Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [40,5,Mod(13,40)] 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("40.13"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 40.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.13479852335\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.2
Character \(\chi\) \(=\) 40.13
Dual form 40.5.i.a.37.2

$q$-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+(-3.98174 + 0.381759i) q^{2} +(-3.10315 + 3.10315i) q^{3} +(15.7085 - 3.04013i) q^{4} +(18.7844 - 16.4969i) q^{5} +(11.1713 - 13.5406i) q^{6} +(-52.9781 + 52.9781i) q^{7} +(-61.3867 + 18.1019i) q^{8} +61.7410i q^{9} +(-68.4966 + 72.8575i) q^{10} +95.9857i q^{11} +(-39.3118 + 58.1798i) q^{12} +(-23.7878 + 23.7878i) q^{13} +(190.720 - 231.170i) q^{14} +(-7.09830 + 109.483i) q^{15} +(237.515 - 95.5120i) q^{16} +(-187.913 + 187.913i) q^{17} +(-23.5702 - 245.837i) q^{18} -312.734 q^{19} +(244.922 - 316.249i) q^{20} -328.798i q^{21} +(-36.6434 - 382.190i) q^{22} +(658.173 + 658.173i) q^{23} +(134.319 - 246.665i) q^{24} +(80.7042 - 619.768i) q^{25} +(85.6356 - 103.798i) q^{26} +(-442.946 - 442.946i) q^{27} +(-671.147 + 993.269i) q^{28} +1301.35 q^{29} +(-13.5325 - 438.642i) q^{30} -215.144 q^{31} +(-909.261 + 470.978i) q^{32} +(-297.858 - 297.858i) q^{33} +(676.484 - 819.959i) q^{34} +(-121.185 + 1869.14i) q^{35} +(187.701 + 969.859i) q^{36} +(-886.034 - 886.034i) q^{37} +(1245.23 - 119.389i) q^{38} -147.634i q^{39} +(-854.484 + 1352.72i) q^{40} -1208.71 q^{41} +(125.522 + 1309.19i) q^{42} +(-413.820 + 413.820i) q^{43} +(291.809 + 1507.79i) q^{44} +(1018.54 + 1159.76i) q^{45} +(-2871.94 - 2369.41i) q^{46} +(738.634 - 738.634i) q^{47} +(-440.657 + 1033.43i) q^{48} -3212.36i q^{49} +(-84.7413 + 2498.56i) q^{50} -1166.24i q^{51} +(-301.353 + 445.989i) q^{52} +(2786.28 - 2786.28i) q^{53} +(1932.79 + 1594.60i) q^{54} +(1583.47 + 1803.03i) q^{55} +(2293.15 - 4211.15i) q^{56} +(970.459 - 970.459i) q^{57} +(-5181.66 + 496.804i) q^{58} -1389.86 q^{59} +(221.339 + 1741.39i) q^{60} +2794.25i q^{61} +(856.647 - 82.1331i) q^{62} +(-3270.92 - 3270.92i) q^{63} +(3440.64 - 2222.43i) q^{64} +(-54.4135 + 839.263i) q^{65} +(1299.70 + 1072.28i) q^{66} +(436.765 + 436.765i) q^{67} +(-2380.56 + 3523.12i) q^{68} -4084.81 q^{69} +(-231.033 - 7488.68i) q^{70} +1913.45 q^{71} +(-1117.63 - 3790.07i) q^{72} +(5030.71 + 5030.71i) q^{73} +(3866.21 + 3189.71i) q^{74} +(1672.79 + 2173.67i) q^{75} +(-4912.59 + 950.753i) q^{76} +(-5085.14 - 5085.14i) q^{77} +(56.3606 + 587.840i) q^{78} -1410.68i q^{79} +(2885.92 - 5712.40i) q^{80} -2251.97 q^{81} +(4812.76 - 461.435i) q^{82} +(7067.43 - 7067.43i) q^{83} +(-999.589 - 5164.92i) q^{84} +(-429.842 + 6629.81i) q^{85} +(1489.74 - 1805.70i) q^{86} +(-4038.29 + 4038.29i) q^{87} +(-1737.52 - 5892.24i) q^{88} +12805.8i q^{89} +(-4498.29 - 4229.05i) q^{90} -2520.47i q^{91} +(12339.9 + 8337.99i) q^{92} +(667.622 - 667.622i) q^{93} +(-2659.07 + 3223.03i) q^{94} +(-5874.51 + 5159.14i) q^{95} +(1360.06 - 4283.08i) q^{96} +(-1957.19 + 1957.19i) q^{97} +(1226.35 + 12790.8i) q^{98} -5926.25 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 2 q^{2} + 32 q^{6} - 4 q^{7} - 92 q^{8} - 154 q^{10} + 196 q^{12} - 4 q^{15} + 72 q^{16} + 236 q^{17} + 682 q^{18} - 904 q^{20} - 740 q^{22} - 4 q^{23} + 332 q^{25} - 2236 q^{26} + 2740 q^{28} + 3940 q^{30}+ \cdots - 65998 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.98174 + 0.381759i −0.995435 + 0.0954398i
\(3\) −3.10315 + 3.10315i −0.344794 + 0.344794i −0.858166 − 0.513372i \(-0.828396\pi\)
0.513372 + 0.858166i \(0.328396\pi\)
\(4\) 15.7085 − 3.04013i 0.981782 − 0.190008i
\(5\) 18.7844 − 16.4969i 0.751374 − 0.659876i
\(6\) 11.1713 − 13.5406i 0.310313 − 0.376127i
\(7\) −52.9781 + 52.9781i −1.08119 + 1.08119i −0.0847872 + 0.996399i \(0.527021\pi\)
−0.996399 + 0.0847872i \(0.972979\pi\)
\(8\) −61.3867 + 18.1019i −0.959166 + 0.282842i
\(9\) 61.7410i 0.762234i
\(10\) −68.4966 + 72.8575i −0.684966 + 0.728575i
\(11\) 95.9857i 0.793271i 0.917976 + 0.396635i \(0.129822\pi\)
−0.917976 + 0.396635i \(0.870178\pi\)
\(12\) −39.3118 + 58.1798i −0.272999 + 0.404026i
\(13\) −23.7878 + 23.7878i −0.140756 + 0.140756i −0.773974 − 0.633218i \(-0.781734\pi\)
0.633218 + 0.773974i \(0.281734\pi\)
\(14\) 190.720 − 231.170i 0.973063 − 1.17944i
\(15\) −7.09830 + 109.483i −0.0315480 + 0.486591i
\(16\) 237.515 − 95.5120i 0.927794 − 0.373094i
\(17\) −187.913 + 187.913i −0.650218 + 0.650218i −0.953045 − 0.302827i \(-0.902069\pi\)
0.302827 + 0.953045i \(0.402069\pi\)
\(18\) −23.5702 − 245.837i −0.0727475 − 0.758755i
\(19\) −312.734 −0.866299 −0.433150 − 0.901322i \(-0.642598\pi\)
−0.433150 + 0.901322i \(0.642598\pi\)
\(20\) 244.922 − 316.249i 0.612304 − 0.790622i
\(21\) − 328.798i − 0.745573i
\(22\) −36.6434 − 382.190i −0.0757096 − 0.789649i
\(23\) 658.173 + 658.173i 1.24418 + 1.24418i 0.958249 + 0.285934i \(0.0923038\pi\)
0.285934 + 0.958249i \(0.407696\pi\)
\(24\) 134.319 − 246.665i 0.233193 − 0.428237i
\(25\) 80.7042 − 619.768i 0.129127 − 0.991628i
\(26\) 85.6356 − 103.798i 0.126680 − 0.153547i
\(27\) −442.946 − 442.946i −0.607608 − 0.607608i
\(28\) −671.147 + 993.269i −0.856055 + 1.26692i
\(29\) 1301.35 1.54739 0.773695 − 0.633559i \(-0.218406\pi\)
0.773695 + 0.633559i \(0.218406\pi\)
\(30\) −13.5325 − 438.642i −0.0150361 − 0.487380i
\(31\) −215.144 −0.223875 −0.111937 − 0.993715i \(-0.535706\pi\)
−0.111937 + 0.993715i \(0.535706\pi\)
\(32\) −909.261 + 470.978i −0.887951 + 0.459939i
\(33\) −297.858 − 297.858i −0.273515 − 0.273515i
\(34\) 676.484 − 819.959i 0.585193 − 0.709307i
\(35\) −121.185 + 1869.14i −0.0989265 + 1.52582i
\(36\) 187.701 + 969.859i 0.144831 + 0.748348i
\(37\) −886.034 − 886.034i −0.647213 − 0.647213i 0.305106 − 0.952318i \(-0.401308\pi\)
−0.952318 + 0.305106i \(0.901308\pi\)
\(38\) 1245.23 − 119.389i 0.862345 − 0.0826795i
\(39\) − 147.634i − 0.0970637i
\(40\) −854.484 + 1352.72i −0.534052 + 0.845451i
\(41\) −1208.71 −0.719040 −0.359520 − 0.933137i \(-0.617060\pi\)
−0.359520 + 0.933137i \(0.617060\pi\)
\(42\) 125.522 + 1309.19i 0.0711573 + 0.742170i
\(43\) −413.820 + 413.820i −0.223807 + 0.223807i −0.810100 − 0.586292i \(-0.800587\pi\)
0.586292 + 0.810100i \(0.300587\pi\)
\(44\) 291.809 + 1507.79i 0.150728 + 0.778819i
\(45\) 1018.54 + 1159.76i 0.502980 + 0.572723i
\(46\) −2871.94 − 2369.41i −1.35725 − 1.11976i
\(47\) 738.634 − 738.634i 0.334375 − 0.334375i −0.519870 − 0.854245i \(-0.674020\pi\)
0.854245 + 0.519870i \(0.174020\pi\)
\(48\) −440.657 + 1033.43i −0.191257 + 0.448538i
\(49\) − 3212.36i − 1.33793i
\(50\) −84.7413 + 2498.56i −0.0338965 + 0.999425i
\(51\) − 1166.24i − 0.448383i
\(52\) −301.353 + 445.989i −0.111447 + 0.164937i
\(53\) 2786.28 − 2786.28i 0.991913 − 0.991913i −0.00805505 − 0.999968i \(-0.502564\pi\)
0.999968 + 0.00805505i \(0.00256403\pi\)
\(54\) 1932.79 + 1594.60i 0.662824 + 0.546844i
\(55\) 1583.47 + 1803.03i 0.523460 + 0.596043i
\(56\) 2293.15 − 4211.15i 0.731233 − 1.34284i
\(57\) 970.459 − 970.459i 0.298695 − 0.298695i
\(58\) −5181.66 + 496.804i −1.54033 + 0.147683i
\(59\) −1389.86 −0.399271 −0.199636 − 0.979870i \(-0.563976\pi\)
−0.199636 + 0.979870i \(0.563976\pi\)
\(60\) 221.339 + 1741.39i 0.0614830 + 0.483721i
\(61\) 2794.25i 0.750940i 0.926835 + 0.375470i \(0.122519\pi\)
−0.926835 + 0.375470i \(0.877481\pi\)
\(62\) 856.647 − 82.1331i 0.222853 − 0.0213666i
\(63\) −3270.92 − 3270.92i −0.824117 − 0.824117i
\(64\) 3440.64 − 2222.43i 0.840001 − 0.542585i
\(65\) −54.4135 + 839.263i −0.0128789 + 0.198642i
\(66\) 1299.70 + 1072.28i 0.298371 + 0.246162i
\(67\) 436.765 + 436.765i 0.0972968 + 0.0972968i 0.754080 − 0.656783i \(-0.228083\pi\)
−0.656783 + 0.754080i \(0.728083\pi\)
\(68\) −2380.56 + 3523.12i −0.514826 + 0.761920i
\(69\) −4084.81 −0.857974
\(70\) −231.033 − 7488.68i −0.0471495 − 1.52830i
\(71\) 1913.45 0.379578 0.189789 − 0.981825i \(-0.439220\pi\)
0.189789 + 0.981825i \(0.439220\pi\)
\(72\) −1117.63 − 3790.07i −0.215592 − 0.731110i
\(73\) 5030.71 + 5030.71i 0.944025 + 0.944025i 0.998514 − 0.0544892i \(-0.0173530\pi\)
−0.0544892 + 0.998514i \(0.517353\pi\)
\(74\) 3866.21 + 3189.71i 0.706028 + 0.582488i
\(75\) 1672.79 + 2173.67i 0.297385 + 0.386429i
\(76\) −4912.59 + 950.753i −0.850518 + 0.164604i
\(77\) −5085.14 − 5085.14i −0.857673 − 0.857673i
\(78\) 56.3606 + 587.840i 0.00926374 + 0.0966207i
\(79\) − 1410.68i − 0.226034i −0.993593 − 0.113017i \(-0.963949\pi\)
0.993593 − 0.113017i \(-0.0360514\pi\)
\(80\) 2885.92 − 5712.40i 0.450925 − 0.892562i
\(81\) −2251.97 −0.343235
\(82\) 4812.76 − 461.435i 0.715758 − 0.0686251i
\(83\) 7067.43 − 7067.43i 1.02590 − 1.02590i 0.0262455 − 0.999656i \(-0.491645\pi\)
0.999656 − 0.0262455i \(-0.00835517\pi\)
\(84\) −999.589 − 5164.92i −0.141665 − 0.731990i
\(85\) −429.842 + 6629.81i −0.0594937 + 0.917621i
\(86\) 1489.74 − 1805.70i 0.201426 − 0.244146i
\(87\) −4038.29 + 4038.29i −0.533530 + 0.533530i
\(88\) −1737.52 − 5892.24i −0.224370 − 0.760879i
\(89\) 12805.8i 1.61669i 0.588710 + 0.808345i \(0.299636\pi\)
−0.588710 + 0.808345i \(0.700364\pi\)
\(90\) −4498.29 − 4229.05i −0.555345 − 0.522105i
\(91\) − 2520.47i − 0.304367i
\(92\) 12339.9 + 8337.99i 1.45792 + 0.985112i
\(93\) 667.622 − 667.622i 0.0771907 − 0.0771907i
\(94\) −2659.07 + 3223.03i −0.300936 + 0.364761i
\(95\) −5874.51 + 5159.14i −0.650915 + 0.571650i
\(96\) 1360.06 − 4283.08i 0.147576 − 0.464744i
\(97\) −1957.19 + 1957.19i −0.208013 + 0.208013i −0.803422 − 0.595409i \(-0.796990\pi\)
0.595409 + 0.803422i \(0.296990\pi\)
\(98\) 1226.35 + 12790.8i 0.127692 + 1.33182i
\(99\) −5926.25 −0.604658
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 40.5.i.a.13.2 ✓ 44
4.3 odd 2 160.5.m.a.113.14 44
5.2 odd 4 inner 40.5.i.a.37.11 yes 44
8.3 odd 2 160.5.m.a.113.9 44
8.5 even 2 inner 40.5.i.a.13.11 yes 44
20.7 even 4 160.5.m.a.17.9 44
40.27 even 4 160.5.m.a.17.14 44
40.37 odd 4 inner 40.5.i.a.37.2 yes 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.i.a.13.2 ✓ 44 1.1 even 1 trivial
40.5.i.a.13.11 yes 44 8.5 even 2 inner
40.5.i.a.37.2 yes 44 40.37 odd 4 inner
40.5.i.a.37.11 yes 44 5.2 odd 4 inner
160.5.m.a.17.9 44 20.7 even 4
160.5.m.a.17.14 44 40.27 even 4
160.5.m.a.113.9 44 8.3 odd 2
160.5.m.a.113.14 44 4.3 odd 2