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 31.7
Character \(\chi\) \(=\) 230.31
Dual form 230.4.g.d.141.7

$q$-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} +(3.62947 + 7.94742i) q^{3} +(-3.83797 - 1.12693i) q^{4} +(-3.27430 - 3.77875i) q^{5} +(-16.7661 + 4.92297i) q^{6} +(25.2655 + 16.2371i) q^{7} +(3.32332 - 7.27706i) q^{8} +(-32.3073 + 37.2846i) q^{9} +(8.41254 - 5.40641i) q^{10} +(4.28654 + 29.8136i) q^{11} +(-4.97360 - 34.5921i) q^{12} +(0.290688 - 0.186814i) q^{13} +(-39.3350 + 45.3950i) q^{14} +(18.1473 - 39.7371i) q^{15} +(13.4601 + 8.65025i) q^{16} +(-103.234 + 30.3124i) q^{17} +(-64.6145 - 74.5691i) q^{18} +(46.0013 + 13.5072i) q^{19} +(8.30830 + 18.1926i) q^{20} +(-37.3432 + 259.727i) q^{21} -60.2403 q^{22} +(-97.4012 - 51.7688i) q^{23} +69.8957 q^{24} +(-3.55787 + 24.7455i) q^{25} +(0.287086 + 0.628631i) q^{26} +(-187.232 - 54.9762i) q^{27} +(-78.6700 - 90.7901i) q^{28} +(94.2652 - 27.6788i) q^{29} +(73.5000 + 47.2356i) q^{30} +(83.1972 - 182.177i) q^{31} +(-20.9555 + 24.1840i) q^{32} +(-221.383 + 142.274i) q^{33} +(-30.6241 - 212.995i) q^{34} +(-21.3708 - 148.637i) q^{35} +(166.011 - 106.689i) q^{36} +(157.647 - 181.935i) q^{37} +(-39.8328 + 87.2217i) q^{38} +(2.53973 + 1.63218i) q^{39} +(-38.3797 + 11.2693i) q^{40} +(56.2970 + 64.9702i) q^{41} +(-503.539 - 147.852i) q^{42} +(-26.4829 - 57.9894i) q^{43} +(17.1462 - 119.254i) q^{44} +246.673 q^{45} +(130.207 - 178.085i) q^{46} +295.396 q^{47} +(-19.8944 + 138.369i) q^{48} +(232.212 + 508.474i) q^{49} +(-47.9746 - 14.0866i) q^{50} +(-615.591 - 710.430i) q^{51} +(-1.32618 + 0.389401i) q^{52} +(-157.301 - 101.091i) q^{53} +(162.125 - 355.004i) q^{54} +(98.6225 - 113.816i) q^{55} +(202.124 - 129.897i) q^{56} +(59.6128 + 414.616i) q^{57} +(27.9634 + 194.490i) q^{58} +(-244.817 + 157.334i) q^{59} +(-114.430 + 132.059i) q^{60} +(-193.538 + 423.788i) q^{61} +(336.964 + 216.554i) q^{62} +(-1421.65 + 417.435i) q^{63} +(-41.9111 - 48.3680i) q^{64} +(-1.65772 - 0.486751i) q^{65} +(-218.640 - 478.755i) q^{66} +(-101.733 + 707.569i) q^{67} +430.371 q^{68} +(57.9146 - 961.981i) q^{69} +300.331 q^{70} +(-112.634 + 783.385i) q^{71} +(163.954 + 359.010i) q^{72} +(-1021.47 - 299.932i) q^{73} +(315.294 + 363.869i) q^{74} +(-209.576 + 61.5372i) q^{75} +(-161.330 - 103.681i) q^{76} +(-375.785 + 822.855i) q^{77} +(-3.95403 + 4.56319i) q^{78} +(836.503 - 537.588i) q^{79} +(-11.3852 - 79.1857i) q^{80} +(-53.0642 - 369.070i) q^{81} +(-144.642 + 92.9555i) q^{82} +(508.090 - 586.367i) q^{83} +(436.017 - 954.744i) q^{84} +(452.564 + 290.845i) q^{85} +(122.336 - 35.9211i) q^{86} +(562.107 + 648.706i) q^{87} +(231.201 + 67.8866i) q^{88} +(488.375 + 1069.39i) q^{89} +(-70.2104 + 488.324i) q^{90} +10.3777 q^{91} +(315.483 + 308.452i) q^{92} +1749.80 q^{93} +(-84.0784 + 584.778i) q^{94} +(-99.5820 - 218.054i) q^{95} +(-268.258 - 78.7676i) q^{96} +(862.474 + 995.348i) q^{97} +(-1072.69 + 314.970i) q^{98} +(-1250.07 - 803.373i) 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
\(3\) 3.62947 + 7.94742i 0.698491 + 1.52948i 0.841791 + 0.539803i \(0.181501\pi\)
−0.143300 + 0.989679i \(0.545771\pi\)
\(4\) −3.83797 − 1.12693i −0.479746 − 0.140866i
\(5\) −3.27430 − 3.77875i −0.292863 − 0.337981i
\(6\) −16.7661 + 4.92297i −1.14079 + 0.334966i
\(7\) 25.2655 + 16.2371i 1.36421 + 0.876723i 0.998540 − 0.0540229i \(-0.0172044\pi\)
0.365667 + 0.930746i \(0.380841\pi\)
\(8\) 3.32332 − 7.27706i 0.146871 − 0.321603i
\(9\) −32.3073 + 37.2846i −1.19657 + 1.38091i
\(10\) 8.41254 − 5.40641i 0.266028 − 0.170966i
\(11\) 4.28654 + 29.8136i 0.117495 + 0.817193i 0.960299 + 0.278973i \(0.0899940\pi\)
−0.842804 + 0.538220i \(0.819097\pi\)
\(12\) −4.97360 − 34.5921i −0.119646 − 0.832158i
\(13\) 0.290688 − 0.186814i 0.00620171 − 0.00398560i −0.537536 − 0.843241i \(-0.680645\pi\)
0.543738 + 0.839255i \(0.317009\pi\)
\(14\) −39.3350 + 45.3950i −0.750909 + 0.866595i
\(15\) 18.1473 − 39.7371i 0.312375 − 0.684005i
\(16\) 13.4601 + 8.65025i 0.210313 + 0.135160i
\(17\) −103.234 + 30.3124i −1.47283 + 0.432461i −0.917016 − 0.398849i \(-0.869410\pi\)
−0.555809 + 0.831310i \(0.687591\pi\)
\(18\) −64.6145 − 74.5691i −0.846099 − 0.976451i
\(19\) 46.0013 + 13.5072i 0.555444 + 0.163093i 0.547395 − 0.836875i \(-0.315620\pi\)
0.00804916 + 0.999968i \(0.497438\pi\)
\(20\) 8.30830 + 18.1926i 0.0928896 + 0.203400i
\(21\) −37.3432 + 259.727i −0.388045 + 2.69891i
\(22\) −60.2403 −0.583785
\(23\) −97.4012 − 51.7688i −0.883024 − 0.469328i
\(24\) 69.8957 0.594475
\(25\) −3.55787 + 24.7455i −0.0284630 + 0.197964i
\(26\) 0.287086 + 0.628631i 0.00216547 + 0.00474172i
\(27\) −187.232 − 54.9762i −1.33455 − 0.391858i
\(28\) −78.6700 − 90.7901i −0.530973 − 0.612775i
\(29\) 94.2652 − 27.6788i 0.603607 − 0.177235i 0.0343710 − 0.999409i \(-0.489057\pi\)
0.569236 + 0.822174i \(0.307239\pi\)
\(30\) 73.5000 + 47.2356i 0.447307 + 0.287467i
\(31\) 83.1972 − 182.177i 0.482022 − 1.05548i −0.499881 − 0.866094i \(-0.666623\pi\)
0.981903 − 0.189386i \(-0.0606497\pi\)
\(32\) −20.9555 + 24.1840i −0.115764 + 0.133599i
\(33\) −221.383 + 142.274i −1.16781 + 0.750508i
\(34\) −30.6241 − 212.995i −0.154470 − 1.07436i
\(35\) −21.3708 − 148.637i −0.103209 − 0.717836i
\(36\) 166.011 − 106.689i 0.768572 − 0.493931i
\(37\) 157.647 − 181.935i 0.700461 − 0.808375i −0.288354 − 0.957524i \(-0.593108\pi\)
0.988815 + 0.149149i \(0.0476535\pi\)
\(38\) −39.8328 + 87.2217i −0.170046 + 0.372348i
\(39\) 2.53973 + 1.63218i 0.0104277 + 0.00670151i
\(40\) −38.3797 + 11.2693i −0.151709 + 0.0445458i
\(41\) 56.2970 + 64.9702i 0.214442 + 0.247479i 0.852772 − 0.522284i \(-0.174920\pi\)
−0.638330 + 0.769763i \(0.720374\pi\)
\(42\) −503.539 − 147.852i −1.84995 − 0.543193i
\(43\) −26.4829 − 57.9894i −0.0939209 − 0.205658i 0.856841 − 0.515581i \(-0.172424\pi\)
−0.950762 + 0.309923i \(0.899697\pi\)
\(44\) 17.1462 − 119.254i 0.0587473 − 0.408597i
\(45\) 246.673 0.817151
\(46\) 130.207 − 178.085i 0.417348 − 0.570807i
\(47\) 295.396 0.916764 0.458382 − 0.888755i \(-0.348429\pi\)
0.458382 + 0.888755i \(0.348429\pi\)
\(48\) −19.8944 + 138.369i −0.0598231 + 0.416079i
\(49\) 232.212 + 508.474i 0.677003 + 1.48243i
\(50\) −47.9746 − 14.0866i −0.135693 − 0.0398430i
\(51\) −615.591 − 710.430i −1.69020 − 1.95059i
\(52\) −1.32618 + 0.389401i −0.00353669 + 0.00103847i
\(53\) −157.301 − 101.091i −0.407678 − 0.261999i 0.320691 − 0.947184i \(-0.396085\pi\)
−0.728369 + 0.685185i \(0.759721\pi\)
\(54\) 162.125 − 355.004i 0.408563 − 0.894629i
\(55\) 98.6225 − 113.816i 0.241786 − 0.279036i
\(56\) 202.124 − 129.897i 0.482320 − 0.309968i
\(57\) 59.6128 + 414.616i 0.138525 + 0.963460i
\(58\) 27.9634 + 194.490i 0.0633064 + 0.440306i
\(59\) −244.817 + 157.334i −0.540211 + 0.347173i −0.782122 − 0.623126i \(-0.785862\pi\)
0.241910 + 0.970299i \(0.422226\pi\)
\(60\) −114.430 + 132.059i −0.246214 + 0.284146i
\(61\) −193.538 + 423.788i −0.406229 + 0.889517i 0.590372 + 0.807131i \(0.298981\pi\)
−0.996601 + 0.0823853i \(0.973746\pi\)
\(62\) 336.964 + 216.554i 0.690234 + 0.443586i
\(63\) −1421.65 + 417.435i −2.84304 + 0.834791i
\(64\) −41.9111 − 48.3680i −0.0818576 − 0.0944687i
\(65\) −1.65772 − 0.486751i −0.00316331 − 0.000928831i
\(66\) −218.640 − 478.755i −0.407769 − 0.892889i
\(67\) −101.733 + 707.569i −0.185502 + 1.29020i 0.657978 + 0.753037i \(0.271412\pi\)
−0.843480 + 0.537160i \(0.819497\pi\)
\(68\) 430.371 0.767502
\(69\) 57.9146 − 961.981i 0.101045 − 1.67839i
\(70\) 300.331 0.512806
\(71\) −112.634 + 783.385i −0.188270 + 1.30945i 0.648216 + 0.761457i \(0.275515\pi\)
−0.836486 + 0.547989i \(0.815394\pi\)
\(72\) 163.954 + 359.010i 0.268364 + 0.587636i
\(73\) −1021.47 − 299.932i −1.63773 − 0.480881i −0.672028 − 0.740526i \(-0.734576\pi\)
−0.965704 + 0.259645i \(0.916395\pi\)
\(74\) 315.294 + 363.869i 0.495300 + 0.571607i
\(75\) −209.576 + 61.5372i −0.322664 + 0.0947427i
\(76\) −161.330 − 103.681i −0.243498 − 0.156487i
\(77\) −375.785 + 822.855i −0.556165 + 1.21783i
\(78\) −3.95403 + 4.56319i −0.00573981 + 0.00662409i
\(79\) 836.503 − 537.588i 1.19132 − 0.765612i 0.213883 − 0.976859i \(-0.431389\pi\)
0.977433 + 0.211247i \(0.0677525\pi\)
\(80\) −11.3852 − 79.1857i −0.0159113 − 0.110665i
\(81\) −53.0642 − 369.070i −0.0727904 − 0.506268i
\(82\) −144.642 + 92.9555i −0.194793 + 0.125186i
\(83\) 508.090 − 586.367i 0.671930 − 0.775448i −0.312747 − 0.949836i \(-0.601249\pi\)
0.984677 + 0.174388i \(0.0557948\pi\)
\(84\) 436.017 − 954.744i 0.566349 − 1.24013i
\(85\) 452.564 + 290.845i 0.577499 + 0.371136i
\(86\) 122.336 − 35.9211i 0.153393 − 0.0450404i
\(87\) 562.107 + 648.706i 0.692692 + 0.799409i
\(88\) 231.201 + 67.8866i 0.280069 + 0.0822356i
\(89\) 488.375 + 1069.39i 0.581659 + 1.27366i 0.940353 + 0.340200i \(0.110495\pi\)
−0.358694 + 0.933455i \(0.616778\pi\)
\(90\) −70.2104 + 488.324i −0.0822314 + 0.571932i
\(91\) 10.3777 0.0119547
\(92\) 315.483 + 308.452i 0.357515 + 0.349547i
\(93\) 1749.80 1.95103
\(94\) −84.0784 + 584.778i −0.0922556 + 0.641652i
\(95\) −99.5820 − 218.054i −0.107546 − 0.235494i
\(96\) −268.258 − 78.7676i −0.285197 − 0.0837415i
\(97\) 862.474 + 995.348i 0.902794 + 1.04188i 0.998918 + 0.0465100i \(0.0148099\pi\)
−0.0961242 + 0.995369i \(0.530645\pi\)
\(98\) −1072.69 + 314.970i −1.10570 + 0.324661i
\(99\) −1250.07 − 803.373i −1.26906 − 0.815576i
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.31.7 ✓ 70
23.3 even 11 inner 230.4.g.d.141.7 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.31.7 ✓ 70 1.1 even 1 trivial
230.4.g.d.141.7 yes 70 23.3 even 11 inner