LMFDB - Embedded newform 4005.2.a.x.1.15

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:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} + \cdots + 24 $ x^17 - 5x^16 - 15x^15 + 105x^14 + 45x^13 - 849x^12 + 320x^11 + 3371x^10 - 2456x^9 - 7002x^8 + 6279x^7 + 7299x^6 - 7119x^5 - 3066x^4 + 3184x^3 + 99x^2 - 231x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$2.50383$ 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)

$f(q)$	$=$	$q+2.50383 q^{2} +4.26915 q^{4} +1.00000 q^{5} -2.88177 q^{7} +5.68155 q^{8} +O(q^{10})$	\(q+2.50383 q^{2} +4.26915 q^{4} +1.00000 q^{5} -2.88177 q^{7} +5.68155 q^{8} +2.50383 q^{10} -3.10609 q^{11} -0.735843 q^{13} -7.21545 q^{14} +5.68733 q^{16} +7.46499 q^{17} +6.34476 q^{19} +4.26915 q^{20} -7.77711 q^{22} +7.29679 q^{23} +1.00000 q^{25} -1.84242 q^{26} -12.3027 q^{28} +4.07403 q^{29} -0.449413 q^{31} +2.87698 q^{32} +18.6910 q^{34} -2.88177 q^{35} +7.81613 q^{37} +15.8862 q^{38} +5.68155 q^{40} -4.57197 q^{41} -2.01288 q^{43} -13.2604 q^{44} +18.2699 q^{46} +0.356029 q^{47} +1.30459 q^{49} +2.50383 q^{50} -3.14142 q^{52} -0.999991 q^{53} -3.10609 q^{55} -16.3729 q^{56} +10.2007 q^{58} +14.7710 q^{59} -4.31056 q^{61} -1.12525 q^{62} -4.17119 q^{64} -0.735843 q^{65} +5.07926 q^{67} +31.8691 q^{68} -7.21545 q^{70} +10.7219 q^{71} -11.1679 q^{73} +19.5702 q^{74} +27.0867 q^{76} +8.95103 q^{77} +5.39673 q^{79} +5.68733 q^{80} -11.4474 q^{82} -13.4991 q^{83} +7.46499 q^{85} -5.03990 q^{86} -17.6474 q^{88} -1.00000 q^{89} +2.12053 q^{91} +31.1511 q^{92} +0.891436 q^{94} +6.34476 q^{95} +10.5296 q^{97} +3.26648 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) $ 17 * q + 5 * q^2 + 21 * q^4 + 17 * q^5 + 12 * q^7 + 15 * q^8	$ 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8} + 5 q^{10} + 2 q^{11} + 8 q^{13} + 4 q^{14} + 33 q^{16} + 10 q^{17} + 32 q^{19} + 21 q^{20} + 8 q^{22} + 15 q^{23} + 17 q^{25} - 15 q^{26} + 24 q^{28} + q^{29} + 18 q^{31} + 25 q^{32} + 14 q^{34} + 12 q^{35} + 12 q^{37} + 22 q^{38} + 15 q^{40} - 7 q^{41} + 28 q^{43} - 14 q^{44} + 4 q^{46} + 26 q^{47} + 41 q^{49} + 5 q^{50} + 10 q^{52} + 12 q^{53} + 2 q^{55} + 13 q^{56} + 16 q^{58} - 23 q^{59} + 26 q^{61} + 10 q^{62} + 59 q^{64} + 8 q^{65} + 31 q^{67} - q^{68} + 4 q^{70} - 2 q^{71} + 33 q^{73} - 10 q^{74} + 66 q^{76} + 12 q^{77} + 33 q^{79} + 33 q^{80} + 30 q^{82} + 13 q^{83} + 10 q^{85} - 20 q^{86} + 12 q^{88} - 17 q^{89} + 40 q^{91} + 16 q^{92} + 38 q^{94} + 32 q^{95} + 45 q^{97} + 2 q^{98}+O(q^{100}) $ 17 * q + 5 * q^2 + 21 * q^4 + 17 * q^5 + 12 * q^7 + 15 * q^8 + 5 * q^10 + 2 * q^11 + 8 * q^13 + 4 * q^14 + 33 * q^16 + 10 * q^17 + 32 * q^19 + 21 * q^20 + 8 * q^22 + 15 * q^23 + 17 * q^25 - 15 * q^26 + 24 * q^28 + q^29 + 18 * q^31 + 25 * q^32 + 14 * q^34 + 12 * q^35 + 12 * q^37 + 22 * q^38 + 15 * q^40 - 7 * q^41 + 28 * q^43 - 14 * q^44 + 4 * q^46 + 26 * q^47 + 41 * q^49 + 5 * q^50 + 10 * q^52 + 12 * q^53 + 2 * q^55 + 13 * q^56 + 16 * q^58 - 23 * q^59 + 26 * q^61 + 10 * q^62 + 59 * q^64 + 8 * q^65 + 31 * q^67 - q^68 + 4 * q^70 - 2 * q^71 + 33 * q^73 - 10 * q^74 + 66 * q^76 + 12 * q^77 + 33 * q^79 + 33 * q^80 + 30 * q^82 + 13 * q^83 + 10 * q^85 - 20 * q^86 + 12 * q^88 - 17 * q^89 + 40 * q^91 + 16 * q^92 + 38 * q^94 + 32 * q^95 + 45 * q^97 + 2 * 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 $

$2$	2.50383	1.77047	0.885236	−	0.465141i	$-0.153996\pi$
$2$	2.50383	1.77047	0.885236	+	0.465141i	$0.153996\pi$
$3$	0	0
$3$	0	0
$4$	4.26915	2.13457
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.88177	−1.08921	−0.544603	−	0.838694i	$-0.683320\pi$
$7$	−2.88177	−1.08921	−0.544603	+	0.838694i	$0.683320\pi$
$8$	5.68155	2.00873
$9$	0	0
$10$	2.50383	0.791780
$11$	−3.10609	−0.936521	−0.468261	−	0.883590i	$-0.655119\pi$
$11$	−3.10609	−0.936521	−0.468261	+	0.883590i	$0.655119\pi$
$12$	0	0
$13$	−0.735843	−0.204086	−0.102043	−	0.994780i	$-0.532538\pi$
$13$	−0.735843	−0.204086	−0.102043	+	0.994780i	$0.532538\pi$
$14$	−7.21545	−1.92841
$15$	0	0
$16$	5.68733	1.42183
$17$	7.46499	1.81052	0.905262	−	0.424853i	$-0.139674\pi$
$17$	7.46499	1.81052	0.905262	+	0.424853i	$0.139674\pi$
$18$	0	0
$19$	6.34476	1.45559	0.727794	−	0.685796i	$-0.240546\pi$
$19$	6.34476	1.45559	0.727794	+	0.685796i	$0.240546\pi$
$20$	4.26915	0.954611
$21$	0	0
$22$	−7.77711	−1.65809
$23$	7.29679	1.52149	0.760743	−	0.649053i	$-0.224835\pi$
$23$	7.29679	1.52149	0.760743	+	0.649053i	$0.224835\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.84242	−0.361329
$27$	0	0
$28$	−12.3027	−2.32499
$29$	4.07403	0.756528	0.378264	−	0.925698i	$-0.376521\pi$
$29$	4.07403	0.756528	0.378264	+	0.925698i	$0.376521\pi$
$30$	0	0
$31$	−0.449413	−0.0807169	−0.0403584	−	0.999185i	$-0.512850\pi$
$31$	−0.449413	−0.0807169	−0.0403584	+	0.999185i	$0.512850\pi$
$32$	2.87698	0.508584
$33$	0	0
$34$	18.6910	3.20549
$35$	−2.88177	−0.487108
$36$	0	0
$37$	7.81613	1.28496	0.642482	−	0.766301i	$-0.277905\pi$
$37$	7.81613	1.28496	0.642482	+	0.766301i	$0.277905\pi$
$38$	15.8862	2.57708
$39$	0	0
$40$	5.68155	0.898333
$41$	−4.57197	−0.714021	−0.357011	−	0.934100i	$-0.616204\pi$
$41$	−4.57197	−0.714021	−0.357011	+	0.934100i	$0.616204\pi$
$42$	0	0
$43$	−2.01288	−0.306961	−0.153481	−	0.988152i	$-0.549048\pi$
$43$	−2.01288	−0.306961	−0.153481	+	0.988152i	$0.549048\pi$
$44$	−13.2604	−1.99907
$45$	0	0
$46$	18.2699	2.69375
$47$	0.356029	0.0519322	0.0259661	−	0.999663i	$-0.491734\pi$
$47$	0.356029	0.0519322	0.0259661	+	0.999663i	$0.491734\pi$
$48$	0	0
$49$	1.30459	0.186371
$50$	2.50383	0.354095
$51$	0	0
$52$	−3.14142	−0.435637
$53$	−0.999991	−0.137359	−0.0686796	−	0.997639i	$-0.521879\pi$
$53$	−0.999991	−0.137359	−0.0686796	+	0.997639i	$0.521879\pi$
$54$	0	0
$55$	−3.10609	−0.418825
$56$	−16.3729	−2.18792
$57$	0	0
$58$	10.2007	1.33941
$59$	14.7710	1.92303	0.961513	−	0.274760i	$-0.0885985\pi$
$59$	14.7710	1.92303	0.961513	+	0.274760i	$0.0885985\pi$
$60$	0	0
$61$	−4.31056	−0.551910	−0.275955	−	0.961171i	$-0.588994\pi$
$61$	−4.31056	−0.551910	−0.275955	+	0.961171i	$0.588994\pi$
$62$	−1.12525	−0.142907
$63$	0	0
$64$	−4.17119	−0.521399
$65$	−0.735843	−0.0912701
$66$	0	0
$67$	5.07926	0.620530	0.310265	−	0.950650i	$-0.399582\pi$
$67$	5.07926	0.620530	0.310265	+	0.950650i	$0.399582\pi$
$68$	31.8691	3.86470
$69$	0	0
$70$	−7.21545	−0.862411
$71$	10.7219	1.27246	0.636230	−	0.771499i	$-0.280493\pi$
$71$	10.7219	1.27246	0.636230	+	0.771499i	$0.280493\pi$
$72$	0	0
$73$	−11.1679	−1.30711	−0.653553	−	0.756881i	$-0.726722\pi$
$73$	−11.1679	−1.30711	−0.653553	+	0.756881i	$0.726722\pi$
$74$	19.5702	2.27499
$75$	0	0
$76$	27.0867	3.10706
$77$	8.95103	1.02006
$78$	0	0
$79$	5.39673	0.607180	0.303590	−	0.952803i	$-0.401815\pi$
$79$	5.39673	0.607180	0.303590	+	0.952803i	$0.401815\pi$
$80$	5.68733	0.635863
$81$	0	0
$82$	−11.4474	−1.26416
$83$	−13.4991	−1.48171	−0.740857	−	0.671662i	$-0.765581\pi$
$83$	−13.4991	−1.48171	−0.740857	+	0.671662i	$0.765581\pi$
$84$	0	0
$85$	7.46499	0.809691
$86$	−5.03990	−0.543467
$87$	0	0
$88$	−17.6474	−1.88122
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	2.12053	0.222292
$92$	31.1511	3.24773
$93$	0	0
$94$	0.891436	0.0919446
$95$	6.34476	0.650959
$96$	0	0
$97$	10.5296	1.06912	0.534562	−	0.845129i	$-0.320477\pi$
$97$	10.5296	1.06912	0.534562	+	0.845129i	$0.320477\pi$
$98$	3.26648	0.329964
$99$	0	0
$100$	4.26915	0.426915
$101$	−18.6744	−1.85817	−0.929084	−	0.369869i	$-0.879403\pi$
$101$	−18.6744	−1.85817	−0.929084	+	0.369869i	$0.879403\pi$
$102$	0	0
$103$	−1.64542	−0.162128	−0.0810640	−	0.996709i	$-0.525832\pi$
$103$	−1.64542	−0.162128	−0.0810640	+	0.996709i	$0.525832\pi$
$104$	−4.18073	−0.409955
$105$	0	0
$106$	−2.50380	−0.243191
$107$	−4.40479	−0.425827	−0.212913	−	0.977071i	$-0.568295\pi$
$107$	−4.40479	−0.425827	−0.212913	+	0.977071i	$0.568295\pi$
$108$	0	0
$109$	−9.90460	−0.948688	−0.474344	−	0.880340i	$-0.657315\pi$
$109$	−9.90460	−0.948688	−0.474344	+	0.880340i	$0.657315\pi$
$110$	−7.77711	−0.741518
$111$	0	0
$112$	−16.3896	−1.54867
$113$	−4.66049	−0.438422	−0.219211	−	0.975677i	$-0.570348\pi$
$113$	−4.66049	−0.438422	−0.219211	+	0.975677i	$0.570348\pi$
$114$	0	0
$115$	7.29679	0.680429
$116$	17.3926	1.61487
$117$	0	0
$118$	36.9841	3.40466
$119$	−21.5124	−1.97204
$120$	0	0
$121$	−1.35221	−0.122928
$122$	−10.7929	−0.977142
$123$	0	0
$124$	−1.91861	−0.172296
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.78477	−0.158373	−0.0791864	−	0.996860i	$-0.525232\pi$
$127$	−1.78477	−0.158373	−0.0791864	+	0.996860i	$0.525232\pi$
$128$	−16.1979	−1.43171
$129$	0	0
$130$	−1.84242	−0.161591
$131$	16.6441	1.45420	0.727099	−	0.686532i	$-0.240868\pi$
$131$	16.6441	1.45420	0.727099	+	0.686532i	$0.240868\pi$
$132$	0	0
$133$	−18.2841	−1.58544
$134$	12.7176	1.09863
$135$	0	0
$136$	42.4127	3.63686
$137$	−23.2407	−1.98558	−0.992792	−	0.119850i	$-0.961758\pi$
$137$	−23.2407	−1.98558	−0.992792	+	0.119850i	$0.961758\pi$
$138$	0	0
$139$	6.69373	0.567755	0.283878	−	0.958861i	$-0.408379\pi$
$139$	6.69373	0.567755	0.283878	+	0.958861i	$0.408379\pi$
$140$	−12.3027	−1.03977
$141$	0	0
$142$	26.8459	2.25286
$143$	2.28559	0.191131
$144$	0	0
$145$	4.07403	0.338330
$146$	−27.9625	−2.31419
$147$	0	0
$148$	33.3682	2.74285
$149$	10.3677	0.849352	0.424676	−	0.905345i	$-0.360388\pi$
$149$	10.3677	0.849352	0.424676	+	0.905345i	$0.360388\pi$
$150$	0	0
$151$	8.47128	0.689383	0.344692	−	0.938716i	$-0.387984\pi$
$151$	8.47128	0.689383	0.344692	+	0.938716i	$0.387984\pi$
$152$	36.0481	2.92389
$153$	0	0
$154$	22.4118	1.80600
$155$	−0.449413	−0.0360977
$156$	0	0
$157$	−4.65923	−0.371847	−0.185924	−	0.982564i	$-0.559528\pi$
$157$	−4.65923	−0.371847	−0.185924	+	0.982564i	$0.559528\pi$
$158$	13.5125	1.07500
$159$	0	0
$160$	2.87698	0.227446
$161$	−21.0277	−1.65721
$162$	0	0
$163$	−15.9763	−1.25136	−0.625679	−	0.780080i	$-0.715178\pi$
$163$	−15.9763	−1.25136	−0.625679	+	0.780080i	$0.715178\pi$
$164$	−19.5184	−1.52413
$165$	0	0
$166$	−33.7993	−2.62334
$167$	0.585677	0.0453211	0.0226605	−	0.999743i	$-0.492786\pi$
$167$	0.585677	0.0453211	0.0226605	+	0.999743i	$0.492786\pi$
$168$	0	0
$169$	−12.4585	−0.958349
$170$	18.6910	1.43354
$171$	0	0
$172$	−8.59329	−0.655232
$173$	2.16214	0.164385	0.0821924	−	0.996616i	$-0.473808\pi$
$173$	2.16214	0.164385	0.0821924	+	0.996616i	$0.473808\pi$
$174$	0	0
$175$	−2.88177	−0.217841
$176$	−17.6654	−1.33158
$177$	0	0
$178$	−2.50383	−0.187670
$179$	3.81948	0.285481	0.142741	−	0.989760i	$-0.454409\pi$
$179$	3.81948	0.285481	0.142741	+	0.989760i	$0.454409\pi$
$180$	0	0
$181$	−12.5751	−0.934703	−0.467352	−	0.884072i	$-0.654792\pi$
$181$	−12.5751	−0.934703	−0.467352	+	0.884072i	$0.654792\pi$
$182$	5.30944	0.393562
$183$	0	0
$184$	41.4571	3.05626
$185$	7.81613	0.574653
$186$	0	0
$187$	−23.1869	−1.69559
$188$	1.51994	0.110853
$189$	0	0
$190$	15.8862	1.15250
$191$	−1.16421	−0.0842396	−0.0421198	−	0.999113i	$-0.513411\pi$
$191$	−1.16421	−0.0842396	−0.0421198	+	0.999113i	$0.513411\pi$
$192$	0	0
$193$	4.65308	0.334936	0.167468	−	0.985878i	$-0.446441\pi$
$193$	4.65308	0.334936	0.167468	+	0.985878i	$0.446441\pi$
$194$	26.3644	1.89285
$195$	0	0
$196$	5.56951	0.397822
$197$	−22.1881	−1.58084	−0.790419	−	0.612567i	$-0.790137\pi$
$197$	−22.1881	−1.58084	−0.790419	+	0.612567i	$0.790137\pi$
$198$	0	0
$199$	−4.42546	−0.313712	−0.156856	−	0.987621i	$-0.550136\pi$
$199$	−4.42546	−0.313712	−0.156856	+	0.987621i	$0.550136\pi$
$200$	5.68155	0.401747
$201$	0	0
$202$	−46.7573	−3.28984
$203$	−11.7404	−0.824016
$204$	0	0
$205$	−4.57197	−0.319320
$206$	−4.11985	−0.287043
$207$	0	0
$208$	−4.18498	−0.290176
$209$	−19.7074	−1.36319
$210$	0	0
$211$	−2.29884	−0.158258	−0.0791292	−	0.996864i	$-0.525214\pi$
$211$	−2.29884	−0.158258	−0.0791292	+	0.996864i	$0.525214\pi$
$212$	−4.26911	−0.293204
$213$	0	0
$214$	−11.0288	−0.753915
$215$	−2.01288	−0.137277
$216$	0	0
$217$	1.29510	0.0879174
$218$	−24.7994	−1.67963
$219$	0	0
$220$	−13.2604	−0.894013
$221$	−5.49306	−0.369503
$222$	0	0
$223$	3.56018	0.238407	0.119204	−	0.992870i	$-0.461966\pi$
$223$	3.56018	0.238407	0.119204	+	0.992870i	$0.461966\pi$
$224$	−8.29081	−0.553953
$225$	0	0
$226$	−11.6691	−0.776215
$227$	−20.9967	−1.39360	−0.696801	−	0.717264i	$-0.745394\pi$
$227$	−20.9967	−1.39360	−0.696801	+	0.717264i	$0.745394\pi$
$228$	0	0
$229$	−16.6634	−1.10115	−0.550574	−	0.834786i	$-0.685591\pi$
$229$	−16.6634	−1.10115	−0.550574	+	0.834786i	$0.685591\pi$
$230$	18.2699	1.20468
$231$	0	0
$232$	23.1468	1.51966
$233$	2.49366	0.163365	0.0816825	−	0.996658i	$-0.473971\pi$
$233$	2.49366	0.163365	0.0816825	+	0.996658i	$0.473971\pi$
$234$	0	0
$235$	0.356029	0.0232248
$236$	63.0598	4.10484
$237$	0	0
$238$	−53.8632	−3.49144
$239$	2.28190	0.147604	0.0738018	−	0.997273i	$-0.476487\pi$
$239$	2.28190	0.147604	0.0738018	+	0.997273i	$0.476487\pi$
$240$	0	0
$241$	12.0069	0.773433	0.386717	−	0.922199i	$-0.373609\pi$
$241$	12.0069	0.773433	0.386717	+	0.922199i	$0.373609\pi$
$242$	−3.38570	−0.217641
$243$	0	0
$244$	−18.4024	−1.17809
$245$	1.30459	0.0833475
$246$	0	0
$247$	−4.66875	−0.297065
$248$	−2.55336	−0.162139
$249$	0	0
$250$	2.50383	0.158356
$251$	3.68964	0.232888	0.116444	−	0.993197i	$-0.462850\pi$
$251$	3.68964	0.232888	0.116444	+	0.993197i	$0.462850\pi$
$252$	0	0
$253$	−22.6645	−1.42490
$254$	−4.46876	−0.280395
$255$	0	0
$256$	−32.2144	−2.01340
$257$	−16.4334	−1.02509	−0.512544	−	0.858661i	$-0.671297\pi$
$257$	−16.4334	−1.02509	−0.512544	+	0.858661i	$0.671297\pi$
$258$	0	0
$259$	−22.5243	−1.39959
$260$	−3.14142	−0.194823
$261$	0	0
$262$	41.6739	2.57462
$263$	23.3494	1.43979	0.719893	−	0.694085i	$-0.244191\pi$
$263$	23.3494	1.43979	0.719893	+	0.694085i	$0.244191\pi$
$264$	0	0
$265$	−0.999991	−0.0614289
$266$	−45.7803	−2.80697
$267$	0	0
$268$	21.6841	1.32457
$269$	−20.7279	−1.26380	−0.631900	−	0.775050i	$-0.717725\pi$
$269$	−20.7279	−1.26380	−0.631900	+	0.775050i	$0.717725\pi$
$270$	0	0
$271$	23.5946	1.43327	0.716634	−	0.697449i	$-0.245682\pi$
$271$	23.5946	1.43327	0.716634	+	0.697449i	$0.245682\pi$
$272$	42.4559	2.57426
$273$	0	0
$274$	−58.1906	−3.51542
$275$	−3.10609	−0.187304
$276$	0	0
$277$	31.1275	1.87027	0.935136	−	0.354288i	$-0.115277\pi$
$277$	31.1275	1.87027	0.935136	+	0.354288i	$0.115277\pi$
$278$	16.7599	1.00519
$279$	0	0
$280$	−16.3729	−0.978470
$281$	−26.4045	−1.57516	−0.787580	−	0.616212i	$-0.788666\pi$
$281$	−26.4045	−1.57516	−0.787580	+	0.616212i	$0.788666\pi$
$282$	0	0
$283$	17.8153	1.05901	0.529506	−	0.848306i	$-0.322377\pi$
$283$	17.8153	1.05901	0.529506	+	0.848306i	$0.322377\pi$
$284$	45.7736	2.71616
$285$	0	0
$286$	5.72273	0.338392
$287$	13.1754	0.777717
$288$	0	0
$289$	38.7260	2.27800
$290$	10.2007	0.599004
$291$	0	0
$292$	−47.6775	−2.79011
$293$	32.5550	1.90189	0.950943	−	0.309367i	$-0.100117\pi$
$293$	32.5550	1.90189	0.950943	+	0.309367i	$0.100117\pi$
$294$	0	0
$295$	14.7710	0.860003
$296$	44.4078	2.58115
$297$	0	0
$298$	25.9588	1.50375
$299$	−5.36930	−0.310514
$300$	0	0
$301$	5.80066	0.334344
$302$	21.2106	1.22053
$303$	0	0
$304$	36.0848	2.06960
$305$	−4.31056	−0.246822
$306$	0	0
$307$	6.95500	0.396943	0.198472	−	0.980107i	$-0.436402\pi$
$307$	6.95500	0.396943	0.198472	+	0.980107i	$0.436402\pi$
$308$	38.2133	2.17740
$309$	0	0
$310$	−1.12525	−0.0639100
$311$	−23.3589	−1.32456	−0.662281	−	0.749255i	$-0.730412\pi$
$311$	−23.3589	−1.32456	−0.662281	+	0.749255i	$0.730412\pi$
$312$	0	0
$313$	4.36202	0.246556	0.123278	−	0.992372i	$-0.460659\pi$
$313$	4.36202	0.246556	0.123278	+	0.992372i	$0.460659\pi$
$314$	−11.6659	−0.658345
$315$	0	0
$316$	23.0394	1.29607
$317$	−1.60640	−0.0902246	−0.0451123	−	0.998982i	$-0.514365\pi$
$317$	−1.60640	−0.0902246	−0.0451123	+	0.998982i	$0.514365\pi$
$318$	0	0
$319$	−12.6543	−0.708505
$320$	−4.17119	−0.233177
$321$	0	0
$322$	−52.6497	−2.93405
$323$	47.3636	2.63538
$324$	0	0
$325$	−0.735843	−0.0408172
$326$	−40.0018	−2.21550
$327$	0	0
$328$	−25.9759	−1.43428
$329$	−1.02599	−0.0565649
$330$	0	0
$331$	21.4379	1.17833	0.589166	−	0.808012i	$-0.299457\pi$
$331$	21.4379	1.17833	0.589166	+	0.808012i	$0.299457\pi$
$332$	−57.6295	−3.16283
$333$	0	0
$334$	1.46643	0.0802397
$335$	5.07926	0.277509
$336$	0	0
$337$	−22.3230	−1.21601	−0.608005	−	0.793933i	$-0.708030\pi$
$337$	−22.3230	−1.21601	−0.608005	+	0.793933i	$0.708030\pi$
$338$	−31.1940	−1.69673
$339$	0	0
$340$	31.8691	1.72835
$341$	1.39592	0.0755931
$342$	0	0
$343$	16.4128	0.886210
$344$	−11.4363	−0.616603
$345$	0	0
$346$	5.41363	0.291039
$347$	14.7378	0.791166	0.395583	−	0.918430i	$-0.370543\pi$
$347$	14.7378	0.791166	0.395583	+	0.918430i	$0.370543\pi$
$348$	0	0
$349$	−12.6132	−0.675171	−0.337586	−	0.941295i	$-0.609610\pi$
$349$	−12.6132	−0.675171	−0.337586	+	0.941295i	$0.609610\pi$
$350$	−7.21545	−0.385682
$351$	0	0
$352$	−8.93617	−0.476300
$353$	29.1611	1.55209	0.776045	−	0.630678i	$-0.217223\pi$
$353$	29.1611	1.55209	0.776045	+	0.630678i	$0.217223\pi$
$354$	0	0
$355$	10.7219	0.569061
$356$	−4.26915	−0.226264
$357$	0	0
$358$	9.56331	0.505437
$359$	−11.8676	−0.626347	−0.313174	−	0.949696i	$-0.601392\pi$
$359$	−11.8676	−0.626347	−0.313174	+	0.949696i	$0.601392\pi$
$360$	0	0
$361$	21.2560	1.11874
$362$	−31.4860	−1.65487
$363$	0	0
$364$	9.05286	0.474499
$365$	−11.1679	−0.584555
$366$	0	0
$367$	−7.73082	−0.403546	−0.201773	−	0.979432i	$-0.564670\pi$
$367$	−7.73082	−0.403546	−0.201773	+	0.979432i	$0.564670\pi$
$368$	41.4993	2.16330
$369$	0	0
$370$	19.5702	1.01741
$371$	2.88174	0.149613
$372$	0	0
$373$	24.1716	1.25156	0.625779	−	0.780000i	$-0.284781\pi$
$373$	24.1716	1.25156	0.625779	+	0.780000i	$0.284781\pi$
$374$	−58.0560	−3.00200
$375$	0	0
$376$	2.02280	0.104318
$377$	−2.99785	−0.154397
$378$	0	0
$379$	−7.12030	−0.365745	−0.182873	−	0.983137i	$-0.558540\pi$
$379$	−7.12030	−0.365745	−0.182873	+	0.983137i	$0.558540\pi$
$380$	27.0867	1.38952
$381$	0	0
$382$	−2.91499	−0.149144
$383$	−22.8123	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$383$	−22.8123	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$384$	0	0
$385$	8.95103	0.456187
$386$	11.6505	0.592995
$387$	0	0
$388$	44.9526	2.28212
$389$	−2.22632	−0.112879	−0.0564395	−	0.998406i	$-0.517975\pi$
$389$	−2.22632	−0.112879	−0.0564395	+	0.998406i	$0.517975\pi$
$390$	0	0
$391$	54.4705	2.75469
$392$	7.41212	0.374369
$393$	0	0
$394$	−55.5552	−2.79883
$395$	5.39673	0.271539
$396$	0	0
$397$	18.7025	0.938652	0.469326	−	0.883025i	$-0.344497\pi$
$397$	18.7025	0.938652	0.469326	+	0.883025i	$0.344497\pi$
$398$	−11.0806	−0.555419
$399$	0	0
$400$	5.68733	0.284367
$401$	−9.15369	−0.457113	−0.228557	−	0.973531i	$-0.573401\pi$
$401$	−9.15369	−0.457113	−0.228557	+	0.973531i	$0.573401\pi$
$402$	0	0
$403$	0.330697	0.0164732
$404$	−79.7236	−3.96640
$405$	0	0
$406$	−29.3960	−1.45890
$407$	−24.2776	−1.20340
$408$	0	0
$409$	28.6053	1.41444	0.707221	−	0.706993i	$-0.249949\pi$
$409$	28.6053	1.41444	0.707221	+	0.706993i	$0.249949\pi$
$410$	−11.4474	−0.565348
$411$	0	0
$412$	−7.02454	−0.346074
$413$	−42.5667	−2.09457
$414$	0	0
$415$	−13.4991	−0.662643
$416$	−2.11701	−0.103795
$417$	0	0
$418$	−49.3439	−2.41349
$419$	−4.81521	−0.235239	−0.117619	−	0.993059i	$-0.537526\pi$
$419$	−4.81521	−0.235239	−0.117619	+	0.993059i	$0.537526\pi$
$420$	0	0
$421$	−40.9570	−1.99612	−0.998062	−	0.0622323i	$-0.980178\pi$
$421$	−40.9570	−1.99612	−0.998062	+	0.0622323i	$0.980178\pi$
$422$	−5.75589	−0.280192
$423$	0	0
$424$	−5.68150	−0.275918
$425$	7.46499	0.362105
$426$	0	0
$427$	12.4220	0.601144
$428$	−18.8047	−0.908959
$429$	0	0
$430$	−5.03990	−0.243046
$431$	−1.01321	−0.0488048	−0.0244024	−	0.999702i	$-0.507768\pi$
$431$	−1.01321	−0.0488048	−0.0244024	+	0.999702i	$0.507768\pi$
$432$	0	0
$433$	35.4827	1.70519	0.852595	−	0.522573i	$-0.175028\pi$
$433$	35.4827	1.70519	0.852595	+	0.522573i	$0.175028\pi$
$434$	3.24271	0.155655
$435$	0	0
$436$	−42.2842	−2.02505
$437$	46.2964	2.21466
$438$	0	0
$439$	−1.14182	−0.0544962	−0.0272481	−	0.999629i	$-0.508674\pi$
$439$	−1.14182	−0.0544962	−0.0272481	+	0.999629i	$0.508674\pi$
$440$	−17.6474	−0.841308
$441$	0	0
$442$	−13.7537	−0.654195
$443$	10.2828	0.488551	0.244275	−	0.969706i	$-0.421450\pi$
$443$	10.2828	0.488551	0.244275	+	0.969706i	$0.421450\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	8.91407	0.422093
$447$	0	0
$448$	12.0204	0.567911
$449$	−28.1874	−1.33024	−0.665122	−	0.746734i	$-0.731621\pi$
$449$	−28.1874	−1.33024	−0.665122	+	0.746734i	$0.731621\pi$
$450$	0	0
$451$	14.2009	0.668696
$452$	−19.8963	−0.935845
$453$	0	0
$454$	−52.5722	−2.46734
$455$	2.12053	0.0994120
$456$	0	0
$457$	−35.2560	−1.64921	−0.824603	−	0.565711i	$-0.808602\pi$
$457$	−35.2560	−1.64921	−0.824603	+	0.565711i	$0.808602\pi$
$458$	−41.7223	−1.94955
$459$	0	0
$460$	31.1511	1.45243
$461$	−16.1287	−0.751188	−0.375594	−	0.926784i	$-0.622561\pi$
$461$	−16.1287	−0.751188	−0.375594	+	0.926784i	$0.622561\pi$
$462$	0	0
$463$	1.16565	0.0541722	0.0270861	−	0.999633i	$-0.491377\pi$
$463$	1.16565	0.0541722	0.0270861	+	0.999633i	$0.491377\pi$
$464$	23.1704	1.07566
$465$	0	0
$466$	6.24369	0.289233
$467$	19.6019	0.907069	0.453534	−	0.891239i	$-0.350163\pi$
$467$	19.6019	0.907069	0.453534	+	0.891239i	$0.350163\pi$
$468$	0	0
$469$	−14.6372	−0.675885
$470$	0.891436	0.0411189
$471$	0	0
$472$	83.9225	3.86284
$473$	6.25219	0.287476
$474$	0	0
$475$	6.34476	0.291118
$476$	−91.8395	−4.20946
$477$	0	0
$478$	5.71347	0.261328
$479$	−6.78625	−0.310072	−0.155036	−	0.987909i	$-0.549549\pi$
$479$	−6.78625	−0.310072	−0.155036	+	0.987909i	$0.549549\pi$
$480$	0	0
$481$	−5.75145	−0.262243
$482$	30.0632	1.36934
$483$	0	0
$484$	−5.77278	−0.262399
$485$	10.5296	0.478127
$486$	0	0
$487$	16.2066	0.734391	0.367195	−	0.930144i	$-0.380318\pi$
$487$	16.2066	0.734391	0.367195	+	0.930144i	$0.380318\pi$
$488$	−24.4907	−1.10864
$489$	0	0
$490$	3.26648	0.147564
$491$	−20.0913	−0.906707	−0.453353	−	0.891331i	$-0.649772\pi$
$491$	−20.0913	−0.906707	−0.453353	+	0.891331i	$0.649772\pi$
$492$	0	0
$493$	30.4126	1.36971
$494$	−11.6897	−0.525946
$495$	0	0
$496$	−2.55596	−0.114766
$497$	−30.8982	−1.38597
$498$	0	0
$499$	−14.9382	−0.668724	−0.334362	−	0.942445i	$-0.608521\pi$
$499$	−14.9382	−0.668724	−0.334362	+	0.942445i	$0.608521\pi$
$500$	4.26915	0.190922
$501$	0	0
$502$	9.23821	0.412321
$503$	−11.1834	−0.498645	−0.249322	−	0.968421i	$-0.580208\pi$
$503$	−11.1834	−0.498645	−0.249322	+	0.968421i	$0.580208\pi$
$504$	0	0
$505$	−18.6744	−0.830998
$506$	−56.7480	−2.52275
$507$	0	0
$508$	−7.61946	−0.338059
$509$	−31.0234	−1.37509	−0.687544	−	0.726143i	$-0.741311\pi$
$509$	−31.0234	−1.37509	−0.687544	+	0.726143i	$0.741311\pi$
$510$	0	0
$511$	32.1833	1.42371
$512$	−48.2634	−2.13296
$513$	0	0
$514$	−41.1464	−1.81489
$515$	−1.64542	−0.0725058
$516$	0	0
$517$	−1.10586	−0.0486356
$518$	−56.3969	−2.47794
$519$	0	0
$520$	−4.18073	−0.183337
$521$	−27.0746	−1.18616	−0.593080	−	0.805143i	$-0.702088\pi$
$521$	−27.0746	−1.18616	−0.593080	+	0.805143i	$0.702088\pi$
$522$	0	0
$523$	41.6597	1.82165	0.910827	−	0.412789i	$-0.135445\pi$
$523$	41.6597	1.82165	0.910827	+	0.412789i	$0.135445\pi$
$524$	71.0560	3.10409
$525$	0	0
$526$	58.4629	2.54910
$527$	−3.35486	−0.146140
$528$	0	0
$529$	30.2432	1.31492
$530$	−2.50380	−0.108758
$531$	0	0
$532$	−78.0577	−3.38423
$533$	3.36425	0.145722
$534$	0	0
$535$	−4.40479	−0.190436
$536$	28.8581	1.24648
$537$	0	0
$538$	−51.8990	−2.23752
$539$	−4.05219	−0.174540
$540$	0	0
$541$	1.31807	0.0566684	0.0283342	−	0.999599i	$-0.490980\pi$
$541$	1.31807	0.0566684	0.0283342	+	0.999599i	$0.490980\pi$
$542$	59.0767	2.53756
$543$	0	0
$544$	21.4766	0.920804
$545$	−9.90460	−0.424266
$546$	0	0
$547$	24.2815	1.03820	0.519100	−	0.854713i	$-0.326267\pi$
$547$	24.2815	1.03820	0.519100	+	0.854713i	$0.326267\pi$
$548$	−99.2178	−4.23838
$549$	0	0
$550$	−7.77711	−0.331617
$551$	25.8488	1.10119
$552$	0	0
$553$	−15.5521	−0.661344
$554$	77.9380	3.31127
$555$	0	0
$556$	28.5765	1.21192
$557$	24.5295	1.03935	0.519673	−	0.854365i	$-0.326054\pi$
$557$	24.5295	1.03935	0.519673	+	0.854365i	$0.326054\pi$
$558$	0	0
$559$	1.48116	0.0626466
$560$	−16.3896	−0.692586
$561$	0	0
$562$	−66.1123	−2.78878
$563$	−19.9347	−0.840146	−0.420073	−	0.907490i	$-0.637996\pi$
$563$	−19.9347	−0.840146	−0.420073	+	0.907490i	$0.637996\pi$
$564$	0	0
$565$	−4.66049	−0.196068
$566$	44.6065	1.87495
$567$	0	0
$568$	60.9173	2.55603
$569$	2.81922	0.118188	0.0590940	−	0.998252i	$-0.481179\pi$
$569$	2.81922	0.118188	0.0590940	+	0.998252i	$0.481179\pi$
$570$	0	0
$571$	41.1313	1.72129	0.860646	−	0.509204i	$-0.170060\pi$
$571$	41.1313	1.72129	0.860646	+	0.509204i	$0.170060\pi$
$572$	9.75754	0.407983
$573$	0	0
$574$	32.9888	1.37693
$575$	7.29679	0.304297
$576$	0	0
$577$	5.26601	0.219227	0.109613	−	0.993974i	$-0.465039\pi$
$577$	5.26601	0.219227	0.109613	+	0.993974i	$0.465039\pi$
$578$	96.9632	4.03314
$579$	0	0
$580$	17.3926	0.722190
$581$	38.9012	1.61389
$582$	0	0
$583$	3.10606	0.128640
$584$	−63.4511	−2.62563
$585$	0	0
$586$	81.5122	3.36724
$587$	−40.4888	−1.67115	−0.835576	−	0.549375i	$-0.814866\pi$
$587$	−40.4888	−1.67115	−0.835576	+	0.549375i	$0.814866\pi$
$588$	0	0
$589$	−2.85142	−0.117491
$590$	36.9841	1.52261
$591$	0	0
$592$	44.4529	1.82700
$593$	−35.5004	−1.45783	−0.728914	−	0.684605i	$-0.759975\pi$
$593$	−35.5004	−1.45783	−0.728914	+	0.684605i	$0.759975\pi$
$594$	0	0
$595$	−21.5124	−0.881921
$596$	44.2611	1.81300
$597$	0	0
$598$	−13.4438	−0.549757
$599$	−23.0368	−0.941259	−0.470630	−	0.882331i	$-0.655973\pi$
$599$	−23.0368	−0.941259	−0.470630	+	0.882331i	$0.655973\pi$
$600$	0	0
$601$	44.4456	1.81297	0.906487	−	0.422233i	$-0.138754\pi$
$601$	44.4456	1.81297	0.906487	+	0.422233i	$0.138754\pi$
$602$	14.5238	0.591948
$603$	0	0
$604$	36.1652	1.47154
$605$	−1.35221	−0.0549751
$606$	0	0
$607$	20.2493	0.821894	0.410947	−	0.911659i	$-0.365198\pi$
$607$	20.2493	0.821894	0.410947	+	0.911659i	$0.365198\pi$
$608$	18.2538	0.740289
$609$	0	0
$610$	−10.7929	−0.436991
$611$	−0.261982	−0.0105987
$612$	0	0
$613$	−12.6758	−0.511971	−0.255985	−	0.966681i	$-0.582400\pi$
$613$	−12.6758	−0.511971	−0.255985	+	0.966681i	$0.582400\pi$
$614$	17.4141	0.702777
$615$	0	0
$616$	50.8558	2.04904
$617$	12.0655	0.485737	0.242869	−	0.970059i	$-0.421912\pi$
$617$	12.0655	0.485737	0.242869	+	0.970059i	$0.421912\pi$
$618$	0	0
$619$	−36.1450	−1.45279	−0.726394	−	0.687278i	$-0.758805\pi$
$619$	−36.1450	−1.45279	−0.726394	+	0.687278i	$0.758805\pi$
$620$	−1.91861	−0.0770532
$621$	0	0
$622$	−58.4867	−2.34510
$623$	2.88177	0.115456
$624$	0	0
$625$	1.00000	0.0400000
$626$	10.9218	0.436521
$627$	0	0
$628$	−19.8910	−0.793736
$629$	58.3473	2.32646
$630$	0	0
$631$	−44.5242	−1.77248	−0.886241	−	0.463225i	$-0.846692\pi$
$631$	−44.5242	−1.77248	−0.886241	+	0.463225i	$0.846692\pi$
$632$	30.6618	1.21966
$633$	0	0
$634$	−4.02215	−0.159740
$635$	−1.78477	−0.0708265
$636$	0	0
$637$	−0.959977	−0.0380357
$638$	−31.6842	−1.25439
$639$	0	0
$640$	−16.1979	−0.640279
$641$	−6.29790	−0.248752	−0.124376	−	0.992235i	$-0.539693\pi$
$641$	−6.29790	−0.248752	−0.124376	+	0.992235i	$0.539693\pi$
$642$	0	0
$643$	−13.9908	−0.551743	−0.275872	−	0.961194i	$-0.588966\pi$
$643$	−13.9908	−0.551743	−0.275872	+	0.961194i	$0.588966\pi$
$644$	−89.7703	−3.53744
$645$	0	0
$646$	118.590	4.66587
$647$	−0.262150	−0.0103062	−0.00515309	−	0.999987i	$-0.501640\pi$
$647$	−0.262150	−0.0103062	−0.00515309	+	0.999987i	$0.501640\pi$
$648$	0	0
$649$	−45.8802	−1.80095
$650$	−1.84242	−0.0722658
$651$	0	0
$652$	−68.2051	−2.67112
$653$	34.2596	1.34068	0.670340	−	0.742054i	$-0.266148\pi$
$653$	34.2596	1.34068	0.670340	+	0.742054i	$0.266148\pi$
$654$	0	0
$655$	16.6441	0.650337
$656$	−26.0023	−1.01522
$657$	0	0
$658$	−2.56891	−0.100147
$659$	−43.8257	−1.70721	−0.853603	−	0.520924i	$-0.825588\pi$
$659$	−43.8257	−1.70721	−0.853603	+	0.520924i	$0.825588\pi$
$660$	0	0
$661$	−13.8152	−0.537350	−0.268675	−	0.963231i	$-0.586586\pi$
$661$	−13.8152	−0.537350	−0.268675	+	0.963231i	$0.586586\pi$
$662$	53.6767	2.08620
$663$	0	0
$664$	−76.6957	−2.97637
$665$	−18.2841	−0.709029
$666$	0	0
$667$	29.7274	1.15105
$668$	2.50034	0.0967411
$669$	0	0
$670$	12.7176	0.491323
$671$	13.3890	0.516876
$672$	0	0
$673$	−6.25122	−0.240967	−0.120483	−	0.992715i	$-0.538444\pi$
$673$	−6.25122	−0.240967	−0.120483	+	0.992715i	$0.538444\pi$
$674$	−55.8928	−2.15291
$675$	0	0
$676$	−53.1873	−2.04567
$677$	−5.55223	−0.213390	−0.106695	−	0.994292i	$-0.534027\pi$
$677$	−5.55223	−0.213390	−0.106695	+	0.994292i	$0.534027\pi$
$678$	0	0
$679$	−30.3440	−1.16450
$680$	42.4127	1.62645
$681$	0	0
$682$	3.49513	0.133835
$683$	−35.4738	−1.35737	−0.678683	−	0.734431i	$-0.737449\pi$
$683$	−35.4738	−1.35737	−0.678683	+	0.734431i	$0.737449\pi$
$684$	0	0
$685$	−23.2407	−0.887980
$686$	41.0949	1.56901
$687$	0	0
$688$	−11.4479	−0.436448
$689$	0.735836	0.0280331
$690$	0	0
$691$	47.8974	1.82210	0.911051	−	0.412293i	$-0.135272\pi$
$691$	47.8974	1.82210	0.911051	+	0.412293i	$0.135272\pi$
$692$	9.23051	0.350891
$693$	0	0
$694$	36.9009	1.40074
$695$	6.69373	0.253908
$696$	0	0
$697$	−34.1297	−1.29275
$698$	−31.5814	−1.19537
$699$	0	0
$700$	−12.3027	−0.464998
$701$	−44.7578	−1.69048	−0.845239	−	0.534389i	$-0.820542\pi$
$701$	−44.7578	−1.69048	−0.845239	+	0.534389i	$0.820542\pi$
$702$	0	0
$703$	49.5915	1.87038
$704$	12.9561	0.488301
$705$	0	0
$706$	73.0144	2.74793
$707$	53.8152	2.02393
$708$	0	0
$709$	32.4227	1.21766	0.608831	−	0.793300i	$-0.291639\pi$
$709$	32.4227	1.21766	0.608831	+	0.793300i	$0.291639\pi$
$710$	26.8459	1.00751
$711$	0	0
$712$	−5.68155	−0.212925
$713$	−3.27927	−0.122810
$714$	0	0
$715$	2.28559	0.0854764
$716$	16.3059	0.609381
$717$	0	0
$718$	−29.7144	−1.10893
$719$	−37.1796	−1.38657	−0.693283	−	0.720666i	$-0.743836\pi$
$719$	−37.1796	−1.38657	−0.693283	+	0.720666i	$0.743836\pi$
$720$	0	0
$721$	4.74172	0.176591
$722$	53.2214	1.98069
$723$	0	0
$724$	−53.6852	−1.99519
$725$	4.07403	0.151306
$726$	0	0
$727$	2.66104	0.0986925	0.0493462	−	0.998782i	$-0.484286\pi$
$727$	2.66104	0.0986925	0.0493462	+	0.998782i	$0.484286\pi$
$728$	12.0479	0.446525
$729$	0	0
$730$	−27.9625	−1.03494
$731$	−15.0261	−0.555761
$732$	0	0
$733$	−7.17004	−0.264831	−0.132416	−	0.991194i	$-0.542273\pi$
$733$	−7.17004	−0.264831	−0.132416	+	0.991194i	$0.542273\pi$
$734$	−19.3566	−0.714467
$735$	0	0
$736$	20.9928	0.773804
$737$	−15.7766	−0.581140
$738$	0	0
$739$	−10.7941	−0.397067	−0.198534	−	0.980094i	$-0.563618\pi$
$739$	−10.7941	−0.397067	−0.198534	+	0.980094i	$0.563618\pi$
$740$	33.3682	1.22664
$741$	0	0
$742$	7.21538	0.264885
$743$	−14.4583	−0.530423	−0.265211	−	0.964190i	$-0.585442\pi$
$743$	−14.4583	−0.530423	−0.265211	+	0.964190i	$0.585442\pi$
$744$	0	0
$745$	10.3677	0.379842
$746$	60.5215	2.21585
$747$	0	0
$748$	−98.9884	−3.61937
$749$	12.6936	0.463813
$750$	0	0
$751$	−38.6718	−1.41115	−0.705576	−	0.708634i	$-0.749312\pi$
$751$	−38.6718	−1.41115	−0.705576	+	0.708634i	$0.749312\pi$
$752$	2.02486	0.0738390
$753$	0	0
$754$	−7.50609	−0.273356
$755$	8.47128	0.308302
$756$	0	0
$757$	−4.91441	−0.178617	−0.0893087	−	0.996004i	$-0.528466\pi$
$757$	−4.91441	−0.178617	−0.0893087	+	0.996004i	$0.528466\pi$
$758$	−17.8280	−0.647542
$759$	0	0
$760$	36.0481	1.30760
$761$	32.5933	1.18151	0.590753	−	0.806852i	$-0.298831\pi$
$761$	32.5933	1.18151	0.590753	+	0.806852i	$0.298831\pi$
$762$	0	0
$763$	28.5428	1.03332
$764$	−4.97020	−0.179816
$765$	0	0
$766$	−57.1181	−2.06376
$767$	−10.8692	−0.392463
$768$	0	0
$769$	−10.9482	−0.394803	−0.197402	−	0.980323i	$-0.563250\pi$
$769$	−10.9482	−0.394803	−0.197402	+	0.980323i	$0.563250\pi$
$770$	22.4118	0.807666
$771$	0	0
$772$	19.8647	0.714945
$773$	25.9688	0.934032	0.467016	−	0.884249i	$-0.345329\pi$
$773$	25.9688	0.934032	0.467016	+	0.884249i	$0.345329\pi$
$774$	0	0
$775$	−0.449413	−0.0161434
$776$	59.8248	2.14758
$777$	0	0
$778$	−5.57433	−0.199849
$779$	−29.0080	−1.03932
$780$	0	0
$781$	−33.3033	−1.19169
$782$	136.385	4.87710
$783$	0	0
$784$	7.41966	0.264988
$785$	−4.65923	−0.166295
$786$	0	0
$787$	−4.33462	−0.154513	−0.0772563	−	0.997011i	$-0.524616\pi$
$787$	−4.33462	−0.154513	−0.0772563	+	0.997011i	$0.524616\pi$
$788$	−94.7244	−3.37442
$789$	0	0
$790$	13.5125	0.480752
$791$	13.4305	0.477532
$792$	0	0
$793$	3.17189	0.112637
$794$	46.8278	1.66186
$795$	0	0
$796$	−18.8929	−0.669643
$797$	25.3700	0.898652	0.449326	−	0.893368i	$-0.351664\pi$
$797$	25.3700	0.898652	0.449326	+	0.893368i	$0.351664\pi$
$798$	0	0
$799$	2.65776	0.0940246
$800$	2.87698	0.101717
$801$	0	0
$802$	−22.9192	−0.809307
$803$	34.6885	1.22413
$804$	0	0
$805$	−21.0277	−0.741128
$806$	0.828009	0.0291654
$807$	0	0
$808$	−106.099	−3.73256
$809$	−17.3605	−0.610363	−0.305181	−	0.952294i	$-0.598717\pi$
$809$	−17.3605	−0.610363	−0.305181	+	0.952294i	$0.598717\pi$
$810$	0	0
$811$	48.6015	1.70663	0.853314	−	0.521397i	$-0.174589\pi$
$811$	48.6015	1.70663	0.853314	+	0.521397i	$0.174589\pi$
$812$	−50.1216	−1.75892
$813$	0	0
$814$	−60.7869	−2.13058
$815$	−15.9763	−0.559624
$816$	0	0
$817$	−12.7712	−0.446809
$818$	71.6227	2.50423
$819$	0	0
$820$	−19.5184	−0.681612
$821$	−7.98818	−0.278789	−0.139395	−	0.990237i	$-0.544516\pi$
$821$	−7.98818	−0.278789	−0.139395	+	0.990237i	$0.544516\pi$
$822$	0	0
$823$	28.5729	0.995990	0.497995	−	0.867180i	$-0.334070\pi$
$823$	28.5729	0.995990	0.497995	+	0.867180i	$0.334070\pi$
$824$	−9.34854	−0.325672
$825$	0	0
$826$	−106.580	−3.70838
$827$	41.2021	1.43274	0.716368	−	0.697722i	$-0.245803\pi$
$827$	41.2021	1.43274	0.716368	+	0.697722i	$0.245803\pi$
$828$	0	0
$829$	−52.2684	−1.81536	−0.907678	−	0.419666i	$-0.862147\pi$
$829$	−52.2684	−1.81536	−0.907678	+	0.419666i	$0.862147\pi$
$830$	−33.7993	−1.17319
$831$	0	0
$832$	3.06934	0.106410
$833$	9.73878	0.337429
$834$	0	0
$835$	0.585677	0.0202682
$836$	−84.1338	−2.90983
$837$	0	0
$838$	−12.0565	−0.416483
$839$	30.2450	1.04417	0.522087	−	0.852892i	$-0.325154\pi$
$839$	30.2450	1.04417	0.522087	+	0.852892i	$0.325154\pi$
$840$	0	0
$841$	−12.4023	−0.427665
$842$	−102.549	−3.53408
$843$	0	0
$844$	−9.81407	−0.337814
$845$	−12.4585	−0.428587
$846$	0	0
$847$	3.89676	0.133894
$848$	−5.68728	−0.195302
$849$	0	0
$850$	18.6910	0.641097
$851$	57.0327	1.95506
$852$	0	0
$853$	−42.5285	−1.45615	−0.728074	−	0.685499i	$-0.759584\pi$
$853$	−42.5285	−1.45615	−0.728074	+	0.685499i	$0.759584\pi$
$854$	31.1026	1.06431
$855$	0	0
$856$	−25.0260	−0.855372
$857$	21.2997	0.727585	0.363792	−	0.931480i	$-0.381482\pi$
$857$	21.2997	0.727585	0.363792	+	0.931480i	$0.381482\pi$
$858$	0	0
$859$	−21.3130	−0.727192	−0.363596	−	0.931557i	$-0.618451\pi$
$859$	−21.3130	−0.727192	−0.363596	+	0.931557i	$0.618451\pi$
$860$	−8.59329	−0.293029
$861$	0	0
$862$	−2.53691	−0.0864075
$863$	10.0504	0.342121	0.171060	−	0.985261i	$-0.445281\pi$
$863$	10.0504	0.342121	0.171060	+	0.985261i	$0.445281\pi$
$864$	0	0
$865$	2.16214	0.0735151
$866$	88.8425	3.01899
$867$	0	0
$868$	5.52899	0.187666
$869$	−16.7627	−0.568637
$870$	0	0
$871$	−3.73754	−0.126642
$872$	−56.2735	−1.90566
$873$	0	0
$874$	115.918	3.92099
$875$	−2.88177	−0.0974216
$876$	0	0
$877$	20.0072	0.675596	0.337798	−	0.941219i	$-0.390318\pi$
$877$	20.0072	0.675596	0.337798	+	0.941219i	$0.390318\pi$
$878$	−2.85893	−0.0964841
$879$	0	0
$880$	−17.6654	−0.595499
$881$	32.3032	1.08832	0.544161	−	0.838981i	$-0.316848\pi$
$881$	32.3032	1.08832	0.544161	+	0.838981i	$0.316848\pi$
$882$	0	0
$883$	−46.4579	−1.56343	−0.781716	−	0.623634i	$-0.785656\pi$
$883$	−46.4579	−1.56343	−0.781716	+	0.623634i	$0.785656\pi$
$884$	−23.4507	−0.788732
$885$	0	0
$886$	25.7464	0.864966
$887$	−23.6229	−0.793181	−0.396590	−	0.917996i	$-0.629807\pi$
$887$	−23.6229	−0.793181	−0.396590	+	0.917996i	$0.629807\pi$
$888$	0	0
$889$	5.14330	0.172501
$890$	−2.50383	−0.0839285
$891$	0	0
$892$	15.1989	0.508898
$893$	2.25892	0.0755920
$894$	0	0
$895$	3.81948	0.127671
$896$	46.6787	1.55942
$897$	0	0
$898$	−70.5763	−2.35516
$899$	−1.83092	−0.0610646
$900$	0	0
$901$	−7.46492	−0.248692
$902$	35.5567	1.18391
$903$	0	0
$904$	−26.4789	−0.880673
$905$	−12.5751	−0.418012
$906$	0	0
$907$	58.0980	1.92911	0.964555	−	0.263880i	$-0.0850024\pi$
$907$	58.0980	1.92911	0.964555	+	0.263880i	$0.0850024\pi$
$908$	−89.6382	−2.97475
$909$	0	0
$910$	5.30944	0.176006
$911$	−47.4685	−1.57270	−0.786351	−	0.617780i	$-0.788032\pi$
$911$	−47.4685	−1.57270	−0.786351	+	0.617780i	$0.788032\pi$
$912$	0	0
$913$	41.9293	1.38766
$914$	−88.2750	−2.91988
$915$	0	0
$916$	−71.1385	−2.35048
$917$	−47.9644	−1.58392
$918$	0	0
$919$	8.19266	0.270251	0.135125	−	0.990829i	$-0.456856\pi$
$919$	8.19266	0.270251	0.135125	+	0.990829i	$0.456856\pi$
$920$	41.4571	1.36680
$921$	0	0
$922$	−40.3834	−1.32996
$923$	−7.88967	−0.259692
$924$	0	0
$925$	7.81613	0.256993
$926$	2.91858	0.0959104
$927$	0	0
$928$	11.7209	0.384758
$929$	34.4424	1.13002	0.565009	−	0.825084i	$-0.308873\pi$
$929$	34.4424	1.13002	0.565009	+	0.825084i	$0.308873\pi$
$930$	0	0
$931$	8.27734	0.271279
$932$	10.6458	0.348715
$933$	0	0
$934$	49.0798	1.60594
$935$	−23.1869	−0.758293
$936$	0	0
$937$	−33.9333	−1.10855	−0.554277	−	0.832332i	$-0.687005\pi$
$937$	−33.9333	−1.10855	−0.554277	+	0.832332i	$0.687005\pi$
$938$	−36.6491	−1.19664
$939$	0	0
$940$	1.51994	0.0495751
$941$	−26.1883	−0.853715	−0.426857	−	0.904319i	$-0.640379\pi$
$941$	−26.1883	−0.853715	−0.426857	+	0.904319i	$0.640379\pi$
$942$	0	0
$943$	−33.3607	−1.08637
$944$	84.0078	2.73422
$945$	0	0
$946$	15.6544	0.508968
$947$	−45.7171	−1.48561	−0.742803	−	0.669510i	$-0.766504\pi$
$947$	−45.7171	−1.48561	−0.742803	+	0.669510i	$0.766504\pi$
$948$	0	0
$949$	8.21783	0.266762
$950$	15.8862	0.515416
$951$	0	0
$952$	−122.224	−3.96129
$953$	20.5498	0.665673	0.332836	−	0.942985i	$-0.391994\pi$
$953$	20.5498	0.665673	0.332836	+	0.942985i	$0.391994\pi$
$954$	0	0
$955$	−1.16421	−0.0376731
$956$	9.74175	0.315071
$957$	0	0
$958$	−16.9916	−0.548973
$959$	66.9742	2.16271
$960$	0	0
$961$	−30.7980	−0.993485
$962$	−14.4006	−0.464295
$963$	0	0
$964$	51.2593	1.65095
$965$	4.65308	0.149788
$966$	0	0
$967$	−32.3722	−1.04102	−0.520510	−	0.853855i	$-0.674258\pi$
$967$	−32.3722	−1.04102	−0.520510	+	0.853855i	$0.674258\pi$
$968$	−7.68265	−0.246930
$969$	0	0
$970$	26.3644	0.846510
$971$	47.9123	1.53758	0.768790	−	0.639501i	$-0.220859\pi$
$971$	47.9123	1.53758	0.768790	+	0.639501i	$0.220859\pi$
$972$	0	0
$973$	−19.2898	−0.618402
$974$	40.5785	1.30022
$975$	0	0
$976$	−24.5156	−0.784724
$977$	12.7029	0.406400	0.203200	−	0.979137i	$-0.434866\pi$
$977$	12.7029	0.406400	0.203200	+	0.979137i	$0.434866\pi$
$978$	0	0
$979$	3.10609	0.0992710
$980$	5.56951	0.177911
$981$	0	0
$982$	−50.3051	−1.60530
$983$	−16.5057	−0.526451	−0.263226	−	0.964734i	$-0.584786\pi$
$983$	−16.5057	−0.526451	−0.263226	+	0.964734i	$0.584786\pi$
$984$	0	0
$985$	−22.1881	−0.706972
$986$	76.1478	2.42504
$987$	0	0
$988$	−19.9316	−0.634108
$989$	−14.6876	−0.467038
$990$	0	0
$991$	16.2113	0.514968	0.257484	−	0.966283i	$-0.417107\pi$
$991$	16.2113	0.514968	0.257484	+	0.966283i	$0.417107\pi$
$992$	−1.29295	−0.0410513
$993$	0	0
$994$	−77.3636	−2.45383
$995$	−4.42546	−0.140296
$996$	0	0
$997$	−14.8403	−0.469998	−0.234999	−	0.971996i	$-0.575509\pi$
$997$	−14.8403	−0.469998	−0.234999	+	0.971996i	$0.575509\pi$
$998$	−37.4026	−1.18396
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.x.1.15	yes	17
3.2	odd	2		4005.2.a.w.1.3	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.w.1.3	✓	17	3.2	odd	2
4005.2.a.x.1.15	yes	17	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+2.50383 q^{2} +4.26915 q^{4} +1.00000 q^{5} -2.88177 q^{7} +5.68155 q^{8} +O(q^{10})\)	\(q+2.50383 q^{2} +4.26915 q^{4} +1.00000 q^{5} -2.88177 q^{7} +5.68155 q^{8} +2.50383 q^{10} -3.10609 q^{11} -0.735843 q^{13} -7.21545 q^{14} +5.68733 q^{16} +7.46499 q^{17} +6.34476 q^{19} +4.26915 q^{20} -7.77711 q^{22} +7.29679 q^{23} +1.00000 q^{25} -1.84242 q^{26} -12.3027 q^{28} +4.07403 q^{29} -0.449413 q^{31} +2.87698 q^{32} +18.6910 q^{34} -2.88177 q^{35} +7.81613 q^{37} +15.8862 q^{38} +5.68155 q^{40} -4.57197 q^{41} -2.01288 q^{43} -13.2604 q^{44} +18.2699 q^{46} +0.356029 q^{47} +1.30459 q^{49} +2.50383 q^{50} -3.14142 q^{52} -0.999991 q^{53} -3.10609 q^{55} -16.3729 q^{56} +10.2007 q^{58} +14.7710 q^{59} -4.31056 q^{61} -1.12525 q^{62} -4.17119 q^{64} -0.735843 q^{65} +5.07926 q^{67} +31.8691 q^{68} -7.21545 q^{70} +10.7219 q^{71} -11.1679 q^{73} +19.5702 q^{74} +27.0867 q^{76} +8.95103 q^{77} +5.39673 q^{79} +5.68733 q^{80} -11.4474 q^{82} -13.4991 q^{83} +7.46499 q^{85} -5.03990 q^{86} -17.6474 q^{88} -1.00000 q^{89} +2.12053 q^{91} +31.1511 q^{92} +0.891436 q^{94} +6.34476 q^{95} +10.5296 q^{97} +3.26648 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) 17 * q + 5 * q^2 + 21 * q^4 + 17 * q^5 + 12 * q^7 + 15 * q^8	\( 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8} + 5 q^{10} + 2 q^{11} + 8 q^{13} + 4 q^{14} + 33 q^{16} + 10 q^{17} + 32 q^{19} + 21 q^{20} + 8 q^{22} + 15 q^{23} + 17 q^{25} - 15 q^{26} + 24 q^{28} + q^{29} + 18 q^{31} + 25 q^{32} + 14 q^{34} + 12 q^{35} + 12 q^{37} + 22 q^{38} + 15 q^{40} - 7 q^{41} + 28 q^{43} - 14 q^{44} + 4 q^{46} + 26 q^{47} + 41 q^{49} + 5 q^{50} + 10 q^{52} + 12 q^{53} + 2 q^{55} + 13 q^{56} + 16 q^{58} - 23 q^{59} + 26 q^{61} + 10 q^{62} + 59 q^{64} + 8 q^{65} + 31 q^{67} - q^{68} + 4 q^{70} - 2 q^{71} + 33 q^{73} - 10 q^{74} + 66 q^{76} + 12 q^{77} + 33 q^{79} + 33 q^{80} + 30 q^{82} + 13 q^{83} + 10 q^{85} - 20 q^{86} + 12 q^{88} - 17 q^{89} + 40 q^{91} + 16 q^{92} + 38 q^{94} + 32 q^{95} + 45 q^{97} + 2 q^{98}+O(q^{100}) \) 17 * q + 5 * q^2 + 21 * q^4 + 17 * q^5 + 12 * q^7 + 15 * q^8 + 5 * q^10 + 2 * q^11 + 8 * q^13 + 4 * q^14 + 33 * q^16 + 10 * q^17 + 32 * q^19 + 21 * q^20 + 8 * q^22 + 15 * q^23 + 17 * q^25 - 15 * q^26 + 24 * q^28 + q^29 + 18 * q^31 + 25 * q^32 + 14 * q^34 + 12 * q^35 + 12 * q^37 + 22 * q^38 + 15 * q^40 - 7 * q^41 + 28 * q^43 - 14 * q^44 + 4 * q^46 + 26 * q^47 + 41 * q^49 + 5 * q^50 + 10 * q^52 + 12 * q^53 + 2 * q^55 + 13 * q^56 + 16 * q^58 - 23 * q^59 + 26 * q^61 + 10 * q^62 + 59 * q^64 + 8 * q^65 + 31 * q^67 - q^68 + 4 * q^70 - 2 * q^71 + 33 * q^73 - 10 * q^74 + 66 * q^76 + 12 * q^77 + 33 * q^79 + 33 * q^80 + 30 * q^82 + 13 * q^83 + 10 * q^85 - 20 * q^86 + 12 * q^88 - 17 * q^89 + 40 * q^91 + 16 * q^92 + 38 * q^94 + 32 * q^95 + 45 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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