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Properties

Label 18.7.d.a.11.5
Level $18$
Weight $7$
Character 18.11
Analytic conductor $4.141$
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] = [18,7,Mod(5,18)] 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("18.5"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 18.d (of order \(6\), degree \(2\), 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: \(4.14097350516\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.5
Root \(-8.88570i\) of defining polynomial
Character \(\chi\) \(=\) 18.11
Dual form 18.7.d.a.5.5

$q$-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+(4.89898 + 2.82843i) q^{2} +(-14.9408 - 22.4894i) q^{3} +(16.0000 + 27.7128i) q^{4} +(202.253 - 116.771i) q^{5} +(-9.58533 - 152.434i) q^{6} +(95.5752 - 165.541i) q^{7} +181.019i q^{8} +(-282.543 + 672.020i) q^{9} +1321.11 q^{10} +(-673.077 - 388.601i) q^{11} +(384.190 - 773.882i) q^{12} +(45.5802 + 78.9472i) q^{13} +(936.442 - 540.655i) q^{14} +(-5647.92 - 2803.88i) q^{15} +(-512.000 + 886.810i) q^{16} +7047.39i q^{17} +(-3284.93 + 2493.06i) q^{18} +2731.10 q^{19} +(6472.09 + 3736.66i) q^{20} +(-5150.89 + 323.897i) q^{21} +(-2198.26 - 3807.50i) q^{22} +(-17228.9 + 9947.14i) q^{23} +(4071.01 - 2704.58i) q^{24} +(19458.3 - 33702.7i) q^{25} +515.681i q^{26} +(19334.7 - 3686.33i) q^{27} +6116.81 q^{28} +(27104.3 + 15648.7i) q^{29} +(-19738.5 - 29710.9i) q^{30} +(6174.50 + 10694.6i) q^{31} +(-5016.55 + 2896.31i) q^{32} +(1316.94 + 20943.1i) q^{33} +(-19933.0 + 34525.0i) q^{34} -44641.5i q^{35} +(-23144.2 + 2922.26i) q^{36} -27972.0 q^{37} +(13379.6 + 7724.70i) q^{38} +(1094.47 - 2204.61i) q^{39} +(21137.7 + 36611.7i) q^{40} +(-37428.2 + 21609.2i) q^{41} +(-26150.2 - 12982.1i) q^{42} +(19256.1 - 33352.5i) q^{43} -24870.5i q^{44} +(21327.2 + 168911. i) q^{45} -112539. q^{46} +(-143771. - 83006.2i) q^{47} +(27593.5 - 1735.13i) q^{48} +(40555.3 + 70243.8i) q^{49} +(190651. - 110073. i) q^{50} +(158491. - 105294. i) q^{51} +(-1458.57 + 2526.31i) q^{52} -54741.5i q^{53} +(105147. + 36627.6i) q^{54} -181509. q^{55} +(29966.1 + 17301.0i) q^{56} +(-40804.8 - 61420.6i) q^{57} +(88522.3 + 153325. i) q^{58} +(-14102.1 + 8141.84i) q^{59} +(-12663.3 - 201382. i) q^{60} +(29443.7 - 50998.0i) q^{61} +69856.5i q^{62} +(84242.8 + 111001. i) q^{63} -32768.0 q^{64} +(18437.4 + 10644.9i) q^{65} +(-52784.4 + 106325. i) q^{66} +(-147998. - 256341. i) q^{67} +(-195303. + 112758. i) q^{68} +(481120. + 238849. i) q^{69} +(126265. - 218698. i) q^{70} +157251. i q^{71} +(-121649. - 51145.7i) q^{72} +80297.0 q^{73} +(-137034. - 79116.8i) q^{74} +(-1.04868e6 + 65942.7i) q^{75} +(43697.5 + 75686.3i) q^{76} +(-128659. + 74281.3i) q^{77} +(11597.3 - 7704.71i) q^{78} +(188424. - 326360. i) q^{79} +239146. i q^{80} +(-371780. - 379749. i) q^{81} -244480. q^{82} +(733992. + 423771. i) q^{83} +(-91390.3 - 137563. i) q^{84} +(822929. + 1.42535e6i) q^{85} +(188670. - 108929. i) q^{86} +(-53032.3 - 843363. i) q^{87} +(70344.3 - 121840. i) q^{88} +1128.91i q^{89} +(-373270. + 887812. i) q^{90} +17425.3 q^{91} +(-551326. - 318308. i) q^{92} +(148261. - 298646. i) q^{93} +(-469554. - 813291. i) q^{94} +(552371. - 318912. i) q^{95} +(140088. + 69545.8i) q^{96} +(675152. - 1.16940e6i) q^{97} +458831. i q^{98} +(451321. - 342525. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 42 q^{3} + 192 q^{4} + 432 q^{5} - 144 q^{6} + 240 q^{7} + 2190 q^{9} + 378 q^{11} + 384 q^{12} + 1680 q^{13} - 4752 q^{14} - 10872 q^{15} - 6144 q^{16} - 2976 q^{18} - 2820 q^{19} + 13824 q^{20}+ \cdots + 4398804 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.89898 + 2.82843i 0.612372 + 0.353553i
\(3\) −14.9408 − 22.4894i −0.553364 − 0.832939i
\(4\) 16.0000 + 27.7128i 0.250000 + 0.433013i
\(5\) 202.253 − 116.771i 1.61802 − 0.934165i 0.630591 − 0.776116i \(-0.282813\pi\)
0.987431 − 0.158050i \(-0.0505206\pi\)
\(6\) −9.58533 − 152.434i −0.0443765 − 0.705713i
\(7\) 95.5752 − 165.541i 0.278645 − 0.482627i −0.692403 − 0.721511i \(-0.743448\pi\)
0.971048 + 0.238884i \(0.0767814\pi\)
\(8\) 181.019i 0.353553i
\(9\) −282.543 + 672.020i −0.387576 + 0.921838i
\(10\) 1321.11 1.32111
\(11\) −673.077 − 388.601i −0.505693 − 0.291962i 0.225369 − 0.974274i \(-0.427641\pi\)
−0.731061 + 0.682312i \(0.760975\pi\)
\(12\) 384.190 − 773.882i 0.222332 − 0.447849i
\(13\) 45.5802 + 78.9472i 0.0207466 + 0.0359341i 0.876212 − 0.481925i \(-0.160062\pi\)
−0.855466 + 0.517859i \(0.826729\pi\)
\(14\) 936.442 − 540.655i 0.341269 − 0.197032i
\(15\) −5647.92 − 2803.88i −1.67346 − 0.830780i
\(16\) −512.000 + 886.810i −0.125000 + 0.216506i
\(17\) 7047.39i 1.43444i 0.696848 + 0.717219i \(0.254585\pi\)
−0.696848 + 0.717219i \(0.745415\pi\)
\(18\) −3284.93 + 2493.06i −0.563260 + 0.427479i
\(19\) 2731.10 0.398177 0.199088 − 0.979982i \(-0.436202\pi\)
0.199088 + 0.979982i \(0.436202\pi\)
\(20\) 6472.09 + 3736.66i 0.809011 + 0.467083i
\(21\) −5150.89 + 323.897i −0.556191 + 0.0349743i
\(22\) −2198.26 − 3807.50i −0.206448 − 0.357579i
\(23\) −17228.9 + 9947.14i −1.41604 + 0.817550i −0.995948 − 0.0899317i \(-0.971335\pi\)
−0.420091 + 0.907482i \(0.638002\pi\)
\(24\) 4071.01 − 2704.58i 0.294489 − 0.195644i
\(25\) 19458.3 − 33702.7i 1.24533 − 2.15697i
\(26\) 515.681i 0.0293401i
\(27\) 19334.7 − 3686.33i 0.982306 − 0.187285i
\(28\) 6116.81 0.278645
\(29\) 27104.3 + 15648.7i 1.11133 + 0.641629i 0.939175 − 0.343440i \(-0.111592\pi\)
0.172159 + 0.985069i \(0.444926\pi\)
\(30\) −19738.5 − 29710.9i −0.731055 − 1.10040i
\(31\) 6174.50 + 10694.6i 0.207261 + 0.358986i 0.950851 − 0.309650i \(-0.100212\pi\)
−0.743590 + 0.668636i \(0.766879\pi\)
\(32\) −5016.55 + 2896.31i −0.153093 + 0.0883883i
\(33\) 1316.94 + 20943.1i 0.0366458 + 0.582773i
\(34\) −19933.0 + 34525.0i −0.507150 + 0.878410i
\(35\) − 44641.5i − 1.04120i
\(36\) −23144.2 + 2922.26i −0.496061 + 0.0626342i
\(37\) −27972.0 −0.552228 −0.276114 − 0.961125i \(-0.589047\pi\)
−0.276114 + 0.961125i \(0.589047\pi\)
\(38\) 13379.6 + 7724.70i 0.243833 + 0.140777i
\(39\) 1094.47 − 2204.61i 0.0184505 − 0.0371653i
\(40\) 21137.7 + 36611.7i 0.330277 + 0.572057i
\(41\) −37428.2 + 21609.2i −0.543059 + 0.313535i −0.746318 − 0.665590i \(-0.768180\pi\)
0.203259 + 0.979125i \(0.434847\pi\)
\(42\) −26150.2 − 12982.1i −0.352961 − 0.175226i
\(43\) 19256.1 − 33352.5i 0.242193 − 0.419491i −0.719146 − 0.694860i \(-0.755467\pi\)
0.961339 + 0.275369i \(0.0887999\pi\)
\(44\) − 24870.5i − 0.291962i
\(45\) 21327.2 + 168911.i 0.234043 + 1.85361i
\(46\) −112539. −1.15619
\(47\) −143771. − 83006.2i −1.38477 − 0.799497i −0.392050 − 0.919944i \(-0.628234\pi\)
−0.992720 + 0.120447i \(0.961567\pi\)
\(48\) 27593.5 − 1735.13i 0.249507 − 0.0156895i
\(49\) 40555.3 + 70243.8i 0.344714 + 0.597062i
\(50\) 190651. − 110073.i 1.52521 − 0.880581i
\(51\) 158491. − 105294.i 1.19480 − 0.793767i
\(52\) −1458.57 + 2526.31i −0.0103733 + 0.0179671i
\(53\) − 54741.5i − 0.367696i −0.982955 − 0.183848i \(-0.941145\pi\)
0.982955 − 0.183848i \(-0.0588554\pi\)
\(54\) 105147. + 36627.6i 0.667752 + 0.232609i
\(55\) −181509. −1.09096
\(56\) 29966.1 + 17301.0i 0.170634 + 0.0985158i
\(57\) −40804.8 − 61420.6i −0.220337 − 0.331657i
\(58\) 88522.3 + 153325.i 0.453700 + 0.785832i
\(59\) −14102.1 + 8141.84i −0.0686637 + 0.0396430i −0.533939 − 0.845523i \(-0.679289\pi\)
0.465275 + 0.885166i \(0.345955\pi\)
\(60\) −12663.3 − 201382.i −0.0586263 − 0.932324i
\(61\) 29443.7 − 50998.0i 0.129719 − 0.224680i −0.793849 − 0.608115i \(-0.791926\pi\)
0.923568 + 0.383436i \(0.125259\pi\)
\(62\) 69856.5i 0.293111i
\(63\) 84242.8 + 111001.i 0.336908 + 0.443920i
\(64\) −32768.0 −0.125000
\(65\) 18437.4 + 10644.9i 0.0671368 + 0.0387614i
\(66\) −52784.4 + 106325.i −0.183600 + 0.369830i
\(67\) −147998. − 256341.i −0.492076 − 0.852301i 0.507882 − 0.861427i \(-0.330429\pi\)
−0.999958 + 0.00912565i \(0.997095\pi\)
\(68\) −195303. + 112758.i −0.621130 + 0.358609i
\(69\) 481120. + 238849.i 1.46456 + 0.727071i
\(70\) 126265. − 218698.i 0.368120 − 0.637603i
\(71\) 157251.i 0.439358i 0.975572 + 0.219679i \(0.0705010\pi\)
−0.975572 + 0.219679i \(0.929499\pi\)
\(72\) −121649. − 51145.7i −0.325919 − 0.137029i
\(73\) 80297.0 0.206410 0.103205 − 0.994660i \(-0.467090\pi\)
0.103205 + 0.994660i \(0.467090\pi\)
\(74\) −137034. − 79116.8i −0.338169 − 0.195242i
\(75\) −1.04868e6 + 65942.7i −2.48575 + 0.156309i
\(76\) 43697.5 + 75686.3i 0.0995442 + 0.172416i
\(77\) −128659. + 74281.3i −0.281817 + 0.162707i
\(78\) 11597.3 − 7704.71i 0.0244385 − 0.0162358i
\(79\) 188424. − 326360.i 0.382169 − 0.661936i −0.609203 − 0.793014i \(-0.708511\pi\)
0.991372 + 0.131078i \(0.0418438\pi\)
\(80\) 239146.i 0.467083i
\(81\) −371780. − 379749.i −0.699570 − 0.714564i
\(82\) −244480. −0.443406
\(83\) 733992. + 423771.i 1.28368 + 0.741134i 0.977519 − 0.210846i \(-0.0676218\pi\)
0.306162 + 0.951980i \(0.400955\pi\)
\(84\) −91390.3 − 137563.i −0.154192 − 0.232094i
\(85\) 822929. + 1.42535e6i 1.34000 + 2.32095i
\(86\) 188670. − 108929.i 0.296625 − 0.171256i
\(87\) −53032.3 − 843363.i −0.0805346 − 1.28073i
\(88\) 70344.3 − 121840.i 0.103224 − 0.178789i
\(89\) 1128.91i 0.00160136i 1.00000 0.000800679i \(0.000254864\pi\)
−1.00000 0.000800679i \(0.999745\pi\)
\(90\) −373270. + 887812.i −0.512030 + 1.21785i
\(91\) 17425.3 0.0231237
\(92\) −551326. − 318308.i −0.708019 − 0.408775i
\(93\) 148261. − 298646.i 0.184323 − 0.371286i
\(94\) −469554. − 813291.i −0.565330 − 0.979180i
\(95\) 552371. − 318912.i 0.644259 − 0.371963i
\(96\) 140088. + 69545.8i 0.158338 + 0.0786063i
\(97\) 675152. − 1.16940e6i 0.739753 − 1.28129i −0.212854 − 0.977084i \(-0.568276\pi\)
0.952607 − 0.304205i \(-0.0983908\pi\)
\(98\) 458831.i 0.487499i
\(99\) 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 18.7.d.a.11.5 yes 12
3.2 odd 2 54.7.d.a.35.1 12
4.3 odd 2 144.7.q.c.65.4 12
9.2 odd 6 162.7.b.c.161.7 12
9.4 even 3 54.7.d.a.17.1 12
9.5 odd 6 inner 18.7.d.a.5.5 ✓ 12
9.7 even 3 162.7.b.c.161.6 12
12.11 even 2 432.7.q.b.305.1 12
36.23 even 6 144.7.q.c.113.4 12
36.31 odd 6 432.7.q.b.17.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.d.a.5.5 ✓ 12 9.5 odd 6 inner
18.7.d.a.11.5 yes 12 1.1 even 1 trivial
54.7.d.a.17.1 12 9.4 even 3
54.7.d.a.35.1 12 3.2 odd 2
144.7.q.c.65.4 12 4.3 odd 2
144.7.q.c.113.4 12 36.23 even 6
162.7.b.c.161.6 12 9.7 even 3
162.7.b.c.161.7 12 9.2 odd 6
432.7.q.b.17.1 12 36.31 odd 6
432.7.q.b.305.1 12 12.11 even 2