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Properties

Label 33.2.e.b.25.1
Level $33$
Weight $2$
Character 33.25
Analytic conductor $0.264$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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,-1] 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 25.1
Root \(0.809017 - 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 33.25
Dual form 33.2.e.b.4.1

$q$-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+(0.309017 - 0.224514i) q^{2} +(-0.309017 - 0.951057i) q^{3} +(-0.572949 + 1.76336i) q^{4} +(-1.30902 - 0.951057i) q^{5} +(-0.309017 - 0.224514i) q^{6} +(-0.309017 + 0.951057i) q^{7} +(0.454915 + 1.40008i) q^{8} +(-0.809017 + 0.587785i) q^{9} -0.618034 q^{10} +(-2.19098 - 2.48990i) q^{11} +1.85410 q^{12} +(3.42705 - 2.48990i) q^{13} +(0.118034 + 0.363271i) q^{14} +(-0.500000 + 1.53884i) q^{15} +(-2.54508 - 1.84911i) q^{16} +(6.35410 + 4.61653i) q^{17} +(-0.118034 + 0.363271i) q^{18} +(-0.263932 - 0.812299i) q^{19} +(2.42705 - 1.76336i) q^{20} +1.00000 q^{21} +(-1.23607 - 0.277515i) q^{22} -4.23607 q^{23} +(1.19098 - 0.865300i) q^{24} +(-0.736068 - 2.26538i) q^{25} +(0.500000 - 1.53884i) q^{26} +(0.809017 + 0.587785i) q^{27} +(-1.50000 - 1.08981i) q^{28} +(-1.85410 + 5.70634i) q^{29} +(0.190983 + 0.587785i) q^{30} +(-4.11803 + 2.99193i) q^{31} -4.14590 q^{32} +(-1.69098 + 2.85317i) q^{33} +3.00000 q^{34} +(1.30902 - 0.951057i) q^{35} +(-0.572949 - 1.76336i) q^{36} +(-0.545085 + 1.67760i) q^{37} +(-0.263932 - 0.191758i) q^{38} +(-3.42705 - 2.48990i) q^{39} +(0.736068 - 2.26538i) q^{40} +(-1.30902 - 4.02874i) q^{41} +(0.309017 - 0.224514i) q^{42} +6.70820 q^{43} +(5.64590 - 2.43690i) q^{44} +1.61803 q^{45} +(-1.30902 + 0.951057i) q^{46} +(0.336881 + 1.03681i) q^{47} +(-0.972136 + 2.99193i) q^{48} +(4.85410 + 3.52671i) q^{49} +(-0.736068 - 0.534785i) q^{50} +(2.42705 - 7.46969i) q^{51} +(2.42705 + 7.46969i) q^{52} +(2.11803 - 1.53884i) q^{53} +0.381966 q^{54} +(0.500000 + 5.34307i) q^{55} -1.47214 q^{56} +(-0.690983 + 0.502029i) q^{57} +(0.708204 + 2.17963i) q^{58} +(2.97214 - 9.14729i) q^{59} +(-2.42705 - 1.76336i) q^{60} +(-6.92705 - 5.03280i) q^{61} +(-0.600813 + 1.84911i) q^{62} +(-0.309017 - 0.951057i) q^{63} +(3.80902 - 2.76741i) q^{64} -6.85410 q^{65} +(0.118034 + 1.26133i) q^{66} -4.85410 q^{67} +(-11.7812 + 8.55951i) q^{68} +(1.30902 + 4.02874i) q^{69} +(0.190983 - 0.587785i) q^{70} +(4.30902 + 3.13068i) q^{71} +(-1.19098 - 0.865300i) q^{72} +(2.38197 - 7.33094i) q^{73} +(0.208204 + 0.640786i) q^{74} +(-1.92705 + 1.40008i) q^{75} +1.58359 q^{76} +(3.04508 - 1.31433i) q^{77} -1.61803 q^{78} +(-8.89919 + 6.46564i) q^{79} +(1.57295 + 4.84104i) q^{80} +(0.309017 - 0.951057i) q^{81} +(-1.30902 - 0.951057i) q^{82} +(6.04508 + 4.39201i) q^{83} +(-0.572949 + 1.76336i) q^{84} +(-3.92705 - 12.0862i) q^{85} +(2.07295 - 1.50609i) q^{86} +6.00000 q^{87} +(2.48936 - 4.20025i) q^{88} -3.76393 q^{89} +(0.500000 - 0.363271i) q^{90} +(1.30902 + 4.02874i) q^{91} +(2.42705 - 7.46969i) q^{92} +(4.11803 + 2.99193i) q^{93} +(0.336881 + 0.244758i) q^{94} +(-0.427051 + 1.31433i) q^{95} +(1.28115 + 3.94298i) q^{96} +(-0.927051 + 0.673542i) q^{97} +2.29180 q^{98} +(3.23607 + 0.726543i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + q^{3} - 9 q^{4} - 3 q^{5} + q^{6} + q^{7} + 13 q^{8} - q^{9} + 2 q^{10} - 11 q^{11} - 6 q^{12} + 7 q^{13} - 4 q^{14} - 2 q^{15} + q^{16} + 12 q^{17} + 4 q^{18} - 10 q^{19} + 3 q^{20}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(13\) \(23\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.309017 − 0.224514i 0.218508 − 0.158755i −0.473147 − 0.880984i \(-0.656882\pi\)
0.691655 + 0.722228i \(0.256882\pi\)
\(3\) −0.309017 − 0.951057i −0.178411 − 0.549093i
\(4\) −0.572949 + 1.76336i −0.286475 + 0.881678i
\(5\) −1.30902 − 0.951057i −0.585410 − 0.425325i 0.255260 − 0.966872i \(-0.417839\pi\)
−0.840670 + 0.541547i \(0.817839\pi\)
\(6\) −0.309017 − 0.224514i −0.126156 − 0.0916575i
\(7\) −0.309017 + 0.951057i −0.116797 + 0.359466i −0.992318 − 0.123716i \(-0.960519\pi\)
0.875520 + 0.483181i \(0.160519\pi\)
\(8\) 0.454915 + 1.40008i 0.160837 + 0.495005i
\(9\) −0.809017 + 0.587785i −0.269672 + 0.195928i
\(10\) −0.618034 −0.195440
\(11\) −2.19098 − 2.48990i −0.660606 − 0.750733i
\(12\) 1.85410 0.535233
\(13\) 3.42705 − 2.48990i 0.950493 − 0.690574i −0.000430477 − 1.00000i \(-0.500137\pi\)
0.950923 + 0.309426i \(0.100137\pi\)
\(14\) 0.118034 + 0.363271i 0.0315459 + 0.0970883i
\(15\) −0.500000 + 1.53884i −0.129099 + 0.397327i
\(16\) −2.54508 − 1.84911i −0.636271 − 0.462278i
\(17\) 6.35410 + 4.61653i 1.54110 + 1.11967i 0.949644 + 0.313332i \(0.101445\pi\)
0.591452 + 0.806340i \(0.298555\pi\)
\(18\) −0.118034 + 0.363271i −0.0278209 + 0.0856239i
\(19\) −0.263932 − 0.812299i −0.0605502 − 0.186354i 0.916206 − 0.400707i \(-0.131236\pi\)
−0.976756 + 0.214353i \(0.931236\pi\)
\(20\) 2.42705 − 1.76336i 0.542705 − 0.394298i
\(21\) 1.00000 0.218218
\(22\) −1.23607 − 0.277515i −0.263531 − 0.0591663i
\(23\) −4.23607 −0.883281 −0.441641 − 0.897192i \(-0.645603\pi\)
−0.441641 + 0.897192i \(0.645603\pi\)
\(24\) 1.19098 − 0.865300i 0.243108 − 0.176629i
\(25\) −0.736068 − 2.26538i −0.147214 − 0.453077i
\(26\) 0.500000 − 1.53884i 0.0980581 − 0.301792i
\(27\) 0.809017 + 0.587785i 0.155695 + 0.113119i
\(28\) −1.50000 − 1.08981i −0.283473 − 0.205955i
\(29\) −1.85410 + 5.70634i −0.344298 + 1.05964i 0.617660 + 0.786445i \(0.288081\pi\)
−0.961958 + 0.273196i \(0.911919\pi\)
\(30\) 0.190983 + 0.587785i 0.0348686 + 0.107314i
\(31\) −4.11803 + 2.99193i −0.739621 + 0.537366i −0.892592 − 0.450865i \(-0.851116\pi\)
0.152972 + 0.988231i \(0.451116\pi\)
\(32\) −4.14590 −0.732898
\(33\) −1.69098 + 2.85317i −0.294362 + 0.496673i
\(34\) 3.00000 0.514496
\(35\) 1.30902 − 0.951057i 0.221264 − 0.160758i
\(36\) −0.572949 − 1.76336i −0.0954915 − 0.293893i
\(37\) −0.545085 + 1.67760i −0.0896114 + 0.275796i −0.985812 − 0.167854i \(-0.946316\pi\)
0.896201 + 0.443649i \(0.146316\pi\)
\(38\) −0.263932 − 0.191758i −0.0428154 − 0.0311072i
\(39\) −3.42705 − 2.48990i −0.548767 − 0.398703i
\(40\) 0.736068 − 2.26538i 0.116383 − 0.358189i
\(41\) −1.30902 − 4.02874i −0.204434 − 0.629183i −0.999736 − 0.0229701i \(-0.992688\pi\)
0.795302 − 0.606213i \(-0.207312\pi\)
\(42\) 0.309017 − 0.224514i 0.0476824 − 0.0346433i
\(43\) 6.70820 1.02299 0.511496 − 0.859286i \(-0.329092\pi\)
0.511496 + 0.859286i \(0.329092\pi\)
\(44\) 5.64590 − 2.43690i 0.851151 − 0.367376i
\(45\) 1.61803 0.241202
\(46\) −1.30902 + 0.951057i −0.193004 + 0.140226i
\(47\) 0.336881 + 1.03681i 0.0491391 + 0.151235i 0.972615 − 0.232421i \(-0.0746648\pi\)
−0.923476 + 0.383656i \(0.874665\pi\)
\(48\) −0.972136 + 2.99193i −0.140316 + 0.431847i
\(49\) 4.85410 + 3.52671i 0.693443 + 0.503816i
\(50\) −0.736068 − 0.534785i −0.104096 − 0.0756300i
\(51\) 2.42705 − 7.46969i 0.339855 − 1.04597i
\(52\) 2.42705 + 7.46969i 0.336571 + 1.03586i
\(53\) 2.11803 − 1.53884i 0.290934 − 0.211376i −0.432738 − 0.901520i \(-0.642453\pi\)
0.723673 + 0.690143i \(0.242453\pi\)
\(54\) 0.381966 0.0519790
\(55\) 0.500000 + 5.34307i 0.0674200 + 0.720459i
\(56\) −1.47214 −0.196722
\(57\) −0.690983 + 0.502029i −0.0915229 + 0.0664953i
\(58\) 0.708204 + 2.17963i 0.0929917 + 0.286199i
\(59\) 2.97214 − 9.14729i 0.386939 − 1.19088i −0.548125 − 0.836397i \(-0.684658\pi\)
0.935064 − 0.354480i \(-0.115342\pi\)
\(60\) −2.42705 − 1.76336i −0.313331 − 0.227648i
\(61\) −6.92705 − 5.03280i −0.886918 − 0.644384i 0.0481546 − 0.998840i \(-0.484666\pi\)
−0.935073 + 0.354456i \(0.884666\pi\)
\(62\) −0.600813 + 1.84911i −0.0763033 + 0.234838i
\(63\) −0.309017 − 0.951057i −0.0389325 − 0.119822i
\(64\) 3.80902 − 2.76741i 0.476127 − 0.345927i
\(65\) −6.85410 −0.850147
\(66\) 0.118034 + 1.26133i 0.0145290 + 0.155259i
\(67\) −4.85410 −0.593023 −0.296511 − 0.955029i \(-0.595823\pi\)
−0.296511 + 0.955029i \(0.595823\pi\)
\(68\) −11.7812 + 8.55951i −1.42867 + 1.03799i
\(69\) 1.30902 + 4.02874i 0.157587 + 0.485003i
\(70\) 0.190983 − 0.587785i 0.0228268 − 0.0702538i
\(71\) 4.30902 + 3.13068i 0.511386 + 0.371544i 0.813349 − 0.581776i \(-0.197642\pi\)
−0.301963 + 0.953320i \(0.597642\pi\)
\(72\) −1.19098 − 0.865300i −0.140359 − 0.101977i
\(73\) 2.38197 − 7.33094i 0.278788 − 0.858021i −0.709404 − 0.704802i \(-0.751036\pi\)
0.988192 − 0.153219i \(-0.0489641\pi\)
\(74\) 0.208204 + 0.640786i 0.0242032 + 0.0744898i
\(75\) −1.92705 + 1.40008i −0.222517 + 0.161668i
\(76\) 1.58359 0.181650
\(77\) 3.04508 − 1.31433i 0.347020 − 0.149782i
\(78\) −1.61803 −0.183206
\(79\) −8.89919 + 6.46564i −1.00124 + 0.727441i −0.962353 − 0.271803i \(-0.912380\pi\)
−0.0388837 + 0.999244i \(0.512380\pi\)
\(80\) 1.57295 + 4.84104i 0.175861 + 0.541245i
\(81\) 0.309017 − 0.951057i 0.0343352 − 0.105673i
\(82\) −1.30902 − 0.951057i −0.144557 − 0.105027i
\(83\) 6.04508 + 4.39201i 0.663534 + 0.482086i 0.867855 − 0.496818i \(-0.165498\pi\)
−0.204320 + 0.978904i \(0.565498\pi\)
\(84\) −0.572949 + 1.76336i −0.0625139 + 0.192398i
\(85\) −3.92705 − 12.0862i −0.425948 − 1.31093i
\(86\) 2.07295 − 1.50609i 0.223532 − 0.162405i
\(87\) 6.00000 0.643268
\(88\) 2.48936 − 4.20025i 0.265366 − 0.447749i
\(89\) −3.76393 −0.398976 −0.199488 − 0.979900i \(-0.563928\pi\)
−0.199488 + 0.979900i \(0.563928\pi\)
\(90\) 0.500000 − 0.363271i 0.0527046 − 0.0382922i
\(91\) 1.30902 + 4.02874i 0.137222 + 0.422327i
\(92\) 2.42705 − 7.46969i 0.253038 − 0.778770i
\(93\) 4.11803 + 2.99193i 0.427020 + 0.310248i
\(94\) 0.336881 + 0.244758i 0.0347466 + 0.0252449i
\(95\) −0.427051 + 1.31433i −0.0438145 + 0.134847i
\(96\) 1.28115 + 3.94298i 0.130757 + 0.402429i
\(97\) −0.927051 + 0.673542i −0.0941278 + 0.0683878i −0.633854 − 0.773453i \(-0.718528\pi\)
0.539726 + 0.841841i \(0.318528\pi\)
\(98\) 2.29180 0.231506
\(99\) 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.b.25.1 yes 4
3.2 odd 2 99.2.f.a.91.1 4
4.3 odd 2 528.2.y.b.289.1 4
5.2 odd 4 825.2.bx.d.124.2 8
5.3 odd 4 825.2.bx.d.124.1 8
5.4 even 2 825.2.n.c.751.1 4
9.2 odd 6 891.2.n.b.190.1 8
9.4 even 3 891.2.n.c.784.1 8
9.5 odd 6 891.2.n.b.784.1 8
9.7 even 3 891.2.n.c.190.1 8
11.2 odd 10 363.2.a.i.1.1 2
11.3 even 5 363.2.e.k.148.1 4
11.4 even 5 inner 33.2.e.b.4.1 ✓ 4
11.5 even 5 363.2.e.k.130.1 4
11.6 odd 10 363.2.e.b.130.1 4
11.7 odd 10 363.2.e.f.202.1 4
11.8 odd 10 363.2.e.b.148.1 4
11.9 even 5 363.2.a.d.1.2 2
11.10 odd 2 363.2.e.f.124.1 4
33.2 even 10 1089.2.a.l.1.2 2
33.20 odd 10 1089.2.a.t.1.1 2
33.26 odd 10 99.2.f.a.37.1 4
44.15 odd 10 528.2.y.b.433.1 4
44.31 odd 10 5808.2.a.cj.1.2 2
44.35 even 10 5808.2.a.ci.1.2 2
55.4 even 10 825.2.n.c.301.1 4
55.9 even 10 9075.2.a.cb.1.1 2
55.24 odd 10 9075.2.a.u.1.2 2
55.37 odd 20 825.2.bx.d.499.1 8
55.48 odd 20 825.2.bx.d.499.2 8
99.4 even 15 891.2.n.c.136.1 8
99.59 odd 30 891.2.n.b.136.1 8
99.70 even 15 891.2.n.c.433.1 8
99.92 odd 30 891.2.n.b.433.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.b.4.1 ✓ 4 11.4 even 5 inner
33.2.e.b.25.1 yes 4 1.1 even 1 trivial
99.2.f.a.37.1 4 33.26 odd 10
99.2.f.a.91.1 4 3.2 odd 2
363.2.a.d.1.2 2 11.9 even 5
363.2.a.i.1.1 2 11.2 odd 10
363.2.e.b.130.1 4 11.6 odd 10
363.2.e.b.148.1 4 11.8 odd 10
363.2.e.f.124.1 4 11.10 odd 2
363.2.e.f.202.1 4 11.7 odd 10
363.2.e.k.130.1 4 11.5 even 5
363.2.e.k.148.1 4 11.3 even 5
528.2.y.b.289.1 4 4.3 odd 2
528.2.y.b.433.1 4 44.15 odd 10
825.2.n.c.301.1 4 55.4 even 10
825.2.n.c.751.1 4 5.4 even 2
825.2.bx.d.124.1 8 5.3 odd 4
825.2.bx.d.124.2 8 5.2 odd 4
825.2.bx.d.499.1 8 55.37 odd 20
825.2.bx.d.499.2 8 55.48 odd 20
891.2.n.b.136.1 8 99.59 odd 30
891.2.n.b.190.1 8 9.2 odd 6
891.2.n.b.433.1 8 99.92 odd 30
891.2.n.b.784.1 8 9.5 odd 6
891.2.n.c.136.1 8 99.4 even 15
891.2.n.c.190.1 8 9.7 even 3
891.2.n.c.433.1 8 99.70 even 15
891.2.n.c.784.1 8 9.4 even 3
1089.2.a.l.1.2 2 33.2 even 10
1089.2.a.t.1.1 2 33.20 odd 10
5808.2.a.ci.1.2 2 44.35 even 10
5808.2.a.cj.1.2 2 44.31 odd 10
9075.2.a.u.1.2 2 55.24 odd 10
9075.2.a.cb.1.1 2 55.9 even 10