# Properties

 Label 1575.4.a.x.1.2 Level $1575$ Weight $4$ Character 1575.1 Self dual yes Analytic conductor $92.928$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$92.9280082590$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 525) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 1575.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.56155 q^{2} +4.68466 q^{4} +7.00000 q^{7} -11.8078 q^{8} +O(q^{10})$$ $$q+3.56155 q^{2} +4.68466 q^{4} +7.00000 q^{7} -11.8078 q^{8} +5.19224 q^{11} -54.5464 q^{13} +24.9309 q^{14} -79.5312 q^{16} +16.1619 q^{17} +87.4470 q^{19} +18.4924 q^{22} +176.477 q^{23} -194.270 q^{26} +32.7926 q^{28} -142.170 q^{29} -94.3002 q^{31} -188.793 q^{32} +57.5616 q^{34} -17.3305 q^{37} +311.447 q^{38} -210.270 q^{41} -521.570 q^{43} +24.3239 q^{44} +628.533 q^{46} -105.417 q^{47} +49.0000 q^{49} -255.531 q^{52} -108.978 q^{53} -82.6543 q^{56} -506.348 q^{58} -210.365 q^{59} -674.304 q^{61} -335.855 q^{62} -36.1449 q^{64} -324.929 q^{67} +75.7131 q^{68} -793.965 q^{71} -315.417 q^{73} -61.7235 q^{74} +409.659 q^{76} +36.3457 q^{77} -425.840 q^{79} -748.887 q^{82} -283.029 q^{83} -1857.60 q^{86} -61.3087 q^{88} +843.131 q^{89} -381.825 q^{91} +826.736 q^{92} -375.447 q^{94} -1537.33 q^{97} +174.516 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 - 3 * q^4 + 14 * q^7 - 3 * q^8 $$2 q + 3 q^{2} - 3 q^{4} + 14 q^{7} - 3 q^{8} + 31 q^{11} - 39 q^{13} + 21 q^{14} - 23 q^{16} - 79 q^{17} - 56 q^{19} + 4 q^{22} + 254 q^{23} - 203 q^{26} - 21 q^{28} + 62 q^{29} - 135 q^{31} - 291 q^{32} + 111 q^{34} - 113 q^{37} + 392 q^{38} - 235 q^{41} - 804 q^{43} - 174 q^{44} + 585 q^{46} + 152 q^{47} + 98 q^{49} - 375 q^{52} + 149 q^{53} - 21 q^{56} - 621 q^{58} + 441 q^{59} - 223 q^{61} - 313 q^{62} - 431 q^{64} - 1157 q^{67} + 807 q^{68} - 619 q^{71} - 268 q^{73} - 8 q^{74} + 1512 q^{76} + 217 q^{77} - 427 q^{79} - 735 q^{82} + 1211 q^{83} - 1699 q^{86} + 166 q^{88} - 466 q^{89} - 273 q^{91} + 231 q^{92} - 520 q^{94} - 172 q^{97} + 147 q^{98}+O(q^{100})$$ 2 * q + 3 * q^2 - 3 * q^4 + 14 * q^7 - 3 * q^8 + 31 * q^11 - 39 * q^13 + 21 * q^14 - 23 * q^16 - 79 * q^17 - 56 * q^19 + 4 * q^22 + 254 * q^23 - 203 * q^26 - 21 * q^28 + 62 * q^29 - 135 * q^31 - 291 * q^32 + 111 * q^34 - 113 * q^37 + 392 * q^38 - 235 * q^41 - 804 * q^43 - 174 * q^44 + 585 * q^46 + 152 * q^47 + 98 * q^49 - 375 * q^52 + 149 * q^53 - 21 * q^56 - 621 * q^58 + 441 * q^59 - 223 * q^61 - 313 * q^62 - 431 * q^64 - 1157 * q^67 + 807 * q^68 - 619 * q^71 - 268 * q^73 - 8 * q^74 + 1512 * q^76 + 217 * q^77 - 427 * q^79 - 735 * q^82 + 1211 * q^83 - 1699 * q^86 + 166 * q^88 - 466 * q^89 - 273 * q^91 + 231 * q^92 - 520 * q^94 - 172 * q^97 + 147 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.56155 1.25920 0.629600 0.776920i $$-0.283219\pi$$
0.629600 + 0.776920i $$0.283219\pi$$
$$3$$ 0 0
$$4$$ 4.68466 0.585582
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ −11.8078 −0.521834
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.19224 0.142320 0.0711599 0.997465i $$-0.477330\pi$$
0.0711599 + 0.997465i $$0.477330\pi$$
$$12$$ 0 0
$$13$$ −54.5464 −1.16373 −0.581863 0.813287i $$-0.697676\pi$$
−0.581863 + 0.813287i $$0.697676\pi$$
$$14$$ 24.9309 0.475933
$$15$$ 0 0
$$16$$ −79.5312 −1.24268
$$17$$ 16.1619 0.230579 0.115289 0.993332i $$-0.463220\pi$$
0.115289 + 0.993332i $$0.463220\pi$$
$$18$$ 0 0
$$19$$ 87.4470 1.05588 0.527940 0.849282i $$-0.322965\pi$$
0.527940 + 0.849282i $$0.322965\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 18.4924 0.179209
$$23$$ 176.477 1.59992 0.799958 0.600056i $$-0.204855\pi$$
0.799958 + 0.600056i $$0.204855\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −194.270 −1.46536
$$27$$ 0 0
$$28$$ 32.7926 0.221329
$$29$$ −142.170 −0.910358 −0.455179 0.890400i $$-0.650425\pi$$
−0.455179 + 0.890400i $$0.650425\pi$$
$$30$$ 0 0
$$31$$ −94.3002 −0.546349 −0.273174 0.961965i $$-0.588074\pi$$
−0.273174 + 0.961965i $$0.588074\pi$$
$$32$$ −188.793 −1.04294
$$33$$ 0 0
$$34$$ 57.5616 0.290345
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −17.3305 −0.0770031 −0.0385016 0.999259i $$-0.512258\pi$$
−0.0385016 + 0.999259i $$0.512258\pi$$
$$38$$ 311.447 1.32956
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −210.270 −0.800942 −0.400471 0.916309i $$-0.631154\pi$$
−0.400471 + 0.916309i $$0.631154\pi$$
$$42$$ 0 0
$$43$$ −521.570 −1.84974 −0.924868 0.380287i $$-0.875825\pi$$
−0.924868 + 0.380287i $$0.875825\pi$$
$$44$$ 24.3239 0.0833400
$$45$$ 0 0
$$46$$ 628.533 2.01461
$$47$$ −105.417 −0.327162 −0.163581 0.986530i $$-0.552304\pi$$
−0.163581 + 0.986530i $$0.552304\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −255.531 −0.681458
$$53$$ −108.978 −0.282440 −0.141220 0.989978i $$-0.545102\pi$$
−0.141220 + 0.989978i $$0.545102\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −82.6543 −0.197235
$$57$$ 0 0
$$58$$ −506.348 −1.14632
$$59$$ −210.365 −0.464189 −0.232094 0.972693i $$-0.574558\pi$$
−0.232094 + 0.972693i $$0.574558\pi$$
$$60$$ 0 0
$$61$$ −674.304 −1.41534 −0.707670 0.706543i $$-0.750254\pi$$
−0.707670 + 0.706543i $$0.750254\pi$$
$$62$$ −335.855 −0.687962
$$63$$ 0 0
$$64$$ −36.1449 −0.0705955
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −324.929 −0.592484 −0.296242 0.955113i $$-0.595733\pi$$
−0.296242 + 0.955113i $$0.595733\pi$$
$$68$$ 75.7131 0.135023
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −793.965 −1.32713 −0.663565 0.748118i $$-0.730958\pi$$
−0.663565 + 0.748118i $$0.730958\pi$$
$$72$$ 0 0
$$73$$ −315.417 −0.505709 −0.252854 0.967504i $$-0.581369\pi$$
−0.252854 + 0.967504i $$0.581369\pi$$
$$74$$ −61.7235 −0.0969623
$$75$$ 0 0
$$76$$ 409.659 0.618304
$$77$$ 36.3457 0.0537918
$$78$$ 0 0
$$79$$ −425.840 −0.606465 −0.303233 0.952917i $$-0.598066\pi$$
−0.303233 + 0.952917i $$0.598066\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −748.887 −1.00855
$$83$$ −283.029 −0.374295 −0.187148 0.982332i $$-0.559924\pi$$
−0.187148 + 0.982332i $$0.559924\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −1857.60 −2.32919
$$87$$ 0 0
$$88$$ −61.3087 −0.0742674
$$89$$ 843.131 1.00418 0.502088 0.864817i $$-0.332565\pi$$
0.502088 + 0.864817i $$0.332565\pi$$
$$90$$ 0 0
$$91$$ −381.825 −0.439847
$$92$$ 826.736 0.936882
$$93$$ 0 0
$$94$$ −375.447 −0.411962
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1537.33 −1.60920 −0.804601 0.593816i $$-0.797621\pi$$
−0.804601 + 0.593816i $$0.797621\pi$$
$$98$$ 174.516 0.179886
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1589.99 1.56644 0.783219 0.621745i $$-0.213576\pi$$
0.783219 + 0.621745i $$0.213576\pi$$
$$102$$ 0 0
$$103$$ 164.793 0.157646 0.0788228 0.996889i $$-0.474884\pi$$
0.0788228 + 0.996889i $$0.474884\pi$$
$$104$$ 644.071 0.607273
$$105$$ 0 0
$$106$$ −388.132 −0.355648
$$107$$ 1184.08 1.06981 0.534904 0.844913i $$-0.320348\pi$$
0.534904 + 0.844913i $$0.320348\pi$$
$$108$$ 0 0
$$109$$ 333.247 0.292837 0.146419 0.989223i $$-0.453225\pi$$
0.146419 + 0.989223i $$0.453225\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −556.719 −0.469687
$$113$$ 1881.49 1.56634 0.783168 0.621810i $$-0.213602\pi$$
0.783168 + 0.621810i $$0.213602\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −666.020 −0.533090
$$117$$ 0 0
$$118$$ −749.224 −0.584506
$$119$$ 113.133 0.0871507
$$120$$ 0 0
$$121$$ −1304.04 −0.979745
$$122$$ −2401.57 −1.78220
$$123$$ 0 0
$$124$$ −441.764 −0.319932
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1638.79 −1.14503 −0.572516 0.819893i $$-0.694033\pi$$
−0.572516 + 0.819893i $$0.694033\pi$$
$$128$$ 1381.61 0.954048
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 598.142 0.398931 0.199465 0.979905i $$-0.436079\pi$$
0.199465 + 0.979905i $$0.436079\pi$$
$$132$$ 0 0
$$133$$ 612.129 0.399085
$$134$$ −1157.25 −0.746055
$$135$$ 0 0
$$136$$ −190.836 −0.120324
$$137$$ −1005.25 −0.626894 −0.313447 0.949606i $$-0.601484\pi$$
−0.313447 + 0.949606i $$0.601484\pi$$
$$138$$ 0 0
$$139$$ −1875.01 −1.14414 −0.572072 0.820204i $$-0.693860\pi$$
−0.572072 + 0.820204i $$0.693860\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −2827.75 −1.67112
$$143$$ −283.218 −0.165621
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −1123.37 −0.636788
$$147$$ 0 0
$$148$$ −81.1875 −0.0450917
$$149$$ 1051.80 0.578299 0.289150 0.957284i $$-0.406627\pi$$
0.289150 + 0.957284i $$0.406627\pi$$
$$150$$ 0 0
$$151$$ −750.383 −0.404406 −0.202203 0.979344i $$-0.564810\pi$$
−0.202203 + 0.979344i $$0.564810\pi$$
$$152$$ −1032.55 −0.550994
$$153$$ 0 0
$$154$$ 129.447 0.0677346
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1453.90 0.739067 0.369533 0.929217i $$-0.379518\pi$$
0.369533 + 0.929217i $$0.379518\pi$$
$$158$$ −1516.65 −0.763660
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1235.34 0.604711
$$162$$ 0 0
$$163$$ −1300.49 −0.624921 −0.312461 0.949931i $$-0.601153\pi$$
−0.312461 + 0.949931i $$0.601153\pi$$
$$164$$ −985.043 −0.469018
$$165$$ 0 0
$$166$$ −1008.02 −0.471312
$$167$$ −2111.46 −0.978381 −0.489191 0.872177i $$-0.662708\pi$$
−0.489191 + 0.872177i $$0.662708\pi$$
$$168$$ 0 0
$$169$$ 778.310 0.354260
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −2443.38 −1.08317
$$173$$ 335.292 0.147351 0.0736756 0.997282i $$-0.476527\pi$$
0.0736756 + 0.997282i $$0.476527\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −412.945 −0.176857
$$177$$ 0 0
$$178$$ 3002.85 1.26446
$$179$$ −2322.23 −0.969672 −0.484836 0.874605i $$-0.661121\pi$$
−0.484836 + 0.874605i $$0.661121\pi$$
$$180$$ 0 0
$$181$$ −1525.59 −0.626500 −0.313250 0.949671i $$-0.601418\pi$$
−0.313250 + 0.949671i $$0.601418\pi$$
$$182$$ −1359.89 −0.553855
$$183$$ 0 0
$$184$$ −2083.80 −0.834891
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 83.9165 0.0328160
$$188$$ −493.841 −0.191580
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −293.912 −0.111344 −0.0556721 0.998449i $$-0.517730\pi$$
−0.0556721 + 0.998449i $$0.517730\pi$$
$$192$$ 0 0
$$193$$ 3664.91 1.36687 0.683435 0.730012i $$-0.260485\pi$$
0.683435 + 0.730012i $$0.260485\pi$$
$$194$$ −5475.29 −2.02630
$$195$$ 0 0
$$196$$ 229.548 0.0836546
$$197$$ −5101.89 −1.84515 −0.922576 0.385816i $$-0.873920\pi$$
−0.922576 + 0.385816i $$0.873920\pi$$
$$198$$ 0 0
$$199$$ 5025.86 1.79032 0.895161 0.445743i $$-0.147061\pi$$
0.895161 + 0.445743i $$0.147061\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 5662.85 1.97246
$$203$$ −995.193 −0.344083
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 586.918 0.198507
$$207$$ 0 0
$$208$$ 4338.14 1.44614
$$209$$ 454.045 0.150273
$$210$$ 0 0
$$211$$ −3267.98 −1.06624 −0.533122 0.846039i $$-0.678981\pi$$
−0.533122 + 0.846039i $$0.678981\pi$$
$$212$$ −510.526 −0.165392
$$213$$ 0 0
$$214$$ 4217.17 1.34710
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −660.101 −0.206500
$$218$$ 1186.88 0.368741
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −881.575 −0.268331
$$222$$ 0 0
$$223$$ −5457.65 −1.63888 −0.819442 0.573162i $$-0.805717\pi$$
−0.819442 + 0.573162i $$0.805717\pi$$
$$224$$ −1321.55 −0.394195
$$225$$ 0 0
$$226$$ 6701.04 1.97233
$$227$$ 281.023 0.0821682 0.0410841 0.999156i $$-0.486919\pi$$
0.0410841 + 0.999156i $$0.486919\pi$$
$$228$$ 0 0
$$229$$ 2776.64 0.801248 0.400624 0.916243i $$-0.368793\pi$$
0.400624 + 0.916243i $$0.368793\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1678.71 0.475056
$$233$$ 5781.09 1.62546 0.812729 0.582642i $$-0.197981\pi$$
0.812729 + 0.582642i $$0.197981\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −985.486 −0.271821
$$237$$ 0 0
$$238$$ 402.931 0.109740
$$239$$ −1588.17 −0.429833 −0.214916 0.976632i $$-0.568948\pi$$
−0.214916 + 0.976632i $$0.568948\pi$$
$$240$$ 0 0
$$241$$ −4330.01 −1.15735 −0.578673 0.815560i $$-0.696429\pi$$
−0.578673 + 0.815560i $$0.696429\pi$$
$$242$$ −4644.41 −1.23369
$$243$$ 0 0
$$244$$ −3158.88 −0.828798
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4769.92 −1.22876
$$248$$ 1113.47 0.285104
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1400.53 −0.352195 −0.176097 0.984373i $$-0.556347\pi$$
−0.176097 + 0.984373i $$0.556347\pi$$
$$252$$ 0 0
$$253$$ 916.312 0.227700
$$254$$ −5836.64 −1.44182
$$255$$ 0 0
$$256$$ 5209.83 1.27193
$$257$$ 4304.86 1.04486 0.522431 0.852681i $$-0.325025\pi$$
0.522431 + 0.852681i $$0.325025\pi$$
$$258$$ 0 0
$$259$$ −121.313 −0.0291045
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 2130.31 0.502333
$$263$$ −1724.69 −0.404369 −0.202184 0.979347i $$-0.564804\pi$$
−0.202184 + 0.979347i $$0.564804\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 2180.13 0.502527
$$267$$ 0 0
$$268$$ −1522.18 −0.346948
$$269$$ −8004.82 −1.81436 −0.907180 0.420744i $$-0.861769\pi$$
−0.907180 + 0.420744i $$0.861769\pi$$
$$270$$ 0 0
$$271$$ −1963.65 −0.440160 −0.220080 0.975482i $$-0.570632\pi$$
−0.220080 + 0.975482i $$0.570632\pi$$
$$272$$ −1285.38 −0.286535
$$273$$ 0 0
$$274$$ −3580.26 −0.789384
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 3278.33 0.711104 0.355552 0.934656i $$-0.384293\pi$$
0.355552 + 0.934656i $$0.384293\pi$$
$$278$$ −6677.93 −1.44070
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2859.04 −0.606961 −0.303480 0.952838i $$-0.598149\pi$$
−0.303480 + 0.952838i $$0.598149\pi$$
$$282$$ 0 0
$$283$$ 5433.66 1.14134 0.570668 0.821181i $$-0.306685\pi$$
0.570668 + 0.821181i $$0.306685\pi$$
$$284$$ −3719.45 −0.777144
$$285$$ 0 0
$$286$$ −1008.70 −0.208550
$$287$$ −1471.89 −0.302728
$$288$$ 0 0
$$289$$ −4651.79 −0.946833
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −1477.62 −0.296134
$$293$$ 8583.43 1.71143 0.855715 0.517447i $$-0.173117\pi$$
0.855715 + 0.517447i $$0.173117\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 204.634 0.0401829
$$297$$ 0 0
$$298$$ 3746.03 0.728194
$$299$$ −9626.20 −1.86186
$$300$$ 0 0
$$301$$ −3650.99 −0.699135
$$302$$ −2672.53 −0.509228
$$303$$ 0 0
$$304$$ −6954.77 −1.31212
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 5269.83 0.979691 0.489846 0.871809i $$-0.337053\pi$$
0.489846 + 0.871809i $$0.337053\pi$$
$$308$$ 170.267 0.0314995
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 4761.43 0.868154 0.434077 0.900876i $$-0.357075\pi$$
0.434077 + 0.900876i $$0.357075\pi$$
$$312$$ 0 0
$$313$$ 7602.95 1.37298 0.686492 0.727137i $$-0.259150\pi$$
0.686492 + 0.727137i $$0.259150\pi$$
$$314$$ 5178.13 0.930632
$$315$$ 0 0
$$316$$ −1994.91 −0.355135
$$317$$ 8064.55 1.42886 0.714432 0.699704i $$-0.246685\pi$$
0.714432 + 0.699704i $$0.246685\pi$$
$$318$$ 0 0
$$319$$ −738.182 −0.129562
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 4399.73 0.761452
$$323$$ 1413.31 0.243464
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −4631.76 −0.786901
$$327$$ 0 0
$$328$$ 2482.82 0.417959
$$329$$ −737.917 −0.123655
$$330$$ 0 0
$$331$$ 6960.79 1.15589 0.577945 0.816076i $$-0.303855\pi$$
0.577945 + 0.816076i $$0.303855\pi$$
$$332$$ −1325.90 −0.219181
$$333$$ 0 0
$$334$$ −7520.08 −1.23198
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −4731.61 −0.764828 −0.382414 0.923991i $$-0.624907\pi$$
−0.382414 + 0.923991i $$0.624907\pi$$
$$338$$ 2771.99 0.446084
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −489.629 −0.0777563
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ 6158.58 0.965256
$$345$$ 0 0
$$346$$ 1194.16 0.185544
$$347$$ 9796.67 1.51560 0.757800 0.652487i $$-0.226274\pi$$
0.757800 + 0.652487i $$0.226274\pi$$
$$348$$ 0 0
$$349$$ 12702.4 1.94827 0.974134 0.225971i $$-0.0725554\pi$$
0.974134 + 0.225971i $$0.0725554\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −980.256 −0.148431
$$353$$ −9970.21 −1.50329 −0.751644 0.659569i $$-0.770739\pi$$
−0.751644 + 0.659569i $$0.770739\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 3949.78 0.588028
$$357$$ 0 0
$$358$$ −8270.73 −1.22101
$$359$$ −4388.21 −0.645128 −0.322564 0.946548i $$-0.604545\pi$$
−0.322564 + 0.946548i $$0.604545\pi$$
$$360$$ 0 0
$$361$$ 787.970 0.114881
$$362$$ −5433.49 −0.788889
$$363$$ 0 0
$$364$$ −1788.72 −0.257567
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 9441.30 1.34287 0.671433 0.741065i $$-0.265679\pi$$
0.671433 + 0.741065i $$0.265679\pi$$
$$368$$ −14035.5 −1.98818
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −762.847 −0.106752
$$372$$ 0 0
$$373$$ 3219.40 0.446901 0.223451 0.974715i $$-0.428268\pi$$
0.223451 + 0.974715i $$0.428268\pi$$
$$374$$ 298.873 0.0413218
$$375$$ 0 0
$$376$$ 1244.73 0.170724
$$377$$ 7754.89 1.05941
$$378$$ 0 0
$$379$$ −14011.4 −1.89899 −0.949495 0.313783i $$-0.898403\pi$$
−0.949495 + 0.313783i $$0.898403\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −1046.78 −0.140204
$$383$$ 5322.87 0.710147 0.355073 0.934838i $$-0.384456\pi$$
0.355073 + 0.934838i $$0.384456\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 13052.8 1.72116
$$387$$ 0 0
$$388$$ −7201.88 −0.942320
$$389$$ 3844.51 0.501091 0.250545 0.968105i $$-0.419390\pi$$
0.250545 + 0.968105i $$0.419390\pi$$
$$390$$ 0 0
$$391$$ 2852.21 0.368907
$$392$$ −578.580 −0.0745478
$$393$$ 0 0
$$394$$ −18170.7 −2.32341
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 8046.40 1.01722 0.508611 0.860996i $$-0.330159\pi$$
0.508611 + 0.860996i $$0.330159\pi$$
$$398$$ 17899.9 2.25437
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −7741.38 −0.964055 −0.482027 0.876156i $$-0.660099\pi$$
−0.482027 + 0.876156i $$0.660099\pi$$
$$402$$ 0 0
$$403$$ 5143.74 0.635801
$$404$$ 7448.58 0.917279
$$405$$ 0 0
$$406$$ −3544.43 −0.433269
$$407$$ −89.9840 −0.0109591
$$408$$ 0 0
$$409$$ −8966.94 −1.08407 −0.542037 0.840354i $$-0.682347\pi$$
−0.542037 + 0.840354i $$0.682347\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 771.997 0.0923145
$$413$$ −1472.55 −0.175447
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 10298.0 1.21370
$$417$$ 0 0
$$418$$ 1617.11 0.189223
$$419$$ 12413.6 1.44736 0.723681 0.690135i $$-0.242449\pi$$
0.723681 + 0.690135i $$0.242449\pi$$
$$420$$ 0 0
$$421$$ −1672.14 −0.193575 −0.0967875 0.995305i $$-0.530857\pi$$
−0.0967875 + 0.995305i $$0.530857\pi$$
$$422$$ −11639.1 −1.34261
$$423$$ 0 0
$$424$$ 1286.79 0.147387
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −4720.13 −0.534948
$$428$$ 5547.02 0.626461
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −16021.8 −1.79059 −0.895296 0.445472i $$-0.853036\pi$$
−0.895296 + 0.445472i $$0.853036\pi$$
$$432$$ 0 0
$$433$$ −10882.7 −1.20782 −0.603912 0.797051i $$-0.706392\pi$$
−0.603912 + 0.797051i $$0.706392\pi$$
$$434$$ −2350.99 −0.260025
$$435$$ 0 0
$$436$$ 1561.15 0.171480
$$437$$ 15432.4 1.68932
$$438$$ 0 0
$$439$$ 7738.40 0.841307 0.420653 0.907221i $$-0.361801\pi$$
0.420653 + 0.907221i $$0.361801\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −3139.78 −0.337882
$$443$$ 8766.56 0.940207 0.470103 0.882611i $$-0.344217\pi$$
0.470103 + 0.882611i $$0.344217\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −19437.7 −2.06368
$$447$$ 0 0
$$448$$ −253.014 −0.0266826
$$449$$ −3099.58 −0.325786 −0.162893 0.986644i $$-0.552083\pi$$
−0.162893 + 0.986644i $$0.552083\pi$$
$$450$$ 0 0
$$451$$ −1091.77 −0.113990
$$452$$ 8814.15 0.917219
$$453$$ 0 0
$$454$$ 1000.88 0.103466
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6122.94 0.626737 0.313369 0.949632i $$-0.398542\pi$$
0.313369 + 0.949632i $$0.398542\pi$$
$$458$$ 9889.16 1.00893
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10412.2 1.05194 0.525970 0.850503i $$-0.323703\pi$$
0.525970 + 0.850503i $$0.323703\pi$$
$$462$$ 0 0
$$463$$ 11278.5 1.13209 0.566043 0.824376i $$-0.308474\pi$$
0.566043 + 0.824376i $$0.308474\pi$$
$$464$$ 11307.0 1.13128
$$465$$ 0 0
$$466$$ 20589.6 2.04677
$$467$$ 14923.2 1.47872 0.739359 0.673311i $$-0.235128\pi$$
0.739359 + 0.673311i $$0.235128\pi$$
$$468$$ 0 0
$$469$$ −2274.50 −0.223938
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 2483.93 0.242230
$$473$$ −2708.11 −0.263254
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 529.992 0.0510339
$$477$$ 0 0
$$478$$ −5656.34 −0.541245
$$479$$ 4674.21 0.445867 0.222933 0.974834i $$-0.428437\pi$$
0.222933 + 0.974834i $$0.428437\pi$$
$$480$$ 0 0
$$481$$ 945.316 0.0896106
$$482$$ −15421.6 −1.45733
$$483$$ 0 0
$$484$$ −6108.99 −0.573721
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −17081.7 −1.58941 −0.794706 0.606994i $$-0.792375\pi$$
−0.794706 + 0.606994i $$0.792375\pi$$
$$488$$ 7962.02 0.738573
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −18203.9 −1.67318 −0.836588 0.547832i $$-0.815453\pi$$
−0.836588 + 0.547832i $$0.815453\pi$$
$$492$$ 0 0
$$493$$ −2297.75 −0.209909
$$494$$ −16988.3 −1.54725
$$495$$ 0 0
$$496$$ 7499.81 0.678934
$$497$$ −5557.75 −0.501608
$$498$$ 0 0
$$499$$ 7109.47 0.637803 0.318901 0.947788i $$-0.396686\pi$$
0.318901 + 0.947788i $$0.396686\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −4988.07 −0.443483
$$503$$ −15402.0 −1.36529 −0.682647 0.730748i $$-0.739171\pi$$
−0.682647 + 0.730748i $$0.739171\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 3263.49 0.286719
$$507$$ 0 0
$$508$$ −7677.17 −0.670511
$$509$$ −6404.72 −0.557730 −0.278865 0.960330i $$-0.589958\pi$$
−0.278865 + 0.960330i $$0.589958\pi$$
$$510$$ 0 0
$$511$$ −2207.92 −0.191140
$$512$$ 7502.22 0.647567
$$513$$ 0 0
$$514$$ 15332.0 1.31569
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −547.348 −0.0465616
$$518$$ −432.064 −0.0366483
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 8916.72 0.749806 0.374903 0.927064i $$-0.377676\pi$$
0.374903 + 0.927064i $$0.377676\pi$$
$$522$$ 0 0
$$523$$ 6929.40 0.579353 0.289677 0.957125i $$-0.406452\pi$$
0.289677 + 0.957125i $$0.406452\pi$$
$$524$$ 2802.09 0.233607
$$525$$ 0 0
$$526$$ −6142.58 −0.509181
$$527$$ −1524.07 −0.125977
$$528$$ 0 0
$$529$$ 18977.2 1.55973
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2867.61 0.233697
$$533$$ 11469.5 0.932078
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 3836.69 0.309178
$$537$$ 0 0
$$538$$ −28509.6 −2.28464
$$539$$ 254.420 0.0203314
$$540$$ 0 0
$$541$$ −6929.23 −0.550667 −0.275334 0.961349i $$-0.588788\pi$$
−0.275334 + 0.961349i $$0.588788\pi$$
$$542$$ −6993.65 −0.554249
$$543$$ 0 0
$$544$$ −3051.25 −0.240480
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8509.95 0.665190 0.332595 0.943070i $$-0.392076\pi$$
0.332595 + 0.943070i $$0.392076\pi$$
$$548$$ −4709.26 −0.367098
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −12432.4 −0.961228
$$552$$ 0 0
$$553$$ −2980.88 −0.229222
$$554$$ 11676.0 0.895422
$$555$$ 0 0
$$556$$ −8783.76 −0.669990
$$557$$ −4043.94 −0.307625 −0.153813 0.988100i $$-0.549155\pi$$
−0.153813 + 0.988100i $$0.549155\pi$$
$$558$$ 0 0
$$559$$ 28449.8 2.15259
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −10182.6 −0.764285
$$563$$ 15878.9 1.18866 0.594331 0.804221i $$-0.297417\pi$$
0.594331 + 0.804221i $$0.297417\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 19352.3 1.43717
$$567$$ 0 0
$$568$$ 9374.95 0.692543
$$569$$ −11611.6 −0.855510 −0.427755 0.903895i $$-0.640695\pi$$
−0.427755 + 0.903895i $$0.640695\pi$$
$$570$$ 0 0
$$571$$ 17395.7 1.27493 0.637466 0.770478i $$-0.279982\pi$$
0.637466 + 0.770478i $$0.279982\pi$$
$$572$$ −1326.78 −0.0969850
$$573$$ 0 0
$$574$$ −5242.21 −0.381195
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 11474.0 0.827848 0.413924 0.910311i $$-0.364158\pi$$
0.413924 + 0.910311i $$0.364158\pi$$
$$578$$ −16567.6 −1.19225
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1981.20 −0.141470
$$582$$ 0 0
$$583$$ −565.841 −0.0401968
$$584$$ 3724.37 0.263896
$$585$$ 0 0
$$586$$ 30570.3 2.15503
$$587$$ 11870.4 0.834659 0.417330 0.908755i $$-0.362966\pi$$
0.417330 + 0.908755i $$0.362966\pi$$
$$588$$ 0 0
$$589$$ −8246.26 −0.576878
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1378.32 0.0956899
$$593$$ −5760.65 −0.398923 −0.199462 0.979906i $$-0.563919\pi$$
−0.199462 + 0.979906i $$0.563919\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 4927.31 0.338642
$$597$$ 0 0
$$598$$ −34284.2 −2.34446
$$599$$ 21696.5 1.47996 0.739978 0.672631i $$-0.234836\pi$$
0.739978 + 0.672631i $$0.234836\pi$$
$$600$$ 0 0
$$601$$ −12403.0 −0.841812 −0.420906 0.907104i $$-0.638288\pi$$
−0.420906 + 0.907104i $$0.638288\pi$$
$$602$$ −13003.2 −0.880350
$$603$$ 0 0
$$604$$ −3515.29 −0.236813
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −17066.5 −1.14120 −0.570600 0.821228i $$-0.693289\pi$$
−0.570600 + 0.821228i $$0.693289\pi$$
$$608$$ −16509.3 −1.10122
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 5750.10 0.380727
$$612$$ 0 0
$$613$$ −2707.50 −0.178393 −0.0891965 0.996014i $$-0.528430\pi$$
−0.0891965 + 0.996014i $$0.528430\pi$$
$$614$$ 18768.8 1.23363
$$615$$ 0 0
$$616$$ −429.161 −0.0280704
$$617$$ −23226.3 −1.51549 −0.757743 0.652553i $$-0.773698\pi$$
−0.757743 + 0.652553i $$0.773698\pi$$
$$618$$ 0 0
$$619$$ −2298.43 −0.149243 −0.0746216 0.997212i $$-0.523775\pi$$
−0.0746216 + 0.997212i $$0.523775\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 16958.1 1.09318
$$623$$ 5901.91 0.379543
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 27078.3 1.72886
$$627$$ 0 0
$$628$$ 6811.01 0.432785
$$629$$ −280.094 −0.0177553
$$630$$ 0 0
$$631$$ −663.913 −0.0418858 −0.0209429 0.999781i $$-0.506667\pi$$
−0.0209429 + 0.999781i $$0.506667\pi$$
$$632$$ 5028.22 0.316474
$$633$$ 0 0
$$634$$ 28722.3 1.79923
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −2672.77 −0.166247
$$638$$ −2629.08 −0.163144
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −15215.6 −0.937566 −0.468783 0.883313i $$-0.655307\pi$$
−0.468783 + 0.883313i $$0.655307\pi$$
$$642$$ 0 0
$$643$$ −12904.0 −0.791420 −0.395710 0.918375i $$-0.629502\pi$$
−0.395710 + 0.918375i $$0.629502\pi$$
$$644$$ 5787.15 0.354108
$$645$$ 0 0
$$646$$ 5033.58 0.306569
$$647$$ −9425.54 −0.572730 −0.286365 0.958121i $$-0.592447\pi$$
−0.286365 + 0.958121i $$0.592447\pi$$
$$648$$ 0 0
$$649$$ −1092.26 −0.0660632
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −6092.35 −0.365943
$$653$$ −29894.7 −1.79153 −0.895765 0.444528i $$-0.853372\pi$$
−0.895765 + 0.444528i $$0.853372\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 16723.0 0.995312
$$657$$ 0 0
$$658$$ −2628.13 −0.155707
$$659$$ 11593.6 0.685313 0.342656 0.939461i $$-0.388673\pi$$
0.342656 + 0.939461i $$0.388673\pi$$
$$660$$ 0 0
$$661$$ −17149.2 −1.00911 −0.504557 0.863378i $$-0.668344\pi$$
−0.504557 + 0.863378i $$0.668344\pi$$
$$662$$ 24791.2 1.45550
$$663$$ 0 0
$$664$$ 3341.94 0.195320
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −25089.9 −1.45650
$$668$$ −9891.47 −0.572923
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −3501.15 −0.201431
$$672$$ 0 0
$$673$$ 16475.0 0.943633 0.471817 0.881697i $$-0.343598\pi$$
0.471817 + 0.881697i $$0.343598\pi$$
$$674$$ −16851.9 −0.963071
$$675$$ 0 0
$$676$$ 3646.11 0.207448
$$677$$ −4559.89 −0.258864 −0.129432 0.991588i $$-0.541315\pi$$
−0.129432 + 0.991588i $$0.541315\pi$$
$$678$$ 0 0
$$679$$ −10761.3 −0.608221
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −1743.84 −0.0979106
$$683$$ −27895.9 −1.56282 −0.781411 0.624017i $$-0.785500\pi$$
−0.781411 + 0.624017i $$0.785500\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1221.61 0.0679904
$$687$$ 0 0
$$688$$ 41481.1 2.29862
$$689$$ 5944.37 0.328683
$$690$$ 0 0
$$691$$ −28178.5 −1.55132 −0.775659 0.631152i $$-0.782582\pi$$
−0.775659 + 0.631152i $$0.782582\pi$$
$$692$$ 1570.73 0.0862862
$$693$$ 0 0
$$694$$ 34891.4 1.90844
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −3398.37 −0.184680
$$698$$ 45240.4 2.45326
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −3912.96 −0.210828 −0.105414 0.994428i $$-0.533617\pi$$
−0.105414 + 0.994428i $$0.533617\pi$$
$$702$$ 0 0
$$703$$ −1515.50 −0.0813060
$$704$$ −187.673 −0.0100471
$$705$$ 0 0
$$706$$ −35509.4 −1.89294
$$707$$ 11130.0 0.592058
$$708$$ 0 0
$$709$$ −7782.72 −0.412251 −0.206126 0.978526i $$-0.566086\pi$$
−0.206126 + 0.978526i $$0.566086\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −9955.49 −0.524014
$$713$$ −16641.8 −0.874112
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −10878.8 −0.567823
$$717$$ 0 0
$$718$$ −15628.9 −0.812345
$$719$$ 27868.8 1.44552 0.722762 0.691097i $$-0.242872\pi$$
0.722762 + 0.691097i $$0.242872\pi$$
$$720$$ 0 0
$$721$$ 1153.55 0.0595844
$$722$$ 2806.40 0.144658
$$723$$ 0 0
$$724$$ −7146.89 −0.366868
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 34202.4 1.74484 0.872419 0.488759i $$-0.162550\pi$$
0.872419 + 0.488759i $$0.162550\pi$$
$$728$$ 4508.50 0.229527
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −8429.58 −0.426510
$$732$$ 0 0
$$733$$ 12544.2 0.632101 0.316051 0.948742i $$-0.397643\pi$$
0.316051 + 0.948742i $$0.397643\pi$$
$$734$$ 33625.7 1.69094
$$735$$ 0 0
$$736$$ −33317.6 −1.66862
$$737$$ −1687.11 −0.0843221
$$738$$ 0 0
$$739$$ 4563.19 0.227144 0.113572 0.993530i $$-0.463771\pi$$
0.113572 + 0.993530i $$0.463771\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −2716.92 −0.134422
$$743$$ 10369.7 0.512017 0.256009 0.966674i $$-0.417592\pi$$
0.256009 + 0.966674i $$0.417592\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 11466.1 0.562738
$$747$$ 0 0
$$748$$ 393.120 0.0192164
$$749$$ 8288.57 0.404349
$$750$$ 0 0
$$751$$ −36808.0 −1.78847 −0.894237 0.447595i $$-0.852281\pi$$
−0.894237 + 0.447595i $$0.852281\pi$$
$$752$$ 8383.92 0.406556
$$753$$ 0 0
$$754$$ 27619.4 1.33401
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 12516.6 0.600955 0.300477 0.953789i $$-0.402854\pi$$
0.300477 + 0.953789i $$0.402854\pi$$
$$758$$ −49902.3 −2.39121
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −11745.1 −0.559473 −0.279736 0.960077i $$-0.590247\pi$$
−0.279736 + 0.960077i $$0.590247\pi$$
$$762$$ 0 0
$$763$$ 2332.73 0.110682
$$764$$ −1376.88 −0.0652011
$$765$$ 0 0
$$766$$ 18957.7 0.894216
$$767$$ 11474.6 0.540189
$$768$$ 0 0
$$769$$ 36497.1 1.71147 0.855735 0.517414i $$-0.173105\pi$$
0.855735 + 0.517414i $$0.173105\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 17168.8 0.800414
$$773$$ −29858.1 −1.38929 −0.694644 0.719353i $$-0.744438\pi$$
−0.694644 + 0.719353i $$0.744438\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 18152.5 0.839737
$$777$$ 0 0
$$778$$ 13692.4 0.630973
$$779$$ −18387.5 −0.845699
$$780$$ 0 0
$$781$$ −4122.45 −0.188877
$$782$$ 10158.3 0.464527
$$783$$ 0 0
$$784$$ −3897.03 −0.177525
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −3168.00 −0.143491 −0.0717453 0.997423i $$-0.522857\pi$$
−0.0717453 + 0.997423i $$0.522857\pi$$
$$788$$ −23900.6 −1.08049
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 13170.5 0.592019
$$792$$ 0 0
$$793$$ 36780.8 1.64707
$$794$$ 28657.7 1.28089
$$795$$ 0 0
$$796$$ 23544.5 1.04838
$$797$$ 29317.0 1.30296 0.651481 0.758665i $$-0.274148\pi$$
0.651481 + 0.758665i $$0.274148\pi$$
$$798$$ 0 0
$$799$$ −1703.74 −0.0754366
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −27571.3 −1.21394
$$803$$ −1637.72 −0.0719724
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 18319.7 0.800600
$$807$$ 0 0
$$808$$ −18774.3 −0.817422
$$809$$ −16657.3 −0.723904 −0.361952 0.932197i $$-0.617890\pi$$
−0.361952 + 0.932197i $$0.617890\pi$$
$$810$$ 0 0
$$811$$ 5144.55 0.222749 0.111375 0.993779i $$-0.464475\pi$$
0.111375 + 0.993779i $$0.464475\pi$$
$$812$$ −4662.14 −0.201489
$$813$$ 0 0
$$814$$ −320.483 −0.0137997
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −45609.7 −1.95310
$$818$$ −31936.2 −1.36507
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 5217.18 0.221779 0.110890 0.993833i $$-0.464630\pi$$
0.110890 + 0.993833i $$0.464630\pi$$
$$822$$ 0 0
$$823$$ −42326.5 −1.79272 −0.896360 0.443327i $$-0.853798\pi$$
−0.896360 + 0.443327i $$0.853798\pi$$
$$824$$ −1945.83 −0.0822649
$$825$$ 0 0
$$826$$ −5244.57 −0.220922
$$827$$ −31675.8 −1.33189 −0.665946 0.746000i $$-0.731972\pi$$
−0.665946 + 0.746000i $$0.731972\pi$$
$$828$$ 0 0
$$829$$ −3471.22 −0.145429 −0.0727144 0.997353i $$-0.523166\pi$$
−0.0727144 + 0.997353i $$0.523166\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1971.57 0.0821539
$$833$$ 791.934 0.0329399
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 2127.05 0.0879970
$$837$$ 0 0
$$838$$ 44211.7 1.82252
$$839$$ 20964.9 0.862682 0.431341 0.902189i $$-0.358041\pi$$
0.431341 + 0.902189i $$0.358041\pi$$
$$840$$ 0 0
$$841$$ −4176.57 −0.171248
$$842$$ −5955.41 −0.243749
$$843$$ 0 0
$$844$$ −15309.4 −0.624373
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −9128.28 −0.370309
$$848$$ 8667.17 0.350981
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −3058.44 −0.123199
$$852$$ 0 0
$$853$$ −1084.77 −0.0435426 −0.0217713 0.999763i $$-0.506931\pi$$
−0.0217713 + 0.999763i $$0.506931\pi$$
$$854$$ −16811.0 −0.673607
$$855$$ 0 0
$$856$$ −13981.4 −0.558263
$$857$$ −12661.6 −0.504679 −0.252340 0.967639i $$-0.581200\pi$$
−0.252340 + 0.967639i $$0.581200\pi$$
$$858$$ 0 0
$$859$$ −39678.7 −1.57604 −0.788021 0.615648i $$-0.788894\pi$$
−0.788021 + 0.615648i $$0.788894\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −57062.6 −2.25471
$$863$$ −41614.0 −1.64143 −0.820717 0.571334i $$-0.806426\pi$$
−0.820717 + 0.571334i $$0.806426\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −38759.2 −1.52089
$$867$$ 0 0
$$868$$ −3092.35 −0.120923
$$869$$ −2211.06 −0.0863120
$$870$$ 0 0
$$871$$ 17723.7 0.689489
$$872$$ −3934.90 −0.152813
$$873$$ 0 0
$$874$$ 54963.3 2.12719
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 37061.0 1.42698 0.713490 0.700665i $$-0.247113\pi$$
0.713490 + 0.700665i $$0.247113\pi$$
$$878$$ 27560.7 1.05937
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 25468.7 0.973962 0.486981 0.873412i $$-0.338098\pi$$
0.486981 + 0.873412i $$0.338098\pi$$
$$882$$ 0 0
$$883$$ −34428.3 −1.31212 −0.656062 0.754707i $$-0.727779\pi$$
−0.656062 + 0.754707i $$0.727779\pi$$
$$884$$ −4129.88 −0.157130
$$885$$ 0 0
$$886$$ 31222.6 1.18391
$$887$$ 41295.4 1.56321 0.781603 0.623777i $$-0.214403\pi$$
0.781603 + 0.623777i $$0.214403\pi$$
$$888$$ 0 0
$$889$$ −11471.5 −0.432782
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −25567.2 −0.959702
$$893$$ −9218.37 −0.345443
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 9671.26 0.360596
$$897$$ 0 0
$$898$$ −11039.3 −0.410230
$$899$$ 13406.7 0.497373
$$900$$ 0 0
$$901$$ −1761.30 −0.0651247
$$902$$ −3888.40 −0.143536
$$903$$ 0 0
$$904$$ −22216.2 −0.817368
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 53733.8 1.96715 0.983573 0.180509i $$-0.0577746\pi$$
0.983573 + 0.180509i $$0.0577746\pi$$
$$908$$ 1316.50 0.0481162
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −24296.8 −0.883634 −0.441817 0.897105i $$-0.645666\pi$$
−0.441817 + 0.897105i $$0.645666\pi$$
$$912$$ 0 0
$$913$$ −1469.55 −0.0532696
$$914$$ 21807.2 0.789187
$$915$$ 0 0
$$916$$ 13007.6 0.469197
$$917$$ 4186.99 0.150782
$$918$$ 0 0
$$919$$ 4280.46 0.153645 0.0768223 0.997045i $$-0.475523\pi$$
0.0768223 + 0.997045i $$0.475523\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 37083.6 1.32460
$$923$$ 43307.9 1.54442
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 40168.9 1.42552
$$927$$ 0 0
$$928$$ 26840.7 0.949450
$$929$$ 31884.5 1.12604 0.563022 0.826442i $$-0.309639\pi$$
0.563022 + 0.826442i $$0.309639\pi$$
$$930$$ 0 0
$$931$$ 4284.90 0.150840
$$932$$ 27082.4 0.951839
$$933$$ 0 0
$$934$$ 53149.6 1.86200
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 44523.1 1.55230 0.776151 0.630548i $$-0.217170\pi$$
0.776151 + 0.630548i $$0.217170\pi$$
$$938$$ −8100.76 −0.281982
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −46374.9 −1.60657 −0.803283 0.595598i $$-0.796915\pi$$
−0.803283 + 0.595598i $$0.796915\pi$$
$$942$$ 0 0
$$943$$ −37107.9 −1.28144
$$944$$ 16730.6 0.576836
$$945$$ 0 0
$$946$$ −9645.09 −0.331489
$$947$$ −20348.2 −0.698234 −0.349117 0.937079i $$-0.613518\pi$$
−0.349117 + 0.937079i $$0.613518\pi$$
$$948$$ 0 0
$$949$$ 17204.8 0.588507
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −1335.85 −0.0454782
$$953$$ −45012.9 −1.53002 −0.765010 0.644018i $$-0.777266\pi$$
−0.765010 + 0.644018i $$0.777266\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −7440.02 −0.251702
$$957$$ 0 0
$$958$$ 16647.4 0.561435
$$959$$ −7036.76 −0.236944
$$960$$ 0 0
$$961$$ −20898.5 −0.701503
$$962$$ 3366.79 0.112838
$$963$$ 0 0
$$964$$ −20284.6 −0.677721
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 40305.8 1.34038 0.670190 0.742190i $$-0.266213\pi$$
0.670190 + 0.742190i $$0.266213\pi$$
$$968$$ 15397.8 0.511265
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −33991.8 −1.12343 −0.561713 0.827332i $$-0.689858\pi$$
−0.561713 + 0.827332i $$0.689858\pi$$
$$972$$ 0 0
$$973$$ −13125.0 −0.432445
$$974$$ −60837.2 −2.00139
$$975$$ 0 0
$$976$$ 53628.2 1.75881
$$977$$ −18219.0 −0.596600 −0.298300 0.954472i $$-0.596420\pi$$
−0.298300 + 0.954472i $$0.596420\pi$$
$$978$$ 0 0
$$979$$ 4377.73 0.142914
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −64834.1 −2.10686
$$983$$ 7676.89 0.249089 0.124545 0.992214i $$-0.460253\pi$$
0.124545 + 0.992214i $$0.460253\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −8183.55 −0.264318
$$987$$ 0 0
$$988$$ −22345.4 −0.719537
$$989$$ −92045.3 −2.95942
$$990$$ 0 0
$$991$$ 50585.6 1.62150 0.810748 0.585395i $$-0.199061\pi$$
0.810748 + 0.585395i $$0.199061\pi$$
$$992$$ 17803.2 0.569810
$$993$$ 0 0
$$994$$ −19794.2 −0.631625
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 53060.5 1.68550 0.842750 0.538305i $$-0.180935\pi$$
0.842750 + 0.538305i $$0.180935\pi$$
$$998$$ 25320.8 0.803121
$$999$$ 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 1575.4.a.x.1.2 2
3.2 odd 2 525.4.a.j.1.1 2
5.4 even 2 1575.4.a.o.1.1 2
15.2 even 4 525.4.d.m.274.1 4
15.8 even 4 525.4.d.m.274.4 4
15.14 odd 2 525.4.a.m.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.j.1.1 2 3.2 odd 2
525.4.a.m.1.2 yes 2 15.14 odd 2
525.4.d.m.274.1 4 15.2 even 4
525.4.d.m.274.4 4 15.8 even 4
1575.4.a.o.1.1 2 5.4 even 2
1575.4.a.x.1.2 2 1.1 even 1 trivial