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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(5,112)] 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("112.5"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 10])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.x (of order \(12\), degree \(4\), 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: \(248\)
Relative dimension: \(62\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 101.2
Character \(\chi\) \(=\) 112.101
Dual form 112.5.x.a.61.2

$q$-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+(-3.98580 + 0.336729i) q^{2} +(1.38046 - 0.369893i) q^{3} +(15.7732 - 2.68427i) q^{4} +(9.20312 + 2.46597i) q^{5} +(-5.37768 + 1.93916i) q^{6} +(-46.4811 - 15.5084i) q^{7} +(-61.9651 + 16.0103i) q^{8} +(-68.3792 + 39.4788i) q^{9} +(-37.5122 - 6.72990i) q^{10} +(162.471 - 43.5340i) q^{11} +(20.7814 - 9.53992i) q^{12} +(50.4780 + 50.4780i) q^{13} +(190.486 + 46.1619i) q^{14} +13.6167 q^{15} +(241.589 - 84.6793i) q^{16} +(-324.584 - 187.399i) q^{17} +(259.252 - 180.380i) q^{18} +(-91.3229 + 340.822i) q^{19} +(151.782 + 14.1926i) q^{20} +(-69.9016 - 4.21569i) q^{21} +(-632.919 + 228.227i) q^{22} +(-722.663 + 417.230i) q^{23} +(-79.6181 + 45.0219i) q^{24} +(-462.650 - 267.111i) q^{25} +(-218.193 - 184.198i) q^{26} +(-161.648 + 161.648i) q^{27} +(-774.785 - 119.850i) q^{28} +(-228.957 + 228.957i) q^{29} +(-54.2733 + 4.58513i) q^{30} +(-170.664 - 98.5330i) q^{31} +(-934.413 + 418.865i) q^{32} +(208.182 - 120.194i) q^{33} +(1356.83 + 637.637i) q^{34} +(-389.527 - 257.346i) q^{35} +(-972.589 + 806.256i) q^{36} +(-123.850 + 462.215i) q^{37} +(249.230 - 1389.20i) q^{38} +(88.3541 + 51.0113i) q^{39} +(-609.753 - 5.45938i) q^{40} -2232.10 q^{41} +(280.033 - 6.73503i) q^{42} +(-552.715 - 552.715i) q^{43} +(2445.84 - 1122.79i) q^{44} +(-726.655 + 194.707i) q^{45} +(2739.90 - 1906.34i) q^{46} +(-1907.70 + 1101.41i) q^{47} +(302.182 - 206.258i) q^{48} +(1919.98 + 1441.69i) q^{49} +(1933.97 + 908.863i) q^{50} +(-517.392 - 138.635i) q^{51} +(931.697 + 660.704i) q^{52} +(-1619.80 + 434.023i) q^{53} +(589.863 - 698.726i) q^{54} +1602.59 q^{55} +(3128.50 + 216.804i) q^{56} +504.270i q^{57} +(835.481 - 989.674i) q^{58} +(997.244 + 3721.77i) q^{59} +(214.779 - 36.5508i) q^{60} +(1646.84 - 6146.11i) q^{61} +(713.412 + 335.265i) q^{62} +(3790.59 - 774.563i) q^{63} +(3583.34 - 1984.16i) q^{64} +(340.078 + 589.032i) q^{65} +(-789.298 + 549.169i) q^{66} +(-812.282 - 3031.48i) q^{67} +(-5622.77 - 2084.61i) q^{68} +(-843.276 + 843.276i) q^{69} +(1639.24 + 894.566i) q^{70} +7296.40i q^{71} +(3605.06 - 3541.07i) q^{72} +(2769.16 - 4796.33i) q^{73} +(338.001 - 1884.00i) q^{74} +(-737.471 - 197.605i) q^{75} +(-525.599 + 5620.99i) q^{76} +(-8226.98 - 496.160i) q^{77} +(-369.339 - 173.569i) q^{78} +(906.266 + 1569.70i) q^{79} +(2432.19 - 183.562i) q^{80} +(3034.42 - 5255.78i) q^{81} +(8896.69 - 751.612i) q^{82} +(9553.72 + 9553.72i) q^{83} +(-1113.89 + 121.140i) q^{84} +(-2525.07 - 2525.07i) q^{85} +(2389.13 + 2016.90i) q^{86} +(-231.376 + 400.755i) q^{87} +(-9370.55 + 5298.80i) q^{88} +(-7159.03 - 12399.8i) q^{89} +(2830.74 - 1020.75i) q^{90} +(-1563.44 - 3129.10i) q^{91} +(-10278.8 + 8520.88i) q^{92} +(-272.041 - 72.8932i) q^{93} +(7232.83 - 5032.38i) q^{94} +(-1680.91 + 2911.42i) q^{95} +(-1134.98 + 923.858i) q^{96} +4391.95i q^{97} +(-8138.12 - 5099.79i) q^{98} +(-9390.98 + 9390.98i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 248 q - 2 q^{2} - 6 q^{3} + 4 q^{4} - 6 q^{5} - 188 q^{8} + 588 q^{10} + 94 q^{11} - 6 q^{12} + 396 q^{14} - 16 q^{15} - 312 q^{16} - 12 q^{17} + 194 q^{18} - 6 q^{19} + 158 q^{21} - 2372 q^{22} - 6 q^{24}+ \cdots - 656 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.98580 + 0.336729i −0.996450 + 0.0841823i
\(3\) 1.38046 − 0.369893i 0.153384 − 0.0410992i −0.181310 − 0.983426i \(-0.558034\pi\)
0.334694 + 0.942327i \(0.391367\pi\)
\(4\) 15.7732 − 2.68427i 0.985827 − 0.167767i
\(5\) 9.20312 + 2.46597i 0.368125 + 0.0986387i 0.438139 − 0.898907i \(-0.355638\pi\)
−0.0700142 + 0.997546i \(0.522304\pi\)
\(6\) −5.37768 + 1.93916i −0.149380 + 0.0538655i
\(7\) −46.4811 − 15.5084i −0.948593 − 0.316498i
\(8\) −61.9651 + 16.0103i −0.968204 + 0.250161i
\(9\) −68.3792 + 39.4788i −0.844188 + 0.487392i
\(10\) −37.5122 − 6.72990i −0.375122 − 0.0672990i
\(11\) 162.471 − 43.5340i 1.34274 − 0.359785i 0.485288 − 0.874354i \(-0.338715\pi\)
0.857449 + 0.514569i \(0.172048\pi\)
\(12\) 20.7814 − 9.53992i 0.144315 − 0.0662495i
\(13\) 50.4780 + 50.4780i 0.298686 + 0.298686i 0.840499 − 0.541813i \(-0.182262\pi\)
−0.541813 + 0.840499i \(0.682262\pi\)
\(14\) 190.486 + 46.1619i 0.971870 + 0.235520i
\(15\) 13.6167 0.0605185
\(16\) 241.589 − 84.6793i 0.943708 − 0.330778i
\(17\) −324.584 − 187.399i −1.12313 − 0.648439i −0.180931 − 0.983496i \(-0.557911\pi\)
−0.942198 + 0.335057i \(0.891244\pi\)
\(18\) 259.252 − 180.380i 0.800161 − 0.556728i
\(19\) −91.3229 + 340.822i −0.252972 + 0.944104i 0.716236 + 0.697859i \(0.245864\pi\)
−0.969207 + 0.246246i \(0.920803\pi\)
\(20\) 151.782 + 14.1926i 0.379455 + 0.0354815i
\(21\) −69.9016 − 4.21569i −0.158507 − 0.00955938i
\(22\) −632.919 + 228.227i −1.30768 + 0.471543i
\(23\) −722.663 + 417.230i −1.36609 + 0.788714i −0.990426 − 0.138041i \(-0.955919\pi\)
−0.375666 + 0.926755i \(0.622586\pi\)
\(24\) −79.6181 + 45.0219i −0.138226 + 0.0781631i
\(25\) −462.650 − 267.111i −0.740239 − 0.427377i
\(26\) −218.193 − 184.198i −0.322770 − 0.272482i
\(27\) −161.648 + 161.648i −0.221739 + 0.221739i
\(28\) −774.785 − 119.850i −0.988246 − 0.152869i
\(29\) −228.957 + 228.957i −0.272244 + 0.272244i −0.830003 − 0.557759i \(-0.811661\pi\)
0.557759 + 0.830003i \(0.311661\pi\)
\(30\) −54.2733 + 4.58513i −0.0603037 + 0.00509459i
\(31\) −170.664 − 98.5330i −0.177590 − 0.102532i 0.408570 − 0.912727i \(-0.366028\pi\)
−0.586160 + 0.810195i \(0.699361\pi\)
\(32\) −934.413 + 418.865i −0.912513 + 0.409048i
\(33\) 208.182 − 120.194i 0.191168 − 0.110371i
\(34\) 1356.83 + 637.637i 1.17373 + 0.551589i
\(35\) −389.527 − 257.346i −0.317982 − 0.210079i
\(36\) −972.589 + 806.256i −0.750455 + 0.622111i
\(37\) −123.850 + 462.215i −0.0904676 + 0.337630i −0.996293 − 0.0860208i \(-0.972585\pi\)
0.905826 + 0.423650i \(0.139252\pi\)
\(38\) 249.230 − 1389.20i 0.172597 − 0.962049i
\(39\) 88.3541 + 51.0113i 0.0580895 + 0.0335380i
\(40\) −609.753 − 5.45938i −0.381095 − 0.00341211i
\(41\) −2232.10 −1.32784 −0.663919 − 0.747805i \(-0.731108\pi\)
−0.663919 + 0.747805i \(0.731108\pi\)
\(42\) 280.033 − 6.73503i 0.158749 − 0.00381804i
\(43\) −552.715 − 552.715i −0.298926 − 0.298926i 0.541667 − 0.840593i \(-0.317793\pi\)
−0.840593 + 0.541667i \(0.817793\pi\)
\(44\) 2445.84 − 1122.79i 1.26335 − 0.579953i
\(45\) −726.655 + 194.707i −0.358842 + 0.0961514i
\(46\) 2739.90 − 1906.34i 1.29485 − 0.900915i
\(47\) −1907.70 + 1101.41i −0.863602 + 0.498601i −0.865217 − 0.501398i \(-0.832819\pi\)
0.00161458 + 0.999999i \(0.499486\pi\)
\(48\) 302.182 − 206.258i 0.131155 − 0.0895218i
\(49\) 1919.98 + 1441.69i 0.799658 + 0.600455i
\(50\) 1933.97 + 908.863i 0.773589 + 0.363545i
\(51\) −517.392 − 138.635i −0.198921 − 0.0533006i
\(52\) 931.697 + 660.704i 0.344563 + 0.244343i
\(53\) −1619.80 + 434.023i −0.576645 + 0.154512i −0.535343 − 0.844635i \(-0.679817\pi\)
−0.0413028 + 0.999147i \(0.513151\pi\)
\(54\) 589.863 − 698.726i 0.202285 − 0.239618i
\(55\) 1602.59 0.529783
\(56\) 3128.50 + 216.804i 0.997607 + 0.0691339i
\(57\) 504.270i 0.155208i
\(58\) 835.481 − 989.674i 0.248359 − 0.294195i
\(59\) 997.244 + 3721.77i 0.286482 + 1.06917i 0.947750 + 0.319015i \(0.103352\pi\)
−0.661267 + 0.750150i \(0.729981\pi\)
\(60\) 214.779 − 36.5508i 0.0596607 − 0.0101530i
\(61\) 1646.84 − 6146.11i 0.442581 − 1.65173i −0.279664 − 0.960098i \(-0.590223\pi\)
0.722245 − 0.691637i \(-0.243110\pi\)
\(62\) 713.412 + 335.265i 0.185591 + 0.0872178i
\(63\) 3790.59 − 774.563i 0.955049 − 0.195153i
\(64\) 3583.34 − 1984.16i 0.874839 − 0.484413i
\(65\) 340.078 + 589.032i 0.0804917 + 0.139416i
\(66\) −789.298 + 549.169i −0.181198 + 0.126072i
\(67\) −812.282 − 3031.48i −0.180949 − 0.675312i −0.995461 − 0.0951658i \(-0.969662\pi\)
0.814512 − 0.580147i \(-0.197005\pi\)
\(68\) −5622.77 − 2084.61i −1.21600 − 0.450824i
\(69\) −843.276 + 843.276i −0.177122 + 0.177122i
\(70\) 1639.24 + 894.566i 0.334538 + 0.182565i
\(71\) 7296.40i 1.44741i 0.690109 + 0.723705i \(0.257563\pi\)
−0.690109 + 0.723705i \(0.742437\pi\)
\(72\) 3605.06 − 3541.07i 0.695420 − 0.683078i
\(73\) 2769.16 − 4796.33i 0.519640 − 0.900043i −0.480099 − 0.877214i \(-0.659399\pi\)
0.999739 − 0.0228287i \(-0.00726722\pi\)
\(74\) 338.001 − 1884.00i 0.0617240 − 0.344047i
\(75\) −737.471 − 197.605i −0.131106 − 0.0351297i
\(76\) −525.599 + 5620.99i −0.0909970 + 0.973163i
\(77\) −8226.98 − 496.160i −1.38758 − 0.0836835i
\(78\) −369.339 − 173.569i −0.0607066 − 0.0285288i
\(79\) 906.266 + 1569.70i 0.145212 + 0.251514i 0.929452 − 0.368943i \(-0.120280\pi\)
−0.784240 + 0.620457i \(0.786947\pi\)
\(80\) 2432.19 − 183.562i 0.380030 − 0.0286815i
\(81\) 3034.42 − 5255.78i 0.462494 − 0.801063i
\(82\) 8896.69 − 751.612i 1.32312 − 0.111780i
\(83\) 9553.72 + 9553.72i 1.38681 + 1.38681i 0.831937 + 0.554871i \(0.187232\pi\)
0.554871 + 0.831937i \(0.312768\pi\)
\(84\) −1113.89 + 121.140i −0.157864 + 0.0171684i
\(85\) −2525.07 − 2525.07i −0.349490 − 0.349490i
\(86\) 2389.13 + 2016.90i 0.323030 + 0.272701i
\(87\) −231.376 + 400.755i −0.0305689 + 0.0529469i
\(88\) −9370.55 + 5298.80i −1.21004 + 0.684246i
\(89\) −7159.03 − 12399.8i −0.903804 − 1.56543i −0.822514 − 0.568744i \(-0.807429\pi\)
−0.0812897 − 0.996691i \(-0.525904\pi\)
\(90\) 2830.74 − 1020.75i 0.349474 − 0.126018i
\(91\) −1563.44 − 3129.10i −0.188798 − 0.377865i
\(92\) −10278.8 + 8520.88i −1.21441 + 1.00672i
\(93\) −272.041 − 72.8932i −0.0314535 − 0.00842794i
\(94\) 7232.83 − 5032.38i 0.818564 − 0.569531i
\(95\) −1680.91 + 2911.42i −0.186250 + 0.322595i
\(96\) −1134.98 + 923.858i −0.123154 + 0.100245i
\(97\) 4391.95i 0.466782i 0.972383 + 0.233391i \(0.0749822\pi\)
−0.972383 + 0.233391i \(0.925018\pi\)
\(98\) −8138.12 − 5099.79i −0.847367 − 0.531007i
\(99\) −9390.98 + 9390.98i −0.958166 + 0.958166i
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.x.a.101.2 yes 248
7.5 odd 6 inner 112.5.x.a.5.44 ✓ 248
16.13 even 4 inner 112.5.x.a.45.44 yes 248
112.61 odd 12 inner 112.5.x.a.61.2 yes 248
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.x.a.5.44 ✓ 248 7.5 odd 6 inner
112.5.x.a.45.44 yes 248 16.13 even 4 inner
112.5.x.a.61.2 yes 248 112.61 odd 12 inner
112.5.x.a.101.2 yes 248 1.1 even 1 trivial