LMFDB - Embedded newform 1287.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(1,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1287.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$10.2767467401$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.8468.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 4 $ x^4 - x^3 - 5x^2 + 3x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.27841$ of defining polynomial
Character	$\chi$	$=$	1287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0872450 q^{2} -1.99239 q^{4} +0.477194 q^{5} +0.435561 q^{7} -0.348316 q^{8} +O(q^{10})$	\(q+0.0872450 q^{2} -1.99239 q^{4} +0.477194 q^{5} +0.435561 q^{7} -0.348316 q^{8} +0.0416328 q^{10} +1.00000 q^{11} +1.00000 q^{13} +0.0380005 q^{14} +3.95439 q^{16} -3.29509 q^{17} +4.43556 q^{19} -0.950756 q^{20} +0.0872450 q^{22} -3.16688 q^{23} -4.77229 q^{25} +0.0872450 q^{26} -0.867807 q^{28} -6.02641 q^{29} +10.6365 q^{31} +1.04163 q^{32} -0.287480 q^{34} +0.207847 q^{35} +3.42795 q^{37} +0.386981 q^{38} -0.166214 q^{40} +7.54922 q^{41} +12.4240 q^{43} -1.99239 q^{44} -0.276294 q^{46} +3.25346 q^{47} -6.81029 q^{49} -0.416358 q^{50} -1.99239 q^{52} +9.81029 q^{53} +0.477194 q^{55} -0.151713 q^{56} -0.525774 q^{58} +12.9239 q^{59} +10.3671 q^{61} +0.927978 q^{62} -7.81790 q^{64} +0.477194 q^{65} +7.50758 q^{67} +6.56510 q^{68} +0.0181336 q^{70} -14.5416 q^{71} +0.344337 q^{73} +0.299072 q^{74} -8.83736 q^{76} +0.435561 q^{77} -7.55285 q^{79} +1.88701 q^{80} +0.658632 q^{82} +9.03039 q^{83} -1.57240 q^{85} +1.08393 q^{86} -0.348316 q^{88} -5.90514 q^{89} +0.435561 q^{91} +6.30965 q^{92} +0.283848 q^{94} +2.11662 q^{95} +12.8906 q^{97} -0.594164 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{4} + 2 q^{7}+O(q^{10}) $ 4 * q + 2 * q^2 + 8 * q^4 + 2 * q^7	$ 4 q + 2 q^{2} + 8 q^{4} + 2 q^{7} - 2 q^{10} + 4 q^{11} + 4 q^{13} - 12 q^{14} + 12 q^{16} + 8 q^{17} + 18 q^{19} + 10 q^{20} + 2 q^{22} + 4 q^{25} + 2 q^{26} - 6 q^{28} + 10 q^{29} + 12 q^{31} + 2 q^{32} + 36 q^{34} - 22 q^{35} - 2 q^{37} - 4 q^{38} + 20 q^{40} - 2 q^{41} + 28 q^{43} + 8 q^{44} - 30 q^{46} - 6 q^{47} + 8 q^{49} + 36 q^{50} + 8 q^{52} + 4 q^{53} - 48 q^{56} - 6 q^{58} - 16 q^{59} - 10 q^{61} + 34 q^{62} - 12 q^{64} + 34 q^{68} - 58 q^{70} - 10 q^{71} - 6 q^{73} - 14 q^{74} + 26 q^{76} + 2 q^{77} - 8 q^{79} + 48 q^{80} + 12 q^{82} + 8 q^{83} - 18 q^{85} + 8 q^{86} - 6 q^{89} + 2 q^{91} + 28 q^{92} - 46 q^{94} - 22 q^{95} + 10 q^{97} + 34 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^4 + 2 * q^7 - 2 * q^10 + 4 * q^11 + 4 * q^13 - 12 * q^14 + 12 * q^16 + 8 * q^17 + 18 * q^19 + 10 * q^20 + 2 * q^22 + 4 * q^25 + 2 * q^26 - 6 * q^28 + 10 * q^29 + 12 * q^31 + 2 * q^32 + 36 * q^34 - 22 * q^35 - 2 * q^37 - 4 * q^38 + 20 * q^40 - 2 * q^41 + 28 * q^43 + 8 * q^44 - 30 * q^46 - 6 * q^47 + 8 * q^49 + 36 * q^50 + 8 * q^52 + 4 * q^53 - 48 * q^56 - 6 * q^58 - 16 * q^59 - 10 * q^61 + 34 * q^62 - 12 * q^64 + 34 * q^68 - 58 * q^70 - 10 * q^71 - 6 * q^73 - 14 * q^74 + 26 * q^76 + 2 * q^77 - 8 * q^79 + 48 * q^80 + 12 * q^82 + 8 * q^83 - 18 * q^85 + 8 * q^86 - 6 * q^89 + 2 * q^91 + 28 * q^92 - 46 * q^94 - 22 * q^95 + 10 * q^97 + 34 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	0.0872450	0.0616916	0.0308458	−	0.999524i	$-0.490180\pi$
$2$	0.0872450	0.0616916	0.0308458	+	0.999524i	$0.490180\pi$
$3$	0	0
$3$	0	0
$4$	−1.99239	−0.996194
$5$	0.477194	0.213408	0.106704	−	0.994291i	$-0.465970\pi$
$5$	0.477194	0.213408	0.106704	+	0.994291i	$0.465970\pi$
$6$	0	0
$7$	0.435561	0.164627	0.0823133	−	0.996607i	$-0.473769\pi$
$7$	0.435561	0.164627	0.0823133	+	0.996607i	$0.473769\pi$
$8$	−0.348316	−0.123148
$9$	0	0
$10$	0.0416328	0.0131654
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.0380005	0.0101561
$15$	0	0
$16$	3.95439	0.988597
$17$	−3.29509	−0.799177	−0.399589	−	0.916695i	$-0.630847\pi$
$17$	−3.29509	−0.799177	−0.399589	+	0.916695i	$0.630847\pi$
$18$	0	0
$19$	4.43556	1.01759	0.508794	−	0.860888i	$-0.330092\pi$
$19$	4.43556	1.01759	0.508794	+	0.860888i	$0.330092\pi$
$20$	−0.950756	−0.212595
$21$	0	0
$22$	0.0872450	0.0186007
$23$	−3.16688	−0.660340	−0.330170	−	0.943922i	$-0.607106\pi$
$23$	−3.16688	−0.660340	−0.330170	+	0.943922i	$0.607106\pi$
$24$	0	0
$25$	−4.77229	−0.954457
$26$	0.0872450	0.0171102
$27$	0	0
$28$	−0.867807	−0.164000
$29$	−6.02641	−1.11908	−0.559538	−	0.828805i	$-0.689021\pi$
$29$	−6.02641	−1.11908	−0.559538	+	0.828805i	$0.689021\pi$
$30$	0	0
$31$	10.6365	1.91036	0.955182	−	0.296018i	$-0.0956588\pi$
$31$	10.6365	1.91036	0.955182	+	0.296018i	$0.0956588\pi$
$32$	1.04163	0.184136
$33$	0	0
$34$	−0.287480	−0.0493025
$35$	0.207847	0.0351326
$36$	0	0
$37$	3.42795	0.563551	0.281776	−	0.959480i	$-0.409077\pi$
$37$	3.42795	0.563551	0.281776	+	0.959480i	$0.409077\pi$
$38$	0.386981	0.0627766
$39$	0	0
$40$	−0.166214	−0.0262808
$41$	7.54922	1.17899	0.589495	−	0.807772i	$-0.299327\pi$
$41$	7.54922	1.17899	0.589495	+	0.807772i	$0.299327\pi$
$42$	0	0
$43$	12.4240	1.89464	0.947319	−	0.320292i	$-0.103781\pi$
$43$	12.4240	1.89464	0.947319	+	0.320292i	$0.103781\pi$
$44$	−1.99239	−0.300364
$45$	0	0
$46$	−0.276294	−0.0407374
$47$	3.25346	0.474566	0.237283	−	0.971441i	$-0.423743\pi$
$47$	3.25346	0.474566	0.237283	+	0.971441i	$0.423743\pi$
$48$	0	0
$49$	−6.81029	−0.972898
$50$	−0.416358	−0.0588819
$51$	0	0
$52$	−1.99239	−0.276295
$53$	9.81029	1.34755	0.673773	−	0.738938i	$-0.264672\pi$
$53$	9.81029	1.34755	0.673773	+	0.738938i	$0.264672\pi$
$54$	0	0
$55$	0.477194	0.0643448
$56$	−0.151713	−0.0202735
$57$	0	0
$58$	−0.525774	−0.0690375
$59$	12.9239	1.68255	0.841277	−	0.540604i	$-0.181804\pi$
$59$	12.9239	1.68255	0.841277	+	0.540604i	$0.181804\pi$
$60$	0	0
$61$	10.3671	1.32737	0.663686	−	0.748011i	$-0.268991\pi$
$61$	10.3671	1.32737	0.663686	+	0.748011i	$0.268991\pi$
$62$	0.927978	0.117853
$63$	0	0
$64$	−7.81790	−0.977237
$65$	0.477194	0.0591886
$66$	0	0
$67$	7.50758	0.917197	0.458599	−	0.888644i	$-0.348352\pi$
$67$	7.50758	0.917197	0.458599	+	0.888644i	$0.348352\pi$
$68$	6.56510	0.796136
$69$	0	0
$70$	0.0181336	0.00216738
$71$	−14.5416	−1.72577	−0.862885	−	0.505400i	$-0.831345\pi$
$71$	−14.5416	−1.72577	−0.862885	+	0.505400i	$0.831345\pi$
$72$	0	0
$73$	0.344337	0.0403016	0.0201508	−	0.999797i	$-0.493585\pi$
$73$	0.344337	0.0403016	0.0201508	+	0.999797i	$0.493585\pi$
$74$	0.299072	0.0347664
$75$	0	0
$76$	−8.83736	−1.01371
$77$	0.435561	0.0496368
$78$	0	0
$79$	−7.55285	−0.849762	−0.424881	−	0.905249i	$-0.639684\pi$
$79$	−7.55285	−0.849762	−0.424881	+	0.905249i	$0.639684\pi$
$80$	1.88701	0.210974
$81$	0	0
$82$	0.658632	0.0727337
$83$	9.03039	0.991214	0.495607	−	0.868547i	$-0.334946\pi$
$83$	9.03039	0.991214	0.495607	+	0.868547i	$0.334946\pi$
$84$	0	0
$85$	−1.57240	−0.170550
$86$	1.08393	0.116883
$87$	0	0
$88$	−0.348316	−0.0371306
$89$	−5.90514	−0.625944	−0.312972	−	0.949762i	$-0.601325\pi$
$89$	−5.90514	−0.625944	−0.312972	+	0.949762i	$0.601325\pi$
$90$	0	0
$91$	0.435561	0.0456592
$92$	6.30965	0.657827
$93$	0	0
$94$	0.283848	0.0292767
$95$	2.11662	0.217161
$96$	0	0
$97$	12.8906	1.30884	0.654420	−	0.756131i	$-0.272913\pi$
$97$	12.8906	1.30884	0.654420	+	0.756131i	$0.272913\pi$
$98$	−0.594164	−0.0600196
$99$	0	0
$100$	9.50825	0.950825
$101$	6.07499	0.604484	0.302242	−	0.953231i	$-0.402265\pi$
$101$	6.07499	0.604484	0.302242	+	0.953231i	$0.402265\pi$
$102$	0	0
$103$	−1.98478	−0.195566	−0.0977829	−	0.995208i	$-0.531175\pi$
$103$	−1.98478	−0.195566	−0.0977829	+	0.995208i	$0.531175\pi$
$104$	−0.348316	−0.0341552
$105$	0	0
$106$	0.855899	0.0831322
$107$	−16.5750	−1.60236	−0.801181	−	0.598422i	$-0.795795\pi$
$107$	−16.5750	−1.60236	−0.801181	+	0.598422i	$0.795795\pi$
$108$	0	0
$109$	5.21546	0.499550	0.249775	−	0.968304i	$-0.419643\pi$
$109$	5.21546	0.499550	0.249775	+	0.968304i	$0.419643\pi$
$110$	0.0416328	0.00396953
$111$	0	0
$112$	1.72238	0.162749
$113$	17.2881	1.62633	0.813166	−	0.582032i	$-0.197742\pi$
$113$	17.2881	1.62633	0.813166	+	0.582032i	$0.197742\pi$
$114$	0	0
$115$	−1.51121	−0.140922
$116$	12.0069	1.11482
$117$	0	0
$118$	1.12755	0.103799
$119$	−1.43521	−0.131566
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.904479	0.0818877
$123$	0	0
$124$	−21.1920	−1.90309
$125$	−4.66328	−0.417096
$126$	0	0
$127$	−4.37539	−0.388253	−0.194127	−	0.980977i	$-0.562187\pi$
$127$	−4.37539	−0.388253	−0.194127	+	0.980977i	$0.562187\pi$
$128$	−2.76534	−0.244424
$129$	0	0
$130$	0.0416328	0.00365144
$131$	−15.8936	−1.38863	−0.694313	−	0.719673i	$-0.744292\pi$
$131$	−15.8936	−1.38863	−0.694313	+	0.719673i	$0.744292\pi$
$132$	0	0
$133$	1.93196	0.167522
$134$	0.654999	0.0565833
$135$	0	0
$136$	1.14773	0.0984173
$137$	−7.01880	−0.599656	−0.299828	−	0.953993i	$-0.596929\pi$
$137$	−7.01880	−0.599656	−0.299828	+	0.953993i	$0.596929\pi$
$138$	0	0
$139$	7.86714	0.667282	0.333641	−	0.942700i	$-0.391723\pi$
$139$	7.86714	0.667282	0.333641	+	0.942700i	$0.391723\pi$
$140$	−0.414112	−0.0349989
$141$	0	0
$142$	−1.26868	−0.106465
$143$	1.00000	0.0836242
$144$	0	0
$145$	−2.87577	−0.238819
$146$	0.0300417	0.00248627
$147$	0	0
$148$	−6.82981	−0.561407
$149$	10.6781	0.874783	0.437392	−	0.899271i	$-0.355902\pi$
$149$	10.6781	0.874783	0.437392	+	0.899271i	$0.355902\pi$
$150$	0	0
$151$	−12.3291	−1.00333	−0.501665	−	0.865062i	$-0.667279\pi$
$151$	−12.3291	−1.00333	−0.501665	+	0.865062i	$0.667279\pi$
$152$	−1.54498	−0.125314
$153$	0	0
$154$	0.0380005	0.00306217
$155$	5.07565	0.407686
$156$	0	0
$157$	2.56741	0.204901	0.102451	−	0.994738i	$-0.467332\pi$
$157$	2.56741	0.204901	0.102451	+	0.994738i	$0.467332\pi$
$158$	−0.658948	−0.0524231
$159$	0	0
$160$	0.497061	0.0392961
$161$	−1.37937	−0.108710
$162$	0	0
$163$	−1.47423	−0.115470	−0.0577351	−	0.998332i	$-0.518388\pi$
$163$	−1.47423	−0.115470	−0.0577351	+	0.998332i	$0.518388\pi$
$164$	−15.0410	−1.17450
$165$	0	0
$166$	0.787857	0.0611495
$167$	19.0984	1.47788	0.738940	−	0.673771i	$-0.235326\pi$
$167$	19.0984	1.47788	0.738940	+	0.673771i	$0.235326\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−0.137184	−0.0105215
$171$	0	0
$172$	−24.7534	−1.88743
$173$	−8.67446	−0.659507	−0.329754	−	0.944067i	$-0.606966\pi$
$173$	−8.67446	−0.659507	−0.329754	+	0.944067i	$0.606966\pi$
$174$	0	0
$175$	−2.07862	−0.157129
$176$	3.95439	0.298073
$177$	0	0
$178$	−0.515194	−0.0386155
$179$	−18.4550	−1.37939	−0.689697	−	0.724098i	$-0.742256\pi$
$179$	−18.4550	−1.37939	−0.689697	+	0.724098i	$0.742256\pi$
$180$	0	0
$181$	6.99239	0.519740	0.259870	−	0.965644i	$-0.416320\pi$
$181$	6.99239	0.519740	0.259870	+	0.965644i	$0.416320\pi$
$182$	0.0380005	0.00281679
$183$	0	0
$184$	1.10307	0.0813197
$185$	1.63580	0.120266
$186$	0	0
$187$	−3.29509	−0.240961
$188$	−6.48215	−0.472760
$189$	0	0
$190$	0.184665	0.0133970
$191$	−21.4461	−1.55178	−0.775892	−	0.630866i	$-0.782700\pi$
$191$	−21.4461	−1.55178	−0.775892	+	0.630866i	$0.782700\pi$
$192$	0	0
$193$	−21.6100	−1.55552	−0.777761	−	0.628561i	$-0.783644\pi$
$193$	−21.6100	−1.55552	−0.777761	+	0.628561i	$0.783644\pi$
$194$	1.12464	0.0807444
$195$	0	0
$196$	13.5687	0.969195
$197$	8.42760	0.600442	0.300221	−	0.953870i	$-0.402940\pi$
$197$	8.42760	0.600442	0.300221	+	0.953870i	$0.402940\pi$
$198$	0	0
$199$	13.0072	0.922056	0.461028	−	0.887385i	$-0.347481\pi$
$199$	13.0072	0.922056	0.461028	+	0.887385i	$0.347481\pi$
$200$	1.66226	0.117540
$201$	0	0
$202$	0.530013	0.0372916
$203$	−2.62487	−0.184230
$204$	0	0
$205$	3.60244	0.251605
$206$	−0.173162	−0.0120648
$207$	0	0
$208$	3.95439	0.274187
$209$	4.43556	0.306814
$210$	0	0
$211$	7.39358	0.508995	0.254498	−	0.967073i	$-0.418090\pi$
$211$	7.39358	0.508995	0.254498	+	0.967073i	$0.418090\pi$
$212$	−19.5459	−1.34242
$213$	0	0
$214$	−1.44608	−0.0988522
$215$	5.92864	0.404330
$216$	0	0
$217$	4.63283	0.314497
$218$	0.455023	0.0308180
$219$	0	0
$220$	−0.950756	−0.0640999
$221$	−3.29509	−0.221652
$222$	0	0
$223$	−15.0826	−1.01001	−0.505003	−	0.863118i	$-0.668509\pi$
$223$	−15.0826	−1.01001	−0.505003	+	0.863118i	$0.668509\pi$
$224$	0.453695	0.0303138
$225$	0	0
$226$	1.50830	0.100331
$227$	−21.3899	−1.41970	−0.709850	−	0.704352i	$-0.751237\pi$
$227$	−21.3899	−1.41970	−0.709850	+	0.704352i	$0.751237\pi$
$228$	0	0
$229$	3.69366	0.244084	0.122042	−	0.992525i	$-0.461056\pi$
$229$	3.69366	0.244084	0.122042	+	0.992525i	$0.461056\pi$
$230$	−0.131846	−0.00869367
$231$	0	0
$232$	2.09910	0.137812
$233$	24.6398	1.61421	0.807103	−	0.590411i	$-0.201034\pi$
$233$	24.6398	1.61421	0.807103	+	0.590411i	$0.201034\pi$
$234$	0	0
$235$	1.55253	0.101276
$236$	−25.7495	−1.67615
$237$	0	0
$238$	−0.125215	−0.00811650
$239$	15.2683	0.987623	0.493811	−	0.869569i	$-0.335603\pi$
$239$	15.2683	0.987623	0.493811	+	0.869569i	$0.335603\pi$
$240$	0	0
$241$	2.50825	0.161570	0.0807852	−	0.996732i	$-0.474257\pi$
$241$	2.50825	0.161570	0.0807852	+	0.996732i	$0.474257\pi$
$242$	0.0872450	0.00560832
$243$	0	0
$244$	−20.6553	−1.32232
$245$	−3.24983	−0.207624
$246$	0	0
$247$	4.43556	0.282228
$248$	−3.70485	−0.235258
$249$	0	0
$250$	−0.406848	−0.0257313
$251$	−11.7343	−0.740662	−0.370331	−	0.928900i	$-0.620756\pi$
$251$	−11.7343	−0.740662	−0.370331	+	0.928900i	$0.620756\pi$
$252$	0	0
$253$	−3.16688	−0.199100
$254$	−0.381731	−0.0239519
$255$	0	0
$256$	15.3945	0.962158
$257$	−12.2273	−0.762719	−0.381359	−	0.924427i	$-0.624544\pi$
$257$	−12.2273	−0.762719	−0.381359	+	0.924427i	$0.624544\pi$
$258$	0	0
$259$	1.49308	0.0927756
$260$	−0.950756	−0.0589634
$261$	0	0
$262$	−1.38663	−0.0856665
$263$	−17.7191	−1.09260	−0.546302	−	0.837588i	$-0.683965\pi$
$263$	−17.7191	−1.09260	−0.546302	+	0.837588i	$0.683965\pi$
$264$	0	0
$265$	4.68141	0.287577
$266$	0.168554	0.0103347
$267$	0	0
$268$	−14.9580	−0.913706
$269$	−17.7038	−1.07942	−0.539711	−	0.841850i	$-0.681467\pi$
$269$	−17.7038	−1.07942	−0.539711	+	0.841850i	$0.681467\pi$
$270$	0	0
$271$	9.45666	0.574451	0.287226	−	0.957863i	$-0.407267\pi$
$271$	9.45666	0.574451	0.287226	+	0.957863i	$0.407267\pi$
$272$	−13.0301	−0.790064
$273$	0	0
$274$	−0.612355	−0.0369937
$275$	−4.77229	−0.287780
$276$	0	0
$277$	−18.4917	−1.11106	−0.555529	−	0.831497i	$-0.687484\pi$
$277$	−18.4917	−1.11106	−0.555529	+	0.831497i	$0.687484\pi$
$278$	0.686369	0.0411657
$279$	0	0
$280$	−0.0723965	−0.00432652
$281$	15.0198	0.896007	0.448003	−	0.894032i	$-0.352135\pi$
$281$	15.0198	0.896007	0.448003	+	0.894032i	$0.352135\pi$
$282$	0	0
$283$	−12.2921	−0.730691	−0.365345	−	0.930872i	$-0.619049\pi$
$283$	−12.2921	−0.730691	−0.365345	+	0.930872i	$0.619049\pi$
$284$	28.9725	1.71920
$285$	0	0
$286$	0.0872450	0.00515891
$287$	3.28814	0.194093
$288$	0	0
$289$	−6.14237	−0.361316
$290$	−0.250896	−0.0147331
$291$	0	0
$292$	−0.686052	−0.0401482
$293$	30.4579	1.77937	0.889686	−	0.456573i	$-0.150923\pi$
$293$	30.4579	1.77937	0.889686	+	0.456573i	$0.150923\pi$
$294$	0	0
$295$	6.16723	0.359070
$296$	−1.19401	−0.0694004
$297$	0	0
$298$	0.931611	0.0539668
$299$	−3.16688	−0.183145
$300$	0	0
$301$	5.41140	0.311908
$302$	−1.07565	−0.0618969
$303$	0	0
$304$	17.5399	1.00598
$305$	4.94712	0.283271
$306$	0	0
$307$	−1.82883	−0.104377	−0.0521883	−	0.998637i	$-0.516620\pi$
$307$	−1.82883	−0.104377	−0.0521883	+	0.998637i	$0.516620\pi$
$308$	−0.867807	−0.0494479
$309$	0	0
$310$	0.442826	0.0251508
$311$	0.144448	0.00819092	0.00409546	−	0.999992i	$-0.498696\pi$
$311$	0.144448	0.00819092	0.00409546	+	0.999992i	$0.498696\pi$
$312$	0	0
$313$	6.69623	0.378493	0.189247	−	0.981930i	$-0.439395\pi$
$313$	6.69623	0.378493	0.189247	+	0.981930i	$0.439395\pi$
$314$	0.223994	0.0126407
$315$	0	0
$316$	15.0482	0.846528
$317$	21.0175	1.18046	0.590229	−	0.807236i	$-0.299037\pi$
$317$	21.0175	1.18046	0.590229	+	0.807236i	$0.299037\pi$
$318$	0	0
$319$	−6.02641	−0.337414
$320$	−3.73065	−0.208550
$321$	0	0
$322$	−0.120343	−0.00670646
$323$	−14.6156	−0.813233
$324$	0	0
$325$	−4.77229	−0.264719
$326$	−0.128619	−0.00712354
$327$	0	0
$328$	−2.62951	−0.145191
$329$	1.41708	0.0781262
$330$	0	0
$331$	2.00363	0.110130	0.0550648	−	0.998483i	$-0.482463\pi$
$331$	2.00363	0.110130	0.0550648	+	0.998483i	$0.482463\pi$
$332$	−17.9920	−0.987442
$333$	0	0
$334$	1.66624	0.0911728
$335$	3.58257	0.195737
$336$	0	0
$337$	−18.5750	−1.01184	−0.505921	−	0.862580i	$-0.668847\pi$
$337$	−18.5750	−1.01184	−0.505921	+	0.862580i	$0.668847\pi$
$338$	0.0872450	0.00474550
$339$	0	0
$340$	3.13283	0.169901
$341$	10.6365	0.575997
$342$	0	0
$343$	−6.01522	−0.324792
$344$	−4.32747	−0.233321
$345$	0	0
$346$	−0.756804	−0.0406860
$347$	31.3063	1.68061	0.840305	−	0.542115i	$-0.182376\pi$
$347$	31.3063	1.68061	0.840305	+	0.542115i	$0.182376\pi$
$348$	0	0
$349$	−28.9847	−1.55152	−0.775758	−	0.631030i	$-0.782632\pi$
$349$	−28.9847	−1.55152	−0.775758	+	0.631030i	$0.782632\pi$
$350$	−0.181349	−0.00969354
$351$	0	0
$352$	1.04163	0.0555192
$353$	1.11305	0.0592416	0.0296208	−	0.999561i	$-0.490570\pi$
$353$	1.11305	0.0592416	0.0296208	+	0.999561i	$0.490570\pi$
$354$	0	0
$355$	−6.93916	−0.368293
$356$	11.7653	0.623562
$357$	0	0
$358$	−1.61011	−0.0850969
$359$	−14.0575	−0.741924	−0.370962	−	0.928648i	$-0.620972\pi$
$359$	−14.0575	−0.741924	−0.370962	+	0.928648i	$0.620972\pi$
$360$	0	0
$361$	0.674202	0.0354843
$362$	0.610051	0.0320636
$363$	0	0
$364$	−0.867807	−0.0454854
$365$	0.164315	0.00860066
$366$	0	0
$367$	11.0456	0.576576	0.288288	−	0.957544i	$-0.406914\pi$
$367$	11.0456	0.576576	0.288288	+	0.957544i	$0.406914\pi$
$368$	−12.5231	−0.652810
$369$	0	0
$370$	0.142715	0.00741941
$371$	4.27298	0.221842
$372$	0	0
$373$	8.41702	0.435817	0.217908	−	0.975969i	$-0.430077\pi$
$373$	8.41702	0.435817	0.217908	+	0.975969i	$0.430077\pi$
$374$	−0.287480	−0.0148653
$375$	0	0
$376$	−1.13323	−0.0584420
$377$	−6.02641	−0.310376
$378$	0	0
$379$	−18.4953	−0.950041	−0.475021	−	0.879975i	$-0.657559\pi$
$379$	−18.4953	−0.950041	−0.475021	+	0.879975i	$0.657559\pi$
$380$	−4.21713	−0.216334
$381$	0	0
$382$	−1.87106	−0.0957320
$383$	6.93916	0.354575	0.177287	−	0.984159i	$-0.443268\pi$
$383$	6.93916	0.354575	0.177287	+	0.984159i	$0.443268\pi$
$384$	0	0
$385$	0.207847	0.0105929
$386$	−1.88536	−0.0959625
$387$	0	0
$388$	−25.6830	−1.30386
$389$	−7.13684	−0.361852	−0.180926	−	0.983497i	$-0.557909\pi$
$389$	−7.13684	−0.361852	−0.180926	+	0.983497i	$0.557909\pi$
$390$	0	0
$391$	10.4352	0.527729
$392$	2.37213	0.119811
$393$	0	0
$394$	0.735266	0.0370422
$395$	−3.60417	−0.181346
$396$	0	0
$397$	−37.2091	−1.86747	−0.933736	−	0.357962i	$-0.883472\pi$
$397$	−37.2091	−1.86747	−0.933736	+	0.357962i	$0.883472\pi$
$398$	1.13481	0.0568831
$399$	0	0
$400$	−18.8715	−0.943573
$401$	6.97022	0.348076	0.174038	−	0.984739i	$-0.444318\pi$
$401$	6.97022	0.348076	0.174038	+	0.984739i	$0.444318\pi$
$402$	0	0
$403$	10.6365	0.529840
$404$	−12.1037	−0.602184
$405$	0	0
$406$	−0.229007	−0.0113654
$407$	3.42795	0.169917
$408$	0	0
$409$	8.66287	0.428351	0.214176	−	0.976795i	$-0.431293\pi$
$409$	8.66287	0.428351	0.214176	+	0.976795i	$0.431293\pi$
$410$	0.314295	0.0155219
$411$	0	0
$412$	3.95445	0.194822
$413$	5.62917	0.276993
$414$	0	0
$415$	4.30925	0.211533
$416$	1.04163	0.0510702
$417$	0	0
$418$	0.386981	0.0189278
$419$	13.3869	0.653994	0.326997	−	0.945025i	$-0.393963\pi$
$419$	13.3869	0.653994	0.326997	+	0.945025i	$0.393963\pi$
$420$	0	0
$421$	23.6053	1.15045	0.575227	−	0.817994i	$-0.304914\pi$
$421$	23.6053	1.15045	0.575227	+	0.817994i	$0.304914\pi$
$422$	0.645053	0.0314007
$423$	0	0
$424$	−3.41708	−0.165948
$425$	15.7251	0.762780
$426$	0	0
$427$	4.51551	0.218521
$428$	33.0238	1.59626
$429$	0	0
$430$	0.517245	0.0249437
$431$	34.4005	1.65701	0.828506	−	0.559980i	$-0.189191\pi$
$431$	34.4005	1.65701	0.828506	+	0.559980i	$0.189191\pi$
$432$	0	0
$433$	7.55980	0.363301	0.181650	−	0.983363i	$-0.441856\pi$
$433$	7.55980	0.363301	0.181650	+	0.983363i	$0.441856\pi$
$434$	0.404191	0.0194018
$435$	0	0
$436$	−10.3912	−0.497649
$437$	−14.0469	−0.671954
$438$	0	0
$439$	−25.4239	−1.21342	−0.606709	−	0.794924i	$-0.707510\pi$
$439$	−25.4239	−1.21342	−0.606709	+	0.794924i	$0.707510\pi$
$440$	−0.166214	−0.00792396
$441$	0	0
$442$	−0.287480	−0.0136740
$443$	−5.04428	−0.239661	−0.119831	−	0.992794i	$-0.538235\pi$
$443$	−5.04428	−0.239661	−0.119831	+	0.992794i	$0.538235\pi$
$444$	0	0
$445$	−2.81790	−0.133581
$446$	−1.31588	−0.0623088
$447$	0	0
$448$	−3.40517	−0.160879
$449$	−4.97615	−0.234839	−0.117420	−	0.993082i	$-0.537462\pi$
$449$	−4.97615	−0.234839	−0.117420	+	0.993082i	$0.537462\pi$
$450$	0	0
$451$	7.54922	0.355479
$452$	−34.4447	−1.62014
$453$	0	0
$454$	−1.86617	−0.0875835
$455$	0.207847	0.00974402
$456$	0	0
$457$	12.6827	0.593273	0.296637	−	0.954990i	$-0.404135\pi$
$457$	12.6827	0.593273	0.296637	+	0.954990i	$0.404135\pi$
$458$	0.322254	0.0150579
$459$	0	0
$460$	3.01093	0.140385
$461$	−10.8605	−0.505826	−0.252913	−	0.967489i	$-0.581389\pi$
$461$	−10.8605	−0.505826	−0.252913	+	0.967489i	$0.581389\pi$
$462$	0	0
$463$	−36.3208	−1.68797	−0.843986	−	0.536365i	$-0.819797\pi$
$463$	−36.3208	−1.68797	−0.843986	+	0.536365i	$0.819797\pi$
$464$	−23.8308	−1.10632
$465$	0	0
$466$	2.14970	0.0995828
$467$	6.54492	0.302863	0.151431	−	0.988468i	$-0.451612\pi$
$467$	6.54492	0.302863	0.151431	+	0.988468i	$0.451612\pi$
$468$	0	0
$469$	3.27001	0.150995
$470$	0.135451	0.00624787
$471$	0	0
$472$	−4.50162	−0.207204
$473$	12.4240	0.571255
$474$	0	0
$475$	−21.1678	−0.971244
$476$	2.85950	0.131065
$477$	0	0
$478$	1.33208	0.0609280
$479$	2.42760	0.110920	0.0554600	−	0.998461i	$-0.482337\pi$
$479$	2.42760	0.110920	0.0554600	+	0.998461i	$0.482337\pi$
$480$	0	0
$481$	3.42795	0.156301
$482$	0.218832	0.00996753
$483$	0	0
$484$	−1.99239	−0.0905631
$485$	6.15131	0.279316
$486$	0	0
$487$	−27.1628	−1.23087	−0.615433	−	0.788189i	$-0.711019\pi$
$487$	−27.1628	−1.23087	−0.615433	+	0.788189i	$0.711019\pi$
$488$	−3.61103	−0.163464
$489$	0	0
$490$	−0.283531	−0.0128086
$491$	−23.0637	−1.04085	−0.520426	−	0.853907i	$-0.674227\pi$
$491$	−23.0637	−1.04085	−0.520426	+	0.853907i	$0.674227\pi$
$492$	0	0
$493$	19.8576	0.894340
$494$	0.386981	0.0174111
$495$	0	0
$496$	42.0607	1.88858
$497$	−6.33376	−0.284108
$498$	0	0
$499$	−10.0978	−0.452038	−0.226019	−	0.974123i	$-0.572571\pi$
$499$	−10.0978	−0.452038	−0.226019	+	0.974123i	$0.572571\pi$
$500$	9.29105	0.415509
$501$	0	0
$502$	−1.02376	−0.0456926
$503$	17.3034	0.771519	0.385760	−	0.922599i	$-0.373939\pi$
$503$	17.3034	0.771519	0.385760	+	0.922599i	$0.373939\pi$
$504$	0	0
$505$	2.89895	0.129001
$506$	−0.276294	−0.0122828
$507$	0	0
$508$	8.71747	0.386775
$509$	−29.3664	−1.30164	−0.650822	−	0.759230i	$-0.725576\pi$
$509$	−29.3664	−1.30164	−0.650822	+	0.759230i	$0.725576\pi$
$510$	0	0
$511$	0.149980	0.00663471
$512$	6.87377	0.303781
$513$	0	0
$514$	−1.06677	−0.0470533
$515$	−0.947123	−0.0417352
$516$	0	0
$517$	3.25346	0.143087
$518$	0.130264	0.00572347
$519$	0	0
$520$	−0.166214	−0.00728898
$521$	26.2881	1.15170	0.575851	−	0.817555i	$-0.304671\pi$
$521$	26.2881	1.15170	0.575851	+	0.817555i	$0.304671\pi$
$522$	0	0
$523$	41.7729	1.82660	0.913301	−	0.407286i	$-0.133525\pi$
$523$	41.7729	1.82660	0.913301	+	0.407286i	$0.133525\pi$
$524$	31.6661	1.38334
$525$	0	0
$526$	−1.54590	−0.0674045
$527$	−35.0481	−1.52672
$528$	0	0
$529$	−12.9709	−0.563951
$530$	0.408430	0.0177411
$531$	0	0
$532$	−3.84921	−0.166884
$533$	7.54922	0.326993
$534$	0	0
$535$	−7.90947	−0.341956
$536$	−2.61501	−0.112951
$537$	0	0
$538$	−1.54457	−0.0665912
$539$	−6.81029	−0.293340
$540$	0	0
$541$	−1.62057	−0.0696739	−0.0348369	−	0.999393i	$-0.511091\pi$
$541$	−1.62057	−0.0696739	−0.0348369	+	0.999393i	$0.511091\pi$
$542$	0.825047	0.0354388
$543$	0	0
$544$	−3.43228	−0.147158
$545$	2.48879	0.106608
$546$	0	0
$547$	−5.74547	−0.245659	−0.122829	−	0.992428i	$-0.539197\pi$
$547$	−5.74547	−0.245659	−0.122829	+	0.992428i	$0.539197\pi$
$548$	13.9842	0.597374
$549$	0	0
$550$	−0.416358	−0.0177536
$551$	−26.7305	−1.13876
$552$	0	0
$553$	−3.28973	−0.139893
$554$	−1.61331	−0.0685429
$555$	0	0
$556$	−15.6744	−0.664743
$557$	15.5399	0.658448	0.329224	−	0.944252i	$-0.393213\pi$
$557$	15.5399	0.658448	0.329224	+	0.944252i	$0.393213\pi$
$558$	0	0
$559$	12.4240	0.525478
$560$	0.821908	0.0347320
$561$	0	0
$562$	1.31040	0.0552760
$563$	29.4533	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$563$	29.4533	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$564$	0	0
$565$	8.24980	0.347072
$566$	−1.07243	−0.0450775
$567$	0	0
$568$	5.06507	0.212526
$569$	−17.1206	−0.717733	−0.358866	−	0.933389i	$-0.616837\pi$
$569$	−17.1206	−0.717733	−0.358866	+	0.933389i	$0.616837\pi$
$570$	0	0
$571$	20.3036	0.849680	0.424840	−	0.905268i	$-0.360330\pi$
$571$	20.3036	0.849680	0.424840	+	0.905268i	$0.360330\pi$
$572$	−1.99239	−0.0833059
$573$	0	0
$574$	0.286874	0.0119739
$575$	15.1132	0.630266
$576$	0	0
$577$	−9.09843	−0.378773	−0.189386	−	0.981903i	$-0.560650\pi$
$577$	−9.09843	−0.378773	−0.189386	+	0.981903i	$0.560650\pi$
$578$	−0.535891	−0.0222901
$579$	0	0
$580$	5.72964	0.237910
$581$	3.93329	0.163180
$582$	0	0
$583$	9.81029	0.406301
$584$	−0.119938	−0.00496307
$585$	0	0
$586$	2.65730	0.109772
$587$	−36.1288	−1.49120	−0.745598	−	0.666396i	$-0.767836\pi$
$587$	−36.1288	−1.49120	−0.745598	+	0.666396i	$0.767836\pi$
$588$	0	0
$589$	47.1787	1.94396
$590$	0.538060	0.0221516
$591$	0	0
$592$	13.5554	0.557125
$593$	7.60203	0.312178	0.156089	−	0.987743i	$-0.450111\pi$
$593$	7.60203	0.312178	0.156089	+	0.987743i	$0.450111\pi$
$594$	0	0
$595$	−0.684875	−0.0280771
$596$	−21.2749	−0.871454
$597$	0	0
$598$	−0.276294	−0.0112985
$599$	−27.5373	−1.12514	−0.562572	−	0.826748i	$-0.690188\pi$
$599$	−27.5373	−1.12514	−0.562572	+	0.826748i	$0.690188\pi$
$600$	0	0
$601$	34.2028	1.39516	0.697581	−	0.716506i	$-0.254260\pi$
$601$	34.2028	1.39516	0.697581	+	0.716506i	$0.254260\pi$
$602$	0.472118	0.0192421
$603$	0	0
$604$	24.5644	0.999510
$605$	0.477194	0.0194007
$606$	0	0
$607$	26.5194	1.07639	0.538195	−	0.842820i	$-0.319106\pi$
$607$	26.5194	1.07639	0.538195	+	0.842820i	$0.319106\pi$
$608$	4.62023	0.187375
$609$	0	0
$610$	0.431612	0.0174755
$611$	3.25346	0.131621
$612$	0	0
$613$	−23.0938	−0.932749	−0.466375	−	0.884587i	$-0.654440\pi$
$613$	−23.0938	−0.932749	−0.466375	+	0.884587i	$0.654440\pi$
$614$	−0.159556	−0.00643915
$615$	0	0
$616$	−0.151713	−0.00611269
$617$	6.14344	0.247325	0.123663	−	0.992324i	$-0.460536\pi$
$617$	6.14344	0.247325	0.123663	+	0.992324i	$0.460536\pi$
$618$	0	0
$619$	11.1767	0.449231	0.224615	−	0.974447i	$-0.427887\pi$
$619$	11.1767	0.449231	0.224615	+	0.974447i	$0.427887\pi$
$620$	−10.1127	−0.406135
$621$	0	0
$622$	0.0126024	0.000505310 0
$623$	−2.57205	−0.103047
$624$	0	0
$625$	21.6361	0.865446
$626$	0.584213	0.0233498
$627$	0	0
$628$	−5.11527	−0.204122
$629$	−11.2954	−0.450377
$630$	0	0
$631$	−22.1325	−0.881079	−0.440540	−	0.897733i	$-0.645213\pi$
$631$	−22.1325	−0.881079	−0.440540	+	0.897733i	$0.645213\pi$
$632$	2.63078	0.104647
$633$	0	0
$634$	1.83367	0.0728243
$635$	−2.08791	−0.0828561
$636$	0	0
$637$	−6.81029	−0.269833
$638$	−0.525774	−0.0208156
$639$	0	0
$640$	−1.31960	−0.0521619
$641$	−11.3562	−0.448542	−0.224271	−	0.974527i	$-0.572000\pi$
$641$	−11.3562	−0.448542	−0.224271	+	0.974527i	$0.572000\pi$
$642$	0	0
$643$	19.8334	0.782152	0.391076	−	0.920358i	$-0.372103\pi$
$643$	19.8334	0.782152	0.391076	+	0.920358i	$0.372103\pi$
$644$	2.74824	0.108296
$645$	0	0
$646$	−1.27514	−0.0501696
$647$	−8.58425	−0.337482	−0.168741	−	0.985660i	$-0.553970\pi$
$647$	−8.58425	−0.337482	−0.168741	+	0.985660i	$0.553970\pi$
$648$	0	0
$649$	12.9239	0.507309
$650$	−0.416358	−0.0163309
$651$	0	0
$652$	2.93723	0.115031
$653$	−19.8783	−0.777899	−0.388950	−	0.921259i	$-0.627162\pi$
$653$	−19.8783	−0.777899	−0.388950	+	0.921259i	$0.627162\pi$
$654$	0	0
$655$	−7.58431	−0.296343
$656$	29.8525	1.16555
$657$	0	0
$658$	0.123633	0.00481972
$659$	29.9818	1.16792	0.583962	−	0.811781i	$-0.301502\pi$
$659$	29.9818	1.16792	0.583962	+	0.811781i	$0.301502\pi$
$660$	0	0
$661$	37.9919	1.47771	0.738857	−	0.673862i	$-0.235366\pi$
$661$	37.9919	1.47771	0.738857	+	0.673862i	$0.235366\pi$
$662$	0.174807	0.00679407
$663$	0	0
$664$	−3.14543	−0.122066
$665$	0.921918	0.0357505
$666$	0	0
$667$	19.0849	0.738970
$668$	−38.0515	−1.47226
$669$	0	0
$670$	0.312562	0.0120753
$671$	10.3671	0.400218
$672$	0	0
$673$	25.5859	0.986263	0.493132	−	0.869955i	$-0.335852\pi$
$673$	25.5859	0.986263	0.493132	+	0.869955i	$0.335852\pi$
$674$	−1.62057	−0.0624221
$675$	0	0
$676$	−1.99239	−0.0766303
$677$	24.4362	0.939158	0.469579	−	0.882890i	$-0.344406\pi$
$677$	24.4362	0.939158	0.469579	+	0.882890i	$0.344406\pi$
$678$	0	0
$679$	5.61464	0.215470
$680$	0.547691	0.0210030
$681$	0	0
$682$	0.927978	0.0355341
$683$	39.0934	1.49587	0.747933	−	0.663774i	$-0.231046\pi$
$683$	39.0934	1.49587	0.747933	+	0.663774i	$0.231046\pi$
$684$	0	0
$685$	−3.34933	−0.127971
$686$	−0.524798	−0.0200369
$687$	0	0
$688$	49.1292	1.87303
$689$	9.81029	0.373742
$690$	0	0
$691$	−40.6394	−1.54599	−0.772997	−	0.634409i	$-0.781243\pi$
$691$	−40.6394	−1.54599	−0.772997	+	0.634409i	$0.781243\pi$
$692$	17.2829	0.656997
$693$	0	0
$694$	2.73132	0.103679
$695$	3.75415	0.142403
$696$	0	0
$697$	−24.8754	−0.942221
$698$	−2.52877	−0.0957155
$699$	0	0
$700$	4.14142	0.156531
$701$	4.06905	0.153686	0.0768430	−	0.997043i	$-0.475516\pi$
$701$	4.06905	0.153686	0.0768430	+	0.997043i	$0.475516\pi$
$702$	0	0
$703$	15.2049	0.573463
$704$	−7.81790	−0.294648
$705$	0	0
$706$	0.0971079	0.00365471
$707$	2.64603	0.0995142
$708$	0	0
$709$	30.7457	1.15468	0.577340	−	0.816504i	$-0.304091\pi$
$709$	30.7457	1.15468	0.577340	+	0.816504i	$0.304091\pi$
$710$	−0.605408	−0.0227205
$711$	0	0
$712$	2.05686	0.0770839
$713$	−33.6844	−1.26149
$714$	0	0
$715$	0.477194	0.0178460
$716$	36.7696	1.37414
$717$	0	0
$718$	−1.22644	−0.0457705
$719$	−25.7422	−0.960024	−0.480012	−	0.877262i	$-0.659368\pi$
$719$	−25.7422	−0.960024	−0.480012	+	0.877262i	$0.659368\pi$
$720$	0	0
$721$	−0.864491	−0.0321953
$722$	0.0588208	0.00218908
$723$	0	0
$724$	−13.9316	−0.517762
$725$	28.7597	1.06811
$726$	0	0
$727$	3.21877	0.119378	0.0596889	−	0.998217i	$-0.480989\pi$
$727$	3.21877	0.119378	0.0596889	+	0.998217i	$0.480989\pi$
$728$	−0.151713	−0.00562285
$729$	0	0
$730$	0.0143357	0.000530588 0
$731$	−40.9381	−1.51415
$732$	0	0
$733$	38.8208	1.43388	0.716940	−	0.697135i	$-0.245542\pi$
$733$	38.8208	1.43388	0.716940	+	0.697135i	$0.245542\pi$
$734$	0.963675	0.0355699
$735$	0	0
$736$	−3.29872	−0.121593
$737$	7.50758	0.276545
$738$	0	0
$739$	14.7389	0.542178	0.271089	−	0.962554i	$-0.412616\pi$
$739$	14.7389	0.542178	0.271089	+	0.962554i	$0.412616\pi$
$740$	−3.25914	−0.119808
$741$	0	0
$742$	0.372796	0.0136858
$743$	21.1064	0.774318	0.387159	−	0.922013i	$-0.373456\pi$
$743$	21.1064	0.774318	0.387159	+	0.922013i	$0.373456\pi$
$744$	0	0
$745$	5.09552	0.186685
$746$	0.734343	0.0268862
$747$	0	0
$748$	6.56510	0.240044
$749$	−7.21941	−0.263791
$750$	0	0
$751$	−13.3701	−0.487881	−0.243941	−	0.969790i	$-0.578440\pi$
$751$	−13.3701	−0.487881	−0.243941	+	0.969790i	$0.578440\pi$
$752$	12.8654	0.469154
$753$	0	0
$754$	−0.525774	−0.0191476
$755$	−5.88338	−0.214118
$756$	0	0
$757$	2.09082	0.0759921	0.0379961	−	0.999278i	$-0.487903\pi$
$757$	2.09082	0.0759921	0.0379961	+	0.999278i	$0.487903\pi$
$758$	−1.61363	−0.0586095
$759$	0	0
$760$	−0.737254	−0.0267430
$761$	−30.7680	−1.11534	−0.557669	−	0.830063i	$-0.688304\pi$
$761$	−30.7680	−1.11534	−0.557669	+	0.830063i	$0.688304\pi$
$762$	0	0
$763$	2.27165	0.0822393
$764$	42.7289	1.54588
$765$	0	0
$766$	0.605408	0.0218743
$767$	12.9239	0.466656
$768$	0	0
$769$	−37.4540	−1.35063	−0.675314	−	0.737531i	$-0.735992\pi$
$769$	−37.4540	−1.35063	−0.675314	+	0.737531i	$0.735992\pi$
$770$	0.0181336	0.000653491 0
$771$	0	0
$772$	43.0555	1.54960
$773$	−40.3657	−1.45185	−0.725927	−	0.687772i	$-0.758589\pi$
$773$	−40.3657	−1.45185	−0.725927	+	0.687772i	$0.758589\pi$
$774$	0	0
$775$	−50.7602	−1.82336
$776$	−4.49000	−0.161182
$777$	0	0
$778$	−0.622654	−0.0223232
$779$	33.4850	1.19972
$780$	0	0
$781$	−14.5416	−0.520339
$782$	0.910416	0.0325564
$783$	0	0
$784$	−26.9305	−0.961804
$785$	1.22515	0.0437275
$786$	0	0
$787$	14.7389	0.525384	0.262692	−	0.964880i	$-0.415390\pi$
$787$	14.7389	0.525384	0.262692	+	0.964880i	$0.415390\pi$
$788$	−16.7911	−0.598157
$789$	0	0
$790$	−0.314446	−0.0111875
$791$	7.53004	0.267738
$792$	0	0
$793$	10.3671	0.368147
$794$	−3.24631	−0.115207
$795$	0	0
$796$	−25.9154	−0.918547
$797$	43.3925	1.53704	0.768520	−	0.639826i	$-0.220994\pi$
$797$	43.3925	1.53704	0.768520	+	0.639826i	$0.220994\pi$
$798$	0	0
$799$	−10.7204	−0.379262
$800$	−4.97097	−0.175750
$801$	0	0
$802$	0.608117	0.0214734
$803$	0.344337	0.0121514
$804$	0	0
$805$	−0.658226	−0.0231994
$806$	0.927978	0.0326866
$807$	0	0
$808$	−2.11602	−0.0744412
$809$	−28.0821	−0.987315	−0.493658	−	0.869656i	$-0.664340\pi$
$809$	−28.0821	−0.987315	−0.493658	+	0.869656i	$0.664340\pi$
$810$	0	0
$811$	42.2927	1.48510	0.742549	−	0.669791i	$-0.233616\pi$
$811$	42.2927	1.48510	0.742549	+	0.669791i	$0.233616\pi$
$812$	5.22976	0.183529
$813$	0	0
$814$	0.299072	0.0104825
$815$	−0.703491	−0.0246422
$816$	0	0
$817$	55.1073	1.92796
$818$	0.755792	0.0264257
$819$	0	0
$820$	−7.17746	−0.250648
$821$	47.6687	1.66365	0.831825	−	0.555037i	$-0.187296\pi$
$821$	47.6687	1.66365	0.831825	+	0.555037i	$0.187296\pi$
$822$	0	0
$823$	−29.8618	−1.04092	−0.520458	−	0.853887i	$-0.674239\pi$
$823$	−29.8618	−1.04092	−0.520458	+	0.853887i	$0.674239\pi$
$824$	0.691330	0.0240836
$825$	0	0
$826$	0.491117	0.0170881
$827$	−53.2133	−1.85041	−0.925204	−	0.379470i	$-0.876107\pi$
$827$	−53.2133	−1.85041	−0.925204	+	0.379470i	$0.876107\pi$
$828$	0	0
$829$	−14.2881	−0.496246	−0.248123	−	0.968729i	$-0.579814\pi$
$829$	−14.2881	−0.496246	−0.248123	+	0.968729i	$0.579814\pi$
$830$	0.375960	0.0130498
$831$	0	0
$832$	−7.81790	−0.271037
$833$	22.4405	0.777518
$834$	0	0
$835$	9.11365	0.315391
$836$	−8.83736	−0.305646
$837$	0	0
$838$	1.16794	0.0403459
$839$	−31.0680	−1.07259	−0.536293	−	0.844032i	$-0.680176\pi$
$839$	−31.0680	−1.07259	−0.536293	+	0.844032i	$0.680176\pi$
$840$	0	0
$841$	7.31761	0.252331
$842$	2.05945	0.0709733
$843$	0	0
$844$	−14.7309	−0.507058
$845$	0.477194	0.0164160
$846$	0	0
$847$	0.435561	0.0149661
$848$	38.7937	1.33218
$849$	0	0
$850$	1.37194	0.0470571
$851$	−10.8559	−0.372135
$852$	0	0
$853$	−16.0515	−0.549593	−0.274796	−	0.961502i	$-0.588610\pi$
$853$	−16.0515	−0.549593	−0.274796	+	0.961502i	$0.588610\pi$
$854$	0.393956	0.0134809
$855$	0	0
$856$	5.77332	0.197328
$857$	11.4725	0.391893	0.195946	−	0.980615i	$-0.437222\pi$
$857$	11.4725	0.391893	0.195946	+	0.980615i	$0.437222\pi$
$858$	0	0
$859$	−26.7956	−0.914255	−0.457128	−	0.889401i	$-0.651122\pi$
$859$	−26.7956	−0.914255	−0.457128	+	0.889401i	$0.651122\pi$
$860$	−11.8122	−0.402791
$861$	0	0
$862$	3.00127	0.102224
$863$	43.5095	1.48108	0.740541	−	0.672012i	$-0.234570\pi$
$863$	43.5095	1.48108	0.740541	+	0.672012i	$0.234570\pi$
$864$	0	0
$865$	−4.13940	−0.140744
$866$	0.659555	0.0224126
$867$	0	0
$868$	−9.23039	−0.313300
$869$	−7.55285	−0.256213
$870$	0	0
$871$	7.50758	0.254385
$872$	−1.81663	−0.0615188
$873$	0	0
$874$	−1.22552	−0.0414539
$875$	−2.03114	−0.0686651
$876$	0	0
$877$	−11.9320	−0.402914	−0.201457	−	0.979497i	$-0.564568\pi$
$877$	−11.9320	−0.402914	−0.201457	+	0.979497i	$0.564568\pi$
$878$	−2.21811	−0.0748576
$879$	0	0
$880$	1.88701	0.0636111
$881$	−13.8176	−0.465525	−0.232763	−	0.972534i	$-0.574777\pi$
$881$	−13.8176	−0.465525	−0.232763	+	0.972534i	$0.574777\pi$
$882$	0	0
$883$	29.8413	1.00424	0.502119	−	0.864799i	$-0.332554\pi$
$883$	29.8413	1.00424	0.502119	+	0.864799i	$0.332554\pi$
$884$	6.56510	0.220808
$885$	0	0
$886$	−0.440089	−0.0147851
$887$	53.2070	1.78652	0.893258	−	0.449545i	$-0.148414\pi$
$887$	53.2070	1.78652	0.893258	+	0.449545i	$0.148414\pi$
$888$	0	0
$889$	−1.90575	−0.0639168
$890$	−0.245848	−0.00824083
$891$	0	0
$892$	30.0504	1.00616
$893$	14.4309	0.482912
$894$	0	0
$895$	−8.80662	−0.294373
$896$	−1.20447	−0.0402386
$897$	0	0
$898$	−0.434145	−0.0144876
$899$	−64.0997	−2.13784
$900$	0	0
$901$	−32.3258	−1.07693
$902$	0.658632	0.0219300
$903$	0	0
$904$	−6.02174	−0.200280
$905$	3.33672	0.110916
$906$	0	0
$907$	−45.8017	−1.52082	−0.760410	−	0.649443i	$-0.775002\pi$
$907$	−45.8017	−1.52082	−0.760410	+	0.649443i	$0.775002\pi$
$908$	42.6171	1.41430
$909$	0	0
$910$	0.0181336	0.000601124 0
$911$	10.9696	0.363439	0.181720	−	0.983350i	$-0.441834\pi$
$911$	10.9696	0.363439	0.181720	+	0.983350i	$0.441834\pi$
$912$	0	0
$913$	9.03039	0.298862
$914$	1.10651	0.0366000
$915$	0	0
$916$	−7.35921	−0.243155
$917$	−6.92261	−0.228605
$918$	0	0
$919$	−19.2008	−0.633377	−0.316689	−	0.948530i	$-0.602571\pi$
$919$	−19.2008	−0.633377	−0.316689	+	0.948530i	$0.602571\pi$
$920$	0.526380	0.0173542
$921$	0	0
$922$	−0.947528	−0.0312052
$923$	−14.5416	−0.478643
$924$	0	0
$925$	−16.3592	−0.537886
$926$	−3.16881	−0.104134
$927$	0	0
$928$	−6.27731	−0.206063
$929$	3.91310	0.128385	0.0641924	−	0.997938i	$-0.479553\pi$
$929$	3.91310	0.128385	0.0641924	+	0.997938i	$0.479553\pi$
$930$	0	0
$931$	−30.2074	−0.990009
$932$	−49.0920	−1.60806
$933$	0	0
$934$	0.571012	0.0186841
$935$	−1.57240	−0.0514229
$936$	0	0
$937$	49.4490	1.61543	0.807714	−	0.589574i	$-0.200704\pi$
$937$	49.4490	1.61543	0.807714	+	0.589574i	$0.200704\pi$
$938$	0.285292	0.00931512
$939$	0	0
$940$	−3.09324	−0.100891
$941$	27.1698	0.885710	0.442855	−	0.896593i	$-0.353966\pi$
$941$	27.1698	0.885710	0.442855	+	0.896593i	$0.353966\pi$
$942$	0	0
$943$	−23.9074	−0.778534
$944$	51.1063	1.66337
$945$	0	0
$946$	1.08393	0.0352416
$947$	−3.67844	−0.119533	−0.0597666	−	0.998212i	$-0.519036\pi$
$947$	−3.67844	−0.119533	−0.0597666	+	0.998212i	$0.519036\pi$
$948$	0	0
$949$	0.344337	0.0111776
$950$	−1.84678	−0.0599175
$951$	0	0
$952$	0.499908	0.0162021
$953$	14.3681	0.465429	0.232715	−	0.972545i	$-0.425239\pi$
$953$	14.3681	0.465429	0.232715	+	0.972545i	$0.425239\pi$
$954$	0	0
$955$	−10.2339	−0.331163
$956$	−30.4203	−0.983864
$957$	0	0
$958$	0.211796	0.00684283
$959$	−3.05712	−0.0987194
$960$	0	0
$961$	82.1343	2.64949
$962$	0.299072	0.00964245
$963$	0	0
$964$	−4.99740	−0.160955
$965$	−10.3122	−0.331960
$966$	0	0
$967$	−15.7904	−0.507786	−0.253893	−	0.967232i	$-0.581711\pi$
$967$	−15.7904	−0.507786	−0.253893	+	0.967232i	$0.581711\pi$
$968$	−0.348316	−0.0111953
$969$	0	0
$970$	0.536671	0.0172315
$971$	−34.6447	−1.11180	−0.555901	−	0.831248i	$-0.687627\pi$
$971$	−34.6447	−1.11180	−0.555901	+	0.831248i	$0.687627\pi$
$972$	0	0
$973$	3.42662	0.109852
$974$	−2.36982	−0.0759340
$975$	0	0
$976$	40.9956	1.31224
$977$	−1.18666	−0.0379645	−0.0189823	−	0.999820i	$-0.506043\pi$
$977$	−1.18666	−0.0379645	−0.0189823	+	0.999820i	$0.506043\pi$
$978$	0	0
$979$	−5.90514	−0.188729
$980$	6.47492	0.206834
$981$	0	0
$982$	−2.01220	−0.0642118
$983$	−0.173162	−0.00552301	−0.00276150	−	0.999996i	$-0.500879\pi$
$983$	−0.173162	−0.00552301	−0.00276150	+	0.999996i	$0.500879\pi$
$984$	0	0
$985$	4.02160	0.128139
$986$	1.73247	0.0551732
$987$	0	0
$988$	−8.83736	−0.281154
$989$	−39.3452	−1.25110
$990$	0	0
$991$	−60.8935	−1.93435	−0.967173	−	0.254120i	$-0.918214\pi$
$991$	−60.8935	−1.93435	−0.967173	+	0.254120i	$0.918214\pi$
$992$	11.0793	0.351768
$993$	0	0
$994$	−0.552589	−0.0175271
$995$	6.20696	0.196774
$996$	0	0
$997$	−50.8776	−1.61131	−0.805655	−	0.592386i	$-0.798186\pi$
$997$	−50.8776	−1.61131	−0.805655	+	0.592386i	$0.798186\pi$
$998$	−0.880980	−0.0278869
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.m.1.2		4
3.2	odd	2		429.2.a.h.1.3	✓	4
12.11	even	2		6864.2.a.bz.1.3		4
33.32	even	2		4719.2.a.z.1.2		4
39.38	odd	2		5577.2.a.m.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.h.1.3	✓	4	3.2	odd	2
1287.2.a.m.1.2		4	1.1	even	1	trivial
4719.2.a.z.1.2		4	33.32	even	2
5577.2.a.m.1.2		4	39.38	odd	2
6864.2.a.bz.1.3		4	12.11	even	2

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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0872450 q^{2} -1.99239 q^{4} +0.477194 q^{5} +0.435561 q^{7} -0.348316 q^{8} +O(q^{10})\)	\(q+0.0872450 q^{2} -1.99239 q^{4} +0.477194 q^{5} +0.435561 q^{7} -0.348316 q^{8} +0.0416328 q^{10} +1.00000 q^{11} +1.00000 q^{13} +0.0380005 q^{14} +3.95439 q^{16} -3.29509 q^{17} +4.43556 q^{19} -0.950756 q^{20} +0.0872450 q^{22} -3.16688 q^{23} -4.77229 q^{25} +0.0872450 q^{26} -0.867807 q^{28} -6.02641 q^{29} +10.6365 q^{31} +1.04163 q^{32} -0.287480 q^{34} +0.207847 q^{35} +3.42795 q^{37} +0.386981 q^{38} -0.166214 q^{40} +7.54922 q^{41} +12.4240 q^{43} -1.99239 q^{44} -0.276294 q^{46} +3.25346 q^{47} -6.81029 q^{49} -0.416358 q^{50} -1.99239 q^{52} +9.81029 q^{53} +0.477194 q^{55} -0.151713 q^{56} -0.525774 q^{58} +12.9239 q^{59} +10.3671 q^{61} +0.927978 q^{62} -7.81790 q^{64} +0.477194 q^{65} +7.50758 q^{67} +6.56510 q^{68} +0.0181336 q^{70} -14.5416 q^{71} +0.344337 q^{73} +0.299072 q^{74} -8.83736 q^{76} +0.435561 q^{77} -7.55285 q^{79} +1.88701 q^{80} +0.658632 q^{82} +9.03039 q^{83} -1.57240 q^{85} +1.08393 q^{86} -0.348316 q^{88} -5.90514 q^{89} +0.435561 q^{91} +6.30965 q^{92} +0.283848 q^{94} +2.11662 q^{95} +12.8906 q^{97} -0.594164 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{4} + 2 q^{7}+O(q^{10}) \) 4 * q + 2 * q^2 + 8 * q^4 + 2 * q^7	\( 4 q + 2 q^{2} + 8 q^{4} + 2 q^{7} - 2 q^{10} + 4 q^{11} + 4 q^{13} - 12 q^{14} + 12 q^{16} + 8 q^{17} + 18 q^{19} + 10 q^{20} + 2 q^{22} + 4 q^{25} + 2 q^{26} - 6 q^{28} + 10 q^{29} + 12 q^{31} + 2 q^{32} + 36 q^{34} - 22 q^{35} - 2 q^{37} - 4 q^{38} + 20 q^{40} - 2 q^{41} + 28 q^{43} + 8 q^{44} - 30 q^{46} - 6 q^{47} + 8 q^{49} + 36 q^{50} + 8 q^{52} + 4 q^{53} - 48 q^{56} - 6 q^{58} - 16 q^{59} - 10 q^{61} + 34 q^{62} - 12 q^{64} + 34 q^{68} - 58 q^{70} - 10 q^{71} - 6 q^{73} - 14 q^{74} + 26 q^{76} + 2 q^{77} - 8 q^{79} + 48 q^{80} + 12 q^{82} + 8 q^{83} - 18 q^{85} + 8 q^{86} - 6 q^{89} + 2 q^{91} + 28 q^{92} - 46 q^{94} - 22 q^{95} + 10 q^{97} + 34 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^4 + 2 * q^7 - 2 * q^10 + 4 * q^11 + 4 * q^13 - 12 * q^14 + 12 * q^16 + 8 * q^17 + 18 * q^19 + 10 * q^20 + 2 * q^22 + 4 * q^25 + 2 * q^26 - 6 * q^28 + 10 * q^29 + 12 * q^31 + 2 * q^32 + 36 * q^34 - 22 * q^35 - 2 * q^37 - 4 * q^38 + 20 * q^40 - 2 * q^41 + 28 * q^43 + 8 * q^44 - 30 * q^46 - 6 * q^47 + 8 * q^49 + 36 * q^50 + 8 * q^52 + 4 * q^53 - 48 * q^56 - 6 * q^58 - 16 * q^59 - 10 * q^61 + 34 * q^62 - 12 * q^64 + 34 * q^68 - 58 * q^70 - 10 * q^71 - 6 * q^73 - 14 * q^74 + 26 * q^76 + 2 * q^77 - 8 * q^79 + 48 * q^80 + 12 * q^82 + 8 * q^83 - 18 * q^85 + 8 * q^86 - 6 * q^89 + 2 * q^91 + 28 * q^92 - 46 * q^94 - 22 * q^95 + 10 * q^97 + 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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