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Properties

Label 203.2.e.a.88.1
Level $203$
Weight $2$
Character 203.88
Analytic conductor $1.621$
Analytic rank $1$
Dimension $2$
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] = [203,2,Mod(30,203)] 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("203.30"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(203, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 203 = 7 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 203.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2] 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: \(1.62096316103\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 88.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 203.88
Dual form 203.2.e.a.30.1

$q$-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+(-1.00000 - 1.73205i) q^{2} +(-1.00000 + 1.73205i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(-0.500000 - 0.866025i) q^{5} +4.00000 q^{6} +(-0.500000 - 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{10} +(-2.00000 + 3.46410i) q^{11} +(-2.00000 - 3.46410i) q^{12} -5.00000 q^{13} +(-4.00000 + 3.46410i) q^{14} +2.00000 q^{15} +(2.00000 + 3.46410i) q^{16} +(-4.00000 + 6.92820i) q^{17} +(-1.00000 + 1.73205i) q^{18} +(-4.00000 - 6.92820i) q^{19} +2.00000 q^{20} +(5.00000 + 1.73205i) q^{21} +8.00000 q^{22} +(2.00000 - 3.46410i) q^{25} +(5.00000 + 8.66025i) q^{26} -4.00000 q^{27} +(5.00000 + 1.73205i) q^{28} +1.00000 q^{29} +(-2.00000 - 3.46410i) q^{30} +(1.00000 - 1.73205i) q^{31} +(4.00000 - 6.92820i) q^{32} +(-4.00000 - 6.92820i) q^{33} +16.0000 q^{34} +(-2.00000 + 1.73205i) q^{35} +2.00000 q^{36} +(2.00000 + 3.46410i) q^{37} +(-8.00000 + 13.8564i) q^{38} +(5.00000 - 8.66025i) q^{39} +6.00000 q^{41} +(-2.00000 - 10.3923i) q^{42} -6.00000 q^{43} +(-4.00000 - 6.92820i) q^{44} +(-0.500000 + 0.866025i) q^{45} +(-2.00000 - 3.46410i) q^{47} -8.00000 q^{48} +(-6.50000 + 2.59808i) q^{49} -8.00000 q^{50} +(-8.00000 - 13.8564i) q^{51} +(5.00000 - 8.66025i) q^{52} +(1.50000 - 2.59808i) q^{53} +(4.00000 + 6.92820i) q^{54} +4.00000 q^{55} +16.0000 q^{57} +(-1.00000 - 1.73205i) q^{58} +(-2.00000 + 3.46410i) q^{60} +(2.00000 + 3.46410i) q^{61} -4.00000 q^{62} +(-2.00000 + 1.73205i) q^{63} -8.00000 q^{64} +(2.50000 + 4.33013i) q^{65} +(-8.00000 + 13.8564i) q^{66} +(6.00000 - 10.3923i) q^{67} +(-8.00000 - 13.8564i) q^{68} +(5.00000 + 1.73205i) q^{70} -7.00000 q^{71} +(2.00000 - 3.46410i) q^{73} +(4.00000 - 6.92820i) q^{74} +(4.00000 + 6.92820i) q^{75} +16.0000 q^{76} +(10.0000 + 3.46410i) q^{77} -20.0000 q^{78} +(6.00000 + 10.3923i) q^{79} +(2.00000 - 3.46410i) q^{80} +(5.50000 - 9.52628i) q^{81} +(-6.00000 - 10.3923i) q^{82} -11.0000 q^{83} +(-8.00000 + 6.92820i) q^{84} +8.00000 q^{85} +(6.00000 + 10.3923i) q^{86} +(-1.00000 + 1.73205i) q^{87} +2.00000 q^{90} +(2.50000 + 12.9904i) q^{91} +(2.00000 + 3.46410i) q^{93} +(-4.00000 + 6.92820i) q^{94} +(-4.00000 + 6.92820i) q^{95} +(8.00000 + 13.8564i) q^{96} -6.00000 q^{97} +(11.0000 + 8.66025i) q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - q^{5} + 8 q^{6} - q^{7} - q^{9} - 2 q^{10} - 4 q^{11} - 4 q^{12} - 10 q^{13} - 8 q^{14} + 4 q^{15} + 4 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} + 4 q^{20} + 10 q^{21}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/203\mathbb{Z}\right)^\times\).

\(n\) \(59\) \(176\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 − 1.73205i −0.707107 − 1.22474i −0.965926 − 0.258819i \(-0.916667\pi\)
0.258819 − 0.965926i \(-0.416667\pi\)
\(3\) −1.00000 + 1.73205i −0.577350 + 1.00000i 0.418432 + 0.908248i \(0.362580\pi\)
−0.995782 + 0.0917517i \(0.970753\pi\)
\(4\) −1.00000 + 1.73205i −0.500000 + 0.866025i
\(5\) −0.500000 − 0.866025i −0.223607 − 0.387298i 0.732294 − 0.680989i \(-0.238450\pi\)
−0.955901 + 0.293691i \(0.905116\pi\)
\(6\) 4.00000 1.63299
\(7\) −0.500000 − 2.59808i −0.188982 − 0.981981i
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) −1.00000 + 1.73205i −0.316228 + 0.547723i
\(11\) −2.00000 + 3.46410i −0.603023 + 1.04447i 0.389338 + 0.921095i \(0.372704\pi\)
−0.992361 + 0.123371i \(0.960630\pi\)
\(12\) −2.00000 − 3.46410i −0.577350 − 1.00000i
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) −4.00000 + 3.46410i −1.06904 + 0.925820i
\(15\) 2.00000 0.516398
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) −4.00000 + 6.92820i −0.970143 + 1.68034i −0.275029 + 0.961436i \(0.588688\pi\)
−0.695113 + 0.718900i \(0.744646\pi\)
\(18\) −1.00000 + 1.73205i −0.235702 + 0.408248i
\(19\) −4.00000 − 6.92820i −0.917663 − 1.58944i −0.802955 − 0.596040i \(-0.796740\pi\)
−0.114708 − 0.993399i \(-0.536593\pi\)
\(20\) 2.00000 0.447214
\(21\) 5.00000 + 1.73205i 1.09109 + 0.377964i
\(22\) 8.00000 1.70561
\(23\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 2.00000 − 3.46410i 0.400000 − 0.692820i
\(26\) 5.00000 + 8.66025i 0.980581 + 1.69842i
\(27\) −4.00000 −0.769800
\(28\) 5.00000 + 1.73205i 0.944911 + 0.327327i
\(29\) 1.00000 0.185695
\(30\) −2.00000 − 3.46410i −0.365148 − 0.632456i
\(31\) 1.00000 − 1.73205i 0.179605 − 0.311086i −0.762140 − 0.647412i \(-0.775851\pi\)
0.941745 + 0.336327i \(0.109185\pi\)
\(32\) 4.00000 − 6.92820i 0.707107 − 1.22474i
\(33\) −4.00000 − 6.92820i −0.696311 − 1.20605i
\(34\) 16.0000 2.74398
\(35\) −2.00000 + 1.73205i −0.338062 + 0.292770i
\(36\) 2.00000 0.333333
\(37\) 2.00000 + 3.46410i 0.328798 + 0.569495i 0.982274 − 0.187453i \(-0.0600231\pi\)
−0.653476 + 0.756948i \(0.726690\pi\)
\(38\) −8.00000 + 13.8564i −1.29777 + 2.24781i
\(39\) 5.00000 − 8.66025i 0.800641 − 1.38675i
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −2.00000 − 10.3923i −0.308607 − 1.60357i
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −4.00000 − 6.92820i −0.603023 − 1.04447i
\(45\) −0.500000 + 0.866025i −0.0745356 + 0.129099i
\(46\) 0 0
\(47\) −2.00000 − 3.46410i −0.291730 − 0.505291i 0.682489 − 0.730896i \(-0.260898\pi\)
−0.974219 + 0.225605i \(0.927564\pi\)
\(48\) −8.00000 −1.15470
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) −8.00000 −1.13137
\(51\) −8.00000 − 13.8564i −1.12022 − 1.94029i
\(52\) 5.00000 − 8.66025i 0.693375 − 1.20096i
\(53\) 1.50000 − 2.59808i 0.206041 − 0.356873i −0.744423 − 0.667708i \(-0.767275\pi\)
0.950464 + 0.310835i \(0.100609\pi\)
\(54\) 4.00000 + 6.92820i 0.544331 + 0.942809i
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 16.0000 2.11925
\(58\) −1.00000 − 1.73205i −0.131306 − 0.227429i
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) −2.00000 + 3.46410i −0.258199 + 0.447214i
\(61\) 2.00000 + 3.46410i 0.256074 + 0.443533i 0.965187 − 0.261562i \(-0.0842377\pi\)
−0.709113 + 0.705095i \(0.750904\pi\)
\(62\) −4.00000 −0.508001
\(63\) −2.00000 + 1.73205i −0.251976 + 0.218218i
\(64\) −8.00000 −1.00000
\(65\) 2.50000 + 4.33013i 0.310087 + 0.537086i
\(66\) −8.00000 + 13.8564i −0.984732 + 1.70561i
\(67\) 6.00000 − 10.3923i 0.733017 − 1.26962i −0.222571 − 0.974916i \(-0.571445\pi\)
0.955588 − 0.294706i \(-0.0952216\pi\)
\(68\) −8.00000 − 13.8564i −0.970143 − 1.68034i
\(69\) 0 0
\(70\) 5.00000 + 1.73205i 0.597614 + 0.207020i
\(71\) −7.00000 −0.830747 −0.415374 − 0.909651i \(-0.636349\pi\)
−0.415374 + 0.909651i \(0.636349\pi\)
\(72\) 0 0
\(73\) 2.00000 − 3.46410i 0.234082 − 0.405442i −0.724923 − 0.688830i \(-0.758125\pi\)
0.959006 + 0.283387i \(0.0914581\pi\)
\(74\) 4.00000 − 6.92820i 0.464991 − 0.805387i
\(75\) 4.00000 + 6.92820i 0.461880 + 0.800000i
\(76\) 16.0000 1.83533
\(77\) 10.0000 + 3.46410i 1.13961 + 0.394771i
\(78\) −20.0000 −2.26455
\(79\) 6.00000 + 10.3923i 0.675053 + 1.16923i 0.976453 + 0.215728i \(0.0692125\pi\)
−0.301401 + 0.953498i \(0.597454\pi\)
\(80\) 2.00000 − 3.46410i 0.223607 − 0.387298i
\(81\) 5.50000 − 9.52628i 0.611111 − 1.05848i
\(82\) −6.00000 − 10.3923i −0.662589 − 1.14764i
\(83\) −11.0000 −1.20741 −0.603703 − 0.797209i \(-0.706309\pi\)
−0.603703 + 0.797209i \(0.706309\pi\)
\(84\) −8.00000 + 6.92820i −0.872872 + 0.755929i
\(85\) 8.00000 0.867722
\(86\) 6.00000 + 10.3923i 0.646997 + 1.12063i
\(87\) −1.00000 + 1.73205i −0.107211 + 0.185695i
\(88\) 0 0
\(89\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 2.00000 0.210819
\(91\) 2.50000 + 12.9904i 0.262071 + 1.36176i
\(92\) 0 0
\(93\) 2.00000 + 3.46410i 0.207390 + 0.359211i
\(94\) −4.00000 + 6.92820i −0.412568 + 0.714590i
\(95\) −4.00000 + 6.92820i −0.410391 + 0.710819i
\(96\) 8.00000 + 13.8564i 0.816497 + 1.41421i
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 11.0000 + 8.66025i 1.11117 + 0.874818i
\(99\) 4.00000 0.402015
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 203.2.e.a.88.1 yes 2
7.2 even 3 inner 203.2.e.a.30.1 ✓ 2
7.3 odd 6 1421.2.a.h.1.1 1
7.4 even 3 1421.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
203.2.e.a.30.1 ✓ 2 7.2 even 3 inner
203.2.e.a.88.1 yes 2 1.1 even 1 trivial
1421.2.a.h.1.1 1 7.3 odd 6
1421.2.a.i.1.1 1 7.4 even 3