Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(137,230)] 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("230.137"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.e (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: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 137.9
Character \(\chi\) \(=\) 230.137
Dual form 230.4.e.a.183.9

$q$-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+(-1.41421 - 1.41421i) q^{2} +(0.422683 - 0.422683i) q^{3} +4.00000i q^{4} +(-11.1421 + 0.923475i) q^{5} -1.19553 q^{6} +(-23.7633 + 23.7633i) q^{7} +(5.65685 - 5.65685i) q^{8} +26.6427i q^{9} +(17.0634 + 14.4514i) q^{10} -53.9779i q^{11} +(1.69073 + 1.69073i) q^{12} +(33.6245 - 33.6245i) q^{13} +67.2128 q^{14} +(-4.31925 + 5.09993i) q^{15} -16.0000 q^{16} +(38.4177 - 38.4177i) q^{17} +(37.6784 - 37.6784i) q^{18} +54.7194 q^{19} +(-3.69390 - 44.5685i) q^{20} +20.0887i q^{21} +(-76.3362 + 76.3362i) q^{22} +(-44.4164 - 100.966i) q^{23} -4.78211i q^{24} +(123.294 - 20.5790i) q^{25} -95.1045 q^{26} +(22.6738 + 22.6738i) q^{27} +(-95.0532 - 95.0532i) q^{28} +42.1915i q^{29} +(13.3207 - 1.10404i) q^{30} +65.9550 q^{31} +(22.6274 + 22.6274i) q^{32} +(-22.8155 - 22.8155i) q^{33} -108.662 q^{34} +(242.829 - 286.719i) q^{35} -106.571 q^{36} +(-146.309 + 146.309i) q^{37} +(-77.3849 - 77.3849i) q^{38} -28.4250i q^{39} +(-57.8055 + 68.2534i) q^{40} +424.129 q^{41} +(28.4097 - 28.4097i) q^{42} +(220.669 + 220.669i) q^{43} +215.911 q^{44} +(-24.6039 - 296.856i) q^{45} +(-79.9735 + 205.602i) q^{46} +(-443.019 - 443.019i) q^{47} +(-6.76293 + 6.76293i) q^{48} -786.390i q^{49} +(-203.468 - 145.262i) q^{50} -32.4770i q^{51} +(134.498 + 134.498i) q^{52} +(-140.488 - 140.488i) q^{53} -64.1313i q^{54} +(49.8472 + 601.429i) q^{55} +268.851i q^{56} +(23.1290 - 23.1290i) q^{57} +(59.6678 - 59.6678i) q^{58} -265.863i q^{59} +(-20.3997 - 17.2770i) q^{60} -437.145i q^{61} +(-93.2745 - 93.2745i) q^{62} +(-633.118 - 633.118i) q^{63} -64.0000i q^{64} +(-343.598 + 405.700i) q^{65} +64.5321i q^{66} +(131.692 - 131.692i) q^{67} +(153.671 + 153.671i) q^{68} +(-61.4508 - 23.9026i) q^{69} +(-748.894 + 62.0694i) q^{70} +1082.16 q^{71} +(150.714 + 150.714i) q^{72} +(-408.810 + 408.810i) q^{73} +413.824 q^{74} +(43.4161 - 60.8128i) q^{75} +218.878i q^{76} +(1282.69 + 1282.69i) q^{77} +(-40.1990 + 40.1990i) q^{78} +465.865 q^{79} +(178.274 - 14.7756i) q^{80} -700.185 q^{81} +(-599.809 - 599.809i) q^{82} +(528.539 + 528.539i) q^{83} -80.3548 q^{84} +(-392.578 + 463.533i) q^{85} -624.146i q^{86} +(17.8336 + 17.8336i) q^{87} +(-305.345 - 305.345i) q^{88} +223.427 q^{89} +(-385.023 + 454.613i) q^{90} +1598.06i q^{91} +(403.865 - 177.666i) q^{92} +(27.8781 - 27.8781i) q^{93} +1253.05i q^{94} +(-609.691 + 50.5320i) q^{95} +19.1284 q^{96} +(418.280 - 418.280i) q^{97} +(-1112.12 + 1112.12i) q^{98} +1438.11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 16 q^{3} - 16 q^{6} - 64 q^{12} + 192 q^{13} - 1152 q^{16} + 32 q^{18} + 276 q^{23} + 880 q^{25} + 304 q^{26} + 728 q^{27} + 608 q^{31} + 688 q^{35} + 2816 q^{36} - 2208 q^{41} - 256 q^{46} + 144 q^{47}+ \cdots - 448 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 − 1.41421i −0.500000 − 0.500000i
\(3\) 0.422683 − 0.422683i 0.0813454 − 0.0813454i −0.665263 − 0.746609i \(-0.731681\pi\)
0.746609 + 0.665263i \(0.231681\pi\)
\(4\) 4.00000i 0.500000i
\(5\) −11.1421 + 0.923475i −0.996583 + 0.0825981i
\(6\) −1.19553 −0.0813454
\(7\) −23.7633 + 23.7633i −1.28310 + 1.28310i −0.344203 + 0.938895i \(0.611851\pi\)
−0.938895 + 0.344203i \(0.888149\pi\)
\(8\) 5.65685 − 5.65685i 0.250000 − 0.250000i
\(9\) 26.6427i 0.986766i
\(10\) 17.0634 + 14.4514i 0.539591 + 0.456992i
\(11\) − 53.9779i − 1.47954i −0.672860 − 0.739770i \(-0.734934\pi\)
0.672860 − 0.739770i \(-0.265066\pi\)
\(12\) 1.69073 + 1.69073i 0.0406727 + 0.0406727i
\(13\) 33.6245 − 33.6245i 0.717366 − 0.717366i −0.250699 − 0.968065i \(-0.580660\pi\)
0.968065 + 0.250699i \(0.0806604\pi\)
\(14\) 67.2128 1.28310
\(15\) −4.31925 + 5.09993i −0.0743484 + 0.0877864i
\(16\) −16.0000 −0.250000
\(17\) 38.4177 − 38.4177i 0.548098 − 0.548098i −0.377792 − 0.925890i \(-0.623317\pi\)
0.925890 + 0.377792i \(0.123317\pi\)
\(18\) 37.6784 − 37.6784i 0.493383 − 0.493383i
\(19\) 54.7194 0.660710 0.330355 − 0.943857i \(-0.392832\pi\)
0.330355 + 0.943857i \(0.392832\pi\)
\(20\) −3.69390 − 44.5685i −0.0412991 − 0.498291i
\(21\) 20.0887i 0.208748i
\(22\) −76.3362 + 76.3362i −0.739770 + 0.739770i
\(23\) −44.4164 − 100.966i −0.402672 − 0.915344i
\(24\) − 4.78211i − 0.0406727i
\(25\) 123.294 − 20.5790i 0.986355 − 0.164632i
\(26\) −95.1045 −0.717366
\(27\) 22.6738 + 22.6738i 0.161614 + 0.161614i
\(28\) −95.0532 − 95.0532i −0.641549 − 0.641549i
\(29\) 42.1915i 0.270164i 0.990834 + 0.135082i \(0.0431298\pi\)
−0.990834 + 0.135082i \(0.956870\pi\)
\(30\) 13.3207 − 1.10404i 0.0810674 − 0.00671898i
\(31\) 65.9550 0.382125 0.191062 − 0.981578i \(-0.438807\pi\)
0.191062 + 0.981578i \(0.438807\pi\)
\(32\) 22.6274 + 22.6274i 0.125000 + 0.125000i
\(33\) −22.8155 − 22.8155i −0.120354 − 0.120354i
\(34\) −108.662 −0.548098
\(35\) 242.829 − 286.719i 1.17273 − 1.38470i
\(36\) −106.571 −0.493383
\(37\) −146.309 + 146.309i −0.650082 + 0.650082i −0.953013 − 0.302930i \(-0.902035\pi\)
0.302930 + 0.953013i \(0.402035\pi\)
\(38\) −77.3849 − 77.3849i −0.330355 − 0.330355i
\(39\) − 28.4250i − 0.116709i
\(40\) −57.8055 + 68.2534i −0.228496 + 0.269795i
\(41\) 424.129 1.61556 0.807779 − 0.589486i \(-0.200670\pi\)
0.807779 + 0.589486i \(0.200670\pi\)
\(42\) 28.4097 − 28.4097i 0.104374 − 0.104374i
\(43\) 220.669 + 220.669i 0.782598 + 0.782598i 0.980268 − 0.197671i \(-0.0633377\pi\)
−0.197671 + 0.980268i \(0.563338\pi\)
\(44\) 215.911 0.739770
\(45\) −24.6039 − 296.856i −0.0815050 − 0.983394i
\(46\) −79.9735 + 205.602i −0.256336 + 0.659008i
\(47\) −443.019 − 443.019i −1.37491 − 1.37491i −0.852994 − 0.521921i \(-0.825216\pi\)
−0.521921 − 0.852994i \(-0.674784\pi\)
\(48\) −6.76293 + 6.76293i −0.0203363 + 0.0203363i
\(49\) − 786.390i − 2.29268i
\(50\) −203.468 − 145.262i −0.575493 − 0.410862i
\(51\) − 32.4770i − 0.0891705i
\(52\) 134.498 + 134.498i 0.358683 + 0.358683i
\(53\) −140.488 − 140.488i −0.364105 − 0.364105i 0.501217 − 0.865322i \(-0.332886\pi\)
−0.865322 + 0.501217i \(0.832886\pi\)
\(54\) − 64.1313i − 0.161614i
\(55\) 49.8472 + 601.429i 0.122207 + 1.47448i
\(56\) 268.851i 0.641549i
\(57\) 23.1290 − 23.1290i 0.0537457 − 0.0537457i
\(58\) 59.6678 − 59.6678i 0.135082 − 0.135082i
\(59\) − 265.863i − 0.586651i −0.956013 − 0.293326i \(-0.905238\pi\)
0.956013 − 0.293326i \(-0.0947620\pi\)
\(60\) −20.3997 − 17.2770i −0.0438932 − 0.0371742i
\(61\) − 437.145i − 0.917553i −0.888552 − 0.458777i \(-0.848288\pi\)
0.888552 − 0.458777i \(-0.151712\pi\)
\(62\) −93.2745 − 93.2745i −0.191062 − 0.191062i
\(63\) −633.118 − 633.118i −1.26612 − 1.26612i
\(64\) − 64.0000i − 0.125000i
\(65\) −343.598 + 405.700i −0.655662 + 0.774168i
\(66\) 64.5321i 0.120354i
\(67\) 131.692 − 131.692i 0.240130 − 0.240130i −0.576774 − 0.816904i \(-0.695689\pi\)
0.816904 + 0.576774i \(0.195689\pi\)
\(68\) 153.671 + 153.671i 0.274049 + 0.274049i
\(69\) −61.4508 − 23.9026i −0.107215 − 0.0417035i
\(70\) −748.894 + 62.0694i −1.27871 + 0.105982i
\(71\) 1082.16 1.80886 0.904429 − 0.426625i \(-0.140298\pi\)
0.904429 + 0.426625i \(0.140298\pi\)
\(72\) 150.714 + 150.714i 0.246691 + 0.246691i
\(73\) −408.810 + 408.810i −0.655446 + 0.655446i −0.954299 − 0.298853i \(-0.903396\pi\)
0.298853 + 0.954299i \(0.403396\pi\)
\(74\) 413.824 0.650082
\(75\) 43.4161 − 60.8128i 0.0668434 − 0.0936275i
\(76\) 218.878i 0.330355i
\(77\) 1282.69 + 1282.69i 1.89839 + 1.89839i
\(78\) −40.1990 + 40.1990i −0.0583544 + 0.0583544i
\(79\) 465.865 0.663467 0.331733 − 0.943373i \(-0.392367\pi\)
0.331733 + 0.943373i \(0.392367\pi\)
\(80\) 178.274 − 14.7756i 0.249146 − 0.0206495i
\(81\) −700.185 −0.960473
\(82\) −599.809 − 599.809i −0.807779 − 0.807779i
\(83\) 528.539 + 528.539i 0.698972 + 0.698972i 0.964189 − 0.265217i \(-0.0854436\pi\)
−0.265217 + 0.964189i \(0.585444\pi\)
\(84\) −80.3548 −0.104374
\(85\) −392.578 + 463.533i −0.500953 + 0.591497i
\(86\) − 624.146i − 0.782598i
\(87\) 17.8336 + 17.8336i 0.0219766 + 0.0219766i
\(88\) −305.345 − 305.345i −0.369885 − 0.369885i
\(89\) 223.427 0.266104 0.133052 − 0.991109i \(-0.457522\pi\)
0.133052 + 0.991109i \(0.457522\pi\)
\(90\) −385.023 + 454.613i −0.450944 + 0.532450i
\(91\) 1598.06i 1.84090i
\(92\) 403.865 − 177.666i 0.457672 − 0.201336i
\(93\) 27.8781 − 27.8781i 0.0310841 − 0.0310841i
\(94\) 1253.05i 1.37491i
\(95\) −609.691 + 50.5320i −0.658452 + 0.0545734i
\(96\) 19.1284 0.0203363
\(97\) 418.280 − 418.280i 0.437834 − 0.437834i −0.453449 − 0.891282i \(-0.649806\pi\)
0.891282 + 0.453449i \(0.149806\pi\)
\(98\) −1112.12 + 1112.12i −1.14634 + 1.14634i
\(99\) 1438.11 1.45996
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 230.4.e.a.137.9 ✓ 72
5.3 odd 4 inner 230.4.e.a.183.10 yes 72
23.22 odd 2 inner 230.4.e.a.137.10 yes 72
115.68 even 4 inner 230.4.e.a.183.9 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.9 ✓ 72 1.1 even 1 trivial
230.4.e.a.137.10 yes 72 23.22 odd 2 inner
230.4.e.a.183.9 yes 72 115.68 even 4 inner
230.4.e.a.183.10 yes 72 5.3 odd 4 inner