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Properties

Label 144.7.q.c.113.4
Level $144$
Weight $7$
Character 144.113
Analytic conductor $33.128$
Analytic rank $0$
Dimension $12$
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] = [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 113.4
Root \(8.88570i\) of defining polynomial
Character \(\chi\) \(=\) 144.113
Dual form 144.7.q.c.65.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(14.9408 - 22.4894i) q^{3} +(202.253 + 116.771i) q^{5} +(-95.5752 - 165.541i) q^{7} +(-282.543 - 672.020i) q^{9} +(673.077 - 388.601i) q^{11} +(45.5802 - 78.9472i) q^{13} +(5647.92 - 2803.88i) q^{15} -7047.39i q^{17} -2731.10 q^{19} +(-5150.89 - 323.897i) q^{21} +(17228.9 + 9947.14i) q^{23} +(19458.3 + 33702.7i) q^{25} +(-19334.7 - 3686.33i) q^{27} +(27104.3 - 15648.7i) q^{29} +(-6174.50 + 10694.6i) q^{31} +(1316.94 - 20943.1i) q^{33} -44641.5i q^{35} -27972.0 q^{37} +(-1094.47 - 2204.61i) q^{39} +(-37428.2 - 21609.2i) q^{41} +(-19256.1 - 33352.5i) q^{43} +(21327.2 - 168911. i) q^{45} +(143771. - 83006.2i) q^{47} +(40555.3 - 70243.8i) q^{49} +(-158491. - 105294. i) q^{51} +54741.5i q^{53} +181509. q^{55} +(-40804.8 + 61420.6i) q^{57} +(14102.1 + 8141.84i) q^{59} +(29443.7 + 50998.0i) q^{61} +(-84242.8 + 111001. i) q^{63} +(18437.4 - 10644.9i) q^{65} +(147998. - 256341. i) q^{67} +(481120. - 238849. i) q^{69} +157251. i q^{71} +80297.0 q^{73} +(1.04868e6 + 65942.7i) q^{75} +(-128659. - 74281.3i) q^{77} +(-188424. - 326360. i) q^{79} +(-371780. + 379749. i) q^{81} +(-733992. + 423771. i) q^{83} +(822929. - 1.42535e6i) q^{85} +(53032.3 - 843363. i) q^{87} -1128.91i q^{89} -17425.3 q^{91} +(148261. + 298646. i) q^{93} +(-552371. - 318912. i) q^{95} +(675152. + 1.16940e6i) q^{97} +(-451321. - 342525. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 42 q^{3} + 432 q^{5} - 240 q^{7} + 2190 q^{9} - 378 q^{11} + 1680 q^{13} + 10872 q^{15} + 2820 q^{19} + 24876 q^{21} + 76248 q^{23} + 8094 q^{25} - 127008 q^{27} + 97092 q^{29} - 21480 q^{31} - 246258 q^{33}+ \cdots - 4398804 q^{99}+O(q^{100}) \) 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
\(3\) 14.9408 − 22.4894i 0.553364 − 0.832939i
\(4\) 0 0
\(5\) 202.253 + 116.771i 1.61802 + 0.934165i 0.987431 + 0.158050i \(0.0505206\pi\)
0.630591 + 0.776116i \(0.282813\pi\)
\(6\) 0 0
\(7\) −95.5752 − 165.541i −0.278645 − 0.482627i 0.692403 − 0.721511i \(-0.256552\pi\)
−0.971048 + 0.238884i \(0.923219\pi\)
\(8\) 0 0
\(9\) −282.543 − 672.020i −0.387576 − 0.921838i
\(10\) 0 0
\(11\) 673.077 − 388.601i 0.505693 − 0.291962i −0.225369 − 0.974274i \(-0.572359\pi\)
0.731061 + 0.682312i \(0.239025\pi\)
\(12\) 0 0
\(13\) 45.5802 − 78.9472i 0.0207466 − 0.0359341i −0.855466 − 0.517859i \(-0.826729\pi\)
0.876212 + 0.481925i \(0.160062\pi\)
\(14\) 0 0
\(15\) 5647.92 − 2803.88i 1.67346 − 0.830780i
\(16\) 0 0
\(17\) − 7047.39i − 1.43444i −0.696848 − 0.717219i \(-0.745415\pi\)
0.696848 − 0.717219i \(-0.254585\pi\)
\(18\) 0 0
\(19\) −2731.10 −0.398177 −0.199088 − 0.979982i \(-0.563798\pi\)
−0.199088 + 0.979982i \(0.563798\pi\)
\(20\) 0 0
\(21\) −5150.89 − 323.897i −0.556191 − 0.0349743i
\(22\) 0 0
\(23\) 17228.9 + 9947.14i 1.41604 + 0.817550i 0.995948 − 0.0899317i \(-0.0286649\pi\)
0.420091 + 0.907482i \(0.361998\pi\)
\(24\) 0 0
\(25\) 19458.3 + 33702.7i 1.24533 + 2.15697i
\(26\) 0 0
\(27\) −19334.7 − 3686.33i −0.982306 − 0.187285i
\(28\) 0 0
\(29\) 27104.3 − 15648.7i 1.11133 − 0.641629i 0.172159 − 0.985069i \(-0.444926\pi\)
0.939175 + 0.343440i \(0.111592\pi\)
\(30\) 0 0
\(31\) −6174.50 + 10694.6i −0.207261 + 0.358986i −0.950851 − 0.309650i \(-0.899788\pi\)
0.743590 + 0.668636i \(0.233121\pi\)
\(32\) 0 0
\(33\) 1316.94 − 20943.1i 0.0366458 − 0.582773i
\(34\) 0 0
\(35\) − 44641.5i − 1.04120i
\(36\) 0 0
\(37\) −27972.0 −0.552228 −0.276114 − 0.961125i \(-0.589047\pi\)
−0.276114 + 0.961125i \(0.589047\pi\)
\(38\) 0 0
\(39\) −1094.47 − 2204.61i −0.0184505 − 0.0371653i
\(40\) 0 0
\(41\) −37428.2 − 21609.2i −0.543059 − 0.313535i 0.203259 − 0.979125i \(-0.434847\pi\)
−0.746318 + 0.665590i \(0.768180\pi\)
\(42\) 0 0
\(43\) −19256.1 − 33352.5i −0.242193 − 0.419491i 0.719146 − 0.694860i \(-0.244533\pi\)
−0.961339 + 0.275369i \(0.911200\pi\)
\(44\) 0 0
\(45\) 21327.2 − 168911.i 0.234043 − 1.85361i
\(46\) 0 0
\(47\) 143771. − 83006.2i 1.38477 − 0.799497i 0.392050 − 0.919944i \(-0.371766\pi\)
0.992720 + 0.120447i \(0.0384327\pi\)
\(48\) 0 0
\(49\) 40555.3 − 70243.8i 0.344714 − 0.597062i
\(50\) 0 0
\(51\) −158491. − 105294.i −1.19480 − 0.793767i
\(52\) 0 0
\(53\) 54741.5i 0.367696i 0.982955 + 0.183848i \(0.0588554\pi\)
−0.982955 + 0.183848i \(0.941145\pi\)
\(54\) 0 0
\(55\) 181509. 1.09096
\(56\) 0 0
\(57\) −40804.8 + 61420.6i −0.220337 + 0.331657i
\(58\) 0 0
\(59\) 14102.1 + 8141.84i 0.0686637 + 0.0396430i 0.533939 − 0.845523i \(-0.320711\pi\)
−0.465275 + 0.885166i \(0.654045\pi\)
\(60\) 0 0
\(61\) 29443.7 + 50998.0i 0.129719 + 0.224680i 0.923568 − 0.383436i \(-0.125259\pi\)
−0.793849 + 0.608115i \(0.791926\pi\)
\(62\) 0 0
\(63\) −84242.8 + 111001.i −0.336908 + 0.443920i
\(64\) 0 0
\(65\) 18437.4 − 10644.9i 0.0671368 − 0.0387614i
\(66\) 0 0
\(67\) 147998. − 256341.i 0.492076 − 0.852301i −0.507882 − 0.861427i \(-0.669571\pi\)
0.999958 + 0.00912565i \(0.00290482\pi\)
\(68\) 0 0
\(69\) 481120. − 238849.i 1.46456 − 0.727071i
\(70\) 0 0
\(71\) 157251.i 0.439358i 0.975572 + 0.219679i \(0.0705010\pi\)
−0.975572 + 0.219679i \(0.929499\pi\)
\(72\) 0 0
\(73\) 80297.0 0.206410 0.103205 − 0.994660i \(-0.467090\pi\)
0.103205 + 0.994660i \(0.467090\pi\)
\(74\) 0 0
\(75\) 1.04868e6 + 65942.7i 2.48575 + 0.156309i
\(76\) 0 0
\(77\) −128659. − 74281.3i −0.281817 − 0.162707i
\(78\) 0 0
\(79\) −188424. − 326360.i −0.382169 − 0.661936i 0.609203 − 0.793014i \(-0.291489\pi\)
−0.991372 + 0.131078i \(0.958156\pi\)
\(80\) 0 0
\(81\) −371780. + 379749.i −0.699570 + 0.714564i
\(82\) 0 0
\(83\) −733992. + 423771.i −1.28368 + 0.741134i −0.977519 − 0.210846i \(-0.932378\pi\)
−0.306162 + 0.951980i \(0.599045\pi\)
\(84\) 0 0
\(85\) 822929. − 1.42535e6i 1.34000 − 2.32095i
\(86\) 0 0
\(87\) 53032.3 − 843363.i 0.0805346 − 1.28073i
\(88\) 0 0
\(89\) − 1128.91i − 0.00160136i −1.00000 0.000800679i \(-0.999745\pi\)
1.00000 0.000800679i \(-0.000254864\pi\)
\(90\) 0 0
\(91\) −17425.3 −0.0231237
\(92\) 0 0
\(93\) 148261. + 298646.i 0.184323 + 0.371286i
\(94\) 0 0
\(95\) −552371. − 318912.i −0.644259 − 0.371963i
\(96\) 0 0
\(97\) 675152. + 1.16940e6i 0.739753 + 1.28129i 0.952607 + 0.304205i \(0.0983908\pi\)
−0.212854 + 0.977084i \(0.568276\pi\)
\(98\) 0 0
\(99\) −451321. − 342525.i −0.465136 − 0.353009i
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.113.4 12
3.2 odd 2 432.7.q.b.17.1 12
4.3 odd 2 18.7.d.a.5.5 ✓ 12
9.2 odd 6 inner 144.7.q.c.65.4 12
9.7 even 3 432.7.q.b.305.1 12
12.11 even 2 54.7.d.a.17.1 12
36.7 odd 6 54.7.d.a.35.1 12
36.11 even 6 18.7.d.a.11.5 yes 12
36.23 even 6 162.7.b.c.161.6 12
36.31 odd 6 162.7.b.c.161.7 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.d.a.5.5 ✓ 12 4.3 odd 2
18.7.d.a.11.5 yes 12 36.11 even 6
54.7.d.a.17.1 12 12.11 even 2
54.7.d.a.35.1 12 36.7 odd 6
144.7.q.c.65.4 12 9.2 odd 6 inner
144.7.q.c.113.4 12 1.1 even 1 trivial
162.7.b.c.161.6 12 36.23 even 6
162.7.b.c.161.7 12 36.31 odd 6
432.7.q.b.17.1 12 3.2 odd 2
432.7.q.b.305.1 12 9.7 even 3