# Properties

 Label 225.4.a.b.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 5) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-4.00000 q^{2} +8.00000 q^{4} -6.00000 q^{7} +O(q^{10})$$ $$q-4.00000 q^{2} +8.00000 q^{4} -6.00000 q^{7} -32.0000 q^{11} +38.0000 q^{13} +24.0000 q^{14} -64.0000 q^{16} +26.0000 q^{17} +100.000 q^{19} +128.000 q^{22} -78.0000 q^{23} -152.000 q^{26} -48.0000 q^{28} +50.0000 q^{29} -108.000 q^{31} +256.000 q^{32} -104.000 q^{34} -266.000 q^{37} -400.000 q^{38} -22.0000 q^{41} -442.000 q^{43} -256.000 q^{44} +312.000 q^{46} -514.000 q^{47} -307.000 q^{49} +304.000 q^{52} +2.00000 q^{53} -200.000 q^{58} -500.000 q^{59} -518.000 q^{61} +432.000 q^{62} -512.000 q^{64} -126.000 q^{67} +208.000 q^{68} -412.000 q^{71} +878.000 q^{73} +1064.00 q^{74} +800.000 q^{76} +192.000 q^{77} +600.000 q^{79} +88.0000 q^{82} +282.000 q^{83} +1768.00 q^{86} +150.000 q^{89} -228.000 q^{91} -624.000 q^{92} +2056.00 q^{94} -386.000 q^{97} +1228.00 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 0 0
$$4$$ 8.00000 1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −6.00000 −0.323970 −0.161985 0.986793i $$-0.551790\pi$$
−0.161985 + 0.986793i $$0.551790\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −32.0000 −0.877124 −0.438562 0.898701i $$-0.644512\pi$$
−0.438562 + 0.898701i $$0.644512\pi$$
$$12$$ 0 0
$$13$$ 38.0000 0.810716 0.405358 0.914158i $$-0.367147\pi$$
0.405358 + 0.914158i $$0.367147\pi$$
$$14$$ 24.0000 0.458162
$$15$$ 0 0
$$16$$ −64.0000 −1.00000
$$17$$ 26.0000 0.370937 0.185468 0.982650i $$-0.440620\pi$$
0.185468 + 0.982650i $$0.440620\pi$$
$$18$$ 0 0
$$19$$ 100.000 1.20745 0.603726 0.797192i $$-0.293682\pi$$
0.603726 + 0.797192i $$0.293682\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 128.000 1.24044
$$23$$ −78.0000 −0.707136 −0.353568 0.935409i $$-0.615032\pi$$
−0.353568 + 0.935409i $$0.615032\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −152.000 −1.14653
$$27$$ 0 0
$$28$$ −48.0000 −0.323970
$$29$$ 50.0000 0.320164 0.160082 0.987104i $$-0.448824\pi$$
0.160082 + 0.987104i $$0.448824\pi$$
$$30$$ 0 0
$$31$$ −108.000 −0.625722 −0.312861 0.949799i $$-0.601287\pi$$
−0.312861 + 0.949799i $$0.601287\pi$$
$$32$$ 256.000 1.41421
$$33$$ 0 0
$$34$$ −104.000 −0.524584
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −266.000 −1.18190 −0.590948 0.806710i $$-0.701246\pi$$
−0.590948 + 0.806710i $$0.701246\pi$$
$$38$$ −400.000 −1.70759
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −22.0000 −0.0838006 −0.0419003 0.999122i $$-0.513341\pi$$
−0.0419003 + 0.999122i $$0.513341\pi$$
$$42$$ 0 0
$$43$$ −442.000 −1.56754 −0.783772 0.621049i $$-0.786707\pi$$
−0.783772 + 0.621049i $$0.786707\pi$$
$$44$$ −256.000 −0.877124
$$45$$ 0 0
$$46$$ 312.000 1.00004
$$47$$ −514.000 −1.59520 −0.797602 0.603184i $$-0.793899\pi$$
−0.797602 + 0.603184i $$0.793899\pi$$
$$48$$ 0 0
$$49$$ −307.000 −0.895044
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 304.000 0.810716
$$53$$ 2.00000 0.00518342 0.00259171 0.999997i $$-0.499175\pi$$
0.00259171 + 0.999997i $$0.499175\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −200.000 −0.452781
$$59$$ −500.000 −1.10330 −0.551648 0.834077i $$-0.686001\pi$$
−0.551648 + 0.834077i $$0.686001\pi$$
$$60$$ 0 0
$$61$$ −518.000 −1.08726 −0.543632 0.839324i $$-0.682951\pi$$
−0.543632 + 0.839324i $$0.682951\pi$$
$$62$$ 432.000 0.884904
$$63$$ 0 0
$$64$$ −512.000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −126.000 −0.229751 −0.114876 0.993380i $$-0.536647\pi$$
−0.114876 + 0.993380i $$0.536647\pi$$
$$68$$ 208.000 0.370937
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −412.000 −0.688668 −0.344334 0.938847i $$-0.611895\pi$$
−0.344334 + 0.938847i $$0.611895\pi$$
$$72$$ 0 0
$$73$$ 878.000 1.40770 0.703850 0.710348i $$-0.251463\pi$$
0.703850 + 0.710348i $$0.251463\pi$$
$$74$$ 1064.00 1.67145
$$75$$ 0 0
$$76$$ 800.000 1.20745
$$77$$ 192.000 0.284161
$$78$$ 0 0
$$79$$ 600.000 0.854497 0.427249 0.904134i $$-0.359483\pi$$
0.427249 + 0.904134i $$0.359483\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 88.0000 0.118512
$$83$$ 282.000 0.372934 0.186467 0.982461i $$-0.440296\pi$$
0.186467 + 0.982461i $$0.440296\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1768.00 2.21684
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 150.000 0.178651 0.0893257 0.996002i $$-0.471529\pi$$
0.0893257 + 0.996002i $$0.471529\pi$$
$$90$$ 0 0
$$91$$ −228.000 −0.262647
$$92$$ −624.000 −0.707136
$$93$$ 0 0
$$94$$ 2056.00 2.25596
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −386.000 −0.404045 −0.202022 0.979381i $$-0.564751\pi$$
−0.202022 + 0.979381i $$0.564751\pi$$
$$98$$ 1228.00 1.26578
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −702.000 −0.691600 −0.345800 0.938308i $$-0.612392\pi$$
−0.345800 + 0.938308i $$0.612392\pi$$
$$102$$ 0 0
$$103$$ 598.000 0.572065 0.286032 0.958220i $$-0.407663\pi$$
0.286032 + 0.958220i $$0.407663\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −8.00000 −0.00733046
$$107$$ −1194.00 −1.07877 −0.539385 0.842059i $$-0.681343\pi$$
−0.539385 + 0.842059i $$0.681343\pi$$
$$108$$ 0 0
$$109$$ −550.000 −0.483307 −0.241653 0.970363i $$-0.577690\pi$$
−0.241653 + 0.970363i $$0.577690\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 384.000 0.323970
$$113$$ 1562.00 1.30036 0.650180 0.759781i $$-0.274694\pi$$
0.650180 + 0.759781i $$0.274694\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 400.000 0.320164
$$117$$ 0 0
$$118$$ 2000.00 1.56030
$$119$$ −156.000 −0.120172
$$120$$ 0 0
$$121$$ −307.000 −0.230654
$$122$$ 2072.00 1.53762
$$123$$ 0 0
$$124$$ −864.000 −0.625722
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −1846.00 −1.28981 −0.644906 0.764262i $$-0.723103\pi$$
−0.644906 + 0.764262i $$0.723103\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 2208.00 1.47262 0.736312 0.676642i $$-0.236565\pi$$
0.736312 + 0.676642i $$0.236565\pi$$
$$132$$ 0 0
$$133$$ −600.000 −0.391177
$$134$$ 504.000 0.324918
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2334.00 −1.45553 −0.727763 0.685829i $$-0.759440\pi$$
−0.727763 + 0.685829i $$0.759440\pi$$
$$138$$ 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
$$142$$ 1648.00 0.973923
$$143$$ −1216.00 −0.711098
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −3512.00 −1.99079
$$147$$ 0 0
$$148$$ −2128.00 −1.18190
$$149$$ −2050.00 −1.12713 −0.563566 0.826071i $$-0.690571\pi$$
−0.563566 + 0.826071i $$0.690571\pi$$
$$150$$ 0 0
$$151$$ 1852.00 0.998103 0.499052 0.866572i $$-0.333682\pi$$
0.499052 + 0.866572i $$0.333682\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ −768.000 −0.401865
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2494.00 1.26779 0.633894 0.773420i $$-0.281455\pi$$
0.633894 + 0.773420i $$0.281455\pi$$
$$158$$ −2400.00 −1.20844
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 468.000 0.229090
$$162$$ 0 0
$$163$$ −2762.00 −1.32722 −0.663609 0.748080i $$-0.730976\pi$$
−0.663609 + 0.748080i $$0.730976\pi$$
$$164$$ −176.000 −0.0838006
$$165$$ 0 0
$$166$$ −1128.00 −0.527408
$$167$$ 3126.00 1.44849 0.724243 0.689545i $$-0.242189\pi$$
0.724243 + 0.689545i $$0.242189\pi$$
$$168$$ 0 0
$$169$$ −753.000 −0.342740
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −3536.00 −1.56754
$$173$$ −78.0000 −0.0342788 −0.0171394 0.999853i $$-0.505456\pi$$
−0.0171394 + 0.999853i $$0.505456\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 2048.00 0.877124
$$177$$ 0 0
$$178$$ −600.000 −0.252651
$$179$$ 1300.00 0.542830 0.271415 0.962462i $$-0.412508\pi$$
0.271415 + 0.962462i $$0.412508\pi$$
$$180$$ 0 0
$$181$$ 1742.00 0.715369 0.357685 0.933842i $$-0.383566\pi$$
0.357685 + 0.933842i $$0.383566\pi$$
$$182$$ 912.000 0.371439
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −832.000 −0.325358
$$188$$ −4112.00 −1.59520
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3772.00 −1.42897 −0.714483 0.699653i $$-0.753338\pi$$
−0.714483 + 0.699653i $$0.753338\pi$$
$$192$$ 0 0
$$193$$ 358.000 0.133520 0.0667601 0.997769i $$-0.478734\pi$$
0.0667601 + 0.997769i $$0.478734\pi$$
$$194$$ 1544.00 0.571406
$$195$$ 0 0
$$196$$ −2456.00 −0.895044
$$197$$ −2214.00 −0.800716 −0.400358 0.916359i $$-0.631114\pi$$
−0.400358 + 0.916359i $$0.631114\pi$$
$$198$$ 0 0
$$199$$ −2600.00 −0.926176 −0.463088 0.886312i $$-0.653259\pi$$
−0.463088 + 0.886312i $$0.653259\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 2808.00 0.978070
$$203$$ −300.000 −0.103724
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −2392.00 −0.809022
$$207$$ 0 0
$$208$$ −2432.00 −0.810716
$$209$$ −3200.00 −1.05908
$$210$$ 0 0
$$211$$ −1168.00 −0.381083 −0.190541 0.981679i $$-0.561024\pi$$
−0.190541 + 0.981679i $$0.561024\pi$$
$$212$$ 16.0000 0.00518342
$$213$$ 0 0
$$214$$ 4776.00 1.52561
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 648.000 0.202715
$$218$$ 2200.00 0.683499
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 988.000 0.300724
$$222$$ 0 0
$$223$$ 6478.00 1.94529 0.972643 0.232303i $$-0.0746262\pi$$
0.972643 + 0.232303i $$0.0746262\pi$$
$$224$$ −1536.00 −0.458162
$$225$$ 0 0
$$226$$ −6248.00 −1.83899
$$227$$ 646.000 0.188883 0.0944417 0.995530i $$-0.469893\pi$$
0.0944417 + 0.995530i $$0.469893\pi$$
$$228$$ 0 0
$$229$$ 3750.00 1.08213 0.541063 0.840982i $$-0.318022\pi$$
0.541063 + 0.840982i $$0.318022\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 1482.00 0.416691 0.208346 0.978055i $$-0.433192\pi$$
0.208346 + 0.978055i $$0.433192\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −4000.00 −1.10330
$$237$$ 0 0
$$238$$ 624.000 0.169949
$$239$$ −1400.00 −0.378906 −0.189453 0.981890i $$-0.560671\pi$$
−0.189453 + 0.981890i $$0.560671\pi$$
$$240$$ 0 0
$$241$$ 3022.00 0.807735 0.403867 0.914817i $$-0.367666\pi$$
0.403867 + 0.914817i $$0.367666\pi$$
$$242$$ 1228.00 0.326194
$$243$$ 0 0
$$244$$ −4144.00 −1.08726
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3800.00 0.978900
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 1248.00 0.313837 0.156918 0.987612i $$-0.449844\pi$$
0.156918 + 0.987612i $$0.449844\pi$$
$$252$$ 0 0
$$253$$ 2496.00 0.620246
$$254$$ 7384.00 1.82407
$$255$$ 0 0
$$256$$ 4096.00 1.00000
$$257$$ 2106.00 0.511162 0.255581 0.966788i $$-0.417733\pi$$
0.255581 + 0.966788i $$0.417733\pi$$
$$258$$ 0 0
$$259$$ 1596.00 0.382898
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −8832.00 −2.08261
$$263$$ −3638.00 −0.852961 −0.426480 0.904497i $$-0.640247\pi$$
−0.426480 + 0.904497i $$0.640247\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 2400.00 0.553208
$$267$$ 0 0
$$268$$ −1008.00 −0.229751
$$269$$ 6550.00 1.48461 0.742306 0.670061i $$-0.233732\pi$$
0.742306 + 0.670061i $$0.233732\pi$$
$$270$$ 0 0
$$271$$ −4388.00 −0.983587 −0.491793 0.870712i $$-0.663658\pi$$
−0.491793 + 0.870712i $$0.663658\pi$$
$$272$$ −1664.00 −0.370937
$$273$$ 0 0
$$274$$ 9336.00 2.05842
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −546.000 −0.118433 −0.0592165 0.998245i $$-0.518860\pi$$
−0.0592165 + 0.998245i $$0.518860\pi$$
$$278$$ 2800.00 0.604075
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 6858.00 1.45592 0.727961 0.685619i $$-0.240468\pi$$
0.727961 + 0.685619i $$0.240468\pi$$
$$282$$ 0 0
$$283$$ −9282.00 −1.94967 −0.974837 0.222920i $$-0.928441\pi$$
−0.974837 + 0.222920i $$0.928441\pi$$
$$284$$ −3296.00 −0.688668
$$285$$ 0 0
$$286$$ 4864.00 1.00564
$$287$$ 132.000 0.0271488
$$288$$ 0 0
$$289$$ −4237.00 −0.862406
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 7024.00 1.40770
$$293$$ 4842.00 0.965436 0.482718 0.875776i $$-0.339650\pi$$
0.482718 + 0.875776i $$0.339650\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 8200.00 1.59400
$$299$$ −2964.00 −0.573286
$$300$$ 0 0
$$301$$ 2652.00 0.507836
$$302$$ −7408.00 −1.41153
$$303$$ 0 0
$$304$$ −6400.00 −1.20745
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2594.00 0.482239 0.241120 0.970495i $$-0.422485\pi$$
0.241120 + 0.970495i $$0.422485\pi$$
$$308$$ 1536.00 0.284161
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −7332.00 −1.33685 −0.668424 0.743781i $$-0.733031\pi$$
−0.668424 + 0.743781i $$0.733031\pi$$
$$312$$ 0 0
$$313$$ −1562.00 −0.282075 −0.141037 0.990004i $$-0.545044\pi$$
−0.141037 + 0.990004i $$0.545044\pi$$
$$314$$ −9976.00 −1.79292
$$315$$ 0 0
$$316$$ 4800.00 0.854497
$$317$$ 1426.00 0.252657 0.126328 0.991988i $$-0.459681\pi$$
0.126328 + 0.991988i $$0.459681\pi$$
$$318$$ 0 0
$$319$$ −1600.00 −0.280824
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −1872.00 −0.323983
$$323$$ 2600.00 0.447888
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 11048.0 1.87697
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 3084.00 0.516798
$$330$$ 0 0
$$331$$ −4008.00 −0.665558 −0.332779 0.943005i $$-0.607986\pi$$
−0.332779 + 0.943005i $$0.607986\pi$$
$$332$$ 2256.00 0.372934
$$333$$ 0 0
$$334$$ −12504.0 −2.04847
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −8866.00 −1.43312 −0.716561 0.697525i $$-0.754285\pi$$
−0.716561 + 0.697525i $$0.754285\pi$$
$$338$$ 3012.00 0.484708
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 3456.00 0.548835
$$342$$ 0 0
$$343$$ 3900.00 0.613936
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 312.000 0.0484775
$$347$$ −1714.00 −0.265165 −0.132583 0.991172i $$-0.542327\pi$$
−0.132583 + 0.991172i $$0.542327\pi$$
$$348$$ 0 0
$$349$$ 1150.00 0.176384 0.0881921 0.996103i $$-0.471891\pi$$
0.0881921 + 0.996103i $$0.471891\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −8192.00 −1.24044
$$353$$ −4398.00 −0.663122 −0.331561 0.943434i $$-0.607575\pi$$
−0.331561 + 0.943434i $$0.607575\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 1200.00 0.178651
$$357$$ 0 0
$$358$$ −5200.00 −0.767677
$$359$$ −1800.00 −0.264625 −0.132312 0.991208i $$-0.542240\pi$$
−0.132312 + 0.991208i $$0.542240\pi$$
$$360$$ 0 0
$$361$$ 3141.00 0.457938
$$362$$ −6968.00 −1.01168
$$363$$ 0 0
$$364$$ −1824.00 −0.262647
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 5874.00 0.835478 0.417739 0.908567i $$-0.362823\pi$$
0.417739 + 0.908567i $$0.362823\pi$$
$$368$$ 4992.00 0.707136
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −12.0000 −0.00167927
$$372$$ 0 0
$$373$$ 2078.00 0.288458 0.144229 0.989544i $$-0.453930\pi$$
0.144229 + 0.989544i $$0.453930\pi$$
$$374$$ 3328.00 0.460125
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1900.00 0.259562
$$378$$ 0 0
$$379$$ 7900.00 1.07070 0.535351 0.844630i $$-0.320179\pi$$
0.535351 + 0.844630i $$0.320179\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 15088.0 2.02086
$$383$$ −7518.00 −1.00301 −0.501504 0.865155i $$-0.667220\pi$$
−0.501504 + 0.865155i $$0.667220\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1432.00 −0.188826
$$387$$ 0 0
$$388$$ −3088.00 −0.404045
$$389$$ 1950.00 0.254162 0.127081 0.991892i $$-0.459439\pi$$
0.127081 + 0.991892i $$0.459439\pi$$
$$390$$ 0 0
$$391$$ −2028.00 −0.262303
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 8856.00 1.13238
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −13786.0 −1.74282 −0.871410 0.490555i $$-0.836794\pi$$
−0.871410 + 0.490555i $$0.836794\pi$$
$$398$$ 10400.0 1.30981
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6402.00 −0.797258 −0.398629 0.917112i $$-0.630514\pi$$
−0.398629 + 0.917112i $$0.630514\pi$$
$$402$$ 0 0
$$403$$ −4104.00 −0.507282
$$404$$ −5616.00 −0.691600
$$405$$ 0 0
$$406$$ 1200.00 0.146687
$$407$$ 8512.00 1.03667
$$408$$ 0 0
$$409$$ 11150.0 1.34800 0.674000 0.738731i $$-0.264575\pi$$
0.674000 + 0.738731i $$0.264575\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 4784.00 0.572065
$$413$$ 3000.00 0.357434
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 9728.00 1.14653
$$417$$ 0 0
$$418$$ 12800.0 1.49777
$$419$$ 13700.0 1.59735 0.798674 0.601764i $$-0.205535\pi$$
0.798674 + 0.601764i $$0.205535\pi$$
$$420$$ 0 0
$$421$$ −5438.00 −0.629529 −0.314765 0.949170i $$-0.601926\pi$$
−0.314765 + 0.949170i $$0.601926\pi$$
$$422$$ 4672.00 0.538932
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3108.00 0.352240
$$428$$ −9552.00 −1.07877
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −7692.00 −0.859653 −0.429827 0.902911i $$-0.641425\pi$$
−0.429827 + 0.902911i $$0.641425\pi$$
$$432$$ 0 0
$$433$$ 1118.00 0.124082 0.0620412 0.998074i $$-0.480239\pi$$
0.0620412 + 0.998074i $$0.480239\pi$$
$$434$$ −2592.00 −0.286682
$$435$$ 0 0
$$436$$ −4400.00 −0.483307
$$437$$ −7800.00 −0.853832
$$438$$ 0 0
$$439$$ −2600.00 −0.282668 −0.141334 0.989962i $$-0.545139\pi$$
−0.141334 + 0.989962i $$0.545139\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −3952.00 −0.425288
$$443$$ −11958.0 −1.28249 −0.641243 0.767337i $$-0.721581\pi$$
−0.641243 + 0.767337i $$0.721581\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −25912.0 −2.75105
$$447$$ 0 0
$$448$$ 3072.00 0.323970
$$449$$ 17050.0 1.79207 0.896035 0.443984i $$-0.146435\pi$$
0.896035 + 0.443984i $$0.146435\pi$$
$$450$$ 0 0
$$451$$ 704.000 0.0735035
$$452$$ 12496.0 1.30036
$$453$$ 0 0
$$454$$ −2584.00 −0.267121
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 9494.00 0.971796 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$458$$ −15000.0 −1.53036
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 11418.0 1.15356 0.576778 0.816901i $$-0.304310\pi$$
0.576778 + 0.816901i $$0.304310\pi$$
$$462$$ 0 0
$$463$$ −7962.00 −0.799191 −0.399596 0.916692i $$-0.630849\pi$$
−0.399596 + 0.916692i $$0.630849\pi$$
$$464$$ −3200.00 −0.320164
$$465$$ 0 0
$$466$$ −5928.00 −0.589290
$$467$$ 6526.00 0.646654 0.323327 0.946287i $$-0.395199\pi$$
0.323327 + 0.946287i $$0.395199\pi$$
$$468$$ 0 0
$$469$$ 756.000 0.0744325
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 14144.0 1.37493
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −1248.00 −0.120172
$$477$$ 0 0
$$478$$ 5600.00 0.535854
$$479$$ −17400.0 −1.65976 −0.829881 0.557940i $$-0.811592\pi$$
−0.829881 + 0.557940i $$0.811592\pi$$
$$480$$ 0 0
$$481$$ −10108.0 −0.958181
$$482$$ −12088.0 −1.14231
$$483$$ 0 0
$$484$$ −2456.00 −0.230654
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −1166.00 −0.108494 −0.0542469 0.998528i $$-0.517276\pi$$
−0.0542469 + 0.998528i $$0.517276\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −7072.00 −0.650010 −0.325005 0.945712i $$-0.605366\pi$$
−0.325005 + 0.945712i $$0.605366\pi$$
$$492$$ 0 0
$$493$$ 1300.00 0.118761
$$494$$ −15200.0 −1.38437
$$495$$ 0 0
$$496$$ 6912.00 0.625722
$$497$$ 2472.00 0.223107
$$498$$ 0 0
$$499$$ 100.000 0.00897117 0.00448559 0.999990i $$-0.498572\pi$$
0.00448559 + 0.999990i $$0.498572\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −4992.00 −0.443832
$$503$$ 2602.00 0.230651 0.115325 0.993328i $$-0.463209\pi$$
0.115325 + 0.993328i $$0.463209\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −9984.00 −0.877160
$$507$$ 0 0
$$508$$ −14768.0 −1.28981
$$509$$ −11150.0 −0.970953 −0.485476 0.874250i $$-0.661354\pi$$
−0.485476 + 0.874250i $$0.661354\pi$$
$$510$$ 0 0
$$511$$ −5268.00 −0.456052
$$512$$ −16384.0 −1.41421
$$513$$ 0 0
$$514$$ −8424.00 −0.722892
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 16448.0 1.39919
$$518$$ −6384.00 −0.541500
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 3638.00 0.305919 0.152959 0.988232i $$-0.451120\pi$$
0.152959 + 0.988232i $$0.451120\pi$$
$$522$$ 0 0
$$523$$ 2078.00 0.173737 0.0868686 0.996220i $$-0.472314\pi$$
0.0868686 + 0.996220i $$0.472314\pi$$
$$524$$ 17664.0 1.47262
$$525$$ 0 0
$$526$$ 14552.0 1.20627
$$527$$ −2808.00 −0.232103
$$528$$ 0 0
$$529$$ −6083.00 −0.499959
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −4800.00 −0.391177
$$533$$ −836.000 −0.0679384
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ −26200.0 −2.09956
$$539$$ 9824.00 0.785064
$$540$$ 0 0
$$541$$ 5622.00 0.446781 0.223391 0.974729i $$-0.428287\pi$$
0.223391 + 0.974729i $$0.428287\pi$$
$$542$$ 17552.0 1.39100
$$543$$ 0 0
$$544$$ 6656.00 0.524584
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −16486.0 −1.28865 −0.644324 0.764753i $$-0.722861\pi$$
−0.644324 + 0.764753i $$0.722861\pi$$
$$548$$ −18672.0 −1.45553
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 5000.00 0.386583
$$552$$ 0 0
$$553$$ −3600.00 −0.276831
$$554$$ 2184.00 0.167490
$$555$$ 0 0
$$556$$ −5600.00 −0.427146
$$557$$ 11706.0 0.890483 0.445242 0.895410i $$-0.353118\pi$$
0.445242 + 0.895410i $$0.353118\pi$$
$$558$$ 0 0
$$559$$ −16796.0 −1.27083
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −27432.0 −2.05898
$$563$$ −25038.0 −1.87429 −0.937146 0.348939i $$-0.886542\pi$$
−0.937146 + 0.348939i $$0.886542\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 37128.0 2.75725
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −17550.0 −1.29303 −0.646515 0.762901i $$-0.723774\pi$$
−0.646515 + 0.762901i $$0.723774\pi$$
$$570$$ 0 0
$$571$$ 10712.0 0.785084 0.392542 0.919734i $$-0.371596\pi$$
0.392542 + 0.919734i $$0.371596\pi$$
$$572$$ −9728.00 −0.711098
$$573$$ 0 0
$$574$$ −528.000 −0.0383942
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 13654.0 0.985136 0.492568 0.870274i $$-0.336058\pi$$
0.492568 + 0.870274i $$0.336058\pi$$
$$578$$ 16948.0 1.21963
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1692.00 −0.120819
$$582$$ 0 0
$$583$$ −64.0000 −0.00454650
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −19368.0 −1.36533
$$587$$ 14166.0 0.996071 0.498035 0.867157i $$-0.334055\pi$$
0.498035 + 0.867157i $$0.334055\pi$$
$$588$$ 0 0
$$589$$ −10800.0 −0.755528
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 17024.0 1.18190
$$593$$ 17842.0 1.23555 0.617777 0.786354i $$-0.288034\pi$$
0.617777 + 0.786354i $$0.288034\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −16400.0 −1.12713
$$597$$ 0 0
$$598$$ 11856.0 0.810749
$$599$$ 17600.0 1.20053 0.600264 0.799802i $$-0.295062\pi$$
0.600264 + 0.799802i $$0.295062\pi$$
$$600$$ 0 0
$$601$$ 27302.0 1.85303 0.926516 0.376256i $$-0.122789\pi$$
0.926516 + 0.376256i $$0.122789\pi$$
$$602$$ −10608.0 −0.718189
$$603$$ 0 0
$$604$$ 14816.0 0.998103
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 3794.00 0.253696 0.126848 0.991922i $$-0.459514\pi$$
0.126848 + 0.991922i $$0.459514\pi$$
$$608$$ 25600.0 1.70759
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −19532.0 −1.29326
$$612$$ 0 0
$$613$$ 13238.0 0.872231 0.436116 0.899891i $$-0.356354\pi$$
0.436116 + 0.899891i $$0.356354\pi$$
$$614$$ −10376.0 −0.681989
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −11574.0 −0.755189 −0.377595 0.925971i $$-0.623249\pi$$
−0.377595 + 0.925971i $$0.623249\pi$$
$$618$$ 0 0
$$619$$ 8300.00 0.538942 0.269471 0.963008i $$-0.413151\pi$$
0.269471 + 0.963008i $$0.413151\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 29328.0 1.89059
$$623$$ −900.000 −0.0578776
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 6248.00 0.398914
$$627$$ 0 0
$$628$$ 19952.0 1.26779
$$629$$ −6916.00 −0.438409
$$630$$ 0 0
$$631$$ −7508.00 −0.473675 −0.236837 0.971549i $$-0.576111\pi$$
−0.236837 + 0.971549i $$0.576111\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ −5704.00 −0.357310
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −11666.0 −0.725626
$$638$$ 6400.00 0.397145
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 27378.0 1.68700 0.843499 0.537130i $$-0.180492\pi$$
0.843499 + 0.537130i $$0.180492\pi$$
$$642$$ 0 0
$$643$$ −1842.00 −0.112973 −0.0564863 0.998403i $$-0.517990\pi$$
−0.0564863 + 0.998403i $$0.517990\pi$$
$$644$$ 3744.00 0.229090
$$645$$ 0 0
$$646$$ −10400.0 −0.633409
$$647$$ −10114.0 −0.614563 −0.307282 0.951619i $$-0.599419\pi$$
−0.307282 + 0.951619i $$0.599419\pi$$
$$648$$ 0 0
$$649$$ 16000.0 0.967727
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −22096.0 −1.32722
$$653$$ 10402.0 0.623372 0.311686 0.950185i $$-0.399106\pi$$
0.311686 + 0.950185i $$0.399106\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 1408.00 0.0838006
$$657$$ 0 0
$$658$$ −12336.0 −0.730862
$$659$$ −7100.00 −0.419692 −0.209846 0.977734i $$-0.567296\pi$$
−0.209846 + 0.977734i $$0.567296\pi$$
$$660$$ 0 0
$$661$$ −7118.00 −0.418847 −0.209424 0.977825i $$-0.567159\pi$$
−0.209424 + 0.977825i $$0.567159\pi$$
$$662$$ 16032.0 0.941241
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −3900.00 −0.226400
$$668$$ 25008.0 1.44849
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 16576.0 0.953665
$$672$$ 0 0
$$673$$ 31278.0 1.79150 0.895749 0.444560i $$-0.146640\pi$$
0.895749 + 0.444560i $$0.146640\pi$$
$$674$$ 35464.0 2.02674
$$675$$ 0 0
$$676$$ −6024.00 −0.342740
$$677$$ −30054.0 −1.70616 −0.853079 0.521782i $$-0.825268\pi$$
−0.853079 + 0.521782i $$0.825268\pi$$
$$678$$ 0 0
$$679$$ 2316.00 0.130898
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −13824.0 −0.776171
$$683$$ −4518.00 −0.253113 −0.126557 0.991959i $$-0.540393\pi$$
−0.126557 + 0.991959i $$0.540393\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −15600.0 −0.868237
$$687$$ 0 0
$$688$$ 28288.0 1.56754
$$689$$ 76.0000 0.00420228
$$690$$ 0 0
$$691$$ 29272.0 1.61152 0.805759 0.592243i $$-0.201758\pi$$
0.805759 + 0.592243i $$0.201758\pi$$
$$692$$ −624.000 −0.0342788
$$693$$ 0 0
$$694$$ 6856.00 0.375000
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −572.000 −0.0310847
$$698$$ −4600.00 −0.249445
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 5798.00 0.312393 0.156196 0.987726i $$-0.450077\pi$$
0.156196 + 0.987726i $$0.450077\pi$$
$$702$$ 0 0
$$703$$ −26600.0 −1.42708
$$704$$ 16384.0 0.877124
$$705$$ 0 0
$$706$$ 17592.0 0.937796
$$707$$ 4212.00 0.224057
$$708$$ 0 0
$$709$$ 8950.00 0.474082 0.237041 0.971500i $$-0.423822\pi$$
0.237041 + 0.971500i $$0.423822\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 8424.00 0.442470
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 10400.0 0.542830
$$717$$ 0 0
$$718$$ 7200.00 0.374236
$$719$$ −7800.00 −0.404577 −0.202289 0.979326i $$-0.564838\pi$$
−0.202289 + 0.979326i $$0.564838\pi$$
$$720$$ 0 0
$$721$$ −3588.00 −0.185332
$$722$$ −12564.0 −0.647623
$$723$$ 0 0
$$724$$ 13936.0 0.715369
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 8554.00 0.436383 0.218191 0.975906i $$-0.429984\pi$$
0.218191 + 0.975906i $$0.429984\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −11492.0 −0.581460
$$732$$ 0 0
$$733$$ −2882.00 −0.145224 −0.0726119 0.997360i $$-0.523133\pi$$
−0.0726119 + 0.997360i $$0.523133\pi$$
$$734$$ −23496.0 −1.18154
$$735$$ 0 0
$$736$$ −19968.0 −1.00004
$$737$$ 4032.00 0.201521
$$738$$ 0 0
$$739$$ 18700.0 0.930840 0.465420 0.885090i $$-0.345903\pi$$
0.465420 + 0.885090i $$0.345903\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 48.0000 0.00237485
$$743$$ 12242.0 0.604462 0.302231 0.953235i $$-0.402269\pi$$
0.302231 + 0.953235i $$0.402269\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −8312.00 −0.407941
$$747$$ 0 0
$$748$$ −6656.00 −0.325358
$$749$$ 7164.00 0.349488
$$750$$ 0 0
$$751$$ −31148.0 −1.51346 −0.756729 0.653729i $$-0.773204\pi$$
−0.756729 + 0.653729i $$0.773204\pi$$
$$752$$ 32896.0 1.59520
$$753$$ 0 0
$$754$$ −7600.00 −0.367076
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 7694.00 0.369410 0.184705 0.982794i $$-0.440867\pi$$
0.184705 + 0.982794i $$0.440867\pi$$
$$758$$ −31600.0 −1.51420
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 4518.00 0.215213 0.107607 0.994194i $$-0.465681\pi$$
0.107607 + 0.994194i $$0.465681\pi$$
$$762$$ 0 0
$$763$$ 3300.00 0.156577
$$764$$ −30176.0 −1.42897
$$765$$ 0 0
$$766$$ 30072.0 1.41847
$$767$$ −19000.0 −0.894459
$$768$$ 0 0
$$769$$ −39550.0 −1.85463 −0.927314 0.374283i $$-0.877889\pi$$
−0.927314 + 0.374283i $$0.877889\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 2864.00 0.133520
$$773$$ 22122.0 1.02933 0.514666 0.857391i $$-0.327916\pi$$
0.514666 + 0.857391i $$0.327916\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −7800.00 −0.359439
$$779$$ −2200.00 −0.101185
$$780$$ 0 0
$$781$$ 13184.0 0.604047
$$782$$ 8112.00 0.370952
$$783$$ 0 0
$$784$$ 19648.0 0.895044
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 16634.0 0.753416 0.376708 0.926332i $$-0.377056\pi$$
0.376708 + 0.926332i $$0.377056\pi$$
$$788$$ −17712.0 −0.800716
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −9372.00 −0.421277
$$792$$ 0 0
$$793$$ −19684.0 −0.881462
$$794$$ 55144.0 2.46472
$$795$$ 0 0
$$796$$ −20800.0 −0.926176
$$797$$ 27586.0 1.22603 0.613015 0.790071i $$-0.289956\pi$$
0.613015 + 0.790071i $$0.289956\pi$$
$$798$$ 0 0
$$799$$ −13364.0 −0.591720
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 25608.0 1.12749
$$803$$ −28096.0 −1.23473
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 16416.0 0.717406
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −3850.00 −0.167316 −0.0836581 0.996495i $$-0.526660\pi$$
−0.0836581 + 0.996495i $$0.526660\pi$$
$$810$$ 0 0
$$811$$ 10032.0 0.434366 0.217183 0.976131i $$-0.430313\pi$$
0.217183 + 0.976131i $$0.430313\pi$$
$$812$$ −2400.00 −0.103724
$$813$$ 0 0
$$814$$ −34048.0 −1.46607
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −44200.0 −1.89273
$$818$$ −44600.0 −1.90636
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −20562.0 −0.874079 −0.437039 0.899442i $$-0.643973\pi$$
−0.437039 + 0.899442i $$0.643973\pi$$
$$822$$ 0 0
$$823$$ −10322.0 −0.437184 −0.218592 0.975816i $$-0.570146\pi$$
−0.218592 + 0.975816i $$0.570146\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ −12000.0 −0.505488
$$827$$ 8846.00 0.371954 0.185977 0.982554i $$-0.440455\pi$$
0.185977 + 0.982554i $$0.440455\pi$$
$$828$$ 0 0
$$829$$ −25350.0 −1.06205 −0.531026 0.847355i $$-0.678194\pi$$
−0.531026 + 0.847355i $$0.678194\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −19456.0 −0.810716
$$833$$ −7982.00 −0.332005
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −25600.0 −1.05908
$$837$$ 0 0
$$838$$ −54800.0 −2.25899
$$839$$ −46000.0 −1.89284 −0.946422 0.322932i $$-0.895331\pi$$
−0.946422 + 0.322932i $$0.895331\pi$$
$$840$$ 0 0
$$841$$ −21889.0 −0.897495
$$842$$ 21752.0 0.890289
$$843$$ 0 0
$$844$$ −9344.00 −0.381083
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 1842.00 0.0747248
$$848$$ −128.000 −0.00518342
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 20748.0 0.835761
$$852$$ 0 0
$$853$$ 16998.0 0.682298 0.341149 0.940009i $$-0.389184\pi$$
0.341149 + 0.940009i $$0.389184\pi$$
$$854$$ −12432.0 −0.498143
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −26494.0 −1.05603 −0.528015 0.849235i $$-0.677064\pi$$
−0.528015 + 0.849235i $$0.677064\pi$$
$$858$$ 0 0
$$859$$ −21500.0 −0.853982 −0.426991 0.904256i $$-0.640426\pi$$
−0.426991 + 0.904256i $$0.640426\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 30768.0 1.21573
$$863$$ 25762.0 1.01616 0.508082 0.861309i $$-0.330355\pi$$
0.508082 + 0.861309i $$0.330355\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −4472.00 −0.175479
$$867$$ 0 0
$$868$$ 5184.00 0.202715
$$869$$ −19200.0 −0.749500
$$870$$ 0 0
$$871$$ −4788.00 −0.186263
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 31200.0 1.20750
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −30546.0 −1.17613 −0.588064 0.808814i $$-0.700110\pi$$
−0.588064 + 0.808814i $$0.700110\pi$$
$$878$$ 10400.0 0.399753
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −32942.0 −1.25976 −0.629878 0.776694i $$-0.716895\pi$$
−0.629878 + 0.776694i $$0.716895\pi$$
$$882$$ 0 0
$$883$$ 27118.0 1.03351 0.516757 0.856132i $$-0.327139\pi$$
0.516757 + 0.856132i $$0.327139\pi$$
$$884$$ 7904.00 0.300724
$$885$$ 0 0
$$886$$ 47832.0 1.81371
$$887$$ −38634.0 −1.46246 −0.731230 0.682131i $$-0.761054\pi$$
−0.731230 + 0.682131i $$0.761054\pi$$
$$888$$ 0 0
$$889$$ 11076.0 0.417860
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 51824.0 1.94529
$$893$$ −51400.0 −1.92613
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −68200.0 −2.53437
$$899$$ −5400.00 −0.200334
$$900$$ 0 0
$$901$$ 52.0000 0.00192272
$$902$$ −2816.00 −0.103950
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 1794.00 0.0656767 0.0328384 0.999461i $$-0.489545\pi$$
0.0328384 + 0.999461i $$0.489545\pi$$
$$908$$ 5168.00 0.188883
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −41732.0 −1.51772 −0.758860 0.651254i $$-0.774243\pi$$
−0.758860 + 0.651254i $$0.774243\pi$$
$$912$$ 0 0
$$913$$ −9024.00 −0.327109
$$914$$ −37976.0 −1.37433
$$915$$ 0 0
$$916$$ 30000.0 1.08213
$$917$$ −13248.0 −0.477086
$$918$$ 0 0
$$919$$ 29200.0 1.04812 0.524058 0.851682i $$-0.324417\pi$$
0.524058 + 0.851682i $$0.324417\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −45672.0 −1.63137
$$923$$ −15656.0 −0.558314
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 31848.0 1.13023
$$927$$ 0 0
$$928$$ 12800.0 0.452781
$$929$$ 48650.0 1.71814 0.859071 0.511856i $$-0.171042\pi$$
0.859071 + 0.511856i $$0.171042\pi$$
$$930$$ 0 0
$$931$$ −30700.0 −1.08072
$$932$$ 11856.0 0.416691
$$933$$ 0 0
$$934$$ −26104.0 −0.914506
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 11334.0 0.395161 0.197580 0.980287i $$-0.436692\pi$$
0.197580 + 0.980287i $$0.436692\pi$$
$$938$$ −3024.00 −0.105263
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 31178.0 1.08010 0.540050 0.841633i $$-0.318405\pi$$
0.540050 + 0.841633i $$0.318405\pi$$
$$942$$ 0 0
$$943$$ 1716.00 0.0592584
$$944$$ 32000.0 1.10330
$$945$$ 0 0
$$946$$ −56576.0 −1.94444
$$947$$ 4686.00 0.160797 0.0803984 0.996763i $$-0.474381\pi$$
0.0803984 + 0.996763i $$0.474381\pi$$
$$948$$ 0 0
$$949$$ 33364.0 1.14124
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −598.000 −0.0203265 −0.0101632 0.999948i $$-0.503235\pi$$
−0.0101632 + 0.999948i $$0.503235\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −11200.0 −0.378906
$$957$$ 0 0
$$958$$ 69600.0 2.34726
$$959$$ 14004.0 0.471546
$$960$$ 0 0
$$961$$ −18127.0 −0.608472
$$962$$ 40432.0 1.35507
$$963$$ 0 0
$$964$$ 24176.0 0.807735
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −41726.0 −1.38761 −0.693804 0.720163i $$-0.744067\pi$$
−0.693804 + 0.720163i $$0.744067\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −24312.0 −0.803511 −0.401756 0.915747i $$-0.631600\pi$$
−0.401756 + 0.915747i $$0.631600\pi$$
$$972$$ 0 0
$$973$$ 4200.00 0.138382
$$974$$ 4664.00 0.153433
$$975$$ 0 0
$$976$$ 33152.0 1.08726
$$977$$ 40946.0 1.34082 0.670409 0.741992i $$-0.266119\pi$$
0.670409 + 0.741992i $$0.266119\pi$$
$$978$$ 0 0
$$979$$ −4800.00 −0.156699
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 28288.0 0.919253
$$983$$ 42282.0 1.37191 0.685954 0.727645i $$-0.259385\pi$$
0.685954 + 0.727645i $$0.259385\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −5200.00 −0.167953
$$987$$ 0 0
$$988$$ 30400.0 0.978900
$$989$$ 34476.0 1.10847
$$990$$ 0 0
$$991$$ 1172.00 0.0375679 0.0187840 0.999824i $$-0.494021\pi$$
0.0187840 + 0.999824i $$0.494021\pi$$
$$992$$ −27648.0 −0.884904
$$993$$ 0 0
$$994$$ −9888.00 −0.315521
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 31614.0 1.00424 0.502119 0.864798i $$-0.332554\pi$$
0.502119 + 0.864798i $$0.332554\pi$$
$$998$$ −400.000 −0.0126872
$$999$$ 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.b.1.1 1
3.2 odd 2 25.4.a.c.1.1 1
5.2 odd 4 225.4.b.c.199.1 2
5.3 odd 4 225.4.b.c.199.2 2
5.4 even 2 45.4.a.d.1.1 1
12.11 even 2 400.4.a.m.1.1 1
15.2 even 4 25.4.b.a.24.2 2
15.8 even 4 25.4.b.a.24.1 2
15.14 odd 2 5.4.a.a.1.1 1
20.19 odd 2 720.4.a.u.1.1 1
21.20 even 2 1225.4.a.k.1.1 1
24.5 odd 2 1600.4.a.bi.1.1 1
24.11 even 2 1600.4.a.s.1.1 1
35.34 odd 2 2205.4.a.q.1.1 1
45.4 even 6 405.4.e.c.136.1 2
45.14 odd 6 405.4.e.l.136.1 2
45.29 odd 6 405.4.e.l.271.1 2
45.34 even 6 405.4.e.c.271.1 2
60.23 odd 4 400.4.c.k.49.2 2
60.47 odd 4 400.4.c.k.49.1 2
60.59 even 2 80.4.a.d.1.1 1
105.44 odd 6 245.4.e.f.116.1 2
105.59 even 6 245.4.e.g.226.1 2
105.74 odd 6 245.4.e.f.226.1 2
105.89 even 6 245.4.e.g.116.1 2
105.104 even 2 245.4.a.a.1.1 1
120.29 odd 2 320.4.a.g.1.1 1
120.59 even 2 320.4.a.h.1.1 1
165.164 even 2 605.4.a.d.1.1 1
195.194 odd 2 845.4.a.b.1.1 1
240.29 odd 4 1280.4.d.e.641.2 2
240.59 even 4 1280.4.d.l.641.2 2
240.149 odd 4 1280.4.d.e.641.1 2
240.179 even 4 1280.4.d.l.641.1 2
255.254 odd 2 1445.4.a.a.1.1 1
285.284 even 2 1805.4.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
5.4.a.a.1.1 1 15.14 odd 2
25.4.a.c.1.1 1 3.2 odd 2
25.4.b.a.24.1 2 15.8 even 4
25.4.b.a.24.2 2 15.2 even 4
45.4.a.d.1.1 1 5.4 even 2
80.4.a.d.1.1 1 60.59 even 2
225.4.a.b.1.1 1 1.1 even 1 trivial
225.4.b.c.199.1 2 5.2 odd 4
225.4.b.c.199.2 2 5.3 odd 4
245.4.a.a.1.1 1 105.104 even 2
245.4.e.f.116.1 2 105.44 odd 6
245.4.e.f.226.1 2 105.74 odd 6
245.4.e.g.116.1 2 105.89 even 6
245.4.e.g.226.1 2 105.59 even 6
320.4.a.g.1.1 1 120.29 odd 2
320.4.a.h.1.1 1 120.59 even 2
400.4.a.m.1.1 1 12.11 even 2
400.4.c.k.49.1 2 60.47 odd 4
400.4.c.k.49.2 2 60.23 odd 4
405.4.e.c.136.1 2 45.4 even 6
405.4.e.c.271.1 2 45.34 even 6
405.4.e.l.136.1 2 45.14 odd 6
405.4.e.l.271.1 2 45.29 odd 6
605.4.a.d.1.1 1 165.164 even 2
720.4.a.u.1.1 1 20.19 odd 2
845.4.a.b.1.1 1 195.194 odd 2
1225.4.a.k.1.1 1 21.20 even 2
1280.4.d.e.641.1 2 240.149 odd 4
1280.4.d.e.641.2 2 240.29 odd 4
1280.4.d.l.641.1 2 240.179 even 4
1280.4.d.l.641.2 2 240.59 even 4
1445.4.a.a.1.1 1 255.254 odd 2
1600.4.a.s.1.1 1 24.11 even 2
1600.4.a.bi.1.1 1 24.5 odd 2
1805.4.a.h.1.1 1 285.284 even 2
2205.4.a.q.1.1 1 35.34 odd 2