LMFDB - Embedded newform 3969.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, 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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$1$
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 81)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{5} -1.73205 q^{8} +O(q^{10})$	$q+1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{5} -1.73205 q^{8} +3.00000 q^{10} -3.46410 q^{11} +1.00000 q^{13} -5.00000 q^{16} -5.19615 q^{17} -2.00000 q^{19} +1.73205 q^{20} -6.00000 q^{22} +3.46410 q^{23} -2.00000 q^{25} +1.73205 q^{26} +1.73205 q^{29} -8.00000 q^{31} -5.19615 q^{32} -9.00000 q^{34} -7.00000 q^{37} -3.46410 q^{38} -3.00000 q^{40} +6.92820 q^{41} +2.00000 q^{43} -3.46410 q^{44} +6.00000 q^{46} -6.92820 q^{47} -3.46410 q^{50} +1.00000 q^{52} -6.00000 q^{55} +3.00000 q^{58} -13.8564 q^{59} +7.00000 q^{61} -13.8564 q^{62} +1.00000 q^{64} +1.73205 q^{65} -10.0000 q^{67} -5.19615 q^{68} -10.3923 q^{71} +7.00000 q^{73} -12.1244 q^{74} -2.00000 q^{76} +2.00000 q^{79} -8.66025 q^{80} +12.0000 q^{82} +13.8564 q^{83} -9.00000 q^{85} +3.46410 q^{86} +6.00000 q^{88} +5.19615 q^{89} +3.46410 q^{92} -12.0000 q^{94} -3.46410 q^{95} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4}+O(q^{10}) $ 2 * q + 2 * q^4	$ 2 q + 2 q^{4} + 6 q^{10} + 2 q^{13} - 10 q^{16} - 4 q^{19} - 12 q^{22} - 4 q^{25} - 16 q^{31} - 18 q^{34} - 14 q^{37} - 6 q^{40} + 4 q^{43} + 12 q^{46} + 2 q^{52} - 12 q^{55} + 6 q^{58} + 14 q^{61} + 2 q^{64} - 20 q^{67} + 14 q^{73} - 4 q^{76} + 4 q^{79} + 24 q^{82} - 18 q^{85} + 12 q^{88} - 24 q^{94} - 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + 6 * q^10 + 2 * q^13 - 10 * q^16 - 4 * q^19 - 12 * q^22 - 4 * q^25 - 16 * q^31 - 18 * q^34 - 14 * q^37 - 6 * q^40 + 4 * q^43 + 12 * q^46 + 2 * q^52 - 12 * q^55 + 6 * q^58 + 14 * q^61 + 2 * q^64 - 20 * q^67 + 14 * q^73 - 4 * q^76 + 4 * q^79 + 24 * q^82 - 18 * q^85 + 12 * q^88 - 24 * q^94 - 4 * q^97

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	$0.290215\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.73205	−0.612372
$9$	0	0
$10$	3.00000	0.948683
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−5.19615	−1.26025	−0.630126	−	0.776493i	$-0.716997\pi$
$17$	−5.19615	−1.26025	−0.630126	+	0.776493i	$0.716997\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	1.73205	0.387298
$21$	0	0
$22$	−6.00000	−1.27920
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	1.73205	0.339683
$27$	0	0
$28$	0	0
$29$	1.73205	0.321634	0.160817	−	0.986984i	$-0.448587\pi$
$29$	1.73205	0.321634	0.160817	+	0.986984i	$0.448587\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	−9.00000	−1.54349
$35$	0	0
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−3.46410	−0.561951
$39$	0	0
$40$	−3.00000	−0.474342
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−3.46410	−0.522233
$45$	0	0
$46$	6.00000	0.884652
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	0	0
$50$	−3.46410	−0.489898
$51$	0	0
$52$	1.00000	0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	0	0
$57$	0	0
$58$	3.00000	0.393919
$59$	−13.8564	−1.80395	−0.901975	−	0.431788i	$-0.857883\pi$
$59$	−13.8564	−1.80395	−0.901975	+	0.431788i	$0.857883\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	−13.8564	−1.75977
$63$	0	0
$64$	1.00000	0.125000
$65$	1.73205	0.214834
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−5.19615	−0.630126
$69$	0	0
$70$	0	0
$71$	−10.3923	−1.23334	−0.616670	−	0.787222i	$-0.711519\pi$
$71$	−10.3923	−1.23334	−0.616670	+	0.787222i	$0.711519\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	−12.1244	−1.40943
$75$	0	0
$76$	−2.00000	−0.229416
$77$	0	0
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	−8.66025	−0.968246
$81$	0	0
$82$	12.0000	1.32518
$83$	13.8564	1.52094	0.760469	−	0.649374i	$-0.224969\pi$
$83$	13.8564	1.52094	0.760469	+	0.649374i	$0.224969\pi$
$84$	0	0
$85$	−9.00000	−0.976187
$86$	3.46410	0.373544
$87$	0	0
$88$	6.00000	0.639602
$89$	5.19615	0.550791	0.275396	−	0.961331i	$-0.411191\pi$
$89$	5.19615	0.550791	0.275396	+	0.961331i	$0.411191\pi$
$90$	0	0
$91$	0	0
$92$	3.46410	0.361158
$93$	0	0
$94$	−12.0000	−1.23771
$95$	−3.46410	−0.355409
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	−2.00000	−0.200000
$101$	−6.92820	−0.689382	−0.344691	−	0.938716i	$-0.612016\pi$
$101$	−6.92820	−0.689382	−0.344691	+	0.938716i	$0.612016\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−1.73205	−0.169842
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	−10.3923	−0.990867
$111$	0	0
$112$	0	0
$113$	−1.73205	−0.162938	−0.0814688	−	0.996676i	$-0.525961\pi$
$113$	−1.73205	−0.162938	−0.0814688	+	0.996676i	$0.525961\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	1.73205	0.160817
$117$	0	0
$118$	−24.0000	−2.20938
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	12.1244	1.09769
$123$	0	0
$124$	−8.00000	−0.718421
$125$	−12.1244	−1.08444
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	3.00000	0.263117
$131$	−3.46410	−0.302660	−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410	−0.302660	−0.151330	+	0.988483i	$0.548356\pi$
$132$	0	0
$133$	0	0
$134$	−17.3205	−1.49626
$135$	0	0
$136$	9.00000	0.771744
$137$	1.73205	0.147979	0.0739895	−	0.997259i	$-0.476427\pi$
$137$	1.73205	0.147979	0.0739895	+	0.997259i	$0.476427\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	−18.0000	−1.51053
$143$	−3.46410	−0.289683
$144$	0	0
$145$	3.00000	0.249136
$146$	12.1244	1.00342
$147$	0	0
$148$	−7.00000	−0.575396
$149$	8.66025	0.709476	0.354738	−	0.934966i	$-0.384570\pi$
$149$	8.66025	0.709476	0.354738	+	0.934966i	$0.384570\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	3.46410	0.280976
$153$	0	0
$154$	0	0
$155$	−13.8564	−1.11297
$156$	0	0
$157$	−17.0000	−1.35675	−0.678374	−	0.734717i	$-0.737315\pi$
$157$	−17.0000	−1.35675	−0.678374	+	0.734717i	$0.737315\pi$
$158$	3.46410	0.275589
$159$	0	0
$160$	−9.00000	−0.711512
$161$	0	0
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	6.92820	0.541002
$165$	0	0
$166$	24.0000	1.86276
$167$	17.3205	1.34030	0.670151	−	0.742225i	$-0.266230\pi$
$167$	17.3205	1.34030	0.670151	+	0.742225i	$0.266230\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−15.5885	−1.19558
$171$	0	0
$172$	2.00000	0.152499
$173$	19.0526	1.44854	0.724270	−	0.689517i	$-0.242177\pi$
$173$	19.0526	1.44854	0.724270	+	0.689517i	$0.242177\pi$
$174$	0	0
$175$	0	0
$176$	17.3205	1.30558
$177$	0	0
$178$	9.00000	0.674579
$179$	20.7846	1.55351	0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846	1.55351	0.776757	+	0.629800i	$0.216863\pi$
$180$	0	0
$181$	−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
$183$	0	0
$184$	−6.00000	−0.442326
$185$	−12.1244	−0.891400
$186$	0	0
$187$	18.0000	1.31629
$188$	−6.92820	−0.505291
$189$	0	0
$190$	−6.00000	−0.435286
$191$	17.3205	1.25327	0.626634	−	0.779314i	$-0.284432\pi$
$191$	17.3205	1.25327	0.626634	+	0.779314i	$0.284432\pi$
$192$	0	0
$193$	−1.00000	−0.0719816	−0.0359908	−	0.999352i	$-0.511459\pi$
$193$	−1.00000	−0.0719816	−0.0359908	+	0.999352i	$0.511459\pi$
$194$	−3.46410	−0.248708
$195$	0	0
$196$	0	0
$197$	−5.19615	−0.370211	−0.185105	−	0.982719i	$-0.559263\pi$
$197$	−5.19615	−0.370211	−0.185105	+	0.982719i	$0.559263\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	3.46410	0.244949
$201$	0	0
$202$	−12.0000	−0.844317
$203$	0	0
$204$	0	0
$205$	12.0000	0.838116
$206$	−13.8564	−0.965422
$207$	0	0
$208$	−5.00000	−0.346688
$209$	6.92820	0.479234
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.46410	0.236250
$216$	0	0
$217$	0	0
$218$	19.0526	1.29040
$219$	0	0
$220$	−6.00000	−0.404520
$221$	−5.19615	−0.349531
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	0	0
$226$	−3.00000	−0.199557
$227$	3.46410	0.229920	0.114960	−	0.993370i	$-0.463326\pi$
$227$	3.46410	0.229920	0.114960	+	0.993370i	$0.463326\pi$
$228$	0	0
$229$	1.00000	0.0660819	0.0330409	−	0.999454i	$-0.489481\pi$
$229$	1.00000	0.0660819	0.0330409	+	0.999454i	$0.489481\pi$
$230$	10.3923	0.685248
$231$	0	0
$232$	−3.00000	−0.196960
$233$	−25.9808	−1.70206	−0.851028	−	0.525120i	$-0.824020\pi$
$233$	−25.9808	−1.70206	−0.851028	+	0.525120i	$0.824020\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	−13.8564	−0.901975
$237$	0	0
$238$	0	0
$239$	−27.7128	−1.79259	−0.896296	−	0.443455i	$-0.853752\pi$
$239$	−27.7128	−1.79259	−0.896296	+	0.443455i	$0.853752\pi$
$240$	0	0
$241$	−29.0000	−1.86805	−0.934027	−	0.357202i	$-0.883731\pi$
$241$	−29.0000	−1.86805	−0.934027	+	0.357202i	$0.883731\pi$
$242$	1.73205	0.111340
$243$	0	0
$244$	7.00000	0.448129
$245$	0	0
$246$	0	0
$247$	−2.00000	−0.127257
$248$	13.8564	0.879883
$249$	0	0
$250$	−21.0000	−1.32816
$251$	−10.3923	−0.655956	−0.327978	−	0.944685i	$-0.606367\pi$
$251$	−10.3923	−0.655956	−0.327978	+	0.944685i	$0.606367\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	3.46410	0.217357
$255$	0	0
$256$	19.0000	1.18750
$257$	−8.66025	−0.540212	−0.270106	−	0.962831i	$-0.587059\pi$
$257$	−8.66025	−0.540212	−0.270106	+	0.962831i	$0.587059\pi$
$258$	0	0
$259$	0	0
$260$	1.73205	0.107417
$261$	0	0
$262$	−6.00000	−0.370681
$263$	6.92820	0.427211	0.213606	−	0.976920i	$-0.431479\pi$
$263$	6.92820	0.427211	0.213606	+	0.976920i	$0.431479\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−10.0000	−0.610847
$269$	15.5885	0.950445	0.475223	−	0.879866i	$-0.342368\pi$
$269$	15.5885	0.950445	0.475223	+	0.879866i	$0.342368\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	25.9808	1.57532
$273$	0	0
$274$	3.00000	0.181237
$275$	6.92820	0.417786
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−13.8564	−0.831052
$279$	0	0
$280$	0	0
$281$	12.1244	0.723278	0.361639	−	0.932318i	$-0.382217\pi$
$281$	12.1244	0.723278	0.361639	+	0.932318i	$0.382217\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	−10.3923	−0.616670
$285$	0	0
$286$	−6.00000	−0.354787
$287$	0	0
$288$	0	0
$289$	10.0000	0.588235
$290$	5.19615	0.305129
$291$	0	0
$292$	7.00000	0.409644
$293$	−19.0526	−1.11306	−0.556531	−	0.830827i	$-0.687868\pi$
$293$	−19.0526	−1.11306	−0.556531	+	0.830827i	$0.687868\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	12.1244	0.704714
$297$	0	0
$298$	15.0000	0.868927
$299$	3.46410	0.200334
$300$	0	0
$301$	0	0
$302$	34.6410	1.99337
$303$	0	0
$304$	10.0000	0.573539
$305$	12.1244	0.694239
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	0	0
$310$	−24.0000	−1.36311
$311$	6.92820	0.392862	0.196431	−	0.980518i	$-0.437065\pi$
$311$	6.92820	0.392862	0.196431	+	0.980518i	$0.437065\pi$
$312$	0	0
$313$	25.0000	1.41308	0.706542	−	0.707671i	$-0.250254\pi$
$313$	25.0000	1.41308	0.706542	+	0.707671i	$0.250254\pi$
$314$	−29.4449	−1.66167
$315$	0	0
$316$	2.00000	0.112509
$317$	−8.66025	−0.486408	−0.243204	−	0.969975i	$-0.578199\pi$
$317$	−8.66025	−0.486408	−0.243204	+	0.969975i	$0.578199\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	1.73205	0.0968246
$321$	0	0
$322$	0	0
$323$	10.3923	0.578243
$324$	0	0
$325$	−2.00000	−0.110940
$326$	−27.7128	−1.53487
$327$	0	0
$328$	−12.0000	−0.662589
$329$	0	0
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	13.8564	0.760469
$333$	0	0
$334$	30.0000	1.64153
$335$	−17.3205	−0.946320
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	−20.7846	−1.13053
$339$	0	0
$340$	−9.00000	−0.488094
$341$	27.7128	1.50073
$342$	0	0
$343$	0	0
$344$	−3.46410	−0.186772
$345$	0	0
$346$	33.0000	1.77409
$347$	3.46410	0.185963	0.0929814	−	0.995668i	$-0.470360\pi$
$347$	3.46410	0.185963	0.0929814	+	0.995668i	$0.470360\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	0	0
$352$	18.0000	0.959403
$353$	13.8564	0.737502	0.368751	−	0.929528i	$-0.379785\pi$
$353$	13.8564	0.737502	0.368751	+	0.929528i	$0.379785\pi$
$354$	0	0
$355$	−18.0000	−0.955341
$356$	5.19615	0.275396
$357$	0	0
$358$	36.0000	1.90266
$359$	10.3923	0.548485	0.274242	−	0.961661i	$-0.411573\pi$
$359$	10.3923	0.548485	0.274242	+	0.961661i	$0.411573\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−3.46410	−0.182069
$363$	0	0
$364$	0	0
$365$	12.1244	0.634618
$366$	0	0
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	−17.3205	−0.902894
$369$	0	0
$370$	−21.0000	−1.09174
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	31.1769	1.61212
$375$	0	0
$376$	12.0000	0.618853
$377$	1.73205	0.0892052
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	−3.46410	−0.177705
$381$	0	0
$382$	30.0000	1.53493
$383$	17.3205	0.885037	0.442518	−	0.896759i	$-0.354085\pi$
$383$	17.3205	0.885037	0.442518	+	0.896759i	$0.354085\pi$
$384$	0	0
$385$	0	0
$386$	−1.73205	−0.0881591
$387$	0	0
$388$	−2.00000	−0.101535
$389$	27.7128	1.40510	0.702548	−	0.711637i	$-0.252046\pi$
$389$	27.7128	1.40510	0.702548	+	0.711637i	$0.252046\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	0	0
$394$	−9.00000	−0.453413
$395$	3.46410	0.174298
$396$	0	0
$397$	−29.0000	−1.45547	−0.727734	−	0.685859i	$-0.759427\pi$
$397$	−29.0000	−1.45547	−0.727734	+	0.685859i	$0.759427\pi$
$398$	−34.6410	−1.73640
$399$	0	0
$400$	10.0000	0.500000
$401$	−12.1244	−0.605461	−0.302731	−	0.953076i	$-0.597898\pi$
$401$	−12.1244	−0.605461	−0.302731	+	0.953076i	$0.597898\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	−6.92820	−0.344691
$405$	0	0
$406$	0	0
$407$	24.2487	1.20196
$408$	0	0
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	20.7846	1.02648
$411$	0	0
$412$	−8.00000	−0.394132
$413$	0	0
$414$	0	0
$415$	24.0000	1.17811
$416$	−5.19615	−0.254762
$417$	0	0
$418$	12.0000	0.586939
$419$	6.92820	0.338465	0.169232	−	0.985576i	$-0.445871\pi$
$419$	6.92820	0.338465	0.169232	+	0.985576i	$0.445871\pi$
$420$	0	0
$421$	−25.0000	−1.21843	−0.609213	−	0.793007i	$-0.708514\pi$
$421$	−25.0000	−1.21843	−0.609213	+	0.793007i	$0.708514\pi$
$422$	−17.3205	−0.843149
$423$	0	0
$424$	0	0
$425$	10.3923	0.504101
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	6.00000	0.289346
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	0	0
$435$	0	0
$436$	11.0000	0.526804
$437$	−6.92820	−0.331421
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	10.3923	0.495434
$441$	0	0
$442$	−9.00000	−0.428086
$443$	−34.6410	−1.64584	−0.822922	−	0.568154i	$-0.807658\pi$
$443$	−34.6410	−1.64584	−0.822922	+	0.568154i	$0.807658\pi$
$444$	0	0
$445$	9.00000	0.426641
$446$	−3.46410	−0.164030
$447$	0	0
$448$	0	0
$449$	20.7846	0.980886	0.490443	−	0.871473i	$-0.336835\pi$
$449$	20.7846	0.980886	0.490443	+	0.871473i	$0.336835\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	−1.73205	−0.0814688
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	29.0000	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$458$	1.73205	0.0809334
$459$	0	0
$460$	6.00000	0.279751
$461$	13.8564	0.645357	0.322679	−	0.946509i	$-0.395417\pi$
$461$	13.8564	0.645357	0.322679	+	0.946509i	$0.395417\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	−8.66025	−0.402042
$465$	0	0
$466$	−45.0000	−2.08458
$467$	20.7846	0.961797	0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846	0.961797	0.480899	+	0.876776i	$0.340311\pi$
$468$	0	0
$469$	0	0
$470$	−20.7846	−0.958723
$471$	0	0
$472$	24.0000	1.10469
$473$	−6.92820	−0.318559
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	0	0
$478$	−48.0000	−2.19547
$479$	24.2487	1.10795	0.553976	−	0.832533i	$-0.313110\pi$
$479$	24.2487	1.10795	0.553976	+	0.832533i	$0.313110\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	−50.2295	−2.28789
$483$	0	0
$484$	1.00000	0.0454545
$485$	−3.46410	−0.157297
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−12.1244	−0.548844
$489$	0	0
$490$	0	0
$491$	−17.3205	−0.781664	−0.390832	−	0.920462i	$-0.627813\pi$
$491$	−17.3205	−0.781664	−0.390832	+	0.920462i	$0.627813\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	−3.46410	−0.155857
$495$	0	0
$496$	40.0000	1.79605
$497$	0	0
$498$	0	0
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	−12.1244	−0.542218
$501$	0	0
$502$	−18.0000	−0.803379
$503$	−20.7846	−0.926740	−0.463370	−	0.886165i	$-0.653360\pi$
$503$	−20.7846	−0.926740	−0.463370	+	0.886165i	$0.653360\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	−20.7846	−0.923989
$507$	0	0
$508$	2.00000	0.0887357
$509$	27.7128	1.22835	0.614174	−	0.789170i	$-0.289489\pi$
$509$	27.7128	1.22835	0.614174	+	0.789170i	$0.289489\pi$
$510$	0	0
$511$	0	0
$512$	8.66025	0.382733
$513$	0	0
$514$	−15.0000	−0.661622
$515$	−13.8564	−0.610586
$516$	0	0
$517$	24.0000	1.05552
$518$	0	0
$519$	0	0
$520$	−3.00000	−0.131559
$521$	20.7846	0.910590	0.455295	−	0.890341i	$-0.349534\pi$
$521$	20.7846	0.910590	0.455295	+	0.890341i	$0.349534\pi$
$522$	0	0
$523$	−38.0000	−1.66162	−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−38.0000	−1.66162	−0.830812	+	0.556553i	$0.812124\pi$
$524$	−3.46410	−0.151330
$525$	0	0
$526$	12.0000	0.523225
$527$	41.5692	1.81078
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.92820	0.300094
$534$	0	0
$535$	0	0
$536$	17.3205	0.748132
$537$	0	0
$538$	27.0000	1.16405
$539$	0	0
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	−3.46410	−0.148796
$543$	0	0
$544$	27.0000	1.15762
$545$	19.0526	0.816122
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	1.73205	0.0739895
$549$	0	0
$550$	12.0000	0.511682
$551$	−3.46410	−0.147576
$552$	0	0
$553$	0	0
$554$	3.46410	0.147176
$555$	0	0
$556$	−8.00000	−0.339276
$557$	36.3731	1.54118	0.770588	−	0.637333i	$-0.219963\pi$
$557$	36.3731	1.54118	0.770588	+	0.637333i	$0.219963\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	0	0
$562$	21.0000	0.885832
$563$	−34.6410	−1.45994	−0.729972	−	0.683477i	$-0.760467\pi$
$563$	−34.6410	−1.45994	−0.729972	+	0.683477i	$0.760467\pi$
$564$	0	0
$565$	−3.00000	−0.126211
$566$	48.4974	2.03850
$567$	0	0
$568$	18.0000	0.755263
$569$	32.9090	1.37962	0.689808	−	0.723993i	$-0.257695\pi$
$569$	32.9090	1.37962	0.689808	+	0.723993i	$0.257695\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	−3.46410	−0.144841
$573$	0	0
$574$	0	0
$575$	−6.92820	−0.288926
$576$	0	0
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	17.3205	0.720438
$579$	0	0
$580$	3.00000	0.124568
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−12.1244	−0.501709
$585$	0	0
$586$	−33.0000	−1.36322
$587$	−38.1051	−1.57277	−0.786383	−	0.617739i	$-0.788049\pi$
$587$	−38.1051	−1.57277	−0.786383	+	0.617739i	$0.788049\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	−41.5692	−1.71138
$591$	0	0
$592$	35.0000	1.43849
$593$	−15.5885	−0.640141	−0.320071	−	0.947394i	$-0.603707\pi$
$593$	−15.5885	−0.640141	−0.320071	+	0.947394i	$0.603707\pi$
$594$	0	0
$595$	0	0
$596$	8.66025	0.354738
$597$	0	0
$598$	6.00000	0.245358
$599$	13.8564	0.566157	0.283079	−	0.959097i	$-0.408644\pi$
$599$	13.8564	0.566157	0.283079	+	0.959097i	$0.408644\pi$
$600$	0	0
$601$	25.0000	1.01977	0.509886	−	0.860242i	$-0.329688\pi$
$601$	25.0000	1.01977	0.509886	+	0.860242i	$0.329688\pi$
$602$	0	0
$603$	0	0
$604$	20.0000	0.813788
$605$	1.73205	0.0704179
$606$	0	0
$607$	−26.0000	−1.05531	−0.527654	−	0.849460i	$-0.676928\pi$
$607$	−26.0000	−1.05531	−0.527654	+	0.849460i	$0.676928\pi$
$608$	10.3923	0.421464
$609$	0	0
$610$	21.0000	0.850265
$611$	−6.92820	−0.280285
$612$	0	0
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	27.7128	1.11840
$615$	0	0
$616$	0	0
$617$	−12.1244	−0.488108	−0.244054	−	0.969762i	$-0.578477\pi$
$617$	−12.1244	−0.488108	−0.244054	+	0.969762i	$0.578477\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	−13.8564	−0.556487
$621$	0	0
$622$	12.0000	0.481156
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	43.3013	1.73067
$627$	0	0
$628$	−17.0000	−0.678374
$629$	36.3731	1.45029
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	−3.46410	−0.137795
$633$	0	0
$634$	−15.0000	−0.595726
$635$	3.46410	0.137469
$636$	0	0
$637$	0	0
$638$	−10.3923	−0.411435
$639$	0	0
$640$	21.0000	0.830098
$641$	22.5167	0.889355	0.444677	−	0.895691i	$-0.353318\pi$
$641$	22.5167	0.889355	0.444677	+	0.895691i	$0.353318\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	0	0
$645$	0	0
$646$	18.0000	0.708201
$647$	31.1769	1.22569	0.612845	−	0.790203i	$-0.290025\pi$
$647$	31.1769	1.22569	0.612845	+	0.790203i	$0.290025\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	−3.46410	−0.135873
$651$	0	0
$652$	−16.0000	−0.626608
$653$	13.8564	0.542243	0.271122	−	0.962545i	$-0.412605\pi$
$653$	13.8564	0.542243	0.271122	+	0.962545i	$0.412605\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	−34.6410	−1.35250
$657$	0	0
$658$	0	0
$659$	−3.46410	−0.134942	−0.0674711	−	0.997721i	$-0.521493\pi$
$659$	−3.46410	−0.134942	−0.0674711	+	0.997721i	$0.521493\pi$
$660$	0	0
$661$	−17.0000	−0.661223	−0.330612	−	0.943767i	$-0.607255\pi$
$661$	−17.0000	−0.661223	−0.330612	+	0.943767i	$0.607255\pi$
$662$	3.46410	0.134636
$663$	0	0
$664$	−24.0000	−0.931381
$665$	0	0
$666$	0	0
$667$	6.00000	0.232321
$668$	17.3205	0.670151
$669$	0	0
$670$	−30.0000	−1.15900
$671$	−24.2487	−0.936111
$672$	0	0
$673$	−25.0000	−0.963679	−0.481840	−	0.876259i	$-0.660031\pi$
$673$	−25.0000	−0.963679	−0.481840	+	0.876259i	$0.660031\pi$
$674$	45.0333	1.73462
$675$	0	0
$676$	−12.0000	−0.461538
$677$	13.8564	0.532545	0.266272	−	0.963898i	$-0.414208\pi$
$677$	13.8564	0.532545	0.266272	+	0.963898i	$0.414208\pi$
$678$	0	0
$679$	0	0
$680$	15.5885	0.597790
$681$	0	0
$682$	48.0000	1.83801
$683$	−20.7846	−0.795301	−0.397650	−	0.917537i	$-0.630174\pi$
$683$	−20.7846	−0.795301	−0.397650	+	0.917537i	$0.630174\pi$
$684$	0	0
$685$	3.00000	0.114624
$686$	0	0
$687$	0	0
$688$	−10.0000	−0.381246
$689$	0	0
$690$	0	0
$691$	−2.00000	−0.0760836	−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000	−0.0760836	−0.0380418	+	0.999276i	$0.512112\pi$
$692$	19.0526	0.724270
$693$	0	0
$694$	6.00000	0.227757
$695$	−13.8564	−0.525603
$696$	0	0
$697$	−36.0000	−1.36360
$698$	−3.46410	−0.131118
$699$	0	0
$700$	0	0
$701$	46.7654	1.76630	0.883152	−	0.469087i	$-0.155417\pi$
$701$	46.7654	1.76630	0.883152	+	0.469087i	$0.155417\pi$
$702$	0	0
$703$	14.0000	0.528020
$704$	−3.46410	−0.130558
$705$	0	0
$706$	24.0000	0.903252
$707$	0	0
$708$	0	0
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	−31.1769	−1.17005
$711$	0	0
$712$	−9.00000	−0.337289
$713$	−27.7128	−1.03785
$714$	0	0
$715$	−6.00000	−0.224387
$716$	20.7846	0.776757
$717$	0	0
$718$	18.0000	0.671754
$719$	10.3923	0.387568	0.193784	−	0.981044i	$-0.437924\pi$
$719$	10.3923	0.387568	0.193784	+	0.981044i	$0.437924\pi$
$720$	0	0
$721$	0	0
$722$	−25.9808	−0.966904
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	−3.46410	−0.128654
$726$	0	0
$727$	34.0000	1.26099	0.630495	−	0.776193i	$-0.282852\pi$
$727$	34.0000	1.26099	0.630495	+	0.776193i	$0.282852\pi$
$728$	0	0
$729$	0	0
$730$	21.0000	0.777245
$731$	−10.3923	−0.384373
$732$	0	0
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	−34.6410	−1.27862
$735$	0	0
$736$	−18.0000	−0.663489
$737$	34.6410	1.27602
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−12.1244	−0.445700
$741$	0	0
$742$	0	0
$743$	−6.92820	−0.254171	−0.127086	−	0.991892i	$-0.540562\pi$
$743$	−6.92820	−0.254171	−0.127086	+	0.991892i	$0.540562\pi$
$744$	0	0
$745$	15.0000	0.549557
$746$	−17.3205	−0.634149
$747$	0	0
$748$	18.0000	0.658145
$749$	0	0
$750$	0	0
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	34.6410	1.26323
$753$	0	0
$754$	3.00000	0.109254
$755$	34.6410	1.26072
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−27.7128	−1.00657
$759$	0	0
$760$	6.00000	0.217643
$761$	−29.4449	−1.06738	−0.533688	−	0.845682i	$-0.679194\pi$
$761$	−29.4449	−1.06738	−0.533688	+	0.845682i	$0.679194\pi$
$762$	0	0
$763$	0	0
$764$	17.3205	0.626634
$765$	0	0
$766$	30.0000	1.08394
$767$	−13.8564	−0.500326
$768$	0	0
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	0	0
$771$	0	0
$772$	−1.00000	−0.0359908
$773$	25.9808	0.934463	0.467232	−	0.884135i	$-0.345251\pi$
$773$	25.9808	0.934463	0.467232	+	0.884135i	$0.345251\pi$
$774$	0	0
$775$	16.0000	0.574737
$776$	3.46410	0.124354
$777$	0	0
$778$	48.0000	1.72088
$779$	−13.8564	−0.496457
$780$	0	0
$781$	36.0000	1.28818
$782$	−31.1769	−1.11488
$783$	0	0
$784$	0	0
$785$	−29.4449	−1.05093
$786$	0	0
$787$	−26.0000	−0.926800	−0.463400	−	0.886149i	$-0.653371\pi$
$787$	−26.0000	−0.926800	−0.463400	+	0.886149i	$0.653371\pi$
$788$	−5.19615	−0.185105
$789$	0	0
$790$	6.00000	0.213470
$791$	0	0
$792$	0	0
$793$	7.00000	0.248577
$794$	−50.2295	−1.78258
$795$	0	0
$796$	−20.0000	−0.708881
$797$	53.6936	1.90192	0.950962	−	0.309308i	$-0.100097\pi$
$797$	53.6936	1.90192	0.950962	+	0.309308i	$0.100097\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	10.3923	0.367423
$801$	0	0
$802$	−21.0000	−0.741536
$803$	−24.2487	−0.855718
$804$	0	0
$805$	0	0
$806$	−13.8564	−0.488071
$807$	0	0
$808$	12.0000	0.422159
$809$	−46.7654	−1.64418	−0.822091	−	0.569355i	$-0.807193\pi$
$809$	−46.7654	−1.64418	−0.822091	+	0.569355i	$0.807193\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	0	0
$814$	42.0000	1.47210
$815$	−27.7128	−0.970737
$816$	0	0
$817$	−4.00000	−0.139942
$818$	32.9090	1.15063
$819$	0	0
$820$	12.0000	0.419058
$821$	12.1244	0.423143	0.211571	−	0.977363i	$-0.432142\pi$
$821$	12.1244	0.423143	0.211571	+	0.977363i	$0.432142\pi$
$822$	0	0
$823$	−28.0000	−0.976019	−0.488009	−	0.872838i	$-0.662277\pi$
$823$	−28.0000	−0.976019	−0.488009	+	0.872838i	$0.662277\pi$
$824$	13.8564	0.482711
$825$	0	0
$826$	0	0
$827$	−10.3923	−0.361376	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923	−0.361376	−0.180688	+	0.983540i	$0.557832\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	41.5692	1.44289
$831$	0	0
$832$	1.00000	0.0346688
$833$	0	0
$834$	0	0
$835$	30.0000	1.03819
$836$	6.92820	0.239617
$837$	0	0
$838$	12.0000	0.414533
$839$	45.0333	1.55472	0.777361	−	0.629054i	$-0.216558\pi$
$839$	45.0333	1.55472	0.777361	+	0.629054i	$0.216558\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	−43.3013	−1.49226
$843$	0	0
$844$	−10.0000	−0.344214
$845$	−20.7846	−0.715012
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	18.0000	0.617395
$851$	−24.2487	−0.831235
$852$	0	0
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.5167	−0.769154	−0.384577	−	0.923093i	$-0.625653\pi$
$857$	−22.5167	−0.769154	−0.384577	+	0.923093i	$0.625653\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	3.46410	0.118125
$861$	0	0
$862$	0	0
$863$	31.1769	1.06127	0.530637	−	0.847599i	$-0.321953\pi$
$863$	31.1769	1.06127	0.530637	+	0.847599i	$0.321953\pi$
$864$	0	0
$865$	33.0000	1.12203
$866$	−19.0526	−0.647432
$867$	0	0
$868$	0	0
$869$	−6.92820	−0.235023
$870$	0	0
$871$	−10.0000	−0.338837
$872$	−19.0526	−0.645201
$873$	0	0
$874$	−12.0000	−0.405906
$875$	0	0
$876$	0	0
$877$	53.0000	1.78968	0.894841	−	0.446384i	$-0.147289\pi$
$877$	53.0000	1.78968	0.894841	+	0.446384i	$0.147289\pi$
$878$	−34.6410	−1.16908
$879$	0	0
$880$	30.0000	1.01130
$881$	−20.7846	−0.700251	−0.350126	−	0.936703i	$-0.613861\pi$
$881$	−20.7846	−0.700251	−0.350126	+	0.936703i	$0.613861\pi$
$882$	0	0
$883$	56.0000	1.88455	0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000	1.88455	0.942275	+	0.334840i	$0.108682\pi$
$884$	−5.19615	−0.174766
$885$	0	0
$886$	−60.0000	−2.01574
$887$	−3.46410	−0.116313	−0.0581566	−	0.998307i	$-0.518522\pi$
$887$	−3.46410	−0.116313	−0.0581566	+	0.998307i	$0.518522\pi$
$888$	0	0
$889$	0	0
$890$	15.5885	0.522526
$891$	0	0
$892$	−2.00000	−0.0669650
$893$	13.8564	0.463687
$894$	0	0
$895$	36.0000	1.20335
$896$	0	0
$897$	0	0
$898$	36.0000	1.20134
$899$	−13.8564	−0.462137
$900$	0	0
$901$	0	0
$902$	−41.5692	−1.38410
$903$	0	0
$904$	3.00000	0.0997785
$905$	−3.46410	−0.115151
$906$	0	0
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	3.46410	0.114960
$909$	0	0
$910$	0	0
$911$	−24.2487	−0.803396	−0.401698	−	0.915772i	$-0.631580\pi$
$911$	−24.2487	−0.803396	−0.401698	+	0.915772i	$0.631580\pi$
$912$	0	0
$913$	−48.0000	−1.58857
$914$	50.2295	1.66144
$915$	0	0
$916$	1.00000	0.0330409
$917$	0	0
$918$	0	0
$919$	2.00000	0.0659739	0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000	0.0659739	0.0329870	+	0.999456i	$0.489498\pi$
$920$	−10.3923	−0.342624
$921$	0	0
$922$	24.0000	0.790398
$923$	−10.3923	−0.342067
$924$	0	0
$925$	14.0000	0.460317
$926$	13.8564	0.455350
$927$	0	0
$928$	−9.00000	−0.295439
$929$	50.2295	1.64798	0.823988	−	0.566608i	$-0.191744\pi$
$929$	50.2295	1.64798	0.823988	+	0.566608i	$0.191744\pi$
$930$	0	0
$931$	0	0
$932$	−25.9808	−0.851028
$933$	0	0
$934$	36.0000	1.17796
$935$	31.1769	1.01959
$936$	0	0
$937$	25.0000	0.816714	0.408357	−	0.912822i	$-0.366102\pi$
$937$	25.0000	0.816714	0.408357	+	0.912822i	$0.366102\pi$
$938$	0	0
$939$	0	0
$940$	−12.0000	−0.391397
$941$	−50.2295	−1.63743	−0.818717	−	0.574197i	$-0.805314\pi$
$941$	−50.2295	−1.63743	−0.818717	+	0.574197i	$0.805314\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	69.2820	2.25494
$945$	0	0
$946$	−12.0000	−0.390154
$947$	17.3205	0.562841	0.281420	−	0.959585i	$-0.409194\pi$
$947$	17.3205	0.562841	0.281420	+	0.959585i	$0.409194\pi$
$948$	0	0
$949$	7.00000	0.227230
$950$	6.92820	0.224781
$951$	0	0
$952$	0	0
$953$	−5.19615	−0.168320	−0.0841599	−	0.996452i	$-0.526821\pi$
$953$	−5.19615	−0.168320	−0.0841599	+	0.996452i	$0.526821\pi$
$954$	0	0
$955$	30.0000	0.970777
$956$	−27.7128	−0.896296
$957$	0	0
$958$	42.0000	1.35696
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	−12.1244	−0.390905
$963$	0	0
$964$	−29.0000	−0.934027
$965$	−1.73205	−0.0557567
$966$	0	0
$967$	−46.0000	−1.47926	−0.739630	−	0.673014i	$-0.765000\pi$
$967$	−46.0000	−1.47926	−0.739630	+	0.673014i	$0.765000\pi$
$968$	−1.73205	−0.0556702
$969$	0	0
$970$	−6.00000	−0.192648
$971$	−31.1769	−1.00051	−0.500257	−	0.865877i	$-0.666761\pi$
$971$	−31.1769	−1.00051	−0.500257	+	0.865877i	$0.666761\pi$
$972$	0	0
$973$	0	0
$974$	−27.7128	−0.887976
$975$	0	0
$976$	−35.0000	−1.12032
$977$	−48.4974	−1.55157	−0.775785	−	0.630997i	$-0.782646\pi$
$977$	−48.4974	−1.55157	−0.775785	+	0.630997i	$0.782646\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	0	0
$982$	−30.0000	−0.957338
$983$	34.6410	1.10488	0.552438	−	0.833554i	$-0.313697\pi$
$983$	34.6410	1.10488	0.552438	+	0.833554i	$0.313697\pi$
$984$	0	0
$985$	−9.00000	−0.286764
$986$	−15.5885	−0.496438
$987$	0	0
$988$	−2.00000	−0.0636285
$989$	6.92820	0.220304
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	41.5692	1.31982
$993$	0	0
$994$	0	0
$995$	−34.6410	−1.09819
$996$	0	0
$997$	7.00000	0.221692	0.110846	−	0.993838i	$-0.464644\pi$
$997$	7.00000	0.221692	0.110846	+	0.993838i	$0.464644\pi$
$998$	−17.3205	−0.548271
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.i.1.2		2
3.2	odd	2	inner	3969.2.a.i.1.1		2
7.6	odd	2		81.2.a.a.1.2	yes	2
21.20	even	2		81.2.a.a.1.1	✓	2
28.27	even	2		1296.2.a.o.1.1		2
35.13	even	4		2025.2.b.k.649.1		4
35.27	even	4		2025.2.b.k.649.4		4
35.34	odd	2		2025.2.a.j.1.1		2
56.13	odd	2		5184.2.a.br.1.2		2
56.27	even	2		5184.2.a.bq.1.2		2
63.13	odd	6		81.2.c.b.55.1		4
63.20	even	6		81.2.c.b.28.2		4
63.34	odd	6		81.2.c.b.28.1		4
63.41	even	6		81.2.c.b.55.2		4
77.76	even	2		9801.2.a.v.1.1		2
84.83	odd	2		1296.2.a.o.1.2		2
105.62	odd	4		2025.2.b.k.649.2		4
105.83	odd	4		2025.2.b.k.649.3		4
105.104	even	2		2025.2.a.j.1.2		2
168.83	odd	2		5184.2.a.bq.1.1		2
168.125	even	2		5184.2.a.br.1.1		2
189.13	odd	18		729.2.e.o.163.1		12
189.20	even	18		729.2.e.o.325.1		12
189.34	odd	18		729.2.e.o.325.2		12
189.41	even	18		729.2.e.o.163.2		12
189.76	odd	18		729.2.e.o.649.2		12
189.83	even	18		729.2.e.o.568.2		12
189.97	odd	18		729.2.e.o.82.2		12
189.104	even	18		729.2.e.o.406.1		12
189.139	odd	18		729.2.e.o.406.2		12
189.146	even	18		729.2.e.o.82.1		12
189.160	odd	18		729.2.e.o.568.1		12
189.167	even	18		729.2.e.o.649.1		12
231.230	odd	2		9801.2.a.v.1.2		2
252.83	odd	6		1296.2.i.s.433.1		4
252.139	even	6		1296.2.i.s.865.2		4
252.167	odd	6		1296.2.i.s.865.1		4
252.223	even	6		1296.2.i.s.433.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.a.a.1.1	✓	2	21.20	even	2
81.2.a.a.1.2	yes	2	7.6	odd	2
81.2.c.b.28.1		4	63.34	odd	6
81.2.c.b.28.2		4	63.20	even	6
81.2.c.b.55.1		4	63.13	odd	6
81.2.c.b.55.2		4	63.41	even	6
729.2.e.o.82.1		12	189.146	even	18
729.2.e.o.82.2		12	189.97	odd	18
729.2.e.o.163.1		12	189.13	odd	18
729.2.e.o.163.2		12	189.41	even	18
729.2.e.o.325.1		12	189.20	even	18
729.2.e.o.325.2		12	189.34	odd	18
729.2.e.o.406.1		12	189.104	even	18
729.2.e.o.406.2		12	189.139	odd	18
729.2.e.o.568.1		12	189.160	odd	18
729.2.e.o.568.2		12	189.83	even	18
729.2.e.o.649.1		12	189.167	even	18
729.2.e.o.649.2		12	189.76	odd	18
1296.2.a.o.1.1		2	28.27	even	2
1296.2.a.o.1.2		2	84.83	odd	2
1296.2.i.s.433.1		4	252.83	odd	6
1296.2.i.s.433.2		4	252.223	even	6
1296.2.i.s.865.1		4	252.167	odd	6
1296.2.i.s.865.2		4	252.139	even	6
2025.2.a.j.1.1		2	35.34	odd	2
2025.2.a.j.1.2		2	105.104	even	2
2025.2.b.k.649.1		4	35.13	even	4
2025.2.b.k.649.2		4	105.62	odd	4
2025.2.b.k.649.3		4	105.83	odd	4
2025.2.b.k.649.4		4	35.27	even	4
3969.2.a.i.1.1		2	3.2	odd	2	inner
3969.2.a.i.1.2		2	1.1	even	1	trivial
5184.2.a.bq.1.1		2	168.83	odd	2
5184.2.a.bq.1.2		2	56.27	even	2
5184.2.a.br.1.1		2	168.125	even	2
5184.2.a.br.1.2		2	56.13	odd	2
9801.2.a.v.1.1		2	77.76	even	2
9801.2.a.v.1.2		2	231.230	odd	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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{5} -1.73205 q^{8} +O(q^{10})\)	\(q+1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{5} -1.73205 q^{8} +3.00000 q^{10} -3.46410 q^{11} +1.00000 q^{13} -5.00000 q^{16} -5.19615 q^{17} -2.00000 q^{19} +1.73205 q^{20} -6.00000 q^{22} +3.46410 q^{23} -2.00000 q^{25} +1.73205 q^{26} +1.73205 q^{29} -8.00000 q^{31} -5.19615 q^{32} -9.00000 q^{34} -7.00000 q^{37} -3.46410 q^{38} -3.00000 q^{40} +6.92820 q^{41} +2.00000 q^{43} -3.46410 q^{44} +6.00000 q^{46} -6.92820 q^{47} -3.46410 q^{50} +1.00000 q^{52} -6.00000 q^{55} +3.00000 q^{58} -13.8564 q^{59} +7.00000 q^{61} -13.8564 q^{62} +1.00000 q^{64} +1.73205 q^{65} -10.0000 q^{67} -5.19615 q^{68} -10.3923 q^{71} +7.00000 q^{73} -12.1244 q^{74} -2.00000 q^{76} +2.00000 q^{79} -8.66025 q^{80} +12.0000 q^{82} +13.8564 q^{83} -9.00000 q^{85} +3.46410 q^{86} +6.00000 q^{88} +5.19615 q^{89} +3.46410 q^{92} -12.0000 q^{94} -3.46410 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4}+O(q^{10}) \) 2 * q + 2 * q^4	\( 2 q + 2 q^{4} + 6 q^{10} + 2 q^{13} - 10 q^{16} - 4 q^{19} - 12 q^{22} - 4 q^{25} - 16 q^{31} - 18 q^{34} - 14 q^{37} - 6 q^{40} + 4 q^{43} + 12 q^{46} + 2 q^{52} - 12 q^{55} + 6 q^{58} + 14 q^{61} + 2 q^{64} - 20 q^{67} + 14 q^{73} - 4 q^{76} + 4 q^{79} + 24 q^{82} - 18 q^{85} + 12 q^{88} - 24 q^{94} - 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + 6 * q^10 + 2 * q^13 - 10 * q^16 - 4 * q^19 - 12 * q^22 - 4 * q^25 - 16 * q^31 - 18 * q^34 - 14 * q^37 - 6 * q^40 + 4 * q^43 + 12 * q^46 + 2 * q^52 - 12 * q^55 + 6 * q^58 + 14 * q^61 + 2 * q^64 - 20 * q^67 + 14 * q^73 - 4 * q^76 + 4 * q^79 + 24 * q^82 - 18 * q^85 + 12 * q^88 - 24 * q^94 - 4 * q^97

Introduction

L-functions

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