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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(31,230)] 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("230.31"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [70,-14,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(70\)
Relative dimension: \(7\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 141.5
Character \(\chi\) \(=\) 230.141
Dual form 230.4.g.d.31.5

$q$-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+(-0.284630 - 1.97964i) q^{2} +(0.934566 - 2.04642i) q^{3} +(-3.83797 + 1.12693i) q^{4} +(-3.27430 + 3.77875i) q^{5} +(-4.31718 - 1.26764i) q^{6} +(13.5404 - 8.70192i) q^{7} +(3.32332 + 7.27706i) q^{8} +(14.3668 + 16.5802i) q^{9} +(8.41254 + 5.40641i) q^{10} +(-6.55528 + 45.5930i) q^{11} +(-1.28067 + 8.90728i) q^{12} +(0.196673 + 0.126394i) q^{13} +(-21.0807 - 24.3284i) q^{14} +(4.67283 + 10.2321i) q^{15} +(13.4601 - 8.65025i) q^{16} +(41.4943 + 12.1838i) q^{17} +(28.7337 - 33.1604i) q^{18} +(-7.67632 + 2.25397i) q^{19} +(8.30830 - 18.1926i) q^{20} +(-5.15329 - 35.8419i) q^{21} +92.1236 q^{22} +(109.182 - 15.6908i) q^{23} +17.9977 q^{24} +(-3.55787 - 24.7455i) q^{25} +(0.194236 - 0.425317i) q^{26} +(105.639 - 31.0183i) q^{27} +(-42.1614 + 48.6569i) q^{28} +(200.172 + 58.7759i) q^{29} +(18.9258 - 12.1629i) q^{30} +(-24.7389 - 54.1706i) q^{31} +(-20.9555 - 24.1840i) q^{32} +(87.1758 + 56.0245i) q^{33} +(12.3091 - 85.6119i) q^{34} +(-11.4532 + 79.6587i) q^{35} +(-73.8243 - 47.4440i) q^{36} +(-57.5416 - 66.4065i) q^{37} +(6.64697 + 14.5548i) q^{38} +(0.442458 - 0.284351i) q^{39} +(-38.3797 - 11.2693i) q^{40} +(-127.024 + 146.594i) q^{41} +(-69.4874 + 20.4033i) q^{42} +(195.299 - 427.645i) q^{43} +(-26.2211 - 182.372i) q^{44} -109.694 q^{45} +(-62.1388 - 211.676i) q^{46} +57.3256 q^{47} +(-5.12269 - 35.6291i) q^{48} +(-34.8670 + 76.3481i) q^{49} +(-47.9746 + 14.0866i) q^{50} +(63.7124 - 73.5281i) q^{51} +(-0.897261 - 0.263460i) q^{52} +(223.990 - 143.950i) q^{53} +(-91.4731 - 200.298i) q^{54} +(-150.820 - 174.056i) q^{55} +(108.324 + 69.6153i) q^{56} +(-2.56147 + 17.8154i) q^{57} +(59.3803 - 412.999i) q^{58} +(169.473 + 108.913i) q^{59} +(-29.4650 - 34.0045i) q^{60} +(301.615 + 660.445i) q^{61} +(-100.197 + 64.3927i) q^{62} +(338.813 + 99.4845i) q^{63} +(-41.9111 + 48.3680i) q^{64} +(-1.12158 + 0.329325i) q^{65} +(86.0956 - 188.523i) q^{66} +(146.532 + 1019.15i) q^{67} -172.984 q^{68} +(69.9283 - 238.097i) q^{69} +160.956 q^{70} +(-129.715 - 902.188i) q^{71} +(-72.9096 + 159.650i) q^{72} +(-471.453 + 138.431i) q^{73} +(-115.083 + 132.813i) q^{74} +(-53.9647 - 15.8455i) q^{75} +(26.9214 - 17.3014i) q^{76} +(307.985 + 674.393i) q^{77} +(-0.688849 - 0.794974i) q^{78} +(-376.329 - 241.852i) q^{79} +(-11.3852 + 79.1857i) q^{80} +(-49.0497 + 341.148i) q^{81} +(326.358 + 209.737i) q^{82} +(163.105 + 188.233i) q^{83} +(60.1695 + 131.753i) q^{84} +(-181.905 + 116.903i) q^{85} +(-902.171 - 264.901i) q^{86} +(307.354 - 354.706i) q^{87} +(-353.568 + 103.817i) q^{88} +(347.465 - 760.843i) q^{89} +(31.2221 + 217.155i) q^{90} +3.76291 q^{91} +(-401.357 + 183.262i) q^{92} -133.976 q^{93} +(-16.3166 - 113.484i) q^{94} +(16.6174 - 36.3871i) q^{95} +(-69.0748 + 20.2822i) q^{96} +(-855.898 + 987.759i) q^{97} +(161.066 + 47.2933i) q^{98} +(-850.120 + 546.339i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 70 q - 14 q^{2} + 3 q^{3} - 28 q^{4} - 35 q^{5} + 6 q^{6} - 78 q^{7} - 56 q^{8} - 24 q^{9} - 70 q^{10} - 15 q^{11} - 120 q^{12} - 270 q^{13} + 64 q^{14} + 15 q^{15} - 112 q^{16} + 114 q^{17} - 48 q^{18}+ \cdots + 11285 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{8}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.284630 − 1.97964i −0.100632 − 0.699909i
\(3\) 0.934566 − 2.04642i 0.179857 − 0.393833i −0.798133 − 0.602481i \(-0.794179\pi\)
0.977991 + 0.208648i \(0.0669063\pi\)
\(4\) −3.83797 + 1.12693i −0.479746 + 0.140866i
\(5\) −3.27430 + 3.77875i −0.292863 + 0.337981i
\(6\) −4.31718 − 1.26764i −0.293747 − 0.0862518i
\(7\) 13.5404 − 8.70192i 0.731115 − 0.469859i −0.121372 − 0.992607i \(-0.538729\pi\)
0.852487 + 0.522748i \(0.175093\pi\)
\(8\) 3.32332 + 7.27706i 0.146871 + 0.321603i
\(9\) 14.3668 + 16.5802i 0.532105 + 0.614082i
\(10\) 8.41254 + 5.40641i 0.266028 + 0.170966i
\(11\) −6.55528 + 45.5930i −0.179681 + 1.24971i 0.677821 + 0.735227i \(0.262925\pi\)
−0.857502 + 0.514481i \(0.827984\pi\)
\(12\) −1.28067 + 8.90728i −0.0308082 + 0.214276i
\(13\) 0.196673 + 0.126394i 0.00419594 + 0.00269657i 0.542737 − 0.839903i \(-0.317388\pi\)
−0.538541 + 0.842599i \(0.681024\pi\)
\(14\) −21.0807 − 24.3284i −0.402432 − 0.464432i
\(15\) 4.67283 + 10.2321i 0.0804347 + 0.176127i
\(16\) 13.4601 − 8.65025i 0.210313 − 0.135160i
\(17\) 41.4943 + 12.1838i 0.591992 + 0.173824i 0.563987 − 0.825783i \(-0.309267\pi\)
0.0280042 + 0.999608i \(0.491085\pi\)
\(18\) 28.7337 − 33.1604i 0.376255 − 0.434222i
\(19\) −7.67632 + 2.25397i −0.0926878 + 0.0272156i −0.327748 − 0.944765i \(-0.606290\pi\)
0.235060 + 0.971981i \(0.424471\pi\)
\(20\) 8.30830 − 18.1926i 0.0928896 − 0.203400i
\(21\) −5.15329 − 35.8419i −0.0535495 − 0.372445i
\(22\) 92.1236 0.892765
\(23\) 109.182 − 15.6908i 0.989831 − 0.142251i
\(24\) 17.9977 0.153074
\(25\) −3.55787 − 24.7455i −0.0284630 − 0.197964i
\(26\) 0.194236 − 0.425317i 0.00146511 − 0.00320814i
\(27\) 105.639 − 31.0183i 0.752969 − 0.221092i
\(28\) −42.1614 + 48.6569i −0.284563 + 0.328403i
\(29\) 200.172 + 58.7759i 1.28176 + 0.376359i 0.850551 − 0.525893i \(-0.176269\pi\)
0.431210 + 0.902252i \(0.358087\pi\)
\(30\) 18.9258 − 12.1629i 0.115179 − 0.0740210i
\(31\) −24.7389 − 54.1706i −0.143330 − 0.313849i 0.824329 − 0.566111i \(-0.191553\pi\)
−0.967659 + 0.252262i \(0.918825\pi\)
\(32\) −20.9555 − 24.1840i −0.115764 − 0.133599i
\(33\) 87.1758 + 56.0245i 0.459859 + 0.295534i
\(34\) 12.3091 − 85.6119i 0.0620882 − 0.431833i
\(35\) −11.4532 + 79.6587i −0.0553126 + 0.384708i
\(36\) −73.8243 − 47.4440i −0.341779 − 0.219648i
\(37\) −57.5416 − 66.4065i −0.255670 − 0.295059i 0.613375 − 0.789792i \(-0.289811\pi\)
−0.869045 + 0.494733i \(0.835266\pi\)
\(38\) 6.64697 + 14.5548i 0.0283758 + 0.0621343i
\(39\) 0.442458 − 0.284351i 0.00181667 − 0.00116750i
\(40\) −38.3797 − 11.2693i −0.151709 − 0.0445458i
\(41\) −127.024 + 146.594i −0.483850 + 0.558392i −0.944212 − 0.329339i \(-0.893174\pi\)
0.460362 + 0.887731i \(0.347720\pi\)
\(42\) −69.4874 + 20.4033i −0.255289 + 0.0749596i
\(43\) 195.299 − 427.645i 0.692623 − 1.51663i −0.156070 − 0.987746i \(-0.549882\pi\)
0.848693 − 0.528886i \(-0.177390\pi\)
\(44\) −26.2211 − 182.372i −0.0898405 − 0.624854i
\(45\) −109.694 −0.363382
\(46\) −62.1388 − 211.676i −0.199171 − 0.678477i
\(47\) 57.3256 0.177911 0.0889554 − 0.996036i \(-0.471647\pi\)
0.0889554 + 0.996036i \(0.471647\pi\)
\(48\) −5.12269 − 35.6291i −0.0154041 − 0.107138i
\(49\) −34.8670 + 76.3481i −0.101653 + 0.222589i
\(50\) −47.9746 + 14.0866i −0.135693 + 0.0398430i
\(51\) 63.7124 − 73.5281i 0.174932 − 0.201882i
\(52\) −0.897261 − 0.263460i −0.00239284 − 0.000702602i
\(53\) 223.990 − 143.950i 0.580517 − 0.373076i −0.217182 − 0.976131i \(-0.569687\pi\)
0.797699 + 0.603055i \(0.206050\pi\)
\(54\) −91.4731 − 200.298i −0.230517 − 0.504761i
\(55\) −150.820 − 174.056i −0.369757 − 0.426722i
\(56\) 108.324 + 69.6153i 0.258488 + 0.166120i
\(57\) −2.56147 + 17.8154i −0.00595220 + 0.0413984i
\(58\) 59.3803 − 412.999i 0.134431 − 0.934990i
\(59\) 169.473 + 108.913i 0.373957 + 0.240328i 0.714096 − 0.700047i \(-0.246838\pi\)
−0.340139 + 0.940375i \(0.610474\pi\)
\(60\) −29.4650 − 34.0045i −0.0633987 − 0.0731660i
\(61\) 301.615 + 660.445i 0.633080 + 1.38625i 0.905613 + 0.424105i \(0.139411\pi\)
−0.272534 + 0.962146i \(0.587862\pi\)
\(62\) −100.197 + 64.3927i −0.205242 + 0.131901i
\(63\) 338.813 + 99.4845i 0.677563 + 0.198950i
\(64\) −41.9111 + 48.3680i −0.0818576 + 0.0944687i
\(65\) −1.12158 + 0.329325i −0.00214022 + 0.000628426i
\(66\) 86.0956 − 188.523i 0.160570 − 0.351600i
\(67\) 146.532 + 1019.15i 0.267190 + 1.85835i 0.474745 + 0.880123i \(0.342540\pi\)
−0.207555 + 0.978223i \(0.566551\pi\)
\(68\) −172.984 −0.308492
\(69\) 69.9283 − 238.097i 0.122005 − 0.415413i
\(70\) 160.956 0.274827
\(71\) −129.715 − 902.188i −0.216822 − 1.50803i −0.749667 − 0.661815i \(-0.769786\pi\)
0.532845 − 0.846213i \(-0.321123\pi\)
\(72\) −72.9096 + 159.650i −0.119340 + 0.261318i
\(73\) −471.453 + 138.431i −0.755882 + 0.221947i −0.636897 − 0.770949i \(-0.719782\pi\)
−0.118985 + 0.992896i \(0.537964\pi\)
\(74\) −115.083 + 132.813i −0.180786 + 0.208638i
\(75\) −53.9647 − 15.8455i −0.0830841 − 0.0243957i
\(76\) 26.9214 − 17.3014i 0.0406329 − 0.0261132i
\(77\) 307.985 + 674.393i 0.455820 + 0.998106i
\(78\) −0.688849 − 0.794974i −0.000999959 − 0.00115401i
\(79\) −376.329 − 241.852i −0.535953 − 0.344436i 0.244501 − 0.969649i \(-0.421376\pi\)
−0.780454 + 0.625213i \(0.785012\pi\)
\(80\) −11.3852 + 79.1857i −0.0159113 + 0.110665i
\(81\) −49.0497 + 341.148i −0.0672836 + 0.467968i
\(82\) 326.358 + 209.737i 0.439515 + 0.282459i
\(83\) 163.105 + 188.233i 0.215700 + 0.248931i 0.853280 − 0.521453i \(-0.174610\pi\)
−0.637580 + 0.770384i \(0.720064\pi\)
\(84\) 60.1695 + 131.753i 0.0781551 + 0.171136i
\(85\) −181.905 + 116.903i −0.232122 + 0.149176i
\(86\) −902.171 − 264.901i −1.13121 − 0.332152i
\(87\) 307.354 − 354.706i 0.378757 − 0.437108i
\(88\) −353.568 + 103.817i −0.428301 + 0.125760i
\(89\) 347.465 − 760.843i 0.413834 − 0.906171i −0.581844 − 0.813300i \(-0.697669\pi\)
0.995678 − 0.0928703i \(-0.0296042\pi\)
\(90\) 31.2221 + 217.155i 0.0365678 + 0.254335i
\(91\) 3.76291 0.00433472
\(92\) −401.357 + 183.262i −0.454829 + 0.207678i
\(93\) −133.976 −0.149383
\(94\) −16.3166 − 113.484i −0.0179035 − 0.124521i
\(95\) 16.6174 − 36.3871i 0.0179464 − 0.0392972i
\(96\) −69.0748 + 20.2822i −0.0734367 + 0.0215630i
\(97\) −855.898 + 987.759i −0.895910 + 1.03394i 0.103318 + 0.994648i \(0.467054\pi\)
−0.999228 + 0.0392872i \(0.987491\pi\)
\(98\) 161.066 + 47.2933i 0.166022 + 0.0487484i
\(99\) −850.120 + 546.339i −0.863033 + 0.554638i
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.g.d.141.5 yes 70
23.8 even 11 inner 230.4.g.d.31.5 ✓ 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.31.5 ✓ 70 23.8 even 11 inner
230.4.g.d.141.5 yes 70 1.1 even 1 trivial