# Properties

 Label 3920.2.a.bv.1.1 Level $3920$ Weight $2$ Character 3920.1 Self dual yes Analytic conductor $31.301$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3920.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.3013575923$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 3920.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} +O(q^{10})$$ $$q-0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} +0.828427 q^{11} +4.82843 q^{13} +0.414214 q^{15} -4.82843 q^{17} +2.82843 q^{19} -0.414214 q^{23} +1.00000 q^{25} +2.41421 q^{27} -1.00000 q^{29} -6.00000 q^{31} -0.343146 q^{33} -2.00000 q^{39} +7.82843 q^{41} -3.58579 q^{43} +2.82843 q^{45} +2.00000 q^{47} +2.00000 q^{51} -1.17157 q^{53} -0.828427 q^{55} -1.17157 q^{57} +4.48528 q^{59} -5.48528 q^{61} -4.82843 q^{65} -9.58579 q^{67} +0.171573 q^{69} -4.48528 q^{71} +0.828427 q^{73} -0.414214 q^{75} -14.8284 q^{79} +7.48528 q^{81} +13.7279 q^{83} +4.82843 q^{85} +0.414214 q^{87} +8.65685 q^{89} +2.48528 q^{93} -2.82843 q^{95} -11.6569 q^{97} -2.34315 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{5} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{5} - 4q^{11} + 4q^{13} - 2q^{15} - 4q^{17} + 2q^{23} + 2q^{25} + 2q^{27} - 2q^{29} - 12q^{31} - 12q^{33} - 4q^{39} + 10q^{41} - 10q^{43} + 4q^{47} + 4q^{51} - 8q^{53} + 4q^{55} - 8q^{57} - 8q^{59} + 6q^{61} - 4q^{65} - 22q^{67} + 6q^{69} + 8q^{71} - 4q^{73} + 2q^{75} - 24q^{79} - 2q^{81} + 2q^{83} + 4q^{85} - 2q^{87} + 6q^{89} - 12q^{93} - 12q^{97} - 16q^{99} + O(q^{100})$$

## 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
$$3$$ −0.414214 −0.239146 −0.119573 0.992825i $$-0.538153\pi$$
−0.119573 + 0.992825i $$0.538153\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −2.82843 −0.942809
$$10$$ 0 0
$$11$$ 0.828427 0.249780 0.124890 0.992171i $$-0.460142\pi$$
0.124890 + 0.992171i $$0.460142\pi$$
$$12$$ 0 0
$$13$$ 4.82843 1.33916 0.669582 0.742738i $$-0.266473\pi$$
0.669582 + 0.742738i $$0.266473\pi$$
$$14$$ 0 0
$$15$$ 0.414214 0.106949
$$16$$ 0 0
$$17$$ −4.82843 −1.17107 −0.585533 0.810649i $$-0.699115\pi$$
−0.585533 + 0.810649i $$0.699115\pi$$
$$18$$ 0 0
$$19$$ 2.82843 0.648886 0.324443 0.945905i $$-0.394823\pi$$
0.324443 + 0.945905i $$0.394823\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −0.414214 −0.0863695 −0.0431847 0.999067i $$-0.513750\pi$$
−0.0431847 + 0.999067i $$0.513750\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 2.41421 0.464616
$$28$$ 0 0
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 0 0
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ 0 0
$$33$$ −0.343146 −0.0597340
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 7.82843 1.22259 0.611297 0.791401i $$-0.290648\pi$$
0.611297 + 0.791401i $$0.290648\pi$$
$$42$$ 0 0
$$43$$ −3.58579 −0.546827 −0.273414 0.961897i $$-0.588153\pi$$
−0.273414 + 0.961897i $$0.588153\pi$$
$$44$$ 0 0
$$45$$ 2.82843 0.421637
$$46$$ 0 0
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ −1.17157 −0.160928 −0.0804640 0.996758i $$-0.525640\pi$$
−0.0804640 + 0.996758i $$0.525640\pi$$
$$54$$ 0 0
$$55$$ −0.828427 −0.111705
$$56$$ 0 0
$$57$$ −1.17157 −0.155179
$$58$$ 0 0
$$59$$ 4.48528 0.583934 0.291967 0.956428i $$-0.405690\pi$$
0.291967 + 0.956428i $$0.405690\pi$$
$$60$$ 0 0
$$61$$ −5.48528 −0.702318 −0.351159 0.936316i $$-0.614212\pi$$
−0.351159 + 0.936316i $$0.614212\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −4.82843 −0.598893
$$66$$ 0 0
$$67$$ −9.58579 −1.17109 −0.585545 0.810640i $$-0.699119\pi$$
−0.585545 + 0.810640i $$0.699119\pi$$
$$68$$ 0 0
$$69$$ 0.171573 0.0206549
$$70$$ 0 0
$$71$$ −4.48528 −0.532305 −0.266152 0.963931i $$-0.585752\pi$$
−0.266152 + 0.963931i $$0.585752\pi$$
$$72$$ 0 0
$$73$$ 0.828427 0.0969601 0.0484800 0.998824i $$-0.484562\pi$$
0.0484800 + 0.998824i $$0.484562\pi$$
$$74$$ 0 0
$$75$$ −0.414214 −0.0478293
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −14.8284 −1.66833 −0.834164 0.551516i $$-0.814049\pi$$
−0.834164 + 0.551516i $$0.814049\pi$$
$$80$$ 0 0
$$81$$ 7.48528 0.831698
$$82$$ 0 0
$$83$$ 13.7279 1.50684 0.753418 0.657542i $$-0.228404\pi$$
0.753418 + 0.657542i $$0.228404\pi$$
$$84$$ 0 0
$$85$$ 4.82843 0.523716
$$86$$ 0 0
$$87$$ 0.414214 0.0444084
$$88$$ 0 0
$$89$$ 8.65685 0.917625 0.458812 0.888533i $$-0.348275\pi$$
0.458812 + 0.888533i $$0.348275\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 2.48528 0.257712
$$94$$ 0 0
$$95$$ −2.82843 −0.290191
$$96$$ 0 0
$$97$$ −11.6569 −1.18357 −0.591787 0.806094i $$-0.701577\pi$$
−0.591787 + 0.806094i $$0.701577\pi$$
$$98$$ 0 0
$$99$$ −2.34315 −0.235495
$$100$$ 0 0
$$101$$ 10.3137 1.02625 0.513126 0.858313i $$-0.328487\pi$$
0.513126 + 0.858313i $$0.328487\pi$$
$$102$$ 0 0
$$103$$ −2.41421 −0.237880 −0.118940 0.992901i $$-0.537950\pi$$
−0.118940 + 0.992901i $$0.537950\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −11.2426 −1.08687 −0.543434 0.839452i $$-0.682876\pi$$
−0.543434 + 0.839452i $$0.682876\pi$$
$$108$$ 0 0
$$109$$ −13.4853 −1.29166 −0.645828 0.763483i $$-0.723488\pi$$
−0.645828 + 0.763483i $$0.723488\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −4.48528 −0.421940 −0.210970 0.977493i $$-0.567662\pi$$
−0.210970 + 0.977493i $$0.567662\pi$$
$$114$$ 0 0
$$115$$ 0.414214 0.0386256
$$116$$ 0 0
$$117$$ −13.6569 −1.26258
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.3137 −0.937610
$$122$$ 0 0
$$123$$ −3.24264 −0.292379
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 9.31371 0.826458 0.413229 0.910627i $$-0.364401\pi$$
0.413229 + 0.910627i $$0.364401\pi$$
$$128$$ 0 0
$$129$$ 1.48528 0.130772
$$130$$ 0 0
$$131$$ −19.3137 −1.68745 −0.843723 0.536778i $$-0.819641\pi$$
−0.843723 + 0.536778i $$0.819641\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −2.41421 −0.207782
$$136$$ 0 0
$$137$$ −9.65685 −0.825041 −0.412520 0.910948i $$-0.635351\pi$$
−0.412520 + 0.910948i $$0.635351\pi$$
$$138$$ 0 0
$$139$$ −16.1421 −1.36916 −0.684579 0.728939i $$-0.740014\pi$$
−0.684579 + 0.728939i $$0.740014\pi$$
$$140$$ 0 0
$$141$$ −0.828427 −0.0697661
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ 1.00000 0.0830455
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −2.17157 −0.177902 −0.0889511 0.996036i $$-0.528351\pi$$
−0.0889511 + 0.996036i $$0.528351\pi$$
$$150$$ 0 0
$$151$$ −11.6569 −0.948621 −0.474311 0.880358i $$-0.657303\pi$$
−0.474311 + 0.880358i $$0.657303\pi$$
$$152$$ 0 0
$$153$$ 13.6569 1.10409
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ −17.3137 −1.38178 −0.690892 0.722958i $$-0.742782\pi$$
−0.690892 + 0.722958i $$0.742782\pi$$
$$158$$ 0 0
$$159$$ 0.485281 0.0384853
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −12.3431 −0.966790 −0.483395 0.875402i $$-0.660596\pi$$
−0.483395 + 0.875402i $$0.660596\pi$$
$$164$$ 0 0
$$165$$ 0.343146 0.0267139
$$166$$ 0 0
$$167$$ 22.4142 1.73446 0.867232 0.497904i $$-0.165897\pi$$
0.867232 + 0.497904i $$0.165897\pi$$
$$168$$ 0 0
$$169$$ 10.3137 0.793362
$$170$$ 0 0
$$171$$ −8.00000 −0.611775
$$172$$ 0 0
$$173$$ −3.31371 −0.251937 −0.125968 0.992034i $$-0.540204\pi$$
−0.125968 + 0.992034i $$0.540204\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −1.85786 −0.139646
$$178$$ 0 0
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ −2.65685 −0.197482 −0.0987412 0.995113i $$-0.531482\pi$$
−0.0987412 + 0.995113i $$0.531482\pi$$
$$182$$ 0 0
$$183$$ 2.27208 0.167957
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 12.8284 0.928232 0.464116 0.885774i $$-0.346372\pi$$
0.464116 + 0.885774i $$0.346372\pi$$
$$192$$ 0 0
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 0 0
$$195$$ 2.00000 0.143223
$$196$$ 0 0
$$197$$ −12.3431 −0.879413 −0.439706 0.898142i $$-0.644917\pi$$
−0.439706 + 0.898142i $$0.644917\pi$$
$$198$$ 0 0
$$199$$ −9.65685 −0.684556 −0.342278 0.939599i $$-0.611199\pi$$
−0.342278 + 0.939599i $$0.611199\pi$$
$$200$$ 0 0
$$201$$ 3.97056 0.280062
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −7.82843 −0.546761
$$206$$ 0 0
$$207$$ 1.17157 0.0814299
$$208$$ 0 0
$$209$$ 2.34315 0.162079
$$210$$ 0 0
$$211$$ −20.4853 −1.41026 −0.705132 0.709076i $$-0.749112\pi$$
−0.705132 + 0.709076i $$0.749112\pi$$
$$212$$ 0 0
$$213$$ 1.85786 0.127299
$$214$$ 0 0
$$215$$ 3.58579 0.244549
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −0.343146 −0.0231876
$$220$$ 0 0
$$221$$ −23.3137 −1.56825
$$222$$ 0 0
$$223$$ 0.343146 0.0229787 0.0114894 0.999934i $$-0.496343\pi$$
0.0114894 + 0.999934i $$0.496343\pi$$
$$224$$ 0 0
$$225$$ −2.82843 −0.188562
$$226$$ 0 0
$$227$$ −6.97056 −0.462652 −0.231326 0.972876i $$-0.574306\pi$$
−0.231326 + 0.972876i $$0.574306\pi$$
$$228$$ 0 0
$$229$$ 11.6569 0.770307 0.385153 0.922853i $$-0.374149\pi$$
0.385153 + 0.922853i $$0.374149\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −16.8284 −1.10247 −0.551233 0.834351i $$-0.685843\pi$$
−0.551233 + 0.834351i $$0.685843\pi$$
$$234$$ 0 0
$$235$$ −2.00000 −0.130466
$$236$$ 0 0
$$237$$ 6.14214 0.398975
$$238$$ 0 0
$$239$$ 21.3137 1.37867 0.689335 0.724443i $$-0.257903\pi$$
0.689335 + 0.724443i $$0.257903\pi$$
$$240$$ 0 0
$$241$$ −27.6569 −1.78153 −0.890767 0.454460i $$-0.849832\pi$$
−0.890767 + 0.454460i $$0.849832\pi$$
$$242$$ 0 0
$$243$$ −10.3431 −0.663513
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 13.6569 0.868965
$$248$$ 0 0
$$249$$ −5.68629 −0.360354
$$250$$ 0 0
$$251$$ −9.31371 −0.587876 −0.293938 0.955824i $$-0.594966\pi$$
−0.293938 + 0.955824i $$0.594966\pi$$
$$252$$ 0 0
$$253$$ −0.343146 −0.0215734
$$254$$ 0 0
$$255$$ −2.00000 −0.125245
$$256$$ 0 0
$$257$$ −6.34315 −0.395675 −0.197837 0.980235i $$-0.563392\pi$$
−0.197837 + 0.980235i $$0.563392\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.82843 0.175075
$$262$$ 0 0
$$263$$ 29.0416 1.79078 0.895392 0.445279i $$-0.146896\pi$$
0.895392 + 0.445279i $$0.146896\pi$$
$$264$$ 0 0
$$265$$ 1.17157 0.0719691
$$266$$ 0 0
$$267$$ −3.58579 −0.219447
$$268$$ 0 0
$$269$$ −20.4558 −1.24721 −0.623607 0.781738i $$-0.714334\pi$$
−0.623607 + 0.781738i $$0.714334\pi$$
$$270$$ 0 0
$$271$$ 16.4853 1.00141 0.500705 0.865618i $$-0.333074\pi$$
0.500705 + 0.865618i $$0.333074\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0.828427 0.0499560
$$276$$ 0 0
$$277$$ 16.1421 0.969887 0.484943 0.874546i $$-0.338840\pi$$
0.484943 + 0.874546i $$0.338840\pi$$
$$278$$ 0 0
$$279$$ 16.9706 1.01600
$$280$$ 0 0
$$281$$ −30.2843 −1.80661 −0.903304 0.429001i $$-0.858866\pi$$
−0.903304 + 0.429001i $$0.858866\pi$$
$$282$$ 0 0
$$283$$ 14.0000 0.832214 0.416107 0.909316i $$-0.363394\pi$$
0.416107 + 0.909316i $$0.363394\pi$$
$$284$$ 0 0
$$285$$ 1.17157 0.0693980
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 6.31371 0.371395
$$290$$ 0 0
$$291$$ 4.82843 0.283047
$$292$$ 0 0
$$293$$ 16.0000 0.934730 0.467365 0.884064i $$-0.345203\pi$$
0.467365 + 0.884064i $$0.345203\pi$$
$$294$$ 0 0
$$295$$ −4.48528 −0.261143
$$296$$ 0 0
$$297$$ 2.00000 0.116052
$$298$$ 0 0
$$299$$ −2.00000 −0.115663
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −4.27208 −0.245424
$$304$$ 0 0
$$305$$ 5.48528 0.314086
$$306$$ 0 0
$$307$$ −4.75736 −0.271517 −0.135758 0.990742i $$-0.543347\pi$$
−0.135758 + 0.990742i $$0.543347\pi$$
$$308$$ 0 0
$$309$$ 1.00000 0.0568880
$$310$$ 0 0
$$311$$ −13.1716 −0.746891 −0.373446 0.927652i $$-0.621824\pi$$
−0.373446 + 0.927652i $$0.621824\pi$$
$$312$$ 0 0
$$313$$ 6.34315 0.358536 0.179268 0.983800i $$-0.442627\pi$$
0.179268 + 0.983800i $$0.442627\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −13.7990 −0.775028 −0.387514 0.921864i $$-0.626666\pi$$
−0.387514 + 0.921864i $$0.626666\pi$$
$$318$$ 0 0
$$319$$ −0.828427 −0.0463830
$$320$$ 0 0
$$321$$ 4.65685 0.259920
$$322$$ 0 0
$$323$$ −13.6569 −0.759888
$$324$$ 0 0
$$325$$ 4.82843 0.267833
$$326$$ 0 0
$$327$$ 5.58579 0.308895
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −22.9706 −1.26258 −0.631288 0.775548i $$-0.717473\pi$$
−0.631288 + 0.775548i $$0.717473\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 9.58579 0.523727
$$336$$ 0 0
$$337$$ 9.17157 0.499607 0.249804 0.968296i $$-0.419634\pi$$
0.249804 + 0.968296i $$0.419634\pi$$
$$338$$ 0 0
$$339$$ 1.85786 0.100905
$$340$$ 0 0
$$341$$ −4.97056 −0.269171
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −0.171573 −0.00923717
$$346$$ 0 0
$$347$$ −7.92893 −0.425647 −0.212824 0.977091i $$-0.568266\pi$$
−0.212824 + 0.977091i $$0.568266\pi$$
$$348$$ 0 0
$$349$$ 15.3431 0.821300 0.410650 0.911793i $$-0.365302\pi$$
0.410650 + 0.911793i $$0.365302\pi$$
$$350$$ 0 0
$$351$$ 11.6569 0.622197
$$352$$ 0 0
$$353$$ 26.8284 1.42793 0.713967 0.700180i $$-0.246897\pi$$
0.713967 + 0.700180i $$0.246897\pi$$
$$354$$ 0 0
$$355$$ 4.48528 0.238054
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 10.0000 0.527780 0.263890 0.964553i $$-0.414994\pi$$
0.263890 + 0.964553i $$0.414994\pi$$
$$360$$ 0 0
$$361$$ −11.0000 −0.578947
$$362$$ 0 0
$$363$$ 4.27208 0.224226
$$364$$ 0 0
$$365$$ −0.828427 −0.0433619
$$366$$ 0 0
$$367$$ −2.75736 −0.143933 −0.0719665 0.997407i $$-0.522927\pi$$
−0.0719665 + 0.997407i $$0.522927\pi$$
$$368$$ 0 0
$$369$$ −22.1421 −1.15267
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −20.9706 −1.08581 −0.542907 0.839793i $$-0.682677\pi$$
−0.542907 + 0.839793i $$0.682677\pi$$
$$374$$ 0 0
$$375$$ 0.414214 0.0213899
$$376$$ 0 0
$$377$$ −4.82843 −0.248677
$$378$$ 0 0
$$379$$ −26.8284 −1.37808 −0.689042 0.724722i $$-0.741968\pi$$
−0.689042 + 0.724722i $$0.741968\pi$$
$$380$$ 0 0
$$381$$ −3.85786 −0.197644
$$382$$ 0 0
$$383$$ −2.89949 −0.148157 −0.0740786 0.997252i $$-0.523602\pi$$
−0.0740786 + 0.997252i $$0.523602\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 10.1421 0.515554
$$388$$ 0 0
$$389$$ 23.6569 1.19945 0.599725 0.800206i $$-0.295277\pi$$
0.599725 + 0.800206i $$0.295277\pi$$
$$390$$ 0 0
$$391$$ 2.00000 0.101144
$$392$$ 0 0
$$393$$ 8.00000 0.403547
$$394$$ 0 0
$$395$$ 14.8284 0.746099
$$396$$ 0 0
$$397$$ 16.6274 0.834506 0.417253 0.908790i $$-0.362993\pi$$
0.417253 + 0.908790i $$0.362993\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 30.3137 1.51379 0.756897 0.653534i $$-0.226714\pi$$
0.756897 + 0.653534i $$0.226714\pi$$
$$402$$ 0 0
$$403$$ −28.9706 −1.44313
$$404$$ 0 0
$$405$$ −7.48528 −0.371947
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 14.7990 0.731763 0.365881 0.930661i $$-0.380768\pi$$
0.365881 + 0.930661i $$0.380768\pi$$
$$410$$ 0 0
$$411$$ 4.00000 0.197305
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −13.7279 −0.673877
$$416$$ 0 0
$$417$$ 6.68629 0.327429
$$418$$ 0 0
$$419$$ −0.686292 −0.0335275 −0.0167638 0.999859i $$-0.505336\pi$$
−0.0167638 + 0.999859i $$0.505336\pi$$
$$420$$ 0 0
$$421$$ 13.4853 0.657232 0.328616 0.944464i $$-0.393418\pi$$
0.328616 + 0.944464i $$0.393418\pi$$
$$422$$ 0 0
$$423$$ −5.65685 −0.275046
$$424$$ 0 0
$$425$$ −4.82843 −0.234213
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −1.65685 −0.0799937
$$430$$ 0 0
$$431$$ −17.7990 −0.857347 −0.428674 0.903459i $$-0.641019\pi$$
−0.428674 + 0.903459i $$0.641019\pi$$
$$432$$ 0 0
$$433$$ −7.79899 −0.374796 −0.187398 0.982284i $$-0.560005\pi$$
−0.187398 + 0.982284i $$0.560005\pi$$
$$434$$ 0 0
$$435$$ −0.414214 −0.0198600
$$436$$ 0 0
$$437$$ −1.17157 −0.0560439
$$438$$ 0 0
$$439$$ −33.9411 −1.61992 −0.809961 0.586484i $$-0.800512\pi$$
−0.809961 + 0.586484i $$0.800512\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −30.2132 −1.43547 −0.717736 0.696315i $$-0.754822\pi$$
−0.717736 + 0.696315i $$0.754822\pi$$
$$444$$ 0 0
$$445$$ −8.65685 −0.410374
$$446$$ 0 0
$$447$$ 0.899495 0.0425447
$$448$$ 0 0
$$449$$ 3.82843 0.180675 0.0903373 0.995911i $$-0.471205\pi$$
0.0903373 + 0.995911i $$0.471205\pi$$
$$450$$ 0 0
$$451$$ 6.48528 0.305380
$$452$$ 0 0
$$453$$ 4.82843 0.226859
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 24.2843 1.13597 0.567985 0.823039i $$-0.307723\pi$$
0.567985 + 0.823039i $$0.307723\pi$$
$$458$$ 0 0
$$459$$ −11.6569 −0.544095
$$460$$ 0 0
$$461$$ −41.3137 −1.92417 −0.962086 0.272748i $$-0.912068\pi$$
−0.962086 + 0.272748i $$0.912068\pi$$
$$462$$ 0 0
$$463$$ 37.0416 1.72147 0.860735 0.509053i $$-0.170004\pi$$
0.860735 + 0.509053i $$0.170004\pi$$
$$464$$ 0 0
$$465$$ −2.48528 −0.115252
$$466$$ 0 0
$$467$$ −3.10051 −0.143474 −0.0717371 0.997424i $$-0.522854\pi$$
−0.0717371 + 0.997424i $$0.522854\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 7.17157 0.330449
$$472$$ 0 0
$$473$$ −2.97056 −0.136587
$$474$$ 0 0
$$475$$ 2.82843 0.129777
$$476$$ 0 0
$$477$$ 3.31371 0.151724
$$478$$ 0 0
$$479$$ −35.6569 −1.62920 −0.814602 0.580021i $$-0.803044\pi$$
−0.814602 + 0.580021i $$0.803044\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 11.6569 0.529310
$$486$$ 0 0
$$487$$ −4.34315 −0.196807 −0.0984034 0.995147i $$-0.531374\pi$$
−0.0984034 + 0.995147i $$0.531374\pi$$
$$488$$ 0 0
$$489$$ 5.11270 0.231204
$$490$$ 0 0
$$491$$ −9.31371 −0.420322 −0.210161 0.977667i $$-0.567399\pi$$
−0.210161 + 0.977667i $$0.567399\pi$$
$$492$$ 0 0
$$493$$ 4.82843 0.217461
$$494$$ 0 0
$$495$$ 2.34315 0.105317
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0.828427 0.0370855 0.0185427 0.999828i $$-0.494097\pi$$
0.0185427 + 0.999828i $$0.494097\pi$$
$$500$$ 0 0
$$501$$ −9.28427 −0.414791
$$502$$ 0 0
$$503$$ −15.8701 −0.707611 −0.353805 0.935319i $$-0.615113\pi$$
−0.353805 + 0.935319i $$0.615113\pi$$
$$504$$ 0 0
$$505$$ −10.3137 −0.458954
$$506$$ 0 0
$$507$$ −4.27208 −0.189730
$$508$$ 0 0
$$509$$ −13.3431 −0.591425 −0.295712 0.955277i $$-0.595557\pi$$
−0.295712 + 0.955277i $$0.595557\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 6.82843 0.301482
$$514$$ 0 0
$$515$$ 2.41421 0.106383
$$516$$ 0 0
$$517$$ 1.65685 0.0728684
$$518$$ 0 0
$$519$$ 1.37258 0.0602497
$$520$$ 0 0
$$521$$ −14.9706 −0.655872 −0.327936 0.944700i $$-0.606353\pi$$
−0.327936 + 0.944700i $$0.606353\pi$$
$$522$$ 0 0
$$523$$ 35.6569 1.55917 0.779583 0.626299i $$-0.215431\pi$$
0.779583 + 0.626299i $$0.215431\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 28.9706 1.26198
$$528$$ 0 0
$$529$$ −22.8284 −0.992540
$$530$$ 0 0
$$531$$ −12.6863 −0.550538
$$532$$ 0 0
$$533$$ 37.7990 1.63726
$$534$$ 0 0
$$535$$ 11.2426 0.486062
$$536$$ 0 0
$$537$$ −4.14214 −0.178746
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 7.34315 0.315706 0.157853 0.987463i $$-0.449543\pi$$
0.157853 + 0.987463i $$0.449543\pi$$
$$542$$ 0 0
$$543$$ 1.10051 0.0472272
$$544$$ 0 0
$$545$$ 13.4853 0.577646
$$546$$ 0 0
$$547$$ −24.8995 −1.06463 −0.532313 0.846548i $$-0.678677\pi$$
−0.532313 + 0.846548i $$0.678677\pi$$
$$548$$ 0 0
$$549$$ 15.5147 0.662152
$$550$$ 0 0
$$551$$ −2.82843 −0.120495
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 22.2843 0.944215 0.472107 0.881541i $$-0.343493\pi$$
0.472107 + 0.881541i $$0.343493\pi$$
$$558$$ 0 0
$$559$$ −17.3137 −0.732292
$$560$$ 0 0
$$561$$ 1.65685 0.0699524
$$562$$ 0 0
$$563$$ 41.7279 1.75862 0.879311 0.476248i $$-0.158003\pi$$
0.879311 + 0.476248i $$0.158003\pi$$
$$564$$ 0 0
$$565$$ 4.48528 0.188697
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 7.65685 0.320992 0.160496 0.987036i $$-0.448691\pi$$
0.160496 + 0.987036i $$0.448691\pi$$
$$570$$ 0 0
$$571$$ −9.17157 −0.383818 −0.191909 0.981413i $$-0.561468\pi$$
−0.191909 + 0.981413i $$0.561468\pi$$
$$572$$ 0 0
$$573$$ −5.31371 −0.221983
$$574$$ 0 0
$$575$$ −0.414214 −0.0172739
$$576$$ 0 0
$$577$$ 43.9411 1.82929 0.914646 0.404255i $$-0.132469\pi$$
0.914646 + 0.404255i $$0.132469\pi$$
$$578$$ 0 0
$$579$$ 0.828427 0.0344283
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −0.970563 −0.0401966
$$584$$ 0 0
$$585$$ 13.6569 0.564641
$$586$$ 0 0
$$587$$ −34.2843 −1.41506 −0.707532 0.706682i $$-0.750191\pi$$
−0.707532 + 0.706682i $$0.750191\pi$$
$$588$$ 0 0
$$589$$ −16.9706 −0.699260
$$590$$ 0 0
$$591$$ 5.11270 0.210308
$$592$$ 0 0
$$593$$ −4.20101 −0.172515 −0.0862574 0.996273i $$-0.527491\pi$$
−0.0862574 + 0.996273i $$0.527491\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 4.00000 0.163709
$$598$$ 0 0
$$599$$ −6.34315 −0.259174 −0.129587 0.991568i $$-0.541365\pi$$
−0.129587 + 0.991568i $$0.541365\pi$$
$$600$$ 0 0
$$601$$ −19.6569 −0.801820 −0.400910 0.916117i $$-0.631306\pi$$
−0.400910 + 0.916117i $$0.631306\pi$$
$$602$$ 0 0
$$603$$ 27.1127 1.10411
$$604$$ 0 0
$$605$$ 10.3137 0.419312
$$606$$ 0 0
$$607$$ 38.2132 1.55103 0.775513 0.631332i $$-0.217491\pi$$
0.775513 + 0.631332i $$0.217491\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 9.65685 0.390675
$$612$$ 0 0
$$613$$ 35.4558 1.43205 0.716024 0.698076i $$-0.245960\pi$$
0.716024 + 0.698076i $$0.245960\pi$$
$$614$$ 0 0
$$615$$ 3.24264 0.130756
$$616$$ 0 0
$$617$$ 11.3137 0.455473 0.227736 0.973723i $$-0.426868\pi$$
0.227736 + 0.973723i $$0.426868\pi$$
$$618$$ 0 0
$$619$$ 25.5147 1.02552 0.512762 0.858531i $$-0.328622\pi$$
0.512762 + 0.858531i $$0.328622\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −0.970563 −0.0387605
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 20.1421 0.801846 0.400923 0.916112i $$-0.368690\pi$$
0.400923 + 0.916112i $$0.368690\pi$$
$$632$$ 0 0
$$633$$ 8.48528 0.337260
$$634$$ 0 0
$$635$$ −9.31371 −0.369603
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 12.6863 0.501862
$$640$$ 0 0
$$641$$ 31.4853 1.24359 0.621797 0.783179i $$-0.286403\pi$$
0.621797 + 0.783179i $$0.286403\pi$$
$$642$$ 0 0
$$643$$ 26.2843 1.03655 0.518275 0.855214i $$-0.326574\pi$$
0.518275 + 0.855214i $$0.326574\pi$$
$$644$$ 0 0
$$645$$ −1.48528 −0.0584829
$$646$$ 0 0
$$647$$ −31.0416 −1.22037 −0.610186 0.792258i $$-0.708905\pi$$
−0.610186 + 0.792258i $$0.708905\pi$$
$$648$$ 0 0
$$649$$ 3.71573 0.145855
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −19.1716 −0.750242 −0.375121 0.926976i $$-0.622399\pi$$
−0.375121 + 0.926976i $$0.622399\pi$$
$$654$$ 0 0
$$655$$ 19.3137 0.754649
$$656$$ 0 0
$$657$$ −2.34315 −0.0914148
$$658$$ 0 0
$$659$$ −21.1716 −0.824727 −0.412364 0.911019i $$-0.635297\pi$$
−0.412364 + 0.911019i $$0.635297\pi$$
$$660$$ 0 0
$$661$$ −31.8284 −1.23798 −0.618991 0.785398i $$-0.712458\pi$$
−0.618991 + 0.785398i $$0.712458\pi$$
$$662$$ 0 0
$$663$$ 9.65685 0.375041
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0.414214 0.0160384
$$668$$ 0 0
$$669$$ −0.142136 −0.00549528
$$670$$ 0 0
$$671$$ −4.54416 −0.175425
$$672$$ 0 0
$$673$$ 29.6569 1.14319 0.571594 0.820537i $$-0.306325\pi$$
0.571594 + 0.820537i $$0.306325\pi$$
$$674$$ 0 0
$$675$$ 2.41421 0.0929231
$$676$$ 0 0
$$677$$ 28.1421 1.08159 0.540795 0.841154i $$-0.318123\pi$$
0.540795 + 0.841154i $$0.318123\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 2.88730 0.110642
$$682$$ 0 0
$$683$$ 34.7574 1.32995 0.664977 0.746864i $$-0.268441\pi$$
0.664977 + 0.746864i $$0.268441\pi$$
$$684$$ 0 0
$$685$$ 9.65685 0.368969
$$686$$ 0 0
$$687$$ −4.82843 −0.184216
$$688$$ 0 0
$$689$$ −5.65685 −0.215509
$$690$$ 0 0
$$691$$ 0.828427 0.0315149 0.0157574 0.999876i $$-0.494984\pi$$
0.0157574 + 0.999876i $$0.494984\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 16.1421 0.612306
$$696$$ 0 0
$$697$$ −37.7990 −1.43174
$$698$$ 0 0
$$699$$ 6.97056 0.263651
$$700$$ 0 0
$$701$$ −3.20101 −0.120900 −0.0604502 0.998171i $$-0.519254\pi$$
−0.0604502 + 0.998171i $$0.519254\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0.828427 0.0312004
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 15.6863 0.589111 0.294556 0.955634i $$-0.404828\pi$$
0.294556 + 0.955634i $$0.404828\pi$$
$$710$$ 0 0
$$711$$ 41.9411 1.57292
$$712$$ 0 0
$$713$$ 2.48528 0.0930745
$$714$$ 0 0
$$715$$ −4.00000 −0.149592
$$716$$ 0 0
$$717$$ −8.82843 −0.329704
$$718$$ 0 0
$$719$$ 21.1127 0.787371 0.393685 0.919245i $$-0.371200\pi$$
0.393685 + 0.919245i $$0.371200\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 11.4558 0.426047
$$724$$ 0 0
$$725$$ −1.00000 −0.0371391
$$726$$ 0 0
$$727$$ −37.5858 −1.39398 −0.696990 0.717081i $$-0.745478\pi$$
−0.696990 + 0.717081i $$0.745478\pi$$
$$728$$ 0 0
$$729$$ −18.1716 −0.673021
$$730$$ 0 0
$$731$$ 17.3137 0.640371
$$732$$ 0 0
$$733$$ 22.0000 0.812589 0.406294 0.913742i $$-0.366821\pi$$
0.406294 + 0.913742i $$0.366821\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −7.94113 −0.292515
$$738$$ 0 0
$$739$$ 21.1127 0.776643 0.388322 0.921524i $$-0.373055\pi$$
0.388322 + 0.921524i $$0.373055\pi$$
$$740$$ 0 0
$$741$$ −5.65685 −0.207810
$$742$$ 0 0
$$743$$ 16.0711 0.589590 0.294795 0.955560i $$-0.404749\pi$$
0.294795 + 0.955560i $$0.404749\pi$$
$$744$$ 0 0
$$745$$ 2.17157 0.0795603
$$746$$ 0 0
$$747$$ −38.8284 −1.42066
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 30.3431 1.10724 0.553619 0.832770i $$-0.313247\pi$$
0.553619 + 0.832770i $$0.313247\pi$$
$$752$$ 0 0
$$753$$ 3.85786 0.140588
$$754$$ 0 0
$$755$$ 11.6569 0.424236
$$756$$ 0 0
$$757$$ −31.4558 −1.14328 −0.571641 0.820504i $$-0.693693\pi$$
−0.571641 + 0.820504i $$0.693693\pi$$
$$758$$ 0 0
$$759$$ 0.142136 0.00515920
$$760$$ 0 0
$$761$$ −9.31371 −0.337622 −0.168811 0.985648i $$-0.553993\pi$$
−0.168811 + 0.985648i $$0.553993\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −13.6569 −0.493765
$$766$$ 0 0
$$767$$ 21.6569 0.781984
$$768$$ 0 0
$$769$$ 0.627417 0.0226252 0.0113126 0.999936i $$-0.496399\pi$$
0.0113126 + 0.999936i $$0.496399\pi$$
$$770$$ 0 0
$$771$$ 2.62742 0.0946241
$$772$$ 0 0
$$773$$ 37.1127 1.33485 0.667425 0.744677i $$-0.267396\pi$$
0.667425 + 0.744677i $$0.267396\pi$$
$$774$$ 0 0
$$775$$ −6.00000 −0.215526
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 22.1421 0.793324
$$780$$ 0 0
$$781$$ −3.71573 −0.132959
$$782$$ 0 0
$$783$$ −2.41421 −0.0862770
$$784$$ 0 0
$$785$$ 17.3137 0.617953
$$786$$ 0 0
$$787$$ 2.55635 0.0911240 0.0455620 0.998962i $$-0.485492\pi$$
0.0455620 + 0.998962i $$0.485492\pi$$
$$788$$ 0 0
$$789$$ −12.0294 −0.428259
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −26.4853 −0.940520
$$794$$ 0 0
$$795$$ −0.485281 −0.0172112
$$796$$ 0 0
$$797$$ 8.00000 0.283375 0.141687 0.989911i $$-0.454747\pi$$
0.141687 + 0.989911i $$0.454747\pi$$
$$798$$ 0 0
$$799$$ −9.65685 −0.341635
$$800$$ 0 0
$$801$$ −24.4853 −0.865145
$$802$$ 0 0
$$803$$ 0.686292 0.0242187
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 8.47309 0.298267
$$808$$ 0 0
$$809$$ 35.6274 1.25259 0.626297 0.779585i $$-0.284570\pi$$
0.626297 + 0.779585i $$0.284570\pi$$
$$810$$ 0 0
$$811$$ −20.6274 −0.724327 −0.362163 0.932115i $$-0.617962\pi$$
−0.362163 + 0.932115i $$0.617962\pi$$
$$812$$ 0 0
$$813$$ −6.82843 −0.239483
$$814$$ 0 0
$$815$$ 12.3431 0.432362
$$816$$ 0 0
$$817$$ −10.1421 −0.354828
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 47.9411 1.67316 0.836578 0.547847i $$-0.184552\pi$$
0.836578 + 0.547847i $$0.184552\pi$$
$$822$$ 0 0
$$823$$ −2.07107 −0.0721929 −0.0360964 0.999348i $$-0.511492\pi$$
−0.0360964 + 0.999348i $$0.511492\pi$$
$$824$$ 0 0
$$825$$ −0.343146 −0.0119468
$$826$$ 0 0
$$827$$ 26.2132 0.911522 0.455761 0.890102i $$-0.349367\pi$$
0.455761 + 0.890102i $$0.349367\pi$$
$$828$$ 0 0
$$829$$ −29.3137 −1.01811 −0.509054 0.860735i $$-0.670005\pi$$
−0.509054 + 0.860735i $$0.670005\pi$$
$$830$$ 0 0
$$831$$ −6.68629 −0.231945
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −22.4142 −0.775676
$$836$$ 0 0
$$837$$ −14.4853 −0.500685
$$838$$ 0 0
$$839$$ 15.1716 0.523781 0.261890 0.965098i $$-0.415654\pi$$
0.261890 + 0.965098i $$0.415654\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 0 0
$$843$$ 12.5442 0.432044
$$844$$ 0 0
$$845$$ −10.3137 −0.354802
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −5.79899 −0.199021
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −2.54416 −0.0871102 −0.0435551 0.999051i $$-0.513868\pi$$
−0.0435551 + 0.999051i $$0.513868\pi$$
$$854$$ 0 0
$$855$$ 8.00000 0.273594
$$856$$ 0 0
$$857$$ −34.2843 −1.17113 −0.585564 0.810626i $$-0.699127\pi$$
−0.585564 + 0.810626i $$0.699127\pi$$
$$858$$ 0 0
$$859$$ 1.37258 0.0468319 0.0234160 0.999726i $$-0.492546\pi$$
0.0234160 + 0.999726i $$0.492546\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −14.5563 −0.495504 −0.247752 0.968823i $$-0.579692\pi$$
−0.247752 + 0.968823i $$0.579692\pi$$
$$864$$ 0 0
$$865$$ 3.31371 0.112669
$$866$$ 0 0
$$867$$ −2.61522 −0.0888177
$$868$$ 0 0
$$869$$ −12.2843 −0.416715
$$870$$ 0 0
$$871$$ −46.2843 −1.56828
$$872$$ 0 0
$$873$$ 32.9706 1.11588
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 25.1716 0.849984 0.424992 0.905197i $$-0.360277\pi$$
0.424992 + 0.905197i $$0.360277\pi$$
$$878$$ 0 0
$$879$$ −6.62742 −0.223537
$$880$$ 0 0
$$881$$ −1.82843 −0.0616013 −0.0308006 0.999526i $$-0.509806\pi$$
−0.0308006 + 0.999526i $$0.509806\pi$$
$$882$$ 0 0
$$883$$ −18.2843 −0.615315 −0.307657 0.951497i $$-0.599545\pi$$
−0.307657 + 0.951497i $$0.599545\pi$$
$$884$$ 0 0
$$885$$ 1.85786 0.0624514
$$886$$ 0 0
$$887$$ 29.9289 1.00492 0.502458 0.864602i $$-0.332429\pi$$
0.502458 + 0.864602i $$0.332429\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 6.20101 0.207742
$$892$$ 0 0
$$893$$ 5.65685 0.189299
$$894$$ 0 0
$$895$$ −10.0000 −0.334263
$$896$$ 0 0
$$897$$ 0.828427 0.0276604
$$898$$ 0 0
$$899$$ 6.00000 0.200111
$$900$$ 0 0
$$901$$ 5.65685 0.188457
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 2.65685 0.0883168
$$906$$ 0 0
$$907$$ 14.2132 0.471942 0.235971 0.971760i $$-0.424173\pi$$
0.235971 + 0.971760i $$0.424173\pi$$
$$908$$ 0 0
$$909$$ −29.1716 −0.967560
$$910$$ 0 0
$$911$$ 10.2010 0.337975 0.168987 0.985618i $$-0.445950\pi$$
0.168987 + 0.985618i $$0.445950\pi$$
$$912$$ 0 0
$$913$$ 11.3726 0.376378
$$914$$ 0 0
$$915$$ −2.27208 −0.0751126
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 43.1127 1.42216 0.711078 0.703113i $$-0.248207\pi$$
0.711078 + 0.703113i $$0.248207\pi$$
$$920$$ 0 0
$$921$$ 1.97056 0.0649323
$$922$$ 0 0
$$923$$ −21.6569 −0.712844
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 6.82843 0.224275
$$928$$ 0 0
$$929$$ −5.48528 −0.179966 −0.0899831 0.995943i $$-0.528681\pi$$
−0.0899831 + 0.995943i $$0.528681\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 5.45584 0.178616
$$934$$ 0 0
$$935$$ 4.00000 0.130814
$$936$$ 0 0
$$937$$ −34.6274 −1.13123 −0.565614 0.824670i $$-0.691361\pi$$
−0.565614 + 0.824670i $$0.691361\pi$$
$$938$$ 0 0
$$939$$ −2.62742 −0.0857425
$$940$$ 0 0
$$941$$ 46.2843 1.50882 0.754412 0.656401i $$-0.227922\pi$$
0.754412 + 0.656401i $$0.227922\pi$$
$$942$$ 0 0
$$943$$ −3.24264 −0.105595
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −33.1838 −1.07833 −0.539164 0.842201i $$-0.681260\pi$$
−0.539164 + 0.842201i $$0.681260\pi$$
$$948$$ 0 0
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ 5.71573 0.185345
$$952$$ 0 0
$$953$$ −13.6569 −0.442389 −0.221194 0.975230i $$-0.570996\pi$$
−0.221194 + 0.975230i $$0.570996\pi$$
$$954$$ 0 0
$$955$$ −12.8284 −0.415118
$$956$$ 0 0
$$957$$ 0.343146 0.0110923
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 31.7990 1.02471
$$964$$ 0 0
$$965$$ 2.00000 0.0643823
$$966$$ 0 0
$$967$$ −37.5269 −1.20678 −0.603392 0.797445i $$-0.706185\pi$$
−0.603392 + 0.797445i $$0.706185\pi$$
$$968$$ 0 0
$$969$$ 5.65685 0.181724
$$970$$ 0 0
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −2.00000 −0.0640513
$$976$$ 0 0
$$977$$ −1.31371 −0.0420293 −0.0210146 0.999779i $$-0.506690\pi$$
−0.0210146 + 0.999779i $$0.506690\pi$$
$$978$$ 0 0
$$979$$ 7.17157 0.229204
$$980$$ 0 0
$$981$$ 38.1421 1.21778
$$982$$ 0 0
$$983$$ 28.2132 0.899861 0.449931 0.893063i $$-0.351449\pi$$
0.449931 + 0.893063i $$0.351449\pi$$
$$984$$ 0 0
$$985$$ 12.3431 0.393285
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.48528 0.0472292
$$990$$ 0 0
$$991$$ 4.34315 0.137965 0.0689823 0.997618i $$-0.478025\pi$$
0.0689823 + 0.997618i $$0.478025\pi$$
$$992$$ 0 0
$$993$$ 9.51472 0.301940
$$994$$ 0 0
$$995$$ 9.65685 0.306143
$$996$$ 0 0
$$997$$ −33.4558 −1.05956 −0.529779 0.848136i $$-0.677725\pi$$
−0.529779 + 0.848136i $$0.677725\pi$$
$$998$$ 0 0
$$999$$ 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 3920.2.a.bv.1.1 2
4.3 odd 2 245.2.a.g.1.2 2
7.3 odd 6 560.2.q.k.401.1 4
7.5 odd 6 560.2.q.k.81.1 4
7.6 odd 2 3920.2.a.bq.1.2 2
12.11 even 2 2205.2.a.q.1.1 2
20.3 even 4 1225.2.b.h.99.1 4
20.7 even 4 1225.2.b.h.99.4 4
20.19 odd 2 1225.2.a.m.1.1 2
28.3 even 6 35.2.e.a.16.1 yes 4
28.11 odd 6 245.2.e.e.226.1 4
28.19 even 6 35.2.e.a.11.1 4
28.23 odd 6 245.2.e.e.116.1 4
28.27 even 2 245.2.a.h.1.2 2
84.47 odd 6 315.2.j.e.46.2 4
84.59 odd 6 315.2.j.e.226.2 4
84.83 odd 2 2205.2.a.n.1.1 2
140.3 odd 12 175.2.k.a.149.4 8
140.19 even 6 175.2.e.c.151.2 4
140.27 odd 4 1225.2.b.g.99.4 4
140.47 odd 12 175.2.k.a.74.4 8
140.59 even 6 175.2.e.c.51.2 4
140.83 odd 4 1225.2.b.g.99.1 4
140.87 odd 12 175.2.k.a.149.1 8
140.103 odd 12 175.2.k.a.74.1 8
140.139 even 2 1225.2.a.k.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.e.a.11.1 4 28.19 even 6
35.2.e.a.16.1 yes 4 28.3 even 6
175.2.e.c.51.2 4 140.59 even 6
175.2.e.c.151.2 4 140.19 even 6
175.2.k.a.74.1 8 140.103 odd 12
175.2.k.a.74.4 8 140.47 odd 12
175.2.k.a.149.1 8 140.87 odd 12
175.2.k.a.149.4 8 140.3 odd 12
245.2.a.g.1.2 2 4.3 odd 2
245.2.a.h.1.2 2 28.27 even 2
245.2.e.e.116.1 4 28.23 odd 6
245.2.e.e.226.1 4 28.11 odd 6
315.2.j.e.46.2 4 84.47 odd 6
315.2.j.e.226.2 4 84.59 odd 6
560.2.q.k.81.1 4 7.5 odd 6
560.2.q.k.401.1 4 7.3 odd 6
1225.2.a.k.1.1 2 140.139 even 2
1225.2.a.m.1.1 2 20.19 odd 2
1225.2.b.g.99.1 4 140.83 odd 4
1225.2.b.g.99.4 4 140.27 odd 4
1225.2.b.h.99.1 4 20.3 even 4
1225.2.b.h.99.4 4 20.7 even 4
2205.2.a.n.1.1 2 84.83 odd 2
2205.2.a.q.1.1 2 12.11 even 2
3920.2.a.bq.1.2 2 7.6 odd 2
3920.2.a.bv.1.1 2 1.1 even 1 trivial