Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,9,Mod(2,27)] 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("27.2"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 27.f (of order \(18\), degree \(6\), 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: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 2.5
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.5

$q$-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+(-21.5826 - 3.80559i) q^{2} +(-80.3561 - 10.1929i) q^{3} +(210.764 + 76.7119i) q^{4} +(129.757 + 154.639i) q^{5} +(1695.50 + 525.792i) q^{6} +(1479.28 - 538.412i) q^{7} +(601.827 + 347.465i) q^{8} +(6353.21 + 1638.13i) q^{9} +(-2212.01 - 3831.31i) q^{10} +(-5647.40 + 6730.30i) q^{11} +(-16154.3 - 8312.57i) q^{12} +(-2496.75 - 14159.8i) q^{13} +(-33975.6 + 5990.81i) q^{14} +(-8850.57 - 13748.8i) q^{15} +(-55651.7 - 46697.3i) q^{16} +(-48327.4 + 27901.8i) q^{17} +(-130885. - 59532.7i) q^{18} +(24124.8 - 41785.3i) q^{19} +(15485.6 + 42546.2i) q^{20} +(-124357. + 28186.6i) q^{21} +(147498. - 123766. i) q^{22} +(-57310.9 + 157460. i) q^{23} +(-44818.8 - 34055.3i) q^{24} +(60755.1 - 344560. i) q^{25} +315106. i q^{26} +(-493822. - 196391. i) q^{27} +353081. q^{28} +(-52796.6 - 9309.47i) q^{29} +(138696. + 330416. i) q^{30} +(971027. + 353425. i) q^{31} +(909043. + 1.08336e6i) q^{32} +(522404. - 483258. i) q^{33} +(1.14921e6 - 418280. i) q^{34} +(275206. + 158890. i) q^{35} +(1.21336e6 + 832625. i) q^{36} +(-932654. - 1.61540e6i) q^{37} +(-679693. + 810026. i) q^{38} +(56299.7 + 1.16327e6i) q^{39} +(24359.9 + 138152. i) q^{40} +(-4.40599e6 + 776896. i) q^{41} +(2.79121e6 - 135088. i) q^{42} +(-3.96290e6 - 3.32527e6i) q^{43} +(-1.70656e6 + 985285. i) q^{44} +(571057. + 1.19501e6i) q^{45} +(1.83615e6 - 3.18030e6i) q^{46} +(-1.82983e6 - 5.02741e6i) q^{47} +(3.99597e6 + 4.31967e6i) q^{48} +(-2.51772e6 + 2.11262e6i) q^{49} +(-2.62251e6 + 7.20528e6i) q^{50} +(4.16780e6 - 1.74949e6i) q^{51} +(559998. - 3.17591e6i) q^{52} -1.03085e7i q^{53} +(9.91057e6 + 6.11792e6i) q^{54} -1.77356e6 q^{55} +(1.07735e6 + 189966. i) q^{56} +(-2.36449e6 + 3.11180e6i) q^{57} +(1.10406e6 + 401845. i) q^{58} +(-555817. - 662397. i) q^{59} +(-810689. - 3.57669e6i) q^{60} +(-1.53716e7 + 5.59480e6i) q^{61} +(-1.96123e7 - 1.13232e7i) q^{62} +(1.02801e7 - 997404. i) q^{63} +(-6.19773e6 - 1.07348e7i) q^{64} +(1.86568e6 - 2.22343e6i) q^{65} +(-1.31139e7 + 8.44189e6i) q^{66} +(1.38317e6 + 7.84434e6i) q^{67} +(-1.23261e7 + 2.17342e6i) q^{68} +(6.21027e6 - 1.20687e7i) q^{69} +(-5.33499e6 - 4.47659e6i) q^{70} +(-1.63960e7 + 9.46623e6i) q^{71} +(3.25434e6 + 3.19339e6i) q^{72} +(2.49290e7 - 4.31782e7i) q^{73} +(1.39815e7 + 3.84139e7i) q^{74} +(-8.39411e6 + 2.70682e7i) q^{75} +(8.29006e6 - 6.95619e6i) q^{76} +(-4.73038e6 + 1.29966e7i) q^{77} +(3.21185e6 - 2.53207e7i) q^{78} +(-2.63044e6 + 1.49180e7i) q^{79} -1.46652e7i q^{80} +(3.76798e7 + 2.08147e7i) q^{81} +9.80493e7 q^{82} +(8.67787e7 + 1.53014e7i) q^{83} +(-2.83722e7 - 3.59893e6i) q^{84} +(-1.05855e7 - 3.85282e6i) q^{85} +(7.28750e7 + 8.68490e7i) q^{86} +(4.14764e6 + 1.28622e6i) q^{87} +(-5.73730e6 + 2.08821e6i) q^{88} +(-6.02765e7 - 3.48007e7i) q^{89} +(-7.77717e6 - 2.79646e7i) q^{90} +(-1.13172e7 - 1.96019e7i) q^{91} +(-2.41582e7 + 2.87906e7i) q^{92} +(-7.44255e7 - 3.82974e7i) q^{93} +(2.03602e7 + 1.15468e8i) q^{94} +(9.59199e6 - 1.69133e6i) q^{95} +(-6.20046e7 - 9.63201e7i) q^{96} +(-2.85952e7 - 2.39942e7i) q^{97} +(6.23788e7 - 3.60144e7i) q^{98} +(-4.69042e7 + 3.35079e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 447 q^{5} - 774 q^{6} - 6 q^{7} - 9 q^{8} - 12960 q^{9} - 3 q^{10} + 28668 q^{11} + 77421 q^{12} - 6 q^{13} - 120975 q^{14} + 105507 q^{15} - 774 q^{16} - 9 q^{17}+ \cdots + 228876057 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −21.5826 − 3.80559i −1.34891 − 0.237850i −0.547923 − 0.836529i \(-0.684581\pi\)
−0.800989 + 0.598679i \(0.795692\pi\)
\(3\) −80.3561 − 10.1929i −0.992051 − 0.125839i
\(4\) 210.764 + 76.7119i 0.823297 + 0.299656i
\(5\) 129.757 + 154.639i 0.207612 + 0.247422i 0.859795 − 0.510639i \(-0.170591\pi\)
−0.652183 + 0.758061i \(0.726147\pi\)
\(6\) 1695.50 + 525.792i 1.30826 + 0.405704i
\(7\) 1479.28 − 538.412i 0.616108 − 0.224245i −0.0150656 − 0.999887i \(-0.504796\pi\)
0.631174 + 0.775641i \(0.282573\pi\)
\(8\) 601.827 + 347.465i 0.146930 + 0.0848304i
\(9\) 6353.21 + 1638.13i 0.968329 + 0.249676i
\(10\) −2212.01 − 3831.31i −0.221201 − 0.383131i
\(11\) −5647.40 + 6730.30i −0.385725 + 0.459689i −0.923612 − 0.383328i \(-0.874778\pi\)
0.537888 + 0.843017i \(0.319222\pi\)
\(12\) −16154.3 − 8312.57i −0.779045 − 0.400876i
\(13\) −2496.75 − 14159.8i −0.0874182 − 0.495773i −0.996809 − 0.0798295i \(-0.974562\pi\)
0.909390 − 0.415944i \(-0.136549\pi\)
\(14\) −33975.6 + 5990.81i −0.884412 + 0.155946i
\(15\) −8850.57 − 13748.8i −0.174826 − 0.271581i
\(16\) −55651.7 − 46697.3i −0.849178 − 0.712545i
\(17\) −48327.4 + 27901.8i −0.578626 + 0.334070i −0.760587 − 0.649236i \(-0.775089\pi\)
0.181961 + 0.983306i \(0.441755\pi\)
\(18\) −130885. − 59532.7i −1.24681 − 0.567108i
\(19\) 24124.8 − 41785.3i 0.185118 − 0.320634i −0.758498 − 0.651675i \(-0.774067\pi\)
0.943616 + 0.331041i \(0.107400\pi\)
\(20\) 15485.6 + 42546.2i 0.0967847 + 0.265914i
\(21\) −124357. + 28186.6i −0.639429 + 0.144932i
\(22\) 147498. − 123766.i 0.629645 − 0.528335i
\(23\) −57310.9 + 157460.i −0.204798 + 0.562678i −0.998987 − 0.0449922i \(-0.985674\pi\)
0.794189 + 0.607671i \(0.207896\pi\)
\(24\) −44818.8 − 34055.3i −0.135088 − 0.102646i
\(25\) 60755.1 − 344560.i 0.155533 − 0.882072i
\(26\) 315106.i 0.689547i
\(27\) −493822. − 196391.i −0.929213 − 0.369545i
\(28\) 353081. 0.574437
\(29\) −52796.6 − 9309.47i −0.0746473 − 0.0131623i 0.136200 − 0.990681i \(-0.456511\pi\)
−0.210847 + 0.977519i \(0.567622\pi\)
\(30\) 138696. + 330416.i 0.171230 + 0.407921i
\(31\) 971027. + 353425.i 1.05144 + 0.382693i 0.809206 − 0.587525i \(-0.199898\pi\)
0.242234 + 0.970218i \(0.422120\pi\)
\(32\) 909043. + 1.08336e6i 0.866931 + 1.03317i
\(33\) 522404. − 483258.i 0.440505 − 0.407496i
\(34\) 1.14921e6 − 418280.i 0.859974 − 0.313005i
\(35\) 275206. + 158890.i 0.183394 + 0.105883i
\(36\) 1.21336e6 + 832625.i 0.722406 + 0.495723i
\(37\) −932654. − 1.61540e6i −0.497638 − 0.861935i 0.502358 − 0.864660i \(-0.332466\pi\)
−0.999996 + 0.00272492i \(0.999133\pi\)
\(38\) −679693. + 810026.i −0.325970 + 0.388476i
\(39\) 56299.7 + 1.16327e6i 0.0243359 + 0.502833i
\(40\) 24359.9 + 138152.i 0.00951559 + 0.0539656i
\(41\) −4.40599e6 + 776896.i −1.55922 + 0.274933i −0.885709 − 0.464241i \(-0.846327\pi\)
−0.673515 + 0.739174i \(0.735216\pi\)
\(42\) 2.79121e6 − 135088.i 0.897006 − 0.0434129i
\(43\) −3.96290e6 − 3.32527e6i −1.15915 − 0.972641i −0.159255 − 0.987238i \(-0.550909\pi\)
−0.999893 + 0.0145968i \(0.995354\pi\)
\(44\) −1.70656e6 + 985285.i −0.455315 + 0.262876i
\(45\) 571057. + 1.19501e6i 0.139261 + 0.291422i
\(46\) 1.83615e6 − 3.18030e6i 0.410087 − 0.710292i
\(47\) −1.82983e6 − 5.02741e6i −0.374990 − 1.03028i −0.973405 − 0.229090i \(-0.926425\pi\)
0.598416 − 0.801186i \(-0.295797\pi\)
\(48\) 3.99597e6 + 4.31967e6i 0.752762 + 0.813740i
\(49\) −2.51772e6 + 2.11262e6i −0.436741 + 0.366469i
\(50\) −2.62251e6 + 7.20528e6i −0.419601 + 1.15284i
\(51\) 4.16780e6 − 1.74949e6i 0.616065 − 0.258601i
\(52\) 559998. − 3.17591e6i 0.0765901 − 0.434364i
\(53\) − 1.03085e7i − 1.30645i −0.757163 − 0.653226i \(-0.773415\pi\)
0.757163 − 0.653226i \(-0.226585\pi\)
\(54\) 9.91057e6 + 6.11792e6i 1.16553 + 0.719496i
\(55\) −1.77356e6 −0.193818
\(56\) 1.07735e6 + 189966.i 0.109548 + 0.0193162i
\(57\) −2.36449e6 + 3.11180e6i −0.223995 + 0.294790i
\(58\) 1.10406e6 + 401845.i 0.0975620 + 0.0355097i
\(59\) −555817. − 662397.i −0.0458695 − 0.0546651i 0.742622 − 0.669711i \(-0.233582\pi\)
−0.788491 + 0.615046i \(0.789138\pi\)
\(60\) −810689. − 3.57669e6i −0.0625532 − 0.275979i
\(61\) −1.53716e7 + 5.59480e6i −1.11020 + 0.404078i −0.831065 − 0.556175i \(-0.812268\pi\)
−0.279130 + 0.960253i \(0.590046\pi\)
\(62\) −1.96123e7 − 1.13232e7i −1.32728 − 0.766303i
\(63\) 1.02801e7 − 997404.i 0.652584 − 0.0633154i
\(64\) −6.19773e6 − 1.07348e7i −0.369414 − 0.639843i
\(65\) 1.86568e6 − 2.22343e6i 0.104516 − 0.124558i
\(66\) −1.31139e7 + 8.44189e6i −0.691125 + 0.444902i
\(67\) 1.38317e6 + 7.84434e6i 0.0686398 + 0.389276i 0.999702 + 0.0244149i \(0.00777228\pi\)
−0.931062 + 0.364861i \(0.881117\pi\)
\(68\) −1.23261e7 + 2.17342e6i −0.576487 + 0.101650i
\(69\) 6.21027e6 − 1.20687e7i 0.273977 − 0.532434i
\(70\) −5.33499e6 − 4.47659e6i −0.222199 − 0.186447i
\(71\) −1.63960e7 + 9.46623e6i −0.645215 + 0.372515i −0.786620 − 0.617437i \(-0.788171\pi\)
0.141406 + 0.989952i \(0.454838\pi\)
\(72\) 3.25434e6 + 3.19339e6i 0.121097 + 0.118829i
\(73\) 2.49290e7 − 4.31782e7i 0.877835 − 1.52045i 0.0241223 − 0.999709i \(-0.492321\pi\)
0.853712 − 0.520745i \(-0.174346\pi\)
\(74\) 1.39815e7 + 3.84139e7i 0.466259 + 1.28104i
\(75\) −8.39411e6 + 2.70682e7i −0.265295 + 0.855488i
\(76\) 8.29006e6 − 6.95619e6i 0.248487 − 0.208505i
\(77\) −4.73038e6 + 1.29966e7i −0.134565 + 0.369715i
\(78\) 3.21185e6 − 2.53207e7i 0.0867716 − 0.684065i
\(79\) −2.63044e6 + 1.49180e7i −0.0675336 + 0.383002i 0.932242 + 0.361834i \(0.117849\pi\)
−0.999776 + 0.0211675i \(0.993262\pi\)
\(80\) − 1.46652e7i − 0.358038i
\(81\) 3.76798e7 + 2.08147e7i 0.875323 + 0.483538i
\(82\) 9.80493e7 2.16865
\(83\) 8.67787e7 + 1.53014e7i 1.82852 + 0.322418i 0.978802 − 0.204811i \(-0.0656580\pi\)
0.849723 + 0.527229i \(0.176769\pi\)
\(84\) −2.83722e7 − 3.59893e6i −0.569870 − 0.0722863i
\(85\) −1.05855e7 − 3.85282e6i −0.202786 − 0.0738080i
\(86\) 7.28750e7 + 8.68490e7i 1.33225 + 1.58771i
\(87\) 4.14764e6 + 1.28622e6i 0.0723976 + 0.0224512i
\(88\) −5.73730e6 + 2.08821e6i −0.0956703 + 0.0348211i
\(89\) −6.02765e7 − 3.48007e7i −0.960701 − 0.554661i −0.0643121 − 0.997930i \(-0.520485\pi\)
−0.896389 + 0.443269i \(0.853819\pi\)
\(90\) −7.77717e6 − 2.79646e7i −0.118536 − 0.426225i
\(91\) −1.13172e7 − 1.96019e7i −0.165034 − 0.285847i
\(92\) −2.41582e7 + 2.87906e7i −0.337220 + 0.401883i
\(93\) −7.44255e7 − 3.82974e7i −0.994924 − 0.511962i
\(94\) 2.03602e7 + 1.15468e8i 0.260777 + 1.47894i
\(95\) 9.59199e6 − 1.69133e6i 0.117765 − 0.0207651i
\(96\) −6.20046e7 − 9.63201e7i −0.730027 − 1.13405i
\(97\) −2.85952e7 − 2.39942e7i −0.323003 − 0.271032i 0.466839 − 0.884343i \(-0.345393\pi\)
−0.789842 + 0.613311i \(0.789837\pi\)
\(98\) 6.23788e7 − 3.60144e7i 0.676289 − 0.390456i
\(99\) −4.69042e7 + 3.35079e7i −0.488282 + 0.348824i
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 27.9.f.a.2.5 ✓ 138
3.2 odd 2 81.9.f.a.8.19 138
27.13 even 9 81.9.f.a.71.19 138
27.14 odd 18 inner 27.9.f.a.14.5 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.5 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.5 yes 138 27.14 odd 18 inner
81.9.f.a.8.19 138 3.2 odd 2
81.9.f.a.71.19 138 27.13 even 9