Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,6,Mod(7,18)] 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("18.7"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 18.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.88690875663\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.47347183152.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 13.3
Root \(0.500000 + 5.23712i\) of defining polynomial
Character \(\chi\) \(=\) 18.13
Dual form 18.6.c.b.7.3

$q$-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+(-2.00000 - 3.46410i) q^{2} +(15.5145 - 1.51695i) q^{3} +(-8.00000 + 13.8564i) q^{4} +(33.0434 - 57.2329i) q^{5} +(-36.2838 - 50.7098i) q^{6} +(-57.0952 - 98.8918i) q^{7} +64.0000 q^{8} +(238.398 - 47.0695i) q^{9} -264.347 q^{10} +(192.812 + 333.961i) q^{11} +(-103.096 + 227.110i) q^{12} +(-516.287 + 894.236i) q^{13} +(-228.381 + 395.567i) q^{14} +(425.832 - 938.063i) q^{15} +(-128.000 - 221.703i) q^{16} +959.020 q^{17} +(-639.849 - 731.695i) q^{18} -464.576 q^{19} +(528.695 + 915.726i) q^{20} +(-1035.82 - 1447.64i) q^{21} +(771.249 - 1335.84i) q^{22} +(-1151.70 + 1994.81i) q^{23} +(992.926 - 97.0850i) q^{24} +(-621.235 - 1076.01i) q^{25} +4130.30 q^{26} +(3627.21 - 1091.90i) q^{27} +1827.05 q^{28} +(-3549.13 - 6147.28i) q^{29} +(-4101.21 + 401.003i) q^{30} +(-3881.53 + 6723.00i) q^{31} +(-512.000 + 886.810i) q^{32} +(3497.98 + 4888.74i) q^{33} +(-1918.04 - 3322.14i) q^{34} -7546.48 q^{35} +(-1254.97 + 3679.89i) q^{36} +9317.57 q^{37} +(929.153 + 1609.34i) q^{38} +(-6653.41 + 14656.8i) q^{39} +(2114.78 - 3662.90i) q^{40} +(6661.64 - 11538.3i) q^{41} +(-2943.15 + 6483.46i) q^{42} +(1056.29 + 1829.55i) q^{43} -6169.99 q^{44} +(5183.55 - 15199.5i) q^{45} +9213.62 q^{46} +(-1247.40 - 2160.56i) q^{47} +(-2322.16 - 3245.43i) q^{48} +(1883.78 - 3262.80i) q^{49} +(-2484.94 + 4304.04i) q^{50} +(14878.7 - 1454.79i) q^{51} +(-8260.60 - 14307.8i) q^{52} -10044.2 q^{53} +(-11036.9 - 10381.2i) q^{54} +25484.7 q^{55} +(-3654.09 - 6329.07i) q^{56} +(-7207.66 + 704.741i) q^{57} +(-14196.5 + 24589.1i) q^{58} +(-2720.33 + 4711.75i) q^{59} +(9591.53 + 13405.0i) q^{60} +(-17094.4 - 29608.4i) q^{61} +31052.2 q^{62} +(-18266.1 - 20888.1i) q^{63} +4096.00 q^{64} +(34119.8 + 59097.2i) q^{65} +(9939.11 - 21894.8i) q^{66} +(-26792.6 + 46406.2i) q^{67} +(-7672.16 + 13288.6i) q^{68} +(-14842.0 + 32695.5i) q^{69} +(15093.0 + 26141.8i) q^{70} -970.010 q^{71} +(15257.5 - 3012.45i) q^{72} -72400.3 q^{73} +(-18635.1 - 32277.0i) q^{74} +(-11270.4 - 15751.3i) q^{75} +(3716.61 - 6437.36i) q^{76} +(22017.3 - 38135.1i) q^{77} +(64079.4 - 6265.47i) q^{78} +(-16098.9 - 27884.0i) q^{79} -16918.2 q^{80} +(54617.9 - 22442.5i) q^{81} -53293.1 q^{82} +(18046.6 + 31257.6i) q^{83} +(28345.7 - 2771.54i) q^{84} +(31689.3 - 54887.5i) q^{85} +(4225.17 - 7318.21i) q^{86} +(-64388.1 - 89987.9i) q^{87} +(12340.0 + 21373.5i) q^{88} +42622.2 q^{89} +(-63019.8 + 12442.7i) q^{90} +117910. q^{91} +(-18427.2 - 31916.9i) q^{92} +(-50021.4 + 110192. i) q^{93} +(-4989.60 + 8642.25i) q^{94} +(-15351.2 + 26589.0i) q^{95} +(-6598.16 + 14535.1i) q^{96} +(21926.9 + 37978.6i) q^{97} -15070.2 q^{98} +(61685.4 + 70539.9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{2} + 9 q^{3} - 48 q^{4} - 54 q^{5} - 12 q^{6} - 132 q^{7} + 384 q^{8} - 177 q^{9} + 432 q^{10} - 315 q^{11} - 96 q^{12} - 744 q^{13} - 528 q^{14} + 2286 q^{15} - 768 q^{16} + 2898 q^{17} + 1056 q^{18}+ \cdots + 282168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 − 3.46410i −0.353553 − 0.612372i
\(3\) 15.5145 − 1.51695i 0.995254 − 0.0973126i
\(4\) −8.00000 + 13.8564i −0.250000 + 0.433013i
\(5\) 33.0434 − 57.2329i 0.591099 − 1.02381i −0.402986 − 0.915206i \(-0.632028\pi\)
0.994085 − 0.108607i \(-0.0346389\pi\)
\(6\) −36.2838 − 50.7098i −0.411467 − 0.575061i
\(7\) −57.0952 − 98.8918i −0.440407 − 0.762808i 0.557312 − 0.830303i \(-0.311833\pi\)
−0.997720 + 0.0674952i \(0.978499\pi\)
\(8\) 64.0000 0.353553
\(9\) 238.398 − 47.0695i 0.981061 − 0.193702i
\(10\) −264.347 −0.835940
\(11\) 192.812 + 333.961i 0.480455 + 0.832173i 0.999749 − 0.0224231i \(-0.00713811\pi\)
−0.519293 + 0.854596i \(0.673805\pi\)
\(12\) −103.096 + 227.110i −0.206676 + 0.455286i
\(13\) −516.287 + 894.236i −0.847292 + 1.46755i 0.0363242 + 0.999340i \(0.488435\pi\)
−0.883616 + 0.468212i \(0.844898\pi\)
\(14\) −228.381 + 395.567i −0.311415 + 0.539386i
\(15\) 425.832 − 938.063i 0.488663 − 1.07648i
\(16\) −128.000 − 221.703i −0.125000 − 0.216506i
\(17\) 959.020 0.804832 0.402416 − 0.915457i \(-0.368171\pi\)
0.402416 + 0.915457i \(0.368171\pi\)
\(18\) −639.849 − 731.695i −0.465475 − 0.532291i
\(19\) −464.576 −0.295239 −0.147619 − 0.989044i \(-0.547161\pi\)
−0.147619 + 0.989044i \(0.547161\pi\)
\(20\) 528.695 + 915.726i 0.295549 + 0.511906i
\(21\) −1035.82 − 1447.64i −0.512548 − 0.716330i
\(22\) 771.249 − 1335.84i 0.339733 − 0.588435i
\(23\) −1151.70 + 1994.81i −0.453963 + 0.786288i −0.998628 − 0.0523662i \(-0.983324\pi\)
0.544664 + 0.838654i \(0.316657\pi\)
\(24\) 992.926 − 97.0850i 0.351875 − 0.0344052i
\(25\) −621.235 − 1076.01i −0.198795 − 0.344323i
\(26\) 4130.30 1.19825
\(27\) 3627.21 − 1091.90i 0.957555 − 0.288252i
\(28\) 1827.05 0.440407
\(29\) −3549.13 − 6147.28i −0.783659 − 1.35734i −0.929797 − 0.368073i \(-0.880018\pi\)
0.146138 − 0.989264i \(-0.453316\pi\)
\(30\) −4101.21 + 401.003i −0.831972 + 0.0813475i
\(31\) −3881.53 + 6723.00i −0.725435 + 1.25649i 0.233360 + 0.972391i \(0.425028\pi\)
−0.958795 + 0.284100i \(0.908305\pi\)
\(32\) −512.000 + 886.810i −0.0883883 + 0.153093i
\(33\) 3497.98 + 4888.74i 0.559156 + 0.781469i
\(34\) −1918.04 − 3322.14i −0.284551 − 0.492857i
\(35\) −7546.48 −1.04130
\(36\) −1254.97 + 3679.89i −0.161390 + 0.473237i
\(37\) 9317.57 1.11892 0.559459 − 0.828858i \(-0.311009\pi\)
0.559459 + 0.828858i \(0.311009\pi\)
\(38\) 929.153 + 1609.34i 0.104383 + 0.180796i
\(39\) −6653.41 + 14656.8i −0.700459 + 1.54304i
\(40\) 2114.78 − 3662.90i 0.208985 − 0.361972i
\(41\) 6661.64 − 11538.3i 0.618901 − 1.07197i −0.370785 − 0.928719i \(-0.620911\pi\)
0.989687 − 0.143250i \(-0.0457552\pi\)
\(42\) −2943.15 + 6483.46i −0.257448 + 0.567131i
\(43\) 1056.29 + 1829.55i 0.0871190 + 0.150895i 0.906292 − 0.422652i \(-0.138901\pi\)
−0.819173 + 0.573546i \(0.805567\pi\)
\(44\) −6169.99 −0.480455
\(45\) 5183.55 − 15199.5i 0.381589 − 1.11892i
\(46\) 9213.62 0.642001
\(47\) −1247.40 − 2160.56i −0.0823686 − 0.142667i 0.821898 − 0.569634i \(-0.192915\pi\)
−0.904267 + 0.426968i \(0.859582\pi\)
\(48\) −2322.16 − 3245.43i −0.145476 − 0.203315i
\(49\) 1883.78 − 3262.80i 0.112083 − 0.194133i
\(50\) −2484.94 + 4304.04i −0.140569 + 0.243473i
\(51\) 14878.7 − 1454.79i 0.801012 − 0.0783203i
\(52\) −8260.60 − 14307.8i −0.423646 − 0.733776i
\(53\) −10044.2 −0.491163 −0.245582 − 0.969376i \(-0.578979\pi\)
−0.245582 + 0.969376i \(0.578979\pi\)
\(54\) −11036.9 − 10381.2i −0.515064 − 0.484468i
\(55\) 25484.7 1.13599
\(56\) −3654.09 − 6329.07i −0.155707 − 0.269693i
\(57\) −7207.66 + 704.741i −0.293837 + 0.0287304i
\(58\) −14196.5 + 24589.1i −0.554131 + 0.959783i
\(59\) −2720.33 + 4711.75i −0.101740 + 0.176219i −0.912402 − 0.409296i \(-0.865774\pi\)
0.810662 + 0.585515i \(0.199108\pi\)
\(60\) 9591.53 + 13405.0i 0.343962 + 0.480716i
\(61\) −17094.4 − 29608.4i −0.588206 − 1.01880i −0.994467 − 0.105046i \(-0.966501\pi\)
0.406261 − 0.913757i \(-0.366832\pi\)
\(62\) 31052.2 1.02592
\(63\) −18266.1 − 20888.1i −0.579823 − 0.663053i
\(64\) 4096.00 0.125000
\(65\) 34119.8 + 59097.2i 1.00167 + 1.73494i
\(66\) 9939.11 − 21894.8i 0.280859 − 0.618703i
\(67\) −26792.6 + 46406.2i −0.729169 + 1.26296i 0.228066 + 0.973646i \(0.426760\pi\)
−0.957235 + 0.289311i \(0.906574\pi\)
\(68\) −7672.16 + 13288.6i −0.201208 + 0.348503i
\(69\) −14842.0 + 32695.5i −0.375293 + 0.826732i
\(70\) 15093.0 + 26141.8i 0.368154 + 0.637661i
\(71\) −970.010 −0.0228365 −0.0114183 − 0.999935i \(-0.503635\pi\)
−0.0114183 + 0.999935i \(0.503635\pi\)
\(72\) 15257.5 − 3012.45i 0.346857 − 0.0684838i
\(73\) −72400.3 −1.59013 −0.795066 − 0.606523i \(-0.792564\pi\)
−0.795066 + 0.606523i \(0.792564\pi\)
\(74\) −18635.1 − 32277.0i −0.395597 − 0.685195i
\(75\) −11270.4 − 15751.3i −0.231359 − 0.323344i
\(76\) 3716.61 − 6437.36i 0.0738097 − 0.127842i
\(77\) 22017.3 − 38135.1i 0.423192 − 0.732990i
\(78\) 64079.4 − 6265.47i 1.19256 − 0.116605i
\(79\) −16098.9 − 27884.0i −0.290220 − 0.502676i 0.683642 − 0.729818i \(-0.260395\pi\)
−0.973862 + 0.227142i \(0.927062\pi\)
\(80\) −16918.2 −0.295549
\(81\) 54617.9 − 22442.5i 0.924959 − 0.380066i
\(82\) −53293.1 −0.875258
\(83\) 18046.6 + 31257.6i 0.287541 + 0.498036i 0.973222 − 0.229866i \(-0.0738289\pi\)
−0.685681 + 0.727902i \(0.740496\pi\)
\(84\) 28345.7 − 2771.54i 0.438317 − 0.0428572i
\(85\) 31689.3 − 54887.5i 0.475735 − 0.823998i
\(86\) 4225.17 − 7318.21i 0.0616024 − 0.106699i
\(87\) −64388.1 − 89987.9i −0.912026 − 1.27464i
\(88\) 12340.0 + 21373.5i 0.169867 + 0.294218i
\(89\) 42622.2 0.570375 0.285188 − 0.958472i \(-0.407944\pi\)
0.285188 + 0.958472i \(0.407944\pi\)
\(90\) −63019.8 + 12442.7i −0.820107 + 0.161923i
\(91\) 117910. 1.49261
\(92\) −18427.2 − 31916.9i −0.226982 − 0.393144i
\(93\) −50021.4 + 110192.i −0.599720 + 1.32112i
\(94\) −4989.60 + 8642.25i −0.0582434 + 0.100880i
\(95\) −15351.2 + 26589.0i −0.174515 + 0.302269i
\(96\) −6598.16 + 14535.1i −0.0730710 + 0.160968i
\(97\) 21926.9 + 37978.6i 0.236618 + 0.409835i 0.959742 − 0.280884i \(-0.0906275\pi\)
−0.723123 + 0.690719i \(0.757294\pi\)
\(98\) −15070.2 −0.158509
\(99\) 61685.4 + 70539.9i 0.632549 + 0.723347i
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 18.6.c.b.13.3 yes 6
3.2 odd 2 54.6.c.b.37.1 6
4.3 odd 2 144.6.i.b.49.1 6
9.2 odd 6 54.6.c.b.19.1 6
9.4 even 3 162.6.a.j.1.1 3
9.5 odd 6 162.6.a.i.1.3 3
9.7 even 3 inner 18.6.c.b.7.3 ✓ 6
12.11 even 2 432.6.i.b.145.1 6
36.7 odd 6 144.6.i.b.97.1 6
36.11 even 6 432.6.i.b.289.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.3 ✓ 6 9.7 even 3 inner
18.6.c.b.13.3 yes 6 1.1 even 1 trivial
54.6.c.b.19.1 6 9.2 odd 6
54.6.c.b.37.1 6 3.2 odd 2
144.6.i.b.49.1 6 4.3 odd 2
144.6.i.b.97.1 6 36.7 odd 6
162.6.a.i.1.3 3 9.5 odd 6
162.6.a.j.1.1 3 9.4 even 3
432.6.i.b.145.1 6 12.11 even 2
432.6.i.b.289.1 6 36.11 even 6