Properties

Label 1150.4.a.q.1.3
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 1.3
Root \(1.53483\) 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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +0.534830 q^{3} +4.00000 q^{4} -1.06966 q^{6} +28.9380 q^{7} -8.00000 q^{8} -26.7140 q^{9} +33.5234 q^{11} +2.13932 q^{12} -10.2917 q^{13} -57.8760 q^{14} +16.0000 q^{16} -56.5854 q^{17} +53.4279 q^{18} -5.26405 q^{19} +15.4769 q^{21} -67.0468 q^{22} -23.0000 q^{23} -4.27864 q^{24} +20.5834 q^{26} -28.7278 q^{27} +115.752 q^{28} -162.334 q^{29} -160.556 q^{31} -32.0000 q^{32} +17.9293 q^{33} +113.171 q^{34} -106.856 q^{36} +16.9178 q^{37} +10.5281 q^{38} -5.50431 q^{39} +22.7758 q^{41} -30.9538 q^{42} -333.620 q^{43} +134.094 q^{44} +46.0000 q^{46} -130.383 q^{47} +8.55728 q^{48} +494.408 q^{49} -30.2636 q^{51} -41.1668 q^{52} -673.206 q^{53} +57.4557 q^{54} -231.504 q^{56} -2.81537 q^{57} +324.669 q^{58} +291.958 q^{59} +454.587 q^{61} +321.113 q^{62} -773.048 q^{63} +64.0000 q^{64} -35.8586 q^{66} -132.389 q^{67} -226.342 q^{68} -12.3011 q^{69} +121.326 q^{71} +213.712 q^{72} -176.482 q^{73} -33.8356 q^{74} -21.0562 q^{76} +970.100 q^{77} +11.0086 q^{78} +563.082 q^{79} +705.912 q^{81} -45.5516 q^{82} -809.068 q^{83} +61.9076 q^{84} +667.241 q^{86} -86.8213 q^{87} -268.187 q^{88} -702.417 q^{89} -297.821 q^{91} -92.0000 q^{92} -85.8703 q^{93} +260.766 q^{94} -17.1146 q^{96} -342.945 q^{97} -988.815 q^{98} -895.543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 10 q^{2} - 5 q^{3} + 20 q^{4} + 10 q^{6} + 3 q^{7} - 40 q^{8} + 84 q^{9} - 26 q^{11} - 20 q^{12} + 61 q^{13} - 6 q^{14} + 80 q^{16} - 231 q^{17} - 168 q^{18} + 74 q^{19} - 88 q^{21} + 52 q^{22} - 115 q^{23}+ \cdots + 2397 q^{99}+O(q^{100}) \) 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
\(3\) 0.534830 0.102928 0.0514640 − 0.998675i \(-0.483611\pi\)
0.0514640 + 0.998675i \(0.483611\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −1.06966 −0.0727812
\(7\) 28.9380 1.56250 0.781252 − 0.624215i \(-0.214581\pi\)
0.781252 + 0.624215i \(0.214581\pi\)
\(8\) −8.00000 −0.353553
\(9\) −26.7140 −0.989406
\(10\) 0 0
\(11\) 33.5234 0.918880 0.459440 − 0.888209i \(-0.348050\pi\)
0.459440 + 0.888209i \(0.348050\pi\)
\(12\) 2.13932 0.0514640
\(13\) −10.2917 −0.219570 −0.109785 − 0.993955i \(-0.535016\pi\)
−0.109785 + 0.993955i \(0.535016\pi\)
\(14\) −57.8760 −1.10486
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −56.5854 −0.807293 −0.403646 − 0.914915i \(-0.632257\pi\)
−0.403646 + 0.914915i \(0.632257\pi\)
\(18\) 53.4279 0.699616
\(19\) −5.26405 −0.0635608 −0.0317804 − 0.999495i \(-0.510118\pi\)
−0.0317804 + 0.999495i \(0.510118\pi\)
\(20\) 0 0
\(21\) 15.4769 0.160826
\(22\) −67.0468 −0.649747
\(23\) −23.0000 −0.208514
\(24\) −4.27864 −0.0363906
\(25\) 0 0
\(26\) 20.5834 0.155259
\(27\) −28.7278 −0.204766
\(28\) 115.752 0.781252
\(29\) −162.334 −1.03947 −0.519737 − 0.854327i \(-0.673970\pi\)
−0.519737 + 0.854327i \(0.673970\pi\)
\(30\) 0 0
\(31\) −160.556 −0.930218 −0.465109 − 0.885253i \(-0.653985\pi\)
−0.465109 + 0.885253i \(0.653985\pi\)
\(32\) −32.0000 −0.176777
\(33\) 17.9293 0.0945786
\(34\) 113.171 0.570842
\(35\) 0 0
\(36\) −106.856 −0.494703
\(37\) 16.9178 0.0751694 0.0375847 − 0.999293i \(-0.488034\pi\)
0.0375847 + 0.999293i \(0.488034\pi\)
\(38\) 10.5281 0.0449443
\(39\) −5.50431 −0.0225999
\(40\) 0 0
\(41\) 22.7758 0.0867557 0.0433779 − 0.999059i \(-0.486188\pi\)
0.0433779 + 0.999059i \(0.486188\pi\)
\(42\) −30.9538 −0.113721
\(43\) −333.620 −1.18318 −0.591589 − 0.806240i \(-0.701499\pi\)
−0.591589 + 0.806240i \(0.701499\pi\)
\(44\) 134.094 0.459440
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) −130.383 −0.404645 −0.202322 − 0.979319i \(-0.564849\pi\)
−0.202322 + 0.979319i \(0.564849\pi\)
\(48\) 8.55728 0.0257320
\(49\) 494.408 1.44142
\(50\) 0 0
\(51\) −30.2636 −0.0830931
\(52\) −41.1668 −0.109785
\(53\) −673.206 −1.74475 −0.872376 − 0.488835i \(-0.837422\pi\)
−0.872376 + 0.488835i \(0.837422\pi\)
\(54\) 57.4557 0.144791
\(55\) 0 0
\(56\) −231.504 −0.552429
\(57\) −2.81537 −0.00654219
\(58\) 324.669 0.735019
\(59\) 291.958 0.644233 0.322116 − 0.946700i \(-0.395606\pi\)
0.322116 + 0.946700i \(0.395606\pi\)
\(60\) 0 0
\(61\) 454.587 0.954162 0.477081 − 0.878859i \(-0.341695\pi\)
0.477081 + 0.878859i \(0.341695\pi\)
\(62\) 321.113 0.657764
\(63\) −773.048 −1.54595
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −35.8586 −0.0668772
\(67\) −132.389 −0.241401 −0.120701 − 0.992689i \(-0.538514\pi\)
−0.120701 + 0.992689i \(0.538514\pi\)
\(68\) −226.342 −0.403646
\(69\) −12.3011 −0.0214620
\(70\) 0 0
\(71\) 121.326 0.202799 0.101399 − 0.994846i \(-0.467668\pi\)
0.101399 + 0.994846i \(0.467668\pi\)
\(72\) 213.712 0.349808
\(73\) −176.482 −0.282955 −0.141477 − 0.989942i \(-0.545185\pi\)
−0.141477 + 0.989942i \(0.545185\pi\)
\(74\) −33.8356 −0.0531528
\(75\) 0 0
\(76\) −21.0562 −0.0317804
\(77\) 970.100 1.43576
\(78\) 11.0086 0.0159805
\(79\) 563.082 0.801920 0.400960 − 0.916096i \(-0.368677\pi\)
0.400960 + 0.916096i \(0.368677\pi\)
\(80\) 0 0
\(81\) 705.912 0.968330
\(82\) −45.5516 −0.0613456
\(83\) −809.068 −1.06996 −0.534981 − 0.844864i \(-0.679681\pi\)
−0.534981 + 0.844864i \(0.679681\pi\)
\(84\) 61.9076 0.0804128
\(85\) 0 0
\(86\) 667.241 0.836633
\(87\) −86.8213 −0.106991
\(88\) −268.187 −0.324873
\(89\) −702.417 −0.836585 −0.418293 − 0.908312i \(-0.637371\pi\)
−0.418293 + 0.908312i \(0.637371\pi\)
\(90\) 0 0
\(91\) −297.821 −0.343079
\(92\) −92.0000 −0.104257
\(93\) −85.8703 −0.0957456
\(94\) 260.766 0.286127
\(95\) 0 0
\(96\) −17.1146 −0.0181953
\(97\) −342.945 −0.358977 −0.179489 − 0.983760i \(-0.557444\pi\)
−0.179489 + 0.983760i \(0.557444\pi\)
\(98\) −988.815 −1.01924
\(99\) −895.543 −0.909146
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.1.3 ✓ 5
5.2 odd 4 1150.4.b.r.599.3 10
5.3 odd 4 1150.4.b.r.599.8 10
5.4 even 2 1150.4.a.v.1.3 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.q.1.3 ✓ 5 1.1 even 1 trivial
1150.4.a.v.1.3 yes 5 5.4 even 2
1150.4.b.r.599.3 10 5.2 odd 4
1150.4.b.r.599.8 10 5.3 odd 4