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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 43.9
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.9

$q$-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 \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.55330 − 1.83686i −0.888326 − 0.459214i
\(3\) 4.91576 + 4.91576i 0.546196 + 0.546196i 0.925338 − 0.379143i \(-0.123781\pi\)
−0.379143 + 0.925338i \(0.623781\pi\)
\(4\) 9.25192 + 13.0538i 0.578245 + 0.815863i
\(5\) −5.29630 − 5.29630i −0.211852 − 0.211852i 0.593202 − 0.805054i \(-0.297864\pi\)
−0.805054 + 0.593202i \(0.797864\pi\)
\(6\) −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 −0.384911 − 0.922954i \(-0.625768\pi\)
0.922954 + 0.384911i \(0.125768\pi\)
\(12\) −18.6892 + 109.650i −0.129786 + 0.761456i
\(13\) 2.97917 − 2.97917i 0.0176282 − 0.0176282i −0.698238 − 0.715866i \(-0.746032\pi\)
0.715866 + 0.698238i \(0.246032\pi\)
\(14\) 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 0.988277 − 0.152668i \(-0.0487866\pi\)
−0.152668 + 0.988277i \(0.548787\pi\)
\(20\) 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 −0.463717 − 0.885984i \(-0.653484\pi\)
0.885984 + 0.463717i \(0.153484\pi\)
\(30\) −95.6463 + 185.023i −0.106274 + 0.205581i
\(31\) − 801.728i − 0.834264i −0.908846 − 0.417132i \(-0.863035\pi\)
0.908846 − 0.417132i \(-0.136965\pi\)
\(32\) 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 0.760261 − 0.649618i \(-0.225071\pi\)
−0.649618 + 0.760261i \(0.725071\pi\)
\(38\) −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 \(-0.924657\pi\)
0.972118 − 0.234494i \(-0.0753433\pi\)
\(42\) 156.267 + 490.726i 0.0885869 + 0.278189i
\(43\) 853.638 − 853.638i 0.461675 − 0.461675i −0.437529 − 0.899204i \(-0.644146\pi\)
0.899204 + 0.437529i \(0.144146\pi\)
\(44\) 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 \(0.181442\pi\)
−0.841893 + 0.539645i \(0.818558\pi\)
\(48\) −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 0.924653 − 0.380812i \(-0.124355\pi\)
−0.380812 + 0.924653i \(0.624355\pi\)
\(54\) −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 0.0394362 − 0.999222i \(-0.487444\pi\)
0.999222 − 0.0394362i \(-0.0125562\pi\)
\(60\) 679.721 − 481.754i 0.188811 − 0.133821i
\(61\) −4237.89 + 4237.89i −1.13891 + 1.13891i −0.150267 + 0.988645i \(0.548013\pi\)
−0.988645 + 0.150267i \(0.951987\pi\)
\(62\) −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 0.301518 − 0.953461i \(-0.402507\pi\)
−0.953461 + 0.301518i \(0.902507\pi\)
\(68\) 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 \(0.0240115\pi\)
−0.997156 + 0.0753630i \(0.975988\pi\)
\(74\) −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 \(-0.690241\pi\)
0.562709 − 0.826655i \(-0.309759\pi\)
\(80\) 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 0.730142 − 0.683296i \(-0.239454\pi\)
−0.683296 + 0.730142i \(0.739454\pi\)
\(84\) 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 \(0.179699\pi\)
−0.844835 + 0.535027i \(0.820301\pi\)
\(90\) 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.43.9 ✓ 96
4.3 odd 2 448.5.k.a.15.16 96
16.3 odd 4 inner 112.5.k.a.99.9 yes 96
16.13 even 4 448.5.k.a.239.16 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.9 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.9 yes 96 16.3 odd 4 inner
448.5.k.a.15.16 96 4.3 odd 2
448.5.k.a.239.16 96 16.13 even 4