Properties

Label 1856.4.a.bk
Level $1856$
Weight $4$
Character orbit 1856.a
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 245 x^{10} + 21294 x^{8} - 755514 x^{6} + 8955005 x^{4} - 27099393 x^{2} + 23710340\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{19} \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -1 + \beta_{2} ) q^{5} -\beta_{7} q^{7} + ( 14 - \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -1 + \beta_{2} ) q^{5} -\beta_{7} q^{7} + ( 14 - \beta_{2} + \beta_{3} ) q^{9} + ( \beta_{7} + \beta_{10} ) q^{11} + ( -3 + 2 \beta_{2} + \beta_{4} ) q^{13} + ( -4 \beta_{1} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{15} + ( 28 - 3 \beta_{2} - \beta_{5} + \beta_{6} ) q^{17} + ( 2 \beta_{1} - \beta_{8} + \beta_{9} ) q^{19} + ( -6 + 4 \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{21} + ( -2 \beta_{1} - 2 \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{23} + ( 24 + 2 \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{25} + ( 18 \beta_{1} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{27} + 29 q^{29} + ( -8 \beta_{1} - 4 \beta_{7} - \beta_{8} - \beta_{10} + 2 \beta_{11} ) q^{31} + ( 19 + 3 \beta_{2} + \beta_{3} + 4 \beta_{4} + 7 \beta_{5} ) q^{33} + ( 15 \beta_{1} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{35} + ( -16 + 8 \beta_{2} + 5 \beta_{3} + \beta_{4} + 4 \beta_{5} - 3 \beta_{6} ) q^{37} + ( -10 \beta_{1} + 5 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 6 \beta_{10} - 3 \beta_{11} ) q^{39} + ( 78 + \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{41} + ( 25 \beta_{1} - 11 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} ) q^{43} + ( -142 + 24 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} ) q^{45} + ( -7 \beta_{1} + 4 \beta_{7} - 6 \beta_{9} + 3 \beta_{10} - 3 \beta_{11} ) q^{47} + ( 81 + 3 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{49} + ( 53 \beta_{1} - 7 \beta_{7} + 4 \beta_{8} - 2 \beta_{10} + 3 \beta_{11} ) q^{51} + ( 35 - 9 \beta_{2} + 5 \beta_{3} + 5 \beta_{4} - \beta_{5} - 5 \beta_{6} ) q^{53} + ( 15 \beta_{1} - 7 \beta_{7} + 2 \beta_{8} + 7 \beta_{9} - 8 \beta_{10} + 6 \beta_{11} ) q^{55} + ( 100 + 21 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} ) q^{57} + ( 51 \beta_{1} - 4 \beta_{7} + 5 \beta_{8} - 6 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{59} + ( -32 - 4 \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 10 \beta_{6} ) q^{61} + ( 6 \beta_{1} - \beta_{7} - 5 \beta_{8} + 12 \beta_{9} - 11 \beta_{10} - \beta_{11} ) q^{63} + ( 247 + 3 \beta_{2} - 5 \beta_{3} - 6 \beta_{4} - 12 \beta_{5} - 4 \beta_{6} ) q^{65} + ( 40 \beta_{1} + 3 \beta_{7} + 3 \beta_{8} + 6 \beta_{9} + 5 \beta_{10} - 3 \beta_{11} ) q^{67} + ( -84 - 16 \beta_{2} - 5 \beta_{3} - 8 \beta_{5} + 2 \beta_{6} ) q^{69} + ( 22 \beta_{1} + 2 \beta_{7} + 6 \beta_{8} - 14 \beta_{9} + 6 \beta_{10} + 2 \beta_{11} ) q^{71} + ( 184 + 28 \beta_{2} + 5 \beta_{3} - \beta_{4} - 5 \beta_{6} ) q^{73} + ( 85 \beta_{1} - 13 \beta_{7} + 5 \beta_{8} + \beta_{9} - 7 \beta_{10} + 2 \beta_{11} ) q^{75} + ( -258 + 6 \beta_{2} - 17 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} + 20 \beta_{6} ) q^{77} + ( 42 \beta_{1} - 4 \beta_{7} - 2 \beta_{8} - 11 \beta_{10} + 4 \beta_{11} ) q^{79} + ( 321 + 26 \beta_{2} + 2 \beta_{3} - 5 \beta_{5} + 4 \beta_{6} ) q^{81} + ( -23 \beta_{1} + 7 \beta_{7} - 12 \beta_{8} + 8 \beta_{9} - 4 \beta_{10} + \beta_{11} ) q^{83} + ( -464 + 32 \beta_{2} + \beta_{4} + 18 \beta_{5} + 5 \beta_{6} ) q^{85} + 29 \beta_{1} q^{87} + ( 430 + 25 \beta_{2} - 5 \beta_{3} - 11 \beta_{4} - \beta_{5} + 10 \beta_{6} ) q^{89} + ( -23 \beta_{1} - 21 \beta_{7} - 18 \beta_{8} - 10 \beta_{10} + 5 \beta_{11} ) q^{91} + ( -311 - 10 \beta_{2} - 23 \beta_{3} - 12 \beta_{4} - 13 \beta_{5} + \beta_{6} ) q^{93} + ( 90 \beta_{1} + 3 \beta_{8} + 8 \beta_{9} - 2 \beta_{11} ) q^{95} + ( 510 - 9 \beta_{2} - 14 \beta_{3} + 12 \beta_{4} - 7 \beta_{5} + 5 \beta_{6} ) q^{97} + ( -15 \beta_{1} + 46 \beta_{7} - \beta_{8} - 15 \beta_{9} + 18 \beta_{10} - 23 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{5} + 166 q^{9} + O(q^{10}) \) \( 12 q - 10 q^{5} + 166 q^{9} - 30 q^{13} + 336 q^{17} - 56 q^{21} + 294 q^{25} + 348 q^{29} + 214 q^{33} - 196 q^{37} + 940 q^{41} - 1672 q^{45} + 972 q^{49} + 406 q^{53} + 1224 q^{57} - 400 q^{61} + 2998 q^{65} - 1004 q^{69} + 2252 q^{73} - 3032 q^{77} + 3932 q^{81} - 5564 q^{85} + 5212 q^{89} - 3722 q^{93} + 6164 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 245 x^{10} + 21294 x^{8} - 755514 x^{6} + 8955005 x^{4} - 27099393 x^{2} + 23710340\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -491 \nu^{10} + 125541 \nu^{8} - 11255035 \nu^{6} + 402225787 \nu^{4} - 4412660266 \nu^{2} + 8041961952 \)\()/ 103502808 \)
\(\beta_{3}\)\(=\)\((\)\( -491 \nu^{10} + 125541 \nu^{8} - 11255035 \nu^{6} + 402225787 \nu^{4} - 4309157458 \nu^{2} + 3798346824 \)\()/ 103502808 \)
\(\beta_{4}\)\(=\)\((\)\( 6265 \nu^{10} - 1235890 \nu^{8} + 79290224 \nu^{6} - 1758043882 \nu^{4} + 8400204463 \nu^{2} + 7986096484 \)\()/ 621016848 \)
\(\beta_{5}\)\(=\)\((\)\( 787 \nu^{10} - 169018 \nu^{8} + 12655964 \nu^{6} - 385175254 \nu^{4} + 4137831661 \nu^{2} - 7570448972 \)\()/51751404\)
\(\beta_{6}\)\(=\)\((\)\( 10809 \nu^{10} - 2602664 \nu^{8} + 220850310 \nu^{6} - 7505285884 \nu^{4} + 78171035497 \nu^{2} - 125081524228 \)\()/ 207005616 \)
\(\beta_{7}\)\(=\)\((\)\( -46274 \nu^{11} + 11740775 \nu^{9} - 1029868159 \nu^{7} + 34119647501 \nu^{5} - 247897005023 \nu^{3} - 616650760628 \nu \)\()/ 55581007896 \)
\(\beta_{8}\)\(=\)\((\)\( 309941 \nu^{11} - 79156292 \nu^{9} + 7073821954 \nu^{7} - 250114895120 \nu^{5} + 2673076575761 \nu^{3} - 7870458399796 \nu \)\()/ 111162015792 \)
\(\beta_{9}\)\(=\)\((\)\( 263196 \nu^{11} - 64165295 \nu^{9} + 5516969457 \nu^{7} - 190373414485 \nu^{5} + 2033891773879 \nu^{3} - 3601582924456 \nu \)\()/ 9263501316 \)
\(\beta_{10}\)\(=\)\((\)\( 549710 \nu^{11} - 132445209 \nu^{9} + 11239206649 \nu^{7} - 381914057707 \nu^{5} + 3980111406817 \nu^{3} - 6042365174700 \nu \)\()/ 18527002632 \)
\(\beta_{11}\)\(=\)\((\)\( 3468293 \nu^{11} - 849139832 \nu^{9} + 73277455438 \nu^{7} - 2534595868940 \nu^{5} + 26968615846517 \nu^{3} - 43085788356244 \nu \)\()/ 111162015792 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - \beta_{2} + 41\)
\(\nu^{3}\)\(=\)\(-\beta_{11} + \beta_{9} + \beta_{8} + 72 \beta_{1}\)
\(\nu^{4}\)\(=\)\(4 \beta_{6} - 5 \beta_{5} + 83 \beta_{3} - 55 \beta_{2} + 2913\)
\(\nu^{5}\)\(=\)\(-109 \beta_{11} - 11 \beta_{10} + 123 \beta_{9} + 61 \beta_{8} - 75 \beta_{7} + 5473 \beta_{1}\)
\(\nu^{6}\)\(=\)\(542 \beta_{6} - 490 \beta_{5} - 198 \beta_{4} + 6607 \beta_{3} - 2633 \beta_{2} + 220481\)
\(\nu^{7}\)\(=\)\(-10487 \beta_{11} - 2116 \beta_{10} + 13197 \beta_{9} + 2279 \beta_{8} - 10416 \beta_{7} + 421882 \beta_{1}\)
\(\nu^{8}\)\(=\)\(58378 \beta_{6} - 40019 \beta_{5} - 33102 \beta_{4} + 526837 \beta_{3} - 82947 \beta_{2} + 16956905\)
\(\nu^{9}\)\(=\)\(-974343 \beta_{11} - 260291 \beta_{10} + 1311371 \beta_{9} - 27801 \beta_{8} - 1130871 \beta_{7} + 32795819 \beta_{1}\)
\(\nu^{10}\)\(=\)\(5779036 \beta_{6} - 3096104 \beta_{5} - 3924972 \beta_{4} + 42260097 \beta_{3} + 2867611 \beta_{2} + 1315829017\)
\(\nu^{11}\)\(=\)\(-88828697 \beta_{11} - 27059108 \beta_{10} + 124349773 \beta_{9} - 18154327 \beta_{8} - 111612348 \beta_{7} + 2568117924 \beta_{1}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−9.17689
−8.55609
−8.30801
−3.82226
−1.50033
−1.30165
1.30165
1.50033
3.82226
8.30801
8.55609
9.17689
0 −9.17689 0 14.0091 0 −13.6574 0 57.2153 0
1.2 0 −8.55609 0 −17.9229 0 −6.79699 0 46.2068 0
1.3 0 −8.30801 0 −11.7163 0 23.1998 0 42.0230 0
1.4 0 −3.82226 0 −3.68395 0 23.1652 0 −12.3903 0
1.5 0 −1.50033 0 −0.787262 0 −27.3506 0 −24.7490 0
1.6 0 −1.30165 0 15.1013 0 −22.0989 0 −25.3057 0
1.7 0 1.30165 0 15.1013 0 22.0989 0 −25.3057 0
1.8 0 1.50033 0 −0.787262 0 27.3506 0 −24.7490 0
1.9 0 3.82226 0 −3.68395 0 −23.1652 0 −12.3903 0
1.10 0 8.30801 0 −11.7163 0 −23.1998 0 42.0230 0
1.11 0 8.55609 0 −17.9229 0 6.79699 0 46.2068 0
1.12 0 9.17689 0 14.0091 0 13.6574 0 57.2153 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(29\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.4.a.bk 12
4.b odd 2 1 inner 1856.4.a.bk 12
8.b even 2 1 928.4.a.i 12
8.d odd 2 1 928.4.a.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.4.a.i 12 8.b even 2 1
928.4.a.i 12 8.d odd 2 1
1856.4.a.bk 12 1.a even 1 1 trivial
1856.4.a.bk 12 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{12} - 245 T_{3}^{10} + 21294 T_{3}^{8} - 755514 T_{3}^{6} + 8955005 T_{3}^{4} - 27099393 T_{3}^{2} + 23710340 \)
\( T_{5}^{6} + 5 T_{5}^{5} - 436 T_{5}^{4} - 1814 T_{5}^{3} + 43849 T_{5}^{2} + 199089 T_{5} + 128842 \)
\( T_{7}^{12} - 2544 T_{7}^{10} + 2529632 T_{7}^{8} - 1231220992 T_{7}^{6} + 297097423104 T_{7}^{4} - \)\(31\!\cdots\!80\)\( T_{7}^{2} + \)\(90\!\cdots\!00\)\( \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 23710340 - 27099393 T^{2} + 8955005 T^{4} - 755514 T^{6} + 21294 T^{8} - 245 T^{10} + T^{12} \)
$5$ \( ( 128842 + 199089 T + 43849 T^{2} - 1814 T^{3} - 436 T^{4} + 5 T^{5} + T^{6} )^{2} \)
$7$ \( 909258653696000 - 31017131233280 T^{2} + 297097423104 T^{4} - 1231220992 T^{6} + 2529632 T^{8} - 2544 T^{10} + T^{12} \)
$11$ \( 734196982995223940 - 39676330971681297 T^{2} + 178073715727517 T^{4} - 181551835386 T^{6} + 75855246 T^{8} - 14245 T^{10} + T^{12} \)
$13$ \( ( -3382882886 - 316457525 T + 20929777 T^{2} + 34918 T^{3} - 9132 T^{4} + 15 T^{5} + T^{6} )^{2} \)
$17$ \( ( 25740107392 - 1544949440 T - 4024384 T^{2} + 1126816 T^{3} - 2692 T^{4} - 168 T^{5} + T^{6} )^{2} \)
$19$ \( \)\(21\!\cdots\!40\)\( - 4718337599374573568 T^{2} + 3913871909371904 T^{4} - 1576285092352 T^{6} + 317593088 T^{8} - 29460 T^{10} + T^{12} \)
$23$ \( \)\(36\!\cdots\!60\)\( - 71190234194159357952 T^{2} + 40211930312278016 T^{4} - 9895667624704 T^{6} + 1151401024 T^{8} - 59076 T^{10} + T^{12} \)
$29$ \( ( -29 + T )^{12} \)
$31$ \( \)\(12\!\cdots\!60\)\( - \)\(93\!\cdots\!13\)\( T^{2} + 2227190845984472141 T^{4} - 197675486608906 T^{6} + 7952463230 T^{8} - 146517 T^{10} + T^{12} \)
$37$ \( ( 40279997186048 - 1098978414592 T + 7999453184 T^{2} + 5816704 T^{3} - 176288 T^{4} + 98 T^{5} + T^{6} )^{2} \)
$41$ \( ( 4616524334080 - 618188718112 T - 1162515824 T^{2} + 39794224 T^{3} - 47184 T^{4} - 470 T^{5} + T^{6} )^{2} \)
$43$ \( \)\(15\!\cdots\!00\)\( - \)\(17\!\cdots\!57\)\( T^{2} + \)\(13\!\cdots\!01\)\( T^{4} - 26767356466492810 T^{6} + 213281445390 T^{8} - 759389 T^{10} + T^{12} \)
$47$ \( \)\(23\!\cdots\!60\)\( - \)\(15\!\cdots\!93\)\( T^{2} + \)\(11\!\cdots\!89\)\( T^{4} - 26793244393564730 T^{6} + 225816554654 T^{8} - 797085 T^{10} + T^{12} \)
$53$ \( ( -1660499916764786 - 1020602626087 T + 48687196873 T^{2} + 28086834 T^{3} - 423512 T^{4} - 203 T^{5} + T^{6} )^{2} \)
$59$ \( \)\(17\!\cdots\!00\)\( - \)\(16\!\cdots\!00\)\( T^{2} + \)\(69\!\cdots\!04\)\( T^{4} - 100547167820623104 T^{6} + 598873587520 T^{8} - 1346500 T^{10} + T^{12} \)
$61$ \( ( -8835655940197760 - 11505177825728 T + 214997417280 T^{2} - 52928480 T^{3} - 877844 T^{4} + 200 T^{5} + T^{6} )^{2} \)
$67$ \( \)\(22\!\cdots\!00\)\( - \)\(54\!\cdots\!20\)\( T^{2} + \)\(26\!\cdots\!36\)\( T^{4} - 281742577942953984 T^{6} + 1132087525376 T^{8} - 1840400 T^{10} + T^{12} \)
$71$ \( \)\(57\!\cdots\!40\)\( - \)\(54\!\cdots\!92\)\( T^{2} + \)\(12\!\cdots\!76\)\( T^{4} - 972416557086442624 T^{6} + 2605331776416 T^{8} - 2775044 T^{10} + T^{12} \)
$73$ \( ( -338596891021312 - 6885547288576 T - 8785972736 T^{2} + 252212160 T^{3} - 27832 T^{4} - 1126 T^{5} + T^{6} )^{2} \)
$79$ \( \)\(44\!\cdots\!60\)\( - \)\(52\!\cdots\!93\)\( T^{2} + \)\(19\!\cdots\!53\)\( T^{4} - 238711573050402746 T^{6} + 1147614461182 T^{8} - 2026749 T^{10} + T^{12} \)
$83$ \( \)\(27\!\cdots\!00\)\( - \)\(51\!\cdots\!20\)\( T^{2} + \)\(12\!\cdots\!96\)\( T^{4} - 973453430750118656 T^{6} + 2867669439936 T^{8} - 3210660 T^{10} + T^{12} \)
$89$ \( ( -3679750127536640 + 156796517511136 T - 1525467801264 T^{2} + 2000030864 T^{3} + 979392 T^{4} - 2606 T^{5} + T^{6} )^{2} \)
$97$ \( ( 363210151505058432 - 450185749854816 T - 1746844444016 T^{2} + 2312694640 T^{3} + 1608600 T^{4} - 3082 T^{5} + T^{6} )^{2} \)
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