LMFDB - Embedded newform 8036.2.a.p.1.1

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 18x^{8} + 32x^{7} + 110x^{6} - 154x^{5} - 282x^{4} + 256x^{3} + 253x^{2} - 126x - 49 $ x^10 - 2x^9 - 18x^8 + 32x^7 + 110x^6 - 154x^5 - 282x^4 + 256x^3 + 253x^2 - 126*x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.96176$ 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)

$f(q)$	$=$	$q-2.96176 q^{3} +4.34462 q^{5} +5.77201 q^{9} +O(q^{10})$	$q-2.96176 q^{3} +4.34462 q^{5} +5.77201 q^{9} -0.777317 q^{11} -0.473242 q^{13} -12.8677 q^{15} +1.81112 q^{17} +6.26526 q^{19} +5.33938 q^{23} +13.8757 q^{25} -8.21003 q^{27} +4.40596 q^{29} -5.82118 q^{31} +2.30222 q^{33} -8.34567 q^{37} +1.40163 q^{39} +1.00000 q^{41} -6.18004 q^{43} +25.0772 q^{45} +6.01779 q^{47} -5.36410 q^{51} +11.5946 q^{53} -3.37714 q^{55} -18.5562 q^{57} -0.805651 q^{59} +11.1497 q^{61} -2.05606 q^{65} -11.2115 q^{67} -15.8140 q^{69} -1.94934 q^{71} +0.839620 q^{73} -41.0965 q^{75} +7.38355 q^{79} +7.00008 q^{81} +9.38726 q^{83} +7.86863 q^{85} -13.0494 q^{87} -1.70669 q^{89} +17.2409 q^{93} +27.2202 q^{95} +14.6859 q^{97} -4.48668 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9} + 6 q^{13} - 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 16 q^{25} + 8 q^{27} + 2 q^{29} + 8 q^{31} - 6 q^{33} - 2 q^{37} - 2 q^{39} + 10 q^{41} - 2 q^{43} + 44 q^{45} - 14 q^{47} + 14 q^{51} + 8 q^{53} + 8 q^{55} - 10 q^{57} + 24 q^{59} + 14 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} + 10 q^{71} + 44 q^{73} - 50 q^{75} + 10 q^{79} - 14 q^{81} + 20 q^{83} + 8 q^{85} + 20 q^{87} + 6 q^{89} + 8 q^{93} + 4 q^{95} + 46 q^{97} + 2 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9 + 6 * q^13 - 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 16 * q^25 + 8 * q^27 + 2 * q^29 + 8 * q^31 - 6 * q^33 - 2 * q^37 - 2 * q^39 + 10 * q^41 - 2 * q^43 + 44 * q^45 - 14 * q^47 + 14 * q^51 + 8 * q^53 + 8 * q^55 - 10 * q^57 + 24 * q^59 + 14 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 + 10 * q^71 + 44 * q^73 - 50 * q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 8 * q^85 + 20 * q^87 + 6 * q^89 + 8 * q^93 + 4 * q^95 + 46 * q^97 + 2 * 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
$3$	−2.96176	−1.70997	−0.854986	−	0.518651i	$-0.826434\pi$
$3$	−2.96176	−1.70997	−0.854986	+	0.518651i	$0.826434\pi$
$4$	0	0
$5$	4.34462	1.94297	0.971486	−	0.237095i	$-0.0761954\pi$
$5$	4.34462	1.94297	0.971486	+	0.237095i	$0.0761954\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.77201	1.92400
$10$	0	0
$11$	−0.777317	−0.234370	−0.117185	−	0.993110i	$-0.537387\pi$
$11$	−0.777317	−0.234370	−0.117185	+	0.993110i	$0.537387\pi$
$12$	0	0
$13$	−0.473242	−0.131254	−0.0656269	−	0.997844i	$-0.520905\pi$
$13$	−0.473242	−0.131254	−0.0656269	+	0.997844i	$0.520905\pi$
$14$	0	0
$15$	−12.8677	−3.32243
$16$	0	0
$17$	1.81112	0.439261	0.219631	−	0.975583i	$-0.429515\pi$
$17$	1.81112	0.439261	0.219631	+	0.975583i	$0.429515\pi$
$18$	0	0
$19$	6.26526	1.43735	0.718675	−	0.695346i	$-0.244749\pi$
$19$	6.26526	1.43735	0.718675	+	0.695346i	$0.244749\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.33938	1.11334	0.556669	−	0.830734i	$-0.312079\pi$
$23$	5.33938	1.11334	0.556669	+	0.830734i	$0.312079\pi$
$24$	0	0
$25$	13.8757	2.77514
$26$	0	0
$27$	−8.21003	−1.58002
$28$	0	0
$29$	4.40596	0.818166	0.409083	−	0.912497i	$-0.365849\pi$
$29$	4.40596	0.818166	0.409083	+	0.912497i	$0.365849\pi$
$30$	0	0
$31$	−5.82118	−1.04551	−0.522757	−	0.852482i	$-0.675097\pi$
$31$	−5.82118	−1.04551	−0.522757	+	0.852482i	$0.675097\pi$
$32$	0	0
$33$	2.30222	0.400766
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.34567	−1.37202	−0.686010	−	0.727592i	$-0.740639\pi$
$37$	−8.34567	−1.37202	−0.686010	+	0.727592i	$0.740639\pi$
$38$	0	0
$39$	1.40163	0.224440
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−6.18004	−0.942448	−0.471224	−	0.882014i	$-0.656188\pi$
$43$	−6.18004	−0.942448	−0.471224	+	0.882014i	$0.656188\pi$
$44$	0	0
$45$	25.0772	3.73829
$46$	0	0
$47$	6.01779	0.877785	0.438893	−	0.898540i	$-0.355371\pi$
$47$	6.01779	0.877785	0.438893	+	0.898540i	$0.355371\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−5.36410	−0.751124
$52$	0	0
$53$	11.5946	1.59263	0.796317	−	0.604879i	$-0.206779\pi$
$53$	11.5946	1.59263	0.796317	+	0.604879i	$0.206779\pi$
$54$	0	0
$55$	−3.37714	−0.455374
$56$	0	0
$57$	−18.5562	−2.45783
$58$	0	0
$59$	−0.805651	−0.104887	−0.0524434	−	0.998624i	$-0.516701\pi$
$59$	−0.805651	−0.104887	−0.0524434	+	0.998624i	$0.516701\pi$
$60$	0	0
$61$	11.1497	1.42757	0.713787	−	0.700363i	$-0.246979\pi$
$61$	11.1497	1.42757	0.713787	+	0.700363i	$0.246979\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.05606	−0.255023
$66$	0	0
$67$	−11.2115	−1.36970	−0.684852	−	0.728682i	$-0.740133\pi$
$67$	−11.2115	−1.36970	−0.684852	+	0.728682i	$0.740133\pi$
$68$	0	0
$69$	−15.8140	−1.90378
$70$	0	0
$71$	−1.94934	−0.231344	−0.115672	−	0.993287i	$-0.536902\pi$
$71$	−1.94934	−0.231344	−0.115672	+	0.993287i	$0.536902\pi$
$72$	0	0
$73$	0.839620	0.0982701	0.0491351	−	0.998792i	$-0.484354\pi$
$73$	0.839620	0.0982701	0.0491351	+	0.998792i	$0.484354\pi$
$74$	0	0
$75$	−41.0965	−4.74542
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.38355	0.830714	0.415357	−	0.909659i	$-0.363657\pi$
$79$	7.38355	0.830714	0.415357	+	0.909659i	$0.363657\pi$
$80$	0	0
$81$	7.00008	0.777787
$82$	0	0
$83$	9.38726	1.03039	0.515193	−	0.857074i	$-0.327720\pi$
$83$	9.38726	1.03039	0.515193	+	0.857074i	$0.327720\pi$
$84$	0	0
$85$	7.86863	0.853473
$86$	0	0
$87$	−13.0494	−1.39904
$88$	0	0
$89$	−1.70669	−0.180909	−0.0904543	−	0.995901i	$-0.528832\pi$
$89$	−1.70669	−0.180909	−0.0904543	+	0.995901i	$0.528832\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	17.2409	1.78780
$94$	0	0
$95$	27.2202	2.79273
$96$	0	0
$97$	14.6859	1.49113	0.745566	−	0.666432i	$-0.232179\pi$
$97$	14.6859	1.49113	0.745566	+	0.666432i	$0.232179\pi$
$98$	0	0
$99$	−4.48668	−0.450928
$100$	0	0
$101$	15.9881	1.59087	0.795437	−	0.606036i	$-0.207241\pi$
$101$	15.9881	1.59087	0.795437	+	0.606036i	$0.207241\pi$
$102$	0	0
$103$	−7.25045	−0.714408	−0.357204	−	0.934026i	$-0.616270\pi$
$103$	−7.25045	−0.714408	−0.357204	+	0.934026i	$0.616270\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.674596	0.0652156	0.0326078	−	0.999468i	$-0.489619\pi$
$107$	0.674596	0.0652156	0.0326078	+	0.999468i	$0.489619\pi$
$108$	0	0
$109$	8.73633	0.836788	0.418394	−	0.908266i	$-0.362593\pi$
$109$	8.73633	0.836788	0.418394	+	0.908266i	$0.362593\pi$
$110$	0	0
$111$	24.7179	2.34612
$112$	0	0
$113$	−12.9348	−1.21681	−0.608403	−	0.793629i	$-0.708189\pi$
$113$	−12.9348	−1.21681	−0.608403	+	0.793629i	$0.708189\pi$
$114$	0	0
$115$	23.1976	2.16319
$116$	0	0
$117$	−2.73156	−0.252533
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.3958	−0.945071
$122$	0	0
$123$	−2.96176	−0.267053
$124$	0	0
$125$	38.5616	3.44906
$126$	0	0
$127$	−1.94499	−0.172590	−0.0862948	−	0.996270i	$-0.527503\pi$
$127$	−1.94499	−0.172590	−0.0862948	+	0.996270i	$0.527503\pi$
$128$	0	0
$129$	18.3038	1.61156
$130$	0	0
$131$	2.79149	0.243894	0.121947	−	0.992537i	$-0.461086\pi$
$131$	2.79149	0.243894	0.121947	+	0.992537i	$0.461086\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−35.6694	−3.06994
$136$	0	0
$137$	−8.57721	−0.732800	−0.366400	−	0.930457i	$-0.619410\pi$
$137$	−8.57721	−0.732800	−0.366400	+	0.930457i	$0.619410\pi$
$138$	0	0
$139$	−9.19507	−0.779916	−0.389958	−	0.920833i	$-0.627510\pi$
$139$	−9.19507	−0.779916	−0.389958	+	0.920833i	$0.627510\pi$
$140$	0	0
$141$	−17.8232	−1.50099
$142$	0	0
$143$	0.367859	0.0307619
$144$	0	0
$145$	19.1422	1.58967
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.9177	−1.38595	−0.692976	−	0.720961i	$-0.743701\pi$
$149$	−16.9177	−1.38595	−0.692976	+	0.720961i	$0.743701\pi$
$150$	0	0
$151$	−3.66377	−0.298153	−0.149077	−	0.988826i	$-0.547630\pi$
$151$	−3.66377	−0.298153	−0.149077	+	0.988826i	$0.547630\pi$
$152$	0	0
$153$	10.4538	0.845140
$154$	0	0
$155$	−25.2908	−2.03141
$156$	0	0
$157$	1.32423	0.105685	0.0528425	−	0.998603i	$-0.483172\pi$
$157$	1.32423	0.105685	0.0528425	+	0.998603i	$0.483172\pi$
$158$	0	0
$159$	−34.3403	−2.72336
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−5.50753	−0.431383	−0.215691	−	0.976462i	$-0.569201\pi$
$163$	−5.50753	−0.431383	−0.215691	+	0.976462i	$0.569201\pi$
$164$	0	0
$165$	10.0023	0.778677
$166$	0	0
$167$	23.2282	1.79745	0.898724	−	0.438514i	$-0.144495\pi$
$167$	23.2282	1.79745	0.898724	+	0.438514i	$0.144495\pi$
$168$	0	0
$169$	−12.7760	−0.982772
$170$	0	0
$171$	36.1632	2.76547
$172$	0	0
$173$	−21.8761	−1.66321	−0.831606	−	0.555366i	$-0.812578\pi$
$173$	−21.8761	−1.66321	−0.831606	+	0.555366i	$0.812578\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.38614	0.179354
$178$	0	0
$179$	−14.6814	−1.09734	−0.548670	−	0.836039i	$-0.684866\pi$
$179$	−14.6814	−1.09734	−0.548670	+	0.836039i	$0.684866\pi$
$180$	0	0
$181$	4.97537	0.369816	0.184908	−	0.982756i	$-0.440801\pi$
$181$	4.97537	0.369816	0.184908	+	0.982756i	$0.440801\pi$
$182$	0	0
$183$	−33.0227	−2.44111
$184$	0	0
$185$	−36.2588	−2.66580
$186$	0	0
$187$	−1.40781	−0.102950
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.3511	1.54491	0.772457	−	0.635067i	$-0.219027\pi$
$191$	21.3511	1.54491	0.772457	+	0.635067i	$0.219027\pi$
$192$	0	0
$193$	−12.5624	−0.904265	−0.452132	−	0.891951i	$-0.649337\pi$
$193$	−12.5624	−0.904265	−0.452132	+	0.891951i	$0.649337\pi$
$194$	0	0
$195$	6.08954	0.436081
$196$	0	0
$197$	20.7339	1.47723	0.738614	−	0.674128i	$-0.235480\pi$
$197$	20.7339	1.47723	0.738614	+	0.674128i	$0.235480\pi$
$198$	0	0
$199$	−15.4142	−1.09268	−0.546341	−	0.837563i	$-0.683980\pi$
$199$	−15.4142	−1.09268	−0.546341	+	0.837563i	$0.683980\pi$
$200$	0	0
$201$	33.2058	2.34216
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.34462	0.303441
$206$	0	0
$207$	30.8190	2.14207
$208$	0	0
$209$	−4.87009	−0.336871
$210$	0	0
$211$	−20.4010	−1.40446	−0.702232	−	0.711948i	$-0.747813\pi$
$211$	−20.4010	−1.40446	−0.702232	+	0.711948i	$0.747813\pi$
$212$	0	0
$213$	5.77346	0.395591
$214$	0	0
$215$	−26.8499	−1.83115
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2.48675	−0.168039
$220$	0	0
$221$	−0.857099	−0.0576547
$222$	0	0
$223$	−27.8030	−1.86182	−0.930912	−	0.365242i	$-0.880986\pi$
$223$	−27.8030	−1.86182	−0.930912	+	0.365242i	$0.880986\pi$
$224$	0	0
$225$	80.0908	5.33939
$226$	0	0
$227$	−10.3315	−0.685727	−0.342864	−	0.939385i	$-0.611397\pi$
$227$	−10.3315	−0.685727	−0.342864	+	0.939385i	$0.611397\pi$
$228$	0	0
$229$	28.5063	1.88375	0.941874	−	0.335965i	$-0.109063\pi$
$229$	28.5063	1.88375	0.941874	+	0.335965i	$0.109063\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.50991	0.0989176	0.0494588	−	0.998776i	$-0.484250\pi$
$233$	1.50991	0.0989176	0.0494588	+	0.998776i	$0.484250\pi$
$234$	0	0
$235$	26.1450	1.70551
$236$	0	0
$237$	−21.8683	−1.42050
$238$	0	0
$239$	−7.04433	−0.455660	−0.227830	−	0.973701i	$-0.573163\pi$
$239$	−7.04433	−0.455660	−0.227830	+	0.973701i	$0.573163\pi$
$240$	0	0
$241$	−15.1907	−0.978521	−0.489261	−	0.872138i	$-0.662733\pi$
$241$	−15.1907	−0.978521	−0.489261	+	0.872138i	$0.662733\pi$
$242$	0	0
$243$	3.89753	0.250027
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.96499	−0.188658
$248$	0	0
$249$	−27.8028	−1.76193
$250$	0	0
$251$	26.7280	1.68705	0.843527	−	0.537086i	$-0.180475\pi$
$251$	26.7280	1.68705	0.843527	+	0.537086i	$0.180475\pi$
$252$	0	0
$253$	−4.15039	−0.260933
$254$	0	0
$255$	−23.3050	−1.45941
$256$	0	0
$257$	1.40136	0.0874142	0.0437071	−	0.999044i	$-0.486083\pi$
$257$	1.40136	0.0874142	0.0437071	+	0.999044i	$0.486083\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	25.4312	1.57415
$262$	0	0
$263$	7.49483	0.462151	0.231075	−	0.972936i	$-0.425776\pi$
$263$	7.49483	0.462151	0.231075	+	0.972936i	$0.425776\pi$
$264$	0	0
$265$	50.3739	3.09445
$266$	0	0
$267$	5.05480	0.309349
$268$	0	0
$269$	−1.15251	−0.0702696	−0.0351348	−	0.999383i	$-0.511186\pi$
$269$	−1.15251	−0.0702696	−0.0351348	+	0.999383i	$0.511186\pi$
$270$	0	0
$271$	22.5833	1.37183	0.685917	−	0.727679i	$-0.259401\pi$
$271$	22.5833	1.37183	0.685917	+	0.727679i	$0.259401\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.7858	−0.650410
$276$	0	0
$277$	−27.5147	−1.65320	−0.826599	−	0.562792i	$-0.809727\pi$
$277$	−27.5147	−1.65320	−0.826599	+	0.562792i	$0.809727\pi$
$278$	0	0
$279$	−33.5999	−2.01157
$280$	0	0
$281$	24.8737	1.48384	0.741919	−	0.670489i	$-0.233916\pi$
$281$	24.8737	1.48384	0.741919	+	0.670489i	$0.233916\pi$
$282$	0	0
$283$	20.3751	1.21117	0.605587	−	0.795779i	$-0.292938\pi$
$283$	20.3751	1.21117	0.605587	+	0.795779i	$0.292938\pi$
$284$	0	0
$285$	−80.6196	−4.77549
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.7198	−0.807050
$290$	0	0
$291$	−43.4962	−2.54979
$292$	0	0
$293$	−30.9980	−1.81092	−0.905461	−	0.424430i	$-0.860475\pi$
$293$	−30.9980	−1.81092	−0.905461	+	0.424430i	$0.860475\pi$
$294$	0	0
$295$	−3.50025	−0.203792
$296$	0	0
$297$	6.38179	0.370309
$298$	0	0
$299$	−2.52682	−0.146130
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−47.3529	−2.72035
$304$	0	0
$305$	48.4412	2.77374
$306$	0	0
$307$	−10.1771	−0.580836	−0.290418	−	0.956900i	$-0.593794\pi$
$307$	−10.1771	−0.580836	−0.290418	+	0.956900i	$0.593794\pi$
$308$	0	0
$309$	21.4741	1.22162
$310$	0	0
$311$	3.14919	0.178574	0.0892872	−	0.996006i	$-0.471541\pi$
$311$	3.14919	0.178574	0.0892872	+	0.996006i	$0.471541\pi$
$312$	0	0
$313$	−8.17767	−0.462229	−0.231115	−	0.972927i	$-0.574237\pi$
$313$	−8.17767	−0.462229	−0.231115	+	0.972927i	$0.574237\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.92361	−0.501200	−0.250600	−	0.968091i	$-0.580628\pi$
$317$	−8.92361	−0.501200	−0.250600	+	0.968091i	$0.580628\pi$
$318$	0	0
$319$	−3.42482	−0.191753
$320$	0	0
$321$	−1.99799	−0.111517
$322$	0	0
$323$	11.3471	0.631372
$324$	0	0
$325$	−6.56658	−0.364248
$326$	0	0
$327$	−25.8749	−1.43088
$328$	0	0
$329$	0	0
$330$	0	0
$331$	22.5865	1.24147	0.620733	−	0.784022i	$-0.286835\pi$
$331$	22.5865	1.24147	0.620733	+	0.784022i	$0.286835\pi$
$332$	0	0
$333$	−48.1713	−2.63977
$334$	0	0
$335$	−48.7098	−2.66130
$336$	0	0
$337$	23.3958	1.27445	0.637226	−	0.770677i	$-0.280082\pi$
$337$	23.3958	1.27445	0.637226	+	0.770677i	$0.280082\pi$
$338$	0	0
$339$	38.3098	2.08070
$340$	0	0
$341$	4.52490	0.245037
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−68.7056	−3.69899
$346$	0	0
$347$	−6.89527	−0.370157	−0.185079	−	0.982724i	$-0.559254\pi$
$347$	−6.89527	−0.370157	−0.185079	+	0.982724i	$0.559254\pi$
$348$	0	0
$349$	16.7273	0.895390	0.447695	−	0.894186i	$-0.352245\pi$
$349$	16.7273	0.895390	0.447695	+	0.894186i	$0.352245\pi$
$350$	0	0
$351$	3.88533	0.207384
$352$	0	0
$353$	−5.57591	−0.296776	−0.148388	−	0.988929i	$-0.547408\pi$
$353$	−5.57591	−0.296776	−0.148388	+	0.988929i	$0.547408\pi$
$354$	0	0
$355$	−8.46912	−0.449494
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.55920	−0.346181	−0.173091	−	0.984906i	$-0.555375\pi$
$359$	−6.55920	−0.346181	−0.173091	+	0.984906i	$0.555375\pi$
$360$	0	0
$361$	20.2535	1.06598
$362$	0	0
$363$	30.7898	1.61604
$364$	0	0
$365$	3.64783	0.190936
$366$	0	0
$367$	13.8404	0.722464	0.361232	−	0.932476i	$-0.382356\pi$
$367$	13.8404	0.722464	0.361232	+	0.932476i	$0.382356\pi$
$368$	0	0
$369$	5.77201	0.300479
$370$	0	0
$371$	0	0
$372$	0	0
$373$	23.6114	1.22255	0.611277	−	0.791417i	$-0.290656\pi$
$373$	23.6114	1.22255	0.611277	+	0.791417i	$0.290656\pi$
$374$	0	0
$375$	−114.210	−5.89779
$376$	0	0
$377$	−2.08509	−0.107387
$378$	0	0
$379$	−28.5591	−1.46698	−0.733490	−	0.679700i	$-0.762110\pi$
$379$	−28.5591	−1.46698	−0.733490	+	0.679700i	$0.762110\pi$
$380$	0	0
$381$	5.76058	0.295123
$382$	0	0
$383$	4.45484	0.227632	0.113816	−	0.993502i	$-0.463693\pi$
$383$	4.45484	0.227632	0.113816	+	0.993502i	$0.463693\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−35.6713	−1.81327
$388$	0	0
$389$	31.2216	1.58300	0.791499	−	0.611171i	$-0.209301\pi$
$389$	31.2216	1.58300	0.791499	+	0.611171i	$0.209301\pi$
$390$	0	0
$391$	9.67026	0.489046
$392$	0	0
$393$	−8.26773	−0.417052
$394$	0	0
$395$	32.0787	1.61405
$396$	0	0
$397$	−3.26537	−0.163884	−0.0819420	−	0.996637i	$-0.526112\pi$
$397$	−3.26537	−0.163884	−0.0819420	+	0.996637i	$0.526112\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.6650	0.832209	0.416104	−	0.909317i	$-0.363395\pi$
$401$	16.6650	0.832209	0.416104	+	0.909317i	$0.363395\pi$
$402$	0	0
$403$	2.75483	0.137228
$404$	0	0
$405$	30.4127	1.51122
$406$	0	0
$407$	6.48723	0.321560
$408$	0	0
$409$	−16.0203	−0.792151	−0.396075	−	0.918218i	$-0.629628\pi$
$409$	−16.0203	−0.792151	−0.396075	+	0.918218i	$0.629628\pi$
$410$	0	0
$411$	25.4036	1.25307
$412$	0	0
$413$	0	0
$414$	0	0
$415$	40.7841	2.00201
$416$	0	0
$417$	27.2336	1.33363
$418$	0	0
$419$	8.46285	0.413437	0.206719	−	0.978400i	$-0.433722\pi$
$419$	8.46285	0.413437	0.206719	+	0.978400i	$0.433722\pi$
$420$	0	0
$421$	−15.6972	−0.765033	−0.382517	−	0.923949i	$-0.624943\pi$
$421$	−15.6972	−0.765033	−0.382517	+	0.923949i	$0.624943\pi$
$422$	0	0
$423$	34.7348	1.68886
$424$	0	0
$425$	25.1306	1.21901
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.08951	−0.0526020
$430$	0	0
$431$	24.8406	1.19653	0.598265	−	0.801298i	$-0.295857\pi$
$431$	24.8406	1.19653	0.598265	+	0.801298i	$0.295857\pi$
$432$	0	0
$433$	−0.240670	−0.0115658	−0.00578292	−	0.999983i	$-0.501841\pi$
$433$	−0.240670	−0.0115658	−0.00578292	+	0.999983i	$0.501841\pi$
$434$	0	0
$435$	−56.6946	−2.71830
$436$	0	0
$437$	33.4526	1.60026
$438$	0	0
$439$	26.0884	1.24513	0.622566	−	0.782567i	$-0.286090\pi$
$439$	26.0884	1.24513	0.622566	+	0.782567i	$0.286090\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−26.6254	−1.26501	−0.632505	−	0.774556i	$-0.717973\pi$
$443$	−26.6254	−1.26501	−0.632505	+	0.774556i	$0.717973\pi$
$444$	0	0
$445$	−7.41491	−0.351501
$446$	0	0
$447$	50.1061	2.36994
$448$	0	0
$449$	13.5727	0.640537	0.320269	−	0.947327i	$-0.396227\pi$
$449$	13.5727	0.640537	0.320269	+	0.947327i	$0.396227\pi$
$450$	0	0
$451$	−0.777317	−0.0366024
$452$	0	0
$453$	10.8512	0.509834
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−15.1675	−0.709503	−0.354752	−	0.934961i	$-0.615435\pi$
$457$	−15.1675	−0.709503	−0.354752	+	0.934961i	$0.615435\pi$
$458$	0	0
$459$	−14.8693	−0.694042
$460$	0	0
$461$	−30.0225	−1.39829	−0.699144	−	0.714981i	$-0.746436\pi$
$461$	−30.0225	−1.39829	−0.699144	+	0.714981i	$0.746436\pi$
$462$	0	0
$463$	18.4379	0.856881	0.428441	−	0.903570i	$-0.359063\pi$
$463$	18.4379	0.856881	0.428441	+	0.903570i	$0.359063\pi$
$464$	0	0
$465$	74.9052	3.47365
$466$	0	0
$467$	14.9313	0.690940	0.345470	−	0.938430i	$-0.387719\pi$
$467$	14.9313	0.690940	0.345470	+	0.938430i	$0.387719\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.92205	−0.180718
$472$	0	0
$473$	4.80385	0.220881
$474$	0	0
$475$	86.9350	3.98885
$476$	0	0
$477$	66.9239	3.06423
$478$	0	0
$479$	1.88129	0.0859585	0.0429792	−	0.999076i	$-0.486315\pi$
$479$	1.88129	0.0859585	0.0429792	+	0.999076i	$0.486315\pi$
$480$	0	0
$481$	3.94952	0.180083
$482$	0	0
$483$	0	0
$484$	0	0
$485$	63.8048	2.89723
$486$	0	0
$487$	−34.5885	−1.56736	−0.783678	−	0.621167i	$-0.786659\pi$
$487$	−34.5885	−1.56736	−0.783678	+	0.621167i	$0.786659\pi$
$488$	0	0
$489$	16.3120	0.737653
$490$	0	0
$491$	10.3562	0.467367	0.233684	−	0.972313i	$-0.424922\pi$
$491$	10.3562	0.467367	0.233684	+	0.972313i	$0.424922\pi$
$492$	0	0
$493$	7.97972	0.359389
$494$	0	0
$495$	−19.4929	−0.876141
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12.6775	0.567523	0.283761	−	0.958895i	$-0.408418\pi$
$499$	12.6775	0.567523	0.283761	+	0.958895i	$0.408418\pi$
$500$	0	0
$501$	−68.7962	−3.07359
$502$	0	0
$503$	9.37790	0.418140	0.209070	−	0.977901i	$-0.432956\pi$
$503$	9.37790	0.418140	0.209070	+	0.977901i	$0.432956\pi$
$504$	0	0
$505$	69.4622	3.09103
$506$	0	0
$507$	37.8395	1.68051
$508$	0	0
$509$	15.4818	0.686217	0.343109	−	0.939296i	$-0.388520\pi$
$509$	15.4818	0.686217	0.343109	+	0.939296i	$0.388520\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−51.4380	−2.27104
$514$	0	0
$515$	−31.5004	−1.38808
$516$	0	0
$517$	−4.67773	−0.205726
$518$	0	0
$519$	64.7918	2.84405
$520$	0	0
$521$	9.75537	0.427391	0.213695	−	0.976900i	$-0.431450\pi$
$521$	9.75537	0.427391	0.213695	+	0.976900i	$0.431450\pi$
$522$	0	0
$523$	−18.1546	−0.793847	−0.396923	−	0.917852i	$-0.629922\pi$
$523$	−18.1546	−0.793847	−0.396923	+	0.917852i	$0.629922\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.5429	−0.459254
$528$	0	0
$529$	5.50900	0.239522
$530$	0	0
$531$	−4.65023	−0.201803
$532$	0	0
$533$	−0.473242	−0.0204984
$534$	0	0
$535$	2.93086	0.126712
$536$	0	0
$537$	43.4828	1.87642
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.1428	0.737026	0.368513	−	0.929623i	$-0.379867\pi$
$541$	17.1428	0.737026	0.368513	+	0.929623i	$0.379867\pi$
$542$	0	0
$543$	−14.7358	−0.632376
$544$	0	0
$545$	37.9560	1.62586
$546$	0	0
$547$	−8.67181	−0.370780	−0.185390	−	0.982665i	$-0.559355\pi$
$547$	−8.67181	−0.370780	−0.185390	+	0.982665i	$0.559355\pi$
$548$	0	0
$549$	64.3562	2.74666
$550$	0	0
$551$	27.6045	1.17599
$552$	0	0
$553$	0	0
$554$	0	0
$555$	107.390	4.55844
$556$	0	0
$557$	32.3112	1.36907	0.684534	−	0.728981i	$-0.260006\pi$
$557$	32.3112	1.36907	0.684534	+	0.728981i	$0.260006\pi$
$558$	0	0
$559$	2.92466	0.123700
$560$	0	0
$561$	4.16960	0.176041
$562$	0	0
$563$	8.65741	0.364866	0.182433	−	0.983218i	$-0.441603\pi$
$563$	8.65741	0.364866	0.182433	+	0.983218i	$0.441603\pi$
$564$	0	0
$565$	−56.1968	−2.36422
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−0.0695216	−0.00291450	−0.00145725	−	0.999999i	$-0.500464\pi$
$569$	−0.0695216	−0.00291450	−0.00145725	+	0.999999i	$0.500464\pi$
$570$	0	0
$571$	22.9988	0.962471	0.481235	−	0.876591i	$-0.340188\pi$
$571$	22.9988	0.962471	0.481235	+	0.876591i	$0.340188\pi$
$572$	0	0
$573$	−63.2369	−2.64176
$574$	0	0
$575$	74.0878	3.08967
$576$	0	0
$577$	11.4503	0.476684	0.238342	−	0.971181i	$-0.423396\pi$
$577$	11.4503	0.476684	0.238342	+	0.971181i	$0.423396\pi$
$578$	0	0
$579$	37.2069	1.54627
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−9.01264	−0.373265
$584$	0	0
$585$	−11.8676	−0.490664
$586$	0	0
$587$	−31.3990	−1.29598	−0.647988	−	0.761650i	$-0.724389\pi$
$587$	−31.3990	−1.29598	−0.647988	+	0.761650i	$0.724389\pi$
$588$	0	0
$589$	−36.4712	−1.50277
$590$	0	0
$591$	−61.4088	−2.52602
$592$	0	0
$593$	−34.2830	−1.40784	−0.703918	−	0.710282i	$-0.748568\pi$
$593$	−34.2830	−1.40784	−0.703918	+	0.710282i	$0.748568\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	45.6531	1.86846
$598$	0	0
$599$	16.7968	0.686299	0.343149	−	0.939281i	$-0.388506\pi$
$599$	16.7968	0.686299	0.343149	+	0.939281i	$0.388506\pi$
$600$	0	0
$601$	33.3700	1.36119	0.680595	−	0.732660i	$-0.261722\pi$
$601$	33.3700	1.36119	0.680595	+	0.732660i	$0.261722\pi$
$602$	0	0
$603$	−64.7130	−2.63532
$604$	0	0
$605$	−45.1657	−1.83625
$606$	0	0
$607$	−34.0401	−1.38165	−0.690823	−	0.723024i	$-0.742752\pi$
$607$	−34.0401	−1.38165	−0.690823	+	0.723024i	$0.742752\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.84787	−0.115213
$612$	0	0
$613$	−3.96034	−0.159957	−0.0799784	−	0.996797i	$-0.525485\pi$
$613$	−3.96034	−0.159957	−0.0799784	+	0.996797i	$0.525485\pi$
$614$	0	0
$615$	−12.8677	−0.518876
$616$	0	0
$617$	−7.50998	−0.302340	−0.151170	−	0.988508i	$-0.548304\pi$
$617$	−7.50998	−0.302340	−0.151170	+	0.988508i	$0.548304\pi$
$618$	0	0
$619$	15.4020	0.619060	0.309530	−	0.950890i	$-0.399828\pi$
$619$	15.4020	0.619060	0.309530	+	0.950890i	$0.399828\pi$
$620$	0	0
$621$	−43.8365	−1.75910
$622$	0	0
$623$	0	0
$624$	0	0
$625$	98.1569	3.92628
$626$	0	0
$627$	14.4240	0.576041
$628$	0	0
$629$	−15.1150	−0.602675
$630$	0	0
$631$	−45.3178	−1.80407	−0.902037	−	0.431659i	$-0.857928\pi$
$631$	−45.3178	−1.80407	−0.902037	+	0.431659i	$0.857928\pi$
$632$	0	0
$633$	60.4229	2.40159
$634$	0	0
$635$	−8.45022	−0.335337
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−11.2516	−0.445106
$640$	0	0
$641$	0.421364	0.0166429	0.00832145	−	0.999965i	$-0.497351\pi$
$641$	0.421364	0.0166429	0.00832145	+	0.999965i	$0.497351\pi$
$642$	0	0
$643$	34.4259	1.35763	0.678813	−	0.734311i	$-0.262495\pi$
$643$	34.4259	1.35763	0.678813	+	0.734311i	$0.262495\pi$
$644$	0	0
$645$	79.5230	3.13122
$646$	0	0
$647$	45.0852	1.77248	0.886241	−	0.463225i	$-0.153308\pi$
$647$	45.0852	1.77248	0.886241	+	0.463225i	$0.153308\pi$
$648$	0	0
$649$	0.626246	0.0245823
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.66039	−0.143242	−0.0716211	−	0.997432i	$-0.522817\pi$
$653$	−3.66039	−0.143242	−0.0716211	+	0.997432i	$0.522817\pi$
$654$	0	0
$655$	12.1280	0.473879
$656$	0	0
$657$	4.84630	0.189072
$658$	0	0
$659$	46.9835	1.83022	0.915110	−	0.403205i	$-0.132104\pi$
$659$	46.9835	1.83022	0.915110	+	0.403205i	$0.132104\pi$
$660$	0	0
$661$	−5.14381	−0.200071	−0.100035	−	0.994984i	$-0.531896\pi$
$661$	−5.14381	−0.200071	−0.100035	+	0.994984i	$0.531896\pi$
$662$	0	0
$663$	2.53852	0.0985879
$664$	0	0
$665$	0	0
$666$	0	0
$667$	23.5251	0.910896
$668$	0	0
$669$	82.3457	3.18367
$670$	0	0
$671$	−8.66685	−0.334580
$672$	0	0
$673$	37.2378	1.43541	0.717705	−	0.696347i	$-0.245193\pi$
$673$	37.2378	1.43541	0.717705	+	0.696347i	$0.245193\pi$
$674$	0	0
$675$	−113.920	−4.38478
$676$	0	0
$677$	38.8501	1.49313	0.746566	−	0.665312i	$-0.231701\pi$
$677$	38.8501	1.49313	0.746566	+	0.665312i	$0.231701\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	30.5995	1.17257
$682$	0	0
$683$	39.8187	1.52362	0.761810	−	0.647800i	$-0.224311\pi$
$683$	39.8187	1.52362	0.761810	+	0.647800i	$0.224311\pi$
$684$	0	0
$685$	−37.2647	−1.42381
$686$	0	0
$687$	−84.4287	−3.22116
$688$	0	0
$689$	−5.48703	−0.209039
$690$	0	0
$691$	0.932852	0.0354874	0.0177437	−	0.999843i	$-0.494352\pi$
$691$	0.932852	0.0354874	0.0177437	+	0.999843i	$0.494352\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−39.9491	−1.51535
$696$	0	0
$697$	1.81112	0.0686011
$698$	0	0
$699$	−4.47199	−0.169146
$700$	0	0
$701$	−5.45593	−0.206068	−0.103034	−	0.994678i	$-0.532855\pi$
$701$	−5.45593	−0.206068	−0.103034	+	0.994678i	$0.532855\pi$
$702$	0	0
$703$	−52.2878	−1.97207
$704$	0	0
$705$	−77.4352	−2.91638
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−45.0496	−1.69187	−0.845936	−	0.533285i	$-0.820957\pi$
$709$	−45.0496	−1.69187	−0.845936	+	0.533285i	$0.820957\pi$
$710$	0	0
$711$	42.6179	1.59830
$712$	0	0
$713$	−31.0815	−1.16401
$714$	0	0
$715$	1.59821	0.0597696
$716$	0	0
$717$	20.8636	0.779165
$718$	0	0
$719$	−28.8202	−1.07481	−0.537406	−	0.843324i	$-0.680596\pi$
$719$	−28.8202	−1.07481	−0.537406	+	0.843324i	$0.680596\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	44.9913	1.67324
$724$	0	0
$725$	61.1358	2.27053
$726$	0	0
$727$	30.8265	1.14329	0.571645	−	0.820501i	$-0.306306\pi$
$727$	30.8265	1.14329	0.571645	+	0.820501i	$0.306306\pi$
$728$	0	0
$729$	−32.5438	−1.20533
$730$	0	0
$731$	−11.1928	−0.413981
$732$	0	0
$733$	53.0576	1.95973	0.979863	−	0.199669i	$-0.0639867\pi$
$733$	53.0576	1.95973	0.979863	+	0.199669i	$0.0639867\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.71489	0.321017
$738$	0	0
$739$	15.2346	0.560415	0.280208	−	0.959939i	$-0.409597\pi$
$739$	15.2346	0.560415	0.280208	+	0.959939i	$0.409597\pi$
$740$	0	0
$741$	8.78158	0.322599
$742$	0	0
$743$	8.89696	0.326398	0.163199	−	0.986593i	$-0.447819\pi$
$743$	8.89696	0.326398	0.163199	+	0.986593i	$0.447819\pi$
$744$	0	0
$745$	−73.5009	−2.69287
$746$	0	0
$747$	54.1834	1.98247
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−27.8216	−1.01523	−0.507613	−	0.861585i	$-0.669472\pi$
$751$	−27.8216	−1.01523	−0.507613	+	0.861585i	$0.669472\pi$
$752$	0	0
$753$	−79.1618	−2.88482
$754$	0	0
$755$	−15.9177	−0.579303
$756$	0	0
$757$	−29.2295	−1.06236	−0.531182	−	0.847258i	$-0.678252\pi$
$757$	−29.2295	−1.06236	−0.531182	+	0.847258i	$0.678252\pi$
$758$	0	0
$759$	12.2925	0.446188
$760$	0	0
$761$	−29.5761	−1.07213	−0.536066	−	0.844176i	$-0.680090\pi$
$761$	−29.5761	−1.07213	−0.536066	+	0.844176i	$0.680090\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	45.4178	1.64208
$766$	0	0
$767$	0.381268	0.0137668
$768$	0	0
$769$	−6.48266	−0.233771	−0.116885	−	0.993145i	$-0.537291\pi$
$769$	−6.48266	−0.233771	−0.116885	+	0.993145i	$0.537291\pi$
$770$	0	0
$771$	−4.15048	−0.149476
$772$	0	0
$773$	18.5208	0.666146	0.333073	−	0.942901i	$-0.391914\pi$
$773$	18.5208	0.666146	0.333073	+	0.942901i	$0.391914\pi$
$774$	0	0
$775$	−80.7730	−2.90145
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.26526	0.224476
$780$	0	0
$781$	1.51525	0.0542199
$782$	0	0
$783$	−36.1730	−1.29272
$784$	0	0
$785$	5.75327	0.205343
$786$	0	0
$787$	−48.6346	−1.73364	−0.866818	−	0.498624i	$-0.833839\pi$
$787$	−48.6346	−1.73364	−0.866818	+	0.498624i	$0.833839\pi$
$788$	0	0
$789$	−22.1979	−0.790265
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5.27651	−0.187374
$794$	0	0
$795$	−149.195	−5.29141
$796$	0	0
$797$	−8.70523	−0.308355	−0.154178	−	0.988043i	$-0.549273\pi$
$797$	−8.70523	−0.308355	−0.154178	+	0.988043i	$0.549273\pi$
$798$	0	0
$799$	10.8989	0.385577
$800$	0	0
$801$	−9.85103	−0.348069
$802$	0	0
$803$	−0.652651	−0.0230315
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.41345	0.120159
$808$	0	0
$809$	8.34063	0.293241	0.146620	−	0.989193i	$-0.453160\pi$
$809$	8.34063	0.293241	0.146620	+	0.989193i	$0.453160\pi$
$810$	0	0
$811$	27.3542	0.960537	0.480269	−	0.877121i	$-0.340539\pi$
$811$	27.3542	0.960537	0.480269	+	0.877121i	$0.340539\pi$
$812$	0	0
$813$	−66.8861	−2.34580
$814$	0	0
$815$	−23.9281	−0.838165
$816$	0	0
$817$	−38.7196	−1.35463
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.23089	0.182559	0.0912797	−	0.995825i	$-0.470904\pi$
$821$	5.23089	0.182559	0.0912797	+	0.995825i	$0.470904\pi$
$822$	0	0
$823$	23.8963	0.832973	0.416487	−	0.909142i	$-0.363261\pi$
$823$	23.8963	0.832973	0.416487	+	0.909142i	$0.363261\pi$
$824$	0	0
$825$	31.9450	1.11218
$826$	0	0
$827$	54.4579	1.89369	0.946843	−	0.321696i	$-0.104253\pi$
$827$	54.4579	1.89369	0.946843	+	0.321696i	$0.104253\pi$
$828$	0	0
$829$	48.6163	1.68851	0.844257	−	0.535939i	$-0.180042\pi$
$829$	48.6163	1.68851	0.844257	+	0.535939i	$0.180042\pi$
$830$	0	0
$831$	81.4919	2.82692
$832$	0	0
$833$	0	0
$834$	0	0
$835$	100.917	3.49239
$836$	0	0
$837$	47.7920	1.65193
$838$	0	0
$839$	−13.5147	−0.466579	−0.233290	−	0.972407i	$-0.574949\pi$
$839$	−13.5147	−0.466579	−0.233290	+	0.972407i	$0.574949\pi$
$840$	0	0
$841$	−9.58753	−0.330604
$842$	0	0
$843$	−73.6698	−2.53732
$844$	0	0
$845$	−55.5070	−1.90950
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−60.3461	−2.07107
$850$	0	0
$851$	−44.5607	−1.52752
$852$	0	0
$853$	−23.0117	−0.787906	−0.393953	−	0.919131i	$-0.628893\pi$
$853$	−23.0117	−0.787906	−0.393953	+	0.919131i	$0.628893\pi$
$854$	0	0
$855$	157.115	5.37323
$856$	0	0
$857$	−32.1888	−1.09955	−0.549775	−	0.835313i	$-0.685287\pi$
$857$	−32.1888	−1.09955	−0.549775	+	0.835313i	$0.685287\pi$
$858$	0	0
$859$	6.42212	0.219120	0.109560	−	0.993980i	$-0.465056\pi$
$859$	6.42212	0.219120	0.109560	+	0.993980i	$0.465056\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	48.5039	1.65109	0.825546	−	0.564335i	$-0.190868\pi$
$863$	48.5039	1.65109	0.825546	+	0.564335i	$0.190868\pi$
$864$	0	0
$865$	−95.0435	−3.23157
$866$	0	0
$867$	40.6349	1.38003
$868$	0	0
$869$	−5.73935	−0.194694
$870$	0	0
$871$	5.30576	0.179779
$872$	0	0
$873$	84.7674	2.86894
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7.65826	−0.258601	−0.129300	−	0.991605i	$-0.541273\pi$
$877$	−7.65826	−0.258601	−0.129300	+	0.991605i	$0.541273\pi$
$878$	0	0
$879$	91.8085	3.09663
$880$	0	0
$881$	5.90364	0.198899	0.0994493	−	0.995043i	$-0.468292\pi$
$881$	5.90364	0.198899	0.0994493	+	0.995043i	$0.468292\pi$
$882$	0	0
$883$	−52.0445	−1.75144	−0.875718	−	0.482823i	$-0.839611\pi$
$883$	−52.0445	−1.75144	−0.875718	+	0.482823i	$0.839611\pi$
$884$	0	0
$885$	10.3669	0.348479
$886$	0	0
$887$	43.2210	1.45122	0.725609	−	0.688107i	$-0.241558\pi$
$887$	43.2210	1.45122	0.725609	+	0.688107i	$0.241558\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−5.44128	−0.182290
$892$	0	0
$893$	37.7031	1.26168
$894$	0	0
$895$	−63.7852	−2.13210
$896$	0	0
$897$	7.48383	0.249878
$898$	0	0
$899$	−25.6479	−0.855404
$900$	0	0
$901$	20.9991	0.699583
$902$	0	0
$903$	0	0
$904$	0	0
$905$	21.6161	0.718543
$906$	0	0
$907$	−38.9644	−1.29379	−0.646896	−	0.762578i	$-0.723933\pi$
$907$	−38.9644	−1.29379	−0.646896	+	0.762578i	$0.723933\pi$
$908$	0	0
$909$	92.2835	3.06085
$910$	0	0
$911$	−6.96047	−0.230611	−0.115305	−	0.993330i	$-0.536785\pi$
$911$	−6.96047	−0.230611	−0.115305	+	0.993330i	$0.536785\pi$
$912$	0	0
$913$	−7.29688	−0.241491
$914$	0	0
$915$	−143.471	−4.74301
$916$	0	0
$917$	0	0
$918$	0	0
$919$	44.8592	1.47977	0.739885	−	0.672734i	$-0.234880\pi$
$919$	44.8592	1.47977	0.739885	+	0.672734i	$0.234880\pi$
$920$	0	0
$921$	30.1420	0.993213
$922$	0	0
$923$	0.922508	0.0303647
$924$	0	0
$925$	−115.802	−3.80755
$926$	0	0
$927$	−41.8497	−1.37452
$928$	0	0
$929$	18.4043	0.603825	0.301912	−	0.953336i	$-0.402375\pi$
$929$	18.4043	0.603825	0.301912	+	0.953336i	$0.402375\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−9.32715	−0.305357
$934$	0	0
$935$	−6.11641	−0.200028
$936$	0	0
$937$	−54.0819	−1.76678	−0.883389	−	0.468640i	$-0.844744\pi$
$937$	−54.0819	−1.76678	−0.883389	+	0.468640i	$0.844744\pi$
$938$	0	0
$939$	24.2203	0.790399
$940$	0	0
$941$	33.4289	1.08975	0.544876	−	0.838517i	$-0.316577\pi$
$941$	33.4289	1.08975	0.544876	+	0.838517i	$0.316577\pi$
$942$	0	0
$943$	5.33938	0.173874
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.2312	1.30734	0.653670	−	0.756780i	$-0.273229\pi$
$947$	40.2312	1.30734	0.653670	+	0.756780i	$0.273229\pi$
$948$	0	0
$949$	−0.397344	−0.0128983
$950$	0	0
$951$	26.4296	0.857038
$952$	0	0
$953$	−27.9614	−0.905758	−0.452879	−	0.891572i	$-0.649603\pi$
$953$	−27.9614	−0.905758	−0.452879	+	0.891572i	$0.649603\pi$
$954$	0	0
$955$	92.7626	3.00173
$956$	0	0
$957$	10.1435	0.327893
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.88610	0.0930999
$962$	0	0
$963$	3.89378	0.125475
$964$	0	0
$965$	−54.5791	−1.75696
$966$	0	0
$967$	−29.5335	−0.949735	−0.474867	−	0.880057i	$-0.657504\pi$
$967$	−29.5335	−0.949735	−0.474867	+	0.880057i	$0.657504\pi$
$968$	0	0
$969$	−33.6075	−1.07963
$970$	0	0
$971$	−30.8254	−0.989234	−0.494617	−	0.869111i	$-0.664692\pi$
$971$	−30.8254	−0.989234	−0.494617	+	0.869111i	$0.664692\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	19.4486	0.622854
$976$	0	0
$977$	10.3591	0.331416	0.165708	−	0.986175i	$-0.447009\pi$
$977$	10.3591	0.331416	0.165708	+	0.986175i	$0.447009\pi$
$978$	0	0
$979$	1.32664	0.0423995
$980$	0	0
$981$	50.4262	1.60998
$982$	0	0
$983$	40.3298	1.28632	0.643160	−	0.765732i	$-0.277623\pi$
$983$	40.3298	1.28632	0.643160	+	0.765732i	$0.277623\pi$
$984$	0	0
$985$	90.0809	2.87022
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−32.9976	−1.04926
$990$	0	0
$991$	9.26788	0.294404	0.147202	−	0.989106i	$-0.452973\pi$
$991$	9.26788	0.294404	0.147202	+	0.989106i	$0.452973\pi$
$992$	0	0
$993$	−66.8957	−2.12287
$994$	0	0
$995$	−66.9688	−2.12305
$996$	0	0
$997$	−59.5037	−1.88450	−0.942251	−	0.334908i	$-0.891295\pi$
$997$	−59.5037	−1.88450	−0.942251	+	0.334908i	$0.891295\pi$
$998$	0	0
$999$	68.5182	2.16782

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.p.1.1	yes	10
7.6	odd	2		8036.2.a.o.1.10	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.o.1.10	✓	10	7.6	odd	2
8036.2.a.p.1.1	yes	10	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)

\(f(q)\)	\(=\)	\(q-2.96176 q^{3} +4.34462 q^{5} +5.77201 q^{9} +O(q^{10})\)	\(q-2.96176 q^{3} +4.34462 q^{5} +5.77201 q^{9} -0.777317 q^{11} -0.473242 q^{13} -12.8677 q^{15} +1.81112 q^{17} +6.26526 q^{19} +5.33938 q^{23} +13.8757 q^{25} -8.21003 q^{27} +4.40596 q^{29} -5.82118 q^{31} +2.30222 q^{33} -8.34567 q^{37} +1.40163 q^{39} +1.00000 q^{41} -6.18004 q^{43} +25.0772 q^{45} +6.01779 q^{47} -5.36410 q^{51} +11.5946 q^{53} -3.37714 q^{55} -18.5562 q^{57} -0.805651 q^{59} +11.1497 q^{61} -2.05606 q^{65} -11.2115 q^{67} -15.8140 q^{69} -1.94934 q^{71} +0.839620 q^{73} -41.0965 q^{75} +7.38355 q^{79} +7.00008 q^{81} +9.38726 q^{83} +7.86863 q^{85} -13.0494 q^{87} -1.70669 q^{89} +17.2409 q^{93} +27.2202 q^{95} +14.6859 q^{97} -4.48668 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9} + 6 q^{13} - 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 16 q^{25} + 8 q^{27} + 2 q^{29} + 8 q^{31} - 6 q^{33} - 2 q^{37} - 2 q^{39} + 10 q^{41} - 2 q^{43} + 44 q^{45} - 14 q^{47} + 14 q^{51} + 8 q^{53} + 8 q^{55} - 10 q^{57} + 24 q^{59} + 14 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} + 10 q^{71} + 44 q^{73} - 50 q^{75} + 10 q^{79} - 14 q^{81} + 20 q^{83} + 8 q^{85} + 20 q^{87} + 6 q^{89} + 8 q^{93} + 4 q^{95} + 46 q^{97} + 2 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9 + 6 * q^13 - 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 16 * q^25 + 8 * q^27 + 2 * q^29 + 8 * q^31 - 6 * q^33 - 2 * q^37 - 2 * q^39 + 10 * q^41 - 2 * q^43 + 44 * q^45 - 14 * q^47 + 14 * q^51 + 8 * q^53 + 8 * q^55 - 10 * q^57 + 24 * q^59 + 14 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 + 10 * q^71 + 44 * q^73 - 50 * q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 8 * q^85 + 20 * q^87 + 6 * q^89 + 8 * q^93 + 4 * q^95 + 46 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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