# Properties

 Label 975.2.c.e.274.2 Level $975$ Weight $2$ Character 975.274 Analytic conductor $7.785$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.78541419707$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 274.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 975.274 Dual form 975.2.c.e.274.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} +1.00000i q^{13} -1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +4.00000i q^{22} +8.00000i q^{23} -3.00000 q^{24} -1.00000 q^{26} -1.00000i q^{27} +2.00000 q^{29} -8.00000 q^{31} +5.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} -6.00000i q^{37} +4.00000i q^{38} -1.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -8.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} +1.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} +4.00000i q^{57} +2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} -8.00000i q^{62} -7.00000 q^{64} -4.00000 q^{66} +4.00000i q^{67} -2.00000i q^{68} -8.00000 q^{69} -3.00000i q^{72} -6.00000i q^{73} +6.00000 q^{74} +4.00000 q^{76} -1.00000i q^{78} -16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} +4.00000 q^{86} +2.00000i q^{87} +12.0000i q^{88} -10.0000 q^{89} +8.00000i q^{92} -8.00000i q^{93} -8.00000 q^{94} -5.00000 q^{96} -18.0000i q^{97} +7.00000i q^{98} -4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} - 2 q^{16} + 8 q^{19} - 6 q^{24} - 2 q^{26} + 4 q^{29} - 16 q^{31} + 4 q^{34} - 2 q^{36} - 2 q^{39} - 12 q^{41} + 8 q^{44} - 16 q^{46} + 14 q^{49} + 4 q^{51} + 2 q^{54} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{66} - 16 q^{69} + 12 q^{74} + 8 q^{76} - 32 q^{79} + 2 q^{81} + 8 q^{86} - 20 q^{89} - 16 q^{94} - 10 q^{96} - 8 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 - 2 * q^16 + 8 * q^19 - 6 * q^24 - 2 * q^26 + 4 * q^29 - 16 * q^31 + 4 * q^34 - 2 * q^36 - 2 * q^39 - 12 * q^41 + 8 * q^44 - 16 * q^46 + 14 * q^49 + 4 * q^51 + 2 * q^54 + 24 * q^59 - 4 * q^61 - 14 * q^64 - 8 * q^66 - 16 * q^69 + 12 * q^74 + 8 * q^76 - 32 * q^79 + 2 * q^81 + 8 * q^86 - 20 * q^89 - 16 * q^94 - 10 * q^96 - 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 3.00000i 1.06066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 1.00000i 0.277350i
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000i 0.852803i
$$23$$ 8.00000i 1.66812i 0.551677 + 0.834058i $$0.313988\pi$$
−0.551677 + 0.834058i $$0.686012\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ − 1.00000i − 0.192450i
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ 4.00000i 0.696311i
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ − 6.00000i − 0.986394i −0.869918 0.493197i $$-0.835828\pi$$
0.869918 0.493197i $$-0.164172\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 1.00000i 0.138675i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.00000i 0.529813i
$$58$$ 2.00000i 0.262613i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ − 8.00000i − 1.01600i
$$63$$ 0 0
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ − 2.00000i − 0.242536i
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ − 3.00000i − 0.353553i
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ − 1.00000i − 0.113228i
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 6.00000i − 0.662589i
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 2.00000i 0.214423i
$$88$$ 12.0000i 1.27920i
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 8.00000i 0.834058i
$$93$$ − 8.00000i − 0.829561i
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ −5.00000 −0.510310
$$97$$ − 18.0000i − 1.82762i −0.406138 0.913812i $$-0.633125\pi$$
0.406138 0.913812i $$-0.366875\pi$$
$$98$$ 7.00000i 0.707107i
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 2.00000i 0.198030i
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ −3.00000 −0.294174
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 0 0
$$113$$ 2.00000i 0.188144i 0.995565 + 0.0940721i $$0.0299884\pi$$
−0.995565 + 0.0940721i $$0.970012\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ − 1.00000i − 0.0924500i
$$118$$ 12.0000i 1.10469i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 2.00000i − 0.181071i
$$123$$ − 6.00000i − 0.541002i
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ − 8.00000i − 0.681005i
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 0 0
$$143$$ 4.00000i 0.334497i
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ 7.00000i 0.577350i
$$148$$ − 6.00000i − 0.493197i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 12.0000i 0.973329i
$$153$$ 2.00000i 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ − 16.0000i − 1.27289i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ − 20.0000i − 1.56652i −0.621694 0.783260i $$-0.713555\pi$$
0.621694 0.783260i $$-0.286445\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ − 16.0000i − 1.23812i −0.785345 0.619059i $$-0.787514\pi$$
0.785345 0.619059i $$-0.212486\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ − 4.00000i − 0.304997i
$$173$$ − 2.00000i − 0.152057i −0.997106 0.0760286i $$-0.975776\pi$$
0.997106 0.0760286i $$-0.0242240\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 12.0000i 0.901975i
$$178$$ − 10.0000i − 0.749532i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 0 0
$$183$$ − 2.00000i − 0.147844i
$$184$$ −24.0000 −1.76930
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ − 8.00000i − 0.585018i
$$188$$ 8.00000i 0.583460i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ − 7.00000i − 0.505181i
$$193$$ − 14.0000i − 1.00774i −0.863779 0.503871i $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ 18.0000 1.29232
$$195$$ 0 0
$$196$$ 7.00000 0.500000
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ − 4.00000i − 0.284268i
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 6.00000i 0.422159i
$$203$$ 0 0
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ − 8.00000i − 0.556038i
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 3.00000 0.204124
$$217$$ 0 0
$$218$$ 2.00000i 0.135457i
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 2.00000 0.134535
$$222$$ 6.00000i 0.402694i
$$223$$ − 24.0000i − 1.60716i −0.595198 0.803579i $$-0.702926\pi$$
0.595198 0.803579i $$-0.297074\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000i 0.393919i
$$233$$ 26.0000i 1.70332i 0.524097 + 0.851658i $$0.324403\pi$$
−0.524097 + 0.851658i $$0.675597\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ − 16.0000i − 1.03931i
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 5.00000i 0.321412i
$$243$$ 1.00000i 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 4.00000i 0.254514i
$$248$$ − 24.0000i − 1.52400i
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 0 0
$$253$$ 32.0000i 2.01182i
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ − 12.0000i − 0.741362i
$$263$$ − 24.0000i − 1.47990i −0.672660 0.739952i $$-0.734848\pi$$
0.672660 0.739952i $$-0.265152\pi$$
$$264$$ −12.0000 −0.738549
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 10.0000i − 0.611990i
$$268$$ 4.00000i 0.244339i
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 2.00000i 0.121268i
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ 10.0000i 0.600842i 0.953807 + 0.300421i $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ 20.0000i 1.19952i
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ − 8.00000i − 0.476393i
$$283$$ − 20.0000i − 1.18888i −0.804141 0.594438i $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 0 0
$$288$$ − 5.00000i − 0.294628i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 18.0000 1.05518
$$292$$ − 6.00000i − 0.351123i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ −7.00000 −0.408248
$$295$$ 0 0
$$296$$ 18.0000 1.04623
$$297$$ − 4.00000i − 0.232104i
$$298$$ 10.0000i 0.579284i
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 6.00000i 0.344691i
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ 4.00000i 0.228292i 0.993464 + 0.114146i $$0.0364132\pi$$
−0.993464 + 0.114146i $$0.963587\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ − 3.00000i − 0.169842i
$$313$$ 26.0000i 1.46961i 0.678280 + 0.734803i $$0.262726\pi$$
−0.678280 + 0.734803i $$0.737274\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ −16.0000 −0.900070
$$317$$ − 30.0000i − 1.68497i −0.538721 0.842484i $$-0.681092\pi$$
0.538721 0.842484i $$-0.318908\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ − 8.00000i − 0.445132i
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ 2.00000i 0.110600i
$$328$$ − 18.0000i − 0.993884i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ − 4.00000i − 0.219529i
$$333$$ 6.00000i 0.328798i
$$334$$ 16.0000 0.875481
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ − 1.00000i − 0.0543928i
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ −32.0000 −1.73290
$$342$$ − 4.00000i − 0.216295i
$$343$$ 0 0
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 2.00000 0.107521
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ 2.00000i 0.107211i
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 20.0000i 1.06600i
$$353$$ − 14.0000i − 0.745145i −0.928003 0.372572i $$-0.878476\pi$$
0.928003 0.372572i $$-0.121524\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ 12.0000i 0.634220i
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 22.0000i 1.15629i
$$363$$ 5.00000i 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ 16.0000i 0.835193i 0.908633 + 0.417597i $$0.137127\pi$$
−0.908633 + 0.417597i $$0.862873\pi$$
$$368$$ − 8.00000i − 0.417029i
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ − 8.00000i − 0.414781i
$$373$$ 22.0000i 1.13912i 0.821951 + 0.569558i $$0.192886\pi$$
−0.821951 + 0.569558i $$0.807114\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ −24.0000 −1.23771
$$377$$ 2.00000i 0.103005i
$$378$$ 0 0
$$379$$ −36.0000 −1.84920 −0.924598 0.380945i $$-0.875599\pi$$
−0.924598 + 0.380945i $$0.875599\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 16.0000i 0.818631i
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 4.00000i 0.203331i
$$388$$ − 18.0000i − 0.913812i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 21.0000i 1.06066i
$$393$$ − 12.0000i − 0.605320i
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ − 14.0000i − 0.702640i −0.936255 0.351320i $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ 8.00000i 0.401004i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ − 4.00000i − 0.199502i
$$403$$ − 8.00000i − 0.398508i
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 24.0000i − 1.18964i
$$408$$ 6.00000i 0.297044i
$$409$$ 38.0000 1.87898 0.939490 0.342578i $$-0.111300\pi$$
0.939490 + 0.342578i $$0.111300\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ − 8.00000i − 0.394132i
$$413$$ 0 0
$$414$$ 8.00000 0.393179
$$415$$ 0 0
$$416$$ −5.00000 −0.245145
$$417$$ 20.0000i 0.979404i
$$418$$ 16.0000i 0.782586i
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 20.0000i 0.973585i
$$423$$ − 8.00000i − 0.388973i
$$424$$ −18.0000 −0.874157
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ − 12.0000i − 0.580042i
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 18.0000i 0.865025i 0.901628 + 0.432512i $$0.142373\pi$$
−0.901628 + 0.432512i $$0.857627\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 32.0000i 1.53077i
$$438$$ 6.00000i 0.286691i
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 2.00000i 0.0951303i
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ 24.0000 1.13643
$$447$$ 10.0000i 0.472984i
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ 2.00000i 0.0940721i
$$453$$ 0 0
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −12.0000 −0.561951
$$457$$ − 26.0000i − 1.21623i −0.793849 0.608114i $$-0.791926\pi$$
0.793849 0.608114i $$-0.208074\pi$$
$$458$$ − 22.0000i − 1.02799i
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ 8.00000i 0.371792i 0.982569 + 0.185896i $$0.0595187\pi$$
−0.982569 + 0.185896i $$0.940481\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −26.0000 −1.20443
$$467$$ − 4.00000i − 0.185098i −0.995708 0.0925490i $$-0.970499\pi$$
0.995708 0.0925490i $$-0.0295015\pi$$
$$468$$ − 1.00000i − 0.0462250i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 36.0000i 1.65703i
$$473$$ − 16.0000i − 0.735681i
$$474$$ 16.0000 0.734904
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 6.00000i − 0.274721i
$$478$$ 24.0000i 1.09773i
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ 6.00000 0.273576
$$482$$ − 14.0000i − 0.637683i
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 16.0000i − 0.725029i −0.931978 0.362515i $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ − 6.00000i − 0.271607i
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ − 4.00000i − 0.180151i
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ 4.00000i 0.179244i
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ 16.0000 0.714827
$$502$$ − 4.00000i − 0.178529i
$$503$$ 24.0000i 1.07011i 0.844818 + 0.535054i $$0.179709\pi$$
−0.844818 + 0.535054i $$0.820291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −32.0000 −1.42257
$$507$$ − 1.00000i − 0.0444116i
$$508$$ 16.0000i 0.709885i
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 11.0000i − 0.486136i
$$513$$ − 4.00000i − 0.176604i
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 32.0000i 1.40736i
$$518$$ 0 0
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ − 2.00000i − 0.0875376i
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 16.0000i 0.696971i
$$528$$ − 4.00000i − 0.174078i
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ − 6.00000i − 0.259889i
$$534$$ 10.0000 0.432742
$$535$$ 0 0
$$536$$ −12.0000 −0.518321
$$537$$ 12.0000i 0.517838i
$$538$$ − 14.0000i − 0.603583i
$$539$$ 28.0000 1.20605
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ − 8.00000i − 0.343629i
$$543$$ 22.0000i 0.944110i
$$544$$ 10.0000 0.428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 36.0000i − 1.53925i −0.638497 0.769624i $$-0.720443\pi$$
0.638497 0.769624i $$-0.279557\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ − 24.0000i − 1.02151i
$$553$$ 0 0
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ 20.0000 0.848189
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 8.00000i 0.338667i
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ − 22.0000i − 0.928014i
$$563$$ 4.00000i 0.168580i 0.996441 + 0.0842900i $$0.0268622\pi$$
−0.996441 + 0.0842900i $$0.973138\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 38.0000 1.59304 0.796521 0.604610i $$-0.206671\pi$$
0.796521 + 0.604610i $$0.206671\pi$$
$$570$$ 0 0
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ 4.00000i 0.167248i
$$573$$ 16.0000i 0.668410i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ − 18.0000i − 0.749350i −0.927156 0.374675i $$-0.877754\pi$$
0.927156 0.374675i $$-0.122246\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 18.0000i 0.746124i
$$583$$ 24.0000i 0.993978i
$$584$$ 18.0000 0.744845
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 7.00000i 0.288675i
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 6.00000i 0.246598i
$$593$$ − 30.0000i − 1.23195i −0.787765 0.615976i $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 8.00000i 0.327418i
$$598$$ − 8.00000i − 0.327144i
$$599$$ −40.0000 −1.63436 −0.817178 0.576386i $$-0.804463\pi$$
−0.817178 + 0.576386i $$0.804463\pi$$
$$600$$ 0 0
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ 0 0
$$603$$ − 4.00000i − 0.162893i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ −6.00000 −0.243733
$$607$$ − 32.0000i − 1.29884i −0.760430 0.649420i $$-0.775012\pi$$
0.760430 0.649420i $$-0.224988\pi$$
$$608$$ 20.0000i 0.811107i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 2.00000i 0.0808452i
$$613$$ 6.00000i 0.242338i 0.992632 + 0.121169i $$0.0386643\pi$$
−0.992632 + 0.121169i $$0.961336\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 38.0000i 1.52982i 0.644136 + 0.764911i $$0.277217\pi$$
−0.644136 + 0.764911i $$0.722783\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 8.00000 0.321029
$$622$$ 24.0000i 0.962312i
$$623$$ 0 0
$$624$$ 1.00000 0.0400320
$$625$$ 0 0
$$626$$ −26.0000 −1.03917
$$627$$ 16.0000i 0.638978i
$$628$$ 2.00000i 0.0798087i
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ − 48.0000i − 1.90934i
$$633$$ 20.0000i 0.794929i
$$634$$ 30.0000 1.19145
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 7.00000i 0.277350i
$$638$$ 8.00000i 0.316723i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ 28.0000i 1.10421i 0.833774 + 0.552106i $$0.186176\pi$$
−0.833774 + 0.552106i $$0.813824\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 8.00000 0.314756
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ 3.00000i 0.117851i
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 20.0000i − 0.783260i
$$653$$ 30.0000i 1.17399i 0.809590 + 0.586995i $$0.199689\pi$$
−0.809590 + 0.586995i $$0.800311\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 6.00000i 0.234082i
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 38.0000 1.47803 0.739014 0.673690i $$-0.235292\pi$$
0.739014 + 0.673690i $$0.235292\pi$$
$$662$$ 20.0000i 0.777322i
$$663$$ 2.00000i 0.0776736i
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 16.0000i 0.619522i
$$668$$ − 16.0000i − 0.619059i
$$669$$ 24.0000 0.927894
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ 34.0000i 1.31060i 0.755367 + 0.655302i $$0.227459\pi$$
−0.755367 + 0.655302i $$0.772541\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ −1.00000 −0.0384615
$$677$$ 10.0000i 0.384331i 0.981363 + 0.192166i $$0.0615511\pi$$
−0.981363 + 0.192166i $$0.938449\pi$$
$$678$$ − 2.00000i − 0.0768095i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ − 32.0000i − 1.22534i
$$683$$ 4.00000i 0.153056i 0.997067 + 0.0765279i $$0.0243834\pi$$
−0.997067 + 0.0765279i $$0.975617\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 22.0000i − 0.839352i
$$688$$ 4.00000i 0.152499i
$$689$$ −6.00000 −0.228582
$$690$$ 0 0
$$691$$ −36.0000 −1.36950 −0.684752 0.728776i $$-0.740090\pi$$
−0.684752 + 0.728776i $$0.740090\pi$$
$$692$$ − 2.00000i − 0.0760286i
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ 12.0000i 0.454532i
$$698$$ − 14.0000i − 0.529908i
$$699$$ −26.0000 −0.983410
$$700$$ 0 0
$$701$$ −34.0000 −1.28416 −0.642081 0.766637i $$-0.721929\pi$$
−0.642081 + 0.766637i $$0.721929\pi$$
$$702$$ 1.00000i 0.0377426i
$$703$$ − 24.0000i − 0.905177i
$$704$$ −28.0000 −1.05529
$$705$$ 0 0
$$706$$ 14.0000 0.526897
$$707$$ 0 0
$$708$$ 12.0000i 0.450988i
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 16.0000 0.600047
$$712$$ − 30.0000i − 1.12430i
$$713$$ − 64.0000i − 2.39682i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 24.0000i 0.896296i
$$718$$ − 16.0000i − 0.597115i
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 3.00000i − 0.111648i
$$723$$ − 14.0000i − 0.520666i
$$724$$ 22.0000 0.817624
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ 24.0000i 0.890111i 0.895503 + 0.445055i $$0.146816\pi$$
−0.895503 + 0.445055i $$0.853184\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ − 2.00000i − 0.0739221i
$$733$$ 30.0000i 1.10808i 0.832492 + 0.554038i $$0.186914\pi$$
−0.832492 + 0.554038i $$0.813086\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 0 0
$$736$$ −40.0000 −1.47442
$$737$$ 16.0000i 0.589368i
$$738$$ 6.00000i 0.220863i
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ −4.00000 −0.146944
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 24.0000 0.879883
$$745$$ 0 0
$$746$$ −22.0000 −0.805477
$$747$$ 4.00000i 0.146352i
$$748$$ − 8.00000i − 0.292509i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ − 8.00000i − 0.291730i
$$753$$ − 4.00000i − 0.145768i
$$754$$ −2.00000 −0.0728357
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 42.0000i 1.52652i 0.646094 + 0.763258i $$0.276401\pi$$
−0.646094 + 0.763258i $$0.723599\pi$$
$$758$$ − 36.0000i − 1.30758i
$$759$$ −32.0000 −1.16153
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ − 16.0000i − 0.579619i
$$763$$ 0 0
$$764$$ 16.0000 0.578860
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ 12.0000i 0.433295i
$$768$$ − 17.0000i − 0.613435i
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ − 14.0000i − 0.503871i
$$773$$ 6.00000i 0.215805i 0.994161 + 0.107903i $$0.0344134\pi$$
−0.994161 + 0.107903i $$0.965587\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 54.0000 1.93849
$$777$$ 0 0
$$778$$ − 6.00000i − 0.215110i
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 16.0000i 0.572159i
$$783$$ − 2.00000i − 0.0714742i
$$784$$ −7.00000 −0.250000
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 4.00000i 0.142585i 0.997455 + 0.0712923i $$0.0227123\pi$$
−0.997455 + 0.0712923i $$0.977288\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 12.0000i − 0.426401i
$$793$$ − 2.00000i − 0.0710221i
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ − 46.0000i − 1.62940i −0.579880 0.814702i $$-0.696901\pi$$
0.579880 0.814702i $$-0.303099\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ − 30.0000i − 1.05934i
$$803$$ − 24.0000i − 0.846942i
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ − 14.0000i − 0.492823i
$$808$$ 18.0000i 0.633238i
$$809$$ −42.0000 −1.47664 −0.738321 0.674450i $$-0.764381\pi$$
−0.738321 + 0.674450i $$0.764381\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ − 8.00000i − 0.280572i
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ − 16.0000i − 0.559769i
$$818$$ 38.0000i 1.32864i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ − 6.00000i − 0.209274i
$$823$$ − 24.0000i − 0.836587i −0.908312 0.418294i $$-0.862628\pi$$
0.908312 0.418294i $$-0.137372\pi$$
$$824$$ 24.0000 0.836080
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ − 8.00000i − 0.278019i
$$829$$ 34.0000 1.18087 0.590434 0.807086i $$-0.298956\pi$$
0.590434 + 0.807086i $$0.298956\pi$$
$$830$$ 0 0
$$831$$ −10.0000 −0.346896
$$832$$ − 7.00000i − 0.242681i
$$833$$ − 14.0000i − 0.485071i
$$834$$ −20.0000 −0.692543
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 8.00000i 0.276520i
$$838$$ − 20.0000i − 0.690889i
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ − 10.0000i − 0.344623i
$$843$$ − 22.0000i − 0.757720i
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ 0 0
$$848$$ − 6.00000i − 0.206041i
$$849$$ 20.0000 0.686398
$$850$$ 0 0
$$851$$ 48.0000 1.64542
$$852$$ 0 0
$$853$$ − 42.0000i − 1.43805i −0.694983 0.719026i $$-0.744588\pi$$
0.694983 0.719026i $$-0.255412\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 36.0000 1.23045
$$857$$ − 10.0000i − 0.341593i −0.985306 0.170797i $$-0.945366\pi$$
0.985306 0.170797i $$-0.0546341\pi$$
$$858$$ − 4.00000i − 0.136558i
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 8.00000i 0.272481i
$$863$$ − 40.0000i − 1.36162i −0.732462 0.680808i $$-0.761629\pi$$
0.732462 0.680808i $$-0.238371\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ −18.0000 −0.611665
$$867$$ 13.0000i 0.441503i
$$868$$ 0 0
$$869$$ −64.0000 −2.17105
$$870$$ 0 0
$$871$$ −4.00000 −0.135535
$$872$$ 6.00000i 0.203186i
$$873$$ 18.0000i 0.609208i
$$874$$ −32.0000 −1.08242
$$875$$ 0 0
$$876$$ 6.00000 0.202721
$$877$$ 50.0000i 1.68838i 0.536044 + 0.844190i $$0.319918\pi$$
−0.536044 + 0.844190i $$0.680082\pi$$
$$878$$ 24.0000i 0.809961i
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ − 7.00000i − 0.235702i
$$883$$ 36.0000i 1.21150i 0.795656 + 0.605748i $$0.207126\pi$$
−0.795656 + 0.605748i $$0.792874\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ − 24.0000i − 0.805841i −0.915235 0.402921i $$-0.867995\pi$$
0.915235 0.402921i $$-0.132005\pi$$
$$888$$ 18.0000i 0.604040i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ − 24.0000i − 0.803579i
$$893$$ 32.0000i 1.07084i
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 8.00000i − 0.267112i
$$898$$ − 18.0000i − 0.600668i
$$899$$ −16.0000 −0.533630
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ − 24.0000i − 0.799113i
$$903$$ 0 0
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 4.00000i 0.132818i 0.997792 + 0.0664089i $$0.0211542\pi$$
−0.997792 + 0.0664089i $$0.978846\pi$$
$$908$$ − 12.0000i − 0.398234i
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ 32.0000 1.06021 0.530104 0.847933i $$-0.322153\pi$$
0.530104 + 0.847933i $$0.322153\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ − 16.0000i − 0.529523i
$$914$$ 26.0000 0.860004
$$915$$ 0 0
$$916$$ −22.0000 −0.726900
$$917$$ 0 0
$$918$$ − 2.00000i − 0.0660098i
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ − 18.0000i − 0.592798i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −8.00000 −0.262896
$$927$$ 8.00000i 0.262754i
$$928$$ 10.0000i 0.328266i
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 28.0000 0.917663
$$932$$ 26.0000i 0.851658i
$$933$$ 24.0000i 0.785725i
$$934$$ 4.00000 0.130884
$$935$$ 0 0
$$936$$ 3.00000 0.0980581
$$937$$ 22.0000i 0.718709i 0.933201 + 0.359354i $$0.117003\pi$$
−0.933201 + 0.359354i $$0.882997\pi$$
$$938$$ 0 0
$$939$$ −26.0000 −0.848478
$$940$$ 0 0
$$941$$ 46.0000 1.49956 0.749779 0.661689i $$-0.230160\pi$$
0.749779 + 0.661689i $$0.230160\pi$$
$$942$$ − 2.00000i − 0.0651635i
$$943$$ − 48.0000i − 1.56310i
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ − 12.0000i − 0.389948i −0.980808 0.194974i $$-0.937538\pi$$
0.980808 0.194974i $$-0.0624622\pi$$
$$948$$ − 16.0000i − 0.519656i
$$949$$ 6.00000 0.194768
$$950$$ 0 0
$$951$$ 30.0000 0.972817
$$952$$ 0 0
$$953$$ 10.0000i 0.323932i 0.986796 + 0.161966i $$0.0517835\pi$$
−0.986796 + 0.161966i $$0.948217\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 24.0000 0.776215
$$957$$ 8.00000i 0.258603i
$$958$$ − 8.00000i − 0.258468i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 6.00000i 0.193448i
$$963$$ 12.0000i 0.386695i
$$964$$ −14.0000 −0.450910
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 16.0000i 0.514525i 0.966342 + 0.257263i $$0.0828206\pi$$
−0.966342 + 0.257263i $$0.917179\pi$$
$$968$$ 15.0000i 0.482118i
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 0 0
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 30.0000i 0.959785i 0.877327 + 0.479893i $$0.159324\pi$$
−0.877327 + 0.479893i $$0.840676\pi$$
$$978$$ 20.0000i 0.639529i
$$979$$ −40.0000 −1.27841
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ − 20.0000i − 0.638226i
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 18.0000 0.573819
$$985$$ 0 0
$$986$$ 4.00000 0.127386
$$987$$ 0 0
$$988$$ 4.00000i 0.127257i
$$989$$ 32.0000 1.01754
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ − 40.0000i − 1.27000i
$$993$$ 20.0000i 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ − 38.0000i − 1.20347i −0.798695 0.601736i $$-0.794476\pi$$
0.798695 0.601736i $$-0.205524\pi$$
$$998$$ − 28.0000i − 0.886325i
$$999$$ −6.00000 −0.189832
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 975.2.c.e.274.2 2
3.2 odd 2 2925.2.c.f.2224.1 2
5.2 odd 4 195.2.a.a.1.1 1
5.3 odd 4 975.2.a.i.1.1 1
5.4 even 2 inner 975.2.c.e.274.1 2
15.2 even 4 585.2.a.g.1.1 1
15.8 even 4 2925.2.a.d.1.1 1
15.14 odd 2 2925.2.c.f.2224.2 2
20.7 even 4 3120.2.a.k.1.1 1
35.27 even 4 9555.2.a.b.1.1 1
60.47 odd 4 9360.2.a.o.1.1 1
65.12 odd 4 2535.2.a.k.1.1 1
195.77 even 4 7605.2.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.a.1.1 1 5.2 odd 4
585.2.a.g.1.1 1 15.2 even 4
975.2.a.i.1.1 1 5.3 odd 4
975.2.c.e.274.1 2 5.4 even 2 inner
975.2.c.e.274.2 2 1.1 even 1 trivial
2535.2.a.k.1.1 1 65.12 odd 4
2925.2.a.d.1.1 1 15.8 even 4
2925.2.c.f.2224.1 2 3.2 odd 2
2925.2.c.f.2224.2 2 15.14 odd 2
3120.2.a.k.1.1 1 20.7 even 4
7605.2.a.h.1.1 1 195.77 even 4
9360.2.a.o.1.1 1 60.47 odd 4
9555.2.a.b.1.1 1 35.27 even 4