# Properties

 Label 240.2.h.b.191.2 Level $240$ Weight $2$ Character 240.191 Analytic conductor $1.916$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$240 = 2^{4} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 240.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.91640964851$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 191.2 Root $$0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 240.191 Dual form 240.2.h.b.191.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.73205 q^{3} +1.00000i q^{5} -3.46410i q^{7} +3.00000 q^{9} +O(q^{10})$$ $$q-1.73205 q^{3} +1.00000i q^{5} -3.46410i q^{7} +3.00000 q^{9} +3.46410 q^{11} +4.00000 q^{13} -1.73205i q^{15} -6.00000i q^{17} +3.46410i q^{19} +6.00000i q^{21} +3.46410 q^{23} -1.00000 q^{25} -5.19615 q^{27} -6.00000i q^{29} -3.46410i q^{31} -6.00000 q^{33} +3.46410 q^{35} -4.00000 q^{37} -6.92820 q^{39} +12.0000i q^{41} +6.92820i q^{43} +3.00000i q^{45} -3.46410 q^{47} -5.00000 q^{49} +10.3923i q^{51} -6.00000i q^{53} +3.46410i q^{55} -6.00000i q^{57} +3.46410 q^{59} -10.0000 q^{61} -10.3923i q^{63} +4.00000i q^{65} +6.92820i q^{67} -6.00000 q^{69} +13.8564 q^{71} -2.00000 q^{73} +1.73205 q^{75} -12.0000i q^{77} -10.3923i q^{79} +9.00000 q^{81} -10.3923 q^{83} +6.00000 q^{85} +10.3923i q^{87} -13.8564i q^{91} +6.00000i q^{93} -3.46410 q^{95} +10.0000 q^{97} +10.3923 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{9}+O(q^{10})$$ 4 * q + 12 * q^9 $$4 q + 12 q^{9} + 16 q^{13} - 4 q^{25} - 24 q^{33} - 16 q^{37} - 20 q^{49} - 40 q^{61} - 24 q^{69} - 8 q^{73} + 36 q^{81} + 24 q^{85} + 40 q^{97}+O(q^{100})$$ 4 * q + 12 * q^9 + 16 * q^13 - 4 * q^25 - 24 * q^33 - 16 * q^37 - 20 * q^49 - 40 * q^61 - 24 * q^69 - 8 * q^73 + 36 * q^81 + 24 * q^85 + 40 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$161$$ $$181$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.73205 −1.00000
$$4$$ 0 0
$$5$$ 1.00000i 0.447214i
$$6$$ 0 0
$$7$$ − 3.46410i − 1.30931i −0.755929 0.654654i $$-0.772814\pi$$
0.755929 0.654654i $$-0.227186\pi$$
$$8$$ 0 0
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ 3.46410 1.04447 0.522233 0.852803i $$-0.325099\pi$$
0.522233 + 0.852803i $$0.325099\pi$$
$$12$$ 0 0
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ − 1.73205i − 0.447214i
$$16$$ 0 0
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 0 0
$$19$$ 3.46410i 0.794719i 0.917663 + 0.397360i $$0.130073\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 0 0
$$21$$ 6.00000i 1.30931i
$$22$$ 0 0
$$23$$ 3.46410 0.722315 0.361158 0.932505i $$-0.382382\pi$$
0.361158 + 0.932505i $$0.382382\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −5.19615 −1.00000
$$28$$ 0 0
$$29$$ − 6.00000i − 1.11417i −0.830455 0.557086i $$-0.811919\pi$$
0.830455 0.557086i $$-0.188081\pi$$
$$30$$ 0 0
$$31$$ − 3.46410i − 0.622171i −0.950382 0.311086i $$-0.899307\pi$$
0.950382 0.311086i $$-0.100693\pi$$
$$32$$ 0 0
$$33$$ −6.00000 −1.04447
$$34$$ 0 0
$$35$$ 3.46410 0.585540
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ −6.92820 −1.10940
$$40$$ 0 0
$$41$$ 12.0000i 1.87409i 0.349215 + 0.937043i $$0.386448\pi$$
−0.349215 + 0.937043i $$0.613552\pi$$
$$42$$ 0 0
$$43$$ 6.92820i 1.05654i 0.849076 + 0.528271i $$0.177159\pi$$
−0.849076 + 0.528271i $$0.822841\pi$$
$$44$$ 0 0
$$45$$ 3.00000i 0.447214i
$$46$$ 0 0
$$47$$ −3.46410 −0.505291 −0.252646 0.967559i $$-0.581301\pi$$
−0.252646 + 0.967559i $$0.581301\pi$$
$$48$$ 0 0
$$49$$ −5.00000 −0.714286
$$50$$ 0 0
$$51$$ 10.3923i 1.45521i
$$52$$ 0 0
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 0 0
$$55$$ 3.46410i 0.467099i
$$56$$ 0 0
$$57$$ − 6.00000i − 0.794719i
$$58$$ 0 0
$$59$$ 3.46410 0.450988 0.225494 0.974245i $$-0.427600\pi$$
0.225494 + 0.974245i $$0.427600\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ − 10.3923i − 1.30931i
$$64$$ 0 0
$$65$$ 4.00000i 0.496139i
$$66$$ 0 0
$$67$$ 6.92820i 0.846415i 0.906033 + 0.423207i $$0.139096\pi$$
−0.906033 + 0.423207i $$0.860904\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 13.8564 1.64445 0.822226 0.569160i $$-0.192732\pi$$
0.822226 + 0.569160i $$0.192732\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ 1.73205 0.200000
$$76$$ 0 0
$$77$$ − 12.0000i − 1.36753i
$$78$$ 0 0
$$79$$ − 10.3923i − 1.16923i −0.811312 0.584613i $$-0.801246\pi$$
0.811312 0.584613i $$-0.198754\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ −10.3923 −1.14070 −0.570352 0.821401i $$-0.693193\pi$$
−0.570352 + 0.821401i $$0.693193\pi$$
$$84$$ 0 0
$$85$$ 6.00000 0.650791
$$86$$ 0 0
$$87$$ 10.3923i 1.11417i
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ − 13.8564i − 1.45255i
$$92$$ 0 0
$$93$$ 6.00000i 0.622171i
$$94$$ 0 0
$$95$$ −3.46410 −0.355409
$$96$$ 0 0
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 0 0
$$99$$ 10.3923 1.04447
$$100$$ 0 0
$$101$$ 6.00000i 0.597022i 0.954406 + 0.298511i $$0.0964900\pi$$
−0.954406 + 0.298511i $$0.903510\pi$$
$$102$$ 0 0
$$103$$ 10.3923i 1.02398i 0.858990 + 0.511992i $$0.171092\pi$$
−0.858990 + 0.511992i $$0.828908\pi$$
$$104$$ 0 0
$$105$$ −6.00000 −0.585540
$$106$$ 0 0
$$107$$ −3.46410 −0.334887 −0.167444 0.985882i $$-0.553551\pi$$
−0.167444 + 0.985882i $$0.553551\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 6.92820 0.657596
$$112$$ 0 0
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 3.46410i 0.323029i
$$116$$ 0 0
$$117$$ 12.0000 1.10940
$$118$$ 0 0
$$119$$ −20.7846 −1.90532
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ − 20.7846i − 1.87409i
$$124$$ 0 0
$$125$$ − 1.00000i − 0.0894427i
$$126$$ 0 0
$$127$$ − 10.3923i − 0.922168i −0.887357 0.461084i $$-0.847461\pi$$
0.887357 0.461084i $$-0.152539\pi$$
$$128$$ 0 0
$$129$$ − 12.0000i − 1.05654i
$$130$$ 0 0
$$131$$ −17.3205 −1.51330 −0.756650 0.653820i $$-0.773165\pi$$
−0.756650 + 0.653820i $$0.773165\pi$$
$$132$$ 0 0
$$133$$ 12.0000 1.04053
$$134$$ 0 0
$$135$$ − 5.19615i − 0.447214i
$$136$$ 0 0
$$137$$ 18.0000i 1.53784i 0.639343 + 0.768922i $$0.279207\pi$$
−0.639343 + 0.768922i $$0.720793\pi$$
$$138$$ 0 0
$$139$$ − 10.3923i − 0.881464i −0.897639 0.440732i $$-0.854719\pi$$
0.897639 0.440732i $$-0.145281\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 0 0
$$143$$ 13.8564 1.15873
$$144$$ 0 0
$$145$$ 6.00000 0.498273
$$146$$ 0 0
$$147$$ 8.66025 0.714286
$$148$$ 0 0
$$149$$ 6.00000i 0.491539i 0.969328 + 0.245770i $$0.0790407\pi$$
−0.969328 + 0.245770i $$0.920959\pi$$
$$150$$ 0 0
$$151$$ 3.46410i 0.281905i 0.990016 + 0.140952i $$0.0450164\pi$$
−0.990016 + 0.140952i $$0.954984\pi$$
$$152$$ 0 0
$$153$$ − 18.0000i − 1.45521i
$$154$$ 0 0
$$155$$ 3.46410 0.278243
$$156$$ 0 0
$$157$$ −16.0000 −1.27694 −0.638470 0.769647i $$-0.720432\pi$$
−0.638470 + 0.769647i $$0.720432\pi$$
$$158$$ 0 0
$$159$$ 10.3923i 0.824163i
$$160$$ 0 0
$$161$$ − 12.0000i − 0.945732i
$$162$$ 0 0
$$163$$ 13.8564i 1.08532i 0.839953 + 0.542659i $$0.182582\pi$$
−0.839953 + 0.542659i $$0.817418\pi$$
$$164$$ 0 0
$$165$$ − 6.00000i − 0.467099i
$$166$$ 0 0
$$167$$ −3.46410 −0.268060 −0.134030 0.990977i $$-0.542792\pi$$
−0.134030 + 0.990977i $$0.542792\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 10.3923i 0.794719i
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 0 0
$$175$$ 3.46410i 0.261861i
$$176$$ 0 0
$$177$$ −6.00000 −0.450988
$$178$$ 0 0
$$179$$ 24.2487 1.81243 0.906217 0.422813i $$-0.138957\pi$$
0.906217 + 0.422813i $$0.138957\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 17.3205 1.28037
$$184$$ 0 0
$$185$$ − 4.00000i − 0.294086i
$$186$$ 0 0
$$187$$ − 20.7846i − 1.51992i
$$188$$ 0 0
$$189$$ 18.0000i 1.30931i
$$190$$ 0 0
$$191$$ −20.7846 −1.50392 −0.751961 0.659208i $$-0.770892\pi$$
−0.751961 + 0.659208i $$0.770892\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ − 6.92820i − 0.496139i
$$196$$ 0 0
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ 24.2487i 1.71895i 0.511182 + 0.859473i $$0.329208\pi$$
−0.511182 + 0.859473i $$0.670792\pi$$
$$200$$ 0 0
$$201$$ − 12.0000i − 0.846415i
$$202$$ 0 0
$$203$$ −20.7846 −1.45879
$$204$$ 0 0
$$205$$ −12.0000 −0.838116
$$206$$ 0 0
$$207$$ 10.3923 0.722315
$$208$$ 0 0
$$209$$ 12.0000i 0.830057i
$$210$$ 0 0
$$211$$ − 17.3205i − 1.19239i −0.802839 0.596196i $$-0.796678\pi$$
0.802839 0.596196i $$-0.203322\pi$$
$$212$$ 0 0
$$213$$ −24.0000 −1.64445
$$214$$ 0 0
$$215$$ −6.92820 −0.472500
$$216$$ 0 0
$$217$$ −12.0000 −0.814613
$$218$$ 0 0
$$219$$ 3.46410 0.234082
$$220$$ 0 0
$$221$$ − 24.0000i − 1.61441i
$$222$$ 0 0
$$223$$ 17.3205i 1.15987i 0.814664 + 0.579934i $$0.196921\pi$$
−0.814664 + 0.579934i $$0.803079\pi$$
$$224$$ 0 0
$$225$$ −3.00000 −0.200000
$$226$$ 0 0
$$227$$ 3.46410 0.229920 0.114960 0.993370i $$-0.463326\pi$$
0.114960 + 0.993370i $$0.463326\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 20.7846i 1.36753i
$$232$$ 0 0
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 0 0
$$235$$ − 3.46410i − 0.225973i
$$236$$ 0 0
$$237$$ 18.0000i 1.16923i
$$238$$ 0 0
$$239$$ −13.8564 −0.896296 −0.448148 0.893959i $$-0.647916\pi$$
−0.448148 + 0.893959i $$0.647916\pi$$
$$240$$ 0 0
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ 0 0
$$243$$ −15.5885 −1.00000
$$244$$ 0 0
$$245$$ − 5.00000i − 0.319438i
$$246$$ 0 0
$$247$$ 13.8564i 0.881662i
$$248$$ 0 0
$$249$$ 18.0000 1.14070
$$250$$ 0 0
$$251$$ 3.46410 0.218652 0.109326 0.994006i $$-0.465131\pi$$
0.109326 + 0.994006i $$0.465131\pi$$
$$252$$ 0 0
$$253$$ 12.0000 0.754434
$$254$$ 0 0
$$255$$ −10.3923 −0.650791
$$256$$ 0 0
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 0 0
$$259$$ 13.8564i 0.860995i
$$260$$ 0 0
$$261$$ − 18.0000i − 1.11417i
$$262$$ 0 0
$$263$$ 3.46410 0.213606 0.106803 0.994280i $$-0.465939\pi$$
0.106803 + 0.994280i $$0.465939\pi$$
$$264$$ 0 0
$$265$$ 6.00000 0.368577
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 6.00000i − 0.365826i −0.983129 0.182913i $$-0.941447\pi$$
0.983129 0.182913i $$-0.0585527\pi$$
$$270$$ 0 0
$$271$$ − 10.3923i − 0.631288i −0.948878 0.315644i $$-0.897780\pi$$
0.948878 0.315644i $$-0.102220\pi$$
$$272$$ 0 0
$$273$$ 24.0000i 1.45255i
$$274$$ 0 0
$$275$$ −3.46410 −0.208893
$$276$$ 0 0
$$277$$ 4.00000 0.240337 0.120168 0.992754i $$-0.461657\pi$$
0.120168 + 0.992754i $$0.461657\pi$$
$$278$$ 0 0
$$279$$ − 10.3923i − 0.622171i
$$280$$ 0 0
$$281$$ − 12.0000i − 0.715860i −0.933748 0.357930i $$-0.883483\pi$$
0.933748 0.357930i $$-0.116517\pi$$
$$282$$ 0 0
$$283$$ − 6.92820i − 0.411839i −0.978569 0.205919i $$-0.933982\pi$$
0.978569 0.205919i $$-0.0660185\pi$$
$$284$$ 0 0
$$285$$ 6.00000 0.355409
$$286$$ 0 0
$$287$$ 41.5692 2.45375
$$288$$ 0 0
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −17.3205 −1.01535
$$292$$ 0 0
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 0 0
$$295$$ 3.46410i 0.201688i
$$296$$ 0 0
$$297$$ −18.0000 −1.04447
$$298$$ 0 0
$$299$$ 13.8564 0.801337
$$300$$ 0 0
$$301$$ 24.0000 1.38334
$$302$$ 0 0
$$303$$ − 10.3923i − 0.597022i
$$304$$ 0 0
$$305$$ − 10.0000i − 0.572598i
$$306$$ 0 0
$$307$$ − 6.92820i − 0.395413i −0.980261 0.197707i $$-0.936651\pi$$
0.980261 0.197707i $$-0.0633494\pi$$
$$308$$ 0 0
$$309$$ − 18.0000i − 1.02398i
$$310$$ 0 0
$$311$$ 13.8564 0.785725 0.392862 0.919597i $$-0.371485\pi$$
0.392862 + 0.919597i $$0.371485\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ 0 0
$$315$$ 10.3923 0.585540
$$316$$ 0 0
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 0 0
$$319$$ − 20.7846i − 1.16371i
$$320$$ 0 0
$$321$$ 6.00000 0.334887
$$322$$ 0 0
$$323$$ 20.7846 1.15649
$$324$$ 0 0
$$325$$ −4.00000 −0.221880
$$326$$ 0 0
$$327$$ −17.3205 −0.957826
$$328$$ 0 0
$$329$$ 12.0000i 0.661581i
$$330$$ 0 0
$$331$$ 10.3923i 0.571213i 0.958347 + 0.285606i $$0.0921950\pi$$
−0.958347 + 0.285606i $$0.907805\pi$$
$$332$$ 0 0
$$333$$ −12.0000 −0.657596
$$334$$ 0 0
$$335$$ −6.92820 −0.378528
$$336$$ 0 0
$$337$$ 34.0000 1.85210 0.926049 0.377403i $$-0.123183\pi$$
0.926049 + 0.377403i $$0.123183\pi$$
$$338$$ 0 0
$$339$$ − 10.3923i − 0.564433i
$$340$$ 0 0
$$341$$ − 12.0000i − 0.649836i
$$342$$ 0 0
$$343$$ − 6.92820i − 0.374088i
$$344$$ 0 0
$$345$$ − 6.00000i − 0.323029i
$$346$$ 0 0
$$347$$ −10.3923 −0.557888 −0.278944 0.960307i $$-0.589984\pi$$
−0.278944 + 0.960307i $$0.589984\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 0 0
$$351$$ −20.7846 −1.10940
$$352$$ 0 0
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 0 0
$$355$$ 13.8564i 0.735422i
$$356$$ 0 0
$$357$$ 36.0000 1.90532
$$358$$ 0 0
$$359$$ 6.92820 0.365657 0.182828 0.983145i $$-0.441475\pi$$
0.182828 + 0.983145i $$0.441475\pi$$
$$360$$ 0 0
$$361$$ 7.00000 0.368421
$$362$$ 0 0
$$363$$ −1.73205 −0.0909091
$$364$$ 0 0
$$365$$ − 2.00000i − 0.104685i
$$366$$ 0 0
$$367$$ 3.46410i 0.180825i 0.995904 + 0.0904123i $$0.0288185\pi$$
−0.995904 + 0.0904123i $$0.971182\pi$$
$$368$$ 0 0
$$369$$ 36.0000i 1.87409i
$$370$$ 0 0
$$371$$ −20.7846 −1.07908
$$372$$ 0 0
$$373$$ −16.0000 −0.828449 −0.414224 0.910175i $$-0.635947\pi$$
−0.414224 + 0.910175i $$0.635947\pi$$
$$374$$ 0 0
$$375$$ 1.73205i 0.0894427i
$$376$$ 0 0
$$377$$ − 24.0000i − 1.23606i
$$378$$ 0 0
$$379$$ − 24.2487i − 1.24557i −0.782392 0.622786i $$-0.786001\pi$$
0.782392 0.622786i $$-0.213999\pi$$
$$380$$ 0 0
$$381$$ 18.0000i 0.922168i
$$382$$ 0 0
$$383$$ 17.3205 0.885037 0.442518 0.896759i $$-0.354085\pi$$
0.442518 + 0.896759i $$0.354085\pi$$
$$384$$ 0 0
$$385$$ 12.0000 0.611577
$$386$$ 0 0
$$387$$ 20.7846i 1.05654i
$$388$$ 0 0
$$389$$ 6.00000i 0.304212i 0.988364 + 0.152106i $$0.0486055\pi$$
−0.988364 + 0.152106i $$0.951394\pi$$
$$390$$ 0 0
$$391$$ − 20.7846i − 1.05112i
$$392$$ 0 0
$$393$$ 30.0000 1.51330
$$394$$ 0 0
$$395$$ 10.3923 0.522894
$$396$$ 0 0
$$397$$ −16.0000 −0.803017 −0.401508 0.915855i $$-0.631514\pi$$
−0.401508 + 0.915855i $$0.631514\pi$$
$$398$$ 0 0
$$399$$ −20.7846 −1.04053
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ − 13.8564i − 0.690237i
$$404$$ 0 0
$$405$$ 9.00000i 0.447214i
$$406$$ 0 0
$$407$$ −13.8564 −0.686837
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ − 31.1769i − 1.53784i
$$412$$ 0 0
$$413$$ − 12.0000i − 0.590481i
$$414$$ 0 0
$$415$$ − 10.3923i − 0.510138i
$$416$$ 0 0
$$417$$ 18.0000i 0.881464i
$$418$$ 0 0
$$419$$ 10.3923 0.507697 0.253849 0.967244i $$-0.418303\pi$$
0.253849 + 0.967244i $$0.418303\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ 0 0
$$423$$ −10.3923 −0.505291
$$424$$ 0 0
$$425$$ 6.00000i 0.291043i
$$426$$ 0 0
$$427$$ 34.6410i 1.67640i
$$428$$ 0 0
$$429$$ −24.0000 −1.15873
$$430$$ 0 0
$$431$$ 6.92820 0.333720 0.166860 0.985981i $$-0.446637\pi$$
0.166860 + 0.985981i $$0.446637\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ −10.3923 −0.498273
$$436$$ 0 0
$$437$$ 12.0000i 0.574038i
$$438$$ 0 0
$$439$$ − 3.46410i − 0.165333i −0.996577 0.0826663i $$-0.973656\pi$$
0.996577 0.0826663i $$-0.0263436\pi$$
$$440$$ 0 0
$$441$$ −15.0000 −0.714286
$$442$$ 0 0
$$443$$ −17.3205 −0.822922 −0.411461 0.911427i $$-0.634981\pi$$
−0.411461 + 0.911427i $$0.634981\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 10.3923i − 0.491539i
$$448$$ 0 0
$$449$$ − 36.0000i − 1.69895i −0.527633 0.849473i $$-0.676920\pi$$
0.527633 0.849473i $$-0.323080\pi$$
$$450$$ 0 0
$$451$$ 41.5692i 1.95742i
$$452$$ 0 0
$$453$$ − 6.00000i − 0.281905i
$$454$$ 0 0
$$455$$ 13.8564 0.649598
$$456$$ 0 0
$$457$$ −2.00000 −0.0935561 −0.0467780 0.998905i $$-0.514895\pi$$
−0.0467780 + 0.998905i $$0.514895\pi$$
$$458$$ 0 0
$$459$$ 31.1769i 1.45521i
$$460$$ 0 0
$$461$$ 18.0000i 0.838344i 0.907907 + 0.419172i $$0.137680\pi$$
−0.907907 + 0.419172i $$0.862320\pi$$
$$462$$ 0 0
$$463$$ − 10.3923i − 0.482971i −0.970404 0.241486i $$-0.922365\pi$$
0.970404 0.241486i $$-0.0776347\pi$$
$$464$$ 0 0
$$465$$ −6.00000 −0.278243
$$466$$ 0 0
$$467$$ 24.2487 1.12210 0.561048 0.827783i $$-0.310398\pi$$
0.561048 + 0.827783i $$0.310398\pi$$
$$468$$ 0 0
$$469$$ 24.0000 1.10822
$$470$$ 0 0
$$471$$ 27.7128 1.27694
$$472$$ 0 0
$$473$$ 24.0000i 1.10352i
$$474$$ 0 0
$$475$$ − 3.46410i − 0.158944i
$$476$$ 0 0
$$477$$ − 18.0000i − 0.824163i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −16.0000 −0.729537
$$482$$ 0 0
$$483$$ 20.7846i 0.945732i
$$484$$ 0 0
$$485$$ 10.0000i 0.454077i
$$486$$ 0 0
$$487$$ − 17.3205i − 0.784867i −0.919780 0.392434i $$-0.871633\pi$$
0.919780 0.392434i $$-0.128367\pi$$
$$488$$ 0 0
$$489$$ − 24.0000i − 1.08532i
$$490$$ 0 0
$$491$$ 31.1769 1.40699 0.703497 0.710698i $$-0.251621\pi$$
0.703497 + 0.710698i $$0.251621\pi$$
$$492$$ 0 0
$$493$$ −36.0000 −1.62136
$$494$$ 0 0
$$495$$ 10.3923i 0.467099i
$$496$$ 0 0
$$497$$ − 48.0000i − 2.15309i
$$498$$ 0 0
$$499$$ 3.46410i 0.155074i 0.996989 + 0.0775372i $$0.0247057\pi$$
−0.996989 + 0.0775372i $$0.975294\pi$$
$$500$$ 0 0
$$501$$ 6.00000 0.268060
$$502$$ 0 0
$$503$$ −10.3923 −0.463370 −0.231685 0.972791i $$-0.574424\pi$$
−0.231685 + 0.972791i $$0.574424\pi$$
$$504$$ 0 0
$$505$$ −6.00000 −0.266996
$$506$$ 0 0
$$507$$ −5.19615 −0.230769
$$508$$ 0 0
$$509$$ − 6.00000i − 0.265945i −0.991120 0.132973i $$-0.957548\pi$$
0.991120 0.132973i $$-0.0424523\pi$$
$$510$$ 0 0
$$511$$ 6.92820i 0.306486i
$$512$$ 0 0
$$513$$ − 18.0000i − 0.794719i
$$514$$ 0 0
$$515$$ −10.3923 −0.457940
$$516$$ 0 0
$$517$$ −12.0000 −0.527759
$$518$$ 0 0
$$519$$ − 10.3923i − 0.456172i
$$520$$ 0 0
$$521$$ 12.0000i 0.525730i 0.964833 + 0.262865i $$0.0846673\pi$$
−0.964833 + 0.262865i $$0.915333\pi$$
$$522$$ 0 0
$$523$$ 34.6410i 1.51475i 0.652983 + 0.757373i $$0.273517\pi$$
−0.652983 + 0.757373i $$0.726483\pi$$
$$524$$ 0 0
$$525$$ − 6.00000i − 0.261861i
$$526$$ 0 0
$$527$$ −20.7846 −0.905392
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ 10.3923 0.450988
$$532$$ 0 0
$$533$$ 48.0000i 2.07911i
$$534$$ 0 0
$$535$$ − 3.46410i − 0.149766i
$$536$$ 0 0
$$537$$ −42.0000 −1.81243
$$538$$ 0 0
$$539$$ −17.3205 −0.746047
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 0 0
$$543$$ 3.46410 0.148659
$$544$$ 0 0
$$545$$ 10.0000i 0.428353i
$$546$$ 0 0
$$547$$ − 20.7846i − 0.888686i −0.895857 0.444343i $$-0.853437\pi$$
0.895857 0.444343i $$-0.146563\pi$$
$$548$$ 0 0
$$549$$ −30.0000 −1.28037
$$550$$ 0 0
$$551$$ 20.7846 0.885454
$$552$$ 0 0
$$553$$ −36.0000 −1.53088
$$554$$ 0 0
$$555$$ 6.92820i 0.294086i
$$556$$ 0 0
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 0 0
$$559$$ 27.7128i 1.17213i
$$560$$ 0 0
$$561$$ 36.0000i 1.51992i
$$562$$ 0 0
$$563$$ −3.46410 −0.145994 −0.0729972 0.997332i $$-0.523256\pi$$
−0.0729972 + 0.997332i $$0.523256\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ − 31.1769i − 1.30931i
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ 24.2487i 1.01478i 0.861717 + 0.507388i $$0.169389\pi$$
−0.861717 + 0.507388i $$0.830611\pi$$
$$572$$ 0 0
$$573$$ 36.0000 1.50392
$$574$$ 0 0
$$575$$ −3.46410 −0.144463
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 0 0
$$579$$ −3.46410 −0.143963
$$580$$ 0 0
$$581$$ 36.0000i 1.49353i
$$582$$ 0 0
$$583$$ − 20.7846i − 0.860811i
$$584$$ 0 0
$$585$$ 12.0000i 0.496139i
$$586$$ 0 0
$$587$$ −31.1769 −1.28681 −0.643404 0.765526i $$-0.722479\pi$$
−0.643404 + 0.765526i $$0.722479\pi$$
$$588$$ 0 0
$$589$$ 12.0000 0.494451
$$590$$ 0 0
$$591$$ − 10.3923i − 0.427482i
$$592$$ 0 0
$$593$$ − 42.0000i − 1.72473i −0.506284 0.862367i $$-0.668981\pi$$
0.506284 0.862367i $$-0.331019\pi$$
$$594$$ 0 0
$$595$$ − 20.7846i − 0.852086i
$$596$$ 0 0
$$597$$ − 42.0000i − 1.71895i
$$598$$ 0 0
$$599$$ −20.7846 −0.849236 −0.424618 0.905373i $$-0.639592\pi$$
−0.424618 + 0.905373i $$0.639592\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ 20.7846i 0.846415i
$$604$$ 0 0
$$605$$ 1.00000i 0.0406558i
$$606$$ 0 0
$$607$$ 31.1769i 1.26543i 0.774384 + 0.632716i $$0.218060\pi$$
−0.774384 + 0.632716i $$0.781940\pi$$
$$608$$ 0 0
$$609$$ 36.0000 1.45879
$$610$$ 0 0
$$611$$ −13.8564 −0.560570
$$612$$ 0 0
$$613$$ 32.0000 1.29247 0.646234 0.763139i $$-0.276343\pi$$
0.646234 + 0.763139i $$0.276343\pi$$
$$614$$ 0 0
$$615$$ 20.7846 0.838116
$$616$$ 0 0
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 0 0
$$619$$ 45.0333i 1.81004i 0.425367 + 0.905021i $$0.360145\pi$$
−0.425367 + 0.905021i $$0.639855\pi$$
$$620$$ 0 0
$$621$$ −18.0000 −0.722315
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ − 20.7846i − 0.830057i
$$628$$ 0 0
$$629$$ 24.0000i 0.956943i
$$630$$ 0 0
$$631$$ − 3.46410i − 0.137904i −0.997620 0.0689519i $$-0.978035\pi$$
0.997620 0.0689519i $$-0.0219655\pi$$
$$632$$ 0 0
$$633$$ 30.0000i 1.19239i
$$634$$ 0 0
$$635$$ 10.3923 0.412406
$$636$$ 0 0
$$637$$ −20.0000 −0.792429
$$638$$ 0 0
$$639$$ 41.5692 1.64445
$$640$$ 0 0
$$641$$ 24.0000i 0.947943i 0.880540 + 0.473972i $$0.157180\pi$$
−0.880540 + 0.473972i $$0.842820\pi$$
$$642$$ 0 0
$$643$$ − 41.5692i − 1.63933i −0.572843 0.819665i $$-0.694160\pi$$
0.572843 0.819665i $$-0.305840\pi$$
$$644$$ 0 0
$$645$$ 12.0000 0.472500
$$646$$ 0 0
$$647$$ −45.0333 −1.77044 −0.885221 0.465170i $$-0.845993\pi$$
−0.885221 + 0.465170i $$0.845993\pi$$
$$648$$ 0 0
$$649$$ 12.0000 0.471041
$$650$$ 0 0
$$651$$ 20.7846 0.814613
$$652$$ 0 0
$$653$$ − 42.0000i − 1.64359i −0.569785 0.821794i $$-0.692974\pi$$
0.569785 0.821794i $$-0.307026\pi$$
$$654$$ 0 0
$$655$$ − 17.3205i − 0.676768i
$$656$$ 0 0
$$657$$ −6.00000 −0.234082
$$658$$ 0 0
$$659$$ −31.1769 −1.21448 −0.607240 0.794518i $$-0.707723\pi$$
−0.607240 + 0.794518i $$0.707723\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 0 0
$$663$$ 41.5692i 1.61441i
$$664$$ 0 0
$$665$$ 12.0000i 0.465340i
$$666$$ 0 0
$$667$$ − 20.7846i − 0.804783i
$$668$$ 0 0
$$669$$ − 30.0000i − 1.15987i
$$670$$ 0 0
$$671$$ −34.6410 −1.33730
$$672$$ 0 0
$$673$$ 2.00000 0.0770943 0.0385472 0.999257i $$-0.487727\pi$$
0.0385472 + 0.999257i $$0.487727\pi$$
$$674$$ 0 0
$$675$$ 5.19615 0.200000
$$676$$ 0 0
$$677$$ − 30.0000i − 1.15299i −0.817099 0.576497i $$-0.804419\pi$$
0.817099 0.576497i $$-0.195581\pi$$
$$678$$ 0 0
$$679$$ − 34.6410i − 1.32940i
$$680$$ 0 0
$$681$$ −6.00000 −0.229920
$$682$$ 0 0
$$683$$ 17.3205 0.662751 0.331375 0.943499i $$-0.392487\pi$$
0.331375 + 0.943499i $$0.392487\pi$$
$$684$$ 0 0
$$685$$ −18.0000 −0.687745
$$686$$ 0 0
$$687$$ 24.2487 0.925146
$$688$$ 0 0
$$689$$ − 24.0000i − 0.914327i
$$690$$ 0 0
$$691$$ − 31.1769i − 1.18603i −0.805193 0.593013i $$-0.797938\pi$$
0.805193 0.593013i $$-0.202062\pi$$
$$692$$ 0 0
$$693$$ − 36.0000i − 1.36753i
$$694$$ 0 0
$$695$$ 10.3923 0.394203
$$696$$ 0 0
$$697$$ 72.0000 2.72719
$$698$$ 0 0
$$699$$ − 10.3923i − 0.393073i
$$700$$ 0 0
$$701$$ 42.0000i 1.58632i 0.609015 + 0.793159i $$0.291565\pi$$
−0.609015 + 0.793159i $$0.708435\pi$$
$$702$$ 0 0
$$703$$ − 13.8564i − 0.522604i
$$704$$ 0 0
$$705$$ 6.00000i 0.225973i
$$706$$ 0 0
$$707$$ 20.7846 0.781686
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ − 31.1769i − 1.16923i
$$712$$ 0 0
$$713$$ − 12.0000i − 0.449404i
$$714$$ 0 0
$$715$$ 13.8564i 0.518200i
$$716$$ 0 0
$$717$$ 24.0000 0.896296
$$718$$ 0 0
$$719$$ 27.7128 1.03351 0.516757 0.856132i $$-0.327139\pi$$
0.516757 + 0.856132i $$0.327139\pi$$
$$720$$ 0 0
$$721$$ 36.0000 1.34071
$$722$$ 0 0
$$723$$ 45.0333 1.67481
$$724$$ 0 0
$$725$$ 6.00000i 0.222834i
$$726$$ 0 0
$$727$$ − 3.46410i − 0.128476i −0.997935 0.0642382i $$-0.979538\pi$$
0.997935 0.0642382i $$-0.0204617\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 41.5692 1.53749
$$732$$ 0 0
$$733$$ 28.0000 1.03420 0.517102 0.855924i $$-0.327011\pi$$
0.517102 + 0.855924i $$0.327011\pi$$
$$734$$ 0 0
$$735$$ 8.66025i 0.319438i
$$736$$ 0 0
$$737$$ 24.0000i 0.884051i
$$738$$ 0 0
$$739$$ − 10.3923i − 0.382287i −0.981562 0.191144i $$-0.938780\pi$$
0.981562 0.191144i $$-0.0612196\pi$$
$$740$$ 0 0
$$741$$ − 24.0000i − 0.881662i
$$742$$ 0 0
$$743$$ −38.1051 −1.39794 −0.698971 0.715150i $$-0.746358\pi$$
−0.698971 + 0.715150i $$0.746358\pi$$
$$744$$ 0 0
$$745$$ −6.00000 −0.219823
$$746$$ 0 0
$$747$$ −31.1769 −1.14070
$$748$$ 0 0
$$749$$ 12.0000i 0.438470i
$$750$$ 0 0
$$751$$ 51.9615i 1.89610i 0.318117 + 0.948051i $$0.396950\pi$$
−0.318117 + 0.948051i $$0.603050\pi$$
$$752$$ 0 0
$$753$$ −6.00000 −0.218652
$$754$$ 0 0
$$755$$ −3.46410 −0.126072
$$756$$ 0 0
$$757$$ −20.0000 −0.726912 −0.363456 0.931611i $$-0.618403\pi$$
−0.363456 + 0.931611i $$0.618403\pi$$
$$758$$ 0 0
$$759$$ −20.7846 −0.754434
$$760$$ 0 0
$$761$$ 12.0000i 0.435000i 0.976060 + 0.217500i $$0.0697902\pi$$
−0.976060 + 0.217500i $$0.930210\pi$$
$$762$$ 0 0
$$763$$ − 34.6410i − 1.25409i
$$764$$ 0 0
$$765$$ 18.0000 0.650791
$$766$$ 0 0
$$767$$ 13.8564 0.500326
$$768$$ 0 0
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ 0 0
$$771$$ − 31.1769i − 1.12281i
$$772$$ 0 0
$$773$$ 54.0000i 1.94225i 0.238581 + 0.971123i $$0.423318\pi$$
−0.238581 + 0.971123i $$0.576682\pi$$
$$774$$ 0 0
$$775$$ 3.46410i 0.124434i
$$776$$ 0 0
$$777$$ − 24.0000i − 0.860995i
$$778$$ 0 0
$$779$$ −41.5692 −1.48937
$$780$$ 0 0
$$781$$ 48.0000 1.71758
$$782$$ 0 0
$$783$$ 31.1769i 1.11417i
$$784$$ 0 0
$$785$$ − 16.0000i − 0.571064i
$$786$$ 0 0
$$787$$ 34.6410i 1.23482i 0.786642 + 0.617409i $$0.211818\pi$$
−0.786642 + 0.617409i $$0.788182\pi$$
$$788$$ 0 0
$$789$$ −6.00000 −0.213606
$$790$$ 0 0
$$791$$ 20.7846 0.739016
$$792$$ 0 0
$$793$$ −40.0000 −1.42044
$$794$$ 0 0
$$795$$ −10.3923 −0.368577
$$796$$ 0 0
$$797$$ − 30.0000i − 1.06265i −0.847167 0.531327i $$-0.821693\pi$$
0.847167 0.531327i $$-0.178307\pi$$
$$798$$ 0 0
$$799$$ 20.7846i 0.735307i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −6.92820 −0.244491
$$804$$ 0 0
$$805$$ 12.0000 0.422944
$$806$$ 0 0
$$807$$ 10.3923i 0.365826i
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ 24.2487i 0.851487i 0.904844 + 0.425744i $$0.139987\pi$$
−0.904844 + 0.425744i $$0.860013\pi$$
$$812$$ 0 0
$$813$$ 18.0000i 0.631288i
$$814$$ 0 0
$$815$$ −13.8564 −0.485369
$$816$$ 0 0
$$817$$ −24.0000 −0.839654
$$818$$ 0 0
$$819$$ − 41.5692i − 1.45255i
$$820$$ 0 0
$$821$$ 6.00000i 0.209401i 0.994504 + 0.104701i $$0.0333885\pi$$
−0.994504 + 0.104701i $$0.966612\pi$$
$$822$$ 0 0
$$823$$ − 45.0333i − 1.56976i −0.619646 0.784881i $$-0.712724\pi$$
0.619646 0.784881i $$-0.287276\pi$$
$$824$$ 0 0
$$825$$ 6.00000 0.208893
$$826$$ 0 0
$$827$$ −38.1051 −1.32504 −0.662522 0.749042i $$-0.730514\pi$$
−0.662522 + 0.749042i $$0.730514\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ −6.92820 −0.240337
$$832$$ 0 0
$$833$$ 30.0000i 1.03944i
$$834$$ 0 0
$$835$$ − 3.46410i − 0.119880i
$$836$$ 0 0
$$837$$ 18.0000i 0.622171i
$$838$$ 0 0
$$839$$ 48.4974 1.67432 0.837158 0.546960i $$-0.184215\pi$$
0.837158 + 0.546960i $$0.184215\pi$$
$$840$$ 0 0
$$841$$ −7.00000 −0.241379
$$842$$ 0 0
$$843$$ 20.7846i 0.715860i
$$844$$ 0 0
$$845$$ 3.00000i 0.103203i
$$846$$ 0 0
$$847$$ − 3.46410i − 0.119028i
$$848$$ 0 0
$$849$$ 12.0000i 0.411839i
$$850$$ 0 0
$$851$$ −13.8564 −0.474991
$$852$$ 0 0
$$853$$ 8.00000 0.273915 0.136957 0.990577i $$-0.456268\pi$$
0.136957 + 0.990577i $$0.456268\pi$$
$$854$$ 0 0
$$855$$ −10.3923 −0.355409
$$856$$ 0 0
$$857$$ − 6.00000i − 0.204956i −0.994735 0.102478i $$-0.967323\pi$$
0.994735 0.102478i $$-0.0326771\pi$$
$$858$$ 0 0
$$859$$ − 51.9615i − 1.77290i −0.462820 0.886452i $$-0.653162\pi$$
0.462820 0.886452i $$-0.346838\pi$$
$$860$$ 0 0
$$861$$ −72.0000 −2.45375
$$862$$ 0 0
$$863$$ −51.9615 −1.76879 −0.884395 0.466738i $$-0.845429\pi$$
−0.884395 + 0.466738i $$0.845429\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ 0 0
$$867$$ 32.9090 1.11765
$$868$$ 0 0
$$869$$ − 36.0000i − 1.22122i
$$870$$ 0 0
$$871$$ 27.7128i 0.939013i
$$872$$ 0 0
$$873$$ 30.0000 1.01535
$$874$$ 0 0
$$875$$ −3.46410 −0.117108
$$876$$ 0 0
$$877$$ 32.0000 1.08056 0.540282 0.841484i $$-0.318318\pi$$
0.540282 + 0.841484i $$0.318318\pi$$
$$878$$ 0 0
$$879$$ 10.3923i 0.350524i
$$880$$ 0 0
$$881$$ − 48.0000i − 1.61716i −0.588386 0.808581i $$-0.700236\pi$$
0.588386 0.808581i $$-0.299764\pi$$
$$882$$ 0 0
$$883$$ − 13.8564i − 0.466305i −0.972440 0.233153i $$-0.925096\pi$$
0.972440 0.233153i $$-0.0749042\pi$$
$$884$$ 0 0
$$885$$ − 6.00000i − 0.201688i
$$886$$ 0 0
$$887$$ 24.2487 0.814192 0.407096 0.913385i $$-0.366541\pi$$
0.407096 + 0.913385i $$0.366541\pi$$
$$888$$ 0 0
$$889$$ −36.0000 −1.20740
$$890$$ 0 0
$$891$$ 31.1769 1.04447
$$892$$ 0 0
$$893$$ − 12.0000i − 0.401565i
$$894$$ 0 0
$$895$$ 24.2487i 0.810545i
$$896$$ 0 0
$$897$$ −24.0000 −0.801337
$$898$$ 0 0
$$899$$ −20.7846 −0.693206
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ −41.5692 −1.38334
$$904$$ 0 0
$$905$$ − 2.00000i − 0.0664822i
$$906$$ 0 0
$$907$$ 41.5692i 1.38028i 0.723674 + 0.690142i $$0.242452\pi$$
−0.723674 + 0.690142i $$0.757548\pi$$
$$908$$ 0 0
$$909$$ 18.0000i 0.597022i
$$910$$ 0 0
$$911$$ −34.6410 −1.14771 −0.573854 0.818958i $$-0.694552\pi$$
−0.573854 + 0.818958i $$0.694552\pi$$
$$912$$ 0 0
$$913$$ −36.0000 −1.19143
$$914$$ 0 0
$$915$$ 17.3205i 0.572598i
$$916$$ 0 0
$$917$$ 60.0000i 1.98137i
$$918$$ 0 0
$$919$$ 45.0333i 1.48551i 0.669562 + 0.742756i $$0.266482\pi$$
−0.669562 + 0.742756i $$0.733518\pi$$
$$920$$ 0 0
$$921$$ 12.0000i 0.395413i
$$922$$ 0 0
$$923$$ 55.4256 1.82436
$$924$$ 0 0
$$925$$ 4.00000 0.131519
$$926$$ 0 0
$$927$$ 31.1769i 1.02398i
$$928$$ 0 0
$$929$$ 12.0000i 0.393707i 0.980433 + 0.196854i $$0.0630724\pi$$
−0.980433 + 0.196854i $$0.936928\pi$$
$$930$$ 0 0
$$931$$ − 17.3205i − 0.567657i
$$932$$ 0 0
$$933$$ −24.0000 −0.785725
$$934$$ 0 0
$$935$$ 20.7846 0.679729
$$936$$ 0 0
$$937$$ 38.0000 1.24141 0.620703 0.784046i $$-0.286847\pi$$
0.620703 + 0.784046i $$0.286847\pi$$
$$938$$ 0 0
$$939$$ −24.2487 −0.791327
$$940$$ 0 0
$$941$$ − 54.0000i − 1.76035i −0.474650 0.880175i $$-0.657425\pi$$
0.474650 0.880175i $$-0.342575\pi$$
$$942$$ 0 0
$$943$$ 41.5692i 1.35368i
$$944$$ 0 0
$$945$$ −18.0000 −0.585540
$$946$$ 0 0
$$947$$ 31.1769 1.01311 0.506557 0.862207i $$-0.330918\pi$$
0.506557 + 0.862207i $$0.330918\pi$$
$$948$$ 0 0
$$949$$ −8.00000 −0.259691
$$950$$ 0 0
$$951$$ − 31.1769i − 1.01098i
$$952$$ 0 0
$$953$$ − 18.0000i − 0.583077i −0.956559 0.291539i $$-0.905833\pi$$
0.956559 0.291539i $$-0.0941672\pi$$
$$954$$ 0 0
$$955$$ − 20.7846i − 0.672574i
$$956$$ 0 0
$$957$$ 36.0000i 1.16371i
$$958$$ 0 0
$$959$$ 62.3538 2.01351
$$960$$ 0 0
$$961$$ 19.0000 0.612903
$$962$$ 0 0
$$963$$ −10.3923 −0.334887
$$964$$ 0 0
$$965$$ 2.00000i 0.0643823i
$$966$$ 0 0
$$967$$ 51.9615i 1.67097i 0.549513 + 0.835485i $$0.314813\pi$$
−0.549513 + 0.835485i $$0.685187\pi$$
$$968$$ 0 0
$$969$$ −36.0000 −1.15649
$$970$$ 0 0
$$971$$ 3.46410 0.111168 0.0555842 0.998454i $$-0.482298\pi$$
0.0555842 + 0.998454i $$0.482298\pi$$
$$972$$ 0 0
$$973$$ −36.0000 −1.15411
$$974$$ 0 0
$$975$$ 6.92820 0.221880
$$976$$ 0 0
$$977$$ − 30.0000i − 0.959785i −0.877327 0.479893i $$-0.840676\pi$$
0.877327 0.479893i $$-0.159324\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 30.0000 0.957826
$$982$$ 0 0
$$983$$ 58.8897 1.87829 0.939145 0.343520i $$-0.111619\pi$$
0.939145 + 0.343520i $$0.111619\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ − 20.7846i − 0.661581i
$$988$$ 0 0
$$989$$ 24.0000i 0.763156i
$$990$$ 0 0
$$991$$ 17.3205i 0.550204i 0.961415 + 0.275102i $$0.0887116\pi$$
−0.961415 + 0.275102i $$0.911288\pi$$
$$992$$ 0 0
$$993$$ − 18.0000i − 0.571213i
$$994$$ 0 0
$$995$$ −24.2487 −0.768736
$$996$$ 0 0
$$997$$ −52.0000 −1.64686 −0.823428 0.567420i $$-0.807941\pi$$
−0.823428 + 0.567420i $$0.807941\pi$$
$$998$$ 0 0
$$999$$ 20.7846 0.657596
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 240.2.h.b.191.2 yes 4
3.2 odd 2 inner 240.2.h.b.191.3 yes 4
4.3 odd 2 inner 240.2.h.b.191.4 yes 4
5.2 odd 4 1200.2.o.b.1199.4 4
5.3 odd 4 1200.2.o.a.1199.1 4
5.4 even 2 1200.2.h.m.1151.4 4
8.3 odd 2 960.2.h.d.191.1 4
8.5 even 2 960.2.h.d.191.3 4
12.11 even 2 inner 240.2.h.b.191.1 4
15.2 even 4 1200.2.o.a.1199.2 4
15.8 even 4 1200.2.o.b.1199.3 4
15.14 odd 2 1200.2.h.m.1151.2 4
20.3 even 4 1200.2.o.a.1199.4 4
20.7 even 4 1200.2.o.b.1199.1 4
20.19 odd 2 1200.2.h.m.1151.1 4
24.5 odd 2 960.2.h.d.191.2 4
24.11 even 2 960.2.h.d.191.4 4
60.23 odd 4 1200.2.o.b.1199.2 4
60.47 odd 4 1200.2.o.a.1199.3 4
60.59 even 2 1200.2.h.m.1151.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
240.2.h.b.191.1 4 12.11 even 2 inner
240.2.h.b.191.2 yes 4 1.1 even 1 trivial
240.2.h.b.191.3 yes 4 3.2 odd 2 inner
240.2.h.b.191.4 yes 4 4.3 odd 2 inner
960.2.h.d.191.1 4 8.3 odd 2
960.2.h.d.191.2 4 24.5 odd 2
960.2.h.d.191.3 4 8.5 even 2
960.2.h.d.191.4 4 24.11 even 2
1200.2.h.m.1151.1 4 20.19 odd 2
1200.2.h.m.1151.2 4 15.14 odd 2
1200.2.h.m.1151.3 4 60.59 even 2
1200.2.h.m.1151.4 4 5.4 even 2
1200.2.o.a.1199.1 4 5.3 odd 4
1200.2.o.a.1199.2 4 15.2 even 4
1200.2.o.a.1199.3 4 60.47 odd 4
1200.2.o.a.1199.4 4 20.3 even 4
1200.2.o.b.1199.1 4 20.7 even 4
1200.2.o.b.1199.2 4 60.23 odd 4
1200.2.o.b.1199.3 4 15.8 even 4
1200.2.o.b.1199.4 4 5.2 odd 4