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Properties

Label 230.4.g.d.31.6
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 31.6
Character \(\chi\) \(=\) 230.31
Dual form 230.4.g.d.141.6

$q$-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} +(3.50164 + 7.66752i) q^{3} +(-3.83797 - 1.12693i) q^{4} +(-3.27430 - 3.77875i) q^{5} +(-16.1756 + 4.74959i) q^{6} +(-21.3660 - 13.7311i) q^{7} +(3.32332 - 7.27706i) q^{8} +(-28.8481 + 33.2925i) q^{9} +(8.41254 - 5.40641i) q^{10} +(-7.27623 - 50.6073i) q^{11} +(-4.79843 - 33.3738i) q^{12} +(39.2580 - 25.2296i) q^{13} +(33.2640 - 38.3887i) q^{14} +(17.5082 - 38.3376i) q^{15} +(13.4601 + 8.65025i) q^{16} +(8.31802 - 2.44239i) q^{17} +(-57.6963 - 66.5850i) q^{18} +(-131.219 - 38.5295i) q^{19} +(8.30830 + 18.1926i) q^{20} +(30.4674 - 211.905i) q^{21} +102.255 q^{22} +(-6.56660 + 110.108i) q^{23} +67.4340 q^{24} +(-3.55787 + 24.7455i) q^{25} +(38.7715 + 84.8978i) q^{26} +(-137.916 - 40.4958i) q^{27} +(66.5280 + 76.7774i) q^{28} +(-289.444 + 84.9884i) q^{29} +(70.9114 + 45.5720i) q^{30} +(18.9813 - 41.5633i) q^{31} +(-20.9555 + 24.1840i) q^{32} +(362.554 - 232.999i) q^{33} +(2.46751 + 17.1619i) q^{34} +(18.0724 + 125.696i) q^{35} +(148.237 - 95.2659i) q^{36} +(-10.9105 + 12.5914i) q^{37} +(113.624 - 248.801i) q^{38} +(330.915 + 212.666i) q^{39} +(-38.3797 + 11.2693i) q^{40} +(135.666 + 156.567i) q^{41} +(410.825 + 120.629i) q^{42} +(-148.002 - 324.080i) q^{43} +(-29.1049 + 202.429i) q^{44} +220.262 q^{45} +(-216.106 - 44.3397i) q^{46} -150.161 q^{47} +(-19.1937 + 133.495i) q^{48} +(125.475 + 274.751i) q^{49} +(-47.9746 - 14.0866i) q^{50} +(47.8538 + 55.2262i) q^{51} +(-179.103 + 52.5894i) q^{52} +(-466.382 - 299.726i) q^{53} +(119.422 - 261.498i) q^{54} +(-167.408 + 193.199i) q^{55} +(-170.928 + 109.849i) q^{56} +(-164.057 - 1141.04i) q^{57} +(-85.8624 - 597.186i) q^{58} +(717.824 - 461.318i) q^{59} +(-110.400 + 127.408i) q^{60} +(-175.458 + 384.200i) q^{61} +(76.8779 + 49.4064i) q^{62} +(1073.51 - 315.211i) q^{63} +(-41.9111 - 48.3680i) q^{64} +(-223.879 - 65.7367i) q^{65} +(358.061 + 784.045i) q^{66} +(3.92576 - 27.3043i) q^{67} -34.6767 q^{68} +(-867.253 + 335.211i) q^{69} -253.978 q^{70} +(104.575 - 727.337i) q^{71} +(146.400 + 320.571i) q^{72} +(-193.571 - 56.8376i) q^{73} +(-21.8210 - 25.1828i) q^{74} +(-202.195 + 59.3699i) q^{75} +(460.197 + 295.750i) q^{76} +(-539.429 + 1181.18i) q^{77} +(-515.192 + 594.563i) q^{78} +(75.4073 - 48.4613i) q^{79} +(-11.3852 - 79.1857i) q^{80} +(-3.15853 - 21.9680i) q^{81} +(-348.562 + 224.007i) q^{82} +(873.182 - 1007.71i) q^{83} +(-355.735 + 778.951i) q^{84} +(-36.4649 - 23.4346i) q^{85} +(683.689 - 200.749i) q^{86} +(-1665.18 - 1921.72i) q^{87} +(-392.453 - 115.235i) q^{88} +(224.791 + 492.223i) q^{89} +(-62.6930 + 436.039i) q^{90} -1185.21 q^{91} +(149.287 - 415.193i) q^{92} +385.154 q^{93} +(42.7402 - 297.264i) q^{94} +(284.059 + 622.003i) q^{95} +(-258.810 - 75.9934i) q^{96} +(-452.994 - 522.783i) q^{97} +(-579.623 + 170.193i) q^{98} +(1894.75 + 1217.68i) 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
\(3\) 3.50164 + 7.66752i 0.673891 + 1.47561i 0.868989 + 0.494831i \(0.164770\pi\)
−0.195099 + 0.980784i \(0.562503\pi\)
\(4\) −3.83797 − 1.12693i −0.479746 − 0.140866i
\(5\) −3.27430 − 3.77875i −0.292863 − 0.337981i
\(6\) −16.1756 + 4.74959i −1.10061 + 0.323169i
\(7\) −21.3660 − 13.7311i −1.15365 − 0.741408i −0.183290 − 0.983059i \(-0.558675\pi\)
−0.970363 + 0.241651i \(0.922311\pi\)
\(8\) 3.32332 − 7.27706i 0.146871 − 0.321603i
\(9\) −28.8481 + 33.2925i −1.06845 + 1.23306i
\(10\) 8.41254 − 5.40641i 0.266028 − 0.170966i
\(11\) −7.27623 − 50.6073i −0.199442 − 1.38715i −0.805907 − 0.592042i \(-0.798322\pi\)
0.606465 − 0.795110i \(-0.292587\pi\)
\(12\) −4.79843 − 33.3738i −0.115432 − 0.802849i
\(13\) 39.2580 − 25.2296i 0.837554 − 0.538263i −0.0501162 − 0.998743i \(-0.515959\pi\)
0.887670 + 0.460480i \(0.152323\pi\)
\(14\) 33.2640 − 38.3887i 0.635013 − 0.732844i
\(15\) 17.5082 − 38.3376i 0.301373 − 0.659915i
\(16\) 13.4601 + 8.65025i 0.210313 + 0.135160i
\(17\) 8.31802 − 2.44239i 0.118672 − 0.0348451i −0.221858 − 0.975079i \(-0.571212\pi\)
0.340529 + 0.940234i \(0.389394\pi\)
\(18\) −57.6963 − 66.5850i −0.755508 − 0.871902i
\(19\) −131.219 − 38.5295i −1.58441 − 0.465225i −0.633257 − 0.773942i \(-0.718282\pi\)
−0.951154 + 0.308716i \(0.900101\pi\)
\(20\) 8.30830 + 18.1926i 0.0928896 + 0.203400i
\(21\) 30.4674 − 211.905i 0.316596 − 2.20198i
\(22\) 102.255 0.990951
\(23\) −6.56660 + 110.108i −0.0595318 + 0.998226i
\(24\) 67.4340 0.573538
\(25\) −3.55787 + 24.7455i −0.0284630 + 0.197964i
\(26\) 38.7715 + 84.8978i 0.292451 + 0.640378i
\(27\) −137.916 − 40.4958i −0.983034 − 0.288645i
\(28\) 66.5280 + 76.7774i 0.449022 + 0.518199i
\(29\) −289.444 + 84.9884i −1.85339 + 0.544205i −0.853667 + 0.520819i \(0.825626\pi\)
−0.999727 + 0.0233859i \(0.992555\pi\)
\(30\) 70.9114 + 45.5720i 0.431553 + 0.277342i
\(31\) 18.9813 − 41.5633i 0.109973 − 0.240806i −0.846642 − 0.532163i \(-0.821379\pi\)
0.956615 + 0.291357i \(0.0941066\pi\)
\(32\) −20.9555 + 24.1840i −0.115764 + 0.133599i
\(33\) 362.554 − 232.999i 1.91250 − 1.22909i
\(34\) 2.46751 + 17.1619i 0.0124463 + 0.0865658i
\(35\) 18.0724 + 125.696i 0.0872798 + 0.607044i
\(36\) 148.237 − 95.2659i 0.686281 − 0.441046i
\(37\) −10.9105 + 12.5914i −0.0484777 + 0.0559462i −0.779471 − 0.626439i \(-0.784512\pi\)
0.730993 + 0.682385i \(0.239057\pi\)
\(38\) 113.624 − 248.801i 0.485058 − 1.06213i
\(39\) 330.915 + 212.666i 1.35869 + 0.873176i
\(40\) −38.3797 + 11.2693i −0.151709 + 0.0445458i
\(41\) 135.666 + 156.567i 0.516769 + 0.596383i 0.952819 − 0.303539i \(-0.0981684\pi\)
−0.436050 + 0.899922i \(0.643623\pi\)
\(42\) 410.825 + 120.629i 1.50932 + 0.443177i
\(43\) −148.002 − 324.080i −0.524888 − 1.14934i −0.967556 − 0.252658i \(-0.918695\pi\)
0.442668 − 0.896685i \(-0.354032\pi\)
\(44\) −29.1049 + 202.429i −0.0997212 + 0.693576i
\(45\) 220.262 0.729659
\(46\) −216.106 − 44.3397i −0.692677 − 0.142120i
\(47\) −150.161 −0.466025 −0.233013 − 0.972474i \(-0.574858\pi\)
−0.233013 + 0.972474i \(0.574858\pi\)
\(48\) −19.1937 + 133.495i −0.0577162 + 0.401425i
\(49\) 125.475 + 274.751i 0.365815 + 0.801024i
\(50\) −47.9746 − 14.0866i −0.135693 − 0.0398430i
\(51\) 47.8538 + 55.2262i 0.131390 + 0.151632i
\(52\) −179.103 + 52.5894i −0.477637 + 0.140247i
\(53\) −466.382 − 299.726i −1.20873 − 0.776801i −0.228282 − 0.973595i \(-0.573311\pi\)
−0.980445 + 0.196794i \(0.936947\pi\)
\(54\) 119.422 − 261.498i 0.300950 − 0.658988i
\(55\) −167.408 + 193.199i −0.410422 + 0.473653i
\(56\) −170.928 + 109.849i −0.407878 + 0.262127i
\(57\) −164.057 − 1141.04i −0.381227 − 2.65149i
\(58\) −85.8624 − 597.186i −0.194384 − 1.35197i
\(59\) 717.824 − 461.318i 1.58395 − 1.01794i 0.609649 − 0.792672i \(-0.291310\pi\)
0.974297 − 0.225268i \(-0.0723259\pi\)
\(60\) −110.400 + 127.408i −0.237542 + 0.274139i
\(61\) −175.458 + 384.200i −0.368281 + 0.806422i 0.631244 + 0.775585i \(0.282545\pi\)
−0.999524 + 0.0308378i \(0.990182\pi\)
\(62\) 76.8779 + 49.4064i 0.157476 + 0.101204i
\(63\) 1073.51 − 315.211i 2.14682 − 0.630363i
\(64\) −41.9111 − 48.3680i −0.0818576 − 0.0944687i
\(65\) −223.879 − 65.7367i −0.427211 − 0.125441i
\(66\) 358.061 + 784.045i 0.667793 + 1.46226i
\(67\) 3.92576 − 27.3043i 0.00715834 − 0.0497873i −0.985929 − 0.167163i \(-0.946539\pi\)
0.993088 + 0.117376i \(0.0374483\pi\)
\(68\) −34.6767 −0.0618407
\(69\) −867.253 + 335.211i −1.51312 + 0.584849i
\(70\) −253.978 −0.433659
\(71\) 104.575 − 727.337i 0.174800 − 1.21576i −0.693771 − 0.720196i \(-0.744052\pi\)
0.868571 − 0.495565i \(-0.165039\pi\)
\(72\) 146.400 + 320.571i 0.239631 + 0.524718i
\(73\) −193.571 − 56.8376i −0.310353 − 0.0911279i 0.122847 − 0.992426i \(-0.460798\pi\)
−0.433200 + 0.901298i \(0.642616\pi\)
\(74\) −21.8210 − 25.1828i −0.0342789 − 0.0395600i
\(75\) −202.195 + 59.3699i −0.311300 + 0.0914059i
\(76\) 460.197 + 295.750i 0.694581 + 0.446380i
\(77\) −539.429 + 1181.18i −0.798358 + 1.74816i
\(78\) −515.192 + 594.563i −0.747871 + 0.863090i
\(79\) 75.4073 − 48.4613i 0.107392 − 0.0690168i −0.485842 − 0.874046i \(-0.661487\pi\)
0.593234 + 0.805030i \(0.297851\pi\)
\(80\) −11.3852 − 79.1857i −0.0159113 − 0.110665i
\(81\) −3.15853 − 21.9680i −0.00433268 − 0.0301345i
\(82\) −348.562 + 224.007i −0.469417 + 0.301676i
\(83\) 873.182 − 1007.71i 1.15475 − 1.33265i 0.220770 − 0.975326i \(-0.429143\pi\)
0.933980 − 0.357326i \(-0.116312\pi\)
\(84\) −355.735 + 778.951i −0.462070 + 1.01179i
\(85\) −36.4649 − 23.4346i −0.0465315 − 0.0299040i
\(86\) 683.689 − 200.749i 0.857257 − 0.251713i
\(87\) −1665.18 − 1921.72i −2.05202 − 2.36816i
\(88\) −392.453 − 115.235i −0.475405 − 0.139592i
\(89\) 224.791 + 492.223i 0.267728 + 0.586242i 0.994974 − 0.100135i \(-0.0319275\pi\)
−0.727246 + 0.686377i \(0.759200\pi\)
\(90\) −62.6930 + 436.039i −0.0734269 + 0.510695i
\(91\) −1185.21 −1.36532
\(92\) 149.287 − 415.193i 0.169177 − 0.470510i
\(93\) 385.154 0.429447
\(94\) 42.7402 − 297.264i 0.0468969 − 0.326175i
\(95\) 284.059 + 622.003i 0.306777 + 0.671749i
\(96\) −258.810 − 75.9934i −0.275153 − 0.0807922i
\(97\) −452.994 − 522.783i −0.474170 − 0.547222i 0.467396 − 0.884048i \(-0.345192\pi\)
−0.941567 + 0.336826i \(0.890647\pi\)
\(98\) −579.623 + 170.193i −0.597457 + 0.175429i
\(99\) 1894.75 + 1217.68i 1.92353 + 1.23618i
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.31.6 ✓ 70
23.3 even 11 inner 230.4.g.d.141.6 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.31.6 ✓ 70 1.1 even 1 trivial
230.4.g.d.141.6 yes 70 23.3 even 11 inner