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Properties

Label 185.2.u.a.8.13
Level $185$
Weight $2$
Character 185.8
Analytic conductor $1.477$
Analytic rank $0$
Dimension $68$
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] = [185,2,Mod(8,185)] 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("185.8"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(12)) 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.u (of order \(12\), degree \(4\), 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: \(68\)
Relative dimension: \(17\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 8.13
Character \(\chi\) \(=\) 185.8
Dual form 185.2.u.a.162.13

$q$-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+(0.971183 - 0.560713i) q^{2} +(0.168506 + 0.628874i) q^{3} +(-0.371203 + 0.642942i) q^{4} +(0.550691 + 2.16720i) q^{5} +(0.516268 + 0.516268i) q^{6} +(-0.377247 - 1.40791i) q^{7} +3.07540i q^{8} +(2.23099 - 1.28806i) q^{9} +(1.75000 + 1.79596i) q^{10} +0.304712i q^{11} +(-0.466879 - 0.125100i) q^{12} +(-0.187550 - 0.108282i) q^{13} +(-1.15581 - 1.15581i) q^{14} +(-1.27010 + 0.711501i) q^{15} +(0.982011 + 1.70089i) q^{16} +(0.836960 + 1.44966i) q^{17} +(1.44446 - 2.50189i) q^{18} +(-1.91513 - 7.14737i) q^{19} +(-1.59780 - 0.450407i) q^{20} +(0.821827 - 0.474482i) q^{21} +(0.170856 + 0.295931i) q^{22} -5.69268i q^{23} +(-1.93404 + 0.518224i) q^{24} +(-4.39348 + 2.38691i) q^{25} -0.242860 q^{26} +(2.56707 + 2.56707i) q^{27} +(1.04524 + 0.280071i) q^{28} +(-1.05734 - 1.05734i) q^{29} +(-0.834549 + 1.40316i) q^{30} +(2.64214 - 2.64214i) q^{31} +(-3.41933 - 1.97415i) q^{32} +(-0.191625 + 0.0513459i) q^{33} +(1.62568 + 0.938587i) q^{34} +(2.84346 - 1.59289i) q^{35} +1.91253i q^{36} +(-6.04562 + 0.671196i) q^{37} +(-5.86756 - 5.86756i) q^{38} +(0.0364924 - 0.136191i) q^{39} +(-6.66500 + 1.69360i) q^{40} +(2.07096 + 1.19567i) q^{41} +(0.532096 - 0.921617i) q^{42} +7.13995i q^{43} +(-0.195912 - 0.113110i) q^{44} +(4.02007 + 4.12567i) q^{45} +(-3.19196 - 5.52863i) q^{46} +(-3.92274 + 3.92274i) q^{47} +(-0.904172 + 0.904172i) q^{48} +(4.22229 - 2.43774i) q^{49} +(-2.92850 + 4.78160i) q^{50} +(-0.770618 + 0.770618i) q^{51} +(0.139238 - 0.0803891i) q^{52} +(2.13924 - 7.98376i) q^{53} +(3.93248 + 1.05370i) q^{54} +(-0.660371 + 0.167802i) q^{55} +(4.32988 - 1.16019i) q^{56} +(4.17208 - 2.40875i) q^{57} +(-1.61973 - 0.434004i) q^{58} +(6.39295 + 1.71299i) q^{59} +(0.0140099 - 1.08071i) q^{60} +(2.09439 + 7.81639i) q^{61} +(1.08452 - 4.04748i) q^{62} +(-2.65511 - 2.65511i) q^{63} -8.35577 q^{64} +(0.131386 - 0.466087i) q^{65} +(-0.157313 + 0.157313i) q^{66} +(-7.56615 + 2.02734i) q^{67} -1.24273 q^{68} +(3.57998 - 0.959252i) q^{69} +(1.86837 - 3.14135i) q^{70} +(5.55820 - 9.62708i) q^{71} +(3.96131 + 6.86119i) q^{72} +(-8.42960 + 8.42960i) q^{73} +(-5.49505 + 4.04171i) q^{74} +(-2.24139 - 2.36073i) q^{75} +(5.30625 + 1.42180i) q^{76} +(0.429006 - 0.114952i) q^{77} +(-0.0409234 - 0.152728i) q^{78} +(-1.51908 - 5.66927i) q^{79} +(-3.14538 + 3.06488i) q^{80} +(2.68239 - 4.64604i) q^{81} +2.68171 q^{82} +(1.06045 - 3.95767i) q^{83} +0.704516i q^{84} +(-2.68078 + 2.61217i) q^{85} +(4.00346 + 6.93419i) q^{86} +(0.486763 - 0.843098i) q^{87} -0.937113 q^{88} +(0.842850 - 3.14556i) q^{89} +(6.21753 + 1.75267i) q^{90} +(-0.0816982 + 0.304902i) q^{91} +(3.66007 + 2.11314i) q^{92} +(2.10679 + 1.21635i) q^{93} +(-1.61017 + 6.00923i) q^{94} +(14.4351 - 8.08646i) q^{95} +(0.665313 - 2.48298i) q^{96} -4.65513 q^{97} +(2.73374 - 4.73498i) q^{98} +(0.392488 + 0.679809i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 68 q - 6 q^{2} - 4 q^{3} + 30 q^{4} - 8 q^{6} - 2 q^{7} - 6 q^{10} - 10 q^{12} - 6 q^{13} - 16 q^{15} - 26 q^{16} - 10 q^{17} - 8 q^{18} - 4 q^{19} - 28 q^{20} - 12 q^{21} - 14 q^{22} + 20 q^{25} - 24 q^{26}+ \cdots - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(76\) \(112\)
\(\chi(n)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.971183 − 0.560713i 0.686730 − 0.396484i −0.115656 − 0.993289i \(-0.536897\pi\)
0.802386 + 0.596806i \(0.203564\pi\)
\(3\) 0.168506 + 0.628874i 0.0972871 + 0.363080i 0.997356 − 0.0726666i \(-0.0231509\pi\)
−0.900069 + 0.435747i \(0.856484\pi\)
\(4\) −0.371203 + 0.642942i −0.185601 + 0.321471i
\(5\) 0.550691 + 2.16720i 0.246276 + 0.969200i
\(6\) 0.516268 + 0.516268i 0.210765 + 0.210765i
\(7\) −0.377247 − 1.40791i −0.142586 − 0.532139i −0.999851 − 0.0172630i \(-0.994505\pi\)
0.857265 − 0.514876i \(-0.172162\pi\)
\(8\) 3.07540i 1.08732i
\(9\) 2.23099 − 1.28806i 0.743663 − 0.429354i
\(10\) 1.75000 + 1.79596i 0.553397 + 0.567934i
\(11\) 0.304712i 0.0918742i 0.998944 + 0.0459371i \(0.0146274\pi\)
−0.998944 + 0.0459371i \(0.985373\pi\)
\(12\) −0.466879 − 0.125100i −0.134776 − 0.0361132i
\(13\) −0.187550 − 0.108282i −0.0520170 − 0.0300320i 0.473766 − 0.880651i \(-0.342894\pi\)
−0.525783 + 0.850619i \(0.676228\pi\)
\(14\) −1.15581 − 1.15581i −0.308902 − 0.308902i
\(15\) −1.27010 + 0.711501i −0.327938 + 0.183709i
\(16\) 0.982011 + 1.70089i 0.245503 + 0.425223i
\(17\) 0.836960 + 1.44966i 0.202993 + 0.351593i 0.949491 − 0.313793i \(-0.101600\pi\)
−0.746499 + 0.665387i \(0.768267\pi\)
\(18\) 1.44446 − 2.50189i 0.340464 − 0.589700i
\(19\) −1.91513 − 7.14737i −0.439361 − 1.63972i −0.730409 − 0.683010i \(-0.760671\pi\)
0.291048 − 0.956708i \(-0.405996\pi\)
\(20\) −1.59780 − 0.450407i −0.357279 − 0.100714i
\(21\) 0.821827 − 0.474482i 0.179337 − 0.103540i
\(22\) 0.170856 + 0.295931i 0.0364266 + 0.0630927i
\(23\) − 5.69268i − 1.18701i −0.804832 − 0.593503i \(-0.797744\pi\)
0.804832 − 0.593503i \(-0.202256\pi\)
\(24\) −1.93404 + 0.518224i −0.394784 + 0.105782i
\(25\) −4.39348 + 2.38691i −0.878696 + 0.477382i
\(26\) −0.242860 −0.0476288
\(27\) 2.56707 + 2.56707i 0.494032 + 0.494032i
\(28\) 1.04524 + 0.280071i 0.197531 + 0.0529284i
\(29\) −1.05734 − 1.05734i −0.196342 − 0.196342i 0.602088 − 0.798430i \(-0.294336\pi\)
−0.798430 + 0.602088i \(0.794336\pi\)
\(30\) −0.834549 + 1.40316i −0.152367 + 0.256180i
\(31\) 2.64214 − 2.64214i 0.474542 − 0.474542i −0.428839 − 0.903381i \(-0.641077\pi\)
0.903381 + 0.428839i \(0.141077\pi\)
\(32\) −3.41933 − 1.97415i −0.604458 − 0.348984i
\(33\) −0.191625 + 0.0513459i −0.0333577 + 0.00893817i
\(34\) 1.62568 + 0.938587i 0.278802 + 0.160966i
\(35\) 2.84346 − 1.59289i 0.480633 − 0.269248i
\(36\) 1.91253i 0.318755i
\(37\) −6.04562 + 0.671196i −0.993893 + 0.110344i
\(38\) −5.86756 − 5.86756i −0.951844 − 0.951844i
\(39\) 0.0364924 − 0.136191i 0.00584345 − 0.0218081i
\(40\) −6.66500 + 1.69360i −1.05383 + 0.267781i
\(41\) 2.07096 + 1.19567i 0.323430 + 0.186732i 0.652920 − 0.757427i \(-0.273544\pi\)
−0.329491 + 0.944159i \(0.606877\pi\)
\(42\) 0.532096 − 0.921617i 0.0821042 − 0.142209i
\(43\) 7.13995i 1.08883i 0.838815 + 0.544416i \(0.183249\pi\)
−0.838815 + 0.544416i \(0.816751\pi\)
\(44\) −0.195912 − 0.113110i −0.0295349 − 0.0170520i
\(45\) 4.02007 + 4.12567i 0.599276 + 0.615018i
\(46\) −3.19196 − 5.52863i −0.470629 − 0.815153i
\(47\) −3.92274 + 3.92274i −0.572191 + 0.572191i −0.932740 − 0.360549i \(-0.882589\pi\)
0.360549 + 0.932740i \(0.382589\pi\)
\(48\) −0.904172 + 0.904172i −0.130506 + 0.130506i
\(49\) 4.22229 − 2.43774i 0.603185 − 0.348249i
\(50\) −2.92850 + 4.78160i −0.414152 + 0.676221i
\(51\) −0.770618 + 0.770618i −0.107908 + 0.107908i
\(52\) 0.139238 − 0.0803891i 0.0193088 − 0.0111480i
\(53\) 2.13924 − 7.98376i 0.293848 − 1.09665i −0.648281 − 0.761401i \(-0.724512\pi\)
0.942128 − 0.335253i \(-0.108822\pi\)
\(54\) 3.93248 + 1.05370i 0.535142 + 0.143391i
\(55\) −0.660371 + 0.167802i −0.0890444 + 0.0226265i
\(56\) 4.32988 − 1.16019i 0.578605 − 0.155037i
\(57\) 4.17208 − 2.40875i 0.552605 − 0.319047i
\(58\) −1.61973 − 0.434004i −0.212681 − 0.0569876i
\(59\) 6.39295 + 1.71299i 0.832291 + 0.223012i 0.649713 − 0.760180i \(-0.274889\pi\)
0.182578 + 0.983191i \(0.441556\pi\)
\(60\) 0.0140099 − 1.08071i 0.00180867 − 0.139519i
\(61\) 2.09439 + 7.81639i 0.268160 + 1.00079i 0.960288 + 0.279011i \(0.0900066\pi\)
−0.692128 + 0.721775i \(0.743327\pi\)
\(62\) 1.08452 − 4.04748i 0.137734 − 0.514030i
\(63\) −2.65511 − 2.65511i −0.334512 − 0.334512i
\(64\) −8.35577 −1.04447
\(65\) 0.131386 − 0.466087i 0.0162965 − 0.0578110i
\(66\) −0.157313 + 0.157313i −0.0193639 + 0.0193639i
\(67\) −7.56615 + 2.02734i −0.924353 + 0.247680i −0.689445 − 0.724338i \(-0.742145\pi\)
−0.234908 + 0.972018i \(0.575479\pi\)
\(68\) −1.24273 −0.150703
\(69\) 3.57998 − 0.959252i 0.430979 − 0.115480i
\(70\) 1.86837 − 3.14135i 0.223313 − 0.375464i
\(71\) 5.55820 − 9.62708i 0.659637 − 1.14252i −0.321073 − 0.947054i \(-0.604044\pi\)
0.980710 − 0.195470i \(-0.0626231\pi\)
\(72\) 3.96131 + 6.86119i 0.466845 + 0.808599i
\(73\) −8.42960 + 8.42960i −0.986611 + 0.986611i −0.999912 − 0.0133009i \(-0.995766\pi\)
0.0133009 + 0.999912i \(0.495766\pi\)
\(74\) −5.49505 + 4.04171i −0.638787 + 0.469839i
\(75\) −2.24139 − 2.36073i −0.258814 − 0.272594i
\(76\) 5.30625 + 1.42180i 0.608668 + 0.163092i
\(77\) 0.429006 − 0.114952i 0.0488898 − 0.0131000i
\(78\) −0.0409234 − 0.152728i −0.00463367 − 0.0172931i
\(79\) −1.51908 − 5.66927i −0.170909 − 0.637843i −0.997212 − 0.0746166i \(-0.976227\pi\)
0.826303 − 0.563226i \(-0.190440\pi\)
\(80\) −3.14538 + 3.06488i −0.351665 + 0.342664i
\(81\) 2.68239 − 4.64604i 0.298044 − 0.516227i
\(82\) 2.68171 0.296145
\(83\) 1.06045 − 3.95767i 0.116400 − 0.434410i −0.882988 − 0.469396i \(-0.844472\pi\)
0.999388 + 0.0349854i \(0.0111385\pi\)
\(84\) 0.704516i 0.0768690i
\(85\) −2.68078 + 2.61217i −0.290772 + 0.283329i
\(86\) 4.00346 + 6.93419i 0.431704 + 0.747733i
\(87\) 0.486763 − 0.843098i 0.0521864 − 0.0903896i
\(88\) −0.937113 −0.0998966
\(89\) 0.842850 − 3.14556i 0.0893419 − 0.333429i −0.906759 − 0.421649i \(-0.861451\pi\)
0.996101 + 0.0882206i \(0.0281181\pi\)
\(90\) 6.21753 + 1.75267i 0.655385 + 0.184748i
\(91\) −0.0816982 + 0.304902i −0.00856430 + 0.0319624i
\(92\) 3.66007 + 2.11314i 0.381588 + 0.220310i
\(93\) 2.10679 + 1.21635i 0.218464 + 0.126130i
\(94\) −1.61017 + 6.00923i −0.166076 + 0.619805i
\(95\) 14.4351 − 8.08646i 1.48101 − 0.829653i
\(96\) 0.665313 − 2.48298i 0.0679032 − 0.253418i
\(97\) −4.65513 −0.472657 −0.236328 − 0.971673i \(-0.575944\pi\)
−0.236328 + 0.971673i \(0.575944\pi\)
\(98\) 2.73374 − 4.73498i 0.276150 − 0.478306i
\(99\) 0.392488 + 0.679809i 0.0394465 + 0.0683234i
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.u.a.8.13 yes 68
5.2 odd 4 185.2.p.a.82.13 ✓ 68
5.3 odd 4 925.2.t.b.82.5 68
5.4 even 2 925.2.y.b.193.5 68
37.14 odd 12 185.2.p.a.88.13 yes 68
185.14 odd 12 925.2.t.b.643.5 68
185.88 even 12 925.2.y.b.532.5 68
185.162 even 12 inner 185.2.u.a.162.13 yes 68
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.p.a.82.13 ✓ 68 5.2 odd 4
185.2.p.a.88.13 yes 68 37.14 odd 12
185.2.u.a.8.13 yes 68 1.1 even 1 trivial
185.2.u.a.162.13 yes 68 185.162 even 12 inner
925.2.t.b.82.5 68 5.3 odd 4
925.2.t.b.643.5 68 185.14 odd 12
925.2.y.b.193.5 68 5.4 even 2
925.2.y.b.532.5 68 185.88 even 12