# Properties

 Label 1521.4.a.v.1.4 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# 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: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 2x^{3} - 25x^{2} + 24x + 78$$ x^4 - 2*x^3 - 25*x^2 + 24*x + 78 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-4.22605$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.22605 q^{2} +9.85953 q^{4} +5.85953 q^{5} -24.1254 q^{7} +7.85849 q^{8} +O(q^{10})$$ $$q+4.22605 q^{2} +9.85953 q^{4} +5.85953 q^{5} -24.1254 q^{7} +7.85849 q^{8} +24.7627 q^{10} +33.8892 q^{11} -101.955 q^{14} -45.6659 q^{16} +49.3956 q^{17} -76.8548 q^{19} +57.7723 q^{20} +143.218 q^{22} -6.29163 q^{23} -90.6659 q^{25} -237.866 q^{28} -100.995 q^{29} -307.580 q^{31} -255.854 q^{32} +208.749 q^{34} -141.364 q^{35} -76.0189 q^{37} -324.793 q^{38} +46.0471 q^{40} +514.418 q^{41} -268.184 q^{43} +334.132 q^{44} -26.5888 q^{46} +460.912 q^{47} +239.037 q^{49} -383.159 q^{50} -67.8057 q^{53} +198.575 q^{55} -189.589 q^{56} -426.812 q^{58} -25.2021 q^{59} -588.832 q^{61} -1299.85 q^{62} -715.927 q^{64} -1004.46 q^{67} +487.018 q^{68} -597.411 q^{70} -895.481 q^{71} +968.599 q^{73} -321.260 q^{74} -757.753 q^{76} -817.592 q^{77} -119.053 q^{79} -267.581 q^{80} +2173.96 q^{82} -480.784 q^{83} +289.435 q^{85} -1133.36 q^{86} +266.318 q^{88} -1085.91 q^{89} -62.0325 q^{92} +1947.84 q^{94} -450.333 q^{95} -16.6552 q^{97} +1010.18 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8 $$4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8} - 62 q^{10} - 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} + 124 q^{19} + 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} - 144 q^{28} - 194 q^{29} + 26 q^{31} - 654 q^{32} + 1062 q^{34} - 88 q^{35} + 102 q^{37} - 332 q^{38} - 998 q^{40} + 1054 q^{41} + 450 q^{43} + 44 q^{44} - 172 q^{46} + 96 q^{47} + 1070 q^{49} - 996 q^{50} - 262 q^{53} + 204 q^{55} - 2164 q^{56} + 722 q^{58} - 308 q^{59} - 928 q^{61} - 2780 q^{62} + 1026 q^{64} - 1134 q^{67} - 1786 q^{68} - 2324 q^{70} - 1064 q^{71} + 952 q^{73} - 1158 q^{74} - 1708 q^{76} - 2508 q^{77} - 746 q^{79} + 2922 q^{80} + 1734 q^{82} + 404 q^{83} - 1394 q^{85} - 3168 q^{86} + 3060 q^{88} - 1620 q^{89} - 332 q^{92} - 772 q^{94} - 2204 q^{95} + 2166 q^{97} + 1906 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8 - 62 * q^10 - 40 * q^11 - 40 * q^14 + 122 * q^16 - 98 * q^17 + 124 * q^19 + 466 * q^20 + 220 * q^22 - 104 * q^23 - 58 * q^25 - 144 * q^28 - 194 * q^29 + 26 * q^31 - 654 * q^32 + 1062 * q^34 - 88 * q^35 + 102 * q^37 - 332 * q^38 - 998 * q^40 + 1054 * q^41 + 450 * q^43 + 44 * q^44 - 172 * q^46 + 96 * q^47 + 1070 * q^49 - 996 * q^50 - 262 * q^53 + 204 * q^55 - 2164 * q^56 + 722 * q^58 - 308 * q^59 - 928 * q^61 - 2780 * q^62 + 1026 * q^64 - 1134 * q^67 - 1786 * q^68 - 2324 * q^70 - 1064 * q^71 + 952 * q^73 - 1158 * q^74 - 1708 * q^76 - 2508 * q^77 - 746 * q^79 + 2922 * q^80 + 1734 * q^82 + 404 * q^83 - 1394 * q^85 - 3168 * q^86 + 3060 * q^88 - 1620 * q^89 - 332 * q^92 - 772 * q^94 - 2204 * q^95 + 2166 * q^97 + 1906 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.22605 1.49414 0.747068 0.664748i $$-0.231461\pi$$
0.747068 + 0.664748i $$0.231461\pi$$
$$3$$ 0 0
$$4$$ 9.85953 1.23244
$$5$$ 5.85953 0.524093 0.262046 0.965055i $$-0.415603\pi$$
0.262046 + 0.965055i $$0.415603\pi$$
$$6$$ 0 0
$$7$$ −24.1254 −1.30265 −0.651326 0.758798i $$-0.725787\pi$$
−0.651326 + 0.758798i $$0.725787\pi$$
$$8$$ 7.85849 0.347299
$$9$$ 0 0
$$10$$ 24.7627 0.783065
$$11$$ 33.8892 0.928907 0.464453 0.885598i $$-0.346251\pi$$
0.464453 + 0.885598i $$0.346251\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −101.955 −1.94634
$$15$$ 0 0
$$16$$ −45.6659 −0.713529
$$17$$ 49.3956 0.704718 0.352359 0.935865i $$-0.385380\pi$$
0.352359 + 0.935865i $$0.385380\pi$$
$$18$$ 0 0
$$19$$ −76.8548 −0.927985 −0.463992 0.885839i $$-0.653583\pi$$
−0.463992 + 0.885839i $$0.653583\pi$$
$$20$$ 57.7723 0.645913
$$21$$ 0 0
$$22$$ 143.218 1.38791
$$23$$ −6.29163 −0.0570389 −0.0285195 0.999593i $$-0.509079\pi$$
−0.0285195 + 0.999593i $$0.509079\pi$$
$$24$$ 0 0
$$25$$ −90.6659 −0.725327
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −237.866 −1.60544
$$29$$ −100.995 −0.646702 −0.323351 0.946279i $$-0.604809\pi$$
−0.323351 + 0.946279i $$0.604809\pi$$
$$30$$ 0 0
$$31$$ −307.580 −1.78203 −0.891016 0.453972i $$-0.850007\pi$$
−0.891016 + 0.453972i $$0.850007\pi$$
$$32$$ −255.854 −1.41341
$$33$$ 0 0
$$34$$ 208.749 1.05294
$$35$$ −141.364 −0.682710
$$36$$ 0 0
$$37$$ −76.0189 −0.337768 −0.168884 0.985636i $$-0.554016\pi$$
−0.168884 + 0.985636i $$0.554016\pi$$
$$38$$ −324.793 −1.38653
$$39$$ 0 0
$$40$$ 46.0471 0.182017
$$41$$ 514.418 1.95948 0.979740 0.200274i $$-0.0641833\pi$$
0.979740 + 0.200274i $$0.0641833\pi$$
$$42$$ 0 0
$$43$$ −268.184 −0.951108 −0.475554 0.879686i $$-0.657752\pi$$
−0.475554 + 0.879686i $$0.657752\pi$$
$$44$$ 334.132 1.14482
$$45$$ 0 0
$$46$$ −26.5888 −0.0852239
$$47$$ 460.912 1.43045 0.715223 0.698896i $$-0.246325\pi$$
0.715223 + 0.698896i $$0.246325\pi$$
$$48$$ 0 0
$$49$$ 239.037 0.696901
$$50$$ −383.159 −1.08374
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −67.8057 −0.175733 −0.0878663 0.996132i $$-0.528005\pi$$
−0.0878663 + 0.996132i $$0.528005\pi$$
$$54$$ 0 0
$$55$$ 198.575 0.486833
$$56$$ −189.589 −0.452410
$$57$$ 0 0
$$58$$ −426.812 −0.966261
$$59$$ −25.2021 −0.0556107 −0.0278053 0.999613i $$-0.508852\pi$$
−0.0278053 + 0.999613i $$0.508852\pi$$
$$60$$ 0 0
$$61$$ −588.832 −1.23594 −0.617969 0.786202i $$-0.712044\pi$$
−0.617969 + 0.786202i $$0.712044\pi$$
$$62$$ −1299.85 −2.66260
$$63$$ 0 0
$$64$$ −715.927 −1.39830
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −1004.46 −1.83156 −0.915778 0.401684i $$-0.868425\pi$$
−0.915778 + 0.401684i $$0.868425\pi$$
$$68$$ 487.018 0.868523
$$69$$ 0 0
$$70$$ −597.411 −1.02006
$$71$$ −895.481 −1.49682 −0.748408 0.663238i $$-0.769182\pi$$
−0.748408 + 0.663238i $$0.769182\pi$$
$$72$$ 0 0
$$73$$ 968.599 1.55296 0.776479 0.630143i $$-0.217004\pi$$
0.776479 + 0.630143i $$0.217004\pi$$
$$74$$ −321.260 −0.504672
$$75$$ 0 0
$$76$$ −757.753 −1.14369
$$77$$ −817.592 −1.21004
$$78$$ 0 0
$$79$$ −119.053 −0.169551 −0.0847755 0.996400i $$-0.527017\pi$$
−0.0847755 + 0.996400i $$0.527017\pi$$
$$80$$ −267.581 −0.373955
$$81$$ 0 0
$$82$$ 2173.96 2.92773
$$83$$ −480.784 −0.635818 −0.317909 0.948121i $$-0.602981\pi$$
−0.317909 + 0.948121i $$0.602981\pi$$
$$84$$ 0 0
$$85$$ 289.435 0.369337
$$86$$ −1133.36 −1.42109
$$87$$ 0 0
$$88$$ 266.318 0.322609
$$89$$ −1085.91 −1.29333 −0.646663 0.762776i $$-0.723836\pi$$
−0.646663 + 0.762776i $$0.723836\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −62.0325 −0.0702972
$$93$$ 0 0
$$94$$ 1947.84 2.13728
$$95$$ −450.333 −0.486350
$$96$$ 0 0
$$97$$ −16.6552 −0.0174338 −0.00871692 0.999962i $$-0.502775\pi$$
−0.00871692 + 0.999962i $$0.502775\pi$$
$$98$$ 1010.18 1.04126
$$99$$ 0 0
$$100$$ −893.923 −0.893923
$$101$$ 958.004 0.943811 0.471906 0.881649i $$-0.343566\pi$$
0.471906 + 0.881649i $$0.343566\pi$$
$$102$$ 0 0
$$103$$ −2.70560 −0.00258826 −0.00129413 0.999999i $$-0.500412\pi$$
−0.00129413 + 0.999999i $$0.500412\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −286.551 −0.262568
$$107$$ −1351.28 −1.22087 −0.610437 0.792065i $$-0.709006\pi$$
−0.610437 + 0.792065i $$0.709006\pi$$
$$108$$ 0 0
$$109$$ 448.455 0.394075 0.197037 0.980396i $$-0.436868\pi$$
0.197037 + 0.980396i $$0.436868\pi$$
$$110$$ 839.188 0.727395
$$111$$ 0 0
$$112$$ 1101.71 0.929480
$$113$$ −1398.85 −1.16453 −0.582267 0.812997i $$-0.697834\pi$$
−0.582267 + 0.812997i $$0.697834\pi$$
$$114$$ 0 0
$$115$$ −36.8660 −0.0298937
$$116$$ −995.767 −0.797023
$$117$$ 0 0
$$118$$ −106.505 −0.0830899
$$119$$ −1191.69 −0.918002
$$120$$ 0 0
$$121$$ −182.523 −0.137132
$$122$$ −2488.44 −1.84666
$$123$$ 0 0
$$124$$ −3032.59 −2.19625
$$125$$ −1263.70 −0.904231
$$126$$ 0 0
$$127$$ −119.504 −0.0834985 −0.0417492 0.999128i $$-0.513293\pi$$
−0.0417492 + 0.999128i $$0.513293\pi$$
$$128$$ −978.713 −0.675834
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2251.70 1.50177 0.750886 0.660432i $$-0.229627\pi$$
0.750886 + 0.660432i $$0.229627\pi$$
$$132$$ 0 0
$$133$$ 1854.16 1.20884
$$134$$ −4244.90 −2.73659
$$135$$ 0 0
$$136$$ 388.175 0.244748
$$137$$ −1130.62 −0.705076 −0.352538 0.935797i $$-0.614681\pi$$
−0.352538 + 0.935797i $$0.614681\pi$$
$$138$$ 0 0
$$139$$ −595.287 −0.363249 −0.181624 0.983368i $$-0.558136\pi$$
−0.181624 + 0.983368i $$0.558136\pi$$
$$140$$ −1393.78 −0.841400
$$141$$ 0 0
$$142$$ −3784.35 −2.23645
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −591.786 −0.338932
$$146$$ 4093.35 2.32033
$$147$$ 0 0
$$148$$ −749.511 −0.416280
$$149$$ 793.175 0.436103 0.218052 0.975937i $$-0.430030\pi$$
0.218052 + 0.975937i $$0.430030\pi$$
$$150$$ 0 0
$$151$$ −134.213 −0.0723317 −0.0361659 0.999346i $$-0.511514\pi$$
−0.0361659 + 0.999346i $$0.511514\pi$$
$$152$$ −603.963 −0.322288
$$153$$ 0 0
$$154$$ −3455.19 −1.80797
$$155$$ −1802.28 −0.933950
$$156$$ 0 0
$$157$$ 1509.07 0.767114 0.383557 0.923517i $$-0.374699\pi$$
0.383557 + 0.923517i $$0.374699\pi$$
$$158$$ −503.125 −0.253332
$$159$$ 0 0
$$160$$ −1499.19 −0.740757
$$161$$ 151.788 0.0743019
$$162$$ 0 0
$$163$$ 1175.08 0.564658 0.282329 0.959318i $$-0.408893\pi$$
0.282329 + 0.959318i $$0.408893\pi$$
$$164$$ 5071.93 2.41494
$$165$$ 0 0
$$166$$ −2031.82 −0.949999
$$167$$ −1474.01 −0.683010 −0.341505 0.939880i $$-0.610937\pi$$
−0.341505 + 0.939880i $$0.610937\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 1223.17 0.551840
$$171$$ 0 0
$$172$$ −2644.17 −1.17219
$$173$$ 2328.31 1.02322 0.511612 0.859216i $$-0.329048\pi$$
0.511612 + 0.859216i $$0.329048\pi$$
$$174$$ 0 0
$$175$$ 2187.35 0.944848
$$176$$ −1547.58 −0.662802
$$177$$ 0 0
$$178$$ −4589.10 −1.93240
$$179$$ 2133.85 0.891015 0.445508 0.895278i $$-0.353023\pi$$
0.445508 + 0.895278i $$0.353023\pi$$
$$180$$ 0 0
$$181$$ −2485.41 −1.02066 −0.510329 0.859979i $$-0.670476\pi$$
−0.510329 + 0.859979i $$0.670476\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −49.4427 −0.0198096
$$185$$ −445.435 −0.177022
$$186$$ 0 0
$$187$$ 1673.98 0.654617
$$188$$ 4544.38 1.76294
$$189$$ 0 0
$$190$$ −1903.13 −0.726673
$$191$$ 2325.07 0.880816 0.440408 0.897798i $$-0.354834\pi$$
0.440408 + 0.897798i $$0.354834\pi$$
$$192$$ 0 0
$$193$$ 3350.12 1.24946 0.624732 0.780839i $$-0.285208\pi$$
0.624732 + 0.780839i $$0.285208\pi$$
$$194$$ −70.3859 −0.0260485
$$195$$ 0 0
$$196$$ 2356.79 0.858890
$$197$$ 3859.30 1.39576 0.697878 0.716217i $$-0.254128\pi$$
0.697878 + 0.716217i $$0.254128\pi$$
$$198$$ 0 0
$$199$$ −4083.60 −1.45467 −0.727333 0.686284i $$-0.759241\pi$$
−0.727333 + 0.686284i $$0.759241\pi$$
$$200$$ −712.497 −0.251906
$$201$$ 0 0
$$202$$ 4048.58 1.41018
$$203$$ 2436.56 0.842428
$$204$$ 0 0
$$205$$ 3014.25 1.02695
$$206$$ −11.4340 −0.00386721
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −2604.55 −0.862011
$$210$$ 0 0
$$211$$ 3027.30 0.987714 0.493857 0.869543i $$-0.335587\pi$$
0.493857 + 0.869543i $$0.335587\pi$$
$$212$$ −668.533 −0.216580
$$213$$ 0 0
$$214$$ −5710.60 −1.82415
$$215$$ −1571.43 −0.498469
$$216$$ 0 0
$$217$$ 7420.50 2.32137
$$218$$ 1895.19 0.588801
$$219$$ 0 0
$$220$$ 1957.86 0.599994
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −1724.76 −0.517930 −0.258965 0.965887i $$-0.583381\pi$$
−0.258965 + 0.965887i $$0.583381\pi$$
$$224$$ 6172.60 1.84118
$$225$$ 0 0
$$226$$ −5911.60 −1.73997
$$227$$ 1923.27 0.562344 0.281172 0.959657i $$-0.409277\pi$$
0.281172 + 0.959657i $$0.409277\pi$$
$$228$$ 0 0
$$229$$ 373.993 0.107922 0.0539610 0.998543i $$-0.482815\pi$$
0.0539610 + 0.998543i $$0.482815\pi$$
$$230$$ −155.798 −0.0446652
$$231$$ 0 0
$$232$$ −793.671 −0.224599
$$233$$ −3094.49 −0.870073 −0.435036 0.900413i $$-0.643265\pi$$
−0.435036 + 0.900413i $$0.643265\pi$$
$$234$$ 0 0
$$235$$ 2700.73 0.749686
$$236$$ −248.481 −0.0685369
$$237$$ 0 0
$$238$$ −5036.15 −1.37162
$$239$$ 1221.18 0.330510 0.165255 0.986251i $$-0.447155\pi$$
0.165255 + 0.986251i $$0.447155\pi$$
$$240$$ 0 0
$$241$$ 145.401 0.0388634 0.0194317 0.999811i $$-0.493814\pi$$
0.0194317 + 0.999811i $$0.493814\pi$$
$$242$$ −771.350 −0.204894
$$243$$ 0 0
$$244$$ −5805.61 −1.52322
$$245$$ 1400.65 0.365241
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −2417.11 −0.618899
$$249$$ 0 0
$$250$$ −5340.47 −1.35104
$$251$$ 985.670 0.247868 0.123934 0.992290i $$-0.460449\pi$$
0.123934 + 0.992290i $$0.460449\pi$$
$$252$$ 0 0
$$253$$ −213.218 −0.0529839
$$254$$ −505.032 −0.124758
$$255$$ 0 0
$$256$$ 1591.33 0.388507
$$257$$ −2929.32 −0.710995 −0.355498 0.934677i $$-0.615689\pi$$
−0.355498 + 0.934677i $$0.615689\pi$$
$$258$$ 0 0
$$259$$ 1833.99 0.439995
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 9515.82 2.24385
$$263$$ −2238.00 −0.524719 −0.262360 0.964970i $$-0.584501\pi$$
−0.262360 + 0.964970i $$0.584501\pi$$
$$264$$ 0 0
$$265$$ −397.310 −0.0921002
$$266$$ 7835.77 1.80617
$$267$$ 0 0
$$268$$ −9903.50 −2.25729
$$269$$ −1925.98 −0.436540 −0.218270 0.975888i $$-0.570041\pi$$
−0.218270 + 0.975888i $$0.570041\pi$$
$$270$$ 0 0
$$271$$ −3562.29 −0.798500 −0.399250 0.916842i $$-0.630729\pi$$
−0.399250 + 0.916842i $$0.630729\pi$$
$$272$$ −2255.69 −0.502837
$$273$$ 0 0
$$274$$ −4778.06 −1.05348
$$275$$ −3072.59 −0.673761
$$276$$ 0 0
$$277$$ −1437.42 −0.311792 −0.155896 0.987773i $$-0.549827\pi$$
−0.155896 + 0.987773i $$0.549827\pi$$
$$278$$ −2515.72 −0.542743
$$279$$ 0 0
$$280$$ −1110.91 −0.237105
$$281$$ 3913.51 0.830820 0.415410 0.909634i $$-0.363638\pi$$
0.415410 + 0.909634i $$0.363638\pi$$
$$282$$ 0 0
$$283$$ −3212.31 −0.674743 −0.337371 0.941372i $$-0.609538\pi$$
−0.337371 + 0.941372i $$0.609538\pi$$
$$284$$ −8829.02 −1.84474
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −12410.6 −2.55252
$$288$$ 0 0
$$289$$ −2473.07 −0.503373
$$290$$ −2500.92 −0.506410
$$291$$ 0 0
$$292$$ 9549.94 1.91393
$$293$$ 4901.77 0.977353 0.488677 0.872465i $$-0.337480\pi$$
0.488677 + 0.872465i $$0.337480\pi$$
$$294$$ 0 0
$$295$$ −147.672 −0.0291452
$$296$$ −597.394 −0.117307
$$297$$ 0 0
$$298$$ 3352.00 0.651598
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 6470.06 1.23896
$$302$$ −567.191 −0.108073
$$303$$ 0 0
$$304$$ 3509.64 0.662144
$$305$$ −3450.28 −0.647746
$$306$$ 0 0
$$307$$ 5800.63 1.07837 0.539185 0.842188i $$-0.318733\pi$$
0.539185 + 0.842188i $$0.318733\pi$$
$$308$$ −8061.07 −1.49131
$$309$$ 0 0
$$310$$ −7616.51 −1.39545
$$311$$ 4913.51 0.895884 0.447942 0.894063i $$-0.352157\pi$$
0.447942 + 0.894063i $$0.352157\pi$$
$$312$$ 0 0
$$313$$ −8104.97 −1.46364 −0.731822 0.681496i $$-0.761330\pi$$
−0.731822 + 0.681496i $$0.761330\pi$$
$$314$$ 6377.41 1.14617
$$315$$ 0 0
$$316$$ −1173.81 −0.208962
$$317$$ −5149.92 −0.912455 −0.456227 0.889863i $$-0.650800\pi$$
−0.456227 + 0.889863i $$0.650800\pi$$
$$318$$ 0 0
$$319$$ −3422.65 −0.600726
$$320$$ −4195.00 −0.732836
$$321$$ 0 0
$$322$$ 641.466 0.111017
$$323$$ −3796.29 −0.653967
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 4965.95 0.843676
$$327$$ 0 0
$$328$$ 4042.55 0.680526
$$329$$ −11119.7 −1.86337
$$330$$ 0 0
$$331$$ 6061.98 1.00664 0.503318 0.864101i $$-0.332112\pi$$
0.503318 + 0.864101i $$0.332112\pi$$
$$332$$ −4740.31 −0.783609
$$333$$ 0 0
$$334$$ −6229.26 −1.02051
$$335$$ −5885.67 −0.959905
$$336$$ 0 0
$$337$$ 3743.50 0.605108 0.302554 0.953132i $$-0.402161\pi$$
0.302554 + 0.953132i $$0.402161\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 2853.70 0.455187
$$341$$ −10423.6 −1.65534
$$342$$ 0 0
$$343$$ 2508.15 0.394832
$$344$$ −2107.52 −0.330319
$$345$$ 0 0
$$346$$ 9839.55 1.52884
$$347$$ 2520.41 0.389921 0.194961 0.980811i $$-0.437542\pi$$
0.194961 + 0.980811i $$0.437542\pi$$
$$348$$ 0 0
$$349$$ 10650.7 1.63359 0.816793 0.576931i $$-0.195750\pi$$
0.816793 + 0.576931i $$0.195750\pi$$
$$350$$ 9243.88 1.41173
$$351$$ 0 0
$$352$$ −8670.70 −1.31293
$$353$$ 9002.82 1.35743 0.678714 0.734403i $$-0.262538\pi$$
0.678714 + 0.734403i $$0.262538\pi$$
$$354$$ 0 0
$$355$$ −5247.10 −0.784471
$$356$$ −10706.5 −1.59395
$$357$$ 0 0
$$358$$ 9017.78 1.33130
$$359$$ 11360.9 1.67021 0.835106 0.550089i $$-0.185406\pi$$
0.835106 + 0.550089i $$0.185406\pi$$
$$360$$ 0 0
$$361$$ −952.336 −0.138845
$$362$$ −10503.5 −1.52500
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 5675.54 0.813894
$$366$$ 0 0
$$367$$ −13938.8 −1.98257 −0.991283 0.131753i $$-0.957939\pi$$
−0.991283 + 0.131753i $$0.957939\pi$$
$$368$$ 287.313 0.0406990
$$369$$ 0 0
$$370$$ −1882.43 −0.264495
$$371$$ 1635.84 0.228918
$$372$$ 0 0
$$373$$ 1593.07 0.221142 0.110571 0.993868i $$-0.464732\pi$$
0.110571 + 0.993868i $$0.464732\pi$$
$$374$$ 7074.32 0.978087
$$375$$ 0 0
$$376$$ 3622.07 0.496793
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 9137.56 1.23843 0.619215 0.785221i $$-0.287451\pi$$
0.619215 + 0.785221i $$0.287451\pi$$
$$380$$ −4440.08 −0.599398
$$381$$ 0 0
$$382$$ 9825.86 1.31606
$$383$$ −9551.07 −1.27425 −0.637124 0.770761i $$-0.719876\pi$$
−0.637124 + 0.770761i $$0.719876\pi$$
$$384$$ 0 0
$$385$$ −4790.71 −0.634174
$$386$$ 14157.8 1.86687
$$387$$ 0 0
$$388$$ −164.213 −0.0214862
$$389$$ −7366.50 −0.960145 −0.480072 0.877229i $$-0.659390\pi$$
−0.480072 + 0.877229i $$0.659390\pi$$
$$390$$ 0 0
$$391$$ −310.779 −0.0401964
$$392$$ 1878.47 0.242033
$$393$$ 0 0
$$394$$ 16309.6 2.08545
$$395$$ −697.596 −0.0888604
$$396$$ 0 0
$$397$$ −11696.5 −1.47866 −0.739332 0.673342i $$-0.764858\pi$$
−0.739332 + 0.673342i $$0.764858\pi$$
$$398$$ −17257.5 −2.17347
$$399$$ 0 0
$$400$$ 4140.34 0.517542
$$401$$ 14167.6 1.76433 0.882167 0.470937i $$-0.156084\pi$$
0.882167 + 0.470937i $$0.156084\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 9445.47 1.16319
$$405$$ 0 0
$$406$$ 10297.0 1.25870
$$407$$ −2576.22 −0.313755
$$408$$ 0 0
$$409$$ 2703.69 0.326868 0.163434 0.986554i $$-0.447743\pi$$
0.163434 + 0.986554i $$0.447743\pi$$
$$410$$ 12738.4 1.53440
$$411$$ 0 0
$$412$$ −26.6759 −0.00318988
$$413$$ 608.011 0.0724414
$$414$$ 0 0
$$415$$ −2817.17 −0.333228
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −11007.0 −1.28796
$$419$$ −7142.52 −0.832781 −0.416390 0.909186i $$-0.636705\pi$$
−0.416390 + 0.909186i $$0.636705\pi$$
$$420$$ 0 0
$$421$$ −3406.45 −0.394347 −0.197174 0.980369i $$-0.563176\pi$$
−0.197174 + 0.980369i $$0.563176\pi$$
$$422$$ 12793.5 1.47578
$$423$$ 0 0
$$424$$ −532.850 −0.0610318
$$425$$ −4478.50 −0.511151
$$426$$ 0 0
$$427$$ 14205.8 1.61000
$$428$$ −13323.0 −1.50466
$$429$$ 0 0
$$430$$ −6640.96 −0.744780
$$431$$ −5172.97 −0.578128 −0.289064 0.957310i $$-0.593344\pi$$
−0.289064 + 0.957310i $$0.593344\pi$$
$$432$$ 0 0
$$433$$ 10955.0 1.21585 0.607924 0.793995i $$-0.292002\pi$$
0.607924 + 0.793995i $$0.292002\pi$$
$$434$$ 31359.4 3.46844
$$435$$ 0 0
$$436$$ 4421.55 0.485674
$$437$$ 483.542 0.0529313
$$438$$ 0 0
$$439$$ 11832.4 1.28640 0.643202 0.765696i $$-0.277605\pi$$
0.643202 + 0.765696i $$0.277605\pi$$
$$440$$ 1560.50 0.169077
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −13479.8 −1.44570 −0.722852 0.691003i $$-0.757169\pi$$
−0.722852 + 0.691003i $$0.757169\pi$$
$$444$$ 0 0
$$445$$ −6362.91 −0.677822
$$446$$ −7288.92 −0.773857
$$447$$ 0 0
$$448$$ 17272.1 1.82149
$$449$$ 6774.34 0.712028 0.356014 0.934481i $$-0.384135\pi$$
0.356014 + 0.934481i $$0.384135\pi$$
$$450$$ 0 0
$$451$$ 17433.2 1.82017
$$452$$ −13792.0 −1.43522
$$453$$ 0 0
$$454$$ 8127.86 0.840219
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −4642.36 −0.475187 −0.237594 0.971365i $$-0.576359\pi$$
−0.237594 + 0.971365i $$0.576359\pi$$
$$458$$ 1580.51 0.161250
$$459$$ 0 0
$$460$$ −363.482 −0.0368422
$$461$$ −2460.37 −0.248570 −0.124285 0.992247i $$-0.539664\pi$$
−0.124285 + 0.992247i $$0.539664\pi$$
$$462$$ 0 0
$$463$$ 4290.01 0.430613 0.215306 0.976547i $$-0.430925\pi$$
0.215306 + 0.976547i $$0.430925\pi$$
$$464$$ 4612.04 0.461441
$$465$$ 0 0
$$466$$ −13077.5 −1.30001
$$467$$ 8798.99 0.871882 0.435941 0.899975i $$-0.356416\pi$$
0.435941 + 0.899975i $$0.356416\pi$$
$$468$$ 0 0
$$469$$ 24233.0 2.38588
$$470$$ 11413.4 1.12013
$$471$$ 0 0
$$472$$ −198.050 −0.0193136
$$473$$ −9088.54 −0.883491
$$474$$ 0 0
$$475$$ 6968.11 0.673092
$$476$$ −11749.5 −1.13138
$$477$$ 0 0
$$478$$ 5160.79 0.493826
$$479$$ 10973.4 1.04673 0.523367 0.852107i $$-0.324676\pi$$
0.523367 + 0.852107i $$0.324676\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 614.470 0.0580671
$$483$$ 0 0
$$484$$ −1799.59 −0.169007
$$485$$ −97.5919 −0.00913694
$$486$$ 0 0
$$487$$ −5209.58 −0.484740 −0.242370 0.970184i $$-0.577925\pi$$
−0.242370 + 0.970184i $$0.577925\pi$$
$$488$$ −4627.33 −0.429241
$$489$$ 0 0
$$490$$ 5919.20 0.545719
$$491$$ 8779.22 0.806926 0.403463 0.914996i $$-0.367806\pi$$
0.403463 + 0.914996i $$0.367806\pi$$
$$492$$ 0 0
$$493$$ −4988.73 −0.455742
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 14045.9 1.27153
$$497$$ 21603.9 1.94983
$$498$$ 0 0
$$499$$ −15590.1 −1.39861 −0.699305 0.714823i $$-0.746507\pi$$
−0.699305 + 0.714823i $$0.746507\pi$$
$$500$$ −12459.5 −1.11441
$$501$$ 0 0
$$502$$ 4165.49 0.370349
$$503$$ 64.3909 0.00570785 0.00285393 0.999996i $$-0.499092\pi$$
0.00285393 + 0.999996i $$0.499092\pi$$
$$504$$ 0 0
$$505$$ 5613.45 0.494644
$$506$$ −901.072 −0.0791651
$$507$$ 0 0
$$508$$ −1178.26 −0.102907
$$509$$ −3214.15 −0.279891 −0.139946 0.990159i $$-0.544693\pi$$
−0.139946 + 0.990159i $$0.544693\pi$$
$$510$$ 0 0
$$511$$ −23367.9 −2.02296
$$512$$ 14554.7 1.25632
$$513$$ 0 0
$$514$$ −12379.5 −1.06232
$$515$$ −15.8535 −0.00135649
$$516$$ 0 0
$$517$$ 15619.9 1.32875
$$518$$ 7750.54 0.657412
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 3053.01 0.256727 0.128363 0.991727i $$-0.459028\pi$$
0.128363 + 0.991727i $$0.459028\pi$$
$$522$$ 0 0
$$523$$ 5096.02 0.426067 0.213034 0.977045i $$-0.431666\pi$$
0.213034 + 0.977045i $$0.431666\pi$$
$$524$$ 22200.7 1.85085
$$525$$ 0 0
$$526$$ −9457.92 −0.784002
$$527$$ −15193.1 −1.25583
$$528$$ 0 0
$$529$$ −12127.4 −0.996747
$$530$$ −1679.05 −0.137610
$$531$$ 0 0
$$532$$ 18281.1 1.48983
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −7917.89 −0.639851
$$536$$ −7893.53 −0.636098
$$537$$ 0 0
$$538$$ −8139.31 −0.652250
$$539$$ 8100.77 0.647356
$$540$$ 0 0
$$541$$ 7861.99 0.624793 0.312397 0.949952i $$-0.398868\pi$$
0.312397 + 0.949952i $$0.398868\pi$$
$$542$$ −15054.4 −1.19307
$$543$$ 0 0
$$544$$ −12638.1 −0.996054
$$545$$ 2627.73 0.206532
$$546$$ 0 0
$$547$$ −6317.48 −0.493814 −0.246907 0.969039i $$-0.579414\pi$$
−0.246907 + 0.969039i $$0.579414\pi$$
$$548$$ −11147.4 −0.868965
$$549$$ 0 0
$$550$$ −12984.9 −1.00669
$$551$$ 7761.98 0.600130
$$552$$ 0 0
$$553$$ 2872.21 0.220866
$$554$$ −6074.63 −0.465860
$$555$$ 0 0
$$556$$ −5869.25 −0.447683
$$557$$ 971.234 0.0738824 0.0369412 0.999317i $$-0.488239\pi$$
0.0369412 + 0.999317i $$0.488239\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 6455.50 0.487134
$$561$$ 0 0
$$562$$ 16538.7 1.24136
$$563$$ −9328.49 −0.698311 −0.349155 0.937065i $$-0.613531\pi$$
−0.349155 + 0.937065i $$0.613531\pi$$
$$564$$ 0 0
$$565$$ −8196.59 −0.610324
$$566$$ −13575.4 −1.00816
$$567$$ 0 0
$$568$$ −7037.12 −0.519843
$$569$$ −17452.2 −1.28582 −0.642911 0.765941i $$-0.722273\pi$$
−0.642911 + 0.765941i $$0.722273\pi$$
$$570$$ 0 0
$$571$$ −20181.4 −1.47910 −0.739548 0.673103i $$-0.764961\pi$$
−0.739548 + 0.673103i $$0.764961\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −52447.8 −3.81381
$$575$$ 570.436 0.0413719
$$576$$ 0 0
$$577$$ 6382.72 0.460513 0.230257 0.973130i $$-0.426043\pi$$
0.230257 + 0.973130i $$0.426043\pi$$
$$578$$ −10451.3 −0.752107
$$579$$ 0 0
$$580$$ −5834.73 −0.417714
$$581$$ 11599.1 0.828250
$$582$$ 0 0
$$583$$ −2297.88 −0.163239
$$584$$ 7611.72 0.539341
$$585$$ 0 0
$$586$$ 20715.2 1.46030
$$587$$ 775.527 0.0545305 0.0272653 0.999628i $$-0.491320\pi$$
0.0272653 + 0.999628i $$0.491320\pi$$
$$588$$ 0 0
$$589$$ 23639.0 1.65370
$$590$$ −624.072 −0.0435468
$$591$$ 0 0
$$592$$ 3471.47 0.241008
$$593$$ 17843.3 1.23564 0.617821 0.786319i $$-0.288016\pi$$
0.617821 + 0.786319i $$0.288016\pi$$
$$594$$ 0 0
$$595$$ −6982.76 −0.481118
$$596$$ 7820.33 0.537472
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 24373.3 1.66255 0.831274 0.555863i $$-0.187612\pi$$
0.831274 + 0.555863i $$0.187612\pi$$
$$600$$ 0 0
$$601$$ −3526.99 −0.239383 −0.119691 0.992811i $$-0.538190\pi$$
−0.119691 + 0.992811i $$0.538190\pi$$
$$602$$ 27342.8 1.85118
$$603$$ 0 0
$$604$$ −1323.28 −0.0891446
$$605$$ −1069.50 −0.0718698
$$606$$ 0 0
$$607$$ 7991.55 0.534377 0.267189 0.963644i $$-0.413905\pi$$
0.267189 + 0.963644i $$0.413905\pi$$
$$608$$ 19663.6 1.31162
$$609$$ 0 0
$$610$$ −14581.1 −0.967821
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −16332.2 −1.07610 −0.538051 0.842912i $$-0.680839\pi$$
−0.538051 + 0.842912i $$0.680839\pi$$
$$614$$ 24513.8 1.61123
$$615$$ 0 0
$$616$$ −6425.04 −0.420247
$$617$$ 19353.6 1.26280 0.631401 0.775457i $$-0.282480\pi$$
0.631401 + 0.775457i $$0.282480\pi$$
$$618$$ 0 0
$$619$$ −9982.52 −0.648193 −0.324096 0.946024i $$-0.605060\pi$$
−0.324096 + 0.946024i $$0.605060\pi$$
$$620$$ −17769.6 −1.15104
$$621$$ 0 0
$$622$$ 20764.8 1.33857
$$623$$ 26198.0 1.68475
$$624$$ 0 0
$$625$$ 3928.53 0.251426
$$626$$ −34252.1 −2.18688
$$627$$ 0 0
$$628$$ 14878.7 0.945423
$$629$$ −3755.00 −0.238031
$$630$$ 0 0
$$631$$ −575.775 −0.0363253 −0.0181626 0.999835i $$-0.505782\pi$$
−0.0181626 + 0.999835i $$0.505782\pi$$
$$632$$ −935.578 −0.0588850
$$633$$ 0 0
$$634$$ −21763.8 −1.36333
$$635$$ −700.240 −0.0437609
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −14464.3 −0.897566
$$639$$ 0 0
$$640$$ −5734.80 −0.354200
$$641$$ 24521.8 1.51100 0.755500 0.655149i $$-0.227394\pi$$
0.755500 + 0.655149i $$0.227394\pi$$
$$642$$ 0 0
$$643$$ −22667.0 −1.39020 −0.695099 0.718914i $$-0.744640\pi$$
−0.695099 + 0.718914i $$0.744640\pi$$
$$644$$ 1496.56 0.0915727
$$645$$ 0 0
$$646$$ −16043.3 −0.977116
$$647$$ 2397.45 0.145678 0.0728389 0.997344i $$-0.476794\pi$$
0.0728389 + 0.997344i $$0.476794\pi$$
$$648$$ 0 0
$$649$$ −854.078 −0.0516572
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 11585.7 0.695908
$$653$$ −20002.1 −1.19868 −0.599342 0.800493i $$-0.704571\pi$$
−0.599342 + 0.800493i $$0.704571\pi$$
$$654$$ 0 0
$$655$$ 13193.9 0.787068
$$656$$ −23491.4 −1.39815
$$657$$ 0 0
$$658$$ −46992.5 −2.78413
$$659$$ 3517.96 0.207952 0.103976 0.994580i $$-0.466844\pi$$
0.103976 + 0.994580i $$0.466844\pi$$
$$660$$ 0 0
$$661$$ 13583.4 0.799294 0.399647 0.916669i $$-0.369133\pi$$
0.399647 + 0.916669i $$0.369133\pi$$
$$662$$ 25618.2 1.50405
$$663$$ 0 0
$$664$$ −3778.24 −0.220819
$$665$$ 10864.5 0.633544
$$666$$ 0 0
$$667$$ 635.425 0.0368872
$$668$$ −14533.1 −0.841770
$$669$$ 0 0
$$670$$ −24873.1 −1.43423
$$671$$ −19955.1 −1.14807
$$672$$ 0 0
$$673$$ −10895.8 −0.624077 −0.312038 0.950069i $$-0.601012\pi$$
−0.312038 + 0.950069i $$0.601012\pi$$
$$674$$ 15820.2 0.904113
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −1449.03 −0.0822609 −0.0411305 0.999154i $$-0.513096\pi$$
−0.0411305 + 0.999154i $$0.513096\pi$$
$$678$$ 0 0
$$679$$ 401.815 0.0227102
$$680$$ 2274.52 0.128271
$$681$$ 0 0
$$682$$ −44050.9 −2.47331
$$683$$ 15366.4 0.860878 0.430439 0.902620i $$-0.358359\pi$$
0.430439 + 0.902620i $$0.358359\pi$$
$$684$$ 0 0
$$685$$ −6624.91 −0.369525
$$686$$ 10599.6 0.589933
$$687$$ 0 0
$$688$$ 12246.9 0.678644
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −2019.71 −0.111191 −0.0555957 0.998453i $$-0.517706\pi$$
−0.0555957 + 0.998453i $$0.517706\pi$$
$$692$$ 22956.0 1.26106
$$693$$ 0 0
$$694$$ 10651.4 0.582595
$$695$$ −3488.10 −0.190376
$$696$$ 0 0
$$697$$ 25410.0 1.38088
$$698$$ 45010.6 2.44080
$$699$$ 0 0
$$700$$ 21566.3 1.16447
$$701$$ −28031.6 −1.51033 −0.755164 0.655536i $$-0.772443\pi$$
−0.755164 + 0.655536i $$0.772443\pi$$
$$702$$ 0 0
$$703$$ 5842.42 0.313444
$$704$$ −24262.2 −1.29889
$$705$$ 0 0
$$706$$ 38046.4 2.02818
$$707$$ −23112.3 −1.22946
$$708$$ 0 0
$$709$$ 19604.4 1.03845 0.519224 0.854638i $$-0.326221\pi$$
0.519224 + 0.854638i $$0.326221\pi$$
$$710$$ −22174.5 −1.17211
$$711$$ 0 0
$$712$$ −8533.59 −0.449171
$$713$$ 1935.18 0.101645
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 21038.8 1.09812
$$717$$ 0 0
$$718$$ 48011.8 2.49552
$$719$$ 14727.4 0.763894 0.381947 0.924184i $$-0.375254\pi$$
0.381947 + 0.924184i $$0.375254\pi$$
$$720$$ 0 0
$$721$$ 65.2737 0.00337160
$$722$$ −4024.62 −0.207453
$$723$$ 0 0
$$724$$ −24505.0 −1.25790
$$725$$ 9156.83 0.469071
$$726$$ 0 0
$$727$$ −16890.5 −0.861668 −0.430834 0.902431i $$-0.641781\pi$$
−0.430834 + 0.902431i $$0.641781\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 23985.1 1.21607
$$731$$ −13247.1 −0.670263
$$732$$ 0 0
$$733$$ 12553.6 0.632578 0.316289 0.948663i $$-0.397563\pi$$
0.316289 + 0.948663i $$0.397563\pi$$
$$734$$ −58906.3 −2.96222
$$735$$ 0 0
$$736$$ 1609.74 0.0806194
$$737$$ −34040.3 −1.70135
$$738$$ 0 0
$$739$$ −37874.8 −1.88531 −0.942656 0.333766i $$-0.891680\pi$$
−0.942656 + 0.333766i $$0.891680\pi$$
$$740$$ −4391.78 −0.218169
$$741$$ 0 0
$$742$$ 6913.16 0.342035
$$743$$ −35882.6 −1.77174 −0.885872 0.463929i $$-0.846439\pi$$
−0.885872 + 0.463929i $$0.846439\pi$$
$$744$$ 0 0
$$745$$ 4647.63 0.228559
$$746$$ 6732.40 0.330416
$$747$$ 0 0
$$748$$ 16504.6 0.806777
$$749$$ 32600.3 1.59037
$$750$$ 0 0
$$751$$ −181.689 −0.00882815 −0.00441407 0.999990i $$-0.501405\pi$$
−0.00441407 + 0.999990i $$0.501405\pi$$
$$752$$ −21048.0 −1.02067
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −786.425 −0.0379085
$$756$$ 0 0
$$757$$ −491.053 −0.0235768 −0.0117884 0.999931i $$-0.503752\pi$$
−0.0117884 + 0.999931i $$0.503752\pi$$
$$758$$ 38615.8 1.85038
$$759$$ 0 0
$$760$$ −3538.94 −0.168909
$$761$$ 8113.01 0.386460 0.193230 0.981153i $$-0.438104\pi$$
0.193230 + 0.981153i $$0.438104\pi$$
$$762$$ 0 0
$$763$$ −10819.2 −0.513342
$$764$$ 22924.1 1.08555
$$765$$ 0 0
$$766$$ −40363.3 −1.90390
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 19864.7 0.931519 0.465759 0.884911i $$-0.345781\pi$$
0.465759 + 0.884911i $$0.345781\pi$$
$$770$$ −20245.8 −0.947542
$$771$$ 0 0
$$772$$ 33030.6 1.53989
$$773$$ −9047.42 −0.420974 −0.210487 0.977597i $$-0.567505\pi$$
−0.210487 + 0.977597i $$0.567505\pi$$
$$774$$ 0 0
$$775$$ 27887.0 1.29256
$$776$$ −130.885 −0.00605476
$$777$$ 0 0
$$778$$ −31131.2 −1.43459
$$779$$ −39535.5 −1.81837
$$780$$ 0 0
$$781$$ −30347.1 −1.39040
$$782$$ −1313.37 −0.0600588
$$783$$ 0 0
$$784$$ −10915.8 −0.497259
$$785$$ 8842.45 0.402039
$$786$$ 0 0
$$787$$ −15018.4 −0.680240 −0.340120 0.940382i $$-0.610468\pi$$
−0.340120 + 0.940382i $$0.610468\pi$$
$$788$$ 38050.9 1.72019
$$789$$ 0 0
$$790$$ −2948.08 −0.132770
$$791$$ 33747.8 1.51698
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −49429.9 −2.20932
$$795$$ 0 0
$$796$$ −40262.4 −1.79279
$$797$$ −31941.1 −1.41959 −0.709794 0.704409i $$-0.751212\pi$$
−0.709794 + 0.704409i $$0.751212\pi$$
$$798$$ 0 0
$$799$$ 22767.1 1.00806
$$800$$ 23197.3 1.02518
$$801$$ 0 0
$$802$$ 59873.2 2.63615
$$803$$ 32825.0 1.44255
$$804$$ 0 0
$$805$$ 889.409 0.0389411
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 7528.46 0.327785
$$809$$ −27260.1 −1.18469 −0.592344 0.805685i $$-0.701797\pi$$
−0.592344 + 0.805685i $$0.701797\pi$$
$$810$$ 0 0
$$811$$ −20707.8 −0.896607 −0.448303 0.893881i $$-0.647972\pi$$
−0.448303 + 0.893881i $$0.647972\pi$$
$$812$$ 24023.3 1.03824
$$813$$ 0 0
$$814$$ −10887.2 −0.468793
$$815$$ 6885.42 0.295933
$$816$$ 0 0
$$817$$ 20611.2 0.882614
$$818$$ 11426.0 0.488385
$$819$$ 0 0
$$820$$ 29719.1 1.26565
$$821$$ −15658.6 −0.665636 −0.332818 0.942991i $$-0.608000\pi$$
−0.332818 + 0.942991i $$0.608000\pi$$
$$822$$ 0 0
$$823$$ −4106.58 −0.173932 −0.0869662 0.996211i $$-0.527717\pi$$
−0.0869662 + 0.996211i $$0.527717\pi$$
$$824$$ −21.2619 −0.000898900 0
$$825$$ 0 0
$$826$$ 2569.49 0.108237
$$827$$ −16747.3 −0.704184 −0.352092 0.935965i $$-0.614530\pi$$
−0.352092 + 0.935965i $$0.614530\pi$$
$$828$$ 0 0
$$829$$ −29157.1 −1.22155 −0.610776 0.791803i $$-0.709142\pi$$
−0.610776 + 0.791803i $$0.709142\pi$$
$$830$$ −11905.5 −0.497887
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 11807.4 0.491119
$$834$$ 0 0
$$835$$ −8637.03 −0.357960
$$836$$ −25679.6 −1.06238
$$837$$ 0 0
$$838$$ −30184.7 −1.24429
$$839$$ −45819.4 −1.88541 −0.942707 0.333621i $$-0.891729\pi$$
−0.942707 + 0.333621i $$0.891729\pi$$
$$840$$ 0 0
$$841$$ −14188.9 −0.581776
$$842$$ −14395.8 −0.589209
$$843$$ 0 0
$$844$$ 29847.7 1.21730
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 4403.44 0.178635
$$848$$ 3096.41 0.125390
$$849$$ 0 0
$$850$$ −18926.4 −0.763729
$$851$$ 478.283 0.0192660
$$852$$ 0 0
$$853$$ 17351.1 0.696471 0.348235 0.937407i $$-0.386781\pi$$
0.348235 + 0.937407i $$0.386781\pi$$
$$854$$ 60034.7 2.40556
$$855$$ 0 0
$$856$$ −10619.0 −0.424009
$$857$$ 21768.1 0.867659 0.433829 0.900995i $$-0.357162\pi$$
0.433829 + 0.900995i $$0.357162\pi$$
$$858$$ 0 0
$$859$$ −29878.4 −1.18677 −0.593387 0.804918i $$-0.702209\pi$$
−0.593387 + 0.804918i $$0.702209\pi$$
$$860$$ −15493.6 −0.614334
$$861$$ 0 0
$$862$$ −21861.2 −0.863801
$$863$$ 15067.7 0.594335 0.297168 0.954825i $$-0.403958\pi$$
0.297168 + 0.954825i $$0.403958\pi$$
$$864$$ 0 0
$$865$$ 13642.8 0.536265
$$866$$ 46296.3 1.81664
$$867$$ 0 0
$$868$$ 73162.7 2.86095
$$869$$ −4034.62 −0.157497
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 3524.17 0.136862
$$873$$ 0 0
$$874$$ 2043.48 0.0790865
$$875$$ 30487.4 1.17790
$$876$$ 0 0
$$877$$ −21119.5 −0.813177 −0.406588 0.913611i $$-0.633282\pi$$
−0.406588 + 0.913611i $$0.633282\pi$$
$$878$$ 50004.5 1.92206
$$879$$ 0 0
$$880$$ −9068.09 −0.347370
$$881$$ −31652.5 −1.21044 −0.605221 0.796057i $$-0.706915\pi$$
−0.605221 + 0.796057i $$0.706915\pi$$
$$882$$ 0 0
$$883$$ 11701.6 0.445969 0.222984 0.974822i $$-0.428420\pi$$
0.222984 + 0.974822i $$0.428420\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −56966.6 −2.16008
$$887$$ −22507.9 −0.852020 −0.426010 0.904719i $$-0.640081\pi$$
−0.426010 + 0.904719i $$0.640081\pi$$
$$888$$ 0 0
$$889$$ 2883.10 0.108769
$$890$$ −26890.0 −1.01276
$$891$$ 0 0
$$892$$ −17005.3 −0.638318
$$893$$ −35423.3 −1.32743
$$894$$ 0 0
$$895$$ 12503.4 0.466974
$$896$$ 23611.9 0.880377
$$897$$ 0 0
$$898$$ 28628.7 1.06387
$$899$$ 31064.1 1.15244
$$900$$ 0 0
$$901$$ −3349.31 −0.123842
$$902$$ 73673.8 2.71959
$$903$$ 0 0
$$904$$ −10992.8 −0.404442
$$905$$ −14563.3 −0.534919
$$906$$ 0 0
$$907$$ 18718.6 0.685273 0.342636 0.939468i $$-0.388680\pi$$
0.342636 + 0.939468i $$0.388680\pi$$
$$908$$ 18962.6 0.693057
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −18616.7 −0.677057 −0.338529 0.940956i $$-0.609929\pi$$
−0.338529 + 0.940956i $$0.609929\pi$$
$$912$$ 0 0
$$913$$ −16293.4 −0.590616
$$914$$ −19618.9 −0.709994
$$915$$ 0 0
$$916$$ 3687.39 0.133008
$$917$$ −54323.3 −1.95629
$$918$$ 0 0
$$919$$ −54764.4 −1.96573 −0.982867 0.184316i $$-0.940993\pi$$
−0.982867 + 0.184316i $$0.940993\pi$$
$$920$$ −289.711 −0.0103821
$$921$$ 0 0
$$922$$ −10397.6 −0.371397
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 6892.32 0.244993
$$926$$ 18129.8 0.643394
$$927$$ 0 0
$$928$$ 25840.1 0.914055
$$929$$ 31832.1 1.12419 0.562097 0.827071i $$-0.309995\pi$$
0.562097 + 0.827071i $$0.309995\pi$$
$$930$$ 0 0
$$931$$ −18371.2 −0.646713
$$932$$ −30510.3 −1.07231
$$933$$ 0 0
$$934$$ 37185.0 1.30271
$$935$$ 9808.73 0.343080
$$936$$ 0 0
$$937$$ −27408.7 −0.955607 −0.477803 0.878467i $$-0.658567\pi$$
−0.477803 + 0.878467i $$0.658567\pi$$
$$938$$ 102410. 3.56483
$$939$$ 0 0
$$940$$ 26627.9 0.923945
$$941$$ −54837.8 −1.89975 −0.949874 0.312634i $$-0.898789\pi$$
−0.949874 + 0.312634i $$0.898789\pi$$
$$942$$ 0 0
$$943$$ −3236.53 −0.111767
$$944$$ 1150.87 0.0396799
$$945$$ 0 0
$$946$$ −38408.6 −1.32006
$$947$$ 39707.8 1.36255 0.681273 0.732030i $$-0.261427\pi$$
0.681273 + 0.732030i $$0.261427\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 29447.6 1.00569
$$951$$ 0 0
$$952$$ −9364.89 −0.318821
$$953$$ 17106.6 0.581468 0.290734 0.956804i $$-0.406101\pi$$
0.290734 + 0.956804i $$0.406101\pi$$
$$954$$ 0 0
$$955$$ 13623.8 0.461629
$$956$$ 12040.3 0.407334
$$957$$ 0 0
$$958$$ 46374.0 1.56396
$$959$$ 27276.7 0.918469
$$960$$ 0 0
$$961$$ 64814.4 2.17564
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 1433.58 0.0478968
$$965$$ 19630.1 0.654835
$$966$$ 0 0
$$967$$ −23417.5 −0.778756 −0.389378 0.921078i $$-0.627310\pi$$
−0.389378 + 0.921078i $$0.627310\pi$$
$$968$$ −1434.35 −0.0476258
$$969$$ 0 0
$$970$$ −412.428 −0.0136518
$$971$$ −16430.6 −0.543032 −0.271516 0.962434i $$-0.587525\pi$$
−0.271516 + 0.962434i $$0.587525\pi$$
$$972$$ 0 0
$$973$$ 14361.6 0.473187
$$974$$ −22015.9 −0.724267
$$975$$ 0 0
$$976$$ 26889.5 0.881879
$$977$$ 10554.6 0.345622 0.172811 0.984955i $$-0.444715\pi$$
0.172811 + 0.984955i $$0.444715\pi$$
$$978$$ 0 0
$$979$$ −36800.5 −1.20138
$$980$$ 13809.7 0.450138
$$981$$ 0 0
$$982$$ 37101.5 1.20566
$$983$$ −1534.33 −0.0497839 −0.0248919 0.999690i $$-0.507924\pi$$
−0.0248919 + 0.999690i $$0.507924\pi$$
$$984$$ 0 0
$$985$$ 22613.7 0.731505
$$986$$ −21082.6 −0.680941
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1687.31 0.0542502
$$990$$ 0 0
$$991$$ −18018.2 −0.577563 −0.288782 0.957395i $$-0.593250\pi$$
−0.288782 + 0.957395i $$0.593250\pi$$
$$992$$ 78695.7 2.51874
$$993$$ 0 0
$$994$$ 91299.1 2.91331
$$995$$ −23928.0 −0.762380
$$996$$ 0 0
$$997$$ 48287.7 1.53389 0.766944 0.641714i $$-0.221776\pi$$
0.766944 + 0.641714i $$0.221776\pi$$
$$998$$ −65884.4 −2.08971
$$999$$ 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.v.1.4 4
3.2 odd 2 507.4.a.m.1.1 4
13.3 even 3 117.4.g.e.100.1 8
13.9 even 3 117.4.g.e.55.1 8
13.12 even 2 1521.4.a.bb.1.1 4
39.5 even 4 507.4.b.h.337.7 8
39.8 even 4 507.4.b.h.337.2 8
39.29 odd 6 39.4.e.c.22.4 yes 8
39.35 odd 6 39.4.e.c.16.4 8
39.38 odd 2 507.4.a.i.1.4 4
156.35 even 6 624.4.q.i.289.2 8
156.107 even 6 624.4.q.i.529.2 8

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.c.16.4 8 39.35 odd 6
39.4.e.c.22.4 yes 8 39.29 odd 6
117.4.g.e.55.1 8 13.9 even 3
117.4.g.e.100.1 8 13.3 even 3
507.4.a.i.1.4 4 39.38 odd 2
507.4.a.m.1.1 4 3.2 odd 2
507.4.b.h.337.2 8 39.8 even 4
507.4.b.h.337.7 8 39.5 even 4
624.4.q.i.289.2 8 156.35 even 6
624.4.q.i.529.2 8 156.107 even 6
1521.4.a.v.1.4 4 1.1 even 1 trivial
1521.4.a.bb.1.1 4 13.12 even 2