Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,3,Mod(3,230)] 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("230.3"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(44)) chi = DirichletCharacter(H, H._module([33, 32])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 230.k (of order \(44\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [240,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.26704608029\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label 13.1
Character \(\chi\) \(=\) 230.13
Dual form 230.3.k.b.177.1

$q$-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+(1.41061 + 0.100889i) q^{2} +(-0.933043 - 4.28913i) q^{3} +(1.97964 + 0.284630i) q^{4} +(4.55004 - 2.07296i) q^{5} +(-0.883435 - 6.14443i) q^{6} +(1.77102 - 0.967050i) q^{7} +(2.76379 + 0.601225i) q^{8} +(-9.33939 + 4.26516i) q^{9} +(6.62747 - 2.46510i) q^{10} +(2.33945 - 2.69987i) q^{11} +(-0.626279 - 8.75652i) q^{12} +(-1.37371 - 0.750105i) q^{13} +(2.59578 - 1.18545i) q^{14} +(-13.1366 - 17.5815i) q^{15} +(3.83797 + 1.12693i) q^{16} +(11.6182 + 8.69724i) q^{17} +(-13.6045 + 5.07423i) q^{18} +(-31.3935 - 4.51370i) q^{19} +(9.59747 - 2.80865i) q^{20} +(-5.80024 - 6.69384i) q^{21} +(3.57244 - 3.57244i) q^{22} +(-17.0970 + 15.3848i) q^{23} -12.4152i q^{24} +(16.4056 - 18.8641i) q^{25} +(-1.86210 - 1.19670i) q^{26} +(3.33341 + 4.45292i) q^{27} +(3.78124 - 1.41033i) q^{28} +(0.538710 - 0.0774548i) q^{29} +(-16.7568 - 26.1260i) q^{30} +(28.5494 - 18.3476i) q^{31} +(5.30019 + 1.97687i) q^{32} +(-13.7629 - 7.51511i) q^{33} +(15.5112 + 13.4406i) q^{34} +(6.05355 - 8.07137i) q^{35} +(-19.7026 + 5.78522i) q^{36} +(35.7313 + 13.3271i) q^{37} +(-43.8286 - 9.53432i) q^{38} +(-1.93556 + 6.59192i) q^{39} +(13.8217 - 2.99364i) q^{40} +(5.05082 - 11.0598i) q^{41} +(-7.50655 - 10.0276i) q^{42} +(12.2732 + 56.4190i) q^{43} +(5.39974 - 4.67890i) q^{44} +(-33.6530 + 38.7668i) q^{45} +(-25.6694 + 19.9770i) q^{46} +(1.16669 - 1.16669i) q^{47} +(1.25256 - 17.5130i) q^{48} +(-24.2901 + 37.7961i) q^{49} +(25.0451 - 24.9548i) q^{50} +(26.4634 - 57.9467i) q^{51} +(-2.50596 - 1.87594i) q^{52} +(46.8004 + 85.7085i) q^{53} +(4.25289 + 6.61763i) q^{54} +(5.04785 - 17.1341i) q^{55} +(5.47614 - 1.60794i) q^{56} +(9.93162 + 138.862i) q^{57} +(0.767725 - 0.0549087i) q^{58} +(7.55612 + 25.7338i) q^{59} +(-21.0015 - 38.5442i) q^{60} +(33.7095 - 21.6638i) q^{61} +(42.1232 - 23.0010i) q^{62} +(-12.4156 + 16.5853i) q^{63} +(7.27706 + 3.32332i) q^{64} +(-7.80539 - 0.565343i) q^{65} +(-18.6559 - 11.9894i) q^{66} +(46.2482 + 3.30774i) q^{67} +(20.5243 + 20.5243i) q^{68} +(81.9396 + 58.9768i) q^{69} +(9.35351 - 10.7748i) q^{70} +(-14.2879 - 16.4891i) q^{71} +(-28.3764 + 6.17291i) q^{72} +(-47.8226 + 35.7996i) q^{73} +(49.0583 + 22.4042i) q^{74} +(-96.2178 - 52.7649i) q^{75} +(-60.8631 - 17.8710i) q^{76} +(1.53230 - 7.04389i) q^{77} +(-3.39538 + 9.10335i) q^{78} +(-27.9896 - 95.3240i) q^{79} +(19.7990 - 2.82840i) q^{80} +(-44.5235 + 51.3828i) q^{81} +(8.24055 - 15.0914i) q^{82} +(21.7650 - 58.3543i) q^{83} +(-9.57715 - 14.9023i) q^{84} +(70.8921 + 15.4887i) q^{85} +(11.6207 + 80.8234i) q^{86} +(-0.834854 - 2.23833i) q^{87} +(8.08897 - 6.05533i) q^{88} +(-20.4604 + 31.8370i) q^{89} +(-51.3825 + 51.2897i) q^{90} -3.15826 q^{91} +(-38.2250 + 25.5900i) q^{92} +(-105.333 - 105.333i) q^{93} +(1.76345 - 1.52803i) q^{94} +(-152.198 + 44.5400i) q^{95} +(3.53374 - 24.5777i) q^{96} +(12.2374 + 32.8097i) q^{97} +(-38.0770 + 50.8650i) q^{98} +(-10.3337 + 35.1932i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 24 q^{2} + 8 q^{3} - 4 q^{5} - 16 q^{6} + 50 q^{7} - 48 q^{8} + 16 q^{10} - 24 q^{11} - 28 q^{12} - 8 q^{13} + 8 q^{15} + 96 q^{16} + 44 q^{17} + 200 q^{18} - 24 q^{20} + 24 q^{21} + 24 q^{22} - 40 q^{23}+ \cdots - 776 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{7}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41061 + 0.100889i 0.705305 + 0.0504444i
\(3\) −0.933043 − 4.28913i −0.311014 − 1.42971i −0.821689 − 0.569936i \(-0.806968\pi\)
0.510674 − 0.859774i \(-0.329396\pi\)
\(4\) 1.97964 + 0.284630i 0.494911 + 0.0711574i
\(5\) 4.55004 − 2.07296i 0.910007 − 0.414593i
\(6\) −0.883435 − 6.14443i −0.147239 − 1.02407i
\(7\) 1.77102 − 0.967050i 0.253003 − 0.138150i −0.347743 − 0.937590i \(-0.613052\pi\)
0.600746 + 0.799440i \(0.294870\pi\)
\(8\) 2.76379 + 0.601225i 0.345474 + 0.0751532i
\(9\) −9.33939 + 4.26516i −1.03771 + 0.473906i
\(10\) 6.62747 − 2.46510i 0.662747 − 0.246510i
\(11\) 2.33945 − 2.69987i 0.212677 − 0.245443i −0.639380 − 0.768891i \(-0.720809\pi\)
0.852058 + 0.523448i \(0.175355\pi\)
\(12\) −0.626279 − 8.75652i −0.0521899 − 0.729710i
\(13\) −1.37371 − 0.750105i −0.105670 − 0.0577004i 0.425545 − 0.904937i \(-0.360083\pi\)
−0.531215 + 0.847237i \(0.678264\pi\)
\(14\) 2.59578 − 1.18545i 0.185413 − 0.0846754i
\(15\) −13.1366 − 17.5815i −0.875773 − 1.17210i
\(16\) 3.83797 + 1.12693i 0.239873 + 0.0704331i
\(17\) 11.6182 + 8.69724i 0.683421 + 0.511602i 0.883651 − 0.468147i \(-0.155078\pi\)
−0.200230 + 0.979749i \(0.564169\pi\)
\(18\) −13.6045 + 5.07423i −0.755808 + 0.281902i
\(19\) −31.3935 − 4.51370i −1.65229 − 0.237563i −0.747757 − 0.663972i \(-0.768869\pi\)
−0.904530 + 0.426409i \(0.859778\pi\)
\(20\) 9.59747 − 2.80865i 0.479874 − 0.140433i
\(21\) −5.80024 − 6.69384i −0.276202 − 0.318754i
\(22\) 3.57244 − 3.57244i 0.162384 − 0.162384i
\(23\) −17.0970 + 15.3848i −0.743350 + 0.668903i
\(24\) − 12.4152i − 0.517301i
\(25\) 16.4056 − 18.8641i 0.656226 − 0.754565i
\(26\) −1.86210 − 1.19670i −0.0716192 − 0.0460268i
\(27\) 3.33341 + 4.45292i 0.123460 + 0.164923i
\(28\) 3.78124 − 1.41033i 0.135044 − 0.0503689i
\(29\) 0.538710 − 0.0774548i 0.0185762 − 0.00267086i −0.133020 − 0.991113i \(-0.542468\pi\)
0.151597 + 0.988442i \(0.451559\pi\)
\(30\) −16.7568 − 26.1260i −0.558561 − 0.870868i
\(31\) 28.5494 − 18.3476i 0.920950 − 0.591859i 0.00801685 − 0.999968i \(-0.497448\pi\)
0.912933 + 0.408109i \(0.133812\pi\)
\(32\) 5.30019 + 1.97687i 0.165631 + 0.0617771i
\(33\) −13.7629 − 7.51511i −0.417058 − 0.227731i
\(34\) 15.5112 + 13.4406i 0.456213 + 0.395310i
\(35\) 6.05355 − 8.07137i 0.172958 − 0.230611i
\(36\) −19.7026 + 5.78522i −0.547296 + 0.160701i
\(37\) 35.7313 + 13.3271i 0.965710 + 0.360191i 0.782339 − 0.622853i \(-0.214026\pi\)
0.183371 + 0.983044i \(0.441299\pi\)
\(38\) −43.8286 − 9.53432i −1.15338 − 0.250903i
\(39\) −1.93556 + 6.59192i −0.0496298 + 0.169024i
\(40\) 13.8217 − 2.99364i 0.345541 − 0.0748409i
\(41\) 5.05082 − 11.0598i 0.123191 − 0.269750i −0.837982 − 0.545698i \(-0.816264\pi\)
0.961173 + 0.275948i \(0.0889918\pi\)
\(42\) −7.50655 − 10.0276i −0.178727 − 0.238752i
\(43\) 12.2732 + 56.4190i 0.285423 + 1.31207i 0.866890 + 0.498499i \(0.166115\pi\)
−0.581467 + 0.813570i \(0.697521\pi\)
\(44\) 5.39974 − 4.67890i 0.122721 − 0.106339i
\(45\) −33.6530 + 38.7668i −0.747845 + 0.861485i
\(46\) −25.6694 + 19.9770i −0.558031 + 0.434283i
\(47\) 1.16669 − 1.16669i 0.0248231 − 0.0248231i −0.694586 − 0.719409i \(-0.744413\pi\)
0.719409 + 0.694586i \(0.244413\pi\)
\(48\) 1.25256 − 17.5130i 0.0260949 − 0.364855i
\(49\) −24.2901 + 37.7961i −0.495716 + 0.771349i
\(50\) 25.0451 − 24.9548i 0.500903 − 0.499095i
\(51\) 26.4634 − 57.9467i 0.518889 − 1.13621i
\(52\) −2.50596 − 1.87594i −0.0481916 − 0.0360757i
\(53\) 46.8004 + 85.7085i 0.883026 + 1.61714i 0.782883 + 0.622168i \(0.213748\pi\)
0.100143 + 0.994973i \(0.468070\pi\)
\(54\) 4.25289 + 6.61763i 0.0787573 + 0.122549i
\(55\) 5.04785 − 17.1341i 0.0917791 − 0.311529i
\(56\) 5.47614 − 1.60794i 0.0977882 − 0.0287132i
\(57\) 9.93162 + 138.862i 0.174239 + 2.43618i
\(58\) 0.767725 − 0.0549087i 0.0132366 − 0.000946703i
\(59\) 7.55612 + 25.7338i 0.128070 + 0.436166i 0.998415 − 0.0562760i \(-0.0179227\pi\)
−0.870345 + 0.492442i \(0.836104\pi\)
\(60\) −21.0015 − 38.5442i −0.350026 − 0.642404i
\(61\) 33.7095 − 21.6638i 0.552615 − 0.355144i −0.234340 − 0.972155i \(-0.575293\pi\)
0.786955 + 0.617011i \(0.211656\pi\)
\(62\) 42.1232 − 23.0010i 0.679407 − 0.370984i
\(63\) −12.4156 + 16.5853i −0.197073 + 0.263259i
\(64\) 7.27706 + 3.32332i 0.113704 + 0.0519269i
\(65\) −7.80539 − 0.565343i −0.120083 − 0.00869759i
\(66\) −18.6559 − 11.9894i −0.282665 − 0.181658i
\(67\) 46.2482 + 3.30774i 0.690271 + 0.0493692i 0.412068 − 0.911153i \(-0.364807\pi\)
0.278203 + 0.960522i \(0.410261\pi\)
\(68\) 20.5243 + 20.5243i 0.301828 + 0.301828i
\(69\) 81.9396 + 58.9768i 1.18753 + 0.854736i
\(70\) 9.35351 − 10.7748i 0.133622 − 0.153926i
\(71\) −14.2879 − 16.4891i −0.201238 − 0.232241i 0.646156 − 0.763205i \(-0.276375\pi\)
−0.847394 + 0.530964i \(0.821830\pi\)
\(72\) −28.3764 + 6.17291i −0.394117 + 0.0857349i
\(73\) −47.8226 + 35.7996i −0.655105 + 0.490405i −0.874370 − 0.485260i \(-0.838725\pi\)
0.219265 + 0.975665i \(0.429634\pi\)
\(74\) 49.0583 + 22.4042i 0.662950 + 0.302759i
\(75\) −96.2178 − 52.7649i −1.28290 − 0.703532i
\(76\) −60.8631 − 17.8710i −0.800830 − 0.235145i
\(77\) 1.53230 − 7.04389i 0.0199001 − 0.0914791i
\(78\) −3.39538 + 9.10335i −0.0435305 + 0.116710i
\(79\) −27.9896 − 95.3240i −0.354299 − 1.20663i −0.923230 − 0.384247i \(-0.874461\pi\)
0.568931 − 0.822385i \(-0.307357\pi\)
\(80\) 19.7990 − 2.82840i 0.247487 − 0.0353551i
\(81\) −44.5235 + 51.3828i −0.549672 + 0.634356i
\(82\) 8.24055 − 15.0914i 0.100495 − 0.184042i
\(83\) 21.7650 − 58.3543i 0.262229 − 0.703064i −0.737390 − 0.675467i \(-0.763942\pi\)
0.999619 − 0.0275965i \(-0.00878536\pi\)
\(84\) −9.57715 − 14.9023i −0.114014 − 0.177409i
\(85\) 70.8921 + 15.4887i 0.834024 + 0.182220i
\(86\) 11.6207 + 80.8234i 0.135124 + 0.939807i
\(87\) −0.834854 − 2.23833i −0.00959602 − 0.0257279i
\(88\) 8.08897 − 6.05533i 0.0919202 − 0.0688106i
\(89\) −20.4604 + 31.8370i −0.229892 + 0.357719i −0.936966 − 0.349421i \(-0.886378\pi\)
0.707074 + 0.707139i \(0.250015\pi\)
\(90\) −51.3825 + 51.2897i −0.570916 + 0.569885i
\(91\) −3.15826 −0.0347062
\(92\) −38.2250 + 25.5900i −0.415489 + 0.278152i
\(93\) −105.333 − 105.333i −1.13262 − 1.13262i
\(94\) 1.76345 − 1.52803i 0.0187601 − 0.0162557i
\(95\) −152.198 + 44.5400i −1.60209 + 0.468842i
\(96\) 3.53374 − 24.5777i 0.0368098 − 0.256018i
\(97\) 12.2374 + 32.8097i 0.126158 + 0.338244i 0.984639 − 0.174602i \(-0.0558639\pi\)
−0.858481 + 0.512846i \(0.828591\pi\)
\(98\) −38.0770 + 50.8650i −0.388541 + 0.519030i
\(99\) −10.3337 + 35.1932i −0.104381 + 0.355487i
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 230.3.k.b.13.1 ✓ 240
5.2 odd 4 inner 230.3.k.b.197.1 yes 240
23.16 even 11 inner 230.3.k.b.223.1 yes 240
115.62 odd 44 inner 230.3.k.b.177.1 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.3.k.b.13.1 ✓ 240 1.1 even 1 trivial
230.3.k.b.177.1 yes 240 115.62 odd 44 inner
230.3.k.b.197.1 yes 240 5.2 odd 4 inner
230.3.k.b.223.1 yes 240 23.16 even 11 inner