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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,7,Mod(65,144)] 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("144.65"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 144.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.1277880413\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{13} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 113.3
Root \(7.20150i\) of defining polynomial
Character \(\chi\) \(=\) 144.113
Dual form 144.7.q.c.65.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(11.1983 - 24.5683i) q^{3} +(-39.5602 - 22.8401i) q^{5} +(245.097 + 424.521i) q^{7} +(-478.198 - 550.243i) q^{9} +(873.336 - 504.221i) q^{11} +(466.801 - 808.523i) q^{13} +(-1004.15 + 716.156i) q^{15} +8090.59i q^{17} +7727.36 q^{19} +(13174.4 - 1267.72i) q^{21} +(11848.0 + 6840.45i) q^{23} +(-6769.16 - 11724.5i) q^{25} +(-18873.5 + 5586.73i) q^{27} +(-1964.70 + 1134.32i) q^{29} +(17062.6 - 29553.2i) q^{31} +(-2607.99 - 27102.7i) q^{33} -22392.2i q^{35} +92058.0 q^{37} +(-14636.6 - 20522.5i) q^{39} +(-31021.6 - 17910.3i) q^{41} +(-34570.9 - 59878.5i) q^{43} +(6350.01 + 32689.8i) q^{45} +(-13211.1 + 7627.46i) q^{47} +(-61321.0 + 106211. i) q^{49} +(198772. + 90600.5i) q^{51} -236591. i q^{53} -46065.9 q^{55} +(86533.0 - 189848. i) q^{57} +(221890. + 128108. i) q^{59} +(-19919.6 - 34501.8i) q^{61} +(116385. - 337868. i) q^{63} +(-36933.5 + 21323.6i) q^{65} +(160204. - 277482. i) q^{67} +(300735. - 214484. i) q^{69} -404593. i q^{71} +393719. q^{73} +(-363854. + 35012.2i) q^{75} +(428105. + 247167. i) q^{77} +(449184. + 778010. i) q^{79} +(-74094.1 + 526251. i) q^{81} +(-154916. + 89441.0i) q^{83} +(184790. - 320066. i) q^{85} +(5867.06 + 60971.8i) q^{87} +826458. i q^{89} +457647. q^{91} +(-535000. - 750142. i) q^{93} +(-305696. - 176494. i) q^{95} +(-317981. - 550760. i) q^{97} +(-695072. - 239430. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 42 q^{3} + 432 q^{5} - 240 q^{7} + 2190 q^{9} - 378 q^{11} + 1680 q^{13} + 10872 q^{15} + 2820 q^{19} + 24876 q^{21} + 76248 q^{23} + 8094 q^{25} - 127008 q^{27} + 97092 q^{29} - 21480 q^{31} - 246258 q^{33}+ \cdots - 4398804 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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 0
\(3\) 11.1983 − 24.5683i 0.414750 − 0.909935i
\(4\) 0 0
\(5\) −39.5602 − 22.8401i −0.316482 − 0.182721i 0.333341 − 0.942806i \(-0.391824\pi\)
−0.649823 + 0.760085i \(0.725157\pi\)
\(6\) 0 0
\(7\) 245.097 + 424.521i 0.714570 + 1.23767i 0.963125 + 0.269053i \(0.0867108\pi\)
−0.248556 + 0.968618i \(0.579956\pi\)
\(8\) 0 0
\(9\) −478.198 − 550.243i −0.655965 − 0.754792i
\(10\) 0 0
\(11\) 873.336 − 504.221i 0.656151 − 0.378829i −0.134658 − 0.990892i \(-0.542994\pi\)
0.790809 + 0.612063i \(0.209660\pi\)
\(12\) 0 0
\(13\) 466.801 − 808.523i 0.212472 − 0.368012i −0.740016 − 0.672590i \(-0.765182\pi\)
0.952488 + 0.304577i \(0.0985152\pi\)
\(14\) 0 0
\(15\) −1004.15 + 716.156i −0.297525 + 0.212194i
\(16\) 0 0
\(17\) 8090.59i 1.64677i 0.567482 + 0.823386i \(0.307918\pi\)
−0.567482 + 0.823386i \(0.692082\pi\)
\(18\) 0 0
\(19\) 7727.36 1.12660 0.563301 − 0.826252i \(-0.309531\pi\)
0.563301 + 0.826252i \(0.309531\pi\)
\(20\) 0 0
\(21\) 13174.4 − 1267.72i 1.42257 − 0.136888i
\(22\) 0 0
\(23\) 11848.0 + 6840.45i 0.973782 + 0.562213i 0.900387 − 0.435090i \(-0.143283\pi\)
0.0733950 + 0.997303i \(0.476617\pi\)
\(24\) 0 0
\(25\) −6769.16 − 11724.5i −0.433226 − 0.750370i
\(26\) 0 0
\(27\) −18873.5 + 5586.73i −0.958873 + 0.283835i
\(28\) 0 0
\(29\) −1964.70 + 1134.32i −0.0805570 + 0.0465096i −0.539737 − 0.841833i \(-0.681476\pi\)
0.459180 + 0.888343i \(0.348143\pi\)
\(30\) 0 0
\(31\) 17062.6 − 29553.2i 0.572742 − 0.992019i −0.423541 − 0.905877i \(-0.639213\pi\)
0.996283 − 0.0861417i \(-0.0274538\pi\)
\(32\) 0 0
\(33\) −2607.99 − 27102.7i −0.0725710 − 0.754174i
\(34\) 0 0
\(35\) − 22392.2i − 0.522267i
\(36\) 0 0
\(37\) 92058.0 1.81743 0.908713 − 0.417422i \(-0.137066\pi\)
0.908713 + 0.417422i \(0.137066\pi\)
\(38\) 0 0
\(39\) −14636.6 − 20522.5i −0.246745 − 0.345969i
\(40\) 0 0
\(41\) −31021.6 − 17910.3i −0.450103 − 0.259867i 0.257771 − 0.966206i \(-0.417012\pi\)
−0.707874 + 0.706339i \(0.750345\pi\)
\(42\) 0 0
\(43\) −34570.9 − 59878.5i −0.434816 − 0.753123i 0.562465 − 0.826821i \(-0.309853\pi\)
−0.997281 + 0.0736985i \(0.976520\pi\)
\(44\) 0 0
\(45\) 6350.01 + 32689.8i 0.0696847 + 0.358736i
\(46\) 0 0
\(47\) −13211.1 + 7627.46i −0.127247 + 0.0734659i −0.562272 − 0.826952i \(-0.690073\pi\)
0.435025 + 0.900418i \(0.356739\pi\)
\(48\) 0 0
\(49\) −61321.0 + 106211.i −0.521220 + 0.902779i
\(50\) 0 0
\(51\) 198772. + 90600.5i 1.49846 + 0.682999i
\(52\) 0 0
\(53\) − 236591.i − 1.58917i −0.607153 − 0.794585i \(-0.707688\pi\)
0.607153 − 0.794585i \(-0.292312\pi\)
\(54\) 0 0
\(55\) −46065.9 −0.276880
\(56\) 0 0
\(57\) 86533.0 − 189848.i 0.467258 − 1.02513i
\(58\) 0 0
\(59\) 221890. + 128108.i 1.08039 + 0.623766i 0.931003 − 0.365012i \(-0.118935\pi\)
0.149392 + 0.988778i \(0.452268\pi\)
\(60\) 0 0
\(61\) −19919.6 − 34501.8i −0.0877589 − 0.152003i 0.818805 − 0.574072i \(-0.194637\pi\)
−0.906563 + 0.422069i \(0.861304\pi\)
\(62\) 0 0
\(63\) 116385. − 337868.i 0.465451 − 1.35122i
\(64\) 0 0
\(65\) −36933.5 + 21323.6i −0.134487 + 0.0776462i
\(66\) 0 0
\(67\) 160204. − 277482.i 0.532660 − 0.922593i −0.466613 − 0.884461i \(-0.654526\pi\)
0.999273 − 0.0381319i \(-0.0121407\pi\)
\(68\) 0 0
\(69\) 300735. − 214484.i 0.915454 − 0.652901i
\(70\) 0 0
\(71\) − 404593.i − 1.13043i −0.824944 − 0.565215i \(-0.808793\pi\)
0.824944 − 0.565215i \(-0.191207\pi\)
\(72\) 0 0
\(73\) 393719. 1.01209 0.506044 − 0.862508i \(-0.331107\pi\)
0.506044 + 0.862508i \(0.331107\pi\)
\(74\) 0 0
\(75\) −363854. + 35012.2i −0.862469 + 0.0829918i
\(76\) 0 0
\(77\) 428105. + 247167.i 0.937731 + 0.541399i
\(78\) 0 0
\(79\) 449184. + 778010.i 0.911052 + 1.57799i 0.812581 + 0.582848i \(0.198062\pi\)
0.0984709 + 0.995140i \(0.468605\pi\)
\(80\) 0 0
\(81\) −74094.1 + 526251.i −0.139421 + 0.990233i
\(82\) 0 0
\(83\) −154916. + 89441.0i −0.270934 + 0.156424i −0.629312 − 0.777153i \(-0.716663\pi\)
0.358378 + 0.933576i \(0.383330\pi\)
\(84\) 0 0
\(85\) 184790. − 320066.i 0.300900 − 0.521173i
\(86\) 0 0
\(87\) 5867.06 + 60971.8i 0.00890970 + 0.0925915i
\(88\) 0 0
\(89\) 826458.i 1.17233i 0.810191 + 0.586166i \(0.199364\pi\)
−0.810191 + 0.586166i \(0.800636\pi\)
\(90\) 0 0
\(91\) 457647. 0.607304
\(92\) 0 0
\(93\) −535000. − 750142.i −0.665128 − 0.932599i
\(94\) 0 0
\(95\) −305696. − 176494.i −0.356549 − 0.205854i
\(96\) 0 0
\(97\) −317981. − 550760.i −0.348406 − 0.603458i 0.637560 − 0.770401i \(-0.279944\pi\)
−0.985967 + 0.166943i \(0.946610\pi\)
\(98\) 0 0
\(99\) −695072. − 239430.i −0.716348 − 0.246759i
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 144.7.q.c.113.3 12
3.2 odd 2 432.7.q.b.17.5 12
4.3 odd 2 18.7.d.a.5.2 ✓ 12
9.2 odd 6 inner 144.7.q.c.65.3 12
9.7 even 3 432.7.q.b.305.5 12
12.11 even 2 54.7.d.a.17.6 12
36.7 odd 6 54.7.d.a.35.6 12
36.11 even 6 18.7.d.a.11.2 yes 12
36.23 even 6 162.7.b.c.161.8 12
36.31 odd 6 162.7.b.c.161.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.d.a.5.2 ✓ 12 4.3 odd 2
18.7.d.a.11.2 yes 12 36.11 even 6
54.7.d.a.17.6 12 12.11 even 2
54.7.d.a.35.6 12 36.7 odd 6
144.7.q.c.65.3 12 9.2 odd 6 inner
144.7.q.c.113.3 12 1.1 even 1 trivial
162.7.b.c.161.5 12 36.31 odd 6
162.7.b.c.161.8 12 36.23 even 6
432.7.q.b.17.5 12 3.2 odd 2
432.7.q.b.305.5 12 9.7 even 3