Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [15,11,Mod(14,15)] 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("15.14"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 15.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.53035879011\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 17880 x^{14} + 140656106 x^{12} + 568287997200 x^{10} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{32}\cdot 5^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 14.8
Root \(61.4341 - 30.7435i\) of defining polynomial
Character \(\chi\) \(=\) 15.14
Dual form 15.11.d.c.14.7

$q$-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-3.37421 q^{2} +(-224.817 + 92.2305i) q^{3} -1012.61 q^{4} +(862.380 + 3003.65i) q^{5} +(758.580 - 311.205i) q^{6} -16006.0i q^{7} +6871.97 q^{8} +(42036.1 - 41469.9i) q^{9} +(-2909.86 - 10135.0i) q^{10} +109087. i q^{11} +(227653. - 93393.9i) q^{12} -572716. i q^{13} +54007.6i q^{14} +(-470906. - 595733. i) q^{15} +1.01373e6 q^{16} +1.54275e6 q^{17} +(-141839. + 139928. i) q^{18} -720399. q^{19} +(-873259. - 3.04154e6i) q^{20} +(1.47624e6 + 3.59841e6i) q^{21} -368084. i q^{22} -2.49708e6 q^{23} +(-1.54493e6 + 633805. i) q^{24} +(-8.27823e6 + 5.18058e6i) q^{25} +1.93247e6i q^{26} +(-5.62563e6 + 1.32001e7i) q^{27} +1.62079e7i q^{28} -1.79908e7i q^{29} +(1.58894e6 + 2.01013e6i) q^{30} +1.83900e7 q^{31} -1.04574e7 q^{32} +(-1.00612e7 - 2.45246e7i) q^{33} -5.20558e6 q^{34} +(4.80764e7 - 1.38032e7i) q^{35} +(-4.25664e7 + 4.19930e7i) q^{36} -6.73581e7i q^{37} +2.43078e6 q^{38} +(5.28219e7 + 1.28756e8i) q^{39} +(5.92625e6 + 2.06410e7i) q^{40} -1.78223e8i q^{41} +(-4.98115e6 - 1.21418e7i) q^{42} -1.66496e8i q^{43} -1.10463e8i q^{44} +(1.60812e8 + 9.04990e7i) q^{45} +8.42568e6 q^{46} +2.99761e8 q^{47} +(-2.27903e8 + 9.34968e7i) q^{48} +2.62835e7 q^{49} +(2.79325e7 - 1.74804e7i) q^{50} +(-3.46837e8 + 1.42289e8i) q^{51} +5.79941e8i q^{52} -6.37040e8 q^{53} +(1.89821e7 - 4.45401e7i) q^{54} +(-3.27660e8 + 9.40747e7i) q^{55} -1.09993e8i q^{56} +(1.61958e8 - 6.64428e7i) q^{57} +6.07047e7i q^{58} +3.05843e7i q^{59} +(4.76846e8 + 6.03248e8i) q^{60} +9.80370e8 q^{61} -6.20518e7 q^{62} +(-6.63767e8 - 6.72829e8i) q^{63} -1.00277e9 q^{64} +(1.72024e9 - 4.93899e8i) q^{65} +(3.39485e7 + 8.27514e7i) q^{66} -6.91391e8i q^{67} -1.56222e9 q^{68} +(5.61385e8 - 2.30307e8i) q^{69} +(-1.62220e8 + 4.65751e7i) q^{70} +2.10199e9i q^{71} +(2.88871e8 - 2.84980e8i) q^{72} -2.47490e9i q^{73} +2.27281e8i q^{74} +(1.38328e9 - 1.92819e9i) q^{75} +7.29487e8 q^{76} +1.74605e9 q^{77} +(-1.78232e8 - 4.34451e8i) q^{78} +1.45740e9 q^{79} +(8.74220e8 + 3.04489e9i) q^{80} +(4.72803e7 - 3.48646e9i) q^{81} +6.01364e8i q^{82} -6.85447e9 q^{83} +(-1.49486e9 - 3.64381e9i) q^{84} +(1.33044e9 + 4.63390e9i) q^{85} +5.61794e8i q^{86} +(1.65930e9 + 4.04462e9i) q^{87} +7.49645e8i q^{88} +5.45554e9i q^{89} +(-5.42615e8 - 3.05363e8i) q^{90} -9.16689e9 q^{91} +2.52858e9 q^{92} +(-4.13438e9 + 1.69612e9i) q^{93} -1.01146e9 q^{94} +(-6.21258e8 - 2.16383e9i) q^{95} +(2.35101e9 - 9.64495e8i) q^{96} +3.04168e9i q^{97} -8.86862e7 q^{98} +(4.52384e9 + 4.58560e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2184 q^{4} + 21516 q^{6} - 63000 q^{9} - 221680 q^{10} + 731640 q^{15} - 4218352 q^{16} + 1487600 q^{19} + 2444616 q^{21} + 28021368 q^{24} - 9324800 q^{25} - 35678700 q^{30} - 77667568 q^{31} + 251882368 q^{34}+ \cdots + 42957756000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.37421 −0.105444 −0.0527221 − 0.998609i \(-0.516790\pi\)
−0.0527221 + 0.998609i \(0.516790\pi\)
\(3\) −224.817 + 92.2305i −0.925172 + 0.379549i
\(4\) −1012.61 −0.988882
\(5\) 862.380 + 3003.65i 0.275962 + 0.961169i
\(6\) 758.580 − 311.205i 0.0975540 − 0.0400213i
\(7\) − 16006.0i − 0.952341i −0.879353 − 0.476170i \(-0.842025\pi\)
0.879353 − 0.476170i \(-0.157975\pi\)
\(8\) 6871.97 0.209716
\(9\) 42036.1 − 41469.9i 0.711885 − 0.702296i
\(10\) −2909.86 − 10135.0i −0.0290986 − 0.101350i
\(11\) 109087.i 0.677346i 0.940904 + 0.338673i \(0.109978\pi\)
−0.940904 + 0.338673i \(0.890022\pi\)
\(12\) 227653. − 93393.9i 0.914885 − 0.375329i
\(13\) − 572716.i − 1.54249i −0.636538 − 0.771245i \(-0.719634\pi\)
0.636538 − 0.771245i \(-0.280366\pi\)
\(14\) 54007.6i 0.100419i
\(15\) −470906. − 595733.i −0.620123 − 0.784505i
\(16\) 1.01373e6 0.966768
\(17\) 1.54275e6 1.08656 0.543278 − 0.839553i \(-0.317183\pi\)
0.543278 + 0.839553i \(0.317183\pi\)
\(18\) −141839. + 139928.i −0.0750641 + 0.0740531i
\(19\) −720399. −0.290941 −0.145471 − 0.989363i \(-0.546470\pi\)
−0.145471 + 0.989363i \(0.546470\pi\)
\(20\) −873259. − 3.04154e6i −0.272893 − 0.950482i
\(21\) 1.47624e6 + 3.59841e6i 0.361460 + 0.881079i
\(22\) − 368084.i − 0.0714222i
\(23\) −2.49708e6 −0.387965 −0.193983 − 0.981005i \(-0.562141\pi\)
−0.193983 + 0.981005i \(0.562141\pi\)
\(24\) −1.54493e6 + 633805.i −0.194023 + 0.0795976i
\(25\) −8.27823e6 + 5.18058e6i −0.847690 + 0.530491i
\(26\) 1.93247e6i 0.162647i
\(27\) −5.62563e6 + 1.32001e7i −0.392060 + 0.919940i
\(28\) 1.62079e7i 0.941752i
\(29\) − 1.79908e7i − 0.877121i −0.898702 − 0.438561i \(-0.855488\pi\)
0.898702 − 0.438561i \(-0.144512\pi\)
\(30\) 1.58894e6 + 2.01013e6i 0.0653883 + 0.0827215i
\(31\) 1.83900e7 0.642352 0.321176 − 0.947020i \(-0.395922\pi\)
0.321176 + 0.947020i \(0.395922\pi\)
\(32\) −1.04574e7 −0.311656
\(33\) −1.00612e7 − 2.45246e7i −0.257086 − 0.626661i
\(34\) −5.20558e6 −0.114571
\(35\) 4.80764e7 − 1.38032e7i 0.915360 − 0.262809i
\(36\) −4.25664e7 + 4.19930e7i −0.703970 + 0.694488i
\(37\) − 6.73581e7i − 0.971362i −0.874136 − 0.485681i \(-0.838572\pi\)
0.874136 − 0.485681i \(-0.161428\pi\)
\(38\) 2.43078e6 0.0306781
\(39\) 5.28219e7 + 1.28756e8i 0.585451 + 1.42707i
\(40\) 5.92625e6 + 2.06410e7i 0.0578736 + 0.201572i
\(41\) − 1.78223e8i − 1.53832i −0.639058 − 0.769158i \(-0.720676\pi\)
0.639058 − 0.769158i \(-0.279324\pi\)
\(42\) −4.98115e6 − 1.21418e7i −0.0381139 − 0.0929046i
\(43\) − 1.66496e8i − 1.13256i −0.824212 − 0.566281i \(-0.808382\pi\)
0.824212 − 0.566281i \(-0.191618\pi\)
\(44\) − 1.10463e8i − 0.669815i
\(45\) 1.60812e8 + 9.04990e7i 0.871478 + 0.490435i
\(46\) 8.42568e6 0.0409087
\(47\) 2.99761e8 1.30703 0.653514 − 0.756914i \(-0.273294\pi\)
0.653514 + 0.756914i \(0.273294\pi\)
\(48\) −2.27903e8 + 9.34968e7i −0.894426 + 0.366936i
\(49\) 2.62835e7 0.0930472
\(50\) 2.79325e7 − 1.74804e7i 0.0893840 − 0.0559372i
\(51\) −3.46837e8 + 1.42289e8i −1.00525 + 0.412402i
\(52\) 5.79941e8i 1.52534i
\(53\) −6.37040e8 −1.52331 −0.761653 − 0.647985i \(-0.775612\pi\)
−0.761653 + 0.647985i \(0.775612\pi\)
\(54\) 1.89821e7 − 4.45401e7i 0.0413404 − 0.0970023i
\(55\) −3.27660e8 + 9.40747e7i −0.651044 + 0.186922i
\(56\) − 1.09993e8i − 0.199721i
\(57\) 1.61958e8 − 6.64428e7i 0.269171 − 0.110427i
\(58\) 6.07047e7i 0.0924873i
\(59\) 3.05843e7i 0.0427798i 0.999771 + 0.0213899i \(0.00680914\pi\)
−0.999771 + 0.0213899i \(0.993191\pi\)
\(60\) 4.76846e8 + 6.03248e8i 0.613228 + 0.775782i
\(61\) 9.80370e8 1.16076 0.580378 − 0.814347i \(-0.302905\pi\)
0.580378 + 0.814347i \(0.302905\pi\)
\(62\) −6.20518e7 −0.0677323
\(63\) −6.63767e8 − 6.72829e8i −0.668825 − 0.677957i
\(64\) −1.00277e9 −0.933906
\(65\) 1.72024e9 − 4.93899e8i 1.48259 − 0.425668i
\(66\) 3.39485e7 + 8.27514e7i 0.0271083 + 0.0660778i
\(67\) − 6.91391e8i − 0.512094i −0.966664 − 0.256047i \(-0.917580\pi\)
0.966664 − 0.256047i \(-0.0824203\pi\)
\(68\) −1.56222e9 −1.07448
\(69\) 5.61385e8 − 2.30307e8i 0.358935 − 0.147252i
\(70\) −1.62220e8 + 4.65751e7i −0.0965194 + 0.0277117i
\(71\) 2.10199e9i 1.16503i 0.812819 + 0.582516i \(0.197932\pi\)
−0.812819 + 0.582516i \(0.802068\pi\)
\(72\) 2.88871e8 − 2.84980e8i 0.149294 − 0.147283i
\(73\) − 2.47490e9i − 1.19383i −0.802304 − 0.596916i \(-0.796393\pi\)
0.802304 − 0.596916i \(-0.203607\pi\)
\(74\) 2.27281e8i 0.102424i
\(75\) 1.38328e9 − 1.92819e9i 0.582911 − 0.812536i
\(76\) 7.29487e8 0.287706
\(77\) 1.74605e9 0.645064
\(78\) −1.78232e8 − 4.34451e8i −0.0617324 − 0.150476i
\(79\) 1.45740e9 0.473635 0.236818 − 0.971554i \(-0.423896\pi\)
0.236818 + 0.971554i \(0.423896\pi\)
\(80\) 8.74220e8 + 3.04489e9i 0.266791 + 0.929227i
\(81\) 4.72803e7 − 3.48646e9i 0.0135599 − 0.999908i
\(82\) 6.01364e8i 0.162207i
\(83\) −6.85447e9 −1.74014 −0.870068 − 0.492931i \(-0.835925\pi\)
−0.870068 + 0.492931i \(0.835925\pi\)
\(84\) −1.49486e9 − 3.64381e9i −0.357441 − 0.871282i
\(85\) 1.33044e9 + 4.63390e9i 0.299848 + 1.04436i
\(86\) 5.61794e8i 0.119422i
\(87\) 1.65930e9 + 4.04462e9i 0.332911 + 0.811487i
\(88\) 7.49645e8i 0.142050i
\(89\) 5.45554e9i 0.976985i 0.872568 + 0.488493i \(0.162453\pi\)
−0.872568 + 0.488493i \(0.837547\pi\)
\(90\) −5.42615e8 − 3.05363e8i −0.0918923 − 0.0517135i
\(91\) −9.16689e9 −1.46898
\(92\) 2.52858e9 0.383652
\(93\) −4.13438e9 + 1.69612e9i −0.594286 + 0.243804i
\(94\) −1.01146e9 −0.137819
\(95\) −6.21258e8 − 2.16383e9i −0.0802886 − 0.279644i
\(96\) 2.35101e9 − 9.64495e8i 0.288335 − 0.118289i
\(97\) 3.04168e9i 0.354205i 0.984192 + 0.177103i \(0.0566725\pi\)
−0.984192 + 0.177103i \(0.943328\pi\)
\(98\) −8.86862e7 −0.00981129
\(99\) 4.52384e9 + 4.58560e9i 0.475698 + 0.482192i
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 15.11.d.c.14.8 yes 16
3.2 odd 2 inner 15.11.d.c.14.10 yes 16
5.2 odd 4 75.11.c.h.26.8 16
5.3 odd 4 75.11.c.h.26.9 16
5.4 even 2 inner 15.11.d.c.14.9 yes 16
15.2 even 4 75.11.c.h.26.10 16
15.8 even 4 75.11.c.h.26.7 16
15.14 odd 2 inner 15.11.d.c.14.7 ✓ 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.11.d.c.14.7 ✓ 16 15.14 odd 2 inner
15.11.d.c.14.8 yes 16 1.1 even 1 trivial
15.11.d.c.14.9 yes 16 5.4 even 2 inner
15.11.d.c.14.10 yes 16 3.2 odd 2 inner
75.11.c.h.26.7 16 15.8 even 4
75.11.c.h.26.8 16 5.2 odd 4
75.11.c.h.26.9 16 5.3 odd 4
75.11.c.h.26.10 16 15.2 even 4