Properties

Label 1150.4.a.q.1.4
Level $1150$
Weight $4$
Character 1150.1
Self dual yes
Analytic conductor $67.852$
Analytic rank $1$
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: \(1\)
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 \(3.88892\) 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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.88892 q^{3} +4.00000 q^{4} -5.77785 q^{6} +8.21839 q^{7} -8.00000 q^{8} -18.6541 q^{9} -71.5466 q^{11} +11.5557 q^{12} +66.1071 q^{13} -16.4368 q^{14} +16.0000 q^{16} +27.7650 q^{17} +37.3082 q^{18} +13.7670 q^{19} +23.7423 q^{21} +143.093 q^{22} -23.0000 q^{23} -23.1114 q^{24} -132.214 q^{26} -131.891 q^{27} +32.8736 q^{28} +274.558 q^{29} +265.877 q^{31} -32.0000 q^{32} -206.693 q^{33} -55.5300 q^{34} -74.6165 q^{36} -204.865 q^{37} -27.5341 q^{38} +190.978 q^{39} -69.5895 q^{41} -47.4846 q^{42} -187.919 q^{43} -286.186 q^{44} +46.0000 q^{46} -135.207 q^{47} +46.2228 q^{48} -275.458 q^{49} +80.2110 q^{51} +264.429 q^{52} -502.564 q^{53} +263.782 q^{54} -65.7472 q^{56} +39.7719 q^{57} -549.116 q^{58} -161.961 q^{59} -103.522 q^{61} -531.754 q^{62} -153.307 q^{63} +64.0000 q^{64} +413.385 q^{66} +985.315 q^{67} +111.060 q^{68} -66.4452 q^{69} -817.825 q^{71} +149.233 q^{72} -1119.51 q^{73} +409.729 q^{74} +55.0681 q^{76} -587.998 q^{77} -381.957 q^{78} +400.301 q^{79} +122.638 q^{81} +139.179 q^{82} -1113.02 q^{83} +94.9692 q^{84} +375.838 q^{86} +793.177 q^{87} +572.373 q^{88} -485.051 q^{89} +543.295 q^{91} -92.0000 q^{92} +768.098 q^{93} +270.413 q^{94} -92.4455 q^{96} +1036.55 q^{97} +550.916 q^{98} +1334.64 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 10 q^{2} - 5 q^{3} + 20 q^{4} + 10 q^{6} + 3 q^{7} - 40 q^{8} + 84 q^{9} - 26 q^{11} - 20 q^{12} + 61 q^{13} - 6 q^{14} + 80 q^{16} - 231 q^{17} - 168 q^{18} + 74 q^{19} - 88 q^{21} + 52 q^{22} - 115 q^{23}+ \cdots + 2397 q^{99}+O(q^{100}) \) 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
\(3\) 2.88892 0.555973 0.277987 − 0.960585i \(-0.410333\pi\)
0.277987 + 0.960585i \(0.410333\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −5.77785 −0.393133
\(7\) 8.21839 0.443752 0.221876 − 0.975075i \(-0.428782\pi\)
0.221876 + 0.975075i \(0.428782\pi\)
\(8\) −8.00000 −0.353553
\(9\) −18.6541 −0.690894
\(10\) 0 0
\(11\) −71.5466 −1.96110 −0.980551 − 0.196265i \(-0.937119\pi\)
−0.980551 + 0.196265i \(0.937119\pi\)
\(12\) 11.5557 0.277987
\(13\) 66.1071 1.41037 0.705185 − 0.709023i \(-0.250864\pi\)
0.705185 + 0.709023i \(0.250864\pi\)
\(14\) −16.4368 −0.313780
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 27.7650 0.396118 0.198059 − 0.980190i \(-0.436536\pi\)
0.198059 + 0.980190i \(0.436536\pi\)
\(18\) 37.3082 0.488535
\(19\) 13.7670 0.166230 0.0831151 − 0.996540i \(-0.473513\pi\)
0.0831151 + 0.996540i \(0.473513\pi\)
\(20\) 0 0
\(21\) 23.7423 0.246714
\(22\) 143.093 1.38671
\(23\) −23.0000 −0.208514
\(24\) −23.1114 −0.196566
\(25\) 0 0
\(26\) −132.214 −0.997283
\(27\) −131.891 −0.940092
\(28\) 32.8736 0.221876
\(29\) 274.558 1.75807 0.879037 − 0.476753i \(-0.158186\pi\)
0.879037 + 0.476753i \(0.158186\pi\)
\(30\) 0 0
\(31\) 265.877 1.54042 0.770208 − 0.637792i \(-0.220152\pi\)
0.770208 + 0.637792i \(0.220152\pi\)
\(32\) −32.0000 −0.176777
\(33\) −206.693 −1.09032
\(34\) −55.5300 −0.280098
\(35\) 0 0
\(36\) −74.6165 −0.345447
\(37\) −204.865 −0.910258 −0.455129 − 0.890425i \(-0.650407\pi\)
−0.455129 + 0.890425i \(0.650407\pi\)
\(38\) −27.5341 −0.117542
\(39\) 190.978 0.784129
\(40\) 0 0
\(41\) −69.5895 −0.265075 −0.132537 − 0.991178i \(-0.542312\pi\)
−0.132537 + 0.991178i \(0.542312\pi\)
\(42\) −47.4846 −0.174453
\(43\) −187.919 −0.666452 −0.333226 − 0.942847i \(-0.608137\pi\)
−0.333226 + 0.942847i \(0.608137\pi\)
\(44\) −286.186 −0.980551
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) −135.207 −0.419615 −0.209808 − 0.977743i \(-0.567284\pi\)
−0.209808 + 0.977743i \(0.567284\pi\)
\(48\) 46.2228 0.138993
\(49\) −275.458 −0.803085
\(50\) 0 0
\(51\) 80.2110 0.220231
\(52\) 264.429 0.705185
\(53\) −502.564 −1.30250 −0.651249 − 0.758864i \(-0.725755\pi\)
−0.651249 + 0.758864i \(0.725755\pi\)
\(54\) 263.782 0.664745
\(55\) 0 0
\(56\) −65.7472 −0.156890
\(57\) 39.7719 0.0924195
\(58\) −549.116 −1.24315
\(59\) −161.961 −0.357382 −0.178691 − 0.983905i \(-0.557186\pi\)
−0.178691 + 0.983905i \(0.557186\pi\)
\(60\) 0 0
\(61\) −103.522 −0.217290 −0.108645 − 0.994081i \(-0.534651\pi\)
−0.108645 + 0.994081i \(0.534651\pi\)
\(62\) −531.754 −1.08924
\(63\) −153.307 −0.306585
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 413.385 0.770973
\(67\) 985.315 1.79665 0.898324 − 0.439334i \(-0.144785\pi\)
0.898324 + 0.439334i \(0.144785\pi\)
\(68\) 111.060 0.198059
\(69\) −66.4452 −0.115928
\(70\) 0 0
\(71\) −817.825 −1.36701 −0.683507 − 0.729944i \(-0.739546\pi\)
−0.683507 + 0.729944i \(0.739546\pi\)
\(72\) 149.233 0.244268
\(73\) −1119.51 −1.79492 −0.897460 − 0.441096i \(-0.854590\pi\)
−0.897460 + 0.441096i \(0.854590\pi\)
\(74\) 409.729 0.643650
\(75\) 0 0
\(76\) 55.0681 0.0831151
\(77\) −587.998 −0.870242
\(78\) −381.957 −0.554463
\(79\) 400.301 0.570094 0.285047 − 0.958514i \(-0.407991\pi\)
0.285047 + 0.958514i \(0.407991\pi\)
\(80\) 0 0
\(81\) 122.638 0.168227
\(82\) 139.179 0.187436
\(83\) −1113.02 −1.47193 −0.735964 − 0.677020i \(-0.763271\pi\)
−0.735964 + 0.677020i \(0.763271\pi\)
\(84\) 94.9692 0.123357
\(85\) 0 0
\(86\) 375.838 0.471252
\(87\) 793.177 0.977443
\(88\) 572.373 0.693354
\(89\) −485.051 −0.577700 −0.288850 − 0.957374i \(-0.593273\pi\)
−0.288850 + 0.957374i \(0.593273\pi\)
\(90\) 0 0
\(91\) 543.295 0.625854
\(92\) −92.0000 −0.104257
\(93\) 768.098 0.856431
\(94\) 270.413 0.296713
\(95\) 0 0
\(96\) −92.4455 −0.0982832
\(97\) 1036.55 1.08500 0.542501 − 0.840055i \(-0.317477\pi\)
0.542501 + 0.840055i \(0.317477\pi\)
\(98\) 550.916 0.567867
\(99\) 1334.64 1.35491
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.q.1.4 ✓ 5
5.2 odd 4 1150.4.b.r.599.2 10
5.3 odd 4 1150.4.b.r.599.9 10
5.4 even 2 1150.4.a.v.1.2 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.q.1.4 ✓ 5 1.1 even 1 trivial
1150.4.a.v.1.2 yes 5 5.4 even 2
1150.4.b.r.599.2 10 5.2 odd 4
1150.4.b.r.599.9 10 5.3 odd 4