Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [42,12,Mod(25,42)] 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("42.25"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 42.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,128] 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: \(32.2704135835\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} + 131491 x^{6} + 44838722 x^{5} + 18152831051 x^{4} + 2926931386118 x^{3} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3}\cdot 7^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.4
Root \(-73.4634 - 127.242i\) of defining polynomial
Character \(\chi\) \(=\) 42.37
Dual form 42.12.e.d.25.4

$q$-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+(16.0000 - 27.7128i) q^{2} +(121.500 + 210.444i) q^{3} +(-512.000 - 886.810i) q^{4} +(6385.96 - 11060.8i) q^{5} +7776.00 q^{6} +(-43747.2 + 7969.08i) q^{7} -32768.0 q^{8} +(-29524.5 + 51137.9i) q^{9} +(-204351. - 353946. i) q^{10} +(-237628. - 411583. i) q^{11} +(124416. - 215495. i) q^{12} -1.02512e6 q^{13} +(-479110. + 1.33986e6i) q^{14} +3.10358e6 q^{15} +(-524288. + 908093. i) q^{16} +(3.50148e6 + 6.06475e6i) q^{17} +(944784. + 1.63641e6i) q^{18} +(1.99041e6 - 3.44749e6i) q^{19} -1.30784e7 q^{20} +(-6.99234e6 - 8.23811e6i) q^{21} -1.52082e7 q^{22} +(-7.54845e6 + 1.30743e7i) q^{23} +(-3.98131e6 - 6.89583e6i) q^{24} +(-5.71469e7 - 9.89814e7i) q^{25} +(-1.64019e7 + 2.84090e7i) q^{26} -1.43489e7 q^{27} +(2.94656e7 + 3.47153e7i) q^{28} -5.56637e7 q^{29} +(4.96572e7 - 8.60088e7i) q^{30} +(-7.95734e7 - 1.37825e8i) q^{31} +(1.67772e7 + 2.90590e7i) q^{32} +(5.77435e7 - 1.00015e8i) q^{33} +2.24095e8 q^{34} +(-1.91224e8 + 5.34770e8i) q^{35} +6.04662e7 q^{36} +(-3.59905e8 + 6.23374e8i) q^{37} +(-6.36930e7 - 1.10320e8i) q^{38} +(-1.24552e8 - 2.15731e8i) q^{39} +(-2.09255e8 + 3.62441e8i) q^{40} -1.45784e8 q^{41} +(-3.40178e8 + 6.19676e7i) q^{42} -1.76669e9 q^{43} +(-2.43331e8 + 4.21461e8i) q^{44} +(3.77085e8 + 6.53130e8i) q^{45} +(2.41550e8 + 4.18377e8i) q^{46} +(7.02041e8 - 1.21597e9i) q^{47} -2.54804e8 q^{48} +(1.85031e9 - 6.97250e8i) q^{49} -3.65740e9 q^{50} +(-8.50861e8 + 1.47373e9i) q^{51} +(5.24862e8 + 9.09088e8i) q^{52} +(6.25567e8 + 1.08351e9i) q^{53} +(-2.29583e8 + 3.97649e8i) q^{54} -6.06992e9 q^{55} +(1.43351e9 - 2.61131e8i) q^{56} +9.67338e8 q^{57} +(-8.90618e8 + 1.54260e9i) q^{58} +(-6.66848e8 - 1.15501e9i) q^{59} +(-1.58903e9 - 2.75228e9i) q^{60} +(4.39590e9 - 7.61393e9i) q^{61} -5.09270e9 q^{62} +(8.84093e8 - 2.47243e9i) q^{63} +1.07374e9 q^{64} +(-6.54639e9 + 1.13387e10i) q^{65} +(-1.84779e9 - 3.20047e9i) q^{66} +(-8.49061e9 - 1.47062e10i) q^{67} +(3.58552e9 - 6.21030e9i) q^{68} -3.66854e9 q^{69} +(1.17604e10 + 1.38557e10i) q^{70} -1.62499e10 q^{71} +(9.67459e8 - 1.67569e9i) q^{72} +(1.06939e10 + 1.85223e10i) q^{73} +(1.15170e10 + 1.99480e10i) q^{74} +(1.38867e10 - 2.40525e10i) q^{75} -4.07635e9 q^{76} +(1.36755e10 + 1.61119e10i) q^{77} -7.97135e9 q^{78} +(2.05351e10 - 3.55678e10i) q^{79} +(6.69616e9 + 1.15981e10i) q^{80} +(-1.74339e9 - 3.01964e9i) q^{81} +(-2.33255e9 + 4.04010e9i) q^{82} +1.96280e10 q^{83} +(-3.72556e9 + 1.04188e10i) q^{84} +8.94413e10 q^{85} +(-2.82670e10 + 4.89598e10i) q^{86} +(-6.76313e9 - 1.17141e10i) q^{87} +(7.78658e9 + 1.34868e10i) q^{88} +(3.88169e10 - 6.72329e10i) q^{89} +2.41334e10 q^{90} +(4.48462e10 - 8.16928e9i) q^{91} +1.54592e10 q^{92} +(1.93363e10 - 3.34915e10i) q^{93} +(-2.24653e10 - 3.89111e10i) q^{94} +(-2.54213e10 - 4.40310e10i) q^{95} +(-4.07686e9 + 7.06133e9i) q^{96} -6.65701e10 q^{97} +(1.02823e10 - 6.24334e10i) q^{98} +2.80633e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 128 q^{2} + 972 q^{3} - 4096 q^{4} - 1420 q^{5} + 62208 q^{6} - 66362 q^{7} - 262144 q^{8} - 236196 q^{9} + 45440 q^{10} - 861962 q^{11} + 995328 q^{12} + 836748 q^{13} - 1361216 q^{14} - 690120 q^{15}+ \cdots + 101795988276 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 − 27.7128i 0.353553 − 0.612372i
\(3\) 121.500 + 210.444i 0.288675 + 0.500000i
\(4\) −512.000 − 886.810i −0.250000 − 0.433013i
\(5\) 6385.96 − 11060.8i 0.913884 − 1.58289i 0.105358 − 0.994434i \(-0.466401\pi\)
0.808527 − 0.588460i \(-0.200265\pi\)
\(6\) 7776.00 0.408248
\(7\) −43747.2 + 7969.08i −0.983810 + 0.179213i
\(8\) −32768.0 −0.353553
\(9\) −29524.5 + 51137.9i −0.166667 + 0.288675i
\(10\) −204351. − 353946.i −0.646214 − 1.11928i
\(11\) −237628. − 411583.i −0.444874 − 0.770544i 0.553169 − 0.833069i \(-0.313418\pi\)
−0.998043 + 0.0625244i \(0.980085\pi\)
\(12\) 124416. − 215495.i 0.144338 − 0.250000i
\(13\) −1.02512e6 −0.765750 −0.382875 − 0.923800i \(-0.625066\pi\)
−0.382875 + 0.923800i \(0.625066\pi\)
\(14\) −479110. + 1.33986e6i −0.238085 + 0.665820i
\(15\) 3.10358e6 1.05526
\(16\) −524288. + 908093.i −0.125000 + 0.216506i
\(17\) 3.50148e6 + 6.06475e6i 0.598113 + 1.03596i 0.993100 + 0.117275i \(0.0374158\pi\)
−0.394987 + 0.918687i \(0.629251\pi\)
\(18\) 944784. + 1.63641e6i 0.117851 + 0.204124i
\(19\) 1.99041e6 − 3.44749e6i 0.184415 − 0.319417i −0.758964 − 0.651133i \(-0.774294\pi\)
0.943379 + 0.331716i \(0.107628\pi\)
\(20\) −1.30784e7 −0.913884
\(21\) −6.99234e6 − 8.23811e6i −0.373608 − 0.440171i
\(22\) −1.52082e7 −0.629147
\(23\) −7.54845e6 + 1.30743e7i −0.244543 + 0.423560i −0.962003 − 0.273039i \(-0.911971\pi\)
0.717460 + 0.696599i \(0.245304\pi\)
\(24\) −3.98131e6 − 6.89583e6i −0.102062 − 0.176777i
\(25\) −5.71469e7 − 9.89814e7i −1.17037 − 2.02714i
\(26\) −1.64019e7 + 2.84090e7i −0.270734 + 0.468924i
\(27\) −1.43489e7 −0.192450
\(28\) 2.94656e7 + 3.47153e7i 0.323554 + 0.381199i
\(29\) −5.56637e7 −0.503945 −0.251972 − 0.967734i \(-0.581079\pi\)
−0.251972 + 0.967734i \(0.581079\pi\)
\(30\) 4.96572e7 − 8.60088e7i 0.373092 − 0.646214i
\(31\) −7.95734e7 − 1.37825e8i −0.499204 − 0.864647i 0.500795 − 0.865566i \(-0.333041\pi\)
−1.00000 0.000918421i \(0.999708\pi\)
\(32\) 1.67772e7 + 2.90590e7i 0.0883883 + 0.153093i
\(33\) 5.77435e7 − 1.00015e8i 0.256848 − 0.444874i
\(34\) 2.24095e8 0.845859
\(35\) −1.91224e8 + 5.34770e8i −0.615414 + 1.72105i
\(36\) 6.04662e7 0.166667
\(37\) −3.59905e8 + 6.23374e8i −0.853254 + 1.47788i 0.0250016 + 0.999687i \(0.492041\pi\)
−0.878256 + 0.478192i \(0.841292\pi\)
\(38\) −6.36930e7 − 1.10320e8i −0.130401 − 0.225862i
\(39\) −1.24552e8 − 2.15731e8i −0.221053 − 0.382875i
\(40\) −2.09255e8 + 3.62441e8i −0.323107 + 0.559638i
\(41\) −1.45784e8 −0.196517 −0.0982584 − 0.995161i \(-0.531327\pi\)
−0.0982584 + 0.995161i \(0.531327\pi\)
\(42\) −3.40178e8 + 6.19676e7i −0.401639 + 0.0731633i
\(43\) −1.76669e9 −1.83266 −0.916332 − 0.400419i \(-0.868865\pi\)
−0.916332 + 0.400419i \(0.868865\pi\)
\(44\) −2.43331e8 + 4.21461e8i −0.222437 + 0.385272i
\(45\) 3.77085e8 + 6.53130e8i 0.304628 + 0.527631i
\(46\) 2.41550e8 + 4.18377e8i 0.172918 + 0.299502i
\(47\) 7.02041e8 − 1.21597e9i 0.446503 − 0.773366i −0.551653 − 0.834074i \(-0.686003\pi\)
0.998156 + 0.0607083i \(0.0193359\pi\)
\(48\) −2.54804e8 −0.144338
\(49\) 1.85031e9 − 6.97250e8i 0.935766 − 0.352623i
\(50\) −3.65740e9 −1.65515
\(51\) −8.50861e8 + 1.47373e9i −0.345321 + 0.598113i
\(52\) 5.24862e8 + 9.09088e8i 0.191438 + 0.331580i
\(53\) 6.25567e8 + 1.08351e9i 0.205474 + 0.355891i 0.950284 − 0.311386i \(-0.100793\pi\)
−0.744810 + 0.667277i \(0.767460\pi\)
\(54\) −2.29583e8 + 3.97649e8i −0.0680414 + 0.117851i
\(55\) −6.06992e9 −1.62625
\(56\) 1.43351e9 − 2.61131e8i 0.347829 − 0.0633613i
\(57\) 9.67338e8 0.212944
\(58\) −8.90618e8 + 1.54260e9i −0.178171 + 0.308602i
\(59\) −6.66848e8 − 1.15501e9i −0.121434 − 0.210330i 0.798899 − 0.601465i \(-0.205416\pi\)
−0.920333 + 0.391135i \(0.872083\pi\)
\(60\) −1.58903e9 − 2.75228e9i −0.263816 − 0.456942i
\(61\) 4.39590e9 − 7.61393e9i 0.666398 − 1.15424i −0.312506 − 0.949916i \(-0.601168\pi\)
0.978904 − 0.204320i \(-0.0654983\pi\)
\(62\) −5.09270e9 −0.705982
\(63\) 8.84093e8 − 2.47243e9i 0.112234 − 0.313870i
\(64\) 1.07374e9 0.125000
\(65\) −6.54639e9 + 1.13387e10i −0.699807 + 1.21210i
\(66\) −1.84779e9 − 3.20047e9i −0.181619 − 0.314573i
\(67\) −8.49061e9 − 1.47062e10i −0.768294 − 1.33072i −0.938488 − 0.345313i \(-0.887773\pi\)
0.170194 − 0.985411i \(-0.445561\pi\)
\(68\) 3.58552e9 − 6.21030e9i 0.299056 − 0.517981i
\(69\) −3.66854e9 −0.282373
\(70\) 1.17604e10 + 1.38557e10i 0.836340 + 0.985345i
\(71\) −1.62499e10 −1.06888 −0.534441 − 0.845206i \(-0.679478\pi\)
−0.534441 + 0.845206i \(0.679478\pi\)
\(72\) 9.67459e8 − 1.67569e9i 0.0589256 − 0.102062i
\(73\) 1.06939e10 + 1.85223e10i 0.603752 + 1.04573i 0.992247 + 0.124279i \(0.0396618\pi\)
−0.388495 + 0.921451i \(0.627005\pi\)
\(74\) 1.15170e10 + 1.99480e10i 0.603342 + 1.04502i
\(75\) 1.38867e10 − 2.40525e10i 0.675713 − 1.17037i
\(76\) −4.07635e9 −0.184415
\(77\) 1.36755e10 + 1.61119e10i 0.575763 + 0.678342i
\(78\) −7.97135e9 −0.312616
\(79\) 2.05351e10 − 3.55678e10i 0.750840 − 1.30049i −0.196577 − 0.980488i \(-0.562982\pi\)
0.947416 − 0.320004i \(-0.103684\pi\)
\(80\) 6.69616e9 + 1.15981e10i 0.228471 + 0.395723i
\(81\) −1.74339e9 − 3.01964e9i −0.0555556 − 0.0962250i
\(82\) −2.33255e9 + 4.04010e9i −0.0694792 + 0.120342i
\(83\) 1.96280e10 0.546947 0.273474 − 0.961879i \(-0.411827\pi\)
0.273474 + 0.961879i \(0.411827\pi\)
\(84\) −3.72556e9 + 1.04188e10i −0.0971976 + 0.271820i
\(85\) 8.94413e10 2.18642
\(86\) −2.82670e10 + 4.89598e10i −0.647945 + 1.12227i
\(87\) −6.76313e9 − 1.17141e10i −0.145476 − 0.251972i
\(88\) 7.78658e9 + 1.34868e10i 0.157287 + 0.272429i
\(89\) 3.88169e10 − 6.72329e10i 0.736845 − 1.27625i −0.217064 − 0.976157i \(-0.569648\pi\)
0.953909 − 0.300095i \(-0.0970185\pi\)
\(90\) 2.41334e10 0.430809
\(91\) 4.48462e10 − 8.16928e9i 0.753353 − 0.137232i
\(92\) 1.54592e10 0.244543
\(93\) 1.93363e10 − 3.34915e10i 0.288216 − 0.499204i
\(94\) −2.24653e10 − 3.89111e10i −0.315725 − 0.546852i
\(95\) −2.54213e10 − 4.40310e10i −0.337069 − 0.583820i
\(96\) −4.07686e9 + 7.06133e9i −0.0510310 + 0.0883883i
\(97\) −6.65701e10 −0.787109 −0.393554 − 0.919301i \(-0.628755\pi\)
−0.393554 + 0.919301i \(0.628755\pi\)
\(98\) 1.02823e10 − 6.24334e10i 0.114907 − 0.697708i
\(99\) 2.80633e10 0.296583
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 42.12.e.d.37.4 yes 8
3.2 odd 2 126.12.g.d.37.1 8
7.4 even 3 inner 42.12.e.d.25.4 ✓ 8
21.11 odd 6 126.12.g.d.109.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.12.e.d.25.4 ✓ 8 7.4 even 3 inner
42.12.e.d.37.4 yes 8 1.1 even 1 trivial
126.12.g.d.37.1 8 3.2 odd 2
126.12.g.d.109.1 8 21.11 odd 6