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Properties

Label 605.6.a.c.1.1
Level $605$
Weight $6$
Character 605.1
Self dual yes
Analytic conductor $97.032$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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] = [605,6,Mod(1,605)] 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("605.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(97.0322109869\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 33x^{2} - 8x + 116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.95665\) of defining polynomial
Character \(\chi\) \(=\) 605.1

$q$-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)
 
\(f(q)\) \(=\) \(q-7.82466 q^{2} -16.3044 q^{3} +29.2253 q^{4} -25.0000 q^{5} +127.576 q^{6} +125.436 q^{7} +21.7111 q^{8} +22.8321 q^{9} +195.616 q^{10} -476.500 q^{12} -532.300 q^{13} -981.491 q^{14} +407.609 q^{15} -1105.09 q^{16} +1373.09 q^{17} -178.654 q^{18} +554.639 q^{19} -730.632 q^{20} -2045.15 q^{21} +4250.72 q^{23} -353.986 q^{24} +625.000 q^{25} +4165.06 q^{26} +3589.70 q^{27} +3665.89 q^{28} +6973.40 q^{29} -3189.40 q^{30} +3130.03 q^{31} +7952.21 q^{32} -10744.0 q^{34} -3135.89 q^{35} +667.276 q^{36} +1384.70 q^{37} -4339.86 q^{38} +8678.81 q^{39} -542.778 q^{40} -679.385 q^{41} +16002.6 q^{42} +1721.06 q^{43} -570.803 q^{45} -33260.4 q^{46} -15143.3 q^{47} +18017.8 q^{48} -1072.91 q^{49} -4890.41 q^{50} -22387.4 q^{51} -15556.6 q^{52} -9544.44 q^{53} -28088.1 q^{54} +2723.35 q^{56} -9043.04 q^{57} -54564.5 q^{58} +27582.7 q^{59} +11912.5 q^{60} +40527.5 q^{61} -24491.5 q^{62} +2863.96 q^{63} -26860.4 q^{64} +13307.5 q^{65} -58726.5 q^{67} +40129.0 q^{68} -69305.2 q^{69} +24537.3 q^{70} -42527.8 q^{71} +495.712 q^{72} -23753.0 q^{73} -10834.8 q^{74} -10190.2 q^{75} +16209.5 q^{76} -67908.7 q^{78} +78690.1 q^{79} +27627.3 q^{80} -64075.9 q^{81} +5315.96 q^{82} -52252.1 q^{83} -59770.0 q^{84} -34327.3 q^{85} -13466.7 q^{86} -113697. q^{87} +8156.09 q^{89} +4466.34 q^{90} -66769.3 q^{91} +124228. q^{92} -51033.2 q^{93} +118491. q^{94} -13866.0 q^{95} -129656. q^{96} +79010.1 q^{97} +8395.17 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} + 61 q^{4} - 100 q^{5} + 157 q^{6} + 90 q^{7} + 135 q^{8} + 22 q^{9} - 125 q^{10} - 795 q^{12} - 820 q^{13} - 1687 q^{14} - 2671 q^{16} + 3800 q^{17} + 1610 q^{18} + 3394 q^{19} - 1525 q^{20}+ \cdots + 393590 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.82466 −1.38322 −0.691609 − 0.722272i \(-0.743098\pi\)
−0.691609 + 0.722272i \(0.743098\pi\)
\(3\) −16.3044 −1.04593 −0.522963 − 0.852356i \(-0.675173\pi\)
−0.522963 + 0.852356i \(0.675173\pi\)
\(4\) 29.2253 0.913290
\(5\) −25.0000 −0.447214
\(6\) 127.576 1.44674
\(7\) 125.436 0.967555 0.483778 − 0.875191i \(-0.339264\pi\)
0.483778 + 0.875191i \(0.339264\pi\)
\(8\) 21.7111 0.119938
\(9\) 22.8321 0.0939594
\(10\) 195.616 0.618594
\(11\) 0 0
\(12\) −476.500 −0.955233
\(13\) −532.300 −0.873570 −0.436785 − 0.899566i \(-0.643883\pi\)
−0.436785 + 0.899566i \(0.643883\pi\)
\(14\) −981.491 −1.33834
\(15\) 407.609 0.467752
\(16\) −1105.09 −1.07919
\(17\) 1373.09 1.15233 0.576166 − 0.817333i \(-0.304548\pi\)
0.576166 + 0.817333i \(0.304548\pi\)
\(18\) −178.654 −0.129966
\(19\) 554.639 0.352474 0.176237 − 0.984348i \(-0.443608\pi\)
0.176237 + 0.984348i \(0.443608\pi\)
\(20\) −730.632 −0.408436
\(21\) −2045.15 −1.01199
\(22\) 0 0
\(23\) 4250.72 1.67549 0.837746 − 0.546060i \(-0.183873\pi\)
0.837746 + 0.546060i \(0.183873\pi\)
\(24\) −353.986 −0.125446
\(25\) 625.000 0.200000
\(26\) 4165.06 1.20834
\(27\) 3589.70 0.947651
\(28\) 3665.89 0.883659
\(29\) 6973.40 1.53975 0.769874 − 0.638196i \(-0.220319\pi\)
0.769874 + 0.638196i \(0.220319\pi\)
\(30\) −3189.40 −0.647003
\(31\) 3130.03 0.584985 0.292493 − 0.956268i \(-0.405515\pi\)
0.292493 + 0.956268i \(0.405515\pi\)
\(32\) 7952.21 1.37282
\(33\) 0 0
\(34\) −10744.0 −1.59392
\(35\) −3135.89 −0.432704
\(36\) 667.276 0.0858122
\(37\) 1384.70 0.166284 0.0831422 − 0.996538i \(-0.473504\pi\)
0.0831422 + 0.996538i \(0.473504\pi\)
\(38\) −4339.86 −0.487548
\(39\) 8678.81 0.913689
\(40\) −542.778 −0.0536380
\(41\) −679.385 −0.0631185 −0.0315592 − 0.999502i \(-0.510047\pi\)
−0.0315592 + 0.999502i \(0.510047\pi\)
\(42\) 16002.6 1.39980
\(43\) 1721.06 0.141946 0.0709732 − 0.997478i \(-0.477390\pi\)
0.0709732 + 0.997478i \(0.477390\pi\)
\(44\) 0 0
\(45\) −570.803 −0.0420199
\(46\) −33260.4 −2.31757
\(47\) −15143.3 −0.999945 −0.499972 − 0.866041i \(-0.666656\pi\)
−0.499972 + 0.866041i \(0.666656\pi\)
\(48\) 18017.8 1.12875
\(49\) −1072.91 −0.0638372
\(50\) −4890.41 −0.276643
\(51\) −22387.4 −1.20525
\(52\) −15556.6 −0.797823
\(53\) −9544.44 −0.466724 −0.233362 − 0.972390i \(-0.574973\pi\)
−0.233362 + 0.972390i \(0.574973\pi\)
\(54\) −28088.1 −1.31081
\(55\) 0 0
\(56\) 2723.35 0.116047
\(57\) −9043.04 −0.368661
\(58\) −54564.5 −2.12981
\(59\) 27582.7 1.03159 0.515794 − 0.856713i \(-0.327497\pi\)
0.515794 + 0.856713i \(0.327497\pi\)
\(60\) 11912.5 0.427193
\(61\) 40527.5 1.39452 0.697261 − 0.716817i \(-0.254402\pi\)
0.697261 + 0.716817i \(0.254402\pi\)
\(62\) −24491.5 −0.809162
\(63\) 2863.96 0.0909109
\(64\) −26860.4 −0.819714
\(65\) 13307.5 0.390672
\(66\) 0 0
\(67\) −58726.5 −1.59826 −0.799130 − 0.601159i \(-0.794706\pi\)
−0.799130 + 0.601159i \(0.794706\pi\)
\(68\) 40129.0 1.05241
\(69\) −69305.2 −1.75244
\(70\) 24537.3 0.598523
\(71\) −42527.8 −1.00121 −0.500607 − 0.865675i \(-0.666890\pi\)
−0.500607 + 0.865675i \(0.666890\pi\)
\(72\) 495.712 0.0112693
\(73\) −23753.0 −0.521690 −0.260845 − 0.965381i \(-0.584001\pi\)
−0.260845 + 0.965381i \(0.584001\pi\)
\(74\) −10834.8 −0.230007
\(75\) −10190.2 −0.209185
\(76\) 16209.5 0.321911
\(77\) 0 0
\(78\) −67908.7 −1.26383
\(79\) 78690.1 1.41857 0.709287 − 0.704919i \(-0.249017\pi\)
0.709287 + 0.704919i \(0.249017\pi\)
\(80\) 27627.3 0.482629
\(81\) −64075.9 −1.08513
\(82\) 5315.96 0.0873066
\(83\) −52252.1 −0.832546 −0.416273 − 0.909240i \(-0.636664\pi\)
−0.416273 + 0.909240i \(0.636664\pi\)
\(84\) −59770.0 −0.924241
\(85\) −34327.3 −0.515338
\(86\) −13466.7 −0.196343
\(87\) −113697. −1.61046
\(88\) 0 0
\(89\) 8156.09 0.109146 0.0545729 − 0.998510i \(-0.482620\pi\)
0.0545729 + 0.998510i \(0.482620\pi\)
\(90\) 4466.34 0.0581227
\(91\) −66769.3 −0.845227
\(92\) 124228. 1.53021
\(93\) −51033.2 −0.611851
\(94\) 118491. 1.38314
\(95\) −13866.0 −0.157631
\(96\) −129656. −1.43586
\(97\) 79010.1 0.852615 0.426308 − 0.904578i \(-0.359814\pi\)
0.426308 + 0.904578i \(0.359814\pi\)
\(98\) 8395.17 0.0883007
\(99\) 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 605.6.a.c.1.1 4
11.10 odd 2 55.6.a.b.1.4 ✓ 4
33.32 even 2 495.6.a.g.1.1 4
44.43 even 2 880.6.a.n.1.4 4
55.32 even 4 275.6.b.d.199.7 8
55.43 even 4 275.6.b.d.199.2 8
55.54 odd 2 275.6.a.d.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.4 ✓ 4 11.10 odd 2
275.6.a.d.1.1 4 55.54 odd 2
275.6.b.d.199.2 8 55.43 even 4
275.6.b.d.199.7 8 55.32 even 4
495.6.a.g.1.1 4 33.32 even 2
605.6.a.c.1.1 4 1.1 even 1 trivial
880.6.a.n.1.4 4 44.43 even 2