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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 137.15
Character \(\chi\) \(=\) 230.137
Dual form 230.4.e.a.183.15

$q$-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} +(33.3466 - 105.143i) 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} +(420.571 + 133.386i) 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 −0.338887 − 0.940827i \(-0.610050\pi\)
0.940827 + 0.338887i \(0.110050\pi\)
\(8\) 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 \(-0.960896\pi\)
0.992464 − 0.122539i \(-0.0391036\pi\)
\(12\) 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 −0.902819 − 0.430022i \(-0.858506\pi\)
0.430022 + 0.902819i \(0.358506\pi\)
\(18\) −33.5277 + 33.5277i −0.439030 + 0.439030i
\(19\) −107.253 −1.29503 −0.647514 − 0.762053i \(-0.724191\pi\)
−0.647514 + 0.762053i \(0.724191\pi\)
\(20\) 35.4826 − 27.2211i 0.396707 − 0.304341i
\(21\) − 112.267i − 1.16660i
\(22\) −12.6447 + 12.6447i −0.122539 + 0.122539i
\(23\) 33.3466 − 105.143i 0.302315 − 0.953208i
\(24\) − 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 −0.907536 − 0.419975i \(-0.862039\pi\)
0.419975 + 0.907536i \(0.362039\pi\)
\(38\) 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 0.849911 − 0.526927i \(-0.176656\pi\)
−0.526927 + 0.849911i \(0.676656\pi\)
\(44\) 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 0.203208 − 0.979136i \(-0.434863\pi\)
−0.979136 + 0.203208i \(0.934863\pi\)
\(54\) − 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 \(0.0135259\pi\)
−0.999097 + 0.0424800i \(0.986474\pi\)
\(62\) −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 −0.356093 − 0.934451i \(-0.615891\pi\)
0.934451 + 0.356093i \(0.115891\pi\)
\(68\) −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 \(-0.0999724\pi\)
0.951083 + 0.308935i \(0.0999724\pi\)
\(80\) 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 0.726814 − 0.686835i \(-0.241000\pi\)
−0.686835 + 0.726814i \(0.741000\pi\)
\(84\) 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 \(-0.431499\pi\)
0.213544 + 0.976933i \(0.431499\pi\)
\(90\) 525.577 + 69.2467i 0.615563 + 0.0811027i
\(91\) − 973.406i − 1.12133i
\(92\) 420.571 + 133.386i 0.476604 + 0.151158i
\(93\) 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 −0.255885 − 0.966707i \(-0.582367\pi\)
0.966707 + 0.255885i \(0.0823668\pi\)
\(98\) 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.137.15 ✓ 72
5.3 odd 4 inner 230.4.e.a.183.16 yes 72
23.22 odd 2 inner 230.4.e.a.137.16 yes 72
115.68 even 4 inner 230.4.e.a.183.15 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.15 ✓ 72 1.1 even 1 trivial
230.4.e.a.137.16 yes 72 23.22 odd 2 inner
230.4.e.a.183.15 yes 72 115.68 even 4 inner
230.4.e.a.183.16 yes 72 5.3 odd 4 inner