# Properties

 Label 3150.2.g.l.2899.2 Level 3150 Weight 2 Character 3150.2899 Analytic conductor 25.153 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 630) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2899.2 Root $$1.00000i$$ Character $$\chi$$ = 3150.2899 Dual form 3150.2.g.l.2899.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} -2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} +2.00000 q^{26} -1.00000i q^{28} +6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} -4.00000i q^{37} -2.00000i q^{38} -6.00000 q^{41} -2.00000i q^{43} +6.00000i q^{47} -1.00000 q^{49} +2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{56} +6.00000i q^{58} +12.0000 q^{59} +8.00000 q^{61} +8.00000i q^{62} -1.00000 q^{64} +2.00000i q^{67} -6.00000 q^{71} -2.00000i q^{73} +4.00000 q^{74} +2.00000 q^{76} +16.0000 q^{79} -6.00000i q^{82} +2.00000 q^{86} +6.00000 q^{89} +2.00000 q^{91} -6.00000 q^{94} -10.0000i q^{97} -1.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{14} + 2q^{16} - 4q^{19} + 4q^{26} + 12q^{29} + 16q^{31} - 12q^{41} - 2q^{49} + 2q^{56} + 24q^{59} + 16q^{61} - 2q^{64} - 12q^{71} + 8q^{74} + 4q^{76} + 32q^{79} + 4q^{86} + 12q^{89} + 4q^{91} - 12q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i
$$8$$ − 1.00000i − 0.353553i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ − 1.00000i − 0.188982i
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 4.00000i − 0.657596i −0.944400 0.328798i $$-0.893356\pi$$
0.944400 0.328798i $$-0.106644\pi$$
$$38$$ − 2.00000i − 0.324443i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ − 2.00000i − 0.304997i −0.988304 0.152499i $$-0.951268\pi$$
0.988304 0.152499i $$-0.0487319\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.00000i 0.875190i 0.899172 + 0.437595i $$0.144170\pi$$
−0.899172 + 0.437595i $$0.855830\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 2.00000i 0.277350i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 6.00000i 0.787839i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.00000i 0.244339i 0.992509 + 0.122169i $$0.0389851\pi$$
−0.992509 + 0.122169i $$0.961015\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 6.00000i − 0.662589i
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ − 1.00000i − 0.101015i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000i 0.0944911i
$$113$$ 18.0000i 1.69330i 0.532152 + 0.846649i $$0.321383\pi$$
−0.532152 + 0.846649i $$0.678617\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ 12.0000i 1.10469i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 8.00000i 0.724286i
$$123$$ 0 0
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 16.0000i − 1.41977i −0.704317 0.709885i $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ − 2.00000i − 0.173422i
$$134$$ −2.00000 −0.172774
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.0000i 1.53784i 0.639343 + 0.768922i $$0.279207\pi$$
−0.639343 + 0.768922i $$0.720793\pi$$
$$138$$ 0 0
$$139$$ 10.0000 0.848189 0.424094 0.905618i $$-0.360592\pi$$
0.424094 + 0.905618i $$0.360592\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 6.00000i − 0.503509i
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ 0 0
$$148$$ 4.00000i 0.328798i
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 2.00000i 0.162221i
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ 16.0000i 1.27289i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 22.0000i 1.72317i 0.507611 + 0.861586i $$0.330529\pi$$
−0.507611 + 0.861586i $$0.669471\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 18.0000i 1.39288i 0.717614 + 0.696441i $$0.245234\pi$$
−0.717614 + 0.696441i $$0.754766\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 2.00000i 0.152499i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 6.00000i 0.449719i
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ 2.00000i 0.148250i
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ − 6.00000i − 0.437595i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 0 0
$$193$$ − 14.0000i − 1.00774i −0.863779 0.503871i $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 6.00000i − 0.422159i
$$203$$ 6.00000i 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ − 2.00000i − 0.138675i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 8.00000i 0.543075i
$$218$$ 10.0000i 0.677285i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −18.0000 −1.19734
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 0 0
$$229$$ −20.0000 −1.32164 −0.660819 0.750546i $$-0.729791\pi$$
−0.660819 + 0.750546i $$0.729791\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ 0 0
$$244$$ −8.00000 −0.512148
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.00000i 0.254514i
$$248$$ − 8.00000i − 0.508001i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 12.0000i 0.748539i 0.927320 + 0.374270i $$0.122107\pi$$
−0.927320 + 0.374270i $$0.877893\pi$$
$$258$$ 0 0
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 12.0000i 0.741362i
$$263$$ − 12.0000i − 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ 0 0
$$268$$ − 2.00000i − 0.122169i
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 32.0000i 1.92269i 0.275340 + 0.961347i $$0.411209\pi$$
−0.275340 + 0.961347i $$0.588791\pi$$
$$278$$ 10.0000i 0.599760i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ − 20.0000i − 1.18888i −0.804141 0.594438i $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 6.00000i − 0.354169i
$$288$$ 0 0
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.00000i 0.117041i
$$293$$ 30.0000i 1.75262i 0.481749 + 0.876309i $$0.340002\pi$$
−0.481749 + 0.876309i $$0.659998\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −4.00000 −0.232495
$$297$$ 0 0
$$298$$ 18.0000i 1.04271i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 2.00000 0.115278
$$302$$ 8.00000i 0.460348i
$$303$$ 0 0
$$304$$ −2.00000 −0.114708
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 4.00000i − 0.228292i −0.993464 0.114146i $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ − 26.0000i − 1.46961i −0.678280 0.734803i $$-0.737274\pi$$
0.678280 0.734803i $$-0.262726\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ −16.0000 −0.900070
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −22.0000 −1.21847
$$327$$ 0 0
$$328$$ 6.00000i 0.331295i
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ −18.0000 −0.984916
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 1.00000i − 0.0539949i
$$344$$ −2.00000 −0.107833
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ 12.0000i 0.644194i 0.946707 + 0.322097i $$0.104388\pi$$
−0.946707 + 0.322097i $$0.895612\pi$$
$$348$$ 0 0
$$349$$ 4.00000 0.214115 0.107058 0.994253i $$-0.465857\pi$$
0.107058 + 0.994253i $$0.465857\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 36.0000i − 1.91609i −0.286623 0.958043i $$-0.592533\pi$$
0.286623 0.958043i $$-0.407467\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ − 12.0000i − 0.634220i
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ − 16.0000i − 0.840941i
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8.00000i 0.417597i 0.977959 + 0.208798i $$0.0669552\pi$$
−0.977959 + 0.208798i $$0.933045\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ 4.00000i 0.207112i 0.994624 + 0.103556i $$0.0330221\pi$$
−0.994624 + 0.103556i $$0.966978\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ − 12.0000i − 0.618031i
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 18.0000i 0.920960i
$$383$$ − 6.00000i − 0.306586i −0.988181 0.153293i $$-0.951012\pi$$
0.988181 0.153293i $$-0.0489878\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 0 0
$$388$$ 10.0000i 0.507673i
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 1.00000i 0.0505076i
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 22.0000i − 1.10415i −0.833795 0.552074i $$-0.813837\pi$$
0.833795 0.552074i $$-0.186163\pi$$
$$398$$ 4.00000i 0.200502i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ − 16.0000i − 0.797017i
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ −6.00000 −0.297775
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 8.00000i 0.394132i
$$413$$ 12.0000i 0.590481i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 8.00000i 0.389434i
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 8.00000i 0.387147i
$$428$$ − 12.0000i − 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −6.00000 −0.289010 −0.144505 0.989504i $$-0.546159\pi$$
−0.144505 + 0.989504i $$0.546159\pi$$
$$432$$ 0 0
$$433$$ − 14.0000i − 0.672797i −0.941720 0.336399i $$-0.890791\pi$$
0.941720 0.336399i $$-0.109209\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 36.0000i − 1.71041i −0.518289 0.855206i $$-0.673431\pi$$
0.518289 0.855206i $$-0.326569\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ 0 0
$$448$$ − 1.00000i − 0.0472456i
$$449$$ −36.0000 −1.69895 −0.849473 0.527633i $$-0.823080\pi$$
−0.849473 + 0.527633i $$0.823080\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 18.0000i − 0.846649i
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 26.0000i 1.21623i 0.793849 + 0.608114i $$0.208074\pi$$
−0.793849 + 0.608114i $$0.791926\pi$$
$$458$$ − 20.0000i − 0.934539i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ − 24.0000i − 1.11059i −0.831654 0.555294i $$-0.812606\pi$$
0.831654 0.555294i $$-0.187394\pi$$
$$468$$ 0 0
$$469$$ −2.00000 −0.0923514
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 12.0000i − 0.552345i
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ − 6.00000i − 0.274434i
$$479$$ 36.0000 1.64488 0.822441 0.568850i $$-0.192612\pi$$
0.822441 + 0.568850i $$0.192612\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 2.00000i 0.0910975i
$$483$$ 0 0
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 16.0000i − 0.725029i −0.931978 0.362515i $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ − 8.00000i − 0.362143i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ − 6.00000i − 0.269137i
$$498$$ 0 0
$$499$$ 16.0000 0.716258 0.358129 0.933672i $$-0.383415\pi$$
0.358129 + 0.933672i $$0.383415\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 12.0000i − 0.535586i
$$503$$ 6.00000i 0.267527i 0.991013 + 0.133763i $$0.0427062\pi$$
−0.991013 + 0.133763i $$0.957294\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 16.0000i 0.709885i
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 2.00000 0.0884748
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ −12.0000 −0.529297
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 4.00000i 0.175750i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ 16.0000i 0.699631i 0.936819 + 0.349816i $$0.113756\pi$$
−0.936819 + 0.349816i $$0.886244\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.00000i 0.0867110i
$$533$$ 12.0000i 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 2.00000 0.0863868
$$537$$ 0 0
$$538$$ 30.0000i 1.29339i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 8.00000i 0.343629i
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 22.0000i − 0.940652i −0.882493 0.470326i $$-0.844136\pi$$
0.882493 0.470326i $$-0.155864\pi$$
$$548$$ − 18.0000i − 0.768922i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ 16.0000i 0.680389i
$$554$$ −32.0000 −1.35955
$$555$$ 0 0
$$556$$ −10.0000 −0.424094
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ 0 0
$$568$$ 6.00000i 0.251754i
$$569$$ 24.0000 1.00613 0.503066 0.864248i $$-0.332205\pi$$
0.503066 + 0.864248i $$0.332205\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 6.00000 0.250435
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ 17.0000i 0.707107i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ −30.0000 −1.23929
$$587$$ − 36.0000i − 1.48588i −0.669359 0.742940i $$-0.733431\pi$$
0.669359 0.742940i $$-0.266569\pi$$
$$588$$ 0 0
$$589$$ −16.0000 −0.659269
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 4.00000i − 0.164399i
$$593$$ − 24.0000i − 0.985562i −0.870153 0.492781i $$-0.835980\pi$$
0.870153 0.492781i $$-0.164020\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −42.0000 −1.71607 −0.858037 0.513588i $$-0.828316\pi$$
−0.858037 + 0.513588i $$0.828316\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 2.00000i 0.0815139i
$$603$$ 0 0
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 40.0000i − 1.62355i −0.583970 0.811775i $$-0.698502\pi$$
0.583970 0.811775i $$-0.301498\pi$$
$$608$$ − 2.00000i − 0.0811107i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ 28.0000i 1.13091i 0.824779 + 0.565455i $$0.191299\pi$$
−0.824779 + 0.565455i $$0.808701\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 18.0000i − 0.724653i −0.932051 0.362326i $$-0.881983\pi$$
0.932051 0.362326i $$-0.118017\pi$$
$$618$$ 0 0
$$619$$ 46.0000 1.84890 0.924448 0.381308i $$-0.124526\pi$$
0.924448 + 0.381308i $$0.124526\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 12.0000i 0.481156i
$$623$$ 6.00000i 0.240385i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 26.0000 1.03917
$$627$$ 0 0
$$628$$ − 14.0000i − 0.558661i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ − 16.0000i − 0.636446i
$$633$$ 0 0
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.00000i 0.0792429i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 12.0000 0.473972 0.236986 0.971513i $$-0.423841\pi$$
0.236986 + 0.971513i $$0.423841\pi$$
$$642$$ 0 0
$$643$$ − 32.0000i − 1.26196i −0.775800 0.630978i $$-0.782654\pi$$
0.775800 0.630978i $$-0.217346\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 42.0000i − 1.65119i −0.564263 0.825595i $$-0.690840\pi$$
0.564263 0.825595i $$-0.309160\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 22.0000i − 0.861586i
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 0 0
$$658$$ − 6.00000i − 0.233904i
$$659$$ −48.0000 −1.86981 −0.934907 0.354892i $$-0.884518\pi$$
−0.934907 + 0.354892i $$0.884518\pi$$
$$660$$ 0 0
$$661$$ −40.0000 −1.55582 −0.777910 0.628376i $$-0.783720\pi$$
−0.777910 + 0.628376i $$0.783720\pi$$
$$662$$ 8.00000i 0.310929i
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ − 18.0000i − 0.696441i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 10.0000i 0.385472i 0.981251 + 0.192736i $$0.0617360\pi$$
−0.981251 + 0.192736i $$0.938264\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 0 0
$$679$$ 10.0000 0.383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ − 2.00000i − 0.0762493i
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 26.0000 0.989087 0.494543 0.869153i $$-0.335335\pi$$
0.494543 + 0.869153i $$0.335335\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 4.00000i 0.151402i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ 8.00000i 0.301726i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 36.0000 1.35488
$$707$$ − 6.00000i − 0.225653i
$$708$$ 0 0
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 6.00000i − 0.224860i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ − 6.00000i − 0.223918i
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ − 15.0000i − 0.558242i
$$723$$ 0 0
$$724$$ 16.0000 0.594635
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 40.0000i − 1.48352i −0.670667 0.741759i $$-0.733992\pi$$
0.670667 0.741759i $$-0.266008\pi$$
$$728$$ − 2.00000i − 0.0741249i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 22.0000i 0.812589i 0.913742 + 0.406294i $$0.133179\pi$$
−0.913742 + 0.406294i $$0.866821\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 6.00000i − 0.220267i
$$743$$ 48.0000i 1.76095i 0.474093 + 0.880475i $$0.342776\pi$$
−0.474093 + 0.880475i $$0.657224\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −4.00000 −0.146450
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 6.00000i 0.218797i
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 20.0000i 0.726912i 0.931611 + 0.363456i $$0.118403\pi$$
−0.931611 + 0.363456i $$0.881597\pi$$
$$758$$ − 8.00000i − 0.290573i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 54.0000 1.95750 0.978749 0.205061i $$-0.0657392\pi$$
0.978749 + 0.205061i $$0.0657392\pi$$
$$762$$ 0 0
$$763$$ 10.0000i 0.362024i
$$764$$ −18.0000 −0.651217
$$765$$ 0 0
$$766$$ 6.00000 0.216789
$$767$$ − 24.0000i − 0.866590i
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 14.0000i 0.503871i
$$773$$ − 42.0000i − 1.51064i −0.655359 0.755318i $$-0.727483\pi$$
0.655359 0.755318i $$-0.272517\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ 0 0
$$778$$ − 30.0000i − 1.07555i
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ 6.00000i 0.213741i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −18.0000 −0.640006
$$792$$ 0 0
$$793$$ − 16.0000i − 0.568177i
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ − 18.0000i − 0.637593i −0.947823 0.318796i $$-0.896721\pi$$
0.947823 0.318796i $$-0.103279\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 16.0000 0.563576
$$807$$ 0 0
$$808$$ 6.00000i 0.211079i
$$809$$ −48.0000 −1.68759 −0.843795 0.536666i $$-0.819684\pi$$
−0.843795 + 0.536666i $$0.819684\pi$$
$$810$$ 0 0
$$811$$ 38.0000 1.33436 0.667180 0.744896i $$-0.267501\pi$$
0.667180 + 0.744896i $$0.267501\pi$$
$$812$$ − 6.00000i − 0.210559i
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 4.00000i 0.139942i
$$818$$ − 2.00000i − 0.0699284i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 0 0
$$823$$ 28.0000i 0.976019i 0.872838 + 0.488009i $$0.162277\pi$$
−0.872838 + 0.488009i $$0.837723\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 0 0
$$829$$ −20.0000 −0.694629 −0.347314 0.937749i $$-0.612906\pi$$
−0.347314 + 0.937749i $$0.612906\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 2.00000i 0.0693375i
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ − 12.0000i − 0.414533i
$$839$$ −48.0000 −1.65714 −0.828572 0.559883i $$-0.810846\pi$$
−0.828572 + 0.559883i $$0.810846\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 10.0000i − 0.344623i
$$843$$ 0 0
$$844$$ −8.00000 −0.275371
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 11.0000i − 0.377964i
$$848$$ 6.00000i 0.206041i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 26.0000i − 0.890223i −0.895475 0.445112i $$-0.853164\pi$$
0.895475 0.445112i $$-0.146836\pi$$
$$854$$ −8.00000 −0.273754
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 12.0000i 0.409912i 0.978771 + 0.204956i $$0.0657052\pi$$
−0.978771 + 0.204956i $$0.934295\pi$$
$$858$$ 0 0
$$859$$ −2.00000 −0.0682391 −0.0341196 0.999418i $$-0.510863\pi$$
−0.0341196 + 0.999418i $$0.510863\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 6.00000i − 0.204361i
$$863$$ − 36.0000i − 1.22545i −0.790295 0.612727i $$-0.790072\pi$$
0.790295 0.612727i $$-0.209928\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 14.0000 0.475739
$$867$$ 0 0
$$868$$ − 8.00000i − 0.271538i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ − 10.0000i − 0.338643i
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 32.0000i 1.08056i 0.841484 + 0.540282i $$0.181682\pi$$
−0.841484 + 0.540282i $$0.818318\pi$$
$$878$$ − 8.00000i − 0.269987i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 0 0
$$883$$ − 2.00000i − 0.0673054i −0.999434 0.0336527i $$-0.989286\pi$$
0.999434 0.0336527i $$-0.0107140\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ 18.0000i 0.604381i 0.953248 + 0.302190i $$0.0977178\pi$$
−0.953248 + 0.302190i $$0.902282\pi$$
$$888$$ 0 0
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 8.00000i 0.267860i
$$893$$ − 12.0000i − 0.401565i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ − 36.0000i − 1.20134i
$$899$$ 48.0000 1.60089
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 2.00000i 0.0664089i 0.999449 + 0.0332045i $$0.0105712\pi$$
−0.999449 + 0.0332045i $$0.989429\pi$$
$$908$$ − 12.0000i − 0.398234i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −26.0000 −0.860004
$$915$$ 0 0
$$916$$ 20.0000 0.660819
$$917$$ 12.0000i 0.396275i
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 30.0000i − 0.987997i
$$923$$ 12.0000i 0.394985i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −4.00000 −0.131448
$$927$$ 0 0
$$928$$ 6.00000i 0.196960i
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ 6.00000i 0.196537i
$$933$$ 0 0
$$934$$ 24.0000 0.785304
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 38.0000i 1.24141i 0.784046 + 0.620703i $$0.213153\pi$$
−0.784046 + 0.620703i $$0.786847\pi$$
$$938$$ − 2.00000i − 0.0653023i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 42.0000 1.36916 0.684580 0.728937i $$-0.259985\pi$$
0.684580 + 0.728937i $$0.259985\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 12.0000i 0.389948i 0.980808 + 0.194974i $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ 0 0
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 6.00000i − 0.194359i −0.995267 0.0971795i $$-0.969018\pi$$
0.995267 0.0971795i $$-0.0309821\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 6.00000 0.194054
$$957$$ 0 0
$$958$$ 36.0000i 1.16311i
$$959$$ −18.0000 −0.581250
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 8.00000i − 0.257930i
$$963$$ 0 0
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.00000i 0.257263i 0.991692 + 0.128631i $$0.0410584\pi$$
−0.991692 + 0.128631i $$0.958942\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 0 0
$$973$$ 10.0000i 0.320585i
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ 6.00000i 0.191957i 0.995383 + 0.0959785i $$0.0305980\pi$$
−0.995383 + 0.0959785i $$0.969402\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 12.0000i 0.382935i
$$983$$ − 54.0000i − 1.72233i −0.508323 0.861166i $$-0.669735\pi$$
0.508323 0.861166i $$-0.330265\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ − 4.00000i − 0.127257i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ 0 0
$$994$$ 6.00000 0.190308
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 46.0000i − 1.45683i −0.685134 0.728417i $$-0.740256\pi$$
0.685134 0.728417i $$-0.259744\pi$$
$$998$$ 16.0000i 0.506471i
$$999$$ 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 3150.2.g.l.2899.2 2
3.2 odd 2 3150.2.g.m.2899.1 2
5.2 odd 4 3150.2.a.g.1.1 1
5.3 odd 4 630.2.a.j.1.1 yes 1
5.4 even 2 inner 3150.2.g.l.2899.1 2
15.2 even 4 3150.2.a.bb.1.1 1
15.8 even 4 630.2.a.c.1.1 1
15.14 odd 2 3150.2.g.m.2899.2 2
20.3 even 4 5040.2.a.z.1.1 1
35.13 even 4 4410.2.a.y.1.1 1
60.23 odd 4 5040.2.a.f.1.1 1
105.83 odd 4 4410.2.a.p.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.a.c.1.1 1 15.8 even 4
630.2.a.j.1.1 yes 1 5.3 odd 4
3150.2.a.g.1.1 1 5.2 odd 4
3150.2.a.bb.1.1 1 15.2 even 4
3150.2.g.l.2899.1 2 5.4 even 2 inner
3150.2.g.l.2899.2 2 1.1 even 1 trivial
3150.2.g.m.2899.1 2 3.2 odd 2
3150.2.g.m.2899.2 2 15.14 odd 2
4410.2.a.p.1.1 1 105.83 odd 4
4410.2.a.y.1.1 1 35.13 even 4
5040.2.a.f.1.1 1 60.23 odd 4
5040.2.a.z.1.1 1 20.3 even 4