# Properties

 Label 576.4.d.c.289.2 Level $576$ Weight $4$ Character 576.289 Analytic conductor $33.985$ 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] = [576,4,Mod(289,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, 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("576.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 576.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: $$33.9851001633$$ 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: $$2^{6}$$ Twist minimal: no (minimal twist has level 192) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 289.2 Root $$-0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 576.289 Dual form 576.4.d.c.289.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.46410i q^{5} +24.2487 q^{7} +O(q^{10})$$ $$q-3.46410i q^{5} +24.2487 q^{7} -48.0000i q^{11} -41.5692i q^{13} -54.0000 q^{17} +4.00000i q^{19} -173.205 q^{23} +113.000 q^{25} +162.813i q^{29} +58.8897 q^{31} -84.0000i q^{35} -325.626i q^{37} +294.000 q^{41} +188.000i q^{43} -505.759 q^{47} +245.000 q^{49} -744.782i q^{53} -166.277 q^{55} -252.000i q^{59} +90.0666i q^{61} -144.000 q^{65} -628.000i q^{67} +6.92820 q^{71} -1006.00 q^{73} -1163.94i q^{77} -1340.61 q^{79} -720.000i q^{83} +187.061i q^{85} +1482.00 q^{89} -1008.00i q^{91} +13.8564 q^{95} +1822.00 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 216 q^{17} + 452 q^{25} + 1176 q^{41} + 980 q^{49} - 576 q^{65} - 4024 q^{73} + 5928 q^{89} + 7288 q^{97}+O(q^{100})$$ 4 * q - 216 * q^17 + 452 * q^25 + 1176 * q^41 + 980 * q^49 - 576 * q^65 - 4024 * q^73 + 5928 * q^89 + 7288 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$325$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 0 0
$$4$$ 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
$$9$$ 0 0
$$10$$ 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 $$-0.853761\pi$$
0.896308 0.443432i $$-0.146239\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −54.0000 −0.770407 −0.385204 0.922832i $$-0.625869\pi$$
−0.385204 + 0.922832i $$0.625869\pi$$
$$18$$ 0 0
$$19$$ 4.00000i 0.0482980i 0.999708 + 0.0241490i $$0.00768762\pi$$
−0.999708 + 0.0241490i $$0.992312\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −173.205 −1.57025 −0.785125 0.619337i $$-0.787401\pi$$
−0.785125 + 0.619337i $$0.787401\pi$$
$$24$$ 0 0
$$25$$ 113.000 0.904000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 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
$$33$$ 0 0
$$34$$ 0 0
$$35$$ − 84.0000i − 0.405674i
$$36$$ 0 0
$$37$$ − 325.626i − 1.44682i −0.690416 0.723412i $$-0.742573\pi$$
0.690416 0.723412i $$-0.257427\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 294.000 1.11988 0.559940 0.828533i $$-0.310824\pi$$
0.559940 + 0.828533i $$0.310824\pi$$
$$42$$ 0 0
$$43$$ 188.000i 0.666738i 0.942796 + 0.333369i $$0.108185\pi$$
−0.942796 + 0.333369i $$0.891815\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −505.759 −1.56963 −0.784814 0.619731i $$-0.787242\pi$$
−0.784814 + 0.619731i $$0.787242\pi$$
$$48$$ 0 0
$$49$$ 245.000 0.714286
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 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
$$57$$ 0 0
$$58$$ 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 $$0.0301327\pi$$
−0.995523 + 0.0945234i $$0.969867\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −144.000 −0.274785
$$66$$ 0 0
$$67$$ − 628.000i − 1.14511i −0.819866 0.572555i $$-0.805952\pi$$
0.819866 0.572555i $$-0.194048\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.92820 0.0115807 0.00579033 0.999983i $$-0.498157\pi$$
0.00579033 + 0.999983i $$0.498157\pi$$
$$72$$ 0 0
$$73$$ −1006.00 −1.61292 −0.806462 0.591286i $$-0.798620\pi$$
−0.806462 + 0.591286i $$0.798620\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 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
$$81$$ 0 0
$$82$$ 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
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 1482.00 1.76508 0.882538 0.470242i $$-0.155833\pi$$
0.882538 + 0.470242i $$0.155833\pi$$
$$90$$ 0 0
$$91$$ − 1008.00i − 1.16118i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 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
$$99$$ 0 0
$$100$$ 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
$$105$$ 0 0
$$106$$ 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 $$-0.933632\pi$$
0.978342 0.206995i $$-0.0663684\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 390.000 0.324674 0.162337 0.986735i $$-0.448097\pi$$
0.162337 + 0.986735i $$0.448097\pi$$
$$114$$ 0 0
$$115$$ 600.000i 0.486524i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −1309.43 −1.00870
$$120$$ 0 0
$$121$$ −973.000 −0.731029
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 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
$$129$$ 0 0
$$130$$ 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
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1158.00 0.722150 0.361075 0.932537i $$-0.382410\pi$$
0.361075 + 0.932537i $$0.382410\pi$$
$$138$$ 0 0
$$139$$ 1180.00i 0.720045i 0.932944 + 0.360023i $$0.117231\pi$$
−0.932944 + 0.360023i $$0.882769\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −1995.32 −1.16683
$$144$$ 0 0
$$145$$ 564.000 0.323018
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 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
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 204.000i − 0.105714i
$$156$$ 0 0
$$157$$ − 1337.14i − 0.679717i −0.940476 0.339859i $$-0.889621\pi$$
0.940476 0.339859i $$-0.110379\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4200.00 −2.05594
$$162$$ 0 0
$$163$$ − 1748.00i − 0.839963i −0.907533 0.419981i $$-0.862037\pi$$
0.907533 0.419981i $$-0.137963\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −13.8564 −0.00642060 −0.00321030 0.999995i $$-0.501022\pi$$
−0.00321030 + 0.999995i $$0.501022\pi$$
$$168$$ 0 0
$$169$$ 469.000 0.213473
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 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
$$177$$ 0 0
$$178$$ 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 $$-0.863645\pi$$
0.909644 0.415390i $$-0.136355\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −1128.00 −0.448282
$$186$$ 0 0
$$187$$ 2592.00i 1.01361i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3477.96 1.31757 0.658786 0.752330i $$-0.271070\pi$$
0.658786 + 0.752330i $$0.271070\pi$$
$$192$$ 0 0
$$193$$ −766.000 −0.285689 −0.142844 0.989745i $$-0.545625\pi$$
−0.142844 + 0.989745i $$0.545625\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 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
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 3948.00i 1.36500i
$$204$$ 0 0
$$205$$ − 1018.45i − 0.346982i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 192.000 0.0635451
$$210$$ 0 0
$$211$$ 1100.00i 0.358896i 0.983767 + 0.179448i $$0.0574312\pi$$
−0.983767 + 0.179448i $$0.942569\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 651.251 0.206581
$$216$$ 0 0
$$217$$ 1428.00 0.446723
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 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
$$225$$ 0 0
$$226$$ 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 $$0.333111\pi$$
−0.500605 + 0.865676i $$0.666889\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −318.000 −0.0894115 −0.0447057 0.999000i $$-0.514235\pi$$
−0.0447057 + 0.999000i $$0.514235\pi$$
$$234$$ 0 0
$$235$$ 1752.00i 0.486331i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 859.097 0.232512 0.116256 0.993219i $$-0.462911\pi$$
0.116256 + 0.993219i $$0.462911\pi$$
$$240$$ 0 0
$$241$$ −2710.00 −0.724342 −0.362171 0.932112i $$-0.617964\pi$$
−0.362171 + 0.932112i $$0.617964\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 848.705i − 0.221313i
$$246$$ 0 0
$$247$$ 166.277 0.0428338
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 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
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 4398.00 1.06747 0.533735 0.845652i $$-0.320788\pi$$
0.533735 + 0.845652i $$0.320788\pi$$
$$258$$ 0 0
$$259$$ − 7896.00i − 1.89434i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 6817.35 1.59839 0.799194 0.601073i $$-0.205260\pi$$
0.799194 + 0.601073i $$0.205260\pi$$
$$264$$ 0 0
$$265$$ −2580.00 −0.598068
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 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
$$273$$ 0 0
$$274$$ 0 0
$$275$$ − 5424.00i − 1.18938i
$$276$$ 0 0
$$277$$ − 1524.20i − 0.330616i −0.986242 0.165308i $$-0.947138\pi$$
0.986242 0.165308i $$-0.0528618\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4398.00 −0.933675 −0.466838 0.884343i $$-0.654607\pi$$
−0.466838 + 0.884343i $$0.654607\pi$$
$$282$$ 0 0
$$283$$ 4372.00i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 7129.12 1.46627
$$288$$ 0 0
$$289$$ −1997.00 −0.406473
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 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
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 7200.00i 1.39260i
$$300$$ 0 0
$$301$$ 4558.76i 0.872965i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 312.000 0.0585740
$$306$$ 0 0
$$307$$ 4172.00i 0.775598i 0.921744 + 0.387799i $$0.126765\pi$$
−0.921744 + 0.387799i $$0.873235\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 6470.94 1.17985 0.589925 0.807458i $$-0.299157\pi$$
0.589925 + 0.807458i $$0.299157\pi$$
$$312$$ 0 0
$$313$$ −74.0000 −0.0133633 −0.00668167 0.999978i $$-0.502127\pi$$
−0.00668167 + 0.999978i $$0.502127\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 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
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 216.000i − 0.0372092i
$$324$$ 0 0
$$325$$ − 4697.32i − 0.801725i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −12264.0 −2.05513
$$330$$ 0 0
$$331$$ − 7556.00i − 1.25473i −0.778726 0.627365i $$-0.784134\pi$$
0.778726 0.627365i $$-0.215866\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 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
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 2826.71i − 0.448900i
$$342$$ 0 0
$$343$$ −2376.37 −0.374088
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 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 $$-0.706713\pi$$
0.604715 0.796442i $$-0.293287\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 3942.00 0.594367 0.297183 0.954820i $$-0.403953\pi$$
0.297183 + 0.954820i $$0.403953\pi$$
$$354$$ 0 0
$$355$$ − 24.0000i − 0.00358813i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −6644.15 −0.976782 −0.488391 0.872625i $$-0.662416\pi$$
−0.488391 + 0.872625i $$0.662416\pi$$
$$360$$ 0 0
$$361$$ 6843.00 0.997667
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 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
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 18060.0i − 2.52730i
$$372$$ 0 0
$$373$$ 10246.8i 1.42241i 0.702983 + 0.711206i $$0.251851\pi$$
−0.702983 + 0.711206i $$0.748149\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6768.00 0.924588
$$378$$ 0 0
$$379$$ 13844.0i 1.87630i 0.346226 + 0.938151i $$0.387463\pi$$
−0.346226 + 0.938151i $$0.612537\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −7163.76 −0.955747 −0.477874 0.878429i $$-0.658592\pi$$
−0.477874 + 0.878429i $$0.658592\pi$$
$$384$$ 0 0
$$385$$ −4032.00 −0.533740
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 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
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 4644.00i 0.591557i
$$396$$ 0 0
$$397$$ 117.779i 0.0148896i 0.999972 + 0.00744481i $$0.00236978\pi$$
−0.999972 + 0.00744481i $$0.997630\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 5418.00 0.674718 0.337359 0.941376i $$-0.390466\pi$$
0.337359 + 0.941376i $$0.390466\pi$$
$$402$$ 0 0
$$403$$ − 2448.00i − 0.302589i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 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
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 6110.68i − 0.728055i
$$414$$ 0 0
$$415$$ −2494.15 −0.295020
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 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 $$-0.802787\pi$$
0.814133 0.580679i $$-0.197213\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −6102.00 −0.696448
$$426$$ 0 0
$$427$$ 2184.00i 0.247520i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 838.313 0.0936893 0.0468447 0.998902i $$-0.485083\pi$$
0.0468447 + 0.998902i $$0.485083\pi$$
$$432$$ 0 0
$$433$$ 4318.00 0.479237 0.239619 0.970867i $$-0.422978\pi$$
0.239619 + 0.970867i $$0.422978\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 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
$$441$$ 0 0
$$442$$ 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
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −726.000 −0.0763075 −0.0381537 0.999272i $$-0.512148\pi$$
−0.0381537 + 0.999272i $$0.512148\pi$$
$$450$$ 0 0
$$451$$ − 14112.0i − 1.47341i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 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
$$459$$ 0 0
$$460$$ 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
$$465$$ 0 0
$$466$$ 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
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 9024.00 0.877218
$$474$$ 0 0
$$475$$ 452.000i 0.0436614i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −10953.5 −1.04484 −0.522419 0.852689i $$-0.674970\pi$$
−0.522419 + 0.852689i $$0.674970\pi$$
$$480$$ 0 0
$$481$$ −13536.0 −1.28314
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 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
$$489$$ 0 0
$$490$$ 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
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 168.000 0.0151626
$$498$$ 0 0
$$499$$ 11156.0i 1.00082i 0.865787 + 0.500412i $$0.166818\pi$$
−0.865787 + 0.500412i $$0.833182\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 14999.6 1.32962 0.664808 0.747014i $$-0.268513\pi$$
0.664808 + 0.747014i $$0.268513\pi$$
$$504$$ 0 0
$$505$$ −3156.00 −0.278099
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 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
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 1572.00i 0.134506i
$$516$$ 0 0
$$517$$ 24276.4i 2.06514i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 2766.00 0.232592 0.116296 0.993215i $$-0.462898\pi$$
0.116296 + 0.993215i $$0.462898\pi$$
$$522$$ 0 0
$$523$$ − 18988.0i − 1.58755i −0.608213 0.793774i $$-0.708113\pi$$
0.608213 0.793774i $$-0.291887\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −3180.05 −0.262856
$$528$$ 0 0
$$529$$ 17833.0 1.46569
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 12221.4i − 0.993181i
$$534$$ 0 0
$$535$$ 4115.35 0.332565
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 11760.0i − 0.939776i
$$540$$ 0 0
$$541$$ 12997.3i 1.03290i 0.856318 + 0.516449i $$0.172746\pi$$
−0.856318 + 0.516449i $$0.827254\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −1632.00 −0.128270
$$546$$ 0 0
$$547$$ − 21188.0i − 1.65619i −0.560591 0.828093i $$-0.689426\pi$$
0.560591 0.828093i $$-0.310574\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −651.251 −0.0503525
$$552$$ 0 0
$$553$$ −32508.0 −2.49978
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 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
$$561$$ 0 0
$$562$$ 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
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 20358.0 1.49992 0.749958 0.661486i $$-0.230074\pi$$
0.749958 + 0.661486i $$0.230074\pi$$
$$570$$ 0 0
$$571$$ 13300.0i 0.974760i 0.873190 + 0.487380i $$0.162047\pi$$
−0.873190 + 0.487380i $$0.837953\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 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
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 17459.1i − 1.24669i
$$582$$ 0 0
$$583$$ −35749.5 −2.53961
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 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
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 12486.0 0.864652 0.432326 0.901717i $$-0.357693\pi$$
0.432326 + 0.901717i $$0.357693\pi$$
$$594$$ 0 0
$$595$$ 4536.00i 0.312534i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −8778.03 −0.598766 −0.299383 0.954133i $$-0.596781\pi$$
−0.299383 + 0.954133i $$0.596781\pi$$
$$600$$ 0 0
$$601$$ 6986.00 0.474151 0.237076 0.971491i $$-0.423811\pi$$
0.237076 + 0.971491i $$0.423811\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 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
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 21024.0i 1.39205i
$$612$$ 0 0
$$613$$ − 11092.1i − 0.730838i −0.930843 0.365419i $$-0.880926\pi$$
0.930843 0.365419i $$-0.119074\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 2850.00 0.185959 0.0929795 0.995668i $$-0.470361\pi$$
0.0929795 + 0.995668i $$0.470361\pi$$
$$618$$ 0 0
$$619$$ 20116.0i 1.30619i 0.757277 + 0.653094i $$0.226529\pi$$
−0.757277 + 0.653094i $$0.773471\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 35936.6 2.31103
$$624$$ 0 0
$$625$$ 11269.0 0.721216
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 2100.00i − 0.131238i
$$636$$ 0 0
$$637$$ − 10184.5i − 0.633474i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −7230.00 −0.445504 −0.222752 0.974875i $$-0.571504\pi$$
−0.222752 + 0.974875i $$0.571504\pi$$
$$642$$ 0 0
$$643$$ − 2948.00i − 0.180805i −0.995905 0.0904026i $$-0.971185\pi$$
0.995905 0.0904026i $$-0.0288154\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −17161.2 −1.04277 −0.521387 0.853320i $$-0.674585\pi$$
−0.521387 + 0.853320i $$0.674585\pi$$
$$648$$ 0 0
$$649$$ −12096.0 −0.731602
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 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
$$657$$ 0 0
$$658$$ 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 $$0.153931\pi$$
−0.885333 + 0.464958i $$0.846069\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 336.000 0.0195933
$$666$$ 0 0
$$667$$ − 28200.0i − 1.63704i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 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
$$675$$ 0 0
$$676$$ 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
$$681$$ 0 0
$$682$$ 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
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −30960.0 −1.71188
$$690$$ 0 0
$$691$$ 9052.00i 0.498342i 0.968460 + 0.249171i $$0.0801581\pi$$
−0.968460 + 0.249171i $$0.919842\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 4087.64 0.223098
$$696$$ 0 0
$$697$$ −15876.0 −0.862764
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 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
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 22092.0i − 1.17518i
$$708$$ 0 0
$$709$$ − 27227.8i − 1.44226i −0.692799 0.721130i $$-0.743623\pi$$
0.692799 0.721130i $$-0.256377\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −10200.0 −0.535755
$$714$$ 0 0
$$715$$ 6912.00i 0.361530i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −685.892 −0.0355764 −0.0177882 0.999842i $$-0.505662\pi$$
−0.0177882 + 0.999842i $$0.505662\pi$$
$$720$$ 0 0
$$721$$ −11004.0 −0.568392
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 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
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 10152.0i − 0.513660i
$$732$$ 0 0
$$733$$ 35236.8i 1.77558i 0.460246 + 0.887792i $$0.347761\pi$$
−0.460246 + 0.887792i $$0.652239\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −30144.0 −1.50661
$$738$$ 0 0
$$739$$ 13940.0i 0.693899i 0.937884 + 0.346949i $$0.112782\pi$$
−0.937884 + 0.346949i $$0.887218\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 11002.0 0.543235 0.271618 0.962405i $$-0.412441\pi$$
0.271618 + 0.962405i $$0.412441\pi$$
$$744$$ 0 0
$$745$$ 7524.00 0.370011
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 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
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 492.000i 0.0237162i
$$756$$ 0 0
$$757$$ − 17445.2i − 0.837592i −0.908080 0.418796i $$-0.862452\pi$$
0.908080 0.418796i $$-0.137548\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30222.0 1.43961 0.719807 0.694174i $$-0.244230\pi$$
0.719807 + 0.694174i $$0.244230\pi$$
$$762$$ 0 0
$$763$$ − 11424.0i − 0.542040i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 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
$$771$$ 0 0
$$772$$ 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
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 1176.00i 0.0540880i
$$780$$ 0 0
$$781$$ − 332.554i − 0.0152365i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −4632.00 −0.210603
$$786$$ 0 0
$$787$$ 31012.0i 1.40465i 0.711857 + 0.702324i $$0.247854\pi$$
−0.711857 + 0.702324i $$0.752146\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 9457.00 0.425097
$$792$$ 0 0
$$793$$ 3744.00 0.167659
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 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
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 48288.0i 2.12210i
$$804$$ 0 0
$$805$$ 14549.2i 0.637010i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −40650.0 −1.76660 −0.883299 0.468810i $$-0.844683\pi$$
−0.883299 + 0.468810i $$0.844683\pi$$
$$810$$ 0 0
$$811$$ 8372.00i 0.362492i 0.983438 + 0.181246i $$0.0580130\pi$$
−0.983438 + 0.181246i $$0.941987\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −6055.25 −0.260253
$$816$$ 0 0
$$817$$ −752.000 −0.0322021
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 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
$$825$$ 0 0
$$826$$ 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 $$-0.829542\pi$$
0.860009 0.510279i $$-0.170458\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −13230.0 −0.550291
$$834$$ 0 0
$$835$$ 48.0000i 0.00198935i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 19613.7 0.807082 0.403541 0.914962i $$-0.367779\pi$$
0.403541 + 0.914962i $$0.367779\pi$$
$$840$$ 0 0
$$841$$ −2119.00 −0.0868834
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 1624.66i − 0.0661422i
$$846$$ 0 0
$$847$$ −23594.0 −0.957142
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 56400.0i 2.27188i
$$852$$ 0 0
$$853$$ 20569.8i 0.825671i 0.910806 + 0.412836i $$0.135462\pi$$
−0.910806 + 0.412836i $$0.864538\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 27222.0 1.08505 0.542524 0.840040i $$-0.317469\pi$$
0.542524 + 0.840040i $$0.317469\pi$$
$$858$$ 0 0
$$859$$ 3548.00i 0.140927i 0.997514 + 0.0704634i $$0.0224478\pi$$
−0.997514 + 0.0704634i $$0.977552\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 45047.2 1.77685 0.888426 0.459019i $$-0.151799\pi$$
0.888426 + 0.459019i $$0.151799\pi$$
$$864$$ 0 0
$$865$$ 2076.00 0.0816024
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 64349.2i 2.51196i
$$870$$ 0 0
$$871$$ −26105.5 −1.01556
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 19992.0i − 0.772403i
$$876$$ 0 0
$$877$$ 29022.2i 1.11746i 0.829350 + 0.558729i $$0.188711\pi$$
−0.829350 + 0.558729i $$0.811289\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 48318.0 1.84776 0.923879 0.382685i $$-0.125000\pi$$
0.923879 + 0.382685i $$0.125000\pi$$
$$882$$ 0 0
$$883$$ 14380.0i 0.548047i 0.961723 + 0.274024i $$0.0883546\pi$$
−0.961723 + 0.274024i $$0.911645\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 34086.8 1.29033 0.645164 0.764044i $$-0.276789\pi$$
0.645164 + 0.764044i $$0.276789\pi$$
$$888$$ 0 0
$$889$$ 14700.0 0.554581
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 2023.04i − 0.0758100i
$$894$$ 0 0
$$895$$ −11182.1 −0.417628
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 9588.00i 0.355704i
$$900$$ 0 0
$$901$$ 40218.2i 1.48708i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −7008.00 −0.257408
$$906$$ 0 0
$$907$$ − 31252.0i − 1.14411i −0.820216 0.572054i $$-0.806147\pi$$
0.820216 0.572054i $$-0.193853\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −13080.4 −0.475713 −0.237857 0.971300i $$-0.576445\pi$$
−0.237857 + 0.971300i $$0.576445\pi$$
$$912$$ 0 0
$$913$$ −34560.0 −1.25276
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 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
$$921$$ 0 0
$$922$$ 0 0
$$923$$ − 288.000i − 0.0102705i
$$924$$ 0 0
$$925$$ − 36795.7i − 1.30793i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −29622.0 −1.04614 −0.523071 0.852289i $$-0.675214\pi$$
−0.523071 + 0.852289i $$0.675214\pi$$
$$930$$ 0 0
$$931$$ 980.000i 0.0344986i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 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
$$939$$ 0 0
$$940$$ 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
$$945$$ 0 0
$$946$$ 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
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −18402.0 −0.625498 −0.312749 0.949836i $$-0.601250\pi$$
−0.312749 + 0.949836i $$0.601250\pi$$
$$954$$ 0 0
$$955$$ − 12048.0i − 0.408235i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 28080.0 0.945517
$$960$$ 0 0
$$961$$ −26323.0 −0.883589
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 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
$$969$$ 0 0
$$970$$ 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
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 23034.0 0.754271 0.377136 0.926158i $$-0.376909\pi$$
0.377136 + 0.926158i $$0.376909\pi$$
$$978$$ 0 0
$$979$$ − 71136.0i − 2.32228i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −17791.6 −0.577278 −0.288639 0.957438i $$-0.593203\pi$$
−0.288639 + 0.957438i $$0.593203\pi$$
$$984$$ 0 0
$$985$$ 10044.0 0.324902
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 6012.00i − 0.191551i
$$996$$ 0 0
$$997$$ − 52273.3i − 1.66049i −0.557396 0.830247i $$-0.688200\pi$$
0.557396 0.830247i $$-0.311800\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 576.4.d.c.289.2 4
3.2 odd 2 192.4.d.c.97.2 yes 4
4.3 odd 2 inner 576.4.d.c.289.1 4
8.3 odd 2 inner 576.4.d.c.289.3 4
8.5 even 2 inner 576.4.d.c.289.4 4
12.11 even 2 192.4.d.c.97.4 yes 4
16.3 odd 4 2304.4.a.bk.1.1 2
16.5 even 4 2304.4.a.bk.1.2 2
16.11 odd 4 2304.4.a.x.1.2 2
16.13 even 4 2304.4.a.x.1.1 2
24.5 odd 2 192.4.d.c.97.3 yes 4
24.11 even 2 192.4.d.c.97.1 4
48.5 odd 4 768.4.a.f.1.1 2
48.11 even 4 768.4.a.o.1.1 2
48.29 odd 4 768.4.a.o.1.2 2
48.35 even 4 768.4.a.f.1.2 2

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