LMFDB - Embedded newform 8036.2.a.t.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$20$
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			1.9
Root			$-0.103746$ 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)

$f(q)$	$=$	$q-0.103746 q^{3} -3.71451 q^{5} -2.98924 q^{9} +O(q^{10})$	$q-0.103746 q^{3} -3.71451 q^{5} -2.98924 q^{9} -1.25890 q^{11} +0.166769 q^{13} +0.385366 q^{15} +5.79993 q^{17} -1.05905 q^{19} -8.98555 q^{23} +8.79757 q^{25} +0.621361 q^{27} -3.21478 q^{29} -9.67227 q^{31} +0.130606 q^{33} +1.70626 q^{37} -0.0173016 q^{39} -1.00000 q^{41} -3.83571 q^{43} +11.1035 q^{45} -7.32109 q^{47} -0.601721 q^{51} +2.82243 q^{53} +4.67619 q^{55} +0.109873 q^{57} -9.74300 q^{59} -4.82191 q^{61} -0.619464 q^{65} -1.21471 q^{67} +0.932217 q^{69} -13.4015 q^{71} +16.5739 q^{73} -0.912716 q^{75} -11.6084 q^{79} +8.90325 q^{81} +6.00621 q^{83} -21.5439 q^{85} +0.333522 q^{87} +11.0453 q^{89} +1.00346 q^{93} +3.93385 q^{95} +4.56071 q^{97} +3.76315 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.103746	−0.0598979	−0.0299490	−	0.999551i	$-0.509534\pi$
$3$	−0.103746	−0.0598979	−0.0299490	+	0.999551i	$0.509534\pi$
$4$	0	0
$5$	−3.71451	−1.66118	−0.830589	−	0.556885i	$-0.811996\pi$
$5$	−3.71451	−1.66118	−0.830589	+	0.556885i	$0.811996\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.98924	−0.996412
$10$	0	0
$11$	−1.25890	−0.379573	−0.189786	−	0.981825i	$-0.560780\pi$
$11$	−1.25890	−0.379573	−0.189786	+	0.981825i	$0.560780\pi$
$12$	0	0
$13$	0.166769	0.0462533	0.0231266	−	0.999733i	$-0.492638\pi$
$13$	0.166769	0.0462533	0.0231266	+	0.999733i	$0.492638\pi$
$14$	0	0
$15$	0.385366	0.0995012
$16$	0	0
$17$	5.79993	1.40669	0.703345	−	0.710849i	$-0.251689\pi$
$17$	5.79993	1.40669	0.703345	+	0.710849i	$0.251689\pi$
$18$	0	0
$19$	−1.05905	−0.242963	−0.121481	−	0.992594i	$-0.538764\pi$
$19$	−1.05905	−0.242963	−0.121481	+	0.992594i	$0.538764\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.98555	−1.87362	−0.936808	−	0.349844i	$-0.886235\pi$
$23$	−8.98555	−1.87362	−0.936808	+	0.349844i	$0.886235\pi$
$24$	0	0
$25$	8.79757	1.75951
$26$	0	0
$27$	0.621361	0.119581
$28$	0	0
$29$	−3.21478	−0.596970	−0.298485	−	0.954414i	$-0.596481\pi$
$29$	−3.21478	−0.596970	−0.298485	+	0.954414i	$0.596481\pi$
$30$	0	0
$31$	−9.67227	−1.73719	−0.868595	−	0.495522i	$-0.834977\pi$
$31$	−9.67227	−1.73719	−0.868595	+	0.495522i	$0.834977\pi$
$32$	0	0
$33$	0.130606	0.0227356
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.70626	0.280507	0.140253	−	0.990116i	$-0.455208\pi$
$37$	1.70626	0.280507	0.140253	+	0.990116i	$0.455208\pi$
$38$	0	0
$39$	−0.0173016	−0.00277048
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−3.83571	−0.584940	−0.292470	−	0.956275i	$-0.594477\pi$
$43$	−3.83571	−0.584940	−0.292470	+	0.956275i	$0.594477\pi$
$44$	0	0
$45$	11.1035	1.65522
$46$	0	0
$47$	−7.32109	−1.06789	−0.533945	−	0.845519i	$-0.679291\pi$
$47$	−7.32109	−1.06789	−0.533945	+	0.845519i	$0.679291\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.601721	−0.0842579
$52$	0	0
$53$	2.82243	0.387690	0.193845	−	0.981032i	$-0.437904\pi$
$53$	2.82243	0.387690	0.193845	+	0.981032i	$0.437904\pi$
$54$	0	0
$55$	4.67619	0.630538
$56$	0	0
$57$	0.109873	0.0145530
$58$	0	0
$59$	−9.74300	−1.26843	−0.634215	−	0.773157i	$-0.718677\pi$
$59$	−9.74300	−1.26843	−0.634215	+	0.773157i	$0.718677\pi$
$60$	0	0
$61$	−4.82191	−0.617383	−0.308691	−	0.951162i	$-0.599891\pi$
$61$	−4.82191	−0.617383	−0.308691	+	0.951162i	$0.599891\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.619464	−0.0768350
$66$	0	0
$67$	−1.21471	−0.148401	−0.0742005	−	0.997243i	$-0.523640\pi$
$67$	−1.21471	−0.148401	−0.0742005	+	0.997243i	$0.523640\pi$
$68$	0	0
$69$	0.932217	0.112226
$70$	0	0
$71$	−13.4015	−1.59046	−0.795232	−	0.606305i	$-0.792651\pi$
$71$	−13.4015	−1.59046	−0.795232	+	0.606305i	$0.792651\pi$
$72$	0	0
$73$	16.5739	1.93982	0.969912	−	0.243456i	$-0.0782811\pi$
$73$	16.5739	1.93982	0.969912	+	0.243456i	$0.0782811\pi$
$74$	0	0
$75$	−0.912716	−0.105391
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.6084	−1.30605	−0.653024	−	0.757337i	$-0.726500\pi$
$79$	−11.6084	−1.30605	−0.653024	+	0.757337i	$0.726500\pi$
$80$	0	0
$81$	8.90325	0.989250
$82$	0	0
$83$	6.00621	0.659267	0.329633	−	0.944109i	$-0.393075\pi$
$83$	6.00621	0.659267	0.329633	+	0.944109i	$0.393075\pi$
$84$	0	0
$85$	−21.5439	−2.33676
$86$	0	0
$87$	0.333522	0.0357573
$88$	0	0
$89$	11.0453	1.17080	0.585402	−	0.810743i	$-0.300937\pi$
$89$	11.0453	1.17080	0.585402	+	0.810743i	$0.300937\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.00346	0.104054
$94$	0	0
$95$	3.93385	0.403605
$96$	0	0
$97$	4.56071	0.463070	0.231535	−	0.972827i	$-0.425625\pi$
$97$	4.56071	0.463070	0.231535	+	0.972827i	$0.425625\pi$
$98$	0	0
$99$	3.76315	0.378211
$100$	0	0
$101$	−16.1605	−1.60803	−0.804015	−	0.594609i	$-0.797307\pi$
$101$	−16.1605	−1.60803	−0.804015	+	0.594609i	$0.797307\pi$
$102$	0	0
$103$	−0.212217	−0.0209103	−0.0104552	−	0.999945i	$-0.503328\pi$
$103$	−0.212217	−0.0209103	−0.0104552	+	0.999945i	$0.503328\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.6381	1.12510	0.562550	−	0.826763i	$-0.309821\pi$
$107$	11.6381	1.12510	0.562550	+	0.826763i	$0.309821\pi$
$108$	0	0
$109$	−16.9891	−1.62726	−0.813632	−	0.581380i	$-0.802513\pi$
$109$	−16.9891	−1.62726	−0.813632	+	0.581380i	$0.802513\pi$
$110$	0	0
$111$	−0.177018	−0.0168018
$112$	0	0
$113$	−5.66265	−0.532697	−0.266348	−	0.963877i	$-0.585817\pi$
$113$	−5.66265	−0.532697	−0.266348	+	0.963877i	$0.585817\pi$
$114$	0	0
$115$	33.3769	3.11241
$116$	0	0
$117$	−0.498511	−0.0460874
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.41517	−0.855925
$122$	0	0
$123$	0.103746	0.00935449
$124$	0	0
$125$	−14.1061	−1.26169
$126$	0	0
$127$	−8.30382	−0.736845	−0.368423	−	0.929658i	$-0.620102\pi$
$127$	−8.30382	−0.736845	−0.368423	+	0.929658i	$0.620102\pi$
$128$	0	0
$129$	0.397941	0.0350367
$130$	0	0
$131$	7.30755	0.638464	0.319232	−	0.947677i	$-0.396575\pi$
$131$	7.30755	0.638464	0.319232	+	0.947677i	$0.396575\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.30805	−0.198645
$136$	0	0
$137$	−10.4686	−0.894395	−0.447198	−	0.894435i	$-0.647578\pi$
$137$	−10.4686	−0.894395	−0.447198	+	0.894435i	$0.647578\pi$
$138$	0	0
$139$	17.4396	1.47921	0.739606	−	0.673040i	$-0.235012\pi$
$139$	17.4396	1.47921	0.739606	+	0.673040i	$0.235012\pi$
$140$	0	0
$141$	0.759536	0.0639644
$142$	0	0
$143$	−0.209945	−0.0175565
$144$	0	0
$145$	11.9413	0.991674
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.2645	−1.33244	−0.666219	−	0.745756i	$-0.732088\pi$
$149$	−16.2645	−1.33244	−0.666219	+	0.745756i	$0.732088\pi$
$150$	0	0
$151$	11.0691	0.900792	0.450396	−	0.892829i	$-0.351283\pi$
$151$	11.0691	0.900792	0.450396	+	0.892829i	$0.351283\pi$
$152$	0	0
$153$	−17.3374	−1.40164
$154$	0	0
$155$	35.9277	2.88578
$156$	0	0
$157$	12.8519	1.02569	0.512845	−	0.858481i	$-0.328592\pi$
$157$	12.8519	1.02569	0.512845	+	0.858481i	$0.328592\pi$
$158$	0	0
$159$	−0.292816	−0.0232219
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.63928	0.206724	0.103362	−	0.994644i	$-0.467040\pi$
$163$	2.63928	0.206724	0.103362	+	0.994644i	$0.467040\pi$
$164$	0	0
$165$	−0.485138	−0.0377679
$166$	0	0
$167$	−11.7990	−0.913031	−0.456516	−	0.889715i	$-0.650903\pi$
$167$	−11.7990	−0.913031	−0.456516	+	0.889715i	$0.650903\pi$
$168$	0	0
$169$	−12.9722	−0.997861
$170$	0	0
$171$	3.16575	0.242091
$172$	0	0
$173$	7.75018	0.589235	0.294618	−	0.955615i	$-0.404808\pi$
$173$	7.75018	0.589235	0.294618	+	0.955615i	$0.404808\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.01080	0.0759764
$178$	0	0
$179$	−3.88020	−0.290019	−0.145010	−	0.989430i	$-0.546321\pi$
$179$	−3.88020	−0.290019	−0.145010	+	0.989430i	$0.546321\pi$
$180$	0	0
$181$	23.1868	1.72346	0.861730	−	0.507366i	$-0.169381\pi$
$181$	23.1868	1.72346	0.861730	+	0.507366i	$0.169381\pi$
$182$	0	0
$183$	0.500255	0.0369799
$184$	0	0
$185$	−6.33791	−0.465972
$186$	0	0
$187$	−7.30153	−0.533941
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.8532	−1.21945	−0.609726	−	0.792613i	$-0.708720\pi$
$191$	−16.8532	−1.21945	−0.609726	+	0.792613i	$0.708720\pi$
$192$	0	0
$193$	11.5599	0.832102	0.416051	−	0.909341i	$-0.363414\pi$
$193$	11.5599	0.832102	0.416051	+	0.909341i	$0.363414\pi$
$194$	0	0
$195$	0.0642670	0.00460226
$196$	0	0
$197$	14.6526	1.04395	0.521977	−	0.852960i	$-0.325195\pi$
$197$	14.6526	1.04395	0.521977	+	0.852960i	$0.325195\pi$
$198$	0	0
$199$	11.3345	0.803480	0.401740	−	0.915754i	$-0.368406\pi$
$199$	11.3345	0.803480	0.401740	+	0.915754i	$0.368406\pi$
$200$	0	0
$201$	0.126022	0.00888891
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.71451	0.259433
$206$	0	0
$207$	26.8599	1.86689
$208$	0	0
$209$	1.33324	0.0922220
$210$	0	0
$211$	−15.4952	−1.06673	−0.533366	−	0.845885i	$-0.679073\pi$
$211$	−15.4952	−1.06673	−0.533366	+	0.845885i	$0.679073\pi$
$212$	0	0
$213$	1.39035	0.0952655
$214$	0	0
$215$	14.2478	0.971690
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.71948	−0.116191
$220$	0	0
$221$	0.967247	0.0650641
$222$	0	0
$223$	19.7332	1.32143	0.660715	−	0.750637i	$-0.270253\pi$
$223$	19.7332	1.32143	0.660715	+	0.750637i	$0.270253\pi$
$224$	0	0
$225$	−26.2980	−1.75320
$226$	0	0
$227$	−3.27003	−0.217040	−0.108520	−	0.994094i	$-0.534611\pi$
$227$	−3.27003	−0.217040	−0.108520	+	0.994094i	$0.534611\pi$
$228$	0	0
$229$	18.1500	1.19938	0.599692	−	0.800231i	$-0.295290\pi$
$229$	18.1500	1.19938	0.599692	+	0.800231i	$0.295290\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.0356	0.788476	0.394238	−	0.919008i	$-0.371009\pi$
$233$	12.0356	0.788476	0.394238	+	0.919008i	$0.371009\pi$
$234$	0	0
$235$	27.1942	1.77396
$236$	0	0
$237$	1.20433	0.0782296
$238$	0	0
$239$	14.9782	0.968861	0.484431	−	0.874830i	$-0.339027\pi$
$239$	14.9782	0.968861	0.484431	+	0.874830i	$0.339027\pi$
$240$	0	0
$241$	−2.73474	−0.176160	−0.0880802	−	0.996113i	$-0.528073\pi$
$241$	−2.73474	−0.176160	−0.0880802	+	0.996113i	$0.528073\pi$
$242$	0	0
$243$	−2.78776	−0.178835
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.176616	−0.0112378
$248$	0	0
$249$	−0.623121	−0.0394887
$250$	0	0
$251$	6.33313	0.399743	0.199872	−	0.979822i	$-0.435947\pi$
$251$	6.33313	0.399743	0.199872	+	0.979822i	$0.435947\pi$
$252$	0	0
$253$	11.3119	0.711173
$254$	0	0
$255$	2.23510	0.139967
$256$	0	0
$257$	29.3821	1.83281	0.916403	−	0.400258i	$-0.131079\pi$
$257$	29.3821	1.83281	0.916403	+	0.400258i	$0.131079\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	9.60974	0.594828
$262$	0	0
$263$	10.1326	0.624804	0.312402	−	0.949950i	$-0.398866\pi$
$263$	10.1326	0.624804	0.312402	+	0.949950i	$0.398866\pi$
$264$	0	0
$265$	−10.4839	−0.644023
$266$	0	0
$267$	−1.14591	−0.0701287
$268$	0	0
$269$	5.47385	0.333746	0.166873	−	0.985978i	$-0.446633\pi$
$269$	5.47385	0.333746	0.166873	+	0.985978i	$0.446633\pi$
$270$	0	0
$271$	10.6123	0.644651	0.322326	−	0.946629i	$-0.395535\pi$
$271$	10.6123	0.644651	0.322326	+	0.946629i	$0.395535\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11.0753	−0.667864
$276$	0	0
$277$	22.1170	1.32888	0.664441	−	0.747341i	$-0.268670\pi$
$277$	22.1170	1.32888	0.664441	+	0.747341i	$0.268670\pi$
$278$	0	0
$279$	28.9127	1.73096
$280$	0	0
$281$	−8.60380	−0.513260	−0.256630	−	0.966510i	$-0.582612\pi$
$281$	−8.60380	−0.513260	−0.256630	+	0.966510i	$0.582612\pi$
$282$	0	0
$283$	−19.4778	−1.15783	−0.578917	−	0.815387i	$-0.696524\pi$
$283$	−19.4778	−1.15783	−0.578917	+	0.815387i	$0.696524\pi$
$284$	0	0
$285$	−0.408123	−0.0241751
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.6392	0.978777
$290$	0	0
$291$	−0.473156	−0.0277369
$292$	0	0
$293$	−0.507171	−0.0296292	−0.0148146	−	0.999890i	$-0.504716\pi$
$293$	−0.507171	−0.0296292	−0.0148146	+	0.999890i	$0.504716\pi$
$294$	0	0
$295$	36.1904	2.10709
$296$	0	0
$297$	−0.782231	−0.0453897
$298$	0	0
$299$	−1.49851	−0.0866609
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.67659	0.0963177
$304$	0	0
$305$	17.9110	1.02558
$306$	0	0
$307$	−19.3930	−1.10682	−0.553410	−	0.832909i	$-0.686674\pi$
$307$	−19.3930	−1.10682	−0.553410	+	0.832909i	$0.686674\pi$
$308$	0	0
$309$	0.0220167	0.00125249
$310$	0	0
$311$	−30.2720	−1.71657	−0.858285	−	0.513174i	$-0.828470\pi$
$311$	−30.2720	−1.71657	−0.858285	+	0.513174i	$0.828470\pi$
$312$	0	0
$313$	−2.36212	−0.133515	−0.0667575	−	0.997769i	$-0.521265\pi$
$313$	−2.36212	−0.133515	−0.0667575	+	0.997769i	$0.521265\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.7324	1.27678	0.638390	−	0.769713i	$-0.279601\pi$
$317$	22.7324	1.27678	0.638390	+	0.769713i	$0.279601\pi$
$318$	0	0
$319$	4.04709	0.226593
$320$	0	0
$321$	−1.20741	−0.0673911
$322$	0	0
$323$	−6.14242	−0.341773
$324$	0	0
$325$	1.46716	0.0813834
$326$	0	0
$327$	1.76256	0.0974698
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.97931	0.273688	0.136844	−	0.990593i	$-0.456304\pi$
$331$	4.97931	0.273688	0.136844	+	0.990593i	$0.456304\pi$
$332$	0	0
$333$	−5.10041	−0.279501
$334$	0	0
$335$	4.51207	0.246521
$336$	0	0
$337$	−9.28389	−0.505726	−0.252863	−	0.967502i	$-0.581372\pi$
$337$	−9.28389	−0.505726	−0.252863	+	0.967502i	$0.581372\pi$
$338$	0	0
$339$	0.587478	0.0319075
$340$	0	0
$341$	12.1764	0.659390
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.46273	−0.186427
$346$	0	0
$347$	27.9523	1.50056	0.750278	−	0.661123i	$-0.229920\pi$
$347$	27.9523	1.50056	0.750278	+	0.661123i	$0.229920\pi$
$348$	0	0
$349$	−2.76798	−0.148166	−0.0740832	−	0.997252i	$-0.523603\pi$
$349$	−2.76798	−0.148166	−0.0740832	+	0.997252i	$0.523603\pi$
$350$	0	0
$351$	0.103624	0.00553102
$352$	0	0
$353$	−26.4071	−1.40551	−0.702754	−	0.711433i	$-0.748046\pi$
$353$	−26.4071	−1.40551	−0.702754	+	0.711433i	$0.748046\pi$
$354$	0	0
$355$	49.7799	2.64204
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.9493	−0.788994	−0.394497	−	0.918897i	$-0.629081\pi$
$359$	−14.9493	−0.788994	−0.394497	+	0.918897i	$0.629081\pi$
$360$	0	0
$361$	−17.8784	−0.940969
$362$	0	0
$363$	0.976789	0.0512681
$364$	0	0
$365$	−61.5638	−3.22239
$366$	0	0
$367$	18.1485	0.947342	0.473671	−	0.880702i	$-0.342929\pi$
$367$	18.1485	0.947342	0.473671	+	0.880702i	$0.342929\pi$
$368$	0	0
$369$	2.98924	0.155613
$370$	0	0
$371$	0	0
$372$	0	0
$373$	13.5184	0.699955	0.349977	−	0.936758i	$-0.386189\pi$
$373$	13.5184	0.699955	0.349977	+	0.936758i	$0.386189\pi$
$374$	0	0
$375$	1.46346	0.0755726
$376$	0	0
$377$	−0.536125	−0.0276118
$378$	0	0
$379$	19.5253	1.00295	0.501474	−	0.865173i	$-0.332791\pi$
$379$	19.5253	1.00295	0.501474	+	0.865173i	$0.332791\pi$
$380$	0	0
$381$	0.861491	0.0441355
$382$	0	0
$383$	32.6971	1.67074	0.835372	−	0.549685i	$-0.185252\pi$
$383$	32.6971	1.67074	0.835372	+	0.549685i	$0.185252\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.4658	0.582842
$388$	0	0
$389$	−18.6007	−0.943092	−0.471546	−	0.881841i	$-0.656304\pi$
$389$	−18.6007	−0.943092	−0.471546	+	0.881841i	$0.656304\pi$
$390$	0	0
$391$	−52.1156	−2.63560
$392$	0	0
$393$	−0.758131	−0.0382427
$394$	0	0
$395$	43.1195	2.16958
$396$	0	0
$397$	16.6724	0.836762	0.418381	−	0.908272i	$-0.362598\pi$
$397$	16.6724	0.836762	0.418381	+	0.908272i	$0.362598\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.1863	−0.908183	−0.454091	−	0.890955i	$-0.650036\pi$
$401$	−18.1863	−0.908183	−0.454091	+	0.890955i	$0.650036\pi$
$402$	0	0
$403$	−1.61303	−0.0803508
$404$	0	0
$405$	−33.0712	−1.64332
$406$	0	0
$407$	−2.14801	−0.106473
$408$	0	0
$409$	21.8086	1.07837	0.539183	−	0.842188i	$-0.318733\pi$
$409$	21.8086	1.07837	0.539183	+	0.842188i	$0.318733\pi$
$410$	0	0
$411$	1.08608	0.0535724
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−22.3101	−1.09516
$416$	0	0
$417$	−1.80930	−0.0886017
$418$	0	0
$419$	−25.3082	−1.23639	−0.618194	−	0.786025i	$-0.712136\pi$
$419$	−25.3082	−1.23639	−0.618194	+	0.786025i	$0.712136\pi$
$420$	0	0
$421$	13.5708	0.661400	0.330700	−	0.943736i	$-0.392715\pi$
$421$	13.5708	0.661400	0.330700	+	0.943736i	$0.392715\pi$
$422$	0	0
$423$	21.8845	1.06406
$424$	0	0
$425$	51.0253	2.47509
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.0217810	0.00105160
$430$	0	0
$431$	32.6571	1.57303	0.786517	−	0.617568i	$-0.211882\pi$
$431$	32.6571	1.57303	0.786517	+	0.617568i	$0.211882\pi$
$432$	0	0
$433$	−19.2443	−0.924820	−0.462410	−	0.886666i	$-0.653015\pi$
$433$	−19.2443	−0.924820	−0.462410	+	0.886666i	$0.653015\pi$
$434$	0	0
$435$	−1.23887	−0.0593992
$436$	0	0
$437$	9.51615	0.455219
$438$	0	0
$439$	−11.3365	−0.541061	−0.270531	−	0.962711i	$-0.587199\pi$
$439$	−11.3365	−0.541061	−0.270531	+	0.962711i	$0.587199\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.47862	−0.307808	−0.153904	−	0.988086i	$-0.549185\pi$
$443$	−6.47862	−0.307808	−0.153904	+	0.988086i	$0.549185\pi$
$444$	0	0
$445$	−41.0280	−1.94491
$446$	0	0
$447$	1.68738	0.0798103
$448$	0	0
$449$	24.8694	1.17366	0.586830	−	0.809710i	$-0.300376\pi$
$449$	24.8694	1.17366	0.586830	+	0.809710i	$0.300376\pi$
$450$	0	0
$451$	1.25890	0.0592793
$452$	0	0
$453$	−1.14838	−0.0539556
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−24.6254	−1.15193	−0.575963	−	0.817476i	$-0.695373\pi$
$457$	−24.6254	−1.15193	−0.575963	+	0.817476i	$0.695373\pi$
$458$	0	0
$459$	3.60385	0.168213
$460$	0	0
$461$	−0.483177	−0.0225038	−0.0112519	−	0.999937i	$-0.503582\pi$
$461$	−0.483177	−0.0225038	−0.0112519	+	0.999937i	$0.503582\pi$
$462$	0	0
$463$	1.07820	0.0501081	0.0250541	−	0.999686i	$-0.492024\pi$
$463$	1.07820	0.0501081	0.0250541	+	0.999686i	$0.492024\pi$
$464$	0	0
$465$	−3.72737	−0.172853
$466$	0	0
$467$	−31.6546	−1.46480	−0.732400	−	0.680875i	$-0.761600\pi$
$467$	−31.6546	−1.46480	−0.732400	+	0.680875i	$0.761600\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.33333	−0.0614367
$472$	0	0
$473$	4.82877	0.222027
$474$	0	0
$475$	−9.31708	−0.427497
$476$	0	0
$477$	−8.43690	−0.386299
$478$	0	0
$479$	1.94043	0.0886605	0.0443303	−	0.999017i	$-0.485885\pi$
$479$	1.94043	0.0886605	0.0443303	+	0.999017i	$0.485885\pi$
$480$	0	0
$481$	0.284550	0.0129744
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16.9408	−0.769242
$486$	0	0
$487$	20.7650	0.940951	0.470476	−	0.882413i	$-0.344082\pi$
$487$	20.7650	0.940951	0.470476	+	0.882413i	$0.344082\pi$
$488$	0	0
$489$	−0.273815	−0.0123823
$490$	0	0
$491$	32.2595	1.45585	0.727925	−	0.685657i	$-0.240485\pi$
$491$	32.2595	1.45585	0.727925	+	0.685657i	$0.240485\pi$
$492$	0	0
$493$	−18.6455	−0.839752
$494$	0	0
$495$	−13.9783	−0.628276
$496$	0	0
$497$	0	0
$498$	0	0
$499$	20.5390	0.919453	0.459726	−	0.888061i	$-0.347948\pi$
$499$	20.5390	0.919453	0.459726	+	0.888061i	$0.347948\pi$
$500$	0	0
$501$	1.22410	0.0546887
$502$	0	0
$503$	18.9911	0.846772	0.423386	−	0.905949i	$-0.360841\pi$
$503$	18.9911	0.846772	0.423386	+	0.905949i	$0.360841\pi$
$504$	0	0
$505$	60.0283	2.67123
$506$	0	0
$507$	1.34582	0.0597698
$508$	0	0
$509$	−19.4017	−0.859964	−0.429982	−	0.902837i	$-0.641480\pi$
$509$	−19.4017	−0.859964	−0.429982	+	0.902837i	$0.641480\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−0.658053	−0.0290537
$514$	0	0
$515$	0.788281	0.0347358
$516$	0	0
$517$	9.21651	0.405342
$518$	0	0
$519$	−0.804053	−0.0352940
$520$	0	0
$521$	−10.6858	−0.468155	−0.234078	−	0.972218i	$-0.575207\pi$
$521$	−10.6858	−0.468155	−0.234078	+	0.972218i	$0.575207\pi$
$522$	0	0
$523$	−24.7762	−1.08339	−0.541694	−	0.840576i	$-0.682217\pi$
$523$	−24.7762	−1.08339	−0.541694	+	0.840576i	$0.682217\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−56.0985	−2.44369
$528$	0	0
$529$	57.7400	2.51044
$530$	0	0
$531$	29.1241	1.26388
$532$	0	0
$533$	−0.166769	−0.00722355
$534$	0	0
$535$	−43.2299	−1.86899
$536$	0	0
$537$	0.402556	0.0173716
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−30.3160	−1.30339	−0.651694	−	0.758482i	$-0.725941\pi$
$541$	−30.3160	−1.30339	−0.651694	+	0.758482i	$0.725941\pi$
$542$	0	0
$543$	−2.40554	−0.103232
$544$	0	0
$545$	63.1063	2.70318
$546$	0	0
$547$	11.8145	0.505152	0.252576	−	0.967577i	$-0.418722\pi$
$547$	11.8145	0.505152	0.252576	+	0.967577i	$0.418722\pi$
$548$	0	0
$549$	14.4138	0.615168
$550$	0	0
$551$	3.40462	0.145042
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.657534	0.0279108
$556$	0	0
$557$	39.7423	1.68394	0.841968	−	0.539528i	$-0.181397\pi$
$557$	39.7423	1.68394	0.841968	+	0.539528i	$0.181397\pi$
$558$	0	0
$559$	−0.639676	−0.0270554
$560$	0	0
$561$	0.757507	0.0319820
$562$	0	0
$563$	−29.5515	−1.24545	−0.622723	−	0.782442i	$-0.713974\pi$
$563$	−29.5515	−1.24545	−0.622723	+	0.782442i	$0.713974\pi$
$564$	0	0
$565$	21.0339	0.884905
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.29222	0.347628	0.173814	−	0.984779i	$-0.444391\pi$
$569$	8.29222	0.347628	0.173814	+	0.984779i	$0.444391\pi$
$570$	0	0
$571$	17.4987	0.732296	0.366148	−	0.930557i	$-0.380676\pi$
$571$	17.4987	0.732296	0.366148	+	0.930557i	$0.380676\pi$
$572$	0	0
$573$	1.74845	0.0730426
$574$	0	0
$575$	−79.0510	−3.29665
$576$	0	0
$577$	44.4641	1.85106	0.925531	−	0.378671i	$-0.123619\pi$
$577$	44.4641	1.85106	0.925531	+	0.378671i	$0.123619\pi$
$578$	0	0
$579$	−1.19930	−0.0498412
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−3.55315	−0.147157
$584$	0	0
$585$	1.85172	0.0765593
$586$	0	0
$587$	−28.3564	−1.17039	−0.585197	−	0.810891i	$-0.698983\pi$
$587$	−28.3564	−1.17039	−0.585197	+	0.810891i	$0.698983\pi$
$588$	0	0
$589$	10.2434	0.422073
$590$	0	0
$591$	−1.52015	−0.0625307
$592$	0	0
$593$	−4.32320	−0.177533	−0.0887663	−	0.996052i	$-0.528292\pi$
$593$	−4.32320	−0.177533	−0.0887663	+	0.996052i	$0.528292\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.17591	−0.0481268
$598$	0	0
$599$	−9.43799	−0.385626	−0.192813	−	0.981236i	$-0.561761\pi$
$599$	−9.43799	−0.385626	−0.192813	+	0.981236i	$0.561761\pi$
$600$	0	0
$601$	−19.7128	−0.804100	−0.402050	−	0.915618i	$-0.631702\pi$
$601$	−19.7128	−0.804100	−0.402050	+	0.915618i	$0.631702\pi$
$602$	0	0
$603$	3.63107	0.147869
$604$	0	0
$605$	34.9727	1.42184
$606$	0	0
$607$	−14.8565	−0.603008	−0.301504	−	0.953465i	$-0.597489\pi$
$607$	−14.8565	−0.603008	−0.301504	+	0.953465i	$0.597489\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.22093	−0.0493934
$612$	0	0
$613$	13.8325	0.558688	0.279344	−	0.960191i	$-0.409883\pi$
$613$	13.8325	0.558688	0.279344	+	0.960191i	$0.409883\pi$
$614$	0	0
$615$	−0.385366	−0.0155395
$616$	0	0
$617$	−21.5481	−0.867492	−0.433746	−	0.901035i	$-0.642808\pi$
$617$	−21.5481	−0.867492	−0.433746	+	0.901035i	$0.642808\pi$
$618$	0	0
$619$	8.11291	0.326085	0.163043	−	0.986619i	$-0.447869\pi$
$619$	8.11291	0.326085	0.163043	+	0.986619i	$0.447869\pi$
$620$	0	0
$621$	−5.58327	−0.224049
$622$	0	0
$623$	0	0
$624$	0	0
$625$	8.40943	0.336377
$626$	0	0
$627$	−0.138319	−0.00552391
$628$	0	0
$629$	9.89617	0.394586
$630$	0	0
$631$	−30.1578	−1.20056	−0.600282	−	0.799788i	$-0.704945\pi$
$631$	−30.1578	−1.20056	−0.600282	+	0.799788i	$0.704945\pi$
$632$	0	0
$633$	1.60757	0.0638951
$634$	0	0
$635$	30.8446	1.22403
$636$	0	0
$637$	0	0
$638$	0	0
$639$	40.0602	1.58476
$640$	0	0
$641$	−43.1257	−1.70336	−0.851681	−	0.524060i	$-0.824417\pi$
$641$	−43.1257	−1.70336	−0.851681	+	0.524060i	$0.824417\pi$
$642$	0	0
$643$	16.9922	0.670107	0.335054	−	0.942199i	$-0.391246\pi$
$643$	16.9922	0.670107	0.335054	+	0.942199i	$0.391246\pi$
$644$	0	0
$645$	−1.47815	−0.0582022
$646$	0	0
$647$	−40.6985	−1.60002	−0.800012	−	0.599985i	$-0.795173\pi$
$647$	−40.6985	−1.60002	−0.800012	+	0.599985i	$0.795173\pi$
$648$	0	0
$649$	12.2655	0.481461
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.0267	−0.392377	−0.196188	−	0.980566i	$-0.562856\pi$
$653$	−10.0267	−0.392377	−0.196188	+	0.980566i	$0.562856\pi$
$654$	0	0
$655$	−27.1440	−1.06060
$656$	0	0
$657$	−49.5432	−1.93286
$658$	0	0
$659$	22.1913	0.864452	0.432226	−	0.901765i	$-0.357728\pi$
$659$	22.1913	0.864452	0.432226	+	0.901765i	$0.357728\pi$
$660$	0	0
$661$	13.9938	0.544298	0.272149	−	0.962255i	$-0.412266\pi$
$661$	13.9938	0.544298	0.272149	+	0.962255i	$0.412266\pi$
$662$	0	0
$663$	−0.100348	−0.00389720
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28.8866	1.11849
$668$	0	0
$669$	−2.04724	−0.0791509
$670$	0	0
$671$	6.07030	0.234342
$672$	0	0
$673$	−9.56955	−0.368879	−0.184440	−	0.982844i	$-0.559047\pi$
$673$	−9.56955	−0.368879	−0.184440	+	0.982844i	$0.559047\pi$
$674$	0	0
$675$	5.46647	0.210405
$676$	0	0
$677$	−33.0781	−1.27130	−0.635648	−	0.771979i	$-0.719267\pi$
$677$	−33.0781	−1.27130	−0.635648	+	0.771979i	$0.719267\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0.339254	0.0130002
$682$	0	0
$683$	12.2408	0.468379	0.234190	−	0.972191i	$-0.424756\pi$
$683$	12.2408	0.468379	0.234190	+	0.972191i	$0.424756\pi$
$684$	0	0
$685$	38.8858	1.48575
$686$	0	0
$687$	−1.88299	−0.0718407
$688$	0	0
$689$	0.470692	0.0179320
$690$	0	0
$691$	−1.86970	−0.0711268	−0.0355634	−	0.999367i	$-0.511323\pi$
$691$	−1.86970	−0.0711268	−0.0355634	+	0.999367i	$0.511323\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−64.7797	−2.45723
$696$	0	0
$697$	−5.79993	−0.219688
$698$	0	0
$699$	−1.24865	−0.0472281
$700$	0	0
$701$	−49.8538	−1.88295	−0.941475	−	0.337082i	$-0.890560\pi$
$701$	−49.8538	−1.88295	−0.941475	+	0.337082i	$0.890560\pi$
$702$	0	0
$703$	−1.80701	−0.0681528
$704$	0	0
$705$	−2.82130	−0.106256
$706$	0	0
$707$	0	0
$708$	0	0
$709$	7.14510	0.268340	0.134170	−	0.990958i	$-0.457163\pi$
$709$	7.14510	0.268340	0.134170	+	0.990958i	$0.457163\pi$
$710$	0	0
$711$	34.7003	1.30136
$712$	0	0
$713$	86.9106	3.25483
$714$	0	0
$715$	0.779843	0.0291645
$716$	0	0
$717$	−1.55394	−0.0580328
$718$	0	0
$719$	41.9999	1.56633	0.783165	−	0.621813i	$-0.213604\pi$
$719$	41.9999	1.56633	0.783165	+	0.621813i	$0.213604\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0.283720	0.0105516
$724$	0	0
$725$	−28.2823	−1.05038
$726$	0	0
$727$	1.39778	0.0518408	0.0259204	−	0.999664i	$-0.491748\pi$
$727$	1.39778	0.0518408	0.0259204	+	0.999664i	$0.491748\pi$
$728$	0	0
$729$	−26.4205	−0.978538
$730$	0	0
$731$	−22.2469	−0.822830
$732$	0	0
$733$	28.1491	1.03971	0.519856	−	0.854254i	$-0.325986\pi$
$733$	28.1491	1.03971	0.519856	+	0.854254i	$0.325986\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.52920	0.0563289
$738$	0	0
$739$	22.6496	0.833179	0.416590	−	0.909095i	$-0.363225\pi$
$739$	22.6496	0.833179	0.416590	+	0.909095i	$0.363225\pi$
$740$	0	0
$741$	0.0183233	0.000673123 0
$742$	0	0
$743$	−16.2233	−0.595175	−0.297587	−	0.954695i	$-0.596182\pi$
$743$	−16.2233	−0.595175	−0.297587	+	0.954695i	$0.596182\pi$
$744$	0	0
$745$	60.4146	2.21342
$746$	0	0
$747$	−17.9540	−0.656901
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−28.6190	−1.04432	−0.522161	−	0.852847i	$-0.674874\pi$
$751$	−28.6190	−1.04432	−0.522161	+	0.852847i	$0.674874\pi$
$752$	0	0
$753$	−0.657038	−0.0239438
$754$	0	0
$755$	−41.1163	−1.49638
$756$	0	0
$757$	40.5494	1.47379	0.736897	−	0.676005i	$-0.236290\pi$
$757$	40.5494	1.47379	0.736897	+	0.676005i	$0.236290\pi$
$758$	0	0
$759$	−1.17357	−0.0425978
$760$	0	0
$761$	−9.16931	−0.332387	−0.166194	−	0.986093i	$-0.553148\pi$
$761$	−9.16931	−0.332387	−0.166194	+	0.986093i	$0.553148\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	64.3998	2.32838
$766$	0	0
$767$	−1.62483	−0.0586691
$768$	0	0
$769$	16.7231	0.603052	0.301526	−	0.953458i	$-0.402504\pi$
$769$	16.7231	0.603052	0.301526	+	0.953458i	$0.402504\pi$
$770$	0	0
$771$	−3.04828	−0.109781
$772$	0	0
$773$	39.9783	1.43792	0.718960	−	0.695052i	$-0.244619\pi$
$773$	39.9783	1.43792	0.718960	+	0.695052i	$0.244619\pi$
$774$	0	0
$775$	−85.0925	−3.05661
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.05905	0.0379444
$780$	0	0
$781$	16.8711	0.603697
$782$	0	0
$783$	−1.99754	−0.0713863
$784$	0	0
$785$	−47.7383	−1.70385
$786$	0	0
$787$	22.2939	0.794693	0.397346	−	0.917669i	$-0.369931\pi$
$787$	22.2939	0.794693	0.397346	+	0.917669i	$0.369931\pi$
$788$	0	0
$789$	−1.05122	−0.0374245
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.804144	−0.0285560
$794$	0	0
$795$	1.08767	0.0385756
$796$	0	0
$797$	30.7525	1.08931	0.544654	−	0.838661i	$-0.316661\pi$
$797$	30.7525	1.08931	0.544654	+	0.838661i	$0.316661\pi$
$798$	0	0
$799$	−42.4618	−1.50219
$800$	0	0
$801$	−33.0171	−1.16660
$802$	0	0
$803$	−20.8648	−0.736304
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.567891	−0.0199907
$808$	0	0
$809$	−34.2468	−1.20405	−0.602027	−	0.798476i	$-0.705640\pi$
$809$	−34.2468	−1.20405	−0.602027	+	0.798476i	$0.705640\pi$
$810$	0	0
$811$	7.17378	0.251905	0.125953	−	0.992036i	$-0.459801\pi$
$811$	7.17378	0.251905	0.125953	+	0.992036i	$0.459801\pi$
$812$	0	0
$813$	−1.10099	−0.0386133
$814$	0	0
$815$	−9.80361	−0.343406
$816$	0	0
$817$	4.06221	0.142119
$818$	0	0
$819$	0	0
$820$	0	0
$821$	49.9117	1.74193	0.870965	−	0.491346i	$-0.163495\pi$
$821$	49.9117	1.74193	0.870965	+	0.491346i	$0.163495\pi$
$822$	0	0
$823$	−28.7584	−1.00246	−0.501228	−	0.865315i	$-0.667118\pi$
$823$	−28.7584	−1.00246	−0.501228	+	0.865315i	$0.667118\pi$
$824$	0	0
$825$	1.14902	0.0400037
$826$	0	0
$827$	10.5598	0.367199	0.183600	−	0.983001i	$-0.441225\pi$
$827$	10.5598	0.367199	0.183600	+	0.983001i	$0.441225\pi$
$828$	0	0
$829$	−1.66400	−0.0577930	−0.0288965	−	0.999582i	$-0.509199\pi$
$829$	−1.66400	−0.0577930	−0.0288965	+	0.999582i	$0.509199\pi$
$830$	0	0
$831$	−2.29456	−0.0795973
$832$	0	0
$833$	0	0
$834$	0	0
$835$	43.8273	1.51671
$836$	0	0
$837$	−6.00997	−0.207735
$838$	0	0
$839$	8.18432	0.282554	0.141277	−	0.989970i	$-0.454879\pi$
$839$	8.18432	0.282554	0.141277	+	0.989970i	$0.454879\pi$
$840$	0	0
$841$	−18.6652	−0.643627
$842$	0	0
$843$	0.892613	0.0307432
$844$	0	0
$845$	48.1853	1.65762
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.02075	0.0693519
$850$	0	0
$851$	−15.3317	−0.525562
$852$	0	0
$853$	14.6872	0.502882	0.251441	−	0.967873i	$-0.419096\pi$
$853$	14.6872	0.502882	0.251441	+	0.967873i	$0.419096\pi$
$854$	0	0
$855$	−11.7592	−0.402157
$856$	0	0
$857$	−8.41864	−0.287575	−0.143788	−	0.989609i	$-0.545928\pi$
$857$	−8.41864	−0.287575	−0.143788	+	0.989609i	$0.545928\pi$
$858$	0	0
$859$	−42.9116	−1.46412	−0.732062	−	0.681238i	$-0.761442\pi$
$859$	−42.9116	−1.46412	−0.732062	+	0.681238i	$0.761442\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.0445	−0.648282	−0.324141	−	0.946009i	$-0.605075\pi$
$863$	−19.0445	−0.648282	−0.324141	+	0.946009i	$0.605075\pi$
$864$	0	0
$865$	−28.7881	−0.978825
$866$	0	0
$867$	−1.72626	−0.0586267
$868$	0	0
$869$	14.6138	0.495740
$870$	0	0
$871$	−0.202576	−0.00686403
$872$	0	0
$873$	−13.6330	−0.461408
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−36.9819	−1.24879	−0.624395	−	0.781109i	$-0.714654\pi$
$877$	−36.9819	−1.24879	−0.624395	+	0.781109i	$0.714654\pi$
$878$	0	0
$879$	0.0526171	0.00177473
$880$	0	0
$881$	38.9419	1.31199	0.655993	−	0.754767i	$-0.272250\pi$
$881$	38.9419	1.31199	0.655993	+	0.754767i	$0.272250\pi$
$882$	0	0
$883$	37.2257	1.25274	0.626371	−	0.779525i	$-0.284539\pi$
$883$	37.2257	1.25274	0.626371	+	0.779525i	$0.284539\pi$
$884$	0	0
$885$	−3.75462	−0.126210
$886$	0	0
$887$	7.85934	0.263891	0.131945	−	0.991257i	$-0.457878\pi$
$887$	7.85934	0.263891	0.131945	+	0.991257i	$0.457878\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−11.2083	−0.375492
$892$	0	0
$893$	7.75340	0.259458
$894$	0	0
$895$	14.4130	0.481774
$896$	0	0
$897$	0.155465	0.00519081
$898$	0	0
$899$	31.0942	1.03705
$900$	0	0
$901$	16.3699	0.545360
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−86.1275	−2.86298
$906$	0	0
$907$	3.40729	0.113137	0.0565686	−	0.998399i	$-0.481984\pi$
$907$	3.40729	0.113137	0.0565686	+	0.998399i	$0.481984\pi$
$908$	0	0
$909$	48.3076	1.60226
$910$	0	0
$911$	−40.5842	−1.34462	−0.672308	−	0.740272i	$-0.734697\pi$
$911$	−40.5842	−1.34462	−0.672308	+	0.740272i	$0.734697\pi$
$912$	0	0
$913$	−7.56121	−0.250240
$914$	0	0
$915$	−1.85820	−0.0614303
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−21.0834	−0.695477	−0.347738	−	0.937592i	$-0.613050\pi$
$919$	−21.0834	−0.695477	−0.347738	+	0.937592i	$0.613050\pi$
$920$	0	0
$921$	2.01196	0.0662962
$922$	0	0
$923$	−2.23495	−0.0735642
$924$	0	0
$925$	15.0109	0.493556
$926$	0	0
$927$	0.634366	0.0208353
$928$	0	0
$929$	−48.8096	−1.60139	−0.800696	−	0.599072i	$-0.795536\pi$
$929$	−48.8096	−1.60139	−0.800696	+	0.599072i	$0.795536\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.14061	0.102819
$934$	0	0
$935$	27.1216	0.886972
$936$	0	0
$937$	3.91878	0.128021	0.0640104	−	0.997949i	$-0.479611\pi$
$937$	3.91878	0.128021	0.0640104	+	0.997949i	$0.479611\pi$
$938$	0	0
$939$	0.245062	0.00799728
$940$	0	0
$941$	−51.6312	−1.68313	−0.841564	−	0.540157i	$-0.818365\pi$
$941$	−51.6312	−1.68313	−0.841564	+	0.540157i	$0.818365\pi$
$942$	0	0
$943$	8.98555	0.292610
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.80509	0.188640	0.0943200	−	0.995542i	$-0.469932\pi$
$947$	5.80509	0.188640	0.0943200	+	0.995542i	$0.469932\pi$
$948$	0	0
$949$	2.76400	0.0897233
$950$	0	0
$951$	−2.35840	−0.0764765
$952$	0	0
$953$	−2.18025	−0.0706253	−0.0353126	−	0.999376i	$-0.511243\pi$
$953$	−2.18025	−0.0706253	−0.0353126	+	0.999376i	$0.511243\pi$
$954$	0	0
$955$	62.6012	2.02573
$956$	0	0
$957$	−0.419870	−0.0135725
$958$	0	0
$959$	0	0
$960$	0	0
$961$	62.5528	2.01783
$962$	0	0
$963$	−34.7891	−1.12106
$964$	0	0
$965$	−42.9394	−1.38227
$966$	0	0
$967$	28.1312	0.904638	0.452319	−	0.891856i	$-0.350597\pi$
$967$	28.1312	0.904638	0.452319	+	0.891856i	$0.350597\pi$
$968$	0	0
$969$	0.637253	0.0204715
$970$	0	0
$971$	4.15681	0.133398	0.0666991	−	0.997773i	$-0.478753\pi$
$971$	4.15681	0.133398	0.0666991	+	0.997773i	$0.478753\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−0.152212	−0.00487470
$976$	0	0
$977$	2.34657	0.0750736	0.0375368	−	0.999295i	$-0.488049\pi$
$977$	2.34657	0.0750736	0.0375368	+	0.999295i	$0.488049\pi$
$978$	0	0
$979$	−13.9050	−0.444405
$980$	0	0
$981$	50.7846	1.62143
$982$	0	0
$983$	2.21649	0.0706950	0.0353475	−	0.999375i	$-0.488746\pi$
$983$	2.21649	0.0706950	0.0353475	+	0.999375i	$0.488746\pi$
$984$	0	0
$985$	−54.4272	−1.73419
$986$	0	0
$987$	0	0
$988$	0	0
$989$	34.4659	1.09595
$990$	0	0
$991$	−43.7956	−1.39121	−0.695607	−	0.718422i	$-0.744865\pi$
$991$	−43.7956	−1.39121	−0.695607	+	0.718422i	$0.744865\pi$
$992$	0	0
$993$	−0.516585	−0.0163933
$994$	0	0
$995$	−42.1020	−1.33472
$996$	0	0
$997$	2.41535	0.0764951	0.0382475	−	0.999268i	$-0.487822\pi$
$997$	2.41535	0.0764951	0.0382475	+	0.999268i	$0.487822\pi$
$998$	0	0
$999$	1.06020	0.0335433

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.t.1.9	yes	20
7.6	odd	2		8036.2.a.s.1.12	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.12	✓	20	7.6	odd	2
8036.2.a.t.1.9	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)

\(f(q)\)	\(=\)	\(q-0.103746 q^{3} -3.71451 q^{5} -2.98924 q^{9} +O(q^{10})\)	\(q-0.103746 q^{3} -3.71451 q^{5} -2.98924 q^{9} -1.25890 q^{11} +0.166769 q^{13} +0.385366 q^{15} +5.79993 q^{17} -1.05905 q^{19} -8.98555 q^{23} +8.79757 q^{25} +0.621361 q^{27} -3.21478 q^{29} -9.67227 q^{31} +0.130606 q^{33} +1.70626 q^{37} -0.0173016 q^{39} -1.00000 q^{41} -3.83571 q^{43} +11.1035 q^{45} -7.32109 q^{47} -0.601721 q^{51} +2.82243 q^{53} +4.67619 q^{55} +0.109873 q^{57} -9.74300 q^{59} -4.82191 q^{61} -0.619464 q^{65} -1.21471 q^{67} +0.932217 q^{69} -13.4015 q^{71} +16.5739 q^{73} -0.912716 q^{75} -11.6084 q^{79} +8.90325 q^{81} +6.00621 q^{83} -21.5439 q^{85} +0.333522 q^{87} +11.0453 q^{89} +1.00346 q^{93} +3.93385 q^{95} +4.56071 q^{97} +3.76315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) \) 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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