# Properties

 Label 784.6.a.ba.1.1 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: $$\Q(\sqrt{37})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 9$$ x^2 - x - 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 7) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-2.54138$$ 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)

 $$f(q)$$ $$=$$ $$q-2.08276 q^{3} +41.8276 q^{5} -238.662 q^{9} +O(q^{10})$$ $$q-2.08276 q^{3} +41.8276 q^{5} -238.662 q^{9} -72.0965 q^{11} -632.317 q^{13} -87.1170 q^{15} -1975.92 q^{17} +1864.93 q^{19} -413.711 q^{23} -1375.45 q^{25} +1003.19 q^{27} +731.934 q^{29} -6123.18 q^{31} +150.160 q^{33} +10350.4 q^{37} +1316.97 q^{39} -3529.84 q^{41} +14515.2 q^{43} -9982.67 q^{45} +21423.3 q^{47} +4115.38 q^{51} +12579.5 q^{53} -3015.62 q^{55} -3884.20 q^{57} +36133.9 q^{59} +4024.80 q^{61} -26448.3 q^{65} -15565.9 q^{67} +861.661 q^{69} -12180.8 q^{71} +19589.1 q^{73} +2864.74 q^{75} -36089.8 q^{79} +55905.5 q^{81} +24572.6 q^{83} -82648.2 q^{85} -1524.44 q^{87} -70243.3 q^{89} +12753.1 q^{93} +78005.4 q^{95} +105758. q^{97} +17206.7 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{3} - 38 q^{5} - 380 q^{9}+O(q^{10})$$ 2 * q + 8 * q^3 - 38 * q^5 - 380 * q^9 $$2 q + 8 q^{3} - 38 q^{5} - 380 q^{9} - 424 q^{11} - 924 q^{13} - 892 q^{15} - 2346 q^{17} + 360 q^{19} + 12 q^{23} + 1872 q^{25} - 2872 q^{27} - 7052 q^{29} - 3548 q^{31} - 3398 q^{33} + 11090 q^{37} - 1624 q^{39} + 3500 q^{41} + 12680 q^{43} + 1300 q^{45} + 22956 q^{47} + 384 q^{51} + 3042 q^{53} + 25076 q^{55} - 19058 q^{57} + 65808 q^{59} - 42486 q^{61} - 3164 q^{65} - 42312 q^{67} + 5154 q^{69} + 2208 q^{71} - 50506 q^{73} + 35608 q^{75} - 9004 q^{79} + 51178 q^{81} + 104328 q^{83} - 53106 q^{85} - 80008 q^{87} - 26666 q^{89} + 38718 q^{93} + 198140 q^{95} + 209132 q^{97} + 66944 q^{99}+O(q^{100})$$ 2 * q + 8 * q^3 - 38 * q^5 - 380 * q^9 - 424 * q^11 - 924 * q^13 - 892 * q^15 - 2346 * q^17 + 360 * q^19 + 12 * q^23 + 1872 * q^25 - 2872 * q^27 - 7052 * q^29 - 3548 * q^31 - 3398 * q^33 + 11090 * q^37 - 1624 * q^39 + 3500 * q^41 + 12680 * q^43 + 1300 * q^45 + 22956 * q^47 + 384 * q^51 + 3042 * q^53 + 25076 * q^55 - 19058 * q^57 + 65808 * q^59 - 42486 * q^61 - 3164 * q^65 - 42312 * q^67 + 5154 * q^69 + 2208 * q^71 - 50506 * q^73 + 35608 * q^75 - 9004 * q^79 + 51178 * q^81 + 104328 * q^83 - 53106 * q^85 - 80008 * q^87 - 26666 * q^89 + 38718 * q^93 + 198140 * q^95 + 209132 * q^97 + 66944 * 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
$$3$$ −2.08276 −0.133609 −0.0668046 0.997766i $$-0.521280\pi$$
−0.0668046 + 0.997766i $$0.521280\pi$$
$$4$$ 0 0
$$5$$ 41.8276 0.748235 0.374118 0.927381i $$-0.377946\pi$$
0.374118 + 0.927381i $$0.377946\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −238.662 −0.982149
$$10$$ 0 0
$$11$$ −72.0965 −0.179652 −0.0898260 0.995957i $$-0.528631\pi$$
−0.0898260 + 0.995957i $$0.528631\pi$$
$$12$$ 0 0
$$13$$ −632.317 −1.03771 −0.518856 0.854862i $$-0.673642\pi$$
−0.518856 + 0.854862i $$0.673642\pi$$
$$14$$ 0 0
$$15$$ −87.1170 −0.0999712
$$16$$ 0 0
$$17$$ −1975.92 −1.65824 −0.829121 0.559069i $$-0.811159\pi$$
−0.829121 + 0.559069i $$0.811159\pi$$
$$18$$ 0 0
$$19$$ 1864.93 1.18516 0.592581 0.805511i $$-0.298109\pi$$
0.592581 + 0.805511i $$0.298109\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −413.711 −0.163071 −0.0815356 0.996670i $$-0.525982\pi$$
−0.0815356 + 0.996670i $$0.525982\pi$$
$$24$$ 0 0
$$25$$ −1375.45 −0.440144
$$26$$ 0 0
$$27$$ 1003.19 0.264833
$$28$$ 0 0
$$29$$ 731.934 0.161613 0.0808066 0.996730i $$-0.474250\pi$$
0.0808066 + 0.996730i $$0.474250\pi$$
$$30$$ 0 0
$$31$$ −6123.18 −1.14439 −0.572193 0.820119i $$-0.693907\pi$$
−0.572193 + 0.820119i $$0.693907\pi$$
$$32$$ 0 0
$$33$$ 150.160 0.0240032
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 10350.4 1.24295 0.621473 0.783436i $$-0.286535\pi$$
0.621473 + 0.783436i $$0.286535\pi$$
$$38$$ 0 0
$$39$$ 1316.97 0.138648
$$40$$ 0 0
$$41$$ −3529.84 −0.327941 −0.163970 0.986465i $$-0.552430\pi$$
−0.163970 + 0.986465i $$0.552430\pi$$
$$42$$ 0 0
$$43$$ 14515.2 1.19716 0.598581 0.801062i $$-0.295731\pi$$
0.598581 + 0.801062i $$0.295731\pi$$
$$44$$ 0 0
$$45$$ −9982.67 −0.734878
$$46$$ 0 0
$$47$$ 21423.3 1.41463 0.707314 0.706900i $$-0.249907\pi$$
0.707314 + 0.706900i $$0.249907\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 4115.38 0.221557
$$52$$ 0 0
$$53$$ 12579.5 0.615138 0.307569 0.951526i $$-0.400485\pi$$
0.307569 + 0.951526i $$0.400485\pi$$
$$54$$ 0 0
$$55$$ −3015.62 −0.134422
$$56$$ 0 0
$$57$$ −3884.20 −0.158349
$$58$$ 0 0
$$59$$ 36133.9 1.35140 0.675702 0.737175i $$-0.263841\pi$$
0.675702 + 0.737175i $$0.263841\pi$$
$$60$$ 0 0
$$61$$ 4024.80 0.138490 0.0692451 0.997600i $$-0.477941\pi$$
0.0692451 + 0.997600i $$0.477941\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −26448.3 −0.776453
$$66$$ 0 0
$$67$$ −15565.9 −0.423632 −0.211816 0.977310i $$-0.567938\pi$$
−0.211816 + 0.977310i $$0.567938\pi$$
$$68$$ 0 0
$$69$$ 861.661 0.0217878
$$70$$ 0 0
$$71$$ −12180.8 −0.286766 −0.143383 0.989667i $$-0.545798\pi$$
−0.143383 + 0.989667i $$0.545798\pi$$
$$72$$ 0 0
$$73$$ 19589.1 0.430237 0.215119 0.976588i $$-0.430986\pi$$
0.215119 + 0.976588i $$0.430986\pi$$
$$74$$ 0 0
$$75$$ 2864.74 0.0588073
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −36089.8 −0.650604 −0.325302 0.945610i $$-0.605466\pi$$
−0.325302 + 0.945610i $$0.605466\pi$$
$$80$$ 0 0
$$81$$ 55905.5 0.946764
$$82$$ 0 0
$$83$$ 24572.6 0.391522 0.195761 0.980652i $$-0.437282\pi$$
0.195761 + 0.980652i $$0.437282\pi$$
$$84$$ 0 0
$$85$$ −82648.2 −1.24076
$$86$$ 0 0
$$87$$ −1524.44 −0.0215930
$$88$$ 0 0
$$89$$ −70243.3 −0.940005 −0.470002 0.882665i $$-0.655747\pi$$
−0.470002 + 0.882665i $$0.655747\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 12753.1 0.152901
$$94$$ 0 0
$$95$$ 78005.4 0.886779
$$96$$ 0 0
$$97$$ 105758. 1.14126 0.570630 0.821207i $$-0.306699\pi$$
0.570630 + 0.821207i $$0.306699\pi$$
$$98$$ 0 0
$$99$$ 17206.7 0.176445
$$100$$ 0 0
$$101$$ 36461.8 0.355660 0.177830 0.984061i $$-0.443092\pi$$
0.177830 + 0.984061i $$0.443092\pi$$
$$102$$ 0 0
$$103$$ −64520.1 −0.599242 −0.299621 0.954058i $$-0.596860\pi$$
−0.299621 + 0.954058i $$0.596860\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −66045.6 −0.557679 −0.278840 0.960338i $$-0.589950\pi$$
−0.278840 + 0.960338i $$0.589950\pi$$
$$108$$ 0 0
$$109$$ −37938.0 −0.305850 −0.152925 0.988238i $$-0.548869\pi$$
−0.152925 + 0.988238i $$0.548869\pi$$
$$110$$ 0 0
$$111$$ −21557.4 −0.166069
$$112$$ 0 0
$$113$$ 123802. 0.912080 0.456040 0.889959i $$-0.349267\pi$$
0.456040 + 0.889959i $$0.349267\pi$$
$$114$$ 0 0
$$115$$ −17304.5 −0.122016
$$116$$ 0 0
$$117$$ 150910. 1.01919
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −155853. −0.967725
$$122$$ 0 0
$$123$$ 7351.81 0.0438159
$$124$$ 0 0
$$125$$ −188243. −1.07757
$$126$$ 0 0
$$127$$ −128724. −0.708189 −0.354095 0.935210i $$-0.615211\pi$$
−0.354095 + 0.935210i $$0.615211\pi$$
$$128$$ 0 0
$$129$$ −30231.8 −0.159952
$$130$$ 0 0
$$131$$ 147902. 0.753003 0.376501 0.926416i $$-0.377127\pi$$
0.376501 + 0.926416i $$0.377127\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 41961.0 0.198158
$$136$$ 0 0
$$137$$ −91157.4 −0.414945 −0.207472 0.978241i $$-0.566524\pi$$
−0.207472 + 0.978241i $$0.566524\pi$$
$$138$$ 0 0
$$139$$ 334657. 1.46914 0.734570 0.678533i $$-0.237384\pi$$
0.734570 + 0.678533i $$0.237384\pi$$
$$140$$ 0 0
$$141$$ −44619.7 −0.189007
$$142$$ 0 0
$$143$$ 45587.8 0.186427
$$144$$ 0 0
$$145$$ 30615.1 0.120925
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 138271. 0.510231 0.255115 0.966911i $$-0.417887\pi$$
0.255115 + 0.966911i $$0.417887\pi$$
$$150$$ 0 0
$$151$$ −111169. −0.396773 −0.198386 0.980124i $$-0.563570\pi$$
−0.198386 + 0.980124i $$0.563570\pi$$
$$152$$ 0 0
$$153$$ 471578. 1.62864
$$154$$ 0 0
$$155$$ −256118. −0.856270
$$156$$ 0 0
$$157$$ −38148.5 −0.123517 −0.0617587 0.998091i $$-0.519671\pi$$
−0.0617587 + 0.998091i $$0.519671\pi$$
$$158$$ 0 0
$$159$$ −26200.0 −0.0821881
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 212905. 0.627648 0.313824 0.949481i $$-0.398390\pi$$
0.313824 + 0.949481i $$0.398390\pi$$
$$164$$ 0 0
$$165$$ 6280.83 0.0179600
$$166$$ 0 0
$$167$$ 120396. 0.334057 0.167028 0.985952i $$-0.446583\pi$$
0.167028 + 0.985952i $$0.446583\pi$$
$$168$$ 0 0
$$169$$ 28532.2 0.0768456
$$170$$ 0 0
$$171$$ −445087. −1.16400
$$172$$ 0 0
$$173$$ 712914. 1.81101 0.905507 0.424331i $$-0.139491\pi$$
0.905507 + 0.424331i $$0.139491\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −75258.4 −0.180560
$$178$$ 0 0
$$179$$ 749738. 1.74895 0.874474 0.485072i $$-0.161207\pi$$
0.874474 + 0.485072i $$0.161207\pi$$
$$180$$ 0 0
$$181$$ 623718. 1.41511 0.707557 0.706656i $$-0.249797\pi$$
0.707557 + 0.706656i $$0.249797\pi$$
$$182$$ 0 0
$$183$$ −8382.69 −0.0185036
$$184$$ 0 0
$$185$$ 432932. 0.930016
$$186$$ 0 0
$$187$$ 142457. 0.297907
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −417726. −0.828530 −0.414265 0.910156i $$-0.635961\pi$$
−0.414265 + 0.910156i $$0.635961\pi$$
$$192$$ 0 0
$$193$$ 770700. 1.48933 0.744667 0.667436i $$-0.232608\pi$$
0.744667 + 0.667436i $$0.232608\pi$$
$$194$$ 0 0
$$195$$ 55085.6 0.103741
$$196$$ 0 0
$$197$$ −479193. −0.879721 −0.439861 0.898066i $$-0.644972\pi$$
−0.439861 + 0.898066i $$0.644972\pi$$
$$198$$ 0 0
$$199$$ 428686. 0.767373 0.383687 0.923463i $$-0.374654\pi$$
0.383687 + 0.923463i $$0.374654\pi$$
$$200$$ 0 0
$$201$$ 32420.2 0.0566011
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −147645. −0.245377
$$206$$ 0 0
$$207$$ 98737.0 0.160160
$$208$$ 0 0
$$209$$ −134455. −0.212917
$$210$$ 0 0
$$211$$ 588544. 0.910066 0.455033 0.890475i $$-0.349627\pi$$
0.455033 + 0.890475i $$0.349627\pi$$
$$212$$ 0 0
$$213$$ 25369.6 0.0383147
$$214$$ 0 0
$$215$$ 607138. 0.895759
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −40799.5 −0.0574837
$$220$$ 0 0
$$221$$ 1.24941e6 1.72078
$$222$$ 0 0
$$223$$ 363249. 0.489151 0.244575 0.969630i $$-0.421351\pi$$
0.244575 + 0.969630i $$0.421351\pi$$
$$224$$ 0 0
$$225$$ 328268. 0.432287
$$226$$ 0 0
$$227$$ 843041. 1.08589 0.542943 0.839770i $$-0.317310\pi$$
0.542943 + 0.839770i $$0.317310\pi$$
$$228$$ 0 0
$$229$$ −568666. −0.716587 −0.358293 0.933609i $$-0.616641\pi$$
−0.358293 + 0.933609i $$0.616641\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1.05651e6 −1.27492 −0.637461 0.770482i $$-0.720015\pi$$
−0.637461 + 0.770482i $$0.720015\pi$$
$$234$$ 0 0
$$235$$ 896086. 1.05847
$$236$$ 0 0
$$237$$ 75166.5 0.0869267
$$238$$ 0 0
$$239$$ −853715. −0.966759 −0.483379 0.875411i $$-0.660591\pi$$
−0.483379 + 0.875411i $$0.660591\pi$$
$$240$$ 0 0
$$241$$ −388888. −0.431302 −0.215651 0.976470i $$-0.569187\pi$$
−0.215651 + 0.976470i $$0.569187\pi$$
$$242$$ 0 0
$$243$$ −360212. −0.391330
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.17922e6 −1.22986
$$248$$ 0 0
$$249$$ −51178.9 −0.0523109
$$250$$ 0 0
$$251$$ 839328. 0.840906 0.420453 0.907314i $$-0.361871\pi$$
0.420453 + 0.907314i $$0.361871\pi$$
$$252$$ 0 0
$$253$$ 29827.1 0.0292961
$$254$$ 0 0
$$255$$ 172137. 0.165776
$$256$$ 0 0
$$257$$ 291986. 0.275759 0.137879 0.990449i $$-0.455971\pi$$
0.137879 + 0.990449i $$0.455971\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −174685. −0.158728
$$262$$ 0 0
$$263$$ 288495. 0.257187 0.128594 0.991697i $$-0.458954\pi$$
0.128594 + 0.991697i $$0.458954\pi$$
$$264$$ 0 0
$$265$$ 526169. 0.460268
$$266$$ 0 0
$$267$$ 146300. 0.125593
$$268$$ 0 0
$$269$$ −259370. −0.218544 −0.109272 0.994012i $$-0.534852\pi$$
−0.109272 + 0.994012i $$0.534852\pi$$
$$270$$ 0 0
$$271$$ 2.19551e6 1.81599 0.907994 0.418984i $$-0.137614\pi$$
0.907994 + 0.418984i $$0.137614\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 99165.1 0.0790728
$$276$$ 0 0
$$277$$ 126991. 0.0994426 0.0497213 0.998763i $$-0.484167\pi$$
0.0497213 + 0.998763i $$0.484167\pi$$
$$278$$ 0 0
$$279$$ 1.46137e6 1.12396
$$280$$ 0 0
$$281$$ −2.22759e6 −1.68294 −0.841472 0.540301i $$-0.818310\pi$$
−0.841472 + 0.540301i $$0.818310\pi$$
$$282$$ 0 0
$$283$$ −1.18895e6 −0.882463 −0.441231 0.897393i $$-0.645458\pi$$
−0.441231 + 0.897393i $$0.645458\pi$$
$$284$$ 0 0
$$285$$ −162467. −0.118482
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2.48442e6 1.74977
$$290$$ 0 0
$$291$$ −220269. −0.152483
$$292$$ 0 0
$$293$$ 1.83223e6 1.24684 0.623421 0.781886i $$-0.285742\pi$$
0.623421 + 0.781886i $$0.285742\pi$$
$$294$$ 0 0
$$295$$ 1.51140e6 1.01117
$$296$$ 0 0
$$297$$ −72326.3 −0.0475779
$$298$$ 0 0
$$299$$ 261596. 0.169221
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −75941.3 −0.0475194
$$304$$ 0 0
$$305$$ 168348. 0.103623
$$306$$ 0 0
$$307$$ 717638. 0.434569 0.217285 0.976108i $$-0.430280\pi$$
0.217285 + 0.976108i $$0.430280\pi$$
$$308$$ 0 0
$$309$$ 134380. 0.0800642
$$310$$ 0 0
$$311$$ 856892. 0.502372 0.251186 0.967939i $$-0.419179\pi$$
0.251186 + 0.967939i $$0.419179\pi$$
$$312$$ 0 0
$$313$$ 1.61699e6 0.932924 0.466462 0.884541i $$-0.345528\pi$$
0.466462 + 0.884541i $$0.345528\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.26559e6 −1.26629 −0.633145 0.774033i $$-0.718236\pi$$
−0.633145 + 0.774033i $$0.718236\pi$$
$$318$$ 0 0
$$319$$ −52769.8 −0.0290341
$$320$$ 0 0
$$321$$ 137557. 0.0745111
$$322$$ 0 0
$$323$$ −3.68495e6 −1.96528
$$324$$ 0 0
$$325$$ 869721. 0.456743
$$326$$ 0 0
$$327$$ 79015.9 0.0408644
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −709650. −0.356020 −0.178010 0.984029i $$-0.556966\pi$$
−0.178010 + 0.984029i $$0.556966\pi$$
$$332$$ 0 0
$$333$$ −2.47024e6 −1.22076
$$334$$ 0 0
$$335$$ −651086. −0.316976
$$336$$ 0 0
$$337$$ 603572. 0.289504 0.144752 0.989468i $$-0.453762\pi$$
0.144752 + 0.989468i $$0.453762\pi$$
$$338$$ 0 0
$$339$$ −257851. −0.121862
$$340$$ 0 0
$$341$$ 441459. 0.205591
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 36041.2 0.0163024
$$346$$ 0 0
$$347$$ −1.75731e6 −0.783474 −0.391737 0.920077i $$-0.628126\pi$$
−0.391737 + 0.920077i $$0.628126\pi$$
$$348$$ 0 0
$$349$$ 391875. 0.172220 0.0861102 0.996286i $$-0.472556\pi$$
0.0861102 + 0.996286i $$0.472556\pi$$
$$350$$ 0 0
$$351$$ −634333. −0.274821
$$352$$ 0 0
$$353$$ 492407. 0.210323 0.105162 0.994455i $$-0.466464\pi$$
0.105162 + 0.994455i $$0.466464\pi$$
$$354$$ 0 0
$$355$$ −509492. −0.214569
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 3.77032e6 1.54398 0.771991 0.635634i $$-0.219261\pi$$
0.771991 + 0.635634i $$0.219261\pi$$
$$360$$ 0 0
$$361$$ 1.00185e6 0.404607
$$362$$ 0 0
$$363$$ 324605. 0.129297
$$364$$ 0 0
$$365$$ 819367. 0.321919
$$366$$ 0 0
$$367$$ 2.19768e6 0.851726 0.425863 0.904788i $$-0.359970\pi$$
0.425863 + 0.904788i $$0.359970\pi$$
$$368$$ 0 0
$$369$$ 842439. 0.322086
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1.65636e6 −0.616427 −0.308213 0.951317i $$-0.599731\pi$$
−0.308213 + 0.951317i $$0.599731\pi$$
$$374$$ 0 0
$$375$$ 392066. 0.143973
$$376$$ 0 0
$$377$$ −462814. −0.167708
$$378$$ 0 0
$$379$$ 2.82050e6 1.00862 0.504310 0.863523i $$-0.331747\pi$$
0.504310 + 0.863523i $$0.331747\pi$$
$$380$$ 0 0
$$381$$ 268101. 0.0946206
$$382$$ 0 0
$$383$$ −3.24845e6 −1.13156 −0.565781 0.824555i $$-0.691425\pi$$
−0.565781 + 0.824555i $$0.691425\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −3.46424e6 −1.17579
$$388$$ 0 0
$$389$$ 4.65348e6 1.55921 0.779604 0.626273i $$-0.215420\pi$$
0.779604 + 0.626273i $$0.215420\pi$$
$$390$$ 0 0
$$391$$ 817461. 0.270411
$$392$$ 0 0
$$393$$ −308045. −0.100608
$$394$$ 0 0
$$395$$ −1.50955e6 −0.486805
$$396$$ 0 0
$$397$$ −1.16361e6 −0.370536 −0.185268 0.982688i $$-0.559315\pi$$
−0.185268 + 0.982688i $$0.559315\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 322380. 0.100117 0.0500584 0.998746i $$-0.484059\pi$$
0.0500584 + 0.998746i $$0.484059\pi$$
$$402$$ 0 0
$$403$$ 3.87179e6 1.18754
$$404$$ 0 0
$$405$$ 2.33839e6 0.708403
$$406$$ 0 0
$$407$$ −746226. −0.223298
$$408$$ 0 0
$$409$$ 1.38690e6 0.409956 0.204978 0.978767i $$-0.434288\pi$$
0.204978 + 0.978767i $$0.434288\pi$$
$$410$$ 0 0
$$411$$ 189859. 0.0554405
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1.02781e6 0.292950
$$416$$ 0 0
$$417$$ −697011. −0.196291
$$418$$ 0 0
$$419$$ −4.90871e6 −1.36594 −0.682971 0.730446i $$-0.739312\pi$$
−0.682971 + 0.730446i $$0.739312\pi$$
$$420$$ 0 0
$$421$$ 2.43924e6 0.670733 0.335367 0.942088i $$-0.391140\pi$$
0.335367 + 0.942088i $$0.391140\pi$$
$$422$$ 0 0
$$423$$ −5.11293e6 −1.38937
$$424$$ 0 0
$$425$$ 2.71779e6 0.729865
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −94948.7 −0.0249084
$$430$$ 0 0
$$431$$ 5.22752e6 1.35551 0.677755 0.735288i $$-0.262953\pi$$
0.677755 + 0.735288i $$0.262953\pi$$
$$432$$ 0 0
$$433$$ −2.63022e6 −0.674174 −0.337087 0.941473i $$-0.609442\pi$$
−0.337087 + 0.941473i $$0.609442\pi$$
$$434$$ 0 0
$$435$$ −63763.9 −0.0161567
$$436$$ 0 0
$$437$$ −771539. −0.193266
$$438$$ 0 0
$$439$$ −2.55412e6 −0.632527 −0.316264 0.948671i $$-0.602428\pi$$
−0.316264 + 0.948671i $$0.602428\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −3.83900e6 −0.929414 −0.464707 0.885465i $$-0.653840\pi$$
−0.464707 + 0.885465i $$0.653840\pi$$
$$444$$ 0 0
$$445$$ −2.93811e6 −0.703345
$$446$$ 0 0
$$447$$ −287986. −0.0681715
$$448$$ 0 0
$$449$$ 1.49369e6 0.349658 0.174829 0.984599i $$-0.444063\pi$$
0.174829 + 0.984599i $$0.444063\pi$$
$$450$$ 0 0
$$451$$ 254489. 0.0589152
$$452$$ 0 0
$$453$$ 231539. 0.0530126
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −2.16221e6 −0.484293 −0.242146 0.970240i $$-0.577851\pi$$
−0.242146 + 0.970240i $$0.577851\pi$$
$$458$$ 0 0
$$459$$ −1.98222e6 −0.439158
$$460$$ 0 0
$$461$$ −6.11949e6 −1.34111 −0.670553 0.741862i $$-0.733943\pi$$
−0.670553 + 0.741862i $$0.733943\pi$$
$$462$$ 0 0
$$463$$ −3.93615e6 −0.853335 −0.426667 0.904409i $$-0.640313\pi$$
−0.426667 + 0.904409i $$0.640313\pi$$
$$464$$ 0 0
$$465$$ 533433. 0.114406
$$466$$ 0 0
$$467$$ 5.47044e6 1.16073 0.580363 0.814358i $$-0.302911\pi$$
0.580363 + 0.814358i $$0.302911\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 79454.3 0.0165031
$$472$$ 0 0
$$473$$ −1.04650e6 −0.215073
$$474$$ 0 0
$$475$$ −2.56511e6 −0.521641
$$476$$ 0 0
$$477$$ −3.00224e6 −0.604157
$$478$$ 0 0
$$479$$ 6.66289e6 1.32686 0.663428 0.748240i $$-0.269101\pi$$
0.663428 + 0.748240i $$0.269101\pi$$
$$480$$ 0 0
$$481$$ −6.54473e6 −1.28982
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4.42362e6 0.853931
$$486$$ 0 0
$$487$$ −9.53693e6 −1.82216 −0.911079 0.412232i $$-0.864749\pi$$
−0.911079 + 0.412232i $$0.864749\pi$$
$$488$$ 0 0
$$489$$ −443430. −0.0838595
$$490$$ 0 0
$$491$$ 8.19294e6 1.53369 0.766843 0.641835i $$-0.221827\pi$$
0.766843 + 0.641835i $$0.221827\pi$$
$$492$$ 0 0
$$493$$ −1.44625e6 −0.267994
$$494$$ 0 0
$$495$$ 719715. 0.132022
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.31437e6 0.775650 0.387825 0.921733i $$-0.373226\pi$$
0.387825 + 0.921733i $$0.373226\pi$$
$$500$$ 0 0
$$501$$ −250756. −0.0446331
$$502$$ 0 0
$$503$$ 1.04015e7 1.83306 0.916529 0.399968i $$-0.130979\pi$$
0.916529 + 0.399968i $$0.130979\pi$$
$$504$$ 0 0
$$505$$ 1.52511e6 0.266117
$$506$$ 0 0
$$507$$ −59425.9 −0.0102673
$$508$$ 0 0
$$509$$ −3.09396e6 −0.529322 −0.264661 0.964342i $$-0.585260\pi$$
−0.264661 + 0.964342i $$0.585260\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 1.87087e6 0.313870
$$514$$ 0 0
$$515$$ −2.69872e6 −0.448374
$$516$$ 0 0
$$517$$ −1.54455e6 −0.254141
$$518$$ 0 0
$$519$$ −1.48483e6 −0.241968
$$520$$ 0 0
$$521$$ −7.60175e6 −1.22693 −0.613464 0.789723i $$-0.710224\pi$$
−0.613464 + 0.789723i $$0.710224\pi$$
$$522$$ 0 0
$$523$$ −4.75669e6 −0.760415 −0.380208 0.924901i $$-0.624147\pi$$
−0.380208 + 0.924901i $$0.624147\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1.20989e7 1.89767
$$528$$ 0 0
$$529$$ −6.26519e6 −0.973408
$$530$$ 0 0
$$531$$ −8.62380e6 −1.32728
$$532$$ 0 0
$$533$$ 2.23198e6 0.340308
$$534$$ 0 0
$$535$$ −2.76253e6 −0.417275
$$536$$ 0 0
$$537$$ −1.56153e6 −0.233676
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 1.10052e7 1.61661 0.808305 0.588764i $$-0.200385\pi$$
0.808305 + 0.588764i $$0.200385\pi$$
$$542$$ 0 0
$$543$$ −1.29906e6 −0.189072
$$544$$ 0 0
$$545$$ −1.58686e6 −0.228848
$$546$$ 0 0
$$547$$ 4.46311e6 0.637778 0.318889 0.947792i $$-0.396690\pi$$
0.318889 + 0.947792i $$0.396690\pi$$
$$548$$ 0 0
$$549$$ −960566. −0.136018
$$550$$ 0 0
$$551$$ 1.36500e6 0.191538
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −901694. −0.124259
$$556$$ 0 0
$$557$$ −6.45222e6 −0.881194 −0.440597 0.897705i $$-0.645233\pi$$
−0.440597 + 0.897705i $$0.645233\pi$$
$$558$$ 0 0
$$559$$ −9.17823e6 −1.24231
$$560$$ 0 0
$$561$$ −296704. −0.0398031
$$562$$ 0 0
$$563$$ −1.74748e6 −0.232349 −0.116175 0.993229i $$-0.537063\pi$$
−0.116175 + 0.993229i $$0.537063\pi$$
$$564$$ 0 0
$$565$$ 5.17836e6 0.682450
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 512789. 0.0663985 0.0331992 0.999449i $$-0.489430\pi$$
0.0331992 + 0.999449i $$0.489430\pi$$
$$570$$ 0 0
$$571$$ −5.22364e6 −0.670475 −0.335238 0.942134i $$-0.608817\pi$$
−0.335238 + 0.942134i $$0.608817\pi$$
$$572$$ 0 0
$$573$$ 870024. 0.110699
$$574$$ 0 0
$$575$$ 569038. 0.0717748
$$576$$ 0 0
$$577$$ 6.63973e6 0.830254 0.415127 0.909763i $$-0.363737\pi$$
0.415127 + 0.909763i $$0.363737\pi$$
$$578$$ 0 0
$$579$$ −1.60519e6 −0.198989
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −906935. −0.110511
$$584$$ 0 0
$$585$$ 6.31221e6 0.762592
$$586$$ 0 0
$$587$$ −774096. −0.0927256 −0.0463628 0.998925i $$-0.514763\pi$$
−0.0463628 + 0.998925i $$0.514763\pi$$
$$588$$ 0 0
$$589$$ −1.14193e7 −1.35628
$$590$$ 0 0
$$591$$ 998046. 0.117539
$$592$$ 0 0
$$593$$ −1.43756e7 −1.67876 −0.839379 0.543546i $$-0.817081\pi$$
−0.839379 + 0.543546i $$0.817081\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −892851. −0.102528
$$598$$ 0 0
$$599$$ −1.20835e7 −1.37602 −0.688010 0.725701i $$-0.741516\pi$$
−0.688010 + 0.725701i $$0.741516\pi$$
$$600$$ 0 0
$$601$$ 5.75607e6 0.650040 0.325020 0.945707i $$-0.394629\pi$$
0.325020 + 0.945707i $$0.394629\pi$$
$$602$$ 0 0
$$603$$ 3.71500e6 0.416069
$$604$$ 0 0
$$605$$ −6.51897e6 −0.724086
$$606$$ 0 0
$$607$$ −4.20121e6 −0.462810 −0.231405 0.972858i $$-0.574332\pi$$
−0.231405 + 0.972858i $$0.574332\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.35463e7 −1.46798
$$612$$ 0 0
$$613$$ −2.64543e6 −0.284344 −0.142172 0.989842i $$-0.545409\pi$$
−0.142172 + 0.989842i $$0.545409\pi$$
$$614$$ 0 0
$$615$$ 307509. 0.0327846
$$616$$ 0 0
$$617$$ 6.43533e6 0.680546 0.340273 0.940327i $$-0.389480\pi$$
0.340273 + 0.940327i $$0.389480\pi$$
$$618$$ 0 0
$$619$$ −1.41177e7 −1.48094 −0.740469 0.672090i $$-0.765397\pi$$
−0.740469 + 0.672090i $$0.765397\pi$$
$$620$$ 0 0
$$621$$ −415029. −0.0431867
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −3.57548e6 −0.366129
$$626$$ 0 0
$$627$$ 280037. 0.0284476
$$628$$ 0 0
$$629$$ −2.04516e7 −2.06111
$$630$$ 0 0
$$631$$ 4.70856e6 0.470777 0.235388 0.971901i $$-0.424364\pi$$
0.235388 + 0.971901i $$0.424364\pi$$
$$632$$ 0 0
$$633$$ −1.22580e6 −0.121593
$$634$$ 0 0
$$635$$ −5.38421e6 −0.529892
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 2.90708e6 0.281647
$$640$$ 0 0
$$641$$ −1.04174e7 −1.00141 −0.500707 0.865617i $$-0.666927\pi$$
−0.500707 + 0.865617i $$0.666927\pi$$
$$642$$ 0 0
$$643$$ −1.27284e7 −1.21407 −0.607037 0.794674i $$-0.707642\pi$$
−0.607037 + 0.794674i $$0.707642\pi$$
$$644$$ 0 0
$$645$$ −1.26452e6 −0.119682
$$646$$ 0 0
$$647$$ 1.61348e7 1.51531 0.757657 0.652653i $$-0.226344\pi$$
0.757657 + 0.652653i $$0.226344\pi$$
$$648$$ 0 0
$$649$$ −2.60513e6 −0.242783
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1.50295e7 −1.37931 −0.689654 0.724139i $$-0.742237\pi$$
−0.689654 + 0.724139i $$0.742237\pi$$
$$654$$ 0 0
$$655$$ 6.18640e6 0.563423
$$656$$ 0 0
$$657$$ −4.67518e6 −0.422557
$$658$$ 0 0
$$659$$ −1.67927e7 −1.50628 −0.753140 0.657860i $$-0.771462\pi$$
−0.753140 + 0.657860i $$0.771462\pi$$
$$660$$ 0 0
$$661$$ 1.08540e7 0.966246 0.483123 0.875552i $$-0.339502\pi$$
0.483123 + 0.875552i $$0.339502\pi$$
$$662$$ 0 0
$$663$$ −2.60223e6 −0.229912
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −302809. −0.0263544
$$668$$ 0 0
$$669$$ −756562. −0.0653551
$$670$$ 0 0
$$671$$ −290174. −0.0248801
$$672$$ 0 0
$$673$$ 1.23697e7 1.05274 0.526371 0.850255i $$-0.323552\pi$$
0.526371 + 0.850255i $$0.323552\pi$$
$$674$$ 0 0
$$675$$ −1.37983e6 −0.116565
$$676$$ 0 0
$$677$$ −1.00501e6 −0.0842746 −0.0421373 0.999112i $$-0.513417\pi$$
−0.0421373 + 0.999112i $$0.513417\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −1.75586e6 −0.145084
$$682$$ 0 0
$$683$$ −1.87019e6 −0.153403 −0.0767014 0.997054i $$-0.524439\pi$$
−0.0767014 + 0.997054i $$0.524439\pi$$
$$684$$ 0 0
$$685$$ −3.81290e6 −0.310476
$$686$$ 0 0
$$687$$ 1.18440e6 0.0957426
$$688$$ 0 0
$$689$$ −7.95421e6 −0.638336
$$690$$ 0 0
$$691$$ −1.93867e7 −1.54457 −0.772286 0.635275i $$-0.780887\pi$$
−0.772286 + 0.635275i $$0.780887\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 1.39979e7 1.09926
$$696$$ 0 0
$$697$$ 6.97469e6 0.543805
$$698$$ 0 0
$$699$$ 2.20046e6 0.170342
$$700$$ 0 0
$$701$$ 1.17488e7 0.903024 0.451512 0.892265i $$-0.350885\pi$$
0.451512 + 0.892265i $$0.350885\pi$$
$$702$$ 0 0
$$703$$ 1.93027e7 1.47309
$$704$$ 0 0
$$705$$ −1.86634e6 −0.141422
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 1.67948e7 1.25475 0.627377 0.778716i $$-0.284129\pi$$
0.627377 + 0.778716i $$0.284129\pi$$
$$710$$ 0 0
$$711$$ 8.61326e6 0.638990
$$712$$ 0 0
$$713$$ 2.53322e6 0.186616
$$714$$ 0 0
$$715$$ 1.90683e6 0.139491
$$716$$ 0 0
$$717$$ 1.77809e6 0.129168
$$718$$ 0 0
$$719$$ 1.65130e7 1.19126 0.595628 0.803261i $$-0.296903\pi$$
0.595628 + 0.803261i $$0.296903\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 809961. 0.0576260
$$724$$ 0 0
$$725$$ −1.00674e6 −0.0711331
$$726$$ 0 0
$$727$$ 1.25756e6 0.0882453 0.0441227 0.999026i $$-0.485951\pi$$
0.0441227 + 0.999026i $$0.485951\pi$$
$$728$$ 0 0
$$729$$ −1.28348e7 −0.894479
$$730$$ 0 0
$$731$$ −2.86810e7 −1.98518
$$732$$ 0 0
$$733$$ −1.98332e6 −0.136343 −0.0681716 0.997674i $$-0.521717\pi$$
−0.0681716 + 0.997674i $$0.521717\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.12225e6 0.0761063
$$738$$ 0 0
$$739$$ 2.38807e7 1.60856 0.804278 0.594253i $$-0.202552\pi$$
0.804278 + 0.594253i $$0.202552\pi$$
$$740$$ 0 0
$$741$$ 2.45604e6 0.164320
$$742$$ 0 0
$$743$$ 1.90819e7 1.26809 0.634043 0.773298i $$-0.281394\pi$$
0.634043 + 0.773298i $$0.281394\pi$$
$$744$$ 0 0
$$745$$ 5.78356e6 0.381773
$$746$$ 0 0
$$747$$ −5.86455e6 −0.384532
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 3.75805e6 0.243144 0.121572 0.992583i $$-0.461206\pi$$
0.121572 + 0.992583i $$0.461206\pi$$
$$752$$ 0 0
$$753$$ −1.74812e6 −0.112353
$$754$$ 0 0
$$755$$ −4.64994e6 −0.296880
$$756$$ 0 0
$$757$$ 1.69904e7 1.07761 0.538807 0.842429i $$-0.318875\pi$$
0.538807 + 0.842429i $$0.318875\pi$$
$$758$$ 0 0
$$759$$ −62122.7 −0.00391423
$$760$$ 0 0
$$761$$ 2.23998e7 1.40211 0.701056 0.713106i $$-0.252712\pi$$
0.701056 + 0.713106i $$0.252712\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 1.97250e7 1.21861
$$766$$ 0 0
$$767$$ −2.28481e7 −1.40237
$$768$$ 0 0
$$769$$ −1.87866e7 −1.14560 −0.572799 0.819696i $$-0.694142\pi$$
−0.572799 + 0.819696i $$0.694142\pi$$
$$770$$ 0 0
$$771$$ −608138. −0.0368439
$$772$$ 0 0
$$773$$ 9.30837e6 0.560306 0.280153 0.959955i $$-0.409615\pi$$
0.280153 + 0.959955i $$0.409615\pi$$
$$774$$ 0 0
$$775$$ 8.42212e6 0.503694
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −6.58288e6 −0.388662
$$780$$ 0 0
$$781$$ 878189. 0.0515182
$$782$$ 0 0
$$783$$ 734267. 0.0428006
$$784$$ 0 0
$$785$$ −1.59566e6 −0.0924201
$$786$$ 0 0
$$787$$ −1.73427e7 −0.998111 −0.499056 0.866570i $$-0.666320\pi$$
−0.499056 + 0.866570i $$0.666320\pi$$
$$788$$ 0 0
$$789$$ −600867. −0.0343626
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −2.54495e6 −0.143713
$$794$$ 0 0
$$795$$ −1.09589e6 −0.0614960
$$796$$ 0 0
$$797$$ 3.10445e7 1.73117 0.865584 0.500764i $$-0.166948\pi$$
0.865584 + 0.500764i $$0.166948\pi$$
$$798$$ 0 0
$$799$$ −4.23309e7 −2.34580
$$800$$ 0 0
$$801$$ 1.67644e7 0.923224
$$802$$ 0 0
$$803$$ −1.41231e6 −0.0772930
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 540206. 0.0291995
$$808$$ 0 0
$$809$$ −2.47038e7 −1.32707 −0.663533 0.748147i $$-0.730944\pi$$
−0.663533 + 0.748147i $$0.730944\pi$$
$$810$$ 0 0
$$811$$ −8.42005e6 −0.449534 −0.224767 0.974413i $$-0.572162\pi$$
−0.224767 + 0.974413i $$0.572162\pi$$
$$812$$ 0 0
$$813$$ −4.57273e6 −0.242633
$$814$$ 0 0
$$815$$ 8.90529e6 0.469628
$$816$$ 0 0
$$817$$ 2.70698e7 1.41883
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −2.58827e7 −1.34014 −0.670071 0.742297i $$-0.733736\pi$$
−0.670071 + 0.742297i $$0.733736\pi$$
$$822$$ 0 0
$$823$$ −1.72004e7 −0.885195 −0.442597 0.896720i $$-0.645943\pi$$
−0.442597 + 0.896720i $$0.645943\pi$$
$$824$$ 0 0
$$825$$ −206537. −0.0105649
$$826$$ 0 0
$$827$$ 2.40337e7 1.22196 0.610979 0.791647i $$-0.290776\pi$$
0.610979 + 0.791647i $$0.290776\pi$$
$$828$$ 0 0
$$829$$ −3.24736e7 −1.64113 −0.820567 0.571550i $$-0.806342\pi$$
−0.820567 + 0.571550i $$0.806342\pi$$
$$830$$ 0 0
$$831$$ −264491. −0.0132865
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 5.03587e6 0.249953
$$836$$ 0 0
$$837$$ −6.14269e6 −0.303072
$$838$$ 0 0
$$839$$ 1.24404e7 0.610139 0.305069 0.952330i $$-0.401320\pi$$
0.305069 + 0.952330i $$0.401320\pi$$
$$840$$ 0 0
$$841$$ −1.99754e7 −0.973881
$$842$$ 0 0
$$843$$ 4.63954e6 0.224857
$$844$$ 0 0
$$845$$ 1.19344e6 0.0574986
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 2.47629e6 0.117905
$$850$$ 0 0
$$851$$ −4.28206e6 −0.202689
$$852$$ 0 0
$$853$$ −999355. −0.0470270 −0.0235135 0.999724i $$-0.507485\pi$$
−0.0235135 + 0.999724i $$0.507485\pi$$
$$854$$ 0 0
$$855$$ −1.86169e7 −0.870949
$$856$$ 0 0
$$857$$ −2.64465e7 −1.23003 −0.615016 0.788514i $$-0.710851\pi$$
−0.615016 + 0.788514i $$0.710851\pi$$
$$858$$ 0 0
$$859$$ 2.86716e7 1.32577 0.662887 0.748719i $$-0.269331\pi$$
0.662887 + 0.748719i $$0.269331\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −4.08173e6 −0.186560 −0.0932798 0.995640i $$-0.529735\pi$$
−0.0932798 + 0.995640i $$0.529735\pi$$
$$864$$ 0 0
$$865$$ 2.98195e7 1.35506
$$866$$ 0 0
$$867$$ −5.17446e6 −0.233785
$$868$$ 0 0
$$869$$ 2.60195e6 0.116882
$$870$$ 0 0
$$871$$ 9.84261e6 0.439608
$$872$$ 0 0
$$873$$ −2.52405e7 −1.12089
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −2.82405e7 −1.23986 −0.619931 0.784657i $$-0.712839\pi$$
−0.619931 + 0.784657i $$0.712839\pi$$
$$878$$ 0 0
$$879$$ −3.81610e6 −0.166590
$$880$$ 0 0
$$881$$ 1.61480e7 0.700936 0.350468 0.936575i $$-0.386022\pi$$
0.350468 + 0.936575i $$0.386022\pi$$
$$882$$ 0 0
$$883$$ 3.86021e7 1.66613 0.833065 0.553174i $$-0.186584\pi$$
0.833065 + 0.553174i $$0.186584\pi$$
$$884$$ 0 0
$$885$$ −3.14788e6 −0.135102
$$886$$ 0 0
$$887$$ −7.29088e6 −0.311151 −0.155575 0.987824i $$-0.549723\pi$$
−0.155575 + 0.987824i $$0.549723\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.03059e6 −0.170088
$$892$$ 0 0
$$893$$ 3.99529e7 1.67656
$$894$$ 0 0
$$895$$ 3.13598e7 1.30862
$$896$$ 0 0
$$897$$ −544843. −0.0226095
$$898$$ 0 0
$$899$$ −4.48176e6 −0.184948
$$900$$ 0 0
$$901$$ −2.48561e7 −1.02005
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 2.60886e7 1.05884
$$906$$ 0 0
$$907$$ 3.73335e7 1.50689 0.753443 0.657514i $$-0.228392\pi$$
0.753443 + 0.657514i $$0.228392\pi$$
$$908$$ 0 0
$$909$$ −8.70205e6 −0.349311
$$910$$ 0 0
$$911$$ 2475.17 9.88120e−5 0 4.94060e−5 1.00000i $$-0.499984\pi$$
4.94060e−5 1.00000i $$0.499984\pi$$
$$912$$ 0 0
$$913$$ −1.77160e6 −0.0703377
$$914$$ 0 0
$$915$$ −350628. −0.0138450
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 4.48238e6 0.175073 0.0875366 0.996161i $$-0.472101\pi$$
0.0875366 + 0.996161i $$0.472101\pi$$
$$920$$ 0 0
$$921$$ −1.49467e6 −0.0580625
$$922$$ 0 0
$$923$$ 7.70210e6 0.297581
$$924$$ 0 0
$$925$$ −1.42364e7 −0.547075
$$926$$ 0 0
$$927$$ 1.53985e7 0.588544
$$928$$ 0 0
$$929$$ 2.12859e7 0.809193 0.404596 0.914495i $$-0.367412\pi$$
0.404596 + 0.914495i $$0.367412\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −1.78470e6 −0.0671215
$$934$$ 0 0
$$935$$ 5.95865e6 0.222904
$$936$$ 0 0
$$937$$ 6.79757e6 0.252932 0.126466 0.991971i $$-0.459636\pi$$
0.126466 + 0.991971i $$0.459636\pi$$
$$938$$ 0 0
$$939$$ −3.36781e6 −0.124647
$$940$$ 0 0
$$941$$ 4.90883e7 1.80719 0.903595 0.428388i $$-0.140918\pi$$
0.903595 + 0.428388i $$0.140918\pi$$
$$942$$ 0 0
$$943$$ 1.46033e6 0.0534776
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −2.45484e7 −0.889505 −0.444753 0.895653i $$-0.646708\pi$$
−0.444753 + 0.895653i $$0.646708\pi$$
$$948$$ 0 0
$$949$$ −1.23865e7 −0.446462
$$950$$ 0 0
$$951$$ 4.71869e6 0.169188
$$952$$ 0 0
$$953$$ −513120. −0.0183015 −0.00915075 0.999958i $$-0.502913\pi$$
−0.00915075 + 0.999958i $$0.502913\pi$$
$$954$$ 0 0
$$955$$ −1.74725e7 −0.619935
$$956$$ 0 0
$$957$$ 109907. 0.00387923
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 8.86412e6 0.309619
$$962$$ 0 0
$$963$$ 1.57626e7 0.547724
$$964$$ 0 0
$$965$$ 3.22366e7 1.11437
$$966$$ 0 0
$$967$$ −3.34818e7 −1.15144 −0.575722 0.817645i $$-0.695279\pi$$
−0.575722 + 0.817645i $$0.695279\pi$$
$$968$$ 0 0
$$969$$ 7.67488e6 0.262580
$$970$$ 0 0
$$971$$ −4.76036e6 −0.162029 −0.0810143 0.996713i $$-0.525816\pi$$
−0.0810143 + 0.996713i $$0.525816\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −1.81142e6 −0.0610250
$$976$$ 0 0
$$977$$ 2.87338e7 0.963067 0.481534 0.876428i $$-0.340080\pi$$
0.481534 + 0.876428i $$0.340080\pi$$
$$978$$ 0 0
$$979$$ 5.06430e6 0.168874
$$980$$ 0 0
$$981$$ 9.05437e6 0.300390
$$982$$ 0 0
$$983$$ 4.97072e7 1.64072 0.820362 0.571845i $$-0.193772\pi$$
0.820362 + 0.571845i $$0.193772\pi$$
$$984$$ 0 0
$$985$$ −2.00435e7 −0.658239
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −6.00511e6 −0.195223
$$990$$ 0 0
$$991$$ 2.91066e6 0.0941471 0.0470736 0.998891i $$-0.485010\pi$$
0.0470736 + 0.998891i $$0.485010\pi$$
$$992$$ 0 0
$$993$$ 1.47803e6 0.0475676
$$994$$ 0 0
$$995$$ 1.79309e7 0.574176
$$996$$ 0 0
$$997$$ 1.43353e7 0.456740 0.228370 0.973574i $$-0.426660\pi$$
0.228370 + 0.973574i $$0.426660\pi$$
$$998$$ 0 0
$$999$$ 1.03834e7 0.329174
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.ba.1.1 2
4.3 odd 2 49.6.a.d.1.1 2
7.2 even 3 112.6.i.c.81.2 4
7.4 even 3 112.6.i.c.65.2 4
7.6 odd 2 784.6.a.t.1.2 2
12.11 even 2 441.6.a.n.1.2 2
28.3 even 6 49.6.c.f.30.2 4
28.11 odd 6 7.6.c.a.2.2 4
28.19 even 6 49.6.c.f.18.2 4
28.23 odd 6 7.6.c.a.4.2 yes 4
28.27 even 2 49.6.a.e.1.1 2
84.11 even 6 63.6.e.d.37.1 4
84.23 even 6 63.6.e.d.46.1 4
84.83 odd 2 441.6.a.m.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.c.a.2.2 4 28.11 odd 6
7.6.c.a.4.2 yes 4 28.23 odd 6
49.6.a.d.1.1 2 4.3 odd 2
49.6.a.e.1.1 2 28.27 even 2
49.6.c.f.18.2 4 28.19 even 6
49.6.c.f.30.2 4 28.3 even 6
63.6.e.d.37.1 4 84.11 even 6
63.6.e.d.46.1 4 84.23 even 6
112.6.i.c.65.2 4 7.4 even 3
112.6.i.c.81.2 4 7.2 even 3
441.6.a.m.1.2 2 84.83 odd 2
441.6.a.n.1.2 2 12.11 even 2
784.6.a.t.1.2 2 7.6 odd 2
784.6.a.ba.1.1 2 1.1 even 1 trivial