LMFDB - Embedded newform 9801.2.a.co.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9801,2,Mod(1,9801)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9801, 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("9801.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9801 = 3^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9801.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:	$78.2613790211$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 2 x^{17} - 22 x^{16} + 42 x^{15} + 198 x^{14} - 357 x^{13} - 944 x^{12} + 1579 x^{11} + \cdots - 55 $ x^18 - 2x^17 - 22x^16 + 42x^15 + 198x^14 - 357x^13 - 944x^12 + 1579x^11 + 2594x^10 - 3897x^9 - 4219x^8 + 5363x^7 + 4035x^6 - 3886x^5 - 2173x^4 + 1303x^3 + 576x^2 - 155*x - 55
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.677669$ of defining polynomial
Character	$\chi$	$=$	9801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.677669 q^{2} -1.54077 q^{4} +0.290795 q^{5} -3.35549 q^{7} +2.39947 q^{8} +O(q^{10})$	$q-0.677669 q^{2} -1.54077 q^{4} +0.290795 q^{5} -3.35549 q^{7} +2.39947 q^{8} -0.197063 q^{10} -4.18671 q^{13} +2.27391 q^{14} +1.45549 q^{16} -3.48327 q^{17} +6.07908 q^{19} -0.448047 q^{20} -6.69090 q^{23} -4.91544 q^{25} +2.83720 q^{26} +5.17002 q^{28} +1.65973 q^{29} +1.96301 q^{31} -5.78527 q^{32} +2.36050 q^{34} -0.975759 q^{35} -2.53648 q^{37} -4.11960 q^{38} +0.697752 q^{40} +6.48635 q^{41} -0.0495959 q^{43} +4.53422 q^{46} +1.34941 q^{47} +4.25930 q^{49} +3.33104 q^{50} +6.45074 q^{52} -1.87766 q^{53} -8.05138 q^{56} -1.12475 q^{58} -14.5741 q^{59} -3.85914 q^{61} -1.33027 q^{62} +1.00952 q^{64} -1.21747 q^{65} -8.93528 q^{67} +5.36690 q^{68} +0.661241 q^{70} -11.1283 q^{71} -1.25869 q^{73} +1.71889 q^{74} -9.36643 q^{76} -2.10713 q^{79} +0.423248 q^{80} -4.39559 q^{82} -9.37527 q^{83} -1.01292 q^{85} +0.0336096 q^{86} -8.41413 q^{89} +14.0485 q^{91} +10.3091 q^{92} -0.914452 q^{94} +1.76776 q^{95} +0.983484 q^{97} -2.88640 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 2 q^{2} + 12 q^{4} - q^{5} - q^{7} + 6 q^{8}+O(q^{10}) $ 18 * q + 2 * q^2 + 12 * q^4 - q^5 - q^7 + 6 * q^8	$ 18 q + 2 q^{2} + 12 q^{4} - q^{5} - q^{7} + 6 q^{8} + 2 q^{10} - 3 q^{13} - 8 q^{16} + 20 q^{17} + 3 q^{19} - 5 q^{20} - 10 q^{23} + 7 q^{25} + 2 q^{26} - 19 q^{28} + 21 q^{29} + 6 q^{31} + 9 q^{32} - 4 q^{34} + 38 q^{35} + 7 q^{37} + 13 q^{38} + 20 q^{41} - 4 q^{43} - 8 q^{46} + 7 q^{47} + 7 q^{49} + 25 q^{50} + 19 q^{52} + 31 q^{53} - 57 q^{56} + 12 q^{58} - 12 q^{59} + 16 q^{61} + 19 q^{62} - 16 q^{64} + 42 q^{65} - 5 q^{67} + 51 q^{68} + 8 q^{70} + 13 q^{71} + 9 q^{74} + 8 q^{76} + 2 q^{79} + 46 q^{80} - 34 q^{82} + 36 q^{83} - 25 q^{85} + 26 q^{86} - 14 q^{89} - 15 q^{91} - 15 q^{92} - 4 q^{94} + 64 q^{95} - 16 q^{97} + 82 q^{98}+O(q^{100}) $ 18 * q + 2 * q^2 + 12 * q^4 - q^5 - q^7 + 6 * q^8 + 2 * q^10 - 3 * q^13 - 8 * q^16 + 20 * q^17 + 3 * q^19 - 5 * q^20 - 10 * q^23 + 7 * q^25 + 2 * q^26 - 19 * q^28 + 21 * q^29 + 6 * q^31 + 9 * q^32 - 4 * q^34 + 38 * q^35 + 7 * q^37 + 13 * q^38 + 20 * q^41 - 4 * q^43 - 8 * q^46 + 7 * q^47 + 7 * q^49 + 25 * q^50 + 19 * q^52 + 31 * q^53 - 57 * q^56 + 12 * q^58 - 12 * q^59 + 16 * q^61 + 19 * q^62 - 16 * q^64 + 42 * q^65 - 5 * q^67 + 51 * q^68 + 8 * q^70 + 13 * q^71 + 9 * q^74 + 8 * q^76 + 2 * q^79 + 46 * q^80 - 34 * q^82 + 36 * q^83 - 25 * q^85 + 26 * q^86 - 14 * q^89 - 15 * q^91 - 15 * q^92 - 4 * q^94 + 64 * q^95 - 16 * q^97 + 82 * 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 $

$2$	−0.677669	−0.479184	−0.239592	−	0.970874i	$-0.577014\pi$
$2$	−0.677669	−0.479184	−0.239592	+	0.970874i	$0.577014\pi$
$3$	0	0
$3$	0	0
$4$	−1.54077	−0.770383
$5$	0.290795	0.130047	0.0650237	−	0.997884i	$-0.479288\pi$
$5$	0.290795	0.130047	0.0650237	+	0.997884i	$0.479288\pi$
$6$	0	0
$7$	−3.35549	−1.26826	−0.634128	−	0.773228i	$-0.718641\pi$
$7$	−3.35549	−1.26826	−0.634128	+	0.773228i	$0.718641\pi$
$8$	2.39947	0.848339
$9$	0	0
$10$	−0.197063	−0.0623166
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.18671	−1.16119	−0.580593	−	0.814194i	$-0.697179\pi$
$13$	−4.18671	−1.16119	−0.580593	+	0.814194i	$0.697179\pi$
$14$	2.27391	0.607728
$15$	0	0
$16$	1.45549	0.363872
$17$	−3.48327	−0.844818	−0.422409	−	0.906405i	$-0.638815\pi$
$17$	−3.48327	−0.844818	−0.422409	+	0.906405i	$0.638815\pi$
$18$	0	0
$19$	6.07908	1.39464	0.697318	−	0.716762i	$-0.254377\pi$
$19$	6.07908	1.39464	0.697318	+	0.716762i	$0.254377\pi$
$20$	−0.448047	−0.100186
$21$	0	0
$22$	0	0
$23$	−6.69090	−1.39515	−0.697575	−	0.716512i	$-0.745738\pi$
$23$	−6.69090	−1.39515	−0.697575	+	0.716512i	$0.745738\pi$
$24$	0	0
$25$	−4.91544	−0.983088
$26$	2.83720	0.556422
$27$	0	0
$28$	5.17002	0.977042
$29$	1.65973	0.308204	0.154102	−	0.988055i	$-0.450752\pi$
$29$	1.65973	0.308204	0.154102	+	0.988055i	$0.450752\pi$
$30$	0	0
$31$	1.96301	0.352567	0.176284	−	0.984339i	$-0.443592\pi$
$31$	1.96301	0.352567	0.176284	+	0.984339i	$0.443592\pi$
$32$	−5.78527	−1.02270
$33$	0	0
$34$	2.36050	0.404823
$35$	−0.975759	−0.164933
$36$	0	0
$37$	−2.53648	−0.416995	−0.208497	−	0.978023i	$-0.566857\pi$
$37$	−2.53648	−0.416995	−0.208497	+	0.978023i	$0.566857\pi$
$38$	−4.11960	−0.668287
$39$	0	0
$40$	0.697752	0.110324
$41$	6.48635	1.01300	0.506499	−	0.862241i	$-0.330940\pi$
$41$	6.48635	1.01300	0.506499	+	0.862241i	$0.330940\pi$
$42$	0	0
$43$	−0.0495959	−0.00756330	−0.00378165	−	0.999993i	$-0.501204\pi$
$43$	−0.0495959	−0.00756330	−0.00378165	+	0.999993i	$0.501204\pi$
$44$	0	0
$45$	0	0
$46$	4.53422	0.668534
$47$	1.34941	0.196831	0.0984157	−	0.995145i	$-0.468623\pi$
$47$	1.34941	0.196831	0.0984157	+	0.995145i	$0.468623\pi$
$48$	0	0
$49$	4.25930	0.608472
$50$	3.33104	0.471080
$51$	0	0
$52$	6.45074	0.894557
$53$	−1.87766	−0.257916	−0.128958	−	0.991650i	$-0.541163\pi$
$53$	−1.87766	−0.257916	−0.128958	+	0.991650i	$0.541163\pi$
$54$	0	0
$55$	0	0
$56$	−8.05138	−1.07591
$57$	0	0
$58$	−1.12475	−0.147686
$59$	−14.5741	−1.89739	−0.948694	−	0.316195i	$-0.897595\pi$
$59$	−14.5741	−1.89739	−0.948694	+	0.316195i	$0.897595\pi$
$60$	0	0
$61$	−3.85914	−0.494113	−0.247056	−	0.969001i	$-0.579463\pi$
$61$	−3.85914	−0.494113	−0.247056	+	0.969001i	$0.579463\pi$
$62$	−1.33027	−0.168945
$63$	0	0
$64$	1.00952	0.126190
$65$	−1.21747	−0.151009
$66$	0	0
$67$	−8.93528	−1.09162	−0.545809	−	0.837909i	$-0.683778\pi$
$67$	−8.93528	−1.09162	−0.545809	+	0.837909i	$0.683778\pi$
$68$	5.36690	0.650833
$69$	0	0
$70$	0.661241	0.0790334
$71$	−11.1283	−1.32068	−0.660340	−	0.750967i	$-0.729588\pi$
$71$	−11.1283	−1.32068	−0.660340	+	0.750967i	$0.729588\pi$
$72$	0	0
$73$	−1.25869	−0.147318	−0.0736591	−	0.997283i	$-0.523468\pi$
$73$	−1.25869	−0.147318	−0.0736591	+	0.997283i	$0.523468\pi$
$74$	1.71889	0.199817
$75$	0	0
$76$	−9.36643	−1.07440
$77$	0	0
$78$	0	0
$79$	−2.10713	−0.237071	−0.118535	−	0.992950i	$-0.537820\pi$
$79$	−2.10713	−0.237071	−0.118535	+	0.992950i	$0.537820\pi$
$80$	0.423248	0.0473206
$81$	0	0
$82$	−4.39559	−0.485412
$83$	−9.37527	−1.02907	−0.514535	−	0.857470i	$-0.672035\pi$
$83$	−9.37527	−1.02907	−0.514535	+	0.857470i	$0.672035\pi$
$84$	0	0
$85$	−1.01292	−0.109866
$86$	0.0336096	0.00362421
$87$	0	0
$88$	0	0
$89$	−8.41413	−0.891896	−0.445948	−	0.895059i	$-0.647133\pi$
$89$	−8.41413	−0.891896	−0.445948	+	0.895059i	$0.647133\pi$
$90$	0	0
$91$	14.0485	1.47268
$92$	10.3091	1.07480
$93$	0	0
$94$	−0.914452	−0.0943185
$95$	1.76776	0.181369
$96$	0	0
$97$	0.983484	0.0998577	0.0499288	−	0.998753i	$-0.484101\pi$
$97$	0.983484	0.0998577	0.0499288	+	0.998753i	$0.484101\pi$
$98$	−2.88640	−0.291570
$99$	0	0
$100$	7.57354	0.757354
$101$	3.91164	0.389223	0.194611	−	0.980880i	$-0.437655\pi$
$101$	3.91164	0.389223	0.194611	+	0.980880i	$0.437655\pi$
$102$	0	0
$103$	15.4623	1.52355	0.761775	−	0.647842i	$-0.224328\pi$
$103$	15.4623	1.52355	0.761775	+	0.647842i	$0.224328\pi$
$104$	−10.0459	−0.985079
$105$	0	0
$106$	1.27243	0.123589
$107$	−6.26446	−0.605609	−0.302804	−	0.953053i	$-0.597923\pi$
$107$	−6.26446	−0.605609	−0.302804	+	0.953053i	$0.597923\pi$
$108$	0	0
$109$	−3.09153	−0.296115	−0.148057	−	0.988979i	$-0.547302\pi$
$109$	−3.09153	−0.296115	−0.148057	+	0.988979i	$0.547302\pi$
$110$	0	0
$111$	0	0
$112$	−4.88387	−0.461482
$113$	15.9441	1.49990	0.749948	−	0.661497i	$-0.230079\pi$
$113$	15.9441	1.49990	0.749948	+	0.661497i	$0.230079\pi$
$114$	0	0
$115$	−1.94568	−0.181436
$116$	−2.55725	−0.237435
$117$	0	0
$118$	9.87642	0.909199
$119$	11.6881	1.07144
$120$	0	0
$121$	0	0
$122$	2.61522	0.236771
$123$	0	0
$124$	−3.02454	−0.271611
$125$	−2.88336	−0.257895
$126$	0	0
$127$	18.5061	1.64215	0.821075	−	0.570821i	$-0.193375\pi$
$127$	18.5061	1.64215	0.821075	+	0.570821i	$0.193375\pi$
$128$	10.8864	0.962233
$129$	0	0
$130$	0.825044	0.0723612
$131$	0.864643	0.0755442	0.0377721	−	0.999286i	$-0.487974\pi$
$131$	0.864643	0.0755442	0.0377721	+	0.999286i	$0.487974\pi$
$132$	0	0
$133$	−20.3983	−1.76875
$134$	6.05516	0.523086
$135$	0	0
$136$	−8.35799	−0.716692
$137$	−12.3782	−1.05754	−0.528769	−	0.848766i	$-0.677346\pi$
$137$	−12.3782	−1.05754	−0.528769	+	0.848766i	$0.677346\pi$
$138$	0	0
$139$	0.665969	0.0564868	0.0282434	−	0.999601i	$-0.491009\pi$
$139$	0.665969	0.0564868	0.0282434	+	0.999601i	$0.491009\pi$
$140$	1.50341	0.127062
$141$	0	0
$142$	7.54127	0.632849
$143$	0	0
$144$	0	0
$145$	0.482640	0.0400811
$146$	0.852974	0.0705926
$147$	0	0
$148$	3.90812	0.321245
$149$	12.0532	0.987440	0.493720	−	0.869621i	$-0.335637\pi$
$149$	12.0532	0.987440	0.493720	+	0.869621i	$0.335637\pi$
$150$	0	0
$151$	−20.5049	−1.66867	−0.834333	−	0.551260i	$-0.814147\pi$
$151$	−20.5049	−1.66867	−0.834333	+	0.551260i	$0.814147\pi$
$152$	14.5865	1.18312
$153$	0	0
$154$	0	0
$155$	0.570833	0.0458504
$156$	0	0
$157$	−8.45523	−0.674801	−0.337401	−	0.941361i	$-0.609548\pi$
$157$	−8.45523	−0.674801	−0.337401	+	0.941361i	$0.609548\pi$
$158$	1.42794	0.113601
$159$	0	0
$160$	−1.68233	−0.133000
$161$	22.4513	1.76941
$162$	0	0
$163$	−23.2731	−1.82289	−0.911446	−	0.411419i	$-0.865033\pi$
$163$	−23.2731	−1.82289	−0.911446	+	0.411419i	$0.865033\pi$
$164$	−9.99394	−0.780395
$165$	0	0
$166$	6.35333	0.493114
$167$	19.5773	1.51494	0.757469	−	0.652871i	$-0.226435\pi$
$167$	19.5773	1.51494	0.757469	+	0.652871i	$0.226435\pi$
$168$	0	0
$169$	4.52856	0.348351
$170$	0.686422	0.0526462
$171$	0	0
$172$	0.0764156	0.00582664
$173$	−17.9102	−1.36169	−0.680843	−	0.732429i	$-0.738386\pi$
$173$	−17.9102	−1.36169	−0.680843	+	0.732429i	$0.738386\pi$
$174$	0	0
$175$	16.4937	1.24681
$176$	0	0
$177$	0	0
$178$	5.70199	0.427382
$179$	9.44944	0.706284	0.353142	−	0.935570i	$-0.385113\pi$
$179$	9.44944	0.706284	0.353142	+	0.935570i	$0.385113\pi$
$180$	0	0
$181$	11.0798	0.823555	0.411777	−	0.911284i	$-0.364908\pi$
$181$	11.0798	0.823555	0.411777	+	0.911284i	$0.364908\pi$
$182$	−9.52021	−0.705685
$183$	0	0
$184$	−16.0546	−1.18356
$185$	−0.737595	−0.0542291
$186$	0	0
$187$	0	0
$188$	−2.07912	−0.151636
$189$	0	0
$190$	−1.19796	−0.0869090
$191$	−4.53719	−0.328300	−0.164150	−	0.986435i	$-0.552488\pi$
$191$	−4.53719	−0.328300	−0.164150	+	0.986435i	$0.552488\pi$
$192$	0	0
$193$	−13.4986	−0.971647	−0.485824	−	0.874057i	$-0.661480\pi$
$193$	−13.4986	−0.971647	−0.485824	+	0.874057i	$0.661480\pi$
$194$	−0.666476	−0.0478502
$195$	0	0
$196$	−6.56259	−0.468756
$197$	−1.70047	−0.121153	−0.0605766	−	0.998164i	$-0.519294\pi$
$197$	−1.70047	−0.121153	−0.0605766	+	0.998164i	$0.519294\pi$
$198$	0	0
$199$	−10.1442	−0.719105	−0.359552	−	0.933125i	$-0.617071\pi$
$199$	−10.1442	−0.719105	−0.359552	+	0.933125i	$0.617071\pi$
$200$	−11.7944	−0.833992
$201$	0	0
$202$	−2.65080	−0.186509
$203$	−5.56920	−0.390881
$204$	0	0
$205$	1.88620	0.131738
$206$	−10.4783	−0.730061
$207$	0	0
$208$	−6.09371	−0.422523
$209$	0	0
$210$	0	0
$211$	10.6754	0.734923	0.367461	−	0.930039i	$-0.380227\pi$
$211$	10.6754	0.734923	0.367461	+	0.930039i	$0.380227\pi$
$212$	2.89303	0.198694
$213$	0	0
$214$	4.24523	0.290198
$215$	−0.0144222	−0.000983588 0
$216$	0	0
$217$	−6.58686	−0.447145
$218$	2.09503	0.141893
$219$	0	0
$220$	0	0
$221$	14.5835	0.980990
$222$	0	0
$223$	2.83739	0.190006	0.0950028	−	0.995477i	$-0.469714\pi$
$223$	2.83739	0.190006	0.0950028	+	0.995477i	$0.469714\pi$
$224$	19.4124	1.29705
$225$	0	0
$226$	−10.8048	−0.718726
$227$	−6.48587	−0.430482	−0.215241	−	0.976561i	$-0.569054\pi$
$227$	−6.48587	−0.430482	−0.215241	+	0.976561i	$0.569054\pi$
$228$	0	0
$229$	−25.6403	−1.69436	−0.847181	−	0.531305i	$-0.821702\pi$
$229$	−25.6403	−1.69436	−0.847181	+	0.531305i	$0.821702\pi$
$230$	1.31853	0.0869411
$231$	0	0
$232$	3.98246	0.261461
$233$	0.649882	0.0425752	0.0212876	−	0.999773i	$-0.493223\pi$
$233$	0.649882	0.0425752	0.0212876	+	0.999773i	$0.493223\pi$
$234$	0	0
$235$	0.392401	0.0255974
$236$	22.4553	1.46172
$237$	0	0
$238$	−7.92065	−0.513419
$239$	16.9346	1.09541	0.547703	−	0.836673i	$-0.315502\pi$
$239$	16.9346	1.09541	0.547703	+	0.836673i	$0.315502\pi$
$240$	0	0
$241$	−24.4993	−1.57814	−0.789069	−	0.614305i	$-0.789437\pi$
$241$	−24.4993	−1.57814	−0.789069	+	0.614305i	$0.789437\pi$
$242$	0	0
$243$	0	0
$244$	5.94603	0.380656
$245$	1.23858	0.0791302
$246$	0	0
$247$	−25.4513	−1.61943
$248$	4.71018	0.299096
$249$	0	0
$250$	1.95396	0.123579
$251$	17.8710	1.12801	0.564005	−	0.825771i	$-0.309260\pi$
$251$	17.8710	1.12801	0.564005	+	0.825771i	$0.309260\pi$
$252$	0	0
$253$	0	0
$254$	−12.5410	−0.786892
$255$	0	0
$256$	−9.39643	−0.587277
$257$	−1.24675	−0.0777702	−0.0388851	−	0.999244i	$-0.512381\pi$
$257$	−1.24675	−0.0777702	−0.0388851	+	0.999244i	$0.512381\pi$
$258$	0	0
$259$	8.51113	0.528856
$260$	1.87584	0.116335
$261$	0	0
$262$	−0.585941	−0.0361996
$263$	16.7807	1.03474	0.517371	−	0.855761i	$-0.326911\pi$
$263$	16.7807	1.03474	0.517371	+	0.855761i	$0.326911\pi$
$264$	0	0
$265$	−0.546013	−0.0335413
$266$	13.8233	0.847559
$267$	0	0
$268$	13.7672	0.840964
$269$	21.9703	1.33955	0.669777	−	0.742563i	$-0.266390\pi$
$269$	21.9703	1.33955	0.669777	+	0.742563i	$0.266390\pi$
$270$	0	0
$271$	−10.1083	−0.614037	−0.307018	−	0.951704i	$-0.599331\pi$
$271$	−10.1083	−0.614037	−0.307018	+	0.951704i	$0.599331\pi$
$272$	−5.06986	−0.307405
$273$	0	0
$274$	8.38829	0.506755
$275$	0	0
$276$	0	0
$277$	−7.77311	−0.467041	−0.233520	−	0.972352i	$-0.575025\pi$
$277$	−7.77311	−0.467041	−0.233520	+	0.972352i	$0.575025\pi$
$278$	−0.451306	−0.0270676
$279$	0	0
$280$	−2.34130	−0.139919
$281$	32.2904	1.92628	0.963142	−	0.268995i	$-0.0866914\pi$
$281$	32.2904	1.92628	0.963142	+	0.268995i	$0.0866914\pi$
$282$	0	0
$283$	12.5231	0.744418	0.372209	−	0.928149i	$-0.378600\pi$
$283$	12.5231	0.744418	0.372209	+	0.928149i	$0.378600\pi$
$284$	17.1460	1.01743
$285$	0	0
$286$	0	0
$287$	−21.7649	−1.28474
$288$	0	0
$289$	−4.86682	−0.286283
$290$	−0.327070	−0.0192062
$291$	0	0
$292$	1.93934	0.113491
$293$	18.3844	1.07403	0.537013	−	0.843574i	$-0.319552\pi$
$293$	18.3844	1.07403	0.537013	+	0.843574i	$0.319552\pi$
$294$	0	0
$295$	−4.23808	−0.246750
$296$	−6.08619	−0.353753
$297$	0	0
$298$	−8.16810	−0.473166
$299$	28.0129	1.62003
$300$	0	0
$301$	0.166418	0.00959220
$302$	13.8955	0.799599
$303$	0	0
$304$	8.84802	0.507469
$305$	−1.12222	−0.0642581
$306$	0	0
$307$	−8.42389	−0.480777	−0.240388	−	0.970677i	$-0.577275\pi$
$307$	−8.42389	−0.480777	−0.240388	+	0.970677i	$0.577275\pi$
$308$	0	0
$309$	0	0
$310$	−0.386836	−0.0219708
$311$	11.3034	0.640954	0.320477	−	0.947256i	$-0.396157\pi$
$311$	11.3034	0.640954	0.320477	+	0.947256i	$0.396157\pi$
$312$	0	0
$313$	6.11064	0.345394	0.172697	−	0.984975i	$-0.444752\pi$
$313$	6.11064	0.345394	0.172697	+	0.984975i	$0.444752\pi$
$314$	5.72985	0.323354
$315$	0	0
$316$	3.24660	0.182635
$317$	−0.828576	−0.0465374	−0.0232687	−	0.999729i	$-0.507407\pi$
$317$	−0.828576	−0.0465374	−0.0232687	+	0.999729i	$0.507407\pi$
$318$	0	0
$319$	0	0
$320$	0.293564	0.0164107
$321$	0	0
$322$	−15.2145	−0.847872
$323$	−21.1751	−1.17821
$324$	0	0
$325$	20.5795	1.14155
$326$	15.7715	0.873501
$327$	0	0
$328$	15.5638	0.859365
$329$	−4.52792	−0.249633
$330$	0	0
$331$	8.18453	0.449863	0.224931	−	0.974375i	$-0.427784\pi$
$331$	8.18453	0.449863	0.224931	+	0.974375i	$0.427784\pi$
$332$	14.4451	0.792777
$333$	0	0
$334$	−13.2669	−0.725935
$335$	−2.59833	−0.141962
$336$	0	0
$337$	2.50485	0.136448	0.0682239	−	0.997670i	$-0.478267\pi$
$337$	2.50485	0.136448	0.0682239	+	0.997670i	$0.478267\pi$
$338$	−3.06887	−0.166924
$339$	0	0
$340$	1.56067	0.0846391
$341$	0	0
$342$	0	0
$343$	9.19637	0.496558
$344$	−0.119004	−0.00641625
$345$	0	0
$346$	12.1372	0.652498
$347$	31.5264	1.69243	0.846213	−	0.532844i	$-0.178877\pi$
$347$	31.5264	1.69243	0.846213	+	0.532844i	$0.178877\pi$
$348$	0	0
$349$	8.75192	0.468480	0.234240	−	0.972179i	$-0.424740\pi$
$349$	8.75192	0.468480	0.234240	+	0.972179i	$0.424740\pi$
$350$	−11.1773	−0.597450
$351$	0	0
$352$	0	0
$353$	0.439863	0.0234115	0.0117058	−	0.999931i	$-0.496274\pi$
$353$	0.439863	0.0234115	0.0117058	+	0.999931i	$0.496274\pi$
$354$	0	0
$355$	−3.23604	−0.171751
$356$	12.9642	0.687101
$357$	0	0
$358$	−6.40359	−0.338440
$359$	−16.7381	−0.883404	−0.441702	−	0.897162i	$-0.645625\pi$
$359$	−16.7381	−0.883404	−0.441702	+	0.897162i	$0.645625\pi$
$360$	0	0
$361$	17.9552	0.945009
$362$	−7.50843	−0.394634
$363$	0	0
$364$	−21.6454	−1.13453
$365$	−0.366020	−0.0191584
$366$	0	0
$367$	−14.7488	−0.769880	−0.384940	−	0.922942i	$-0.625778\pi$
$367$	−14.7488	−0.769880	−0.384940	+	0.922942i	$0.625778\pi$
$368$	−9.73853	−0.507656
$369$	0	0
$370$	0.499845	0.0259857
$371$	6.30046	0.327103
$372$	0	0
$373$	−23.6508	−1.22459	−0.612296	−	0.790629i	$-0.709754\pi$
$373$	−23.6508	−1.22459	−0.612296	+	0.790629i	$0.709754\pi$
$374$	0	0
$375$	0	0
$376$	3.23786	0.166980
$377$	−6.94880	−0.357882
$378$	0	0
$379$	−11.4766	−0.589514	−0.294757	−	0.955572i	$-0.595239\pi$
$379$	−11.4766	−0.589514	−0.294757	+	0.955572i	$0.595239\pi$
$380$	−2.72371	−0.139723
$381$	0	0
$382$	3.07471	0.157316
$383$	−7.92867	−0.405136	−0.202568	−	0.979268i	$-0.564929\pi$
$383$	−7.92867	−0.405136	−0.202568	+	0.979268i	$0.564929\pi$
$384$	0	0
$385$	0	0
$386$	9.14755	0.465598
$387$	0	0
$388$	−1.51532	−0.0769286
$389$	9.05099	0.458903	0.229452	−	0.973320i	$-0.426307\pi$
$389$	9.05099	0.458903	0.229452	+	0.973320i	$0.426307\pi$
$390$	0	0
$391$	23.3062	1.17865
$392$	10.2201	0.516191
$393$	0	0
$394$	1.15235	0.0580547
$395$	−0.612743	−0.0308305
$396$	0	0
$397$	5.38182	0.270106	0.135053	−	0.990838i	$-0.456880\pi$
$397$	5.38182	0.270106	0.135053	+	0.990838i	$0.456880\pi$
$398$	6.87442	0.344584
$399$	0	0
$400$	−7.15436	−0.357718
$401$	−15.6683	−0.782438	−0.391219	−	0.920298i	$-0.627947\pi$
$401$	−15.6683	−0.782438	−0.391219	+	0.920298i	$0.627947\pi$
$402$	0	0
$403$	−8.21856	−0.409396
$404$	−6.02692	−0.299850
$405$	0	0
$406$	3.77407	0.187304
$407$	0	0
$408$	0	0
$409$	22.7413	1.12448	0.562242	−	0.826973i	$-0.309939\pi$
$409$	22.7413	1.12448	0.562242	+	0.826973i	$0.309939\pi$
$410$	−1.27822	−0.0631266
$411$	0	0
$412$	−23.8238	−1.17372
$413$	48.9033	2.40637
$414$	0	0
$415$	−2.72628	−0.133828
$416$	24.2213	1.18755
$417$	0	0
$418$	0	0
$419$	−39.0263	−1.90656	−0.953279	−	0.302090i	$-0.902316\pi$
$419$	−39.0263	−1.90656	−0.953279	+	0.302090i	$0.902316\pi$
$420$	0	0
$421$	24.1875	1.17883	0.589414	−	0.807831i	$-0.299359\pi$
$421$	24.1875	1.17883	0.589414	+	0.807831i	$0.299359\pi$
$422$	−7.23436	−0.352163
$423$	0	0
$424$	−4.50537	−0.218800
$425$	17.1218	0.830530
$426$	0	0
$427$	12.9493	0.626661
$428$	9.65207	0.466550
$429$	0	0
$430$	0.00977349	0.000471320 0
$431$	12.2455	0.589844	0.294922	−	0.955521i	$-0.404706\pi$
$431$	12.2455	0.589844	0.294922	+	0.955521i	$0.404706\pi$
$432$	0	0
$433$	−7.13183	−0.342734	−0.171367	−	0.985207i	$-0.554818\pi$
$433$	−7.13183	−0.342734	−0.171367	+	0.985207i	$0.554818\pi$
$434$	4.46371	0.214265
$435$	0	0
$436$	4.76332	0.228122
$437$	−40.6745	−1.94573
$438$	0	0
$439$	19.1944	0.916099	0.458049	−	0.888927i	$-0.348548\pi$
$439$	19.1944	0.916099	0.458049	+	0.888927i	$0.348548\pi$
$440$	0	0
$441$	0	0
$442$	−9.88275	−0.470075
$443$	−9.65055	−0.458511	−0.229256	−	0.973366i	$-0.573629\pi$
$443$	−9.65055	−0.458511	−0.229256	+	0.973366i	$0.573629\pi$
$444$	0	0
$445$	−2.44678	−0.115989
$446$	−1.92281	−0.0910477
$447$	0	0
$448$	−3.38744	−0.160041
$449$	−15.3818	−0.725912	−0.362956	−	0.931806i	$-0.618233\pi$
$449$	−15.3818	−0.725912	−0.362956	+	0.931806i	$0.618233\pi$
$450$	0	0
$451$	0	0
$452$	−24.5661	−1.15549
$453$	0	0
$454$	4.39527	0.206280
$455$	4.08522	0.191518
$456$	0	0
$457$	19.0977	0.893355	0.446678	−	0.894695i	$-0.352607\pi$
$457$	19.0977	0.893355	0.446678	+	0.894695i	$0.352607\pi$
$458$	17.3757	0.811911
$459$	0	0
$460$	2.99784	0.139775
$461$	−8.37180	−0.389914	−0.194957	−	0.980812i	$-0.562457\pi$
$461$	−8.37180	−0.389914	−0.194957	+	0.980812i	$0.562457\pi$
$462$	0	0
$463$	28.2595	1.31333	0.656665	−	0.754182i	$-0.271966\pi$
$463$	28.2595	1.31333	0.656665	+	0.754182i	$0.271966\pi$
$464$	2.41571	0.112147
$465$	0	0
$466$	−0.440405	−0.0204014
$467$	−8.22408	−0.380565	−0.190282	−	0.981729i	$-0.560940\pi$
$467$	−8.22408	−0.380565	−0.190282	+	0.981729i	$0.560940\pi$
$468$	0	0
$469$	29.9822	1.38445
$470$	−0.265918	−0.0122659
$471$	0	0
$472$	−34.9701	−1.60963
$473$	0	0
$474$	0	0
$475$	−29.8813	−1.37105
$476$	−18.0086	−0.825422
$477$	0	0
$478$	−11.4760	−0.524901
$479$	30.4221	1.39002	0.695012	−	0.718999i	$-0.255399\pi$
$479$	30.4221	1.39002	0.695012	+	0.718999i	$0.255399\pi$
$480$	0	0
$481$	10.6195	0.484208
$482$	16.6024	0.756219
$483$	0	0
$484$	0	0
$485$	0.285992	0.0129862
$486$	0	0
$487$	−22.2893	−1.01002	−0.505012	−	0.863112i	$-0.668512\pi$
$487$	−22.2893	−1.01002	−0.505012	+	0.863112i	$0.668512\pi$
$488$	−9.25988	−0.419175
$489$	0	0
$490$	−0.839349	−0.0379179
$491$	32.3658	1.46065	0.730325	−	0.683100i	$-0.239369\pi$
$491$	32.3658	1.46065	0.730325	+	0.683100i	$0.239369\pi$
$492$	0	0
$493$	−5.78128	−0.260376
$494$	17.2476	0.776005
$495$	0	0
$496$	2.85714	0.128289
$497$	37.3407	1.67496
$498$	0	0
$499$	0.261774	0.0117186	0.00585931	−	0.999983i	$-0.498135\pi$
$499$	0.261774	0.0117186	0.00585931	+	0.999983i	$0.498135\pi$
$500$	4.44258	0.198678
$501$	0	0
$502$	−12.1106	−0.540524
$503$	18.0436	0.804525	0.402263	−	0.915524i	$-0.368224\pi$
$503$	18.0436	0.804525	0.402263	+	0.915524i	$0.368224\pi$
$504$	0	0
$505$	1.13748	0.0506174
$506$	0	0
$507$	0	0
$508$	−28.5135	−1.26508
$509$	−4.25278	−0.188501	−0.0942506	−	0.995549i	$-0.530045\pi$
$509$	−4.25278	−0.188501	−0.0942506	+	0.995549i	$0.530045\pi$
$510$	0	0
$511$	4.22351	0.186837
$512$	−15.4052	−0.680819
$513$	0	0
$514$	0.844884	0.0372662
$515$	4.49637	0.198134
$516$	0	0
$517$	0	0
$518$	−5.76772	−0.253419
$519$	0	0
$520$	−2.92129	−0.128107
$521$	13.7326	0.601635	0.300817	−	0.953682i	$-0.402741\pi$
$521$	13.7326	0.601635	0.300817	+	0.953682i	$0.402741\pi$
$522$	0	0
$523$	−26.0362	−1.13849	−0.569243	−	0.822170i	$-0.692764\pi$
$523$	−26.0362	−1.13849	−0.569243	+	0.822170i	$0.692764\pi$
$524$	−1.33221	−0.0581979
$525$	0	0
$526$	−11.3718	−0.495832
$527$	−6.83770	−0.297855
$528$	0	0
$529$	21.7682	0.946444
$530$	0.370016	0.0160725
$531$	0	0
$532$	31.4289	1.36262
$533$	−27.1565	−1.17628
$534$	0	0
$535$	−1.82167	−0.0787578
$536$	−21.4399	−0.926063
$537$	0	0
$538$	−14.8886	−0.641893
$539$	0	0
$540$	0	0
$541$	−15.1235	−0.650211	−0.325106	−	0.945678i	$-0.605400\pi$
$541$	−15.1235	−0.650211	−0.325106	+	0.945678i	$0.605400\pi$
$542$	6.85009	0.294237
$543$	0	0
$544$	20.1517	0.863996
$545$	−0.899000	−0.0385089
$546$	0	0
$547$	22.2430	0.951042	0.475521	−	0.879704i	$-0.342260\pi$
$547$	22.2430	0.951042	0.475521	+	0.879704i	$0.342260\pi$
$548$	19.0718	0.814709
$549$	0	0
$550$	0	0
$551$	10.0896	0.429832
$552$	0	0
$553$	7.07046	0.300667
$554$	5.26759	0.223798
$555$	0	0
$556$	−1.02610	−0.0435164
$557$	35.8076	1.51722	0.758609	−	0.651546i	$-0.225879\pi$
$557$	35.8076	1.51722	0.758609	+	0.651546i	$0.225879\pi$
$558$	0	0
$559$	0.207644	0.00878239
$560$	−1.42020	−0.0600146
$561$	0	0
$562$	−21.8822	−0.923044
$563$	−33.6120	−1.41658	−0.708289	−	0.705922i	$-0.750533\pi$
$563$	−33.6120	−1.41658	−0.708289	+	0.705922i	$0.750533\pi$
$564$	0	0
$565$	4.63646	0.195058
$566$	−8.48648	−0.356713
$567$	0	0
$568$	−26.7019	−1.12039
$569$	22.7477	0.953635	0.476817	−	0.879002i	$-0.341790\pi$
$569$	22.7477	0.953635	0.476817	+	0.879002i	$0.341790\pi$
$570$	0	0
$571$	35.3194	1.47807	0.739035	−	0.673667i	$-0.235282\pi$
$571$	35.3194	1.47807	0.739035	+	0.673667i	$0.235282\pi$
$572$	0	0
$573$	0	0
$574$	14.7494	0.615627
$575$	32.8887	1.37155
$576$	0	0
$577$	29.5174	1.22883	0.614413	−	0.788984i	$-0.289393\pi$
$577$	29.5174	1.22883	0.614413	+	0.788984i	$0.289393\pi$
$578$	3.29809	0.137182
$579$	0	0
$580$	−0.743635	−0.0308778
$581$	31.4586	1.30512
$582$	0	0
$583$	0	0
$584$	−3.02018	−0.124976
$585$	0	0
$586$	−12.4585	−0.514657
$587$	−33.0777	−1.36526	−0.682632	−	0.730762i	$-0.739165\pi$
$587$	−33.0777	−1.36526	−0.682632	+	0.730762i	$0.739165\pi$
$588$	0	0
$589$	11.9333	0.491703
$590$	2.87201	0.118239
$591$	0	0
$592$	−3.69181	−0.151733
$593$	9.74358	0.400121	0.200060	−	0.979784i	$-0.435886\pi$
$593$	9.74358	0.400121	0.200060	+	0.979784i	$0.435886\pi$
$594$	0	0
$595$	3.39883	0.139339
$596$	−18.5712	−0.760706
$597$	0	0
$598$	−18.9835	−0.776292
$599$	27.8578	1.13824	0.569120	−	0.822255i	$-0.307284\pi$
$599$	27.8578	1.13824	0.569120	+	0.822255i	$0.307284\pi$
$600$	0	0
$601$	21.3192	0.869630	0.434815	−	0.900520i	$-0.356814\pi$
$601$	21.3192	0.869630	0.434815	+	0.900520i	$0.356814\pi$
$602$	−0.112777	−0.00459643
$603$	0	0
$604$	31.5933	1.28551
$605$	0	0
$606$	0	0
$607$	−19.1431	−0.776994	−0.388497	−	0.921450i	$-0.627006\pi$
$607$	−19.1431	−0.776994	−0.388497	+	0.921450i	$0.627006\pi$
$608$	−35.1691	−1.42630
$609$	0	0
$610$	0.760492	0.0307914
$611$	−5.64959	−0.228558
$612$	0	0
$613$	−6.05388	−0.244514	−0.122257	−	0.992498i	$-0.539013\pi$
$613$	−6.05388	−0.244514	−0.122257	+	0.992498i	$0.539013\pi$
$614$	5.70860	0.230380
$615$	0	0
$616$	0	0
$617$	−30.3264	−1.22090	−0.610448	−	0.792057i	$-0.709010\pi$
$617$	−30.3264	−1.22090	−0.610448	+	0.792057i	$0.709010\pi$
$618$	0	0
$619$	−4.66068	−0.187328	−0.0936642	−	0.995604i	$-0.529858\pi$
$619$	−4.66068	−0.187328	−0.0936642	+	0.995604i	$0.529858\pi$
$620$	−0.879520	−0.0353224
$621$	0	0
$622$	−7.65993	−0.307135
$623$	28.2335	1.13115
$624$	0	0
$625$	23.7387	0.949549
$626$	−4.14099	−0.165507
$627$	0	0
$628$	13.0275	0.519855
$629$	8.83525	0.352284
$630$	0	0
$631$	49.0740	1.95360	0.976802	−	0.214146i	$-0.0686968\pi$
$631$	49.0740	1.95360	0.976802	+	0.214146i	$0.0686968\pi$
$632$	−5.05599	−0.201117
$633$	0	0
$634$	0.561500	0.0223000
$635$	5.38147	0.213557
$636$	0	0
$637$	−17.8325	−0.706549
$638$	0	0
$639$	0	0
$640$	3.16571	0.125136
$641$	28.7227	1.13448	0.567239	−	0.823553i	$-0.308012\pi$
$641$	28.7227	1.13448	0.567239	+	0.823553i	$0.308012\pi$
$642$	0	0
$643$	13.0211	0.513501	0.256751	−	0.966478i	$-0.417348\pi$
$643$	13.0211	0.513501	0.256751	+	0.966478i	$0.417348\pi$
$644$	−34.5921	−1.36312
$645$	0	0
$646$	14.3497	0.564581
$647$	36.6127	1.43939	0.719696	−	0.694289i	$-0.244281\pi$
$647$	36.6127	1.43939	0.719696	+	0.694289i	$0.244281\pi$
$648$	0	0
$649$	0	0
$650$	−13.9461	−0.547011
$651$	0	0
$652$	35.8584	1.40432
$653$	−11.6694	−0.456658	−0.228329	−	0.973584i	$-0.573326\pi$
$653$	−11.6694	−0.456658	−0.228329	+	0.973584i	$0.573326\pi$
$654$	0	0
$655$	0.251434	0.00982432
$656$	9.44080	0.368601
$657$	0	0
$658$	3.06843	0.119620
$659$	39.7322	1.54775	0.773873	−	0.633341i	$-0.218317\pi$
$659$	39.7322	1.54775	0.773873	+	0.633341i	$0.218317\pi$
$660$	0	0
$661$	21.4819	0.835550	0.417775	−	0.908551i	$-0.362810\pi$
$661$	21.4819	0.835550	0.417775	+	0.908551i	$0.362810\pi$
$662$	−5.54640	−0.215567
$663$	0	0
$664$	−22.4956	−0.873000
$665$	−5.93171	−0.230022
$666$	0	0
$667$	−11.1051	−0.429990
$668$	−30.1641	−1.16708
$669$	0	0
$670$	1.76081	0.0680260
$671$	0	0
$672$	0	0
$673$	20.5108	0.790632	0.395316	−	0.918545i	$-0.370635\pi$
$673$	20.5108	0.790632	0.395316	+	0.918545i	$0.370635\pi$
$674$	−1.69746	−0.0653836
$675$	0	0
$676$	−6.97745	−0.268364
$677$	−1.18298	−0.0454658	−0.0227329	−	0.999742i	$-0.507237\pi$
$677$	−1.18298	−0.0454658	−0.0227329	+	0.999742i	$0.507237\pi$
$678$	0	0
$679$	−3.30007	−0.126645
$680$	−2.43046	−0.0932039
$681$	0	0
$682$	0	0
$683$	−6.06013	−0.231884	−0.115942	−	0.993256i	$-0.536989\pi$
$683$	−6.06013	−0.231884	−0.115942	+	0.993256i	$0.536989\pi$
$684$	0	0
$685$	−3.59951	−0.137530
$686$	−6.23210	−0.237943
$687$	0	0
$688$	−0.0721862	−0.00275207
$689$	7.86121	0.299488
$690$	0	0
$691$	42.8292	1.62930	0.814650	−	0.579953i	$-0.196929\pi$
$691$	42.8292	1.62930	0.814650	+	0.579953i	$0.196929\pi$
$692$	27.5954	1.04902
$693$	0	0
$694$	−21.3645	−0.810984
$695$	0.193660	0.00734596
$696$	0	0
$697$	−22.5937	−0.855798
$698$	−5.93091	−0.224488
$699$	0	0
$700$	−25.4129	−0.960518
$701$	3.89209	0.147002	0.0735011	−	0.997295i	$-0.476583\pi$
$701$	3.89209	0.147002	0.0735011	+	0.997295i	$0.476583\pi$
$702$	0	0
$703$	−15.4195	−0.581556
$704$	0	0
$705$	0	0
$706$	−0.298081	−0.0112184
$707$	−13.1255	−0.493634
$708$	0	0
$709$	22.1493	0.831833	0.415917	−	0.909403i	$-0.363461\pi$
$709$	22.1493	0.831833	0.415917	+	0.909403i	$0.363461\pi$
$710$	2.19296	0.0823004
$711$	0	0
$712$	−20.1894	−0.756630
$713$	−13.1343	−0.491884
$714$	0	0
$715$	0	0
$716$	−14.5594	−0.544109
$717$	0	0
$718$	11.3429	0.423313
$719$	−37.5592	−1.40072	−0.700362	−	0.713788i	$-0.746978\pi$
$719$	−37.5592	−1.40072	−0.700362	+	0.713788i	$0.746978\pi$
$720$	0	0
$721$	−51.8837	−1.93225
$722$	−12.1677	−0.452833
$723$	0	0
$724$	−17.0714	−0.634452
$725$	−8.15829	−0.302991
$726$	0	0
$727$	−28.4777	−1.05618	−0.528090	−	0.849189i	$-0.677092\pi$
$727$	−28.4777	−1.05618	−0.528090	+	0.849189i	$0.677092\pi$
$728$	33.7088	1.24933
$729$	0	0
$730$	0.248040	0.00918038
$731$	0.172756	0.00638961
$732$	0	0
$733$	−15.8964	−0.587148	−0.293574	−	0.955936i	$-0.594845\pi$
$733$	−15.8964	−0.587148	−0.293574	+	0.955936i	$0.594845\pi$
$734$	9.99478	0.368914
$735$	0	0
$736$	38.7087	1.42682
$737$	0	0
$738$	0	0
$739$	−36.3058	−1.33553	−0.667766	−	0.744371i	$-0.732749\pi$
$739$	−36.3058	−1.33553	−0.667766	+	0.744371i	$0.732749\pi$
$740$	1.13646	0.0417771
$741$	0	0
$742$	−4.26962	−0.156743
$743$	−41.8306	−1.53462	−0.767308	−	0.641279i	$-0.778404\pi$
$743$	−41.8306	−1.53462	−0.767308	+	0.641279i	$0.778404\pi$
$744$	0	0
$745$	3.50502	0.128414
$746$	16.0274	0.586805
$747$	0	0
$748$	0	0
$749$	21.0203	0.768066
$750$	0	0
$751$	17.9957	0.656673	0.328337	−	0.944561i	$-0.393512\pi$
$751$	17.9957	0.656673	0.328337	+	0.944561i	$0.393512\pi$
$752$	1.96405	0.0716214
$753$	0	0
$754$	4.70899	0.171491
$755$	−5.96272	−0.217006
$756$	0	0
$757$	27.2640	0.990928	0.495464	−	0.868628i	$-0.334998\pi$
$757$	27.2640	0.990928	0.495464	+	0.868628i	$0.334998\pi$
$758$	7.77735	0.282486
$759$	0	0
$760$	4.24169	0.153862
$761$	0.524438	0.0190109	0.00950543	−	0.999955i	$-0.496974\pi$
$761$	0.524438	0.0190109	0.00950543	+	0.999955i	$0.496974\pi$
$762$	0	0
$763$	10.3736	0.375549
$764$	6.99075	0.252916
$765$	0	0
$766$	5.37301	0.194135
$767$	61.0177	2.20322
$768$	0	0
$769$	1.19450	0.0430748	0.0215374	−	0.999768i	$-0.493144\pi$
$769$	1.19450	0.0430748	0.0215374	+	0.999768i	$0.493144\pi$
$770$	0	0
$771$	0	0
$772$	20.7981	0.748540
$773$	28.6486	1.03042	0.515209	−	0.857065i	$-0.327714\pi$
$773$	28.6486	1.03042	0.515209	+	0.857065i	$0.327714\pi$
$774$	0	0
$775$	−9.64906	−0.346604
$776$	2.35984	0.0847132
$777$	0	0
$778$	−6.13357	−0.219899
$779$	39.4310	1.41276
$780$	0	0
$781$	0	0
$782$	−15.7939	−0.564789
$783$	0	0
$784$	6.19936	0.221406
$785$	−2.45874	−0.0877561
$786$	0	0
$787$	−11.5076	−0.410202	−0.205101	−	0.978741i	$-0.565752\pi$
$787$	−11.5076	−0.410202	−0.205101	+	0.978741i	$0.565752\pi$
$788$	2.62002	0.0933342
$789$	0	0
$790$	0.415237	0.0147735
$791$	−53.5003	−1.90225
$792$	0	0
$793$	16.1571	0.573756
$794$	−3.64709	−0.129430
$795$	0	0
$796$	15.6299	0.553986
$797$	1.34078	0.0474930	0.0237465	−	0.999718i	$-0.492441\pi$
$797$	1.34078	0.0474930	0.0237465	+	0.999718i	$0.492441\pi$
$798$	0	0
$799$	−4.70036	−0.166287
$800$	28.4371	1.00540
$801$	0	0
$802$	10.6179	0.374932
$803$	0	0
$804$	0	0
$805$	6.52871	0.230107
$806$	5.56946	0.196176
$807$	0	0
$808$	9.38584	0.330193
$809$	29.7375	1.04551	0.522757	−	0.852482i	$-0.324904\pi$
$809$	29.7375	1.04551	0.522757	+	0.852482i	$0.324904\pi$
$810$	0	0
$811$	−28.5904	−1.00395	−0.501973	−	0.864883i	$-0.667392\pi$
$811$	−28.5904	−1.00395	−0.501973	+	0.864883i	$0.667392\pi$
$812$	8.58082	0.301128
$813$	0	0
$814$	0	0
$815$	−6.76771	−0.237062
$816$	0	0
$817$	−0.301497	−0.0105481
$818$	−15.4111	−0.538835
$819$	0	0
$820$	−2.90619	−0.101488
$821$	15.5398	0.542341	0.271171	−	0.962531i	$-0.412589\pi$
$821$	15.5398	0.542341	0.271171	+	0.962531i	$0.412589\pi$
$822$	0	0
$823$	−40.5777	−1.41445	−0.707225	−	0.706989i	$-0.750053\pi$
$823$	−40.5777	−1.41445	−0.707225	+	0.706989i	$0.750053\pi$
$824$	37.1014	1.29249
$825$	0	0
$826$	−33.1402	−1.15310
$827$	2.54353	0.0884472	0.0442236	−	0.999022i	$-0.485919\pi$
$827$	2.54353	0.0884472	0.0442236	+	0.999022i	$0.485919\pi$
$828$	0	0
$829$	−14.3710	−0.499127	−0.249564	−	0.968358i	$-0.580287\pi$
$829$	−14.3710	−0.499127	−0.249564	+	0.968358i	$0.580287\pi$
$830$	1.84751	0.0641282
$831$	0	0
$832$	−4.22658	−0.146530
$833$	−14.8363	−0.514048
$834$	0	0
$835$	5.69298	0.197014
$836$	0	0
$837$	0	0
$838$	26.4469	0.913593
$839$	37.5810	1.29744	0.648721	−	0.761026i	$-0.275304\pi$
$839$	37.5810	1.29744	0.648721	+	0.761026i	$0.275304\pi$
$840$	0	0
$841$	−26.2453	−0.905010
$842$	−16.3911	−0.564876
$843$	0	0
$844$	−16.4482	−0.566172
$845$	1.31688	0.0453022
$846$	0	0
$847$	0	0
$848$	−2.73291	−0.0938484
$849$	0	0
$850$	−11.6029	−0.397977
$851$	16.9713	0.581770
$852$	0	0
$853$	−35.1157	−1.20234	−0.601170	−	0.799121i	$-0.705298\pi$
$853$	−35.1157	−1.20234	−0.601170	+	0.799121i	$0.705298\pi$
$854$	−8.77534	−0.300286
$855$	0	0
$856$	−15.0314	−0.513762
$857$	−29.0102	−0.990971	−0.495486	−	0.868616i	$-0.665010\pi$
$857$	−29.0102	−0.990971	−0.495486	+	0.868616i	$0.665010\pi$
$858$	0	0
$859$	−3.91781	−0.133674	−0.0668370	−	0.997764i	$-0.521291\pi$
$859$	−3.91781	−0.133674	−0.0668370	+	0.997764i	$0.521291\pi$
$860$	0.0222213	0.000757739 0
$861$	0	0
$862$	−8.29838	−0.282644
$863$	−15.5259	−0.528507	−0.264253	−	0.964453i	$-0.585126\pi$
$863$	−15.5259	−0.528507	−0.264253	+	0.964453i	$0.585126\pi$
$864$	0	0
$865$	−5.20819	−0.177084
$866$	4.83302	0.164233
$867$	0	0
$868$	10.1488	0.344473
$869$	0	0
$870$	0	0
$871$	37.4095	1.26757
$872$	−7.41801	−0.251206
$873$	0	0
$874$	27.5639	0.932361
$875$	9.67508	0.327077
$876$	0	0
$877$	34.0243	1.14892	0.574460	−	0.818533i	$-0.305212\pi$
$877$	34.0243	1.14892	0.574460	+	0.818533i	$0.305212\pi$
$878$	−13.0074	−0.438980
$879$	0	0
$880$	0	0
$881$	−24.8513	−0.837261	−0.418630	−	0.908157i	$-0.637490\pi$
$881$	−24.8513	−0.837261	−0.418630	+	0.908157i	$0.637490\pi$
$882$	0	0
$883$	23.5034	0.790952	0.395476	−	0.918476i	$-0.370580\pi$
$883$	23.5034	0.790952	0.395476	+	0.918476i	$0.370580\pi$
$884$	−22.4697	−0.755737
$885$	0	0
$886$	6.53988	0.219711
$887$	−37.9743	−1.27505	−0.637527	−	0.770428i	$-0.720042\pi$
$887$	−37.9743	−1.27505	−0.637527	+	0.770428i	$0.720042\pi$
$888$	0	0
$889$	−62.0970	−2.08266
$890$	1.65811	0.0555800
$891$	0	0
$892$	−4.37175	−0.146377
$893$	8.20316	0.274508
$894$	0	0
$895$	2.74785	0.0918504
$896$	−36.5293	−1.22036
$897$	0	0
$898$	10.4238	0.347846
$899$	3.25806	0.108662
$900$	0	0
$901$	6.54039	0.217892
$902$	0	0
$903$	0	0
$904$	38.2573	1.27242
$905$	3.22195	0.107101
$906$	0	0
$907$	−18.3515	−0.609352	−0.304676	−	0.952456i	$-0.598548\pi$
$907$	−18.3515	−0.609352	−0.304676	+	0.952456i	$0.598548\pi$
$908$	9.99321	0.331636
$909$	0	0
$910$	−2.76843	−0.0917725
$911$	2.10960	0.0698943	0.0349472	−	0.999389i	$-0.488874\pi$
$911$	2.10960	0.0698943	0.0349472	+	0.999389i	$0.488874\pi$
$912$	0	0
$913$	0	0
$914$	−12.9419	−0.428082
$915$	0	0
$916$	39.5057	1.30531
$917$	−2.90130	−0.0958093
$918$	0	0
$919$	−11.0435	−0.364290	−0.182145	−	0.983272i	$-0.558304\pi$
$919$	−11.0435	−0.364290	−0.182145	+	0.983272i	$0.558304\pi$
$920$	−4.66859	−0.153919
$921$	0	0
$922$	5.67331	0.186841
$923$	46.5908	1.53355
$924$	0	0
$925$	12.4679	0.409942
$926$	−19.1506	−0.629327
$927$	0	0
$928$	−9.60197	−0.315200
$929$	15.5418	0.509912	0.254956	−	0.966953i	$-0.417939\pi$
$929$	15.5418	0.509912	0.254956	+	0.966953i	$0.417939\pi$
$930$	0	0
$931$	25.8926	0.848597
$932$	−1.00132	−0.0327992
$933$	0	0
$934$	5.57320	0.182361
$935$	0	0
$936$	0	0
$937$	−22.4663	−0.733941	−0.366971	−	0.930232i	$-0.619605\pi$
$937$	−22.4663	−0.733941	−0.366971	+	0.930232i	$0.619605\pi$
$938$	−20.3180	−0.663407
$939$	0	0
$940$	−0.604598	−0.0197198
$941$	5.21088	0.169870	0.0849350	−	0.996386i	$-0.472932\pi$
$941$	5.21088	0.169870	0.0849350	+	0.996386i	$0.472932\pi$
$942$	0	0
$943$	−43.3995	−1.41328
$944$	−21.2124	−0.690406
$945$	0	0
$946$	0	0
$947$	−17.6216	−0.572625	−0.286312	−	0.958136i	$-0.592430\pi$
$947$	−17.6216	−0.572625	−0.286312	+	0.958136i	$0.592430\pi$
$948$	0	0
$949$	5.26977	0.171064
$950$	20.2496	0.656985
$951$	0	0
$952$	28.0451	0.908948
$953$	46.6714	1.51183	0.755917	−	0.654667i	$-0.227191\pi$
$953$	46.6714	1.51183	0.755917	+	0.654667i	$0.227191\pi$
$954$	0	0
$955$	−1.31939	−0.0426945
$956$	−26.0922	−0.843882
$957$	0	0
$958$	−20.6161	−0.666077
$959$	41.5348	1.34123
$960$	0	0
$961$	−27.1466	−0.875696
$962$	−7.19651	−0.232025
$963$	0	0
$964$	37.7476	1.21577
$965$	−3.92531	−0.126360
$966$	0	0
$967$	0.572210	0.0184010	0.00920051	−	0.999958i	$-0.497071\pi$
$967$	0.572210	0.0184010	0.00920051	+	0.999958i	$0.497071\pi$
$968$	0	0
$969$	0	0
$970$	−0.193808	−0.00622280
$971$	37.1703	1.19285	0.596426	−	0.802668i	$-0.296587\pi$
$971$	37.1703	1.19285	0.596426	+	0.802668i	$0.296587\pi$
$972$	0	0
$973$	−2.23465	−0.0716397
$974$	15.1048	0.483988
$975$	0	0
$976$	−5.61693	−0.179794
$977$	−16.0481	−0.513424	−0.256712	−	0.966488i	$-0.582639\pi$
$977$	−16.0481	−0.513424	−0.256712	+	0.966488i	$0.582639\pi$
$978$	0	0
$979$	0	0
$980$	−1.90837	−0.0609605
$981$	0	0
$982$	−21.9333	−0.699920
$983$	45.8928	1.46375	0.731877	−	0.681437i	$-0.238645\pi$
$983$	45.8928	1.46375	0.731877	+	0.681437i	$0.238645\pi$
$984$	0	0
$985$	−0.494486	−0.0157556
$986$	3.91779	0.124768
$987$	0	0
$988$	39.2145	1.24758
$989$	0.331841	0.0105519
$990$	0	0
$991$	−28.3187	−0.899573	−0.449786	−	0.893136i	$-0.648500\pi$
$991$	−28.3187	−0.899573	−0.449786	+	0.893136i	$0.648500\pi$
$992$	−11.3565	−0.360571
$993$	0	0
$994$	−25.3046	−0.802614
$995$	−2.94989	−0.0935177
$996$	0	0
$997$	35.1470	1.11312	0.556558	−	0.830809i	$-0.312122\pi$
$997$	35.1470	1.11312	0.556558	+	0.830809i	$0.312122\pi$
$998$	−0.177396	−0.00561538
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.co.1.7		18
3.2	odd	2		9801.2.a.cn.1.12		18
9.4	even	3		1089.2.e.o.727.12		36
9.7	even	3		1089.2.e.o.364.12		36
11.7	odd	10		891.2.f.f.82.6		36
11.8	odd	10		891.2.f.f.163.6		36
11.10	odd	2		9801.2.a.cm.1.12		18
33.8	even	10		891.2.f.e.163.4		36
33.29	even	10		891.2.f.e.82.4		36
33.32	even	2		9801.2.a.cp.1.7		18
99.7	odd	30		99.2.m.b.49.6	yes	72
99.29	even	30		297.2.n.b.280.4		72
99.40	odd	30		99.2.m.b.16.4	✓	72
99.41	even	30		297.2.n.b.262.4		72
99.43	odd	6		1089.2.e.p.364.7		36
99.52	odd	30		99.2.m.b.31.4	yes	72
99.74	even	30		297.2.n.b.64.6		72
99.76	odd	6		1089.2.e.p.727.7		36
99.85	odd	30		99.2.m.b.97.6	yes	72
99.95	even	30		297.2.n.b.181.6		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.m.b.16.4	✓	72	99.40	odd	30
99.2.m.b.31.4	yes	72	99.52	odd	30
99.2.m.b.49.6	yes	72	99.7	odd	30
99.2.m.b.97.6	yes	72	99.85	odd	30
297.2.n.b.64.6		72	99.74	even	30
297.2.n.b.181.6		72	99.95	even	30
297.2.n.b.262.4		72	99.41	even	30
297.2.n.b.280.4		72	99.29	even	30
891.2.f.e.82.4		36	33.29	even	10
891.2.f.e.163.4		36	33.8	even	10
891.2.f.f.82.6		36	11.7	odd	10
891.2.f.f.163.6		36	11.8	odd	10
1089.2.e.o.364.12		36	9.7	even	3
1089.2.e.o.727.12		36	9.4	even	3
1089.2.e.p.364.7		36	99.43	odd	6
1089.2.e.p.727.7		36	99.76	odd	6
9801.2.a.cm.1.12		18	11.10	odd	2
9801.2.a.cn.1.12		18	3.2	odd	2
9801.2.a.co.1.7		18	1.1	even	1	trivial
9801.2.a.cp.1.7		18	33.32	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:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

\(f(q)\)	\(=\)	\(q-0.677669 q^{2} -1.54077 q^{4} +0.290795 q^{5} -3.35549 q^{7} +2.39947 q^{8} +O(q^{10})\)	\(q-0.677669 q^{2} -1.54077 q^{4} +0.290795 q^{5} -3.35549 q^{7} +2.39947 q^{8} -0.197063 q^{10} -4.18671 q^{13} +2.27391 q^{14} +1.45549 q^{16} -3.48327 q^{17} +6.07908 q^{19} -0.448047 q^{20} -6.69090 q^{23} -4.91544 q^{25} +2.83720 q^{26} +5.17002 q^{28} +1.65973 q^{29} +1.96301 q^{31} -5.78527 q^{32} +2.36050 q^{34} -0.975759 q^{35} -2.53648 q^{37} -4.11960 q^{38} +0.697752 q^{40} +6.48635 q^{41} -0.0495959 q^{43} +4.53422 q^{46} +1.34941 q^{47} +4.25930 q^{49} +3.33104 q^{50} +6.45074 q^{52} -1.87766 q^{53} -8.05138 q^{56} -1.12475 q^{58} -14.5741 q^{59} -3.85914 q^{61} -1.33027 q^{62} +1.00952 q^{64} -1.21747 q^{65} -8.93528 q^{67} +5.36690 q^{68} +0.661241 q^{70} -11.1283 q^{71} -1.25869 q^{73} +1.71889 q^{74} -9.36643 q^{76} -2.10713 q^{79} +0.423248 q^{80} -4.39559 q^{82} -9.37527 q^{83} -1.01292 q^{85} +0.0336096 q^{86} -8.41413 q^{89} +14.0485 q^{91} +10.3091 q^{92} -0.914452 q^{94} +1.76776 q^{95} +0.983484 q^{97} -2.88640 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 2 q^{2} + 12 q^{4} - q^{5} - q^{7} + 6 q^{8}+O(q^{10}) \) 18 * q + 2 * q^2 + 12 * q^4 - q^5 - q^7 + 6 * q^8	\( 18 q + 2 q^{2} + 12 q^{4} - q^{5} - q^{7} + 6 q^{8} + 2 q^{10} - 3 q^{13} - 8 q^{16} + 20 q^{17} + 3 q^{19} - 5 q^{20} - 10 q^{23} + 7 q^{25} + 2 q^{26} - 19 q^{28} + 21 q^{29} + 6 q^{31} + 9 q^{32} - 4 q^{34} + 38 q^{35} + 7 q^{37} + 13 q^{38} + 20 q^{41} - 4 q^{43} - 8 q^{46} + 7 q^{47} + 7 q^{49} + 25 q^{50} + 19 q^{52} + 31 q^{53} - 57 q^{56} + 12 q^{58} - 12 q^{59} + 16 q^{61} + 19 q^{62} - 16 q^{64} + 42 q^{65} - 5 q^{67} + 51 q^{68} + 8 q^{70} + 13 q^{71} + 9 q^{74} + 8 q^{76} + 2 q^{79} + 46 q^{80} - 34 q^{82} + 36 q^{83} - 25 q^{85} + 26 q^{86} - 14 q^{89} - 15 q^{91} - 15 q^{92} - 4 q^{94} + 64 q^{95} - 16 q^{97} + 82 q^{98}+O(q^{100}) \) 18 * q + 2 * q^2 + 12 * q^4 - q^5 - q^7 + 6 * q^8 + 2 * q^10 - 3 * q^13 - 8 * q^16 + 20 * q^17 + 3 * q^19 - 5 * q^20 - 10 * q^23 + 7 * q^25 + 2 * q^26 - 19 * q^28 + 21 * q^29 + 6 * q^31 + 9 * q^32 - 4 * q^34 + 38 * q^35 + 7 * q^37 + 13 * q^38 + 20 * q^41 - 4 * q^43 - 8 * q^46 + 7 * q^47 + 7 * q^49 + 25 * q^50 + 19 * q^52 + 31 * q^53 - 57 * q^56 + 12 * q^58 - 12 * q^59 + 16 * q^61 + 19 * q^62 - 16 * q^64 + 42 * q^65 - 5 * q^67 + 51 * q^68 + 8 * q^70 + 13 * q^71 + 9 * q^74 + 8 * q^76 + 2 * q^79 + 46 * q^80 - 34 * q^82 + 36 * q^83 - 25 * q^85 + 26 * q^86 - 14 * q^89 - 15 * q^91 - 15 * q^92 - 4 * q^94 + 64 * q^95 - 16 * q^97 + 82 * q^98

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