# Properties

 Label 975.2.c.c.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: $$\Z[a_1, a_2, a_3]$$ 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.c.274.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} -1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+2.00000i q^{2} -1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} -1.00000 q^{9} +5.00000 q^{11} +2.00000i q^{12} +1.00000i q^{13} +2.00000 q^{14} -4.00000 q^{16} -7.00000i q^{17} -2.00000i q^{18} +6.00000 q^{19} -1.00000 q^{21} +10.0000i q^{22} -3.00000i q^{23} -2.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -2.00000 q^{29} +2.00000 q^{31} -8.00000i q^{32} -5.00000i q^{33} +14.0000 q^{34} +2.00000 q^{36} +7.00000i q^{37} +12.0000i q^{38} +1.00000 q^{39} +9.00000 q^{41} -2.00000i q^{42} +8.00000i q^{43} -10.0000 q^{44} +6.00000 q^{46} +10.0000i q^{47} +4.00000i q^{48} +6.00000 q^{49} -7.00000 q^{51} -2.00000i q^{52} -5.00000i q^{53} -2.00000 q^{54} -6.00000i q^{57} -4.00000i q^{58} +5.00000 q^{61} +4.00000i q^{62} +1.00000i q^{63} +8.00000 q^{64} +10.0000 q^{66} -4.00000i q^{67} +14.0000i q^{68} -3.00000 q^{69} +9.00000 q^{71} +6.00000i q^{73} -14.0000 q^{74} -12.0000 q^{76} -5.00000i q^{77} +2.00000i q^{78} +3.00000 q^{79} +1.00000 q^{81} +18.0000i q^{82} +4.00000i q^{83} +2.00000 q^{84} -16.0000 q^{86} +2.00000i q^{87} -11.0000 q^{89} +1.00000 q^{91} +6.00000i q^{92} -2.00000i q^{93} -20.0000 q^{94} -8.00000 q^{96} -11.0000i q^{97} +12.0000i q^{98} -5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 + 4 * q^6 - 2 * q^9 $$2 q - 4 q^{4} + 4 q^{6} - 2 q^{9} + 10 q^{11} + 4 q^{14} - 8 q^{16} + 12 q^{19} - 2 q^{21} - 4 q^{26} - 4 q^{29} + 4 q^{31} + 28 q^{34} + 4 q^{36} + 2 q^{39} + 18 q^{41} - 20 q^{44} + 12 q^{46} + 12 q^{49} - 14 q^{51} - 4 q^{54} + 10 q^{61} + 16 q^{64} + 20 q^{66} - 6 q^{69} + 18 q^{71} - 28 q^{74} - 24 q^{76} + 6 q^{79} + 2 q^{81} + 4 q^{84} - 32 q^{86} - 22 q^{89} + 2 q^{91} - 40 q^{94} - 16 q^{96} - 10 q^{99}+O(q^{100})$$ 2 * q - 4 * q^4 + 4 * q^6 - 2 * q^9 + 10 * q^11 + 4 * q^14 - 8 * q^16 + 12 * q^19 - 2 * q^21 - 4 * q^26 - 4 * q^29 + 4 * q^31 + 28 * q^34 + 4 * q^36 + 2 * q^39 + 18 * q^41 - 20 * q^44 + 12 * q^46 + 12 * q^49 - 14 * q^51 - 4 * q^54 + 10 * q^61 + 16 * q^64 + 20 * q^66 - 6 * q^69 + 18 * q^71 - 28 * q^74 - 24 * q^76 + 6 * q^79 + 2 * q^81 + 4 * q^84 - 32 * q^86 - 22 * q^89 + 2 * q^91 - 40 * q^94 - 16 * q^96 - 10 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000i 1.41421i 0.707107 + 0.707107i $$0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ −2.00000 −1.00000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ − 1.00000i − 0.377964i −0.981981 0.188982i $$-0.939481\pi$$
0.981981 0.188982i $$-0.0605189\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 2.00000i 0.577350i
$$13$$ 1.00000i 0.277350i
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ − 7.00000i − 1.69775i −0.528594 0.848875i $$-0.677281\pi$$
0.528594 0.848875i $$-0.322719\pi$$
$$18$$ − 2.00000i − 0.471405i
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 10.0000i 2.13201i
$$23$$ − 3.00000i − 0.625543i −0.949828 0.312772i $$-0.898743\pi$$
0.949828 0.312772i $$-0.101257\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ 2.00000i 0.377964i
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ − 8.00000i − 1.41421i
$$33$$ − 5.00000i − 0.870388i
$$34$$ 14.0000 2.40098
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ 7.00000i 1.15079i 0.817875 + 0.575396i $$0.195152\pi$$
−0.817875 + 0.575396i $$0.804848\pi$$
$$38$$ 12.0000i 1.94666i
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 9.00000 1.40556 0.702782 0.711405i $$-0.251941\pi$$
0.702782 + 0.711405i $$0.251941\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ −10.0000 −1.50756
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ 10.0000i 1.45865i 0.684167 + 0.729325i $$0.260166\pi$$
−0.684167 + 0.729325i $$0.739834\pi$$
$$48$$ 4.00000i 0.577350i
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ −7.00000 −0.980196
$$52$$ − 2.00000i − 0.277350i
$$53$$ − 5.00000i − 0.686803i −0.939189 0.343401i $$-0.888421\pi$$
0.939189 0.343401i $$-0.111579\pi$$
$$54$$ −2.00000 −0.272166
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 6.00000i − 0.794719i
$$58$$ − 4.00000i − 0.525226i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ 1.00000i 0.125988i
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 10.0000 1.23091
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 14.0000i 1.69775i
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ 9.00000 1.06810 0.534052 0.845452i $$-0.320669\pi$$
0.534052 + 0.845452i $$0.320669\pi$$
$$72$$ 0 0
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ −14.0000 −1.62747
$$75$$ 0 0
$$76$$ −12.0000 −1.37649
$$77$$ − 5.00000i − 0.569803i
$$78$$ 2.00000i 0.226455i
$$79$$ 3.00000 0.337526 0.168763 0.985657i $$-0.446023\pi$$
0.168763 + 0.985657i $$0.446023\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 18.0000i 1.98777i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ −16.0000 −1.72532
$$87$$ 2.00000i 0.214423i
$$88$$ 0 0
$$89$$ −11.0000 −1.16600 −0.582999 0.812473i $$-0.698121\pi$$
−0.582999 + 0.812473i $$0.698121\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 6.00000i 0.625543i
$$93$$ − 2.00000i − 0.207390i
$$94$$ −20.0000 −2.06284
$$95$$ 0 0
$$96$$ −8.00000 −0.816497
$$97$$ − 11.0000i − 1.11688i −0.829545 0.558440i $$-0.811400\pi$$
0.829545 0.558440i $$-0.188600\pi$$
$$98$$ 12.0000i 1.21218i
$$99$$ −5.00000 −0.502519
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ − 14.0000i − 1.38621i
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 10.0000 0.971286
$$107$$ − 17.0000i − 1.64345i −0.569883 0.821726i $$-0.693011\pi$$
0.569883 0.821726i $$-0.306989\pi$$
$$108$$ − 2.00000i − 0.192450i
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 7.00000 0.664411
$$112$$ 4.00000i 0.377964i
$$113$$ − 10.0000i − 0.940721i −0.882474 0.470360i $$-0.844124\pi$$
0.882474 0.470360i $$-0.155876\pi$$
$$114$$ 12.0000 1.12390
$$115$$ 0 0
$$116$$ 4.00000 0.371391
$$117$$ − 1.00000i − 0.0924500i
$$118$$ 0 0
$$119$$ −7.00000 −0.641689
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 10.0000i 0.905357i
$$123$$ − 9.00000i − 0.811503i
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ − 2.00000i − 0.177471i −0.996055 0.0887357i $$-0.971717\pi$$
0.996055 0.0887357i $$-0.0282826\pi$$
$$128$$ 0 0
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ −22.0000 −1.92215 −0.961074 0.276289i $$-0.910895\pi$$
−0.961074 + 0.276289i $$0.910895\pi$$
$$132$$ 10.0000i 0.870388i
$$133$$ − 6.00000i − 0.520266i
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 14.0000i − 1.19610i −0.801459 0.598050i $$-0.795942\pi$$
0.801459 0.598050i $$-0.204058\pi$$
$$138$$ − 6.00000i − 0.510754i
$$139$$ −15.0000 −1.27228 −0.636142 0.771572i $$-0.719471\pi$$
−0.636142 + 0.771572i $$0.719471\pi$$
$$140$$ 0 0
$$141$$ 10.0000 0.842152
$$142$$ 18.0000i 1.51053i
$$143$$ 5.00000i 0.418121i
$$144$$ 4.00000 0.333333
$$145$$ 0 0
$$146$$ −12.0000 −0.993127
$$147$$ − 6.00000i − 0.494872i
$$148$$ − 14.0000i − 1.15079i
$$149$$ −15.0000 −1.22885 −0.614424 0.788976i $$-0.710612\pi$$
−0.614424 + 0.788976i $$0.710612\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 7.00000i 0.565916i
$$154$$ 10.0000 0.805823
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ 18.0000i 1.43656i 0.695756 + 0.718278i $$0.255069\pi$$
−0.695756 + 0.718278i $$0.744931\pi$$
$$158$$ 6.00000i 0.477334i
$$159$$ −5.00000 −0.396526
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 2.00000i 0.157135i
$$163$$ − 15.0000i − 1.17489i −0.809264 0.587445i $$-0.800134\pi$$
0.809264 0.587445i $$-0.199866\pi$$
$$164$$ −18.0000 −1.40556
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ − 24.0000i − 1.85718i −0.371113 0.928588i $$-0.621024\pi$$
0.371113 0.928588i $$-0.378976\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ −6.00000 −0.458831
$$172$$ − 16.0000i − 1.21999i
$$173$$ 18.0000i 1.36851i 0.729241 + 0.684257i $$0.239873\pi$$
−0.729241 + 0.684257i $$0.760127\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ 0 0
$$176$$ −20.0000 −1.50756
$$177$$ 0 0
$$178$$ − 22.0000i − 1.64897i
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ 2.00000i 0.148250i
$$183$$ − 5.00000i − 0.369611i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 4.00000 0.293294
$$187$$ − 35.0000i − 2.55945i
$$188$$ − 20.0000i − 1.45865i
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ − 8.00000i − 0.577350i
$$193$$ 17.0000i 1.22369i 0.790979 + 0.611843i $$0.209572\pi$$
−0.790979 + 0.611843i $$0.790428\pi$$
$$194$$ 22.0000 1.57951
$$195$$ 0 0
$$196$$ −12.0000 −0.857143
$$197$$ 24.0000i 1.70993i 0.518686 + 0.854965i $$0.326421\pi$$
−0.518686 + 0.854965i $$0.673579\pi$$
$$198$$ − 10.0000i − 0.710669i
$$199$$ 28.0000 1.98487 0.992434 0.122782i $$-0.0391815\pi$$
0.992434 + 0.122782i $$0.0391815\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ 2.00000i 0.140372i
$$204$$ 14.0000 0.980196
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 3.00000i 0.208514i
$$208$$ − 4.00000i − 0.277350i
$$209$$ 30.0000 2.07514
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 10.0000i 0.686803i
$$213$$ − 9.00000i − 0.616670i
$$214$$ 34.0000 2.32419
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 2.00000i − 0.135769i
$$218$$ − 8.00000i − 0.541828i
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 7.00000 0.470871
$$222$$ 14.0000i 0.939618i
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ −8.00000 −0.534522
$$225$$ 0 0
$$226$$ 20.0000 1.33038
$$227$$ − 2.00000i − 0.132745i −0.997795 0.0663723i $$-0.978857\pi$$
0.997795 0.0663723i $$-0.0211425\pi$$
$$228$$ 12.0000i 0.794719i
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ −5.00000 −0.328976
$$232$$ 0 0
$$233$$ − 19.0000i − 1.24473i −0.782727 0.622366i $$-0.786172\pi$$
0.782727 0.622366i $$-0.213828\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 3.00000i − 0.194871i
$$238$$ − 14.0000i − 0.907485i
$$239$$ −9.00000 −0.582162 −0.291081 0.956698i $$-0.594015\pi$$
−0.291081 + 0.956698i $$0.594015\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ 28.0000i 1.79991i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 18.0000 1.14764
$$247$$ 6.00000i 0.381771i
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ − 15.0000i − 0.943042i
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 2.00000i 0.124757i 0.998053 + 0.0623783i $$0.0198685\pi$$
−0.998053 + 0.0623783i $$0.980131\pi$$
$$258$$ 16.0000i 0.996116i
$$259$$ 7.00000 0.434959
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ − 44.0000i − 2.71833i
$$263$$ 16.0000i 0.986602i 0.869859 + 0.493301i $$0.164210\pi$$
−0.869859 + 0.493301i $$0.835790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 12.0000 0.735767
$$267$$ 11.0000i 0.673189i
$$268$$ 8.00000i 0.488678i
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ −22.0000 −1.33640 −0.668202 0.743980i $$-0.732936\pi$$
−0.668202 + 0.743980i $$0.732936\pi$$
$$272$$ 28.0000i 1.69775i
$$273$$ − 1.00000i − 0.0605228i
$$274$$ 28.0000 1.69154
$$275$$ 0 0
$$276$$ 6.00000 0.361158
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ − 30.0000i − 1.79928i
$$279$$ −2.00000 −0.119737
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 20.0000i 1.19098i
$$283$$ − 20.0000i − 1.18888i −0.804141 0.594438i $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ −18.0000 −1.06810
$$285$$ 0 0
$$286$$ −10.0000 −0.591312
$$287$$ − 9.00000i − 0.531253i
$$288$$ 8.00000i 0.471405i
$$289$$ −32.0000 −1.88235
$$290$$ 0 0
$$291$$ −11.0000 −0.644831
$$292$$ − 12.0000i − 0.702247i
$$293$$ − 4.00000i − 0.233682i −0.993151 0.116841i $$-0.962723\pi$$
0.993151 0.116841i $$-0.0372769\pi$$
$$294$$ 12.0000 0.699854
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 5.00000i 0.290129i
$$298$$ − 30.0000i − 1.73785i
$$299$$ 3.00000 0.173494
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ − 16.0000i − 0.920697i
$$303$$ 0 0
$$304$$ −24.0000 −1.37649
$$305$$ 0 0
$$306$$ −14.0000 −0.800327
$$307$$ 23.0000i 1.31268i 0.754466 + 0.656340i $$0.227896\pi$$
−0.754466 + 0.656340i $$0.772104\pi$$
$$308$$ 10.0000i 0.569803i
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ 20.0000 1.13410 0.567048 0.823685i $$-0.308085\pi$$
0.567048 + 0.823685i $$0.308085\pi$$
$$312$$ 0 0
$$313$$ 22.0000i 1.24351i 0.783210 + 0.621757i $$0.213581\pi$$
−0.783210 + 0.621757i $$0.786419\pi$$
$$314$$ −36.0000 −2.03160
$$315$$ 0 0
$$316$$ −6.00000 −0.337526
$$317$$ − 24.0000i − 1.34797i −0.738743 0.673987i $$-0.764580\pi$$
0.738743 0.673987i $$-0.235420\pi$$
$$318$$ − 10.0000i − 0.560772i
$$319$$ −10.0000 −0.559893
$$320$$ 0 0
$$321$$ −17.0000 −0.948847
$$322$$ − 6.00000i − 0.334367i
$$323$$ − 42.0000i − 2.33694i
$$324$$ −2.00000 −0.111111
$$325$$ 0 0
$$326$$ 30.0000 1.66155
$$327$$ 4.00000i 0.221201i
$$328$$ 0 0
$$329$$ 10.0000 0.551318
$$330$$ 0 0
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ − 8.00000i − 0.439057i
$$333$$ − 7.00000i − 0.383598i
$$334$$ 48.0000 2.62644
$$335$$ 0 0
$$336$$ 4.00000 0.218218
$$337$$ 8.00000i 0.435788i 0.975972 + 0.217894i $$0.0699187\pi$$
−0.975972 + 0.217894i $$0.930081\pi$$
$$338$$ − 2.00000i − 0.108786i
$$339$$ −10.0000 −0.543125
$$340$$ 0 0
$$341$$ 10.0000 0.541530
$$342$$ − 12.0000i − 0.648886i
$$343$$ − 13.0000i − 0.701934i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −36.0000 −1.93537
$$347$$ 11.0000i 0.590511i 0.955418 + 0.295255i $$0.0954048\pi$$
−0.955418 + 0.295255i $$0.904595\pi$$
$$348$$ − 4.00000i − 0.214423i
$$349$$ −24.0000 −1.28469 −0.642345 0.766415i $$-0.722038\pi$$
−0.642345 + 0.766415i $$0.722038\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ − 40.0000i − 2.13201i
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 22.0000 1.16600
$$357$$ 7.00000i 0.370479i
$$358$$ − 12.0000i − 0.634220i
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ − 14.0000i − 0.735824i
$$363$$ − 14.0000i − 0.734809i
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 10.0000 0.522708
$$367$$ 4.00000i 0.208798i 0.994535 + 0.104399i $$0.0332919\pi$$
−0.994535 + 0.104399i $$0.966708\pi$$
$$368$$ 12.0000i 0.625543i
$$369$$ −9.00000 −0.468521
$$370$$ 0 0
$$371$$ −5.00000 −0.259587
$$372$$ 4.00000i 0.207390i
$$373$$ 32.0000i 1.65690i 0.560065 + 0.828449i $$0.310776\pi$$
−0.560065 + 0.828449i $$0.689224\pi$$
$$374$$ 70.0000 3.61961
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 2.00000i − 0.103005i
$$378$$ 2.00000i 0.102869i
$$379$$ −10.0000 −0.513665 −0.256833 0.966456i $$-0.582679\pi$$
−0.256833 + 0.966456i $$0.582679\pi$$
$$380$$ 0 0
$$381$$ −2.00000 −0.102463
$$382$$ − 24.0000i − 1.22795i
$$383$$ 6.00000i 0.306586i 0.988181 + 0.153293i $$0.0489878\pi$$
−0.988181 + 0.153293i $$0.951012\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −34.0000 −1.73055
$$387$$ − 8.00000i − 0.406663i
$$388$$ 22.0000i 1.11688i
$$389$$ 4.00000 0.202808 0.101404 0.994845i $$-0.467667\pi$$
0.101404 + 0.994845i $$0.467667\pi$$
$$390$$ 0 0
$$391$$ −21.0000 −1.06202
$$392$$ 0 0
$$393$$ 22.0000i 1.10975i
$$394$$ −48.0000 −2.41821
$$395$$ 0 0
$$396$$ 10.0000 0.502519
$$397$$ 15.0000i 0.752828i 0.926451 + 0.376414i $$0.122843\pi$$
−0.926451 + 0.376414i $$0.877157\pi$$
$$398$$ 56.0000i 2.80703i
$$399$$ −6.00000 −0.300376
$$400$$ 0 0
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ − 8.00000i − 0.399004i
$$403$$ 2.00000i 0.0996271i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ −4.00000 −0.198517
$$407$$ 35.0000i 1.73489i
$$408$$ 0 0
$$409$$ −30.0000 −1.48340 −0.741702 0.670729i $$-0.765981\pi$$
−0.741702 + 0.670729i $$0.765981\pi$$
$$410$$ 0 0
$$411$$ −14.0000 −0.690569
$$412$$ − 8.00000i − 0.394132i
$$413$$ 0 0
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ 8.00000 0.392232
$$417$$ 15.0000i 0.734553i
$$418$$ 60.0000i 2.93470i
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ 0 0
$$421$$ −20.0000 −0.974740 −0.487370 0.873195i $$-0.662044\pi$$
−0.487370 + 0.873195i $$0.662044\pi$$
$$422$$ − 24.0000i − 1.16830i
$$423$$ − 10.0000i − 0.486217i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 18.0000 0.872103
$$427$$ − 5.00000i − 0.241967i
$$428$$ 34.0000i 1.64345i
$$429$$ 5.00000 0.241402
$$430$$ 0 0
$$431$$ −40.0000 −1.92673 −0.963366 0.268190i $$-0.913575\pi$$
−0.963366 + 0.268190i $$0.913575\pi$$
$$432$$ − 4.00000i − 0.192450i
$$433$$ 20.0000i 0.961139i 0.876957 + 0.480569i $$0.159570\pi$$
−0.876957 + 0.480569i $$0.840430\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 0 0
$$436$$ 8.00000 0.383131
$$437$$ − 18.0000i − 0.861057i
$$438$$ 12.0000i 0.573382i
$$439$$ −15.0000 −0.715911 −0.357955 0.933739i $$-0.616526\pi$$
−0.357955 + 0.933739i $$0.616526\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 14.0000i 0.665912i
$$443$$ 1.00000i 0.0475114i 0.999718 + 0.0237557i $$0.00756239\pi$$
−0.999718 + 0.0237557i $$0.992438\pi$$
$$444$$ −14.0000 −0.664411
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 15.0000i 0.709476i
$$448$$ − 8.00000i − 0.377964i
$$449$$ −9.00000 −0.424736 −0.212368 0.977190i $$-0.568118\pi$$
−0.212368 + 0.977190i $$0.568118\pi$$
$$450$$ 0 0
$$451$$ 45.0000 2.11897
$$452$$ 20.0000i 0.940721i
$$453$$ 8.00000i 0.375873i
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 7.00000i − 0.327446i −0.986506 0.163723i $$-0.947650\pi$$
0.986506 0.163723i $$-0.0523504\pi$$
$$458$$ − 28.0000i − 1.30835i
$$459$$ 7.00000 0.326732
$$460$$ 0 0
$$461$$ 37.0000 1.72326 0.861631 0.507535i $$-0.169443\pi$$
0.861631 + 0.507535i $$0.169443\pi$$
$$462$$ − 10.0000i − 0.465242i
$$463$$ − 15.0000i − 0.697109i −0.937288 0.348555i $$-0.886673\pi$$
0.937288 0.348555i $$-0.113327\pi$$
$$464$$ 8.00000 0.371391
$$465$$ 0 0
$$466$$ 38.0000 1.76032
$$467$$ − 1.00000i − 0.0462745i −0.999732 0.0231372i $$-0.992635\pi$$
0.999732 0.0231372i $$-0.00736547\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 0 0
$$473$$ 40.0000i 1.83920i
$$474$$ 6.00000 0.275589
$$475$$ 0 0
$$476$$ 14.0000 0.641689
$$477$$ 5.00000i 0.228934i
$$478$$ − 18.0000i − 0.823301i
$$479$$ −3.00000 −0.137073 −0.0685367 0.997649i $$-0.521833\pi$$
−0.0685367 + 0.997649i $$0.521833\pi$$
$$480$$ 0 0
$$481$$ −7.00000 −0.319173
$$482$$ 44.0000i 2.00415i
$$483$$ 3.00000i 0.136505i
$$484$$ −28.0000 −1.27273
$$485$$ 0 0
$$486$$ 2.00000 0.0907218
$$487$$ 5.00000i 0.226572i 0.993562 + 0.113286i $$0.0361376\pi$$
−0.993562 + 0.113286i $$0.963862\pi$$
$$488$$ 0 0
$$489$$ −15.0000 −0.678323
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 18.0000i 0.811503i
$$493$$ 14.0000i 0.630528i
$$494$$ −12.0000 −0.539906
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ − 9.00000i − 0.403705i
$$498$$ 8.00000i 0.358489i
$$499$$ 14.0000 0.626726 0.313363 0.949633i $$-0.398544\pi$$
0.313363 + 0.949633i $$0.398544\pi$$
$$500$$ 0 0
$$501$$ −24.0000 −1.07224
$$502$$ 0 0
$$503$$ − 36.0000i − 1.60516i −0.596544 0.802580i $$-0.703460\pi$$
0.596544 0.802580i $$-0.296540\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 30.0000 1.33366
$$507$$ 1.00000i 0.0444116i
$$508$$ 4.00000i 0.177471i
$$509$$ −27.0000 −1.19675 −0.598377 0.801215i $$-0.704187\pi$$
−0.598377 + 0.801215i $$0.704187\pi$$
$$510$$ 0 0
$$511$$ 6.00000 0.265424
$$512$$ 32.0000i 1.41421i
$$513$$ 6.00000i 0.264906i
$$514$$ −4.00000 −0.176432
$$515$$ 0 0
$$516$$ −16.0000 −0.704361
$$517$$ 50.0000i 2.19900i
$$518$$ 14.0000i 0.615125i
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 4.00000i 0.175075i
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 44.0000 1.92215
$$525$$ 0 0
$$526$$ −32.0000 −1.39527
$$527$$ − 14.0000i − 0.609850i
$$528$$ 20.0000i 0.870388i
$$529$$ 14.0000 0.608696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 12.0000i 0.520266i
$$533$$ 9.00000i 0.389833i
$$534$$ −22.0000 −0.952033
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 6.00000i 0.258919i
$$538$$ 48.0000i 2.06943i
$$539$$ 30.0000 1.29219
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ − 44.0000i − 1.88996i
$$543$$ 7.00000i 0.300399i
$$544$$ −56.0000 −2.40098
$$545$$ 0 0
$$546$$ 2.00000 0.0855921
$$547$$ 36.0000i 1.53925i 0.638497 + 0.769624i $$0.279557\pi$$
−0.638497 + 0.769624i $$0.720443\pi$$
$$548$$ 28.0000i 1.19610i
$$549$$ −5.00000 −0.213395
$$550$$ 0 0
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ − 3.00000i − 0.127573i
$$554$$ 4.00000 0.169944
$$555$$ 0 0
$$556$$ 30.0000 1.27228
$$557$$ − 2.00000i − 0.0847427i −0.999102 0.0423714i $$-0.986509\pi$$
0.999102 0.0423714i $$-0.0134913\pi$$
$$558$$ − 4.00000i − 0.169334i
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ −35.0000 −1.47770
$$562$$ 36.0000i 1.51857i
$$563$$ − 11.0000i − 0.463595i −0.972764 0.231797i $$-0.925539\pi$$
0.972764 0.231797i $$-0.0744606\pi$$
$$564$$ −20.0000 −0.842152
$$565$$ 0 0
$$566$$ 40.0000 1.68133
$$567$$ − 1.00000i − 0.0419961i
$$568$$ 0 0
$$569$$ 20.0000 0.838444 0.419222 0.907884i $$-0.362303\pi$$
0.419222 + 0.907884i $$0.362303\pi$$
$$570$$ 0 0
$$571$$ −39.0000 −1.63210 −0.816050 0.577982i $$-0.803840\pi$$
−0.816050 + 0.577982i $$0.803840\pi$$
$$572$$ − 10.0000i − 0.418121i
$$573$$ 12.0000i 0.501307i
$$574$$ 18.0000 0.751305
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ − 7.00000i − 0.291414i −0.989328 0.145707i $$-0.953454\pi$$
0.989328 0.145707i $$-0.0465456\pi$$
$$578$$ − 64.0000i − 2.66205i
$$579$$ 17.0000 0.706496
$$580$$ 0 0
$$581$$ 4.00000 0.165948
$$582$$ − 22.0000i − 0.911929i
$$583$$ − 25.0000i − 1.03539i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 8.00000 0.330477
$$587$$ − 2.00000i − 0.0825488i −0.999148 0.0412744i $$-0.986858\pi$$
0.999148 0.0412744i $$-0.0131418\pi$$
$$588$$ 12.0000i 0.494872i
$$589$$ 12.0000 0.494451
$$590$$ 0 0
$$591$$ 24.0000 0.987228
$$592$$ − 28.0000i − 1.15079i
$$593$$ 16.0000i 0.657041i 0.944497 + 0.328521i $$0.106550\pi$$
−0.944497 + 0.328521i $$0.893450\pi$$
$$594$$ −10.0000 −0.410305
$$595$$ 0 0
$$596$$ 30.0000 1.22885
$$597$$ − 28.0000i − 1.14596i
$$598$$ 6.00000i 0.245358i
$$599$$ −20.0000 −0.817178 −0.408589 0.912719i $$-0.633979\pi$$
−0.408589 + 0.912719i $$0.633979\pi$$
$$600$$ 0 0
$$601$$ −5.00000 −0.203954 −0.101977 0.994787i $$-0.532517\pi$$
−0.101977 + 0.994787i $$0.532517\pi$$
$$602$$ 16.0000i 0.652111i
$$603$$ 4.00000i 0.162893i
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 32.0000i − 1.29884i −0.760430 0.649420i $$-0.775012\pi$$
0.760430 0.649420i $$-0.224988\pi$$
$$608$$ − 48.0000i − 1.94666i
$$609$$ 2.00000 0.0810441
$$610$$ 0 0
$$611$$ −10.0000 −0.404557
$$612$$ − 14.0000i − 0.565916i
$$613$$ − 9.00000i − 0.363507i −0.983344 0.181753i $$-0.941823\pi$$
0.983344 0.181753i $$-0.0581772\pi$$
$$614$$ −46.0000 −1.85641
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ 38.0000 1.52735 0.763674 0.645601i $$-0.223393\pi$$
0.763674 + 0.645601i $$0.223393\pi$$
$$620$$ 0 0
$$621$$ 3.00000 0.120386
$$622$$ 40.0000i 1.60385i
$$623$$ 11.0000i 0.440706i
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ −44.0000 −1.75859
$$627$$ − 30.0000i − 1.19808i
$$628$$ − 36.0000i − 1.43656i
$$629$$ 49.0000 1.95376
$$630$$ 0 0
$$631$$ −20.0000 −0.796187 −0.398094 0.917345i $$-0.630328\pi$$
−0.398094 + 0.917345i $$0.630328\pi$$
$$632$$ 0 0
$$633$$ 12.0000i 0.476957i
$$634$$ 48.0000 1.90632
$$635$$ 0 0
$$636$$ 10.0000 0.396526
$$637$$ 6.00000i 0.237729i
$$638$$ − 20.0000i − 0.791808i
$$639$$ −9.00000 −0.356034
$$640$$ 0 0
$$641$$ 20.0000 0.789953 0.394976 0.918691i $$-0.370753\pi$$
0.394976 + 0.918691i $$0.370753\pi$$
$$642$$ − 34.0000i − 1.34187i
$$643$$ 37.0000i 1.45914i 0.683907 + 0.729569i $$0.260279\pi$$
−0.683907 + 0.729569i $$0.739721\pi$$
$$644$$ 6.00000 0.236433
$$645$$ 0 0
$$646$$ 84.0000 3.30494
$$647$$ 17.0000i 0.668339i 0.942513 + 0.334169i $$0.108456\pi$$
−0.942513 + 0.334169i $$0.891544\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −2.00000 −0.0783862
$$652$$ 30.0000i 1.17489i
$$653$$ − 18.0000i − 0.704394i −0.935926 0.352197i $$-0.885435\pi$$
0.935926 0.352197i $$-0.114565\pi$$
$$654$$ −8.00000 −0.312825
$$655$$ 0 0
$$656$$ −36.0000 −1.40556
$$657$$ − 6.00000i − 0.234082i
$$658$$ 20.0000i 0.779681i
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ −40.0000 −1.55582 −0.777910 0.628376i $$-0.783720\pi$$
−0.777910 + 0.628376i $$0.783720\pi$$
$$662$$ 0 0
$$663$$ − 7.00000i − 0.271857i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 14.0000 0.542489
$$667$$ 6.00000i 0.232321i
$$668$$ 48.0000i 1.85718i
$$669$$ 8.00000 0.309298
$$670$$ 0 0
$$671$$ 25.0000 0.965114
$$672$$ 8.00000i 0.308607i
$$673$$ − 42.0000i − 1.61898i −0.587133 0.809491i $$-0.699743\pi$$
0.587133 0.809491i $$-0.300257\pi$$
$$674$$ −16.0000 −0.616297
$$675$$ 0 0
$$676$$ 2.00000 0.0769231
$$677$$ 21.0000i 0.807096i 0.914959 + 0.403548i $$0.132223\pi$$
−0.914959 + 0.403548i $$0.867777\pi$$
$$678$$ − 20.0000i − 0.768095i
$$679$$ −11.0000 −0.422141
$$680$$ 0 0
$$681$$ −2.00000 −0.0766402
$$682$$ 20.0000i 0.765840i
$$683$$ − 16.0000i − 0.612223i −0.951996 0.306111i $$-0.900972\pi$$
0.951996 0.306111i $$-0.0990280\pi$$
$$684$$ 12.0000 0.458831
$$685$$ 0 0
$$686$$ 26.0000 0.992685
$$687$$ 14.0000i 0.534133i
$$688$$ − 32.0000i − 1.21999i
$$689$$ 5.00000 0.190485
$$690$$ 0 0
$$691$$ 10.0000 0.380418 0.190209 0.981744i $$-0.439083\pi$$
0.190209 + 0.981744i $$0.439083\pi$$
$$692$$ − 36.0000i − 1.36851i
$$693$$ 5.00000i 0.189934i
$$694$$ −22.0000 −0.835109
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 63.0000i − 2.38630i
$$698$$ − 48.0000i − 1.81683i
$$699$$ −19.0000 −0.718646
$$700$$ 0 0
$$701$$ 28.0000 1.05755 0.528773 0.848763i $$-0.322652\pi$$
0.528773 + 0.848763i $$0.322652\pi$$
$$702$$ − 2.00000i − 0.0754851i
$$703$$ 42.0000i 1.58406i
$$704$$ 40.0000 1.50756
$$705$$ 0 0
$$706$$ −12.0000 −0.451626
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 44.0000 1.65245 0.826227 0.563337i $$-0.190483\pi$$
0.826227 + 0.563337i $$0.190483\pi$$
$$710$$ 0 0
$$711$$ −3.00000 −0.112509
$$712$$ 0 0
$$713$$ − 6.00000i − 0.224702i
$$714$$ −14.0000 −0.523937
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 9.00000i 0.336111i
$$718$$ 32.0000i 1.19423i
$$719$$ −16.0000 −0.596699 −0.298350 0.954457i $$-0.596436\pi$$
−0.298350 + 0.954457i $$0.596436\pi$$
$$720$$ 0 0
$$721$$ 4.00000 0.148968
$$722$$ 34.0000i 1.26535i
$$723$$ − 22.0000i − 0.818189i
$$724$$ 14.0000 0.520306
$$725$$ 0 0
$$726$$ 28.0000 1.03918
$$727$$ − 18.0000i − 0.667583i −0.942647 0.333792i $$-0.891672\pi$$
0.942647 0.333792i $$-0.108328\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 56.0000 2.07123
$$732$$ 10.0000i 0.369611i
$$733$$ − 43.0000i − 1.58824i −0.607760 0.794121i $$-0.707932\pi$$
0.607760 0.794121i $$-0.292068\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ −24.0000 −0.884652
$$737$$ − 20.0000i − 0.736709i
$$738$$ − 18.0000i − 0.662589i
$$739$$ 6.00000 0.220714 0.110357 0.993892i $$-0.464801\pi$$
0.110357 + 0.993892i $$0.464801\pi$$
$$740$$ 0 0
$$741$$ 6.00000 0.220416
$$742$$ − 10.0000i − 0.367112i
$$743$$ − 30.0000i − 1.10059i −0.834969 0.550297i $$-0.814515\pi$$
0.834969 0.550297i $$-0.185485\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −64.0000 −2.34321
$$747$$ − 4.00000i − 0.146352i
$$748$$ 70.0000i 2.55945i
$$749$$ −17.0000 −0.621166
$$750$$ 0 0
$$751$$ −13.0000 −0.474377 −0.237188 0.971464i $$-0.576226\pi$$
−0.237188 + 0.971464i $$0.576226\pi$$
$$752$$ − 40.0000i − 1.45865i
$$753$$ 0 0
$$754$$ 4.00000 0.145671
$$755$$ 0 0
$$756$$ −2.00000 −0.0727393
$$757$$ 20.0000i 0.726912i 0.931611 + 0.363456i $$0.118403\pi$$
−0.931611 + 0.363456i $$0.881597\pi$$
$$758$$ − 20.0000i − 0.726433i
$$759$$ −15.0000 −0.544466
$$760$$ 0 0
$$761$$ −42.0000 −1.52250 −0.761249 0.648459i $$-0.775414\pi$$
−0.761249 + 0.648459i $$0.775414\pi$$
$$762$$ − 4.00000i − 0.144905i
$$763$$ 4.00000i 0.144810i
$$764$$ 24.0000 0.868290
$$765$$ 0 0
$$766$$ −12.0000 −0.433578
$$767$$ 0 0
$$768$$ − 16.0000i − 0.577350i
$$769$$ 28.0000 1.00971 0.504853 0.863205i $$-0.331547\pi$$
0.504853 + 0.863205i $$0.331547\pi$$
$$770$$ 0 0
$$771$$ 2.00000 0.0720282
$$772$$ − 34.0000i − 1.22369i
$$773$$ 20.0000i 0.719350i 0.933078 + 0.359675i $$0.117112\pi$$
−0.933078 + 0.359675i $$0.882888\pi$$
$$774$$ 16.0000 0.575108
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 7.00000i − 0.251124i
$$778$$ 8.00000i 0.286814i
$$779$$ 54.0000 1.93475
$$780$$ 0 0
$$781$$ 45.0000 1.61023
$$782$$ − 42.0000i − 1.50192i
$$783$$ − 2.00000i − 0.0714742i
$$784$$ −24.0000 −0.857143
$$785$$ 0 0
$$786$$ −44.0000 −1.56943
$$787$$ 44.0000i 1.56843i 0.620489 + 0.784215i $$0.286934\pi$$
−0.620489 + 0.784215i $$0.713066\pi$$
$$788$$ − 48.0000i − 1.70993i
$$789$$ 16.0000 0.569615
$$790$$ 0 0
$$791$$ −10.0000 −0.355559
$$792$$ 0 0
$$793$$ 5.00000i 0.177555i
$$794$$ −30.0000 −1.06466
$$795$$ 0 0
$$796$$ −56.0000 −1.98487
$$797$$ 39.0000i 1.38145i 0.723117 + 0.690725i $$0.242709\pi$$
−0.723117 + 0.690725i $$0.757291\pi$$
$$798$$ − 12.0000i − 0.424795i
$$799$$ 70.0000 2.47642
$$800$$ 0 0
$$801$$ 11.0000 0.388666
$$802$$ − 28.0000i − 0.988714i
$$803$$ 30.0000i 1.05868i
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ − 24.0000i − 0.844840i
$$808$$ 0 0
$$809$$ 50.0000 1.75791 0.878953 0.476908i $$-0.158243\pi$$
0.878953 + 0.476908i $$0.158243\pi$$
$$810$$ 0 0
$$811$$ 24.0000 0.842754 0.421377 0.906886i $$-0.361547\pi$$
0.421377 + 0.906886i $$0.361547\pi$$
$$812$$ − 4.00000i − 0.140372i
$$813$$ 22.0000i 0.771574i
$$814$$ −70.0000 −2.45350
$$815$$ 0 0
$$816$$ 28.0000 0.980196
$$817$$ 48.0000i 1.67931i
$$818$$ − 60.0000i − 2.09785i
$$819$$ −1.00000 −0.0349428
$$820$$ 0 0
$$821$$ 17.0000 0.593304 0.296652 0.954986i $$-0.404130\pi$$
0.296652 + 0.954986i $$0.404130\pi$$
$$822$$ − 28.0000i − 0.976612i
$$823$$ 4.00000i 0.139431i 0.997567 + 0.0697156i $$0.0222092\pi$$
−0.997567 + 0.0697156i $$0.977791\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 26.0000i − 0.904109i −0.891990 0.452054i $$-0.850691\pi$$
0.891990 0.452054i $$-0.149309\pi$$
$$828$$ − 6.00000i − 0.208514i
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 8.00000i 0.277350i
$$833$$ − 42.0000i − 1.45521i
$$834$$ −30.0000 −1.03882
$$835$$ 0 0
$$836$$ −60.0000 −2.07514
$$837$$ 2.00000i 0.0691301i
$$838$$ − 52.0000i − 1.79631i
$$839$$ −17.0000 −0.586905 −0.293453 0.955974i $$-0.594804\pi$$
−0.293453 + 0.955974i $$0.594804\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ − 40.0000i − 1.37849i
$$843$$ − 18.0000i − 0.619953i
$$844$$ 24.0000 0.826114
$$845$$ 0 0
$$846$$ 20.0000 0.687614
$$847$$ − 14.0000i − 0.481046i
$$848$$ 20.0000i 0.686803i
$$849$$ −20.0000 −0.686398
$$850$$ 0 0
$$851$$ 21.0000 0.719871
$$852$$ 18.0000i 0.616670i
$$853$$ 39.0000i 1.33533i 0.744460 + 0.667667i $$0.232707\pi$$
−0.744460 + 0.667667i $$0.767293\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 45.0000i − 1.53717i −0.639747 0.768585i $$-0.720961\pi$$
0.639747 0.768585i $$-0.279039\pi$$
$$858$$ 10.0000i 0.341394i
$$859$$ −19.0000 −0.648272 −0.324136 0.946011i $$-0.605073\pi$$
−0.324136 + 0.946011i $$0.605073\pi$$
$$860$$ 0 0
$$861$$ −9.00000 −0.306719
$$862$$ − 80.0000i − 2.72481i
$$863$$ − 6.00000i − 0.204242i −0.994772 0.102121i $$-0.967437\pi$$
0.994772 0.102121i $$-0.0325630\pi$$
$$864$$ 8.00000 0.272166
$$865$$ 0 0
$$866$$ −40.0000 −1.35926
$$867$$ 32.0000i 1.08678i
$$868$$ 4.00000i 0.135769i
$$869$$ 15.0000 0.508840
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ 0 0
$$873$$ 11.0000i 0.372294i
$$874$$ 36.0000 1.21772
$$875$$ 0 0
$$876$$ −12.0000 −0.405442
$$877$$ 18.0000i 0.607817i 0.952701 + 0.303908i $$0.0982917\pi$$
−0.952701 + 0.303908i $$0.901708\pi$$
$$878$$ − 30.0000i − 1.01245i
$$879$$ −4.00000 −0.134917
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ − 12.0000i − 0.404061i
$$883$$ − 44.0000i − 1.48072i −0.672212 0.740359i $$-0.734656\pi$$
0.672212 0.740359i $$-0.265344\pi$$
$$884$$ −14.0000 −0.470871
$$885$$ 0 0
$$886$$ −2.00000 −0.0671913
$$887$$ 13.0000i 0.436497i 0.975893 + 0.218249i $$0.0700344\pi$$
−0.975893 + 0.218249i $$0.929966\pi$$
$$888$$ 0 0
$$889$$ −2.00000 −0.0670778
$$890$$ 0 0
$$891$$ 5.00000 0.167506
$$892$$ − 16.0000i − 0.535720i
$$893$$ 60.0000i 2.00782i
$$894$$ −30.0000 −1.00335
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 3.00000i − 0.100167i
$$898$$ − 18.0000i − 0.600668i
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ −35.0000 −1.16602
$$902$$ 90.0000i 2.99667i
$$903$$ − 8.00000i − 0.266223i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ −16.0000 −0.531564
$$907$$ 38.0000i 1.26177i 0.775877 + 0.630885i $$0.217308\pi$$
−0.775877 + 0.630885i $$0.782692\pi$$
$$908$$ 4.00000i 0.132745i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 24.0000i 0.794719i
$$913$$ 20.0000i 0.661903i
$$914$$ 14.0000 0.463079
$$915$$ 0 0
$$916$$ 28.0000 0.925146
$$917$$ 22.0000i 0.726504i
$$918$$ 14.0000i 0.462069i
$$919$$ 19.0000 0.626752 0.313376 0.949629i $$-0.398540\pi$$
0.313376 + 0.949629i $$0.398540\pi$$
$$920$$ 0 0
$$921$$ 23.0000 0.757876
$$922$$ 74.0000i 2.43706i
$$923$$ 9.00000i 0.296239i
$$924$$ 10.0000 0.328976
$$925$$ 0 0
$$926$$ 30.0000 0.985861
$$927$$ − 4.00000i − 0.131377i
$$928$$ 16.0000i 0.525226i
$$929$$ 29.0000 0.951459 0.475730 0.879592i $$-0.342184\pi$$
0.475730 + 0.879592i $$0.342184\pi$$
$$930$$ 0 0
$$931$$ 36.0000 1.17985
$$932$$ 38.0000i 1.24473i
$$933$$ − 20.0000i − 0.654771i
$$934$$ 2.00000 0.0654420
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 6.00000i − 0.196011i −0.995186 0.0980057i $$-0.968754\pi$$
0.995186 0.0980057i $$-0.0312463\pi$$
$$938$$ − 8.00000i − 0.261209i
$$939$$ 22.0000 0.717943
$$940$$ 0 0
$$941$$ 15.0000 0.488986 0.244493 0.969651i $$-0.421378\pi$$
0.244493 + 0.969651i $$0.421378\pi$$
$$942$$ 36.0000i 1.17294i
$$943$$ − 27.0000i − 0.879241i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −80.0000 −2.60102
$$947$$ 12.0000i 0.389948i 0.980808 + 0.194974i $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ 6.00000i 0.194871i
$$949$$ −6.00000 −0.194768
$$950$$ 0 0
$$951$$ −24.0000 −0.778253
$$952$$ 0 0
$$953$$ 15.0000i 0.485898i 0.970039 + 0.242949i $$0.0781147\pi$$
−0.970039 + 0.242949i $$0.921885\pi$$
$$954$$ −10.0000 −0.323762
$$955$$ 0 0
$$956$$ 18.0000 0.582162
$$957$$ 10.0000i 0.323254i
$$958$$ − 6.00000i − 0.193851i
$$959$$ −14.0000 −0.452084
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ − 14.0000i − 0.451378i
$$963$$ 17.0000i 0.547817i
$$964$$ −44.0000 −1.41714
$$965$$ 0 0
$$966$$ −6.00000 −0.193047
$$967$$ − 8.00000i − 0.257263i −0.991692 0.128631i $$-0.958942\pi$$
0.991692 0.128631i $$-0.0410584\pi$$
$$968$$ 0 0
$$969$$ −42.0000 −1.34923
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 2.00000i 0.0641500i
$$973$$ 15.0000i 0.480878i
$$974$$ −10.0000 −0.320421
$$975$$ 0 0
$$976$$ −20.0000 −0.640184
$$977$$ − 12.0000i − 0.383914i −0.981403 0.191957i $$-0.938517\pi$$
0.981403 0.191957i $$-0.0614834\pi$$
$$978$$ − 30.0000i − 0.959294i
$$979$$ −55.0000 −1.75781
$$980$$ 0 0
$$981$$ 4.00000 0.127710
$$982$$ 0 0
$$983$$ 4.00000i 0.127580i 0.997963 + 0.0637901i $$0.0203188\pi$$
−0.997963 + 0.0637901i $$0.979681\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −28.0000 −0.891702
$$987$$ − 10.0000i − 0.318304i
$$988$$ − 12.0000i − 0.381771i
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ −57.0000 −1.81066 −0.905332 0.424704i $$-0.860378\pi$$
−0.905332 + 0.424704i $$0.860378\pi$$
$$992$$ − 16.0000i − 0.508001i
$$993$$ 0 0
$$994$$ 18.0000 0.570925
$$995$$ 0 0
$$996$$ −8.00000 −0.253490
$$997$$ 8.00000i 0.253363i 0.991943 + 0.126681i $$0.0404325\pi$$
−0.991943 + 0.126681i $$0.959567\pi$$
$$998$$ 28.0000i 0.886325i
$$999$$ −7.00000 −0.221470
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.c.274.2 2
3.2 odd 2 2925.2.c.a.2224.1 2
5.2 odd 4 975.2.a.a.1.1 1
5.3 odd 4 195.2.a.c.1.1 1
5.4 even 2 inner 975.2.c.c.274.1 2
15.2 even 4 2925.2.a.s.1.1 1
15.8 even 4 585.2.a.c.1.1 1
15.14 odd 2 2925.2.c.a.2224.2 2
20.3 even 4 3120.2.a.d.1.1 1
35.13 even 4 9555.2.a.u.1.1 1
60.23 odd 4 9360.2.a.bv.1.1 1
65.38 odd 4 2535.2.a.d.1.1 1
195.38 even 4 7605.2.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.c.1.1 1 5.3 odd 4
585.2.a.c.1.1 1 15.8 even 4
975.2.a.a.1.1 1 5.2 odd 4
975.2.c.c.274.1 2 5.4 even 2 inner
975.2.c.c.274.2 2 1.1 even 1 trivial
2535.2.a.d.1.1 1 65.38 odd 4
2925.2.a.s.1.1 1 15.2 even 4
2925.2.c.a.2224.1 2 3.2 odd 2
2925.2.c.a.2224.2 2 15.14 odd 2
3120.2.a.d.1.1 1 20.3 even 4
7605.2.a.t.1.1 1 195.38 even 4
9360.2.a.bv.1.1 1 60.23 odd 4
9555.2.a.u.1.1 1 35.13 even 4