# Properties

 Label 25.4.b.b.24.2 Level $25$ Weight $4$ Character 25.24 Analytic conductor $1.475$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,4,Mod(24,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("25.24");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (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: $$1.47504775014$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 24.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 25.24 Dual form 25.4.b.b.24.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -7.00000i q^{3} +7.00000 q^{4} +7.00000 q^{6} +6.00000i q^{7} +15.0000i q^{8} -22.0000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -7.00000i q^{3} +7.00000 q^{4} +7.00000 q^{6} +6.00000i q^{7} +15.0000i q^{8} -22.0000 q^{9} -43.0000 q^{11} -49.0000i q^{12} +28.0000i q^{13} -6.00000 q^{14} +41.0000 q^{16} +91.0000i q^{17} -22.0000i q^{18} +35.0000 q^{19} +42.0000 q^{21} -43.0000i q^{22} -162.000i q^{23} +105.000 q^{24} -28.0000 q^{26} -35.0000i q^{27} +42.0000i q^{28} -160.000 q^{29} +42.0000 q^{31} +161.000i q^{32} +301.000i q^{33} -91.0000 q^{34} -154.000 q^{36} -314.000i q^{37} +35.0000i q^{38} +196.000 q^{39} -203.000 q^{41} +42.0000i q^{42} -92.0000i q^{43} -301.000 q^{44} +162.000 q^{46} +196.000i q^{47} -287.000i q^{48} +307.000 q^{49} +637.000 q^{51} +196.000i q^{52} -82.0000i q^{53} +35.0000 q^{54} -90.0000 q^{56} -245.000i q^{57} -160.000i q^{58} +280.000 q^{59} -518.000 q^{61} +42.0000i q^{62} -132.000i q^{63} +167.000 q^{64} -301.000 q^{66} +141.000i q^{67} +637.000i q^{68} -1134.00 q^{69} +412.000 q^{71} -330.000i q^{72} +763.000i q^{73} +314.000 q^{74} +245.000 q^{76} -258.000i q^{77} +196.000i q^{78} -510.000 q^{79} -839.000 q^{81} -203.000i q^{82} -777.000i q^{83} +294.000 q^{84} +92.0000 q^{86} +1120.00i q^{87} -645.000i q^{88} +945.000 q^{89} -168.000 q^{91} -1134.00i q^{92} -294.000i q^{93} -196.000 q^{94} +1127.00 q^{96} +1246.00i q^{97} +307.000i q^{98} +946.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{4} + 14 q^{6} - 44 q^{9}+O(q^{10})$$ 2 * q + 14 * q^4 + 14 * q^6 - 44 * q^9 $$2 q + 14 q^{4} + 14 q^{6} - 44 q^{9} - 86 q^{11} - 12 q^{14} + 82 q^{16} + 70 q^{19} + 84 q^{21} + 210 q^{24} - 56 q^{26} - 320 q^{29} + 84 q^{31} - 182 q^{34} - 308 q^{36} + 392 q^{39} - 406 q^{41} - 602 q^{44} + 324 q^{46} + 614 q^{49} + 1274 q^{51} + 70 q^{54} - 180 q^{56} + 560 q^{59} - 1036 q^{61} + 334 q^{64} - 602 q^{66} - 2268 q^{69} + 824 q^{71} + 628 q^{74} + 490 q^{76} - 1020 q^{79} - 1678 q^{81} + 588 q^{84} + 184 q^{86} + 1890 q^{89} - 336 q^{91} - 392 q^{94} + 2254 q^{96} + 1892 q^{99}+O(q^{100})$$ 2 * q + 14 * q^4 + 14 * q^6 - 44 * q^9 - 86 * q^11 - 12 * q^14 + 82 * q^16 + 70 * q^19 + 84 * q^21 + 210 * q^24 - 56 * q^26 - 320 * q^29 + 84 * q^31 - 182 * q^34 - 308 * q^36 + 392 * q^39 - 406 * q^41 - 602 * q^44 + 324 * q^46 + 614 * q^49 + 1274 * q^51 + 70 * q^54 - 180 * q^56 + 560 * q^59 - 1036 * q^61 + 334 * q^64 - 602 * q^66 - 2268 * q^69 + 824 * q^71 + 628 * q^74 + 490 * q^76 - 1020 * q^79 - 1678 * q^81 + 588 * q^84 + 184 * q^86 + 1890 * q^89 - 336 * q^91 - 392 * q^94 + 2254 * q^96 + 1892 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.353553i 0.984251 + 0.176777i $$0.0565670\pi$$
−0.984251 + 0.176777i $$0.943433\pi$$
$$3$$ − 7.00000i − 1.34715i −0.739119 0.673575i $$-0.764758\pi$$
0.739119 0.673575i $$-0.235242\pi$$
$$4$$ 7.00000 0.875000
$$5$$ 0 0
$$6$$ 7.00000 0.476290
$$7$$ 6.00000i 0.323970i 0.986793 + 0.161985i $$0.0517895\pi$$
−0.986793 + 0.161985i $$0.948210\pi$$
$$8$$ 15.0000i 0.662913i
$$9$$ −22.0000 −0.814815
$$10$$ 0 0
$$11$$ −43.0000 −1.17864 −0.589318 0.807901i $$-0.700603\pi$$
−0.589318 + 0.807901i $$0.700603\pi$$
$$12$$ − 49.0000i − 1.17876i
$$13$$ 28.0000i 0.597369i 0.954352 + 0.298685i $$0.0965479\pi$$
−0.954352 + 0.298685i $$0.903452\pi$$
$$14$$ −6.00000 −0.114541
$$15$$ 0 0
$$16$$ 41.0000 0.640625
$$17$$ 91.0000i 1.29828i 0.760669 + 0.649139i $$0.224871\pi$$
−0.760669 + 0.649139i $$0.775129\pi$$
$$18$$ − 22.0000i − 0.288081i
$$19$$ 35.0000 0.422608 0.211304 0.977420i $$-0.432229\pi$$
0.211304 + 0.977420i $$0.432229\pi$$
$$20$$ 0 0
$$21$$ 42.0000 0.436436
$$22$$ − 43.0000i − 0.416710i
$$23$$ − 162.000i − 1.46867i −0.678789 0.734333i $$-0.737495\pi$$
0.678789 0.734333i $$-0.262505\pi$$
$$24$$ 105.000 0.893043
$$25$$ 0 0
$$26$$ −28.0000 −0.211202
$$27$$ − 35.0000i − 0.249472i
$$28$$ 42.0000i 0.283473i
$$29$$ −160.000 −1.02453 −0.512263 0.858829i $$-0.671193\pi$$
−0.512263 + 0.858829i $$0.671193\pi$$
$$30$$ 0 0
$$31$$ 42.0000 0.243336 0.121668 0.992571i $$-0.461176\pi$$
0.121668 + 0.992571i $$0.461176\pi$$
$$32$$ 161.000i 0.889408i
$$33$$ 301.000i 1.58780i
$$34$$ −91.0000 −0.459011
$$35$$ 0 0
$$36$$ −154.000 −0.712963
$$37$$ − 314.000i − 1.39517i −0.716502 0.697585i $$-0.754258\pi$$
0.716502 0.697585i $$-0.245742\pi$$
$$38$$ 35.0000i 0.149414i
$$39$$ 196.000 0.804747
$$40$$ 0 0
$$41$$ −203.000 −0.773251 −0.386625 0.922237i $$-0.626359\pi$$
−0.386625 + 0.922237i $$0.626359\pi$$
$$42$$ 42.0000i 0.154303i
$$43$$ − 92.0000i − 0.326276i −0.986603 0.163138i $$-0.947838\pi$$
0.986603 0.163138i $$-0.0521616\pi$$
$$44$$ −301.000 −1.03131
$$45$$ 0 0
$$46$$ 162.000 0.519252
$$47$$ 196.000i 0.608288i 0.952626 + 0.304144i $$0.0983704\pi$$
−0.952626 + 0.304144i $$0.901630\pi$$
$$48$$ − 287.000i − 0.863018i
$$49$$ 307.000 0.895044
$$50$$ 0 0
$$51$$ 637.000 1.74898
$$52$$ 196.000i 0.522698i
$$53$$ − 82.0000i − 0.212520i −0.994338 0.106260i $$-0.966112\pi$$
0.994338 0.106260i $$-0.0338876\pi$$
$$54$$ 35.0000 0.0882018
$$55$$ 0 0
$$56$$ −90.0000 −0.214763
$$57$$ − 245.000i − 0.569317i
$$58$$ − 160.000i − 0.362225i
$$59$$ 280.000 0.617846 0.308923 0.951087i $$-0.400032\pi$$
0.308923 + 0.951087i $$0.400032\pi$$
$$60$$ 0 0
$$61$$ −518.000 −1.08726 −0.543632 0.839324i $$-0.682951\pi$$
−0.543632 + 0.839324i $$0.682951\pi$$
$$62$$ 42.0000i 0.0860323i
$$63$$ − 132.000i − 0.263975i
$$64$$ 167.000 0.326172
$$65$$ 0 0
$$66$$ −301.000 −0.561372
$$67$$ 141.000i 0.257103i 0.991703 + 0.128551i $$0.0410327\pi$$
−0.991703 + 0.128551i $$0.958967\pi$$
$$68$$ 637.000i 1.13599i
$$69$$ −1134.00 −1.97852
$$70$$ 0 0
$$71$$ 412.000 0.688668 0.344334 0.938847i $$-0.388105\pi$$
0.344334 + 0.938847i $$0.388105\pi$$
$$72$$ − 330.000i − 0.540151i
$$73$$ 763.000i 1.22332i 0.791121 + 0.611660i $$0.209498\pi$$
−0.791121 + 0.611660i $$0.790502\pi$$
$$74$$ 314.000 0.493267
$$75$$ 0 0
$$76$$ 245.000 0.369782
$$77$$ − 258.000i − 0.381842i
$$78$$ 196.000i 0.284521i
$$79$$ −510.000 −0.726323 −0.363161 0.931726i $$-0.618303\pi$$
−0.363161 + 0.931726i $$0.618303\pi$$
$$80$$ 0 0
$$81$$ −839.000 −1.15089
$$82$$ − 203.000i − 0.273385i
$$83$$ − 777.000i − 1.02755i −0.857924 0.513776i $$-0.828246\pi$$
0.857924 0.513776i $$-0.171754\pi$$
$$84$$ 294.000 0.381881
$$85$$ 0 0
$$86$$ 92.0000 0.115356
$$87$$ 1120.00i 1.38019i
$$88$$ − 645.000i − 0.781332i
$$89$$ 945.000 1.12550 0.562752 0.826626i $$-0.309743\pi$$
0.562752 + 0.826626i $$0.309743\pi$$
$$90$$ 0 0
$$91$$ −168.000 −0.193530
$$92$$ − 1134.00i − 1.28508i
$$93$$ − 294.000i − 0.327811i
$$94$$ −196.000 −0.215062
$$95$$ 0 0
$$96$$ 1127.00 1.19817
$$97$$ 1246.00i 1.30425i 0.758112 + 0.652124i $$0.226122\pi$$
−0.758112 + 0.652124i $$0.773878\pi$$
$$98$$ 307.000i 0.316446i
$$99$$ 946.000 0.960369
$$100$$ 0 0
$$101$$ 1302.00 1.28271 0.641356 0.767244i $$-0.278372\pi$$
0.641356 + 0.767244i $$0.278372\pi$$
$$102$$ 637.000i 0.618357i
$$103$$ − 532.000i − 0.508927i −0.967082 0.254464i $$-0.918101\pi$$
0.967082 0.254464i $$-0.0818989\pi$$
$$104$$ −420.000 −0.396004
$$105$$ 0 0
$$106$$ 82.0000 0.0751372
$$107$$ − 1269.00i − 1.14653i −0.819370 0.573266i $$-0.805676\pi$$
0.819370 0.573266i $$-0.194324\pi$$
$$108$$ − 245.000i − 0.218288i
$$109$$ −1070.00 −0.940251 −0.470126 0.882599i $$-0.655791\pi$$
−0.470126 + 0.882599i $$0.655791\pi$$
$$110$$ 0 0
$$111$$ −2198.00 −1.87950
$$112$$ 246.000i 0.207543i
$$113$$ 503.000i 0.418746i 0.977836 + 0.209373i $$0.0671422\pi$$
−0.977836 + 0.209373i $$0.932858\pi$$
$$114$$ 245.000 0.201284
$$115$$ 0 0
$$116$$ −1120.00 −0.896460
$$117$$ − 616.000i − 0.486745i
$$118$$ 280.000i 0.218441i
$$119$$ −546.000 −0.420603
$$120$$ 0 0
$$121$$ 518.000 0.389181
$$122$$ − 518.000i − 0.384406i
$$123$$ 1421.00i 1.04169i
$$124$$ 294.000 0.212919
$$125$$ 0 0
$$126$$ 132.000 0.0933293
$$127$$ − 874.000i − 0.610669i −0.952245 0.305334i $$-0.901232\pi$$
0.952245 0.305334i $$-0.0987683\pi$$
$$128$$ 1455.00i 1.00473i
$$129$$ −644.000 −0.439543
$$130$$ 0 0
$$131$$ 1092.00 0.728309 0.364155 0.931339i $$-0.381358\pi$$
0.364155 + 0.931339i $$0.381358\pi$$
$$132$$ 2107.00i 1.38932i
$$133$$ 210.000i 0.136912i
$$134$$ −141.000 −0.0908996
$$135$$ 0 0
$$136$$ −1365.00 −0.860645
$$137$$ 411.000i 0.256307i 0.991754 + 0.128154i $$0.0409051\pi$$
−0.991754 + 0.128154i $$0.959095\pi$$
$$138$$ − 1134.00i − 0.699511i
$$139$$ 595.000 0.363074 0.181537 0.983384i $$-0.441893\pi$$
0.181537 + 0.983384i $$0.441893\pi$$
$$140$$ 0 0
$$141$$ 1372.00 0.819456
$$142$$ 412.000i 0.243481i
$$143$$ − 1204.00i − 0.704081i
$$144$$ −902.000 −0.521991
$$145$$ 0 0
$$146$$ −763.000 −0.432509
$$147$$ − 2149.00i − 1.20576i
$$148$$ − 2198.00i − 1.22077i
$$149$$ 3200.00 1.75942 0.879712 0.475507i $$-0.157735\pi$$
0.879712 + 0.475507i $$0.157735\pi$$
$$150$$ 0 0
$$151$$ 202.000 0.108864 0.0544322 0.998517i $$-0.482665\pi$$
0.0544322 + 0.998517i $$0.482665\pi$$
$$152$$ 525.000i 0.280152i
$$153$$ − 2002.00i − 1.05786i
$$154$$ 258.000 0.135002
$$155$$ 0 0
$$156$$ 1372.00 0.704153
$$157$$ 406.000i 0.206384i 0.994661 + 0.103192i $$0.0329057\pi$$
−0.994661 + 0.103192i $$0.967094\pi$$
$$158$$ − 510.000i − 0.256794i
$$159$$ −574.000 −0.286297
$$160$$ 0 0
$$161$$ 972.000 0.475803
$$162$$ − 839.000i − 0.406902i
$$163$$ 3803.00i 1.82745i 0.406336 + 0.913724i $$0.366806\pi$$
−0.406336 + 0.913724i $$0.633194\pi$$
$$164$$ −1421.00 −0.676594
$$165$$ 0 0
$$166$$ 777.000 0.363295
$$167$$ 4116.00i 1.90722i 0.301046 + 0.953610i $$0.402664\pi$$
−0.301046 + 0.953610i $$0.597336\pi$$
$$168$$ 630.000i 0.289319i
$$169$$ 1413.00 0.643150
$$170$$ 0 0
$$171$$ −770.000 −0.344347
$$172$$ − 644.000i − 0.285492i
$$173$$ − 1512.00i − 0.664481i −0.943195 0.332241i $$-0.892195\pi$$
0.943195 0.332241i $$-0.107805\pi$$
$$174$$ −1120.00 −0.487971
$$175$$ 0 0
$$176$$ −1763.00 −0.755063
$$177$$ − 1960.00i − 0.832331i
$$178$$ 945.000i 0.397926i
$$179$$ −2585.00 −1.07940 −0.539698 0.841859i $$-0.681462\pi$$
−0.539698 + 0.841859i $$0.681462\pi$$
$$180$$ 0 0
$$181$$ −2758.00 −1.13260 −0.566300 0.824199i $$-0.691626\pi$$
−0.566300 + 0.824199i $$0.691626\pi$$
$$182$$ − 168.000i − 0.0684230i
$$183$$ 3626.00i 1.46471i
$$184$$ 2430.00 0.973598
$$185$$ 0 0
$$186$$ 294.000 0.115899
$$187$$ − 3913.00i − 1.53020i
$$188$$ 1372.00i 0.532252i
$$189$$ 210.000 0.0808214
$$190$$ 0 0
$$191$$ −2378.00 −0.900869 −0.450435 0.892809i $$-0.648731\pi$$
−0.450435 + 0.892809i $$0.648731\pi$$
$$192$$ − 1169.00i − 0.439403i
$$193$$ − 3067.00i − 1.14387i −0.820298 0.571937i $$-0.806192\pi$$
0.820298 0.571937i $$-0.193808\pi$$
$$194$$ −1246.00 −0.461122
$$195$$ 0 0
$$196$$ 2149.00 0.783163
$$197$$ 2346.00i 0.848455i 0.905556 + 0.424227i $$0.139454\pi$$
−0.905556 + 0.424227i $$0.860546\pi$$
$$198$$ 946.000i 0.339542i
$$199$$ −4900.00 −1.74549 −0.872743 0.488180i $$-0.837661\pi$$
−0.872743 + 0.488180i $$0.837661\pi$$
$$200$$ 0 0
$$201$$ 987.000 0.346356
$$202$$ 1302.00i 0.453507i
$$203$$ − 960.000i − 0.331915i
$$204$$ 4459.00 1.53036
$$205$$ 0 0
$$206$$ 532.000 0.179933
$$207$$ 3564.00i 1.19669i
$$208$$ 1148.00i 0.382690i
$$209$$ −1505.00 −0.498101
$$210$$ 0 0
$$211$$ 4307.00 1.40524 0.702621 0.711564i $$-0.252013\pi$$
0.702621 + 0.711564i $$0.252013\pi$$
$$212$$ − 574.000i − 0.185955i
$$213$$ − 2884.00i − 0.927739i
$$214$$ 1269.00 0.405360
$$215$$ 0 0
$$216$$ 525.000 0.165378
$$217$$ 252.000i 0.0788335i
$$218$$ − 1070.00i − 0.332429i
$$219$$ 5341.00 1.64800
$$220$$ 0 0
$$221$$ −2548.00 −0.775552
$$222$$ − 2198.00i − 0.664505i
$$223$$ − 2212.00i − 0.664244i −0.943236 0.332122i $$-0.892235\pi$$
0.943236 0.332122i $$-0.107765\pi$$
$$224$$ −966.000 −0.288141
$$225$$ 0 0
$$226$$ −503.000 −0.148049
$$227$$ 476.000i 0.139177i 0.997576 + 0.0695886i $$0.0221687\pi$$
−0.997576 + 0.0695886i $$0.977831\pi$$
$$228$$ − 1715.00i − 0.498152i
$$229$$ 2940.00 0.848387 0.424194 0.905572i $$-0.360558\pi$$
0.424194 + 0.905572i $$0.360558\pi$$
$$230$$ 0 0
$$231$$ −1806.00 −0.514399
$$232$$ − 2400.00i − 0.679171i
$$233$$ − 1002.00i − 0.281730i −0.990029 0.140865i $$-0.955012\pi$$
0.990029 0.140865i $$-0.0449884\pi$$
$$234$$ 616.000 0.172091
$$235$$ 0 0
$$236$$ 1960.00 0.540615
$$237$$ 3570.00i 0.978466i
$$238$$ − 546.000i − 0.148706i
$$239$$ −2480.00 −0.671204 −0.335602 0.942004i $$-0.608940\pi$$
−0.335602 + 0.942004i $$0.608940\pi$$
$$240$$ 0 0
$$241$$ 1897.00 0.507039 0.253520 0.967330i $$-0.418412\pi$$
0.253520 + 0.967330i $$0.418412\pi$$
$$242$$ 518.000i 0.137596i
$$243$$ 4928.00i 1.30095i
$$244$$ −3626.00 −0.951356
$$245$$ 0 0
$$246$$ −1421.00 −0.368291
$$247$$ 980.000i 0.252453i
$$248$$ 630.000i 0.161311i
$$249$$ −5439.00 −1.38427
$$250$$ 0 0
$$251$$ −2373.00 −0.596743 −0.298371 0.954450i $$-0.596443\pi$$
−0.298371 + 0.954450i $$0.596443\pi$$
$$252$$ − 924.000i − 0.230978i
$$253$$ 6966.00i 1.73102i
$$254$$ 874.000 0.215904
$$255$$ 0 0
$$256$$ −119.000 −0.0290527
$$257$$ − 4494.00i − 1.09077i −0.838185 0.545385i $$-0.816383\pi$$
0.838185 0.545385i $$-0.183617\pi$$
$$258$$ − 644.000i − 0.155402i
$$259$$ 1884.00 0.451993
$$260$$ 0 0
$$261$$ 3520.00 0.834799
$$262$$ 1092.00i 0.257496i
$$263$$ − 722.000i − 0.169279i −0.996412 0.0846396i $$-0.973026\pi$$
0.996412 0.0846396i $$-0.0269739\pi$$
$$264$$ −4515.00 −1.05257
$$265$$ 0 0
$$266$$ −210.000 −0.0484057
$$267$$ − 6615.00i − 1.51622i
$$268$$ 987.000i 0.224965i
$$269$$ 6160.00 1.39621 0.698107 0.715993i $$-0.254026\pi$$
0.698107 + 0.715993i $$0.254026\pi$$
$$270$$ 0 0
$$271$$ −7238.00 −1.62243 −0.811213 0.584751i $$-0.801192\pi$$
−0.811213 + 0.584751i $$0.801192\pi$$
$$272$$ 3731.00i 0.831710i
$$273$$ 1176.00i 0.260713i
$$274$$ −411.000 −0.0906183
$$275$$ 0 0
$$276$$ −7938.00 −1.73120
$$277$$ 1776.00i 0.385233i 0.981274 + 0.192616i $$0.0616973\pi$$
−0.981274 + 0.192616i $$0.938303\pi$$
$$278$$ 595.000i 0.128366i
$$279$$ −924.000 −0.198274
$$280$$ 0 0
$$281$$ 4542.00 0.964246 0.482123 0.876104i $$-0.339866\pi$$
0.482123 + 0.876104i $$0.339866\pi$$
$$282$$ 1372.00i 0.289721i
$$283$$ − 7077.00i − 1.48652i −0.669005 0.743258i $$-0.733280\pi$$
0.669005 0.743258i $$-0.266720\pi$$
$$284$$ 2884.00 0.602584
$$285$$ 0 0
$$286$$ 1204.00 0.248930
$$287$$ − 1218.00i − 0.250510i
$$288$$ − 3542.00i − 0.724703i
$$289$$ −3368.00 −0.685528
$$290$$ 0 0
$$291$$ 8722.00 1.75702
$$292$$ 5341.00i 1.07041i
$$293$$ 4158.00i 0.829054i 0.910037 + 0.414527i $$0.136053\pi$$
−0.910037 + 0.414527i $$0.863947\pi$$
$$294$$ 2149.00 0.426300
$$295$$ 0 0
$$296$$ 4710.00 0.924876
$$297$$ 1505.00i 0.294037i
$$298$$ 3200.00i 0.622050i
$$299$$ 4536.00 0.877337
$$300$$ 0 0
$$301$$ 552.000 0.105703
$$302$$ 202.000i 0.0384894i
$$303$$ − 9114.00i − 1.72801i
$$304$$ 1435.00 0.270733
$$305$$ 0 0
$$306$$ 2002.00 0.374009
$$307$$ − 2569.00i − 0.477591i −0.971070 0.238796i $$-0.923247\pi$$
0.971070 0.238796i $$-0.0767526\pi$$
$$308$$ − 1806.00i − 0.334112i
$$309$$ −3724.00 −0.685602
$$310$$ 0 0
$$311$$ 2982.00 0.543710 0.271855 0.962338i $$-0.412363\pi$$
0.271855 + 0.962338i $$0.412363\pi$$
$$312$$ 2940.00i 0.533477i
$$313$$ − 2422.00i − 0.437379i −0.975795 0.218689i $$-0.929822\pi$$
0.975795 0.218689i $$-0.0701781\pi$$
$$314$$ −406.000 −0.0729679
$$315$$ 0 0
$$316$$ −3570.00 −0.635532
$$317$$ − 9484.00i − 1.68036i −0.542307 0.840181i $$-0.682449\pi$$
0.542307 0.840181i $$-0.317551\pi$$
$$318$$ − 574.000i − 0.101221i
$$319$$ 6880.00 1.20754
$$320$$ 0 0
$$321$$ −8883.00 −1.54455
$$322$$ 972.000i 0.168222i
$$323$$ 3185.00i 0.548663i
$$324$$ −5873.00 −1.00703
$$325$$ 0 0
$$326$$ −3803.00 −0.646100
$$327$$ 7490.00i 1.26666i
$$328$$ − 3045.00i − 0.512598i
$$329$$ −1176.00 −0.197067
$$330$$ 0 0
$$331$$ −183.000 −0.0303885 −0.0151942 0.999885i $$-0.504837\pi$$
−0.0151942 + 0.999885i $$0.504837\pi$$
$$332$$ − 5439.00i − 0.899108i
$$333$$ 6908.00i 1.13681i
$$334$$ −4116.00 −0.674304
$$335$$ 0 0
$$336$$ 1722.00 0.279592
$$337$$ 2861.00i 0.462459i 0.972899 + 0.231229i $$0.0742748\pi$$
−0.972899 + 0.231229i $$0.925725\pi$$
$$338$$ 1413.00i 0.227388i
$$339$$ 3521.00 0.564113
$$340$$ 0 0
$$341$$ −1806.00 −0.286805
$$342$$ − 770.000i − 0.121745i
$$343$$ 3900.00i 0.613936i
$$344$$ 1380.00 0.216292
$$345$$ 0 0
$$346$$ 1512.00 0.234930
$$347$$ − 629.000i − 0.0973098i −0.998816 0.0486549i $$-0.984507\pi$$
0.998816 0.0486549i $$-0.0154934\pi$$
$$348$$ 7840.00i 1.20767i
$$349$$ −5950.00 −0.912597 −0.456298 0.889827i $$-0.650825\pi$$
−0.456298 + 0.889827i $$0.650825\pi$$
$$350$$ 0 0
$$351$$ 980.000 0.149027
$$352$$ − 6923.00i − 1.04829i
$$353$$ 11718.0i 1.76682i 0.468604 + 0.883408i $$0.344757\pi$$
−0.468604 + 0.883408i $$0.655243\pi$$
$$354$$ 1960.00 0.294274
$$355$$ 0 0
$$356$$ 6615.00 0.984815
$$357$$ 3822.00i 0.566615i
$$358$$ − 2585.00i − 0.381624i
$$359$$ −8070.00 −1.18640 −0.593201 0.805054i $$-0.702136\pi$$
−0.593201 + 0.805054i $$0.702136\pi$$
$$360$$ 0 0
$$361$$ −5634.00 −0.821403
$$362$$ − 2758.00i − 0.400434i
$$363$$ − 3626.00i − 0.524286i
$$364$$ −1176.00 −0.169338
$$365$$ 0 0
$$366$$ −3626.00 −0.517853
$$367$$ 8316.00i 1.18281i 0.806374 + 0.591406i $$0.201427\pi$$
−0.806374 + 0.591406i $$0.798573\pi$$
$$368$$ − 6642.00i − 0.940865i
$$369$$ 4466.00 0.630056
$$370$$ 0 0
$$371$$ 492.000 0.0688500
$$372$$ − 2058.00i − 0.286834i
$$373$$ − 12062.0i − 1.67439i −0.546906 0.837194i $$-0.684195\pi$$
0.546906 0.837194i $$-0.315805\pi$$
$$374$$ 3913.00 0.541006
$$375$$ 0 0
$$376$$ −2940.00 −0.403242
$$377$$ − 4480.00i − 0.612021i
$$378$$ 210.000i 0.0285747i
$$379$$ −1735.00 −0.235148 −0.117574 0.993064i $$-0.537512\pi$$
−0.117574 + 0.993064i $$0.537512\pi$$
$$380$$ 0 0
$$381$$ −6118.00 −0.822663
$$382$$ − 2378.00i − 0.318505i
$$383$$ − 7602.00i − 1.01421i −0.861883 0.507107i $$-0.830715\pi$$
0.861883 0.507107i $$-0.169285\pi$$
$$384$$ 10185.0 1.35352
$$385$$ 0 0
$$386$$ 3067.00 0.404420
$$387$$ 2024.00i 0.265855i
$$388$$ 8722.00i 1.14122i
$$389$$ −3030.00 −0.394928 −0.197464 0.980310i $$-0.563271\pi$$
−0.197464 + 0.980310i $$0.563271\pi$$
$$390$$ 0 0
$$391$$ 14742.0 1.90674
$$392$$ 4605.00i 0.593336i
$$393$$ − 7644.00i − 0.981142i
$$394$$ −2346.00 −0.299974
$$395$$ 0 0
$$396$$ 6622.00 0.840323
$$397$$ − 1204.00i − 0.152209i −0.997100 0.0761046i $$-0.975752\pi$$
0.997100 0.0761046i $$-0.0242483\pi$$
$$398$$ − 4900.00i − 0.617123i
$$399$$ 1470.00 0.184441
$$400$$ 0 0
$$401$$ 1077.00 0.134122 0.0670609 0.997749i $$-0.478638\pi$$
0.0670609 + 0.997749i $$0.478638\pi$$
$$402$$ 987.000i 0.122455i
$$403$$ 1176.00i 0.145362i
$$404$$ 9114.00 1.12237
$$405$$ 0 0
$$406$$ 960.000 0.117350
$$407$$ 13502.0i 1.64440i
$$408$$ 9555.00i 1.15942i
$$409$$ 3955.00 0.478147 0.239074 0.971001i $$-0.423156\pi$$
0.239074 + 0.971001i $$0.423156\pi$$
$$410$$ 0 0
$$411$$ 2877.00 0.345285
$$412$$ − 3724.00i − 0.445311i
$$413$$ 1680.00i 0.200163i
$$414$$ −3564.00 −0.423094
$$415$$ 0 0
$$416$$ −4508.00 −0.531305
$$417$$ − 4165.00i − 0.489115i
$$418$$ − 1505.00i − 0.176105i
$$419$$ −6265.00 −0.730466 −0.365233 0.930916i $$-0.619011\pi$$
−0.365233 + 0.930916i $$0.619011\pi$$
$$420$$ 0 0
$$421$$ −3788.00 −0.438517 −0.219259 0.975667i $$-0.570364\pi$$
−0.219259 + 0.975667i $$0.570364\pi$$
$$422$$ 4307.00i 0.496828i
$$423$$ − 4312.00i − 0.495642i
$$424$$ 1230.00 0.140882
$$425$$ 0 0
$$426$$ 2884.00 0.328005
$$427$$ − 3108.00i − 0.352240i
$$428$$ − 8883.00i − 1.00321i
$$429$$ −8428.00 −0.948503
$$430$$ 0 0
$$431$$ −15258.0 −1.70523 −0.852613 0.522544i $$-0.824983\pi$$
−0.852613 + 0.522544i $$0.824983\pi$$
$$432$$ − 1435.00i − 0.159818i
$$433$$ 13573.0i 1.50641i 0.657784 + 0.753206i $$0.271494\pi$$
−0.657784 + 0.753206i $$0.728506\pi$$
$$434$$ −252.000 −0.0278719
$$435$$ 0 0
$$436$$ −7490.00 −0.822720
$$437$$ − 5670.00i − 0.620670i
$$438$$ 5341.00i 0.582655i
$$439$$ 8120.00 0.882794 0.441397 0.897312i $$-0.354483\pi$$
0.441397 + 0.897312i $$0.354483\pi$$
$$440$$ 0 0
$$441$$ −6754.00 −0.729295
$$442$$ − 2548.00i − 0.274199i
$$443$$ 6183.00i 0.663122i 0.943434 + 0.331561i $$0.107575\pi$$
−0.943434 + 0.331561i $$0.892425\pi$$
$$444$$ −15386.0 −1.64457
$$445$$ 0 0
$$446$$ 2212.00 0.234846
$$447$$ − 22400.0i − 2.37021i
$$448$$ 1002.00i 0.105670i
$$449$$ 1975.00 0.207586 0.103793 0.994599i $$-0.466902\pi$$
0.103793 + 0.994599i $$0.466902\pi$$
$$450$$ 0 0
$$451$$ 8729.00 0.911380
$$452$$ 3521.00i 0.366402i
$$453$$ − 1414.00i − 0.146657i
$$454$$ −476.000 −0.0492066
$$455$$ 0 0
$$456$$ 3675.00 0.377407
$$457$$ 11831.0i 1.21101i 0.795842 + 0.605504i $$0.207029\pi$$
−0.795842 + 0.605504i $$0.792971\pi$$
$$458$$ 2940.00i 0.299950i
$$459$$ 3185.00 0.323885
$$460$$ 0 0
$$461$$ 1932.00 0.195189 0.0975946 0.995226i $$-0.468885\pi$$
0.0975946 + 0.995226i $$0.468885\pi$$
$$462$$ − 1806.00i − 0.181867i
$$463$$ 9228.00i 0.926267i 0.886289 + 0.463133i $$0.153275\pi$$
−0.886289 + 0.463133i $$0.846725\pi$$
$$464$$ −6560.00 −0.656337
$$465$$ 0 0
$$466$$ 1002.00 0.0996068
$$467$$ 13916.0i 1.37892i 0.724324 + 0.689460i $$0.242152\pi$$
−0.724324 + 0.689460i $$0.757848\pi$$
$$468$$ − 4312.00i − 0.425902i
$$469$$ −846.000 −0.0832935
$$470$$ 0 0
$$471$$ 2842.00 0.278031
$$472$$ 4200.00i 0.409578i
$$473$$ 3956.00i 0.384560i
$$474$$ −3570.00 −0.345940
$$475$$ 0 0
$$476$$ −3822.00 −0.368027
$$477$$ 1804.00i 0.173165i
$$478$$ − 2480.00i − 0.237307i
$$479$$ −2310.00 −0.220348 −0.110174 0.993912i $$-0.535141\pi$$
−0.110174 + 0.993912i $$0.535141\pi$$
$$480$$ 0 0
$$481$$ 8792.00 0.833432
$$482$$ 1897.00i 0.179266i
$$483$$ − 6804.00i − 0.640979i
$$484$$ 3626.00 0.340533
$$485$$ 0 0
$$486$$ −4928.00 −0.459956
$$487$$ − 17114.0i − 1.59242i −0.605019 0.796211i $$-0.706835\pi$$
0.605019 0.796211i $$-0.293165\pi$$
$$488$$ − 7770.00i − 0.720761i
$$489$$ 26621.0 2.46185
$$490$$ 0 0
$$491$$ −17228.0 −1.58348 −0.791740 0.610858i $$-0.790825\pi$$
−0.791740 + 0.610858i $$0.790825\pi$$
$$492$$ 9947.00i 0.911474i
$$493$$ − 14560.0i − 1.33012i
$$494$$ −980.000 −0.0892556
$$495$$ 0 0
$$496$$ 1722.00 0.155887
$$497$$ 2472.00i 0.223107i
$$498$$ − 5439.00i − 0.489412i
$$499$$ 12500.0 1.12140 0.560698 0.828020i $$-0.310533\pi$$
0.560698 + 0.828020i $$0.310533\pi$$
$$500$$ 0 0
$$501$$ 28812.0 2.56931
$$502$$ − 2373.00i − 0.210980i
$$503$$ 868.000i 0.0769428i 0.999260 + 0.0384714i $$0.0122488\pi$$
−0.999260 + 0.0384714i $$0.987751\pi$$
$$504$$ 1980.00 0.174992
$$505$$ 0 0
$$506$$ −6966.00 −0.612009
$$507$$ − 9891.00i − 0.866420i
$$508$$ − 6118.00i − 0.534335i
$$509$$ −13370.0 −1.16427 −0.582136 0.813091i $$-0.697783\pi$$
−0.582136 + 0.813091i $$0.697783\pi$$
$$510$$ 0 0
$$511$$ −4578.00 −0.396319
$$512$$ 11521.0i 0.994455i
$$513$$ − 1225.00i − 0.105429i
$$514$$ 4494.00 0.385646
$$515$$ 0 0
$$516$$ −4508.00 −0.384600
$$517$$ − 8428.00i − 0.716950i
$$518$$ 1884.00i 0.159803i
$$519$$ −10584.0 −0.895156
$$520$$ 0 0
$$521$$ 21637.0 1.81945 0.909726 0.415210i $$-0.136292\pi$$
0.909726 + 0.415210i $$0.136292\pi$$
$$522$$ 3520.00i 0.295146i
$$523$$ − 287.000i − 0.0239955i −0.999928 0.0119977i $$-0.996181\pi$$
0.999928 0.0119977i $$-0.00381909\pi$$
$$524$$ 7644.00 0.637270
$$525$$ 0 0
$$526$$ 722.000 0.0598492
$$527$$ 3822.00i 0.315918i
$$528$$ 12341.0i 1.01718i
$$529$$ −14077.0 −1.15698
$$530$$ 0 0
$$531$$ −6160.00 −0.503430
$$532$$ 1470.00i 0.119798i
$$533$$ − 5684.00i − 0.461916i
$$534$$ 6615.00 0.536066
$$535$$ 0 0
$$536$$ −2115.00 −0.170437
$$537$$ 18095.0i 1.45411i
$$538$$ 6160.00i 0.493637i
$$539$$ −13201.0 −1.05493
$$540$$ 0 0
$$541$$ −5328.00 −0.423417 −0.211709 0.977333i $$-0.567903\pi$$
−0.211709 + 0.977333i $$0.567903\pi$$
$$542$$ − 7238.00i − 0.573614i
$$543$$ 19306.0i 1.52578i
$$544$$ −14651.0 −1.15470
$$545$$ 0 0
$$546$$ −1176.00 −0.0921761
$$547$$ 71.0000i 0.00554980i 0.999996 + 0.00277490i $$0.000883279\pi$$
−0.999996 + 0.00277490i $$0.999117\pi$$
$$548$$ 2877.00i 0.224269i
$$549$$ 11396.0 0.885919
$$550$$ 0 0
$$551$$ −5600.00 −0.432973
$$552$$ − 17010.0i − 1.31158i
$$553$$ − 3060.00i − 0.235306i
$$554$$ −1776.00 −0.136200
$$555$$ 0 0
$$556$$ 4165.00 0.317689
$$557$$ − 18444.0i − 1.40305i −0.712646 0.701524i $$-0.752503\pi$$
0.712646 0.701524i $$-0.247497\pi$$
$$558$$ − 924.000i − 0.0701004i
$$559$$ 2576.00 0.194907
$$560$$ 0 0
$$561$$ −27391.0 −2.06141
$$562$$ 4542.00i 0.340912i
$$563$$ − 672.000i − 0.0503045i −0.999684 0.0251522i $$-0.991993\pi$$
0.999684 0.0251522i $$-0.00800705\pi$$
$$564$$ 9604.00 0.717024
$$565$$ 0 0
$$566$$ 7077.00 0.525563
$$567$$ − 5034.00i − 0.372854i
$$568$$ 6180.00i 0.456526i
$$569$$ 10935.0 0.805657 0.402829 0.915275i $$-0.368027\pi$$
0.402829 + 0.915275i $$0.368027\pi$$
$$570$$ 0 0
$$571$$ −13588.0 −0.995867 −0.497934 0.867215i $$-0.665908\pi$$
−0.497934 + 0.867215i $$0.665908\pi$$
$$572$$ − 8428.00i − 0.616071i
$$573$$ 16646.0i 1.21361i
$$574$$ 1218.00 0.0885685
$$575$$ 0 0
$$576$$ −3674.00 −0.265770
$$577$$ 8701.00i 0.627777i 0.949460 + 0.313889i $$0.101632\pi$$
−0.949460 + 0.313889i $$0.898368\pi$$
$$578$$ − 3368.00i − 0.242371i
$$579$$ −21469.0 −1.54097
$$580$$ 0 0
$$581$$ 4662.00 0.332896
$$582$$ 8722.00i 0.621200i
$$583$$ 3526.00i 0.250484i
$$584$$ −11445.0 −0.810955
$$585$$ 0 0
$$586$$ −4158.00 −0.293115
$$587$$ 11361.0i 0.798839i 0.916768 + 0.399420i $$0.130788\pi$$
−0.916768 + 0.399420i $$0.869212\pi$$
$$588$$ − 15043.0i − 1.05504i
$$589$$ 1470.00 0.102836
$$590$$ 0 0
$$591$$ 16422.0 1.14300
$$592$$ − 12874.0i − 0.893781i
$$593$$ − 11417.0i − 0.790624i −0.918547 0.395312i $$-0.870636\pi$$
0.918547 0.395312i $$-0.129364\pi$$
$$594$$ −1505.00 −0.103958
$$595$$ 0 0
$$596$$ 22400.0 1.53950
$$597$$ 34300.0i 2.35143i
$$598$$ 4536.00i 0.310185i
$$599$$ 21050.0 1.43586 0.717930 0.696116i $$-0.245090\pi$$
0.717930 + 0.696116i $$0.245090\pi$$
$$600$$ 0 0
$$601$$ 7427.00 0.504083 0.252041 0.967716i $$-0.418898\pi$$
0.252041 + 0.967716i $$0.418898\pi$$
$$602$$ 552.000i 0.0373718i
$$603$$ − 3102.00i − 0.209491i
$$604$$ 1414.00 0.0952564
$$605$$ 0 0
$$606$$ 9114.00 0.610942
$$607$$ − 4144.00i − 0.277100i −0.990355 0.138550i $$-0.955756\pi$$
0.990355 0.138550i $$-0.0442442\pi$$
$$608$$ 5635.00i 0.375871i
$$609$$ −6720.00 −0.447140
$$610$$ 0 0
$$611$$ −5488.00 −0.363373
$$612$$ − 14014.0i − 0.925625i
$$613$$ − 30122.0i − 1.98469i −0.123489 0.992346i $$-0.539408\pi$$
0.123489 0.992346i $$-0.460592\pi$$
$$614$$ 2569.00 0.168854
$$615$$ 0 0
$$616$$ 3870.00 0.253128
$$617$$ − 11934.0i − 0.778679i −0.921094 0.389339i $$-0.872703\pi$$
0.921094 0.389339i $$-0.127297\pi$$
$$618$$ − 3724.00i − 0.242397i
$$619$$ −8540.00 −0.554526 −0.277263 0.960794i $$-0.589427\pi$$
−0.277263 + 0.960794i $$0.589427\pi$$
$$620$$ 0 0
$$621$$ −5670.00 −0.366392
$$622$$ 2982.00i 0.192230i
$$623$$ 5670.00i 0.364629i
$$624$$ 8036.00 0.515541
$$625$$ 0 0
$$626$$ 2422.00 0.154637
$$627$$ 10535.0i 0.671017i
$$628$$ 2842.00i 0.180586i
$$629$$ 28574.0 1.81132
$$630$$ 0 0
$$631$$ −3158.00 −0.199236 −0.0996181 0.995026i $$-0.531762\pi$$
−0.0996181 + 0.995026i $$0.531762\pi$$
$$632$$ − 7650.00i − 0.481488i
$$633$$ − 30149.0i − 1.89307i
$$634$$ 9484.00 0.594097
$$635$$ 0 0
$$636$$ −4018.00 −0.250510
$$637$$ 8596.00i 0.534672i
$$638$$ 6880.00i 0.426931i
$$639$$ −9064.00 −0.561137
$$640$$ 0 0
$$641$$ −4278.00 −0.263605 −0.131803 0.991276i $$-0.542076\pi$$
−0.131803 + 0.991276i $$0.542076\pi$$
$$642$$ − 8883.00i − 0.546081i
$$643$$ 11508.0i 0.705803i 0.935661 + 0.352901i $$0.114805\pi$$
−0.935661 + 0.352901i $$0.885195\pi$$
$$644$$ 6804.00 0.416328
$$645$$ 0 0
$$646$$ −3185.00 −0.193982
$$647$$ − 8204.00i − 0.498505i −0.968439 0.249252i $$-0.919815\pi$$
0.968439 0.249252i $$-0.0801849\pi$$
$$648$$ − 12585.0i − 0.762941i
$$649$$ −12040.0 −0.728215
$$650$$ 0 0
$$651$$ 1764.00 0.106201
$$652$$ 26621.0i 1.59902i
$$653$$ 5518.00i 0.330683i 0.986236 + 0.165342i $$0.0528726\pi$$
−0.986236 + 0.165342i $$0.947127\pi$$
$$654$$ −7490.00 −0.447832
$$655$$ 0 0
$$656$$ −8323.00 −0.495364
$$657$$ − 16786.0i − 0.996780i
$$658$$ − 1176.00i − 0.0696736i
$$659$$ −13295.0 −0.785887 −0.392944 0.919563i $$-0.628543\pi$$
−0.392944 + 0.919563i $$0.628543\pi$$
$$660$$ 0 0
$$661$$ −9968.00 −0.586551 −0.293276 0.956028i $$-0.594745\pi$$
−0.293276 + 0.956028i $$0.594745\pi$$
$$662$$ − 183.000i − 0.0107440i
$$663$$ 17836.0i 1.04479i
$$664$$ 11655.0 0.681177
$$665$$ 0 0
$$666$$ −6908.00 −0.401921
$$667$$ 25920.0i 1.50469i
$$668$$ 28812.0i 1.66882i
$$669$$ −15484.0 −0.894837
$$670$$ 0 0
$$671$$ 22274.0 1.28149
$$672$$ 6762.00i 0.388169i
$$673$$ 15738.0i 0.901419i 0.892671 + 0.450710i $$0.148829\pi$$
−0.892671 + 0.450710i $$0.851171\pi$$
$$674$$ −2861.00 −0.163504
$$675$$ 0 0
$$676$$ 9891.00 0.562756
$$677$$ − 19824.0i − 1.12540i −0.826660 0.562702i $$-0.809762\pi$$
0.826660 0.562702i $$-0.190238\pi$$
$$678$$ 3521.00i 0.199444i
$$679$$ −7476.00 −0.422537
$$680$$ 0 0
$$681$$ 3332.00 0.187493
$$682$$ − 1806.00i − 0.101401i
$$683$$ 11073.0i 0.620346i 0.950680 + 0.310173i $$0.100387\pi$$
−0.950680 + 0.310173i $$0.899613\pi$$
$$684$$ −5390.00 −0.301304
$$685$$ 0 0
$$686$$ −3900.00 −0.217059
$$687$$ − 20580.0i − 1.14291i
$$688$$ − 3772.00i − 0.209021i
$$689$$ 2296.00 0.126953
$$690$$ 0 0
$$691$$ −6503.00 −0.358011 −0.179006 0.983848i $$-0.557288\pi$$
−0.179006 + 0.983848i $$0.557288\pi$$
$$692$$ − 10584.0i − 0.581421i
$$693$$ 5676.00i 0.311130i
$$694$$ 629.000 0.0344042
$$695$$ 0 0
$$696$$ −16800.0 −0.914946
$$697$$ − 18473.0i − 1.00389i
$$698$$ − 5950.00i − 0.322652i
$$699$$ −7014.00 −0.379533
$$700$$ 0 0
$$701$$ −10148.0 −0.546768 −0.273384 0.961905i $$-0.588143\pi$$
−0.273384 + 0.961905i $$0.588143\pi$$
$$702$$ 980.000i 0.0526891i
$$703$$ − 10990.0i − 0.589610i
$$704$$ −7181.00 −0.384438
$$705$$ 0 0
$$706$$ −11718.0 −0.624664
$$707$$ 7812.00i 0.415559i
$$708$$ − 13720.0i − 0.728290i
$$709$$ 9980.00 0.528641 0.264321 0.964435i $$-0.414852\pi$$
0.264321 + 0.964435i $$0.414852\pi$$
$$710$$ 0 0
$$711$$ 11220.0 0.591818
$$712$$ 14175.0i 0.746110i
$$713$$ − 6804.00i − 0.357380i
$$714$$ −3822.00 −0.200329
$$715$$ 0 0
$$716$$ −18095.0 −0.944472
$$717$$ 17360.0i 0.904214i
$$718$$ − 8070.00i − 0.419456i
$$719$$ 27510.0 1.42691 0.713456 0.700700i $$-0.247129\pi$$
0.713456 + 0.700700i $$0.247129\pi$$
$$720$$ 0 0
$$721$$ 3192.00 0.164877
$$722$$ − 5634.00i − 0.290410i
$$723$$ − 13279.0i − 0.683059i
$$724$$ −19306.0 −0.991025
$$725$$ 0 0
$$726$$ 3626.00 0.185363
$$727$$ − 17024.0i − 0.868480i −0.900797 0.434240i $$-0.857017\pi$$
0.900797 0.434240i $$-0.142983\pi$$
$$728$$ − 2520.00i − 0.128293i
$$729$$ 11843.0 0.601687
$$730$$ 0 0
$$731$$ 8372.00 0.423597
$$732$$ 25382.0i 1.28162i
$$733$$ 34748.0i 1.75095i 0.483263 + 0.875475i $$0.339451\pi$$
−0.483263 + 0.875475i $$0.660549\pi$$
$$734$$ −8316.00 −0.418187
$$735$$ 0 0
$$736$$ 26082.0 1.30624
$$737$$ − 6063.00i − 0.303030i
$$738$$ 4466.00i 0.222758i
$$739$$ 12020.0 0.598326 0.299163 0.954202i $$-0.403293\pi$$
0.299163 + 0.954202i $$0.403293\pi$$
$$740$$ 0 0
$$741$$ 6860.00 0.340092
$$742$$ 492.000i 0.0243422i
$$743$$ − 28642.0i − 1.41423i −0.707098 0.707115i $$-0.749996\pi$$
0.707098 0.707115i $$-0.250004\pi$$
$$744$$ 4410.00 0.217310
$$745$$ 0 0
$$746$$ 12062.0 0.591986
$$747$$ 17094.0i 0.837265i
$$748$$ − 27391.0i − 1.33892i
$$749$$ 7614.00 0.371441
$$750$$ 0 0
$$751$$ 8752.00 0.425253 0.212627 0.977134i $$-0.431798\pi$$
0.212627 + 0.977134i $$0.431798\pi$$
$$752$$ 8036.00i 0.389685i
$$753$$ 16611.0i 0.803902i
$$754$$ 4480.00 0.216382
$$755$$ 0 0
$$756$$ 1470.00 0.0707188
$$757$$ 10256.0i 0.492418i 0.969217 + 0.246209i $$0.0791850\pi$$
−0.969217 + 0.246209i $$0.920815\pi$$
$$758$$ − 1735.00i − 0.0831373i
$$759$$ 48762.0 2.33195
$$760$$ 0 0
$$761$$ 33957.0 1.61753 0.808765 0.588132i $$-0.200136\pi$$
0.808765 + 0.588132i $$0.200136\pi$$
$$762$$ − 6118.00i − 0.290855i
$$763$$ − 6420.00i − 0.304613i
$$764$$ −16646.0 −0.788261
$$765$$ 0 0
$$766$$ 7602.00 0.358579
$$767$$ 7840.00i 0.369082i
$$768$$ 833.000i 0.0391384i
$$769$$ −27965.0 −1.31137 −0.655685 0.755034i $$-0.727620\pi$$
−0.655685 + 0.755034i $$0.727620\pi$$
$$770$$ 0 0
$$771$$ −31458.0 −1.46943
$$772$$ − 21469.0i − 1.00089i
$$773$$ − 9912.00i − 0.461203i −0.973048 0.230601i $$-0.925931\pi$$
0.973048 0.230601i $$-0.0740694\pi$$
$$774$$ −2024.00 −0.0939938
$$775$$ 0 0
$$776$$ −18690.0 −0.864603
$$777$$ − 13188.0i − 0.608902i
$$778$$ − 3030.00i − 0.139628i
$$779$$ −7105.00 −0.326782
$$780$$ 0 0
$$781$$ −17716.0 −0.811688
$$782$$ 14742.0i 0.674134i
$$783$$ 5600.00i 0.255591i
$$784$$ 12587.0 0.573387
$$785$$ 0 0
$$786$$ 7644.00 0.346886
$$787$$ − 25564.0i − 1.15789i −0.815367 0.578944i $$-0.803465\pi$$
0.815367 0.578944i $$-0.196535\pi$$
$$788$$ 16422.0i 0.742398i
$$789$$ −5054.00 −0.228045
$$790$$ 0 0
$$791$$ −3018.00 −0.135661
$$792$$ 14190.0i 0.636641i
$$793$$ − 14504.0i − 0.649498i
$$794$$ 1204.00 0.0538141
$$795$$ 0 0
$$796$$ −34300.0 −1.52730
$$797$$ 12446.0i 0.553149i 0.960992 + 0.276575i $$0.0891993\pi$$
−0.960992 + 0.276575i $$0.910801\pi$$
$$798$$ 1470.00i 0.0652098i
$$799$$ −17836.0 −0.789728
$$800$$ 0 0
$$801$$ −20790.0 −0.917077
$$802$$ 1077.00i 0.0474192i
$$803$$ − 32809.0i − 1.44185i
$$804$$ 6909.00 0.303062
$$805$$ 0 0
$$806$$ −1176.00 −0.0513931
$$807$$ − 43120.0i − 1.88091i
$$808$$ 19530.0i 0.850325i
$$809$$ −33970.0 −1.47629 −0.738147 0.674640i $$-0.764299\pi$$
−0.738147 + 0.674640i $$0.764299\pi$$
$$810$$ 0 0
$$811$$ 18732.0 0.811060 0.405530 0.914082i $$-0.367087\pi$$
0.405530 + 0.914082i $$0.367087\pi$$
$$812$$ − 6720.00i − 0.290426i
$$813$$ 50666.0i 2.18565i
$$814$$ −13502.0 −0.581382
$$815$$ 0 0
$$816$$ 26117.0 1.12044
$$817$$ − 3220.00i − 0.137887i
$$818$$ 3955.00i 0.169051i
$$819$$ 3696.00 0.157691
$$820$$ 0 0
$$821$$ 6162.00 0.261943 0.130972 0.991386i $$-0.458190\pi$$
0.130972 + 0.991386i $$0.458190\pi$$
$$822$$ 2877.00i 0.122077i
$$823$$ 25388.0i 1.07530i 0.843169 + 0.537649i $$0.180687\pi$$
−0.843169 + 0.537649i $$0.819313\pi$$
$$824$$ 7980.00 0.337374
$$825$$ 0 0
$$826$$ −1680.00 −0.0707684
$$827$$ 25201.0i 1.05964i 0.848109 + 0.529821i $$0.177741\pi$$
−0.848109 + 0.529821i $$0.822259\pi$$
$$828$$ 24948.0i 1.04710i
$$829$$ 19740.0 0.827019 0.413509 0.910500i $$-0.364303\pi$$
0.413509 + 0.910500i $$0.364303\pi$$
$$830$$ 0 0
$$831$$ 12432.0 0.518967
$$832$$ 4676.00i 0.194845i
$$833$$ 27937.0i 1.16202i
$$834$$ 4165.00 0.172928
$$835$$ 0 0
$$836$$ −10535.0 −0.435838
$$837$$ − 1470.00i − 0.0607057i
$$838$$ − 6265.00i − 0.258259i
$$839$$ −29680.0 −1.22130 −0.610648 0.791902i $$-0.709091\pi$$
−0.610648 + 0.791902i $$0.709091\pi$$
$$840$$ 0 0
$$841$$ 1211.00 0.0496535
$$842$$ − 3788.00i − 0.155039i
$$843$$ − 31794.0i − 1.29898i
$$844$$ 30149.0 1.22959
$$845$$ 0 0
$$846$$ 4312.00 0.175236
$$847$$ 3108.00i 0.126083i
$$848$$ − 3362.00i − 0.136146i
$$849$$ −49539.0 −2.00256
$$850$$ 0 0
$$851$$ −50868.0 −2.04904
$$852$$ − 20188.0i − 0.811772i
$$853$$ 1218.00i 0.0488904i 0.999701 + 0.0244452i $$0.00778193\pi$$
−0.999701 + 0.0244452i $$0.992218\pi$$
$$854$$ 3108.00 0.124536
$$855$$ 0 0
$$856$$ 19035.0 0.760050
$$857$$ 38731.0i 1.54379i 0.635752 + 0.771894i $$0.280690\pi$$
−0.635752 + 0.771894i $$0.719310\pi$$
$$858$$ − 8428.00i − 0.335346i
$$859$$ 23555.0 0.935607 0.467803 0.883833i $$-0.345046\pi$$
0.467803 + 0.883833i $$0.345046\pi$$
$$860$$ 0 0
$$861$$ −8526.00 −0.337474
$$862$$ − 15258.0i − 0.602888i
$$863$$ − 24872.0i − 0.981058i −0.871425 0.490529i $$-0.836804\pi$$
0.871425 0.490529i $$-0.163196\pi$$
$$864$$ 5635.00 0.221883
$$865$$ 0 0
$$866$$ −13573.0 −0.532597
$$867$$ 23576.0i 0.923510i
$$868$$ 1764.00i 0.0689793i
$$869$$ 21930.0 0.856069
$$870$$ 0 0
$$871$$ −3948.00 −0.153585
$$872$$ − 16050.0i − 0.623305i
$$873$$ − 27412.0i − 1.06272i
$$874$$ 5670.00 0.219440
$$875$$ 0 0
$$876$$ 37387.0 1.44200
$$877$$ − 17124.0i − 0.659335i −0.944097 0.329667i $$-0.893063\pi$$
0.944097 0.329667i $$-0.106937\pi$$
$$878$$ 8120.00i 0.312115i
$$879$$ 29106.0 1.11686
$$880$$ 0 0
$$881$$ −658.000 −0.0251630 −0.0125815 0.999921i $$-0.504005\pi$$
−0.0125815 + 0.999921i $$0.504005\pi$$
$$882$$ − 6754.00i − 0.257845i
$$883$$ − 33727.0i − 1.28540i −0.766120 0.642698i $$-0.777815\pi$$
0.766120 0.642698i $$-0.222185\pi$$
$$884$$ −17836.0 −0.678608
$$885$$ 0 0
$$886$$ −6183.00 −0.234449
$$887$$ 36036.0i 1.36412i 0.731298 + 0.682058i $$0.238915\pi$$
−0.731298 + 0.682058i $$0.761085\pi$$
$$888$$ − 32970.0i − 1.24595i
$$889$$ 5244.00 0.197838
$$890$$ 0 0
$$891$$ 36077.0 1.35648
$$892$$ − 15484.0i − 0.581214i
$$893$$ 6860.00i 0.257067i
$$894$$ 22400.0 0.837996
$$895$$ 0 0
$$896$$ −8730.00 −0.325501
$$897$$ − 31752.0i − 1.18190i
$$898$$ 1975.00i 0.0733927i
$$899$$ −6720.00 −0.249304
$$900$$ 0 0
$$901$$ 7462.00 0.275910
$$902$$ 8729.00i 0.322222i
$$903$$ − 3864.00i − 0.142399i
$$904$$ −7545.00 −0.277592
$$905$$ 0 0
$$906$$ 1414.00 0.0518510
$$907$$ 39156.0i 1.43347i 0.697348 + 0.716733i $$0.254363\pi$$
−0.697348 + 0.716733i $$0.745637\pi$$
$$908$$ 3332.00i 0.121780i
$$909$$ −28644.0 −1.04517
$$910$$ 0 0
$$911$$ 43532.0 1.58318 0.791591 0.611051i $$-0.209253\pi$$
0.791591 + 0.611051i $$0.209253\pi$$
$$912$$ − 10045.0i − 0.364718i
$$913$$ 33411.0i 1.21111i
$$914$$ −11831.0 −0.428156
$$915$$ 0 0
$$916$$ 20580.0 0.742339
$$917$$ 6552.00i 0.235950i
$$918$$ 3185.00i 0.114511i
$$919$$ 28610.0 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$920$$ 0 0
$$921$$ −17983.0 −0.643388
$$922$$ 1932.00i 0.0690098i
$$923$$ 11536.0i 0.411389i
$$924$$ −12642.0 −0.450099
$$925$$ 0 0
$$926$$ −9228.00 −0.327485
$$927$$ 11704.0i 0.414682i
$$928$$ − 25760.0i − 0.911221i
$$929$$ 24290.0 0.857835 0.428918 0.903344i $$-0.358895\pi$$
0.428918 + 0.903344i $$0.358895\pi$$
$$930$$ 0 0
$$931$$ 10745.0 0.378253
$$932$$ − 7014.00i − 0.246514i
$$933$$ − 20874.0i − 0.732459i
$$934$$ −13916.0 −0.487522
$$935$$ 0 0
$$936$$ 9240.00 0.322670
$$937$$ 34461.0i 1.20149i 0.799442 + 0.600743i $$0.205128\pi$$
−0.799442 + 0.600743i $$0.794872\pi$$
$$938$$ − 846.000i − 0.0294487i
$$939$$ −16954.0 −0.589215
$$940$$ 0 0
$$941$$ −40628.0 −1.40748 −0.703738 0.710460i $$-0.748487\pi$$
−0.703738 + 0.710460i $$0.748487\pi$$
$$942$$ 2842.00i 0.0982987i
$$943$$ 32886.0i 1.13565i
$$944$$ 11480.0 0.395807
$$945$$ 0 0
$$946$$ −3956.00 −0.135963
$$947$$ − 20904.0i − 0.717306i −0.933471 0.358653i $$-0.883236\pi$$
0.933471 0.358653i $$-0.116764\pi$$
$$948$$ 24990.0i 0.856158i
$$949$$ −21364.0 −0.730774
$$950$$ 0 0
$$951$$ −66388.0 −2.26370
$$952$$ − 8190.00i − 0.278823i
$$953$$ − 1807.00i − 0.0614213i −0.999528 0.0307106i $$-0.990223\pi$$
0.999528 0.0307106i $$-0.00977704\pi$$
$$954$$ −1804.00 −0.0612229
$$955$$ 0 0
$$956$$ −17360.0 −0.587304
$$957$$ − 48160.0i − 1.62674i
$$958$$ − 2310.00i − 0.0779047i
$$959$$ −2466.00 −0.0830358
$$960$$ 0 0
$$961$$ −28027.0 −0.940787
$$962$$ 8792.00i 0.294663i
$$963$$ 27918.0i 0.934211i
$$964$$ 13279.0 0.443660
$$965$$ 0 0
$$966$$ 6804.00 0.226620
$$967$$ − 57584.0i − 1.91497i −0.288482 0.957485i $$-0.593151\pi$$
0.288482 0.957485i $$-0.406849\pi$$
$$968$$ 7770.00i 0.257993i
$$969$$ 22295.0 0.739132
$$970$$ 0 0
$$971$$ 27237.0 0.900182 0.450091 0.892983i $$-0.351392\pi$$
0.450091 + 0.892983i $$0.351392\pi$$
$$972$$ 34496.0i 1.13833i
$$973$$ 3570.00i 0.117625i
$$974$$ 17114.0 0.563006
$$975$$ 0 0
$$976$$ −21238.0 −0.696528
$$977$$ − 13649.0i − 0.446950i −0.974710 0.223475i $$-0.928260\pi$$
0.974710 0.223475i $$-0.0717401\pi$$
$$978$$ 26621.0i 0.870394i
$$979$$ −40635.0 −1.32656
$$980$$ 0 0
$$981$$ 23540.0 0.766131
$$982$$ − 17228.0i − 0.559845i
$$983$$ − 16002.0i − 0.519211i −0.965715 0.259606i $$-0.916407\pi$$
0.965715 0.259606i $$-0.0835925\pi$$
$$984$$ −21315.0 −0.690546
$$985$$ 0 0
$$986$$ 14560.0 0.470269
$$987$$ 8232.00i 0.265479i
$$988$$ 6860.00i 0.220896i
$$989$$ −14904.0 −0.479191
$$990$$ 0 0
$$991$$ 37022.0 1.18672 0.593362 0.804936i $$-0.297800\pi$$
0.593362 + 0.804936i $$0.297800\pi$$
$$992$$ 6762.00i 0.216425i
$$993$$ 1281.00i 0.0409379i
$$994$$ −2472.00 −0.0788804
$$995$$ 0 0
$$996$$ −38073.0 −1.21123
$$997$$ 18396.0i 0.584360i 0.956363 + 0.292180i $$0.0943807\pi$$
−0.956363 + 0.292180i $$0.905619\pi$$
$$998$$ 12500.0i 0.396474i
$$999$$ −10990.0 −0.348056
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 25.4.b.b.24.2 2
3.2 odd 2 225.4.b.f.199.1 2
4.3 odd 2 400.4.c.e.49.2 2
5.2 odd 4 25.4.a.a.1.1 1
5.3 odd 4 25.4.a.b.1.1 yes 1
5.4 even 2 inner 25.4.b.b.24.1 2
15.2 even 4 225.4.a.e.1.1 1
15.8 even 4 225.4.a.c.1.1 1
15.14 odd 2 225.4.b.f.199.2 2
20.3 even 4 400.4.a.c.1.1 1
20.7 even 4 400.4.a.s.1.1 1
20.19 odd 2 400.4.c.e.49.1 2
35.13 even 4 1225.4.a.i.1.1 1
35.27 even 4 1225.4.a.h.1.1 1
40.3 even 4 1600.4.a.bs.1.1 1
40.13 odd 4 1600.4.a.i.1.1 1
40.27 even 4 1600.4.a.h.1.1 1
40.37 odd 4 1600.4.a.bt.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
25.4.a.a.1.1 1 5.2 odd 4
25.4.a.b.1.1 yes 1 5.3 odd 4
25.4.b.b.24.1 2 5.4 even 2 inner
25.4.b.b.24.2 2 1.1 even 1 trivial
225.4.a.c.1.1 1 15.8 even 4
225.4.a.e.1.1 1 15.2 even 4
225.4.b.f.199.1 2 3.2 odd 2
225.4.b.f.199.2 2 15.14 odd 2
400.4.a.c.1.1 1 20.3 even 4
400.4.a.s.1.1 1 20.7 even 4
400.4.c.e.49.1 2 20.19 odd 2
400.4.c.e.49.2 2 4.3 odd 2
1225.4.a.h.1.1 1 35.27 even 4
1225.4.a.i.1.1 1 35.13 even 4
1600.4.a.h.1.1 1 40.27 even 4
1600.4.a.i.1.1 1 40.13 odd 4
1600.4.a.bs.1.1 1 40.3 even 4
1600.4.a.bt.1.1 1 40.37 odd 4