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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 99.9
Character \(\chi\) \(=\) 112.99
Dual form 112.5.k.a.43.9

$q$-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+(-3.55330 + 1.83686i) q^{2} +(4.91576 - 4.91576i) q^{3} +(9.25192 - 13.0538i) q^{4} +(-5.29630 + 5.29630i) q^{5} +(-8.43764 + 26.4967i) q^{6} -18.5203 q^{7} +(-8.89692 + 63.3786i) q^{8} +32.6706i q^{9} +(9.09082 - 28.5479i) q^{10} +(65.1032 + 65.1032i) q^{11} +(-18.6892 - 109.650i) q^{12} +(2.97917 + 2.97917i) q^{13} +(65.8081 - 34.0190i) q^{14} +52.0707i q^{15} +(-84.8039 - 241.546i) q^{16} +212.501 q^{17} +(-60.0112 - 116.089i) q^{18} +(301.655 - 301.655i) q^{19} +(20.1359 + 118.138i) q^{20} +(-91.0412 + 91.0412i) q^{21} +(-350.916 - 111.746i) q^{22} +474.668 q^{23} +(267.819 + 355.289i) q^{24} +568.898i q^{25} +(-16.0582 - 5.11359i) q^{26} +(558.777 + 558.777i) q^{27} +(-171.348 + 241.760i) q^{28} +(355.127 + 355.127i) q^{29} +(-95.6463 - 185.023i) q^{30} +801.728i q^{31} +(745.018 + 702.512i) q^{32} +640.063 q^{33} +(-755.080 + 390.333i) q^{34} +(98.0888 - 98.0888i) q^{35} +(426.476 + 302.266i) q^{36} +(151.471 - 151.471i) q^{37} +(-517.775 + 1625.97i) q^{38} +29.2898 q^{39} +(-288.551 - 382.793i) q^{40} +788.369i q^{41} +(156.267 - 490.726i) q^{42} +(853.638 + 853.638i) q^{43} +(1452.17 - 247.515i) q^{44} +(-173.033 - 173.033i) q^{45} +(-1686.64 + 871.896i) q^{46} -2384.15i q^{47} +(-1604.26 - 770.505i) q^{48} +343.000 q^{49} +(-1044.98 - 2021.47i) q^{50} +(1044.60 - 1044.60i) q^{51} +(66.4526 - 11.3265i) q^{52} +(1527.65 - 1527.65i) q^{53} +(-3011.90 - 959.112i) q^{54} -689.612 q^{55} +(164.773 - 1173.79i) q^{56} -2965.73i q^{57} +(-1914.19 - 609.556i) q^{58} +(3615.57 + 3615.57i) q^{59} +(679.721 + 481.754i) q^{60} +(-4237.89 - 4237.89i) q^{61} +(-1472.66 - 2848.78i) q^{62} -605.068i q^{63} +(-3937.69 - 1127.75i) q^{64} -31.5572 q^{65} +(-2274.34 + 1175.70i) q^{66} +(-2926.57 + 2926.57i) q^{67} +(1966.04 - 2773.94i) q^{68} +(2333.35 - 2333.35i) q^{69} +(-168.364 + 528.714i) q^{70} -2711.36 q^{71} +(-2070.62 - 290.668i) q^{72} -803.219i q^{73} +(-259.992 + 816.453i) q^{74} +(2796.57 + 2796.57i) q^{75} +(-1146.86 - 6728.63i) q^{76} +(-1205.73 - 1205.73i) q^{77} +(-104.076 + 53.8011i) q^{78} +10318.3i q^{79} +(1728.45 + 830.151i) q^{80} +2847.31 q^{81} +(-1448.12 - 2801.31i) q^{82} +(322.720 - 322.720i) q^{83} +(346.128 + 2030.74i) q^{84} +(-1125.47 + 1125.47i) q^{85} +(-4601.24 - 1465.22i) q^{86} +3491.43 q^{87} +(-4705.36 + 3546.93i) q^{88} -8475.90i q^{89} +(932.677 + 297.002i) q^{90} +(-55.1751 - 55.1751i) q^{91} +(4391.59 - 6196.22i) q^{92} +(3941.10 + 3941.10i) q^{93} +(4379.34 + 8471.61i) q^{94} +3195.31i q^{95} +(7115.71 - 208.949i) q^{96} +3166.40 q^{97} +(-1218.78 + 630.042i) q^{98} +(-2126.96 + 2126.96i) 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 \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.55330 + 1.83686i −0.888326 + 0.459214i
\(3\) 4.91576 − 4.91576i 0.546196 − 0.546196i −0.379143 − 0.925338i \(-0.623781\pi\)
0.925338 + 0.379143i \(0.123781\pi\)
\(4\) 9.25192 − 13.0538i 0.578245 − 0.815863i
\(5\) −5.29630 + 5.29630i −0.211852 + 0.211852i −0.805054 − 0.593202i \(-0.797864\pi\)
0.593202 + 0.805054i \(0.297864\pi\)
\(6\) −8.43764 + 26.4967i −0.234379 + 0.736020i
\(7\) −18.5203 −0.377964
\(8\) −8.89692 + 63.3786i −0.139014 + 0.990290i
\(9\) 32.6706i 0.403341i
\(10\) 9.09082 − 28.5479i 0.0909082 − 0.285479i
\(11\) 65.1032 + 65.1032i 0.538043 + 0.538043i 0.922954 − 0.384911i \(-0.125768\pi\)
−0.384911 + 0.922954i \(0.625768\pi\)
\(12\) −18.6892 − 109.650i −0.129786 − 0.761456i
\(13\) 2.97917 + 2.97917i 0.0176282 + 0.0176282i 0.715866 − 0.698238i \(-0.246032\pi\)
−0.698238 + 0.715866i \(0.746032\pi\)
\(14\) 65.8081 − 34.0190i 0.335756 − 0.173567i
\(15\) 52.0707i 0.231425i
\(16\) −84.8039 − 241.546i −0.331265 − 0.943538i
\(17\) 212.501 0.735297 0.367648 − 0.929965i \(-0.380163\pi\)
0.367648 + 0.929965i \(0.380163\pi\)
\(18\) −60.0112 − 116.089i −0.185220 − 0.358298i
\(19\) 301.655 − 301.655i 0.835609 − 0.835609i −0.152668 − 0.988277i \(-0.548787\pi\)
0.988277 + 0.152668i \(0.0487866\pi\)
\(20\) 20.1359 + 118.138i 0.0503398 + 0.295345i
\(21\) −91.0412 + 91.0412i −0.206443 + 0.206443i
\(22\) −350.916 − 111.746i −0.725034 − 0.230880i
\(23\) 474.668 0.897293 0.448646 − 0.893709i \(-0.351906\pi\)
0.448646 + 0.893709i \(0.351906\pi\)
\(24\) 267.819 + 355.289i 0.464963 + 0.616821i
\(25\) 568.898i 0.910237i
\(26\) −16.0582 − 5.11359i −0.0237548 − 0.00756449i
\(27\) 558.777 + 558.777i 0.766499 + 0.766499i
\(28\) −171.348 + 241.760i −0.218556 + 0.308367i
\(29\) 355.127 + 355.127i 0.422267 + 0.422267i 0.885984 − 0.463717i \(-0.153484\pi\)
−0.463717 + 0.885984i \(0.653484\pi\)
\(30\) −95.6463 − 185.023i −0.106274 − 0.205581i
\(31\) 801.728i 0.834264i 0.908846 + 0.417132i \(0.136965\pi\)
−0.908846 + 0.417132i \(0.863035\pi\)
\(32\) 745.018 + 702.512i 0.727557 + 0.686047i
\(33\) 640.063 0.587753
\(34\) −755.080 + 390.333i −0.653183 + 0.337659i
\(35\) 98.0888 − 98.0888i 0.0800725 − 0.0800725i
\(36\) 426.476 + 302.266i 0.329071 + 0.233230i
\(37\) 151.471 − 151.471i 0.110644 − 0.110644i −0.649618 − 0.760261i \(-0.725071\pi\)
0.760261 + 0.649618i \(0.225071\pi\)
\(38\) −517.775 + 1625.97i −0.358570 + 1.12602i
\(39\) 29.2898 0.0192569
\(40\) −288.551 − 382.793i −0.180345 − 0.239245i
\(41\) 788.369i 0.468988i 0.972118 + 0.234494i \(0.0753433\pi\)
−0.972118 + 0.234494i \(0.924657\pi\)
\(42\) 156.267 − 490.726i 0.0885869 − 0.278189i
\(43\) 853.638 + 853.638i 0.461675 + 0.461675i 0.899204 − 0.437529i \(-0.144146\pi\)
−0.437529 + 0.899204i \(0.644146\pi\)
\(44\) 1452.17 − 247.515i 0.750090 − 0.127849i
\(45\) −173.033 − 173.033i −0.0854485 − 0.0854485i
\(46\) −1686.64 + 871.896i −0.797088 + 0.412049i
\(47\) − 2384.15i − 1.07929i −0.841893 − 0.539645i \(-0.818558\pi\)
0.841893 − 0.539645i \(-0.181442\pi\)
\(48\) −1604.26 − 770.505i −0.696292 − 0.334421i
\(49\) 343.000 0.142857
\(50\) −1044.98 − 2021.47i −0.417994 − 0.808587i
\(51\) 1044.60 − 1044.60i 0.401616 − 0.401616i
\(52\) 66.4526 − 11.3265i 0.0245757 − 0.00418879i
\(53\) 1527.65 − 1527.65i 0.543841 − 0.543841i −0.380812 − 0.924653i \(-0.624355\pi\)
0.924653 + 0.380812i \(0.124355\pi\)
\(54\) −3011.90 − 959.112i −1.03289 − 0.328914i
\(55\) −689.612 −0.227971
\(56\) 164.773 − 1173.79i 0.0525425 − 0.374295i
\(57\) − 2965.73i − 0.912812i
\(58\) −1914.19 − 609.556i −0.569021 − 0.181200i
\(59\) 3615.57 + 3615.57i 1.03866 + 1.03866i 0.999222 + 0.0394362i \(0.0125562\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(60\) 679.721 + 481.754i 0.188811 + 0.133821i
\(61\) −4237.89 − 4237.89i −1.13891 − 1.13891i −0.988645 − 0.150267i \(-0.951987\pi\)
−0.150267 − 0.988645i \(-0.548013\pi\)
\(62\) −1472.66 − 2848.78i −0.383106 − 0.741098i
\(63\) − 605.068i − 0.152448i
\(64\) −3937.69 − 1127.75i −0.961350 − 0.275329i
\(65\) −31.5572 −0.00746916
\(66\) −2274.34 + 1175.70i −0.522116 + 0.269904i
\(67\) −2926.57 + 2926.57i −0.651943 + 0.651943i −0.953461 − 0.301518i \(-0.902507\pi\)
0.301518 + 0.953461i \(0.402507\pi\)
\(68\) 1966.04 − 2773.94i 0.425182 − 0.599902i
\(69\) 2333.35 − 2333.35i 0.490097 − 0.490097i
\(70\) −168.364 + 528.714i −0.0343601 + 0.107901i
\(71\) −2711.36 −0.537861 −0.268930 − 0.963160i \(-0.586670\pi\)
−0.268930 + 0.963160i \(0.586670\pi\)
\(72\) −2070.62 − 290.668i −0.399424 − 0.0560701i
\(73\) − 803.219i − 0.150726i −0.997156 − 0.0753630i \(-0.975988\pi\)
0.997156 − 0.0753630i \(-0.0240115\pi\)
\(74\) −259.992 + 816.453i −0.0474785 + 0.149097i
\(75\) 2796.57 + 2796.57i 0.497168 + 0.497168i
\(76\) −1146.86 − 6728.63i −0.198556 − 1.16493i
\(77\) −1205.73 − 1205.73i −0.203361 − 0.203361i
\(78\) −104.076 + 53.8011i −0.0171064 + 0.00884305i
\(79\) 10318.3i 1.65331i 0.562709 + 0.826655i \(0.309759\pi\)
−0.562709 + 0.826655i \(0.690241\pi\)
\(80\) 1728.45 + 830.151i 0.270070 + 0.129711i
\(81\) 2847.31 0.433976
\(82\) −1448.12 − 2801.31i −0.215366 − 0.416614i
\(83\) 322.720 − 322.720i 0.0468458 − 0.0468458i −0.683296 − 0.730142i \(-0.739454\pi\)
0.730142 + 0.683296i \(0.239454\pi\)
\(84\) 346.128 + 2030.74i 0.0490545 + 0.287803i
\(85\) −1125.47 + 1125.47i −0.155774 + 0.155774i
\(86\) −4601.24 − 1465.22i −0.622126 − 0.198110i
\(87\) 3491.43 0.461281
\(88\) −4705.36 + 3546.93i −0.607614 + 0.458023i
\(89\) − 8475.90i − 1.07005i −0.844835 − 0.535027i \(-0.820301\pi\)
0.844835 − 0.535027i \(-0.179699\pi\)
\(90\) 932.677 + 297.002i 0.115145 + 0.0366670i
\(91\) −55.1751 − 55.1751i −0.00666285 − 0.00666285i
\(92\) 4391.59 − 6196.22i 0.518855 − 0.732068i
\(93\) 3941.10 + 3941.10i 0.455671 + 0.455671i
\(94\) 4379.34 + 8471.61i 0.495625 + 0.958761i
\(95\) 3195.31i 0.354051i
\(96\) 7115.71 − 208.949i 0.772104 − 0.0226725i
\(97\) 3166.40 0.336528 0.168264 − 0.985742i \(-0.446184\pi\)
0.168264 + 0.985742i \(0.446184\pi\)
\(98\) −1218.78 + 630.042i −0.126904 + 0.0656020i
\(99\) −2126.96 + 2126.96i −0.217015 + 0.217015i
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.99.9 yes 96
4.3 odd 2 448.5.k.a.239.16 96
16.5 even 4 448.5.k.a.15.16 96
16.11 odd 4 inner 112.5.k.a.43.9 ✓ 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.9 ✓ 96 16.11 odd 4 inner
112.5.k.a.99.9 yes 96 1.1 even 1 trivial
448.5.k.a.15.16 96 16.5 even 4
448.5.k.a.239.16 96 4.3 odd 2