LMFDB - Embedded newform 4005.2.a.q.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
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} - 4x^{8} - 6x^{7} + 31x^{6} + 13x^{5} - 75x^{4} - 17x^{3} + 52x^{2} + 11x - 1 $ x^9 - 4x^8 - 6x^7 + 31x^6 + 13x^5 - 75x^4 - 17x^3 + 52x^2 + 11x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.09195$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0919478 q^{2} -1.99155 q^{4} +1.00000 q^{5} -4.79397 q^{7} -0.367014 q^{8} +O(q^{10})$	\(q+0.0919478 q^{2} -1.99155 q^{4} +1.00000 q^{5} -4.79397 q^{7} -0.367014 q^{8} +0.0919478 q^{10} +6.16745 q^{11} -2.11965 q^{13} -0.440795 q^{14} +3.94935 q^{16} -2.39119 q^{17} -4.72975 q^{19} -1.99155 q^{20} +0.567084 q^{22} +5.37376 q^{23} +1.00000 q^{25} -0.194897 q^{26} +9.54742 q^{28} +9.61148 q^{29} -0.532712 q^{31} +1.09716 q^{32} -0.219865 q^{34} -4.79397 q^{35} +1.10150 q^{37} -0.434890 q^{38} -0.367014 q^{40} -8.65227 q^{41} +3.89418 q^{43} -12.2828 q^{44} +0.494105 q^{46} -4.32725 q^{47} +15.9822 q^{49} +0.0919478 q^{50} +4.22138 q^{52} -8.00873 q^{53} +6.16745 q^{55} +1.75945 q^{56} +0.883755 q^{58} +11.5839 q^{59} -11.1641 q^{61} -0.0489817 q^{62} -7.79781 q^{64} -2.11965 q^{65} -8.48241 q^{67} +4.76217 q^{68} -0.440795 q^{70} -0.486638 q^{71} +6.00859 q^{73} +0.101280 q^{74} +9.41951 q^{76} -29.5666 q^{77} -1.40360 q^{79} +3.94935 q^{80} -0.795558 q^{82} -2.03691 q^{83} -2.39119 q^{85} +0.358061 q^{86} -2.26354 q^{88} -1.00000 q^{89} +10.1615 q^{91} -10.7021 q^{92} -0.397882 q^{94} -4.72975 q^{95} -17.8326 q^{97} +1.46953 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8}+O(q^{10}) $ 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8	$ 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8} - 5 q^{10} - 4 q^{11} + 5 q^{13} + q^{14} + 15 q^{16} - 21 q^{17} - 18 q^{19} + 11 q^{20} + 6 q^{22} - 16 q^{23} + 9 q^{25} - 8 q^{26} + 6 q^{28} - 3 q^{29} - 6 q^{31} - 46 q^{32} + 12 q^{34} - 3 q^{35} + 11 q^{37} - 20 q^{38} - 18 q^{40} + q^{41} - 3 q^{43} - 38 q^{44} + 16 q^{46} - 27 q^{47} + 24 q^{49} - 5 q^{50} + 17 q^{52} - 43 q^{53} - 4 q^{55} - 5 q^{56} + 34 q^{58} - 3 q^{59} - 30 q^{61} - 36 q^{62} + 50 q^{64} + 5 q^{65} - 12 q^{67} - 64 q^{68} + q^{70} + 4 q^{71} + 26 q^{73} - 2 q^{74} - 12 q^{76} - 34 q^{77} + q^{79} + 15 q^{80} - 51 q^{82} - 24 q^{83} - 21 q^{85} - 18 q^{86} + 64 q^{88} - 9 q^{89} - 50 q^{91} - 10 q^{92} - 11 q^{94} - 18 q^{95} - 4 q^{97} - 75 q^{98}+O(q^{100}) $ 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8 - 5 * q^10 - 4 * q^11 + 5 * q^13 + q^14 + 15 * q^16 - 21 * q^17 - 18 * q^19 + 11 * q^20 + 6 * q^22 - 16 * q^23 + 9 * q^25 - 8 * q^26 + 6 * q^28 - 3 * q^29 - 6 * q^31 - 46 * q^32 + 12 * q^34 - 3 * q^35 + 11 * q^37 - 20 * q^38 - 18 * q^40 + q^41 - 3 * q^43 - 38 * q^44 + 16 * q^46 - 27 * q^47 + 24 * q^49 - 5 * q^50 + 17 * q^52 - 43 * q^53 - 4 * q^55 - 5 * q^56 + 34 * q^58 - 3 * q^59 - 30 * q^61 - 36 * q^62 + 50 * q^64 + 5 * q^65 - 12 * q^67 - 64 * q^68 + q^70 + 4 * q^71 + 26 * q^73 - 2 * q^74 - 12 * q^76 - 34 * q^77 + q^79 + 15 * q^80 - 51 * q^82 - 24 * q^83 - 21 * q^85 - 18 * q^86 + 64 * q^88 - 9 * q^89 - 50 * q^91 - 10 * q^92 - 11 * q^94 - 18 * q^95 - 4 * q^97 - 75 * 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 $

$2$	0.0919478	0.0650169	0.0325085	−	0.999471i	$-0.489650\pi$
$2$	0.0919478	0.0650169	0.0325085	+	0.999471i	$0.489650\pi$
$3$	0	0
$3$	0	0
$4$	−1.99155	−0.995773
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.79397	−1.81195	−0.905976	−	0.423330i	$-0.860861\pi$
$7$	−4.79397	−1.81195	−0.905976	+	0.423330i	$0.860861\pi$
$8$	−0.367014	−0.129759
$9$	0	0
$10$	0.0919478	0.0290765
$11$	6.16745	1.85956	0.929779	−	0.368119i	$-0.119998\pi$
$11$	6.16745	1.85956	0.929779	+	0.368119i	$0.119998\pi$
$12$	0	0
$13$	−2.11965	−0.587885	−0.293943	−	0.955823i	$-0.594967\pi$
$13$	−2.11965	−0.587885	−0.293943	+	0.955823i	$0.594967\pi$
$14$	−0.440795	−0.117808
$15$	0	0
$16$	3.94935	0.987336
$17$	−2.39119	−0.579949	−0.289975	−	0.957034i	$-0.593647\pi$
$17$	−2.39119	−0.579949	−0.289975	+	0.957034i	$0.593647\pi$
$18$	0	0
$19$	−4.72975	−1.08508	−0.542539	−	0.840030i	$-0.682537\pi$
$19$	−4.72975	−1.08508	−0.542539	+	0.840030i	$0.682537\pi$
$20$	−1.99155	−0.445323
$21$	0	0
$22$	0.567084	0.120903
$23$	5.37376	1.12051	0.560253	−	0.828322i	$-0.310704\pi$
$23$	5.37376	1.12051	0.560253	+	0.828322i	$0.310704\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.194897	−0.0382225
$27$	0	0
$28$	9.54742	1.80429
$29$	9.61148	1.78481	0.892404	−	0.451237i	$-0.149017\pi$
$29$	9.61148	1.78481	0.892404	+	0.451237i	$0.149017\pi$
$30$	0	0
$31$	−0.532712	−0.0956778	−0.0478389	−	0.998855i	$-0.515233\pi$
$31$	−0.532712	−0.0956778	−0.0478389	+	0.998855i	$0.515233\pi$
$32$	1.09716	0.193953
$33$	0	0
$34$	−0.219865	−0.0377065
$35$	−4.79397	−0.810329
$36$	0	0
$37$	1.10150	0.181085	0.0905427	−	0.995893i	$-0.471140\pi$
$37$	1.10150	0.181085	0.0905427	+	0.995893i	$0.471140\pi$
$38$	−0.434890	−0.0705485
$39$	0	0
$40$	−0.367014	−0.0580300
$41$	−8.65227	−1.35126	−0.675629	−	0.737242i	$-0.736128\pi$
$41$	−8.65227	−1.35126	−0.675629	+	0.737242i	$0.736128\pi$
$42$	0	0
$43$	3.89418	0.593857	0.296928	−	0.954900i	$-0.404038\pi$
$43$	3.89418	0.593857	0.296928	+	0.954900i	$0.404038\pi$
$44$	−12.2828	−1.85170
$45$	0	0
$46$	0.494105	0.0728518
$47$	−4.32725	−0.631195	−0.315597	−	0.948893i	$-0.602205\pi$
$47$	−4.32725	−0.631195	−0.315597	+	0.948893i	$0.602205\pi$
$48$	0	0
$49$	15.9822	2.28317
$50$	0.0919478	0.0130034
$51$	0	0
$52$	4.22138	0.585400
$53$	−8.00873	−1.10008	−0.550042	−	0.835137i	$-0.685388\pi$
$53$	−8.00873	−1.10008	−0.550042	+	0.835137i	$0.685388\pi$
$54$	0	0
$55$	6.16745	0.831619
$56$	1.75945	0.235117
$57$	0	0
$58$	0.883755	0.116043
$59$	11.5839	1.50810	0.754050	−	0.656817i	$-0.228098\pi$
$59$	11.5839	1.50810	0.754050	+	0.656817i	$0.228098\pi$
$60$	0	0
$61$	−11.1641	−1.42942	−0.714710	−	0.699420i	$-0.753442\pi$
$61$	−11.1641	−1.42942	−0.714710	+	0.699420i	$0.753442\pi$
$62$	−0.0489817	−0.00622068
$63$	0	0
$64$	−7.79781	−0.974726
$65$	−2.11965	−0.262910
$66$	0	0
$67$	−8.48241	−1.03629	−0.518146	−	0.855292i	$-0.673377\pi$
$67$	−8.48241	−1.03629	−0.518146	+	0.855292i	$0.673377\pi$
$68$	4.76217	0.577498
$69$	0	0
$70$	−0.440795	−0.0526851
$71$	−0.486638	−0.0577533	−0.0288767	−	0.999583i	$-0.509193\pi$
$71$	−0.486638	−0.0577533	−0.0288767	+	0.999583i	$0.509193\pi$
$72$	0	0
$73$	6.00859	0.703253	0.351626	−	0.936140i	$-0.385629\pi$
$73$	6.00859	0.703253	0.351626	+	0.936140i	$0.385629\pi$
$74$	0.101280	0.0117736
$75$	0	0
$76$	9.41951	1.08049
$77$	−29.5666	−3.36943
$78$	0	0
$79$	−1.40360	−0.157918	−0.0789588	−	0.996878i	$-0.525160\pi$
$79$	−1.40360	−0.157918	−0.0789588	+	0.996878i	$0.525160\pi$
$80$	3.94935	0.441550
$81$	0	0
$82$	−0.795558	−0.0878546
$83$	−2.03691	−0.223580	−0.111790	−	0.993732i	$-0.535658\pi$
$83$	−2.03691	−0.223580	−0.111790	+	0.993732i	$0.535658\pi$
$84$	0	0
$85$	−2.39119	−0.259361
$86$	0.358061	0.0386107
$87$	0	0
$88$	−2.26354	−0.241294
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	10.1615	1.06522
$92$	−10.7021	−1.11577
$93$	0	0
$94$	−0.397882	−0.0410384
$95$	−4.72975	−0.485262
$96$	0	0
$97$	−17.8326	−1.81062	−0.905311	−	0.424750i	$-0.860362\pi$
$97$	−17.8326	−1.81062	−0.905311	+	0.424750i	$0.860362\pi$
$98$	1.46953	0.148445
$99$	0	0
$100$	−1.99155	−0.199155
$101$	5.42912	0.540217	0.270109	−	0.962830i	$-0.412940\pi$
$101$	5.42912	0.540217	0.270109	+	0.962830i	$0.412940\pi$
$102$	0	0
$103$	9.77556	0.963214	0.481607	−	0.876387i	$-0.340053\pi$
$103$	9.77556	0.963214	0.481607	+	0.876387i	$0.340053\pi$
$104$	0.777941	0.0762834
$105$	0	0
$106$	−0.736385	−0.0715241
$107$	−9.50951	−0.919319	−0.459660	−	0.888095i	$-0.652029\pi$
$107$	−9.50951	−0.919319	−0.459660	+	0.888095i	$0.652029\pi$
$108$	0	0
$109$	−7.41592	−0.710316	−0.355158	−	0.934806i	$-0.615573\pi$
$109$	−7.41592	−0.710316	−0.355158	+	0.934806i	$0.615573\pi$
$110$	0.567084	0.0540693
$111$	0	0
$112$	−18.9331	−1.78901
$113$	−2.13635	−0.200971	−0.100486	−	0.994939i	$-0.532040\pi$
$113$	−2.13635	−0.200971	−0.100486	+	0.994939i	$0.532040\pi$
$114$	0	0
$115$	5.37376	0.501105
$116$	−19.1417	−1.77726
$117$	0	0
$118$	1.06512	0.0980520
$119$	11.4633	1.05084
$120$	0	0
$121$	27.0375	2.45795
$122$	−1.02652	−0.0929366
$123$	0	0
$124$	1.06092	0.0952734
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.32417	−0.738651	−0.369325	−	0.929300i	$-0.620411\pi$
$127$	−8.32417	−0.738651	−0.369325	+	0.929300i	$0.620411\pi$
$128$	−2.91131	−0.257326
$129$	0	0
$130$	−0.194897	−0.0170936
$131$	−12.9778	−1.13387	−0.566937	−	0.823761i	$-0.691872\pi$
$131$	−12.9778	−1.13387	−0.566937	+	0.823761i	$0.691872\pi$
$132$	0	0
$133$	22.6743	1.96611
$134$	−0.779939	−0.0673765
$135$	0	0
$136$	0.877601	0.0752537
$137$	−3.92023	−0.334928	−0.167464	−	0.985878i	$-0.553558\pi$
$137$	−3.92023	−0.334928	−0.167464	+	0.985878i	$0.553558\pi$
$138$	0	0
$139$	−1.22785	−0.104145	−0.0520726	−	0.998643i	$-0.516583\pi$
$139$	−1.22785	−0.104145	−0.0520726	+	0.998643i	$0.516583\pi$
$140$	9.54742	0.806904
$141$	0	0
$142$	−0.0447453	−0.00375495
$143$	−13.0728	−1.09321
$144$	0	0
$145$	9.61148	0.798190
$146$	0.552477	0.0457233
$147$	0	0
$148$	−2.19369	−0.180320
$149$	16.5865	1.35882	0.679410	−	0.733759i	$-0.262236\pi$
$149$	16.5865	1.35882	0.679410	+	0.733759i	$0.262236\pi$
$150$	0	0
$151$	−16.9111	−1.37621	−0.688104	−	0.725612i	$-0.741557\pi$
$151$	−16.9111	−1.37621	−0.688104	+	0.725612i	$0.741557\pi$
$152$	1.73588	0.140799
$153$	0	0
$154$	−2.71859	−0.219070
$155$	−0.532712	−0.0427884
$156$	0	0
$157$	−12.1586	−0.970365	−0.485182	−	0.874413i	$-0.661247\pi$
$157$	−12.1586	−0.970365	−0.485182	+	0.874413i	$0.661247\pi$
$158$	−0.129058	−0.0102673
$159$	0	0
$160$	1.09716	0.0867382
$161$	−25.7616	−2.03030
$162$	0	0
$163$	−19.7034	−1.54329	−0.771643	−	0.636055i	$-0.780565\pi$
$163$	−19.7034	−1.54329	−0.771643	+	0.636055i	$0.780565\pi$
$164$	17.2314	1.34555
$165$	0	0
$166$	−0.187290	−0.0145365
$167$	−6.73896	−0.521477	−0.260738	−	0.965410i	$-0.583966\pi$
$167$	−6.73896	−0.521477	−0.260738	+	0.965410i	$0.583966\pi$
$168$	0	0
$169$	−8.50709	−0.654391
$170$	−0.219865	−0.0168629
$171$	0	0
$172$	−7.75544	−0.591346
$173$	−19.3801	−1.47344	−0.736721	−	0.676196i	$-0.763627\pi$
$173$	−19.3801	−1.47344	−0.736721	+	0.676196i	$0.763627\pi$
$174$	0	0
$175$	−4.79397	−0.362390
$176$	24.3574	1.83601
$177$	0	0
$178$	−0.0919478	−0.00689178
$179$	1.95592	0.146192	0.0730962	−	0.997325i	$-0.476712\pi$
$179$	1.95592	0.146192	0.0730962	+	0.997325i	$0.476712\pi$
$180$	0	0
$181$	19.7042	1.46460	0.732301	−	0.680982i	$-0.238447\pi$
$181$	19.7042	1.46460	0.732301	+	0.680982i	$0.238447\pi$
$182$	0.934332	0.0692573
$183$	0	0
$184$	−1.97224	−0.145396
$185$	1.10150	0.0809838
$186$	0	0
$187$	−14.7476	−1.07845
$188$	8.61793	0.628527
$189$	0	0
$190$	−0.434890	−0.0315502
$191$	−5.25568	−0.380288	−0.190144	−	0.981756i	$-0.560895\pi$
$191$	−5.25568	−0.380288	−0.190144	+	0.981756i	$0.560895\pi$
$192$	0	0
$193$	11.6256	0.836826	0.418413	−	0.908257i	$-0.362587\pi$
$193$	11.6256	0.836826	0.418413	+	0.908257i	$0.362587\pi$
$194$	−1.63966	−0.117721
$195$	0	0
$196$	−31.8292	−2.27352
$197$	25.7226	1.83266	0.916330	−	0.400424i	$-0.131137\pi$
$197$	25.7226	1.83266	0.916330	+	0.400424i	$0.131137\pi$
$198$	0	0
$199$	−11.2210	−0.795438	−0.397719	−	0.917507i	$-0.630198\pi$
$199$	−11.2210	−0.795438	−0.397719	+	0.917507i	$0.630198\pi$
$200$	−0.367014	−0.0259518
$201$	0	0
$202$	0.499195	0.0351233
$203$	−46.0772	−3.23399
$204$	0	0
$205$	−8.65227	−0.604301
$206$	0.898841	0.0626252
$207$	0	0
$208$	−8.37123	−0.580440
$209$	−29.1705	−2.01777
$210$	0	0
$211$	−23.8477	−1.64174	−0.820870	−	0.571115i	$-0.806511\pi$
$211$	−23.8477	−1.64174	−0.820870	+	0.571115i	$0.806511\pi$
$212$	15.9498	1.09543
$213$	0	0
$214$	−0.874379	−0.0597713
$215$	3.89418	0.265581
$216$	0	0
$217$	2.55381	0.173364
$218$	−0.681878	−0.0461826
$219$	0	0
$220$	−12.2828	−0.828104
$221$	5.06849	0.340944
$222$	0	0
$223$	−9.53956	−0.638816	−0.319408	−	0.947617i	$-0.603484\pi$
$223$	−9.53956	−0.638816	−0.319408	+	0.947617i	$0.603484\pi$
$224$	−5.25976	−0.351433
$225$	0	0
$226$	−0.196433	−0.0130665
$227$	−19.7990	−1.31411	−0.657053	−	0.753844i	$-0.728197\pi$
$227$	−19.7990	−1.31411	−0.657053	+	0.753844i	$0.728197\pi$
$228$	0	0
$229$	−17.5065	−1.15686	−0.578431	−	0.815731i	$-0.696335\pi$
$229$	−17.5065	−1.15686	−0.578431	+	0.815731i	$0.696335\pi$
$230$	0.494105	0.0325803
$231$	0	0
$232$	−3.52755	−0.231595
$233$	−15.2223	−0.997248	−0.498624	−	0.866818i	$-0.666161\pi$
$233$	−15.2223	−0.997248	−0.498624	+	0.866818i	$0.666161\pi$
$234$	0	0
$235$	−4.32725	−0.282279
$236$	−23.0699	−1.50173
$237$	0	0
$238$	1.05403	0.0683224
$239$	−9.35413	−0.605068	−0.302534	−	0.953139i	$-0.597833\pi$
$239$	−9.35413	−0.605068	−0.302534	+	0.953139i	$0.597833\pi$
$240$	0	0
$241$	−16.4310	−1.05841	−0.529206	−	0.848493i	$-0.677510\pi$
$241$	−16.4310	−1.05841	−0.529206	+	0.848493i	$0.677510\pi$
$242$	2.48604	0.159809
$243$	0	0
$244$	22.2339	1.42338
$245$	15.9822	1.02106
$246$	0	0
$247$	10.0254	0.637902
$248$	0.195513	0.0124151
$249$	0	0
$250$	0.0919478	0.00581529
$251$	2.45547	0.154988	0.0774940	−	0.996993i	$-0.475308\pi$
$251$	2.45547	0.154988	0.0774940	+	0.996993i	$0.475308\pi$
$252$	0	0
$253$	33.1424	2.08364
$254$	−0.765389	−0.0480248
$255$	0	0
$256$	15.3279	0.957995
$257$	4.11961	0.256974	0.128487	−	0.991711i	$-0.458988\pi$
$257$	4.11961	0.256974	0.128487	+	0.991711i	$0.458988\pi$
$258$	0	0
$259$	−5.28056	−0.328118
$260$	4.22138	0.261799
$261$	0	0
$262$	−1.19328	−0.0737210
$263$	9.35485	0.576845	0.288422	−	0.957503i	$-0.406869\pi$
$263$	9.35485	0.576845	0.288422	+	0.957503i	$0.406869\pi$
$264$	0	0
$265$	−8.00873	−0.491972
$266$	2.08485	0.127830
$267$	0	0
$268$	16.8931	1.03191
$269$	18.5615	1.13171	0.565856	−	0.824504i	$-0.308546\pi$
$269$	18.5615	1.13171	0.565856	+	0.824504i	$0.308546\pi$
$270$	0	0
$271$	24.0667	1.46195	0.730975	−	0.682404i	$-0.239065\pi$
$271$	24.0667	1.46195	0.730975	+	0.682404i	$0.239065\pi$
$272$	−9.44365	−0.572605
$273$	0	0
$274$	−0.360456	−0.0217760
$275$	6.16745	0.371911
$276$	0	0
$277$	0.204015	0.0122581	0.00612905	−	0.999981i	$-0.498049\pi$
$277$	0.204015	0.0122581	0.00612905	+	0.999981i	$0.498049\pi$
$278$	−0.112898	−0.00677120
$279$	0	0
$280$	1.75945	0.105148
$281$	4.39781	0.262351	0.131176	−	0.991359i	$-0.458125\pi$
$281$	4.39781	0.262351	0.131176	+	0.991359i	$0.458125\pi$
$282$	0	0
$283$	−20.0280	−1.19054	−0.595271	−	0.803525i	$-0.702955\pi$
$283$	−20.0280	−1.19054	−0.595271	+	0.803525i	$0.702955\pi$
$284$	0.969162	0.0575092
$285$	0	0
$286$	−1.20202	−0.0710769
$287$	41.4788	2.44841
$288$	0	0
$289$	−11.2822	−0.663659
$290$	0.883755	0.0518959
$291$	0	0
$292$	−11.9664	−0.700280
$293$	8.53207	0.498449	0.249224	−	0.968446i	$-0.419824\pi$
$293$	8.53207	0.498449	0.249224	+	0.968446i	$0.419824\pi$
$294$	0	0
$295$	11.5839	0.674443
$296$	−0.404266	−0.0234975
$297$	0	0
$298$	1.52509	0.0883463
$299$	−11.3905	−0.658729
$300$	0	0
$301$	−18.6686	−1.07604
$302$	−1.55494	−0.0894768
$303$	0	0
$304$	−18.6794	−1.07134
$305$	−11.1641	−0.639256
$306$	0	0
$307$	25.5346	1.45734	0.728668	−	0.684867i	$-0.240140\pi$
$307$	25.5346	1.45734	0.728668	+	0.684867i	$0.240140\pi$
$308$	58.8832	3.35518
$309$	0	0
$310$	−0.0489817	−0.00278197
$311$	20.2965	1.15091	0.575454	−	0.817834i	$-0.304825\pi$
$311$	20.2965	1.15091	0.575454	+	0.817834i	$0.304825\pi$
$312$	0	0
$313$	4.91708	0.277930	0.138965	−	0.990297i	$-0.455623\pi$
$313$	4.91708	0.277930	0.138965	+	0.990297i	$0.455623\pi$
$314$	−1.11796	−0.0630901
$315$	0	0
$316$	2.79534	0.157250
$317$	6.08503	0.341769	0.170885	−	0.985291i	$-0.445337\pi$
$317$	6.08503	0.341769	0.170885	+	0.985291i	$0.445337\pi$
$318$	0	0
$319$	59.2784	3.31895
$320$	−7.79781	−0.435911
$321$	0	0
$322$	−2.36873	−0.132004
$323$	11.3097	0.629291
$324$	0	0
$325$	−2.11965	−0.117577
$326$	−1.81168	−0.100340
$327$	0	0
$328$	3.17550	0.175338
$329$	20.7447	1.14369
$330$	0	0
$331$	13.6082	0.747975	0.373988	−	0.927434i	$-0.377990\pi$
$331$	13.6082	0.747975	0.373988	+	0.927434i	$0.377990\pi$
$332$	4.05660	0.222635
$333$	0	0
$334$	−0.619633	−0.0339048
$335$	−8.48241	−0.463444
$336$	0	0
$337$	−13.1440	−0.715999	−0.358000	−	0.933722i	$-0.616541\pi$
$337$	−13.1440	−0.715999	−0.358000	+	0.933722i	$0.616541\pi$
$338$	−0.782208	−0.0425465
$339$	0	0
$340$	4.76217	0.258265
$341$	−3.28547	−0.177918
$342$	0	0
$343$	−43.0603	−2.32504
$344$	−1.42922	−0.0770583
$345$	0	0
$346$	−1.78196	−0.0957987
$347$	−16.0098	−0.859449	−0.429725	−	0.902960i	$-0.641389\pi$
$347$	−16.0098	−0.859449	−0.429725	+	0.902960i	$0.641389\pi$
$348$	0	0
$349$	−12.9042	−0.690747	−0.345374	−	0.938465i	$-0.612248\pi$
$349$	−12.9042	−0.690747	−0.345374	+	0.938465i	$0.612248\pi$
$350$	−0.440795	−0.0235615
$351$	0	0
$352$	6.76669	0.360666
$353$	21.6511	1.15237	0.576186	−	0.817318i	$-0.304540\pi$
$353$	21.6511	1.15237	0.576186	+	0.817318i	$0.304540\pi$
$354$	0	0
$355$	−0.486638	−0.0258281
$356$	1.99155	0.105552
$357$	0	0
$358$	0.179843	0.00950498
$359$	−23.3217	−1.23087	−0.615436	−	0.788187i	$-0.711020\pi$
$359$	−23.3217	−1.23087	−0.615436	+	0.788187i	$0.711020\pi$
$360$	0	0
$361$	3.37053	0.177396
$362$	1.81176	0.0952239
$363$	0	0
$364$	−20.2372	−1.06072
$365$	6.00859	0.314504
$366$	0	0
$367$	7.27313	0.379654	0.189827	−	0.981818i	$-0.439207\pi$
$367$	7.27313	0.379654	0.189827	+	0.981818i	$0.439207\pi$
$368$	21.2228	1.10632
$369$	0	0
$370$	0.101280	0.00526532
$371$	38.3936	1.99330
$372$	0	0
$373$	25.3635	1.31327	0.656636	−	0.754208i	$-0.271979\pi$
$373$	25.3635	1.31327	0.656636	+	0.754208i	$0.271979\pi$
$374$	−1.35601	−0.0701175
$375$	0	0
$376$	1.58816	0.0819032
$377$	−20.3730	−1.04926
$378$	0	0
$379$	12.9738	0.666421	0.333210	−	0.942852i	$-0.391868\pi$
$379$	12.9738	0.666421	0.333210	+	0.942852i	$0.391868\pi$
$380$	9.41951	0.483211
$381$	0	0
$382$	−0.483249	−0.0247252
$383$	−37.8998	−1.93659	−0.968295	−	0.249809i	$-0.919632\pi$
$383$	−37.8998	−1.93659	−0.968295	+	0.249809i	$0.919632\pi$
$384$	0	0
$385$	−29.5666	−1.50685
$386$	1.06894	0.0544078
$387$	0	0
$388$	35.5143	1.80297
$389$	−16.5797	−0.840626	−0.420313	−	0.907379i	$-0.638080\pi$
$389$	−16.5797	−0.840626	−0.420313	+	0.907379i	$0.638080\pi$
$390$	0	0
$391$	−12.8497	−0.649837
$392$	−5.86568	−0.296262
$393$	0	0
$394$	2.36514	0.119154
$395$	−1.40360	−0.0706229
$396$	0	0
$397$	−2.04963	−0.102868	−0.0514341	−	0.998676i	$-0.516379\pi$
$397$	−2.04963	−0.102868	−0.0514341	+	0.998676i	$0.516379\pi$
$398$	−1.03175	−0.0517169
$399$	0	0
$400$	3.94935	0.197467
$401$	−25.5924	−1.27802	−0.639012	−	0.769197i	$-0.720657\pi$
$401$	−25.5924	−1.27802	−0.639012	+	0.769197i	$0.720657\pi$
$402$	0	0
$403$	1.12916	0.0562476
$404$	−10.8123	−0.537934
$405$	0	0
$406$	−4.23670	−0.210264
$407$	6.79345	0.336739
$408$	0	0
$409$	−21.2704	−1.05175	−0.525877	−	0.850561i	$-0.676263\pi$
$409$	−21.2704	−1.05175	−0.525877	+	0.850561i	$0.676263\pi$
$410$	−0.795558	−0.0392898
$411$	0	0
$412$	−19.4685	−0.959142
$413$	−55.5331	−2.73260
$414$	0	0
$415$	−2.03691	−0.0999880
$416$	−2.32560	−0.114022
$417$	0	0
$418$	−2.68217	−0.131189
$419$	−14.5113	−0.708924	−0.354462	−	0.935070i	$-0.615336\pi$
$419$	−14.5113	−0.708924	−0.354462	+	0.935070i	$0.615336\pi$
$420$	0	0
$421$	−6.85379	−0.334033	−0.167017	−	0.985954i	$-0.553413\pi$
$421$	−6.85379	−0.334033	−0.167017	+	0.985954i	$0.553413\pi$
$422$	−2.19274	−0.106741
$423$	0	0
$424$	2.93932	0.142746
$425$	−2.39119	−0.115990
$426$	0	0
$427$	53.5206	2.59004
$428$	18.9386	0.915433
$429$	0	0
$430$	0.358061	0.0172673
$431$	21.9953	1.05948	0.529739	−	0.848161i	$-0.322290\pi$
$431$	21.9953	1.05948	0.529739	+	0.848161i	$0.322290\pi$
$432$	0	0
$433$	17.9342	0.861864	0.430932	−	0.902384i	$-0.358185\pi$
$433$	17.9342	0.861864	0.430932	+	0.902384i	$0.358185\pi$
$434$	0.234817	0.0112716
$435$	0	0
$436$	14.7691	0.707314
$437$	−25.4165	−1.21584
$438$	0	0
$439$	−30.0842	−1.43584	−0.717920	−	0.696126i	$-0.754906\pi$
$439$	−30.0842	−1.43584	−0.717920	+	0.696126i	$0.754906\pi$
$440$	−2.26354	−0.107910
$441$	0	0
$442$	0.466037	0.0221671
$443$	−18.5994	−0.883684	−0.441842	−	0.897093i	$-0.645675\pi$
$443$	−18.5994	−0.883684	−0.441842	+	0.897093i	$0.645675\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−0.877142	−0.0415339
$447$	0	0
$448$	37.3825	1.76616
$449$	28.0764	1.32501	0.662504	−	0.749059i	$-0.269494\pi$
$449$	28.0764	1.32501	0.662504	+	0.749059i	$0.269494\pi$
$450$	0	0
$451$	−53.3625	−2.51274
$452$	4.25465	0.200122
$453$	0	0
$454$	−1.82048	−0.0854392
$455$	10.1615	0.476380
$456$	0	0
$457$	35.5024	1.66073	0.830366	−	0.557218i	$-0.188131\pi$
$457$	35.5024	1.66073	0.830366	+	0.557218i	$0.188131\pi$
$458$	−1.60968	−0.0752156
$459$	0	0
$460$	−10.7021	−0.498987
$461$	5.48768	0.255587	0.127793	−	0.991801i	$-0.459211\pi$
$461$	5.48768	0.255587	0.127793	+	0.991801i	$0.459211\pi$
$462$	0	0
$463$	−28.1315	−1.30738	−0.653692	−	0.756761i	$-0.726781\pi$
$463$	−28.1315	−1.30738	−0.653692	+	0.756761i	$0.726781\pi$
$464$	37.9591	1.76221
$465$	0	0
$466$	−1.39966	−0.0648380
$467$	−9.50922	−0.440034	−0.220017	−	0.975496i	$-0.570611\pi$
$467$	−9.50922	−0.440034	−0.220017	+	0.975496i	$0.570611\pi$
$468$	0	0
$469$	40.6644	1.87771
$470$	−0.397882	−0.0183529
$471$	0	0
$472$	−4.25147	−0.195690
$473$	24.0172	1.10431
$474$	0	0
$475$	−4.72975	−0.217016
$476$	−22.8297	−1.04640
$477$	0	0
$478$	−0.860092	−0.0393397
$479$	−30.5454	−1.39566	−0.697828	−	0.716265i	$-0.745850\pi$
$479$	−30.5454	−1.39566	−0.697828	+	0.716265i	$0.745850\pi$
$480$	0	0
$481$	−2.33479	−0.106457
$482$	−1.51079	−0.0688147
$483$	0	0
$484$	−53.8464	−2.44756
$485$	−17.8326	−0.809735
$486$	0	0
$487$	35.6518	1.61554	0.807768	−	0.589501i	$-0.200675\pi$
$487$	35.6518	1.61554	0.807768	+	0.589501i	$0.200675\pi$
$488$	4.09739	0.185480
$489$	0	0
$490$	1.46953	0.0663864
$491$	−5.05449	−0.228106	−0.114053	−	0.993475i	$-0.536383\pi$
$491$	−5.05449	−0.228106	−0.114053	+	0.993475i	$0.536383\pi$
$492$	0	0
$493$	−22.9829	−1.03510
$494$	0.921815	0.0414744
$495$	0	0
$496$	−2.10386	−0.0944662
$497$	2.33293	0.104646
$498$	0	0
$499$	−13.3093	−0.595805	−0.297902	−	0.954596i	$-0.596287\pi$
$499$	−13.3093	−0.595805	−0.297902	+	0.954596i	$0.596287\pi$
$500$	−1.99155	−0.0890646
$501$	0	0
$502$	0.225775	0.0100768
$503$	−2.49740	−0.111353	−0.0556767	−	0.998449i	$-0.517732\pi$
$503$	−2.49740	−0.111353	−0.0556767	+	0.998449i	$0.517732\pi$
$504$	0	0
$505$	5.42912	0.241593
$506$	3.04737	0.135472
$507$	0	0
$508$	16.5780	0.735528
$509$	6.20818	0.275173	0.137586	−	0.990490i	$-0.456066\pi$
$509$	6.20818	0.275173	0.137586	+	0.990490i	$0.456066\pi$
$510$	0	0
$511$	−28.8050	−1.27426
$512$	7.23200	0.319612
$513$	0	0
$514$	0.378789	0.0167077
$515$	9.77556	0.430762
$516$	0	0
$517$	−26.6881	−1.17374
$518$	−0.485536	−0.0213332
$519$	0	0
$520$	0.777941	0.0341150
$521$	−4.18450	−0.183326	−0.0916632	−	0.995790i	$-0.529218\pi$
$521$	−4.18450	−0.183326	−0.0916632	+	0.995790i	$0.529218\pi$
$522$	0	0
$523$	40.5211	1.77187	0.885933	−	0.463813i	$-0.153519\pi$
$523$	40.5211	1.77187	0.885933	+	0.463813i	$0.153519\pi$
$524$	25.8459	1.12908
$525$	0	0
$526$	0.860158	0.0375047
$527$	1.27382	0.0554883
$528$	0	0
$529$	5.87726	0.255533
$530$	−0.736385	−0.0319865
$531$	0	0
$532$	−45.1569	−1.95780
$533$	18.3398	0.794384
$534$	0	0
$535$	−9.50951	−0.411132
$536$	3.11316	0.134468
$537$	0	0
$538$	1.70669	0.0735805
$539$	98.5693	4.24568
$540$	0	0
$541$	33.3315	1.43303	0.716516	−	0.697571i	$-0.245736\pi$
$541$	33.3315	1.43303	0.716516	+	0.697571i	$0.245736\pi$
$542$	2.21288	0.0950515
$543$	0	0
$544$	−2.62352	−0.112483
$545$	−7.41592	−0.317663
$546$	0	0
$547$	15.8529	0.677820	0.338910	−	0.940819i	$-0.389942\pi$
$547$	15.8529	0.677820	0.338910	+	0.940819i	$0.389942\pi$
$548$	7.80731	0.333512
$549$	0	0
$550$	0.567084	0.0241805
$551$	−45.4599	−1.93666
$552$	0	0
$553$	6.72883	0.286139
$554$	0.0187588	0.000796984 0
$555$	0	0
$556$	2.44533	0.103705
$557$	−38.6303	−1.63682	−0.818410	−	0.574635i	$-0.805144\pi$
$557$	−38.6303	−1.63682	−0.818410	+	0.574635i	$0.805144\pi$
$558$	0	0
$559$	−8.25430	−0.349120
$560$	−18.9331	−0.800067
$561$	0	0
$562$	0.404369	0.0170573
$563$	−36.8929	−1.55485	−0.777424	−	0.628977i	$-0.783474\pi$
$563$	−36.8929	−1.55485	−0.777424	+	0.628977i	$0.783474\pi$
$564$	0	0
$565$	−2.13635	−0.0898771
$566$	−1.84153	−0.0774054
$567$	0	0
$568$	0.178603	0.00749402
$569$	−25.8400	−1.08327	−0.541634	−	0.840614i	$-0.682194\pi$
$569$	−25.8400	−1.08327	−0.541634	+	0.840614i	$0.682194\pi$
$570$	0	0
$571$	41.6364	1.74243	0.871214	−	0.490903i	$-0.163333\pi$
$571$	41.6364	1.74243	0.871214	+	0.490903i	$0.163333\pi$
$572$	26.0352	1.08858
$573$	0	0
$574$	3.81388	0.159188
$575$	5.37376	0.224101
$576$	0	0
$577$	−28.8919	−1.20278	−0.601392	−	0.798954i	$-0.705387\pi$
$577$	−28.8919	−1.20278	−0.601392	+	0.798954i	$0.705387\pi$
$578$	−1.03737	−0.0431490
$579$	0	0
$580$	−19.1417	−0.794816
$581$	9.76490	0.405116
$582$	0	0
$583$	−49.3935	−2.04567
$584$	−2.20524	−0.0912534
$585$	0	0
$586$	0.784505	0.0324076
$587$	16.5177	0.681760	0.340880	−	0.940107i	$-0.389275\pi$
$587$	16.5177	0.681760	0.340880	+	0.940107i	$0.389275\pi$
$588$	0	0
$589$	2.51959	0.103818
$590$	1.06512	0.0438502
$591$	0	0
$592$	4.35020	0.178792
$593$	36.7955	1.51101	0.755506	−	0.655142i	$-0.227391\pi$
$593$	36.7955	1.51101	0.755506	+	0.655142i	$0.227391\pi$
$594$	0	0
$595$	11.4633	0.469950
$596$	−33.0328	−1.35308
$597$	0	0
$598$	−1.04733	−0.0428285
$599$	18.3522	0.749852	0.374926	−	0.927055i	$-0.377668\pi$
$599$	18.3522	0.749852	0.374926	+	0.927055i	$0.377668\pi$
$600$	0	0
$601$	−36.0758	−1.47156	−0.735781	−	0.677219i	$-0.763185\pi$
$601$	−36.0758	−1.47156	−0.735781	+	0.677219i	$0.763185\pi$
$602$	−1.71654	−0.0699608
$603$	0	0
$604$	33.6793	1.37039
$605$	27.0375	1.09923
$606$	0	0
$607$	−24.2426	−0.983978	−0.491989	−	0.870601i	$-0.663730\pi$
$607$	−24.2426	−0.983978	−0.491989	+	0.870601i	$0.663730\pi$
$608$	−5.18930	−0.210454
$609$	0	0
$610$	−1.02652	−0.0415625
$611$	9.17226	0.371070
$612$	0	0
$613$	−8.72223	−0.352288	−0.176144	−	0.984364i	$-0.556362\pi$
$613$	−8.72223	−0.352288	−0.176144	+	0.984364i	$0.556362\pi$
$614$	2.34785	0.0947515
$615$	0	0
$616$	10.8514	0.437214
$617$	−6.90271	−0.277893	−0.138946	−	0.990300i	$-0.544372\pi$
$617$	−6.90271	−0.277893	−0.138946	+	0.990300i	$0.544372\pi$
$618$	0	0
$619$	−0.531769	−0.0213736	−0.0106868	−	0.999943i	$-0.503402\pi$
$619$	−0.531769	−0.0213736	−0.0106868	+	0.999943i	$0.503402\pi$
$620$	1.06092	0.0426076
$621$	0	0
$622$	1.86622	0.0748285
$623$	4.79397	0.192066
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.452115	0.0180701
$627$	0	0
$628$	24.2145	0.966263
$629$	−2.63390	−0.105020
$630$	0	0
$631$	6.64892	0.264689	0.132345	−	0.991204i	$-0.457749\pi$
$631$	6.64892	0.264689	0.132345	+	0.991204i	$0.457749\pi$
$632$	0.515142	0.0204912
$633$	0	0
$634$	0.559505	0.0222208
$635$	−8.32417	−0.330335
$636$	0	0
$637$	−33.8766	−1.34224
$638$	5.45052	0.215788
$639$	0	0
$640$	−2.91131	−0.115080
$641$	32.4598	1.28209	0.641043	−	0.767505i	$-0.278502\pi$
$641$	32.4598	1.28209	0.641043	+	0.767505i	$0.278502\pi$
$642$	0	0
$643$	−42.1675	−1.66293	−0.831463	−	0.555580i	$-0.812496\pi$
$643$	−42.1675	−1.66293	−0.831463	+	0.555580i	$0.812496\pi$
$644$	51.3055	2.02172
$645$	0	0
$646$	1.03991	0.0409146
$647$	−17.3037	−0.680277	−0.340138	−	0.940375i	$-0.610474\pi$
$647$	−17.3037	−0.680277	−0.340138	+	0.940375i	$0.610474\pi$
$648$	0	0
$649$	71.4434	2.80440
$650$	−0.194897	−0.00764450
$651$	0	0
$652$	39.2402	1.53676
$653$	3.49104	0.136615	0.0683076	−	0.997664i	$-0.478240\pi$
$653$	3.49104	0.136615	0.0683076	+	0.997664i	$0.478240\pi$
$654$	0	0
$655$	−12.9778	−0.507084
$656$	−34.1708	−1.33415
$657$	0	0
$658$	1.90743	0.0743595
$659$	−35.1370	−1.36874	−0.684372	−	0.729133i	$-0.739923\pi$
$659$	−35.1370	−1.36874	−0.684372	+	0.729133i	$0.739923\pi$
$660$	0	0
$661$	49.7878	1.93652	0.968260	−	0.249946i	$-0.0804129\pi$
$661$	49.7878	1.93652	0.968260	+	0.249946i	$0.0804129\pi$
$662$	1.25125	0.0486310
$663$	0	0
$664$	0.747575	0.0290115
$665$	22.6743	0.879271
$666$	0	0
$667$	51.6498	1.99989
$668$	13.4210	0.519272
$669$	0	0
$670$	−0.779939	−0.0301317
$671$	−68.8543	−2.65809
$672$	0	0
$673$	12.4179	0.478673	0.239337	−	0.970937i	$-0.423070\pi$
$673$	12.4179	0.478673	0.239337	+	0.970937i	$0.423070\pi$
$674$	−1.20856	−0.0465521
$675$	0	0
$676$	16.9422	0.651625
$677$	−22.7851	−0.875701	−0.437850	−	0.899048i	$-0.644260\pi$
$677$	−22.7851	−0.875701	−0.437850	+	0.899048i	$0.644260\pi$
$678$	0	0
$679$	85.4888	3.28076
$680$	0.877601	0.0336545
$681$	0	0
$682$	−0.302092	−0.0115677
$683$	7.97263	0.305064	0.152532	−	0.988299i	$-0.451257\pi$
$683$	7.97263	0.305064	0.152532	+	0.988299i	$0.451257\pi$
$684$	0	0
$685$	−3.92023	−0.149784
$686$	−3.95930	−0.151167
$687$	0	0
$688$	15.3795	0.586336
$689$	16.9757	0.646723
$690$	0	0
$691$	−38.9495	−1.48171	−0.740854	−	0.671666i	$-0.765579\pi$
$691$	−38.9495	−1.48171	−0.740854	+	0.671666i	$0.765579\pi$
$692$	38.5964	1.46721
$693$	0	0
$694$	−1.47206	−0.0558788
$695$	−1.22785	−0.0465751
$696$	0	0
$697$	20.6893	0.783661
$698$	−1.18652	−0.0449103
$699$	0	0
$700$	9.54742	0.360858
$701$	20.4866	0.773769	0.386885	−	0.922128i	$-0.373551\pi$
$701$	20.4866	0.773769	0.386885	+	0.922128i	$0.373551\pi$
$702$	0	0
$703$	−5.20981	−0.196492
$704$	−48.0926	−1.81256
$705$	0	0
$706$	1.99077	0.0749237
$707$	−26.0270	−0.978847
$708$	0	0
$709$	32.8597	1.23407	0.617036	−	0.786935i	$-0.288333\pi$
$709$	32.8597	1.23407	0.617036	+	0.786935i	$0.288333\pi$
$710$	−0.0447453	−0.00167926
$711$	0	0
$712$	0.367014	0.0137544
$713$	−2.86266	−0.107208
$714$	0	0
$715$	−13.0728	−0.488897
$716$	−3.89531	−0.145574
$717$	0	0
$718$	−2.14438	−0.0800275
$719$	3.47204	0.129485	0.0647425	−	0.997902i	$-0.479377\pi$
$719$	3.47204	0.129485	0.0647425	+	0.997902i	$0.479377\pi$
$720$	0	0
$721$	−46.8637	−1.74530
$722$	0.309913	0.0115338
$723$	0	0
$724$	−39.2418	−1.45841
$725$	9.61148	0.356962
$726$	0	0
$727$	−26.6971	−0.990141	−0.495070	−	0.868853i	$-0.664858\pi$
$727$	−26.6971	−0.990141	−0.495070	+	0.868853i	$0.664858\pi$
$728$	−3.72943	−0.138222
$729$	0	0
$730$	0.552477	0.0204481
$731$	−9.31173	−0.344407
$732$	0	0
$733$	46.5607	1.71976	0.859879	−	0.510498i	$-0.170539\pi$
$733$	46.5607	1.71976	0.859879	+	0.510498i	$0.170539\pi$
$734$	0.668749	0.0246840
$735$	0	0
$736$	5.89588	0.217325
$737$	−52.3149	−1.92704
$738$	0	0
$739$	−14.5595	−0.535581	−0.267791	−	0.963477i	$-0.586294\pi$
$739$	−14.5595	−0.535581	−0.267791	+	0.963477i	$0.586294\pi$
$740$	−2.19369	−0.0806415
$741$	0	0
$742$	3.53021	0.129598
$743$	4.66908	0.171292	0.0856460	−	0.996326i	$-0.472705\pi$
$743$	4.66908	0.171292	0.0856460	+	0.996326i	$0.472705\pi$
$744$	0	0
$745$	16.5865	0.607683
$746$	2.33212	0.0853849
$747$	0	0
$748$	29.3705	1.07389
$749$	45.5883	1.66576
$750$	0	0
$751$	20.8291	0.760064	0.380032	−	0.924973i	$-0.375913\pi$
$751$	20.8291	0.760064	0.380032	+	0.924973i	$0.375913\pi$
$752$	−17.0898	−0.623202
$753$	0	0
$754$	−1.87325	−0.0682198
$755$	−16.9111	−0.615459
$756$	0	0
$757$	−1.91852	−0.0697298	−0.0348649	−	0.999392i	$-0.511100\pi$
$757$	−1.91852	−0.0697298	−0.0348649	+	0.999392i	$0.511100\pi$
$758$	1.19292	0.0433286
$759$	0	0
$760$	1.73588	0.0629671
$761$	−22.0805	−0.800416	−0.400208	−	0.916424i	$-0.631062\pi$
$761$	−22.0805	−0.800416	−0.400208	+	0.916424i	$0.631062\pi$
$762$	0	0
$763$	35.5517	1.28706
$764$	10.4669	0.378680
$765$	0	0
$766$	−3.48481	−0.125911
$767$	−24.5539	−0.886589
$768$	0	0
$769$	50.9279	1.83651	0.918254	−	0.395991i	$-0.129599\pi$
$769$	50.9279	1.83651	0.918254	+	0.395991i	$0.129599\pi$
$770$	−2.71859	−0.0979710
$771$	0	0
$772$	−23.1528	−0.833288
$773$	39.8382	1.43288	0.716441	−	0.697648i	$-0.245770\pi$
$773$	39.8382	1.43288	0.716441	+	0.697648i	$0.245770\pi$
$774$	0	0
$775$	−0.532712	−0.0191356
$776$	6.54480	0.234944
$777$	0	0
$778$	−1.52447	−0.0546549
$779$	40.9231	1.46622
$780$	0	0
$781$	−3.00132	−0.107396
$782$	−1.18150	−0.0422504
$783$	0	0
$784$	63.1191	2.25425
$785$	−12.1586	−0.433960
$786$	0	0
$787$	−26.2156	−0.934485	−0.467242	−	0.884129i	$-0.654752\pi$
$787$	−26.2156	−0.934485	−0.467242	+	0.884129i	$0.654752\pi$
$788$	−51.2278	−1.82491
$789$	0	0
$790$	−0.129058	−0.00459168
$791$	10.2416	0.364150
$792$	0	0
$793$	23.6641	0.840335
$794$	−0.188459	−0.00668817
$795$	0	0
$796$	22.3472	0.792076
$797$	25.3270	0.897129	0.448564	−	0.893750i	$-0.351935\pi$
$797$	25.3270	0.897129	0.448564	+	0.893750i	$0.351935\pi$
$798$	0	0
$799$	10.3473	0.366061
$800$	1.09716	0.0387905
$801$	0	0
$802$	−2.35317	−0.0830932
$803$	37.0577	1.30774
$804$	0	0
$805$	−25.7616	−0.907979
$806$	0.103824	0.00365704
$807$	0	0
$808$	−1.99256	−0.0700981
$809$	−41.2080	−1.44880	−0.724398	−	0.689382i	$-0.757882\pi$
$809$	−41.2080	−1.44880	−0.724398	+	0.689382i	$0.757882\pi$
$810$	0	0
$811$	−8.54640	−0.300105	−0.150052	−	0.988678i	$-0.547944\pi$
$811$	−8.54640	−0.300105	−0.150052	+	0.988678i	$0.547944\pi$
$812$	91.7648	3.22031
$813$	0	0
$814$	0.624643	0.0218937
$815$	−19.7034	−0.690179
$816$	0	0
$817$	−18.4185	−0.644381
$818$	−1.95577	−0.0683818
$819$	0	0
$820$	17.2314	0.601746
$821$	29.3268	1.02351	0.511756	−	0.859131i	$-0.328995\pi$
$821$	29.3268	1.02351	0.511756	+	0.859131i	$0.328995\pi$
$822$	0	0
$823$	−23.8233	−0.830426	−0.415213	−	0.909724i	$-0.636293\pi$
$823$	−23.8233	−0.830426	−0.415213	+	0.909724i	$0.636293\pi$
$824$	−3.58777	−0.124986
$825$	0	0
$826$	−5.10615	−0.177666
$827$	6.44764	0.224206	0.112103	−	0.993697i	$-0.464241\pi$
$827$	6.44764	0.224206	0.112103	+	0.993697i	$0.464241\pi$
$828$	0	0
$829$	−4.07114	−0.141397	−0.0706983	−	0.997498i	$-0.522523\pi$
$829$	−4.07114	−0.141397	−0.0706983	+	0.997498i	$0.522523\pi$
$830$	−0.187290	−0.00650092
$831$	0	0
$832$	16.5286	0.573027
$833$	−38.2165	−1.32412
$834$	0	0
$835$	−6.73896	−0.233211
$836$	58.0944	2.00924
$837$	0	0
$838$	−1.33428	−0.0460920
$839$	−32.3664	−1.11741	−0.558706	−	0.829366i	$-0.688702\pi$
$839$	−32.3664	−1.11741	−0.558706	+	0.829366i	$0.688702\pi$
$840$	0	0
$841$	63.3806	2.18554
$842$	−0.630191	−0.0217178
$843$	0	0
$844$	47.4937	1.63480
$845$	−8.50709	−0.292653
$846$	0	0
$847$	−129.617	−4.45369
$848$	−31.6292	−1.08615
$849$	0	0
$850$	−0.219865	−0.00754131
$851$	5.91919	0.202907
$852$	0	0
$853$	13.8516	0.474269	0.237135	−	0.971477i	$-0.423792\pi$
$853$	13.8516	0.474269	0.237135	+	0.971477i	$0.423792\pi$
$854$	4.92110	0.168397
$855$	0	0
$856$	3.49012	0.119290
$857$	−10.1588	−0.347017	−0.173509	−	0.984832i	$-0.555510\pi$
$857$	−10.1588	−0.347017	−0.173509	+	0.984832i	$0.555510\pi$
$858$	0	0
$859$	38.8090	1.32415	0.662074	−	0.749439i	$-0.269677\pi$
$859$	38.8090	1.32415	0.662074	+	0.749439i	$0.269677\pi$
$860$	−7.75544	−0.264458
$861$	0	0
$862$	2.02242	0.0688840
$863$	49.1925	1.67453	0.837266	−	0.546795i	$-0.184152\pi$
$863$	49.1925	1.67453	0.837266	+	0.546795i	$0.184152\pi$
$864$	0	0
$865$	−19.3801	−0.658944
$866$	1.64901	0.0560358
$867$	0	0
$868$	−5.08602	−0.172631
$869$	−8.65665	−0.293657
$870$	0	0
$871$	17.9797	0.609220
$872$	2.72175	0.0921700
$873$	0	0
$874$	−2.33699	−0.0790500
$875$	−4.79397	−0.162066
$876$	0	0
$877$	16.9837	0.573498	0.286749	−	0.958006i	$-0.407425\pi$
$877$	16.9837	0.573498	0.286749	+	0.958006i	$0.407425\pi$
$878$	−2.76618	−0.0933539
$879$	0	0
$880$	24.3574	0.821088
$881$	23.5754	0.794274	0.397137	−	0.917759i	$-0.370004\pi$
$881$	23.5754	0.794274	0.397137	+	0.917759i	$0.370004\pi$
$882$	0	0
$883$	−16.9375	−0.569993	−0.284997	−	0.958529i	$-0.591993\pi$
$883$	−16.9375	−0.569993	−0.284997	+	0.958529i	$0.591993\pi$
$884$	−10.0941	−0.339502
$885$	0	0
$886$	−1.71017	−0.0574544
$887$	−1.28945	−0.0432955	−0.0216477	−	0.999766i	$-0.506891\pi$
$887$	−1.28945	−0.0432955	−0.0216477	+	0.999766i	$0.506891\pi$
$888$	0	0
$889$	39.9058	1.33840
$890$	−0.0919478	−0.00308210
$891$	0	0
$892$	18.9985	0.636116
$893$	20.4668	0.684896
$894$	0	0
$895$	1.95592	0.0653793
$896$	13.9568	0.466263
$897$	0	0
$898$	2.58156	0.0861479
$899$	−5.12015	−0.170767
$900$	0	0
$901$	19.1504	0.637993
$902$	−4.90656	−0.163371
$903$	0	0
$904$	0.784072	0.0260778
$905$	19.7042	0.654990
$906$	0	0
$907$	−3.15327	−0.104703	−0.0523513	−	0.998629i	$-0.516672\pi$
$907$	−3.15327	−0.104703	−0.0523513	+	0.998629i	$0.516672\pi$
$908$	39.4306	1.30855
$909$	0	0
$910$	0.934332	0.0309728
$911$	−21.0849	−0.698574	−0.349287	−	0.937016i	$-0.613576\pi$
$911$	−21.0849	−0.698574	−0.349287	+	0.937016i	$0.613576\pi$
$912$	0	0
$913$	−12.5626	−0.415760
$914$	3.26437	0.107976
$915$	0	0
$916$	34.8650	1.15197
$917$	62.2152	2.05453
$918$	0	0
$919$	−11.1278	−0.367074	−0.183537	−	0.983013i	$-0.558755\pi$
$919$	−11.1278	−0.367074	−0.183537	+	0.983013i	$0.558755\pi$
$920$	−1.97224	−0.0650229
$921$	0	0
$922$	0.504580	0.0166175
$923$	1.03150	0.0339523
$924$	0	0
$925$	1.10150	0.0362171
$926$	−2.58663	−0.0850021
$927$	0	0
$928$	10.5454	0.346168
$929$	−0.447381	−0.0146781	−0.00733904	−	0.999973i	$-0.502336\pi$
$929$	−0.447381	−0.0146781	−0.00733904	+	0.999973i	$0.502336\pi$
$930$	0	0
$931$	−75.5917	−2.47742
$932$	30.3160	0.993032
$933$	0	0
$934$	−0.874352	−0.0286097
$935$	−14.7476	−0.482297
$936$	0	0
$937$	−42.8454	−1.39970	−0.699849	−	0.714290i	$-0.746750\pi$
$937$	−42.8454	−1.39970	−0.699849	+	0.714290i	$0.746750\pi$
$938$	3.73901	0.122083
$939$	0	0
$940$	8.61793	0.281086
$941$	18.9070	0.616349	0.308175	−	0.951330i	$-0.400282\pi$
$941$	18.9070	0.616349	0.308175	+	0.951330i	$0.400282\pi$
$942$	0	0
$943$	−46.4952	−1.51409
$944$	45.7490	1.48900
$945$	0	0
$946$	2.20833	0.0717989
$947$	41.2020	1.33889	0.669443	−	0.742864i	$-0.266533\pi$
$947$	41.2020	1.33889	0.669443	+	0.742864i	$0.266533\pi$
$948$	0	0
$949$	−12.7361	−0.413432
$950$	−0.434890	−0.0141097
$951$	0	0
$952$	−4.20720	−0.136356
$953$	−53.6539	−1.73802	−0.869011	−	0.494793i	$-0.835244\pi$
$953$	−53.6539	−1.73802	−0.869011	+	0.494793i	$0.835244\pi$
$954$	0	0
$955$	−5.25568	−0.170070
$956$	18.6292	0.602510
$957$	0	0
$958$	−2.80858	−0.0907413
$959$	18.7935	0.606873
$960$	0	0
$961$	−30.7162	−0.990846
$962$	−0.214679	−0.00692153
$963$	0	0
$964$	32.7230	1.05394
$965$	11.6256	0.374240
$966$	0	0
$967$	41.5614	1.33652	0.668262	−	0.743926i	$-0.267039\pi$
$967$	41.5614	1.33652	0.668262	+	0.743926i	$0.267039\pi$
$968$	−9.92314	−0.318942
$969$	0	0
$970$	−1.63966	−0.0526465
$971$	10.0203	0.321567	0.160783	−	0.986990i	$-0.448598\pi$
$971$	10.0203	0.321567	0.160783	+	0.986990i	$0.448598\pi$
$972$	0	0
$973$	5.88630	0.188706
$974$	3.27810	0.105037
$975$	0	0
$976$	−44.0910	−1.41132
$977$	−13.5361	−0.433057	−0.216528	−	0.976276i	$-0.569473\pi$
$977$	−13.5361	−0.433057	−0.216528	+	0.976276i	$0.569473\pi$
$978$	0	0
$979$	−6.16745	−0.197113
$980$	−31.8292	−1.01675
$981$	0	0
$982$	−0.464749	−0.0148307
$983$	13.2125	0.421413	0.210707	−	0.977549i	$-0.432424\pi$
$983$	13.2125	0.421413	0.210707	+	0.977549i	$0.432424\pi$
$984$	0	0
$985$	25.7226	0.819591
$986$	−2.11323	−0.0672989
$987$	0	0
$988$	−19.9661	−0.635205
$989$	20.9264	0.665420
$990$	0	0
$991$	−16.8148	−0.534139	−0.267070	−	0.963677i	$-0.586055\pi$
$991$	−16.8148	−0.534139	−0.267070	+	0.963677i	$0.586055\pi$
$992$	−0.584471	−0.0185570
$993$	0	0
$994$	0.214508	0.00680378
$995$	−11.2210	−0.355731
$996$	0	0
$997$	−2.29897	−0.0728091	−0.0364045	−	0.999337i	$-0.511590\pi$
$997$	−2.29897	−0.0728091	−0.0364045	+	0.999337i	$0.511590\pi$
$998$	−1.22376	−0.0387374
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.q.1.6		9
3.2	odd	2		1335.2.a.h.1.4	✓	9
15.14	odd	2		6675.2.a.x.1.6		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.h.1.4	✓	9	3.2	odd	2
4005.2.a.q.1.6		9	1.1	even	1	trivial
6675.2.a.x.1.6		9	15.14	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0919478 q^{2} -1.99155 q^{4} +1.00000 q^{5} -4.79397 q^{7} -0.367014 q^{8} +O(q^{10})\)	\(q+0.0919478 q^{2} -1.99155 q^{4} +1.00000 q^{5} -4.79397 q^{7} -0.367014 q^{8} +0.0919478 q^{10} +6.16745 q^{11} -2.11965 q^{13} -0.440795 q^{14} +3.94935 q^{16} -2.39119 q^{17} -4.72975 q^{19} -1.99155 q^{20} +0.567084 q^{22} +5.37376 q^{23} +1.00000 q^{25} -0.194897 q^{26} +9.54742 q^{28} +9.61148 q^{29} -0.532712 q^{31} +1.09716 q^{32} -0.219865 q^{34} -4.79397 q^{35} +1.10150 q^{37} -0.434890 q^{38} -0.367014 q^{40} -8.65227 q^{41} +3.89418 q^{43} -12.2828 q^{44} +0.494105 q^{46} -4.32725 q^{47} +15.9822 q^{49} +0.0919478 q^{50} +4.22138 q^{52} -8.00873 q^{53} +6.16745 q^{55} +1.75945 q^{56} +0.883755 q^{58} +11.5839 q^{59} -11.1641 q^{61} -0.0489817 q^{62} -7.79781 q^{64} -2.11965 q^{65} -8.48241 q^{67} +4.76217 q^{68} -0.440795 q^{70} -0.486638 q^{71} +6.00859 q^{73} +0.101280 q^{74} +9.41951 q^{76} -29.5666 q^{77} -1.40360 q^{79} +3.94935 q^{80} -0.795558 q^{82} -2.03691 q^{83} -2.39119 q^{85} +0.358061 q^{86} -2.26354 q^{88} -1.00000 q^{89} +10.1615 q^{91} -10.7021 q^{92} -0.397882 q^{94} -4.72975 q^{95} -17.8326 q^{97} +1.46953 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8}+O(q^{10}) \) 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8	\( 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8} - 5 q^{10} - 4 q^{11} + 5 q^{13} + q^{14} + 15 q^{16} - 21 q^{17} - 18 q^{19} + 11 q^{20} + 6 q^{22} - 16 q^{23} + 9 q^{25} - 8 q^{26} + 6 q^{28} - 3 q^{29} - 6 q^{31} - 46 q^{32} + 12 q^{34} - 3 q^{35} + 11 q^{37} - 20 q^{38} - 18 q^{40} + q^{41} - 3 q^{43} - 38 q^{44} + 16 q^{46} - 27 q^{47} + 24 q^{49} - 5 q^{50} + 17 q^{52} - 43 q^{53} - 4 q^{55} - 5 q^{56} + 34 q^{58} - 3 q^{59} - 30 q^{61} - 36 q^{62} + 50 q^{64} + 5 q^{65} - 12 q^{67} - 64 q^{68} + q^{70} + 4 q^{71} + 26 q^{73} - 2 q^{74} - 12 q^{76} - 34 q^{77} + q^{79} + 15 q^{80} - 51 q^{82} - 24 q^{83} - 21 q^{85} - 18 q^{86} + 64 q^{88} - 9 q^{89} - 50 q^{91} - 10 q^{92} - 11 q^{94} - 18 q^{95} - 4 q^{97} - 75 q^{98}+O(q^{100}) \) 9 * q - 5 * q^2 + 11 * q^4 + 9 * q^5 - 3 * q^7 - 18 * q^8 - 5 * q^10 - 4 * q^11 + 5 * q^13 + q^14 + 15 * q^16 - 21 * q^17 - 18 * q^19 + 11 * q^20 + 6 * q^22 - 16 * q^23 + 9 * q^25 - 8 * q^26 + 6 * q^28 - 3 * q^29 - 6 * q^31 - 46 * q^32 + 12 * q^34 - 3 * q^35 + 11 * q^37 - 20 * q^38 - 18 * q^40 + q^41 - 3 * q^43 - 38 * q^44 + 16 * q^46 - 27 * q^47 + 24 * q^49 - 5 * q^50 + 17 * q^52 - 43 * q^53 - 4 * q^55 - 5 * q^56 + 34 * q^58 - 3 * q^59 - 30 * q^61 - 36 * q^62 + 50 * q^64 + 5 * q^65 - 12 * q^67 - 64 * q^68 + q^70 + 4 * q^71 + 26 * q^73 - 2 * q^74 - 12 * q^76 - 34 * q^77 + q^79 + 15 * q^80 - 51 * q^82 - 24 * q^83 - 21 * q^85 - 18 * q^86 + 64 * q^88 - 9 * q^89 - 50 * q^91 - 10 * q^92 - 11 * q^94 - 18 * q^95 - 4 * q^97 - 75 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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