# Properties

 Label 770.2.n.h Level $770$ Weight $2$ Character orbit 770.n Analytic conductor $6.148$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.14848095564$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 3 x^{11} + 7 x^{10} - 9 x^{9} + 55 x^{8} - 32 x^{7} + 287 x^{6} - 302 x^{5} + 1175 x^{4} - 1639 x^{3} + 1692 x^{2} - 988 x + 361$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} - \beta_{3} - \beta_{10} ) q^{2} + ( -\beta_{5} + \beta_{8} - \beta_{10} ) q^{3} + \beta_{3} q^{4} -\beta_{2} q^{5} + ( -\beta_{2} - \beta_{7} ) q^{6} + \beta_{3} q^{7} + \beta_{10} q^{8} + ( -2 - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{10} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{2} - \beta_{3} - \beta_{10} ) q^{2} + ( -\beta_{5} + \beta_{8} - \beta_{10} ) q^{3} + \beta_{3} q^{4} -\beta_{2} q^{5} + ( -\beta_{2} - \beta_{7} ) q^{6} + \beta_{3} q^{7} + \beta_{10} q^{8} + ( -2 - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{10} - \beta_{11} ) q^{9} - q^{10} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{8} + 2 \beta_{10} ) q^{11} + ( -1 - \beta_{8} ) q^{12} + ( -2 \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{13} + \beta_{10} q^{14} + ( \beta_{1} + \beta_{3} ) q^{15} + \beta_{2} q^{16} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{17} + ( -1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{18} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{10} - \beta_{11} ) q^{19} + ( 1 + \beta_{2} + \beta_{3} + \beta_{10} ) q^{20} + ( -1 - \beta_{8} ) q^{21} + ( \beta_{2} - \beta_{3} + \beta_{5} - \beta_{9} ) q^{22} -2 \beta_{11} q^{23} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{10} ) q^{24} + \beta_{10} q^{25} + ( \beta_{1} - \beta_{4} ) q^{26} + ( -2 - 7 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{27} + \beta_{2} q^{28} + ( 3 - \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{9} ) q^{29} + ( \beta_{5} - \beta_{8} + \beta_{10} ) q^{30} + ( 2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{8} + \beta_{10} + \beta_{11} ) q^{31} + q^{32} + ( -1 + \beta_{1} + 3 \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{33} + ( -1 - 3 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{34} + ( 1 + \beta_{2} + \beta_{3} + \beta_{10} ) q^{35} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{36} + ( -2 - 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{37} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{38} + ( 1 - 2 \beta_{1} - 5 \beta_{2} - \beta_{7} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{39} -\beta_{3} q^{40} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 4 \beta_{10} ) q^{41} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{10} ) q^{42} + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{43} + ( 1 + \beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{44} + ( -2 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{45} + ( 2 \beta_{1} - 2 \beta_{6} + 2 \beta_{8} ) q^{46} + ( 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 2 \beta_{6} + 2 \beta_{8} - 6 \beta_{10} - 2 \beta_{11} ) q^{47} + ( -\beta_{1} - \beta_{3} ) q^{48} + \beta_{2} q^{49} + \beta_{2} q^{50} + ( 5 - 2 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{51} + ( -\beta_{1} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{52} + ( -6 + 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 3 \beta_{10} - 2 \beta_{11} ) q^{53} + ( -5 + 4 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{54} + ( 1 + \beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{55} + q^{56} + ( -1 - 4 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{10} ) q^{57} + ( -3 \beta_{2} - 3 \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{10} - \beta_{11} ) q^{58} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{7} - \beta_{8} ) q^{59} + ( \beta_{2} + \beta_{7} ) q^{60} + ( 5 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 5 \beta_{10} ) q^{61} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{62} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{63} + ( -1 - \beta_{2} - \beta_{3} - \beta_{10} ) q^{64} + ( -\beta_{8} - \beta_{11} ) q^{65} + ( 4 - 2 \beta_{1} + 5 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{66} + ( -6 - 4 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{67} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{68} + ( -6 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - 4 \beta_{9} + 2 \beta_{11} ) q^{69} -\beta_{3} q^{70} + ( -1 - 2 \beta_{1} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{10} ) q^{71} + ( 1 - \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{72} + ( 4 + 4 \beta_{1} + 4 \beta_{2} + \beta_{3} + \beta_{4} + 4 \beta_{7} + 4 \beta_{8} ) q^{73} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{74} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{10} ) q^{75} + ( -3 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{76} + ( 1 + \beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{77} + ( -6 + 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{78} + ( 3 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + 3 \beta_{10} ) q^{79} -\beta_{10} q^{80} + ( -6 + 3 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{81} + ( -2 + 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{82} + ( 3 - 5 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{83} + ( -\beta_{1} - \beta_{3} ) q^{84} + ( \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{85} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{86} + ( -1 - 5 \beta_{1} - 7 \beta_{3} - \beta_{4} - 4 \beta_{5} - \beta_{7} + 4 \beta_{8} - \beta_{9} - 7 \beta_{10} ) q^{87} + ( -1 - 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{88} + ( -4 - 3 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} - 4 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{89} + ( 2 + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{90} + ( -\beta_{1} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{91} + 2 \beta_{4} q^{92} + ( 7 + \beta_{1} + 6 \beta_{2} - \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 7 \beta_{10} - 2 \beta_{11} ) q^{93} + ( -4 + 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{8} - 4 \beta_{10} ) q^{94} + ( 1 + \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{95} + ( -\beta_{5} + \beta_{8} - \beta_{10} ) q^{96} + ( 6 - 3 \beta_{1} - \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + \beta_{8} - \beta_{10} + 2 \beta_{11} ) q^{97} + q^{98} + ( 8 - 8 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 5 \beta_{8} - 3 \beta_{9} + 5 \beta_{10} + 3 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 3q^{2} - 3q^{4} + 3q^{5} + 5q^{6} - 3q^{7} - 3q^{8} - 3q^{9} + O(q^{10})$$ $$12q - 3q^{2} - 3q^{4} + 3q^{5} + 5q^{6} - 3q^{7} - 3q^{8} - 3q^{9} - 12q^{10} - q^{11} - 10q^{12} - 3q^{14} - 3q^{16} - 8q^{18} - q^{19} + 3q^{20} - 10q^{21} - q^{22} - 4q^{23} + 5q^{24} - 3q^{25} + 3q^{27} - 3q^{28} + 22q^{29} + 6q^{31} + 12q^{32} - 29q^{33} - 30q^{34} + 3q^{35} - 8q^{36} - 10q^{37} + 14q^{38} + 20q^{39} + 3q^{40} + 16q^{41} + 5q^{42} + 30q^{43} + 14q^{44} - 22q^{45} - 4q^{46} + 34q^{47} - 3q^{49} - 3q^{50} + 37q^{51} - 26q^{53} - 52q^{54} + 11q^{55} + 12q^{56} - 19q^{57} + 22q^{58} + q^{59} - 5q^{60} + 40q^{61} - 4q^{62} - 8q^{63} - 3q^{64} + 16q^{66} - 58q^{67} + 14q^{69} + 3q^{70} - 14q^{71} - 3q^{72} + 32q^{73} - 10q^{74} + 5q^{75} - 26q^{76} + 14q^{77} - 60q^{78} + 16q^{79} + 3q^{80} - 46q^{81} + q^{82} + 35q^{83} - 15q^{85} + 5q^{86} - q^{88} - 58q^{89} + 3q^{90} + 6q^{92} + 46q^{93} - 16q^{94} + q^{95} + 57q^{97} + 12q^{98} + 69q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 7 x^{10} - 9 x^{9} + 55 x^{8} - 32 x^{7} + 287 x^{6} - 302 x^{5} + 1175 x^{4} - 1639 x^{3} + 1692 x^{2} - 988 x + 361$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-66027685671987 \nu^{11} + 300911576247391 \nu^{10} - 495079952903319 \nu^{9} + 724241193428737 \nu^{8} - 3260947044739288 \nu^{7} + 6750113796886901 \nu^{6} - 8713698622449008 \nu^{5} + 52435682633669063 \nu^{4} - 33814714622500081 \nu^{3} + 232570016274416890 \nu^{2} - 14926597030314910 \nu + 13496248724727067$$$$)/ 120459493170841451$$ $$\beta_{3}$$ $$=$$ $$($$$$78194599018136 \nu^{11} - 583919559521768 \nu^{10} + 1278661624498476 \nu^{9} - 2350491056102053 \nu^{8} + 5472660828569196 \nu^{7} - 18873216932592696 \nu^{6} + 16323963624711196 \nu^{5} - 122550411817090002 \nu^{4} + 105943143477850444 \nu^{3} - 474517422887083464 \nu^{2} + 369734251015507748 \nu - 255976547516660537$$$$)/ 120459493170841451$$ $$\beta_{4}$$ $$=$$ $$($$$$-13935225036164 \nu^{11} + 14741542109035 \nu^{10} + 14427641195505 \nu^{9} - 37810372429471 \nu^{8} - 564162885948498 \nu^{7} - 839377530222322 \nu^{6} - 1896214951086632 \nu^{5} + 1227414327246411 \nu^{4} + 2727594952765321 \nu^{3} + 8691122527482570 \nu^{2} + 38198972987889553 \nu + 7067693098651702$$$$)/ 10950863015531041$$ $$\beta_{5}$$ $$=$$ $$($$$$12974065433470 \nu^{11} - 36758593588006 \nu^{10} + 79828823292525 \nu^{9} - 81185976304827 \nu^{8} + 617566416243633 \nu^{7} - 216755835364075 \nu^{6} + 3496859018572839 \nu^{5} - 3043805561769752 \nu^{4} + 11684517665175177 \nu^{3} - 17590591400184670 \nu^{2} + 5789468170268253 \nu + 231171353576831$$$$)/ 6339973324781129$$ $$\beta_{6}$$ $$=$$ $$($$$$-274108974347116 \nu^{11} - 675019018801603 \nu^{10} + 1152514856355359 \nu^{9} - 3816738876886027 \nu^{8} - 10353185974674761 \nu^{7} - 64168027866190223 \nu^{6} - 102113038847938484 \nu^{5} - 308594139224901378 \nu^{4} - 227319719527129720 \nu^{3} - 903561215038369461 \nu^{2} + 612989098365220141 \nu - 243476676021953225$$$$)/ 120459493170841451$$ $$\beta_{7}$$ $$=$$ $$($$$$-18386092761440 \nu^{11} + 38489443756396 \nu^{10} - 86670508680991 \nu^{9} + 61681993819564 \nu^{8} - 861631040211176 \nu^{7} - 321994015447044 \nu^{6} - 5207139100716470 \nu^{5} + 740236296396876 \nu^{4} - 18229288162966240 \nu^{3} + 12496262604043244 \nu^{2} - 9406330720354851 \nu - 1485697381344584$$$$)/ 6339973324781129$$ $$\beta_{8}$$ $$=$$ $$($$$$22736202141892 \nu^{11} - 64882405687006 \nu^{10} + 140403633386251 \nu^{9} - 162442932162717 \nu^{8} + 1167426573924417 \nu^{7} - 524257843603323 \nu^{6} + 6173590630330414 \nu^{5} - 6201809314945028 \nu^{4} + 22585229868384763 \nu^{3} - 33666982825971444 \nu^{2} + 23138265291543190 \nu - 10138628105218865$$$$)/ 6339973324781129$$ $$\beta_{9}$$ $$=$$ $$($$$$-519081844887701 \nu^{11} + 1944634258482198 \nu^{10} - 4835370161920450 \nu^{9} + 7061612132481793 \nu^{8} - 31246387855001735 \nu^{7} + 36704603512010767 \nu^{6} - 161000855663723496 \nu^{5} + 245845175111403657 \nu^{4} - 729318047737200709 \nu^{3} + 1242423443280485409 \nu^{2} - 1448259403008892547 \nu + 901555205608557022$$$$)/ 120459493170841451$$ $$\beta_{10}$$ $$=$$ $$($$$$533612005537835 \nu^{11} - 1168848175917557 \nu^{10} + 2502518330711731 \nu^{9} - 2134839015501746 \nu^{8} + 26262244593489302 \nu^{7} + 5105520727353203 \nu^{6} + 143185746560895508 \nu^{5} - 43852603696148304 \nu^{4} + 509159729523000593 \nu^{3} - 445470709577201068 \nu^{2} + 263198839676559384 \nu - 87581620932060370$$$$)/ 120459493170841451$$ $$\beta_{11}$$ $$=$$ $$($$$$-99158490471442 \nu^{11} + 290212999539605 \nu^{10} - 612416220981491 \nu^{9} + 713489977101633 \nu^{8} - 5153117320376611 \nu^{7} + 2612037001816903 \nu^{6} - 25271768881891621 \nu^{5} + 28614037398765184 \nu^{4} - 100172280587257599 \nu^{3} + 148048209140776684 \nu^{2} - 101468473199949026 \nu + 46901907355242216$$$$)/ 6339973324781129$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} + 5 \beta_{2} - \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} + 3 \beta_{10} - 7 \beta_{7} + 2 \beta_{6} - 7 \beta_{5} - \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 7 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$8 \beta_{11} + 35 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} + 8 \beta_{6} - 12 \beta_{5} + 13 \beta_{3} + 13 \beta_{2} - 10 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$10 \beta_{11} + 24 \beta_{10} + 14 \beta_{9} + 57 \beta_{8} + 14 \beta_{7} - 25 \beta_{5} + 14 \beta_{4} + 24 \beta_{3} - 11 \beta_{1} - 24$$ $$\nu^{6}$$ $$=$$ $$67 \beta_{9} + 130 \beta_{8} + 130 \beta_{7} - 67 \beta_{6} + 96 \beta_{4} + 145 \beta_{3} - 137 \beta_{2} + 68 \beta_{1} - 137$$ $$\nu^{7}$$ $$=$$ $$-101 \beta_{11} - 323 \beta_{10} + 101 \beta_{9} + 259 \beta_{8} + 608 \beta_{7} - 259 \beta_{6} + 267 \beta_{5} + 259 \beta_{4} - 574 \beta_{2} + 360 \beta_{1} - 323$$ $$\nu^{8}$$ $$=$$ $$-608 \beta_{11} - 2486 \beta_{10} + 184 \beta_{8} + 1340 \beta_{7} - 932 \beta_{6} + 1340 \beta_{5} + 608 \beta_{4} - 1113 \beta_{3} - 2486 \beta_{2} + 1524 \beta_{1} - 1113$$ $$\nu^{9}$$ $$=$$ $$-1664 \beta_{11} - 6232 \beta_{10} - 1032 \beta_{9} - 3094 \beta_{8} + 1697 \beta_{7} - 1664 \beta_{6} + 4758 \beta_{5} - 3684 \beta_{3} - 3684 \beta_{2} + 3361 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-3361 \beta_{11} - 13610 \beta_{10} - 5790 \beta_{9} - 13686 \beta_{8} - 5790 \beta_{7} + 8077 \beta_{5} - 5790 \beta_{4} - 13610 \beta_{3} + 2287 \beta_{1} + 9581$$ $$\nu^{11}$$ $$=$$ $$-17047 \beta_{9} - 44524 \beta_{8} - 44524 \beta_{7} + 17047 \beta_{6} - 27553 \beta_{4} - 25655 \beta_{3} + 39272 \beta_{2} - 15543 \beta_{1} + 39272$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/770\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$617$$ $$661$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1
 2.55917 − 1.85934i 0.609000 − 0.442464i −1.85915 + 1.35075i 2.55917 + 1.85934i 0.609000 + 0.442464i −1.85915 − 1.35075i 0.603111 − 1.85618i 0.254744 − 0.784022i −0.666872 + 2.05242i 0.603111 + 1.85618i 0.254744 + 0.784022i −0.666872 − 2.05242i
0.309017 + 0.951057i −1.75015 1.27156i −0.809017 + 0.587785i −0.309017 + 0.951057i 0.668497 2.05742i −0.809017 + 0.587785i −0.809017 0.587785i 0.519111 + 1.59766i −1.00000
71.2 0.309017 + 0.951057i 0.200017 + 0.145321i −0.809017 + 0.587785i −0.309017 + 0.951057i −0.0763997 + 0.235134i −0.809017 + 0.587785i −0.809017 0.587785i −0.908162 2.79504i −1.00000
71.3 0.309017 + 0.951057i 2.66817 + 1.93854i −0.809017 + 0.587785i −0.309017 + 0.951057i −1.01915 + 3.13662i −0.809017 + 0.587785i −0.809017 0.587785i 2.43414 + 7.49150i −1.00000
141.1 0.309017 0.951057i −1.75015 + 1.27156i −0.809017 0.587785i −0.309017 0.951057i 0.668497 + 2.05742i −0.809017 0.587785i −0.809017 + 0.587785i 0.519111 1.59766i −1.00000
141.2 0.309017 0.951057i 0.200017 0.145321i −0.809017 0.587785i −0.309017 0.951057i −0.0763997 0.235134i −0.809017 0.587785i −0.809017 + 0.587785i −0.908162 + 2.79504i −1.00000
141.3 0.309017 0.951057i 2.66817 1.93854i −0.809017 0.587785i −0.309017 0.951057i −1.01915 3.13662i −0.809017 0.587785i −0.809017 + 0.587785i 2.43414 7.49150i −1.00000
421.1 −0.809017 + 0.587785i −0.912128 2.80724i 0.309017 0.951057i 0.809017 + 0.587785i 2.38798 + 1.73497i 0.309017 0.951057i 0.309017 + 0.951057i −4.62157 + 3.35777i −1.00000
421.2 −0.809017 + 0.587785i −0.563761 1.73508i 0.309017 0.951057i 0.809017 + 0.587785i 1.47595 + 1.07234i 0.309017 0.951057i 0.309017 + 0.951057i −0.265619 + 0.192984i −1.00000
421.3 −0.809017 + 0.587785i 0.357855 + 1.10136i 0.309017 0.951057i 0.809017 + 0.587785i −0.936877 0.680681i 0.309017 0.951057i 0.309017 + 0.951057i 1.34211 0.975098i −1.00000
631.1 −0.809017 0.587785i −0.912128 + 2.80724i 0.309017 + 0.951057i 0.809017 0.587785i 2.38798 1.73497i 0.309017 + 0.951057i 0.309017 0.951057i −4.62157 3.35777i −1.00000
631.2 −0.809017 0.587785i −0.563761 + 1.73508i 0.309017 + 0.951057i 0.809017 0.587785i 1.47595 1.07234i 0.309017 + 0.951057i 0.309017 0.951057i −0.265619 0.192984i −1.00000
631.3 −0.809017 0.587785i 0.357855 1.10136i 0.309017 + 0.951057i 0.809017 0.587785i −0.936877 + 0.680681i 0.309017 + 0.951057i 0.309017 0.951057i 1.34211 + 0.975098i −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 631.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.n.h 12
11.c even 5 1 inner 770.2.n.h 12
11.c even 5 1 8470.2.a.da 6
11.d odd 10 1 8470.2.a.cu 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.h 12 1.a even 1 1 trivial
770.2.n.h 12 11.c even 5 1 inner
8470.2.a.cu 6 11.d odd 10 1
8470.2.a.da 6 11.c even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(770, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{3}$$
$3$ $$121 - 759 T + 1871 T^{2} - 105 T^{3} + 1429 T^{4} + 622 T^{5} + 657 T^{6} + 168 T^{7} + 61 T^{8} - 11 T^{9} + 6 T^{10} + T^{12}$$
$5$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{3}$$
$7$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{3}$$
$11$ $$1771561 + 161051 T + 29282 T^{2} + 15972 T^{3} + 1573 T^{4} + 3872 T^{5} + 1583 T^{6} + 352 T^{7} + 13 T^{8} + 12 T^{9} + 2 T^{10} + T^{11} + T^{12}$$
$13$ $$309136 - 473712 T + 592624 T^{2} - 395720 T^{3} + 179784 T^{4} - 53864 T^{5} + 13568 T^{6} - 2264 T^{7} + 496 T^{8} - 48 T^{9} + 4 T^{10} + T^{12}$$
$17$ $$297025 - 1098175 T + 1876135 T^{2} - 1169915 T^{3} + 895891 T^{4} + 380960 T^{5} + 120683 T^{6} + 13750 T^{7} + 1899 T^{8} - 135 T^{9} + 32 T^{10} + T^{12}$$
$19$ $$160801 - 69774 T + 132988 T^{2} - 139807 T^{3} + 111547 T^{4} + 51216 T^{5} + 22319 T^{6} + 3092 T^{7} + 503 T^{8} - 51 T^{9} + 19 T^{10} + T^{11} + T^{12}$$
$23$ $$( 3520 - 4640 T + 1664 T^{2} + 24 T^{3} - 80 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$29$ $$234256 + 238128 T + 446912 T^{2} + 70144 T^{3} + 10000 T^{4} - 11408 T^{5} + 31832 T^{6} - 19348 T^{7} + 7040 T^{8} - 1566 T^{9} + 242 T^{10} - 22 T^{11} + T^{12}$$
$31$ $$5856400 + 7453600 T + 42233840 T^{2} - 20015160 T^{3} + 6199656 T^{4} - 1316424 T^{5} + 385312 T^{6} - 37792 T^{7} + 5460 T^{8} + 132 T^{9} + 2 T^{10} - 6 T^{11} + T^{12}$$
$37$ $$46840336 + 28033024 T + 11585536 T^{2} + 2982720 T^{3} + 819544 T^{4} + 143008 T^{5} + 40592 T^{6} + 6392 T^{7} + 1836 T^{8} + 346 T^{9} + 76 T^{10} + 10 T^{11} + T^{12}$$
$41$ $$12313081 - 4052895 T + 5257571 T^{2} + 83831 T^{3} + 486381 T^{4} - 48178 T^{5} + 152797 T^{6} - 64032 T^{7} + 13429 T^{8} - 1515 T^{9} + 166 T^{10} - 16 T^{11} + T^{12}$$
$43$ $$( -29 - 449 T - 396 T^{2} + 197 T^{3} + 36 T^{4} - 15 T^{5} + T^{6} )^{2}$$
$47$ $$296390656 - 12120064 T + 63920128 T^{2} + 9840128 T^{3} + 5531392 T^{4} - 68864 T^{5} + 396864 T^{6} - 68288 T^{7} + 18768 T^{8} - 4056 T^{9} + 524 T^{10} - 34 T^{11} + T^{12}$$
$53$ $$3254416 + 3593568 T + 30746736 T^{2} - 16673288 T^{3} + 43801512 T^{4} - 4639160 T^{5} + 2190840 T^{6} + 212936 T^{7} - 7252 T^{8} - 1224 T^{9} + 210 T^{10} + 26 T^{11} + T^{12}$$
$59$ $$2929082641 - 2899045486 T + 2995021188 T^{2} - 1143625633 T^{3} + 236821247 T^{4} - 22908716 T^{5} + 2177839 T^{6} - 21732 T^{7} + 14183 T^{8} - 1189 T^{9} + 99 T^{10} - T^{11} + T^{12}$$
$61$ $$487703056 + 199639360 T + 1423641600 T^{2} - 1164891384 T^{3} + 432300880 T^{4} - 93108800 T^{5} + 14214096 T^{6} - 1734880 T^{7} + 182040 T^{8} - 14964 T^{9} + 940 T^{10} - 40 T^{11} + T^{12}$$
$67$ $$( -15679 - 28301 T - 14630 T^{2} - 1465 T^{3} + 170 T^{4} + 29 T^{5} + T^{6} )^{2}$$
$71$ $$1936 + 152416 T + 4542928 T^{2} - 5357768 T^{3} + 13465360 T^{4} - 3180376 T^{5} + 839888 T^{6} - 43904 T^{7} + 80 T^{8} + 428 T^{9} + 188 T^{10} + 14 T^{11} + T^{12}$$
$73$ $$85539285841 + 8477271935 T - 1303174441 T^{2} - 689382363 T^{3} + 341867991 T^{4} - 68358384 T^{5} + 11118623 T^{6} - 1380054 T^{7} + 145639 T^{8} - 11765 T^{9} + 764 T^{10} - 32 T^{11} + T^{12}$$
$79$ $$2448666256 - 722466400 T + 226878976 T^{2} - 20066584 T^{3} + 4882816 T^{4} - 2178848 T^{5} + 937992 T^{6} - 191552 T^{7} + 30504 T^{8} - 2880 T^{9} + 236 T^{10} - 16 T^{11} + T^{12}$$
$83$ $$2927521 + 9845094 T + 31413906 T^{2} + 38823265 T^{3} + 19995119 T^{4} - 7056782 T^{5} + 4794797 T^{6} - 916858 T^{7} + 100521 T^{8} - 7679 T^{9} + 631 T^{10} - 35 T^{11} + T^{12}$$
$89$ $$( 370271 + 15771 T - 25312 T^{2} - 3011 T^{3} + 120 T^{4} + 29 T^{5} + T^{6} )^{2}$$
$97$ $$5679189441025 - 3471802688200 T + 1051745416040 T^{2} - 201628044905 T^{3} + 27796745171 T^{4} - 2944780334 T^{5} + 252101199 T^{6} - 17681168 T^{7} + 1048739 T^{8} - 51021 T^{9} + 2011 T^{10} - 57 T^{11} + T^{12}$$
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