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Properties

Label 35.3.d.a.6.2
Level $35$
Weight $3$
Character 35.6
Analytic conductor $0.954$
Analytic rank $0$
Dimension $2$
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] = [35,3,Mod(6,35)] 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("35.6"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 35.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2] 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: no
Analytic conductor: \(0.953680925261\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 6.2
Root \(2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 35.6
Dual form 35.3.d.a.6.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +4.47214i q^{3} -3.00000 q^{4} +2.23607i q^{5} -4.47214i q^{6} +7.00000 q^{7} +7.00000 q^{8} -11.0000 q^{9} -2.23607i q^{10} +2.00000 q^{11} -13.4164i q^{12} -13.4164i q^{13} -7.00000 q^{14} -10.0000 q^{15} +5.00000 q^{16} +26.8328i q^{17} +11.0000 q^{18} -13.4164i q^{19} -6.70820i q^{20} +31.3050i q^{21} -2.00000 q^{22} +26.0000 q^{23} +31.3050i q^{24} -5.00000 q^{25} +13.4164i q^{26} -8.94427i q^{27} -21.0000 q^{28} -22.0000 q^{29} +10.0000 q^{30} -53.6656i q^{31} -33.0000 q^{32} +8.94427i q^{33} -26.8328i q^{34} +15.6525i q^{35} +33.0000 q^{36} +14.0000 q^{37} +13.4164i q^{38} +60.0000 q^{39} +15.6525i q^{40} -26.8328i q^{41} -31.3050i q^{42} -34.0000 q^{43} -6.00000 q^{44} -24.5967i q^{45} -26.0000 q^{46} -26.8328i q^{47} +22.3607i q^{48} +49.0000 q^{49} +5.00000 q^{50} -120.000 q^{51} +40.2492i q^{52} -34.0000 q^{53} +8.94427i q^{54} +4.47214i q^{55} +49.0000 q^{56} +60.0000 q^{57} +22.0000 q^{58} -40.2492i q^{59} +30.0000 q^{60} +93.9149i q^{61} +53.6656i q^{62} -77.0000 q^{63} +13.0000 q^{64} +30.0000 q^{65} -8.94427i q^{66} +14.0000 q^{67} -80.4984i q^{68} +116.276i q^{69} -15.6525i q^{70} +62.0000 q^{71} -77.0000 q^{72} -53.6656i q^{73} -14.0000 q^{74} -22.3607i q^{75} +40.2492i q^{76} +14.0000 q^{77} -60.0000 q^{78} +38.0000 q^{79} +11.1803i q^{80} -59.0000 q^{81} +26.8328i q^{82} +40.2492i q^{83} -93.9149i q^{84} -60.0000 q^{85} +34.0000 q^{86} -98.3870i q^{87} +14.0000 q^{88} -26.8328i q^{89} +24.5967i q^{90} -93.9149i q^{91} -78.0000 q^{92} +240.000 q^{93} +26.8328i q^{94} +30.0000 q^{95} -147.580i q^{96} -26.8328i q^{97} -49.0000 q^{98} -22.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 6 q^{4} + 14 q^{7} + 14 q^{8} - 22 q^{9} + 4 q^{11} - 14 q^{14} - 20 q^{15} + 10 q^{16} + 22 q^{18} - 4 q^{22} + 52 q^{23} - 10 q^{25} - 42 q^{28} - 44 q^{29} + 20 q^{30} - 66 q^{32} + 66 q^{36}+ \cdots - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.500000 −0.250000 − 0.968246i \(-0.580431\pi\)
−0.250000 + 0.968246i \(0.580431\pi\)
\(3\) 4.47214i 1.49071i 0.666667 + 0.745356i \(0.267720\pi\)
−0.666667 + 0.745356i \(0.732280\pi\)
\(4\) −3.00000 −0.750000
\(5\) 2.23607i 0.447214i
\(6\) − 4.47214i − 0.745356i
\(7\) 7.00000 1.00000
\(8\) 7.00000 0.875000
\(9\) −11.0000 −1.22222
\(10\) − 2.23607i − 0.223607i
\(11\) 2.00000 0.181818 0.0909091 − 0.995859i \(-0.471023\pi\)
0.0909091 + 0.995859i \(0.471023\pi\)
\(12\) − 13.4164i − 1.11803i
\(13\) − 13.4164i − 1.03203i −0.856579 − 0.516016i \(-0.827415\pi\)
0.856579 − 0.516016i \(-0.172585\pi\)
\(14\) −7.00000 −0.500000
\(15\) −10.0000 −0.666667
\(16\) 5.00000 0.312500
\(17\) 26.8328i 1.57840i 0.614136 + 0.789200i \(0.289505\pi\)
−0.614136 + 0.789200i \(0.710495\pi\)
\(18\) 11.0000 0.611111
\(19\) − 13.4164i − 0.706127i −0.935599 − 0.353063i \(-0.885140\pi\)
0.935599 − 0.353063i \(-0.114860\pi\)
\(20\) − 6.70820i − 0.335410i
\(21\) 31.3050i 1.49071i
\(22\) −2.00000 −0.0909091
\(23\) 26.0000 1.13043 0.565217 − 0.824942i \(-0.308792\pi\)
0.565217 + 0.824942i \(0.308792\pi\)
\(24\) 31.3050i 1.30437i
\(25\) −5.00000 −0.200000
\(26\) 13.4164i 0.516016i
\(27\) − 8.94427i − 0.331269i
\(28\) −21.0000 −0.750000
\(29\) −22.0000 −0.758621 −0.379310 − 0.925270i \(-0.623839\pi\)
−0.379310 + 0.925270i \(0.623839\pi\)
\(30\) 10.0000 0.333333
\(31\) − 53.6656i − 1.73115i −0.500780 − 0.865575i \(-0.666953\pi\)
0.500780 − 0.865575i \(-0.333047\pi\)
\(32\) −33.0000 −1.03125
\(33\) 8.94427i 0.271039i
\(34\) − 26.8328i − 0.789200i
\(35\) 15.6525i 0.447214i
\(36\) 33.0000 0.916667
\(37\) 14.0000 0.378378 0.189189 − 0.981941i \(-0.439414\pi\)
0.189189 + 0.981941i \(0.439414\pi\)
\(38\) 13.4164i 0.353063i
\(39\) 60.0000 1.53846
\(40\) 15.6525i 0.391312i
\(41\) − 26.8328i − 0.654459i −0.944945 − 0.327229i \(-0.893885\pi\)
0.944945 − 0.327229i \(-0.106115\pi\)
\(42\) − 31.3050i − 0.745356i
\(43\) −34.0000 −0.790698 −0.395349 − 0.918531i \(-0.629376\pi\)
−0.395349 + 0.918531i \(0.629376\pi\)
\(44\) −6.00000 −0.136364
\(45\) − 24.5967i − 0.546594i
\(46\) −26.0000 −0.565217
\(47\) − 26.8328i − 0.570911i −0.958392 − 0.285455i \(-0.907855\pi\)
0.958392 − 0.285455i \(-0.0921449\pi\)
\(48\) 22.3607i 0.465847i
\(49\) 49.0000 1.00000
\(50\) 5.00000 0.100000
\(51\) −120.000 −2.35294
\(52\) 40.2492i 0.774024i
\(53\) −34.0000 −0.641509 −0.320755 − 0.947162i \(-0.603937\pi\)
−0.320755 + 0.947162i \(0.603937\pi\)
\(54\) 8.94427i 0.165635i
\(55\) 4.47214i 0.0813116i
\(56\) 49.0000 0.875000
\(57\) 60.0000 1.05263
\(58\) 22.0000 0.379310
\(59\) − 40.2492i − 0.682190i −0.940029 − 0.341095i \(-0.889202\pi\)
0.940029 − 0.341095i \(-0.110798\pi\)
\(60\) 30.0000 0.500000
\(61\) 93.9149i 1.53959i 0.638293 + 0.769794i \(0.279641\pi\)
−0.638293 + 0.769794i \(0.720359\pi\)
\(62\) 53.6656i 0.865575i
\(63\) −77.0000 −1.22222
\(64\) 13.0000 0.203125
\(65\) 30.0000 0.461538
\(66\) − 8.94427i − 0.135519i
\(67\) 14.0000 0.208955 0.104478 − 0.994527i \(-0.466683\pi\)
0.104478 + 0.994527i \(0.466683\pi\)
\(68\) − 80.4984i − 1.18380i
\(69\) 116.276i 1.68515i
\(70\) − 15.6525i − 0.223607i
\(71\) 62.0000 0.873239 0.436620 − 0.899646i \(-0.356176\pi\)
0.436620 + 0.899646i \(0.356176\pi\)
\(72\) −77.0000 −1.06944
\(73\) − 53.6656i − 0.735146i −0.929995 − 0.367573i \(-0.880189\pi\)
0.929995 − 0.367573i \(-0.119811\pi\)
\(74\) −14.0000 −0.189189
\(75\) − 22.3607i − 0.298142i
\(76\) 40.2492i 0.529595i
\(77\) 14.0000 0.181818
\(78\) −60.0000 −0.769231
\(79\) 38.0000 0.481013 0.240506 − 0.970648i \(-0.422687\pi\)
0.240506 + 0.970648i \(0.422687\pi\)
\(80\) 11.1803i 0.139754i
\(81\) −59.0000 −0.728395
\(82\) 26.8328i 0.327229i
\(83\) 40.2492i 0.484930i 0.970160 + 0.242465i \(0.0779560\pi\)
−0.970160 + 0.242465i \(0.922044\pi\)
\(84\) − 93.9149i − 1.11803i
\(85\) −60.0000 −0.705882
\(86\) 34.0000 0.395349
\(87\) − 98.3870i − 1.13088i
\(88\) 14.0000 0.159091
\(89\) − 26.8328i − 0.301492i −0.988573 − 0.150746i \(-0.951832\pi\)
0.988573 − 0.150746i \(-0.0481676\pi\)
\(90\) 24.5967i 0.273297i
\(91\) − 93.9149i − 1.03203i
\(92\) −78.0000 −0.847826
\(93\) 240.000 2.58065
\(94\) 26.8328i 0.285455i
\(95\) 30.0000 0.315789
\(96\) − 147.580i − 1.53730i
\(97\) − 26.8328i − 0.276627i −0.990388 − 0.138313i \(-0.955832\pi\)
0.990388 − 0.138313i \(-0.0441681\pi\)
\(98\) −49.0000 −0.500000
\(99\) −22.0000 −0.222222
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 35.3.d.a.6.2 yes 2
3.2 odd 2 315.3.h.b.181.1 2
4.3 odd 2 560.3.f.a.321.1 2
5.2 odd 4 175.3.c.d.174.2 4
5.3 odd 4 175.3.c.d.174.3 4
5.4 even 2 175.3.d.f.76.1 2
7.2 even 3 245.3.h.b.31.1 4
7.3 odd 6 245.3.h.b.166.1 4
7.4 even 3 245.3.h.b.166.2 4
7.5 odd 6 245.3.h.b.31.2 4
7.6 odd 2 inner 35.3.d.a.6.1 ✓ 2
21.20 even 2 315.3.h.b.181.2 2
28.27 even 2 560.3.f.a.321.2 2
35.13 even 4 175.3.c.d.174.4 4
35.27 even 4 175.3.c.d.174.1 4
35.34 odd 2 175.3.d.f.76.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.d.a.6.1 ✓ 2 7.6 odd 2 inner
35.3.d.a.6.2 yes 2 1.1 even 1 trivial
175.3.c.d.174.1 4 35.27 even 4
175.3.c.d.174.2 4 5.2 odd 4
175.3.c.d.174.3 4 5.3 odd 4
175.3.c.d.174.4 4 35.13 even 4
175.3.d.f.76.1 2 5.4 even 2
175.3.d.f.76.2 2 35.34 odd 2
245.3.h.b.31.1 4 7.2 even 3
245.3.h.b.31.2 4 7.5 odd 6
245.3.h.b.166.1 4 7.3 odd 6
245.3.h.b.166.2 4 7.4 even 3
315.3.h.b.181.1 2 3.2 odd 2
315.3.h.b.181.2 2 21.20 even 2
560.3.f.a.321.1 2 4.3 odd 2
560.3.f.a.321.2 2 28.27 even 2