LMFDB - Embedded newform 6003.2.a.q.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 22 x^{14} + 68 x^{13} + 187 x^{12} - 597 x^{11} - 795 x^{10} + 2592 x^{9} + \cdots + 64 $ x^16 - 3x^15 - 22x^14 + 68x^13 + 187x^12 - 597x^11 - 795x^10 + 2592x^9 + 1860x^8 - 5877x^7 - 2496x^6 + 6612x^5 + 1842x^4 - 3011x^3 - 505x^2 + 336*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.745705$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.745705 q^{2} -1.44392 q^{4} +2.81935 q^{5} +3.27962 q^{7} +2.56815 q^{8} +O(q^{10})$	\(q-0.745705 q^{2} -1.44392 q^{4} +2.81935 q^{5} +3.27962 q^{7} +2.56815 q^{8} -2.10240 q^{10} -4.74029 q^{11} +5.26065 q^{13} -2.44563 q^{14} +0.972761 q^{16} -3.78228 q^{17} -5.67624 q^{19} -4.07092 q^{20} +3.53486 q^{22} -1.00000 q^{23} +2.94872 q^{25} -3.92289 q^{26} -4.73553 q^{28} +1.00000 q^{29} +3.34398 q^{31} -5.86170 q^{32} +2.82046 q^{34} +9.24640 q^{35} -7.39041 q^{37} +4.23281 q^{38} +7.24052 q^{40} -2.57493 q^{41} -2.01679 q^{43} +6.84462 q^{44} +0.745705 q^{46} -1.24351 q^{47} +3.75593 q^{49} -2.19888 q^{50} -7.59597 q^{52} -12.2633 q^{53} -13.3645 q^{55} +8.42257 q^{56} -0.745705 q^{58} +0.582654 q^{59} -12.5729 q^{61} -2.49362 q^{62} +2.42558 q^{64} +14.8316 q^{65} -6.34363 q^{67} +5.46132 q^{68} -6.89509 q^{70} +1.00328 q^{71} -15.9306 q^{73} +5.51107 q^{74} +8.19606 q^{76} -15.5464 q^{77} -14.7676 q^{79} +2.74255 q^{80} +1.92014 q^{82} -13.9060 q^{83} -10.6636 q^{85} +1.50393 q^{86} -12.1738 q^{88} +11.3752 q^{89} +17.2529 q^{91} +1.44392 q^{92} +0.927294 q^{94} -16.0033 q^{95} -12.8661 q^{97} -2.80082 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) $ 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8	$ 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8} - 14 q^{10} - 4 q^{11} + 15 q^{13} - 8 q^{14} + 23 q^{16} - 20 q^{17} - 4 q^{19} - 25 q^{20} + 13 q^{22} - 16 q^{23} + 30 q^{25} - 25 q^{26} - 13 q^{28} + 16 q^{29} + 19 q^{32} - 23 q^{34} - 5 q^{35} + 5 q^{37} - 38 q^{38} - 20 q^{40} - 7 q^{41} - 17 q^{43} + 21 q^{44} + 3 q^{46} - 29 q^{47} + 31 q^{49} + 44 q^{50} + 20 q^{52} - 63 q^{53} + q^{55} + 19 q^{56} - 3 q^{58} - 11 q^{59} - 33 q^{62} + 29 q^{64} - 53 q^{65} - 13 q^{67} - 63 q^{68} - 46 q^{70} + 23 q^{71} - 38 q^{73} + 47 q^{74} - 56 q^{76} - 97 q^{77} - 27 q^{79} - 8 q^{80} + 9 q^{82} - 36 q^{83} + 6 q^{85} + 11 q^{86} - 24 q^{88} + 16 q^{89} - 47 q^{91} - 21 q^{92} + 37 q^{94} + 12 q^{95} - 30 q^{97} + 27 q^{98}+O(q^{100}) $ 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8 - 14 * q^10 - 4 * q^11 + 15 * q^13 - 8 * q^14 + 23 * q^16 - 20 * q^17 - 4 * q^19 - 25 * q^20 + 13 * q^22 - 16 * q^23 + 30 * q^25 - 25 * q^26 - 13 * q^28 + 16 * q^29 + 19 * q^32 - 23 * q^34 - 5 * q^35 + 5 * q^37 - 38 * q^38 - 20 * q^40 - 7 * q^41 - 17 * q^43 + 21 * q^44 + 3 * q^46 - 29 * q^47 + 31 * q^49 + 44 * q^50 + 20 * q^52 - 63 * q^53 + q^55 + 19 * q^56 - 3 * q^58 - 11 * q^59 - 33 * q^62 + 29 * q^64 - 53 * q^65 - 13 * q^67 - 63 * q^68 - 46 * q^70 + 23 * q^71 - 38 * q^73 + 47 * q^74 - 56 * q^76 - 97 * q^77 - 27 * q^79 - 8 * q^80 + 9 * q^82 - 36 * q^83 + 6 * q^85 + 11 * q^86 - 24 * q^88 + 16 * q^89 - 47 * q^91 - 21 * q^92 + 37 * q^94 + 12 * q^95 - 30 * q^97 + 27 * q^98

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 $

$2$	−0.745705	−0.527293	−0.263647	−	0.964619i	$-0.584925\pi$
$2$	−0.745705	−0.527293	−0.263647	+	0.964619i	$0.584925\pi$
$3$	0	0
$3$	0	0
$4$	−1.44392	−0.721962
$5$	2.81935	1.26085	0.630425	−	0.776250i	$-0.282880\pi$
$5$	2.81935	1.26085	0.630425	+	0.776250i	$0.282880\pi$
$6$	0	0
$7$	3.27962	1.23958	0.619791	−	0.784767i	$-0.287217\pi$
$7$	3.27962	1.23958	0.619791	+	0.784767i	$0.287217\pi$
$8$	2.56815	0.907979
$9$	0	0
$10$	−2.10240	−0.664838
$11$	−4.74029	−1.42925	−0.714626	−	0.699507i	$-0.753403\pi$
$11$	−4.74029	−1.42925	−0.714626	+	0.699507i	$0.753403\pi$
$12$	0	0
$13$	5.26065	1.45904	0.729521	−	0.683959i	$-0.239743\pi$
$13$	5.26065	1.45904	0.729521	+	0.683959i	$0.239743\pi$
$14$	−2.44563	−0.653623
$15$	0	0
$16$	0.972761	0.243190
$17$	−3.78228	−0.917337	−0.458668	−	0.888608i	$-0.651673\pi$
$17$	−3.78228	−0.917337	−0.458668	+	0.888608i	$0.651673\pi$
$18$	0	0
$19$	−5.67624	−1.30222	−0.651110	−	0.758983i	$-0.725696\pi$
$19$	−5.67624	−1.30222	−0.651110	+	0.758983i	$0.725696\pi$
$20$	−4.07092	−0.910286
$21$	0	0
$22$	3.53486	0.753635
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.94872	0.589745
$26$	−3.92289	−0.769343
$27$	0	0
$28$	−4.73553	−0.894930
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	3.34398	0.600597	0.300298	−	0.953845i	$-0.402914\pi$
$31$	3.34398	0.600597	0.300298	+	0.953845i	$0.402914\pi$
$32$	−5.86170	−1.03621
$33$	0	0
$34$	2.82046	0.483705
$35$	9.24640	1.56293
$36$	0	0
$37$	−7.39041	−1.21498	−0.607488	−	0.794329i	$-0.707823\pi$
$37$	−7.39041	−1.21498	−0.607488	+	0.794329i	$0.707823\pi$
$38$	4.23281	0.686652
$39$	0	0
$40$	7.24052	1.14483
$41$	−2.57493	−0.402137	−0.201068	−	0.979577i	$-0.564441\pi$
$41$	−2.57493	−0.402137	−0.201068	+	0.979577i	$0.564441\pi$
$42$	0	0
$43$	−2.01679	−0.307557	−0.153778	−	0.988105i	$-0.549144\pi$
$43$	−2.01679	−0.307557	−0.153778	+	0.988105i	$0.549144\pi$
$44$	6.84462	1.03187
$45$	0	0
$46$	0.745705	0.109948
$47$	−1.24351	−0.181385	−0.0906925	−	0.995879i	$-0.528908\pi$
$47$	−1.24351	−0.181385	−0.0906925	+	0.995879i	$0.528908\pi$
$48$	0	0
$49$	3.75593	0.536562
$50$	−2.19888	−0.310969
$51$	0	0
$52$	−7.59597	−1.05337
$53$	−12.2633	−1.68449	−0.842247	−	0.539092i	$-0.818767\pi$
$53$	−12.2633	−1.68449	−0.842247	+	0.539092i	$0.818767\pi$
$54$	0	0
$55$	−13.3645	−1.80207
$56$	8.42257	1.12551
$57$	0	0
$58$	−0.745705	−0.0979159
$59$	0.582654	0.0758551	0.0379276	−	0.999280i	$-0.487924\pi$
$59$	0.582654	0.0758551	0.0379276	+	0.999280i	$0.487924\pi$
$60$	0	0
$61$	−12.5729	−1.60980	−0.804900	−	0.593410i	$-0.797781\pi$
$61$	−12.5729	−1.60980	−0.804900	+	0.593410i	$0.797781\pi$
$62$	−2.49362	−0.316691
$63$	0	0
$64$	2.42558	0.303197
$65$	14.8316	1.83963
$66$	0	0
$67$	−6.34363	−0.774998	−0.387499	−	0.921870i	$-0.626661\pi$
$67$	−6.34363	−0.774998	−0.387499	+	0.921870i	$0.626661\pi$
$68$	5.46132	0.662282
$69$	0	0
$70$	−6.89509	−0.824121
$71$	1.00328	0.119067	0.0595335	−	0.998226i	$-0.481039\pi$
$71$	1.00328	0.119067	0.0595335	+	0.998226i	$0.481039\pi$
$72$	0	0
$73$	−15.9306	−1.86454	−0.932270	−	0.361764i	$-0.882175\pi$
$73$	−15.9306	−1.86454	−0.932270	+	0.361764i	$0.882175\pi$
$74$	5.51107	0.640649
$75$	0	0
$76$	8.19606	0.940153
$77$	−15.5464	−1.77167
$78$	0	0
$79$	−14.7676	−1.66149	−0.830743	−	0.556656i	$-0.812084\pi$
$79$	−14.7676	−1.66149	−0.830743	+	0.556656i	$0.812084\pi$
$80$	2.74255	0.306627
$81$	0	0
$82$	1.92014	0.212044
$83$	−13.9060	−1.52638	−0.763189	−	0.646176i	$-0.776367\pi$
$83$	−13.9060	−1.52638	−0.763189	+	0.646176i	$0.776367\pi$
$84$	0	0
$85$	−10.6636	−1.15662
$86$	1.50393	0.162173
$87$	0	0
$88$	−12.1738	−1.29773
$89$	11.3752	1.20577	0.602883	−	0.797830i	$-0.294019\pi$
$89$	11.3752	1.20577	0.602883	+	0.797830i	$0.294019\pi$
$90$	0	0
$91$	17.2529	1.80860
$92$	1.44392	0.150539
$93$	0	0
$94$	0.927294	0.0956431
$95$	−16.0033	−1.64190
$96$	0	0
$97$	−12.8661	−1.30635	−0.653176	−	0.757206i	$-0.726564\pi$
$97$	−12.8661	−1.30635	−0.653176	+	0.757206i	$0.726564\pi$
$98$	−2.80082	−0.282926
$99$	0	0
$100$	−4.25773	−0.425773
$101$	−12.1573	−1.20969	−0.604847	−	0.796342i	$-0.706766\pi$
$101$	−12.1573	−1.20969	−0.604847	+	0.796342i	$0.706766\pi$
$102$	0	0
$103$	4.81864	0.474795	0.237397	−	0.971413i	$-0.423706\pi$
$103$	4.81864	0.474795	0.237397	+	0.971413i	$0.423706\pi$
$104$	13.5101	1.32478
$105$	0	0
$106$	9.14481	0.888222
$107$	5.97132	0.577269	0.288635	−	0.957439i	$-0.406799\pi$
$107$	5.97132	0.577269	0.288635	+	0.957439i	$0.406799\pi$
$108$	0	0
$109$	7.82520	0.749518	0.374759	−	0.927122i	$-0.377725\pi$
$109$	7.82520	0.749518	0.374759	+	0.927122i	$0.377725\pi$
$110$	9.96601	0.950222
$111$	0	0
$112$	3.19029	0.301454
$113$	14.3887	1.35358	0.676789	−	0.736177i	$-0.263371\pi$
$113$	14.3887	1.35358	0.676789	+	0.736177i	$0.263371\pi$
$114$	0	0
$115$	−2.81935	−0.262906
$116$	−1.44392	−0.134065
$117$	0	0
$118$	−0.434488	−0.0399979
$119$	−12.4044	−1.13711
$120$	0	0
$121$	11.4704	1.04276
$122$	9.37571	0.848837
$123$	0	0
$124$	−4.82845	−0.433608
$125$	−5.78326	−0.517270
$126$	0	0
$127$	19.2021	1.70391	0.851954	−	0.523617i	$-0.175418\pi$
$127$	19.2021	1.70391	0.851954	+	0.523617i	$0.175418\pi$
$128$	9.91463	0.876338
$129$	0	0
$130$	−11.0600	−0.970027
$131$	6.22530	0.543907	0.271954	−	0.962310i	$-0.412330\pi$
$131$	6.22530	0.543907	0.271954	+	0.962310i	$0.412330\pi$
$132$	0	0
$133$	−18.6159	−1.61421
$134$	4.73048	0.408651
$135$	0	0
$136$	−9.71346	−0.832922
$137$	6.97582	0.595984	0.297992	−	0.954568i	$-0.403683\pi$
$137$	6.97582	0.595984	0.297992	+	0.954568i	$0.403683\pi$
$138$	0	0
$139$	14.6331	1.24116	0.620580	−	0.784143i	$-0.286897\pi$
$139$	14.6331	1.24116	0.620580	+	0.784143i	$0.286897\pi$
$140$	−13.3511	−1.12837
$141$	0	0
$142$	−0.748148	−0.0627832
$143$	−24.9370	−2.08534
$144$	0	0
$145$	2.81935	0.234134
$146$	11.8796	0.983159
$147$	0	0
$148$	10.6712	0.877166
$149$	−3.56243	−0.291845	−0.145923	−	0.989296i	$-0.546615\pi$
$149$	−3.56243	−0.291845	−0.145923	+	0.989296i	$0.546615\pi$
$150$	0	0
$151$	9.71624	0.790696	0.395348	−	0.918531i	$-0.370624\pi$
$151$	9.71624	0.790696	0.395348	+	0.918531i	$0.370624\pi$
$152$	−14.5775	−1.18239
$153$	0	0
$154$	11.5930	0.934192
$155$	9.42785	0.757263
$156$	0	0
$157$	17.6117	1.40557	0.702785	−	0.711403i	$-0.251940\pi$
$157$	17.6117	1.40557	0.702785	+	0.711403i	$0.251940\pi$
$158$	11.0123	0.876090
$159$	0	0
$160$	−16.5262	−1.30651
$161$	−3.27962	−0.258471
$162$	0	0
$163$	0.409752	0.0320942	0.0160471	−	0.999871i	$-0.494892\pi$
$163$	0.409752	0.0320942	0.0160471	+	0.999871i	$0.494892\pi$
$164$	3.71800	0.290327
$165$	0	0
$166$	10.3697	0.804849
$167$	8.82190	0.682659	0.341329	−	0.939944i	$-0.389123\pi$
$167$	8.82190	0.682659	0.341329	+	0.939944i	$0.389123\pi$
$168$	0	0
$169$	14.6744	1.12880
$170$	7.95187	0.609880
$171$	0	0
$172$	2.91208	0.222044
$173$	−6.02690	−0.458216	−0.229108	−	0.973401i	$-0.573581\pi$
$173$	−6.02690	−0.458216	−0.229108	+	0.973401i	$0.573581\pi$
$174$	0	0
$175$	9.67071	0.731037
$176$	−4.61118	−0.347580
$177$	0	0
$178$	−8.48253	−0.635792
$179$	−15.3815	−1.14967	−0.574834	−	0.818270i	$-0.694933\pi$
$179$	−15.3815	−1.14967	−0.574834	+	0.818270i	$0.694933\pi$
$180$	0	0
$181$	−16.2761	−1.20979	−0.604897	−	0.796304i	$-0.706786\pi$
$181$	−16.2761	−1.20979	−0.604897	+	0.796304i	$0.706786\pi$
$182$	−12.8656	−0.953663
$183$	0	0
$184$	−2.56815	−0.189327
$185$	−20.8361	−1.53190
$186$	0	0
$187$	17.9291	1.31111
$188$	1.79554	0.130953
$189$	0	0
$190$	11.9338	0.865766
$191$	19.7942	1.43226	0.716129	−	0.697968i	$-0.245912\pi$
$191$	19.7942	1.43226	0.716129	+	0.697968i	$0.245912\pi$
$192$	0	0
$193$	−13.3338	−0.959787	−0.479893	−	0.877327i	$-0.659325\pi$
$193$	−13.3338	−0.959787	−0.479893	+	0.877327i	$0.659325\pi$
$194$	9.59430	0.688831
$195$	0	0
$196$	−5.42328	−0.387377
$197$	15.0077	1.06926	0.534628	−	0.845088i	$-0.320452\pi$
$197$	15.0077	1.06926	0.534628	+	0.845088i	$0.320452\pi$
$198$	0	0
$199$	−14.7649	−1.04666	−0.523329	−	0.852130i	$-0.675310\pi$
$199$	−14.7649	−1.04666	−0.523329	+	0.852130i	$0.675310\pi$
$200$	7.57277	0.535476
$201$	0	0
$202$	9.06575	0.637864
$203$	3.27962	0.230184
$204$	0	0
$205$	−7.25963	−0.507034
$206$	−3.59329	−0.250356
$207$	0	0
$208$	5.11736	0.354825
$209$	26.9071	1.86120
$210$	0	0
$211$	1.23566	0.0850661	0.0425331	−	0.999095i	$-0.486457\pi$
$211$	1.23566	0.0850661	0.0425331	+	0.999095i	$0.486457\pi$
$212$	17.7073	1.21614
$213$	0	0
$214$	−4.45285	−0.304390
$215$	−5.68602	−0.387783
$216$	0	0
$217$	10.9670	0.744488
$218$	−5.83529	−0.395216
$219$	0	0
$220$	19.2974	1.30103
$221$	−19.8972	−1.33843
$222$	0	0
$223$	8.41801	0.563712	0.281856	−	0.959457i	$-0.409050\pi$
$223$	8.41801	0.563712	0.281856	+	0.959457i	$0.409050\pi$
$224$	−19.2242	−1.28447
$225$	0	0
$226$	−10.7298	−0.713733
$227$	4.12583	0.273841	0.136920	−	0.990582i	$-0.456280\pi$
$227$	4.12583	0.273841	0.136920	+	0.990582i	$0.456280\pi$
$228$	0	0
$229$	−24.7522	−1.63567	−0.817836	−	0.575452i	$-0.804826\pi$
$229$	−24.7522	−1.63567	−0.817836	+	0.575452i	$0.804826\pi$
$230$	2.10240	0.138628
$231$	0	0
$232$	2.56815	0.168607
$233$	−1.60657	−0.105250	−0.0526249	−	0.998614i	$-0.516759\pi$
$233$	−1.60657	−0.105250	−0.0526249	+	0.998614i	$0.516759\pi$
$234$	0	0
$235$	−3.50589	−0.228699
$236$	−0.841308	−0.0547645
$237$	0	0
$238$	9.25006	0.599592
$239$	−8.40977	−0.543983	−0.271991	−	0.962300i	$-0.587682\pi$
$239$	−8.40977	−0.543983	−0.271991	+	0.962300i	$0.587682\pi$
$240$	0	0
$241$	9.41350	0.606377	0.303188	−	0.952931i	$-0.401949\pi$
$241$	9.41350	0.606377	0.303188	+	0.952931i	$0.401949\pi$
$242$	−8.55353	−0.549842
$243$	0	0
$244$	18.1544	1.16221
$245$	10.5893	0.676525
$246$	0	0
$247$	−29.8607	−1.89999
$248$	8.58785	0.545329
$249$	0	0
$250$	4.31261	0.272753
$251$	−4.29577	−0.271147	−0.135573	−	0.990767i	$-0.543288\pi$
$251$	−4.29577	−0.271147	−0.135573	+	0.990767i	$0.543288\pi$
$252$	0	0
$253$	4.74029	0.298020
$254$	−14.3191	−0.898459
$255$	0	0
$256$	−12.2445	−0.765284
$257$	−2.32487	−0.145021	−0.0725106	−	0.997368i	$-0.523101\pi$
$257$	−2.32487	−0.145021	−0.0725106	+	0.997368i	$0.523101\pi$
$258$	0	0
$259$	−24.2378	−1.50606
$260$	−21.4157	−1.32814
$261$	0	0
$262$	−4.64224	−0.286799
$263$	23.0125	1.41901	0.709507	−	0.704698i	$-0.248918\pi$
$263$	23.0125	1.41901	0.709507	+	0.704698i	$0.248918\pi$
$264$	0	0
$265$	−34.5745	−2.12389
$266$	13.8820	0.851161
$267$	0	0
$268$	9.15972	0.559519
$269$	−21.0938	−1.28611	−0.643055	−	0.765820i	$-0.722334\pi$
$269$	−21.0938	−1.28611	−0.643055	+	0.765820i	$0.722334\pi$
$270$	0	0
$271$	30.5448	1.85547	0.927733	−	0.373245i	$-0.121755\pi$
$271$	30.5448	1.85547	0.927733	+	0.373245i	$0.121755\pi$
$272$	−3.67925	−0.223087
$273$	0	0
$274$	−5.20190	−0.314259
$275$	−13.9778	−0.842894
$276$	0	0
$277$	12.8468	0.771889	0.385944	−	0.922522i	$-0.373876\pi$
$277$	12.8468	0.771889	0.385944	+	0.922522i	$0.373876\pi$
$278$	−10.9120	−0.654456
$279$	0	0
$280$	23.7462	1.41911
$281$	−27.5782	−1.64518	−0.822590	−	0.568635i	$-0.807472\pi$
$281$	−27.5782	−1.64518	−0.822590	+	0.568635i	$0.807472\pi$
$282$	0	0
$283$	−8.83683	−0.525295	−0.262647	−	0.964892i	$-0.584596\pi$
$283$	−8.83683	−0.525295	−0.262647	+	0.964892i	$0.584596\pi$
$284$	−1.44865	−0.0859618
$285$	0	0
$286$	18.5957	1.09959
$287$	−8.44481	−0.498481
$288$	0	0
$289$	−2.69439	−0.158494
$290$	−2.10240	−0.123457
$291$	0	0
$292$	23.0026	1.34613
$293$	−9.10701	−0.532037	−0.266019	−	0.963968i	$-0.585708\pi$
$293$	−9.10701	−0.532037	−0.266019	+	0.963968i	$0.585708\pi$
$294$	0	0
$295$	1.64271	0.0956420
$296$	−18.9797	−1.10317
$297$	0	0
$298$	2.65652	0.153888
$299$	−5.26065	−0.304231
$300$	0	0
$301$	−6.61430	−0.381242
$302$	−7.24545	−0.416929
$303$	0	0
$304$	−5.52163	−0.316687
$305$	−35.4475	−2.02972
$306$	0	0
$307$	−7.67967	−0.438302	−0.219151	−	0.975691i	$-0.570329\pi$
$307$	−7.67967	−0.438302	−0.219151	+	0.975691i	$0.570329\pi$
$308$	22.4478	1.27908
$309$	0	0
$310$	−7.03040	−0.399300
$311$	0.298680	0.0169366	0.00846828	−	0.999964i	$-0.497304\pi$
$311$	0.298680	0.0169366	0.00846828	+	0.999964i	$0.497304\pi$
$312$	0	0
$313$	−6.69575	−0.378466	−0.189233	−	0.981932i	$-0.560600\pi$
$313$	−6.69575	−0.378466	−0.189233	+	0.981932i	$0.560600\pi$
$314$	−13.1332	−0.741147
$315$	0	0
$316$	21.3233	1.19953
$317$	10.8920	0.611757	0.305879	−	0.952070i	$-0.401050\pi$
$317$	10.8920	0.611757	0.305879	+	0.952070i	$0.401050\pi$
$318$	0	0
$319$	−4.74029	−0.265406
$320$	6.83855	0.382286
$321$	0	0
$322$	2.44563	0.136290
$323$	21.4691	1.19457
$324$	0	0
$325$	15.5122	0.860462
$326$	−0.305554	−0.0169231
$327$	0	0
$328$	−6.61282	−0.365132
$329$	−4.07825	−0.224841
$330$	0	0
$331$	34.7209	1.90843	0.954215	−	0.299121i	$-0.0966933\pi$
$331$	34.7209	1.90843	0.954215	+	0.299121i	$0.0966933\pi$
$332$	20.0791	1.10199
$333$	0	0
$334$	−6.57854	−0.359961
$335$	−17.8849	−0.977157
$336$	0	0
$337$	26.1652	1.42531	0.712655	−	0.701515i	$-0.247493\pi$
$337$	26.1652	1.42531	0.712655	+	0.701515i	$0.247493\pi$
$338$	−10.9428	−0.595209
$339$	0	0
$340$	15.3974	0.835039
$341$	−15.8514	−0.858404
$342$	0	0
$343$	−10.6393	−0.574469
$344$	−5.17941	−0.279255
$345$	0	0
$346$	4.49429	0.241615
$347$	28.4182	1.52557	0.762785	−	0.646652i	$-0.223832\pi$
$347$	28.4182	1.52557	0.762785	+	0.646652i	$0.223832\pi$
$348$	0	0
$349$	14.0179	0.750359	0.375180	−	0.926952i	$-0.377581\pi$
$349$	14.0179	0.750359	0.375180	+	0.926952i	$0.377581\pi$
$350$	−7.21150	−0.385471
$351$	0	0
$352$	27.7862	1.48101
$353$	22.0953	1.17601	0.588007	−	0.808856i	$-0.299913\pi$
$353$	22.0953	1.17601	0.588007	+	0.808856i	$0.299913\pi$
$354$	0	0
$355$	2.82858	0.150126
$356$	−16.4249	−0.870517
$357$	0	0
$358$	11.4701	0.606212
$359$	−15.7837	−0.833030	−0.416515	−	0.909129i	$-0.636749\pi$
$359$	−15.7837	−0.833030	−0.416515	+	0.909129i	$0.636749\pi$
$360$	0	0
$361$	13.2197	0.695776
$362$	12.1372	0.637917
$363$	0	0
$364$	−24.9119	−1.30574
$365$	−44.9140	−2.35091
$366$	0	0
$367$	3.85433	0.201194	0.100597	−	0.994927i	$-0.467925\pi$
$367$	3.85433	0.201194	0.100597	+	0.994927i	$0.467925\pi$
$368$	−0.972761	−0.0507087
$369$	0	0
$370$	15.5376	0.807763
$371$	−40.2190	−2.08807
$372$	0	0
$373$	−21.1675	−1.09601	−0.548007	−	0.836474i	$-0.684613\pi$
$373$	−21.1675	−1.09601	−0.548007	+	0.836474i	$0.684613\pi$
$374$	−13.3698	−0.691337
$375$	0	0
$376$	−3.19353	−0.164694
$377$	5.26065	0.270937
$378$	0	0
$379$	−17.0072	−0.873602	−0.436801	−	0.899558i	$-0.643889\pi$
$379$	−17.0072	−0.873602	−0.436801	+	0.899558i	$0.643889\pi$
$380$	23.1076	1.18539
$381$	0	0
$382$	−14.7606	−0.755220
$383$	−5.72087	−0.292323	−0.146161	−	0.989261i	$-0.546692\pi$
$383$	−5.72087	−0.292323	−0.146161	+	0.989261i	$0.546692\pi$
$384$	0	0
$385$	−43.8307	−2.23382
$386$	9.94308	0.506089
$387$	0	0
$388$	18.5776	0.943136
$389$	−13.7727	−0.698304	−0.349152	−	0.937066i	$-0.613530\pi$
$389$	−13.7727	−0.698304	−0.349152	+	0.937066i	$0.613530\pi$
$390$	0	0
$391$	3.78228	0.191278
$392$	9.64581	0.487187
$393$	0	0
$394$	−11.1913	−0.563811
$395$	−41.6350	−2.09489
$396$	0	0
$397$	−19.0097	−0.954071	−0.477035	−	0.878884i	$-0.658289\pi$
$397$	−19.0097	−0.954071	−0.477035	+	0.878884i	$0.658289\pi$
$398$	11.0103	0.551896
$399$	0	0
$400$	2.86841	0.143420
$401$	−19.5467	−0.976116	−0.488058	−	0.872811i	$-0.662294\pi$
$401$	−19.5467	−0.976116	−0.488058	+	0.872811i	$0.662294\pi$
$402$	0	0
$403$	17.5915	0.876295
$404$	17.5542	0.873353
$405$	0	0
$406$	−2.44563	−0.121375
$407$	35.0327	1.73651
$408$	0	0
$409$	5.43515	0.268751	0.134375	−	0.990931i	$-0.457097\pi$
$409$	5.43515	0.268751	0.134375	+	0.990931i	$0.457097\pi$
$410$	5.41354	0.267356
$411$	0	0
$412$	−6.95775	−0.342784
$413$	1.91089	0.0940286
$414$	0	0
$415$	−39.2057	−1.92453
$416$	−30.8363	−1.51188
$417$	0	0
$418$	−20.0647	−0.981399
$419$	−0.635042	−0.0310238	−0.0155119	−	0.999880i	$-0.504938\pi$
$419$	−0.635042	−0.0310238	−0.0155119	+	0.999880i	$0.504938\pi$
$420$	0	0
$421$	−38.7954	−1.89077	−0.945385	−	0.325956i	$-0.894314\pi$
$421$	−38.7954	−1.89077	−0.945385	+	0.325956i	$0.894314\pi$
$422$	−0.921436	−0.0448548
$423$	0	0
$424$	−31.4940	−1.52948
$425$	−11.1529	−0.540995
$426$	0	0
$427$	−41.2345	−1.99548
$428$	−8.62213	−0.416766
$429$	0	0
$430$	4.24010	0.204476
$431$	19.2802	0.928695	0.464348	−	0.885653i	$-0.346289\pi$
$431$	19.2802	0.928695	0.464348	+	0.885653i	$0.346289\pi$
$432$	0	0
$433$	−17.9157	−0.860975	−0.430488	−	0.902596i	$-0.641658\pi$
$433$	−17.9157	−0.860975	−0.430488	+	0.902596i	$0.641658\pi$
$434$	−8.17815	−0.392564
$435$	0	0
$436$	−11.2990	−0.541123
$437$	5.67624	0.271532
$438$	0	0
$439$	8.35812	0.398911	0.199456	−	0.979907i	$-0.436083\pi$
$439$	8.35812	0.398911	0.199456	+	0.979907i	$0.436083\pi$
$440$	−34.3222	−1.63625
$441$	0	0
$442$	14.8375	0.705746
$443$	16.2571	0.772399	0.386200	−	0.922415i	$-0.373788\pi$
$443$	16.2571	0.772399	0.386200	+	0.922415i	$0.373788\pi$
$444$	0	0
$445$	32.0706	1.52029
$446$	−6.27736	−0.297242
$447$	0	0
$448$	7.95498	0.375838
$449$	27.3794	1.29211	0.646057	−	0.763289i	$-0.276417\pi$
$449$	27.3794	1.29211	0.646057	+	0.763289i	$0.276417\pi$
$450$	0	0
$451$	12.2059	0.574755
$452$	−20.7762	−0.977232
$453$	0	0
$454$	−3.07665	−0.144394
$455$	48.6421	2.28038
$456$	0	0
$457$	−15.4647	−0.723406	−0.361703	−	0.932293i	$-0.617805\pi$
$457$	−15.4647	−0.723406	−0.361703	+	0.932293i	$0.617805\pi$
$458$	18.4578	0.862479
$459$	0	0
$460$	4.07092	0.189808
$461$	−18.3026	−0.852435	−0.426218	−	0.904621i	$-0.640154\pi$
$461$	−18.3026	−0.852435	−0.426218	+	0.904621i	$0.640154\pi$
$462$	0	0
$463$	5.81224	0.270118	0.135059	−	0.990838i	$-0.456878\pi$
$463$	5.81224	0.270118	0.135059	+	0.990838i	$0.456878\pi$
$464$	0.972761	0.0451593
$465$	0	0
$466$	1.19803	0.0554975
$467$	25.4048	1.17559	0.587797	−	0.809008i	$-0.299995\pi$
$467$	25.4048	1.17559	0.587797	+	0.809008i	$0.299995\pi$
$468$	0	0
$469$	−20.8047	−0.960673
$470$	2.61436	0.120592
$471$	0	0
$472$	1.49634	0.0688749
$473$	9.56016	0.439576
$474$	0	0
$475$	−16.7377	−0.767977
$476$	17.9111	0.820952
$477$	0	0
$478$	6.27121	0.286838
$479$	7.68321	0.351055	0.175527	−	0.984475i	$-0.443837\pi$
$479$	7.68321	0.351055	0.175527	+	0.984475i	$0.443837\pi$
$480$	0	0
$481$	−38.8784	−1.77270
$482$	−7.01970	−0.319739
$483$	0	0
$484$	−16.5624	−0.752835
$485$	−36.2740	−1.64712
$486$	0	0
$487$	−14.8557	−0.673174	−0.336587	−	0.941652i	$-0.609273\pi$
$487$	−14.8557	−0.673174	−0.336587	+	0.941652i	$0.609273\pi$
$488$	−32.2892	−1.46166
$489$	0	0
$490$	−7.89649	−0.356727
$491$	−15.8512	−0.715353	−0.357676	−	0.933846i	$-0.616431\pi$
$491$	−15.8512	−0.715353	−0.357676	+	0.933846i	$0.616431\pi$
$492$	0	0
$493$	−3.78228	−0.170345
$494$	22.2673	1.00185
$495$	0	0
$496$	3.25289	0.146059
$497$	3.29037	0.147593
$498$	0	0
$499$	15.1100	0.676416	0.338208	−	0.941071i	$-0.390179\pi$
$499$	15.1100	0.676416	0.338208	+	0.941071i	$0.390179\pi$
$500$	8.35058	0.373449
$501$	0	0
$502$	3.20338	0.142974
$503$	−27.5406	−1.22798	−0.613988	−	0.789315i	$-0.710436\pi$
$503$	−27.5406	−1.22798	−0.613988	+	0.789315i	$0.710436\pi$
$504$	0	0
$505$	−34.2756	−1.52524
$506$	−3.53486	−0.157144
$507$	0	0
$508$	−27.7263	−1.23016
$509$	13.8531	0.614026	0.307013	−	0.951705i	$-0.400671\pi$
$509$	13.8531	0.614026	0.307013	+	0.951705i	$0.400671\pi$
$510$	0	0
$511$	−52.2465	−2.31125
$512$	−10.6984	−0.472808
$513$	0	0
$514$	1.73366	0.0764687
$515$	13.5854	0.598646
$516$	0	0
$517$	5.89461	0.259245
$518$	18.0742	0.794136
$519$	0	0
$520$	38.0898	1.67035
$521$	−12.3185	−0.539682	−0.269841	−	0.962905i	$-0.586971\pi$
$521$	−12.3185	−0.539682	−0.269841	+	0.962905i	$0.586971\pi$
$522$	0	0
$523$	17.6875	0.773422	0.386711	−	0.922201i	$-0.373611\pi$
$523$	17.6875	0.773422	0.386711	+	0.922201i	$0.373611\pi$
$524$	−8.98886	−0.392680
$525$	0	0
$526$	−17.1606	−0.748237
$527$	−12.6479	−0.550949
$528$	0	0
$529$	1.00000	0.0434783
$530$	25.7824	1.11992
$531$	0	0
$532$	26.8800	1.16540
$533$	−13.5458	−0.586734
$534$	0	0
$535$	16.8352	0.727851
$536$	−16.2914	−0.703682
$537$	0	0
$538$	15.7298	0.678158
$539$	−17.8042	−0.766883
$540$	0	0
$541$	27.3029	1.17384	0.586922	−	0.809643i	$-0.300340\pi$
$541$	27.3029	1.17384	0.586922	+	0.809643i	$0.300340\pi$
$542$	−22.7774	−0.978375
$543$	0	0
$544$	22.1706	0.950555
$545$	22.0620	0.945031
$546$	0	0
$547$	0.929830	0.0397567	0.0198783	−	0.999802i	$-0.493672\pi$
$547$	0.929830	0.0397567	0.0198783	+	0.999802i	$0.493672\pi$
$548$	−10.0725	−0.430278
$549$	0	0
$550$	10.4233	0.444453
$551$	−5.67624	−0.241816
$552$	0	0
$553$	−48.4322	−2.05955
$554$	−9.57992	−0.407012
$555$	0	0
$556$	−21.1290	−0.896070
$557$	−35.6377	−1.51002	−0.755010	−	0.655713i	$-0.772368\pi$
$557$	−35.6377	−1.51002	−0.755010	+	0.655713i	$0.772368\pi$
$558$	0	0
$559$	−10.6096	−0.448738
$560$	8.99454	0.380089
$561$	0	0
$562$	20.5652	0.867493
$563$	11.9646	0.504249	0.252124	−	0.967695i	$-0.418871\pi$
$563$	11.9646	0.504249	0.252124	+	0.967695i	$0.418871\pi$
$564$	0	0
$565$	40.5669	1.70666
$566$	6.58967	0.276985
$567$	0	0
$568$	2.57657	0.108110
$569$	−9.64998	−0.404548	−0.202274	−	0.979329i	$-0.564833\pi$
$569$	−9.64998	−0.404548	−0.202274	+	0.979329i	$0.564833\pi$
$570$	0	0
$571$	16.2945	0.681906	0.340953	−	0.940080i	$-0.389250\pi$
$571$	16.2945	0.681906	0.340953	+	0.940080i	$0.389250\pi$
$572$	36.0071	1.50553
$573$	0	0
$574$	6.29734	0.262846
$575$	−2.94872	−0.122970
$576$	0	0
$577$	−2.04221	−0.0850184	−0.0425092	−	0.999096i	$-0.513535\pi$
$577$	−2.04221	−0.0850184	−0.0425092	+	0.999096i	$0.513535\pi$
$578$	2.00922	0.0835727
$579$	0	0
$580$	−4.07092	−0.169036
$581$	−45.6063	−1.89207
$582$	0	0
$583$	58.1316	2.40757
$584$	−40.9123	−1.69296
$585$	0	0
$586$	6.79115	0.280540
$587$	−5.51573	−0.227659	−0.113829	−	0.993500i	$-0.536312\pi$
$587$	−5.51573	−0.227659	−0.113829	+	0.993500i	$0.536312\pi$
$588$	0	0
$589$	−18.9812	−0.782109
$590$	−1.22497	−0.0504314
$591$	0	0
$592$	−7.18911	−0.295470
$593$	32.7365	1.34433	0.672163	−	0.740403i	$-0.265365\pi$
$593$	32.7365	1.34433	0.672163	+	0.740403i	$0.265365\pi$
$594$	0	0
$595$	−34.9724	−1.43373
$596$	5.14387	0.210701
$597$	0	0
$598$	3.92289	0.160419
$599$	−12.0571	−0.492641	−0.246321	−	0.969188i	$-0.579222\pi$
$599$	−12.0571	−0.492641	−0.246321	+	0.969188i	$0.579222\pi$
$600$	0	0
$601$	1.42188	0.0579996	0.0289998	−	0.999579i	$-0.490768\pi$
$601$	1.42188	0.0579996	0.0289998	+	0.999579i	$0.490768\pi$
$602$	4.93232	0.201026
$603$	0	0
$604$	−14.0295	−0.570852
$605$	32.3390	1.31477
$606$	0	0
$607$	−11.5014	−0.466829	−0.233415	−	0.972377i	$-0.574990\pi$
$607$	−11.5014	−0.466829	−0.233415	+	0.972377i	$0.574990\pi$
$608$	33.2724	1.34938
$609$	0	0
$610$	26.4334	1.07026
$611$	−6.54168	−0.264648
$612$	0	0
$613$	6.72128	0.271470	0.135735	−	0.990745i	$-0.456660\pi$
$613$	6.72128	0.271470	0.135735	+	0.990745i	$0.456660\pi$
$614$	5.72677	0.231114
$615$	0	0
$616$	−39.9255	−1.60864
$617$	−3.79947	−0.152961	−0.0764805	−	0.997071i	$-0.524368\pi$
$617$	−3.79947	−0.152961	−0.0764805	+	0.997071i	$0.524368\pi$
$618$	0	0
$619$	−25.5907	−1.02858	−0.514288	−	0.857618i	$-0.671944\pi$
$619$	−25.5907	−1.02858	−0.514288	+	0.857618i	$0.671944\pi$
$620$	−13.6131	−0.546715
$621$	0	0
$622$	−0.222727	−0.00893054
$623$	37.3063	1.49464
$624$	0	0
$625$	−31.0486	−1.24195
$626$	4.99306	0.199563
$627$	0	0
$628$	−25.4300	−1.01477
$629$	27.9526	1.11454
$630$	0	0
$631$	29.8982	1.19023	0.595113	−	0.803642i	$-0.297107\pi$
$631$	29.8982	1.19023	0.595113	+	0.803642i	$0.297107\pi$
$632$	−37.9255	−1.50859
$633$	0	0
$634$	−8.12225	−0.322576
$635$	54.1373	2.14837
$636$	0	0
$637$	19.7587	0.782866
$638$	3.53486	0.139947
$639$	0	0
$640$	27.9528	1.10493
$641$	19.3111	0.762743	0.381371	−	0.924422i	$-0.375452\pi$
$641$	19.3111	0.762743	0.381371	+	0.924422i	$0.375452\pi$
$642$	0	0
$643$	24.2854	0.957721	0.478861	−	0.877891i	$-0.341050\pi$
$643$	24.2854	0.957721	0.478861	+	0.877891i	$0.341050\pi$
$644$	4.73553	0.186606
$645$	0	0
$646$	−16.0096	−0.629891
$647$	−14.0122	−0.550876	−0.275438	−	0.961319i	$-0.588823\pi$
$647$	−14.0122	−0.550876	−0.275438	+	0.961319i	$0.588823\pi$
$648$	0	0
$649$	−2.76195	−0.108416
$650$	−11.5675	−0.453716
$651$	0	0
$652$	−0.591650	−0.0231708
$653$	−10.7049	−0.418915	−0.209458	−	0.977818i	$-0.567170\pi$
$653$	−10.7049	−0.418915	−0.209458	+	0.977818i	$0.567170\pi$
$654$	0	0
$655$	17.5513	0.685786
$656$	−2.50479	−0.0977958
$657$	0	0
$658$	3.04118	0.118557
$659$	−43.1485	−1.68083	−0.840413	−	0.541947i	$-0.817687\pi$
$659$	−43.1485	−1.68083	−0.840413	+	0.541947i	$0.817687\pi$
$660$	0	0
$661$	1.22019	0.0474599	0.0237299	−	0.999718i	$-0.492446\pi$
$661$	1.22019	0.0474599	0.0237299	+	0.999718i	$0.492446\pi$
$662$	−25.8915	−1.00630
$663$	0	0
$664$	−35.7126	−1.38592
$665$	−52.4848	−2.03527
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	−12.7381	−0.492854
$669$	0	0
$670$	13.3369	0.515248
$671$	59.5994	2.30081
$672$	0	0
$673$	0.712833	0.0274777	0.0137388	−	0.999906i	$-0.495627\pi$
$673$	0.712833	0.0274777	0.0137388	+	0.999906i	$0.495627\pi$
$674$	−19.5115	−0.751556
$675$	0	0
$676$	−21.1887	−0.814951
$677$	−29.3600	−1.12840	−0.564198	−	0.825640i	$-0.690815\pi$
$677$	−29.3600	−1.12840	−0.564198	+	0.825640i	$0.690815\pi$
$678$	0	0
$679$	−42.1959	−1.61933
$680$	−27.3856	−1.05019
$681$	0	0
$682$	11.8205	0.452631
$683$	−27.5282	−1.05334	−0.526668	−	0.850071i	$-0.676559\pi$
$683$	−27.5282	−1.05334	−0.526668	+	0.850071i	$0.676559\pi$
$684$	0	0
$685$	19.6673	0.751447
$686$	7.93379	0.302914
$687$	0	0
$688$	−1.96185	−0.0747949
$689$	−64.5129	−2.45775
$690$	0	0
$691$	−40.1502	−1.52739	−0.763694	−	0.645579i	$-0.776616\pi$
$691$	−40.1502	−1.52739	−0.763694	+	0.645579i	$0.776616\pi$
$692$	8.70238	0.330815
$693$	0	0
$694$	−21.1916	−0.804423
$695$	41.2557	1.56492
$696$	0	0
$697$	9.73910	0.368895
$698$	−10.4532	−0.395660
$699$	0	0
$700$	−13.9638	−0.527781
$701$	44.3429	1.67481	0.837404	−	0.546585i	$-0.184073\pi$
$701$	44.3429	1.67481	0.837404	+	0.546585i	$0.184073\pi$
$702$	0	0
$703$	41.9498	1.58217
$704$	−11.4980	−0.433345
$705$	0	0
$706$	−16.4766	−0.620104
$707$	−39.8713	−1.49951
$708$	0	0
$709$	−4.68949	−0.176118	−0.0880588	−	0.996115i	$-0.528066\pi$
$709$	−4.68949	−0.176118	−0.0880588	+	0.996115i	$0.528066\pi$
$710$	−2.10929	−0.0791603
$711$	0	0
$712$	29.2132	1.09481
$713$	−3.34398	−0.125233
$714$	0	0
$715$	−70.3061	−2.62930
$716$	22.2097	0.830016
$717$	0	0
$718$	11.7700	0.439251
$719$	−34.3501	−1.28104	−0.640521	−	0.767940i	$-0.721282\pi$
$719$	−34.3501	−1.28104	−0.640521	+	0.767940i	$0.721282\pi$
$720$	0	0
$721$	15.8033	0.588547
$722$	−9.85804	−0.366878
$723$	0	0
$724$	23.5015	0.873425
$725$	2.94872	0.109513
$726$	0	0
$727$	19.5770	0.726072	0.363036	−	0.931775i	$-0.381740\pi$
$727$	19.5770	0.726072	0.363036	+	0.931775i	$0.381740\pi$
$728$	44.3082	1.64217
$729$	0	0
$730$	33.4926	1.23962
$731$	7.62804	0.282133
$732$	0	0
$733$	23.0325	0.850727	0.425363	−	0.905023i	$-0.360146\pi$
$733$	23.0325	0.850727	0.425363	+	0.905023i	$0.360146\pi$
$734$	−2.87420	−0.106088
$735$	0	0
$736$	5.86170	0.216065
$737$	30.0707	1.10767
$738$	0	0
$739$	−20.3767	−0.749569	−0.374784	−	0.927112i	$-0.622283\pi$
$739$	−20.3767	−0.749569	−0.374784	+	0.927112i	$0.622283\pi$
$740$	30.0858	1.10598
$741$	0	0
$742$	29.9915	1.10102
$743$	18.2969	0.671249	0.335625	−	0.941996i	$-0.391053\pi$
$743$	18.2969	0.671249	0.335625	+	0.941996i	$0.391053\pi$
$744$	0	0
$745$	−10.0437	−0.367973
$746$	15.7848	0.577921
$747$	0	0
$748$	−25.8882	−0.946568
$749$	19.5837	0.715573
$750$	0	0
$751$	32.5971	1.18948	0.594742	−	0.803917i	$-0.297254\pi$
$751$	32.5971	1.18948	0.594742	+	0.803917i	$0.297254\pi$
$752$	−1.20964	−0.0441111
$753$	0	0
$754$	−3.92289	−0.142863
$755$	27.3935	0.996950
$756$	0	0
$757$	46.8094	1.70132	0.850659	−	0.525718i	$-0.176203\pi$
$757$	46.8094	1.70132	0.850659	+	0.525718i	$0.176203\pi$
$758$	12.6824	0.460645
$759$	0	0
$760$	−41.0989	−1.49082
$761$	−45.5881	−1.65257	−0.826283	−	0.563254i	$-0.809549\pi$
$761$	−45.5881	−1.65257	−0.826283	+	0.563254i	$0.809549\pi$
$762$	0	0
$763$	25.6637	0.929089
$764$	−28.5813	−1.03403
$765$	0	0
$766$	4.26609	0.154140
$767$	3.06514	0.110676
$768$	0	0
$769$	−45.0615	−1.62496	−0.812479	−	0.582990i	$-0.801883\pi$
$769$	−45.0615	−1.62496	−0.812479	+	0.582990i	$0.801883\pi$
$770$	32.6848	1.17788
$771$	0	0
$772$	19.2530	0.692929
$773$	−27.4818	−0.988451	−0.494225	−	0.869334i	$-0.664548\pi$
$773$	−27.4818	−0.988451	−0.494225	+	0.869334i	$0.664548\pi$
$774$	0	0
$775$	9.86048	0.354199
$776$	−33.0420	−1.18614
$777$	0	0
$778$	10.2704	0.368211
$779$	14.6159	0.523670
$780$	0	0
$781$	−4.75582	−0.170177
$782$	−2.82046	−0.100860
$783$	0	0
$784$	3.65363	0.130487
$785$	49.6536	1.77221
$786$	0	0
$787$	−30.4090	−1.08396	−0.541981	−	0.840391i	$-0.682326\pi$
$787$	−30.4090	−1.08396	−0.541981	+	0.840391i	$0.682326\pi$
$788$	−21.6700	−0.771961
$789$	0	0
$790$	31.0475	1.10462
$791$	47.1896	1.67787
$792$	0	0
$793$	−66.1418	−2.34876
$794$	14.1757	0.503075
$795$	0	0
$796$	21.3194	0.755648
$797$	−37.4295	−1.32582	−0.662911	−	0.748698i	$-0.730679\pi$
$797$	−37.4295	−1.32582	−0.662911	+	0.748698i	$0.730679\pi$
$798$	0	0
$799$	4.70331	0.166391
$800$	−17.2845	−0.611101
$801$	0	0
$802$	14.5761	0.514699
$803$	75.5159	2.66490
$804$	0	0
$805$	−9.24640	−0.325893
$806$	−13.1181	−0.462065
$807$	0	0
$808$	−31.2217	−1.09838
$809$	0.0861528	0.00302897	0.00151449	−	0.999999i	$-0.499518\pi$
$809$	0.0861528	0.00302897	0.00151449	+	0.999999i	$0.499518\pi$
$810$	0	0
$811$	−11.0489	−0.387979	−0.193990	−	0.981004i	$-0.562143\pi$
$811$	−11.0489	−0.387979	−0.193990	+	0.981004i	$0.562143\pi$
$812$	−4.73553	−0.166184
$813$	0	0
$814$	−26.1241	−0.915649
$815$	1.15523	0.0404661
$816$	0	0
$817$	11.4478	0.400507
$818$	−4.05302	−0.141711
$819$	0	0
$820$	10.4823	0.366059
$821$	53.1206	1.85392	0.926960	−	0.375160i	$-0.122412\pi$
$821$	53.1206	1.85392	0.926960	+	0.375160i	$0.122412\pi$
$822$	0	0
$823$	−4.06031	−0.141534	−0.0707668	−	0.997493i	$-0.522545\pi$
$823$	−4.06031	−0.141534	−0.0707668	+	0.997493i	$0.522545\pi$
$824$	12.3750	0.431104
$825$	0	0
$826$	−1.42496	−0.0495807
$827$	−0.603295	−0.0209786	−0.0104893	−	0.999945i	$-0.503339\pi$
$827$	−0.603295	−0.0209786	−0.0104893	+	0.999945i	$0.503339\pi$
$828$	0	0
$829$	27.3478	0.949828	0.474914	−	0.880032i	$-0.342479\pi$
$829$	27.3478	0.949828	0.474914	+	0.880032i	$0.342479\pi$
$830$	29.2359	1.01479
$831$	0	0
$832$	12.7601	0.442377
$833$	−14.2060	−0.492208
$834$	0	0
$835$	24.8720	0.860731
$836$	−38.8517	−1.34372
$837$	0	0
$838$	0.473554	0.0163587
$839$	45.0600	1.55565	0.777823	−	0.628484i	$-0.216324\pi$
$839$	45.0600	1.55565	0.777823	+	0.628484i	$0.216324\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	28.9299	0.996991
$843$	0	0
$844$	−1.78419	−0.0614145
$845$	41.3723	1.42325
$846$	0	0
$847$	37.6186	1.29259
$848$	−11.9293	−0.409653
$849$	0	0
$850$	8.31677	0.285263
$851$	7.39041	0.253340
$852$	0	0
$853$	16.1362	0.552492	0.276246	−	0.961087i	$-0.410909\pi$
$853$	16.1362	0.552492	0.276246	+	0.961087i	$0.410909\pi$
$854$	30.7488	1.05220
$855$	0	0
$856$	15.3353	0.524149
$857$	6.87649	0.234896	0.117448	−	0.993079i	$-0.462529\pi$
$857$	6.87649	0.234896	0.117448	+	0.993079i	$0.462529\pi$
$858$	0	0
$859$	8.84475	0.301779	0.150890	−	0.988551i	$-0.451786\pi$
$859$	8.84475	0.301779	0.150890	+	0.988551i	$0.451786\pi$
$860$	8.21018	0.279965
$861$	0	0
$862$	−14.3774	−0.489695
$863$	29.2923	0.997122	0.498561	−	0.866855i	$-0.333862\pi$
$863$	29.2923	0.997122	0.498561	+	0.866855i	$0.333862\pi$
$864$	0	0
$865$	−16.9919	−0.577743
$866$	13.3599	0.453987
$867$	0	0
$868$	−15.8355	−0.537492
$869$	70.0028	2.37468
$870$	0	0
$871$	−33.3716	−1.13075
$872$	20.0963	0.680547
$873$	0	0
$874$	−4.23281	−0.143177
$875$	−18.9669	−0.641199
$876$	0	0
$877$	20.5455	0.693774	0.346887	−	0.937907i	$-0.387239\pi$
$877$	20.5455	0.693774	0.346887	+	0.937907i	$0.387239\pi$
$878$	−6.23269	−0.210343
$879$	0	0
$880$	−13.0005	−0.438247
$881$	0.803678	0.0270766	0.0135383	−	0.999908i	$-0.495690\pi$
$881$	0.803678	0.0270766	0.0135383	+	0.999908i	$0.495690\pi$
$882$	0	0
$883$	−39.2541	−1.32101	−0.660503	−	0.750823i	$-0.729657\pi$
$883$	−39.2541	−1.32101	−0.660503	+	0.750823i	$0.729657\pi$
$884$	28.7301	0.966296
$885$	0	0
$886$	−12.1230	−0.407281
$887$	−24.3250	−0.816755	−0.408378	−	0.912813i	$-0.633905\pi$
$887$	−24.3250	−0.816755	−0.408378	+	0.912813i	$0.633905\pi$
$888$	0	0
$889$	62.9755	2.11213
$890$	−23.9152	−0.801639
$891$	0	0
$892$	−12.1550	−0.406978
$893$	7.05848	0.236203
$894$	0	0
$895$	−43.3658	−1.44956
$896$	32.5163	1.08629
$897$	0	0
$898$	−20.4170	−0.681323
$899$	3.34398	0.111528
$900$	0	0
$901$	46.3832	1.54525
$902$	−9.10203	−0.303064
$903$	0	0
$904$	36.9525	1.22902
$905$	−45.8881	−1.52537
$906$	0	0
$907$	3.04078	0.100967	0.0504837	−	0.998725i	$-0.483924\pi$
$907$	3.04078	0.100967	0.0504837	+	0.998725i	$0.483924\pi$
$908$	−5.95738	−0.197703
$909$	0	0
$910$	−36.2727	−1.20243
$911$	40.5839	1.34460	0.672302	−	0.740277i	$-0.265306\pi$
$911$	40.5839	1.34460	0.672302	+	0.740277i	$0.265306\pi$
$912$	0	0
$913$	65.9183	2.18158
$914$	11.5321	0.381447
$915$	0	0
$916$	35.7403	1.18089
$917$	20.4167	0.674217
$918$	0	0
$919$	16.8456	0.555684	0.277842	−	0.960627i	$-0.410381\pi$
$919$	16.8456	0.555684	0.277842	+	0.960627i	$0.410381\pi$
$920$	−7.24052	−0.238713
$921$	0	0
$922$	13.6483	0.449483
$923$	5.27788	0.173724
$924$	0	0
$925$	−21.7923	−0.716526
$926$	−4.33422	−0.142431
$927$	0	0
$928$	−5.86170	−0.192420
$929$	12.9053	0.423408	0.211704	−	0.977334i	$-0.432099\pi$
$929$	12.9053	0.423408	0.211704	+	0.977334i	$0.432099\pi$
$930$	0	0
$931$	−21.3196	−0.698722
$932$	2.31976	0.0759863
$933$	0	0
$934$	−18.9445	−0.619883
$935$	50.5484	1.65311
$936$	0	0
$937$	31.3663	1.02469	0.512346	−	0.858779i	$-0.328777\pi$
$937$	31.3663	1.02469	0.512346	+	0.858779i	$0.328777\pi$
$938$	15.5142	0.506556
$939$	0	0
$940$	5.06224	0.165112
$941$	−1.74601	−0.0569182	−0.0284591	−	0.999595i	$-0.509060\pi$
$941$	−1.74601	−0.0569182	−0.0284591	+	0.999595i	$0.509060\pi$
$942$	0	0
$943$	2.57493	0.0838513
$944$	0.566784	0.0184472
$945$	0	0
$946$	−7.12906	−0.231786
$947$	28.0994	0.913108	0.456554	−	0.889696i	$-0.349083\pi$
$947$	28.0994	0.913108	0.456554	+	0.889696i	$0.349083\pi$
$948$	0	0
$949$	−83.8054	−2.72044
$950$	12.4814	0.404949
$951$	0	0
$952$	−31.8565	−1.03248
$953$	−20.9344	−0.678133	−0.339066	−	0.940762i	$-0.610111\pi$
$953$	−20.9344	−0.678133	−0.339066	+	0.940762i	$0.610111\pi$
$954$	0	0
$955$	55.8067	1.80586
$956$	12.1431	0.392735
$957$	0	0
$958$	−5.72941	−0.185109
$959$	22.8781	0.738771
$960$	0	0
$961$	−19.8178	−0.639284
$962$	28.9918	0.934733
$963$	0	0
$964$	−13.5924	−0.437781
$965$	−37.5926	−1.21015
$966$	0	0
$967$	−58.5319	−1.88226	−0.941130	−	0.338045i	$-0.890234\pi$
$967$	−58.5319	−1.88226	−0.941130	+	0.338045i	$0.890234\pi$
$968$	29.4577	0.946806
$969$	0	0
$970$	27.0497	0.868513
$971$	−18.2934	−0.587064	−0.293532	−	0.955949i	$-0.594831\pi$
$971$	−18.2934	−0.587064	−0.293532	+	0.955949i	$0.594831\pi$
$972$	0	0
$973$	47.9910	1.53852
$974$	11.0779	0.354960
$975$	0	0
$976$	−12.2305	−0.391488
$977$	−19.6460	−0.628530	−0.314265	−	0.949335i	$-0.601758\pi$
$977$	−19.6460	−0.628530	−0.314265	+	0.949335i	$0.601758\pi$
$978$	0	0
$979$	−53.9216	−1.72334
$980$	−15.2901	−0.488425
$981$	0	0
$982$	11.8203	0.377201
$983$	−5.00069	−0.159497	−0.0797486	−	0.996815i	$-0.525412\pi$
$983$	−5.00069	−0.159497	−0.0797486	+	0.996815i	$0.525412\pi$
$984$	0	0
$985$	42.3120	1.34817
$986$	2.82046	0.0898219
$987$	0	0
$988$	43.1166	1.37172
$989$	2.01679	0.0641300
$990$	0	0
$991$	−30.0105	−0.953314	−0.476657	−	0.879089i	$-0.658152\pi$
$991$	−30.0105	−0.953314	−0.476657	+	0.879089i	$0.658152\pi$
$992$	−19.6014	−0.622345
$993$	0	0
$994$	−2.45365	−0.0778249
$995$	−41.6275	−1.31968
$996$	0	0
$997$	−11.1032	−0.351641	−0.175820	−	0.984422i	$-0.556258\pi$
$997$	−11.1032	−0.351641	−0.175820	+	0.984422i	$0.556258\pi$
$998$	−11.2676	−0.356670
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.q.1.7		16
3.2	odd	2		667.2.a.d.1.10	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.d.1.10	✓	16	3.2	odd	2
6003.2.a.q.1.7		16	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-0.745705 q^{2} -1.44392 q^{4} +2.81935 q^{5} +3.27962 q^{7} +2.56815 q^{8} +O(q^{10})\)	\(q-0.745705 q^{2} -1.44392 q^{4} +2.81935 q^{5} +3.27962 q^{7} +2.56815 q^{8} -2.10240 q^{10} -4.74029 q^{11} +5.26065 q^{13} -2.44563 q^{14} +0.972761 q^{16} -3.78228 q^{17} -5.67624 q^{19} -4.07092 q^{20} +3.53486 q^{22} -1.00000 q^{23} +2.94872 q^{25} -3.92289 q^{26} -4.73553 q^{28} +1.00000 q^{29} +3.34398 q^{31} -5.86170 q^{32} +2.82046 q^{34} +9.24640 q^{35} -7.39041 q^{37} +4.23281 q^{38} +7.24052 q^{40} -2.57493 q^{41} -2.01679 q^{43} +6.84462 q^{44} +0.745705 q^{46} -1.24351 q^{47} +3.75593 q^{49} -2.19888 q^{50} -7.59597 q^{52} -12.2633 q^{53} -13.3645 q^{55} +8.42257 q^{56} -0.745705 q^{58} +0.582654 q^{59} -12.5729 q^{61} -2.49362 q^{62} +2.42558 q^{64} +14.8316 q^{65} -6.34363 q^{67} +5.46132 q^{68} -6.89509 q^{70} +1.00328 q^{71} -15.9306 q^{73} +5.51107 q^{74} +8.19606 q^{76} -15.5464 q^{77} -14.7676 q^{79} +2.74255 q^{80} +1.92014 q^{82} -13.9060 q^{83} -10.6636 q^{85} +1.50393 q^{86} -12.1738 q^{88} +11.3752 q^{89} +17.2529 q^{91} +1.44392 q^{92} +0.927294 q^{94} -16.0033 q^{95} -12.8661 q^{97} -2.80082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8}+O(q^{10}) \) 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8	\( 16 q - 3 q^{2} + 21 q^{4} - 16 q^{5} + q^{7} - 9 q^{8} - 14 q^{10} - 4 q^{11} + 15 q^{13} - 8 q^{14} + 23 q^{16} - 20 q^{17} - 4 q^{19} - 25 q^{20} + 13 q^{22} - 16 q^{23} + 30 q^{25} - 25 q^{26} - 13 q^{28} + 16 q^{29} + 19 q^{32} - 23 q^{34} - 5 q^{35} + 5 q^{37} - 38 q^{38} - 20 q^{40} - 7 q^{41} - 17 q^{43} + 21 q^{44} + 3 q^{46} - 29 q^{47} + 31 q^{49} + 44 q^{50} + 20 q^{52} - 63 q^{53} + q^{55} + 19 q^{56} - 3 q^{58} - 11 q^{59} - 33 q^{62} + 29 q^{64} - 53 q^{65} - 13 q^{67} - 63 q^{68} - 46 q^{70} + 23 q^{71} - 38 q^{73} + 47 q^{74} - 56 q^{76} - 97 q^{77} - 27 q^{79} - 8 q^{80} + 9 q^{82} - 36 q^{83} + 6 q^{85} + 11 q^{86} - 24 q^{88} + 16 q^{89} - 47 q^{91} - 21 q^{92} + 37 q^{94} + 12 q^{95} - 30 q^{97} + 27 q^{98}+O(q^{100}) \) 16 * q - 3 * q^2 + 21 * q^4 - 16 * q^5 + q^7 - 9 * q^8 - 14 * q^10 - 4 * q^11 + 15 * q^13 - 8 * q^14 + 23 * q^16 - 20 * q^17 - 4 * q^19 - 25 * q^20 + 13 * q^22 - 16 * q^23 + 30 * q^25 - 25 * q^26 - 13 * q^28 + 16 * q^29 + 19 * q^32 - 23 * q^34 - 5 * q^35 + 5 * q^37 - 38 * q^38 - 20 * q^40 - 7 * q^41 - 17 * q^43 + 21 * q^44 + 3 * q^46 - 29 * q^47 + 31 * q^49 + 44 * q^50 + 20 * q^52 - 63 * q^53 + q^55 + 19 * q^56 - 3 * q^58 - 11 * q^59 - 33 * q^62 + 29 * q^64 - 53 * q^65 - 13 * q^67 - 63 * q^68 - 46 * q^70 + 23 * q^71 - 38 * q^73 + 47 * q^74 - 56 * q^76 - 97 * q^77 - 27 * q^79 - 8 * q^80 + 9 * q^82 - 36 * q^83 + 6 * q^85 + 11 * q^86 - 24 * q^88 + 16 * q^89 - 47 * q^91 - 21 * q^92 + 37 * q^94 + 12 * q^95 - 30 * q^97 + 27 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more