# Properties

 Label 2475.2.c.k.199.3 Level $2475$ Weight $2$ Character 2475.199 Analytic conductor $19.763$ 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] = [2475,2,Mod(199,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 275) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.3 Root $$1.30278i$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.199 Dual form 2475.2.c.k.199.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.30278i q^{2} +0.302776 q^{4} +0.697224i q^{7} +3.00000i q^{8} +O(q^{10})$$ $$q+1.30278i q^{2} +0.302776 q^{4} +0.697224i q^{7} +3.00000i q^{8} +1.00000 q^{11} -5.00000i q^{13} -0.908327 q^{14} -3.30278 q^{16} -6.90833i q^{17} +1.00000 q^{19} +1.30278i q^{22} -7.30278i q^{23} +6.51388 q^{26} +0.211103i q^{28} +0.908327 q^{29} +10.2111 q^{31} +1.69722i q^{32} +9.00000 q^{34} +2.39445i q^{37} +1.30278i q^{38} +5.60555 q^{41} -7.21110i q^{43} +0.302776 q^{44} +9.51388 q^{46} +3.00000i q^{47} +6.51388 q^{49} -1.51388i q^{52} -1.30278i q^{53} -2.09167 q^{56} +1.18335i q^{58} -14.2111 q^{59} -7.90833 q^{61} +13.3028i q^{62} -8.81665 q^{64} -4.00000i q^{67} -2.09167i q^{68} +2.60555 q^{71} +7.90833i q^{73} -3.11943 q^{74} +0.302776 q^{76} +0.697224i q^{77} +10.9083 q^{79} +7.30278i q^{82} -3.51388i q^{83} +9.39445 q^{86} +3.00000i q^{88} +1.69722 q^{89} +3.48612 q^{91} -2.21110i q^{92} -3.90833 q^{94} +15.3028i q^{97} +8.48612i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{4}+O(q^{10})$$ 4 * q - 6 * q^4 $$4 q - 6 q^{4} + 4 q^{11} + 18 q^{14} - 6 q^{16} + 4 q^{19} - 10 q^{26} - 18 q^{29} + 12 q^{31} + 36 q^{34} + 8 q^{41} - 6 q^{44} + 2 q^{46} - 10 q^{49} - 30 q^{56} - 28 q^{59} - 10 q^{61} + 8 q^{64} - 4 q^{71} + 38 q^{74} - 6 q^{76} + 22 q^{79} + 52 q^{86} + 14 q^{89} + 50 q^{91} + 6 q^{94}+O(q^{100})$$ 4 * q - 6 * q^4 + 4 * q^11 + 18 * q^14 - 6 * q^16 + 4 * q^19 - 10 * q^26 - 18 * q^29 + 12 * q^31 + 36 * q^34 + 8 * q^41 - 6 * q^44 + 2 * q^46 - 10 * q^49 - 30 * q^56 - 28 * q^59 - 10 * q^61 + 8 * q^64 - 4 * q^71 + 38 * q^74 - 6 * q^76 + 22 * q^79 + 52 * q^86 + 14 * q^89 + 50 * q^91 + 6 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2026$$ $$2377$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.30278i 0.921201i 0.887607 + 0.460601i $$0.152366\pi$$
−0.887607 + 0.460601i $$0.847634\pi$$
$$3$$ 0 0
$$4$$ 0.302776 0.151388
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.697224i 0.263526i 0.991281 + 0.131763i $$0.0420638\pi$$
−0.991281 + 0.131763i $$0.957936\pi$$
$$8$$ 3.00000i 1.06066i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ − 5.00000i − 1.38675i −0.720577 0.693375i $$-0.756123\pi$$
0.720577 0.693375i $$-0.243877\pi$$
$$14$$ −0.908327 −0.242761
$$15$$ 0 0
$$16$$ −3.30278 −0.825694
$$17$$ − 6.90833i − 1.67552i −0.546042 0.837758i $$-0.683866\pi$$
0.546042 0.837758i $$-0.316134\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 1.30278i 0.277753i
$$23$$ − 7.30278i − 1.52273i −0.648321 0.761367i $$-0.724529\pi$$
0.648321 0.761367i $$-0.275471\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 6.51388 1.27748
$$27$$ 0 0
$$28$$ 0.211103i 0.0398946i
$$29$$ 0.908327 0.168672 0.0843360 0.996437i $$-0.473123\pi$$
0.0843360 + 0.996437i $$0.473123\pi$$
$$30$$ 0 0
$$31$$ 10.2111 1.83397 0.916984 0.398924i $$-0.130616\pi$$
0.916984 + 0.398924i $$0.130616\pi$$
$$32$$ 1.69722i 0.300030i
$$33$$ 0 0
$$34$$ 9.00000 1.54349
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.39445i 0.393645i 0.980439 + 0.196822i $$0.0630623\pi$$
−0.980439 + 0.196822i $$0.936938\pi$$
$$38$$ 1.30278i 0.211338i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 5.60555 0.875440 0.437720 0.899111i $$-0.355786\pi$$
0.437720 + 0.899111i $$0.355786\pi$$
$$42$$ 0 0
$$43$$ − 7.21110i − 1.09968i −0.835269 0.549841i $$-0.814688\pi$$
0.835269 0.549841i $$-0.185312\pi$$
$$44$$ 0.302776 0.0456451
$$45$$ 0 0
$$46$$ 9.51388 1.40274
$$47$$ 3.00000i 0.437595i 0.975770 + 0.218797i $$0.0702134\pi$$
−0.975770 + 0.218797i $$0.929787\pi$$
$$48$$ 0 0
$$49$$ 6.51388 0.930554
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 1.51388i − 0.209937i
$$53$$ − 1.30278i − 0.178950i −0.995989 0.0894750i $$-0.971481\pi$$
0.995989 0.0894750i $$-0.0285189\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −2.09167 −0.279512
$$57$$ 0 0
$$58$$ 1.18335i 0.155381i
$$59$$ −14.2111 −1.85013 −0.925064 0.379811i $$-0.875989\pi$$
−0.925064 + 0.379811i $$0.875989\pi$$
$$60$$ 0 0
$$61$$ −7.90833 −1.01256 −0.506279 0.862370i $$-0.668979\pi$$
−0.506279 + 0.862370i $$0.668979\pi$$
$$62$$ 13.3028i 1.68945i
$$63$$ 0 0
$$64$$ −8.81665 −1.10208
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ − 2.09167i − 0.253653i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 2.60555 0.309222 0.154611 0.987975i $$-0.450588\pi$$
0.154611 + 0.987975i $$0.450588\pi$$
$$72$$ 0 0
$$73$$ 7.90833i 0.925600i 0.886463 + 0.462800i $$0.153155\pi$$
−0.886463 + 0.462800i $$0.846845\pi$$
$$74$$ −3.11943 −0.362626
$$75$$ 0 0
$$76$$ 0.302776 0.0347307
$$77$$ 0.697224i 0.0794561i
$$78$$ 0 0
$$79$$ 10.9083 1.22728 0.613641 0.789585i $$-0.289704\pi$$
0.613641 + 0.789585i $$0.289704\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 7.30278i 0.806457i
$$83$$ − 3.51388i − 0.385698i −0.981228 0.192849i $$-0.938227\pi$$
0.981228 0.192849i $$-0.0617728\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 9.39445 1.01303
$$87$$ 0 0
$$88$$ 3.00000i 0.319801i
$$89$$ 1.69722 0.179905 0.0899527 0.995946i $$-0.471328\pi$$
0.0899527 + 0.995946i $$0.471328\pi$$
$$90$$ 0 0
$$91$$ 3.48612 0.365445
$$92$$ − 2.21110i − 0.230523i
$$93$$ 0 0
$$94$$ −3.90833 −0.403113
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 15.3028i 1.55376i 0.629648 + 0.776881i $$0.283199\pi$$
−0.629648 + 0.776881i $$0.716801\pi$$
$$98$$ 8.48612i 0.857228i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0.513878 0.0511328 0.0255664 0.999673i $$-0.491861\pi$$
0.0255664 + 0.999673i $$0.491861\pi$$
$$102$$ 0 0
$$103$$ − 2.90833i − 0.286566i −0.989682 0.143283i $$-0.954234\pi$$
0.989682 0.143283i $$-0.0457659\pi$$
$$104$$ 15.0000 1.47087
$$105$$ 0 0
$$106$$ 1.69722 0.164849
$$107$$ 3.00000i 0.290021i 0.989430 + 0.145010i $$0.0463216\pi$$
−0.989430 + 0.145010i $$0.953678\pi$$
$$108$$ 0 0
$$109$$ −11.5139 −1.10283 −0.551415 0.834231i $$-0.685912\pi$$
−0.551415 + 0.834231i $$0.685912\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 2.30278i − 0.217592i
$$113$$ − 10.8167i − 1.01755i −0.860901 0.508773i $$-0.830099\pi$$
0.860901 0.508773i $$-0.169901\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0.275019 0.0255349
$$117$$ 0 0
$$118$$ − 18.5139i − 1.70434i
$$119$$ 4.81665 0.441542
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 10.3028i − 0.932769i
$$123$$ 0 0
$$124$$ 3.09167 0.277640
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.11943i 0.720483i 0.932859 + 0.360241i $$0.117306\pi$$
−0.932859 + 0.360241i $$0.882694\pi$$
$$128$$ − 8.09167i − 0.715210i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 9.90833 0.865695 0.432847 0.901467i $$-0.357509\pi$$
0.432847 + 0.901467i $$0.357509\pi$$
$$132$$ 0 0
$$133$$ 0.697224i 0.0604570i
$$134$$ 5.21110 0.450171
$$135$$ 0 0
$$136$$ 20.7250 1.77715
$$137$$ 12.9083i 1.10283i 0.834230 + 0.551416i $$0.185912\pi$$
−0.834230 + 0.551416i $$0.814088\pi$$
$$138$$ 0 0
$$139$$ 6.21110 0.526819 0.263409 0.964684i $$-0.415153\pi$$
0.263409 + 0.964684i $$0.415153\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 3.39445i 0.284856i
$$143$$ − 5.00000i − 0.418121i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −10.3028 −0.852664
$$147$$ 0 0
$$148$$ 0.724981i 0.0595930i
$$149$$ 17.2111 1.40999 0.704994 0.709213i $$-0.250950\pi$$
0.704994 + 0.709213i $$0.250950\pi$$
$$150$$ 0 0
$$151$$ 0.816654 0.0664583 0.0332292 0.999448i $$-0.489421\pi$$
0.0332292 + 0.999448i $$0.489421\pi$$
$$152$$ 3.00000i 0.243332i
$$153$$ 0 0
$$154$$ −0.908327 −0.0731951
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 19.2111i 1.53321i 0.642117 + 0.766606i $$0.278056\pi$$
−0.642117 + 0.766606i $$0.721944\pi$$
$$158$$ 14.2111i 1.13057i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 5.09167 0.401280
$$162$$ 0 0
$$163$$ − 9.30278i − 0.728650i −0.931272 0.364325i $$-0.881300\pi$$
0.931272 0.364325i $$-0.118700\pi$$
$$164$$ 1.69722 0.132531
$$165$$ 0 0
$$166$$ 4.57779 0.355306
$$167$$ − 13.4222i − 1.03864i −0.854579 0.519321i $$-0.826185\pi$$
0.854579 0.519321i $$-0.173815\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 2.18335i − 0.166479i
$$173$$ − 4.81665i − 0.366203i −0.983094 0.183102i $$-0.941386\pi$$
0.983094 0.183102i $$-0.0586138\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.30278 −0.248956
$$177$$ 0 0
$$178$$ 2.21110i 0.165729i
$$179$$ 12.5139 0.935331 0.467666 0.883905i $$-0.345095\pi$$
0.467666 + 0.883905i $$0.345095\pi$$
$$180$$ 0 0
$$181$$ −19.9083 −1.47977 −0.739887 0.672731i $$-0.765121\pi$$
−0.739887 + 0.672731i $$0.765121\pi$$
$$182$$ 4.54163i 0.336648i
$$183$$ 0 0
$$184$$ 21.9083 1.61510
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 6.90833i − 0.505187i
$$188$$ 0.908327i 0.0662465i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −10.3028 −0.745483 −0.372741 0.927935i $$-0.621582\pi$$
−0.372741 + 0.927935i $$0.621582\pi$$
$$192$$ 0 0
$$193$$ − 13.2111i − 0.950956i −0.879728 0.475478i $$-0.842275\pi$$
0.879728 0.475478i $$-0.157725\pi$$
$$194$$ −19.9361 −1.43133
$$195$$ 0 0
$$196$$ 1.97224 0.140875
$$197$$ − 13.3028i − 0.947784i −0.880583 0.473892i $$-0.842849\pi$$
0.880583 0.473892i $$-0.157151\pi$$
$$198$$ 0 0
$$199$$ 6.48612 0.459789 0.229894 0.973216i $$-0.426162\pi$$
0.229894 + 0.973216i $$0.426162\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0.669468i 0.0471036i
$$203$$ 0.633308i 0.0444495i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 3.78890 0.263985
$$207$$ 0 0
$$208$$ 16.5139i 1.14503i
$$209$$ 1.00000 0.0691714
$$210$$ 0 0
$$211$$ −25.2389 −1.73751 −0.868757 0.495238i $$-0.835081\pi$$
−0.868757 + 0.495238i $$0.835081\pi$$
$$212$$ − 0.394449i − 0.0270908i
$$213$$ 0 0
$$214$$ −3.90833 −0.267168
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 7.11943i 0.483298i
$$218$$ − 15.0000i − 1.01593i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −34.5416 −2.32352
$$222$$ 0 0
$$223$$ 22.6333i 1.51564i 0.652465 + 0.757819i $$0.273735\pi$$
−0.652465 + 0.757819i $$0.726265\pi$$
$$224$$ −1.18335 −0.0790656
$$225$$ 0 0
$$226$$ 14.0917 0.937364
$$227$$ − 1.69722i − 0.112649i −0.998413 0.0563244i $$-0.982062\pi$$
0.998413 0.0563244i $$-0.0179381\pi$$
$$228$$ 0 0
$$229$$ 18.7250 1.23738 0.618691 0.785635i $$-0.287663\pi$$
0.618691 + 0.785635i $$0.287663\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.72498i 0.178904i
$$233$$ 15.9083i 1.04219i 0.853499 + 0.521095i $$0.174476\pi$$
−0.853499 + 0.521095i $$0.825524\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −4.30278 −0.280087
$$237$$ 0 0
$$238$$ 6.27502i 0.406749i
$$239$$ 21.1194 1.36610 0.683051 0.730371i $$-0.260653\pi$$
0.683051 + 0.730371i $$0.260653\pi$$
$$240$$ 0 0
$$241$$ 21.9361 1.41303 0.706514 0.707699i $$-0.250267\pi$$
0.706514 + 0.707699i $$0.250267\pi$$
$$242$$ 1.30278i 0.0837456i
$$243$$ 0 0
$$244$$ −2.39445 −0.153289
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 5.00000i − 0.318142i
$$248$$ 30.6333i 1.94522i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −6.90833 −0.436050 −0.218025 0.975943i $$-0.569961\pi$$
−0.218025 + 0.975943i $$0.569961\pi$$
$$252$$ 0 0
$$253$$ − 7.30278i − 0.459122i
$$254$$ −10.5778 −0.663710
$$255$$ 0 0
$$256$$ −7.09167 −0.443230
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 0 0
$$259$$ −1.66947 −0.103736
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 12.9083i 0.797479i
$$263$$ − 22.8167i − 1.40694i −0.710727 0.703468i $$-0.751634\pi$$
0.710727 0.703468i $$-0.248366\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −0.908327 −0.0556931
$$267$$ 0 0
$$268$$ − 1.21110i − 0.0739799i
$$269$$ 8.72498 0.531971 0.265986 0.963977i $$-0.414303\pi$$
0.265986 + 0.963977i $$0.414303\pi$$
$$270$$ 0 0
$$271$$ −0.211103 −0.0128236 −0.00641178 0.999979i $$-0.502041\pi$$
−0.00641178 + 0.999979i $$0.502041\pi$$
$$272$$ 22.8167i 1.38346i
$$273$$ 0 0
$$274$$ −16.8167 −1.01593
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 14.3944i 0.864879i 0.901663 + 0.432439i $$0.142347\pi$$
−0.901663 + 0.432439i $$0.857653\pi$$
$$278$$ 8.09167i 0.485306i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1.18335 0.0705925 0.0352963 0.999377i $$-0.488763\pi$$
0.0352963 + 0.999377i $$0.488763\pi$$
$$282$$ 0 0
$$283$$ − 6.30278i − 0.374661i −0.982297 0.187331i $$-0.940016\pi$$
0.982297 0.187331i $$-0.0599836\pi$$
$$284$$ 0.788897 0.0468125
$$285$$ 0 0
$$286$$ 6.51388 0.385174
$$287$$ 3.90833i 0.230701i
$$288$$ 0 0
$$289$$ −30.7250 −1.80735
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.39445i 0.140125i
$$293$$ 0.788897i 0.0460879i 0.999734 + 0.0230439i $$0.00733576\pi$$
−0.999734 + 0.0230439i $$0.992664\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −7.18335 −0.417524
$$297$$ 0 0
$$298$$ 22.4222i 1.29888i
$$299$$ −36.5139 −2.11165
$$300$$ 0 0
$$301$$ 5.02776 0.289795
$$302$$ 1.06392i 0.0612215i
$$303$$ 0 0
$$304$$ −3.30278 −0.189427
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 16.9083i − 0.965009i −0.875893 0.482505i $$-0.839727\pi$$
0.875893 0.482505i $$-0.160273\pi$$
$$308$$ 0.211103i 0.0120287i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −4.81665 −0.273127 −0.136564 0.990631i $$-0.543606\pi$$
−0.136564 + 0.990631i $$0.543606\pi$$
$$312$$ 0 0
$$313$$ − 0.183346i − 0.0103633i −0.999987 0.00518167i $$-0.998351\pi$$
0.999987 0.00518167i $$-0.00164938\pi$$
$$314$$ −25.0278 −1.41240
$$315$$ 0 0
$$316$$ 3.30278 0.185796
$$317$$ 0.908327i 0.0510167i 0.999675 + 0.0255084i $$0.00812044\pi$$
−0.999675 + 0.0255084i $$0.991880\pi$$
$$318$$ 0 0
$$319$$ 0.908327 0.0508565
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 6.63331i 0.369660i
$$323$$ − 6.90833i − 0.384390i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 12.1194 0.671233
$$327$$ 0 0
$$328$$ 16.8167i 0.928544i
$$329$$ −2.09167 −0.115318
$$330$$ 0 0
$$331$$ −21.6056 −1.18755 −0.593774 0.804632i $$-0.702363\pi$$
−0.593774 + 0.804632i $$0.702363\pi$$
$$332$$ − 1.06392i − 0.0583900i
$$333$$ 0 0
$$334$$ 17.4861 0.956798
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 30.8444i − 1.68020i −0.542430 0.840101i $$-0.682496\pi$$
0.542430 0.840101i $$-0.317504\pi$$
$$338$$ − 15.6333i − 0.850340i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 10.2111 0.552962
$$342$$ 0 0
$$343$$ 9.42221i 0.508751i
$$344$$ 21.6333 1.16639
$$345$$ 0 0
$$346$$ 6.27502 0.337347
$$347$$ 12.5139i 0.671780i 0.941901 + 0.335890i $$0.109037\pi$$
−0.941901 + 0.335890i $$0.890963\pi$$
$$348$$ 0 0
$$349$$ 5.18335 0.277458 0.138729 0.990330i $$-0.455698\pi$$
0.138729 + 0.990330i $$0.455698\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1.69722i 0.0904624i
$$353$$ 18.6333i 0.991751i 0.868394 + 0.495875i $$0.165153\pi$$
−0.868394 + 0.495875i $$0.834847\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0.513878 0.0272355
$$357$$ 0 0
$$358$$ 16.3028i 0.861628i
$$359$$ 0.788897 0.0416364 0.0208182 0.999783i $$-0.493373\pi$$
0.0208182 + 0.999783i $$0.493373\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ − 25.9361i − 1.36317i
$$363$$ 0 0
$$364$$ 1.05551 0.0553239
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 20.6972i − 1.08039i −0.841541 0.540193i $$-0.818351\pi$$
0.841541 0.540193i $$-0.181649\pi$$
$$368$$ 24.1194i 1.25731i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0.908327 0.0471580
$$372$$ 0 0
$$373$$ − 27.4222i − 1.41987i −0.704268 0.709934i $$-0.748725\pi$$
0.704268 0.709934i $$-0.251275\pi$$
$$374$$ 9.00000 0.465379
$$375$$ 0 0
$$376$$ −9.00000 −0.464140
$$377$$ − 4.54163i − 0.233906i
$$378$$ 0 0
$$379$$ −3.18335 −0.163518 −0.0817588 0.996652i $$-0.526054\pi$$
−0.0817588 + 0.996652i $$0.526054\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 13.4222i − 0.686740i
$$383$$ 21.6333i 1.10541i 0.833377 + 0.552705i $$0.186404\pi$$
−0.833377 + 0.552705i $$0.813596\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 17.2111 0.876022
$$387$$ 0 0
$$388$$ 4.63331i 0.235221i
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 0 0
$$391$$ −50.4500 −2.55136
$$392$$ 19.5416i 0.987002i
$$393$$ 0 0
$$394$$ 17.3305 0.873100
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 21.6972i 1.08895i 0.838776 + 0.544476i $$0.183272\pi$$
−0.838776 + 0.544476i $$0.816728\pi$$
$$398$$ 8.44996i 0.423558i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 12.7889 0.638647 0.319324 0.947646i $$-0.396544\pi$$
0.319324 + 0.947646i $$0.396544\pi$$
$$402$$ 0 0
$$403$$ − 51.0555i − 2.54326i
$$404$$ 0.155590 0.00774088
$$405$$ 0 0
$$406$$ −0.825058 −0.0409469
$$407$$ 2.39445i 0.118688i
$$408$$ 0 0
$$409$$ 6.21110 0.307119 0.153560 0.988139i $$-0.450926\pi$$
0.153560 + 0.988139i $$0.450926\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 0.880571i − 0.0433826i
$$413$$ − 9.90833i − 0.487557i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 8.48612 0.416066
$$417$$ 0 0
$$418$$ 1.30278i 0.0637208i
$$419$$ 6.39445 0.312389 0.156195 0.987726i $$-0.450077\pi$$
0.156195 + 0.987726i $$0.450077\pi$$
$$420$$ 0 0
$$421$$ 0.697224 0.0339806 0.0169903 0.999856i $$-0.494592\pi$$
0.0169903 + 0.999856i $$0.494592\pi$$
$$422$$ − 32.8806i − 1.60060i
$$423$$ 0 0
$$424$$ 3.90833 0.189805
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 5.51388i − 0.266835i
$$428$$ 0.908327i 0.0439056i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −33.0000 −1.58955 −0.794777 0.606902i $$-0.792412\pi$$
−0.794777 + 0.606902i $$0.792412\pi$$
$$432$$ 0 0
$$433$$ − 5.00000i − 0.240285i −0.992757 0.120142i $$-0.961665\pi$$
0.992757 0.120142i $$-0.0383351\pi$$
$$434$$ −9.27502 −0.445215
$$435$$ 0 0
$$436$$ −3.48612 −0.166955
$$437$$ − 7.30278i − 0.349339i
$$438$$ 0 0
$$439$$ −24.3028 −1.15991 −0.579954 0.814649i $$-0.696930\pi$$
−0.579954 + 0.814649i $$0.696930\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 45.0000i − 2.14043i
$$443$$ − 8.60555i − 0.408862i −0.978881 0.204431i $$-0.934466\pi$$
0.978881 0.204431i $$-0.0655344\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −29.4861 −1.39621
$$447$$ 0 0
$$448$$ − 6.14719i − 0.290427i
$$449$$ 23.4861 1.10838 0.554189 0.832391i $$-0.313028\pi$$
0.554189 + 0.832391i $$0.313028\pi$$
$$450$$ 0 0
$$451$$ 5.60555 0.263955
$$452$$ − 3.27502i − 0.154044i
$$453$$ 0 0
$$454$$ 2.21110 0.103772
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 20.6972i − 0.968175i −0.875020 0.484088i $$-0.839152\pi$$
0.875020 0.484088i $$-0.160848\pi$$
$$458$$ 24.3944i 1.13988i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −32.2111 −1.50022 −0.750110 0.661313i $$-0.770000\pi$$
−0.750110 + 0.661313i $$0.770000\pi$$
$$462$$ 0 0
$$463$$ − 11.7889i − 0.547877i −0.961747 0.273938i $$-0.911674\pi$$
0.961747 0.273938i $$-0.0883264\pi$$
$$464$$ −3.00000 −0.139272
$$465$$ 0 0
$$466$$ −20.7250 −0.960066
$$467$$ 18.6333i 0.862247i 0.902293 + 0.431123i $$0.141883\pi$$
−0.902293 + 0.431123i $$0.858117\pi$$
$$468$$ 0 0
$$469$$ 2.78890 0.128779
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 42.6333i − 1.96236i
$$473$$ − 7.21110i − 0.331567i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 1.45837 0.0668441
$$477$$ 0 0
$$478$$ 27.5139i 1.25846i
$$479$$ −34.8167 −1.59081 −0.795407 0.606076i $$-0.792743\pi$$
−0.795407 + 0.606076i $$0.792743\pi$$
$$480$$ 0 0
$$481$$ 11.9722 0.545887
$$482$$ 28.5778i 1.30168i
$$483$$ 0 0
$$484$$ 0.302776 0.0137625
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 4.21110i 0.190823i 0.995438 + 0.0954116i $$0.0304167\pi$$
−0.995438 + 0.0954116i $$0.969583\pi$$
$$488$$ − 23.7250i − 1.07398i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 9.78890 0.441767 0.220883 0.975300i $$-0.429106\pi$$
0.220883 + 0.975300i $$0.429106\pi$$
$$492$$ 0 0
$$493$$ − 6.27502i − 0.282613i
$$494$$ 6.51388 0.293073
$$495$$ 0 0
$$496$$ −33.7250 −1.51430
$$497$$ 1.81665i 0.0814881i
$$498$$ 0 0
$$499$$ 3.48612 0.156060 0.0780301 0.996951i $$-0.475137\pi$$
0.0780301 + 0.996951i $$0.475137\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 9.00000i − 0.401690i
$$503$$ 9.39445i 0.418878i 0.977822 + 0.209439i $$0.0671637\pi$$
−0.977822 + 0.209439i $$0.932836\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 9.51388 0.422943
$$507$$ 0 0
$$508$$ 2.45837i 0.109072i
$$509$$ −22.6972 −1.00604 −0.503018 0.864276i $$-0.667777\pi$$
−0.503018 + 0.864276i $$0.667777\pi$$
$$510$$ 0 0
$$511$$ −5.51388 −0.243920
$$512$$ − 25.4222i − 1.12351i
$$513$$ 0 0
$$514$$ −23.4500 −1.03433
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 3.00000i 0.131940i
$$518$$ − 2.17494i − 0.0955615i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 41.4500 1.81596 0.907978 0.419018i $$-0.137626\pi$$
0.907978 + 0.419018i $$0.137626\pi$$
$$522$$ 0 0
$$523$$ 32.4222i 1.41772i 0.705347 + 0.708862i $$0.250791\pi$$
−0.705347 + 0.708862i $$0.749209\pi$$
$$524$$ 3.00000 0.131056
$$525$$ 0 0
$$526$$ 29.7250 1.29607
$$527$$ − 70.5416i − 3.07284i
$$528$$ 0 0
$$529$$ −30.3305 −1.31872
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0.211103i 0.00915246i
$$533$$ − 28.0278i − 1.21402i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ 11.3667i 0.490053i
$$539$$ 6.51388 0.280573
$$540$$ 0 0
$$541$$ 25.7250 1.10600 0.553002 0.833180i $$-0.313482\pi$$
0.553002 + 0.833180i $$0.313482\pi$$
$$542$$ − 0.275019i − 0.0118131i
$$543$$ 0 0
$$544$$ 11.7250 0.502704
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 7.11943i − 0.304405i −0.988349 0.152202i $$-0.951363\pi$$
0.988349 0.152202i $$-0.0486366\pi$$
$$548$$ 3.90833i 0.166955i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0.908327 0.0386960
$$552$$ 0 0
$$553$$ 7.60555i 0.323421i
$$554$$ −18.7527 −0.796727
$$555$$ 0 0
$$556$$ 1.88057 0.0797540
$$557$$ − 19.4222i − 0.822945i −0.911422 0.411473i $$-0.865015\pi$$
0.911422 0.411473i $$-0.134985\pi$$
$$558$$ 0 0
$$559$$ −36.0555 −1.52499
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 1.54163i 0.0650299i
$$563$$ 8.09167i 0.341023i 0.985356 + 0.170512i $$0.0545421\pi$$
−0.985356 + 0.170512i $$0.945458\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 8.21110 0.345138
$$567$$ 0 0
$$568$$ 7.81665i 0.327980i
$$569$$ −46.1472 −1.93459 −0.967295 0.253653i $$-0.918368\pi$$
−0.967295 + 0.253653i $$0.918368\pi$$
$$570$$ 0 0
$$571$$ 22.3305 0.934504 0.467252 0.884124i $$-0.345244\pi$$
0.467252 + 0.884124i $$0.345244\pi$$
$$572$$ − 1.51388i − 0.0632984i
$$573$$ 0 0
$$574$$ −5.09167 −0.212522
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 44.3583i 1.84666i 0.384008 + 0.923330i $$0.374544\pi$$
−0.384008 + 0.923330i $$0.625456\pi$$
$$578$$ − 40.0278i − 1.66494i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 2.44996 0.101642
$$582$$ 0 0
$$583$$ − 1.30278i − 0.0539555i
$$584$$ −23.7250 −0.981747
$$585$$ 0 0
$$586$$ −1.02776 −0.0424562
$$587$$ − 16.5416i − 0.682746i −0.939928 0.341373i $$-0.889108\pi$$
0.939928 0.341373i $$-0.110892\pi$$
$$588$$ 0 0
$$589$$ 10.2111 0.420741
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 7.90833i − 0.325030i
$$593$$ − 6.39445i − 0.262589i −0.991343 0.131294i $$-0.958087\pi$$
0.991343 0.131294i $$-0.0419133\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 5.21110 0.213455
$$597$$ 0 0
$$598$$ − 47.5694i − 1.94526i
$$599$$ −24.9083 −1.01773 −0.508863 0.860847i $$-0.669934\pi$$
−0.508863 + 0.860847i $$0.669934\pi$$
$$600$$ 0 0
$$601$$ −1.90833 −0.0778423 −0.0389211 0.999242i $$-0.512392\pi$$
−0.0389211 + 0.999242i $$0.512392\pi$$
$$602$$ 6.55004i 0.266960i
$$603$$ 0 0
$$604$$ 0.247263 0.0100610
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 7.21110i 0.292690i 0.989234 + 0.146345i $$0.0467509\pi$$
−0.989234 + 0.146345i $$0.953249\pi$$
$$608$$ 1.69722i 0.0688315i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 15.0000 0.606835
$$612$$ 0 0
$$613$$ 15.8806i 0.641410i 0.947179 + 0.320705i $$0.103920\pi$$
−0.947179 + 0.320705i $$0.896080\pi$$
$$614$$ 22.0278 0.888968
$$615$$ 0 0
$$616$$ −2.09167 −0.0842759
$$617$$ − 3.39445i − 0.136655i −0.997663 0.0683277i $$-0.978234\pi$$
0.997663 0.0683277i $$-0.0217663\pi$$
$$618$$ 0 0
$$619$$ 11.4222 0.459097 0.229549 0.973297i $$-0.426275\pi$$
0.229549 + 0.973297i $$0.426275\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 6.27502i − 0.251605i
$$623$$ 1.18335i 0.0474098i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0.238859 0.00954672
$$627$$ 0 0
$$628$$ 5.81665i 0.232110i
$$629$$ 16.5416 0.659558
$$630$$ 0 0
$$631$$ 6.93608 0.276121 0.138061 0.990424i $$-0.455913\pi$$
0.138061 + 0.990424i $$0.455913\pi$$
$$632$$ 32.7250i 1.30173i
$$633$$ 0 0
$$634$$ −1.18335 −0.0469967
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 32.5694i − 1.29045i
$$638$$ 1.18335i 0.0468491i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −27.7889 −1.09760 −0.548798 0.835955i $$-0.684914\pi$$
−0.548798 + 0.835955i $$0.684914\pi$$
$$642$$ 0 0
$$643$$ 22.0000i 0.867595i 0.901010 + 0.433798i $$0.142827\pi$$
−0.901010 + 0.433798i $$0.857173\pi$$
$$644$$ 1.54163 0.0607489
$$645$$ 0 0
$$646$$ 9.00000 0.354100
$$647$$ 33.2389i 1.30675i 0.757033 + 0.653377i $$0.226648\pi$$
−0.757033 + 0.653377i $$0.773352\pi$$
$$648$$ 0 0
$$649$$ −14.2111 −0.557835
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 2.81665i − 0.110309i
$$653$$ 6.11943i 0.239472i 0.992806 + 0.119736i $$0.0382048\pi$$
−0.992806 + 0.119736i $$0.961795\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −18.5139 −0.722846
$$657$$ 0 0
$$658$$ − 2.72498i − 0.106231i
$$659$$ −30.9083 −1.20402 −0.602009 0.798489i $$-0.705633\pi$$
−0.602009 + 0.798489i $$0.705633\pi$$
$$660$$ 0 0
$$661$$ −8.81665 −0.342928 −0.171464 0.985190i $$-0.554850\pi$$
−0.171464 + 0.985190i $$0.554850\pi$$
$$662$$ − 28.1472i − 1.09397i
$$663$$ 0 0
$$664$$ 10.5416 0.409095
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 6.63331i − 0.256843i
$$668$$ − 4.06392i − 0.157238i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −7.90833 −0.305298
$$672$$ 0 0
$$673$$ − 30.0278i − 1.15748i −0.815510 0.578742i $$-0.803544\pi$$
0.815510 0.578742i $$-0.196456\pi$$
$$674$$ 40.1833 1.54780
$$675$$ 0 0
$$676$$ −3.63331 −0.139743
$$677$$ − 24.2389i − 0.931575i −0.884897 0.465788i $$-0.845771\pi$$
0.884897 0.465788i $$-0.154229\pi$$
$$678$$ 0 0
$$679$$ −10.6695 −0.409457
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 13.3028i 0.509390i
$$683$$ 47.8444i 1.83072i 0.402642 + 0.915358i $$0.368092\pi$$
−0.402642 + 0.915358i $$0.631908\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −12.2750 −0.468662
$$687$$ 0 0
$$688$$ 23.8167i 0.908001i
$$689$$ −6.51388 −0.248159
$$690$$ 0 0
$$691$$ 27.5416 1.04773 0.523867 0.851800i $$-0.324489\pi$$
0.523867 + 0.851800i $$0.324489\pi$$
$$692$$ − 1.45837i − 0.0554387i
$$693$$ 0 0
$$694$$ −16.3028 −0.618845
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 38.7250i − 1.46681i
$$698$$ 6.75274i 0.255595i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 41.2111 1.55652 0.778261 0.627941i $$-0.216102\pi$$
0.778261 + 0.627941i $$0.216102\pi$$
$$702$$ 0 0
$$703$$ 2.39445i 0.0903083i
$$704$$ −8.81665 −0.332290
$$705$$ 0 0
$$706$$ −24.2750 −0.913602
$$707$$ 0.358288i 0.0134748i
$$708$$ 0 0
$$709$$ 31.6333 1.18801 0.594007 0.804460i $$-0.297545\pi$$
0.594007 + 0.804460i $$0.297545\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 5.09167i 0.190819i
$$713$$ − 74.5694i − 2.79265i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 3.78890 0.141598
$$717$$ 0 0
$$718$$ 1.02776i 0.0383555i
$$719$$ 7.18335 0.267894 0.133947 0.990989i $$-0.457235\pi$$
0.133947 + 0.990989i $$0.457235\pi$$
$$720$$ 0 0
$$721$$ 2.02776 0.0755176
$$722$$ − 23.4500i − 0.872717i
$$723$$ 0 0
$$724$$ −6.02776 −0.224020
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 39.3305i − 1.45869i −0.684147 0.729344i $$-0.739825\pi$$
0.684147 0.729344i $$-0.260175\pi$$
$$728$$ 10.4584i 0.387613i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −49.8167 −1.84254
$$732$$ 0 0
$$733$$ − 19.6056i − 0.724148i −0.932149 0.362074i $$-0.882069\pi$$
0.932149 0.362074i $$-0.117931\pi$$
$$734$$ 26.9638 0.995253
$$735$$ 0 0
$$736$$ 12.3944 0.456865
$$737$$ − 4.00000i − 0.147342i
$$738$$ 0 0
$$739$$ −35.1194 −1.29189 −0.645945 0.763384i $$-0.723536\pi$$
−0.645945 + 0.763384i $$0.723536\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 1.18335i 0.0434420i
$$743$$ 40.6972i 1.49304i 0.665365 + 0.746518i $$0.268276\pi$$
−0.665365 + 0.746518i $$0.731724\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 35.7250 1.30798
$$747$$ 0 0
$$748$$ − 2.09167i − 0.0764791i
$$749$$ −2.09167 −0.0764281
$$750$$ 0 0
$$751$$ −45.3305 −1.65413 −0.827067 0.562103i $$-0.809992\pi$$
−0.827067 + 0.562103i $$0.809992\pi$$
$$752$$ − 9.90833i − 0.361320i
$$753$$ 0 0
$$754$$ 5.91673 0.215475
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 49.0555i 1.78295i 0.453067 + 0.891476i $$0.350330\pi$$
−0.453067 + 0.891476i $$0.649670\pi$$
$$758$$ − 4.14719i − 0.150633i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 13.5778 0.492195 0.246097 0.969245i $$-0.420852\pi$$
0.246097 + 0.969245i $$0.420852\pi$$
$$762$$ 0 0
$$763$$ − 8.02776i − 0.290624i
$$764$$ −3.11943 −0.112857
$$765$$ 0 0
$$766$$ −28.1833 −1.01831
$$767$$ 71.0555i 2.56567i
$$768$$ 0 0
$$769$$ 5.18335 0.186916 0.0934581 0.995623i $$-0.470208\pi$$
0.0934581 + 0.995623i $$0.470208\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 4.00000i − 0.143963i
$$773$$ 3.11943i 0.112198i 0.998425 + 0.0560990i $$0.0178663\pi$$
−0.998425 + 0.0560990i $$0.982134\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −45.9083 −1.64801
$$777$$ 0 0
$$778$$ 15.6333i 0.560481i
$$779$$ 5.60555 0.200840
$$780$$ 0 0
$$781$$ 2.60555 0.0932340
$$782$$ − 65.7250i − 2.35032i
$$783$$ 0 0
$$784$$ −21.5139 −0.768353
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 10.2111i 0.363986i 0.983300 + 0.181993i $$0.0582549\pi$$
−0.983300 + 0.181993i $$0.941745\pi$$
$$788$$ − 4.02776i − 0.143483i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 7.54163 0.268150
$$792$$ 0 0
$$793$$ 39.5416i 1.40416i
$$794$$ −28.2666 −1.00314
$$795$$ 0 0
$$796$$ 1.96384 0.0696065
$$797$$ − 3.51388i − 0.124468i −0.998062 0.0622340i $$-0.980178\pi$$
0.998062 0.0622340i $$-0.0198225\pi$$
$$798$$ 0 0
$$799$$ 20.7250 0.733197
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 16.6611i 0.588323i
$$803$$ 7.90833i 0.279079i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 66.5139 2.34285
$$807$$ 0 0
$$808$$ 1.54163i 0.0542345i
$$809$$ 39.6333 1.39343 0.696716 0.717347i $$-0.254644\pi$$
0.696716 + 0.717347i $$0.254644\pi$$
$$810$$ 0 0
$$811$$ 38.8722 1.36499 0.682493 0.730892i $$-0.260896\pi$$
0.682493 + 0.730892i $$0.260896\pi$$
$$812$$ 0.191750i 0.00672911i
$$813$$ 0 0
$$814$$ −3.11943 −0.109336
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 7.21110i − 0.252285i
$$818$$ 8.09167i 0.282919i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −12.0000 −0.418803 −0.209401 0.977830i $$-0.567152\pi$$
−0.209401 + 0.977830i $$0.567152\pi$$
$$822$$ 0 0
$$823$$ − 18.4222i − 0.642158i −0.947052 0.321079i $$-0.895955\pi$$
0.947052 0.321079i $$-0.104045\pi$$
$$824$$ 8.72498 0.303949
$$825$$ 0 0
$$826$$ 12.9083 0.449138
$$827$$ − 13.8167i − 0.480452i −0.970717 0.240226i $$-0.922778\pi$$
0.970717 0.240226i $$-0.0772216\pi$$
$$828$$ 0 0
$$829$$ −29.7527 −1.03336 −0.516678 0.856180i $$-0.672831\pi$$
−0.516678 + 0.856180i $$0.672831\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 44.0833i 1.52831i
$$833$$ − 45.0000i − 1.55916i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0.302776 0.0104717
$$837$$ 0 0
$$838$$ 8.33053i 0.287773i
$$839$$ −9.11943 −0.314838 −0.157419 0.987532i $$-0.550317\pi$$
−0.157419 + 0.987532i $$0.550317\pi$$
$$840$$ 0 0
$$841$$ −28.1749 −0.971550
$$842$$ 0.908327i 0.0313030i
$$843$$ 0 0
$$844$$ −7.64171 −0.263039
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0.697224i 0.0239569i
$$848$$ 4.30278i 0.147758i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 17.4861 0.599417
$$852$$ 0 0
$$853$$ 12.7250i 0.435695i 0.975983 + 0.217848i $$0.0699035\pi$$
−0.975983 + 0.217848i $$0.930096\pi$$
$$854$$ 7.18335 0.245809
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ 3.00000i 0.102478i 0.998686 + 0.0512390i $$0.0163170\pi$$
−0.998686 + 0.0512390i $$0.983683\pi$$
$$858$$ 0 0
$$859$$ −41.3944 −1.41236 −0.706180 0.708032i $$-0.749583\pi$$
−0.706180 + 0.708032i $$0.749583\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 42.9916i − 1.46430i
$$863$$ − 12.3944i − 0.421912i −0.977496 0.210956i $$-0.932342\pi$$
0.977496 0.210956i $$-0.0676577\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 6.51388 0.221351
$$867$$ 0 0
$$868$$ 2.15559i 0.0731655i
$$869$$ 10.9083 0.370040
$$870$$ 0 0
$$871$$ −20.0000 −0.677674
$$872$$ − 34.5416i − 1.16973i
$$873$$ 0 0
$$874$$ 9.51388 0.321812
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 20.0000i 0.675352i 0.941262 + 0.337676i $$0.109641\pi$$
−0.941262 + 0.337676i $$0.890359\pi$$
$$878$$ − 31.6611i − 1.06851i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −19.5416 −0.658374 −0.329187 0.944265i $$-0.606775\pi$$
−0.329187 + 0.944265i $$0.606775\pi$$
$$882$$ 0 0
$$883$$ − 52.4500i − 1.76508i −0.470236 0.882541i $$-0.655831\pi$$
0.470236 0.882541i $$-0.344169\pi$$
$$884$$ −10.4584 −0.351753
$$885$$ 0 0
$$886$$ 11.2111 0.376644
$$887$$ 3.23886i 0.108750i 0.998521 + 0.0543751i $$0.0173167\pi$$
−0.998521 + 0.0543751i $$0.982683\pi$$
$$888$$ 0 0
$$889$$ −5.66106 −0.189866
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 6.85281i 0.229449i
$$893$$ 3.00000i 0.100391i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 5.64171 0.188476
$$897$$ 0 0
$$898$$ 30.5971i 1.02104i
$$899$$ 9.27502 0.309339
$$900$$ 0 0
$$901$$ −9.00000 −0.299833
$$902$$ 7.30278i 0.243156i
$$903$$ 0 0
$$904$$ 32.4500 1.07927
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 4.00000i − 0.132818i −0.997792 0.0664089i $$-0.978846\pi$$
0.997792 0.0664089i $$-0.0211542\pi$$
$$908$$ − 0.513878i − 0.0170536i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 24.7889 0.821293 0.410646 0.911795i $$-0.365303\pi$$
0.410646 + 0.911795i $$0.365303\pi$$
$$912$$ 0 0
$$913$$ − 3.51388i − 0.116292i
$$914$$ 26.9638 0.891885
$$915$$ 0 0
$$916$$ 5.66947 0.187324
$$917$$ 6.90833i 0.228133i
$$918$$ 0 0
$$919$$ −26.7889 −0.883684 −0.441842 0.897093i $$-0.645675\pi$$
−0.441842 + 0.897093i $$0.645675\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 41.9638i − 1.38201i
$$923$$ − 13.0278i − 0.428814i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 15.3583 0.504705
$$927$$ 0 0
$$928$$ 1.54163i 0.0506066i
$$929$$ 53.6056 1.75874 0.879371 0.476138i $$-0.157964\pi$$
0.879371 + 0.476138i $$0.157964\pi$$
$$930$$ 0 0
$$931$$ 6.51388 0.213484
$$932$$ 4.81665i 0.157775i
$$933$$ 0 0
$$934$$ −24.2750 −0.794303
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 9.21110i − 0.300914i −0.988617 0.150457i $$-0.951926\pi$$
0.988617 0.150457i $$-0.0480744\pi$$
$$938$$ 3.63331i 0.118632i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −59.6056 −1.94309 −0.971543 0.236864i $$-0.923880\pi$$
−0.971543 + 0.236864i $$0.923880\pi$$
$$942$$ 0 0
$$943$$ − 40.9361i − 1.33306i
$$944$$ 46.9361 1.52764
$$945$$ 0 0
$$946$$ 9.39445 0.305440
$$947$$ − 6.63331i − 0.215554i −0.994175 0.107777i $$-0.965627\pi$$
0.994175 0.107777i $$-0.0343732\pi$$
$$948$$ 0 0
$$949$$ 39.5416 1.28358
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 14.4500i 0.468326i
$$953$$ 37.2666i 1.20718i 0.797293 + 0.603592i $$0.206264\pi$$
−0.797293 + 0.603592i $$0.793736\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 6.39445 0.206811
$$957$$ 0 0
$$958$$ − 45.3583i − 1.46546i
$$959$$ −9.00000 −0.290625
$$960$$ 0 0
$$961$$ 73.2666 2.36344
$$962$$ 15.5971i 0.502872i
$$963$$ 0 0
$$964$$ 6.64171 0.213915
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 14.9083i 0.479419i 0.970845 + 0.239710i $$0.0770523\pi$$
−0.970845 + 0.239710i $$0.922948\pi$$
$$968$$ 3.00000i 0.0964237i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −45.3583 −1.45562 −0.727808 0.685781i $$-0.759461\pi$$
−0.727808 + 0.685781i $$0.759461\pi$$
$$972$$ 0 0
$$973$$ 4.33053i 0.138830i
$$974$$ −5.48612 −0.175787
$$975$$ 0 0
$$976$$ 26.1194 0.836063
$$977$$ 52.0278i 1.66452i 0.554389 + 0.832258i $$0.312952\pi$$
−0.554389 + 0.832258i $$0.687048\pi$$
$$978$$ 0 0
$$979$$ 1.69722 0.0542435
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 12.7527i 0.406956i
$$983$$ 8.84441i 0.282093i 0.990003 + 0.141046i $$0.0450467\pi$$
−0.990003 + 0.141046i $$0.954953\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 8.17494 0.260343
$$987$$ 0 0
$$988$$ − 1.51388i − 0.0481629i
$$989$$ −52.6611 −1.67452
$$990$$ 0 0
$$991$$ −16.9083 −0.537111 −0.268555 0.963264i $$-0.586546\pi$$
−0.268555 + 0.963264i $$0.586546\pi$$
$$992$$ 17.3305i 0.550245i
$$993$$ 0 0
$$994$$ −2.36669 −0.0750669
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 46.7250i 1.47979i 0.672720 + 0.739897i $$0.265126\pi$$
−0.672720 + 0.739897i $$0.734874\pi$$
$$998$$ 4.54163i 0.143763i
$$999$$ 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 2475.2.c.k.199.3 4
3.2 odd 2 275.2.b.c.199.2 4
5.2 odd 4 2475.2.a.t.1.1 2
5.3 odd 4 2475.2.a.o.1.2 2
5.4 even 2 inner 2475.2.c.k.199.2 4
12.11 even 2 4400.2.b.y.4049.4 4
15.2 even 4 275.2.a.e.1.2 2
15.8 even 4 275.2.a.f.1.1 yes 2
15.14 odd 2 275.2.b.c.199.3 4
60.23 odd 4 4400.2.a.bh.1.1 2
60.47 odd 4 4400.2.a.bs.1.2 2
60.59 even 2 4400.2.b.y.4049.1 4
165.32 odd 4 3025.2.a.n.1.1 2
165.98 odd 4 3025.2.a.h.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.2 2 15.2 even 4
275.2.a.f.1.1 yes 2 15.8 even 4
275.2.b.c.199.2 4 3.2 odd 2
275.2.b.c.199.3 4 15.14 odd 2
2475.2.a.o.1.2 2 5.3 odd 4
2475.2.a.t.1.1 2 5.2 odd 4
2475.2.c.k.199.2 4 5.4 even 2 inner
2475.2.c.k.199.3 4 1.1 even 1 trivial
3025.2.a.h.1.2 2 165.98 odd 4
3025.2.a.n.1.1 2 165.32 odd 4
4400.2.a.bh.1.1 2 60.23 odd 4
4400.2.a.bs.1.2 2 60.47 odd 4
4400.2.b.y.4049.1 4 60.59 even 2
4400.2.b.y.4049.4 4 12.11 even 2