Properties

 Label 1521.4.a.ba.1.1 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$89.7419051187$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.1520092.1 Defining polynomial: $$x^{4} - x^{3} - 40x^{2} - 9x + 81$$ x^4 - x^3 - 40*x^2 - 9*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 117) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Root $$1.32145$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-5.48928 q^{2} +22.1322 q^{4} +14.0859 q^{5} -24.2643 q^{7} -77.5754 q^{8} +O(q^{10})$$ $$q-5.48928 q^{2} +22.1322 q^{4} +14.0859 q^{5} -24.2643 q^{7} -77.5754 q^{8} -77.3217 q^{10} +3.10739 q^{11} +133.194 q^{14} +248.776 q^{16} -43.9142 q^{17} +85.8504 q^{19} +311.753 q^{20} -17.0574 q^{22} -203.829 q^{23} +73.4140 q^{25} -537.022 q^{28} +31.0739 q^{29} +135.736 q^{31} -744.995 q^{32} +241.057 q^{34} -341.786 q^{35} -290.229 q^{37} -471.256 q^{38} -1092.72 q^{40} -148.731 q^{41} +281.057 q^{43} +68.7734 q^{44} +1118.87 q^{46} -225.991 q^{47} +245.758 q^{49} -402.990 q^{50} -172.755 q^{53} +43.7706 q^{55} +1882.32 q^{56} -170.574 q^{58} -41.2175 q^{59} +499.815 q^{61} -745.091 q^{62} +2099.28 q^{64} -503.506 q^{67} -971.917 q^{68} +1876.16 q^{70} +946.442 q^{71} +1115.97 q^{73} +1593.15 q^{74} +1900.05 q^{76} -75.3989 q^{77} +674.803 q^{79} +3504.24 q^{80} +816.424 q^{82} +59.4512 q^{83} -618.574 q^{85} -1542.80 q^{86} -241.057 q^{88} -1218.41 q^{89} -4511.17 q^{92} +1240.53 q^{94} +1209.28 q^{95} +879.746 q^{97} -1349.03 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 58 q^{4} - 36 q^{7}+O(q^{10})$$ 4 * q + 58 * q^4 - 36 * q^7 $$4 q + 58 q^{4} - 36 q^{7} - 4 q^{10} + 354 q^{16} - 84 q^{19} + 176 q^{22} + 660 q^{25} - 988 q^{28} + 604 q^{31} + 720 q^{34} - 184 q^{37} - 2356 q^{40} + 880 q^{43} + 2888 q^{46} - 116 q^{49} + 1152 q^{55} + 1760 q^{58} + 656 q^{61} + 3482 q^{64} - 3052 q^{67} + 4696 q^{70} + 312 q^{73} + 2044 q^{76} - 720 q^{79} + 396 q^{82} - 32 q^{85} - 720 q^{88} + 4840 q^{94} + 344 q^{97}+O(q^{100})$$ 4 * q + 58 * q^4 - 36 * q^7 - 4 * q^10 + 354 * q^16 - 84 * q^19 + 176 * q^22 + 660 * q^25 - 988 * q^28 + 604 * q^31 + 720 * q^34 - 184 * q^37 - 2356 * q^40 + 880 * q^43 + 2888 * q^46 - 116 * q^49 + 1152 * q^55 + 1760 * q^58 + 656 * q^61 + 3482 * q^64 - 3052 * q^67 + 4696 * q^70 + 312 * q^73 + 2044 * q^76 - 720 * q^79 + 396 * q^82 - 32 * q^85 - 720 * q^88 + 4840 * q^94 + 344 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −5.48928 −1.94075 −0.970376 0.241598i $$-0.922328\pi$$
−0.970376 + 0.241598i $$0.922328\pi$$
$$3$$ 0 0
$$4$$ 22.1322 2.76652
$$5$$ 14.0859 1.25989 0.629943 0.776642i $$-0.283078\pi$$
0.629943 + 0.776642i $$0.283078\pi$$
$$6$$ 0 0
$$7$$ −24.2643 −1.31015 −0.655076 0.755563i $$-0.727363\pi$$
−0.655076 + 0.755563i $$0.727363\pi$$
$$8$$ −77.5754 −3.42838
$$9$$ 0 0
$$10$$ −77.3217 −2.44513
$$11$$ 3.10739 0.0851741 0.0425870 0.999093i $$-0.486440\pi$$
0.0425870 + 0.999093i $$0.486440\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 133.194 2.54268
$$15$$ 0 0
$$16$$ 248.776 3.88712
$$17$$ −43.9142 −0.626515 −0.313258 0.949668i $$-0.601420\pi$$
−0.313258 + 0.949668i $$0.601420\pi$$
$$18$$ 0 0
$$19$$ 85.8504 1.03660 0.518301 0.855198i $$-0.326565\pi$$
0.518301 + 0.855198i $$0.326565\pi$$
$$20$$ 311.753 3.48550
$$21$$ 0 0
$$22$$ −17.0574 −0.165302
$$23$$ −203.829 −1.84788 −0.923940 0.382537i $$-0.875050\pi$$
−0.923940 + 0.382537i $$0.875050\pi$$
$$24$$ 0 0
$$25$$ 73.4140 0.587312
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −537.022 −3.62456
$$29$$ 31.0739 0.198975 0.0994877 0.995039i $$-0.468280\pi$$
0.0994877 + 0.995039i $$0.468280\pi$$
$$30$$ 0 0
$$31$$ 135.736 0.786414 0.393207 0.919450i $$-0.371366\pi$$
0.393207 + 0.919450i $$0.371366\pi$$
$$32$$ −744.995 −4.11555
$$33$$ 0 0
$$34$$ 241.057 1.21591
$$35$$ −341.786 −1.65064
$$36$$ 0 0
$$37$$ −290.229 −1.28955 −0.644776 0.764372i $$-0.723049\pi$$
−0.644776 + 0.764372i $$0.723049\pi$$
$$38$$ −471.256 −2.01179
$$39$$ 0 0
$$40$$ −1092.72 −4.31937
$$41$$ −148.731 −0.566532 −0.283266 0.959041i $$-0.591418\pi$$
−0.283266 + 0.959041i $$0.591418\pi$$
$$42$$ 0 0
$$43$$ 281.057 0.996764 0.498382 0.866958i $$-0.333928\pi$$
0.498382 + 0.866958i $$0.333928\pi$$
$$44$$ 68.7734 0.235636
$$45$$ 0 0
$$46$$ 1118.87 3.58628
$$47$$ −225.991 −0.701366 −0.350683 0.936494i $$-0.614051\pi$$
−0.350683 + 0.936494i $$0.614051\pi$$
$$48$$ 0 0
$$49$$ 245.758 0.716496
$$50$$ −402.990 −1.13983
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −172.755 −0.447730 −0.223865 0.974620i $$-0.571868\pi$$
−0.223865 + 0.974620i $$0.571868\pi$$
$$54$$ 0 0
$$55$$ 43.7706 0.107310
$$56$$ 1882.32 4.49170
$$57$$ 0 0
$$58$$ −170.574 −0.386162
$$59$$ −41.2175 −0.0909503 −0.0454751 0.998965i $$-0.514480\pi$$
−0.0454751 + 0.998965i $$0.514480\pi$$
$$60$$ 0 0
$$61$$ 499.815 1.04910 0.524548 0.851381i $$-0.324234\pi$$
0.524548 + 0.851381i $$0.324234\pi$$
$$62$$ −745.091 −1.52624
$$63$$ 0 0
$$64$$ 2099.28 4.10015
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −503.506 −0.918106 −0.459053 0.888409i $$-0.651811\pi$$
−0.459053 + 0.888409i $$0.651811\pi$$
$$68$$ −971.917 −1.73327
$$69$$ 0 0
$$70$$ 1876.16 3.20349
$$71$$ 946.442 1.58200 0.791000 0.611816i $$-0.209561\pi$$
0.791000 + 0.611816i $$0.209561\pi$$
$$72$$ 0 0
$$73$$ 1115.97 1.78925 0.894623 0.446821i $$-0.147444\pi$$
0.894623 + 0.446821i $$0.147444\pi$$
$$74$$ 1593.15 2.50270
$$75$$ 0 0
$$76$$ 1900.05 2.86778
$$77$$ −75.3989 −0.111591
$$78$$ 0 0
$$79$$ 674.803 0.961029 0.480514 0.876987i $$-0.340450\pi$$
0.480514 + 0.876987i $$0.340450\pi$$
$$80$$ 3504.24 4.89732
$$81$$ 0 0
$$82$$ 816.424 1.09950
$$83$$ 59.4512 0.0786219 0.0393109 0.999227i $$-0.487484\pi$$
0.0393109 + 0.999227i $$0.487484\pi$$
$$84$$ 0 0
$$85$$ −618.574 −0.789338
$$86$$ −1542.80 −1.93447
$$87$$ 0 0
$$88$$ −241.057 −0.292009
$$89$$ −1218.41 −1.45114 −0.725571 0.688147i $$-0.758424\pi$$
−0.725571 + 0.688147i $$0.758424\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −4511.17 −5.11220
$$93$$ 0 0
$$94$$ 1240.53 1.36118
$$95$$ 1209.28 1.30600
$$96$$ 0 0
$$97$$ 879.746 0.920872 0.460436 0.887693i $$-0.347693\pi$$
0.460436 + 0.887693i $$0.347693\pi$$
$$98$$ −1349.03 −1.39054
$$99$$ 0 0
$$100$$ 1624.81 1.62481
$$101$$ 255.628 0.251841 0.125920 0.992040i $$-0.459812\pi$$
0.125920 + 0.992040i $$0.459812\pi$$
$$102$$ 0 0
$$103$$ 176.369 0.168720 0.0843600 0.996435i $$-0.473115\pi$$
0.0843600 + 0.996435i $$0.473115\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 948.299 0.868934
$$107$$ 816.164 0.737397 0.368699 0.929549i $$-0.379803\pi$$
0.368699 + 0.929549i $$0.379803\pi$$
$$108$$ 0 0
$$109$$ −174.369 −0.153225 −0.0766125 0.997061i $$-0.524410\pi$$
−0.0766125 + 0.997061i $$0.524410\pi$$
$$110$$ −240.269 −0.208261
$$111$$ 0 0
$$112$$ −6036.37 −5.09271
$$113$$ 199.667 0.166222 0.0831112 0.996540i $$-0.473514\pi$$
0.0831112 + 0.996540i $$0.473514\pi$$
$$114$$ 0 0
$$115$$ −2871.12 −2.32812
$$116$$ 687.734 0.550470
$$117$$ 0 0
$$118$$ 226.254 0.176512
$$119$$ 1065.55 0.820830
$$120$$ 0 0
$$121$$ −1321.34 −0.992745
$$122$$ −2743.63 −2.03603
$$123$$ 0 0
$$124$$ 3004.12 2.17563
$$125$$ −726.638 −0.519940
$$126$$ 0 0
$$127$$ 407.840 0.284961 0.142480 0.989798i $$-0.454492\pi$$
0.142480 + 0.989798i $$0.454492\pi$$
$$128$$ −5563.57 −3.84183
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2177.50 1.45229 0.726143 0.687544i $$-0.241311\pi$$
0.726143 + 0.687544i $$0.241311\pi$$
$$132$$ 0 0
$$133$$ −2083.10 −1.35810
$$134$$ 2763.89 1.78182
$$135$$ 0 0
$$136$$ 3406.66 2.14793
$$137$$ 723.750 0.451344 0.225672 0.974203i $$-0.427542\pi$$
0.225672 + 0.974203i $$0.427542\pi$$
$$138$$ 0 0
$$139$$ −1520.85 −0.928033 −0.464017 0.885826i $$-0.653592\pi$$
−0.464017 + 0.885826i $$0.653592\pi$$
$$140$$ −7564.47 −4.56653
$$141$$ 0 0
$$142$$ −5195.28 −3.07027
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 437.706 0.250686
$$146$$ −6125.90 −3.47248
$$147$$ 0 0
$$148$$ −6423.41 −3.56757
$$149$$ −2027.51 −1.11477 −0.557384 0.830255i $$-0.688195\pi$$
−0.557384 + 0.830255i $$0.688195\pi$$
$$150$$ 0 0
$$151$$ 11.8952 0.00641071 0.00320535 0.999995i $$-0.498980\pi$$
0.00320535 + 0.999995i $$0.498980\pi$$
$$152$$ −6659.88 −3.55386
$$153$$ 0 0
$$154$$ 413.885 0.216570
$$155$$ 1911.97 0.990792
$$156$$ 0 0
$$157$$ 1369.10 0.695963 0.347982 0.937501i $$-0.386867\pi$$
0.347982 + 0.937501i $$0.386867\pi$$
$$158$$ −3704.18 −1.86512
$$159$$ 0 0
$$160$$ −10494.0 −5.18513
$$161$$ 4945.77 2.42100
$$162$$ 0 0
$$163$$ −1650.54 −0.793130 −0.396565 0.918007i $$-0.629798\pi$$
−0.396565 + 0.918007i $$0.629798\pi$$
$$164$$ −3291.73 −1.56732
$$165$$ 0 0
$$166$$ −326.344 −0.152586
$$167$$ 872.686 0.404374 0.202187 0.979347i $$-0.435195\pi$$
0.202187 + 0.979347i $$0.435195\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 3395.52 1.53191
$$171$$ 0 0
$$172$$ 6220.41 2.75757
$$173$$ −732.880 −0.322080 −0.161040 0.986948i $$-0.551485\pi$$
−0.161040 + 0.986948i $$0.551485\pi$$
$$174$$ 0 0
$$175$$ −1781.34 −0.769467
$$176$$ 773.044 0.331082
$$177$$ 0 0
$$178$$ 6688.21 2.81631
$$179$$ 1530.81 0.639207 0.319604 0.947551i $$-0.396450\pi$$
0.319604 + 0.947551i $$0.396450\pi$$
$$180$$ 0 0
$$181$$ −1198.21 −0.492056 −0.246028 0.969263i $$-0.579126\pi$$
−0.246028 + 0.969263i $$0.579126\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 15812.1 6.33524
$$185$$ −4088.16 −1.62469
$$186$$ 0 0
$$187$$ −136.459 −0.0533629
$$188$$ −5001.68 −1.94034
$$189$$ 0 0
$$190$$ −6638.09 −2.53462
$$191$$ −2465.44 −0.933994 −0.466997 0.884259i $$-0.654664\pi$$
−0.466997 + 0.884259i $$0.654664\pi$$
$$192$$ 0 0
$$193$$ 4588.52 1.71134 0.855671 0.517520i $$-0.173145\pi$$
0.855671 + 0.517520i $$0.173145\pi$$
$$194$$ −4829.17 −1.78719
$$195$$ 0 0
$$196$$ 5439.16 1.98220
$$197$$ 3862.82 1.39703 0.698514 0.715596i $$-0.253845\pi$$
0.698514 + 0.715596i $$0.253845\pi$$
$$198$$ 0 0
$$199$$ 1434.59 0.511033 0.255517 0.966805i $$-0.417754\pi$$
0.255517 + 0.966805i $$0.417754\pi$$
$$200$$ −5695.12 −2.01353
$$201$$ 0 0
$$202$$ −1403.21 −0.488761
$$203$$ −753.989 −0.260688
$$204$$ 0 0
$$205$$ −2095.01 −0.713766
$$206$$ −968.139 −0.327444
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 266.771 0.0882915
$$210$$ 0 0
$$211$$ −1577.24 −0.514604 −0.257302 0.966331i $$-0.582834\pi$$
−0.257302 + 0.966331i $$0.582834\pi$$
$$212$$ −3823.44 −1.23866
$$213$$ 0 0
$$214$$ −4480.15 −1.43111
$$215$$ 3958.96 1.25581
$$216$$ 0 0
$$217$$ −3293.54 −1.03032
$$218$$ 957.161 0.297372
$$219$$ 0 0
$$220$$ 968.738 0.296874
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 3123.69 0.938016 0.469008 0.883194i $$-0.344612\pi$$
0.469008 + 0.883194i $$0.344612\pi$$
$$224$$ 18076.8 5.39200
$$225$$ 0 0
$$226$$ −1096.03 −0.322597
$$227$$ 3006.22 0.878987 0.439493 0.898246i $$-0.355158\pi$$
0.439493 + 0.898246i $$0.355158\pi$$
$$228$$ 0 0
$$229$$ 2344.43 0.676526 0.338263 0.941052i $$-0.390161\pi$$
0.338263 + 0.941052i $$0.390161\pi$$
$$230$$ 15760.4 4.51830
$$231$$ 0 0
$$232$$ −2410.57 −0.682163
$$233$$ 5913.50 1.66269 0.831344 0.555759i $$-0.187572\pi$$
0.831344 + 0.555759i $$0.187572\pi$$
$$234$$ 0 0
$$235$$ −3183.30 −0.883641
$$236$$ −912.233 −0.251616
$$237$$ 0 0
$$238$$ −5849.10 −1.59303
$$239$$ 250.440 0.0677807 0.0338904 0.999426i $$-0.489210\pi$$
0.0338904 + 0.999426i $$0.489210\pi$$
$$240$$ 0 0
$$241$$ −1112.43 −0.297337 −0.148668 0.988887i $$-0.547499\pi$$
−0.148668 + 0.988887i $$0.547499\pi$$
$$242$$ 7253.22 1.92667
$$243$$ 0 0
$$244$$ 11062.0 2.90234
$$245$$ 3461.74 0.902703
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −10529.7 −2.69613
$$249$$ 0 0
$$250$$ 3988.72 1.00907
$$251$$ 3496.19 0.879194 0.439597 0.898195i $$-0.355121\pi$$
0.439597 + 0.898195i $$0.355121\pi$$
$$252$$ 0 0
$$253$$ −633.376 −0.157391
$$254$$ −2238.75 −0.553038
$$255$$ 0 0
$$256$$ 13745.7 3.35589
$$257$$ 4644.20 1.12723 0.563614 0.826039i $$-0.309411\pi$$
0.563614 + 0.826039i $$0.309411\pi$$
$$258$$ 0 0
$$259$$ 7042.22 1.68951
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −11952.9 −2.81853
$$263$$ −4930.41 −1.15598 −0.577989 0.816045i $$-0.696162\pi$$
−0.577989 + 0.816045i $$0.696162\pi$$
$$264$$ 0 0
$$265$$ −2433.42 −0.564089
$$266$$ 11434.7 2.63574
$$267$$ 0 0
$$268$$ −11143.7 −2.53996
$$269$$ −1276.44 −0.289316 −0.144658 0.989482i $$-0.546208\pi$$
−0.144658 + 0.989482i $$0.546208\pi$$
$$270$$ 0 0
$$271$$ 2519.91 0.564848 0.282424 0.959290i $$-0.408861\pi$$
0.282424 + 0.959290i $$0.408861\pi$$
$$272$$ −10924.8 −2.43534
$$273$$ 0 0
$$274$$ −3972.86 −0.875947
$$275$$ 228.126 0.0500237
$$276$$ 0 0
$$277$$ −2462.00 −0.534033 −0.267017 0.963692i $$-0.586038\pi$$
−0.267017 + 0.963692i $$0.586038\pi$$
$$278$$ 8348.36 1.80108
$$279$$ 0 0
$$280$$ 26514.2 5.65902
$$281$$ −5058.09 −1.07381 −0.536904 0.843643i $$-0.680406\pi$$
−0.536904 + 0.843643i $$0.680406\pi$$
$$282$$ 0 0
$$283$$ 3078.55 0.646647 0.323323 0.946289i $$-0.395200\pi$$
0.323323 + 0.946289i $$0.395200\pi$$
$$284$$ 20946.8 4.37664
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 3608.85 0.742243
$$288$$ 0 0
$$289$$ −2984.54 −0.607478
$$290$$ −2402.69 −0.486520
$$291$$ 0 0
$$292$$ 24698.9 4.94999
$$293$$ −8626.31 −1.71998 −0.859990 0.510310i $$-0.829531\pi$$
−0.859990 + 0.510310i $$0.829531\pi$$
$$294$$ 0 0
$$295$$ −580.588 −0.114587
$$296$$ 22514.7 4.42107
$$297$$ 0 0
$$298$$ 11129.6 2.16349
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −6819.67 −1.30591
$$302$$ −65.2960 −0.0124416
$$303$$ 0 0
$$304$$ 21357.5 4.02939
$$305$$ 7040.37 1.32174
$$306$$ 0 0
$$307$$ −545.591 −0.101428 −0.0507142 0.998713i $$-0.516150\pi$$
−0.0507142 + 0.998713i $$0.516150\pi$$
$$308$$ −1668.74 −0.308719
$$309$$ 0 0
$$310$$ −10495.3 −1.92288
$$311$$ 3024.25 0.551413 0.275707 0.961242i $$-0.411088\pi$$
0.275707 + 0.961242i $$0.411088\pi$$
$$312$$ 0 0
$$313$$ −10082.4 −1.82074 −0.910370 0.413796i $$-0.864203\pi$$
−0.910370 + 0.413796i $$0.864203\pi$$
$$314$$ −7515.38 −1.35069
$$315$$ 0 0
$$316$$ 14934.9 2.65871
$$317$$ 8498.67 1.50578 0.752891 0.658145i $$-0.228659\pi$$
0.752891 + 0.658145i $$0.228659\pi$$
$$318$$ 0 0
$$319$$ 96.5590 0.0169475
$$320$$ 29570.3 5.16573
$$321$$ 0 0
$$322$$ −27148.7 −4.69857
$$323$$ −3770.05 −0.649447
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 9060.26 1.53927
$$327$$ 0 0
$$328$$ 11537.8 1.94229
$$329$$ 5483.53 0.918896
$$330$$ 0 0
$$331$$ −9969.94 −1.65558 −0.827791 0.561037i $$-0.810403\pi$$
−0.827791 + 0.561037i $$0.810403\pi$$
$$332$$ 1315.78 0.217509
$$333$$ 0 0
$$334$$ −4790.41 −0.784790
$$335$$ −7092.36 −1.15671
$$336$$ 0 0
$$337$$ −3231.96 −0.522422 −0.261211 0.965282i $$-0.584122\pi$$
−0.261211 + 0.965282i $$0.584122\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −13690.4 −2.18372
$$341$$ 421.784 0.0669821
$$342$$ 0 0
$$343$$ 2359.51 0.371433
$$344$$ −21803.1 −3.41729
$$345$$ 0 0
$$346$$ 4022.98 0.625078
$$347$$ 5989.75 0.926647 0.463323 0.886189i $$-0.346657\pi$$
0.463323 + 0.886189i $$0.346657\pi$$
$$348$$ 0 0
$$349$$ 9974.91 1.52993 0.764964 0.644074i $$-0.222757\pi$$
0.764964 + 0.644074i $$0.222757\pi$$
$$350$$ 9778.28 1.49335
$$351$$ 0 0
$$352$$ −2314.99 −0.350538
$$353$$ −10834.0 −1.63352 −0.816761 0.576976i $$-0.804233\pi$$
−0.816761 + 0.576976i $$0.804233\pi$$
$$354$$ 0 0
$$355$$ 13331.5 1.99314
$$356$$ −26966.2 −4.01462
$$357$$ 0 0
$$358$$ −8403.04 −1.24054
$$359$$ −4315.76 −0.634477 −0.317239 0.948346i $$-0.602756\pi$$
−0.317239 + 0.948346i $$0.602756\pi$$
$$360$$ 0 0
$$361$$ 511.285 0.0745422
$$362$$ 6577.31 0.954960
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 15719.6 2.25425
$$366$$ 0 0
$$367$$ 10505.3 1.49420 0.747101 0.664711i $$-0.231445\pi$$
0.747101 + 0.664711i $$0.231445\pi$$
$$368$$ −50707.6 −7.18293
$$369$$ 0 0
$$370$$ 22441.0 3.15312
$$371$$ 4191.78 0.586594
$$372$$ 0 0
$$373$$ 5869.44 0.814767 0.407384 0.913257i $$-0.366441\pi$$
0.407384 + 0.913257i $$0.366441\pi$$
$$374$$ 749.060 0.103564
$$375$$ 0 0
$$376$$ 17531.4 2.40455
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 6525.75 0.884447 0.442223 0.896905i $$-0.354190\pi$$
0.442223 + 0.896905i $$0.354190\pi$$
$$380$$ 26764.1 3.61307
$$381$$ 0 0
$$382$$ 13533.5 1.81265
$$383$$ 7042.66 0.939590 0.469795 0.882775i $$-0.344328\pi$$
0.469795 + 0.882775i $$0.344328\pi$$
$$384$$ 0 0
$$385$$ −1062.06 −0.140592
$$386$$ −25187.7 −3.32129
$$387$$ 0 0
$$388$$ 19470.7 2.54761
$$389$$ −10339.4 −1.34763 −0.673815 0.738900i $$-0.735345\pi$$
−0.673815 + 0.738900i $$0.735345\pi$$
$$390$$ 0 0
$$391$$ 8950.98 1.15773
$$392$$ −19064.8 −2.45642
$$393$$ 0 0
$$394$$ −21204.1 −2.71129
$$395$$ 9505.24 1.21079
$$396$$ 0 0
$$397$$ 7465.86 0.943831 0.471915 0.881644i $$-0.343563\pi$$
0.471915 + 0.881644i $$0.343563\pi$$
$$398$$ −7874.88 −0.991789
$$399$$ 0 0
$$400$$ 18263.6 2.28295
$$401$$ 3224.37 0.401539 0.200770 0.979638i $$-0.435656\pi$$
0.200770 + 0.979638i $$0.435656\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 5657.60 0.696723
$$405$$ 0 0
$$406$$ 4138.85 0.505931
$$407$$ −901.857 −0.109836
$$408$$ 0 0
$$409$$ −12391.9 −1.49814 −0.749072 0.662489i $$-0.769500\pi$$
−0.749072 + 0.662489i $$0.769500\pi$$
$$410$$ 11500.1 1.38524
$$411$$ 0 0
$$412$$ 3903.43 0.466768
$$413$$ 1000.12 0.119159
$$414$$ 0 0
$$415$$ 837.426 0.0990546
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −1464.38 −0.171352
$$419$$ −13464.4 −1.56988 −0.784938 0.619574i $$-0.787306\pi$$
−0.784938 + 0.619574i $$0.787306\pi$$
$$420$$ 0 0
$$421$$ 6265.49 0.725324 0.362662 0.931921i $$-0.381868\pi$$
0.362662 + 0.931921i $$0.381868\pi$$
$$422$$ 8657.89 0.998719
$$423$$ 0 0
$$424$$ 13401.5 1.53499
$$425$$ −3223.92 −0.367960
$$426$$ 0 0
$$427$$ −12127.7 −1.37447
$$428$$ 18063.5 2.04003
$$429$$ 0 0
$$430$$ −21731.8 −2.43721
$$431$$ −9499.09 −1.06161 −0.530806 0.847493i $$-0.678111\pi$$
−0.530806 + 0.847493i $$0.678111\pi$$
$$432$$ 0 0
$$433$$ 5002.67 0.555226 0.277613 0.960693i $$-0.410457\pi$$
0.277613 + 0.960693i $$0.410457\pi$$
$$434$$ 18079.1 1.99960
$$435$$ 0 0
$$436$$ −3859.17 −0.423900
$$437$$ −17498.8 −1.91551
$$438$$ 0 0
$$439$$ 13664.9 1.48563 0.742816 0.669496i $$-0.233490\pi$$
0.742816 + 0.669496i $$0.233490\pi$$
$$440$$ −3395.52 −0.367898
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 10780.6 1.15621 0.578105 0.815962i $$-0.303792\pi$$
0.578105 + 0.815962i $$0.303792\pi$$
$$444$$ 0 0
$$445$$ −17162.5 −1.82827
$$446$$ −17146.8 −1.82046
$$447$$ 0 0
$$448$$ −50937.6 −5.37182
$$449$$ 11959.1 1.25699 0.628493 0.777815i $$-0.283672\pi$$
0.628493 + 0.777815i $$0.283672\pi$$
$$450$$ 0 0
$$451$$ −462.165 −0.0482539
$$452$$ 4419.07 0.459858
$$453$$ 0 0
$$454$$ −16502.0 −1.70590
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 2120.66 0.217069 0.108534 0.994093i $$-0.465384\pi$$
0.108534 + 0.994093i $$0.465384\pi$$
$$458$$ −12869.2 −1.31297
$$459$$ 0 0
$$460$$ −63544.2 −6.44079
$$461$$ 15462.9 1.56221 0.781103 0.624402i $$-0.214657\pi$$
0.781103 + 0.624402i $$0.214657\pi$$
$$462$$ 0 0
$$463$$ 10395.4 1.04344 0.521722 0.853116i $$-0.325290\pi$$
0.521722 + 0.853116i $$0.325290\pi$$
$$464$$ 7730.44 0.773441
$$465$$ 0 0
$$466$$ −32460.8 −3.22687
$$467$$ 11845.8 1.17379 0.586893 0.809664i $$-0.300351\pi$$
0.586893 + 0.809664i $$0.300351\pi$$
$$468$$ 0 0
$$469$$ 12217.2 1.20286
$$470$$ 17474.0 1.71493
$$471$$ 0 0
$$472$$ 3197.47 0.311812
$$473$$ 873.356 0.0848984
$$474$$ 0 0
$$475$$ 6302.62 0.608808
$$476$$ 23582.9 2.27084
$$477$$ 0 0
$$478$$ −1374.73 −0.131546
$$479$$ 12879.9 1.22860 0.614299 0.789074i $$-0.289439\pi$$
0.614299 + 0.789074i $$0.289439\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 6106.46 0.577057
$$483$$ 0 0
$$484$$ −29244.2 −2.74645
$$485$$ 12392.1 1.16019
$$486$$ 0 0
$$487$$ 4946.81 0.460290 0.230145 0.973156i $$-0.426080\pi$$
0.230145 + 0.973156i $$0.426080\pi$$
$$488$$ −38773.4 −3.59670
$$489$$ 0 0
$$490$$ −19002.4 −1.75192
$$491$$ −14390.3 −1.32266 −0.661330 0.750095i $$-0.730008\pi$$
−0.661330 + 0.750095i $$0.730008\pi$$
$$492$$ 0 0
$$493$$ −1364.59 −0.124661
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 33767.7 3.05689
$$497$$ −22964.8 −2.07266
$$498$$ 0 0
$$499$$ 1427.01 0.128019 0.0640096 0.997949i $$-0.479611\pi$$
0.0640096 + 0.997949i $$0.479611\pi$$
$$500$$ −16082.1 −1.43842
$$501$$ 0 0
$$502$$ −19191.6 −1.70630
$$503$$ 18033.6 1.59857 0.799283 0.600955i $$-0.205213\pi$$
0.799283 + 0.600955i $$0.205213\pi$$
$$504$$ 0 0
$$505$$ 3600.76 0.317290
$$506$$ 3476.78 0.305458
$$507$$ 0 0
$$508$$ 9026.39 0.788349
$$509$$ 7512.52 0.654198 0.327099 0.944990i $$-0.393929\pi$$
0.327099 + 0.944990i $$0.393929\pi$$
$$510$$ 0 0
$$511$$ −27078.4 −2.34418
$$512$$ −30945.6 −2.67112
$$513$$ 0 0
$$514$$ −25493.3 −2.18767
$$515$$ 2484.33 0.212568
$$516$$ 0 0
$$517$$ −702.244 −0.0597382
$$518$$ −38656.7 −3.27892
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −21355.8 −1.79580 −0.897901 0.440198i $$-0.854908\pi$$
−0.897901 + 0.440198i $$0.854908\pi$$
$$522$$ 0 0
$$523$$ 3086.57 0.258062 0.129031 0.991641i $$-0.458813\pi$$
0.129031 + 0.991641i $$0.458813\pi$$
$$524$$ 48192.9 4.01778
$$525$$ 0 0
$$526$$ 27064.4 2.24347
$$527$$ −5960.73 −0.492701
$$528$$ 0 0
$$529$$ 29379.2 2.41466
$$530$$ 13357.7 1.09476
$$531$$ 0 0
$$532$$ −46103.6 −3.75722
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 11496.4 0.929036
$$536$$ 39059.7 3.14762
$$537$$ 0 0
$$538$$ 7006.74 0.561491
$$539$$ 763.667 0.0610269
$$540$$ 0 0
$$541$$ 2029.46 0.161282 0.0806408 0.996743i $$-0.474303\pi$$
0.0806408 + 0.996743i $$0.474303\pi$$
$$542$$ −13832.5 −1.09623
$$543$$ 0 0
$$544$$ 32715.9 2.57846
$$545$$ −2456.16 −0.193046
$$546$$ 0 0
$$547$$ 22144.8 1.73098 0.865488 0.500929i $$-0.167008\pi$$
0.865488 + 0.500929i $$0.167008\pi$$
$$548$$ 16018.1 1.24865
$$549$$ 0 0
$$550$$ −1252.25 −0.0970837
$$551$$ 2667.71 0.206258
$$552$$ 0 0
$$553$$ −16373.6 −1.25909
$$554$$ 13514.6 1.03643
$$555$$ 0 0
$$556$$ −33659.7 −2.56742
$$557$$ 23136.5 1.76001 0.880004 0.474967i $$-0.157540\pi$$
0.880004 + 0.474967i $$0.157540\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −85028.1 −6.41623
$$561$$ 0 0
$$562$$ 27765.2 2.08400
$$563$$ 3978.29 0.297806 0.148903 0.988852i $$-0.452426\pi$$
0.148903 + 0.988852i $$0.452426\pi$$
$$564$$ 0 0
$$565$$ 2812.51 0.209421
$$566$$ −16899.0 −1.25498
$$567$$ 0 0
$$568$$ −73420.6 −5.42370
$$569$$ 25871.5 1.90613 0.953065 0.302764i $$-0.0979095\pi$$
0.953065 + 0.302764i $$0.0979095\pi$$
$$570$$ 0 0
$$571$$ −7241.31 −0.530717 −0.265358 0.964150i $$-0.585490\pi$$
−0.265358 + 0.964150i $$0.585490\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −19810.0 −1.44051
$$575$$ −14963.9 −1.08528
$$576$$ 0 0
$$577$$ −11537.4 −0.832426 −0.416213 0.909267i $$-0.636643\pi$$
−0.416213 + 0.909267i $$0.636643\pi$$
$$578$$ 16383.0 1.17897
$$579$$ 0 0
$$580$$ 9687.38 0.693529
$$581$$ −1442.54 −0.103007
$$582$$ 0 0
$$583$$ −536.817 −0.0381350
$$584$$ −86572.2 −6.13422
$$585$$ 0 0
$$586$$ 47352.2 3.33806
$$587$$ 16195.2 1.13875 0.569376 0.822078i $$-0.307185\pi$$
0.569376 + 0.822078i $$0.307185\pi$$
$$588$$ 0 0
$$589$$ 11653.0 0.815198
$$590$$ 3187.01 0.222385
$$591$$ 0 0
$$592$$ −72202.0 −5.01264
$$593$$ −14885.6 −1.03083 −0.515413 0.856942i $$-0.672362\pi$$
−0.515413 + 0.856942i $$0.672362\pi$$
$$594$$ 0 0
$$595$$ 15009.3 1.03415
$$596$$ −44873.3 −3.08403
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 21516.2 1.46766 0.733829 0.679334i $$-0.237731\pi$$
0.733829 + 0.679334i $$0.237731\pi$$
$$600$$ 0 0
$$601$$ −675.727 −0.0458627 −0.0229313 0.999737i $$-0.507300\pi$$
−0.0229313 + 0.999737i $$0.507300\pi$$
$$602$$ 37435.1 2.53445
$$603$$ 0 0
$$604$$ 263.266 0.0177354
$$605$$ −18612.4 −1.25075
$$606$$ 0 0
$$607$$ −12166.0 −0.813512 −0.406756 0.913537i $$-0.633340\pi$$
−0.406756 + 0.913537i $$0.633340\pi$$
$$608$$ −63958.1 −4.26619
$$609$$ 0 0
$$610$$ −38646.6 −2.56517
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 22799.0 1.50219 0.751095 0.660194i $$-0.229526\pi$$
0.751095 + 0.660194i $$0.229526\pi$$
$$614$$ 2994.90 0.196847
$$615$$ 0 0
$$616$$ 5849.10 0.382576
$$617$$ −1635.30 −0.106701 −0.0533506 0.998576i $$-0.516990\pi$$
−0.0533506 + 0.998576i $$0.516990\pi$$
$$618$$ 0 0
$$619$$ 4435.20 0.287990 0.143995 0.989578i $$-0.454005\pi$$
0.143995 + 0.989578i $$0.454005\pi$$
$$620$$ 42315.9 2.74105
$$621$$ 0 0
$$622$$ −16600.9 −1.07016
$$623$$ 29564.0 1.90122
$$624$$ 0 0
$$625$$ −19412.1 −1.24238
$$626$$ 55345.1 3.53360
$$627$$ 0 0
$$628$$ 30301.2 1.92540
$$629$$ 12745.2 0.807924
$$630$$ 0 0
$$631$$ 17645.7 1.11325 0.556626 0.830763i $$-0.312096\pi$$
0.556626 + 0.830763i $$0.312096\pi$$
$$632$$ −52348.1 −3.29477
$$633$$ 0 0
$$634$$ −46651.6 −2.92235
$$635$$ 5744.82 0.359018
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −530.039 −0.0328910
$$639$$ 0 0
$$640$$ −78368.1 −4.84027
$$641$$ −1482.41 −0.0913440 −0.0456720 0.998956i $$-0.514543\pi$$
−0.0456720 + 0.998956i $$0.514543\pi$$
$$642$$ 0 0
$$643$$ −9629.03 −0.590563 −0.295281 0.955410i $$-0.595413\pi$$
−0.295281 + 0.955410i $$0.595413\pi$$
$$644$$ 109461. 6.69775
$$645$$ 0 0
$$646$$ 20694.9 1.26042
$$647$$ −15991.6 −0.971709 −0.485854 0.874040i $$-0.661491\pi$$
−0.485854 + 0.874040i $$0.661491\pi$$
$$648$$ 0 0
$$649$$ −128.079 −0.00774660
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −36530.0 −2.19421
$$653$$ −20321.2 −1.21781 −0.608907 0.793242i $$-0.708392\pi$$
−0.608907 + 0.793242i $$0.708392\pi$$
$$654$$ 0 0
$$655$$ 30672.2 1.82971
$$656$$ −37000.5 −2.20218
$$657$$ 0 0
$$658$$ −30100.6 −1.78335
$$659$$ 21597.1 1.27664 0.638318 0.769773i $$-0.279631\pi$$
0.638318 + 0.769773i $$0.279631\pi$$
$$660$$ 0 0
$$661$$ −513.172 −0.0301968 −0.0150984 0.999886i $$-0.504806\pi$$
−0.0150984 + 0.999886i $$0.504806\pi$$
$$662$$ 54727.8 3.21307
$$663$$ 0 0
$$664$$ −4611.95 −0.269546
$$665$$ −29342.5 −1.71106
$$666$$ 0 0
$$667$$ −6333.76 −0.367683
$$668$$ 19314.4 1.11871
$$669$$ 0 0
$$670$$ 38932.0 2.24488
$$671$$ 1553.12 0.0893557
$$672$$ 0 0
$$673$$ 11983.8 0.686389 0.343195 0.939264i $$-0.388491\pi$$
0.343195 + 0.939264i $$0.388491\pi$$
$$674$$ 17741.1 1.01389
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 21876.5 1.24192 0.620960 0.783842i $$-0.286743\pi$$
0.620960 + 0.783842i $$0.286743\pi$$
$$678$$ 0 0
$$679$$ −21346.4 −1.20648
$$680$$ 47986.1 2.70615
$$681$$ 0 0
$$682$$ −2315.29 −0.129996
$$683$$ 2551.17 0.142925 0.0714625 0.997443i $$-0.477233\pi$$
0.0714625 + 0.997443i $$0.477233\pi$$
$$684$$ 0 0
$$685$$ 10194.7 0.568642
$$686$$ −12952.0 −0.720860
$$687$$ 0 0
$$688$$ 69920.2 3.87454
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 15321.2 0.843481 0.421741 0.906717i $$-0.361419\pi$$
0.421741 + 0.906717i $$0.361419\pi$$
$$692$$ −16220.2 −0.891041
$$693$$ 0 0
$$694$$ −32879.4 −1.79839
$$695$$ −21422.6 −1.16922
$$696$$ 0 0
$$697$$ 6531.39 0.354941
$$698$$ −54755.0 −2.96921
$$699$$ 0 0
$$700$$ −39424.9 −2.12875
$$701$$ −558.289 −0.0300803 −0.0150401 0.999887i $$-0.504788\pi$$
−0.0150401 + 0.999887i $$0.504788\pi$$
$$702$$ 0 0
$$703$$ −24916.3 −1.33675
$$704$$ 6523.29 0.349227
$$705$$ 0 0
$$706$$ 59470.6 3.17026
$$707$$ −6202.64 −0.329949
$$708$$ 0 0
$$709$$ 7058.31 0.373879 0.186940 0.982371i $$-0.440143\pi$$
0.186940 + 0.982371i $$0.440143\pi$$
$$710$$ −73180.5 −3.86819
$$711$$ 0 0
$$712$$ 94519.0 4.97507
$$713$$ −27666.8 −1.45320
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 33880.1 1.76838
$$717$$ 0 0
$$718$$ 23690.4 1.23136
$$719$$ 14673.0 0.761073 0.380536 0.924766i $$-0.375739\pi$$
0.380536 + 0.924766i $$0.375739\pi$$
$$720$$ 0 0
$$721$$ −4279.48 −0.221049
$$722$$ −2806.58 −0.144668
$$723$$ 0 0
$$724$$ −26519.0 −1.36128
$$725$$ 2281.26 0.116861
$$726$$ 0 0
$$727$$ 35867.8 1.82980 0.914899 0.403684i $$-0.132270\pi$$
0.914899 + 0.403684i $$0.132270\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −86289.1 −4.37493
$$731$$ −12342.4 −0.624488
$$732$$ 0 0
$$733$$ −27817.7 −1.40173 −0.700865 0.713294i $$-0.747203\pi$$
−0.700865 + 0.713294i $$0.747203\pi$$
$$734$$ −57666.5 −2.89988
$$735$$ 0 0
$$736$$ 151851. 7.60505
$$737$$ −1564.59 −0.0781988
$$738$$ 0 0
$$739$$ 7162.20 0.356517 0.178258 0.983984i $$-0.442954\pi$$
0.178258 + 0.983984i $$0.442954\pi$$
$$740$$ −90479.8 −4.49473
$$741$$ 0 0
$$742$$ −23009.9 −1.13843
$$743$$ 12529.8 0.618671 0.309336 0.950953i $$-0.399893\pi$$
0.309336 + 0.950953i $$0.399893\pi$$
$$744$$ 0 0
$$745$$ −28559.5 −1.40448
$$746$$ −32219.0 −1.58126
$$747$$ 0 0
$$748$$ −3020.13 −0.147629
$$749$$ −19803.7 −0.966102
$$750$$ 0 0
$$751$$ 8282.62 0.402446 0.201223 0.979545i $$-0.435508\pi$$
0.201223 + 0.979545i $$0.435508\pi$$
$$752$$ −56221.1 −2.72629
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 167.555 0.00807676
$$756$$ 0 0
$$757$$ −22044.3 −1.05840 −0.529202 0.848496i $$-0.677509\pi$$
−0.529202 + 0.848496i $$0.677509\pi$$
$$758$$ −35821.7 −1.71649
$$759$$ 0 0
$$760$$ −93810.7 −4.47746
$$761$$ 5710.64 0.272025 0.136012 0.990707i $$-0.456571\pi$$
0.136012 + 0.990707i $$0.456571\pi$$
$$762$$ 0 0
$$763$$ 4230.95 0.200748
$$764$$ −54565.5 −2.58391
$$765$$ 0 0
$$766$$ −38659.1 −1.82351
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −16851.6 −0.790228 −0.395114 0.918632i $$-0.629295\pi$$
−0.395114 + 0.918632i $$0.629295\pi$$
$$770$$ 5829.97 0.272854
$$771$$ 0 0
$$772$$ 101554. 4.73447
$$773$$ 593.103 0.0275970 0.0137985 0.999905i $$-0.495608\pi$$
0.0137985 + 0.999905i $$0.495608\pi$$
$$774$$ 0 0
$$775$$ 9964.89 0.461870
$$776$$ −68246.6 −3.15710
$$777$$ 0 0
$$778$$ 56755.8 2.61542
$$779$$ −12768.6 −0.587268
$$780$$ 0 0
$$781$$ 2940.97 0.134745
$$782$$ −49134.4 −2.24686
$$783$$ 0 0
$$784$$ 61138.6 2.78510
$$785$$ 19285.1 0.876834
$$786$$ 0 0
$$787$$ −18564.0 −0.840834 −0.420417 0.907331i $$-0.638116\pi$$
−0.420417 + 0.907331i $$0.638116\pi$$
$$788$$ 85492.6 3.86491
$$789$$ 0 0
$$790$$ −52176.9 −2.34984
$$791$$ −4844.80 −0.217777
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −40982.2 −1.83174
$$795$$ 0 0
$$796$$ 31750.7 1.41378
$$797$$ −5227.16 −0.232316 −0.116158 0.993231i $$-0.537058\pi$$
−0.116158 + 0.993231i $$0.537058\pi$$
$$798$$ 0 0
$$799$$ 9924.23 0.439417
$$800$$ −54693.0 −2.41711
$$801$$ 0 0
$$802$$ −17699.5 −0.779289
$$803$$ 3467.77 0.152397
$$804$$ 0 0
$$805$$ 69665.9 3.05019
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −19830.4 −0.863406
$$809$$ 8907.33 0.387101 0.193551 0.981090i $$-0.438000\pi$$
0.193551 + 0.981090i $$0.438000\pi$$
$$810$$ 0 0
$$811$$ 15352.3 0.664725 0.332363 0.943152i $$-0.392154\pi$$
0.332363 + 0.943152i $$0.392154\pi$$
$$812$$ −16687.4 −0.721198
$$813$$ 0 0
$$814$$ 4950.54 0.213165
$$815$$ −23249.4 −0.999253
$$816$$ 0 0
$$817$$ 24128.9 1.03325
$$818$$ 68022.7 2.90753
$$819$$ 0 0
$$820$$ −46367.2 −1.97465
$$821$$ −12869.6 −0.547080 −0.273540 0.961861i $$-0.588195\pi$$
−0.273540 + 0.961861i $$0.588195\pi$$
$$822$$ 0 0
$$823$$ −402.065 −0.0170293 −0.00851464 0.999964i $$-0.502710\pi$$
−0.00851464 + 0.999964i $$0.502710\pi$$
$$824$$ −13681.9 −0.578437
$$825$$ 0 0
$$826$$ −5489.91 −0.231257
$$827$$ 42772.7 1.79849 0.899245 0.437446i $$-0.144117\pi$$
0.899245 + 0.437446i $$0.144117\pi$$
$$828$$ 0 0
$$829$$ −22933.5 −0.960814 −0.480407 0.877046i $$-0.659511\pi$$
−0.480407 + 0.877046i $$0.659511\pi$$
$$830$$ −4596.87 −0.192240
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −10792.3 −0.448896
$$834$$ 0 0
$$835$$ 12292.6 0.509465
$$836$$ 5904.22 0.244260
$$837$$ 0 0
$$838$$ 73909.8 3.04674
$$839$$ 42510.1 1.74924 0.874619 0.484811i $$-0.161112\pi$$
0.874619 + 0.484811i $$0.161112\pi$$
$$840$$ 0 0
$$841$$ −23423.4 −0.960409
$$842$$ −34393.0 −1.40767
$$843$$ 0 0
$$844$$ −34907.7 −1.42366
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 32061.5 1.30065
$$848$$ −42977.2 −1.74038
$$849$$ 0 0
$$850$$ 17697.0 0.714119
$$851$$ 59157.1 2.38294
$$852$$ 0 0
$$853$$ −40464.6 −1.62425 −0.812123 0.583487i $$-0.801688\pi$$
−0.812123 + 0.583487i $$0.801688\pi$$
$$854$$ 66572.3 2.66751
$$855$$ 0 0
$$856$$ −63314.2 −2.52808
$$857$$ −38964.3 −1.55308 −0.776542 0.630065i $$-0.783028\pi$$
−0.776542 + 0.630065i $$0.783028\pi$$
$$858$$ 0 0
$$859$$ −6379.47 −0.253393 −0.126697 0.991942i $$-0.540437\pi$$
−0.126697 + 0.991942i $$0.540437\pi$$
$$860$$ 87620.4 3.47422
$$861$$ 0 0
$$862$$ 52143.1 2.06033
$$863$$ −33744.0 −1.33101 −0.665504 0.746395i $$-0.731783\pi$$
−0.665504 + 0.746395i $$0.731783\pi$$
$$864$$ 0 0
$$865$$ −10323.3 −0.405784
$$866$$ −27461.0 −1.07756
$$867$$ 0 0
$$868$$ −72893.1 −2.85041
$$869$$ 2096.88 0.0818547
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 13526.8 0.525314
$$873$$ 0 0
$$874$$ 96055.6 3.71754
$$875$$ 17631.4 0.681200
$$876$$ 0 0
$$877$$ −42898.6 −1.65175 −0.825874 0.563854i $$-0.809318\pi$$
−0.825874 + 0.563854i $$0.809318\pi$$
$$878$$ −75010.7 −2.88324
$$879$$ 0 0
$$880$$ 10889.1 0.417125
$$881$$ −1750.90 −0.0669572 −0.0334786 0.999439i $$-0.510659\pi$$
−0.0334786 + 0.999439i $$0.510659\pi$$
$$882$$ 0 0
$$883$$ 33196.7 1.26518 0.632591 0.774486i $$-0.281991\pi$$
0.632591 + 0.774486i $$0.281991\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −59177.6 −2.24392
$$887$$ 29998.9 1.13559 0.567794 0.823171i $$-0.307797\pi$$
0.567794 + 0.823171i $$0.307797\pi$$
$$888$$ 0 0
$$889$$ −9895.98 −0.373341
$$890$$ 94209.8 3.54823
$$891$$ 0 0
$$892$$ 69133.9 2.59504
$$893$$ −19401.4 −0.727037
$$894$$ 0 0
$$895$$ 21562.9 0.805328
$$896$$ 134996. 5.03338
$$897$$ 0 0
$$898$$ −65647.0 −2.43950
$$899$$ 4217.84 0.156477
$$900$$ 0 0
$$901$$ 7586.39 0.280510
$$902$$ 2536.95 0.0936488
$$903$$ 0 0
$$904$$ −15489.3 −0.569874
$$905$$ −16877.9 −0.619935
$$906$$ 0 0
$$907$$ 24629.3 0.901657 0.450829 0.892611i $$-0.351129\pi$$
0.450829 + 0.892611i $$0.351129\pi$$
$$908$$ 66534.2 2.43173
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −7338.72 −0.266896 −0.133448 0.991056i $$-0.542605\pi$$
−0.133448 + 0.991056i $$0.542605\pi$$
$$912$$ 0 0
$$913$$ 184.738 0.00669654
$$914$$ −11640.9 −0.421277
$$915$$ 0 0
$$916$$ 51887.4 1.87162
$$917$$ −52835.7 −1.90271
$$918$$ 0 0
$$919$$ −1449.13 −0.0520155 −0.0260078 0.999662i $$-0.508279\pi$$
−0.0260078 + 0.999662i $$0.508279\pi$$
$$920$$ 222728. 7.98167
$$921$$ 0 0
$$922$$ −84879.9 −3.03186
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −21306.9 −0.757369
$$926$$ −57063.2 −2.02507
$$927$$ 0 0
$$928$$ −23149.9 −0.818894
$$929$$ −6478.07 −0.228782 −0.114391 0.993436i $$-0.536492\pi$$
−0.114391 + 0.993436i $$0.536492\pi$$
$$930$$ 0 0
$$931$$ 21098.4 0.742720
$$932$$ 130879. 4.59986
$$933$$ 0 0
$$934$$ −65024.9 −2.27803
$$935$$ −1922.15 −0.0672311
$$936$$ 0 0
$$937$$ 4679.24 0.163142 0.0815710 0.996668i $$-0.474006\pi$$
0.0815710 + 0.996668i $$0.474006\pi$$
$$938$$ −67063.9 −2.33445
$$939$$ 0 0
$$940$$ −70453.4 −2.44461
$$941$$ 47081.7 1.63105 0.815525 0.578722i $$-0.196448\pi$$
0.815525 + 0.578722i $$0.196448\pi$$
$$942$$ 0 0
$$943$$ 30315.6 1.04688
$$944$$ −10253.9 −0.353534
$$945$$ 0 0
$$946$$ −4794.09 −0.164767
$$947$$ 4264.62 0.146338 0.0731688 0.997320i $$-0.476689\pi$$
0.0731688 + 0.997320i $$0.476689\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −34596.8 −1.18155
$$951$$ 0 0
$$952$$ −82660.4 −2.81412
$$953$$ −24368.1 −0.828292 −0.414146 0.910211i $$-0.635920\pi$$
−0.414146 + 0.910211i $$0.635920\pi$$
$$954$$ 0 0
$$955$$ −34728.0 −1.17673
$$956$$ 5542.77 0.187517
$$957$$ 0 0
$$958$$ −70701.4 −2.38440
$$959$$ −17561.3 −0.591329
$$960$$ 0 0
$$961$$ −11366.8 −0.381552
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −24620.6 −0.822588
$$965$$ 64633.7 2.15610
$$966$$ 0 0
$$967$$ −11355.1 −0.377616 −0.188808 0.982014i $$-0.560462\pi$$
−0.188808 + 0.982014i $$0.560462\pi$$
$$968$$ 102504. 3.40351
$$969$$ 0 0
$$970$$ −68023.4 −2.25165
$$971$$ −3024.06 −0.0999451 −0.0499726 0.998751i $$-0.515913\pi$$
−0.0499726 + 0.998751i $$0.515913\pi$$
$$972$$ 0 0
$$973$$ 36902.4 1.21586
$$974$$ −27154.4 −0.893309
$$975$$ 0 0
$$976$$ 124342. 4.07796
$$977$$ 41218.9 1.34975 0.674877 0.737930i $$-0.264197\pi$$
0.674877 + 0.737930i $$0.264197\pi$$
$$978$$ 0 0
$$979$$ −3786.09 −0.123600
$$980$$ 76615.7 2.49735
$$981$$ 0 0
$$982$$ 78992.4 2.56695
$$983$$ −21579.3 −0.700177 −0.350088 0.936717i $$-0.613848\pi$$
−0.350088 + 0.936717i $$0.613848\pi$$
$$984$$ 0 0
$$985$$ 54411.5 1.76010
$$986$$ 7490.60 0.241936
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −57287.6 −1.84190
$$990$$ 0 0
$$991$$ −34613.5 −1.10952 −0.554760 0.832010i $$-0.687190\pi$$
−0.554760 + 0.832010i $$0.687190\pi$$
$$992$$ −101122. −3.23653
$$993$$ 0 0
$$994$$ 126060. 4.02252
$$995$$ 20207.6 0.643843
$$996$$ 0 0
$$997$$ 20841.5 0.662043 0.331021 0.943623i $$-0.392607\pi$$
0.331021 + 0.943623i $$0.392607\pi$$
$$998$$ −7833.24 −0.248454
$$999$$ 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 1521.4.a.ba.1.1 4
3.2 odd 2 inner 1521.4.a.ba.1.4 4
13.12 even 2 117.4.a.g.1.4 yes 4
39.38 odd 2 117.4.a.g.1.1 4
52.51 odd 2 1872.4.a.bo.1.2 4
156.155 even 2 1872.4.a.bo.1.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.a.g.1.1 4 39.38 odd 2
117.4.a.g.1.4 yes 4 13.12 even 2
1521.4.a.ba.1.1 4 1.1 even 1 trivial
1521.4.a.ba.1.4 4 3.2 odd 2 inner
1872.4.a.bo.1.2 4 52.51 odd 2
1872.4.a.bo.1.3 4 156.155 even 2