LMFDB - Embedded newform 4008.2.a.i.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, 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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$1$
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} - 3x^{8} - 16x^{7} + 45x^{6} + 67x^{5} - 166x^{4} - 83x^{3} + 152x^{2} + 51x - 10 $ x^9 - 3x^8 - 16x^7 + 45x^6 + 67x^5 - 166x^4 - 83x^3 + 152x^2 + 51x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.58856$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.588564 q^{5} -1.23518 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.588564 q^{5} -1.23518 q^{7} +1.00000 q^{9} -5.01763 q^{11} +4.86230 q^{13} +0.588564 q^{15} -6.70724 q^{17} -1.72560 q^{19} -1.23518 q^{21} +3.90691 q^{23} -4.65359 q^{25} +1.00000 q^{27} +8.98846 q^{29} -5.36898 q^{31} -5.01763 q^{33} -0.726984 q^{35} +1.07031 q^{37} +4.86230 q^{39} +6.22415 q^{41} -9.74169 q^{43} +0.588564 q^{45} +3.85840 q^{47} -5.47432 q^{49} -6.70724 q^{51} -7.57769 q^{53} -2.95320 q^{55} -1.72560 q^{57} -8.99297 q^{59} -4.59852 q^{61} -1.23518 q^{63} +2.86177 q^{65} -15.2918 q^{67} +3.90691 q^{69} -6.37962 q^{71} +8.12499 q^{73} -4.65359 q^{75} +6.19770 q^{77} -6.10983 q^{79} +1.00000 q^{81} +3.46966 q^{83} -3.94764 q^{85} +8.98846 q^{87} +5.15759 q^{89} -6.00583 q^{91} -5.36898 q^{93} -1.01563 q^{95} -13.6764 q^{97} -5.01763 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} - 6 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 - 6 * q^5 - 11 * q^7 + 9 * q^9	$ 9 q + 9 q^{3} - 6 q^{5} - 11 q^{7} + 9 q^{9} + q^{11} - 4 q^{13} - 6 q^{15} - 9 q^{17} - 8 q^{19} - 11 q^{21} - 7 q^{23} - q^{25} + 9 q^{27} - 9 q^{29} - 25 q^{31} + q^{33} + 5 q^{35} - 6 q^{37} - 4 q^{39} - 4 q^{41} - 24 q^{43} - 6 q^{45} - 16 q^{47} + 4 q^{49} - 9 q^{51} - 26 q^{53} - 29 q^{55} - 8 q^{57} - 4 q^{59} - 20 q^{61} - 11 q^{63} - 8 q^{65} - 25 q^{67} - 7 q^{69} - 15 q^{71} - 10 q^{73} - q^{75} - 20 q^{77} - 34 q^{79} + 9 q^{81} - 4 q^{83} - 13 q^{85} - 9 q^{87} + 13 q^{89} - 21 q^{91} - 25 q^{93} - 7 q^{95} - 4 q^{97} + q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 - 6 * q^5 - 11 * q^7 + 9 * q^9 + q^11 - 4 * q^13 - 6 * q^15 - 9 * q^17 - 8 * q^19 - 11 * q^21 - 7 * q^23 - q^25 + 9 * q^27 - 9 * q^29 - 25 * q^31 + q^33 + 5 * q^35 - 6 * q^37 - 4 * q^39 - 4 * q^41 - 24 * q^43 - 6 * q^45 - 16 * q^47 + 4 * q^49 - 9 * q^51 - 26 * q^53 - 29 * q^55 - 8 * q^57 - 4 * q^59 - 20 * q^61 - 11 * q^63 - 8 * q^65 - 25 * q^67 - 7 * q^69 - 15 * q^71 - 10 * q^73 - q^75 - 20 * q^77 - 34 * q^79 + 9 * q^81 - 4 * q^83 - 13 * q^85 - 9 * q^87 + 13 * q^89 - 21 * q^91 - 25 * q^93 - 7 * q^95 - 4 * q^97 + 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
$5$	0.588564	0.263214	0.131607	−	0.991302i	$-0.457986\pi$
$5$	0.588564	0.263214	0.131607	+	0.991302i	$0.457986\pi$
$6$	0	0
$7$	−1.23518	−0.466855	−0.233428	−	0.972374i	$-0.574994\pi$
$7$	−1.23518	−0.466855	−0.233428	+	0.972374i	$0.574994\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.01763	−1.51287	−0.756437	−	0.654067i	$-0.773061\pi$
$11$	−5.01763	−1.51287	−0.756437	+	0.654067i	$0.773061\pi$
$12$	0	0
$13$	4.86230	1.34856	0.674280	−	0.738476i	$-0.264454\pi$
$13$	4.86230	1.34856	0.674280	+	0.738476i	$0.264454\pi$
$14$	0	0
$15$	0.588564	0.151966
$16$	0	0
$17$	−6.70724	−1.62675	−0.813373	−	0.581743i	$-0.802371\pi$
$17$	−6.70724	−1.62675	−0.813373	+	0.581743i	$0.802371\pi$
$18$	0	0
$19$	−1.72560	−0.395880	−0.197940	−	0.980214i	$-0.563425\pi$
$19$	−1.72560	−0.395880	−0.197940	+	0.980214i	$0.563425\pi$
$20$	0	0
$21$	−1.23518	−0.269539
$22$	0	0
$23$	3.90691	0.814648	0.407324	−	0.913284i	$-0.366462\pi$
$23$	3.90691	0.814648	0.407324	+	0.913284i	$0.366462\pi$
$24$	0	0
$25$	−4.65359	−0.930719
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.98846	1.66912	0.834558	−	0.550920i	$-0.185723\pi$
$29$	8.98846	1.66912	0.834558	+	0.550920i	$0.185723\pi$
$30$	0	0
$31$	−5.36898	−0.964297	−0.482149	−	0.876089i	$-0.660144\pi$
$31$	−5.36898	−0.964297	−0.482149	+	0.876089i	$0.660144\pi$
$32$	0	0
$33$	−5.01763	−0.873458
$34$	0	0
$35$	−0.726984	−0.122883
$36$	0	0
$37$	1.07031	0.175957	0.0879786	−	0.996122i	$-0.471959\pi$
$37$	1.07031	0.175957	0.0879786	+	0.996122i	$0.471959\pi$
$38$	0	0
$39$	4.86230	0.778591
$40$	0	0
$41$	6.22415	0.972050	0.486025	−	0.873945i	$-0.338446\pi$
$41$	6.22415	0.972050	0.486025	+	0.873945i	$0.338446\pi$
$42$	0	0
$43$	−9.74169	−1.48559	−0.742797	−	0.669517i	$-0.766501\pi$
$43$	−9.74169	−1.48559	−0.742797	+	0.669517i	$0.766501\pi$
$44$	0	0
$45$	0.588564	0.0877379
$46$	0	0
$47$	3.85840	0.562806	0.281403	−	0.959590i	$-0.409200\pi$
$47$	3.85840	0.562806	0.281403	+	0.959590i	$0.409200\pi$
$48$	0	0
$49$	−5.47432	−0.782046
$50$	0	0
$51$	−6.70724	−0.939202
$52$	0	0
$53$	−7.57769	−1.04088	−0.520438	−	0.853900i	$-0.674231\pi$
$53$	−7.57769	−1.04088	−0.520438	+	0.853900i	$0.674231\pi$
$54$	0	0
$55$	−2.95320	−0.398209
$56$	0	0
$57$	−1.72560	−0.228562
$58$	0	0
$59$	−8.99297	−1.17079	−0.585393	−	0.810750i	$-0.699060\pi$
$59$	−8.99297	−1.17079	−0.585393	+	0.810750i	$0.699060\pi$
$60$	0	0
$61$	−4.59852	−0.588780	−0.294390	−	0.955685i	$-0.595116\pi$
$61$	−4.59852	−0.588780	−0.294390	+	0.955685i	$0.595116\pi$
$62$	0	0
$63$	−1.23518	−0.155618
$64$	0	0
$65$	2.86177	0.354959
$66$	0	0
$67$	−15.2918	−1.86819	−0.934093	−	0.357029i	$-0.883790\pi$
$67$	−15.2918	−1.86819	−0.934093	+	0.357029i	$0.883790\pi$
$68$	0	0
$69$	3.90691	0.470337
$70$	0	0
$71$	−6.37962	−0.757122	−0.378561	−	0.925576i	$-0.623581\pi$
$71$	−6.37962	−0.757122	−0.378561	+	0.925576i	$0.623581\pi$
$72$	0	0
$73$	8.12499	0.950959	0.475479	−	0.879727i	$-0.342275\pi$
$73$	8.12499	0.950959	0.475479	+	0.879727i	$0.342275\pi$
$74$	0	0
$75$	−4.65359	−0.537351
$76$	0	0
$77$	6.19770	0.706293
$78$	0	0
$79$	−6.10983	−0.687410	−0.343705	−	0.939078i	$-0.611682\pi$
$79$	−6.10983	−0.687410	−0.343705	+	0.939078i	$0.611682\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.46966	0.380844	0.190422	−	0.981702i	$-0.439014\pi$
$83$	3.46966	0.380844	0.190422	+	0.981702i	$0.439014\pi$
$84$	0	0
$85$	−3.94764	−0.428182
$86$	0	0
$87$	8.98846	0.963664
$88$	0	0
$89$	5.15759	0.546703	0.273351	−	0.961914i	$-0.411868\pi$
$89$	5.15759	0.546703	0.273351	+	0.961914i	$0.411868\pi$
$90$	0	0
$91$	−6.00583	−0.629582
$92$	0	0
$93$	−5.36898	−0.556737
$94$	0	0
$95$	−1.01563	−0.104201
$96$	0	0
$97$	−13.6764	−1.38863	−0.694315	−	0.719671i	$-0.744293\pi$
$97$	−13.6764	−1.38863	−0.694315	+	0.719671i	$0.744293\pi$
$98$	0	0
$99$	−5.01763	−0.504291
$100$	0	0
$101$	−3.33213	−0.331559	−0.165780	−	0.986163i	$-0.553014\pi$
$101$	−3.33213	−0.331559	−0.165780	+	0.986163i	$0.553014\pi$
$102$	0	0
$103$	−1.59690	−0.157347	−0.0786737	−	0.996900i	$-0.525069\pi$
$103$	−1.59690	−0.157347	−0.0786737	+	0.996900i	$0.525069\pi$
$104$	0	0
$105$	−0.726984	−0.0709464
$106$	0	0
$107$	12.8858	1.24571	0.622857	−	0.782335i	$-0.285972\pi$
$107$	12.8858	1.24571	0.622857	+	0.782335i	$0.285972\pi$
$108$	0	0
$109$	−1.41783	−0.135804	−0.0679018	−	0.997692i	$-0.521630\pi$
$109$	−1.41783	−0.135804	−0.0679018	+	0.997692i	$0.521630\pi$
$110$	0	0
$111$	1.07031	0.101589
$112$	0	0
$113$	−18.0186	−1.69505	−0.847524	−	0.530757i	$-0.821908\pi$
$113$	−18.0186	−1.69505	−0.847524	+	0.530757i	$0.821908\pi$
$114$	0	0
$115$	2.29947	0.214426
$116$	0	0
$117$	4.86230	0.449520
$118$	0	0
$119$	8.28468	0.759455
$120$	0	0
$121$	14.1766	1.28879
$122$	0	0
$123$	6.22415	0.561213
$124$	0	0
$125$	−5.68175	−0.508191
$126$	0	0
$127$	19.7958	1.75659	0.878295	−	0.478120i	$-0.158682\pi$
$127$	19.7958	1.75659	0.878295	+	0.478120i	$0.158682\pi$
$128$	0	0
$129$	−9.74169	−0.857708
$130$	0	0
$131$	−1.29378	−0.113038	−0.0565191	−	0.998402i	$-0.518000\pi$
$131$	−1.29378	−0.113038	−0.0565191	+	0.998402i	$0.518000\pi$
$132$	0	0
$133$	2.13144	0.184819
$134$	0	0
$135$	0.588564	0.0506555
$136$	0	0
$137$	15.2142	1.29983	0.649917	−	0.760005i	$-0.274804\pi$
$137$	15.2142	1.29983	0.649917	+	0.760005i	$0.274804\pi$
$138$	0	0
$139$	−15.1674	−1.28649	−0.643243	−	0.765662i	$-0.722411\pi$
$139$	−15.1674	−1.28649	−0.643243	+	0.765662i	$0.722411\pi$
$140$	0	0
$141$	3.85840	0.324936
$142$	0	0
$143$	−24.3972	−2.04020
$144$	0	0
$145$	5.29028	0.439334
$146$	0	0
$147$	−5.47432	−0.451514
$148$	0	0
$149$	−14.4870	−1.18682	−0.593412	−	0.804899i	$-0.702220\pi$
$149$	−14.4870	−1.18682	−0.593412	+	0.804899i	$0.702220\pi$
$150$	0	0
$151$	−4.68695	−0.381419	−0.190709	−	0.981647i	$-0.561079\pi$
$151$	−4.68695	−0.381419	−0.190709	+	0.981647i	$0.561079\pi$
$152$	0	0
$153$	−6.70724	−0.542249
$154$	0	0
$155$	−3.15999	−0.253816
$156$	0	0
$157$	−22.2647	−1.77692	−0.888458	−	0.458958i	$-0.848223\pi$
$157$	−22.2647	−1.77692	−0.888458	+	0.458958i	$0.848223\pi$
$158$	0	0
$159$	−7.57769	−0.600950
$160$	0	0
$161$	−4.82576	−0.380323
$162$	0	0
$163$	−18.8338	−1.47517	−0.737587	−	0.675252i	$-0.764035\pi$
$163$	−18.8338	−1.47517	−0.737587	+	0.675252i	$0.764035\pi$
$164$	0	0
$165$	−2.95320	−0.229906
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	10.6420	0.818612
$170$	0	0
$171$	−1.72560	−0.131960
$172$	0	0
$173$	−0.471351	−0.0358362	−0.0179181	−	0.999839i	$-0.505704\pi$
$173$	−0.471351	−0.0358362	−0.0179181	+	0.999839i	$0.505704\pi$
$174$	0	0
$175$	5.74804	0.434511
$176$	0	0
$177$	−8.99297	−0.675953
$178$	0	0
$179$	15.7303	1.17574	0.587868	−	0.808957i	$-0.299967\pi$
$179$	15.7303	1.17574	0.587868	+	0.808957i	$0.299967\pi$
$180$	0	0
$181$	13.6079	1.01147	0.505735	−	0.862689i	$-0.331221\pi$
$181$	13.6079	1.01147	0.505735	+	0.862689i	$0.331221\pi$
$182$	0	0
$183$	−4.59852	−0.339932
$184$	0	0
$185$	0.629943	0.0463143
$186$	0	0
$187$	33.6545	2.46106
$188$	0	0
$189$	−1.23518	−0.0898464
$190$	0	0
$191$	−9.61701	−0.695862	−0.347931	−	0.937520i	$-0.613116\pi$
$191$	−9.61701	−0.695862	−0.347931	+	0.937520i	$0.613116\pi$
$192$	0	0
$193$	3.01313	0.216890	0.108445	−	0.994102i	$-0.465413\pi$
$193$	3.01313	0.216890	0.108445	+	0.994102i	$0.465413\pi$
$194$	0	0
$195$	2.86177	0.204936
$196$	0	0
$197$	2.78043	0.198097	0.0990486	−	0.995083i	$-0.468420\pi$
$197$	2.78043	0.198097	0.0990486	+	0.995083i	$0.468420\pi$
$198$	0	0
$199$	12.7220	0.901835	0.450918	−	0.892566i	$-0.351097\pi$
$199$	12.7220	0.901835	0.450918	+	0.892566i	$0.351097\pi$
$200$	0	0
$201$	−15.2918	−1.07860
$202$	0	0
$203$	−11.1024	−0.779236
$204$	0	0
$205$	3.66331	0.255857
$206$	0	0
$207$	3.90691	0.271549
$208$	0	0
$209$	8.65844	0.598917
$210$	0	0
$211$	7.37467	0.507693	0.253847	−	0.967244i	$-0.418304\pi$
$211$	7.37467	0.507693	0.253847	+	0.967244i	$0.418304\pi$
$212$	0	0
$213$	−6.37962	−0.437124
$214$	0	0
$215$	−5.73360	−0.391028
$216$	0	0
$217$	6.63168	0.450187
$218$	0	0
$219$	8.12499	0.549036
$220$	0	0
$221$	−32.6126	−2.19376
$222$	0	0
$223$	6.83212	0.457513	0.228756	−	0.973484i	$-0.426534\pi$
$223$	6.83212	0.457513	0.228756	+	0.973484i	$0.426534\pi$
$224$	0	0
$225$	−4.65359	−0.310240
$226$	0	0
$227$	6.63738	0.440538	0.220269	−	0.975439i	$-0.429306\pi$
$227$	6.63738	0.440538	0.220269	+	0.975439i	$0.429306\pi$
$228$	0	0
$229$	22.4406	1.48292	0.741458	−	0.670999i	$-0.234135\pi$
$229$	22.4406	1.48292	0.741458	+	0.670999i	$0.234135\pi$
$230$	0	0
$231$	6.19770	0.407779
$232$	0	0
$233$	−26.1458	−1.71287	−0.856435	−	0.516255i	$-0.827326\pi$
$233$	−26.1458	−1.71287	−0.856435	+	0.516255i	$0.827326\pi$
$234$	0	0
$235$	2.27091	0.148138
$236$	0	0
$237$	−6.10983	−0.396876
$238$	0	0
$239$	−20.5154	−1.32703	−0.663517	−	0.748161i	$-0.730937\pi$
$239$	−20.5154	−1.32703	−0.663517	+	0.748161i	$0.730937\pi$
$240$	0	0
$241$	−20.5369	−1.32290	−0.661449	−	0.749990i	$-0.730058\pi$
$241$	−20.5369	−1.32290	−0.661449	+	0.749990i	$0.730058\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.22199	−0.205845
$246$	0	0
$247$	−8.39040	−0.533868
$248$	0	0
$249$	3.46966	0.219881
$250$	0	0
$251$	5.12657	0.323586	0.161793	−	0.986825i	$-0.448272\pi$
$251$	5.12657	0.323586	0.161793	+	0.986825i	$0.448272\pi$
$252$	0	0
$253$	−19.6035	−1.23246
$254$	0	0
$255$	−3.94764	−0.247211
$256$	0	0
$257$	26.5939	1.65888	0.829442	−	0.558593i	$-0.188659\pi$
$257$	26.5939	1.65888	0.829442	+	0.558593i	$0.188659\pi$
$258$	0	0
$259$	−1.32202	−0.0821466
$260$	0	0
$261$	8.98846	0.556372
$262$	0	0
$263$	10.5800	0.652393	0.326196	−	0.945302i	$-0.394233\pi$
$263$	10.5800	0.652393	0.326196	+	0.945302i	$0.394233\pi$
$264$	0	0
$265$	−4.45995	−0.273973
$266$	0	0
$267$	5.15759	0.315639
$268$	0	0
$269$	10.0877	0.615056	0.307528	−	0.951539i	$-0.400498\pi$
$269$	10.0877	0.615056	0.307528	+	0.951539i	$0.400498\pi$
$270$	0	0
$271$	−20.1257	−1.22255	−0.611275	−	0.791418i	$-0.709343\pi$
$271$	−20.1257	−1.22255	−0.611275	+	0.791418i	$0.709343\pi$
$272$	0	0
$273$	−6.00583	−0.363489
$274$	0	0
$275$	23.3500	1.40806
$276$	0	0
$277$	−4.99188	−0.299933	−0.149966	−	0.988691i	$-0.547917\pi$
$277$	−4.99188	−0.299933	−0.149966	+	0.988691i	$0.547917\pi$
$278$	0	0
$279$	−5.36898	−0.321432
$280$	0	0
$281$	22.7616	1.35784	0.678922	−	0.734210i	$-0.262447\pi$
$281$	22.7616	1.35784	0.678922	+	0.734210i	$0.262447\pi$
$282$	0	0
$283$	−23.4891	−1.39628	−0.698141	−	0.715961i	$-0.745989\pi$
$283$	−23.4891	−1.39628	−0.698141	+	0.715961i	$0.745989\pi$
$284$	0	0
$285$	−1.01563	−0.0601605
$286$	0	0
$287$	−7.68797	−0.453807
$288$	0	0
$289$	27.9871	1.64630
$290$	0	0
$291$	−13.6764	−0.801726
$292$	0	0
$293$	28.5327	1.66690	0.833450	−	0.552595i	$-0.186363\pi$
$293$	28.5327	1.66690	0.833450	+	0.552595i	$0.186363\pi$
$294$	0	0
$295$	−5.29294	−0.308167
$296$	0	0
$297$	−5.01763	−0.291153
$298$	0	0
$299$	18.9966	1.09860
$300$	0	0
$301$	12.0328	0.693557
$302$	0	0
$303$	−3.33213	−0.191426
$304$	0	0
$305$	−2.70652	−0.154975
$306$	0	0
$307$	1.14107	0.0651242	0.0325621	−	0.999470i	$-0.489633\pi$
$307$	1.14107	0.0651242	0.0325621	+	0.999470i	$0.489633\pi$
$308$	0	0
$309$	−1.59690	−0.0908445
$310$	0	0
$311$	−27.8320	−1.57821	−0.789103	−	0.614261i	$-0.789454\pi$
$311$	−27.8320	−1.57821	−0.789103	+	0.614261i	$0.789454\pi$
$312$	0	0
$313$	9.27809	0.524428	0.262214	−	0.965010i	$-0.415547\pi$
$313$	9.27809	0.524428	0.262214	+	0.965010i	$0.415547\pi$
$314$	0	0
$315$	−0.726984	−0.0409609
$316$	0	0
$317$	11.0846	0.622572	0.311286	−	0.950316i	$-0.399240\pi$
$317$	11.0846	0.622572	0.311286	+	0.950316i	$0.399240\pi$
$318$	0	0
$319$	−45.1008	−2.52516
$320$	0	0
$321$	12.8858	0.719214
$322$	0	0
$323$	11.5740	0.643997
$324$	0	0
$325$	−22.6272	−1.25513
$326$	0	0
$327$	−1.41783	−0.0784063
$328$	0	0
$329$	−4.76583	−0.262749
$330$	0	0
$331$	28.7641	1.58102	0.790510	−	0.612449i	$-0.209816\pi$
$331$	28.7641	1.58102	0.790510	+	0.612449i	$0.209816\pi$
$332$	0	0
$333$	1.07031	0.0586524
$334$	0	0
$335$	−9.00018	−0.491732
$336$	0	0
$337$	4.07043	0.221730	0.110865	−	0.993835i	$-0.464638\pi$
$337$	4.07043	0.221730	0.110865	+	0.993835i	$0.464638\pi$
$338$	0	0
$339$	−18.0186	−0.978636
$340$	0	0
$341$	26.9396	1.45886
$342$	0	0
$343$	15.4081	0.831958
$344$	0	0
$345$	2.29947	0.123799
$346$	0	0
$347$	−2.34196	−0.125723	−0.0628615	−	0.998022i	$-0.520023\pi$
$347$	−2.34196	−0.125723	−0.0628615	+	0.998022i	$0.520023\pi$
$348$	0	0
$349$	20.0906	1.07542	0.537711	−	0.843129i	$-0.319289\pi$
$349$	20.0906	1.07542	0.537711	+	0.843129i	$0.319289\pi$
$350$	0	0
$351$	4.86230	0.259530
$352$	0	0
$353$	1.08213	0.0575960	0.0287980	−	0.999585i	$-0.490832\pi$
$353$	1.08213	0.0575960	0.0287980	+	0.999585i	$0.490832\pi$
$354$	0	0
$355$	−3.75481	−0.199285
$356$	0	0
$357$	8.28468	0.438472
$358$	0	0
$359$	−4.25451	−0.224545	−0.112272	−	0.993677i	$-0.535813\pi$
$359$	−4.25451	−0.224545	−0.112272	+	0.993677i	$0.535813\pi$
$360$	0	0
$361$	−16.0223	−0.843279
$362$	0	0
$363$	14.1766	0.744081
$364$	0	0
$365$	4.78208	0.250305
$366$	0	0
$367$	−21.5891	−1.12694	−0.563472	−	0.826135i	$-0.690535\pi$
$367$	−21.5891	−1.12694	−0.563472	+	0.826135i	$0.690535\pi$
$368$	0	0
$369$	6.22415	0.324017
$370$	0	0
$371$	9.35983	0.485938
$372$	0	0
$373$	−2.46856	−0.127817	−0.0639087	−	0.997956i	$-0.520357\pi$
$373$	−2.46856	−0.127817	−0.0639087	+	0.997956i	$0.520357\pi$
$374$	0	0
$375$	−5.68175	−0.293404
$376$	0	0
$377$	43.7046	2.25090
$378$	0	0
$379$	−20.7402	−1.06535	−0.532675	−	0.846320i	$-0.678813\pi$
$379$	−20.7402	−1.06535	−0.532675	+	0.846320i	$0.678813\pi$
$380$	0	0
$381$	19.7958	1.01417
$382$	0	0
$383$	−12.1308	−0.619855	−0.309928	−	0.950760i	$-0.600305\pi$
$383$	−12.1308	−0.619855	−0.309928	+	0.950760i	$0.600305\pi$
$384$	0	0
$385$	3.64774	0.185906
$386$	0	0
$387$	−9.74169	−0.495198
$388$	0	0
$389$	−26.2089	−1.32884	−0.664422	−	0.747358i	$-0.731322\pi$
$389$	−26.2089	−1.32884	−0.664422	+	0.747358i	$0.731322\pi$
$390$	0	0
$391$	−26.2046	−1.32523
$392$	0	0
$393$	−1.29378	−0.0652626
$394$	0	0
$395$	−3.59602	−0.180936
$396$	0	0
$397$	19.9952	1.00353	0.501766	−	0.865003i	$-0.332684\pi$
$397$	19.9952	1.00353	0.501766	+	0.865003i	$0.332684\pi$
$398$	0	0
$399$	2.13144	0.106705
$400$	0	0
$401$	19.4819	0.972878	0.486439	−	0.873715i	$-0.338296\pi$
$401$	19.4819	0.972878	0.486439	+	0.873715i	$0.338296\pi$
$402$	0	0
$403$	−26.1056	−1.30041
$404$	0	0
$405$	0.588564	0.0292460
$406$	0	0
$407$	−5.37040	−0.266201
$408$	0	0
$409$	−10.9296	−0.540433	−0.270217	−	0.962800i	$-0.587095\pi$
$409$	−10.9296	−0.540433	−0.270217	+	0.962800i	$0.587095\pi$
$410$	0	0
$411$	15.2142	0.750460
$412$	0	0
$413$	11.1080	0.546588
$414$	0	0
$415$	2.04211	0.100243
$416$	0	0
$417$	−15.1674	−0.742753
$418$	0	0
$419$	−4.80239	−0.234612	−0.117306	−	0.993096i	$-0.537426\pi$
$419$	−4.80239	−0.234612	−0.117306	+	0.993096i	$0.537426\pi$
$420$	0	0
$421$	−12.5947	−0.613829	−0.306915	−	0.951737i	$-0.599297\pi$
$421$	−12.5947	−0.613829	−0.306915	+	0.951737i	$0.599297\pi$
$422$	0	0
$423$	3.85840	0.187602
$424$	0	0
$425$	31.2128	1.51404
$426$	0	0
$427$	5.68002	0.274875
$428$	0	0
$429$	−24.3972	−1.17791
$430$	0	0
$431$	38.3341	1.84649	0.923243	−	0.384216i	$-0.125528\pi$
$431$	38.3341	1.84649	0.923243	+	0.384216i	$0.125528\pi$
$432$	0	0
$433$	−21.6211	−1.03904	−0.519521	−	0.854458i	$-0.673890\pi$
$433$	−21.6211	−1.03904	−0.519521	+	0.854458i	$0.673890\pi$
$434$	0	0
$435$	5.29028	0.253650
$436$	0	0
$437$	−6.74178	−0.322503
$438$	0	0
$439$	−11.0736	−0.528512	−0.264256	−	0.964453i	$-0.585126\pi$
$439$	−11.0736	−0.528512	−0.264256	+	0.964453i	$0.585126\pi$
$440$	0	0
$441$	−5.47432	−0.260682
$442$	0	0
$443$	−22.7154	−1.07924	−0.539620	−	0.841909i	$-0.681432\pi$
$443$	−22.7154	−1.07924	−0.539620	+	0.841909i	$0.681432\pi$
$444$	0	0
$445$	3.03557	0.143900
$446$	0	0
$447$	−14.4870	−0.685213
$448$	0	0
$449$	−4.64041	−0.218994	−0.109497	−	0.993987i	$-0.534924\pi$
$449$	−4.64041	−0.218994	−0.109497	+	0.993987i	$0.534924\pi$
$450$	0	0
$451$	−31.2305	−1.47059
$452$	0	0
$453$	−4.68695	−0.220212
$454$	0	0
$455$	−3.53481	−0.165715
$456$	0	0
$457$	7.63494	0.357147	0.178574	−	0.983927i	$-0.442852\pi$
$457$	7.63494	0.357147	0.178574	+	0.983927i	$0.442852\pi$
$458$	0	0
$459$	−6.70724	−0.313067
$460$	0	0
$461$	−1.87870	−0.0874999	−0.0437500	−	0.999043i	$-0.513930\pi$
$461$	−1.87870	−0.0874999	−0.0437500	+	0.999043i	$0.513930\pi$
$462$	0	0
$463$	−10.1989	−0.473985	−0.236993	−	0.971511i	$-0.576162\pi$
$463$	−10.1989	−0.473985	−0.236993	+	0.971511i	$0.576162\pi$
$464$	0	0
$465$	−3.15999	−0.146541
$466$	0	0
$467$	27.4879	1.27199	0.635995	−	0.771694i	$-0.280590\pi$
$467$	27.4879	1.27199	0.635995	+	0.771694i	$0.280590\pi$
$468$	0	0
$469$	18.8881	0.872173
$470$	0	0
$471$	−22.2647	−1.02590
$472$	0	0
$473$	48.8802	2.24752
$474$	0	0
$475$	8.03025	0.368453
$476$	0	0
$477$	−7.57769	−0.346958
$478$	0	0
$479$	4.84027	0.221158	0.110579	−	0.993867i	$-0.464730\pi$
$479$	4.84027	0.221158	0.110579	+	0.993867i	$0.464730\pi$
$480$	0	0
$481$	5.20415	0.237289
$482$	0	0
$483$	−4.82576	−0.219580
$484$	0	0
$485$	−8.04944	−0.365506
$486$	0	0
$487$	5.00720	0.226898	0.113449	−	0.993544i	$-0.463810\pi$
$487$	5.00720	0.226898	0.113449	+	0.993544i	$0.463810\pi$
$488$	0	0
$489$	−18.8338	−0.851692
$490$	0	0
$491$	−1.70630	−0.0770041	−0.0385021	−	0.999259i	$-0.512259\pi$
$491$	−1.70630	−0.0770041	−0.0385021	+	0.999259i	$0.512259\pi$
$492$	0	0
$493$	−60.2878	−2.71523
$494$	0	0
$495$	−2.95320	−0.132736
$496$	0	0
$497$	7.88000	0.353466
$498$	0	0
$499$	−22.2962	−0.998114	−0.499057	−	0.866569i	$-0.666320\pi$
$499$	−22.2962	−0.998114	−0.499057	+	0.866569i	$0.666320\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−1.40579	−0.0626811	−0.0313406	−	0.999509i	$-0.509978\pi$
$503$	−1.40579	−0.0626811	−0.0313406	+	0.999509i	$0.509978\pi$
$504$	0	0
$505$	−1.96117	−0.0872709
$506$	0	0
$507$	10.6420	0.472626
$508$	0	0
$509$	−43.1248	−1.91147	−0.955737	−	0.294223i	$-0.904939\pi$
$509$	−43.1248	−1.91147	−0.955737	+	0.294223i	$0.904939\pi$
$510$	0	0
$511$	−10.0359	−0.443960
$512$	0	0
$513$	−1.72560	−0.0761872
$514$	0	0
$515$	−0.939878	−0.0414160
$516$	0	0
$517$	−19.3600	−0.851454
$518$	0	0
$519$	−0.471351	−0.0206900
$520$	0	0
$521$	18.5381	0.812169	0.406085	−	0.913836i	$-0.366894\pi$
$521$	18.5381	0.812169	0.406085	+	0.913836i	$0.366894\pi$
$522$	0	0
$523$	−24.6425	−1.07754	−0.538771	−	0.842452i	$-0.681111\pi$
$523$	−24.6425	−1.07754	−0.538771	+	0.842452i	$0.681111\pi$
$524$	0	0
$525$	5.74804	0.250865
$526$	0	0
$527$	36.0111	1.56867
$528$	0	0
$529$	−7.73602	−0.336349
$530$	0	0
$531$	−8.99297	−0.390262
$532$	0	0
$533$	30.2637	1.31087
$534$	0	0
$535$	7.58410	0.327889
$536$	0	0
$537$	15.7303	0.678812
$538$	0	0
$539$	27.4681	1.18314
$540$	0	0
$541$	41.0312	1.76407	0.882034	−	0.471186i	$-0.156174\pi$
$541$	41.0312	1.76407	0.882034	+	0.471186i	$0.156174\pi$
$542$	0	0
$543$	13.6079	0.583973
$544$	0	0
$545$	−0.834484	−0.0357454
$546$	0	0
$547$	33.5441	1.43424	0.717121	−	0.696948i	$-0.245459\pi$
$547$	33.5441	1.43424	0.717121	+	0.696948i	$0.245459\pi$
$548$	0	0
$549$	−4.59852	−0.196260
$550$	0	0
$551$	−15.5105	−0.660770
$552$	0	0
$553$	7.54676	0.320921
$554$	0	0
$555$	0.629943	0.0267396
$556$	0	0
$557$	−4.42968	−0.187692	−0.0938458	−	0.995587i	$-0.529916\pi$
$557$	−4.42968	−0.187692	−0.0938458	+	0.995587i	$0.529916\pi$
$558$	0	0
$559$	−47.3670	−2.00341
$560$	0	0
$561$	33.6545	1.42089
$562$	0	0
$563$	4.13466	0.174255	0.0871276	−	0.996197i	$-0.472231\pi$
$563$	4.13466	0.174255	0.0871276	+	0.996197i	$0.472231\pi$
$564$	0	0
$565$	−10.6051	−0.446160
$566$	0	0
$567$	−1.23518	−0.0518728
$568$	0	0
$569$	35.5937	1.49217	0.746083	−	0.665852i	$-0.231932\pi$
$569$	35.5937	1.49217	0.746083	+	0.665852i	$0.231932\pi$
$570$	0	0
$571$	−39.2825	−1.64392	−0.821962	−	0.569543i	$-0.807120\pi$
$571$	−39.2825	−1.64392	−0.821962	+	0.569543i	$0.807120\pi$
$572$	0	0
$573$	−9.61701	−0.401756
$574$	0	0
$575$	−18.1812	−0.758208
$576$	0	0
$577$	25.1884	1.04860	0.524302	−	0.851532i	$-0.324326\pi$
$577$	25.1884	1.04860	0.524302	+	0.851532i	$0.324326\pi$
$578$	0	0
$579$	3.01313	0.125222
$580$	0	0
$581$	−4.28566	−0.177799
$582$	0	0
$583$	38.0221	1.57471
$584$	0	0
$585$	2.86177	0.118320
$586$	0	0
$587$	30.6309	1.26427	0.632136	−	0.774857i	$-0.282178\pi$
$587$	30.6309	1.26427	0.632136	+	0.774857i	$0.282178\pi$
$588$	0	0
$589$	9.26473	0.381746
$590$	0	0
$591$	2.78043	0.114371
$592$	0	0
$593$	−16.8460	−0.691781	−0.345890	−	0.938275i	$-0.612423\pi$
$593$	−16.8460	−0.691781	−0.345890	+	0.938275i	$0.612423\pi$
$594$	0	0
$595$	4.87606	0.199899
$596$	0	0
$597$	12.7220	0.520675
$598$	0	0
$599$	12.4465	0.508549	0.254275	−	0.967132i	$-0.418163\pi$
$599$	12.4465	0.508549	0.254275	+	0.967132i	$0.418163\pi$
$600$	0	0
$601$	3.96091	0.161569	0.0807844	−	0.996732i	$-0.474257\pi$
$601$	3.96091	0.161569	0.0807844	+	0.996732i	$0.474257\pi$
$602$	0	0
$603$	−15.2918	−0.622729
$604$	0	0
$605$	8.34386	0.339226
$606$	0	0
$607$	−5.93264	−0.240799	−0.120399	−	0.992726i	$-0.538417\pi$
$607$	−5.93264	−0.240799	−0.120399	+	0.992726i	$0.538417\pi$
$608$	0	0
$609$	−11.1024	−0.449892
$610$	0	0
$611$	18.7607	0.758977
$612$	0	0
$613$	13.9450	0.563232	0.281616	−	0.959527i	$-0.409130\pi$
$613$	13.9450	0.563232	0.281616	+	0.959527i	$0.409130\pi$
$614$	0	0
$615$	3.66331	0.147719
$616$	0	0
$617$	−24.0063	−0.966457	−0.483229	−	0.875494i	$-0.660536\pi$
$617$	−24.0063	−0.966457	−0.483229	+	0.875494i	$0.660536\pi$
$618$	0	0
$619$	45.8501	1.84287	0.921435	−	0.388533i	$-0.127018\pi$
$619$	45.8501	1.84287	0.921435	+	0.388533i	$0.127018\pi$
$620$	0	0
$621$	3.90691	0.156779
$622$	0	0
$623$	−6.37056	−0.255231
$624$	0	0
$625$	19.9239	0.796956
$626$	0	0
$627$	8.65844	0.345785
$628$	0	0
$629$	−7.17880	−0.286238
$630$	0	0
$631$	−19.7197	−0.785030	−0.392515	−	0.919746i	$-0.628395\pi$
$631$	−19.7197	−0.785030	−0.392515	+	0.919746i	$0.628395\pi$
$632$	0	0
$633$	7.37467	0.293117
$634$	0	0
$635$	11.6511	0.462358
$636$	0	0
$637$	−26.6178	−1.05464
$638$	0	0
$639$	−6.37962	−0.252374
$640$	0	0
$641$	−34.2968	−1.35464	−0.677322	−	0.735687i	$-0.736859\pi$
$641$	−34.2968	−1.35464	−0.677322	+	0.735687i	$0.736859\pi$
$642$	0	0
$643$	−12.4494	−0.490957	−0.245478	−	0.969402i	$-0.578945\pi$
$643$	−12.4494	−0.490957	−0.245478	+	0.969402i	$0.578945\pi$
$644$	0	0
$645$	−5.73360	−0.225760
$646$	0	0
$647$	38.4089	1.51001	0.755005	−	0.655719i	$-0.227635\pi$
$647$	38.4089	1.51001	0.755005	+	0.655719i	$0.227635\pi$
$648$	0	0
$649$	45.1234	1.77125
$650$	0	0
$651$	6.63168	0.259916
$652$	0	0
$653$	18.6064	0.728125	0.364062	−	0.931375i	$-0.381390\pi$
$653$	18.6064	0.728125	0.364062	+	0.931375i	$0.381390\pi$
$654$	0	0
$655$	−0.761472	−0.0297532
$656$	0	0
$657$	8.12499	0.316986
$658$	0	0
$659$	13.7868	0.537059	0.268530	−	0.963271i	$-0.413462\pi$
$659$	13.7868	0.537059	0.268530	+	0.963271i	$0.413462\pi$
$660$	0	0
$661$	−15.0955	−0.587149	−0.293574	−	0.955936i	$-0.594845\pi$
$661$	−15.0955	−0.587149	−0.293574	+	0.955936i	$0.594845\pi$
$662$	0	0
$663$	−32.6126	−1.26657
$664$	0	0
$665$	1.25449	0.0486468
$666$	0	0
$667$	35.1172	1.35974
$668$	0	0
$669$	6.83212	0.264145
$670$	0	0
$671$	23.0737	0.890750
$672$	0	0
$673$	25.1028	0.967642	0.483821	−	0.875167i	$-0.339249\pi$
$673$	25.1028	0.967642	0.483821	+	0.875167i	$0.339249\pi$
$674$	0	0
$675$	−4.65359	−0.179117
$676$	0	0
$677$	−9.75143	−0.374778	−0.187389	−	0.982286i	$-0.560002\pi$
$677$	−9.75143	−0.374778	−0.187389	+	0.982286i	$0.560002\pi$
$678$	0	0
$679$	16.8929	0.648289
$680$	0	0
$681$	6.63738	0.254345
$682$	0	0
$683$	13.0994	0.501233	0.250616	−	0.968086i	$-0.419367\pi$
$683$	13.0994	0.501233	0.250616	+	0.968086i	$0.419367\pi$
$684$	0	0
$685$	8.95450	0.342134
$686$	0	0
$687$	22.4406	0.856162
$688$	0	0
$689$	−36.8450	−1.40368
$690$	0	0
$691$	−12.4983	−0.475459	−0.237729	−	0.971331i	$-0.576403\pi$
$691$	−12.4983	−0.475459	−0.237729	+	0.971331i	$0.576403\pi$
$692$	0	0
$693$	6.19770	0.235431
$694$	0	0
$695$	−8.92700	−0.338621
$696$	0	0
$697$	−41.7469	−1.58128
$698$	0	0
$699$	−26.1458	−0.988926
$700$	0	0
$701$	8.26205	0.312053	0.156027	−	0.987753i	$-0.450131\pi$
$701$	8.26205	0.312053	0.156027	+	0.987753i	$0.450131\pi$
$702$	0	0
$703$	−1.84692	−0.0696580
$704$	0	0
$705$	2.27091	0.0855276
$706$	0	0
$707$	4.11579	0.154790
$708$	0	0
$709$	−16.6255	−0.624382	−0.312191	−	0.950019i	$-0.601063\pi$
$709$	−16.6255	−0.624382	−0.312191	+	0.950019i	$0.601063\pi$
$710$	0	0
$711$	−6.10983	−0.229137
$712$	0	0
$713$	−20.9761	−0.785563
$714$	0	0
$715$	−14.3593	−0.537008
$716$	0	0
$717$	−20.5154	−0.766163
$718$	0	0
$719$	15.7611	0.587788	0.293894	−	0.955838i	$-0.405049\pi$
$719$	15.7611	0.587788	0.293894	+	0.955838i	$0.405049\pi$
$720$	0	0
$721$	1.97247	0.0734585
$722$	0	0
$723$	−20.5369	−0.763775
$724$	0	0
$725$	−41.8287	−1.55348
$726$	0	0
$727$	5.66621	0.210148	0.105074	−	0.994464i	$-0.466492\pi$
$727$	5.66621	0.210148	0.105074	+	0.994464i	$0.466492\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	65.3399	2.41668
$732$	0	0
$733$	16.3946	0.605549	0.302775	−	0.953062i	$-0.402087\pi$
$733$	16.3946	0.605549	0.302775	+	0.953062i	$0.402087\pi$
$734$	0	0
$735$	−3.22199	−0.118845
$736$	0	0
$737$	76.7285	2.82633
$738$	0	0
$739$	14.6648	0.539453	0.269726	−	0.962937i	$-0.413067\pi$
$739$	14.6648	0.539453	0.269726	+	0.962937i	$0.413067\pi$
$740$	0	0
$741$	−8.39040	−0.308229
$742$	0	0
$743$	−4.63935	−0.170201	−0.0851006	−	0.996372i	$-0.527121\pi$
$743$	−4.63935	−0.170201	−0.0851006	+	0.996372i	$0.527121\pi$
$744$	0	0
$745$	−8.52654	−0.312388
$746$	0	0
$747$	3.46966	0.126948
$748$	0	0
$749$	−15.9163	−0.581569
$750$	0	0
$751$	7.52507	0.274594	0.137297	−	0.990530i	$-0.456159\pi$
$751$	7.52507	0.274594	0.137297	+	0.990530i	$0.456159\pi$
$752$	0	0
$753$	5.12657	0.186823
$754$	0	0
$755$	−2.75857	−0.100395
$756$	0	0
$757$	49.4148	1.79601	0.898006	−	0.439984i	$-0.145016\pi$
$757$	49.4148	1.79601	0.898006	+	0.439984i	$0.145016\pi$
$758$	0	0
$759$	−19.6035	−0.711561
$760$	0	0
$761$	−35.2793	−1.27887	−0.639437	−	0.768843i	$-0.720833\pi$
$761$	−35.2793	−1.27887	−0.639437	+	0.768843i	$0.720833\pi$
$762$	0	0
$763$	1.75128	0.0634007
$764$	0	0
$765$	−3.94764	−0.142727
$766$	0	0
$767$	−43.7265	−1.57887
$768$	0	0
$769$	27.7698	1.00140	0.500702	−	0.865620i	$-0.333075\pi$
$769$	27.7698	1.00140	0.500702	+	0.865620i	$0.333075\pi$
$770$	0	0
$771$	26.5939	0.957757
$772$	0	0
$773$	−3.62922	−0.130534	−0.0652669	−	0.997868i	$-0.520790\pi$
$773$	−3.62922	−0.130534	−0.0652669	+	0.997868i	$0.520790\pi$
$774$	0	0
$775$	24.9851	0.897490
$776$	0	0
$777$	−1.32202	−0.0474274
$778$	0	0
$779$	−10.7404	−0.384815
$780$	0	0
$781$	32.0106	1.14543
$782$	0	0
$783$	8.98846	0.321221
$784$	0	0
$785$	−13.1042	−0.467709
$786$	0	0
$787$	−38.3837	−1.36823	−0.684116	−	0.729374i	$-0.739812\pi$
$787$	−38.3837	−1.36823	−0.684116	+	0.729374i	$0.739812\pi$
$788$	0	0
$789$	10.5800	0.376659
$790$	0	0
$791$	22.2563	0.791342
$792$	0	0
$793$	−22.3594	−0.794005
$794$	0	0
$795$	−4.45995	−0.158178
$796$	0	0
$797$	32.6530	1.15663	0.578315	−	0.815814i	$-0.303711\pi$
$797$	32.6530	1.15663	0.578315	+	0.815814i	$0.303711\pi$
$798$	0	0
$799$	−25.8792	−0.915542
$800$	0	0
$801$	5.15759	0.182234
$802$	0	0
$803$	−40.7682	−1.43868
$804$	0	0
$805$	−2.84026	−0.100106
$806$	0	0
$807$	10.0877	0.355103
$808$	0	0
$809$	−27.3338	−0.961007	−0.480503	−	0.876993i	$-0.659546\pi$
$809$	−27.3338	−0.961007	−0.480503	+	0.876993i	$0.659546\pi$
$810$	0	0
$811$	−36.5168	−1.28228	−0.641140	−	0.767424i	$-0.721538\pi$
$811$	−36.5168	−1.28228	−0.641140	+	0.767424i	$0.721538\pi$
$812$	0	0
$813$	−20.1257	−0.705839
$814$	0	0
$815$	−11.0849	−0.388286
$816$	0	0
$817$	16.8103	0.588117
$818$	0	0
$819$	−6.00583	−0.209861
$820$	0	0
$821$	−34.8331	−1.21568	−0.607842	−	0.794058i	$-0.707964\pi$
$821$	−34.8331	−1.21568	−0.607842	+	0.794058i	$0.707964\pi$
$822$	0	0
$823$	27.8026	0.969139	0.484570	−	0.874753i	$-0.338976\pi$
$823$	27.8026	0.969139	0.484570	+	0.874753i	$0.338976\pi$
$824$	0	0
$825$	23.3500	0.812944
$826$	0	0
$827$	−8.08198	−0.281038	−0.140519	−	0.990078i	$-0.544877\pi$
$827$	−8.08198	−0.281038	−0.140519	+	0.990078i	$0.544877\pi$
$828$	0	0
$829$	8.93224	0.310229	0.155115	−	0.987896i	$-0.450425\pi$
$829$	8.93224	0.310229	0.155115	+	0.987896i	$0.450425\pi$
$830$	0	0
$831$	−4.99188	−0.173166
$832$	0	0
$833$	36.7176	1.27219
$834$	0	0
$835$	−0.588564	−0.0203681
$836$	0	0
$837$	−5.36898	−0.185579
$838$	0	0
$839$	−6.89723	−0.238119	−0.119059	−	0.992887i	$-0.537988\pi$
$839$	−6.89723	−0.238119	−0.119059	+	0.992887i	$0.537988\pi$
$840$	0	0
$841$	51.7925	1.78595
$842$	0	0
$843$	22.7616	0.783952
$844$	0	0
$845$	6.26347	0.215470
$846$	0	0
$847$	−17.5108	−0.601677
$848$	0	0
$849$	−23.4891	−0.806143
$850$	0	0
$851$	4.18159	0.143343
$852$	0	0
$853$	34.7250	1.18896	0.594481	−	0.804109i	$-0.297357\pi$
$853$	34.7250	1.18896	0.594481	+	0.804109i	$0.297357\pi$
$854$	0	0
$855$	−1.01563	−0.0347337
$856$	0	0
$857$	−35.9067	−1.22655	−0.613276	−	0.789869i	$-0.710148\pi$
$857$	−35.9067	−1.22655	−0.613276	+	0.789869i	$0.710148\pi$
$858$	0	0
$859$	56.0190	1.91134	0.955671	−	0.294436i	$-0.0951319\pi$
$859$	56.0190	1.91134	0.955671	+	0.294436i	$0.0951319\pi$
$860$	0	0
$861$	−7.68797	−0.262005
$862$	0	0
$863$	−3.47561	−0.118311	−0.0591556	−	0.998249i	$-0.518841\pi$
$863$	−3.47561	−0.118311	−0.0591556	+	0.998249i	$0.518841\pi$
$864$	0	0
$865$	−0.277420	−0.00943257
$866$	0	0
$867$	27.9871	0.950492
$868$	0	0
$869$	30.6569	1.03996
$870$	0	0
$871$	−74.3532	−2.51936
$872$	0	0
$873$	−13.6764	−0.462877
$874$	0	0
$875$	7.01801	0.237252
$876$	0	0
$877$	−5.27529	−0.178134	−0.0890670	−	0.996026i	$-0.528389\pi$
$877$	−5.27529	−0.178134	−0.0890670	+	0.996026i	$0.528389\pi$
$878$	0	0
$879$	28.5327	0.962385
$880$	0	0
$881$	10.5890	0.356751	0.178376	−	0.983962i	$-0.442916\pi$
$881$	10.5890	0.356751	0.178376	+	0.983962i	$0.442916\pi$
$882$	0	0
$883$	−28.3144	−0.952855	−0.476427	−	0.879214i	$-0.658068\pi$
$883$	−28.3144	−0.952855	−0.476427	+	0.879214i	$0.658068\pi$
$884$	0	0
$885$	−5.29294	−0.177920
$886$	0	0
$887$	−32.9274	−1.10560	−0.552798	−	0.833316i	$-0.686440\pi$
$887$	−32.9274	−1.10560	−0.552798	+	0.833316i	$0.686440\pi$
$888$	0	0
$889$	−24.4514	−0.820073
$890$	0	0
$891$	−5.01763	−0.168097
$892$	0	0
$893$	−6.65807	−0.222804
$894$	0	0
$895$	9.25827	0.309470
$896$	0	0
$897$	18.9966	0.634278
$898$	0	0
$899$	−48.2589	−1.60952
$900$	0	0
$901$	50.8254	1.69324
$902$	0	0
$903$	12.0328	0.400426
$904$	0	0
$905$	8.00914	0.266233
$906$	0	0
$907$	30.3164	1.00664	0.503320	−	0.864100i	$-0.332112\pi$
$907$	30.3164	1.00664	0.503320	+	0.864100i	$0.332112\pi$
$908$	0	0
$909$	−3.33213	−0.110520
$910$	0	0
$911$	57.0150	1.88899	0.944496	−	0.328524i	$-0.106551\pi$
$911$	57.0150	1.88899	0.944496	+	0.328524i	$0.106551\pi$
$912$	0	0
$913$	−17.4095	−0.576169
$914$	0	0
$915$	−2.70652	−0.0894748
$916$	0	0
$917$	1.59806	0.0527725
$918$	0	0
$919$	−39.9071	−1.31641	−0.658206	−	0.752838i	$-0.728685\pi$
$919$	−39.9071	−1.31641	−0.658206	+	0.752838i	$0.728685\pi$
$920$	0	0
$921$	1.14107	0.0375995
$922$	0	0
$923$	−31.0196	−1.02102
$924$	0	0
$925$	−4.98077	−0.163767
$926$	0	0
$927$	−1.59690	−0.0524491
$928$	0	0
$929$	13.0857	0.429327	0.214663	−	0.976688i	$-0.431135\pi$
$929$	13.0857	0.429327	0.214663	+	0.976688i	$0.431135\pi$
$930$	0	0
$931$	9.44650	0.309597
$932$	0	0
$933$	−27.8320	−0.911178
$934$	0	0
$935$	19.8078	0.647785
$936$	0	0
$937$	−25.4531	−0.831516	−0.415758	−	0.909475i	$-0.636484\pi$
$937$	−25.4531	−0.831516	−0.415758	+	0.909475i	$0.636484\pi$
$938$	0	0
$939$	9.27809	0.302779
$940$	0	0
$941$	46.6123	1.51952	0.759759	−	0.650205i	$-0.225317\pi$
$941$	46.6123	1.51952	0.759759	+	0.650205i	$0.225317\pi$
$942$	0	0
$943$	24.3172	0.791878
$944$	0	0
$945$	−0.726984	−0.0236488
$946$	0	0
$947$	28.4433	0.924284	0.462142	−	0.886806i	$-0.347081\pi$
$947$	28.4433	0.924284	0.462142	+	0.886806i	$0.347081\pi$
$948$	0	0
$949$	39.5062	1.28242
$950$	0	0
$951$	11.0846	0.359442
$952$	0	0
$953$	25.2469	0.817829	0.408914	−	0.912573i	$-0.365908\pi$
$953$	25.2469	0.817829	0.408914	+	0.912573i	$0.365908\pi$
$954$	0	0
$955$	−5.66022	−0.183160
$956$	0	0
$957$	−45.1008	−1.45790
$958$	0	0
$959$	−18.7923	−0.606835
$960$	0	0
$961$	−2.17405	−0.0701305
$962$	0	0
$963$	12.8858	0.415238
$964$	0	0
$965$	1.77342	0.0570884
$966$	0	0
$967$	9.76071	0.313883	0.156942	−	0.987608i	$-0.449837\pi$
$967$	9.76071	0.313883	0.156942	+	0.987608i	$0.449837\pi$
$968$	0	0
$969$	11.5740	0.371812
$970$	0	0
$971$	47.6643	1.52962	0.764810	−	0.644256i	$-0.222833\pi$
$971$	47.6643	1.52962	0.764810	+	0.644256i	$0.222833\pi$
$972$	0	0
$973$	18.7346	0.600603
$974$	0	0
$975$	−22.6272	−0.724649
$976$	0	0
$977$	−42.6661	−1.36501	−0.682505	−	0.730881i	$-0.739110\pi$
$977$	−42.6661	−1.36501	−0.682505	+	0.730881i	$0.739110\pi$
$978$	0	0
$979$	−25.8789	−0.827092
$980$	0	0
$981$	−1.41783	−0.0452679
$982$	0	0
$983$	14.1240	0.450486	0.225243	−	0.974303i	$-0.427682\pi$
$983$	14.1240	0.450486	0.225243	+	0.974303i	$0.427682\pi$
$984$	0	0
$985$	1.63646	0.0521419
$986$	0	0
$987$	−4.76583	−0.151698
$988$	0	0
$989$	−38.0599	−1.21024
$990$	0	0
$991$	10.5986	0.336677	0.168339	−	0.985729i	$-0.446160\pi$
$991$	10.5986	0.336677	0.168339	+	0.985729i	$0.446160\pi$
$992$	0	0
$993$	28.7641	0.912802
$994$	0	0
$995$	7.48768	0.237375
$996$	0	0
$997$	−26.7128	−0.846002	−0.423001	−	0.906129i	$-0.639023\pi$
$997$	−26.7128	−0.846002	−0.423001	+	0.906129i	$0.639023\pi$
$998$	0	0
$999$	1.07031	0.0338630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.i.1.7	✓	9
4.3	odd	2		8016.2.a.ba.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.i.1.7	✓	9	1.1	even	1	trivial
8016.2.a.ba.1.7		9	4.3	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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.588564 q^{5} -1.23518 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.588564 q^{5} -1.23518 q^{7} +1.00000 q^{9} -5.01763 q^{11} +4.86230 q^{13} +0.588564 q^{15} -6.70724 q^{17} -1.72560 q^{19} -1.23518 q^{21} +3.90691 q^{23} -4.65359 q^{25} +1.00000 q^{27} +8.98846 q^{29} -5.36898 q^{31} -5.01763 q^{33} -0.726984 q^{35} +1.07031 q^{37} +4.86230 q^{39} +6.22415 q^{41} -9.74169 q^{43} +0.588564 q^{45} +3.85840 q^{47} -5.47432 q^{49} -6.70724 q^{51} -7.57769 q^{53} -2.95320 q^{55} -1.72560 q^{57} -8.99297 q^{59} -4.59852 q^{61} -1.23518 q^{63} +2.86177 q^{65} -15.2918 q^{67} +3.90691 q^{69} -6.37962 q^{71} +8.12499 q^{73} -4.65359 q^{75} +6.19770 q^{77} -6.10983 q^{79} +1.00000 q^{81} +3.46966 q^{83} -3.94764 q^{85} +8.98846 q^{87} +5.15759 q^{89} -6.00583 q^{91} -5.36898 q^{93} -1.01563 q^{95} -13.6764 q^{97} -5.01763 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} - 6 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 - 6 * q^5 - 11 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} - 6 q^{5} - 11 q^{7} + 9 q^{9} + q^{11} - 4 q^{13} - 6 q^{15} - 9 q^{17} - 8 q^{19} - 11 q^{21} - 7 q^{23} - q^{25} + 9 q^{27} - 9 q^{29} - 25 q^{31} + q^{33} + 5 q^{35} - 6 q^{37} - 4 q^{39} - 4 q^{41} - 24 q^{43} - 6 q^{45} - 16 q^{47} + 4 q^{49} - 9 q^{51} - 26 q^{53} - 29 q^{55} - 8 q^{57} - 4 q^{59} - 20 q^{61} - 11 q^{63} - 8 q^{65} - 25 q^{67} - 7 q^{69} - 15 q^{71} - 10 q^{73} - q^{75} - 20 q^{77} - 34 q^{79} + 9 q^{81} - 4 q^{83} - 13 q^{85} - 9 q^{87} + 13 q^{89} - 21 q^{91} - 25 q^{93} - 7 q^{95} - 4 q^{97} + q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 - 6 * q^5 - 11 * q^7 + 9 * q^9 + q^11 - 4 * q^13 - 6 * q^15 - 9 * q^17 - 8 * q^19 - 11 * q^21 - 7 * q^23 - q^25 + 9 * q^27 - 9 * q^29 - 25 * q^31 + q^33 + 5 * q^35 - 6 * q^37 - 4 * q^39 - 4 * q^41 - 24 * q^43 - 6 * q^45 - 16 * q^47 + 4 * q^49 - 9 * q^51 - 26 * q^53 - 29 * q^55 - 8 * q^57 - 4 * q^59 - 20 * q^61 - 11 * q^63 - 8 * q^65 - 25 * q^67 - 7 * q^69 - 15 * q^71 - 10 * q^73 - q^75 - 20 * q^77 - 34 * q^79 + 9 * q^81 - 4 * q^83 - 13 * q^85 - 9 * q^87 + 13 * q^89 - 21 * q^91 - 25 * q^93 - 7 * q^95 - 4 * q^97 + q^99

Introduction

L-functions

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Varieties

Fields

Representations

Groups

Database

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