# Properties

 Label 784.6.a.q.1.2 Level $784$ Weight $6$ Character 784.1 Self dual yes Analytic conductor $125.741$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,Mod(1,784)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 784.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$125.740914733$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{193})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 48$$ x^2 - x - 48 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 56) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-6.44622$$ of defining polynomial Character $$\chi$$ $$=$$ 784.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+6.89244 q^{3} +48.4622 q^{5} -195.494 q^{9} +O(q^{10})$$ $$q+6.89244 q^{3} +48.4622 q^{5} -195.494 q^{9} +163.506 q^{11} +120.828 q^{13} +334.023 q^{15} -78.1920 q^{17} -2265.00 q^{19} +2451.95 q^{23} -776.413 q^{25} -3022.30 q^{27} +6985.27 q^{29} +2794.03 q^{31} +1126.95 q^{33} +9459.44 q^{37} +832.803 q^{39} -10088.5 q^{41} +6934.29 q^{43} -9474.08 q^{45} +1165.76 q^{47} -538.934 q^{51} +8562.42 q^{53} +7923.85 q^{55} -15611.4 q^{57} +6220.26 q^{59} +41928.9 q^{61} +5855.62 q^{65} -1812.09 q^{67} +16900.0 q^{69} +56823.3 q^{71} +44299.5 q^{73} -5351.38 q^{75} -34912.4 q^{79} +26674.1 q^{81} -39652.9 q^{83} -3789.36 q^{85} +48145.6 q^{87} +126299. q^{89} +19257.7 q^{93} -109767. q^{95} +145513. q^{97} -31964.4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 14 q^{3} - 42 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q - 14 * q^3 - 42 * q^5 - 2 * q^9 $$2 q - 14 q^{3} - 42 q^{5} - 2 q^{9} + 716 q^{11} + 714 q^{13} + 2224 q^{15} + 1344 q^{17} - 1946 q^{19} + 1792 q^{23} + 4282 q^{25} - 1988 q^{27} - 1200 q^{29} - 6804 q^{31} - 10416 q^{33} + 14640 q^{37} - 11560 q^{39} - 7896 q^{41} - 524 q^{43} - 26978 q^{45} - 18396 q^{47} - 30252 q^{51} + 45132 q^{53} - 42056 q^{55} - 22276 q^{57} + 22582 q^{59} + 52822 q^{61} - 47804 q^{65} - 9848 q^{67} + 30688 q^{69} + 840 q^{71} + 122052 q^{73} - 111034 q^{75} - 31704 q^{79} - 41954 q^{81} + 36974 q^{83} - 132444 q^{85} + 219156 q^{87} + 210588 q^{89} + 219784 q^{93} - 138624 q^{95} + 44240 q^{97} + 74940 q^{99}+O(q^{100})$$ 2 * q - 14 * q^3 - 42 * q^5 - 2 * q^9 + 716 * q^11 + 714 * q^13 + 2224 * q^15 + 1344 * q^17 - 1946 * q^19 + 1792 * q^23 + 4282 * q^25 - 1988 * q^27 - 1200 * q^29 - 6804 * q^31 - 10416 * q^33 + 14640 * q^37 - 11560 * q^39 - 7896 * q^41 - 524 * q^43 - 26978 * q^45 - 18396 * q^47 - 30252 * q^51 + 45132 * q^53 - 42056 * q^55 - 22276 * q^57 + 22582 * q^59 + 52822 * q^61 - 47804 * q^65 - 9848 * q^67 + 30688 * q^69 + 840 * q^71 + 122052 * q^73 - 111034 * q^75 - 31704 * q^79 - 41954 * q^81 + 36974 * q^83 - 132444 * q^85 + 219156 * q^87 + 210588 * q^89 + 219784 * q^93 - 138624 * q^95 + 44240 * q^97 + 74940 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 6.89244 0.442150 0.221075 0.975257i $$-0.429043\pi$$
0.221075 + 0.975257i $$0.429043\pi$$
$$4$$ 0 0
$$5$$ 48.4622 0.866919 0.433459 0.901173i $$-0.357293\pi$$
0.433459 + 0.901173i $$0.357293\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −195.494 −0.804503
$$10$$ 0 0
$$11$$ 163.506 0.407429 0.203714 0.979030i $$-0.434699\pi$$
0.203714 + 0.979030i $$0.434699\pi$$
$$12$$ 0 0
$$13$$ 120.828 0.198295 0.0991473 0.995073i $$-0.468389\pi$$
0.0991473 + 0.995073i $$0.468389\pi$$
$$14$$ 0 0
$$15$$ 334.023 0.383308
$$16$$ 0 0
$$17$$ −78.1920 −0.0656206 −0.0328103 0.999462i $$-0.510446\pi$$
−0.0328103 + 0.999462i $$0.510446\pi$$
$$18$$ 0 0
$$19$$ −2265.00 −1.43941 −0.719704 0.694281i $$-0.755722\pi$$
−0.719704 + 0.694281i $$0.755722\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 2451.95 0.966480 0.483240 0.875488i $$-0.339460\pi$$
0.483240 + 0.875488i $$0.339460\pi$$
$$24$$ 0 0
$$25$$ −776.413 −0.248452
$$26$$ 0 0
$$27$$ −3022.30 −0.797862
$$28$$ 0 0
$$29$$ 6985.27 1.54237 0.771185 0.636611i $$-0.219664\pi$$
0.771185 + 0.636611i $$0.219664\pi$$
$$30$$ 0 0
$$31$$ 2794.03 0.522188 0.261094 0.965313i $$-0.415917\pi$$
0.261094 + 0.965313i $$0.415917\pi$$
$$32$$ 0 0
$$33$$ 1126.95 0.180145
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9459.44 1.13595 0.567977 0.823044i $$-0.307726\pi$$
0.567977 + 0.823044i $$0.307726\pi$$
$$38$$ 0 0
$$39$$ 832.803 0.0876760
$$40$$ 0 0
$$41$$ −10088.5 −0.937271 −0.468636 0.883392i $$-0.655254\pi$$
−0.468636 + 0.883392i $$0.655254\pi$$
$$42$$ 0 0
$$43$$ 6934.29 0.571914 0.285957 0.958242i $$-0.407689\pi$$
0.285957 + 0.958242i $$0.407689\pi$$
$$44$$ 0 0
$$45$$ −9474.08 −0.697439
$$46$$ 0 0
$$47$$ 1165.76 0.0769778 0.0384889 0.999259i $$-0.487746\pi$$
0.0384889 + 0.999259i $$0.487746\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −538.934 −0.0290142
$$52$$ 0 0
$$53$$ 8562.42 0.418704 0.209352 0.977840i $$-0.432865\pi$$
0.209352 + 0.977840i $$0.432865\pi$$
$$54$$ 0 0
$$55$$ 7923.85 0.353207
$$56$$ 0 0
$$57$$ −15611.4 −0.636435
$$58$$ 0 0
$$59$$ 6220.26 0.232637 0.116318 0.993212i $$-0.462891\pi$$
0.116318 + 0.993212i $$0.462891\pi$$
$$60$$ 0 0
$$61$$ 41928.9 1.44274 0.721371 0.692549i $$-0.243512\pi$$
0.721371 + 0.692549i $$0.243512\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 5855.62 0.171905
$$66$$ 0 0
$$67$$ −1812.09 −0.0493166 −0.0246583 0.999696i $$-0.507850\pi$$
−0.0246583 + 0.999696i $$0.507850\pi$$
$$68$$ 0 0
$$69$$ 16900.0 0.427329
$$70$$ 0 0
$$71$$ 56823.3 1.33777 0.668884 0.743367i $$-0.266772\pi$$
0.668884 + 0.743367i $$0.266772\pi$$
$$72$$ 0 0
$$73$$ 44299.5 0.972953 0.486476 0.873694i $$-0.338282\pi$$
0.486476 + 0.873694i $$0.338282\pi$$
$$74$$ 0 0
$$75$$ −5351.38 −0.109853
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −34912.4 −0.629379 −0.314690 0.949195i $$-0.601900\pi$$
−0.314690 + 0.949195i $$0.601900\pi$$
$$80$$ 0 0
$$81$$ 26674.1 0.451728
$$82$$ 0 0
$$83$$ −39652.9 −0.631800 −0.315900 0.948793i $$-0.602306\pi$$
−0.315900 + 0.948793i $$0.602306\pi$$
$$84$$ 0 0
$$85$$ −3789.36 −0.0568877
$$86$$ 0 0
$$87$$ 48145.6 0.681960
$$88$$ 0 0
$$89$$ 126299. 1.69015 0.845077 0.534645i $$-0.179555\pi$$
0.845077 + 0.534645i $$0.179555\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 19257.7 0.230886
$$94$$ 0 0
$$95$$ −109767. −1.24785
$$96$$ 0 0
$$97$$ 145513. 1.57026 0.785130 0.619331i $$-0.212596\pi$$
0.785130 + 0.619331i $$0.212596\pi$$
$$98$$ 0 0
$$99$$ −31964.4 −0.327777
$$100$$ 0 0
$$101$$ −99545.5 −0.970998 −0.485499 0.874237i $$-0.661362\pi$$
−0.485499 + 0.874237i $$0.661362\pi$$
$$102$$ 0 0
$$103$$ −129791. −1.20546 −0.602729 0.797946i $$-0.705920\pi$$
−0.602729 + 0.797946i $$0.705920\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −218811. −1.84761 −0.923804 0.382865i $$-0.874938\pi$$
−0.923804 + 0.382865i $$0.874938\pi$$
$$108$$ 0 0
$$109$$ 201627. 1.62549 0.812743 0.582623i $$-0.197974\pi$$
0.812743 + 0.582623i $$0.197974\pi$$
$$110$$ 0 0
$$111$$ 65198.6 0.502263
$$112$$ 0 0
$$113$$ 31803.9 0.234306 0.117153 0.993114i $$-0.462623\pi$$
0.117153 + 0.993114i $$0.462623\pi$$
$$114$$ 0 0
$$115$$ 118827. 0.837859
$$116$$ 0 0
$$117$$ −23621.3 −0.159529
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −134317. −0.834002
$$122$$ 0 0
$$123$$ −69534.1 −0.414415
$$124$$ 0 0
$$125$$ −189071. −1.08231
$$126$$ 0 0
$$127$$ 318115. 1.75015 0.875075 0.483988i $$-0.160812\pi$$
0.875075 + 0.483988i $$0.160812\pi$$
$$128$$ 0 0
$$129$$ 47794.2 0.252872
$$130$$ 0 0
$$131$$ −305194. −1.55381 −0.776905 0.629618i $$-0.783212\pi$$
−0.776905 + 0.629618i $$0.783212\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −146467. −0.691681
$$136$$ 0 0
$$137$$ 180045. 0.819556 0.409778 0.912185i $$-0.365606\pi$$
0.409778 + 0.912185i $$0.365606\pi$$
$$138$$ 0 0
$$139$$ −323204. −1.41886 −0.709430 0.704776i $$-0.751047\pi$$
−0.709430 + 0.704776i $$0.751047\pi$$
$$140$$ 0 0
$$141$$ 8034.96 0.0340358
$$142$$ 0 0
$$143$$ 19756.2 0.0807909
$$144$$ 0 0
$$145$$ 338522. 1.33711
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 294159. 1.08547 0.542733 0.839905i $$-0.317390\pi$$
0.542733 + 0.839905i $$0.317390\pi$$
$$150$$ 0 0
$$151$$ 334962. 1.19551 0.597754 0.801679i $$-0.296060\pi$$
0.597754 + 0.801679i $$0.296060\pi$$
$$152$$ 0 0
$$153$$ 15286.1 0.0527919
$$154$$ 0 0
$$155$$ 135405. 0.452694
$$156$$ 0 0
$$157$$ 145150. 0.469967 0.234984 0.971999i $$-0.424496\pi$$
0.234984 + 0.971999i $$0.424496\pi$$
$$158$$ 0 0
$$159$$ 59016.0 0.185130
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 176221. 0.519504 0.259752 0.965675i $$-0.416359\pi$$
0.259752 + 0.965675i $$0.416359\pi$$
$$164$$ 0 0
$$165$$ 54614.7 0.156171
$$166$$ 0 0
$$167$$ −45460.7 −0.126138 −0.0630689 0.998009i $$-0.520089\pi$$
−0.0630689 + 0.998009i $$0.520089\pi$$
$$168$$ 0 0
$$169$$ −356693. −0.960679
$$170$$ 0 0
$$171$$ 442794. 1.15801
$$172$$ 0 0
$$173$$ 205842. 0.522900 0.261450 0.965217i $$-0.415799\pi$$
0.261450 + 0.965217i $$0.415799\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 42872.8 0.102860
$$178$$ 0 0
$$179$$ 199205. 0.464695 0.232348 0.972633i $$-0.425359\pi$$
0.232348 + 0.972633i $$0.425359\pi$$
$$180$$ 0 0
$$181$$ −198411. −0.450162 −0.225081 0.974340i $$-0.572265\pi$$
−0.225081 + 0.974340i $$0.572265\pi$$
$$182$$ 0 0
$$183$$ 288992. 0.637909
$$184$$ 0 0
$$185$$ 458425. 0.984780
$$186$$ 0 0
$$187$$ −12784.8 −0.0267357
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −372947. −0.739714 −0.369857 0.929089i $$-0.620593\pi$$
−0.369857 + 0.929089i $$0.620593\pi$$
$$192$$ 0 0
$$193$$ 175020. 0.338215 0.169108 0.985598i $$-0.445911\pi$$
0.169108 + 0.985598i $$0.445911\pi$$
$$194$$ 0 0
$$195$$ 40359.5 0.0760080
$$196$$ 0 0
$$197$$ 883770. 1.62246 0.811229 0.584728i $$-0.198799\pi$$
0.811229 + 0.584728i $$0.198799\pi$$
$$198$$ 0 0
$$199$$ 785843. 1.40671 0.703353 0.710841i $$-0.251686\pi$$
0.703353 + 0.710841i $$0.251686\pi$$
$$200$$ 0 0
$$201$$ −12489.7 −0.0218054
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −488909. −0.812538
$$206$$ 0 0
$$207$$ −479343. −0.777536
$$208$$ 0 0
$$209$$ −370340. −0.586456
$$210$$ 0 0
$$211$$ −763861. −1.18116 −0.590579 0.806980i $$-0.701101\pi$$
−0.590579 + 0.806980i $$0.701101\pi$$
$$212$$ 0 0
$$213$$ 391652. 0.591495
$$214$$ 0 0
$$215$$ 336051. 0.495803
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 305332. 0.430191
$$220$$ 0 0
$$221$$ −9447.82 −0.0130122
$$222$$ 0 0
$$223$$ −204400. −0.275245 −0.137623 0.990485i $$-0.543946\pi$$
−0.137623 + 0.990485i $$0.543946\pi$$
$$224$$ 0 0
$$225$$ 151784. 0.199881
$$226$$ 0 0
$$227$$ 1.00806e6 1.29844 0.649218 0.760603i $$-0.275096\pi$$
0.649218 + 0.760603i $$0.275096\pi$$
$$228$$ 0 0
$$229$$ 244324. 0.307877 0.153938 0.988080i $$-0.450804\pi$$
0.153938 + 0.988080i $$0.450804\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 20822.9 0.0251276 0.0125638 0.999921i $$-0.496001\pi$$
0.0125638 + 0.999921i $$0.496001\pi$$
$$234$$ 0 0
$$235$$ 56495.5 0.0667335
$$236$$ 0 0
$$237$$ −240632. −0.278280
$$238$$ 0 0
$$239$$ 474033. 0.536801 0.268401 0.963307i $$-0.413505\pi$$
0.268401 + 0.963307i $$0.413505\pi$$
$$240$$ 0 0
$$241$$ 1.70390e6 1.88974 0.944869 0.327450i $$-0.106189\pi$$
0.944869 + 0.327450i $$0.106189\pi$$
$$242$$ 0 0
$$243$$ 918268. 0.997594
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −273676. −0.285427
$$248$$ 0 0
$$249$$ −273305. −0.279351
$$250$$ 0 0
$$251$$ −810918. −0.812442 −0.406221 0.913775i $$-0.633154\pi$$
−0.406221 + 0.913775i $$0.633154\pi$$
$$252$$ 0 0
$$253$$ 400909. 0.393771
$$254$$ 0 0
$$255$$ −26117.9 −0.0251529
$$256$$ 0 0
$$257$$ −1.26657e6 −1.19618 −0.598088 0.801431i $$-0.704073\pi$$
−0.598088 + 0.801431i $$0.704073\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −1.36558e6 −1.24084
$$262$$ 0 0
$$263$$ 1.96048e6 1.74773 0.873863 0.486172i $$-0.161607\pi$$
0.873863 + 0.486172i $$0.161607\pi$$
$$264$$ 0 0
$$265$$ 414954. 0.362982
$$266$$ 0 0
$$267$$ 870511. 0.747302
$$268$$ 0 0
$$269$$ 1.49358e6 1.25848 0.629242 0.777209i $$-0.283365\pi$$
0.629242 + 0.777209i $$0.283365\pi$$
$$270$$ 0 0
$$271$$ 1.23626e6 1.02255 0.511275 0.859417i $$-0.329173\pi$$
0.511275 + 0.859417i $$0.329173\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −126948. −0.101227
$$276$$ 0 0
$$277$$ −170150. −0.133239 −0.0666197 0.997778i $$-0.521221\pi$$
−0.0666197 + 0.997778i $$0.521221\pi$$
$$278$$ 0 0
$$279$$ −546217. −0.420102
$$280$$ 0 0
$$281$$ 1.29957e6 0.981824 0.490912 0.871209i $$-0.336664\pi$$
0.490912 + 0.871209i $$0.336664\pi$$
$$282$$ 0 0
$$283$$ −374032. −0.277615 −0.138807 0.990319i $$-0.544327\pi$$
−0.138807 + 0.990319i $$0.544327\pi$$
$$284$$ 0 0
$$285$$ −756561. −0.551737
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1.41374e6 −0.995694
$$290$$ 0 0
$$291$$ 1.00294e6 0.694291
$$292$$ 0 0
$$293$$ −1.15022e6 −0.782728 −0.391364 0.920236i $$-0.627997\pi$$
−0.391364 + 0.920236i $$0.627997\pi$$
$$294$$ 0 0
$$295$$ 301448. 0.201677
$$296$$ 0 0
$$297$$ −494163. −0.325072
$$298$$ 0 0
$$299$$ 296266. 0.191648
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −686112. −0.429327
$$304$$ 0 0
$$305$$ 2.03197e6 1.25074
$$306$$ 0 0
$$307$$ 2.61214e6 1.58179 0.790897 0.611949i $$-0.209614\pi$$
0.790897 + 0.611949i $$0.209614\pi$$
$$308$$ 0 0
$$309$$ −894578. −0.532994
$$310$$ 0 0
$$311$$ −1.40646e6 −0.824566 −0.412283 0.911056i $$-0.635269\pi$$
−0.412283 + 0.911056i $$0.635269\pi$$
$$312$$ 0 0
$$313$$ −2.37930e6 −1.37274 −0.686369 0.727253i $$-0.740797\pi$$
−0.686369 + 0.727253i $$0.740797\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −563199. −0.314785 −0.157393 0.987536i $$-0.550309\pi$$
−0.157393 + 0.987536i $$0.550309\pi$$
$$318$$ 0 0
$$319$$ 1.14213e6 0.628405
$$320$$ 0 0
$$321$$ −1.50814e6 −0.816921
$$322$$ 0 0
$$323$$ 177105. 0.0944547
$$324$$ 0 0
$$325$$ −93812.8 −0.0492667
$$326$$ 0 0
$$327$$ 1.38970e6 0.718709
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 850649. 0.426757 0.213378 0.976970i $$-0.431553\pi$$
0.213378 + 0.976970i $$0.431553\pi$$
$$332$$ 0 0
$$333$$ −1.84927e6 −0.913879
$$334$$ 0 0
$$335$$ −87818.0 −0.0427535
$$336$$ 0 0
$$337$$ 4.01506e6 1.92582 0.962912 0.269814i $$-0.0869623\pi$$
0.962912 + 0.269814i $$0.0869623\pi$$
$$338$$ 0 0
$$339$$ 219207. 0.103599
$$340$$ 0 0
$$341$$ 456840. 0.212754
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 819009. 0.370460
$$346$$ 0 0
$$347$$ 2.39961e6 1.06984 0.534918 0.844904i $$-0.320343\pi$$
0.534918 + 0.844904i $$0.320343\pi$$
$$348$$ 0 0
$$349$$ 919171. 0.403955 0.201977 0.979390i $$-0.435263\pi$$
0.201977 + 0.979390i $$0.435263\pi$$
$$350$$ 0 0
$$351$$ −365179. −0.158212
$$352$$ 0 0
$$353$$ −1.07684e6 −0.459953 −0.229976 0.973196i $$-0.573865\pi$$
−0.229976 + 0.973196i $$0.573865\pi$$
$$354$$ 0 0
$$355$$ 2.75378e6 1.15974
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 1.24219e6 0.508688 0.254344 0.967114i $$-0.418140\pi$$
0.254344 + 0.967114i $$0.418140\pi$$
$$360$$ 0 0
$$361$$ 2.65411e6 1.07189
$$362$$ 0 0
$$363$$ −925771. −0.368754
$$364$$ 0 0
$$365$$ 2.14685e6 0.843471
$$366$$ 0 0
$$367$$ 4.17293e6 1.61725 0.808624 0.588326i $$-0.200213\pi$$
0.808624 + 0.588326i $$0.200213\pi$$
$$368$$ 0 0
$$369$$ 1.97224e6 0.754037
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.72238e6 −0.640999 −0.320500 0.947249i $$-0.603851\pi$$
−0.320500 + 0.947249i $$0.603851\pi$$
$$374$$ 0 0
$$375$$ −1.30316e6 −0.478542
$$376$$ 0 0
$$377$$ 844020. 0.305844
$$378$$ 0 0
$$379$$ −1.72998e6 −0.618647 −0.309324 0.950957i $$-0.600103\pi$$
−0.309324 + 0.950957i $$0.600103\pi$$
$$380$$ 0 0
$$381$$ 2.19259e6 0.773830
$$382$$ 0 0
$$383$$ −3.34881e6 −1.16652 −0.583262 0.812284i $$-0.698224\pi$$
−0.583262 + 0.812284i $$0.698224\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1.35561e6 −0.460106
$$388$$ 0 0
$$389$$ −5.18691e6 −1.73794 −0.868970 0.494864i $$-0.835218\pi$$
−0.868970 + 0.494864i $$0.835218\pi$$
$$390$$ 0 0
$$391$$ −191723. −0.0634209
$$392$$ 0 0
$$393$$ −2.10353e6 −0.687018
$$394$$ 0 0
$$395$$ −1.69193e6 −0.545621
$$396$$ 0 0
$$397$$ 1.30271e6 0.414830 0.207415 0.978253i $$-0.433495\pi$$
0.207415 + 0.978253i $$0.433495\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −1.53036e6 −0.475261 −0.237631 0.971356i $$-0.576371\pi$$
−0.237631 + 0.971356i $$0.576371\pi$$
$$402$$ 0 0
$$403$$ 337598. 0.103547
$$404$$ 0 0
$$405$$ 1.29269e6 0.391611
$$406$$ 0 0
$$407$$ 1.54667e6 0.462820
$$408$$ 0 0
$$409$$ 2.93690e6 0.868123 0.434062 0.900883i $$-0.357080\pi$$
0.434062 + 0.900883i $$0.357080\pi$$
$$410$$ 0 0
$$411$$ 1.24095e6 0.362367
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −1.92167e6 −0.547719
$$416$$ 0 0
$$417$$ −2.22766e6 −0.627349
$$418$$ 0 0
$$419$$ −1.16465e6 −0.324085 −0.162043 0.986784i $$-0.551808\pi$$
−0.162043 + 0.986784i $$0.551808\pi$$
$$420$$ 0 0
$$421$$ −1.58204e6 −0.435022 −0.217511 0.976058i $$-0.569794\pi$$
−0.217511 + 0.976058i $$0.569794\pi$$
$$422$$ 0 0
$$423$$ −227900. −0.0619289
$$424$$ 0 0
$$425$$ 60709.3 0.0163036
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 136168. 0.0357217
$$430$$ 0 0
$$431$$ −6.10233e6 −1.58235 −0.791174 0.611591i $$-0.790530\pi$$
−0.791174 + 0.611591i $$0.790530\pi$$
$$432$$ 0 0
$$433$$ −1.08115e6 −0.277118 −0.138559 0.990354i $$-0.544247\pi$$
−0.138559 + 0.990354i $$0.544247\pi$$
$$434$$ 0 0
$$435$$ 2.33324e6 0.591203
$$436$$ 0 0
$$437$$ −5.55367e6 −1.39116
$$438$$ 0 0
$$439$$ −7.04807e6 −1.74546 −0.872728 0.488207i $$-0.837651\pi$$
−0.872728 + 0.488207i $$0.837651\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 2.32031e6 0.561742 0.280871 0.959746i $$-0.409377\pi$$
0.280871 + 0.959746i $$0.409377\pi$$
$$444$$ 0 0
$$445$$ 6.12075e6 1.46523
$$446$$ 0 0
$$447$$ 2.02747e6 0.479939
$$448$$ 0 0
$$449$$ −5.45039e6 −1.27589 −0.637943 0.770084i $$-0.720214\pi$$
−0.637943 + 0.770084i $$0.720214\pi$$
$$450$$ 0 0
$$451$$ −1.64952e6 −0.381871
$$452$$ 0 0
$$453$$ 2.30871e6 0.528595
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6.62676e6 1.48426 0.742132 0.670254i $$-0.233815\pi$$
0.742132 + 0.670254i $$0.233815\pi$$
$$458$$ 0 0
$$459$$ 236319. 0.0523561
$$460$$ 0 0
$$461$$ −606397. −0.132894 −0.0664469 0.997790i $$-0.521166\pi$$
−0.0664469 + 0.997790i $$0.521166\pi$$
$$462$$ 0 0
$$463$$ −3.65855e6 −0.793153 −0.396576 0.918002i $$-0.629802\pi$$
−0.396576 + 0.918002i $$0.629802\pi$$
$$464$$ 0 0
$$465$$ 933271. 0.200159
$$466$$ 0 0
$$467$$ 5.62712e6 1.19397 0.596986 0.802252i $$-0.296365\pi$$
0.596986 + 0.802252i $$0.296365\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 1.00044e6 0.207796
$$472$$ 0 0
$$473$$ 1.13380e6 0.233014
$$474$$ 0 0
$$475$$ 1.75857e6 0.357624
$$476$$ 0 0
$$477$$ −1.67390e6 −0.336848
$$478$$ 0 0
$$479$$ 4.25107e6 0.846563 0.423281 0.905998i $$-0.360878\pi$$
0.423281 + 0.905998i $$0.360878\pi$$
$$480$$ 0 0
$$481$$ 1.14297e6 0.225254
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 7.05187e6 1.36129
$$486$$ 0 0
$$487$$ −5.28756e6 −1.01026 −0.505130 0.863043i $$-0.668555\pi$$
−0.505130 + 0.863043i $$0.668555\pi$$
$$488$$ 0 0
$$489$$ 1.21459e6 0.229699
$$490$$ 0 0
$$491$$ 3.33842e6 0.624939 0.312469 0.949928i $$-0.398844\pi$$
0.312469 + 0.949928i $$0.398844\pi$$
$$492$$ 0 0
$$493$$ −546192. −0.101211
$$494$$ 0 0
$$495$$ −1.54907e6 −0.284156
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −2.52847e6 −0.454576 −0.227288 0.973828i $$-0.572986\pi$$
−0.227288 + 0.973828i $$0.572986\pi$$
$$500$$ 0 0
$$501$$ −313335. −0.0557719
$$502$$ 0 0
$$503$$ −5.94498e6 −1.04768 −0.523842 0.851815i $$-0.675502\pi$$
−0.523842 + 0.851815i $$0.675502\pi$$
$$504$$ 0 0
$$505$$ −4.82420e6 −0.841776
$$506$$ 0 0
$$507$$ −2.45849e6 −0.424765
$$508$$ 0 0
$$509$$ 603525. 0.103253 0.0516263 0.998666i $$-0.483560\pi$$
0.0516263 + 0.998666i $$0.483560\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 6.84549e6 1.14845
$$514$$ 0 0
$$515$$ −6.28997e6 −1.04503
$$516$$ 0 0
$$517$$ 190609. 0.0313630
$$518$$ 0 0
$$519$$ 1.41875e6 0.231200
$$520$$ 0 0
$$521$$ −1.03389e7 −1.66870 −0.834350 0.551235i $$-0.814157\pi$$
−0.834350 + 0.551235i $$0.814157\pi$$
$$522$$ 0 0
$$523$$ −8.82310e6 −1.41048 −0.705240 0.708968i $$-0.749161\pi$$
−0.705240 + 0.708968i $$0.749161\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −218471. −0.0342663
$$528$$ 0 0
$$529$$ −424266. −0.0659172
$$530$$ 0 0
$$531$$ −1.21602e6 −0.187157
$$532$$ 0 0
$$533$$ −1.21897e6 −0.185856
$$534$$ 0 0
$$535$$ −1.06041e7 −1.60173
$$536$$ 0 0
$$537$$ 1.37301e6 0.205465
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 1.16811e6 0.171590 0.0857949 0.996313i $$-0.472657\pi$$
0.0857949 + 0.996313i $$0.472657\pi$$
$$542$$ 0 0
$$543$$ −1.36753e6 −0.199039
$$544$$ 0 0
$$545$$ 9.77130e6 1.40916
$$546$$ 0 0
$$547$$ −9.17075e6 −1.31050 −0.655249 0.755413i $$-0.727436\pi$$
−0.655249 + 0.755413i $$0.727436\pi$$
$$548$$ 0 0
$$549$$ −8.19685e6 −1.16069
$$550$$ 0 0
$$551$$ −1.58216e7 −2.22010
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 3.15967e6 0.435421
$$556$$ 0 0
$$557$$ 6.22421e6 0.850054 0.425027 0.905181i $$-0.360265\pi$$
0.425027 + 0.905181i $$0.360265\pi$$
$$558$$ 0 0
$$559$$ 837859. 0.113407
$$560$$ 0 0
$$561$$ −88118.8 −0.0118212
$$562$$ 0 0
$$563$$ −5.64042e6 −0.749964 −0.374982 0.927032i $$-0.622351\pi$$
−0.374982 + 0.927032i $$0.622351\pi$$
$$564$$ 0 0
$$565$$ 1.54129e6 0.203125
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −1.05286e7 −1.36330 −0.681650 0.731678i $$-0.738737\pi$$
−0.681650 + 0.731678i $$0.738737\pi$$
$$570$$ 0 0
$$571$$ 7.31395e6 0.938776 0.469388 0.882992i $$-0.344475\pi$$
0.469388 + 0.882992i $$0.344475\pi$$
$$572$$ 0 0
$$573$$ −2.57052e6 −0.327065
$$574$$ 0 0
$$575$$ −1.90373e6 −0.240124
$$576$$ 0 0
$$577$$ −2.66534e6 −0.333282 −0.166641 0.986018i $$-0.553292\pi$$
−0.166641 + 0.986018i $$0.553292\pi$$
$$578$$ 0 0
$$579$$ 1.20631e6 0.149542
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 1.40000e6 0.170592
$$584$$ 0 0
$$585$$ −1.14474e6 −0.138298
$$586$$ 0 0
$$587$$ 4.92431e6 0.589861 0.294931 0.955519i $$-0.404703\pi$$
0.294931 + 0.955519i $$0.404703\pi$$
$$588$$ 0 0
$$589$$ −6.32847e6 −0.751641
$$590$$ 0 0
$$591$$ 6.09133e6 0.717371
$$592$$ 0 0
$$593$$ 5.08735e6 0.594094 0.297047 0.954863i $$-0.403998\pi$$
0.297047 + 0.954863i $$0.403998\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 5.41638e6 0.621975
$$598$$ 0 0
$$599$$ −4.64678e6 −0.529157 −0.264579 0.964364i $$-0.585233\pi$$
−0.264579 + 0.964364i $$0.585233\pi$$
$$600$$ 0 0
$$601$$ 4.35424e6 0.491730 0.245865 0.969304i $$-0.420928\pi$$
0.245865 + 0.969304i $$0.420928\pi$$
$$602$$ 0 0
$$603$$ 354254. 0.0396754
$$604$$ 0 0
$$605$$ −6.50929e6 −0.723012
$$606$$ 0 0
$$607$$ −1.25461e7 −1.38209 −0.691043 0.722813i $$-0.742849\pi$$
−0.691043 + 0.722813i $$0.742849\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 140857. 0.0152643
$$612$$ 0 0
$$613$$ −4.07637e6 −0.438149 −0.219075 0.975708i $$-0.570304\pi$$
−0.219075 + 0.975708i $$0.570304\pi$$
$$614$$ 0 0
$$615$$ −3.36978e6 −0.359264
$$616$$ 0 0
$$617$$ −8.88533e6 −0.939639 −0.469819 0.882763i $$-0.655681\pi$$
−0.469819 + 0.882763i $$0.655681\pi$$
$$618$$ 0 0
$$619$$ −1.28530e7 −1.34827 −0.674136 0.738607i $$-0.735484\pi$$
−0.674136 + 0.738607i $$0.735484\pi$$
$$620$$ 0 0
$$621$$ −7.41053e6 −0.771117
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −6.73652e6 −0.689819
$$626$$ 0 0
$$627$$ −2.55255e6 −0.259302
$$628$$ 0 0
$$629$$ −739652. −0.0745420
$$630$$ 0 0
$$631$$ 288616. 0.0288567 0.0144283 0.999896i $$-0.495407\pi$$
0.0144283 + 0.999896i $$0.495407\pi$$
$$632$$ 0 0
$$633$$ −5.26487e6 −0.522250
$$634$$ 0 0
$$635$$ 1.54166e7 1.51724
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −1.11086e7 −1.07624
$$640$$ 0 0
$$641$$ −1.94591e7 −1.87058 −0.935291 0.353880i $$-0.884862\pi$$
−0.935291 + 0.353880i $$0.884862\pi$$
$$642$$ 0 0
$$643$$ 1.30171e7 1.24161 0.620805 0.783965i $$-0.286806\pi$$
0.620805 + 0.783965i $$0.286806\pi$$
$$644$$ 0 0
$$645$$ 2.31621e6 0.219219
$$646$$ 0 0
$$647$$ 6.42864e6 0.603751 0.301876 0.953347i $$-0.402387\pi$$
0.301876 + 0.953347i $$0.402387\pi$$
$$648$$ 0 0
$$649$$ 1.01705e6 0.0947829
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1.00202e7 0.919592 0.459796 0.888025i $$-0.347923\pi$$
0.459796 + 0.888025i $$0.347923\pi$$
$$654$$ 0 0
$$655$$ −1.47904e7 −1.34703
$$656$$ 0 0
$$657$$ −8.66030e6 −0.782743
$$658$$ 0 0
$$659$$ −1.49356e7 −1.33971 −0.669854 0.742493i $$-0.733643\pi$$
−0.669854 + 0.742493i $$0.733643\pi$$
$$660$$ 0 0
$$661$$ 1.35055e6 0.120229 0.0601143 0.998191i $$-0.480853\pi$$
0.0601143 + 0.998191i $$0.480853\pi$$
$$662$$ 0 0
$$663$$ −65118.5 −0.00575335
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 1.71276e7 1.49067
$$668$$ 0 0
$$669$$ −1.40882e6 −0.121700
$$670$$ 0 0
$$671$$ 6.85561e6 0.587814
$$672$$ 0 0
$$673$$ 6.59401e6 0.561193 0.280596 0.959826i $$-0.409468\pi$$
0.280596 + 0.959826i $$0.409468\pi$$
$$674$$ 0 0
$$675$$ 2.34655e6 0.198231
$$676$$ 0 0
$$677$$ −1.42517e7 −1.19508 −0.597538 0.801840i $$-0.703854\pi$$
−0.597538 + 0.801840i $$0.703854\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 6.94797e6 0.574104
$$682$$ 0 0
$$683$$ 7.65006e6 0.627499 0.313749 0.949506i $$-0.398415\pi$$
0.313749 + 0.949506i $$0.398415\pi$$
$$684$$ 0 0
$$685$$ 8.72536e6 0.710489
$$686$$ 0 0
$$687$$ 1.68399e6 0.136128
$$688$$ 0 0
$$689$$ 1.03458e6 0.0830266
$$690$$ 0 0
$$691$$ 1.38655e7 1.10469 0.552344 0.833616i $$-0.313733\pi$$
0.552344 + 0.833616i $$0.313733\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −1.56632e7 −1.23004
$$696$$ 0 0
$$697$$ 788837. 0.0615042
$$698$$ 0 0
$$699$$ 143521. 0.0111102
$$700$$ 0 0
$$701$$ −2.38825e6 −0.183563 −0.0917814 0.995779i $$-0.529256\pi$$
−0.0917814 + 0.995779i $$0.529256\pi$$
$$702$$ 0 0
$$703$$ −2.14256e7 −1.63510
$$704$$ 0 0
$$705$$ 389392. 0.0295063
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −1.55641e7 −1.16281 −0.581404 0.813615i $$-0.697496\pi$$
−0.581404 + 0.813615i $$0.697496\pi$$
$$710$$ 0 0
$$711$$ 6.82518e6 0.506337
$$712$$ 0 0
$$713$$ 6.85083e6 0.504684
$$714$$ 0 0
$$715$$ 957427. 0.0700391
$$716$$ 0 0
$$717$$ 3.26724e6 0.237347
$$718$$ 0 0
$$719$$ 1.82555e7 1.31696 0.658478 0.752600i $$-0.271200\pi$$
0.658478 + 0.752600i $$0.271200\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 1.17440e7 0.835548
$$724$$ 0 0
$$725$$ −5.42346e6 −0.383205
$$726$$ 0 0
$$727$$ −2.11427e6 −0.148362 −0.0741812 0.997245i $$-0.523634\pi$$
−0.0741812 + 0.997245i $$0.523634\pi$$
$$728$$ 0 0
$$729$$ −152693. −0.0106414
$$730$$ 0 0
$$731$$ −542206. −0.0375293
$$732$$ 0 0
$$733$$ −2.12932e6 −0.146380 −0.0731898 0.997318i $$-0.523318\pi$$
−0.0731898 + 0.997318i $$0.523318\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −296288. −0.0200930
$$738$$ 0 0
$$739$$ 8.08671e6 0.544704 0.272352 0.962198i $$-0.412198\pi$$
0.272352 + 0.962198i $$0.412198\pi$$
$$740$$ 0 0
$$741$$ −1.88630e6 −0.126202
$$742$$ 0 0
$$743$$ 1.18944e7 0.790442 0.395221 0.918586i $$-0.370668\pi$$
0.395221 + 0.918586i $$0.370668\pi$$
$$744$$ 0 0
$$745$$ 1.42556e7 0.941010
$$746$$ 0 0
$$747$$ 7.75191e6 0.508285
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 1.85538e7 1.20042 0.600211 0.799842i $$-0.295083\pi$$
0.600211 + 0.799842i $$0.295083\pi$$
$$752$$ 0 0
$$753$$ −5.58921e6 −0.359222
$$754$$ 0 0
$$755$$ 1.62330e7 1.03641
$$756$$ 0 0
$$757$$ 9.32534e6 0.591459 0.295730 0.955272i $$-0.404437\pi$$
0.295730 + 0.955272i $$0.404437\pi$$
$$758$$ 0 0
$$759$$ 2.76324e6 0.174106
$$760$$ 0 0
$$761$$ −8.00756e6 −0.501232 −0.250616 0.968087i $$-0.580633\pi$$
−0.250616 + 0.968087i $$0.580633\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 740797. 0.0457663
$$766$$ 0 0
$$767$$ 751584. 0.0461306
$$768$$ 0 0
$$769$$ −2.02720e7 −1.23618 −0.618089 0.786108i $$-0.712093\pi$$
−0.618089 + 0.786108i $$0.712093\pi$$
$$770$$ 0 0
$$771$$ −8.72973e6 −0.528889
$$772$$ 0 0
$$773$$ −2.87881e6 −0.173286 −0.0866431 0.996239i $$-0.527614\pi$$
−0.0866431 + 0.996239i $$0.527614\pi$$
$$774$$ 0 0
$$775$$ −2.16932e6 −0.129739
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 2.28503e7 1.34911
$$780$$ 0 0
$$781$$ 9.29094e6 0.545045
$$782$$ 0 0
$$783$$ −2.11116e7 −1.23060
$$784$$ 0 0
$$785$$ 7.03428e6 0.407423
$$786$$ 0 0
$$787$$ −2.03508e7 −1.17124 −0.585618 0.810587i $$-0.699148\pi$$
−0.585618 + 0.810587i $$0.699148\pi$$
$$788$$ 0 0
$$789$$ 1.35125e7 0.772758
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 5.06620e6 0.286088
$$794$$ 0 0
$$795$$ 2.86005e6 0.160493
$$796$$ 0 0
$$797$$ 2.24765e7 1.25338 0.626689 0.779269i $$-0.284410\pi$$
0.626689 + 0.779269i $$0.284410\pi$$
$$798$$ 0 0
$$799$$ −91153.3 −0.00505133
$$800$$ 0 0
$$801$$ −2.46908e7 −1.35973
$$802$$ 0 0
$$803$$ 7.24322e6 0.396409
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 1.02944e7 0.556439
$$808$$ 0 0
$$809$$ −1.15427e6 −0.0620062 −0.0310031 0.999519i $$-0.509870\pi$$
−0.0310031 + 0.999519i $$0.509870\pi$$
$$810$$ 0 0
$$811$$ −2.26698e7 −1.21031 −0.605154 0.796108i $$-0.706888\pi$$
−0.605154 + 0.796108i $$0.706888\pi$$
$$812$$ 0 0
$$813$$ 8.52082e6 0.452121
$$814$$ 0 0
$$815$$ 8.54007e6 0.450368
$$816$$ 0 0
$$817$$ −1.57061e7 −0.823217
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −2.41657e6 −0.125124 −0.0625622 0.998041i $$-0.519927\pi$$
−0.0625622 + 0.998041i $$0.519927\pi$$
$$822$$ 0 0
$$823$$ 2.10169e6 0.108161 0.0540804 0.998537i $$-0.482777\pi$$
0.0540804 + 0.998537i $$0.482777\pi$$
$$824$$ 0 0
$$825$$ −874982. −0.0447574
$$826$$ 0 0
$$827$$ −1.53997e7 −0.782977 −0.391489 0.920183i $$-0.628040\pi$$
−0.391489 + 0.920183i $$0.628040\pi$$
$$828$$ 0 0
$$829$$ 2.33839e7 1.18177 0.590883 0.806757i $$-0.298779\pi$$
0.590883 + 0.806757i $$0.298779\pi$$
$$830$$ 0 0
$$831$$ −1.17275e6 −0.0589118
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −2.20313e6 −0.109351
$$836$$ 0 0
$$837$$ −8.44439e6 −0.416634
$$838$$ 0 0
$$839$$ 2.03016e7 0.995695 0.497848 0.867265i $$-0.334124\pi$$
0.497848 + 0.867265i $$0.334124\pi$$
$$840$$ 0 0
$$841$$ 2.82829e7 1.37890
$$842$$ 0 0
$$843$$ 8.95721e6 0.434114
$$844$$ 0 0
$$845$$ −1.72862e7 −0.832831
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −2.57800e6 −0.122748
$$850$$ 0 0
$$851$$ 2.31941e7 1.09788
$$852$$ 0 0
$$853$$ −3.35163e7 −1.57719 −0.788594 0.614914i $$-0.789191\pi$$
−0.788594 + 0.614914i $$0.789191\pi$$
$$854$$ 0 0
$$855$$ 2.14588e7 1.00390
$$856$$ 0 0
$$857$$ 2.30661e7 1.07281 0.536405 0.843961i $$-0.319782\pi$$
0.536405 + 0.843961i $$0.319782\pi$$
$$858$$ 0 0
$$859$$ 3.85609e7 1.78305 0.891527 0.452968i $$-0.149635\pi$$
0.891527 + 0.452968i $$0.149635\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −3.86892e7 −1.76833 −0.884163 0.467179i $$-0.845271\pi$$
−0.884163 + 0.467179i $$0.845271\pi$$
$$864$$ 0 0
$$865$$ 9.97555e6 0.453311
$$866$$ 0 0
$$867$$ −9.74414e6 −0.440247
$$868$$ 0 0
$$869$$ −5.70838e6 −0.256427
$$870$$ 0 0
$$871$$ −218952. −0.00977922
$$872$$ 0 0
$$873$$ −2.84469e7 −1.26328
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −2.17369e7 −0.954332 −0.477166 0.878813i $$-0.658336\pi$$
−0.477166 + 0.878813i $$0.658336\pi$$
$$878$$ 0 0
$$879$$ −7.92781e6 −0.346084
$$880$$ 0 0
$$881$$ −8.42086e6 −0.365525 −0.182762 0.983157i $$-0.558504\pi$$
−0.182762 + 0.983157i $$0.558504\pi$$
$$882$$ 0 0
$$883$$ −3.22954e7 −1.39392 −0.696962 0.717108i $$-0.745465\pi$$
−0.696962 + 0.717108i $$0.745465\pi$$
$$884$$ 0 0
$$885$$ 2.07771e6 0.0891716
$$886$$ 0 0
$$887$$ −4.21800e7 −1.80010 −0.900052 0.435782i $$-0.856472\pi$$
−0.900052 + 0.435782i $$0.856472\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.36137e6 0.184047
$$892$$ 0 0
$$893$$ −2.64045e6 −0.110802
$$894$$ 0 0
$$895$$ 9.65393e6 0.402853
$$896$$ 0 0
$$897$$ 2.04200e6 0.0847371
$$898$$ 0 0
$$899$$ 1.95171e7 0.805407
$$900$$ 0 0
$$901$$ −669512. −0.0274756
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −9.61542e6 −0.390254
$$906$$ 0 0
$$907$$ 7.64517e6 0.308581 0.154291 0.988026i $$-0.450691\pi$$
0.154291 + 0.988026i $$0.450691\pi$$
$$908$$ 0 0
$$909$$ 1.94606e7 0.781170
$$910$$ 0 0
$$911$$ 3.93310e7 1.57014 0.785072 0.619405i $$-0.212626\pi$$
0.785072 + 0.619405i $$0.212626\pi$$
$$912$$ 0 0
$$913$$ −6.48347e6 −0.257413
$$914$$ 0 0
$$915$$ 1.40052e7 0.553015
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −3.52204e7 −1.37564 −0.687822 0.725880i $$-0.741433\pi$$
−0.687822 + 0.725880i $$0.741433\pi$$
$$920$$ 0 0
$$921$$ 1.80040e7 0.699391
$$922$$ 0 0
$$923$$ 6.86587e6 0.265272
$$924$$ 0 0
$$925$$ −7.34443e6 −0.282230
$$926$$ 0 0
$$927$$ 2.53734e7 0.969794
$$928$$ 0 0
$$929$$ 1.66300e7 0.632199 0.316099 0.948726i $$-0.397627\pi$$
0.316099 + 0.948726i $$0.397627\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −9.69393e6 −0.364582
$$934$$ 0 0
$$935$$ −619582. −0.0231777
$$936$$ 0 0
$$937$$ −2.07121e7 −0.770681 −0.385341 0.922774i $$-0.625916\pi$$
−0.385341 + 0.922774i $$0.625916\pi$$
$$938$$ 0 0
$$939$$ −1.63992e7 −0.606957
$$940$$ 0 0
$$941$$ 1.09797e7 0.404219 0.202110 0.979363i $$-0.435220\pi$$
0.202110 + 0.979363i $$0.435220\pi$$
$$942$$ 0 0
$$943$$ −2.47364e7 −0.905853
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −2.71723e6 −0.0984581 −0.0492290 0.998788i $$-0.515676\pi$$
−0.0492290 + 0.998788i $$0.515676\pi$$
$$948$$ 0 0
$$949$$ 5.35264e6 0.192931
$$950$$ 0 0
$$951$$ −3.88182e6 −0.139182
$$952$$ 0 0
$$953$$ −9.24083e6 −0.329594 −0.164797 0.986328i $$-0.552697\pi$$
−0.164797 + 0.986328i $$0.552697\pi$$
$$954$$ 0 0
$$955$$ −1.80739e7 −0.641272
$$956$$ 0 0
$$957$$ 7.87209e6 0.277850
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2.08225e7 −0.727320
$$962$$ 0 0
$$963$$ 4.27763e7 1.48641
$$964$$ 0 0
$$965$$ 8.48183e6 0.293205
$$966$$ 0 0
$$967$$ 1.50444e7 0.517379 0.258690 0.965960i $$-0.416709\pi$$
0.258690 + 0.965960i $$0.416709\pi$$
$$968$$ 0 0
$$969$$ 1.22068e6 0.0417632
$$970$$ 0 0
$$971$$ 2.27894e7 0.775685 0.387842 0.921726i $$-0.373220\pi$$
0.387842 + 0.921726i $$0.373220\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −646600. −0.0217833
$$976$$ 0 0
$$977$$ −3.96682e7 −1.32955 −0.664777 0.747042i $$-0.731474\pi$$
−0.664777 + 0.747042i $$0.731474\pi$$
$$978$$ 0 0
$$979$$ 2.06507e7 0.688617
$$980$$ 0 0
$$981$$ −3.94170e7 −1.30771
$$982$$ 0 0
$$983$$ 2.77093e7 0.914621 0.457310 0.889307i $$-0.348813\pi$$
0.457310 + 0.889307i $$0.348813\pi$$
$$984$$ 0 0
$$985$$ 4.28294e7 1.40654
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.70025e7 0.552743
$$990$$ 0 0
$$991$$ −4.78535e7 −1.54785 −0.773926 0.633276i $$-0.781710\pi$$
−0.773926 + 0.633276i $$0.781710\pi$$
$$992$$ 0 0
$$993$$ 5.86305e6 0.188691
$$994$$ 0 0
$$995$$ 3.80837e7 1.21950
$$996$$ 0 0
$$997$$ 5.48289e7 1.74691 0.873457 0.486902i $$-0.161873\pi$$
0.873457 + 0.486902i $$0.161873\pi$$
$$998$$ 0 0
$$999$$ −2.85892e7 −0.906335
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 784.6.a.q.1.2 2
4.3 odd 2 392.6.a.e.1.1 2
7.6 odd 2 112.6.a.j.1.1 2
21.20 even 2 1008.6.a.bi.1.2 2
28.3 even 6 392.6.i.k.177.1 4
28.11 odd 6 392.6.i.h.177.2 4
28.19 even 6 392.6.i.k.361.1 4
28.23 odd 6 392.6.i.h.361.2 4
28.27 even 2 56.6.a.d.1.2 2
56.13 odd 2 448.6.a.r.1.2 2
56.27 even 2 448.6.a.x.1.1 2
84.83 odd 2 504.6.a.m.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.a.d.1.2 2 28.27 even 2
112.6.a.j.1.1 2 7.6 odd 2
392.6.a.e.1.1 2 4.3 odd 2
392.6.i.h.177.2 4 28.11 odd 6
392.6.i.h.361.2 4 28.23 odd 6
392.6.i.k.177.1 4 28.3 even 6
392.6.i.k.361.1 4 28.19 even 6
448.6.a.r.1.2 2 56.13 odd 2
448.6.a.x.1.1 2 56.27 even 2
504.6.a.m.1.2 2 84.83 odd 2
784.6.a.q.1.2 2 1.1 even 1 trivial
1008.6.a.bi.1.2 2 21.20 even 2