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Properties

Label 35.5.d.a.6.3
Level $35$
Weight $5$
Character 35.6
Analytic conductor $3.618$
Analytic rank $0$
Dimension $12$
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] = [35,5,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, 5); 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, 5, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 35.d (of order \(2\), degree \(1\), 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: \(3.61794870793\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 109 x^{10} + 570 x^{9} + 5814 x^{8} - 22512 x^{7} - 151120 x^{6} + 300288 x^{5} + \cdots + 205833600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 5^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 6.3
Root \(-3.62973 + 2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 35.6
Dual form 35.5.d.a.6.4

$q$-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)
 
\(f(q)\) \(=\) \(q-4.62973 q^{2} -0.296012i q^{3} +5.43443 q^{4} +11.1803i q^{5} +1.37046i q^{6} +(15.2405 - 46.5696i) q^{7} +48.9158 q^{8} +80.9124 q^{9} -51.7620i q^{10} +140.880 q^{11} -1.60866i q^{12} -27.4854i q^{13} +(-70.5596 + 215.605i) q^{14} +3.30952 q^{15} -313.418 q^{16} -469.738i q^{17} -374.603 q^{18} +681.353i q^{19} +60.7587i q^{20} +(-13.7852 - 4.51139i) q^{21} -652.236 q^{22} +222.532 q^{23} -14.4797i q^{24} -125.000 q^{25} +127.250i q^{26} -47.9281i q^{27} +(82.8236 - 253.079i) q^{28} +1012.14 q^{29} -15.3222 q^{30} -1420.40i q^{31} +668.388 q^{32} -41.7022i q^{33} +2174.76i q^{34} +(520.664 + 170.394i) q^{35} +439.712 q^{36} +459.262 q^{37} -3154.48i q^{38} -8.13602 q^{39} +546.895i q^{40} +695.715i q^{41} +(63.8217 + 20.8865i) q^{42} -1553.82 q^{43} +765.601 q^{44} +904.628i q^{45} -1030.26 q^{46} +1027.61i q^{47} +92.7756i q^{48} +(-1936.45 - 1419.49i) q^{49} +578.717 q^{50} -139.048 q^{51} -149.367i q^{52} +2463.31 q^{53} +221.894i q^{54} +1575.09i q^{55} +(745.503 - 2277.99i) q^{56} +201.689 q^{57} -4685.96 q^{58} +215.017i q^{59} +17.9853 q^{60} -480.976i q^{61} +6576.07i q^{62} +(1233.15 - 3768.06i) q^{63} +1920.23 q^{64} +307.296 q^{65} +193.070i q^{66} -4835.01 q^{67} -2552.76i q^{68} -65.8722i q^{69} +(-2410.53 - 788.881i) q^{70} -5797.92 q^{71} +3957.89 q^{72} +6606.91i q^{73} -2126.26 q^{74} +37.0016i q^{75} +3702.76i q^{76} +(2147.09 - 6560.72i) q^{77} +37.6676 q^{78} -1195.42 q^{79} -3504.12i q^{80} +6539.72 q^{81} -3220.97i q^{82} +7004.34i q^{83} +(-74.9145 - 24.5168i) q^{84} +5251.83 q^{85} +7193.77 q^{86} -299.607i q^{87} +6891.25 q^{88} -2619.72i q^{89} -4188.19i q^{90} +(-1279.98 - 418.892i) q^{91} +1209.33 q^{92} -420.456 q^{93} -4757.56i q^{94} -7617.75 q^{95} -197.851i q^{96} -12481.2i q^{97} +(8965.26 + 6571.86i) q^{98} +11398.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 122 q^{4} - 50 q^{7} - 186 q^{8} - 434 q^{9} + 126 q^{11} + 78 q^{14} + 50 q^{15} + 578 q^{16} + 734 q^{18} - 642 q^{21} + 2264 q^{22} - 756 q^{23} - 1500 q^{25} + 1414 q^{28} - 2190 q^{29}+ \cdots + 23084 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.62973 −1.15743 −0.578717 − 0.815529i \(-0.696446\pi\)
−0.578717 + 0.815529i \(0.696446\pi\)
\(3\) − 0.296012i − 0.0328903i −0.999865 − 0.0164451i \(-0.994765\pi\)
0.999865 − 0.0164451i \(-0.00523489\pi\)
\(4\) 5.43443 0.339652
\(5\) 11.1803i 0.447214i
\(6\) 1.37046i 0.0380683i
\(7\) 15.2405 − 46.5696i 0.311031 − 0.950400i
\(8\) 48.9158 0.764309
\(9\) 80.9124 0.998918
\(10\) − 51.7620i − 0.517620i
\(11\) 140.880 1.16430 0.582148 − 0.813083i \(-0.302212\pi\)
0.582148 + 0.813083i \(0.302212\pi\)
\(12\) − 1.60866i − 0.0111712i
\(13\) − 27.4854i − 0.162635i −0.996688 − 0.0813177i \(-0.974087\pi\)
0.996688 − 0.0813177i \(-0.0259129\pi\)
\(14\) −70.5596 + 215.605i −0.359998 + 1.10002i
\(15\) 3.30952 0.0147090
\(16\) −313.418 −1.22429
\(17\) − 469.738i − 1.62539i −0.582689 − 0.812695i \(-0.698000\pi\)
0.582689 − 0.812695i \(-0.302000\pi\)
\(18\) −374.603 −1.15618
\(19\) 681.353i 1.88740i 0.330798 + 0.943702i \(0.392682\pi\)
−0.330798 + 0.943702i \(0.607318\pi\)
\(20\) 60.7587i 0.151897i
\(21\) −13.7852 − 4.51139i −0.0312589 − 0.0102299i
\(22\) −652.236 −1.34760
\(23\) 222.532 0.420665 0.210333 − 0.977630i \(-0.432545\pi\)
0.210333 + 0.977630i \(0.432545\pi\)
\(24\) − 14.4797i − 0.0251383i
\(25\) −125.000 −0.200000
\(26\) 127.250i 0.188240i
\(27\) − 47.9281i − 0.0657450i
\(28\) 82.8236 − 253.079i 0.105642 − 0.322805i
\(29\) 1012.14 1.20350 0.601750 − 0.798684i \(-0.294470\pi\)
0.601750 + 0.798684i \(0.294470\pi\)
\(30\) −15.3222 −0.0170247
\(31\) − 1420.40i − 1.47804i −0.673681 − 0.739022i \(-0.735288\pi\)
0.673681 − 0.739022i \(-0.264712\pi\)
\(32\) 668.388 0.652723
\(33\) − 41.7022i − 0.0382940i
\(34\) 2174.76i 1.88128i
\(35\) 520.664 + 170.394i 0.425032 + 0.139097i
\(36\) 439.712 0.339284
\(37\) 459.262 0.335473 0.167736 − 0.985832i \(-0.446354\pi\)
0.167736 + 0.985832i \(0.446354\pi\)
\(38\) − 3154.48i − 2.18454i
\(39\) −8.13602 −0.00534913
\(40\) 546.895i 0.341809i
\(41\) 695.715i 0.413869i 0.978355 + 0.206935i \(0.0663488\pi\)
−0.978355 + 0.206935i \(0.933651\pi\)
\(42\) 63.8217 + 20.8865i 0.0361801 + 0.0118404i
\(43\) −1553.82 −0.840357 −0.420179 − 0.907441i \(-0.638033\pi\)
−0.420179 + 0.907441i \(0.638033\pi\)
\(44\) 765.601 0.395455
\(45\) 904.628i 0.446730i
\(46\) −1030.26 −0.486892
\(47\) 1027.61i 0.465192i 0.972573 + 0.232596i \(0.0747220\pi\)
−0.972573 + 0.232596i \(0.925278\pi\)
\(48\) 92.7756i 0.0402672i
\(49\) −1936.45 − 1419.49i −0.806519 − 0.591208i
\(50\) 578.717 0.231487
\(51\) −139.048 −0.0534595
\(52\) − 149.367i − 0.0552394i
\(53\) 2463.31 0.876934 0.438467 − 0.898747i \(-0.355522\pi\)
0.438467 + 0.898747i \(0.355522\pi\)
\(54\) 221.894i 0.0760954i
\(55\) 1575.09i 0.520689i
\(56\) 745.503 − 2277.99i 0.237724 − 0.726399i
\(57\) 201.689 0.0620772
\(58\) −4685.96 −1.39297
\(59\) 215.017i 0.0617687i 0.999523 + 0.0308844i \(0.00983236\pi\)
−0.999523 + 0.0308844i \(0.990168\pi\)
\(60\) 17.9853 0.00499593
\(61\) − 480.976i − 0.129260i −0.997909 − 0.0646299i \(-0.979413\pi\)
0.997909 − 0.0646299i \(-0.0205867\pi\)
\(62\) 6576.07i 1.71074i
\(63\) 1233.15 − 3768.06i 0.310695 − 0.949372i
\(64\) 1920.23 0.468805
\(65\) 307.296 0.0727328
\(66\) 193.070i 0.0443228i
\(67\) −4835.01 −1.07708 −0.538539 − 0.842600i \(-0.681024\pi\)
−0.538539 + 0.842600i \(0.681024\pi\)
\(68\) − 2552.76i − 0.552066i
\(69\) − 65.8722i − 0.0138358i
\(70\) −2410.53 − 788.881i −0.491946 − 0.160996i
\(71\) −5797.92 −1.15015 −0.575076 − 0.818100i \(-0.695028\pi\)
−0.575076 + 0.818100i \(0.695028\pi\)
\(72\) 3957.89 0.763482
\(73\) 6606.91i 1.23980i 0.784680 + 0.619901i \(0.212827\pi\)
−0.784680 + 0.619901i \(0.787173\pi\)
\(74\) −2126.26 −0.388287
\(75\) 37.0016i 0.00657805i
\(76\) 3702.76i 0.641060i
\(77\) 2147.09 − 6560.72i 0.362133 − 1.10655i
\(78\) 37.6676 0.00619126
\(79\) −1195.42 −0.191543 −0.0957716 − 0.995403i \(-0.530532\pi\)
−0.0957716 + 0.995403i \(0.530532\pi\)
\(80\) − 3504.12i − 0.547518i
\(81\) 6539.72 0.996756
\(82\) − 3220.97i − 0.479026i
\(83\) 7004.34i 1.01674i 0.861138 + 0.508371i \(0.169752\pi\)
−0.861138 + 0.508371i \(0.830248\pi\)
\(84\) −74.9145 − 24.5168i −0.0106171 − 0.00347460i
\(85\) 5251.83 0.726896
\(86\) 7193.77 0.972657
\(87\) − 299.607i − 0.0395835i
\(88\) 6891.25 0.889883
\(89\) − 2619.72i − 0.330730i −0.986232 − 0.165365i \(-0.947120\pi\)
0.986232 − 0.165365i \(-0.0528803\pi\)
\(90\) − 4188.19i − 0.517060i
\(91\) −1279.98 − 418.892i −0.154569 − 0.0505847i
\(92\) 1209.33 0.142880
\(93\) −420.456 −0.0486133
\(94\) − 4757.56i − 0.538429i
\(95\) −7617.75 −0.844072
\(96\) − 197.851i − 0.0214682i
\(97\) − 12481.2i − 1.32651i −0.748392 − 0.663257i \(-0.769174\pi\)
0.748392 − 0.663257i \(-0.230826\pi\)
\(98\) 8965.26 + 6571.86i 0.933492 + 0.684284i
\(99\) 11398.9 1.16304
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 35.5.d.a.6.3 ✓ 12
3.2 odd 2 315.5.h.a.181.9 12
4.3 odd 2 560.5.f.b.321.7 12
5.2 odd 4 175.5.c.d.174.11 24
5.3 odd 4 175.5.c.d.174.14 24
5.4 even 2 175.5.d.i.76.10 12
7.6 odd 2 inner 35.5.d.a.6.4 yes 12
21.20 even 2 315.5.h.a.181.10 12
28.27 even 2 560.5.f.b.321.6 12
35.13 even 4 175.5.c.d.174.12 24
35.27 even 4 175.5.c.d.174.13 24
35.34 odd 2 175.5.d.i.76.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.d.a.6.3 ✓ 12 1.1 even 1 trivial
35.5.d.a.6.4 yes 12 7.6 odd 2 inner
175.5.c.d.174.11 24 5.2 odd 4
175.5.c.d.174.12 24 35.13 even 4
175.5.c.d.174.13 24 35.27 even 4
175.5.c.d.174.14 24 5.3 odd 4
175.5.d.i.76.9 12 35.34 odd 2
175.5.d.i.76.10 12 5.4 even 2
315.5.h.a.181.9 12 3.2 odd 2
315.5.h.a.181.10 12 21.20 even 2
560.5.f.b.321.6 12 28.27 even 2
560.5.f.b.321.7 12 4.3 odd 2