LMFDB - Embedded newform 1089.3.c.m.604.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.c (of order $2$, degree $1$, 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:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 77x^12 + 88x^11 - 577x^10 + 578x^9 + 1520x^8 + 1868x^7 - 1619x^6 - 16804x^5 + 32427x^4 + 43316x^3 - 71672x^2 + 83521
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.4
Root			$-1.95510 - 0.109518i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.m.604.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47556i q^{2} -2.12839 q^{4} +2.55350 q^{5} -0.170400i q^{7} -4.63328i q^{8} +O(q^{10})$	$q-2.47556i q^{2} -2.12839 q^{4} +2.55350 q^{5} -0.170400i q^{7} -4.63328i q^{8} -6.32135i q^{10} +19.2744i q^{13} -0.421835 q^{14} -19.9835 q^{16} -20.2947i q^{17} -14.5860i q^{19} -5.43486 q^{20} -7.74583 q^{23} -18.4796 q^{25} +47.7149 q^{26} +0.362678i q^{28} -38.2630i q^{29} -42.3732 q^{31} +30.9373i q^{32} -50.2408 q^{34} -0.435117i q^{35} +51.6438 q^{37} -36.1085 q^{38} -11.8311i q^{40} -46.2448i q^{41} -59.7836i q^{43} +19.1753i q^{46} -34.1325 q^{47} +48.9710 q^{49} +45.7474i q^{50} -41.0234i q^{52} +15.2253 q^{53} -0.789510 q^{56} -94.7223 q^{58} +26.3418 q^{59} -63.3714i q^{61} +104.897i q^{62} -3.34707 q^{64} +49.2172i q^{65} -2.91469 q^{67} +43.1951i q^{68} -1.07716 q^{70} -96.7843 q^{71} -17.3281i q^{73} -127.847i q^{74} +31.0447i q^{76} -52.7777i q^{79} -51.0280 q^{80} -114.482 q^{82} +23.4757i q^{83} -51.8227i q^{85} -147.998 q^{86} -97.1861 q^{89} +3.28435 q^{91} +16.4862 q^{92} +84.4969i q^{94} -37.2454i q^{95} -50.6148 q^{97} -121.230i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) $ 16 * q - 20 * q^4 + 4 * q^5	$ 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) $ 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$-1$	$1$

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 2.47556i	− 1.23778i	−0.785478	−	0.618890i	$-0.787583\pi$
$2$	− 2.47556i	− 1.23778i	0.785478	−	0.618890i	$-0.212417\pi$
$3$	0	0
$3$	0	0
$4$	−2.12839	−0.532098
$5$	2.55350	0.510701	0.255350	−	0.966849i	$-0.417809\pi$
$5$	2.55350	0.510701	0.255350	+	0.966849i	$0.417809\pi$
$6$	0	0
$7$	− 0.170400i	− 0.0243428i	−0.999926	−	0.0121714i	$-0.996126\pi$
$7$	− 0.170400i	− 0.0243428i	0.999926	−	0.0121714i	$-0.00387438\pi$
$8$	− 4.63328i	− 0.579160i
$9$	0	0
$10$	− 6.32135i	− 0.632135i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	19.2744i	1.48265i	0.671149	+	0.741323i	$0.265801\pi$
$13$	19.2744i	1.48265i	−0.671149	+	0.741323i	$0.734199\pi$
$14$	−0.421835	−0.0301311
$15$	0	0
$16$	−19.9835	−1.24897
$17$	− 20.2947i	− 1.19381i	−0.802313	−	0.596904i	$-0.796397\pi$
$17$	− 20.2947i	− 1.19381i	0.802313	−	0.596904i	$-0.203603\pi$
$18$	0	0
$19$	− 14.5860i	− 0.767683i	−0.923399	−	0.383842i	$-0.874601\pi$
$19$	− 14.5860i	− 0.767683i	0.923399	−	0.383842i	$-0.125399\pi$
$20$	−5.43486	−0.271743
$21$	0	0
$22$	0	0
$23$	−7.74583	−0.336775	−0.168388	−	0.985721i	$-0.553856\pi$
$23$	−7.74583	−0.336775	−0.168388	+	0.985721i	$0.553856\pi$
$24$	0	0
$25$	−18.4796	−0.739185
$26$	47.7149	1.83519
$27$	0	0
$28$	0.362678i	0.0129528i
$29$	− 38.2630i	− 1.31941i	−0.751523	−	0.659707i	$-0.770680\pi$
$29$	− 38.2630i	− 1.31941i	0.751523	−	0.659707i	$-0.229320\pi$
$30$	0	0
$31$	−42.3732	−1.36688	−0.683439	−	0.730008i	$-0.739517\pi$
$31$	−42.3732	−1.36688	−0.683439	+	0.730008i	$0.739517\pi$
$32$	30.9373i	0.966789i
$33$	0	0
$34$	−50.2408	−1.47767
$35$	− 0.435117i	− 0.0124319i
$36$	0	0
$37$	51.6438	1.39578	0.697889	−	0.716206i	$-0.254123\pi$
$37$	51.6438	1.39578	0.697889	+	0.716206i	$0.254123\pi$
$38$	−36.1085	−0.950223
$39$	0	0
$40$	− 11.8311i	− 0.295777i
$41$	− 46.2448i	− 1.12792i	−0.825802	−	0.563961i	$-0.809277\pi$
$41$	− 46.2448i	− 1.12792i	0.825802	−	0.563961i	$-0.190723\pi$
$42$	0	0
$43$	− 59.7836i	− 1.39032i	−0.718857	−	0.695158i	$-0.755334\pi$
$43$	− 59.7836i	− 1.39032i	0.718857	−	0.695158i	$-0.244666\pi$
$44$	0	0
$45$	0	0
$46$	19.1753i	0.416853i
$47$	−34.1325	−0.726223	−0.363111	−	0.931746i	$-0.618286\pi$
$47$	−34.1325	−0.726223	−0.363111	+	0.931746i	$0.618286\pi$
$48$	0	0
$49$	48.9710	0.999407
$50$	45.7474i	0.914947i
$51$	0	0
$52$	− 41.0234i	− 0.788912i
$53$	15.2253	0.287270	0.143635	−	0.989631i	$-0.454121\pi$
$53$	15.2253	0.287270	0.143635	+	0.989631i	$0.454121\pi$
$54$	0	0
$55$	0	0
$56$	−0.789510	−0.0140984
$57$	0	0
$58$	−94.7223	−1.63314
$59$	26.3418	0.446472	0.223236	−	0.974764i	$-0.428338\pi$
$59$	26.3418	0.446472	0.223236	+	0.974764i	$0.428338\pi$
$60$	0	0
$61$	− 63.3714i	− 1.03887i	−0.854508	−	0.519437i	$-0.826142\pi$
$61$	− 63.3714i	− 1.03887i	0.854508	−	0.519437i	$-0.173858\pi$
$62$	104.897i	1.69189i
$63$	0	0
$64$	−3.34707	−0.0522979
$65$	49.2172i	0.757188i
$66$	0	0
$67$	−2.91469	−0.0435028	−0.0217514	−	0.999763i	$-0.506924\pi$
$67$	−2.91469	−0.0435028	−0.0217514	+	0.999763i	$0.506924\pi$
$68$	43.1951i	0.635223i
$69$	0	0
$70$	−1.07716	−0.0153880
$71$	−96.7843	−1.36316	−0.681579	−	0.731744i	$-0.738707\pi$
$71$	−96.7843	−1.36316	−0.681579	+	0.731744i	$0.738707\pi$
$72$	0	0
$73$	− 17.3281i	− 0.237371i	−0.992932	−	0.118685i	$-0.962132\pi$
$73$	− 17.3281i	− 0.237371i	0.992932	−	0.118685i	$-0.0378680\pi$
$74$	− 127.847i	− 1.72766i
$75$	0	0
$76$	31.0447i	0.408483i
$77$	0	0
$78$	0	0
$79$	− 52.7777i	− 0.668072i	−0.942560	−	0.334036i	$-0.891589\pi$
$79$	− 52.7777i	− 0.668072i	0.942560	−	0.334036i	$-0.108411\pi$
$80$	−51.0280	−0.637850
$81$	0	0
$82$	−114.482	−1.39612
$83$	23.4757i	0.282839i	0.989950	+	0.141420i	$0.0451667\pi$
$83$	23.4757i	0.282839i	−0.989950	+	0.141420i	$0.954833\pi$
$84$	0	0
$85$	− 51.8227i	− 0.609679i
$86$	−147.998	−1.72090
$87$	0	0
$88$	0	0
$89$	−97.1861	−1.09198	−0.545989	−	0.837792i	$-0.683846\pi$
$89$	−97.1861	−1.09198	−0.545989	+	0.837792i	$0.683846\pi$
$90$	0	0
$91$	3.28435	0.0360918
$92$	16.4862	0.179197
$93$	0	0
$94$	84.4969i	0.898903i
$95$	− 37.2454i	− 0.392057i
$96$	0	0
$97$	−50.6148	−0.521802	−0.260901	−	0.965366i	$-0.584020\pi$
$97$	−50.6148	−0.521802	−0.260901	+	0.965366i	$0.584020\pi$
$98$	− 121.230i	− 1.23705i
$99$	0	0
$100$	39.3318	0.393318
$101$	− 65.8066i	− 0.651551i	−0.945447	−	0.325775i	$-0.894375\pi$
$101$	− 65.8066i	− 0.651551i	0.945447	−	0.325775i	$-0.105625\pi$
$102$	0	0
$103$	139.180	1.35127	0.675633	−	0.737238i	$-0.263870\pi$
$103$	139.180	1.35127	0.675633	+	0.737238i	$0.263870\pi$
$104$	89.3036	0.858688
$105$	0	0
$106$	− 37.6912i	− 0.355577i
$107$	− 12.8951i	− 0.120515i	−0.998183	−	0.0602573i	$-0.980808\pi$
$107$	− 12.8951i	− 0.120515i	0.998183	−	0.0602573i	$-0.0191921\pi$
$108$	0	0
$109$	117.681i	1.07964i	0.841780	+	0.539821i	$0.181508\pi$
$109$	117.681i	1.07964i	−0.841780	+	0.539821i	$0.818492\pi$
$110$	0	0
$111$	0	0
$112$	3.40519i	0.0304035i
$113$	−175.640	−1.55433	−0.777166	−	0.629295i	$-0.783344\pi$
$113$	−175.640	−1.55433	−0.777166	+	0.629295i	$0.783344\pi$
$114$	0	0
$115$	−19.7790	−0.171991
$116$	81.4386i	0.702057i
$117$	0	0
$118$	− 65.2108i	− 0.552634i
$119$	−3.45822	−0.0290607
$120$	0	0
$121$	0	0
$122$	−156.880	−1.28590
$123$	0	0
$124$	90.1868	0.727313
$125$	−111.025	−0.888203
$126$	0	0
$127$	59.8470i	0.471236i	0.971846	+	0.235618i	$0.0757114\pi$
$127$	59.8470i	0.471236i	−0.971846	+	0.235618i	$0.924289\pi$
$128$	132.035i	1.03152i
$129$	0	0
$130$	121.840	0.937232
$131$	27.1623i	0.207346i	0.994611	+	0.103673i	$0.0330595\pi$
$131$	27.1623i	0.207346i	−0.994611	+	0.103673i	$0.966940\pi$
$132$	0	0
$133$	−2.48545	−0.0186876
$134$	7.21548i	0.0538469i
$135$	0	0
$136$	−94.0312	−0.691406
$137$	81.1683	0.592469	0.296235	−	0.955115i	$-0.404269\pi$
$137$	81.1683	0.592469	0.296235	+	0.955115i	$0.404269\pi$
$138$	0	0
$139$	194.413i	1.39865i	0.714802	+	0.699327i	$0.246517\pi$
$139$	194.413i	1.39865i	−0.714802	+	0.699327i	$0.753483\pi$
$140$	0.926099i	0.00661499i
$141$	0	0
$142$	239.595i	1.68729i
$143$	0	0
$144$	0	0
$145$	− 97.7048i	− 0.673826i
$146$	−42.8967	−0.293813
$147$	0	0
$148$	−109.918	−0.742690
$149$	121.294i	0.814055i	0.913416	+	0.407027i	$0.133435\pi$
$149$	121.294i	0.814055i	−0.913416	+	0.407027i	$0.866565\pi$
$150$	0	0
$151$	102.557i	0.679186i	0.940572	+	0.339593i	$0.110289\pi$
$151$	102.557i	0.679186i	−0.940572	+	0.339593i	$0.889711\pi$
$152$	−67.5809	−0.444611
$153$	0	0
$154$	0	0
$155$	−108.200	−0.698066
$156$	0	0
$157$	177.331	1.12950	0.564748	−	0.825264i	$-0.308974\pi$
$157$	177.331	1.12950	0.564748	+	0.825264i	$0.308974\pi$
$158$	−130.654	−0.826925
$159$	0	0
$160$	78.9984i	0.493740i
$161$	1.31989i	0.00819807i
$162$	0	0
$163$	105.183	0.645294	0.322647	−	0.946519i	$-0.395427\pi$
$163$	105.183	0.645294	0.322647	+	0.946519i	$0.395427\pi$
$164$	98.4270i	0.600165i
$165$	0	0
$166$	58.1154	0.350093
$167$	129.407i	0.774891i	0.921893	+	0.387446i	$0.126642\pi$
$167$	129.407i	0.774891i	−0.921893	+	0.387446i	$0.873358\pi$
$168$	0	0
$169$	−202.502	−1.19824
$170$	−128.290	−0.754648
$171$	0	0
$172$	127.243i	0.739784i
$173$	44.7971i	0.258943i	0.991583	+	0.129471i	$0.0413281\pi$
$173$	44.7971i	0.258943i	−0.991583	+	0.129471i	$0.958672\pi$
$174$	0	0
$175$	3.14892i	0.0179939i
$176$	0	0
$177$	0	0
$178$	240.590i	1.35163i
$179$	317.726	1.77501	0.887504	−	0.460800i	$-0.152437\pi$
$179$	317.726	1.77501	0.887504	+	0.460800i	$0.152437\pi$
$180$	0	0
$181$	330.778	1.82750	0.913750	−	0.406276i	$-0.133173\pi$
$181$	330.778	1.82750	0.913750	+	0.406276i	$0.133173\pi$
$182$	− 8.13061i	− 0.0446737i
$183$	0	0
$184$	35.8886i	0.195047i
$185$	131.873	0.712825
$186$	0	0
$187$	0	0
$188$	72.6472	0.386421
$189$	0	0
$190$	−92.2031	−0.485280
$191$	127.040	0.665132	0.332566	−	0.943080i	$-0.392086\pi$
$191$	127.040	0.665132	0.332566	+	0.943080i	$0.392086\pi$
$192$	0	0
$193$	− 211.396i	− 1.09531i	−0.836703	−	0.547657i	$-0.815520\pi$
$193$	− 211.396i	− 1.09531i	0.836703	−	0.547657i	$-0.184480\pi$
$194$	125.300i	0.645876i
$195$	0	0
$196$	−104.229	−0.531782
$197$	− 2.87439i	− 0.0145908i	−0.999973	−	0.00729541i	$-0.997678\pi$
$197$	− 2.87439i	− 0.0145908i	0.999973	−	0.00729541i	$-0.00232222\pi$
$198$	0	0
$199$	−158.546	−0.796713	−0.398357	−	0.917231i	$-0.630419\pi$
$199$	−158.546	−0.796713	−0.398357	+	0.917231i	$0.630419\pi$
$200$	85.6212i	0.428106i
$201$	0	0
$202$	−162.908	−0.806476
$203$	−6.52001	−0.0321183
$204$	0	0
$205$	− 118.086i	− 0.576031i
$206$	− 344.549i	− 1.67257i
$207$	0	0
$208$	− 385.170i	− 1.85178i
$209$	0	0
$210$	0	0
$211$	− 355.990i	− 1.68715i	−0.537008	−	0.843577i	$-0.680446\pi$
$211$	− 355.990i	− 1.68715i	0.537008	−	0.843577i	$-0.319554\pi$
$212$	−32.4055	−0.152856
$213$	0	0
$214$	−31.9225	−0.149171
$215$	− 152.658i	− 0.710036i
$216$	0	0
$217$	7.22039i	0.0332737i
$218$	291.326	1.33636
$219$	0	0
$220$	0	0
$221$	391.169	1.76999
$222$	0	0
$223$	−213.373	−0.956831	−0.478415	−	0.878134i	$-0.658789\pi$
$223$	−213.373	−0.956831	−0.478415	+	0.878134i	$0.658789\pi$
$224$	5.27171	0.0235344
$225$	0	0
$226$	434.806i	1.92392i
$227$	− 176.748i	− 0.778628i	−0.921105	−	0.389314i	$-0.872712\pi$
$227$	− 176.748i	− 0.778628i	0.921105	−	0.389314i	$-0.127288\pi$
$228$	0	0
$229$	−62.9641	−0.274953	−0.137476	−	0.990505i	$-0.543899\pi$
$229$	−62.9641	−0.274953	−0.137476	+	0.990505i	$0.543899\pi$
$230$	48.9641i	0.212887i
$231$	0	0
$232$	−177.283	−0.764151
$233$	− 307.347i	− 1.31909i	−0.751667	−	0.659543i	$-0.770750\pi$
$233$	− 307.347i	− 1.31909i	0.751667	−	0.659543i	$-0.229250\pi$
$234$	0	0
$235$	−87.1574	−0.370883
$236$	−56.0658	−0.237567
$237$	0	0
$238$	8.56103i	0.0359707i
$239$	373.207i	1.56153i	0.624822	+	0.780767i	$0.285171\pi$
$239$	373.207i	1.56153i	−0.624822	+	0.780767i	$0.714829\pi$
$240$	0	0
$241$	− 13.0743i	− 0.0542501i	−0.999632	−	0.0271250i	$-0.991365\pi$
$241$	− 13.0743i	− 0.0542501i	0.999632	−	0.0271250i	$-0.00863523\pi$
$242$	0	0
$243$	0	0
$244$	134.879i	0.552783i
$245$	125.048	0.510398
$246$	0	0
$247$	281.136	1.13820
$248$	196.327i	0.791641i
$249$	0	0
$250$	274.850i	1.09940i
$251$	−138.413	−0.551446	−0.275723	−	0.961237i	$-0.588917\pi$
$251$	−138.413	−0.551446	−0.275723	+	0.961237i	$0.588917\pi$
$252$	0	0
$253$	0	0
$254$	148.155	0.583286
$255$	0	0
$256$	313.472	1.22450
$257$	287.892	1.12020	0.560102	−	0.828424i	$-0.310762\pi$
$257$	287.892	1.12020	0.560102	+	0.828424i	$0.310762\pi$
$258$	0	0
$259$	− 8.80009i	− 0.0339772i
$260$	− 104.754i	− 0.402898i
$261$	0	0
$262$	67.2419	0.256648
$263$	338.296i	1.28629i	0.765742	+	0.643147i	$0.222372\pi$
$263$	338.296i	1.28629i	−0.765742	+	0.643147i	$0.777628\pi$
$264$	0	0
$265$	38.8779	0.146709
$266$	6.15288i	0.0231311i
$267$	0	0
$268$	6.20359	0.0231477
$269$	63.1555	0.234779	0.117389	−	0.993086i	$-0.462547\pi$
$269$	63.1555	0.234779	0.117389	+	0.993086i	$0.462547\pi$
$270$	0	0
$271$	− 166.707i	− 0.615153i	−0.951523	−	0.307577i	$-0.900482\pi$
$271$	− 166.707i	− 0.615153i	0.951523	−	0.307577i	$-0.0995181\pi$
$272$	405.560i	1.49103i
$273$	0	0
$274$	− 200.937i	− 0.733346i
$275$	0	0
$276$	0	0
$277$	42.3827i	0.153006i	0.997069	+	0.0765030i	$0.0243755\pi$
$277$	42.3827i	0.153006i	−0.997069	+	0.0765030i	$0.975625\pi$
$278$	481.280	1.73122
$279$	0	0
$280$	−2.01602	−0.00720006
$281$	− 142.796i	− 0.508169i	−0.967182	−	0.254085i	$-0.918226\pi$
$281$	− 142.796i	− 0.508169i	0.967182	−	0.254085i	$-0.0817742\pi$
$282$	0	0
$283$	− 259.085i	− 0.915494i	−0.889082	−	0.457747i	$-0.848657\pi$
$283$	− 259.085i	− 0.915494i	0.889082	−	0.457747i	$-0.151343\pi$
$284$	205.995	0.725334
$285$	0	0
$286$	0	0
$287$	−7.88011	−0.0274568
$288$	0	0
$289$	−122.877	−0.425178
$290$	−241.874	−0.834048
$291$	0	0
$292$	36.8809i	0.126304i
$293$	− 115.020i	− 0.392561i	−0.980548	−	0.196280i	$-0.937114\pi$
$293$	− 115.020i	− 0.392561i	0.980548	−	0.196280i	$-0.0628862\pi$
$294$	0	0
$295$	67.2640	0.228014
$296$	− 239.280i	− 0.808378i
$297$	0	0
$298$	300.271	1.00762
$299$	− 149.296i	− 0.499318i
$300$	0	0
$301$	−10.1871	−0.0338442
$302$	253.886	0.840683
$303$	0	0
$304$	291.479i	0.958813i
$305$	− 161.819i	− 0.530554i
$306$	0	0
$307$	− 22.9819i	− 0.0748596i	−0.999299	−	0.0374298i	$-0.988083\pi$
$307$	− 22.9819i	− 0.0748596i	0.999299	−	0.0374298i	$-0.0119171\pi$
$308$	0	0
$309$	0	0
$310$	267.856i	0.864052i
$311$	−594.646	−1.91205	−0.956023	−	0.293291i	$-0.905250\pi$
$311$	−594.646	−1.91205	−0.956023	+	0.293291i	$0.905250\pi$
$312$	0	0
$313$	−521.860	−1.66728	−0.833642	−	0.552305i	$-0.813748\pi$
$313$	−521.860	−1.66728	−0.833642	+	0.552305i	$0.813748\pi$
$314$	− 438.993i	− 1.39807i
$315$	0	0
$316$	112.331i	0.355479i
$317$	−153.642	−0.484675	−0.242337	−	0.970192i	$-0.577914\pi$
$317$	−153.642	−0.484675	−0.242337	+	0.970192i	$0.577914\pi$
$318$	0	0
$319$	0	0
$320$	−8.54675	−0.0267086
$321$	0	0
$322$	3.26746	0.0101474
$323$	−296.019	−0.916467
$324$	0	0
$325$	− 356.183i	− 1.09595i
$326$	− 260.386i	− 0.798731i
$327$	0	0
$328$	−214.265	−0.653247
$329$	5.81617i	0.0176783i
$330$	0	0
$331$	297.441	0.898613	0.449306	−	0.893378i	$-0.351671\pi$
$331$	297.441	0.898613	0.449306	+	0.893378i	$0.351671\pi$
$332$	− 49.9654i	− 0.150498i
$333$	0	0
$334$	320.354	0.959144
$335$	−7.44267	−0.0222169
$336$	0	0
$337$	4.17943i	0.0124019i	0.999981	+	0.00620094i	$0.00197383\pi$
$337$	4.17943i	0.0124019i	−0.999981	+	0.00620094i	$0.998026\pi$
$338$	501.306i	1.48315i
$339$	0	0
$340$	110.299i	0.324409i
$341$	0	0
$342$	0	0
$343$	− 16.6942i	− 0.0486713i
$344$	−276.994	−0.805215
$345$	0	0
$346$	110.898	0.320514
$347$	− 585.875i	− 1.68840i	−0.536028	−	0.844200i	$-0.680076\pi$
$347$	− 585.875i	− 1.68840i	0.536028	−	0.844200i	$-0.319924\pi$
$348$	0	0
$349$	231.222i	0.662527i	0.943538	+	0.331263i	$0.107475\pi$
$349$	231.222i	0.662527i	−0.943538	+	0.331263i	$0.892525\pi$
$350$	7.79535	0.0222724
$351$	0	0
$352$	0	0
$353$	416.133	1.17885	0.589423	−	0.807824i	$-0.299355\pi$
$353$	416.133	1.17885	0.589423	+	0.807824i	$0.299355\pi$
$354$	0	0
$355$	−247.139	−0.696166
$356$	206.850	0.581040
$357$	0	0
$358$	− 786.551i	− 2.19707i
$359$	− 37.9041i	− 0.105582i	−0.998606	−	0.0527912i	$-0.983188\pi$
$359$	− 37.9041i	− 0.105582i	0.998606	−	0.0527912i	$-0.0168118\pi$
$360$	0	0
$361$	148.249	0.410662
$362$	− 818.859i	− 2.26204i
$363$	0	0
$364$	−6.99039	−0.0192044
$365$	− 44.2473i	− 0.121225i
$366$	0	0
$367$	219.116	0.597047	0.298523	−	0.954402i	$-0.403506\pi$
$367$	219.116	0.597047	0.298523	+	0.954402i	$0.403506\pi$
$368$	154.789	0.420622
$369$	0	0
$370$	− 326.458i	− 0.882320i
$371$	− 2.59439i	− 0.00699298i
$372$	0	0
$373$	− 17.9491i	− 0.0481209i	−0.999711	−	0.0240605i	$-0.992341\pi$
$373$	− 17.9491i	− 0.0481209i	0.999711	−	0.0240605i	$-0.00765942\pi$
$374$	0	0
$375$	0	0
$376$	158.145i	0.420599i
$377$	737.496	1.95622
$378$	0	0
$379$	−45.6414	−0.120426	−0.0602129	−	0.998186i	$-0.519178\pi$
$379$	−45.6414	−0.120426	−0.0602129	+	0.998186i	$0.519178\pi$
$380$	79.2727i	0.208612i
$381$	0	0
$382$	− 314.495i	− 0.823286i
$383$	28.1815	0.0735810	0.0367905	−	0.999323i	$-0.488287\pi$
$383$	28.1815	0.0735810	0.0367905	+	0.999323i	$0.488287\pi$
$384$	0	0
$385$	0	0
$386$	−523.322	−1.35576
$387$	0	0
$388$	107.728	0.277650
$389$	564.821	1.45198	0.725992	−	0.687704i	$-0.241381\pi$
$389$	564.821	1.45198	0.725992	+	0.687704i	$0.241381\pi$
$390$	0	0
$391$	157.200i	0.402045i
$392$	− 226.896i	− 0.578817i
$393$	0	0
$394$	−7.11572	−0.0180602
$395$	− 134.768i	− 0.341185i
$396$	0	0
$397$	6.35093	0.0159973	0.00799866	−	0.999968i	$-0.497454\pi$
$397$	6.35093	0.0159973	0.00799866	+	0.999968i	$0.497454\pi$
$398$	392.490i	0.986155i
$399$	0	0
$400$	369.288	0.923219
$401$	165.822	0.413521	0.206761	−	0.978392i	$-0.433708\pi$
$401$	165.822	0.413521	0.206761	+	0.978392i	$0.433708\pi$
$402$	0	0
$403$	− 816.718i	− 2.02660i
$404$	140.062i	0.346689i
$405$	0	0
$406$	16.1407i	0.0397554i
$407$	0	0
$408$	0	0
$409$	38.0311i	0.0929856i	0.998919	+	0.0464928i	$0.0148045\pi$
$409$	38.0311i	0.0929856i	−0.998919	+	0.0464928i	$0.985196\pi$
$410$	−292.330	−0.712999
$411$	0	0
$412$	−296.230	−0.719006
$413$	− 4.48865i	− 0.0108684i
$414$	0	0
$415$	59.9452i	0.144446i
$416$	−596.297	−1.43341
$417$	0	0
$418$	0	0
$419$	254.746	0.607985	0.303993	−	0.952674i	$-0.401680\pi$
$419$	254.746	0.607985	0.303993	+	0.952674i	$0.401680\pi$
$420$	0	0
$421$	554.739	1.31767	0.658835	−	0.752288i	$-0.271050\pi$
$421$	554.739	1.31767	0.658835	+	0.752288i	$0.271050\pi$
$422$	−881.273	−2.08832
$423$	0	0
$424$	− 70.5432i	− 0.166375i
$425$	375.039i	0.882445i
$426$	0	0
$427$	−10.7985	−0.0252892
$428$	27.4457i	0.0641256i
$429$	0	0
$430$	−377.913	−0.878867
$431$	− 31.4905i	− 0.0730639i	−0.999332	−	0.0365319i	$-0.988369\pi$
$431$	− 31.4905i	− 0.0730639i	0.999332	−	0.0365319i	$-0.0116311\pi$
$432$	0	0
$433$	633.975	1.46415	0.732073	−	0.681226i	$-0.238553\pi$
$433$	633.975	1.46415	0.732073	+	0.681226i	$0.238553\pi$
$434$	17.8745	0.0411855
$435$	0	0
$436$	− 250.471i	− 0.574475i
$437$	112.981i	0.258537i
$438$	0	0
$439$	336.283i	0.766021i	0.923744	+	0.383010i	$0.125113\pi$
$439$	336.283i	0.766021i	−0.923744	+	0.383010i	$0.874887\pi$
$440$	0	0
$441$	0	0
$442$	− 968.361i	− 2.19086i
$443$	487.743	1.10100	0.550500	−	0.834835i	$-0.314437\pi$
$443$	487.743	1.10100	0.550500	+	0.834835i	$0.314437\pi$
$444$	0	0
$445$	−248.165	−0.557675
$446$	528.218i	1.18435i
$447$	0	0
$448$	0.570340i	0.00127308i
$449$	668.379	1.48860	0.744298	−	0.667848i	$-0.232784\pi$
$449$	668.379	1.48860	0.744298	+	0.667848i	$0.232784\pi$
$450$	0	0
$451$	0	0
$452$	373.830	0.827057
$453$	0	0
$454$	−437.551	−0.963769
$455$	8.38662	0.0184321
$456$	0	0
$457$	− 511.486i	− 1.11923i	−0.828754	−	0.559613i	$-0.810950\pi$
$457$	− 511.486i	− 1.11923i	0.828754	−	0.559613i	$-0.189050\pi$
$458$	155.871i	0.340331i
$459$	0	0
$460$	42.0975	0.0915162
$461$	75.7082i	0.164226i	0.996623	+	0.0821131i	$0.0261669\pi$
$461$	75.7082i	0.164226i	−0.996623	+	0.0821131i	$0.973833\pi$
$462$	0	0
$463$	292.607	0.631980	0.315990	−	0.948763i	$-0.397663\pi$
$463$	292.607	0.631980	0.315990	+	0.948763i	$0.397663\pi$
$464$	764.629i	1.64791i
$465$	0	0
$466$	−760.856	−1.63274
$467$	−774.519	−1.65850	−0.829249	−	0.558879i	$-0.811232\pi$
$467$	−774.519	−1.65850	−0.829249	+	0.558879i	$0.811232\pi$
$468$	0	0
$469$	0.496662i	0.00105898i
$470$	215.763i	0.459071i
$471$	0	0
$472$	− 122.049i	− 0.258579i
$473$	0	0
$474$	0	0
$475$	269.543i	0.567460i
$476$	7.36045	0.0154631
$477$	0	0
$478$	923.895	1.93283
$479$	65.6365i	0.137028i	0.997650	+	0.0685141i	$0.0218258\pi$
$479$	65.6365i	0.137028i	−0.997650	+	0.0685141i	$0.978174\pi$
$480$	0	0
$481$	995.402i	2.06944i
$482$	−32.3661	−0.0671496
$483$	0	0
$484$	0	0
$485$	−129.245	−0.266485
$486$	0	0
$487$	−18.1162	−0.0371995	−0.0185998	−	0.999827i	$-0.505921\pi$
$487$	−18.1162	−0.0371995	−0.0185998	+	0.999827i	$0.505921\pi$
$488$	−293.617	−0.601674
$489$	0	0
$490$	− 309.563i	− 0.631760i
$491$	− 426.903i	− 0.869455i	−0.900562	−	0.434728i	$-0.856845\pi$
$491$	− 426.903i	− 0.869455i	0.900562	−	0.434728i	$-0.143155\pi$
$492$	0	0
$493$	−776.538	−1.57513
$494$	− 695.969i	− 1.40884i
$495$	0	0
$496$	846.766	1.70719
$497$	16.4920i	0.0331832i
$498$	0	0
$499$	285.373	0.571890	0.285945	−	0.958246i	$-0.407693\pi$
$499$	285.373	0.571890	0.285945	+	0.958246i	$0.407693\pi$
$500$	236.305	0.472611
$501$	0	0
$502$	342.649i	0.682568i
$503$	− 543.584i	− 1.08068i	−0.841446	−	0.540342i	$-0.818295\pi$
$503$	− 543.584i	− 1.08068i	0.841446	−	0.540342i	$-0.181705\pi$
$504$	0	0
$505$	− 168.038i	− 0.332748i
$506$	0	0
$507$	0	0
$508$	− 127.378i	− 0.250744i
$509$	−196.501	−0.386054	−0.193027	−	0.981193i	$-0.561830\pi$
$509$	−196.501	−0.386054	−0.193027	+	0.981193i	$0.561830\pi$
$510$	0	0
$511$	−2.95270	−0.00577828
$512$	− 247.878i	− 0.484138i
$513$	0	0
$514$	− 712.695i	− 1.38657i
$515$	355.398	0.690093
$516$	0	0
$517$	0	0
$518$	−21.7851	−0.0420563
$519$	0	0
$520$	228.037	0.438533
$521$	573.267	1.10032	0.550160	−	0.835059i	$-0.314567\pi$
$521$	573.267	1.10032	0.550160	+	0.835059i	$0.314567\pi$
$522$	0	0
$523$	993.636i	1.89988i	0.312437	+	0.949938i	$0.398855\pi$
$523$	993.636i	1.89988i	−0.312437	+	0.949938i	$0.601145\pi$
$524$	− 57.8120i	− 0.110328i
$525$	0	0
$526$	837.471	1.59215
$527$	859.954i	1.63179i
$528$	0	0
$529$	−469.002	−0.886583
$530$	− 96.2446i	− 0.181594i
$531$	0	0
$532$	5.29001	0.00994363
$533$	891.340	1.67231
$534$	0	0
$535$	− 32.9276i	− 0.0615469i
$536$	13.5046i	0.0251951i
$537$	0	0
$538$	− 156.345i	− 0.290604i
$539$	0	0
$540$	0	0
$541$	226.110i	0.417949i	0.977921	+	0.208974i	$0.0670125\pi$
$541$	226.110i	0.417949i	−0.977921	+	0.208974i	$0.932987\pi$
$542$	−412.692	−0.761424
$543$	0	0
$544$	627.864	1.15416
$545$	300.499i	0.551374i
$546$	0	0
$547$	− 457.431i	− 0.836254i	−0.908389	−	0.418127i	$-0.862687\pi$
$547$	− 457.431i	− 0.836254i	0.908389	−	0.418127i	$-0.137313\pi$
$548$	−172.758	−0.315252
$549$	0	0
$550$	0	0
$551$	−558.104	−1.01289
$552$	0	0
$553$	−8.99331	−0.0162628
$554$	104.921	0.189388
$555$	0	0
$556$	− 413.787i	− 0.744220i
$557$	281.137i	0.504735i	0.967631	+	0.252368i	$0.0812092\pi$
$557$	281.137i	0.504735i	−0.967631	+	0.252368i	$0.918791\pi$
$558$	0	0
$559$	1152.29	2.06135
$560$	8.69517i	0.0155271i
$561$	0	0
$562$	−353.499	−0.629002
$563$	− 225.243i	− 0.400076i	−0.979788	−	0.200038i	$-0.935893\pi$
$563$	− 225.243i	− 0.400076i	0.979788	−	0.200038i	$-0.0641066\pi$
$564$	0	0
$565$	−448.496	−0.793799
$566$	−641.380	−1.13318
$567$	0	0
$568$	448.428i	0.789487i
$569$	− 148.348i	− 0.260717i	−0.991467	−	0.130358i	$-0.958387\pi$
$569$	− 148.348i	− 0.260717i	0.991467	−	0.130358i	$-0.0416128\pi$
$570$	0	0
$571$	111.405i	0.195106i	0.995230	+	0.0975528i	$0.0311015\pi$
$571$	111.405i	0.195106i	−0.995230	+	0.0975528i	$0.968899\pi$
$572$	0	0
$573$	0	0
$574$	19.5077i	0.0339855i
$575$	143.140	0.248939
$576$	0	0
$577$	−540.605	−0.936925	−0.468462	−	0.883484i	$-0.655192\pi$
$577$	−540.605	−0.936925	−0.468462	+	0.883484i	$0.655192\pi$
$578$	304.188i	0.526277i
$579$	0	0
$580$	207.954i	0.358541i
$581$	4.00025	0.00688511
$582$	0	0
$583$	0	0
$584$	−80.2858	−0.137476
$585$	0	0
$586$	−284.739	−0.485903
$587$	−344.226	−0.586416	−0.293208	−	0.956049i	$-0.594723\pi$
$587$	−344.226	−0.586416	−0.293208	+	0.956049i	$0.594723\pi$
$588$	0	0
$589$	618.055i	1.04933i
$590$	− 166.516i	− 0.282231i
$591$	0	0
$592$	−1032.02	−1.74328
$593$	− 723.311i	− 1.21975i	−0.792498	−	0.609874i	$-0.791220\pi$
$593$	− 723.311i	− 1.21975i	0.792498	−	0.609874i	$-0.208780\pi$
$594$	0	0
$595$	−8.83059	−0.0148413
$596$	− 258.161i	− 0.433157i
$597$	0	0
$598$	−369.591	−0.618046
$599$	283.768	0.473737	0.236868	−	0.971542i	$-0.423879\pi$
$599$	283.768	0.473737	0.236868	+	0.971542i	$0.423879\pi$
$600$	0	0
$601$	1028.06i	1.71059i	0.518144	+	0.855294i	$0.326623\pi$
$601$	1028.06i	1.71059i	−0.518144	+	0.855294i	$0.673377\pi$
$602$	25.2188i	0.0418917i
$603$	0	0
$604$	− 218.282i	− 0.361393i
$605$	0	0
$606$	0	0
$607$	630.478i	1.03868i	0.854568	+	0.519339i	$0.173822\pi$
$607$	630.478i	1.03868i	−0.854568	+	0.519339i	$0.826178\pi$
$608$	451.250	0.742188
$609$	0	0
$610$	−400.593	−0.656709
$611$	− 657.882i	− 1.07673i
$612$	0	0
$613$	− 562.321i	− 0.917325i	−0.888610	−	0.458663i	$-0.848329\pi$
$613$	− 562.321i	− 0.917325i	0.888610	−	0.458663i	$-0.151671\pi$
$614$	−56.8931	−0.0926597
$615$	0	0
$616$	0	0
$617$	892.792	1.44699	0.723494	−	0.690331i	$-0.242535\pi$
$617$	892.792	1.44699	0.723494	+	0.690331i	$0.242535\pi$
$618$	0	0
$619$	648.381	1.04746	0.523732	−	0.851883i	$-0.324539\pi$
$619$	648.381	1.04746	0.523732	+	0.851883i	$0.324539\pi$
$620$	230.292	0.371439
$621$	0	0
$622$	1472.08i	2.36669i
$623$	16.5605i	0.0265819i
$624$	0	0
$625$	178.486	0.285578
$626$	1291.89i	2.06373i
$627$	0	0
$628$	−377.429	−0.601002
$629$	− 1048.10i	− 1.66629i
$630$	0	0
$631$	−1019.41	−1.61554	−0.807772	−	0.589495i	$-0.799327\pi$
$631$	−1019.41	−1.61554	−0.807772	+	0.589495i	$0.799327\pi$
$632$	−244.534	−0.386920
$633$	0	0
$634$	380.349i	0.599920i
$635$	152.820i	0.240661i
$636$	0	0
$637$	943.885i	1.48177i
$638$	0	0
$639$	0	0
$640$	337.152i	0.526800i
$641$	−436.401	−0.680813	−0.340406	−	0.940278i	$-0.610565\pi$
$641$	−436.401	−0.680813	−0.340406	+	0.940278i	$0.610565\pi$
$642$	0	0
$643$	−514.698	−0.800463	−0.400231	−	0.916414i	$-0.631070\pi$
$643$	−514.698	−0.800463	−0.400231	+	0.916414i	$0.631070\pi$
$644$	− 2.80924i	− 0.00436217i
$645$	0	0
$646$	732.812i	1.13438i
$647$	−241.038	−0.372547	−0.186273	−	0.982498i	$-0.559641\pi$
$647$	−241.038	−0.372547	−0.186273	+	0.982498i	$0.559641\pi$
$648$	0	0
$649$	0	0
$650$	−881.753	−1.35654
$651$	0	0
$652$	−223.870	−0.343359
$653$	162.540	0.248913	0.124456	−	0.992225i	$-0.460281\pi$
$653$	162.540	0.248913	0.124456	+	0.992225i	$0.460281\pi$
$654$	0	0
$655$	69.3591i	0.105892i
$656$	924.133i	1.40874i
$657$	0	0
$658$	14.3983	0.0218819
$659$	1078.94i	1.63724i	0.574332	+	0.818622i	$0.305262\pi$
$659$	1078.94i	1.63724i	−0.574332	+	0.818622i	$0.694738\pi$
$660$	0	0
$661$	428.584	0.648387	0.324193	−	0.945991i	$-0.394907\pi$
$661$	428.584	0.648387	0.324193	+	0.945991i	$0.394907\pi$
$662$	− 736.332i	− 1.11228i
$663$	0	0
$664$	108.769	0.163809
$665$	−6.34661	−0.00954377
$666$	0	0
$667$	296.379i	0.444346i
$668$	− 275.428i	− 0.412318i
$669$	0	0
$670$	18.4248i	0.0274996i
$671$	0	0
$672$	0	0
$673$	− 262.693i	− 0.390331i	−0.980770	−	0.195165i	$-0.937476\pi$
$673$	− 262.693i	− 0.390331i	0.980770	−	0.195165i	$-0.0625244\pi$
$674$	10.3464	0.0153508
$675$	0	0
$676$	431.004	0.637580
$677$	149.879i	0.221387i	0.993855	+	0.110694i	$0.0353073\pi$
$677$	149.879i	0.221387i	−0.993855	+	0.110694i	$0.964693\pi$
$678$	0	0
$679$	8.62476i	0.0127021i
$680$	−240.109	−0.353102
$681$	0	0
$682$	0	0
$683$	52.4276	0.0767608	0.0383804	−	0.999263i	$-0.487780\pi$
$683$	52.4276	0.0767608	0.0383804	+	0.999263i	$0.487780\pi$
$684$	0	0
$685$	207.264	0.302575
$686$	−41.3276	−0.0602443
$687$	0	0
$688$	1194.69i	1.73646i
$689$	293.459i	0.425920i
$690$	0	0
$691$	−196.722	−0.284692	−0.142346	−	0.989817i	$-0.545465\pi$
$691$	−196.722	−0.284692	−0.142346	+	0.989817i	$0.545465\pi$
$692$	− 95.3458i	− 0.137783i
$693$	0	0
$694$	−1450.37	−2.08987
$695$	496.434i	0.714294i
$696$	0	0
$697$	−938.526	−1.34652
$698$	572.403	0.820062
$699$	0	0
$700$	− 6.70214i	− 0.00957449i
$701$	− 886.407i	− 1.26449i	−0.774769	−	0.632245i	$-0.782134\pi$
$701$	− 886.407i	− 1.26449i	0.774769	−	0.632245i	$-0.217866\pi$
$702$	0	0
$703$	− 753.275i	− 1.07152i
$704$	0	0
$705$	0	0
$706$	− 1030.16i	− 1.45915i
$707$	−11.2134	−0.0158606
$708$	0	0
$709$	310.329	0.437700	0.218850	−	0.975758i	$-0.429769\pi$
$709$	310.329	0.437700	0.218850	+	0.975758i	$0.429769\pi$
$710$	611.807i	0.861700i
$711$	0	0
$712$	450.290i	0.632430i
$713$	328.216	0.460331
$714$	0	0
$715$	0	0
$716$	−676.246	−0.944478
$717$	0	0
$718$	−93.8337	−0.130688
$719$	−451.160	−0.627482	−0.313741	−	0.949509i	$-0.601582\pi$
$719$	−451.160	−0.627482	−0.313741	+	0.949509i	$0.601582\pi$
$720$	0	0
$721$	− 23.7163i	− 0.0328937i
$722$	− 366.999i	− 0.508309i
$723$	0	0
$724$	−704.024	−0.972409
$725$	707.086i	0.975290i
$726$	0	0
$727$	−89.6851	−0.123363	−0.0616817	−	0.998096i	$-0.519646\pi$
$727$	−89.6851	−0.123363	−0.0616817	+	0.998096i	$0.519646\pi$
$728$	− 15.2173i	− 0.0209029i
$729$	0	0
$730$	−109.537	−0.150050
$731$	−1213.29	−1.65977
$732$	0	0
$733$	− 557.486i	− 0.760554i	−0.924873	−	0.380277i	$-0.875829\pi$
$733$	− 557.486i	− 0.760554i	0.924873	−	0.380277i	$-0.124171\pi$
$734$	− 542.435i	− 0.739012i
$735$	0	0
$736$	− 239.635i	− 0.325591i
$737$	0	0
$738$	0	0
$739$	95.7304i	0.129541i	0.997900	+	0.0647703i	$0.0206315\pi$
$739$	95.7304i	0.129541i	−0.997900	+	0.0647703i	$0.979369\pi$
$740$	−280.676	−0.379292
$741$	0	0
$742$	−6.42258	−0.00865576
$743$	1121.35i	1.50921i	0.656177	+	0.754607i	$0.272172\pi$
$743$	1121.35i	1.50921i	−0.656177	+	0.754607i	$0.727828\pi$
$744$	0	0
$745$	309.725i	0.415738i
$746$	−44.4341	−0.0595631
$747$	0	0
$748$	0	0
$749$	−2.19732	−0.00293367
$750$	0	0
$751$	−255.391	−0.340068	−0.170034	−	0.985438i	$-0.554388\pi$
$751$	−255.391	−0.340068	−0.170034	+	0.985438i	$0.554388\pi$
$752$	682.087	0.907030
$753$	0	0
$754$	− 1825.71i	− 2.42137i
$755$	261.880i	0.346861i
$756$	0	0
$757$	123.853	0.163610	0.0818052	−	0.996648i	$-0.473931\pi$
$757$	123.853	0.163610	0.0818052	+	0.996648i	$0.473931\pi$
$758$	112.988i	0.149061i
$759$	0	0
$760$	−172.568	−0.227063
$761$	349.009i	0.458619i	0.973354	+	0.229309i	$0.0736467\pi$
$761$	349.009i	0.458619i	−0.973354	+	0.229309i	$0.926353\pi$
$762$	0	0
$763$	20.0528	0.0262815
$764$	−270.391	−0.353915
$765$	0	0
$766$	− 69.7650i	− 0.0910770i
$767$	507.723i	0.661960i
$768$	0	0
$769$	867.000i	1.12744i	0.825966	+	0.563719i	$0.190630\pi$
$769$	867.000i	1.12744i	−0.825966	+	0.563719i	$0.809370\pi$
$770$	0	0
$771$	0	0
$772$	449.932i	0.582814i
$773$	173.732	0.224751	0.112375	−	0.993666i	$-0.464154\pi$
$773$	173.732	0.224751	0.112375	+	0.993666i	$0.464154\pi$
$774$	0	0
$775$	783.041	1.01038
$776$	234.512i	0.302207i
$777$	0	0
$778$	− 1398.25i	− 1.79723i
$779$	−674.526	−0.865887
$780$	0	0
$781$	0	0
$782$	389.157	0.497643
$783$	0	0
$784$	−978.612	−1.24823
$785$	452.815	0.576834
$786$	0	0
$787$	− 45.2594i	− 0.0575087i	−0.999587	−	0.0287544i	$-0.990846\pi$
$787$	− 45.2594i	− 0.0575087i	0.999587	−	0.0287544i	$-0.00915406\pi$
$788$	6.11783i	0.00776374i
$789$	0	0
$790$	−333.626	−0.422311
$791$	29.9290i	0.0378369i
$792$	0	0
$793$	1221.44	1.54028
$794$	− 15.7221i	− 0.0198011i
$795$	0	0
$796$	337.448	0.423929
$797$	−943.478	−1.18379	−0.591894	−	0.806016i	$-0.701619\pi$
$797$	−943.478	−1.18379	−0.591894	+	0.806016i	$0.701619\pi$
$798$	0	0
$799$	692.710i	0.866971i
$800$	− 571.709i	− 0.714636i
$801$	0	0
$802$	− 410.502i	− 0.511848i
$803$	0	0
$804$	0	0
$805$	3.37034i	0.00418676i
$806$	−2021.83	−2.50848
$807$	0	0
$808$	−304.900	−0.377352
$809$	528.501i	0.653277i	0.945149	+	0.326639i	$0.105916\pi$
$809$	528.501i	0.653277i	−0.945149	+	0.326639i	$0.894084\pi$
$810$	0	0
$811$	− 731.685i	− 0.902201i	−0.892473	−	0.451101i	$-0.851032\pi$
$811$	− 731.685i	− 0.902201i	0.892473	−	0.451101i	$-0.148968\pi$
$812$	13.8771	0.0170901
$813$	0	0
$814$	0	0
$815$	268.585	0.329552
$816$	0	0
$817$	−872.002	−1.06732
$818$	94.1482	0.115096
$819$	0	0
$820$	251.334i	0.306505i
$821$	1124.91i	1.37017i	0.728463	+	0.685085i	$0.240235\pi$
$821$	1124.91i	1.37017i	−0.728463	+	0.685085i	$0.759765\pi$
$822$	0	0
$823$	238.410	0.289684	0.144842	−	0.989455i	$-0.453733\pi$
$823$	238.410	0.289684	0.144842	+	0.989455i	$0.453733\pi$
$824$	− 644.861i	− 0.782599i
$825$	0	0
$826$	−11.1119	−0.0134527
$827$	1536.19i	1.85755i	0.370647	+	0.928774i	$0.379136\pi$
$827$	1536.19i	1.85755i	−0.370647	+	0.928774i	$0.620864\pi$
$828$	0	0
$829$	1576.52	1.90172	0.950859	−	0.309624i	$-0.100203\pi$
$829$	1576.52	1.90172	0.950859	+	0.309624i	$0.100203\pi$
$830$	148.398	0.178793
$831$	0	0
$832$	− 64.5126i	− 0.0775392i
$833$	− 993.853i	− 1.19310i
$834$	0	0
$835$	330.441i	0.395738i
$836$	0	0
$837$	0	0
$838$	− 630.638i	− 0.752552i
$839$	47.4743	0.0565844	0.0282922	−	0.999600i	$-0.490993\pi$
$839$	47.4743	0.0565844	0.0282922	+	0.999600i	$0.490993\pi$
$840$	0	0
$841$	−623.058	−0.740853
$842$	− 1373.29i	− 1.63098i
$843$	0	0
$844$	757.685i	0.897731i
$845$	−517.090	−0.611941
$846$	0	0
$847$	0	0
$848$	−304.256	−0.358792
$849$	0	0
$850$	928.431	1.09227
$851$	−400.024	−0.470063
$852$	0	0
$853$	− 743.273i	− 0.871364i	−0.900101	−	0.435682i	$-0.856507\pi$
$853$	− 743.273i	− 0.871364i	0.900101	−	0.435682i	$-0.143493\pi$
$854$	26.7323i	0.0313024i
$855$	0	0
$856$	−59.7464	−0.0697972
$857$	− 572.262i	− 0.667750i	−0.942617	−	0.333875i	$-0.891644\pi$
$857$	− 572.262i	− 0.667750i	0.942617	−	0.333875i	$-0.108356\pi$
$858$	0	0
$859$	−792.198	−0.922233	−0.461117	−	0.887340i	$-0.652551\pi$
$859$	−792.198	−0.922233	−0.461117	+	0.887340i	$0.652551\pi$
$860$	324.915i	0.377808i
$861$	0	0
$862$	−77.9566	−0.0904369
$863$	−1052.87	−1.22001	−0.610007	−	0.792396i	$-0.708833\pi$
$863$	−1052.87	−1.22001	−0.610007	+	0.792396i	$0.708833\pi$
$864$	0	0
$865$	114.390i	0.132242i
$866$	− 1569.44i	− 1.81229i
$867$	0	0
$868$	− 15.3678i	− 0.0177049i
$869$	0	0
$870$	0	0
$871$	− 56.1788i	− 0.0644992i
$872$	545.248	0.625285
$873$	0	0
$874$	279.690	0.320011
$875$	18.9187i	0.0216214i
$876$	0	0
$877$	− 1419.86i	− 1.61899i	−0.587124	−	0.809497i	$-0.699740\pi$
$877$	− 1419.86i	− 1.61899i	0.587124	−	0.809497i	$-0.300260\pi$
$878$	832.489	0.948165
$879$	0	0
$880$	0	0
$881$	−952.550	−1.08121	−0.540607	−	0.841275i	$-0.681805\pi$
$881$	−952.550	−1.08121	−0.540607	+	0.841275i	$0.681805\pi$
$882$	0	0
$883$	355.803	0.402948	0.201474	−	0.979494i	$-0.435427\pi$
$883$	355.803	0.402948	0.201474	+	0.979494i	$0.435427\pi$
$884$	−832.560	−0.941810
$885$	0	0
$886$	− 1207.44i	− 1.36280i
$887$	− 1032.34i	− 1.16385i	−0.813241	−	0.581927i	$-0.802299\pi$
$887$	− 1032.34i	− 1.16385i	0.813241	−	0.581927i	$-0.197701\pi$
$888$	0	0
$889$	10.1979	0.0114712
$890$	614.348i	0.690278i
$891$	0	0
$892$	454.142	0.509128
$893$	497.856i	0.557509i
$894$	0	0
$895$	811.316	0.906498
$896$	22.4987	0.0251102
$897$	0	0
$898$	− 1654.61i	− 1.84255i
$899$	1621.33i	1.80348i
$900$	0	0
$901$	− 308.994i	− 0.342946i
$902$	0	0
$903$	0	0
$904$	813.787i	0.900207i
$905$	844.642	0.933306
$906$	0	0
$907$	−1087.06	−1.19853	−0.599263	−	0.800552i	$-0.704540\pi$
$907$	−1087.06	−1.19853	−0.599263	+	0.800552i	$0.704540\pi$
$908$	376.190i	0.414306i
$909$	0	0
$910$	− 20.7616i	− 0.0228149i
$911$	−400.130	−0.439221	−0.219610	−	0.975588i	$-0.570479\pi$
$911$	−400.130	−0.439221	−0.219610	+	0.975588i	$0.570479\pi$
$912$	0	0
$913$	0	0
$914$	−1266.21	−1.38536
$915$	0	0
$916$	134.012	0.146302
$917$	4.62846	0.00504739
$918$	0	0
$919$	1347.43i	1.46619i	0.680124	+	0.733097i	$0.261926\pi$
$919$	1347.43i	1.46619i	−0.680124	+	0.733097i	$0.738074\pi$
$920$	91.6416i	0.0996105i
$921$	0	0
$922$	187.420	0.203276
$923$	− 1865.46i	− 2.02108i
$924$	0	0
$925$	−954.357	−1.03174
$926$	− 724.365i	− 0.782252i
$927$	0	0
$928$	1183.75	1.27560
$929$	1493.12	1.60723	0.803616	−	0.595148i	$-0.202907\pi$
$929$	1493.12	1.60723	0.803616	+	0.595148i	$0.202907\pi$
$930$	0	0
$931$	− 714.290i	− 0.767229i
$932$	654.155i	0.701883i
$933$	0	0
$934$	1917.37i	2.05286i
$935$	0	0
$936$	0	0
$937$	− 556.584i	− 0.594007i	−0.954876	−	0.297003i	$-0.904013\pi$
$937$	− 556.584i	− 0.594007i	0.954876	−	0.297003i	$-0.0959873\pi$
$938$	1.22952	0.00131079
$939$	0	0
$940$	185.505	0.197346
$941$	842.698i	0.895535i	0.894150	+	0.447767i	$0.147781\pi$
$941$	842.698i	0.895535i	−0.894150	+	0.447767i	$0.852219\pi$
$942$	0	0
$943$	358.204i	0.379856i
$944$	−526.403	−0.557630
$945$	0	0
$946$	0	0
$947$	−789.960	−0.834171	−0.417085	−	0.908867i	$-0.636948\pi$
$947$	−789.960	−0.834171	−0.417085	+	0.908867i	$0.636948\pi$
$948$	0	0
$949$	333.988	0.351937
$950$	667.270	0.702390
$951$	0	0
$952$	16.0229i	0.0168308i
$953$	− 1167.43i	− 1.22501i	−0.790468	−	0.612503i	$-0.790163\pi$
$953$	− 1167.43i	− 1.22501i	0.790468	−	0.612503i	$-0.209837\pi$
$954$	0	0
$955$	324.398	0.339683
$956$	− 794.330i	− 0.830889i
$957$	0	0
$958$	162.487	0.169611
$959$	− 13.8311i	− 0.0144224i
$960$	0	0
$961$	834.490	0.868356
$962$	2464.18	2.56151
$963$	0	0
$964$	27.8272i	0.0288664i
$965$	− 539.800i	− 0.559378i
$966$	0	0
$967$	− 840.359i	− 0.869037i	−0.900663	−	0.434518i	$-0.856919\pi$
$967$	− 840.359i	− 0.869037i	0.900663	−	0.434518i	$-0.143081\pi$
$968$	0	0
$969$	0	0
$970$	319.954i	0.329849i
$971$	−1250.52	−1.28787	−0.643937	−	0.765079i	$-0.722700\pi$
$971$	−1250.52	−1.28787	−0.643937	+	0.765079i	$0.722700\pi$
$972$	0	0
$973$	33.1279	0.0340472
$974$	44.8477i	0.0460448i
$975$	0	0
$976$	1266.38i	1.29752i
$977$	−114.574	−0.117271	−0.0586356	−	0.998279i	$-0.518675\pi$
$977$	−114.574	−0.117271	−0.0586356	+	0.998279i	$0.518675\pi$
$978$	0	0
$979$	0	0
$980$	−266.150	−0.271582
$981$	0	0
$982$	−1056.82	−1.07619
$983$	854.035	0.868804	0.434402	−	0.900719i	$-0.356960\pi$
$983$	854.035	0.868804	0.434402	+	0.900719i	$0.356960\pi$
$984$	0	0
$985$	− 7.33977i	− 0.00745154i
$986$	1922.36i	1.94966i
$987$	0	0
$988$	−598.367	−0.605635
$989$	463.073i	0.468224i
$990$	0	0
$991$	1236.21	1.24743	0.623716	−	0.781651i	$-0.285622\pi$
$991$	1236.21	1.24743	0.623716	+	0.781651i	$0.285622\pi$
$992$	− 1310.91i	− 1.32148i
$993$	0	0
$994$	40.8270	0.0410734
$995$	−404.848	−0.406882
$996$	0	0
$997$	704.988i	0.707109i	0.935414	+	0.353555i	$0.115027\pi$
$997$	704.988i	0.707109i	−0.935414	+	0.353555i	$0.884973\pi$
$998$	− 706.458i	− 0.707874i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.m.604.4		16
3.2	odd	2		363.3.c.e.241.13		16
11.3	even	5		99.3.k.c.46.1		16
11.7	odd	10		99.3.k.c.28.1		16
11.10	odd	2	inner	1089.3.c.m.604.13		16
33.2	even	10		363.3.g.g.40.1		16
33.5	odd	10		363.3.g.g.118.1		16
33.8	even	10		363.3.g.f.112.1		16
33.14	odd	10		33.3.g.a.13.4	✓	16
33.17	even	10		363.3.g.a.118.4		16
33.20	odd	10		363.3.g.a.40.4		16
33.26	odd	10		363.3.g.f.94.1		16
33.29	even	10		33.3.g.a.28.4	yes	16
33.32	even	2		363.3.c.e.241.4		16
132.47	even	10		528.3.bf.b.145.3		16
132.95	odd	10		528.3.bf.b.193.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.13.4	✓	16	33.14	odd	10
33.3.g.a.28.4	yes	16	33.29	even	10
99.3.k.c.28.1		16	11.7	odd	10
99.3.k.c.46.1		16	11.3	even	5
363.3.c.e.241.4		16	33.32	even	2
363.3.c.e.241.13		16	3.2	odd	2
363.3.g.a.40.4		16	33.20	odd	10
363.3.g.a.118.4		16	33.17	even	10
363.3.g.f.94.1		16	33.26	odd	10
363.3.g.f.112.1		16	33.8	even	10
363.3.g.g.40.1		16	33.2	even	10
363.3.g.g.118.1		16	33.5	odd	10
528.3.bf.b.145.3		16	132.47	even	10
528.3.bf.b.193.3		16	132.95	odd	10
1089.3.c.m.604.4		16	1.1	even	1	trivial
1089.3.c.m.604.13		16	11.10	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.47556i q^{2} -2.12839 q^{4} +2.55350 q^{5} -0.170400i q^{7} -4.63328i q^{8} +O(q^{10})\)	\(q-2.47556i q^{2} -2.12839 q^{4} +2.55350 q^{5} -0.170400i q^{7} -4.63328i q^{8} -6.32135i q^{10} +19.2744i q^{13} -0.421835 q^{14} -19.9835 q^{16} -20.2947i q^{17} -14.5860i q^{19} -5.43486 q^{20} -7.74583 q^{23} -18.4796 q^{25} +47.7149 q^{26} +0.362678i q^{28} -38.2630i q^{29} -42.3732 q^{31} +30.9373i q^{32} -50.2408 q^{34} -0.435117i q^{35} +51.6438 q^{37} -36.1085 q^{38} -11.8311i q^{40} -46.2448i q^{41} -59.7836i q^{43} +19.1753i q^{46} -34.1325 q^{47} +48.9710 q^{49} +45.7474i q^{50} -41.0234i q^{52} +15.2253 q^{53} -0.789510 q^{56} -94.7223 q^{58} +26.3418 q^{59} -63.3714i q^{61} +104.897i q^{62} -3.34707 q^{64} +49.2172i q^{65} -2.91469 q^{67} +43.1951i q^{68} -1.07716 q^{70} -96.7843 q^{71} -17.3281i q^{73} -127.847i q^{74} +31.0447i q^{76} -52.7777i q^{79} -51.0280 q^{80} -114.482 q^{82} +23.4757i q^{83} -51.8227i q^{85} -147.998 q^{86} -97.1861 q^{89} +3.28435 q^{91} +16.4862 q^{92} +84.4969i q^{94} -37.2454i q^{95} -50.6148 q^{97} -121.230i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) \) 16 * q - 20 * q^4 + 4 * q^5	\( 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) \) 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * q^97

Introduction

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