# Properties

 Label 1840.2.e.g Level $1840$ Weight $2$ Character orbit 1840.e Analytic conductor $14.692$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - 2 x^{11} + 39 x^{10} - 10 x^{9} + 2 x^{8} - 26 x^{7} + 297 x^{6} - 116 x^{5} + 24 x^{4} - 20 x^{3} + 64 x^{2} - 32 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 920) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{8} q^{3} -\beta_{5} q^{5} + ( \beta_{7} - \beta_{13} ) q^{7} + ( 1 - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{10} - \beta_{12} ) q^{9} +O(q^{10})$$ $$q -\beta_{8} q^{3} -\beta_{5} q^{5} + ( \beta_{7} - \beta_{13} ) q^{7} + ( 1 - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{10} - \beta_{12} ) q^{9} + ( \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{10} + \beta_{12} ) q^{11} + ( -\beta_{5} - \beta_{8} + \beta_{13} ) q^{13} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} - \beta_{12} - \beta_{13} ) q^{15} + ( -\beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{13} ) q^{17} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} + 2 \beta_{12} ) q^{19} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{10} - \beta_{12} ) q^{21} + \beta_{4} q^{23} + ( \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{25} + ( \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} ) q^{27} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{9} + \beta_{10} + 3 \beta_{12} ) q^{29} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} - 3 \beta_{6} - \beta_{9} + \beta_{12} ) q^{31} + ( -2 \beta_{3} - 5 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{13} ) q^{33} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{35} + ( \beta_{4} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{37} + ( \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{10} - 2 \beta_{12} ) q^{39} + ( -1 + \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{41} + ( \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{13} ) q^{43} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} - \beta_{11} - \beta_{12} ) q^{45} + ( 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + 3 \beta_{11} ) q^{47} + ( 2 + \beta_{2} + \beta_{9} - \beta_{12} ) q^{49} + ( 4 + 3 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{10} - 2 \beta_{12} ) q^{51} + ( -2 \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{11} + 3 \beta_{13} ) q^{53} + ( 3 + \beta_{1} + 2 \beta_{3} - 3 \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{12} + \beta_{13} ) q^{55} + ( -2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + 2 \beta_{11} - 4 \beta_{13} ) q^{57} + ( -5 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 3 \beta_{6} + \beta_{9} - \beta_{10} - 2 \beta_{12} ) q^{59} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - \beta_{6} - 4 \beta_{9} + \beta_{10} + 5 \beta_{12} ) q^{61} + ( \beta_{3} - 4 \beta_{4} + \beta_{7} + 3 \beta_{8} - \beta_{11} ) q^{63} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} - \beta_{12} - \beta_{13} ) q^{65} + ( -2 \beta_{3} + 3 \beta_{4} - 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + 2 \beta_{13} ) q^{67} + \beta_{2} q^{69} + ( 5 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} + 3 \beta_{6} - \beta_{9} - \beta_{10} ) q^{71} + ( \beta_{3} - 3 \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{73} + ( 3 + \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{75} + ( -\beta_{3} + 2 \beta_{4} - 2 \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{13} ) q^{77} + ( -6 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} - 3 \beta_{9} + \beta_{10} + 4 \beta_{12} ) q^{79} + ( -3 - 4 \beta_{1} + \beta_{10} + \beta_{12} ) q^{81} + ( \beta_{3} - 4 \beta_{4} + \beta_{5} + \beta_{7} + 5 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{83} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{9} + 2 \beta_{11} + \beta_{12} + \beta_{13} ) q^{85} + ( \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} - 2 \beta_{13} ) q^{87} + ( 2 - \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{9} - \beta_{10} - 2 \beta_{12} ) q^{89} + ( 2 - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{9} - \beta_{10} + \beta_{12} ) q^{91} + ( 2 \beta_{3} - \beta_{4} - 4 \beta_{5} + 3 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} + 3 \beta_{13} ) q^{93} + ( 2 - \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 3 \beta_{11} + 3 \beta_{12} + 4 \beta_{13} ) q^{95} + ( \beta_{3} - 4 \beta_{7} + \beta_{8} - 3 \beta_{11} + 5 \beta_{13} ) q^{97} + ( -2 + 3 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} + \beta_{6} - 4 \beta_{10} - 4 \beta_{12} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q + 2q^{5} - 4q^{9} + O(q^{10})$$ $$14q + 2q^{5} - 4q^{9} + 14q^{11} + 6q^{15} - 14q^{19} - 12q^{21} - 14q^{25} + 22q^{29} + 20q^{31} + 2q^{35} - 48q^{39} - 32q^{41} - 26q^{45} + 34q^{49} + 14q^{51} + 38q^{55} - 22q^{59} + 10q^{61} - 38q^{65} + 6q^{69} + 28q^{71} + 24q^{75} - 64q^{79} - 10q^{81} - 50q^{85} + 48q^{89} + 14q^{91} + 30q^{95} - 122q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 2 x^{11} + 39 x^{10} - 10 x^{9} + 2 x^{8} - 26 x^{7} + 297 x^{6} - 116 x^{5} + 24 x^{4} - 20 x^{3} + 64 x^{2} - 32 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$224809861 \nu^{13} + 40698207136 \nu^{12} + 13894530424 \nu^{11} + 2084956408 \nu^{10} - 68154663521 \nu^{9} + 1559311271490 \nu^{8} + 134953581114 \nu^{7} + 28178713856 \nu^{6} - 814144658387 \nu^{5} + 11737106209660 \nu^{4} - 402160406420 \nu^{3} + 22675714854 \nu^{2} + 32069661208 \nu + 1689305356572$$$$)/ 732246302876$$ $$\beta_{2}$$ $$=$$ $$($$$$9599816455 \nu^{13} - 39917569898 \nu^{12} - 9824626316 \nu^{11} - 19980270148 \nu^{10} + 450158077433 \nu^{9} - 1634888076832 \nu^{8} + 36371917978 \nu^{7} - 253853562284 \nu^{6} + 3717676805391 \nu^{5} - 12743422120058 \nu^{4} + 1957301556084 \nu^{3} - 143702096254 \nu^{2} - 142401498584 \nu - 2008645915756$$$$)/ 732246302876$$ $$\beta_{3}$$ $$=$$ $$($$$$-14185734445 \nu^{13} + 33377065138 \nu^{12} + 22691432254 \nu^{11} + 49350875674 \nu^{10} - 612714646635 \nu^{9} + 1401214875276 \nu^{8} + 503279347384 \nu^{7} + 1031632771786 \nu^{6} - 4962097019689 \nu^{5} + 11022845347922 \nu^{4} + 2505098929650 \nu^{3} + 4594222559616 \nu^{2} - 1305274242300 \nu + 2294924313380$$$$)/ 732246302876$$ $$\beta_{4}$$ $$=$$ $$($$$$19958784949 \nu^{13} + 4912313158 \nu^{12} + 390318619 \nu^{11} - 37882617844 \nu^{10} + 769444956141 \nu^{9} - 8586142534 \nu^{8} + 2129167227 \nu^{7} - 433265659128 \nu^{6} + 5814921705639 \nu^{5} - 863452980582 \nu^{4} - 24147116423 \nu^{3} + 378394875852 \nu^{2} + 1216849046036 \nu - 327723885618$$$$)/ 366123151438$$ $$\beta_{5}$$ $$=$$ $$($$$$-40039329007 \nu^{13} - 48607321956 \nu^{12} - 16903078854 \nu^{11} + 91091509614 \nu^{10} - 1462932942489 \nu^{9} - 1462473090366 \nu^{8} - 274689612780 \nu^{7} + 1553791371930 \nu^{6} - 10634500106239 \nu^{5} - 9375688488996 \nu^{4} - 615006277590 \nu^{3} + 4857036973804 \nu^{2} - 1092034671500 \nu - 1596402425868$$$$)/ 732246302876$$ $$\beta_{6}$$ $$=$$ $$($$$$51158604007 \nu^{13} + 95455044155 \nu^{12} + 57732941236 \nu^{11} - 86278427068 \nu^{10} + 1799215942067 \nu^{9} + 3072231013943 \nu^{8} + 1361220760372 \nu^{7} - 1082405618534 \nu^{6} + 12498302486205 \nu^{5} + 20232610940731 \nu^{4} + 6809297955588 \nu^{3} - 557261024458 \nu^{2} - 472073315000 \nu + 982823726512$$$$)/ 732246302876$$ $$\beta_{7}$$ $$=$$ $$($$$$-61621117035 \nu^{13} - 58787127771 \nu^{12} - 866985826 \nu^{11} + 134629395734 \nu^{10} - 2271165378531 \nu^{9} - 1671410135475 \nu^{8} + 385087447514 \nu^{7} + 1922577441112 \nu^{6} - 16317016456265 \nu^{5} - 10243935308455 \nu^{4} + 4086990094642 \nu^{3} + 3333018633676 \nu^{2} - 65148969984 \nu - 1737381486820$$$$)/ 732246302876$$ $$\beta_{8}$$ $$=$$ $$($$$$-66772094683 \nu^{13} - 15046471791 \nu^{12} - 4521994539 \nu^{11} + 135188822801 \nu^{10} - 2575176731663 \nu^{9} + 89277697645 \nu^{8} - 162746449503 \nu^{7} + 1819608044077 \nu^{6} - 19510883885937 \nu^{5} + 3382407351091 \nu^{4} - 1242235247009 \nu^{3} + 2137159584915 \nu^{2} - 4712322126264 \nu + 1230815216570$$$$)/ 366123151438$$ $$\beta_{9}$$ $$=$$ $$($$$$134719081377 \nu^{13} + 166717918 \nu^{12} - 18595581648 \nu^{11} - 277803595150 \nu^{10} + 5252053841299 \nu^{9} - 1301055484020 \nu^{8} - 460738908234 \nu^{7} - 3645608595402 \nu^{6} + 39984967995781 \nu^{5} - 14884213113334 \nu^{4} - 2574515483268 \nu^{3} - 3062961226104 \nu^{2} + 8601416001444 \nu - 2143125728988$$$$)/ 732246302876$$ $$\beta_{10}$$ $$=$$ $$($$$$-159011401463 \nu^{13} - 74011296796 \nu^{12} - 17491051676 \nu^{11} + 317230204266 \nu^{10} - 6056933221517 \nu^{9} - 1259279988702 \nu^{8} - 257699180354 \nu^{7} + 4139003457850 \nu^{6} - 45390689597815 \nu^{5} - 2875620487424 \nu^{4} - 394657434132 \nu^{3} + 3312052402844 \nu^{2} - 8398081464788 \nu + 2488082785940$$$$)/ 732246302876$$ $$\beta_{11}$$ $$=$$ $$($$$$161362783524 \nu^{13} + 45302004725 \nu^{12} - 17266584944 \nu^{11} - 359941008954 \nu^{10} + 6182243364690 \nu^{9} + 182163846443 \nu^{8} - 731148740058 \nu^{7} - 5300077616756 \nu^{6} + 46296345417282 \nu^{5} - 4899429442643 \nu^{4} - 5587980665236 \nu^{3} - 9490182700870 \nu^{2} + 7131200437916 \nu - 2076944162144$$$$)/ 732246302876$$ $$\beta_{12}$$ $$=$$ $$($$$$189780871574 \nu^{13} + 115971282613 \nu^{12} + 47907054048 \nu^{11} - 369867084278 \nu^{10} + 7164617072456 \nu^{9} + 2529556762223 \nu^{8} + 1049309031078 \nu^{7} - 4799923558176 \nu^{6} + 53149974193276 \nu^{5} + 11212902114921 \nu^{4} + 4612113196920 \nu^{3} - 3654530344182 \nu^{2} + 8104580101084 \nu - 88794449364$$$$)/ 732246302876$$ $$\beta_{13}$$ $$=$$ $$($$$$-225258249983 \nu^{13} - 97924060431 \nu^{12} - 6621708034 \nu^{11} + 462876963284 \nu^{10} - 8578405340227 \nu^{9} - 1551259260459 \nu^{8} + 226661713146 \nu^{7} + 6202649999094 \nu^{6} - 64065095492413 \nu^{5} - 2758530555507 \nu^{4} + 3205954983134 \nu^{3} + 6729521670402 \nu^{2} - 10569330546208 \nu + 1060356158372$$$$)/ 732246302876$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{13} - \beta_{8} - \beta_{7} + \beta_{4} - \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{10} - \beta_{9} - \beta_{5} + 6 \beta_{4} - \beta_{3}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{13} - \beta_{10} + \beta_{9} - 3 \beta_{8} - 5 \beta_{7} + \beta_{5} + 3 \beta_{4} - \beta_{3} + 3 \beta_{2} + 5 \beta_{1} + 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{12} + 5 \beta_{10} + 7 \beta_{9} + 2 \beta_{6} - 5 \beta_{5} + 5 \beta_{3} - 26$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-26 \beta_{13} + 2 \beta_{12} + 9 \beta_{10} - 9 \beta_{9} + 12 \beta_{8} + 24 \beta_{7} + 9 \beta_{5} - 10 \beta_{4} - 9 \beta_{3} + 12 \beta_{2} + 24 \beta_{1} + 14$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$10 \beta_{13} + 9 \beta_{11} - 13 \beta_{10} + 13 \beta_{9} - \beta_{8} - \beta_{7} + 12 \beta_{5} - 61 \beta_{4} + 21 \beta_{3}$$ $$\nu^{7}$$ $$=$$ $$($$$$-139 \beta_{13} - 20 \beta_{12} - 2 \beta_{11} + 44 \beta_{10} - 40 \beta_{9} + 53 \beta_{8} + 119 \beta_{7} + 2 \beta_{6} - 44 \beta_{5} - 29 \beta_{4} + 40 \beta_{3} - 53 \beta_{2} - 119 \beta_{1} - 90$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$124 \beta_{12} - 117 \beta_{10} - 241 \beta_{9} - 120 \beta_{6} + 145 \beta_{5} - 145 \beta_{3} + 22 \beta_{2} + 30 \beta_{1} + 630$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$757 \beta_{13} - 148 \beta_{12} + 28 \beta_{11} - 361 \beta_{10} + 419 \beta_{9} - 251 \beta_{8} - 609 \beta_{7} + 28 \beta_{6} - 361 \beta_{5} + 45 \beta_{4} + 419 \beta_{3} - 251 \beta_{2} - 609 \beta_{1} - 564$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-1072 \beta_{13} - 722 \beta_{11} + 843 \beta_{10} - 843 \beta_{9} + 178 \beta_{8} + 294 \beta_{7} - 581 \beta_{5} + 3056 \beta_{4} - 1359 \beta_{3}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$4180 \beta_{13} + 984 \beta_{12} + 262 \beta_{11} - 1653 \beta_{10} + 1097 \beta_{9} - 1256 \beta_{8} - 3196 \beta_{7} - 262 \beta_{6} + 1653 \beta_{5} - 288 \beta_{4} - 1097 \beta_{3} + 1256 \beta_{2} + 3196 \beta_{1} + 3484$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$-2343 \beta_{12} + 1467 \beta_{10} + 3810 \beta_{9} + 2081 \beta_{6} - 2499 \beta_{5} + 2499 \beta_{3} - 641 \beta_{2} - 1197 \beta_{1} - 9188$$ $$\nu^{13}$$ $$=$$ $$($$$$-23331 \beta_{13} + 6226 \beta_{12} - 2064 \beta_{11} + 11782 \beta_{10} - 16240 \beta_{9} + 6551 \beta_{8} + 17105 \beta_{7} - 2064 \beta_{6} + 11782 \beta_{5} + 4205 \beta_{4} - 16240 \beta_{3} + 6551 \beta_{2} + 17105 \beta_{1} + 21310$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$$.

 $$n$$ $$737$$ $$1151$$ $$1201$$ $$1381$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
369.1
 −0.503254 − 0.503254i 0.285770 − 0.285770i −1.27121 − 1.27121i −1.71470 − 1.71470i 0.416087 − 0.416087i 1.55369 + 1.55369i 1.23362 + 1.23362i 1.23362 − 1.23362i 1.55369 − 1.55369i 0.416087 + 0.416087i −1.71470 + 1.71470i −1.27121 + 1.27121i 0.285770 + 0.285770i −0.503254 + 0.503254i
0 2.98707i 0 0.274289 2.21918i 0 0.980560i 0 −5.92257 0
369.2 0 2.49931i 0 2.11714 + 0.719533i 0 2.92777i 0 −3.24657 0
369.3 0 1.78665i 0 1.46089 + 1.69287i 0 1.75578i 0 −0.192116 0
369.4 0 1.58319i 0 −2.09277 + 0.787606i 0 2.84620i 0 0.493499 0
369.5 0 1.40334i 0 0.466981 + 2.18676i 0 1.57117i 0 1.03063 0
369.6 0 0.356372i 0 0.376266 + 2.20418i 0 2.46376i 0 2.87300 0
369.7 0 0.189375i 0 −1.60280 1.55918i 0 1.65661i 0 2.96414 0
369.8 0 0.189375i 0 −1.60280 + 1.55918i 0 1.65661i 0 2.96414 0
369.9 0 0.356372i 0 0.376266 2.20418i 0 2.46376i 0 2.87300 0
369.10 0 1.40334i 0 0.466981 2.18676i 0 1.57117i 0 1.03063 0
369.11 0 1.58319i 0 −2.09277 0.787606i 0 2.84620i 0 0.493499 0
369.12 0 1.78665i 0 1.46089 1.69287i 0 1.75578i 0 −0.192116 0
369.13 0 2.49931i 0 2.11714 0.719533i 0 2.92777i 0 −3.24657 0
369.14 0 2.98707i 0 0.274289 + 2.21918i 0 0.980560i 0 −5.92257 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 369.14 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.e.g 14
4.b odd 2 1 920.2.e.b 14
5.b even 2 1 inner 1840.2.e.g 14
5.c odd 4 1 9200.2.a.cz 7
5.c odd 4 1 9200.2.a.dc 7
20.d odd 2 1 920.2.e.b 14
20.e even 4 1 4600.2.a.bh 7
20.e even 4 1 4600.2.a.bi 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.e.b 14 4.b odd 2 1
920.2.e.b 14 20.d odd 2 1
1840.2.e.g 14 1.a even 1 1 trivial
1840.2.e.g 14 5.b even 2 1 inner
4600.2.a.bh 7 20.e even 4 1
4600.2.a.bi 7 20.e even 4 1
9200.2.a.cz 7 5.c odd 4 1
9200.2.a.dc 7 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1840, [\chi])$$:

 $$T_{3}^{14} + 23 T_{3}^{12} + 195 T_{3}^{10} + 766 T_{3}^{8} + 1431 T_{3}^{6} + 1095 T_{3}^{4} + 149 T_{3}^{2} + 4$$ $$T_{7}^{14} + 32 T_{7}^{12} + 412 T_{7}^{10} + 2745 T_{7}^{8} + 10156 T_{7}^{6} + 20760 T_{7}^{4} + 21488 T_{7}^{2} + 8464$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14}$$
$3$ $$4 + 149 T^{2} + 1095 T^{4} + 1431 T^{6} + 766 T^{8} + 195 T^{10} + 23 T^{12} + T^{14}$$
$5$ $$78125 - 31250 T + 28125 T^{2} - 2500 T^{3} + 875 T^{4} + 50 T^{5} + 635 T^{6} - 344 T^{7} + 127 T^{8} + 2 T^{9} + 7 T^{10} - 4 T^{11} + 9 T^{12} - 2 T^{13} + T^{14}$$
$7$ $$8464 + 21488 T^{2} + 20760 T^{4} + 10156 T^{6} + 2745 T^{8} + 412 T^{10} + 32 T^{12} + T^{14}$$
$11$ $$( -128 - 1780 T - 1224 T^{2} + 248 T^{3} + 198 T^{4} - 27 T^{5} - 7 T^{6} + T^{7} )^{2}$$
$13$ $$4 + 26461 T^{2} + 60131 T^{4} + 45695 T^{6} + 14030 T^{8} + 1715 T^{10} + 75 T^{12} + T^{14}$$
$17$ $$2704 + 143056 T^{2} + 1298156 T^{4} + 493848 T^{6} + 69625 T^{8} + 4368 T^{10} + 116 T^{12} + T^{14}$$
$19$ $$( 1936 - 3828 T + 1104 T^{2} + 1030 T^{3} - 238 T^{4} - 53 T^{5} + 7 T^{6} + T^{7} )^{2}$$
$23$ $$( 1 + T^{2} )^{7}$$
$29$ $$( -9244 + 38290 T - 15243 T^{2} - 971 T^{3} + 930 T^{4} - 56 T^{5} - 11 T^{6} + T^{7} )^{2}$$
$31$ $$( 225251 + 21800 T - 38532 T^{2} + 2165 T^{3} + 1243 T^{4} - 110 T^{5} - 10 T^{6} + T^{7} )^{2}$$
$37$ $$1834580224 + 1118268096 T^{2} + 216564752 T^{4} + 18884160 T^{6} + 840264 T^{8} + 19560 T^{10} + 225 T^{12} + T^{14}$$
$41$ $$( 110153 + 76850 T + 8034 T^{2} - 4287 T^{3} - 955 T^{4} + 12 T^{5} + 16 T^{6} + T^{7} )^{2}$$
$43$ $$4096 + 134400 T^{2} + 162752 T^{4} + 77776 T^{6} + 18064 T^{8} + 2044 T^{10} + 96 T^{12} + T^{14}$$
$47$ $$33085504 + 933046080 T^{2} + 1893275152 T^{4} + 142748497 T^{6} + 4182872 T^{8} + 58638 T^{10} + 392 T^{12} + T^{14}$$
$53$ $$24445947904 + 11896952832 T^{2} + 1786705664 T^{4} + 105707008 T^{6} + 3057008 T^{8} + 46072 T^{10} + 345 T^{12} + T^{14}$$
$59$ $$( 486592 - 430528 T + 70592 T^{2} + 16480 T^{3} - 2288 T^{4} - 252 T^{5} + 11 T^{6} + T^{7} )^{2}$$
$61$ $$( 2669336 - 157108 T - 209468 T^{2} + 25754 T^{3} + 2048 T^{4} - 315 T^{5} - 5 T^{6} + T^{7} )^{2}$$
$67$ $$214213460224 + 79965683136 T^{2} + 9534585104 T^{4} + 455222336 T^{6} + 10321640 T^{8} + 114456 T^{10} + 569 T^{12} + T^{14}$$
$71$ $$( -632317 + 554654 T - 127100 T^{2} - 211 T^{3} + 2711 T^{4} - 156 T^{5} - 14 T^{6} + T^{7} )^{2}$$
$73$ $$256 + 2051088 T^{2} + 5173624 T^{4} + 2585073 T^{6} + 246936 T^{8} + 9438 T^{10} + 160 T^{12} + T^{14}$$
$79$ $$( 473312 + 422784 T + 29312 T^{2} - 22724 T^{3} - 2876 T^{4} + 166 T^{5} + 32 T^{6} + T^{7} )^{2}$$
$83$ $$143616256 + 4222250432 T^{2} + 1182090704 T^{4} + 107153968 T^{6} + 3863004 T^{8} + 62864 T^{10} + 445 T^{12} + T^{14}$$
$89$ $$( -311456 + 288384 T - 48656 T^{2} - 10836 T^{3} + 2260 T^{4} + 46 T^{5} - 24 T^{6} + T^{7} )^{2}$$
$97$ $$16312357788736 + 2646852894608 T^{2} + 131116091648 T^{4} + 3027822056 T^{6} + 36892664 T^{8} + 241097 T^{10} + 787 T^{12} + T^{14}$$