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 1.2
Root \(-5.17669\) 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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -5.17669 q^{3} +4.00000 q^{4} -10.3534 q^{6} -22.9133 q^{7} +8.00000 q^{8} -0.201881 q^{9} -40.2129 q^{11} -20.7068 q^{12} -33.8976 q^{13} -45.8265 q^{14} +16.0000 q^{16} +40.1358 q^{17} -0.403761 q^{18} -89.5256 q^{19} +118.615 q^{21} -80.4257 q^{22} -23.0000 q^{23} -41.4135 q^{24} -67.7952 q^{26} +140.816 q^{27} -91.6530 q^{28} -180.633 q^{29} +67.2935 q^{31} +32.0000 q^{32} +208.170 q^{33} +80.2717 q^{34} -0.807522 q^{36} -31.8860 q^{37} -179.051 q^{38} +175.477 q^{39} -214.595 q^{41} +237.230 q^{42} +165.783 q^{43} -160.851 q^{44} -46.0000 q^{46} +409.051 q^{47} -82.8270 q^{48} +182.017 q^{49} -207.771 q^{51} -135.590 q^{52} +179.473 q^{53} +281.631 q^{54} -183.306 q^{56} +463.446 q^{57} -361.266 q^{58} +595.328 q^{59} -388.999 q^{61} +134.587 q^{62} +4.62574 q^{63} +64.0000 q^{64} +416.339 q^{66} +506.930 q^{67} +160.543 q^{68} +119.064 q^{69} -317.777 q^{71} -1.61504 q^{72} -396.653 q^{73} -63.7720 q^{74} -358.103 q^{76} +921.408 q^{77} +350.955 q^{78} +610.474 q^{79} -723.508 q^{81} -429.190 q^{82} -709.836 q^{83} +474.459 q^{84} +331.566 q^{86} +935.081 q^{87} -321.703 q^{88} -178.052 q^{89} +776.705 q^{91} -92.0000 q^{92} -348.358 q^{93} +818.102 q^{94} -165.654 q^{96} -711.093 q^{97} +364.035 q^{98} +8.11819 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 10 q^{2} + 20 q^{4} + 20 q^{7} + 40 q^{8} + 11 q^{9} + 16 q^{11} + 56 q^{13} + 40 q^{14} + 80 q^{16} + 70 q^{17} + 22 q^{18} + 32 q^{19} + 204 q^{21} + 32 q^{22} - 115 q^{23} + 112 q^{26} + 219 q^{27}+ \cdots + 2098 q^{99}+O(q^{100}) \) 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
\(3\) −5.17669 −0.996254 −0.498127 − 0.867104i \(-0.665979\pi\)
−0.498127 + 0.867104i \(0.665979\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −10.3534 −0.704458
\(7\) −22.9133 −1.23720 −0.618600 − 0.785706i \(-0.712300\pi\)
−0.618600 + 0.785706i \(0.712300\pi\)
\(8\) 8.00000 0.353553
\(9\) −0.201881 −0.00747706
\(10\) 0 0
\(11\) −40.2129 −1.10224 −0.551120 − 0.834426i \(-0.685799\pi\)
−0.551120 + 0.834426i \(0.685799\pi\)
\(12\) −20.7068 −0.498127
\(13\) −33.8976 −0.723193 −0.361596 − 0.932335i \(-0.617768\pi\)
−0.361596 + 0.932335i \(0.617768\pi\)
\(14\) −45.8265 −0.874832
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 40.1358 0.572610 0.286305 − 0.958139i \(-0.407573\pi\)
0.286305 + 0.958139i \(0.407573\pi\)
\(18\) −0.403761 −0.00528708
\(19\) −89.5256 −1.08098 −0.540489 − 0.841351i \(-0.681761\pi\)
−0.540489 + 0.841351i \(0.681761\pi\)
\(20\) 0 0
\(21\) 118.615 1.23257
\(22\) −80.4257 −0.779401
\(23\) −23.0000 −0.208514
\(24\) −41.4135 −0.352229
\(25\) 0 0
\(26\) −67.7952 −0.511374
\(27\) 140.816 1.00370
\(28\) −91.6530 −0.618600
\(29\) −180.633 −1.15664 −0.578322 − 0.815808i \(-0.696292\pi\)
−0.578322 + 0.815808i \(0.696292\pi\)
\(30\) 0 0
\(31\) 67.2935 0.389880 0.194940 − 0.980815i \(-0.437549\pi\)
0.194940 + 0.980815i \(0.437549\pi\)
\(32\) 32.0000 0.176777
\(33\) 208.170 1.09811
\(34\) 80.2717 0.404897
\(35\) 0 0
\(36\) −0.807522 −0.00373853
\(37\) −31.8860 −0.141676 −0.0708382 − 0.997488i \(-0.522567\pi\)
−0.0708382 + 0.997488i \(0.522567\pi\)
\(38\) −179.051 −0.764367
\(39\) 175.477 0.720484
\(40\) 0 0
\(41\) −214.595 −0.817417 −0.408708 − 0.912665i \(-0.634021\pi\)
−0.408708 + 0.912665i \(0.634021\pi\)
\(42\) 237.230 0.871556
\(43\) 165.783 0.587946 0.293973 − 0.955814i \(-0.405022\pi\)
0.293973 + 0.955814i \(0.405022\pi\)
\(44\) −160.851 −0.551120
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) 409.051 1.26949 0.634747 − 0.772720i \(-0.281104\pi\)
0.634747 + 0.772720i \(0.281104\pi\)
\(48\) −82.8270 −0.249064
\(49\) 182.017 0.530663
\(50\) 0 0
\(51\) −207.771 −0.570465
\(52\) −135.590 −0.361596
\(53\) 179.473 0.465141 0.232570 − 0.972580i \(-0.425286\pi\)
0.232570 + 0.972580i \(0.425286\pi\)
\(54\) 281.631 0.709726
\(55\) 0 0
\(56\) −183.306 −0.437416
\(57\) 463.446 1.07693
\(58\) −361.266 −0.817871
\(59\) 595.328 1.31365 0.656823 − 0.754045i \(-0.271900\pi\)
0.656823 + 0.754045i \(0.271900\pi\)
\(60\) 0 0
\(61\) −388.999 −0.816494 −0.408247 − 0.912871i \(-0.633860\pi\)
−0.408247 + 0.912871i \(0.633860\pi\)
\(62\) 134.587 0.275687
\(63\) 4.62574 0.00925061
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 416.339 0.776482
\(67\) 506.930 0.924348 0.462174 − 0.886789i \(-0.347070\pi\)
0.462174 + 0.886789i \(0.347070\pi\)
\(68\) 160.543 0.286305
\(69\) 119.064 0.207733
\(70\) 0 0
\(71\) −317.777 −0.531172 −0.265586 − 0.964087i \(-0.585565\pi\)
−0.265586 + 0.964087i \(0.585565\pi\)
\(72\) −1.61504 −0.00264354
\(73\) −396.653 −0.635955 −0.317978 − 0.948098i \(-0.603004\pi\)
−0.317978 + 0.948098i \(0.603004\pi\)
\(74\) −63.7720 −0.100180
\(75\) 0 0
\(76\) −358.103 −0.540489
\(77\) 921.408 1.36369
\(78\) 350.955 0.509459
\(79\) 610.474 0.869414 0.434707 − 0.900572i \(-0.356852\pi\)
0.434707 + 0.900572i \(0.356852\pi\)
\(80\) 0 0
\(81\) −723.508 −0.992467
\(82\) −429.190 −0.578001
\(83\) −709.836 −0.938730 −0.469365 − 0.883004i \(-0.655517\pi\)
−0.469365 + 0.883004i \(0.655517\pi\)
\(84\) 474.459 0.616283
\(85\) 0 0
\(86\) 331.566 0.415741
\(87\) 935.081 1.15231
\(88\) −321.703 −0.389701
\(89\) −178.052 −0.212061 −0.106031 − 0.994363i \(-0.533814\pi\)
−0.106031 + 0.994363i \(0.533814\pi\)
\(90\) 0 0
\(91\) 776.705 0.894734
\(92\) −92.0000 −0.104257
\(93\) −348.358 −0.388419
\(94\) 818.102 0.897667
\(95\) 0 0
\(96\) −165.654 −0.176115
\(97\) −711.093 −0.744336 −0.372168 − 0.928165i \(-0.621385\pi\)
−0.372168 + 0.928165i \(0.621385\pi\)
\(98\) 364.035 0.375235
\(99\) 8.11819 0.00824151
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.1.2 yes 5
5.2 odd 4 1150.4.b.q.599.9 10
5.3 odd 4 1150.4.b.q.599.2 10
5.4 even 2 1150.4.a.r.1.4 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.r.1.4 ✓ 5 5.4 even 2
1150.4.a.u.1.2 yes 5 1.1 even 1 trivial
1150.4.b.q.599.2 10 5.3 odd 4
1150.4.b.q.599.9 10 5.2 odd 4