# Properties

 Label 2352.4.a.u.1.1 Level $2352$ Weight $4$ Character 2352.1 Self dual yes Analytic conductor $138.772$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2352.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$138.772492334$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -15.0000 q^{5} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -15.0000 q^{5} +9.00000 q^{9} +9.00000 q^{11} -88.0000 q^{13} -45.0000 q^{15} -84.0000 q^{17} -104.000 q^{19} +84.0000 q^{23} +100.000 q^{25} +27.0000 q^{27} +51.0000 q^{29} -185.000 q^{31} +27.0000 q^{33} +44.0000 q^{37} -264.000 q^{39} -168.000 q^{41} -326.000 q^{43} -135.000 q^{45} +138.000 q^{47} -252.000 q^{51} +639.000 q^{53} -135.000 q^{55} -312.000 q^{57} -159.000 q^{59} +722.000 q^{61} +1320.00 q^{65} +166.000 q^{67} +252.000 q^{69} -1086.00 q^{71} +218.000 q^{73} +300.000 q^{75} +583.000 q^{79} +81.0000 q^{81} +597.000 q^{83} +1260.00 q^{85} +153.000 q^{87} -1038.00 q^{89} -555.000 q^{93} +1560.00 q^{95} -169.000 q^{97} +81.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −15.0000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 9.00000 0.246691 0.123346 0.992364i $$-0.460638\pi$$
0.123346 + 0.992364i $$0.460638\pi$$
$$12$$ 0 0
$$13$$ −88.0000 −1.87745 −0.938723 0.344671i $$-0.887990\pi$$
−0.938723 + 0.344671i $$0.887990\pi$$
$$14$$ 0 0
$$15$$ −45.0000 −0.774597
$$16$$ 0 0
$$17$$ −84.0000 −1.19841 −0.599206 0.800595i $$-0.704517\pi$$
−0.599206 + 0.800595i $$0.704517\pi$$
$$18$$ 0 0
$$19$$ −104.000 −1.25575 −0.627875 0.778314i $$-0.716075\pi$$
−0.627875 + 0.778314i $$0.716075\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 84.0000 0.761531 0.380765 0.924672i $$-0.375661\pi$$
0.380765 + 0.924672i $$0.375661\pi$$
$$24$$ 0 0
$$25$$ 100.000 0.800000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 51.0000 0.326568 0.163284 0.986579i $$-0.447791\pi$$
0.163284 + 0.986579i $$0.447791\pi$$
$$30$$ 0 0
$$31$$ −185.000 −1.07184 −0.535919 0.844269i $$-0.680035\pi$$
−0.535919 + 0.844269i $$0.680035\pi$$
$$32$$ 0 0
$$33$$ 27.0000 0.142427
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 44.0000 0.195501 0.0977507 0.995211i $$-0.468835\pi$$
0.0977507 + 0.995211i $$0.468835\pi$$
$$38$$ 0 0
$$39$$ −264.000 −1.08394
$$40$$ 0 0
$$41$$ −168.000 −0.639932 −0.319966 0.947429i $$-0.603671\pi$$
−0.319966 + 0.947429i $$0.603671\pi$$
$$42$$ 0 0
$$43$$ −326.000 −1.15615 −0.578076 0.815983i $$-0.696196\pi$$
−0.578076 + 0.815983i $$0.696196\pi$$
$$44$$ 0 0
$$45$$ −135.000 −0.447214
$$46$$ 0 0
$$47$$ 138.000 0.428284 0.214142 0.976802i $$-0.431304\pi$$
0.214142 + 0.976802i $$0.431304\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −252.000 −0.691903
$$52$$ 0 0
$$53$$ 639.000 1.65610 0.828051 0.560653i $$-0.189450\pi$$
0.828051 + 0.560653i $$0.189450\pi$$
$$54$$ 0 0
$$55$$ −135.000 −0.330971
$$56$$ 0 0
$$57$$ −312.000 −0.725007
$$58$$ 0 0
$$59$$ −159.000 −0.350848 −0.175424 0.984493i $$-0.556130\pi$$
−0.175424 + 0.984493i $$0.556130\pi$$
$$60$$ 0 0
$$61$$ 722.000 1.51545 0.757726 0.652572i $$-0.226310\pi$$
0.757726 + 0.652572i $$0.226310\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 1320.00 2.51886
$$66$$ 0 0
$$67$$ 166.000 0.302688 0.151344 0.988481i $$-0.451640\pi$$
0.151344 + 0.988481i $$0.451640\pi$$
$$68$$ 0 0
$$69$$ 252.000 0.439670
$$70$$ 0 0
$$71$$ −1086.00 −1.81527 −0.907637 0.419755i $$-0.862116\pi$$
−0.907637 + 0.419755i $$0.862116\pi$$
$$72$$ 0 0
$$73$$ 218.000 0.349520 0.174760 0.984611i $$-0.444085\pi$$
0.174760 + 0.984611i $$0.444085\pi$$
$$74$$ 0 0
$$75$$ 300.000 0.461880
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 583.000 0.830286 0.415143 0.909756i $$-0.363731\pi$$
0.415143 + 0.909756i $$0.363731\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 597.000 0.789509 0.394755 0.918787i $$-0.370830\pi$$
0.394755 + 0.918787i $$0.370830\pi$$
$$84$$ 0 0
$$85$$ 1260.00 1.60784
$$86$$ 0 0
$$87$$ 153.000 0.188544
$$88$$ 0 0
$$89$$ −1038.00 −1.23627 −0.618134 0.786073i $$-0.712111\pi$$
−0.618134 + 0.786073i $$0.712111\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −555.000 −0.618826
$$94$$ 0 0
$$95$$ 1560.00 1.68476
$$96$$ 0 0
$$97$$ −169.000 −0.176901 −0.0884503 0.996081i $$-0.528191\pi$$
−0.0884503 + 0.996081i $$0.528191\pi$$
$$98$$ 0 0
$$99$$ 81.0000 0.0822304
$$100$$ 0 0
$$101$$ 642.000 0.632489 0.316244 0.948678i $$-0.397578\pi$$
0.316244 + 0.948678i $$0.397578\pi$$
$$102$$ 0 0
$$103$$ −464.000 −0.443876 −0.221938 0.975061i $$-0.571238\pi$$
−0.221938 + 0.975061i $$0.571238\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −393.000 −0.355072 −0.177536 0.984114i $$-0.556813\pi$$
−0.177536 + 0.984114i $$0.556813\pi$$
$$108$$ 0 0
$$109$$ 14.0000 0.0123024 0.00615118 0.999981i $$-0.498042\pi$$
0.00615118 + 0.999981i $$0.498042\pi$$
$$110$$ 0 0
$$111$$ 132.000 0.112873
$$112$$ 0 0
$$113$$ −2184.00 −1.81817 −0.909086 0.416608i $$-0.863219\pi$$
−0.909086 + 0.416608i $$0.863219\pi$$
$$114$$ 0 0
$$115$$ −1260.00 −1.02170
$$116$$ 0 0
$$117$$ −792.000 −0.625816
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1250.00 −0.939144
$$122$$ 0 0
$$123$$ −504.000 −0.369465
$$124$$ 0 0
$$125$$ 375.000 0.268328
$$126$$ 0 0
$$127$$ 373.000 0.260617 0.130309 0.991473i $$-0.458403\pi$$
0.130309 + 0.991473i $$0.458403\pi$$
$$128$$ 0 0
$$129$$ −978.000 −0.667505
$$130$$ 0 0
$$131$$ 1173.00 0.782332 0.391166 0.920320i $$-0.372072\pi$$
0.391166 + 0.920320i $$0.372072\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −405.000 −0.258199
$$136$$ 0 0
$$137$$ 30.0000 0.0187086 0.00935428 0.999956i $$-0.497022\pi$$
0.00935428 + 0.999956i $$0.497022\pi$$
$$138$$ 0 0
$$139$$ 82.0000 0.0500370 0.0250185 0.999687i $$-0.492036\pi$$
0.0250185 + 0.999687i $$0.492036\pi$$
$$140$$ 0 0
$$141$$ 414.000 0.247270
$$142$$ 0 0
$$143$$ −792.000 −0.463149
$$144$$ 0 0
$$145$$ −765.000 −0.438136
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1434.00 −0.788442 −0.394221 0.919016i $$-0.628986\pi$$
−0.394221 + 0.919016i $$0.628986\pi$$
$$150$$ 0 0
$$151$$ 2671.00 1.43949 0.719745 0.694239i $$-0.244259\pi$$
0.719745 + 0.694239i $$0.244259\pi$$
$$152$$ 0 0
$$153$$ −756.000 −0.399470
$$154$$ 0 0
$$155$$ 2775.00 1.43802
$$156$$ 0 0
$$157$$ 2252.00 1.14477 0.572386 0.819984i $$-0.306018\pi$$
0.572386 + 0.819984i $$0.306018\pi$$
$$158$$ 0 0
$$159$$ 1917.00 0.956151
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −1676.00 −0.805365 −0.402682 0.915340i $$-0.631922\pi$$
−0.402682 + 0.915340i $$0.631922\pi$$
$$164$$ 0 0
$$165$$ −405.000 −0.191086
$$166$$ 0 0
$$167$$ −3030.00 −1.40400 −0.702001 0.712176i $$-0.747710\pi$$
−0.702001 + 0.712176i $$0.747710\pi$$
$$168$$ 0 0
$$169$$ 5547.00 2.52481
$$170$$ 0 0
$$171$$ −936.000 −0.418583
$$172$$ 0 0
$$173$$ 3438.00 1.51090 0.755452 0.655204i $$-0.227417\pi$$
0.755452 + 0.655204i $$0.227417\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −477.000 −0.202562
$$178$$ 0 0
$$179$$ 1212.00 0.506085 0.253042 0.967455i $$-0.418569\pi$$
0.253042 + 0.967455i $$0.418569\pi$$
$$180$$ 0 0
$$181$$ 3032.00 1.24512 0.622560 0.782572i $$-0.286093\pi$$
0.622560 + 0.782572i $$0.286093\pi$$
$$182$$ 0 0
$$183$$ 2166.00 0.874947
$$184$$ 0 0
$$185$$ −660.000 −0.262293
$$186$$ 0 0
$$187$$ −756.000 −0.295637
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2520.00 −0.954664 −0.477332 0.878723i $$-0.658396\pi$$
−0.477332 + 0.878723i $$0.658396\pi$$
$$192$$ 0 0
$$193$$ 365.000 0.136131 0.0680655 0.997681i $$-0.478317\pi$$
0.0680655 + 0.997681i $$0.478317\pi$$
$$194$$ 0 0
$$195$$ 3960.00 1.45426
$$196$$ 0 0
$$197$$ −1590.00 −0.575040 −0.287520 0.957775i $$-0.592831\pi$$
−0.287520 + 0.957775i $$0.592831\pi$$
$$198$$ 0 0
$$199$$ 5380.00 1.91647 0.958236 0.285977i $$-0.0923182\pi$$
0.958236 + 0.285977i $$0.0923182\pi$$
$$200$$ 0 0
$$201$$ 498.000 0.174757
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 2520.00 0.858558
$$206$$ 0 0
$$207$$ 756.000 0.253844
$$208$$ 0 0
$$209$$ −936.000 −0.309782
$$210$$ 0 0
$$211$$ 5362.00 1.74946 0.874728 0.484614i $$-0.161040\pi$$
0.874728 + 0.484614i $$0.161040\pi$$
$$212$$ 0 0
$$213$$ −3258.00 −1.04805
$$214$$ 0 0
$$215$$ 4890.00 1.55114
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 654.000 0.201796
$$220$$ 0 0
$$221$$ 7392.00 2.24995
$$222$$ 0 0
$$223$$ 1573.00 0.472358 0.236179 0.971710i $$-0.424105\pi$$
0.236179 + 0.971710i $$0.424105\pi$$
$$224$$ 0 0
$$225$$ 900.000 0.266667
$$226$$ 0 0
$$227$$ −921.000 −0.269290 −0.134645 0.990894i $$-0.542989\pi$$
−0.134645 + 0.990894i $$0.542989\pi$$
$$228$$ 0 0
$$229$$ 4052.00 1.16927 0.584637 0.811295i $$-0.301237\pi$$
0.584637 + 0.811295i $$0.301237\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −468.000 −0.131587 −0.0657933 0.997833i $$-0.520958\pi$$
−0.0657933 + 0.997833i $$0.520958\pi$$
$$234$$ 0 0
$$235$$ −2070.00 −0.574604
$$236$$ 0 0
$$237$$ 1749.00 0.479366
$$238$$ 0 0
$$239$$ −4932.00 −1.33483 −0.667415 0.744686i $$-0.732599\pi$$
−0.667415 + 0.744686i $$0.732599\pi$$
$$240$$ 0 0
$$241$$ −1537.00 −0.410817 −0.205408 0.978676i $$-0.565852\pi$$
−0.205408 + 0.978676i $$0.565852\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 9152.00 2.35760
$$248$$ 0 0
$$249$$ 1791.00 0.455823
$$250$$ 0 0
$$251$$ −5319.00 −1.33758 −0.668789 0.743452i $$-0.733187\pi$$
−0.668789 + 0.743452i $$0.733187\pi$$
$$252$$ 0 0
$$253$$ 756.000 0.187863
$$254$$ 0 0
$$255$$ 3780.00 0.928285
$$256$$ 0 0
$$257$$ 5346.00 1.29757 0.648783 0.760974i $$-0.275278\pi$$
0.648783 + 0.760974i $$0.275278\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 459.000 0.108856
$$262$$ 0 0
$$263$$ −774.000 −0.181471 −0.0907355 0.995875i $$-0.528922\pi$$
−0.0907355 + 0.995875i $$0.528922\pi$$
$$264$$ 0 0
$$265$$ −9585.00 −2.22189
$$266$$ 0 0
$$267$$ −3114.00 −0.713759
$$268$$ 0 0
$$269$$ 2415.00 0.547380 0.273690 0.961818i $$-0.411756\pi$$
0.273690 + 0.961818i $$0.411756\pi$$
$$270$$ 0 0
$$271$$ 475.000 0.106473 0.0532365 0.998582i $$-0.483046\pi$$
0.0532365 + 0.998582i $$0.483046\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 900.000 0.197353
$$276$$ 0 0
$$277$$ 3728.00 0.808642 0.404321 0.914617i $$-0.367508\pi$$
0.404321 + 0.914617i $$0.367508\pi$$
$$278$$ 0 0
$$279$$ −1665.00 −0.357279
$$280$$ 0 0
$$281$$ 1602.00 0.340097 0.170049 0.985436i $$-0.445608\pi$$
0.170049 + 0.985436i $$0.445608\pi$$
$$282$$ 0 0
$$283$$ −686.000 −0.144094 −0.0720468 0.997401i $$-0.522953\pi$$
−0.0720468 + 0.997401i $$0.522953\pi$$
$$284$$ 0 0
$$285$$ 4680.00 0.972699
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2143.00 0.436190
$$290$$ 0 0
$$291$$ −507.000 −0.102134
$$292$$ 0 0
$$293$$ −1101.00 −0.219526 −0.109763 0.993958i $$-0.535009\pi$$
−0.109763 + 0.993958i $$0.535009\pi$$
$$294$$ 0 0
$$295$$ 2385.00 0.470712
$$296$$ 0 0
$$297$$ 243.000 0.0474757
$$298$$ 0 0
$$299$$ −7392.00 −1.42973
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 1926.00 0.365168
$$304$$ 0 0
$$305$$ −10830.0 −2.03319
$$306$$ 0 0
$$307$$ −2780.00 −0.516818 −0.258409 0.966036i $$-0.583198\pi$$
−0.258409 + 0.966036i $$0.583198\pi$$
$$308$$ 0 0
$$309$$ −1392.00 −0.256272
$$310$$ 0 0
$$311$$ −4296.00 −0.783292 −0.391646 0.920116i $$-0.628094\pi$$
−0.391646 + 0.920116i $$0.628094\pi$$
$$312$$ 0 0
$$313$$ 5489.00 0.991235 0.495618 0.868541i $$-0.334942\pi$$
0.495618 + 0.868541i $$0.334942\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 4491.00 0.795709 0.397854 0.917449i $$-0.369755\pi$$
0.397854 + 0.917449i $$0.369755\pi$$
$$318$$ 0 0
$$319$$ 459.000 0.0805613
$$320$$ 0 0
$$321$$ −1179.00 −0.205001
$$322$$ 0 0
$$323$$ 8736.00 1.50490
$$324$$ 0 0
$$325$$ −8800.00 −1.50196
$$326$$ 0 0
$$327$$ 42.0000 0.00710277
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 3964.00 0.658251 0.329126 0.944286i $$-0.393246\pi$$
0.329126 + 0.944286i $$0.393246\pi$$
$$332$$ 0 0
$$333$$ 396.000 0.0651672
$$334$$ 0 0
$$335$$ −2490.00 −0.406099
$$336$$ 0 0
$$337$$ 161.000 0.0260244 0.0130122 0.999915i $$-0.495858\pi$$
0.0130122 + 0.999915i $$0.495858\pi$$
$$338$$ 0 0
$$339$$ −6552.00 −1.04972
$$340$$ 0 0
$$341$$ −1665.00 −0.264413
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −3780.00 −0.589879
$$346$$ 0 0
$$347$$ 5916.00 0.915238 0.457619 0.889148i $$-0.348702\pi$$
0.457619 + 0.889148i $$0.348702\pi$$
$$348$$ 0 0
$$349$$ −142.000 −0.0217796 −0.0108898 0.999941i $$-0.503466\pi$$
−0.0108898 + 0.999941i $$0.503466\pi$$
$$350$$ 0 0
$$351$$ −2376.00 −0.361315
$$352$$ 0 0
$$353$$ −4440.00 −0.669454 −0.334727 0.942315i $$-0.608644\pi$$
−0.334727 + 0.942315i $$0.608644\pi$$
$$354$$ 0 0
$$355$$ 16290.0 2.43545
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −2286.00 −0.336074 −0.168037 0.985781i $$-0.553743\pi$$
−0.168037 + 0.985781i $$0.553743\pi$$
$$360$$ 0 0
$$361$$ 3957.00 0.576906
$$362$$ 0 0
$$363$$ −3750.00 −0.542215
$$364$$ 0 0
$$365$$ −3270.00 −0.468930
$$366$$ 0 0
$$367$$ 2869.00 0.408067 0.204033 0.978964i $$-0.434595\pi$$
0.204033 + 0.978964i $$0.434595\pi$$
$$368$$ 0 0
$$369$$ −1512.00 −0.213311
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −3064.00 −0.425330 −0.212665 0.977125i $$-0.568214\pi$$
−0.212665 + 0.977125i $$0.568214\pi$$
$$374$$ 0 0
$$375$$ 1125.00 0.154919
$$376$$ 0 0
$$377$$ −4488.00 −0.613113
$$378$$ 0 0
$$379$$ 6040.00 0.818612 0.409306 0.912397i $$-0.365771\pi$$
0.409306 + 0.912397i $$0.365771\pi$$
$$380$$ 0 0
$$381$$ 1119.00 0.150467
$$382$$ 0 0
$$383$$ −1842.00 −0.245749 −0.122874 0.992422i $$-0.539211\pi$$
−0.122874 + 0.992422i $$0.539211\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −2934.00 −0.385384
$$388$$ 0 0
$$389$$ 7830.00 1.02056 0.510279 0.860009i $$-0.329542\pi$$
0.510279 + 0.860009i $$0.329542\pi$$
$$390$$ 0 0
$$391$$ −7056.00 −0.912627
$$392$$ 0 0
$$393$$ 3519.00 0.451680
$$394$$ 0 0
$$395$$ −8745.00 −1.11395
$$396$$ 0 0
$$397$$ −14764.0 −1.86646 −0.933229 0.359282i $$-0.883022\pi$$
−0.933229 + 0.359282i $$0.883022\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6264.00 0.780073 0.390036 0.920799i $$-0.372462\pi$$
0.390036 + 0.920799i $$0.372462\pi$$
$$402$$ 0 0
$$403$$ 16280.0 2.01232
$$404$$ 0 0
$$405$$ −1215.00 −0.149071
$$406$$ 0 0
$$407$$ 396.000 0.0482285
$$408$$ 0 0
$$409$$ 4751.00 0.574381 0.287191 0.957873i $$-0.407279\pi$$
0.287191 + 0.957873i $$0.407279\pi$$
$$410$$ 0 0
$$411$$ 90.0000 0.0108014
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −8955.00 −1.05924
$$416$$ 0 0
$$417$$ 246.000 0.0288889
$$418$$ 0 0
$$419$$ 4704.00 0.548462 0.274231 0.961664i $$-0.411577\pi$$
0.274231 + 0.961664i $$0.411577\pi$$
$$420$$ 0 0
$$421$$ −4474.00 −0.517932 −0.258966 0.965886i $$-0.583382\pi$$
−0.258966 + 0.965886i $$0.583382\pi$$
$$422$$ 0 0
$$423$$ 1242.00 0.142761
$$424$$ 0 0
$$425$$ −8400.00 −0.958729
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −2376.00 −0.267399
$$430$$ 0 0
$$431$$ 12804.0 1.43097 0.715484 0.698629i $$-0.246206\pi$$
0.715484 + 0.698629i $$0.246206\pi$$
$$432$$ 0 0
$$433$$ −5074.00 −0.563143 −0.281571 0.959540i $$-0.590856\pi$$
−0.281571 + 0.959540i $$0.590856\pi$$
$$434$$ 0 0
$$435$$ −2295.00 −0.252958
$$436$$ 0 0
$$437$$ −8736.00 −0.956292
$$438$$ 0 0
$$439$$ 1267.00 0.137746 0.0688731 0.997625i $$-0.478060\pi$$
0.0688731 + 0.997625i $$0.478060\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 6933.00 0.743559 0.371780 0.928321i $$-0.378748\pi$$
0.371780 + 0.928321i $$0.378748\pi$$
$$444$$ 0 0
$$445$$ 15570.0 1.65863
$$446$$ 0 0
$$447$$ −4302.00 −0.455207
$$448$$ 0 0
$$449$$ 11688.0 1.22849 0.614244 0.789116i $$-0.289461\pi$$
0.614244 + 0.789116i $$0.289461\pi$$
$$450$$ 0 0
$$451$$ −1512.00 −0.157865
$$452$$ 0 0
$$453$$ 8013.00 0.831090
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 551.000 0.0563998 0.0281999 0.999602i $$-0.491023\pi$$
0.0281999 + 0.999602i $$0.491023\pi$$
$$458$$ 0 0
$$459$$ −2268.00 −0.230634
$$460$$ 0 0
$$461$$ 13386.0 1.35238 0.676191 0.736726i $$-0.263629\pi$$
0.676191 + 0.736726i $$0.263629\pi$$
$$462$$ 0 0
$$463$$ 6376.00 0.639995 0.319998 0.947418i $$-0.396318\pi$$
0.319998 + 0.947418i $$0.396318\pi$$
$$464$$ 0 0
$$465$$ 8325.00 0.830242
$$466$$ 0 0
$$467$$ −5700.00 −0.564806 −0.282403 0.959296i $$-0.591132\pi$$
−0.282403 + 0.959296i $$0.591132\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 6756.00 0.660934
$$472$$ 0 0
$$473$$ −2934.00 −0.285212
$$474$$ 0 0
$$475$$ −10400.0 −1.00460
$$476$$ 0 0
$$477$$ 5751.00 0.552034
$$478$$ 0 0
$$479$$ −19794.0 −1.88812 −0.944062 0.329769i $$-0.893029\pi$$
−0.944062 + 0.329769i $$0.893029\pi$$
$$480$$ 0 0
$$481$$ −3872.00 −0.367044
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 2535.00 0.237337
$$486$$ 0 0
$$487$$ −15935.0 −1.48272 −0.741359 0.671109i $$-0.765818\pi$$
−0.741359 + 0.671109i $$0.765818\pi$$
$$488$$ 0 0
$$489$$ −5028.00 −0.464978
$$490$$ 0 0
$$491$$ −9963.00 −0.915731 −0.457865 0.889021i $$-0.651386\pi$$
−0.457865 + 0.889021i $$0.651386\pi$$
$$492$$ 0 0
$$493$$ −4284.00 −0.391362
$$494$$ 0 0
$$495$$ −1215.00 −0.110324
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −19142.0 −1.71726 −0.858631 0.512594i $$-0.828684\pi$$
−0.858631 + 0.512594i $$0.828684\pi$$
$$500$$ 0 0
$$501$$ −9090.00 −0.810601
$$502$$ 0 0
$$503$$ 12192.0 1.08074 0.540372 0.841426i $$-0.318283\pi$$
0.540372 + 0.841426i $$0.318283\pi$$
$$504$$ 0 0
$$505$$ −9630.00 −0.848573
$$506$$ 0 0
$$507$$ 16641.0 1.45770
$$508$$ 0 0
$$509$$ −19809.0 −1.72499 −0.862494 0.506068i $$-0.831098\pi$$
−0.862494 + 0.506068i $$0.831098\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −2808.00 −0.241669
$$514$$ 0 0
$$515$$ 6960.00 0.595523
$$516$$ 0 0
$$517$$ 1242.00 0.105654
$$518$$ 0 0
$$519$$ 10314.0 0.872321
$$520$$ 0 0
$$521$$ −1794.00 −0.150857 −0.0754286 0.997151i $$-0.524032\pi$$
−0.0754286 + 0.997151i $$0.524032\pi$$
$$522$$ 0 0
$$523$$ 6448.00 0.539104 0.269552 0.962986i $$-0.413124\pi$$
0.269552 + 0.962986i $$0.413124\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 15540.0 1.28450
$$528$$ 0 0
$$529$$ −5111.00 −0.420071
$$530$$ 0 0
$$531$$ −1431.00 −0.116949
$$532$$ 0 0
$$533$$ 14784.0 1.20144
$$534$$ 0 0
$$535$$ 5895.00 0.476380
$$536$$ 0 0
$$537$$ 3636.00 0.292188
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 7262.00 0.577112 0.288556 0.957463i $$-0.406825\pi$$
0.288556 + 0.957463i $$0.406825\pi$$
$$542$$ 0 0
$$543$$ 9096.00 0.718871
$$544$$ 0 0
$$545$$ −210.000 −0.0165053
$$546$$ 0 0
$$547$$ −14204.0 −1.11027 −0.555136 0.831759i $$-0.687334\pi$$
−0.555136 + 0.831759i $$0.687334\pi$$
$$548$$ 0 0
$$549$$ 6498.00 0.505151
$$550$$ 0 0
$$551$$ −5304.00 −0.410087
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −1980.00 −0.151435
$$556$$ 0 0
$$557$$ 15825.0 1.20382 0.601909 0.798565i $$-0.294407\pi$$
0.601909 + 0.798565i $$0.294407\pi$$
$$558$$ 0 0
$$559$$ 28688.0 2.17061
$$560$$ 0 0
$$561$$ −2268.00 −0.170686
$$562$$ 0 0
$$563$$ −1059.00 −0.0792745 −0.0396372 0.999214i $$-0.512620\pi$$
−0.0396372 + 0.999214i $$0.512620\pi$$
$$564$$ 0 0
$$565$$ 32760.0 2.43933
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 3960.00 0.291761 0.145880 0.989302i $$-0.453399\pi$$
0.145880 + 0.989302i $$0.453399\pi$$
$$570$$ 0 0
$$571$$ 2530.00 0.185424 0.0927121 0.995693i $$-0.470446\pi$$
0.0927121 + 0.995693i $$0.470446\pi$$
$$572$$ 0 0
$$573$$ −7560.00 −0.551175
$$574$$ 0 0
$$575$$ 8400.00 0.609225
$$576$$ 0 0
$$577$$ 11831.0 0.853607 0.426803 0.904344i $$-0.359640\pi$$
0.426803 + 0.904344i $$0.359640\pi$$
$$578$$ 0 0
$$579$$ 1095.00 0.0785952
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 5751.00 0.408546
$$584$$ 0 0
$$585$$ 11880.0 0.839620
$$586$$ 0 0
$$587$$ −4809.00 −0.338141 −0.169070 0.985604i $$-0.554077\pi$$
−0.169070 + 0.985604i $$0.554077\pi$$
$$588$$ 0 0
$$589$$ 19240.0 1.34596
$$590$$ 0 0
$$591$$ −4770.00 −0.331999
$$592$$ 0 0
$$593$$ 21804.0 1.50992 0.754960 0.655770i $$-0.227656\pi$$
0.754960 + 0.655770i $$0.227656\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 16140.0 1.10648
$$598$$ 0 0
$$599$$ −14166.0 −0.966289 −0.483144 0.875541i $$-0.660505\pi$$
−0.483144 + 0.875541i $$0.660505\pi$$
$$600$$ 0 0
$$601$$ 5891.00 0.399832 0.199916 0.979813i $$-0.435933\pi$$
0.199916 + 0.979813i $$0.435933\pi$$
$$602$$ 0 0
$$603$$ 1494.00 0.100896
$$604$$ 0 0
$$605$$ 18750.0 1.25999
$$606$$ 0 0
$$607$$ 2737.00 0.183017 0.0915086 0.995804i $$-0.470831\pi$$
0.0915086 + 0.995804i $$0.470831\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −12144.0 −0.804081
$$612$$ 0 0
$$613$$ −26188.0 −1.72549 −0.862743 0.505642i $$-0.831256\pi$$
−0.862743 + 0.505642i $$0.831256\pi$$
$$614$$ 0 0
$$615$$ 7560.00 0.495689
$$616$$ 0 0
$$617$$ −2358.00 −0.153857 −0.0769283 0.997037i $$-0.524511\pi$$
−0.0769283 + 0.997037i $$0.524511\pi$$
$$618$$ 0 0
$$619$$ −13766.0 −0.893865 −0.446932 0.894568i $$-0.647484\pi$$
−0.446932 + 0.894568i $$0.647484\pi$$
$$620$$ 0 0
$$621$$ 2268.00 0.146557
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −18125.0 −1.16000
$$626$$ 0 0
$$627$$ −2808.00 −0.178853
$$628$$ 0 0
$$629$$ −3696.00 −0.234291
$$630$$ 0 0
$$631$$ −21287.0 −1.34298 −0.671491 0.741012i $$-0.734346\pi$$
−0.671491 + 0.741012i $$0.734346\pi$$
$$632$$ 0 0
$$633$$ 16086.0 1.01005
$$634$$ 0 0
$$635$$ −5595.00 −0.349655
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −9774.00 −0.605091
$$640$$ 0 0
$$641$$ 21426.0 1.32024 0.660122 0.751159i $$-0.270505\pi$$
0.660122 + 0.751159i $$0.270505\pi$$
$$642$$ 0 0
$$643$$ −9962.00 −0.610984 −0.305492 0.952195i $$-0.598821\pi$$
−0.305492 + 0.952195i $$0.598821\pi$$
$$644$$ 0 0
$$645$$ 14670.0 0.895551
$$646$$ 0 0
$$647$$ 18174.0 1.10432 0.552159 0.833739i $$-0.313804\pi$$
0.552159 + 0.833739i $$0.313804\pi$$
$$648$$ 0 0
$$649$$ −1431.00 −0.0865511
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −19167.0 −1.14864 −0.574321 0.818630i $$-0.694734\pi$$
−0.574321 + 0.818630i $$0.694734\pi$$
$$654$$ 0 0
$$655$$ −17595.0 −1.04961
$$656$$ 0 0
$$657$$ 1962.00 0.116507
$$658$$ 0 0
$$659$$ −13080.0 −0.773178 −0.386589 0.922252i $$-0.626347\pi$$
−0.386589 + 0.922252i $$0.626347\pi$$
$$660$$ 0 0
$$661$$ −15190.0 −0.893831 −0.446916 0.894576i $$-0.647478\pi$$
−0.446916 + 0.894576i $$0.647478\pi$$
$$662$$ 0 0
$$663$$ 22176.0 1.29901
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 4284.00 0.248691
$$668$$ 0 0
$$669$$ 4719.00 0.272716
$$670$$ 0 0
$$671$$ 6498.00 0.373849
$$672$$ 0 0
$$673$$ 4397.00 0.251845 0.125923 0.992040i $$-0.459811\pi$$
0.125923 + 0.992040i $$0.459811\pi$$
$$674$$ 0 0
$$675$$ 2700.00 0.153960
$$676$$ 0 0
$$677$$ 4029.00 0.228725 0.114363 0.993439i $$-0.463517\pi$$
0.114363 + 0.993439i $$0.463517\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −2763.00 −0.155475
$$682$$ 0 0
$$683$$ −15021.0 −0.841526 −0.420763 0.907170i $$-0.638238\pi$$
−0.420763 + 0.907170i $$0.638238\pi$$
$$684$$ 0 0
$$685$$ −450.000 −0.0251002
$$686$$ 0 0
$$687$$ 12156.0 0.675081
$$688$$ 0 0
$$689$$ −56232.0 −3.10924
$$690$$ 0 0
$$691$$ 13984.0 0.769865 0.384932 0.922945i $$-0.374225\pi$$
0.384932 + 0.922945i $$0.374225\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −1230.00 −0.0671317
$$696$$ 0 0
$$697$$ 14112.0 0.766901
$$698$$ 0 0
$$699$$ −1404.00 −0.0759716
$$700$$ 0 0
$$701$$ −31053.0 −1.67312 −0.836559 0.547877i $$-0.815436\pi$$
−0.836559 + 0.547877i $$0.815436\pi$$
$$702$$ 0 0
$$703$$ −4576.00 −0.245501
$$704$$ 0 0
$$705$$ −6210.00 −0.331748
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 15086.0 0.799107 0.399553 0.916710i $$-0.369165\pi$$
0.399553 + 0.916710i $$0.369165\pi$$
$$710$$ 0 0
$$711$$ 5247.00 0.276762
$$712$$ 0 0
$$713$$ −15540.0 −0.816238
$$714$$ 0 0
$$715$$ 11880.0 0.621380
$$716$$ 0 0
$$717$$ −14796.0 −0.770665
$$718$$ 0 0
$$719$$ 6378.00 0.330820 0.165410 0.986225i $$-0.447105\pi$$
0.165410 + 0.986225i $$0.447105\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −4611.00 −0.237185
$$724$$ 0 0
$$725$$ 5100.00 0.261254
$$726$$ 0 0
$$727$$ 7363.00 0.375624 0.187812 0.982205i $$-0.439860\pi$$
0.187812 + 0.982205i $$0.439860\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 27384.0 1.38555
$$732$$ 0 0
$$733$$ 32810.0 1.65329 0.826647 0.562720i $$-0.190245\pi$$
0.826647 + 0.562720i $$0.190245\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1494.00 0.0746706
$$738$$ 0 0
$$739$$ 24034.0 1.19635 0.598177 0.801364i $$-0.295892\pi$$
0.598177 + 0.801364i $$0.295892\pi$$
$$740$$ 0 0
$$741$$ 27456.0 1.36116
$$742$$ 0 0
$$743$$ 8022.00 0.396095 0.198048 0.980192i $$-0.436540\pi$$
0.198048 + 0.980192i $$0.436540\pi$$
$$744$$ 0 0
$$745$$ 21510.0 1.05781
$$746$$ 0 0
$$747$$ 5373.00 0.263170
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −29519.0 −1.43431 −0.717153 0.696916i $$-0.754555\pi$$
−0.717153 + 0.696916i $$0.754555\pi$$
$$752$$ 0 0
$$753$$ −15957.0 −0.772252
$$754$$ 0 0
$$755$$ −40065.0 −1.93128
$$756$$ 0 0
$$757$$ −3742.00 −0.179664 −0.0898318 0.995957i $$-0.528633\pi$$
−0.0898318 + 0.995957i $$0.528633\pi$$
$$758$$ 0 0
$$759$$ 2268.00 0.108463
$$760$$ 0 0
$$761$$ −10896.0 −0.519027 −0.259514 0.965739i $$-0.583562\pi$$
−0.259514 + 0.965739i $$0.583562\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 11340.0 0.535946
$$766$$ 0 0
$$767$$ 13992.0 0.658699
$$768$$ 0 0
$$769$$ 17285.0 0.810550 0.405275 0.914195i $$-0.367176\pi$$
0.405275 + 0.914195i $$0.367176\pi$$
$$770$$ 0 0
$$771$$ 16038.0 0.749150
$$772$$ 0 0
$$773$$ 11826.0 0.550261 0.275130 0.961407i $$-0.411279\pi$$
0.275130 + 0.961407i $$0.411279\pi$$
$$774$$ 0 0
$$775$$ −18500.0 −0.857470
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 17472.0 0.803594
$$780$$ 0 0
$$781$$ −9774.00 −0.447812
$$782$$ 0 0
$$783$$ 1377.00 0.0628480
$$784$$ 0 0
$$785$$ −33780.0 −1.53587
$$786$$ 0 0
$$787$$ −17714.0 −0.802333 −0.401166 0.916005i $$-0.631395\pi$$
−0.401166 + 0.916005i $$0.631395\pi$$
$$788$$ 0 0
$$789$$ −2322.00 −0.104772
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −63536.0 −2.84518
$$794$$ 0 0
$$795$$ −28755.0 −1.28281
$$796$$ 0 0
$$797$$ −24939.0 −1.10839 −0.554194 0.832388i $$-0.686973\pi$$
−0.554194 + 0.832388i $$0.686973\pi$$
$$798$$ 0 0
$$799$$ −11592.0 −0.513261
$$800$$ 0 0
$$801$$ −9342.00 −0.412089
$$802$$ 0 0
$$803$$ 1962.00 0.0862235
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 7245.00 0.316030
$$808$$ 0 0
$$809$$ −29064.0 −1.26309 −0.631543 0.775341i $$-0.717578\pi$$
−0.631543 + 0.775341i $$0.717578\pi$$
$$810$$ 0 0
$$811$$ 15370.0 0.665492 0.332746 0.943017i $$-0.392025\pi$$
0.332746 + 0.943017i $$0.392025\pi$$
$$812$$ 0 0
$$813$$ 1425.00 0.0614722
$$814$$ 0 0
$$815$$ 25140.0 1.08051
$$816$$ 0 0
$$817$$ 33904.0 1.45184
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 44031.0 1.87173 0.935866 0.352355i $$-0.114619\pi$$
0.935866 + 0.352355i $$0.114619\pi$$
$$822$$ 0 0
$$823$$ 4192.00 0.177550 0.0887752 0.996052i $$-0.471705\pi$$
0.0887752 + 0.996052i $$0.471705\pi$$
$$824$$ 0 0
$$825$$ 2700.00 0.113942
$$826$$ 0 0
$$827$$ −33195.0 −1.39577 −0.697886 0.716209i $$-0.745876\pi$$
−0.697886 + 0.716209i $$0.745876\pi$$
$$828$$ 0 0
$$829$$ 16448.0 0.689098 0.344549 0.938768i $$-0.388032\pi$$
0.344549 + 0.938768i $$0.388032\pi$$
$$830$$ 0 0
$$831$$ 11184.0 0.466870
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 45450.0 1.88367
$$836$$ 0 0
$$837$$ −4995.00 −0.206275
$$838$$ 0 0
$$839$$ −16860.0 −0.693769 −0.346884 0.937908i $$-0.612760\pi$$
−0.346884 + 0.937908i $$0.612760\pi$$
$$840$$ 0 0
$$841$$ −21788.0 −0.893354
$$842$$ 0 0
$$843$$ 4806.00 0.196355
$$844$$ 0 0
$$845$$ −83205.0 −3.38738
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −2058.00 −0.0831924
$$850$$ 0 0
$$851$$ 3696.00 0.148880
$$852$$ 0 0
$$853$$ 29054.0 1.16623 0.583113 0.812391i $$-0.301835\pi$$
0.583113 + 0.812391i $$0.301835\pi$$
$$854$$ 0 0
$$855$$ 14040.0 0.561588
$$856$$ 0 0
$$857$$ 41958.0 1.67241 0.836207 0.548415i $$-0.184768\pi$$
0.836207 + 0.548415i $$0.184768\pi$$
$$858$$ 0 0
$$859$$ −5546.00 −0.220288 −0.110144 0.993916i $$-0.535131\pi$$
−0.110144 + 0.993916i $$0.535131\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 32538.0 1.28344 0.641719 0.766940i $$-0.278222\pi$$
0.641719 + 0.766940i $$0.278222\pi$$
$$864$$ 0 0
$$865$$ −51570.0 −2.02709
$$866$$ 0 0
$$867$$ 6429.00 0.251834
$$868$$ 0 0
$$869$$ 5247.00 0.204824
$$870$$ 0 0
$$871$$ −14608.0 −0.568282
$$872$$ 0 0
$$873$$ −1521.00 −0.0589668
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 32096.0 1.23581 0.617905 0.786253i $$-0.287982\pi$$
0.617905 + 0.786253i $$0.287982\pi$$
$$878$$ 0 0
$$879$$ −3303.00 −0.126743
$$880$$ 0 0
$$881$$ −8490.00 −0.324671 −0.162336 0.986736i $$-0.551903\pi$$
−0.162336 + 0.986736i $$0.551903\pi$$
$$882$$ 0 0
$$883$$ 48352.0 1.84278 0.921390 0.388640i $$-0.127055\pi$$
0.921390 + 0.388640i $$0.127055\pi$$
$$884$$ 0 0
$$885$$ 7155.00 0.271766
$$886$$ 0 0
$$887$$ −15492.0 −0.586438 −0.293219 0.956045i $$-0.594726\pi$$
−0.293219 + 0.956045i $$0.594726\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 729.000 0.0274101
$$892$$ 0 0
$$893$$ −14352.0 −0.537818
$$894$$ 0 0
$$895$$ −18180.0 −0.678984
$$896$$ 0 0
$$897$$ −22176.0 −0.825457
$$898$$ 0 0
$$899$$ −9435.00 −0.350028
$$900$$ 0 0
$$901$$ −53676.0 −1.98469
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −45480.0 −1.67050
$$906$$ 0 0
$$907$$ 8116.00 0.297119 0.148560 0.988903i $$-0.452536\pi$$
0.148560 + 0.988903i $$0.452536\pi$$
$$908$$ 0 0
$$909$$ 5778.00 0.210830
$$910$$ 0 0
$$911$$ 4446.00 0.161693 0.0808466 0.996727i $$-0.474238\pi$$
0.0808466 + 0.996727i $$0.474238\pi$$
$$912$$ 0 0
$$913$$ 5373.00 0.194765
$$914$$ 0 0
$$915$$ −32490.0 −1.17386
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −26504.0 −0.951345 −0.475673 0.879622i $$-0.657795\pi$$
−0.475673 + 0.879622i $$0.657795\pi$$
$$920$$ 0 0
$$921$$ −8340.00 −0.298385
$$922$$ 0 0
$$923$$ 95568.0 3.40808
$$924$$ 0 0
$$925$$ 4400.00 0.156401
$$926$$ 0 0
$$927$$ −4176.00 −0.147959
$$928$$ 0 0
$$929$$ −5430.00 −0.191768 −0.0958840 0.995393i $$-0.530568\pi$$
−0.0958840 + 0.995393i $$0.530568\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −12888.0 −0.452234
$$934$$ 0 0
$$935$$ 11340.0 0.396639
$$936$$ 0 0
$$937$$ 33803.0 1.17854 0.589272 0.807935i $$-0.299415\pi$$
0.589272 + 0.807935i $$0.299415\pi$$
$$938$$ 0 0
$$939$$ 16467.0 0.572290
$$940$$ 0 0
$$941$$ −48483.0 −1.67960 −0.839798 0.542898i $$-0.817327\pi$$
−0.839798 + 0.542898i $$0.817327\pi$$
$$942$$ 0 0
$$943$$ −14112.0 −0.487328
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −37296.0 −1.27979 −0.639893 0.768464i $$-0.721021\pi$$
−0.639893 + 0.768464i $$0.721021\pi$$
$$948$$ 0 0
$$949$$ −19184.0 −0.656205
$$950$$ 0 0
$$951$$ 13473.0 0.459403
$$952$$ 0 0
$$953$$ −38478.0 −1.30790 −0.653948 0.756540i $$-0.726888\pi$$
−0.653948 + 0.756540i $$0.726888\pi$$
$$954$$ 0 0
$$955$$ 37800.0 1.28082
$$956$$ 0 0
$$957$$ 1377.00 0.0465121
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 4434.00 0.148837
$$962$$ 0 0
$$963$$ −3537.00 −0.118357
$$964$$ 0 0
$$965$$ −5475.00 −0.182639
$$966$$ 0 0
$$967$$ −27257.0 −0.906438 −0.453219 0.891399i $$-0.649725\pi$$
−0.453219 + 0.891399i $$0.649725\pi$$
$$968$$ 0 0
$$969$$ 26208.0 0.868857
$$970$$ 0 0
$$971$$ −34341.0 −1.13497 −0.567485 0.823384i $$-0.692083\pi$$
−0.567485 + 0.823384i $$0.692083\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −26400.0 −0.867156
$$976$$ 0 0
$$977$$ 27426.0 0.898092 0.449046 0.893509i $$-0.351764\pi$$
0.449046 + 0.893509i $$0.351764\pi$$
$$978$$ 0 0
$$979$$ −9342.00 −0.304976
$$980$$ 0 0
$$981$$ 126.000 0.00410079
$$982$$ 0 0
$$983$$ −12324.0 −0.399872 −0.199936 0.979809i $$-0.564073\pi$$
−0.199936 + 0.979809i $$0.564073\pi$$
$$984$$ 0 0
$$985$$ 23850.0 0.771497
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −27384.0 −0.880445
$$990$$ 0 0
$$991$$ −47597.0 −1.52570 −0.762850 0.646576i $$-0.776200\pi$$
−0.762850 + 0.646576i $$0.776200\pi$$
$$992$$ 0 0
$$993$$ 11892.0 0.380042
$$994$$ 0 0
$$995$$ −80700.0 −2.57122
$$996$$ 0 0
$$997$$ −11242.0 −0.357109 −0.178555 0.983930i $$-0.557142\pi$$
−0.178555 + 0.983930i $$0.557142\pi$$
$$998$$ 0 0
$$999$$ 1188.00 0.0376243
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 2352.4.a.u.1.1 1
4.3 odd 2 294.4.a.a.1.1 1
7.2 even 3 336.4.q.d.193.1 2
7.4 even 3 336.4.q.d.289.1 2
7.6 odd 2 2352.4.a.q.1.1 1
12.11 even 2 882.4.a.r.1.1 1
28.3 even 6 294.4.e.e.79.1 2
28.11 odd 6 42.4.e.b.37.1 yes 2
28.19 even 6 294.4.e.e.67.1 2
28.23 odd 6 42.4.e.b.25.1 2
28.27 even 2 294.4.a.g.1.1 1
84.11 even 6 126.4.g.a.37.1 2
84.23 even 6 126.4.g.a.109.1 2
84.47 odd 6 882.4.g.l.361.1 2
84.59 odd 6 882.4.g.l.667.1 2
84.83 odd 2 882.4.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
42.4.e.b.25.1 2 28.23 odd 6
42.4.e.b.37.1 yes 2 28.11 odd 6
126.4.g.a.37.1 2 84.11 even 6
126.4.g.a.109.1 2 84.23 even 6
294.4.a.a.1.1 1 4.3 odd 2
294.4.a.g.1.1 1 28.27 even 2
294.4.e.e.67.1 2 28.19 even 6
294.4.e.e.79.1 2 28.3 even 6
336.4.q.d.193.1 2 7.2 even 3
336.4.q.d.289.1 2 7.4 even 3
882.4.a.h.1.1 1 84.83 odd 2
882.4.a.r.1.1 1 12.11 even 2
882.4.g.l.361.1 2 84.47 odd 6
882.4.g.l.667.1 2 84.59 odd 6
2352.4.a.q.1.1 1 7.6 odd 2
2352.4.a.u.1.1 1 1.1 even 1 trivial