# Properties

 Label 450.4.a.m.1.1 Level $450$ Weight $4$ Character 450.1 Self dual yes Analytic conductor $26.551$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.5508595026$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 90) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +4.00000 q^{4} -14.0000 q^{7} +8.00000 q^{8} +O(q^{10})$$ $$q+2.00000 q^{2} +4.00000 q^{4} -14.0000 q^{7} +8.00000 q^{8} +6.00000 q^{11} -68.0000 q^{13} -28.0000 q^{14} +16.0000 q^{16} -78.0000 q^{17} +44.0000 q^{19} +12.0000 q^{22} -120.000 q^{23} -136.000 q^{26} -56.0000 q^{28} +126.000 q^{29} -244.000 q^{31} +32.0000 q^{32} -156.000 q^{34} +304.000 q^{37} +88.0000 q^{38} -480.000 q^{41} -104.000 q^{43} +24.0000 q^{44} -240.000 q^{46} -600.000 q^{47} -147.000 q^{49} -272.000 q^{52} +258.000 q^{53} -112.000 q^{56} +252.000 q^{58} +534.000 q^{59} +362.000 q^{61} -488.000 q^{62} +64.0000 q^{64} +268.000 q^{67} -312.000 q^{68} -972.000 q^{71} -470.000 q^{73} +608.000 q^{74} +176.000 q^{76} -84.0000 q^{77} +1244.00 q^{79} -960.000 q^{82} -396.000 q^{83} -208.000 q^{86} +48.0000 q^{88} -972.000 q^{89} +952.000 q^{91} -480.000 q^{92} -1200.00 q^{94} +46.0000 q^{97} -294.000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 0.707107
$$3$$ 0 0
$$4$$ 4.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −14.0000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 8.00000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 6.00000 0.164461 0.0822304 0.996613i $$-0.473796\pi$$
0.0822304 + 0.996613i $$0.473796\pi$$
$$12$$ 0 0
$$13$$ −68.0000 −1.45075 −0.725377 0.688352i $$-0.758335\pi$$
−0.725377 + 0.688352i $$0.758335\pi$$
$$14$$ −28.0000 −0.534522
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ −78.0000 −1.11281 −0.556405 0.830911i $$-0.687820\pi$$
−0.556405 + 0.830911i $$0.687820\pi$$
$$18$$ 0 0
$$19$$ 44.0000 0.531279 0.265639 0.964072i $$-0.414417\pi$$
0.265639 + 0.964072i $$0.414417\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 12.0000 0.116291
$$23$$ −120.000 −1.08790 −0.543951 0.839117i $$-0.683072\pi$$
−0.543951 + 0.839117i $$0.683072\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −136.000 −1.02584
$$27$$ 0 0
$$28$$ −56.0000 −0.377964
$$29$$ 126.000 0.806814 0.403407 0.915021i $$-0.367826\pi$$
0.403407 + 0.915021i $$0.367826\pi$$
$$30$$ 0 0
$$31$$ −244.000 −1.41367 −0.706834 0.707380i $$-0.749877\pi$$
−0.706834 + 0.707380i $$0.749877\pi$$
$$32$$ 32.0000 0.176777
$$33$$ 0 0
$$34$$ −156.000 −0.786876
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 304.000 1.35074 0.675369 0.737480i $$-0.263984\pi$$
0.675369 + 0.737480i $$0.263984\pi$$
$$38$$ 88.0000 0.375671
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −480.000 −1.82838 −0.914188 0.405291i $$-0.867170\pi$$
−0.914188 + 0.405291i $$0.867170\pi$$
$$42$$ 0 0
$$43$$ −104.000 −0.368834 −0.184417 0.982848i $$-0.559040\pi$$
−0.184417 + 0.982848i $$0.559040\pi$$
$$44$$ 24.0000 0.0822304
$$45$$ 0 0
$$46$$ −240.000 −0.769262
$$47$$ −600.000 −1.86211 −0.931053 0.364884i $$-0.881109\pi$$
−0.931053 + 0.364884i $$0.881109\pi$$
$$48$$ 0 0
$$49$$ −147.000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −272.000 −0.725377
$$53$$ 258.000 0.668661 0.334330 0.942456i $$-0.391490\pi$$
0.334330 + 0.942456i $$0.391490\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −112.000 −0.267261
$$57$$ 0 0
$$58$$ 252.000 0.570504
$$59$$ 534.000 1.17832 0.589160 0.808016i $$-0.299459\pi$$
0.589160 + 0.808016i $$0.299459\pi$$
$$60$$ 0 0
$$61$$ 362.000 0.759825 0.379913 0.925022i $$-0.375954\pi$$
0.379913 + 0.925022i $$0.375954\pi$$
$$62$$ −488.000 −0.999614
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 268.000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ −312.000 −0.556405
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −972.000 −1.62472 −0.812360 0.583156i $$-0.801818\pi$$
−0.812360 + 0.583156i $$0.801818\pi$$
$$72$$ 0 0
$$73$$ −470.000 −0.753553 −0.376776 0.926304i $$-0.622967\pi$$
−0.376776 + 0.926304i $$0.622967\pi$$
$$74$$ 608.000 0.955116
$$75$$ 0 0
$$76$$ 176.000 0.265639
$$77$$ −84.0000 −0.124321
$$78$$ 0 0
$$79$$ 1244.00 1.77166 0.885829 0.464012i $$-0.153591\pi$$
0.885829 + 0.464012i $$0.153591\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −960.000 −1.29286
$$83$$ −396.000 −0.523695 −0.261847 0.965109i $$-0.584332\pi$$
−0.261847 + 0.965109i $$0.584332\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −208.000 −0.260805
$$87$$ 0 0
$$88$$ 48.0000 0.0581456
$$89$$ −972.000 −1.15766 −0.578830 0.815448i $$-0.696491\pi$$
−0.578830 + 0.815448i $$0.696491\pi$$
$$90$$ 0 0
$$91$$ 952.000 1.09667
$$92$$ −480.000 −0.543951
$$93$$ 0 0
$$94$$ −1200.00 −1.31671
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 46.0000 0.0481504 0.0240752 0.999710i $$-0.492336\pi$$
0.0240752 + 0.999710i $$0.492336\pi$$
$$98$$ −294.000 −0.303046
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1506.00 −1.48369 −0.741845 0.670572i $$-0.766049\pi$$
−0.741845 + 0.670572i $$0.766049\pi$$
$$102$$ 0 0
$$103$$ 1474.00 1.41007 0.705037 0.709171i $$-0.250931\pi$$
0.705037 + 0.709171i $$0.250931\pi$$
$$104$$ −544.000 −0.512919
$$105$$ 0 0
$$106$$ 516.000 0.472815
$$107$$ −924.000 −0.834827 −0.417413 0.908717i $$-0.637063\pi$$
−0.417413 + 0.908717i $$0.637063\pi$$
$$108$$ 0 0
$$109$$ 698.000 0.613360 0.306680 0.951813i $$-0.400782\pi$$
0.306680 + 0.951813i $$0.400782\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −224.000 −0.188982
$$113$$ 222.000 0.184814 0.0924071 0.995721i $$-0.470544\pi$$
0.0924071 + 0.995721i $$0.470544\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 504.000 0.403407
$$117$$ 0 0
$$118$$ 1068.00 0.833198
$$119$$ 1092.00 0.841206
$$120$$ 0 0
$$121$$ −1295.00 −0.972953
$$122$$ 724.000 0.537278
$$123$$ 0 0
$$124$$ −976.000 −0.706834
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1906.00 1.33173 0.665867 0.746071i $$-0.268062\pi$$
0.665867 + 0.746071i $$0.268062\pi$$
$$128$$ 128.000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2874.00 1.91681 0.958407 0.285406i $$-0.0921285\pi$$
0.958407 + 0.285406i $$0.0921285\pi$$
$$132$$ 0 0
$$133$$ −616.000 −0.401609
$$134$$ 536.000 0.345547
$$135$$ 0 0
$$136$$ −624.000 −0.393438
$$137$$ 798.000 0.497648 0.248824 0.968549i $$-0.419956\pi$$
0.248824 + 0.968549i $$0.419956\pi$$
$$138$$ 0 0
$$139$$ −700.000 −0.427146 −0.213573 0.976927i $$-0.568510\pi$$
−0.213573 + 0.976927i $$0.568510\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −1944.00 −1.14885
$$143$$ −408.000 −0.238592
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −940.000 −0.532842
$$147$$ 0 0
$$148$$ 1216.00 0.675369
$$149$$ 114.000 0.0626795 0.0313397 0.999509i $$-0.490023\pi$$
0.0313397 + 0.999509i $$0.490023\pi$$
$$150$$ 0 0
$$151$$ 1064.00 0.573424 0.286712 0.958017i $$-0.407438\pi$$
0.286712 + 0.958017i $$0.407438\pi$$
$$152$$ 352.000 0.187835
$$153$$ 0 0
$$154$$ −168.000 −0.0879080
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1948.00 0.990238 0.495119 0.868825i $$-0.335125\pi$$
0.495119 + 0.868825i $$0.335125\pi$$
$$158$$ 2488.00 1.25275
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1680.00 0.822376
$$162$$ 0 0
$$163$$ −2060.00 −0.989887 −0.494944 0.868925i $$-0.664811\pi$$
−0.494944 + 0.868925i $$0.664811\pi$$
$$164$$ −1920.00 −0.914188
$$165$$ 0 0
$$166$$ −792.000 −0.370308
$$167$$ 1248.00 0.578282 0.289141 0.957286i $$-0.406630\pi$$
0.289141 + 0.957286i $$0.406630\pi$$
$$168$$ 0 0
$$169$$ 2427.00 1.10469
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −416.000 −0.184417
$$173$$ 1146.00 0.503634 0.251817 0.967775i $$-0.418972\pi$$
0.251817 + 0.967775i $$0.418972\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 96.0000 0.0411152
$$177$$ 0 0
$$178$$ −1944.00 −0.818590
$$179$$ −1146.00 −0.478525 −0.239263 0.970955i $$-0.576906\pi$$
−0.239263 + 0.970955i $$0.576906\pi$$
$$180$$ 0 0
$$181$$ −118.000 −0.0484579 −0.0242289 0.999706i $$-0.507713\pi$$
−0.0242289 + 0.999706i $$0.507713\pi$$
$$182$$ 1904.00 0.775461
$$183$$ 0 0
$$184$$ −960.000 −0.384631
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −468.000 −0.183014
$$188$$ −2400.00 −0.931053
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1692.00 0.640989 0.320494 0.947250i $$-0.396151\pi$$
0.320494 + 0.947250i $$0.396151\pi$$
$$192$$ 0 0
$$193$$ −3350.00 −1.24942 −0.624711 0.780856i $$-0.714783\pi$$
−0.624711 + 0.780856i $$0.714783\pi$$
$$194$$ 92.0000 0.0340475
$$195$$ 0 0
$$196$$ −588.000 −0.214286
$$197$$ 3606.00 1.30415 0.652073 0.758156i $$-0.273899\pi$$
0.652073 + 0.758156i $$0.273899\pi$$
$$198$$ 0 0
$$199$$ 2696.00 0.960374 0.480187 0.877166i $$-0.340569\pi$$
0.480187 + 0.877166i $$0.340569\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −3012.00 −1.04913
$$203$$ −1764.00 −0.609894
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 2948.00 0.997072
$$207$$ 0 0
$$208$$ −1088.00 −0.362689
$$209$$ 264.000 0.0873745
$$210$$ 0 0
$$211$$ −4.00000 −0.00130508 −0.000652539 1.00000i $$-0.500208\pi$$
−0.000652539 1.00000i $$0.500208\pi$$
$$212$$ 1032.00 0.334330
$$213$$ 0 0
$$214$$ −1848.00 −0.590312
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3416.00 1.06863
$$218$$ 1396.00 0.433711
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 5304.00 1.61441
$$222$$ 0 0
$$223$$ 1162.00 0.348938 0.174469 0.984663i $$-0.444179\pi$$
0.174469 + 0.984663i $$0.444179\pi$$
$$224$$ −448.000 −0.133631
$$225$$ 0 0
$$226$$ 444.000 0.130683
$$227$$ 2400.00 0.701734 0.350867 0.936425i $$-0.385887\pi$$
0.350867 + 0.936425i $$0.385887\pi$$
$$228$$ 0 0
$$229$$ −2314.00 −0.667744 −0.333872 0.942618i $$-0.608355\pi$$
−0.333872 + 0.942618i $$0.608355\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1008.00 0.285252
$$233$$ 18.0000 0.00506103 0.00253051 0.999997i $$-0.499195\pi$$
0.00253051 + 0.999997i $$0.499195\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 2136.00 0.589160
$$237$$ 0 0
$$238$$ 2184.00 0.594822
$$239$$ −5868.00 −1.58816 −0.794078 0.607816i $$-0.792046\pi$$
−0.794078 + 0.607816i $$0.792046\pi$$
$$240$$ 0 0
$$241$$ −4330.00 −1.15734 −0.578672 0.815560i $$-0.696429\pi$$
−0.578672 + 0.815560i $$0.696429\pi$$
$$242$$ −2590.00 −0.687981
$$243$$ 0 0
$$244$$ 1448.00 0.379913
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2992.00 −0.770755
$$248$$ −1952.00 −0.499807
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 498.000 0.125233 0.0626165 0.998038i $$-0.480056\pi$$
0.0626165 + 0.998038i $$0.480056\pi$$
$$252$$ 0 0
$$253$$ −720.000 −0.178917
$$254$$ 3812.00 0.941678
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ −642.000 −0.155824 −0.0779122 0.996960i $$-0.524825\pi$$
−0.0779122 + 0.996960i $$0.524825\pi$$
$$258$$ 0 0
$$259$$ −4256.00 −1.02106
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 5748.00 1.35539
$$263$$ 7968.00 1.86817 0.934084 0.357055i $$-0.116219\pi$$
0.934084 + 0.357055i $$0.116219\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −1232.00 −0.283980
$$267$$ 0 0
$$268$$ 1072.00 0.244339
$$269$$ −4218.00 −0.956045 −0.478022 0.878348i $$-0.658646\pi$$
−0.478022 + 0.878348i $$0.658646\pi$$
$$270$$ 0 0
$$271$$ 848.000 0.190082 0.0950412 0.995473i $$-0.469702\pi$$
0.0950412 + 0.995473i $$0.469702\pi$$
$$272$$ −1248.00 −0.278203
$$273$$ 0 0
$$274$$ 1596.00 0.351890
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1504.00 0.326233 0.163117 0.986607i $$-0.447845\pi$$
0.163117 + 0.986607i $$0.447845\pi$$
$$278$$ −1400.00 −0.302037
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1308.00 −0.277682 −0.138841 0.990315i $$-0.544338\pi$$
−0.138841 + 0.990315i $$0.544338\pi$$
$$282$$ 0 0
$$283$$ 5932.00 1.24601 0.623005 0.782218i $$-0.285912\pi$$
0.623005 + 0.782218i $$0.285912\pi$$
$$284$$ −3888.00 −0.812360
$$285$$ 0 0
$$286$$ −816.000 −0.168710
$$287$$ 6720.00 1.38212
$$288$$ 0 0
$$289$$ 1171.00 0.238347
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −1880.00 −0.376776
$$293$$ −5226.00 −1.04200 −0.521000 0.853556i $$-0.674441\pi$$
−0.521000 + 0.853556i $$0.674441\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 2432.00 0.477558
$$297$$ 0 0
$$298$$ 228.000 0.0443211
$$299$$ 8160.00 1.57828
$$300$$ 0 0
$$301$$ 1456.00 0.278812
$$302$$ 2128.00 0.405472
$$303$$ 0 0
$$304$$ 704.000 0.132820
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −4448.00 −0.826908 −0.413454 0.910525i $$-0.635678\pi$$
−0.413454 + 0.910525i $$0.635678\pi$$
$$308$$ −336.000 −0.0621603
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −9132.00 −1.66504 −0.832521 0.553993i $$-0.813103\pi$$
−0.832521 + 0.553993i $$0.813103\pi$$
$$312$$ 0 0
$$313$$ 2170.00 0.391871 0.195936 0.980617i $$-0.437226\pi$$
0.195936 + 0.980617i $$0.437226\pi$$
$$314$$ 3896.00 0.700204
$$315$$ 0 0
$$316$$ 4976.00 0.885829
$$317$$ −7674.00 −1.35967 −0.679834 0.733366i $$-0.737948\pi$$
−0.679834 + 0.733366i $$0.737948\pi$$
$$318$$ 0 0
$$319$$ 756.000 0.132689
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 3360.00 0.581508
$$323$$ −3432.00 −0.591212
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −4120.00 −0.699956
$$327$$ 0 0
$$328$$ −3840.00 −0.646428
$$329$$ 8400.00 1.40762
$$330$$ 0 0
$$331$$ 9596.00 1.59349 0.796743 0.604318i $$-0.206554\pi$$
0.796743 + 0.604318i $$0.206554\pi$$
$$332$$ −1584.00 −0.261847
$$333$$ 0 0
$$334$$ 2496.00 0.408907
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −12158.0 −1.96525 −0.982624 0.185608i $$-0.940574\pi$$
−0.982624 + 0.185608i $$0.940574\pi$$
$$338$$ 4854.00 0.781133
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −1464.00 −0.232493
$$342$$ 0 0
$$343$$ 6860.00 1.07990
$$344$$ −832.000 −0.130402
$$345$$ 0 0
$$346$$ 2292.00 0.356123
$$347$$ −10320.0 −1.59656 −0.798280 0.602286i $$-0.794257\pi$$
−0.798280 + 0.602286i $$0.794257\pi$$
$$348$$ 0 0
$$349$$ −2158.00 −0.330989 −0.165494 0.986211i $$-0.552922\pi$$
−0.165494 + 0.986211i $$0.552922\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 192.000 0.0290728
$$353$$ 330.000 0.0497567 0.0248784 0.999690i $$-0.492080\pi$$
0.0248784 + 0.999690i $$0.492080\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −3888.00 −0.578830
$$357$$ 0 0
$$358$$ −2292.00 −0.338369
$$359$$ −8664.00 −1.27373 −0.636864 0.770976i $$-0.719769\pi$$
−0.636864 + 0.770976i $$0.719769\pi$$
$$360$$ 0 0
$$361$$ −4923.00 −0.717743
$$362$$ −236.000 −0.0342649
$$363$$ 0 0
$$364$$ 3808.00 0.548334
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −3782.00 −0.537926 −0.268963 0.963151i $$-0.586681\pi$$
−0.268963 + 0.963151i $$0.586681\pi$$
$$368$$ −1920.00 −0.271975
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −3612.00 −0.505460
$$372$$ 0 0
$$373$$ −11276.0 −1.56528 −0.782640 0.622475i $$-0.786127\pi$$
−0.782640 + 0.622475i $$0.786127\pi$$
$$374$$ −936.000 −0.129410
$$375$$ 0 0
$$376$$ −4800.00 −0.658354
$$377$$ −8568.00 −1.17049
$$378$$ 0 0
$$379$$ 980.000 0.132821 0.0664106 0.997792i $$-0.478845\pi$$
0.0664106 + 0.997792i $$0.478845\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 3384.00 0.453247
$$383$$ 4200.00 0.560339 0.280170 0.959950i $$-0.409609\pi$$
0.280170 + 0.959950i $$0.409609\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −6700.00 −0.883474
$$387$$ 0 0
$$388$$ 184.000 0.0240752
$$389$$ 13338.0 1.73847 0.869233 0.494402i $$-0.164613\pi$$
0.869233 + 0.494402i $$0.164613\pi$$
$$390$$ 0 0
$$391$$ 9360.00 1.21063
$$392$$ −1176.00 −0.151523
$$393$$ 0 0
$$394$$ 7212.00 0.922171
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 7192.00 0.909209 0.454605 0.890693i $$-0.349781\pi$$
0.454605 + 0.890693i $$0.349781\pi$$
$$398$$ 5392.00 0.679087
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −2316.00 −0.288418 −0.144209 0.989547i $$-0.546064\pi$$
−0.144209 + 0.989547i $$0.546064\pi$$
$$402$$ 0 0
$$403$$ 16592.0 2.05088
$$404$$ −6024.00 −0.741845
$$405$$ 0 0
$$406$$ −3528.00 −0.431260
$$407$$ 1824.00 0.222143
$$408$$ 0 0
$$409$$ −12358.0 −1.49404 −0.747022 0.664800i $$-0.768517\pi$$
−0.747022 + 0.664800i $$0.768517\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 5896.00 0.705037
$$413$$ −7476.00 −0.890726
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2176.00 −0.256460
$$417$$ 0 0
$$418$$ 528.000 0.0617831
$$419$$ 3306.00 0.385462 0.192731 0.981252i $$-0.438265\pi$$
0.192731 + 0.981252i $$0.438265\pi$$
$$420$$ 0 0
$$421$$ −14506.0 −1.67929 −0.839643 0.543139i $$-0.817236\pi$$
−0.839643 + 0.543139i $$0.817236\pi$$
$$422$$ −8.00000 −0.000922829 0
$$423$$ 0 0
$$424$$ 2064.00 0.236407
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −5068.00 −0.574374
$$428$$ −3696.00 −0.417413
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −6480.00 −0.724201 −0.362100 0.932139i $$-0.617940\pi$$
−0.362100 + 0.932139i $$0.617940\pi$$
$$432$$ 0 0
$$433$$ −11894.0 −1.32007 −0.660034 0.751236i $$-0.729458\pi$$
−0.660034 + 0.751236i $$0.729458\pi$$
$$434$$ 6832.00 0.755637
$$435$$ 0 0
$$436$$ 2792.00 0.306680
$$437$$ −5280.00 −0.577979
$$438$$ 0 0
$$439$$ −12688.0 −1.37942 −0.689710 0.724086i $$-0.742262\pi$$
−0.689710 + 0.724086i $$0.742262\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 10608.0 1.14156
$$443$$ −4968.00 −0.532814 −0.266407 0.963861i $$-0.585837\pi$$
−0.266407 + 0.963861i $$0.585837\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 2324.00 0.246737
$$447$$ 0 0
$$448$$ −896.000 −0.0944911
$$449$$ −11508.0 −1.20957 −0.604784 0.796389i $$-0.706741\pi$$
−0.604784 + 0.796389i $$0.706741\pi$$
$$450$$ 0 0
$$451$$ −2880.00 −0.300696
$$452$$ 888.000 0.0924071
$$453$$ 0 0
$$454$$ 4800.00 0.496201
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1082.00 −0.110752 −0.0553762 0.998466i $$-0.517636\pi$$
−0.0553762 + 0.998466i $$0.517636\pi$$
$$458$$ −4628.00 −0.472166
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −11238.0 −1.13537 −0.567685 0.823246i $$-0.692161\pi$$
−0.567685 + 0.823246i $$0.692161\pi$$
$$462$$ 0 0
$$463$$ 2302.00 0.231065 0.115532 0.993304i $$-0.463143\pi$$
0.115532 + 0.993304i $$0.463143\pi$$
$$464$$ 2016.00 0.201704
$$465$$ 0 0
$$466$$ 36.0000 0.00357869
$$467$$ −15876.0 −1.57313 −0.786567 0.617505i $$-0.788144\pi$$
−0.786567 + 0.617505i $$0.788144\pi$$
$$468$$ 0 0
$$469$$ −3752.00 −0.369406
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 4272.00 0.416599
$$473$$ −624.000 −0.0606587
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 4368.00 0.420603
$$477$$ 0 0
$$478$$ −11736.0 −1.12300
$$479$$ 4644.00 0.442985 0.221492 0.975162i $$-0.428907\pi$$
0.221492 + 0.975162i $$0.428907\pi$$
$$480$$ 0 0
$$481$$ −20672.0 −1.95959
$$482$$ −8660.00 −0.818366
$$483$$ 0 0
$$484$$ −5180.00 −0.486476
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −2426.00 −0.225734 −0.112867 0.993610i $$-0.536003\pi$$
−0.112867 + 0.993610i $$0.536003\pi$$
$$488$$ 2896.00 0.268639
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 234.000 0.0215077 0.0107538 0.999942i $$-0.496577\pi$$
0.0107538 + 0.999942i $$0.496577\pi$$
$$492$$ 0 0
$$493$$ −9828.00 −0.897831
$$494$$ −5984.00 −0.545006
$$495$$ 0 0
$$496$$ −3904.00 −0.353417
$$497$$ 13608.0 1.22817
$$498$$ 0 0
$$499$$ 14204.0 1.27427 0.637133 0.770754i $$-0.280120\pi$$
0.637133 + 0.770754i $$0.280120\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 996.000 0.0885531
$$503$$ −4920.00 −0.436127 −0.218064 0.975935i $$-0.569974\pi$$
−0.218064 + 0.975935i $$0.569974\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −1440.00 −0.126513
$$507$$ 0 0
$$508$$ 7624.00 0.665867
$$509$$ −4458.00 −0.388207 −0.194104 0.980981i $$-0.562180\pi$$
−0.194104 + 0.980981i $$0.562180\pi$$
$$510$$ 0 0
$$511$$ 6580.00 0.569632
$$512$$ 512.000 0.0441942
$$513$$ 0 0
$$514$$ −1284.00 −0.110184
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −3600.00 −0.306243
$$518$$ −8512.00 −0.722000
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 4212.00 0.354186 0.177093 0.984194i $$-0.443331\pi$$
0.177093 + 0.984194i $$0.443331\pi$$
$$522$$ 0 0
$$523$$ 11212.0 0.937412 0.468706 0.883354i $$-0.344720\pi$$
0.468706 + 0.883354i $$0.344720\pi$$
$$524$$ 11496.0 0.958407
$$525$$ 0 0
$$526$$ 15936.0 1.32099
$$527$$ 19032.0 1.57314
$$528$$ 0 0
$$529$$ 2233.00 0.183529
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −2464.00 −0.200804
$$533$$ 32640.0 2.65252
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 2144.00 0.172774
$$537$$ 0 0
$$538$$ −8436.00 −0.676026
$$539$$ −882.000 −0.0704832
$$540$$ 0 0
$$541$$ 14018.0 1.11401 0.557006 0.830508i $$-0.311950\pi$$
0.557006 + 0.830508i $$0.311950\pi$$
$$542$$ 1696.00 0.134409
$$543$$ 0 0
$$544$$ −2496.00 −0.196719
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −18200.0 −1.42262 −0.711312 0.702876i $$-0.751899\pi$$
−0.711312 + 0.702876i $$0.751899\pi$$
$$548$$ 3192.00 0.248824
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 5544.00 0.428643
$$552$$ 0 0
$$553$$ −17416.0 −1.33925
$$554$$ 3008.00 0.230682
$$555$$ 0 0
$$556$$ −2800.00 −0.213573
$$557$$ 11826.0 0.899612 0.449806 0.893126i $$-0.351493\pi$$
0.449806 + 0.893126i $$0.351493\pi$$
$$558$$ 0 0
$$559$$ 7072.00 0.535087
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −2616.00 −0.196351
$$563$$ 2952.00 0.220980 0.110490 0.993877i $$-0.464758\pi$$
0.110490 + 0.993877i $$0.464758\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 11864.0 0.881062
$$567$$ 0 0
$$568$$ −7776.00 −0.574426
$$569$$ 3084.00 0.227220 0.113610 0.993525i $$-0.463759\pi$$
0.113610 + 0.993525i $$0.463759\pi$$
$$570$$ 0 0
$$571$$ −4756.00 −0.348568 −0.174284 0.984695i $$-0.555761\pi$$
−0.174284 + 0.984695i $$0.555761\pi$$
$$572$$ −1632.00 −0.119296
$$573$$ 0 0
$$574$$ 13440.0 0.977308
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 11014.0 0.794660 0.397330 0.917676i $$-0.369937\pi$$
0.397330 + 0.917676i $$0.369937\pi$$
$$578$$ 2342.00 0.168537
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 5544.00 0.395876
$$582$$ 0 0
$$583$$ 1548.00 0.109968
$$584$$ −3760.00 −0.266421
$$585$$ 0 0
$$586$$ −10452.0 −0.736806
$$587$$ 852.000 0.0599077 0.0299538 0.999551i $$-0.490464\pi$$
0.0299538 + 0.999551i $$0.490464\pi$$
$$588$$ 0 0
$$589$$ −10736.0 −0.751051
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 4864.00 0.337684
$$593$$ −15546.0 −1.07656 −0.538278 0.842767i $$-0.680925\pi$$
−0.538278 + 0.842767i $$0.680925\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 456.000 0.0313397
$$597$$ 0 0
$$598$$ 16320.0 1.11601
$$599$$ −8616.00 −0.587713 −0.293857 0.955850i $$-0.594939\pi$$
−0.293857 + 0.955850i $$0.594939\pi$$
$$600$$ 0 0
$$601$$ 17510.0 1.18843 0.594216 0.804305i $$-0.297462\pi$$
0.594216 + 0.804305i $$0.297462\pi$$
$$602$$ 2912.00 0.197150
$$603$$ 0 0
$$604$$ 4256.00 0.286712
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 13894.0 0.929061 0.464531 0.885557i $$-0.346223\pi$$
0.464531 + 0.885557i $$0.346223\pi$$
$$608$$ 1408.00 0.0939177
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 40800.0 2.70146
$$612$$ 0 0
$$613$$ 6496.00 0.428011 0.214006 0.976832i $$-0.431349\pi$$
0.214006 + 0.976832i $$0.431349\pi$$
$$614$$ −8896.00 −0.584712
$$615$$ 0 0
$$616$$ −672.000 −0.0439540
$$617$$ 570.000 0.0371918 0.0185959 0.999827i $$-0.494080\pi$$
0.0185959 + 0.999827i $$0.494080\pi$$
$$618$$ 0 0
$$619$$ −2140.00 −0.138956 −0.0694781 0.997583i $$-0.522133\pi$$
−0.0694781 + 0.997583i $$0.522133\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −18264.0 −1.17736
$$623$$ 13608.0 0.875109
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 4340.00 0.277095
$$627$$ 0 0
$$628$$ 7792.00 0.495119
$$629$$ −23712.0 −1.50312
$$630$$ 0 0
$$631$$ 14660.0 0.924890 0.462445 0.886648i $$-0.346972\pi$$
0.462445 + 0.886648i $$0.346972\pi$$
$$632$$ 9952.00 0.626375
$$633$$ 0 0
$$634$$ −15348.0 −0.961431
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 9996.00 0.621752
$$638$$ 1512.00 0.0938255
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −456.000 −0.0280982 −0.0140491 0.999901i $$-0.504472\pi$$
−0.0140491 + 0.999901i $$0.504472\pi$$
$$642$$ 0 0
$$643$$ 23452.0 1.43835 0.719173 0.694831i $$-0.244521\pi$$
0.719173 + 0.694831i $$0.244521\pi$$
$$644$$ 6720.00 0.411188
$$645$$ 0 0
$$646$$ −6864.00 −0.418050
$$647$$ −7224.00 −0.438956 −0.219478 0.975617i $$-0.570435\pi$$
−0.219478 + 0.975617i $$0.570435\pi$$
$$648$$ 0 0
$$649$$ 3204.00 0.193787
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −8240.00 −0.494944
$$653$$ −19146.0 −1.14738 −0.573691 0.819072i $$-0.694489\pi$$
−0.573691 + 0.819072i $$0.694489\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −7680.00 −0.457094
$$657$$ 0 0
$$658$$ 16800.0 0.995338
$$659$$ 27810.0 1.64389 0.821945 0.569567i $$-0.192889\pi$$
0.821945 + 0.569567i $$0.192889\pi$$
$$660$$ 0 0
$$661$$ −30598.0 −1.80049 −0.900245 0.435383i $$-0.856613\pi$$
−0.900245 + 0.435383i $$0.856613\pi$$
$$662$$ 19192.0 1.12676
$$663$$ 0 0
$$664$$ −3168.00 −0.185154
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −15120.0 −0.877734
$$668$$ 4992.00 0.289141
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 2172.00 0.124961
$$672$$ 0 0
$$673$$ 3778.00 0.216391 0.108196 0.994130i $$-0.465493\pi$$
0.108196 + 0.994130i $$0.465493\pi$$
$$674$$ −24316.0 −1.38964
$$675$$ 0 0
$$676$$ 9708.00 0.552344
$$677$$ 27198.0 1.54402 0.772012 0.635608i $$-0.219251\pi$$
0.772012 + 0.635608i $$0.219251\pi$$
$$678$$ 0 0
$$679$$ −644.000 −0.0363983
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −2928.00 −0.164397
$$683$$ −32316.0 −1.81045 −0.905225 0.424933i $$-0.860298\pi$$
−0.905225 + 0.424933i $$0.860298\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 13720.0 0.763604
$$687$$ 0 0
$$688$$ −1664.00 −0.0922084
$$689$$ −17544.0 −0.970063
$$690$$ 0 0
$$691$$ 29324.0 1.61438 0.807191 0.590291i $$-0.200987\pi$$
0.807191 + 0.590291i $$0.200987\pi$$
$$692$$ 4584.00 0.251817
$$693$$ 0 0
$$694$$ −20640.0 −1.12894
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 37440.0 2.03464
$$698$$ −4316.00 −0.234044
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 22782.0 1.22748 0.613741 0.789508i $$-0.289664\pi$$
0.613741 + 0.789508i $$0.289664\pi$$
$$702$$ 0 0
$$703$$ 13376.0 0.717618
$$704$$ 384.000 0.0205576
$$705$$ 0 0
$$706$$ 660.000 0.0351833
$$707$$ 21084.0 1.12156
$$708$$ 0 0
$$709$$ 26054.0 1.38008 0.690041 0.723770i $$-0.257592\pi$$
0.690041 + 0.723770i $$0.257592\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −7776.00 −0.409295
$$713$$ 29280.0 1.53793
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −4584.00 −0.239263
$$717$$ 0 0
$$718$$ −17328.0 −0.900662
$$719$$ −5976.00 −0.309968 −0.154984 0.987917i $$-0.549533\pi$$
−0.154984 + 0.987917i $$0.549533\pi$$
$$720$$ 0 0
$$721$$ −20636.0 −1.06592
$$722$$ −9846.00 −0.507521
$$723$$ 0 0
$$724$$ −472.000 −0.0242289
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 5110.00 0.260687 0.130343 0.991469i $$-0.458392\pi$$
0.130343 + 0.991469i $$0.458392\pi$$
$$728$$ 7616.00 0.387730
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 8112.00 0.410442
$$732$$ 0 0
$$733$$ −17336.0 −0.873560 −0.436780 0.899568i $$-0.643881\pi$$
−0.436780 + 0.899568i $$0.643881\pi$$
$$734$$ −7564.00 −0.380371
$$735$$ 0 0
$$736$$ −3840.00 −0.192316
$$737$$ 1608.00 0.0803683
$$738$$ 0 0
$$739$$ −13660.0 −0.679961 −0.339981 0.940432i $$-0.610420\pi$$
−0.339981 + 0.940432i $$0.610420\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −7224.00 −0.357414
$$743$$ −1320.00 −0.0651765 −0.0325882 0.999469i $$-0.510375\pi$$
−0.0325882 + 0.999469i $$0.510375\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −22552.0 −1.10682
$$747$$ 0 0
$$748$$ −1872.00 −0.0915068
$$749$$ 12936.0 0.631070
$$750$$ 0 0
$$751$$ 15860.0 0.770625 0.385313 0.922786i $$-0.374094\pi$$
0.385313 + 0.922786i $$0.374094\pi$$
$$752$$ −9600.00 −0.465527
$$753$$ 0 0
$$754$$ −17136.0 −0.827661
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −22160.0 −1.06396 −0.531981 0.846756i $$-0.678552\pi$$
−0.531981 + 0.846756i $$0.678552\pi$$
$$758$$ 1960.00 0.0939187
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 13116.0 0.624776 0.312388 0.949955i $$-0.398871\pi$$
0.312388 + 0.949955i $$0.398871\pi$$
$$762$$ 0 0
$$763$$ −9772.00 −0.463657
$$764$$ 6768.00 0.320494
$$765$$ 0 0
$$766$$ 8400.00 0.396220
$$767$$ −36312.0 −1.70945
$$768$$ 0 0
$$769$$ 32846.0 1.54026 0.770128 0.637889i $$-0.220192\pi$$
0.770128 + 0.637889i $$0.220192\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −13400.0 −0.624711
$$773$$ 11982.0 0.557520 0.278760 0.960361i $$-0.410077\pi$$
0.278760 + 0.960361i $$0.410077\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 368.000 0.0170238
$$777$$ 0 0
$$778$$ 26676.0 1.22928
$$779$$ −21120.0 −0.971377
$$780$$ 0 0
$$781$$ −5832.00 −0.267203
$$782$$ 18720.0 0.856043
$$783$$ 0 0
$$784$$ −2352.00 −0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 21076.0 0.954610 0.477305 0.878738i $$-0.341614\pi$$
0.477305 + 0.878738i $$0.341614\pi$$
$$788$$ 14424.0 0.652073
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −3108.00 −0.139706
$$792$$ 0 0
$$793$$ −24616.0 −1.10232
$$794$$ 14384.0 0.642908
$$795$$ 0 0
$$796$$ 10784.0 0.480187
$$797$$ −22086.0 −0.981589 −0.490794 0.871275i $$-0.663293\pi$$
−0.490794 + 0.871275i $$0.663293\pi$$
$$798$$ 0 0
$$799$$ 46800.0 2.07217
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −4632.00 −0.203942
$$803$$ −2820.00 −0.123930
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 33184.0 1.45019
$$807$$ 0 0
$$808$$ −12048.0 −0.524563
$$809$$ 21384.0 0.929322 0.464661 0.885489i $$-0.346176\pi$$
0.464661 + 0.885489i $$0.346176\pi$$
$$810$$ 0 0
$$811$$ 5228.00 0.226362 0.113181 0.993574i $$-0.463896\pi$$
0.113181 + 0.993574i $$0.463896\pi$$
$$812$$ −7056.00 −0.304947
$$813$$ 0 0
$$814$$ 3648.00 0.157079
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −4576.00 −0.195953
$$818$$ −24716.0 −1.05645
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −38010.0 −1.61578 −0.807892 0.589331i $$-0.799391\pi$$
−0.807892 + 0.589331i $$0.799391\pi$$
$$822$$ 0 0
$$823$$ −38642.0 −1.63667 −0.818333 0.574745i $$-0.805101\pi$$
−0.818333 + 0.574745i $$0.805101\pi$$
$$824$$ 11792.0 0.498536
$$825$$ 0 0
$$826$$ −14952.0 −0.629839
$$827$$ 15432.0 0.648879 0.324440 0.945906i $$-0.394824\pi$$
0.324440 + 0.945906i $$0.394824\pi$$
$$828$$ 0 0
$$829$$ −3886.00 −0.162806 −0.0814031 0.996681i $$-0.525940\pi$$
−0.0814031 + 0.996681i $$0.525940\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −4352.00 −0.181344
$$833$$ 11466.0 0.476919
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 1056.00 0.0436872
$$837$$ 0 0
$$838$$ 6612.00 0.272563
$$839$$ 27552.0 1.13373 0.566866 0.823810i $$-0.308156\pi$$
0.566866 + 0.823810i $$0.308156\pi$$
$$840$$ 0 0
$$841$$ −8513.00 −0.349051
$$842$$ −29012.0 −1.18743
$$843$$ 0 0
$$844$$ −16.0000 −0.000652539 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 18130.0 0.735483
$$848$$ 4128.00 0.167165
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −36480.0 −1.46947
$$852$$ 0 0
$$853$$ −15104.0 −0.606273 −0.303137 0.952947i $$-0.598034\pi$$
−0.303137 + 0.952947i $$0.598034\pi$$
$$854$$ −10136.0 −0.406144
$$855$$ 0 0
$$856$$ −7392.00 −0.295156
$$857$$ 12306.0 0.490508 0.245254 0.969459i $$-0.421129\pi$$
0.245254 + 0.969459i $$0.421129\pi$$
$$858$$ 0 0
$$859$$ −47500.0 −1.88670 −0.943352 0.331793i $$-0.892346\pi$$
−0.943352 + 0.331793i $$0.892346\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −12960.0 −0.512087
$$863$$ −4272.00 −0.168506 −0.0842529 0.996444i $$-0.526850\pi$$
−0.0842529 + 0.996444i $$0.526850\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −23788.0 −0.933429
$$867$$ 0 0
$$868$$ 13664.0 0.534316
$$869$$ 7464.00 0.291368
$$870$$ 0 0
$$871$$ −18224.0 −0.708951
$$872$$ 5584.00 0.216856
$$873$$ 0 0
$$874$$ −10560.0 −0.408693
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 27796.0 1.07024 0.535122 0.844775i $$-0.320266\pi$$
0.535122 + 0.844775i $$0.320266\pi$$
$$878$$ −25376.0 −0.975397
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −39996.0 −1.52951 −0.764756 0.644320i $$-0.777140\pi$$
−0.764756 + 0.644320i $$0.777140\pi$$
$$882$$ 0 0
$$883$$ 3772.00 0.143758 0.0718788 0.997413i $$-0.477101\pi$$
0.0718788 + 0.997413i $$0.477101\pi$$
$$884$$ 21216.0 0.807207
$$885$$ 0 0
$$886$$ −9936.00 −0.376757
$$887$$ 5784.00 0.218949 0.109474 0.993990i $$-0.465083\pi$$
0.109474 + 0.993990i $$0.465083\pi$$
$$888$$ 0 0
$$889$$ −26684.0 −1.00670
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 4648.00 0.174469
$$893$$ −26400.0 −0.989297
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −1792.00 −0.0668153
$$897$$ 0 0
$$898$$ −23016.0 −0.855294
$$899$$ −30744.0 −1.14057
$$900$$ 0 0
$$901$$ −20124.0 −0.744093
$$902$$ −5760.00 −0.212624
$$903$$ 0 0
$$904$$ 1776.00 0.0653417
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 8440.00 0.308981 0.154490 0.987994i $$-0.450626\pi$$
0.154490 + 0.987994i $$0.450626\pi$$
$$908$$ 9600.00 0.350867
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −31920.0 −1.16087 −0.580437 0.814305i $$-0.697118\pi$$
−0.580437 + 0.814305i $$0.697118\pi$$
$$912$$ 0 0
$$913$$ −2376.00 −0.0861272
$$914$$ −2164.00 −0.0783137
$$915$$ 0 0
$$916$$ −9256.00 −0.333872
$$917$$ −40236.0 −1.44897
$$918$$ 0 0
$$919$$ 34652.0 1.24381 0.621906 0.783092i $$-0.286358\pi$$
0.621906 + 0.783092i $$0.286358\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −22476.0 −0.802828
$$923$$ 66096.0 2.35707
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 4604.00 0.163388
$$927$$ 0 0
$$928$$ 4032.00 0.142626
$$929$$ 1404.00 0.0495842 0.0247921 0.999693i $$-0.492108\pi$$
0.0247921 + 0.999693i $$0.492108\pi$$
$$930$$ 0 0
$$931$$ −6468.00 −0.227691
$$932$$ 72.0000 0.00253051
$$933$$ 0 0
$$934$$ −31752.0 −1.11237
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 7654.00 0.266857 0.133429 0.991058i $$-0.457401\pi$$
0.133429 + 0.991058i $$0.457401\pi$$
$$938$$ −7504.00 −0.261209
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 11298.0 0.391397 0.195698 0.980664i $$-0.437303\pi$$
0.195698 + 0.980664i $$0.437303\pi$$
$$942$$ 0 0
$$943$$ 57600.0 1.98909
$$944$$ 8544.00 0.294580
$$945$$ 0 0
$$946$$ −1248.00 −0.0428922
$$947$$ −28968.0 −0.994016 −0.497008 0.867746i $$-0.665568\pi$$
−0.497008 + 0.867746i $$0.665568\pi$$
$$948$$ 0 0
$$949$$ 31960.0 1.09322
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 8736.00 0.297411
$$953$$ −46410.0 −1.57751 −0.788755 0.614707i $$-0.789274\pi$$
−0.788755 + 0.614707i $$0.789274\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −23472.0 −0.794078
$$957$$ 0 0
$$958$$ 9288.00 0.313238
$$959$$ −11172.0 −0.376186
$$960$$ 0 0
$$961$$ 29745.0 0.998456
$$962$$ −41344.0 −1.38564
$$963$$ 0 0
$$964$$ −17320.0 −0.578672
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 41506.0 1.38029 0.690146 0.723670i $$-0.257546\pi$$
0.690146 + 0.723670i $$0.257546\pi$$
$$968$$ −10360.0 −0.343991
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −18246.0 −0.603030 −0.301515 0.953461i $$-0.597492\pi$$
−0.301515 + 0.953461i $$0.597492\pi$$
$$972$$ 0 0
$$973$$ 9800.00 0.322892
$$974$$ −4852.00 −0.159618
$$975$$ 0 0
$$976$$ 5792.00 0.189956
$$977$$ −25998.0 −0.851330 −0.425665 0.904881i $$-0.639960\pi$$
−0.425665 + 0.904881i $$0.639960\pi$$
$$978$$ 0 0
$$979$$ −5832.00 −0.190390
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 468.000 0.0152082
$$983$$ −14616.0 −0.474240 −0.237120 0.971480i $$-0.576203\pi$$
−0.237120 + 0.971480i $$0.576203\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −19656.0 −0.634863
$$987$$ 0 0
$$988$$ −11968.0 −0.385377
$$989$$ 12480.0 0.401255
$$990$$ 0 0
$$991$$ −2968.00 −0.0951379 −0.0475689 0.998868i $$-0.515147\pi$$
−0.0475689 + 0.998868i $$0.515147\pi$$
$$992$$ −7808.00 −0.249903
$$993$$ 0 0
$$994$$ 27216.0 0.868450
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 9052.00 0.287542 0.143771 0.989611i $$-0.454077\pi$$
0.143771 + 0.989611i $$0.454077\pi$$
$$998$$ 28408.0 0.901042
$$999$$ 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 450.4.a.m.1.1 1
3.2 odd 2 450.4.a.c.1.1 1
5.2 odd 4 450.4.c.g.199.2 2
5.3 odd 4 450.4.c.g.199.1 2
5.4 even 2 90.4.a.b.1.1 1
15.2 even 4 450.4.c.f.199.1 2
15.8 even 4 450.4.c.f.199.2 2
15.14 odd 2 90.4.a.e.1.1 yes 1
20.19 odd 2 720.4.a.e.1.1 1
45.4 even 6 810.4.e.u.541.1 2
45.14 odd 6 810.4.e.a.541.1 2
45.29 odd 6 810.4.e.a.271.1 2
45.34 even 6 810.4.e.u.271.1 2
60.59 even 2 720.4.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
90.4.a.b.1.1 1 5.4 even 2
90.4.a.e.1.1 yes 1 15.14 odd 2
450.4.a.c.1.1 1 3.2 odd 2
450.4.a.m.1.1 1 1.1 even 1 trivial
450.4.c.f.199.1 2 15.2 even 4
450.4.c.f.199.2 2 15.8 even 4
450.4.c.g.199.1 2 5.3 odd 4
450.4.c.g.199.2 2 5.2 odd 4
720.4.a.e.1.1 1 20.19 odd 2
720.4.a.t.1.1 1 60.59 even 2
810.4.e.a.271.1 2 45.29 odd 6
810.4.e.a.541.1 2 45.14 odd 6
810.4.e.u.271.1 2 45.34 even 6
810.4.e.u.541.1 2 45.4 even 6