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Properties

Label 230.4.g.d.41.1
Level $230$
Weight $4$
Character 230.41
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 41.1
Character \(\chi\) \(=\) 230.41
Dual form 230.4.g.d.101.1

$q$-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+(-1.91899 - 0.563465i) q^{2} +(-5.62703 + 6.49394i) q^{3} +(3.36501 + 2.16256i) q^{4} +(-0.711574 + 4.94911i) q^{5} +(14.4573 - 9.29114i) q^{6} +(-0.0746076 - 0.163368i) q^{7} +(-5.23889 - 6.04600i) q^{8} +(-6.66528 - 46.3580i) q^{9} +(4.15415 - 9.09632i) q^{10} +(51.8665 - 15.2294i) q^{11} +(-32.9786 + 9.68339i) q^{12} +(37.2430 - 81.5507i) q^{13} +(0.0511189 + 0.355540i) q^{14} +(-28.1352 - 32.4697i) q^{15} +(6.64664 + 14.5541i) q^{16} +(-86.8661 + 55.8255i) q^{17} +(-13.3306 + 92.7160i) q^{18} +(81.6511 + 52.4740i) q^{19} +(-13.0972 + 15.1150i) q^{20} +(1.48072 + 0.434779i) q^{21} -108.112 q^{22} +(-12.7248 - 109.568i) q^{23} +68.7417 q^{24} +(-23.9873 - 7.04331i) q^{25} +(-117.420 + 135.510i) q^{26} +(143.378 + 92.1437i) q^{27} +(0.102238 - 0.711079i) q^{28} +(153.037 - 98.3509i) q^{29} +(35.6954 + 78.1621i) q^{30} +(43.4943 + 50.1951i) q^{31} +(-4.55407 - 31.6743i) q^{32} +(-192.956 + 422.514i) q^{33} +(198.151 - 58.1823i) q^{34} +(0.861614 - 0.252993i) q^{35} +(77.8234 - 170.409i) q^{36} +(33.2814 + 231.477i) q^{37} +(-127.120 - 146.704i) q^{38} +(320.018 + 700.742i) q^{39} +(33.6501 - 21.6256i) q^{40} +(-24.5499 + 170.748i) q^{41} +(-2.59650 - 1.66867i) q^{42} +(96.1754 - 110.992i) q^{43} +(207.466 + 60.9175i) q^{44} +234.174 q^{45} +(-37.3188 + 217.429i) q^{46} +208.249 q^{47} +(-131.914 - 38.7336i) q^{48} +(224.596 - 259.198i) q^{49} +(42.0627 + 27.0320i) q^{50} +(126.271 - 878.235i) q^{51} +(301.682 - 193.879i) q^{52} +(282.526 + 618.647i) q^{53} +(-223.221 - 257.611i) q^{54} +(38.4650 + 267.530i) q^{55} +(-0.596861 + 1.30694i) q^{56} +(-800.217 + 234.965i) q^{57} +(-349.093 + 102.503i) q^{58} +(-11.9734 + 26.2182i) q^{59} +(-24.4574 - 170.105i) q^{60} +(-518.154 - 597.981i) q^{61} +(-55.1817 - 120.831i) q^{62} +(-7.07613 + 4.54755i) q^{63} +(-9.10815 + 63.3486i) q^{64} +(377.102 + 242.349i) q^{65} +(608.351 - 702.075i) q^{66} +(-487.239 - 143.066i) q^{67} -413.032 q^{68} +(783.129 + 533.907i) q^{69} -1.79598 q^{70} +(-121.441 - 35.6584i) q^{71} +(-245.362 + 283.163i) q^{72} +(464.206 + 298.327i) q^{73} +(66.5628 - 462.955i) q^{74} +(180.716 - 116.139i) q^{75} +(161.279 + 353.152i) q^{76} +(-6.35763 - 7.33709i) q^{77} +(-219.267 - 1525.03i) q^{78} +(341.178 - 747.075i) q^{79} +(-76.7594 + 22.5386i) q^{80} +(-191.855 + 56.3336i) q^{81} +(143.321 - 313.830i) q^{82} +(-62.9775 - 438.018i) q^{83} +(4.04241 + 4.66519i) q^{84} +(-214.475 - 469.634i) q^{85} +(-247.099 + 158.801i) q^{86} +(-222.459 + 1547.24i) q^{87} +(-363.799 - 233.800i) q^{88} +(-868.185 + 1001.94i) q^{89} +(-449.376 - 131.949i) q^{90} -16.1014 q^{91} +(194.128 - 396.215i) q^{92} -570.707 q^{93} +(-399.627 - 117.341i) q^{94} +(-317.800 + 366.761i) q^{95} +(231.317 + 148.658i) q^{96} +(-30.1631 + 209.789i) q^{97} +(-577.046 + 370.845i) q^{98} +(-1051.71 - 2302.92i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{6}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.91899 − 0.563465i −0.678464 − 0.199215i
\(3\) −5.62703 + 6.49394i −1.08292 + 1.24976i −0.116393 + 0.993203i \(0.537133\pi\)
−0.966529 + 0.256556i \(0.917412\pi\)
\(4\) 3.36501 + 2.16256i 0.420627 + 0.270320i
\(5\) −0.711574 + 4.94911i −0.0636451 + 0.442662i
\(6\) 14.4573 − 9.29114i 0.983695 − 0.632182i
\(7\) −0.0746076 − 0.163368i −0.00402843 − 0.00882104i 0.907607 − 0.419821i \(-0.137907\pi\)
−0.911636 + 0.411000i \(0.865180\pi\)
\(8\) −5.23889 − 6.04600i −0.231528 − 0.267198i
\(9\) −6.66528 − 46.3580i −0.246862 − 1.71696i
\(10\) 4.15415 − 9.09632i 0.131366 − 0.287651i
\(11\) 51.8665 − 15.2294i 1.42167 − 0.417439i 0.521600 − 0.853190i \(-0.325335\pi\)
0.900067 + 0.435751i \(0.143517\pi\)
\(12\) −32.9786 + 9.68339i −0.793342 + 0.232946i
\(13\) 37.2430 − 81.5507i 0.794565 − 1.73985i 0.131468 − 0.991320i \(-0.458031\pi\)
0.663096 − 0.748534i \(-0.269242\pi\)
\(14\) 0.0511189 + 0.355540i 0.000975864 + 0.00678728i
\(15\) −28.1352 − 32.4697i −0.484298 − 0.558909i
\(16\) 6.64664 + 14.5541i 0.103854 + 0.227408i
\(17\) −86.8661 + 55.8255i −1.23930 + 0.796451i −0.985313 − 0.170756i \(-0.945379\pi\)
−0.253989 + 0.967207i \(0.581743\pi\)
\(18\) −13.3306 + 92.7160i −0.174558 + 1.21408i
\(19\) 81.6511 + 52.4740i 0.985898 + 0.633598i 0.931048 − 0.364897i \(-0.118896\pi\)
0.0548497 + 0.998495i \(0.482532\pi\)
\(20\) −13.0972 + 15.1150i −0.146431 + 0.168991i
\(21\) 1.48072 + 0.434779i 0.0153867 + 0.00451793i
\(22\) −108.112 −1.04771
\(23\) −12.7248 − 109.568i −0.115361 − 0.993324i
\(24\) 68.7417 0.584660
\(25\) −23.9873 − 7.04331i −0.191899 − 0.0563465i
\(26\) −117.420 + 135.510i −0.885689 + 1.02214i
\(27\) 143.378 + 92.1437i 1.02197 + 0.656780i
\(28\) 0.102238 − 0.711079i 0.000690040 − 0.00479933i
\(29\) 153.037 − 98.3509i 0.979940 − 0.629769i 0.0504926 − 0.998724i \(-0.483921\pi\)
0.929447 + 0.368955i \(0.120285\pi\)
\(30\) 35.6954 + 78.1621i 0.217235 + 0.475679i
\(31\) 43.4943 + 50.1951i 0.251994 + 0.290816i 0.867626 − 0.497217i \(-0.165645\pi\)
−0.615632 + 0.788033i \(0.711099\pi\)
\(32\) −4.55407 − 31.6743i −0.0251579 − 0.174977i
\(33\) −192.956 + 422.514i −1.01786 + 2.22880i
\(34\) 198.151 − 58.1823i 0.999487 − 0.293476i
\(35\) 0.861614 − 0.252993i 0.00416113 − 0.00122182i
\(36\) 77.8234 − 170.409i 0.360294 − 0.788933i
\(37\) 33.2814 + 231.477i 0.147876 + 1.02850i 0.919687 + 0.392651i \(0.128442\pi\)
−0.771811 + 0.635852i \(0.780649\pi\)
\(38\) −127.120 − 146.704i −0.542674 − 0.626279i
\(39\) 320.018 + 700.742i 1.31395 + 2.87714i
\(40\) 33.6501 − 21.6256i 0.133014 − 0.0854828i
\(41\) −24.5499 + 170.748i −0.0935133 + 0.650399i 0.888119 + 0.459614i \(0.152012\pi\)
−0.981632 + 0.190785i \(0.938897\pi\)
\(42\) −2.59650 − 1.66867i −0.00953925 − 0.00613051i
\(43\) 96.1754 − 110.992i 0.341084 − 0.393632i −0.559130 − 0.829080i \(-0.688865\pi\)
0.900214 + 0.435448i \(0.143410\pi\)
\(44\) 207.466 + 60.9175i 0.710833 + 0.208720i
\(45\) 234.174 0.775745
\(46\) −37.3188 + 217.429i −0.119616 + 0.696916i
\(47\) 208.249 0.646304 0.323152 − 0.946347i \(-0.395258\pi\)
0.323152 + 0.946347i \(0.395258\pi\)
\(48\) −131.914 − 38.7336i −0.396671 − 0.116473i
\(49\) 224.596 − 259.198i 0.654799 − 0.755679i
\(50\) 42.0627 + 27.0320i 0.118971 + 0.0764582i
\(51\) 126.271 − 878.235i 0.346696 − 2.41132i
\(52\) 301.682 − 193.879i 0.804533 − 0.517042i
\(53\) 282.526 + 618.647i 0.732226 + 1.60335i 0.795934 + 0.605383i \(0.206980\pi\)
−0.0637078 + 0.997969i \(0.520293\pi\)
\(54\) −223.221 − 257.611i −0.562529 − 0.649193i
\(55\) 38.4650 + 267.530i 0.0943021 + 0.655885i
\(56\) −0.596861 + 1.30694i −0.00142427 + 0.00311871i
\(57\) −800.217 + 234.965i −1.85950 + 0.545997i
\(58\) −349.093 + 102.503i −0.790313 + 0.232057i
\(59\) −11.9734 + 26.2182i −0.0264205 + 0.0578528i −0.922382 − 0.386279i \(-0.873760\pi\)
0.895962 + 0.444131i \(0.146488\pi\)
\(60\) −24.4574 − 170.105i −0.0526240 − 0.366008i
\(61\) −518.154 − 597.981i −1.08759 − 1.25514i −0.964878 − 0.262699i \(-0.915387\pi\)
−0.122708 − 0.992443i \(-0.539158\pi\)
\(62\) −55.1817 − 120.831i −0.113034 − 0.247509i
\(63\) −7.07613 + 4.54755i −0.0141509 + 0.00909425i
\(64\) −9.10815 + 63.3486i −0.0177894 + 0.123728i
\(65\) 377.102 + 242.349i 0.719597 + 0.462457i
\(66\) 608.351 − 702.075i 1.13459 − 1.30939i
\(67\) −487.239 − 143.066i −0.888443 − 0.260871i −0.194502 − 0.980902i \(-0.562309\pi\)
−0.693941 + 0.720032i \(0.744127\pi\)
\(68\) −413.032 −0.736581
\(69\) 783.129 + 533.907i 1.36634 + 0.931519i
\(70\) −1.79598 −0.00306658
\(71\) −121.441 − 35.6584i −0.202992 − 0.0596038i 0.178656 − 0.983912i \(-0.442825\pi\)
−0.381648 + 0.924308i \(0.624643\pi\)
\(72\) −245.362 + 283.163i −0.401613 + 0.463487i
\(73\) 464.206 + 298.327i 0.744263 + 0.478309i 0.857000 − 0.515316i \(-0.172325\pi\)
−0.112737 + 0.993625i \(0.535962\pi\)
\(74\) 66.5628 − 462.955i 0.104564 − 0.727262i
\(75\) 180.716 − 116.139i 0.278231 − 0.178808i
\(76\) 161.279 + 353.152i 0.243421 + 0.533017i
\(77\) −6.35763 − 7.33709i −0.00940934 − 0.0108590i
\(78\) −219.267 − 1525.03i −0.318296 − 2.21380i
\(79\) 341.178 − 747.075i 0.485893 − 1.06396i −0.494908 − 0.868945i \(-0.664798\pi\)
0.980801 − 0.195011i \(-0.0624743\pi\)
\(80\) −76.7594 + 22.5386i −0.107275 + 0.0314987i
\(81\) −191.855 + 56.3336i −0.263175 + 0.0772751i
\(82\) 143.321 − 313.830i 0.193015 − 0.422643i
\(83\) −62.9775 − 438.018i −0.0832853 − 0.579262i −0.988142 − 0.153545i \(-0.950931\pi\)
0.904856 − 0.425717i \(-0.139978\pi\)
\(84\) 4.04241 + 4.66519i 0.00525075 + 0.00605969i
\(85\) −214.475 − 469.634i −0.273683 − 0.599282i
\(86\) −247.099 + 158.801i −0.309830 + 0.199116i
\(87\) −222.459 + 1547.24i −0.274139 + 1.90668i
\(88\) −363.799 − 233.800i −0.440695 − 0.283217i
\(89\) −868.185 + 1001.94i −1.03402 + 1.19332i −0.0531595 + 0.998586i \(0.516929\pi\)
−0.980856 + 0.194732i \(0.937616\pi\)
\(90\) −449.376 − 131.949i −0.526315 − 0.154540i
\(91\) −16.1014 −0.0185482
\(92\) 194.128 − 396.215i 0.219992 − 0.449003i
\(93\) −570.707 −0.636340
\(94\) −399.627 − 117.341i −0.438494 − 0.128753i
\(95\) −317.800 + 366.761i −0.343217 + 0.396094i
\(96\) 231.317 + 148.658i 0.245924 + 0.158046i
\(97\) −30.1631 + 209.789i −0.0315731 + 0.219596i −0.999499 − 0.0316385i \(-0.989927\pi\)
0.967926 + 0.251235i \(0.0808366\pi\)
\(98\) −577.046 + 370.845i −0.594800 + 0.382255i
\(99\) −1051.71 − 2302.92i −1.06768 − 2.33790i
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.41.1 ✓ 70
23.9 even 11 inner 230.4.g.d.101.1 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.41.1 ✓ 70 1.1 even 1 trivial
230.4.g.d.101.1 yes 70 23.9 even 11 inner