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 = [5,10,0] 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: \( x^{5} - 73x^{3} - 73x^{2} + 810x - 260 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(8.40587\) 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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +8.40587 q^{3} +4.00000 q^{4} +16.8117 q^{6} +13.1917 q^{7} +8.00000 q^{8} +43.6586 q^{9} +21.7482 q^{11} +33.6235 q^{12} +46.5095 q^{13} +26.3834 q^{14} +16.0000 q^{16} +12.5291 q^{17} +87.3172 q^{18} -40.0158 q^{19} +110.888 q^{21} +43.4963 q^{22} -23.0000 q^{23} +67.2469 q^{24} +93.0190 q^{26} +140.030 q^{27} +52.7669 q^{28} +16.8513 q^{29} -77.4447 q^{31} +32.0000 q^{32} +182.812 q^{33} +25.0581 q^{34} +174.634 q^{36} -114.056 q^{37} -80.0316 q^{38} +390.953 q^{39} -62.7218 q^{41} +221.776 q^{42} +221.576 q^{43} +86.9927 q^{44} -46.0000 q^{46} -521.567 q^{47} +134.494 q^{48} -168.978 q^{49} +105.318 q^{51} +186.038 q^{52} -64.8860 q^{53} +280.060 q^{54} +105.534 q^{56} -336.367 q^{57} +33.7026 q^{58} +563.031 q^{59} +251.960 q^{61} -154.889 q^{62} +575.932 q^{63} +64.0000 q^{64} +365.624 q^{66} -702.025 q^{67} +50.1163 q^{68} -193.335 q^{69} +617.651 q^{71} +349.269 q^{72} -469.215 q^{73} -228.113 q^{74} -160.063 q^{76} +286.896 q^{77} +781.905 q^{78} +782.656 q^{79} -1.70968 q^{81} -125.444 q^{82} -342.619 q^{83} +443.551 q^{84} +443.151 q^{86} +141.650 q^{87} +173.985 q^{88} +572.765 q^{89} +613.541 q^{91} -92.0000 q^{92} -650.990 q^{93} -1043.13 q^{94} +268.988 q^{96} +959.780 q^{97} -337.957 q^{98} +949.494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 10 q^{2} + 20 q^{4} + 20 q^{7} + 40 q^{8} + 11 q^{9} + 16 q^{11} + 56 q^{13} + 40 q^{14} + 80 q^{16} + 70 q^{17} + 22 q^{18} + 32 q^{19} + 204 q^{21} + 32 q^{22} - 115 q^{23} + 112 q^{26} + 219 q^{27}+ \cdots + 2098 q^{99}+O(q^{100}) \) 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
\(3\) 8.40587 1.61771 0.808855 − 0.588008i \(-0.200088\pi\)
0.808855 + 0.588008i \(0.200088\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 16.8117 1.14389
\(7\) 13.1917 0.712286 0.356143 − 0.934431i \(-0.384092\pi\)
0.356143 + 0.934431i \(0.384092\pi\)
\(8\) 8.00000 0.353553
\(9\) 43.6586 1.61698
\(10\) 0 0
\(11\) 21.7482 0.596120 0.298060 − 0.954547i \(-0.403660\pi\)
0.298060 + 0.954547i \(0.403660\pi\)
\(12\) 33.6235 0.808855
\(13\) 46.5095 0.992263 0.496131 − 0.868247i \(-0.334753\pi\)
0.496131 + 0.868247i \(0.334753\pi\)
\(14\) 26.3834 0.503662
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 12.5291 0.178750 0.0893748 − 0.995998i \(-0.471513\pi\)
0.0893748 + 0.995998i \(0.471513\pi\)
\(18\) 87.3172 1.14338
\(19\) −40.0158 −0.483171 −0.241586 − 0.970380i \(-0.577667\pi\)
−0.241586 + 0.970380i \(0.577667\pi\)
\(20\) 0 0
\(21\) 110.888 1.15227
\(22\) 43.4963 0.421520
\(23\) −23.0000 −0.208514
\(24\) 67.2469 0.571947
\(25\) 0 0
\(26\) 93.0190 0.701636
\(27\) 140.030 0.998102
\(28\) 52.7669 0.356143
\(29\) 16.8513 0.107904 0.0539519 − 0.998544i \(-0.482818\pi\)
0.0539519 + 0.998544i \(0.482818\pi\)
\(30\) 0 0
\(31\) −77.4447 −0.448693 −0.224347 − 0.974509i \(-0.572025\pi\)
−0.224347 + 0.974509i \(0.572025\pi\)
\(32\) 32.0000 0.176777
\(33\) 182.812 0.964349
\(34\) 25.0581 0.126395
\(35\) 0 0
\(36\) 174.634 0.808492
\(37\) −114.056 −0.506777 −0.253389 − 0.967365i \(-0.581545\pi\)
−0.253389 + 0.967365i \(0.581545\pi\)
\(38\) −80.0316 −0.341654
\(39\) 390.953 1.60519
\(40\) 0 0
\(41\) −62.7218 −0.238915 −0.119457 − 0.992839i \(-0.538115\pi\)
−0.119457 + 0.992839i \(0.538115\pi\)
\(42\) 221.776 0.814779
\(43\) 221.576 0.785813 0.392907 − 0.919578i \(-0.371470\pi\)
0.392907 + 0.919578i \(0.371470\pi\)
\(44\) 86.9927 0.298060
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) −521.567 −1.61869 −0.809344 − 0.587334i \(-0.800177\pi\)
−0.809344 + 0.587334i \(0.800177\pi\)
\(48\) 134.494 0.404427
\(49\) −168.978 −0.492649
\(50\) 0 0
\(51\) 105.318 0.289165
\(52\) 186.038 0.496131
\(53\) −64.8860 −0.168166 −0.0840828 − 0.996459i \(-0.526796\pi\)
−0.0840828 + 0.996459i \(0.526796\pi\)
\(54\) 280.060 0.705765
\(55\) 0 0
\(56\) 105.534 0.251831
\(57\) −336.367 −0.781631
\(58\) 33.7026 0.0762994
\(59\) 563.031 1.24238 0.621190 − 0.783660i \(-0.286650\pi\)
0.621190 + 0.783660i \(0.286650\pi\)
\(60\) 0 0
\(61\) 251.960 0.528855 0.264428 − 0.964406i \(-0.414817\pi\)
0.264428 + 0.964406i \(0.414817\pi\)
\(62\) −154.889 −0.317274
\(63\) 575.932 1.15176
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 365.624 0.681898
\(67\) −702.025 −1.28009 −0.640045 − 0.768337i \(-0.721084\pi\)
−0.640045 + 0.768337i \(0.721084\pi\)
\(68\) 50.1163 0.0893748
\(69\) −193.335 −0.337316
\(70\) 0 0
\(71\) 617.651 1.03242 0.516209 − 0.856462i \(-0.327343\pi\)
0.516209 + 0.856462i \(0.327343\pi\)
\(72\) 349.269 0.571690
\(73\) −469.215 −0.752294 −0.376147 − 0.926560i \(-0.622751\pi\)
−0.376147 + 0.926560i \(0.622751\pi\)
\(74\) −228.113 −0.358346
\(75\) 0 0
\(76\) −160.063 −0.241586
\(77\) 286.896 0.424608
\(78\) 781.905 1.13504
\(79\) 782.656 1.11463 0.557315 − 0.830301i \(-0.311832\pi\)
0.557315 + 0.830301i \(0.311832\pi\)
\(80\) 0 0
\(81\) −1.70968 −0.00234524
\(82\) −125.444 −0.168938
\(83\) −342.619 −0.453101 −0.226550 − 0.973999i \(-0.572745\pi\)
−0.226550 + 0.973999i \(0.572745\pi\)
\(84\) 443.551 0.576136
\(85\) 0 0
\(86\) 443.151 0.555654
\(87\) 141.650 0.174557
\(88\) 173.985 0.210760
\(89\) 572.765 0.682168 0.341084 − 0.940033i \(-0.389206\pi\)
0.341084 + 0.940033i \(0.389206\pi\)
\(90\) 0 0
\(91\) 613.541 0.706775
\(92\) −92.0000 −0.104257
\(93\) −650.990 −0.725855
\(94\) −1043.13 −1.14459
\(95\) 0 0
\(96\) 268.988 0.285973
\(97\) 959.780 1.00465 0.502324 − 0.864679i \(-0.332478\pi\)
0.502324 + 0.864679i \(0.332478\pi\)
\(98\) −337.957 −0.348355
\(99\) 949.494 0.963917
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.u.1.5 yes 5
5.2 odd 4 1150.4.b.q.599.6 10
5.3 odd 4 1150.4.b.q.599.5 10
5.4 even 2 1150.4.a.r.1.1 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.r.1.1 ✓ 5 5.4 even 2
1150.4.a.u.1.5 yes 5 1.1 even 1 trivial
1150.4.b.q.599.5 10 5.3 odd 4
1150.4.b.q.599.6 10 5.2 odd 4