# Properties

 Label 2304.4.a.bk.1.2 Level $2304$ Weight $4$ Character 2304.1 Self dual yes Analytic conductor $135.940$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,4,Mod(1,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2304.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: $$135.940400653$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 192) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 2304.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.46410 q^{5} -24.2487 q^{7} +O(q^{10})$$ $$q+3.46410 q^{5} -24.2487 q^{7} +48.0000 q^{11} -41.5692 q^{13} -54.0000 q^{17} +4.00000 q^{19} +173.205 q^{23} -113.000 q^{25} +162.813 q^{29} +58.8897 q^{31} -84.0000 q^{35} +325.626 q^{37} -294.000 q^{41} -188.000 q^{43} -505.759 q^{47} +245.000 q^{49} +744.782 q^{53} +166.277 q^{55} +252.000 q^{59} +90.0666 q^{61} -144.000 q^{65} -628.000 q^{67} -6.92820 q^{71} +1006.00 q^{73} -1163.94 q^{77} -1340.61 q^{79} -720.000 q^{83} -187.061 q^{85} -1482.00 q^{89} +1008.00 q^{91} +13.8564 q^{95} +1822.00 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 96 q^{11} - 108 q^{17} + 8 q^{19} - 226 q^{25} - 168 q^{35} - 588 q^{41} - 376 q^{43} + 490 q^{49} + 504 q^{59} - 288 q^{65} - 1256 q^{67} + 2012 q^{73} - 1440 q^{83} - 2964 q^{89} + 2016 q^{91} + 3644 q^{97}+O(q^{100})$$ 2 * q + 96 * q^11 - 108 * q^17 + 8 * q^19 - 226 * q^25 - 168 * q^35 - 588 * q^41 - 376 * q^43 + 490 * q^49 + 504 * q^59 - 288 * q^65 - 1256 * q^67 + 2012 * q^73 - 1440 * q^83 - 2964 * q^89 + 2016 * q^91 + 3644 * q^97

## 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
$$5$$ 3.46410 0.309839 0.154919 0.987927i $$-0.450488\pi$$
0.154919 + 0.987927i $$0.450488\pi$$
$$6$$ 0 0
$$7$$ −24.2487 −1.30931 −0.654654 0.755929i $$-0.727186\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 48.0000 1.31569 0.657843 0.753155i $$-0.271469\pi$$
0.657843 + 0.753155i $$0.271469\pi$$
$$12$$ 0 0
$$13$$ −41.5692 −0.886864 −0.443432 0.896308i $$-0.646239\pi$$
−0.443432 + 0.896308i $$0.646239\pi$$
$$14$$ 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
$$19$$ 4.00000 0.0482980 0.0241490 0.999708i $$-0.492312\pi$$
0.0241490 + 0.999708i $$0.492312\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 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
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 162.813 1.04254 0.521269 0.853393i $$-0.325459\pi$$
0.521269 + 0.853393i $$0.325459\pi$$
$$30$$ 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
$$35$$ −84.0000 −0.405674
$$36$$ 0 0
$$37$$ 325.626 1.44682 0.723412 0.690416i $$-0.242573\pi$$
0.723412 + 0.690416i $$0.242573\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −294.000 −1.11988 −0.559940 0.828533i $$-0.689176\pi$$
−0.559940 + 0.828533i $$0.689176\pi$$
$$42$$ 0 0
$$43$$ −188.000 −0.666738 −0.333369 0.942796i $$-0.608185\pi$$
−0.333369 + 0.942796i $$0.608185\pi$$
$$44$$ 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
$$53$$ 744.782 1.93026 0.965129 0.261775i $$-0.0843080\pi$$
0.965129 + 0.261775i $$0.0843080\pi$$
$$54$$ 0 0
$$55$$ 166.277 0.407650
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 252.000 0.556061 0.278031 0.960572i $$-0.410318\pi$$
0.278031 + 0.960572i $$0.410318\pi$$
$$60$$ 0 0
$$61$$ 90.0666 0.189047 0.0945234 0.995523i $$-0.469867\pi$$
0.0945234 + 0.995523i $$0.469867\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −144.000 −0.274785
$$66$$ 0 0
$$67$$ −628.000 −1.14511 −0.572555 0.819866i $$-0.694048\pi$$
−0.572555 + 0.819866i $$0.694048\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 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 $$-0.201380\pi$$
0.806462 + 0.591286i $$0.201380\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1163.94 −1.72264
$$78$$ 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
$$83$$ −720.000 −0.952172 −0.476086 0.879399i $$-0.657945\pi$$
−0.476086 + 0.879399i $$0.657945\pi$$
$$84$$ 0 0
$$85$$ −187.061 −0.238702
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −1482.00 −1.76508 −0.882538 0.470242i $$-0.844167\pi$$
−0.882538 + 0.470242i $$0.844167\pi$$
$$90$$ 0 0
$$91$$ 1008.00 1.16118
$$92$$ 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
$$101$$ 911.059 0.897562 0.448781 0.893642i $$-0.351858\pi$$
0.448781 + 0.893642i $$0.351858\pi$$
$$102$$ 0 0
$$103$$ 453.797 0.434116 0.217058 0.976159i $$-0.430354\pi$$
0.217058 + 0.976159i $$0.430354\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1188.00 −1.07335 −0.536674 0.843790i $$-0.680320\pi$$
−0.536674 + 0.843790i $$0.680320\pi$$
$$108$$ 0 0
$$109$$ −471.118 −0.413990 −0.206995 0.978342i $$-0.566368\pi$$
−0.206995 + 0.978342i $$0.566368\pi$$
$$110$$ 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
$$115$$ 600.000 0.486524
$$116$$ 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
$$125$$ −824.456 −0.589933
$$126$$ 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
$$131$$ −1380.00 −0.920391 −0.460195 0.887818i $$-0.652221\pi$$
−0.460195 + 0.887818i $$0.652221\pi$$
$$132$$ 0 0
$$133$$ −96.9948 −0.0632370
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1158.00 −0.722150 −0.361075 0.932537i $$-0.617590\pi$$
−0.361075 + 0.932537i $$0.617590\pi$$
$$138$$ 0 0
$$139$$ −1180.00 −0.720045 −0.360023 0.932944i $$-0.617231\pi$$
−0.360023 + 0.932944i $$0.617231\pi$$
$$140$$ 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
$$149$$ −2171.99 −1.19420 −0.597102 0.802165i $$-0.703681\pi$$
−0.597102 + 0.802165i $$0.703681\pi$$
$$150$$ 0 0
$$151$$ 142.028 0.0765436 0.0382718 0.999267i $$-0.487815\pi$$
0.0382718 + 0.999267i $$0.487815\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 204.000 0.105714
$$156$$ 0 0
$$157$$ −1337.14 −0.679717 −0.339859 0.940476i $$-0.610379\pi$$
−0.339859 + 0.940476i $$0.610379\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4200.00 −2.05594
$$162$$ 0 0
$$163$$ −1748.00 −0.839963 −0.419981 0.907533i $$-0.637963\pi$$
−0.419981 + 0.907533i $$0.637963\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 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
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 599.290 0.263371 0.131685 0.991292i $$-0.457961\pi$$
0.131685 + 0.991292i $$0.457961\pi$$
$$174$$ 0 0
$$175$$ 2740.10 1.18361
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −3228.00 −1.34789 −0.673944 0.738782i $$-0.735401\pi$$
−0.673944 + 0.738782i $$0.735401\pi$$
$$180$$ 0 0
$$181$$ 2023.04 0.830779 0.415390 0.909644i $$-0.363645\pi$$
0.415390 + 0.909644i $$0.363645\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1128.00 0.448282
$$186$$ 0 0
$$187$$ −2592.00 −1.01361
$$188$$ 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
$$197$$ −2899.45 −1.04862 −0.524308 0.851529i $$-0.675676\pi$$
−0.524308 + 0.851529i $$0.675676\pi$$
$$198$$ 0 0
$$199$$ −1735.51 −0.618228 −0.309114 0.951025i $$-0.600032\pi$$
−0.309114 + 0.951025i $$0.600032\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −3948.00 −1.36500
$$204$$ 0 0
$$205$$ −1018.45 −0.346982
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 192.000 0.0635451
$$210$$ 0 0
$$211$$ 1100.00 0.358896 0.179448 0.983767i $$-0.442569\pi$$
0.179448 + 0.983767i $$0.442569\pi$$
$$212$$ 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
$$221$$ 2244.74 0.683246
$$222$$ 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
$$227$$ 3336.00 0.975410 0.487705 0.873008i $$-0.337834\pi$$
0.487705 + 0.873008i $$0.337834\pi$$
$$228$$ 0 0
$$229$$ −5999.82 −1.73135 −0.865676 0.500605i $$-0.833111\pi$$
−0.865676 + 0.500605i $$0.833111\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 318.000 0.0894115 0.0447057 0.999000i $$-0.485765\pi$$
0.0447057 + 0.999000i $$0.485765\pi$$
$$234$$ 0 0
$$235$$ −1752.00 −0.486331
$$236$$ 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
$$245$$ 848.705 0.221313
$$246$$ 0 0
$$247$$ −166.277 −0.0428338
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −5136.00 −1.29156 −0.645780 0.763524i $$-0.723468\pi$$
−0.645780 + 0.763524i $$0.723468\pi$$
$$252$$ 0 0
$$253$$ 8313.84 2.06596
$$254$$ 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
$$259$$ −7896.00 −1.89434
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 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
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 4624.58 1.04820 0.524099 0.851657i $$-0.324402\pi$$
0.524099 + 0.851657i $$0.324402\pi$$
$$270$$ 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
$$275$$ −5424.00 −1.18938
$$276$$ 0 0
$$277$$ 1524.20 0.330616 0.165308 0.986242i $$-0.447138\pi$$
0.165308 + 0.986242i $$0.447138\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 4398.00 0.933675 0.466838 0.884343i $$-0.345393\pi$$
0.466838 + 0.884343i $$0.345393\pi$$
$$282$$ 0 0
$$283$$ −4372.00 −0.918334 −0.459167 0.888350i $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$284$$ 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
$$293$$ 3571.49 0.712111 0.356056 0.934465i $$-0.384121\pi$$
0.356056 + 0.934465i $$0.384121\pi$$
$$294$$ 0 0
$$295$$ 872.954 0.172289
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −7200.00 −1.39260
$$300$$ 0 0
$$301$$ 4558.76 0.872965
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 312.000 0.0585740
$$306$$ 0 0
$$307$$ 4172.00 0.775598 0.387799 0.921744i $$-0.373235\pi$$
0.387799 + 0.921744i $$0.373235\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 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 $$-0.497873\pi$$
0.00668167 + 0.999978i $$0.497873\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1964.15 0.348004 0.174002 0.984745i $$-0.444330\pi$$
0.174002 + 0.984745i $$0.444330\pi$$
$$318$$ 0 0
$$319$$ 7815.01 1.37165
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −216.000 −0.0372092
$$324$$ 0 0
$$325$$ 4697.32 0.801725
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 12264.0 2.05513
$$330$$ 0 0
$$331$$ 7556.00 1.25473 0.627365 0.778726i $$-0.284134\pi$$
0.627365 + 0.778726i $$0.284134\pi$$
$$332$$ 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
$$341$$ 2826.71 0.448900
$$342$$ 0 0
$$343$$ 2376.37 0.374088
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −5256.00 −0.813132 −0.406566 0.913621i $$-0.633274\pi$$
−0.406566 + 0.913621i $$0.633274\pi$$
$$348$$ 0 0
$$349$$ −10385.4 −1.59288 −0.796442 0.604715i $$-0.793287\pi$$
−0.796442 + 0.604715i $$0.793287\pi$$
$$350$$ 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
$$355$$ −24.0000 −0.00358813
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 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
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3484.89 0.499746
$$366$$ 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
$$371$$ −18060.0 −2.52730
$$372$$ 0 0
$$373$$ −10246.8 −1.42241 −0.711206 0.702983i $$-0.751851\pi$$
−0.711206 + 0.702983i $$0.751851\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6768.00 −0.924588
$$378$$ 0 0
$$379$$ −13844.0 −1.87630 −0.938151 0.346226i $$-0.887463\pi$$
−0.938151 + 0.346226i $$0.887463\pi$$
$$380$$ 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
$$389$$ −12993.8 −1.69361 −0.846805 0.531904i $$-0.821477\pi$$
−0.846805 + 0.531904i $$0.821477\pi$$
$$390$$ 0 0
$$391$$ −9353.07 −1.20973
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −4644.00 −0.591557
$$396$$ 0 0
$$397$$ 117.779 0.0148896 0.00744481 0.999972i $$-0.497630\pi$$
0.00744481 + 0.999972i $$0.497630\pi$$
$$398$$ 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
$$403$$ −2448.00 −0.302589
$$404$$ 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 $$-0.743330\pi$$
−0.692135 + 0.721768i $$0.743330\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −6110.68 −0.728055
$$414$$ 0 0
$$415$$ −2494.15 −0.295020
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 1176.00 0.137115 0.0685577 0.997647i $$-0.478160\pi$$
0.0685577 + 0.997647i $$0.478160\pi$$
$$420$$ 0 0
$$421$$ 10032.0 1.16136 0.580679 0.814133i $$-0.302787\pi$$
0.580679 + 0.814133i $$0.302787\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 6102.00 0.696448
$$426$$ 0 0
$$427$$ −2184.00 −0.247520
$$428$$ 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
$$437$$ 692.820 0.0758400
$$438$$ 0 0
$$439$$ −1610.81 −0.175124 −0.0875622 0.996159i $$-0.527908\pi$$
−0.0875622 + 0.996159i $$0.527908\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −1032.00 −0.110681 −0.0553406 0.998468i $$-0.517624\pi$$
−0.0553406 + 0.998468i $$0.517624\pi$$
$$444$$ 0 0
$$445$$ −5133.80 −0.546889
$$446$$ 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
$$451$$ −14112.0 −1.47341
$$452$$ 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 $$-0.646271\pi$$
−0.443521 + 0.896264i $$0.646271\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −14684.3 −1.48355 −0.741776 0.670648i $$-0.766016\pi$$
−0.741776 + 0.670648i $$0.766016\pi$$
$$462$$ 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
$$467$$ −16824.0 −1.66707 −0.833535 0.552466i $$-0.813687\pi$$
−0.833535 + 0.552466i $$0.813687\pi$$
$$468$$ 0 0
$$469$$ 15228.2 1.49930
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −9024.00 −0.877218
$$474$$ 0 0
$$475$$ −452.000 −0.0436614
$$476$$ 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
$$485$$ 6311.59 0.590917
$$486$$ 0 0
$$487$$ 10714.5 0.996959 0.498479 0.866902i $$-0.333892\pi$$
0.498479 + 0.866902i $$0.333892\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 852.000 0.0783100 0.0391550 0.999233i $$-0.487533\pi$$
0.0391550 + 0.999233i $$0.487533\pi$$
$$492$$ 0 0
$$493$$ −8791.89 −0.803178
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 168.000 0.0151626
$$498$$ 0 0
$$499$$ 11156.0 1.00082 0.500412 0.865787i $$-0.333182\pi$$
0.500412 + 0.865787i $$0.333182\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 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
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −9287.26 −0.808743 −0.404372 0.914595i $$-0.632510\pi$$
−0.404372 + 0.914595i $$0.632510\pi$$
$$510$$ 0 0
$$511$$ −24394.2 −2.11181
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 1572.00 0.134506
$$516$$ 0 0
$$517$$ −24276.4 −2.06514
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −2766.00 −0.232592 −0.116296 0.993215i $$-0.537102\pi$$
−0.116296 + 0.993215i $$0.537102\pi$$
$$522$$ 0 0
$$523$$ 18988.0 1.58755 0.793774 0.608213i $$-0.208113\pi$$
0.793774 + 0.608213i $$0.208113\pi$$
$$524$$ 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
$$533$$ 12221.4 0.993181
$$534$$ 0 0
$$535$$ −4115.35 −0.332565
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 11760.0 0.939776
$$540$$ 0 0
$$541$$ 12997.3 1.03290 0.516449 0.856318i $$-0.327254\pi$$
0.516449 + 0.856318i $$0.327254\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −1632.00 −0.128270
$$546$$ 0 0
$$547$$ −21188.0 −1.65619 −0.828093 0.560591i $$-0.810574\pi$$
−0.828093 + 0.560591i $$0.810574\pi$$
$$548$$ 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
$$557$$ −12231.7 −0.930477 −0.465238 0.885185i $$-0.654031\pi$$
−0.465238 + 0.885185i $$0.654031\pi$$
$$558$$ 0 0
$$559$$ 7815.01 0.591306
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 504.000 0.0377284 0.0188642 0.999822i $$-0.493995\pi$$
0.0188642 + 0.999822i $$0.493995\pi$$
$$564$$ 0 0
$$565$$ 1351.00 0.100596
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −20358.0 −1.49992 −0.749958 0.661486i $$-0.769926\pi$$
−0.749958 + 0.661486i $$0.769926\pi$$
$$570$$ 0 0
$$571$$ −13300.0 −0.974760 −0.487380 0.873190i $$-0.662047\pi$$
−0.487380 + 0.873190i $$0.662047\pi$$
$$572$$ 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
$$581$$ 17459.1 1.24669
$$582$$ 0 0
$$583$$ 35749.5 2.53961
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −13980.0 −0.982992 −0.491496 0.870880i $$-0.663550\pi$$
−0.491496 + 0.870880i $$0.663550\pi$$
$$588$$ 0 0
$$589$$ 235.559 0.0164788
$$590$$ 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
$$595$$ 4536.00 0.312534
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 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 $$-0.576189\pi$$
−0.237076 + 0.971491i $$0.576189\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 3370.57 0.226501
$$606$$ 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
$$611$$ 21024.0 1.39205
$$612$$ 0 0
$$613$$ 11092.1 0.730838 0.365419 0.930843i $$-0.380926\pi$$
0.365419 + 0.930843i $$0.380926\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2850.00 −0.185959 −0.0929795 0.995668i $$-0.529639\pi$$
−0.0929795 + 0.995668i $$0.529639\pi$$
$$618$$ 0 0
$$619$$ −20116.0 −1.30619 −0.653094 0.757277i $$-0.726529\pi$$
−0.653094 + 0.757277i $$0.726529\pi$$
$$620$$ 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
$$629$$ −17583.8 −1.11464
$$630$$ 0 0
$$631$$ −7271.15 −0.458732 −0.229366 0.973340i $$-0.573665\pi$$
−0.229366 + 0.973340i $$0.573665\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 2100.00 0.131238
$$636$$ 0 0
$$637$$ −10184.5 −0.633474
$$638$$ 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
$$643$$ −2948.00 −0.180805 −0.0904026 0.995905i $$-0.528815\pi$$
−0.0904026 + 0.995905i $$0.528815\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 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
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 10381.9 0.622168 0.311084 0.950382i $$-0.399308\pi$$
0.311084 + 0.950382i $$0.399308\pi$$
$$654$$ 0 0
$$655$$ −4780.46 −0.285173
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −10308.0 −0.609321 −0.304661 0.952461i $$-0.598543\pi$$
−0.304661 + 0.952461i $$0.598543\pi$$
$$660$$ 0 0
$$661$$ −15803.2 −0.929916 −0.464958 0.885333i $$-0.653931\pi$$
−0.464958 + 0.885333i $$0.653931\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −336.000 −0.0195933
$$666$$ 0 0
$$667$$ 28200.0 1.63704
$$668$$ 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
$$677$$ 14802.1 0.840312 0.420156 0.907452i $$-0.361975\pi$$
0.420156 + 0.907452i $$0.361975\pi$$
$$678$$ 0 0
$$679$$ −44181.2 −2.49708
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 528.000 0.0295803 0.0147902 0.999891i $$-0.495292\pi$$
0.0147902 + 0.999891i $$0.495292\pi$$
$$684$$ 0 0
$$685$$ −4011.43 −0.223750
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −30960.0 −1.71188
$$690$$ 0 0
$$691$$ 9052.00 0.498342 0.249171 0.968460i $$-0.419842\pi$$
0.249171 + 0.968460i $$0.419842\pi$$
$$692$$ 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
$$701$$ 32600.7 1.75650 0.878252 0.478197i $$-0.158710\pi$$
0.878252 + 0.478197i $$0.158710\pi$$
$$702$$ 0 0
$$703$$ 1302.50 0.0698788
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −22092.0 −1.17518
$$708$$ 0 0
$$709$$ 27227.8 1.44226 0.721130 0.692799i $$-0.243623\pi$$
0.721130 + 0.692799i $$0.243623\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 10200.0 0.535755
$$714$$ 0 0
$$715$$ −6912.00 −0.361530
$$716$$ 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
$$725$$ −18397.8 −0.942453
$$726$$ 0 0
$$727$$ 20192.2 1.03011 0.515054 0.857158i $$-0.327772\pi$$
0.515054 + 0.857158i $$0.327772\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 10152.0 0.513660
$$732$$ 0 0
$$733$$ 35236.8 1.77558 0.887792 0.460246i $$-0.152239\pi$$
0.887792 + 0.460246i $$0.152239\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −30144.0 −1.50661
$$738$$ 0 0
$$739$$ 13940.0 0.693899 0.346949 0.937884i $$-0.387218\pi$$
0.346949 + 0.937884i $$0.387218\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 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
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 28807.5 1.40534
$$750$$ 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
$$755$$ 492.000 0.0237162
$$756$$ 0 0
$$757$$ 17445.2 0.837592 0.418796 0.908080i $$-0.362452\pi$$
0.418796 + 0.908080i $$0.362452\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30222.0 −1.43961 −0.719807 0.694174i $$-0.755770\pi$$
−0.719807 + 0.694174i $$0.755770\pi$$
$$762$$ 0 0
$$763$$ 11424.0 0.542040
$$764$$ 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
$$773$$ −1874.08 −0.0872004 −0.0436002 0.999049i $$-0.513883\pi$$
−0.0436002 + 0.999049i $$0.513883\pi$$
$$774$$ 0 0
$$775$$ −6654.54 −0.308436
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −1176.00 −0.0540880
$$780$$ 0 0
$$781$$ −332.554 −0.0152365
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −4632.00 −0.210603
$$786$$ 0 0
$$787$$ 31012.0 1.40465 0.702324 0.711857i $$-0.252146\pi$$
0.702324 + 0.711857i $$0.252146\pi$$
$$788$$ 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
$$797$$ −7091.02 −0.315153 −0.157576 0.987507i $$-0.550368\pi$$
−0.157576 + 0.987507i $$0.550368\pi$$
$$798$$ 0 0
$$799$$ 27311.0 1.20925
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 48288.0 2.12210
$$804$$ 0 0
$$805$$ −14549.2 −0.637010
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 40650.0 1.76660 0.883299 0.468810i $$-0.155317\pi$$
0.883299 + 0.468810i $$0.155317\pi$$
$$810$$ 0 0
$$811$$ −8372.00 −0.362492 −0.181246 0.983438i $$-0.558013\pi$$
−0.181246 + 0.983438i $$0.558013\pi$$
$$812$$ 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
$$821$$ −9370.39 −0.398330 −0.199165 0.979966i $$-0.563823\pi$$
−0.199165 + 0.979966i $$0.563823\pi$$
$$822$$ 0 0
$$823$$ −21668.0 −0.917737 −0.458868 0.888504i $$-0.651745\pi$$
−0.458868 + 0.888504i $$0.651745\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 6684.00 0.281046 0.140523 0.990077i $$-0.455122\pi$$
0.140523 + 0.990077i $$0.455122\pi$$
$$828$$ 0 0
$$829$$ −24359.6 −1.02056 −0.510279 0.860009i $$-0.670458\pi$$
−0.510279 + 0.860009i $$0.670458\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −13230.0 −0.550291
$$834$$ 0 0
$$835$$ 48.0000 0.00198935
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 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
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −1624.66 −0.0661422
$$846$$ 0 0
$$847$$ −23594.0 −0.957142
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 56400.0 2.27188
$$852$$ 0 0
$$853$$ −20569.8 −0.825671 −0.412836 0.910806i $$-0.635462\pi$$
−0.412836 + 0.910806i $$0.635462\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −27222.0 −1.08505 −0.542524 0.840040i $$-0.682531\pi$$
−0.542524 + 0.840040i $$0.682531\pi$$
$$858$$ 0 0
$$859$$ −3548.00 −0.140927 −0.0704634 0.997514i $$-0.522448\pi$$
−0.0704634 + 0.997514i $$0.522448\pi$$
$$860$$ 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
$$869$$ −64349.2 −2.51196
$$870$$ 0 0
$$871$$ 26105.5 1.01556
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 19992.0 0.772403
$$876$$ 0 0
$$877$$ 29022.2 1.11746 0.558729 0.829350i $$-0.311289\pi$$
0.558729 + 0.829350i $$0.311289\pi$$
$$878$$ 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
$$883$$ 14380.0 0.548047 0.274024 0.961723i $$-0.411645\pi$$
0.274024 + 0.961723i $$0.411645\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 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
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −2023.04 −0.0758100
$$894$$ 0 0
$$895$$ −11182.1 −0.417628
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 9588.00 0.355704
$$900$$ 0 0
$$901$$ −40218.2 −1.48708
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 7008.00 0.257408
$$906$$ 0 0
$$907$$ 31252.0 1.14411 0.572054 0.820216i $$-0.306147\pi$$
0.572054 + 0.820216i $$0.306147\pi$$
$$908$$ 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
$$917$$ 33463.2 1.20507
$$918$$ 0 0
$$919$$ 8843.85 0.317445 0.158722 0.987323i $$-0.449263\pi$$
0.158722 + 0.987323i $$0.449263\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 288.000 0.0102705
$$924$$ 0 0
$$925$$ −36795.7 −1.30793
$$926$$ 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
$$931$$ 980.000 0.0344986
$$932$$ 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 $$-0.367408\pi$$
0.404609 + 0.914490i $$0.367408\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −19728.1 −0.683439 −0.341720 0.939802i $$-0.611009\pi$$
−0.341720 + 0.939802i $$0.611009\pi$$
$$942$$ 0 0
$$943$$ −50922.3 −1.75849
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1236.00 0.0424125 0.0212062 0.999775i $$-0.493249\pi$$
0.0212062 + 0.999775i $$0.493249\pi$$
$$948$$ 0 0
$$949$$ −41818.6 −1.43044
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 18402.0 0.625498 0.312749 0.949836i $$-0.398750\pi$$
0.312749 + 0.949836i $$0.398750\pi$$
$$954$$ 0 0
$$955$$ 12048.0 0.408235
$$956$$ 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
$$965$$ −2653.50 −0.0885174
$$966$$ 0 0
$$967$$ −41836.0 −1.39127 −0.695633 0.718398i $$-0.744876\pi$$
−0.695633 + 0.718398i $$0.744876\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −8832.00 −0.291897 −0.145949 0.989292i $$-0.546623\pi$$
−0.145949 + 0.989292i $$0.546623\pi$$
$$972$$ 0 0
$$973$$ 28613.5 0.942761
$$974$$ 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
$$979$$ −71136.0 −2.32228
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 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
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −32562.6 −1.04695
$$990$$ 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
$$995$$ −6012.00 −0.191551
$$996$$ 0 0
$$997$$ 52273.3 1.66049 0.830247 0.557396i $$-0.188200\pi$$
0.830247 + 0.557396i $$0.188200\pi$$
$$998$$ 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 2304.4.a.bk.1.2 2
3.2 odd 2 768.4.a.f.1.1 2
4.3 odd 2 2304.4.a.x.1.2 2
8.3 odd 2 inner 2304.4.a.bk.1.1 2
8.5 even 2 2304.4.a.x.1.1 2
12.11 even 2 768.4.a.o.1.1 2
16.3 odd 4 576.4.d.c.289.1 4
16.5 even 4 576.4.d.c.289.4 4
16.11 odd 4 576.4.d.c.289.3 4
16.13 even 4 576.4.d.c.289.2 4
24.5 odd 2 768.4.a.o.1.2 2
24.11 even 2 768.4.a.f.1.2 2
48.5 odd 4 192.4.d.c.97.3 yes 4
48.11 even 4 192.4.d.c.97.1 4
48.29 odd 4 192.4.d.c.97.2 yes 4
48.35 even 4 192.4.d.c.97.4 yes 4

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