LMFDB - Embedded newform 8036.2.a.s.1.20

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:	$1$
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			1.20
Root			$-3.15720$ 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)

$f(q)$	$=$	$q+3.15720 q^{3} -2.22019 q^{5} +6.96793 q^{9} +O(q^{10})$	$q+3.15720 q^{3} -2.22019 q^{5} +6.96793 q^{9} -4.12500 q^{11} -2.30989 q^{13} -7.00960 q^{15} +2.24833 q^{17} -1.92368 q^{19} -2.59856 q^{23} -0.0707440 q^{25} +12.5275 q^{27} +4.54657 q^{29} -0.881932 q^{31} -13.0235 q^{33} +7.08817 q^{37} -7.29279 q^{39} +1.00000 q^{41} -12.1276 q^{43} -15.4701 q^{45} +5.83253 q^{47} +7.09843 q^{51} -8.01354 q^{53} +9.15830 q^{55} -6.07345 q^{57} -11.7512 q^{59} -7.46459 q^{61} +5.12840 q^{65} +1.58808 q^{67} -8.20417 q^{69} +8.75853 q^{71} -2.18255 q^{73} -0.223353 q^{75} -0.403174 q^{79} +18.6482 q^{81} -5.26975 q^{83} -4.99173 q^{85} +14.3544 q^{87} -13.2977 q^{89} -2.78444 q^{93} +4.27095 q^{95} -3.49549 q^{97} -28.7427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.15720	1.82281	0.911406	−	0.411509i	$-0.134998\pi$
$3$	3.15720	1.82281	0.911406	+	0.411509i	$0.134998\pi$
$4$	0	0
$5$	−2.22019	−0.992900	−0.496450	−	0.868065i	$-0.665363\pi$
$5$	−2.22019	−0.992900	−0.496450	+	0.868065i	$0.665363\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.96793	2.32264
$10$	0	0
$11$	−4.12500	−1.24374	−0.621868	−	0.783122i	$-0.713626\pi$
$11$	−4.12500	−1.24374	−0.621868	+	0.783122i	$0.713626\pi$
$12$	0	0
$13$	−2.30989	−0.640649	−0.320324	−	0.947308i	$-0.603792\pi$
$13$	−2.30989	−0.640649	−0.320324	+	0.947308i	$0.603792\pi$
$14$	0	0
$15$	−7.00960	−1.80987
$16$	0	0
$17$	2.24833	0.545300	0.272650	−	0.962113i	$-0.412100\pi$
$17$	2.24833	0.545300	0.272650	+	0.962113i	$0.412100\pi$
$18$	0	0
$19$	−1.92368	−0.441323	−0.220661	−	0.975350i	$-0.570822\pi$
$19$	−1.92368	−0.441323	−0.220661	+	0.975350i	$0.570822\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.59856	−0.541837	−0.270918	−	0.962602i	$-0.587327\pi$
$23$	−2.59856	−0.541837	−0.270918	+	0.962602i	$0.587327\pi$
$24$	0	0
$25$	−0.0707440	−0.0141488
$26$	0	0
$27$	12.5275	2.41093
$28$	0	0
$29$	4.54657	0.844277	0.422138	−	0.906531i	$-0.361280\pi$
$29$	4.54657	0.844277	0.422138	+	0.906531i	$0.361280\pi$
$30$	0	0
$31$	−0.881932	−0.158400	−0.0791998	−	0.996859i	$-0.525237\pi$
$31$	−0.881932	−0.158400	−0.0791998	+	0.996859i	$0.525237\pi$
$32$	0	0
$33$	−13.0235	−2.26709
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.08817	1.16529	0.582644	−	0.812727i	$-0.302018\pi$
$37$	7.08817	1.16529	0.582644	+	0.812727i	$0.302018\pi$
$38$	0	0
$39$	−7.29279	−1.16778
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−12.1276	−1.84945	−0.924724	−	0.380637i	$-0.875705\pi$
$43$	−12.1276	−1.84945	−0.924724	+	0.380637i	$0.875705\pi$
$44$	0	0
$45$	−15.4701	−2.30615
$46$	0	0
$47$	5.83253	0.850763	0.425381	−	0.905014i	$-0.360140\pi$
$47$	5.83253	0.850763	0.425381	+	0.905014i	$0.360140\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	7.09843	0.993980
$52$	0	0
$53$	−8.01354	−1.10074	−0.550372	−	0.834920i	$-0.685514\pi$
$53$	−8.01354	−1.10074	−0.550372	+	0.834920i	$0.685514\pi$
$54$	0	0
$55$	9.15830	1.23491
$56$	0	0
$57$	−6.07345	−0.804449
$58$	0	0
$59$	−11.7512	−1.52988	−0.764939	−	0.644103i	$-0.777231\pi$
$59$	−11.7512	−1.52988	−0.764939	+	0.644103i	$0.777231\pi$
$60$	0	0
$61$	−7.46459	−0.955743	−0.477872	−	0.878430i	$-0.658592\pi$
$61$	−7.46459	−0.955743	−0.477872	+	0.878430i	$0.658592\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.12840	0.636100
$66$	0	0
$67$	1.58808	0.194014	0.0970072	−	0.995284i	$-0.469073\pi$
$67$	1.58808	0.194014	0.0970072	+	0.995284i	$0.469073\pi$
$68$	0	0
$69$	−8.20417	−0.987666
$70$	0	0
$71$	8.75853	1.03945	0.519723	−	0.854335i	$-0.326035\pi$
$71$	8.75853	1.03945	0.519723	+	0.854335i	$0.326035\pi$
$72$	0	0
$73$	−2.18255	−0.255448	−0.127724	−	0.991810i	$-0.540767\pi$
$73$	−2.18255	−0.255448	−0.127724	+	0.991810i	$0.540767\pi$
$74$	0	0
$75$	−0.223353	−0.0257906
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.403174	−0.0453606	−0.0226803	−	0.999743i	$-0.507220\pi$
$79$	−0.403174	−0.0453606	−0.0226803	+	0.999743i	$0.507220\pi$
$80$	0	0
$81$	18.6482	2.07202
$82$	0	0
$83$	−5.26975	−0.578430	−0.289215	−	0.957264i	$-0.593394\pi$
$83$	−5.26975	−0.578430	−0.289215	+	0.957264i	$0.593394\pi$
$84$	0	0
$85$	−4.99173	−0.541429
$86$	0	0
$87$	14.3544	1.53896
$88$	0	0
$89$	−13.2977	−1.40956	−0.704778	−	0.709428i	$-0.748953\pi$
$89$	−13.2977	−1.40956	−0.704778	+	0.709428i	$0.748953\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.78444	−0.288733
$94$	0	0
$95$	4.27095	0.438190
$96$	0	0
$97$	−3.49549	−0.354913	−0.177457	−	0.984129i	$-0.556787\pi$
$97$	−3.49549	−0.354913	−0.177457	+	0.984129i	$0.556787\pi$
$98$	0	0
$99$	−28.7427	−2.88875
$100$	0	0
$101$	2.26651	0.225526	0.112763	−	0.993622i	$-0.464030\pi$
$101$	2.26651	0.225526	0.112763	+	0.993622i	$0.464030\pi$
$102$	0	0
$103$	−10.2779	−1.01271	−0.506355	−	0.862325i	$-0.669007\pi$
$103$	−10.2779	−1.01271	−0.506355	+	0.862325i	$0.669007\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.6855	−1.12968	−0.564839	−	0.825201i	$-0.691062\pi$
$107$	−11.6855	−1.12968	−0.564839	+	0.825201i	$0.691062\pi$
$108$	0	0
$109$	−14.5791	−1.39642	−0.698212	−	0.715891i	$-0.746021\pi$
$109$	−14.5791	−1.39642	−0.698212	+	0.715891i	$0.746021\pi$
$110$	0	0
$111$	22.3788	2.12410
$112$	0	0
$113$	4.30076	0.404582	0.202291	−	0.979326i	$-0.435161\pi$
$113$	4.30076	0.404582	0.202291	+	0.979326i	$0.435161\pi$
$114$	0	0
$115$	5.76930	0.537990
$116$	0	0
$117$	−16.0952	−1.48800
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.01565	0.546877
$122$	0	0
$123$	3.15720	0.284675
$124$	0	0
$125$	11.2580	1.00695
$126$	0	0
$127$	17.3254	1.53738	0.768688	−	0.639623i	$-0.220910\pi$
$127$	17.3254	1.53738	0.768688	+	0.639623i	$0.220910\pi$
$128$	0	0
$129$	−38.2894	−3.37120
$130$	0	0
$131$	−8.51120	−0.743627	−0.371814	−	0.928307i	$-0.621264\pi$
$131$	−8.51120	−0.743627	−0.371814	+	0.928307i	$0.621264\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−27.8136	−2.39381
$136$	0	0
$137$	−19.7219	−1.68495	−0.842477	−	0.538732i	$-0.818903\pi$
$137$	−19.7219	−1.68495	−0.842477	+	0.538732i	$0.818903\pi$
$138$	0	0
$139$	2.78918	0.236575	0.118288	−	0.992979i	$-0.462260\pi$
$139$	2.78918	0.236575	0.118288	+	0.992979i	$0.462260\pi$
$140$	0	0
$141$	18.4145	1.55078
$142$	0	0
$143$	9.52831	0.796797
$144$	0	0
$145$	−10.0943	−0.838283
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.5417	−1.60092	−0.800458	−	0.599389i	$-0.795410\pi$
$149$	−19.5417	−1.60092	−0.800458	+	0.599389i	$0.795410\pi$
$150$	0	0
$151$	5.15883	0.419820	0.209910	−	0.977721i	$-0.432683\pi$
$151$	5.15883	0.419820	0.209910	+	0.977721i	$0.432683\pi$
$152$	0	0
$153$	15.6662	1.26654
$154$	0	0
$155$	1.95806	0.157275
$156$	0	0
$157$	−18.7616	−1.49734	−0.748670	−	0.662943i	$-0.769307\pi$
$157$	−18.7616	−1.49734	−0.748670	+	0.662943i	$0.769307\pi$
$158$	0	0
$159$	−25.3004	−2.00645
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.9239	−1.63888	−0.819442	−	0.573163i	$-0.805716\pi$
$163$	−20.9239	−1.63888	−0.819442	+	0.573163i	$0.805716\pi$
$164$	0	0
$165$	28.9146	2.25100
$166$	0	0
$167$	11.9531	0.924957	0.462479	−	0.886630i	$-0.346960\pi$
$167$	11.9531	0.924957	0.462479	+	0.886630i	$0.346960\pi$
$168$	0	0
$169$	−7.66440	−0.589569
$170$	0	0
$171$	−13.4041	−1.02504
$172$	0	0
$173$	−5.88830	−0.447679	−0.223840	−	0.974626i	$-0.571859\pi$
$173$	−5.88830	−0.447679	−0.223840	+	0.974626i	$0.571859\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−37.1010	−2.78868
$178$	0	0
$179$	11.1919	0.836524	0.418262	−	0.908326i	$-0.362639\pi$
$179$	11.1919	0.836524	0.418262	+	0.908326i	$0.362639\pi$
$180$	0	0
$181$	−14.6950	−1.09227	−0.546134	−	0.837698i	$-0.683901\pi$
$181$	−14.6950	−1.09227	−0.546134	+	0.837698i	$0.683901\pi$
$182$	0	0
$183$	−23.5672	−1.74214
$184$	0	0
$185$	−15.7371	−1.15702
$186$	0	0
$187$	−9.27437	−0.678209
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.75392	0.126909	0.0634547	−	0.997985i	$-0.479788\pi$
$191$	1.75392	0.126909	0.0634547	+	0.997985i	$0.479788\pi$
$192$	0	0
$193$	10.3137	0.742398	0.371199	−	0.928553i	$-0.378947\pi$
$193$	10.3137	0.742398	0.371199	+	0.928553i	$0.378947\pi$
$194$	0	0
$195$	16.1914	1.15949
$196$	0	0
$197$	−6.69132	−0.476737	−0.238368	−	0.971175i	$-0.576613\pi$
$197$	−6.69132	−0.476737	−0.238368	+	0.971175i	$0.576613\pi$
$198$	0	0
$199$	12.1873	0.863932	0.431966	−	0.901890i	$-0.357820\pi$
$199$	12.1873	0.863932	0.431966	+	0.901890i	$0.357820\pi$
$200$	0	0
$201$	5.01388	0.353652
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.22019	−0.155065
$206$	0	0
$207$	−18.1066	−1.25849
$208$	0	0
$209$	7.93519	0.548889
$210$	0	0
$211$	18.7973	1.29406	0.647028	−	0.762466i	$-0.276012\pi$
$211$	18.7973	1.29406	0.647028	+	0.762466i	$0.276012\pi$
$212$	0	0
$213$	27.6525	1.89472
$214$	0	0
$215$	26.9257	1.83632
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−6.89074	−0.465633
$220$	0	0
$221$	−5.19340	−0.349346
$222$	0	0
$223$	−9.36899	−0.627394	−0.313697	−	0.949523i	$-0.601568\pi$
$223$	−9.36899	−0.627394	−0.313697	+	0.949523i	$0.601568\pi$
$224$	0	0
$225$	−0.492939	−0.0328626
$226$	0	0
$227$	−5.87731	−0.390091	−0.195045	−	0.980794i	$-0.562485\pi$
$227$	−5.87731	−0.390091	−0.195045	+	0.980794i	$0.562485\pi$
$228$	0	0
$229$	12.4528	0.822903	0.411452	−	0.911432i	$-0.365022\pi$
$229$	12.4528	0.822903	0.411452	+	0.911432i	$0.365022\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.5448	−0.887350	−0.443675	−	0.896188i	$-0.646325\pi$
$233$	−13.5448	−0.887350	−0.443675	+	0.896188i	$0.646325\pi$
$234$	0	0
$235$	−12.9493	−0.844722
$236$	0	0
$237$	−1.27290	−0.0826838
$238$	0	0
$239$	7.28427	0.471180	0.235590	−	0.971853i	$-0.424298\pi$
$239$	7.28427	0.471180	0.235590	+	0.971853i	$0.424298\pi$
$240$	0	0
$241$	−3.31710	−0.213673	−0.106837	−	0.994277i	$-0.534072\pi$
$241$	−3.31710	−0.213673	−0.106837	+	0.994277i	$0.534072\pi$
$242$	0	0
$243$	21.2935	1.36598
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.44350	0.282733
$248$	0	0
$249$	−16.6377	−1.05437
$250$	0	0
$251$	14.3120	0.903367	0.451683	−	0.892178i	$-0.350824\pi$
$251$	14.3120	0.903367	0.451683	+	0.892178i	$0.350824\pi$
$252$	0	0
$253$	10.7191	0.673901
$254$	0	0
$255$	−15.7599	−0.986923
$256$	0	0
$257$	29.6916	1.85211	0.926056	−	0.377387i	$-0.123177\pi$
$257$	29.6916	1.85211	0.926056	+	0.377387i	$0.123177\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	31.6802	1.96095
$262$	0	0
$263$	11.9460	0.736624	0.368312	−	0.929702i	$-0.379936\pi$
$263$	11.9460	0.736624	0.368312	+	0.929702i	$0.379936\pi$
$264$	0	0
$265$	17.7916	1.09293
$266$	0	0
$267$	−41.9836	−2.56936
$268$	0	0
$269$	−8.76168	−0.534209	−0.267105	−	0.963668i	$-0.586067\pi$
$269$	−8.76168	−0.534209	−0.267105	+	0.963668i	$0.586067\pi$
$270$	0	0
$271$	−9.34066	−0.567405	−0.283702	−	0.958912i	$-0.591563\pi$
$271$	−9.34066	−0.567405	−0.283702	+	0.958912i	$0.591563\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.291819	0.0175974
$276$	0	0
$277$	32.2328	1.93668	0.968340	−	0.249635i	$-0.0803106\pi$
$277$	32.2328	1.93668	0.968340	+	0.249635i	$0.0803106\pi$
$278$	0	0
$279$	−6.14524	−0.367906
$280$	0	0
$281$	−11.0271	−0.657822	−0.328911	−	0.944361i	$-0.606682\pi$
$281$	−11.0271	−0.657822	−0.328911	+	0.944361i	$0.606682\pi$
$282$	0	0
$283$	7.96162	0.473269	0.236635	−	0.971599i	$-0.423956\pi$
$283$	7.96162	0.473269	0.236635	+	0.971599i	$0.423956\pi$
$284$	0	0
$285$	13.4842	0.798737
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−11.9450	−0.702648
$290$	0	0
$291$	−11.0360	−0.646940
$292$	0	0
$293$	17.5374	1.02455	0.512273	−	0.858823i	$-0.328804\pi$
$293$	17.5374	1.02455	0.512273	+	0.858823i	$0.328804\pi$
$294$	0	0
$295$	26.0900	1.51902
$296$	0	0
$297$	−51.6761	−2.99855
$298$	0	0
$299$	6.00239	0.347127
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7.15582	0.411091
$304$	0	0
$305$	16.5728	0.948958
$306$	0	0
$307$	−28.8642	−1.64737	−0.823683	−	0.567051i	$-0.808084\pi$
$307$	−28.8642	−1.64737	−0.823683	+	0.567051i	$0.808084\pi$
$308$	0	0
$309$	−32.4493	−1.84598
$310$	0	0
$311$	12.3112	0.698106	0.349053	−	0.937103i	$-0.386503\pi$
$311$	12.3112	0.698106	0.349053	+	0.937103i	$0.386503\pi$
$312$	0	0
$313$	12.6368	0.714273	0.357136	−	0.934052i	$-0.383753\pi$
$313$	12.6368	0.714273	0.357136	+	0.934052i	$0.383753\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.6687	1.66636	0.833181	−	0.553000i	$-0.186517\pi$
$317$	29.6687	1.66636	0.833181	+	0.553000i	$0.186517\pi$
$318$	0	0
$319$	−18.7546	−1.05006
$320$	0	0
$321$	−36.8934	−2.05919
$322$	0	0
$323$	−4.32507	−0.240654
$324$	0	0
$325$	0.163411	0.00906441
$326$	0	0
$327$	−46.0292	−2.54542
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.94401	−0.161817	−0.0809087	−	0.996722i	$-0.525782\pi$
$331$	−2.94401	−0.161817	−0.0809087	+	0.996722i	$0.525782\pi$
$332$	0	0
$333$	49.3899	2.70655
$334$	0	0
$335$	−3.52584	−0.192637
$336$	0	0
$337$	21.3836	1.16484	0.582418	−	0.812889i	$-0.302107\pi$
$337$	21.3836	1.16484	0.582418	+	0.812889i	$0.302107\pi$
$338$	0	0
$339$	13.5784	0.737476
$340$	0	0
$341$	3.63797	0.197007
$342$	0	0
$343$	0	0
$344$	0	0
$345$	18.2148	0.980654
$346$	0	0
$347$	25.3614	1.36147	0.680736	−	0.732529i	$-0.261660\pi$
$347$	25.3614	1.36147	0.680736	+	0.732529i	$0.261660\pi$
$348$	0	0
$349$	−6.63010	−0.354901	−0.177451	−	0.984130i	$-0.556785\pi$
$349$	−6.63010	−0.354901	−0.177451	+	0.984130i	$0.556785\pi$
$350$	0	0
$351$	−28.9373	−1.54456
$352$	0	0
$353$	−27.0903	−1.44187	−0.720936	−	0.693001i	$-0.756288\pi$
$353$	−27.0903	−1.44187	−0.720936	+	0.693001i	$0.756288\pi$
$354$	0	0
$355$	−19.4456	−1.03207
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.6399	0.983774	0.491887	−	0.870659i	$-0.336307\pi$
$359$	18.6399	0.983774	0.491887	+	0.870659i	$0.336307\pi$
$360$	0	0
$361$	−15.2994	−0.805234
$362$	0	0
$363$	18.9926	0.996853
$364$	0	0
$365$	4.84567	0.253634
$366$	0	0
$367$	1.23109	0.0642624	0.0321312	−	0.999484i	$-0.489771\pi$
$367$	1.23109	0.0642624	0.0321312	+	0.999484i	$0.489771\pi$
$368$	0	0
$369$	6.96793	0.362736
$370$	0	0
$371$	0	0
$372$	0	0
$373$	13.7085	0.709798	0.354899	−	0.934905i	$-0.384515\pi$
$373$	13.7085	0.709798	0.354899	+	0.934905i	$0.384515\pi$
$374$	0	0
$375$	35.5439	1.83548
$376$	0	0
$377$	−10.5021	−0.540885
$378$	0	0
$379$	−2.85583	−0.146694	−0.0733471	−	0.997306i	$-0.523368\pi$
$379$	−2.85583	−0.146694	−0.0733471	+	0.997306i	$0.523368\pi$
$380$	0	0
$381$	54.6997	2.80235
$382$	0	0
$383$	−10.5276	−0.537938	−0.268969	−	0.963149i	$-0.586683\pi$
$383$	−10.5276	−0.537938	−0.268969	+	0.963149i	$0.586683\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−84.5045	−4.29561
$388$	0	0
$389$	−10.5333	−0.534060	−0.267030	−	0.963688i	$-0.586042\pi$
$389$	−10.5333	−0.534060	−0.267030	+	0.963688i	$0.586042\pi$
$390$	0	0
$391$	−5.84242	−0.295464
$392$	0	0
$393$	−26.8716	−1.35549
$394$	0	0
$395$	0.895123	0.0450385
$396$	0	0
$397$	−14.0280	−0.704043	−0.352022	−	0.935992i	$-0.614506\pi$
$397$	−14.0280	−0.704043	−0.352022	+	0.935992i	$0.614506\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	25.8036	1.28857	0.644286	−	0.764785i	$-0.277155\pi$
$401$	25.8036	1.28857	0.644286	+	0.764785i	$0.277155\pi$
$402$	0	0
$403$	2.03717	0.101479
$404$	0	0
$405$	−41.4026	−2.05731
$406$	0	0
$407$	−29.2387	−1.44931
$408$	0	0
$409$	−34.9885	−1.73007	−0.865036	−	0.501710i	$-0.832704\pi$
$409$	−34.9885	−1.73007	−0.865036	+	0.501710i	$0.832704\pi$
$410$	0	0
$411$	−62.2660	−3.07135
$412$	0	0
$413$	0	0
$414$	0	0
$415$	11.6999	0.574324
$416$	0	0
$417$	8.80600	0.431232
$418$	0	0
$419$	11.9556	0.584068	0.292034	−	0.956408i	$-0.405668\pi$
$419$	11.9556	0.584068	0.292034	+	0.956408i	$0.405668\pi$
$420$	0	0
$421$	9.73209	0.474313	0.237157	−	0.971471i	$-0.423785\pi$
$421$	9.73209	0.474313	0.237157	+	0.971471i	$0.423785\pi$
$422$	0	0
$423$	40.6407	1.97602
$424$	0	0
$425$	−0.159056	−0.00771534
$426$	0	0
$427$	0	0
$428$	0	0
$429$	30.0828	1.45241
$430$	0	0
$431$	−10.4751	−0.504570	−0.252285	−	0.967653i	$-0.581182\pi$
$431$	−10.4751	−0.504570	−0.252285	+	0.967653i	$0.581182\pi$
$432$	0	0
$433$	−14.7632	−0.709476	−0.354738	−	0.934966i	$-0.615430\pi$
$433$	−14.7632	−0.709476	−0.354738	+	0.934966i	$0.615430\pi$
$434$	0	0
$435$	−31.8696	−1.52803
$436$	0	0
$437$	4.99880	0.239125
$438$	0	0
$439$	25.1915	1.20232	0.601161	−	0.799128i	$-0.294705\pi$
$439$	25.1915	1.20232	0.601161	+	0.799128i	$0.294705\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.308041	0.0146355	0.00731774	−	0.999973i	$-0.497671\pi$
$443$	0.308041	0.0146355	0.00731774	+	0.999973i	$0.497671\pi$
$444$	0	0
$445$	29.5235	1.39955
$446$	0	0
$447$	−61.6970	−2.91817
$448$	0	0
$449$	36.1781	1.70735	0.853676	−	0.520804i	$-0.174368\pi$
$449$	36.1781	1.70735	0.853676	+	0.520804i	$0.174368\pi$
$450$	0	0
$451$	−4.12500	−0.194239
$452$	0	0
$453$	16.2875	0.765252
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−24.1807	−1.13113	−0.565563	−	0.824705i	$-0.691341\pi$
$457$	−24.1807	−1.13113	−0.565563	+	0.824705i	$0.691341\pi$
$458$	0	0
$459$	28.1661	1.31468
$460$	0	0
$461$	−15.3173	−0.713398	−0.356699	−	0.934219i	$-0.616098\pi$
$461$	−15.3173	−0.713398	−0.356699	+	0.934219i	$0.616098\pi$
$462$	0	0
$463$	4.79929	0.223042	0.111521	−	0.993762i	$-0.464428\pi$
$463$	4.79929	0.223042	0.111521	+	0.993762i	$0.464428\pi$
$464$	0	0
$465$	6.18199	0.286683
$466$	0	0
$467$	−22.3349	−1.03353	−0.516767	−	0.856126i	$-0.672865\pi$
$467$	−22.3349	−1.03353	−0.516767	+	0.856126i	$0.672865\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−59.2342	−2.72937
$472$	0	0
$473$	50.0266	2.30022
$474$	0	0
$475$	0.136089	0.00624419
$476$	0	0
$477$	−55.8377	−2.55663
$478$	0	0
$479$	−22.1981	−1.01426	−0.507128	−	0.861871i	$-0.669293\pi$
$479$	−22.1981	−1.01426	−0.507128	+	0.861871i	$0.669293\pi$
$480$	0	0
$481$	−16.3729	−0.746540
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.76066	0.352393
$486$	0	0
$487$	−33.4536	−1.51593	−0.757963	−	0.652297i	$-0.773805\pi$
$487$	−33.4536	−1.51593	−0.757963	+	0.652297i	$0.773805\pi$
$488$	0	0
$489$	−66.0609	−2.98738
$490$	0	0
$491$	36.3600	1.64090	0.820451	−	0.571717i	$-0.193722\pi$
$491$	36.3600	1.64090	0.820451	+	0.571717i	$0.193722\pi$
$492$	0	0
$493$	10.2222	0.460384
$494$	0	0
$495$	63.8144	2.86824
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0.134195	0.00600738	0.00300369	−	0.999995i	$-0.499044\pi$
$499$	0.134195	0.00600738	0.00300369	+	0.999995i	$0.499044\pi$
$500$	0	0
$501$	37.7383	1.68602
$502$	0	0
$503$	−4.13202	−0.184238	−0.0921189	−	0.995748i	$-0.529364\pi$
$503$	−4.13202	−0.184238	−0.0921189	+	0.995748i	$0.529364\pi$
$504$	0	0
$505$	−5.03208	−0.223925
$506$	0	0
$507$	−24.1981	−1.07467
$508$	0	0
$509$	−42.7673	−1.89563	−0.947815	−	0.318822i	$-0.896713\pi$
$509$	−42.7673	−1.89563	−0.947815	+	0.318822i	$0.896713\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−24.0990	−1.06400
$514$	0	0
$515$	22.8189	1.00552
$516$	0	0
$517$	−24.0592	−1.05812
$518$	0	0
$519$	−18.5906	−0.816035
$520$	0	0
$521$	2.13363	0.0934762	0.0467381	−	0.998907i	$-0.485117\pi$
$521$	2.13363	0.0934762	0.0467381	+	0.998907i	$0.485117\pi$
$522$	0	0
$523$	−43.5611	−1.90479	−0.952396	−	0.304863i	$-0.901389\pi$
$523$	−43.5611	−1.90479	−0.952396	+	0.304863i	$0.901389\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.98287	−0.0863754
$528$	0	0
$529$	−16.2475	−0.706413
$530$	0	0
$531$	−81.8816	−3.55336
$532$	0	0
$533$	−2.30989	−0.100053
$534$	0	0
$535$	25.9440	1.12166
$536$	0	0
$537$	35.3352	1.52483
$538$	0	0
$539$	0	0
$540$	0	0
$541$	40.0196	1.72058	0.860289	−	0.509806i	$-0.170283\pi$
$541$	40.0196	1.72058	0.860289	+	0.509806i	$0.170283\pi$
$542$	0	0
$543$	−46.3950	−1.99100
$544$	0	0
$545$	32.3684	1.38651
$546$	0	0
$547$	23.5014	1.00485	0.502423	−	0.864622i	$-0.332442\pi$
$547$	23.5014	1.00485	0.502423	+	0.864622i	$0.332442\pi$
$548$	0	0
$549$	−52.0127	−2.21985
$550$	0	0
$551$	−8.74615	−0.372599
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−49.6852	−2.10902
$556$	0	0
$557$	33.9635	1.43908	0.719539	−	0.694452i	$-0.244353\pi$
$557$	33.9635	1.43908	0.719539	+	0.694452i	$0.244353\pi$
$558$	0	0
$559$	28.0135	1.18485
$560$	0	0
$561$	−29.2811	−1.23625
$562$	0	0
$563$	45.3615	1.91176	0.955880	−	0.293757i	$-0.0949057\pi$
$563$	45.3615	1.91176	0.955880	+	0.293757i	$0.0949057\pi$
$564$	0	0
$565$	−9.54852	−0.401709
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.7354	0.827350	0.413675	−	0.910425i	$-0.364245\pi$
$569$	19.7354	0.827350	0.413675	+	0.910425i	$0.364245\pi$
$570$	0	0
$571$	20.8619	0.873043	0.436522	−	0.899694i	$-0.356210\pi$
$571$	20.8619	0.873043	0.436522	+	0.899694i	$0.356210\pi$
$572$	0	0
$573$	5.53749	0.231332
$574$	0	0
$575$	0.183832	0.00766634
$576$	0	0
$577$	11.1897	0.465832	0.232916	−	0.972497i	$-0.425173\pi$
$577$	11.1897	0.465832	0.232916	+	0.972497i	$0.425173\pi$
$578$	0	0
$579$	32.5625	1.35325
$580$	0	0
$581$	0	0
$582$	0	0
$583$	33.0559	1.36903
$584$	0	0
$585$	35.7343	1.47743
$586$	0	0
$587$	−2.93067	−0.120962	−0.0604808	−	0.998169i	$-0.519263\pi$
$587$	−2.93067	−0.120962	−0.0604808	+	0.998169i	$0.519263\pi$
$588$	0	0
$589$	1.69656	0.0699054
$590$	0	0
$591$	−21.1258	−0.869001
$592$	0	0
$593$	38.4675	1.57967	0.789835	−	0.613320i	$-0.210166\pi$
$593$	38.4675	1.57967	0.789835	+	0.613320i	$0.210166\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	38.4777	1.57479
$598$	0	0
$599$	34.8868	1.42544	0.712719	−	0.701450i	$-0.247464\pi$
$599$	34.8868	1.42544	0.712719	+	0.701450i	$0.247464\pi$
$600$	0	0
$601$	−21.5920	−0.880757	−0.440379	−	0.897812i	$-0.645156\pi$
$601$	−21.5920	−0.880757	−0.440379	+	0.897812i	$0.645156\pi$
$602$	0	0
$603$	11.0656	0.450626
$604$	0	0
$605$	−13.3559	−0.542994
$606$	0	0
$607$	31.5921	1.28228	0.641142	−	0.767422i	$-0.278461\pi$
$607$	31.5921	1.28228	0.641142	+	0.767422i	$0.278461\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13.4725	−0.545040
$612$	0	0
$613$	−18.7893	−0.758894	−0.379447	−	0.925213i	$-0.623886\pi$
$613$	−18.7893	−0.758894	−0.379447	+	0.925213i	$0.623886\pi$
$614$	0	0
$615$	−7.00960	−0.282654
$616$	0	0
$617$	15.4135	0.620526	0.310263	−	0.950651i	$-0.399583\pi$
$617$	15.4135	0.620526	0.310263	+	0.950651i	$0.399583\pi$
$618$	0	0
$619$	−3.61805	−0.145422	−0.0727108	−	0.997353i	$-0.523165\pi$
$619$	−3.61805	−0.145422	−0.0727108	+	0.997353i	$0.523165\pi$
$620$	0	0
$621$	−32.5535	−1.30633
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−24.6413	−0.985651
$626$	0	0
$627$	25.0530	1.00052
$628$	0	0
$629$	15.9366	0.635432
$630$	0	0
$631$	35.0390	1.39488	0.697441	−	0.716642i	$-0.254322\pi$
$631$	35.0390	1.39488	0.697441	+	0.716642i	$0.254322\pi$
$632$	0	0
$633$	59.3467	2.35882
$634$	0	0
$635$	−38.4656	−1.52646
$636$	0	0
$637$	0	0
$638$	0	0
$639$	61.0288	2.41426
$640$	0	0
$641$	1.26732	0.0500562	0.0250281	−	0.999687i	$-0.492032\pi$
$641$	1.26732	0.0500562	0.0250281	+	0.999687i	$0.492032\pi$
$642$	0	0
$643$	3.98105	0.156997	0.0784986	−	0.996914i	$-0.474987\pi$
$643$	3.98105	0.156997	0.0784986	+	0.996914i	$0.474987\pi$
$644$	0	0
$645$	85.0099	3.34726
$646$	0	0
$647$	−24.0142	−0.944097	−0.472049	−	0.881573i	$-0.656485\pi$
$647$	−24.0142	−0.944097	−0.472049	+	0.881573i	$0.656485\pi$
$648$	0	0
$649$	48.4738	1.90276
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25.6864	1.00518	0.502592	−	0.864523i	$-0.332380\pi$
$653$	25.6864	1.00518	0.502592	+	0.864523i	$0.332380\pi$
$654$	0	0
$655$	18.8965	0.738348
$656$	0	0
$657$	−15.2078	−0.593313
$658$	0	0
$659$	17.4949	0.681505	0.340753	−	0.940153i	$-0.389318\pi$
$659$	17.4949	0.681505	0.340753	+	0.940153i	$0.389318\pi$
$660$	0	0
$661$	−32.6256	−1.26899	−0.634494	−	0.772928i	$-0.718791\pi$
$661$	−32.6256	−1.26899	−0.634494	+	0.772928i	$0.718791\pi$
$662$	0	0
$663$	−16.3966	−0.636792
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.8145	−0.457460
$668$	0	0
$669$	−29.5798	−1.14362
$670$	0	0
$671$	30.7915	1.18869
$672$	0	0
$673$	−7.92133	−0.305345	−0.152672	−	0.988277i	$-0.548788\pi$
$673$	−7.92133	−0.305345	−0.152672	+	0.988277i	$0.548788\pi$
$674$	0	0
$675$	−0.886248	−0.0341117
$676$	0	0
$677$	−30.1753	−1.15973	−0.579866	−	0.814712i	$-0.696895\pi$
$677$	−30.1753	−1.15973	−0.579866	+	0.814712i	$0.696895\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−18.5558	−0.711062
$682$	0	0
$683$	−16.4402	−0.629066	−0.314533	−	0.949247i	$-0.601848\pi$
$683$	−16.4402	−0.629066	−0.314533	+	0.949247i	$0.601848\pi$
$684$	0	0
$685$	43.7864	1.67299
$686$	0	0
$687$	39.3160	1.50000
$688$	0	0
$689$	18.5104	0.705190
$690$	0	0
$691$	−51.1349	−1.94526	−0.972631	−	0.232354i	$-0.925357\pi$
$691$	−51.1349	−1.94526	−0.972631	+	0.232354i	$0.925357\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.19252	−0.234896
$696$	0	0
$697$	2.24833	0.0851616
$698$	0	0
$699$	−42.7637	−1.61747
$700$	0	0
$701$	7.11382	0.268685	0.134343	−	0.990935i	$-0.457108\pi$
$701$	7.11382	0.268685	0.134343	+	0.990935i	$0.457108\pi$
$702$	0	0
$703$	−13.6354	−0.514268
$704$	0	0
$705$	−40.8837	−1.53977
$706$	0	0
$707$	0	0
$708$	0	0
$709$	29.9782	1.12585	0.562927	−	0.826507i	$-0.309675\pi$
$709$	29.9782	1.12585	0.562927	+	0.826507i	$0.309675\pi$
$710$	0	0
$711$	−2.80928	−0.105356
$712$	0	0
$713$	2.29175	0.0858267
$714$	0	0
$715$	−21.1547	−0.791140
$716$	0	0
$717$	22.9979	0.858872
$718$	0	0
$719$	20.9726	0.782147	0.391074	−	0.920359i	$-0.372104\pi$
$719$	20.9726	0.782147	0.391074	+	0.920359i	$0.372104\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−10.4728	−0.389486
$724$	0	0
$725$	−0.321642	−0.0119455
$726$	0	0
$727$	−26.7303	−0.991372	−0.495686	−	0.868502i	$-0.665083\pi$
$727$	−26.7303	−0.991372	−0.495686	+	0.868502i	$0.665083\pi$
$728$	0	0
$729$	11.2834	0.417903
$730$	0	0
$731$	−27.2670	−1.00850
$732$	0	0
$733$	−20.1849	−0.745545	−0.372772	−	0.927923i	$-0.621593\pi$
$733$	−20.1849	−0.745545	−0.372772	+	0.927923i	$0.621593\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.55082	−0.241303
$738$	0	0
$739$	−47.8248	−1.75926	−0.879632	−	0.475655i	$-0.842211\pi$
$739$	−47.8248	−1.75926	−0.879632	+	0.475655i	$0.842211\pi$
$740$	0	0
$741$	14.0290	0.515369
$742$	0	0
$743$	−42.2673	−1.55064	−0.775318	−	0.631571i	$-0.782411\pi$
$743$	−42.2673	−1.55064	−0.775318	+	0.631571i	$0.782411\pi$
$744$	0	0
$745$	43.3863	1.58955
$746$	0	0
$747$	−36.7192	−1.34349
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−29.6944	−1.08357	−0.541783	−	0.840519i	$-0.682250\pi$
$751$	−29.6944	−1.08357	−0.541783	+	0.840519i	$0.682250\pi$
$752$	0	0
$753$	45.1859	1.64667
$754$	0	0
$755$	−11.4536	−0.416839
$756$	0	0
$757$	7.17713	0.260857	0.130429	−	0.991458i	$-0.458365\pi$
$757$	7.17713	0.260857	0.130429	+	0.991458i	$0.458365\pi$
$758$	0	0
$759$	33.8422	1.22840
$760$	0	0
$761$	26.3861	0.956496	0.478248	−	0.878225i	$-0.341272\pi$
$761$	26.3861	0.956496	0.478248	+	0.878225i	$0.341272\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−34.7820	−1.25755
$766$	0	0
$767$	27.1440	0.980114
$768$	0	0
$769$	41.9486	1.51270	0.756352	−	0.654164i	$-0.226980\pi$
$769$	41.9486	1.51270	0.756352	+	0.654164i	$0.226980\pi$
$770$	0	0
$771$	93.7424	3.37605
$772$	0	0
$773$	22.6024	0.812951	0.406476	−	0.913662i	$-0.366758\pi$
$773$	22.6024	0.812951	0.406476	+	0.913662i	$0.366758\pi$
$774$	0	0
$775$	0.0623914	0.00224116
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.92368	−0.0689231
$780$	0	0
$781$	−36.1290	−1.29280
$782$	0	0
$783$	56.9573	2.03549
$784$	0	0
$785$	41.6544	1.48671
$786$	0	0
$787$	−7.45290	−0.265667	−0.132834	−	0.991138i	$-0.542408\pi$
$787$	−7.45290	−0.265667	−0.132834	+	0.991138i	$0.542408\pi$
$788$	0	0
$789$	37.7161	1.34273
$790$	0	0
$791$	0	0
$792$	0	0
$793$	17.2424	0.612296
$794$	0	0
$795$	56.1717	1.99220
$796$	0	0
$797$	38.3124	1.35710	0.678548	−	0.734556i	$-0.262609\pi$
$797$	38.3124	1.35710	0.678548	+	0.734556i	$0.262609\pi$
$798$	0	0
$799$	13.1135	0.463921
$800$	0	0
$801$	−92.6576	−3.27389
$802$	0	0
$803$	9.00300	0.317709
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−27.6624	−0.973762
$808$	0	0
$809$	−13.5594	−0.476723	−0.238362	−	0.971176i	$-0.576610\pi$
$809$	−13.5594	−0.476723	−0.238362	+	0.971176i	$0.576610\pi$
$810$	0	0
$811$	−10.3823	−0.364572	−0.182286	−	0.983246i	$-0.558350\pi$
$811$	−10.3823	−0.364572	−0.182286	+	0.983246i	$0.558350\pi$
$812$	0	0
$813$	−29.4904	−1.03427
$814$	0	0
$815$	46.4550	1.62725
$816$	0	0
$817$	23.3297	0.816204
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23.4003	−0.816675	−0.408337	−	0.912831i	$-0.633891\pi$
$821$	−23.4003	−0.816675	−0.408337	+	0.912831i	$0.633891\pi$
$822$	0	0
$823$	−1.66810	−0.0581465	−0.0290732	−	0.999577i	$-0.509256\pi$
$823$	−1.66810	−0.0581465	−0.0290732	+	0.999577i	$0.509256\pi$
$824$	0	0
$825$	0.921332	0.0320767
$826$	0	0
$827$	−13.7281	−0.477373	−0.238686	−	0.971097i	$-0.576717\pi$
$827$	−13.7281	−0.477373	−0.238686	+	0.971097i	$0.576717\pi$
$828$	0	0
$829$	17.6716	0.613761	0.306880	−	0.951748i	$-0.400715\pi$
$829$	17.6716	0.613761	0.306880	+	0.951748i	$0.400715\pi$
$830$	0	0
$831$	101.765	3.53020
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−26.5381	−0.918390
$836$	0	0
$837$	−11.0484	−0.381890
$838$	0	0
$839$	−7.26376	−0.250773	−0.125386	−	0.992108i	$-0.540017\pi$
$839$	−7.26376	−0.250773	−0.125386	+	0.992108i	$0.540017\pi$
$840$	0	0
$841$	−8.32871	−0.287197
$842$	0	0
$843$	−34.8148	−1.19908
$844$	0	0
$845$	17.0164	0.585384
$846$	0	0
$847$	0	0
$848$	0	0
$849$	25.1365	0.862681
$850$	0	0
$851$	−18.4190	−0.631396
$852$	0	0
$853$	−44.2376	−1.51467	−0.757333	−	0.653029i	$-0.773498\pi$
$853$	−44.2376	−1.51467	−0.757333	+	0.653029i	$0.773498\pi$
$854$	0	0
$855$	29.7596	1.01776
$856$	0	0
$857$	44.7732	1.52942	0.764711	−	0.644373i	$-0.222881\pi$
$857$	44.7732	1.52942	0.764711	+	0.644373i	$0.222881\pi$
$858$	0	0
$859$	−32.7417	−1.11713	−0.558567	−	0.829460i	$-0.688649\pi$
$859$	−32.7417	−1.11713	−0.558567	+	0.829460i	$0.688649\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0.500831	0.0170485	0.00852424	−	0.999964i	$-0.497287\pi$
$863$	0.500831	0.0170485	0.00852424	+	0.999964i	$0.497287\pi$
$864$	0	0
$865$	13.0732	0.444501
$866$	0	0
$867$	−37.7128	−1.28079
$868$	0	0
$869$	1.66309	0.0564165
$870$	0	0
$871$	−3.66829	−0.124295
$872$	0	0
$873$	−24.3563	−0.824336
$874$	0	0
$875$	0	0
$876$	0	0
$877$	37.6696	1.27201	0.636006	−	0.771684i	$-0.280585\pi$
$877$	37.6696	1.27201	0.636006	+	0.771684i	$0.280585\pi$
$878$	0	0
$879$	55.3691	1.86755
$880$	0	0
$881$	27.4049	0.923293	0.461647	−	0.887064i	$-0.347259\pi$
$881$	27.4049	0.923293	0.461647	+	0.887064i	$0.347259\pi$
$882$	0	0
$883$	−19.0131	−0.639842	−0.319921	−	0.947444i	$-0.603656\pi$
$883$	−19.0131	−0.639842	−0.319921	+	0.947444i	$0.603656\pi$
$884$	0	0
$885$	82.3713	2.76888
$886$	0	0
$887$	27.3297	0.917643	0.458821	−	0.888529i	$-0.348272\pi$
$887$	27.3297	0.917643	0.458821	+	0.888529i	$0.348272\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−76.9239	−2.57705
$892$	0	0
$893$	−11.2199	−0.375461
$894$	0	0
$895$	−24.8482	−0.830585
$896$	0	0
$897$	18.9507	0.632747
$898$	0	0
$899$	−4.00976	−0.133733
$900$	0	0
$901$	−18.0171	−0.600236
$902$	0	0
$903$	0	0
$904$	0	0
$905$	32.6257	1.08451
$906$	0	0
$907$	52.7038	1.75000	0.875001	−	0.484121i	$-0.160860\pi$
$907$	52.7038	1.75000	0.875001	+	0.484121i	$0.160860\pi$
$908$	0	0
$909$	15.7929	0.523816
$910$	0	0
$911$	55.8154	1.84925	0.924623	−	0.380883i	$-0.124380\pi$
$911$	55.8154	1.84925	0.924623	+	0.380883i	$0.124380\pi$
$912$	0	0
$913$	21.7377	0.719414
$914$	0	0
$915$	52.3238	1.72977
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−56.6479	−1.86864	−0.934321	−	0.356434i	$-0.883992\pi$
$919$	−56.6479	−1.86864	−0.934321	+	0.356434i	$0.883992\pi$
$920$	0	0
$921$	−91.1300	−3.00284
$922$	0	0
$923$	−20.2313	−0.665920
$924$	0	0
$925$	−0.501445	−0.0164874
$926$	0	0
$927$	−71.6155	−2.35216
$928$	0	0
$929$	27.9807	0.918016	0.459008	−	0.888432i	$-0.348205\pi$
$929$	27.9807	0.918016	0.459008	+	0.888432i	$0.348205\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	38.8691	1.27252
$934$	0	0
$935$	20.5909	0.673394
$936$	0	0
$937$	17.6255	0.575799	0.287900	−	0.957661i	$-0.407043\pi$
$937$	17.6255	0.575799	0.287900	+	0.957661i	$0.407043\pi$
$938$	0	0
$939$	39.8969	1.30198
$940$	0	0
$941$	5.16930	0.168514	0.0842572	−	0.996444i	$-0.473148\pi$
$941$	5.16930	0.168514	0.0842572	+	0.996444i	$0.473148\pi$
$942$	0	0
$943$	−2.59856	−0.0846207
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.1393	0.654441	0.327220	−	0.944948i	$-0.393888\pi$
$947$	20.1393	0.654441	0.327220	+	0.944948i	$0.393888\pi$
$948$	0	0
$949$	5.04144	0.163652
$950$	0	0
$951$	93.6702	3.03747
$952$	0	0
$953$	−44.3472	−1.43655	−0.718273	−	0.695761i	$-0.755067\pi$
$953$	−44.3472	−1.43655	−0.718273	+	0.695761i	$0.755067\pi$
$954$	0	0
$955$	−3.89405	−0.126008
$956$	0	0
$957$	−59.2121	−1.91406
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.2222	−0.974910
$962$	0	0
$963$	−81.4235	−2.62384
$964$	0	0
$965$	−22.8985	−0.737128
$966$	0	0
$967$	60.6168	1.94931	0.974653	−	0.223722i	$-0.0718208\pi$
$967$	60.6168	1.94931	0.974653	+	0.223722i	$0.0718208\pi$
$968$	0	0
$969$	−13.6551	−0.438666
$970$	0	0
$971$	−22.0331	−0.707075	−0.353538	−	0.935420i	$-0.615021\pi$
$971$	−22.0331	−0.707075	−0.353538	+	0.935420i	$0.615021\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0.515921	0.0165227
$976$	0	0
$977$	−33.2759	−1.06459	−0.532296	−	0.846558i	$-0.678671\pi$
$977$	−33.2759	−1.06459	−0.532296	+	0.846558i	$0.678671\pi$
$978$	0	0
$979$	54.8532	1.75311
$980$	0	0
$981$	−101.586	−3.24339
$982$	0	0
$983$	61.0791	1.94812	0.974061	−	0.226285i	$-0.0726582\pi$
$983$	61.0791	1.94812	0.974061	+	0.226285i	$0.0726582\pi$
$984$	0	0
$985$	14.8560	0.473352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	31.5144	1.00210
$990$	0	0
$991$	31.0442	0.986151	0.493075	−	0.869987i	$-0.335873\pi$
$991$	31.0442	0.986151	0.493075	+	0.869987i	$0.335873\pi$
$992$	0	0
$993$	−9.29483	−0.294963
$994$	0	0
$995$	−27.0581	−0.857799
$996$	0	0
$997$	−26.4932	−0.839049	−0.419525	−	0.907744i	$-0.637803\pi$
$997$	−26.4932	−0.839049	−0.419525	+	0.907744i	$0.637803\pi$
$998$	0	0
$999$	88.7974	2.80942

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.s.1.20	✓	20
7.6	odd	2		8036.2.a.t.1.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.20	✓	20	1.1	even	1	trivial
8036.2.a.t.1.1	yes	20	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+3.15720 q^{3} -2.22019 q^{5} +6.96793 q^{9} +O(q^{10})\)	\(q+3.15720 q^{3} -2.22019 q^{5} +6.96793 q^{9} -4.12500 q^{11} -2.30989 q^{13} -7.00960 q^{15} +2.24833 q^{17} -1.92368 q^{19} -2.59856 q^{23} -0.0707440 q^{25} +12.5275 q^{27} +4.54657 q^{29} -0.881932 q^{31} -13.0235 q^{33} +7.08817 q^{37} -7.29279 q^{39} +1.00000 q^{41} -12.1276 q^{43} -15.4701 q^{45} +5.83253 q^{47} +7.09843 q^{51} -8.01354 q^{53} +9.15830 q^{55} -6.07345 q^{57} -11.7512 q^{59} -7.46459 q^{61} +5.12840 q^{65} +1.58808 q^{67} -8.20417 q^{69} +8.75853 q^{71} -2.18255 q^{73} -0.223353 q^{75} -0.403174 q^{79} +18.6482 q^{81} -5.26975 q^{83} -4.99173 q^{85} +14.3544 q^{87} -13.2977 q^{89} -2.78444 q^{93} +4.27095 q^{95} -3.49549 q^{97} -28.7427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9}+O(q^{10}) \) 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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