Properties

Label 10.15.c.a
Level 10
Weight 15
Character orbit 10.c
Analytic conductor 12.433
Analytic rank 0
Dimension 6
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 15 \)
Character orbit: \([\chi]\) = 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.4328968152\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -64 + 64 \beta_{1} ) q^{2} + ( 485 + 485 \beta_{1} + \beta_{3} ) q^{3} -8192 \beta_{1} q^{4} + ( 13752 - 18634 \beta_{1} - 7 \beta_{2} + 4 \beta_{3} - 17 \beta_{4} - 6 \beta_{5} ) q^{5} + ( -62080 - 64 \beta_{3} - 64 \beta_{4} ) q^{6} + ( 157124 - 157124 \beta_{1} - 79 \beta_{2} + 113 \beta_{4} + 79 \beta_{5} ) q^{7} + ( 524288 + 524288 \beta_{1} ) q^{8} + ( 862987 \beta_{1} - 149 \beta_{3} + 149 \beta_{4} + 246 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( -64 + 64 \beta_{1} ) q^{2} + ( 485 + 485 \beta_{1} + \beta_{3} ) q^{3} -8192 \beta_{1} q^{4} + ( 13752 - 18634 \beta_{1} - 7 \beta_{2} + 4 \beta_{3} - 17 \beta_{4} - 6 \beta_{5} ) q^{5} + ( -62080 - 64 \beta_{3} - 64 \beta_{4} ) q^{6} + ( 157124 - 157124 \beta_{1} - 79 \beta_{2} + 113 \beta_{4} + 79 \beta_{5} ) q^{7} + ( 524288 + 524288 \beta_{1} ) q^{8} + ( 862987 \beta_{1} - 149 \beta_{3} + 149 \beta_{4} + 246 \beta_{5} ) q^{9} + ( 312448 + 2072704 \beta_{1} + 832 \beta_{2} - 1344 \beta_{3} + 832 \beta_{4} - 64 \beta_{5} ) q^{10} + ( -7594208 - 542 \beta_{2} - 601 \beta_{3} - 601 \beta_{4} ) q^{11} + ( 3973120 - 3973120 \beta_{1} + 8192 \beta_{4} ) q^{12} + ( -8701134 - 8701134 \beta_{1} - 9113 \beta_{2} + 19630 \beta_{3} - 9113 \beta_{5} ) q^{13} + ( 20111872 \beta_{1} + 7232 \beta_{3} - 7232 \beta_{4} - 10112 \beta_{5} ) q^{14} + ( -74582968 - 11651234 \beta_{1} - 5787 \beta_{2} + 19719 \beta_{3} + 33898 \beta_{4} - 19881 \beta_{5} ) q^{15} -67108864 q^{16} + ( -49114757 + 49114757 \beta_{1} + 14878 \beta_{2} + 184678 \beta_{4} - 14878 \beta_{5} ) q^{17} + ( -55231168 - 55231168 \beta_{1} - 15744 \beta_{2} + 19072 \beta_{3} - 15744 \beta_{5} ) q^{18} + ( 636309218 \beta_{1} - 20294 \beta_{3} + 20294 \beta_{4} - 78634 \beta_{5} ) q^{19} + ( -152649728 - 112656384 \beta_{1} - 49152 \beta_{2} + 139264 \beta_{3} + 32768 \beta_{4} + 57344 \beta_{5} ) q^{20} + ( 372803450 - 225792 \beta_{2} + 59807 \beta_{3} + 59807 \beta_{4} ) q^{21} + ( 486029312 - 486029312 \beta_{1} + 34688 \beta_{2} + 76928 \beta_{4} - 34688 \beta_{5} ) q^{22} + ( -1571403368 - 1571403368 \beta_{1} + 26493 \beta_{2} - 1345107 \beta_{3} + 26493 \beta_{5} ) q^{23} + ( 508559360 \beta_{1} + 524288 \beta_{3} - 524288 \beta_{4} ) q^{24} + ( 773798375 + 1857548250 \beta_{1} + 32250 \beta_{2} + 2302375 \beta_{3} - 1149625 \beta_{4} - 153250 \beta_{5} ) q^{25} + ( 1113745152 + 1166464 \beta_{2} - 1256320 \beta_{3} - 1256320 \beta_{4} ) q^{26} + ( 2104923968 - 2104923968 \beta_{1} - 358176 \beta_{2} + 3651100 \beta_{4} + 358176 \beta_{5} ) q^{27} + ( -1287159808 - 1287159808 \beta_{1} + 647168 \beta_{2} - 925696 \beta_{3} + 647168 \beta_{5} ) q^{28} + ( 5918169182 \beta_{1} + 322060 \beta_{3} - 322060 \beta_{4} + 1967306 \beta_{5} ) q^{29} + ( 5518988928 - 4027630976 \beta_{1} + 1642752 \beta_{2} + 907456 \beta_{3} - 3431488 \beta_{4} + 902016 \beta_{5} ) q^{30} + ( 625659100 - 1335122 \beta_{2} - 8804863 \beta_{3} - 8804863 \beta_{4} ) q^{31} + ( 4294967296 - 4294967296 \beta_{1} ) q^{32} + ( -8043834178 - 8043834178 \beta_{1} - 1017756 \beta_{2} - 7008290 \beta_{3} - 1017756 \beta_{5} ) q^{33} + ( -6286688896 \beta_{1} + 11819392 \beta_{3} - 11819392 \beta_{4} + 1904384 \beta_{5} ) q^{34} + ( 46869144229 - 4938445583 \beta_{1} - 4233664 \beta_{2} + 8070538 \beta_{3} + 13238611 \beta_{4} + 332178 \beta_{5} ) q^{35} + ( 7069589504 + 2015232 \beta_{2} - 1220608 \beta_{3} - 1220608 \beta_{4} ) q^{36} + ( -71658052212 + 71658052212 \beta_{1} - 856227 \beta_{2} + 24285408 \beta_{4} + 856227 \beta_{5} ) q^{37} + ( -40723789952 - 40723789952 \beta_{1} + 5032576 \beta_{2} + 2597632 \beta_{3} + 5032576 \beta_{5} ) q^{38} + ( 51115428624 \beta_{1} - 24108279 \beta_{3} + 24108279 \beta_{4} - 24423750 \beta_{5} ) q^{39} + ( 16979591168 - 2559574016 \beta_{1} - 524288 \beta_{2} - 6815744 \beta_{3} - 11010048 \beta_{4} - 6815744 \beta_{5} ) q^{40} + ( 7578435160 - 12833134 \beta_{2} + 46195171 \beta_{3} + 46195171 \beta_{4} ) q^{41} + ( -23859420800 + 23859420800 \beta_{1} + 14450688 \beta_{2} - 7655296 \beta_{4} - 14450688 \beta_{5} ) q^{42} + ( -155943192417 - 155943192417 \beta_{1} - 27289402 \beta_{2} + 69513665 \beta_{3} - 27289402 \beta_{5} ) q^{43} + ( 62211751936 \beta_{1} + 4923392 \beta_{3} - 4923392 \beta_{4} + 4440064 \beta_{5} ) q^{44} + ( 88300449196 - 64748367612 \beta_{1} + 2261964 \beta_{2} - 29146448 \beta_{3} + 28254949 \beta_{4} - 2865483 \beta_{5} ) q^{45} + ( 201139631104 - 3391104 \beta_{2} + 86086848 \beta_{3} + 86086848 \beta_{4} ) q^{46} + ( -160952988042 + 160952988042 \beta_{1} - 14126433 \beta_{2} - 268955403 \beta_{4} + 14126433 \beta_{5} ) q^{47} + ( -32547799040 - 32547799040 \beta_{1} - 67108864 \beta_{3} ) q^{48} + ( 35693907665 \beta_{1} - 150923555 \beta_{3} + 150923555 \beta_{4} + 90572102 \beta_{5} ) q^{49} + ( -168406184000 - 69359992000 \beta_{1} + 7744000 \beta_{2} - 220928000 \beta_{3} - 73776000 \beta_{4} + 11872000 \beta_{5} ) q^{50} + ( 976795258234 + 93189168 \beta_{2} - 161365259 \beta_{3} - 161365259 \beta_{4} ) q^{51} + ( -71279689728 + 71279689728 \beta_{1} - 74653696 \beta_{2} + 160808960 \beta_{4} + 74653696 \beta_{5} ) q^{52} + ( -311388421006 - 311388421006 \beta_{1} + 145594479 \beta_{2} + 219814968 \beta_{3} + 145594479 \beta_{5} ) q^{53} + ( 269430267904 \beta_{1} + 233670400 \beta_{3} - 233670400 \beta_{4} - 45846528 \beta_{5} ) q^{54} + ( 156901419864 + 225350052962 \beta_{1} + 63706226 \beta_{2} + 56263403 \beta_{3} + 95845731 \beta_{4} + 46829558 \beta_{5} ) q^{55} + ( 164756455424 - 82837504 \beta_{2} + 59244544 \beta_{3} + 59244544 \beta_{4} ) q^{56} + ( -22202954782 + 22202954782 \beta_{1} + 131199894 \beta_{2} - 636485960 \beta_{4} - 131199894 \beta_{5} ) q^{57} + ( -378762827648 - 378762827648 \beta_{1} - 125907584 \beta_{2} - 41223680 \beta_{3} - 125907584 \beta_{5} ) q^{58} + ( -1919025468826 \beta_{1} + 92485290 \beta_{3} - 92485290 \beta_{4} - 57902718 \beta_{5} ) q^{59} + ( -95446908928 + 610983673856 \beta_{1} - 162865152 \beta_{2} - 277692416 \beta_{3} + 161538048 \beta_{4} + 47407104 \beta_{5} ) q^{60} + ( 352029635808 - 94802414 \beta_{2} - 316872301 \beta_{3} - 316872301 \beta_{4} ) q^{61} + ( -40042182400 + 40042182400 \beta_{1} + 85447808 \beta_{2} + 1127022464 \beta_{4} - 85447808 \beta_{5} ) q^{62} + ( -781984468756 - 781984468756 \beta_{1} + 30170913 \beta_{2} - 316889723 \beta_{3} + 30170913 \beta_{5} ) q^{63} + 549755813888 \beta_{1} q^{64} + ( -894577115913 + 5476304399631 \beta_{1} - 46458542 \beta_{2} + 2047498479 \beta_{3} - 76899407 \beta_{4} - 229165846 \beta_{5} ) q^{65} + ( 1029610774784 + 130272768 \beta_{2} + 448530560 \beta_{3} + 448530560 \beta_{4} ) q^{66} + ( -1414236925873 + 1414236925873 \beta_{1} - 261369138 \beta_{2} + 2081684625 \beta_{4} + 261369138 \beta_{5} ) q^{67} + ( 402348089344 + 402348089344 \beta_{1} - 121880576 \beta_{2} - 1512882176 \beta_{3} - 121880576 \beta_{5} ) q^{68} + ( -8363654380166 \beta_{1} - 709995305 \beta_{3} + 709995305 \beta_{4} - 245853792 \beta_{5} ) q^{69} + ( -2683564713344 + 3315685747968 \beta_{1} + 249695104 \beta_{2} + 330756672 \beta_{3} - 1363785536 \beta_{4} - 292213888 \beta_{5} ) q^{70} + ( 3722591433356 - 40746482 \beta_{2} - 495159247 \beta_{3} - 495159247 \beta_{4} ) q^{71} + ( -452453728256 + 452453728256 \beta_{1} - 128974848 \beta_{2} + 156237824 \beta_{4} + 128974848 \beta_{5} ) q^{72} + ( -1166111112881 - 1166111112881 \beta_{1} - 1140373166 \beta_{2} + 39115288 \beta_{3} - 1140373166 \beta_{5} ) q^{73} + ( -9172230683136 \beta_{1} + 1554266112 \beta_{3} - 1554266112 \beta_{4} - 109597056 \beta_{5} ) q^{74} + ( -6047639926625 + 12913063043875 \beta_{1} + 14919750 \beta_{2} + 53436125 \beta_{3} + 380826000 \beta_{4} + 372179250 \beta_{5} ) q^{75} + ( 5212645113856 - 644169728 \beta_{2} - 166248448 \beta_{3} - 166248448 \beta_{4} ) q^{76} + ( 623620659116 - 623620659116 \beta_{1} + 521873398 \beta_{2} + 45601558 \beta_{4} - 521873398 \beta_{5} ) q^{77} + ( -3271387431936 - 3271387431936 \beta_{1} + 1563120000 \beta_{2} + 3085859712 \beta_{3} + 1563120000 \beta_{5} ) q^{78} + ( -3321768540888 \beta_{1} - 57359552 \beta_{3} + 57359552 \beta_{4} + 489899192 \beta_{5} ) q^{79} + ( -922881097728 + 1250506571776 \beta_{1} + 469762048 \beta_{2} - 268435456 \beta_{3} + 1140850688 \beta_{4} + 402653184 \beta_{5} ) q^{80} + ( 23413385943211 + 925036014 \beta_{2} - 1038943013 \beta_{3} - 1038943013 \beta_{4} ) q^{81} + ( -485019850240 + 485019850240 \beta_{1} + 821320576 \beta_{2} - 5912981888 \beta_{4} - 821320576 \beta_{5} ) q^{82} + ( -10085519551347 - 10085519551347 \beta_{1} + 2381105704 \beta_{2} - 1956036779 \beta_{3} + 2381105704 \beta_{5} ) q^{83} + ( -3054005862400 \beta_{1} - 489938944 \beta_{3} + 489938944 \beta_{4} + 1849688064 \beta_{5} ) q^{84} + ( -10643524261696 + 20158440621992 \beta_{1} - 1744971239 \beta_{2} - 11009214637 \beta_{3} + 3120474611 \beta_{4} + 1009913103 \beta_{5} ) q^{85} + ( 19960728629376 + 3493043456 \beta_{2} - 4448874560 \beta_{3} - 4448874560 \beta_{4} ) q^{86} + ( -9074876319886 + 9074876319886 \beta_{1} - 3236752890 \beta_{2} - 6149171552 \beta_{4} + 3236752890 \beta_{5} ) q^{87} + ( -3981552123904 - 3981552123904 \beta_{1} - 284164096 \beta_{2} - 630194176 \beta_{3} - 284164096 \beta_{5} ) q^{88} + ( -21005367910460 \beta_{1} + 2948444520 \beta_{3} - 2948444520 \beta_{4} - 437255700 \beta_{5} ) q^{89} + ( -1507333221376 + 9795124275712 \beta_{1} + 38625216 \beta_{2} + 3673689408 \beta_{3} + 57055936 \beta_{4} + 328156608 \beta_{5} ) q^{90} + ( 67142024555502 - 7193087876 \beta_{2} + 14411341217 \beta_{3} + 14411341217 \beta_{4} ) q^{91} + ( -12872936390656 + 12872936390656 \beta_{1} + 217030656 \beta_{2} - 11019116544 \beta_{4} - 217030656 \beta_{5} ) q^{92} + ( -48345736984450 - 48345736984450 \beta_{1} - 4308867108 \beta_{2} + 11356310734 \beta_{3} - 4308867108 \beta_{5} ) q^{93} + ( -20601982469376 \beta_{1} - 17213145792 \beta_{3} + 17213145792 \beta_{4} - 1808183424 \beta_{5} ) q^{94} + ( -5725891583380 + 40004144607510 \beta_{1} + 4187956580 \beta_{2} - 10951904610 \beta_{3} - 13906961170 \beta_{4} - 4186960910 \beta_{5} ) q^{95} + ( 4166118277120 + 4294967296 \beta_{3} + 4294967296 \beta_{4} ) q^{96} + ( -51218932804839 + 51218932804839 \beta_{1} + 7585255592 \beta_{2} + 10698613328 \beta_{4} - 7585255592 \beta_{5} ) q^{97} + ( -2284410090560 - 2284410090560 \beta_{1} - 5796614528 \beta_{2} + 19318215040 \beta_{3} - 5796614528 \beta_{5} ) q^{98} + ( -12446167780760 \beta_{1} - 1056784649 \beta_{3} + 1056784649 \beta_{4} - 2398666902 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 384q^{2} + 2912q^{3} + 82500q^{5} - 372736q^{6} + 943128q^{7} + 3145728q^{8} + O(q^{10}) \) \( 6q - 384q^{2} + 2912q^{3} + 82500q^{5} - 372736q^{6} + 943128q^{7} + 3145728q^{8} + 1872000q^{10} - 45566568q^{11} + 23855104q^{12} - 52149318q^{13} - 447379000q^{15} - 402653184q^{16} - 294348942q^{17} - 331317376q^{18} - 915456000q^{20} + 2237511512q^{21} + 2916260352q^{22} - 9431163408q^{23} + 4645031250q^{25} + 6675112704q^{26} + 12637562360q^{27} - 7726104576q^{28} + 33105600000q^{30} + 3721405392q^{31} + 25769803776q^{32} - 48274986136q^{33} + 281265951000q^{35} + 42408624128q^{36} - 429898030002q^{37} - 244347609600q^{38} + 101842944000q^{40} + 45681057912q^{41} - 143200736768q^{42} - 935465548368q^{43} + 529796388250q^{45} + 1207188916224q^{46} - 966227586192q^{47} - 195421011968q^{48} - 1011042000000q^{50} + 5859939710032q^{51} - 427207213056q^{52} - 1868182085058q^{53} + 941585325000q^{55} + 988941385728q^{56} - 134753100400q^{57} - 2272407598080q^{58} - 572588032000q^{60} + 2111099930472q^{61} - 238169945088q^{62} - 4692600933808q^{63} - 5363428580250q^{65} + 6179198225408q^{66} - 8480735447712q^{67} + 2411306532864q^{68} - 16103953728000q^{70} + 22333649456112q^{71} - 2714151944192q^{72} - 6994307700378q^{73} - 36285000875000q^{75} + 31276494028800q^{76} + 3740771411016q^{77} - 19625279112192q^{78} - 5536481280000q^{80} + 140474309815186q^{81} - 2923587706368q^{82} - 60521791593048q^{83} - 63873433107750q^{85} + 119739590191104q^{86} - 54455082756640q^{87} - 23890004803584q^{88} - 9036615088000q^{90} + 402924178873632q^{91} - 77260090638336q^{92} - 290043091551016q^{93} - 34413443145000q^{95} + 25013889531904q^{96} - 307307370113562q^{97} - 13656230884224q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} - 11690 x^{3} + 819025 x^{2} - 12217500 x + 91125000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -27556 \nu^{5} - 148963 \nu^{4} - 962087 \nu^{3} + 137033615 \nu^{2} - 21376234525 \nu + 177209808750 \)\()/ 162210633750 \)
\(\beta_{2}\)\(=\)\((\)\( -222 \nu^{5} - 24026 \nu^{4} - 201354 \nu^{3} + 1297590 \nu^{2} + 2997000 \nu - 9607187345 \)\()/1044835\)
\(\beta_{3}\)\(=\)\((\)\( -43521437 \nu^{5} + 619265974 \nu^{4} + 34470074426 \nu^{3} + 991491950230 \nu^{2} - 38755988958425 \nu + 524538144697500 \)\()/ 162210633750 \)
\(\beta_{4}\)\(=\)\((\)\( 91365437 \nu^{5} + 1381798526 \nu^{4} + 8924433574 \nu^{3} - 1271140130230 \nu^{2} + 38110094958425 \nu - 139134133350000 \)\()/ 162210633750 \)
\(\beta_{5}\)\(=\)\((\)\( 325457492 \nu^{5} + 1767216041 \nu^{4} + 4244146309 \nu^{3} - 4855565916805 \nu^{2} + 245935421194175 \nu - 2045037534536250 \)\()/ 162210633750 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - 4 \beta_{4} - \beta_{2} - 201 \beta_{1} + 201\)\()/600\)
\(\nu^{2}\)\(=\)\((\)\(-17 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} - 181203 \beta_{1}\)\()/300\)
\(\nu^{3}\)\(=\)\((\)\(871 \beta_{5} + 3604 \beta_{3} + 871 \beta_{2} + 3505689 \beta_{1} + 3505689\)\()/600\)
\(\nu^{4}\)\(=\)\((\)\(-2553 \beta_{4} - 2553 \beta_{3} - 3544 \beta_{2} - 26521046\)\()/50\)
\(\nu^{5}\)\(=\)\((\)\(-975227 \beta_{5} + 3308348 \beta_{4} + 975227 \beta_{2} - 5300636493 \beta_{1} + 5300636493\)\()/600\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
16.0869 16.0869i
8.78753 8.78753i
−23.8745 + 23.8745i
16.0869 + 16.0869i
8.78753 + 8.78753i
−23.8745 23.8745i
−64.0000 + 64.0000i −1760.67 1760.67i 8192.00i 46225.0 62982.2i 225366. −59627.2 + 59627.2i 524288. + 524288.i 1.41695e6i 1.07246e6 + 6.98927e6i
3.2 −64.0000 + 64.0000i 1295.91 + 1295.91i 8192.00i 61416.3 + 48286.1i −165877. 905655. 905655.i 524288. + 524288.i 1.42420e6i −7.02096e6 + 840331.i
3.3 −64.0000 + 64.0000i 1920.76 + 1920.76i 8192.00i −66391.4 41178.9i −245857. −374464. + 374464.i 524288. + 524288.i 2.59566e6i 6.88450e6 1.61360e6i
7.1 −64.0000 64.0000i −1760.67 + 1760.67i 8192.00i 46225.0 + 62982.2i 225366. −59627.2 59627.2i 524288. 524288.i 1.41695e6i 1.07246e6 6.98927e6i
7.2 −64.0000 64.0000i 1295.91 1295.91i 8192.00i 61416.3 48286.1i −165877. 905655. + 905655.i 524288. 524288.i 1.42420e6i −7.02096e6 840331.i
7.3 −64.0000 64.0000i 1920.76 1920.76i 8192.00i −66391.4 + 41178.9i −245857. −374464. 374464.i 524288. 524288.i 2.59566e6i 6.88450e6 + 1.61360e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.15.c.a 6
3.b odd 2 1 90.15.g.a 6
4.b odd 2 1 80.15.p.a 6
5.b even 2 1 50.15.c.b 6
5.c odd 4 1 inner 10.15.c.a 6
5.c odd 4 1 50.15.c.b 6
15.e even 4 1 90.15.g.a 6
20.e even 4 1 80.15.p.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.15.c.a 6 1.a even 1 1 trivial
10.15.c.a 6 5.c odd 4 1 inner
50.15.c.b 6 5.b even 2 1
50.15.c.b 6 5.c odd 4 1
80.15.p.a 6 4.b odd 2 1
80.15.p.a 6 20.e even 4 1
90.15.g.a 6 3.b odd 2 1
90.15.g.a 6 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 2912 T_{3}^{5} + 4239872 T_{3}^{4} + 957313944 T_{3}^{3} + \)\(40\!\cdots\!76\)\( T_{3}^{2} - \)\(11\!\cdots\!68\)\( T_{3} + \)\(15\!\cdots\!12\)\( \) acting on \(S_{15}^{\mathrm{new}}(10, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 128 T + 8192 T^{2} )^{3} \)
$3$ \( 1 - 2912 T + 4239872 T^{2} - 12970691784 T^{3} - 12526339402017 T^{4} + 95435235141696072 T^{5} - \)\(14\!\cdots\!12\)\( T^{6} + \)\(45\!\cdots\!68\)\( T^{7} - \)\(28\!\cdots\!37\)\( T^{8} - \)\(14\!\cdots\!56\)\( T^{9} + \)\(22\!\cdots\!12\)\( T^{10} - \)\(72\!\cdots\!88\)\( T^{11} + \)\(11\!\cdots\!81\)\( T^{12} \)
$5$ \( 1 - 82500 T + 1080609375 T^{2} + 500785546875000 T^{3} + 6595516204833984375 T^{4} - \)\(30\!\cdots\!00\)\( T^{5} + \)\(22\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 943128 T + 444745212192 T^{2} + 140675682243197024 T^{3} - \)\(81\!\cdots\!57\)\( T^{4} - \)\(49\!\cdots\!32\)\( T^{5} + \)\(51\!\cdots\!28\)\( T^{6} - \)\(33\!\cdots\!68\)\( T^{7} - \)\(37\!\cdots\!57\)\( T^{8} + \)\(43\!\cdots\!76\)\( T^{9} + \)\(94\!\cdots\!92\)\( T^{10} - \)\(13\!\cdots\!72\)\( T^{11} + \)\(97\!\cdots\!01\)\( T^{12} \)
$11$ \( ( 1 + 22783284 T + 1280642975322975 T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(48\!\cdots\!75\)\( T^{4} + \)\(32\!\cdots\!04\)\( T^{5} + \)\(54\!\cdots\!21\)\( T^{6} )^{2} \)
$13$ \( 1 + 52149318 T + 1359775683932562 T^{2} + \)\(54\!\cdots\!66\)\( T^{3} + \)\(34\!\cdots\!03\)\( T^{4} - \)\(48\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!68\)\( T^{6} - \)\(19\!\cdots\!52\)\( T^{7} + \)\(53\!\cdots\!63\)\( T^{8} + \)\(33\!\cdots\!54\)\( T^{9} + \)\(32\!\cdots\!42\)\( T^{10} + \)\(49\!\cdots\!82\)\( T^{11} + \)\(37\!\cdots\!61\)\( T^{12} \)
$17$ \( 1 + 294348942 T + 43320649828259682 T^{2} - \)\(29\!\cdots\!06\)\( T^{3} - \)\(28\!\cdots\!37\)\( T^{4} - \)\(19\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!08\)\( T^{6} - \)\(33\!\cdots\!28\)\( T^{7} - \)\(79\!\cdots\!17\)\( T^{8} - \)\(14\!\cdots\!34\)\( T^{9} + \)\(34\!\cdots\!42\)\( T^{10} + \)\(39\!\cdots\!58\)\( T^{11} + \)\(22\!\cdots\!21\)\( T^{12} \)
$19$ \( 1 - 2616540268763896326 T^{2} + \)\(35\!\cdots\!15\)\( T^{4} - \)\(33\!\cdots\!20\)\( T^{6} + \)\(22\!\cdots\!15\)\( T^{8} - \)\(10\!\cdots\!06\)\( T^{10} + \)\(26\!\cdots\!21\)\( T^{12} \)
$23$ \( 1 + 9431163408 T + 44473421614199087232 T^{2} + \)\(17\!\cdots\!96\)\( T^{3} + \)\(45\!\cdots\!63\)\( T^{4} + \)\(71\!\cdots\!52\)\( T^{5} + \)\(15\!\cdots\!08\)\( T^{6} + \)\(82\!\cdots\!68\)\( T^{7} + \)\(61\!\cdots\!03\)\( T^{8} + \)\(27\!\cdots\!84\)\( T^{9} + \)\(80\!\cdots\!52\)\( T^{10} + \)\(19\!\cdots\!92\)\( T^{11} + \)\(24\!\cdots\!41\)\( T^{12} \)
$29$ \( 1 - \)\(10\!\cdots\!86\)\( T^{2} + \)\(56\!\cdots\!15\)\( T^{4} - \)\(19\!\cdots\!20\)\( T^{6} + \)\(50\!\cdots\!15\)\( T^{8} - \)\(85\!\cdots\!06\)\( T^{10} + \)\(69\!\cdots\!81\)\( T^{12} \)
$31$ \( ( 1 - 1860702696 T + \)\(77\!\cdots\!35\)\( T^{2} - \)\(66\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!35\)\( T^{4} - \)\(10\!\cdots\!36\)\( T^{5} + \)\(43\!\cdots\!61\)\( T^{6} )^{2} \)
$37$ \( 1 + 429898030002 T + \)\(92\!\cdots\!02\)\( T^{2} + \)\(14\!\cdots\!94\)\( T^{3} + \)\(18\!\cdots\!23\)\( T^{4} + \)\(22\!\cdots\!88\)\( T^{5} + \)\(23\!\cdots\!48\)\( T^{6} + \)\(20\!\cdots\!32\)\( T^{7} + \)\(15\!\cdots\!83\)\( T^{8} + \)\(10\!\cdots\!86\)\( T^{9} + \)\(60\!\cdots\!82\)\( T^{10} + \)\(25\!\cdots\!98\)\( T^{11} + \)\(53\!\cdots\!61\)\( T^{12} \)
$41$ \( ( 1 - 22840528956 T + \)\(76\!\cdots\!95\)\( T^{2} - \)\(72\!\cdots\!40\)\( T^{3} + \)\(29\!\cdots\!95\)\( T^{4} - \)\(32\!\cdots\!76\)\( T^{5} + \)\(54\!\cdots\!81\)\( T^{6} )^{2} \)
$43$ \( 1 + 935465548368 T + \)\(43\!\cdots\!12\)\( T^{2} + \)\(13\!\cdots\!96\)\( T^{3} + \)\(24\!\cdots\!03\)\( T^{4} + \)\(83\!\cdots\!72\)\( T^{5} - \)\(43\!\cdots\!32\)\( T^{6} + \)\(61\!\cdots\!28\)\( T^{7} + \)\(13\!\cdots\!03\)\( T^{8} + \)\(55\!\cdots\!04\)\( T^{9} + \)\(13\!\cdots\!12\)\( T^{10} + \)\(20\!\cdots\!32\)\( T^{11} + \)\(16\!\cdots\!01\)\( T^{12} \)
$47$ \( 1 + 966227586192 T + \)\(46\!\cdots\!32\)\( T^{2} + \)\(92\!\cdots\!24\)\( T^{3} - \)\(65\!\cdots\!37\)\( T^{4} - \)\(82\!\cdots\!92\)\( T^{5} - \)\(45\!\cdots\!92\)\( T^{6} - \)\(21\!\cdots\!48\)\( T^{7} - \)\(42\!\cdots\!57\)\( T^{8} + \)\(15\!\cdots\!16\)\( T^{9} + \)\(20\!\cdots\!72\)\( T^{10} + \)\(10\!\cdots\!08\)\( T^{11} + \)\(28\!\cdots\!81\)\( T^{12} \)
$53$ \( 1 + 1868182085058 T + \)\(17\!\cdots\!82\)\( T^{2} + \)\(23\!\cdots\!26\)\( T^{3} - \)\(15\!\cdots\!37\)\( T^{4} - \)\(63\!\cdots\!08\)\( T^{5} - \)\(62\!\cdots\!92\)\( T^{6} - \)\(87\!\cdots\!52\)\( T^{7} - \)\(29\!\cdots\!57\)\( T^{8} + \)\(61\!\cdots\!34\)\( T^{9} + \)\(63\!\cdots\!22\)\( T^{10} + \)\(93\!\cdots\!42\)\( T^{11} + \)\(69\!\cdots\!81\)\( T^{12} \)
$59$ \( 1 - \)\(25\!\cdots\!66\)\( T^{2} + \)\(32\!\cdots\!15\)\( T^{4} - \)\(25\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!15\)\( T^{8} - \)\(37\!\cdots\!06\)\( T^{10} + \)\(56\!\cdots\!61\)\( T^{12} \)
$61$ \( ( 1 - 1055549965236 T + \)\(27\!\cdots\!55\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!55\)\( T^{4} - \)\(10\!\cdots\!16\)\( T^{5} + \)\(96\!\cdots\!21\)\( T^{6} )^{2} \)
$67$ \( 1 + 8480735447712 T + \)\(35\!\cdots\!72\)\( T^{2} + \)\(14\!\cdots\!04\)\( T^{3} + \)\(10\!\cdots\!83\)\( T^{4} + \)\(11\!\cdots\!88\)\( T^{5} + \)\(71\!\cdots\!88\)\( T^{6} + \)\(43\!\cdots\!52\)\( T^{7} + \)\(14\!\cdots\!03\)\( T^{8} + \)\(73\!\cdots\!56\)\( T^{9} + \)\(65\!\cdots\!32\)\( T^{10} + \)\(56\!\cdots\!88\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} \)
$71$ \( ( 1 - 11166824728056 T + \)\(28\!\cdots\!55\)\( T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + \)\(23\!\cdots\!55\)\( T^{4} - \)\(76\!\cdots\!16\)\( T^{5} + \)\(56\!\cdots\!41\)\( T^{6} )^{2} \)
$73$ \( 1 + 6994307700378 T + \)\(24\!\cdots\!42\)\( T^{2} - \)\(15\!\cdots\!94\)\( T^{3} - \)\(11\!\cdots\!57\)\( T^{4} + \)\(39\!\cdots\!72\)\( T^{5} + \)\(30\!\cdots\!28\)\( T^{6} + \)\(47\!\cdots\!48\)\( T^{7} - \)\(16\!\cdots\!17\)\( T^{8} - \)\(27\!\cdots\!26\)\( T^{9} + \)\(54\!\cdots\!62\)\( T^{10} + \)\(18\!\cdots\!22\)\( T^{11} + \)\(33\!\cdots\!41\)\( T^{12} \)
$79$ \( 1 - \)\(21\!\cdots\!86\)\( T^{2} + \)\(19\!\cdots\!15\)\( T^{4} - \)\(94\!\cdots\!20\)\( T^{6} + \)\(26\!\cdots\!15\)\( T^{8} - \)\(39\!\cdots\!06\)\( T^{10} + \)\(25\!\cdots\!81\)\( T^{12} \)
$83$ \( 1 + 60521791593048 T + \)\(18\!\cdots\!52\)\( T^{2} + \)\(40\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!23\)\( T^{4} + \)\(43\!\cdots\!72\)\( T^{5} + \)\(13\!\cdots\!48\)\( T^{6} + \)\(32\!\cdots\!88\)\( T^{7} + \)\(60\!\cdots\!43\)\( T^{8} + \)\(16\!\cdots\!64\)\( T^{9} + \)\(53\!\cdots\!12\)\( T^{10} + \)\(13\!\cdots\!52\)\( T^{11} + \)\(15\!\cdots\!21\)\( T^{12} \)
$89$ \( 1 - \)\(10\!\cdots\!46\)\( T^{2} + \)\(45\!\cdots\!15\)\( T^{4} - \)\(11\!\cdots\!20\)\( T^{6} + \)\(17\!\cdots\!15\)\( T^{8} - \)\(14\!\cdots\!06\)\( T^{10} + \)\(56\!\cdots\!41\)\( T^{12} \)
$97$ \( 1 + 307307370113562 T + \)\(47\!\cdots\!22\)\( T^{2} + \)\(46\!\cdots\!34\)\( T^{3} + \)\(29\!\cdots\!83\)\( T^{4} + \)\(11\!\cdots\!28\)\( T^{5} + \)\(44\!\cdots\!88\)\( T^{6} + \)\(74\!\cdots\!32\)\( T^{7} + \)\(12\!\cdots\!63\)\( T^{8} + \)\(12\!\cdots\!06\)\( T^{9} + \)\(85\!\cdots\!62\)\( T^{10} + \)\(36\!\cdots\!38\)\( T^{11} + \)\(77\!\cdots\!81\)\( T^{12} \)
show more
show less