# Properties

 Label 1900.2.c.e.1749.4 Level $1900$ Weight $2$ Character 1900.1749 Analytic conductor $15.172$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 380) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1749.4 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1900.1749 Dual form 1900.2.c.e.1749.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.73205i q^{3} -2.00000i q^{7} -4.46410 q^{9} +O(q^{10})$$ $$q+2.73205i q^{3} -2.00000i q^{7} -4.46410 q^{9} -3.46410 q^{11} -2.73205i q^{13} +3.46410i q^{17} -1.00000 q^{19} +5.46410 q^{21} -3.46410i q^{23} -4.00000i q^{27} -3.46410 q^{29} -1.46410 q^{31} -9.46410i q^{33} -6.73205i q^{37} +7.46410 q^{39} -6.00000 q^{41} -4.92820i q^{43} -12.9282i q^{47} +3.00000 q^{49} -9.46410 q^{51} -10.7321i q^{53} -2.73205i q^{57} -6.92820 q^{59} +12.3923 q^{61} +8.92820i q^{63} -6.73205i q^{67} +9.46410 q^{69} -2.53590 q^{71} -0.535898i q^{73} +6.92820i q^{77} -2.92820 q^{79} -2.46410 q^{81} +3.46410i q^{83} -9.46410i q^{87} +15.4641 q^{89} -5.46410 q^{91} -4.00000i q^{93} +16.5885i q^{97} +15.4641 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 4 q^{19} + 8 q^{21} + 8 q^{31} + 16 q^{39} - 24 q^{41} + 12 q^{49} - 24 q^{51} + 8 q^{61} + 24 q^{69} - 24 q^{71} + 16 q^{79} + 4 q^{81} + 48 q^{89} - 8 q^{91} + 48 q^{99}+O(q^{100})$$ 4 * q - 4 * q^9 - 4 * q^19 + 8 * q^21 + 8 * q^31 + 16 * q^39 - 24 * q^41 + 12 * q^49 - 24 * q^51 + 8 * q^61 + 24 * q^69 - 24 * q^71 + 16 * q^79 + 4 * q^81 + 48 * q^89 - 8 * q^91 + 48 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.73205i 1.57735i 0.614810 + 0.788675i $$0.289233\pi$$
−0.614810 + 0.788675i $$0.710767\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 0 0
$$9$$ −4.46410 −1.48803
$$10$$ 0 0
$$11$$ −3.46410 −1.04447 −0.522233 0.852803i $$-0.674901\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ 0 0
$$13$$ − 2.73205i − 0.757735i −0.925451 0.378867i $$-0.876314\pi$$
0.925451 0.378867i $$-0.123686\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.46410i 0.840168i 0.907485 + 0.420084i $$0.137999\pi$$
−0.907485 + 0.420084i $$0.862001\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 5.46410 1.19236
$$22$$ 0 0
$$23$$ − 3.46410i − 0.722315i −0.932505 0.361158i $$-0.882382\pi$$
0.932505 0.361158i $$-0.117618\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 4.00000i − 0.769800i
$$28$$ 0 0
$$29$$ −3.46410 −0.643268 −0.321634 0.946864i $$-0.604232\pi$$
−0.321634 + 0.946864i $$0.604232\pi$$
$$30$$ 0 0
$$31$$ −1.46410 −0.262960 −0.131480 0.991319i $$-0.541973\pi$$
−0.131480 + 0.991319i $$0.541973\pi$$
$$32$$ 0 0
$$33$$ − 9.46410i − 1.64749i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 6.73205i − 1.10674i −0.832935 0.553371i $$-0.813341\pi$$
0.832935 0.553371i $$-0.186659\pi$$
$$38$$ 0 0
$$39$$ 7.46410 1.19521
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ − 4.92820i − 0.751544i −0.926712 0.375772i $$-0.877378\pi$$
0.926712 0.375772i $$-0.122622\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 12.9282i − 1.88577i −0.333115 0.942886i $$-0.608100\pi$$
0.333115 0.942886i $$-0.391900\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −9.46410 −1.32524
$$52$$ 0 0
$$53$$ − 10.7321i − 1.47416i −0.675805 0.737080i $$-0.736204\pi$$
0.675805 0.737080i $$-0.263796\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 2.73205i − 0.361869i
$$58$$ 0 0
$$59$$ −6.92820 −0.901975 −0.450988 0.892530i $$-0.648928\pi$$
−0.450988 + 0.892530i $$0.648928\pi$$
$$60$$ 0 0
$$61$$ 12.3923 1.58667 0.793336 0.608784i $$-0.208342\pi$$
0.793336 + 0.608784i $$0.208342\pi$$
$$62$$ 0 0
$$63$$ 8.92820i 1.12485i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 6.73205i − 0.822451i −0.911534 0.411225i $$-0.865101\pi$$
0.911534 0.411225i $$-0.134899\pi$$
$$68$$ 0 0
$$69$$ 9.46410 1.13934
$$70$$ 0 0
$$71$$ −2.53590 −0.300956 −0.150478 0.988613i $$-0.548081\pi$$
−0.150478 + 0.988613i $$0.548081\pi$$
$$72$$ 0 0
$$73$$ − 0.535898i − 0.0627222i −0.999508 0.0313611i $$-0.990016\pi$$
0.999508 0.0313611i $$-0.00998418\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.92820i 0.789542i
$$78$$ 0 0
$$79$$ −2.92820 −0.329449 −0.164724 0.986340i $$-0.552673\pi$$
−0.164724 + 0.986340i $$0.552673\pi$$
$$80$$ 0 0
$$81$$ −2.46410 −0.273789
$$82$$ 0 0
$$83$$ 3.46410i 0.380235i 0.981761 + 0.190117i $$0.0608868\pi$$
−0.981761 + 0.190117i $$0.939113\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 9.46410i − 1.01466i
$$88$$ 0 0
$$89$$ 15.4641 1.63919 0.819596 0.572942i $$-0.194198\pi$$
0.819596 + 0.572942i $$0.194198\pi$$
$$90$$ 0 0
$$91$$ −5.46410 −0.572793
$$92$$ 0 0
$$93$$ − 4.00000i − 0.414781i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 16.5885i 1.68430i 0.539241 + 0.842151i $$0.318711\pi$$
−0.539241 + 0.842151i $$0.681289\pi$$
$$98$$ 0 0
$$99$$ 15.4641 1.55420
$$100$$ 0 0
$$101$$ −16.3923 −1.63110 −0.815548 0.578690i $$-0.803564\pi$$
−0.815548 + 0.578690i $$0.803564\pi$$
$$102$$ 0 0
$$103$$ 13.6603i 1.34598i 0.739649 + 0.672992i $$0.234991\pi$$
−0.739649 + 0.672992i $$0.765009\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 5.66025i 0.547197i 0.961844 + 0.273599i $$0.0882140\pi$$
−0.961844 + 0.273599i $$0.911786\pi$$
$$108$$ 0 0
$$109$$ 14.3923 1.37853 0.689266 0.724508i $$-0.257933\pi$$
0.689266 + 0.724508i $$0.257933\pi$$
$$110$$ 0 0
$$111$$ 18.3923 1.74572
$$112$$ 0 0
$$113$$ 12.5885i 1.18422i 0.805856 + 0.592111i $$0.201705\pi$$
−0.805856 + 0.592111i $$0.798295\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 12.1962i 1.12753i
$$118$$ 0 0
$$119$$ 6.92820 0.635107
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ − 16.3923i − 1.47804i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 20.5885i − 1.82693i −0.406917 0.913465i $$-0.633396\pi$$
0.406917 0.913465i $$-0.366604\pi$$
$$128$$ 0 0
$$129$$ 13.4641 1.18545
$$130$$ 0 0
$$131$$ −18.9282 −1.65376 −0.826882 0.562375i $$-0.809888\pi$$
−0.826882 + 0.562375i $$0.809888\pi$$
$$132$$ 0 0
$$133$$ 2.00000i 0.173422i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 17.3205i − 1.47979i −0.672722 0.739895i $$-0.734875\pi$$
0.672722 0.739895i $$-0.265125\pi$$
$$138$$ 0 0
$$139$$ −11.4641 −0.972372 −0.486186 0.873855i $$-0.661612\pi$$
−0.486186 + 0.873855i $$0.661612\pi$$
$$140$$ 0 0
$$141$$ 35.3205 2.97452
$$142$$ 0 0
$$143$$ 9.46410i 0.791428i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 8.19615i 0.676007i
$$148$$ 0 0
$$149$$ 4.39230 0.359832 0.179916 0.983682i $$-0.442417\pi$$
0.179916 + 0.983682i $$0.442417\pi$$
$$150$$ 0 0
$$151$$ −8.39230 −0.682956 −0.341478 0.939890i $$-0.610927\pi$$
−0.341478 + 0.939890i $$0.610927\pi$$
$$152$$ 0 0
$$153$$ − 15.4641i − 1.25020i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 23.4641i − 1.87264i −0.351149 0.936320i $$-0.614209\pi$$
0.351149 0.936320i $$-0.385791\pi$$
$$158$$ 0 0
$$159$$ 29.3205 2.32527
$$160$$ 0 0
$$161$$ −6.92820 −0.546019
$$162$$ 0 0
$$163$$ 7.07180i 0.553906i 0.960883 + 0.276953i $$0.0893246\pi$$
−0.960883 + 0.276953i $$0.910675\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 10.7321i 0.830471i 0.909714 + 0.415236i $$0.136301\pi$$
−0.909714 + 0.415236i $$0.863699\pi$$
$$168$$ 0 0
$$169$$ 5.53590 0.425838
$$170$$ 0 0
$$171$$ 4.46410 0.341378
$$172$$ 0 0
$$173$$ 3.80385i 0.289201i 0.989490 + 0.144601i $$0.0461897\pi$$
−0.989490 + 0.144601i $$0.953810\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 18.9282i − 1.42273i
$$178$$ 0 0
$$179$$ −20.7846 −1.55351 −0.776757 0.629800i $$-0.783137\pi$$
−0.776757 + 0.629800i $$0.783137\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 33.8564i 2.50274i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 12.0000i − 0.877527i
$$188$$ 0 0
$$189$$ −8.00000 −0.581914
$$190$$ 0 0
$$191$$ 6.92820 0.501307 0.250654 0.968077i $$-0.419354\pi$$
0.250654 + 0.968077i $$0.419354\pi$$
$$192$$ 0 0
$$193$$ − 7.80385i − 0.561733i −0.959747 0.280867i $$-0.909378\pi$$
0.959747 0.280867i $$-0.0906219\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 0.928203i − 0.0661317i −0.999453 0.0330659i $$-0.989473\pi$$
0.999453 0.0330659i $$-0.0105271\pi$$
$$198$$ 0 0
$$199$$ −26.9282 −1.90889 −0.954445 0.298387i $$-0.903551\pi$$
−0.954445 + 0.298387i $$0.903551\pi$$
$$200$$ 0 0
$$201$$ 18.3923 1.29729
$$202$$ 0 0
$$203$$ 6.92820i 0.486265i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 15.4641i 1.07483i
$$208$$ 0 0
$$209$$ 3.46410 0.239617
$$210$$ 0 0
$$211$$ 19.3205 1.33008 0.665039 0.746808i $$-0.268415\pi$$
0.665039 + 0.746808i $$0.268415\pi$$
$$212$$ 0 0
$$213$$ − 6.92820i − 0.474713i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 2.92820i 0.198779i
$$218$$ 0 0
$$219$$ 1.46410 0.0989348
$$220$$ 0 0
$$221$$ 9.46410 0.636624
$$222$$ 0 0
$$223$$ − 9.66025i − 0.646898i −0.946246 0.323449i $$-0.895158\pi$$
0.946246 0.323449i $$-0.104842\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 8.19615i − 0.543998i −0.962298 0.271999i $$-0.912315\pi$$
0.962298 0.271999i $$-0.0876847\pi$$
$$228$$ 0 0
$$229$$ −17.4641 −1.15406 −0.577030 0.816723i $$-0.695788\pi$$
−0.577030 + 0.816723i $$0.695788\pi$$
$$230$$ 0 0
$$231$$ −18.9282 −1.24538
$$232$$ 0 0
$$233$$ − 0.928203i − 0.0608086i −0.999538 0.0304043i $$-0.990321\pi$$
0.999538 0.0304043i $$-0.00967948\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 8.00000i − 0.519656i
$$238$$ 0 0
$$239$$ 13.8564 0.896296 0.448148 0.893959i $$-0.352084\pi$$
0.448148 + 0.893959i $$0.352084\pi$$
$$240$$ 0 0
$$241$$ −11.8564 −0.763738 −0.381869 0.924216i $$-0.624719\pi$$
−0.381869 + 0.924216i $$0.624719\pi$$
$$242$$ 0 0
$$243$$ − 18.7321i − 1.20166i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.73205i 0.173836i
$$248$$ 0 0
$$249$$ −9.46410 −0.599763
$$250$$ 0 0
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 0 0
$$253$$ 12.0000i 0.754434i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 15.1244i 0.943431i 0.881751 + 0.471716i $$0.156365\pi$$
−0.881751 + 0.471716i $$0.843635\pi$$
$$258$$ 0 0
$$259$$ −13.4641 −0.836619
$$260$$ 0 0
$$261$$ 15.4641 0.957204
$$262$$ 0 0
$$263$$ 8.53590i 0.526346i 0.964749 + 0.263173i $$0.0847690\pi$$
−0.964749 + 0.263173i $$0.915231\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 42.2487i 2.58558i
$$268$$ 0 0
$$269$$ −10.3923 −0.633630 −0.316815 0.948487i $$-0.602613\pi$$
−0.316815 + 0.948487i $$0.602613\pi$$
$$270$$ 0 0
$$271$$ −2.39230 −0.145322 −0.0726611 0.997357i $$-0.523149\pi$$
−0.0726611 + 0.997357i $$0.523149\pi$$
$$272$$ 0 0
$$273$$ − 14.9282i − 0.903496i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 18.3923i − 1.10509i −0.833484 0.552543i $$-0.813657\pi$$
0.833484 0.552543i $$-0.186343\pi$$
$$278$$ 0 0
$$279$$ 6.53590 0.391294
$$280$$ 0 0
$$281$$ 0.928203 0.0553720 0.0276860 0.999617i $$-0.491186\pi$$
0.0276860 + 0.999617i $$0.491186\pi$$
$$282$$ 0 0
$$283$$ 18.3923i 1.09331i 0.837358 + 0.546655i $$0.184099\pi$$
−0.837358 + 0.546655i $$0.815901\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 12.0000i 0.708338i
$$288$$ 0 0
$$289$$ 5.00000 0.294118
$$290$$ 0 0
$$291$$ −45.3205 −2.65674
$$292$$ 0 0
$$293$$ − 5.66025i − 0.330676i −0.986237 0.165338i $$-0.947129\pi$$
0.986237 0.165338i $$-0.0528714\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 13.8564i 0.804030i
$$298$$ 0 0
$$299$$ −9.46410 −0.547323
$$300$$ 0 0
$$301$$ −9.85641 −0.568114
$$302$$ 0 0
$$303$$ − 44.7846i − 2.57281i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 7.80385i 0.445389i 0.974888 + 0.222695i $$0.0714853\pi$$
−0.974888 + 0.222695i $$0.928515\pi$$
$$308$$ 0 0
$$309$$ −37.3205 −2.12309
$$310$$ 0 0
$$311$$ −1.60770 −0.0911640 −0.0455820 0.998961i $$-0.514514\pi$$
−0.0455820 + 0.998961i $$0.514514\pi$$
$$312$$ 0 0
$$313$$ 10.7846i 0.609582i 0.952419 + 0.304791i $$0.0985866\pi$$
−0.952419 + 0.304791i $$0.901413\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 3.12436i − 0.175481i −0.996143 0.0877406i $$-0.972035\pi$$
0.996143 0.0877406i $$-0.0279647\pi$$
$$318$$ 0 0
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ −15.4641 −0.863122
$$322$$ 0 0
$$323$$ − 3.46410i − 0.192748i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 39.3205i 2.17443i
$$328$$ 0 0
$$329$$ −25.8564 −1.42551
$$330$$ 0 0
$$331$$ −22.2487 −1.22290 −0.611450 0.791283i $$-0.709413\pi$$
−0.611450 + 0.791283i $$0.709413\pi$$
$$332$$ 0 0
$$333$$ 30.0526i 1.64687i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 17.2679i 0.940645i 0.882495 + 0.470323i $$0.155862\pi$$
−0.882495 + 0.470323i $$0.844138\pi$$
$$338$$ 0 0
$$339$$ −34.3923 −1.86793
$$340$$ 0 0
$$341$$ 5.07180 0.274653
$$342$$ 0 0
$$343$$ − 20.0000i − 1.07990i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 31.1769i − 1.67366i −0.547459 0.836832i $$-0.684405\pi$$
0.547459 0.836832i $$-0.315595\pi$$
$$348$$ 0 0
$$349$$ −20.9282 −1.12026 −0.560131 0.828404i $$-0.689249\pi$$
−0.560131 + 0.828404i $$0.689249\pi$$
$$350$$ 0 0
$$351$$ −10.9282 −0.583304
$$352$$ 0 0
$$353$$ 1.60770i 0.0855690i 0.999084 + 0.0427845i $$0.0136229\pi$$
−0.999084 + 0.0427845i $$0.986377\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 18.9282i 1.00179i
$$358$$ 0 0
$$359$$ 8.53590 0.450507 0.225254 0.974300i $$-0.427679\pi$$
0.225254 + 0.974300i $$0.427679\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 2.73205i 0.143395i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 3.85641i − 0.201303i −0.994922 0.100651i $$-0.967907\pi$$
0.994922 0.100651i $$-0.0320927\pi$$
$$368$$ 0 0
$$369$$ 26.7846 1.39435
$$370$$ 0 0
$$371$$ −21.4641 −1.11436
$$372$$ 0 0
$$373$$ − 16.5885i − 0.858918i −0.903086 0.429459i $$-0.858704\pi$$
0.903086 0.429459i $$-0.141296\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 9.46410i 0.487426i
$$378$$ 0 0
$$379$$ −14.9282 −0.766810 −0.383405 0.923580i $$-0.625249\pi$$
−0.383405 + 0.923580i $$0.625249\pi$$
$$380$$ 0 0
$$381$$ 56.2487 2.88171
$$382$$ 0 0
$$383$$ 6.33975i 0.323946i 0.986795 + 0.161973i $$0.0517857\pi$$
−0.986795 + 0.161973i $$0.948214\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 22.0000i 1.11832i
$$388$$ 0 0
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ − 51.7128i − 2.60857i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 19.4641i 0.976875i 0.872599 + 0.488438i $$0.162433\pi$$
−0.872599 + 0.488438i $$0.837567\pi$$
$$398$$ 0 0
$$399$$ −5.46410 −0.273547
$$400$$ 0 0
$$401$$ 38.7846 1.93681 0.968405 0.249381i $$-0.0802271\pi$$
0.968405 + 0.249381i $$0.0802271\pi$$
$$402$$ 0 0
$$403$$ 4.00000i 0.199254i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 23.3205i 1.15595i
$$408$$ 0 0
$$409$$ 14.3923 0.711654 0.355827 0.934552i $$-0.384199\pi$$
0.355827 + 0.934552i $$0.384199\pi$$
$$410$$ 0 0
$$411$$ 47.3205 2.33415
$$412$$ 0 0
$$413$$ 13.8564i 0.681829i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 31.3205i − 1.53377i
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ 0 0
$$423$$ 57.7128i 2.80609i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 24.7846i − 1.19941i
$$428$$ 0 0
$$429$$ −25.8564 −1.24836
$$430$$ 0 0
$$431$$ −33.4641 −1.61191 −0.805955 0.591977i $$-0.798347\pi$$
−0.805955 + 0.591977i $$0.798347\pi$$
$$432$$ 0 0
$$433$$ − 22.3397i − 1.07358i −0.843716 0.536790i $$-0.819637\pi$$
0.843716 0.536790i $$-0.180363\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.46410i 0.165710i
$$438$$ 0 0
$$439$$ 31.7128 1.51357 0.756785 0.653664i $$-0.226769\pi$$
0.756785 + 0.653664i $$0.226769\pi$$
$$440$$ 0 0
$$441$$ −13.3923 −0.637729
$$442$$ 0 0
$$443$$ 13.6077i 0.646521i 0.946310 + 0.323261i $$0.104779\pi$$
−0.946310 + 0.323261i $$0.895221\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 12.0000i 0.567581i
$$448$$ 0 0
$$449$$ 3.46410 0.163481 0.0817405 0.996654i $$-0.473952\pi$$
0.0817405 + 0.996654i $$0.473952\pi$$
$$450$$ 0 0
$$451$$ 20.7846 0.978709
$$452$$ 0 0
$$453$$ − 22.9282i − 1.07726i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 34.7846i − 1.62716i −0.581456 0.813578i $$-0.697517\pi$$
0.581456 0.813578i $$-0.302483\pi$$
$$458$$ 0 0
$$459$$ 13.8564 0.646762
$$460$$ 0 0
$$461$$ −21.7128 −1.01127 −0.505633 0.862749i $$-0.668741\pi$$
−0.505633 + 0.862749i $$0.668741\pi$$
$$462$$ 0 0
$$463$$ 4.53590i 0.210801i 0.994430 + 0.105401i $$0.0336125\pi$$
−0.994430 + 0.105401i $$0.966388\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 29.3205i 1.35679i 0.734697 + 0.678396i $$0.237324\pi$$
−0.734697 + 0.678396i $$0.762676\pi$$
$$468$$ 0 0
$$469$$ −13.4641 −0.621714
$$470$$ 0 0
$$471$$ 64.1051 2.95381
$$472$$ 0 0
$$473$$ 17.0718i 0.784962i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 47.9090i 2.19360i
$$478$$ 0 0
$$479$$ 22.3923 1.02313 0.511565 0.859244i $$-0.329066\pi$$
0.511565 + 0.859244i $$0.329066\pi$$
$$480$$ 0 0
$$481$$ −18.3923 −0.838617
$$482$$ 0 0
$$483$$ − 18.9282i − 0.861263i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 12.1962i 0.552660i 0.961063 + 0.276330i $$0.0891183\pi$$
−0.961063 + 0.276330i $$0.910882\pi$$
$$488$$ 0 0
$$489$$ −19.3205 −0.873704
$$490$$ 0 0
$$491$$ 18.9282 0.854218 0.427109 0.904200i $$-0.359532\pi$$
0.427109 + 0.904200i $$0.359532\pi$$
$$492$$ 0 0
$$493$$ − 12.0000i − 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 5.07180i 0.227501i
$$498$$ 0 0
$$499$$ −23.4641 −1.05040 −0.525199 0.850980i $$-0.676009\pi$$
−0.525199 + 0.850980i $$0.676009\pi$$
$$500$$ 0 0
$$501$$ −29.3205 −1.30994
$$502$$ 0 0
$$503$$ 15.4641i 0.689510i 0.938693 + 0.344755i $$0.112038\pi$$
−0.938693 + 0.344755i $$0.887962\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 15.1244i 0.671696i
$$508$$ 0 0
$$509$$ 5.32051 0.235827 0.117914 0.993024i $$-0.462379\pi$$
0.117914 + 0.993024i $$0.462379\pi$$
$$510$$ 0 0
$$511$$ −1.07180 −0.0474135
$$512$$ 0 0
$$513$$ 4.00000i 0.176604i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 44.7846i 1.96962i
$$518$$ 0 0
$$519$$ −10.3923 −0.456172
$$520$$ 0 0
$$521$$ −14.7846 −0.647726 −0.323863 0.946104i $$-0.604982\pi$$
−0.323863 + 0.946104i $$0.604982\pi$$
$$522$$ 0 0
$$523$$ − 37.3731i − 1.63421i −0.576489 0.817105i $$-0.695578\pi$$
0.576489 0.817105i $$-0.304422\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 5.07180i − 0.220931i
$$528$$ 0 0
$$529$$ 11.0000 0.478261
$$530$$ 0 0
$$531$$ 30.9282 1.34217
$$532$$ 0 0
$$533$$ 16.3923i 0.710030i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 56.7846i − 2.45044i
$$538$$ 0 0
$$539$$ −10.3923 −0.447628
$$540$$ 0 0
$$541$$ −29.1769 −1.25441 −0.627207 0.778853i $$-0.715802\pi$$
−0.627207 + 0.778853i $$0.715802\pi$$
$$542$$ 0 0
$$543$$ 38.2487i 1.64141i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 6.73205i − 0.287842i −0.989589 0.143921i $$-0.954029\pi$$
0.989589 0.143921i $$-0.0459711\pi$$
$$548$$ 0 0
$$549$$ −55.3205 −2.36102
$$550$$ 0 0
$$551$$ 3.46410 0.147576
$$552$$ 0 0
$$553$$ 5.85641i 0.249040i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 5.32051i − 0.225437i −0.993627 0.112719i $$-0.964044\pi$$
0.993627 0.112719i $$-0.0359559\pi$$
$$558$$ 0 0
$$559$$ −13.4641 −0.569471
$$560$$ 0 0
$$561$$ 32.7846 1.38417
$$562$$ 0 0
$$563$$ 44.1962i 1.86265i 0.364195 + 0.931323i $$0.381344\pi$$
−0.364195 + 0.931323i $$0.618656\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 4.92820i 0.206965i
$$568$$ 0 0
$$569$$ 29.3205 1.22918 0.614590 0.788847i $$-0.289322\pi$$
0.614590 + 0.788847i $$0.289322\pi$$
$$570$$ 0 0
$$571$$ 18.3923 0.769694 0.384847 0.922980i $$-0.374254\pi$$
0.384847 + 0.922980i $$0.374254\pi$$
$$572$$ 0 0
$$573$$ 18.9282i 0.790737i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 6.78461i 0.282447i 0.989978 + 0.141223i $$0.0451036\pi$$
−0.989978 + 0.141223i $$0.954896\pi$$
$$578$$ 0 0
$$579$$ 21.3205 0.886050
$$580$$ 0 0
$$581$$ 6.92820 0.287430
$$582$$ 0 0
$$583$$ 37.1769i 1.53971i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 43.1769i − 1.78210i −0.453903 0.891051i $$-0.649969\pi$$
0.453903 0.891051i $$-0.350031\pi$$
$$588$$ 0 0
$$589$$ 1.46410 0.0603273
$$590$$ 0 0
$$591$$ 2.53590 0.104313
$$592$$ 0 0
$$593$$ 31.8564i 1.30819i 0.756414 + 0.654093i $$0.226949\pi$$
−0.756414 + 0.654093i $$0.773051\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 73.5692i − 3.01099i
$$598$$ 0 0
$$599$$ 20.7846 0.849236 0.424618 0.905373i $$-0.360408\pi$$
0.424618 + 0.905373i $$0.360408\pi$$
$$600$$ 0 0
$$601$$ 24.6410 1.00513 0.502564 0.864540i $$-0.332390\pi$$
0.502564 + 0.864540i $$0.332390\pi$$
$$602$$ 0 0
$$603$$ 30.0526i 1.22383i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 39.9090i 1.61985i 0.586530 + 0.809927i $$0.300494\pi$$
−0.586530 + 0.809927i $$0.699506\pi$$
$$608$$ 0 0
$$609$$ −18.9282 −0.767010
$$610$$ 0 0
$$611$$ −35.3205 −1.42891
$$612$$ 0 0
$$613$$ 6.39230i 0.258183i 0.991633 + 0.129091i $$0.0412061\pi$$
−0.991633 + 0.129091i $$0.958794\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 1.60770i − 0.0647234i −0.999476 0.0323617i $$-0.989697\pi$$
0.999476 0.0323617i $$-0.0103028\pi$$
$$618$$ 0 0
$$619$$ 2.39230 0.0961549 0.0480774 0.998844i $$-0.484691\pi$$
0.0480774 + 0.998844i $$0.484691\pi$$
$$620$$ 0 0
$$621$$ −13.8564 −0.556038
$$622$$ 0 0
$$623$$ − 30.9282i − 1.23911i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 9.46410i 0.377960i
$$628$$ 0 0
$$629$$ 23.3205 0.929850
$$630$$ 0 0
$$631$$ 11.4641 0.456379 0.228189 0.973617i $$-0.426719\pi$$
0.228189 + 0.973617i $$0.426719\pi$$
$$632$$ 0 0
$$633$$ 52.7846i 2.09800i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 8.19615i − 0.324743i
$$638$$ 0 0
$$639$$ 11.3205 0.447832
$$640$$ 0 0
$$641$$ −24.9282 −0.984605 −0.492302 0.870424i $$-0.663845\pi$$
−0.492302 + 0.870424i $$0.663845\pi$$
$$642$$ 0 0
$$643$$ − 14.3923i − 0.567577i −0.958887 0.283789i $$-0.908409\pi$$
0.958887 0.283789i $$-0.0915914\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 15.4641i − 0.607957i −0.952679 0.303978i $$-0.901685\pi$$
0.952679 0.303978i $$-0.0983150\pi$$
$$648$$ 0 0
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ −8.00000 −0.313545
$$652$$ 0 0
$$653$$ 27.4641i 1.07475i 0.843342 + 0.537377i $$0.180585\pi$$
−0.843342 + 0.537377i $$0.819415\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2.39230i 0.0933327i
$$658$$ 0 0
$$659$$ −8.78461 −0.342200 −0.171100 0.985254i $$-0.554732\pi$$
−0.171100 + 0.985254i $$0.554732\pi$$
$$660$$ 0 0
$$661$$ 5.21539 0.202855 0.101428 0.994843i $$-0.467659\pi$$
0.101428 + 0.994843i $$0.467659\pi$$
$$662$$ 0 0
$$663$$ 25.8564i 1.00418i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 12.0000i 0.464642i
$$668$$ 0 0
$$669$$ 26.3923 1.02039
$$670$$ 0 0
$$671$$ −42.9282 −1.65722
$$672$$ 0 0
$$673$$ − 50.7321i − 1.95558i −0.209594 0.977788i $$-0.567214\pi$$
0.209594 0.977788i $$-0.432786\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 16.9808i − 0.652624i −0.945262 0.326312i $$-0.894194\pi$$
0.945262 0.326312i $$-0.105806\pi$$
$$678$$ 0 0
$$679$$ 33.1769 1.27321
$$680$$ 0 0
$$681$$ 22.3923 0.858075
$$682$$ 0 0
$$683$$ − 17.6603i − 0.675751i −0.941191 0.337875i $$-0.890292\pi$$
0.941191 0.337875i $$-0.109708\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 47.7128i − 1.82036i
$$688$$ 0 0
$$689$$ −29.3205 −1.11702
$$690$$ 0 0
$$691$$ 51.1769 1.94686 0.973431 0.228981i $$-0.0735395\pi$$
0.973431 + 0.228981i $$0.0735395\pi$$
$$692$$ 0 0
$$693$$ − 30.9282i − 1.17487i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 20.7846i − 0.787273i
$$698$$ 0 0
$$699$$ 2.53590 0.0959165
$$700$$ 0 0
$$701$$ −26.5359 −1.00225 −0.501124 0.865376i $$-0.667080\pi$$
−0.501124 + 0.865376i $$0.667080\pi$$
$$702$$ 0 0
$$703$$ 6.73205i 0.253904i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 32.7846i 1.23299i
$$708$$ 0 0
$$709$$ 30.7846 1.15614 0.578070 0.815987i $$-0.303806\pi$$
0.578070 + 0.815987i $$0.303806\pi$$
$$710$$ 0 0
$$711$$ 13.0718 0.490231
$$712$$ 0 0
$$713$$ 5.07180i 0.189940i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 37.8564i 1.41377i
$$718$$ 0 0
$$719$$ −17.3205 −0.645946 −0.322973 0.946408i $$-0.604682\pi$$
−0.322973 + 0.946408i $$0.604682\pi$$
$$720$$ 0 0
$$721$$ 27.3205 1.01747
$$722$$ 0 0
$$723$$ − 32.3923i − 1.20468i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 0.143594i − 0.00532559i −0.999996 0.00266279i $$-0.999152\pi$$
0.999996 0.00266279i $$-0.000847595\pi$$
$$728$$ 0 0
$$729$$ 43.7846 1.62165
$$730$$ 0 0
$$731$$ 17.0718 0.631423
$$732$$ 0 0
$$733$$ 10.7846i 0.398339i 0.979965 + 0.199169i $$0.0638244\pi$$
−0.979965 + 0.199169i $$0.936176\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 23.3205i 0.859022i
$$738$$ 0 0
$$739$$ 28.0000 1.03000 0.514998 0.857191i $$-0.327793\pi$$
0.514998 + 0.857191i $$0.327793\pi$$
$$740$$ 0 0
$$741$$ −7.46410 −0.274201
$$742$$ 0 0
$$743$$ 20.8756i 0.765853i 0.923779 + 0.382927i $$0.125084\pi$$
−0.923779 + 0.382927i $$0.874916\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 15.4641i − 0.565802i
$$748$$ 0 0
$$749$$ 11.3205 0.413642
$$750$$ 0 0
$$751$$ −25.4641 −0.929198 −0.464599 0.885521i $$-0.653802\pi$$
−0.464599 + 0.885521i $$0.653802\pi$$
$$752$$ 0 0
$$753$$ − 65.5692i − 2.38948i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10.0000i 0.363456i 0.983349 + 0.181728i $$0.0581691\pi$$
−0.983349 + 0.181728i $$0.941831\pi$$
$$758$$ 0 0
$$759$$ −32.7846 −1.19001
$$760$$ 0 0
$$761$$ 19.6077 0.710778 0.355389 0.934718i $$-0.384348\pi$$
0.355389 + 0.934718i $$0.384348\pi$$
$$762$$ 0 0
$$763$$ − 28.7846i − 1.04207i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 18.9282i 0.683458i
$$768$$ 0 0
$$769$$ 30.5359 1.10115 0.550576 0.834785i $$-0.314408\pi$$
0.550576 + 0.834785i $$0.314408\pi$$
$$770$$ 0 0
$$771$$ −41.3205 −1.48812
$$772$$ 0 0
$$773$$ 22.0526i 0.793175i 0.917997 + 0.396588i $$0.129806\pi$$
−0.917997 + 0.396588i $$0.870194\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 36.7846i − 1.31964i
$$778$$ 0 0
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 8.78461 0.314338
$$782$$ 0 0
$$783$$ 13.8564i 0.495188i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 12.1962i 0.434746i 0.976089 + 0.217373i $$0.0697488\pi$$
−0.976089 + 0.217373i $$0.930251\pi$$
$$788$$ 0 0
$$789$$ −23.3205 −0.830232
$$790$$ 0 0
$$791$$ 25.1769 0.895188
$$792$$ 0 0
$$793$$ − 33.8564i − 1.20228i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 5.66025i 0.200496i 0.994962 + 0.100248i $$0.0319637\pi$$
−0.994962 + 0.100248i $$0.968036\pi$$
$$798$$ 0 0
$$799$$ 44.7846 1.58437
$$800$$ 0 0
$$801$$ −69.0333 −2.43917
$$802$$ 0 0
$$803$$ 1.85641i 0.0655112i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 28.3923i − 0.999456i
$$808$$ 0 0
$$809$$ −9.71281 −0.341484 −0.170742 0.985316i $$-0.554617\pi$$
−0.170742 + 0.985316i $$0.554617\pi$$
$$810$$ 0 0
$$811$$ 15.6077 0.548060 0.274030 0.961721i $$-0.411643\pi$$
0.274030 + 0.961721i $$0.411643\pi$$
$$812$$ 0 0
$$813$$ − 6.53590i − 0.229224i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 4.92820i 0.172416i
$$818$$ 0 0
$$819$$ 24.3923 0.852336
$$820$$ 0 0
$$821$$ −16.1436 −0.563415 −0.281708 0.959500i $$-0.590901\pi$$
−0.281708 + 0.959500i $$0.590901\pi$$
$$822$$ 0 0
$$823$$ − 39.5692i − 1.37930i −0.724145 0.689648i $$-0.757765\pi$$
0.724145 0.689648i $$-0.242235\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 29.6603i 1.03139i 0.856773 + 0.515694i $$0.172466\pi$$
−0.856773 + 0.515694i $$0.827534\pi$$
$$828$$ 0 0
$$829$$ −8.24871 −0.286490 −0.143245 0.989687i $$-0.545754\pi$$
−0.143245 + 0.989687i $$0.545754\pi$$
$$830$$ 0 0
$$831$$ 50.2487 1.74311
$$832$$ 0 0
$$833$$ 10.3923i 0.360072i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 5.85641i 0.202427i
$$838$$ 0 0
$$839$$ −18.9282 −0.653474 −0.326737 0.945115i $$-0.605949\pi$$
−0.326737 + 0.945115i $$0.605949\pi$$
$$840$$ 0 0
$$841$$ −17.0000 −0.586207
$$842$$ 0 0
$$843$$ 2.53590i 0.0873410i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 2.00000i − 0.0687208i
$$848$$ 0 0
$$849$$ −50.2487 −1.72453
$$850$$ 0 0
$$851$$ −23.3205 −0.799417
$$852$$ 0 0
$$853$$ − 44.6410i − 1.52848i −0.644932 0.764240i $$-0.723114\pi$$
0.644932 0.764240i $$-0.276886\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 3.80385i − 0.129937i −0.997887 0.0649685i $$-0.979305\pi$$
0.997887 0.0649685i $$-0.0206947\pi$$
$$858$$ 0 0
$$859$$ −47.7128 −1.62794 −0.813970 0.580907i $$-0.802698\pi$$
−0.813970 + 0.580907i $$0.802698\pi$$
$$860$$ 0 0
$$861$$ −32.7846 −1.11730
$$862$$ 0 0
$$863$$ − 37.2679i − 1.26862i −0.773081 0.634308i $$-0.781285\pi$$
0.773081 0.634308i $$-0.218715\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 13.6603i 0.463927i
$$868$$ 0 0
$$869$$ 10.1436 0.344098
$$870$$ 0 0
$$871$$ −18.3923 −0.623199
$$872$$ 0 0
$$873$$ − 74.0526i − 2.50630i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 0.980762i − 0.0331180i −0.999863 0.0165590i $$-0.994729\pi$$
0.999863 0.0165590i $$-0.00527113\pi$$
$$878$$ 0 0
$$879$$ 15.4641 0.521591
$$880$$ 0 0
$$881$$ −0.679492 −0.0228927 −0.0114463 0.999934i $$-0.503644\pi$$
−0.0114463 + 0.999934i $$0.503644\pi$$
$$882$$ 0 0
$$883$$ − 22.0000i − 0.740359i −0.928960 0.370179i $$-0.879296\pi$$
0.928960 0.370179i $$-0.120704\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 23.4115i − 0.786083i −0.919521 0.393041i $$-0.871423\pi$$
0.919521 0.393041i $$-0.128577\pi$$
$$888$$ 0 0
$$889$$ −41.1769 −1.38103
$$890$$ 0 0
$$891$$ 8.53590 0.285963
$$892$$ 0 0
$$893$$ 12.9282i 0.432626i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 25.8564i − 0.863320i
$$898$$ 0 0
$$899$$ 5.07180 0.169154
$$900$$ 0 0
$$901$$ 37.1769 1.23854
$$902$$ 0 0
$$903$$ − 26.9282i − 0.896114i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 7.41154i − 0.246096i −0.992401 0.123048i $$-0.960733\pi$$
0.992401 0.123048i $$-0.0392670\pi$$
$$908$$ 0 0
$$909$$ 73.1769 2.42713
$$910$$ 0 0
$$911$$ 14.5359 0.481596 0.240798 0.970575i $$-0.422591\pi$$
0.240798 + 0.970575i $$0.422591\pi$$
$$912$$ 0 0
$$913$$ − 12.0000i − 0.397142i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 37.8564i 1.25013i
$$918$$ 0 0
$$919$$ −44.4974 −1.46783 −0.733917 0.679239i $$-0.762310\pi$$
−0.733917 + 0.679239i $$0.762310\pi$$
$$920$$ 0 0
$$921$$ −21.3205 −0.702535
$$922$$ 0 0
$$923$$ 6.92820i 0.228045i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 60.9808i − 2.00287i
$$928$$ 0 0
$$929$$ 47.5692 1.56070 0.780348 0.625346i $$-0.215042\pi$$
0.780348 + 0.625346i $$0.215042\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ 0 0
$$933$$ − 4.39230i − 0.143798i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 51.1769i − 1.67188i −0.548823 0.835938i $$-0.684924\pi$$
0.548823 0.835938i $$-0.315076\pi$$
$$938$$ 0 0
$$939$$ −29.4641 −0.961525
$$940$$ 0 0
$$941$$ 52.6410 1.71605 0.858024 0.513610i $$-0.171692\pi$$
0.858024 + 0.513610i $$0.171692\pi$$
$$942$$ 0 0
$$943$$ 20.7846i 0.676840i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 6.67949i − 0.217054i −0.994093 0.108527i $$-0.965387\pi$$
0.994093 0.108527i $$-0.0346134\pi$$
$$948$$ 0 0
$$949$$ −1.46410 −0.0475267
$$950$$ 0 0
$$951$$ 8.53590 0.276795
$$952$$ 0 0
$$953$$ 60.5885i 1.96265i 0.192350 + 0.981326i $$0.438389\pi$$
−0.192350 + 0.981326i $$0.561611\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 32.7846i 1.05978i
$$958$$ 0 0
$$959$$ −34.6410 −1.11862
$$960$$ 0 0
$$961$$ −28.8564 −0.930852
$$962$$ 0 0
$$963$$ − 25.2679i − 0.814248i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 45.3205i 1.45741i 0.684828 + 0.728705i $$0.259877\pi$$
−0.684828 + 0.728705i $$0.740123\pi$$
$$968$$ 0 0
$$969$$ 9.46410 0.304031
$$970$$ 0 0
$$971$$ −14.5359 −0.466479 −0.233240 0.972419i $$-0.574933\pi$$
−0.233240 + 0.972419i $$0.574933\pi$$
$$972$$ 0 0
$$973$$ 22.9282i 0.735044i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 32.1962i − 1.03005i −0.857176 0.515023i $$-0.827783\pi$$
0.857176 0.515023i $$-0.172217\pi$$
$$978$$ 0 0
$$979$$ −53.5692 −1.71208
$$980$$ 0 0
$$981$$ −64.2487 −2.05130
$$982$$ 0 0
$$983$$ − 2.44486i − 0.0779790i −0.999240 0.0389895i $$-0.987586\pi$$
0.999240 0.0389895i $$-0.0124139\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 70.6410i − 2.24853i
$$988$$ 0 0
$$989$$ −17.0718 −0.542852
$$990$$ 0 0
$$991$$ −8.39230 −0.266590 −0.133295 0.991076i $$-0.542556\pi$$
−0.133295 + 0.991076i $$0.542556\pi$$
$$992$$ 0 0
$$993$$ − 60.7846i − 1.92894i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 30.3923i − 0.962534i −0.876574 0.481267i $$-0.840177\pi$$
0.876574 0.481267i $$-0.159823\pi$$
$$998$$ 0 0
$$999$$ −26.9282 −0.851971
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 1900.2.c.e.1749.4 4
5.2 odd 4 380.2.a.d.1.2 2
5.3 odd 4 1900.2.a.d.1.1 2
5.4 even 2 inner 1900.2.c.e.1749.1 4
15.2 even 4 3420.2.a.h.1.2 2
20.3 even 4 7600.2.a.bf.1.2 2
20.7 even 4 1520.2.a.l.1.1 2
40.27 even 4 6080.2.a.bj.1.2 2
40.37 odd 4 6080.2.a.z.1.1 2
95.37 even 4 7220.2.a.h.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.d.1.2 2 5.2 odd 4
1520.2.a.l.1.1 2 20.7 even 4
1900.2.a.d.1.1 2 5.3 odd 4
1900.2.c.e.1749.1 4 5.4 even 2 inner
1900.2.c.e.1749.4 4 1.1 even 1 trivial
3420.2.a.h.1.2 2 15.2 even 4
6080.2.a.z.1.1 2 40.37 odd 4
6080.2.a.bj.1.2 2 40.27 even 4
7220.2.a.h.1.1 2 95.37 even 4
7600.2.a.bf.1.2 2 20.3 even 4