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Properties

Label 114.2.i.d.55.1
Level $114$
Weight $2$
Character 114.55
Analytic conductor $0.910$
Analytic rank $0$
Dimension $6$
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] = [114,2,Mod(25,114)] 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("114.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(114, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 14])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 114.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.910294583043\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 55.1
Root \(-0.173648 + 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 114.55
Dual form 114.2.i.d.85.1

$q$-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+(-0.173648 + 0.984808i) q^{2} +(-0.766044 - 0.642788i) q^{3} +(-0.939693 - 0.342020i) q^{4} +(3.20574 - 1.16679i) q^{5} +(0.766044 - 0.642788i) q^{6} +(1.43969 + 2.49362i) q^{7} +(0.500000 - 0.866025i) q^{8} +(0.173648 + 0.984808i) q^{9} +(0.592396 + 3.35965i) q^{10} +(0.173648 - 0.300767i) q^{11} +(0.500000 + 0.866025i) q^{12} +(-1.26604 + 1.06234i) q^{13} +(-2.70574 + 0.984808i) q^{14} +(-3.20574 - 1.16679i) q^{15} +(0.766044 + 0.642788i) q^{16} +(1.20574 - 6.83807i) q^{17} -1.00000 q^{18} +(-2.82635 + 3.31839i) q^{19} -3.41147 q^{20} +(0.500000 - 2.83564i) q^{21} +(0.266044 + 0.223238i) q^{22} +(-6.39053 - 2.32596i) q^{23} +(-0.939693 + 0.342020i) q^{24} +(5.08512 - 4.26692i) q^{25} +(-0.826352 - 1.43128i) q^{26} +(0.500000 - 0.866025i) q^{27} +(-0.500000 - 2.83564i) q^{28} +(1.10354 + 6.25849i) q^{29} +(1.70574 - 2.95442i) q^{30} +(-0.798133 - 1.38241i) q^{31} +(-0.766044 + 0.642788i) q^{32} +(-0.326352 + 0.118782i) q^{33} +(6.52481 + 2.37484i) q^{34} +(7.52481 + 6.31407i) q^{35} +(0.173648 - 0.984808i) q^{36} -11.2121 q^{37} +(-2.77719 - 3.35965i) q^{38} +1.65270 q^{39} +(0.592396 - 3.35965i) q^{40} +(-2.67365 - 2.24346i) q^{41} +(2.70574 + 0.984808i) q^{42} +(-2.14543 + 0.780873i) q^{43} +(-0.266044 + 0.223238i) q^{44} +(1.70574 + 2.95442i) q^{45} +(3.40033 - 5.88954i) q^{46} +(0.971782 + 5.51125i) q^{47} +(-0.173648 - 0.984808i) q^{48} +(-0.645430 + 1.11792i) q^{49} +(3.31908 + 5.74881i) q^{50} +(-5.31908 + 4.46324i) q^{51} +(1.55303 - 0.565258i) q^{52} +(-1.86097 - 0.677337i) q^{53} +(0.766044 + 0.642788i) q^{54} +(0.205737 - 1.16679i) q^{55} +2.87939 q^{56} +(4.29813 - 0.725293i) q^{57} -6.35504 q^{58} +(0.0773815 - 0.438852i) q^{59} +(2.61334 + 2.19285i) q^{60} +(11.7763 + 4.28623i) q^{61} +(1.50000 - 0.545955i) q^{62} +(-2.20574 + 1.85083i) q^{63} +(-0.500000 - 0.866025i) q^{64} +(-2.81908 + 4.88279i) q^{65} +(-0.0603074 - 0.342020i) q^{66} +(-0.187319 - 1.06234i) q^{67} +(-3.47178 + 6.01330i) q^{68} +(3.40033 + 5.88954i) q^{69} +(-7.52481 + 6.31407i) q^{70} +(15.6211 - 5.68561i) q^{71} +(0.939693 + 0.342020i) q^{72} +(9.51367 + 7.98292i) q^{73} +(1.94697 - 11.0418i) q^{74} -6.63816 q^{75} +(3.79086 - 2.15160i) q^{76} +1.00000 q^{77} +(-0.286989 + 1.62760i) q^{78} +(-8.36824 - 7.02179i) q^{79} +(3.20574 + 1.16679i) q^{80} +(-0.939693 + 0.342020i) q^{81} +(2.67365 - 2.24346i) q^{82} +(-5.85844 - 10.1471i) q^{83} +(-1.43969 + 2.49362i) q^{84} +(-4.11334 - 23.3279i) q^{85} +(-0.396459 - 2.24843i) q^{86} +(3.17752 - 5.50362i) q^{87} +(-0.173648 - 0.300767i) q^{88} +(1.37346 - 1.15247i) q^{89} +(-3.20574 + 1.16679i) q^{90} +(-4.47178 - 1.62760i) q^{91} +(5.20961 + 4.37138i) q^{92} +(-0.277189 + 1.57202i) q^{93} -5.59627 q^{94} +(-5.18866 + 13.9357i) q^{95} +1.00000 q^{96} +(0.634285 - 3.59721i) q^{97} +(-0.988856 - 0.829748i) q^{98} +(0.326352 + 0.118782i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{5} + 3 q^{7} + 3 q^{8} + 3 q^{12} - 3 q^{13} - 6 q^{14} - 9 q^{15} - 3 q^{17} - 6 q^{18} - 18 q^{19} + 3 q^{21} - 3 q^{22} - 21 q^{23} + 9 q^{25} - 6 q^{26} + 3 q^{27} - 3 q^{28} - 3 q^{29} + 9 q^{31}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.173648 + 0.984808i −0.122788 + 0.696364i
\(3\) −0.766044 − 0.642788i −0.442276 − 0.371114i
\(4\) −0.939693 − 0.342020i −0.469846 − 0.171010i
\(5\) 3.20574 − 1.16679i 1.43365 − 0.521806i 0.495674 − 0.868509i \(-0.334921\pi\)
0.937975 + 0.346703i \(0.112699\pi\)
\(6\) 0.766044 − 0.642788i 0.312736 − 0.262417i
\(7\) 1.43969 + 2.49362i 0.544153 + 0.942500i 0.998660 + 0.0517569i \(0.0164821\pi\)
−0.454507 + 0.890743i \(0.650185\pi\)
\(8\) 0.500000 − 0.866025i 0.176777 − 0.306186i
\(9\) 0.173648 + 0.984808i 0.0578827 + 0.328269i
\(10\) 0.592396 + 3.35965i 0.187332 + 1.06241i
\(11\) 0.173648 − 0.300767i 0.0523569 − 0.0906848i −0.838659 − 0.544657i \(-0.816660\pi\)
0.891016 + 0.453972i \(0.149993\pi\)
\(12\) 0.500000 + 0.866025i 0.144338 + 0.250000i
\(13\) −1.26604 + 1.06234i −0.351138 + 0.294639i −0.801247 − 0.598334i \(-0.795830\pi\)
0.450109 + 0.892974i \(0.351385\pi\)
\(14\) −2.70574 + 0.984808i −0.723139 + 0.263201i
\(15\) −3.20574 − 1.16679i −0.827718 − 0.301265i
\(16\) 0.766044 + 0.642788i 0.191511 + 0.160697i
\(17\) 1.20574 − 6.83807i 0.292434 − 1.65848i −0.385017 − 0.922909i \(-0.625805\pi\)
0.677452 − 0.735567i \(-0.263084\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.82635 + 3.31839i −0.648410 + 0.761292i
\(20\) −3.41147 −0.762829
\(21\) 0.500000 − 2.83564i 0.109109 − 0.618788i
\(22\) 0.266044 + 0.223238i 0.0567209 + 0.0475945i
\(23\) −6.39053 − 2.32596i −1.33252 − 0.484997i −0.425069 − 0.905161i \(-0.639750\pi\)
−0.907448 + 0.420164i \(0.861973\pi\)
\(24\) −0.939693 + 0.342020i −0.191814 + 0.0698146i
\(25\) 5.08512 − 4.26692i 1.01702 − 0.853385i
\(26\) −0.826352 − 1.43128i −0.162061 − 0.280698i
\(27\) 0.500000 − 0.866025i 0.0962250 − 0.166667i
\(28\) −0.500000 − 2.83564i −0.0944911 − 0.535886i
\(29\) 1.10354 + 6.25849i 0.204922 + 1.16217i 0.897562 + 0.440888i \(0.145337\pi\)
−0.692640 + 0.721284i \(0.743552\pi\)
\(30\) 1.70574 − 2.95442i 0.311424 − 0.539401i
\(31\) −0.798133 − 1.38241i −0.143349 − 0.248288i 0.785407 − 0.618980i \(-0.212454\pi\)
−0.928756 + 0.370692i \(0.879120\pi\)
\(32\) −0.766044 + 0.642788i −0.135419 + 0.113630i
\(33\) −0.326352 + 0.118782i −0.0568106 + 0.0206774i
\(34\) 6.52481 + 2.37484i 1.11900 + 0.407281i
\(35\) 7.52481 + 6.31407i 1.27193 + 1.06727i
\(36\) 0.173648 − 0.984808i 0.0289414 − 0.164135i
\(37\) −11.2121 −1.84326 −0.921632 − 0.388066i \(-0.873143\pi\)
−0.921632 + 0.388066i \(0.873143\pi\)
\(38\) −2.77719 − 3.35965i −0.450520 − 0.545007i
\(39\) 1.65270 0.264644
\(40\) 0.592396 − 3.35965i 0.0936661 − 0.531207i
\(41\) −2.67365 − 2.24346i −0.417554 − 0.350369i 0.409678 − 0.912230i \(-0.365641\pi\)
−0.827232 + 0.561861i \(0.810086\pi\)
\(42\) 2.70574 + 0.984808i 0.417504 + 0.151959i
\(43\) −2.14543 + 0.780873i −0.327175 + 0.119082i −0.500385 − 0.865803i \(-0.666808\pi\)
0.173210 + 0.984885i \(0.444586\pi\)
\(44\) −0.266044 + 0.223238i −0.0401077 + 0.0336544i
\(45\) 1.70574 + 2.95442i 0.254276 + 0.440419i
\(46\) 3.40033 − 5.88954i 0.501351 − 0.868366i
\(47\) 0.971782 + 5.51125i 0.141749 + 0.803898i 0.969920 + 0.243423i \(0.0782703\pi\)
−0.828171 + 0.560475i \(0.810619\pi\)
\(48\) −0.173648 − 0.984808i −0.0250640 − 0.142145i
\(49\) −0.645430 + 1.11792i −0.0922042 + 0.159702i
\(50\) 3.31908 + 5.74881i 0.469388 + 0.813005i
\(51\) −5.31908 + 4.46324i −0.744820 + 0.624978i
\(52\) 1.55303 − 0.565258i 0.215367 − 0.0783872i
\(53\) −1.86097 − 0.677337i −0.255623 − 0.0930393i 0.211030 − 0.977480i \(-0.432318\pi\)
−0.466653 + 0.884440i \(0.654540\pi\)
\(54\) 0.766044 + 0.642788i 0.104245 + 0.0874723i
\(55\) 0.205737 − 1.16679i 0.0277416 − 0.157330i
\(56\) 2.87939 0.384774
\(57\) 4.29813 − 0.725293i 0.569302 − 0.0960674i
\(58\) −6.35504 −0.834457
\(59\) 0.0773815 − 0.438852i 0.0100742 − 0.0571337i −0.979356 − 0.202142i \(-0.935210\pi\)
0.989430 + 0.145008i \(0.0463209\pi\)
\(60\) 2.61334 + 2.19285i 0.337381 + 0.283096i
\(61\) 11.7763 + 4.28623i 1.50780 + 0.548795i 0.958067 − 0.286544i \(-0.0925064\pi\)
0.549735 + 0.835339i \(0.314729\pi\)
\(62\) 1.50000 − 0.545955i 0.190500 − 0.0693364i
\(63\) −2.20574 + 1.85083i −0.277897 + 0.233183i
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) −2.81908 + 4.88279i −0.349664 + 0.605635i
\(66\) −0.0603074 − 0.342020i −0.00742333 − 0.0420998i
\(67\) −0.187319 − 1.06234i −0.0228846 − 0.129785i 0.971225 − 0.238163i \(-0.0765454\pi\)
−0.994110 + 0.108378i \(0.965434\pi\)
\(68\) −3.47178 + 6.01330i −0.421015 + 0.729220i
\(69\) 3.40033 + 5.88954i 0.409352 + 0.709018i
\(70\) −7.52481 + 6.31407i −0.899387 + 0.754676i
\(71\) 15.6211 − 5.68561i 1.85388 − 0.674758i 0.870786 − 0.491662i \(-0.163610\pi\)
0.983095 − 0.183096i \(-0.0586119\pi\)
\(72\) 0.939693 + 0.342020i 0.110744 + 0.0403075i
\(73\) 9.51367 + 7.98292i 1.11349 + 0.934330i 0.998257 − 0.0590086i \(-0.0187939\pi\)
0.115233 + 0.993338i \(0.463238\pi\)
\(74\) 1.94697 − 11.0418i 0.226330 − 1.28358i
\(75\) −6.63816 −0.766508
\(76\) 3.79086 − 2.15160i 0.434841 − 0.246806i
\(77\) 1.00000 0.113961
\(78\) −0.286989 + 1.62760i −0.0324951 + 0.184289i
\(79\) −8.36824 − 7.02179i −0.941501 − 0.790013i 0.0363452 − 0.999339i \(-0.488428\pi\)
−0.977846 + 0.209326i \(0.932873\pi\)
\(80\) 3.20574 + 1.16679i 0.358412 + 0.130451i
\(81\) −0.939693 + 0.342020i −0.104410 + 0.0380022i
\(82\) 2.67365 − 2.24346i 0.295255 − 0.247748i
\(83\) −5.85844 − 10.1471i −0.643047 − 1.11379i −0.984749 − 0.173982i \(-0.944336\pi\)
0.341701 − 0.939809i \(-0.388997\pi\)
\(84\) −1.43969 + 2.49362i −0.157083 + 0.272076i
\(85\) −4.11334 − 23.3279i −0.446154 − 2.53027i
\(86\) −0.396459 − 2.24843i −0.0427513 − 0.242455i
\(87\) 3.17752 − 5.50362i 0.340666 − 0.590050i
\(88\) −0.173648 − 0.300767i −0.0185110 − 0.0320619i
\(89\) 1.37346 − 1.15247i 0.145586 − 0.122161i −0.567086 − 0.823659i \(-0.691929\pi\)
0.712672 + 0.701497i \(0.247485\pi\)
\(90\) −3.20574 + 1.16679i −0.337914 + 0.122991i
\(91\) −4.47178 − 1.62760i −0.468770 − 0.170618i
\(92\) 5.20961 + 4.37138i 0.543139 + 0.455748i
\(93\) −0.277189 + 1.57202i −0.0287431 + 0.163010i
\(94\) −5.59627 −0.577211
\(95\) −5.18866 + 13.9357i −0.532346 + 1.42977i
\(96\) 1.00000 0.102062
\(97\) 0.634285 − 3.59721i 0.0644019 − 0.365241i −0.935526 − 0.353257i \(-0.885074\pi\)
0.999928 − 0.0119843i \(-0.00381481\pi\)
\(98\) −0.988856 − 0.829748i −0.0998895 − 0.0838172i
\(99\) 0.326352 + 0.118782i 0.0327996 + 0.0119381i
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 114.2.i.d.55.1 ✓ 6
3.2 odd 2 342.2.u.a.55.1 6
4.3 odd 2 912.2.bo.f.625.1 6
19.3 odd 18 2166.2.a.u.1.1 3
19.9 even 9 inner 114.2.i.d.85.1 yes 6
19.16 even 9 2166.2.a.o.1.1 3
57.35 odd 18 6498.2.a.bs.1.3 3
57.41 even 18 6498.2.a.bn.1.3 3
57.47 odd 18 342.2.u.a.199.1 6
76.47 odd 18 912.2.bo.f.769.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.55.1 ✓ 6 1.1 even 1 trivial
114.2.i.d.85.1 yes 6 19.9 even 9 inner
342.2.u.a.55.1 6 3.2 odd 2
342.2.u.a.199.1 6 57.47 odd 18
912.2.bo.f.625.1 6 4.3 odd 2
912.2.bo.f.769.1 6 76.47 odd 18
2166.2.a.o.1.1 3 19.16 even 9
2166.2.a.u.1.1 3 19.3 odd 18
6498.2.a.bn.1.3 3 57.41 even 18
6498.2.a.bs.1.3 3 57.35 odd 18