Properties

 Label 1323.4.a.x.1.2 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$78.0595269376$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+4.73205 q^{2} +14.3923 q^{4} -9.92820 q^{5} +30.2487 q^{8} +O(q^{10})$$ $$q+4.73205 q^{2} +14.3923 q^{4} -9.92820 q^{5} +30.2487 q^{8} -46.9808 q^{10} -3.71281 q^{11} +15.5692 q^{13} +28.0000 q^{16} +33.4974 q^{17} -135.923 q^{19} -142.890 q^{20} -17.5692 q^{22} -87.7795 q^{23} -26.4308 q^{25} +73.6743 q^{26} +242.354 q^{29} +194.708 q^{31} -109.492 q^{32} +158.512 q^{34} -239.708 q^{37} -643.195 q^{38} -300.315 q^{40} -470.338 q^{41} -448.215 q^{43} -53.4359 q^{44} -415.377 q^{46} -4.15906 q^{47} -125.072 q^{50} +224.077 q^{52} -736.543 q^{53} +36.8616 q^{55} +1146.83 q^{58} -279.564 q^{59} +514.831 q^{61} +921.367 q^{62} -742.123 q^{64} -154.574 q^{65} -102.123 q^{67} +482.105 q^{68} -44.1539 q^{71} +901.615 q^{73} -1134.31 q^{74} -1956.25 q^{76} +1054.31 q^{79} -277.990 q^{80} -2225.67 q^{82} -487.246 q^{83} -332.569 q^{85} -2120.98 q^{86} -112.308 q^{88} -963.682 q^{89} -1263.35 q^{92} -19.6809 q^{94} +1349.47 q^{95} -726.908 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{2} + 8 q^{4} - 6 q^{5} + 12 q^{8}+O(q^{10})$$ 2 * q + 6 * q^2 + 8 * q^4 - 6 * q^5 + 12 * q^8 $$2 q + 6 q^{2} + 8 q^{4} - 6 q^{5} + 12 q^{8} - 42 q^{10} + 48 q^{11} - 52 q^{13} + 56 q^{16} - 30 q^{17} - 64 q^{19} - 168 q^{20} + 48 q^{22} + 60 q^{23} - 136 q^{25} - 12 q^{26} + 360 q^{29} + 140 q^{31} + 72 q^{32} + 78 q^{34} - 230 q^{37} - 552 q^{38} - 372 q^{40} - 234 q^{41} - 938 q^{43} - 384 q^{44} - 228 q^{46} - 618 q^{47} - 264 q^{50} + 656 q^{52} - 420 q^{53} + 240 q^{55} + 1296 q^{58} - 282 q^{59} + 32 q^{61} + 852 q^{62} - 736 q^{64} - 420 q^{65} + 544 q^{67} + 888 q^{68} - 504 q^{71} + 764 q^{73} - 1122 q^{74} - 2416 q^{76} + 238 q^{79} - 168 q^{80} - 1926 q^{82} + 522 q^{83} - 582 q^{85} - 2742 q^{86} - 1056 q^{88} - 708 q^{89} - 2208 q^{92} - 798 q^{94} + 1632 q^{95} - 664 q^{97}+O(q^{100})$$ 2 * q + 6 * q^2 + 8 * q^4 - 6 * q^5 + 12 * q^8 - 42 * q^10 + 48 * q^11 - 52 * q^13 + 56 * q^16 - 30 * q^17 - 64 * q^19 - 168 * q^20 + 48 * q^22 + 60 * q^23 - 136 * q^25 - 12 * q^26 + 360 * q^29 + 140 * q^31 + 72 * q^32 + 78 * q^34 - 230 * q^37 - 552 * q^38 - 372 * q^40 - 234 * q^41 - 938 * q^43 - 384 * q^44 - 228 * q^46 - 618 * q^47 - 264 * q^50 + 656 * q^52 - 420 * q^53 + 240 * q^55 + 1296 * q^58 - 282 * q^59 + 32 * q^61 + 852 * q^62 - 736 * q^64 - 420 * q^65 + 544 * q^67 + 888 * q^68 - 504 * q^71 + 764 * q^73 - 1122 * q^74 - 2416 * q^76 + 238 * q^79 - 168 * q^80 - 1926 * q^82 + 522 * q^83 - 582 * q^85 - 2742 * q^86 - 1056 * q^88 - 708 * q^89 - 2208 * q^92 - 798 * q^94 + 1632 * q^95 - 664 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.73205 1.67303 0.836516 0.547942i $$-0.184589\pi$$
0.836516 + 0.547942i $$0.184589\pi$$
$$3$$ 0 0
$$4$$ 14.3923 1.79904
$$5$$ −9.92820 −0.888005 −0.444003 0.896025i $$-0.646442\pi$$
−0.444003 + 0.896025i $$0.646442\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 30.2487 1.33682
$$9$$ 0 0
$$10$$ −46.9808 −1.48566
$$11$$ −3.71281 −0.101769 −0.0508843 0.998705i $$-0.516204\pi$$
−0.0508843 + 0.998705i $$0.516204\pi$$
$$12$$ 0 0
$$13$$ 15.5692 0.332163 0.166082 0.986112i $$-0.446888\pi$$
0.166082 + 0.986112i $$0.446888\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 28.0000 0.437500
$$17$$ 33.4974 0.477901 0.238951 0.971032i $$-0.423197\pi$$
0.238951 + 0.971032i $$0.423197\pi$$
$$18$$ 0 0
$$19$$ −135.923 −1.64120 −0.820602 0.571500i $$-0.806362\pi$$
−0.820602 + 0.571500i $$0.806362\pi$$
$$20$$ −142.890 −1.59756
$$21$$ 0 0
$$22$$ −17.5692 −0.170262
$$23$$ −87.7795 −0.795795 −0.397897 0.917430i $$-0.630260\pi$$
−0.397897 + 0.917430i $$0.630260\pi$$
$$24$$ 0 0
$$25$$ −26.4308 −0.211446
$$26$$ 73.6743 0.555720
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 242.354 1.55186 0.775931 0.630818i $$-0.217281\pi$$
0.775931 + 0.630818i $$0.217281\pi$$
$$30$$ 0 0
$$31$$ 194.708 1.12808 0.564041 0.825747i $$-0.309246\pi$$
0.564041 + 0.825747i $$0.309246\pi$$
$$32$$ −109.492 −0.604865
$$33$$ 0 0
$$34$$ 158.512 0.799544
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −239.708 −1.06507 −0.532536 0.846407i $$-0.678761\pi$$
−0.532536 + 0.846407i $$0.678761\pi$$
$$38$$ −643.195 −2.74579
$$39$$ 0 0
$$40$$ −300.315 −1.18710
$$41$$ −470.338 −1.79157 −0.895787 0.444484i $$-0.853387\pi$$
−0.895787 + 0.444484i $$0.853387\pi$$
$$42$$ 0 0
$$43$$ −448.215 −1.58959 −0.794793 0.606880i $$-0.792421\pi$$
−0.794793 + 0.606880i $$0.792421\pi$$
$$44$$ −53.4359 −0.183086
$$45$$ 0 0
$$46$$ −415.377 −1.33139
$$47$$ −4.15906 −0.0129077 −0.00645384 0.999979i $$-0.502054\pi$$
−0.00645384 + 0.999979i $$0.502054\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −125.072 −0.353756
$$51$$ 0 0
$$52$$ 224.077 0.597575
$$53$$ −736.543 −1.90891 −0.954453 0.298361i $$-0.903560\pi$$
−0.954453 + 0.298361i $$0.903560\pi$$
$$54$$ 0 0
$$55$$ 36.8616 0.0903711
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 1146.83 2.59631
$$59$$ −279.564 −0.616884 −0.308442 0.951243i $$-0.599808\pi$$
−0.308442 + 0.951243i $$0.599808\pi$$
$$60$$ 0 0
$$61$$ 514.831 1.08061 0.540306 0.841469i $$-0.318309\pi$$
0.540306 + 0.841469i $$0.318309\pi$$
$$62$$ 921.367 1.88732
$$63$$ 0 0
$$64$$ −742.123 −1.44946
$$65$$ −154.574 −0.294963
$$66$$ 0 0
$$67$$ −102.123 −0.186214 −0.0931068 0.995656i $$-0.529680\pi$$
−0.0931068 + 0.995656i $$0.529680\pi$$
$$68$$ 482.105 0.859762
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −44.1539 −0.0738043 −0.0369021 0.999319i $$-0.511749\pi$$
−0.0369021 + 0.999319i $$0.511749\pi$$
$$72$$ 0 0
$$73$$ 901.615 1.44556 0.722781 0.691077i $$-0.242863\pi$$
0.722781 + 0.691077i $$0.242863\pi$$
$$74$$ −1134.31 −1.78190
$$75$$ 0 0
$$76$$ −1956.25 −2.95259
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1054.31 1.50150 0.750752 0.660584i $$-0.229691\pi$$
0.750752 + 0.660584i $$0.229691\pi$$
$$80$$ −277.990 −0.388502
$$81$$ 0 0
$$82$$ −2225.67 −2.99736
$$83$$ −487.246 −0.644364 −0.322182 0.946678i $$-0.604416\pi$$
−0.322182 + 0.946678i $$0.604416\pi$$
$$84$$ 0 0
$$85$$ −332.569 −0.424379
$$86$$ −2120.98 −2.65943
$$87$$ 0 0
$$88$$ −112.308 −0.136046
$$89$$ −963.682 −1.14775 −0.573877 0.818942i $$-0.694561\pi$$
−0.573877 + 0.818942i $$0.694561\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −1263.35 −1.43167
$$93$$ 0 0
$$94$$ −19.6809 −0.0215950
$$95$$ 1349.47 1.45740
$$96$$ 0 0
$$97$$ −726.908 −0.760890 −0.380445 0.924804i $$-0.624229\pi$$
−0.380445 + 0.924804i $$0.624229\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −380.400 −0.380400
$$101$$ −268.872 −0.264889 −0.132444 0.991190i $$-0.542283\pi$$
−0.132444 + 0.991190i $$0.542283\pi$$
$$102$$ 0 0
$$103$$ −1108.35 −1.06028 −0.530142 0.847909i $$-0.677862\pi$$
−0.530142 + 0.847909i $$0.677862\pi$$
$$104$$ 470.949 0.444042
$$105$$ 0 0
$$106$$ −3485.36 −3.19366
$$107$$ 84.4665 0.0763148 0.0381574 0.999272i $$-0.487851\pi$$
0.0381574 + 0.999272i $$0.487851\pi$$
$$108$$ 0 0
$$109$$ −174.169 −0.153050 −0.0765248 0.997068i $$-0.524382\pi$$
−0.0765248 + 0.997068i $$0.524382\pi$$
$$110$$ 174.431 0.151194
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1049.77 0.873929 0.436964 0.899479i $$-0.356054\pi$$
0.436964 + 0.899479i $$0.356054\pi$$
$$114$$ 0 0
$$115$$ 871.492 0.706670
$$116$$ 3488.03 2.79186
$$117$$ 0 0
$$118$$ −1322.91 −1.03207
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1317.22 −0.989643
$$122$$ 2436.20 1.80790
$$123$$ 0 0
$$124$$ 2802.29 2.02946
$$125$$ 1503.44 1.07577
$$126$$ 0 0
$$127$$ −293.969 −0.205398 −0.102699 0.994712i $$-0.532748\pi$$
−0.102699 + 0.994712i $$0.532748\pi$$
$$128$$ −2635.83 −1.82013
$$129$$ 0 0
$$130$$ −731.454 −0.493483
$$131$$ −1110.76 −0.740820 −0.370410 0.928868i $$-0.620783\pi$$
−0.370410 + 0.928868i $$0.620783\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −483.251 −0.311541
$$135$$ 0 0
$$136$$ 1013.25 0.638866
$$137$$ −238.113 −0.148492 −0.0742458 0.997240i $$-0.523655\pi$$
−0.0742458 + 0.997240i $$0.523655\pi$$
$$138$$ 0 0
$$139$$ 189.108 0.115395 0.0576974 0.998334i $$-0.481624\pi$$
0.0576974 + 0.998334i $$0.481624\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −208.939 −0.123477
$$143$$ −57.8056 −0.0338038
$$144$$ 0 0
$$145$$ −2406.14 −1.37806
$$146$$ 4266.49 2.41847
$$147$$ 0 0
$$148$$ −3449.95 −1.91611
$$149$$ 1107.59 0.608975 0.304487 0.952516i $$-0.401515\pi$$
0.304487 + 0.952516i $$0.401515\pi$$
$$150$$ 0 0
$$151$$ 3512.55 1.89303 0.946515 0.322660i $$-0.104577\pi$$
0.946515 + 0.322660i $$0.104577\pi$$
$$152$$ −4111.50 −2.19399
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1933.10 −1.00174
$$156$$ 0 0
$$157$$ −524.169 −0.266454 −0.133227 0.991086i $$-0.542534\pi$$
−0.133227 + 0.991086i $$0.542534\pi$$
$$158$$ 4989.04 2.51207
$$159$$ 0 0
$$160$$ 1087.06 0.537123
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 245.693 0.118062 0.0590311 0.998256i $$-0.481199\pi$$
0.0590311 + 0.998256i $$0.481199\pi$$
$$164$$ −6769.25 −3.22311
$$165$$ 0 0
$$166$$ −2305.67 −1.07804
$$167$$ −586.785 −0.271897 −0.135948 0.990716i $$-0.543408\pi$$
−0.135948 + 0.990716i $$0.543408\pi$$
$$168$$ 0 0
$$169$$ −1954.60 −0.889667
$$170$$ −1573.73 −0.710000
$$171$$ 0 0
$$172$$ −6450.85 −2.85973
$$173$$ −2957.80 −1.29987 −0.649936 0.759989i $$-0.725204\pi$$
−0.649936 + 0.759989i $$0.725204\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −103.959 −0.0445238
$$177$$ 0 0
$$178$$ −4560.19 −1.92023
$$179$$ 2724.73 1.13774 0.568872 0.822426i $$-0.307380\pi$$
0.568872 + 0.822426i $$0.307380\pi$$
$$180$$ 0 0
$$181$$ −446.138 −0.183211 −0.0916055 0.995795i $$-0.529200\pi$$
−0.0916055 + 0.995795i $$0.529200\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −2655.22 −1.06383
$$185$$ 2379.87 0.945791
$$186$$ 0 0
$$187$$ −124.370 −0.0486354
$$188$$ −59.8584 −0.0232214
$$189$$ 0 0
$$190$$ 6385.77 2.43828
$$191$$ 3517.90 1.33270 0.666351 0.745638i $$-0.267855\pi$$
0.666351 + 0.745638i $$0.267855\pi$$
$$192$$ 0 0
$$193$$ 4869.61 1.81618 0.908090 0.418776i $$-0.137541\pi$$
0.908090 + 0.418776i $$0.137541\pi$$
$$194$$ −3439.76 −1.27299
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3427.46 −1.23958 −0.619788 0.784769i $$-0.712781\pi$$
−0.619788 + 0.784769i $$0.712781\pi$$
$$198$$ 0 0
$$199$$ 4012.51 1.42934 0.714671 0.699461i $$-0.246576\pi$$
0.714671 + 0.699461i $$0.246576\pi$$
$$200$$ −799.497 −0.282665
$$201$$ 0 0
$$202$$ −1272.32 −0.443167
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 4669.61 1.59093
$$206$$ −5244.79 −1.77389
$$207$$ 0 0
$$208$$ 435.938 0.145321
$$209$$ 504.657 0.167023
$$210$$ 0 0
$$211$$ 4281.08 1.39678 0.698392 0.715715i $$-0.253899\pi$$
0.698392 + 0.715715i $$0.253899\pi$$
$$212$$ −10600.6 −3.43419
$$213$$ 0 0
$$214$$ 399.700 0.127677
$$215$$ 4449.97 1.41156
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −824.178 −0.256057
$$219$$ 0 0
$$220$$ 530.523 0.162581
$$221$$ 521.529 0.158741
$$222$$ 0 0
$$223$$ 764.153 0.229469 0.114734 0.993396i $$-0.463398\pi$$
0.114734 + 0.993396i $$0.463398\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 4967.56 1.46211
$$227$$ −3872.35 −1.13223 −0.566116 0.824325i $$-0.691555\pi$$
−0.566116 + 0.824325i $$0.691555\pi$$
$$228$$ 0 0
$$229$$ 3866.11 1.11563 0.557816 0.829965i $$-0.311640\pi$$
0.557816 + 0.829965i $$0.311640\pi$$
$$230$$ 4123.95 1.18228
$$231$$ 0 0
$$232$$ 7330.89 2.07455
$$233$$ −5378.29 −1.51220 −0.756102 0.654454i $$-0.772898\pi$$
−0.756102 + 0.654454i $$0.772898\pi$$
$$234$$ 0 0
$$235$$ 41.2920 0.0114621
$$236$$ −4023.57 −1.10980
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 4763.15 1.28913 0.644566 0.764548i $$-0.277038\pi$$
0.644566 + 0.764548i $$0.277038\pi$$
$$240$$ 0 0
$$241$$ 318.030 0.0850047 0.0425023 0.999096i $$-0.486467\pi$$
0.0425023 + 0.999096i $$0.486467\pi$$
$$242$$ −6233.13 −1.65571
$$243$$ 0 0
$$244$$ 7409.60 1.94406
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2116.22 −0.545148
$$248$$ 5889.66 1.50804
$$249$$ 0 0
$$250$$ 7114.33 1.79980
$$251$$ −1577.00 −0.396571 −0.198285 0.980144i $$-0.563537\pi$$
−0.198285 + 0.980144i $$0.563537\pi$$
$$252$$ 0 0
$$253$$ 325.909 0.0809870
$$254$$ −1391.08 −0.343638
$$255$$ 0 0
$$256$$ −6535.88 −1.59567
$$257$$ 6069.46 1.47316 0.736580 0.676350i $$-0.236439\pi$$
0.736580 + 0.676350i $$0.236439\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −2224.68 −0.530650
$$261$$ 0 0
$$262$$ −5256.17 −1.23942
$$263$$ −3654.73 −0.856883 −0.428441 0.903570i $$-0.640937\pi$$
−0.428441 + 0.903570i $$0.640937\pi$$
$$264$$ 0 0
$$265$$ 7312.55 1.69512
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −1469.78 −0.335005
$$269$$ 435.652 0.0987441 0.0493721 0.998780i $$-0.484278\pi$$
0.0493721 + 0.998780i $$0.484278\pi$$
$$270$$ 0 0
$$271$$ −1230.98 −0.275930 −0.137965 0.990437i $$-0.544056\pi$$
−0.137965 + 0.990437i $$0.544056\pi$$
$$272$$ 937.928 0.209082
$$273$$ 0 0
$$274$$ −1126.76 −0.248431
$$275$$ 98.1325 0.0215186
$$276$$ 0 0
$$277$$ 3994.17 0.866377 0.433188 0.901303i $$-0.357388\pi$$
0.433188 + 0.901303i $$0.357388\pi$$
$$278$$ 894.866 0.193059
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3615.71 −0.767598 −0.383799 0.923417i $$-0.625384\pi$$
−0.383799 + 0.923417i $$0.625384\pi$$
$$282$$ 0 0
$$283$$ 2353.42 0.494332 0.247166 0.968973i $$-0.420501\pi$$
0.247166 + 0.968973i $$0.420501\pi$$
$$284$$ −635.476 −0.132777
$$285$$ 0 0
$$286$$ −273.539 −0.0565549
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −3790.92 −0.771611
$$290$$ −11386.0 −2.30554
$$291$$ 0 0
$$292$$ 12976.3 2.60062
$$293$$ −3580.68 −0.713943 −0.356972 0.934115i $$-0.616191\pi$$
−0.356972 + 0.934115i $$0.616191\pi$$
$$294$$ 0 0
$$295$$ 2775.57 0.547796
$$296$$ −7250.85 −1.42381
$$297$$ 0 0
$$298$$ 5241.17 1.01883
$$299$$ −1366.66 −0.264334
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 16621.6 3.16710
$$303$$ 0 0
$$304$$ −3805.85 −0.718027
$$305$$ −5111.34 −0.959589
$$306$$ 0 0
$$307$$ 9077.92 1.68764 0.843818 0.536629i $$-0.180303\pi$$
0.843818 + 0.536629i $$0.180303\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −9147.51 −1.67595
$$311$$ 8966.68 1.63490 0.817450 0.576000i $$-0.195387\pi$$
0.817450 + 0.576000i $$0.195387\pi$$
$$312$$ 0 0
$$313$$ −7191.74 −1.29873 −0.649363 0.760479i $$-0.724964\pi$$
−0.649363 + 0.760479i $$0.724964\pi$$
$$314$$ −2480.39 −0.445786
$$315$$ 0 0
$$316$$ 15173.9 2.70126
$$317$$ −2288.95 −0.405553 −0.202776 0.979225i $$-0.564996\pi$$
−0.202776 + 0.979225i $$0.564996\pi$$
$$318$$ 0 0
$$319$$ −899.814 −0.157931
$$320$$ 7367.95 1.28713
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −4553.07 −0.784333
$$324$$ 0 0
$$325$$ −411.507 −0.0702347
$$326$$ 1162.63 0.197522
$$327$$ 0 0
$$328$$ −14227.1 −2.39501
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −6298.61 −1.04593 −0.522965 0.852354i $$-0.675174\pi$$
−0.522965 + 0.852354i $$0.675174\pi$$
$$332$$ −7012.59 −1.15923
$$333$$ 0 0
$$334$$ −2776.70 −0.454892
$$335$$ 1013.90 0.165359
$$336$$ 0 0
$$337$$ 189.615 0.0306498 0.0153249 0.999883i $$-0.495122\pi$$
0.0153249 + 0.999883i $$0.495122\pi$$
$$338$$ −9249.26 −1.48844
$$339$$ 0 0
$$340$$ −4786.44 −0.763474
$$341$$ −722.913 −0.114803
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −13557.9 −2.12499
$$345$$ 0 0
$$346$$ −13996.5 −2.17473
$$347$$ 50.4728 0.00780841 0.00390421 0.999992i $$-0.498757\pi$$
0.00390421 + 0.999992i $$0.498757\pi$$
$$348$$ 0 0
$$349$$ 1338.71 0.205328 0.102664 0.994716i $$-0.467263\pi$$
0.102664 + 0.994716i $$0.467263\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 406.524 0.0615563
$$353$$ −3192.99 −0.481433 −0.240717 0.970595i $$-0.577382\pi$$
−0.240717 + 0.970595i $$0.577382\pi$$
$$354$$ 0 0
$$355$$ 438.369 0.0655386
$$356$$ −13869.6 −2.06485
$$357$$ 0 0
$$358$$ 12893.6 1.90348
$$359$$ 6661.22 0.979292 0.489646 0.871921i $$-0.337126\pi$$
0.489646 + 0.871921i $$0.337126\pi$$
$$360$$ 0 0
$$361$$ 11616.1 1.69355
$$362$$ −2111.15 −0.306518
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −8951.42 −1.28367
$$366$$ 0 0
$$367$$ −4592.77 −0.653244 −0.326622 0.945155i $$-0.605910\pi$$
−0.326622 + 0.945155i $$0.605910\pi$$
$$368$$ −2457.82 −0.348160
$$369$$ 0 0
$$370$$ 11261.6 1.58234
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 6982.57 0.969286 0.484643 0.874712i $$-0.338950\pi$$
0.484643 + 0.874712i $$0.338950\pi$$
$$374$$ −588.524 −0.0813685
$$375$$ 0 0
$$376$$ −125.806 −0.0172552
$$377$$ 3773.26 0.515472
$$378$$ 0 0
$$379$$ −12060.4 −1.63457 −0.817286 0.576233i $$-0.804522\pi$$
−0.817286 + 0.576233i $$0.804522\pi$$
$$380$$ 19422.0 2.62192
$$381$$ 0 0
$$382$$ 16646.9 2.22965
$$383$$ −9674.46 −1.29071 −0.645355 0.763883i $$-0.723291\pi$$
−0.645355 + 0.763883i $$0.723291\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 23043.3 3.03853
$$387$$ 0 0
$$388$$ −10461.9 −1.36887
$$389$$ −11377.9 −1.48299 −0.741496 0.670957i $$-0.765883\pi$$
−0.741496 + 0.670957i $$0.765883\pi$$
$$390$$ 0 0
$$391$$ −2940.39 −0.380311
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −16218.9 −2.07385
$$395$$ −10467.4 −1.33334
$$396$$ 0 0
$$397$$ 15202.7 1.92192 0.960958 0.276693i $$-0.0892384\pi$$
0.960958 + 0.276693i $$0.0892384\pi$$
$$398$$ 18987.4 2.39134
$$399$$ 0 0
$$400$$ −740.062 −0.0925077
$$401$$ −1396.02 −0.173851 −0.0869254 0.996215i $$-0.527704\pi$$
−0.0869254 + 0.996215i $$0.527704\pi$$
$$402$$ 0 0
$$403$$ 3031.45 0.374707
$$404$$ −3869.69 −0.476545
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 889.990 0.108391
$$408$$ 0 0
$$409$$ 8673.54 1.04860 0.524302 0.851533i $$-0.324326\pi$$
0.524302 + 0.851533i $$0.324326\pi$$
$$410$$ 22096.9 2.66167
$$411$$ 0 0
$$412$$ −15951.8 −1.90749
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 4837.48 0.572199
$$416$$ −1704.71 −0.200914
$$417$$ 0 0
$$418$$ 2388.06 0.279435
$$419$$ −7313.92 −0.852765 −0.426382 0.904543i $$-0.640212\pi$$
−0.426382 + 0.904543i $$0.640212\pi$$
$$420$$ 0 0
$$421$$ −7695.26 −0.890841 −0.445420 0.895322i $$-0.646946\pi$$
−0.445420 + 0.895322i $$0.646946\pi$$
$$422$$ 20258.3 2.33687
$$423$$ 0 0
$$424$$ −22279.5 −2.55186
$$425$$ −885.363 −0.101050
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 1215.67 0.137293
$$429$$ 0 0
$$430$$ 21057.5 2.36159
$$431$$ 6258.10 0.699402 0.349701 0.936861i $$-0.386283\pi$$
0.349701 + 0.936861i $$0.386283\pi$$
$$432$$ 0 0
$$433$$ 11923.3 1.32332 0.661660 0.749804i $$-0.269852\pi$$
0.661660 + 0.749804i $$0.269852\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2506.70 −0.275342
$$437$$ 11931.3 1.30606
$$438$$ 0 0
$$439$$ −17115.6 −1.86079 −0.930393 0.366564i $$-0.880534\pi$$
−0.930393 + 0.366564i $$0.880534\pi$$
$$440$$ 1115.01 0.120810
$$441$$ 0 0
$$442$$ 2467.90 0.265579
$$443$$ −8095.04 −0.868187 −0.434094 0.900868i $$-0.642931\pi$$
−0.434094 + 0.900868i $$0.642931\pi$$
$$444$$ 0 0
$$445$$ 9567.63 1.01921
$$446$$ 3616.01 0.383908
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −6156.71 −0.647111 −0.323556 0.946209i $$-0.604878\pi$$
−0.323556 + 0.946209i $$0.604878\pi$$
$$450$$ 0 0
$$451$$ 1746.28 0.182326
$$452$$ 15108.6 1.57223
$$453$$ 0 0
$$454$$ −18324.2 −1.89426
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5657.23 0.579068 0.289534 0.957168i $$-0.406500\pi$$
0.289534 + 0.957168i $$0.406500\pi$$
$$458$$ 18294.6 1.86649
$$459$$ 0 0
$$460$$ 12542.8 1.27133
$$461$$ 1414.61 0.142918 0.0714589 0.997444i $$-0.477235\pi$$
0.0714589 + 0.997444i $$0.477235\pi$$
$$462$$ 0 0
$$463$$ −5404.92 −0.542523 −0.271261 0.962506i $$-0.587441\pi$$
−0.271261 + 0.962506i $$0.587441\pi$$
$$464$$ 6785.91 0.678939
$$465$$ 0 0
$$466$$ −25450.3 −2.52996
$$467$$ −16952.8 −1.67983 −0.839916 0.542717i $$-0.817396\pi$$
−0.839916 + 0.542717i $$0.817396\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 195.396 0.0191765
$$471$$ 0 0
$$472$$ −8456.45 −0.824661
$$473$$ 1664.14 0.161770
$$474$$ 0 0
$$475$$ 3592.55 0.347027
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 22539.5 2.15676
$$479$$ 18372.7 1.75255 0.876276 0.481810i $$-0.160021\pi$$
0.876276 + 0.481810i $$0.160021\pi$$
$$480$$ 0 0
$$481$$ −3732.06 −0.353778
$$482$$ 1504.94 0.142216
$$483$$ 0 0
$$484$$ −18957.8 −1.78041
$$485$$ 7216.89 0.675674
$$486$$ 0 0
$$487$$ −42.5560 −0.00395974 −0.00197987 0.999998i $$-0.500630\pi$$
−0.00197987 + 0.999998i $$0.500630\pi$$
$$488$$ 15573.0 1.44458
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −6092.09 −0.559943 −0.279972 0.960008i $$-0.590325\pi$$
−0.279972 + 0.960008i $$0.590325\pi$$
$$492$$ 0 0
$$493$$ 8118.23 0.741636
$$494$$ −10014.0 −0.912051
$$495$$ 0 0
$$496$$ 5451.81 0.493536
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 6128.64 0.549811 0.274906 0.961471i $$-0.411353\pi$$
0.274906 + 0.961471i $$0.411353\pi$$
$$500$$ 21637.9 1.93535
$$501$$ 0 0
$$502$$ −7462.44 −0.663476
$$503$$ 3606.14 0.319661 0.159831 0.987144i $$-0.448905\pi$$
0.159831 + 0.987144i $$0.448905\pi$$
$$504$$ 0 0
$$505$$ 2669.41 0.235223
$$506$$ 1542.22 0.135494
$$507$$ 0 0
$$508$$ −4230.90 −0.369519
$$509$$ 15207.2 1.32426 0.662128 0.749391i $$-0.269654\pi$$
0.662128 + 0.749391i $$0.269654\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −9841.49 −0.849486
$$513$$ 0 0
$$514$$ 28721.0 2.46465
$$515$$ 11004.0 0.941539
$$516$$ 0 0
$$517$$ 15.4418 0.00131360
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −4675.68 −0.394311
$$521$$ 5497.74 0.462304 0.231152 0.972918i $$-0.425750\pi$$
0.231152 + 0.972918i $$0.425750\pi$$
$$522$$ 0 0
$$523$$ −15158.7 −1.26739 −0.633693 0.773584i $$-0.718462\pi$$
−0.633693 + 0.773584i $$0.718462\pi$$
$$524$$ −15986.4 −1.33276
$$525$$ 0 0
$$526$$ −17294.4 −1.43359
$$527$$ 6522.20 0.539111
$$528$$ 0 0
$$529$$ −4461.77 −0.366711
$$530$$ 34603.4 2.83599
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −7322.80 −0.595095
$$534$$ 0 0
$$535$$ −838.601 −0.0677680
$$536$$ −3089.09 −0.248933
$$537$$ 0 0
$$538$$ 2061.53 0.165202
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 21418.8 1.70216 0.851080 0.525037i $$-0.175948\pi$$
0.851080 + 0.525037i $$0.175948\pi$$
$$542$$ −5825.08 −0.461639
$$543$$ 0 0
$$544$$ −3667.71 −0.289066
$$545$$ 1729.19 0.135909
$$546$$ 0 0
$$547$$ 15536.1 1.21439 0.607197 0.794551i $$-0.292294\pi$$
0.607197 + 0.794551i $$0.292294\pi$$
$$548$$ −3426.99 −0.267142
$$549$$ 0 0
$$550$$ 464.368 0.0360013
$$551$$ −32941.5 −2.54692
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 18900.6 1.44948
$$555$$ 0 0
$$556$$ 2721.69 0.207600
$$557$$ 13975.1 1.06309 0.531547 0.847029i $$-0.321611\pi$$
0.531547 + 0.847029i $$0.321611\pi$$
$$558$$ 0 0
$$559$$ −6978.36 −0.528002
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −17109.7 −1.28422
$$563$$ −1117.89 −0.0836825 −0.0418412 0.999124i $$-0.513322\pi$$
−0.0418412 + 0.999124i $$0.513322\pi$$
$$564$$ 0 0
$$565$$ −10422.3 −0.776054
$$566$$ 11136.5 0.827034
$$567$$ 0 0
$$568$$ −1335.60 −0.0986628
$$569$$ 14452.1 1.06478 0.532392 0.846498i $$-0.321293\pi$$
0.532392 + 0.846498i $$0.321293\pi$$
$$570$$ 0 0
$$571$$ 17396.8 1.27501 0.637506 0.770445i $$-0.279966\pi$$
0.637506 + 0.770445i $$0.279966\pi$$
$$572$$ −831.956 −0.0608144
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 2320.08 0.168268
$$576$$ 0 0
$$577$$ −12251.2 −0.883923 −0.441961 0.897034i $$-0.645717\pi$$
−0.441961 + 0.897034i $$0.645717\pi$$
$$578$$ −17938.8 −1.29093
$$579$$ 0 0
$$580$$ −34629.9 −2.47918
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 2734.65 0.194267
$$584$$ 27272.7 1.93245
$$585$$ 0 0
$$586$$ −16943.9 −1.19445
$$587$$ −24620.6 −1.73118 −0.865589 0.500755i $$-0.833056\pi$$
−0.865589 + 0.500755i $$0.833056\pi$$
$$588$$ 0 0
$$589$$ −26465.3 −1.85141
$$590$$ 13134.1 0.916481
$$591$$ 0 0
$$592$$ −6711.81 −0.465969
$$593$$ 3020.20 0.209148 0.104574 0.994517i $$-0.466652\pi$$
0.104574 + 0.994517i $$0.466652\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 15940.8 1.09557
$$597$$ 0 0
$$598$$ −6467.09 −0.442239
$$599$$ −16830.3 −1.14803 −0.574013 0.818846i $$-0.694614\pi$$
−0.574013 + 0.818846i $$0.694614\pi$$
$$600$$ 0 0
$$601$$ −8783.94 −0.596180 −0.298090 0.954538i $$-0.596350\pi$$
−0.298090 + 0.954538i $$0.596350\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 50553.7 3.40563
$$605$$ 13077.6 0.878809
$$606$$ 0 0
$$607$$ −16803.3 −1.12360 −0.561799 0.827274i $$-0.689891\pi$$
−0.561799 + 0.827274i $$0.689891\pi$$
$$608$$ 14882.5 0.992707
$$609$$ 0 0
$$610$$ −24187.1 −1.60542
$$611$$ −64.7533 −0.00428746
$$612$$ 0 0
$$613$$ 1319.88 0.0869647 0.0434823 0.999054i $$-0.486155\pi$$
0.0434823 + 0.999054i $$0.486155\pi$$
$$614$$ 42957.2 2.82347
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2586.39 −0.168759 −0.0843793 0.996434i $$-0.526891\pi$$
−0.0843793 + 0.996434i $$0.526891\pi$$
$$618$$ 0 0
$$619$$ −5296.63 −0.343925 −0.171962 0.985104i $$-0.555011\pi$$
−0.171962 + 0.985104i $$0.555011\pi$$
$$620$$ −27821.7 −1.80217
$$621$$ 0 0
$$622$$ 42430.8 2.73524
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −11622.6 −0.743844
$$626$$ −34031.7 −2.17281
$$627$$ 0 0
$$628$$ −7544.00 −0.479360
$$629$$ −8029.59 −0.508999
$$630$$ 0 0
$$631$$ −15857.8 −1.00046 −0.500228 0.865894i $$-0.666751\pi$$
−0.500228 + 0.865894i $$0.666751\pi$$
$$632$$ 31891.4 2.00724
$$633$$ 0 0
$$634$$ −10831.4 −0.678503
$$635$$ 2918.59 0.182395
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −4257.97 −0.264223
$$639$$ 0 0
$$640$$ 26169.0 1.61628
$$641$$ −30365.3 −1.87107 −0.935537 0.353228i $$-0.885084\pi$$
−0.935537 + 0.353228i $$0.885084\pi$$
$$642$$ 0 0
$$643$$ −3746.71 −0.229791 −0.114896 0.993378i $$-0.536653\pi$$
−0.114896 + 0.993378i $$0.536653\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −21545.4 −1.31222
$$647$$ −13032.7 −0.791915 −0.395957 0.918269i $$-0.629587\pi$$
−0.395957 + 0.918269i $$0.629587\pi$$
$$648$$ 0 0
$$649$$ 1037.97 0.0627794
$$650$$ −1947.27 −0.117505
$$651$$ 0 0
$$652$$ 3536.08 0.212398
$$653$$ −3096.91 −0.185592 −0.0927960 0.995685i $$-0.529580\pi$$
−0.0927960 + 0.995685i $$0.529580\pi$$
$$654$$ 0 0
$$655$$ 11027.8 0.657853
$$656$$ −13169.5 −0.783813
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 11759.2 0.695103 0.347551 0.937661i $$-0.387013\pi$$
0.347551 + 0.937661i $$0.387013\pi$$
$$660$$ 0 0
$$661$$ −2363.12 −0.139054 −0.0695270 0.997580i $$-0.522149\pi$$
−0.0695270 + 0.997580i $$0.522149\pi$$
$$662$$ −29805.4 −1.74988
$$663$$ 0 0
$$664$$ −14738.6 −0.861396
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −21273.7 −1.23496
$$668$$ −8445.19 −0.489153
$$669$$ 0 0
$$670$$ 4797.82 0.276650
$$671$$ −1911.47 −0.109972
$$672$$ 0 0
$$673$$ −19358.2 −1.10877 −0.554387 0.832259i $$-0.687047\pi$$
−0.554387 + 0.832259i $$0.687047\pi$$
$$674$$ 897.267 0.0512781
$$675$$ 0 0
$$676$$ −28131.2 −1.60055
$$677$$ −7931.13 −0.450248 −0.225124 0.974330i $$-0.572279\pi$$
−0.225124 + 0.974330i $$0.572279\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −10059.8 −0.567317
$$681$$ 0 0
$$682$$ −3420.86 −0.192070
$$683$$ 7289.89 0.408404 0.204202 0.978929i $$-0.434540\pi$$
0.204202 + 0.978929i $$0.434540\pi$$
$$684$$ 0 0
$$685$$ 2364.03 0.131861
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −12550.0 −0.695444
$$689$$ −11467.4 −0.634069
$$690$$ 0 0
$$691$$ 13338.6 0.734335 0.367167 0.930155i $$-0.380328\pi$$
0.367167 + 0.930155i $$0.380328\pi$$
$$692$$ −42569.6 −2.33852
$$693$$ 0 0
$$694$$ 238.840 0.0130637
$$695$$ −1877.50 −0.102471
$$696$$ 0 0
$$697$$ −15755.1 −0.856195
$$698$$ 6334.84 0.343520
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −12918.5 −0.696042 −0.348021 0.937487i $$-0.613146\pi$$
−0.348021 + 0.937487i $$0.613146\pi$$
$$702$$ 0 0
$$703$$ 32581.8 1.74800
$$704$$ 2755.36 0.147509
$$705$$ 0 0
$$706$$ −15109.4 −0.805453
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2282.78 0.120919 0.0604596 0.998171i $$-0.480743\pi$$
0.0604596 + 0.998171i $$0.480743\pi$$
$$710$$ 2074.38 0.109648
$$711$$ 0 0
$$712$$ −29150.1 −1.53434
$$713$$ −17091.3 −0.897721
$$714$$ 0 0
$$715$$ 573.906 0.0300180
$$716$$ 39215.2 2.04684
$$717$$ 0 0
$$718$$ 31521.2 1.63839
$$719$$ 29360.8 1.52291 0.761456 0.648217i $$-0.224485\pi$$
0.761456 + 0.648217i $$0.224485\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 54967.9 2.83337
$$723$$ 0 0
$$724$$ −6420.96 −0.329603
$$725$$ −6405.60 −0.328135
$$726$$ 0 0
$$727$$ 18139.1 0.925368 0.462684 0.886523i $$-0.346886\pi$$
0.462684 + 0.886523i $$0.346886\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −42358.6 −2.14762
$$731$$ −15014.1 −0.759665
$$732$$ 0 0
$$733$$ 5292.54 0.266691 0.133345 0.991070i $$-0.457428\pi$$
0.133345 + 0.991070i $$0.457428\pi$$
$$734$$ −21733.2 −1.09290
$$735$$ 0 0
$$736$$ 9611.17 0.481348
$$737$$ 379.163 0.0189507
$$738$$ 0 0
$$739$$ −25955.7 −1.29201 −0.646007 0.763332i $$-0.723562\pi$$
−0.646007 + 0.763332i $$0.723562\pi$$
$$740$$ 34251.8 1.70151
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −25972.0 −1.28240 −0.641199 0.767375i $$-0.721563\pi$$
−0.641199 + 0.767375i $$0.721563\pi$$
$$744$$ 0 0
$$745$$ −10996.4 −0.540773
$$746$$ 33041.9 1.62165
$$747$$ 0 0
$$748$$ −1789.97 −0.0874968
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 6487.01 0.315199 0.157599 0.987503i $$-0.449625\pi$$
0.157599 + 0.987503i $$0.449625\pi$$
$$752$$ −116.454 −0.00564711
$$753$$ 0 0
$$754$$ 17855.3 0.862401
$$755$$ −34873.3 −1.68102
$$756$$ 0 0
$$757$$ 22221.6 1.06692 0.533459 0.845826i $$-0.320892\pi$$
0.533459 + 0.845826i $$0.320892\pi$$
$$758$$ −57070.6 −2.73469
$$759$$ 0 0
$$760$$ 40819.8 1.94827
$$761$$ −19820.0 −0.944120 −0.472060 0.881567i $$-0.656489\pi$$
−0.472060 + 0.881567i $$0.656489\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 50630.6 2.39758
$$765$$ 0 0
$$766$$ −45780.1 −2.15940
$$767$$ −4352.59 −0.204906
$$768$$ 0 0
$$769$$ 3307.64 0.155106 0.0775531 0.996988i $$-0.475289\pi$$
0.0775531 + 0.996988i $$0.475289\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 70085.0 3.26738
$$773$$ −15964.0 −0.742801 −0.371401 0.928473i $$-0.621122\pi$$
−0.371401 + 0.928473i $$0.621122\pi$$
$$774$$ 0 0
$$775$$ −5146.28 −0.238529
$$776$$ −21988.0 −1.01717
$$777$$ 0 0
$$778$$ −53840.9 −2.48109
$$779$$ 63929.8 2.94034
$$780$$ 0 0
$$781$$ 163.935 0.00751096
$$782$$ −13914.1 −0.636273
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 5204.06 0.236612
$$786$$ 0 0
$$787$$ 36394.0 1.64842 0.824210 0.566284i $$-0.191619\pi$$
0.824210 + 0.566284i $$0.191619\pi$$
$$788$$ −49329.1 −2.23004
$$789$$ 0 0
$$790$$ −49532.2 −2.23073
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 8015.51 0.358940
$$794$$ 71939.9 3.21543
$$795$$ 0 0
$$796$$ 57749.2 2.57144
$$797$$ 13384.0 0.594836 0.297418 0.954747i $$-0.403874\pi$$
0.297418 + 0.954747i $$0.403874\pi$$
$$798$$ 0 0
$$799$$ −139.318 −0.00616859
$$800$$ 2893.97 0.127896
$$801$$ 0 0
$$802$$ −6606.06 −0.290858
$$803$$ −3347.53 −0.147113
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 14345.0 0.626898
$$807$$ 0 0
$$808$$ −8133.03 −0.354108
$$809$$ 13971.0 0.607161 0.303580 0.952806i $$-0.401818\pi$$
0.303580 + 0.952806i $$0.401818\pi$$
$$810$$ 0 0
$$811$$ −18471.8 −0.799793 −0.399897 0.916560i $$-0.630954\pi$$
−0.399897 + 0.916560i $$0.630954\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 4211.48 0.181342
$$815$$ −2439.29 −0.104840
$$816$$ 0 0
$$817$$ 60922.8 2.60884
$$818$$ 41043.6 1.75435
$$819$$ 0 0
$$820$$ 67206.5 2.86214
$$821$$ −28188.3 −1.19827 −0.599135 0.800648i $$-0.704489\pi$$
−0.599135 + 0.800648i $$0.704489\pi$$
$$822$$ 0 0
$$823$$ −8071.14 −0.341849 −0.170925 0.985284i $$-0.554675\pi$$
−0.170925 + 0.985284i $$0.554675\pi$$
$$824$$ −33526.3 −1.41741
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −41952.8 −1.76402 −0.882009 0.471233i $$-0.843809\pi$$
−0.882009 + 0.471233i $$0.843809\pi$$
$$828$$ 0 0
$$829$$ −14443.2 −0.605108 −0.302554 0.953132i $$-0.597839\pi$$
−0.302554 + 0.953132i $$0.597839\pi$$
$$830$$ 22891.2 0.957307
$$831$$ 0 0
$$832$$ −11554.3 −0.481457
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 5825.72 0.241446
$$836$$ 7263.18 0.300481
$$837$$ 0 0
$$838$$ −34609.9 −1.42670
$$839$$ 26360.6 1.08471 0.542353 0.840151i $$-0.317534\pi$$
0.542353 + 0.840151i $$0.317534\pi$$
$$840$$ 0 0
$$841$$ 34346.4 1.40827
$$842$$ −36414.4 −1.49041
$$843$$ 0 0
$$844$$ 61614.5 2.51287
$$845$$ 19405.7 0.790030
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −20623.2 −0.835146
$$849$$ 0 0
$$850$$ −4189.58 −0.169061
$$851$$ 21041.4 0.847580
$$852$$ 0 0
$$853$$ 18939.8 0.760241 0.380120 0.924937i $$-0.375883\pi$$
0.380120 + 0.924937i $$0.375883\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 2555.00 0.102019
$$857$$ 44583.6 1.77707 0.888533 0.458812i $$-0.151725\pi$$
0.888533 + 0.458812i $$0.151725\pi$$
$$858$$ 0 0
$$859$$ −7975.35 −0.316782 −0.158391 0.987376i $$-0.550631\pi$$
−0.158391 + 0.987376i $$0.550631\pi$$
$$860$$ 64045.4 2.53945
$$861$$ 0 0
$$862$$ 29613.7 1.17012
$$863$$ 8258.42 0.325747 0.162874 0.986647i $$-0.447924\pi$$
0.162874 + 0.986647i $$0.447924\pi$$
$$864$$ 0 0
$$865$$ 29365.7 1.15429
$$866$$ 56421.7 2.21396
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −3914.45 −0.152806
$$870$$ 0 0
$$871$$ −1589.97 −0.0618533
$$872$$ −5268.40 −0.204599
$$873$$ 0 0
$$874$$ 56459.3 2.18508
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −43106.2 −1.65974 −0.829871 0.557956i $$-0.811586\pi$$
−0.829871 + 0.557956i $$0.811586\pi$$
$$878$$ −80992.1 −3.11316
$$879$$ 0 0
$$880$$ 1032.12 0.0395374
$$881$$ −662.616 −0.0253395 −0.0126697 0.999920i $$-0.504033\pi$$
−0.0126697 + 0.999920i $$0.504033\pi$$
$$882$$ 0 0
$$883$$ 32497.0 1.23852 0.619258 0.785187i $$-0.287433\pi$$
0.619258 + 0.785187i $$0.287433\pi$$
$$884$$ 7506.00 0.285582
$$885$$ 0 0
$$886$$ −38306.1 −1.45251
$$887$$ −3176.34 −0.120238 −0.0601190 0.998191i $$-0.519148\pi$$
−0.0601190 + 0.998191i $$0.519148\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 45274.5 1.70517
$$891$$ 0 0
$$892$$ 10997.9 0.412823
$$893$$ 565.312 0.0211841
$$894$$ 0 0
$$895$$ −27051.7 −1.01032
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −29133.8 −1.08264
$$899$$ 47188.1 1.75063
$$900$$ 0 0
$$901$$ −24672.3 −0.912268
$$902$$ 8263.48 0.305037
$$903$$ 0 0
$$904$$ 31754.2 1.16828
$$905$$ 4429.35 0.162692
$$906$$ 0 0
$$907$$ −31726.2 −1.16147 −0.580733 0.814094i $$-0.697234\pi$$
−0.580733 + 0.814094i $$0.697234\pi$$
$$908$$ −55732.0 −2.03693
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −10882.3 −0.395772 −0.197886 0.980225i $$-0.563408\pi$$
−0.197886 + 0.980225i $$0.563408\pi$$
$$912$$ 0 0
$$913$$ 1809.05 0.0655760
$$914$$ 26770.3 0.968799
$$915$$ 0 0
$$916$$ 55642.2 2.00706
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −29610.9 −1.06286 −0.531432 0.847101i $$-0.678346\pi$$
−0.531432 + 0.847101i $$0.678346\pi$$
$$920$$ 26361.5 0.944689
$$921$$ 0 0
$$922$$ 6694.02 0.239106
$$923$$ −687.442 −0.0245151
$$924$$ 0 0
$$925$$ 6335.66 0.225206
$$926$$ −25576.4 −0.907658
$$927$$ 0 0
$$928$$ −26535.9 −0.938667
$$929$$ −52496.6 −1.85399 −0.926995 0.375074i $$-0.877617\pi$$
−0.926995 + 0.375074i $$0.877617\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −77405.9 −2.72051
$$933$$ 0 0
$$934$$ −80221.5 −2.81041
$$935$$ 1234.77 0.0431885
$$936$$ 0 0
$$937$$ 1661.46 0.0579269 0.0289634 0.999580i $$-0.490779\pi$$
0.0289634 + 0.999580i $$0.490779\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 594.287 0.0206207
$$941$$ −43066.9 −1.49197 −0.745983 0.665965i $$-0.768020\pi$$
−0.745983 + 0.665965i $$0.768020\pi$$
$$942$$ 0 0
$$943$$ 41286.0 1.42572
$$944$$ −7827.79 −0.269887
$$945$$ 0 0
$$946$$ 7874.79 0.270647
$$947$$ −10024.6 −0.343986 −0.171993 0.985098i $$-0.555021\pi$$
−0.171993 + 0.985098i $$0.555021\pi$$
$$948$$ 0 0
$$949$$ 14037.4 0.480163
$$950$$ 17000.1 0.580587
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −36134.8 −1.22825 −0.614125 0.789209i $$-0.710491\pi$$
−0.614125 + 0.789209i $$0.710491\pi$$
$$954$$ 0 0
$$955$$ −34926.4 −1.18345
$$956$$ 68552.7 2.31920
$$957$$ 0 0
$$958$$ 86940.7 2.93208
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 8120.07 0.272568
$$962$$ −17660.3 −0.591883
$$963$$ 0 0
$$964$$ 4577.19 0.152927
$$965$$ −48346.5 −1.61278
$$966$$ 0 0
$$967$$ 22016.7 0.732171 0.366086 0.930581i $$-0.380698\pi$$
0.366086 + 0.930581i $$0.380698\pi$$
$$968$$ −39844.1 −1.32297
$$969$$ 0 0
$$970$$ 34150.7 1.13042
$$971$$ −8087.76 −0.267300 −0.133650 0.991029i $$-0.542670\pi$$
−0.133650 + 0.991029i $$0.542670\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −201.377 −0.00662478
$$975$$ 0 0
$$976$$ 14415.3 0.472768
$$977$$ −19050.6 −0.623830 −0.311915 0.950110i $$-0.600970\pi$$
−0.311915 + 0.950110i $$0.600970\pi$$
$$978$$ 0 0
$$979$$ 3577.97 0.116805
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −28828.1 −0.936803
$$983$$ 35176.7 1.14136 0.570682 0.821171i $$-0.306679\pi$$
0.570682 + 0.821171i $$0.306679\pi$$
$$984$$ 0 0
$$985$$ 34028.5 1.10075
$$986$$ 38415.9 1.24078
$$987$$ 0 0
$$988$$ −30457.2 −0.980742
$$989$$ 39344.1 1.26498
$$990$$ 0 0
$$991$$ −26830.6 −0.860041 −0.430021 0.902819i $$-0.641494\pi$$
−0.430021 + 0.902819i $$0.641494\pi$$
$$992$$ −21319.0 −0.682337
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −39837.0 −1.26926
$$996$$ 0 0
$$997$$ −27316.3 −0.867721 −0.433860 0.900980i $$-0.642849\pi$$
−0.433860 + 0.900980i $$0.642849\pi$$
$$998$$ 29001.1 0.919852
$$999$$ 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 1323.4.a.x.1.2 2
3.2 odd 2 1323.4.a.o.1.1 2
7.6 odd 2 189.4.a.i.1.2 yes 2
21.20 even 2 189.4.a.e.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.a.e.1.1 2 21.20 even 2
189.4.a.i.1.2 yes 2 7.6 odd 2
1323.4.a.o.1.1 2 3.2 odd 2
1323.4.a.x.1.2 2 1.1 even 1 trivial