# Properties

 Label 225.4.a.g.1.1 Level $225$ Weight $4$ Character 225.1 Self dual yes Analytic conductor $13.275$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.2754297513$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 225.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{2} +1.00000 q^{4} -20.0000 q^{7} -21.0000 q^{8} +O(q^{10})$$ $$q+3.00000 q^{2} +1.00000 q^{4} -20.0000 q^{7} -21.0000 q^{8} +24.0000 q^{11} -74.0000 q^{13} -60.0000 q^{14} -71.0000 q^{16} +54.0000 q^{17} -124.000 q^{19} +72.0000 q^{22} -120.000 q^{23} -222.000 q^{26} -20.0000 q^{28} +78.0000 q^{29} +200.000 q^{31} -45.0000 q^{32} +162.000 q^{34} +70.0000 q^{37} -372.000 q^{38} -330.000 q^{41} -92.0000 q^{43} +24.0000 q^{44} -360.000 q^{46} -24.0000 q^{47} +57.0000 q^{49} -74.0000 q^{52} +450.000 q^{53} +420.000 q^{56} +234.000 q^{58} -24.0000 q^{59} -322.000 q^{61} +600.000 q^{62} +433.000 q^{64} +196.000 q^{67} +54.0000 q^{68} +288.000 q^{71} +430.000 q^{73} +210.000 q^{74} -124.000 q^{76} -480.000 q^{77} -520.000 q^{79} -990.000 q^{82} +156.000 q^{83} -276.000 q^{86} -504.000 q^{88} -1026.00 q^{89} +1480.00 q^{91} -120.000 q^{92} -72.0000 q^{94} +286.000 q^{97} +171.000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.00000 1.06066 0.530330 0.847791i $$-0.322068\pi$$
0.530330 + 0.847791i $$0.322068\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.125000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −20.0000 −1.07990 −0.539949 0.841698i $$-0.681557\pi$$
−0.539949 + 0.841698i $$0.681557\pi$$
$$8$$ −21.0000 −0.928078
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 24.0000 0.657843 0.328921 0.944357i $$-0.393315\pi$$
0.328921 + 0.944357i $$0.393315\pi$$
$$12$$ 0 0
$$13$$ −74.0000 −1.57876 −0.789381 0.613904i $$-0.789598\pi$$
−0.789381 + 0.613904i $$0.789598\pi$$
$$14$$ −60.0000 −1.14541
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ 54.0000 0.770407 0.385204 0.922832i $$-0.374131\pi$$
0.385204 + 0.922832i $$0.374131\pi$$
$$18$$ 0 0
$$19$$ −124.000 −1.49724 −0.748620 0.663000i $$-0.769283\pi$$
−0.748620 + 0.663000i $$0.769283\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 72.0000 0.697748
$$23$$ −120.000 −1.08790 −0.543951 0.839117i $$-0.683072\pi$$
−0.543951 + 0.839117i $$0.683072\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −222.000 −1.67453
$$27$$ 0 0
$$28$$ −20.0000 −0.134987
$$29$$ 78.0000 0.499456 0.249728 0.968316i $$-0.419659\pi$$
0.249728 + 0.968316i $$0.419659\pi$$
$$30$$ 0 0
$$31$$ 200.000 1.15874 0.579372 0.815063i $$-0.303298\pi$$
0.579372 + 0.815063i $$0.303298\pi$$
$$32$$ −45.0000 −0.248592
$$33$$ 0 0
$$34$$ 162.000 0.817140
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 70.0000 0.311025 0.155513 0.987834i $$-0.450297\pi$$
0.155513 + 0.987834i $$0.450297\pi$$
$$38$$ −372.000 −1.58806
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −330.000 −1.25701 −0.628504 0.777806i $$-0.716332\pi$$
−0.628504 + 0.777806i $$0.716332\pi$$
$$42$$ 0 0
$$43$$ −92.0000 −0.326276 −0.163138 0.986603i $$-0.552162\pi$$
−0.163138 + 0.986603i $$0.552162\pi$$
$$44$$ 24.0000 0.0822304
$$45$$ 0 0
$$46$$ −360.000 −1.15389
$$47$$ −24.0000 −0.0744843 −0.0372421 0.999306i $$-0.511857\pi$$
−0.0372421 + 0.999306i $$0.511857\pi$$
$$48$$ 0 0
$$49$$ 57.0000 0.166181
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −74.0000 −0.197345
$$53$$ 450.000 1.16627 0.583134 0.812376i $$-0.301826\pi$$
0.583134 + 0.812376i $$0.301826\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 420.000 1.00223
$$57$$ 0 0
$$58$$ 234.000 0.529754
$$59$$ −24.0000 −0.0529582 −0.0264791 0.999649i $$-0.508430\pi$$
−0.0264791 + 0.999649i $$0.508430\pi$$
$$60$$ 0 0
$$61$$ −322.000 −0.675867 −0.337933 0.941170i $$-0.609728\pi$$
−0.337933 + 0.941170i $$0.609728\pi$$
$$62$$ 600.000 1.22903
$$63$$ 0 0
$$64$$ 433.000 0.845703
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 196.000 0.357391 0.178696 0.983904i $$-0.442812\pi$$
0.178696 + 0.983904i $$0.442812\pi$$
$$68$$ 54.0000 0.0963009
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 288.000 0.481399 0.240699 0.970600i $$-0.422623\pi$$
0.240699 + 0.970600i $$0.422623\pi$$
$$72$$ 0 0
$$73$$ 430.000 0.689420 0.344710 0.938709i $$-0.387977\pi$$
0.344710 + 0.938709i $$0.387977\pi$$
$$74$$ 210.000 0.329892
$$75$$ 0 0
$$76$$ −124.000 −0.187155
$$77$$ −480.000 −0.710404
$$78$$ 0 0
$$79$$ −520.000 −0.740564 −0.370282 0.928919i $$-0.620739\pi$$
−0.370282 + 0.928919i $$0.620739\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −990.000 −1.33326
$$83$$ 156.000 0.206304 0.103152 0.994666i $$-0.467107\pi$$
0.103152 + 0.994666i $$0.467107\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −276.000 −0.346068
$$87$$ 0 0
$$88$$ −504.000 −0.610529
$$89$$ −1026.00 −1.22198 −0.610988 0.791640i $$-0.709227\pi$$
−0.610988 + 0.791640i $$0.709227\pi$$
$$90$$ 0 0
$$91$$ 1480.00 1.70490
$$92$$ −120.000 −0.135988
$$93$$ 0 0
$$94$$ −72.0000 −0.0790025
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 286.000 0.299370 0.149685 0.988734i $$-0.452174\pi$$
0.149685 + 0.988734i $$0.452174\pi$$
$$98$$ 171.000 0.176261
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1734.00 1.70831 0.854156 0.520017i $$-0.174075\pi$$
0.854156 + 0.520017i $$0.174075\pi$$
$$102$$ 0 0
$$103$$ −452.000 −0.432397 −0.216198 0.976349i $$-0.569366\pi$$
−0.216198 + 0.976349i $$0.569366\pi$$
$$104$$ 1554.00 1.46521
$$105$$ 0 0
$$106$$ 1350.00 1.23702
$$107$$ −1404.00 −1.26850 −0.634251 0.773127i $$-0.718692\pi$$
−0.634251 + 0.773127i $$0.718692\pi$$
$$108$$ 0 0
$$109$$ −1474.00 −1.29526 −0.647631 0.761954i $$-0.724240\pi$$
−0.647631 + 0.761954i $$0.724240\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1420.00 1.19801
$$113$$ 1086.00 0.904091 0.452046 0.891995i $$-0.350694\pi$$
0.452046 + 0.891995i $$0.350694\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 78.0000 0.0624321
$$117$$ 0 0
$$118$$ −72.0000 −0.0561707
$$119$$ −1080.00 −0.831962
$$120$$ 0 0
$$121$$ −755.000 −0.567243
$$122$$ −966.000 −0.716865
$$123$$ 0 0
$$124$$ 200.000 0.144843
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1244.00 −0.869190 −0.434595 0.900626i $$-0.643109\pi$$
−0.434595 + 0.900626i $$0.643109\pi$$
$$128$$ 1659.00 1.14560
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2328.00 −1.55266 −0.776329 0.630327i $$-0.782921\pi$$
−0.776329 + 0.630327i $$0.782921\pi$$
$$132$$ 0 0
$$133$$ 2480.00 1.61687
$$134$$ 588.000 0.379071
$$135$$ 0 0
$$136$$ −1134.00 −0.714998
$$137$$ 2118.00 1.32082 0.660412 0.750903i $$-0.270382\pi$$
0.660412 + 0.750903i $$0.270382\pi$$
$$138$$ 0 0
$$139$$ 2324.00 1.41812 0.709062 0.705147i $$-0.249119\pi$$
0.709062 + 0.705147i $$0.249119\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 864.000 0.510600
$$143$$ −1776.00 −1.03858
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 1290.00 0.731241
$$147$$ 0 0
$$148$$ 70.0000 0.0388781
$$149$$ −258.000 −0.141854 −0.0709268 0.997482i $$-0.522596\pi$$
−0.0709268 + 0.997482i $$0.522596\pi$$
$$150$$ 0 0
$$151$$ −808.000 −0.435458 −0.217729 0.976009i $$-0.569865\pi$$
−0.217729 + 0.976009i $$0.569865\pi$$
$$152$$ 2604.00 1.38955
$$153$$ 0 0
$$154$$ −1440.00 −0.753497
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −2378.00 −1.20882 −0.604411 0.796673i $$-0.706592\pi$$
−0.604411 + 0.796673i $$0.706592\pi$$
$$158$$ −1560.00 −0.785487
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 2400.00 1.17482
$$162$$ 0 0
$$163$$ 52.0000 0.0249874 0.0124937 0.999922i $$-0.496023\pi$$
0.0124937 + 0.999922i $$0.496023\pi$$
$$164$$ −330.000 −0.157126
$$165$$ 0 0
$$166$$ 468.000 0.218818
$$167$$ −3720.00 −1.72373 −0.861863 0.507141i $$-0.830702\pi$$
−0.861863 + 0.507141i $$0.830702\pi$$
$$168$$ 0 0
$$169$$ 3279.00 1.49249
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −92.0000 −0.0407845
$$173$$ 426.000 0.187215 0.0936075 0.995609i $$-0.470160\pi$$
0.0936075 + 0.995609i $$0.470160\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1704.00 −0.729795
$$177$$ 0 0
$$178$$ −3078.00 −1.29610
$$179$$ 1440.00 0.601289 0.300644 0.953736i $$-0.402798\pi$$
0.300644 + 0.953736i $$0.402798\pi$$
$$180$$ 0 0
$$181$$ −3130.00 −1.28537 −0.642683 0.766133i $$-0.722179\pi$$
−0.642683 + 0.766133i $$0.722179\pi$$
$$182$$ 4440.00 1.80832
$$183$$ 0 0
$$184$$ 2520.00 1.00966
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1296.00 0.506807
$$188$$ −24.0000 −0.00931053
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3576.00 −1.35471 −0.677357 0.735655i $$-0.736875\pi$$
−0.677357 + 0.735655i $$0.736875\pi$$
$$192$$ 0 0
$$193$$ −2666.00 −0.994315 −0.497158 0.867660i $$-0.665623\pi$$
−0.497158 + 0.867660i $$0.665623\pi$$
$$194$$ 858.000 0.317530
$$195$$ 0 0
$$196$$ 57.0000 0.0207726
$$197$$ −2718.00 −0.982992 −0.491496 0.870880i $$-0.663550\pi$$
−0.491496 + 0.870880i $$0.663550\pi$$
$$198$$ 0 0
$$199$$ −3832.00 −1.36504 −0.682521 0.730866i $$-0.739116\pi$$
−0.682521 + 0.730866i $$0.739116\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 5202.00 1.81194
$$203$$ −1560.00 −0.539362
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −1356.00 −0.458626
$$207$$ 0 0
$$208$$ 5254.00 1.75144
$$209$$ −2976.00 −0.984948
$$210$$ 0 0
$$211$$ 1100.00 0.358896 0.179448 0.983767i $$-0.442569\pi$$
0.179448 + 0.983767i $$0.442569\pi$$
$$212$$ 450.000 0.145784
$$213$$ 0 0
$$214$$ −4212.00 −1.34545
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −4000.00 −1.25133
$$218$$ −4422.00 −1.37383
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −3996.00 −1.21629
$$222$$ 0 0
$$223$$ −1964.00 −0.589772 −0.294886 0.955532i $$-0.595282\pi$$
−0.294886 + 0.955532i $$0.595282\pi$$
$$224$$ 900.000 0.268454
$$225$$ 0 0
$$226$$ 3258.00 0.958933
$$227$$ 660.000 0.192977 0.0964884 0.995334i $$-0.469239\pi$$
0.0964884 + 0.995334i $$0.469239\pi$$
$$228$$ 0 0
$$229$$ −1906.00 −0.550009 −0.275004 0.961443i $$-0.588679\pi$$
−0.275004 + 0.961443i $$0.588679\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −1638.00 −0.463534
$$233$$ −1458.00 −0.409943 −0.204972 0.978768i $$-0.565710\pi$$
−0.204972 + 0.978768i $$0.565710\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −24.0000 −0.00661978
$$237$$ 0 0
$$238$$ −3240.00 −0.882429
$$239$$ −1176.00 −0.318281 −0.159140 0.987256i $$-0.550872\pi$$
−0.159140 + 0.987256i $$0.550872\pi$$
$$240$$ 0 0
$$241$$ 866.000 0.231469 0.115734 0.993280i $$-0.463078\pi$$
0.115734 + 0.993280i $$0.463078\pi$$
$$242$$ −2265.00 −0.601652
$$243$$ 0 0
$$244$$ −322.000 −0.0844834
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 9176.00 2.36379
$$248$$ −4200.00 −1.07540
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −432.000 −0.108636 −0.0543179 0.998524i $$-0.517298\pi$$
−0.0543179 + 0.998524i $$0.517298\pi$$
$$252$$ 0 0
$$253$$ −2880.00 −0.715668
$$254$$ −3732.00 −0.921915
$$255$$ 0 0
$$256$$ 1513.00 0.369385
$$257$$ 2526.00 0.613103 0.306552 0.951854i $$-0.400825\pi$$
0.306552 + 0.951854i $$0.400825\pi$$
$$258$$ 0 0
$$259$$ −1400.00 −0.335876
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −6984.00 −1.64684
$$263$$ 5448.00 1.27733 0.638666 0.769484i $$-0.279487\pi$$
0.638666 + 0.769484i $$0.279487\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 7440.00 1.71495
$$267$$ 0 0
$$268$$ 196.000 0.0446739
$$269$$ 2574.00 0.583418 0.291709 0.956507i $$-0.405776\pi$$
0.291709 + 0.956507i $$0.405776\pi$$
$$270$$ 0 0
$$271$$ −3184.00 −0.713706 −0.356853 0.934161i $$-0.616150\pi$$
−0.356853 + 0.934161i $$0.616150\pi$$
$$272$$ −3834.00 −0.854671
$$273$$ 0 0
$$274$$ 6354.00 1.40095
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −3962.00 −0.859399 −0.429699 0.902972i $$-0.641380\pi$$
−0.429699 + 0.902972i $$0.641380\pi$$
$$278$$ 6972.00 1.50415
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 8286.00 1.75908 0.879540 0.475825i $$-0.157851\pi$$
0.879540 + 0.475825i $$0.157851\pi$$
$$282$$ 0 0
$$283$$ 2716.00 0.570493 0.285246 0.958454i $$-0.407925\pi$$
0.285246 + 0.958454i $$0.407925\pi$$
$$284$$ 288.000 0.0601748
$$285$$ 0 0
$$286$$ −5328.00 −1.10158
$$287$$ 6600.00 1.35744
$$288$$ 0 0
$$289$$ −1997.00 −0.406473
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 430.000 0.0861776
$$293$$ 6018.00 1.19992 0.599958 0.800032i $$-0.295184\pi$$
0.599958 + 0.800032i $$0.295184\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −1470.00 −0.288655
$$297$$ 0 0
$$298$$ −774.000 −0.150458
$$299$$ 8880.00 1.71754
$$300$$ 0 0
$$301$$ 1840.00 0.352345
$$302$$ −2424.00 −0.461873
$$303$$ 0 0
$$304$$ 8804.00 1.66100
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −9236.00 −1.71702 −0.858512 0.512793i $$-0.828611\pi$$
−0.858512 + 0.512793i $$0.828611\pi$$
$$308$$ −480.000 −0.0888004
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −1536.00 −0.280060 −0.140030 0.990147i $$-0.544720\pi$$
−0.140030 + 0.990147i $$0.544720\pi$$
$$312$$ 0 0
$$313$$ 7342.00 1.32586 0.662930 0.748681i $$-0.269313\pi$$
0.662930 + 0.748681i $$0.269313\pi$$
$$314$$ −7134.00 −1.28215
$$315$$ 0 0
$$316$$ −520.000 −0.0925705
$$317$$ −3894.00 −0.689933 −0.344967 0.938615i $$-0.612110\pi$$
−0.344967 + 0.938615i $$0.612110\pi$$
$$318$$ 0 0
$$319$$ 1872.00 0.328564
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 7200.00 1.24609
$$323$$ −6696.00 −1.15348
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 156.000 0.0265032
$$327$$ 0 0
$$328$$ 6930.00 1.16660
$$329$$ 480.000 0.0804354
$$330$$ 0 0
$$331$$ 3692.00 0.613084 0.306542 0.951857i $$-0.400828\pi$$
0.306542 + 0.951857i $$0.400828\pi$$
$$332$$ 156.000 0.0257880
$$333$$ 0 0
$$334$$ −11160.0 −1.82829
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 8998.00 1.45446 0.727229 0.686395i $$-0.240808\pi$$
0.727229 + 0.686395i $$0.240808\pi$$
$$338$$ 9837.00 1.58302
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 4800.00 0.762271
$$342$$ 0 0
$$343$$ 5720.00 0.900440
$$344$$ 1932.00 0.302809
$$345$$ 0 0
$$346$$ 1278.00 0.198571
$$347$$ 5244.00 0.811276 0.405638 0.914034i $$-0.367049\pi$$
0.405638 + 0.914034i $$0.367049\pi$$
$$348$$ 0 0
$$349$$ 6302.00 0.966585 0.483293 0.875459i $$-0.339441\pi$$
0.483293 + 0.875459i $$0.339441\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1080.00 −0.163535
$$353$$ 3414.00 0.514756 0.257378 0.966311i $$-0.417141\pi$$
0.257378 + 0.966311i $$0.417141\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −1026.00 −0.152747
$$357$$ 0 0
$$358$$ 4320.00 0.637763
$$359$$ −4824.00 −0.709195 −0.354597 0.935019i $$-0.615382\pi$$
−0.354597 + 0.935019i $$0.615382\pi$$
$$360$$ 0 0
$$361$$ 8517.00 1.24173
$$362$$ −9390.00 −1.36334
$$363$$ 0 0
$$364$$ 1480.00 0.213113
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 3508.00 0.498954 0.249477 0.968381i $$-0.419741\pi$$
0.249477 + 0.968381i $$0.419741\pi$$
$$368$$ 8520.00 1.20689
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −9000.00 −1.25945
$$372$$ 0 0
$$373$$ −10802.0 −1.49948 −0.749740 0.661732i $$-0.769822\pi$$
−0.749740 + 0.661732i $$0.769822\pi$$
$$374$$ 3888.00 0.537550
$$375$$ 0 0
$$376$$ 504.000 0.0691272
$$377$$ −5772.00 −0.788523
$$378$$ 0 0
$$379$$ 1460.00 0.197876 0.0989382 0.995094i $$-0.468455\pi$$
0.0989382 + 0.995094i $$0.468455\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −10728.0 −1.43689
$$383$$ −4872.00 −0.649994 −0.324997 0.945715i $$-0.605363\pi$$
−0.324997 + 0.945715i $$0.605363\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −7998.00 −1.05463
$$387$$ 0 0
$$388$$ 286.000 0.0374213
$$389$$ 14046.0 1.83075 0.915373 0.402606i $$-0.131896\pi$$
0.915373 + 0.402606i $$0.131896\pi$$
$$390$$ 0 0
$$391$$ −6480.00 −0.838127
$$392$$ −1197.00 −0.154229
$$393$$ 0 0
$$394$$ −8154.00 −1.04262
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 2734.00 0.345631 0.172816 0.984954i $$-0.444714\pi$$
0.172816 + 0.984954i $$0.444714\pi$$
$$398$$ −11496.0 −1.44785
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 15942.0 1.98530 0.992650 0.121019i $$-0.0386161\pi$$
0.992650 + 0.121019i $$0.0386161\pi$$
$$402$$ 0 0
$$403$$ −14800.0 −1.82938
$$404$$ 1734.00 0.213539
$$405$$ 0 0
$$406$$ −4680.00 −0.572080
$$407$$ 1680.00 0.204606
$$408$$ 0 0
$$409$$ 8714.00 1.05350 0.526748 0.850022i $$-0.323411\pi$$
0.526748 + 0.850022i $$0.323411\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −452.000 −0.0540496
$$413$$ 480.000 0.0571895
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 3330.00 0.392468
$$417$$ 0 0
$$418$$ −8928.00 −1.04470
$$419$$ −11976.0 −1.39634 −0.698169 0.715933i $$-0.746002\pi$$
−0.698169 + 0.715933i $$0.746002\pi$$
$$420$$ 0 0
$$421$$ 11054.0 1.27967 0.639833 0.768514i $$-0.279004\pi$$
0.639833 + 0.768514i $$0.279004\pi$$
$$422$$ 3300.00 0.380667
$$423$$ 0 0
$$424$$ −9450.00 −1.08239
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 6440.00 0.729868
$$428$$ −1404.00 −0.158563
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −720.000 −0.0804668 −0.0402334 0.999190i $$-0.512810\pi$$
−0.0402334 + 0.999190i $$0.512810\pi$$
$$432$$ 0 0
$$433$$ 15622.0 1.73382 0.866912 0.498462i $$-0.166102\pi$$
0.866912 + 0.498462i $$0.166102\pi$$
$$434$$ −12000.0 −1.32723
$$435$$ 0 0
$$436$$ −1474.00 −0.161908
$$437$$ 14880.0 1.62885
$$438$$ 0 0
$$439$$ −9880.00 −1.07414 −0.537069 0.843538i $$-0.680469\pi$$
−0.537069 + 0.843538i $$0.680469\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −11988.0 −1.29007
$$443$$ −16116.0 −1.72843 −0.864215 0.503123i $$-0.832184\pi$$
−0.864215 + 0.503123i $$0.832184\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −5892.00 −0.625548
$$447$$ 0 0
$$448$$ −8660.00 −0.913274
$$449$$ −9018.00 −0.947852 −0.473926 0.880565i $$-0.657164\pi$$
−0.473926 + 0.880565i $$0.657164\pi$$
$$450$$ 0 0
$$451$$ −7920.00 −0.826914
$$452$$ 1086.00 0.113011
$$453$$ 0 0
$$454$$ 1980.00 0.204683
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3670.00 0.375657 0.187829 0.982202i $$-0.439855\pi$$
0.187829 + 0.982202i $$0.439855\pi$$
$$458$$ −5718.00 −0.583372
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −17562.0 −1.77428 −0.887141 0.461499i $$-0.847312\pi$$
−0.887141 + 0.461499i $$0.847312\pi$$
$$462$$ 0 0
$$463$$ −1172.00 −0.117640 −0.0588202 0.998269i $$-0.518734\pi$$
−0.0588202 + 0.998269i $$0.518734\pi$$
$$464$$ −5538.00 −0.554084
$$465$$ 0 0
$$466$$ −4374.00 −0.434810
$$467$$ 6876.00 0.681335 0.340667 0.940184i $$-0.389347\pi$$
0.340667 + 0.940184i $$0.389347\pi$$
$$468$$ 0 0
$$469$$ −3920.00 −0.385946
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 504.000 0.0491493
$$473$$ −2208.00 −0.214638
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −1080.00 −0.103995
$$477$$ 0 0
$$478$$ −3528.00 −0.337588
$$479$$ −2280.00 −0.217486 −0.108743 0.994070i $$-0.534683\pi$$
−0.108743 + 0.994070i $$0.534683\pi$$
$$480$$ 0 0
$$481$$ −5180.00 −0.491035
$$482$$ 2598.00 0.245510
$$483$$ 0 0
$$484$$ −755.000 −0.0709053
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 3076.00 0.286215 0.143108 0.989707i $$-0.454290\pi$$
0.143108 + 0.989707i $$0.454290\pi$$
$$488$$ 6762.00 0.627257
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 18912.0 1.73826 0.869131 0.494582i $$-0.164679\pi$$
0.869131 + 0.494582i $$0.164679\pi$$
$$492$$ 0 0
$$493$$ 4212.00 0.384785
$$494$$ 27528.0 2.50717
$$495$$ 0 0
$$496$$ −14200.0 −1.28548
$$497$$ −5760.00 −0.519862
$$498$$ 0 0
$$499$$ 9956.00 0.893170 0.446585 0.894741i $$-0.352640\pi$$
0.446585 + 0.894741i $$0.352640\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −1296.00 −0.115226
$$503$$ −10656.0 −0.944588 −0.472294 0.881441i $$-0.656574\pi$$
−0.472294 + 0.881441i $$0.656574\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −8640.00 −0.759081
$$507$$ 0 0
$$508$$ −1244.00 −0.108649
$$509$$ 2766.00 0.240866 0.120433 0.992721i $$-0.461572\pi$$
0.120433 + 0.992721i $$0.461572\pi$$
$$510$$ 0 0
$$511$$ −8600.00 −0.744504
$$512$$ −8733.00 −0.753804
$$513$$ 0 0
$$514$$ 7578.00 0.650294
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −576.000 −0.0489989
$$518$$ −4200.00 −0.356250
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −10530.0 −0.885466 −0.442733 0.896654i $$-0.645991\pi$$
−0.442733 + 0.896654i $$0.645991\pi$$
$$522$$ 0 0
$$523$$ −12692.0 −1.06115 −0.530576 0.847637i $$-0.678024\pi$$
−0.530576 + 0.847637i $$0.678024\pi$$
$$524$$ −2328.00 −0.194082
$$525$$ 0 0
$$526$$ 16344.0 1.35481
$$527$$ 10800.0 0.892705
$$528$$ 0 0
$$529$$ 2233.00 0.183529
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2480.00 0.202108
$$533$$ 24420.0 1.98452
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −4116.00 −0.331687
$$537$$ 0 0
$$538$$ 7722.00 0.618809
$$539$$ 1368.00 0.109321
$$540$$ 0 0
$$541$$ 18110.0 1.43920 0.719602 0.694386i $$-0.244324\pi$$
0.719602 + 0.694386i $$0.244324\pi$$
$$542$$ −9552.00 −0.756999
$$543$$ 0 0
$$544$$ −2430.00 −0.191517
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −3620.00 −0.282962 −0.141481 0.989941i $$-0.545186\pi$$
−0.141481 + 0.989941i $$0.545186\pi$$
$$548$$ 2118.00 0.165103
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −9672.00 −0.747806
$$552$$ 0 0
$$553$$ 10400.0 0.799734
$$554$$ −11886.0 −0.911530
$$555$$ 0 0
$$556$$ 2324.00 0.177265
$$557$$ −14166.0 −1.07762 −0.538809 0.842428i $$-0.681125\pi$$
−0.538809 + 0.842428i $$0.681125\pi$$
$$558$$ 0 0
$$559$$ 6808.00 0.515112
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 24858.0 1.86579
$$563$$ −13404.0 −1.00339 −0.501697 0.865043i $$-0.667291\pi$$
−0.501697 + 0.865043i $$0.667291\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 8148.00 0.605099
$$567$$ 0 0
$$568$$ −6048.00 −0.446775
$$569$$ 18654.0 1.37437 0.687185 0.726483i $$-0.258846\pi$$
0.687185 + 0.726483i $$0.258846\pi$$
$$570$$ 0 0
$$571$$ −7684.00 −0.563162 −0.281581 0.959537i $$-0.590859\pi$$
−0.281581 + 0.959537i $$0.590859\pi$$
$$572$$ −1776.00 −0.129822
$$573$$ 0 0
$$574$$ 19800.0 1.43978
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 1726.00 0.124531 0.0622654 0.998060i $$-0.480167\pi$$
0.0622654 + 0.998060i $$0.480167\pi$$
$$578$$ −5991.00 −0.431129
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −3120.00 −0.222787
$$582$$ 0 0
$$583$$ 10800.0 0.767222
$$584$$ −9030.00 −0.639836
$$585$$ 0 0
$$586$$ 18054.0 1.27270
$$587$$ 10596.0 0.745049 0.372524 0.928022i $$-0.378492\pi$$
0.372524 + 0.928022i $$0.378492\pi$$
$$588$$ 0 0
$$589$$ −24800.0 −1.73492
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −4970.00 −0.345043
$$593$$ 2862.00 0.198193 0.0990963 0.995078i $$-0.468405\pi$$
0.0990963 + 0.995078i $$0.468405\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −258.000 −0.0177317
$$597$$ 0 0
$$598$$ 26640.0 1.82172
$$599$$ 23592.0 1.60925 0.804627 0.593781i $$-0.202365\pi$$
0.804627 + 0.593781i $$0.202365\pi$$
$$600$$ 0 0
$$601$$ −9574.00 −0.649803 −0.324902 0.945748i $$-0.605331\pi$$
−0.324902 + 0.945748i $$0.605331\pi$$
$$602$$ 5520.00 0.373718
$$603$$ 0 0
$$604$$ −808.000 −0.0544322
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −17444.0 −1.16644 −0.583221 0.812314i $$-0.698208\pi$$
−0.583221 + 0.812314i $$0.698208\pi$$
$$608$$ 5580.00 0.372202
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1776.00 0.117593
$$612$$ 0 0
$$613$$ 2374.00 0.156419 0.0782096 0.996937i $$-0.475080\pi$$
0.0782096 + 0.996937i $$0.475080\pi$$
$$614$$ −27708.0 −1.82118
$$615$$ 0 0
$$616$$ 10080.0 0.659310
$$617$$ −12162.0 −0.793555 −0.396778 0.917915i $$-0.629872\pi$$
−0.396778 + 0.917915i $$0.629872\pi$$
$$618$$ 0 0
$$619$$ 8804.00 0.571668 0.285834 0.958279i $$-0.407729\pi$$
0.285834 + 0.958279i $$0.407729\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −4608.00 −0.297048
$$623$$ 20520.0 1.31961
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 22026.0 1.40629
$$627$$ 0 0
$$628$$ −2378.00 −0.151103
$$629$$ 3780.00 0.239616
$$630$$ 0 0
$$631$$ −12688.0 −0.800478 −0.400239 0.916411i $$-0.631073\pi$$
−0.400239 + 0.916411i $$0.631073\pi$$
$$632$$ 10920.0 0.687301
$$633$$ 0 0
$$634$$ −11682.0 −0.731785
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −4218.00 −0.262360
$$638$$ 5616.00 0.348495
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 9150.00 0.563812 0.281906 0.959442i $$-0.409033\pi$$
0.281906 + 0.959442i $$0.409033\pi$$
$$642$$ 0 0
$$643$$ −25292.0 −1.55120 −0.775598 0.631227i $$-0.782552\pi$$
−0.775598 + 0.631227i $$0.782552\pi$$
$$644$$ 2400.00 0.146853
$$645$$ 0 0
$$646$$ −20088.0 −1.22345
$$647$$ −2736.00 −0.166249 −0.0831246 0.996539i $$-0.526490\pi$$
−0.0831246 + 0.996539i $$0.526490\pi$$
$$648$$ 0 0
$$649$$ −576.000 −0.0348382
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 52.0000 0.00312343
$$653$$ 22218.0 1.33148 0.665741 0.746183i $$-0.268116\pi$$
0.665741 + 0.746183i $$0.268116\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 23430.0 1.39449
$$657$$ 0 0
$$658$$ 1440.00 0.0853147
$$659$$ −14520.0 −0.858299 −0.429149 0.903234i $$-0.641187\pi$$
−0.429149 + 0.903234i $$0.641187\pi$$
$$660$$ 0 0
$$661$$ −10618.0 −0.624799 −0.312400 0.949951i $$-0.601133\pi$$
−0.312400 + 0.949951i $$0.601133\pi$$
$$662$$ 11076.0 0.650273
$$663$$ 0 0
$$664$$ −3276.00 −0.191466
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −9360.00 −0.543359
$$668$$ −3720.00 −0.215466
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −7728.00 −0.444614
$$672$$ 0 0
$$673$$ −1370.00 −0.0784690 −0.0392345 0.999230i $$-0.512492\pi$$
−0.0392345 + 0.999230i $$0.512492\pi$$
$$674$$ 26994.0 1.54269
$$675$$ 0 0
$$676$$ 3279.00 0.186561
$$677$$ −13758.0 −0.781038 −0.390519 0.920595i $$-0.627704\pi$$
−0.390519 + 0.920595i $$0.627704\pi$$
$$678$$ 0 0
$$679$$ −5720.00 −0.323289
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 14400.0 0.808511
$$683$$ 11988.0 0.671608 0.335804 0.941932i $$-0.390992\pi$$
0.335804 + 0.941932i $$0.390992\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 17160.0 0.955061
$$687$$ 0 0
$$688$$ 6532.00 0.361962
$$689$$ −33300.0 −1.84126
$$690$$ 0 0
$$691$$ 32996.0 1.81654 0.908268 0.418388i $$-0.137405\pi$$
0.908268 + 0.418388i $$0.137405\pi$$
$$692$$ 426.000 0.0234019
$$693$$ 0 0
$$694$$ 15732.0 0.860488
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −17820.0 −0.968408
$$698$$ 18906.0 1.02522
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 25902.0 1.39558 0.697792 0.716300i $$-0.254166\pi$$
0.697792 + 0.716300i $$0.254166\pi$$
$$702$$ 0 0
$$703$$ −8680.00 −0.465679
$$704$$ 10392.0 0.556340
$$705$$ 0 0
$$706$$ 10242.0 0.545981
$$707$$ −34680.0 −1.84480
$$708$$ 0 0
$$709$$ −27394.0 −1.45106 −0.725531 0.688189i $$-0.758406\pi$$
−0.725531 + 0.688189i $$0.758406\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 21546.0 1.13409
$$713$$ −24000.0 −1.26060
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 1440.00 0.0751611
$$717$$ 0 0
$$718$$ −14472.0 −0.752215
$$719$$ −34848.0 −1.80753 −0.903763 0.428033i $$-0.859207\pi$$
−0.903763 + 0.428033i $$0.859207\pi$$
$$720$$ 0 0
$$721$$ 9040.00 0.466945
$$722$$ 25551.0 1.31705
$$723$$ 0 0
$$724$$ −3130.00 −0.160671
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −28028.0 −1.42985 −0.714925 0.699201i $$-0.753539\pi$$
−0.714925 + 0.699201i $$0.753539\pi$$
$$728$$ −31080.0 −1.58228
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −4968.00 −0.251365
$$732$$ 0 0
$$733$$ −18002.0 −0.907120 −0.453560 0.891226i $$-0.649846\pi$$
−0.453560 + 0.891226i $$0.649846\pi$$
$$734$$ 10524.0 0.529221
$$735$$ 0 0
$$736$$ 5400.00 0.270444
$$737$$ 4704.00 0.235107
$$738$$ 0 0
$$739$$ 15284.0 0.760800 0.380400 0.924822i $$-0.375786\pi$$
0.380400 + 0.924822i $$0.375786\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −27000.0 −1.33585
$$743$$ −18768.0 −0.926691 −0.463345 0.886178i $$-0.653351\pi$$
−0.463345 + 0.886178i $$0.653351\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −32406.0 −1.59044
$$747$$ 0 0
$$748$$ 1296.00 0.0633509
$$749$$ 28080.0 1.36985
$$750$$ 0 0
$$751$$ 8696.00 0.422532 0.211266 0.977429i $$-0.432241\pi$$
0.211266 + 0.977429i $$0.432241\pi$$
$$752$$ 1704.00 0.0826310
$$753$$ 0 0
$$754$$ −17316.0 −0.836355
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 38662.0 1.85627 0.928134 0.372247i $$-0.121413\pi$$
0.928134 + 0.372247i $$0.121413\pi$$
$$758$$ 4380.00 0.209880
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −23874.0 −1.13723 −0.568615 0.822604i $$-0.692521\pi$$
−0.568615 + 0.822604i $$0.692521\pi$$
$$762$$ 0 0
$$763$$ 29480.0 1.39875
$$764$$ −3576.00 −0.169339
$$765$$ 0 0
$$766$$ −14616.0 −0.689422
$$767$$ 1776.00 0.0836084
$$768$$ 0 0
$$769$$ 23618.0 1.10753 0.553763 0.832675i $$-0.313192\pi$$
0.553763 + 0.832675i $$0.313192\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −2666.00 −0.124289
$$773$$ 11538.0 0.536860 0.268430 0.963299i $$-0.413495\pi$$
0.268430 + 0.963299i $$0.413495\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −6006.00 −0.277839
$$777$$ 0 0
$$778$$ 42138.0 1.94180
$$779$$ 40920.0 1.88204
$$780$$ 0 0
$$781$$ 6912.00 0.316685
$$782$$ −19440.0 −0.888968
$$783$$ 0 0
$$784$$ −4047.00 −0.184357
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 14884.0 0.674152 0.337076 0.941478i $$-0.390562\pi$$
0.337076 + 0.941478i $$0.390562\pi$$
$$788$$ −2718.00 −0.122874
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −21720.0 −0.976327
$$792$$ 0 0
$$793$$ 23828.0 1.06703
$$794$$ 8202.00 0.366597
$$795$$ 0 0
$$796$$ −3832.00 −0.170630
$$797$$ −11334.0 −0.503728 −0.251864 0.967763i $$-0.581043\pi$$
−0.251864 + 0.967763i $$0.581043\pi$$
$$798$$ 0 0
$$799$$ −1296.00 −0.0573832
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 47826.0 2.10573
$$803$$ 10320.0 0.453530
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −44400.0 −1.94035
$$807$$ 0 0
$$808$$ −36414.0 −1.58545
$$809$$ −44730.0 −1.94391 −0.971955 0.235167i $$-0.924436\pi$$
−0.971955 + 0.235167i $$0.924436\pi$$
$$810$$ 0 0
$$811$$ −42748.0 −1.85091 −0.925453 0.378862i $$-0.876316\pi$$
−0.925453 + 0.378862i $$0.876316\pi$$
$$812$$ −1560.00 −0.0674203
$$813$$ 0 0
$$814$$ 5040.00 0.217017
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 11408.0 0.488513
$$818$$ 26142.0 1.11740
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 31686.0 1.34695 0.673477 0.739208i $$-0.264800\pi$$
0.673477 + 0.739208i $$0.264800\pi$$
$$822$$ 0 0
$$823$$ −11036.0 −0.467425 −0.233713 0.972306i $$-0.575087\pi$$
−0.233713 + 0.972306i $$0.575087\pi$$
$$824$$ 9492.00 0.401298
$$825$$ 0 0
$$826$$ 1440.00 0.0606586
$$827$$ 25884.0 1.08836 0.544181 0.838968i $$-0.316841\pi$$
0.544181 + 0.838968i $$0.316841\pi$$
$$828$$ 0 0
$$829$$ 15950.0 0.668234 0.334117 0.942532i $$-0.391562\pi$$
0.334117 + 0.942532i $$0.391562\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −32042.0 −1.33516
$$833$$ 3078.00 0.128027
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −2976.00 −0.123119
$$837$$ 0 0
$$838$$ −35928.0 −1.48104
$$839$$ −13800.0 −0.567853 −0.283927 0.958846i $$-0.591637\pi$$
−0.283927 + 0.958846i $$0.591637\pi$$
$$840$$ 0 0
$$841$$ −18305.0 −0.750543
$$842$$ 33162.0 1.35729
$$843$$ 0 0
$$844$$ 1100.00 0.0448620
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 15100.0 0.612565
$$848$$ −31950.0 −1.29383
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −8400.00 −0.338365
$$852$$ 0 0
$$853$$ 27862.0 1.11838 0.559189 0.829040i $$-0.311113\pi$$
0.559189 + 0.829040i $$0.311113\pi$$
$$854$$ 19320.0 0.774141
$$855$$ 0 0
$$856$$ 29484.0 1.17727
$$857$$ −7314.00 −0.291530 −0.145765 0.989319i $$-0.546564\pi$$
−0.145765 + 0.989319i $$0.546564\pi$$
$$858$$ 0 0
$$859$$ −28780.0 −1.14314 −0.571572 0.820552i $$-0.693666\pi$$
−0.571572 + 0.820552i $$0.693666\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −2160.00 −0.0853479
$$863$$ −32688.0 −1.28935 −0.644677 0.764455i $$-0.723008\pi$$
−0.644677 + 0.764455i $$0.723008\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 46866.0 1.83900
$$867$$ 0 0
$$868$$ −4000.00 −0.156416
$$869$$ −12480.0 −0.487175
$$870$$ 0 0
$$871$$ −14504.0 −0.564236
$$872$$ 30954.0 1.20210
$$873$$ 0 0
$$874$$ 44640.0 1.72766
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −36650.0 −1.41115 −0.705577 0.708633i $$-0.749312\pi$$
−0.705577 + 0.708633i $$0.749312\pi$$
$$878$$ −29640.0 −1.13930
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 2646.00 0.101187 0.0505936 0.998719i $$-0.483889\pi$$
0.0505936 + 0.998719i $$0.483889\pi$$
$$882$$ 0 0
$$883$$ −10892.0 −0.415113 −0.207557 0.978223i $$-0.566551\pi$$
−0.207557 + 0.978223i $$0.566551\pi$$
$$884$$ −3996.00 −0.152036
$$885$$ 0 0
$$886$$ −48348.0 −1.83328
$$887$$ −43464.0 −1.64530 −0.822648 0.568550i $$-0.807504\pi$$
−0.822648 + 0.568550i $$0.807504\pi$$
$$888$$ 0 0
$$889$$ 24880.0 0.938637
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −1964.00 −0.0737215
$$893$$ 2976.00 0.111521
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −33180.0 −1.23713
$$897$$ 0 0
$$898$$ −27054.0 −1.00535
$$899$$ 15600.0 0.578742
$$900$$ 0 0
$$901$$ 24300.0 0.898502
$$902$$ −23760.0 −0.877075
$$903$$ 0 0
$$904$$ −22806.0 −0.839067
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 14884.0 0.544890 0.272445 0.962171i $$-0.412168\pi$$
0.272445 + 0.962171i $$0.412168\pi$$
$$908$$ 660.000 0.0241221
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 1248.00 0.0453876 0.0226938 0.999742i $$-0.492776\pi$$
0.0226938 + 0.999742i $$0.492776\pi$$
$$912$$ 0 0
$$913$$ 3744.00 0.135716
$$914$$ 11010.0 0.398445
$$915$$ 0 0
$$916$$ −1906.00 −0.0687511
$$917$$ 46560.0 1.67671
$$918$$ 0 0
$$919$$ −6640.00 −0.238339 −0.119169 0.992874i $$-0.538023\pi$$
−0.119169 + 0.992874i $$0.538023\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −52686.0 −1.88191
$$923$$ −21312.0 −0.760014
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −3516.00 −0.124776
$$927$$ 0 0
$$928$$ −3510.00 −0.124161
$$929$$ −29946.0 −1.05758 −0.528792 0.848751i $$-0.677355\pi$$
−0.528792 + 0.848751i $$0.677355\pi$$
$$930$$ 0 0
$$931$$ −7068.00 −0.248812
$$932$$ −1458.00 −0.0512429
$$933$$ 0 0
$$934$$ 20628.0 0.722665
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −45002.0 −1.56900 −0.784499 0.620130i $$-0.787080\pi$$
−0.784499 + 0.620130i $$0.787080\pi$$
$$938$$ −11760.0 −0.409358
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −6090.00 −0.210976 −0.105488 0.994421i $$-0.533640\pi$$
−0.105488 + 0.994421i $$0.533640\pi$$
$$942$$ 0 0
$$943$$ 39600.0 1.36750
$$944$$ 1704.00 0.0587505
$$945$$ 0 0
$$946$$ −6624.00 −0.227658
$$947$$ 56388.0 1.93491 0.967457 0.253035i $$-0.0814288\pi$$
0.967457 + 0.253035i $$0.0814288\pi$$
$$948$$ 0 0
$$949$$ −31820.0 −1.08843
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 22680.0 0.772125
$$953$$ 10854.0 0.368936 0.184468 0.982839i $$-0.440944\pi$$
0.184468 + 0.982839i $$0.440944\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −1176.00 −0.0397851
$$957$$ 0 0
$$958$$ −6840.00 −0.230679
$$959$$ −42360.0 −1.42636
$$960$$ 0 0
$$961$$ 10209.0 0.342687
$$962$$ −15540.0 −0.520821
$$963$$ 0 0
$$964$$ 866.000 0.0289336
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 42316.0 1.40723 0.703615 0.710582i $$-0.251568\pi$$
0.703615 + 0.710582i $$0.251568\pi$$
$$968$$ 15855.0 0.526445
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −24480.0 −0.809063 −0.404532 0.914524i $$-0.632565\pi$$
−0.404532 + 0.914524i $$0.632565\pi$$
$$972$$ 0 0
$$973$$ −46480.0 −1.53143
$$974$$ 9228.00 0.303577
$$975$$ 0 0
$$976$$ 22862.0 0.749790
$$977$$ −6906.00 −0.226144 −0.113072 0.993587i $$-0.536069\pi$$
−0.113072 + 0.993587i $$0.536069\pi$$
$$978$$ 0 0
$$979$$ −24624.0 −0.803868
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 56736.0 1.84371
$$983$$ 6960.00 0.225829 0.112914 0.993605i $$-0.463981\pi$$
0.112914 + 0.993605i $$0.463981\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 12636.0 0.408126
$$987$$ 0 0
$$988$$ 9176.00 0.295473
$$989$$ 11040.0 0.354956
$$990$$ 0 0
$$991$$ 47792.0 1.53195 0.765975 0.642870i $$-0.222256\pi$$
0.765975 + 0.642870i $$0.222256\pi$$
$$992$$ −9000.00 −0.288055
$$993$$ 0 0
$$994$$ −17280.0 −0.551397
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −9938.00 −0.315687 −0.157843 0.987464i $$-0.550454\pi$$
−0.157843 + 0.987464i $$0.550454\pi$$
$$998$$ 29868.0 0.947350
$$999$$ 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 225.4.a.g.1.1 1
3.2 odd 2 75.4.a.a.1.1 1
5.2 odd 4 225.4.b.d.199.2 2
5.3 odd 4 225.4.b.d.199.1 2
5.4 even 2 45.4.a.b.1.1 1
12.11 even 2 1200.4.a.o.1.1 1
15.2 even 4 75.4.b.a.49.1 2
15.8 even 4 75.4.b.a.49.2 2
15.14 odd 2 15.4.a.b.1.1 1
20.19 odd 2 720.4.a.r.1.1 1
35.34 odd 2 2205.4.a.c.1.1 1
45.4 even 6 405.4.e.k.136.1 2
45.14 odd 6 405.4.e.d.136.1 2
45.29 odd 6 405.4.e.d.271.1 2
45.34 even 6 405.4.e.k.271.1 2
60.23 odd 4 1200.4.f.m.49.1 2
60.47 odd 4 1200.4.f.m.49.2 2
60.59 even 2 240.4.a.f.1.1 1
105.104 even 2 735.4.a.i.1.1 1
120.29 odd 2 960.4.a.bi.1.1 1
120.59 even 2 960.4.a.l.1.1 1
165.164 even 2 1815.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 15.14 odd 2
45.4.a.b.1.1 1 5.4 even 2
75.4.a.a.1.1 1 3.2 odd 2
75.4.b.a.49.1 2 15.2 even 4
75.4.b.a.49.2 2 15.8 even 4
225.4.a.g.1.1 1 1.1 even 1 trivial
225.4.b.d.199.1 2 5.3 odd 4
225.4.b.d.199.2 2 5.2 odd 4
240.4.a.f.1.1 1 60.59 even 2
405.4.e.d.136.1 2 45.14 odd 6
405.4.e.d.271.1 2 45.29 odd 6
405.4.e.k.136.1 2 45.4 even 6
405.4.e.k.271.1 2 45.34 even 6
720.4.a.r.1.1 1 20.19 odd 2
735.4.a.i.1.1 1 105.104 even 2
960.4.a.l.1.1 1 120.59 even 2
960.4.a.bi.1.1 1 120.29 odd 2
1200.4.a.o.1.1 1 12.11 even 2
1200.4.f.m.49.1 2 60.23 odd 4
1200.4.f.m.49.2 2 60.47 odd 4
1815.4.a.a.1.1 1 165.164 even 2
2205.4.a.c.1.1 1 35.34 odd 2