.r.599.4 10 5.3 odd 4
1150.4.b.r.599.7 10 5.2 odd 4