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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 = [60,12,-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: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 31.2
Character \(\chi\) \(=\) 230.31
Dual form 230.4.g.b.141.2

$q$-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} +(-1.76914 - 3.87387i) q^{3} +(-3.83797 - 1.12693i) q^{4} +(-3.27430 - 3.77875i) q^{5} +(-8.17242 + 2.39964i) q^{6} +(-26.5942 - 17.0910i) q^{7} +(-3.32332 + 7.27706i) q^{8} +(5.80424 - 6.69844i) q^{9} +(-8.41254 + 5.40641i) q^{10} +(2.31578 + 16.1066i) q^{11} +(2.42432 + 16.8615i) q^{12} +(-2.85088 + 1.83215i) q^{13} +(-41.4036 + 47.7823i) q^{14} +(-8.84568 + 19.3693i) q^{15} +(13.4601 + 8.65025i) q^{16} +(19.9821 - 5.86727i) q^{17} +(-11.6085 - 13.3969i) q^{18} +(52.6683 + 15.4648i) q^{19} +(8.30830 + 18.1926i) q^{20} +(-19.1597 + 133.259i) q^{21} +32.5444 q^{22} +(-77.0352 + 78.9467i) q^{23} +34.0698 q^{24} +(-3.55787 + 24.7455i) q^{25} +(2.81556 + 6.16522i) q^{26} +(-146.545 - 43.0295i) q^{27} +(82.8072 + 95.5646i) q^{28} +(-13.8771 + 4.07468i) q^{29} +(35.8266 + 23.0244i) q^{30} +(-3.19871 + 7.00419i) q^{31} +(20.9555 - 24.1840i) q^{32} +(58.2979 - 37.4658i) q^{33} +(-5.92760 - 41.2274i) q^{34} +(22.4947 + 156.454i) q^{35} +(-29.8252 + 19.1675i) q^{36} +(87.6813 - 101.190i) q^{37} +(45.6057 - 99.8627i) q^{38} +(12.1411 + 7.80262i) q^{39} +(38.3797 - 11.2693i) q^{40} +(177.057 + 204.335i) q^{41} +(258.351 + 75.8587i) q^{42} +(80.9235 + 177.198i) q^{43} +(9.26311 - 64.4264i) q^{44} -44.3166 q^{45} +(134.360 + 174.973i) q^{46} -374.802 q^{47} +(9.69726 - 67.4459i) q^{48} +(272.659 + 597.039i) q^{49} +(47.9746 + 14.0866i) q^{50} +(-58.0800 - 67.0279i) q^{51} +(13.0063 - 3.81900i) q^{52} +(-249.519 - 160.356i) q^{53} +(-126.894 + 277.859i) q^{54} +(53.2802 - 61.4886i) q^{55} +(212.753 - 136.728i) q^{56} +(-33.2688 - 231.389i) q^{57} +(4.11658 + 28.6315i) q^{58} +(-211.554 + 135.958i) q^{59} +(55.7774 - 64.3705i) q^{60} +(305.937 - 669.909i) q^{61} +(12.9554 + 8.32590i) q^{62} +(-268.842 + 78.9391i) q^{63} +(-41.9111 - 48.3680i) q^{64} +(16.2579 + 4.77375i) q^{65} +(-57.5755 - 126.073i) q^{66} +(-15.3596 + 106.828i) q^{67} -83.3026 q^{68} +(442.115 + 158.757i) q^{69} +316.125 q^{70} +(67.3543 - 468.460i) q^{71} +(29.4556 + 64.4988i) q^{72} +(-1137.06 - 333.871i) q^{73} +(-175.363 - 202.379i) q^{74} +(102.155 - 29.9955i) q^{75} +(-184.712 - 118.707i) q^{76} +(213.692 - 467.920i) q^{77} +(18.9021 - 21.8142i) q^{78} +(-1134.53 + 729.116i) q^{79} +(-11.3852 - 79.1857i) q^{80} +(58.5102 + 406.948i) q^{81} +(454.905 - 292.350i) q^{82} +(-358.620 + 413.870i) q^{83} +(223.707 - 489.851i) q^{84} +(-87.5983 - 56.2960i) q^{85} +(373.821 - 109.764i) q^{86} +(40.3353 + 46.5494i) q^{87} +(-124.905 - 36.6753i) q^{88} +(-74.3745 - 162.857i) q^{89} +(-12.6138 + 87.7310i) q^{90} +107.130 q^{91} +(384.626 - 216.182i) q^{92} +32.7923 q^{93} +(-106.680 + 741.975i) q^{94} +(-114.014 - 249.657i) q^{95} +(-130.759 - 38.3942i) q^{96} +(-673.651 - 777.434i) q^{97} +(1259.53 - 369.832i) q^{98} +(121.330 + 77.9743i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q + 12 q^{2} - 3 q^{3} - 24 q^{4} - 30 q^{5} + 6 q^{6} + 100 q^{7} + 48 q^{8} + 69 q^{9} + 60 q^{10} - 51 q^{11} + 120 q^{12} + 184 q^{13} + 20 q^{14} - 15 q^{15} - 96 q^{16} - 334 q^{17} - 138 q^{18}+ \cdots - 10589 q^{99}+O(q^{100}) \) 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
\(3\) −1.76914 − 3.87387i −0.340470 − 0.745526i 0.659511 − 0.751695i \(-0.270764\pi\)
−0.999981 + 0.00616918i \(0.998036\pi\)
\(4\) −3.83797 − 1.12693i −0.479746 − 0.140866i
\(5\) −3.27430 − 3.77875i −0.292863 − 0.337981i
\(6\) −8.17242 + 2.39964i −0.556063 + 0.163275i
\(7\) −26.5942 − 17.0910i −1.43595 − 0.922829i −0.999736 − 0.0229855i \(-0.992683\pi\)
−0.436214 − 0.899843i \(-0.643681\pi\)
\(8\) −3.32332 + 7.27706i −0.146871 + 0.321603i
\(9\) 5.80424 − 6.69844i 0.214972 − 0.248091i
\(10\) −8.41254 + 5.40641i −0.266028 + 0.170966i
\(11\) 2.31578 + 16.1066i 0.0634758 + 0.441484i 0.996631 + 0.0820111i \(0.0261343\pi\)
−0.933156 + 0.359473i \(0.882957\pi\)
\(12\) 2.42432 + 16.8615i 0.0583200 + 0.405624i
\(13\) −2.85088 + 1.83215i −0.0608225 + 0.0390883i −0.570699 − 0.821160i \(-0.693328\pi\)
0.509876 + 0.860248i \(0.329691\pi\)
\(14\) −41.4036 + 47.7823i −0.790399 + 0.912169i
\(15\) −8.84568 + 19.3693i −0.152263 + 0.333409i
\(16\) 13.4601 + 8.65025i 0.210313 + 0.135160i
\(17\) 19.9821 − 5.86727i 0.285080 − 0.0837071i −0.136066 − 0.990700i \(-0.543446\pi\)
0.421146 + 0.906993i \(0.361628\pi\)
\(18\) −11.6085 − 13.3969i −0.152008 − 0.175427i
\(19\) 52.6683 + 15.4648i 0.635944 + 0.186730i 0.583784 − 0.811909i \(-0.301571\pi\)
0.0521596 + 0.998639i \(0.483390\pi\)
\(20\) 8.30830 + 18.1926i 0.0928896 + 0.203400i
\(21\) −19.1597 + 133.259i −0.199095 + 1.38473i
\(22\) 32.5444 0.315386
\(23\) −77.0352 + 78.9467i −0.698389 + 0.715718i
\(24\) 34.0698 0.289769
\(25\) −3.55787 + 24.7455i −0.0284630 + 0.197964i
\(26\) 2.81556 + 6.16522i 0.0212376 + 0.0465038i
\(27\) −146.545 − 43.0295i −1.04454 − 0.306705i
\(28\) 82.8072 + 95.5646i 0.558896 + 0.645001i
\(29\) −13.8771 + 4.07468i −0.0888591 + 0.0260914i −0.325860 − 0.945418i \(-0.605654\pi\)
0.237001 + 0.971509i \(0.423836\pi\)
\(30\) 35.8266 + 23.0244i 0.218034 + 0.140122i
\(31\) −3.19871 + 7.00419i −0.0185324 + 0.0405803i −0.918672 − 0.395021i \(-0.870737\pi\)
0.900139 + 0.435602i \(0.143464\pi\)
\(32\) 20.9555 − 24.1840i 0.115764 − 0.133599i
\(33\) 58.2979 − 37.4658i 0.307526 − 0.197635i
\(34\) −5.92760 − 41.2274i −0.0298993 − 0.207954i
\(35\) 22.4947 + 156.454i 0.108637 + 0.755586i
\(36\) −29.8252 + 19.1675i −0.138080 + 0.0887383i
\(37\) 87.6813 − 101.190i 0.389587 − 0.449608i −0.526747 − 0.850022i \(-0.676588\pi\)
0.916334 + 0.400415i \(0.131134\pi\)
\(38\) 45.6057 − 99.8627i 0.194690 − 0.426312i
\(39\) 12.1411 + 7.80262i 0.0498496 + 0.0320364i
\(40\) 38.3797 − 11.2693i 0.151709 − 0.0445458i
\(41\) 177.057 + 204.335i 0.674431 + 0.778335i 0.985063 − 0.172196i \(-0.0550861\pi\)
−0.310632 + 0.950530i \(0.600541\pi\)
\(42\) 258.351 + 75.8587i 0.949153 + 0.278696i
\(43\) 80.9235 + 177.198i 0.286993 + 0.628428i 0.997136 − 0.0756301i \(-0.0240968\pi\)
−0.710143 + 0.704058i \(0.751370\pi\)
\(44\) 9.26311 − 64.4264i 0.0317379 − 0.220742i
\(45\) −44.3166 −0.146807
\(46\) 134.360 + 174.973i 0.430658 + 0.560833i
\(47\) −374.802 −1.16320 −0.581601 − 0.813474i \(-0.697574\pi\)
−0.581601 + 0.813474i \(0.697574\pi\)
\(48\) 9.69726 − 67.4459i 0.0291600 − 0.202812i
\(49\) 272.659 + 597.039i 0.794923 + 1.74064i
\(50\) 47.9746 + 14.0866i 0.135693 + 0.0398430i
\(51\) −58.0800 − 67.0279i −0.159467 − 0.184035i
\(52\) 13.0063 − 3.81900i 0.0346856 − 0.0101846i
\(53\) −249.519 − 160.356i −0.646681 − 0.415596i 0.175771 − 0.984431i \(-0.443758\pi\)
−0.822452 + 0.568835i \(0.807394\pi\)
\(54\) −126.894 + 277.859i −0.319780 + 0.700220i
\(55\) 53.2802 − 61.4886i 0.130624 − 0.150748i
\(56\) 212.753 − 136.728i 0.507685 − 0.326269i
\(57\) −33.2688 − 231.389i −0.0773080 − 0.537689i
\(58\) 4.11658 + 28.6315i 0.00931955 + 0.0648189i
\(59\) −211.554 + 135.958i −0.466814 + 0.300003i −0.752822 − 0.658224i \(-0.771308\pi\)
0.286008 + 0.958227i \(0.407672\pi\)
\(60\) 55.7774 − 64.3705i 0.120014 − 0.138503i
\(61\) 305.937 − 669.909i 0.642152 − 1.40612i −0.256107 − 0.966648i \(-0.582440\pi\)
0.898259 − 0.439467i \(-0.144833\pi\)
\(62\) 12.9554 + 8.32590i 0.0265376 + 0.0170547i
\(63\) −268.842 + 78.9391i −0.537634 + 0.157863i
\(64\) −41.9111 − 48.3680i −0.0818576 − 0.0944687i
\(65\) 16.2579 + 4.77375i 0.0310238 + 0.00910940i
\(66\) −57.5755 − 126.073i −0.107380 − 0.235129i
\(67\) −15.3596 + 106.828i −0.0280071 + 0.194794i −0.999021 − 0.0442274i \(-0.985917\pi\)
0.971014 + 0.239021i \(0.0768265\pi\)
\(68\) −83.3026 −0.148558
\(69\) 442.115 + 158.757i 0.771367 + 0.276986i
\(70\) 316.125 0.539774
\(71\) 67.3543 − 468.460i 0.112584 − 0.783041i −0.852805 − 0.522229i \(-0.825101\pi\)
0.965390 − 0.260812i \(-0.0839903\pi\)
\(72\) 29.4556 + 64.4988i 0.0482136 + 0.105573i
\(73\) −1137.06 − 333.871i −1.82305 − 0.535297i −0.823563 − 0.567224i \(-0.808017\pi\)
−0.999490 + 0.0319274i \(0.989835\pi\)
\(74\) −175.363 − 202.379i −0.275480 − 0.317921i
\(75\) 102.155 − 29.9955i 0.157278 − 0.0461811i
\(76\) −184.712 − 118.707i −0.278788 − 0.179166i
\(77\) 213.692 − 467.920i 0.316266 − 0.692525i
\(78\) 18.9021 − 21.8142i 0.0274390 − 0.0316663i
\(79\) −1134.53 + 729.116i −1.61575 + 1.03838i −0.657096 + 0.753807i \(0.728215\pi\)
−0.958655 + 0.284572i \(0.908148\pi\)
\(80\) −11.3852 − 79.1857i −0.0159113 − 0.110665i
\(81\) 58.5102 + 406.948i 0.0802609 + 0.558227i
\(82\) 454.905 − 292.350i 0.612633 − 0.393715i
\(83\) −358.620 + 413.870i −0.474261 + 0.547327i −0.941592 − 0.336756i \(-0.890670\pi\)
0.467331 + 0.884083i \(0.345216\pi\)
\(84\) 223.707 − 489.851i 0.290577 − 0.636275i
\(85\) −87.5983 − 56.2960i −0.111781 − 0.0718372i
\(86\) 373.821 − 109.764i 0.468723 − 0.137630i
\(87\) 40.3353 + 46.5494i 0.0497057 + 0.0573634i
\(88\) −124.905 − 36.6753i −0.151305 − 0.0444273i
\(89\) −74.3745 − 162.857i −0.0885807 − 0.193965i 0.860157 − 0.510030i \(-0.170366\pi\)
−0.948738 + 0.316065i \(0.897638\pi\)
\(90\) −12.6138 + 87.7310i −0.0147735 + 0.102752i
\(91\) 107.130 0.123410
\(92\) 384.626 − 216.182i 0.435870 − 0.244984i
\(93\) 32.7923 0.0365634
\(94\) −106.680 + 741.975i −0.117055 + 0.814137i
\(95\) −114.014 − 249.657i −0.123133 − 0.269624i
\(96\) −130.759 − 38.3942i −0.139016 − 0.0408187i
\(97\) −673.651 − 777.434i −0.705143 − 0.813778i 0.284295 − 0.958737i \(-0.408241\pi\)
−0.989438 + 0.144959i \(0.953695\pi\)
\(98\) 1259.53 − 369.832i 1.29828 − 0.381211i
\(99\) 121.330 + 77.9743i 0.123173 + 0.0791587i
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.b.31.2 ✓ 60
23.3 even 11 inner 230.4.g.b.141.2 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.b.31.2 ✓ 60 1.1 even 1 trivial
230.4.g.b.141.2 yes 60 23.3 even 11 inner