# Properties

 Label 153.4.a.g.1.1 Level $153$ Weight $4$ Character 153.1 Self dual yes Analytic conductor $9.027$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [153,4,Mod(1,153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(153, 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("153.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$153 = 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 153.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: $$9.02729223088$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.2636.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 14x - 4$$ x^3 - 14*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.287410$$ of defining polynomial Character $$\chi$$ $$=$$ 153.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.67129 q^{2} +13.8209 q^{4} +11.9174 q^{5} +26.1222 q^{7} -27.1912 q^{8} +O(q^{10})$$ $$q-4.67129 q^{2} +13.8209 q^{4} +11.9174 q^{5} +26.1222 q^{7} -27.1912 q^{8} -55.6696 q^{10} +3.24412 q^{11} -20.0515 q^{13} -122.024 q^{14} +16.4506 q^{16} +17.0000 q^{17} +57.3466 q^{19} +164.709 q^{20} -15.1542 q^{22} -77.0438 q^{23} +17.0243 q^{25} +93.6662 q^{26} +361.033 q^{28} +286.162 q^{29} -8.54816 q^{31} +140.684 q^{32} -79.4119 q^{34} +311.309 q^{35} +357.982 q^{37} -267.882 q^{38} -324.049 q^{40} -194.467 q^{41} -74.2619 q^{43} +44.8367 q^{44} +359.894 q^{46} -23.6130 q^{47} +339.369 q^{49} -79.5255 q^{50} -277.130 q^{52} -104.330 q^{53} +38.6614 q^{55} -710.295 q^{56} -1336.75 q^{58} -249.363 q^{59} -370.384 q^{61} +39.9309 q^{62} -788.781 q^{64} -238.961 q^{65} +939.650 q^{67} +234.956 q^{68} -1454.21 q^{70} +520.197 q^{71} +348.741 q^{73} -1672.24 q^{74} +792.583 q^{76} +84.7434 q^{77} -953.827 q^{79} +196.049 q^{80} +908.412 q^{82} +1414.28 q^{83} +202.596 q^{85} +346.899 q^{86} -88.2115 q^{88} +486.132 q^{89} -523.788 q^{91} -1064.82 q^{92} +110.303 q^{94} +683.422 q^{95} -685.281 q^{97} -1585.29 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8}+O(q^{10})$$ 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8 $$3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8} - 56 q^{10} + 28 q^{11} + 30 q^{13} - 92 q^{14} + 137 q^{16} + 51 q^{17} + 80 q^{19} + 168 q^{20} + 286 q^{22} - 142 q^{23} - 223 q^{25} - 26 q^{26} + 476 q^{28} + 456 q^{29} + 230 q^{31} + 71 q^{32} - 17 q^{34} + 332 q^{35} + 356 q^{37} - 724 q^{38} - 424 q^{40} + 294 q^{41} + 556 q^{43} + 1122 q^{44} - 704 q^{46} - 640 q^{47} - 269 q^{49} - 547 q^{50} - 774 q^{52} - 302 q^{53} + 76 q^{55} - 684 q^{56} - 1304 q^{58} - 636 q^{59} - 84 q^{61} - 508 q^{62} - 919 q^{64} - 408 q^{65} + 1008 q^{67} + 425 q^{68} - 1504 q^{70} + 402 q^{71} + 838 q^{73} - 836 q^{74} - 908 q^{76} + 504 q^{77} - 594 q^{79} + 40 q^{80} + 358 q^{82} + 2396 q^{83} + 136 q^{85} + 1264 q^{86} + 1838 q^{88} + 170 q^{89} - 1016 q^{91} - 4896 q^{92} - 2016 q^{94} + 472 q^{95} - 270 q^{97} - 2857 q^{98}+O(q^{100})$$ 3 * q - q^2 + 25 * q^4 + 8 * q^5 + 22 * q^7 + 39 * q^8 - 56 * q^10 + 28 * q^11 + 30 * q^13 - 92 * q^14 + 137 * q^16 + 51 * q^17 + 80 * q^19 + 168 * q^20 + 286 * q^22 - 142 * q^23 - 223 * q^25 - 26 * q^26 + 476 * q^28 + 456 * q^29 + 230 * q^31 + 71 * q^32 - 17 * q^34 + 332 * q^35 + 356 * q^37 - 724 * q^38 - 424 * q^40 + 294 * q^41 + 556 * q^43 + 1122 * q^44 - 704 * q^46 - 640 * q^47 - 269 * q^49 - 547 * q^50 - 774 * q^52 - 302 * q^53 + 76 * q^55 - 684 * q^56 - 1304 * q^58 - 636 * q^59 - 84 * q^61 - 508 * q^62 - 919 * q^64 - 408 * q^65 + 1008 * q^67 + 425 * q^68 - 1504 * q^70 + 402 * q^71 + 838 * q^73 - 836 * q^74 - 908 * q^76 + 504 * q^77 - 594 * q^79 + 40 * q^80 + 358 * q^82 + 2396 * q^83 + 136 * q^85 + 1264 * q^86 + 1838 * q^88 + 170 * q^89 - 1016 * q^91 - 4896 * q^92 - 2016 * q^94 + 472 * q^95 - 270 * q^97 - 2857 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.67129 −1.65155 −0.825775 0.564000i $$-0.809262\pi$$
−0.825775 + 0.564000i $$0.809262\pi$$
$$3$$ 0 0
$$4$$ 13.8209 1.72762
$$5$$ 11.9174 1.06592 0.532962 0.846139i $$-0.321079\pi$$
0.532962 + 0.846139i $$0.321079\pi$$
$$6$$ 0 0
$$7$$ 26.1222 1.41047 0.705233 0.708975i $$-0.250842\pi$$
0.705233 + 0.708975i $$0.250842\pi$$
$$8$$ −27.1912 −1.20169
$$9$$ 0 0
$$10$$ −55.6696 −1.76043
$$11$$ 3.24412 0.0889216 0.0444608 0.999011i $$-0.485843\pi$$
0.0444608 + 0.999011i $$0.485843\pi$$
$$12$$ 0 0
$$13$$ −20.0515 −0.427790 −0.213895 0.976857i $$-0.568615\pi$$
−0.213895 + 0.976857i $$0.568615\pi$$
$$14$$ −122.024 −2.32945
$$15$$ 0 0
$$16$$ 16.4506 0.257041
$$17$$ 17.0000 0.242536
$$18$$ 0 0
$$19$$ 57.3466 0.692432 0.346216 0.938155i $$-0.387466\pi$$
0.346216 + 0.938155i $$0.387466\pi$$
$$20$$ 164.709 1.84151
$$21$$ 0 0
$$22$$ −15.1542 −0.146858
$$23$$ −77.0438 −0.698467 −0.349233 0.937036i $$-0.613558\pi$$
−0.349233 + 0.937036i $$0.613558\pi$$
$$24$$ 0 0
$$25$$ 17.0243 0.136195
$$26$$ 93.6662 0.706517
$$27$$ 0 0
$$28$$ 361.033 2.43674
$$29$$ 286.162 1.83238 0.916190 0.400744i $$-0.131248\pi$$
0.916190 + 0.400744i $$0.131248\pi$$
$$30$$ 0 0
$$31$$ −8.54816 −0.0495256 −0.0247628 0.999693i $$-0.507883\pi$$
−0.0247628 + 0.999693i $$0.507883\pi$$
$$32$$ 140.684 0.777178
$$33$$ 0 0
$$34$$ −79.4119 −0.400560
$$35$$ 311.309 1.50345
$$36$$ 0 0
$$37$$ 357.982 1.59059 0.795296 0.606221i $$-0.207315\pi$$
0.795296 + 0.606221i $$0.207315\pi$$
$$38$$ −267.882 −1.14359
$$39$$ 0 0
$$40$$ −324.049 −1.28091
$$41$$ −194.467 −0.740748 −0.370374 0.928883i $$-0.620770\pi$$
−0.370374 + 0.928883i $$0.620770\pi$$
$$42$$ 0 0
$$43$$ −74.2619 −0.263368 −0.131684 0.991292i $$-0.542038\pi$$
−0.131684 + 0.991292i $$0.542038\pi$$
$$44$$ 44.8367 0.153622
$$45$$ 0 0
$$46$$ 359.894 1.15355
$$47$$ −23.6130 −0.0732831 −0.0366416 0.999328i $$-0.511666\pi$$
−0.0366416 + 0.999328i $$0.511666\pi$$
$$48$$ 0 0
$$49$$ 339.369 0.989415
$$50$$ −79.5255 −0.224932
$$51$$ 0 0
$$52$$ −277.130 −0.739058
$$53$$ −104.330 −0.270393 −0.135197 0.990819i $$-0.543167\pi$$
−0.135197 + 0.990819i $$0.543167\pi$$
$$54$$ 0 0
$$55$$ 38.6614 0.0947837
$$56$$ −710.295 −1.69495
$$57$$ 0 0
$$58$$ −1336.75 −3.02627
$$59$$ −249.363 −0.550243 −0.275122 0.961409i $$-0.588718\pi$$
−0.275122 + 0.961409i $$0.588718\pi$$
$$60$$ 0 0
$$61$$ −370.384 −0.777424 −0.388712 0.921359i $$-0.627080\pi$$
−0.388712 + 0.921359i $$0.627080\pi$$
$$62$$ 39.9309 0.0817940
$$63$$ 0 0
$$64$$ −788.781 −1.54059
$$65$$ −238.961 −0.455992
$$66$$ 0 0
$$67$$ 939.650 1.71338 0.856691 0.515830i $$-0.172517\pi$$
0.856691 + 0.515830i $$0.172517\pi$$
$$68$$ 234.956 0.419008
$$69$$ 0 0
$$70$$ −1454.21 −2.48302
$$71$$ 520.197 0.869522 0.434761 0.900546i $$-0.356833\pi$$
0.434761 + 0.900546i $$0.356833\pi$$
$$72$$ 0 0
$$73$$ 348.741 0.559137 0.279568 0.960126i $$-0.409809\pi$$
0.279568 + 0.960126i $$0.409809\pi$$
$$74$$ −1672.24 −2.62694
$$75$$ 0 0
$$76$$ 792.583 1.19626
$$77$$ 84.7434 0.125421
$$78$$ 0 0
$$79$$ −953.827 −1.35840 −0.679202 0.733951i $$-0.737674\pi$$
−0.679202 + 0.733951i $$0.737674\pi$$
$$80$$ 196.049 0.273986
$$81$$ 0 0
$$82$$ 908.412 1.22338
$$83$$ 1414.28 1.87033 0.935166 0.354211i $$-0.115250\pi$$
0.935166 + 0.354211i $$0.115250\pi$$
$$84$$ 0 0
$$85$$ 202.596 0.258525
$$86$$ 346.899 0.434966
$$87$$ 0 0
$$88$$ −88.2115 −0.106857
$$89$$ 486.132 0.578987 0.289493 0.957180i $$-0.406513\pi$$
0.289493 + 0.957180i $$0.406513\pi$$
$$90$$ 0 0
$$91$$ −523.788 −0.603384
$$92$$ −1064.82 −1.20668
$$93$$ 0 0
$$94$$ 110.303 0.121031
$$95$$ 683.422 0.738080
$$96$$ 0 0
$$97$$ −685.281 −0.717317 −0.358659 0.933469i $$-0.616766\pi$$
−0.358659 + 0.933469i $$0.616766\pi$$
$$98$$ −1585.29 −1.63407
$$99$$ 0 0
$$100$$ 235.292 0.235292
$$101$$ −864.755 −0.851944 −0.425972 0.904736i $$-0.640068\pi$$
−0.425972 + 0.904736i $$0.640068\pi$$
$$102$$ 0 0
$$103$$ 1880.91 1.79933 0.899665 0.436580i $$-0.143810\pi$$
0.899665 + 0.436580i $$0.143810\pi$$
$$104$$ 545.224 0.514073
$$105$$ 0 0
$$106$$ 487.355 0.446567
$$107$$ 32.8149 0.0296480 0.0148240 0.999890i $$-0.495281\pi$$
0.0148240 + 0.999890i $$0.495281\pi$$
$$108$$ 0 0
$$109$$ 528.727 0.464613 0.232307 0.972643i $$-0.425373\pi$$
0.232307 + 0.972643i $$0.425373\pi$$
$$110$$ −180.599 −0.156540
$$111$$ 0 0
$$112$$ 429.727 0.362548
$$113$$ −414.691 −0.345229 −0.172614 0.984989i $$-0.555221\pi$$
−0.172614 + 0.984989i $$0.555221\pi$$
$$114$$ 0 0
$$115$$ −918.161 −0.744513
$$116$$ 3955.03 3.16565
$$117$$ 0 0
$$118$$ 1164.85 0.908754
$$119$$ 444.077 0.342088
$$120$$ 0 0
$$121$$ −1320.48 −0.992093
$$122$$ 1730.17 1.28395
$$123$$ 0 0
$$124$$ −118.143 −0.0855613
$$125$$ −1286.79 −0.920751
$$126$$ 0 0
$$127$$ 596.093 0.416494 0.208247 0.978076i $$-0.433224\pi$$
0.208247 + 0.978076i $$0.433224\pi$$
$$128$$ 2559.15 1.76718
$$129$$ 0 0
$$130$$ 1116.26 0.753094
$$131$$ −121.819 −0.0812472 −0.0406236 0.999175i $$-0.512934\pi$$
−0.0406236 + 0.999175i $$0.512934\pi$$
$$132$$ 0 0
$$133$$ 1498.02 0.976652
$$134$$ −4389.38 −2.82973
$$135$$ 0 0
$$136$$ −462.251 −0.291454
$$137$$ 897.365 0.559614 0.279807 0.960056i $$-0.409730\pi$$
0.279807 + 0.960056i $$0.409730\pi$$
$$138$$ 0 0
$$139$$ −2113.61 −1.28974 −0.644871 0.764292i $$-0.723089\pi$$
−0.644871 + 0.764292i $$0.723089\pi$$
$$140$$ 4302.57 2.59738
$$141$$ 0 0
$$142$$ −2429.99 −1.43606
$$143$$ −65.0493 −0.0380398
$$144$$ 0 0
$$145$$ 3410.31 1.95318
$$146$$ −1629.07 −0.923442
$$147$$ 0 0
$$148$$ 4947.65 2.74793
$$149$$ −2580.76 −1.41895 −0.709476 0.704729i $$-0.751068\pi$$
−0.709476 + 0.704729i $$0.751068\pi$$
$$150$$ 0 0
$$151$$ 1342.77 0.723662 0.361831 0.932244i $$-0.382152\pi$$
0.361831 + 0.932244i $$0.382152\pi$$
$$152$$ −1559.32 −0.832091
$$153$$ 0 0
$$154$$ −395.861 −0.207139
$$155$$ −101.872 −0.0527906
$$156$$ 0 0
$$157$$ −2495.82 −1.26871 −0.634357 0.773041i $$-0.718735\pi$$
−0.634357 + 0.773041i $$0.718735\pi$$
$$158$$ 4455.60 2.24347
$$159$$ 0 0
$$160$$ 1676.59 0.828413
$$161$$ −2012.55 −0.985164
$$162$$ 0 0
$$163$$ 1961.58 0.942595 0.471297 0.881974i $$-0.343786\pi$$
0.471297 + 0.881974i $$0.343786\pi$$
$$164$$ −2687.72 −1.27973
$$165$$ 0 0
$$166$$ −6606.51 −3.08894
$$167$$ −2179.24 −1.00979 −0.504894 0.863182i $$-0.668468\pi$$
−0.504894 + 0.863182i $$0.668468\pi$$
$$168$$ 0 0
$$169$$ −1794.94 −0.816995
$$170$$ −946.383 −0.426966
$$171$$ 0 0
$$172$$ −1026.37 −0.454999
$$173$$ −3111.45 −1.36739 −0.683697 0.729766i $$-0.739629\pi$$
−0.683697 + 0.729766i $$0.739629\pi$$
$$174$$ 0 0
$$175$$ 444.713 0.192098
$$176$$ 53.3677 0.0228565
$$177$$ 0 0
$$178$$ −2270.86 −0.956226
$$179$$ −810.106 −0.338269 −0.169135 0.985593i $$-0.554097\pi$$
−0.169135 + 0.985593i $$0.554097\pi$$
$$180$$ 0 0
$$181$$ −3356.23 −1.37827 −0.689134 0.724634i $$-0.742009\pi$$
−0.689134 + 0.724634i $$0.742009\pi$$
$$182$$ 2446.77 0.996519
$$183$$ 0 0
$$184$$ 2094.92 0.839343
$$185$$ 4266.22 1.69545
$$186$$ 0 0
$$187$$ 55.1500 0.0215667
$$188$$ −326.353 −0.126605
$$189$$ 0 0
$$190$$ −3192.46 −1.21898
$$191$$ −1338.41 −0.507038 −0.253519 0.967330i $$-0.581588\pi$$
−0.253519 + 0.967330i $$0.581588\pi$$
$$192$$ 0 0
$$193$$ −227.465 −0.0848358 −0.0424179 0.999100i $$-0.513506\pi$$
−0.0424179 + 0.999100i $$0.513506\pi$$
$$194$$ 3201.15 1.18468
$$195$$ 0 0
$$196$$ 4690.40 1.70933
$$197$$ −815.549 −0.294952 −0.147476 0.989066i $$-0.547115\pi$$
−0.147476 + 0.989066i $$0.547115\pi$$
$$198$$ 0 0
$$199$$ 1866.90 0.665030 0.332515 0.943098i $$-0.392103\pi$$
0.332515 + 0.943098i $$0.392103\pi$$
$$200$$ −462.912 −0.163664
$$201$$ 0 0
$$202$$ 4039.52 1.40703
$$203$$ 7475.19 2.58451
$$204$$ 0 0
$$205$$ −2317.54 −0.789581
$$206$$ −8786.25 −2.97168
$$207$$ 0 0
$$208$$ −329.859 −0.109960
$$209$$ 186.039 0.0615721
$$210$$ 0 0
$$211$$ −1102.88 −0.359836 −0.179918 0.983682i $$-0.557583\pi$$
−0.179918 + 0.983682i $$0.557583\pi$$
$$212$$ −1441.94 −0.467135
$$213$$ 0 0
$$214$$ −153.288 −0.0489651
$$215$$ −885.008 −0.280731
$$216$$ 0 0
$$217$$ −223.297 −0.0698542
$$218$$ −2469.84 −0.767332
$$219$$ 0 0
$$220$$ 534.337 0.163750
$$221$$ −340.875 −0.103754
$$222$$ 0 0
$$223$$ −568.848 −0.170820 −0.0854100 0.996346i $$-0.527220\pi$$
−0.0854100 + 0.996346i $$0.527220\pi$$
$$224$$ 3674.98 1.09618
$$225$$ 0 0
$$226$$ 1937.14 0.570163
$$227$$ −2106.99 −0.616061 −0.308030 0.951377i $$-0.599670\pi$$
−0.308030 + 0.951377i $$0.599670\pi$$
$$228$$ 0 0
$$229$$ 4336.30 1.25131 0.625656 0.780099i $$-0.284831\pi$$
0.625656 + 0.780099i $$0.284831\pi$$
$$230$$ 4289.00 1.22960
$$231$$ 0 0
$$232$$ −7781.11 −2.20196
$$233$$ 4517.39 1.27014 0.635072 0.772453i $$-0.280970\pi$$
0.635072 + 0.772453i $$0.280970\pi$$
$$234$$ 0 0
$$235$$ −281.405 −0.0781143
$$236$$ −3446.43 −0.950609
$$237$$ 0 0
$$238$$ −2074.41 −0.564976
$$239$$ −5300.88 −1.43467 −0.717333 0.696731i $$-0.754637\pi$$
−0.717333 + 0.696731i $$0.754637\pi$$
$$240$$ 0 0
$$241$$ −1368.82 −0.365864 −0.182932 0.983126i $$-0.558559\pi$$
−0.182932 + 0.983126i $$0.558559\pi$$
$$242$$ 6168.32 1.63849
$$243$$ 0 0
$$244$$ −5119.05 −1.34309
$$245$$ 4044.40 1.05464
$$246$$ 0 0
$$247$$ −1149.88 −0.296216
$$248$$ 232.435 0.0595146
$$249$$ 0 0
$$250$$ 6010.96 1.52067
$$251$$ 5547.63 1.39507 0.697536 0.716549i $$-0.254280\pi$$
0.697536 + 0.716549i $$0.254280\pi$$
$$252$$ 0 0
$$253$$ −249.939 −0.0621088
$$254$$ −2784.52 −0.687860
$$255$$ 0 0
$$256$$ −5644.28 −1.37800
$$257$$ 193.949 0.0470748 0.0235374 0.999723i $$-0.492507\pi$$
0.0235374 + 0.999723i $$0.492507\pi$$
$$258$$ 0 0
$$259$$ 9351.29 2.24348
$$260$$ −3302.67 −0.787780
$$261$$ 0 0
$$262$$ 569.052 0.134184
$$263$$ 1345.63 0.315494 0.157747 0.987480i $$-0.449577\pi$$
0.157747 + 0.987480i $$0.449577\pi$$
$$264$$ 0 0
$$265$$ −1243.34 −0.288218
$$266$$ −6997.67 −1.61299
$$267$$ 0 0
$$268$$ 12986.8 2.96007
$$269$$ 3083.04 0.698797 0.349398 0.936974i $$-0.386386\pi$$
0.349398 + 0.936974i $$0.386386\pi$$
$$270$$ 0 0
$$271$$ −422.163 −0.0946294 −0.0473147 0.998880i $$-0.515066\pi$$
−0.0473147 + 0.998880i $$0.515066\pi$$
$$272$$ 279.661 0.0623416
$$273$$ 0 0
$$274$$ −4191.85 −0.924230
$$275$$ 55.2288 0.0121106
$$276$$ 0 0
$$277$$ −8260.00 −1.79168 −0.895840 0.444377i $$-0.853425\pi$$
−0.895840 + 0.444377i $$0.853425\pi$$
$$278$$ 9873.28 2.13007
$$279$$ 0 0
$$280$$ −8464.86 −1.80669
$$281$$ −3321.91 −0.705226 −0.352613 0.935769i $$-0.614707\pi$$
−0.352613 + 0.935769i $$0.614707\pi$$
$$282$$ 0 0
$$283$$ −7954.43 −1.67082 −0.835409 0.549629i $$-0.814769\pi$$
−0.835409 + 0.549629i $$0.814769\pi$$
$$284$$ 7189.61 1.50220
$$285$$ 0 0
$$286$$ 303.864 0.0628246
$$287$$ −5079.91 −1.04480
$$288$$ 0 0
$$289$$ 289.000 0.0588235
$$290$$ −15930.5 −3.22577
$$291$$ 0 0
$$292$$ 4819.92 0.965974
$$293$$ 1171.99 0.233681 0.116841 0.993151i $$-0.462723\pi$$
0.116841 + 0.993151i $$0.462723\pi$$
$$294$$ 0 0
$$295$$ −2971.76 −0.586517
$$296$$ −9733.98 −1.91141
$$297$$ 0 0
$$298$$ 12055.5 2.34347
$$299$$ 1544.84 0.298798
$$300$$ 0 0
$$301$$ −1939.88 −0.371472
$$302$$ −6272.46 −1.19516
$$303$$ 0 0
$$304$$ 943.387 0.177983
$$305$$ −4414.01 −0.828675
$$306$$ 0 0
$$307$$ 865.763 0.160950 0.0804751 0.996757i $$-0.474356\pi$$
0.0804751 + 0.996757i $$0.474356\pi$$
$$308$$ 1171.23 0.216679
$$309$$ 0 0
$$310$$ 475.872 0.0871862
$$311$$ −6994.83 −1.27537 −0.637685 0.770297i $$-0.720108\pi$$
−0.637685 + 0.770297i $$0.720108\pi$$
$$312$$ 0 0
$$313$$ 3442.33 0.621635 0.310818 0.950470i $$-0.399397\pi$$
0.310818 + 0.950470i $$0.399397\pi$$
$$314$$ 11658.7 2.09534
$$315$$ 0 0
$$316$$ −13182.8 −2.34680
$$317$$ −2066.15 −0.366078 −0.183039 0.983106i $$-0.558593\pi$$
−0.183039 + 0.983106i $$0.558593\pi$$
$$318$$ 0 0
$$319$$ 928.344 0.162938
$$320$$ −9400.22 −1.64215
$$321$$ 0 0
$$322$$ 9401.21 1.62705
$$323$$ 974.892 0.167939
$$324$$ 0 0
$$325$$ −341.362 −0.0582627
$$326$$ −9163.11 −1.55674
$$327$$ 0 0
$$328$$ 5287.80 0.890152
$$329$$ −616.823 −0.103363
$$330$$ 0 0
$$331$$ 9027.44 1.49907 0.749536 0.661964i $$-0.230277\pi$$
0.749536 + 0.661964i $$0.230277\pi$$
$$332$$ 19546.7 3.23121
$$333$$ 0 0
$$334$$ 10179.8 1.66771
$$335$$ 11198.2 1.82633
$$336$$ 0 0
$$337$$ 204.309 0.0330250 0.0165125 0.999864i $$-0.494744\pi$$
0.0165125 + 0.999864i $$0.494744\pi$$
$$338$$ 8384.67 1.34931
$$339$$ 0 0
$$340$$ 2800.06 0.446631
$$341$$ −27.7312 −0.00440390
$$342$$ 0 0
$$343$$ −94.8397 −0.0149296
$$344$$ 2019.27 0.316488
$$345$$ 0 0
$$346$$ 14534.5 2.25832
$$347$$ −143.063 −0.0221326 −0.0110663 0.999939i $$-0.503523\pi$$
−0.0110663 + 0.999939i $$0.503523\pi$$
$$348$$ 0 0
$$349$$ −3998.42 −0.613268 −0.306634 0.951828i $$-0.599203\pi$$
−0.306634 + 0.951828i $$0.599203\pi$$
$$350$$ −2077.38 −0.317259
$$351$$ 0 0
$$352$$ 456.396 0.0691079
$$353$$ 5809.57 0.875956 0.437978 0.898986i $$-0.355695\pi$$
0.437978 + 0.898986i $$0.355695\pi$$
$$354$$ 0 0
$$355$$ 6199.39 0.926844
$$356$$ 6718.79 1.00027
$$357$$ 0 0
$$358$$ 3784.24 0.558668
$$359$$ 4895.37 0.719687 0.359844 0.933013i $$-0.382830\pi$$
0.359844 + 0.933013i $$0.382830\pi$$
$$360$$ 0 0
$$361$$ −3570.37 −0.520538
$$362$$ 15677.9 2.27628
$$363$$ 0 0
$$364$$ −7239.24 −1.04242
$$365$$ 4156.08 0.595998
$$366$$ 0 0
$$367$$ 528.151 0.0751206 0.0375603 0.999294i $$-0.488041\pi$$
0.0375603 + 0.999294i $$0.488041\pi$$
$$368$$ −1267.42 −0.179535
$$369$$ 0 0
$$370$$ −19928.7 −2.80012
$$371$$ −2725.33 −0.381380
$$372$$ 0 0
$$373$$ −10113.5 −1.40390 −0.701950 0.712226i $$-0.747687\pi$$
−0.701950 + 0.712226i $$0.747687\pi$$
$$374$$ −257.621 −0.0356184
$$375$$ 0 0
$$376$$ 642.066 0.0880639
$$377$$ −5737.98 −0.783875
$$378$$ 0 0
$$379$$ 729.385 0.0988548 0.0494274 0.998778i $$-0.484260\pi$$
0.0494274 + 0.998778i $$0.484260\pi$$
$$380$$ 9445.52 1.27512
$$381$$ 0 0
$$382$$ 6252.12 0.837399
$$383$$ 1608.08 0.214540 0.107270 0.994230i $$-0.465789\pi$$
0.107270 + 0.994230i $$0.465789\pi$$
$$384$$ 0 0
$$385$$ 1009.92 0.133689
$$386$$ 1062.56 0.140111
$$387$$ 0 0
$$388$$ −9471.22 −1.23925
$$389$$ −9824.09 −1.28047 −0.640233 0.768181i $$-0.721162\pi$$
−0.640233 + 0.768181i $$0.721162\pi$$
$$390$$ 0 0
$$391$$ −1309.74 −0.169403
$$392$$ −9227.87 −1.18897
$$393$$ 0 0
$$394$$ 3809.66 0.487127
$$395$$ −11367.1 −1.44796
$$396$$ 0 0
$$397$$ 2876.88 0.363694 0.181847 0.983327i $$-0.441792\pi$$
0.181847 + 0.983327i $$0.441792\pi$$
$$398$$ −8720.82 −1.09833
$$399$$ 0 0
$$400$$ 280.061 0.0350076
$$401$$ 6515.91 0.811444 0.405722 0.913996i $$-0.367020\pi$$
0.405722 + 0.913996i $$0.367020\pi$$
$$402$$ 0 0
$$403$$ 171.403 0.0211866
$$404$$ −11951.7 −1.47183
$$405$$ 0 0
$$406$$ −34918.8 −4.26845
$$407$$ 1161.34 0.141438
$$408$$ 0 0
$$409$$ −8870.10 −1.07237 −0.536183 0.844101i $$-0.680134\pi$$
−0.536183 + 0.844101i $$0.680134\pi$$
$$410$$ 10825.9 1.30403
$$411$$ 0 0
$$412$$ 25995.9 3.10855
$$413$$ −6513.92 −0.776099
$$414$$ 0 0
$$415$$ 16854.5 1.99363
$$416$$ −2820.93 −0.332469
$$417$$ 0 0
$$418$$ −869.041 −0.101689
$$419$$ 1009.53 0.117706 0.0588531 0.998267i $$-0.481256\pi$$
0.0588531 + 0.998267i $$0.481256\pi$$
$$420$$ 0 0
$$421$$ −3253.60 −0.376652 −0.188326 0.982107i $$-0.560306\pi$$
−0.188326 + 0.982107i $$0.560306\pi$$
$$422$$ 5151.87 0.594287
$$423$$ 0 0
$$424$$ 2836.86 0.324930
$$425$$ 289.413 0.0330320
$$426$$ 0 0
$$427$$ −9675.25 −1.09653
$$428$$ 453.532 0.0512203
$$429$$ 0 0
$$430$$ 4134.13 0.463640
$$431$$ 2352.51 0.262915 0.131457 0.991322i $$-0.458034\pi$$
0.131457 + 0.991322i $$0.458034\pi$$
$$432$$ 0 0
$$433$$ −5860.51 −0.650434 −0.325217 0.945639i $$-0.605437\pi$$
−0.325217 + 0.945639i $$0.605437\pi$$
$$434$$ 1043.08 0.115368
$$435$$ 0 0
$$436$$ 7307.50 0.802674
$$437$$ −4418.20 −0.483641
$$438$$ 0 0
$$439$$ 2894.17 0.314650 0.157325 0.987547i $$-0.449713\pi$$
0.157325 + 0.987547i $$0.449713\pi$$
$$440$$ −1051.25 −0.113901
$$441$$ 0 0
$$442$$ 1592.32 0.171356
$$443$$ 8256.85 0.885541 0.442771 0.896635i $$-0.353996\pi$$
0.442771 + 0.896635i $$0.353996\pi$$
$$444$$ 0 0
$$445$$ 5793.42 0.617156
$$446$$ 2657.25 0.282118
$$447$$ 0 0
$$448$$ −20604.7 −2.17295
$$449$$ −15487.1 −1.62779 −0.813897 0.581009i $$-0.802658\pi$$
−0.813897 + 0.581009i $$0.802658\pi$$
$$450$$ 0 0
$$451$$ −630.874 −0.0658685
$$452$$ −5731.42 −0.596423
$$453$$ 0 0
$$454$$ 9842.35 1.01745
$$455$$ −6242.19 −0.643162
$$456$$ 0 0
$$457$$ −16055.6 −1.64343 −0.821716 0.569897i $$-0.806983\pi$$
−0.821716 + 0.569897i $$0.806983\pi$$
$$458$$ −20256.1 −2.06661
$$459$$ 0 0
$$460$$ −12689.8 −1.28623
$$461$$ −14064.0 −1.42088 −0.710440 0.703758i $$-0.751504\pi$$
−0.710440 + 0.703758i $$0.751504\pi$$
$$462$$ 0 0
$$463$$ −8071.30 −0.810162 −0.405081 0.914281i $$-0.632757\pi$$
−0.405081 + 0.914281i $$0.632757\pi$$
$$464$$ 4707.55 0.470997
$$465$$ 0 0
$$466$$ −21102.0 −2.09771
$$467$$ −8582.41 −0.850421 −0.425211 0.905094i $$-0.639800\pi$$
−0.425211 + 0.905094i $$0.639800\pi$$
$$468$$ 0 0
$$469$$ 24545.7 2.41667
$$470$$ 1314.53 0.129010
$$471$$ 0 0
$$472$$ 6780.49 0.661224
$$473$$ −240.914 −0.0234191
$$474$$ 0 0
$$475$$ 976.286 0.0943054
$$476$$ 6137.56 0.590997
$$477$$ 0 0
$$478$$ 24761.9 2.36942
$$479$$ 6320.96 0.602948 0.301474 0.953474i $$-0.402521\pi$$
0.301474 + 0.953474i $$0.402521\pi$$
$$480$$ 0 0
$$481$$ −7178.07 −0.680441
$$482$$ 6394.13 0.604242
$$483$$ 0 0
$$484$$ −18250.2 −1.71396
$$485$$ −8166.77 −0.764606
$$486$$ 0 0
$$487$$ 7336.47 0.682643 0.341321 0.939947i $$-0.389126\pi$$
0.341321 + 0.939947i $$0.389126\pi$$
$$488$$ 10071.2 0.934225
$$489$$ 0 0
$$490$$ −18892.6 −1.74179
$$491$$ −6672.53 −0.613294 −0.306647 0.951823i $$-0.599207\pi$$
−0.306647 + 0.951823i $$0.599207\pi$$
$$492$$ 0 0
$$493$$ 4864.76 0.444417
$$494$$ 5371.43 0.489215
$$495$$ 0 0
$$496$$ −140.623 −0.0127301
$$497$$ 13588.7 1.22643
$$498$$ 0 0
$$499$$ 17920.9 1.60772 0.803858 0.594821i $$-0.202777\pi$$
0.803858 + 0.594821i $$0.202777\pi$$
$$500$$ −17784.6 −1.59070
$$501$$ 0 0
$$502$$ −25914.6 −2.30403
$$503$$ 11325.3 1.00392 0.501959 0.864891i $$-0.332613\pi$$
0.501959 + 0.864891i $$0.332613\pi$$
$$504$$ 0 0
$$505$$ −10305.6 −0.908108
$$506$$ 1167.54 0.102576
$$507$$ 0 0
$$508$$ 8238.56 0.719541
$$509$$ 8313.78 0.723972 0.361986 0.932184i $$-0.382099\pi$$
0.361986 + 0.932184i $$0.382099\pi$$
$$510$$ 0 0
$$511$$ 9109.87 0.788644
$$512$$ 5892.85 0.508651
$$513$$ 0 0
$$514$$ −905.993 −0.0777464
$$515$$ 22415.5 1.91795
$$516$$ 0 0
$$517$$ −76.6033 −0.00651646
$$518$$ −43682.6 −3.70521
$$519$$ 0 0
$$520$$ 6497.65 0.547963
$$521$$ −5121.64 −0.430677 −0.215339 0.976539i $$-0.569086\pi$$
−0.215339 + 0.976539i $$0.569086\pi$$
$$522$$ 0 0
$$523$$ 13378.5 1.11855 0.559275 0.828982i $$-0.311080\pi$$
0.559275 + 0.828982i $$0.311080\pi$$
$$524$$ −1683.65 −0.140364
$$525$$ 0 0
$$526$$ −6285.81 −0.521054
$$527$$ −145.319 −0.0120117
$$528$$ 0 0
$$529$$ −6231.26 −0.512144
$$530$$ 5808.01 0.476007
$$531$$ 0 0
$$532$$ 20704.0 1.68728
$$533$$ 3899.35 0.316885
$$534$$ 0 0
$$535$$ 391.068 0.0316025
$$536$$ −25550.2 −2.05896
$$537$$ 0 0
$$538$$ −14401.8 −1.15410
$$539$$ 1100.95 0.0879804
$$540$$ 0 0
$$541$$ 9906.81 0.787296 0.393648 0.919261i $$-0.371213\pi$$
0.393648 + 0.919261i $$0.371213\pi$$
$$542$$ 1972.04 0.156285
$$543$$ 0 0
$$544$$ 2391.63 0.188493
$$545$$ 6301.05 0.495243
$$546$$ 0 0
$$547$$ 16399.6 1.28189 0.640947 0.767585i $$-0.278542\pi$$
0.640947 + 0.767585i $$0.278542\pi$$
$$548$$ 12402.4 0.966798
$$549$$ 0 0
$$550$$ −257.990 −0.0200013
$$551$$ 16410.4 1.26880
$$552$$ 0 0
$$553$$ −24916.1 −1.91598
$$554$$ 38584.8 2.95905
$$555$$ 0 0
$$556$$ −29212.0 −2.22818
$$557$$ −22044.3 −1.67692 −0.838461 0.544962i $$-0.816544\pi$$
−0.838461 + 0.544962i $$0.816544\pi$$
$$558$$ 0 0
$$559$$ 1489.06 0.112666
$$560$$ 5121.22 0.386448
$$561$$ 0 0
$$562$$ 15517.6 1.16472
$$563$$ −12048.8 −0.901947 −0.450973 0.892537i $$-0.648923\pi$$
−0.450973 + 0.892537i $$0.648923\pi$$
$$564$$ 0 0
$$565$$ −4942.04 −0.367988
$$566$$ 37157.4 2.75944
$$567$$ 0 0
$$568$$ −14144.8 −1.04490
$$569$$ 23785.4 1.75243 0.876217 0.481916i $$-0.160059\pi$$
0.876217 + 0.481916i $$0.160059\pi$$
$$570$$ 0 0
$$571$$ −10878.3 −0.797271 −0.398635 0.917110i $$-0.630516\pi$$
−0.398635 + 0.917110i $$0.630516\pi$$
$$572$$ −899.041 −0.0657182
$$573$$ 0 0
$$574$$ 23729.7 1.72554
$$575$$ −1311.62 −0.0951274
$$576$$ 0 0
$$577$$ 6315.86 0.455689 0.227845 0.973698i $$-0.426832\pi$$
0.227845 + 0.973698i $$0.426832\pi$$
$$578$$ −1350.00 −0.0971500
$$579$$ 0 0
$$580$$ 47133.7 3.37434
$$581$$ 36944.1 2.63804
$$582$$ 0 0
$$583$$ −338.459 −0.0240438
$$584$$ −9482.69 −0.671912
$$585$$ 0 0
$$586$$ −5474.72 −0.385936
$$587$$ −18192.1 −1.27916 −0.639581 0.768724i $$-0.720892\pi$$
−0.639581 + 0.768724i $$0.720892\pi$$
$$588$$ 0 0
$$589$$ −490.207 −0.0342931
$$590$$ 13882.0 0.968663
$$591$$ 0 0
$$592$$ 5889.03 0.408848
$$593$$ 9828.72 0.680636 0.340318 0.940310i $$-0.389465\pi$$
0.340318 + 0.940310i $$0.389465\pi$$
$$594$$ 0 0
$$595$$ 5292.25 0.364640
$$596$$ −35668.5 −2.45140
$$597$$ 0 0
$$598$$ −7216.40 −0.493479
$$599$$ −4662.57 −0.318043 −0.159021 0.987275i $$-0.550834\pi$$
−0.159021 + 0.987275i $$0.550834\pi$$
$$600$$ 0 0
$$601$$ 21658.6 1.47000 0.735001 0.678066i $$-0.237182\pi$$
0.735001 + 0.678066i $$0.237182\pi$$
$$602$$ 9061.76 0.613504
$$603$$ 0 0
$$604$$ 18558.3 1.25021
$$605$$ −15736.6 −1.05750
$$606$$ 0 0
$$607$$ 25764.7 1.72283 0.861415 0.507902i $$-0.169579\pi$$
0.861415 + 0.507902i $$0.169579\pi$$
$$608$$ 8067.76 0.538143
$$609$$ 0 0
$$610$$ 20619.1 1.36860
$$611$$ 473.475 0.0313498
$$612$$ 0 0
$$613$$ 16018.1 1.05541 0.527705 0.849428i $$-0.323053\pi$$
0.527705 + 0.849428i $$0.323053\pi$$
$$614$$ −4044.23 −0.265817
$$615$$ 0 0
$$616$$ −2304.28 −0.150718
$$617$$ −22250.3 −1.45180 −0.725902 0.687798i $$-0.758578\pi$$
−0.725902 + 0.687798i $$0.758578\pi$$
$$618$$ 0 0
$$619$$ −3765.95 −0.244534 −0.122267 0.992497i $$-0.539016\pi$$
−0.122267 + 0.992497i $$0.539016\pi$$
$$620$$ −1407.96 −0.0912018
$$621$$ 0 0
$$622$$ 32674.9 2.10634
$$623$$ 12698.8 0.816642
$$624$$ 0 0
$$625$$ −17463.2 −1.11765
$$626$$ −16080.1 −1.02666
$$627$$ 0 0
$$628$$ −34494.5 −2.19185
$$629$$ 6085.70 0.385775
$$630$$ 0 0
$$631$$ −20806.5 −1.31267 −0.656334 0.754470i $$-0.727894\pi$$
−0.656334 + 0.754470i $$0.727894\pi$$
$$632$$ 25935.7 1.63239
$$633$$ 0 0
$$634$$ 9651.60 0.604596
$$635$$ 7103.88 0.443951
$$636$$ 0 0
$$637$$ −6804.85 −0.423262
$$638$$ −4336.56 −0.269100
$$639$$ 0 0
$$640$$ 30498.4 1.88368
$$641$$ −2439.58 −0.150324 −0.0751620 0.997171i $$-0.523947\pi$$
−0.0751620 + 0.997171i $$0.523947\pi$$
$$642$$ 0 0
$$643$$ −19320.1 −1.18493 −0.592466 0.805595i $$-0.701846\pi$$
−0.592466 + 0.805595i $$0.701846\pi$$
$$644$$ −27815.4 −1.70199
$$645$$ 0 0
$$646$$ −4554.00 −0.277360
$$647$$ 14067.1 0.854766 0.427383 0.904071i $$-0.359436\pi$$
0.427383 + 0.904071i $$0.359436\pi$$
$$648$$ 0 0
$$649$$ −808.963 −0.0489285
$$650$$ 1594.60 0.0962238
$$651$$ 0 0
$$652$$ 27110.9 1.62844
$$653$$ −15893.7 −0.952478 −0.476239 0.879316i $$-0.658000\pi$$
−0.476239 + 0.879316i $$0.658000\pi$$
$$654$$ 0 0
$$655$$ −1451.77 −0.0866033
$$656$$ −3199.11 −0.190403
$$657$$ 0 0
$$658$$ 2881.36 0.170710
$$659$$ 9653.54 0.570635 0.285318 0.958433i $$-0.407901\pi$$
0.285318 + 0.958433i $$0.407901\pi$$
$$660$$ 0 0
$$661$$ 5389.06 0.317111 0.158555 0.987350i $$-0.449316\pi$$
0.158555 + 0.987350i $$0.449316\pi$$
$$662$$ −42169.7 −2.47579
$$663$$ 0 0
$$664$$ −38456.0 −2.24757
$$665$$ 17852.5 1.04104
$$666$$ 0 0
$$667$$ −22047.0 −1.27986
$$668$$ −30119.1 −1.74453
$$669$$ 0 0
$$670$$ −52309.9 −3.01628
$$671$$ −1201.57 −0.0691297
$$672$$ 0 0
$$673$$ −3032.18 −0.173673 −0.0868366 0.996223i $$-0.527676\pi$$
−0.0868366 + 0.996223i $$0.527676\pi$$
$$674$$ −954.386 −0.0545424
$$675$$ 0 0
$$676$$ −24807.7 −1.41145
$$677$$ 22029.2 1.25059 0.625295 0.780388i $$-0.284979\pi$$
0.625295 + 0.780388i $$0.284979\pi$$
$$678$$ 0 0
$$679$$ −17901.1 −1.01175
$$680$$ −5508.83 −0.310667
$$681$$ 0 0
$$682$$ 129.540 0.00727326
$$683$$ 9040.72 0.506491 0.253246 0.967402i $$-0.418502\pi$$
0.253246 + 0.967402i $$0.418502\pi$$
$$684$$ 0 0
$$685$$ 10694.3 0.596506
$$686$$ 443.023 0.0246570
$$687$$ 0 0
$$688$$ −1221.65 −0.0676964
$$689$$ 2091.97 0.115672
$$690$$ 0 0
$$691$$ −22863.5 −1.25871 −0.629355 0.777118i $$-0.716681\pi$$
−0.629355 + 0.777118i $$0.716681\pi$$
$$692$$ −43003.1 −2.36233
$$693$$ 0 0
$$694$$ 668.288 0.0365531
$$695$$ −25188.7 −1.37477
$$696$$ 0 0
$$697$$ −3305.94 −0.179658
$$698$$ 18677.8 1.01284
$$699$$ 0 0
$$700$$ 6146.34 0.331871
$$701$$ −1753.00 −0.0944507 −0.0472253 0.998884i $$-0.515038\pi$$
−0.0472253 + 0.998884i $$0.515038\pi$$
$$702$$ 0 0
$$703$$ 20529.1 1.10138
$$704$$ −2558.90 −0.136992
$$705$$ 0 0
$$706$$ −27138.2 −1.44668
$$707$$ −22589.3 −1.20164
$$708$$ 0 0
$$709$$ 11547.0 0.611645 0.305823 0.952089i $$-0.401069\pi$$
0.305823 + 0.952089i $$0.401069\pi$$
$$710$$ −28959.2 −1.53073
$$711$$ 0 0
$$712$$ −13218.5 −0.695765
$$713$$ 658.582 0.0345920
$$714$$ 0 0
$$715$$ −775.218 −0.0405476
$$716$$ −11196.4 −0.584399
$$717$$ 0 0
$$718$$ −22867.7 −1.18860
$$719$$ 10289.8 0.533720 0.266860 0.963735i $$-0.414014\pi$$
0.266860 + 0.963735i $$0.414014\pi$$
$$720$$ 0 0
$$721$$ 49133.4 2.53790
$$722$$ 16678.2 0.859695
$$723$$ 0 0
$$724$$ −46386.2 −2.38112
$$725$$ 4871.72 0.249560
$$726$$ 0 0
$$727$$ 2950.10 0.150499 0.0752497 0.997165i $$-0.476025\pi$$
0.0752497 + 0.997165i $$0.476025\pi$$
$$728$$ 14242.5 0.725083
$$729$$ 0 0
$$730$$ −19414.2 −0.984320
$$731$$ −1262.45 −0.0638762
$$732$$ 0 0
$$733$$ 24348.2 1.22691 0.613453 0.789731i $$-0.289780\pi$$
0.613453 + 0.789731i $$0.289780\pi$$
$$734$$ −2467.15 −0.124065
$$735$$ 0 0
$$736$$ −10838.8 −0.542833
$$737$$ 3048.33 0.152357
$$738$$ 0 0
$$739$$ −29233.5 −1.45517 −0.727585 0.686017i $$-0.759358\pi$$
−0.727585 + 0.686017i $$0.759358\pi$$
$$740$$ 58963.1 2.92909
$$741$$ 0 0
$$742$$ 12730.8 0.629868
$$743$$ −15340.6 −0.757457 −0.378729 0.925508i $$-0.623639\pi$$
−0.378729 + 0.925508i $$0.623639\pi$$
$$744$$ 0 0
$$745$$ −30755.9 −1.51250
$$746$$ 47242.9 2.31861
$$747$$ 0 0
$$748$$ 762.224 0.0372589
$$749$$ 857.197 0.0418175
$$750$$ 0 0
$$751$$ 39862.6 1.93689 0.968446 0.249223i $$-0.0801752\pi$$
0.968446 + 0.249223i $$0.0801752\pi$$
$$752$$ −388.448 −0.0188368
$$753$$ 0 0
$$754$$ 26803.7 1.29461
$$755$$ 16002.3 0.771369
$$756$$ 0 0
$$757$$ 26375.1 1.26634 0.633169 0.774013i $$-0.281754\pi$$
0.633169 + 0.774013i $$0.281754\pi$$
$$758$$ −3407.17 −0.163264
$$759$$ 0 0
$$760$$ −18583.1 −0.886946
$$761$$ −7848.63 −0.373867 −0.186933 0.982373i $$-0.559855\pi$$
−0.186933 + 0.982373i $$0.559855\pi$$
$$762$$ 0 0
$$763$$ 13811.5 0.655322
$$764$$ −18498.1 −0.875967
$$765$$ 0 0
$$766$$ −7511.78 −0.354323
$$767$$ 5000.10 0.235389
$$768$$ 0 0
$$769$$ 31818.9 1.49209 0.746046 0.665895i $$-0.231950\pi$$
0.746046 + 0.665895i $$0.231950\pi$$
$$770$$ −4717.63 −0.220794
$$771$$ 0 0
$$772$$ −3143.78 −0.146564
$$773$$ 29559.8 1.37541 0.687706 0.725990i $$-0.258618\pi$$
0.687706 + 0.725990i $$0.258618\pi$$
$$774$$ 0 0
$$775$$ −145.527 −0.00674512
$$776$$ 18633.6 0.861996
$$777$$ 0 0
$$778$$ 45891.2 2.11475
$$779$$ −11152.0 −0.512917
$$780$$ 0 0
$$781$$ 1687.58 0.0773193
$$782$$ 6118.19 0.279778
$$783$$ 0 0
$$784$$ 5582.84 0.254320
$$785$$ −29743.6 −1.35235
$$786$$ 0 0
$$787$$ −28038.7 −1.26998 −0.634989 0.772521i $$-0.718995\pi$$
−0.634989 + 0.772521i $$0.718995\pi$$
$$788$$ −11271.6 −0.509563
$$789$$ 0 0
$$790$$ 53099.2 2.39137
$$791$$ −10832.6 −0.486934
$$792$$ 0 0
$$793$$ 7426.75 0.332574
$$794$$ −13438.7 −0.600659
$$795$$ 0 0
$$796$$ 25802.3 1.14892
$$797$$ −5320.45 −0.236462 −0.118231 0.992986i $$-0.537722\pi$$
−0.118231 + 0.992986i $$0.537722\pi$$
$$798$$ 0 0
$$799$$ −401.421 −0.0177738
$$800$$ 2395.05 0.105847
$$801$$ 0 0
$$802$$ −30437.7 −1.34014
$$803$$ 1131.35 0.0497194
$$804$$ 0 0
$$805$$ −23984.4 −1.05011
$$806$$ −800.673 −0.0349907
$$807$$ 0 0
$$808$$ 23513.8 1.02378
$$809$$ −30934.0 −1.34435 −0.672177 0.740390i $$-0.734641\pi$$
−0.672177 + 0.740390i $$0.734641\pi$$
$$810$$ 0 0
$$811$$ −40364.5 −1.74771 −0.873854 0.486189i $$-0.838387\pi$$
−0.873854 + 0.486189i $$0.838387\pi$$
$$812$$ 103314. 4.46504
$$813$$ 0 0
$$814$$ −5424.94 −0.233592
$$815$$ 23376.9 1.00473
$$816$$ 0 0
$$817$$ −4258.66 −0.182364
$$818$$ 41434.8 1.77107
$$819$$ 0 0
$$820$$ −32030.6 −1.36409
$$821$$ −19799.7 −0.841672 −0.420836 0.907137i $$-0.638263\pi$$
−0.420836 + 0.907137i $$0.638263\pi$$
$$822$$ 0 0
$$823$$ 18756.4 0.794419 0.397210 0.917728i $$-0.369979\pi$$
0.397210 + 0.917728i $$0.369979\pi$$
$$824$$ −51144.1 −2.16225
$$825$$ 0 0
$$826$$ 30428.4 1.28177
$$827$$ −20958.0 −0.881234 −0.440617 0.897695i $$-0.645240\pi$$
−0.440617 + 0.897695i $$0.645240\pi$$
$$828$$ 0 0
$$829$$ −31320.3 −1.31218 −0.656091 0.754682i $$-0.727791\pi$$
−0.656091 + 0.754682i $$0.727791\pi$$
$$830$$ −78732.4 −3.29258
$$831$$ 0 0
$$832$$ 15816.2 0.659049
$$833$$ 5769.28 0.239968
$$834$$ 0 0
$$835$$ −25970.8 −1.07636
$$836$$ 2571.23 0.106373
$$837$$ 0 0
$$838$$ −4715.82 −0.194398
$$839$$ 30290.6 1.24642 0.623212 0.782053i $$-0.285827\pi$$
0.623212 + 0.782053i $$0.285827\pi$$
$$840$$ 0 0
$$841$$ 57499.9 2.35762
$$842$$ 15198.5 0.622060
$$843$$ 0 0
$$844$$ −15242.8 −0.621659
$$845$$ −21391.0 −0.870855
$$846$$ 0 0
$$847$$ −34493.7 −1.39931
$$848$$ −1716.29 −0.0695021
$$849$$ 0 0
$$850$$ −1351.93 −0.0545540
$$851$$ −27580.3 −1.11098
$$852$$ 0 0
$$853$$ −21111.8 −0.847425 −0.423712 0.905797i $$-0.639273\pi$$
−0.423712 + 0.905797i $$0.639273\pi$$
$$854$$ 45195.9 1.81097
$$855$$ 0 0
$$856$$ −892.277 −0.0356278
$$857$$ 39983.0 1.59369 0.796845 0.604184i $$-0.206501\pi$$
0.796845 + 0.604184i $$0.206501\pi$$
$$858$$ 0 0
$$859$$ −39503.3 −1.56907 −0.784537 0.620082i $$-0.787099\pi$$
−0.784537 + 0.620082i $$0.787099\pi$$
$$860$$ −12231.6 −0.484995
$$861$$ 0 0
$$862$$ −10989.2 −0.434217
$$863$$ −26019.9 −1.02634 −0.513168 0.858288i $$-0.671528\pi$$
−0.513168 + 0.858288i $$0.671528\pi$$
$$864$$ 0 0
$$865$$ −37080.4 −1.45754
$$866$$ 27376.1 1.07422
$$867$$ 0 0
$$868$$ −3086.17 −0.120681
$$869$$ −3094.33 −0.120791
$$870$$ 0 0
$$871$$ −18841.4 −0.732968
$$872$$ −14376.7 −0.558323
$$873$$ 0 0
$$874$$ 20638.7 0.798757
$$875$$ −33613.8 −1.29869
$$876$$ 0 0
$$877$$ 15038.3 0.579027 0.289514 0.957174i $$-0.406506\pi$$
0.289514 + 0.957174i $$0.406506\pi$$
$$878$$ −13519.5 −0.519660
$$879$$ 0 0
$$880$$ 636.004 0.0243633
$$881$$ 18334.3 0.701133 0.350567 0.936538i $$-0.385989\pi$$
0.350567 + 0.936538i $$0.385989\pi$$
$$882$$ 0 0
$$883$$ 26659.5 1.01604 0.508020 0.861345i $$-0.330378\pi$$
0.508020 + 0.861345i $$0.330378\pi$$
$$884$$ −4711.21 −0.179248
$$885$$ 0 0
$$886$$ −38570.1 −1.46251
$$887$$ −11473.7 −0.434327 −0.217163 0.976135i $$-0.569680\pi$$
−0.217163 + 0.976135i $$0.569680\pi$$
$$888$$ 0 0
$$889$$ 15571.3 0.587450
$$890$$ −27062.7 −1.01926
$$891$$ 0 0
$$892$$ −7862.01 −0.295111
$$893$$ −1354.12 −0.0507436
$$894$$ 0 0
$$895$$ −9654.36 −0.360569
$$896$$ 66850.7 2.49255
$$897$$ 0 0
$$898$$ 72344.6 2.68838
$$899$$ −2446.16 −0.0907498
$$900$$ 0 0
$$901$$ −1773.61 −0.0655799
$$902$$ 2946.99 0.108785
$$903$$ 0 0
$$904$$ 11276.0 0.414860
$$905$$ −39997.5 −1.46913
$$906$$ 0 0
$$907$$ −20361.6 −0.745421 −0.372710 0.927948i $$-0.621571\pi$$
−0.372710 + 0.927948i $$0.621571\pi$$
$$908$$ −29120.5 −1.06432
$$909$$ 0 0
$$910$$ 29159.1 1.06221
$$911$$ 19261.9 0.700523 0.350262 0.936652i $$-0.386093\pi$$
0.350262 + 0.936652i $$0.386093\pi$$
$$912$$ 0 0
$$913$$ 4588.09 0.166313
$$914$$ 75000.3 2.71421
$$915$$ 0 0
$$916$$ 59931.7 2.16179
$$917$$ −3182.18 −0.114596
$$918$$ 0 0
$$919$$ −21191.8 −0.760666 −0.380333 0.924850i $$-0.624191\pi$$
−0.380333 + 0.924850i $$0.624191\pi$$
$$920$$ 24965.9 0.894677
$$921$$ 0 0
$$922$$ 65697.0 2.34665
$$923$$ −10430.7 −0.371973
$$924$$ 0 0
$$925$$ 6094.40 0.216630
$$926$$ 37703.4 1.33802
$$927$$ 0 0
$$928$$ 40258.5 1.42409
$$929$$ −40815.0 −1.44144 −0.720719 0.693227i $$-0.756188\pi$$
−0.720719 + 0.693227i $$0.756188\pi$$
$$930$$ 0 0
$$931$$ 19461.7 0.685102
$$932$$ 62434.5 2.19432
$$933$$ 0 0
$$934$$ 40090.9 1.40451
$$935$$ 657.244 0.0229884
$$936$$ 0 0
$$937$$ −38439.1 −1.34018 −0.670092 0.742278i $$-0.733745\pi$$
−0.670092 + 0.742278i $$0.733745\pi$$
$$938$$ −114660. −3.99124
$$939$$ 0 0
$$940$$ −3889.28 −0.134951
$$941$$ 2244.08 0.0777415 0.0388708 0.999244i $$-0.487624\pi$$
0.0388708 + 0.999244i $$0.487624\pi$$
$$942$$ 0 0
$$943$$ 14982.5 0.517388
$$944$$ −4102.18 −0.141435
$$945$$ 0 0
$$946$$ 1125.38 0.0386778
$$947$$ 42289.0 1.45112 0.725559 0.688160i $$-0.241581\pi$$
0.725559 + 0.688160i $$0.241581\pi$$
$$948$$ 0 0
$$949$$ −6992.76 −0.239193
$$950$$ −4560.51 −0.155750
$$951$$ 0 0
$$952$$ −12075.0 −0.411085
$$953$$ −37426.2 −1.27214 −0.636072 0.771629i $$-0.719442\pi$$
−0.636072 + 0.771629i $$0.719442\pi$$
$$954$$ 0 0
$$955$$ −15950.4 −0.540464
$$956$$ −73263.0 −2.47855
$$957$$ 0 0
$$958$$ −29527.0 −0.995799
$$959$$ 23441.2 0.789317
$$960$$ 0 0
$$961$$ −29717.9 −0.997547
$$962$$ 33530.8 1.12378
$$963$$ 0 0
$$964$$ −18918.3 −0.632072
$$965$$ −2710.80 −0.0904286
$$966$$ 0 0
$$967$$ 1088.56 0.0362003 0.0181001 0.999836i $$-0.494238\pi$$
0.0181001 + 0.999836i $$0.494238\pi$$
$$968$$ 35905.4 1.19219
$$969$$ 0 0
$$970$$ 38149.3 1.26278
$$971$$ −39506.5 −1.30569 −0.652845 0.757491i $$-0.726425\pi$$
−0.652845 + 0.757491i $$0.726425\pi$$
$$972$$ 0 0
$$973$$ −55212.1 −1.81914
$$974$$ −34270.8 −1.12742
$$975$$ 0 0
$$976$$ −6093.05 −0.199830
$$977$$ 43326.8 1.41878 0.709389 0.704817i $$-0.248971\pi$$
0.709389 + 0.704817i $$0.248971\pi$$
$$978$$ 0 0
$$979$$ 1577.07 0.0514845
$$980$$ 55897.3 1.82202
$$981$$ 0 0
$$982$$ 31169.3 1.01288
$$983$$ −10664.1 −0.346014 −0.173007 0.984921i $$-0.555348\pi$$
−0.173007 + 0.984921i $$0.555348\pi$$
$$984$$ 0 0
$$985$$ −9719.22 −0.314396
$$986$$ −22724.7 −0.733977
$$987$$ 0 0
$$988$$ −15892.4 −0.511747
$$989$$ 5721.42 0.183954
$$990$$ 0 0
$$991$$ 15461.4 0.495609 0.247804 0.968810i $$-0.420291\pi$$
0.247804 + 0.968810i $$0.420291\pi$$
$$992$$ −1202.59 −0.0384902
$$993$$ 0 0
$$994$$ −63476.7 −2.02551
$$995$$ 22248.6 0.708871
$$996$$ 0 0
$$997$$ 46061.1 1.46316 0.731579 0.681756i $$-0.238784\pi$$
0.731579 + 0.681756i $$0.238784\pi$$
$$998$$ −83713.7 −2.65522
$$999$$ 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 153.4.a.g.1.1 3
3.2 odd 2 17.4.a.b.1.3 3
4.3 odd 2 2448.4.a.bi.1.3 3
12.11 even 2 272.4.a.h.1.3 3
15.2 even 4 425.4.b.f.324.5 6
15.8 even 4 425.4.b.f.324.2 6
15.14 odd 2 425.4.a.g.1.1 3
21.20 even 2 833.4.a.d.1.3 3
24.5 odd 2 1088.4.a.v.1.3 3
24.11 even 2 1088.4.a.x.1.1 3
33.32 even 2 2057.4.a.e.1.1 3
51.38 odd 4 289.4.b.b.288.2 6
51.47 odd 4 289.4.b.b.288.1 6
51.50 odd 2 289.4.a.b.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.b.1.3 3 3.2 odd 2
153.4.a.g.1.1 3 1.1 even 1 trivial
272.4.a.h.1.3 3 12.11 even 2
289.4.a.b.1.3 3 51.50 odd 2
289.4.b.b.288.1 6 51.47 odd 4
289.4.b.b.288.2 6 51.38 odd 4
425.4.a.g.1.1 3 15.14 odd 2
425.4.b.f.324.2 6 15.8 even 4
425.4.b.f.324.5 6 15.2 even 4
833.4.a.d.1.3 3 21.20 even 2
1088.4.a.v.1.3 3 24.5 odd 2
1088.4.a.x.1.1 3 24.11 even 2
2057.4.a.e.1.1 3 33.32 even 2
2448.4.a.bi.1.3 3 4.3 odd 2