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Properties

Label 18.7.d.a.5.5
Level $18$
Weight $7$
Character 18.5
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 5.5
Root \(8.88570i\) of defining polynomial
Character \(\chi\) \(=\) 18.5
Dual form 18.7.d.a.11.5

$q$-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 \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\)

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 0.987431 + 0.158050i \(0.0505206\pi\)
0.630591 + 0.776116i \(0.282813\pi\)
\(6\) −9.58533 + 152.434i −0.0443765 + 0.705713i
\(7\) 95.5752 + 165.541i 0.278645 + 0.482627i 0.971048 − 0.238884i \(-0.0767814\pi\)
−0.692403 + 0.721511i \(0.743448\pi\)
\(8\) − 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 −0.731061 − 0.682312i \(-0.760975\pi\)
0.225369 + 0.974274i \(0.427641\pi\)
\(12\) 384.190 + 773.882i 0.222332 + 0.447849i
\(13\) 45.5802 − 78.9472i 0.0207466 − 0.0359341i −0.855466 − 0.517859i \(-0.826729\pi\)
0.876212 + 0.481925i \(0.160062\pi\)
\(14\) 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 \(-0.745415\pi\)
0.696848 − 0.717219i \(-0.254585\pi\)
\(18\) −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 −0.420091 − 0.907482i \(-0.638002\pi\)
−0.995948 + 0.0899317i \(0.971335\pi\)
\(24\) 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 0.172159 − 0.985069i \(-0.444926\pi\)
0.939175 + 0.343440i \(0.111592\pi\)
\(30\) −19738.5 + 29710.9i −0.731055 + 1.10040i
\(31\) 6174.50 − 10694.6i 0.207261 − 0.358986i −0.743590 − 0.668636i \(-0.766879\pi\)
0.950851 + 0.309650i \(0.100212\pi\)
\(32\) −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 0.203259 − 0.979125i \(-0.434847\pi\)
−0.746318 + 0.665590i \(0.768180\pi\)
\(42\) −26150.2 + 12982.1i −0.352961 + 0.175226i
\(43\) 19256.1 + 33352.5i 0.242193 + 0.419491i 0.961339 − 0.275369i \(-0.0887999\pi\)
−0.719146 + 0.694860i \(0.755467\pi\)
\(44\) 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 −0.992720 − 0.120447i \(-0.961567\pi\)
−0.392050 + 0.919944i \(0.628234\pi\)
\(48\) 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 \(0.0588554\pi\)
−0.982955 + 0.183848i \(0.941145\pi\)
\(54\) 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 0.465275 − 0.885166i \(-0.345955\pi\)
−0.533939 + 0.845523i \(0.679289\pi\)
\(60\) −12663.3 + 201382.i −0.0586263 + 0.932324i
\(61\) 29443.7 + 50998.0i 0.129719 + 0.224680i 0.923568 − 0.383436i \(-0.125259\pi\)
−0.793849 + 0.608115i \(0.791926\pi\)
\(62\) − 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 −0.999958 − 0.00912565i \(-0.997095\pi\)
0.507882 + 0.861427i \(0.330429\pi\)
\(68\) −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 \(-0.929499\pi\)
0.975572 − 0.219679i \(-0.0705010\pi\)
\(72\) −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 0.991372 − 0.131078i \(-0.0418438\pi\)
−0.609203 + 0.793014i \(0.708511\pi\)
\(80\) − 239146.i − 0.467083i
\(81\) −371780. + 379749.i −0.699570 + 0.714564i
\(82\) −244480. −0.443406
\(83\) 733992. − 423771.i 1.28368 − 0.741134i 0.306162 − 0.951980i \(-0.400955\pi\)
0.977519 + 0.210846i \(0.0676218\pi\)
\(84\) −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 \(-0.999745\pi\)
1.00000 0.000800679i \(-0.000254864\pi\)
\(90\) −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 0.952607 + 0.304205i \(0.0983908\pi\)
−0.212854 + 0.977084i \(0.568276\pi\)
\(98\) − 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.5.5 ✓ 12
3.2 odd 2 54.7.d.a.17.1 12
4.3 odd 2 144.7.q.c.113.4 12
9.2 odd 6 inner 18.7.d.a.11.5 yes 12
9.4 even 3 162.7.b.c.161.7 12
9.5 odd 6 162.7.b.c.161.6 12
9.7 even 3 54.7.d.a.35.1 12
12.11 even 2 432.7.q.b.17.1 12
36.7 odd 6 432.7.q.b.305.1 12
36.11 even 6 144.7.q.c.65.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.d.a.5.5 ✓ 12 1.1 even 1 trivial
18.7.d.a.11.5 yes 12 9.2 odd 6 inner
54.7.d.a.17.1 12 3.2 odd 2
54.7.d.a.35.1 12 9.7 even 3
144.7.q.c.65.4 12 36.11 even 6
144.7.q.c.113.4 12 4.3 odd 2
162.7.b.c.161.6 12 9.5 odd 6
162.7.b.c.161.7 12 9.4 even 3
432.7.q.b.17.1 12 12.11 even 2
432.7.q.b.305.1 12 36.7 odd 6