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 = [6,12,-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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 109x^{4} + 94x^{3} + 2808x^{2} + 81x - 9774 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4.90184\) 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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -5.90184 q^{3} +4.00000 q^{4} -11.8037 q^{6} -1.63515 q^{7} +8.00000 q^{8} +7.83172 q^{9} +42.3658 q^{11} -23.6074 q^{12} -18.1952 q^{13} -3.27031 q^{14} +16.0000 q^{16} -122.548 q^{17} +15.6634 q^{18} +87.0852 q^{19} +9.65041 q^{21} +84.7316 q^{22} -23.0000 q^{23} -47.2147 q^{24} -36.3904 q^{26} +113.128 q^{27} -6.54061 q^{28} -187.490 q^{29} +167.702 q^{31} +32.0000 q^{32} -250.036 q^{33} -245.096 q^{34} +31.3269 q^{36} +405.079 q^{37} +174.170 q^{38} +107.385 q^{39} -111.070 q^{41} +19.3008 q^{42} -192.424 q^{43} +169.463 q^{44} -46.0000 q^{46} -245.437 q^{47} -94.4294 q^{48} -340.326 q^{49} +723.258 q^{51} -72.7809 q^{52} +52.0761 q^{53} +226.256 q^{54} -13.0812 q^{56} -513.963 q^{57} -374.979 q^{58} -425.068 q^{59} +669.437 q^{61} +335.404 q^{62} -12.8061 q^{63} +64.0000 q^{64} -500.072 q^{66} +661.392 q^{67} -490.191 q^{68} +135.742 q^{69} -425.237 q^{71} +62.6537 q^{72} -1109.13 q^{73} +810.158 q^{74} +348.341 q^{76} -69.2745 q^{77} +214.771 q^{78} +1195.16 q^{79} -879.121 q^{81} -222.139 q^{82} -1475.59 q^{83} +38.6016 q^{84} -384.849 q^{86} +1106.53 q^{87} +338.926 q^{88} -74.2100 q^{89} +29.7520 q^{91} -92.0000 q^{92} -989.750 q^{93} -490.874 q^{94} -188.859 q^{96} -937.166 q^{97} -680.653 q^{98} +331.797 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 12 q^{2} - 5 q^{3} + 24 q^{4} - 10 q^{6} - 42 q^{7} + 48 q^{8} + 61 q^{9} - 49 q^{11} - 20 q^{12} - 16 q^{13} - 84 q^{14} + 96 q^{16} - 175 q^{17} + 122 q^{18} - 229 q^{19} + 92 q^{21} - 98 q^{22}+ \cdots - 142 q^{99}+O(q^{100}) \) 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
\(3\) −5.90184 −1.13581 −0.567905 − 0.823094i \(-0.692246\pi\)
−0.567905 + 0.823094i \(0.692246\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) −11.8037 −0.803139
\(7\) −1.63515 −0.0882900 −0.0441450 − 0.999025i \(-0.514056\pi\)
−0.0441450 + 0.999025i \(0.514056\pi\)
\(8\) 8.00000 0.353553
\(9\) 7.83172 0.290064
\(10\) 0 0
\(11\) 42.3658 1.16125 0.580626 − 0.814171i \(-0.302808\pi\)
0.580626 + 0.814171i \(0.302808\pi\)
\(12\) −23.6074 −0.567905
\(13\) −18.1952 −0.388188 −0.194094 − 0.980983i \(-0.562177\pi\)
−0.194094 + 0.980983i \(0.562177\pi\)
\(14\) −3.27031 −0.0624304
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −122.548 −1.74837 −0.874183 − 0.485597i \(-0.838602\pi\)
−0.874183 + 0.485597i \(0.838602\pi\)
\(18\) 15.6634 0.205106
\(19\) 87.0852 1.05151 0.525756 − 0.850636i \(-0.323783\pi\)
0.525756 + 0.850636i \(0.323783\pi\)
\(20\) 0 0
\(21\) 9.65041 0.100281
\(22\) 84.7316 0.821129
\(23\) −23.0000 −0.208514
\(24\) −47.2147 −0.401569
\(25\) 0 0
\(26\) −36.3904 −0.274490
\(27\) 113.128 0.806353
\(28\) −6.54061 −0.0441450
\(29\) −187.490 −1.20055 −0.600275 − 0.799793i \(-0.704942\pi\)
−0.600275 + 0.799793i \(0.704942\pi\)
\(30\) 0 0
\(31\) 167.702 0.971618 0.485809 − 0.874065i \(-0.338525\pi\)
0.485809 + 0.874065i \(0.338525\pi\)
\(32\) 32.0000 0.176777
\(33\) −250.036 −1.31896
\(34\) −245.096 −1.23628
\(35\) 0 0
\(36\) 31.3269 0.145032
\(37\) 405.079 1.79985 0.899926 − 0.436042i \(-0.143620\pi\)
0.899926 + 0.436042i \(0.143620\pi\)
\(38\) 174.170 0.743531
\(39\) 107.385 0.440908
\(40\) 0 0
\(41\) −111.070 −0.423078 −0.211539 − 0.977370i \(-0.567847\pi\)
−0.211539 + 0.977370i \(0.567847\pi\)
\(42\) 19.3008 0.0709091
\(43\) −192.424 −0.682429 −0.341214 − 0.939986i \(-0.610838\pi\)
−0.341214 + 0.939986i \(0.610838\pi\)
\(44\) 169.463 0.580626
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) −245.437 −0.761716 −0.380858 − 0.924634i \(-0.624371\pi\)
−0.380858 + 0.924634i \(0.624371\pi\)
\(48\) −94.4294 −0.283952
\(49\) −340.326 −0.992205
\(50\) 0 0
\(51\) 723.258 1.98581
\(52\) −72.7809 −0.194094
\(53\) 52.0761 0.134966 0.0674831 − 0.997720i \(-0.478503\pi\)
0.0674831 + 0.997720i \(0.478503\pi\)
\(54\) 226.256 0.570177
\(55\) 0 0
\(56\) −13.0812 −0.0312152
\(57\) −513.963 −1.19432
\(58\) −374.979 −0.848918
\(59\) −425.068 −0.937952 −0.468976 − 0.883211i \(-0.655377\pi\)
−0.468976 + 0.883211i \(0.655377\pi\)
\(60\) 0 0
\(61\) 669.437 1.40513 0.702563 − 0.711622i \(-0.252039\pi\)
0.702563 + 0.711622i \(0.252039\pi\)
\(62\) 335.404 0.687037
\(63\) −12.8061 −0.0256097
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −500.072 −0.932646
\(67\) 661.392 1.20600 0.602999 − 0.797742i \(-0.293972\pi\)
0.602999 + 0.797742i \(0.293972\pi\)
\(68\) −490.191 −0.874183
\(69\) 135.742 0.236833
\(70\) 0 0
\(71\) −425.237 −0.710793 −0.355396 − 0.934716i \(-0.615654\pi\)
−0.355396 + 0.934716i \(0.615654\pi\)
\(72\) 62.6537 0.102553
\(73\) −1109.13 −1.77827 −0.889133 − 0.457649i \(-0.848692\pi\)
−0.889133 + 0.457649i \(0.848692\pi\)
\(74\) 810.158 1.27269
\(75\) 0 0
\(76\) 348.341 0.525756
\(77\) −69.2745 −0.102527
\(78\) 214.771 0.311769
\(79\) 1195.16 1.70210 0.851052 − 0.525082i \(-0.175965\pi\)
0.851052 + 0.525082i \(0.175965\pi\)
\(80\) 0 0
\(81\) −879.121 −1.20593
\(82\) −222.139 −0.299161
\(83\) −1475.59 −1.95141 −0.975704 − 0.219095i \(-0.929690\pi\)
−0.975704 + 0.219095i \(0.929690\pi\)
\(84\) 38.6016 0.0501403
\(85\) 0 0
\(86\) −384.849 −0.482550
\(87\) 1106.53 1.36360
\(88\) 338.926 0.410564
\(89\) −74.2100 −0.0883847 −0.0441924 − 0.999023i \(-0.514071\pi\)
−0.0441924 + 0.999023i \(0.514071\pi\)
\(90\) 0 0
\(91\) 29.7520 0.0342731
\(92\) −92.0000 −0.104257
\(93\) −989.750 −1.10357
\(94\) −490.874 −0.538614
\(95\) 0 0
\(96\) −188.859 −0.200785
\(97\) −937.166 −0.980977 −0.490489 − 0.871448i \(-0.663182\pi\)
−0.490489 + 0.871448i \(0.663182\pi\)
\(98\) −680.653 −0.701595
\(99\) 331.797 0.336837
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.x.1.2 yes 6
5.2 odd 4 1150.4.b.s.599.11 12
5.3 odd 4 1150.4.b.s.599.2 12
5.4 even 2 1150.4.a.w.1.5 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.w.1.5 ✓ 6 5.4 even 2
1150.4.a.x.1.2 yes 6 1.1 even 1 trivial
1150.4.b.s.599.2 12 5.3 odd 4
1150.4.b.s.599.11 12 5.2 odd 4