LMFDB - Embedded newform 8036.2.a.t.1.5

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 $ x^20 - 4x^19 - 30x^18 + 128x^17 + 348x^16 - 1644x^15 - 1934x^14 + 10948x^13 + 4748x^12 - 40524x^11 - 220x^10 + 82500x^9 - 21992x^8 - 84720x^7 + 37544x^6 + 34656x^5 - 18823x^4 - 2276x^3 + 1130x^2 + 204*x + 9
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			1.5
Root			$-1.77641$ 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)

$f(q)$	$=$	$q-1.77641 q^{3} -0.716761 q^{5} +0.155650 q^{9} +O(q^{10})$	$q-1.77641 q^{3} -0.716761 q^{5} +0.155650 q^{9} +1.69774 q^{11} -4.41385 q^{13} +1.27327 q^{15} +1.35813 q^{17} +6.61457 q^{19} -4.94432 q^{23} -4.48625 q^{25} +5.05275 q^{27} -0.711000 q^{29} +3.41573 q^{31} -3.01588 q^{33} -5.21830 q^{37} +7.84083 q^{39} -1.00000 q^{41} -6.26787 q^{43} -0.111564 q^{45} +7.29375 q^{47} -2.41260 q^{51} -12.1698 q^{53} -1.21687 q^{55} -11.7502 q^{57} -1.69715 q^{59} +7.76380 q^{61} +3.16368 q^{65} +3.13992 q^{67} +8.78317 q^{69} +3.54572 q^{71} +11.4366 q^{73} +7.96945 q^{75} +8.90590 q^{79} -9.44272 q^{81} -12.4174 q^{83} -0.973454 q^{85} +1.26303 q^{87} -6.08357 q^{89} -6.06775 q^{93} -4.74107 q^{95} -15.9366 q^{97} +0.264252 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.77641	−1.02561	−0.512807	−	0.858504i	$-0.671394\pi$
$3$	−1.77641	−1.02561	−0.512807	+	0.858504i	$0.671394\pi$
$4$	0	0
$5$	−0.716761	−0.320545	−0.160273	−	0.987073i	$-0.551237\pi$
$5$	−0.716761	−0.320545	−0.160273	+	0.987073i	$0.551237\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.155650	0.0518832
$10$	0	0
$11$	1.69774	0.511887	0.255943	−	0.966692i	$-0.417614\pi$
$11$	1.69774	0.511887	0.255943	+	0.966692i	$0.417614\pi$
$12$	0	0
$13$	−4.41385	−1.22418	−0.612091	−	0.790787i	$-0.709671\pi$
$13$	−4.41385	−1.22418	−0.612091	+	0.790787i	$0.709671\pi$
$14$	0	0
$15$	1.27327	0.328756
$16$	0	0
$17$	1.35813	0.329394	0.164697	−	0.986344i	$-0.447335\pi$
$17$	1.35813	0.329394	0.164697	+	0.986344i	$0.447335\pi$
$18$	0	0
$19$	6.61457	1.51749	0.758743	−	0.651390i	$-0.225814\pi$
$19$	6.61457	1.51749	0.758743	+	0.651390i	$0.225814\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.94432	−1.03096	−0.515481	−	0.856901i	$-0.672387\pi$
$23$	−4.94432	−1.03096	−0.515481	+	0.856901i	$0.672387\pi$
$24$	0	0
$25$	−4.48625	−0.897251
$26$	0	0
$27$	5.05275	0.972401
$28$	0	0
$29$	−0.711000	−0.132029	−0.0660147	−	0.997819i	$-0.521028\pi$
$29$	−0.711000	−0.132029	−0.0660147	+	0.997819i	$0.521028\pi$
$30$	0	0
$31$	3.41573	0.613483	0.306742	−	0.951793i	$-0.400761\pi$
$31$	3.41573	0.613483	0.306742	+	0.951793i	$0.400761\pi$
$32$	0	0
$33$	−3.01588	−0.524998
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.21830	−0.857883	−0.428942	−	0.903332i	$-0.641113\pi$
$37$	−5.21830	−0.857883	−0.428942	+	0.903332i	$0.641113\pi$
$38$	0	0
$39$	7.84083	1.25554
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−6.26787	−0.955841	−0.477921	−	0.878403i	$-0.658609\pi$
$43$	−6.26787	−0.955841	−0.477921	+	0.878403i	$0.658609\pi$
$44$	0	0
$45$	−0.111564	−0.0166309
$46$	0	0
$47$	7.29375	1.06390	0.531952	−	0.846775i	$-0.321459\pi$
$47$	7.29375	1.06390	0.531952	+	0.846775i	$0.321459\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.41260	−0.337831
$52$	0	0
$53$	−12.1698	−1.67165	−0.835827	−	0.548994i	$-0.815011\pi$
$53$	−12.1698	−1.67165	−0.835827	+	0.548994i	$0.815011\pi$
$54$	0	0
$55$	−1.21687	−0.164083
$56$	0	0
$57$	−11.7502	−1.55635
$58$	0	0
$59$	−1.69715	−0.220950	−0.110475	−	0.993879i	$-0.535237\pi$
$59$	−1.69715	−0.220950	−0.110475	+	0.993879i	$0.535237\pi$
$60$	0	0
$61$	7.76380	0.994053	0.497027	−	0.867735i	$-0.334425\pi$
$61$	7.76380	0.994053	0.497027	+	0.867735i	$0.334425\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.16368	0.392406
$66$	0	0
$67$	3.13992	0.383602	0.191801	−	0.981434i	$-0.438567\pi$
$67$	3.13992	0.383602	0.191801	+	0.981434i	$0.438567\pi$
$68$	0	0
$69$	8.78317	1.05737
$70$	0	0
$71$	3.54572	0.420799	0.210400	−	0.977615i	$-0.432523\pi$
$71$	3.54572	0.420799	0.210400	+	0.977615i	$0.432523\pi$
$72$	0	0
$73$	11.4366	1.33856	0.669278	−	0.743012i	$-0.266603\pi$
$73$	11.4366	1.33856	0.669278	+	0.743012i	$0.266603\pi$
$74$	0	0
$75$	7.96945	0.920232
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.90590	1.00199	0.500996	−	0.865450i	$-0.332967\pi$
$79$	8.90590	1.00199	0.500996	+	0.865450i	$0.332967\pi$
$80$	0	0
$81$	−9.44272	−1.04919
$82$	0	0
$83$	−12.4174	−1.36299	−0.681494	−	0.731824i	$-0.738669\pi$
$83$	−12.4174	−1.36299	−0.681494	+	0.731824i	$0.738669\pi$
$84$	0	0
$85$	−0.973454	−0.105586
$86$	0	0
$87$	1.26303	0.135411
$88$	0	0
$89$	−6.08357	−0.644857	−0.322429	−	0.946594i	$-0.604499\pi$
$89$	−6.08357	−0.644857	−0.322429	+	0.946594i	$0.604499\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.06775	−0.629197
$94$	0	0
$95$	−4.74107	−0.486423
$96$	0	0
$97$	−15.9366	−1.61812	−0.809059	−	0.587728i	$-0.800023\pi$
$97$	−15.9366	−1.61812	−0.809059	+	0.587728i	$0.800023\pi$
$98$	0	0
$99$	0.264252	0.0265583
$100$	0	0
$101$	15.1504	1.50752	0.753759	−	0.657151i	$-0.228239\pi$
$101$	15.1504	1.50752	0.753759	+	0.657151i	$0.228239\pi$
$102$	0	0
$103$	−18.4235	−1.81532	−0.907662	−	0.419702i	$-0.862134\pi$
$103$	−18.4235	−1.81532	−0.907662	+	0.419702i	$0.862134\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.0184	−1.54855	−0.774277	−	0.632847i	$-0.781886\pi$
$107$	−16.0184	−1.54855	−0.774277	+	0.632847i	$0.781886\pi$
$108$	0	0
$109$	5.59424	0.535831	0.267916	−	0.963442i	$-0.413665\pi$
$109$	5.59424	0.535831	0.267916	+	0.963442i	$0.413665\pi$
$110$	0	0
$111$	9.26986	0.879856
$112$	0	0
$113$	−14.8004	−1.39231	−0.696153	−	0.717893i	$-0.745107\pi$
$113$	−14.8004	−1.39231	−0.696153	+	0.717893i	$0.745107\pi$
$114$	0	0
$115$	3.54390	0.330470
$116$	0	0
$117$	−0.687014	−0.0635145
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.11769	−0.737972
$122$	0	0
$123$	1.77641	0.160174
$124$	0	0
$125$	6.79938	0.608155
$126$	0	0
$127$	−2.17883	−0.193340	−0.0966700	−	0.995316i	$-0.530819\pi$
$127$	−2.17883	−0.193340	−0.0966700	+	0.995316i	$0.530819\pi$
$128$	0	0
$129$	11.1343	0.980324
$130$	0	0
$131$	6.33225	0.553251	0.276625	−	0.960978i	$-0.410784\pi$
$131$	6.33225	0.553251	0.276625	+	0.960978i	$0.410784\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.62161	−0.311699
$136$	0	0
$137$	21.0536	1.79873	0.899365	−	0.437198i	$-0.144029\pi$
$137$	21.0536	1.79873	0.899365	+	0.437198i	$0.144029\pi$
$138$	0	0
$139$	−7.86444	−0.667053	−0.333526	−	0.942741i	$-0.608239\pi$
$139$	−7.86444	−0.667053	−0.333526	+	0.942741i	$0.608239\pi$
$140$	0	0
$141$	−12.9567	−1.09115
$142$	0	0
$143$	−7.49356	−0.626643
$144$	0	0
$145$	0.509617	0.0423214
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.33573	0.764813	0.382407	−	0.923994i	$-0.375095\pi$
$149$	9.33573	0.764813	0.382407	+	0.923994i	$0.375095\pi$
$150$	0	0
$151$	−1.50195	−0.122227	−0.0611135	−	0.998131i	$-0.519465\pi$
$151$	−1.50195	−0.122227	−0.0611135	+	0.998131i	$0.519465\pi$
$152$	0	0
$153$	0.211392	0.0170900
$154$	0	0
$155$	−2.44826	−0.196649
$156$	0	0
$157$	13.5894	1.08455	0.542276	−	0.840201i	$-0.317563\pi$
$157$	13.5894	1.08455	0.542276	+	0.840201i	$0.317563\pi$
$158$	0	0
$159$	21.6186	1.71447
$160$	0	0
$161$	0	0
$162$	0	0
$163$	14.1671	1.10966	0.554828	−	0.831965i	$-0.312784\pi$
$163$	14.1671	1.10966	0.554828	+	0.831965i	$0.312784\pi$
$164$	0	0
$165$	2.16167	0.168286
$166$	0	0
$167$	−7.61946	−0.589611	−0.294806	−	0.955557i	$-0.595255\pi$
$167$	−7.61946	−0.589611	−0.294806	+	0.955557i	$0.595255\pi$
$168$	0	0
$169$	6.48208	0.498621
$170$	0	0
$171$	1.02955	0.0787320
$172$	0	0
$173$	6.78468	0.515830	0.257915	−	0.966168i	$-0.416965\pi$
$173$	6.78468	0.515830	0.257915	+	0.966168i	$0.416965\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.01484	0.226609
$178$	0	0
$179$	13.4212	1.00315	0.501575	−	0.865114i	$-0.332754\pi$
$179$	13.4212	1.00315	0.501575	+	0.865114i	$0.332754\pi$
$180$	0	0
$181$	−23.4891	−1.74593	−0.872964	−	0.487784i	$-0.837805\pi$
$181$	−23.4891	−1.74593	−0.872964	+	0.487784i	$0.837805\pi$
$182$	0	0
$183$	−13.7917	−1.01951
$184$	0	0
$185$	3.74027	0.274990
$186$	0	0
$187$	2.30574	0.168613
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.74296	0.270831	0.135416	−	0.990789i	$-0.456763\pi$
$191$	3.74296	0.270831	0.135416	+	0.990789i	$0.456763\pi$
$192$	0	0
$193$	22.3013	1.60528	0.802640	−	0.596464i	$-0.203428\pi$
$193$	22.3013	1.60528	0.802640	+	0.596464i	$0.203428\pi$
$194$	0	0
$195$	−5.62000	−0.402457
$196$	0	0
$197$	12.1467	0.865415	0.432708	−	0.901534i	$-0.357558\pi$
$197$	12.1467	0.865415	0.432708	+	0.901534i	$0.357558\pi$
$198$	0	0
$199$	12.7447	0.903445	0.451722	−	0.892159i	$-0.350810\pi$
$199$	12.7447	0.903445	0.451722	+	0.892159i	$0.350810\pi$
$200$	0	0
$201$	−5.57780	−0.393427
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.716761	0.0500608
$206$	0	0
$207$	−0.769582	−0.0534896
$208$	0	0
$209$	11.2298	0.776781
$210$	0	0
$211$	2.56860	0.176830	0.0884149	−	0.996084i	$-0.471820\pi$
$211$	2.56860	0.176830	0.0884149	+	0.996084i	$0.471820\pi$
$212$	0	0
$213$	−6.29866	−0.431577
$214$	0	0
$215$	4.49257	0.306391
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−20.3162	−1.37284
$220$	0	0
$221$	−5.99458	−0.403239
$222$	0	0
$223$	−20.3812	−1.36483	−0.682413	−	0.730967i	$-0.739069\pi$
$223$	−20.3812	−1.36483	−0.682413	+	0.730967i	$0.739069\pi$
$224$	0	0
$225$	−0.698283	−0.0465522
$226$	0	0
$227$	−25.8696	−1.71703	−0.858513	−	0.512792i	$-0.828611\pi$
$227$	−25.8696	−1.71703	−0.858513	+	0.512792i	$0.828611\pi$
$228$	0	0
$229$	23.2042	1.53338	0.766689	−	0.642018i	$-0.221903\pi$
$229$	23.2042	1.53338	0.766689	+	0.642018i	$0.221903\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.8849	1.17168	0.585838	−	0.810428i	$-0.300765\pi$
$233$	17.8849	1.17168	0.585838	+	0.810428i	$0.300765\pi$
$234$	0	0
$235$	−5.22788	−0.341029
$236$	0	0
$237$	−15.8206	−1.02766
$238$	0	0
$239$	−9.03234	−0.584254	−0.292127	−	0.956380i	$-0.594363\pi$
$239$	−9.03234	−0.584254	−0.292127	+	0.956380i	$0.594363\pi$
$240$	0	0
$241$	−3.99359	−0.257250	−0.128625	−	0.991693i	$-0.541056\pi$
$241$	−3.99359	−0.257250	−0.128625	+	0.991693i	$0.541056\pi$
$242$	0	0
$243$	1.61595	0.103663
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−29.1957	−1.85768
$248$	0	0
$249$	22.0585	1.39790
$250$	0	0
$251$	25.9887	1.64039	0.820195	−	0.572084i	$-0.193865\pi$
$251$	25.9887	1.64039	0.820195	+	0.572084i	$0.193865\pi$
$252$	0	0
$253$	−8.39416	−0.527736
$254$	0	0
$255$	1.72926	0.108290
$256$	0	0
$257$	24.1578	1.50692	0.753460	−	0.657494i	$-0.228384\pi$
$257$	24.1578	1.50692	0.753460	+	0.657494i	$0.228384\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.110667	−0.00685010
$262$	0	0
$263$	−15.3693	−0.947712	−0.473856	−	0.880602i	$-0.657138\pi$
$263$	−15.3693	−0.947712	−0.473856	+	0.880602i	$0.657138\pi$
$264$	0	0
$265$	8.72286	0.535841
$266$	0	0
$267$	10.8069	0.661374
$268$	0	0
$269$	−13.8438	−0.844074	−0.422037	−	0.906579i	$-0.638685\pi$
$269$	−13.8438	−0.844074	−0.422037	+	0.906579i	$0.638685\pi$
$270$	0	0
$271$	17.5459	1.06584	0.532919	−	0.846167i	$-0.321095\pi$
$271$	17.5459	1.06584	0.532919	+	0.846167i	$0.321095\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.61648	−0.459291
$276$	0	0
$277$	−14.9385	−0.897568	−0.448784	−	0.893640i	$-0.648143\pi$
$277$	−14.9385	−0.897568	−0.448784	+	0.893640i	$0.648143\pi$
$278$	0	0
$279$	0.531657	0.0318295
$280$	0	0
$281$	−17.8574	−1.06529	−0.532643	−	0.846340i	$-0.678801\pi$
$281$	−17.8574	−1.06529	−0.532643	+	0.846340i	$0.678801\pi$
$282$	0	0
$283$	10.5195	0.625317	0.312658	−	0.949866i	$-0.398780\pi$
$283$	10.5195	0.625317	0.312658	+	0.949866i	$0.398780\pi$
$284$	0	0
$285$	8.42210	0.498882
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.1555	−0.891499
$290$	0	0
$291$	28.3100	1.65956
$292$	0	0
$293$	−2.08524	−0.121821	−0.0609105	−	0.998143i	$-0.519400\pi$
$293$	−2.08524	−0.121821	−0.0609105	+	0.998143i	$0.519400\pi$
$294$	0	0
$295$	1.21645	0.0708244
$296$	0	0
$297$	8.57823	0.497760
$298$	0	0
$299$	21.8235	1.26209
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−26.9133	−1.54613
$304$	0	0
$305$	−5.56480	−0.318639
$306$	0	0
$307$	31.1415	1.77734	0.888669	−	0.458549i	$-0.151631\pi$
$307$	31.1415	1.77734	0.888669	+	0.458549i	$0.151631\pi$
$308$	0	0
$309$	32.7278	1.86182
$310$	0	0
$311$	1.29468	0.0734144	0.0367072	−	0.999326i	$-0.488313\pi$
$311$	1.29468	0.0734144	0.0367072	+	0.999326i	$0.488313\pi$
$312$	0	0
$313$	3.26088	0.184316	0.0921580	−	0.995744i	$-0.470624\pi$
$313$	3.26088	0.184316	0.0921580	+	0.995744i	$0.470624\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.6758	1.49826	0.749131	−	0.662422i	$-0.230472\pi$
$317$	26.6758	1.49826	0.749131	+	0.662422i	$0.230472\pi$
$318$	0	0
$319$	−1.20709	−0.0675841
$320$	0	0
$321$	28.4553	1.58822
$322$	0	0
$323$	8.98343	0.499852
$324$	0	0
$325$	19.8017	1.09840
$326$	0	0
$327$	−9.93769	−0.549556
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−15.3959	−0.846237	−0.423118	−	0.906074i	$-0.639065\pi$
$331$	−15.3959	−0.846237	−0.423118	+	0.906074i	$0.639065\pi$
$332$	0	0
$333$	−0.812226	−0.0445097
$334$	0	0
$335$	−2.25057	−0.122962
$336$	0	0
$337$	9.41514	0.512875	0.256438	−	0.966561i	$-0.417451\pi$
$337$	9.41514	0.512875	0.256438	+	0.966561i	$0.417451\pi$
$338$	0	0
$339$	26.2917	1.42797
$340$	0	0
$341$	5.79901	0.314034
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−6.29544	−0.338935
$346$	0	0
$347$	17.8029	0.955707	0.477854	−	0.878440i	$-0.341415\pi$
$347$	17.8029	0.955707	0.477854	+	0.878440i	$0.341415\pi$
$348$	0	0
$349$	8.62781	0.461836	0.230918	−	0.972973i	$-0.425827\pi$
$349$	8.62781	0.461836	0.230918	+	0.972973i	$0.425827\pi$
$350$	0	0
$351$	−22.3021	−1.19040
$352$	0	0
$353$	5.87764	0.312835	0.156418	−	0.987691i	$-0.450005\pi$
$353$	5.87764	0.312835	0.156418	+	0.987691i	$0.450005\pi$
$354$	0	0
$355$	−2.54143	−0.134885
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.18050	0.220638	0.110319	−	0.993896i	$-0.464813\pi$
$359$	4.18050	0.220638	0.110319	+	0.993896i	$0.464813\pi$
$360$	0	0
$361$	24.7525	1.30276
$362$	0	0
$363$	14.4204	0.756874
$364$	0	0
$365$	−8.19734	−0.429068
$366$	0	0
$367$	−12.3369	−0.643980	−0.321990	−	0.946743i	$-0.604352\pi$
$367$	−12.3369	−0.643980	−0.321990	+	0.946743i	$0.604352\pi$
$368$	0	0
$369$	−0.155650	−0.00810279
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.6356	−0.550690	−0.275345	−	0.961345i	$-0.588792\pi$
$373$	−10.6356	−0.550690	−0.275345	+	0.961345i	$0.588792\pi$
$374$	0	0
$375$	−12.0785	−0.623732
$376$	0	0
$377$	3.13825	0.161628
$378$	0	0
$379$	10.0037	0.513857	0.256929	−	0.966430i	$-0.417289\pi$
$379$	10.0037	0.513857	0.256929	+	0.966430i	$0.417289\pi$
$380$	0	0
$381$	3.87051	0.198292
$382$	0	0
$383$	8.42654	0.430576	0.215288	−	0.976551i	$-0.430931\pi$
$383$	8.42654	0.430576	0.215288	+	0.976551i	$0.430931\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.975591	−0.0495921
$388$	0	0
$389$	21.0941	1.06951	0.534756	−	0.845007i	$-0.320404\pi$
$389$	21.0941	1.06951	0.534756	+	0.845007i	$0.320404\pi$
$390$	0	0
$391$	−6.71502	−0.339593
$392$	0	0
$393$	−11.2487	−0.567422
$394$	0	0
$395$	−6.38340	−0.321184
$396$	0	0
$397$	−7.46045	−0.374429	−0.187215	−	0.982319i	$-0.559946\pi$
$397$	−7.46045	−0.374429	−0.187215	+	0.982319i	$0.559946\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.2028	−0.958940	−0.479470	−	0.877558i	$-0.659171\pi$
$401$	−19.2028	−0.958940	−0.479470	+	0.877558i	$0.659171\pi$
$402$	0	0
$403$	−15.0765	−0.751015
$404$	0	0
$405$	6.76818	0.336313
$406$	0	0
$407$	−8.85930	−0.439139
$408$	0	0
$409$	18.7835	0.928784	0.464392	−	0.885630i	$-0.346273\pi$
$409$	18.7835	0.928784	0.464392	+	0.885630i	$0.346273\pi$
$410$	0	0
$411$	−37.3999	−1.84480
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.90031	0.436899
$416$	0	0
$417$	13.9705	0.684139
$418$	0	0
$419$	−2.94543	−0.143894	−0.0719469	−	0.997408i	$-0.522921\pi$
$419$	−2.94543	−0.143894	−0.0719469	+	0.997408i	$0.522921\pi$
$420$	0	0
$421$	−24.1692	−1.17793	−0.588967	−	0.808157i	$-0.700465\pi$
$421$	−24.1692	−1.17793	−0.588967	+	0.808157i	$0.700465\pi$
$422$	0	0
$423$	1.13527	0.0551987
$424$	0	0
$425$	−6.09291	−0.295549
$426$	0	0
$427$	0	0
$428$	0	0
$429$	13.3117	0.642693
$430$	0	0
$431$	19.2317	0.926359	0.463179	−	0.886265i	$-0.346709\pi$
$431$	19.2317	0.926359	0.463179	+	0.886265i	$0.346709\pi$
$432$	0	0
$433$	25.3115	1.21639	0.608196	−	0.793787i	$-0.291894\pi$
$433$	25.3115	1.21639	0.608196	+	0.793787i	$0.291894\pi$
$434$	0	0
$435$	−0.905291	−0.0434054
$436$	0	0
$437$	−32.7046	−1.56447
$438$	0	0
$439$	21.2047	1.01205	0.506023	−	0.862520i	$-0.331115\pi$
$439$	21.2047	1.01205	0.506023	+	0.862520i	$0.331115\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.2976	−1.43948	−0.719741	−	0.694242i	$-0.755740\pi$
$443$	−30.2976	−1.43948	−0.719741	+	0.694242i	$0.755740\pi$
$444$	0	0
$445$	4.36047	0.206706
$446$	0	0
$447$	−16.5841	−0.784403
$448$	0	0
$449$	−7.56127	−0.356838	−0.178419	−	0.983955i	$-0.557098\pi$
$449$	−7.56127	−0.356838	−0.178419	+	0.983955i	$0.557098\pi$
$450$	0	0
$451$	−1.69774	−0.0799433
$452$	0	0
$453$	2.66809	0.125358
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.5520	1.52272	0.761360	−	0.648329i	$-0.224532\pi$
$457$	32.5520	1.52272	0.761360	+	0.648329i	$0.224532\pi$
$458$	0	0
$459$	6.86228	0.320304
$460$	0	0
$461$	20.1030	0.936289	0.468145	−	0.883652i	$-0.344923\pi$
$461$	20.1030	0.936289	0.468145	+	0.883652i	$0.344923\pi$
$462$	0	0
$463$	−9.92239	−0.461133	−0.230566	−	0.973057i	$-0.574058\pi$
$463$	−9.92239	−0.461133	−0.230566	+	0.973057i	$0.574058\pi$
$464$	0	0
$465$	4.34913	0.201686
$466$	0	0
$467$	−5.88181	−0.272178	−0.136089	−	0.990697i	$-0.543453\pi$
$467$	−5.88181	−0.272178	−0.136089	+	0.990697i	$0.543453\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−24.1404	−1.11233
$472$	0	0
$473$	−10.6412	−0.489283
$474$	0	0
$475$	−29.6746	−1.36157
$476$	0	0
$477$	−1.89423	−0.0867307
$478$	0	0
$479$	−14.5129	−0.663109	−0.331555	−	0.943436i	$-0.607573\pi$
$479$	−14.5129	−0.663109	−0.331555	+	0.943436i	$0.607573\pi$
$480$	0	0
$481$	23.0328	1.05020
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.4227	0.518680
$486$	0	0
$487$	31.8829	1.44475	0.722375	−	0.691501i	$-0.243050\pi$
$487$	31.8829	1.44475	0.722375	+	0.691501i	$0.243050\pi$
$488$	0	0
$489$	−25.1667	−1.13808
$490$	0	0
$491$	29.3079	1.32265	0.661323	−	0.750101i	$-0.269995\pi$
$491$	29.3079	1.32265	0.661323	+	0.750101i	$0.269995\pi$
$492$	0	0
$493$	−0.965629	−0.0434897
$494$	0	0
$495$	−0.189406	−0.00851315
$496$	0	0
$497$	0	0
$498$	0	0
$499$	21.4677	0.961024	0.480512	−	0.876988i	$-0.340451\pi$
$499$	21.4677	0.961024	0.480512	+	0.876988i	$0.340451\pi$
$500$	0	0
$501$	13.5353	0.604713
$502$	0	0
$503$	−4.79250	−0.213687	−0.106844	−	0.994276i	$-0.534074\pi$
$503$	−4.79250	−0.213687	−0.106844	+	0.994276i	$0.534074\pi$
$504$	0	0
$505$	−10.8592	−0.483228
$506$	0	0
$507$	−11.5149	−0.511393
$508$	0	0
$509$	5.25958	0.233127	0.116563	−	0.993183i	$-0.462812\pi$
$509$	5.25958	0.233127	0.116563	+	0.993183i	$0.462812\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	33.4217	1.47561
$514$	0	0
$515$	13.2053	0.581894
$516$	0	0
$517$	12.3829	0.544598
$518$	0	0
$519$	−12.0524	−0.529042
$520$	0	0
$521$	−0.492973	−0.0215976	−0.0107988	−	0.999942i	$-0.503437\pi$
$521$	−0.492973	−0.0215976	−0.0107988	+	0.999942i	$0.503437\pi$
$522$	0	0
$523$	10.1976	0.445908	0.222954	−	0.974829i	$-0.428430\pi$
$523$	10.1976	0.445908	0.222954	+	0.974829i	$0.428430\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.63900	0.202078
$528$	0	0
$529$	1.44633	0.0628840
$530$	0	0
$531$	−0.264160	−0.0114636
$532$	0	0
$533$	4.41385	0.191185
$534$	0	0
$535$	11.4813	0.496382
$536$	0	0
$537$	−23.8417	−1.02884
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5.80961	−0.249775	−0.124887	−	0.992171i	$-0.539857\pi$
$541$	−5.80961	−0.249775	−0.124887	+	0.992171i	$0.539857\pi$
$542$	0	0
$543$	41.7263	1.79065
$544$	0	0
$545$	−4.00974	−0.171758
$546$	0	0
$547$	44.3728	1.89724	0.948621	−	0.316413i	$-0.102479\pi$
$547$	44.3728	1.89724	0.948621	+	0.316413i	$0.102479\pi$
$548$	0	0
$549$	1.20843	0.0515746
$550$	0	0
$551$	−4.70296	−0.200353
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−6.64428	−0.282034
$556$	0	0
$557$	28.2088	1.19525	0.597624	−	0.801777i	$-0.296112\pi$
$557$	28.2088	1.19525	0.597624	+	0.801777i	$0.296112\pi$
$558$	0	0
$559$	27.6654	1.17012
$560$	0	0
$561$	−4.09596	−0.172931
$562$	0	0
$563$	43.1476	1.81845	0.909226	−	0.416302i	$-0.136674\pi$
$563$	43.1476	1.81845	0.909226	+	0.416302i	$0.136674\pi$
$564$	0	0
$565$	10.6084	0.446298
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23.4484	0.983009	0.491504	−	0.870875i	$-0.336447\pi$
$569$	23.4484	0.983009	0.491504	+	0.870875i	$0.336447\pi$
$570$	0	0
$571$	15.7679	0.659865	0.329933	−	0.944005i	$-0.392974\pi$
$571$	15.7679	0.659865	0.329933	+	0.944005i	$0.392974\pi$
$572$	0	0
$573$	−6.64905	−0.277768
$574$	0	0
$575$	22.1815	0.925032
$576$	0	0
$577$	34.7488	1.44661	0.723306	−	0.690528i	$-0.242622\pi$
$577$	34.7488	1.44661	0.723306	+	0.690528i	$0.242622\pi$
$578$	0	0
$579$	−39.6163	−1.64640
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−20.6611	−0.855697
$584$	0	0
$585$	0.492425	0.0203593
$586$	0	0
$587$	9.23176	0.381036	0.190518	−	0.981684i	$-0.438983\pi$
$587$	9.23176	0.381036	0.190518	+	0.981684i	$0.438983\pi$
$588$	0	0
$589$	22.5936	0.930952
$590$	0	0
$591$	−21.5775	−0.887581
$592$	0	0
$593$	42.6848	1.75285	0.876427	−	0.481535i	$-0.159920\pi$
$593$	42.6848	1.75285	0.876427	+	0.481535i	$0.159920\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−22.6398	−0.926585
$598$	0	0
$599$	−30.3442	−1.23983	−0.619915	−	0.784669i	$-0.712833\pi$
$599$	−30.3442	−1.23983	−0.619915	+	0.784669i	$0.712833\pi$
$600$	0	0
$601$	18.9452	0.772792	0.386396	−	0.922333i	$-0.373720\pi$
$601$	18.9452	0.772792	0.386396	+	0.922333i	$0.373720\pi$
$602$	0	0
$603$	0.488727	0.0199025
$604$	0	0
$605$	5.81845	0.236554
$606$	0	0
$607$	24.6141	0.999057	0.499529	−	0.866297i	$-0.333507\pi$
$607$	24.6141	0.999057	0.499529	+	0.866297i	$0.333507\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−32.1935	−1.30241
$612$	0	0
$613$	−26.3723	−1.06517	−0.532583	−	0.846378i	$-0.678779\pi$
$613$	−26.3723	−1.06517	−0.532583	+	0.846378i	$0.678779\pi$
$614$	0	0
$615$	−1.27327	−0.0513430
$616$	0	0
$617$	17.8189	0.717361	0.358681	−	0.933460i	$-0.383227\pi$
$617$	17.8189	0.717361	0.358681	+	0.933460i	$0.383227\pi$
$618$	0	0
$619$	−7.78205	−0.312787	−0.156393	−	0.987695i	$-0.549987\pi$
$619$	−7.78205	−0.312787	−0.156393	+	0.987695i	$0.549987\pi$
$620$	0	0
$621$	−24.9824	−1.00251
$622$	0	0
$623$	0	0
$624$	0	0
$625$	17.5577	0.702309
$626$	0	0
$627$	−19.9488	−0.796677
$628$	0	0
$629$	−7.08712	−0.282582
$630$	0	0
$631$	38.4583	1.53100	0.765499	−	0.643437i	$-0.222492\pi$
$631$	38.4583	1.53100	0.765499	+	0.643437i	$0.222492\pi$
$632$	0	0
$633$	−4.56290	−0.181359
$634$	0	0
$635$	1.56170	0.0619743
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.551889	0.0218324
$640$	0	0
$641$	27.2535	1.07645	0.538224	−	0.842802i	$-0.319096\pi$
$641$	27.2535	1.07645	0.538224	+	0.842802i	$0.319096\pi$
$642$	0	0
$643$	24.3283	0.959414	0.479707	−	0.877429i	$-0.340743\pi$
$643$	24.3283	0.959414	0.479707	+	0.877429i	$0.340743\pi$
$644$	0	0
$645$	−7.98066	−0.314238
$646$	0	0
$647$	−7.86154	−0.309069	−0.154535	−	0.987987i	$-0.549388\pi$
$647$	−7.86154	−0.309069	−0.154535	+	0.987987i	$0.549388\pi$
$648$	0	0
$649$	−2.88131	−0.113101
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−44.8914	−1.75674	−0.878368	−	0.477986i	$-0.841367\pi$
$653$	−44.8914	−1.75674	−0.878368	+	0.477986i	$0.841367\pi$
$654$	0	0
$655$	−4.53871	−0.177342
$656$	0	0
$657$	1.78011	0.0694486
$658$	0	0
$659$	−32.8974	−1.28150	−0.640750	−	0.767750i	$-0.721376\pi$
$659$	−32.8974	−1.28150	−0.640750	+	0.767750i	$0.721376\pi$
$660$	0	0
$661$	−27.4082	−1.06606	−0.533028	−	0.846098i	$-0.678946\pi$
$661$	−27.4082	−1.06606	−0.533028	+	0.846098i	$0.678946\pi$
$662$	0	0
$663$	10.6489	0.413567
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.51541	0.136117
$668$	0	0
$669$	36.2055	1.39978
$670$	0	0
$671$	13.1809	0.508843
$672$	0	0
$673$	−11.7433	−0.452673	−0.226336	−	0.974049i	$-0.572675\pi$
$673$	−11.7433	−0.452673	−0.226336	+	0.974049i	$0.572675\pi$
$674$	0	0
$675$	−22.6679	−0.872488
$676$	0	0
$677$	34.8790	1.34051	0.670255	−	0.742131i	$-0.266185\pi$
$677$	34.8790	1.34051	0.670255	+	0.742131i	$0.266185\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	45.9552	1.76101
$682$	0	0
$683$	28.1265	1.07623	0.538115	−	0.842871i	$-0.319137\pi$
$683$	28.1265	1.07623	0.538115	+	0.842871i	$0.319137\pi$
$684$	0	0
$685$	−15.0904	−0.576575
$686$	0	0
$687$	−41.2203	−1.57265
$688$	0	0
$689$	53.7158	2.04641
$690$	0	0
$691$	−22.0866	−0.840216	−0.420108	−	0.907474i	$-0.638008\pi$
$691$	−22.0866	−0.840216	−0.420108	+	0.907474i	$0.638008\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.63693	0.213821
$696$	0	0
$697$	−1.35813	−0.0514428
$698$	0	0
$699$	−31.7709	−1.20169
$700$	0	0
$701$	19.4355	0.734067	0.367034	−	0.930208i	$-0.380373\pi$
$701$	19.4355	0.734067	0.367034	+	0.930208i	$0.380373\pi$
$702$	0	0
$703$	−34.5168	−1.30183
$704$	0	0
$705$	9.28688	0.349764
$706$	0	0
$707$	0	0
$708$	0	0
$709$	28.3317	1.06402	0.532010	−	0.846738i	$-0.321437\pi$
$709$	28.3317	1.06402	0.532010	+	0.846738i	$0.321437\pi$
$710$	0	0
$711$	1.38620	0.0519865
$712$	0	0
$713$	−16.8885	−0.632478
$714$	0	0
$715$	5.37109	0.200867
$716$	0	0
$717$	16.0452	0.599218
$718$	0	0
$719$	−39.2419	−1.46348	−0.731738	−	0.681586i	$-0.761291\pi$
$719$	−39.2419	−1.46348	−0.731738	+	0.681586i	$0.761291\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	7.09428	0.263839
$724$	0	0
$725$	3.18972	0.118463
$726$	0	0
$727$	34.1234	1.26557	0.632784	−	0.774328i	$-0.281912\pi$
$727$	34.1234	1.26557	0.632784	+	0.774328i	$0.281912\pi$
$728$	0	0
$729$	25.4576	0.942873
$730$	0	0
$731$	−8.51257	−0.314849
$732$	0	0
$733$	20.4573	0.755607	0.377804	−	0.925886i	$-0.376679\pi$
$733$	20.4573	0.755607	0.377804	+	0.925886i	$0.376679\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.33075	0.196361
$738$	0	0
$739$	17.2786	0.635603	0.317801	−	0.948157i	$-0.397056\pi$
$739$	17.2786	0.635603	0.317801	+	0.948157i	$0.397056\pi$
$740$	0	0
$741$	51.8637	1.90526
$742$	0	0
$743$	−40.0023	−1.46754	−0.733771	−	0.679397i	$-0.762241\pi$
$743$	−40.0023	−1.46754	−0.733771	+	0.679397i	$0.762241\pi$
$744$	0	0
$745$	−6.69149	−0.245157
$746$	0	0
$747$	−1.93276	−0.0707161
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−23.8954	−0.871955	−0.435977	−	0.899958i	$-0.643597\pi$
$751$	−23.8954	−0.871955	−0.435977	+	0.899958i	$0.643597\pi$
$752$	0	0
$753$	−46.1667	−1.68241
$754$	0	0
$755$	1.07654	0.0391793
$756$	0	0
$757$	18.8809	0.686239	0.343120	−	0.939292i	$-0.388516\pi$
$757$	18.8809	0.686239	0.343120	+	0.939292i	$0.388516\pi$
$758$	0	0
$759$	14.9115	0.541253
$760$	0	0
$761$	−36.3709	−1.31844	−0.659222	−	0.751949i	$-0.729114\pi$
$761$	−36.3709	−1.31844	−0.659222	+	0.751949i	$0.729114\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.151518	−0.00547813
$766$	0	0
$767$	7.49095	0.270483
$768$	0	0
$769$	19.8688	0.716488	0.358244	−	0.933628i	$-0.383376\pi$
$769$	19.8688	0.716488	0.358244	+	0.933628i	$0.383376\pi$
$770$	0	0
$771$	−42.9142	−1.54552
$772$	0	0
$773$	−52.9289	−1.90372	−0.951860	−	0.306533i	$-0.900831\pi$
$773$	−52.9289	−1.90372	−0.951860	+	0.306533i	$0.900831\pi$
$774$	0	0
$775$	−15.3238	−0.550448
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.61457	−0.236992
$780$	0	0
$781$	6.01969	0.215402
$782$	0	0
$783$	−3.59250	−0.128386
$784$	0	0
$785$	−9.74035	−0.347648
$786$	0	0
$787$	−33.5636	−1.19641	−0.598207	−	0.801341i	$-0.704120\pi$
$787$	−33.5636	−1.19641	−0.598207	+	0.801341i	$0.704120\pi$
$788$	0	0
$789$	27.3023	0.971986
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−34.2683	−1.21690
$794$	0	0
$795$	−15.4954	−0.549566
$796$	0	0
$797$	−1.71633	−0.0607957	−0.0303978	−	0.999538i	$-0.509677\pi$
$797$	−1.71633	−0.0607957	−0.0303978	+	0.999538i	$0.509677\pi$
$798$	0	0
$799$	9.90585	0.350444
$800$	0	0
$801$	−0.946905	−0.0334572
$802$	0	0
$803$	19.4164	0.685189
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24.5924	0.865694
$808$	0	0
$809$	19.5588	0.687650	0.343825	−	0.939034i	$-0.388277\pi$
$809$	19.5588	0.687650	0.343825	+	0.939034i	$0.388277\pi$
$810$	0	0
$811$	−9.68156	−0.339966	−0.169983	−	0.985447i	$-0.554371\pi$
$811$	−9.68156	−0.339966	−0.169983	+	0.985447i	$0.554371\pi$
$812$	0	0
$813$	−31.1688	−1.09314
$814$	0	0
$815$	−10.1545	−0.355695
$816$	0	0
$817$	−41.4593	−1.45048
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.1857	1.08839	0.544195	−	0.838959i	$-0.316835\pi$
$821$	31.1857	1.08839	0.544195	+	0.838959i	$0.316835\pi$
$822$	0	0
$823$	19.1768	0.668462	0.334231	−	0.942491i	$-0.391523\pi$
$823$	19.1768	0.668462	0.334231	+	0.942491i	$0.391523\pi$
$824$	0	0
$825$	13.5300	0.471055
$826$	0	0
$827$	11.5616	0.402037	0.201018	−	0.979587i	$-0.435575\pi$
$827$	11.5616	0.402037	0.201018	+	0.979587i	$0.435575\pi$
$828$	0	0
$829$	33.7231	1.17125	0.585626	−	0.810582i	$-0.300849\pi$
$829$	33.7231	1.17125	0.585626	+	0.810582i	$0.300849\pi$
$830$	0	0
$831$	26.5370	0.920557
$832$	0	0
$833$	0	0
$834$	0	0
$835$	5.46133	0.188997
$836$	0	0
$837$	17.2588	0.596552
$838$	0	0
$839$	6.17987	0.213353	0.106676	−	0.994294i	$-0.465979\pi$
$839$	6.17987	0.213353	0.106676	+	0.994294i	$0.465979\pi$
$840$	0	0
$841$	−28.4945	−0.982568
$842$	0	0
$843$	31.7222	1.09257
$844$	0	0
$845$	−4.64610	−0.159831
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−18.6869	−0.641333
$850$	0	0
$851$	25.8010	0.884445
$852$	0	0
$853$	−1.00381	−0.0343698	−0.0171849	−	0.999852i	$-0.505470\pi$
$853$	−1.00381	−0.0343698	−0.0171849	+	0.999852i	$0.505470\pi$
$854$	0	0
$855$	−0.737945	−0.0252372
$856$	0	0
$857$	38.1622	1.30360	0.651798	−	0.758393i	$-0.274015\pi$
$857$	38.1622	1.30360	0.651798	+	0.758393i	$0.274015\pi$
$858$	0	0
$859$	26.2597	0.895970	0.447985	−	0.894041i	$-0.352142\pi$
$859$	26.2597	0.895970	0.447985	+	0.894041i	$0.352142\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.57453	−0.189759	−0.0948796	−	0.995489i	$-0.530247\pi$
$863$	−5.57453	−0.189759	−0.0948796	+	0.995489i	$0.530247\pi$
$864$	0	0
$865$	−4.86300	−0.165347
$866$	0	0
$867$	26.9224	0.914334
$868$	0	0
$869$	15.1199	0.512906
$870$	0	0
$871$	−13.8591	−0.469599
$872$	0	0
$873$	−2.48053	−0.0839531
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−21.2734	−0.718353	−0.359177	−	0.933270i	$-0.616942\pi$
$877$	−21.2734	−0.718353	−0.359177	+	0.933270i	$0.616942\pi$
$878$	0	0
$879$	3.70425	0.124941
$880$	0	0
$881$	15.3317	0.516538	0.258269	−	0.966073i	$-0.416848\pi$
$881$	15.3317	0.516538	0.258269	+	0.966073i	$0.416848\pi$
$882$	0	0
$883$	46.6633	1.57034	0.785172	−	0.619278i	$-0.212574\pi$
$883$	46.6633	1.57034	0.785172	+	0.619278i	$0.212574\pi$
$884$	0	0
$885$	−2.16092	−0.0726385
$886$	0	0
$887$	−48.3270	−1.62266	−0.811332	−	0.584586i	$-0.801257\pi$
$887$	−48.3270	−1.62266	−0.811332	+	0.584586i	$0.801257\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−16.0313	−0.537067
$892$	0	0
$893$	48.2450	1.61446
$894$	0	0
$895$	−9.61982	−0.321555
$896$	0	0
$897$	−38.7676	−1.29441
$898$	0	0
$899$	−2.42858	−0.0809978
$900$	0	0
$901$	−16.5282	−0.550633
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.8361	0.559649
$906$	0	0
$907$	−41.4253	−1.37550	−0.687752	−	0.725946i	$-0.741402\pi$
$907$	−41.4253	−1.37550	−0.687752	+	0.725946i	$0.741402\pi$
$908$	0	0
$909$	2.35815	0.0782148
$910$	0	0
$911$	16.6815	0.552682	0.276341	−	0.961060i	$-0.410878\pi$
$911$	16.6815	0.552682	0.276341	+	0.961060i	$0.410878\pi$
$912$	0	0
$913$	−21.0815	−0.697695
$914$	0	0
$915$	9.88538	0.326801
$916$	0	0
$917$	0	0
$918$	0	0
$919$	26.0748	0.860128	0.430064	−	0.902798i	$-0.358491\pi$
$919$	26.0748	0.860128	0.430064	+	0.902798i	$0.358491\pi$
$920$	0	0
$921$	−55.3202	−1.82286
$922$	0	0
$923$	−15.6503	−0.515135
$924$	0	0
$925$	23.4106	0.769736
$926$	0	0
$927$	−2.86761	−0.0941848
$928$	0	0
$929$	−5.10830	−0.167598	−0.0837990	−	0.996483i	$-0.526705\pi$
$929$	−5.10830	−0.167598	−0.0837990	+	0.996483i	$0.526705\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−2.29988	−0.0752948
$934$	0	0
$935$	−1.65267	−0.0540480
$936$	0	0
$937$	28.5517	0.932743	0.466372	−	0.884589i	$-0.345561\pi$
$937$	28.5517	0.932743	0.466372	+	0.884589i	$0.345561\pi$
$938$	0	0
$939$	−5.79268	−0.189037
$940$	0	0
$941$	25.2333	0.822582	0.411291	−	0.911504i	$-0.365078\pi$
$941$	25.2333	0.822582	0.411291	+	0.911504i	$0.365078\pi$
$942$	0	0
$943$	4.94432	0.161009
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−14.8090	−0.481226	−0.240613	−	0.970621i	$-0.577349\pi$
$947$	−14.8090	−0.481226	−0.240613	+	0.970621i	$0.577349\pi$
$948$	0	0
$949$	−50.4796	−1.63864
$950$	0	0
$951$	−47.3872	−1.53664
$952$	0	0
$953$	−33.0779	−1.07150	−0.535750	−	0.844377i	$-0.679971\pi$
$953$	−33.0779	−1.07150	−0.535750	+	0.844377i	$0.679971\pi$
$954$	0	0
$955$	−2.68281	−0.0868137
$956$	0	0
$957$	2.14429	0.0693152
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−19.3328	−0.623638
$962$	0	0
$963$	−2.49325	−0.0803439
$964$	0	0
$965$	−15.9847	−0.514565
$966$	0	0
$967$	−28.0778	−0.902920	−0.451460	−	0.892291i	$-0.649097\pi$
$967$	−28.0778	−0.902920	−0.451460	+	0.892291i	$0.649097\pi$
$968$	0	0
$969$	−15.9583	−0.512655
$970$	0	0
$971$	37.1577	1.19245	0.596224	−	0.802818i	$-0.296667\pi$
$971$	37.1577	1.19245	0.596224	+	0.802818i	$0.296667\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−35.1759	−1.12653
$976$	0	0
$977$	−53.9199	−1.72505	−0.862526	−	0.506013i	$-0.831119\pi$
$977$	−53.9199	−1.72505	−0.862526	+	0.506013i	$0.831119\pi$
$978$	0	0
$979$	−10.3283	−0.330094
$980$	0	0
$981$	0.870741	0.0278006
$982$	0	0
$983$	−18.1739	−0.579658	−0.289829	−	0.957078i	$-0.593598\pi$
$983$	−18.1739	−0.579658	−0.289829	+	0.957078i	$0.593598\pi$
$984$	0	0
$985$	−8.70627	−0.277405
$986$	0	0
$987$	0	0
$988$	0	0
$989$	30.9904	0.985437
$990$	0	0
$991$	−61.2275	−1.94496	−0.972478	−	0.232996i	$-0.925147\pi$
$991$	−61.2275	−1.94496	−0.972478	+	0.232996i	$0.925147\pi$
$992$	0	0
$993$	27.3496	0.867912
$994$	0	0
$995$	−9.13488	−0.289595
$996$	0	0
$997$	38.0328	1.20451	0.602256	−	0.798303i	$-0.294269\pi$
$997$	38.0328	1.20451	0.602256	+	0.798303i	$0.294269\pi$
$998$	0	0
$999$	−26.3667	−0.834207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.t.1.5	yes	20
7.6	odd	2		8036.2.a.s.1.16	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.16	✓	20	7.6	odd	2
8036.2.a.t.1.5	yes	20	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)

\(f(q)\)	\(=\)	\(q-1.77641 q^{3} -0.716761 q^{5} +0.155650 q^{9} +O(q^{10})\)	\(q-1.77641 q^{3} -0.716761 q^{5} +0.155650 q^{9} +1.69774 q^{11} -4.41385 q^{13} +1.27327 q^{15} +1.35813 q^{17} +6.61457 q^{19} -4.94432 q^{23} -4.48625 q^{25} +5.05275 q^{27} -0.711000 q^{29} +3.41573 q^{31} -3.01588 q^{33} -5.21830 q^{37} +7.84083 q^{39} -1.00000 q^{41} -6.26787 q^{43} -0.111564 q^{45} +7.29375 q^{47} -2.41260 q^{51} -12.1698 q^{53} -1.21687 q^{55} -11.7502 q^{57} -1.69715 q^{59} +7.76380 q^{61} +3.16368 q^{65} +3.13992 q^{67} +8.78317 q^{69} +3.54572 q^{71} +11.4366 q^{73} +7.96945 q^{75} +8.90590 q^{79} -9.44272 q^{81} -12.4174 q^{83} -0.973454 q^{85} +1.26303 q^{87} -6.08357 q^{89} -6.06775 q^{93} -4.74107 q^{95} -15.9366 q^{97} +0.264252 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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