# Properties

 Label 784.6.a.bf.1.4 Level $784$ Weight $6$ Character 784.1 Self dual yes Analytic conductor $125.741$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{113})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 59x^{2} + 60x + 674$$ x^4 - 2*x^3 - 59*x^2 + 60*x + 674 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}\cdot 7$$ Twist minimal: no (minimal twist has level 49) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-6.22929$$ 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)

 $$f(q)$$ $$=$$ $$q+23.5186 q^{3} +74.2753 q^{5} +310.123 q^{9} +O(q^{10})$$ $$q+23.5186 q^{3} +74.2753 q^{5} +310.123 q^{9} +424.219 q^{11} +252.233 q^{13} +1746.85 q^{15} +1104.35 q^{17} -6.47100 q^{19} +3612.39 q^{23} +2391.82 q^{25} +1578.65 q^{27} -5005.02 q^{29} +2821.69 q^{31} +9977.03 q^{33} -2046.88 q^{37} +5932.17 q^{39} -9393.81 q^{41} -10320.8 q^{43} +23034.5 q^{45} +17035.6 q^{47} +25972.7 q^{51} -39506.7 q^{53} +31509.0 q^{55} -152.189 q^{57} +33949.8 q^{59} -28295.2 q^{61} +18734.7 q^{65} -56100.9 q^{67} +84958.2 q^{69} +15537.4 q^{71} +78219.5 q^{73} +56252.3 q^{75} +45335.5 q^{79} -38232.4 q^{81} +1381.82 q^{83} +82025.7 q^{85} -117711. q^{87} +68879.4 q^{89} +66362.1 q^{93} -480.635 q^{95} -108857. q^{97} +131560. q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 220 q^{9}+O(q^{10})$$ 4 * q + 220 * q^9 $$4 q + 220 q^{9} + 1952 q^{11} + 4096 q^{15} + 7136 q^{23} + 2764 q^{25} - 3352 q^{29} - 9208 q^{37} - 2464 q^{39} - 20448 q^{43} + 67408 q^{51} - 102920 q^{53} - 15576 q^{57} - 63168 q^{65} + 22896 q^{67} + 153824 q^{71} + 90688 q^{79} - 17204 q^{81} + 272656 q^{85} + 247760 q^{93} - 108224 q^{95} + 42272 q^{99}+O(q^{100})$$ 4 * q + 220 * q^9 + 1952 * q^11 + 4096 * q^15 + 7136 * q^23 + 2764 * q^25 - 3352 * q^29 - 9208 * q^37 - 2464 * q^39 - 20448 * q^43 + 67408 * q^51 - 102920 * q^53 - 15576 * q^57 - 63168 * q^65 + 22896 * q^67 + 153824 * q^71 + 90688 * q^79 - 17204 * q^81 + 272656 * q^85 + 247760 * q^93 - 108224 * q^95 + 42272 * 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
$$3$$ 23.5186 1.50872 0.754359 0.656462i $$-0.227948\pi$$
0.754359 + 0.656462i $$0.227948\pi$$
$$4$$ 0 0
$$5$$ 74.2753 1.32868 0.664339 0.747432i $$-0.268713\pi$$
0.664339 + 0.747432i $$0.268713\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 310.123 1.27623
$$10$$ 0 0
$$11$$ 424.219 1.05708 0.528541 0.848908i $$-0.322739\pi$$
0.528541 + 0.848908i $$0.322739\pi$$
$$12$$ 0 0
$$13$$ 252.233 0.413946 0.206973 0.978347i $$-0.433639\pi$$
0.206973 + 0.978347i $$0.433639\pi$$
$$14$$ 0 0
$$15$$ 1746.85 2.00460
$$16$$ 0 0
$$17$$ 1104.35 0.926794 0.463397 0.886151i $$-0.346630\pi$$
0.463397 + 0.886151i $$0.346630\pi$$
$$18$$ 0 0
$$19$$ −6.47100 −0.00411232 −0.00205616 0.999998i $$-0.500654\pi$$
−0.00205616 + 0.999998i $$0.500654\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3612.39 1.42388 0.711942 0.702239i $$-0.247816\pi$$
0.711942 + 0.702239i $$0.247816\pi$$
$$24$$ 0 0
$$25$$ 2391.82 0.765383
$$26$$ 0 0
$$27$$ 1578.65 0.416751
$$28$$ 0 0
$$29$$ −5005.02 −1.10512 −0.552561 0.833472i $$-0.686350\pi$$
−0.552561 + 0.833472i $$0.686350\pi$$
$$30$$ 0 0
$$31$$ 2821.69 0.527357 0.263679 0.964611i $$-0.415064\pi$$
0.263679 + 0.964611i $$0.415064\pi$$
$$32$$ 0 0
$$33$$ 9977.03 1.59484
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2046.88 −0.245803 −0.122902 0.992419i $$-0.539220\pi$$
−0.122902 + 0.992419i $$0.539220\pi$$
$$38$$ 0 0
$$39$$ 5932.17 0.624528
$$40$$ 0 0
$$41$$ −9393.81 −0.872734 −0.436367 0.899769i $$-0.643735\pi$$
−0.436367 + 0.899769i $$0.643735\pi$$
$$42$$ 0 0
$$43$$ −10320.8 −0.851218 −0.425609 0.904907i $$-0.639940\pi$$
−0.425609 + 0.904907i $$0.639940\pi$$
$$44$$ 0 0
$$45$$ 23034.5 1.69570
$$46$$ 0 0
$$47$$ 17035.6 1.12490 0.562448 0.826833i $$-0.309860\pi$$
0.562448 + 0.826833i $$0.309860\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 25972.7 1.39827
$$52$$ 0 0
$$53$$ −39506.7 −1.93188 −0.965941 0.258761i $$-0.916686\pi$$
−0.965941 + 0.258761i $$0.916686\pi$$
$$54$$ 0 0
$$55$$ 31509.0 1.40452
$$56$$ 0 0
$$57$$ −152.189 −0.00620433
$$58$$ 0 0
$$59$$ 33949.8 1.26972 0.634859 0.772628i $$-0.281058\pi$$
0.634859 + 0.772628i $$0.281058\pi$$
$$60$$ 0 0
$$61$$ −28295.2 −0.973618 −0.486809 0.873508i $$-0.661839\pi$$
−0.486809 + 0.873508i $$0.661839\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 18734.7 0.550001
$$66$$ 0 0
$$67$$ −56100.9 −1.52680 −0.763402 0.645924i $$-0.776472\pi$$
−0.763402 + 0.645924i $$0.776472\pi$$
$$68$$ 0 0
$$69$$ 84958.2 2.14824
$$70$$ 0 0
$$71$$ 15537.4 0.365791 0.182895 0.983132i $$-0.441453\pi$$
0.182895 + 0.983132i $$0.441453\pi$$
$$72$$ 0 0
$$73$$ 78219.5 1.71794 0.858970 0.512025i $$-0.171105\pi$$
0.858970 + 0.512025i $$0.171105\pi$$
$$74$$ 0 0
$$75$$ 56252.3 1.15475
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 45335.5 0.817279 0.408640 0.912696i $$-0.366003\pi$$
0.408640 + 0.912696i $$0.366003\pi$$
$$80$$ 0 0
$$81$$ −38232.4 −0.647469
$$82$$ 0 0
$$83$$ 1381.82 0.0220169 0.0110085 0.999939i $$-0.496496\pi$$
0.0110085 + 0.999939i $$0.496496\pi$$
$$84$$ 0 0
$$85$$ 82025.7 1.23141
$$86$$ 0 0
$$87$$ −117711. −1.66732
$$88$$ 0 0
$$89$$ 68879.4 0.921753 0.460876 0.887464i $$-0.347535\pi$$
0.460876 + 0.887464i $$0.347535\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 66362.1 0.795633
$$94$$ 0 0
$$95$$ −480.635 −0.00546395
$$96$$ 0 0
$$97$$ −108857. −1.17470 −0.587351 0.809332i $$-0.699829\pi$$
−0.587351 + 0.809332i $$0.699829\pi$$
$$98$$ 0 0
$$99$$ 131560. 1.34908
$$100$$ 0 0
$$101$$ −17972.3 −0.175307 −0.0876535 0.996151i $$-0.527937\pi$$
−0.0876535 + 0.996151i $$0.527937\pi$$
$$102$$ 0 0
$$103$$ 31773.7 0.295103 0.147552 0.989054i $$-0.452861\pi$$
0.147552 + 0.989054i $$0.452861\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −8229.36 −0.0694875 −0.0347438 0.999396i $$-0.511062\pi$$
−0.0347438 + 0.999396i $$0.511062\pi$$
$$108$$ 0 0
$$109$$ −11068.7 −0.0892338 −0.0446169 0.999004i $$-0.514207\pi$$
−0.0446169 + 0.999004i $$0.514207\pi$$
$$110$$ 0 0
$$111$$ −48139.6 −0.370847
$$112$$ 0 0
$$113$$ 65184.3 0.480228 0.240114 0.970745i $$-0.422815\pi$$
0.240114 + 0.970745i $$0.422815\pi$$
$$114$$ 0 0
$$115$$ 268311. 1.89188
$$116$$ 0 0
$$117$$ 78223.5 0.528290
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 18910.9 0.117422
$$122$$ 0 0
$$123$$ −220929. −1.31671
$$124$$ 0 0
$$125$$ −54456.9 −0.311730
$$126$$ 0 0
$$127$$ 194777. 1.07159 0.535796 0.844348i $$-0.320012\pi$$
0.535796 + 0.844348i $$0.320012\pi$$
$$128$$ 0 0
$$129$$ −242730. −1.28425
$$130$$ 0 0
$$131$$ −236503. −1.20409 −0.602046 0.798462i $$-0.705647\pi$$
−0.602046 + 0.798462i $$0.705647\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 117255. 0.553727
$$136$$ 0 0
$$137$$ −200903. −0.914503 −0.457252 0.889337i $$-0.651166\pi$$
−0.457252 + 0.889337i $$0.651166\pi$$
$$138$$ 0 0
$$139$$ 52985.2 0.232604 0.116302 0.993214i $$-0.462896\pi$$
0.116302 + 0.993214i $$0.462896\pi$$
$$140$$ 0 0
$$141$$ 400653. 1.69715
$$142$$ 0 0
$$143$$ 107002. 0.437575
$$144$$ 0 0
$$145$$ −371749. −1.46835
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −100770. −0.371849 −0.185925 0.982564i $$-0.559528\pi$$
−0.185925 + 0.982564i $$0.559528\pi$$
$$150$$ 0 0
$$151$$ 457904. 1.63430 0.817150 0.576425i $$-0.195553\pi$$
0.817150 + 0.576425i $$0.195553\pi$$
$$152$$ 0 0
$$153$$ 342484. 1.18280
$$154$$ 0 0
$$155$$ 209582. 0.700688
$$156$$ 0 0
$$157$$ 179037. 0.579688 0.289844 0.957074i $$-0.406397\pi$$
0.289844 + 0.957074i $$0.406397\pi$$
$$158$$ 0 0
$$159$$ −929141. −2.91467
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −243610. −0.718168 −0.359084 0.933305i $$-0.616911\pi$$
−0.359084 + 0.933305i $$0.616911\pi$$
$$164$$ 0 0
$$165$$ 741047. 2.11902
$$166$$ 0 0
$$167$$ 117033. 0.324725 0.162362 0.986731i $$-0.448089\pi$$
0.162362 + 0.986731i $$0.448089\pi$$
$$168$$ 0 0
$$169$$ −307671. −0.828648
$$170$$ 0 0
$$171$$ −2006.81 −0.00524826
$$172$$ 0 0
$$173$$ −269733. −0.685203 −0.342602 0.939481i $$-0.611308\pi$$
−0.342602 + 0.939481i $$0.611308\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 798451. 1.91564
$$178$$ 0 0
$$179$$ −376525. −0.878336 −0.439168 0.898405i $$-0.644727\pi$$
−0.439168 + 0.898405i $$0.644727\pi$$
$$180$$ 0 0
$$181$$ 434641. 0.986131 0.493065 0.869992i $$-0.335876\pi$$
0.493065 + 0.869992i $$0.335876\pi$$
$$182$$ 0 0
$$183$$ −665464. −1.46891
$$184$$ 0 0
$$185$$ −152032. −0.326593
$$186$$ 0 0
$$187$$ 468485. 0.979697
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −565940. −1.12250 −0.561250 0.827646i $$-0.689680\pi$$
−0.561250 + 0.827646i $$0.689680\pi$$
$$192$$ 0 0
$$193$$ 514461. 0.994167 0.497084 0.867703i $$-0.334404\pi$$
0.497084 + 0.867703i $$0.334404\pi$$
$$194$$ 0 0
$$195$$ 440614. 0.829796
$$196$$ 0 0
$$197$$ −298541. −0.548073 −0.274037 0.961719i $$-0.588359\pi$$
−0.274037 + 0.961719i $$0.588359\pi$$
$$198$$ 0 0
$$199$$ −591919. −1.05957 −0.529785 0.848132i $$-0.677727\pi$$
−0.529785 + 0.848132i $$0.677727\pi$$
$$200$$ 0 0
$$201$$ −1.31941e6 −2.30351
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −697728. −1.15958
$$206$$ 0 0
$$207$$ 1.12029e6 1.81720
$$208$$ 0 0
$$209$$ −2745.12 −0.00434706
$$210$$ 0 0
$$211$$ 140535. 0.217309 0.108655 0.994080i $$-0.465346\pi$$
0.108655 + 0.994080i $$0.465346\pi$$
$$212$$ 0 0
$$213$$ 365418. 0.551875
$$214$$ 0 0
$$215$$ −766579. −1.13099
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 1.83961e6 2.59189
$$220$$ 0 0
$$221$$ 278553. 0.383643
$$222$$ 0 0
$$223$$ 490.526 0.000660541 0 0.000330271 1.00000i $$-0.499895\pi$$
0.000330271 1.00000i $$0.499895\pi$$
$$224$$ 0 0
$$225$$ 741761. 0.976804
$$226$$ 0 0
$$227$$ 593898. 0.764975 0.382488 0.923961i $$-0.375067\pi$$
0.382488 + 0.923961i $$0.375067\pi$$
$$228$$ 0 0
$$229$$ 35880.3 0.0452135 0.0226067 0.999744i $$-0.492803\pi$$
0.0226067 + 0.999744i $$0.492803\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 1.24822e6 1.50626 0.753131 0.657871i $$-0.228543\pi$$
0.753131 + 0.657871i $$0.228543\pi$$
$$234$$ 0 0
$$235$$ 1.26532e6 1.49462
$$236$$ 0 0
$$237$$ 1.06623e6 1.23304
$$238$$ 0 0
$$239$$ 576943. 0.653339 0.326669 0.945139i $$-0.394074\pi$$
0.326669 + 0.945139i $$0.394074\pi$$
$$240$$ 0 0
$$241$$ 1.38241e6 1.53318 0.766592 0.642135i $$-0.221951\pi$$
0.766592 + 0.642135i $$0.221951\pi$$
$$242$$ 0 0
$$243$$ −1.28278e6 −1.39360
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1632.20 −0.00170228
$$248$$ 0 0
$$249$$ 32498.5 0.0332173
$$250$$ 0 0
$$251$$ 323217. 0.323824 0.161912 0.986805i $$-0.448234\pi$$
0.161912 + 0.986805i $$0.448234\pi$$
$$252$$ 0 0
$$253$$ 1.53244e6 1.50516
$$254$$ 0 0
$$255$$ 1.92913e6 1.85785
$$256$$ 0 0
$$257$$ −1.84601e6 −1.74342 −0.871711 0.490021i $$-0.836989\pi$$
−0.871711 + 0.490021i $$0.836989\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −1.55217e6 −1.41039
$$262$$ 0 0
$$263$$ 458222. 0.408495 0.204248 0.978919i $$-0.434525\pi$$
0.204248 + 0.978919i $$0.434525\pi$$
$$264$$ 0 0
$$265$$ −2.93437e6 −2.56685
$$266$$ 0 0
$$267$$ 1.61995e6 1.39066
$$268$$ 0 0
$$269$$ −416958. −0.351327 −0.175663 0.984450i $$-0.556207\pi$$
−0.175663 + 0.984450i $$0.556207\pi$$
$$270$$ 0 0
$$271$$ −900379. −0.744735 −0.372368 0.928085i $$-0.621454\pi$$
−0.372368 + 0.928085i $$0.621454\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.01466e6 0.809073
$$276$$ 0 0
$$277$$ −447641. −0.350535 −0.175267 0.984521i $$-0.556079\pi$$
−0.175267 + 0.984521i $$0.556079\pi$$
$$278$$ 0 0
$$279$$ 875072. 0.673029
$$280$$ 0 0
$$281$$ −768521. −0.580617 −0.290309 0.956933i $$-0.593758\pi$$
−0.290309 + 0.956933i $$0.593758\pi$$
$$282$$ 0 0
$$283$$ −2.13220e6 −1.58256 −0.791282 0.611452i $$-0.790586\pi$$
−0.791282 + 0.611452i $$0.790586\pi$$
$$284$$ 0 0
$$285$$ −11303.9 −0.00824356
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −200275. −0.141053
$$290$$ 0 0
$$291$$ −2.56017e6 −1.77229
$$292$$ 0 0
$$293$$ 2.42669e6 1.65138 0.825688 0.564128i $$-0.190787\pi$$
0.825688 + 0.564128i $$0.190787\pi$$
$$294$$ 0 0
$$295$$ 2.52163e6 1.68704
$$296$$ 0 0
$$297$$ 669693. 0.440539
$$298$$ 0 0
$$299$$ 911164. 0.589411
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −422682. −0.264489
$$304$$ 0 0
$$305$$ −2.10164e6 −1.29362
$$306$$ 0 0
$$307$$ 2.44328e6 1.47954 0.739772 0.672857i $$-0.234933\pi$$
0.739772 + 0.672857i $$0.234933\pi$$
$$308$$ 0 0
$$309$$ 747271. 0.445228
$$310$$ 0 0
$$311$$ −1.15465e6 −0.676938 −0.338469 0.940978i $$-0.609909\pi$$
−0.338469 + 0.940978i $$0.609909\pi$$
$$312$$ 0 0
$$313$$ 1.65706e6 0.956044 0.478022 0.878348i $$-0.341354\pi$$
0.478022 + 0.878348i $$0.341354\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 821361. 0.459077 0.229539 0.973300i $$-0.426278\pi$$
0.229539 + 0.973300i $$0.426278\pi$$
$$318$$ 0 0
$$319$$ −2.12322e6 −1.16821
$$320$$ 0 0
$$321$$ −193543. −0.104837
$$322$$ 0 0
$$323$$ −7146.23 −0.00381128
$$324$$ 0 0
$$325$$ 603298. 0.316828
$$326$$ 0 0
$$327$$ −260319. −0.134629
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 95670.7 0.0479964 0.0239982 0.999712i $$-0.492360\pi$$
0.0239982 + 0.999712i $$0.492360\pi$$
$$332$$ 0 0
$$333$$ −634785. −0.313701
$$334$$ 0 0
$$335$$ −4.16691e6 −2.02863
$$336$$ 0 0
$$337$$ 2.37020e6 1.13687 0.568435 0.822728i $$-0.307549\pi$$
0.568435 + 0.822728i $$0.307549\pi$$
$$338$$ 0 0
$$339$$ 1.53304e6 0.724528
$$340$$ 0 0
$$341$$ 1.19701e6 0.557460
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 6.31029e6 2.85431
$$346$$ 0 0
$$347$$ −490571. −0.218715 −0.109358 0.994002i $$-0.534879\pi$$
−0.109358 + 0.994002i $$0.534879\pi$$
$$348$$ 0 0
$$349$$ 4.21208e6 1.85111 0.925557 0.378607i $$-0.123597\pi$$
0.925557 + 0.378607i $$0.123597\pi$$
$$350$$ 0 0
$$351$$ 398188. 0.172512
$$352$$ 0 0
$$353$$ −3.17378e6 −1.35563 −0.677814 0.735234i $$-0.737072\pi$$
−0.677814 + 0.735234i $$0.737072\pi$$
$$354$$ 0 0
$$355$$ 1.15405e6 0.486018
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 3.40098e6 1.39273 0.696366 0.717687i $$-0.254799\pi$$
0.696366 + 0.717687i $$0.254799\pi$$
$$360$$ 0 0
$$361$$ −2.47606e6 −0.999983
$$362$$ 0 0
$$363$$ 444757. 0.177156
$$364$$ 0 0
$$365$$ 5.80978e6 2.28259
$$366$$ 0 0
$$367$$ 1.96872e6 0.762988 0.381494 0.924371i $$-0.375410\pi$$
0.381494 + 0.924371i $$0.375410\pi$$
$$368$$ 0 0
$$369$$ −2.91324e6 −1.11381
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −3.47889e6 −1.29470 −0.647349 0.762194i $$-0.724122\pi$$
−0.647349 + 0.762194i $$0.724122\pi$$
$$374$$ 0 0
$$375$$ −1.28075e6 −0.470312
$$376$$ 0 0
$$377$$ −1.26243e6 −0.457462
$$378$$ 0 0
$$379$$ 421294. 0.150656 0.0753281 0.997159i $$-0.476000\pi$$
0.0753281 + 0.997159i $$0.476000\pi$$
$$380$$ 0 0
$$381$$ 4.58089e6 1.61673
$$382$$ 0 0
$$383$$ 2.66910e6 0.929754 0.464877 0.885375i $$-0.346099\pi$$
0.464877 + 0.885375i $$0.346099\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −3.20071e6 −1.08635
$$388$$ 0 0
$$389$$ 3.10178e6 1.03929 0.519645 0.854382i $$-0.326064\pi$$
0.519645 + 0.854382i $$0.326064\pi$$
$$390$$ 0 0
$$391$$ 3.98933e6 1.31965
$$392$$ 0 0
$$393$$ −5.56223e6 −1.81663
$$394$$ 0 0
$$395$$ 3.36731e6 1.08590
$$396$$ 0 0
$$397$$ −613257. −0.195284 −0.0976419 0.995222i $$-0.531130\pi$$
−0.0976419 + 0.995222i $$0.531130\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 2.82223e6 0.876459 0.438229 0.898863i $$-0.355606\pi$$
0.438229 + 0.898863i $$0.355606\pi$$
$$402$$ 0 0
$$403$$ 711724. 0.218298
$$404$$ 0 0
$$405$$ −2.83973e6 −0.860278
$$406$$ 0 0
$$407$$ −868324. −0.259834
$$408$$ 0 0
$$409$$ −2.28350e6 −0.674983 −0.337492 0.941329i $$-0.609578\pi$$
−0.337492 + 0.941329i $$0.609578\pi$$
$$410$$ 0 0
$$411$$ −4.72496e6 −1.37973
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 102635. 0.0292534
$$416$$ 0 0
$$417$$ 1.24614e6 0.350934
$$418$$ 0 0
$$419$$ −2.65270e6 −0.738163 −0.369082 0.929397i $$-0.620328\pi$$
−0.369082 + 0.929397i $$0.620328\pi$$
$$420$$ 0 0
$$421$$ 2.93674e6 0.807532 0.403766 0.914862i $$-0.367701\pi$$
0.403766 + 0.914862i $$0.367701\pi$$
$$422$$ 0 0
$$423$$ 5.28314e6 1.43562
$$424$$ 0 0
$$425$$ 2.64140e6 0.709353
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 2.51654e6 0.660177
$$430$$ 0 0
$$431$$ 2.44565e6 0.634164 0.317082 0.948398i $$-0.397297\pi$$
0.317082 + 0.948398i $$0.397297\pi$$
$$432$$ 0 0
$$433$$ 2.11718e6 0.542673 0.271336 0.962485i $$-0.412534\pi$$
0.271336 + 0.962485i $$0.412534\pi$$
$$434$$ 0 0
$$435$$ −8.74301e6 −2.21533
$$436$$ 0 0
$$437$$ −23375.7 −0.00585547
$$438$$ 0 0
$$439$$ 4.64764e6 1.15099 0.575495 0.817806i $$-0.304810\pi$$
0.575495 + 0.817806i $$0.304810\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −4.42925e6 −1.07231 −0.536155 0.844119i $$-0.680124\pi$$
−0.536155 + 0.844119i $$0.680124\pi$$
$$444$$ 0 0
$$445$$ 5.11604e6 1.22471
$$446$$ 0 0
$$447$$ −2.36998e6 −0.561016
$$448$$ 0 0
$$449$$ −6.70171e6 −1.56881 −0.784404 0.620250i $$-0.787031\pi$$
−0.784404 + 0.620250i $$0.787031\pi$$
$$450$$ 0 0
$$451$$ −3.98503e6 −0.922551
$$452$$ 0 0
$$453$$ 1.07692e7 2.46570
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −5.88344e6 −1.31777 −0.658887 0.752242i $$-0.728972\pi$$
−0.658887 + 0.752242i $$0.728972\pi$$
$$458$$ 0 0
$$459$$ 1.74338e6 0.386242
$$460$$ 0 0
$$461$$ −1.54764e6 −0.339171 −0.169585 0.985515i $$-0.554243\pi$$
−0.169585 + 0.985515i $$0.554243\pi$$
$$462$$ 0 0
$$463$$ 3.93764e6 0.853656 0.426828 0.904333i $$-0.359631\pi$$
0.426828 + 0.904333i $$0.359631\pi$$
$$464$$ 0 0
$$465$$ 4.92907e6 1.05714
$$466$$ 0 0
$$467$$ −6.81586e6 −1.44620 −0.723100 0.690743i $$-0.757284\pi$$
−0.723100 + 0.690743i $$0.757284\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 4.21070e6 0.874585
$$472$$ 0 0
$$473$$ −4.37827e6 −0.899807
$$474$$ 0 0
$$475$$ −15477.5 −0.00314750
$$476$$ 0 0
$$477$$ −1.22519e7 −2.46552
$$478$$ 0 0
$$479$$ −5.20406e6 −1.03634 −0.518171 0.855277i $$-0.673387\pi$$
−0.518171 + 0.855277i $$0.673387\pi$$
$$480$$ 0 0
$$481$$ −516291. −0.101749
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −8.08540e6 −1.56080
$$486$$ 0 0
$$487$$ 154998. 0.0296145 0.0148073 0.999890i $$-0.495287\pi$$
0.0148073 + 0.999890i $$0.495287\pi$$
$$488$$ 0 0
$$489$$ −5.72936e6 −1.08351
$$490$$ 0 0
$$491$$ 1.61951e6 0.303165 0.151583 0.988445i $$-0.451563\pi$$
0.151583 + 0.988445i $$0.451563\pi$$
$$492$$ 0 0
$$493$$ −5.52727e6 −1.02422
$$494$$ 0 0
$$495$$ 9.77168e6 1.79249
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −4.10674e6 −0.738322 −0.369161 0.929366i $$-0.620355\pi$$
−0.369161 + 0.929366i $$0.620355\pi$$
$$500$$ 0 0
$$501$$ 2.75244e6 0.489918
$$502$$ 0 0
$$503$$ −3.40748e6 −0.600501 −0.300250 0.953860i $$-0.597070\pi$$
−0.300250 + 0.953860i $$0.597070\pi$$
$$504$$ 0 0
$$505$$ −1.33490e6 −0.232927
$$506$$ 0 0
$$507$$ −7.23599e6 −1.25020
$$508$$ 0 0
$$509$$ −1.07091e7 −1.83213 −0.916067 0.401025i $$-0.868654\pi$$
−0.916067 + 0.401025i $$0.868654\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −10215.4 −0.00171381
$$514$$ 0 0
$$515$$ 2.36000e6 0.392097
$$516$$ 0 0
$$517$$ 7.22682e6 1.18911
$$518$$ 0 0
$$519$$ −6.34374e6 −1.03378
$$520$$ 0 0
$$521$$ −7.92001e6 −1.27830 −0.639148 0.769084i $$-0.720713\pi$$
−0.639148 + 0.769084i $$0.720713\pi$$
$$522$$ 0 0
$$523$$ −8.32746e6 −1.33125 −0.665623 0.746288i $$-0.731834\pi$$
−0.665623 + 0.746288i $$0.731834\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 3.11612e6 0.488752
$$528$$ 0 0
$$529$$ 6.61298e6 1.02744
$$530$$ 0 0
$$531$$ 1.05286e7 1.62045
$$532$$ 0 0
$$533$$ −2.36943e6 −0.361265
$$534$$ 0 0
$$535$$ −611239. −0.0923265
$$536$$ 0 0
$$537$$ −8.85532e6 −1.32516
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 623261. 0.0915539 0.0457770 0.998952i $$-0.485424\pi$$
0.0457770 + 0.998952i $$0.485424\pi$$
$$542$$ 0 0
$$543$$ 1.02221e7 1.48779
$$544$$ 0 0
$$545$$ −822129. −0.118563
$$546$$ 0 0
$$547$$ −1.05691e7 −1.51032 −0.755159 0.655541i $$-0.772441\pi$$
−0.755159 + 0.655541i $$0.772441\pi$$
$$548$$ 0 0
$$549$$ −8.77502e6 −1.24256
$$550$$ 0 0
$$551$$ 32387.5 0.00454462
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −3.57559e6 −0.492737
$$556$$ 0 0
$$557$$ −1.35398e7 −1.84916 −0.924579 0.380991i $$-0.875583\pi$$
−0.924579 + 0.380991i $$0.875583\pi$$
$$558$$ 0 0
$$559$$ −2.60324e6 −0.352359
$$560$$ 0 0
$$561$$ 1.10181e7 1.47809
$$562$$ 0 0
$$563$$ −1.39757e7 −1.85824 −0.929122 0.369774i $$-0.879435\pi$$
−0.929122 + 0.369774i $$0.879435\pi$$
$$564$$ 0 0
$$565$$ 4.84159e6 0.638068
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −5.61993e6 −0.727697 −0.363848 0.931458i $$-0.618537\pi$$
−0.363848 + 0.931458i $$0.618537\pi$$
$$570$$ 0 0
$$571$$ −8.70790e6 −1.11769 −0.558847 0.829271i $$-0.688756\pi$$
−0.558847 + 0.829271i $$0.688756\pi$$
$$572$$ 0 0
$$573$$ −1.33101e7 −1.69354
$$574$$ 0 0
$$575$$ 8.64019e6 1.08982
$$576$$ 0 0
$$577$$ 6.63992e6 0.830278 0.415139 0.909758i $$-0.363733\pi$$
0.415139 + 0.909758i $$0.363733\pi$$
$$578$$ 0 0
$$579$$ 1.20994e7 1.49992
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −1.67595e7 −2.04216
$$584$$ 0 0
$$585$$ 5.81007e6 0.701927
$$586$$ 0 0
$$587$$ −1.36652e7 −1.63689 −0.818446 0.574583i $$-0.805164\pi$$
−0.818446 + 0.574583i $$0.805164\pi$$
$$588$$ 0 0
$$589$$ −18259.2 −0.00216866
$$590$$ 0 0
$$591$$ −7.02126e6 −0.826888
$$592$$ 0 0
$$593$$ 7.02589e6 0.820474 0.410237 0.911979i $$-0.365446\pi$$
0.410237 + 0.911979i $$0.365446\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −1.39211e7 −1.59859
$$598$$ 0 0
$$599$$ 3.58663e6 0.408432 0.204216 0.978926i $$-0.434536\pi$$
0.204216 + 0.978926i $$0.434536\pi$$
$$600$$ 0 0
$$601$$ −1.58600e7 −1.79108 −0.895542 0.444977i $$-0.853212\pi$$
−0.895542 + 0.444977i $$0.853212\pi$$
$$602$$ 0 0
$$603$$ −1.73982e7 −1.94855
$$604$$ 0 0
$$605$$ 1.40461e6 0.156015
$$606$$ 0 0
$$607$$ 6.44170e6 0.709625 0.354812 0.934938i $$-0.384545\pi$$
0.354812 + 0.934938i $$0.384545\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 4.29694e6 0.465647
$$612$$ 0 0
$$613$$ −4.58865e6 −0.493212 −0.246606 0.969116i $$-0.579315\pi$$
−0.246606 + 0.969116i $$0.579315\pi$$
$$614$$ 0 0
$$615$$ −1.64096e7 −1.74948
$$616$$ 0 0
$$617$$ 1.47104e6 0.155565 0.0777825 0.996970i $$-0.475216\pi$$
0.0777825 + 0.996970i $$0.475216\pi$$
$$618$$ 0 0
$$619$$ −3.13569e6 −0.328932 −0.164466 0.986383i $$-0.552590\pi$$
−0.164466 + 0.986383i $$0.552590\pi$$
$$620$$ 0 0
$$621$$ 5.70269e6 0.593404
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −1.15193e7 −1.17957
$$626$$ 0 0
$$627$$ −64561.3 −0.00655849
$$628$$ 0 0
$$629$$ −2.26046e6 −0.227809
$$630$$ 0 0
$$631$$ −484547. −0.0484465 −0.0242233 0.999707i $$-0.507711\pi$$
−0.0242233 + 0.999707i $$0.507711\pi$$
$$632$$ 0 0
$$633$$ 3.30518e6 0.327858
$$634$$ 0 0
$$635$$ 1.44672e7 1.42380
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 4.81851e6 0.466832
$$640$$ 0 0
$$641$$ 3.04085e6 0.292314 0.146157 0.989261i $$-0.453310\pi$$
0.146157 + 0.989261i $$0.453310\pi$$
$$642$$ 0 0
$$643$$ 5.25888e6 0.501609 0.250805 0.968038i $$-0.419305\pi$$
0.250805 + 0.968038i $$0.419305\pi$$
$$644$$ 0 0
$$645$$ −1.80288e7 −1.70635
$$646$$ 0 0
$$647$$ 2.11970e7 1.99074 0.995368 0.0961386i $$-0.0306492\pi$$
0.995368 + 0.0961386i $$0.0306492\pi$$
$$648$$ 0 0
$$649$$ 1.44021e7 1.34219
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1.30106e7 1.19403 0.597013 0.802232i $$-0.296354\pi$$
0.597013 + 0.802232i $$0.296354\pi$$
$$654$$ 0 0
$$655$$ −1.75664e7 −1.59985
$$656$$ 0 0
$$657$$ 2.42577e7 2.19248
$$658$$ 0 0
$$659$$ 1.59874e7 1.43405 0.717024 0.697049i $$-0.245504\pi$$
0.717024 + 0.697049i $$0.245504\pi$$
$$660$$ 0 0
$$661$$ −4.03142e6 −0.358884 −0.179442 0.983769i $$-0.557429\pi$$
−0.179442 + 0.983769i $$0.557429\pi$$
$$662$$ 0 0
$$663$$ 6.55117e6 0.578809
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −1.80800e7 −1.57357
$$668$$ 0 0
$$669$$ 11536.5 0.000996570 0
$$670$$ 0 0
$$671$$ −1.20034e7 −1.02919
$$672$$ 0 0
$$673$$ 2.98234e6 0.253816 0.126908 0.991914i $$-0.459495\pi$$
0.126908 + 0.991914i $$0.459495\pi$$
$$674$$ 0 0
$$675$$ 3.77585e6 0.318974
$$676$$ 0 0
$$677$$ −1.94696e6 −0.163262 −0.0816309 0.996663i $$-0.526013\pi$$
−0.0816309 + 0.996663i $$0.526013\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 1.39676e7 1.15413
$$682$$ 0 0
$$683$$ −1.19940e7 −0.983812 −0.491906 0.870648i $$-0.663700\pi$$
−0.491906 + 0.870648i $$0.663700\pi$$
$$684$$ 0 0
$$685$$ −1.49221e7 −1.21508
$$686$$ 0 0
$$687$$ 843854. 0.0682143
$$688$$ 0 0
$$689$$ −9.96490e6 −0.799696
$$690$$ 0 0
$$691$$ −8.66304e6 −0.690201 −0.345100 0.938566i $$-0.612155\pi$$
−0.345100 + 0.938566i $$0.612155\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 3.93549e6 0.309056
$$696$$ 0 0
$$697$$ −1.03740e7 −0.808845
$$698$$ 0 0
$$699$$ 2.93563e7 2.27252
$$700$$ 0 0
$$701$$ −8.13382e6 −0.625172 −0.312586 0.949889i $$-0.601195\pi$$
−0.312586 + 0.949889i $$0.601195\pi$$
$$702$$ 0 0
$$703$$ 13245.3 0.00101082
$$704$$ 0 0
$$705$$ 2.97586e7 2.25497
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2.21326e7 1.65355 0.826773 0.562535i $$-0.190174\pi$$
0.826773 + 0.562535i $$0.190174\pi$$
$$710$$ 0 0
$$711$$ 1.40596e7 1.04303
$$712$$ 0 0
$$713$$ 1.01930e7 0.750896
$$714$$ 0 0
$$715$$ 7.94762e6 0.581396
$$716$$ 0 0
$$717$$ 1.35689e7 0.985704
$$718$$ 0 0
$$719$$ −7.23196e6 −0.521716 −0.260858 0.965377i $$-0.584005\pi$$
−0.260858 + 0.965377i $$0.584005\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 3.25123e7 2.31314
$$724$$ 0 0
$$725$$ −1.19711e7 −0.845843
$$726$$ 0 0
$$727$$ −1.70200e7 −1.19433 −0.597163 0.802120i $$-0.703705\pi$$
−0.597163 + 0.802120i $$0.703705\pi$$
$$728$$ 0 0
$$729$$ −2.08788e7 −1.45508
$$730$$ 0 0
$$731$$ −1.13977e7 −0.788904
$$732$$ 0 0
$$733$$ −1.00011e6 −0.0687525 −0.0343763 0.999409i $$-0.510944\pi$$
−0.0343763 + 0.999409i $$0.510944\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −2.37991e7 −1.61396
$$738$$ 0 0
$$739$$ −3.25979e6 −0.219572 −0.109786 0.993955i $$-0.535017\pi$$
−0.109786 + 0.993955i $$0.535017\pi$$
$$740$$ 0 0
$$741$$ −38387.1 −0.00256826
$$742$$ 0 0
$$743$$ −1.36125e7 −0.904617 −0.452309 0.891861i $$-0.649399\pi$$
−0.452309 + 0.891861i $$0.649399\pi$$
$$744$$ 0 0
$$745$$ −7.48475e6 −0.494068
$$746$$ 0 0
$$747$$ 428536. 0.0280986
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −6.56544e6 −0.424780 −0.212390 0.977185i $$-0.568125\pi$$
−0.212390 + 0.977185i $$0.568125\pi$$
$$752$$ 0 0
$$753$$ 7.60160e6 0.488560
$$754$$ 0 0
$$755$$ 3.40110e7 2.17146
$$756$$ 0 0
$$757$$ 2.62531e7 1.66510 0.832551 0.553948i $$-0.186879\pi$$
0.832551 + 0.553948i $$0.186879\pi$$
$$758$$ 0 0
$$759$$ 3.60409e7 2.27086
$$760$$ 0 0
$$761$$ 5.25111e6 0.328692 0.164346 0.986403i $$-0.447449\pi$$
0.164346 + 0.986403i $$0.447449\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 2.54381e7 1.57156
$$766$$ 0 0
$$767$$ 8.56327e6 0.525595
$$768$$ 0 0
$$769$$ 1.77307e7 1.08121 0.540605 0.841277i $$-0.318195\pi$$
0.540605 + 0.841277i $$0.318195\pi$$
$$770$$ 0 0
$$771$$ −4.34156e7 −2.63033
$$772$$ 0 0
$$773$$ −3.82592e6 −0.230296 −0.115148 0.993348i $$-0.536734\pi$$
−0.115148 + 0.993348i $$0.536734\pi$$
$$774$$ 0 0
$$775$$ 6.74898e6 0.403631
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 60787.3 0.00358896
$$780$$ 0 0
$$781$$ 6.59126e6 0.386671
$$782$$ 0 0
$$783$$ −7.90117e6 −0.460561
$$784$$ 0 0
$$785$$ 1.32980e7 0.770218
$$786$$ 0 0
$$787$$ 2.94263e7 1.69355 0.846777 0.531948i $$-0.178540\pi$$
0.846777 + 0.531948i $$0.178540\pi$$
$$788$$ 0 0
$$789$$ 1.07767e7 0.616304
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −7.13700e6 −0.403026
$$794$$ 0 0
$$795$$ −6.90122e7 −3.87265
$$796$$ 0 0
$$797$$ −1.35805e7 −0.757305 −0.378653 0.925539i $$-0.623613\pi$$
−0.378653 + 0.925539i $$0.623613\pi$$
$$798$$ 0 0
$$799$$ 1.88132e7 1.04255
$$800$$ 0 0
$$801$$ 2.13611e7 1.17637
$$802$$ 0 0
$$803$$ 3.31822e7 1.81600
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −9.80625e6 −0.530053
$$808$$ 0 0
$$809$$ −1.15714e7 −0.621604 −0.310802 0.950475i $$-0.600598\pi$$
−0.310802 + 0.950475i $$0.600598\pi$$
$$810$$ 0 0
$$811$$ −3.52530e6 −0.188210 −0.0941052 0.995562i $$-0.529999\pi$$
−0.0941052 + 0.995562i $$0.529999\pi$$
$$812$$ 0 0
$$813$$ −2.11756e7 −1.12360
$$814$$ 0 0
$$815$$ −1.80942e7 −0.954214
$$816$$ 0 0
$$817$$ 66785.7 0.00350049
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 1.78951e7 0.926567 0.463283 0.886210i $$-0.346671\pi$$
0.463283 + 0.886210i $$0.346671\pi$$
$$822$$ 0 0
$$823$$ 3.61421e7 1.86000 0.930001 0.367557i $$-0.119806\pi$$
0.930001 + 0.367557i $$0.119806\pi$$
$$824$$ 0 0
$$825$$ 2.38633e7 1.22066
$$826$$ 0 0
$$827$$ 1.00605e7 0.511512 0.255756 0.966741i $$-0.417676\pi$$
0.255756 + 0.966741i $$0.417676\pi$$
$$828$$ 0 0
$$829$$ −2.03654e7 −1.02921 −0.514607 0.857426i $$-0.672062\pi$$
−0.514607 + 0.857426i $$0.672062\pi$$
$$830$$ 0 0
$$831$$ −1.05279e7 −0.528858
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 8.69263e6 0.431454
$$836$$ 0 0
$$837$$ 4.45446e6 0.219777
$$838$$ 0 0
$$839$$ −5.95014e6 −0.291825 −0.145912 0.989298i $$-0.546612\pi$$
−0.145912 + 0.989298i $$0.546612\pi$$
$$840$$ 0 0
$$841$$ 4.53905e6 0.221297
$$842$$ 0 0
$$843$$ −1.80745e7 −0.875987
$$844$$ 0 0
$$845$$ −2.28524e7 −1.10101
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −5.01462e7 −2.38764
$$850$$ 0 0
$$851$$ −7.39411e6 −0.349995
$$852$$ 0 0
$$853$$ −1.59836e7 −0.752146 −0.376073 0.926590i $$-0.622726\pi$$
−0.376073 + 0.926590i $$0.622726\pi$$
$$854$$ 0 0
$$855$$ −149056. −0.00697325
$$856$$ 0 0
$$857$$ 2.34591e6 0.109109 0.0545544 0.998511i $$-0.482626\pi$$
0.0545544 + 0.998511i $$0.482626\pi$$
$$858$$ 0 0
$$859$$ −1.30223e7 −0.602152 −0.301076 0.953600i $$-0.597346\pi$$
−0.301076 + 0.953600i $$0.597346\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −1.96688e7 −0.898983 −0.449492 0.893285i $$-0.648395\pi$$
−0.449492 + 0.893285i $$0.648395\pi$$
$$864$$ 0 0
$$865$$ −2.00345e7 −0.910414
$$866$$ 0 0
$$867$$ −4.71019e6 −0.212809
$$868$$ 0 0
$$869$$ 1.92322e7 0.863931
$$870$$ 0 0
$$871$$ −1.41505e7 −0.632015
$$872$$ 0 0
$$873$$ −3.37592e7 −1.49919
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.34581e7 1.02990 0.514950 0.857221i $$-0.327811\pi$$
0.514950 + 0.857221i $$0.327811\pi$$
$$878$$ 0 0
$$879$$ 5.70724e7 2.49146
$$880$$ 0 0
$$881$$ −4.59257e6 −0.199350 −0.0996750 0.995020i $$-0.531780\pi$$
−0.0996750 + 0.995020i $$0.531780\pi$$
$$882$$ 0 0
$$883$$ 1.23402e7 0.532622 0.266311 0.963887i $$-0.414195\pi$$
0.266311 + 0.963887i $$0.414195\pi$$
$$884$$ 0 0
$$885$$ 5.93052e7 2.54527
$$886$$ 0 0
$$887$$ 1.36554e7 0.582769 0.291384 0.956606i $$-0.405884\pi$$
0.291384 + 0.956606i $$0.405884\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −1.62189e7 −0.684428
$$892$$ 0 0
$$893$$ −110237. −0.00462594
$$894$$ 0 0
$$895$$ −2.79665e7 −1.16703
$$896$$ 0 0
$$897$$ 2.14293e7 0.889255
$$898$$ 0 0
$$899$$ −1.41226e7 −0.582795
$$900$$ 0 0
$$901$$ −4.36291e7 −1.79046
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 3.22831e7 1.31025
$$906$$ 0 0
$$907$$ −7.39599e6 −0.298523 −0.149262 0.988798i $$-0.547690\pi$$
−0.149262 + 0.988798i $$0.547690\pi$$
$$908$$ 0 0
$$909$$ −5.57362e6 −0.223732
$$910$$ 0 0
$$911$$ 3.51041e7 1.40140 0.700699 0.713457i $$-0.252871\pi$$
0.700699 + 0.713457i $$0.252871\pi$$
$$912$$ 0 0
$$913$$ 586195. 0.0232737
$$914$$ 0 0
$$915$$ −4.94275e7 −1.95171
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 4.81337e7 1.88001 0.940005 0.341159i $$-0.110820\pi$$
0.940005 + 0.341159i $$0.110820\pi$$
$$920$$ 0 0
$$921$$ 5.74626e7 2.23221
$$922$$ 0 0
$$923$$ 3.91905e6 0.151418
$$924$$ 0 0
$$925$$ −4.89577e6 −0.188134
$$926$$ 0 0
$$927$$ 9.85376e6 0.376619
$$928$$ 0 0
$$929$$ 1.83602e7 0.697971 0.348986 0.937128i $$-0.386526\pi$$
0.348986 + 0.937128i $$0.386526\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −2.71557e7 −1.02131
$$934$$ 0 0
$$935$$ 3.47969e7 1.30170
$$936$$ 0 0
$$937$$ −2.27081e7 −0.844951 −0.422475 0.906374i $$-0.638839\pi$$
−0.422475 + 0.906374i $$0.638839\pi$$
$$938$$ 0 0
$$939$$ 3.89718e7 1.44240
$$940$$ 0 0
$$941$$ −4.15801e7 −1.53078 −0.765389 0.643568i $$-0.777453\pi$$
−0.765389 + 0.643568i $$0.777453\pi$$
$$942$$ 0 0
$$943$$ −3.39340e7 −1.24267
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1.55341e7 −0.562874 −0.281437 0.959580i $$-0.590811\pi$$
−0.281437 + 0.959580i $$0.590811\pi$$
$$948$$ 0 0
$$949$$ 1.97296e7 0.711135
$$950$$ 0 0
$$951$$ 1.93172e7 0.692618
$$952$$ 0 0
$$953$$ 3.94908e7 1.40852 0.704262 0.709940i $$-0.251278\pi$$
0.704262 + 0.709940i $$0.251278\pi$$
$$954$$ 0 0
$$955$$ −4.20354e7 −1.49144
$$956$$ 0 0
$$957$$ −4.99352e7 −1.76249
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2.06672e7 −0.721894
$$962$$ 0 0
$$963$$ −2.55212e6 −0.0886819
$$964$$ 0 0
$$965$$ 3.82118e7 1.32093
$$966$$ 0 0
$$967$$ 1.87472e7 0.644718 0.322359 0.946617i $$-0.395524\pi$$
0.322359 + 0.946617i $$0.395524\pi$$
$$968$$ 0 0
$$969$$ −168069. −0.00575014
$$970$$ 0 0
$$971$$ −5.35423e7 −1.82242 −0.911211 0.411940i $$-0.864851\pi$$
−0.911211 + 0.411940i $$0.864851\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 1.41887e7 0.478004
$$976$$ 0 0
$$977$$ 4.46104e7 1.49520 0.747600 0.664149i $$-0.231206\pi$$
0.747600 + 0.664149i $$0.231206\pi$$
$$978$$ 0 0
$$979$$ 2.92200e7 0.974368
$$980$$ 0 0
$$981$$ −3.43265e6 −0.113883
$$982$$ 0 0
$$983$$ −2.00068e7 −0.660380 −0.330190 0.943914i $$-0.607113\pi$$
−0.330190 + 0.943914i $$0.607113\pi$$
$$984$$ 0 0
$$985$$ −2.21742e7 −0.728213
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −3.72826e7 −1.21204
$$990$$ 0 0
$$991$$ −3.24635e7 −1.05005 −0.525026 0.851086i $$-0.675944\pi$$
−0.525026 + 0.851086i $$0.675944\pi$$
$$992$$ 0 0
$$993$$ 2.25004e6 0.0724130
$$994$$ 0 0
$$995$$ −4.39649e7 −1.40783
$$996$$ 0 0
$$997$$ −2.78940e7 −0.888736 −0.444368 0.895844i $$-0.646572\pi$$
−0.444368 + 0.895844i $$0.646572\pi$$
$$998$$ 0 0
$$999$$ −3.23130e6 −0.102439
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.bf.1.4 4
4.3 odd 2 49.6.a.g.1.1 4
7.6 odd 2 inner 784.6.a.bf.1.1 4
12.11 even 2 441.6.a.z.1.3 4
28.3 even 6 49.6.c.h.30.3 8
28.11 odd 6 49.6.c.h.30.4 8
28.19 even 6 49.6.c.h.18.3 8
28.23 odd 6 49.6.c.h.18.4 8
28.27 even 2 49.6.a.g.1.2 yes 4
84.83 odd 2 441.6.a.z.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.g.1.1 4 4.3 odd 2
49.6.a.g.1.2 yes 4 28.27 even 2
49.6.c.h.18.3 8 28.19 even 6
49.6.c.h.18.4 8 28.23 odd 6
49.6.c.h.30.3 8 28.3 even 6
49.6.c.h.30.4 8 28.11 odd 6
441.6.a.z.1.3 4 12.11 even 2
441.6.a.z.1.4 4 84.83 odd 2
784.6.a.bf.1.1 4 7.6 odd 2 inner
784.6.a.bf.1.4 4 1.1 even 1 trivial