# Properties

 Label 845.2.c.g.506.7 Level $845$ Weight $2$ Character 845.506 Analytic conductor $6.747$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$845 = 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 845.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.74735897080$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.22581504.2 Defining polynomial: $$x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 506.7 Root $$1.20036 - 0.747754i$$ of defining polynomial Character $$\chi$$ $$=$$ 845.506 Dual form 845.2.c.g.506.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.49551i q^{2} +0.0947876 q^{3} -0.236543 q^{4} +1.00000i q^{5} +0.141756i q^{6} +4.82684i q^{7} +2.63726i q^{8} -2.99102 q^{9} +O(q^{10})$$ $$q+1.49551i q^{2} +0.0947876 q^{3} -0.236543 q^{4} +1.00000i q^{5} +0.141756i q^{6} +4.82684i q^{7} +2.63726i q^{8} -2.99102 q^{9} -1.49551 q^{10} -1.06939i q^{11} -0.0224214 q^{12} -7.21857 q^{14} +0.0947876i q^{15} -4.41713 q^{16} -3.55889 q^{17} -4.47309i q^{18} -5.73205i q^{19} -0.236543i q^{20} +0.457524i q^{21} +1.59928 q^{22} +7.08580 q^{23} +0.249980i q^{24} -1.00000 q^{25} -0.567874 q^{27} -1.14176i q^{28} +1.47309 q^{29} -0.141756 q^{30} +1.46410i q^{31} -1.33133i q^{32} -0.101365i q^{33} -5.32235i q^{34} -4.82684 q^{35} +0.707504 q^{36} -0.0253983i q^{37} +8.57233 q^{38} -2.63726 q^{40} -0.267949i q^{41} -0.684231 q^{42} -3.55889 q^{43} +0.252957i q^{44} -2.99102i q^{45} +10.5969i q^{46} +6.51793i q^{47} -0.418689 q^{48} -16.2984 q^{49} -1.49551i q^{50} -0.337339 q^{51} +0.991015 q^{53} -0.849260i q^{54} +1.06939 q^{55} -12.7296 q^{56} -0.543327i q^{57} +2.20301i q^{58} +8.72307i q^{59} -0.0224214i q^{60} +6.33734 q^{61} -2.18958 q^{62} -14.4371i q^{63} -6.84325 q^{64} +0.151592 q^{66} -5.17316i q^{67} +0.841831 q^{68} +0.671646 q^{69} -7.21857i q^{70} +7.76488i q^{71} -7.88809i q^{72} +10.1088i q^{73} +0.0379833 q^{74} -0.0947876 q^{75} +1.35588i q^{76} +5.16177 q^{77} +8.78347 q^{79} -4.41713i q^{80} +8.91922 q^{81} +0.400720 q^{82} +0.725474i q^{83} -0.108224i q^{84} -3.55889i q^{85} -5.32235i q^{86} +0.139630 q^{87} +2.82026 q^{88} +13.5065i q^{89} +4.47309 q^{90} -1.67610 q^{92} +0.138779i q^{93} -9.74761 q^{94} +5.73205 q^{95} -0.126194i q^{96} +3.43870i q^{97} -24.3743i q^{98} +3.19856i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{3} - 4q^{4} + 8q^{9} + O(q^{10})$$ $$8q - 4q^{3} - 4q^{4} + 8q^{9} + 4q^{10} + 20q^{12} + 4q^{14} + 4q^{16} + 4q^{17} + 24q^{22} + 20q^{23} - 8q^{25} - 4q^{27} + 16q^{29} - 8q^{30} - 20q^{35} - 40q^{36} - 16q^{38} - 12q^{40} - 8q^{42} + 4q^{43} - 56q^{48} - 24q^{49} - 8q^{51} - 24q^{53} - 24q^{56} + 56q^{61} - 8q^{62} - 8q^{64} + 12q^{66} + 28q^{68} + 32q^{69} - 20q^{74} + 4q^{75} - 36q^{77} - 16q^{79} - 16q^{81} - 8q^{82} - 44q^{87} + 36q^{88} + 40q^{90} + 44q^{92} - 64q^{94} + 32q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.49551i 1.05748i 0.848783 + 0.528742i $$0.177336\pi$$
−0.848783 + 0.528742i $$0.822664\pi$$
$$3$$ 0.0947876 0.0547256 0.0273628 0.999626i $$-0.491289\pi$$
0.0273628 + 0.999626i $$0.491289\pi$$
$$4$$ −0.236543 −0.118272
$$5$$ 1.00000i 0.447214i
$$6$$ 0.141756i 0.0578715i
$$7$$ 4.82684i 1.82437i 0.409775 + 0.912187i $$0.365607\pi$$
−0.409775 + 0.912187i $$0.634393\pi$$
$$8$$ 2.63726i 0.932413i
$$9$$ −2.99102 −0.997005
$$10$$ −1.49551 −0.472921
$$11$$ − 1.06939i − 0.322433i −0.986919 0.161217i $$-0.948458\pi$$
0.986919 0.161217i $$-0.0515417\pi$$
$$12$$ −0.0224214 −0.00647249
$$13$$ 0 0
$$14$$ −7.21857 −1.92924
$$15$$ 0.0947876i 0.0244740i
$$16$$ −4.41713 −1.10428
$$17$$ −3.55889 −0.863157 −0.431579 0.902075i $$-0.642043\pi$$
−0.431579 + 0.902075i $$0.642043\pi$$
$$18$$ − 4.47309i − 1.05432i
$$19$$ − 5.73205i − 1.31502i −0.753445 0.657511i $$-0.771609\pi$$
0.753445 0.657511i $$-0.228391\pi$$
$$20$$ − 0.236543i − 0.0528927i
$$21$$ 0.457524i 0.0998400i
$$22$$ 1.59928 0.340968
$$23$$ 7.08580 1.47749 0.738746 0.673984i $$-0.235418\pi$$
0.738746 + 0.673984i $$0.235418\pi$$
$$24$$ 0.249980i 0.0510269i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −0.567874 −0.109287
$$28$$ − 1.14176i − 0.215772i
$$29$$ 1.47309 0.273545 0.136773 0.990602i $$-0.456327\pi$$
0.136773 + 0.990602i $$0.456327\pi$$
$$30$$ −0.141756 −0.0258809
$$31$$ 1.46410i 0.262960i 0.991319 + 0.131480i $$0.0419730\pi$$
−0.991319 + 0.131480i $$0.958027\pi$$
$$32$$ − 1.33133i − 0.235348i
$$33$$ − 0.101365i − 0.0176454i
$$34$$ − 5.32235i − 0.912775i
$$35$$ −4.82684 −0.815885
$$36$$ 0.707504 0.117917
$$37$$ − 0.0253983i − 0.00417545i −0.999998 0.00208772i $$-0.999335\pi$$
0.999998 0.00208772i $$-0.000664544\pi$$
$$38$$ 8.57233 1.39061
$$39$$ 0 0
$$40$$ −2.63726 −0.416988
$$41$$ − 0.267949i − 0.0418466i −0.999781 0.0209233i $$-0.993339\pi$$
0.999781 0.0209233i $$-0.00666058\pi$$
$$42$$ −0.684231 −0.105579
$$43$$ −3.55889 −0.542726 −0.271363 0.962477i $$-0.587474\pi$$
−0.271363 + 0.962477i $$0.587474\pi$$
$$44$$ 0.252957i 0.0381347i
$$45$$ − 2.99102i − 0.445874i
$$46$$ 10.5969i 1.56242i
$$47$$ 6.51793i 0.950738i 0.879787 + 0.475369i $$0.157685\pi$$
−0.879787 + 0.475369i $$0.842315\pi$$
$$48$$ −0.418689 −0.0604326
$$49$$ −16.2984 −2.32834
$$50$$ − 1.49551i − 0.211497i
$$51$$ −0.337339 −0.0472368
$$52$$ 0 0
$$53$$ 0.991015 0.136126 0.0680632 0.997681i $$-0.478318\pi$$
0.0680632 + 0.997681i $$0.478318\pi$$
$$54$$ − 0.849260i − 0.115570i
$$55$$ 1.06939 0.144196
$$56$$ −12.7296 −1.70107
$$57$$ − 0.543327i − 0.0719655i
$$58$$ 2.20301i 0.289270i
$$59$$ 8.72307i 1.13565i 0.823151 + 0.567823i $$0.192214\pi$$
−0.823151 + 0.567823i $$0.807786\pi$$
$$60$$ − 0.0224214i − 0.00289458i
$$61$$ 6.33734 0.811413 0.405707 0.914003i $$-0.367026\pi$$
0.405707 + 0.914003i $$0.367026\pi$$
$$62$$ −2.18958 −0.278076
$$63$$ − 14.4371i − 1.81891i
$$64$$ −6.84325 −0.855406
$$65$$ 0 0
$$66$$ 0.151592 0.0186597
$$67$$ − 5.17316i − 0.632002i −0.948759 0.316001i $$-0.897660\pi$$
0.948759 0.316001i $$-0.102340\pi$$
$$68$$ 0.841831 0.102087
$$69$$ 0.671646 0.0808567
$$70$$ − 7.21857i − 0.862785i
$$71$$ 7.76488i 0.921521i 0.887524 + 0.460761i $$0.152423\pi$$
−0.887524 + 0.460761i $$0.847577\pi$$
$$72$$ − 7.88809i − 0.929621i
$$73$$ 10.1088i 1.18314i 0.806252 + 0.591572i $$0.201493\pi$$
−0.806252 + 0.591572i $$0.798507\pi$$
$$74$$ 0.0379833 0.00441547
$$75$$ −0.0947876 −0.0109451
$$76$$ 1.35588i 0.155530i
$$77$$ 5.16177 0.588238
$$78$$ 0 0
$$79$$ 8.78347 0.988218 0.494109 0.869400i $$-0.335494\pi$$
0.494109 + 0.869400i $$0.335494\pi$$
$$80$$ − 4.41713i − 0.493851i
$$81$$ 8.91922 0.991024
$$82$$ 0.400720 0.0442521
$$83$$ 0.725474i 0.0796311i 0.999207 + 0.0398155i $$0.0126770\pi$$
−0.999207 + 0.0398155i $$0.987323\pi$$
$$84$$ − 0.108224i − 0.0118082i
$$85$$ − 3.55889i − 0.386016i
$$86$$ − 5.32235i − 0.573923i
$$87$$ 0.139630 0.0149699
$$88$$ 2.82026 0.300641
$$89$$ 13.5065i 1.43169i 0.698259 + 0.715845i $$0.253958\pi$$
−0.698259 + 0.715845i $$0.746042\pi$$
$$90$$ 4.47309 0.471505
$$91$$ 0 0
$$92$$ −1.67610 −0.174745
$$93$$ 0.138779i 0.0143907i
$$94$$ −9.74761 −1.00539
$$95$$ 5.73205 0.588096
$$96$$ − 0.126194i − 0.0128796i
$$97$$ 3.43870i 0.349147i 0.984644 + 0.174574i $$0.0558547\pi$$
−0.984644 + 0.174574i $$0.944145\pi$$
$$98$$ − 24.3743i − 2.46218i
$$99$$ 3.19856i 0.321467i
$$100$$ 0.236543 0.0236543
$$101$$ −2.85527 −0.284110 −0.142055 0.989859i $$-0.545371\pi$$
−0.142055 + 0.989859i $$0.545371\pi$$
$$102$$ − 0.504492i − 0.0499522i
$$103$$ 5.54488 0.546354 0.273177 0.961964i $$-0.411926\pi$$
0.273177 + 0.961964i $$0.411926\pi$$
$$104$$ 0 0
$$105$$ −0.457524 −0.0446498
$$106$$ 1.48207i 0.143951i
$$107$$ −4.44111 −0.429338 −0.214669 0.976687i $$-0.568867\pi$$
−0.214669 + 0.976687i $$0.568867\pi$$
$$108$$ 0.134327 0.0129256
$$109$$ − 13.7804i − 1.31993i −0.751298 0.659963i $$-0.770572\pi$$
0.751298 0.659963i $$-0.229428\pi$$
$$110$$ 1.59928i 0.152485i
$$111$$ − 0.00240744i 0 0.000228504i
$$112$$ − 21.3208i − 2.01463i
$$113$$ −8.04399 −0.756715 −0.378358 0.925660i $$-0.623511\pi$$
−0.378358 + 0.925660i $$0.623511\pi$$
$$114$$ 0.812550 0.0761023
$$115$$ 7.08580i 0.660755i
$$116$$ −0.348448 −0.0323526
$$117$$ 0 0
$$118$$ −13.0454 −1.20093
$$119$$ − 17.1782i − 1.57472i
$$120$$ −0.249980 −0.0228199
$$121$$ 9.85641 0.896037
$$122$$ 9.47754i 0.858056i
$$123$$ − 0.0253983i − 0.00229008i
$$124$$ − 0.346323i − 0.0311007i
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 21.5909 1.92347
$$127$$ −0.706653 −0.0627053 −0.0313526 0.999508i $$-0.509981\pi$$
−0.0313526 + 0.999508i $$0.509981\pi$$
$$128$$ − 12.8968i − 1.13993i
$$129$$ −0.337339 −0.0297010
$$130$$ 0 0
$$131$$ 6.26554 0.547423 0.273711 0.961812i $$-0.411749\pi$$
0.273711 + 0.961812i $$0.411749\pi$$
$$132$$ 0.0239772i 0.00208694i
$$133$$ 27.6677 2.39909
$$134$$ 7.73650 0.668332
$$135$$ − 0.567874i − 0.0488748i
$$136$$ − 9.38573i − 0.804820i
$$137$$ 16.3058i 1.39310i 0.717509 + 0.696549i $$0.245282\pi$$
−0.717509 + 0.696549i $$0.754718\pi$$
$$138$$ 1.00445i 0.0855046i
$$139$$ −6.82528 −0.578913 −0.289456 0.957191i $$-0.593475\pi$$
−0.289456 + 0.957191i $$0.593475\pi$$
$$140$$ 1.14176 0.0964960
$$141$$ 0.617819i 0.0520297i
$$142$$ −11.6124 −0.974494
$$143$$ 0 0
$$144$$ 13.2117 1.10098
$$145$$ 1.47309i 0.122333i
$$146$$ −15.1178 −1.25116
$$147$$ −1.54488 −0.127420
$$148$$ 0.00600778i 0 0.000493837i
$$149$$ 8.43955i 0.691395i 0.938346 + 0.345698i $$0.112358\pi$$
−0.938346 + 0.345698i $$0.887642\pi$$
$$150$$ − 0.141756i − 0.0115743i
$$151$$ − 1.37017i − 0.111503i −0.998445 0.0557513i $$-0.982245\pi$$
0.998445 0.0557513i $$-0.0177554\pi$$
$$152$$ 15.1169 1.22614
$$153$$ 10.6447 0.860572
$$154$$ 7.71947i 0.622052i
$$155$$ −1.46410 −0.117599
$$156$$ 0 0
$$157$$ 11.9700 0.955311 0.477656 0.878547i $$-0.341487\pi$$
0.477656 + 0.878547i $$0.341487\pi$$
$$158$$ 13.1357i 1.04502i
$$159$$ 0.0939360 0.00744961
$$160$$ 1.33133 0.105251
$$161$$ 34.2020i 2.69550i
$$162$$ 13.3388i 1.04799i
$$163$$ − 22.5713i − 1.76792i −0.467559 0.883962i $$-0.654866\pi$$
0.467559 0.883962i $$-0.345134\pi$$
$$164$$ 0.0633815i 0.00494927i
$$165$$ 0.101365 0.00789124
$$166$$ −1.08495 −0.0842085
$$167$$ 8.19700i 0.634303i 0.948375 + 0.317152i $$0.102726\pi$$
−0.948375 + 0.317152i $$0.897274\pi$$
$$168$$ −1.20661 −0.0930922
$$169$$ 0 0
$$170$$ 5.32235 0.408205
$$171$$ 17.1447i 1.31108i
$$172$$ 0.841831 0.0641890
$$173$$ 9.16772 0.697009 0.348505 0.937307i $$-0.386690\pi$$
0.348505 + 0.937307i $$0.386690\pi$$
$$174$$ 0.208818i 0.0158305i
$$175$$ − 4.82684i − 0.364875i
$$176$$ 4.72364i 0.356057i
$$177$$ 0.826838i 0.0621490i
$$178$$ −20.1991 −1.51399
$$179$$ 10.0370 0.750200 0.375100 0.926984i $$-0.377608\pi$$
0.375100 + 0.926984i $$0.377608\pi$$
$$180$$ 0.707504i 0.0527342i
$$181$$ −17.0238 −1.26537 −0.632686 0.774408i $$-0.718048\pi$$
−0.632686 + 0.774408i $$0.718048\pi$$
$$182$$ 0 0
$$183$$ 0.600701 0.0444051
$$184$$ 18.6871i 1.37763i
$$185$$ 0.0253983 0.00186732
$$186$$ −0.207545 −0.0152179
$$187$$ 3.80584i 0.278310i
$$188$$ − 1.54177i − 0.112445i
$$189$$ − 2.74104i − 0.199381i
$$190$$ 8.57233i 0.621902i
$$191$$ −3.87741 −0.280559 −0.140280 0.990112i $$-0.544800\pi$$
−0.140280 + 0.990112i $$0.544800\pi$$
$$192$$ −0.648655 −0.0468127
$$193$$ 1.25394i 0.0902608i 0.998981 + 0.0451304i $$0.0143703\pi$$
−0.998981 + 0.0451304i $$0.985630\pi$$
$$194$$ −5.14261 −0.369218
$$195$$ 0 0
$$196$$ 3.85527 0.275376
$$197$$ 15.2820i 1.08879i 0.838828 + 0.544397i $$0.183242\pi$$
−0.838828 + 0.544397i $$0.816758\pi$$
$$198$$ −4.78347 −0.339946
$$199$$ 13.2296 0.937822 0.468911 0.883246i $$-0.344647\pi$$
0.468911 + 0.883246i $$0.344647\pi$$
$$200$$ − 2.63726i − 0.186483i
$$201$$ − 0.490352i − 0.0345867i
$$202$$ − 4.27007i − 0.300441i
$$203$$ 7.11035i 0.499049i
$$204$$ 0.0797951 0.00558678
$$205$$ 0.267949 0.0187144
$$206$$ 8.29242i 0.577760i
$$207$$ −21.1937 −1.47307
$$208$$ 0 0
$$209$$ −6.12979 −0.424007
$$210$$ − 0.684231i − 0.0472164i
$$211$$ −4.81042 −0.331163 −0.165582 0.986196i $$-0.552950\pi$$
−0.165582 + 0.986196i $$0.552950\pi$$
$$212$$ −0.234418 −0.0160999
$$213$$ 0.736014i 0.0504309i
$$214$$ − 6.64172i − 0.454018i
$$215$$ − 3.55889i − 0.242714i
$$216$$ − 1.49763i − 0.101901i
$$217$$ −7.06698 −0.479738
$$218$$ 20.6088 1.39580
$$219$$ 0.958188i 0.0647484i
$$220$$ −0.252957 −0.0170543
$$221$$ 0 0
$$222$$ 0.00360034 0.000241639 0
$$223$$ − 14.7132i − 0.985271i −0.870236 0.492635i $$-0.836034\pi$$
0.870236 0.492635i $$-0.163966\pi$$
$$224$$ 6.42612 0.429363
$$225$$ 2.99102 0.199401
$$226$$ − 12.0299i − 0.800214i
$$227$$ − 14.9028i − 0.989134i −0.869140 0.494567i $$-0.835327\pi$$
0.869140 0.494567i $$-0.164673\pi$$
$$228$$ 0.128520i 0.00851147i
$$229$$ − 19.3074i − 1.27587i −0.770092 0.637933i $$-0.779790\pi$$
0.770092 0.637933i $$-0.220210\pi$$
$$230$$ −10.5969 −0.698737
$$231$$ 0.489272 0.0321917
$$232$$ 3.88492i 0.255057i
$$233$$ 21.1937 1.38845 0.694224 0.719759i $$-0.255748\pi$$
0.694224 + 0.719759i $$0.255748\pi$$
$$234$$ 0 0
$$235$$ −6.51793 −0.425183
$$236$$ − 2.06338i − 0.134315i
$$237$$ 0.832564 0.0540808
$$238$$ 25.6901 1.66524
$$239$$ 14.8971i 0.963612i 0.876278 + 0.481806i $$0.160019\pi$$
−0.876278 + 0.481806i $$0.839981\pi$$
$$240$$ − 0.418689i − 0.0270263i
$$241$$ 9.39168i 0.604971i 0.953154 + 0.302486i $$0.0978164\pi$$
−0.953154 + 0.302486i $$0.902184\pi$$
$$242$$ 14.7403i 0.947544i
$$243$$ 2.54905 0.163522
$$244$$ −1.49905 −0.0959671
$$245$$ − 16.2984i − 1.04126i
$$246$$ 0.0379833 0.00242173
$$247$$ 0 0
$$248$$ −3.86122 −0.245188
$$249$$ 0.0687659i 0.00435786i
$$250$$ 1.49551 0.0945842
$$251$$ −11.3163 −0.714281 −0.357140 0.934051i $$-0.616248\pi$$
−0.357140 + 0.934051i $$0.616248\pi$$
$$252$$ 3.41501i 0.215125i
$$253$$ − 7.57748i − 0.476392i
$$254$$ − 1.05680i − 0.0663098i
$$255$$ − 0.337339i − 0.0211250i
$$256$$ 5.60076 0.350047
$$257$$ 26.5319 1.65502 0.827508 0.561453i $$-0.189758\pi$$
0.827508 + 0.561453i $$0.189758\pi$$
$$258$$ − 0.504492i − 0.0314083i
$$259$$ 0.122593 0.00761758
$$260$$ 0 0
$$261$$ −4.40602 −0.272726
$$262$$ 9.37017i 0.578891i
$$263$$ −14.1408 −0.871957 −0.435979 0.899957i $$-0.643598\pi$$
−0.435979 + 0.899957i $$0.643598\pi$$
$$264$$ 0.267326 0.0164528
$$265$$ 0.991015i 0.0608776i
$$266$$ 41.3772i 2.53700i
$$267$$ 1.28025i 0.0783502i
$$268$$ 1.22368i 0.0747479i
$$269$$ 24.7745 1.51053 0.755264 0.655421i $$-0.227509\pi$$
0.755264 + 0.655421i $$0.227509\pi$$
$$270$$ 0.849260 0.0516843
$$271$$ 18.7171i 1.13698i 0.822689 + 0.568492i $$0.192473\pi$$
−0.822689 + 0.568492i $$0.807527\pi$$
$$272$$ 15.7201 0.953170
$$273$$ 0 0
$$274$$ −24.3854 −1.47318
$$275$$ 1.06939i 0.0644866i
$$276$$ −0.158873 −0.00956305
$$277$$ 22.6647 1.36179 0.680893 0.732382i $$-0.261592\pi$$
0.680893 + 0.732382i $$0.261592\pi$$
$$278$$ − 10.2073i − 0.612191i
$$279$$ − 4.37915i − 0.262173i
$$280$$ − 12.7296i − 0.760742i
$$281$$ − 27.8384i − 1.66070i −0.557241 0.830351i $$-0.688140\pi$$
0.557241 0.830351i $$-0.311860\pi$$
$$282$$ −0.923953 −0.0550206
$$283$$ −7.92007 −0.470799 −0.235400 0.971899i $$-0.575640\pi$$
−0.235400 + 0.971899i $$0.575640\pi$$
$$284$$ − 1.83673i − 0.108990i
$$285$$ 0.543327 0.0321839
$$286$$ 0 0
$$287$$ 1.29335 0.0763439
$$288$$ 3.98203i 0.234643i
$$289$$ −4.33431 −0.254959
$$290$$ −2.20301 −0.129365
$$291$$ 0.325946i 0.0191073i
$$292$$ − 2.39117i − 0.139932i
$$293$$ − 0.272971i − 0.0159471i −0.999968 0.00797356i $$-0.997462\pi$$
0.999968 0.00797356i $$-0.00253809\pi$$
$$294$$ − 2.31038i − 0.134744i
$$295$$ −8.72307 −0.507877
$$296$$ 0.0669819 0.00389324
$$297$$ 0.607278i 0.0352379i
$$298$$ −12.6214 −0.731139
$$299$$ 0 0
$$300$$ 0.0224214 0.00129450
$$301$$ − 17.1782i − 0.990134i
$$302$$ 2.04909 0.117912
$$303$$ −0.270644 −0.0155481
$$304$$ 25.3192i 1.45216i
$$305$$ 6.33734i 0.362875i
$$306$$ 15.9192i 0.910041i
$$307$$ 6.85224i 0.391078i 0.980696 + 0.195539i $$0.0626456\pi$$
−0.980696 + 0.195539i $$0.937354\pi$$
$$308$$ −1.22098 −0.0695719
$$309$$ 0.525586 0.0298995
$$310$$ − 2.18958i − 0.124360i
$$311$$ −10.6447 −0.603605 −0.301803 0.953370i $$-0.597588\pi$$
−0.301803 + 0.953370i $$0.597588\pi$$
$$312$$ 0 0
$$313$$ 17.8236 1.00745 0.503724 0.863865i $$-0.331963\pi$$
0.503724 + 0.863865i $$0.331963\pi$$
$$314$$ 17.9012i 1.01023i
$$315$$ 14.4371 0.813441
$$316$$ −2.07767 −0.116878
$$317$$ − 8.17161i − 0.458963i −0.973313 0.229482i $$-0.926297\pi$$
0.973313 0.229482i $$-0.0737031\pi$$
$$318$$ 0.140482i 0.00787784i
$$319$$ − 1.57530i − 0.0882000i
$$320$$ − 6.84325i − 0.382549i
$$321$$ −0.420962 −0.0234958
$$322$$ −51.1494 −2.85044
$$323$$ 20.3997i 1.13507i
$$324$$ −2.10978 −0.117210
$$325$$ 0 0
$$326$$ 33.7556 1.86955
$$327$$ − 1.30621i − 0.0722338i
$$328$$ 0.706653 0.0390184
$$329$$ −31.4610 −1.73450
$$330$$ 0.151592i 0.00834486i
$$331$$ − 24.9395i − 1.37080i −0.728167 0.685400i $$-0.759627\pi$$
0.728167 0.685400i $$-0.240373\pi$$
$$332$$ − 0.171606i − 0.00941809i
$$333$$ 0.0759666i 0.00416294i
$$334$$ −12.2587 −0.670765
$$335$$ 5.17316 0.282640
$$336$$ − 2.02095i − 0.110252i
$$337$$ 19.6057 1.06799 0.533996 0.845487i $$-0.320690\pi$$
0.533996 + 0.845487i $$0.320690\pi$$
$$338$$ 0 0
$$339$$ −0.762471 −0.0414117
$$340$$ 0.841831i 0.0456547i
$$341$$ 1.56569 0.0847871
$$342$$ −25.6400 −1.38645
$$343$$ − 44.8817i − 2.42339i
$$344$$ − 9.38573i − 0.506045i
$$345$$ 0.671646i 0.0361602i
$$346$$ 13.7104i 0.737076i
$$347$$ 17.0810 0.916955 0.458478 0.888706i $$-0.348395\pi$$
0.458478 + 0.888706i $$0.348395\pi$$
$$348$$ −0.0330286 −0.00177052
$$349$$ − 28.3719i − 1.51871i −0.650674 0.759357i $$-0.725514\pi$$
0.650674 0.759357i $$-0.274486\pi$$
$$350$$ 7.21857 0.385849
$$351$$ 0 0
$$352$$ −1.42371 −0.0758840
$$353$$ − 21.2520i − 1.13113i −0.824704 0.565564i $$-0.808658\pi$$
0.824704 0.565564i $$-0.191342\pi$$
$$354$$ −1.23654 −0.0657215
$$355$$ −7.76488 −0.412117
$$356$$ − 3.19488i − 0.169328i
$$357$$ − 1.62828i − 0.0861776i
$$358$$ 15.0104i 0.793325i
$$359$$ 32.6519i 1.72330i 0.507502 + 0.861650i $$0.330569\pi$$
−0.507502 + 0.861650i $$0.669431\pi$$
$$360$$ 7.88809 0.415739
$$361$$ −13.8564 −0.729285
$$362$$ − 25.4593i − 1.33811i
$$363$$ 0.934265 0.0490362
$$364$$ 0 0
$$365$$ −10.1088 −0.529118
$$366$$ 0.898353i 0.0469577i
$$367$$ −5.91837 −0.308936 −0.154468 0.987998i $$-0.549366\pi$$
−0.154468 + 0.987998i $$0.549366\pi$$
$$368$$ −31.2989 −1.63157
$$369$$ 0.801440i 0.0417213i
$$370$$ 0.0379833i 0.00197466i
$$371$$ 4.78347i 0.248345i
$$372$$ − 0.0328271i − 0.00170201i
$$373$$ −13.3185 −0.689607 −0.344803 0.938675i $$-0.612054\pi$$
−0.344803 + 0.938675i $$0.612054\pi$$
$$374$$ −5.69166 −0.294309
$$375$$ − 0.0947876i − 0.00489481i
$$376$$ −17.1895 −0.886480
$$377$$ 0 0
$$378$$ 4.09924 0.210842
$$379$$ − 25.4186i − 1.30566i −0.757503 0.652832i $$-0.773581\pi$$
0.757503 0.652832i $$-0.226419\pi$$
$$380$$ −1.35588 −0.0695550
$$381$$ −0.0669819 −0.00343159
$$382$$ − 5.79869i − 0.296687i
$$383$$ 10.8268i 0.553226i 0.960981 + 0.276613i $$0.0892119\pi$$
−0.960981 + 0.276613i $$0.910788\pi$$
$$384$$ − 1.22246i − 0.0623832i
$$385$$ 5.16177i 0.263068i
$$386$$ −1.87528 −0.0954493
$$387$$ 10.6447 0.541100
$$388$$ − 0.813402i − 0.0412942i
$$389$$ 23.0370 1.16802 0.584011 0.811746i $$-0.301482\pi$$
0.584011 + 0.811746i $$0.301482\pi$$
$$390$$ 0 0
$$391$$ −25.2176 −1.27531
$$392$$ − 42.9831i − 2.17097i
$$393$$ 0.593896 0.0299581
$$394$$ −22.8543 −1.15138
$$395$$ 8.78347i 0.441944i
$$396$$ − 0.756597i − 0.0380205i
$$397$$ 21.0864i 1.05830i 0.848529 + 0.529149i $$0.177489\pi$$
−0.848529 + 0.529149i $$0.822511\pi$$
$$398$$ 19.7850i 0.991731i
$$399$$ 2.62255 0.131292
$$400$$ 4.41713 0.220857
$$401$$ − 19.7769i − 0.987611i −0.869572 0.493805i $$-0.835605\pi$$
0.869572 0.493805i $$-0.164395\pi$$
$$402$$ 0.733324 0.0365749
$$403$$ 0 0
$$404$$ 0.675394 0.0336021
$$405$$ 8.91922i 0.443200i
$$406$$ −10.6336 −0.527736
$$407$$ −0.0271606 −0.00134630
$$408$$ − 0.889650i − 0.0440443i
$$409$$ 31.8809i 1.57641i 0.615414 + 0.788204i $$0.288989\pi$$
−0.615414 + 0.788204i $$0.711011\pi$$
$$410$$ 0.400720i 0.0197902i
$$411$$ 1.54559i 0.0762382i
$$412$$ −1.31160 −0.0646181
$$413$$ −42.1048 −2.07184
$$414$$ − 31.6954i − 1.55774i
$$415$$ −0.725474 −0.0356121
$$416$$ 0 0
$$417$$ −0.646952 −0.0316814
$$418$$ − 9.16715i − 0.448380i
$$419$$ 30.7296 1.50124 0.750621 0.660733i $$-0.229755\pi$$
0.750621 + 0.660733i $$0.229755\pi$$
$$420$$ 0.108224 0.00528080
$$421$$ − 17.9820i − 0.876391i −0.898880 0.438195i $$-0.855618\pi$$
0.898880 0.438195i $$-0.144382\pi$$
$$422$$ − 7.19403i − 0.350200i
$$423$$ − 19.4952i − 0.947890i
$$424$$ 2.61357i 0.126926i
$$425$$ 3.55889 0.172631
$$426$$ −1.10071 −0.0533298
$$427$$ 30.5893i 1.48032i
$$428$$ 1.05051 0.0507785
$$429$$ 0 0
$$430$$ 5.32235 0.256666
$$431$$ − 4.89949i − 0.236000i −0.993014 0.118000i $$-0.962352\pi$$
0.993014 0.118000i $$-0.0376483\pi$$
$$432$$ 2.50837 0.120684
$$433$$ 19.2394 0.924588 0.462294 0.886727i $$-0.347026\pi$$
0.462294 + 0.886727i $$0.347026\pi$$
$$434$$ − 10.5687i − 0.507315i
$$435$$ 0.139630i 0.00669476i
$$436$$ 3.25967i 0.156110i
$$437$$ − 40.6162i − 1.94294i
$$438$$ −1.43298 −0.0684703
$$439$$ 8.55974 0.408534 0.204267 0.978915i $$-0.434519\pi$$
0.204267 + 0.978915i $$0.434519\pi$$
$$440$$ 2.82026i 0.134451i
$$441$$ 48.7487 2.32137
$$442$$ 0 0
$$443$$ −37.9652 −1.80378 −0.901891 0.431965i $$-0.857821\pi$$
−0.901891 + 0.431965i $$0.857821\pi$$
$$444$$ 0 0.000569463i 0 2.70255e-5i
$$445$$ −13.5065 −0.640271
$$446$$ 22.0037 1.04191
$$447$$ 0.799965i 0.0378370i
$$448$$ − 33.0313i − 1.56058i
$$449$$ − 26.7062i − 1.26035i −0.776455 0.630173i $$-0.782984\pi$$
0.776455 0.630173i $$-0.217016\pi$$
$$450$$ 4.47309i 0.210863i
$$451$$ −0.286542 −0.0134927
$$452$$ 1.90275 0.0894979
$$453$$ − 0.129875i − 0.00610205i
$$454$$ 22.2873 1.04599
$$455$$ 0 0
$$456$$ 1.43290 0.0671016
$$457$$ 4.27127i 0.199801i 0.994997 + 0.0999007i $$0.0318525\pi$$
−0.994997 + 0.0999007i $$0.968147\pi$$
$$458$$ 28.8743 1.34921
$$459$$ 2.02100 0.0943322
$$460$$ − 1.67610i − 0.0781485i
$$461$$ − 20.6423i − 0.961407i −0.876883 0.480704i $$-0.840381\pi$$
0.876883 0.480704i $$-0.159619\pi$$
$$462$$ 0.731710i 0.0340422i
$$463$$ − 32.1040i − 1.49200i −0.665947 0.745999i $$-0.731972\pi$$
0.665947 0.745999i $$-0.268028\pi$$
$$464$$ −6.50682 −0.302071
$$465$$ −0.138779 −0.00643571
$$466$$ 31.6954i 1.46826i
$$467$$ −23.3774 −1.08178 −0.540888 0.841095i $$-0.681912\pi$$
−0.540888 + 0.841095i $$0.681912\pi$$
$$468$$ 0 0
$$469$$ 24.9700 1.15301
$$470$$ − 9.74761i − 0.449624i
$$471$$ 1.13461 0.0522800
$$472$$ −23.0050 −1.05889
$$473$$ 3.80584i 0.174993i
$$474$$ 1.24511i 0.0571896i
$$475$$ 5.73205i 0.263005i
$$476$$ 4.06338i 0.186245i
$$477$$ −2.96414 −0.135719
$$478$$ −22.2787 −1.01900
$$479$$ − 5.17534i − 0.236467i −0.992986 0.118234i $$-0.962277\pi$$
0.992986 0.118234i $$-0.0377232\pi$$
$$480$$ 0.126194 0.00575992
$$481$$ 0 0
$$482$$ −14.0453 −0.639747
$$483$$ 3.24193i 0.147513i
$$484$$ −2.33147 −0.105976
$$485$$ −3.43870 −0.156143
$$486$$ 3.81213i 0.172922i
$$487$$ 30.7729i 1.39445i 0.716850 + 0.697227i $$0.245583\pi$$
−0.716850 + 0.697227i $$0.754417\pi$$
$$488$$ 16.7132i 0.756572i
$$489$$ − 2.13948i − 0.0967508i
$$490$$ 24.3743 1.10112
$$491$$ −35.7983 −1.61556 −0.807778 0.589487i $$-0.799330\pi$$
−0.807778 + 0.589487i $$0.799330\pi$$
$$492$$ 0.00600778i 0 0.000270852i
$$493$$ −5.24255 −0.236113
$$494$$ 0 0
$$495$$ −3.19856 −0.143765
$$496$$ − 6.46713i − 0.290383i
$$497$$ −37.4798 −1.68120
$$498$$ −0.102840 −0.00460837
$$499$$ − 28.8971i − 1.29361i −0.762655 0.646805i $$-0.776105\pi$$
0.762655 0.646805i $$-0.223895\pi$$
$$500$$ 0.236543i 0.0105785i
$$501$$ 0.776974i 0.0347126i
$$502$$ − 16.9237i − 0.755340i
$$503$$ −7.86321 −0.350603 −0.175302 0.984515i $$-0.556090\pi$$
−0.175302 + 0.984515i $$0.556090\pi$$
$$504$$ 38.0746 1.69598
$$505$$ − 2.85527i − 0.127058i
$$506$$ 11.3322 0.503777
$$507$$ 0 0
$$508$$ 0.167154 0.00741625
$$509$$ 28.0113i 1.24158i 0.783977 + 0.620790i $$0.213188\pi$$
−0.783977 + 0.620790i $$0.786812\pi$$
$$510$$ 0.504492 0.0223393
$$511$$ −48.7935 −2.15850
$$512$$ − 17.4176i − 0.769757i
$$513$$ 3.25508i 0.143715i
$$514$$ 39.6787i 1.75015i
$$515$$ 5.54488i 0.244337i
$$516$$ 0.0797951 0.00351278
$$517$$ 6.97020 0.306549
$$518$$ 0.183339i 0.00805546i
$$519$$ 0.868986 0.0381443
$$520$$ 0 0
$$521$$ −37.5609 −1.64557 −0.822786 0.568351i $$-0.807581\pi$$
−0.822786 + 0.568351i $$0.807581\pi$$
$$522$$ − 6.58924i − 0.288403i
$$523$$ −45.3106 −1.98129 −0.990647 0.136450i $$-0.956431\pi$$
−0.990647 + 0.136450i $$0.956431\pi$$
$$524$$ −1.48207 −0.0647446
$$525$$ − 0.457524i − 0.0199680i
$$526$$ − 21.1476i − 0.922080i
$$527$$ − 5.21058i − 0.226976i
$$528$$ 0.447742i 0.0194855i
$$529$$ 27.2086 1.18298
$$530$$ −1.48207 −0.0643770
$$531$$ − 26.0908i − 1.13225i
$$532$$ −6.54460 −0.283744
$$533$$ 0 0
$$534$$ −1.91463 −0.0828540
$$535$$ − 4.44111i − 0.192006i
$$536$$ 13.6430 0.589287
$$537$$ 0.951383 0.0410552
$$538$$ 37.0504i 1.59736i
$$539$$ 17.4293i 0.750733i
$$540$$ 0.134327i 0.00578050i
$$541$$ 19.7445i 0.848882i 0.905456 + 0.424441i $$0.139529\pi$$
−0.905456 + 0.424441i $$0.860471\pi$$
$$542$$ −27.9916 −1.20234
$$543$$ −1.61365 −0.0692483
$$544$$ 4.73806i 0.203143i
$$545$$ 13.7804 0.590289
$$546$$ 0 0
$$547$$ −11.8312 −0.505867 −0.252934 0.967484i $$-0.581395\pi$$
−0.252934 + 0.967484i $$0.581395\pi$$
$$548$$ − 3.85702i − 0.164764i
$$549$$ −18.9551 −0.808983
$$550$$ −1.59928 −0.0681935
$$551$$ − 8.44381i − 0.359718i
$$552$$ 1.77131i 0.0753919i
$$553$$ 42.3964i 1.80288i
$$554$$ 33.8952i 1.44007i
$$555$$ 0.00240744 0.000102190 0
$$556$$ 1.61447 0.0684689
$$557$$ − 4.04621i − 0.171443i −0.996319 0.0857217i $$-0.972680\pi$$
0.996319 0.0857217i $$-0.0273196\pi$$
$$558$$ 6.54905 0.277244
$$559$$ 0 0
$$560$$ 21.3208 0.900968
$$561$$ 0.360746i 0.0152307i
$$562$$ 41.6326 1.75617
$$563$$ −3.89926 −0.164334 −0.0821671 0.996619i $$-0.526184\pi$$
−0.0821671 + 0.996619i $$0.526184\pi$$
$$564$$ − 0.146141i − 0.00615364i
$$565$$ − 8.04399i − 0.338413i
$$566$$ − 11.8445i − 0.497863i
$$567$$ 43.0516i 1.80800i
$$568$$ −20.4780 −0.859239
$$569$$ 17.3356 0.726744 0.363372 0.931644i $$-0.381625\pi$$
0.363372 + 0.931644i $$0.381625\pi$$
$$570$$ 0.812550i 0.0340340i
$$571$$ −29.5118 −1.23503 −0.617515 0.786559i $$-0.711860\pi$$
−0.617515 + 0.786559i $$0.711860\pi$$
$$572$$ 0 0
$$573$$ −0.367530 −0.0153538
$$574$$ 1.93421i 0.0807324i
$$575$$ −7.08580 −0.295498
$$576$$ 20.4683 0.852845
$$577$$ 28.3684i 1.18099i 0.807041 + 0.590496i $$0.201068\pi$$
−0.807041 + 0.590496i $$0.798932\pi$$
$$578$$ − 6.48199i − 0.269615i
$$579$$ 0.118858i 0.00493958i
$$580$$ − 0.348448i − 0.0144685i
$$581$$ −3.50174 −0.145277
$$582$$ −0.487455 −0.0202057
$$583$$ − 1.05978i − 0.0438917i
$$584$$ −26.6595 −1.10318
$$585$$ 0 0
$$586$$ 0.408230 0.0168638
$$587$$ 34.3877i 1.41933i 0.704538 + 0.709667i $$0.251154\pi$$
−0.704538 + 0.709667i $$0.748846\pi$$
$$588$$ 0.365432 0.0150701
$$589$$ 8.39230 0.345799
$$590$$ − 13.0454i − 0.537071i
$$591$$ 1.44854i 0.0595850i
$$592$$ 0.112187i 0.00461088i
$$593$$ − 5.47612i − 0.224877i −0.993659 0.112439i $$-0.964134\pi$$
0.993659 0.112439i $$-0.0358662\pi$$
$$594$$ −0.908189 −0.0372635
$$595$$ 17.1782 0.704237
$$596$$ − 1.99632i − 0.0817724i
$$597$$ 1.25400 0.0513229
$$598$$ 0 0
$$599$$ 38.6039 1.57731 0.788657 0.614833i $$-0.210777\pi$$
0.788657 + 0.614833i $$0.210777\pi$$
$$600$$ − 0.249980i − 0.0102054i
$$601$$ 6.57896 0.268361 0.134181 0.990957i $$-0.457160\pi$$
0.134181 + 0.990957i $$0.457160\pi$$
$$602$$ 25.6901 1.04705
$$603$$ 15.4730i 0.630109i
$$604$$ 0.324103i 0.0131876i
$$605$$ 9.85641i 0.400720i
$$606$$ − 0.404750i − 0.0164418i
$$607$$ −16.7664 −0.680525 −0.340263 0.940330i $$-0.610516\pi$$
−0.340263 + 0.940330i $$0.610516\pi$$
$$608$$ −7.63126 −0.309488
$$609$$ 0.673973i 0.0273108i
$$610$$ −9.47754 −0.383734
$$611$$ 0 0
$$612$$ −2.51793 −0.101781
$$613$$ 28.7789i 1.16237i 0.813772 + 0.581184i $$0.197410\pi$$
−0.813772 + 0.581184i $$0.802590\pi$$
$$614$$ −10.2476 −0.413558
$$615$$ 0.0253983 0.00102416
$$616$$ 13.6129i 0.548481i
$$617$$ − 37.3291i − 1.50281i −0.659840 0.751406i $$-0.729376\pi$$
0.659840 0.751406i $$-0.270624\pi$$
$$618$$ 0.786018i 0.0316183i
$$619$$ − 12.7535i − 0.512606i −0.966597 0.256303i $$-0.917496\pi$$
0.966597 0.256303i $$-0.0825045\pi$$
$$620$$ 0.346323 0.0139087
$$621$$ −4.02384 −0.161471
$$622$$ − 15.9192i − 0.638303i
$$623$$ −65.1939 −2.61194
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 26.6553i 1.06536i
$$627$$ −0.581028 −0.0232040
$$628$$ −2.83143 −0.112986
$$629$$ 0.0903896i 0.00360407i
$$630$$ 21.5909i 0.860201i
$$631$$ − 28.7242i − 1.14349i −0.820430 0.571746i $$-0.806266\pi$$
0.820430 0.571746i $$-0.193734\pi$$
$$632$$ 23.1643i 0.921427i
$$633$$ −0.455969 −0.0181231
$$634$$ 12.2207 0.485346
$$635$$ − 0.706653i − 0.0280427i
$$636$$ −0.0222199 −0.000881077 0
$$637$$ 0 0
$$638$$ 2.35588 0.0932701
$$639$$ − 23.2249i − 0.918762i
$$640$$ 12.8968 0.509791
$$641$$ 22.3970 0.884630 0.442315 0.896860i $$-0.354157\pi$$
0.442315 + 0.896860i $$0.354157\pi$$
$$642$$ − 0.629552i − 0.0248464i
$$643$$ − 14.1642i − 0.558581i −0.960207 0.279290i $$-0.909901\pi$$
0.960207 0.279290i $$-0.0900992\pi$$
$$644$$ − 8.09025i − 0.318801i
$$645$$ − 0.337339i − 0.0132827i
$$646$$ −30.5080 −1.20032
$$647$$ 23.6097 0.928193 0.464096 0.885785i $$-0.346379\pi$$
0.464096 + 0.885785i $$0.346379\pi$$
$$648$$ 23.5223i 0.924044i
$$649$$ 9.32835 0.366170
$$650$$ 0 0
$$651$$ −0.669862 −0.0262540
$$652$$ 5.33910i 0.209095i
$$653$$ 33.2765 1.30221 0.651105 0.758987i $$-0.274306\pi$$
0.651105 + 0.758987i $$0.274306\pi$$
$$654$$ 1.95345 0.0763861
$$655$$ 6.26554i 0.244815i
$$656$$ 1.18357i 0.0462105i
$$657$$ − 30.2356i − 1.17960i
$$658$$ − 47.0502i − 1.83421i
$$659$$ 23.0908 0.899491 0.449745 0.893157i $$-0.351515\pi$$
0.449745 + 0.893157i $$0.351515\pi$$
$$660$$ −0.0239772 −0.000933310 0
$$661$$ 13.4365i 0.522620i 0.965255 + 0.261310i $$0.0841545\pi$$
−0.965255 + 0.261310i $$0.915845\pi$$
$$662$$ 37.2972 1.44960
$$663$$ 0 0
$$664$$ −1.91326 −0.0742491
$$665$$ 27.6677i 1.07291i
$$666$$ −0.113609 −0.00440224
$$667$$ 10.4380 0.404161
$$668$$ − 1.93895i − 0.0750200i
$$669$$ − 1.39463i − 0.0539196i
$$670$$ 7.73650i 0.298887i
$$671$$ − 6.77708i − 0.261626i
$$672$$ 0.609116 0.0234972
$$673$$ 1.94524 0.0749835 0.0374918 0.999297i $$-0.488063\pi$$
0.0374918 + 0.999297i $$0.488063\pi$$
$$674$$ 29.3205i 1.12938i
$$675$$ 0.567874 0.0218575
$$676$$ 0 0
$$677$$ 24.8683 0.955768 0.477884 0.878423i $$-0.341404\pi$$
0.477884 + 0.878423i $$0.341404\pi$$
$$678$$ − 1.14028i − 0.0437922i
$$679$$ −16.5981 −0.636975
$$680$$ 9.38573 0.359926
$$681$$ − 1.41260i − 0.0541310i
$$682$$ 2.34151i 0.0896610i
$$683$$ 14.6221i 0.559500i 0.960073 + 0.279750i $$0.0902517\pi$$
−0.960073 + 0.279750i $$0.909748\pi$$
$$684$$ − 4.05545i − 0.155064i
$$685$$ −16.3058 −0.623013
$$686$$ 67.1210 2.56269
$$687$$ − 1.83010i − 0.0698226i
$$688$$ 15.7201 0.599323
$$689$$ 0 0
$$690$$ −1.00445 −0.0382388
$$691$$ − 3.52451i − 0.134079i −0.997750 0.0670393i $$-0.978645\pi$$
0.997750 0.0670393i $$-0.0213553\pi$$
$$692$$ −2.16856 −0.0824364
$$693$$ −15.4389 −0.586477
$$694$$ 25.5447i 0.969665i
$$695$$ − 6.82528i − 0.258898i
$$696$$ 0.368242i 0.0139582i
$$697$$ 0.953601i 0.0361202i
$$698$$ 42.4304 1.60602
$$699$$ 2.00890 0.0759837
$$700$$ 1.14176i 0.0431543i
$$701$$ 1.53457 0.0579599 0.0289800 0.999580i $$-0.490774\pi$$
0.0289800 + 0.999580i $$0.490774\pi$$
$$702$$ 0 0
$$703$$ −0.145584 −0.00549081
$$704$$ 7.31810i 0.275811i
$$705$$ −0.617819 −0.0232684
$$706$$ 31.7825 1.19615
$$707$$ − 13.7819i − 0.518322i
$$708$$ − 0.195583i − 0.00735046i
$$709$$ − 14.0052i − 0.525978i −0.964799 0.262989i $$-0.915292\pi$$
0.964799 0.262989i $$-0.0847083\pi$$
$$710$$ − 11.6124i − 0.435807i
$$711$$ −26.2715 −0.985258
$$712$$ −35.6203 −1.33493
$$713$$ 10.3743i 0.388522i
$$714$$ 2.43510 0.0911314
$$715$$ 0 0
$$716$$ −2.37418 −0.0887274
$$717$$ 1.41206i 0.0527343i
$$718$$ −48.8312 −1.82236
$$719$$ −22.4761 −0.838218 −0.419109 0.907936i $$-0.637657\pi$$
−0.419109 + 0.907936i $$0.637657\pi$$
$$720$$ 13.2117i 0.492372i
$$721$$ 26.7643i 0.996753i
$$722$$ − 20.7224i − 0.771206i
$$723$$ 0.890215i 0.0331074i
$$724$$ 4.02687 0.149658
$$725$$ −1.47309 −0.0547091
$$726$$ 1.39720i 0.0518550i
$$727$$ 10.3421 0.383566 0.191783 0.981437i $$-0.438573\pi$$
0.191783 + 0.981437i $$0.438573\pi$$
$$728$$ 0 0
$$729$$ −26.5160 −0.982075
$$730$$ − 15.1178i − 0.559534i
$$731$$ 12.6657 0.468458
$$732$$ −0.142092 −0.00525186
$$733$$ 27.3533i 1.01032i 0.863026 + 0.505159i $$0.168566\pi$$
−0.863026 + 0.505159i $$0.831434\pi$$
$$734$$ − 8.85096i − 0.326695i
$$735$$ − 1.54488i − 0.0569839i
$$736$$ − 9.43355i − 0.347725i
$$737$$ −5.53212 −0.203778
$$738$$ −1.19856 −0.0441196
$$739$$ 13.4517i 0.494827i 0.968910 + 0.247413i $$0.0795806\pi$$
−0.968910 + 0.247413i $$0.920419\pi$$
$$740$$ −0.00600778 −0.000220851 0
$$741$$ 0 0
$$742$$ −7.15372 −0.262621
$$743$$ 16.3926i 0.601388i 0.953721 + 0.300694i $$0.0972182\pi$$
−0.953721 + 0.300694i $$0.902782\pi$$
$$744$$ −0.365996 −0.0134181
$$745$$ −8.43955 −0.309201
$$746$$ − 19.9179i − 0.729248i
$$747$$ − 2.16990i − 0.0793926i
$$748$$ − 0.900245i − 0.0329162i
$$749$$ − 21.4365i − 0.783274i
$$750$$ 0.141756 0.00517618
$$751$$ 27.6655 1.00953 0.504764 0.863257i $$-0.331579\pi$$
0.504764 + 0.863257i $$0.331579\pi$$
$$752$$ − 28.7906i − 1.04988i
$$753$$ −1.07265 −0.0390895
$$754$$ 0 0
$$755$$ 1.37017 0.0498654
$$756$$ 0.648373i 0.0235811i
$$757$$ 22.9978 0.835870 0.417935 0.908477i $$-0.362754\pi$$
0.417935 + 0.908477i $$0.362754\pi$$
$$758$$ 38.0136 1.38072
$$759$$ − 0.718251i − 0.0260709i
$$760$$ 15.1169i 0.548349i
$$761$$ − 7.66442i − 0.277835i −0.990304 0.138918i $$-0.955638\pi$$
0.990304 0.138918i $$-0.0443623\pi$$
$$762$$ − 0.100172i − 0.00362885i
$$763$$ 66.5160 2.40804
$$764$$ 0.917174 0.0331822
$$765$$ 10.6447i 0.384860i
$$766$$ −16.1916 −0.585027
$$767$$ 0 0
$$768$$ 0.530882 0.0191566
$$769$$ 7.23095i 0.260755i 0.991464 + 0.130377i $$0.0416189\pi$$
−0.991464 + 0.130377i $$0.958381\pi$$
$$770$$ −7.71947 −0.278190
$$771$$ 2.51490 0.0905718
$$772$$ − 0.296612i − 0.0106753i
$$773$$ 33.5995i 1.20849i 0.796798 + 0.604246i $$0.206525\pi$$
−0.796798 + 0.604246i $$0.793475\pi$$
$$774$$ 15.9192i 0.572204i
$$775$$ − 1.46410i − 0.0525921i
$$776$$ −9.06877 −0.325550
$$777$$ 0.0116203 0.000416877 0
$$778$$ 34.4520i 1.23516i
$$779$$ −1.53590 −0.0550293
$$780$$ 0 0
$$781$$ 8.30368 0.297129
$$782$$ − 37.7131i − 1.34862i
$$783$$ −0.836527 −0.0298950
$$784$$ 71.9921 2.57115
$$785$$ 11.9700i 0.427228i
$$786$$ 0.888175i 0.0316802i
$$787$$ − 2.97168i − 0.105929i −0.998596 0.0529645i $$-0.983133\pi$$
0.998596 0.0529645i $$-0.0168670\pi$$
$$788$$ − 3.61484i − 0.128773i
$$789$$ −1.34037 −0.0477184
$$790$$ −13.1357 −0.467349
$$791$$ − 38.8270i − 1.38053i
$$792$$ −8.43544 −0.299740
$$793$$ 0 0
$$794$$ −31.5349 −1.11913
$$795$$ 0.0939360i 0.00333156i
$$796$$ −3.12937 −0.110918
$$797$$ 22.5751 0.799650 0.399825 0.916591i $$-0.369071\pi$$
0.399825 + 0.916591i $$0.369071\pi$$
$$798$$ 3.92205i 0.138839i
$$799$$ − 23.1966i − 0.820636i
$$800$$ 1.33133i 0.0470696i
$$801$$ − 40.3983i − 1.42740i
$$802$$ 29.5765 1.04438
$$803$$ 10.8102 0.381485
$$804$$ 0.115989i 0.00409063i
$$805$$ −34.2020 −1.20546
$$806$$ 0 0
$$807$$ 2.34831 0.0826646
$$808$$ − 7.53009i − 0.264908i
$$809$$ 13.6584 0.480204 0.240102 0.970748i $$-0.422819\pi$$
0.240102 + 0.970748i $$0.422819\pi$$
$$810$$ −13.3388 −0.468676
$$811$$ − 14.1147i − 0.495636i −0.968807 0.247818i $$-0.920287\pi$$
0.968807 0.247818i $$-0.0797135\pi$$
$$812$$ − 1.68190i − 0.0590233i
$$813$$ 1.77415i 0.0622222i
$$814$$ − 0.0406189i − 0.00142369i
$$815$$ 22.5713 0.790640
$$816$$ 1.49007 0.0521629
$$817$$ 20.3997i 0.713696i
$$818$$ −47.6781 −1.66703
$$819$$ 0 0
$$820$$ −0.0633815 −0.00221338
$$821$$ − 1.58562i − 0.0553383i −0.999617 0.0276692i $$-0.991192\pi$$
0.999617 0.0276692i $$-0.00880850\pi$$
$$822$$ −2.31144 −0.0806206
$$823$$ −18.5648 −0.647127 −0.323563 0.946206i $$-0.604881\pi$$
−0.323563 + 0.946206i $$0.604881\pi$$
$$824$$ 14.6233i 0.509427i
$$825$$ 0.101365i 0.00352907i
$$826$$ − 62.9681i − 2.19094i
$$827$$ 9.01023i 0.313316i 0.987653 + 0.156658i $$0.0500721\pi$$
−0.987653 + 0.156658i $$0.949928\pi$$
$$828$$ 5.01324 0.174222
$$829$$ −47.1177 −1.63647 −0.818233 0.574887i $$-0.805046\pi$$
−0.818233 + 0.574887i $$0.805046\pi$$
$$830$$ − 1.08495i − 0.0376592i
$$831$$ 2.14833 0.0745247
$$832$$ 0 0
$$833$$ 58.0041 2.00972
$$834$$ − 0.967522i − 0.0335025i
$$835$$ −8.19700 −0.283669
$$836$$ 1.44996 0.0501479
$$837$$ − 0.831425i − 0.0287383i
$$838$$ 45.9564i 1.58754i
$$839$$ − 53.4766i − 1.84622i −0.384541 0.923108i $$-0.625640\pi$$
0.384541 0.923108i $$-0.374360\pi$$
$$840$$ − 1.20661i − 0.0416321i
$$841$$ −26.8300 −0.925173
$$842$$ 26.8923 0.926769
$$843$$ − 2.63874i − 0.0908830i
$$844$$ 1.13787 0.0391672
$$845$$ 0 0
$$846$$ 29.1553 1.00238
$$847$$ 47.5753i 1.63471i
$$848$$ −4.37745 −0.150322
$$849$$ −0.750724 −0.0257648
$$850$$ 5.32235i 0.182555i
$$851$$ − 0.179967i − 0.00616919i
$$852$$ − 0.174099i − 0.00596454i
$$853$$ 27.7756i 0.951019i 0.879711 + 0.475510i $$0.157736\pi$$
−0.879711 + 0.475510i $$0.842264\pi$$
$$854$$ −45.7465 −1.56541
$$855$$ −17.1447 −0.586335
$$856$$ − 11.7124i − 0.400321i
$$857$$ −53.6917 −1.83407 −0.917037 0.398801i $$-0.869426\pi$$
−0.917037 + 0.398801i $$0.869426\pi$$
$$858$$ 0 0
$$859$$ 2.08958 0.0712955 0.0356477 0.999364i $$-0.488651\pi$$
0.0356477 + 0.999364i $$0.488651\pi$$
$$860$$ 0.841831i 0.0287062i
$$861$$ 0.122593 0.00417797
$$862$$ 7.32722 0.249566
$$863$$ − 1.75413i − 0.0597113i −0.999554 0.0298557i $$-0.990495\pi$$
0.999554 0.0298557i $$-0.00950476\pi$$
$$864$$ 0.756028i 0.0257206i
$$865$$ 9.16772i 0.311712i
$$866$$ 28.7727i 0.977737i
$$867$$ −0.410839 −0.0139528
$$868$$ 1.67165 0.0567394
$$869$$ − 9.39295i − 0.318634i
$$870$$ −0.208818 −0.00707960
$$871$$ 0 0
$$872$$ 36.3426 1.23072
$$873$$ − 10.2852i − 0.348102i
$$874$$ 60.7418 2.05462
$$875$$ 4.82684 0.163177
$$876$$ − 0.226653i − 0.00765789i
$$877$$ 21.5672i 0.728272i 0.931346 + 0.364136i $$0.118636\pi$$
−0.931346 + 0.364136i $$0.881364\pi$$
$$878$$ 12.8012i 0.432018i
$$879$$ − 0.0258742i 0 0.000872716i
$$880$$ −4.72364 −0.159234
$$881$$ −25.0263 −0.843158 −0.421579 0.906792i $$-0.638524\pi$$
−0.421579 + 0.906792i $$0.638524\pi$$
$$882$$ 72.9040i 2.45481i
$$883$$ 48.7832 1.64169 0.820843 0.571154i $$-0.193504\pi$$
0.820843 + 0.571154i $$0.193504\pi$$
$$884$$ 0 0
$$885$$ −0.826838 −0.0277939
$$886$$ − 56.7772i − 1.90747i
$$887$$ −33.7933 −1.13467 −0.567334 0.823488i $$-0.692025\pi$$
−0.567334 + 0.823488i $$0.692025\pi$$
$$888$$ 0.00634905 0.000213060 0
$$889$$ − 3.41090i − 0.114398i
$$890$$ − 20.1991i − 0.677076i
$$891$$ − 9.53812i − 0.319539i
$$892$$ 3.48031i 0.116530i
$$893$$ 37.3611 1.25024
$$894$$ −1.19635 −0.0400121
$$895$$ 10.0370i 0.335500i
$$896$$ 62.2508 2.07965
$$897$$ 0 0
$$898$$ 39.9394 1.33279
$$899$$ 2.15675i 0.0719316i
$$900$$ −0.707504 −0.0235835
$$901$$ −3.52691 −0.117499
$$902$$ − 0.428526i − 0.0142683i
$$903$$ − 1.62828i − 0.0541857i
$$904$$ − 21.2141i − 0.705571i
$$905$$ − 17.0238i − 0.565892i
$$906$$ 0.194229 0.00645281
$$907$$ 34.6270 1.14977 0.574885 0.818234i $$-0.305047\pi$$
0.574885 + 0.818234i $$0.305047\pi$$
$$908$$ 3.52516i 0.116986i
$$909$$ 8.54015 0.283259
$$910$$ 0 0
$$911$$ 31.1865 1.03326 0.516628 0.856210i $$-0.327187\pi$$
0.516628 + 0.856210i $$0.327187\pi$$
$$912$$ 2.39995i 0.0794703i
$$913$$ 0.775814 0.0256757
$$914$$ −6.38771 −0.211287
$$915$$ 0.600701i 0.0198586i
$$916$$ 4.56702i 0.150899i
$$917$$ 30.2428i 0.998704i
$$918$$ 3.02242i 0.0997548i
$$919$$ −51.9220 −1.71275 −0.856374 0.516356i $$-0.827288\pi$$
−0.856374 + 0.516356i $$0.827288\pi$$
$$920$$ −18.6871 −0.616096
$$921$$ 0.649507i 0.0214020i
$$922$$ 30.8707 1.01667
$$923$$ 0 0
$$924$$ −0.115734 −0.00380736
$$925$$ 0.0253983i 0 0.000835090i
$$926$$ 48.0117 1.57776
$$927$$ −16.5848 −0.544717
$$928$$ − 1.96117i − 0.0643784i
$$929$$ 20.4915i 0.672304i 0.941808 + 0.336152i $$0.109126\pi$$
−0.941808 + 0.336152i $$0.890874\pi$$
$$930$$ − 0.207545i − 0.00680565i
$$931$$ 93.4231i 3.06182i
$$932$$ −5.01324 −0.164214
$$933$$ −1.00898 −0.0330327
$$934$$ − 34.9610i − 1.14396i
$$935$$ −3.80584 −0.124464
$$936$$ 0 0
$$937$$ −39.6806 −1.29631 −0.648154 0.761510i $$-0.724459\pi$$
−0.648154 + 0.761510i $$0.724459\pi$$
$$938$$ 37.3428i 1.21929i
$$939$$ 1.68945 0.0551333
$$940$$ 1.54177 0.0502870
$$941$$ − 19.6189i − 0.639557i −0.947492 0.319779i $$-0.896391\pi$$
0.947492 0.319779i $$-0.103609\pi$$
$$942$$ 1.69682i 0.0552853i
$$943$$ − 1.89864i − 0.0618281i
$$944$$ − 38.5309i − 1.25408i
$$945$$ 2.74104 0.0891659
$$946$$ −5.69166 −0.185052
$$947$$ − 57.2124i − 1.85915i −0.368631 0.929576i $$-0.620173\pi$$
0.368631 0.929576i $$-0.379827\pi$$
$$948$$ −0.196937 −0.00639623
$$949$$ 0 0
$$950$$ −8.57233 −0.278123
$$951$$ − 0.774567i − 0.0251170i
$$952$$ 45.3034 1.46829
$$953$$ −27.4770 −0.890066 −0.445033 0.895514i $$-0.646808\pi$$
−0.445033 + 0.895514i $$0.646808\pi$$
$$954$$ − 4.43290i − 0.143520i
$$955$$ − 3.87741i − 0.125470i
$$956$$ − 3.52380i − 0.113968i
$$957$$ − 0.149319i − 0.00482680i
$$958$$ 7.73976 0.250060
$$959$$ −78.7055 −2.54153
$$960$$ − 0.648655i − 0.0209353i
$$961$$ 28.8564 0.930852
$$962$$ 0 0
$$963$$ 13.2834 0.428053
$$964$$ − 2.22154i − 0.0715509i
$$965$$ −1.25394 −0.0403659
$$966$$ −4.84833 −0.155992
$$967$$ − 10.3643i − 0.333293i −0.986017 0.166647i $$-0.946706\pi$$
0.986017 0.166647i $$-0.0532939\pi$$
$$968$$ 25.9939i 0.835477i
$$969$$ 1.93364i 0.0621175i
$$970$$ − 5.14261i − 0.165119i
$$971$$ 41.7515 1.33987 0.669935 0.742420i $$-0.266322\pi$$
0.669935 + 0.742420i $$0.266322\pi$$
$$972$$ −0.602961 −0.0193400
$$973$$ − 32.9445i − 1.05615i
$$974$$ −46.0212 −1.47461
$$975$$ 0 0
$$976$$ −27.9929 −0.896030
$$977$$ − 13.7938i − 0.441303i −0.975353 0.220652i $$-0.929182\pi$$
0.975353 0.220652i $$-0.0708184\pi$$
$$978$$ 3.19961 0.102312
$$979$$ 14.4437 0.461624
$$980$$ 3.85527i 0.123152i
$$981$$ 41.2175i 1.31597i
$$982$$ − 53.5367i − 1.70842i
$$983$$ − 37.9997i − 1.21200i −0.795463 0.606002i $$-0.792773\pi$$
0.795463 0.606002i $$-0.207227\pi$$
$$984$$ 0.0669819 0.00213530
$$985$$ −15.2820 −0.486924
$$986$$ − 7.84028i − 0.249685i
$$987$$ −2.98211 −0.0949217
$$988$$ 0 0
$$989$$ −25.2176 −0.801873
$$990$$ − 4.78347i − 0.152029i
$$991$$ −52.5530 −1.66940 −0.834700 0.550705i $$-0.814359\pi$$
−0.834700 + 0.550705i $$0.814359\pi$$
$$992$$ 1.94920 0.0618873
$$993$$ − 2.36396i − 0.0750179i
$$994$$ − 56.0513i − 1.77784i
$$995$$ 13.2296i 0.419407i
$$996$$ − 0.0162661i 0 0.000515411i
$$997$$ 18.5899 0.588749 0.294375 0.955690i $$-0.404889\pi$$
0.294375 + 0.955690i $$0.404889\pi$$
$$998$$ 43.2158 1.36797
$$999$$ 0.0144230i 0 0.000456324i
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 845.2.c.g.506.7 8
13.2 odd 12 845.2.e.n.191.1 8
13.3 even 3 845.2.m.g.316.1 8
13.4 even 6 845.2.m.g.361.1 8
13.5 odd 4 845.2.a.l.1.4 4
13.6 odd 12 845.2.e.n.146.1 8
13.7 odd 12 845.2.e.m.146.4 8
13.8 odd 4 845.2.a.m.1.1 4
13.9 even 3 65.2.m.a.36.4 8
13.10 even 6 65.2.m.a.56.4 yes 8
13.11 odd 12 845.2.e.m.191.4 8
13.12 even 2 inner 845.2.c.g.506.2 8
39.5 even 4 7605.2.a.cj.1.1 4
39.8 even 4 7605.2.a.cf.1.4 4
39.23 odd 6 585.2.bu.c.316.1 8
39.35 odd 6 585.2.bu.c.361.1 8
52.23 odd 6 1040.2.da.b.641.3 8
52.35 odd 6 1040.2.da.b.881.3 8
65.9 even 6 325.2.n.d.101.1 8
65.22 odd 12 325.2.m.b.49.4 8
65.23 odd 12 325.2.m.b.199.4 8
65.34 odd 4 4225.2.a.bi.1.4 4
65.44 odd 4 4225.2.a.bl.1.1 4
65.48 odd 12 325.2.m.c.49.1 8
65.49 even 6 325.2.n.d.251.1 8
65.62 odd 12 325.2.m.c.199.1 8

By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.m.a.36.4 8 13.9 even 3
65.2.m.a.56.4 yes 8 13.10 even 6
325.2.m.b.49.4 8 65.22 odd 12
325.2.m.b.199.4 8 65.23 odd 12
325.2.m.c.49.1 8 65.48 odd 12
325.2.m.c.199.1 8 65.62 odd 12
325.2.n.d.101.1 8 65.9 even 6
325.2.n.d.251.1 8 65.49 even 6
585.2.bu.c.316.1 8 39.23 odd 6
585.2.bu.c.361.1 8 39.35 odd 6
845.2.a.l.1.4 4 13.5 odd 4
845.2.a.m.1.1 4 13.8 odd 4
845.2.c.g.506.2 8 13.12 even 2 inner
845.2.c.g.506.7 8 1.1 even 1 trivial
845.2.e.m.146.4 8 13.7 odd 12
845.2.e.m.191.4 8 13.11 odd 12
845.2.e.n.146.1 8 13.6 odd 12
845.2.e.n.191.1 8 13.2 odd 12
845.2.m.g.316.1 8 13.3 even 3
845.2.m.g.361.1 8 13.4 even 6
1040.2.da.b.641.3 8 52.23 odd 6
1040.2.da.b.881.3 8 52.35 odd 6
4225.2.a.bi.1.4 4 65.34 odd 4
4225.2.a.bl.1.1 4 65.44 odd 4
7605.2.a.cf.1.4 4 39.8 even 4
7605.2.a.cj.1.1 4 39.5 even 4