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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(43,112)] 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("112.43"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.3
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.3

$q$-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+(-3.93459 + 0.720428i) q^{2} +(-2.13007 - 2.13007i) q^{3} +(14.9620 - 5.66917i) q^{4} +(-28.8248 - 28.8248i) q^{5} +(9.91550 + 6.84638i) q^{6} -18.5203 q^{7} +(-54.7850 + 33.0849i) q^{8} -71.9256i q^{9} +(134.180 + 92.6475i) q^{10} +(-29.6316 + 29.6316i) q^{11} +(-43.9458 - 19.7943i) q^{12} +(46.3300 - 46.3300i) q^{13} +(72.8696 - 13.3425i) q^{14} +122.798i q^{15} +(191.721 - 169.644i) q^{16} +45.3403 q^{17} +(51.8172 + 282.998i) q^{18} +(144.548 + 144.548i) q^{19} +(-594.688 - 267.863i) q^{20} +(39.4494 + 39.4494i) q^{21} +(95.2407 - 137.935i) q^{22} -385.214 q^{23} +(187.169 + 46.2227i) q^{24} +1036.74i q^{25} +(-148.912 + 215.667i) q^{26} +(-325.742 + 325.742i) q^{27} +(-277.100 + 104.995i) q^{28} +(-450.417 + 450.417i) q^{29} +(-88.4667 - 483.158i) q^{30} +1671.76i q^{31} +(-632.127 + 805.600i) q^{32} +126.235 q^{33} +(-178.395 + 32.6644i) q^{34} +(533.842 + 533.842i) q^{35} +(-407.759 - 1076.15i) q^{36} +(-468.062 - 468.062i) q^{37} +(-672.873 - 464.601i) q^{38} -197.372 q^{39} +(2532.83 + 625.500i) q^{40} +2367.38i q^{41} +(-183.638 - 126.797i) q^{42} +(2107.70 - 2107.70i) q^{43} +(-275.360 + 611.333i) q^{44} +(-2073.24 + 2073.24i) q^{45} +(1515.66 - 277.519i) q^{46} +85.0320i q^{47} +(-769.732 - 47.0257i) q^{48} +343.000 q^{49} +(-746.893 - 4079.13i) q^{50} +(-96.5780 - 96.5780i) q^{51} +(430.536 - 955.842i) q^{52} +(-3289.93 - 3289.93i) q^{53} +(1046.99 - 1516.33i) q^{54} +1708.25 q^{55} +(1014.63 - 612.740i) q^{56} -615.794i q^{57} +(1447.71 - 2096.70i) q^{58} +(1596.21 - 1596.21i) q^{59} +(696.160 + 1837.29i) q^{60} +(719.735 - 719.735i) q^{61} +(-1204.38 - 6577.68i) q^{62} +1332.08i q^{63} +(1906.78 - 3625.11i) q^{64} -2670.91 q^{65} +(-496.681 + 90.9429i) q^{66} +(193.349 + 193.349i) q^{67} +(678.380 - 257.042i) q^{68} +(820.533 + 820.533i) q^{69} +(-2485.05 - 1715.86i) q^{70} +4819.87 q^{71} +(2379.65 + 3940.44i) q^{72} +9412.63i q^{73} +(2178.84 + 1504.43i) q^{74} +(2208.32 - 2208.32i) q^{75} +(2982.19 + 1343.26i) q^{76} +(548.785 - 548.785i) q^{77} +(776.579 - 142.192i) q^{78} +4966.98i q^{79} +(-10416.3 - 636.367i) q^{80} -4438.27 q^{81} +(-1705.53 - 9314.68i) q^{82} +(1913.79 + 1913.79i) q^{83} +(813.887 + 366.596i) q^{84} +(-1306.92 - 1306.92i) q^{85} +(-6774.50 + 9811.40i) q^{86} +1918.84 q^{87} +(643.008 - 2603.72i) q^{88} -12809.5i q^{89} +(6663.73 - 9650.97i) q^{90} +(-858.044 + 858.044i) q^{91} +(-5763.56 + 2183.85i) q^{92} +(3560.96 - 3560.96i) q^{93} +(-61.2594 - 334.566i) q^{94} -8333.13i q^{95} +(3062.46 - 369.509i) q^{96} -3220.13 q^{97} +(-1349.56 + 247.107i) q^{98} +(2131.27 + 2131.27i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 6 q^{4} - 132 q^{6} - 200 q^{10} - 96 q^{11} + 660 q^{12} - 294 q^{14} + 642 q^{16} - 810 q^{18} + 1408 q^{19} - 2604 q^{20} - 106 q^{22} - 1152 q^{23} + 3880 q^{24} + 6828 q^{26} + 3648 q^{27} - 864 q^{29}+ \cdots - 59552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.93459 + 0.720428i −0.983647 + 0.180107i
\(3\) −2.13007 − 2.13007i −0.236674 − 0.236674i 0.578797 − 0.815472i \(-0.303522\pi\)
−0.815472 + 0.578797i \(0.803522\pi\)
\(4\) 14.9620 − 5.66917i 0.935123 − 0.354323i
\(5\) −28.8248 − 28.8248i −1.15299 − 1.15299i −0.985950 − 0.167041i \(-0.946579\pi\)
−0.167041 − 0.985950i \(-0.553421\pi\)
\(6\) 9.91550 + 6.84638i 0.275431 + 0.190177i
\(7\) −18.5203 −0.377964
\(8\) −54.7850 + 33.0849i −0.856015 + 0.516951i
\(9\) − 71.9256i − 0.887971i
\(10\) 134.180 + 92.6475i 1.34180 + 0.926475i
\(11\) −29.6316 + 29.6316i −0.244889 + 0.244889i −0.818869 − 0.573980i \(-0.805399\pi\)
0.573980 + 0.818869i \(0.305399\pi\)
\(12\) −43.9458 − 19.7943i −0.305179 − 0.137460i
\(13\) 46.3300 − 46.3300i 0.274142 − 0.274142i −0.556623 − 0.830765i \(-0.687903\pi\)
0.830765 + 0.556623i \(0.187903\pi\)
\(14\) 72.8696 − 13.3425i 0.371784 − 0.0680740i
\(15\) 122.798i 0.545767i
\(16\) 191.721 − 169.644i 0.748910 − 0.662672i
\(17\) 45.3403 0.156887 0.0784435 − 0.996919i \(-0.475005\pi\)
0.0784435 + 0.996919i \(0.475005\pi\)
\(18\) 51.8172 + 282.998i 0.159930 + 0.873450i
\(19\) 144.548 + 144.548i 0.400410 + 0.400410i 0.878378 − 0.477968i \(-0.158626\pi\)
−0.477968 + 0.878378i \(0.658626\pi\)
\(20\) −594.688 − 267.863i −1.48672 − 0.669657i
\(21\) 39.4494 + 39.4494i 0.0894545 + 0.0894545i
\(22\) 95.2407 − 137.935i 0.196778 − 0.284991i
\(23\) −385.214 −0.728193 −0.364097 − 0.931361i \(-0.618622\pi\)
−0.364097 + 0.931361i \(0.618622\pi\)
\(24\) 187.169 + 46.2227i 0.324946 + 0.0802477i
\(25\) 1036.74i 1.65878i
\(26\) −148.912 + 215.667i −0.220284 + 0.319034i
\(27\) −325.742 + 325.742i −0.446834 + 0.446834i
\(28\) −277.100 + 104.995i −0.353443 + 0.133922i
\(29\) −450.417 + 450.417i −0.535573 + 0.535573i −0.922225 − 0.386653i \(-0.873631\pi\)
0.386653 + 0.922225i \(0.373631\pi\)
\(30\) −88.4667 − 483.158i −0.0982964 − 0.536842i
\(31\) 1671.76i 1.73960i 0.493402 + 0.869801i \(0.335753\pi\)
−0.493402 + 0.869801i \(0.664247\pi\)
\(32\) −632.127 + 805.600i −0.617312 + 0.786719i
\(33\) 126.235 0.115918
\(34\) −178.395 + 32.6644i −0.154321 + 0.0282564i
\(35\) 533.842 + 533.842i 0.435790 + 0.435790i
\(36\) −407.759 − 1076.15i −0.314629 − 0.830362i
\(37\) −468.062 − 468.062i −0.341901 − 0.341901i 0.515181 − 0.857082i \(-0.327725\pi\)
−0.857082 + 0.515181i \(0.827725\pi\)
\(38\) −672.873 − 464.601i −0.465979 − 0.321746i
\(39\) −197.372 −0.129765
\(40\) 2532.83 + 625.500i 1.58302 + 0.390938i
\(41\) 2367.38i 1.40832i 0.710042 + 0.704159i \(0.248676\pi\)
−0.710042 + 0.704159i \(0.751324\pi\)
\(42\) −183.638 − 126.797i −0.104103 − 0.0718803i
\(43\) 2107.70 − 2107.70i 1.13992 − 1.13992i 0.151451 − 0.988465i \(-0.451605\pi\)
0.988465 − 0.151451i \(-0.0483946\pi\)
\(44\) −275.360 + 611.333i −0.142232 + 0.315771i
\(45\) −2073.24 + 2073.24i −1.02382 + 1.02382i
\(46\) 1515.66 − 277.519i 0.716285 − 0.131153i
\(47\) 85.0320i 0.0384934i 0.999815 + 0.0192467i \(0.00612680\pi\)
−0.999815 + 0.0192467i \(0.993873\pi\)
\(48\) −769.732 − 47.0257i −0.334085 − 0.0204105i
\(49\) 343.000 0.142857
\(50\) −746.893 − 4079.13i −0.298757 − 1.63165i
\(51\) −96.5780 − 96.5780i −0.0371311 − 0.0371311i
\(52\) 430.536 − 955.842i 0.159222 − 0.353492i
\(53\) −3289.93 − 3289.93i −1.17121 − 1.17121i −0.981921 − 0.189290i \(-0.939381\pi\)
−0.189290 − 0.981921i \(-0.560619\pi\)
\(54\) 1046.99 − 1516.33i 0.359049 − 0.520005i
\(55\) 1708.25 0.564710
\(56\) 1014.63 − 612.740i 0.323543 − 0.195389i
\(57\) − 615.794i − 0.189534i
\(58\) 1447.71 − 2096.70i 0.430354 − 0.623275i
\(59\) 1596.21 − 1596.21i 0.458548 − 0.458548i −0.439631 − 0.898179i \(-0.644891\pi\)
0.898179 + 0.439631i \(0.144891\pi\)
\(60\) 696.160 + 1837.29i 0.193378 + 0.510359i
\(61\) 719.735 − 719.735i 0.193425 − 0.193425i −0.603749 − 0.797174i \(-0.706327\pi\)
0.797174 + 0.603749i \(0.206327\pi\)
\(62\) −1204.38 − 6577.68i −0.313314 − 1.71116i
\(63\) 1332.08i 0.335621i
\(64\) 1906.78 − 3625.11i 0.465523 − 0.885036i
\(65\) −2670.91 −0.632167
\(66\) −496.681 + 90.9429i −0.114022 + 0.0208776i
\(67\) 193.349 + 193.349i 0.0430717 + 0.0430717i 0.728315 − 0.685243i \(-0.240304\pi\)
−0.685243 + 0.728315i \(0.740304\pi\)
\(68\) 678.380 − 257.042i 0.146709 − 0.0555887i
\(69\) 820.533 + 820.533i 0.172345 + 0.172345i
\(70\) −2485.05 − 1715.86i −0.507152 − 0.350175i
\(71\) 4819.87 0.956133 0.478067 − 0.878324i \(-0.341338\pi\)
0.478067 + 0.878324i \(0.341338\pi\)
\(72\) 2379.65 + 3940.44i 0.459037 + 0.760116i
\(73\) 9412.63i 1.76630i 0.469087 + 0.883152i \(0.344583\pi\)
−0.469087 + 0.883152i \(0.655417\pi\)
\(74\) 2178.84 + 1504.43i 0.397888 + 0.274731i
\(75\) 2208.32 − 2208.32i 0.392590 − 0.392590i
\(76\) 2982.19 + 1343.26i 0.516307 + 0.232558i
\(77\) 548.785 − 548.785i 0.0925594 − 0.0925594i
\(78\) 776.579 − 142.192i 0.127643 − 0.0233715i
\(79\) 4966.98i 0.795863i 0.917415 + 0.397931i \(0.130272\pi\)
−0.917415 + 0.397931i \(0.869728\pi\)
\(80\) −10416.3 − 636.367i −1.62754 − 0.0994323i
\(81\) −4438.27 −0.676462
\(82\) −1705.53 − 9314.68i −0.253648 − 1.38529i
\(83\) 1913.79 + 1913.79i 0.277803 + 0.277803i 0.832231 − 0.554428i \(-0.187063\pi\)
−0.554428 + 0.832231i \(0.687063\pi\)
\(84\) 813.887 + 366.596i 0.115347 + 0.0519552i
\(85\) −1306.92 − 1306.92i −0.180889 − 0.180889i
\(86\) −6774.50 + 9811.40i −0.915968 + 1.32658i
\(87\) 1918.84 0.253513
\(88\) 643.008 − 2603.72i 0.0830330 − 0.336224i
\(89\) − 12809.5i − 1.61716i −0.588386 − 0.808580i \(-0.700237\pi\)
0.588386 − 0.808580i \(-0.299763\pi\)
\(90\) 6663.73 − 9650.97i 0.822682 − 1.19148i
\(91\) −858.044 + 858.044i −0.103616 + 0.103616i
\(92\) −5763.56 + 2183.85i −0.680950 + 0.258016i
\(93\) 3560.96 − 3560.96i 0.411719 − 0.411719i
\(94\) −61.2594 − 334.566i −0.00693293 − 0.0378640i
\(95\) − 8333.13i − 0.923339i
\(96\) 3062.46 − 369.509i 0.332298 − 0.0400943i
\(97\) −3220.13 −0.342239 −0.171120 − 0.985250i \(-0.554738\pi\)
−0.171120 + 0.985250i \(0.554738\pi\)
\(98\) −1349.56 + 247.107i −0.140521 + 0.0257296i
\(99\) 2131.27 + 2131.27i 0.217454 + 0.217454i
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 112.5.k.a.43.3 ✓ 96
4.3 odd 2 448.5.k.a.15.28 96
16.3 odd 4 inner 112.5.k.a.99.3 yes 96
16.13 even 4 448.5.k.a.239.28 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.3 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.3 yes 96 16.3 odd 4 inner
448.5.k.a.15.28 96 4.3 odd 2
448.5.k.a.239.28 96 16.13 even 4