# Properties

 Label 5.14.a.a Level $5$ Weight $14$ Character orbit 5.a Self dual yes Analytic conductor $5.362$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$14$$ Character orbit: $$[\chi]$$ $$=$$ 5.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.36154644760$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{499})$$ Defining polynomial: $$x^{2} - 499$$ x^2 - 499 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4\sqrt{499}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 40) q^{2} + ( - 12 \beta + 390) q^{3} + ( - 80 \beta + 1392) q^{4} - 15625 q^{5} + (870 \beta - 111408) q^{6} + (1636 \beta - 308150) q^{7} + ( - 3600 \beta - 366720) q^{8} + ( - 9360 \beta - 292527) q^{9}+O(q^{10})$$ q + (b - 40) * q^2 + (-12*b + 390) * q^3 + (-80*b + 1392) * q^4 - 15625 * q^5 + (870*b - 111408) * q^6 + (1636*b - 308150) * q^7 + (-3600*b - 366720) * q^8 + (-9360*b - 292527) * q^9 $$q + (\beta - 40) q^{2} + ( - 12 \beta + 390) q^{3} + ( - 80 \beta + 1392) q^{4} - 15625 q^{5} + (870 \beta - 111408) q^{6} + (1636 \beta - 308150) q^{7} + ( - 3600 \beta - 366720) q^{8} + ( - 9360 \beta - 292527) q^{9} + ( - 15625 \beta + 625000) q^{10} + (79000 \beta - 1233568) q^{11} + ( - 47904 \beta + 8207520) q^{12} + (133328 \beta + 3258930) q^{13} + ( - 373590 \beta + 25387824) q^{14} + (187500 \beta - 6093750) q^{15} + (432640 \beta - 25476864) q^{16} + ( - 1020944 \beta + 316730) q^{17} + (81873 \beta - 63029160) q^{18} + (440480 \beta - 187031700) q^{19} + (1250000 \beta - 21750000) q^{20} + (4335840 \beta - 276920388) q^{21} + ( - 4393568 \beta + 680078720) q^{22} + ( - 9519732 \beta + 310991070) q^{23} + (2996640 \beta + 201888000) q^{24} + 244140625 q^{25} + ( - 2074190 \beta + 934133552) q^{26} + (18991800 \beta + 160891380) q^{27} + (26929312 \beta - 1473890720) q^{28} + ( - 18112480 \beta - 3897567050) q^{29} + ( - 13593750 \beta + 1740750000) q^{30} + ( - 59933000 \beta - 2478909908) q^{31} + ( - 13291264 \beta + 7477442560) q^{32} + (45612816 \beta - 8049923520) q^{33} + (41154490 \beta - 8163886096) q^{34} + ( - 25562500 \beta + 4814843750) q^{35} + (10373040 \beta + 5571221616) q^{36} + (199286496 \beta + 10916535890) q^{37} + ( - 204650900 \beta + 10998060320) q^{38} + (12890760 \beta - 11502906324) q^{39} + (56250000 \beta + 5730000000) q^{40} + (9362000 \beta + 2782906822) q^{41} + ( - 450353988 \beta + 45694162080) q^{42} + ( - 44127052 \beta - 26255438850) q^{43} + (208653440 \beta - 52176006656) q^{44} + (146250000 \beta + 4570734375) q^{45} + (691780350 \beta - 88445183088) q^{46} + ( - 310471484 \beta + 46427943170) q^{47} + (474451968 \beta - 51386350080) q^{48} + ( - 1008266800 \beta + 19436556157) q^{49} + (244140625 \beta - 9765625000) q^{50} + ( - 401968920 \beta + 97938127452) q^{51} + ( - 75121824 \beta - 80622829600) q^{52} + (823152688 \beta + 133124438090) q^{53} + ( - 598780620 \beta + 145194876000) q^{54} + ( - 1234375000 \beta + 19274500000) q^{55} + (509386080 \beta + 65982201600) q^{56} + (2416167600 \beta - 115143870840) q^{57} + ( - 3173067850 \beta + 11292641680) q^{58} + (2843365840 \beta + 126750803900) q^{59} + (748500000 \beta - 128242500000) q^{60} + ( - 2095600000 \beta - 188729619918) q^{61} + ( - 81589908 \beta - 379348675680) q^{62} + (2405709828 \beta - 32116477590) q^{63} + (4464906240 \beta - 196508684288) q^{64} + ( - 2083250000 \beta - 50920781250) q^{65} + ( - 9874436160 \beta + 686169663744) q^{66} + ( - 36295844 \beta - 1187891156870) q^{67} + ( - 1446492448 \beta + 652538239840) q^{68} + ( - 7444588320 \beta + 1033353000756) q^{69} + (5837343750 \beta - 396684750000) q^{70} + (10496675000 \beta - 278095051388) q^{71} + (4485596400 \beta + 376304365440) q^{72} + (7446633968 \beta - 278232691230) q^{73} + (2945076050 \beta + 1154441948464) q^{74} + ( - 2929687500 \beta + 95214843750) q^{75} + (15575684160 \beta - 541691512000) q^{76} + ( - 26361967248 \beta + 1412008075200) q^{77} + ( - 12018536724 \beta + 563036080800) q^{78} + ( - 28106767280 \beta - 1204115283800) q^{79} + ( - 6760000000 \beta + 398076000000) q^{80} + (20398968720 \beta - 1290436211979) q^{81} + (2408426822 \beta - 36570064880) q^{82} + (12892649508 \beta - 1949147801490) q^{83} + (28189120320 \beta - 3154860904896) q^{84} + (15952250000 \beta - 4948906250) q^{85} + ( - 24490356770 \beta + 697907170832) q^{86} + (39706937400 \beta + 215269334340) q^{87} + ( - 24530035200 \beta - 1818275543040) q^{88} + ( - 3668003040 \beta + 53423837850) q^{89} + ( - 1279265625 \beta + 984830625000) q^{90} + ( - 35753413720 \beta + 737267590772) q^{91} + ( - 38130752544 \beta + 6513342792480) q^{92} + (6373048896 \beta + 4775285999880) q^{93} + (58846802530 \beta - 4335922055056) q^{94} + ( - 6882500000 \beta + 2922370312500) q^{95} + ( - 94912903680 \beta + 4189612019712) q^{96} + (28322321616 \beta + 9372204725570) q^{97} + (59767228157 \beta - 8827464377480) q^{98} + ( - 11563436520 \beta - 5542837013664) q^{99}+O(q^{100})$$ q + (b - 40) * q^2 + (-12*b + 390) * q^3 + (-80*b + 1392) * q^4 - 15625 * q^5 + (870*b - 111408) * q^6 + (1636*b - 308150) * q^7 + (-3600*b - 366720) * q^8 + (-9360*b - 292527) * q^9 + (-15625*b + 625000) * q^10 + (79000*b - 1233568) * q^11 + (-47904*b + 8207520) * q^12 + (133328*b + 3258930) * q^13 + (-373590*b + 25387824) * q^14 + (187500*b - 6093750) * q^15 + (432640*b - 25476864) * q^16 + (-1020944*b + 316730) * q^17 + (81873*b - 63029160) * q^18 + (440480*b - 187031700) * q^19 + (1250000*b - 21750000) * q^20 + (4335840*b - 276920388) * q^21 + (-4393568*b + 680078720) * q^22 + (-9519732*b + 310991070) * q^23 + (2996640*b + 201888000) * q^24 + 244140625 * q^25 + (-2074190*b + 934133552) * q^26 + (18991800*b + 160891380) * q^27 + (26929312*b - 1473890720) * q^28 + (-18112480*b - 3897567050) * q^29 + (-13593750*b + 1740750000) * q^30 + (-59933000*b - 2478909908) * q^31 + (-13291264*b + 7477442560) * q^32 + (45612816*b - 8049923520) * q^33 + (41154490*b - 8163886096) * q^34 + (-25562500*b + 4814843750) * q^35 + (10373040*b + 5571221616) * q^36 + (199286496*b + 10916535890) * q^37 + (-204650900*b + 10998060320) * q^38 + (12890760*b - 11502906324) * q^39 + (56250000*b + 5730000000) * q^40 + (9362000*b + 2782906822) * q^41 + (-450353988*b + 45694162080) * q^42 + (-44127052*b - 26255438850) * q^43 + (208653440*b - 52176006656) * q^44 + (146250000*b + 4570734375) * q^45 + (691780350*b - 88445183088) * q^46 + (-310471484*b + 46427943170) * q^47 + (474451968*b - 51386350080) * q^48 + (-1008266800*b + 19436556157) * q^49 + (244140625*b - 9765625000) * q^50 + (-401968920*b + 97938127452) * q^51 + (-75121824*b - 80622829600) * q^52 + (823152688*b + 133124438090) * q^53 + (-598780620*b + 145194876000) * q^54 + (-1234375000*b + 19274500000) * q^55 + (509386080*b + 65982201600) * q^56 + (2416167600*b - 115143870840) * q^57 + (-3173067850*b + 11292641680) * q^58 + (2843365840*b + 126750803900) * q^59 + (748500000*b - 128242500000) * q^60 + (-2095600000*b - 188729619918) * q^61 + (-81589908*b - 379348675680) * q^62 + (2405709828*b - 32116477590) * q^63 + (4464906240*b - 196508684288) * q^64 + (-2083250000*b - 50920781250) * q^65 + (-9874436160*b + 686169663744) * q^66 + (-36295844*b - 1187891156870) * q^67 + (-1446492448*b + 652538239840) * q^68 + (-7444588320*b + 1033353000756) * q^69 + (5837343750*b - 396684750000) * q^70 + (10496675000*b - 278095051388) * q^71 + (4485596400*b + 376304365440) * q^72 + (7446633968*b - 278232691230) * q^73 + (2945076050*b + 1154441948464) * q^74 + (-2929687500*b + 95214843750) * q^75 + (15575684160*b - 541691512000) * q^76 + (-26361967248*b + 1412008075200) * q^77 + (-12018536724*b + 563036080800) * q^78 + (-28106767280*b - 1204115283800) * q^79 + (-6760000000*b + 398076000000) * q^80 + (20398968720*b - 1290436211979) * q^81 + (2408426822*b - 36570064880) * q^82 + (12892649508*b - 1949147801490) * q^83 + (28189120320*b - 3154860904896) * q^84 + (15952250000*b - 4948906250) * q^85 + (-24490356770*b + 697907170832) * q^86 + (39706937400*b + 215269334340) * q^87 + (-24530035200*b - 1818275543040) * q^88 + (-3668003040*b + 53423837850) * q^89 + (-1279265625*b + 984830625000) * q^90 + (-35753413720*b + 737267590772) * q^91 + (-38130752544*b + 6513342792480) * q^92 + (6373048896*b + 4775285999880) * q^93 + (58846802530*b - 4335922055056) * q^94 + (-6882500000*b + 2922370312500) * q^95 + (-94912903680*b + 4189612019712) * q^96 + (28322321616*b + 9372204725570) * q^97 + (59767228157*b - 8827464377480) * q^98 + (-11563436520*b - 5542837013664) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 80 q^{2} + 780 q^{3} + 2784 q^{4} - 31250 q^{5} - 222816 q^{6} - 616300 q^{7} - 733440 q^{8} - 585054 q^{9}+O(q^{10})$$ 2 * q - 80 * q^2 + 780 * q^3 + 2784 * q^4 - 31250 * q^5 - 222816 * q^6 - 616300 * q^7 - 733440 * q^8 - 585054 * q^9 $$2 q - 80 q^{2} + 780 q^{3} + 2784 q^{4} - 31250 q^{5} - 222816 q^{6} - 616300 q^{7} - 733440 q^{8} - 585054 q^{9} + 1250000 q^{10} - 2467136 q^{11} + 16415040 q^{12} + 6517860 q^{13} + 50775648 q^{14} - 12187500 q^{15} - 50953728 q^{16} + 633460 q^{17} - 126058320 q^{18} - 374063400 q^{19} - 43500000 q^{20} - 553840776 q^{21} + 1360157440 q^{22} + 621982140 q^{23} + 403776000 q^{24} + 488281250 q^{25} + 1868267104 q^{26} + 321782760 q^{27} - 2947781440 q^{28} - 7795134100 q^{29} + 3481500000 q^{30} - 4957819816 q^{31} + 14954885120 q^{32} - 16099847040 q^{33} - 16327772192 q^{34} + 9629687500 q^{35} + 11142443232 q^{36} + 21833071780 q^{37} + 21996120640 q^{38} - 23005812648 q^{39} + 11460000000 q^{40} + 5565813644 q^{41} + 91388324160 q^{42} - 52510877700 q^{43} - 104352013312 q^{44} + 9141468750 q^{45} - 176890366176 q^{46} + 92855886340 q^{47} - 102772700160 q^{48} + 38873112314 q^{49} - 19531250000 q^{50} + 195876254904 q^{51} - 161245659200 q^{52} + 266248876180 q^{53} + 290389752000 q^{54} + 38549000000 q^{55} + 131964403200 q^{56} - 230287741680 q^{57} + 22585283360 q^{58} + 253501607800 q^{59} - 256485000000 q^{60} - 377459239836 q^{61} - 758697351360 q^{62} - 64232955180 q^{63} - 393017368576 q^{64} - 101841562500 q^{65} + 1372339327488 q^{66} - 2375782313740 q^{67} + 1305076479680 q^{68} + 2066706001512 q^{69} - 793369500000 q^{70} - 556190102776 q^{71} + 752608730880 q^{72} - 556465382460 q^{73} + 2308883896928 q^{74} + 190429687500 q^{75} - 1083383024000 q^{76} + 2824016150400 q^{77} + 1126072161600 q^{78} - 2408230567600 q^{79} + 796152000000 q^{80} - 2580872423958 q^{81} - 73140129760 q^{82} - 3898295602980 q^{83} - 6309721809792 q^{84} - 9897812500 q^{85} + 1395814341664 q^{86} + 430538668680 q^{87} - 3636551086080 q^{88} + 106847675700 q^{89} + 1969661250000 q^{90} + 1474535181544 q^{91} + 13026685584960 q^{92} + 9550571999760 q^{93} - 8671844110112 q^{94} + 5844740625000 q^{95} + 8379224039424 q^{96} + 18744409451140 q^{97} - 17654928754960 q^{98} - 11085674027328 q^{99}+O(q^{100})$$ 2 * q - 80 * q^2 + 780 * q^3 + 2784 * q^4 - 31250 * q^5 - 222816 * q^6 - 616300 * q^7 - 733440 * q^8 - 585054 * q^9 + 1250000 * q^10 - 2467136 * q^11 + 16415040 * q^12 + 6517860 * q^13 + 50775648 * q^14 - 12187500 * q^15 - 50953728 * q^16 + 633460 * q^17 - 126058320 * q^18 - 374063400 * q^19 - 43500000 * q^20 - 553840776 * q^21 + 1360157440 * q^22 + 621982140 * q^23 + 403776000 * q^24 + 488281250 * q^25 + 1868267104 * q^26 + 321782760 * q^27 - 2947781440 * q^28 - 7795134100 * q^29 + 3481500000 * q^30 - 4957819816 * q^31 + 14954885120 * q^32 - 16099847040 * q^33 - 16327772192 * q^34 + 9629687500 * q^35 + 11142443232 * q^36 + 21833071780 * q^37 + 21996120640 * q^38 - 23005812648 * q^39 + 11460000000 * q^40 + 5565813644 * q^41 + 91388324160 * q^42 - 52510877700 * q^43 - 104352013312 * q^44 + 9141468750 * q^45 - 176890366176 * q^46 + 92855886340 * q^47 - 102772700160 * q^48 + 38873112314 * q^49 - 19531250000 * q^50 + 195876254904 * q^51 - 161245659200 * q^52 + 266248876180 * q^53 + 290389752000 * q^54 + 38549000000 * q^55 + 131964403200 * q^56 - 230287741680 * q^57 + 22585283360 * q^58 + 253501607800 * q^59 - 256485000000 * q^60 - 377459239836 * q^61 - 758697351360 * q^62 - 64232955180 * q^63 - 393017368576 * q^64 - 101841562500 * q^65 + 1372339327488 * q^66 - 2375782313740 * q^67 + 1305076479680 * q^68 + 2066706001512 * q^69 - 793369500000 * q^70 - 556190102776 * q^71 + 752608730880 * q^72 - 556465382460 * q^73 + 2308883896928 * q^74 + 190429687500 * q^75 - 1083383024000 * q^76 + 2824016150400 * q^77 + 1126072161600 * q^78 - 2408230567600 * q^79 + 796152000000 * q^80 - 2580872423958 * q^81 - 73140129760 * q^82 - 3898295602980 * q^83 - 6309721809792 * q^84 - 9897812500 * q^85 + 1395814341664 * q^86 + 430538668680 * q^87 - 3636551086080 * q^88 + 106847675700 * q^89 + 1969661250000 * q^90 + 1474535181544 * q^91 + 13026685584960 * q^92 + 9550571999760 * q^93 - 8671844110112 * q^94 + 5844740625000 * q^95 + 8379224039424 * q^96 + 18744409451140 * q^97 - 17654928754960 * q^98 - 11085674027328 * q^99

## 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
 −22.3383 22.3383
−129.353 1462.24 8540.26 −15625.0 −189145. −454332. −45048.4 543819. 2.02114e6
1.2 49.3532 −682.239 −5756.26 −15625.0 −33670.7 −161968. −688392. −1.12887e6 −771144.
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$

## 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 5.14.a.a 2
3.b odd 2 1 45.14.a.d 2
4.b odd 2 1 80.14.a.d 2
5.b even 2 1 25.14.a.a 2
5.c odd 4 2 25.14.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.14.a.a 2 1.a even 1 1 trivial
25.14.a.a 2 5.b even 2 1
25.14.b.a 4 5.c odd 4 2
45.14.a.d 2 3.b odd 2 1
80.14.a.d 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 80T_{2} - 6384$$ acting on $$S_{14}^{\mathrm{new}}(\Gamma_0(5))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 80T - 6384$$
$3$ $$T^{2} - 780T - 997596$$
$5$ $$(T + 15625)^{2}$$
$7$ $$T^{2} + 616300 T + 73587278436$$
$11$ $$T^{2} + 2467136 T - 48306453989376$$
$13$ $$T^{2} + \cdots - 131305798237756$$
$17$ $$T^{2} - 633460 T - 83\!\cdots\!24$$
$19$ $$T^{2} + 374063400 T + 33\!\cdots\!00$$
$23$ $$T^{2} - 621982140 T - 62\!\cdots\!16$$
$29$ $$T^{2} + 7795134100 T + 12\!\cdots\!00$$
$31$ $$T^{2} + 4957819816 T - 22\!\cdots\!36$$
$37$ $$T^{2} - 21833071780 T - 19\!\cdots\!44$$
$41$ $$T^{2} - 5565813644 T + 70\!\cdots\!84$$
$43$ $$T^{2} + 52510877700 T + 67\!\cdots\!64$$
$47$ $$T^{2} - 92855886340 T + 13\!\cdots\!96$$
$53$ $$T^{2} - 266248876180 T + 12\!\cdots\!04$$
$59$ $$T^{2} - 253501607800 T - 48\!\cdots\!00$$
$61$ $$T^{2} + 377459239836 T + 55\!\cdots\!24$$
$67$ $$T^{2} + 2375782313740 T + 14\!\cdots\!76$$
$71$ $$T^{2} + 556190102776 T - 80\!\cdots\!56$$
$73$ $$T^{2} + 556465382460 T - 36\!\cdots\!16$$
$79$ $$T^{2} + 2408230567600 T - 48\!\cdots\!00$$
$83$ $$T^{2} + 3898295602980 T + 24\!\cdots\!24$$
$89$ $$T^{2} - 106847675700 T - 10\!\cdots\!00$$
$97$ $$T^{2} - 18744409451140 T + 81\!\cdots\!96$$