↑

Properties

Label 2205.4.a.u.1.2
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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] = [2205,4,Mod(1,2205)] 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("2205.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2205, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-8,0,20,-10,0,0,-48,0,40,14,0,-50] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.58579 q^{2} -1.31371 q^{4} -5.00000 q^{5} +24.0833 q^{8} +12.9289 q^{10} -38.2548 q^{11} -19.3431 q^{13} -51.7645 q^{16} -87.2254 q^{17} +44.2254 q^{19} +6.56854 q^{20} +98.9188 q^{22} -218.167 q^{23} +25.0000 q^{25} +50.0172 q^{26} +46.9411 q^{29} -194.558 q^{31} -58.8141 q^{32} +225.546 q^{34} +366.853 q^{37} -114.357 q^{38} -120.416 q^{40} -339.362 q^{41} -226.167 q^{43} +50.2557 q^{44} +564.132 q^{46} +11.6762 q^{47} -64.6447 q^{50} +25.4113 q^{52} +209.019 q^{53} +191.274 q^{55} -121.380 q^{58} -616.000 q^{59} -320.735 q^{61} +503.087 q^{62} +566.197 q^{64} +96.7157 q^{65} +14.5097 q^{67} +114.589 q^{68} +952.000 q^{71} -824.489 q^{73} -948.603 q^{74} -58.0993 q^{76} +156.275 q^{79} +258.823 q^{80} +877.519 q^{82} -1036.53 q^{83} +436.127 q^{85} +584.818 q^{86} -921.301 q^{88} -170.225 q^{89} +286.607 q^{92} -30.1921 q^{94} -221.127 q^{95} -1059.87 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 20 q^{4} - 10 q^{5} - 48 q^{8} + 40 q^{10} + 14 q^{11} - 50 q^{13} + 168 q^{16} - 50 q^{17} - 36 q^{19} - 100 q^{20} - 184 q^{22} - 244 q^{23} + 50 q^{25} + 216 q^{26} + 26 q^{29} + 120 q^{31}+ \cdots - 2742 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.58579 −0.914214 −0.457107 − 0.889412i \(-0.651114\pi\)
−0.457107 + 0.889412i \(0.651114\pi\)
\(3\) 0 0
\(4\) −1.31371 −0.164214
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 24.0833 1.06434
\(9\) 0 0
\(10\) 12.9289 0.408849
\(11\) −38.2548 −1.04857 −0.524285 − 0.851543i \(-0.675667\pi\)
−0.524285 + 0.851543i \(0.675667\pi\)
\(12\) 0 0
\(13\) −19.3431 −0.412679 −0.206339 − 0.978480i \(-0.566155\pi\)
−0.206339 + 0.978480i \(0.566155\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −51.7645 −0.808820
\(17\) −87.2254 −1.24443 −0.622214 − 0.782847i \(-0.713767\pi\)
−0.622214 + 0.782847i \(0.713767\pi\)
\(18\) 0 0
\(19\) 44.2254 0.534000 0.267000 − 0.963697i \(-0.413968\pi\)
0.267000 + 0.963697i \(0.413968\pi\)
\(20\) 6.56854 0.0734385
\(21\) 0 0
\(22\) 98.9188 0.958617
\(23\) −218.167 −1.97786 −0.988932 − 0.148371i \(-0.952597\pi\)
−0.988932 + 0.148371i \(0.952597\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 50.0172 0.377276
\(27\) 0 0
\(28\) 0 0
\(29\) 46.9411 0.300578 0.150289 − 0.988642i \(-0.451980\pi\)
0.150289 + 0.988642i \(0.451980\pi\)
\(30\) 0 0
\(31\) −194.558 −1.12722 −0.563609 − 0.826042i \(-0.690587\pi\)
−0.563609 + 0.826042i \(0.690587\pi\)
\(32\) −58.8141 −0.324905
\(33\) 0 0
\(34\) 225.546 1.13767
\(35\) 0 0
\(36\) 0 0
\(37\) 366.853 1.63001 0.815003 − 0.579457i \(-0.196735\pi\)
0.815003 + 0.579457i \(0.196735\pi\)
\(38\) −114.357 −0.488190
\(39\) 0 0
\(40\) −120.416 −0.475987
\(41\) −339.362 −1.29267 −0.646336 − 0.763053i \(-0.723699\pi\)
−0.646336 + 0.763053i \(0.723699\pi\)
\(42\) 0 0
\(43\) −226.167 −0.802095 −0.401047 − 0.916057i \(-0.631354\pi\)
−0.401047 + 0.916057i \(0.631354\pi\)
\(44\) 50.2557 0.172189
\(45\) 0 0
\(46\) 564.132 1.80819
\(47\) 11.6762 0.0362372 0.0181186 − 0.999836i \(-0.494232\pi\)
0.0181186 + 0.999836i \(0.494232\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −64.6447 −0.182843
\(51\) 0 0
\(52\) 25.4113 0.0677674
\(53\) 209.019 0.541717 0.270859 − 0.962619i \(-0.412692\pi\)
0.270859 + 0.962619i \(0.412692\pi\)
\(54\) 0 0
\(55\) 191.274 0.468935
\(56\) 0 0
\(57\) 0 0
\(58\) −121.380 −0.274792
\(59\) −616.000 −1.35926 −0.679630 − 0.733555i \(-0.737860\pi\)
−0.679630 + 0.733555i \(0.737860\pi\)
\(60\) 0 0
\(61\) −320.735 −0.673212 −0.336606 − 0.941646i \(-0.609279\pi\)
−0.336606 + 0.941646i \(0.609279\pi\)
\(62\) 503.087 1.03052
\(63\) 0 0
\(64\) 566.197 1.10585
\(65\) 96.7157 0.184556
\(66\) 0 0
\(67\) 14.5097 0.0264573 0.0132286 − 0.999912i \(-0.495789\pi\)
0.0132286 + 0.999912i \(0.495789\pi\)
\(68\) 114.589 0.204352
\(69\) 0 0
\(70\) 0 0
\(71\) 952.000 1.59129 0.795645 − 0.605763i \(-0.207132\pi\)
0.795645 + 0.605763i \(0.207132\pi\)
\(72\) 0 0
\(73\) −824.489 −1.32191 −0.660953 − 0.750427i \(-0.729848\pi\)
−0.660953 + 0.750427i \(0.729848\pi\)
\(74\) −948.603 −1.49017
\(75\) 0 0
\(76\) −58.0993 −0.0876901
\(77\) 0 0
\(78\) 0 0
\(79\) 156.275 0.222561 0.111280 − 0.993789i \(-0.464505\pi\)
0.111280 + 0.993789i \(0.464505\pi\)
\(80\) 258.823 0.361715
\(81\) 0 0
\(82\) 877.519 1.18178
\(83\) −1036.53 −1.37077 −0.685384 − 0.728182i \(-0.740366\pi\)
−0.685384 + 0.728182i \(0.740366\pi\)
\(84\) 0 0
\(85\) 436.127 0.556525
\(86\) 584.818 0.733286
\(87\) 0 0
\(88\) −921.301 −1.11603
\(89\) −170.225 −0.202740 −0.101370 − 0.994849i \(-0.532323\pi\)
−0.101370 + 0.994849i \(0.532323\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 286.607 0.324792
\(93\) 0 0
\(94\) −30.1921 −0.0331285
\(95\) −221.127 −0.238812
\(96\) 0 0
\(97\) −1059.87 −1.10942 −0.554710 − 0.832044i \(-0.687171\pi\)
−0.554710 + 0.832044i \(0.687171\pi\)
\(98\) 0 0
\(99\) 0 0
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 2205.4.a.u.1.2 2
3.2 odd 2 245.4.a.k.1.1 2
7.6 odd 2 315.4.a.f.1.2 2
15.14 odd 2 1225.4.a.m.1.2 2
21.2 odd 6 245.4.e.i.116.2 4
21.5 even 6 245.4.e.h.116.2 4
21.11 odd 6 245.4.e.i.226.2 4
21.17 even 6 245.4.e.h.226.2 4
21.20 even 2 35.4.a.b.1.1 ✓ 2
35.34 odd 2 1575.4.a.z.1.1 2
84.83 odd 2 560.4.a.r.1.1 2
105.62 odd 4 175.4.b.c.99.3 4
105.83 odd 4 175.4.b.c.99.2 4
105.104 even 2 175.4.a.c.1.2 2
168.83 odd 2 2240.4.a.bo.1.2 2
168.125 even 2 2240.4.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.1 ✓ 2 21.20 even 2
175.4.a.c.1.2 2 105.104 even 2
175.4.b.c.99.2 4 105.83 odd 4
175.4.b.c.99.3 4 105.62 odd 4
245.4.a.k.1.1 2 3.2 odd 2
245.4.e.h.116.2 4 21.5 even 6
245.4.e.h.226.2 4 21.17 even 6
245.4.e.i.116.2 4 21.2 odd 6
245.4.e.i.226.2 4 21.11 odd 6
315.4.a.f.1.2 2 7.6 odd 2
560.4.a.r.1.1 2 84.83 odd 2
1225.4.a.m.1.2 2 15.14 odd 2
1575.4.a.z.1.1 2 35.34 odd 2
2205.4.a.u.1.2 2 1.1 even 1 trivial
2240.4.a.bn.1.1 2 168.125 even 2
2240.4.a.bo.1.2 2 168.83 odd 2