# Properties

 Label 4012.2.a.h Level 4012 Weight 2 Character orbit 4012.a Self dual yes Analytic conductor 32.036 Analytic rank 1 Dimension 15 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4012 = 2^{2} \cdot 17 \cdot 59$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4012.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0359812909$$ Analytic rank: $$1$$ Dimension: $$15$$ Coefficient field: $$\mathbb{Q}[x]/(x^{15} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{14}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{3} -\beta_{8} q^{5} + ( -1 - \beta_{9} ) q^{7} + ( 1 + \beta_{10} - \beta_{12} + \beta_{14} ) q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{3} -\beta_{8} q^{5} + ( -1 - \beta_{9} ) q^{7} + ( 1 + \beta_{10} - \beta_{12} + \beta_{14} ) q^{9} + ( -1 + \beta_{1} - \beta_{5} + \beta_{7} + \beta_{12} - \beta_{14} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} - \beta_{5} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{13} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{9} - \beta_{12} ) q^{15} + q^{17} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{12} ) q^{19} + ( -2 \beta_{1} + \beta_{4} - \beta_{6} + \beta_{8} - \beta_{13} ) q^{21} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + 3 \beta_{8} + \beta_{9} - \beta_{13} + \beta_{14} ) q^{23} + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{14} ) q^{25} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} ) q^{27} + ( 3 - \beta_{1} + \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{29} + ( -1 - 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{31} + ( -1 - \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{13} ) q^{33} + ( -2 - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{13} ) q^{35} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{37} + ( \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{39} + ( -3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{41} + ( -1 + \beta_{1} + \beta_{3} - \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{43} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{8} + \beta_{11} ) q^{45} + ( -4 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{47} + ( 2 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{11} - \beta_{13} - \beta_{14} ) q^{49} + \beta_{5} q^{51} + ( -1 + 5 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{8} + 3 \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{53} + ( -1 - 3 \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{9} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{55} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} - 4 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{57} - q^{59} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{12} + 2 \beta_{13} ) q^{61} + ( 5 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} + 4 \beta_{12} ) q^{63} + ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{65} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} - 3 \beta_{6} - \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} ) q^{67} + ( -2 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{69} + ( 1 + 3 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} + \beta_{12} - 2 \beta_{14} ) q^{71} + ( -2 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} + \beta_{4} - 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + \beta_{10} - 5 \beta_{12} + \beta_{14} ) q^{73} + ( -3 + 4 \beta_{1} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{13} ) q^{75} + ( 1 - 3 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + 3 \beta_{8} + 4 \beta_{9} - \beta_{10} - \beta_{11} - 6 \beta_{12} + 2 \beta_{14} ) q^{77} + ( -2 + 3 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{10} + 2 \beta_{12} + \beta_{13} - 3 \beta_{14} ) q^{79} + ( -1 + 5 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{12} - \beta_{14} ) q^{81} + ( -3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - \beta_{9} + 3 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{83} -\beta_{8} q^{85} + ( -3 - 4 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{87} + ( -3 + 2 \beta_{1} - 4 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{12} ) q^{89} + ( -4 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - 3 \beta_{12} + 2 \beta_{14} ) q^{91} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{12} + 3 \beta_{13} + \beta_{14} ) q^{93} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} + 3 \beta_{7} - \beta_{8} + \beta_{9} + 5 \beta_{10} - 2 \beta_{11} - 3 \beta_{13} ) q^{95} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{97} + ( -6 - \beta_{1} - 3 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$15q - q^{3} + q^{5} - 11q^{7} + 16q^{9} + O(q^{10})$$ $$15q - q^{3} + q^{5} - 11q^{7} + 16q^{9} - 12q^{11} - 10q^{13} - 13q^{15} + 15q^{17} - 4q^{21} - 21q^{23} + 4q^{25} - 37q^{27} + 23q^{29} - 31q^{31} - 11q^{33} - 35q^{35} - 10q^{37} - 2q^{39} - 15q^{41} - 3q^{43} + 15q^{45} - 47q^{47} + 18q^{49} - q^{51} - 7q^{53} - 20q^{55} - 30q^{57} - 15q^{59} + q^{61} + 19q^{63} + 11q^{65} - 20q^{67} - 24q^{69} - 13q^{71} - 24q^{73} - 10q^{75} + 21q^{77} - 34q^{79} - 21q^{81} - 40q^{83} + q^{85} - 60q^{87} - 46q^{89} - 19q^{91} + 33q^{93} - 4q^{95} - 5q^{97} - 74q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} - 249 x^{6} + 2736 x^{5} - 801 x^{4} - 900 x^{3} + 429 x^{2} - 36 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-1642110 \nu^{14} + 5493041 \nu^{13} + 44856541 \nu^{12} - 160648719 \nu^{11} - 341454762 \nu^{10} + 1316309295 \nu^{9} + 1206761264 \nu^{8} - 4604137302 \nu^{7} - 2481282245 \nu^{6} + 7168124400 \nu^{5} + 3230047412 \nu^{4} - 3872105642 \nu^{3} - 2078341617 \nu^{2} - 113490130 \nu + 189641702$$$$)/95351182$$ $$\beta_{3}$$ $$=$$ $$($$$$2499271 \nu^{14} + 5394424 \nu^{13} - 96824570 \nu^{12} - 88043849 \nu^{11} + 1318014655 \nu^{10} - 95090619 \nu^{9} - 6916869158 \nu^{8} + 2968463153 \nu^{7} + 16654248791 \nu^{6} - 9186606266 \nu^{5} - 18331801638 \nu^{4} + 10082491657 \nu^{3} + 7142637423 \nu^{2} - 3199335814 \nu + 131157754$$$$)/95351182$$ $$\beta_{4}$$ $$=$$ $$($$$$-6589235 \nu^{14} + 21534662 \nu^{13} + 149175107 \nu^{12} - 577550108 \nu^{11} - 627835958 \nu^{10} + 3703715580 \nu^{9} + 438222586 \nu^{8} - 9886195276 \nu^{7} + 1359291740 \nu^{6} + 12036309335 \nu^{5} - 1339292667 \nu^{4} - 5866163863 \nu^{3} - 410465505 \nu^{2} + 485967518 \nu + 11524587$$$$)/47675591$$ $$\beta_{5}$$ $$=$$ $$($$$$29448859 \nu^{14} - 118865162 \nu^{13} - 583179274 \nu^{12} + 3043456985 \nu^{11} + 653319903 \nu^{10} - 17499504791 \nu^{9} + 10520388562 \nu^{8} + 39503795341 \nu^{7} - 36007368575 \nu^{6} - 36114958670 \nu^{5} + 39200943698 \nu^{4} + 8415289279 \nu^{3} - 12848394501 \nu^{2} + 2319111788 \nu + 129215570$$$$)/95351182$$ $$\beta_{6}$$ $$=$$ $$($$$$-36873004 \nu^{14} + 144992745 \nu^{13} + 729566385 \nu^{12} - 3708988537 \nu^{11} - 818596384 \nu^{10} + 21093381481 \nu^{9} - 13361532648 \nu^{8} - 46378126822 \nu^{7} + 46596966433 \nu^{6} + 39124095638 \nu^{5} - 52573837040 \nu^{4} - 4706759198 \nu^{3} + 18396452745 \nu^{2} - 4564703394 \nu - 133290618$$$$)/95351182$$ $$\beta_{7}$$ $$=$$ $$($$$$40746102 \nu^{14} - 141104307 \nu^{13} - 875338947 \nu^{12} + 3685347851 \nu^{11} + 2730849730 \nu^{10} - 22011930647 \nu^{9} + 3355203376 \nu^{8} + 53261736674 \nu^{7} - 21911047283 \nu^{6} - 56230036218 \nu^{5} + 23460592692 \nu^{4} + 21085698418 \nu^{3} - 4854984431 \nu^{2} + 398646228 \nu - 338210566$$$$)/95351182$$ $$\beta_{8}$$ $$=$$ $$($$$$95633725 \nu^{14} - 359609000 \nu^{13} - 1951460100 \nu^{12} + 9276232291 \nu^{11} + 3654449555 \nu^{10} - 53869328967 \nu^{9} + 25740619298 \nu^{8} + 123383929169 \nu^{7} - 100008537973 \nu^{6} - 115872881722 \nu^{5} + 115167642186 \nu^{4} + 30532653209 \nu^{3} - 39836754771 \nu^{2} + 5001928278 \nu + 443495554$$$$)/ 190702364$$ $$\beta_{9}$$ $$=$$ $$($$$$-97204111 \nu^{14} + 408284470 \nu^{13} + 1824797366 \nu^{12} - 10322410111 \nu^{11} + 386823059 \nu^{10} + 56972906955 \nu^{9} - 50219018070 \nu^{8} - 118538250583 \nu^{7} + 158702692833 \nu^{6} + 87957399718 \nu^{5} - 176258359614 \nu^{4} + 646244605 \nu^{3} + 62525219875 \nu^{2} - 14572316814 \nu - 167261742$$$$)/ 190702364$$ $$\beta_{10}$$ $$=$$ $$($$$$-25842475 \nu^{14} + 101169385 \nu^{13} + 521083651 \nu^{12} - 2604048863 \nu^{11} - 816126103 \nu^{10} + 15153025831 \nu^{9} - 7864783050 \nu^{8} - 34957337633 \nu^{7} + 28573976493 \nu^{6} + 33373023966 \nu^{5} - 31339622646 \nu^{4} - 8963008460 \nu^{3} + 9722679174 \nu^{2} - 1839898369 \nu + 106478707$$$$)/47675591$$ $$\beta_{11}$$ $$=$$ $$($$$$-58215653 \nu^{14} + 202536060 \nu^{13} + 1239397782 \nu^{12} - 5282011523 \nu^{11} - 3579431491 \nu^{10} + 31339492215 \nu^{9} - 7527204066 \nu^{8} - 74223474393 \nu^{7} + 40946834245 \nu^{6} + 73844820286 \nu^{5} - 48459183198 \nu^{4} - 22486916493 \nu^{3} + 15280826425 \nu^{2} - 2796392008 \nu + 74376318$$$$)/95351182$$ $$\beta_{12}$$ $$=$$ $$($$$$-121421599 \nu^{14} + 480687854 \nu^{13} + 2412644590 \nu^{12} - 12328562947 \nu^{11} - 2924433725 \nu^{10} + 70935144335 \nu^{9} - 42646712690 \nu^{8} - 160312566091 \nu^{7} + 149689508201 \nu^{6} + 146614988770 \nu^{5} - 167221293562 \nu^{4} - 33298989487 \nu^{3} + 55815466299 \nu^{2} - 10559674518 \nu + 210174674$$$$)/ 190702364$$ $$\beta_{13}$$ $$=$$ $$($$$$-166794261 \nu^{14} + 629928074 \nu^{13} + 3417073022 \nu^{12} - 16278312837 \nu^{11} - 6694066567 \nu^{10} + 95271285377 \nu^{9} - 42808440098 \nu^{8} - 221681072765 \nu^{7} + 167971062563 \nu^{6} + 215663490182 \nu^{5} - 190929742802 \nu^{4} - 63378890185 \nu^{3} + 63563825561 \nu^{2} - 8886145330 \nu - 204973042$$$$)/ 190702364$$ $$\beta_{14}$$ $$=$$ $$($$$$-169653351 \nu^{14} + 605403962 \nu^{13} + 3601868958 \nu^{12} - 15798177543 \nu^{11} - 10130008829 \nu^{10} + 94608841675 \nu^{9} - 23155286746 \nu^{8} - 229847533475 \nu^{7} + 120212213389 \nu^{6} + 244549054426 \nu^{5} - 139871902806 \nu^{4} - 94024452139 \nu^{3} + 44401361523 \nu^{2} - 227999114 \nu - 898759938$$$$)/ 190702364$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{14} - \beta_{13} - \beta_{12} + \beta_{9} + \beta_{4} + \beta_{3} - \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{13} - \beta_{11} - 3 \beta_{10} - \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + 4 \beta_{3} - 3 \beta_{2} + 9 \beta_{1} - 5$$ $$\nu^{4}$$ $$=$$ $$11 \beta_{14} - 17 \beta_{13} - 9 \beta_{12} + 2 \beta_{11} - \beta_{10} + 15 \beta_{9} - 6 \beta_{8} + 3 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} + 12 \beta_{4} + 6 \beta_{3} + 11 \beta_{2} - 17 \beta_{1} + 46$$ $$\nu^{5}$$ $$=$$ $$-9 \beta_{14} + 64 \beta_{13} + 5 \beta_{12} - 21 \beta_{11} - 52 \beta_{10} - 28 \beta_{9} + 32 \beta_{8} - 56 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 9 \beta_{4} + 65 \beta_{3} - 56 \beta_{2} + 134 \beta_{1} - 120$$ $$\nu^{6}$$ $$=$$ $$159 \beta_{14} - 293 \beta_{13} - 127 \beta_{12} + 44 \beta_{11} + 7 \beta_{10} + 244 \beta_{9} - 129 \beta_{8} + 82 \beta_{7} - 72 \beta_{6} - 36 \beta_{5} + 177 \beta_{4} + 30 \beta_{3} + 235 \beta_{2} - 338 \beta_{1} + 726$$ $$\nu^{7}$$ $$=$$ $$-251 \beta_{14} + 1195 \beta_{13} + 149 \beta_{12} - 368 \beta_{11} - 819 \beta_{10} - 593 \beta_{9} + 552 \beta_{8} - 937 \beta_{7} + 75 \beta_{6} + 62 \beta_{5} - 261 \beta_{4} + 993 \beta_{3} - 1000 \beta_{2} + 2231 \beta_{1} - 2367$$ $$\nu^{8}$$ $$=$$ $$2546 \beta_{14} - 5203 \beta_{13} - 2027 \beta_{12} + 893 \beta_{11} + 592 \beta_{10} + 4113 \beta_{9} - 2393 \beta_{8} + 1838 \beta_{7} - 1179 \beta_{6} - 590 \beta_{5} + 2837 \beta_{4} - 224 \beta_{3} + 4348 \beta_{2} - 6583 \beta_{1} + 12440$$ $$\nu^{9}$$ $$=$$ $$-5510 \beta_{14} + 21747 \beta_{13} + 3494 \beta_{12} - 6292 \beta_{11} - 12997 \beta_{10} - 11707 \beta_{9} + 9878 \beta_{8} - 15645 \beta_{7} + 1882 \beta_{6} + 1381 \beta_{5} - 5848 \beta_{4} + 15431 \beta_{3} - 18007 \beta_{2} + 38305 \beta_{1} - 44481$$ $$\nu^{10}$$ $$=$$ $$42402 \beta_{14} - 93352 \beta_{13} - 33426 \beta_{12} + 17563 \beta_{11} + 17148 \beta_{10} + 70572 \beta_{9} - 43240 \beta_{8} + 37953 \beta_{7} - 19397 \beta_{6} - 9858 \beta_{5} + 47106 \beta_{4} - 13249 \beta_{3} + 78536 \beta_{2} - 125154 \beta_{1} + 218298$$ $$\nu^{11}$$ $$=$$ $$-111435 \beta_{14} + 393291 \beta_{13} + 74135 \beta_{12} - 108079 \beta_{11} - 210704 \beta_{10} - 224050 \beta_{9} + 178411 \beta_{8} - 264708 \beta_{7} + 40958 \beta_{6} + 27717 \beta_{5} - 119611 \beta_{4} + 245731 \beta_{3} - 325439 \beta_{2} + 666991 \beta_{1} - 821757$$ $$\nu^{12}$$ $$=$$ $$721157 \beta_{14} - 1680103 \beta_{13} - 561369 \beta_{12} + 336971 \beta_{11} + 394848 \beta_{10} + 1225651 \beta_{9} - 777610 \beta_{8} + 748242 \beta_{7} - 324384 \beta_{6} - 168384 \beta_{5} + 798611 \beta_{4} - 357149 \beta_{3} + 1412830 \beta_{2} - 2340970 \beta_{1} + 3867318$$ $$\nu^{13}$$ $$=$$ $$-2164334 \beta_{14} + 7100193 \beta_{13} + 1487194 \beta_{12} - 1874470 \beta_{11} - 3488558 \beta_{10} - 4209635 \beta_{9} + 3226956 \beta_{8} - 4541011 \beta_{7} + 831376 \beta_{6} + 532924 \beta_{5} - 2338680 \beta_{4} + 4004710 \beta_{3} - 5884307 \beta_{2} + 11720405 \beta_{1} - 15058170$$ $$\nu^{14}$$ $$=$$ $$12443114 \beta_{14} - 30267026 \beta_{13} - 9571250 \beta_{12} + 6348517 \beta_{11} + 8244380 \beta_{10} + 21483762 \beta_{9} - 13979884 \beta_{8} + 14335354 \beta_{7} - 5512014 \beta_{6} - 2919908 \beta_{5} + 13743120 \beta_{4} - 7984739 \beta_{3} + 25411016 \beta_{2} - 43314563 \beta_{1} + 68870090$$

## 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
 1.28051 2.10632 −4.24628 −1.49960 −1.20072 2.23095 1.18786 −0.0375146 0.346366 −1.64466 3.62663 −0.829690 1.66914 0.186961 1.82373
0 −3.40908 0 1.59062 0 2.94577 0 8.62180 0
1.2 0 −2.74294 0 −0.560055 0 1.03063 0 4.52374 0
1.3 0 −2.34833 0 3.93023 0 −4.81217 0 2.51465 0
1.4 0 −2.30933 0 2.18242 0 −0.264202 0 2.33302 0
1.5 0 −2.21367 0 −3.70336 0 −3.08925 0 1.90033 0
1.6 0 −0.598056 0 −0.722695 0 −0.897746 0 −2.64233 0
1.7 0 −0.460784 0 −0.730164 0 3.58746 0 −2.78768 0
1.8 0 0.249278 0 −1.04042 0 −3.44893 0 −2.93786 0
1.9 0 0.915888 0 1.84475 0 −3.15913 0 −2.16115 0
1.10 0 1.48779 0 1.28557 0 0.0101628 0 −0.786472 0
1.11 0 1.57548 0 3.44709 0 −4.23195 0 −0.517858 0
1.12 0 1.84628 0 −2.98774 0 −0.920155 0 0.408737 0
1.13 0 2.17000 0 −3.20913 0 1.70252 0 1.70890 0
1.14 0 2.17219 0 −1.55586 0 3.67456 0 1.71840 0
1.15 0 2.66529 0 1.22875 0 −3.12758 0 4.10376 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.15 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4012.2.a.h 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4012.2.a.h 15 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$17$$ $$-1$$
$$59$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4012))$$:

 $$T_{3}^{15} + \cdots$$ $$T_{5}^{15} - \cdots$$