# Properties

 Label 2475.2.a.o.1.2 Level $2475$ Weight $2$ Character 2475.1 Self dual yes Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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, 0, 0]))

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

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.a (trivial)

## 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: yes Analytic conductor: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 275) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.30278 q^{2} -0.302776 q^{4} +0.697224 q^{7} -3.00000 q^{8} +O(q^{10})$$ $$q+1.30278 q^{2} -0.302776 q^{4} +0.697224 q^{7} -3.00000 q^{8} +1.00000 q^{11} +5.00000 q^{13} +0.908327 q^{14} -3.30278 q^{16} -6.90833 q^{17} -1.00000 q^{19} +1.30278 q^{22} +7.30278 q^{23} +6.51388 q^{26} -0.211103 q^{28} -0.908327 q^{29} +10.2111 q^{31} +1.69722 q^{32} -9.00000 q^{34} +2.39445 q^{37} -1.30278 q^{38} +5.60555 q^{41} +7.21110 q^{43} -0.302776 q^{44} +9.51388 q^{46} +3.00000 q^{47} -6.51388 q^{49} -1.51388 q^{52} +1.30278 q^{53} -2.09167 q^{56} -1.18335 q^{58} +14.2111 q^{59} -7.90833 q^{61} +13.3028 q^{62} +8.81665 q^{64} -4.00000 q^{67} +2.09167 q^{68} +2.60555 q^{71} -7.90833 q^{73} +3.11943 q^{74} +0.302776 q^{76} +0.697224 q^{77} -10.9083 q^{79} +7.30278 q^{82} +3.51388 q^{83} +9.39445 q^{86} -3.00000 q^{88} -1.69722 q^{89} +3.48612 q^{91} -2.21110 q^{92} +3.90833 q^{94} +15.3028 q^{97} -8.48612 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - q^2 + 3 * q^4 + 5 * q^7 - 6 * q^8 $$2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8} + 2 q^{11} + 10 q^{13} - 9 q^{14} - 3 q^{16} - 3 q^{17} - 2 q^{19} - q^{22} + 11 q^{23} - 5 q^{26} + 14 q^{28} + 9 q^{29} + 6 q^{31} + 7 q^{32} - 18 q^{34} + 12 q^{37} + q^{38} + 4 q^{41} + 3 q^{44} + q^{46} + 6 q^{47} + 5 q^{49} + 15 q^{52} - q^{53} - 15 q^{56} - 24 q^{58} + 14 q^{59} - 5 q^{61} + 23 q^{62} - 4 q^{64} - 8 q^{67} + 15 q^{68} - 2 q^{71} - 5 q^{73} - 19 q^{74} - 3 q^{76} + 5 q^{77} - 11 q^{79} + 11 q^{82} - 11 q^{83} + 26 q^{86} - 6 q^{88} - 7 q^{89} + 25 q^{91} + 10 q^{92} - 3 q^{94} + 27 q^{97} - 35 q^{98}+O(q^{100})$$ 2 * q - q^2 + 3 * q^4 + 5 * q^7 - 6 * q^8 + 2 * q^11 + 10 * q^13 - 9 * q^14 - 3 * q^16 - 3 * q^17 - 2 * q^19 - q^22 + 11 * q^23 - 5 * q^26 + 14 * q^28 + 9 * q^29 + 6 * q^31 + 7 * q^32 - 18 * q^34 + 12 * q^37 + q^38 + 4 * q^41 + 3 * q^44 + q^46 + 6 * q^47 + 5 * q^49 + 15 * q^52 - q^53 - 15 * q^56 - 24 * q^58 + 14 * q^59 - 5 * q^61 + 23 * q^62 - 4 * q^64 - 8 * q^67 + 15 * q^68 - 2 * q^71 - 5 * q^73 - 19 * q^74 - 3 * q^76 + 5 * q^77 - 11 * q^79 + 11 * q^82 - 11 * q^83 + 26 * q^86 - 6 * q^88 - 7 * q^89 + 25 * q^91 + 10 * q^92 - 3 * q^94 + 27 * q^97 - 35 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.30278 0.921201 0.460601 0.887607i $$-0.347634\pi$$
0.460601 + 0.887607i $$0.347634\pi$$
$$3$$ 0 0
$$4$$ −0.302776 −0.151388
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.697224 0.263526 0.131763 0.991281i $$-0.457936\pi$$
0.131763 + 0.991281i $$0.457936\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ 0.908327 0.242761
$$15$$ 0 0
$$16$$ −3.30278 −0.825694
$$17$$ −6.90833 −1.67552 −0.837758 0.546042i $$-0.816134\pi$$
−0.837758 + 0.546042i $$0.816134\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 1.30278 0.277753
$$23$$ 7.30278 1.52273 0.761367 0.648321i $$-0.224529\pi$$
0.761367 + 0.648321i $$0.224529\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 6.51388 1.27748
$$27$$ 0 0
$$28$$ −0.211103 −0.0398946
$$29$$ −0.908327 −0.168672 −0.0843360 0.996437i $$-0.526877\pi$$
−0.0843360 + 0.996437i $$0.526877\pi$$
$$30$$ 0 0
$$31$$ 10.2111 1.83397 0.916984 0.398924i $$-0.130616\pi$$
0.916984 + 0.398924i $$0.130616\pi$$
$$32$$ 1.69722 0.300030
$$33$$ 0 0
$$34$$ −9.00000 −1.54349
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.39445 0.393645 0.196822 0.980439i $$-0.436938\pi$$
0.196822 + 0.980439i $$0.436938\pi$$
$$38$$ −1.30278 −0.211338
$$39$$ 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
$$43$$ 7.21110 1.09968 0.549841 0.835269i $$-0.314688\pi$$
0.549841 + 0.835269i $$0.314688\pi$$
$$44$$ −0.302776 −0.0456451
$$45$$ 0 0
$$46$$ 9.51388 1.40274
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ 0 0
$$49$$ −6.51388 −0.930554
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −1.51388 −0.209937
$$53$$ 1.30278 0.178950 0.0894750 0.995989i $$-0.471481\pi$$
0.0894750 + 0.995989i $$0.471481\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −2.09167 −0.279512
$$57$$ 0 0
$$58$$ −1.18335 −0.155381
$$59$$ 14.2111 1.85013 0.925064 0.379811i $$-0.124011\pi$$
0.925064 + 0.379811i $$0.124011\pi$$
$$60$$ 0 0
$$61$$ −7.90833 −1.01256 −0.506279 0.862370i $$-0.668979\pi$$
−0.506279 + 0.862370i $$0.668979\pi$$
$$62$$ 13.3028 1.68945
$$63$$ 0 0
$$64$$ 8.81665 1.10208
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 2.09167 0.253653
$$69$$ 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
$$73$$ −7.90833 −0.925600 −0.462800 0.886463i $$-0.653155\pi$$
−0.462800 + 0.886463i $$0.653155\pi$$
$$74$$ 3.11943 0.362626
$$75$$ 0 0
$$76$$ 0.302776 0.0347307
$$77$$ 0.697224 0.0794561
$$78$$ 0 0
$$79$$ −10.9083 −1.22728 −0.613641 0.789585i $$-0.710296\pi$$
−0.613641 + 0.789585i $$0.710296\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 7.30278 0.806457
$$83$$ 3.51388 0.385698 0.192849 0.981228i $$-0.438227\pi$$
0.192849 + 0.981228i $$0.438227\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 9.39445 1.01303
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ −1.69722 −0.179905 −0.0899527 0.995946i $$-0.528672\pi$$
−0.0899527 + 0.995946i $$0.528672\pi$$
$$90$$ 0 0
$$91$$ 3.48612 0.365445
$$92$$ −2.21110 −0.230523
$$93$$ 0 0
$$94$$ 3.90833 0.403113
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 15.3028 1.55376 0.776881 0.629648i $$-0.216801\pi$$
0.776881 + 0.629648i $$0.216801\pi$$
$$98$$ −8.48612 −0.857228
$$99$$ 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
$$103$$ 2.90833 0.286566 0.143283 0.989682i $$-0.454234\pi$$
0.143283 + 0.989682i $$0.454234\pi$$
$$104$$ −15.0000 −1.47087
$$105$$ 0 0
$$106$$ 1.69722 0.164849
$$107$$ 3.00000 0.290021 0.145010 0.989430i $$-0.453678\pi$$
0.145010 + 0.989430i $$0.453678\pi$$
$$108$$ 0 0
$$109$$ 11.5139 1.10283 0.551415 0.834231i $$-0.314088\pi$$
0.551415 + 0.834231i $$0.314088\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −2.30278 −0.217592
$$113$$ 10.8167 1.01755 0.508773 0.860901i $$-0.330099\pi$$
0.508773 + 0.860901i $$0.330099\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0.275019 0.0255349
$$117$$ 0 0
$$118$$ 18.5139 1.70434
$$119$$ −4.81665 −0.441542
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −10.3028 −0.932769
$$123$$ 0 0
$$124$$ −3.09167 −0.277640
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.11943 0.720483 0.360241 0.932859i $$-0.382694\pi$$
0.360241 + 0.932859i $$0.382694\pi$$
$$128$$ 8.09167 0.715210
$$129$$ 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
$$133$$ −0.697224 −0.0604570
$$134$$ −5.21110 −0.450171
$$135$$ 0 0
$$136$$ 20.7250 1.77715
$$137$$ 12.9083 1.10283 0.551416 0.834230i $$-0.314088\pi$$
0.551416 + 0.834230i $$0.314088\pi$$
$$138$$ 0 0
$$139$$ −6.21110 −0.526819 −0.263409 0.964684i $$-0.584847\pi$$
−0.263409 + 0.964684i $$0.584847\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 3.39445 0.284856
$$143$$ 5.00000 0.418121
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −10.3028 −0.852664
$$147$$ 0 0
$$148$$ −0.724981 −0.0595930
$$149$$ −17.2111 −1.40999 −0.704994 0.709213i $$-0.749050\pi$$
−0.704994 + 0.709213i $$0.749050\pi$$
$$150$$ 0 0
$$151$$ 0.816654 0.0664583 0.0332292 0.999448i $$-0.489421\pi$$
0.0332292 + 0.999448i $$0.489421\pi$$
$$152$$ 3.00000 0.243332
$$153$$ 0 0
$$154$$ 0.908327 0.0731951
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 19.2111 1.53321 0.766606 0.642117i $$-0.221944\pi$$
0.766606 + 0.642117i $$0.221944\pi$$
$$158$$ −14.2111 −1.13057
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 5.09167 0.401280
$$162$$ 0 0
$$163$$ 9.30278 0.728650 0.364325 0.931272i $$-0.381300\pi$$
0.364325 + 0.931272i $$0.381300\pi$$
$$164$$ −1.69722 −0.132531
$$165$$ 0 0
$$166$$ 4.57779 0.355306
$$167$$ −13.4222 −1.03864 −0.519321 0.854579i $$-0.673815\pi$$
−0.519321 + 0.854579i $$0.673815\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −2.18335 −0.166479
$$173$$ 4.81665 0.366203 0.183102 0.983094i $$-0.441386\pi$$
0.183102 + 0.983094i $$0.441386\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.30278 −0.248956
$$177$$ 0 0
$$178$$ −2.21110 −0.165729
$$179$$ −12.5139 −0.935331 −0.467666 0.883905i $$-0.654905\pi$$
−0.467666 + 0.883905i $$0.654905\pi$$
$$180$$ 0 0
$$181$$ −19.9083 −1.47977 −0.739887 0.672731i $$-0.765121\pi$$
−0.739887 + 0.672731i $$0.765121\pi$$
$$182$$ 4.54163 0.336648
$$183$$ 0 0
$$184$$ −21.9083 −1.61510
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −6.90833 −0.505187
$$188$$ −0.908327 −0.0662465
$$189$$ 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
$$193$$ 13.2111 0.950956 0.475478 0.879728i $$-0.342275\pi$$
0.475478 + 0.879728i $$0.342275\pi$$
$$194$$ 19.9361 1.43133
$$195$$ 0 0
$$196$$ 1.97224 0.140875
$$197$$ −13.3028 −0.947784 −0.473892 0.880583i $$-0.657151\pi$$
−0.473892 + 0.880583i $$0.657151\pi$$
$$198$$ 0 0
$$199$$ −6.48612 −0.459789 −0.229894 0.973216i $$-0.573838\pi$$
−0.229894 + 0.973216i $$0.573838\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0.669468 0.0471036
$$203$$ −0.633308 −0.0444495
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 3.78890 0.263985
$$207$$ 0 0
$$208$$ −16.5139 −1.14503
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ −25.2389 −1.73751 −0.868757 0.495238i $$-0.835081\pi$$
−0.868757 + 0.495238i $$0.835081\pi$$
$$212$$ −0.394449 −0.0270908
$$213$$ 0 0
$$214$$ 3.90833 0.267168
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 7.11943 0.483298
$$218$$ 15.0000 1.01593
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −34.5416 −2.32352
$$222$$ 0 0
$$223$$ −22.6333 −1.51564 −0.757819 0.652465i $$-0.773735\pi$$
−0.757819 + 0.652465i $$0.773735\pi$$
$$224$$ 1.18335 0.0790656
$$225$$ 0 0
$$226$$ 14.0917 0.937364
$$227$$ −1.69722 −0.112649 −0.0563244 0.998413i $$-0.517938\pi$$
−0.0563244 + 0.998413i $$0.517938\pi$$
$$228$$ 0 0
$$229$$ −18.7250 −1.23738 −0.618691 0.785635i $$-0.712337\pi$$
−0.618691 + 0.785635i $$0.712337\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.72498 0.178904
$$233$$ −15.9083 −1.04219 −0.521095 0.853499i $$-0.674476\pi$$
−0.521095 + 0.853499i $$0.674476\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −4.30278 −0.280087
$$237$$ 0 0
$$238$$ −6.27502 −0.406749
$$239$$ −21.1194 −1.36610 −0.683051 0.730371i $$-0.739347\pi$$
−0.683051 + 0.730371i $$0.739347\pi$$
$$240$$ 0 0
$$241$$ 21.9361 1.41303 0.706514 0.707699i $$-0.250267\pi$$
0.706514 + 0.707699i $$0.250267\pi$$
$$242$$ 1.30278 0.0837456
$$243$$ 0 0
$$244$$ 2.39445 0.153289
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −5.00000 −0.318142
$$248$$ −30.6333 −1.94522
$$249$$ 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
$$253$$ 7.30278 0.459122
$$254$$ 10.5778 0.663710
$$255$$ 0 0
$$256$$ −7.09167 −0.443230
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 1.66947 0.103736
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 12.9083 0.797479
$$263$$ 22.8167 1.40694 0.703468 0.710727i $$-0.251634\pi$$
0.703468 + 0.710727i $$0.251634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −0.908327 −0.0556931
$$267$$ 0 0
$$268$$ 1.21110 0.0739799
$$269$$ −8.72498 −0.531971 −0.265986 0.963977i $$-0.585697\pi$$
−0.265986 + 0.963977i $$0.585697\pi$$
$$270$$ 0 0
$$271$$ −0.211103 −0.0128236 −0.00641178 0.999979i $$-0.502041\pi$$
−0.00641178 + 0.999979i $$0.502041\pi$$
$$272$$ 22.8167 1.38346
$$273$$ 0 0
$$274$$ 16.8167 1.01593
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 14.3944 0.864879 0.432439 0.901663i $$-0.357653\pi$$
0.432439 + 0.901663i $$0.357653\pi$$
$$278$$ −8.09167 −0.485306
$$279$$ 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
$$283$$ 6.30278 0.374661 0.187331 0.982297i $$-0.440016\pi$$
0.187331 + 0.982297i $$0.440016\pi$$
$$284$$ −0.788897 −0.0468125
$$285$$ 0 0
$$286$$ 6.51388 0.385174
$$287$$ 3.90833 0.230701
$$288$$ 0 0
$$289$$ 30.7250 1.80735
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 2.39445 0.140125
$$293$$ −0.788897 −0.0460879 −0.0230439 0.999734i $$-0.507336\pi$$
−0.0230439 + 0.999734i $$0.507336\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −7.18335 −0.417524
$$297$$ 0 0
$$298$$ −22.4222 −1.29888
$$299$$ 36.5139 2.11165
$$300$$ 0 0
$$301$$ 5.02776 0.289795
$$302$$ 1.06392 0.0612215
$$303$$ 0 0
$$304$$ 3.30278 0.189427
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −16.9083 −0.965009 −0.482505 0.875893i $$-0.660273\pi$$
−0.482505 + 0.875893i $$0.660273\pi$$
$$308$$ −0.211103 −0.0120287
$$309$$ 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
$$313$$ 0.183346 0.0103633 0.00518167 0.999987i $$-0.498351\pi$$
0.00518167 + 0.999987i $$0.498351\pi$$
$$314$$ 25.0278 1.41240
$$315$$ 0 0
$$316$$ 3.30278 0.185796
$$317$$ 0.908327 0.0510167 0.0255084 0.999675i $$-0.491880\pi$$
0.0255084 + 0.999675i $$0.491880\pi$$
$$318$$ 0 0
$$319$$ −0.908327 −0.0508565
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 6.63331 0.369660
$$323$$ 6.90833 0.384390
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 12.1194 0.671233
$$327$$ 0 0
$$328$$ −16.8167 −0.928544
$$329$$ 2.09167 0.115318
$$330$$ 0 0
$$331$$ −21.6056 −1.18755 −0.593774 0.804632i $$-0.702363\pi$$
−0.593774 + 0.804632i $$0.702363\pi$$
$$332$$ −1.06392 −0.0583900
$$333$$ 0 0
$$334$$ −17.4861 −0.956798
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −30.8444 −1.68020 −0.840101 0.542430i $$-0.817504\pi$$
−0.840101 + 0.542430i $$0.817504\pi$$
$$338$$ 15.6333 0.850340
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 10.2111 0.552962
$$342$$ 0 0
$$343$$ −9.42221 −0.508751
$$344$$ −21.6333 −1.16639
$$345$$ 0 0
$$346$$ 6.27502 0.337347
$$347$$ 12.5139 0.671780 0.335890 0.941901i $$-0.390963\pi$$
0.335890 + 0.941901i $$0.390963\pi$$
$$348$$ 0 0
$$349$$ −5.18335 −0.277458 −0.138729 0.990330i $$-0.544302\pi$$
−0.138729 + 0.990330i $$0.544302\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1.69722 0.0904624
$$353$$ −18.6333 −0.991751 −0.495875 0.868394i $$-0.665153\pi$$
−0.495875 + 0.868394i $$0.665153\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0.513878 0.0272355
$$357$$ 0 0
$$358$$ −16.3028 −0.861628
$$359$$ −0.788897 −0.0416364 −0.0208182 0.999783i $$-0.506627\pi$$
−0.0208182 + 0.999783i $$0.506627\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ −25.9361 −1.36317
$$363$$ 0 0
$$364$$ −1.05551 −0.0553239
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −20.6972 −1.08039 −0.540193 0.841541i $$-0.681649\pi$$
−0.540193 + 0.841541i $$0.681649\pi$$
$$368$$ −24.1194 −1.25731
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0.908327 0.0471580
$$372$$ 0 0
$$373$$ 27.4222 1.41987 0.709934 0.704268i $$-0.248725\pi$$
0.709934 + 0.704268i $$0.248725\pi$$
$$374$$ −9.00000 −0.465379
$$375$$ 0 0
$$376$$ −9.00000 −0.464140
$$377$$ −4.54163 −0.233906
$$378$$ 0 0
$$379$$ 3.18335 0.163518 0.0817588 0.996652i $$-0.473946\pi$$
0.0817588 + 0.996652i $$0.473946\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −13.4222 −0.686740
$$383$$ −21.6333 −1.10541 −0.552705 0.833377i $$-0.686404\pi$$
−0.552705 + 0.833377i $$0.686404\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 17.2111 0.876022
$$387$$ 0 0
$$388$$ −4.63331 −0.235221
$$389$$ −12.0000 −0.608424 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$390$$ 0 0
$$391$$ −50.4500 −2.55136
$$392$$ 19.5416 0.987002
$$393$$ 0 0
$$394$$ −17.3305 −0.873100
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 21.6972 1.08895 0.544476 0.838776i $$-0.316728\pi$$
0.544476 + 0.838776i $$0.316728\pi$$
$$398$$ −8.44996 −0.423558
$$399$$ 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
$$403$$ 51.0555 2.54326
$$404$$ −0.155590 −0.00774088
$$405$$ 0 0
$$406$$ −0.825058 −0.0409469
$$407$$ 2.39445 0.118688
$$408$$ 0 0
$$409$$ −6.21110 −0.307119 −0.153560 0.988139i $$-0.549074\pi$$
−0.153560 + 0.988139i $$0.549074\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −0.880571 −0.0433826
$$413$$ 9.90833 0.487557
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 8.48612 0.416066
$$417$$ 0 0
$$418$$ −1.30278 −0.0637208
$$419$$ −6.39445 −0.312389 −0.156195 0.987726i $$-0.549923\pi$$
−0.156195 + 0.987726i $$0.549923\pi$$
$$420$$ 0 0
$$421$$ 0.697224 0.0339806 0.0169903 0.999856i $$-0.494592\pi$$
0.0169903 + 0.999856i $$0.494592\pi$$
$$422$$ −32.8806 −1.60060
$$423$$ 0 0
$$424$$ −3.90833 −0.189805
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −5.51388 −0.266835
$$428$$ −0.908327 −0.0439056
$$429$$ 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
$$433$$ 5.00000 0.240285 0.120142 0.992757i $$-0.461665\pi$$
0.120142 + 0.992757i $$0.461665\pi$$
$$434$$ 9.27502 0.445215
$$435$$ 0 0
$$436$$ −3.48612 −0.166955
$$437$$ −7.30278 −0.349339
$$438$$ 0 0
$$439$$ 24.3028 1.15991 0.579954 0.814649i $$-0.303070\pi$$
0.579954 + 0.814649i $$0.303070\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −45.0000 −2.14043
$$443$$ 8.60555 0.408862 0.204431 0.978881i $$-0.434466\pi$$
0.204431 + 0.978881i $$0.434466\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −29.4861 −1.39621
$$447$$ 0 0
$$448$$ 6.14719 0.290427
$$449$$ −23.4861 −1.10838 −0.554189 0.832391i $$-0.686972\pi$$
−0.554189 + 0.832391i $$0.686972\pi$$
$$450$$ 0 0
$$451$$ 5.60555 0.263955
$$452$$ −3.27502 −0.154044
$$453$$ 0 0
$$454$$ −2.21110 −0.103772
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −20.6972 −0.968175 −0.484088 0.875020i $$-0.660848\pi$$
−0.484088 + 0.875020i $$0.660848\pi$$
$$458$$ −24.3944 −1.13988
$$459$$ 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
$$463$$ 11.7889 0.547877 0.273938 0.961747i $$-0.411674\pi$$
0.273938 + 0.961747i $$0.411674\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 0 0
$$466$$ −20.7250 −0.960066
$$467$$ 18.6333 0.862247 0.431123 0.902293i $$-0.358117\pi$$
0.431123 + 0.902293i $$0.358117\pi$$
$$468$$ 0 0
$$469$$ −2.78890 −0.128779
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −42.6333 −1.96236
$$473$$ 7.21110 0.331567
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 1.45837 0.0668441
$$477$$ 0 0
$$478$$ −27.5139 −1.25846
$$479$$ 34.8167 1.59081 0.795407 0.606076i $$-0.207257\pi$$
0.795407 + 0.606076i $$0.207257\pi$$
$$480$$ 0 0
$$481$$ 11.9722 0.545887
$$482$$ 28.5778 1.30168
$$483$$ 0 0
$$484$$ −0.302776 −0.0137625
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 4.21110 0.190823 0.0954116 0.995438i $$-0.469583\pi$$
0.0954116 + 0.995438i $$0.469583\pi$$
$$488$$ 23.7250 1.07398
$$489$$ 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
$$493$$ 6.27502 0.282613
$$494$$ −6.51388 −0.293073
$$495$$ 0 0
$$496$$ −33.7250 −1.51430
$$497$$ 1.81665 0.0814881
$$498$$ 0 0
$$499$$ −3.48612 −0.156060 −0.0780301 0.996951i $$-0.524863\pi$$
−0.0780301 + 0.996951i $$0.524863\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −9.00000 −0.401690
$$503$$ −9.39445 −0.418878 −0.209439 0.977822i $$-0.567164\pi$$
−0.209439 + 0.977822i $$0.567164\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 9.51388 0.422943
$$507$$ 0 0
$$508$$ −2.45837 −0.109072
$$509$$ 22.6972 1.00604 0.503018 0.864276i $$-0.332223\pi$$
0.503018 + 0.864276i $$0.332223\pi$$
$$510$$ 0 0
$$511$$ −5.51388 −0.243920
$$512$$ −25.4222 −1.12351
$$513$$ 0 0
$$514$$ 23.4500 1.03433
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 3.00000 0.131940
$$518$$ 2.17494 0.0955615
$$519$$ 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
$$523$$ −32.4222 −1.41772 −0.708862 0.705347i $$-0.750791\pi$$
−0.708862 + 0.705347i $$0.750791\pi$$
$$524$$ −3.00000 −0.131056
$$525$$ 0 0
$$526$$ 29.7250 1.29607
$$527$$ −70.5416 −3.07284
$$528$$ 0 0
$$529$$ 30.3305 1.31872
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0.211103 0.00915246
$$533$$ 28.0278 1.21402
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ −11.3667 −0.490053
$$539$$ −6.51388 −0.280573
$$540$$ 0 0
$$541$$ 25.7250 1.10600 0.553002 0.833180i $$-0.313482\pi$$
0.553002 + 0.833180i $$0.313482\pi$$
$$542$$ −0.275019 −0.0118131
$$543$$ 0 0
$$544$$ −11.7250 −0.502704
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −7.11943 −0.304405 −0.152202 0.988349i $$-0.548637\pi$$
−0.152202 + 0.988349i $$0.548637\pi$$
$$548$$ −3.90833 −0.166955
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0.908327 0.0386960
$$552$$ 0 0
$$553$$ −7.60555 −0.323421
$$554$$ 18.7527 0.796727
$$555$$ 0 0
$$556$$ 1.88057 0.0797540
$$557$$ −19.4222 −0.822945 −0.411473 0.911422i $$-0.634985\pi$$
−0.411473 + 0.911422i $$0.634985\pi$$
$$558$$ 0 0
$$559$$ 36.0555 1.52499
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 1.54163 0.0650299
$$563$$ −8.09167 −0.341023 −0.170512 0.985356i $$-0.554542\pi$$
−0.170512 + 0.985356i $$0.554542\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 8.21110 0.345138
$$567$$ 0 0
$$568$$ −7.81665 −0.327980
$$569$$ 46.1472 1.93459 0.967295 0.253653i $$-0.0816321\pi$$
0.967295 + 0.253653i $$0.0816321\pi$$
$$570$$ 0 0
$$571$$ 22.3305 0.934504 0.467252 0.884124i $$-0.345244\pi$$
0.467252 + 0.884124i $$0.345244\pi$$
$$572$$ −1.51388 −0.0632984
$$573$$ 0 0
$$574$$ 5.09167 0.212522
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 44.3583 1.84666 0.923330 0.384008i $$-0.125456\pi$$
0.923330 + 0.384008i $$0.125456\pi$$
$$578$$ 40.0278 1.66494
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 2.44996 0.101642
$$582$$ 0 0
$$583$$ 1.30278 0.0539555
$$584$$ 23.7250 0.981747
$$585$$ 0 0
$$586$$ −1.02776 −0.0424562
$$587$$ −16.5416 −0.682746 −0.341373 0.939928i $$-0.610892\pi$$
−0.341373 + 0.939928i $$0.610892\pi$$
$$588$$ 0 0
$$589$$ −10.2111 −0.420741
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −7.90833 −0.325030
$$593$$ 6.39445 0.262589 0.131294 0.991343i $$-0.458087\pi$$
0.131294 + 0.991343i $$0.458087\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 5.21110 0.213455
$$597$$ 0 0
$$598$$ 47.5694 1.94526
$$599$$ 24.9083 1.01773 0.508863 0.860847i $$-0.330066\pi$$
0.508863 + 0.860847i $$0.330066\pi$$
$$600$$ 0 0
$$601$$ −1.90833 −0.0778423 −0.0389211 0.999242i $$-0.512392\pi$$
−0.0389211 + 0.999242i $$0.512392\pi$$
$$602$$ 6.55004 0.266960
$$603$$ 0 0
$$604$$ −0.247263 −0.0100610
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 7.21110 0.292690 0.146345 0.989234i $$-0.453249\pi$$
0.146345 + 0.989234i $$0.453249\pi$$
$$608$$ −1.69722 −0.0688315
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 15.0000 0.606835
$$612$$ 0 0
$$613$$ −15.8806 −0.641410 −0.320705 0.947179i $$-0.603920\pi$$
−0.320705 + 0.947179i $$0.603920\pi$$
$$614$$ −22.0278 −0.888968
$$615$$ 0 0
$$616$$ −2.09167 −0.0842759
$$617$$ −3.39445 −0.136655 −0.0683277 0.997663i $$-0.521766\pi$$
−0.0683277 + 0.997663i $$0.521766\pi$$
$$618$$ 0 0
$$619$$ −11.4222 −0.459097 −0.229549 0.973297i $$-0.573725\pi$$
−0.229549 + 0.973297i $$0.573725\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −6.27502 −0.251605
$$623$$ −1.18335 −0.0474098
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0.238859 0.00954672
$$627$$ 0 0
$$628$$ −5.81665 −0.232110
$$629$$ −16.5416 −0.659558
$$630$$ 0 0
$$631$$ 6.93608 0.276121 0.138061 0.990424i $$-0.455913\pi$$
0.138061 + 0.990424i $$0.455913\pi$$
$$632$$ 32.7250 1.30173
$$633$$ 0 0
$$634$$ 1.18335 0.0469967
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −32.5694 −1.29045
$$638$$ −1.18335 −0.0468491
$$639$$ 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
$$643$$ −22.0000 −0.867595 −0.433798 0.901010i $$-0.642827\pi$$
−0.433798 + 0.901010i $$0.642827\pi$$
$$644$$ −1.54163 −0.0607489
$$645$$ 0 0
$$646$$ 9.00000 0.354100
$$647$$ 33.2389 1.30675 0.653377 0.757033i $$-0.273352\pi$$
0.653377 + 0.757033i $$0.273352\pi$$
$$648$$ 0 0
$$649$$ 14.2111 0.557835
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −2.81665 −0.110309
$$653$$ −6.11943 −0.239472 −0.119736 0.992806i $$-0.538205\pi$$
−0.119736 + 0.992806i $$0.538205\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −18.5139 −0.722846
$$657$$ 0 0
$$658$$ 2.72498 0.106231
$$659$$ 30.9083 1.20402 0.602009 0.798489i $$-0.294367\pi$$
0.602009 + 0.798489i $$0.294367\pi$$
$$660$$ 0 0
$$661$$ −8.81665 −0.342928 −0.171464 0.985190i $$-0.554850\pi$$
−0.171464 + 0.985190i $$0.554850\pi$$
$$662$$ −28.1472 −1.09397
$$663$$ 0 0
$$664$$ −10.5416 −0.409095
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.63331 −0.256843
$$668$$ 4.06392 0.157238
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −7.90833 −0.305298
$$672$$ 0 0
$$673$$ 30.0278 1.15748 0.578742 0.815510i $$-0.303544\pi$$
0.578742 + 0.815510i $$0.303544\pi$$
$$674$$ −40.1833 −1.54780
$$675$$ 0 0
$$676$$ −3.63331 −0.139743
$$677$$ −24.2389 −0.931575 −0.465788 0.884897i $$-0.654229\pi$$
−0.465788 + 0.884897i $$0.654229\pi$$
$$678$$ 0 0
$$679$$ 10.6695 0.409457
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 13.3028 0.509390
$$683$$ −47.8444 −1.83072 −0.915358 0.402642i $$-0.868092\pi$$
−0.915358 + 0.402642i $$0.868092\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −12.2750 −0.468662
$$687$$ 0 0
$$688$$ −23.8167 −0.908001
$$689$$ 6.51388 0.248159
$$690$$ 0 0
$$691$$ 27.5416 1.04773 0.523867 0.851800i $$-0.324489\pi$$
0.523867 + 0.851800i $$0.324489\pi$$
$$692$$ −1.45837 −0.0554387
$$693$$ 0 0
$$694$$ 16.3028 0.618845
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −38.7250 −1.46681
$$698$$ −6.75274 −0.255595
$$699$$ 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
$$703$$ −2.39445 −0.0903083
$$704$$ 8.81665 0.332290
$$705$$ 0 0
$$706$$ −24.2750 −0.913602
$$707$$ 0.358288 0.0134748
$$708$$ 0 0
$$709$$ −31.6333 −1.18801 −0.594007 0.804460i $$-0.702455\pi$$
−0.594007 + 0.804460i $$0.702455\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 5.09167 0.190819
$$713$$ 74.5694 2.79265
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 3.78890 0.141598
$$717$$ 0 0
$$718$$ −1.02776 −0.0383555
$$719$$ −7.18335 −0.267894 −0.133947 0.990989i $$-0.542765\pi$$
−0.133947 + 0.990989i $$0.542765\pi$$
$$720$$ 0 0
$$721$$ 2.02776 0.0755176
$$722$$ −23.4500 −0.872717
$$723$$ 0 0
$$724$$ 6.02776 0.224020
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −39.3305 −1.45869 −0.729344 0.684147i $$-0.760175\pi$$
−0.729344 + 0.684147i $$0.760175\pi$$
$$728$$ −10.4584 −0.387613
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −49.8167 −1.84254
$$732$$ 0 0
$$733$$ 19.6056 0.724148 0.362074 0.932149i $$-0.382069\pi$$
0.362074 + 0.932149i $$0.382069\pi$$
$$734$$ −26.9638 −0.995253
$$735$$ 0 0
$$736$$ 12.3944 0.456865
$$737$$ −4.00000 −0.147342
$$738$$ 0 0
$$739$$ 35.1194 1.29189 0.645945 0.763384i $$-0.276464\pi$$
0.645945 + 0.763384i $$0.276464\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 1.18335 0.0434420
$$743$$ −40.6972 −1.49304 −0.746518 0.665365i $$-0.768276\pi$$
−0.746518 + 0.665365i $$0.768276\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 35.7250 1.30798
$$747$$ 0 0
$$748$$ 2.09167 0.0764791
$$749$$ 2.09167 0.0764281
$$750$$ 0 0
$$751$$ −45.3305 −1.65413 −0.827067 0.562103i $$-0.809992\pi$$
−0.827067 + 0.562103i $$0.809992\pi$$
$$752$$ −9.90833 −0.361320
$$753$$ 0 0
$$754$$ −5.91673 −0.215475
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 49.0555 1.78295 0.891476 0.453067i $$-0.149670\pi$$
0.891476 + 0.453067i $$0.149670\pi$$
$$758$$ 4.14719 0.150633
$$759$$ 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
$$763$$ 8.02776 0.290624
$$764$$ 3.11943 0.112857
$$765$$ 0 0
$$766$$ −28.1833 −1.01831
$$767$$ 71.0555 2.56567
$$768$$ 0 0
$$769$$ −5.18335 −0.186916 −0.0934581 0.995623i $$-0.529792\pi$$
−0.0934581 + 0.995623i $$0.529792\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −4.00000 −0.143963
$$773$$ −3.11943 −0.112198 −0.0560990 0.998425i $$-0.517866\pi$$
−0.0560990 + 0.998425i $$0.517866\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −45.9083 −1.64801
$$777$$ 0 0
$$778$$ −15.6333 −0.560481
$$779$$ −5.60555 −0.200840
$$780$$ 0 0
$$781$$ 2.60555 0.0932340
$$782$$ −65.7250 −2.35032
$$783$$ 0 0
$$784$$ 21.5139 0.768353
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 10.2111 0.363986 0.181993 0.983300i $$-0.441745\pi$$
0.181993 + 0.983300i $$0.441745\pi$$
$$788$$ 4.02776 0.143483
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 7.54163 0.268150
$$792$$ 0 0
$$793$$ −39.5416 −1.40416
$$794$$ 28.2666 1.00314
$$795$$ 0 0
$$796$$ 1.96384 0.0696065
$$797$$ −3.51388 −0.124468 −0.0622340 0.998062i $$-0.519822\pi$$
−0.0622340 + 0.998062i $$0.519822\pi$$
$$798$$ 0 0
$$799$$ −20.7250 −0.733197
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 16.6611 0.588323
$$803$$ −7.90833 −0.279079
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 66.5139 2.34285
$$807$$ 0 0
$$808$$ −1.54163 −0.0542345
$$809$$ −39.6333 −1.39343 −0.696716 0.717347i $$-0.745356\pi$$
−0.696716 + 0.717347i $$0.745356\pi$$
$$810$$ 0 0
$$811$$ 38.8722 1.36499 0.682493 0.730892i $$-0.260896\pi$$
0.682493 + 0.730892i $$0.260896\pi$$
$$812$$ 0.191750 0.00672911
$$813$$ 0 0
$$814$$ 3.11943 0.109336
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −7.21110 −0.252285
$$818$$ −8.09167 −0.282919
$$819$$ 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
$$823$$ 18.4222 0.642158 0.321079 0.947052i $$-0.395955\pi$$
0.321079 + 0.947052i $$0.395955\pi$$
$$824$$ −8.72498 −0.303949
$$825$$ 0 0
$$826$$ 12.9083 0.449138
$$827$$ −13.8167 −0.480452 −0.240226 0.970717i $$-0.577222\pi$$
−0.240226 + 0.970717i $$0.577222\pi$$
$$828$$ 0 0
$$829$$ 29.7527 1.03336 0.516678 0.856180i $$-0.327169\pi$$
0.516678 + 0.856180i $$0.327169\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 44.0833 1.52831
$$833$$ 45.0000 1.55916
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0.302776 0.0104717
$$837$$ 0 0
$$838$$ −8.33053 −0.287773
$$839$$ 9.11943 0.314838 0.157419 0.987532i $$-0.449683\pi$$
0.157419 + 0.987532i $$0.449683\pi$$
$$840$$ 0 0
$$841$$ −28.1749 −0.971550
$$842$$ 0.908327 0.0313030
$$843$$ 0 0
$$844$$ 7.64171 0.263039
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0.697224 0.0239569
$$848$$ −4.30278 −0.147758
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 17.4861 0.599417
$$852$$ 0 0
$$853$$ −12.7250 −0.435695 −0.217848 0.975983i $$-0.569904\pi$$
−0.217848 + 0.975983i $$0.569904\pi$$
$$854$$ −7.18335 −0.245809
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ 3.00000 0.102478 0.0512390 0.998686i $$-0.483683\pi$$
0.0512390 + 0.998686i $$0.483683\pi$$
$$858$$ 0 0
$$859$$ 41.3944 1.41236 0.706180 0.708032i $$-0.250417\pi$$
0.706180 + 0.708032i $$0.250417\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −42.9916 −1.46430
$$863$$ 12.3944 0.421912 0.210956 0.977496i $$-0.432342\pi$$
0.210956 + 0.977496i $$0.432342\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 6.51388 0.221351
$$867$$ 0 0
$$868$$ −2.15559 −0.0731655
$$869$$ −10.9083 −0.370040
$$870$$ 0 0
$$871$$ −20.0000 −0.677674
$$872$$ −34.5416 −1.16973
$$873$$ 0 0
$$874$$ −9.51388 −0.321812
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 20.0000 0.675352 0.337676 0.941262i $$-0.390359\pi$$
0.337676 + 0.941262i $$0.390359\pi$$
$$878$$ 31.6611 1.06851
$$879$$ 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
$$883$$ 52.4500 1.76508 0.882541 0.470236i $$-0.155831\pi$$
0.882541 + 0.470236i $$0.155831\pi$$
$$884$$ 10.4584 0.351753
$$885$$ 0 0
$$886$$ 11.2111 0.376644
$$887$$ 3.23886 0.108750 0.0543751 0.998521i $$-0.482683\pi$$
0.0543751 + 0.998521i $$0.482683\pi$$
$$888$$ 0 0
$$889$$ 5.66106 0.189866
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 6.85281 0.229449
$$893$$ −3.00000 −0.100391
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 5.64171 0.188476
$$897$$ 0 0
$$898$$ −30.5971 −1.02104
$$899$$ −9.27502 −0.309339
$$900$$ 0 0
$$901$$ −9.00000 −0.299833
$$902$$ 7.30278 0.243156
$$903$$ 0 0
$$904$$ −32.4500 −1.07927
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −4.00000 −0.132818 −0.0664089 0.997792i $$-0.521154\pi$$
−0.0664089 + 0.997792i $$0.521154\pi$$
$$908$$ 0.513878 0.0170536
$$909$$ 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
$$913$$ 3.51388 0.116292
$$914$$ −26.9638 −0.891885
$$915$$ 0 0
$$916$$ 5.66947 0.187324
$$917$$ 6.90833 0.228133
$$918$$ 0 0
$$919$$ 26.7889 0.883684 0.441842 0.897093i $$-0.354325\pi$$
0.441842 + 0.897093i $$0.354325\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −41.9638 −1.38201
$$923$$ 13.0278 0.428814
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 15.3583 0.504705
$$927$$ 0 0
$$928$$ −1.54163 −0.0506066
$$929$$ −53.6056 −1.75874 −0.879371 0.476138i $$-0.842036\pi$$
−0.879371 + 0.476138i $$0.842036\pi$$
$$930$$ 0 0
$$931$$ 6.51388 0.213484
$$932$$ 4.81665 0.157775
$$933$$ 0 0
$$934$$ 24.2750 0.794303
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −9.21110 −0.300914 −0.150457 0.988617i $$-0.548074\pi$$
−0.150457 + 0.988617i $$0.548074\pi$$
$$938$$ −3.63331 −0.118632
$$939$$ 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
$$943$$ 40.9361 1.33306
$$944$$ −46.9361 −1.52764
$$945$$ 0 0
$$946$$ 9.39445 0.305440
$$947$$ −6.63331 −0.215554 −0.107777 0.994175i $$-0.534373\pi$$
−0.107777 + 0.994175i $$0.534373\pi$$
$$948$$ 0 0
$$949$$ −39.5416 −1.28358
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 14.4500 0.468326
$$953$$ −37.2666 −1.20718 −0.603592 0.797293i $$-0.706264\pi$$
−0.603592 + 0.797293i $$0.706264\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 6.39445 0.206811
$$957$$ 0 0
$$958$$ 45.3583 1.46546
$$959$$ 9.00000 0.290625
$$960$$ 0 0
$$961$$ 73.2666 2.36344
$$962$$ 15.5971 0.502872
$$963$$ 0 0
$$964$$ −6.64171 −0.213915
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 14.9083 0.479419 0.239710 0.970845i $$-0.422948\pi$$
0.239710 + 0.970845i $$0.422948\pi$$
$$968$$ −3.00000 −0.0964237
$$969$$ 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
$$973$$ −4.33053 −0.138830
$$974$$ 5.48612 0.175787
$$975$$ 0 0
$$976$$ 26.1194 0.836063
$$977$$ 52.0278 1.66452 0.832258 0.554389i $$-0.187048\pi$$
0.832258 + 0.554389i $$0.187048\pi$$
$$978$$ 0 0
$$979$$ −1.69722 −0.0542435
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 12.7527 0.406956
$$983$$ −8.84441 −0.282093 −0.141046 0.990003i $$-0.545047\pi$$
−0.141046 + 0.990003i $$0.545047\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 8.17494 0.260343
$$987$$ 0 0
$$988$$ 1.51388 0.0481629
$$989$$ 52.6611 1.67452
$$990$$ 0 0
$$991$$ −16.9083 −0.537111 −0.268555 0.963264i $$-0.586546\pi$$
−0.268555 + 0.963264i $$0.586546\pi$$
$$992$$ 17.3305 0.550245
$$993$$ 0 0
$$994$$ 2.36669 0.0750669
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 46.7250 1.47979 0.739897 0.672720i $$-0.234874\pi$$
0.739897 + 0.672720i $$0.234874\pi$$
$$998$$ −4.54163 −0.143763
$$999$$ 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.a.o.1.2 2
3.2 odd 2 275.2.a.f.1.1 yes 2
5.2 odd 4 2475.2.c.k.199.3 4
5.3 odd 4 2475.2.c.k.199.2 4
5.4 even 2 2475.2.a.t.1.1 2
12.11 even 2 4400.2.a.bh.1.1 2
15.2 even 4 275.2.b.c.199.2 4
15.8 even 4 275.2.b.c.199.3 4
15.14 odd 2 275.2.a.e.1.2 2
33.32 even 2 3025.2.a.h.1.2 2
60.23 odd 4 4400.2.b.y.4049.1 4
60.47 odd 4 4400.2.b.y.4049.4 4
60.59 even 2 4400.2.a.bs.1.2 2
165.164 even 2 3025.2.a.n.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.2 2 15.14 odd 2
275.2.a.f.1.1 yes 2 3.2 odd 2
275.2.b.c.199.2 4 15.2 even 4
275.2.b.c.199.3 4 15.8 even 4
2475.2.a.o.1.2 2 1.1 even 1 trivial
2475.2.a.t.1.1 2 5.4 even 2
2475.2.c.k.199.2 4 5.3 odd 4
2475.2.c.k.199.3 4 5.2 odd 4
3025.2.a.h.1.2 2 33.32 even 2
3025.2.a.n.1.1 2 165.164 even 2
4400.2.a.bh.1.1 2 12.11 even 2
4400.2.a.bs.1.2 2 60.59 even 2
4400.2.b.y.4049.1 4 60.23 odd 4
4400.2.b.y.4049.4 4 60.47 odd 4