Properties

Label 252.9.d.c
Level $252$
Weight $9$
Character orbit 252.d
Analytic conductor $102.659$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} + 229371 x^{6} - 1944986 x^{5} + 13911336469 x^{4} - 182147783748 x^{3} + 180227670127764 x^{2} - 1401469286341824 x + 451631142832842756\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{6}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} + ( 756 - 7 \beta_{2} ) q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} + ( 756 - 7 \beta_{2} ) q^{7} -\beta_{5} q^{11} + ( 23 + 34 \beta_{2} + 23 \beta_{3} + 11 \beta_{4} ) q^{13} + ( -15 \beta_{1} - \beta_{7} ) q^{17} + ( 160 - 133 \beta_{2} + 160 \beta_{3} - 293 \beta_{4} ) q^{19} + ( -2 \beta_{5} - \beta_{6} ) q^{23} + ( -269300 - 1429 \beta_{2} + 1429 \beta_{3} ) q^{25} + ( -\beta_{5} - 3 \beta_{6} ) q^{29} + ( 393 + 188 \beta_{2} + 393 \beta_{3} - 205 \beta_{4} ) q^{31} + ( 126 \beta_{1} - 28 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} ) q^{35} + ( 311887 + 1233 \beta_{2} - 1233 \beta_{3} ) q^{37} + ( -407 \beta_{1} - 33 \beta_{7} ) q^{41} + ( 2292529 + 10795 \beta_{2} - 10795 \beta_{3} ) q^{43} + ( -616 \beta_{1} - 18 \beta_{7} ) q^{47} + ( -1753759 - 9261 \beta_{2} + 4802 \beta_{3} + 7203 \beta_{4} ) q^{49} + ( -195 \beta_{5} + 19 \beta_{6} ) q^{53} + ( -4002 + 12861 \beta_{2} - 4002 \beta_{3} + 16863 \beta_{4} ) q^{55} + ( 1820 \beta_{1} + 152 \beta_{7} ) q^{59} + ( 16759 - 21090 \beta_{2} + 16759 \beta_{3} - 37849 \beta_{4} ) q^{61} + ( 448 \beta_{5} + 68 \beta_{6} ) q^{65} + ( 1540031 - 17819 \beta_{2} + 17819 \beta_{3} ) q^{67} + ( 1936 \beta_{5} + 11 \beta_{6} ) q^{71} + ( 61714 + 59028 \beta_{2} + 61714 \beta_{3} - 2686 \beta_{4} ) q^{73} + ( 10871 \beta_{1} - 1463 \beta_{5} + 63 \beta_{6} + 280 \beta_{7} ) q^{77} + ( 6654737 - 41241 \beta_{2} + 41241 \beta_{3} ) q^{79} + ( 6904 \beta_{1} - 474 \beta_{7} ) q^{83} + ( -9918405 - 170629 \beta_{2} + 170629 \beta_{3} ) q^{85} + ( -33053 \beta_{1} - 49 \beta_{7} ) q^{89} + ( 32173687 + 64078 \beta_{2} - 72373 \beta_{3} - 16121 \beta_{4} ) q^{91} + ( -5752 \beta_{5} - 266 \beta_{6} ) q^{95} + ( 157742 + 137176 \beta_{2} + 157742 \beta_{3} - 20566 \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6076 q^{7} + O(q^{10}) \) \( 8 q + 6076 q^{7} - 2154400 q^{25} + 2495096 q^{37} + 18340232 q^{43} - 13983424 q^{49} + 12320248 q^{67} + 53237896 q^{79} - 79347240 q^{85} + 257358192 q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 229371 x^{6} - 1944986 x^{5} + 13911336469 x^{4} - 182147783748 x^{3} + 180227670127764 x^{2} - 1401469286341824 x + 451631142832842756\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-720077573955026783001493240607 \nu^{7} + 18215170490158256899656654829947 \nu^{6} - 208018082659907333798337660912780015 \nu^{5} + 4285262182875923233810664691881691667 \nu^{4} - 18472771808346877484136941864888416227117 \nu^{3} + 236529670616777830453575084898569426721371 \nu^{2} - 358161904769986565287250066959177082188087488 \nu + 1537910357468552127300799176429439972937570670\)\()/ \)\(41\!\cdots\!24\)\( \)
\(\beta_{2}\)\(=\)\((\)\(189699550476117947687792567461 \nu^{7} + 3969630763275069291351740197383 \nu^{6} + 41020070466922619690673149377534701 \nu^{5} + 442014073098124763006552846501995897 \nu^{4} + 2193083132476837319924796956293869792039 \nu^{3} - 8915299713348400261284593326762002075993 \nu^{2} + 17051577366428812224914582831459801549488398 \nu - 530544763781669265322530705178937727242056464\)\()/ \)\(29\!\cdots\!66\)\( \)
\(\beta_{3}\)\(=\)\((\)\(192711519527165837106405933037 \nu^{7} - 1265899070505511897769098405845 \nu^{6} + 41564494916694195035178568427715813 \nu^{5} - 487714187554375215785581016204678855 \nu^{4} + 2215088737286668567930540502700327361383 \nu^{3} - 22993873000590601627469071398839666900361 \nu^{2} + 17187847448830283809761170479013241165768750 \nu + 444968765772701634070368793513383118383850050\)\()/ \)\(29\!\cdots\!66\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-535343211830403247161411314047 \nu^{7} - 4716533644464718452437007198309 \nu^{6} - 120877953853746107229854282946288555 \nu^{5} - 133814521780983908907020491549600411 \nu^{4} - 6856871467533328865508913303669570809993 \nu^{3} + 48688227262798560996012050201186662462791 \nu^{2} - 54073154480282688694921782607599579392475474 \nu + 636790468536179189314515016758854957907476064\)\()/ \)\(29\!\cdots\!66\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-9767501609504201064860 \nu^{7} + 25871608943424978463433083 \nu^{6} - 1944832673576807126112507796 \nu^{5} + 5547558021347203211412939104243 \nu^{4} - 114264207734441043290893439472084 \nu^{3} + 290071600624345371453467274094086897 \nu^{2} - 2549726456564236105962439523611996050 \nu + 1901409377734877582805722103741502666878\)\()/ \)\(46\!\cdots\!92\)\( \)
\(\beta_{6}\)\(=\)\((\)\(39666979023762651388300 \nu^{7} - 118495487879962465889353185 \nu^{6} + 9030711328253446122757732652 \nu^{5} - 31560325151732148059990849974145 \nu^{4} + 632741148324848565932404694190588 \nu^{3} - 2094227410161216635552031769526806579 \nu^{2} + 18297619982273396263601138222833092726 \nu - 14426743333918237980365425977451631738586\)\()/ \)\(46\!\cdots\!92\)\( \)
\(\beta_{7}\)\(=\)\((\)\(185207577596691785740526313703025 \nu^{7} - 237354949260218464238341127828277 \nu^{6} + 44151354729840720898662548927863651617 \nu^{5} - 301826168307958088792436479367181329469 \nu^{4} + 2939262407230590749677890154949473501608771 \nu^{3} - 28618925906523389963947332282752796256878645 \nu^{2} + 53715693890067460759271625868489903666117204032 \nu - 207363655476060288995398913894082603157662558258\)\()/ \)\(20\!\cdots\!62\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - 40 \beta_{3} - 56 \beta_{2} + 158 \beta_{1} + 120\)\()/672\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + 48 \beta_{6} + 528 \beta_{5} + 336 \beta_{4} + 96616 \beta_{3} - 96376 \beta_{2} - 1502 \beta_{1} - 38534040\)\()/672\)
\(\nu^{3}\)\(=\)\((\)\(-17445 \beta_{7} - 162 \beta_{6} - 774 \beta_{5} + 1506834 \beta_{4} + 1593146 \beta_{3} + 3751972 \beta_{2} - 4848918 \beta_{1} + 95552682\)\()/168\)
\(\nu^{4}\)\(=\)\((\)\(936745 \beta_{7} - 11011968 \beta_{6} - 88649856 \beta_{5} - 107494464 \beta_{4} - 12106325672 \beta_{3} + 11802233576 \beta_{2} + 302182670 \beta_{1} + 4165348752600\)\()/672\)
\(\nu^{5}\)\(=\)\((\)\(7916434675 \beta_{7} + 213031800 \beta_{6} + 1613268840 \beta_{5} - 1163053659432 \beta_{4} - 1051983615904 \beta_{3} - 2681163098936 \beta_{2} + 2338466048570 \beta_{1} - 79644537554976\)\()/672\)
\(\nu^{6}\)\(=\)\((\)\(-11204031696 \beta_{7} + 113615590707 \beta_{6} + 891173402649 \beta_{5} + 1687624637769 \beta_{4} + 94848981993652 \beta_{3} - 89529282998365 \beta_{2} - 3358744137408 \beta_{1} - 31848548189423820\)\()/42\)
\(\nu^{7}\)\(=\)\((\)\(-993477844279543 \beta_{7} - 48699501105024 \beta_{6} - 380058484240512 \beta_{5} + 181515537198840432 \beta_{4} + 158393515018998824 \beta_{3} + 419728953673038184 \beta_{2} - 295285372379607218 \beta_{1} + 13818856556787206952\)\()/672\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
−3.81971 + 274.634i
−3.81971 + 370.997i
4.31971 114.319i
4.31971 57.4847i
4.31971 + 57.4847i
4.31971 + 114.319i
−3.81971 370.997i
−3.81971 274.634i
0 0 0 1071.65i 0 −436.993 2360.90i 0 0 0
181.2 0 0 0 1071.65i 0 −436.993 + 2360.90i 0 0 0
181.3 0 0 0 414.022i 0 1955.99 1392.44i 0 0 0
181.4 0 0 0 414.022i 0 1955.99 + 1392.44i 0 0 0
181.5 0 0 0 414.022i 0 1955.99 1392.44i 0 0 0
181.6 0 0 0 414.022i 0 1955.99 + 1392.44i 0 0 0
181.7 0 0 0 1071.65i 0 −436.993 2360.90i 0 0 0
181.8 0 0 0 1071.65i 0 −436.993 + 2360.90i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.9.d.c 8
3.b odd 2 1 inner 252.9.d.c 8
7.b odd 2 1 inner 252.9.d.c 8
21.c even 2 1 inner 252.9.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.9.d.c 8 1.a even 1 1 trivial
252.9.d.c 8 3.b odd 2 1 inner
252.9.d.c 8 7.b odd 2 1 inner
252.9.d.c 8 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 1319850 T_{5}^{2} + 196857884160 \) acting on \(S_{9}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 196857884160 + 1319850 T^{2} + T^{4} )^{2} \)
$7$ \( ( 33232930569601 - 17513465438 T + 8110578 T^{2} - 3038 T^{3} + T^{4} )^{2} \)
$11$ \( ( 51896585889973440 - 575486730 T^{2} + T^{4} )^{2} \)
$13$ \( ( 11647705437080064 + 627852336 T^{2} + T^{4} )^{2} \)
$17$ \( ( 55455388822412328960 + 17728753770 T^{2} + T^{4} )^{2} \)
$19$ \( ( \)\(67\!\cdots\!64\)\( + 54927628836 T^{2} + T^{4} )^{2} \)
$23$ \( ( \)\(18\!\cdots\!40\)\( - 131516168490 T^{2} + T^{4} )^{2} \)
$29$ \( ( \)\(21\!\cdots\!40\)\( - 1171023204960 T^{2} + T^{4} )^{2} \)
$31$ \( ( \)\(53\!\cdots\!84\)\( + 69117363936 T^{2} + T^{4} )^{2} \)
$37$ \( ( -80395073216 - 623774 T + T^{2} )^{4} \)
$41$ \( ( \)\(61\!\cdots\!00\)\( + 19201621581450 T^{2} + T^{4} )^{2} \)
$43$ \( ( -8362825885784 - 4585058 T + T^{2} )^{4} \)
$47$ \( ( \)\(88\!\cdots\!60\)\( + 6149210689440 T^{2} + T^{4} )^{2} \)
$53$ \( ( \)\(87\!\cdots\!40\)\( - 70748538134400 T^{2} + T^{4} )^{2} \)
$59$ \( ( \)\(27\!\cdots\!60\)\( + 407110427851680 T^{2} + T^{4} )^{2} \)
$61$ \( ( \)\(21\!\cdots\!44\)\( + 941902228939056 T^{2} + T^{4} )^{2} \)
$67$ \( ( -34734900793304 - 3080062 T + T^{2} )^{4} \)
$71$ \( ( \)\(91\!\cdots\!40\)\( - 2162059746829290 T^{2} + T^{4} )^{2} \)
$73$ \( ( \)\(10\!\cdots\!00\)\( + 2213214904109760 T^{2} + T^{4} )^{2} \)
$79$ \( ( -154480814226896 - 13309474 T + T^{2} )^{4} \)
$83$ \( ( \)\(86\!\cdots\!00\)\( + 3978896305456800 T^{2} + T^{4} )^{2} \)
$89$ \( ( \)\(83\!\cdots\!60\)\( + 1483914624020010 T^{2} + T^{4} )^{2} \)
$97$ \( ( \)\(40\!\cdots\!04\)\( + 13110083500166016 T^{2} + T^{4} )^{2} \)
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