# Properties

 Label 605.2.b.h Level $605$ Weight $2$ Character orbit 605.b Analytic conductor $4.831$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 4 x^{11} + x^{10} + 34 x^{9} - 123 x^{8} - 20 x^{7} + 516 x^{6} - 668 x^{5} - 67 x^{4} + 3848 x^{3} + 6697 x^{2} + 4398 x + 1089$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{9} q^{2} -\beta_{8} q^{3} + ( -2 - \beta_{6} ) q^{4} + \beta_{4} q^{5} + ( -2 \beta_{1} - \beta_{3} + \beta_{7} ) q^{6} + ( \beta_{7} + \beta_{9} ) q^{7} + ( -2 \beta_{9} - \beta_{10} ) q^{8} + ( -3 + \beta_{2} - \beta_{4} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + \beta_{9} q^{2} -\beta_{8} q^{3} + ( -2 - \beta_{6} ) q^{4} + \beta_{4} q^{5} + ( -2 \beta_{1} - \beta_{3} + \beta_{7} ) q^{6} + ( \beta_{7} + \beta_{9} ) q^{7} + ( -2 \beta_{9} - \beta_{10} ) q^{8} + ( -3 + \beta_{2} - \beta_{4} - \beta_{6} ) q^{9} + ( \beta_{1} + \beta_{3} - \beta_{9} - \beta_{10} ) q^{10} + ( \beta_{2} + \beta_{4} + 2 \beta_{8} - \beta_{11} ) q^{12} + ( -\beta_{7} - \beta_{9} - \beta_{10} ) q^{13} + ( -4 + \beta_{2} - \beta_{4} - \beta_{6} ) q^{14} + ( 3 - \beta_{2} + \beta_{4} + \beta_{6} - \beta_{11} ) q^{15} + ( 1 + 2 \beta_{2} - 2 \beta_{4} ) q^{16} + \beta_{7} q^{17} + ( -\beta_{7} - 3 \beta_{9} + \beta_{10} ) q^{18} + ( -\beta_{1} - \beta_{5} + \beta_{7} ) q^{19} + ( 1 + 2 \beta_{2} - \beta_{4} + \beta_{6} - 2 \beta_{8} ) q^{20} + ( -2 \beta_{1} - 3 \beta_{3} + \beta_{7} ) q^{21} + ( \beta_{2} + \beta_{4} - \beta_{8} + \beta_{11} ) q^{23} + ( 3 \beta_{1} - \beta_{3} + \beta_{5} - 2 \beta_{7} ) q^{24} + ( -1 - \beta_{4} + \beta_{6} + \beta_{8} + \beta_{11} ) q^{25} + ( 1 + \beta_{2} - \beta_{4} + \beta_{6} ) q^{26} + ( \beta_{2} + \beta_{4} + \beta_{11} ) q^{27} + ( \beta_{7} - 2 \beta_{9} + \beta_{10} ) q^{28} + ( -2 \beta_{1} + \beta_{7} ) q^{29} + ( \beta_{1} - 3 \beta_{3} + \beta_{5} + 3 \beta_{9} - \beta_{10} ) q^{30} + ( 3 - 2 \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{31} + ( -2 \beta_{7} + \beta_{9} + 2 \beta_{10} ) q^{32} + ( \beta_{2} - \beta_{4} ) q^{34} + ( \beta_{1} - \beta_{5} - \beta_{10} ) q^{35} + ( 9 - \beta_{2} + \beta_{4} + \beta_{6} ) q^{36} + ( -\beta_{2} - \beta_{4} + 2 \beta_{8} ) q^{37} + ( \beta_{2} + \beta_{4} + \beta_{8} - 3 \beta_{11} ) q^{38} + ( \beta_{1} + \beta_{5} - \beta_{7} ) q^{39} + ( -\beta_{1} + \beta_{3} + 4 \beta_{9} + 2 \beta_{10} ) q^{40} + ( \beta_{1} + 3 \beta_{3} + \beta_{5} - \beta_{7} ) q^{41} + ( \beta_{2} + \beta_{4} + 4 \beta_{8} + \beta_{11} ) q^{42} + ( \beta_{7} + \beta_{9} - 2 \beta_{10} ) q^{43} + ( -3 + 2 \beta_{2} - \beta_{4} - 3 \beta_{8} - \beta_{11} ) q^{45} + ( -\beta_{1} + 4 \beta_{3} - \beta_{5} + \beta_{7} ) q^{46} -\beta_{8} q^{47} + ( -\beta_{8} + 4 \beta_{11} ) q^{48} + ( -1 + \beta_{2} - \beta_{4} + \beta_{6} ) q^{49} + ( 3 \beta_{3} - \beta_{5} + 2 \beta_{9} + 2 \beta_{10} ) q^{50} -2 \beta_{3} q^{51} + ( -3 \beta_{7} + 3 \beta_{9} + \beta_{10} ) q^{52} + ( -\beta_{8} - \beta_{11} ) q^{53} + ( \beta_{1} + 5 \beta_{3} - \beta_{5} ) q^{54} + ( 3 + \beta_{2} - \beta_{4} ) q^{56} + ( -3 \beta_{9} - 3 \beta_{10} ) q^{57} + ( \beta_{2} + \beta_{4} + 4 \beta_{8} - 2 \beta_{11} ) q^{58} + ( -1 - 2 \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{59} + ( -9 - 2 \beta_{4} - \beta_{6} - \beta_{8} + 4 \beta_{11} ) q^{60} + ( \beta_{1} + \beta_{3} - \beta_{5} ) q^{61} + ( 2 \beta_{7} + \beta_{9} - 3 \beta_{10} ) q^{62} + ( -3 \beta_{9} + 3 \beta_{10} ) q^{63} + ( 4 - 2 \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{64} + ( 3 \beta_{3} + \beta_{5} + 2 \beta_{9} + \beta_{10} ) q^{65} + ( -2 \beta_{2} - 2 \beta_{4} + \beta_{8} + 2 \beta_{11} ) q^{67} + ( \beta_{7} + 2 \beta_{9} + 2 \beta_{10} ) q^{68} + ( 3 - 4 \beta_{2} + 4 \beta_{4} + \beta_{6} ) q^{69} + ( -3 + 2 \beta_{2} - 2 \beta_{4} - 3 \beta_{8} - \beta_{11} ) q^{70} + ( 11 - 2 \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{71} + ( -\beta_{7} + 3 \beta_{9} + \beta_{10} ) q^{72} + ( 2 \beta_{7} - 4 \beta_{9} ) q^{73} + ( 2 \beta_{1} - \beta_{7} ) q^{74} + ( 6 - 4 \beta_{2} + 2 \beta_{4} + \beta_{8} + 2 \beta_{11} ) q^{75} + ( 5 \beta_{1} - 6 \beta_{3} + \beta_{5} - 3 \beta_{7} ) q^{76} + ( -\beta_{2} - \beta_{4} - \beta_{8} + 3 \beta_{11} ) q^{78} + ( -4 \beta_{1} - 4 \beta_{3} + 2 \beta_{7} ) q^{79} + ( -8 + 3 \beta_{4} - 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{11} ) q^{80} + ( -2 \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{81} + ( -\beta_{2} - \beta_{4} - \beta_{8} ) q^{82} + ( -4 \beta_{7} - \beta_{9} + \beta_{10} ) q^{83} + ( 5 \beta_{1} + 3 \beta_{3} - \beta_{5} - 2 \beta_{7} ) q^{84} + ( -\beta_{3} - \beta_{5} + \beta_{9} ) q^{85} + ( -10 + 5 \beta_{2} - 5 \beta_{4} - \beta_{6} ) q^{86} -6 \beta_{9} q^{87} -9 q^{89} + ( -4 \beta_{1} - 5 \beta_{3} + \beta_{5} + 3 \beta_{10} ) q^{90} + ( 3 + 2 \beta_{2} - 2 \beta_{4} - 3 \beta_{6} ) q^{91} + ( 3 \beta_{2} + 3 \beta_{4} - \beta_{8} - 5 \beta_{11} ) q^{92} + ( -\beta_{2} - \beta_{4} - 3 \beta_{8} - 3 \beta_{11} ) q^{93} + ( -2 \beta_{1} - \beta_{3} + \beta_{7} ) q^{94} + ( 3 \beta_{1} + \beta_{5} - 5 \beta_{7} ) q^{95} + ( 9 \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{96} + ( -2 \beta_{2} - 2 \beta_{4} + \beta_{8} - 3 \beta_{11} ) q^{97} + ( -\beta_{7} + 3 \beta_{9} + 3 \beta_{10} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 20q^{4} - 6q^{5} - 20q^{9} + O(q^{10})$$ $$12q - 20q^{4} - 6q^{5} - 20q^{9} - 32q^{14} + 20q^{15} + 36q^{16} + 26q^{20} - 10q^{25} + 20q^{26} + 8q^{31} + 12q^{34} + 92q^{36} - 18q^{45} - 4q^{49} + 48q^{56} - 32q^{59} - 92q^{60} + 28q^{64} - 16q^{69} - 12q^{70} + 112q^{71} + 36q^{75} - 106q^{80} - 20q^{81} - 56q^{86} - 108q^{89} + 72q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} + x^{10} + 34 x^{9} - 123 x^{8} - 20 x^{7} + 516 x^{6} - 668 x^{5} - 67 x^{4} + 3848 x^{3} + 6697 x^{2} + 4398 x + 1089$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4845953018 \nu^{11} + 963958706563 \nu^{10} - 4797665688052 \nu^{9} + 4902239486888 \nu^{8} + 29463765100965 \nu^{7} - 144716933147575 \nu^{6} + 100455369195159 \nu^{5} + 451532147574218 \nu^{4} - 1024544568879806 \nu^{3} + 830338876364632 \nu^{2} + 3214832172020702 \nu + 2640803187617808$$$$)/ 308731269222837$$ $$\beta_{2}$$ $$=$$ $$($$$$-3079103815 \nu^{11} + 42281880539 \nu^{10} - 121107594570 \nu^{9} - 71811869159 \nu^{8} + 1451122751103 \nu^{7} - 4422506999162 \nu^{6} - 861459944397 \nu^{5} + 20698222998830 \nu^{4} - 34901727549676 \nu^{3} - 2633367965615 \nu^{2} + 150711385091199 \nu + 156378153801710$$$$)/ 34303474358093$$ $$\beta_{3}$$ $$=$$ $$($$$$66674 \nu^{11} - 682376 \nu^{10} + 2180609 \nu^{9} + 29294 \nu^{8} - 21496194 \nu^{7} + 62985308 \nu^{6} - 15490998 \nu^{5} - 237821503 \nu^{4} + 480592660 \nu^{3} - 125236496 \nu^{2} - 731111689 \nu - 625607220$$$$)/ 167498433$$ $$\beta_{4}$$ $$=$$ $$($$$$17157797861 \nu^{11} - 108022004876 \nu^{10} + 240501508768 \nu^{9} + 116703163235 \nu^{8} - 2752803664949 \nu^{7} + 6509800106364 \nu^{6} - 2252818110113 \nu^{5} - 17932503852766 \nu^{4} + 56194969010624 \nu^{3} - 21972965776276 \nu^{2} + 20581144429809 \nu - 48489496588954$$$$)/ 34303474358093$$ $$\beta_{5}$$ $$=$$ $$($$$$344078139086 \nu^{11} - 1840864299167 \nu^{10} + 2869558436669 \nu^{9} + 7607666777144 \nu^{8} - 56284018568505 \nu^{7} + 88901551533944 \nu^{6} + 60092547218757 \nu^{5} - 469211901841258 \nu^{4} + 953521102895434 \nu^{3} + 178655400257203 \nu^{2} + 72155206793867 \nu + 993528750572394$$$$)/ 308731269222837$$ $$\beta_{6}$$ $$=$$ $$($$$$-38582738806 \nu^{11} + 312787367384 \nu^{10} - 856308492056 \nu^{9} - 228508949095 \nu^{8} + 9183041720844 \nu^{7} - 24844432486591 \nu^{6} + 5452626654420 \nu^{5} + 90363515007482 \nu^{4} - 197964026974916 \nu^{3} + 40017270005661 \nu^{2} + 283470208403922 \nu + 227364595507030$$$$)/ 34303474358093$$ $$\beta_{7}$$ $$=$$ $$($$$$403353306232 \nu^{11} - 1768122412744 \nu^{10} + 1528746626863 \nu^{9} + 11168136370855 \nu^{8} - 53716346622306 \nu^{7} + 30570924246736 \nu^{6} + 134603366679414 \nu^{5} - 293221491288314 \nu^{4} + 306364476596648 \nu^{3} + 1009945784211140 \nu^{2} + 2783110159517131 \nu + 1354116402969555$$$$)/ 308731269222837$$ $$\beta_{8}$$ $$=$$ $$($$$$231664336675 \nu^{11} - 1055801231693 \nu^{10} + 791158199715 \nu^{9} + 7565116724057 \nu^{8} - 32910179910835 \nu^{7} + 13051921442207 \nu^{6} + 116252132958211 \nu^{5} - 222965796707818 \nu^{4} + 101050327463435 \nu^{3} + 862720765054912 \nu^{2} + 1031755067491947 \nu + 364041010452140$$$$)/ 34303474358093$$ $$\beta_{9}$$ $$=$$ $$($$$$-2127198780859 \nu^{11} + 9706814485894 \nu^{10} - 7553443773796 \nu^{9} - 68392001388256 \nu^{8} + 301382108617977 \nu^{7} - 126582594542236 \nu^{6} - 1038270617338185 \nu^{5} + 2033937285480269 \nu^{4} - 985435816509719 \nu^{3} - 7809222367564952 \nu^{2} - 9870775459126630 \nu - 3609546395800872$$$$)/ 308731269222837$$ $$\beta_{10}$$ $$=$$ $$($$$$-255788416792 \nu^{11} + 1158943341170 \nu^{10} - 884228369606 \nu^{9} - 8239329204561 \nu^{8} + 36235143420730 \nu^{7} - 14668544866693 \nu^{6} - 127645956272930 \nu^{5} + 249275773018073 \nu^{4} - 117080896543899 \nu^{3} - 980356433060835 \nu^{2} - 1124029667366819 \nu - 386058541069797$$$$)/ 34303474358093$$ $$\beta_{11}$$ $$=$$ $$($$$$-446829251302 \nu^{11} + 2049366765782 \nu^{10} - 1565285873757 \nu^{9} - 14631348692510 \nu^{8} + 63423307459104 \nu^{7} - 25443799656918 \nu^{6} - 223082461775778 \nu^{5} + 426935913231974 \nu^{4} - 186144655855487 \nu^{3} - 1621686956953874 \nu^{2} - 2009074080357948 \nu - 724073294811309$$$$)/ 34303474358093$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{9} - \beta_{8} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + 2$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 3 \beta_{1} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{11} - 2 \beta_{10} + 3 \beta_{9} - 3 \beta_{8} - 6 \beta_{7} + \beta_{6} + 5 \beta_{4} + 15 \beta_{3} - 3 \beta_{2} + 12 \beta_{1} - 4$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{11} - 4 \beta_{10} + 16 \beta_{9} + 5 \beta_{8} + 3 \beta_{7} - \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + 11 \beta_{3} - 5 \beta_{2} + 10 \beta_{1} + 16$$ $$\nu^{5}$$ $$=$$ $$($$$$-19 \beta_{11} - 45 \beta_{10} + 78 \beta_{9} - 10 \beta_{8} + 17 \beta_{7} + 50 \beta_{6} - 5 \beta_{5} + 68 \beta_{4} + 59 \beta_{3} - 44 \beta_{2} + 5 \beta_{1} + 195$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-50 \beta_{11} - 100 \beta_{10} + 90 \beta_{9} - 132 \beta_{8} + 141 \beta_{7} + 101 \beta_{6} - 39 \beta_{5} + 112 \beta_{4} + 30 \beta_{3} - 224 \beta_{2} - 39 \beta_{1} + 735$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-261 \beta_{11} - 325 \beta_{10} - 44 \beta_{9} - 914 \beta_{8} - 15 \beta_{7} + 248 \beta_{6} - 21 \beta_{5} + 360 \beta_{4} + 349 \beta_{3} - 518 \beta_{2} + 175 \beta_{1} + 1185$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$-491 \beta_{11} - 396 \beta_{10} + 348 \beta_{9} - 1087 \beta_{8} + 92 \beta_{7} - 91 \beta_{6} - 128 \beta_{5} - 56 \beta_{4} + 700 \beta_{3} - 539 \beta_{2} + 720 \beta_{1} + 229$$ $$\nu^{9}$$ $$=$$ $$($$$$-3350 \beta_{11} - 4108 \beta_{10} + 5953 \beta_{9} - 5265 \beta_{8} + 154 \beta_{7} - 539 \beta_{6} - 576 \beta_{5} - 13 \beta_{4} + 5175 \beta_{3} - 213 \beta_{2} + 4284 \beta_{1} - 3082$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-8420 \beta_{11} - 14920 \beta_{10} + 25746 \beta_{9} - 8578 \beta_{8} + 11169 \beta_{7} + 93 \beta_{6} - 1651 \beta_{5} - 2558 \beta_{4} + 6958 \beta_{3} - 2188 \beta_{2} + 3801 \beta_{1} + 1945$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-24494 \beta_{11} - 52886 \beta_{10} + 65207 \beta_{9} - 44377 \beta_{8} + 37878 \beta_{7} + 9567 \beta_{6} + 550 \beta_{5} + 3817 \beta_{4} - 10669 \beta_{3} - 9343 \beta_{2} - 16940 \beta_{1} + 42832$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$486$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
364.1
 −0.576643 + 0.233980i 1.15541 + 2.37379i 3.11392 − 0.445974i 1.38187 + 2.46878i −2.40330 − 0.316825i −0.671252 + 0.645175i −0.671252 − 0.645175i −2.40330 + 0.316825i 1.38187 − 2.46878i 3.11392 + 0.445974i 1.15541 − 2.37379i −0.576643 − 0.233980i
2.60777i 2.13980i −4.80044 −2.11084 + 0.737808i −5.58011 0.988879i 7.30289i −1.57876 1.92403 + 5.50457i
364.2 2.60777i 2.13980i −4.80044 −2.11084 0.737808i 5.58011 0.988879i 7.30289i −1.57876 −1.92403 + 5.50457i
364.3 2.02281i 2.91475i −2.09174 1.20202 + 1.88551i −5.89598 3.21128i 0.185581i −5.49579 3.81402 2.43146i
364.4 2.02281i 2.91475i −2.09174 1.20202 1.88551i 5.89598 3.21128i 0.185581i −5.49579 −3.81402 2.43146i
364.5 0.328351i 0.962000i 1.89219 −0.591185 2.15650i −0.315873 3.27259i 1.27800i 2.07456 −0.708089 + 0.194116i
364.6 0.328351i 0.962000i 1.89219 −0.591185 + 2.15650i 0.315873 3.27259i 1.27800i 2.07456 0.708089 + 0.194116i
364.7 0.328351i 0.962000i 1.89219 −0.591185 2.15650i 0.315873 3.27259i 1.27800i 2.07456 0.708089 0.194116i
364.8 0.328351i 0.962000i 1.89219 −0.591185 + 2.15650i −0.315873 3.27259i 1.27800i 2.07456 −0.708089 0.194116i
364.9 2.02281i 2.91475i −2.09174 1.20202 + 1.88551i 5.89598 3.21128i 0.185581i −5.49579 −3.81402 + 2.43146i
364.10 2.02281i 2.91475i −2.09174 1.20202 1.88551i −5.89598 3.21128i 0.185581i −5.49579 3.81402 + 2.43146i
364.11 2.60777i 2.13980i −4.80044 −2.11084 + 0.737808i 5.58011 0.988879i 7.30289i −1.57876 −1.92403 5.50457i
364.12 2.60777i 2.13980i −4.80044 −2.11084 0.737808i −5.58011 0.988879i 7.30289i −1.57876 1.92403 5.50457i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 364.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
55.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.b.h 12
5.b even 2 1 inner 605.2.b.h 12
5.c odd 4 2 3025.2.a.bo 12
11.b odd 2 1 inner 605.2.b.h 12
11.c even 5 4 605.2.j.k 48
11.d odd 10 4 605.2.j.k 48
55.d odd 2 1 inner 605.2.b.h 12
55.e even 4 2 3025.2.a.bo 12
55.h odd 10 4 605.2.j.k 48
55.j even 10 4 605.2.j.k 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.h 12 1.a even 1 1 trivial
605.2.b.h 12 5.b even 2 1 inner
605.2.b.h 12 11.b odd 2 1 inner
605.2.b.h 12 55.d odd 2 1 inner
605.2.j.k 48 11.c even 5 4
605.2.j.k 48 11.d odd 10 4
605.2.j.k 48 55.h odd 10 4
605.2.j.k 48 55.j even 10 4
3025.2.a.bo 12 5.c odd 4 2
3025.2.a.bo 12 55.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{6} + 11 T_{2}^{4} + 29 T_{2}^{2} + 3$$ $$T_{19}^{6} - 72 T_{19}^{4} + 1332 T_{19}^{2} - 3888$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 3 + 29 T^{2} + 11 T^{4} + T^{6} )^{2}$$
$3$ $$( 36 + 51 T^{2} + 14 T^{4} + T^{6} )^{2}$$
$5$ $$( 125 + 75 T + 35 T^{2} + 18 T^{3} + 7 T^{4} + 3 T^{5} + T^{6} )^{2}$$
$7$ $$( 108 + 131 T^{2} + 22 T^{4} + T^{6} )^{2}$$
$11$ $$T^{12}$$
$13$ $$( 108 + 272 T^{2} + 37 T^{4} + T^{6} )^{2}$$
$17$ $$( 48 + 56 T^{2} + 17 T^{4} + T^{6} )^{2}$$
$19$ $$( -3888 + 1332 T^{2} - 72 T^{4} + T^{6} )^{2}$$
$23$ $$( 17424 + 2076 T^{2} + 80 T^{4} + T^{6} )^{2}$$
$29$ $$( -3888 + 1224 T^{2} - 75 T^{4} + T^{6} )^{2}$$
$31$ $$( -68 - 38 T - 2 T^{2} + T^{3} )^{4}$$
$37$ $$( 1296 + 504 T^{2} + 51 T^{4} + T^{6} )^{2}$$
$41$ $$( -2187 + 1359 T^{2} - 117 T^{4} + T^{6} )^{2}$$
$43$ $$( 432 + 1259 T^{2} + 130 T^{4} + T^{6} )^{2}$$
$47$ $$( 36 + 51 T^{2} + 14 T^{4} + T^{6} )^{2}$$
$53$ $$( 36 + 240 T^{2} + 47 T^{4} + T^{6} )^{2}$$
$59$ $$( 24 - 50 T + 8 T^{2} + T^{3} )^{4}$$
$61$ $$( -1728 + 1809 T^{2} - 90 T^{4} + T^{6} )^{2}$$
$67$ $$( 39204 + 7875 T^{2} + 174 T^{4} + T^{6} )^{2}$$
$71$ $$( 48 + 190 T - 28 T^{2} + T^{3} )^{4}$$
$73$ $$( 442368 + 22016 T^{2} + 292 T^{4} + T^{6} )^{2}$$
$79$ $$( -110592 + 16128 T^{2} - 300 T^{4} + T^{6} )^{2}$$
$83$ $$( 591408 + 25916 T^{2} + 296 T^{4} + T^{6} )^{2}$$
$89$ $$( 9 + T )^{12}$$
$97$ $$( 4743684 + 87120 T^{2} + 519 T^{4} + T^{6} )^{2}$$