LMFDB - Embedded newform 2873.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2873,2,Mod(1,2873)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2873, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2873.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2873 = 13^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2873.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:	$22.9410205007$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 221)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	2873.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +2.00000 q^{3} +1.00000 q^{4} +3.46410 q^{5} +3.46410 q^{6} +3.46410 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.73205 q^{2} +2.00000 q^{3} +1.00000 q^{4} +3.46410 q^{5} +3.46410 q^{6} +3.46410 q^{7} -1.73205 q^{8} +1.00000 q^{9} +6.00000 q^{10} +3.46410 q^{11} +2.00000 q^{12} +6.00000 q^{14} +6.92820 q^{15} -5.00000 q^{16} -1.00000 q^{17} +1.73205 q^{18} +3.46410 q^{19} +3.46410 q^{20} +6.92820 q^{21} +6.00000 q^{22} -6.00000 q^{23} -3.46410 q^{24} +7.00000 q^{25} -4.00000 q^{27} +3.46410 q^{28} -6.00000 q^{29} +12.0000 q^{30} -3.46410 q^{31} -5.19615 q^{32} +6.92820 q^{33} -1.73205 q^{34} +12.0000 q^{35} +1.00000 q^{36} -3.46410 q^{37} +6.00000 q^{38} -6.00000 q^{40} -3.46410 q^{41} +12.0000 q^{42} -8.00000 q^{43} +3.46410 q^{44} +3.46410 q^{45} -10.3923 q^{46} -10.3923 q^{47} -10.0000 q^{48} +5.00000 q^{49} +12.1244 q^{50} -2.00000 q^{51} -6.00000 q^{53} -6.92820 q^{54} +12.0000 q^{55} -6.00000 q^{56} +6.92820 q^{57} -10.3923 q^{58} +10.3923 q^{59} +6.92820 q^{60} +10.0000 q^{61} -6.00000 q^{62} +3.46410 q^{63} +1.00000 q^{64} +12.0000 q^{66} -3.46410 q^{67} -1.00000 q^{68} -12.0000 q^{69} +20.7846 q^{70} +10.3923 q^{71} -1.73205 q^{72} +10.3923 q^{73} -6.00000 q^{74} +14.0000 q^{75} +3.46410 q^{76} +12.0000 q^{77} -2.00000 q^{79} -17.3205 q^{80} -11.0000 q^{81} -6.00000 q^{82} -3.46410 q^{83} +6.92820 q^{84} -3.46410 q^{85} -13.8564 q^{86} -12.0000 q^{87} -6.00000 q^{88} -13.8564 q^{89} +6.00000 q^{90} -6.00000 q^{92} -6.92820 q^{93} -18.0000 q^{94} +12.0000 q^{95} -10.3923 q^{96} +3.46410 q^{97} +8.66025 q^{98} +3.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 + 2 * q^4 + 2 * q^9	$ 2 q + 4 q^{3} + 2 q^{4} + 2 q^{9} + 12 q^{10} + 4 q^{12} + 12 q^{14} - 10 q^{16} - 2 q^{17} + 12 q^{22} - 12 q^{23} + 14 q^{25} - 8 q^{27} - 12 q^{29} + 24 q^{30} + 24 q^{35} + 2 q^{36} + 12 q^{38} - 12 q^{40} + 24 q^{42} - 16 q^{43} - 20 q^{48} + 10 q^{49} - 4 q^{51} - 12 q^{53} + 24 q^{55} - 12 q^{56} + 20 q^{61} - 12 q^{62} + 2 q^{64} + 24 q^{66} - 2 q^{68} - 24 q^{69} - 12 q^{74} + 28 q^{75} + 24 q^{77} - 4 q^{79} - 22 q^{81} - 12 q^{82} - 24 q^{87} - 12 q^{88} + 12 q^{90} - 12 q^{92} - 36 q^{94} + 24 q^{95}+O(q^{100}) $ 2 * q + 4 * q^3 + 2 * q^4 + 2 * q^9 + 12 * q^10 + 4 * q^12 + 12 * q^14 - 10 * q^16 - 2 * q^17 + 12 * q^22 - 12 * q^23 + 14 * q^25 - 8 * q^27 - 12 * q^29 + 24 * q^30 + 24 * q^35 + 2 * q^36 + 12 * q^38 - 12 * q^40 + 24 * q^42 - 16 * q^43 - 20 * q^48 + 10 * q^49 - 4 * q^51 - 12 * q^53 + 24 * q^55 - 12 * q^56 + 20 * q^61 - 12 * q^62 + 2 * q^64 + 24 * q^66 - 2 * q^68 - 24 * q^69 - 12 * q^74 + 28 * q^75 + 24 * q^77 - 4 * q^79 - 22 * q^81 - 12 * q^82 - 24 * q^87 - 12 * q^88 + 12 * q^90 - 12 * q^92 - 36 * q^94 + 24 * q^95

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	1.00000	0.500000
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	3.46410	1.41421
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	−1.73205	−0.612372
$9$	1.00000	0.333333
$10$	6.00000	1.89737
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	2.00000	0.577350
$13$	0	0
$13$	0	0
$14$	6.00000	1.60357
$15$	6.92820	1.78885
$16$	−5.00000	−1.25000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.73205	0.408248
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	3.46410	0.774597
$21$	6.92820	1.51186
$22$	6.00000	1.27920
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−3.46410	−0.707107
$25$	7.00000	1.40000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	3.46410	0.654654
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	12.0000	2.19089
$31$	−3.46410	−0.622171	−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410	−0.622171	−0.311086	+	0.950382i	$0.600693\pi$
$32$	−5.19615	−0.918559
$33$	6.92820	1.20605
$34$	−1.73205	−0.297044
$35$	12.0000	2.02837
$36$	1.00000	0.166667
$37$	−3.46410	−0.569495	−0.284747	−	0.958603i	$-0.591910\pi$
$37$	−3.46410	−0.569495	−0.284747	+	0.958603i	$0.591910\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	−6.00000	−0.948683
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	12.0000	1.85164
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	3.46410	0.522233
$45$	3.46410	0.516398
$46$	−10.3923	−1.53226
$47$	−10.3923	−1.51587	−0.757937	−	0.652328i	$-0.773792\pi$
$47$	−10.3923	−1.51587	−0.757937	+	0.652328i	$0.773792\pi$
$48$	−10.0000	−1.44338
$49$	5.00000	0.714286
$50$	12.1244	1.71464
$51$	−2.00000	−0.280056
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−6.92820	−0.942809
$55$	12.0000	1.61808
$56$	−6.00000	−0.801784
$57$	6.92820	0.917663
$58$	−10.3923	−1.36458
$59$	10.3923	1.35296	0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923	1.35296	0.676481	+	0.736460i	$0.263504\pi$
$60$	6.92820	0.894427
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−6.00000	−0.762001
$63$	3.46410	0.436436
$64$	1.00000	0.125000
$65$	0	0
$66$	12.0000	1.47710
$67$	−3.46410	−0.423207	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	−3.46410	−0.423207	−0.211604	+	0.977356i	$0.567869\pi$
$68$	−1.00000	−0.121268
$69$	−12.0000	−1.44463
$70$	20.7846	2.48424
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	−1.73205	−0.204124
$73$	10.3923	1.21633	0.608164	−	0.793812i	$-0.291906\pi$
$73$	10.3923	1.21633	0.608164	+	0.793812i	$0.291906\pi$
$74$	−6.00000	−0.697486
$75$	14.0000	1.61658
$76$	3.46410	0.397360
$77$	12.0000	1.36753
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	−17.3205	−1.93649
$81$	−11.0000	−1.22222
$82$	−6.00000	−0.662589
$83$	−3.46410	−0.380235	−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410	−0.380235	−0.190117	+	0.981761i	$0.560887\pi$
$84$	6.92820	0.755929
$85$	−3.46410	−0.375735
$86$	−13.8564	−1.49417
$87$	−12.0000	−1.28654
$88$	−6.00000	−0.639602
$89$	−13.8564	−1.46878	−0.734388	−	0.678730i	$-0.762531\pi$
$89$	−13.8564	−1.46878	−0.734388	+	0.678730i	$0.762531\pi$
$90$	6.00000	0.632456
$91$	0	0
$92$	−6.00000	−0.625543
$93$	−6.92820	−0.718421
$94$	−18.0000	−1.85656
$95$	12.0000	1.23117
$96$	−10.3923	−1.06066
$97$	3.46410	0.351726	0.175863	−	0.984415i	$-0.443728\pi$
$97$	3.46410	0.351726	0.175863	+	0.984415i	$0.443728\pi$
$98$	8.66025	0.874818
$99$	3.46410	0.348155
$100$	7.00000	0.700000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−3.46410	−0.342997
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	24.0000	2.34216
$106$	−10.3923	−1.00939
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−4.00000	−0.384900
$109$	10.3923	0.995402	0.497701	−	0.867349i	$-0.334178\pi$
$109$	10.3923	0.995402	0.497701	+	0.867349i	$0.334178\pi$
$110$	20.7846	1.98173
$111$	−6.92820	−0.657596
$112$	−17.3205	−1.63663
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	12.0000	1.12390
$115$	−20.7846	−1.93817
$116$	−6.00000	−0.557086
$117$	0	0
$118$	18.0000	1.65703
$119$	−3.46410	−0.317554
$120$	−12.0000	−1.09545
$121$	1.00000	0.0909091
$122$	17.3205	1.56813
$123$	−6.92820	−0.624695
$124$	−3.46410	−0.311086
$125$	6.92820	0.619677
$126$	6.00000	0.534522
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	12.1244	1.07165
$129$	−16.0000	−1.40872
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	6.92820	0.603023
$133$	12.0000	1.04053
$134$	−6.00000	−0.518321
$135$	−13.8564	−1.19257
$136$	1.73205	0.148522
$137$	13.8564	1.18383	0.591916	−	0.805999i	$-0.298372\pi$
$137$	13.8564	1.18383	0.591916	+	0.805999i	$0.298372\pi$
$138$	−20.7846	−1.76930
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	12.0000	1.01419
$141$	−20.7846	−1.75038
$142$	18.0000	1.51053
$143$	0	0
$144$	−5.00000	−0.416667
$145$	−20.7846	−1.72607
$146$	18.0000	1.48969
$147$	10.0000	0.824786
$148$	−3.46410	−0.284747
$149$	−6.92820	−0.567581	−0.283790	−	0.958886i	$-0.591592\pi$
$149$	−6.92820	−0.567581	−0.283790	+	0.958886i	$0.591592\pi$
$150$	24.2487	1.97990
$151$	−3.46410	−0.281905	−0.140952	−	0.990016i	$-0.545016\pi$
$151$	−3.46410	−0.281905	−0.140952	+	0.990016i	$0.545016\pi$
$152$	−6.00000	−0.486664
$153$	−1.00000	−0.0808452
$154$	20.7846	1.67487
$155$	−12.0000	−0.963863
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−3.46410	−0.275589
$159$	−12.0000	−0.951662
$160$	−18.0000	−1.42302
$161$	−20.7846	−1.63806
$162$	−19.0526	−1.49691
$163$	−10.3923	−0.813988	−0.406994	−	0.913431i	$-0.633423\pi$
$163$	−10.3923	−0.813988	−0.406994	+	0.913431i	$0.633423\pi$
$164$	−3.46410	−0.270501
$165$	24.0000	1.86840
$166$	−6.00000	−0.465690
$167$	−10.3923	−0.804181	−0.402090	−	0.915600i	$-0.631716\pi$
$167$	−10.3923	−0.804181	−0.402090	+	0.915600i	$0.631716\pi$
$168$	−12.0000	−0.925820
$169$	0	0
$170$	−6.00000	−0.460179
$171$	3.46410	0.264906
$172$	−8.00000	−0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−20.7846	−1.57568
$175$	24.2487	1.83303
$176$	−17.3205	−1.30558
$177$	20.7846	1.56227
$178$	−24.0000	−1.79888
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	3.46410	0.258199
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	20.0000	1.47844
$184$	10.3923	0.766131
$185$	−12.0000	−0.882258
$186$	−12.0000	−0.879883
$187$	−3.46410	−0.253320
$188$	−10.3923	−0.757937
$189$	−13.8564	−1.00791
$190$	20.7846	1.50787
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	2.00000	0.144338
$193$	3.46410	0.249351	0.124676	−	0.992198i	$-0.460211\pi$
$193$	3.46410	0.249351	0.124676	+	0.992198i	$0.460211\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	5.00000	0.357143
$197$	24.2487	1.72765	0.863825	−	0.503793i	$-0.168062\pi$
$197$	24.2487	1.72765	0.863825	+	0.503793i	$0.168062\pi$
$198$	6.00000	0.426401
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	−12.1244	−0.857321
$201$	−6.92820	−0.488678
$202$	10.3923	0.731200
$203$	−20.7846	−1.45879
$204$	−2.00000	−0.140028
$205$	−12.0000	−0.838116
$206$	27.7128	1.93084
$207$	−6.00000	−0.417029
$208$	0	0
$209$	12.0000	0.830057
$210$	41.5692	2.86855
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	−6.00000	−0.412082
$213$	20.7846	1.42414
$214$	−10.3923	−0.710403
$215$	−27.7128	−1.89000
$216$	6.92820	0.471405
$217$	−12.0000	−0.814613
$218$	18.0000	1.21911
$219$	20.7846	1.40449
$220$	12.0000	0.809040
$221$	0	0
$222$	−12.0000	−0.805387
$223$	−10.3923	−0.695920	−0.347960	−	0.937509i	$-0.613126\pi$
$223$	−10.3923	−0.695920	−0.347960	+	0.937509i	$0.613126\pi$
$224$	−18.0000	−1.20268
$225$	7.00000	0.466667
$226$	−10.3923	−0.691286
$227$	10.3923	0.689761	0.344881	−	0.938647i	$-0.387919\pi$
$227$	10.3923	0.689761	0.344881	+	0.938647i	$0.387919\pi$
$228$	6.92820	0.458831
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	−36.0000	−2.37377
$231$	24.0000	1.57908
$232$	10.3923	0.682288
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−36.0000	−2.34838
$236$	10.3923	0.676481
$237$	−4.00000	−0.259828
$238$	−6.00000	−0.388922
$239$	3.46410	0.224074	0.112037	−	0.993704i	$-0.464262\pi$
$239$	3.46410	0.224074	0.112037	+	0.993704i	$0.464262\pi$
$240$	−34.6410	−2.23607
$241$	10.3923	0.669427	0.334714	−	0.942320i	$-0.391360\pi$
$241$	10.3923	0.669427	0.334714	+	0.942320i	$0.391360\pi$
$242$	1.73205	0.111340
$243$	−10.0000	−0.641500
$244$	10.0000	0.640184
$245$	17.3205	1.10657
$246$	−12.0000	−0.765092
$247$	0	0
$248$	6.00000	0.381000
$249$	−6.92820	−0.439057
$250$	12.0000	0.758947
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	3.46410	0.218218
$253$	−20.7846	−1.30672
$254$	34.6410	2.17357
$255$	−6.92820	−0.433861
$256$	19.0000	1.18750
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	−27.7128	−1.72532
$259$	−12.0000	−0.745644
$260$	0	0
$261$	−6.00000	−0.371391
$262$	31.1769	1.92612
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	−12.0000	−0.738549
$265$	−20.7846	−1.27679
$266$	20.7846	1.27439
$267$	−27.7128	−1.69600
$268$	−3.46410	−0.211604
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−24.0000	−1.46059
$271$	−3.46410	−0.210429	−0.105215	−	0.994450i	$-0.533553\pi$
$271$	−3.46410	−0.210429	−0.105215	+	0.994450i	$0.533553\pi$
$272$	5.00000	0.303170
$273$	0	0
$274$	24.0000	1.44989
$275$	24.2487	1.46225
$276$	−12.0000	−0.722315
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	3.46410	0.207763
$279$	−3.46410	−0.207390
$280$	−20.7846	−1.24212
$281$	20.7846	1.23991	0.619953	−	0.784639i	$-0.287152\pi$
$281$	20.7846	1.23991	0.619953	+	0.784639i	$0.287152\pi$
$282$	−36.0000	−2.14377
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	10.3923	0.616670
$285$	24.0000	1.42164
$286$	0	0
$287$	−12.0000	−0.708338
$288$	−5.19615	−0.306186
$289$	1.00000	0.0588235
$290$	−36.0000	−2.11399
$291$	6.92820	0.406138
$292$	10.3923	0.608164
$293$	20.7846	1.21425	0.607125	−	0.794606i	$-0.292323\pi$
$293$	20.7846	1.21425	0.607125	+	0.794606i	$0.292323\pi$
$294$	17.3205	1.01015
$295$	36.0000	2.09600
$296$	6.00000	0.348743
$297$	−13.8564	−0.804030
$298$	−12.0000	−0.695141
$299$	0	0
$300$	14.0000	0.808290
$301$	−27.7128	−1.59734
$302$	−6.00000	−0.345261
$303$	12.0000	0.689382
$304$	−17.3205	−0.993399
$305$	34.6410	1.98354
$306$	−1.73205	−0.0990148
$307$	10.3923	0.593120	0.296560	−	0.955014i	$-0.404160\pi$
$307$	10.3923	0.593120	0.296560	+	0.955014i	$0.404160\pi$
$308$	12.0000	0.683763
$309$	32.0000	1.82042
$310$	−20.7846	−1.18049
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	3.46410	0.195491
$315$	12.0000	0.676123
$316$	−2.00000	−0.112509
$317$	10.3923	0.583690	0.291845	−	0.956466i	$-0.405731\pi$
$317$	10.3923	0.583690	0.291845	+	0.956466i	$0.405731\pi$
$318$	−20.7846	−1.16554
$319$	−20.7846	−1.16371
$320$	3.46410	0.193649
$321$	−12.0000	−0.669775
$322$	−36.0000	−2.00620
$323$	−3.46410	−0.192748
$324$	−11.0000	−0.611111
$325$	0	0
$326$	−18.0000	−0.996928
$327$	20.7846	1.14939
$328$	6.00000	0.331295
$329$	−36.0000	−1.98474
$330$	41.5692	2.28831
$331$	−17.3205	−0.952021	−0.476011	−	0.879440i	$-0.657918\pi$
$331$	−17.3205	−0.952021	−0.476011	+	0.879440i	$0.657918\pi$
$332$	−3.46410	−0.190117
$333$	−3.46410	−0.189832
$334$	−18.0000	−0.984916
$335$	−12.0000	−0.655630
$336$	−34.6410	−1.88982
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	−3.46410	−0.187867
$341$	−12.0000	−0.649836
$342$	6.00000	0.324443
$343$	−6.92820	−0.374088
$344$	13.8564	0.747087
$345$	−41.5692	−2.23801
$346$	10.3923	0.558694
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	−12.0000	−0.643268
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	42.0000	2.24499
$351$	0	0
$352$	−18.0000	−0.959403
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	36.0000	1.91338
$355$	36.0000	1.91068
$356$	−13.8564	−0.734388
$357$	−6.92820	−0.366679
$358$	−41.5692	−2.19700
$359$	−24.2487	−1.27980	−0.639899	−	0.768459i	$-0.721024\pi$
$359$	−24.2487	−1.27980	−0.639899	+	0.768459i	$0.721024\pi$
$360$	−6.00000	−0.316228
$361$	−7.00000	−0.368421
$362$	−3.46410	−0.182069
$363$	2.00000	0.104973
$364$	0	0
$365$	36.0000	1.88433
$366$	34.6410	1.81071
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	30.0000	1.56386
$369$	−3.46410	−0.180334
$370$	−20.7846	−1.08054
$371$	−20.7846	−1.07908
$372$	−6.92820	−0.359211
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−6.00000	−0.310253
$375$	13.8564	0.715542
$376$	18.0000	0.928279
$377$	0	0
$378$	−24.0000	−1.23443
$379$	−17.3205	−0.889695	−0.444847	−	0.895606i	$-0.646742\pi$
$379$	−17.3205	−0.889695	−0.444847	+	0.895606i	$0.646742\pi$
$380$	12.0000	0.615587
$381$	40.0000	2.04926
$382$	0	0
$383$	−17.3205	−0.885037	−0.442518	−	0.896759i	$-0.645915\pi$
$383$	−17.3205	−0.885037	−0.442518	+	0.896759i	$0.645915\pi$
$384$	24.2487	1.23744
$385$	41.5692	2.11856
$386$	6.00000	0.305392
$387$	−8.00000	−0.406663
$388$	3.46410	0.175863
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	−8.66025	−0.437409
$393$	36.0000	1.81596
$394$	42.0000	2.11593
$395$	−6.92820	−0.348596
$396$	3.46410	0.174078
$397$	31.1769	1.56472	0.782362	−	0.622824i	$-0.214015\pi$
$397$	31.1769	1.56472	0.782362	+	0.622824i	$0.214015\pi$
$398$	24.2487	1.21548
$399$	24.0000	1.20150
$400$	−35.0000	−1.75000
$401$	−10.3923	−0.518967	−0.259483	−	0.965748i	$-0.583552\pi$
$401$	−10.3923	−0.518967	−0.259483	+	0.965748i	$0.583552\pi$
$402$	−12.0000	−0.598506
$403$	0	0
$404$	6.00000	0.298511
$405$	−38.1051	−1.89346
$406$	−36.0000	−1.78665
$407$	−12.0000	−0.594818
$408$	3.46410	0.171499
$409$	−27.7128	−1.37031	−0.685155	−	0.728397i	$-0.740266\pi$
$409$	−27.7128	−1.37031	−0.685155	+	0.728397i	$0.740266\pi$
$410$	−20.7846	−1.02648
$411$	27.7128	1.36697
$412$	16.0000	0.788263
$413$	36.0000	1.77144
$414$	−10.3923	−0.510754
$415$	−12.0000	−0.589057
$416$	0	0
$417$	4.00000	0.195881
$418$	20.7846	1.01661
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	24.0000	1.17108
$421$	−6.92820	−0.337660	−0.168830	−	0.985645i	$-0.553999\pi$
$421$	−6.92820	−0.337660	−0.168830	+	0.985645i	$0.553999\pi$
$422$	38.1051	1.85493
$423$	−10.3923	−0.505291
$424$	10.3923	0.504695
$425$	−7.00000	−0.339550
$426$	36.0000	1.74421
$427$	34.6410	1.67640
$428$	−6.00000	−0.290021
$429$	0	0
$430$	−48.0000	−2.31477
$431$	−3.46410	−0.166860	−0.0834300	−	0.996514i	$-0.526587\pi$
$431$	−3.46410	−0.166860	−0.0834300	+	0.996514i	$0.526587\pi$
$432$	20.0000	0.962250
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	−20.7846	−0.997693
$435$	−41.5692	−1.99309
$436$	10.3923	0.497701
$437$	−20.7846	−0.994263
$438$	36.0000	1.72015
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	−20.7846	−0.990867
$441$	5.00000	0.238095
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−6.92820	−0.328798
$445$	−48.0000	−2.27542
$446$	−18.0000	−0.852325
$447$	−13.8564	−0.655386
$448$	3.46410	0.163663
$449$	−10.3923	−0.490443	−0.245222	−	0.969467i	$-0.578861\pi$
$449$	−10.3923	−0.490443	−0.245222	+	0.969467i	$0.578861\pi$
$450$	12.1244	0.571548
$451$	−12.0000	−0.565058
$452$	−6.00000	−0.282216
$453$	−6.92820	−0.325515
$454$	18.0000	0.844782
$455$	0	0
$456$	−12.0000	−0.561951
$457$	20.7846	0.972263	0.486132	−	0.873886i	$-0.338408\pi$
$457$	20.7846	0.972263	0.486132	+	0.873886i	$0.338408\pi$
$458$	0	0
$459$	4.00000	0.186704
$460$	−20.7846	−0.969087
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	41.5692	1.93398
$463$	−38.1051	−1.77090	−0.885448	−	0.464739i	$-0.846148\pi$
$463$	−38.1051	−1.77090	−0.885448	+	0.464739i	$0.846148\pi$
$464$	30.0000	1.39272
$465$	−24.0000	−1.11297
$466$	−31.1769	−1.44424
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	−62.3538	−2.87617
$471$	4.00000	0.184310
$472$	−18.0000	−0.828517
$473$	−27.7128	−1.27424
$474$	−6.92820	−0.318223
$475$	24.2487	1.11261
$476$	−3.46410	−0.158777
$477$	−6.00000	−0.274721
$478$	6.00000	0.274434
$479$	17.3205	0.791394	0.395697	−	0.918381i	$-0.370503\pi$
$479$	17.3205	0.791394	0.395697	+	0.918381i	$0.370503\pi$
$480$	−36.0000	−1.64317
$481$	0	0
$482$	18.0000	0.819878
$483$	−41.5692	−1.89146
$484$	1.00000	0.0454545
$485$	12.0000	0.544892
$486$	−17.3205	−0.785674
$487$	24.2487	1.09881	0.549407	−	0.835555i	$-0.314854\pi$
$487$	24.2487	1.09881	0.549407	+	0.835555i	$0.314854\pi$
$488$	−17.3205	−0.784063
$489$	−20.7846	−0.939913
$490$	30.0000	1.35526
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	−6.92820	−0.312348
$493$	6.00000	0.270226
$494$	0	0
$495$	12.0000	0.539360
$496$	17.3205	0.777714
$497$	36.0000	1.61482
$498$	−12.0000	−0.537733
$499$	24.2487	1.08552	0.542761	−	0.839887i	$-0.317379\pi$
$499$	24.2487	1.08552	0.542761	+	0.839887i	$0.317379\pi$
$500$	6.92820	0.309839
$501$	−20.7846	−0.928588
$502$	−20.7846	−0.927663
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	−6.00000	−0.267261
$505$	20.7846	0.924903
$506$	−36.0000	−1.60040
$507$	0	0
$508$	20.0000	0.887357
$509$	6.92820	0.307087	0.153544	−	0.988142i	$-0.450931\pi$
$509$	6.92820	0.307087	0.153544	+	0.988142i	$0.450931\pi$
$510$	−12.0000	−0.531369
$511$	36.0000	1.59255
$512$	8.66025	0.382733
$513$	−13.8564	−0.611775
$514$	51.9615	2.29192
$515$	55.4256	2.44234
$516$	−16.0000	−0.704361
$517$	−36.0000	−1.58328
$518$	−20.7846	−0.913223
$519$	12.0000	0.526742
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−10.3923	−0.454859
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	18.0000	0.786334
$525$	48.4974	2.11660
$526$	−20.7846	−0.906252
$527$	3.46410	0.150899
$528$	−34.6410	−1.50756
$529$	13.0000	0.565217
$530$	−36.0000	−1.56374
$531$	10.3923	0.450988
$532$	12.0000	0.520266
$533$	0	0
$534$	−48.0000	−2.07716
$535$	−20.7846	−0.898597
$536$	6.00000	0.259161
$537$	−48.0000	−2.07135
$538$	31.1769	1.34413
$539$	17.3205	0.746047
$540$	−13.8564	−0.596285
$541$	−31.1769	−1.34040	−0.670200	−	0.742180i	$-0.733792\pi$
$541$	−31.1769	−1.34040	−0.670200	+	0.742180i	$0.733792\pi$
$542$	−6.00000	−0.257722
$543$	−4.00000	−0.171656
$544$	5.19615	0.222783
$545$	36.0000	1.54207
$546$	0	0
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	13.8564	0.591916
$549$	10.0000	0.426790
$550$	42.0000	1.79089
$551$	−20.7846	−0.885454
$552$	20.7846	0.884652
$553$	−6.92820	−0.294617
$554$	−45.0333	−1.91328
$555$	−24.0000	−1.01874
$556$	2.00000	0.0848189
$557$	6.92820	0.293557	0.146779	−	0.989169i	$-0.453109\pi$
$557$	6.92820	0.293557	0.146779	+	0.989169i	$0.453109\pi$
$558$	−6.00000	−0.254000
$559$	0	0
$560$	−60.0000	−2.53546
$561$	−6.92820	−0.292509
$562$	36.0000	1.51857
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	−20.7846	−0.875190
$565$	−20.7846	−0.874415
$566$	3.46410	0.145607
$567$	−38.1051	−1.60026
$568$	−18.0000	−0.755263
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	41.5692	1.74114
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	0	0
$574$	−20.7846	−0.867533
$575$	−42.0000	−1.75152
$576$	1.00000	0.0416667
$577$	−27.7128	−1.15370	−0.576850	−	0.816850i	$-0.695718\pi$
$577$	−27.7128	−1.15370	−0.576850	+	0.816850i	$0.695718\pi$
$578$	1.73205	0.0720438
$579$	6.92820	0.287926
$580$	−20.7846	−0.863034
$581$	−12.0000	−0.497844
$582$	12.0000	0.497416
$583$	−20.7846	−0.860811
$584$	−18.0000	−0.744845
$585$	0	0
$586$	36.0000	1.48715
$587$	45.0333	1.85872	0.929362	−	0.369170i	$-0.120358\pi$
$587$	45.0333	1.85872	0.929362	+	0.369170i	$0.120358\pi$
$588$	10.0000	0.412393
$589$	−12.0000	−0.494451
$590$	62.3538	2.56707
$591$	48.4974	1.99492
$592$	17.3205	0.711868
$593$	−20.7846	−0.853522	−0.426761	−	0.904365i	$-0.640345\pi$
$593$	−20.7846	−0.853522	−0.426761	+	0.904365i	$0.640345\pi$
$594$	−24.0000	−0.984732
$595$	−12.0000	−0.491952
$596$	−6.92820	−0.283790
$597$	28.0000	1.14596
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	−24.2487	−0.989949
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	−48.0000	−1.95633
$603$	−3.46410	−0.141069
$604$	−3.46410	−0.140952
$605$	3.46410	0.140836
$606$	20.7846	0.844317
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−18.0000	−0.729996
$609$	−41.5692	−1.68447
$610$	60.0000	2.42933
$611$	0	0
$612$	−1.00000	−0.0404226
$613$	−34.6410	−1.39914	−0.699569	−	0.714565i	$-0.746625\pi$
$613$	−34.6410	−1.39914	−0.699569	+	0.714565i	$0.746625\pi$
$614$	18.0000	0.726421
$615$	−24.0000	−0.967773
$616$	−20.7846	−0.837436
$617$	31.1769	1.25514	0.627568	−	0.778562i	$-0.284051\pi$
$617$	31.1769	1.25514	0.627568	+	0.778562i	$0.284051\pi$
$618$	55.4256	2.22955
$619$	31.1769	1.25311	0.626553	−	0.779379i	$-0.284465\pi$
$619$	31.1769	1.25311	0.626553	+	0.779379i	$0.284465\pi$
$620$	−12.0000	−0.481932
$621$	24.0000	0.963087
$622$	10.3923	0.416693
$623$	−48.0000	−1.92308
$624$	0	0
$625$	−11.0000	−0.440000
$626$	17.3205	0.692267
$627$	24.0000	0.958468
$628$	2.00000	0.0798087
$629$	3.46410	0.138123
$630$	20.7846	0.828079
$631$	3.46410	0.137904	0.0689519	−	0.997620i	$-0.478035\pi$
$631$	3.46410	0.137904	0.0689519	+	0.997620i	$0.478035\pi$
$632$	3.46410	0.137795
$633$	44.0000	1.74884
$634$	18.0000	0.714871
$635$	69.2820	2.74937
$636$	−12.0000	−0.475831
$637$	0	0
$638$	−36.0000	−1.42525
$639$	10.3923	0.411113
$640$	42.0000	1.66020
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	−20.7846	−0.820303
$643$	−3.46410	−0.136611	−0.0683054	−	0.997664i	$-0.521759\pi$
$643$	−3.46410	−0.136611	−0.0683054	+	0.997664i	$0.521759\pi$
$644$	−20.7846	−0.819028
$645$	−55.4256	−2.18238
$646$	−6.00000	−0.236067
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	19.0526	0.748455
$649$	36.0000	1.41312
$650$	0	0
$651$	−24.0000	−0.940634
$652$	−10.3923	−0.406994
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	36.0000	1.40771
$655$	62.3538	2.43637
$656$	17.3205	0.676252
$657$	10.3923	0.405442
$658$	−62.3538	−2.43081
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	24.0000	0.934199
$661$	−13.8564	−0.538952	−0.269476	−	0.963007i	$-0.586850\pi$
$661$	−13.8564	−0.538952	−0.269476	+	0.963007i	$0.586850\pi$
$662$	−30.0000	−1.16598
$663$	0	0
$664$	6.00000	0.232845
$665$	41.5692	1.61199
$666$	−6.00000	−0.232495
$667$	36.0000	1.39393
$668$	−10.3923	−0.402090
$669$	−20.7846	−0.803579
$670$	−20.7846	−0.802980
$671$	34.6410	1.33730
$672$	−36.0000	−1.38873
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	24.2487	0.934025
$675$	−28.0000	−1.07772
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	−20.7846	−0.798228
$679$	12.0000	0.460518
$680$	6.00000	0.230089
$681$	20.7846	0.796468
$682$	−20.7846	−0.795884
$683$	3.46410	0.132550	0.0662751	−	0.997801i	$-0.478889\pi$
$683$	3.46410	0.132550	0.0662751	+	0.997801i	$0.478889\pi$
$684$	3.46410	0.132453
$685$	48.0000	1.83399
$686$	−12.0000	−0.458162
$687$	0	0
$688$	40.0000	1.52499
$689$	0	0
$690$	−72.0000	−2.74099
$691$	−3.46410	−0.131781	−0.0658903	−	0.997827i	$-0.520989\pi$
$691$	−3.46410	−0.131781	−0.0658903	+	0.997827i	$0.520989\pi$
$692$	6.00000	0.228086
$693$	12.0000	0.455842
$694$	10.3923	0.394486
$695$	6.92820	0.262802
$696$	20.7846	0.787839
$697$	3.46410	0.131212
$698$	0	0
$699$	−36.0000	−1.36165
$700$	24.2487	0.916515
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	3.46410	0.130558
$705$	−72.0000	−2.71168
$706$	0	0
$707$	20.7846	0.781686
$708$	20.7846	0.781133
$709$	−3.46410	−0.130097	−0.0650485	−	0.997882i	$-0.520720\pi$
$709$	−3.46410	−0.130097	−0.0650485	+	0.997882i	$0.520720\pi$
$710$	62.3538	2.34010
$711$	−2.00000	−0.0750059
$712$	24.0000	0.899438
$713$	20.7846	0.778390
$714$	−12.0000	−0.449089
$715$	0	0
$716$	−24.0000	−0.896922
$717$	6.92820	0.258738
$718$	−42.0000	−1.56743
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	−17.3205	−0.645497
$721$	55.4256	2.06416
$722$	−12.1244	−0.451222
$723$	20.7846	0.772988
$724$	−2.00000	−0.0743294
$725$	−42.0000	−1.55984
$726$	3.46410	0.128565
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	62.3538	2.30782
$731$	8.00000	0.295891
$732$	20.0000	0.739221
$733$	13.8564	0.511798	0.255899	−	0.966704i	$-0.417629\pi$
$733$	13.8564	0.511798	0.255899	+	0.966704i	$0.417629\pi$
$734$	24.2487	0.895036
$735$	34.6410	1.27775
$736$	31.1769	1.14920
$737$	−12.0000	−0.442026
$738$	−6.00000	−0.220863
$739$	−24.2487	−0.892003	−0.446002	−	0.895032i	$-0.647152\pi$
$739$	−24.2487	−0.892003	−0.446002	+	0.895032i	$0.647152\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	−36.0000	−1.32160
$743$	24.2487	0.889599	0.444799	−	0.895630i	$-0.353275\pi$
$743$	24.2487	0.889599	0.444799	+	0.895630i	$0.353275\pi$
$744$	12.0000	0.439941
$745$	−24.0000	−0.879292
$746$	17.3205	0.634149
$747$	−3.46410	−0.126745
$748$	−3.46410	−0.126660
$749$	−20.7846	−0.759453
$750$	24.0000	0.876356
$751$	−50.0000	−1.82453	−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000	−1.82453	−0.912263	+	0.409605i	$0.865667\pi$
$752$	51.9615	1.89484
$753$	−24.0000	−0.874609
$754$	0	0
$755$	−12.0000	−0.436725
$756$	−13.8564	−0.503953
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	−30.0000	−1.08965
$759$	−41.5692	−1.50887
$760$	−20.7846	−0.753937
$761$	−27.7128	−1.00459	−0.502294	−	0.864697i	$-0.667511\pi$
$761$	−27.7128	−1.00459	−0.502294	+	0.864697i	$0.667511\pi$
$762$	69.2820	2.50982
$763$	36.0000	1.30329
$764$	0	0
$765$	−3.46410	−0.125245
$766$	−30.0000	−1.08394
$767$	0	0
$768$	38.0000	1.37121
$769$	41.5692	1.49902	0.749512	−	0.661991i	$-0.230288\pi$
$769$	41.5692	1.49902	0.749512	+	0.661991i	$0.230288\pi$
$770$	72.0000	2.59470
$771$	60.0000	2.16085
$772$	3.46410	0.124676
$773$	−13.8564	−0.498380	−0.249190	−	0.968455i	$-0.580164\pi$
$773$	−13.8564	−0.498380	−0.249190	+	0.968455i	$0.580164\pi$
$774$	−13.8564	−0.498058
$775$	−24.2487	−0.871039
$776$	−6.00000	−0.215387
$777$	−24.0000	−0.860995
$778$	−51.9615	−1.86291
$779$	−12.0000	−0.429945
$780$	0	0
$781$	36.0000	1.28818
$782$	10.3923	0.371628
$783$	24.0000	0.857690
$784$	−25.0000	−0.892857
$785$	6.92820	0.247278
$786$	62.3538	2.22409
$787$	3.46410	0.123482	0.0617409	−	0.998092i	$-0.480335\pi$
$787$	3.46410	0.123482	0.0617409	+	0.998092i	$0.480335\pi$
$788$	24.2487	0.863825
$789$	−24.0000	−0.854423
$790$	−12.0000	−0.426941
$791$	−20.7846	−0.739016
$792$	−6.00000	−0.213201
$793$	0	0
$794$	54.0000	1.91639
$795$	−41.5692	−1.47431
$796$	14.0000	0.496217
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	41.5692	1.47153
$799$	10.3923	0.367653
$800$	−36.3731	−1.28598
$801$	−13.8564	−0.489592
$802$	−18.0000	−0.635602
$803$	36.0000	1.27041
$804$	−6.92820	−0.244339
$805$	−72.0000	−2.53767
$806$	0	0
$807$	36.0000	1.26726
$808$	−10.3923	−0.365600
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	−66.0000	−2.31900
$811$	−17.3205	−0.608205	−0.304103	−	0.952639i	$-0.598357\pi$
$811$	−17.3205	−0.608205	−0.304103	+	0.952639i	$0.598357\pi$
$812$	−20.7846	−0.729397
$813$	−6.92820	−0.242983
$814$	−20.7846	−0.728500
$815$	−36.0000	−1.26102
$816$	10.0000	0.350070
$817$	−27.7128	−0.969549
$818$	−48.0000	−1.67828
$819$	0	0
$820$	−12.0000	−0.419058
$821$	−24.2487	−0.846286	−0.423143	−	0.906063i	$-0.639073\pi$
$821$	−24.2487	−0.846286	−0.423143	+	0.906063i	$0.639073\pi$
$822$	48.0000	1.67419
$823$	−14.0000	−0.488009	−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000	−0.488009	−0.244005	+	0.969774i	$0.578461\pi$
$824$	−27.7128	−0.965422
$825$	48.4974	1.68846
$826$	62.3538	2.16957
$827$	31.1769	1.08413	0.542064	−	0.840337i	$-0.317643\pi$
$827$	31.1769	1.08413	0.542064	+	0.840337i	$0.317643\pi$
$828$	−6.00000	−0.208514
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	−20.7846	−0.721444
$831$	−52.0000	−1.80386
$832$	0	0
$833$	−5.00000	−0.173240
$834$	6.92820	0.239904
$835$	−36.0000	−1.24583
$836$	12.0000	0.415029
$837$	13.8564	0.478947
$838$	−31.1769	−1.07699
$839$	45.0333	1.55472	0.777361	−	0.629054i	$-0.216558\pi$
$839$	45.0333	1.55472	0.777361	+	0.629054i	$0.216558\pi$
$840$	−41.5692	−1.43427
$841$	7.00000	0.241379
$842$	−12.0000	−0.413547
$843$	41.5692	1.43172
$844$	22.0000	0.757271
$845$	0	0
$846$	−18.0000	−0.618853
$847$	3.46410	0.119028
$848$	30.0000	1.03020
$849$	4.00000	0.137280
$850$	−12.1244	−0.415862
$851$	20.7846	0.712487
$852$	20.7846	0.712069
$853$	38.1051	1.30469	0.652347	−	0.757920i	$-0.273784\pi$
$853$	38.1051	1.30469	0.652347	+	0.757920i	$0.273784\pi$
$854$	60.0000	2.05316
$855$	12.0000	0.410391
$856$	10.3923	0.355202
$857$	30.0000	1.02478	0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000	1.02478	0.512390	+	0.858753i	$0.328760\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−27.7128	−0.944999
$861$	−24.0000	−0.817918
$862$	−6.00000	−0.204361
$863$	3.46410	0.117919	0.0589597	−	0.998260i	$-0.481222\pi$
$863$	3.46410	0.117919	0.0589597	+	0.998260i	$0.481222\pi$
$864$	20.7846	0.707107
$865$	20.7846	0.706698
$866$	45.0333	1.53029
$867$	2.00000	0.0679236
$868$	−12.0000	−0.407307
$869$	−6.92820	−0.235023
$870$	−72.0000	−2.44103
$871$	0	0
$872$	−18.0000	−0.609557
$873$	3.46410	0.117242
$874$	−36.0000	−1.21772
$875$	24.0000	0.811348
$876$	20.7846	0.702247
$877$	−31.1769	−1.05277	−0.526385	−	0.850246i	$-0.676453\pi$
$877$	−31.1769	−1.05277	−0.526385	+	0.850246i	$0.676453\pi$
$878$	−45.0333	−1.51980
$879$	41.5692	1.40209
$880$	−60.0000	−2.02260
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	8.66025	0.291606
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	72.0000	2.42025
$886$	−20.7846	−0.698273
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	12.0000	0.402694
$889$	69.2820	2.32364
$890$	−83.1384	−2.78681
$891$	−38.1051	−1.27657
$892$	−10.3923	−0.347960
$893$	−36.0000	−1.20469
$894$	−24.0000	−0.802680
$895$	−83.1384	−2.77901
$896$	42.0000	1.40312
$897$	0	0
$898$	−18.0000	−0.600668
$899$	20.7846	0.693206
$900$	7.00000	0.233333
$901$	6.00000	0.199889
$902$	−20.7846	−0.692052
$903$	−55.4256	−1.84445
$904$	10.3923	0.345643
$905$	−6.92820	−0.230301
$906$	−12.0000	−0.398673
$907$	26.0000	0.863316	0.431658	−	0.902037i	$-0.357929\pi$
$907$	26.0000	0.863316	0.431658	+	0.902037i	$0.357929\pi$
$908$	10.3923	0.344881
$909$	6.00000	0.199007
$910$	0	0
$911$	−42.0000	−1.39152	−0.695761	−	0.718273i	$-0.744933\pi$
$911$	−42.0000	−1.39152	−0.695761	+	0.718273i	$0.744933\pi$
$912$	−34.6410	−1.14708
$913$	−12.0000	−0.397142
$914$	36.0000	1.19077
$915$	69.2820	2.29039
$916$	0	0
$917$	62.3538	2.05910
$918$	6.92820	0.228665
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	36.0000	1.18688
$921$	20.7846	0.684876
$922$	0	0
$923$	0	0
$924$	24.0000	0.789542
$925$	−24.2487	−0.797293
$926$	−66.0000	−2.16889
$927$	16.0000	0.525509
$928$	31.1769	1.02343
$929$	−17.3205	−0.568267	−0.284134	−	0.958785i	$-0.591706\pi$
$929$	−17.3205	−0.568267	−0.284134	+	0.958785i	$0.591706\pi$
$930$	−41.5692	−1.36311
$931$	17.3205	0.567657
$932$	−18.0000	−0.589610
$933$	12.0000	0.392862
$934$	−41.5692	−1.36019
$935$	−12.0000	−0.392442
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	−20.7846	−0.678642
$939$	20.0000	0.652675
$940$	−36.0000	−1.17419
$941$	−17.3205	−0.564632	−0.282316	−	0.959321i	$-0.591103\pi$
$941$	−17.3205	−0.564632	−0.282316	+	0.959321i	$0.591103\pi$
$942$	6.92820	0.225733
$943$	20.7846	0.676840
$944$	−51.9615	−1.69120
$945$	−48.0000	−1.56144
$946$	−48.0000	−1.56061
$947$	3.46410	0.112568	0.0562841	−	0.998415i	$-0.482075\pi$
$947$	3.46410	0.112568	0.0562841	+	0.998415i	$0.482075\pi$
$948$	−4.00000	−0.129914
$949$	0	0
$950$	42.0000	1.36266
$951$	20.7846	0.673987
$952$	6.00000	0.194461
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	−10.3923	−0.336463
$955$	0	0
$956$	3.46410	0.112037
$957$	−41.5692	−1.34374
$958$	30.0000	0.969256
$959$	48.0000	1.55000
$960$	6.92820	0.223607
$961$	−19.0000	−0.612903
$962$	0	0
$963$	−6.00000	−0.193347
$964$	10.3923	0.334714
$965$	12.0000	0.386294
$966$	−72.0000	−2.31656
$967$	−31.1769	−1.00258	−0.501291	−	0.865279i	$-0.667141\pi$
$967$	−31.1769	−1.00258	−0.501291	+	0.865279i	$0.667141\pi$
$968$	−1.73205	−0.0556702
$969$	−6.92820	−0.222566
$970$	20.7846	0.667354
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	−10.0000	−0.320750
$973$	6.92820	0.222108
$974$	42.0000	1.34577
$975$	0	0
$976$	−50.0000	−1.60046
$977$	−48.4974	−1.55157	−0.775785	−	0.630997i	$-0.782646\pi$
$977$	−48.4974	−1.55157	−0.775785	+	0.630997i	$0.782646\pi$
$978$	−36.0000	−1.15115
$979$	−48.0000	−1.53409
$980$	17.3205	0.553283
$981$	10.3923	0.331801
$982$	−41.5692	−1.32653
$983$	−24.2487	−0.773414	−0.386707	−	0.922203i	$-0.626387\pi$
$983$	−24.2487	−0.773414	−0.386707	+	0.922203i	$0.626387\pi$
$984$	12.0000	0.382546
$985$	84.0000	2.67646
$986$	10.3923	0.330958
$987$	−72.0000	−2.29179
$988$	0	0
$989$	48.0000	1.52631
$990$	20.7846	0.660578
$991$	50.0000	1.58830	0.794151	−	0.607720i	$-0.207916\pi$
$991$	50.0000	1.58830	0.794151	+	0.607720i	$0.207916\pi$
$992$	18.0000	0.571501
$993$	−34.6410	−1.09930
$994$	62.3538	1.97774
$995$	48.4974	1.53747
$996$	−6.92820	−0.219529
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	42.0000	1.32949
$999$	13.8564	0.438397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2873.2.a.g.1.2		2
13.5	odd	4		221.2.c.a.103.1	✓	2
13.8	odd	4		221.2.c.a.103.2	yes	2
13.12	even	2	inner	2873.2.a.g.1.1		2
39.5	even	4		1989.2.b.a.766.2		2
39.8	even	4		1989.2.b.a.766.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
221.2.c.a.103.1	✓	2	13.5	odd	4
221.2.c.a.103.2	yes	2	13.8	odd	4
1989.2.b.a.766.1		2	39.8	even	4
1989.2.b.a.766.2		2	39.5	even	4
2873.2.a.g.1.1		2	13.12	even	2	inner
2873.2.a.g.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2873 = 13^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2873.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +2.00000 q^{3} +1.00000 q^{4} +3.46410 q^{5} +3.46410 q^{6} +3.46410 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{2} +2.00000 q^{3} +1.00000 q^{4} +3.46410 q^{5} +3.46410 q^{6} +3.46410 q^{7} -1.73205 q^{8} +1.00000 q^{9} +6.00000 q^{10} +3.46410 q^{11} +2.00000 q^{12} +6.00000 q^{14} +6.92820 q^{15} -5.00000 q^{16} -1.00000 q^{17} +1.73205 q^{18} +3.46410 q^{19} +3.46410 q^{20} +6.92820 q^{21} +6.00000 q^{22} -6.00000 q^{23} -3.46410 q^{24} +7.00000 q^{25} -4.00000 q^{27} +3.46410 q^{28} -6.00000 q^{29} +12.0000 q^{30} -3.46410 q^{31} -5.19615 q^{32} +6.92820 q^{33} -1.73205 q^{34} +12.0000 q^{35} +1.00000 q^{36} -3.46410 q^{37} +6.00000 q^{38} -6.00000 q^{40} -3.46410 q^{41} +12.0000 q^{42} -8.00000 q^{43} +3.46410 q^{44} +3.46410 q^{45} -10.3923 q^{46} -10.3923 q^{47} -10.0000 q^{48} +5.00000 q^{49} +12.1244 q^{50} -2.00000 q^{51} -6.00000 q^{53} -6.92820 q^{54} +12.0000 q^{55} -6.00000 q^{56} +6.92820 q^{57} -10.3923 q^{58} +10.3923 q^{59} +6.92820 q^{60} +10.0000 q^{61} -6.00000 q^{62} +3.46410 q^{63} +1.00000 q^{64} +12.0000 q^{66} -3.46410 q^{67} -1.00000 q^{68} -12.0000 q^{69} +20.7846 q^{70} +10.3923 q^{71} -1.73205 q^{72} +10.3923 q^{73} -6.00000 q^{74} +14.0000 q^{75} +3.46410 q^{76} +12.0000 q^{77} -2.00000 q^{79} -17.3205 q^{80} -11.0000 q^{81} -6.00000 q^{82} -3.46410 q^{83} +6.92820 q^{84} -3.46410 q^{85} -13.8564 q^{86} -12.0000 q^{87} -6.00000 q^{88} -13.8564 q^{89} +6.00000 q^{90} -6.00000 q^{92} -6.92820 q^{93} -18.0000 q^{94} +12.0000 q^{95} -10.3923 q^{96} +3.46410 q^{97} +8.66025 q^{98} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 + 2 * q^4 + 2 * q^9	\( 2 q + 4 q^{3} + 2 q^{4} + 2 q^{9} + 12 q^{10} + 4 q^{12} + 12 q^{14} - 10 q^{16} - 2 q^{17} + 12 q^{22} - 12 q^{23} + 14 q^{25} - 8 q^{27} - 12 q^{29} + 24 q^{30} + 24 q^{35} + 2 q^{36} + 12 q^{38} - 12 q^{40} + 24 q^{42} - 16 q^{43} - 20 q^{48} + 10 q^{49} - 4 q^{51} - 12 q^{53} + 24 q^{55} - 12 q^{56} + 20 q^{61} - 12 q^{62} + 2 q^{64} + 24 q^{66} - 2 q^{68} - 24 q^{69} - 12 q^{74} + 28 q^{75} + 24 q^{77} - 4 q^{79} - 22 q^{81} - 12 q^{82} - 24 q^{87} - 12 q^{88} + 12 q^{90} - 12 q^{92} - 36 q^{94} + 24 q^{95}+O(q^{100}) \) 2 * q + 4 * q^3 + 2 * q^4 + 2 * q^9 + 12 * q^10 + 4 * q^12 + 12 * q^14 - 10 * q^16 - 2 * q^17 + 12 * q^22 - 12 * q^23 + 14 * q^25 - 8 * q^27 - 12 * q^29 + 24 * q^30 + 24 * q^35 + 2 * q^36 + 12 * q^38 - 12 * q^40 + 24 * q^42 - 16 * q^43 - 20 * q^48 + 10 * q^49 - 4 * q^51 - 12 * q^53 + 24 * q^55 - 12 * q^56 + 20 * q^61 - 12 * q^62 + 2 * q^64 + 24 * q^66 - 2 * q^68 - 24 * q^69 - 12 * q^74 + 28 * q^75 + 24 * q^77 - 4 * q^79 - 22 * q^81 - 12 * q^82 - 24 * q^87 - 12 * q^88 + 12 * q^90 - 12 * q^92 - 36 * q^94 + 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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