# Properties

 Label 920.2.a.h.1.3 Level $920$ Weight $2$ Character 920.1 Self dual yes Analytic conductor $7.346$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [920,2,Mod(1,920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(920, 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("920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$920 = 2^{3} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 920.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: $$7.34623698596$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2597.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 9x + 8$$ x^3 - x^2 - 9*x + 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$3.07912$$ of defining polynomial Character $$\chi$$ $$=$$ 920.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.07912 q^{3} +1.00000 q^{5} -2.07912 q^{7} +6.48097 q^{9} +O(q^{10})$$ $$q+3.07912 q^{3} +1.00000 q^{5} -2.07912 q^{7} +6.48097 q^{9} +5.07912 q^{11} -3.48097 q^{13} +3.07912 q^{15} +6.48097 q^{17} -7.48097 q^{19} -6.40185 q^{21} +1.00000 q^{23} +1.00000 q^{25} +10.7183 q^{27} -1.56009 q^{29} +0.0791189 q^{31} +15.6392 q^{33} -2.07912 q^{35} -9.71833 q^{37} -10.7183 q^{39} -0.480973 q^{41} +8.00000 q^{43} +6.48097 q^{45} +6.96195 q^{47} -2.67726 q^{49} +19.9557 q^{51} +11.7183 q^{53} +5.07912 q^{55} -23.0348 q^{57} -11.5601 q^{59} +7.88283 q^{61} -13.4747 q^{63} -3.48097 q^{65} -9.71833 q^{67} +3.07912 q^{69} -9.67726 q^{71} -13.2784 q^{73} +3.07912 q^{75} -10.5601 q^{77} -12.3165 q^{79} +13.5601 q^{81} -4.59815 q^{83} +6.48097 q^{85} -4.80371 q^{87} +8.31648 q^{89} +7.23736 q^{91} +0.243616 q^{93} -7.48097 q^{95} +7.23736 q^{97} +32.9176 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} + 3 q^{5} + 2 q^{7} + 10 q^{9}+O(q^{10})$$ 3 * q + q^3 + 3 * q^5 + 2 * q^7 + 10 * q^9 $$3 q + q^{3} + 3 q^{5} + 2 q^{7} + 10 q^{9} + 7 q^{11} - q^{13} + q^{15} + 10 q^{17} - 13 q^{19} - 18 q^{21} + 3 q^{23} + 3 q^{25} - 2 q^{27} + 13 q^{29} - 8 q^{31} + 21 q^{33} + 2 q^{35} + 5 q^{37} + 2 q^{39} + 8 q^{41} + 24 q^{43} + 10 q^{45} + 2 q^{47} - q^{49} + q^{51} + q^{53} + 7 q^{55} - 2 q^{57} - 17 q^{59} + 13 q^{61} + 9 q^{63} - q^{65} + 5 q^{67} + q^{69} - 22 q^{71} + 12 q^{73} + q^{75} - 14 q^{77} - 4 q^{79} + 23 q^{81} - 15 q^{83} + 10 q^{85} - 12 q^{87} - 8 q^{89} - 3 q^{91} + 16 q^{93} - 13 q^{95} - 3 q^{97} + 21 q^{99}+O(q^{100})$$ 3 * q + q^3 + 3 * q^5 + 2 * q^7 + 10 * q^9 + 7 * q^11 - q^13 + q^15 + 10 * q^17 - 13 * q^19 - 18 * q^21 + 3 * q^23 + 3 * q^25 - 2 * q^27 + 13 * q^29 - 8 * q^31 + 21 * q^33 + 2 * q^35 + 5 * q^37 + 2 * q^39 + 8 * q^41 + 24 * q^43 + 10 * q^45 + 2 * q^47 - q^49 + q^51 + q^53 + 7 * q^55 - 2 * q^57 - 17 * q^59 + 13 * q^61 + 9 * q^63 - q^65 + 5 * q^67 + q^69 - 22 * q^71 + 12 * q^73 + q^75 - 14 * q^77 - 4 * q^79 + 23 * q^81 - 15 * q^83 + 10 * q^85 - 12 * q^87 - 8 * q^89 - 3 * q^91 + 16 * q^93 - 13 * q^95 - 3 * q^97 + 21 * 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
$$3$$ 3.07912 1.77773 0.888865 0.458169i $$-0.151495\pi$$
0.888865 + 0.458169i $$0.151495\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −2.07912 −0.785833 −0.392917 0.919574i $$-0.628534\pi$$
−0.392917 + 0.919574i $$0.628534\pi$$
$$8$$ 0 0
$$9$$ 6.48097 2.16032
$$10$$ 0 0
$$11$$ 5.07912 1.53141 0.765706 0.643191i $$-0.222390\pi$$
0.765706 + 0.643191i $$0.222390\pi$$
$$12$$ 0 0
$$13$$ −3.48097 −0.965448 −0.482724 0.875772i $$-0.660353\pi$$
−0.482724 + 0.875772i $$0.660353\pi$$
$$14$$ 0 0
$$15$$ 3.07912 0.795025
$$16$$ 0 0
$$17$$ 6.48097 1.57187 0.785933 0.618311i $$-0.212183\pi$$
0.785933 + 0.618311i $$0.212183\pi$$
$$18$$ 0 0
$$19$$ −7.48097 −1.71625 −0.858126 0.513438i $$-0.828371\pi$$
−0.858126 + 0.513438i $$0.828371\pi$$
$$20$$ 0 0
$$21$$ −6.40185 −1.39700
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 10.7183 2.06274
$$28$$ 0 0
$$29$$ −1.56009 −0.289702 −0.144851 0.989453i $$-0.546270\pi$$
−0.144851 + 0.989453i $$0.546270\pi$$
$$30$$ 0 0
$$31$$ 0.0791189 0.0142102 0.00710508 0.999975i $$-0.497738\pi$$
0.00710508 + 0.999975i $$0.497738\pi$$
$$32$$ 0 0
$$33$$ 15.6392 2.72244
$$34$$ 0 0
$$35$$ −2.07912 −0.351435
$$36$$ 0 0
$$37$$ −9.71833 −1.59768 −0.798842 0.601541i $$-0.794554\pi$$
−0.798842 + 0.601541i $$0.794554\pi$$
$$38$$ 0 0
$$39$$ −10.7183 −1.71631
$$40$$ 0 0
$$41$$ −0.480973 −0.0751154 −0.0375577 0.999294i $$-0.511958\pi$$
−0.0375577 + 0.999294i $$0.511958\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 6.48097 0.966126
$$46$$ 0 0
$$47$$ 6.96195 1.01550 0.507752 0.861503i $$-0.330477\pi$$
0.507752 + 0.861503i $$0.330477\pi$$
$$48$$ 0 0
$$49$$ −2.67726 −0.382466
$$50$$ 0 0
$$51$$ 19.9557 2.79435
$$52$$ 0 0
$$53$$ 11.7183 1.60964 0.804818 0.593521i $$-0.202263\pi$$
0.804818 + 0.593521i $$0.202263\pi$$
$$54$$ 0 0
$$55$$ 5.07912 0.684868
$$56$$ 0 0
$$57$$ −23.0348 −3.05103
$$58$$ 0 0
$$59$$ −11.5601 −1.50500 −0.752498 0.658595i $$-0.771151\pi$$
−0.752498 + 0.658595i $$0.771151\pi$$
$$60$$ 0 0
$$61$$ 7.88283 1.00929 0.504646 0.863326i $$-0.331623\pi$$
0.504646 + 0.863326i $$0.331623\pi$$
$$62$$ 0 0
$$63$$ −13.4747 −1.69765
$$64$$ 0 0
$$65$$ −3.48097 −0.431762
$$66$$ 0 0
$$67$$ −9.71833 −1.18728 −0.593641 0.804730i $$-0.702310\pi$$
−0.593641 + 0.804730i $$0.702310\pi$$
$$68$$ 0 0
$$69$$ 3.07912 0.370682
$$70$$ 0 0
$$71$$ −9.67726 −1.14848 −0.574240 0.818687i $$-0.694702\pi$$
−0.574240 + 0.818687i $$0.694702\pi$$
$$72$$ 0 0
$$73$$ −13.2784 −1.55412 −0.777061 0.629425i $$-0.783290\pi$$
−0.777061 + 0.629425i $$0.783290\pi$$
$$74$$ 0 0
$$75$$ 3.07912 0.355546
$$76$$ 0 0
$$77$$ −10.5601 −1.20343
$$78$$ 0 0
$$79$$ −12.3165 −1.38571 −0.692856 0.721076i $$-0.743648\pi$$
−0.692856 + 0.721076i $$0.743648\pi$$
$$80$$ 0 0
$$81$$ 13.5601 1.50668
$$82$$ 0 0
$$83$$ −4.59815 −0.504712 −0.252356 0.967634i $$-0.581205\pi$$
−0.252356 + 0.967634i $$0.581205\pi$$
$$84$$ 0 0
$$85$$ 6.48097 0.702960
$$86$$ 0 0
$$87$$ −4.80371 −0.515012
$$88$$ 0 0
$$89$$ 8.31648 0.881545 0.440772 0.897619i $$-0.354705\pi$$
0.440772 + 0.897619i $$0.354705\pi$$
$$90$$ 0 0
$$91$$ 7.23736 0.758681
$$92$$ 0 0
$$93$$ 0.243616 0.0252618
$$94$$ 0 0
$$95$$ −7.48097 −0.767532
$$96$$ 0 0
$$97$$ 7.23736 0.734842 0.367421 0.930055i $$-0.380241\pi$$
0.367421 + 0.930055i $$0.380241\pi$$
$$98$$ 0 0
$$99$$ 32.9176 3.30835
$$100$$ 0 0
$$101$$ 2.43991 0.242780 0.121390 0.992605i $$-0.461265\pi$$
0.121390 + 0.992605i $$0.461265\pi$$
$$102$$ 0 0
$$103$$ 7.88283 0.776718 0.388359 0.921508i $$-0.373042\pi$$
0.388359 + 0.921508i $$0.373042\pi$$
$$104$$ 0 0
$$105$$ −6.40185 −0.624757
$$106$$ 0 0
$$107$$ −16.6803 −1.61254 −0.806272 0.591546i $$-0.798518\pi$$
−0.806272 + 0.591546i $$0.798518\pi$$
$$108$$ 0 0
$$109$$ 5.32274 0.509826 0.254913 0.966964i $$-0.417953\pi$$
0.254913 + 0.966964i $$0.417953\pi$$
$$110$$ 0 0
$$111$$ −29.9239 −2.84025
$$112$$ 0 0
$$113$$ −2.20556 −0.207482 −0.103741 0.994604i $$-0.533081\pi$$
−0.103741 + 0.994604i $$0.533081\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ −22.5601 −2.08568
$$118$$ 0 0
$$119$$ −13.4747 −1.23522
$$120$$ 0 0
$$121$$ 14.7974 1.34522
$$122$$ 0 0
$$123$$ −1.48097 −0.133535
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 6.15824 0.546455 0.273228 0.961949i $$-0.411909\pi$$
0.273228 + 0.961949i $$0.411909\pi$$
$$128$$ 0 0
$$129$$ 24.6330 2.16881
$$130$$ 0 0
$$131$$ −7.19629 −0.628743 −0.314371 0.949300i $$-0.601794\pi$$
−0.314371 + 0.949300i $$0.601794\pi$$
$$132$$ 0 0
$$133$$ 15.5538 1.34869
$$134$$ 0 0
$$135$$ 10.7183 0.922487
$$136$$ 0 0
$$137$$ 12.6012 1.07659 0.538295 0.842757i $$-0.319069\pi$$
0.538295 + 0.842757i $$0.319069\pi$$
$$138$$ 0 0
$$139$$ −8.75638 −0.742707 −0.371353 0.928492i $$-0.621106\pi$$
−0.371353 + 0.928492i $$0.621106\pi$$
$$140$$ 0 0
$$141$$ 21.4367 1.80529
$$142$$ 0 0
$$143$$ −17.6803 −1.47850
$$144$$ 0 0
$$145$$ −1.56009 −0.129559
$$146$$ 0 0
$$147$$ −8.24362 −0.679922
$$148$$ 0 0
$$149$$ 1.79745 0.147253 0.0736264 0.997286i $$-0.476543\pi$$
0.0736264 + 0.997286i $$0.476543\pi$$
$$150$$ 0 0
$$151$$ −19.3956 −1.57839 −0.789196 0.614142i $$-0.789502\pi$$
−0.789196 + 0.614142i $$0.789502\pi$$
$$152$$ 0 0
$$153$$ 42.0030 3.39574
$$154$$ 0 0
$$155$$ 0.0791189 0.00635498
$$156$$ 0 0
$$157$$ 4.36380 0.348269 0.174135 0.984722i $$-0.444287\pi$$
0.174135 + 0.984722i $$0.444287\pi$$
$$158$$ 0 0
$$159$$ 36.0821 2.86150
$$160$$ 0 0
$$161$$ −2.07912 −0.163858
$$162$$ 0 0
$$163$$ 5.32274 0.416909 0.208454 0.978032i $$-0.433157\pi$$
0.208454 + 0.978032i $$0.433157\pi$$
$$164$$ 0 0
$$165$$ 15.6392 1.21751
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ −0.882827 −0.0679098
$$170$$ 0 0
$$171$$ −48.4840 −3.70766
$$172$$ 0 0
$$173$$ −9.23736 −0.702303 −0.351152 0.936319i $$-0.614210\pi$$
−0.351152 + 0.936319i $$0.614210\pi$$
$$174$$ 0 0
$$175$$ −2.07912 −0.157167
$$176$$ 0 0
$$177$$ −35.5949 −2.67548
$$178$$ 0 0
$$179$$ 2.96195 0.221386 0.110693 0.993855i $$-0.464693\pi$$
0.110693 + 0.993855i $$0.464693\pi$$
$$180$$ 0 0
$$181$$ 10.8355 0.805397 0.402698 0.915333i $$-0.368072\pi$$
0.402698 + 0.915333i $$0.368072\pi$$
$$182$$ 0 0
$$183$$ 24.2722 1.79425
$$184$$ 0 0
$$185$$ −9.71833 −0.714506
$$186$$ 0 0
$$187$$ 32.9176 2.40718
$$188$$ 0 0
$$189$$ −22.2847 −1.62097
$$190$$ 0 0
$$191$$ 7.76565 0.561903 0.280952 0.959722i $$-0.409350\pi$$
0.280952 + 0.959722i $$0.409350\pi$$
$$192$$ 0 0
$$193$$ −2.15824 −0.155353 −0.0776767 0.996979i $$-0.524750\pi$$
−0.0776767 + 0.996979i $$0.524750\pi$$
$$194$$ 0 0
$$195$$ −10.7183 −0.767556
$$196$$ 0 0
$$197$$ 12.0411 0.857890 0.428945 0.903331i $$-0.358885\pi$$
0.428945 + 0.903331i $$0.358885\pi$$
$$198$$ 0 0
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 0 0
$$201$$ −29.9239 −2.11067
$$202$$ 0 0
$$203$$ 3.24362 0.227657
$$204$$ 0 0
$$205$$ −0.480973 −0.0335926
$$206$$ 0 0
$$207$$ 6.48097 0.450459
$$208$$ 0 0
$$209$$ −37.9968 −2.62829
$$210$$ 0 0
$$211$$ 3.40185 0.234193 0.117097 0.993121i $$-0.462641\pi$$
0.117097 + 0.993121i $$0.462641\pi$$
$$212$$ 0 0
$$213$$ −29.7974 −2.04169
$$214$$ 0 0
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ −0.164498 −0.0111668
$$218$$ 0 0
$$219$$ −40.8858 −2.76281
$$220$$ 0 0
$$221$$ −22.5601 −1.51756
$$222$$ 0 0
$$223$$ 5.19629 0.347969 0.173985 0.984748i $$-0.444336\pi$$
0.173985 + 0.984748i $$0.444336\pi$$
$$224$$ 0 0
$$225$$ 6.48097 0.432065
$$226$$ 0 0
$$227$$ 5.35453 0.355393 0.177696 0.984085i $$-0.443136\pi$$
0.177696 + 0.984085i $$0.443136\pi$$
$$228$$ 0 0
$$229$$ −0.803708 −0.0531105 −0.0265553 0.999647i $$-0.508454\pi$$
−0.0265553 + 0.999647i $$0.508454\pi$$
$$230$$ 0 0
$$231$$ −32.5158 −2.13938
$$232$$ 0 0
$$233$$ 14.1582 0.927537 0.463768 0.885956i $$-0.346497\pi$$
0.463768 + 0.885956i $$0.346497\pi$$
$$234$$ 0 0
$$235$$ 6.96195 0.454147
$$236$$ 0 0
$$237$$ −37.9239 −2.46342
$$238$$ 0 0
$$239$$ 10.9146 0.706008 0.353004 0.935622i $$-0.385160\pi$$
0.353004 + 0.935622i $$0.385160\pi$$
$$240$$ 0 0
$$241$$ −27.4367 −1.76735 −0.883675 0.468100i $$-0.844939\pi$$
−0.883675 + 0.468100i $$0.844939\pi$$
$$242$$ 0 0
$$243$$ 9.59815 0.615721
$$244$$ 0 0
$$245$$ −2.67726 −0.171044
$$246$$ 0 0
$$247$$ 26.0411 1.65695
$$248$$ 0 0
$$249$$ −14.1582 −0.897242
$$250$$ 0 0
$$251$$ −16.5190 −1.04267 −0.521336 0.853352i $$-0.674566\pi$$
−0.521336 + 0.853352i $$0.674566\pi$$
$$252$$ 0 0
$$253$$ 5.07912 0.319321
$$254$$ 0 0
$$255$$ 19.9557 1.24967
$$256$$ 0 0
$$257$$ 19.7657 1.23295 0.616474 0.787375i $$-0.288561\pi$$
0.616474 + 0.787375i $$0.288561\pi$$
$$258$$ 0 0
$$259$$ 20.2056 1.25551
$$260$$ 0 0
$$261$$ −10.1109 −0.625850
$$262$$ 0 0
$$263$$ 7.44292 0.458950 0.229475 0.973315i $$-0.426299\pi$$
0.229475 + 0.973315i $$0.426299\pi$$
$$264$$ 0 0
$$265$$ 11.7183 0.719851
$$266$$ 0 0
$$267$$ 25.6074 1.56715
$$268$$ 0 0
$$269$$ −22.7564 −1.38748 −0.693741 0.720225i $$-0.744039\pi$$
−0.693741 + 0.720225i $$0.744039\pi$$
$$270$$ 0 0
$$271$$ −19.0411 −1.15666 −0.578331 0.815802i $$-0.696296\pi$$
−0.578331 + 0.815802i $$0.696296\pi$$
$$272$$ 0 0
$$273$$ 22.2847 1.34873
$$274$$ 0 0
$$275$$ 5.07912 0.306282
$$276$$ 0 0
$$277$$ −18.3165 −1.10053 −0.550265 0.834990i $$-0.685473\pi$$
−0.550265 + 0.834990i $$0.685473\pi$$
$$278$$ 0 0
$$279$$ 0.512767 0.0306986
$$280$$ 0 0
$$281$$ 26.0821 1.55593 0.777965 0.628308i $$-0.216252\pi$$
0.777965 + 0.628308i $$0.216252\pi$$
$$282$$ 0 0
$$283$$ −25.6422 −1.52427 −0.762136 0.647417i $$-0.775849\pi$$
−0.762136 + 0.647417i $$0.775849\pi$$
$$284$$ 0 0
$$285$$ −23.0348 −1.36446
$$286$$ 0 0
$$287$$ 1.00000 0.0590281
$$288$$ 0 0
$$289$$ 25.0030 1.47077
$$290$$ 0 0
$$291$$ 22.2847 1.30635
$$292$$ 0 0
$$293$$ 30.8385 1.80161 0.900803 0.434229i $$-0.142979\pi$$
0.900803 + 0.434229i $$0.142979\pi$$
$$294$$ 0 0
$$295$$ −11.5601 −0.673055
$$296$$ 0 0
$$297$$ 54.4397 3.15891
$$298$$ 0 0
$$299$$ −3.48097 −0.201310
$$300$$ 0 0
$$301$$ −16.6330 −0.958707
$$302$$ 0 0
$$303$$ 7.51277 0.431597
$$304$$ 0 0
$$305$$ 7.88283 0.451369
$$306$$ 0 0
$$307$$ −1.88283 −0.107459 −0.0537293 0.998556i $$-0.517111\pi$$
−0.0537293 + 0.998556i $$0.517111\pi$$
$$308$$ 0 0
$$309$$ 24.2722 1.38080
$$310$$ 0 0
$$311$$ 1.60742 0.0911482 0.0455741 0.998961i $$-0.485488\pi$$
0.0455741 + 0.998961i $$0.485488\pi$$
$$312$$ 0 0
$$313$$ −1.68654 −0.0953286 −0.0476643 0.998863i $$-0.515178\pi$$
−0.0476643 + 0.998863i $$0.515178\pi$$
$$314$$ 0 0
$$315$$ −13.4747 −0.759214
$$316$$ 0 0
$$317$$ −12.6773 −0.712026 −0.356013 0.934481i $$-0.615864\pi$$
−0.356013 + 0.934481i $$0.615864\pi$$
$$318$$ 0 0
$$319$$ −7.92389 −0.443653
$$320$$ 0 0
$$321$$ −51.3606 −2.86667
$$322$$ 0 0
$$323$$ −48.4840 −2.69772
$$324$$ 0 0
$$325$$ −3.48097 −0.193090
$$326$$ 0 0
$$327$$ 16.3893 0.906332
$$328$$ 0 0
$$329$$ −14.4747 −0.798017
$$330$$ 0 0
$$331$$ 33.3893 1.83524 0.917622 0.397454i $$-0.130106\pi$$
0.917622 + 0.397454i $$0.130106\pi$$
$$332$$ 0 0
$$333$$ −62.9842 −3.45151
$$334$$ 0 0
$$335$$ −9.71833 −0.530969
$$336$$ 0 0
$$337$$ −17.6392 −0.960869 −0.480435 0.877031i $$-0.659521\pi$$
−0.480435 + 0.877031i $$0.659521\pi$$
$$338$$ 0 0
$$339$$ −6.79119 −0.368847
$$340$$ 0 0
$$341$$ 0.401854 0.0217616
$$342$$ 0 0
$$343$$ 20.1202 1.08639
$$344$$ 0 0
$$345$$ 3.07912 0.165774
$$346$$ 0 0
$$347$$ 19.7121 1.05820 0.529100 0.848560i $$-0.322530\pi$$
0.529100 + 0.848560i $$0.322530\pi$$
$$348$$ 0 0
$$349$$ −16.0348 −0.858323 −0.429162 0.903228i $$-0.641191\pi$$
−0.429162 + 0.903228i $$0.641191\pi$$
$$350$$ 0 0
$$351$$ −37.3102 −1.99147
$$352$$ 0 0
$$353$$ −8.00000 −0.425797 −0.212899 0.977074i $$-0.568290\pi$$
−0.212899 + 0.977074i $$0.568290\pi$$
$$354$$ 0 0
$$355$$ −9.67726 −0.513616
$$356$$ 0 0
$$357$$ −41.4902 −2.19590
$$358$$ 0 0
$$359$$ −13.1202 −0.692457 −0.346228 0.938150i $$-0.612538\pi$$
−0.346228 + 0.938150i $$0.612538\pi$$
$$360$$ 0 0
$$361$$ 36.9650 1.94552
$$362$$ 0 0
$$363$$ 45.5631 2.39144
$$364$$ 0 0
$$365$$ −13.2784 −0.695024
$$366$$ 0 0
$$367$$ 3.47796 0.181548 0.0907741 0.995872i $$-0.471066\pi$$
0.0907741 + 0.995872i $$0.471066\pi$$
$$368$$ 0 0
$$369$$ −3.11717 −0.162274
$$370$$ 0 0
$$371$$ −24.3638 −1.26491
$$372$$ 0 0
$$373$$ 21.5949 1.11814 0.559071 0.829120i $$-0.311158\pi$$
0.559071 + 0.829120i $$0.311158\pi$$
$$374$$ 0 0
$$375$$ 3.07912 0.159005
$$376$$ 0 0
$$377$$ 5.43064 0.279692
$$378$$ 0 0
$$379$$ 7.80070 0.400695 0.200347 0.979725i $$-0.435793\pi$$
0.200347 + 0.979725i $$0.435793\pi$$
$$380$$ 0 0
$$381$$ 18.9619 0.971450
$$382$$ 0 0
$$383$$ 32.4274 1.65696 0.828481 0.560017i $$-0.189205\pi$$
0.828481 + 0.560017i $$0.189205\pi$$
$$384$$ 0 0
$$385$$ −10.5601 −0.538192
$$386$$ 0 0
$$387$$ 51.8478 2.63557
$$388$$ 0 0
$$389$$ 23.5538 1.19423 0.597113 0.802157i $$-0.296314\pi$$
0.597113 + 0.802157i $$0.296314\pi$$
$$390$$ 0 0
$$391$$ 6.48097 0.327757
$$392$$ 0 0
$$393$$ −22.1582 −1.11774
$$394$$ 0 0
$$395$$ −12.3165 −0.619709
$$396$$ 0 0
$$397$$ −0.370060 −0.0185728 −0.00928639 0.999957i $$-0.502956\pi$$
−0.00928639 + 0.999957i $$0.502956\pi$$
$$398$$ 0 0
$$399$$ 47.8921 2.39760
$$400$$ 0 0
$$401$$ 11.3545 0.567018 0.283509 0.958970i $$-0.408501\pi$$
0.283509 + 0.958970i $$0.408501\pi$$
$$402$$ 0 0
$$403$$ −0.275411 −0.0137192
$$404$$ 0 0
$$405$$ 13.5601 0.673806
$$406$$ 0 0
$$407$$ −49.3606 −2.44671
$$408$$ 0 0
$$409$$ 4.88283 0.241440 0.120720 0.992687i $$-0.461480\pi$$
0.120720 + 0.992687i $$0.461480\pi$$
$$410$$ 0 0
$$411$$ 38.8005 1.91389
$$412$$ 0 0
$$413$$ 24.0348 1.18268
$$414$$ 0 0
$$415$$ −4.59815 −0.225714
$$416$$ 0 0
$$417$$ −26.9619 −1.32033
$$418$$ 0 0
$$419$$ 1.51277 0.0739035 0.0369518 0.999317i $$-0.488235\pi$$
0.0369518 + 0.999317i $$0.488235\pi$$
$$420$$ 0 0
$$421$$ 22.2722 1.08548 0.542739 0.839901i $$-0.317387\pi$$
0.542739 + 0.839901i $$0.317387\pi$$
$$422$$ 0 0
$$423$$ 45.1202 2.19382
$$424$$ 0 0
$$425$$ 6.48097 0.314373
$$426$$ 0 0
$$427$$ −16.3893 −0.793135
$$428$$ 0 0
$$429$$ −54.4397 −2.62837
$$430$$ 0 0
$$431$$ −16.8858 −0.813362 −0.406681 0.913570i $$-0.633314\pi$$
−0.406681 + 0.913570i $$0.633314\pi$$
$$432$$ 0 0
$$433$$ 24.3956 1.17238 0.586189 0.810175i $$-0.300628\pi$$
0.586189 + 0.810175i $$0.300628\pi$$
$$434$$ 0 0
$$435$$ −4.80371 −0.230320
$$436$$ 0 0
$$437$$ −7.48097 −0.357863
$$438$$ 0 0
$$439$$ 40.0378 1.91090 0.955450 0.295152i $$-0.0953703\pi$$
0.955450 + 0.295152i $$0.0953703\pi$$
$$440$$ 0 0
$$441$$ −17.3513 −0.826251
$$442$$ 0 0
$$443$$ −5.22809 −0.248394 −0.124197 0.992258i $$-0.539635\pi$$
−0.124197 + 0.992258i $$0.539635\pi$$
$$444$$ 0 0
$$445$$ 8.31648 0.394239
$$446$$ 0 0
$$447$$ 5.53456 0.261776
$$448$$ 0 0
$$449$$ 4.72459 0.222967 0.111484 0.993766i $$-0.464440\pi$$
0.111484 + 0.993766i $$0.464440\pi$$
$$450$$ 0 0
$$451$$ −2.44292 −0.115033
$$452$$ 0 0
$$453$$ −59.7213 −2.80595
$$454$$ 0 0
$$455$$ 7.23736 0.339293
$$456$$ 0 0
$$457$$ 33.2311 1.55449 0.777243 0.629201i $$-0.216618\pi$$
0.777243 + 0.629201i $$0.216618\pi$$
$$458$$ 0 0
$$459$$ 69.4652 3.24236
$$460$$ 0 0
$$461$$ −28.9619 −1.34889 −0.674446 0.738324i $$-0.735618\pi$$
−0.674446 + 0.738324i $$0.735618\pi$$
$$462$$ 0 0
$$463$$ 4.39258 0.204141 0.102070 0.994777i $$-0.467453\pi$$
0.102070 + 0.994777i $$0.467453\pi$$
$$464$$ 0 0
$$465$$ 0.243616 0.0112974
$$466$$ 0 0
$$467$$ 17.0729 0.790038 0.395019 0.918673i $$-0.370738\pi$$
0.395019 + 0.918673i $$0.370738\pi$$
$$468$$ 0 0
$$469$$ 20.2056 0.933006
$$470$$ 0 0
$$471$$ 13.4367 0.619129
$$472$$ 0 0
$$473$$ 40.6330 1.86831
$$474$$ 0 0
$$475$$ −7.48097 −0.343251
$$476$$ 0 0
$$477$$ 75.9462 3.47734
$$478$$ 0 0
$$479$$ 27.0441 1.23568 0.617838 0.786306i $$-0.288009\pi$$
0.617838 + 0.786306i $$0.288009\pi$$
$$480$$ 0 0
$$481$$ 33.8292 1.54248
$$482$$ 0 0
$$483$$ −6.40185 −0.291294
$$484$$ 0 0
$$485$$ 7.23736 0.328631
$$486$$ 0 0
$$487$$ −28.0821 −1.27252 −0.636261 0.771474i $$-0.719520\pi$$
−0.636261 + 0.771474i $$0.719520\pi$$
$$488$$ 0 0
$$489$$ 16.3893 0.741151
$$490$$ 0 0
$$491$$ −7.48398 −0.337747 −0.168874 0.985638i $$-0.554013\pi$$
−0.168874 + 0.985638i $$0.554013\pi$$
$$492$$ 0 0
$$493$$ −10.1109 −0.455373
$$494$$ 0 0
$$495$$ 32.9176 1.47954
$$496$$ 0 0
$$497$$ 20.1202 0.902514
$$498$$ 0 0
$$499$$ 21.7183 0.972246 0.486123 0.873890i $$-0.338411\pi$$
0.486123 + 0.873890i $$0.338411\pi$$
$$500$$ 0 0
$$501$$ −24.6330 −1.10052
$$502$$ 0 0
$$503$$ −7.27541 −0.324395 −0.162197 0.986758i $$-0.551858\pi$$
−0.162197 + 0.986758i $$0.551858\pi$$
$$504$$ 0 0
$$505$$ 2.43991 0.108574
$$506$$ 0 0
$$507$$ −2.71833 −0.120725
$$508$$ 0 0
$$509$$ 22.7151 1.00683 0.503414 0.864045i $$-0.332077\pi$$
0.503414 + 0.864045i $$0.332077\pi$$
$$510$$ 0 0
$$511$$ 27.6074 1.22128
$$512$$ 0 0
$$513$$ −80.1835 −3.54019
$$514$$ 0 0
$$515$$ 7.88283 0.347359
$$516$$ 0 0
$$517$$ 35.3606 1.55516
$$518$$ 0 0
$$519$$ −28.4429 −1.24851
$$520$$ 0 0
$$521$$ −16.5568 −0.725368 −0.362684 0.931912i $$-0.618140\pi$$
−0.362684 + 0.931912i $$0.618140\pi$$
$$522$$ 0 0
$$523$$ 24.8037 1.08459 0.542295 0.840188i $$-0.317555\pi$$
0.542295 + 0.840188i $$0.317555\pi$$
$$524$$ 0 0
$$525$$ −6.40185 −0.279400
$$526$$ 0 0
$$527$$ 0.512767 0.0223365
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −74.9206 −3.25128
$$532$$ 0 0
$$533$$ 1.67425 0.0725200
$$534$$ 0 0
$$535$$ −16.6803 −0.721151
$$536$$ 0 0
$$537$$ 9.12018 0.393565
$$538$$ 0 0
$$539$$ −13.5981 −0.585714
$$540$$ 0 0
$$541$$ −21.8292 −0.938512 −0.469256 0.883062i $$-0.655478\pi$$
−0.469256 + 0.883062i $$0.655478\pi$$
$$542$$ 0 0
$$543$$ 33.3638 1.43178
$$544$$ 0 0
$$545$$ 5.32274 0.228001
$$546$$ 0 0
$$547$$ −29.0759 −1.24319 −0.621597 0.783337i $$-0.713516\pi$$
−0.621597 + 0.783337i $$0.713516\pi$$
$$548$$ 0 0
$$549$$ 51.0884 2.18040
$$550$$ 0 0
$$551$$ 11.6710 0.497202
$$552$$ 0 0
$$553$$ 25.6074 1.08894
$$554$$ 0 0
$$555$$ −29.9239 −1.27020
$$556$$ 0 0
$$557$$ −22.0348 −0.933645 −0.466822 0.884351i $$-0.654601\pi$$
−0.466822 + 0.884351i $$0.654601\pi$$
$$558$$ 0 0
$$559$$ −27.8478 −1.17784
$$560$$ 0 0
$$561$$ 101.357 4.27931
$$562$$ 0 0
$$563$$ 1.40185 0.0590811 0.0295406 0.999564i $$-0.490596\pi$$
0.0295406 + 0.999564i $$0.490596\pi$$
$$564$$ 0 0
$$565$$ −2.20556 −0.0927887
$$566$$ 0 0
$$567$$ −28.1930 −1.18400
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −18.3575 −0.768239 −0.384120 0.923283i $$-0.625495\pi$$
−0.384120 + 0.923283i $$0.625495\pi$$
$$572$$ 0 0
$$573$$ 23.9114 0.998912
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ −11.5128 −0.479283 −0.239641 0.970861i $$-0.577030\pi$$
−0.239641 + 0.970861i $$0.577030\pi$$
$$578$$ 0 0
$$579$$ −6.64547 −0.276176
$$580$$ 0 0
$$581$$ 9.56009 0.396619
$$582$$ 0 0
$$583$$ 59.5188 2.46502
$$584$$ 0 0
$$585$$ −22.5601 −0.932745
$$586$$ 0 0
$$587$$ 40.1232 1.65606 0.828031 0.560683i $$-0.189461\pi$$
0.828031 + 0.560683i $$0.189461\pi$$
$$588$$ 0 0
$$589$$ −0.591886 −0.0243882
$$590$$ 0 0
$$591$$ 37.0759 1.52510
$$592$$ 0 0
$$593$$ 18.7912 0.771662 0.385831 0.922570i $$-0.373915\pi$$
0.385831 + 0.922570i $$0.373915\pi$$
$$594$$ 0 0
$$595$$ −13.4747 −0.552409
$$596$$ 0 0
$$597$$ −43.1077 −1.76428
$$598$$ 0 0
$$599$$ −27.4777 −1.12271 −0.561355 0.827575i $$-0.689720\pi$$
−0.561355 + 0.827575i $$0.689720\pi$$
$$600$$ 0 0
$$601$$ −19.5283 −0.796576 −0.398288 0.917260i $$-0.630396\pi$$
−0.398288 + 0.917260i $$0.630396\pi$$
$$602$$ 0 0
$$603$$ −62.9842 −2.56492
$$604$$ 0 0
$$605$$ 14.7974 0.601602
$$606$$ 0 0
$$607$$ 45.9239 1.86399 0.931997 0.362467i $$-0.118065\pi$$
0.931997 + 0.362467i $$0.118065\pi$$
$$608$$ 0 0
$$609$$ 9.98748 0.404713
$$610$$ 0 0
$$611$$ −24.2343 −0.980417
$$612$$ 0 0
$$613$$ 12.7091 0.513314 0.256657 0.966503i $$-0.417379\pi$$
0.256657 + 0.966503i $$0.417379\pi$$
$$614$$ 0 0
$$615$$ −1.48097 −0.0597186
$$616$$ 0 0
$$617$$ 48.3102 1.94490 0.972448 0.233120i $$-0.0748934\pi$$
0.972448 + 0.233120i $$0.0748934\pi$$
$$618$$ 0 0
$$619$$ 0.677265 0.0272216 0.0136108 0.999907i $$-0.495667\pi$$
0.0136108 + 0.999907i $$0.495667\pi$$
$$620$$ 0 0
$$621$$ 10.7183 0.430112
$$622$$ 0 0
$$623$$ −17.2909 −0.692747
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −116.997 −4.67239
$$628$$ 0 0
$$629$$ −62.9842 −2.51135
$$630$$ 0 0
$$631$$ 44.3986 1.76748 0.883740 0.467978i $$-0.155017\pi$$
0.883740 + 0.467978i $$0.155017\pi$$
$$632$$ 0 0
$$633$$ 10.4747 0.416332
$$634$$ 0 0
$$635$$ 6.15824 0.244382
$$636$$ 0 0
$$637$$ 9.31949 0.369251
$$638$$ 0 0
$$639$$ −62.7181 −2.48109
$$640$$ 0 0
$$641$$ 16.8798 0.666713 0.333356 0.942801i $$-0.391819\pi$$
0.333356 + 0.942801i $$0.391819\pi$$
$$642$$ 0 0
$$643$$ −26.7564 −1.05517 −0.527584 0.849503i $$-0.676902\pi$$
−0.527584 + 0.849503i $$0.676902\pi$$
$$644$$ 0 0
$$645$$ 24.6330 0.969921
$$646$$ 0 0
$$647$$ 32.5568 1.27994 0.639971 0.768399i $$-0.278946\pi$$
0.639971 + 0.768399i $$0.278946\pi$$
$$648$$ 0 0
$$649$$ −58.7151 −2.30477
$$650$$ 0 0
$$651$$ −0.506507 −0.0198516
$$652$$ 0 0
$$653$$ −9.87356 −0.386382 −0.193191 0.981161i $$-0.561884\pi$$
−0.193191 + 0.981161i $$0.561884\pi$$
$$654$$ 0 0
$$655$$ −7.19629 −0.281182
$$656$$ 0 0
$$657$$ −86.0571 −3.35741
$$658$$ 0 0
$$659$$ −13.5128 −0.526383 −0.263191 0.964744i $$-0.584775\pi$$
−0.263191 + 0.964744i $$0.584775\pi$$
$$660$$ 0 0
$$661$$ −26.7687 −1.04118 −0.520590 0.853807i $$-0.674288\pi$$
−0.520590 + 0.853807i $$0.674288\pi$$
$$662$$ 0 0
$$663$$ −69.4652 −2.69780
$$664$$ 0 0
$$665$$ 15.5538 0.603152
$$666$$ 0 0
$$667$$ −1.56009 −0.0604070
$$668$$ 0 0
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ 40.0378 1.54564
$$672$$ 0 0
$$673$$ −0.803708 −0.0309807 −0.0154903 0.999880i $$-0.504931\pi$$
−0.0154903 + 0.999880i $$0.504931\pi$$
$$674$$ 0 0
$$675$$ 10.7183 0.412549
$$676$$ 0 0
$$677$$ −33.6422 −1.29298 −0.646488 0.762924i $$-0.723763\pi$$
−0.646488 + 0.762924i $$0.723763\pi$$
$$678$$ 0 0
$$679$$ −15.0473 −0.577463
$$680$$ 0 0
$$681$$ 16.4872 0.631792
$$682$$ 0 0
$$683$$ −11.8736 −0.454329 −0.227165 0.973856i $$-0.572946\pi$$
−0.227165 + 0.973856i $$0.572946\pi$$
$$684$$ 0 0
$$685$$ 12.6012 0.481465
$$686$$ 0 0
$$687$$ −2.47471 −0.0944162
$$688$$ 0 0
$$689$$ −40.7912 −1.55402
$$690$$ 0 0
$$691$$ 27.9114 1.06180 0.530899 0.847435i $$-0.321854\pi$$
0.530899 + 0.847435i $$0.321854\pi$$
$$692$$ 0 0
$$693$$ −68.4397 −2.59981
$$694$$ 0 0
$$695$$ −8.75638 −0.332149
$$696$$ 0 0
$$697$$ −3.11717 −0.118071
$$698$$ 0 0
$$699$$ 43.5949 1.64891
$$700$$ 0 0
$$701$$ −26.3668 −0.995861 −0.497930 0.867217i $$-0.665906\pi$$
−0.497930 + 0.867217i $$0.665906\pi$$
$$702$$ 0 0
$$703$$ 72.7026 2.74203
$$704$$ 0 0
$$705$$ 21.4367 0.807351
$$706$$ 0 0
$$707$$ −5.07286 −0.190785
$$708$$ 0 0
$$709$$ 11.8007 0.443184 0.221592 0.975139i $$-0.428875\pi$$
0.221592 + 0.975139i $$0.428875\pi$$
$$710$$ 0 0
$$711$$ −79.8227 −2.99359
$$712$$ 0 0
$$713$$ 0.0791189 0.00296302
$$714$$ 0 0
$$715$$ −17.6803 −0.661205
$$716$$ 0 0
$$717$$ 33.6074 1.25509
$$718$$ 0 0
$$719$$ −12.9589 −0.483287 −0.241643 0.970365i $$-0.577686\pi$$
−0.241643 + 0.970365i $$0.577686\pi$$
$$720$$ 0 0
$$721$$ −16.3893 −0.610371
$$722$$ 0 0
$$723$$ −84.4807 −3.14187
$$724$$ 0 0
$$725$$ −1.56009 −0.0579404
$$726$$ 0 0
$$727$$ −19.0318 −0.705850 −0.352925 0.935652i $$-0.614813\pi$$
−0.352925 + 0.935652i $$0.614813\pi$$
$$728$$ 0 0
$$729$$ −11.1264 −0.412090
$$730$$ 0 0
$$731$$ 51.8478 1.91766
$$732$$ 0 0
$$733$$ −30.5220 −1.12736 −0.563679 0.825994i $$-0.690614\pi$$
−0.563679 + 0.825994i $$0.690614\pi$$
$$734$$ 0 0
$$735$$ −8.24362 −0.304070
$$736$$ 0 0
$$737$$ −49.3606 −1.81822
$$738$$ 0 0
$$739$$ −6.18702 −0.227593 −0.113797 0.993504i $$-0.536301\pi$$
−0.113797 + 0.993504i $$0.536301\pi$$
$$740$$ 0 0
$$741$$ 80.1835 2.94562
$$742$$ 0 0
$$743$$ 16.2722 0.596968 0.298484 0.954415i $$-0.403519\pi$$
0.298484 + 0.954415i $$0.403519\pi$$
$$744$$ 0 0
$$745$$ 1.79745 0.0658534
$$746$$ 0 0
$$747$$ −29.8005 −1.09034
$$748$$ 0 0
$$749$$ 34.6803 1.26719
$$750$$ 0 0
$$751$$ −17.5949 −0.642047 −0.321023 0.947071i $$-0.604027\pi$$
−0.321023 + 0.947071i $$0.604027\pi$$
$$752$$ 0 0
$$753$$ −50.8640 −1.85359
$$754$$ 0 0
$$755$$ −19.3956 −0.705878
$$756$$ 0 0
$$757$$ 41.2496 1.49924 0.749622 0.661866i $$-0.230235\pi$$
0.749622 + 0.661866i $$0.230235\pi$$
$$758$$ 0 0
$$759$$ 15.6392 0.567667
$$760$$ 0 0
$$761$$ 26.8067 0.971743 0.485871 0.874030i $$-0.338502\pi$$
0.485871 + 0.874030i $$0.338502\pi$$
$$762$$ 0 0
$$763$$ −11.0666 −0.400638
$$764$$ 0 0
$$765$$ 42.0030 1.51862
$$766$$ 0 0
$$767$$ 40.2404 1.45300
$$768$$ 0 0
$$769$$ −26.8798 −0.969311 −0.484655 0.874705i $$-0.661055\pi$$
−0.484655 + 0.874705i $$0.661055\pi$$
$$770$$ 0 0
$$771$$ 60.8608 2.19185
$$772$$ 0 0
$$773$$ 41.0441 1.47625 0.738126 0.674662i $$-0.235711\pi$$
0.738126 + 0.674662i $$0.235711\pi$$
$$774$$ 0 0
$$775$$ 0.0791189 0.00284203
$$776$$ 0 0
$$777$$ 62.2153 2.23196
$$778$$ 0 0
$$779$$ 3.59815 0.128917
$$780$$ 0 0
$$781$$ −49.1520 −1.75880
$$782$$ 0 0
$$783$$ −16.7216 −0.597580
$$784$$ 0 0
$$785$$ 4.36380 0.155751
$$786$$ 0 0
$$787$$ −21.4840 −0.765821 −0.382911 0.923785i $$-0.625078\pi$$
−0.382911 + 0.923785i $$0.625078\pi$$
$$788$$ 0 0
$$789$$ 22.9176 0.815889
$$790$$ 0 0
$$791$$ 4.58563 0.163046
$$792$$ 0 0
$$793$$ −27.4399 −0.974420
$$794$$ 0 0
$$795$$ 36.0821 1.27970
$$796$$ 0 0
$$797$$ 38.2692 1.35556 0.677781 0.735263i $$-0.262942\pi$$
0.677781 + 0.735263i $$0.262942\pi$$
$$798$$ 0 0
$$799$$ 45.1202 1.59624
$$800$$ 0 0
$$801$$ 53.8989 1.90442
$$802$$ 0 0
$$803$$ −67.4427 −2.38000
$$804$$ 0 0
$$805$$ −2.07912 −0.0732793
$$806$$ 0 0
$$807$$ −70.0696 −2.46657
$$808$$ 0 0
$$809$$ −27.2026 −0.956391 −0.478195 0.878253i $$-0.658709\pi$$
−0.478195 + 0.878253i $$0.658709\pi$$
$$810$$ 0 0
$$811$$ 38.5095 1.35225 0.676126 0.736786i $$-0.263657\pi$$
0.676126 + 0.736786i $$0.263657\pi$$
$$812$$ 0 0
$$813$$ −58.6297 −2.05623
$$814$$ 0 0
$$815$$ 5.32274 0.186447
$$816$$ 0 0
$$817$$ −59.8478 −2.09381
$$818$$ 0 0
$$819$$ 46.9051 1.63900
$$820$$ 0 0
$$821$$ 10.0000 0.349002 0.174501 0.984657i $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 0 0
$$823$$ 40.1457 1.39939 0.699696 0.714441i $$-0.253319\pi$$
0.699696 + 0.714441i $$0.253319\pi$$
$$824$$ 0 0
$$825$$ 15.6392 0.544487
$$826$$ 0 0
$$827$$ 0.851033 0.0295933 0.0147967 0.999891i $$-0.495290\pi$$
0.0147967 + 0.999891i $$0.495290\pi$$
$$828$$ 0 0
$$829$$ −2.83851 −0.0985856 −0.0492928 0.998784i $$-0.515697\pi$$
−0.0492928 + 0.998784i $$0.515697\pi$$
$$830$$ 0 0
$$831$$ −56.3986 −1.95645
$$832$$ 0 0
$$833$$ −17.3513 −0.601186
$$834$$ 0 0
$$835$$ −8.00000 −0.276851
$$836$$ 0 0
$$837$$ 0.848022 0.0293119
$$838$$ 0 0
$$839$$ −10.2343 −0.353329 −0.176664 0.984271i $$-0.556531\pi$$
−0.176664 + 0.984271i $$0.556531\pi$$
$$840$$ 0 0
$$841$$ −26.5661 −0.916073
$$842$$ 0 0
$$843$$ 80.3100 2.76602
$$844$$ 0 0
$$845$$ −0.882827 −0.0303702
$$846$$ 0 0
$$847$$ −30.7657 −1.05712
$$848$$ 0 0
$$849$$ −78.9554 −2.70974
$$850$$ 0 0
$$851$$ −9.71833 −0.333140
$$852$$ 0 0
$$853$$ 17.7213 0.606767 0.303384 0.952869i $$-0.401884\pi$$
0.303384 + 0.952869i $$0.401884\pi$$
$$854$$ 0 0
$$855$$ −48.4840 −1.65812
$$856$$ 0 0
$$857$$ 17.6835 0.604058 0.302029 0.953299i $$-0.402336\pi$$
0.302029 + 0.953299i $$0.402336\pi$$
$$858$$ 0 0
$$859$$ −26.6042 −0.907722 −0.453861 0.891072i $$-0.649954\pi$$
−0.453861 + 0.891072i $$0.649954\pi$$
$$860$$ 0 0
$$861$$ 3.07912 0.104936
$$862$$ 0 0
$$863$$ −53.5949 −1.82439 −0.912196 0.409755i $$-0.865614\pi$$
−0.912196 + 0.409755i $$0.865614\pi$$
$$864$$ 0 0
$$865$$ −9.23736 −0.314080
$$866$$ 0 0
$$867$$ 76.9872 2.61462
$$868$$ 0 0
$$869$$ −62.5568 −2.12210
$$870$$ 0 0
$$871$$ 33.8292 1.14626
$$872$$ 0 0
$$873$$ 46.9051 1.58750
$$874$$ 0 0
$$875$$ −2.07912 −0.0702870
$$876$$ 0 0
$$877$$ 25.9650 0.876774 0.438387 0.898786i $$-0.355550\pi$$
0.438387 + 0.898786i $$0.355550\pi$$
$$878$$ 0 0
$$879$$ 94.9554 3.20277
$$880$$ 0 0
$$881$$ 29.9239 1.00816 0.504081 0.863657i $$-0.331831\pi$$
0.504081 + 0.863657i $$0.331831\pi$$
$$882$$ 0 0
$$883$$ −4.75939 −0.160166 −0.0800832 0.996788i $$-0.525519\pi$$
−0.0800832 + 0.996788i $$0.525519\pi$$
$$884$$ 0 0
$$885$$ −35.5949 −1.19651
$$886$$ 0 0
$$887$$ 38.8037 1.30290 0.651451 0.758691i $$-0.274161\pi$$
0.651451 + 0.758691i $$0.274161\pi$$
$$888$$ 0 0
$$889$$ −12.8037 −0.429423
$$890$$ 0 0
$$891$$ 68.8733 2.30734
$$892$$ 0 0
$$893$$ −52.0821 −1.74286
$$894$$ 0 0
$$895$$ 2.96195 0.0990069
$$896$$ 0 0
$$897$$ −10.7183 −0.357875
$$898$$ 0 0
$$899$$ −0.123433 −0.00411671
$$900$$ 0 0
$$901$$ 75.9462 2.53013
$$902$$ 0 0
$$903$$ −51.2148 −1.70432
$$904$$ 0 0
$$905$$ 10.8355 0.360184
$$906$$ 0 0
$$907$$ −51.0729 −1.69585 −0.847923 0.530119i $$-0.822147\pi$$
−0.847923 + 0.530119i $$0.822147\pi$$
$$908$$ 0 0
$$909$$ 15.8130 0.524483
$$910$$ 0 0
$$911$$ −30.4933 −1.01029 −0.505143 0.863035i $$-0.668560\pi$$
−0.505143 + 0.863035i $$0.668560\pi$$
$$912$$ 0 0
$$913$$ −23.3545 −0.772922
$$914$$ 0 0
$$915$$ 24.2722 0.802413
$$916$$ 0 0
$$917$$ 14.9619 0.494087
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ −5.79745 −0.191032
$$922$$ 0 0
$$923$$ 33.6863 1.10880
$$924$$ 0 0
$$925$$ −9.71833 −0.319537
$$926$$ 0 0
$$927$$ 51.0884 1.67796
$$928$$ 0 0
$$929$$ −17.3257 −0.568439 −0.284220 0.958759i $$-0.591734\pi$$
−0.284220 + 0.958759i $$0.591734\pi$$
$$930$$ 0 0
$$931$$ 20.0285 0.656409
$$932$$ 0 0
$$933$$ 4.94943 0.162037
$$934$$ 0 0
$$935$$ 32.9176 1.07652
$$936$$ 0 0
$$937$$ −47.5631 −1.55382 −0.776909 0.629612i $$-0.783214\pi$$
−0.776909 + 0.629612i $$0.783214\pi$$
$$938$$ 0 0
$$939$$ −5.19304 −0.169469
$$940$$ 0 0
$$941$$ 14.2754 0.465365 0.232683 0.972553i $$-0.425250\pi$$
0.232683 + 0.972553i $$0.425250\pi$$
$$942$$ 0 0
$$943$$ −0.480973 −0.0156626
$$944$$ 0 0
$$945$$ −22.2847 −0.724921
$$946$$ 0 0
$$947$$ 36.1900 1.17602 0.588009 0.808854i $$-0.299912\pi$$
0.588009 + 0.808854i $$0.299912\pi$$
$$948$$ 0 0
$$949$$ 46.2218 1.50042
$$950$$ 0 0
$$951$$ −39.0348 −1.26579
$$952$$ 0 0
$$953$$ 37.7942 1.22427 0.612137 0.790752i $$-0.290310\pi$$
0.612137 + 0.790752i $$0.290310\pi$$
$$954$$ 0 0
$$955$$ 7.76565 0.251291
$$956$$ 0 0
$$957$$ −24.3986 −0.788695
$$958$$ 0 0
$$959$$ −26.1993 −0.846020
$$960$$ 0 0
$$961$$ −30.9937 −0.999798
$$962$$ 0 0
$$963$$ −108.104 −3.48362
$$964$$ 0 0
$$965$$ −2.15824 −0.0694761
$$966$$ 0 0
$$967$$ 50.6390 1.62844 0.814220 0.580557i $$-0.197165\pi$$
0.814220 + 0.580557i $$0.197165\pi$$
$$968$$ 0 0
$$969$$ −149.288 −4.79582
$$970$$ 0 0
$$971$$ 2.82623 0.0906981 0.0453490 0.998971i $$-0.485560\pi$$
0.0453490 + 0.998971i $$0.485560\pi$$
$$972$$ 0 0
$$973$$ 18.2056 0.583644
$$974$$ 0 0
$$975$$ −10.7183 −0.343261
$$976$$ 0 0
$$977$$ 25.8448 0.826848 0.413424 0.910539i $$-0.364333\pi$$
0.413424 + 0.910539i $$0.364333\pi$$
$$978$$ 0 0
$$979$$ 42.2404 1.35001
$$980$$ 0 0
$$981$$ 34.4965 1.10139
$$982$$ 0 0
$$983$$ −41.3668 −1.31940 −0.659698 0.751531i $$-0.729316\pi$$
−0.659698 + 0.751531i $$0.729316\pi$$
$$984$$ 0 0
$$985$$ 12.0411 0.383660
$$986$$ 0 0
$$987$$ −44.5694 −1.41866
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 0 0
$$991$$ 54.4745 1.73044 0.865219 0.501394i $$-0.167179\pi$$
0.865219 + 0.501394i $$0.167179\pi$$
$$992$$ 0 0
$$993$$ 102.810 3.26257
$$994$$ 0 0
$$995$$ −14.0000 −0.443830
$$996$$ 0 0
$$997$$ −15.5128 −0.491294 −0.245647 0.969359i $$-0.579000\pi$$
−0.245647 + 0.969359i $$0.579000\pi$$
$$998$$ 0 0
$$999$$ −104.164 −3.29561
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 920.2.a.h.1.3 3
3.2 odd 2 8280.2.a.bj.1.1 3
4.3 odd 2 1840.2.a.s.1.1 3
5.2 odd 4 4600.2.e.p.4049.1 6
5.3 odd 4 4600.2.e.p.4049.6 6
5.4 even 2 4600.2.a.x.1.1 3
8.3 odd 2 7360.2.a.cc.1.3 3
8.5 even 2 7360.2.a.by.1.1 3
20.19 odd 2 9200.2.a.ce.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.h.1.3 3 1.1 even 1 trivial
1840.2.a.s.1.1 3 4.3 odd 2
4600.2.a.x.1.1 3 5.4 even 2
4600.2.e.p.4049.1 6 5.2 odd 4
4600.2.e.p.4049.6 6 5.3 odd 4
7360.2.a.by.1.1 3 8.5 even 2
7360.2.a.cc.1.3 3 8.3 odd 2
8280.2.a.bj.1.1 3 3.2 odd 2
9200.2.a.ce.1.3 3 20.19 odd 2