Properties

Label 450.6.f.f
Level $450$
Weight $6$
Character orbit 450.f
Analytic conductor $72.173$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 252 x^{14} + 27174 x^{12} - 1635700 x^{10} + 60061815 x^{8} - 1376564028 x^{6} + 19220200150 x^{4} - 149540021784 x^{2} + 498214340649\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{8}\cdot 5^{16} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \beta_{1} + 2 \beta_{5} ) q^{2} -16 \beta_{4} q^{4} + ( -33 - 33 \beta_{4} + \beta_{12} ) q^{7} + ( -32 \beta_{1} - 32 \beta_{5} ) q^{8} +O(q^{10})\) \( q + ( -2 \beta_{1} + 2 \beta_{5} ) q^{2} -16 \beta_{4} q^{4} + ( -33 - 33 \beta_{4} + \beta_{12} ) q^{7} + ( -32 \beta_{1} - 32 \beta_{5} ) q^{8} + ( -42 \beta_{1} - \beta_{3} - \beta_{10} + \beta_{11} ) q^{11} + ( 12 + 2 \beta_{2} - 12 \beta_{4} + 5 \beta_{9} - 2 \beta_{13} - 2 \beta_{14} ) q^{13} + ( -132 \beta_{5} - 2 \beta_{10} - 2 \beta_{11} ) q^{14} -256 q^{16} + ( 123 \beta_{1} + \beta_{3} - 123 \beta_{5} + 11 \beta_{6} + \beta_{15} ) q^{17} + ( -28 \beta_{4} - 11 \beta_{8} + 11 \beta_{9} + 6 \beta_{12} - 6 \beta_{13} + \beta_{14} ) q^{19} + ( -168 + 4 \beta_{2} - 168 \beta_{4} - 8 \beta_{12} + 4 \beta_{14} ) q^{22} + ( 348 \beta_{1} + 3 \beta_{3} + 348 \beta_{5} + 18 \beta_{7} + 3 \beta_{11} - 3 \beta_{15} ) q^{23} + ( -48 \beta_{1} + 8 \beta_{3} + 10 \beta_{6} + 10 \beta_{7} + 4 \beta_{10} - 4 \beta_{11} ) q^{26} + ( -528 + 528 \beta_{4} + 16 \beta_{13} ) q^{28} + ( -1581 \beta_{5} - 33 \beta_{6} + 33 \beta_{7} + \beta_{10} + \beta_{11} - 2 \beta_{15} ) q^{29} + ( 814 + 3 \beta_{2} + 27 \beta_{8} + 27 \beta_{9} + 6 \beta_{12} + 6 \beta_{13} ) q^{31} + ( 512 \beta_{1} - 512 \beta_{5} ) q^{32} + ( 984 \beta_{4} - 44 \beta_{8} + 44 \beta_{9} - 8 \beta_{14} ) q^{34} + ( -2958 - 6 \beta_{2} - 2958 \beta_{4} - 46 \beta_{8} + 16 \beta_{12} - 6 \beta_{14} ) q^{37} + ( -56 \beta_{1} - 2 \beta_{3} - 56 \beta_{5} + 44 \beta_{7} - 24 \beta_{11} + 2 \beta_{15} ) q^{38} + ( 2472 \beta_{1} - 20 \beta_{3} - 39 \beta_{6} - 39 \beta_{7} + \beta_{10} - \beta_{11} ) q^{41} + ( -3465 - 16 \beta_{2} + 3465 \beta_{4} - 238 \beta_{9} - 39 \beta_{13} + 16 \beta_{14} ) q^{43} + ( -672 \beta_{5} + 16 \beta_{10} + 16 \beta_{11} + 16 \beta_{15} ) q^{44} + ( 2784 - 24 \beta_{2} - 72 \beta_{8} - 72 \beta_{9} - 12 \beta_{12} - 12 \beta_{13} ) q^{46} + ( 1305 \beta_{1} - 12 \beta_{3} - 1305 \beta_{5} + 72 \beta_{6} + 3 \beta_{10} - 12 \beta_{15} ) q^{47} + ( 13796 \beta_{4} - 154 \beta_{8} + 154 \beta_{9} - 66 \beta_{12} + 66 \beta_{13} + 20 \beta_{14} ) q^{49} + ( -192 - 32 \beta_{2} - 192 \beta_{4} - 80 \beta_{8} + 32 \beta_{12} - 32 \beta_{14} ) q^{52} + ( 6087 \beta_{1} - 31 \beta_{3} + 6087 \beta_{5} + 250 \beta_{7} + 60 \beta_{11} + 31 \beta_{15} ) q^{53} + ( 2112 \beta_{1} - 32 \beta_{10} + 32 \beta_{11} ) q^{56} + ( -6324 - 8 \beta_{2} + 6324 \beta_{4} - 264 \beta_{9} - 8 \beta_{13} + 8 \beta_{14} ) q^{58} + ( -17328 \beta_{5} - 99 \beta_{6} + 99 \beta_{7} - 4 \beta_{10} - 4 \beta_{11} - 37 \beta_{15} ) q^{59} + ( 1775 + 58 \beta_{2} - 386 \beta_{8} - 386 \beta_{9} - 48 \beta_{12} - 48 \beta_{13} ) q^{61} + ( -1628 \beta_{1} + 6 \beta_{3} + 1628 \beta_{5} + 108 \beta_{6} - 24 \beta_{10} + 6 \beta_{15} ) q^{62} + 4096 \beta_{4} q^{64} + ( 15141 + 46 \beta_{2} + 15141 \beta_{4} - 558 \beta_{8} - 143 \beta_{12} + 46 \beta_{14} ) q^{67} + ( 1968 \beta_{1} + 16 \beta_{3} + 1968 \beta_{5} + 176 \beta_{7} - 16 \beta_{15} ) q^{68} + ( -5394 \beta_{1} + 70 \beta_{3} - 369 \beta_{6} - 369 \beta_{7} + 49 \beta_{10} - 49 \beta_{11} ) q^{71} + ( -26934 + 34 \beta_{2} + 26934 \beta_{4} - 158 \beta_{9} + 160 \beta_{13} - 34 \beta_{14} ) q^{73} + ( -11832 \beta_{5} - 92 \beta_{6} + 92 \beta_{7} - 32 \beta_{10} - 32 \beta_{11} - 24 \beta_{15} ) q^{74} + ( -448 + 16 \beta_{2} - 176 \beta_{8} - 176 \beta_{9} + 96 \beta_{12} + 96 \beta_{13} ) q^{76} + ( -14589 \beta_{1} + 43 \beta_{3} + 14589 \beta_{5} + 797 \beta_{6} + 54 \beta_{10} + 43 \beta_{15} ) q^{77} + ( 18592 \beta_{4} - 134 \beta_{8} + 134 \beta_{9} + 264 \beta_{12} - 264 \beta_{13} - 74 \beta_{14} ) q^{79} + ( 9888 + 80 \beta_{2} + 9888 \beta_{4} + 312 \beta_{8} + 8 \beta_{12} + 80 \beta_{14} ) q^{82} + ( 14649 \beta_{1} + 124 \beta_{3} + 14649 \beta_{5} + 758 \beta_{7} - 219 \beta_{11} - 124 \beta_{15} ) q^{83} + ( 13860 \beta_{1} - 64 \beta_{3} - 476 \beta_{6} - 476 \beta_{7} + 78 \beta_{10} - 78 \beta_{11} ) q^{86} + ( -2688 + 64 \beta_{2} + 2688 \beta_{4} - 128 \beta_{13} - 64 \beta_{14} ) q^{88} + ( -29298 \beta_{5} - 24 \beta_{6} + 24 \beta_{7} - 16 \beta_{10} - 16 \beta_{11} + 200 \beta_{15} ) q^{89} + ( -11592 - 257 \beta_{2} - 2117 \beta_{8} - 2117 \beta_{9} + 138 \beta_{12} + 138 \beta_{13} ) q^{91} + ( -5568 \beta_{1} - 48 \beta_{3} + 5568 \beta_{5} - 288 \beta_{6} + 48 \beta_{10} - 48 \beta_{15} ) q^{92} + ( 10440 \beta_{4} - 288 \beta_{8} + 288 \beta_{9} + 12 \beta_{12} - 12 \beta_{13} + 96 \beta_{14} ) q^{94} + ( 28572 - 104 \beta_{2} + 28572 \beta_{4} - 229 \beta_{8} + 476 \beta_{12} - 104 \beta_{14} ) q^{97} + ( 27592 \beta_{1} - 40 \beta_{3} + 27592 \beta_{5} + 616 \beta_{7} + 264 \beta_{11} + 40 \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 528q^{7} + O(q^{10}) \) \( 16q - 528q^{7} + 192q^{13} - 4096q^{16} - 2688q^{22} - 8448q^{28} + 13024q^{31} - 47328q^{37} - 55440q^{43} + 44544q^{46} - 3072q^{52} - 101184q^{58} + 28400q^{61} + 242256q^{67} - 430944q^{73} - 7168q^{76} + 158208q^{82} - 43008q^{88} - 185472q^{91} + 457152q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 252 x^{14} + 27174 x^{12} - 1635700 x^{10} + 60061815 x^{8} - 1376564028 x^{6} + 19220200150 x^{4} - 149540021784 x^{2} + 498214340649\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(1243 \nu^{14} - 299263 \nu^{12} + 29877040 \nu^{10} - 1593636821 \nu^{8} + 48749934596 \nu^{6} - 850502188925 \nu^{4} + 7807706068941 \nu^{2} - 29261576101626\)\()/ 112420373040 \)
\(\beta_{2}\)\(=\)\((\)\(24977 \nu^{14} - 5856811 \nu^{12} + 573008898 \nu^{10} - 30267472987 \nu^{8} + 931137236870 \nu^{6} - 16647263693019 \nu^{4} + 159420486277953 \nu^{2} - 629954627005836\)\()/ 19448681904 \)
\(\beta_{3}\)\(=\)\((\)\(163510355 \nu^{14} - 40270160735 \nu^{12} + 4115640904256 \nu^{10} - 225374410231405 \nu^{8} + 7116631539657652 \nu^{6} - 129323565323281285 \nu^{4} + 1253546679325162581 \nu^{2} - 5046266306555779554\)\()/ 51286174180848 \)
\(\beta_{4}\)\(=\)\((\)\(13005769 \nu^{15} - 2922257905 \nu^{13} + 274239495022 \nu^{11} - 13908239540510 \nu^{9} + 411296237681312 \nu^{7} - 7087408750009418 \nu^{5} + 65965034851293429 \nu^{3} - 257547598373810223 \nu\)\()/ 3431928995291268 \)
\(\beta_{5}\)\(=\)\((\)\(4342753 \nu^{15} - 939402004 \nu^{13} + 83950770889 \nu^{11} - 4002387230393 \nu^{9} + 109723224072929 \nu^{7} - 1730047326394301 \nu^{5} + 14604057301696524 \nu^{3} - 51741237981797217 \nu\)\()/ 1018954446593040 \)
\(\beta_{6}\)\(=\)\((\)\(-35476130685 \nu^{15} - 763584094480 \nu^{14} + 8478443606160 \nu^{13} + 169116797879530 \nu^{12} - 847604575634655 \nu^{11} - 15579660732155590 \nu^{10} + 45837615905666520 \nu^{9} + 771588700584996110 \nu^{8} - 1444273690738881705 \nu^{7} - 22127804932109933090 \nu^{6} + 26394311595139439100 \nu^{5} + 366231722721457066250 \nu^{4} - 257036793014447433450 \nu^{3} - 3230642544959119765050 \nu^{2} + 1022287821928865814690 \nu + 11765042771871181236120\)\()/ 1005555195620341524 \)
\(\beta_{7}\)\(=\)\((\)\(35476130685 \nu^{15} - 763584094480 \nu^{14} - 8478443606160 \nu^{13} + 169116797879530 \nu^{12} + 847604575634655 \nu^{11} - 15579660732155590 \nu^{10} - 45837615905666520 \nu^{9} + 771588700584996110 \nu^{8} + 1444273690738881705 \nu^{7} - 22127804932109933090 \nu^{6} - 26394311595139439100 \nu^{5} + 366231722721457066250 \nu^{4} + 257036793014447433450 \nu^{3} - 3230642544959119765050 \nu^{2} - 1022287821928865814690 \nu + 11765042771871181236120\)\()/ 1005555195620341524 \)
\(\beta_{8}\)\(=\)\((\)\(2072048995 \nu^{15} + 19070701455 \nu^{14} - 491744708950 \nu^{13} - 4207239550965 \nu^{12} + 48657332485945 \nu^{11} + 385641856914870 \nu^{10} - 2593149892873775 \nu^{9} - 18988491404348205 \nu^{8} + 80162616118324505 \nu^{7} + 541766611309834050 \nu^{6} - 1435078362807418355 \nu^{5} - 8954852543815712685 \nu^{4} + 13789454263549266330 \nu^{3} + 79505501363599487895 \nu^{2} - 55292629689268300845 \nu - 293642169956678167740\)\()/ 27455431962330144 \)
\(\beta_{9}\)\(=\)\((\)\(-2072048995 \nu^{15} + 19070701455 \nu^{14} + 491744708950 \nu^{13} - 4207239550965 \nu^{12} - 48657332485945 \nu^{11} + 385641856914870 \nu^{10} + 2593149892873775 \nu^{9} - 18988491404348205 \nu^{8} - 80162616118324505 \nu^{7} + 541766611309834050 \nu^{6} + 1435078362807418355 \nu^{5} - 8954852543815712685 \nu^{4} - 13789454263549266330 \nu^{3} + 79505501363599487895 \nu^{2} + 55292629689268300845 \nu - 293642169956678167740\)\()/ 27455431962330144 \)
\(\beta_{10}\)\(=\)\((\)\(429018357807 \nu^{15} + 10599864475589 \nu^{14} - 93430716258888 \nu^{13} - 2352749061201449 \nu^{12} + 8435772096579891 \nu^{11} + 218002089612412016 \nu^{10} - 408152015971899291 \nu^{9} - 10914253010827358923 \nu^{8} + 11416571773837453251 \nu^{7} + 318718407927911548540 \nu^{6} - 184803579516075422391 \nu^{5} - 5427322210989097504435 \nu^{4} + 1613127280753375470864 \nu^{3} + 49967475773438741371299 \nu^{2} - 6076343329940034415503 \nu - 193236481422756177806142\)\()/ 4022220782481366096 \)
\(\beta_{11}\)\(=\)\((\)\(429018357807 \nu^{15} - 10599864475589 \nu^{14} - 93430716258888 \nu^{13} + 2352749061201449 \nu^{12} + 8435772096579891 \nu^{11} - 218002089612412016 \nu^{10} - 408152015971899291 \nu^{9} + 10914253010827358923 \nu^{8} + 11416571773837453251 \nu^{7} - 318718407927911548540 \nu^{6} - 184803579516075422391 \nu^{5} + 5427322210989097504435 \nu^{4} + 1613127280753375470864 \nu^{3} - 49967475773438741371299 \nu^{2} - 6076343329940034415503 \nu + 193236481422756177806142\)\()/ 4022220782481366096 \)
\(\beta_{12}\)\(=\)\((\)\(3041597975 \nu^{15} + 52730001315 \nu^{14} - 738765803630 \nu^{13} - 11534685140745 \nu^{12} + 74042619080045 \nu^{11} + 1044373667039550 \nu^{10} - 3940572100341235 \nu^{9} - 50559994550210265 \nu^{8} + 119339329319558125 \nu^{7} + 1410929255124996570 \nu^{6} - 2040756026005557175 \nu^{5} - 22693248907073932905 \nu^{4} + 18121783233242965050 \nu^{3} + 195459092552968990875 \nu^{2} - 64627814447659785465 \nu - 703830181889124818100\)\()/ 27455431962330144 \)
\(\beta_{13}\)\(=\)\((\)\(-3041597975 \nu^{15} + 52730001315 \nu^{14} + 738765803630 \nu^{13} - 11534685140745 \nu^{12} - 74042619080045 \nu^{11} + 1044373667039550 \nu^{10} + 3940572100341235 \nu^{9} - 50559994550210265 \nu^{8} - 119339329319558125 \nu^{7} + 1410929255124996570 \nu^{6} + 2040756026005557175 \nu^{5} - 22693248907073932905 \nu^{4} - 18121783233242965050 \nu^{3} + 195459092552968990875 \nu^{2} + 64627814447659785465 \nu - 703830181889124818100\)\()/ 27455431962330144 \)
\(\beta_{14}\)\(=\)\((\)\(-11804228221 \nu^{15} + 2602717150930 \nu^{13} - 238555634297431 \nu^{11} + 11746269286389005 \nu^{9} - 334640537583005279 \nu^{7} + 5496485765099469449 \nu^{5} - 48061890418857044454 \nu^{3} + 173305687483144264671 \nu\)\()/ 13727715981165072 \)
\(\beta_{15}\)\(=\)\((\)\(-710861580817 \nu^{15} + 156264690672860 \nu^{13} - 14265987240139401 \nu^{11} + 699313112450076185 \nu^{9} - 19855932280902046081 \nu^{7} + 326496697419587059533 \nu^{5} - 2882356343192526090732 \nu^{3} + 10573311463608984643593 \nu\)\()/ 446913420275707344 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-5 \beta_{11} - 5 \beta_{10} - 2 \beta_{7} + 2 \beta_{6} + 150 \beta_{5} + 150 \beta_{4}\)\()/300\)
\(\nu^{2}\)\(=\)\((\)\(10 \beta_{13} + 10 \beta_{12} - 5 \beta_{11} + 5 \beta_{10} - 50 \beta_{9} - 50 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 150 \beta_{1} + 9450\)\()/300\)
\(\nu^{3}\)\(=\)\((\)\(45 \beta_{15} - 15 \beta_{13} + 15 \beta_{12} - 160 \beta_{11} - 160 \beta_{10} + 75 \beta_{9} - 75 \beta_{8} - \beta_{7} + \beta_{6} + 14025 \beta_{5} + 14250 \beta_{4}\)\()/300\)
\(\nu^{4}\)\(=\)\((\)\(620 \beta_{13} + 620 \beta_{12} - 325 \beta_{11} + 325 \beta_{10} - 3136 \beta_{9} - 3136 \beta_{8} - 4 \beta_{7} - 4 \beta_{6} - 90 \beta_{3} + 180 \beta_{2} + 28200 \beta_{1} + 343350\)\()/300\)
\(\nu^{5}\)\(=\)\((\)\(2925 \beta_{15} + 450 \beta_{14} - 1575 \beta_{13} + 1575 \beta_{12} - 5870 \beta_{11} - 5870 \beta_{10} + 7965 \beta_{9} - 7965 \beta_{8} + 4852 \beta_{7} - 4852 \beta_{6} + 811725 \beta_{5} + 882150 \beta_{4}\)\()/300\)
\(\nu^{6}\)\(=\)\((\)\(6428 \beta_{13} + 6428 \beta_{12} - 3685 \beta_{11} + 3685 \beta_{10} - 31510 \beta_{9} - 31510 \beta_{8} + 2909 \beta_{7} + 2909 \beta_{6} - 1800 \beta_{3} + 3330 \beta_{2} + 501150 \beta_{1} + 2749230\)\()/60\)
\(\nu^{7}\)\(=\)\((\)\(151380 \beta_{15} + 59850 \beta_{14} - 118020 \beta_{13} + 118020 \beta_{12} - 235960 \beta_{11} - 235960 \beta_{10} + 579390 \beta_{9} - 579390 \beta_{8} + 406583 \beta_{7} - 406583 \beta_{6} + 42494700 \beta_{5} + 51215700 \beta_{4}\)\()/300\)
\(\nu^{8}\)\(=\)\((\)\(1593440 \beta_{13} + 1593440 \beta_{12} - 1030585 \beta_{11} + 1030585 \beta_{10} - 7401064 \beta_{9} - 7401064 \beta_{8} + 1694198 \beta_{7} + 1694198 \beta_{6} - 647730 \beta_{3} + 1057320 \beta_{2} + 181738200 \beta_{1} + 581929650\)\()/300\)
\(\nu^{9}\)\(=\)\((\)\(7255935 \beta_{15} + 5118930 \beta_{14} - 7885245 \beta_{13} + 7885245 \beta_{12} - 9980255 \beta_{11} - 9980255 \beta_{10} + 36814731 \beta_{9} - 36814731 \beta_{8} + 24582772 \beta_{7} - 24582772 \beta_{6} + 2122160325 \beta_{5} + 2929711200 \beta_{4}\)\()/300\)
\(\nu^{10}\)\(=\)\((\)\(76776310 \beta_{13} + 76776310 \beta_{12} - 57761270 \beta_{11} + 57761270 \beta_{10} - 336779420 \beta_{9} - 336779420 \beta_{8} + 135722137 \beta_{7} + 135722137 \beta_{6} - 41201100 \beta_{3} + 57158550 \beta_{2} + 11991519600 \beta_{1} + 25247590200\)\()/300\)
\(\nu^{11}\)\(=\)\((\)\(333273195 \beta_{15} + 361958850 \beta_{14} - 495866745 \beta_{13} + 495866745 \beta_{12} - 430300690 \beta_{11} - 430300690 \beta_{10} + 2196216495 \beta_{9} - 2196216495 \beta_{8} + 1288408472 \beta_{7} - 1288408472 \beta_{6} + 101755038675 \beta_{5} + 166290586950 \beta_{4}\)\()/300\)
\(\nu^{12}\)\(=\)\((\)\(893550485 \beta_{13} + 893550485 \beta_{12} - 808648375 \beta_{11} + 808648375 \beta_{10} - 3728999989 \beta_{9} - 3728999989 \beta_{8} + 2312917130 \beta_{7} + 2312917130 \beta_{6} - 615934575 \beta_{3} + 702172620 \beta_{2} + 186372805125 \beta_{1} + 273843706725\)\()/75\)
\(\nu^{13}\)\(=\)\((\)\(14730106860 \beta_{15} + 23086418940 \beta_{14} - 29871401820 \beta_{13} + 29871401820 \beta_{12} - 18426051335 \beta_{11} - 18426051335 \beta_{10} + 126406247448 \beta_{9} - 126406247448 \beta_{8} + 61670251540 \beta_{7} - 61670251540 \beta_{6} + 4663275766350 \beta_{5} + 9353669469450 \beta_{4}\)\()/300\)
\(\nu^{14}\)\(=\)\((\)\(158932875970 \beta_{13} + 158932875970 \beta_{12} - 180043217735 \beta_{11} + 180043217735 \beta_{10} - 637715336054 \beta_{9} - 637715336054 \beta_{8} + 576659919400 \beta_{7} + 576659919400 \beta_{6} - 141899409360 \beta_{3} + 128074755420 \beta_{2} + 44362911494850 \beta_{1} + 46466673841350\)\()/300\)
\(\nu^{15}\)\(=\)\((\)\(620186330205 \beta_{15} + 1381665070800 \beta_{14} - 1738273372215 \beta_{13} + 1738273372215 \beta_{12} - 763889393140 \beta_{11} - 763889393140 \beta_{10} + 7099415389875 \beta_{9} - 7099415389875 \beta_{8} + 2729549614211 \beta_{7} - 2729549614211 \beta_{6} + 201921728340225 \beta_{5} + 520115803250250 \beta_{4}\)\()/300\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
5.79464 + 0.500000i
3.79037 + 0.500000i
−5.20458 + 0.500000i
−7.20885 + 0.500000i
−5.79464 + 0.500000i
−3.79037 + 0.500000i
5.20458 + 0.500000i
7.20885 + 0.500000i
5.79464 0.500000i
3.79037 0.500000i
−5.20458 0.500000i
−7.20885 0.500000i
−5.79464 0.500000i
−3.79037 0.500000i
5.20458 0.500000i
7.20885 0.500000i
−2.82843 + 2.82843i 0 16.0000i 0 0 −158.675 158.675i 45.2548 + 45.2548i 0 0
107.2 −2.82843 + 2.82843i 0 16.0000i 0 0 −140.653 140.653i 45.2548 + 45.2548i 0 0
107.3 −2.82843 + 2.82843i 0 16.0000i 0 0 50.1584 + 50.1584i 45.2548 + 45.2548i 0 0
107.4 −2.82843 + 2.82843i 0 16.0000i 0 0 117.170 + 117.170i 45.2548 + 45.2548i 0 0
107.5 2.82843 2.82843i 0 16.0000i 0 0 −158.675 158.675i −45.2548 45.2548i 0 0
107.6 2.82843 2.82843i 0 16.0000i 0 0 −140.653 140.653i −45.2548 45.2548i 0 0
107.7 2.82843 2.82843i 0 16.0000i 0 0 50.1584 + 50.1584i −45.2548 45.2548i 0 0
107.8 2.82843 2.82843i 0 16.0000i 0 0 117.170 + 117.170i −45.2548 45.2548i 0 0
143.1 −2.82843 2.82843i 0 16.0000i 0 0 −158.675 + 158.675i 45.2548 45.2548i 0 0
143.2 −2.82843 2.82843i 0 16.0000i 0 0 −140.653 + 140.653i 45.2548 45.2548i 0 0
143.3 −2.82843 2.82843i 0 16.0000i 0 0 50.1584 50.1584i 45.2548 45.2548i 0 0
143.4 −2.82843 2.82843i 0 16.0000i 0 0 117.170 117.170i 45.2548 45.2548i 0 0
143.5 2.82843 + 2.82843i 0 16.0000i 0 0 −158.675 + 158.675i −45.2548 + 45.2548i 0 0
143.6 2.82843 + 2.82843i 0 16.0000i 0 0 −140.653 + 140.653i −45.2548 + 45.2548i 0 0
143.7 2.82843 + 2.82843i 0 16.0000i 0 0 50.1584 50.1584i −45.2548 + 45.2548i 0 0
143.8 2.82843 + 2.82843i 0 16.0000i 0 0 117.170 117.170i −45.2548 + 45.2548i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.6.f.f 16
3.b odd 2 1 inner 450.6.f.f 16
5.b even 2 1 450.6.f.g yes 16
5.c odd 4 1 inner 450.6.f.f 16
5.c odd 4 1 450.6.f.g yes 16
15.d odd 2 1 450.6.f.g yes 16
15.e even 4 1 inner 450.6.f.f 16
15.e even 4 1 450.6.f.g yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.6.f.f 16 1.a even 1 1 trivial
450.6.f.f 16 3.b odd 2 1 inner
450.6.f.f 16 5.c odd 4 1 inner
450.6.f.f 16 15.e even 4 1 inner
450.6.f.g yes 16 5.b even 2 1
450.6.f.g yes 16 5.c odd 4 1
450.6.f.g yes 16 15.d odd 2 1
450.6.f.g yes 16 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(31\!\cdots\!32\)\( T_{7}^{2} - \)\(41\!\cdots\!72\)\( T_{7} + \)\(27\!\cdots\!81\)\( \)">\(T_{7}^{8} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(450, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 256 + T^{4} )^{4} \)
$3$ \( T^{16} \)
$5$ \( T^{16} \)
$7$ \( ( 275272322339952081 - 4145479465222872 T + 31214543929632 T^{2} + 207419390280 T^{3} + 867535506 T^{4} - 3657240 T^{5} + 34848 T^{6} + 264 T^{7} + T^{8} )^{2} \)
$11$ \( ( 2202827329562090256 + 2869366023055008 T^{2} + 143710409304 T^{4} + 750312 T^{6} + T^{8} )^{2} \)
$13$ \( ( \)\(58\!\cdots\!61\)\( - 53498702309017766112 T + 24452964190167552 T^{2} + 299794937093280 T^{3} + 1080225725346 T^{4} + 341206560 T^{5} + 4608 T^{6} - 96 T^{7} + T^{8} )^{2} \)
$17$ \( \)\(26\!\cdots\!96\)\( + \)\(44\!\cdots\!64\)\( T^{4} + \)\(29\!\cdots\!16\)\( T^{8} + 6230890356624 T^{12} + T^{16} \)
$19$ \( ( \)\(50\!\cdots\!61\)\( + 34813732986594665716 T^{2} + 40474130593686 T^{4} + 12463636 T^{6} + T^{8} )^{2} \)
$23$ \( \)\(18\!\cdots\!96\)\( + \)\(13\!\cdots\!64\)\( T^{4} + \)\(15\!\cdots\!16\)\( T^{8} + 389863996519824 T^{12} + T^{16} \)
$29$ \( ( \)\(14\!\cdots\!56\)\( - \)\(26\!\cdots\!92\)\( T^{2} + 823065547874904 T^{4} - 54261288 T^{6} + T^{8} )^{2} \)
$31$ \( ( -32693228586959 + 36287406824 T - 7449474 T^{2} - 3256 T^{3} + T^{4} )^{4} \)
$37$ \( ( \)\(93\!\cdots\!56\)\( + \)\(20\!\cdots\!28\)\( T + \)\(21\!\cdots\!32\)\( T^{2} + 14313551369447796480 T^{3} + 6091731916858656 T^{4} + 1666650836160 T^{5} + 279992448 T^{6} + 23664 T^{7} + T^{8} )^{2} \)
$41$ \( ( \)\(56\!\cdots\!76\)\( + \)\(15\!\cdots\!28\)\( T^{2} + 40722348408822144 T^{4} + 355203072 T^{6} + T^{8} )^{2} \)
$43$ \( ( \)\(17\!\cdots\!25\)\( + \)\(45\!\cdots\!00\)\( T + \)\(58\!\cdots\!00\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{3} + 148211262823781250 T^{4} + 3791981817000 T^{5} + 384199200 T^{6} + 27720 T^{7} + T^{8} )^{2} \)
$47$ \( \)\(26\!\cdots\!00\)\( + \)\(11\!\cdots\!00\)\( T^{4} + \)\(24\!\cdots\!00\)\( T^{8} + 79434374011290000 T^{12} + T^{16} \)
$53$ \( \)\(57\!\cdots\!76\)\( + \)\(54\!\cdots\!04\)\( T^{4} + \)\(17\!\cdots\!56\)\( T^{8} + 2313003202667426304 T^{12} + T^{16} \)
$59$ \( ( \)\(71\!\cdots\!76\)\( - \)\(78\!\cdots\!28\)\( T^{2} + 3351097683309702744 T^{4} - 3496658472 T^{6} + T^{8} )^{2} \)
$61$ \( ( -16778221777724375 + 28715262272500 T - 1678116450 T^{2} - 7100 T^{3} + T^{4} )^{4} \)
$67$ \( ( \)\(54\!\cdots\!61\)\( - \)\(37\!\cdots\!84\)\( T + \)\(12\!\cdots\!48\)\( T^{2} - \)\(21\!\cdots\!60\)\( T^{3} + 4852369026937042146 T^{4} - 213905707492680 T^{5} + 7335996192 T^{6} - 121128 T^{7} + T^{8} )^{2} \)
$71$ \( ( \)\(97\!\cdots\!56\)\( + \)\(96\!\cdots\!92\)\( T^{2} + 16771747678338386304 T^{4} + 7560050688 T^{6} + T^{8} )^{2} \)
$73$ \( ( \)\(11\!\cdots\!36\)\( + \)\(11\!\cdots\!16\)\( T + \)\(51\!\cdots\!48\)\( T^{2} + \)\(54\!\cdots\!40\)\( T^{3} + 44251059717530616096 T^{4} + 1349346664681920 T^{5} + 23214091392 T^{6} + 215472 T^{7} + T^{8} )^{2} \)
$79$ \( ( \)\(34\!\cdots\!16\)\( + \)\(15\!\cdots\!76\)\( T^{2} + 82788848706226510176 T^{4} + 15776101456 T^{6} + T^{8} )^{2} \)
$83$ \( \)\(58\!\cdots\!36\)\( + \)\(10\!\cdots\!84\)\( T^{4} + \)\(32\!\cdots\!36\)\( T^{8} + \)\(33\!\cdots\!64\)\( T^{12} + T^{16} \)
$89$ \( ( \)\(12\!\cdots\!96\)\( - \)\(11\!\cdots\!48\)\( T^{2} + \)\(38\!\cdots\!84\)\( T^{4} - 35591987232 T^{6} + T^{8} )^{2} \)
$97$ \( ( \)\(11\!\cdots\!01\)\( - \)\(13\!\cdots\!32\)\( T + \)\(87\!\cdots\!12\)\( T^{2} - \)\(53\!\cdots\!20\)\( T^{3} + 25867898018211951666 T^{4} - 817289052467040 T^{5} + 26123493888 T^{6} - 228576 T^{7} + T^{8} )^{2} \)
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