# Properties

 Label 525.4.d.l.274.4 Level 525 Weight 4 Character 525.274 Analytic conductor 30.976 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 525.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.9760027530$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 274.4 Root $$0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 525.274 Dual form 525.4.d.l.274.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.82843i q^{2} -3.00000i q^{3} -6.65685 q^{4} +11.4853 q^{6} +7.00000i q^{7} +5.14214i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q+3.82843i q^{2} -3.00000i q^{3} -6.65685 q^{4} +11.4853 q^{6} +7.00000i q^{7} +5.14214i q^{8} -9.00000 q^{9} +48.5685 q^{11} +19.9706i q^{12} -43.6569i q^{13} -26.7990 q^{14} -72.9411 q^{16} +67.6569i q^{17} -34.4558i q^{18} +93.2548 q^{19} +21.0000 q^{21} +185.941i q^{22} -104.167i q^{23} +15.4264 q^{24} +167.137 q^{26} +27.0000i q^{27} -46.5980i q^{28} +58.7351 q^{29} -9.08831 q^{31} -238.113i q^{32} -145.706i q^{33} -259.019 q^{34} +59.9117 q^{36} +252.676i q^{37} +357.019i q^{38} -130.971 q^{39} +276.274 q^{41} +80.3970i q^{42} -92.6375i q^{43} -323.314 q^{44} +398.794 q^{46} +582.794i q^{47} +218.823i q^{48} -49.0000 q^{49} +202.971 q^{51} +290.617i q^{52} +623.019i q^{53} -103.368 q^{54} -35.9949 q^{56} -279.765i q^{57} +224.863i q^{58} +524.999 q^{59} -352.794 q^{61} -34.7939i q^{62} -63.0000i q^{63} +328.068 q^{64} +557.823 q^{66} +736.520i q^{67} -450.382i q^{68} -312.500 q^{69} -492.264 q^{71} -46.2792i q^{72} +1164.75i q^{73} -967.352 q^{74} -620.784 q^{76} +339.980i q^{77} -501.411i q^{78} +872.195 q^{79} +81.0000 q^{81} +1057.70i q^{82} -529.588i q^{83} -139.794 q^{84} +354.656 q^{86} -176.205i q^{87} +249.746i q^{88} +385.216 q^{89} +305.598 q^{91} +693.421i q^{92} +27.2649i q^{93} -2231.18 q^{94} -714.338 q^{96} +463.892i q^{97} -187.593i q^{98} -437.117 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 12q^{6} - 36q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 12q^{6} - 36q^{9} - 32q^{11} - 28q^{14} - 156q^{16} + 192q^{19} + 84q^{21} - 108q^{24} + 216q^{26} - 376q^{29} - 240q^{31} - 312q^{34} + 36q^{36} - 456q^{39} + 200q^{41} - 1248q^{44} + 1120q^{46} - 196q^{49} + 744q^{51} - 108q^{54} + 252q^{56} - 208q^{59} - 936q^{61} - 68q^{64} + 1824q^{66} - 96q^{69} - 272q^{71} - 2376q^{74} - 1216q^{76} + 864q^{79} + 324q^{81} - 84q^{84} - 912q^{86} + 2808q^{89} + 1064q^{91} - 3200q^{94} - 2484q^{96} + 288q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.82843i 1.35355i 0.736188 + 0.676777i $$0.236624\pi$$
−0.736188 + 0.676777i $$0.763376\pi$$
$$3$$ − 3.00000i − 0.577350i
$$4$$ −6.65685 −0.832107
$$5$$ 0 0
$$6$$ 11.4853 0.781474
$$7$$ 7.00000i 0.377964i
$$8$$ 5.14214i 0.227252i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 48.5685 1.33127 0.665635 0.746278i $$-0.268161\pi$$
0.665635 + 0.746278i $$0.268161\pi$$
$$12$$ 19.9706i 0.480417i
$$13$$ − 43.6569i − 0.931403i −0.884942 0.465701i $$-0.845802\pi$$
0.884942 0.465701i $$-0.154198\pi$$
$$14$$ −26.7990 −0.511595
$$15$$ 0 0
$$16$$ −72.9411 −1.13971
$$17$$ 67.6569i 0.965247i 0.875828 + 0.482623i $$0.160316\pi$$
−0.875828 + 0.482623i $$0.839684\pi$$
$$18$$ − 34.4558i − 0.451184i
$$19$$ 93.2548 1.12601 0.563003 0.826455i $$-0.309646\pi$$
0.563003 + 0.826455i $$0.309646\pi$$
$$20$$ 0 0
$$21$$ 21.0000 0.218218
$$22$$ 185.941i 1.80194i
$$23$$ − 104.167i − 0.944357i −0.881503 0.472179i $$-0.843468\pi$$
0.881503 0.472179i $$-0.156532\pi$$
$$24$$ 15.4264 0.131204
$$25$$ 0 0
$$26$$ 167.137 1.26070
$$27$$ 27.0000i 0.192450i
$$28$$ − 46.5980i − 0.314507i
$$29$$ 58.7351 0.376098 0.188049 0.982160i $$-0.439784\pi$$
0.188049 + 0.982160i $$0.439784\pi$$
$$30$$ 0 0
$$31$$ −9.08831 −0.0526551 −0.0263276 0.999653i $$-0.508381\pi$$
−0.0263276 + 0.999653i $$0.508381\pi$$
$$32$$ − 238.113i − 1.31540i
$$33$$ − 145.706i − 0.768609i
$$34$$ −259.019 −1.30651
$$35$$ 0 0
$$36$$ 59.9117 0.277369
$$37$$ 252.676i 1.12269i 0.827580 + 0.561347i $$0.189717\pi$$
−0.827580 + 0.561347i $$0.810283\pi$$
$$38$$ 357.019i 1.52411i
$$39$$ −130.971 −0.537745
$$40$$ 0 0
$$41$$ 276.274 1.05236 0.526180 0.850373i $$-0.323624\pi$$
0.526180 + 0.850373i $$0.323624\pi$$
$$42$$ 80.3970i 0.295370i
$$43$$ − 92.6375i − 0.328537i −0.986416 0.164268i $$-0.947474\pi$$
0.986416 0.164268i $$-0.0525263\pi$$
$$44$$ −323.314 −1.10776
$$45$$ 0 0
$$46$$ 398.794 1.27824
$$47$$ 582.794i 1.80871i 0.426784 + 0.904354i $$0.359646\pi$$
−0.426784 + 0.904354i $$0.640354\pi$$
$$48$$ 218.823i 0.658009i
$$49$$ −49.0000 −0.142857
$$50$$ 0 0
$$51$$ 202.971 0.557286
$$52$$ 290.617i 0.775026i
$$53$$ 623.019i 1.61468i 0.590083 + 0.807342i $$0.299095\pi$$
−0.590083 + 0.807342i $$0.700905\pi$$
$$54$$ −103.368 −0.260491
$$55$$ 0 0
$$56$$ −35.9949 −0.0858933
$$57$$ − 279.765i − 0.650100i
$$58$$ 224.863i 0.509068i
$$59$$ 524.999 1.15846 0.579229 0.815165i $$-0.303354\pi$$
0.579229 + 0.815165i $$0.303354\pi$$
$$60$$ 0 0
$$61$$ −352.794 −0.740502 −0.370251 0.928932i $$-0.620728\pi$$
−0.370251 + 0.928932i $$0.620728\pi$$
$$62$$ − 34.7939i − 0.0712715i
$$63$$ − 63.0000i − 0.125988i
$$64$$ 328.068 0.640758
$$65$$ 0 0
$$66$$ 557.823 1.04035
$$67$$ 736.520i 1.34299i 0.741010 + 0.671494i $$0.234347\pi$$
−0.741010 + 0.671494i $$0.765653\pi$$
$$68$$ − 450.382i − 0.803188i
$$69$$ −312.500 −0.545225
$$70$$ 0 0
$$71$$ −492.264 −0.822831 −0.411415 0.911448i $$-0.634965\pi$$
−0.411415 + 0.911448i $$0.634965\pi$$
$$72$$ − 46.2792i − 0.0757508i
$$73$$ 1164.75i 1.86745i 0.357987 + 0.933727i $$0.383463\pi$$
−0.357987 + 0.933727i $$0.616537\pi$$
$$74$$ −967.352 −1.51963
$$75$$ 0 0
$$76$$ −620.784 −0.936958
$$77$$ 339.980i 0.503173i
$$78$$ − 501.411i − 0.727867i
$$79$$ 872.195 1.24215 0.621074 0.783752i $$-0.286697\pi$$
0.621074 + 0.783752i $$0.286697\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 1057.70i 1.42443i
$$83$$ − 529.588i − 0.700359i −0.936683 0.350180i $$-0.886121\pi$$
0.936683 0.350180i $$-0.113879\pi$$
$$84$$ −139.794 −0.181581
$$85$$ 0 0
$$86$$ 354.656 0.444692
$$87$$ − 176.205i − 0.217140i
$$88$$ 249.746i 0.302534i
$$89$$ 385.216 0.458796 0.229398 0.973333i $$-0.426324\pi$$
0.229398 + 0.973333i $$0.426324\pi$$
$$90$$ 0 0
$$91$$ 305.598 0.352037
$$92$$ 693.421i 0.785806i
$$93$$ 27.2649i 0.0304005i
$$94$$ −2231.18 −2.44818
$$95$$ 0 0
$$96$$ −714.338 −0.759446
$$97$$ 463.892i 0.485579i 0.970079 + 0.242789i $$0.0780624\pi$$
−0.970079 + 0.242789i $$0.921938\pi$$
$$98$$ − 187.593i − 0.193365i
$$99$$ −437.117 −0.443757
$$100$$ 0 0
$$101$$ −432.725 −0.426314 −0.213157 0.977018i $$-0.568375\pi$$
−0.213157 + 0.977018i $$0.568375\pi$$
$$102$$ 777.058i 0.754316i
$$103$$ 512.626i 0.490393i 0.969473 + 0.245197i $$0.0788525\pi$$
−0.969473 + 0.245197i $$0.921147\pi$$
$$104$$ 224.489 0.211663
$$105$$ 0 0
$$106$$ −2385.18 −2.18556
$$107$$ − 1963.09i − 1.77363i −0.462122 0.886817i $$-0.652912\pi$$
0.462122 0.886817i $$-0.347088\pi$$
$$108$$ − 179.735i − 0.160139i
$$109$$ −545.176 −0.479068 −0.239534 0.970888i $$-0.576995\pi$$
−0.239534 + 0.970888i $$0.576995\pi$$
$$110$$ 0 0
$$111$$ 758.029 0.648188
$$112$$ − 510.588i − 0.430768i
$$113$$ − 231.823i − 0.192992i −0.995333 0.0964961i $$-0.969236\pi$$
0.995333 0.0964961i $$-0.0307635\pi$$
$$114$$ 1071.06 0.879945
$$115$$ 0 0
$$116$$ −390.991 −0.312953
$$117$$ 392.912i 0.310468i
$$118$$ 2009.92i 1.56804i
$$119$$ −473.598 −0.364829
$$120$$ 0 0
$$121$$ 1027.90 0.772279
$$122$$ − 1350.65i − 1.00231i
$$123$$ − 828.823i − 0.607581i
$$124$$ 60.4996 0.0438147
$$125$$ 0 0
$$126$$ 241.191 0.170532
$$127$$ − 2372.90i − 1.65796i −0.559280 0.828979i $$-0.688922\pi$$
0.559280 0.828979i $$-0.311078\pi$$
$$128$$ − 648.917i − 0.448099i
$$129$$ −277.913 −0.189681
$$130$$ 0 0
$$131$$ 1200.04 0.800364 0.400182 0.916436i $$-0.368947\pi$$
0.400182 + 0.916436i $$0.368947\pi$$
$$132$$ 969.941i 0.639565i
$$133$$ 652.784i 0.425591i
$$134$$ −2819.71 −1.81781
$$135$$ 0 0
$$136$$ −347.901 −0.219355
$$137$$ − 2781.25i − 1.73444i −0.497924 0.867221i $$-0.665904\pi$$
0.497924 0.867221i $$-0.334096\pi$$
$$138$$ − 1196.38i − 0.737991i
$$139$$ 1245.60 0.760078 0.380039 0.924971i $$-0.375911\pi$$
0.380039 + 0.924971i $$0.375911\pi$$
$$140$$ 0 0
$$141$$ 1748.38 1.04426
$$142$$ − 1884.60i − 1.11375i
$$143$$ − 2120.35i − 1.23995i
$$144$$ 656.470 0.379902
$$145$$ 0 0
$$146$$ −4459.17 −2.52770
$$147$$ 147.000i 0.0824786i
$$148$$ − 1682.03i − 0.934202i
$$149$$ −19.4046 −0.0106690 −0.00533452 0.999986i $$-0.501698\pi$$
−0.00533452 + 0.999986i $$0.501698\pi$$
$$150$$ 0 0
$$151$$ −2349.80 −1.26638 −0.633192 0.773995i $$-0.718256\pi$$
−0.633192 + 0.773995i $$0.718256\pi$$
$$152$$ 479.529i 0.255888i
$$153$$ − 608.912i − 0.321749i
$$154$$ −1301.59 −0.681071
$$155$$ 0 0
$$156$$ 871.852 0.447462
$$157$$ 3898.46i 1.98172i 0.134880 + 0.990862i $$0.456935\pi$$
−0.134880 + 0.990862i $$0.543065\pi$$
$$158$$ 3339.14i 1.68131i
$$159$$ 1869.06 0.932239
$$160$$ 0 0
$$161$$ 729.166 0.356934
$$162$$ 310.103i 0.150395i
$$163$$ 1527.54i 0.734024i 0.930216 + 0.367012i $$0.119619\pi$$
−0.930216 + 0.367012i $$0.880381\pi$$
$$164$$ −1839.12 −0.875676
$$165$$ 0 0
$$166$$ 2027.49 0.947974
$$167$$ 998.518i 0.462681i 0.972873 + 0.231340i $$0.0743111\pi$$
−0.972873 + 0.231340i $$0.925689\pi$$
$$168$$ 107.985i 0.0495905i
$$169$$ 291.079 0.132489
$$170$$ 0 0
$$171$$ −839.294 −0.375336
$$172$$ 616.674i 0.273378i
$$173$$ 685.253i 0.301149i 0.988599 + 0.150575i $$0.0481124\pi$$
−0.988599 + 0.150575i $$0.951888\pi$$
$$174$$ 674.589 0.293911
$$175$$ 0 0
$$176$$ −3542.64 −1.51725
$$177$$ − 1575.00i − 0.668836i
$$178$$ 1474.77i 0.621005i
$$179$$ −1025.58 −0.428245 −0.214122 0.976807i $$-0.568689\pi$$
−0.214122 + 0.976807i $$0.568689\pi$$
$$180$$ 0 0
$$181$$ 2899.40 1.19067 0.595333 0.803479i $$-0.297020\pi$$
0.595333 + 0.803479i $$0.297020\pi$$
$$182$$ 1169.96i 0.476501i
$$183$$ 1058.38i 0.427529i
$$184$$ 535.638 0.214608
$$185$$ 0 0
$$186$$ −104.382 −0.0411486
$$187$$ 3285.99i 1.28500i
$$188$$ − 3879.57i − 1.50504i
$$189$$ −189.000 −0.0727393
$$190$$ 0 0
$$191$$ 1074.18 0.406939 0.203469 0.979081i $$-0.434778\pi$$
0.203469 + 0.979081i $$0.434778\pi$$
$$192$$ − 984.204i − 0.369942i
$$193$$ 898.999i 0.335292i 0.985847 + 0.167646i $$0.0536166\pi$$
−0.985847 + 0.167646i $$0.946383\pi$$
$$194$$ −1775.98 −0.657257
$$195$$ 0 0
$$196$$ 326.186 0.118872
$$197$$ − 3063.63i − 1.10799i −0.832519 0.553996i $$-0.813102\pi$$
0.832519 0.553996i $$-0.186898\pi$$
$$198$$ − 1673.47i − 0.600648i
$$199$$ 949.522 0.338240 0.169120 0.985595i $$-0.445907\pi$$
0.169120 + 0.985595i $$0.445907\pi$$
$$200$$ 0 0
$$201$$ 2209.56 0.775375
$$202$$ − 1656.66i − 0.577039i
$$203$$ 411.145i 0.142151i
$$204$$ −1351.15 −0.463721
$$205$$ 0 0
$$206$$ −1962.55 −0.663774
$$207$$ 937.499i 0.314786i
$$208$$ 3184.38i 1.06152i
$$209$$ 4529.25 1.49902
$$210$$ 0 0
$$211$$ 2306.64 0.752587 0.376294 0.926500i $$-0.377198\pi$$
0.376294 + 0.926500i $$0.377198\pi$$
$$212$$ − 4147.35i − 1.34359i
$$213$$ 1476.79i 0.475062i
$$214$$ 7515.53 2.40071
$$215$$ 0 0
$$216$$ −138.838 −0.0437348
$$217$$ − 63.6182i − 0.0199018i
$$218$$ − 2087.17i − 0.648444i
$$219$$ 3494.26 1.07817
$$220$$ 0 0
$$221$$ 2953.69 0.899033
$$222$$ 2902.06i 0.877357i
$$223$$ − 3227.61i − 0.969222i −0.874730 0.484611i $$-0.838961\pi$$
0.874730 0.484611i $$-0.161039\pi$$
$$224$$ 1666.79 0.497174
$$225$$ 0 0
$$226$$ 887.519 0.261225
$$227$$ 637.820i 0.186492i 0.995643 + 0.0932458i $$0.0297242\pi$$
−0.995643 + 0.0932458i $$0.970276\pi$$
$$228$$ 1862.35i 0.540953i
$$229$$ −544.774 −0.157204 −0.0786019 0.996906i $$-0.525046\pi$$
−0.0786019 + 0.996906i $$0.525046\pi$$
$$230$$ 0 0
$$231$$ 1019.94 0.290507
$$232$$ 302.024i 0.0854691i
$$233$$ − 5748.54i − 1.61631i −0.588972 0.808154i $$-0.700467\pi$$
0.588972 0.808154i $$-0.299533\pi$$
$$234$$ −1504.23 −0.420234
$$235$$ 0 0
$$236$$ −3494.84 −0.963961
$$237$$ − 2616.59i − 0.717154i
$$238$$ − 1813.14i − 0.493816i
$$239$$ 2678.10 0.724820 0.362410 0.932019i $$-0.381954\pi$$
0.362410 + 0.932019i $$0.381954\pi$$
$$240$$ 0 0
$$241$$ −2202.16 −0.588604 −0.294302 0.955713i $$-0.595087\pi$$
−0.294302 + 0.955713i $$0.595087\pi$$
$$242$$ 3935.25i 1.04532i
$$243$$ − 243.000i − 0.0641500i
$$244$$ 2348.50 0.616177
$$245$$ 0 0
$$246$$ 3173.09 0.822393
$$247$$ − 4071.21i − 1.04877i
$$248$$ − 46.7333i − 0.0119660i
$$249$$ −1588.76 −0.404353
$$250$$ 0 0
$$251$$ −5716.90 −1.43764 −0.718820 0.695196i $$-0.755317\pi$$
−0.718820 + 0.695196i $$0.755317\pi$$
$$252$$ 419.382i 0.104836i
$$253$$ − 5059.22i − 1.25719i
$$254$$ 9084.47 2.24413
$$255$$ 0 0
$$256$$ 5108.88 1.24728
$$257$$ − 4724.29i − 1.14666i −0.819323 0.573332i $$-0.805650\pi$$
0.819323 0.573332i $$-0.194350\pi$$
$$258$$ − 1063.97i − 0.256743i
$$259$$ −1768.73 −0.424339
$$260$$ 0 0
$$261$$ −528.616 −0.125366
$$262$$ 4594.25i 1.08334i
$$263$$ 5975.36i 1.40097i 0.713665 + 0.700487i $$0.247034\pi$$
−0.713665 + 0.700487i $$0.752966\pi$$
$$264$$ 749.238 0.174668
$$265$$ 0 0
$$266$$ −2499.14 −0.576059
$$267$$ − 1155.65i − 0.264886i
$$268$$ − 4902.90i − 1.11751i
$$269$$ −4486.11 −1.01681 −0.508407 0.861117i $$-0.669766\pi$$
−0.508407 + 0.861117i $$0.669766\pi$$
$$270$$ 0 0
$$271$$ 3827.68 0.857989 0.428994 0.903307i $$-0.358868\pi$$
0.428994 + 0.903307i $$0.358868\pi$$
$$272$$ − 4934.97i − 1.10010i
$$273$$ − 916.794i − 0.203249i
$$274$$ 10647.8 2.34766
$$275$$ 0 0
$$276$$ 2080.26 0.453685
$$277$$ 3420.54i 0.741950i 0.928643 + 0.370975i $$0.120976\pi$$
−0.928643 + 0.370975i $$0.879024\pi$$
$$278$$ 4768.71i 1.02881i
$$279$$ 81.7948 0.0175517
$$280$$ 0 0
$$281$$ 5235.92 1.11156 0.555781 0.831329i $$-0.312419\pi$$
0.555781 + 0.831329i $$0.312419\pi$$
$$282$$ 6693.55i 1.41346i
$$283$$ − 6985.88i − 1.46738i −0.679486 0.733688i $$-0.737797\pi$$
0.679486 0.733688i $$-0.262203\pi$$
$$284$$ 3276.93 0.684683
$$285$$ 0 0
$$286$$ 8117.60 1.67834
$$287$$ 1933.92i 0.397755i
$$288$$ 2143.01i 0.438466i
$$289$$ 335.550 0.0682984
$$290$$ 0 0
$$291$$ 1391.68 0.280349
$$292$$ − 7753.59i − 1.55392i
$$293$$ 7399.70i 1.47541i 0.675123 + 0.737705i $$0.264090\pi$$
−0.675123 + 0.737705i $$0.735910\pi$$
$$294$$ −562.779 −0.111639
$$295$$ 0 0
$$296$$ −1299.30 −0.255135
$$297$$ 1311.35i 0.256203i
$$298$$ − 74.2892i − 0.0144411i
$$299$$ −4547.58 −0.879577
$$300$$ 0 0
$$301$$ 648.463 0.124175
$$302$$ − 8996.04i − 1.71412i
$$303$$ 1298.17i 0.246133i
$$304$$ −6802.11 −1.28332
$$305$$ 0 0
$$306$$ 2331.17 0.435504
$$307$$ − 2668.64i − 0.496116i −0.968745 0.248058i $$-0.920208\pi$$
0.968745 0.248058i $$-0.0797923\pi$$
$$308$$ − 2263.20i − 0.418693i
$$309$$ 1537.88 0.283129
$$310$$ 0 0
$$311$$ 6189.25 1.12849 0.564244 0.825608i $$-0.309168\pi$$
0.564244 + 0.825608i $$0.309168\pi$$
$$312$$ − 673.468i − 0.122204i
$$313$$ − 2921.59i − 0.527598i −0.964578 0.263799i $$-0.915024\pi$$
0.964578 0.263799i $$-0.0849755\pi$$
$$314$$ −14925.0 −2.68237
$$315$$ 0 0
$$316$$ −5806.08 −1.03360
$$317$$ − 9825.56i − 1.74088i −0.492276 0.870439i $$-0.663835\pi$$
0.492276 0.870439i $$-0.336165\pi$$
$$318$$ 7155.55i 1.26183i
$$319$$ 2852.68 0.500687
$$320$$ 0 0
$$321$$ −5889.26 −1.02401
$$322$$ 2791.56i 0.483129i
$$323$$ 6309.33i 1.08687i
$$324$$ −539.205 −0.0924563
$$325$$ 0 0
$$326$$ −5848.07 −0.993541
$$327$$ 1635.53i 0.276590i
$$328$$ 1420.64i 0.239151i
$$329$$ −4079.56 −0.683627
$$330$$ 0 0
$$331$$ −9258.17 −1.53739 −0.768693 0.639618i $$-0.779093\pi$$
−0.768693 + 0.639618i $$0.779093\pi$$
$$332$$ 3525.39i 0.582774i
$$333$$ − 2274.09i − 0.374232i
$$334$$ −3822.75 −0.626263
$$335$$ 0 0
$$336$$ −1531.76 −0.248704
$$337$$ 3693.98i 0.597103i 0.954394 + 0.298552i $$0.0965035\pi$$
−0.954394 + 0.298552i $$0.903497\pi$$
$$338$$ 1114.38i 0.179331i
$$339$$ −695.470 −0.111424
$$340$$ 0 0
$$341$$ −441.406 −0.0700982
$$342$$ − 3213.17i − 0.508037i
$$343$$ − 343.000i − 0.0539949i
$$344$$ 476.355 0.0746608
$$345$$ 0 0
$$346$$ −2623.44 −0.407622
$$347$$ − 3832.83i − 0.592960i −0.955039 0.296480i $$-0.904187\pi$$
0.955039 0.296480i $$-0.0958128\pi$$
$$348$$ 1172.97i 0.180684i
$$349$$ −8325.22 −1.27690 −0.638451 0.769662i $$-0.720425\pi$$
−0.638451 + 0.769662i $$0.720425\pi$$
$$350$$ 0 0
$$351$$ 1178.74 0.179248
$$352$$ − 11564.8i − 1.75115i
$$353$$ − 8991.52i − 1.35572i −0.735189 0.677862i $$-0.762907\pi$$
0.735189 0.677862i $$-0.237093\pi$$
$$354$$ 6029.76 0.905306
$$355$$ 0 0
$$356$$ −2564.33 −0.381767
$$357$$ 1420.79i 0.210634i
$$358$$ − 3926.38i − 0.579652i
$$359$$ 12893.8 1.89557 0.947783 0.318917i $$-0.103319\pi$$
0.947783 + 0.318917i $$0.103319\pi$$
$$360$$ 0 0
$$361$$ 1837.46 0.267891
$$362$$ 11100.1i 1.61163i
$$363$$ − 3083.71i − 0.445875i
$$364$$ −2034.32 −0.292932
$$365$$ 0 0
$$366$$ −4051.94 −0.578684
$$367$$ 7480.17i 1.06393i 0.846767 + 0.531964i $$0.178546\pi$$
−0.846767 + 0.531964i $$0.821454\pi$$
$$368$$ 7598.02i 1.07629i
$$369$$ −2486.47 −0.350787
$$370$$ 0 0
$$371$$ −4361.14 −0.610293
$$372$$ − 181.499i − 0.0252964i
$$373$$ − 3523.32i − 0.489090i −0.969638 0.244545i $$-0.921361\pi$$
0.969638 0.244545i $$-0.0786386\pi$$
$$374$$ −12580.2 −1.73932
$$375$$ 0 0
$$376$$ −2996.81 −0.411033
$$377$$ − 2564.19i − 0.350298i
$$378$$ − 723.573i − 0.0984565i
$$379$$ −13515.4 −1.83177 −0.915886 0.401438i $$-0.868510\pi$$
−0.915886 + 0.401438i $$0.868510\pi$$
$$380$$ 0 0
$$381$$ −7118.69 −0.957222
$$382$$ 4112.44i 0.550813i
$$383$$ − 657.182i − 0.0876774i −0.999039 0.0438387i $$-0.986041\pi$$
0.999039 0.0438387i $$-0.0139588\pi$$
$$384$$ −1946.75 −0.258710
$$385$$ 0 0
$$386$$ −3441.75 −0.453836
$$387$$ 833.738i 0.109512i
$$388$$ − 3088.06i − 0.404053i
$$389$$ 9741.87 1.26975 0.634875 0.772615i $$-0.281052\pi$$
0.634875 + 0.772615i $$0.281052\pi$$
$$390$$ 0 0
$$391$$ 7047.58 0.911538
$$392$$ − 251.965i − 0.0324646i
$$393$$ − 3600.11i − 0.462091i
$$394$$ 11728.9 1.49973
$$395$$ 0 0
$$396$$ 2909.82 0.369253
$$397$$ − 4407.42i − 0.557184i −0.960410 0.278592i $$-0.910132\pi$$
0.960410 0.278592i $$-0.0898678\pi$$
$$398$$ 3635.18i 0.457827i
$$399$$ 1958.35 0.245715
$$400$$ 0 0
$$401$$ −11569.5 −1.44078 −0.720391 0.693568i $$-0.756037\pi$$
−0.720391 + 0.693568i $$0.756037\pi$$
$$402$$ 8459.14i 1.04951i
$$403$$ 396.767i 0.0490431i
$$404$$ 2880.59 0.354739
$$405$$ 0 0
$$406$$ −1574.04 −0.192410
$$407$$ 12272.1i 1.49461i
$$408$$ 1043.70i 0.126645i
$$409$$ 3083.03 0.372729 0.186364 0.982481i $$-0.440330\pi$$
0.186364 + 0.982481i $$0.440330\pi$$
$$410$$ 0 0
$$411$$ −8343.76 −1.00138
$$412$$ − 3412.47i − 0.408060i
$$413$$ 3674.99i 0.437856i
$$414$$ −3589.15 −0.426079
$$415$$ 0 0
$$416$$ −10395.3 −1.22517
$$417$$ − 3736.81i − 0.438831i
$$418$$ 17339.9i 2.02900i
$$419$$ 5415.21 0.631385 0.315692 0.948862i $$-0.397763\pi$$
0.315692 + 0.948862i $$0.397763\pi$$
$$420$$ 0 0
$$421$$ 4188.34 0.484863 0.242432 0.970168i $$-0.422055\pi$$
0.242432 + 0.970168i $$0.422055\pi$$
$$422$$ 8830.82i 1.01867i
$$423$$ − 5245.15i − 0.602902i
$$424$$ −3203.65 −0.366941
$$425$$ 0 0
$$426$$ −5653.79 −0.643021
$$427$$ − 2469.56i − 0.279884i
$$428$$ 13068.0i 1.47585i
$$429$$ −6361.05 −0.715884
$$430$$ 0 0
$$431$$ −9108.41 −1.01795 −0.508975 0.860781i $$-0.669976\pi$$
−0.508975 + 0.860781i $$0.669976\pi$$
$$432$$ − 1969.41i − 0.219336i
$$433$$ − 16847.0i − 1.86979i −0.354930 0.934893i $$-0.615495\pi$$
0.354930 0.934893i $$-0.384505\pi$$
$$434$$ 243.558 0.0269381
$$435$$ 0 0
$$436$$ 3629.16 0.398635
$$437$$ − 9714.03i − 1.06335i
$$438$$ 13377.5i 1.45937i
$$439$$ −8434.14 −0.916946 −0.458473 0.888708i $$-0.651603\pi$$
−0.458473 + 0.888708i $$0.651603\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 11308.0i 1.21689i
$$443$$ − 4298.49i − 0.461010i −0.973071 0.230505i $$-0.925962\pi$$
0.973071 0.230505i $$-0.0740378\pi$$
$$444$$ −5046.09 −0.539362
$$445$$ 0 0
$$446$$ 12356.7 1.31189
$$447$$ 58.2139i 0.00615978i
$$448$$ 2296.48i 0.242184i
$$449$$ −10545.0 −1.10835 −0.554173 0.832402i $$-0.686965\pi$$
−0.554173 + 0.832402i $$0.686965\pi$$
$$450$$ 0 0
$$451$$ 13418.2 1.40098
$$452$$ 1543.21i 0.160590i
$$453$$ 7049.40i 0.731147i
$$454$$ −2441.85 −0.252426
$$455$$ 0 0
$$456$$ 1438.59 0.147737
$$457$$ − 11952.4i − 1.22344i −0.791075 0.611719i $$-0.790478\pi$$
0.791075 0.611719i $$-0.209522\pi$$
$$458$$ − 2085.63i − 0.212784i
$$459$$ −1826.74 −0.185762
$$460$$ 0 0
$$461$$ −17200.9 −1.73780 −0.868900 0.494988i $$-0.835173\pi$$
−0.868900 + 0.494988i $$0.835173\pi$$
$$462$$ 3904.76i 0.393217i
$$463$$ − 10368.7i − 1.04076i −0.853934 0.520381i $$-0.825790\pi$$
0.853934 0.520381i $$-0.174210\pi$$
$$464$$ −4284.20 −0.428640
$$465$$ 0 0
$$466$$ 22007.9 2.18776
$$467$$ 16879.5i 1.67257i 0.548296 + 0.836284i $$0.315277\pi$$
−0.548296 + 0.836284i $$0.684723\pi$$
$$468$$ − 2615.56i − 0.258342i
$$469$$ −5155.64 −0.507602
$$470$$ 0 0
$$471$$ 11695.4 1.14415
$$472$$ 2699.62i 0.263263i
$$473$$ − 4499.27i − 0.437371i
$$474$$ 10017.4 0.970706
$$475$$ 0 0
$$476$$ 3152.67 0.303577
$$477$$ − 5607.17i − 0.538228i
$$478$$ 10252.9i 0.981082i
$$479$$ −7329.12 −0.699115 −0.349558 0.936915i $$-0.613668\pi$$
−0.349558 + 0.936915i $$0.613668\pi$$
$$480$$ 0 0
$$481$$ 11031.0 1.04568
$$482$$ − 8430.80i − 0.796706i
$$483$$ − 2187.50i − 0.206076i
$$484$$ −6842.60 −0.642619
$$485$$ 0 0
$$486$$ 930.308 0.0868305
$$487$$ 17209.5i 1.60131i 0.599125 + 0.800655i $$0.295515\pi$$
−0.599125 + 0.800655i $$0.704485\pi$$
$$488$$ − 1814.11i − 0.168281i
$$489$$ 4582.61 0.423789
$$490$$ 0 0
$$491$$ 11392.5 1.04712 0.523560 0.851989i $$-0.324604\pi$$
0.523560 + 0.851989i $$0.324604\pi$$
$$492$$ 5517.35i 0.505572i
$$493$$ 3973.83i 0.363027i
$$494$$ 15586.3 1.41956
$$495$$ 0 0
$$496$$ 662.912 0.0600113
$$497$$ − 3445.85i − 0.311001i
$$498$$ − 6082.47i − 0.547313i
$$499$$ −19079.4 −1.71164 −0.855822 0.517271i $$-0.826948\pi$$
−0.855822 + 0.517271i $$0.826948\pi$$
$$500$$ 0 0
$$501$$ 2995.55 0.267129
$$502$$ − 21886.7i − 1.94592i
$$503$$ − 13499.4i − 1.19663i −0.801259 0.598317i $$-0.795836\pi$$
0.801259 0.598317i $$-0.204164\pi$$
$$504$$ 323.955 0.0286311
$$505$$ 0 0
$$506$$ 19368.8 1.70168
$$507$$ − 873.237i − 0.0764928i
$$508$$ 15796.0i 1.37960i
$$509$$ 4328.85 0.376960 0.188480 0.982077i $$-0.439644\pi$$
0.188480 + 0.982077i $$0.439644\pi$$
$$510$$ 0 0
$$511$$ −8153.27 −0.705831
$$512$$ 14367.6i 1.24017i
$$513$$ 2517.88i 0.216700i
$$514$$ 18086.6 1.55207
$$515$$ 0 0
$$516$$ 1850.02 0.157835
$$517$$ 28305.5i 2.40788i
$$518$$ − 6771.47i − 0.574365i
$$519$$ 2055.76 0.173869
$$520$$ 0 0
$$521$$ 19395.7 1.63098 0.815490 0.578771i $$-0.196468\pi$$
0.815490 + 0.578771i $$0.196468\pi$$
$$522$$ − 2023.77i − 0.169689i
$$523$$ 20413.8i 1.70675i 0.521294 + 0.853377i $$0.325450\pi$$
−0.521294 + 0.853377i $$0.674550\pi$$
$$524$$ −7988.47 −0.665989
$$525$$ 0 0
$$526$$ −22876.2 −1.89629
$$527$$ − 614.887i − 0.0508252i
$$528$$ 10627.9i 0.875987i
$$529$$ 1316.34 0.108189
$$530$$ 0 0
$$531$$ −4724.99 −0.386153
$$532$$ − 4345.49i − 0.354137i
$$533$$ − 12061.3i − 0.980171i
$$534$$ 4424.32 0.358537
$$535$$ 0 0
$$536$$ −3787.28 −0.305197
$$537$$ 3076.75i 0.247247i
$$538$$ − 17174.8i − 1.37631i
$$539$$ −2379.86 −0.190181
$$540$$ 0 0
$$541$$ −4919.18 −0.390928 −0.195464 0.980711i $$-0.562621\pi$$
−0.195464 + 0.980711i $$0.562621\pi$$
$$542$$ 14654.0i 1.16133i
$$543$$ − 8698.19i − 0.687431i
$$544$$ 16110.0 1.26969
$$545$$ 0 0
$$546$$ 3509.88 0.275108
$$547$$ − 15334.2i − 1.19862i −0.800518 0.599308i $$-0.795442\pi$$
0.800518 0.599308i $$-0.204558\pi$$
$$548$$ 18514.4i 1.44324i
$$549$$ 3175.15 0.246834
$$550$$ 0 0
$$551$$ 5477.33 0.423488
$$552$$ − 1606.92i − 0.123904i
$$553$$ 6105.37i 0.469487i
$$554$$ −13095.3 −1.00427
$$555$$ 0 0
$$556$$ −8291.81 −0.632466
$$557$$ 8613.78i 0.655256i 0.944807 + 0.327628i $$0.106249\pi$$
−0.944807 + 0.327628i $$0.893751\pi$$
$$558$$ 313.145i 0.0237572i
$$559$$ −4044.26 −0.306000
$$560$$ 0 0
$$561$$ 9857.98 0.741897
$$562$$ 20045.3i 1.50456i
$$563$$ − 2320.81i − 0.173731i −0.996220 0.0868654i $$-0.972315\pi$$
0.996220 0.0868654i $$-0.0276850\pi$$
$$564$$ −11638.7 −0.868934
$$565$$ 0 0
$$566$$ 26744.9 1.98617
$$567$$ 567.000i 0.0419961i
$$568$$ − 2531.29i − 0.186990i
$$569$$ 1736.04 0.127906 0.0639529 0.997953i $$-0.479629\pi$$
0.0639529 + 0.997953i $$0.479629\pi$$
$$570$$ 0 0
$$571$$ 23897.8 1.75148 0.875738 0.482786i $$-0.160375\pi$$
0.875738 + 0.482786i $$0.160375\pi$$
$$572$$ 14114.9i 1.03177i
$$573$$ − 3222.55i − 0.234946i
$$574$$ −7403.87 −0.538382
$$575$$ 0 0
$$576$$ −2952.61 −0.213586
$$577$$ − 8029.26i − 0.579311i −0.957131 0.289655i $$-0.906459\pi$$
0.957131 0.289655i $$-0.0935407\pi$$
$$578$$ 1284.63i 0.0924455i
$$579$$ 2697.00 0.193581
$$580$$ 0 0
$$581$$ 3707.12 0.264711
$$582$$ 5327.93i 0.379467i
$$583$$ 30259.1i 2.14958i
$$584$$ −5989.32 −0.424383
$$585$$ 0 0
$$586$$ −28329.2 −1.99705
$$587$$ 8015.14i 0.563578i 0.959476 + 0.281789i $$0.0909278\pi$$
−0.959476 + 0.281789i $$0.909072\pi$$
$$588$$ − 978.558i − 0.0686310i
$$589$$ −847.529 −0.0592900
$$590$$ 0 0
$$591$$ −9190.88 −0.639700
$$592$$ − 18430.5i − 1.27954i
$$593$$ 12820.3i 0.887801i 0.896076 + 0.443901i $$0.146406\pi$$
−0.896076 + 0.443901i $$0.853594\pi$$
$$594$$ −5020.41 −0.346784
$$595$$ 0 0
$$596$$ 129.174 0.00887779
$$597$$ − 2848.57i − 0.195283i
$$598$$ − 17410.1i − 1.19055i
$$599$$ 15330.5 1.04572 0.522860 0.852418i $$-0.324865\pi$$
0.522860 + 0.852418i $$0.324865\pi$$
$$600$$ 0 0
$$601$$ 107.658 0.00730694 0.00365347 0.999993i $$-0.498837\pi$$
0.00365347 + 0.999993i $$0.498837\pi$$
$$602$$ 2482.59i 0.168078i
$$603$$ − 6628.68i − 0.447663i
$$604$$ 15642.3 1.05377
$$605$$ 0 0
$$606$$ −4969.97 −0.333154
$$607$$ 8213.06i 0.549189i 0.961560 + 0.274595i $$0.0885436\pi$$
−0.961560 + 0.274595i $$0.911456\pi$$
$$608$$ − 22205.2i − 1.48115i
$$609$$ 1233.44 0.0820712
$$610$$ 0 0
$$611$$ 25443.0 1.68463
$$612$$ 4053.44i 0.267729i
$$613$$ 1242.60i 0.0818732i 0.999162 + 0.0409366i $$0.0130342\pi$$
−0.999162 + 0.0409366i $$0.986966\pi$$
$$614$$ 10216.7 0.671519
$$615$$ 0 0
$$616$$ −1748.22 −0.114347
$$617$$ 13170.6i 0.859367i 0.902980 + 0.429683i $$0.141375\pi$$
−0.902980 + 0.429683i $$0.858625\pi$$
$$618$$ 5887.65i 0.383230i
$$619$$ −19774.1 −1.28399 −0.641993 0.766711i $$-0.721892\pi$$
−0.641993 + 0.766711i $$0.721892\pi$$
$$620$$ 0 0
$$621$$ 2812.50 0.181742
$$622$$ 23695.1i 1.52747i
$$623$$ 2696.51i 0.173409i
$$624$$ 9553.14 0.612871
$$625$$ 0 0
$$626$$ 11185.1 0.714132
$$627$$ − 13587.8i − 0.865459i
$$628$$ − 25951.5i − 1.64901i
$$629$$ −17095.3 −1.08368
$$630$$ 0 0
$$631$$ −14308.8 −0.902735 −0.451367 0.892338i $$-0.649064\pi$$
−0.451367 + 0.892338i $$0.649064\pi$$
$$632$$ 4484.95i 0.282281i
$$633$$ − 6919.93i − 0.434507i
$$634$$ 37616.4 2.35637
$$635$$ 0 0
$$636$$ −12442.0 −0.775722
$$637$$ 2139.19i 0.133058i
$$638$$ 10921.3i 0.677707i
$$639$$ 4430.38 0.274277
$$640$$ 0 0
$$641$$ 11537.5 0.710925 0.355463 0.934691i $$-0.384323\pi$$
0.355463 + 0.934691i $$0.384323\pi$$
$$642$$ − 22546.6i − 1.38605i
$$643$$ 19603.0i 1.20228i 0.799144 + 0.601139i $$0.205286\pi$$
−0.799144 + 0.601139i $$0.794714\pi$$
$$644$$ −4853.95 −0.297007
$$645$$ 0 0
$$646$$ −24154.8 −1.47114
$$647$$ − 21650.0i − 1.31553i −0.753223 0.657765i $$-0.771502\pi$$
0.753223 0.657765i $$-0.228498\pi$$
$$648$$ 416.513i 0.0252503i
$$649$$ 25498.4 1.54222
$$650$$ 0 0
$$651$$ −190.855 −0.0114903
$$652$$ − 10168.6i − 0.610787i
$$653$$ − 2927.33i − 0.175429i −0.996146 0.0877145i $$-0.972044\pi$$
0.996146 0.0877145i $$-0.0279563\pi$$
$$654$$ −6261.50 −0.374379
$$655$$ 0 0
$$656$$ −20151.7 −1.19938
$$657$$ − 10482.8i − 0.622484i
$$658$$ − 15618.3i − 0.925326i
$$659$$ 4778.76 0.282480 0.141240 0.989975i $$-0.454891\pi$$
0.141240 + 0.989975i $$0.454891\pi$$
$$660$$ 0 0
$$661$$ −31510.3 −1.85417 −0.927086 0.374849i $$-0.877695\pi$$
−0.927086 + 0.374849i $$0.877695\pi$$
$$662$$ − 35444.2i − 2.08093i
$$663$$ − 8861.06i − 0.519057i
$$664$$ 2723.21 0.159158
$$665$$ 0 0
$$666$$ 8706.17 0.506542
$$667$$ − 6118.23i − 0.355170i
$$668$$ − 6646.99i − 0.385000i
$$669$$ −9682.82 −0.559581
$$670$$ 0 0
$$671$$ −17134.7 −0.985808
$$672$$ − 5000.37i − 0.287044i
$$673$$ − 8992.15i − 0.515040i −0.966273 0.257520i $$-0.917095\pi$$
0.966273 0.257520i $$-0.0829054\pi$$
$$674$$ −14142.1 −0.808211
$$675$$ 0 0
$$676$$ −1937.67 −0.110245
$$677$$ − 19340.8i − 1.09797i −0.835832 0.548985i $$-0.815014\pi$$
0.835832 0.548985i $$-0.184986\pi$$
$$678$$ − 2662.56i − 0.150818i
$$679$$ −3247.25 −0.183531
$$680$$ 0 0
$$681$$ 1913.46 0.107671
$$682$$ − 1689.89i − 0.0948816i
$$683$$ − 4255.14i − 0.238387i −0.992871 0.119194i $$-0.961969\pi$$
0.992871 0.119194i $$-0.0380309\pi$$
$$684$$ 5587.05 0.312319
$$685$$ 0 0
$$686$$ 1313.15 0.0730850
$$687$$ 1634.32i 0.0907616i
$$688$$ 6757.08i 0.374435i
$$689$$ 27199.1 1.50392
$$690$$ 0 0
$$691$$ −17505.5 −0.963733 −0.481867 0.876245i $$-0.660041\pi$$
−0.481867 + 0.876245i $$0.660041\pi$$
$$692$$ − 4561.63i − 0.250588i
$$693$$ − 3059.82i − 0.167724i
$$694$$ 14673.7 0.802603
$$695$$ 0 0
$$696$$ 906.071 0.0493456
$$697$$ 18691.8i 1.01579i
$$698$$ − 31872.5i − 1.72836i
$$699$$ −17245.6 −0.933175
$$700$$ 0 0
$$701$$ −3240.77 −0.174611 −0.0873054 0.996182i $$-0.527826\pi$$
−0.0873054 + 0.996182i $$0.527826\pi$$
$$702$$ 4512.70i 0.242622i
$$703$$ 23563.3i 1.26416i
$$704$$ 15933.8 0.853022
$$705$$ 0 0
$$706$$ 34423.4 1.83504
$$707$$ − 3029.07i − 0.161132i
$$708$$ 10484.5i 0.556543i
$$709$$ −19949.3 −1.05672 −0.528358 0.849022i $$-0.677192\pi$$
−0.528358 + 0.849022i $$0.677192\pi$$
$$710$$ 0 0
$$711$$ −7849.76 −0.414049
$$712$$ 1980.83i 0.104262i
$$713$$ 946.698i 0.0497253i
$$714$$ −5439.41 −0.285105
$$715$$ 0 0
$$716$$ 6827.17 0.356345
$$717$$ − 8034.30i − 0.418475i
$$718$$ 49362.9i 2.56575i
$$719$$ 11259.4 0.584011 0.292006 0.956417i $$-0.405677\pi$$
0.292006 + 0.956417i $$0.405677\pi$$
$$720$$ 0 0
$$721$$ −3588.38 −0.185351
$$722$$ 7034.60i 0.362605i
$$723$$ 6606.47i 0.339830i
$$724$$ −19300.9 −0.990761
$$725$$ 0 0
$$726$$ 11805.8 0.603516
$$727$$ 12228.5i 0.623840i 0.950108 + 0.311920i $$0.100972\pi$$
−0.950108 + 0.311920i $$0.899028\pi$$
$$728$$ 1571.43i 0.0800013i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 6267.56 0.317119
$$732$$ − 7045.49i − 0.355750i
$$733$$ − 26635.1i − 1.34214i −0.741392 0.671072i $$-0.765834\pi$$
0.741392 0.671072i $$-0.234166\pi$$
$$734$$ −28637.3 −1.44008
$$735$$ 0 0
$$736$$ −24803.4 −1.24221
$$737$$ 35771.7i 1.78788i
$$738$$ − 9519.26i − 0.474809i
$$739$$ 6074.00 0.302349 0.151174 0.988507i $$-0.451694\pi$$
0.151174 + 0.988507i $$0.451694\pi$$
$$740$$ 0 0
$$741$$ −12213.6 −0.605505
$$742$$ − 16696.3i − 0.826065i
$$743$$ − 4016.87i − 0.198337i −0.995071 0.0991686i $$-0.968382\pi$$
0.995071 0.0991686i $$-0.0316183\pi$$
$$744$$ −140.200 −0.00690858
$$745$$ 0 0
$$746$$ 13488.8 0.662010
$$747$$ 4766.29i 0.233453i
$$748$$ − 21874.4i − 1.06926i
$$749$$ 13741.6 0.670370
$$750$$ 0 0
$$751$$ −23913.2 −1.16192 −0.580962 0.813931i $$-0.697324\pi$$
−0.580962 + 0.813931i $$0.697324\pi$$
$$752$$ − 42509.6i − 2.06139i
$$753$$ 17150.7i 0.830022i
$$754$$ 9816.81 0.474147
$$755$$ 0 0
$$756$$ 1258.15 0.0605269
$$757$$ − 31044.9i − 1.49055i −0.666758 0.745275i $$-0.732318\pi$$
0.666758 0.745275i $$-0.267682\pi$$
$$758$$ − 51742.9i − 2.47940i
$$759$$ −15177.6 −0.725842
$$760$$ 0 0
$$761$$ 14011.8 0.667446 0.333723 0.942671i $$-0.391695\pi$$
0.333723 + 0.942671i $$0.391695\pi$$
$$762$$ − 27253.4i − 1.29565i
$$763$$ − 3816.23i − 0.181071i
$$764$$ −7150.69 −0.338616
$$765$$ 0 0
$$766$$ 2515.97 0.118676
$$767$$ − 22919.8i − 1.07899i
$$768$$ − 15326.6i − 0.720120i
$$769$$ −3342.49 −0.156740 −0.0783701 0.996924i $$-0.524972\pi$$
−0.0783701 + 0.996924i $$0.524972\pi$$
$$770$$ 0 0
$$771$$ −14172.9 −0.662027
$$772$$ − 5984.51i − 0.278999i
$$773$$ − 21074.6i − 0.980594i −0.871555 0.490297i $$-0.836888\pi$$
0.871555 0.490297i $$-0.163112\pi$$
$$774$$ −3191.90 −0.148231
$$775$$ 0 0
$$776$$ −2385.40 −0.110349
$$777$$ 5306.20i 0.244992i
$$778$$ 37296.0i 1.71867i
$$779$$ 25763.9 1.18496
$$780$$ 0 0
$$781$$ −23908.5 −1.09541
$$782$$ 26981.1i 1.23382i
$$783$$ 1585.85i 0.0723800i
$$784$$ 3574.12 0.162815
$$785$$ 0 0
$$786$$ 13782.8 0.625464
$$787$$ − 21394.8i − 0.969048i −0.874778 0.484524i $$-0.838993\pi$$
0.874778 0.484524i $$-0.161007\pi$$
$$788$$ 20394.1i 0.921968i
$$789$$ 17926.1 0.808853
$$790$$ 0 0
$$791$$ 1622.76 0.0729442
$$792$$ − 2247.71i − 0.100845i
$$793$$ 15401.9i 0.689706i
$$794$$ 16873.5 0.754179
$$795$$ 0 0
$$796$$ −6320.83 −0.281452
$$797$$ − 20645.0i − 0.917547i −0.888553 0.458773i $$-0.848289\pi$$
0.888553 0.458773i $$-0.151711\pi$$
$$798$$ 7497.41i 0.332588i
$$799$$ −39430.0 −1.74585
$$800$$ 0 0
$$801$$ −3466.95 −0.152932
$$802$$ − 44293.0i − 1.95017i
$$803$$ 56570.4i 2.48608i
$$804$$ −14708.7 −0.645194
$$805$$ 0 0
$$806$$ −1518.99 −0.0663825
$$807$$ 13458.3i 0.587058i
$$808$$ − 2225.13i − 0.0968810i
$$809$$ 15939.0 0.692688 0.346344 0.938108i $$-0.387423\pi$$
0.346344 + 0.938108i $$0.387423\pi$$
$$810$$ 0 0
$$811$$ 22829.2 0.988460 0.494230 0.869331i $$-0.335450\pi$$
0.494230 + 0.869331i $$0.335450\pi$$
$$812$$ − 2736.94i − 0.118285i
$$813$$ − 11483.0i − 0.495360i
$$814$$ −46982.9 −2.02303
$$815$$ 0 0
$$816$$ −14804.9 −0.635141
$$817$$ − 8638.90i − 0.369935i
$$818$$ 11803.2i 0.504508i
$$819$$ −2750.38 −0.117346
$$820$$ 0 0
$$821$$ −5700.22 −0.242313 −0.121157 0.992633i $$-0.538660\pi$$
−0.121157 + 0.992633i $$0.538660\pi$$
$$822$$ − 31943.5i − 1.35542i
$$823$$ − 32438.0i − 1.37390i −0.726707 0.686948i $$-0.758950\pi$$
0.726707 0.686948i $$-0.241050\pi$$
$$824$$ −2635.99 −0.111443
$$825$$ 0 0
$$826$$ −14069.4 −0.592662
$$827$$ 12762.6i 0.536638i 0.963330 + 0.268319i $$0.0864681\pi$$
−0.963330 + 0.268319i $$0.913532\pi$$
$$828$$ − 6240.79i − 0.261935i
$$829$$ 30766.7 1.28899 0.644494 0.764609i $$-0.277068\pi$$
0.644494 + 0.764609i $$0.277068\pi$$
$$830$$ 0 0
$$831$$ 10261.6 0.428365
$$832$$ − 14322.4i − 0.596804i
$$833$$ − 3315.19i − 0.137892i
$$834$$ 14306.1 0.593981
$$835$$ 0 0
$$836$$ −30150.6 −1.24734
$$837$$ − 245.384i − 0.0101335i
$$838$$ 20731.7i 0.854613i
$$839$$ 9779.71 0.402423 0.201212 0.979548i $$-0.435512\pi$$
0.201212 + 0.979548i $$0.435512\pi$$
$$840$$ 0 0
$$841$$ −20939.2 −0.858551
$$842$$ 16034.8i 0.656288i
$$843$$ − 15707.8i − 0.641760i
$$844$$ −15355.0 −0.626233
$$845$$ 0 0
$$846$$ 20080.7 0.816061
$$847$$ 7195.32i 0.291894i
$$848$$ − 45443.7i − 1.84026i
$$849$$ −20957.6 −0.847190
$$850$$ 0 0
$$851$$ 26320.4 1.06023
$$852$$ − 9830.79i − 0.395302i
$$853$$ 24201.5i 0.971445i 0.874113 + 0.485723i $$0.161444\pi$$
−0.874113 + 0.485723i $$0.838556\pi$$
$$854$$ 9454.52 0.378837
$$855$$ 0 0
$$856$$ 10094.5 0.403062
$$857$$ − 21036.7i − 0.838507i −0.907869 0.419254i $$-0.862292\pi$$
0.907869 0.419254i $$-0.137708\pi$$
$$858$$ − 24352.8i − 0.968988i
$$859$$ 6179.19 0.245438 0.122719 0.992441i $$-0.460839\pi$$
0.122719 + 0.992441i $$0.460839\pi$$
$$860$$ 0 0
$$861$$ 5801.76 0.229644
$$862$$ − 34870.9i − 1.37785i
$$863$$ 50256.2i 1.98232i 0.132671 + 0.991160i $$0.457644\pi$$
−0.132671 + 0.991160i $$0.542356\pi$$
$$864$$ 6429.04 0.253149
$$865$$ 0 0
$$866$$ 64497.7 2.53085
$$867$$ − 1006.65i − 0.0394321i
$$868$$ 423.497i 0.0165604i
$$869$$ 42361.2 1.65363
$$870$$ 0 0
$$871$$ 32154.1 1.25086
$$872$$ − 2803.37i − 0.108869i
$$873$$ − 4175.03i − 0.161860i
$$874$$ 37189.5 1.43930
$$875$$ 0 0
$$876$$ −23260.8 −0.897156
$$877$$ 9175.95i 0.353306i 0.984273 + 0.176653i $$0.0565271\pi$$
−0.984273 + 0.176653i $$0.943473\pi$$
$$878$$ − 32289.5i − 1.24114i
$$879$$ 22199.1 0.851828
$$880$$ 0 0
$$881$$ −26172.7 −1.00089 −0.500444 0.865769i $$-0.666830\pi$$
−0.500444 + 0.865769i $$0.666830\pi$$
$$882$$ 1688.34i 0.0644549i
$$883$$ 18615.5i 0.709471i 0.934967 + 0.354736i $$0.115429\pi$$
−0.934967 + 0.354736i $$0.884571\pi$$
$$884$$ −19662.3 −0.748092
$$885$$ 0 0
$$886$$ 16456.4 0.624001
$$887$$ 12837.8i 0.485964i 0.970031 + 0.242982i $$0.0781256\pi$$
−0.970031 + 0.242982i $$0.921874\pi$$
$$888$$ 3897.89i 0.147302i
$$889$$ 16610.3 0.626649
$$890$$ 0 0
$$891$$ 3934.05 0.147919
$$892$$ 21485.7i 0.806496i
$$893$$ 54348.4i 2.03662i
$$894$$ −222.868 −0.00833759
$$895$$ 0 0
$$896$$ 4542.42 0.169366
$$897$$ 13642.7i 0.507824i
$$898$$ − 40370.6i − 1.50020i
$$899$$ −533.803 −0.0198035
$$900$$ 0 0
$$901$$ −42151.5 −1.55857
$$902$$ 51370.7i 1.89630i
$$903$$ − 1945.39i − 0.0716926i
$$904$$ 1192.07 0.0438579
$$905$$ 0 0
$$906$$ −26988.1 −0.989647
$$907$$ 26766.1i 0.979885i 0.871755 + 0.489942i $$0.162982\pi$$
−0.871755 + 0.489942i $$0.837018\pi$$
$$908$$ − 4245.87i − 0.155181i
$$909$$ 3894.52 0.142105
$$910$$ 0 0
$$911$$ 5022.67 0.182666 0.0913328 0.995820i $$-0.470887\pi$$
0.0913328 + 0.995820i $$0.470887\pi$$
$$912$$ 20406.3i 0.740923i
$$913$$ − 25721.3i − 0.932367i
$$914$$ 45759.0 1.65599
$$915$$ 0 0
$$916$$ 3626.48 0.130810
$$917$$ 8400.26i 0.302509i
$$918$$ − 6993.52i − 0.251439i
$$919$$ −9541.79 −0.342497 −0.171248 0.985228i $$-0.554780\pi$$
−0.171248 + 0.985228i $$0.554780\pi$$
$$920$$ 0 0
$$921$$ −8005.93 −0.286432
$$922$$ − 65852.4i − 2.35220i
$$923$$ 21490.7i 0.766387i
$$924$$ −6789.59 −0.241733
$$925$$ 0 0
$$926$$ 39695.7 1.40873
$$927$$ − 4613.63i − 0.163464i
$$928$$ − 13985.6i − 0.494718i
$$929$$ 25479.6 0.899846 0.449923 0.893067i $$-0.351451\pi$$
0.449923 + 0.893067i $$0.351451\pi$$
$$930$$ 0 0
$$931$$ −4569.49 −0.160858
$$932$$ 38267.2i 1.34494i
$$933$$ − 18567.7i − 0.651533i
$$934$$ −64621.9 −2.26391
$$935$$ 0 0
$$936$$ −2020.41 −0.0705545
$$937$$ 33608.3i 1.17176i 0.810399 + 0.585878i $$0.199250\pi$$
−0.810399 + 0.585878i $$0.800750\pi$$
$$938$$ − 19738.0i − 0.687066i
$$939$$ −8764.77 −0.304609
$$940$$ 0 0
$$941$$ −19173.6 −0.664232 −0.332116 0.943239i $$-0.607762\pi$$
−0.332116 + 0.943239i $$0.607762\pi$$
$$942$$ 44774.9i 1.54867i
$$943$$ − 28778.5i − 0.993804i
$$944$$ −38294.0 −1.32030
$$945$$ 0 0
$$946$$ 17225.1 0.592005
$$947$$ 979.315i 0.0336045i 0.999859 + 0.0168023i $$0.00534857\pi$$
−0.999859 + 0.0168023i $$0.994651\pi$$
$$948$$ 17418.2i 0.596749i
$$949$$ 50849.5 1.73935
$$950$$ 0 0
$$951$$ −29476.7 −1.00510
$$952$$ − 2435.31i − 0.0829083i
$$953$$ 3048.61i 0.103624i 0.998657 + 0.0518122i $$0.0164997\pi$$
−0.998657 + 0.0518122i $$0.983500\pi$$
$$954$$ 21466.7 0.728521
$$955$$ 0 0
$$956$$ −17827.7 −0.603127
$$957$$ − 8558.03i − 0.289072i
$$958$$ − 28059.0i − 0.946290i
$$959$$ 19468.8 0.655557
$$960$$ 0 0
$$961$$ −29708.4 −0.997227
$$962$$ 42231.6i 1.41538i
$$963$$ 17667.8i 0.591211i
$$964$$ 14659.4 0.489781
$$965$$ 0 0
$$966$$ 8374.67 0.278934
$$967$$ 14467.9i 0.481133i 0.970633 + 0.240567i $$0.0773332\pi$$
−0.970633 + 0.240567i $$0.922667\pi$$
$$968$$ 5285.62i 0.175502i
$$969$$ 18928.0 0.627507
$$970$$ 0 0
$$971$$ 12952.8 0.428090 0.214045 0.976824i $$-0.431336\pi$$
0.214045 + 0.976824i $$0.431336\pi$$
$$972$$ 1617.62i 0.0533797i
$$973$$ 8719.23i 0.287282i
$$974$$ −65885.4 −2.16746
$$975$$ 0 0
$$976$$ 25733.2 0.843954
$$977$$ − 47244.2i − 1.54706i −0.633760 0.773529i $$-0.718489\pi$$
0.633760 0.773529i $$-0.281511\pi$$
$$978$$ 17544.2i 0.573621i
$$979$$ 18709.4 0.610781
$$980$$ 0 0
$$981$$ 4906.58 0.159689
$$982$$ 43615.3i 1.41733i
$$983$$ − 1536.84i − 0.0498651i −0.999689 0.0249326i $$-0.992063\pi$$
0.999689 0.0249326i $$-0.00793711\pi$$
$$984$$ 4261.92 0.138074
$$985$$ 0 0
$$986$$ −15213.5 −0.491376
$$987$$ 12238.7i 0.394692i
$$988$$ 27101.5i 0.872685i
$$989$$ −9649.73 −0.310256
$$990$$ 0 0
$$991$$ 3785.22 0.121334 0.0606668 0.998158i $$-0.480677\pi$$
0.0606668 + 0.998158i $$0.480677\pi$$
$$992$$ 2164.04i 0.0692625i
$$993$$ 27774.5i 0.887610i
$$994$$ 13192.2 0.420956
$$995$$ 0 0
$$996$$ 10576.2 0.336465
$$997$$ 25894.9i 0.822566i 0.911508 + 0.411283i $$0.134919\pi$$
−0.911508 + 0.411283i $$0.865081\pi$$
$$998$$ − 73044.0i − 2.31680i
$$999$$ −6822.26 −0.216063
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 525.4.d.l.274.4 4
5.2 odd 4 105.4.a.e.1.1 2
5.3 odd 4 525.4.a.l.1.2 2
5.4 even 2 inner 525.4.d.l.274.1 4
15.2 even 4 315.4.a.k.1.2 2
15.8 even 4 1575.4.a.q.1.1 2
20.7 even 4 1680.4.a.bo.1.1 2
35.27 even 4 735.4.a.o.1.1 2
105.62 odd 4 2205.4.a.bb.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.e.1.1 2 5.2 odd 4
315.4.a.k.1.2 2 15.2 even 4
525.4.a.l.1.2 2 5.3 odd 4
525.4.d.l.274.1 4 5.4 even 2 inner
525.4.d.l.274.4 4 1.1 even 1 trivial
735.4.a.o.1.1 2 35.27 even 4
1575.4.a.q.1.1 2 15.8 even 4
1680.4.a.bo.1.1 2 20.7 even 4
2205.4.a.bb.1.2 2 105.62 odd 4