# Properties

 Label 9680.2.a.da.1.5 Level $9680$ Weight $2$ Character 9680.1 Self dual yes Analytic conductor $77.295$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9680 = 2^{4} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9680.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: $$77.2951891566$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.22733568.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2$$ x^6 - 8*x^4 - 2*x^3 + 16*x^2 + 8*x - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4840) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.5 Root $$-0.821728$$ of defining polynomial Character $$\chi$$ $$=$$ 9680.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.32476 q^{3} +1.00000 q^{5} -4.78768 q^{7} +2.40452 q^{9} +O(q^{10})$$ $$q+2.32476 q^{3} +1.00000 q^{5} -4.78768 q^{7} +2.40452 q^{9} -0.601547 q^{13} +2.32476 q^{15} -2.64953 q^{17} -2.60155 q^{19} -11.1302 q^{21} -5.17083 q^{23} +1.00000 q^{25} -1.38434 q^{27} +5.44643 q^{29} +10.4577 q^{31} -4.78768 q^{35} +9.39816 q^{37} -1.39845 q^{39} -4.54633 q^{41} +3.48547 q^{43} +2.40452 q^{45} -5.07340 q^{47} +15.9218 q^{49} -6.15952 q^{51} +3.76714 q^{53} -6.04798 q^{57} -11.7599 q^{59} +12.6454 q^{61} -11.5121 q^{63} -0.601547 q^{65} +15.0452 q^{67} -12.0210 q^{69} -0.346489 q^{71} +3.68096 q^{73} +2.32476 q^{75} +8.05322 q^{79} -10.4318 q^{81} +13.0094 q^{83} -2.64953 q^{85} +12.6617 q^{87} +0.462692 q^{89} +2.88001 q^{91} +24.3118 q^{93} -2.60155 q^{95} +7.97011 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10})$$ 6 * q + 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 $$6 q + 2 q^{3} + 6 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{15} + 8 q^{17} - 12 q^{19} - 8 q^{21} + 8 q^{23} + 6 q^{25} + 14 q^{27} + 16 q^{29} + 4 q^{31} - 4 q^{35} + 8 q^{37} - 12 q^{39} + 32 q^{41} + 4 q^{43} + 4 q^{45} + 6 q^{47} + 16 q^{49} - 40 q^{51} + 8 q^{53} - 16 q^{57} - 4 q^{59} + 16 q^{61} - 28 q^{63} + 2 q^{67} + 8 q^{69} + 28 q^{71} + 16 q^{73} + 2 q^{75} - 10 q^{81} - 12 q^{83} + 8 q^{85} + 24 q^{87} + 18 q^{89} + 24 q^{91} + 20 q^{93} - 12 q^{95}+O(q^{100})$$ 6 * q + 2 * q^3 + 6 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^15 + 8 * q^17 - 12 * q^19 - 8 * q^21 + 8 * q^23 + 6 * q^25 + 14 * q^27 + 16 * q^29 + 4 * q^31 - 4 * q^35 + 8 * q^37 - 12 * q^39 + 32 * q^41 + 4 * q^43 + 4 * q^45 + 6 * q^47 + 16 * q^49 - 40 * q^51 + 8 * q^53 - 16 * q^57 - 4 * q^59 + 16 * q^61 - 28 * q^63 + 2 * q^67 + 8 * q^69 + 28 * q^71 + 16 * q^73 + 2 * q^75 - 10 * q^81 - 12 * q^83 + 8 * q^85 + 24 * q^87 + 18 * q^89 + 24 * q^91 + 20 * q^93 - 12 * q^95

## 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
$$3$$ 2.32476 1.34220 0.671101 0.741366i $$-0.265822\pi$$
0.671101 + 0.741366i $$0.265822\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −4.78768 −1.80957 −0.904786 0.425867i $$-0.859969\pi$$
−0.904786 + 0.425867i $$0.859969\pi$$
$$8$$ 0 0
$$9$$ 2.40452 0.801508
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −0.601547 −0.166839 −0.0834195 0.996515i $$-0.526584\pi$$
−0.0834195 + 0.996515i $$0.526584\pi$$
$$14$$ 0 0
$$15$$ 2.32476 0.600251
$$16$$ 0 0
$$17$$ −2.64953 −0.642605 −0.321302 0.946977i $$-0.604121\pi$$
−0.321302 + 0.946977i $$0.604121\pi$$
$$18$$ 0 0
$$19$$ −2.60155 −0.596836 −0.298418 0.954435i $$-0.596459\pi$$
−0.298418 + 0.954435i $$0.596459\pi$$
$$20$$ 0 0
$$21$$ −11.1302 −2.42881
$$22$$ 0 0
$$23$$ −5.17083 −1.07819 −0.539096 0.842244i $$-0.681234\pi$$
−0.539096 + 0.842244i $$0.681234\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.38434 −0.266417
$$28$$ 0 0
$$29$$ 5.44643 1.01138 0.505689 0.862716i $$-0.331238\pi$$
0.505689 + 0.862716i $$0.331238\pi$$
$$30$$ 0 0
$$31$$ 10.4577 1.87827 0.939133 0.343554i $$-0.111631\pi$$
0.939133 + 0.343554i $$0.111631\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −4.78768 −0.809265
$$36$$ 0 0
$$37$$ 9.39816 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$38$$ 0 0
$$39$$ −1.39845 −0.223932
$$40$$ 0 0
$$41$$ −4.54633 −0.710018 −0.355009 0.934863i $$-0.615522\pi$$
−0.355009 + 0.934863i $$0.615522\pi$$
$$42$$ 0 0
$$43$$ 3.48547 0.531530 0.265765 0.964038i $$-0.414376\pi$$
0.265765 + 0.964038i $$0.414376\pi$$
$$44$$ 0 0
$$45$$ 2.40452 0.358445
$$46$$ 0 0
$$47$$ −5.07340 −0.740031 −0.370016 0.929026i $$-0.620648\pi$$
−0.370016 + 0.929026i $$0.620648\pi$$
$$48$$ 0 0
$$49$$ 15.9218 2.27455
$$50$$ 0 0
$$51$$ −6.15952 −0.862506
$$52$$ 0 0
$$53$$ 3.76714 0.517456 0.258728 0.965950i $$-0.416697\pi$$
0.258728 + 0.965950i $$0.416697\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −6.04798 −0.801074
$$58$$ 0 0
$$59$$ −11.7599 −1.53101 −0.765507 0.643428i $$-0.777512\pi$$
−0.765507 + 0.643428i $$0.777512\pi$$
$$60$$ 0 0
$$61$$ 12.6454 1.61908 0.809542 0.587063i $$-0.199716\pi$$
0.809542 + 0.587063i $$0.199716\pi$$
$$62$$ 0 0
$$63$$ −11.5121 −1.45039
$$64$$ 0 0
$$65$$ −0.601547 −0.0746127
$$66$$ 0 0
$$67$$ 15.0452 1.83806 0.919030 0.394187i $$-0.128974\pi$$
0.919030 + 0.394187i $$0.128974\pi$$
$$68$$ 0 0
$$69$$ −12.0210 −1.44715
$$70$$ 0 0
$$71$$ −0.346489 −0.0411207 −0.0205604 0.999789i $$-0.506545\pi$$
−0.0205604 + 0.999789i $$0.506545\pi$$
$$72$$ 0 0
$$73$$ 3.68096 0.430823 0.215412 0.976523i $$-0.430891\pi$$
0.215412 + 0.976523i $$0.430891\pi$$
$$74$$ 0 0
$$75$$ 2.32476 0.268441
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 8.05322 0.906058 0.453029 0.891496i $$-0.350343\pi$$
0.453029 + 0.891496i $$0.350343\pi$$
$$80$$ 0 0
$$81$$ −10.4318 −1.15909
$$82$$ 0 0
$$83$$ 13.0094 1.42796 0.713981 0.700165i $$-0.246890\pi$$
0.713981 + 0.700165i $$0.246890\pi$$
$$84$$ 0 0
$$85$$ −2.64953 −0.287381
$$86$$ 0 0
$$87$$ 12.6617 1.35747
$$88$$ 0 0
$$89$$ 0.462692 0.0490453 0.0245226 0.999699i $$-0.492193\pi$$
0.0245226 + 0.999699i $$0.492193\pi$$
$$90$$ 0 0
$$91$$ 2.88001 0.301907
$$92$$ 0 0
$$93$$ 24.3118 2.52101
$$94$$ 0 0
$$95$$ −2.60155 −0.266913
$$96$$ 0 0
$$97$$ 7.97011 0.809242 0.404621 0.914484i $$-0.367403\pi$$
0.404621 + 0.914484i $$0.367403\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −10.9134 −1.08592 −0.542960 0.839759i $$-0.682696\pi$$
−0.542960 + 0.839759i $$0.682696\pi$$
$$102$$ 0 0
$$103$$ 16.9081 1.66600 0.833001 0.553271i $$-0.186621\pi$$
0.833001 + 0.553271i $$0.186621\pi$$
$$104$$ 0 0
$$105$$ −11.1302 −1.08620
$$106$$ 0 0
$$107$$ −8.94029 −0.864291 −0.432145 0.901804i $$-0.642243\pi$$
−0.432145 + 0.901804i $$0.642243\pi$$
$$108$$ 0 0
$$109$$ 8.76627 0.839656 0.419828 0.907604i $$-0.362090\pi$$
0.419828 + 0.907604i $$0.362090\pi$$
$$110$$ 0 0
$$111$$ 21.8485 2.07377
$$112$$ 0 0
$$113$$ −2.49000 −0.234240 −0.117120 0.993118i $$-0.537366\pi$$
−0.117120 + 0.993118i $$0.537366\pi$$
$$114$$ 0 0
$$115$$ −5.17083 −0.482182
$$116$$ 0 0
$$117$$ −1.44643 −0.133723
$$118$$ 0 0
$$119$$ 12.6851 1.16284
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −10.5692 −0.952988
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 9.22655 0.818724 0.409362 0.912372i $$-0.365751\pi$$
0.409362 + 0.912372i $$0.365751\pi$$
$$128$$ 0 0
$$129$$ 8.10290 0.713421
$$130$$ 0 0
$$131$$ −8.09882 −0.707597 −0.353799 0.935322i $$-0.615110\pi$$
−0.353799 + 0.935322i $$0.615110\pi$$
$$132$$ 0 0
$$133$$ 12.4554 1.08002
$$134$$ 0 0
$$135$$ −1.38434 −0.119145
$$136$$ 0 0
$$137$$ 6.57195 0.561480 0.280740 0.959784i $$-0.409420\pi$$
0.280740 + 0.959784i $$0.409420\pi$$
$$138$$ 0 0
$$139$$ 3.61119 0.306297 0.153149 0.988203i $$-0.451059\pi$$
0.153149 + 0.988203i $$0.451059\pi$$
$$140$$ 0 0
$$141$$ −11.7945 −0.993272
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 5.44643 0.452302
$$146$$ 0 0
$$147$$ 37.0145 3.05291
$$148$$ 0 0
$$149$$ 12.1685 0.996886 0.498443 0.866922i $$-0.333905\pi$$
0.498443 + 0.866922i $$0.333905\pi$$
$$150$$ 0 0
$$151$$ −21.5830 −1.75640 −0.878198 0.478296i $$-0.841254\pi$$
−0.878198 + 0.478296i $$0.841254\pi$$
$$152$$ 0 0
$$153$$ −6.37085 −0.515053
$$154$$ 0 0
$$155$$ 10.4577 0.839986
$$156$$ 0 0
$$157$$ 7.68096 0.613007 0.306504 0.951870i $$-0.400841\pi$$
0.306504 + 0.951870i $$0.400841\pi$$
$$158$$ 0 0
$$159$$ 8.75771 0.694531
$$160$$ 0 0
$$161$$ 24.7562 1.95107
$$162$$ 0 0
$$163$$ 5.21691 0.408620 0.204310 0.978906i $$-0.434505\pi$$
0.204310 + 0.978906i $$0.434505\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 19.2767 1.49168 0.745838 0.666127i $$-0.232049\pi$$
0.745838 + 0.666127i $$0.232049\pi$$
$$168$$ 0 0
$$169$$ −12.6381 −0.972165
$$170$$ 0 0
$$171$$ −6.25548 −0.478369
$$172$$ 0 0
$$173$$ 19.6210 1.49175 0.745877 0.666084i $$-0.232031\pi$$
0.745877 + 0.666084i $$0.232031\pi$$
$$174$$ 0 0
$$175$$ −4.78768 −0.361914
$$176$$ 0 0
$$177$$ −27.3391 −2.05493
$$178$$ 0 0
$$179$$ 7.26184 0.542775 0.271388 0.962470i $$-0.412517\pi$$
0.271388 + 0.962470i $$0.412517\pi$$
$$180$$ 0 0
$$181$$ −8.78809 −0.653214 −0.326607 0.945160i $$-0.605905\pi$$
−0.326607 + 0.945160i $$0.605905\pi$$
$$182$$ 0 0
$$183$$ 29.3977 2.17314
$$184$$ 0 0
$$185$$ 9.39816 0.690967
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 6.62778 0.482100
$$190$$ 0 0
$$191$$ −24.9236 −1.80341 −0.901703 0.432355i $$-0.857683\pi$$
−0.901703 + 0.432355i $$0.857683\pi$$
$$192$$ 0 0
$$193$$ −6.66623 −0.479845 −0.239923 0.970792i $$-0.577122\pi$$
−0.239923 + 0.970792i $$0.577122\pi$$
$$194$$ 0 0
$$195$$ −1.39845 −0.100145
$$196$$ 0 0
$$197$$ −1.66329 −0.118504 −0.0592522 0.998243i $$-0.518872\pi$$
−0.0592522 + 0.998243i $$0.518872\pi$$
$$198$$ 0 0
$$199$$ 21.0008 1.48870 0.744352 0.667787i $$-0.232758\pi$$
0.744352 + 0.667787i $$0.232758\pi$$
$$200$$ 0 0
$$201$$ 34.9765 2.46705
$$202$$ 0 0
$$203$$ −26.0758 −1.83016
$$204$$ 0 0
$$205$$ −4.54633 −0.317530
$$206$$ 0 0
$$207$$ −12.4334 −0.864180
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −11.3170 −0.779095 −0.389548 0.921006i $$-0.627369\pi$$
−0.389548 + 0.921006i $$0.627369\pi$$
$$212$$ 0 0
$$213$$ −0.805506 −0.0551924
$$214$$ 0 0
$$215$$ 3.48547 0.237707
$$216$$ 0 0
$$217$$ −50.0683 −3.39886
$$218$$ 0 0
$$219$$ 8.55735 0.578252
$$220$$ 0 0
$$221$$ 1.59381 0.107212
$$222$$ 0 0
$$223$$ −0.746956 −0.0500198 −0.0250099 0.999687i $$-0.507962\pi$$
−0.0250099 + 0.999687i $$0.507962\pi$$
$$224$$ 0 0
$$225$$ 2.40452 0.160302
$$226$$ 0 0
$$227$$ 13.5057 0.896405 0.448203 0.893932i $$-0.352064\pi$$
0.448203 + 0.893932i $$0.352064\pi$$
$$228$$ 0 0
$$229$$ 2.36737 0.156440 0.0782201 0.996936i $$-0.475076\pi$$
0.0782201 + 0.996936i $$0.475076\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2.15499 −0.141178 −0.0705892 0.997505i $$-0.522488\pi$$
−0.0705892 + 0.997505i $$0.522488\pi$$
$$234$$ 0 0
$$235$$ −5.07340 −0.330952
$$236$$ 0 0
$$237$$ 18.7218 1.21611
$$238$$ 0 0
$$239$$ −28.5539 −1.84700 −0.923500 0.383600i $$-0.874684\pi$$
−0.923500 + 0.383600i $$0.874684\pi$$
$$240$$ 0 0
$$241$$ 5.62289 0.362202 0.181101 0.983465i $$-0.442034\pi$$
0.181101 + 0.983465i $$0.442034\pi$$
$$242$$ 0 0
$$243$$ −20.0985 −1.28932
$$244$$ 0 0
$$245$$ 15.9218 1.01721
$$246$$ 0 0
$$247$$ 1.56495 0.0995755
$$248$$ 0 0
$$249$$ 30.2437 1.91661
$$250$$ 0 0
$$251$$ −20.5337 −1.29607 −0.648036 0.761609i $$-0.724410\pi$$
−0.648036 + 0.761609i $$0.724410\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −6.15952 −0.385724
$$256$$ 0 0
$$257$$ −15.7592 −0.983030 −0.491515 0.870869i $$-0.663557\pi$$
−0.491515 + 0.870869i $$0.663557\pi$$
$$258$$ 0 0
$$259$$ −44.9954 −2.79588
$$260$$ 0 0
$$261$$ 13.0961 0.810627
$$262$$ 0 0
$$263$$ −15.8194 −0.975464 −0.487732 0.872993i $$-0.662176\pi$$
−0.487732 + 0.872993i $$0.662176\pi$$
$$264$$ 0 0
$$265$$ 3.76714 0.231413
$$266$$ 0 0
$$267$$ 1.07565 0.0658287
$$268$$ 0 0
$$269$$ 18.1701 1.10785 0.553924 0.832567i $$-0.313130\pi$$
0.553924 + 0.832567i $$0.313130\pi$$
$$270$$ 0 0
$$271$$ −24.8599 −1.51013 −0.755065 0.655650i $$-0.772395\pi$$
−0.755065 + 0.655650i $$0.772395\pi$$
$$272$$ 0 0
$$273$$ 6.69534 0.405221
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 23.6352 1.42010 0.710050 0.704151i $$-0.248672\pi$$
0.710050 + 0.704151i $$0.248672\pi$$
$$278$$ 0 0
$$279$$ 25.1459 1.50544
$$280$$ 0 0
$$281$$ −6.70036 −0.399710 −0.199855 0.979825i $$-0.564047\pi$$
−0.199855 + 0.979825i $$0.564047\pi$$
$$282$$ 0 0
$$283$$ −7.11292 −0.422819 −0.211410 0.977398i $$-0.567805\pi$$
−0.211410 + 0.977398i $$0.567805\pi$$
$$284$$ 0 0
$$285$$ −6.04798 −0.358251
$$286$$ 0 0
$$287$$ 21.7664 1.28483
$$288$$ 0 0
$$289$$ −9.98001 −0.587059
$$290$$ 0 0
$$291$$ 18.5286 1.08617
$$292$$ 0 0
$$293$$ 27.4950 1.60628 0.803138 0.595793i $$-0.203162\pi$$
0.803138 + 0.595793i $$0.203162\pi$$
$$294$$ 0 0
$$295$$ −11.7599 −0.684690
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 3.11049 0.179885
$$300$$ 0 0
$$301$$ −16.6873 −0.961841
$$302$$ 0 0
$$303$$ −25.3710 −1.45752
$$304$$ 0 0
$$305$$ 12.6454 0.724076
$$306$$ 0 0
$$307$$ 23.0888 1.31775 0.658875 0.752253i $$-0.271033\pi$$
0.658875 + 0.752253i $$0.271033\pi$$
$$308$$ 0 0
$$309$$ 39.3073 2.23611
$$310$$ 0 0
$$311$$ 7.07508 0.401191 0.200595 0.979674i $$-0.435712\pi$$
0.200595 + 0.979674i $$0.435712\pi$$
$$312$$ 0 0
$$313$$ −28.1646 −1.59196 −0.795979 0.605325i $$-0.793043\pi$$
−0.795979 + 0.605325i $$0.793043\pi$$
$$314$$ 0 0
$$315$$ −11.5121 −0.648632
$$316$$ 0 0
$$317$$ 23.4408 1.31657 0.658284 0.752770i $$-0.271283\pi$$
0.658284 + 0.752770i $$0.271283\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −20.7841 −1.16005
$$322$$ 0 0
$$323$$ 6.89287 0.383529
$$324$$ 0 0
$$325$$ −0.601547 −0.0333678
$$326$$ 0 0
$$327$$ 20.3795 1.12699
$$328$$ 0 0
$$329$$ 24.2898 1.33914
$$330$$ 0 0
$$331$$ −5.08798 −0.279661 −0.139830 0.990175i $$-0.544656\pi$$
−0.139830 + 0.990175i $$0.544656\pi$$
$$332$$ 0 0
$$333$$ 22.5981 1.23837
$$334$$ 0 0
$$335$$ 15.0452 0.822006
$$336$$ 0 0
$$337$$ 25.4172 1.38456 0.692280 0.721629i $$-0.256606\pi$$
0.692280 + 0.721629i $$0.256606\pi$$
$$338$$ 0 0
$$339$$ −5.78867 −0.314398
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −42.7149 −2.30639
$$344$$ 0 0
$$345$$ −12.0210 −0.647186
$$346$$ 0 0
$$347$$ 17.8827 0.959995 0.479997 0.877270i $$-0.340638\pi$$
0.479997 + 0.877270i $$0.340638\pi$$
$$348$$ 0 0
$$349$$ 6.29515 0.336971 0.168486 0.985704i $$-0.446112\pi$$
0.168486 + 0.985704i $$0.446112\pi$$
$$350$$ 0 0
$$351$$ 0.832746 0.0444487
$$352$$ 0 0
$$353$$ 3.17636 0.169060 0.0845302 0.996421i $$-0.473061\pi$$
0.0845302 + 0.996421i $$0.473061\pi$$
$$354$$ 0 0
$$355$$ −0.346489 −0.0183898
$$356$$ 0 0
$$357$$ 29.4898 1.56077
$$358$$ 0 0
$$359$$ 5.12020 0.270234 0.135117 0.990830i $$-0.456859\pi$$
0.135117 + 0.990830i $$0.456859\pi$$
$$360$$ 0 0
$$361$$ −12.2320 −0.643787
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3.68096 0.192670
$$366$$ 0 0
$$367$$ 10.8346 0.565564 0.282782 0.959184i $$-0.408743\pi$$
0.282782 + 0.959184i $$0.408743\pi$$
$$368$$ 0 0
$$369$$ −10.9318 −0.569085
$$370$$ 0 0
$$371$$ −18.0358 −0.936374
$$372$$ 0 0
$$373$$ 36.5605 1.89303 0.946516 0.322656i $$-0.104576\pi$$
0.946516 + 0.322656i $$0.104576\pi$$
$$374$$ 0 0
$$375$$ 2.32476 0.120050
$$376$$ 0 0
$$377$$ −3.27628 −0.168737
$$378$$ 0 0
$$379$$ −0.681538 −0.0350083 −0.0175041 0.999847i $$-0.505572\pi$$
−0.0175041 + 0.999847i $$0.505572\pi$$
$$380$$ 0 0
$$381$$ 21.4495 1.09889
$$382$$ 0 0
$$383$$ −36.6876 −1.87465 −0.937323 0.348461i $$-0.886705\pi$$
−0.937323 + 0.348461i $$0.886705\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 8.38090 0.426025
$$388$$ 0 0
$$389$$ 36.9171 1.87177 0.935887 0.352301i $$-0.114601\pi$$
0.935887 + 0.352301i $$0.114601\pi$$
$$390$$ 0 0
$$391$$ 13.7002 0.692851
$$392$$ 0 0
$$393$$ −18.8278 −0.949739
$$394$$ 0 0
$$395$$ 8.05322 0.405201
$$396$$ 0 0
$$397$$ −29.6085 −1.48601 −0.743003 0.669288i $$-0.766599\pi$$
−0.743003 + 0.669288i $$0.766599\pi$$
$$398$$ 0 0
$$399$$ 28.9558 1.44960
$$400$$ 0 0
$$401$$ −17.1919 −0.858523 −0.429262 0.903180i $$-0.641226\pi$$
−0.429262 + 0.903180i $$0.641226\pi$$
$$402$$ 0 0
$$403$$ −6.29082 −0.313368
$$404$$ 0 0
$$405$$ −10.4318 −0.518362
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 7.36760 0.364304 0.182152 0.983270i $$-0.441694\pi$$
0.182152 + 0.983270i $$0.441694\pi$$
$$410$$ 0 0
$$411$$ 15.2782 0.753619
$$412$$ 0 0
$$413$$ 56.3028 2.77048
$$414$$ 0 0
$$415$$ 13.0094 0.638604
$$416$$ 0 0
$$417$$ 8.39516 0.411113
$$418$$ 0 0
$$419$$ −19.3728 −0.946425 −0.473212 0.880948i $$-0.656906\pi$$
−0.473212 + 0.880948i $$0.656906\pi$$
$$420$$ 0 0
$$421$$ 17.2101 0.838769 0.419384 0.907809i $$-0.362246\pi$$
0.419384 + 0.907809i $$0.362246\pi$$
$$422$$ 0 0
$$423$$ −12.1991 −0.593141
$$424$$ 0 0
$$425$$ −2.64953 −0.128521
$$426$$ 0 0
$$427$$ −60.5423 −2.92985
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −30.8744 −1.48717 −0.743584 0.668643i $$-0.766876\pi$$
−0.743584 + 0.668643i $$0.766876\pi$$
$$432$$ 0 0
$$433$$ 2.48737 0.119536 0.0597678 0.998212i $$-0.480964\pi$$
0.0597678 + 0.998212i $$0.480964\pi$$
$$434$$ 0 0
$$435$$ 12.6617 0.607080
$$436$$ 0 0
$$437$$ 13.4522 0.643504
$$438$$ 0 0
$$439$$ 14.6203 0.697790 0.348895 0.937162i $$-0.386557\pi$$
0.348895 + 0.937162i $$0.386557\pi$$
$$440$$ 0 0
$$441$$ 38.2844 1.82307
$$442$$ 0 0
$$443$$ 32.0885 1.52457 0.762285 0.647241i $$-0.224077\pi$$
0.762285 + 0.647241i $$0.224077\pi$$
$$444$$ 0 0
$$445$$ 0.462692 0.0219337
$$446$$ 0 0
$$447$$ 28.2890 1.33802
$$448$$ 0 0
$$449$$ 11.2208 0.529544 0.264772 0.964311i $$-0.414703\pi$$
0.264772 + 0.964311i $$0.414703\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −50.1753 −2.35744
$$454$$ 0 0
$$455$$ 2.88001 0.135017
$$456$$ 0 0
$$457$$ 7.28861 0.340947 0.170473 0.985362i $$-0.445470\pi$$
0.170473 + 0.985362i $$0.445470\pi$$
$$458$$ 0 0
$$459$$ 3.66785 0.171200
$$460$$ 0 0
$$461$$ −8.56686 −0.398999 −0.199499 0.979898i $$-0.563932\pi$$
−0.199499 + 0.979898i $$0.563932\pi$$
$$462$$ 0 0
$$463$$ 22.6373 1.05204 0.526022 0.850471i $$-0.323683\pi$$
0.526022 + 0.850471i $$0.323683\pi$$
$$464$$ 0 0
$$465$$ 24.3118 1.12743
$$466$$ 0 0
$$467$$ 31.1970 1.44363 0.721813 0.692089i $$-0.243309\pi$$
0.721813 + 0.692089i $$0.243309\pi$$
$$468$$ 0 0
$$469$$ −72.0314 −3.32610
$$470$$ 0 0
$$471$$ 17.8564 0.822780
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −2.60155 −0.119367
$$476$$ 0 0
$$477$$ 9.05817 0.414745
$$478$$ 0 0
$$479$$ −33.1292 −1.51371 −0.756856 0.653582i $$-0.773265\pi$$
−0.756856 + 0.653582i $$0.773265\pi$$
$$480$$ 0 0
$$481$$ −5.65343 −0.257774
$$482$$ 0 0
$$483$$ 57.5524 2.61873
$$484$$ 0 0
$$485$$ 7.97011 0.361904
$$486$$ 0 0
$$487$$ 6.89985 0.312662 0.156331 0.987705i $$-0.450033\pi$$
0.156331 + 0.987705i $$0.450033\pi$$
$$488$$ 0 0
$$489$$ 12.1281 0.548451
$$490$$ 0 0
$$491$$ −3.53178 −0.159387 −0.0796935 0.996819i $$-0.525394\pi$$
−0.0796935 + 0.996819i $$0.525394\pi$$
$$492$$ 0 0
$$493$$ −14.4305 −0.649916
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 1.65888 0.0744109
$$498$$ 0 0
$$499$$ 4.67978 0.209496 0.104748 0.994499i $$-0.466596\pi$$
0.104748 + 0.994499i $$0.466596\pi$$
$$500$$ 0 0
$$501$$ 44.8138 2.00213
$$502$$ 0 0
$$503$$ −35.5622 −1.58564 −0.792821 0.609455i $$-0.791388\pi$$
−0.792821 + 0.609455i $$0.791388\pi$$
$$504$$ 0 0
$$505$$ −10.9134 −0.485638
$$506$$ 0 0
$$507$$ −29.3807 −1.30484
$$508$$ 0 0
$$509$$ −10.7168 −0.475012 −0.237506 0.971386i $$-0.576330\pi$$
−0.237506 + 0.971386i $$0.576330\pi$$
$$510$$ 0 0
$$511$$ −17.6232 −0.779606
$$512$$ 0 0
$$513$$ 3.60143 0.159007
$$514$$ 0 0
$$515$$ 16.9081 0.745059
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 45.6141 2.00224
$$520$$ 0 0
$$521$$ −2.19519 −0.0961729 −0.0480865 0.998843i $$-0.515312\pi$$
−0.0480865 + 0.998843i $$0.515312\pi$$
$$522$$ 0 0
$$523$$ −2.21848 −0.0970075 −0.0485038 0.998823i $$-0.515445\pi$$
−0.0485038 + 0.998823i $$0.515445\pi$$
$$524$$ 0 0
$$525$$ −11.1302 −0.485762
$$526$$ 0 0
$$527$$ −27.7081 −1.20698
$$528$$ 0 0
$$529$$ 3.73746 0.162498
$$530$$ 0 0
$$531$$ −28.2771 −1.22712
$$532$$ 0 0
$$533$$ 2.73483 0.118459
$$534$$ 0 0
$$535$$ −8.94029 −0.386523
$$536$$ 0 0
$$537$$ 16.8821 0.728514
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 38.8481 1.67021 0.835106 0.550089i $$-0.185406\pi$$
0.835106 + 0.550089i $$0.185406\pi$$
$$542$$ 0 0
$$543$$ −20.4302 −0.876745
$$544$$ 0 0
$$545$$ 8.76627 0.375506
$$546$$ 0 0
$$547$$ 12.4573 0.532637 0.266319 0.963885i $$-0.414193\pi$$
0.266319 + 0.963885i $$0.414193\pi$$
$$548$$ 0 0
$$549$$ 30.4063 1.29771
$$550$$ 0 0
$$551$$ −14.1691 −0.603626
$$552$$ 0 0
$$553$$ −38.5562 −1.63958
$$554$$ 0 0
$$555$$ 21.8485 0.927417
$$556$$ 0 0
$$557$$ −33.9789 −1.43973 −0.719867 0.694112i $$-0.755797\pi$$
−0.719867 + 0.694112i $$0.755797\pi$$
$$558$$ 0 0
$$559$$ −2.09667 −0.0886799
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 23.9594 1.00977 0.504884 0.863187i $$-0.331535\pi$$
0.504884 + 0.863187i $$0.331535\pi$$
$$564$$ 0 0
$$565$$ −2.49000 −0.104755
$$566$$ 0 0
$$567$$ 49.9443 2.09746
$$568$$ 0 0
$$569$$ 7.52879 0.315623 0.157812 0.987469i $$-0.449556\pi$$
0.157812 + 0.987469i $$0.449556\pi$$
$$570$$ 0 0
$$571$$ −13.5353 −0.566433 −0.283216 0.959056i $$-0.591401\pi$$
−0.283216 + 0.959056i $$0.591401\pi$$
$$572$$ 0 0
$$573$$ −57.9414 −2.42054
$$574$$ 0 0
$$575$$ −5.17083 −0.215638
$$576$$ 0 0
$$577$$ 27.3683 1.13936 0.569678 0.821868i $$-0.307068\pi$$
0.569678 + 0.821868i $$0.307068\pi$$
$$578$$ 0 0
$$579$$ −15.4974 −0.644050
$$580$$ 0 0
$$581$$ −62.2846 −2.58400
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −1.44643 −0.0598027
$$586$$ 0 0
$$587$$ −17.1927 −0.709618 −0.354809 0.934939i $$-0.615454\pi$$
−0.354809 + 0.934939i $$0.615454\pi$$
$$588$$ 0 0
$$589$$ −27.2063 −1.12102
$$590$$ 0 0
$$591$$ −3.86675 −0.159057
$$592$$ 0 0
$$593$$ 38.4574 1.57925 0.789627 0.613587i $$-0.210274\pi$$
0.789627 + 0.613587i $$0.210274\pi$$
$$594$$ 0 0
$$595$$ 12.6851 0.520037
$$596$$ 0 0
$$597$$ 48.8218 1.99814
$$598$$ 0 0
$$599$$ 9.63410 0.393639 0.196819 0.980440i $$-0.436939\pi$$
0.196819 + 0.980440i $$0.436939\pi$$
$$600$$ 0 0
$$601$$ −20.0426 −0.817555 −0.408777 0.912634i $$-0.634045\pi$$
−0.408777 + 0.912634i $$0.634045\pi$$
$$602$$ 0 0
$$603$$ 36.1765 1.47322
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 17.4596 0.708662 0.354331 0.935120i $$-0.384709\pi$$
0.354331 + 0.935120i $$0.384709\pi$$
$$608$$ 0 0
$$609$$ −60.6200 −2.45644
$$610$$ 0 0
$$611$$ 3.05189 0.123466
$$612$$ 0 0
$$613$$ −20.9105 −0.844569 −0.422284 0.906463i $$-0.638772\pi$$
−0.422284 + 0.906463i $$0.638772\pi$$
$$614$$ 0 0
$$615$$ −10.5692 −0.426189
$$616$$ 0 0
$$617$$ −41.2590 −1.66103 −0.830513 0.557000i $$-0.811952\pi$$
−0.830513 + 0.557000i $$0.811952\pi$$
$$618$$ 0 0
$$619$$ −0.151406 −0.00608551 −0.00304276 0.999995i $$-0.500969\pi$$
−0.00304276 + 0.999995i $$0.500969\pi$$
$$620$$ 0 0
$$621$$ 7.15819 0.287248
$$622$$ 0 0
$$623$$ −2.21522 −0.0887510
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −24.9007 −0.992855
$$630$$ 0 0
$$631$$ −9.21832 −0.366976 −0.183488 0.983022i $$-0.558739\pi$$
−0.183488 + 0.983022i $$0.558739\pi$$
$$632$$ 0 0
$$633$$ −26.3094 −1.04570
$$634$$ 0 0
$$635$$ 9.22655 0.366145
$$636$$ 0 0
$$637$$ −9.57773 −0.379483
$$638$$ 0 0
$$639$$ −0.833142 −0.0329586
$$640$$ 0 0
$$641$$ −19.8665 −0.784678 −0.392339 0.919821i $$-0.628334\pi$$
−0.392339 + 0.919821i $$0.628334\pi$$
$$642$$ 0 0
$$643$$ 17.8982 0.705838 0.352919 0.935654i $$-0.385189\pi$$
0.352919 + 0.935654i $$0.385189\pi$$
$$644$$ 0 0
$$645$$ 8.10290 0.319051
$$646$$ 0 0
$$647$$ 22.6373 0.889964 0.444982 0.895539i $$-0.353210\pi$$
0.444982 + 0.895539i $$0.353210\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −116.397 −4.56195
$$652$$ 0 0
$$653$$ 11.1498 0.436325 0.218163 0.975912i $$-0.429994\pi$$
0.218163 + 0.975912i $$0.429994\pi$$
$$654$$ 0 0
$$655$$ −8.09882 −0.316447
$$656$$ 0 0
$$657$$ 8.85095 0.345308
$$658$$ 0 0
$$659$$ 18.0304 0.702366 0.351183 0.936307i $$-0.385779\pi$$
0.351183 + 0.936307i $$0.385779\pi$$
$$660$$ 0 0
$$661$$ −16.1373 −0.627668 −0.313834 0.949478i $$-0.601613\pi$$
−0.313834 + 0.949478i $$0.601613\pi$$
$$662$$ 0 0
$$663$$ 3.70524 0.143900
$$664$$ 0 0
$$665$$ 12.4554 0.482998
$$666$$ 0 0
$$667$$ −28.1626 −1.09046
$$668$$ 0 0
$$669$$ −1.73649 −0.0671368
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −35.5210 −1.36923 −0.684617 0.728903i $$-0.740031\pi$$
−0.684617 + 0.728903i $$0.740031\pi$$
$$674$$ 0 0
$$675$$ −1.38434 −0.0532833
$$676$$ 0 0
$$677$$ −32.4138 −1.24576 −0.622882 0.782315i $$-0.714039\pi$$
−0.622882 + 0.782315i $$0.714039\pi$$
$$678$$ 0 0
$$679$$ −38.1583 −1.46438
$$680$$ 0 0
$$681$$ 31.3976 1.20316
$$682$$ 0 0
$$683$$ −7.36687 −0.281885 −0.140943 0.990018i $$-0.545013\pi$$
−0.140943 + 0.990018i $$0.545013\pi$$
$$684$$ 0 0
$$685$$ 6.57195 0.251101
$$686$$ 0 0
$$687$$ 5.50357 0.209974
$$688$$ 0 0
$$689$$ −2.26611 −0.0863319
$$690$$ 0 0
$$691$$ 0.768429 0.0292324 0.0146162 0.999893i $$-0.495347\pi$$
0.0146162 + 0.999893i $$0.495347\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 3.61119 0.136980
$$696$$ 0 0
$$697$$ 12.0456 0.456261
$$698$$ 0 0
$$699$$ −5.00985 −0.189490
$$700$$ 0 0
$$701$$ −7.95905 −0.300609 −0.150305 0.988640i $$-0.548025\pi$$
−0.150305 + 0.988640i $$0.548025\pi$$
$$702$$ 0 0
$$703$$ −24.4498 −0.922140
$$704$$ 0 0
$$705$$ −11.7945 −0.444205
$$706$$ 0 0
$$707$$ 52.2496 1.96505
$$708$$ 0 0
$$709$$ 15.3307 0.575755 0.287877 0.957667i $$-0.407050\pi$$
0.287877 + 0.957667i $$0.407050\pi$$
$$710$$ 0 0
$$711$$ 19.3642 0.726212
$$712$$ 0 0
$$713$$ −54.0752 −2.02513
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −66.3811 −2.47905
$$718$$ 0 0
$$719$$ −46.2627 −1.72531 −0.862654 0.505794i $$-0.831200\pi$$
−0.862654 + 0.505794i $$0.831200\pi$$
$$720$$ 0 0
$$721$$ −80.9504 −3.01475
$$722$$ 0 0
$$723$$ 13.0719 0.486148
$$724$$ 0 0
$$725$$ 5.44643 0.202275
$$726$$ 0 0
$$727$$ −24.0817 −0.893140 −0.446570 0.894749i $$-0.647355\pi$$
−0.446570 + 0.894749i $$0.647355\pi$$
$$728$$ 0 0
$$729$$ −15.4288 −0.571437
$$730$$ 0 0
$$731$$ −9.23485 −0.341563
$$732$$ 0 0
$$733$$ −13.0668 −0.482632 −0.241316 0.970447i $$-0.577579\pi$$
−0.241316 + 0.970447i $$0.577579\pi$$
$$734$$ 0 0
$$735$$ 37.0145 1.36530
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 33.5963 1.23586 0.617930 0.786233i $$-0.287972\pi$$
0.617930 + 0.786233i $$0.287972\pi$$
$$740$$ 0 0
$$741$$ 3.63814 0.133650
$$742$$ 0 0
$$743$$ 27.1661 0.996626 0.498313 0.866997i $$-0.333953\pi$$
0.498313 + 0.866997i $$0.333953\pi$$
$$744$$ 0 0
$$745$$ 12.1685 0.445821
$$746$$ 0 0
$$747$$ 31.2813 1.14452
$$748$$ 0 0
$$749$$ 42.8032 1.56400
$$750$$ 0 0
$$751$$ −13.5581 −0.494742 −0.247371 0.968921i $$-0.579567\pi$$
−0.247371 + 0.968921i $$0.579567\pi$$
$$752$$ 0 0
$$753$$ −47.7359 −1.73959
$$754$$ 0 0
$$755$$ −21.5830 −0.785485
$$756$$ 0 0
$$757$$ 32.1498 1.16850 0.584252 0.811572i $$-0.301388\pi$$
0.584252 + 0.811572i $$0.301388\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 9.70988 0.351983 0.175991 0.984392i $$-0.443687\pi$$
0.175991 + 0.984392i $$0.443687\pi$$
$$762$$ 0 0
$$763$$ −41.9701 −1.51942
$$764$$ 0 0
$$765$$ −6.37085 −0.230339
$$766$$ 0 0
$$767$$ 7.07415 0.255433
$$768$$ 0 0
$$769$$ 15.6081 0.562841 0.281421 0.959584i $$-0.409194\pi$$
0.281421 + 0.959584i $$0.409194\pi$$
$$770$$ 0 0
$$771$$ −36.6363 −1.31943
$$772$$ 0 0
$$773$$ −21.4874 −0.772849 −0.386425 0.922321i $$-0.626290\pi$$
−0.386425 + 0.922321i $$0.626290\pi$$
$$774$$ 0 0
$$775$$ 10.4577 0.375653
$$776$$ 0 0
$$777$$ −104.604 −3.75263
$$778$$ 0 0
$$779$$ 11.8275 0.423764
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −7.53972 −0.269448
$$784$$ 0 0
$$785$$ 7.68096 0.274145
$$786$$ 0 0
$$787$$ −6.04684 −0.215547 −0.107773 0.994175i $$-0.534372\pi$$
−0.107773 + 0.994175i $$0.534372\pi$$
$$788$$ 0 0
$$789$$ −36.7763 −1.30927
$$790$$ 0 0
$$791$$ 11.9213 0.423874
$$792$$ 0 0
$$793$$ −7.60682 −0.270126
$$794$$ 0 0
$$795$$ 8.75771 0.310604
$$796$$ 0 0
$$797$$ 32.0918 1.13675 0.568376 0.822769i $$-0.307572\pi$$
0.568376 + 0.822769i $$0.307572\pi$$
$$798$$ 0 0
$$799$$ 13.4421 0.475547
$$800$$ 0 0
$$801$$ 1.11255 0.0393102
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 24.7562 0.872543
$$806$$ 0 0
$$807$$ 42.2411 1.48696
$$808$$ 0 0
$$809$$ −2.20394 −0.0774865 −0.0387432 0.999249i $$-0.512335\pi$$
−0.0387432 + 0.999249i $$0.512335\pi$$
$$810$$ 0 0
$$811$$ 9.59190 0.336817 0.168409 0.985717i $$-0.446137\pi$$
0.168409 + 0.985717i $$0.446137\pi$$
$$812$$ 0 0
$$813$$ −57.7933 −2.02690
$$814$$ 0 0
$$815$$ 5.21691 0.182741
$$816$$ 0 0
$$817$$ −9.06762 −0.317236
$$818$$ 0 0
$$819$$ 6.92505 0.241981
$$820$$ 0 0
$$821$$ 14.2376 0.496894 0.248447 0.968645i $$-0.420080\pi$$
0.248447 + 0.968645i $$0.420080\pi$$
$$822$$ 0 0
$$823$$ 0.430172 0.0149949 0.00749743 0.999972i $$-0.497613\pi$$
0.00749743 + 0.999972i $$0.497613\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −35.6560 −1.23988 −0.619940 0.784649i $$-0.712843\pi$$
−0.619940 + 0.784649i $$0.712843\pi$$
$$828$$ 0 0
$$829$$ 10.9221 0.379339 0.189669 0.981848i $$-0.439258\pi$$
0.189669 + 0.981848i $$0.439258\pi$$
$$830$$ 0 0
$$831$$ 54.9462 1.90606
$$832$$ 0 0
$$833$$ −42.1853 −1.46164
$$834$$ 0 0
$$835$$ 19.2767 0.667098
$$836$$ 0 0
$$837$$ −14.4771 −0.500401
$$838$$ 0 0
$$839$$ −30.0181 −1.03634 −0.518169 0.855278i $$-0.673386\pi$$
−0.518169 + 0.855278i $$0.673386\pi$$
$$840$$ 0 0
$$841$$ 0.663634 0.0228839
$$842$$ 0 0
$$843$$ −15.5768 −0.536492
$$844$$ 0 0
$$845$$ −12.6381 −0.434765
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −16.5359 −0.567509
$$850$$ 0 0
$$851$$ −48.5963 −1.66586
$$852$$ 0 0
$$853$$ 20.5878 0.704912 0.352456 0.935828i $$-0.385347\pi$$
0.352456 + 0.935828i $$0.385347\pi$$
$$854$$ 0 0
$$855$$ −6.25548 −0.213933
$$856$$ 0 0
$$857$$ −7.44802 −0.254419 −0.127210 0.991876i $$-0.540602\pi$$
−0.127210 + 0.991876i $$0.540602\pi$$
$$858$$ 0 0
$$859$$ 53.0056 1.80853 0.904263 0.426975i $$-0.140421\pi$$
0.904263 + 0.426975i $$0.140421\pi$$
$$860$$ 0 0
$$861$$ 50.6017 1.72450
$$862$$ 0 0
$$863$$ 29.1040 0.990713 0.495356 0.868690i $$-0.335037\pi$$
0.495356 + 0.868690i $$0.335037\pi$$
$$864$$ 0 0
$$865$$ 19.6210 0.667132
$$866$$ 0 0
$$867$$ −23.2012 −0.787953
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −9.05037 −0.306660
$$872$$ 0 0
$$873$$ 19.1643 0.648614
$$874$$ 0 0
$$875$$ −4.78768 −0.161853
$$876$$ 0 0
$$877$$ −28.1013 −0.948912 −0.474456 0.880279i $$-0.657355\pi$$
−0.474456 + 0.880279i $$0.657355\pi$$
$$878$$ 0 0
$$879$$ 63.9194 2.15595
$$880$$ 0 0
$$881$$ 29.5508 0.995591 0.497796 0.867294i $$-0.334143\pi$$
0.497796 + 0.867294i $$0.334143\pi$$
$$882$$ 0 0
$$883$$ −38.9274 −1.31001 −0.655005 0.755624i $$-0.727334\pi$$
−0.655005 + 0.755624i $$0.727334\pi$$
$$884$$ 0 0
$$885$$ −27.3391 −0.918993
$$886$$ 0 0
$$887$$ 26.2960 0.882933 0.441467 0.897278i $$-0.354458\pi$$
0.441467 + 0.897278i $$0.354458\pi$$
$$888$$ 0 0
$$889$$ −44.1737 −1.48154
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 13.1987 0.441677
$$894$$ 0 0
$$895$$ 7.26184 0.242736
$$896$$ 0 0
$$897$$ 7.23116 0.241441
$$898$$ 0 0
$$899$$ 56.9574 1.89964
$$900$$ 0 0
$$901$$ −9.98113 −0.332520
$$902$$ 0 0
$$903$$ −38.7941 −1.29099
$$904$$ 0 0
$$905$$ −8.78809 −0.292126
$$906$$ 0 0
$$907$$ −25.5540 −0.848505 −0.424253 0.905544i $$-0.639463\pi$$
−0.424253 + 0.905544i $$0.639463\pi$$
$$908$$ 0 0
$$909$$ −26.2414 −0.870373
$$910$$ 0 0
$$911$$ 48.0753 1.59281 0.796403 0.604766i $$-0.206733\pi$$
0.796403 + 0.604766i $$0.206733\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 29.3977 0.971857
$$916$$ 0 0
$$917$$ 38.7745 1.28045
$$918$$ 0 0
$$919$$ −8.02354 −0.264672 −0.132336 0.991205i $$-0.542248\pi$$
−0.132336 + 0.991205i $$0.542248\pi$$
$$920$$ 0 0
$$921$$ 53.6761 1.76869
$$922$$ 0 0
$$923$$ 0.208430 0.00686054
$$924$$ 0 0
$$925$$ 9.39816 0.309010
$$926$$ 0 0
$$927$$ 40.6559 1.33531
$$928$$ 0 0
$$929$$ 47.7554 1.56680 0.783402 0.621516i $$-0.213483\pi$$
0.783402 + 0.621516i $$0.213483\pi$$
$$930$$ 0 0
$$931$$ −41.4214 −1.35753
$$932$$ 0 0
$$933$$ 16.4479 0.538480
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −13.5515 −0.442709 −0.221354 0.975193i $$-0.571048\pi$$
−0.221354 + 0.975193i $$0.571048\pi$$
$$938$$ 0 0
$$939$$ −65.4760 −2.13673
$$940$$ 0 0
$$941$$ −11.7661 −0.383564 −0.191782 0.981438i $$-0.561427\pi$$
−0.191782 + 0.981438i $$0.561427\pi$$
$$942$$ 0 0
$$943$$ 23.5083 0.765536
$$944$$ 0 0
$$945$$ 6.62778 0.215602
$$946$$ 0 0
$$947$$ −7.85540 −0.255266 −0.127633 0.991821i $$-0.540738\pi$$
−0.127633 + 0.991821i $$0.540738\pi$$
$$948$$ 0 0
$$949$$ −2.21427 −0.0718782
$$950$$ 0 0
$$951$$ 54.4944 1.76710
$$952$$ 0 0
$$953$$ 35.7034 1.15655 0.578274 0.815843i $$-0.303726\pi$$
0.578274 + 0.815843i $$0.303726\pi$$
$$954$$ 0 0
$$955$$ −24.9236 −0.806508
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −31.4644 −1.01604
$$960$$ 0 0
$$961$$ 78.3643 2.52788
$$962$$ 0 0
$$963$$ −21.4971 −0.692736
$$964$$ 0 0
$$965$$ −6.66623 −0.214593
$$966$$ 0 0
$$967$$ 12.5681 0.404162 0.202081 0.979369i $$-0.435230\pi$$
0.202081 + 0.979369i $$0.435230\pi$$
$$968$$ 0 0
$$969$$ 16.0243 0.514774
$$970$$ 0 0
$$971$$ −1.33283 −0.0427724 −0.0213862 0.999771i $$-0.506808\pi$$
−0.0213862 + 0.999771i $$0.506808\pi$$
$$972$$ 0 0
$$973$$ −17.2892 −0.554267
$$974$$ 0 0
$$975$$ −1.39845 −0.0447864
$$976$$ 0 0
$$977$$ −44.0415 −1.40901 −0.704506 0.709698i $$-0.748831\pi$$
−0.704506 + 0.709698i $$0.748831\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 21.0787 0.672991
$$982$$ 0 0
$$983$$ 13.3425 0.425560 0.212780 0.977100i $$-0.431748\pi$$
0.212780 + 0.977100i $$0.431748\pi$$
$$984$$ 0 0
$$985$$ −1.66329 −0.0529968
$$986$$ 0 0
$$987$$ 56.4680 1.79740
$$988$$ 0 0
$$989$$ −18.0228 −0.573091
$$990$$ 0 0
$$991$$ −8.69302 −0.276143 −0.138071 0.990422i $$-0.544090\pi$$
−0.138071 + 0.990422i $$0.544090\pi$$
$$992$$ 0 0
$$993$$ −11.8283 −0.375361
$$994$$ 0 0
$$995$$ 21.0008 0.665769
$$996$$ 0 0
$$997$$ −19.1952 −0.607918 −0.303959 0.952685i $$-0.598309\pi$$
−0.303959 + 0.952685i $$0.598309\pi$$
$$998$$ 0 0
$$999$$ −13.0103 −0.411626
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 9680.2.a.da.1.5 6
4.3 odd 2 4840.2.a.bd.1.2 yes 6
11.10 odd 2 9680.2.a.db.1.5 6
44.43 even 2 4840.2.a.bc.1.2 6

By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.bc.1.2 6 44.43 even 2
4840.2.a.bd.1.2 yes 6 4.3 odd 2
9680.2.a.da.1.5 6 1.1 even 1 trivial
9680.2.a.db.1.5 6 11.10 odd 2