Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [33,2,Mod(4,33)] 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("33.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(33, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 33.e (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-3] 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: \(0.263506326670\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 16.1
Root \(-0.309017 - 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 33.16
Dual form 33.2.e.a.31.1

$q$-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+(-0.190983 - 0.587785i) q^{2} +(-0.809017 - 0.587785i) q^{3} +(1.30902 - 0.951057i) q^{4} +(-0.809017 + 2.48990i) q^{5} +(-0.190983 + 0.587785i) q^{6} +(-2.42705 + 1.76336i) q^{7} +(-1.80902 - 1.31433i) q^{8} +(0.309017 + 0.951057i) q^{9} +1.61803 q^{10} +(1.69098 - 2.85317i) q^{11} -1.61803 q^{12} +(0.545085 + 1.67760i) q^{13} +(1.50000 + 1.08981i) q^{14} +(2.11803 - 1.53884i) q^{15} +(0.572949 - 1.76336i) q^{16} +(0.500000 - 1.53884i) q^{17} +(0.500000 - 0.363271i) q^{18} +(-4.73607 - 3.44095i) q^{19} +(1.30902 + 4.02874i) q^{20} +3.00000 q^{21} +(-2.00000 - 0.449028i) q^{22} +3.47214 q^{23} +(0.690983 + 2.12663i) q^{24} +(-1.50000 - 1.08981i) q^{25} +(0.881966 - 0.640786i) q^{26} +(0.309017 - 0.951057i) q^{27} +(-1.50000 + 4.61653i) q^{28} +(-3.61803 + 2.62866i) q^{29} +(-1.30902 - 0.951057i) q^{30} +(0.881966 + 2.71441i) q^{31} -5.61803 q^{32} +(-3.04508 + 1.31433i) q^{33} -1.00000 q^{34} +(-2.42705 - 7.46969i) q^{35} +(1.30902 + 0.951057i) q^{36} +(-0.190983 + 0.138757i) q^{37} +(-1.11803 + 3.44095i) q^{38} +(0.545085 - 1.67760i) q^{39} +(4.73607 - 3.44095i) q^{40} +(9.66312 + 7.02067i) q^{41} +(-0.572949 - 1.76336i) q^{42} +6.23607 q^{43} +(-0.500000 - 5.34307i) q^{44} -2.61803 q^{45} +(-0.663119 - 2.04087i) q^{46} +(-1.30902 - 0.951057i) q^{47} +(-1.50000 + 1.08981i) q^{48} +(0.618034 - 1.90211i) q^{49} +(-0.354102 + 1.08981i) q^{50} +(-1.30902 + 0.951057i) q^{51} +(2.30902 + 1.67760i) q^{52} +(-2.97214 - 9.14729i) q^{53} -0.618034 q^{54} +(5.73607 + 6.51864i) q^{55} +6.70820 q^{56} +(1.80902 + 5.56758i) q^{57} +(2.23607 + 1.62460i) q^{58} +(-8.35410 + 6.06961i) q^{59} +(1.30902 - 4.02874i) q^{60} +(2.42705 - 7.46969i) q^{61} +(1.42705 - 1.03681i) q^{62} +(-2.42705 - 1.76336i) q^{63} +(-0.0729490 - 0.224514i) q^{64} -4.61803 q^{65} +(1.35410 + 1.53884i) q^{66} -9.56231 q^{67} +(-0.809017 - 2.48990i) q^{68} +(-2.80902 - 2.04087i) q^{69} +(-3.92705 + 2.85317i) q^{70} +(-1.71885 + 5.29007i) q^{71} +(0.690983 - 2.12663i) q^{72} +(2.61803 - 1.90211i) q^{73} +(0.118034 + 0.0857567i) q^{74} +(0.572949 + 1.76336i) q^{75} -9.47214 q^{76} +(0.927051 + 9.90659i) q^{77} -1.09017 q^{78} +(2.92705 + 9.00854i) q^{79} +(3.92705 + 2.85317i) q^{80} +(-0.809017 + 0.587785i) q^{81} +(2.28115 - 7.02067i) q^{82} +(0.218847 - 0.673542i) q^{83} +(3.92705 - 2.85317i) q^{84} +(3.42705 + 2.48990i) q^{85} +(-1.19098 - 3.66547i) q^{86} +4.47214 q^{87} +(-6.80902 + 2.93893i) q^{88} +0.527864 q^{89} +(0.500000 + 1.53884i) q^{90} +(-4.28115 - 3.11044i) q^{91} +(4.54508 - 3.30220i) q^{92} +(0.881966 - 2.71441i) q^{93} +(-0.309017 + 0.951057i) q^{94} +(12.3992 - 9.00854i) q^{95} +(4.54508 + 3.30220i) q^{96} +(-4.33688 - 13.3475i) q^{97} -1.23607 q^{98} +(3.23607 + 0.726543i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - 3 q^{6} - 3 q^{7} - 5 q^{8} - q^{9} + 2 q^{10} + 9 q^{11} - 2 q^{12} - 9 q^{13} + 6 q^{14} + 4 q^{15} + 9 q^{16} + 2 q^{17} + 2 q^{18} - 10 q^{19} + 3 q^{20}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(e\left(\frac{2}{5}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.190983 − 0.587785i −0.135045 − 0.415627i 0.860552 − 0.509363i \(-0.170119\pi\)
−0.995597 + 0.0937362i \(0.970119\pi\)
\(3\) −0.809017 − 0.587785i −0.467086 − 0.339358i
\(4\) 1.30902 − 0.951057i 0.654508 − 0.475528i
\(5\) −0.809017 + 2.48990i −0.361803 + 1.11352i 0.590155 + 0.807290i \(0.299067\pi\)
−0.951959 + 0.306227i \(0.900933\pi\)
\(6\) −0.190983 + 0.587785i −0.0779685 + 0.239962i
\(7\) −2.42705 + 1.76336i −0.917339 + 0.666486i −0.942860 − 0.333188i \(-0.891875\pi\)
0.0255212 + 0.999674i \(0.491875\pi\)
\(8\) −1.80902 − 1.31433i −0.639584 − 0.464685i
\(9\) 0.309017 + 0.951057i 0.103006 + 0.317019i
\(10\) 1.61803 0.511667
\(11\) 1.69098 − 2.85317i 0.509851 − 0.860263i
\(12\) −1.61803 −0.467086
\(13\) 0.545085 + 1.67760i 0.151179 + 0.465282i 0.997754 − 0.0669881i \(-0.0213390\pi\)
−0.846574 + 0.532270i \(0.821339\pi\)
\(14\) 1.50000 + 1.08981i 0.400892 + 0.291265i
\(15\) 2.11803 − 1.53884i 0.546874 − 0.397327i
\(16\) 0.572949 − 1.76336i 0.143237 − 0.440839i
\(17\) 0.500000 − 1.53884i 0.121268 − 0.373224i −0.871935 − 0.489622i \(-0.837135\pi\)
0.993203 + 0.116398i \(0.0371348\pi\)
\(18\) 0.500000 − 0.363271i 0.117851 − 0.0856239i
\(19\) −4.73607 − 3.44095i −1.08653 − 0.789409i −0.107719 − 0.994181i \(-0.534355\pi\)
−0.978810 + 0.204772i \(0.934355\pi\)
\(20\) 1.30902 + 4.02874i 0.292705 + 0.900854i
\(21\) 3.00000 0.654654
\(22\) −2.00000 − 0.449028i −0.426401 − 0.0957331i
\(23\) 3.47214 0.723990 0.361995 − 0.932180i \(-0.382096\pi\)
0.361995 + 0.932180i \(0.382096\pi\)
\(24\) 0.690983 + 2.12663i 0.141046 + 0.434096i
\(25\) −1.50000 − 1.08981i −0.300000 − 0.217963i
\(26\) 0.881966 − 0.640786i 0.172968 − 0.125668i
\(27\) 0.309017 − 0.951057i 0.0594703 − 0.183031i
\(28\) −1.50000 + 4.61653i −0.283473 + 0.872441i
\(29\) −3.61803 + 2.62866i −0.671852 + 0.488129i −0.870645 − 0.491912i \(-0.836298\pi\)
0.198793 + 0.980042i \(0.436298\pi\)
\(30\) −1.30902 − 0.951057i −0.238993 − 0.173638i
\(31\) 0.881966 + 2.71441i 0.158406 + 0.487523i 0.998490 − 0.0549331i \(-0.0174946\pi\)
−0.840084 + 0.542456i \(0.817495\pi\)
\(32\) −5.61803 −0.993137
\(33\) −3.04508 + 1.31433i −0.530081 + 0.228795i
\(34\) −1.00000 −0.171499
\(35\) −2.42705 − 7.46969i −0.410246 − 1.26261i
\(36\) 1.30902 + 0.951057i 0.218169 + 0.158509i
\(37\) −0.190983 + 0.138757i −0.0313974 + 0.0228116i −0.603373 − 0.797459i \(-0.706177\pi\)
0.571976 + 0.820270i \(0.306177\pi\)
\(38\) −1.11803 + 3.44095i −0.181369 + 0.558197i
\(39\) 0.545085 − 1.67760i 0.0872835 − 0.268631i
\(40\) 4.73607 − 3.44095i 0.748838 − 0.544063i
\(41\) 9.66312 + 7.02067i 1.50913 + 1.09644i 0.966563 + 0.256428i \(0.0825458\pi\)
0.542562 + 0.840015i \(0.317454\pi\)
\(42\) −0.572949 − 1.76336i −0.0884080 − 0.272092i
\(43\) 6.23607 0.950991 0.475496 − 0.879718i \(-0.342269\pi\)
0.475496 + 0.879718i \(0.342269\pi\)
\(44\) −0.500000 − 5.34307i −0.0753778 − 0.805498i
\(45\) −2.61803 −0.390273
\(46\) −0.663119 − 2.04087i −0.0977716 − 0.300910i
\(47\) −1.30902 − 0.951057i −0.190940 − 0.138726i 0.488208 − 0.872727i \(-0.337651\pi\)
−0.679148 + 0.734001i \(0.737651\pi\)
\(48\) −1.50000 + 1.08981i −0.216506 + 0.157301i
\(49\) 0.618034 − 1.90211i 0.0882906 − 0.271730i
\(50\) −0.354102 + 1.08981i −0.0500776 + 0.154123i
\(51\) −1.30902 + 0.951057i −0.183299 + 0.133175i
\(52\) 2.30902 + 1.67760i 0.320203 + 0.232641i
\(53\) −2.97214 − 9.14729i −0.408254 − 1.25648i −0.918147 − 0.396240i \(-0.870315\pi\)
0.509893 − 0.860238i \(-0.329685\pi\)
\(54\) −0.618034 −0.0841038
\(55\) 5.73607 + 6.51864i 0.773451 + 0.878973i
\(56\) 6.70820 0.896421
\(57\) 1.80902 + 5.56758i 0.239610 + 0.737444i
\(58\) 2.23607 + 1.62460i 0.293610 + 0.213320i
\(59\) −8.35410 + 6.06961i −1.08761 + 0.790196i −0.978994 − 0.203888i \(-0.934642\pi\)
−0.108617 + 0.994084i \(0.534642\pi\)
\(60\) 1.30902 − 4.02874i 0.168993 − 0.520108i
\(61\) 2.42705 − 7.46969i 0.310752 − 0.956396i −0.666716 − 0.745312i \(-0.732301\pi\)
0.977468 − 0.211084i \(-0.0676995\pi\)
\(62\) 1.42705 − 1.03681i 0.181236 − 0.131675i
\(63\) −2.42705 − 1.76336i −0.305780 − 0.222162i
\(64\) −0.0729490 − 0.224514i −0.00911863 − 0.0280642i
\(65\) −4.61803 −0.572797
\(66\) 1.35410 + 1.53884i 0.166678 + 0.189418i
\(67\) −9.56231 −1.16822 −0.584111 − 0.811674i \(-0.698557\pi\)
−0.584111 + 0.811674i \(0.698557\pi\)
\(68\) −0.809017 − 2.48990i −0.0981077 − 0.301945i
\(69\) −2.80902 − 2.04087i −0.338166 − 0.245692i
\(70\) −3.92705 + 2.85317i −0.469372 + 0.341019i
\(71\) −1.71885 + 5.29007i −0.203990 + 0.627815i 0.795764 + 0.605607i \(0.207070\pi\)
−0.999753 + 0.0222083i \(0.992930\pi\)
\(72\) 0.690983 − 2.12663i 0.0814331 − 0.250625i
\(73\) 2.61803 − 1.90211i 0.306418 − 0.222625i −0.423940 − 0.905690i \(-0.639353\pi\)
0.730358 + 0.683065i \(0.239353\pi\)
\(74\) 0.118034 + 0.0857567i 0.0137212 + 0.00996902i
\(75\) 0.572949 + 1.76336i 0.0661585 + 0.203615i
\(76\) −9.47214 −1.08653
\(77\) 0.927051 + 9.90659i 0.105647 + 1.12896i
\(78\) −1.09017 −0.123437
\(79\) 2.92705 + 9.00854i 0.329319 + 1.01354i 0.969453 + 0.245276i \(0.0788787\pi\)
−0.640134 + 0.768263i \(0.721121\pi\)
\(80\) 3.92705 + 2.85317i 0.439058 + 0.318994i
\(81\) −0.809017 + 0.587785i −0.0898908 + 0.0653095i
\(82\) 2.28115 − 7.02067i 0.251911 − 0.775303i
\(83\) 0.218847 − 0.673542i 0.0240216 − 0.0739308i −0.938327 − 0.345749i \(-0.887625\pi\)
0.962349 + 0.271818i \(0.0876249\pi\)
\(84\) 3.92705 − 2.85317i 0.428476 − 0.311306i
\(85\) 3.42705 + 2.48990i 0.371716 + 0.270067i
\(86\) −1.19098 − 3.66547i −0.128427 − 0.395258i
\(87\) 4.47214 0.479463
\(88\) −6.80902 + 2.93893i −0.725844 + 0.313291i
\(89\) 0.527864 0.0559535 0.0279767 − 0.999609i \(-0.491094\pi\)
0.0279767 + 0.999609i \(0.491094\pi\)
\(90\) 0.500000 + 1.53884i 0.0527046 + 0.162208i
\(91\) −4.28115 − 3.11044i −0.448787 − 0.326063i
\(92\) 4.54508 − 3.30220i 0.473858 − 0.344278i
\(93\) 0.881966 − 2.71441i 0.0914556 − 0.281471i
\(94\) −0.309017 + 0.951057i −0.0318727 + 0.0980940i
\(95\) 12.3992 − 9.00854i 1.27213 − 0.924256i
\(96\) 4.54508 + 3.30220i 0.463881 + 0.337029i
\(97\) −4.33688 − 13.3475i −0.440344 − 1.35524i −0.887510 − 0.460788i \(-0.847567\pi\)
0.447167 − 0.894451i \(-0.352433\pi\)
\(98\) −1.23607 −0.124862
\(99\) 3.23607 + 0.726543i 0.325237 + 0.0730203i
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 33.2.e.a.16.1 ✓ 4
3.2 odd 2 99.2.f.b.82.1 4
4.3 odd 2 528.2.y.f.49.1 4
5.2 odd 4 825.2.bx.b.49.2 8
5.3 odd 4 825.2.bx.b.49.1 8
5.4 even 2 825.2.n.f.676.1 4
9.2 odd 6 891.2.n.a.676.1 8
9.4 even 3 891.2.n.d.379.1 8
9.5 odd 6 891.2.n.a.379.1 8
9.7 even 3 891.2.n.d.676.1 8
11.2 odd 10 363.2.e.j.130.1 4
11.3 even 5 363.2.a.h.1.1 2
11.4 even 5 363.2.e.h.124.1 4
11.5 even 5 363.2.e.h.202.1 4
11.6 odd 10 363.2.e.c.202.1 4
11.7 odd 10 363.2.e.c.124.1 4
11.8 odd 10 363.2.a.e.1.2 2
11.9 even 5 inner 33.2.e.a.31.1 yes 4
11.10 odd 2 363.2.e.j.148.1 4
33.8 even 10 1089.2.a.s.1.1 2
33.14 odd 10 1089.2.a.m.1.2 2
33.20 odd 10 99.2.f.b.64.1 4
44.3 odd 10 5808.2.a.bl.1.1 2
44.19 even 10 5808.2.a.bm.1.1 2
44.31 odd 10 528.2.y.f.97.1 4
55.9 even 10 825.2.n.f.526.1 4
55.14 even 10 9075.2.a.x.1.2 2
55.19 odd 10 9075.2.a.bv.1.1 2
55.42 odd 20 825.2.bx.b.724.1 8
55.53 odd 20 825.2.bx.b.724.2 8
99.20 odd 30 891.2.n.a.757.1 8
99.31 even 15 891.2.n.d.460.1 8
99.86 odd 30 891.2.n.a.460.1 8
99.97 even 15 891.2.n.d.757.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.a.16.1 ✓ 4 1.1 even 1 trivial
33.2.e.a.31.1 yes 4 11.9 even 5 inner
99.2.f.b.64.1 4 33.20 odd 10
99.2.f.b.82.1 4 3.2 odd 2
363.2.a.e.1.2 2 11.8 odd 10
363.2.a.h.1.1 2 11.3 even 5
363.2.e.c.124.1 4 11.7 odd 10
363.2.e.c.202.1 4 11.6 odd 10
363.2.e.h.124.1 4 11.4 even 5
363.2.e.h.202.1 4 11.5 even 5
363.2.e.j.130.1 4 11.2 odd 10
363.2.e.j.148.1 4 11.10 odd 2
528.2.y.f.49.1 4 4.3 odd 2
528.2.y.f.97.1 4 44.31 odd 10
825.2.n.f.526.1 4 55.9 even 10
825.2.n.f.676.1 4 5.4 even 2
825.2.bx.b.49.1 8 5.3 odd 4
825.2.bx.b.49.2 8 5.2 odd 4
825.2.bx.b.724.1 8 55.42 odd 20
825.2.bx.b.724.2 8 55.53 odd 20
891.2.n.a.379.1 8 9.5 odd 6
891.2.n.a.460.1 8 99.86 odd 30
891.2.n.a.676.1 8 9.2 odd 6
891.2.n.a.757.1 8 99.20 odd 30
891.2.n.d.379.1 8 9.4 even 3
891.2.n.d.460.1 8 99.31 even 15
891.2.n.d.676.1 8 9.7 even 3
891.2.n.d.757.1 8 99.97 even 15
1089.2.a.m.1.2 2 33.14 odd 10
1089.2.a.s.1.1 2 33.8 even 10
5808.2.a.bl.1.1 2 44.3 odd 10
5808.2.a.bm.1.1 2 44.19 even 10
9075.2.a.x.1.2 2 55.14 even 10
9075.2.a.bv.1.1 2 55.19 odd 10