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 137.16
Character \(\chi\) \(=\) 230.137
Dual form 230.4.e.a.183.16

$q$-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} +(5.03526 - 5.03526i) q^{3} +4.00000i q^{4} +(6.80528 + 8.87064i) q^{5} -14.2419 q^{6} +(-11.1481 + 11.1481i) q^{7} +(5.65685 - 5.65685i) q^{8} -23.7076i q^{9} +(2.92086 - 22.1691i) q^{10} +8.94113i q^{11} +(20.1410 + 20.1410i) q^{12} +(43.6579 - 43.6579i) q^{13} +31.5316 q^{14} +(78.9323 + 10.3996i) q^{15} -16.0000 q^{16} +(33.1397 - 33.1397i) q^{17} +(-33.5277 + 33.5277i) q^{18} +107.253 q^{19} +(-35.4826 + 27.2211i) q^{20} +112.267i q^{21} +(12.6447 - 12.6447i) q^{22} +(105.143 - 33.3466i) q^{23} -56.9674i q^{24} +(-32.3764 + 120.734i) q^{25} -123.483 q^{26} +(16.5779 + 16.5779i) q^{27} +(-44.5924 - 44.5924i) q^{28} -25.7005i q^{29} +(-96.9198 - 126.334i) q^{30} +81.0455 q^{31} +(22.6274 + 22.6274i) q^{32} +(45.0209 + 45.0209i) q^{33} -93.7331 q^{34} +(-174.757 - 23.0248i) q^{35} +94.8305 q^{36} +(109.732 - 109.732i) q^{37} +(-151.679 - 151.679i) q^{38} -439.658i q^{39} +(88.6764 + 11.6834i) q^{40} -276.972 q^{41} +(158.770 - 158.770i) q^{42} +(-91.0717 - 91.0717i) q^{43} -35.7645 q^{44} +(210.302 - 161.337i) q^{45} +(-195.854 - 101.535i) q^{46} +(147.078 + 147.078i) q^{47} +(-80.5641 + 80.5641i) q^{48} +94.4399i q^{49} +(216.531 - 124.957i) q^{50} -333.733i q^{51} +(174.632 + 174.632i) q^{52} +(299.388 + 299.388i) q^{53} -46.8894i q^{54} +(-79.3135 + 60.8469i) q^{55} +126.126i q^{56} +(540.047 - 540.047i) q^{57} +(-36.3460 + 36.3460i) q^{58} -613.321i q^{59} +(-41.5985 + 315.729i) q^{60} -40.4771i q^{61} +(-114.616 - 114.616i) q^{62} +(264.295 + 264.295i) q^{63} -64.0000i q^{64} +(684.378 + 90.1694i) q^{65} -127.338i q^{66} +(-317.182 + 317.182i) q^{67} +(132.559 + 132.559i) q^{68} +(361.512 - 697.330i) q^{69} +(214.581 + 279.705i) q^{70} -748.659 q^{71} +(-134.111 - 134.111i) q^{72} +(-142.851 + 142.851i) q^{73} -310.368 q^{74} +(444.905 + 770.952i) q^{75} +429.012i q^{76} +(-99.6766 - 99.6766i) q^{77} +(-621.770 + 621.770i) q^{78} -1335.64 q^{79} +(-108.884 - 141.930i) q^{80} +807.054 q^{81} +(391.698 + 391.698i) q^{82} +(-30.2308 - 30.2308i) q^{83} -449.068 q^{84} +(519.495 + 68.4454i) q^{85} +257.590i q^{86} +(-129.408 - 129.408i) q^{87} +(50.5787 + 50.5787i) q^{88} -358.594 q^{89} +(-525.577 - 69.2467i) q^{90} +973.406i q^{91} +(133.386 + 420.571i) q^{92} +(408.085 - 408.085i) q^{93} -415.999i q^{94} +(729.887 + 951.403i) q^{95} +227.870 q^{96} +(-679.077 + 679.077i) q^{97} +(133.558 - 133.558i) q^{98} +211.973 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
\(3\) 5.03526 − 5.03526i 0.969036 − 0.969036i −0.0304990 − 0.999535i \(-0.509710\pi\)
0.999535 + 0.0304990i \(0.00970965\pi\)
\(4\) 4.00000i 0.500000i
\(5\) 6.80528 + 8.87064i 0.608682 + 0.793414i
\(6\) −14.2419 −0.969036
\(7\) −11.1481 + 11.1481i −0.601941 + 0.601941i −0.940827 − 0.338887i \(-0.889950\pi\)
0.338887 + 0.940827i \(0.389950\pi\)
\(8\) 5.65685 − 5.65685i 0.250000 − 0.250000i
\(9\) − 23.7076i − 0.878061i
\(10\) 2.92086 − 22.1691i 0.0923658 − 0.701048i
\(11\) 8.94113i 0.245078i 0.992464 + 0.122539i \(0.0391036\pi\)
−0.992464 + 0.122539i \(0.960896\pi\)
\(12\) 20.1410 + 20.1410i 0.484518 + 0.484518i
\(13\) 43.6579 − 43.6579i 0.931426 − 0.931426i −0.0663693 − 0.997795i \(-0.521142\pi\)
0.997795 + 0.0663693i \(0.0211415\pi\)
\(14\) 31.5316 0.601941
\(15\) 78.9323 + 10.3996i 1.35868 + 0.179011i
\(16\) −16.0000 −0.250000
\(17\) 33.1397 − 33.1397i 0.472797 − 0.472797i −0.430022 − 0.902819i \(-0.641494\pi\)
0.902819 + 0.430022i \(0.141494\pi\)
\(18\) −33.5277 + 33.5277i −0.439030 + 0.439030i
\(19\) 107.253 1.29503 0.647514 − 0.762053i \(-0.275809\pi\)
0.647514 + 0.762053i \(0.275809\pi\)
\(20\) −35.4826 + 27.2211i −0.396707 + 0.304341i
\(21\) 112.267i 1.16660i
\(22\) 12.6447 − 12.6447i 0.122539 − 0.122539i
\(23\) 105.143 − 33.3466i 0.953208 − 0.302315i
\(24\) − 56.9674i − 0.484518i
\(25\) −32.3764 + 120.734i −0.259011 + 0.965874i
\(26\) −123.483 −0.931426
\(27\) 16.5779 + 16.5779i 0.118164 + 0.118164i
\(28\) −44.5924 − 44.5924i −0.300970 − 0.300970i
\(29\) − 25.7005i − 0.164568i −0.996609 − 0.0822838i \(-0.973779\pi\)
0.996609 − 0.0822838i \(-0.0262214\pi\)
\(30\) −96.9198 − 126.334i −0.589835 − 0.768846i
\(31\) 81.0455 0.469555 0.234777 − 0.972049i \(-0.424564\pi\)
0.234777 + 0.972049i \(0.424564\pi\)
\(32\) 22.6274 + 22.6274i 0.125000 + 0.125000i
\(33\) 45.0209 + 45.0209i 0.237489 + 0.237489i
\(34\) −93.7331 −0.472797
\(35\) −174.757 − 23.0248i −0.843979 − 0.111197i
\(36\) 94.8305 0.439030
\(37\) 109.732 − 109.732i 0.487561 − 0.487561i −0.419975 − 0.907536i \(-0.637961\pi\)
0.907536 + 0.419975i \(0.137961\pi\)
\(38\) −151.679 − 151.679i −0.647514 − 0.647514i
\(39\) − 439.658i − 1.80517i
\(40\) 88.6764 + 11.6834i 0.350524 + 0.0461829i
\(41\) −276.972 −1.05502 −0.527510 − 0.849549i \(-0.676874\pi\)
−0.527510 + 0.849549i \(0.676874\pi\)
\(42\) 158.770 − 158.770i 0.583302 − 0.583302i
\(43\) −91.0717 − 91.0717i −0.322984 − 0.322984i 0.526927 − 0.849911i \(-0.323344\pi\)
−0.849911 + 0.526927i \(0.823344\pi\)
\(44\) −35.7645 −0.122539
\(45\) 210.302 − 161.337i 0.696666 − 0.534460i
\(46\) −195.854 − 101.535i −0.627762 − 0.325446i
\(47\) 147.078 + 147.078i 0.456458 + 0.456458i 0.897491 − 0.441033i \(-0.145388\pi\)
−0.441033 + 0.897491i \(0.645388\pi\)
\(48\) −80.5641 + 80.5641i −0.242259 + 0.242259i
\(49\) 94.4399i 0.275335i
\(50\) 216.531 − 124.957i 0.612443 − 0.353431i
\(51\) − 333.733i − 0.916314i
\(52\) 174.632 + 174.632i 0.465713 + 0.465713i
\(53\) 299.388 + 299.388i 0.775928 + 0.775928i 0.979136 − 0.203208i \(-0.0651368\pi\)
−0.203208 + 0.979136i \(0.565137\pi\)
\(54\) − 46.8894i − 0.118164i
\(55\) −79.3135 + 60.8469i −0.194448 + 0.149174i
\(56\) 126.126i 0.300970i
\(57\) 540.047 − 540.047i 1.25493 − 1.25493i
\(58\) −36.3460 + 36.3460i −0.0822838 + 0.0822838i
\(59\) − 613.321i − 1.35335i −0.736283 − 0.676674i \(-0.763421\pi\)
0.736283 − 0.676674i \(-0.236579\pi\)
\(60\) −41.5985 + 315.729i −0.0895057 + 0.679341i
\(61\) − 40.4771i − 0.0849600i −0.999097 − 0.0424800i \(-0.986474\pi\)
0.999097 − 0.0424800i \(-0.0135259\pi\)
\(62\) −114.616 − 114.616i −0.234777 − 0.234777i
\(63\) 264.295 + 264.295i 0.528540 + 0.528540i
\(64\) − 64.0000i − 0.125000i
\(65\) 684.378 + 90.1694i 1.30595 + 0.172064i
\(66\) − 127.338i − 0.237489i
\(67\) −317.182 + 317.182i −0.578358 + 0.578358i −0.934451 − 0.356093i \(-0.884109\pi\)
0.356093 + 0.934451i \(0.384109\pi\)
\(68\) 132.559 + 132.559i 0.236399 + 0.236399i
\(69\) 361.512 − 697.330i 0.630739 − 1.21665i
\(70\) 214.581 + 279.705i 0.366391 + 0.477588i
\(71\) −748.659 −1.25140 −0.625701 − 0.780063i \(-0.715187\pi\)
−0.625701 + 0.780063i \(0.715187\pi\)
\(72\) −134.111 − 134.111i −0.219515 − 0.219515i
\(73\) −142.851 + 142.851i −0.229033 + 0.229033i −0.812289 − 0.583256i \(-0.801779\pi\)
0.583256 + 0.812289i \(0.301779\pi\)
\(74\) −310.368 −0.487561
\(75\) 444.905 + 770.952i 0.684975 + 1.18696i
\(76\) 429.012i 0.647514i
\(77\) −99.6766 − 99.6766i −0.147522 − 0.147522i
\(78\) −621.770 + 621.770i −0.902585 + 0.902585i
\(79\) −1335.64 −1.90217 −0.951083 − 0.308935i \(-0.900028\pi\)
−0.951083 + 0.308935i \(0.900028\pi\)
\(80\) −108.884 − 141.930i −0.152171 − 0.198353i
\(81\) 807.054 1.10707
\(82\) 391.698 + 391.698i 0.527510 + 0.527510i
\(83\) −30.2308 − 30.2308i −0.0399790 − 0.0399790i 0.686835 − 0.726814i \(-0.259000\pi\)
−0.726814 + 0.686835i \(0.759000\pi\)
\(84\) −449.068 −0.583302
\(85\) 519.495 + 68.4454i 0.662907 + 0.0873405i
\(86\) 257.590i 0.322984i
\(87\) −129.408 − 129.408i −0.159472 − 0.159472i
\(88\) 50.5787 + 50.5787i 0.0612694 + 0.0612694i
\(89\) −358.594 −0.427088 −0.213544 − 0.976933i \(-0.568501\pi\)
−0.213544 + 0.976933i \(0.568501\pi\)
\(90\) −525.577 − 69.2467i −0.615563 − 0.0811027i
\(91\) 973.406i 1.12133i
\(92\) 133.386 + 420.571i 0.151158 + 0.476604i
\(93\) 408.085 − 408.085i 0.455015 − 0.455015i
\(94\) − 415.999i − 0.456458i
\(95\) 729.887 + 951.403i 0.788261 + 1.02749i
\(96\) 227.870 0.242259
\(97\) −679.077 + 679.077i −0.710822 + 0.710822i −0.966707 − 0.255885i \(-0.917633\pi\)
0.255885 + 0.966707i \(0.417633\pi\)
\(98\) 133.558 − 133.558i 0.137668 − 0.137668i
\(99\) 211.973 0.215193
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.137.16 yes 72
5.3 odd 4 inner 230.4.e.a.183.15 yes 72
23.22 odd 2 inner 230.4.e.a.137.15 ✓ 72
115.68 even 4 inner 230.4.e.a.183.16 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.15 ✓ 72 23.22 odd 2 inner
230.4.e.a.137.16 yes 72 1.1 even 1 trivial
230.4.e.a.183.15 yes 72 5.3 odd 4 inner
230.4.e.a.183.16 yes 72 115.68 even 4 inner