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Properties

Label 230.4.g.d.31.5
Level $230$
Weight $4$
Character 230.31
Analytic conductor $13.570$
Analytic rank $0$
Dimension $70$
Inner twists $2$

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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(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 31.5
Character \(\chi\) \(=\) 230.31
Dual form 230.4.g.d.141.5

$q$-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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{11}\right)\)

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 0.977991 − 0.208648i \(-0.0669063\pi\)
−0.798133 + 0.602481i \(0.794179\pi\)
\(4\) −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 0.852487 − 0.522748i \(-0.175093\pi\)
−0.121372 + 0.992607i \(0.538729\pi\)
\(8\) 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 −0.857502 − 0.514481i \(-0.827984\pi\)
0.677821 − 0.735227i \(-0.262925\pi\)
\(12\) −1.28067 − 8.90728i −0.0308082 − 0.214276i
\(13\) 0.196673 − 0.126394i 0.00419594 − 0.00269657i −0.538541 − 0.842599i \(-0.681024\pi\)
0.542737 + 0.839903i \(0.317388\pi\)
\(14\) −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 0.0280042 − 0.999608i \(-0.491085\pi\)
0.563987 + 0.825783i \(0.309267\pi\)
\(18\) 28.7337 + 33.1604i 0.376255 + 0.434222i
\(19\) −7.67632 − 2.25397i −0.0926878 − 0.0272156i 0.235060 − 0.971981i \(-0.424471\pi\)
−0.327748 + 0.944765i \(0.606290\pi\)
\(20\) 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 0.431210 − 0.902252i \(-0.358087\pi\)
0.850551 + 0.525893i \(0.176269\pi\)
\(30\) 18.9258 + 12.1629i 0.115179 + 0.0740210i
\(31\) −24.7389 + 54.1706i −0.143330 + 0.313849i −0.967659 − 0.252262i \(-0.918825\pi\)
0.824329 + 0.566111i \(0.191553\pi\)
\(32\) −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 −0.869045 − 0.494733i \(-0.835266\pi\)
0.613375 + 0.789792i \(0.289811\pi\)
\(38\) 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 0.460362 − 0.887731i \(-0.347720\pi\)
−0.944212 + 0.329339i \(0.893174\pi\)
\(42\) −69.4874 − 20.4033i −0.255289 − 0.0749596i
\(43\) 195.299 + 427.645i 0.692623 + 1.51663i 0.848693 + 0.528886i \(0.177390\pi\)
−0.156070 + 0.987746i \(0.549882\pi\)
\(44\) −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 0.797699 − 0.603055i \(-0.206050\pi\)
−0.217182 + 0.976131i \(0.569687\pi\)
\(54\) −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 −0.340139 − 0.940375i \(-0.610474\pi\)
0.714096 + 0.700047i \(0.246838\pi\)
\(60\) −29.4650 + 34.0045i −0.0633987 + 0.0731660i
\(61\) 301.615 − 660.445i 0.633080 − 1.38625i −0.272534 − 0.962146i \(-0.587862\pi\)
0.905613 − 0.424105i \(-0.139411\pi\)
\(62\) −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 −0.207555 − 0.978223i \(-0.566551\pi\)
0.474745 − 0.880123i \(-0.342540\pi\)
\(68\) −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 0.532845 + 0.846213i \(0.321123\pi\)
−0.749667 + 0.661815i \(0.769786\pi\)
\(72\) −72.9096 − 159.650i −0.119340 − 0.261318i
\(73\) −471.453 − 138.431i −0.755882 − 0.221947i −0.118985 − 0.992896i \(-0.537964\pi\)
−0.636897 + 0.770949i \(0.719782\pi\)
\(74\) −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 −0.780454 − 0.625213i \(-0.785012\pi\)
0.244501 + 0.969649i \(0.421376\pi\)
\(80\) −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 −0.637580 − 0.770384i \(-0.720064\pi\)
0.853280 + 0.521453i \(0.174610\pi\)
\(84\) 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 0.995678 + 0.0928703i \(0.0296042\pi\)
−0.581844 + 0.813300i \(0.697669\pi\)
\(90\) 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 −0.999228 − 0.0392872i \(-0.987491\pi\)
0.103318 − 0.994648i \(-0.467054\pi\)
\(98\) 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.31.5 ✓ 70
23.3 even 11 inner 230.4.g.d.141.5 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.31.5 ✓ 70 1.1 even 1 trivial
230.4.g.d.141.5 yes 70 23.3 even 11 inner