LMFDB - Embedded newform 8016.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 31x^{7} + 24x^{6} + 293x^{5} - 101x^{4} - 864x^{3} - 278x^{2} + 24x + 2 $ x^9 - x^8 - 31x^7 + 24x^6 + 293x^5 - 101x^4 - 864x^3 - 278x^2 + 24*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.06168$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.06168 q^{5} +1.52198 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.06168 q^{5} +1.52198 q^{7} +1.00000 q^{9} +3.75016 q^{11} -0.808587 q^{13} -4.06168 q^{15} -1.48425 q^{17} +5.96105 q^{19} +1.52198 q^{21} +8.26017 q^{23} +11.4972 q^{25} +1.00000 q^{27} -5.45397 q^{29} -8.18417 q^{31} +3.75016 q^{33} -6.18178 q^{35} -5.32365 q^{37} -0.808587 q^{39} +5.14543 q^{41} +5.00707 q^{43} -4.06168 q^{45} -3.25196 q^{47} -4.68358 q^{49} -1.48425 q^{51} +7.56769 q^{53} -15.2319 q^{55} +5.96105 q^{57} +14.2066 q^{59} +2.91923 q^{61} +1.52198 q^{63} +3.28422 q^{65} -15.6127 q^{67} +8.26017 q^{69} -0.362319 q^{71} -12.1669 q^{73} +11.4972 q^{75} +5.70765 q^{77} -11.8747 q^{79} +1.00000 q^{81} +2.55930 q^{83} +6.02855 q^{85} -5.45397 q^{87} +11.6827 q^{89} -1.23065 q^{91} -8.18417 q^{93} -24.2119 q^{95} +6.54466 q^{97} +3.75016 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} + q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 + q^5 - 2 * q^7 + 9 * q^9	$ 9 q + 9 q^{3} + q^{5} - 2 q^{7} + 9 q^{9} + 9 q^{11} + 10 q^{13} + q^{15} + 7 q^{17} + 2 q^{19} - 2 q^{21} + 3 q^{23} + 18 q^{25} + 9 q^{27} + 5 q^{29} - 12 q^{31} + 9 q^{33} + 6 q^{35} + 15 q^{37} + 10 q^{39} + 14 q^{41} - 6 q^{43} + q^{45} + 3 q^{47} + 27 q^{49} + 7 q^{51} + 9 q^{53} - 19 q^{55} + 2 q^{57} + 9 q^{59} + 30 q^{61} - 2 q^{63} + 28 q^{65} - 16 q^{67} + 3 q^{69} + 3 q^{71} + 32 q^{73} + 18 q^{75} + 18 q^{77} - 24 q^{79} + 9 q^{81} + 3 q^{83} + 37 q^{85} + 5 q^{87} + 46 q^{89} - 33 q^{91} - 12 q^{93} - 11 q^{95} + 43 q^{97} + 9 q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 + q^5 - 2 * q^7 + 9 * q^9 + 9 * q^11 + 10 * q^13 + q^15 + 7 * q^17 + 2 * q^19 - 2 * q^21 + 3 * q^23 + 18 * q^25 + 9 * q^27 + 5 * q^29 - 12 * q^31 + 9 * q^33 + 6 * q^35 + 15 * q^37 + 10 * q^39 + 14 * q^41 - 6 * q^43 + q^45 + 3 * q^47 + 27 * q^49 + 7 * q^51 + 9 * q^53 - 19 * q^55 + 2 * q^57 + 9 * q^59 + 30 * q^61 - 2 * q^63 + 28 * q^65 - 16 * q^67 + 3 * q^69 + 3 * q^71 + 32 * q^73 + 18 * q^75 + 18 * q^77 - 24 * q^79 + 9 * q^81 + 3 * q^83 + 37 * q^85 + 5 * q^87 + 46 * q^89 - 33 * q^91 - 12 * q^93 - 11 * q^95 + 43 * q^97 + 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−4.06168	−1.81644	−0.908219	−	0.418496i	$-0.862558\pi$
$5$	−4.06168	−1.81644	−0.908219	+	0.418496i	$0.862558\pi$
$6$	0	0
$7$	1.52198	0.575253	0.287627	−	0.957743i	$-0.407134\pi$
$7$	1.52198	0.575253	0.287627	+	0.957743i	$0.407134\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.75016	1.13071	0.565357	−	0.824846i	$-0.308738\pi$
$11$	3.75016	1.13071	0.565357	+	0.824846i	$0.308738\pi$
$12$	0	0
$13$	−0.808587	−0.224262	−0.112131	−	0.993693i	$-0.535768\pi$
$13$	−0.808587	−0.224262	−0.112131	+	0.993693i	$0.535768\pi$
$14$	0	0
$15$	−4.06168	−1.04872
$16$	0	0
$17$	−1.48425	−0.359984	−0.179992	−	0.983668i	$-0.557607\pi$
$17$	−1.48425	−0.359984	−0.179992	+	0.983668i	$0.557607\pi$
$18$	0	0
$19$	5.96105	1.36756	0.683779	−	0.729689i	$-0.260335\pi$
$19$	5.96105	1.36756	0.683779	+	0.729689i	$0.260335\pi$
$20$	0	0
$21$	1.52198	0.332123
$22$	0	0
$23$	8.26017	1.72236	0.861182	−	0.508297i	$-0.169725\pi$
$23$	8.26017	1.72236	0.861182	+	0.508297i	$0.169725\pi$
$24$	0	0
$25$	11.4972	2.29945
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.45397	−1.01278	−0.506388	−	0.862306i	$-0.669020\pi$
$29$	−5.45397	−1.01278	−0.506388	+	0.862306i	$0.669020\pi$
$30$	0	0
$31$	−8.18417	−1.46992	−0.734960	−	0.678110i	$-0.762799\pi$
$31$	−8.18417	−1.46992	−0.734960	+	0.678110i	$0.762799\pi$
$32$	0	0
$33$	3.75016	0.652818
$34$	0	0
$35$	−6.18178	−1.04491
$36$	0	0
$37$	−5.32365	−0.875203	−0.437602	−	0.899169i	$-0.644172\pi$
$37$	−5.32365	−0.875203	−0.437602	+	0.899169i	$0.644172\pi$
$38$	0	0
$39$	−0.808587	−0.129478
$40$	0	0
$41$	5.14543	0.803581	0.401790	−	0.915732i	$-0.368388\pi$
$41$	5.14543	0.803581	0.401790	+	0.915732i	$0.368388\pi$
$42$	0	0
$43$	5.00707	0.763572	0.381786	−	0.924251i	$-0.375309\pi$
$43$	5.00707	0.763572	0.381786	+	0.924251i	$0.375309\pi$
$44$	0	0
$45$	−4.06168	−0.605479
$46$	0	0
$47$	−3.25196	−0.474347	−0.237173	−	0.971467i	$-0.576221\pi$
$47$	−3.25196	−0.474347	−0.237173	+	0.971467i	$0.576221\pi$
$48$	0	0
$49$	−4.68358	−0.669084
$50$	0	0
$51$	−1.48425	−0.207837
$52$	0	0
$53$	7.56769	1.03950	0.519751	−	0.854318i	$-0.326025\pi$
$53$	7.56769	1.03950	0.519751	+	0.854318i	$0.326025\pi$
$54$	0	0
$55$	−15.2319	−2.05387
$56$	0	0
$57$	5.96105	0.789560
$58$	0	0
$59$	14.2066	1.84954	0.924770	−	0.380527i	$-0.124258\pi$
$59$	14.2066	1.84954	0.924770	+	0.380527i	$0.124258\pi$
$60$	0	0
$61$	2.91923	0.373769	0.186885	−	0.982382i	$-0.440161\pi$
$61$	2.91923	0.373769	0.186885	+	0.982382i	$0.440161\pi$
$62$	0	0
$63$	1.52198	0.191751
$64$	0	0
$65$	3.28422	0.407357
$66$	0	0
$67$	−15.6127	−1.90739	−0.953695	−	0.300775i	$-0.902755\pi$
$67$	−15.6127	−1.90739	−0.953695	+	0.300775i	$0.902755\pi$
$68$	0	0
$69$	8.26017	0.994407
$70$	0	0
$71$	−0.362319	−0.0429994	−0.0214997	−	0.999769i	$-0.506844\pi$
$71$	−0.362319	−0.0429994	−0.0214997	+	0.999769i	$0.506844\pi$
$72$	0	0
$73$	−12.1669	−1.42403	−0.712014	−	0.702165i	$-0.752217\pi$
$73$	−12.1669	−1.42403	−0.712014	+	0.702165i	$0.752217\pi$
$74$	0	0
$75$	11.4972	1.32759
$76$	0	0
$77$	5.70765	0.650448
$78$	0	0
$79$	−11.8747	−1.33601	−0.668006	−	0.744156i	$-0.732852\pi$
$79$	−11.8747	−1.33601	−0.668006	+	0.744156i	$0.732852\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.55930	0.280919	0.140460	−	0.990086i	$-0.455142\pi$
$83$	2.55930	0.280919	0.140460	+	0.990086i	$0.455142\pi$
$84$	0	0
$85$	6.02855	0.653888
$86$	0	0
$87$	−5.45397	−0.584727
$88$	0	0
$89$	11.6827	1.23836	0.619179	−	0.785250i	$-0.287465\pi$
$89$	11.6827	1.23836	0.619179	+	0.785250i	$0.287465\pi$
$90$	0	0
$91$	−1.23065	−0.129007
$92$	0	0
$93$	−8.18417	−0.848659
$94$	0	0
$95$	−24.2119	−2.48409
$96$	0	0
$97$	6.54466	0.664510	0.332255	−	0.943190i	$-0.392191\pi$
$97$	6.54466	0.664510	0.332255	+	0.943190i	$0.392191\pi$
$98$	0	0
$99$	3.75016	0.376905
$100$	0	0
$101$	17.5654	1.74782	0.873912	−	0.486085i	$-0.161575\pi$
$101$	17.5654	1.74782	0.873912	+	0.486085i	$0.161575\pi$
$102$	0	0
$103$	−17.8844	−1.76220	−0.881100	−	0.472930i	$-0.843196\pi$
$103$	−17.8844	−1.76220	−0.881100	+	0.472930i	$0.843196\pi$
$104$	0	0
$105$	−6.18178	−0.603280
$106$	0	0
$107$	8.38542	0.810649	0.405325	−	0.914173i	$-0.367159\pi$
$107$	8.38542	0.810649	0.405325	+	0.914173i	$0.367159\pi$
$108$	0	0
$109$	0.932415	0.0893092	0.0446546	−	0.999002i	$-0.485781\pi$
$109$	0.932415	0.0893092	0.0446546	+	0.999002i	$0.485781\pi$
$110$	0	0
$111$	−5.32365	−0.505299
$112$	0	0
$113$	−13.4832	−1.26839	−0.634196	−	0.773172i	$-0.718669\pi$
$113$	−13.4832	−1.26839	−0.634196	+	0.773172i	$0.718669\pi$
$114$	0	0
$115$	−33.5501	−3.12857
$116$	0	0
$117$	−0.808587	−0.0747539
$118$	0	0
$119$	−2.25900	−0.207082
$120$	0	0
$121$	3.06368	0.278516
$122$	0	0
$123$	5.14543	0.463948
$124$	0	0
$125$	−26.3897	−2.36036
$126$	0	0
$127$	19.8720	1.76335	0.881675	−	0.471857i	$-0.156416\pi$
$127$	19.8720	1.76335	0.881675	+	0.471857i	$0.156416\pi$
$128$	0	0
$129$	5.00707	0.440848
$130$	0	0
$131$	20.4780	1.78918	0.894588	−	0.446893i	$-0.147469\pi$
$131$	20.4780	1.78918	0.894588	+	0.446893i	$0.147469\pi$
$132$	0	0
$133$	9.07259	0.786693
$134$	0	0
$135$	−4.06168	−0.349574
$136$	0	0
$137$	−1.96583	−0.167952	−0.0839761	−	0.996468i	$-0.526762\pi$
$137$	−1.96583	−0.167952	−0.0839761	+	0.996468i	$0.526762\pi$
$138$	0	0
$139$	−3.35402	−0.284485	−0.142242	−	0.989832i	$-0.545431\pi$
$139$	−3.35402	−0.284485	−0.142242	+	0.989832i	$0.545431\pi$
$140$	0	0
$141$	−3.25196	−0.273864
$142$	0	0
$143$	−3.03233	−0.253576
$144$	0	0
$145$	22.1523	1.83965
$146$	0	0
$147$	−4.68358	−0.386296
$148$	0	0
$149$	10.8386	0.887932	0.443966	−	0.896044i	$-0.353571\pi$
$149$	10.8386	0.887932	0.443966	+	0.896044i	$0.353571\pi$
$150$	0	0
$151$	9.05641	0.737001	0.368500	−	0.929628i	$-0.379871\pi$
$151$	9.05641	0.737001	0.368500	+	0.929628i	$0.379871\pi$
$152$	0	0
$153$	−1.48425	−0.119995
$154$	0	0
$155$	33.2415	2.67002
$156$	0	0
$157$	16.1306	1.28736	0.643682	−	0.765293i	$-0.277406\pi$
$157$	16.1306	1.28736	0.643682	+	0.765293i	$0.277406\pi$
$158$	0	0
$159$	7.56769	0.600157
$160$	0	0
$161$	12.5718	0.990796
$162$	0	0
$163$	−5.30832	−0.415780	−0.207890	−	0.978152i	$-0.566660\pi$
$163$	−5.30832	−0.415780	−0.207890	+	0.978152i	$0.566660\pi$
$164$	0	0
$165$	−15.2319	−1.18580
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.3462	−0.949707
$170$	0	0
$171$	5.96105	0.455853
$172$	0	0
$173$	8.04182	0.611408	0.305704	−	0.952127i	$-0.401108\pi$
$173$	8.04182	0.611408	0.305704	+	0.952127i	$0.401108\pi$
$174$	0	0
$175$	17.4985	1.32276
$176$	0	0
$177$	14.2066	1.06783
$178$	0	0
$179$	−13.8439	−1.03474	−0.517371	−	0.855761i	$-0.673089\pi$
$179$	−13.8439	−1.03474	−0.517371	+	0.855761i	$0.673089\pi$
$180$	0	0
$181$	−0.189929	−0.0141173	−0.00705864	−	0.999975i	$-0.502247\pi$
$181$	−0.189929	−0.0141173	−0.00705864	+	0.999975i	$0.502247\pi$
$182$	0	0
$183$	2.91923	0.215796
$184$	0	0
$185$	21.6230	1.58975
$186$	0	0
$187$	−5.56618	−0.407039
$188$	0	0
$189$	1.52198	0.110708
$190$	0	0
$191$	−5.16497	−0.373724	−0.186862	−	0.982386i	$-0.559832\pi$
$191$	−5.16497	−0.373724	−0.186862	+	0.982386i	$0.559832\pi$
$192$	0	0
$193$	14.4877	1.04284	0.521422	−	0.853299i	$-0.325402\pi$
$193$	14.4877	1.04284	0.521422	+	0.853299i	$0.325402\pi$
$194$	0	0
$195$	3.28422	0.235188
$196$	0	0
$197$	−7.95294	−0.566624	−0.283312	−	0.959028i	$-0.591433\pi$
$197$	−7.95294	−0.566624	−0.283312	+	0.959028i	$0.591433\pi$
$198$	0	0
$199$	13.0286	0.923575	0.461788	−	0.886991i	$-0.347208\pi$
$199$	13.0286	0.923575	0.461788	+	0.886991i	$0.347208\pi$
$200$	0	0
$201$	−15.6127	−1.10123
$202$	0	0
$203$	−8.30082	−0.582603
$204$	0	0
$205$	−20.8991	−1.45965
$206$	0	0
$207$	8.26017	0.574121
$208$	0	0
$209$	22.3549	1.54632
$210$	0	0
$211$	16.3340	1.12448	0.562239	−	0.826975i	$-0.309940\pi$
$211$	16.3340	1.12448	0.562239	+	0.826975i	$0.309940\pi$
$212$	0	0
$213$	−0.362319	−0.0248257
$214$	0	0
$215$	−20.3371	−1.38698
$216$	0	0
$217$	−12.4561	−0.845577
$218$	0	0
$219$	−12.1669	−0.822163
$220$	0	0
$221$	1.20015	0.0807306
$222$	0	0
$223$	29.4549	1.97245	0.986224	−	0.165414i	$-0.0528959\pi$
$223$	29.4549	1.97245	0.986224	+	0.165414i	$0.0528959\pi$
$224$	0	0
$225$	11.4972	0.766482
$226$	0	0
$227$	11.2229	0.744889	0.372445	−	0.928054i	$-0.378520\pi$
$227$	11.2229	0.744889	0.372445	+	0.928054i	$0.378520\pi$
$228$	0	0
$229$	−2.35856	−0.155858	−0.0779291	−	0.996959i	$-0.524831\pi$
$229$	−2.35856	−0.155858	−0.0779291	+	0.996959i	$0.524831\pi$
$230$	0	0
$231$	5.70765	0.375536
$232$	0	0
$233$	−19.6121	−1.28483	−0.642415	−	0.766357i	$-0.722067\pi$
$233$	−19.6121	−1.28483	−0.642415	+	0.766357i	$0.722067\pi$
$234$	0	0
$235$	13.2084	0.861622
$236$	0	0
$237$	−11.8747	−0.771347
$238$	0	0
$239$	5.61893	0.363458	0.181729	−	0.983349i	$-0.441831\pi$
$239$	5.61893	0.363458	0.181729	+	0.983349i	$0.441831\pi$
$240$	0	0
$241$	22.5680	1.45373	0.726867	−	0.686778i	$-0.240976\pi$
$241$	22.5680	1.45373	0.726867	+	0.686778i	$0.240976\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	19.0232	1.21535
$246$	0	0
$247$	−4.82003	−0.306691
$248$	0	0
$249$	2.55930	0.162189
$250$	0	0
$251$	−2.49552	−0.157516	−0.0787578	−	0.996894i	$-0.525095\pi$
$251$	−2.49552	−0.157516	−0.0787578	+	0.996894i	$0.525095\pi$
$252$	0	0
$253$	30.9769	1.94750
$254$	0	0
$255$	6.02855	0.377523
$256$	0	0
$257$	−25.7511	−1.60631	−0.803153	−	0.595772i	$-0.796846\pi$
$257$	−25.7511	−1.60631	−0.803153	+	0.595772i	$0.796846\pi$
$258$	0	0
$259$	−8.10248	−0.503464
$260$	0	0
$261$	−5.45397	−0.337592
$262$	0	0
$263$	19.8069	1.22135	0.610673	−	0.791883i	$-0.290899\pi$
$263$	19.8069	1.22135	0.610673	+	0.791883i	$0.290899\pi$
$264$	0	0
$265$	−30.7375	−1.88819
$266$	0	0
$267$	11.6827	0.714967
$268$	0	0
$269$	3.54672	0.216247	0.108124	−	0.994137i	$-0.465516\pi$
$269$	3.54672	0.216247	0.108124	+	0.994137i	$0.465516\pi$
$270$	0	0
$271$	−13.6862	−0.831380	−0.415690	−	0.909506i	$-0.636460\pi$
$271$	−13.6862	−0.831380	−0.415690	+	0.909506i	$0.636460\pi$
$272$	0	0
$273$	−1.23065	−0.0744824
$274$	0	0
$275$	43.1164	2.60002
$276$	0	0
$277$	−2.28135	−0.137073	−0.0685366	−	0.997649i	$-0.521833\pi$
$277$	−2.28135	−0.137073	−0.0685366	+	0.997649i	$0.521833\pi$
$278$	0	0
$279$	−8.18417	−0.489974
$280$	0	0
$281$	24.9176	1.48646	0.743230	−	0.669036i	$-0.233293\pi$
$281$	24.9176	1.48646	0.743230	+	0.669036i	$0.233293\pi$
$282$	0	0
$283$	12.7901	0.760295	0.380147	−	0.924926i	$-0.375873\pi$
$283$	12.7901	0.760295	0.380147	+	0.924926i	$0.375873\pi$
$284$	0	0
$285$	−24.2119	−1.43419
$286$	0	0
$287$	7.83122	0.462263
$288$	0	0
$289$	−14.7970	−0.870412
$290$	0	0
$291$	6.54466	0.383655
$292$	0	0
$293$	10.7252	0.626571	0.313285	−	0.949659i	$-0.398570\pi$
$293$	10.7252	0.626571	0.313285	+	0.949659i	$0.398570\pi$
$294$	0	0
$295$	−57.7026	−3.35957
$296$	0	0
$297$	3.75016	0.217606
$298$	0	0
$299$	−6.67906	−0.386260
$300$	0	0
$301$	7.62065	0.439247
$302$	0	0
$303$	17.5654	1.00911
$304$	0	0
$305$	−11.8570	−0.678929
$306$	0	0
$307$	15.8870	0.906720	0.453360	−	0.891328i	$-0.350225\pi$
$307$	15.8870	0.906720	0.453360	+	0.891328i	$0.350225\pi$
$308$	0	0
$309$	−17.8844	−1.01741
$310$	0	0
$311$	10.4982	0.595297	0.297649	−	0.954675i	$-0.403798\pi$
$311$	10.4982	0.595297	0.297649	+	0.954675i	$0.403798\pi$
$312$	0	0
$313$	24.7386	1.39831	0.699154	−	0.714971i	$-0.253560\pi$
$313$	24.7386	1.39831	0.699154	+	0.714971i	$0.253560\pi$
$314$	0	0
$315$	−6.18178	−0.348304
$316$	0	0
$317$	3.85429	0.216478	0.108239	−	0.994125i	$-0.465479\pi$
$317$	3.85429	0.216478	0.108239	+	0.994125i	$0.465479\pi$
$318$	0	0
$319$	−20.4532	−1.14516
$320$	0	0
$321$	8.38542	0.468029
$322$	0	0
$323$	−8.84770	−0.492299
$324$	0	0
$325$	−9.29651	−0.515678
$326$	0	0
$327$	0.932415	0.0515627
$328$	0	0
$329$	−4.94941	−0.272870
$330$	0	0
$331$	−14.4529	−0.794405	−0.397203	−	0.917731i	$-0.630019\pi$
$331$	−14.4529	−0.794405	−0.397203	+	0.917731i	$0.630019\pi$
$332$	0	0
$333$	−5.32365	−0.291734
$334$	0	0
$335$	63.4136	3.46466
$336$	0	0
$337$	−20.1456	−1.09740	−0.548702	−	0.836018i	$-0.684878\pi$
$337$	−20.1456	−1.09740	−0.548702	+	0.836018i	$0.684878\pi$
$338$	0	0
$339$	−13.4832	−0.732307
$340$	0	0
$341$	−30.6919	−1.66206
$342$	0	0
$343$	−17.7822	−0.960146
$344$	0	0
$345$	−33.5501	−1.80628
$346$	0	0
$347$	2.01744	0.108302	0.0541510	−	0.998533i	$-0.482755\pi$
$347$	2.01744	0.108302	0.0541510	+	0.998533i	$0.482755\pi$
$348$	0	0
$349$	−8.37224	−0.448156	−0.224078	−	0.974571i	$-0.571937\pi$
$349$	−8.37224	−0.448156	−0.224078	+	0.974571i	$0.571937\pi$
$350$	0	0
$351$	−0.808587	−0.0431592
$352$	0	0
$353$	18.4168	0.980227	0.490113	−	0.871659i	$-0.336955\pi$
$353$	18.4168	0.980227	0.490113	+	0.871659i	$0.336955\pi$
$354$	0	0
$355$	1.47162	0.0781057
$356$	0	0
$357$	−2.25900	−0.119559
$358$	0	0
$359$	−3.18725	−0.168217	−0.0841083	−	0.996457i	$-0.526804\pi$
$359$	−3.18725	−0.168217	−0.0841083	+	0.996457i	$0.526804\pi$
$360$	0	0
$361$	16.5341	0.870217
$362$	0	0
$363$	3.06368	0.160801
$364$	0	0
$365$	49.4180	2.58666
$366$	0	0
$367$	−4.27091	−0.222940	−0.111470	−	0.993768i	$-0.535556\pi$
$367$	−4.27091	−0.222940	−0.111470	+	0.993768i	$0.535556\pi$
$368$	0	0
$369$	5.14543	0.267860
$370$	0	0
$371$	11.5178	0.597977
$372$	0	0
$373$	13.1400	0.680361	0.340181	−	0.940360i	$-0.389512\pi$
$373$	13.1400	0.680361	0.340181	+	0.940360i	$0.389512\pi$
$374$	0	0
$375$	−26.3897	−1.36276
$376$	0	0
$377$	4.41001	0.227127
$378$	0	0
$379$	20.8851	1.07280	0.536398	−	0.843965i	$-0.319785\pi$
$379$	20.8851	1.07280	0.536398	+	0.843965i	$0.319785\pi$
$380$	0	0
$381$	19.8720	1.01807
$382$	0	0
$383$	−29.8312	−1.52430	−0.762152	−	0.647398i	$-0.775857\pi$
$383$	−29.8312	−1.52430	−0.762152	+	0.647398i	$0.775857\pi$
$384$	0	0
$385$	−23.1827	−1.18150
$386$	0	0
$387$	5.00707	0.254524
$388$	0	0
$389$	−30.6104	−1.55201	−0.776004	−	0.630728i	$-0.782756\pi$
$389$	−30.6104	−1.55201	−0.776004	+	0.630728i	$0.782756\pi$
$390$	0	0
$391$	−12.2602	−0.620023
$392$	0	0
$393$	20.4780	1.03298
$394$	0	0
$395$	48.2313	2.42678
$396$	0	0
$397$	22.7310	1.14083	0.570417	−	0.821355i	$-0.306782\pi$
$397$	22.7310	1.14083	0.570417	+	0.821355i	$0.306782\pi$
$398$	0	0
$399$	9.07259	0.454197
$400$	0	0
$401$	14.1345	0.705841	0.352920	−	0.935653i	$-0.385189\pi$
$401$	14.1345	0.705841	0.352920	+	0.935653i	$0.385189\pi$
$402$	0	0
$403$	6.61762	0.329647
$404$	0	0
$405$	−4.06168	−0.201826
$406$	0	0
$407$	−19.9645	−0.989605
$408$	0	0
$409$	21.7975	1.07782	0.538908	−	0.842365i	$-0.318837\pi$
$409$	21.7975	1.07782	0.538908	+	0.842365i	$0.318837\pi$
$410$	0	0
$411$	−1.96583	−0.0969673
$412$	0	0
$413$	21.6221	1.06395
$414$	0	0
$415$	−10.3950	−0.510272
$416$	0	0
$417$	−3.35402	−0.164247
$418$	0	0
$419$	−39.5721	−1.93322	−0.966612	−	0.256245i	$-0.917515\pi$
$419$	−39.5721	−1.93322	−0.966612	+	0.256245i	$0.917515\pi$
$420$	0	0
$421$	20.4056	0.994510	0.497255	−	0.867604i	$-0.334341\pi$
$421$	20.4056	0.994510	0.497255	+	0.867604i	$0.334341\pi$
$422$	0	0
$423$	−3.25196	−0.158116
$424$	0	0
$425$	−17.0648	−0.827764
$426$	0	0
$427$	4.44301	0.215012
$428$	0	0
$429$	−3.03233	−0.146402
$430$	0	0
$431$	37.2385	1.79371	0.896857	−	0.442321i	$-0.145845\pi$
$431$	37.2385	1.79371	0.896857	+	0.442321i	$0.145845\pi$
$432$	0	0
$433$	−11.6179	−0.558322	−0.279161	−	0.960244i	$-0.590056\pi$
$433$	−11.6179	−0.558322	−0.279161	+	0.960244i	$0.590056\pi$
$434$	0	0
$435$	22.1523	1.06212
$436$	0	0
$437$	49.2393	2.35543
$438$	0	0
$439$	29.9288	1.42843	0.714213	−	0.699929i	$-0.246785\pi$
$439$	29.9288	1.42843	0.714213	+	0.699929i	$0.246785\pi$
$440$	0	0
$441$	−4.68358	−0.223028
$442$	0	0
$443$	26.7037	1.26873	0.634366	−	0.773033i	$-0.281261\pi$
$443$	26.7037	1.26873	0.634366	+	0.773033i	$0.281261\pi$
$444$	0	0
$445$	−47.4512	−2.24940
$446$	0	0
$447$	10.8386	0.512648
$448$	0	0
$449$	−9.31122	−0.439423	−0.219712	−	0.975565i	$-0.570512\pi$
$449$	−9.31122	−0.439423	−0.219712	+	0.975565i	$0.570512\pi$
$450$	0	0
$451$	19.2962	0.908621
$452$	0	0
$453$	9.05641	0.425507
$454$	0	0
$455$	4.99851	0.234334
$456$	0	0
$457$	−0.552255	−0.0258334	−0.0129167	−	0.999917i	$-0.504112\pi$
$457$	−0.552255	−0.0258334	−0.0129167	+	0.999917i	$0.504112\pi$
$458$	0	0
$459$	−1.48425	−0.0692789
$460$	0	0
$461$	20.1939	0.940525	0.470263	−	0.882527i	$-0.344159\pi$
$461$	20.1939	0.940525	0.470263	+	0.882527i	$0.344159\pi$
$462$	0	0
$463$	−25.7307	−1.19581	−0.597903	−	0.801568i	$-0.703999\pi$
$463$	−25.7307	−1.19581	−0.597903	+	0.801568i	$0.703999\pi$
$464$	0	0
$465$	33.2415	1.54154
$466$	0	0
$467$	9.08232	0.420280	0.210140	−	0.977671i	$-0.432608\pi$
$467$	9.08232	0.420280	0.210140	+	0.977671i	$0.432608\pi$
$468$	0	0
$469$	−23.7621	−1.09723
$470$	0	0
$471$	16.1306	0.743259
$472$	0	0
$473$	18.7773	0.863382
$474$	0	0
$475$	68.5356	3.14463
$476$	0	0
$477$	7.56769	0.346501
$478$	0	0
$479$	−19.2393	−0.879064	−0.439532	−	0.898227i	$-0.644856\pi$
$479$	−19.2393	−0.879064	−0.439532	+	0.898227i	$0.644856\pi$
$480$	0	0
$481$	4.30464	0.196275
$482$	0	0
$483$	12.5718	0.572036
$484$	0	0
$485$	−26.5823	−1.20704
$486$	0	0
$487$	3.48245	0.157805	0.0789024	−	0.996882i	$-0.474858\pi$
$487$	3.48245	0.157805	0.0789024	+	0.996882i	$0.474858\pi$
$488$	0	0
$489$	−5.30832	−0.240051
$490$	0	0
$491$	25.3570	1.14435	0.572173	−	0.820133i	$-0.306101\pi$
$491$	25.3570	1.14435	0.572173	+	0.820133i	$0.306101\pi$
$492$	0	0
$493$	8.09506	0.364583
$494$	0	0
$495$	−15.2319	−0.684624
$496$	0	0
$497$	−0.551441	−0.0247355
$498$	0	0
$499$	11.6852	0.523102	0.261551	−	0.965190i	$-0.415766\pi$
$499$	11.6852	0.523102	0.261551	+	0.965190i	$0.415766\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	9.88462	0.440734	0.220367	−	0.975417i	$-0.429275\pi$
$503$	9.88462	0.440734	0.220367	+	0.975417i	$0.429275\pi$
$504$	0	0
$505$	−71.3450	−3.17481
$506$	0	0
$507$	−12.3462	−0.548313
$508$	0	0
$509$	7.67964	0.340394	0.170197	−	0.985410i	$-0.445560\pi$
$509$	7.67964	0.340394	0.170197	+	0.985410i	$0.445560\pi$
$510$	0	0
$511$	−18.5177	−0.819177
$512$	0	0
$513$	5.96105	0.263187
$514$	0	0
$515$	72.6406	3.20093
$516$	0	0
$517$	−12.1954	−0.536351
$518$	0	0
$519$	8.04182	0.352997
$520$	0	0
$521$	−17.9007	−0.784242	−0.392121	−	0.919914i	$-0.628259\pi$
$521$	−17.9007	−0.784242	−0.392121	+	0.919914i	$0.628259\pi$
$522$	0	0
$523$	43.7100	1.91131	0.955653	−	0.294495i	$-0.0951516\pi$
$523$	43.7100	1.91131	0.955653	+	0.294495i	$0.0951516\pi$
$524$	0	0
$525$	17.4985	0.763698
$526$	0	0
$527$	12.1474	0.529148
$528$	0	0
$529$	45.2303	1.96654
$530$	0	0
$531$	14.2066	0.616513
$532$	0	0
$533$	−4.16053	−0.180212
$534$	0	0
$535$	−34.0589	−1.47249
$536$	0	0
$537$	−13.8439	−0.597409
$538$	0	0
$539$	−17.5642	−0.756543
$540$	0	0
$541$	6.15894	0.264794	0.132397	−	0.991197i	$-0.457733\pi$
$541$	6.15894	0.264794	0.132397	+	0.991197i	$0.457733\pi$
$542$	0	0
$543$	−0.189929	−0.00815062
$544$	0	0
$545$	−3.78717	−0.162225
$546$	0	0
$547$	−35.9938	−1.53898	−0.769492	−	0.638656i	$-0.779491\pi$
$547$	−35.9938	−1.53898	−0.769492	+	0.638656i	$0.779491\pi$
$548$	0	0
$549$	2.91923	0.124590
$550$	0	0
$551$	−32.5114	−1.38503
$552$	0	0
$553$	−18.0731	−0.768545
$554$	0	0
$555$	21.6230	0.917844
$556$	0	0
$557$	−16.1340	−0.683620	−0.341810	−	0.939769i	$-0.611040\pi$
$557$	−16.1340	−0.683620	−0.341810	+	0.939769i	$0.611040\pi$
$558$	0	0
$559$	−4.04866	−0.171240
$560$	0	0
$561$	−5.56618	−0.235004
$562$	0	0
$563$	1.71176	0.0721421	0.0360711	−	0.999349i	$-0.488516\pi$
$563$	1.71176	0.0721421	0.0360711	+	0.999349i	$0.488516\pi$
$564$	0	0
$565$	54.7644	2.30396
$566$	0	0
$567$	1.52198	0.0639170
$568$	0	0
$569$	23.0289	0.965422	0.482711	−	0.875780i	$-0.339652\pi$
$569$	23.0289	0.965422	0.482711	+	0.875780i	$0.339652\pi$
$570$	0	0
$571$	−6.81798	−0.285324	−0.142662	−	0.989771i	$-0.545566\pi$
$571$	−6.81798	−0.285324	−0.142662	+	0.989771i	$0.545566\pi$
$572$	0	0
$573$	−5.16497	−0.215770
$574$	0	0
$575$	94.9690	3.96048
$576$	0	0
$577$	13.5267	0.563124	0.281562	−	0.959543i	$-0.409147\pi$
$577$	13.5267	0.563124	0.281562	+	0.959543i	$0.409147\pi$
$578$	0	0
$579$	14.4877	0.602086
$580$	0	0
$581$	3.89519	0.161600
$582$	0	0
$583$	28.3800	1.17538
$584$	0	0
$585$	3.28422	0.135786
$586$	0	0
$587$	5.89395	0.243269	0.121635	−	0.992575i	$-0.461186\pi$
$587$	5.89395	0.243269	0.121635	+	0.992575i	$0.461186\pi$
$588$	0	0
$589$	−48.7863	−2.01020
$590$	0	0
$591$	−7.95294	−0.327140
$592$	0	0
$593$	−23.5539	−0.967244	−0.483622	−	0.875277i	$-0.660679\pi$
$593$	−23.5539	−0.967244	−0.483622	+	0.875277i	$0.660679\pi$
$594$	0	0
$595$	9.17532	0.376152
$596$	0	0
$597$	13.0286	0.533226
$598$	0	0
$599$	−10.0084	−0.408931	−0.204466	−	0.978874i	$-0.565546\pi$
$599$	−10.0084	−0.408931	−0.204466	+	0.978874i	$0.565546\pi$
$600$	0	0
$601$	−6.20219	−0.252993	−0.126496	−	0.991967i	$-0.540373\pi$
$601$	−6.20219	−0.252993	−0.126496	+	0.991967i	$0.540373\pi$
$602$	0	0
$603$	−15.6127	−0.635797
$604$	0	0
$605$	−12.4437	−0.505907
$606$	0	0
$607$	20.7183	0.840931	0.420466	−	0.907308i	$-0.361867\pi$
$607$	20.7183	0.840931	0.420466	+	0.907308i	$0.361867\pi$
$608$	0	0
$609$	−8.30082	−0.336366
$610$	0	0
$611$	2.62949	0.106378
$612$	0	0
$613$	−38.1939	−1.54264	−0.771319	−	0.636449i	$-0.780403\pi$
$613$	−38.1939	−1.54264	−0.771319	+	0.636449i	$0.780403\pi$
$614$	0	0
$615$	−20.8991	−0.842732
$616$	0	0
$617$	22.1762	0.892781	0.446391	−	0.894838i	$-0.352709\pi$
$617$	22.1762	0.892781	0.446391	+	0.894838i	$0.352709\pi$
$618$	0	0
$619$	20.0203	0.804684	0.402342	−	0.915489i	$-0.368196\pi$
$619$	20.0203	0.804684	0.402342	+	0.915489i	$0.368196\pi$
$620$	0	0
$621$	8.26017	0.331469
$622$	0	0
$623$	17.7807	0.712370
$624$	0	0
$625$	49.7001	1.98801
$626$	0	0
$627$	22.3549	0.892768
$628$	0	0
$629$	7.90164	0.315059
$630$	0	0
$631$	−6.74136	−0.268369	−0.134185	−	0.990956i	$-0.542842\pi$
$631$	−6.74136	−0.268369	−0.134185	+	0.990956i	$0.542842\pi$
$632$	0	0
$633$	16.3340	0.649217
$634$	0	0
$635$	−80.7135	−3.20302
$636$	0	0
$637$	3.78709	0.150050
$638$	0	0
$639$	−0.362319	−0.0143331
$640$	0	0
$641$	−14.5423	−0.574386	−0.287193	−	0.957873i	$-0.592722\pi$
$641$	−14.5423	−0.574386	−0.287193	+	0.957873i	$0.592722\pi$
$642$	0	0
$643$	15.1380	0.596985	0.298493	−	0.954412i	$-0.403516\pi$
$643$	15.1380	0.596985	0.298493	+	0.954412i	$0.403516\pi$
$644$	0	0
$645$	−20.3371	−0.800774
$646$	0	0
$647$	−36.0389	−1.41683	−0.708417	−	0.705794i	$-0.750591\pi$
$647$	−36.0389	−1.41683	−0.708417	+	0.705794i	$0.750591\pi$
$648$	0	0
$649$	53.2769	2.09130
$650$	0	0
$651$	−12.4561	−0.488194
$652$	0	0
$653$	12.9667	0.507428	0.253714	−	0.967279i	$-0.418348\pi$
$653$	12.9667	0.507428	0.253714	+	0.967279i	$0.418348\pi$
$654$	0	0
$655$	−83.1752	−3.24993
$656$	0	0
$657$	−12.1669	−0.474676
$658$	0	0
$659$	−47.8284	−1.86313	−0.931564	−	0.363577i	$-0.881556\pi$
$659$	−47.8284	−1.86313	−0.931564	+	0.363577i	$0.881556\pi$
$660$	0	0
$661$	17.8577	0.694582	0.347291	−	0.937757i	$-0.387102\pi$
$661$	17.8577	0.694582	0.347291	+	0.937757i	$0.387102\pi$
$662$	0	0
$663$	1.20015	0.0466098
$664$	0	0
$665$	−36.8499	−1.42898
$666$	0	0
$667$	−45.0507	−1.74437
$668$	0	0
$669$	29.4549	1.13879
$670$	0	0
$671$	10.9476	0.422627
$672$	0	0
$673$	−42.6031	−1.64223	−0.821115	−	0.570763i	$-0.806647\pi$
$673$	−42.6031	−1.64223	−0.821115	+	0.570763i	$0.806647\pi$
$674$	0	0
$675$	11.4972	0.442529
$676$	0	0
$677$	12.8028	0.492053	0.246027	−	0.969263i	$-0.420875\pi$
$677$	12.8028	0.492053	0.246027	+	0.969263i	$0.420875\pi$
$678$	0	0
$679$	9.96083	0.382261
$680$	0	0
$681$	11.2229	0.430062
$682$	0	0
$683$	29.7459	1.13819	0.569097	−	0.822270i	$-0.307293\pi$
$683$	29.7459	1.13819	0.569097	+	0.822270i	$0.307293\pi$
$684$	0	0
$685$	7.98457	0.305075
$686$	0	0
$687$	−2.35856	−0.0899848
$688$	0	0
$689$	−6.11913	−0.233120
$690$	0	0
$691$	−41.7414	−1.58792	−0.793959	−	0.607972i	$-0.791983\pi$
$691$	−41.7414	−1.58792	−0.793959	+	0.607972i	$0.791983\pi$
$692$	0	0
$693$	5.70765	0.216816
$694$	0	0
$695$	13.6230	0.516748
$696$	0	0
$697$	−7.63711	−0.289276
$698$	0	0
$699$	−19.6121	−0.741797
$700$	0	0
$701$	−30.4690	−1.15080	−0.575399	−	0.817873i	$-0.695153\pi$
$701$	−30.4690	−1.15080	−0.575399	+	0.817873i	$0.695153\pi$
$702$	0	0
$703$	−31.7346	−1.19689
$704$	0	0
$705$	13.2084	0.497457
$706$	0	0
$707$	26.7342	1.00544
$708$	0	0
$709$	36.5858	1.37401	0.687005	−	0.726653i	$-0.258925\pi$
$709$	36.5858	1.37401	0.687005	+	0.726653i	$0.258925\pi$
$710$	0	0
$711$	−11.8747	−0.445337
$712$	0	0
$713$	−67.6026	−2.53174
$714$	0	0
$715$	12.3163	0.460605
$716$	0	0
$717$	5.61893	0.209843
$718$	0	0
$719$	−43.8911	−1.63686	−0.818430	−	0.574606i	$-0.805155\pi$
$719$	−43.8911	−1.63686	−0.818430	+	0.574606i	$0.805155\pi$
$720$	0	0
$721$	−27.2196	−1.01371
$722$	0	0
$723$	22.5680	0.839314
$724$	0	0
$725$	−62.7055	−2.32882
$726$	0	0
$727$	−19.9724	−0.740737	−0.370368	−	0.928885i	$-0.620769\pi$
$727$	−19.9724	−0.740737	−0.370368	+	0.928885i	$0.620769\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.43176	−0.274874
$732$	0	0
$733$	−6.84512	−0.252830	−0.126415	−	0.991977i	$-0.540347\pi$
$733$	−6.84512	−0.252830	−0.126415	+	0.991977i	$0.540347\pi$
$734$	0	0
$735$	19.0232	0.701682
$736$	0	0
$737$	−58.5499	−2.15671
$738$	0	0
$739$	24.6138	0.905434	0.452717	−	0.891654i	$-0.350455\pi$
$739$	24.6138	0.905434	0.452717	+	0.891654i	$0.350455\pi$
$740$	0	0
$741$	−4.82003	−0.177068
$742$	0	0
$743$	5.20444	0.190932	0.0954661	−	0.995433i	$-0.469566\pi$
$743$	5.20444	0.190932	0.0954661	+	0.995433i	$0.469566\pi$
$744$	0	0
$745$	−44.0229	−1.61287
$746$	0	0
$747$	2.55930	0.0936398
$748$	0	0
$749$	12.7624	0.466329
$750$	0	0
$751$	−29.2658	−1.06792	−0.533962	−	0.845508i	$-0.679298\pi$
$751$	−29.2658	−1.06792	−0.533962	+	0.845508i	$0.679298\pi$
$752$	0	0
$753$	−2.49552	−0.0909417
$754$	0	0
$755$	−36.7842	−1.33872
$756$	0	0
$757$	−33.9464	−1.23380	−0.616901	−	0.787041i	$-0.711612\pi$
$757$	−33.9464	−1.23380	−0.616901	+	0.787041i	$0.711612\pi$
$758$	0	0
$759$	30.9769	1.12439
$760$	0	0
$761$	23.6786	0.858348	0.429174	−	0.903222i	$-0.358805\pi$
$761$	23.6786	0.858348	0.429174	+	0.903222i	$0.358805\pi$
$762$	0	0
$763$	1.41911	0.0513754
$764$	0	0
$765$	6.02855	0.217963
$766$	0	0
$767$	−11.4873	−0.414781
$768$	0	0
$769$	18.8397	0.679378	0.339689	−	0.940538i	$-0.389678\pi$
$769$	18.8397	0.679378	0.339689	+	0.940538i	$0.389678\pi$
$770$	0	0
$771$	−25.7511	−0.927402
$772$	0	0
$773$	11.0546	0.397605	0.198803	−	0.980040i	$-0.436295\pi$
$773$	11.0546	0.397605	0.198803	+	0.980040i	$0.436295\pi$
$774$	0	0
$775$	−94.0953	−3.38000
$776$	0	0
$777$	−8.10248	−0.290675
$778$	0	0
$779$	30.6722	1.09894
$780$	0	0
$781$	−1.35875	−0.0486200
$782$	0	0
$783$	−5.45397	−0.194909
$784$	0	0
$785$	−65.5174	−2.33841
$786$	0	0
$787$	−20.9181	−0.745651	−0.372825	−	0.927902i	$-0.621611\pi$
$787$	−20.9181	−0.745651	−0.372825	+	0.927902i	$0.621611\pi$
$788$	0	0
$789$	19.8069	0.705144
$790$	0	0
$791$	−20.5211	−0.729647
$792$	0	0
$793$	−2.36045	−0.0838222
$794$	0	0
$795$	−30.7375	−1.09015
$796$	0	0
$797$	42.6335	1.51016	0.755078	−	0.655635i	$-0.227599\pi$
$797$	42.6335	1.51016	0.755078	+	0.655635i	$0.227599\pi$
$798$	0	0
$799$	4.82673	0.170757
$800$	0	0
$801$	11.6827	0.412786
$802$	0	0
$803$	−45.6278	−1.61017
$804$	0	0
$805$	−51.0626	−1.79972
$806$	0	0
$807$	3.54672	0.124851
$808$	0	0
$809$	5.21586	0.183380	0.0916899	−	0.995788i	$-0.470773\pi$
$809$	5.21586	0.183380	0.0916899	+	0.995788i	$0.470773\pi$
$810$	0	0
$811$	−25.3972	−0.891817	−0.445908	−	0.895079i	$-0.647119\pi$
$811$	−25.3972	−0.891817	−0.445908	+	0.895079i	$0.647119\pi$
$812$	0	0
$813$	−13.6862	−0.479997
$814$	0	0
$815$	21.5607	0.755238
$816$	0	0
$817$	29.8474	1.04423
$818$	0	0
$819$	−1.23065	−0.0430024
$820$	0	0
$821$	−8.54482	−0.298216	−0.149108	−	0.988821i	$-0.547640\pi$
$821$	−8.54482	−0.298216	−0.149108	+	0.988821i	$0.547640\pi$
$822$	0	0
$823$	−15.5315	−0.541393	−0.270697	−	0.962665i	$-0.587254\pi$
$823$	−15.5315	−0.541393	−0.270697	+	0.962665i	$0.587254\pi$
$824$	0	0
$825$	43.1164	1.50112
$826$	0	0
$827$	−31.4424	−1.09336	−0.546680	−	0.837341i	$-0.684109\pi$
$827$	−31.4424	−1.09336	−0.546680	+	0.837341i	$0.684109\pi$
$828$	0	0
$829$	−29.1091	−1.01100	−0.505501	−	0.862826i	$-0.668692\pi$
$829$	−29.1091	−1.01100	−0.505501	+	0.862826i	$0.668692\pi$
$830$	0	0
$831$	−2.28135	−0.0791393
$832$	0	0
$833$	6.95162	0.240859
$834$	0	0
$835$	−4.06168	−0.140560
$836$	0	0
$837$	−8.18417	−0.282886
$838$	0	0
$839$	−1.39747	−0.0482461	−0.0241231	−	0.999709i	$-0.507679\pi$
$839$	−1.39747	−0.0482461	−0.0241231	+	0.999709i	$0.507679\pi$
$840$	0	0
$841$	0.745762	0.0257159
$842$	0	0
$843$	24.9176	0.858208
$844$	0	0
$845$	50.1462	1.72508
$846$	0	0
$847$	4.66285	0.160217
$848$	0	0
$849$	12.7901	0.438956
$850$	0	0
$851$	−43.9743	−1.50742
$852$	0	0
$853$	−31.5987	−1.08192	−0.540959	−	0.841049i	$-0.681938\pi$
$853$	−31.5987	−1.08192	−0.540959	+	0.841049i	$0.681938\pi$
$854$	0	0
$855$	−24.2119	−0.828029
$856$	0	0
$857$	7.76282	0.265173	0.132586	−	0.991171i	$-0.457672\pi$
$857$	7.76282	0.265173	0.132586	+	0.991171i	$0.457672\pi$
$858$	0	0
$859$	−20.9777	−0.715750	−0.357875	−	0.933769i	$-0.616499\pi$
$859$	−20.9777	−0.715750	−0.357875	+	0.933769i	$0.616499\pi$
$860$	0	0
$861$	7.83122	0.266887
$862$	0	0
$863$	−4.46954	−0.152145	−0.0760725	−	0.997102i	$-0.524238\pi$
$863$	−4.46954	−0.152145	−0.0760725	+	0.997102i	$0.524238\pi$
$864$	0	0
$865$	−32.6633	−1.11059
$866$	0	0
$867$	−14.7970	−0.502532
$868$	0	0
$869$	−44.5321	−1.51065
$870$	0	0
$871$	12.6242	0.427755
$872$	0	0
$873$	6.54466	0.221503
$874$	0	0
$875$	−40.1645	−1.35781
$876$	0	0
$877$	−6.68629	−0.225780	−0.112890	−	0.993607i	$-0.536011\pi$
$877$	−6.68629	−0.225780	−0.112890	+	0.993607i	$0.536011\pi$
$878$	0	0
$879$	10.7252	0.361751
$880$	0	0
$881$	35.7196	1.20342	0.601711	−	0.798714i	$-0.294486\pi$
$881$	35.7196	1.20342	0.601711	+	0.798714i	$0.294486\pi$
$882$	0	0
$883$	1.00066	0.0336747	0.0168374	−	0.999858i	$-0.494640\pi$
$883$	1.00066	0.0336747	0.0168374	+	0.999858i	$0.494640\pi$
$884$	0	0
$885$	−57.7026	−1.93965
$886$	0	0
$887$	2.70835	0.0909374	0.0454687	−	0.998966i	$-0.485522\pi$
$887$	2.70835	0.0909374	0.0454687	+	0.998966i	$0.485522\pi$
$888$	0	0
$889$	30.2447	1.01437
$890$	0	0
$891$	3.75016	0.125635
$892$	0	0
$893$	−19.3851	−0.648697
$894$	0	0
$895$	56.2295	1.87955
$896$	0	0
$897$	−6.67906	−0.223007
$898$	0	0
$899$	44.6362	1.48870
$900$	0	0
$901$	−11.2324	−0.374204
$902$	0	0
$903$	7.62065	0.253600
$904$	0	0
$905$	0.771429	0.0256432
$906$	0	0
$907$	21.5715	0.716268	0.358134	−	0.933670i	$-0.383413\pi$
$907$	21.5715	0.716268	0.358134	+	0.933670i	$0.383413\pi$
$908$	0	0
$909$	17.5654	0.582608
$910$	0	0
$911$	−59.3148	−1.96519	−0.982594	−	0.185767i	$-0.940523\pi$
$911$	−59.3148	−1.96519	−0.982594	+	0.185767i	$0.940523\pi$
$912$	0	0
$913$	9.59776	0.317640
$914$	0	0
$915$	−11.8570	−0.391980
$916$	0	0
$917$	31.1671	1.02923
$918$	0	0
$919$	16.6875	0.550471	0.275236	−	0.961377i	$-0.411244\pi$
$919$	16.6875	0.550471	0.275236	+	0.961377i	$0.411244\pi$
$920$	0	0
$921$	15.8870	0.523495
$922$	0	0
$923$	0.292967	0.00964311
$924$	0	0
$925$	−61.2073	−2.01248
$926$	0	0
$927$	−17.8844	−0.587400
$928$	0	0
$929$	25.5294	0.837592	0.418796	−	0.908080i	$-0.362452\pi$
$929$	25.5294	0.837592	0.418796	+	0.908080i	$0.362452\pi$
$930$	0	0
$931$	−27.9191	−0.915011
$932$	0	0
$933$	10.4982	0.343695
$934$	0	0
$935$	22.6080	0.739361
$936$	0	0
$937$	5.94170	0.194107	0.0970535	−	0.995279i	$-0.469058\pi$
$937$	5.94170	0.194107	0.0970535	+	0.995279i	$0.469058\pi$
$938$	0	0
$939$	24.7386	0.807314
$940$	0	0
$941$	57.9869	1.89032	0.945159	−	0.326609i	$-0.105906\pi$
$941$	57.9869	1.89032	0.945159	+	0.326609i	$0.105906\pi$
$942$	0	0
$943$	42.5021	1.38406
$944$	0	0
$945$	−6.18178	−0.201093
$946$	0	0
$947$	−31.4862	−1.02316	−0.511582	−	0.859235i	$-0.670940\pi$
$947$	−31.4862	−1.02316	−0.511582	+	0.859235i	$0.670940\pi$
$948$	0	0
$949$	9.83800	0.319355
$950$	0	0
$951$	3.85429	0.124984
$952$	0	0
$953$	−3.63227	−0.117661	−0.0588304	−	0.998268i	$-0.518737\pi$
$953$	−3.63227	−0.117661	−0.0588304	+	0.998268i	$0.518737\pi$
$954$	0	0
$955$	20.9785	0.678847
$956$	0	0
$957$	−20.4532	−0.661159
$958$	0	0
$959$	−2.99195	−0.0966151
$960$	0	0
$961$	35.9807	1.16067
$962$	0	0
$963$	8.38542	0.270216
$964$	0	0
$965$	−58.8442	−1.89426
$966$	0	0
$967$	41.2944	1.32794	0.663970	−	0.747760i	$-0.268870\pi$
$967$	41.2944	1.32794	0.663970	+	0.747760i	$0.268870\pi$
$968$	0	0
$969$	−8.84770	−0.284229
$970$	0	0
$971$	−11.5508	−0.370684	−0.185342	−	0.982674i	$-0.559339\pi$
$971$	−11.5508	−0.370684	−0.185342	+	0.982674i	$0.559339\pi$
$972$	0	0
$973$	−5.10475	−0.163651
$974$	0	0
$975$	−9.29651	−0.297727
$976$	0	0
$977$	−3.65595	−0.116964	−0.0584821	−	0.998288i	$-0.518626\pi$
$977$	−3.65595	−0.116964	−0.0584821	+	0.998288i	$0.518626\pi$
$978$	0	0
$979$	43.8118	1.40023
$980$	0	0
$981$	0.932415	0.0297697
$982$	0	0
$983$	12.3408	0.393610	0.196805	−	0.980443i	$-0.436943\pi$
$983$	12.3408	0.393610	0.196805	+	0.980443i	$0.436943\pi$
$984$	0	0
$985$	32.3023	1.02924
$986$	0	0
$987$	−4.94941	−0.157541
$988$	0	0
$989$	41.3593	1.31515
$990$	0	0
$991$	−34.7724	−1.10458	−0.552291	−	0.833651i	$-0.686246\pi$
$991$	−34.7724	−1.10458	−0.552291	+	0.833651i	$0.686246\pi$
$992$	0	0
$993$	−14.4529	−0.458650
$994$	0	0
$995$	−52.9181	−1.67762
$996$	0	0
$997$	47.6009	1.50754	0.753768	−	0.657141i	$-0.228234\pi$
$997$	47.6009	1.50754	0.753768	+	0.657141i	$0.228234\pi$
$998$	0	0
$999$	−5.32365	−0.168433

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bc.1.1		9
4.3	odd	2		2004.2.a.c.1.1	✓	9
12.11	even	2		6012.2.a.i.1.9		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.c.1.1	✓	9	4.3	odd	2
6012.2.a.i.1.9		9	12.11	even	2
8016.2.a.bc.1.1		9	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.06168 q^{5} +1.52198 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.06168 q^{5} +1.52198 q^{7} +1.00000 q^{9} +3.75016 q^{11} -0.808587 q^{13} -4.06168 q^{15} -1.48425 q^{17} +5.96105 q^{19} +1.52198 q^{21} +8.26017 q^{23} +11.4972 q^{25} +1.00000 q^{27} -5.45397 q^{29} -8.18417 q^{31} +3.75016 q^{33} -6.18178 q^{35} -5.32365 q^{37} -0.808587 q^{39} +5.14543 q^{41} +5.00707 q^{43} -4.06168 q^{45} -3.25196 q^{47} -4.68358 q^{49} -1.48425 q^{51} +7.56769 q^{53} -15.2319 q^{55} +5.96105 q^{57} +14.2066 q^{59} +2.91923 q^{61} +1.52198 q^{63} +3.28422 q^{65} -15.6127 q^{67} +8.26017 q^{69} -0.362319 q^{71} -12.1669 q^{73} +11.4972 q^{75} +5.70765 q^{77} -11.8747 q^{79} +1.00000 q^{81} +2.55930 q^{83} +6.02855 q^{85} -5.45397 q^{87} +11.6827 q^{89} -1.23065 q^{91} -8.18417 q^{93} -24.2119 q^{95} +6.54466 q^{97} +3.75016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} + q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 + q^5 - 2 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} + q^{5} - 2 q^{7} + 9 q^{9} + 9 q^{11} + 10 q^{13} + q^{15} + 7 q^{17} + 2 q^{19} - 2 q^{21} + 3 q^{23} + 18 q^{25} + 9 q^{27} + 5 q^{29} - 12 q^{31} + 9 q^{33} + 6 q^{35} + 15 q^{37} + 10 q^{39} + 14 q^{41} - 6 q^{43} + q^{45} + 3 q^{47} + 27 q^{49} + 7 q^{51} + 9 q^{53} - 19 q^{55} + 2 q^{57} + 9 q^{59} + 30 q^{61} - 2 q^{63} + 28 q^{65} - 16 q^{67} + 3 q^{69} + 3 q^{71} + 32 q^{73} + 18 q^{75} + 18 q^{77} - 24 q^{79} + 9 q^{81} + 3 q^{83} + 37 q^{85} + 5 q^{87} + 46 q^{89} - 33 q^{91} - 12 q^{93} - 11 q^{95} + 43 q^{97} + 9 q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 + q^5 - 2 * q^7 + 9 * q^9 + 9 * q^11 + 10 * q^13 + q^15 + 7 * q^17 + 2 * q^19 - 2 * q^21 + 3 * q^23 + 18 * q^25 + 9 * q^27 + 5 * q^29 - 12 * q^31 + 9 * q^33 + 6 * q^35 + 15 * q^37 + 10 * q^39 + 14 * q^41 - 6 * q^43 + q^45 + 3 * q^47 + 27 * q^49 + 7 * q^51 + 9 * q^53 - 19 * q^55 + 2 * q^57 + 9 * q^59 + 30 * q^61 - 2 * q^63 + 28 * q^65 - 16 * q^67 + 3 * q^69 + 3 * q^71 + 32 * q^73 + 18 * q^75 + 18 * q^77 - 24 * q^79 + 9 * q^81 + 3 * q^83 + 37 * q^85 + 5 * q^87 + 46 * q^89 - 33 * q^91 - 12 * q^93 - 11 * q^95 + 43 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more