# Properties

 Label 76.5.h.a Level $76$ Weight $5$ Character orbit 76.h Analytic conductor $7.856$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 76.h (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.85611719437$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 6 x^{11} + 631 x^{10} - 3100 x^{9} + 142264 x^{8} - 550522 x^{7} + 14083117 x^{6} - 40335478 x^{5} + 638031136 x^{4} - 1209472584 x^{3} + 12784623475 x^{2} - 12186518934 x + 90728724573$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{3} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} ) q^{5} + ( -4 + \beta_{3} + \beta_{4} ) q^{7} + ( -1 + 2 \beta_{1} - \beta_{3} + 22 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{3} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} ) q^{5} + ( -4 + \beta_{3} + \beta_{4} ) q^{7} + ( -1 + 2 \beta_{1} - \beta_{3} + 22 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{9} + ( 1 - \beta_{2} + \beta_{3} - \beta_{10} + \beta_{11} ) q^{11} + ( -11 - \beta_{3} + 5 \beta_{5} + \beta_{7} - \beta_{10} + \beta_{11} ) q^{13} + ( -79 + 4 \beta_{1} - 6 \beta_{2} + 11 \beta_{3} + 2 \beta_{4} + 41 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{15} + ( -81 + 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + 81 \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - 3 \beta_{10} - 2 \beta_{11} ) q^{17} + ( -18 + \beta_{1} + 10 \beta_{3} + 5 \beta_{4} - 45 \beta_{5} + 3 \beta_{7} - 3 \beta_{9} + 5 \beta_{11} ) q^{19} + ( 55 + \beta_{1} + 8 \beta_{2} - \beta_{3} + 3 \beta_{4} + 55 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} + 8 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{21} + ( -6 + 4 \beta_{1} - 6 \beta_{3} + 88 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} - 8 \beta_{8} - 6 \beta_{10} - 3 \beta_{11} ) q^{23} + ( 10 - 34 \beta_{1} + 10 \beta_{3} - 20 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 9 \beta_{8} + 5 \beta_{9} + 2 \beta_{10} ) q^{25} + ( 37 - 22 \beta_{1} + 7 \beta_{2} - 14 \beta_{3} - 7 \beta_{4} - 80 \beta_{5} - \beta_{6} + \beta_{7} - 14 \beta_{8} + 14 \beta_{9} ) q^{27} + ( 213 - 5 \beta_{1} + 16 \beta_{2} - 37 \beta_{3} + 6 \beta_{4} - 120 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} - 8 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{29} + ( 19 - 11 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + 8 \beta_{4} - 17 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} - 6 \beta_{8} - 16 \beta_{9} - \beta_{10} - \beta_{11} ) q^{31} + ( 11 - 20 \beta_{1} + 7 \beta_{2} + 20 \beta_{3} - 3 \beta_{4} + 11 \beta_{5} - 3 \beta_{6} + 5 \beta_{7} + 7 \beta_{8} - 3 \beta_{9} + 8 \beta_{10} - 3 \beta_{11} ) q^{33} + ( -183 + 52 \beta_{1} - 28 \beta_{2} + 52 \beta_{3} - 18 \beta_{4} + 183 \beta_{5} + 7 \beta_{6} - \beta_{7} + 28 \beta_{8} + 18 \beta_{9} - 6 \beta_{10} - 7 \beta_{11} ) q^{35} + ( 146 + 11 \beta_{1} - 3 \beta_{2} + 15 \beta_{3} + 12 \beta_{4} - 273 \beta_{5} - 11 \beta_{6} + 11 \beta_{7} + 6 \beta_{8} - 24 \beta_{9} + 20 \beta_{10} + 20 \beta_{11} ) q^{37} + ( -200 - 2 \beta_{1} + 18 \beta_{2} - 23 \beta_{3} - 9 \beta_{4} + 2 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + 5 \beta_{10} - 5 \beta_{11} ) q^{39} + ( -112 + 27 \beta_{1} - 21 \beta_{2} - 27 \beta_{3} - 21 \beta_{4} - 112 \beta_{5} + 7 \beta_{6} + 14 \beta_{7} - 21 \beta_{8} - 21 \beta_{9} + 7 \beta_{10} + 7 \beta_{11} ) q^{41} + ( 24 + 45 \beta_{1} - 72 \beta_{2} + 45 \beta_{3} - 3 \beta_{4} - 24 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} + 72 \beta_{8} + 3 \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{43} + ( 583 + 73 \beta_{1} + 25 \beta_{2} - 188 \beta_{3} - 17 \beta_{4} - 73 \beta_{5} - 19 \beta_{6} - 19 \beta_{7} + 9 \beta_{10} - 9 \beta_{11} ) q^{45} + ( -57 + 58 \beta_{1} - 57 \beta_{3} - 621 \beta_{5} + 4 \beta_{6} + 14 \beta_{7} - 29 \beta_{8} + 27 \beta_{9} + 14 \beta_{10} + 18 \beta_{11} ) q^{47} + ( 78 - 32 \beta_{1} - 36 \beta_{2} + 64 \beta_{3} - 36 \beta_{4} + 32 \beta_{5} - 10 \beta_{6} - 10 \beta_{7} + 7 \beta_{10} - 7 \beta_{11} ) q^{49} + ( 638 - 59 \beta_{1} + 92 \beta_{2} - 298 \beta_{3} - 26 \beta_{4} - 409 \beta_{5} - 6 \beta_{6} - 5 \beta_{7} - 46 \beta_{8} + 13 \beta_{9} + 5 \beta_{10} + \beta_{11} ) q^{51} + ( -354 + 76 \beta_{1} - 110 \beta_{2} + 134 \beta_{3} + 42 \beta_{4} + 168 \beta_{5} + \beta_{6} - 7 \beta_{7} + 55 \beta_{8} - 21 \beta_{9} + 7 \beta_{10} - 8 \beta_{11} ) q^{53} + ( -366 - 15 \beta_{1} + 54 \beta_{2} - 15 \beta_{3} - 7 \beta_{4} + 366 \beta_{5} + 10 \beta_{6} - 8 \beta_{7} - 54 \beta_{8} + 7 \beta_{9} - 2 \beta_{10} - 10 \beta_{11} ) q^{55} + ( 572 + 173 \beta_{1} + 57 \beta_{2} + 58 \beta_{3} + 29 \beta_{4} + 62 \beta_{5} - 19 \beta_{6} - 13 \beta_{7} + 57 \beta_{8} - 25 \beta_{9} + 19 \beta_{10} - 9 \beta_{11} ) q^{57} + ( -363 - 248 \beta_{1} + 73 \beta_{2} + 248 \beta_{3} + 15 \beta_{4} - 363 \beta_{5} + 7 \beta_{6} + 8 \beta_{7} + 73 \beta_{8} + 15 \beta_{9} + \beta_{10} + 7 \beta_{11} ) q^{59} + ( -58 + 21 \beta_{1} - 58 \beta_{3} + 195 \beta_{5} - 10 \beta_{6} + 16 \beta_{7} - 117 \beta_{8} - 22 \beta_{9} + 16 \beta_{10} + 6 \beta_{11} ) q^{61} + ( 190 - 476 \beta_{1} + 190 \beta_{3} + 488 \beta_{5} + 8 \beta_{6} - 6 \beta_{7} - 88 \beta_{8} + 8 \beta_{9} - 6 \beta_{10} + 2 \beta_{11} ) q^{63} + ( 155 + 79 \beta_{1} + 34 \beta_{2} - 46 \beta_{3} - 12 \beta_{4} - 481 \beta_{5} + 11 \beta_{6} - 11 \beta_{7} - 68 \beta_{8} + 24 \beta_{9} - 16 \beta_{10} - 16 \beta_{11} ) q^{65} + ( 737 + 2 \beta_{1} + 12 \beta_{2} - 267 \beta_{3} + 16 \beta_{4} - 504 \beta_{5} + 25 \beta_{6} + 21 \beta_{7} - 6 \beta_{8} - 8 \beta_{9} - 21 \beta_{10} - 4 \beta_{11} ) q^{67} + ( 563 - 290 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + 8 \beta_{4} - 826 \beta_{5} - 36 \beta_{6} + 36 \beta_{7} - 6 \beta_{8} - 16 \beta_{9} + 20 \beta_{10} + 20 \beta_{11} ) q^{69} + ( 204 - 237 \beta_{1} - 12 \beta_{2} + 237 \beta_{3} + 9 \beta_{4} + 204 \beta_{5} + 30 \beta_{6} - 15 \beta_{7} - 12 \beta_{8} + 9 \beta_{9} - 45 \beta_{10} + 30 \beta_{11} ) q^{71} + ( -452 + 352 \beta_{1} - 36 \beta_{2} + 352 \beta_{3} + 452 \beta_{5} - 19 \beta_{6} + 13 \beta_{7} + 36 \beta_{8} + 6 \beta_{10} + 19 \beta_{11} ) q^{73} + ( -1771 + 178 \beta_{1} - 69 \beta_{2} + 49 \beta_{3} - 20 \beta_{4} + 3462 \beta_{5} + 36 \beta_{6} - 36 \beta_{7} + 138 \beta_{8} + 40 \beta_{9} - 77 \beta_{10} - 77 \beta_{11} ) q^{75} + ( -1283 + 18 \beta_{1} - 14 \beta_{2} + 5 \beta_{3} + 27 \beta_{4} - 18 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} - 33 \beta_{10} + 33 \beta_{11} ) q^{77} + ( 412 + 305 \beta_{1} + 30 \beta_{2} - 305 \beta_{3} + 53 \beta_{4} + 412 \beta_{5} - 22 \beta_{6} - 55 \beta_{7} + 30 \beta_{8} + 53 \beta_{9} - 33 \beta_{10} - 22 \beta_{11} ) q^{79} + ( 422 + 83 \beta_{1} - 125 \beta_{2} + 83 \beta_{3} + 37 \beta_{4} - 422 \beta_{5} - 15 \beta_{6} + 16 \beta_{7} + 125 \beta_{8} - 37 \beta_{9} - \beta_{10} + 15 \beta_{11} ) q^{81} + ( -1375 + 183 \beta_{1} - 52 \beta_{2} - 323 \beta_{3} - 9 \beta_{4} - 183 \beta_{5} + 63 \beta_{6} + 63 \beta_{7} - 57 \beta_{10} + 57 \beta_{11} ) q^{83} + ( -371 + 812 \beta_{1} - 371 \beta_{3} + 1205 \beta_{5} - 20 \beta_{6} - 57 \beta_{7} + 90 \beta_{8} + 20 \beta_{9} - 57 \beta_{10} - 77 \beta_{11} ) q^{85} + ( -2964 - 252 \beta_{1} - 138 \beta_{2} + 708 \beta_{3} + 66 \beta_{4} + 252 \beta_{5} + 29 \beta_{6} + 29 \beta_{7} - 13 \beta_{10} + 13 \beta_{11} ) q^{87} + ( 441 - 26 \beta_{1} + 136 \beta_{2} - 727 \beta_{3} + 84 \beta_{4} - 558 \beta_{5} + 43 \beta_{6} + 14 \beta_{7} - 68 \beta_{8} - 42 \beta_{9} - 14 \beta_{10} - 29 \beta_{11} ) q^{89} + ( -860 + 16 \beta_{1} - 72 \beta_{2} - 44 \beta_{3} - 40 \beta_{4} + 392 \beta_{5} - 10 \beta_{6} + 20 \beta_{7} + 36 \beta_{8} + 20 \beta_{9} - 20 \beta_{10} + 30 \beta_{11} ) q^{91} + ( -869 + 52 \beta_{1} + 75 \beta_{2} + 52 \beta_{3} + 90 \beta_{4} + 869 \beta_{5} - 71 \beta_{6} + 22 \beta_{7} - 75 \beta_{8} - 90 \beta_{9} + 49 \beta_{10} + 71 \beta_{11} ) q^{93} + ( -39 + 132 \beta_{1} - 19 \beta_{2} + 484 \beta_{3} - 81 \beta_{4} + 577 \beta_{5} + 95 \beta_{6} + 16 \beta_{7} + 117 \beta_{9} - 76 \beta_{10} - 43 \beta_{11} ) q^{95} + ( -1235 - 957 \beta_{1} + 39 \beta_{2} + 957 \beta_{3} - 59 \beta_{4} - 1235 \beta_{5} + 8 \beta_{6} + 17 \beta_{7} + 39 \beta_{8} - 59 \beta_{9} + 9 \beta_{10} + 8 \beta_{11} ) q^{97} + ( -170 + 256 \beta_{1} - 170 \beta_{3} + 2069 \beta_{5} - 29 \beta_{6} + 29 \beta_{7} - 46 \beta_{8} + 38 \beta_{9} + 29 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 12q^{3} + 9q^{5} - 52q^{7} + 136q^{9} + O(q^{10})$$ $$12q - 12q^{3} + 9q^{5} - 52q^{7} + 136q^{9} + 6q^{11} - 93q^{13} - 741q^{15} - 483q^{17} - 533q^{19} + 972q^{21} + 531q^{23} - 217q^{25} + 2025q^{29} - 75q^{33} - 1128q^{35} - 2250q^{39} - 1692q^{41} - 63q^{43} + 7976q^{45} - 3471q^{47} + 420q^{49} + 6741q^{51} - 3771q^{53} - 2014q^{55} + 7617q^{57} - 9594q^{59} + 1229q^{61} + 1514q^{63} + 7590q^{67} + 963q^{71} - 2838q^{73} - 15408q^{77} + 11073q^{79} + 2086q^{81} - 14202q^{83} + 9455q^{85} - 39510q^{87} + 6525q^{89} - 7686q^{91} - 5316q^{93} + 1521q^{95} - 34110q^{97} + 13220q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 631 x^{10} - 3100 x^{9} + 142264 x^{8} - 550522 x^{7} + 14083117 x^{6} - 40335478 x^{5} + 638031136 x^{4} - 1209472584 x^{3} + 12784623475 x^{2} - 12186518934 x + 90728724573$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-101041319944 \nu^{11} - 35886079446026 \nu^{10} + 120597901725160 \nu^{9} - 21688944900716313 \nu^{8} + 73651816175397896 \nu^{7} - 4528618732116717019 \nu^{6} + 12258768447274812411 \nu^{5} - 385183615682462895791 \nu^{4} + 712454459463362422549 \nu^{3} - 12754824374811348645172 \nu^{2} + 12053250662873660603679 \nu - 132390992838065377320378$$$$)/$$$$16\!\cdots\!42$$ $$\beta_{3}$$ $$=$$ $$($$$$-162445852 \nu^{11} - 5113227722 \nu^{10} - 69019867025 \nu^{9} - 3210361481925 \nu^{8} - 6648576376768 \nu^{7} - 708698187766006 \nu^{6} + 429118260440970 \nu^{5} - 65796779401944092 \nu^{4} + 70473352267918468 \nu^{3} - 2463175718751630814 \nu^{2} + 1869527130172466331 \nu - 28887748399394727957$$$$)/ 259266510437924061$$ $$\beta_{4}$$ $$=$$ $$($$$$303123959832 \nu^{11} - 2063916106978 \nu^{10} + 186817067049800 \nu^{9} + 1178718425842685 \nu^{8} + 31305351945679648 \nu^{7} + 993378222949727206 \nu^{6} + 120519941968623820 \nu^{5} + 188839536189947845115 \nu^{4} - 265142258435854616554 \nu^{3} + 11848614265145615440610 \nu^{2} - 10673896097312549109474 \nu + 188876972636206396639209$$$$)/$$$$48\!\cdots\!26$$ $$\beta_{5}$$ $$=$$ $$($$$$324891704 \nu^{11} - 1786904372 \nu^{10} + 198106533130 \nu^{9} - 878077616295 \nu^{8} + 42131954279636 \nu^{7} - 143376652766620 \nu^{6} + 3723413039228742 \nu^{5} - 8952130879404800 \nu^{4} + 132542112698746852 \nu^{3} - 189932143790377816 \nu^{2} + 1500510243249142770 \nu - 589261528873914435$$$$)/ 259266510437924061$$ $$\beta_{6}$$ $$=$$ $$($$$$-1328468937428 \nu^{11} - 47554498066674 \nu^{10} - 659027444905734 \nu^{9} - 27798859742239349 \nu^{8} - 123128209041708237 \nu^{7} - 5443126967355294744 \nu^{6} - 13143780252585735025 \nu^{5} - 406498317644182363992 \nu^{4} - 941570426054674109019 \nu^{3} - 10225945830638540587309 \nu^{2} - 25213733145995204565456 \nu - 60829507421154934066440$$$$)/$$$$48\!\cdots\!26$$ $$\beta_{7}$$ $$=$$ $$($$$$1328468937428 \nu^{11} - 62167656378382 \nu^{10} + 1207638217131014 \nu^{9} - 36089256534066905 \nu^{8} + 375389009513581573 \nu^{7} - 7149351006045129107 \nu^{6} + 50040605536378796078 \nu^{5} - 560212993213258478266 \nu^{4} + 2813791546008906760112 \nu^{3} - 15706121720172723609771 \nu^{2} + 50215797728826470969193 \nu - 97635993683686672579407$$$$)/$$$$48\!\cdots\!26$$ $$\beta_{8}$$ $$=$$ $$($$$$4824263493526 \nu^{11} - 42886275115864 \nu^{10} + 2963447158019165 \nu^{9} - 22614622788674679 \nu^{8} + 631293150975272089 \nu^{7} - 4057901665651781165 \nu^{6} + 55328058002096853825 \nu^{5} - 290454000420088732366 \nu^{4} + 1928599196221796103584 \nu^{3} - 7885471842605540229521 \nu^{2} + 21311325751298831041470 \nu - 64849978983939510773565$$$$)/$$$$16\!\cdots\!42$$ $$\beta_{9}$$ $$=$$ $$($$$$22328146096228 \nu^{11} - 128607403047369 \nu^{10} + 13445636462718138 \nu^{9} - 61859677616319311 \nu^{8} + 2790874729577260122 \nu^{7} - 9622923120465176787 \nu^{6} + 235113756453940102304 \nu^{5} - 530093569333369367955 \nu^{4} + 7694049347122175076054 \nu^{3} - 7456703875559540792869 \nu^{2} + 81032889998841690174888 \nu + 27243925171299701922915$$$$)/$$$$48\!\cdots\!26$$ $$\beta_{10}$$ $$=$$ $$($$$$43277988338263 \nu^{11} + 40046604044395 \nu^{10} + 24746522487182347 \nu^{9} + 51933269161744916 \nu^{8} + 4796153501585748617 \nu^{7} + 16533603265390663442 \nu^{6} + 362854584926757284788 \nu^{5} + 1906566726043641916024 \nu^{4} + 9709564623681771293464 \nu^{3} + 82355608237050146880789 \nu^{2} + 61066854468680585947722 \nu + 1091016435863124801390951$$$$)/$$$$48\!\cdots\!26$$ $$\beta_{11}$$ $$=$$ $$($$$$43277988338263 \nu^{11} - 516104475765288 \nu^{10} + 27527277886230762 \nu^{9} - 283594936804197209 \nu^{8} + 6121581793055226627 \nu^{7} - 53667921419404679013 \nu^{6} + 568831839133676142461 \nu^{5} - 4143485154411497562771 \nu^{4} + 21467913732409768449114 \nu^{3} - 126903320774871653802172 \nu^{2} + 264480718398117582663993 \nu - 1246439291023390922435718$$$$)/$$$$48\!\cdots\!26$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta_{4} - \beta_{2} + \beta_{1} - 103$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{11} - 3 \beta_{10} + 17 \beta_{9} - 11 \beta_{8} + \beta_{7} + 2 \beta_{6} + 67 \beta_{5} - 10 \beta_{4} - 14 \beta_{3} + 4 \beta_{2} - 181 \beta_{1} - 111$$ $$\nu^{4}$$ $$=$$ $$-26 \beta_{11} + 14 \beta_{10} + 34 \beta_{9} - 22 \beta_{8} - 250 \beta_{7} - 248 \beta_{6} + 279 \beta_{5} + 234 \beta_{4} + 184 \beta_{3} + 340 \beta_{2} - 508 \beta_{1} + 17950$$ $$\nu^{5}$$ $$=$$ $$964 \beta_{11} + 1064 \beta_{10} - 4760 \beta_{9} + 2966 \beta_{8} - 212 \beta_{7} - 1038 \beta_{6} - 41715 \beta_{5} + 3010 \beta_{4} + 4393 \beta_{3} - 658 \beta_{2} + 37256 \beta_{1} + 51008$$ $$\nu^{6}$$ $$=$$ $$9523 \beta_{11} - 3409 \beta_{10} - 14365 \beta_{9} + 8953 \beta_{8} + 56977 \beta_{7} + 54494 \beta_{6} - 183628 \beta_{5} - 49772 \beta_{4} - 80211 \beta_{3} - 83682 \beta_{2} + 170824 \beta_{1} - 3639920$$ $$\nu^{7}$$ $$=$$ $$-258177 \beta_{11} - 303789 \beta_{10} + 1135779 \beta_{9} - 644067 \beta_{8} + 65256 \beta_{7} + 329271 \beta_{6} + 14039235 \beta_{5} - 769437 \beta_{4} - 1213323 \beta_{3} + 41925 \beta_{2} - 7861850 \beta_{1} - 16938933$$ $$\nu^{8}$$ $$=$$ $$-2726304 \beta_{11} + 449880 \beta_{10} + 4610232 \beta_{9} - 2618100 \beta_{8} - 12798344 \beta_{7} - 11730692 \beta_{6} + 74171253 \beta_{5} + 10627604 \beta_{4} + 25167300 \beta_{3} + 18699164 \beta_{2} - 49402496 \beta_{1} + 765988175$$ $$\nu^{9}$$ $$=$$ $$65592264 \beta_{11} + 80159184 \beta_{10} - 260711524 \beta_{9} + 128003272 \beta_{8} - 21438716 \beta_{7} - 91314364 \beta_{6} - 3935592755 \beta_{5} + 189764936 \beta_{4} + 325816621 \beta_{3} + 15937996 \beta_{2} + 1665475967 \beta_{1} + 4926306888$$ $$\nu^{10}$$ $$=$$ $$730967377 \beta_{11} + 14905901 \beta_{10} - 1338235085 \beta_{9} + 659714891 \beta_{8} + 2863528856 \beta_{7} + 2506125835 \beta_{6} - 24856554570 \beta_{5} - 2279919123 \beta_{4} - 6991857647 \beta_{3} - 4020244031 \beta_{2} + 13320116471 \beta_{1} - 162574704089$$ $$\nu^{11}$$ $$=$$ $$-16201138664 \beta_{11} - 20273509714 \beta_{10} + 58997836558 \beta_{9} - 24135753784 \beta_{8} + 6631724131 \beta_{7} + 23939300133 \beta_{6} + 1021105964508 \beta_{5} - 46265427995 \beta_{4} - 86724980771 \beta_{3} - 8965119565 \beta_{2} - 352054103833 \beta_{1} - 1340587725328$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$\beta_{5}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.5 + 14.3199i 0.5 + 9.58497i 0.5 + 4.44379i 0.5 − 4.96177i 0.5 − 6.42649i 0.5 − 15.2283i 0.5 − 14.3199i 0.5 − 9.58497i 0.5 − 4.44379i 0.5 + 4.96177i 0.5 + 6.42649i 0.5 + 15.2283i
0 −13.1514 7.59296i 0 19.9265 34.5138i 0 −53.3663 0 74.8062 + 129.568i 0
65.2 0 −9.05083 5.22550i 0 −12.7505 + 22.0845i 0 27.7589 0 14.1116 + 24.4421i 0
65.3 0 −4.59843 2.65491i 0 −1.89169 + 3.27650i 0 36.8385 0 −26.4029 45.7312i 0
65.4 0 3.54702 + 2.04787i 0 15.5891 27.0012i 0 −2.67956 0 −32.1124 55.6204i 0
65.5 0 4.81550 + 2.78023i 0 −13.2594 + 22.9660i 0 −84.4930 0 −25.0406 43.3717i 0
65.6 0 12.4381 + 7.18116i 0 −3.11411 + 5.39380i 0 49.9415 0 62.6382 + 108.493i 0
69.1 0 −13.1514 + 7.59296i 0 19.9265 + 34.5138i 0 −53.3663 0 74.8062 129.568i 0
69.2 0 −9.05083 + 5.22550i 0 −12.7505 22.0845i 0 27.7589 0 14.1116 24.4421i 0
69.3 0 −4.59843 + 2.65491i 0 −1.89169 3.27650i 0 36.8385 0 −26.4029 + 45.7312i 0
69.4 0 3.54702 2.04787i 0 15.5891 + 27.0012i 0 −2.67956 0 −32.1124 + 55.6204i 0
69.5 0 4.81550 2.78023i 0 −13.2594 22.9660i 0 −84.4930 0 −25.0406 + 43.3717i 0
69.6 0 12.4381 7.18116i 0 −3.11411 5.39380i 0 49.9415 0 62.6382 108.493i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 69.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.5.h.a 12
3.b odd 2 1 684.5.y.c 12
4.b odd 2 1 304.5.r.b 12
19.d odd 6 1 inner 76.5.h.a 12
57.f even 6 1 684.5.y.c 12
76.f even 6 1 304.5.r.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.5.h.a 12 1.a even 1 1 trivial
76.5.h.a 12 19.d odd 6 1 inner
304.5.r.b 12 4.b odd 2 1
304.5.r.b 12 76.f even 6 1
684.5.y.c 12 3.b odd 2 1
684.5.y.c 12 57.f even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$75979063449 - 18918481662 T - 3182152311 T^{2} + 1183318794 T^{3} + 155960136 T^{4} - 43553322 T^{5} - 2208909 T^{6} + 711432 T^{7} + 54256 T^{8} - 3444 T^{9} - 239 T^{10} + 12 T^{11} + T^{12}$$
$5$ $$392044989615876 + 174315280071996 T + 58406029288380 T^{2} + 8556919136964 T^{3} + 983335559548 T^{4} + 32461763778 T^{5} + 1995775355 T^{6} + 29332341 T^{7} + 2795952 T^{8} + 20745 T^{9} + 2024 T^{10} - 9 T^{11} + T^{12}$$
$7$ $$( -617046128 - 197809864 T + 12165932 T^{2} - 658 T^{3} - 6970 T^{4} + 26 T^{5} + T^{6} )^{2}$$
$11$ $$( -781272216 + 544855932 T - 74734322 T^{2} + 2988867 T^{3} - 37505 T^{4} - 3 T^{5} + T^{6} )^{2}$$
$13$ $$50\!\cdots\!44$$$$+$$$$16\!\cdots\!80$$$$T +$$$$20\!\cdots\!44$$$$T^{2} + 7852258527055532280 T^{3} + 101635890763384008 T^{4} - 585008381898450 T^{5} - 14049986051007 T^{6} + 49672026171 T^{7} + 1403686134 T^{8} - 3936969 T^{9} - 39450 T^{10} + 93 T^{11} + T^{12}$$
$17$ $$15\!\cdots\!76$$$$-$$$$50\!\cdots\!28$$$$T +$$$$17\!\cdots\!56$$$$T^{2} -$$$$61\!\cdots\!96$$$$T^{3} + 16988087741215595824 T^{4} + 32265244906203378 T^{5} + 1651018271594651 T^{6} + 3663664629669 T^{7} + 29655336060 T^{8} + 34619013 T^{9} + 335492 T^{10} + 483 T^{11} + T^{12}$$
$19$ $$48\!\cdots\!21$$$$+$$$$20\!\cdots\!33$$$$T -$$$$59\!\cdots\!21$$$$T^{2} -$$$$28\!\cdots\!50$$$$T^{3} +$$$$41\!\cdots\!93$$$$T^{4} + 1356750428630436649 T^{5} - 1524620852430386 T^{6} + 10410835004569 T^{7} + 24562401373 T^{8} - 126656850 T^{9} - 207841 T^{10} + 533 T^{11} + T^{12}$$
$23$ $$48\!\cdots\!24$$$$-$$$$32\!\cdots\!32$$$$T +$$$$21\!\cdots\!36$$$$T^{2} -$$$$67\!\cdots\!20$$$$T^{3} +$$$$28\!\cdots\!38$$$$T^{4} - 75056480395714595610 T^{5} + 246094906792609613 T^{6} - 427691701026261 T^{7} + 810889003236 T^{8} - 637284771 T^{9} + 1010870 T^{10} - 531 T^{11} + T^{12}$$
$29$ $$43\!\cdots\!64$$$$+$$$$16\!\cdots\!88$$$$T +$$$$18\!\cdots\!20$$$$T^{2} -$$$$82\!\cdots\!04$$$$T^{3} -$$$$41\!\cdots\!24$$$$T^{4} + 18513841814256553056 T^{5} + 85661960352023187 T^{6} - 286466374898325 T^{7} - 163179592308 T^{8} + 1057371975 T^{9} + 844716 T^{10} - 2025 T^{11} + T^{12}$$
$31$ $$20\!\cdots\!44$$$$+$$$$23\!\cdots\!68$$$$T^{2} +$$$$54\!\cdots\!04$$$$T^{4} + 3469414352479974252 T^{6} + 7675476031116 T^{8} + 5102232 T^{10} + T^{12}$$
$37$ $$14\!\cdots\!96$$$$+$$$$79\!\cdots\!56$$$$T^{2} +$$$$15\!\cdots\!16$$$$T^{4} +$$$$13\!\cdots\!12$$$$T^{6} + 63431766900576 T^{8} + 13373304 T^{10} + T^{12}$$
$41$ $$48\!\cdots\!81$$$$-$$$$11\!\cdots\!16$$$$T +$$$$51\!\cdots\!51$$$$T^{2} +$$$$87\!\cdots\!76$$$$T^{3} -$$$$45\!\cdots\!18$$$$T^{4} -$$$$65\!\cdots\!40$$$$T^{5} + 45414196073520434451 T^{6} + 120422881727589204 T^{7} + 88052466627306 T^{8} - 16538902380 T^{9} - 8820477 T^{10} + 1692 T^{11} + T^{12}$$
$43$ $$68\!\cdots\!36$$$$+$$$$13\!\cdots\!96$$$$T +$$$$51\!\cdots\!68$$$$T^{2} -$$$$41\!\cdots\!96$$$$T^{3} +$$$$75\!\cdots\!52$$$$T^{4} -$$$$25\!\cdots\!22$$$$T^{5} +$$$$27\!\cdots\!55$$$$T^{6} - 53508616996454343 T^{7} + 74200136841234 T^{8} - 7174865381 T^{9} + 10125594 T^{10} + 63 T^{11} + T^{12}$$
$47$ $$24\!\cdots\!96$$$$-$$$$64\!\cdots\!28$$$$T +$$$$19\!\cdots\!00$$$$T^{2} +$$$$33\!\cdots\!64$$$$T^{3} +$$$$85\!\cdots\!62$$$$T^{4} +$$$$92\!\cdots\!84$$$$T^{5} +$$$$22\!\cdots\!59$$$$T^{6} + 348933347175398859 T^{7} + 296316335088234 T^{8} + 39251107161 T^{9} + 26253656 T^{10} + 3471 T^{11} + T^{12}$$
$53$ $$80\!\cdots\!96$$$$+$$$$13\!\cdots\!40$$$$T -$$$$22\!\cdots\!36$$$$T^{2} -$$$$15\!\cdots\!40$$$$T^{3} +$$$$10\!\cdots\!96$$$$T^{4} -$$$$82\!\cdots\!28$$$$T^{5} -$$$$97\!\cdots\!21$$$$T^{6} + 1412614156954926015 T^{7} + 768913715772936 T^{8} - 127038485025 T^{9} - 28948128 T^{10} + 3771 T^{11} + T^{12}$$
$59$ $$48\!\cdots\!09$$$$+$$$$79\!\cdots\!44$$$$T +$$$$48\!\cdots\!69$$$$T^{2} +$$$$66\!\cdots\!16$$$$T^{3} -$$$$38\!\cdots\!68$$$$T^{4} -$$$$97\!\cdots\!40$$$$T^{5} +$$$$35\!\cdots\!79$$$$T^{6} + 13205686234211590518 T^{7} + 72604255272012 T^{8} - 383844033846 T^{9} - 9327147 T^{10} + 9594 T^{11} + T^{12}$$
$61$ $$74\!\cdots\!96$$$$+$$$$11\!\cdots\!68$$$$T +$$$$27\!\cdots\!56$$$$T^{2} +$$$$71\!\cdots\!60$$$$T^{3} +$$$$64\!\cdots\!80$$$$T^{4} +$$$$86\!\cdots\!94$$$$T^{5} +$$$$80\!\cdots\!79$$$$T^{6} - 224465306920546267 T^{7} + 707747425061884 T^{8} - 14630441391 T^{9} + 32692388 T^{10} - 1229 T^{11} + T^{12}$$
$67$ $$27\!\cdots\!41$$$$-$$$$44\!\cdots\!08$$$$T +$$$$26\!\cdots\!57$$$$T^{2} -$$$$42\!\cdots\!32$$$$T^{3} -$$$$18\!\cdots\!96$$$$T^{4} +$$$$60\!\cdots\!16$$$$T^{5} +$$$$14\!\cdots\!91$$$$T^{6} - 7620900631756827066 T^{7} + 258204721394808 T^{8} + 260501118570 T^{9} - 15118923 T^{10} - 7590 T^{11} + T^{12}$$
$71$ $$22\!\cdots\!36$$$$-$$$$68\!\cdots\!64$$$$T +$$$$80\!\cdots\!76$$$$T^{2} -$$$$28\!\cdots\!36$$$$T^{3} -$$$$13\!\cdots\!96$$$$T^{4} +$$$$24\!\cdots\!34$$$$T^{5} +$$$$42\!\cdots\!01$$$$T^{6} - 24896117015842580361 T^{7} + 4678996233688452 T^{8} + 70885269585 T^{9} - 73299672 T^{10} - 963 T^{11} + T^{12}$$
$73$ $$11\!\cdots\!89$$$$-$$$$40\!\cdots\!14$$$$T +$$$$39\!\cdots\!45$$$$T^{2} +$$$$33\!\cdots\!50$$$$T^{3} +$$$$57\!\cdots\!74$$$$T^{4} +$$$$99\!\cdots\!90$$$$T^{5} +$$$$29\!\cdots\!73$$$$T^{6} + 19085978951544061782 T^{7} + 9291621628967382 T^{8} + 257388792874 T^{9} + 112498485 T^{10} + 2838 T^{11} + T^{12}$$
$79$ $$19\!\cdots\!64$$$$+$$$$36\!\cdots\!56$$$$T +$$$$19\!\cdots\!60$$$$T^{2} -$$$$83\!\cdots\!92$$$$T^{3} -$$$$16\!\cdots\!24$$$$T^{4} +$$$$74\!\cdots\!56$$$$T^{5} +$$$$11\!\cdots\!95$$$$T^{6} - 53608607731360384605 T^{7} + 2026392578840214 T^{8} + 856319614533 T^{9} - 36463578 T^{10} - 11073 T^{11} + T^{12}$$
$83$ $$($$$$48\!\cdots\!04$$$$+ 31042070063670676824 T + 3251047576647658 T^{2} - 1011890823219 T^{3} - 147951455 T^{4} + 7101 T^{5} + T^{6} )^{2}$$
$89$ $$52\!\cdots\!56$$$$-$$$$46\!\cdots\!12$$$$T +$$$$16\!\cdots\!48$$$$T^{2} -$$$$25\!\cdots\!80$$$$T^{3} +$$$$25\!\cdots\!44$$$$T^{4} +$$$$35\!\cdots\!20$$$$T^{5} -$$$$15\!\cdots\!69$$$$T^{6} -$$$$51\!\cdots\!01$$$$T^{7} + 49028741503966656 T^{8} + 1694524301775 T^{9} - 245505336 T^{10} - 6525 T^{11} + T^{12}$$
$97$ $$49\!\cdots\!69$$$$-$$$$18\!\cdots\!94$$$$T +$$$$13\!\cdots\!17$$$$T^{2} +$$$$40\!\cdots\!06$$$$T^{3} -$$$$43\!\cdots\!18$$$$T^{4} -$$$$64\!\cdots\!94$$$$T^{5} +$$$$76\!\cdots\!29$$$$T^{6} +$$$$76\!\cdots\!42$$$$T^{7} - 52551528614574102 T^{8} - 5919954739650 T^{9} + 214275885 T^{10} + 34110 T^{11} + T^{12}$$