# Properties

 Label 1900.2.a.j.1.2 Level $1900$ Weight $2$ Character 1900.1 Self dual yes Analytic conductor $15.172$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.1715763840$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{5})$$ Defining polynomial: $$x^{4} - 6 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 380) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.874032$$ of defining polynomial Character $$\chi$$ $$=$$ 1900.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.874032 q^{3} -2.82843 q^{7} -2.23607 q^{9} +O(q^{10})$$ $$q-0.874032 q^{3} -2.82843 q^{7} -2.23607 q^{9} -0.763932 q^{11} +5.45052 q^{13} +7.40492 q^{17} -1.00000 q^{19} +2.47214 q^{21} +1.08036 q^{23} +4.57649 q^{27} -4.47214 q^{29} -4.00000 q^{31} +0.667701 q^{33} +2.62210 q^{37} -4.76393 q^{39} -6.00000 q^{41} -8.48528 q^{43} -8.48528 q^{47} +1.00000 q^{49} -6.47214 q^{51} -2.62210 q^{53} +0.874032 q^{57} -1.52786 q^{59} -11.7082 q^{61} +6.32456 q^{63} -11.1074 q^{67} -0.944272 q^{69} -10.4721 q^{71} -5.24419 q^{73} +2.16073 q^{77} +15.4164 q^{79} +2.70820 q^{81} -13.7295 q^{83} +3.90879 q^{87} +2.94427 q^{89} -15.4164 q^{91} +3.49613 q^{93} -13.9358 q^{97} +1.70820 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q - 12 q^{11} - 4 q^{19} - 8 q^{21} - 16 q^{31} - 28 q^{39} - 24 q^{41} + 4 q^{49} - 8 q^{51} - 24 q^{59} - 20 q^{61} + 32 q^{69} - 24 q^{71} + 8 q^{79} - 16 q^{81} - 24 q^{89} - 8 q^{91} - 20 q^{99} + O(q^{100})$$

## 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
$$3$$ −0.874032 −0.504623 −0.252311 0.967646i $$-0.581191\pi$$
−0.252311 + 0.967646i $$0.581191\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.82843 −1.06904 −0.534522 0.845154i $$-0.679509\pi$$
−0.534522 + 0.845154i $$0.679509\pi$$
$$8$$ 0 0
$$9$$ −2.23607 −0.745356
$$10$$ 0 0
$$11$$ −0.763932 −0.230334 −0.115167 0.993346i $$-0.536740\pi$$
−0.115167 + 0.993346i $$0.536740\pi$$
$$12$$ 0 0
$$13$$ 5.45052 1.51170 0.755852 0.654743i $$-0.227223\pi$$
0.755852 + 0.654743i $$0.227223\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 7.40492 1.79596 0.897978 0.440040i $$-0.145036\pi$$
0.897978 + 0.440040i $$0.145036\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 2.47214 0.539464
$$22$$ 0 0
$$23$$ 1.08036 0.225271 0.112636 0.993636i $$-0.464071\pi$$
0.112636 + 0.993636i $$0.464071\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4.57649 0.880746
$$28$$ 0 0
$$29$$ −4.47214 −0.830455 −0.415227 0.909718i $$-0.636298\pi$$
−0.415227 + 0.909718i $$0.636298\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 0.667701 0.116232
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.62210 0.431070 0.215535 0.976496i $$-0.430850\pi$$
0.215535 + 0.976496i $$0.430850\pi$$
$$38$$ 0 0
$$39$$ −4.76393 −0.762840
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ −8.48528 −1.29399 −0.646997 0.762493i $$-0.723975\pi$$
−0.646997 + 0.762493i $$0.723975\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −8.48528 −1.23771 −0.618853 0.785507i $$-0.712402\pi$$
−0.618853 + 0.785507i $$0.712402\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −6.47214 −0.906280
$$52$$ 0 0
$$53$$ −2.62210 −0.360173 −0.180086 0.983651i $$-0.557638\pi$$
−0.180086 + 0.983651i $$0.557638\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0.874032 0.115768
$$58$$ 0 0
$$59$$ −1.52786 −0.198911 −0.0994555 0.995042i $$-0.531710\pi$$
−0.0994555 + 0.995042i $$0.531710\pi$$
$$60$$ 0 0
$$61$$ −11.7082 −1.49908 −0.749541 0.661958i $$-0.769726\pi$$
−0.749541 + 0.661958i $$0.769726\pi$$
$$62$$ 0 0
$$63$$ 6.32456 0.796819
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −11.1074 −1.35698 −0.678491 0.734609i $$-0.737366\pi$$
−0.678491 + 0.734609i $$0.737366\pi$$
$$68$$ 0 0
$$69$$ −0.944272 −0.113677
$$70$$ 0 0
$$71$$ −10.4721 −1.24281 −0.621407 0.783488i $$-0.713439\pi$$
−0.621407 + 0.783488i $$0.713439\pi$$
$$72$$ 0 0
$$73$$ −5.24419 −0.613786 −0.306893 0.951744i $$-0.599289\pi$$
−0.306893 + 0.951744i $$0.599289\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.16073 0.246238
$$78$$ 0 0
$$79$$ 15.4164 1.73448 0.867241 0.497889i $$-0.165891\pi$$
0.867241 + 0.497889i $$0.165891\pi$$
$$80$$ 0 0
$$81$$ 2.70820 0.300912
$$82$$ 0 0
$$83$$ −13.7295 −1.50701 −0.753503 0.657445i $$-0.771637\pi$$
−0.753503 + 0.657445i $$0.771637\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 3.90879 0.419066
$$88$$ 0 0
$$89$$ 2.94427 0.312092 0.156046 0.987750i $$-0.450125\pi$$
0.156046 + 0.987750i $$0.450125\pi$$
$$90$$ 0 0
$$91$$ −15.4164 −1.61608
$$92$$ 0 0
$$93$$ 3.49613 0.362532
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −13.9358 −1.41497 −0.707483 0.706730i $$-0.750170\pi$$
−0.707483 + 0.706730i $$0.750170\pi$$
$$98$$ 0 0
$$99$$ 1.70820 0.171681
$$100$$ 0 0
$$101$$ 5.23607 0.521008 0.260504 0.965473i $$-0.416111\pi$$
0.260504 + 0.965473i $$0.416111\pi$$
$$102$$ 0 0
$$103$$ 5.86319 0.577717 0.288858 0.957372i $$-0.406724\pi$$
0.288858 + 0.957372i $$0.406724\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 13.2681 1.28268 0.641338 0.767258i $$-0.278380\pi$$
0.641338 + 0.767258i $$0.278380\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −2.29180 −0.217528
$$112$$ 0 0
$$113$$ 16.3516 1.53823 0.769113 0.639113i $$-0.220698\pi$$
0.769113 + 0.639113i $$0.220698\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −12.1877 −1.12676
$$118$$ 0 0
$$119$$ −20.9443 −1.91996
$$120$$ 0 0
$$121$$ −10.4164 −0.946946
$$122$$ 0 0
$$123$$ 5.24419 0.472853
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 19.5927 1.73857 0.869284 0.494314i $$-0.164581\pi$$
0.869284 + 0.494314i $$0.164581\pi$$
$$128$$ 0 0
$$129$$ 7.41641 0.652978
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 2.82843 0.245256
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 7.40492 0.632645 0.316322 0.948652i $$-0.397552\pi$$
0.316322 + 0.948652i $$0.397552\pi$$
$$138$$ 0 0
$$139$$ 2.29180 0.194388 0.0971938 0.995265i $$-0.469013\pi$$
0.0971938 + 0.995265i $$0.469013\pi$$
$$140$$ 0 0
$$141$$ 7.41641 0.624574
$$142$$ 0 0
$$143$$ −4.16383 −0.348197
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −0.874032 −0.0720889
$$148$$ 0 0
$$149$$ −12.6525 −1.03653 −0.518266 0.855220i $$-0.673422\pi$$
−0.518266 + 0.855220i $$0.673422\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ −16.5579 −1.33863
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0.412662 0.0329340 0.0164670 0.999864i $$-0.494758\pi$$
0.0164670 + 0.999864i $$0.494758\pi$$
$$158$$ 0 0
$$159$$ 2.29180 0.181751
$$160$$ 0 0
$$161$$ −3.05573 −0.240825
$$162$$ 0 0
$$163$$ 8.48528 0.664619 0.332309 0.943170i $$-0.392172\pi$$
0.332309 + 0.943170i $$0.392172\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −17.4319 −1.34892 −0.674462 0.738310i $$-0.735624\pi$$
−0.674462 + 0.738310i $$0.735624\pi$$
$$168$$ 0 0
$$169$$ 16.7082 1.28525
$$170$$ 0 0
$$171$$ 2.23607 0.170996
$$172$$ 0 0
$$173$$ −8.94665 −0.680201 −0.340101 0.940389i $$-0.610461\pi$$
−0.340101 + 0.940389i $$0.610461\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1.33540 0.100375
$$178$$ 0 0
$$179$$ −22.4721 −1.67965 −0.839823 0.542860i $$-0.817341\pi$$
−0.839823 + 0.542860i $$0.817341\pi$$
$$180$$ 0 0
$$181$$ 6.00000 0.445976 0.222988 0.974821i $$-0.428419\pi$$
0.222988 + 0.974821i $$0.428419\pi$$
$$182$$ 0 0
$$183$$ 10.2333 0.756471
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −5.65685 −0.413670
$$188$$ 0 0
$$189$$ −12.9443 −0.941557
$$190$$ 0 0
$$191$$ −3.05573 −0.221105 −0.110552 0.993870i $$-0.535262\pi$$
−0.110552 + 0.993870i $$0.535262\pi$$
$$192$$ 0 0
$$193$$ −13.9358 −1.00312 −0.501561 0.865123i $$-0.667241\pi$$
−0.501561 + 0.865123i $$0.667241\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −12.6491 −0.901212 −0.450606 0.892723i $$-0.648792\pi$$
−0.450606 + 0.892723i $$0.648792\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 9.70820 0.684764
$$202$$ 0 0
$$203$$ 12.6491 0.887794
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −2.41577 −0.167907
$$208$$ 0 0
$$209$$ 0.763932 0.0528423
$$210$$ 0 0
$$211$$ 15.4164 1.06131 0.530655 0.847588i $$-0.321946\pi$$
0.530655 + 0.847588i $$0.321946\pi$$
$$212$$ 0 0
$$213$$ 9.15298 0.627152
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 11.3137 0.768025
$$218$$ 0 0
$$219$$ 4.58359 0.309730
$$220$$ 0 0
$$221$$ 40.3607 2.71495
$$222$$ 0 0
$$223$$ −18.7673 −1.25675 −0.628377 0.777909i $$-0.716280\pi$$
−0.628377 + 0.777909i $$0.716280\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −5.86319 −0.389153 −0.194577 0.980887i $$-0.562333\pi$$
−0.194577 + 0.980887i $$0.562333\pi$$
$$228$$ 0 0
$$229$$ 7.70820 0.509372 0.254686 0.967024i $$-0.418028\pi$$
0.254686 + 0.967024i $$0.418028\pi$$
$$230$$ 0 0
$$231$$ −1.88854 −0.124257
$$232$$ 0 0
$$233$$ 12.6491 0.828671 0.414335 0.910124i $$-0.364014\pi$$
0.414335 + 0.910124i $$0.364014\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −13.4744 −0.875259
$$238$$ 0 0
$$239$$ −29.8885 −1.93333 −0.966665 0.256046i $$-0.917580\pi$$
−0.966665 + 0.256046i $$0.917580\pi$$
$$240$$ 0 0
$$241$$ −28.8328 −1.85728 −0.928642 0.370976i $$-0.879023\pi$$
−0.928642 + 0.370976i $$0.879023\pi$$
$$242$$ 0 0
$$243$$ −16.0965 −1.03259
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −5.45052 −0.346808
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ −5.88854 −0.371682 −0.185841 0.982580i $$-0.559501\pi$$
−0.185841 + 0.982580i $$0.559501\pi$$
$$252$$ 0 0
$$253$$ −0.825324 −0.0518877
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 17.4319 1.08737 0.543687 0.839288i $$-0.317028\pi$$
0.543687 + 0.839288i $$0.317028\pi$$
$$258$$ 0 0
$$259$$ −7.41641 −0.460833
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ 0 0
$$263$$ −15.8902 −0.979832 −0.489916 0.871770i $$-0.662973\pi$$
−0.489916 + 0.871770i $$0.662973\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −2.57339 −0.157489
$$268$$ 0 0
$$269$$ 14.9443 0.911168 0.455584 0.890193i $$-0.349430\pi$$
0.455584 + 0.890193i $$0.349430\pi$$
$$270$$ 0 0
$$271$$ 21.1246 1.28323 0.641614 0.767027i $$-0.278265\pi$$
0.641614 + 0.767027i $$0.278265\pi$$
$$272$$ 0 0
$$273$$ 13.4744 0.815510
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.2148 1.33476 0.667378 0.744719i $$-0.267417\pi$$
0.667378 + 0.744719i $$0.267417\pi$$
$$278$$ 0 0
$$279$$ 8.94427 0.535480
$$280$$ 0 0
$$281$$ −25.4164 −1.51622 −0.758108 0.652129i $$-0.773876\pi$$
−0.758108 + 0.652129i $$0.773876\pi$$
$$282$$ 0 0
$$283$$ 2.41577 0.143602 0.0718012 0.997419i $$-0.477125\pi$$
0.0718012 + 0.997419i $$0.477125\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 16.9706 1.00174
$$288$$ 0 0
$$289$$ 37.8328 2.22546
$$290$$ 0 0
$$291$$ 12.1803 0.714024
$$292$$ 0 0
$$293$$ −2.62210 −0.153184 −0.0765922 0.997062i $$-0.524404\pi$$
−0.0765922 + 0.997062i $$0.524404\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −3.49613 −0.202866
$$298$$ 0 0
$$299$$ 5.88854 0.340543
$$300$$ 0 0
$$301$$ 24.0000 1.38334
$$302$$ 0 0
$$303$$ −4.57649 −0.262913
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 5.45052 0.311078 0.155539 0.987830i $$-0.450289\pi$$
0.155539 + 0.987830i $$0.450289\pi$$
$$308$$ 0 0
$$309$$ −5.12461 −0.291529
$$310$$ 0 0
$$311$$ −9.70820 −0.550502 −0.275251 0.961372i $$-0.588761\pi$$
−0.275251 + 0.961372i $$0.588761\pi$$
$$312$$ 0 0
$$313$$ 10.4884 0.592839 0.296419 0.955058i $$-0.404207\pi$$
0.296419 + 0.955058i $$0.404207\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.94665 0.502494 0.251247 0.967923i $$-0.419159\pi$$
0.251247 + 0.967923i $$0.419159\pi$$
$$318$$ 0 0
$$319$$ 3.41641 0.191282
$$320$$ 0 0
$$321$$ −11.5967 −0.647267
$$322$$ 0 0
$$323$$ −7.40492 −0.412021
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −1.74806 −0.0966682
$$328$$ 0 0
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ 7.41641 0.407643 0.203821 0.979008i $$-0.434664\pi$$
0.203821 + 0.979008i $$0.434664\pi$$
$$332$$ 0 0
$$333$$ −5.86319 −0.321301
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −16.7642 −0.913206 −0.456603 0.889671i $$-0.650934\pi$$
−0.456603 + 0.889671i $$0.650934\pi$$
$$338$$ 0 0
$$339$$ −14.2918 −0.776224
$$340$$ 0 0
$$341$$ 3.05573 0.165477
$$342$$ 0 0
$$343$$ 16.9706 0.916324
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 9.40802 0.505049 0.252525 0.967590i $$-0.418739\pi$$
0.252525 + 0.967590i $$0.418739\pi$$
$$348$$ 0 0
$$349$$ 25.4164 1.36051 0.680255 0.732976i $$-0.261869\pi$$
0.680255 + 0.732976i $$0.261869\pi$$
$$350$$ 0 0
$$351$$ 24.9443 1.33143
$$352$$ 0 0
$$353$$ −9.56564 −0.509128 −0.254564 0.967056i $$-0.581932\pi$$
−0.254564 + 0.967056i $$0.581932\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 18.3060 0.968854
$$358$$ 0 0
$$359$$ 29.1246 1.53714 0.768569 0.639767i $$-0.220969\pi$$
0.768569 + 0.639767i $$0.220969\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 9.10427 0.477850
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.48528 −0.442928 −0.221464 0.975169i $$-0.571084\pi$$
−0.221464 + 0.975169i $$0.571084\pi$$
$$368$$ 0 0
$$369$$ 13.4164 0.698430
$$370$$ 0 0
$$371$$ 7.41641 0.385041
$$372$$ 0 0
$$373$$ −13.1105 −0.678835 −0.339417 0.940636i $$-0.610230\pi$$
−0.339417 + 0.940636i $$0.610230\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −24.3755 −1.25540
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ −17.1246 −0.877320
$$382$$ 0 0
$$383$$ 36.4056 1.86024 0.930120 0.367257i $$-0.119703\pi$$
0.930120 + 0.367257i $$0.119703\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 18.9737 0.964486
$$388$$ 0 0
$$389$$ −14.9443 −0.757705 −0.378852 0.925457i $$-0.623681\pi$$
−0.378852 + 0.925457i $$0.623681\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −11.7264 −0.588530 −0.294265 0.955724i $$-0.595075\pi$$
−0.294265 + 0.955724i $$0.595075\pi$$
$$398$$ 0 0
$$399$$ −2.47214 −0.123762
$$400$$ 0 0
$$401$$ −4.47214 −0.223328 −0.111664 0.993746i $$-0.535618\pi$$
−0.111664 + 0.993746i $$0.535618\pi$$
$$402$$ 0 0
$$403$$ −21.8021 −1.08604
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2.00310 −0.0992901
$$408$$ 0 0
$$409$$ 17.4164 0.861186 0.430593 0.902546i $$-0.358304\pi$$
0.430593 + 0.902546i $$0.358304\pi$$
$$410$$ 0 0
$$411$$ −6.47214 −0.319247
$$412$$ 0 0
$$413$$ 4.32145 0.212645
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −2.00310 −0.0980924
$$418$$ 0 0
$$419$$ −20.9443 −1.02319 −0.511597 0.859225i $$-0.670946\pi$$
−0.511597 + 0.859225i $$0.670946\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ 0 0
$$423$$ 18.9737 0.922531
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 33.1158 1.60259
$$428$$ 0 0
$$429$$ 3.63932 0.175708
$$430$$ 0 0
$$431$$ 1.52786 0.0735946 0.0367973 0.999323i $$-0.488284\pi$$
0.0367973 + 0.999323i $$0.488284\pi$$
$$432$$ 0 0
$$433$$ 28.4906 1.36917 0.684585 0.728933i $$-0.259983\pi$$
0.684585 + 0.728933i $$0.259983\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −1.08036 −0.0516808
$$438$$ 0 0
$$439$$ −18.8328 −0.898841 −0.449421 0.893320i $$-0.648370\pi$$
−0.449421 + 0.893320i $$0.648370\pi$$
$$440$$ 0 0
$$441$$ −2.23607 −0.106479
$$442$$ 0 0
$$443$$ 13.7295 0.652307 0.326153 0.945317i $$-0.394247\pi$$
0.326153 + 0.945317i $$0.394247\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 11.0587 0.523057
$$448$$ 0 0
$$449$$ −16.4721 −0.777368 −0.388684 0.921371i $$-0.627070\pi$$
−0.388684 + 0.921371i $$0.627070\pi$$
$$450$$ 0 0
$$451$$ 4.58359 0.215833
$$452$$ 0 0
$$453$$ 6.99226 0.328525
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −37.9473 −1.77510 −0.887551 0.460710i $$-0.847595\pi$$
−0.887551 + 0.460710i $$0.847595\pi$$
$$458$$ 0 0
$$459$$ 33.8885 1.58178
$$460$$ 0 0
$$461$$ 23.8885 1.11260 0.556300 0.830981i $$-0.312220\pi$$
0.556300 + 0.830981i $$0.312220\pi$$
$$462$$ 0 0
$$463$$ −14.5548 −0.676419 −0.338209 0.941071i $$-0.609821\pi$$
−0.338209 + 0.941071i $$0.609821\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −30.7000 −1.42063 −0.710314 0.703885i $$-0.751447\pi$$
−0.710314 + 0.703885i $$0.751447\pi$$
$$468$$ 0 0
$$469$$ 31.4164 1.45067
$$470$$ 0 0
$$471$$ −0.360680 −0.0166192
$$472$$ 0 0
$$473$$ 6.48218 0.298051
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 5.86319 0.268457
$$478$$ 0 0
$$479$$ 11.2361 0.513389 0.256695 0.966493i $$-0.417367\pi$$
0.256695 + 0.966493i $$0.417367\pi$$
$$480$$ 0 0
$$481$$ 14.2918 0.651650
$$482$$ 0 0
$$483$$ 2.67080 0.121526
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 10.6947 0.484624 0.242312 0.970198i $$-0.422094\pi$$
0.242312 + 0.970198i $$0.422094\pi$$
$$488$$ 0 0
$$489$$ −7.41641 −0.335382
$$490$$ 0 0
$$491$$ 5.88854 0.265746 0.132873 0.991133i $$-0.457580\pi$$
0.132873 + 0.991133i $$0.457580\pi$$
$$492$$ 0 0
$$493$$ −33.1158 −1.49146
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 29.6197 1.32862
$$498$$ 0 0
$$499$$ −17.1246 −0.766603 −0.383301 0.923623i $$-0.625213\pi$$
−0.383301 + 0.923623i $$0.625213\pi$$
$$500$$ 0 0
$$501$$ 15.2361 0.680697
$$502$$ 0 0
$$503$$ −20.2117 −0.901193 −0.450597 0.892728i $$-0.648789\pi$$
−0.450597 + 0.892728i $$0.648789\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −14.6035 −0.648564
$$508$$ 0 0
$$509$$ −10.5836 −0.469109 −0.234555 0.972103i $$-0.575363\pi$$
−0.234555 + 0.972103i $$0.575363\pi$$
$$510$$ 0 0
$$511$$ 14.8328 0.656165
$$512$$ 0 0
$$513$$ −4.57649 −0.202057
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 6.48218 0.285086
$$518$$ 0 0
$$519$$ 7.81966 0.343245
$$520$$ 0 0
$$521$$ 0.111456 0.00488298 0.00244149 0.999997i $$-0.499223\pi$$
0.00244149 + 0.999997i $$0.499223\pi$$
$$522$$ 0 0
$$523$$ 24.8369 1.08604 0.543020 0.839720i $$-0.317281\pi$$
0.543020 + 0.839720i $$0.317281\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −29.6197 −1.29025
$$528$$ 0 0
$$529$$ −21.8328 −0.949253
$$530$$ 0 0
$$531$$ 3.41641 0.148259
$$532$$ 0 0
$$533$$ −32.7031 −1.41653
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 19.6414 0.847588
$$538$$ 0 0
$$539$$ −0.763932 −0.0329049
$$540$$ 0 0
$$541$$ 11.7082 0.503375 0.251688 0.967809i $$-0.419014\pi$$
0.251688 + 0.967809i $$0.419014\pi$$
$$542$$ 0 0
$$543$$ −5.24419 −0.225050
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8.27895 0.353982 0.176991 0.984212i $$-0.443364\pi$$
0.176991 + 0.984212i $$0.443364\pi$$
$$548$$ 0 0
$$549$$ 26.1803 1.11735
$$550$$ 0 0
$$551$$ 4.47214 0.190519
$$552$$ 0 0
$$553$$ −43.6042 −1.85424
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −20.0540 −0.849716 −0.424858 0.905260i $$-0.639676\pi$$
−0.424858 + 0.905260i $$0.639676\pi$$
$$558$$ 0 0
$$559$$ −46.2492 −1.95613
$$560$$ 0 0
$$561$$ 4.94427 0.208747
$$562$$ 0 0
$$563$$ −30.0810 −1.26776 −0.633882 0.773429i $$-0.718540\pi$$
−0.633882 + 0.773429i $$0.718540\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −7.65996 −0.321688
$$568$$ 0 0
$$569$$ −38.9443 −1.63263 −0.816314 0.577608i $$-0.803986\pi$$
−0.816314 + 0.577608i $$0.803986\pi$$
$$570$$ 0 0
$$571$$ 25.1246 1.05143 0.525716 0.850660i $$-0.323797\pi$$
0.525716 + 0.850660i $$0.323797\pi$$
$$572$$ 0 0
$$573$$ 2.67080 0.111574
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 5.65685 0.235498 0.117749 0.993043i $$-0.462432\pi$$
0.117749 + 0.993043i $$0.462432\pi$$
$$578$$ 0 0
$$579$$ 12.1803 0.506198
$$580$$ 0 0
$$581$$ 38.8328 1.61106
$$582$$ 0 0
$$583$$ 2.00310 0.0829601
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −5.40182 −0.222957 −0.111478 0.993767i $$-0.535559\pi$$
−0.111478 + 0.993767i $$0.535559\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 11.0557 0.454772
$$592$$ 0 0
$$593$$ 2.16073 0.0887304 0.0443652 0.999015i $$-0.485873\pi$$
0.0443652 + 0.999015i $$0.485873\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 13.9845 0.572348
$$598$$ 0 0
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ 16.8328 0.686625 0.343312 0.939221i $$-0.388451\pi$$
0.343312 + 0.939221i $$0.388451\pi$$
$$602$$ 0 0
$$603$$ 24.8369 1.01143
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −22.4211 −0.910044 −0.455022 0.890480i $$-0.650369\pi$$
−0.455022 + 0.890480i $$0.650369\pi$$
$$608$$ 0 0
$$609$$ −11.0557 −0.448001
$$610$$ 0 0
$$611$$ −46.2492 −1.87104
$$612$$ 0 0
$$613$$ 39.1853 1.58268 0.791340 0.611376i $$-0.209384\pi$$
0.791340 + 0.611376i $$0.209384\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −3.08347 −0.124136 −0.0620678 0.998072i $$-0.519770\pi$$
−0.0620678 + 0.998072i $$0.519770\pi$$
$$618$$ 0 0
$$619$$ 41.1246 1.65294 0.826469 0.562982i $$-0.190346\pi$$
0.826469 + 0.562982i $$0.190346\pi$$
$$620$$ 0 0
$$621$$ 4.94427 0.198407
$$622$$ 0 0
$$623$$ −8.32766 −0.333641
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −0.667701 −0.0266654
$$628$$ 0 0
$$629$$ 19.4164 0.774183
$$630$$ 0 0
$$631$$ −1.70820 −0.0680025 −0.0340013 0.999422i $$-0.510825\pi$$
−0.0340013 + 0.999422i $$0.510825\pi$$
$$632$$ 0 0
$$633$$ −13.4744 −0.535561
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 5.45052 0.215958
$$638$$ 0 0
$$639$$ 23.4164 0.926339
$$640$$ 0 0
$$641$$ −12.1115 −0.478374 −0.239187 0.970974i $$-0.576881\pi$$
−0.239187 + 0.970974i $$0.576881\pi$$
$$642$$ 0 0
$$643$$ 30.7000 1.21069 0.605346 0.795963i $$-0.293035\pi$$
0.605346 + 0.795963i $$0.293035\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 9.40802 0.369867 0.184934 0.982751i $$-0.440793\pi$$
0.184934 + 0.982751i $$0.440793\pi$$
$$648$$ 0 0
$$649$$ 1.16718 0.0458160
$$650$$ 0 0
$$651$$ −9.88854 −0.387563
$$652$$ 0 0
$$653$$ 22.2148 0.869331 0.434665 0.900592i $$-0.356867\pi$$
0.434665 + 0.900592i $$0.356867\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 11.7264 0.457489
$$658$$ 0 0
$$659$$ −44.9443 −1.75078 −0.875390 0.483417i $$-0.839395\pi$$
−0.875390 + 0.483417i $$0.839395\pi$$
$$660$$ 0 0
$$661$$ −26.0000 −1.01128 −0.505641 0.862744i $$-0.668744\pi$$
−0.505641 + 0.862744i $$0.668744\pi$$
$$662$$ 0 0
$$663$$ −35.2765 −1.37003
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −4.83153 −0.187078
$$668$$ 0 0
$$669$$ 16.4033 0.634186
$$670$$ 0 0
$$671$$ 8.94427 0.345290
$$672$$ 0 0
$$673$$ 4.62520 0.178288 0.0891442 0.996019i $$-0.471587\pi$$
0.0891442 + 0.996019i $$0.471587\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −42.7302 −1.64225 −0.821127 0.570746i $$-0.806654\pi$$
−0.821127 + 0.570746i $$0.806654\pi$$
$$678$$ 0 0
$$679$$ 39.4164 1.51266
$$680$$ 0 0
$$681$$ 5.12461 0.196376
$$682$$ 0 0
$$683$$ −19.5927 −0.749692 −0.374846 0.927087i $$-0.622304\pi$$
−0.374846 + 0.927087i $$0.622304\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −6.73722 −0.257041
$$688$$ 0 0
$$689$$ −14.2918 −0.544474
$$690$$ 0 0
$$691$$ 25.1246 0.955785 0.477893 0.878418i $$-0.341401\pi$$
0.477893 + 0.878418i $$0.341401\pi$$
$$692$$ 0 0
$$693$$ −4.83153 −0.183535
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −44.4295 −1.68289
$$698$$ 0 0
$$699$$ −11.0557 −0.418166
$$700$$ 0 0
$$701$$ 0.875388 0.0330630 0.0165315 0.999863i $$-0.494738\pi$$
0.0165315 + 0.999863i $$0.494738\pi$$
$$702$$ 0 0
$$703$$ −2.62210 −0.0988942
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −14.8098 −0.556981
$$708$$ 0 0
$$709$$ 33.4164 1.25498 0.627490 0.778625i $$-0.284082\pi$$
0.627490 + 0.778625i $$0.284082\pi$$
$$710$$ 0 0
$$711$$ −34.4721 −1.29281
$$712$$ 0 0
$$713$$ −4.32145 −0.161840
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 26.1235 0.975602
$$718$$ 0 0
$$719$$ 27.5967 1.02919 0.514593 0.857435i $$-0.327943\pi$$
0.514593 + 0.857435i $$0.327943\pi$$
$$720$$ 0 0
$$721$$ −16.5836 −0.617605
$$722$$ 0 0
$$723$$ 25.2008 0.937228
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −8.48528 −0.314702 −0.157351 0.987543i $$-0.550295\pi$$
−0.157351 + 0.987543i $$0.550295\pi$$
$$728$$ 0 0
$$729$$ 5.94427 0.220158
$$730$$ 0 0
$$731$$ −62.8328 −2.32396
$$732$$ 0 0
$$733$$ 33.9411 1.25364 0.626822 0.779162i $$-0.284355\pi$$
0.626822 + 0.779162i $$0.284355\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 8.48528 0.312559
$$738$$ 0 0
$$739$$ −18.8328 −0.692776 −0.346388 0.938091i $$-0.612592\pi$$
−0.346388 + 0.938091i $$0.612592\pi$$
$$740$$ 0 0
$$741$$ 4.76393 0.175007
$$742$$ 0 0
$$743$$ 2.62210 0.0961954 0.0480977 0.998843i $$-0.484684\pi$$
0.0480977 + 0.998843i $$0.484684\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 30.7000 1.12326
$$748$$ 0 0
$$749$$ −37.5279 −1.37124
$$750$$ 0 0
$$751$$ 20.5836 0.751106 0.375553 0.926801i $$-0.377453\pi$$
0.375553 + 0.926801i $$0.377453\pi$$
$$752$$ 0 0
$$753$$ 5.14678 0.187559
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 38.7727 1.40922 0.704608 0.709597i $$-0.251123\pi$$
0.704608 + 0.709597i $$0.251123\pi$$
$$758$$ 0 0
$$759$$ 0.721360 0.0261837
$$760$$ 0 0
$$761$$ −42.5410 −1.54211 −0.771055 0.636768i $$-0.780271\pi$$
−0.771055 + 0.636768i $$0.780271\pi$$
$$762$$ 0 0
$$763$$ −5.65685 −0.204792
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −8.32766 −0.300694
$$768$$ 0 0
$$769$$ −38.5410 −1.38982 −0.694912 0.719094i $$-0.744557\pi$$
−0.694912 + 0.719094i $$0.744557\pi$$
$$770$$ 0 0
$$771$$ −15.2361 −0.548714
$$772$$ 0 0
$$773$$ −5.86319 −0.210884 −0.105442 0.994425i $$-0.533626\pi$$
−0.105442 + 0.994425i $$0.533626\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 6.48218 0.232547
$$778$$ 0 0
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ 0 0
$$783$$ −20.4667 −0.731420
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −24.8369 −0.885338 −0.442669 0.896685i $$-0.645968\pi$$
−0.442669 + 0.896685i $$0.645968\pi$$
$$788$$ 0 0
$$789$$ 13.8885 0.494445
$$790$$ 0 0
$$791$$ −46.2492 −1.64443
$$792$$ 0 0
$$793$$ −63.8158 −2.26617
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 52.2958 1.85241 0.926206 0.377018i $$-0.123050\pi$$
0.926206 + 0.377018i $$0.123050\pi$$
$$798$$ 0 0
$$799$$ −62.8328 −2.22287
$$800$$ 0 0
$$801$$ −6.58359 −0.232620
$$802$$ 0 0
$$803$$ 4.00621 0.141376
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −13.0618 −0.459796
$$808$$ 0 0
$$809$$ −7.52786 −0.264666 −0.132333 0.991205i $$-0.542247\pi$$
−0.132333 + 0.991205i $$0.542247\pi$$
$$810$$ 0 0
$$811$$ 8.00000 0.280918 0.140459 0.990086i $$-0.455142\pi$$
0.140459 + 0.990086i $$0.455142\pi$$
$$812$$ 0 0
$$813$$ −18.4636 −0.647546
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 8.48528 0.296862
$$818$$ 0 0
$$819$$ 34.4721 1.20455
$$820$$ 0 0
$$821$$ 38.9443 1.35916 0.679582 0.733599i $$-0.262161\pi$$
0.679582 + 0.733599i $$0.262161\pi$$
$$822$$ 0 0
$$823$$ 35.9442 1.25294 0.626469 0.779447i $$-0.284500\pi$$
0.626469 + 0.779447i $$0.284500\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 7.86629 0.273538 0.136769 0.990603i $$-0.456328\pi$$
0.136769 + 0.990603i $$0.456328\pi$$
$$828$$ 0 0
$$829$$ 18.0000 0.625166 0.312583 0.949890i $$-0.398806\pi$$
0.312583 + 0.949890i $$0.398806\pi$$
$$830$$ 0 0
$$831$$ −19.4164 −0.673548
$$832$$ 0 0
$$833$$ 7.40492 0.256565
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −18.3060 −0.632747
$$838$$ 0 0
$$839$$ 49.3050 1.70220 0.851098 0.525007i $$-0.175937\pi$$
0.851098 + 0.525007i $$0.175937\pi$$
$$840$$ 0 0
$$841$$ −9.00000 −0.310345
$$842$$ 0 0
$$843$$ 22.2148 0.765117
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 29.4621 1.01233
$$848$$ 0 0
$$849$$ −2.11146 −0.0724650
$$850$$ 0 0
$$851$$ 2.83282 0.0971077
$$852$$ 0 0
$$853$$ 27.4589 0.940176 0.470088 0.882619i $$-0.344222\pi$$
0.470088 + 0.882619i $$0.344222\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −31.3190 −1.06984 −0.534919 0.844903i $$-0.679658\pi$$
−0.534919 + 0.844903i $$0.679658\pi$$
$$858$$ 0 0
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ −14.8328 −0.505501
$$862$$ 0 0
$$863$$ −7.86629 −0.267772 −0.133886 0.990997i $$-0.542746\pi$$
−0.133886 + 0.990997i $$0.542746\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −33.0671 −1.12302
$$868$$ 0 0
$$869$$ −11.7771 −0.399510
$$870$$ 0 0
$$871$$ −60.5410 −2.05135
$$872$$ 0 0
$$873$$ 31.1614 1.05465
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −13.9358 −0.470579 −0.235289 0.971925i $$-0.575604\pi$$
−0.235289 + 0.971925i $$0.575604\pi$$
$$878$$ 0 0
$$879$$ 2.29180 0.0773004
$$880$$ 0 0
$$881$$ 0.652476 0.0219825 0.0109912 0.999940i $$-0.496501\pi$$
0.0109912 + 0.999940i $$0.496501\pi$$
$$882$$ 0 0
$$883$$ 52.9148 1.78072 0.890362 0.455253i $$-0.150451\pi$$
0.890362 + 0.455253i $$0.150451\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 49.0547 1.64710 0.823548 0.567247i $$-0.191991\pi$$
0.823548 + 0.567247i $$0.191991\pi$$
$$888$$ 0 0
$$889$$ −55.4164 −1.85861
$$890$$ 0 0
$$891$$ −2.06888 −0.0693102
$$892$$ 0 0
$$893$$ 8.48528 0.283949
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −5.14678 −0.171846
$$898$$ 0 0
$$899$$ 17.8885 0.596616
$$900$$ 0 0
$$901$$ −19.4164 −0.646854
$$902$$ 0 0
$$903$$ −20.9768 −0.698063
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 38.5663 1.28057 0.640287 0.768136i $$-0.278815\pi$$
0.640287 + 0.768136i $$0.278815\pi$$
$$908$$ 0 0
$$909$$ −11.7082 −0.388337
$$910$$ 0 0
$$911$$ −46.4721 −1.53969 −0.769845 0.638231i $$-0.779667\pi$$
−0.769845 + 0.638231i $$0.779667\pi$$
$$912$$ 0 0
$$913$$ 10.4884 0.347115
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 26.8328 0.885133 0.442566 0.896736i $$-0.354068\pi$$
0.442566 + 0.896736i $$0.354068\pi$$
$$920$$ 0 0
$$921$$ −4.76393 −0.156977
$$922$$ 0 0
$$923$$ −57.0786 −1.87877
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −13.1105 −0.430605
$$928$$ 0 0
$$929$$ 7.52786 0.246981 0.123491 0.992346i $$-0.460591\pi$$
0.123491 + 0.992346i $$0.460591\pi$$
$$930$$ 0 0
$$931$$ −1.00000 −0.0327737
$$932$$ 0 0
$$933$$ 8.48528 0.277796
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 43.1915 1.41101 0.705503 0.708707i $$-0.250721\pi$$
0.705503 + 0.708707i $$0.250721\pi$$
$$938$$ 0 0
$$939$$ −9.16718 −0.299160
$$940$$ 0 0
$$941$$ −38.9443 −1.26955 −0.634773 0.772698i $$-0.718907\pi$$
−0.634773 + 0.772698i $$0.718907\pi$$
$$942$$ 0 0
$$943$$ −6.48218 −0.211089
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 47.6706 1.54909 0.774543 0.632521i $$-0.217980\pi$$
0.774543 + 0.632521i $$0.217980\pi$$
$$948$$ 0 0
$$949$$ −28.5836 −0.927863
$$950$$ 0 0
$$951$$ −7.81966 −0.253570
$$952$$ 0 0
$$953$$ 36.5632 1.18440 0.592199 0.805791i $$-0.298260\pi$$
0.592199 + 0.805791i $$0.298260\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −2.98605 −0.0965253
$$958$$ 0 0
$$959$$ −20.9443 −0.676326
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −29.6684 −0.956050
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 2.41577 0.0776858 0.0388429 0.999245i $$-0.487633\pi$$
0.0388429 + 0.999245i $$0.487633\pi$$
$$968$$ 0 0
$$969$$ 6.47214 0.207915
$$970$$ 0 0
$$971$$ −40.3607 −1.29524 −0.647618 0.761965i $$-0.724235\pi$$
−0.647618 + 0.761965i $$0.724235\pi$$
$$972$$ 0 0
$$973$$ −6.48218 −0.207809
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −24.9945 −0.799644 −0.399822 0.916593i $$-0.630928\pi$$
−0.399822 + 0.916593i $$0.630928\pi$$
$$978$$ 0 0
$$979$$ −2.24922 −0.0718855
$$980$$ 0 0
$$981$$ −4.47214 −0.142784
$$982$$ 0 0
$$983$$ 30.2387 0.964464 0.482232 0.876044i $$-0.339826\pi$$
0.482232 + 0.876044i $$0.339826\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −20.9768 −0.667698
$$988$$ 0 0
$$989$$ −9.16718 −0.291500
$$990$$ 0 0
$$991$$ 38.8328 1.23357 0.616783 0.787134i $$-0.288436\pi$$
0.616783 + 0.787134i $$0.288436\pi$$
$$992$$ 0 0
$$993$$ −6.48218 −0.205706
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 11.7264 0.371378 0.185689 0.982609i $$-0.440548\pi$$
0.185689 + 0.982609i $$0.440548\pi$$
$$998$$ 0 0
$$999$$ 12.0000 0.379663
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 1900.2.a.j.1.2 4
4.3 odd 2 7600.2.a.ce.1.3 4
5.2 odd 4 380.2.c.a.229.3 yes 4
5.3 odd 4 380.2.c.a.229.2 4
5.4 even 2 inner 1900.2.a.j.1.3 4
15.2 even 4 3420.2.f.a.1369.1 4
15.8 even 4 3420.2.f.a.1369.2 4
20.3 even 4 1520.2.d.f.609.3 4
20.7 even 4 1520.2.d.f.609.2 4
20.19 odd 2 7600.2.a.ce.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.c.a.229.2 4 5.3 odd 4
380.2.c.a.229.3 yes 4 5.2 odd 4
1520.2.d.f.609.2 4 20.7 even 4
1520.2.d.f.609.3 4 20.3 even 4
1900.2.a.j.1.2 4 1.1 even 1 trivial
1900.2.a.j.1.3 4 5.4 even 2 inner
3420.2.f.a.1369.1 4 15.2 even 4
3420.2.f.a.1369.2 4 15.8 even 4
7600.2.a.ce.1.2 4 20.19 odd 2
7600.2.a.ce.1.3 4 4.3 odd 2