↑

Properties

Label 185.2.v.a.139.3
Level $185$
Weight $2$
Character 185.139
Analytic conductor $1.477$
Analytic rank $0$
Dimension $96$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [185,2,Mod(4,185)] 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("185.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.v (of order \(18\), degree \(6\), 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: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 139.3
Character \(\chi\) \(=\) 185.139
Dual form 185.2.v.a.4.3

$q$-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+(-0.391541 - 2.22054i) q^{2} +(2.59680 + 0.457886i) q^{3} +(-2.89811 + 1.05483i) q^{4} +(1.60069 + 1.56134i) q^{5} -5.94558i q^{6} +(-0.639059 + 0.761600i) q^{7} +(1.22222 + 2.11694i) q^{8} +(3.71464 + 1.35202i) q^{9} +(2.84028 - 4.16573i) q^{10} +(-2.23710 - 3.87477i) q^{11} +(-8.00880 + 1.41217i) q^{12} +(0.430863 - 0.156821i) q^{13} +(1.94138 + 1.12086i) q^{14} +(3.44176 + 4.78743i) q^{15} +(-0.502918 + 0.421999i) q^{16} +(3.61628 + 1.31622i) q^{17} +(1.54777 - 8.77787i) q^{18} +(-6.33129 - 1.11638i) q^{19} +(-6.28592 - 2.83649i) q^{20} +(-2.00823 + 1.68511i) q^{21} +(-7.72816 + 6.48469i) q^{22} +(-0.923384 + 1.59935i) q^{23} +(2.20453 + 6.05691i) q^{24} +(0.124429 + 4.99845i) q^{25} +(-0.516929 - 0.895348i) q^{26} +(2.17633 + 1.25651i) q^{27} +(1.04871 - 2.88130i) q^{28} +(-6.71853 + 3.87895i) q^{29} +(9.28308 - 9.51704i) q^{30} -1.28733i q^{31} +(4.87906 + 4.09402i) q^{32} +(-4.03509 - 11.0863i) q^{33} +(1.50679 - 8.54544i) q^{34} +(-2.21205 + 0.221299i) q^{35} -12.1916 q^{36} +(-2.79140 - 5.40445i) q^{37} +14.4960i q^{38} +(1.19067 - 0.209948i) q^{39} +(-1.34887 + 5.29686i) q^{40} +(10.2048 - 3.71426i) q^{41} +(4.52816 + 3.79957i) q^{42} -4.21419 q^{43} +(10.5706 + 8.86975i) q^{44} +(3.83503 + 7.96397i) q^{45} +(3.91296 + 1.42420i) q^{46} +(10.5851 + 6.11134i) q^{47} +(-1.49921 + 0.865567i) q^{48} +(1.04390 + 5.92024i) q^{49} +(11.0505 - 2.23340i) q^{50} +(8.78807 + 5.07379i) q^{51} +(-1.08327 + 0.908971i) q^{52} +(1.83762 + 2.19000i) q^{53} +(1.93800 - 5.32461i) q^{54} +(2.46893 - 9.69518i) q^{55} +(-2.39333 - 0.422009i) q^{56} +(-15.9299 - 5.79802i) q^{57} +(11.2439 + 13.4000i) q^{58} +(-4.99958 - 5.95827i) q^{59} +(-15.0245 - 10.2440i) q^{60} +(-0.980592 - 2.69415i) q^{61} +(-2.85858 + 0.504044i) q^{62} +(-3.40357 + 1.96505i) q^{63} +(6.52407 - 11.3000i) q^{64} +(0.934531 + 0.421702i) q^{65} +(-23.0377 + 13.3008i) q^{66} +(1.13608 - 1.35393i) q^{67} -11.8687 q^{68} +(-3.13016 + 3.73038i) q^{69} +(1.35751 + 4.82531i) q^{70} +(-1.98450 + 11.2547i) q^{71} +(1.67795 + 9.51611i) q^{72} -12.7316i q^{73} +(-10.9079 + 8.31448i) q^{74} +(-1.96560 + 13.0370i) q^{75} +(19.5263 - 3.44302i) q^{76} +(4.38066 + 0.772428i) q^{77} +(-0.932395 - 2.56173i) q^{78} +(9.83583 - 11.7219i) q^{79} +(-1.46390 - 0.109737i) q^{80} +(-4.00843 - 3.36347i) q^{81} +(-12.2433 - 21.2060i) q^{82} +(0.819933 - 2.25275i) q^{83} +(4.04259 - 7.00196i) q^{84} +(3.73348 + 7.75310i) q^{85} +(1.65003 + 9.35777i) q^{86} +(-19.2228 + 6.99653i) q^{87} +(5.46843 - 9.47160i) q^{88} +(-0.0452057 - 0.0538740i) q^{89} +(16.1828 - 11.6341i) q^{90} +(-0.155912 + 0.428364i) q^{91} +(0.989035 - 5.60910i) q^{92} +(0.589452 - 3.34295i) q^{93} +(9.42595 - 25.8976i) q^{94} +(-8.39140 - 11.6723i) q^{95} +(10.7954 + 12.8654i) q^{96} +(-6.55858 + 11.3598i) q^{97} +(12.7374 - 4.63604i) q^{98} +(-3.07125 - 17.4179i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q - 12 q^{4} - 12 q^{5} - 6 q^{9} - 3 q^{10} - 30 q^{11} - 36 q^{14} - 21 q^{15} - 18 q^{19} - 39 q^{20} - 24 q^{21} + 96 q^{24} + 36 q^{25} + 48 q^{26} - 18 q^{29} - 30 q^{30} - 54 q^{34} + 6 q^{35}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.391541 − 2.22054i −0.276861 − 1.57016i −0.732986 − 0.680243i \(-0.761874\pi\)
0.456125 − 0.889916i \(-0.349237\pi\)
\(3\) 2.59680 + 0.457886i 1.49926 + 0.264361i 0.862246 − 0.506489i \(-0.169057\pi\)
0.637017 + 0.770850i \(0.280168\pi\)
\(4\) −2.89811 + 1.05483i −1.44905 + 0.527413i
\(5\) 1.60069 + 1.56134i 0.715851 + 0.698253i
\(6\) − 5.94558i − 2.42727i
\(7\) −0.639059 + 0.761600i −0.241541 + 0.287858i −0.873173 − 0.487411i \(-0.837941\pi\)
0.631631 + 0.775269i \(0.282386\pi\)
\(8\) 1.22222 + 2.11694i 0.432119 + 0.748451i
\(9\) 3.71464 + 1.35202i 1.23821 + 0.450672i
\(10\) 2.84028 − 4.16573i 0.898177 − 1.31732i
\(11\) −2.23710 − 3.87477i −0.674510 − 1.16829i −0.976612 − 0.215010i \(-0.931022\pi\)
0.302102 − 0.953276i \(-0.402312\pi\)
\(12\) −8.00880 + 1.41217i −2.31194 + 0.407658i
\(13\) 0.430863 − 0.156821i 0.119500 − 0.0434945i −0.281578 − 0.959538i \(-0.590858\pi\)
0.401078 + 0.916044i \(0.368636\pi\)
\(14\) 1.94138 + 1.12086i 0.518856 + 0.299562i
\(15\) 3.44176 + 4.78743i 0.888659 + 1.23611i
\(16\) −0.502918 + 0.421999i −0.125730 + 0.105500i
\(17\) 3.61628 + 1.31622i 0.877076 + 0.319229i 0.741029 − 0.671473i \(-0.234338\pi\)
0.136047 + 0.990702i \(0.456560\pi\)
\(18\) 1.54777 − 8.77787i 0.364814 − 2.06896i
\(19\) −6.33129 − 1.11638i −1.45250 − 0.256114i −0.608967 − 0.793196i \(-0.708416\pi\)
−0.843530 + 0.537081i \(0.819527\pi\)
\(20\) −6.28592 − 2.83649i −1.40557 − 0.634258i
\(21\) −2.00823 + 1.68511i −0.438232 + 0.367721i
\(22\) −7.72816 + 6.48469i −1.64765 + 1.38254i
\(23\) −0.923384 + 1.59935i −0.192539 + 0.333487i −0.946091 − 0.323901i \(-0.895006\pi\)
0.753552 + 0.657388i \(0.228339\pi\)
\(24\) 2.20453 + 6.05691i 0.449998 + 1.23636i
\(25\) 0.124429 + 4.99845i 0.0248858 + 0.999690i
\(26\) −0.516929 − 0.895348i −0.101378 − 0.175592i
\(27\) 2.17633 + 1.25651i 0.418835 + 0.241815i
\(28\) 1.04871 − 2.88130i 0.198187 − 0.544514i
\(29\) −6.71853 + 3.87895i −1.24760 + 0.720302i −0.970630 − 0.240576i \(-0.922664\pi\)
−0.276970 + 0.960879i \(0.589330\pi\)
\(30\) 9.28308 − 9.51704i 1.69485 − 1.73757i
\(31\) − 1.28733i − 0.231212i −0.993295 − 0.115606i \(-0.963119\pi\)
0.993295 − 0.115606i \(-0.0368810\pi\)
\(32\) 4.87906 + 4.09402i 0.862505 + 0.723728i
\(33\) −4.03509 − 11.0863i −0.702420 − 1.92988i
\(34\) 1.50679 − 8.54544i 0.258413 − 1.46553i
\(35\) −2.21205 + 0.221299i −0.373905 + 0.0374064i
\(36\) −12.1916 −2.03193
\(37\) −2.79140 − 5.40445i −0.458903 − 0.888486i
\(38\) 14.4960i 2.35156i
\(39\) 1.19067 − 0.209948i 0.190660 − 0.0336185i
\(40\) −1.34887 + 5.29686i −0.213276 + 0.837508i
\(41\) 10.2048 − 3.71426i 1.59373 − 0.580070i 0.615598 − 0.788060i \(-0.288915\pi\)
0.978131 + 0.207990i \(0.0666923\pi\)
\(42\) 4.52816 + 3.79957i 0.698710 + 0.586287i
\(43\) −4.21419 −0.642658 −0.321329 − 0.946968i \(-0.604129\pi\)
−0.321329 + 0.946968i \(0.604129\pi\)
\(44\) 10.5706 + 8.86975i 1.59357 + 1.33716i
\(45\) 3.83503 + 7.96397i 0.571692 + 1.18720i
\(46\) 3.91296 + 1.42420i 0.576935 + 0.209987i
\(47\) 10.5851 + 6.11134i 1.54400 + 0.891430i 0.998580 + 0.0532682i \(0.0169638\pi\)
0.545422 + 0.838162i \(0.316370\pi\)
\(48\) −1.49921 + 0.865567i −0.216392 + 0.124934i
\(49\) 1.04390 + 5.92024i 0.149128 + 0.845749i
\(50\) 11.0505 − 2.23340i 1.56278 − 0.315850i
\(51\) 8.78807 + 5.07379i 1.23058 + 0.710473i
\(52\) −1.08327 + 0.908971i −0.150223 + 0.126052i
\(53\) 1.83762 + 2.19000i 0.252417 + 0.300819i 0.877342 − 0.479866i \(-0.159315\pi\)
−0.624925 + 0.780685i \(0.714870\pi\)
\(54\) 1.93800 − 5.32461i 0.263728 − 0.724587i
\(55\) 2.46893 − 9.69518i 0.332910 − 1.30730i
\(56\) −2.39333 − 0.422009i −0.319822 − 0.0563933i
\(57\) −15.9299 − 5.79802i −2.10997 − 0.767966i
\(58\) 11.2439 + 13.4000i 1.47640 + 1.75951i
\(59\) −4.99958 − 5.95827i −0.650890 − 0.775700i 0.335158 − 0.942162i \(-0.391210\pi\)
−0.986048 + 0.166462i \(0.946766\pi\)
\(60\) −15.0245 − 10.2440i −1.93965 − 1.32250i
\(61\) −0.980592 − 2.69415i −0.125552 − 0.344951i 0.860953 − 0.508685i \(-0.169868\pi\)
−0.986504 + 0.163734i \(0.947646\pi\)
\(62\) −2.85858 + 0.504044i −0.363040 + 0.0640137i
\(63\) −3.40357 + 1.96505i −0.428809 + 0.247573i
\(64\) 6.52407 − 11.3000i 0.815509 − 1.41250i
\(65\) 0.934531 + 0.421702i 0.115914 + 0.0523057i
\(66\) −23.0377 + 13.3008i −2.83575 + 1.63722i
\(67\) 1.13608 − 1.35393i 0.138794 − 0.165409i −0.692170 − 0.721735i \(-0.743345\pi\)
0.830964 + 0.556326i \(0.187789\pi\)
\(68\) −11.8687 −1.43930
\(69\) −3.13016 + 3.73038i −0.376828 + 0.449086i
\(70\) 1.35751 + 4.82531i 0.162254 + 0.576734i
\(71\) −1.98450 + 11.2547i −0.235517 + 1.33569i 0.606004 + 0.795461i \(0.292771\pi\)
−0.841521 + 0.540224i \(0.818340\pi\)
\(72\) 1.67795 + 9.51611i 0.197748 + 1.12148i
\(73\) − 12.7316i − 1.49012i −0.666999 − 0.745059i \(-0.732422\pi\)
0.666999 − 0.745059i \(-0.267578\pi\)
\(74\) −10.9079 + 8.31448i −1.26801 + 0.966539i
\(75\) −1.96560 + 13.0370i −0.226968 + 1.50538i
\(76\) 19.5263 − 3.44302i 2.23983 − 0.394942i
\(77\) 4.38066 + 0.772428i 0.499222 + 0.0880264i
\(78\) −0.932395 − 2.56173i −0.105573 − 0.290059i
\(79\) 9.83583 − 11.7219i 1.10662 − 1.31881i 0.163429 − 0.986555i \(-0.447745\pi\)
0.943188 − 0.332260i \(-0.107811\pi\)
\(80\) −1.46390 − 0.109737i −0.163669 − 0.0122690i
\(81\) −4.00843 − 3.36347i −0.445381 − 0.373719i
\(82\) −12.2433 − 21.2060i −1.35204 − 2.34181i
\(83\) 0.819933 − 2.25275i 0.0899993 − 0.247271i −0.886524 − 0.462682i \(-0.846887\pi\)
0.976523 + 0.215411i \(0.0691092\pi\)
\(84\) 4.04259 − 7.00196i 0.441082 − 0.763977i
\(85\) 3.73348 + 7.75310i 0.404953 + 0.840941i
\(86\) 1.65003 + 9.35777i 0.177927 + 1.00907i
\(87\) −19.2228 + 6.99653i −2.06090 + 0.750107i
\(88\) 5.46843 − 9.47160i 0.582937 − 1.00968i
\(89\) −0.0452057 − 0.0538740i −0.00479179 − 0.00571064i 0.763643 − 0.645638i \(-0.223409\pi\)
−0.768435 + 0.639928i \(0.778964\pi\)
\(90\) 16.1828 − 11.6341i 1.70581 − 1.22634i
\(91\) −0.155912 + 0.428364i −0.0163440 + 0.0449047i
\(92\) 0.989035 − 5.60910i 0.103114 − 0.584789i
\(93\) 0.589452 − 3.34295i 0.0611234 − 0.346648i
\(94\) 9.42595 − 25.8976i 0.972212 − 2.67113i
\(95\) −8.39140 − 11.6723i −0.860939 − 1.19755i
\(96\) 10.7954 + 12.8654i 1.10180 + 1.31307i
\(97\) −6.55858 + 11.3598i −0.665923 + 1.15341i 0.313111 + 0.949717i \(0.398629\pi\)
−0.979034 + 0.203696i \(0.934704\pi\)
\(98\) 12.7374 − 4.63604i 1.28667 − 0.468310i
\(99\) −3.07125 − 17.4179i −0.308672 − 1.75057i
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 185.2.v.a.139.3 yes 96
5.2 odd 4 925.2.bb.e.176.14 96
5.3 odd 4 925.2.bb.e.176.3 96
5.4 even 2 inner 185.2.v.a.139.14 yes 96
37.4 even 18 inner 185.2.v.a.4.14 yes 96
185.4 even 18 inner 185.2.v.a.4.3 ✓ 96
185.78 odd 36 925.2.bb.e.226.3 96
185.152 odd 36 925.2.bb.e.226.14 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.v.a.4.3 ✓ 96 185.4 even 18 inner
185.2.v.a.4.14 yes 96 37.4 even 18 inner
185.2.v.a.139.3 yes 96 1.1 even 1 trivial
185.2.v.a.139.14 yes 96 5.4 even 2 inner
925.2.bb.e.176.3 96 5.3 odd 4
925.2.bb.e.176.14 96 5.2 odd 4
925.2.bb.e.226.3 96 185.78 odd 36
925.2.bb.e.226.14 96 185.152 odd 36