# Properties

 Label 825.4.c.b.199.2 Level $825$ Weight $4$ Character 825.199 Analytic conductor $48.677$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.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: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.4.c.b.199.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000i q^{2} -3.00000i q^{3} -17.0000 q^{4} +15.0000 q^{6} +3.00000i q^{7} -45.0000i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q+5.00000i q^{2} -3.00000i q^{3} -17.0000 q^{4} +15.0000 q^{6} +3.00000i q^{7} -45.0000i q^{8} -9.00000 q^{9} -11.0000 q^{11} +51.0000i q^{12} -32.0000i q^{13} -15.0000 q^{14} +89.0000 q^{16} +33.0000i q^{17} -45.0000i q^{18} -47.0000 q^{19} +9.00000 q^{21} -55.0000i q^{22} -113.000i q^{23} -135.000 q^{24} +160.000 q^{26} +27.0000i q^{27} -51.0000i q^{28} +54.0000 q^{29} +178.000 q^{31} +85.0000i q^{32} +33.0000i q^{33} -165.000 q^{34} +153.000 q^{36} +19.0000i q^{37} -235.000i q^{38} -96.0000 q^{39} +139.000 q^{41} +45.0000i q^{42} +308.000i q^{43} +187.000 q^{44} +565.000 q^{46} +195.000i q^{47} -267.000i q^{48} +334.000 q^{49} +99.0000 q^{51} +544.000i q^{52} -152.000i q^{53} -135.000 q^{54} +135.000 q^{56} +141.000i q^{57} +270.000i q^{58} +625.000 q^{59} +320.000 q^{61} +890.000i q^{62} -27.0000i q^{63} +287.000 q^{64} -165.000 q^{66} +200.000i q^{67} -561.000i q^{68} -339.000 q^{69} -947.000 q^{71} +405.000i q^{72} +448.000i q^{73} -95.0000 q^{74} +799.000 q^{76} -33.0000i q^{77} -480.000i q^{78} +721.000 q^{79} +81.0000 q^{81} +695.000i q^{82} -142.000i q^{83} -153.000 q^{84} -1540.00 q^{86} -162.000i q^{87} +495.000i q^{88} -404.000 q^{89} +96.0000 q^{91} +1921.00i q^{92} -534.000i q^{93} -975.000 q^{94} +255.000 q^{96} +79.0000i q^{97} +1670.00i q^{98} +99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 34 q^{4} + 30 q^{6} - 18 q^{9}+O(q^{10})$$ 2 * q - 34 * q^4 + 30 * q^6 - 18 * q^9 $$2 q - 34 q^{4} + 30 q^{6} - 18 q^{9} - 22 q^{11} - 30 q^{14} + 178 q^{16} - 94 q^{19} + 18 q^{21} - 270 q^{24} + 320 q^{26} + 108 q^{29} + 356 q^{31} - 330 q^{34} + 306 q^{36} - 192 q^{39} + 278 q^{41} + 374 q^{44} + 1130 q^{46} + 668 q^{49} + 198 q^{51} - 270 q^{54} + 270 q^{56} + 1250 q^{59} + 640 q^{61} + 574 q^{64} - 330 q^{66} - 678 q^{69} - 1894 q^{71} - 190 q^{74} + 1598 q^{76} + 1442 q^{79} + 162 q^{81} - 306 q^{84} - 3080 q^{86} - 808 q^{89} + 192 q^{91} - 1950 q^{94} + 510 q^{96} + 198 q^{99}+O(q^{100})$$ 2 * q - 34 * q^4 + 30 * q^6 - 18 * q^9 - 22 * q^11 - 30 * q^14 + 178 * q^16 - 94 * q^19 + 18 * q^21 - 270 * q^24 + 320 * q^26 + 108 * q^29 + 356 * q^31 - 330 * q^34 + 306 * q^36 - 192 * q^39 + 278 * q^41 + 374 * q^44 + 1130 * q^46 + 668 * q^49 + 198 * q^51 - 270 * q^54 + 270 * q^56 + 1250 * q^59 + 640 * q^61 + 574 * q^64 - 330 * q^66 - 678 * q^69 - 1894 * q^71 - 190 * q^74 + 1598 * q^76 + 1442 * q^79 + 162 * q^81 - 306 * q^84 - 3080 * q^86 - 808 * q^89 + 192 * q^91 - 1950 * q^94 + 510 * q^96 + 198 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.00000i 1.76777i 0.467707 + 0.883883i $$0.345080\pi$$
−0.467707 + 0.883883i $$0.654920\pi$$
$$3$$ − 3.00000i − 0.577350i
$$4$$ −17.0000 −2.12500
$$5$$ 0 0
$$6$$ 15.0000 1.02062
$$7$$ 3.00000i 0.161985i 0.996715 + 0.0809924i $$0.0258089\pi$$
−0.996715 + 0.0809924i $$0.974191\pi$$
$$8$$ − 45.0000i − 1.98874i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 51.0000i 1.22687i
$$13$$ − 32.0000i − 0.682708i −0.939935 0.341354i $$-0.889115\pi$$
0.939935 0.341354i $$-0.110885\pi$$
$$14$$ −15.0000 −0.286351
$$15$$ 0 0
$$16$$ 89.0000 1.39062
$$17$$ 33.0000i 0.470804i 0.971898 + 0.235402i $$0.0756407\pi$$
−0.971898 + 0.235402i $$0.924359\pi$$
$$18$$ − 45.0000i − 0.589256i
$$19$$ −47.0000 −0.567502 −0.283751 0.958898i $$-0.591579\pi$$
−0.283751 + 0.958898i $$0.591579\pi$$
$$20$$ 0 0
$$21$$ 9.00000 0.0935220
$$22$$ − 55.0000i − 0.533002i
$$23$$ − 113.000i − 1.02444i −0.858854 0.512220i $$-0.828823\pi$$
0.858854 0.512220i $$-0.171177\pi$$
$$24$$ −135.000 −1.14820
$$25$$ 0 0
$$26$$ 160.000 1.20687
$$27$$ 27.0000i 0.192450i
$$28$$ − 51.0000i − 0.344218i
$$29$$ 54.0000 0.345778 0.172889 0.984941i $$-0.444690\pi$$
0.172889 + 0.984941i $$0.444690\pi$$
$$30$$ 0 0
$$31$$ 178.000 1.03128 0.515641 0.856805i $$-0.327554\pi$$
0.515641 + 0.856805i $$0.327554\pi$$
$$32$$ 85.0000i 0.469563i
$$33$$ 33.0000i 0.174078i
$$34$$ −165.000 −0.832273
$$35$$ 0 0
$$36$$ 153.000 0.708333
$$37$$ 19.0000i 0.0844211i 0.999109 + 0.0422106i $$0.0134400\pi$$
−0.999109 + 0.0422106i $$0.986560\pi$$
$$38$$ − 235.000i − 1.00321i
$$39$$ −96.0000 −0.394162
$$40$$ 0 0
$$41$$ 139.000 0.529467 0.264734 0.964322i $$-0.414716\pi$$
0.264734 + 0.964322i $$0.414716\pi$$
$$42$$ 45.0000i 0.165325i
$$43$$ 308.000i 1.09232i 0.837682 + 0.546158i $$0.183910\pi$$
−0.837682 + 0.546158i $$0.816090\pi$$
$$44$$ 187.000 0.640712
$$45$$ 0 0
$$46$$ 565.000 1.81097
$$47$$ 195.000i 0.605185i 0.953120 + 0.302592i $$0.0978520\pi$$
−0.953120 + 0.302592i $$0.902148\pi$$
$$48$$ − 267.000i − 0.802878i
$$49$$ 334.000 0.973761
$$50$$ 0 0
$$51$$ 99.0000 0.271819
$$52$$ 544.000i 1.45075i
$$53$$ − 152.000i − 0.393940i −0.980410 0.196970i $$-0.936890\pi$$
0.980410 0.196970i $$-0.0631101\pi$$
$$54$$ −135.000 −0.340207
$$55$$ 0 0
$$56$$ 135.000 0.322145
$$57$$ 141.000i 0.327647i
$$58$$ 270.000i 0.611254i
$$59$$ 625.000 1.37912 0.689560 0.724229i $$-0.257804\pi$$
0.689560 + 0.724229i $$0.257804\pi$$
$$60$$ 0 0
$$61$$ 320.000 0.671669 0.335834 0.941921i $$-0.390982\pi$$
0.335834 + 0.941921i $$0.390982\pi$$
$$62$$ 890.000i 1.82307i
$$63$$ − 27.0000i − 0.0539949i
$$64$$ 287.000 0.560547
$$65$$ 0 0
$$66$$ −165.000 −0.307729
$$67$$ 200.000i 0.364685i 0.983235 + 0.182342i $$0.0583679\pi$$
−0.983235 + 0.182342i $$0.941632\pi$$
$$68$$ − 561.000i − 1.00046i
$$69$$ −339.000 −0.591461
$$70$$ 0 0
$$71$$ −947.000 −1.58293 −0.791466 0.611213i $$-0.790682\pi$$
−0.791466 + 0.611213i $$0.790682\pi$$
$$72$$ 405.000i 0.662913i
$$73$$ 448.000i 0.718280i 0.933284 + 0.359140i $$0.116930\pi$$
−0.933284 + 0.359140i $$0.883070\pi$$
$$74$$ −95.0000 −0.149237
$$75$$ 0 0
$$76$$ 799.000 1.20594
$$77$$ − 33.0000i − 0.0488402i
$$78$$ − 480.000i − 0.696786i
$$79$$ 721.000 1.02682 0.513410 0.858143i $$-0.328382\pi$$
0.513410 + 0.858143i $$0.328382\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 695.000i 0.935975i
$$83$$ − 142.000i − 0.187789i −0.995582 0.0938947i $$-0.970068\pi$$
0.995582 0.0938947i $$-0.0299317\pi$$
$$84$$ −153.000 −0.198734
$$85$$ 0 0
$$86$$ −1540.00 −1.93096
$$87$$ − 162.000i − 0.199635i
$$88$$ 495.000i 0.599627i
$$89$$ −404.000 −0.481168 −0.240584 0.970628i $$-0.577339\pi$$
−0.240584 + 0.970628i $$0.577339\pi$$
$$90$$ 0 0
$$91$$ 96.0000 0.110588
$$92$$ 1921.00i 2.17694i
$$93$$ − 534.000i − 0.595411i
$$94$$ −975.000 −1.06983
$$95$$ 0 0
$$96$$ 255.000 0.271102
$$97$$ 79.0000i 0.0826931i 0.999145 + 0.0413466i $$0.0131648\pi$$
−0.999145 + 0.0413466i $$0.986835\pi$$
$$98$$ 1670.00i 1.72138i
$$99$$ 99.0000 0.100504
$$100$$ 0 0
$$101$$ −545.000 −0.536926 −0.268463 0.963290i $$-0.586516\pi$$
−0.268463 + 0.963290i $$0.586516\pi$$
$$102$$ 495.000i 0.480513i
$$103$$ 1306.00i 1.24936i 0.780881 + 0.624680i $$0.214770\pi$$
−0.780881 + 0.624680i $$0.785230\pi$$
$$104$$ −1440.00 −1.35773
$$105$$ 0 0
$$106$$ 760.000 0.696394
$$107$$ 1938.00i 1.75097i 0.483247 + 0.875484i $$0.339457\pi$$
−0.483247 + 0.875484i $$0.660543\pi$$
$$108$$ − 459.000i − 0.408956i
$$109$$ 576.000 0.506154 0.253077 0.967446i $$-0.418557\pi$$
0.253077 + 0.967446i $$0.418557\pi$$
$$110$$ 0 0
$$111$$ 57.0000 0.0487405
$$112$$ 267.000i 0.225260i
$$113$$ 1104.00i 0.919076i 0.888158 + 0.459538i $$0.151985\pi$$
−0.888158 + 0.459538i $$0.848015\pi$$
$$114$$ −705.000 −0.579204
$$115$$ 0 0
$$116$$ −918.000 −0.734777
$$117$$ 288.000i 0.227569i
$$118$$ 3125.00i 2.43796i
$$119$$ −99.0000 −0.0762632
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 1600.00i 1.18735i
$$123$$ − 417.000i − 0.305688i
$$124$$ −3026.00 −2.19147
$$125$$ 0 0
$$126$$ 135.000 0.0954504
$$127$$ − 1739.00i − 1.21505i −0.794301 0.607525i $$-0.792163\pi$$
0.794301 0.607525i $$-0.207837\pi$$
$$128$$ 2115.00i 1.46048i
$$129$$ 924.000 0.630649
$$130$$ 0 0
$$131$$ 1818.00 1.21251 0.606257 0.795269i $$-0.292670\pi$$
0.606257 + 0.795269i $$0.292670\pi$$
$$132$$ − 561.000i − 0.369915i
$$133$$ − 141.000i − 0.0919267i
$$134$$ −1000.00 −0.644678
$$135$$ 0 0
$$136$$ 1485.00 0.936307
$$137$$ 870.000i 0.542548i 0.962502 + 0.271274i $$0.0874450\pi$$
−0.962502 + 0.271274i $$0.912555\pi$$
$$138$$ − 1695.00i − 1.04557i
$$139$$ 636.000 0.388092 0.194046 0.980992i $$-0.437839\pi$$
0.194046 + 0.980992i $$0.437839\pi$$
$$140$$ 0 0
$$141$$ 585.000 0.349403
$$142$$ − 4735.00i − 2.79826i
$$143$$ 352.000i 0.205844i
$$144$$ −801.000 −0.463542
$$145$$ 0 0
$$146$$ −2240.00 −1.26975
$$147$$ − 1002.00i − 0.562201i
$$148$$ − 323.000i − 0.179395i
$$149$$ 239.000 0.131407 0.0657035 0.997839i $$-0.479071\pi$$
0.0657035 + 0.997839i $$0.479071\pi$$
$$150$$ 0 0
$$151$$ 1208.00 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 2115.00i 1.12861i
$$153$$ − 297.000i − 0.156935i
$$154$$ 165.000 0.0863382
$$155$$ 0 0
$$156$$ 1632.00 0.837593
$$157$$ − 1874.00i − 0.952621i −0.879277 0.476310i $$-0.841974\pi$$
0.879277 0.476310i $$-0.158026\pi$$
$$158$$ 3605.00i 1.81518i
$$159$$ −456.000 −0.227441
$$160$$ 0 0
$$161$$ 339.000 0.165944
$$162$$ 405.000i 0.196419i
$$163$$ 1904.00i 0.914925i 0.889229 + 0.457463i $$0.151242\pi$$
−0.889229 + 0.457463i $$0.848758\pi$$
$$164$$ −2363.00 −1.12512
$$165$$ 0 0
$$166$$ 710.000 0.331968
$$167$$ − 1180.00i − 0.546773i −0.961904 0.273387i $$-0.911856\pi$$
0.961904 0.273387i $$-0.0881438\pi$$
$$168$$ − 405.000i − 0.185991i
$$169$$ 1173.00 0.533910
$$170$$ 0 0
$$171$$ 423.000 0.189167
$$172$$ − 5236.00i − 2.32117i
$$173$$ 3177.00i 1.39620i 0.716000 + 0.698101i $$0.245971\pi$$
−0.716000 + 0.698101i $$0.754029\pi$$
$$174$$ 810.000 0.352908
$$175$$ 0 0
$$176$$ −979.000 −0.419289
$$177$$ − 1875.00i − 0.796235i
$$178$$ − 2020.00i − 0.850592i
$$179$$ −1787.00 −0.746182 −0.373091 0.927795i $$-0.621702\pi$$
−0.373091 + 0.927795i $$0.621702\pi$$
$$180$$ 0 0
$$181$$ −835.000 −0.342901 −0.171450 0.985193i $$-0.554845\pi$$
−0.171450 + 0.985193i $$0.554845\pi$$
$$182$$ 480.000i 0.195494i
$$183$$ − 960.000i − 0.387788i
$$184$$ −5085.00 −2.03734
$$185$$ 0 0
$$186$$ 2670.00 1.05255
$$187$$ − 363.000i − 0.141953i
$$188$$ − 3315.00i − 1.28602i
$$189$$ −81.0000 −0.0311740
$$190$$ 0 0
$$191$$ 3613.00 1.36873 0.684365 0.729139i $$-0.260079\pi$$
0.684365 + 0.729139i $$0.260079\pi$$
$$192$$ − 861.000i − 0.323632i
$$193$$ − 4204.00i − 1.56793i −0.620805 0.783965i $$-0.713194\pi$$
0.620805 0.783965i $$-0.286806\pi$$
$$194$$ −395.000 −0.146182
$$195$$ 0 0
$$196$$ −5678.00 −2.06924
$$197$$ − 4517.00i − 1.63362i −0.576908 0.816809i $$-0.695741\pi$$
0.576908 0.816809i $$-0.304259\pi$$
$$198$$ 495.000i 0.177667i
$$199$$ −4164.00 −1.48331 −0.741654 0.670783i $$-0.765958\pi$$
−0.741654 + 0.670783i $$0.765958\pi$$
$$200$$ 0 0
$$201$$ 600.000 0.210551
$$202$$ − 2725.00i − 0.949160i
$$203$$ 162.000i 0.0560107i
$$204$$ −1683.00 −0.577616
$$205$$ 0 0
$$206$$ −6530.00 −2.20858
$$207$$ 1017.00i 0.341480i
$$208$$ − 2848.00i − 0.949391i
$$209$$ 517.000 0.171108
$$210$$ 0 0
$$211$$ 4660.00 1.52042 0.760208 0.649680i $$-0.225097\pi$$
0.760208 + 0.649680i $$0.225097\pi$$
$$212$$ 2584.00i 0.837122i
$$213$$ 2841.00i 0.913907i
$$214$$ −9690.00 −3.09530
$$215$$ 0 0
$$216$$ 1215.00 0.382733
$$217$$ 534.000i 0.167052i
$$218$$ 2880.00i 0.894762i
$$219$$ 1344.00 0.414699
$$220$$ 0 0
$$221$$ 1056.00 0.321422
$$222$$ 285.000i 0.0861619i
$$223$$ − 3560.00i − 1.06904i −0.845157 0.534518i $$-0.820493\pi$$
0.845157 0.534518i $$-0.179507\pi$$
$$224$$ −255.000 −0.0760621
$$225$$ 0 0
$$226$$ −5520.00 −1.62471
$$227$$ − 4678.00i − 1.36780i −0.729577 0.683898i $$-0.760283\pi$$
0.729577 0.683898i $$-0.239717\pi$$
$$228$$ − 2397.00i − 0.696251i
$$229$$ 4447.00 1.28326 0.641629 0.767015i $$-0.278259\pi$$
0.641629 + 0.767015i $$0.278259\pi$$
$$230$$ 0 0
$$231$$ −99.0000 −0.0281979
$$232$$ − 2430.00i − 0.687661i
$$233$$ − 411.000i − 0.115560i −0.998329 0.0577801i $$-0.981598\pi$$
0.998329 0.0577801i $$-0.0184022\pi$$
$$234$$ −1440.00 −0.402290
$$235$$ 0 0
$$236$$ −10625.0 −2.93063
$$237$$ − 2163.00i − 0.592835i
$$238$$ − 495.000i − 0.134815i
$$239$$ 6380.00 1.72673 0.863364 0.504582i $$-0.168353\pi$$
0.863364 + 0.504582i $$0.168353\pi$$
$$240$$ 0 0
$$241$$ 7282.00 1.94637 0.973184 0.230027i $$-0.0738813\pi$$
0.973184 + 0.230027i $$0.0738813\pi$$
$$242$$ 605.000i 0.160706i
$$243$$ − 243.000i − 0.0641500i
$$244$$ −5440.00 −1.42730
$$245$$ 0 0
$$246$$ 2085.00 0.540385
$$247$$ 1504.00i 0.387438i
$$248$$ − 8010.00i − 2.05095i
$$249$$ −426.000 −0.108420
$$250$$ 0 0
$$251$$ −4728.00 −1.18896 −0.594480 0.804111i $$-0.702642\pi$$
−0.594480 + 0.804111i $$0.702642\pi$$
$$252$$ 459.000i 0.114739i
$$253$$ 1243.00i 0.308880i
$$254$$ 8695.00 2.14792
$$255$$ 0 0
$$256$$ −8279.00 −2.02124
$$257$$ 5418.00i 1.31504i 0.753437 + 0.657521i $$0.228395\pi$$
−0.753437 + 0.657521i $$0.771605\pi$$
$$258$$ 4620.00i 1.11484i
$$259$$ −57.0000 −0.0136749
$$260$$ 0 0
$$261$$ −486.000 −0.115259
$$262$$ 9090.00i 2.14344i
$$263$$ 3354.00i 0.786375i 0.919458 + 0.393187i $$0.128628\pi$$
−0.919458 + 0.393187i $$0.871372\pi$$
$$264$$ 1485.00 0.346195
$$265$$ 0 0
$$266$$ 705.000 0.162505
$$267$$ 1212.00i 0.277802i
$$268$$ − 3400.00i − 0.774955i
$$269$$ −1062.00 −0.240711 −0.120356 0.992731i $$-0.538403\pi$$
−0.120356 + 0.992731i $$0.538403\pi$$
$$270$$ 0 0
$$271$$ −4821.00 −1.08065 −0.540323 0.841458i $$-0.681698\pi$$
−0.540323 + 0.841458i $$0.681698\pi$$
$$272$$ 2937.00i 0.654712i
$$273$$ − 288.000i − 0.0638482i
$$274$$ −4350.00 −0.959099
$$275$$ 0 0
$$276$$ 5763.00 1.25685
$$277$$ 4.00000i 0 0.000867642i 1.00000 0.000433821i $$0.000138089\pi$$
−1.00000 0.000433821i $$0.999862\pi$$
$$278$$ 3180.00i 0.686057i
$$279$$ −1602.00 −0.343761
$$280$$ 0 0
$$281$$ 4647.00 0.986537 0.493268 0.869877i $$-0.335802\pi$$
0.493268 + 0.869877i $$0.335802\pi$$
$$282$$ 2925.00i 0.617664i
$$283$$ 4283.00i 0.899639i 0.893119 + 0.449820i $$0.148512\pi$$
−0.893119 + 0.449820i $$0.851488\pi$$
$$284$$ 16099.0 3.36373
$$285$$ 0 0
$$286$$ −1760.00 −0.363885
$$287$$ 417.000i 0.0857656i
$$288$$ − 765.000i − 0.156521i
$$289$$ 3824.00 0.778343
$$290$$ 0 0
$$291$$ 237.000 0.0477429
$$292$$ − 7616.00i − 1.52634i
$$293$$ 6811.00i 1.35803i 0.734124 + 0.679015i $$0.237593\pi$$
−0.734124 + 0.679015i $$0.762407\pi$$
$$294$$ 5010.00 0.993841
$$295$$ 0 0
$$296$$ 855.000 0.167891
$$297$$ − 297.000i − 0.0580259i
$$298$$ 1195.00i 0.232297i
$$299$$ −3616.00 −0.699394
$$300$$ 0 0
$$301$$ −924.000 −0.176938
$$302$$ 6040.00i 1.15087i
$$303$$ 1635.00i 0.309994i
$$304$$ −4183.00 −0.789183
$$305$$ 0 0
$$306$$ 1485.00 0.277424
$$307$$ − 460.000i − 0.0855166i −0.999085 0.0427583i $$-0.986385\pi$$
0.999085 0.0427583i $$-0.0136145\pi$$
$$308$$ 561.000i 0.103786i
$$309$$ 3918.00 0.721318
$$310$$ 0 0
$$311$$ 8328.00 1.51845 0.759224 0.650829i $$-0.225579\pi$$
0.759224 + 0.650829i $$0.225579\pi$$
$$312$$ 4320.00i 0.783884i
$$313$$ 5929.00i 1.07069i 0.844633 + 0.535346i $$0.179819\pi$$
−0.844633 + 0.535346i $$0.820181\pi$$
$$314$$ 9370.00 1.68401
$$315$$ 0 0
$$316$$ −12257.0 −2.18199
$$317$$ 5040.00i 0.892980i 0.894789 + 0.446490i $$0.147326\pi$$
−0.894789 + 0.446490i $$0.852674\pi$$
$$318$$ − 2280.00i − 0.402063i
$$319$$ −594.000 −0.104256
$$320$$ 0 0
$$321$$ 5814.00 1.01092
$$322$$ 1695.00i 0.293350i
$$323$$ − 1551.00i − 0.267183i
$$324$$ −1377.00 −0.236111
$$325$$ 0 0
$$326$$ −9520.00 −1.61737
$$327$$ − 1728.00i − 0.292228i
$$328$$ − 6255.00i − 1.05297i
$$329$$ −585.000 −0.0980307
$$330$$ 0 0
$$331$$ 10396.0 1.72633 0.863166 0.504920i $$-0.168478\pi$$
0.863166 + 0.504920i $$0.168478\pi$$
$$332$$ 2414.00i 0.399053i
$$333$$ − 171.000i − 0.0281404i
$$334$$ 5900.00 0.966568
$$335$$ 0 0
$$336$$ 801.000 0.130054
$$337$$ − 7236.00i − 1.16964i −0.811162 0.584822i $$-0.801164\pi$$
0.811162 0.584822i $$-0.198836\pi$$
$$338$$ 5865.00i 0.943828i
$$339$$ 3312.00 0.530629
$$340$$ 0 0
$$341$$ −1958.00 −0.310943
$$342$$ 2115.00i 0.334404i
$$343$$ 2031.00i 0.319719i
$$344$$ 13860.0 2.17233
$$345$$ 0 0
$$346$$ −15885.0 −2.46816
$$347$$ 1468.00i 0.227108i 0.993532 + 0.113554i $$0.0362234\pi$$
−0.993532 + 0.113554i $$0.963777\pi$$
$$348$$ 2754.00i 0.424224i
$$349$$ −5690.00 −0.872718 −0.436359 0.899773i $$-0.643732\pi$$
−0.436359 + 0.899773i $$0.643732\pi$$
$$350$$ 0 0
$$351$$ 864.000 0.131387
$$352$$ − 935.000i − 0.141579i
$$353$$ − 5376.00i − 0.810582i −0.914188 0.405291i $$-0.867170\pi$$
0.914188 0.405291i $$-0.132830\pi$$
$$354$$ 9375.00 1.40756
$$355$$ 0 0
$$356$$ 6868.00 1.02248
$$357$$ 297.000i 0.0440306i
$$358$$ − 8935.00i − 1.31908i
$$359$$ −3734.00 −0.548950 −0.274475 0.961594i $$-0.588504\pi$$
−0.274475 + 0.961594i $$0.588504\pi$$
$$360$$ 0 0
$$361$$ −4650.00 −0.677941
$$362$$ − 4175.00i − 0.606169i
$$363$$ − 363.000i − 0.0524864i
$$364$$ −1632.00 −0.235000
$$365$$ 0 0
$$366$$ 4800.00 0.685519
$$367$$ − 10274.0i − 1.46130i −0.682750 0.730652i $$-0.739216\pi$$
0.682750 0.730652i $$-0.260784\pi$$
$$368$$ − 10057.0i − 1.42461i
$$369$$ −1251.00 −0.176489
$$370$$ 0 0
$$371$$ 456.000 0.0638122
$$372$$ 9078.00i 1.26525i
$$373$$ − 13662.0i − 1.89649i −0.317537 0.948246i $$-0.602856\pi$$
0.317537 0.948246i $$-0.397144\pi$$
$$374$$ 1815.00 0.250940
$$375$$ 0 0
$$376$$ 8775.00 1.20355
$$377$$ − 1728.00i − 0.236065i
$$378$$ − 405.000i − 0.0551083i
$$379$$ 7906.00 1.07151 0.535757 0.844372i $$-0.320026\pi$$
0.535757 + 0.844372i $$0.320026\pi$$
$$380$$ 0 0
$$381$$ −5217.00 −0.701509
$$382$$ 18065.0i 2.41960i
$$383$$ 3168.00i 0.422656i 0.977415 + 0.211328i $$0.0677788\pi$$
−0.977415 + 0.211328i $$0.932221\pi$$
$$384$$ 6345.00 0.843208
$$385$$ 0 0
$$386$$ 21020.0 2.77174
$$387$$ − 2772.00i − 0.364105i
$$388$$ − 1343.00i − 0.175723i
$$389$$ −10770.0 −1.40375 −0.701877 0.712298i $$-0.747655\pi$$
−0.701877 + 0.712298i $$0.747655\pi$$
$$390$$ 0 0
$$391$$ 3729.00 0.482311
$$392$$ − 15030.0i − 1.93656i
$$393$$ − 5454.00i − 0.700046i
$$394$$ 22585.0 2.88786
$$395$$ 0 0
$$396$$ −1683.00 −0.213571
$$397$$ 5670.00i 0.716799i 0.933568 + 0.358399i $$0.116677\pi$$
−0.933568 + 0.358399i $$0.883323\pi$$
$$398$$ − 20820.0i − 2.62214i
$$399$$ −423.000 −0.0530739
$$400$$ 0 0
$$401$$ 832.000 0.103611 0.0518056 0.998657i $$-0.483502\pi$$
0.0518056 + 0.998657i $$0.483502\pi$$
$$402$$ 3000.00i 0.372205i
$$403$$ − 5696.00i − 0.704064i
$$404$$ 9265.00 1.14097
$$405$$ 0 0
$$406$$ −810.000 −0.0990139
$$407$$ − 209.000i − 0.0254539i
$$408$$ − 4455.00i − 0.540577i
$$409$$ 5712.00 0.690563 0.345281 0.938499i $$-0.387783\pi$$
0.345281 + 0.938499i $$0.387783\pi$$
$$410$$ 0 0
$$411$$ 2610.00 0.313240
$$412$$ − 22202.0i − 2.65489i
$$413$$ 1875.00i 0.223396i
$$414$$ −5085.00 −0.603657
$$415$$ 0 0
$$416$$ 2720.00 0.320574
$$417$$ − 1908.00i − 0.224065i
$$418$$ 2585.00i 0.302480i
$$419$$ 4559.00 0.531555 0.265778 0.964034i $$-0.414371\pi$$
0.265778 + 0.964034i $$0.414371\pi$$
$$420$$ 0 0
$$421$$ 6855.00 0.793568 0.396784 0.917912i $$-0.370126\pi$$
0.396784 + 0.917912i $$0.370126\pi$$
$$422$$ 23300.0i 2.68774i
$$423$$ − 1755.00i − 0.201728i
$$424$$ −6840.00 −0.783443
$$425$$ 0 0
$$426$$ −14205.0 −1.61557
$$427$$ 960.000i 0.108800i
$$428$$ − 32946.0i − 3.72081i
$$429$$ 1056.00 0.118844
$$430$$ 0 0
$$431$$ 10770.0 1.20365 0.601824 0.798628i $$-0.294441\pi$$
0.601824 + 0.798628i $$0.294441\pi$$
$$432$$ 2403.00i 0.267626i
$$433$$ − 8498.00i − 0.943159i −0.881824 0.471579i $$-0.843684\pi$$
0.881824 0.471579i $$-0.156316\pi$$
$$434$$ −2670.00 −0.295309
$$435$$ 0 0
$$436$$ −9792.00 −1.07558
$$437$$ 5311.00i 0.581372i
$$438$$ 6720.00i 0.733091i
$$439$$ −9835.00 −1.06925 −0.534623 0.845091i $$-0.679546\pi$$
−0.534623 + 0.845091i $$0.679546\pi$$
$$440$$ 0 0
$$441$$ −3006.00 −0.324587
$$442$$ 5280.00i 0.568199i
$$443$$ 10745.0i 1.15239i 0.817311 + 0.576197i $$0.195464\pi$$
−0.817311 + 0.576197i $$0.804536\pi$$
$$444$$ −969.000 −0.103574
$$445$$ 0 0
$$446$$ 17800.0 1.88981
$$447$$ − 717.000i − 0.0758679i
$$448$$ 861.000i 0.0908001i
$$449$$ −8356.00 −0.878272 −0.439136 0.898421i $$-0.644715\pi$$
−0.439136 + 0.898421i $$0.644715\pi$$
$$450$$ 0 0
$$451$$ −1529.00 −0.159640
$$452$$ − 18768.0i − 1.95304i
$$453$$ − 3624.00i − 0.375873i
$$454$$ 23390.0 2.41795
$$455$$ 0 0
$$456$$ 6345.00 0.651605
$$457$$ − 7058.00i − 0.722449i −0.932479 0.361225i $$-0.882359\pi$$
0.932479 0.361225i $$-0.117641\pi$$
$$458$$ 22235.0i 2.26850i
$$459$$ −891.000 −0.0906064
$$460$$ 0 0
$$461$$ 646.000 0.0652651 0.0326326 0.999467i $$-0.489611\pi$$
0.0326326 + 0.999467i $$0.489611\pi$$
$$462$$ − 495.000i − 0.0498474i
$$463$$ 8982.00i 0.901574i 0.892631 + 0.450787i $$0.148857\pi$$
−0.892631 + 0.450787i $$0.851143\pi$$
$$464$$ 4806.00 0.480847
$$465$$ 0 0
$$466$$ 2055.00 0.204283
$$467$$ − 13476.0i − 1.33532i −0.744466 0.667661i $$-0.767296\pi$$
0.744466 0.667661i $$-0.232704\pi$$
$$468$$ − 4896.00i − 0.483585i
$$469$$ −600.000 −0.0590734
$$470$$ 0 0
$$471$$ −5622.00 −0.549996
$$472$$ − 28125.0i − 2.74271i
$$473$$ − 3388.00i − 0.329345i
$$474$$ 10815.0 1.04799
$$475$$ 0 0
$$476$$ 1683.00 0.162059
$$477$$ 1368.00i 0.131313i
$$478$$ 31900.0i 3.05245i
$$479$$ −12996.0 −1.23967 −0.619835 0.784732i $$-0.712801\pi$$
−0.619835 + 0.784732i $$0.712801\pi$$
$$480$$ 0 0
$$481$$ 608.000 0.0576350
$$482$$ 36410.0i 3.44073i
$$483$$ − 1017.00i − 0.0958077i
$$484$$ −2057.00 −0.193182
$$485$$ 0 0
$$486$$ 1215.00 0.113402
$$487$$ − 6026.00i − 0.560707i −0.959897 0.280353i $$-0.909548\pi$$
0.959897 0.280353i $$-0.0904516\pi$$
$$488$$ − 14400.0i − 1.33577i
$$489$$ 5712.00 0.528232
$$490$$ 0 0
$$491$$ 11698.0 1.07520 0.537600 0.843200i $$-0.319331\pi$$
0.537600 + 0.843200i $$0.319331\pi$$
$$492$$ 7089.00i 0.649587i
$$493$$ 1782.00i 0.162794i
$$494$$ −7520.00 −0.684900
$$495$$ 0 0
$$496$$ 15842.0 1.43413
$$497$$ − 2841.00i − 0.256411i
$$498$$ − 2130.00i − 0.191662i
$$499$$ 17052.0 1.52976 0.764882 0.644170i $$-0.222797\pi$$
0.764882 + 0.644170i $$0.222797\pi$$
$$500$$ 0 0
$$501$$ −3540.00 −0.315680
$$502$$ − 23640.0i − 2.10180i
$$503$$ 932.000i 0.0826160i 0.999146 + 0.0413080i $$0.0131525\pi$$
−0.999146 + 0.0413080i $$0.986848\pi$$
$$504$$ −1215.00 −0.107382
$$505$$ 0 0
$$506$$ −6215.00 −0.546029
$$507$$ − 3519.00i − 0.308253i
$$508$$ 29563.0i 2.58198i
$$509$$ −4384.00 −0.381763 −0.190882 0.981613i $$-0.561135\pi$$
−0.190882 + 0.981613i $$0.561135\pi$$
$$510$$ 0 0
$$511$$ −1344.00 −0.116350
$$512$$ − 24475.0i − 2.11260i
$$513$$ − 1269.00i − 0.109216i
$$514$$ −27090.0 −2.32469
$$515$$ 0 0
$$516$$ −15708.0 −1.34013
$$517$$ − 2145.00i − 0.182470i
$$518$$ − 285.000i − 0.0241741i
$$519$$ 9531.00 0.806097
$$520$$ 0 0
$$521$$ −2322.00 −0.195257 −0.0976283 0.995223i $$-0.531126\pi$$
−0.0976283 + 0.995223i $$0.531126\pi$$
$$522$$ − 2430.00i − 0.203751i
$$523$$ 9749.00i 0.815094i 0.913184 + 0.407547i $$0.133616\pi$$
−0.913184 + 0.407547i $$0.866384\pi$$
$$524$$ −30906.0 −2.57659
$$525$$ 0 0
$$526$$ −16770.0 −1.39013
$$527$$ 5874.00i 0.485532i
$$528$$ 2937.00i 0.242077i
$$529$$ −602.000 −0.0494781
$$530$$ 0 0
$$531$$ −5625.00 −0.459707
$$532$$ 2397.00i 0.195344i
$$533$$ − 4448.00i − 0.361471i
$$534$$ −6060.00 −0.491090
$$535$$ 0 0
$$536$$ 9000.00 0.725263
$$537$$ 5361.00i 0.430809i
$$538$$ − 5310.00i − 0.425521i
$$539$$ −3674.00 −0.293600
$$540$$ 0 0
$$541$$ 4208.00 0.334410 0.167205 0.985922i $$-0.446526\pi$$
0.167205 + 0.985922i $$0.446526\pi$$
$$542$$ − 24105.0i − 1.91033i
$$543$$ 2505.00i 0.197974i
$$544$$ −2805.00 −0.221072
$$545$$ 0 0
$$546$$ 1440.00 0.112869
$$547$$ 10179.0i 0.795654i 0.917461 + 0.397827i $$0.130236\pi$$
−0.917461 + 0.397827i $$0.869764\pi$$
$$548$$ − 14790.0i − 1.15292i
$$549$$ −2880.00 −0.223890
$$550$$ 0 0
$$551$$ −2538.00 −0.196229
$$552$$ 15255.0i 1.17626i
$$553$$ 2163.00i 0.166329i
$$554$$ −20.0000 −0.00153379
$$555$$ 0 0
$$556$$ −10812.0 −0.824696
$$557$$ − 2314.00i − 0.176028i −0.996119 0.0880138i $$-0.971948\pi$$
0.996119 0.0880138i $$-0.0280519\pi$$
$$558$$ − 8010.00i − 0.607689i
$$559$$ 9856.00 0.745732
$$560$$ 0 0
$$561$$ −1089.00 −0.0819565
$$562$$ 23235.0i 1.74397i
$$563$$ − 24330.0i − 1.82129i −0.413188 0.910646i $$-0.635585\pi$$
0.413188 0.910646i $$-0.364415\pi$$
$$564$$ −9945.00 −0.742482
$$565$$ 0 0
$$566$$ −21415.0 −1.59035
$$567$$ 243.000i 0.0179983i
$$568$$ 42615.0i 3.14804i
$$569$$ −3445.00 −0.253817 −0.126909 0.991914i $$-0.540505\pi$$
−0.126909 + 0.991914i $$0.540505\pi$$
$$570$$ 0 0
$$571$$ −13056.0 −0.956877 −0.478438 0.878121i $$-0.658797\pi$$
−0.478438 + 0.878121i $$0.658797\pi$$
$$572$$ − 5984.00i − 0.437419i
$$573$$ − 10839.0i − 0.790237i
$$574$$ −2085.00 −0.151614
$$575$$ 0 0
$$576$$ −2583.00 −0.186849
$$577$$ − 17347.0i − 1.25159i −0.779989 0.625793i $$-0.784775\pi$$
0.779989 0.625793i $$-0.215225\pi$$
$$578$$ 19120.0i 1.37593i
$$579$$ −12612.0 −0.905245
$$580$$ 0 0
$$581$$ 426.000 0.0304190
$$582$$ 1185.00i 0.0843983i
$$583$$ 1672.00i 0.118777i
$$584$$ 20160.0 1.42847
$$585$$ 0 0
$$586$$ −34055.0 −2.40068
$$587$$ 8379.00i 0.589162i 0.955626 + 0.294581i $$0.0951801\pi$$
−0.955626 + 0.294581i $$0.904820\pi$$
$$588$$ 17034.0i 1.19468i
$$589$$ −8366.00 −0.585255
$$590$$ 0 0
$$591$$ −13551.0 −0.943170
$$592$$ 1691.00i 0.117398i
$$593$$ − 1958.00i − 0.135591i −0.997699 0.0677955i $$-0.978403\pi$$
0.997699 0.0677955i $$-0.0215965\pi$$
$$594$$ 1485.00 0.102576
$$595$$ 0 0
$$596$$ −4063.00 −0.279240
$$597$$ 12492.0i 0.856388i
$$598$$ − 18080.0i − 1.23636i
$$599$$ −23583.0 −1.60864 −0.804320 0.594196i $$-0.797470\pi$$
−0.804320 + 0.594196i $$0.797470\pi$$
$$600$$ 0 0
$$601$$ −15328.0 −1.04034 −0.520168 0.854064i $$-0.674131\pi$$
−0.520168 + 0.854064i $$0.674131\pi$$
$$602$$ − 4620.00i − 0.312786i
$$603$$ − 1800.00i − 0.121562i
$$604$$ −20536.0 −1.38344
$$605$$ 0 0
$$606$$ −8175.00 −0.547998
$$607$$ 160.000i 0.0106988i 0.999986 + 0.00534942i $$0.00170278\pi$$
−0.999986 + 0.00534942i $$0.998297\pi$$
$$608$$ − 3995.00i − 0.266478i
$$609$$ 486.000 0.0323378
$$610$$ 0 0
$$611$$ 6240.00 0.413164
$$612$$ 5049.00i 0.333486i
$$613$$ 5948.00i 0.391904i 0.980613 + 0.195952i $$0.0627798\pi$$
−0.980613 + 0.195952i $$0.937220\pi$$
$$614$$ 2300.00 0.151173
$$615$$ 0 0
$$616$$ −1485.00 −0.0971304
$$617$$ 334.000i 0.0217931i 0.999941 + 0.0108965i $$0.00346855\pi$$
−0.999941 + 0.0108965i $$0.996531\pi$$
$$618$$ 19590.0i 1.27512i
$$619$$ 7202.00 0.467646 0.233823 0.972279i $$-0.424876\pi$$
0.233823 + 0.972279i $$0.424876\pi$$
$$620$$ 0 0
$$621$$ 3051.00 0.197154
$$622$$ 41640.0i 2.68426i
$$623$$ − 1212.00i − 0.0779418i
$$624$$ −8544.00 −0.548131
$$625$$ 0 0
$$626$$ −29645.0 −1.89274
$$627$$ − 1551.00i − 0.0987894i
$$628$$ 31858.0i 2.02432i
$$629$$ −627.000 −0.0397458
$$630$$ 0 0
$$631$$ 10306.0 0.650199 0.325099 0.945680i $$-0.394602\pi$$
0.325099 + 0.945680i $$0.394602\pi$$
$$632$$ − 32445.0i − 2.04208i
$$633$$ − 13980.0i − 0.877812i
$$634$$ −25200.0 −1.57858
$$635$$ 0 0
$$636$$ 7752.00 0.483313
$$637$$ − 10688.0i − 0.664794i
$$638$$ − 2970.00i − 0.184300i
$$639$$ 8523.00 0.527644
$$640$$ 0 0
$$641$$ −1228.00 −0.0756678 −0.0378339 0.999284i $$-0.512046\pi$$
−0.0378339 + 0.999284i $$0.512046\pi$$
$$642$$ 29070.0i 1.78707i
$$643$$ 18454.0i 1.13181i 0.824470 + 0.565906i $$0.191473\pi$$
−0.824470 + 0.565906i $$0.808527\pi$$
$$644$$ −5763.00 −0.352630
$$645$$ 0 0
$$646$$ 7755.00 0.472316
$$647$$ 17647.0i 1.07230i 0.844124 + 0.536148i $$0.180121\pi$$
−0.844124 + 0.536148i $$0.819879\pi$$
$$648$$ − 3645.00i − 0.220971i
$$649$$ −6875.00 −0.415820
$$650$$ 0 0
$$651$$ 1602.00 0.0964475
$$652$$ − 32368.0i − 1.94422i
$$653$$ − 25918.0i − 1.55322i −0.629984 0.776608i $$-0.716939\pi$$
0.629984 0.776608i $$-0.283061\pi$$
$$654$$ 8640.00 0.516591
$$655$$ 0 0
$$656$$ 12371.0 0.736290
$$657$$ − 4032.00i − 0.239427i
$$658$$ − 2925.00i − 0.173295i
$$659$$ −12864.0 −0.760410 −0.380205 0.924902i $$-0.624147\pi$$
−0.380205 + 0.924902i $$0.624147\pi$$
$$660$$ 0 0
$$661$$ −11419.0 −0.671933 −0.335966 0.941874i $$-0.609063\pi$$
−0.335966 + 0.941874i $$0.609063\pi$$
$$662$$ 51980.0i 3.05175i
$$663$$ − 3168.00i − 0.185573i
$$664$$ −6390.00 −0.373464
$$665$$ 0 0
$$666$$ 855.000 0.0497456
$$667$$ − 6102.00i − 0.354228i
$$668$$ 20060.0i 1.16189i
$$669$$ −10680.0 −0.617209
$$670$$ 0 0
$$671$$ −3520.00 −0.202516
$$672$$ 765.000i 0.0439145i
$$673$$ − 15784.0i − 0.904054i −0.892004 0.452027i $$-0.850701\pi$$
0.892004 0.452027i $$-0.149299\pi$$
$$674$$ 36180.0 2.06766
$$675$$ 0 0
$$676$$ −19941.0 −1.13456
$$677$$ − 26050.0i − 1.47885i −0.673238 0.739426i $$-0.735097\pi$$
0.673238 0.739426i $$-0.264903\pi$$
$$678$$ 16560.0i 0.938028i
$$679$$ −237.000 −0.0133950
$$680$$ 0 0
$$681$$ −14034.0 −0.789698
$$682$$ − 9790.00i − 0.549675i
$$683$$ 15095.0i 0.845672i 0.906206 + 0.422836i $$0.138965\pi$$
−0.906206 + 0.422836i $$0.861035\pi$$
$$684$$ −7191.00 −0.401981
$$685$$ 0 0
$$686$$ −10155.0 −0.565189
$$687$$ − 13341.0i − 0.740889i
$$688$$ 27412.0i 1.51900i
$$689$$ −4864.00 −0.268946
$$690$$ 0 0
$$691$$ 15896.0 0.875126 0.437563 0.899188i $$-0.355842\pi$$
0.437563 + 0.899188i $$0.355842\pi$$
$$692$$ − 54009.0i − 2.96693i
$$693$$ 297.000i 0.0162801i
$$694$$ −7340.00 −0.401473
$$695$$ 0 0
$$696$$ −7290.00 −0.397021
$$697$$ 4587.00i 0.249275i
$$698$$ − 28450.0i − 1.54276i
$$699$$ −1233.00 −0.0667187
$$700$$ 0 0
$$701$$ 10529.0 0.567296 0.283648 0.958928i $$-0.408455\pi$$
0.283648 + 0.958928i $$0.408455\pi$$
$$702$$ 4320.00i 0.232262i
$$703$$ − 893.000i − 0.0479092i
$$704$$ −3157.00 −0.169011
$$705$$ 0 0
$$706$$ 26880.0 1.43292
$$707$$ − 1635.00i − 0.0869738i
$$708$$ 31875.0i 1.69200i
$$709$$ 16087.0 0.852130 0.426065 0.904693i $$-0.359900\pi$$
0.426065 + 0.904693i $$0.359900\pi$$
$$710$$ 0 0
$$711$$ −6489.00 −0.342274
$$712$$ 18180.0i 0.956916i
$$713$$ − 20114.0i − 1.05649i
$$714$$ −1485.00 −0.0778358
$$715$$ 0 0
$$716$$ 30379.0 1.58564
$$717$$ − 19140.0i − 0.996927i
$$718$$ − 18670.0i − 0.970415i
$$719$$ −24336.0 −1.26228 −0.631140 0.775669i $$-0.717413\pi$$
−0.631140 + 0.775669i $$0.717413\pi$$
$$720$$ 0 0
$$721$$ −3918.00 −0.202377
$$722$$ − 23250.0i − 1.19844i
$$723$$ − 21846.0i − 1.12374i
$$724$$ 14195.0 0.728664
$$725$$ 0 0
$$726$$ 1815.00 0.0927837
$$727$$ 13960.0i 0.712170i 0.934454 + 0.356085i $$0.115889\pi$$
−0.934454 + 0.356085i $$0.884111\pi$$
$$728$$ − 4320.00i − 0.219931i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ −10164.0 −0.514267
$$732$$ 16320.0i 0.824050i
$$733$$ − 9252.00i − 0.466208i −0.972452 0.233104i $$-0.925112\pi$$
0.972452 0.233104i $$-0.0748882\pi$$
$$734$$ 51370.0 2.58324
$$735$$ 0 0
$$736$$ 9605.00 0.481039
$$737$$ − 2200.00i − 0.109957i
$$738$$ − 6255.00i − 0.311992i
$$739$$ −28453.0 −1.41632 −0.708160 0.706052i $$-0.750474\pi$$
−0.708160 + 0.706052i $$0.750474\pi$$
$$740$$ 0 0
$$741$$ 4512.00 0.223688
$$742$$ 2280.00i 0.112805i
$$743$$ − 512.000i − 0.0252806i −0.999920 0.0126403i $$-0.995976\pi$$
0.999920 0.0126403i $$-0.00402363\pi$$
$$744$$ −24030.0 −1.18412
$$745$$ 0 0
$$746$$ 68310.0 3.35256
$$747$$ 1278.00i 0.0625965i
$$748$$ 6171.00i 0.301650i
$$749$$ −5814.00 −0.283630
$$750$$ 0 0
$$751$$ 772.000 0.0375109 0.0187554 0.999824i $$-0.494030\pi$$
0.0187554 + 0.999824i $$0.494030\pi$$
$$752$$ 17355.0i 0.841585i
$$753$$ 14184.0i 0.686446i
$$754$$ 8640.00 0.417308
$$755$$ 0 0
$$756$$ 1377.00 0.0662447
$$757$$ 8058.00i 0.386886i 0.981111 + 0.193443i $$0.0619655\pi$$
−0.981111 + 0.193443i $$0.938034\pi$$
$$758$$ 39530.0i 1.89419i
$$759$$ 3729.00 0.178332
$$760$$ 0 0
$$761$$ 18650.0 0.888386 0.444193 0.895931i $$-0.353490\pi$$
0.444193 + 0.895931i $$0.353490\pi$$
$$762$$ − 26085.0i − 1.24010i
$$763$$ 1728.00i 0.0819893i
$$764$$ −61421.0 −2.90855
$$765$$ 0 0
$$766$$ −15840.0 −0.747157
$$767$$ − 20000.0i − 0.941536i
$$768$$ 24837.0i 1.16696i
$$769$$ 7144.00 0.335005 0.167503 0.985872i $$-0.446430\pi$$
0.167503 + 0.985872i $$0.446430\pi$$
$$770$$ 0 0
$$771$$ 16254.0 0.759239
$$772$$ 71468.0i 3.33185i
$$773$$ 1904.00i 0.0885927i 0.999018 + 0.0442963i $$0.0141046\pi$$
−0.999018 + 0.0442963i $$0.985895\pi$$
$$774$$ 13860.0 0.643653
$$775$$ 0 0
$$776$$ 3555.00 0.164455
$$777$$ 171.000i 0.00789523i
$$778$$ − 53850.0i − 2.48151i
$$779$$ −6533.00 −0.300474
$$780$$ 0 0
$$781$$ 10417.0 0.477272
$$782$$ 18645.0i 0.852614i
$$783$$ 1458.00i 0.0665449i
$$784$$ 29726.0 1.35414
$$785$$ 0 0
$$786$$ 27270.0 1.23752
$$787$$ 7555.00i 0.342194i 0.985254 + 0.171097i $$0.0547311\pi$$
−0.985254 + 0.171097i $$0.945269\pi$$
$$788$$ 76789.0i 3.47144i
$$789$$ 10062.0 0.454014
$$790$$ 0 0
$$791$$ −3312.00 −0.148876
$$792$$ − 4455.00i − 0.199876i
$$793$$ − 10240.0i − 0.458554i
$$794$$ −28350.0 −1.26713
$$795$$ 0 0
$$796$$ 70788.0 3.15203
$$797$$ 24950.0i 1.10888i 0.832225 + 0.554438i $$0.187067\pi$$
−0.832225 + 0.554438i $$0.812933\pi$$
$$798$$ − 2115.00i − 0.0938223i
$$799$$ −6435.00 −0.284924
$$800$$ 0 0
$$801$$ 3636.00 0.160389
$$802$$ 4160.00i 0.183160i
$$803$$ − 4928.00i − 0.216570i
$$804$$ −10200.0 −0.447421
$$805$$ 0 0
$$806$$ 28480.0 1.24462
$$807$$ 3186.00i 0.138975i
$$808$$ 24525.0i 1.06781i
$$809$$ −19893.0 −0.864525 −0.432262 0.901748i $$-0.642285\pi$$
−0.432262 + 0.901748i $$0.642285\pi$$
$$810$$ 0 0
$$811$$ 34503.0 1.49391 0.746957 0.664872i $$-0.231514\pi$$
0.746957 + 0.664872i $$0.231514\pi$$
$$812$$ − 2754.00i − 0.119023i
$$813$$ 14463.0i 0.623911i
$$814$$ 1045.00 0.0449966
$$815$$ 0 0
$$816$$ 8811.00 0.377998
$$817$$ − 14476.0i − 0.619891i
$$818$$ 28560.0i 1.22075i
$$819$$ −864.000 −0.0368628
$$820$$ 0 0
$$821$$ 16890.0 0.717984 0.358992 0.933341i $$-0.383120\pi$$
0.358992 + 0.933341i $$0.383120\pi$$
$$822$$ 13050.0i 0.553736i
$$823$$ − 34692.0i − 1.46936i −0.678411 0.734682i $$-0.737331\pi$$
0.678411 0.734682i $$-0.262669\pi$$
$$824$$ 58770.0 2.48465
$$825$$ 0 0
$$826$$ −9375.00 −0.394913
$$827$$ 41424.0i 1.74178i 0.491476 + 0.870891i $$0.336457\pi$$
−0.491476 + 0.870891i $$0.663543\pi$$
$$828$$ − 17289.0i − 0.725645i
$$829$$ 18494.0 0.774817 0.387408 0.921908i $$-0.373370\pi$$
0.387408 + 0.921908i $$0.373370\pi$$
$$830$$ 0 0
$$831$$ 12.0000 0.000500933 0
$$832$$ − 9184.00i − 0.382690i
$$833$$ 11022.0i 0.458451i
$$834$$ 9540.00 0.396095
$$835$$ 0 0
$$836$$ −8789.00 −0.363605
$$837$$ 4806.00i 0.198470i
$$838$$ 22795.0i 0.939666i
$$839$$ −6680.00 −0.274874 −0.137437 0.990511i $$-0.543886\pi$$
−0.137437 + 0.990511i $$0.543886\pi$$
$$840$$ 0 0
$$841$$ −21473.0 −0.880438
$$842$$ 34275.0i 1.40284i
$$843$$ − 13941.0i − 0.569577i
$$844$$ −79220.0 −3.23088
$$845$$ 0 0
$$846$$ 8775.00 0.356608
$$847$$ 363.000i 0.0147259i
$$848$$ − 13528.0i − 0.547822i
$$849$$ 12849.0 0.519407
$$850$$ 0 0
$$851$$ 2147.00 0.0864844
$$852$$ − 48297.0i − 1.94205i
$$853$$ − 43358.0i − 1.74039i −0.492711 0.870193i $$-0.663994\pi$$
0.492711 0.870193i $$-0.336006\pi$$
$$854$$ −4800.00 −0.192333
$$855$$ 0 0
$$856$$ 87210.0 3.48222
$$857$$ 15585.0i 0.621206i 0.950540 + 0.310603i $$0.100531\pi$$
−0.950540 + 0.310603i $$0.899469\pi$$
$$858$$ 5280.00i 0.210089i
$$859$$ 17036.0 0.676672 0.338336 0.941025i $$-0.390136\pi$$
0.338336 + 0.941025i $$0.390136\pi$$
$$860$$ 0 0
$$861$$ 1251.00 0.0495168
$$862$$ 53850.0i 2.12777i
$$863$$ − 28064.0i − 1.10696i −0.832861 0.553482i $$-0.813299\pi$$
0.832861 0.553482i $$-0.186701\pi$$
$$864$$ −2295.00 −0.0903675
$$865$$ 0 0
$$866$$ 42490.0 1.66729
$$867$$ − 11472.0i − 0.449377i
$$868$$ − 9078.00i − 0.354985i
$$869$$ −7931.00 −0.309598
$$870$$ 0 0
$$871$$ 6400.00 0.248973
$$872$$ − 25920.0i − 1.00661i
$$873$$ − 711.000i − 0.0275644i
$$874$$ −26555.0 −1.02773
$$875$$ 0 0
$$876$$ −22848.0 −0.881236
$$877$$ − 22654.0i − 0.872259i −0.899884 0.436130i $$-0.856349\pi$$
0.899884 0.436130i $$-0.143651\pi$$
$$878$$ − 49175.0i − 1.89018i
$$879$$ 20433.0 0.784059
$$880$$ 0 0
$$881$$ −22380.0 −0.855847 −0.427924 0.903815i $$-0.640755\pi$$
−0.427924 + 0.903815i $$0.640755\pi$$
$$882$$ − 15030.0i − 0.573794i
$$883$$ − 35174.0i − 1.34054i −0.742116 0.670271i $$-0.766178\pi$$
0.742116 0.670271i $$-0.233822\pi$$
$$884$$ −17952.0 −0.683022
$$885$$ 0 0
$$886$$ −53725.0 −2.03716
$$887$$ 30868.0i 1.16848i 0.811579 + 0.584242i $$0.198608\pi$$
−0.811579 + 0.584242i $$0.801392\pi$$
$$888$$ − 2565.00i − 0.0969322i
$$889$$ 5217.00 0.196820
$$890$$ 0 0
$$891$$ −891.000 −0.0335013
$$892$$ 60520.0i 2.27170i
$$893$$ − 9165.00i − 0.343443i
$$894$$ 3585.00 0.134117
$$895$$ 0 0
$$896$$ −6345.00 −0.236575
$$897$$ 10848.0i 0.403795i
$$898$$ − 41780.0i − 1.55258i
$$899$$ 9612.00 0.356594
$$900$$ 0 0
$$901$$ 5016.00 0.185469
$$902$$ − 7645.00i − 0.282207i
$$903$$ 2772.00i 0.102155i
$$904$$ 49680.0 1.82780
$$905$$ 0 0
$$906$$ 18120.0 0.664455
$$907$$ 10070.0i 0.368654i 0.982865 + 0.184327i $$0.0590105\pi$$
−0.982865 + 0.184327i $$0.940990\pi$$
$$908$$ 79526.0i 2.90657i
$$909$$ 4905.00 0.178975
$$910$$ 0 0
$$911$$ 1885.00 0.0685542 0.0342771 0.999412i $$-0.489087\pi$$
0.0342771 + 0.999412i $$0.489087\pi$$
$$912$$ 12549.0i 0.455635i
$$913$$ 1562.00i 0.0566207i
$$914$$ 35290.0 1.27712
$$915$$ 0 0
$$916$$ −75599.0 −2.72692
$$917$$ 5454.00i 0.196409i
$$918$$ − 4455.00i − 0.160171i
$$919$$ −23703.0 −0.850805 −0.425403 0.905004i $$-0.639867\pi$$
−0.425403 + 0.905004i $$0.639867\pi$$
$$920$$ 0 0
$$921$$ −1380.00 −0.0493730
$$922$$ 3230.00i 0.115374i
$$923$$ 30304.0i 1.08068i
$$924$$ 1683.00 0.0599206
$$925$$ 0 0
$$926$$ −44910.0 −1.59377
$$927$$ − 11754.0i − 0.416453i
$$928$$ 4590.00i 0.162364i
$$929$$ −53804.0 −1.90016 −0.950082 0.312001i $$-0.899001\pi$$
−0.950082 + 0.312001i $$0.899001\pi$$
$$930$$ 0 0
$$931$$ −15698.0 −0.552611
$$932$$ 6987.00i 0.245565i
$$933$$ − 24984.0i − 0.876677i
$$934$$ 67380.0 2.36054
$$935$$ 0 0
$$936$$ 12960.0 0.452576
$$937$$ 1326.00i 0.0462311i 0.999733 + 0.0231155i $$0.00735856\pi$$
−0.999733 + 0.0231155i $$0.992641\pi$$
$$938$$ − 3000.00i − 0.104428i
$$939$$ 17787.0 0.618165
$$940$$ 0 0
$$941$$ −27109.0 −0.939137 −0.469569 0.882896i $$-0.655591\pi$$
−0.469569 + 0.882896i $$0.655591\pi$$
$$942$$ − 28110.0i − 0.972265i
$$943$$ − 15707.0i − 0.542408i
$$944$$ 55625.0 1.91784
$$945$$ 0 0
$$946$$ 16940.0 0.582206
$$947$$ 31143.0i 1.06865i 0.845279 + 0.534325i $$0.179434\pi$$
−0.845279 + 0.534325i $$0.820566\pi$$
$$948$$ 36771.0i 1.25977i
$$949$$ 14336.0 0.490375
$$950$$ 0 0
$$951$$ 15120.0 0.515562
$$952$$ 4455.00i 0.151667i
$$953$$ 879.000i 0.0298779i 0.999888 + 0.0149389i $$0.00475539\pi$$
−0.999888 + 0.0149389i $$0.995245\pi$$
$$954$$ −6840.00 −0.232131
$$955$$ 0 0
$$956$$ −108460. −3.66930
$$957$$ 1782.00i 0.0601921i
$$958$$ − 64980.0i − 2.19145i
$$959$$ −2610.00 −0.0878846
$$960$$ 0 0
$$961$$ 1893.00 0.0635427
$$962$$ 3040.00i 0.101885i
$$963$$ − 17442.0i − 0.583656i
$$964$$ −123794. −4.13603
$$965$$ 0 0
$$966$$ 5085.00 0.169366
$$967$$ − 14824.0i − 0.492976i −0.969146 0.246488i $$-0.920723\pi$$
0.969146 0.246488i $$-0.0792766\pi$$
$$968$$ − 5445.00i − 0.180794i
$$969$$ −4653.00 −0.154258
$$970$$ 0 0
$$971$$ 34089.0 1.12664 0.563320 0.826239i $$-0.309524\pi$$
0.563320 + 0.826239i $$0.309524\pi$$
$$972$$ 4131.00i 0.136319i
$$973$$ 1908.00i 0.0628650i
$$974$$ 30130.0 0.991199
$$975$$ 0 0
$$976$$ 28480.0 0.934040
$$977$$ 33446.0i 1.09522i 0.836733 + 0.547611i $$0.184463\pi$$
−0.836733 + 0.547611i $$0.815537\pi$$
$$978$$ 28560.0i 0.933792i
$$979$$ 4444.00 0.145077
$$980$$ 0 0
$$981$$ −5184.00 −0.168718
$$982$$ 58490.0i 1.90070i
$$983$$ 52025.0i 1.68804i 0.536315 + 0.844018i $$0.319816\pi$$
−0.536315 + 0.844018i $$0.680184\pi$$
$$984$$ −18765.0 −0.607933
$$985$$ 0 0
$$986$$ −8910.00 −0.287781
$$987$$ 1755.00i 0.0565980i
$$988$$ − 25568.0i − 0.823306i
$$989$$ 34804.0 1.11901
$$990$$ 0 0
$$991$$ −41260.0 −1.32257 −0.661285 0.750135i $$-0.729989\pi$$
−0.661285 + 0.750135i $$0.729989\pi$$
$$992$$ 15130.0i 0.484252i
$$993$$ − 31188.0i − 0.996698i
$$994$$ 14205.0 0.453275
$$995$$ 0 0
$$996$$ 7242.00 0.230393
$$997$$ − 190.000i − 0.00603547i −0.999995 0.00301773i $$-0.999039\pi$$
0.999995 0.00301773i $$-0.000960576\pi$$
$$998$$ 85260.0i 2.70427i
$$999$$ −513.000 −0.0162468
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 825.4.c.b.199.2 2
5.2 odd 4 825.4.a.a.1.1 1
5.3 odd 4 825.4.a.j.1.1 yes 1
5.4 even 2 inner 825.4.c.b.199.1 2
15.2 even 4 2475.4.a.k.1.1 1
15.8 even 4 2475.4.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.a.1.1 1 5.2 odd 4
825.4.a.j.1.1 yes 1 5.3 odd 4
825.4.c.b.199.1 2 5.4 even 2 inner
825.4.c.b.199.2 2 1.1 even 1 trivial
2475.4.a.a.1.1 1 15.8 even 4
2475.4.a.k.1.1 1 15.2 even 4