# Properties

 Label 7200.2.o.e.7199.1 Level $7200$ Weight $2$ Character 7200.7199 Analytic conductor $57.492$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7200 = 2^{5} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7200.o (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$57.4922894553$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1440) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 7199.1 Root $$0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 7200.7199 Dual form 7200.2.o.e.7199.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.41421 q^{7} +O(q^{10})$$ $$q-3.41421 q^{7} +2.58579 q^{11} -3.41421i q^{13} -1.17157 q^{17} +4.82843i q^{23} +6.00000i q^{29} -6.48528i q^{31} +9.07107i q^{37} -11.0711i q^{41} -6.82843 q^{43} +5.65685i q^{47} +4.65685 q^{49} +1.17157 q^{53} -6.58579 q^{59} +12.8284 q^{61} -8.00000 q^{67} +5.65685 q^{71} -10.4853i q^{73} -8.82843 q^{77} +14.4853i q^{79} +9.17157i q^{83} -4.24264i q^{89} +11.6569i q^{91} -2.48528i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{7} + O(q^{10})$$ $$4 q - 8 q^{7} + 16 q^{11} - 16 q^{17} - 16 q^{43} - 4 q^{49} + 16 q^{53} - 32 q^{59} + 40 q^{61} - 32 q^{67} - 24 q^{77} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$6401$$ $$6751$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.41421 −1.29045 −0.645226 0.763992i $$-0.723237\pi$$
−0.645226 + 0.763992i $$0.723237\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.58579 0.779644 0.389822 0.920890i $$-0.372537\pi$$
0.389822 + 0.920890i $$0.372537\pi$$
$$12$$ 0 0
$$13$$ − 3.41421i − 0.946932i −0.880812 0.473466i $$-0.843003\pi$$
0.880812 0.473466i $$-0.156997\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.17157 −0.284148 −0.142074 0.989856i $$-0.545377\pi$$
−0.142074 + 0.989856i $$0.545377\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.82843i 1.00680i 0.864054 + 0.503398i $$0.167917\pi$$
−0.864054 + 0.503398i $$0.832083\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.00000i 1.11417i 0.830455 + 0.557086i $$0.188081\pi$$
−0.830455 + 0.557086i $$0.811919\pi$$
$$30$$ 0 0
$$31$$ − 6.48528i − 1.16479i −0.812906 0.582395i $$-0.802116\pi$$
0.812906 0.582395i $$-0.197884\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9.07107i 1.49127i 0.666352 + 0.745637i $$0.267855\pi$$
−0.666352 + 0.745637i $$0.732145\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 11.0711i − 1.72901i −0.502624 0.864505i $$-0.667632\pi$$
0.502624 0.864505i $$-0.332368\pi$$
$$42$$ 0 0
$$43$$ −6.82843 −1.04133 −0.520663 0.853762i $$-0.674315\pi$$
−0.520663 + 0.853762i $$0.674315\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 5.65685i 0.825137i 0.910927 + 0.412568i $$0.135368\pi$$
−0.910927 + 0.412568i $$0.864632\pi$$
$$48$$ 0 0
$$49$$ 4.65685 0.665265
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.17157 0.160928 0.0804640 0.996758i $$-0.474360\pi$$
0.0804640 + 0.996758i $$0.474360\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −6.58579 −0.857396 −0.428698 0.903448i $$-0.641028\pi$$
−0.428698 + 0.903448i $$0.641028\pi$$
$$60$$ 0 0
$$61$$ 12.8284 1.64251 0.821256 0.570560i $$-0.193274\pi$$
0.821256 + 0.570560i $$0.193274\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 5.65685 0.671345 0.335673 0.941979i $$-0.391036\pi$$
0.335673 + 0.941979i $$0.391036\pi$$
$$72$$ 0 0
$$73$$ − 10.4853i − 1.22721i −0.789613 0.613605i $$-0.789719\pi$$
0.789613 0.613605i $$-0.210281\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −8.82843 −1.00609
$$78$$ 0 0
$$79$$ 14.4853i 1.62972i 0.579657 + 0.814861i $$0.303187\pi$$
−0.579657 + 0.814861i $$0.696813\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 9.17157i 1.00671i 0.864079 + 0.503355i $$0.167901\pi$$
−0.864079 + 0.503355i $$0.832099\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 4.24264i − 0.449719i −0.974391 0.224860i $$-0.927808\pi$$
0.974391 0.224860i $$-0.0721923\pi$$
$$90$$ 0 0
$$91$$ 11.6569i 1.22197i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 2.48528i − 0.252342i −0.992009 0.126171i $$-0.959731\pi$$
0.992009 0.126171i $$-0.0402688\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 4.82843i − 0.480446i −0.970718 0.240223i $$-0.922779\pi$$
0.970718 0.240223i $$-0.0772206\pi$$
$$102$$ 0 0
$$103$$ 17.5563 1.72988 0.864939 0.501877i $$-0.167357\pi$$
0.864939 + 0.501877i $$0.167357\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 14.8284i − 1.43352i −0.697321 0.716759i $$-0.745625\pi$$
0.697321 0.716759i $$-0.254375\pi$$
$$108$$ 0 0
$$109$$ 7.17157 0.686912 0.343456 0.939169i $$-0.388402\pi$$
0.343456 + 0.939169i $$0.388402\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1.65685 −0.155864 −0.0779319 0.996959i $$-0.524832\pi$$
−0.0779319 + 0.996959i $$0.524832\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −4.31371 −0.392155
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −10.7279 −0.951949 −0.475975 0.879459i $$-0.657905\pi$$
−0.475975 + 0.879459i $$0.657905\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.7279 1.11204 0.556022 0.831168i $$-0.312327\pi$$
0.556022 + 0.831168i $$0.312327\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 19.3137 1.65008 0.825041 0.565073i $$-0.191152\pi$$
0.825041 + 0.565073i $$0.191152\pi$$
$$138$$ 0 0
$$139$$ − 3.65685i − 0.310170i −0.987901 0.155085i $$-0.950435\pi$$
0.987901 0.155085i $$-0.0495652\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 8.82843i − 0.738270i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 8.82843i − 0.723253i −0.932323 0.361626i $$-0.882222\pi$$
0.932323 0.361626i $$-0.117778\pi$$
$$150$$ 0 0
$$151$$ 15.6569i 1.27414i 0.770807 + 0.637068i $$0.219853\pi$$
−0.770807 + 0.637068i $$0.780147\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 9.75736i 0.778722i 0.921085 + 0.389361i $$0.127304\pi$$
−0.921085 + 0.389361i $$0.872696\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ − 16.4853i − 1.29922i
$$162$$ 0 0
$$163$$ −0.485281 −0.0380102 −0.0190051 0.999819i $$-0.506050\pi$$
−0.0190051 + 0.999819i $$0.506050\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 21.7990i 1.68686i 0.537242 + 0.843428i $$0.319466\pi$$
−0.537242 + 0.843428i $$0.680534\pi$$
$$168$$ 0 0
$$169$$ 1.34315 0.103319
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 8.48528 0.645124 0.322562 0.946548i $$-0.395456\pi$$
0.322562 + 0.946548i $$0.395456\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 24.0416 1.79696 0.898478 0.439019i $$-0.144674\pi$$
0.898478 + 0.439019i $$0.144674\pi$$
$$180$$ 0 0
$$181$$ 1.51472 0.112588 0.0562941 0.998414i $$-0.482072\pi$$
0.0562941 + 0.998414i $$0.482072\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −3.02944 −0.221534
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3.51472 −0.254316 −0.127158 0.991882i $$-0.540586\pi$$
−0.127158 + 0.991882i $$0.540586\pi$$
$$192$$ 0 0
$$193$$ 18.9706i 1.36553i 0.730638 + 0.682765i $$0.239223\pi$$
−0.730638 + 0.682765i $$0.760777\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 21.3137 1.51854 0.759269 0.650776i $$-0.225556\pi$$
0.759269 + 0.650776i $$0.225556\pi$$
$$198$$ 0 0
$$199$$ 15.6569i 1.10988i 0.831889 + 0.554942i $$0.187260\pi$$
−0.831889 + 0.554942i $$0.812740\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 20.4853i − 1.43778i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 26.0000i 1.78991i 0.446153 + 0.894957i $$0.352794\pi$$
−0.446153 + 0.894957i $$0.647206\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 22.1421i 1.50311i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.00000i 0.269069i
$$222$$ 0 0
$$223$$ −2.72792 −0.182675 −0.0913376 0.995820i $$-0.529114\pi$$
−0.0913376 + 0.995820i $$0.529114\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 5.65685i − 0.375459i −0.982221 0.187729i $$-0.939887\pi$$
0.982221 0.187729i $$-0.0601128\pi$$
$$228$$ 0 0
$$229$$ 14.9706 0.989283 0.494641 0.869097i $$-0.335299\pi$$
0.494641 + 0.869097i $$0.335299\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 2.14214 0.140336 0.0701680 0.997535i $$-0.477646\pi$$
0.0701680 + 0.997535i $$0.477646\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −25.4558 −1.64660 −0.823301 0.567605i $$-0.807870\pi$$
−0.823301 + 0.567605i $$0.807870\pi$$
$$240$$ 0 0
$$241$$ 28.9706 1.86616 0.933079 0.359671i $$-0.117111\pi$$
0.933079 + 0.359671i $$0.117111\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 13.8995 0.877328 0.438664 0.898651i $$-0.355452\pi$$
0.438664 + 0.898651i $$0.355452\pi$$
$$252$$ 0 0
$$253$$ 12.4853i 0.784943i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ − 30.9706i − 1.92442i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 9.51472i − 0.586703i −0.956005 0.293351i $$-0.905229\pi$$
0.956005 0.293351i $$-0.0947706\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 7.65685i − 0.466847i −0.972375 0.233423i $$-0.925007\pi$$
0.972375 0.233423i $$-0.0749928\pi$$
$$270$$ 0 0
$$271$$ 22.0000i 1.33640i 0.743980 + 0.668202i $$0.232936\pi$$
−0.743980 + 0.668202i $$0.767064\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 5.27208i 0.316768i 0.987378 + 0.158384i $$0.0506285\pi$$
−0.987378 + 0.158384i $$0.949372\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 22.5858i 1.34736i 0.739025 + 0.673678i $$0.235286\pi$$
−0.739025 + 0.673678i $$0.764714\pi$$
$$282$$ 0 0
$$283$$ 7.31371 0.434755 0.217377 0.976088i $$-0.430250\pi$$
0.217377 + 0.976088i $$0.430250\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 37.7990i 2.23120i
$$288$$ 0 0
$$289$$ −15.6274 −0.919260
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −29.3137 −1.71253 −0.856263 0.516541i $$-0.827219\pi$$
−0.856263 + 0.516541i $$0.827219\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 16.4853 0.953368
$$300$$ 0 0
$$301$$ 23.3137 1.34378
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −24.0000 −1.36975 −0.684876 0.728659i $$-0.740144\pi$$
−0.684876 + 0.728659i $$0.740144\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 26.1421 1.48238 0.741192 0.671293i $$-0.234261\pi$$
0.741192 + 0.671293i $$0.234261\pi$$
$$312$$ 0 0
$$313$$ 7.65685i 0.432791i 0.976306 + 0.216395i $$0.0694301\pi$$
−0.976306 + 0.216395i $$0.930570\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 16.6274 0.933889 0.466944 0.884287i $$-0.345355\pi$$
0.466944 + 0.884287i $$0.345355\pi$$
$$318$$ 0 0
$$319$$ 15.5147i 0.868657i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ − 19.3137i − 1.06480i
$$330$$ 0 0
$$331$$ 28.9706i 1.59237i 0.605056 + 0.796183i $$0.293151\pi$$
−0.605056 + 0.796183i $$0.706849\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 26.4853i 1.44275i 0.692547 + 0.721373i $$0.256488\pi$$
−0.692547 + 0.721373i $$0.743512\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 16.7696i − 0.908122i
$$342$$ 0 0
$$343$$ 8.00000 0.431959
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 22.3431i 1.19944i 0.800209 + 0.599721i $$0.204722\pi$$
−0.800209 + 0.599721i $$0.795278\pi$$
$$348$$ 0 0
$$349$$ −31.4558 −1.68379 −0.841896 0.539639i $$-0.818561\pi$$
−0.841896 + 0.539639i $$0.818561\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 1.85786 0.0988841 0.0494421 0.998777i $$-0.484256\pi$$
0.0494421 + 0.998777i $$0.484256\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 18.8284 0.993726 0.496863 0.867829i $$-0.334485\pi$$
0.496863 + 0.867829i $$0.334485\pi$$
$$360$$ 0 0
$$361$$ 19.0000 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 24.8701 1.29821 0.649103 0.760700i $$-0.275144\pi$$
0.649103 + 0.760700i $$0.275144\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ 0 0
$$373$$ − 10.7279i − 0.555471i −0.960658 0.277735i $$-0.910416\pi$$
0.960658 0.277735i $$-0.0895838\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 20.4853 1.05505
$$378$$ 0 0
$$379$$ 9.31371i 0.478413i 0.970969 + 0.239207i $$0.0768873\pi$$
−0.970969 + 0.239207i $$0.923113\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 10.3431i 0.528510i 0.964453 + 0.264255i $$0.0851261\pi$$
−0.964453 + 0.264255i $$0.914874\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 12.3431i − 0.625822i −0.949782 0.312911i $$-0.898696\pi$$
0.949782 0.312911i $$-0.101304\pi$$
$$390$$ 0 0
$$391$$ − 5.65685i − 0.286079i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 14.7279i − 0.739173i −0.929196 0.369587i $$-0.879499\pi$$
0.929196 0.369587i $$-0.120501\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 0.727922i − 0.0363507i −0.999835 0.0181753i $$-0.994214\pi$$
0.999835 0.0181753i $$-0.00578571\pi$$
$$402$$ 0 0
$$403$$ −22.1421 −1.10298
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 23.4558i 1.16266i
$$408$$ 0 0
$$409$$ 38.2843 1.89304 0.946518 0.322652i $$-0.104574\pi$$
0.946518 + 0.322652i $$0.104574\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 22.4853 1.10643
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 38.3848 1.87522 0.937610 0.347690i $$-0.113034\pi$$
0.937610 + 0.347690i $$0.113034\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −43.7990 −2.11958
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 2.14214 0.103183 0.0515915 0.998668i $$-0.483571\pi$$
0.0515915 + 0.998668i $$0.483571\pi$$
$$432$$ 0 0
$$433$$ − 14.4853i − 0.696118i −0.937473 0.348059i $$-0.886841\pi$$
0.937473 0.348059i $$-0.113159\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ − 18.2843i − 0.872661i −0.899787 0.436330i $$-0.856278\pi$$
0.899787 0.436330i $$-0.143722\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 26.6274i − 1.26511i −0.774517 0.632553i $$-0.782007\pi$$
0.774517 0.632553i $$-0.217993\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 18.5858i 0.877117i 0.898703 + 0.438559i $$0.144511\pi$$
−0.898703 + 0.438559i $$0.855489\pi$$
$$450$$ 0 0
$$451$$ − 28.6274i − 1.34801i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 2.48528i − 0.116257i −0.998309 0.0581283i $$-0.981487\pi$$
0.998309 0.0581283i $$-0.0185132\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 24.8284i − 1.15638i −0.815904 0.578188i $$-0.803760\pi$$
0.815904 0.578188i $$-0.196240\pi$$
$$462$$ 0 0
$$463$$ −17.0711 −0.793360 −0.396680 0.917957i $$-0.629838\pi$$
−0.396680 + 0.917957i $$0.629838\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 31.1127i − 1.43972i −0.694117 0.719862i $$-0.744205\pi$$
0.694117 0.719862i $$-0.255795\pi$$
$$468$$ 0 0
$$469$$ 27.3137 1.26123
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −17.6569 −0.811863
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −27.3137 −1.24800 −0.623998 0.781426i $$-0.714493\pi$$
−0.623998 + 0.781426i $$0.714493\pi$$
$$480$$ 0 0
$$481$$ 30.9706 1.41214
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 10.7279 0.486129 0.243064 0.970010i $$-0.421847\pi$$
0.243064 + 0.970010i $$0.421847\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −31.0711 −1.40222 −0.701109 0.713054i $$-0.747311\pi$$
−0.701109 + 0.713054i $$0.747311\pi$$
$$492$$ 0 0
$$493$$ − 7.02944i − 0.316590i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −19.3137 −0.866338
$$498$$ 0 0
$$499$$ 18.6274i 0.833878i 0.908935 + 0.416939i $$0.136897\pi$$
−0.908935 + 0.416939i $$0.863103\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 36.2843i 1.61784i 0.587922 + 0.808918i $$0.299946\pi$$
−0.587922 + 0.808918i $$0.700054\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 8.14214i − 0.360894i −0.983585 0.180447i $$-0.942246\pi$$
0.983585 0.180447i $$-0.0577544\pi$$
$$510$$ 0 0
$$511$$ 35.7990i 1.58365i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 14.6274i 0.643313i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0.727922i 0.0318908i 0.999873 + 0.0159454i $$0.00507580\pi$$
−0.999873 + 0.0159454i $$0.994924\pi$$
$$522$$ 0 0
$$523$$ −7.51472 −0.328596 −0.164298 0.986411i $$-0.552536\pi$$
−0.164298 + 0.986411i $$0.552536\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 7.59798i 0.330973i
$$528$$ 0 0
$$529$$ −0.313708 −0.0136395
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −37.7990 −1.63726
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 12.0416 0.518670
$$540$$ 0 0
$$541$$ 28.8284 1.23943 0.619715 0.784827i $$-0.287248\pi$$
0.619715 + 0.784827i $$0.287248\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −16.4853 −0.704860 −0.352430 0.935838i $$-0.614644\pi$$
−0.352430 + 0.935838i $$0.614644\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ − 49.4558i − 2.10308i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 20.4853 0.867989 0.433995 0.900915i $$-0.357104\pi$$
0.433995 + 0.900915i $$0.357104\pi$$
$$558$$ 0 0
$$559$$ 23.3137i 0.986065i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 6.34315i − 0.267332i −0.991026 0.133666i $$-0.957325\pi$$
0.991026 0.133666i $$-0.0426749\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 9.21320i 0.386238i 0.981175 + 0.193119i $$0.0618603\pi$$
−0.981175 + 0.193119i $$0.938140\pi$$
$$570$$ 0 0
$$571$$ − 4.97056i − 0.208012i −0.994577 0.104006i $$-0.966834\pi$$
0.994577 0.104006i $$-0.0331660\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 15.4558i − 0.643435i −0.946836 0.321718i $$-0.895740\pi$$
0.946836 0.321718i $$-0.104260\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 31.3137i − 1.29911i
$$582$$ 0 0
$$583$$ 3.02944 0.125466
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 26.1421i − 1.07900i −0.841985 0.539501i $$-0.818613\pi$$
0.841985 0.539501i $$-0.181387\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −10.3431 −0.424742 −0.212371 0.977189i $$-0.568119\pi$$
−0.212371 + 0.977189i $$0.568119\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −42.1421 −1.72188 −0.860940 0.508706i $$-0.830124\pi$$
−0.860940 + 0.508706i $$0.830124\pi$$
$$600$$ 0 0
$$601$$ 16.6863 0.680648 0.340324 0.940308i $$-0.389463\pi$$
0.340324 + 0.940308i $$0.389463\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −3.21320 −0.130420 −0.0652100 0.997872i $$-0.520772\pi$$
−0.0652100 + 0.997872i $$0.520772\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 19.3137 0.781349
$$612$$ 0 0
$$613$$ − 3.21320i − 0.129780i −0.997892 0.0648900i $$-0.979330\pi$$
0.997892 0.0648900i $$-0.0206697\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −26.8284 −1.08007 −0.540036 0.841642i $$-0.681589\pi$$
−0.540036 + 0.841642i $$0.681589\pi$$
$$618$$ 0 0
$$619$$ 22.9706i 0.923265i 0.887071 + 0.461632i $$0.152736\pi$$
−0.887071 + 0.461632i $$0.847264\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 14.4853i 0.580341i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 10.6274i − 0.423743i
$$630$$ 0 0
$$631$$ − 7.17157i − 0.285496i −0.989759 0.142748i $$-0.954406\pi$$
0.989759 0.142748i $$-0.0455938\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 15.8995i − 0.629961i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 43.5563i − 1.72037i −0.509980 0.860186i $$-0.670347\pi$$
0.509980 0.860186i $$-0.329653\pi$$
$$642$$ 0 0
$$643$$ 35.1127 1.38471 0.692355 0.721557i $$-0.256573\pi$$
0.692355 + 0.721557i $$0.256573\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 12.6863i − 0.498750i −0.968407 0.249375i $$-0.919775\pi$$
0.968407 0.249375i $$-0.0802251\pi$$
$$648$$ 0 0
$$649$$ −17.0294 −0.668464
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 19.6569 0.769232 0.384616 0.923077i $$-0.374334\pi$$
0.384616 + 0.923077i $$0.374334\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −6.58579 −0.256546 −0.128273 0.991739i $$-0.540943\pi$$
−0.128273 + 0.991739i $$0.540943\pi$$
$$660$$ 0 0
$$661$$ −40.4264 −1.57240 −0.786202 0.617969i $$-0.787956\pi$$
−0.786202 + 0.617969i $$0.787956\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −28.9706 −1.12174
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 33.1716 1.28057
$$672$$ 0 0
$$673$$ 51.4558i 1.98348i 0.128276 + 0.991739i $$0.459056\pi$$
−0.128276 + 0.991739i $$0.540944\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6.68629 0.256975 0.128488 0.991711i $$-0.458988\pi$$
0.128488 + 0.991711i $$0.458988\pi$$
$$678$$ 0 0
$$679$$ 8.48528i 0.325635i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 10.8284i 0.414338i 0.978305 + 0.207169i $$0.0664250\pi$$
−0.978305 + 0.207169i $$0.933575\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ − 4.00000i − 0.152388i
$$690$$ 0 0
$$691$$ − 8.00000i − 0.304334i −0.988355 0.152167i $$-0.951375\pi$$
0.988355 0.152167i $$-0.0486252\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 12.9706i 0.491295i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 9.51472i − 0.359366i −0.983725 0.179683i $$-0.942493\pi$$
0.983725 0.179683i $$-0.0575072\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 16.4853i 0.619993i
$$708$$ 0 0
$$709$$ −6.97056 −0.261785 −0.130892 0.991397i $$-0.541784\pi$$
−0.130892 + 0.991397i $$0.541784\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 31.3137 1.17271
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 10.8284 0.403832 0.201916 0.979403i $$-0.435283\pi$$
0.201916 + 0.979403i $$0.435283\pi$$
$$720$$ 0 0
$$721$$ −59.9411 −2.23232
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 26.9289 0.998739 0.499369 0.866389i $$-0.333565\pi$$
0.499369 + 0.866389i $$0.333565\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ − 12.1005i − 0.446942i −0.974711 0.223471i $$-0.928261\pi$$
0.974711 0.223471i $$-0.0717389\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −20.6863 −0.761989
$$738$$ 0 0
$$739$$ − 22.3431i − 0.821906i −0.911657 0.410953i $$-0.865196\pi$$
0.911657 0.410953i $$-0.134804\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 48.0000i 1.76095i 0.474093 + 0.880475i $$0.342776\pi$$
−0.474093 + 0.880475i $$0.657224\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 50.6274i 1.84989i
$$750$$ 0 0
$$751$$ − 28.8284i − 1.05196i −0.850496 0.525982i $$-0.823698\pi$$
0.850496 0.525982i $$-0.176302\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 52.8701i 1.92159i 0.277251 + 0.960797i $$0.410577\pi$$
−0.277251 + 0.960797i $$0.589423\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30.8701i 1.11904i 0.828817 + 0.559519i $$0.189014\pi$$
−0.828817 + 0.559519i $$0.810986\pi$$
$$762$$ 0 0
$$763$$ −24.4853 −0.886427
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 22.4853i 0.811896i
$$768$$ 0 0
$$769$$ 2.34315 0.0844960 0.0422480 0.999107i $$-0.486548\pi$$
0.0422480 + 0.999107i $$0.486548\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 41.3137 1.48595 0.742975 0.669319i $$-0.233414\pi$$
0.742975 + 0.669319i $$0.233414\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 14.6274 0.523410
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −42.1421 −1.50220 −0.751102 0.660186i $$-0.770478\pi$$
−0.751102 + 0.660186i $$0.770478\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 5.65685 0.201135
$$792$$ 0 0
$$793$$ − 43.7990i − 1.55535i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −18.8284 −0.666937 −0.333469 0.942761i $$-0.608219\pi$$
−0.333469 + 0.942761i $$0.608219\pi$$
$$798$$ 0 0
$$799$$ − 6.62742i − 0.234461i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 27.1127i − 0.956786i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ − 36.0416i − 1.26716i −0.773679 0.633578i $$-0.781586\pi$$
0.773679 0.633578i $$-0.218414\pi$$
$$810$$ 0 0
$$811$$ 3.37258i 0.118427i 0.998245 + 0.0592137i $$0.0188593\pi$$
−0.998245 + 0.0592137i $$0.981141\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 29.3137i 1.02306i 0.859267 + 0.511528i $$0.170920\pi$$
−0.859267 + 0.511528i $$0.829080\pi$$
$$822$$ 0 0
$$823$$ −6.24264 −0.217605 −0.108802 0.994063i $$-0.534702\pi$$
−0.108802 + 0.994063i $$0.534702\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 42.6274i − 1.48230i −0.671339 0.741150i $$-0.734281\pi$$
0.671339 0.741150i $$-0.265719\pi$$
$$828$$ 0 0
$$829$$ 19.4558 0.675729 0.337865 0.941195i $$-0.390295\pi$$
0.337865 + 0.941195i $$0.390295\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −5.45584 −0.189034
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −42.4264 −1.46472 −0.732361 0.680916i $$-0.761582\pi$$
−0.732361 + 0.680916i $$0.761582\pi$$
$$840$$ 0 0
$$841$$ −7.00000 −0.241379
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 14.7279 0.506057
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −43.7990 −1.50141
$$852$$ 0 0
$$853$$ 21.7574i 0.744958i 0.928041 + 0.372479i $$0.121492\pi$$
−0.928041 + 0.372479i $$0.878508\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 35.7990 1.22287 0.611435 0.791295i $$-0.290593\pi$$
0.611435 + 0.791295i $$0.290593\pi$$
$$858$$ 0 0
$$859$$ − 26.0000i − 0.887109i −0.896248 0.443554i $$-0.853717\pi$$
0.896248 0.443554i $$-0.146283\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 6.20101i − 0.211085i −0.994415 0.105542i $$-0.966342\pi$$
0.994415 0.105542i $$-0.0336579\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 37.4558i 1.27060i
$$870$$ 0 0
$$871$$ 27.3137i 0.925490i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 1.55635i 0.0525542i 0.999655 + 0.0262771i $$0.00836522\pi$$
−0.999655 + 0.0262771i $$0.991635\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ − 35.3553i − 1.19115i −0.803299 0.595576i $$-0.796924\pi$$
0.803299 0.595576i $$-0.203076\pi$$
$$882$$ 0 0
$$883$$ 42.4264 1.42776 0.713881 0.700267i $$-0.246936\pi$$
0.713881 + 0.700267i $$0.246936\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 12.8284i 0.430736i 0.976533 + 0.215368i $$0.0690952\pi$$
−0.976533 + 0.215368i $$0.930905\pi$$
$$888$$ 0 0
$$889$$ 36.6274 1.22844
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 38.9117 1.29778
$$900$$ 0 0
$$901$$ −1.37258 −0.0457274
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −1.85786 −0.0616894 −0.0308447 0.999524i $$-0.509820\pi$$
−0.0308447 + 0.999524i $$0.509820\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 18.3431 0.607736 0.303868 0.952714i $$-0.401722\pi$$
0.303868 + 0.952714i $$0.401722\pi$$
$$912$$ 0 0
$$913$$ 23.7157i 0.784876i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −43.4558 −1.43504
$$918$$ 0 0
$$919$$ 12.8284i 0.423171i 0.977360 + 0.211585i $$0.0678626\pi$$
−0.977360 + 0.211585i $$0.932137\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ − 19.3137i − 0.635718i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 45.8995i 1.50591i 0.658070 + 0.752957i $$0.271373\pi$$
−0.658070 + 0.752957i $$0.728627\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 14.9706i 0.489067i 0.969641 + 0.244533i $$0.0786348\pi$$
−0.969641 + 0.244533i $$0.921365\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 42.2843i 1.37843i 0.724558 + 0.689214i $$0.242044\pi$$
−0.724558 + 0.689214i $$0.757956\pi$$
$$942$$ 0 0
$$943$$ 53.4558 1.74076
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 51.5980i − 1.67671i −0.545125 0.838355i $$-0.683518\pi$$
0.545125 0.838355i $$-0.316482\pi$$
$$948$$ 0 0
$$949$$ −35.7990 −1.16208
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 45.2548 1.46595 0.732974 0.680257i $$-0.238132\pi$$
0.732974 + 0.680257i $$0.238132\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −65.9411 −2.12935
$$960$$ 0 0
$$961$$ −11.0589 −0.356738
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 26.7279 0.859512 0.429756 0.902945i $$-0.358600\pi$$
0.429756 + 0.902945i $$0.358600\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 27.0711 0.868752 0.434376 0.900732i $$-0.356969\pi$$
0.434376 + 0.900732i $$0.356969\pi$$
$$972$$ 0 0
$$973$$ 12.4853i 0.400260i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 37.1716 1.18922 0.594612 0.804013i $$-0.297306\pi$$
0.594612 + 0.804013i $$0.297306\pi$$
$$978$$ 0 0
$$979$$ − 10.9706i − 0.350621i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ − 39.5980i − 1.26298i −0.775384 0.631490i $$-0.782444\pi$$
0.775384 0.631490i $$-0.217556\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ − 32.9706i − 1.04840i
$$990$$ 0 0
$$991$$ − 16.3431i − 0.519157i −0.965722 0.259579i $$-0.916416\pi$$
0.965722 0.259579i $$-0.0835837\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 5.27208i − 0.166968i −0.996509 0.0834842i $$-0.973395\pi$$
0.996509 0.0834842i $$-0.0266048\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7200.2.o.e.7199.1 4
3.2 odd 2 7200.2.o.b.7199.1 4
4.3 odd 2 7200.2.o.j.7199.3 4
5.2 odd 4 1440.2.h.d.1151.3 yes 4
5.3 odd 4 7200.2.h.i.1151.4 4
5.4 even 2 7200.2.o.m.7199.4 4
12.11 even 2 7200.2.o.m.7199.3 4
15.2 even 4 1440.2.h.a.1151.1 4
15.8 even 4 7200.2.h.c.1151.4 4
15.14 odd 2 7200.2.o.j.7199.4 4
20.3 even 4 7200.2.h.c.1151.1 4
20.7 even 4 1440.2.h.a.1151.4 yes 4
20.19 odd 2 7200.2.o.b.7199.2 4
40.27 even 4 2880.2.h.d.1151.2 4
40.37 odd 4 2880.2.h.a.1151.1 4
60.23 odd 4 7200.2.h.i.1151.1 4
60.47 odd 4 1440.2.h.d.1151.2 yes 4
60.59 even 2 inner 7200.2.o.e.7199.2 4
120.77 even 4 2880.2.h.d.1151.3 4
120.107 odd 4 2880.2.h.a.1151.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1440.2.h.a.1151.1 4 15.2 even 4
1440.2.h.a.1151.4 yes 4 20.7 even 4
1440.2.h.d.1151.2 yes 4 60.47 odd 4
1440.2.h.d.1151.3 yes 4 5.2 odd 4
2880.2.h.a.1151.1 4 40.37 odd 4
2880.2.h.a.1151.4 4 120.107 odd 4
2880.2.h.d.1151.2 4 40.27 even 4
2880.2.h.d.1151.3 4 120.77 even 4
7200.2.h.c.1151.1 4 20.3 even 4
7200.2.h.c.1151.4 4 15.8 even 4
7200.2.h.i.1151.1 4 60.23 odd 4
7200.2.h.i.1151.4 4 5.3 odd 4
7200.2.o.b.7199.1 4 3.2 odd 2
7200.2.o.b.7199.2 4 20.19 odd 2
7200.2.o.e.7199.1 4 1.1 even 1 trivial
7200.2.o.e.7199.2 4 60.59 even 2 inner
7200.2.o.j.7199.3 4 4.3 odd 2
7200.2.o.j.7199.4 4 15.14 odd 2
7200.2.o.m.7199.3 4 12.11 even 2
7200.2.o.m.7199.4 4 5.4 even 2