LMFDB - Embedded newform 2008.2.a.d.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2008,2,Mod(1,2008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2008, 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("2008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2008 = 2^{3} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2008.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:	$16.0339607259$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	2008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.259421 q^{3} -1.96371 q^{5} -4.76268 q^{7} -2.93270 q^{9} +O(q^{10})$	$q+0.259421 q^{3} -1.96371 q^{5} -4.76268 q^{7} -2.93270 q^{9} +3.31268 q^{11} +0.493039 q^{13} -0.509426 q^{15} -3.02691 q^{17} +1.99518 q^{19} -1.23554 q^{21} +3.65140 q^{23} -1.14386 q^{25} -1.53906 q^{27} +5.29536 q^{29} -8.88848 q^{31} +0.859377 q^{33} +9.35250 q^{35} +6.88114 q^{37} +0.127904 q^{39} -3.94408 q^{41} +10.3912 q^{43} +5.75896 q^{45} -12.7776 q^{47} +15.6831 q^{49} -0.785243 q^{51} +5.75231 q^{53} -6.50513 q^{55} +0.517590 q^{57} -12.3248 q^{59} +8.92476 q^{61} +13.9675 q^{63} -0.968183 q^{65} +14.3282 q^{67} +0.947248 q^{69} +10.5818 q^{71} +8.88283 q^{73} -0.296741 q^{75} -15.7772 q^{77} +8.79170 q^{79} +8.39884 q^{81} +8.70243 q^{83} +5.94396 q^{85} +1.37372 q^{87} +10.7568 q^{89} -2.34819 q^{91} -2.30585 q^{93} -3.91794 q^{95} +13.1731 q^{97} -9.71510 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * 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
$3$	0.259421	0.149777	0.0748883	−	0.997192i	$-0.476140\pi$
$3$	0.259421	0.149777	0.0748883	+	0.997192i	$0.476140\pi$
$4$	0	0
$5$	−1.96371	−0.878196	−0.439098	−	0.898439i	$-0.644702\pi$
$5$	−1.96371	−0.878196	−0.439098	+	0.898439i	$0.644702\pi$
$6$	0	0
$7$	−4.76268	−1.80012	−0.900062	−	0.435762i	$-0.856479\pi$
$7$	−4.76268	−1.80012	−0.900062	+	0.435762i	$0.856479\pi$
$8$	0	0
$9$	−2.93270	−0.977567
$10$	0	0
$11$	3.31268	0.998811	0.499405	−	0.866368i	$-0.333552\pi$
$11$	3.31268	0.998811	0.499405	+	0.866368i	$0.333552\pi$
$12$	0	0
$13$	0.493039	0.136744	0.0683722	−	0.997660i	$-0.478219\pi$
$13$	0.493039	0.136744	0.0683722	+	0.997660i	$0.478219\pi$
$14$	0	0
$15$	−0.509426	−0.131533
$16$	0	0
$17$	−3.02691	−0.734134	−0.367067	−	0.930195i	$-0.619638\pi$
$17$	−3.02691	−0.734134	−0.367067	+	0.930195i	$0.619638\pi$
$18$	0	0
$19$	1.99518	0.457725	0.228863	−	0.973459i	$-0.426499\pi$
$19$	1.99518	0.457725	0.228863	+	0.973459i	$0.426499\pi$
$20$	0	0
$21$	−1.23554	−0.269616
$22$	0	0
$23$	3.65140	0.761370	0.380685	−	0.924705i	$-0.375688\pi$
$23$	3.65140	0.761370	0.380685	+	0.924705i	$0.375688\pi$
$24$	0	0
$25$	−1.14386	−0.228772
$26$	0	0
$27$	−1.53906	−0.296193
$28$	0	0
$29$	5.29536	0.983324	0.491662	−	0.870786i	$-0.336390\pi$
$29$	5.29536	0.983324	0.491662	+	0.870786i	$0.336390\pi$
$30$	0	0
$31$	−8.88848	−1.59642	−0.798209	−	0.602380i	$-0.794219\pi$
$31$	−8.88848	−1.59642	−0.798209	+	0.602380i	$0.794219\pi$
$32$	0	0
$33$	0.859377	0.149598
$34$	0	0
$35$	9.35250	1.58086
$36$	0	0
$37$	6.88114	1.13125	0.565626	−	0.824662i	$-0.308635\pi$
$37$	6.88114	1.13125	0.565626	+	0.824662i	$0.308635\pi$
$38$	0	0
$39$	0.127904	0.0204811
$40$	0	0
$41$	−3.94408	−0.615961	−0.307981	−	0.951393i	$-0.599653\pi$
$41$	−3.94408	−0.615961	−0.307981	+	0.951393i	$0.599653\pi$
$42$	0	0
$43$	10.3912	1.58465	0.792323	−	0.610102i	$-0.208872\pi$
$43$	10.3912	1.58465	0.792323	+	0.610102i	$0.208872\pi$
$44$	0	0
$45$	5.75896	0.858495
$46$	0	0
$47$	−12.7776	−1.86381	−0.931905	−	0.362703i	$-0.881854\pi$
$47$	−12.7776	−1.86381	−0.931905	+	0.362703i	$0.881854\pi$
$48$	0	0
$49$	15.6831	2.24045
$50$	0	0
$51$	−0.785243	−0.109956
$52$	0	0
$53$	5.75231	0.790140	0.395070	−	0.918651i	$-0.370720\pi$
$53$	5.75231	0.790140	0.395070	+	0.918651i	$0.370720\pi$
$54$	0	0
$55$	−6.50513	−0.877151
$56$	0	0
$57$	0.517590	0.0685565
$58$	0	0
$59$	−12.3248	−1.60456	−0.802278	−	0.596950i	$-0.796379\pi$
$59$	−12.3248	−1.60456	−0.802278	+	0.596950i	$0.796379\pi$
$60$	0	0
$61$	8.92476	1.14270	0.571349	−	0.820707i	$-0.306420\pi$
$61$	8.92476	1.14270	0.571349	+	0.820707i	$0.306420\pi$
$62$	0	0
$63$	13.9675	1.75974
$64$	0	0
$65$	−0.968183	−0.120088
$66$	0	0
$67$	14.3282	1.75047	0.875237	−	0.483695i	$-0.160706\pi$
$67$	14.3282	1.75047	0.875237	+	0.483695i	$0.160706\pi$
$68$	0	0
$69$	0.947248	0.114035
$70$	0	0
$71$	10.5818	1.25583	0.627916	−	0.778281i	$-0.283908\pi$
$71$	10.5818	1.25583	0.627916	+	0.778281i	$0.283908\pi$
$72$	0	0
$73$	8.88283	1.03966	0.519828	−	0.854271i	$-0.325996\pi$
$73$	8.88283	1.03966	0.519828	+	0.854271i	$0.325996\pi$
$74$	0	0
$75$	−0.296741	−0.0342647
$76$	0	0
$77$	−15.7772	−1.79798
$78$	0	0
$79$	8.79170	0.989143	0.494572	−	0.869137i	$-0.335325\pi$
$79$	8.79170	0.989143	0.494572	+	0.869137i	$0.335325\pi$
$80$	0	0
$81$	8.39884	0.933204
$82$	0	0
$83$	8.70243	0.955216	0.477608	−	0.878573i	$-0.341504\pi$
$83$	8.70243	0.955216	0.477608	+	0.878573i	$0.341504\pi$
$84$	0	0
$85$	5.94396	0.644713
$86$	0	0
$87$	1.37372	0.147279
$88$	0	0
$89$	10.7568	1.14022	0.570108	−	0.821570i	$-0.306901\pi$
$89$	10.7568	1.14022	0.570108	+	0.821570i	$0.306901\pi$
$90$	0	0
$91$	−2.34819	−0.246157
$92$	0	0
$93$	−2.30585	−0.239106
$94$	0	0
$95$	−3.91794	−0.401972
$96$	0	0
$97$	13.1731	1.33753	0.668764	−	0.743475i	$-0.266824\pi$
$97$	13.1731	1.33753	0.668764	+	0.743475i	$0.266824\pi$
$98$	0	0
$99$	−9.71510	−0.976404
$100$	0	0
$101$	−9.38904	−0.934245	−0.467122	−	0.884193i	$-0.654709\pi$
$101$	−9.38904	−0.934245	−0.467122	+	0.884193i	$0.654709\pi$
$102$	0	0
$103$	−9.79842	−0.965467	−0.482733	−	0.875767i	$-0.660356\pi$
$103$	−9.79842	−0.965467	−0.482733	+	0.875767i	$0.660356\pi$
$104$	0	0
$105$	2.42623	0.236776
$106$	0	0
$107$	4.35926	0.421425	0.210713	−	0.977548i	$-0.432422\pi$
$107$	4.35926	0.421425	0.210713	+	0.977548i	$0.432422\pi$
$108$	0	0
$109$	−5.37187	−0.514532	−0.257266	−	0.966341i	$-0.582822\pi$
$109$	−5.37187	−0.514532	−0.257266	+	0.966341i	$0.582822\pi$
$110$	0	0
$111$	1.78511	0.169435
$112$	0	0
$113$	−0.594998	−0.0559727	−0.0279863	−	0.999608i	$-0.508909\pi$
$113$	−0.594998	−0.0559727	−0.0279863	+	0.999608i	$0.508909\pi$
$114$	0	0
$115$	−7.17028	−0.668632
$116$	0	0
$117$	−1.44594	−0.133677
$118$	0	0
$119$	14.4162	1.32153
$120$	0	0
$121$	−0.0261504	−0.00237731
$122$	0	0
$123$	−1.02317	−0.0922565
$124$	0	0
$125$	12.0647	1.07910
$126$	0	0
$127$	−21.0814	−1.87067	−0.935337	−	0.353759i	$-0.884903\pi$
$127$	−21.0814	−1.87067	−0.935337	+	0.353759i	$0.884903\pi$
$128$	0	0
$129$	2.69569	0.237343
$130$	0	0
$131$	6.41979	0.560899	0.280450	−	0.959869i	$-0.409516\pi$
$131$	6.41979	0.560899	0.280450	+	0.959869i	$0.409516\pi$
$132$	0	0
$133$	−9.50240	−0.823962
$134$	0	0
$135$	3.02227	0.260116
$136$	0	0
$137$	10.2575	0.876354	0.438177	−	0.898889i	$-0.355624\pi$
$137$	10.2575	0.876354	0.438177	+	0.898889i	$0.355624\pi$
$138$	0	0
$139$	−15.9338	−1.35149	−0.675743	−	0.737137i	$-0.736177\pi$
$139$	−15.9338	−1.35149	−0.675743	+	0.737137i	$0.736177\pi$
$140$	0	0
$141$	−3.31478	−0.279155
$142$	0	0
$143$	1.63328	0.136582
$144$	0	0
$145$	−10.3985	−0.863551
$146$	0	0
$147$	4.06852	0.335566
$148$	0	0
$149$	10.0095	0.820008	0.410004	−	0.912084i	$-0.365527\pi$
$149$	10.0095	0.820008	0.410004	+	0.912084i	$0.365527\pi$
$150$	0	0
$151$	−21.5890	−1.75689	−0.878443	−	0.477847i	$-0.841418\pi$
$151$	−21.5890	−1.75689	−0.878443	+	0.477847i	$0.841418\pi$
$152$	0	0
$153$	8.87703	0.717665
$154$	0	0
$155$	17.4544	1.40197
$156$	0	0
$157$	−9.38585	−0.749072	−0.374536	−	0.927212i	$-0.622198\pi$
$157$	−9.38585	−0.749072	−0.374536	+	0.927212i	$0.622198\pi$
$158$	0	0
$159$	1.49227	0.118344
$160$	0	0
$161$	−17.3905	−1.37056
$162$	0	0
$163$	20.3983	1.59772	0.798860	−	0.601517i	$-0.205437\pi$
$163$	20.3983	1.59772	0.798860	+	0.601517i	$0.205437\pi$
$164$	0	0
$165$	−1.68756	−0.131377
$166$	0	0
$167$	−8.88676	−0.687678	−0.343839	−	0.939029i	$-0.611728\pi$
$167$	−8.88676	−0.687678	−0.343839	+	0.939029i	$0.611728\pi$
$168$	0	0
$169$	−12.7569	−0.981301
$170$	0	0
$171$	−5.85126	−0.447457
$172$	0	0
$173$	−9.23441	−0.702079	−0.351039	−	0.936361i	$-0.614172\pi$
$173$	−9.23441	−0.702079	−0.351039	+	0.936361i	$0.614172\pi$
$174$	0	0
$175$	5.44784	0.411818
$176$	0	0
$177$	−3.19731	−0.240325
$178$	0	0
$179$	0.666728	0.0498336	0.0249168	−	0.999690i	$-0.492068\pi$
$179$	0.666728	0.0498336	0.0249168	+	0.999690i	$0.492068\pi$
$180$	0	0
$181$	−6.34647	−0.471729	−0.235865	−	0.971786i	$-0.575792\pi$
$181$	−6.34647	−0.471729	−0.235865	+	0.971786i	$0.575792\pi$
$182$	0	0
$183$	2.31527	0.171149
$184$	0	0
$185$	−13.5125	−0.993461
$186$	0	0
$187$	−10.0272	−0.733261
$188$	0	0
$189$	7.33007	0.533184
$190$	0	0
$191$	−0.222688	−0.0161131	−0.00805656	−	0.999968i	$-0.502565\pi$
$191$	−0.222688	−0.0161131	−0.00805656	+	0.999968i	$0.502565\pi$
$192$	0	0
$193$	20.5186	1.47696	0.738480	−	0.674276i	$-0.235544\pi$
$193$	20.5186	1.47696	0.738480	+	0.674276i	$0.235544\pi$
$194$	0	0
$195$	−0.251167	−0.0179864
$196$	0	0
$197$	4.39141	0.312875	0.156437	−	0.987688i	$-0.449999\pi$
$197$	4.39141	0.312875	0.156437	+	0.987688i	$0.449999\pi$
$198$	0	0
$199$	−23.7138	−1.68103	−0.840514	−	0.541790i	$-0.817747\pi$
$199$	−23.7138	−1.68103	−0.840514	+	0.541790i	$0.817747\pi$
$200$	0	0
$201$	3.71704	0.262180
$202$	0	0
$203$	−25.2201	−1.77010
$204$	0	0
$205$	7.74500	0.540935
$206$	0	0
$207$	−10.7085	−0.744290
$208$	0	0
$209$	6.60939	0.457181
$210$	0	0
$211$	−10.7520	−0.740195	−0.370097	−	0.928993i	$-0.620676\pi$
$211$	−10.7520	−0.740195	−0.370097	+	0.928993i	$0.620676\pi$
$212$	0	0
$213$	2.74515	0.188094
$214$	0	0
$215$	−20.4053	−1.39163
$216$	0	0
$217$	42.3330	2.87375
$218$	0	0
$219$	2.30439	0.155716
$220$	0	0
$221$	−1.49238	−0.100389
$222$	0	0
$223$	−27.2728	−1.82632	−0.913160	−	0.407600i	$-0.866366\pi$
$223$	−27.2728	−1.82632	−0.913160	+	0.407600i	$0.866366\pi$
$224$	0	0
$225$	3.35460	0.223640
$226$	0	0
$227$	24.5283	1.62800	0.814000	−	0.580865i	$-0.197285\pi$
$227$	24.5283	1.62800	0.814000	+	0.580865i	$0.197285\pi$
$228$	0	0
$229$	6.80771	0.449866	0.224933	−	0.974374i	$-0.427784\pi$
$229$	6.80771	0.449866	0.224933	+	0.974374i	$0.427784\pi$
$230$	0	0
$231$	−4.09294	−0.269296
$232$	0	0
$233$	23.6993	1.55259	0.776296	−	0.630369i	$-0.217096\pi$
$233$	23.6993	1.55259	0.776296	+	0.630369i	$0.217096\pi$
$234$	0	0
$235$	25.0915	1.63679
$236$	0	0
$237$	2.28075	0.148150
$238$	0	0
$239$	26.9335	1.74218	0.871092	−	0.491119i	$-0.163412\pi$
$239$	26.9335	1.74218	0.871092	+	0.491119i	$0.163412\pi$
$240$	0	0
$241$	3.54758	0.228520	0.114260	−	0.993451i	$-0.463550\pi$
$241$	3.54758	0.228520	0.114260	+	0.993451i	$0.463550\pi$
$242$	0	0
$243$	6.79602	0.435965
$244$	0	0
$245$	−30.7970	−1.96755
$246$	0	0
$247$	0.983700	0.0625914
$248$	0	0
$249$	2.25759	0.143069
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	12.0959	0.760464
$254$	0	0
$255$	1.54199	0.0965629
$256$	0	0
$257$	9.72022	0.606331	0.303165	−	0.952938i	$-0.401957\pi$
$257$	9.72022	0.606331	0.303165	+	0.952938i	$0.401957\pi$
$258$	0	0
$259$	−32.7727	−2.03639
$260$	0	0
$261$	−15.5297	−0.961265
$262$	0	0
$263$	0.264944	0.0163371	0.00816857	−	0.999967i	$-0.497400\pi$
$263$	0.264944	0.0163371	0.00816857	+	0.999967i	$0.497400\pi$
$264$	0	0
$265$	−11.2958	−0.693898
$266$	0	0
$267$	2.79053	0.170777
$268$	0	0
$269$	1.29684	0.0790698	0.0395349	−	0.999218i	$-0.487412\pi$
$269$	1.29684	0.0790698	0.0395349	+	0.999218i	$0.487412\pi$
$270$	0	0
$271$	−4.32607	−0.262790	−0.131395	−	0.991330i	$-0.541946\pi$
$271$	−4.32607	−0.262790	−0.131395	+	0.991330i	$0.541946\pi$
$272$	0	0
$273$	−0.609168	−0.0368685
$274$	0	0
$275$	−3.78924	−0.228500
$276$	0	0
$277$	27.3368	1.64251	0.821253	−	0.570564i	$-0.193275\pi$
$277$	27.3368	1.64251	0.821253	+	0.570564i	$0.193275\pi$
$278$	0	0
$279$	26.0673	1.56061
$280$	0	0
$281$	1.11608	0.0665799	0.0332899	−	0.999446i	$-0.489402\pi$
$281$	1.11608	0.0665799	0.0332899	+	0.999446i	$0.489402\pi$
$282$	0	0
$283$	−3.36117	−0.199801	−0.0999004	−	0.994997i	$-0.531852\pi$
$283$	−3.36117	−0.199801	−0.0999004	+	0.994997i	$0.531852\pi$
$284$	0	0
$285$	−1.01639	−0.0602060
$286$	0	0
$287$	18.7844	1.10881
$288$	0	0
$289$	−7.83781	−0.461048
$290$	0	0
$291$	3.41738	0.200330
$292$	0	0
$293$	15.8718	0.927241	0.463621	−	0.886034i	$-0.346550\pi$
$293$	15.8718	0.927241	0.463621	+	0.886034i	$0.346550\pi$
$294$	0	0
$295$	24.2023	1.40911
$296$	0	0
$297$	−5.09843	−0.295841
$298$	0	0
$299$	1.80028	0.104113
$300$	0	0
$301$	−49.4900	−2.85256
$302$	0	0
$303$	−2.43571	−0.139928
$304$	0	0
$305$	−17.5256	−1.00351
$306$	0	0
$307$	4.54729	0.259528	0.129764	−	0.991545i	$-0.458578\pi$
$307$	4.54729	0.259528	0.129764	+	0.991545i	$0.458578\pi$
$308$	0	0
$309$	−2.54191	−0.144604
$310$	0	0
$311$	−1.05931	−0.0600680	−0.0300340	−	0.999549i	$-0.509562\pi$
$311$	−1.05931	−0.0600680	−0.0300340	+	0.999549i	$0.509562\pi$
$312$	0	0
$313$	24.7080	1.39658	0.698290	−	0.715815i	$-0.253945\pi$
$313$	24.7080	1.39658	0.698290	+	0.715815i	$0.253945\pi$
$314$	0	0
$315$	−27.4281	−1.54540
$316$	0	0
$317$	13.0254	0.731580	0.365790	−	0.930697i	$-0.380799\pi$
$317$	13.0254	0.731580	0.365790	+	0.930697i	$0.380799\pi$
$318$	0	0
$319$	17.5418	0.982154
$320$	0	0
$321$	1.13088	0.0631196
$322$	0	0
$323$	−6.03923	−0.336032
$324$	0	0
$325$	−0.563968	−0.0312833
$326$	0	0
$327$	−1.39357	−0.0770647
$328$	0	0
$329$	60.8558	3.35509
$330$	0	0
$331$	10.6130	0.583345	0.291673	−	0.956518i	$-0.405788\pi$
$331$	10.6130	0.583345	0.291673	+	0.956518i	$0.405788\pi$
$332$	0	0
$333$	−20.1803	−1.10587
$334$	0	0
$335$	−28.1364	−1.53726
$336$	0	0
$337$	−21.4890	−1.17058	−0.585291	−	0.810823i	$-0.699020\pi$
$337$	−21.4890	−1.17058	−0.585291	+	0.810823i	$0.699020\pi$
$338$	0	0
$339$	−0.154355	−0.00838339
$340$	0	0
$341$	−29.4447	−1.59452
$342$	0	0
$343$	−41.3550	−2.23296
$344$	0	0
$345$	−1.86012	−0.100145
$346$	0	0
$347$	−30.5178	−1.63828	−0.819140	−	0.573594i	$-0.805549\pi$
$347$	−30.5178	−1.63828	−0.819140	+	0.573594i	$0.805549\pi$
$348$	0	0
$349$	22.4297	1.20063	0.600317	−	0.799762i	$-0.295041\pi$
$349$	22.4297	1.20063	0.600317	+	0.799762i	$0.295041\pi$
$350$	0	0
$351$	−0.758819	−0.0405027
$352$	0	0
$353$	−12.2019	−0.649441	−0.324720	−	0.945810i	$-0.605270\pi$
$353$	−12.2019	−0.649441	−0.324720	+	0.945810i	$0.605270\pi$
$354$	0	0
$355$	−20.7796	−1.10287
$356$	0	0
$357$	3.73986	0.197934
$358$	0	0
$359$	−2.54125	−0.134122	−0.0670611	−	0.997749i	$-0.521362\pi$
$359$	−2.54125	−0.134122	−0.0670611	+	0.997749i	$0.521362\pi$
$360$	0	0
$361$	−15.0193	−0.790488
$362$	0	0
$363$	−0.00678395	−0.000356065 0
$364$	0	0
$365$	−17.4433	−0.913022
$366$	0	0
$367$	21.9765	1.14716	0.573582	−	0.819148i	$-0.305553\pi$
$367$	21.9765	1.14716	0.573582	+	0.819148i	$0.305553\pi$
$368$	0	0
$369$	11.5668	0.602143
$370$	0	0
$371$	−27.3964	−1.42235
$372$	0	0
$373$	−22.4020	−1.15993	−0.579965	−	0.814641i	$-0.696934\pi$
$373$	−22.4020	−1.15993	−0.579965	+	0.814641i	$0.696934\pi$
$374$	0	0
$375$	3.12984	0.161624
$376$	0	0
$377$	2.61082	0.134464
$378$	0	0
$379$	−27.5019	−1.41268	−0.706339	−	0.707873i	$-0.749655\pi$
$379$	−27.5019	−1.41268	−0.706339	+	0.707873i	$0.749655\pi$
$380$	0	0
$381$	−5.46895	−0.280183
$382$	0	0
$383$	22.7920	1.16462	0.582310	−	0.812967i	$-0.302149\pi$
$383$	22.7920	1.16462	0.582310	+	0.812967i	$0.302149\pi$
$384$	0	0
$385$	30.9818	1.57898
$386$	0	0
$387$	−30.4743	−1.54910
$388$	0	0
$389$	−12.4333	−0.630394	−0.315197	−	0.949026i	$-0.602071\pi$
$389$	−12.4333	−0.630394	−0.315197	+	0.949026i	$0.602071\pi$
$390$	0	0
$391$	−11.0525	−0.558947
$392$	0	0
$393$	1.66542	0.0840096
$394$	0	0
$395$	−17.2643	−0.868661
$396$	0	0
$397$	13.6525	0.685197	0.342599	−	0.939482i	$-0.388693\pi$
$397$	13.6525	0.685197	0.342599	+	0.939482i	$0.388693\pi$
$398$	0	0
$399$	−2.46512	−0.123410
$400$	0	0
$401$	−23.6015	−1.17860	−0.589300	−	0.807914i	$-0.700597\pi$
$401$	−23.6015	−1.17860	−0.589300	+	0.807914i	$0.700597\pi$
$402$	0	0
$403$	−4.38237	−0.218301
$404$	0	0
$405$	−16.4928	−0.819536
$406$	0	0
$407$	22.7950	1.12991
$408$	0	0
$409$	16.0047	0.791381	0.395690	−	0.918384i	$-0.370505\pi$
$409$	16.0047	0.791381	0.395690	+	0.918384i	$0.370505\pi$
$410$	0	0
$411$	2.66100	0.131257
$412$	0	0
$413$	58.6992	2.88840
$414$	0	0
$415$	−17.0890	−0.838866
$416$	0	0
$417$	−4.13355	−0.202421
$418$	0	0
$419$	9.90724	0.484000	0.242000	−	0.970276i	$-0.422197\pi$
$419$	9.90724	0.484000	0.242000	+	0.970276i	$0.422197\pi$
$420$	0	0
$421$	19.3614	0.943615	0.471808	−	0.881702i	$-0.343602\pi$
$421$	19.3614	0.943615	0.471808	+	0.881702i	$0.343602\pi$
$422$	0	0
$423$	37.4730	1.82200
$424$	0	0
$425$	3.46236	0.167949
$426$	0	0
$427$	−42.5058	−2.05700
$428$	0	0
$429$	0.423706	0.0204567
$430$	0	0
$431$	−5.24269	−0.252531	−0.126266	−	0.991996i	$-0.540299\pi$
$431$	−5.24269	−0.252531	−0.126266	+	0.991996i	$0.540299\pi$
$432$	0	0
$433$	−1.85655	−0.0892200	−0.0446100	−	0.999004i	$-0.514205\pi$
$433$	−1.85655	−0.0892200	−0.0446100	+	0.999004i	$0.514205\pi$
$434$	0	0
$435$	−2.69759	−0.129340
$436$	0	0
$437$	7.28519	0.348498
$438$	0	0
$439$	35.5187	1.69522	0.847609	−	0.530622i	$-0.178042\pi$
$439$	35.5187	1.69522	0.847609	+	0.530622i	$0.178042\pi$
$440$	0	0
$441$	−45.9939	−2.19019
$442$	0	0
$443$	−14.5959	−0.693470	−0.346735	−	0.937963i	$-0.612710\pi$
$443$	−14.5959	−0.693470	−0.346735	+	0.937963i	$0.612710\pi$
$444$	0	0
$445$	−21.1231	−1.00133
$446$	0	0
$447$	2.59666	0.122818
$448$	0	0
$449$	−25.6909	−1.21243	−0.606215	−	0.795301i	$-0.707313\pi$
$449$	−25.6909	−1.21243	−0.606215	+	0.795301i	$0.707313\pi$
$450$	0	0
$451$	−13.0655	−0.615229
$452$	0	0
$453$	−5.60063	−0.263140
$454$	0	0
$455$	4.61115	0.216174
$456$	0	0
$457$	18.7278	0.876049	0.438025	−	0.898963i	$-0.355678\pi$
$457$	18.7278	0.876049	0.438025	+	0.898963i	$0.355678\pi$
$458$	0	0
$459$	4.65861	0.217445
$460$	0	0
$461$	−9.56918	−0.445681	−0.222840	−	0.974855i	$-0.571533\pi$
$461$	−9.56918	−0.445681	−0.222840	+	0.974855i	$0.571533\pi$
$462$	0	0
$463$	−13.0202	−0.605101	−0.302550	−	0.953133i	$-0.597838\pi$
$463$	−13.0202	−0.605101	−0.302550	+	0.953133i	$0.597838\pi$
$464$	0	0
$465$	4.52802	0.209982
$466$	0	0
$467$	38.6859	1.79017	0.895085	−	0.445896i	$-0.147115\pi$
$467$	38.6859	1.79017	0.895085	+	0.445896i	$0.147115\pi$
$468$	0	0
$469$	−68.2408	−3.15107
$470$	0	0
$471$	−2.43488	−0.112193
$472$	0	0
$473$	34.4228	1.58276
$474$	0	0
$475$	−2.28220	−0.104715
$476$	0	0
$477$	−16.8698	−0.772415
$478$	0	0
$479$	−0.216817	−0.00990662	−0.00495331	−	0.999988i	$-0.501577\pi$
$479$	−0.216817	−0.00990662	−0.00495331	+	0.999988i	$0.501577\pi$
$480$	0	0
$481$	3.39267	0.154692
$482$	0	0
$483$	−4.51144	−0.205278
$484$	0	0
$485$	−25.8681	−1.17461
$486$	0	0
$487$	−42.9856	−1.94786	−0.973932	−	0.226839i	$-0.927161\pi$
$487$	−42.9856	−1.94786	−0.973932	+	0.226839i	$0.927161\pi$
$488$	0	0
$489$	5.29175	0.239301
$490$	0	0
$491$	20.8521	0.941042	0.470521	−	0.882389i	$-0.344066\pi$
$491$	20.8521	0.941042	0.470521	+	0.882389i	$0.344066\pi$
$492$	0	0
$493$	−16.0286	−0.721891
$494$	0	0
$495$	19.0776	0.857474
$496$	0	0
$497$	−50.3979	−2.26066
$498$	0	0
$499$	25.2821	1.13178	0.565891	−	0.824480i	$-0.308532\pi$
$499$	25.2821	1.13178	0.565891	+	0.824480i	$0.308532\pi$
$500$	0	0
$501$	−2.30541	−0.102998
$502$	0	0
$503$	38.0521	1.69666	0.848331	−	0.529467i	$-0.177608\pi$
$503$	38.0521	1.69666	0.848331	+	0.529467i	$0.177608\pi$
$504$	0	0
$505$	18.4373	0.820450
$506$	0	0
$507$	−3.30940	−0.146976
$508$	0	0
$509$	19.4293	0.861189	0.430595	−	0.902545i	$-0.358304\pi$
$509$	19.4293	0.861189	0.430595	+	0.902545i	$0.358304\pi$
$510$	0	0
$511$	−42.3061	−1.87151
$512$	0	0
$513$	−3.07071	−0.135575
$514$	0	0
$515$	19.2412	0.847869
$516$	0	0
$517$	−42.3282	−1.86159
$518$	0	0
$519$	−2.39559	−0.105155
$520$	0	0
$521$	−5.72389	−0.250768	−0.125384	−	0.992108i	$-0.540016\pi$
$521$	−5.72389	−0.250768	−0.125384	+	0.992108i	$0.540016\pi$
$522$	0	0
$523$	−5.17502	−0.226288	−0.113144	−	0.993579i	$-0.536092\pi$
$523$	−5.17502	−0.226288	−0.113144	+	0.993579i	$0.536092\pi$
$524$	0	0
$525$	1.41328	0.0616807
$526$	0	0
$527$	26.9046	1.17198
$528$	0	0
$529$	−9.66727	−0.420316
$530$	0	0
$531$	36.1450	1.56856
$532$	0	0
$533$	−1.94458	−0.0842292
$534$	0	0
$535$	−8.56029	−0.370094
$536$	0	0
$537$	0.172963	0.00746391
$538$	0	0
$539$	51.9532	2.23778
$540$	0	0
$541$	12.0026	0.516034	0.258017	−	0.966140i	$-0.416931\pi$
$541$	12.0026	0.516034	0.258017	+	0.966140i	$0.416931\pi$
$542$	0	0
$543$	−1.64640	−0.0706540
$544$	0	0
$545$	10.5488	0.451860
$546$	0	0
$547$	11.7088	0.500632	0.250316	−	0.968164i	$-0.419465\pi$
$547$	11.7088	0.500632	0.250316	+	0.968164i	$0.419465\pi$
$548$	0	0
$549$	−26.1737	−1.11706
$550$	0	0
$551$	10.5652	0.450092
$552$	0	0
$553$	−41.8720	−1.78058
$554$	0	0
$555$	−3.50543	−0.148797
$556$	0	0
$557$	34.8425	1.47633	0.738163	−	0.674623i	$-0.235694\pi$
$557$	34.8425	1.47633	0.738163	+	0.674623i	$0.235694\pi$
$558$	0	0
$559$	5.12327	0.216691
$560$	0	0
$561$	−2.60126	−0.109825
$562$	0	0
$563$	−4.65603	−0.196228	−0.0981141	−	0.995175i	$-0.531281\pi$
$563$	−4.65603	−0.196228	−0.0981141	+	0.995175i	$0.531281\pi$
$564$	0	0
$565$	1.16840	0.0491550
$566$	0	0
$567$	−40.0010	−1.67988
$568$	0	0
$569$	42.6440	1.78773	0.893865	−	0.448336i	$-0.147983\pi$
$569$	42.6440	1.78773	0.893865	+	0.448336i	$0.147983\pi$
$570$	0	0
$571$	11.6374	0.487011	0.243506	−	0.969899i	$-0.421703\pi$
$571$	11.6374	0.487011	0.243506	+	0.969899i	$0.421703\pi$
$572$	0	0
$573$	−0.0577698	−0.00241337
$574$	0	0
$575$	−4.17669	−0.174180
$576$	0	0
$577$	−6.32219	−0.263196	−0.131598	−	0.991303i	$-0.542011\pi$
$577$	−6.32219	−0.263196	−0.131598	+	0.991303i	$0.542011\pi$
$578$	0	0
$579$	5.32294	0.221214
$580$	0	0
$581$	−41.4469	−1.71951
$582$	0	0
$583$	19.0556	0.789201
$584$	0	0
$585$	2.83939	0.117394
$586$	0	0
$587$	11.5968	0.478652	0.239326	−	0.970939i	$-0.423073\pi$
$587$	11.5968	0.478652	0.239326	+	0.970939i	$0.423073\pi$
$588$	0	0
$589$	−17.7341	−0.730721
$590$	0	0
$591$	1.13922	0.0468613
$592$	0	0
$593$	28.2933	1.16187	0.580934	−	0.813951i	$-0.302688\pi$
$593$	28.2933	1.16187	0.580934	+	0.813951i	$0.302688\pi$
$594$	0	0
$595$	−28.3092	−1.16056
$596$	0	0
$597$	−6.15185	−0.251779
$598$	0	0
$599$	35.9025	1.46694	0.733468	−	0.679724i	$-0.237900\pi$
$599$	35.9025	1.46694	0.733468	+	0.679724i	$0.237900\pi$
$600$	0	0
$601$	−4.84654	−0.197694	−0.0988472	−	0.995103i	$-0.531515\pi$
$601$	−4.84654	−0.197694	−0.0988472	+	0.995103i	$0.531515\pi$
$602$	0	0
$603$	−42.0204	−1.71120
$604$	0	0
$605$	0.0513517	0.00208774
$606$	0	0
$607$	30.4540	1.23609	0.618045	−	0.786143i	$-0.287925\pi$
$607$	30.4540	1.23609	0.618045	+	0.786143i	$0.287925\pi$
$608$	0	0
$609$	−6.54261	−0.265120
$610$	0	0
$611$	−6.29987	−0.254865
$612$	0	0
$613$	10.3728	0.418952	0.209476	−	0.977814i	$-0.432824\pi$
$613$	10.3728	0.418952	0.209476	+	0.977814i	$0.432824\pi$
$614$	0	0
$615$	2.00921	0.0810193
$616$	0	0
$617$	−9.15320	−0.368494	−0.184247	−	0.982880i	$-0.558985\pi$
$617$	−9.15320	−0.368494	−0.184247	+	0.982880i	$0.558985\pi$
$618$	0	0
$619$	−32.8453	−1.32016	−0.660081	−	0.751194i	$-0.729478\pi$
$619$	−32.8453	−1.32016	−0.660081	+	0.751194i	$0.729478\pi$
$620$	0	0
$621$	−5.61974	−0.225512
$622$	0	0
$623$	−51.2311	−2.05253
$624$	0	0
$625$	−17.9723	−0.718891
$626$	0	0
$627$	1.71461	0.0684749
$628$	0	0
$629$	−20.8286	−0.830490
$630$	0	0
$631$	38.1274	1.51783	0.758914	−	0.651191i	$-0.225730\pi$
$631$	38.1274	1.51783	0.758914	+	0.651191i	$0.225730\pi$
$632$	0	0
$633$	−2.78928	−0.110864
$634$	0	0
$635$	41.3977	1.64282
$636$	0	0
$637$	7.73239	0.306368
$638$	0	0
$639$	−31.0334	−1.22766
$640$	0	0
$641$	−5.22739	−0.206470	−0.103235	−	0.994657i	$-0.532919\pi$
$641$	−5.22739	−0.206470	−0.103235	+	0.994657i	$0.532919\pi$
$642$	0	0
$643$	11.5907	0.457093	0.228546	−	0.973533i	$-0.426603\pi$
$643$	11.5907	0.457093	0.228546	+	0.973533i	$0.426603\pi$
$644$	0	0
$645$	−5.29355	−0.208433
$646$	0	0
$647$	19.5074	0.766915	0.383457	−	0.923559i	$-0.374733\pi$
$647$	19.5074	0.766915	0.383457	+	0.923559i	$0.374733\pi$
$648$	0	0
$649$	−40.8282	−1.60265
$650$	0	0
$651$	10.9821	0.430420
$652$	0	0
$653$	37.6926	1.47503	0.737513	−	0.675333i	$-0.236000\pi$
$653$	37.6926	1.47503	0.737513	+	0.675333i	$0.236000\pi$
$654$	0	0
$655$	−12.6066	−0.492580
$656$	0	0
$657$	−26.0507	−1.01633
$658$	0	0
$659$	−2.55314	−0.0994562	−0.0497281	−	0.998763i	$-0.515835\pi$
$659$	−2.55314	−0.0994562	−0.0497281	+	0.998763i	$0.515835\pi$
$660$	0	0
$661$	41.5397	1.61571	0.807853	−	0.589384i	$-0.200629\pi$
$661$	41.5397	1.61571	0.807853	+	0.589384i	$0.200629\pi$
$662$	0	0
$663$	−0.387155	−0.0150359
$664$	0	0
$665$	18.6599	0.723600
$666$	0	0
$667$	19.3355	0.748673
$668$	0	0
$669$	−7.07512	−0.273540
$670$	0	0
$671$	29.5649	1.14134
$672$	0	0
$673$	−42.7327	−1.64722	−0.823612	−	0.567154i	$-0.808044\pi$
$673$	−42.7327	−1.64722	−0.823612	+	0.567154i	$0.808044\pi$
$674$	0	0
$675$	1.76047	0.0677607
$676$	0	0
$677$	−23.9024	−0.918646	−0.459323	−	0.888269i	$-0.651908\pi$
$677$	−23.9024	−0.918646	−0.459323	+	0.888269i	$0.651908\pi$
$678$	0	0
$679$	−62.7394	−2.40772
$680$	0	0
$681$	6.36314	0.243836
$682$	0	0
$683$	−41.1877	−1.57600	−0.788002	−	0.615672i	$-0.788885\pi$
$683$	−41.1877	−1.57600	−0.788002	+	0.615672i	$0.788885\pi$
$684$	0	0
$685$	−20.1426	−0.769611
$686$	0	0
$687$	1.76606	0.0673794
$688$	0	0
$689$	2.83611	0.108047
$690$	0	0
$691$	−6.38418	−0.242866	−0.121433	−	0.992600i	$-0.538749\pi$
$691$	−6.38418	−0.242866	−0.121433	+	0.992600i	$0.538749\pi$
$692$	0	0
$693$	46.2699	1.75765
$694$	0	0
$695$	31.2893	1.18687
$696$	0	0
$697$	11.9384	0.452198
$698$	0	0
$699$	6.14808	0.232542
$700$	0	0
$701$	−51.4514	−1.94329	−0.971647	−	0.236438i	$-0.924020\pi$
$701$	−51.4514	−1.94329	−0.971647	+	0.236438i	$0.924020\pi$
$702$	0	0
$703$	13.7291	0.517803
$704$	0	0
$705$	6.50925	0.245153
$706$	0	0
$707$	44.7170	1.68176
$708$	0	0
$709$	28.8547	1.08366	0.541830	−	0.840488i	$-0.317732\pi$
$709$	28.8547	1.08366	0.541830	+	0.840488i	$0.317732\pi$
$710$	0	0
$711$	−25.7834	−0.966954
$712$	0	0
$713$	−32.4554	−1.21546
$714$	0	0
$715$	−3.20728	−0.119946
$716$	0	0
$717$	6.98711	0.260938
$718$	0	0
$719$	−10.4747	−0.390639	−0.195319	−	0.980740i	$-0.562574\pi$
$719$	−10.4747	−0.390639	−0.195319	+	0.980740i	$0.562574\pi$
$720$	0	0
$721$	46.6667	1.73796
$722$	0	0
$723$	0.920315	0.0342269
$724$	0	0
$725$	−6.05715	−0.224957
$726$	0	0
$727$	2.21072	0.0819909	0.0409955	−	0.999159i	$-0.486947\pi$
$727$	2.21072	0.0819909	0.0409955	+	0.999159i	$0.486947\pi$
$728$	0	0
$729$	−23.4335	−0.867907
$730$	0	0
$731$	−31.4533	−1.16334
$732$	0	0
$733$	5.00026	0.184689	0.0923445	−	0.995727i	$-0.470564\pi$
$733$	5.00026	0.184689	0.0923445	+	0.995727i	$0.470564\pi$
$734$	0	0
$735$	−7.98938	−0.294693
$736$	0	0
$737$	47.4649	1.74839
$738$	0	0
$739$	−6.03692	−0.222072	−0.111036	−	0.993816i	$-0.535417\pi$
$739$	−6.03692	−0.222072	−0.111036	+	0.993816i	$0.535417\pi$
$740$	0	0
$741$	0.255192	0.00937471
$742$	0	0
$743$	−1.69036	−0.0620134	−0.0310067	−	0.999519i	$-0.509871\pi$
$743$	−1.69036	−0.0620134	−0.0310067	+	0.999519i	$0.509871\pi$
$744$	0	0
$745$	−19.6557	−0.720128
$746$	0	0
$747$	−25.5216	−0.933787
$748$	0	0
$749$	−20.7617	−0.758617
$750$	0	0
$751$	−3.48556	−0.127190	−0.0635949	−	0.997976i	$-0.520257\pi$
$751$	−3.48556	−0.127190	−0.0635949	+	0.997976i	$0.520257\pi$
$752$	0	0
$753$	−0.259421	−0.00945381
$754$	0	0
$755$	42.3944	1.54289
$756$	0	0
$757$	6.91788	0.251435	0.125717	−	0.992066i	$-0.459877\pi$
$757$	6.91788	0.251435	0.125717	+	0.992066i	$0.459877\pi$
$758$	0	0
$759$	3.13793	0.113900
$760$	0	0
$761$	37.4085	1.35606	0.678028	−	0.735036i	$-0.262835\pi$
$761$	37.4085	1.35606	0.678028	+	0.735036i	$0.262835\pi$
$762$	0	0
$763$	25.5845	0.926221
$764$	0	0
$765$	−17.4319	−0.630250
$766$	0	0
$767$	−6.07662	−0.219414
$768$	0	0
$769$	−27.3839	−0.987489	−0.493745	−	0.869607i	$-0.664372\pi$
$769$	−27.3839	−0.987489	−0.493745	+	0.869607i	$0.664372\pi$
$770$	0	0
$771$	2.52162	0.0908141
$772$	0	0
$773$	20.8686	0.750590	0.375295	−	0.926905i	$-0.377541\pi$
$773$	20.8686	0.750590	0.375295	+	0.926905i	$0.377541\pi$
$774$	0	0
$775$	10.1672	0.365216
$776$	0	0
$777$	−8.50190	−0.305004
$778$	0	0
$779$	−7.86913	−0.281941
$780$	0	0
$781$	35.0542	1.25434
$782$	0	0
$783$	−8.14990	−0.291254
$784$	0	0
$785$	18.4310	0.657832
$786$	0	0
$787$	17.8159	0.635069	0.317535	−	0.948247i	$-0.397145\pi$
$787$	17.8159	0.635069	0.317535	+	0.948247i	$0.397145\pi$
$788$	0	0
$789$	0.0687319	0.00244692
$790$	0	0
$791$	2.83378	0.100758
$792$	0	0
$793$	4.40025	0.156258
$794$	0	0
$795$	−2.93037	−0.103930
$796$	0	0
$797$	−22.7141	−0.804574	−0.402287	−	0.915514i	$-0.631785\pi$
$797$	−22.7141	−0.804574	−0.402287	+	0.915514i	$0.631785\pi$
$798$	0	0
$799$	38.6768	1.36829
$800$	0	0
$801$	−31.5464	−1.11464
$802$	0	0
$803$	29.4260	1.03842
$804$	0	0
$805$	34.1497	1.20362
$806$	0	0
$807$	0.336427	0.0118428
$808$	0	0
$809$	1.91286	0.0672527	0.0336264	−	0.999434i	$-0.489294\pi$
$809$	1.91286	0.0672527	0.0336264	+	0.999434i	$0.489294\pi$
$810$	0	0
$811$	−11.7428	−0.412345	−0.206172	−	0.978516i	$-0.566101\pi$
$811$	−11.7428	−0.412345	−0.206172	+	0.978516i	$0.566101\pi$
$812$	0	0
$813$	−1.12227	−0.0393598
$814$	0	0
$815$	−40.0563	−1.40311
$816$	0	0
$817$	20.7323	0.725332
$818$	0	0
$819$	6.88653	0.240635
$820$	0	0
$821$	55.0109	1.91989	0.959947	−	0.280183i	$-0.0903950\pi$
$821$	55.0109	1.91989	0.959947	+	0.280183i	$0.0903950\pi$
$822$	0	0
$823$	16.2665	0.567015	0.283507	−	0.958970i	$-0.408502\pi$
$823$	16.2665	0.567015	0.283507	+	0.958970i	$0.408502\pi$
$824$	0	0
$825$	−0.983007	−0.0342239
$826$	0	0
$827$	−51.4976	−1.79075	−0.895374	−	0.445315i	$-0.853092\pi$
$827$	−51.4976	−1.79075	−0.895374	+	0.445315i	$0.853092\pi$
$828$	0	0
$829$	−17.0310	−0.591512	−0.295756	−	0.955263i	$-0.595572\pi$
$829$	−17.0310	−0.591512	−0.295756	+	0.955263i	$0.595572\pi$
$830$	0	0
$831$	7.09172	0.246009
$832$	0	0
$833$	−47.4714	−1.64479
$834$	0	0
$835$	17.4510	0.603916
$836$	0	0
$837$	13.6799	0.472848
$838$	0	0
$839$	0.140399	0.00484713	0.00242356	−	0.999997i	$-0.499229\pi$
$839$	0.140399	0.00484713	0.00242356	+	0.999997i	$0.499229\pi$
$840$	0	0
$841$	−0.959168	−0.0330748
$842$	0	0
$843$	0.289535	0.00997210
$844$	0	0
$845$	25.0508	0.861774
$846$	0	0
$847$	0.124546	0.00427945
$848$	0	0
$849$	−0.871957	−0.0299255
$850$	0	0
$851$	25.1258	0.861301
$852$	0	0
$853$	−37.4221	−1.28131	−0.640654	−	0.767829i	$-0.721337\pi$
$853$	−37.4221	−1.28131	−0.640654	+	0.767829i	$0.721337\pi$
$854$	0	0
$855$	11.4902	0.392955
$856$	0	0
$857$	30.7419	1.05012	0.525062	−	0.851064i	$-0.324042\pi$
$857$	30.7419	1.05012	0.525062	+	0.851064i	$0.324042\pi$
$858$	0	0
$859$	−17.2923	−0.590004	−0.295002	−	0.955497i	$-0.595320\pi$
$859$	−17.2923	−0.590004	−0.295002	+	0.955497i	$0.595320\pi$
$860$	0	0
$861$	4.87305	0.166073
$862$	0	0
$863$	21.6653	0.737497	0.368748	−	0.929529i	$-0.379786\pi$
$863$	21.6653	0.737497	0.368748	+	0.929529i	$0.379786\pi$
$864$	0	0
$865$	18.1337	0.616563
$866$	0	0
$867$	−2.03329	−0.0690541
$868$	0	0
$869$	29.1241	0.987967
$870$	0	0
$871$	7.06438	0.239367
$872$	0	0
$873$	−38.6328	−1.30752
$874$	0	0
$875$	−57.4605	−1.94252
$876$	0	0
$877$	−17.7954	−0.600909	−0.300455	−	0.953796i	$-0.597138\pi$
$877$	−17.7954	−0.600909	−0.300455	+	0.953796i	$0.597138\pi$
$878$	0	0
$879$	4.11747	0.138879
$880$	0	0
$881$	14.2543	0.480238	0.240119	−	0.970743i	$-0.422813\pi$
$881$	14.2543	0.480238	0.240119	+	0.970743i	$0.422813\pi$
$882$	0	0
$883$	46.6843	1.57105	0.785526	−	0.618828i	$-0.212392\pi$
$883$	46.6843	1.57105	0.785526	+	0.618828i	$0.212392\pi$
$884$	0	0
$885$	6.27858	0.211052
$886$	0	0
$887$	−34.7724	−1.16754	−0.583772	−	0.811918i	$-0.698424\pi$
$887$	−34.7724	−1.16754	−0.583772	+	0.811918i	$0.698424\pi$
$888$	0	0
$889$	100.404	3.36744
$890$	0	0
$891$	27.8227	0.932094
$892$	0	0
$893$	−25.4937	−0.853113
$894$	0	0
$895$	−1.30926	−0.0437637
$896$	0	0
$897$	0.467030	0.0155937
$898$	0	0
$899$	−47.0677	−1.56980
$900$	0	0
$901$	−17.4117	−0.580069
$902$	0	0
$903$	−12.8387	−0.427246
$904$	0	0
$905$	12.4626	0.414271
$906$	0	0
$907$	−11.2349	−0.373050	−0.186525	−	0.982450i	$-0.559722\pi$
$907$	−11.2349	−0.373050	−0.186525	+	0.982450i	$0.559722\pi$
$908$	0	0
$909$	27.5353	0.913287
$910$	0	0
$911$	12.7604	0.422771	0.211385	−	0.977403i	$-0.432203\pi$
$911$	12.7604	0.422771	0.211385	+	0.977403i	$0.432203\pi$
$912$	0	0
$913$	28.8284	0.954080
$914$	0	0
$915$	−4.54650	−0.150303
$916$	0	0
$917$	−30.5754	−1.00969
$918$	0	0
$919$	3.03946	0.100263	0.0501313	−	0.998743i	$-0.484036\pi$
$919$	3.03946	0.100263	0.0501313	+	0.998743i	$0.484036\pi$
$920$	0	0
$921$	1.17966	0.0388712
$922$	0	0
$923$	5.21726	0.171728
$924$	0	0
$925$	−7.87106	−0.258799
$926$	0	0
$927$	28.7358	0.943808
$928$	0	0
$929$	24.2895	0.796911	0.398456	−	0.917188i	$-0.369546\pi$
$929$	24.2895	0.796911	0.398456	+	0.917188i	$0.369546\pi$
$930$	0	0
$931$	31.2906	1.02551
$932$	0	0
$933$	−0.274807	−0.00899678
$934$	0	0
$935$	19.6904	0.643946
$936$	0	0
$937$	−20.8628	−0.681556	−0.340778	−	0.940144i	$-0.610691\pi$
$937$	−20.8628	−0.681556	−0.340778	+	0.940144i	$0.610691\pi$
$938$	0	0
$939$	6.40977	0.209175
$940$	0	0
$941$	−47.9126	−1.56190	−0.780952	−	0.624590i	$-0.785266\pi$
$941$	−47.9126	−1.56190	−0.780952	+	0.624590i	$0.785266\pi$
$942$	0	0
$943$	−14.4014	−0.468974
$944$	0	0
$945$	−14.3941	−0.468240
$946$	0	0
$947$	−28.1670	−0.915306	−0.457653	−	0.889131i	$-0.651310\pi$
$947$	−28.1670	−0.915306	−0.457653	+	0.889131i	$0.651310\pi$
$948$	0	0
$949$	4.37958	0.142167
$950$	0	0
$951$	3.37906	0.109573
$952$	0	0
$953$	18.5877	0.602115	0.301057	−	0.953606i	$-0.402660\pi$
$953$	18.5877	0.602115	0.301057	+	0.953606i	$0.402660\pi$
$954$	0	0
$955$	0.437293	0.0141505
$956$	0	0
$957$	4.55071	0.147104
$958$	0	0
$959$	−48.8530	−1.57755
$960$	0	0
$961$	48.0051	1.54855
$962$	0	0
$963$	−12.7844	−0.411971
$964$	0	0
$965$	−40.2924	−1.29706
$966$	0	0
$967$	15.4754	0.497654	0.248827	−	0.968548i	$-0.419955\pi$
$967$	15.4754	0.497654	0.248827	+	0.968548i	$0.419955\pi$
$968$	0	0
$969$	−1.56670	−0.0503296
$970$	0	0
$971$	33.8048	1.08485	0.542424	−	0.840105i	$-0.317507\pi$
$971$	33.8048	1.08485	0.542424	+	0.840105i	$0.317507\pi$
$972$	0	0
$973$	75.8876	2.43284
$974$	0	0
$975$	−0.146305	−0.00468550
$976$	0	0
$977$	12.0997	0.387104	0.193552	−	0.981090i	$-0.437999\pi$
$977$	12.0997	0.387104	0.193552	+	0.981090i	$0.437999\pi$
$978$	0	0
$979$	35.6337	1.13886
$980$	0	0
$981$	15.7541	0.502989
$982$	0	0
$983$	18.4774	0.589337	0.294668	−	0.955600i	$-0.404791\pi$
$983$	18.4774	0.589337	0.294668	+	0.955600i	$0.404791\pi$
$984$	0	0
$985$	−8.62343	−0.274765
$986$	0	0
$987$	15.7872	0.502513
$988$	0	0
$989$	37.9425	1.20650
$990$	0	0
$991$	−20.7131	−0.657974	−0.328987	−	0.944334i	$-0.606707\pi$
$991$	−20.7131	−0.657974	−0.328987	+	0.944334i	$0.606707\pi$
$992$	0	0
$993$	2.75324	0.0873714
$994$	0	0
$995$	46.5670	1.47627
$996$	0	0
$997$	10.7767	0.341302	0.170651	−	0.985332i	$-0.445413\pi$
$997$	10.7767	0.341302	0.170651	+	0.985332i	$0.445413\pi$
$998$	0	0
$999$	−10.5905	−0.335069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.2.a.d.1.13	✓	23
4.3	odd	2		4016.2.a.m.1.11		23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.13	✓	23	1.1	even	1	trivial
4016.2.a.m.1.11		23	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:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.259421 q^{3} -1.96371 q^{5} -4.76268 q^{7} -2.93270 q^{9} +O(q^{10})\)	\(q+0.259421 q^{3} -1.96371 q^{5} -4.76268 q^{7} -2.93270 q^{9} +3.31268 q^{11} +0.493039 q^{13} -0.509426 q^{15} -3.02691 q^{17} +1.99518 q^{19} -1.23554 q^{21} +3.65140 q^{23} -1.14386 q^{25} -1.53906 q^{27} +5.29536 q^{29} -8.88848 q^{31} +0.859377 q^{33} +9.35250 q^{35} +6.88114 q^{37} +0.127904 q^{39} -3.94408 q^{41} +10.3912 q^{43} +5.75896 q^{45} -12.7776 q^{47} +15.6831 q^{49} -0.785243 q^{51} +5.75231 q^{53} -6.50513 q^{55} +0.517590 q^{57} -12.3248 q^{59} +8.92476 q^{61} +13.9675 q^{63} -0.968183 q^{65} +14.3282 q^{67} +0.947248 q^{69} +10.5818 q^{71} +8.88283 q^{73} -0.296741 q^{75} -15.7772 q^{77} +8.79170 q^{79} +8.39884 q^{81} +8.70243 q^{83} +5.94396 q^{85} +1.37372 q^{87} +10.7568 q^{89} -2.34819 q^{91} -2.30585 q^{93} -3.91794 q^{95} +13.1731 q^{97} -9.71510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * q^99

Introduction

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