LMFDB - Embedded newform 8036.2.a.r.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + \cdots + 3381 $ x^15 - x^14 - 37x^13 + 39x^12 + 537x^11 - 616x^10 - 3853x^9 + 4929x^8 + 13971x^7 - 20311x^6 - 22309x^5 + 38415x^4 + 8429x^3 - 22584x^2 - 399*x + 3381
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.27444$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.27444 q^{3} -4.37429 q^{5} -1.37580 q^{9} +O(q^{10})$	$q+1.27444 q^{3} -4.37429 q^{5} -1.37580 q^{9} +6.01391 q^{11} +5.59853 q^{13} -5.57477 q^{15} -6.15244 q^{17} -5.17601 q^{19} -4.66413 q^{23} +14.1344 q^{25} -5.57670 q^{27} -1.77771 q^{29} +4.02499 q^{31} +7.66437 q^{33} +8.25323 q^{37} +7.13499 q^{39} +1.00000 q^{41} +0.831106 q^{43} +6.01815 q^{45} -2.72099 q^{47} -7.84091 q^{51} -0.906871 q^{53} -26.3066 q^{55} -6.59652 q^{57} -7.92624 q^{59} +5.68627 q^{61} -24.4896 q^{65} -0.0872911 q^{67} -5.94416 q^{69} -8.10863 q^{71} +5.60866 q^{73} +18.0134 q^{75} +0.401351 q^{79} -2.97976 q^{81} +7.91362 q^{83} +26.9125 q^{85} -2.26558 q^{87} -13.4751 q^{89} +5.12960 q^{93} +22.6414 q^{95} +7.58525 q^{97} -8.27395 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * q^97 - 18 * q^99

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	0
$2$	0	0
$3$	1.27444	0.735798	0.367899	−	0.929866i	$-0.380077\pi$
$3$	1.27444	0.735798	0.367899	+	0.929866i	$0.380077\pi$
$4$	0	0
$5$	−4.37429	−1.95624	−0.978120	−	0.208040i	$-0.933292\pi$
$5$	−4.37429	−1.95624	−0.978120	+	0.208040i	$0.933292\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.37580	−0.458601
$10$	0	0
$11$	6.01391	1.81326	0.906631	−	0.421925i	$-0.138646\pi$
$11$	6.01391	1.81326	0.906631	+	0.421925i	$0.138646\pi$
$12$	0	0
$13$	5.59853	1.55275	0.776376	−	0.630270i	$-0.217056\pi$
$13$	5.59853	1.55275	0.776376	+	0.630270i	$0.217056\pi$
$14$	0	0
$15$	−5.57477	−1.43940
$16$	0	0
$17$	−6.15244	−1.49218	−0.746092	−	0.665842i	$-0.768072\pi$
$17$	−6.15244	−1.49218	−0.746092	+	0.665842i	$0.768072\pi$
$18$	0	0
$19$	−5.17601	−1.18746	−0.593729	−	0.804665i	$-0.702345\pi$
$19$	−5.17601	−1.18746	−0.593729	+	0.804665i	$0.702345\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.66413	−0.972539	−0.486270	−	0.873809i	$-0.661643\pi$
$23$	−4.66413	−0.972539	−0.486270	+	0.873809i	$0.661643\pi$
$24$	0	0
$25$	14.1344	2.82688
$26$	0	0
$27$	−5.57670	−1.07324
$28$	0	0
$29$	−1.77771	−0.330112	−0.165056	−	0.986284i	$-0.552781\pi$
$29$	−1.77771	−0.330112	−0.165056	+	0.986284i	$0.552781\pi$
$30$	0	0
$31$	4.02499	0.722909	0.361454	−	0.932390i	$-0.382280\pi$
$31$	4.02499	0.722909	0.361454	+	0.932390i	$0.382280\pi$
$32$	0	0
$33$	7.66437	1.33419
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.25323	1.35682	0.678411	−	0.734683i	$-0.262669\pi$
$37$	8.25323	1.35682	0.678411	+	0.734683i	$0.262669\pi$
$38$	0	0
$39$	7.13499	1.14251
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	0.831106	0.126743	0.0633713	−	0.997990i	$-0.479815\pi$
$43$	0.831106	0.126743	0.0633713	+	0.997990i	$0.479815\pi$
$44$	0	0
$45$	6.01815	0.897133
$46$	0	0
$47$	−2.72099	−0.396896	−0.198448	−	0.980111i	$-0.563590\pi$
$47$	−2.72099	−0.396896	−0.198448	+	0.980111i	$0.563590\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−7.84091	−1.09795
$52$	0	0
$53$	−0.906871	−0.124568	−0.0622842	−	0.998058i	$-0.519839\pi$
$53$	−0.906871	−0.124568	−0.0622842	+	0.998058i	$0.519839\pi$
$54$	0	0
$55$	−26.3066	−3.54718
$56$	0	0
$57$	−6.59652	−0.873730
$58$	0	0
$59$	−7.92624	−1.03191	−0.515954	−	0.856616i	$-0.672563\pi$
$59$	−7.92624	−1.03191	−0.515954	+	0.856616i	$0.672563\pi$
$60$	0	0
$61$	5.68627	0.728053	0.364026	−	0.931389i	$-0.381402\pi$
$61$	5.68627	0.728053	0.364026	+	0.931389i	$0.381402\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−24.4896	−3.03756
$66$	0	0
$67$	−0.0872911	−0.0106643	−0.00533215	−	0.999986i	$-0.501697\pi$
$67$	−0.0872911	−0.0106643	−0.00533215	+	0.999986i	$0.501697\pi$
$68$	0	0
$69$	−5.94416	−0.715593
$70$	0	0
$71$	−8.10863	−0.962317	−0.481158	−	0.876634i	$-0.659784\pi$
$71$	−8.10863	−0.962317	−0.481158	+	0.876634i	$0.659784\pi$
$72$	0	0
$73$	5.60866	0.656444	0.328222	−	0.944601i	$-0.393551\pi$
$73$	5.60866	0.656444	0.328222	+	0.944601i	$0.393551\pi$
$74$	0	0
$75$	18.0134	2.08001
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.401351	0.0451555	0.0225778	−	0.999745i	$-0.492813\pi$
$79$	0.401351	0.0451555	0.0225778	+	0.999745i	$0.492813\pi$
$80$	0	0
$81$	−2.97976	−0.331085
$82$	0	0
$83$	7.91362	0.868633	0.434317	−	0.900760i	$-0.356990\pi$
$83$	7.91362	0.868633	0.434317	+	0.900760i	$0.356990\pi$
$84$	0	0
$85$	26.9125	2.91907
$86$	0	0
$87$	−2.26558	−0.242896
$88$	0	0
$89$	−13.4751	−1.42836	−0.714181	−	0.699961i	$-0.753201\pi$
$89$	−13.4751	−1.42836	−0.714181	+	0.699961i	$0.753201\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.12960	0.531915
$94$	0	0
$95$	22.6414	2.32296
$96$	0	0
$97$	7.58525	0.770165	0.385083	−	0.922882i	$-0.374173\pi$
$97$	7.58525	0.770165	0.385083	+	0.922882i	$0.374173\pi$
$98$	0	0
$99$	−8.27395	−0.831563
$100$	0	0
$101$	−8.69249	−0.864935	−0.432467	−	0.901650i	$-0.642357\pi$
$101$	−8.69249	−0.864935	−0.432467	+	0.901650i	$0.642357\pi$
$102$	0	0
$103$	−0.936858	−0.0923113	−0.0461557	−	0.998934i	$-0.514697\pi$
$103$	−0.936858	−0.0923113	−0.0461557	+	0.998934i	$0.514697\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.950579	0.0918960	0.0459480	−	0.998944i	$-0.485369\pi$
$107$	0.950579	0.0918960	0.0459480	+	0.998944i	$0.485369\pi$
$108$	0	0
$109$	−4.22000	−0.404203	−0.202102	−	0.979365i	$-0.564777\pi$
$109$	−4.22000	−0.404203	−0.202102	+	0.979365i	$0.564777\pi$
$110$	0	0
$111$	10.5182	0.998348
$112$	0	0
$113$	18.5506	1.74509	0.872546	−	0.488532i	$-0.162468\pi$
$113$	18.5506	1.74509	0.872546	+	0.488532i	$0.162468\pi$
$114$	0	0
$115$	20.4023	1.90252
$116$	0	0
$117$	−7.70246	−0.712093
$118$	0	0
$119$	0	0
$120$	0	0
$121$	25.1671	2.28792
$122$	0	0
$123$	1.27444	0.114912
$124$	0	0
$125$	−39.9564	−3.57381
$126$	0	0
$127$	2.38798	0.211899	0.105949	−	0.994372i	$-0.466212\pi$
$127$	2.38798	0.211899	0.105949	+	0.994372i	$0.466212\pi$
$128$	0	0
$129$	1.05920	0.0932570
$130$	0	0
$131$	4.47978	0.391400	0.195700	−	0.980664i	$-0.437302\pi$
$131$	4.47978	0.391400	0.195700	+	0.980664i	$0.437302\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	24.3941	2.09951
$136$	0	0
$137$	21.2025	1.81145	0.905726	−	0.423864i	$-0.139327\pi$
$137$	21.2025	1.81145	0.905726	+	0.423864i	$0.139327\pi$
$138$	0	0
$139$	6.76140	0.573494	0.286747	−	0.958006i	$-0.407426\pi$
$139$	6.76140	0.573494	0.286747	+	0.958006i	$0.407426\pi$
$140$	0	0
$141$	−3.46773	−0.292036
$142$	0	0
$143$	33.6690	2.81554
$144$	0	0
$145$	7.77621	0.645779
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.88005	−0.481712	−0.240856	−	0.970561i	$-0.577428\pi$
$149$	−5.88005	−0.481712	−0.240856	+	0.970561i	$0.577428\pi$
$150$	0	0
$151$	−6.62898	−0.539459	−0.269729	−	0.962936i	$-0.586934\pi$
$151$	−6.62898	−0.539459	−0.269729	+	0.962936i	$0.586934\pi$
$152$	0	0
$153$	8.46453	0.684317
$154$	0	0
$155$	−17.6064	−1.41418
$156$	0	0
$157$	−9.09576	−0.725921	−0.362960	−	0.931805i	$-0.618234\pi$
$157$	−9.09576	−0.725921	−0.362960	+	0.931805i	$0.618234\pi$
$158$	0	0
$159$	−1.15575	−0.0916572
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−9.02020	−0.706517	−0.353258	−	0.935526i	$-0.614926\pi$
$163$	−9.02020	−0.706517	−0.353258	+	0.935526i	$0.614926\pi$
$164$	0	0
$165$	−33.5261	−2.61001
$166$	0	0
$167$	24.7420	1.91459	0.957297	−	0.289105i	$-0.0933577\pi$
$167$	24.7420	1.91459	0.957297	+	0.289105i	$0.0933577\pi$
$168$	0	0
$169$	18.3435	1.41104
$170$	0	0
$171$	7.12117	0.544570
$172$	0	0
$173$	6.16366	0.468615	0.234307	−	0.972163i	$-0.424718\pi$
$173$	6.16366	0.468615	0.234307	+	0.972163i	$0.424718\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.1015	−0.759277
$178$	0	0
$179$	14.8825	1.11237	0.556184	−	0.831059i	$-0.312265\pi$
$179$	14.8825	1.11237	0.556184	+	0.831059i	$0.312265\pi$
$180$	0	0
$181$	−6.30244	−0.468457	−0.234228	−	0.972182i	$-0.575256\pi$
$181$	−6.30244	−0.468457	−0.234228	+	0.972182i	$0.575256\pi$
$182$	0	0
$183$	7.24682	0.535700
$184$	0	0
$185$	−36.1020	−2.65427
$186$	0	0
$187$	−37.0002	−2.70572
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.42229	−0.102913	−0.0514566	−	0.998675i	$-0.516386\pi$
$191$	−1.42229	−0.102913	−0.0514566	+	0.998675i	$0.516386\pi$
$192$	0	0
$193$	−3.69210	−0.265763	−0.132882	−	0.991132i	$-0.542423\pi$
$193$	−3.69210	−0.265763	−0.132882	+	0.991132i	$0.542423\pi$
$194$	0	0
$195$	−31.2105	−2.23503
$196$	0	0
$197$	−0.537968	−0.0383286	−0.0191643	−	0.999816i	$-0.506101\pi$
$197$	−0.537968	−0.0383286	−0.0191643	+	0.999816i	$0.506101\pi$
$198$	0	0
$199$	17.9853	1.27494	0.637471	−	0.770474i	$-0.279981\pi$
$199$	17.9853	1.27494	0.637471	+	0.770474i	$0.279981\pi$
$200$	0	0
$201$	−0.111247	−0.00784678
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.37429	−0.305513
$206$	0	0
$207$	6.41693	0.446007
$208$	0	0
$209$	−31.1281	−2.15317
$210$	0	0
$211$	4.08723	0.281377	0.140688	−	0.990054i	$-0.455068\pi$
$211$	4.08723	0.281377	0.140688	+	0.990054i	$0.455068\pi$
$212$	0	0
$213$	−10.3340	−0.708071
$214$	0	0
$215$	−3.63550	−0.247939
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.14791	0.483011
$220$	0	0
$221$	−34.4446	−2.31699
$222$	0	0
$223$	−5.20162	−0.348326	−0.174163	−	0.984717i	$-0.555722\pi$
$223$	−5.20162	−0.348326	−0.174163	+	0.984717i	$0.555722\pi$
$224$	0	0
$225$	−19.4461	−1.29641
$226$	0	0
$227$	−8.82361	−0.585644	−0.292822	−	0.956167i	$-0.594594\pi$
$227$	−8.82361	−0.585644	−0.292822	+	0.956167i	$0.594594\pi$
$228$	0	0
$229$	11.2314	0.742194	0.371097	−	0.928594i	$-0.378982\pi$
$229$	11.2314	0.742194	0.371097	+	0.928594i	$0.378982\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.1088	1.44840	0.724199	−	0.689591i	$-0.242210\pi$
$233$	22.1088	1.44840	0.724199	+	0.689591i	$0.242210\pi$
$234$	0	0
$235$	11.9024	0.776425
$236$	0	0
$237$	0.511498	0.0332254
$238$	0	0
$239$	8.28588	0.535969	0.267984	−	0.963423i	$-0.413642\pi$
$239$	8.28588	0.535969	0.267984	+	0.963423i	$0.413642\pi$
$240$	0	0
$241$	5.86206	0.377608	0.188804	−	0.982015i	$-0.439539\pi$
$241$	5.86206	0.377608	0.188804	+	0.982015i	$0.439539\pi$
$242$	0	0
$243$	12.9326	0.829625
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−28.9780	−1.84383
$248$	0	0
$249$	10.0854	0.639139
$250$	0	0
$251$	−0.584676	−0.0369044	−0.0184522	−	0.999830i	$-0.505874\pi$
$251$	−0.584676	−0.0369044	−0.0184522	+	0.999830i	$0.505874\pi$
$252$	0	0
$253$	−28.0497	−1.76347
$254$	0	0
$255$	34.2984	2.14785
$256$	0	0
$257$	7.48023	0.466604	0.233302	−	0.972404i	$-0.425047\pi$
$257$	7.48023	0.466604	0.233302	+	0.972404i	$0.425047\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.44578	0.151390
$262$	0	0
$263$	1.93364	0.119233	0.0596167	−	0.998221i	$-0.481012\pi$
$263$	1.93364	0.119233	0.0596167	+	0.998221i	$0.481012\pi$
$264$	0	0
$265$	3.96691	0.243686
$266$	0	0
$267$	−17.1733	−1.05099
$268$	0	0
$269$	−4.18817	−0.255357	−0.127679	−	0.991816i	$-0.540753\pi$
$269$	−4.18817	−0.255357	−0.127679	+	0.991816i	$0.540753\pi$
$270$	0	0
$271$	22.5286	1.36851	0.684256	−	0.729241i	$-0.260127\pi$
$271$	22.5286	1.36851	0.684256	+	0.729241i	$0.260127\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	85.0029	5.12587
$276$	0	0
$277$	5.70409	0.342726	0.171363	−	0.985208i	$-0.445183\pi$
$277$	5.70409	0.342726	0.171363	+	0.985208i	$0.445183\pi$
$278$	0	0
$279$	−5.53758	−0.331526
$280$	0	0
$281$	16.7441	0.998871	0.499436	−	0.866351i	$-0.333541\pi$
$281$	16.7441	0.998871	0.499436	+	0.866351i	$0.333541\pi$
$282$	0	0
$283$	9.51746	0.565754	0.282877	−	0.959156i	$-0.408711\pi$
$283$	9.51746	0.565754	0.282877	+	0.959156i	$0.408711\pi$
$284$	0	0
$285$	28.8551	1.70923
$286$	0	0
$287$	0	0
$288$	0	0
$289$	20.8525	1.22662
$290$	0	0
$291$	9.66695	0.566686
$292$	0	0
$293$	27.4692	1.60477	0.802383	−	0.596809i	$-0.203565\pi$
$293$	27.4692	1.60477	0.802383	+	0.596809i	$0.203565\pi$
$294$	0	0
$295$	34.6717	2.01866
$296$	0	0
$297$	−33.5377	−1.94606
$298$	0	0
$299$	−26.1123	−1.51011
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−11.0781	−0.636418
$304$	0	0
$305$	−24.8734	−1.42425
$306$	0	0
$307$	27.7003	1.58094	0.790470	−	0.612500i	$-0.209836\pi$
$307$	27.7003	1.58094	0.790470	+	0.612500i	$0.209836\pi$
$308$	0	0
$309$	−1.19397	−0.0679225
$310$	0	0
$311$	−0.0840665	−0.00476697	−0.00238349	−	0.999997i	$-0.500759\pi$
$311$	−0.0840665	−0.00476697	−0.00238349	+	0.999997i	$0.500759\pi$
$312$	0	0
$313$	20.8627	1.17923	0.589615	−	0.807684i	$-0.299279\pi$
$313$	20.8627	1.17923	0.589615	+	0.807684i	$0.299279\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.1200	0.624563	0.312282	−	0.949990i	$-0.398907\pi$
$317$	11.1200	0.624563	0.312282	+	0.949990i	$0.398907\pi$
$318$	0	0
$319$	−10.6910	−0.598580
$320$	0	0
$321$	1.21146	0.0676169
$322$	0	0
$323$	31.8451	1.77191
$324$	0	0
$325$	79.1317	4.38944
$326$	0	0
$327$	−5.37814	−0.297412
$328$	0	0
$329$	0	0
$330$	0	0
$331$	23.5550	1.29470	0.647350	−	0.762193i	$-0.275877\pi$
$331$	23.5550	1.29470	0.647350	+	0.762193i	$0.275877\pi$
$332$	0	0
$333$	−11.3548	−0.622240
$334$	0	0
$335$	0.381836	0.0208620
$336$	0	0
$337$	14.4817	0.788867	0.394433	−	0.918924i	$-0.370941\pi$
$337$	14.4817	0.788867	0.394433	+	0.918924i	$0.370941\pi$
$338$	0	0
$339$	23.6416	1.28404
$340$	0	0
$341$	24.2059	1.31082
$342$	0	0
$343$	0	0
$344$	0	0
$345$	26.0015	1.39987
$346$	0	0
$347$	−11.8736	−0.637409	−0.318704	−	0.947854i	$-0.603248\pi$
$347$	−11.8736	−0.637409	−0.318704	+	0.947854i	$0.603248\pi$
$348$	0	0
$349$	−1.47735	−0.0790805	−0.0395403	−	0.999218i	$-0.512589\pi$
$349$	−1.47735	−0.0790805	−0.0395403	+	0.999218i	$0.512589\pi$
$350$	0	0
$351$	−31.2213	−1.66647
$352$	0	0
$353$	−20.5859	−1.09568	−0.547838	−	0.836585i	$-0.684549\pi$
$353$	−20.5859	−1.09568	−0.547838	+	0.836585i	$0.684549\pi$
$354$	0	0
$355$	35.4695	1.88252
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.0198	1.16216	0.581080	−	0.813847i	$-0.302630\pi$
$359$	22.0198	1.16216	0.581080	+	0.813847i	$0.302630\pi$
$360$	0	0
$361$	7.79112	0.410059
$362$	0	0
$363$	32.0739	1.68345
$364$	0	0
$365$	−24.5339	−1.28416
$366$	0	0
$367$	11.1637	0.582742	0.291371	−	0.956610i	$-0.405889\pi$
$367$	11.1637	0.582742	0.291371	+	0.956610i	$0.405889\pi$
$368$	0	0
$369$	−1.37580	−0.0716214
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−27.1833	−1.40750	−0.703750	−	0.710448i	$-0.748493\pi$
$373$	−27.1833	−1.40750	−0.703750	+	0.710448i	$0.748493\pi$
$374$	0	0
$375$	−50.9221	−2.62960
$376$	0	0
$377$	−9.95255	−0.512582
$378$	0	0
$379$	−17.6910	−0.908724	−0.454362	−	0.890817i	$-0.650133\pi$
$379$	−17.6910	−0.908724	−0.454362	+	0.890817i	$0.650133\pi$
$380$	0	0
$381$	3.04333	0.155915
$382$	0	0
$383$	−2.22193	−0.113535	−0.0567676	−	0.998387i	$-0.518079\pi$
$383$	−2.22193	−0.113535	−0.0567676	+	0.998387i	$0.518079\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.14344	−0.0581242
$388$	0	0
$389$	8.04637	0.407967	0.203983	−	0.978974i	$-0.434611\pi$
$389$	8.04637	0.407967	0.203983	+	0.978974i	$0.434611\pi$
$390$	0	0
$391$	28.6958	1.45121
$392$	0	0
$393$	5.70921	0.287992
$394$	0	0
$395$	−1.75563	−0.0883351
$396$	0	0
$397$	−31.8213	−1.59707	−0.798533	−	0.601951i	$-0.794390\pi$
$397$	−31.8213	−1.59707	−0.798533	+	0.601951i	$0.794390\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.3619	1.36639	0.683194	−	0.730237i	$-0.260590\pi$
$401$	27.3619	1.36639	0.683194	+	0.730237i	$0.260590\pi$
$402$	0	0
$403$	22.5340	1.12250
$404$	0	0
$405$	13.0343	0.647681
$406$	0	0
$407$	49.6341	2.46027
$408$	0	0
$409$	−18.8393	−0.931542	−0.465771	−	0.884905i	$-0.654223\pi$
$409$	−18.8393	−0.931542	−0.465771	+	0.884905i	$0.654223\pi$
$410$	0	0
$411$	27.0213	1.33286
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−34.6165	−1.69926
$416$	0	0
$417$	8.61700	0.421976
$418$	0	0
$419$	10.0834	0.492606	0.246303	−	0.969193i	$-0.420784\pi$
$419$	10.0834	0.492606	0.246303	+	0.969193i	$0.420784\pi$
$420$	0	0
$421$	5.09516	0.248323	0.124161	−	0.992262i	$-0.460376\pi$
$421$	5.09516	0.248323	0.124161	+	0.992262i	$0.460376\pi$
$422$	0	0
$423$	3.74354	0.182017
$424$	0	0
$425$	−86.9609	−4.21822
$426$	0	0
$427$	0	0
$428$	0	0
$429$	42.9091	2.07167
$430$	0	0
$431$	−0.0144149	−0.000694344 0	−0.000347172	−	1.00000i	$-0.500111\pi$
$431$	−0.0144149	−0.000694344 0	−0.000347172		1.00000i	$0.500111\pi$
$432$	0	0
$433$	16.1368	0.775483	0.387741	−	0.921768i	$-0.373255\pi$
$433$	16.1368	0.775483	0.387741	+	0.921768i	$0.373255\pi$
$434$	0	0
$435$	9.91031	0.475163
$436$	0	0
$437$	24.1416	1.15485
$438$	0	0
$439$	−4.51306	−0.215397	−0.107698	−	0.994184i	$-0.534348\pi$
$439$	−4.51306	−0.215397	−0.107698	+	0.994184i	$0.534348\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11.3960	0.541438	0.270719	−	0.962658i	$-0.412739\pi$
$443$	11.3960	0.541438	0.270719	+	0.962658i	$0.412739\pi$
$444$	0	0
$445$	58.9441	2.79422
$446$	0	0
$447$	−7.49377	−0.354443
$448$	0	0
$449$	26.9657	1.27259	0.636295	−	0.771446i	$-0.280466\pi$
$449$	26.9657	1.27259	0.636295	+	0.771446i	$0.280466\pi$
$450$	0	0
$451$	6.01391	0.283184
$452$	0	0
$453$	−8.44824	−0.396933
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−15.5826	−0.728922	−0.364461	−	0.931219i	$-0.618747\pi$
$457$	−15.5826	−0.728922	−0.364461	+	0.931219i	$0.618747\pi$
$458$	0	0
$459$	34.3103	1.60147
$460$	0	0
$461$	−31.6708	−1.47506	−0.737529	−	0.675316i	$-0.764007\pi$
$461$	−31.6708	−1.47506	−0.737529	+	0.675316i	$0.764007\pi$
$462$	0	0
$463$	−15.8220	−0.735312	−0.367656	−	0.929962i	$-0.619840\pi$
$463$	−15.8220	−0.735312	−0.367656	+	0.929962i	$0.619840\pi$
$464$	0	0
$465$	−22.4384	−1.04055
$466$	0	0
$467$	−11.4140	−0.528179	−0.264089	−	0.964498i	$-0.585071\pi$
$467$	−11.4140	−0.528179	−0.264089	+	0.964498i	$0.585071\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−11.5920	−0.534131
$472$	0	0
$473$	4.99820	0.229817
$474$	0	0
$475$	−73.1598	−3.35680
$476$	0	0
$477$	1.24768	0.0571271
$478$	0	0
$479$	−18.2230	−0.832631	−0.416316	−	0.909220i	$-0.636679\pi$
$479$	−18.2230	−0.832631	−0.416316	+	0.909220i	$0.636679\pi$
$480$	0	0
$481$	46.2059	2.10681
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−33.1801	−1.50663
$486$	0	0
$487$	−35.8062	−1.62254	−0.811268	−	0.584675i	$-0.801222\pi$
$487$	−35.8062	−1.62254	−0.811268	+	0.584675i	$0.801222\pi$
$488$	0	0
$489$	−11.4957	−0.519854
$490$	0	0
$491$	34.2707	1.54662	0.773309	−	0.634030i	$-0.218600\pi$
$491$	34.2707	1.54662	0.773309	+	0.634030i	$0.218600\pi$
$492$	0	0
$493$	10.9372	0.492589
$494$	0	0
$495$	36.1926	1.62674
$496$	0	0
$497$	0	0
$498$	0	0
$499$	30.9556	1.38576	0.692881	−	0.721052i	$-0.256341\pi$
$499$	30.9556	1.38576	0.692881	+	0.721052i	$0.256341\pi$
$500$	0	0
$501$	31.5322	1.40876
$502$	0	0
$503$	−34.2582	−1.52750	−0.763748	−	0.645515i	$-0.776643\pi$
$503$	−34.2582	−1.52750	−0.763748	+	0.645515i	$0.776643\pi$
$504$	0	0
$505$	38.0234	1.69202
$506$	0	0
$507$	23.3777	1.03824
$508$	0	0
$509$	27.4575	1.21703	0.608517	−	0.793541i	$-0.291765\pi$
$509$	27.4575	1.21703	0.608517	+	0.793541i	$0.291765\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	28.8651	1.27442
$514$	0	0
$515$	4.09808	0.180583
$516$	0	0
$517$	−16.3638	−0.719677
$518$	0	0
$519$	7.85522	0.344806
$520$	0	0
$521$	−12.1471	−0.532174	−0.266087	−	0.963949i	$-0.585731\pi$
$521$	−12.1471	−0.532174	−0.266087	+	0.963949i	$0.585731\pi$
$522$	0	0
$523$	21.0692	0.921291	0.460646	−	0.887584i	$-0.347618\pi$
$523$	21.0692	0.921291	0.460646	+	0.887584i	$0.347618\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24.7635	−1.07871
$528$	0	0
$529$	−1.24586	−0.0541678
$530$	0	0
$531$	10.9049	0.473234
$532$	0	0
$533$	5.59853	0.242499
$534$	0	0
$535$	−4.15811	−0.179771
$536$	0	0
$537$	18.9668	0.818479
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22.3830	−0.962320	−0.481160	−	0.876633i	$-0.659784\pi$
$541$	−22.3830	−0.962320	−0.481160	+	0.876633i	$0.659784\pi$
$542$	0	0
$543$	−8.03208	−0.344690
$544$	0	0
$545$	18.4595	0.790718
$546$	0	0
$547$	−41.7440	−1.78485	−0.892423	−	0.451200i	$-0.850996\pi$
$547$	−41.7440	−1.78485	−0.892423	+	0.451200i	$0.850996\pi$
$548$	0	0
$549$	−7.82319	−0.333886
$550$	0	0
$551$	9.20145	0.391995
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−46.0098	−1.95301
$556$	0	0
$557$	21.6176	0.915965	0.457983	−	0.888961i	$-0.348572\pi$
$557$	21.6176	0.915965	0.457983	+	0.888961i	$0.348572\pi$
$558$	0	0
$559$	4.65297	0.196800
$560$	0	0
$561$	−47.1545	−1.99087
$562$	0	0
$563$	37.7846	1.59243	0.796215	−	0.605014i	$-0.206833\pi$
$563$	37.7846	1.59243	0.796215	+	0.605014i	$0.206833\pi$
$564$	0	0
$565$	−81.1456	−3.41382
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.43234	0.227735	0.113868	−	0.993496i	$-0.463676\pi$
$569$	5.43234	0.227735	0.113868	+	0.993496i	$0.463676\pi$
$570$	0	0
$571$	38.9281	1.62909	0.814545	−	0.580101i	$-0.196987\pi$
$571$	38.9281	1.62909	0.814545	+	0.580101i	$0.196987\pi$
$572$	0	0
$573$	−1.81262	−0.0757234
$574$	0	0
$575$	−65.9247	−2.74925
$576$	0	0
$577$	−12.0099	−0.499978	−0.249989	−	0.968249i	$-0.580427\pi$
$577$	−12.0099	−0.499978	−0.249989	+	0.968249i	$0.580427\pi$
$578$	0	0
$579$	−4.70536	−0.195548
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5.45384	−0.225875
$584$	0	0
$585$	33.6928	1.39303
$586$	0	0
$587$	24.6756	1.01847	0.509235	−	0.860628i	$-0.329929\pi$
$587$	24.6756	1.01847	0.509235	+	0.860628i	$0.329929\pi$
$588$	0	0
$589$	−20.8334	−0.858425
$590$	0	0
$591$	−0.685608	−0.0282021
$592$	0	0
$593$	12.2525	0.503151	0.251575	−	0.967838i	$-0.419051\pi$
$593$	12.2525	0.503151	0.251575	+	0.967838i	$0.419051\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.9211	0.938100
$598$	0	0
$599$	7.84464	0.320523	0.160262	−	0.987075i	$-0.448766\pi$
$599$	7.84464	0.320523	0.160262	+	0.987075i	$0.448766\pi$
$600$	0	0
$601$	−14.4302	−0.588619	−0.294310	−	0.955710i	$-0.595090\pi$
$601$	−14.4302	−0.588619	−0.294310	+	0.955710i	$0.595090\pi$
$602$	0	0
$603$	0.120095	0.00489066
$604$	0	0
$605$	−110.088	−4.47572
$606$	0	0
$607$	−13.4212	−0.544751	−0.272376	−	0.962191i	$-0.587809\pi$
$607$	−13.4212	−0.544751	−0.272376	+	0.962191i	$0.587809\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.2335	−0.616282
$612$	0	0
$613$	11.3835	0.459776	0.229888	−	0.973217i	$-0.426164\pi$
$613$	11.3835	0.459776	0.229888	+	0.973217i	$0.426164\pi$
$614$	0	0
$615$	−5.57477	−0.224796
$616$	0	0
$617$	−7.68338	−0.309321	−0.154661	−	0.987968i	$-0.549428\pi$
$617$	−7.68338	−0.309321	−0.154661	+	0.987968i	$0.549428\pi$
$618$	0	0
$619$	−1.76746	−0.0710401	−0.0355201	−	0.999369i	$-0.511309\pi$
$619$	−1.76746	−0.0710401	−0.0355201	+	0.999369i	$0.511309\pi$
$620$	0	0
$621$	26.0105	1.04376
$622$	0	0
$623$	0	0
$624$	0	0
$625$	104.109	4.16436
$626$	0	0
$627$	−39.6709	−1.58430
$628$	0	0
$629$	−50.7774	−2.02463
$630$	0	0
$631$	26.4941	1.05471	0.527356	−	0.849645i	$-0.323184\pi$
$631$	26.4941	1.05471	0.527356	+	0.849645i	$0.323184\pi$
$632$	0	0
$633$	5.20893	0.207036
$634$	0	0
$635$	−10.4457	−0.414525
$636$	0	0
$637$	0	0
$638$	0	0
$639$	11.1559	0.441319
$640$	0	0
$641$	−3.10746	−0.122737	−0.0613687	−	0.998115i	$-0.519547\pi$
$641$	−3.10746	−0.122737	−0.0613687	+	0.998115i	$0.519547\pi$
$642$	0	0
$643$	−20.4230	−0.805405	−0.402702	−	0.915331i	$-0.631929\pi$
$643$	−20.4230	−0.805405	−0.402702	+	0.915331i	$0.631929\pi$
$644$	0	0
$645$	−4.63322	−0.182433
$646$	0	0
$647$	−27.3520	−1.07532	−0.537660	−	0.843162i	$-0.680692\pi$
$647$	−27.3520	−1.07532	−0.537660	+	0.843162i	$0.680692\pi$
$648$	0	0
$649$	−47.6677	−1.87112
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.1311	0.709525	0.354763	−	0.934956i	$-0.384562\pi$
$653$	18.1311	0.709525	0.354763	+	0.934956i	$0.384562\pi$
$654$	0	0
$655$	−19.5958	−0.765673
$656$	0	0
$657$	−7.71641	−0.301046
$658$	0	0
$659$	−3.67331	−0.143092	−0.0715458	−	0.997437i	$-0.522793\pi$
$659$	−3.67331	−0.143092	−0.0715458	+	0.997437i	$0.522793\pi$
$660$	0	0
$661$	4.34291	0.168919	0.0844597	−	0.996427i	$-0.473084\pi$
$661$	4.34291	0.168919	0.0844597	+	0.996427i	$0.473084\pi$
$662$	0	0
$663$	−43.8975	−1.70484
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.29147	0.321047
$668$	0	0
$669$	−6.62916	−0.256298
$670$	0	0
$671$	34.1967	1.32015
$672$	0	0
$673$	−17.9495	−0.691904	−0.345952	−	0.938252i	$-0.612444\pi$
$673$	−17.9495	−0.691904	−0.345952	+	0.938252i	$0.612444\pi$
$674$	0	0
$675$	−78.8232	−3.03391
$676$	0	0
$677$	47.2419	1.81565	0.907826	−	0.419347i	$-0.137741\pi$
$677$	47.2419	1.81565	0.907826	+	0.419347i	$0.137741\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−11.2452	−0.430916
$682$	0	0
$683$	−32.2394	−1.23361	−0.616803	−	0.787118i	$-0.711572\pi$
$683$	−32.2394	−1.23361	−0.616803	+	0.787118i	$0.711572\pi$
$684$	0	0
$685$	−92.7458	−3.54363
$686$	0	0
$687$	14.3138	0.546105
$688$	0	0
$689$	−5.07714	−0.193424
$690$	0	0
$691$	43.8664	1.66876	0.834378	−	0.551193i	$-0.185827\pi$
$691$	43.8664	1.66876	0.834378	+	0.551193i	$0.185827\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−29.5763	−1.12189
$696$	0	0
$697$	−6.15244	−0.233040
$698$	0	0
$699$	28.1764	1.06573
$700$	0	0
$701$	17.1256	0.646826	0.323413	−	0.946258i	$-0.395170\pi$
$701$	17.1256	0.646826	0.323413	+	0.946258i	$0.395170\pi$
$702$	0	0
$703$	−42.7188	−1.61117
$704$	0	0
$705$	15.1689	0.571292
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−4.59653	−0.172626	−0.0863131	−	0.996268i	$-0.527509\pi$
$709$	−4.59653	−0.172626	−0.0863131	+	0.996268i	$0.527509\pi$
$710$	0	0
$711$	−0.552180	−0.0207084
$712$	0	0
$713$	−18.7731	−0.703057
$714$	0	0
$715$	−147.278	−5.50788
$716$	0	0
$717$	10.5599	0.394365
$718$	0	0
$719$	−48.2702	−1.80018	−0.900088	−	0.435708i	$-0.856498\pi$
$719$	−48.2702	−1.80018	−0.900088	+	0.435708i	$0.856498\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	7.47084	0.277844
$724$	0	0
$725$	−25.1268	−0.933187
$726$	0	0
$727$	−18.6128	−0.690309	−0.345154	−	0.938546i	$-0.612173\pi$
$727$	−18.6128	−0.690309	−0.345154	+	0.938546i	$0.612173\pi$
$728$	0	0
$729$	25.4211	0.941521
$730$	0	0
$731$	−5.11333	−0.189123
$732$	0	0
$733$	20.7445	0.766217	0.383109	−	0.923703i	$-0.374853\pi$
$733$	20.7445	0.766217	0.383109	+	0.923703i	$0.374853\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.524961	−0.0193372
$738$	0	0
$739$	31.5512	1.16063	0.580316	−	0.814392i	$-0.302929\pi$
$739$	31.5512	1.16063	0.580316	+	0.814392i	$0.302929\pi$
$740$	0	0
$741$	−36.9308	−1.35669
$742$	0	0
$743$	25.9780	0.953041	0.476520	−	0.879163i	$-0.341898\pi$
$743$	25.9780	0.953041	0.476520	+	0.879163i	$0.341898\pi$
$744$	0	0
$745$	25.7210	0.942345
$746$	0	0
$747$	−10.8876	−0.398356
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.58963	−0.167478	−0.0837389	−	0.996488i	$-0.526686\pi$
$751$	−4.58963	−0.167478	−0.0837389	+	0.996488i	$0.526686\pi$
$752$	0	0
$753$	−0.745134	−0.0271542
$754$	0	0
$755$	28.9971	1.05531
$756$	0	0
$757$	−1.12358	−0.0408371	−0.0204185	−	0.999792i	$-0.506500\pi$
$757$	−1.12358	−0.0408371	−0.0204185	+	0.999792i	$0.506500\pi$
$758$	0	0
$759$	−35.7476	−1.29756
$760$	0	0
$761$	21.1890	0.768099	0.384049	−	0.923313i	$-0.374529\pi$
$761$	21.1890	0.768099	0.384049	+	0.923313i	$0.374529\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−37.0263	−1.33869
$766$	0	0
$767$	−44.3753	−1.60230
$768$	0	0
$769$	−17.2834	−0.623257	−0.311628	−	0.950204i	$-0.600874\pi$
$769$	−17.2834	−0.623257	−0.311628	+	0.950204i	$0.600874\pi$
$770$	0	0
$771$	9.53310	0.343326
$772$	0	0
$773$	−5.95952	−0.214349	−0.107175	−	0.994240i	$-0.534180\pi$
$773$	−5.95952	−0.214349	−0.107175	+	0.994240i	$0.534180\pi$
$774$	0	0
$775$	56.8907	2.04357
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.17601	−0.185450
$780$	0	0
$781$	−48.7645	−1.74493
$782$	0	0
$783$	9.91375	0.354288
$784$	0	0
$785$	39.7875	1.42008
$786$	0	0
$787$	−48.0534	−1.71292	−0.856459	−	0.516216i	$-0.827340\pi$
$787$	−48.0534	−1.71292	−0.856459	+	0.516216i	$0.827340\pi$
$788$	0	0
$789$	2.46431	0.0877317
$790$	0	0
$791$	0	0
$792$	0	0
$793$	31.8348	1.13049
$794$	0	0
$795$	5.05559	0.179303
$796$	0	0
$797$	−12.3574	−0.437723	−0.218862	−	0.975756i	$-0.570234\pi$
$797$	−12.3574	−0.437723	−0.218862	+	0.975756i	$0.570234\pi$
$798$	0	0
$799$	16.7407	0.592243
$800$	0	0
$801$	18.5391	0.655048
$802$	0	0
$803$	33.7300	1.19031
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.33757	−0.187891
$808$	0	0
$809$	31.4203	1.10468	0.552340	−	0.833619i	$-0.313735\pi$
$809$	31.4203	1.10468	0.552340	+	0.833619i	$0.313735\pi$
$810$	0	0
$811$	−50.3267	−1.76721	−0.883604	−	0.468235i	$-0.844890\pi$
$811$	−50.3267	−1.76721	−0.883604	+	0.468235i	$0.844890\pi$
$812$	0	0
$813$	28.7113	1.00695
$814$	0	0
$815$	39.4569	1.38212
$816$	0	0
$817$	−4.30182	−0.150502
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−35.0005	−1.22153	−0.610763	−	0.791813i	$-0.709137\pi$
$821$	−35.0005	−1.22153	−0.610763	+	0.791813i	$0.709137\pi$
$822$	0	0
$823$	42.6664	1.48726	0.743628	−	0.668594i	$-0.233104\pi$
$823$	42.6664	1.48726	0.743628	+	0.668594i	$0.233104\pi$
$824$	0	0
$825$	108.331	3.77160
$826$	0	0
$827$	−21.3960	−0.744013	−0.372007	−	0.928230i	$-0.621330\pi$
$827$	−21.3960	−0.744013	−0.372007	+	0.928230i	$0.621330\pi$
$828$	0	0
$829$	45.6668	1.58607	0.793036	−	0.609174i	$-0.208499\pi$
$829$	45.6668	1.58607	0.793036	+	0.609174i	$0.208499\pi$
$830$	0	0
$831$	7.26952	0.252177
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−108.229	−3.74541
$836$	0	0
$837$	−22.4461	−0.775852
$838$	0	0
$839$	−19.2945	−0.666121	−0.333060	−	0.942906i	$-0.608081\pi$
$839$	−19.2945	−0.666121	−0.333060	+	0.942906i	$0.608081\pi$
$840$	0	0
$841$	−25.8398	−0.891026
$842$	0	0
$843$	21.3394	0.734968
$844$	0	0
$845$	−80.2397	−2.76033
$846$	0	0
$847$	0	0
$848$	0	0
$849$	12.1294	0.416281
$850$	0	0
$851$	−38.4942	−1.31956
$852$	0	0
$853$	15.5000	0.530709	0.265355	−	0.964151i	$-0.414511\pi$
$853$	15.5000	0.530709	0.265355	+	0.964151i	$0.414511\pi$
$854$	0	0
$855$	−31.1500	−1.06531
$856$	0	0
$857$	−36.0081	−1.23001	−0.615007	−	0.788522i	$-0.710847\pi$
$857$	−36.0081	−1.23001	−0.615007	+	0.788522i	$0.710847\pi$
$858$	0	0
$859$	−20.2200	−0.689898	−0.344949	−	0.938621i	$-0.612104\pi$
$859$	−20.2200	−0.689898	−0.344949	+	0.938621i	$0.612104\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22.1025	0.752377	0.376189	−	0.926543i	$-0.377234\pi$
$863$	22.1025	0.752377	0.376189	+	0.926543i	$0.377234\pi$
$864$	0	0
$865$	−26.9616	−0.916723
$866$	0	0
$867$	26.5752	0.902542
$868$	0	0
$869$	2.41369	0.0818788
$870$	0	0
$871$	−0.488702	−0.0165590
$872$	0	0
$873$	−10.4358	−0.353198
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13.8589	−0.467981	−0.233991	−	0.972239i	$-0.575178\pi$
$877$	−13.8589	−0.467981	−0.233991	+	0.972239i	$0.575178\pi$
$878$	0	0
$879$	35.0078	1.18078
$880$	0	0
$881$	−36.0284	−1.21383	−0.606914	−	0.794768i	$-0.707593\pi$
$881$	−36.0284	−1.21383	−0.606914	+	0.794768i	$0.707593\pi$
$882$	0	0
$883$	10.3770	0.349213	0.174606	−	0.984638i	$-0.444135\pi$
$883$	10.3770	0.349213	0.174606	+	0.984638i	$0.444135\pi$
$884$	0	0
$885$	44.1869	1.48533
$886$	0	0
$887$	−5.36696	−0.180205	−0.0901025	−	0.995932i	$-0.528719\pi$
$887$	−5.36696	−0.180205	−0.0901025	+	0.995932i	$0.528719\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−17.9200	−0.600343
$892$	0	0
$893$	14.0839	0.471298
$894$	0	0
$895$	−65.1002	−2.17606
$896$	0	0
$897$	−33.2785	−1.11114
$898$	0	0
$899$	−7.15525	−0.238641
$900$	0	0
$901$	5.57947	0.185879
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27.5687	0.916414
$906$	0	0
$907$	6.71146	0.222850	0.111425	−	0.993773i	$-0.464458\pi$
$907$	6.71146	0.222850	0.111425	+	0.993773i	$0.464458\pi$
$908$	0	0
$909$	11.9591	0.396660
$910$	0	0
$911$	−37.7515	−1.25076	−0.625381	−	0.780319i	$-0.715057\pi$
$911$	−37.7515	−1.25076	−0.625381	+	0.780319i	$0.715057\pi$
$912$	0	0
$913$	47.5918	1.57506
$914$	0	0
$915$	−31.6997	−1.04796
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−27.9965	−0.923520	−0.461760	−	0.887005i	$-0.652782\pi$
$919$	−27.9965	−0.923520	−0.461760	+	0.887005i	$0.652782\pi$
$920$	0	0
$921$	35.3024	1.16325
$922$	0	0
$923$	−45.3964	−1.49424
$924$	0	0
$925$	116.654	3.83557
$926$	0	0
$927$	1.28893	0.0423340
$928$	0	0
$929$	0.392973	0.0128930	0.00644652	−	0.999979i	$-0.497948\pi$
$929$	0.392973	0.0128930	0.00644652	+	0.999979i	$0.497948\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−0.107138	−0.00350753
$934$	0	0
$935$	161.849	5.29304
$936$	0	0
$937$	48.7090	1.59126	0.795628	−	0.605786i	$-0.207141\pi$
$937$	48.7090	1.59126	0.795628	+	0.605786i	$0.207141\pi$
$938$	0	0
$939$	26.5883	0.867675
$940$	0	0
$941$	−50.3207	−1.64041	−0.820204	−	0.572071i	$-0.806140\pi$
$941$	−50.3207	−1.64041	−0.820204	+	0.572071i	$0.806140\pi$
$942$	0	0
$943$	−4.66413	−0.151885
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−60.6072	−1.96947	−0.984735	−	0.174062i	$-0.944311\pi$
$947$	−60.6072	−1.96947	−0.984735	+	0.174062i	$0.944311\pi$
$948$	0	0
$949$	31.4003	1.01930
$950$	0	0
$951$	14.1718	0.459553
$952$	0	0
$953$	49.8014	1.61322	0.806612	−	0.591081i	$-0.201299\pi$
$953$	49.8014	1.61322	0.806612	+	0.591081i	$0.201299\pi$
$954$	0	0
$955$	6.22150	0.201323
$956$	0	0
$957$	−13.6250	−0.440434
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−14.7995	−0.477403
$962$	0	0
$963$	−1.30781	−0.0421436
$964$	0	0
$965$	16.1503	0.519896
$966$	0	0
$967$	−54.0824	−1.73917	−0.869587	−	0.493780i	$-0.835615\pi$
$967$	−54.0824	−1.73917	−0.869587	+	0.493780i	$0.835615\pi$
$968$	0	0
$969$	40.5847	1.30377
$970$	0	0
$971$	40.6144	1.30338	0.651690	−	0.758486i	$-0.274060\pi$
$971$	40.6144	1.30338	0.651690	+	0.758486i	$0.274060\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	100.849	3.22974
$976$	0	0
$977$	−8.05354	−0.257656	−0.128828	−	0.991667i	$-0.541121\pi$
$977$	−8.05354	−0.257656	−0.128828	+	0.991667i	$0.541121\pi$
$978$	0	0
$979$	−81.0382	−2.58999
$980$	0	0
$981$	5.80589	0.185368
$982$	0	0
$983$	39.9017	1.27267	0.636333	−	0.771415i	$-0.280451\pi$
$983$	39.9017	1.27267	0.636333	+	0.771415i	$0.280451\pi$
$984$	0	0
$985$	2.35323	0.0749800
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.87639	−0.123262
$990$	0	0
$991$	−31.0321	−0.985768	−0.492884	−	0.870095i	$-0.664057\pi$
$991$	−31.0321	−0.985768	−0.492884	+	0.870095i	$0.664057\pi$
$992$	0	0
$993$	30.0194	0.952638
$994$	0	0
$995$	−78.6727	−2.49409
$996$	0	0
$997$	46.2267	1.46401	0.732007	−	0.681297i	$-0.238584\pi$
$997$	46.2267	1.46401	0.732007	+	0.681297i	$0.238584\pi$
$998$	0	0
$999$	−46.0258	−1.45619

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.r.1.9		15
7.3	odd	6		1148.2.i.e.821.9	yes	30
7.5	odd	6		1148.2.i.e.165.9	✓	30
7.6	odd	2		8036.2.a.q.1.7		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.e.165.9	✓	30	7.5	odd	6
1148.2.i.e.821.9	yes	30	7.3	odd	6
8036.2.a.q.1.7		15	7.6	odd	2
8036.2.a.r.1.9		15	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.27444 q^{3} -4.37429 q^{5} -1.37580 q^{9} +O(q^{10})\)	\(q+1.27444 q^{3} -4.37429 q^{5} -1.37580 q^{9} +6.01391 q^{11} +5.59853 q^{13} -5.57477 q^{15} -6.15244 q^{17} -5.17601 q^{19} -4.66413 q^{23} +14.1344 q^{25} -5.57670 q^{27} -1.77771 q^{29} +4.02499 q^{31} +7.66437 q^{33} +8.25323 q^{37} +7.13499 q^{39} +1.00000 q^{41} +0.831106 q^{43} +6.01815 q^{45} -2.72099 q^{47} -7.84091 q^{51} -0.906871 q^{53} -26.3066 q^{55} -6.59652 q^{57} -7.92624 q^{59} +5.68627 q^{61} -24.4896 q^{65} -0.0872911 q^{67} -5.94416 q^{69} -8.10863 q^{71} +5.60866 q^{73} +18.0134 q^{75} +0.401351 q^{79} -2.97976 q^{81} +7.91362 q^{83} +26.9125 q^{85} -2.26558 q^{87} -13.4751 q^{89} +5.12960 q^{93} +22.6414 q^{95} +7.58525 q^{97} -8.27395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

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