# Properties

 Label 192.4.d.c.97.3 Level $192$ Weight $4$ Character 192.97 Analytic conductor $11.328$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,4,Mod(97,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("192.97");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 192.d (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$11.3283667211$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 97.3 Root $$-0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 192.97 Dual form 192.4.d.c.97.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000i q^{3} -3.46410i q^{5} +24.2487 q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q+3.00000i q^{3} -3.46410i q^{5} +24.2487 q^{7} -9.00000 q^{9} -48.0000i q^{11} +41.5692i q^{13} +10.3923 q^{15} +54.0000 q^{17} -4.00000i q^{19} +72.7461i q^{21} +173.205 q^{23} +113.000 q^{25} -27.0000i q^{27} +162.813i q^{29} +58.8897 q^{31} +144.000 q^{33} -84.0000i q^{35} +325.626i q^{37} -124.708 q^{39} -294.000 q^{41} -188.000i q^{43} +31.1769i q^{45} +505.759 q^{47} +245.000 q^{49} +162.000i q^{51} -744.782i q^{53} -166.277 q^{55} +12.0000 q^{57} -252.000i q^{59} -90.0666i q^{61} -218.238 q^{63} +144.000 q^{65} +628.000i q^{67} +519.615i q^{69} -6.92820 q^{71} -1006.00 q^{73} +339.000i q^{75} -1163.94i q^{77} -1340.61 q^{79} +81.0000 q^{81} -720.000i q^{83} -187.061i q^{85} -488.438 q^{87} -1482.00 q^{89} +1008.00i q^{91} +176.669i q^{93} -13.8564 q^{95} +1822.00 q^{97} +432.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 36 q^{9}+O(q^{10})$$ 4 * q - 36 * q^9 $$4 q - 36 q^{9} + 216 q^{17} + 452 q^{25} + 576 q^{33} - 1176 q^{41} + 980 q^{49} + 48 q^{57} + 576 q^{65} - 4024 q^{73} + 324 q^{81} - 5928 q^{89} + 7288 q^{97}+O(q^{100})$$ 4 * q - 36 * q^9 + 216 * q^17 + 452 * q^25 + 576 * q^33 - 1176 * q^41 + 980 * q^49 + 48 * q^57 + 576 * q^65 - 4024 * q^73 + 324 * q^81 - 5928 * q^89 + 7288 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 3.00000i 0.577350i
$$4$$ 0 0
$$5$$ − 3.46410i − 0.309839i −0.987927 0.154919i $$-0.950488\pi$$
0.987927 0.154919i $$-0.0495118\pi$$
$$6$$ 0 0
$$7$$ 24.2487 1.30931 0.654654 0.755929i $$-0.272814\pi$$
0.654654 + 0.755929i $$0.272814\pi$$
$$8$$ 0 0
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ − 48.0000i − 1.31569i −0.753155 0.657843i $$-0.771469\pi$$
0.753155 0.657843i $$-0.228531\pi$$
$$12$$ 0 0
$$13$$ 41.5692i 0.886864i 0.896308 + 0.443432i $$0.146239\pi$$
−0.896308 + 0.443432i $$0.853761\pi$$
$$14$$ 0 0
$$15$$ 10.3923 0.178885
$$16$$ 0 0
$$17$$ 54.0000 0.770407 0.385204 0.922832i $$-0.374131\pi$$
0.385204 + 0.922832i $$0.374131\pi$$
$$18$$ 0 0
$$19$$ − 4.00000i − 0.0482980i −0.999708 0.0241490i $$-0.992312\pi$$
0.999708 0.0241490i $$-0.00768762\pi$$
$$20$$ 0 0
$$21$$ 72.7461i 0.755929i
$$22$$ 0 0
$$23$$ 173.205 1.57025 0.785125 0.619337i $$-0.212599\pi$$
0.785125 + 0.619337i $$0.212599\pi$$
$$24$$ 0 0
$$25$$ 113.000 0.904000
$$26$$ 0 0
$$27$$ − 27.0000i − 0.192450i
$$28$$ 0 0
$$29$$ 162.813i 1.04254i 0.853393 + 0.521269i $$0.174541\pi$$
−0.853393 + 0.521269i $$0.825459\pi$$
$$30$$ 0 0
$$31$$ 58.8897 0.341191 0.170595 0.985341i $$-0.445431\pi$$
0.170595 + 0.985341i $$0.445431\pi$$
$$32$$ 0 0
$$33$$ 144.000 0.759612
$$34$$ 0 0
$$35$$ − 84.0000i − 0.405674i
$$36$$ 0 0
$$37$$ 325.626i 1.44682i 0.690416 + 0.723412i $$0.257427\pi$$
−0.690416 + 0.723412i $$0.742573\pi$$
$$38$$ 0 0
$$39$$ −124.708 −0.512031
$$40$$ 0 0
$$41$$ −294.000 −1.11988 −0.559940 0.828533i $$-0.689176\pi$$
−0.559940 + 0.828533i $$0.689176\pi$$
$$42$$ 0 0
$$43$$ − 188.000i − 0.666738i −0.942796 0.333369i $$-0.891815\pi$$
0.942796 0.333369i $$-0.108185\pi$$
$$44$$ 0 0
$$45$$ 31.1769i 0.103280i
$$46$$ 0 0
$$47$$ 505.759 1.56963 0.784814 0.619731i $$-0.212758\pi$$
0.784814 + 0.619731i $$0.212758\pi$$
$$48$$ 0 0
$$49$$ 245.000 0.714286
$$50$$ 0 0
$$51$$ 162.000i 0.444795i
$$52$$ 0 0
$$53$$ − 744.782i − 1.93026i −0.261775 0.965129i $$-0.584308\pi$$
0.261775 0.965129i $$-0.415692\pi$$
$$54$$ 0 0
$$55$$ −166.277 −0.407650
$$56$$ 0 0
$$57$$ 12.0000 0.0278849
$$58$$ 0 0
$$59$$ − 252.000i − 0.556061i −0.960572 0.278031i $$-0.910318\pi$$
0.960572 0.278031i $$-0.0896817\pi$$
$$60$$ 0 0
$$61$$ − 90.0666i − 0.189047i −0.995523 0.0945234i $$-0.969867\pi$$
0.995523 0.0945234i $$-0.0301327\pi$$
$$62$$ 0 0
$$63$$ −218.238 −0.436436
$$64$$ 0 0
$$65$$ 144.000 0.274785
$$66$$ 0 0
$$67$$ 628.000i 1.14511i 0.819866 + 0.572555i $$0.194048\pi$$
−0.819866 + 0.572555i $$0.805952\pi$$
$$68$$ 0 0
$$69$$ 519.615i 0.906584i
$$70$$ 0 0
$$71$$ −6.92820 −0.0115807 −0.00579033 0.999983i $$-0.501843\pi$$
−0.00579033 + 0.999983i $$0.501843\pi$$
$$72$$ 0 0
$$73$$ −1006.00 −1.61292 −0.806462 0.591286i $$-0.798620\pi$$
−0.806462 + 0.591286i $$0.798620\pi$$
$$74$$ 0 0
$$75$$ 339.000i 0.521925i
$$76$$ 0 0
$$77$$ − 1163.94i − 1.72264i
$$78$$ 0 0
$$79$$ −1340.61 −1.90924 −0.954621 0.297824i $$-0.903739\pi$$
−0.954621 + 0.297824i $$0.903739\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ − 720.000i − 0.952172i −0.879399 0.476086i $$-0.842055\pi$$
0.879399 0.476086i $$-0.157945\pi$$
$$84$$ 0 0
$$85$$ − 187.061i − 0.238702i
$$86$$ 0 0
$$87$$ −488.438 −0.601909
$$88$$ 0 0
$$89$$ −1482.00 −1.76508 −0.882538 0.470242i $$-0.844167\pi$$
−0.882538 + 0.470242i $$0.844167\pi$$
$$90$$ 0 0
$$91$$ 1008.00i 1.16118i
$$92$$ 0 0
$$93$$ 176.669i 0.196986i
$$94$$ 0 0
$$95$$ −13.8564 −0.0149646
$$96$$ 0 0
$$97$$ 1822.00 1.90718 0.953588 0.301114i $$-0.0973586\pi$$
0.953588 + 0.301114i $$0.0973586\pi$$
$$98$$ 0 0
$$99$$ 432.000i 0.438562i
$$100$$ 0 0
$$101$$ − 911.059i − 0.897562i −0.893642 0.448781i $$-0.851858\pi$$
0.893642 0.448781i $$-0.148142\pi$$
$$102$$ 0 0
$$103$$ −453.797 −0.434116 −0.217058 0.976159i $$-0.569646\pi$$
−0.217058 + 0.976159i $$0.569646\pi$$
$$104$$ 0 0
$$105$$ 252.000 0.234216
$$106$$ 0 0
$$107$$ 1188.00i 1.07335i 0.843790 + 0.536674i $$0.180320\pi$$
−0.843790 + 0.536674i $$0.819680\pi$$
$$108$$ 0 0
$$109$$ 471.118i 0.413990i 0.978342 + 0.206995i $$0.0663684\pi$$
−0.978342 + 0.206995i $$0.933632\pi$$
$$110$$ 0 0
$$111$$ −976.877 −0.835325
$$112$$ 0 0
$$113$$ −390.000 −0.324674 −0.162337 0.986735i $$-0.551903\pi$$
−0.162337 + 0.986735i $$0.551903\pi$$
$$114$$ 0 0
$$115$$ − 600.000i − 0.486524i
$$116$$ 0 0
$$117$$ − 374.123i − 0.295621i
$$118$$ 0 0
$$119$$ 1309.43 1.00870
$$120$$ 0 0
$$121$$ −973.000 −0.731029
$$122$$ 0 0
$$123$$ − 882.000i − 0.646563i
$$124$$ 0 0
$$125$$ − 824.456i − 0.589933i
$$126$$ 0 0
$$127$$ 606.218 0.423568 0.211784 0.977317i $$-0.432073\pi$$
0.211784 + 0.977317i $$0.432073\pi$$
$$128$$ 0 0
$$129$$ 564.000 0.384941
$$130$$ 0 0
$$131$$ − 1380.00i − 0.920391i −0.887818 0.460195i $$-0.847779\pi$$
0.887818 0.460195i $$-0.152221\pi$$
$$132$$ 0 0
$$133$$ − 96.9948i − 0.0632370i
$$134$$ 0 0
$$135$$ −93.5307 −0.0596285
$$136$$ 0 0
$$137$$ −1158.00 −0.722150 −0.361075 0.932537i $$-0.617590\pi$$
−0.361075 + 0.932537i $$0.617590\pi$$
$$138$$ 0 0
$$139$$ − 1180.00i − 0.720045i −0.932944 0.360023i $$-0.882769\pi$$
0.932944 0.360023i $$-0.117231\pi$$
$$140$$ 0 0
$$141$$ 1517.28i 0.906225i
$$142$$ 0 0
$$143$$ 1995.32 1.16683
$$144$$ 0 0
$$145$$ 564.000 0.323018
$$146$$ 0 0
$$147$$ 735.000i 0.412393i
$$148$$ 0 0
$$149$$ 2171.99i 1.19420i 0.802165 + 0.597102i $$0.203681\pi$$
−0.802165 + 0.597102i $$0.796319\pi$$
$$150$$ 0 0
$$151$$ −142.028 −0.0765436 −0.0382718 0.999267i $$-0.512185\pi$$
−0.0382718 + 0.999267i $$0.512185\pi$$
$$152$$ 0 0
$$153$$ −486.000 −0.256802
$$154$$ 0 0
$$155$$ − 204.000i − 0.105714i
$$156$$ 0 0
$$157$$ 1337.14i 0.679717i 0.940476 + 0.339859i $$0.110379\pi$$
−0.940476 + 0.339859i $$0.889621\pi$$
$$158$$ 0 0
$$159$$ 2234.35 1.11443
$$160$$ 0 0
$$161$$ 4200.00 2.05594
$$162$$ 0 0
$$163$$ 1748.00i 0.839963i 0.907533 + 0.419981i $$0.137963\pi$$
−0.907533 + 0.419981i $$0.862037\pi$$
$$164$$ 0 0
$$165$$ − 498.831i − 0.235357i
$$166$$ 0 0
$$167$$ 13.8564 0.00642060 0.00321030 0.999995i $$-0.498978\pi$$
0.00321030 + 0.999995i $$0.498978\pi$$
$$168$$ 0 0
$$169$$ 469.000 0.213473
$$170$$ 0 0
$$171$$ 36.0000i 0.0160993i
$$172$$ 0 0
$$173$$ 599.290i 0.263371i 0.991292 + 0.131685i $$0.0420389\pi$$
−0.991292 + 0.131685i $$0.957961\pi$$
$$174$$ 0 0
$$175$$ 2740.10 1.18361
$$176$$ 0 0
$$177$$ 756.000 0.321042
$$178$$ 0 0
$$179$$ − 3228.00i − 1.34789i −0.738782 0.673944i $$-0.764599\pi$$
0.738782 0.673944i $$-0.235401\pi$$
$$180$$ 0 0
$$181$$ 2023.04i 0.830779i 0.909644 + 0.415390i $$0.136355\pi$$
−0.909644 + 0.415390i $$0.863645\pi$$
$$182$$ 0 0
$$183$$ 270.200 0.109146
$$184$$ 0 0
$$185$$ 1128.00 0.448282
$$186$$ 0 0
$$187$$ − 2592.00i − 1.01361i
$$188$$ 0 0
$$189$$ − 654.715i − 0.251976i
$$190$$ 0 0
$$191$$ −3477.96 −1.31757 −0.658786 0.752330i $$-0.728930\pi$$
−0.658786 + 0.752330i $$0.728930\pi$$
$$192$$ 0 0
$$193$$ −766.000 −0.285689 −0.142844 0.989745i $$-0.545625\pi$$
−0.142844 + 0.989745i $$0.545625\pi$$
$$194$$ 0 0
$$195$$ 432.000i 0.158647i
$$196$$ 0 0
$$197$$ 2899.45i 1.04862i 0.851529 + 0.524308i $$0.175676\pi$$
−0.851529 + 0.524308i $$0.824324\pi$$
$$198$$ 0 0
$$199$$ 1735.51 0.618228 0.309114 0.951025i $$-0.399968\pi$$
0.309114 + 0.951025i $$0.399968\pi$$
$$200$$ 0 0
$$201$$ −1884.00 −0.661130
$$202$$ 0 0
$$203$$ 3948.00i 1.36500i
$$204$$ 0 0
$$205$$ 1018.45i 0.346982i
$$206$$ 0 0
$$207$$ −1558.85 −0.523417
$$208$$ 0 0
$$209$$ −192.000 −0.0635451
$$210$$ 0 0
$$211$$ − 1100.00i − 0.358896i −0.983767 0.179448i $$-0.942569\pi$$
0.983767 0.179448i $$-0.0574312\pi$$
$$212$$ 0 0
$$213$$ − 20.7846i − 0.00668609i
$$214$$ 0 0
$$215$$ −651.251 −0.206581
$$216$$ 0 0
$$217$$ 1428.00 0.446723
$$218$$ 0 0
$$219$$ − 3018.00i − 0.931222i
$$220$$ 0 0
$$221$$ 2244.74i 0.683246i
$$222$$ 0 0
$$223$$ −391.443 −0.117547 −0.0587735 0.998271i $$-0.518719\pi$$
−0.0587735 + 0.998271i $$0.518719\pi$$
$$224$$ 0 0
$$225$$ −1017.00 −0.301333
$$226$$ 0 0
$$227$$ 3336.00i 0.975410i 0.873008 + 0.487705i $$0.162166\pi$$
−0.873008 + 0.487705i $$0.837834\pi$$
$$228$$ 0 0
$$229$$ − 5999.82i − 1.73135i −0.500605 0.865676i $$-0.666889\pi$$
0.500605 0.865676i $$-0.333111\pi$$
$$230$$ 0 0
$$231$$ 3491.81 0.994565
$$232$$ 0 0
$$233$$ 318.000 0.0894115 0.0447057 0.999000i $$-0.485765\pi$$
0.0447057 + 0.999000i $$0.485765\pi$$
$$234$$ 0 0
$$235$$ − 1752.00i − 0.486331i
$$236$$ 0 0
$$237$$ − 4021.82i − 1.10230i
$$238$$ 0 0
$$239$$ −859.097 −0.232512 −0.116256 0.993219i $$-0.537089\pi$$
−0.116256 + 0.993219i $$0.537089\pi$$
$$240$$ 0 0
$$241$$ −2710.00 −0.724342 −0.362171 0.932112i $$-0.617964\pi$$
−0.362171 + 0.932112i $$0.617964\pi$$
$$242$$ 0 0
$$243$$ 243.000i 0.0641500i
$$244$$ 0 0
$$245$$ − 848.705i − 0.221313i
$$246$$ 0 0
$$247$$ 166.277 0.0428338
$$248$$ 0 0
$$249$$ 2160.00 0.549737
$$250$$ 0 0
$$251$$ 5136.00i 1.29156i 0.763524 + 0.645780i $$0.223468\pi$$
−0.763524 + 0.645780i $$0.776532\pi$$
$$252$$ 0 0
$$253$$ − 8313.84i − 2.06596i
$$254$$ 0 0
$$255$$ 561.184 0.137815
$$256$$ 0 0
$$257$$ −4398.00 −1.06747 −0.533735 0.845652i $$-0.679212\pi$$
−0.533735 + 0.845652i $$0.679212\pi$$
$$258$$ 0 0
$$259$$ 7896.00i 1.89434i
$$260$$ 0 0
$$261$$ − 1465.31i − 0.347512i
$$262$$ 0 0
$$263$$ −6817.35 −1.59839 −0.799194 0.601073i $$-0.794740\pi$$
−0.799194 + 0.601073i $$0.794740\pi$$
$$264$$ 0 0
$$265$$ −2580.00 −0.598068
$$266$$ 0 0
$$267$$ − 4446.00i − 1.01907i
$$268$$ 0 0
$$269$$ 4624.58i 1.04820i 0.851657 + 0.524099i $$0.175598\pi$$
−0.851657 + 0.524099i $$0.824402\pi$$
$$270$$ 0 0
$$271$$ −3883.26 −0.870447 −0.435223 0.900322i $$-0.643331\pi$$
−0.435223 + 0.900322i $$0.643331\pi$$
$$272$$ 0 0
$$273$$ −3024.00 −0.670406
$$274$$ 0 0
$$275$$ − 5424.00i − 1.18938i
$$276$$ 0 0
$$277$$ 1524.20i 0.330616i 0.986242 + 0.165308i $$0.0528618\pi$$
−0.986242 + 0.165308i $$0.947138\pi$$
$$278$$ 0 0
$$279$$ −530.008 −0.113730
$$280$$ 0 0
$$281$$ 4398.00 0.933675 0.466838 0.884343i $$-0.345393\pi$$
0.466838 + 0.884343i $$0.345393\pi$$
$$282$$ 0 0
$$283$$ − 4372.00i − 0.918334i −0.888350 0.459167i $$-0.848148\pi$$
0.888350 0.459167i $$-0.151852\pi$$
$$284$$ 0 0
$$285$$ − 41.5692i − 0.00863982i
$$286$$ 0 0
$$287$$ −7129.12 −1.46627
$$288$$ 0 0
$$289$$ −1997.00 −0.406473
$$290$$ 0 0
$$291$$ 5466.00i 1.10111i
$$292$$ 0 0
$$293$$ − 3571.49i − 0.712111i −0.934465 0.356056i $$-0.884121\pi$$
0.934465 0.356056i $$-0.115879\pi$$
$$294$$ 0 0
$$295$$ −872.954 −0.172289
$$296$$ 0 0
$$297$$ −1296.00 −0.253204
$$298$$ 0 0
$$299$$ 7200.00i 1.39260i
$$300$$ 0 0
$$301$$ − 4558.76i − 0.872965i
$$302$$ 0 0
$$303$$ 2733.18 0.518207
$$304$$ 0 0
$$305$$ −312.000 −0.0585740
$$306$$ 0 0
$$307$$ − 4172.00i − 0.775598i −0.921744 0.387799i $$-0.873235\pi$$
0.921744 0.387799i $$-0.126765\pi$$
$$308$$ 0 0
$$309$$ − 1361.39i − 0.250637i
$$310$$ 0 0
$$311$$ −6470.94 −1.17985 −0.589925 0.807458i $$-0.700843\pi$$
−0.589925 + 0.807458i $$0.700843\pi$$
$$312$$ 0 0
$$313$$ −74.0000 −0.0133633 −0.00668167 0.999978i $$-0.502127\pi$$
−0.00668167 + 0.999978i $$0.502127\pi$$
$$314$$ 0 0
$$315$$ 756.000i 0.135225i
$$316$$ 0 0
$$317$$ 1964.15i 0.348004i 0.984745 + 0.174002i $$0.0556700\pi$$
−0.984745 + 0.174002i $$0.944330\pi$$
$$318$$ 0 0
$$319$$ 7815.01 1.37165
$$320$$ 0 0
$$321$$ −3564.00 −0.619698
$$322$$ 0 0
$$323$$ − 216.000i − 0.0372092i
$$324$$ 0 0
$$325$$ 4697.32i 0.801725i
$$326$$ 0 0
$$327$$ −1413.35 −0.239017
$$328$$ 0 0
$$329$$ 12264.0 2.05513
$$330$$ 0 0
$$331$$ 7556.00i 1.25473i 0.778726 + 0.627365i $$0.215866\pi$$
−0.778726 + 0.627365i $$0.784134\pi$$
$$332$$ 0 0
$$333$$ − 2930.63i − 0.482275i
$$334$$ 0 0
$$335$$ 2175.46 0.354800
$$336$$ 0 0
$$337$$ −4106.00 −0.663703 −0.331852 0.943332i $$-0.607673\pi$$
−0.331852 + 0.943332i $$0.607673\pi$$
$$338$$ 0 0
$$339$$ − 1170.00i − 0.187450i
$$340$$ 0 0
$$341$$ − 2826.71i − 0.448900i
$$342$$ 0 0
$$343$$ −2376.37 −0.374088
$$344$$ 0 0
$$345$$ 1800.00 0.280895
$$346$$ 0 0
$$347$$ 5256.00i 0.813132i 0.913621 + 0.406566i $$0.133274\pi$$
−0.913621 + 0.406566i $$0.866726\pi$$
$$348$$ 0 0
$$349$$ 10385.4i 1.59288i 0.604715 + 0.796442i $$0.293287\pi$$
−0.604715 + 0.796442i $$0.706713\pi$$
$$350$$ 0 0
$$351$$ 1122.37 0.170677
$$352$$ 0 0
$$353$$ −3942.00 −0.594367 −0.297183 0.954820i $$-0.596047\pi$$
−0.297183 + 0.954820i $$0.596047\pi$$
$$354$$ 0 0
$$355$$ 24.0000i 0.00358813i
$$356$$ 0 0
$$357$$ 3928.29i 0.582373i
$$358$$ 0 0
$$359$$ 6644.15 0.976782 0.488391 0.872625i $$-0.337584\pi$$
0.488391 + 0.872625i $$0.337584\pi$$
$$360$$ 0 0
$$361$$ 6843.00 0.997667
$$362$$ 0 0
$$363$$ − 2919.00i − 0.422060i
$$364$$ 0 0
$$365$$ 3484.89i 0.499746i
$$366$$ 0 0
$$367$$ −2906.38 −0.413384 −0.206692 0.978406i $$-0.566270\pi$$
−0.206692 + 0.978406i $$0.566270\pi$$
$$368$$ 0 0
$$369$$ 2646.00 0.373293
$$370$$ 0 0
$$371$$ − 18060.0i − 2.52730i
$$372$$ 0 0
$$373$$ − 10246.8i − 1.42241i −0.702983 0.711206i $$-0.748149\pi$$
0.702983 0.711206i $$-0.251851\pi$$
$$374$$ 0 0
$$375$$ 2473.37 0.340598
$$376$$ 0 0
$$377$$ −6768.00 −0.924588
$$378$$ 0 0
$$379$$ − 13844.0i − 1.87630i −0.346226 0.938151i $$-0.612537\pi$$
0.346226 0.938151i $$-0.387463\pi$$
$$380$$ 0 0
$$381$$ 1818.65i 0.244547i
$$382$$ 0 0
$$383$$ 7163.76 0.955747 0.477874 0.878429i $$-0.341408\pi$$
0.477874 + 0.878429i $$0.341408\pi$$
$$384$$ 0 0
$$385$$ −4032.00 −0.533740
$$386$$ 0 0
$$387$$ 1692.00i 0.222246i
$$388$$ 0 0
$$389$$ 12993.8i 1.69361i 0.531904 + 0.846805i $$0.321477\pi$$
−0.531904 + 0.846805i $$0.678523\pi$$
$$390$$ 0 0
$$391$$ 9353.07 1.20973
$$392$$ 0 0
$$393$$ 4140.00 0.531388
$$394$$ 0 0
$$395$$ 4644.00i 0.591557i
$$396$$ 0 0
$$397$$ − 117.779i − 0.0148896i −0.999972 0.00744481i $$-0.997630\pi$$
0.999972 0.00744481i $$-0.00236978\pi$$
$$398$$ 0 0
$$399$$ 290.985 0.0365099
$$400$$ 0 0
$$401$$ −5418.00 −0.674718 −0.337359 0.941376i $$-0.609534\pi$$
−0.337359 + 0.941376i $$0.609534\pi$$
$$402$$ 0 0
$$403$$ 2448.00i 0.302589i
$$404$$ 0 0
$$405$$ − 280.592i − 0.0344265i
$$406$$ 0 0
$$407$$ 15630.0 1.90357
$$408$$ 0 0
$$409$$ 11450.0 1.38427 0.692135 0.721768i $$-0.256670\pi$$
0.692135 + 0.721768i $$0.256670\pi$$
$$410$$ 0 0
$$411$$ − 3474.00i − 0.416934i
$$412$$ 0 0
$$413$$ − 6110.68i − 0.728055i
$$414$$ 0 0
$$415$$ −2494.15 −0.295020
$$416$$ 0 0
$$417$$ 3540.00 0.415718
$$418$$ 0 0
$$419$$ 1176.00i 0.137115i 0.997647 + 0.0685577i $$0.0218397\pi$$
−0.997647 + 0.0685577i $$0.978160\pi$$
$$420$$ 0 0
$$421$$ 10032.0i 1.16136i 0.814133 + 0.580679i $$0.197213\pi$$
−0.814133 + 0.580679i $$0.802787\pi$$
$$422$$ 0 0
$$423$$ −4551.83 −0.523209
$$424$$ 0 0
$$425$$ 6102.00 0.696448
$$426$$ 0 0
$$427$$ − 2184.00i − 0.247520i
$$428$$ 0 0
$$429$$ 5985.97i 0.673672i
$$430$$ 0 0
$$431$$ −838.313 −0.0936893 −0.0468447 0.998902i $$-0.514917\pi$$
−0.0468447 + 0.998902i $$0.514917\pi$$
$$432$$ 0 0
$$433$$ 4318.00 0.479237 0.239619 0.970867i $$-0.422978\pi$$
0.239619 + 0.970867i $$0.422978\pi$$
$$434$$ 0 0
$$435$$ 1692.00i 0.186495i
$$436$$ 0 0
$$437$$ − 692.820i − 0.0758400i
$$438$$ 0 0
$$439$$ 1610.81 0.175124 0.0875622 0.996159i $$-0.472092\pi$$
0.0875622 + 0.996159i $$0.472092\pi$$
$$440$$ 0 0
$$441$$ −2205.00 −0.238095
$$442$$ 0 0
$$443$$ 1032.00i 0.110681i 0.998468 + 0.0553406i $$0.0176245\pi$$
−0.998468 + 0.0553406i $$0.982376\pi$$
$$444$$ 0 0
$$445$$ 5133.80i 0.546889i
$$446$$ 0 0
$$447$$ −6515.98 −0.689474
$$448$$ 0 0
$$449$$ 726.000 0.0763075 0.0381537 0.999272i $$-0.487852\pi$$
0.0381537 + 0.999272i $$0.487852\pi$$
$$450$$ 0 0
$$451$$ 14112.0i 1.47341i
$$452$$ 0 0
$$453$$ − 426.084i − 0.0441925i
$$454$$ 0 0
$$455$$ 3491.81 0.359778
$$456$$ 0 0
$$457$$ 8666.00 0.887042 0.443521 0.896264i $$-0.353729\pi$$
0.443521 + 0.896264i $$0.353729\pi$$
$$458$$ 0 0
$$459$$ − 1458.00i − 0.148265i
$$460$$ 0 0
$$461$$ − 14684.3i − 1.48355i −0.670648 0.741776i $$-0.733984\pi$$
0.670648 0.741776i $$-0.266016\pi$$
$$462$$ 0 0
$$463$$ 4998.70 0.501748 0.250874 0.968020i $$-0.419282\pi$$
0.250874 + 0.968020i $$0.419282\pi$$
$$464$$ 0 0
$$465$$ 612.000 0.0610340
$$466$$ 0 0
$$467$$ − 16824.0i − 1.66707i −0.552466 0.833535i $$-0.686313\pi$$
0.552466 0.833535i $$-0.313687\pi$$
$$468$$ 0 0
$$469$$ 15228.2i 1.49930i
$$470$$ 0 0
$$471$$ −4011.43 −0.392435
$$472$$ 0 0
$$473$$ −9024.00 −0.877218
$$474$$ 0 0
$$475$$ − 452.000i − 0.0436614i
$$476$$ 0 0
$$477$$ 6703.04i 0.643419i
$$478$$ 0 0
$$479$$ 10953.5 1.04484 0.522419 0.852689i $$-0.325030\pi$$
0.522419 + 0.852689i $$0.325030\pi$$
$$480$$ 0 0
$$481$$ −13536.0 −1.28314
$$482$$ 0 0
$$483$$ 12600.0i 1.18700i
$$484$$ 0 0
$$485$$ − 6311.59i − 0.590917i
$$486$$ 0 0
$$487$$ −10714.5 −0.996959 −0.498479 0.866902i $$-0.666108\pi$$
−0.498479 + 0.866902i $$0.666108\pi$$
$$488$$ 0 0
$$489$$ −5244.00 −0.484953
$$490$$ 0 0
$$491$$ − 852.000i − 0.0783100i −0.999233 0.0391550i $$-0.987533\pi$$
0.999233 0.0391550i $$-0.0124666\pi$$
$$492$$ 0 0
$$493$$ 8791.89i 0.803178i
$$494$$ 0 0
$$495$$ 1496.49 0.135883
$$496$$ 0 0
$$497$$ −168.000 −0.0151626
$$498$$ 0 0
$$499$$ − 11156.0i − 1.00082i −0.865787 0.500412i $$-0.833182\pi$$
0.865787 0.500412i $$-0.166818\pi$$
$$500$$ 0 0
$$501$$ 41.5692i 0.00370694i
$$502$$ 0 0
$$503$$ −14999.6 −1.32962 −0.664808 0.747014i $$-0.731487\pi$$
−0.664808 + 0.747014i $$0.731487\pi$$
$$504$$ 0 0
$$505$$ −3156.00 −0.278099
$$506$$ 0 0
$$507$$ 1407.00i 0.123249i
$$508$$ 0 0
$$509$$ − 9287.26i − 0.808743i −0.914595 0.404372i $$-0.867490\pi$$
0.914595 0.404372i $$-0.132510\pi$$
$$510$$ 0 0
$$511$$ −24394.2 −2.11181
$$512$$ 0 0
$$513$$ −108.000 −0.00929496
$$514$$ 0 0
$$515$$ 1572.00i 0.134506i
$$516$$ 0 0
$$517$$ − 24276.4i − 2.06514i
$$518$$ 0 0
$$519$$ −1797.87 −0.152057
$$520$$ 0 0
$$521$$ −2766.00 −0.232592 −0.116296 0.993215i $$-0.537102\pi$$
−0.116296 + 0.993215i $$0.537102\pi$$
$$522$$ 0 0
$$523$$ 18988.0i 1.58755i 0.608213 + 0.793774i $$0.291887\pi$$
−0.608213 + 0.793774i $$0.708113\pi$$
$$524$$ 0 0
$$525$$ 8220.31i 0.683360i
$$526$$ 0 0
$$527$$ 3180.05 0.262856
$$528$$ 0 0
$$529$$ 17833.0 1.46569
$$530$$ 0 0
$$531$$ 2268.00i 0.185354i
$$532$$ 0 0
$$533$$ − 12221.4i − 0.993181i
$$534$$ 0 0
$$535$$ 4115.35 0.332565
$$536$$ 0 0
$$537$$ 9684.00 0.778204
$$538$$ 0 0
$$539$$ − 11760.0i − 0.939776i
$$540$$ 0 0
$$541$$ − 12997.3i − 1.03290i −0.856318 0.516449i $$-0.827254\pi$$
0.856318 0.516449i $$-0.172746\pi$$
$$542$$ 0 0
$$543$$ −6069.11 −0.479651
$$544$$ 0 0
$$545$$ 1632.00 0.128270
$$546$$ 0 0
$$547$$ 21188.0i 1.65619i 0.560591 + 0.828093i $$0.310574\pi$$
−0.560591 + 0.828093i $$0.689426\pi$$
$$548$$ 0 0
$$549$$ 810.600i 0.0630156i
$$550$$ 0 0
$$551$$ 651.251 0.0503525
$$552$$ 0 0
$$553$$ −32508.0 −2.49978
$$554$$ 0 0
$$555$$ 3384.00i 0.258816i
$$556$$ 0 0
$$557$$ − 12231.7i − 0.930477i −0.885185 0.465238i $$-0.845969\pi$$
0.885185 0.465238i $$-0.154031\pi$$
$$558$$ 0 0
$$559$$ 7815.01 0.591306
$$560$$ 0 0
$$561$$ 7776.00 0.585210
$$562$$ 0 0
$$563$$ 504.000i 0.0377284i 0.999822 + 0.0188642i $$0.00600501\pi$$
−0.999822 + 0.0188642i $$0.993995\pi$$
$$564$$ 0 0
$$565$$ 1351.00i 0.100596i
$$566$$ 0 0
$$567$$ 1964.15 0.145479
$$568$$ 0 0
$$569$$ −20358.0 −1.49992 −0.749958 0.661486i $$-0.769926\pi$$
−0.749958 + 0.661486i $$0.769926\pi$$
$$570$$ 0 0
$$571$$ − 13300.0i − 0.974760i −0.873190 0.487380i $$-0.837953\pi$$
0.873190 0.487380i $$-0.162047\pi$$
$$572$$ 0 0
$$573$$ − 10433.9i − 0.760700i
$$574$$ 0 0
$$575$$ 19572.2 1.41951
$$576$$ 0 0
$$577$$ 4606.00 0.332323 0.166161 0.986099i $$-0.446863\pi$$
0.166161 + 0.986099i $$0.446863\pi$$
$$578$$ 0 0
$$579$$ − 2298.00i − 0.164942i
$$580$$ 0 0
$$581$$ − 17459.1i − 1.24669i
$$582$$ 0 0
$$583$$ −35749.5 −2.53961
$$584$$ 0 0
$$585$$ −1296.00 −0.0915949
$$586$$ 0 0
$$587$$ 13980.0i 0.982992i 0.870880 + 0.491496i $$0.163550\pi$$
−0.870880 + 0.491496i $$0.836450\pi$$
$$588$$ 0 0
$$589$$ − 235.559i − 0.0164788i
$$590$$ 0 0
$$591$$ −8698.36 −0.605419
$$592$$ 0 0
$$593$$ −12486.0 −0.864652 −0.432326 0.901717i $$-0.642307\pi$$
−0.432326 + 0.901717i $$0.642307\pi$$
$$594$$ 0 0
$$595$$ − 4536.00i − 0.312534i
$$596$$ 0 0
$$597$$ 5206.54i 0.356934i
$$598$$ 0 0
$$599$$ 8778.03 0.598766 0.299383 0.954133i $$-0.403219\pi$$
0.299383 + 0.954133i $$0.403219\pi$$
$$600$$ 0 0
$$601$$ 6986.00 0.474151 0.237076 0.971491i $$-0.423811\pi$$
0.237076 + 0.971491i $$0.423811\pi$$
$$602$$ 0 0
$$603$$ − 5652.00i − 0.381704i
$$604$$ 0 0
$$605$$ 3370.57i 0.226501i
$$606$$ 0 0
$$607$$ 4596.86 0.307382 0.153691 0.988119i $$-0.450884\pi$$
0.153691 + 0.988119i $$0.450884\pi$$
$$608$$ 0 0
$$609$$ −11844.0 −0.788084
$$610$$ 0 0
$$611$$ 21024.0i 1.39205i
$$612$$ 0 0
$$613$$ 11092.1i 0.730838i 0.930843 + 0.365419i $$0.119074\pi$$
−0.930843 + 0.365419i $$0.880926\pi$$
$$614$$ 0 0
$$615$$ −3055.34 −0.200330
$$616$$ 0 0
$$617$$ −2850.00 −0.185959 −0.0929795 0.995668i $$-0.529639\pi$$
−0.0929795 + 0.995668i $$0.529639\pi$$
$$618$$ 0 0
$$619$$ − 20116.0i − 1.30619i −0.757277 0.653094i $$-0.773471\pi$$
0.757277 0.653094i $$-0.226529\pi$$
$$620$$ 0 0
$$621$$ − 4676.54i − 0.302195i
$$622$$ 0 0
$$623$$ −35936.6 −2.31103
$$624$$ 0 0
$$625$$ 11269.0 0.721216
$$626$$ 0 0
$$627$$ − 576.000i − 0.0366878i
$$628$$ 0 0
$$629$$ 17583.8i 1.11464i
$$630$$ 0 0
$$631$$ 7271.15 0.458732 0.229366 0.973340i $$-0.426335\pi$$
0.229366 + 0.973340i $$0.426335\pi$$
$$632$$ 0 0
$$633$$ 3300.00 0.207209
$$634$$ 0 0
$$635$$ − 2100.00i − 0.131238i
$$636$$ 0 0
$$637$$ 10184.5i 0.633474i
$$638$$ 0 0
$$639$$ 62.3538 0.00386022
$$640$$ 0 0
$$641$$ 7230.00 0.445504 0.222752 0.974875i $$-0.428496\pi$$
0.222752 + 0.974875i $$0.428496\pi$$
$$642$$ 0 0
$$643$$ 2948.00i 0.180805i 0.995905 + 0.0904026i $$0.0288154\pi$$
−0.995905 + 0.0904026i $$0.971185\pi$$
$$644$$ 0 0
$$645$$ − 1953.75i − 0.119270i
$$646$$ 0 0
$$647$$ 17161.2 1.04277 0.521387 0.853320i $$-0.325415\pi$$
0.521387 + 0.853320i $$0.325415\pi$$
$$648$$ 0 0
$$649$$ −12096.0 −0.731602
$$650$$ 0 0
$$651$$ 4284.00i 0.257916i
$$652$$ 0 0
$$653$$ 10381.9i 0.622168i 0.950382 + 0.311084i $$0.100692\pi$$
−0.950382 + 0.311084i $$0.899308\pi$$
$$654$$ 0 0
$$655$$ −4780.46 −0.285173
$$656$$ 0 0
$$657$$ 9054.00 0.537641
$$658$$ 0 0
$$659$$ − 10308.0i − 0.609321i −0.952461 0.304661i $$-0.901457\pi$$
0.952461 0.304661i $$-0.0985430\pi$$
$$660$$ 0 0
$$661$$ − 15803.2i − 0.929916i −0.885333 0.464958i $$-0.846069\pi$$
0.885333 0.464958i $$-0.153931\pi$$
$$662$$ 0 0
$$663$$ −6734.21 −0.394472
$$664$$ 0 0
$$665$$ −336.000 −0.0195933
$$666$$ 0 0
$$667$$ 28200.0i 1.63704i
$$668$$ 0 0
$$669$$ − 1174.33i − 0.0678658i
$$670$$ 0 0
$$671$$ −4323.20 −0.248726
$$672$$ 0 0
$$673$$ 30910.0 1.77042 0.885210 0.465191i $$-0.154014\pi$$
0.885210 + 0.465191i $$0.154014\pi$$
$$674$$ 0 0
$$675$$ − 3051.00i − 0.173975i
$$676$$ 0 0
$$677$$ − 14802.1i − 0.840312i −0.907452 0.420156i $$-0.861975\pi$$
0.907452 0.420156i $$-0.138025\pi$$
$$678$$ 0 0
$$679$$ 44181.2 2.49708
$$680$$ 0 0
$$681$$ −10008.0 −0.563153
$$682$$ 0 0
$$683$$ − 528.000i − 0.0295803i −0.999891 0.0147902i $$-0.995292\pi$$
0.999891 0.0147902i $$-0.00470803\pi$$
$$684$$ 0 0
$$685$$ 4011.43i 0.223750i
$$686$$ 0 0
$$687$$ 17999.5 0.999596
$$688$$ 0 0
$$689$$ 30960.0 1.71188
$$690$$ 0 0
$$691$$ − 9052.00i − 0.498342i −0.968460 0.249171i $$-0.919842\pi$$
0.968460 0.249171i $$-0.0801581\pi$$
$$692$$ 0 0
$$693$$ 10475.4i 0.574212i
$$694$$ 0 0
$$695$$ −4087.64 −0.223098
$$696$$ 0 0
$$697$$ −15876.0 −0.862764
$$698$$ 0 0
$$699$$ 954.000i 0.0516217i
$$700$$ 0 0
$$701$$ 32600.7i 1.75650i 0.478197 + 0.878252i $$0.341290\pi$$
−0.478197 + 0.878252i $$0.658710\pi$$
$$702$$ 0 0
$$703$$ 1302.50 0.0698788
$$704$$ 0 0
$$705$$ 5256.00 0.280784
$$706$$ 0 0
$$707$$ − 22092.0i − 1.17518i
$$708$$ 0 0
$$709$$ 27227.8i 1.44226i 0.692799 + 0.721130i $$0.256377\pi$$
−0.692799 + 0.721130i $$0.743623\pi$$
$$710$$ 0 0
$$711$$ 12065.5 0.636414
$$712$$ 0 0
$$713$$ 10200.0 0.535755
$$714$$ 0 0
$$715$$ − 6912.00i − 0.361530i
$$716$$ 0 0
$$717$$ − 2577.29i − 0.134241i
$$718$$ 0 0
$$719$$ 685.892 0.0355764 0.0177882 0.999842i $$-0.494338\pi$$
0.0177882 + 0.999842i $$0.494338\pi$$
$$720$$ 0 0
$$721$$ −11004.0 −0.568392
$$722$$ 0 0
$$723$$ − 8130.00i − 0.418199i
$$724$$ 0 0
$$725$$ 18397.8i 0.942453i
$$726$$ 0 0
$$727$$ −20192.2 −1.03011 −0.515054 0.857158i $$-0.672228\pi$$
−0.515054 + 0.857158i $$0.672228\pi$$
$$728$$ 0 0
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ − 10152.0i − 0.513660i
$$732$$ 0 0
$$733$$ − 35236.8i − 1.77558i −0.460246 0.887792i $$-0.652239\pi$$
0.460246 0.887792i $$-0.347761\pi$$
$$734$$ 0 0
$$735$$ 2546.11 0.127775
$$736$$ 0 0
$$737$$ 30144.0 1.50661
$$738$$ 0 0
$$739$$ − 13940.0i − 0.693899i −0.937884 0.346949i $$-0.887218\pi$$
0.937884 0.346949i $$-0.112782\pi$$
$$740$$ 0 0
$$741$$ 498.831i 0.0247301i
$$742$$ 0 0
$$743$$ −11002.0 −0.543235 −0.271618 0.962405i $$-0.587559\pi$$
−0.271618 + 0.962405i $$0.587559\pi$$
$$744$$ 0 0
$$745$$ 7524.00 0.370011
$$746$$ 0 0
$$747$$ 6480.00i 0.317391i
$$748$$ 0 0
$$749$$ 28807.5i 1.40534i
$$750$$ 0 0
$$751$$ 33342.0 1.62006 0.810031 0.586388i $$-0.199450\pi$$
0.810031 + 0.586388i $$0.199450\pi$$
$$752$$ 0 0
$$753$$ −15408.0 −0.745682
$$754$$ 0 0
$$755$$ 492.000i 0.0237162i
$$756$$ 0 0
$$757$$ 17445.2i 0.837592i 0.908080 + 0.418796i $$0.137548\pi$$
−0.908080 + 0.418796i $$0.862452\pi$$
$$758$$ 0 0
$$759$$ 24941.5 1.19278
$$760$$ 0 0
$$761$$ −30222.0 −1.43961 −0.719807 0.694174i $$-0.755770\pi$$
−0.719807 + 0.694174i $$0.755770\pi$$
$$762$$ 0 0
$$763$$ 11424.0i 0.542040i
$$764$$ 0 0
$$765$$ 1683.55i 0.0795673i
$$766$$ 0 0
$$767$$ 10475.4 0.493150
$$768$$ 0 0
$$769$$ −11758.0 −0.551371 −0.275686 0.961248i $$-0.588905\pi$$
−0.275686 + 0.961248i $$0.588905\pi$$
$$770$$ 0 0
$$771$$ − 13194.0i − 0.616304i
$$772$$ 0 0
$$773$$ 1874.08i 0.0872004i 0.999049 + 0.0436002i $$0.0138828\pi$$
−0.999049 + 0.0436002i $$0.986117\pi$$
$$774$$ 0 0
$$775$$ 6654.54 0.308436
$$776$$ 0 0
$$777$$ −23688.0 −1.09370
$$778$$ 0 0
$$779$$ 1176.00i 0.0540880i
$$780$$ 0 0
$$781$$ 332.554i 0.0152365i
$$782$$ 0 0
$$783$$ 4395.94 0.200636
$$784$$ 0 0
$$785$$ 4632.00 0.210603
$$786$$ 0 0
$$787$$ − 31012.0i − 1.40465i −0.711857 0.702324i $$-0.752146\pi$$
0.711857 0.702324i $$-0.247854\pi$$
$$788$$ 0 0
$$789$$ − 20452.1i − 0.922829i
$$790$$ 0 0
$$791$$ −9457.00 −0.425097
$$792$$ 0 0
$$793$$ 3744.00 0.167659
$$794$$ 0 0
$$795$$ − 7740.00i − 0.345295i
$$796$$ 0 0
$$797$$ − 7091.02i − 0.315153i −0.987507 0.157576i $$-0.949632\pi$$
0.987507 0.157576i $$-0.0503680\pi$$
$$798$$ 0 0
$$799$$ 27311.0 1.20925
$$800$$ 0 0
$$801$$ 13338.0 0.588358
$$802$$ 0 0
$$803$$ 48288.0i 2.12210i
$$804$$ 0 0
$$805$$ − 14549.2i − 0.637010i
$$806$$ 0 0
$$807$$ −13873.7 −0.605178
$$808$$ 0 0
$$809$$ 40650.0 1.76660 0.883299 0.468810i $$-0.155317\pi$$
0.883299 + 0.468810i $$0.155317\pi$$
$$810$$ 0 0
$$811$$ − 8372.00i − 0.362492i −0.983438 0.181246i $$-0.941987\pi$$
0.983438 0.181246i $$-0.0580130\pi$$
$$812$$ 0 0
$$813$$ − 11649.8i − 0.502553i
$$814$$ 0 0
$$815$$ 6055.25 0.260253
$$816$$ 0 0
$$817$$ −752.000 −0.0322021
$$818$$ 0 0
$$819$$ − 9072.00i − 0.387059i
$$820$$ 0 0
$$821$$ 9370.39i 0.398330i 0.979966 + 0.199165i $$0.0638230\pi$$
−0.979966 + 0.199165i $$0.936177\pi$$
$$822$$ 0 0
$$823$$ 21668.0 0.917737 0.458868 0.888504i $$-0.348255\pi$$
0.458868 + 0.888504i $$0.348255\pi$$
$$824$$ 0 0
$$825$$ 16272.0 0.686689
$$826$$ 0 0
$$827$$ − 6684.00i − 0.281046i −0.990077 0.140523i $$-0.955122\pi$$
0.990077 0.140523i $$-0.0448785\pi$$
$$828$$ 0 0
$$829$$ 24359.6i 1.02056i 0.860009 + 0.510279i $$0.170458\pi$$
−0.860009 + 0.510279i $$0.829542\pi$$
$$830$$ 0 0
$$831$$ −4572.61 −0.190881
$$832$$ 0 0
$$833$$ 13230.0 0.550291
$$834$$ 0 0
$$835$$ − 48.0000i − 0.00198935i
$$836$$ 0 0
$$837$$ − 1590.02i − 0.0656622i
$$838$$ 0 0
$$839$$ −19613.7 −0.807082 −0.403541 0.914962i $$-0.632221\pi$$
−0.403541 + 0.914962i $$0.632221\pi$$
$$840$$ 0 0
$$841$$ −2119.00 −0.0868834
$$842$$ 0 0
$$843$$ 13194.0i 0.539058i
$$844$$ 0 0
$$845$$ − 1624.66i − 0.0661422i
$$846$$ 0 0
$$847$$ −23594.0 −0.957142
$$848$$ 0 0
$$849$$ 13116.0 0.530200
$$850$$ 0 0
$$851$$ 56400.0i 2.27188i
$$852$$ 0 0
$$853$$ − 20569.8i − 0.825671i −0.910806 0.412836i $$-0.864538\pi$$
0.910806 0.412836i $$-0.135462\pi$$
$$854$$ 0 0
$$855$$ 124.708 0.00498820
$$856$$ 0 0
$$857$$ −27222.0 −1.08505 −0.542524 0.840040i $$-0.682531\pi$$
−0.542524 + 0.840040i $$0.682531\pi$$
$$858$$ 0 0
$$859$$ − 3548.00i − 0.140927i −0.997514 0.0704634i $$-0.977552\pi$$
0.997514 0.0704634i $$-0.0224478\pi$$
$$860$$ 0 0
$$861$$ − 21387.4i − 0.846550i
$$862$$ 0 0
$$863$$ −45047.2 −1.77685 −0.888426 0.459019i $$-0.848201\pi$$
−0.888426 + 0.459019i $$0.848201\pi$$
$$864$$ 0 0
$$865$$ 2076.00 0.0816024
$$866$$ 0 0
$$867$$ − 5991.00i − 0.234677i
$$868$$ 0 0
$$869$$ 64349.2i 2.51196i
$$870$$ 0 0
$$871$$ −26105.5 −1.01556
$$872$$ 0 0
$$873$$ −16398.0 −0.635725
$$874$$ 0 0
$$875$$ − 19992.0i − 0.772403i
$$876$$ 0 0
$$877$$ − 29022.2i − 1.11746i −0.829350 0.558729i $$-0.811289\pi$$
0.829350 0.558729i $$-0.188711\pi$$
$$878$$ 0 0
$$879$$ 10714.5 0.411138
$$880$$ 0 0
$$881$$ −48318.0 −1.84776 −0.923879 0.382685i $$-0.875000\pi$$
−0.923879 + 0.382685i $$0.875000\pi$$
$$882$$ 0 0
$$883$$ − 14380.0i − 0.548047i −0.961723 0.274024i $$-0.911645\pi$$
0.961723 0.274024i $$-0.0883546\pi$$
$$884$$ 0 0
$$885$$ − 2618.86i − 0.0994712i
$$886$$ 0 0
$$887$$ −34086.8 −1.29033 −0.645164 0.764044i $$-0.723211\pi$$
−0.645164 + 0.764044i $$0.723211\pi$$
$$888$$ 0 0
$$889$$ 14700.0 0.554581
$$890$$ 0 0
$$891$$ − 3888.00i − 0.146187i
$$892$$ 0 0
$$893$$ − 2023.04i − 0.0758100i
$$894$$ 0 0
$$895$$ −11182.1 −0.417628
$$896$$ 0 0
$$897$$ −21600.0 −0.804017
$$898$$ 0 0
$$899$$ 9588.00i 0.355704i
$$900$$ 0 0
$$901$$ − 40218.2i − 1.48708i
$$902$$ 0 0
$$903$$ 13676.3 0.504007
$$904$$ 0 0
$$905$$ 7008.00 0.257408
$$906$$ 0 0
$$907$$ 31252.0i 1.14411i 0.820216 + 0.572054i $$0.193853\pi$$
−0.820216 + 0.572054i $$0.806147\pi$$
$$908$$ 0 0
$$909$$ 8199.53i 0.299187i
$$910$$ 0 0
$$911$$ 13080.4 0.475713 0.237857 0.971300i $$-0.423555\pi$$
0.237857 + 0.971300i $$0.423555\pi$$
$$912$$ 0 0
$$913$$ −34560.0 −1.25276
$$914$$ 0 0
$$915$$ − 936.000i − 0.0338177i
$$916$$ 0 0
$$917$$ − 33463.2i − 1.20507i
$$918$$ 0 0
$$919$$ −8843.85 −0.317445 −0.158722 0.987323i $$-0.550737\pi$$
−0.158722 + 0.987323i $$0.550737\pi$$
$$920$$ 0 0
$$921$$ 12516.0 0.447792
$$922$$ 0 0
$$923$$ − 288.000i − 0.0102705i
$$924$$ 0 0
$$925$$ 36795.7i 1.30793i
$$926$$ 0 0
$$927$$ 4084.18 0.144705
$$928$$ 0 0
$$929$$ 29622.0 1.04614 0.523071 0.852289i $$-0.324786\pi$$
0.523071 + 0.852289i $$0.324786\pi$$
$$930$$ 0 0
$$931$$ − 980.000i − 0.0344986i
$$932$$ 0 0
$$933$$ − 19412.8i − 0.681187i
$$934$$ 0 0
$$935$$ −8978.95 −0.314057
$$936$$ 0 0
$$937$$ −23210.0 −0.809218 −0.404609 0.914490i $$-0.632592\pi$$
−0.404609 + 0.914490i $$0.632592\pi$$
$$938$$ 0 0
$$939$$ − 222.000i − 0.00771533i
$$940$$ 0 0
$$941$$ − 19728.1i − 0.683439i −0.939802 0.341720i $$-0.888991\pi$$
0.939802 0.341720i $$-0.111009\pi$$
$$942$$ 0 0
$$943$$ −50922.3 −1.75849
$$944$$ 0 0
$$945$$ −2268.00 −0.0780720
$$946$$ 0 0
$$947$$ 1236.00i 0.0424125i 0.999775 + 0.0212062i $$0.00675066\pi$$
−0.999775 + 0.0212062i $$0.993249\pi$$
$$948$$ 0 0
$$949$$ − 41818.6i − 1.43044i
$$950$$ 0 0
$$951$$ −5892.44 −0.200920
$$952$$ 0 0
$$953$$ 18402.0 0.625498 0.312749 0.949836i $$-0.398750\pi$$
0.312749 + 0.949836i $$0.398750\pi$$
$$954$$ 0 0
$$955$$ 12048.0i 0.408235i
$$956$$ 0 0
$$957$$ 23445.0i 0.791923i
$$958$$ 0 0
$$959$$ −28080.0 −0.945517
$$960$$ 0 0
$$961$$ −26323.0 −0.883589
$$962$$ 0 0
$$963$$ − 10692.0i − 0.357783i
$$964$$ 0 0
$$965$$ 2653.50i 0.0885174i
$$966$$ 0 0
$$967$$ 41836.0 1.39127 0.695633 0.718398i $$-0.255124\pi$$
0.695633 + 0.718398i $$0.255124\pi$$
$$968$$ 0 0
$$969$$ 648.000 0.0214827
$$970$$ 0 0
$$971$$ 8832.00i 0.291897i 0.989292 + 0.145949i $$0.0466234\pi$$
−0.989292 + 0.145949i $$0.953377\pi$$
$$972$$ 0 0
$$973$$ − 28613.5i − 0.942761i
$$974$$ 0 0
$$975$$ −14092.0 −0.462876
$$976$$ 0 0
$$977$$ −23034.0 −0.754271 −0.377136 0.926158i $$-0.623091\pi$$
−0.377136 + 0.926158i $$0.623091\pi$$
$$978$$ 0 0
$$979$$ 71136.0i 2.32228i
$$980$$ 0 0
$$981$$ − 4240.06i − 0.137997i
$$982$$ 0 0
$$983$$ 17791.6 0.577278 0.288639 0.957438i $$-0.406797\pi$$
0.288639 + 0.957438i $$0.406797\pi$$
$$984$$ 0 0
$$985$$ 10044.0 0.324902
$$986$$ 0 0
$$987$$ 36792.0i 1.18653i
$$988$$ 0 0
$$989$$ − 32562.6i − 1.04695i
$$990$$ 0 0
$$991$$ −3072.66 −0.0984926 −0.0492463 0.998787i $$-0.515682\pi$$
−0.0492463 + 0.998787i $$0.515682\pi$$
$$992$$ 0 0
$$993$$ −22668.0 −0.724418
$$994$$ 0 0
$$995$$ − 6012.00i − 0.191551i
$$996$$ 0 0
$$997$$ 52273.3i 1.66049i 0.557396 + 0.830247i $$0.311800\pi$$
−0.557396 + 0.830247i $$0.688200\pi$$
$$998$$ 0 0
$$999$$ 8791.89 0.278442
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 192.4.d.c.97.3 yes 4
3.2 odd 2 576.4.d.c.289.4 4
4.3 odd 2 inner 192.4.d.c.97.1 4
8.3 odd 2 inner 192.4.d.c.97.4 yes 4
8.5 even 2 inner 192.4.d.c.97.2 yes 4
12.11 even 2 576.4.d.c.289.3 4
16.3 odd 4 768.4.a.o.1.1 2
16.5 even 4 768.4.a.o.1.2 2
16.11 odd 4 768.4.a.f.1.2 2
16.13 even 4 768.4.a.f.1.1 2
24.5 odd 2 576.4.d.c.289.2 4
24.11 even 2 576.4.d.c.289.1 4
48.5 odd 4 2304.4.a.x.1.1 2
48.11 even 4 2304.4.a.bk.1.1 2
48.29 odd 4 2304.4.a.bk.1.2 2
48.35 even 4 2304.4.a.x.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
192.4.d.c.97.1 4 4.3 odd 2 inner
192.4.d.c.97.2 yes 4 8.5 even 2 inner
192.4.d.c.97.3 yes 4 1.1 even 1 trivial
192.4.d.c.97.4 yes 4 8.3 odd 2 inner
576.4.d.c.289.1 4 24.11 even 2
576.4.d.c.289.2 4 24.5 odd 2
576.4.d.c.289.3 4 12.11 even 2
576.4.d.c.289.4 4 3.2 odd 2
768.4.a.f.1.1 2 16.13 even 4
768.4.a.f.1.2 2 16.11 odd 4
768.4.a.o.1.1 2 16.3 odd 4
768.4.a.o.1.2 2 16.5 even 4
2304.4.a.x.1.1 2 48.5 odd 4
2304.4.a.x.1.2 2 48.35 even 4
2304.4.a.bk.1.1 2 48.11 even 4
2304.4.a.bk.1.2 2 48.29 odd 4