LMFDB - Embedded newform 4004.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.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:	$31.9721009693$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.463341.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x + 1 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.09892$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.09892 q^{3} -2.18167 q^{5} -1.00000 q^{7} +1.40545 q^{9} +O(q^{10})$	$q-2.09892 q^{3} -2.18167 q^{5} -1.00000 q^{7} +1.40545 q^{9} -1.00000 q^{11} +1.00000 q^{13} +4.57915 q^{15} -0.842458 q^{17} +1.25389 q^{19} +2.09892 q^{21} -3.10946 q^{23} -0.240299 q^{25} +3.34683 q^{27} +2.98203 q^{29} +9.84322 q^{31} +2.09892 q^{33} +2.18167 q^{35} +4.05681 q^{37} -2.09892 q^{39} -2.29599 q^{41} -8.19526 q^{43} -3.06624 q^{45} +7.75485 q^{47} +1.00000 q^{49} +1.76825 q^{51} +7.37693 q^{53} +2.18167 q^{55} -2.63180 q^{57} -0.236896 q^{59} -13.7710 q^{61} -1.40545 q^{63} -2.18167 q^{65} +3.89166 q^{67} +6.52650 q^{69} +8.81605 q^{71} +6.01728 q^{73} +0.504367 q^{75} +1.00000 q^{77} -12.1005 q^{79} -11.2411 q^{81} +9.98424 q^{83} +1.83797 q^{85} -6.25903 q^{87} +10.1233 q^{89} -1.00000 q^{91} -20.6601 q^{93} -2.73557 q^{95} -10.2565 q^{97} -1.40545 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - 5 q^{7} + 2 q^{9}+O(q^{10}) $ 5 * q + 3 * q^3 - 5 * q^7 + 2 * q^9	$ 5 q + 3 q^{3} - 5 q^{7} + 2 q^{9} - 5 q^{11} + 5 q^{13} - 3 q^{15} - 9 q^{17} - 4 q^{19} - 3 q^{21} - 4 q^{23} + q^{25} + 3 q^{27} - 8 q^{29} + 7 q^{31} - 3 q^{33} - 10 q^{37} + 3 q^{39} - 18 q^{41} - 22 q^{43} - 26 q^{45} + 12 q^{47} + 5 q^{49} - 7 q^{51} + 7 q^{53} - 6 q^{57} - q^{59} - 26 q^{61} - 2 q^{63} - 8 q^{67} - 12 q^{69} + 18 q^{71} - 25 q^{73} - 16 q^{75} + 5 q^{77} - 6 q^{79} - 7 q^{81} + 11 q^{83} - 7 q^{85} - 5 q^{87} - 14 q^{89} - 5 q^{91} - 41 q^{93} - 22 q^{95} - 33 q^{97} - 2 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - 5 * q^7 + 2 * q^9 - 5 * q^11 + 5 * q^13 - 3 * q^15 - 9 * q^17 - 4 * q^19 - 3 * q^21 - 4 * q^23 + q^25 + 3 * q^27 - 8 * q^29 + 7 * q^31 - 3 * q^33 - 10 * q^37 + 3 * q^39 - 18 * q^41 - 22 * q^43 - 26 * q^45 + 12 * q^47 + 5 * q^49 - 7 * q^51 + 7 * q^53 - 6 * q^57 - q^59 - 26 * q^61 - 2 * q^63 - 8 * q^67 - 12 * q^69 + 18 * q^71 - 25 * q^73 - 16 * q^75 + 5 * q^77 - 6 * q^79 - 7 * q^81 + 11 * q^83 - 7 * q^85 - 5 * q^87 - 14 * q^89 - 5 * q^91 - 41 * q^93 - 22 * q^95 - 33 * q^97 - 2 * 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
$3$	−2.09892	−1.21181	−0.605905	−	0.795537i	$-0.707189\pi$
$3$	−2.09892	−1.21181	−0.605905	+	0.795537i	$0.707189\pi$
$4$	0	0
$5$	−2.18167	−0.975674	−0.487837	−	0.872935i	$-0.662214\pi$
$5$	−2.18167	−0.975674	−0.487837	+	0.872935i	$0.662214\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.40545	0.468484
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	4.57915	1.18233
$16$	0	0
$17$	−0.842458	−0.204326	−0.102163	−	0.994768i	$-0.532576\pi$
$17$	−0.842458	−0.204326	−0.102163	+	0.994768i	$0.532576\pi$
$18$	0	0
$19$	1.25389	0.287661	0.143831	−	0.989602i	$-0.454058\pi$
$19$	1.25389	0.287661	0.143831	+	0.989602i	$0.454058\pi$
$20$	0	0
$21$	2.09892	0.458021
$22$	0	0
$23$	−3.10946	−0.648368	−0.324184	−	0.945994i	$-0.605090\pi$
$23$	−3.10946	−0.648368	−0.324184	+	0.945994i	$0.605090\pi$
$24$	0	0
$25$	−0.240299	−0.0480598
$26$	0	0
$27$	3.34683	0.644097
$28$	0	0
$29$	2.98203	0.553749	0.276874	−	0.960906i	$-0.410701\pi$
$29$	2.98203	0.553749	0.276874	+	0.960906i	$0.410701\pi$
$30$	0	0
$31$	9.84322	1.76789	0.883947	−	0.467587i	$-0.154876\pi$
$31$	9.84322	1.76789	0.883947	+	0.467587i	$0.154876\pi$
$32$	0	0
$33$	2.09892	0.365374
$34$	0	0
$35$	2.18167	0.368770
$36$	0	0
$37$	4.05681	0.666936	0.333468	−	0.942761i	$-0.391781\pi$
$37$	4.05681	0.666936	0.333468	+	0.942761i	$0.391781\pi$
$38$	0	0
$39$	−2.09892	−0.336096
$40$	0	0
$41$	−2.29599	−0.358573	−0.179287	−	0.983797i	$-0.557379\pi$
$41$	−2.29599	−0.358573	−0.179287	+	0.983797i	$0.557379\pi$
$42$	0	0
$43$	−8.19526	−1.24977	−0.624883	−	0.780719i	$-0.714853\pi$
$43$	−8.19526	−1.24977	−0.624883	+	0.780719i	$0.714853\pi$
$44$	0	0
$45$	−3.06624	−0.457087
$46$	0	0
$47$	7.75485	1.13116	0.565581	−	0.824693i	$-0.308652\pi$
$47$	7.75485	1.13116	0.565581	+	0.824693i	$0.308652\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.76825	0.247605
$52$	0	0
$53$	7.37693	1.01330	0.506650	−	0.862152i	$-0.330884\pi$
$53$	7.37693	1.01330	0.506650	+	0.862152i	$0.330884\pi$
$54$	0	0
$55$	2.18167	0.294177
$56$	0	0
$57$	−2.63180	−0.348590
$58$	0	0
$59$	−0.236896	−0.0308412	−0.0154206	−	0.999881i	$-0.504909\pi$
$59$	−0.236896	−0.0308412	−0.0154206	+	0.999881i	$0.504909\pi$
$60$	0	0
$61$	−13.7710	−1.76320	−0.881598	−	0.472000i	$-0.843532\pi$
$61$	−13.7710	−1.76320	−0.881598	+	0.472000i	$0.843532\pi$
$62$	0	0
$63$	−1.40545	−0.177070
$64$	0	0
$65$	−2.18167	−0.270603
$66$	0	0
$67$	3.89166	0.475442	0.237721	−	0.971333i	$-0.423600\pi$
$67$	3.89166	0.475442	0.237721	+	0.971333i	$0.423600\pi$
$68$	0	0
$69$	6.52650	0.785699
$70$	0	0
$71$	8.81605	1.04627	0.523136	−	0.852249i	$-0.324762\pi$
$71$	8.81605	1.04627	0.523136	+	0.852249i	$0.324762\pi$
$72$	0	0
$73$	6.01728	0.704270	0.352135	−	0.935949i	$-0.385456\pi$
$73$	6.01728	0.704270	0.352135	+	0.935949i	$0.385456\pi$
$74$	0	0
$75$	0.504367	0.0582393
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−12.1005	−1.36141	−0.680707	−	0.732556i	$-0.738327\pi$
$79$	−12.1005	−1.36141	−0.680707	+	0.732556i	$0.738327\pi$
$80$	0	0
$81$	−11.2411	−1.24901
$82$	0	0
$83$	9.98424	1.09591	0.547956	−	0.836507i	$-0.315406\pi$
$83$	9.98424	1.09591	0.547956	+	0.836507i	$0.315406\pi$
$84$	0	0
$85$	1.83797	0.199356
$86$	0	0
$87$	−6.25903	−0.671039
$88$	0	0
$89$	10.1233	1.07307	0.536536	−	0.843877i	$-0.319733\pi$
$89$	10.1233	1.07307	0.536536	+	0.843877i	$0.319733\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−20.6601	−2.14235
$94$	0	0
$95$	−2.73557	−0.280663
$96$	0	0
$97$	−10.2565	−1.04139	−0.520693	−	0.853744i	$-0.674326\pi$
$97$	−10.2565	−1.04139	−0.520693	+	0.853744i	$0.674326\pi$
$98$	0	0
$99$	−1.40545	−0.141253
$100$	0	0
$101$	−4.31472	−0.429331	−0.214665	−	0.976688i	$-0.568866\pi$
$101$	−4.31472	−0.429331	−0.214665	+	0.976688i	$0.568866\pi$
$102$	0	0
$103$	−11.0764	−1.09139	−0.545694	−	0.837985i	$-0.683734\pi$
$103$	−11.0764	−1.09139	−0.545694	+	0.837985i	$0.683734\pi$
$104$	0	0
$105$	−4.57915	−0.446879
$106$	0	0
$107$	13.6732	1.32184	0.660920	−	0.750456i	$-0.270166\pi$
$107$	13.6732	1.32184	0.660920	+	0.750456i	$0.270166\pi$
$108$	0	0
$109$	−15.2945	−1.46495	−0.732476	−	0.680793i	$-0.761635\pi$
$109$	−15.2945	−1.46495	−0.732476	+	0.680793i	$0.761635\pi$
$110$	0	0
$111$	−8.51491	−0.808200
$112$	0	0
$113$	19.9854	1.88006	0.940032	−	0.341085i	$-0.110794\pi$
$113$	19.9854	1.88006	0.940032	+	0.341085i	$0.110794\pi$
$114$	0	0
$115$	6.78383	0.632596
$116$	0	0
$117$	1.40545	0.129934
$118$	0	0
$119$	0.842458	0.0772280
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.81909	0.434522
$124$	0	0
$125$	11.4326	1.02256
$126$	0	0
$127$	−0.572708	−0.0508196	−0.0254098	−	0.999677i	$-0.508089\pi$
$127$	−0.572708	−0.0508196	−0.0254098	+	0.999677i	$0.508089\pi$
$128$	0	0
$129$	17.2012	1.51448
$130$	0	0
$131$	0.654190	0.0571569	0.0285784	−	0.999592i	$-0.490902\pi$
$131$	0.654190	0.0571569	0.0285784	+	0.999592i	$0.490902\pi$
$132$	0	0
$133$	−1.25389	−0.108726
$134$	0	0
$135$	−7.30168	−0.628429
$136$	0	0
$137$	−13.4122	−1.14588	−0.572940	−	0.819597i	$-0.694197\pi$
$137$	−13.4122	−1.14588	−0.572940	+	0.819597i	$0.694197\pi$
$138$	0	0
$139$	15.7161	1.33302	0.666510	−	0.745496i	$-0.267787\pi$
$139$	15.7161	1.33302	0.666510	+	0.745496i	$0.267787\pi$
$140$	0	0
$141$	−16.2768	−1.37075
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−6.50582	−0.540279
$146$	0	0
$147$	−2.09892	−0.173116
$148$	0	0
$149$	−16.3577	−1.34007	−0.670036	−	0.742329i	$-0.733721\pi$
$149$	−16.3577	−1.34007	−0.670036	+	0.742329i	$0.733721\pi$
$150$	0	0
$151$	−8.12515	−0.661216	−0.330608	−	0.943768i	$-0.607254\pi$
$151$	−8.12515	−0.661216	−0.330608	+	0.943768i	$0.607254\pi$
$152$	0	0
$153$	−1.18403	−0.0957235
$154$	0	0
$155$	−21.4747	−1.72489
$156$	0	0
$157$	−17.5702	−1.40225	−0.701127	−	0.713036i	$-0.747319\pi$
$157$	−17.5702	−1.40225	−0.701127	+	0.713036i	$0.747319\pi$
$158$	0	0
$159$	−15.4836	−1.22793
$160$	0	0
$161$	3.10946	0.245060
$162$	0	0
$163$	−22.4556	−1.75886	−0.879429	−	0.476030i	$-0.842075\pi$
$163$	−22.4556	−1.75886	−0.879429	+	0.476030i	$0.842075\pi$
$164$	0	0
$165$	−4.57915	−0.356486
$166$	0	0
$167$	19.5810	1.51523	0.757613	−	0.652704i	$-0.226365\pi$
$167$	19.5810	1.51523	0.757613	+	0.652704i	$0.226365\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.76227	0.134764
$172$	0	0
$173$	−20.7002	−1.57381	−0.786904	−	0.617076i	$-0.788317\pi$
$173$	−20.7002	−1.57381	−0.786904	+	0.617076i	$0.788317\pi$
$174$	0	0
$175$	0.240299	0.0181649
$176$	0	0
$177$	0.497224	0.0373737
$178$	0	0
$179$	−18.1998	−1.36032	−0.680158	−	0.733066i	$-0.738089\pi$
$179$	−18.1998	−1.36032	−0.680158	+	0.733066i	$0.738089\pi$
$180$	0	0
$181$	−17.6463	−1.31164	−0.655821	−	0.754917i	$-0.727677\pi$
$181$	−17.6463	−1.31164	−0.655821	+	0.754917i	$0.727677\pi$
$182$	0	0
$183$	28.9042	2.13666
$184$	0	0
$185$	−8.85064	−0.650712
$186$	0	0
$187$	0.842458	0.0616067
$188$	0	0
$189$	−3.34683	−0.243446
$190$	0	0
$191$	−14.4021	−1.04210	−0.521050	−	0.853526i	$-0.674460\pi$
$191$	−14.4021	−1.04210	−0.521050	+	0.853526i	$0.674460\pi$
$192$	0	0
$193$	20.2415	1.45701	0.728506	−	0.685039i	$-0.240215\pi$
$193$	20.2415	1.45701	0.728506	+	0.685039i	$0.240215\pi$
$194$	0	0
$195$	4.57915	0.327920
$196$	0	0
$197$	−12.1745	−0.867395	−0.433697	−	0.901059i	$-0.642791\pi$
$197$	−12.1745	−0.867395	−0.433697	+	0.901059i	$0.642791\pi$
$198$	0	0
$199$	26.0488	1.84655	0.923274	−	0.384141i	$-0.125503\pi$
$199$	26.0488	1.84655	0.923274	+	0.384141i	$0.125503\pi$
$200$	0	0
$201$	−8.16827	−0.576146
$202$	0	0
$203$	−2.98203	−0.209297
$204$	0	0
$205$	5.00910	0.349851
$206$	0	0
$207$	−4.37020	−0.303750
$208$	0	0
$209$	−1.25389	−0.0867331
$210$	0	0
$211$	−18.1988	−1.25286	−0.626428	−	0.779479i	$-0.715484\pi$
$211$	−18.1988	−1.25286	−0.626428	+	0.779479i	$0.715484\pi$
$212$	0	0
$213$	−18.5041	−1.26788
$214$	0	0
$215$	17.8794	1.21936
$216$	0	0
$217$	−9.84322	−0.668201
$218$	0	0
$219$	−12.6298	−0.853441
$220$	0	0
$221$	−0.842458	−0.0566699
$222$	0	0
$223$	11.3800	0.762059	0.381030	−	0.924563i	$-0.375570\pi$
$223$	11.3800	0.762059	0.381030	+	0.924563i	$0.375570\pi$
$224$	0	0
$225$	−0.337728	−0.0225152
$226$	0	0
$227$	−19.6457	−1.30393	−0.651965	−	0.758249i	$-0.726055\pi$
$227$	−19.6457	−1.30393	−0.651965	+	0.758249i	$0.726055\pi$
$228$	0	0
$229$	−2.68538	−0.177455	−0.0887276	−	0.996056i	$-0.528280\pi$
$229$	−2.68538	−0.177455	−0.0887276	+	0.996056i	$0.528280\pi$
$230$	0	0
$231$	−2.09892	−0.138099
$232$	0	0
$233$	−23.8471	−1.56228	−0.781138	−	0.624359i	$-0.785360\pi$
$233$	−23.8471	−1.56228	−0.781138	+	0.624359i	$0.785360\pi$
$234$	0	0
$235$	−16.9186	−1.10364
$236$	0	0
$237$	25.3980	1.64977
$238$	0	0
$239$	−13.1520	−0.850734	−0.425367	−	0.905021i	$-0.639855\pi$
$239$	−13.1520	−0.850734	−0.425367	+	0.905021i	$0.639855\pi$
$240$	0	0
$241$	−22.4051	−1.44324	−0.721621	−	0.692289i	$-0.756602\pi$
$241$	−22.4051	−1.44324	−0.721621	+	0.692289i	$0.756602\pi$
$242$	0	0
$243$	13.5536	0.869462
$244$	0	0
$245$	−2.18167	−0.139382
$246$	0	0
$247$	1.25389	0.0797828
$248$	0	0
$249$	−20.9561	−1.32804
$250$	0	0
$251$	23.5767	1.48815	0.744075	−	0.668096i	$-0.232891\pi$
$251$	23.5767	1.48815	0.744075	+	0.668096i	$0.232891\pi$
$252$	0	0
$253$	3.10946	0.195490
$254$	0	0
$255$	−3.85774	−0.241581
$256$	0	0
$257$	29.9326	1.86715	0.933574	−	0.358386i	$-0.116673\pi$
$257$	29.9326	1.86715	0.933574	+	0.358386i	$0.116673\pi$
$258$	0	0
$259$	−4.05681	−0.252078
$260$	0	0
$261$	4.19110	0.259422
$262$	0	0
$263$	−5.03935	−0.310740	−0.155370	−	0.987856i	$-0.549657\pi$
$263$	−5.03935	−0.310740	−0.155370	+	0.987856i	$0.549657\pi$
$264$	0	0
$265$	−16.0941	−0.988651
$266$	0	0
$267$	−21.2481	−1.30036
$268$	0	0
$269$	23.2048	1.41482	0.707412	−	0.706802i	$-0.249863\pi$
$269$	23.2048	1.41482	0.707412	+	0.706802i	$0.249863\pi$
$270$	0	0
$271$	−5.82945	−0.354114	−0.177057	−	0.984201i	$-0.556658\pi$
$271$	−5.82945	−0.354114	−0.177057	+	0.984201i	$0.556658\pi$
$272$	0	0
$273$	2.09892	0.127032
$274$	0	0
$275$	0.240299	0.0144906
$276$	0	0
$277$	6.92359	0.415998	0.207999	−	0.978129i	$-0.433305\pi$
$277$	6.92359	0.415998	0.207999	+	0.978129i	$0.433305\pi$
$278$	0	0
$279$	13.8342	0.828230
$280$	0	0
$281$	−14.5701	−0.869179	−0.434590	−	0.900629i	$-0.643107\pi$
$281$	−14.5701	−0.869179	−0.434590	+	0.900629i	$0.643107\pi$
$282$	0	0
$283$	16.1390	0.959366	0.479683	−	0.877442i	$-0.340752\pi$
$283$	16.1390	0.959366	0.479683	+	0.877442i	$0.340752\pi$
$284$	0	0
$285$	5.74173	0.340111
$286$	0	0
$287$	2.29599	0.135528
$288$	0	0
$289$	−16.2903	−0.958251
$290$	0	0
$291$	21.5274	1.26196
$292$	0	0
$293$	−16.5819	−0.968723	−0.484362	−	0.874868i	$-0.660948\pi$
$293$	−16.5819	−0.968723	−0.484362	+	0.874868i	$0.660948\pi$
$294$	0	0
$295$	0.516829	0.0300910
$296$	0	0
$297$	−3.34683	−0.194203
$298$	0	0
$299$	−3.10946	−0.179825
$300$	0	0
$301$	8.19526	0.472367
$302$	0	0
$303$	9.05624	0.520267
$304$	0	0
$305$	30.0439	1.72031
$306$	0	0
$307$	−1.85239	−0.105721	−0.0528606	−	0.998602i	$-0.516834\pi$
$307$	−1.85239	−0.105721	−0.0528606	+	0.998602i	$0.516834\pi$
$308$	0	0
$309$	23.2484	1.32255
$310$	0	0
$311$	7.48005	0.424155	0.212077	−	0.977253i	$-0.431977\pi$
$311$	7.48005	0.424155	0.212077	+	0.977253i	$0.431977\pi$
$312$	0	0
$313$	−3.21703	−0.181837	−0.0909187	−	0.995858i	$-0.528980\pi$
$313$	−3.21703	−0.181837	−0.0909187	+	0.995858i	$0.528980\pi$
$314$	0	0
$315$	3.06624	0.172763
$316$	0	0
$317$	−27.6127	−1.55088	−0.775442	−	0.631419i	$-0.782473\pi$
$317$	−27.6127	−1.55088	−0.775442	+	0.631419i	$0.782473\pi$
$318$	0	0
$319$	−2.98203	−0.166962
$320$	0	0
$321$	−28.6989	−1.60182
$322$	0	0
$323$	−1.05635	−0.0587767
$324$	0	0
$325$	−0.240299	−0.0133294
$326$	0	0
$327$	32.1020	1.77524
$328$	0	0
$329$	−7.75485	−0.427539
$330$	0	0
$331$	5.17146	0.284249	0.142125	−	0.989849i	$-0.454607\pi$
$331$	5.17146	0.284249	0.142125	+	0.989849i	$0.454607\pi$
$332$	0	0
$333$	5.70165	0.312449
$334$	0	0
$335$	−8.49034	−0.463877
$336$	0	0
$337$	−1.15395	−0.0628598	−0.0314299	−	0.999506i	$-0.510006\pi$
$337$	−1.15395	−0.0628598	−0.0314299	+	0.999506i	$0.510006\pi$
$338$	0	0
$339$	−41.9476	−2.27828
$340$	0	0
$341$	−9.84322	−0.533040
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−14.2387	−0.766586
$346$	0	0
$347$	19.4914	1.04635	0.523176	−	0.852225i	$-0.324747\pi$
$347$	19.4914	1.04635	0.523176	+	0.852225i	$0.324747\pi$
$348$	0	0
$349$	3.69104	0.197577	0.0987883	−	0.995108i	$-0.468503\pi$
$349$	3.69104	0.197577	0.0987883	+	0.995108i	$0.468503\pi$
$350$	0	0
$351$	3.34683	0.178640
$352$	0	0
$353$	11.0945	0.590499	0.295249	−	0.955420i	$-0.404597\pi$
$353$	11.0945	0.590499	0.295249	+	0.955420i	$0.404597\pi$
$354$	0	0
$355$	−19.2337	−1.02082
$356$	0	0
$357$	−1.76825	−0.0935857
$358$	0	0
$359$	7.43958	0.392646	0.196323	−	0.980539i	$-0.437100\pi$
$359$	7.43958	0.392646	0.196323	+	0.980539i	$0.437100\pi$
$360$	0	0
$361$	−17.4278	−0.917251
$362$	0	0
$363$	−2.09892	−0.110165
$364$	0	0
$365$	−13.1278	−0.687138
$366$	0	0
$367$	20.6734	1.07914	0.539571	−	0.841940i	$-0.318586\pi$
$367$	20.6734	1.07914	0.539571	+	0.841940i	$0.318586\pi$
$368$	0	0
$369$	−3.22690	−0.167986
$370$	0	0
$371$	−7.37693	−0.382991
$372$	0	0
$373$	0.213045	0.0110311	0.00551553	−	0.999985i	$-0.498244\pi$
$373$	0.213045	0.0110311	0.00551553	+	0.999985i	$0.498244\pi$
$374$	0	0
$375$	−23.9961	−1.23915
$376$	0	0
$377$	2.98203	0.153582
$378$	0	0
$379$	−37.3920	−1.92070	−0.960348	−	0.278803i	$-0.910062\pi$
$379$	−37.3920	−1.92070	−0.960348	+	0.278803i	$0.910062\pi$
$380$	0	0
$381$	1.20207	0.0615837
$382$	0	0
$383$	−15.5511	−0.794622	−0.397311	−	0.917684i	$-0.630057\pi$
$383$	−15.5511	−0.794622	−0.397311	+	0.917684i	$0.630057\pi$
$384$	0	0
$385$	−2.18167	−0.111188
$386$	0	0
$387$	−11.5180	−0.585495
$388$	0	0
$389$	18.1100	0.918212	0.459106	−	0.888382i	$-0.348170\pi$
$389$	18.1100	0.918212	0.459106	+	0.888382i	$0.348170\pi$
$390$	0	0
$391$	2.61959	0.132479
$392$	0	0
$393$	−1.37309	−0.0692633
$394$	0	0
$395$	26.3994	1.32830
$396$	0	0
$397$	−31.2580	−1.56879	−0.784397	−	0.620259i	$-0.787027\pi$
$397$	−31.2580	−1.56879	−0.784397	+	0.620259i	$0.787027\pi$
$398$	0	0
$399$	2.63180	0.131755
$400$	0	0
$401$	−10.3604	−0.517372	−0.258686	−	0.965961i	$-0.583290\pi$
$401$	−10.3604	−0.517372	−0.258686	+	0.965961i	$0.583290\pi$
$402$	0	0
$403$	9.84322	0.490326
$404$	0	0
$405$	24.5243	1.21862
$406$	0	0
$407$	−4.05681	−0.201089
$408$	0	0
$409$	17.4661	0.863645	0.431823	−	0.901959i	$-0.357871\pi$
$409$	17.4661	0.863645	0.431823	+	0.901959i	$0.357871\pi$
$410$	0	0
$411$	28.1511	1.38859
$412$	0	0
$413$	0.236896	0.0116569
$414$	0	0
$415$	−21.7824	−1.06925
$416$	0	0
$417$	−32.9867	−1.61537
$418$	0	0
$419$	−10.8450	−0.529815	−0.264908	−	0.964274i	$-0.585341\pi$
$419$	−10.8450	−0.529815	−0.264908	+	0.964274i	$0.585341\pi$
$420$	0	0
$421$	1.79517	0.0874911	0.0437455	−	0.999043i	$-0.486071\pi$
$421$	1.79517	0.0874911	0.0437455	+	0.999043i	$0.486071\pi$
$422$	0	0
$423$	10.8991	0.529930
$424$	0	0
$425$	0.202442	0.00981987
$426$	0	0
$427$	13.7710	0.666426
$428$	0	0
$429$	2.09892	0.101337
$430$	0	0
$431$	−29.3351	−1.41302	−0.706512	−	0.707701i	$-0.749732\pi$
$431$	−29.3351	−1.41302	−0.706512	+	0.707701i	$0.749732\pi$
$432$	0	0
$433$	−10.8762	−0.522676	−0.261338	−	0.965247i	$-0.584164\pi$
$433$	−10.8762	−0.522676	−0.261338	+	0.965247i	$0.584164\pi$
$434$	0	0
$435$	13.6552	0.654715
$436$	0	0
$437$	−3.89891	−0.186510
$438$	0	0
$439$	−4.83532	−0.230777	−0.115389	−	0.993320i	$-0.536811\pi$
$439$	−4.83532	−0.230777	−0.115389	+	0.993320i	$0.536811\pi$
$440$	0	0
$441$	1.40545	0.0669262
$442$	0	0
$443$	1.36903	0.0650446	0.0325223	−	0.999471i	$-0.489646\pi$
$443$	1.36903	0.0650446	0.0325223	+	0.999471i	$0.489646\pi$
$444$	0	0
$445$	−22.0858	−1.04697
$446$	0	0
$447$	34.3334	1.62391
$448$	0	0
$449$	−24.9115	−1.17565	−0.587823	−	0.808990i	$-0.700015\pi$
$449$	−24.9115	−1.17565	−0.587823	+	0.808990i	$0.700015\pi$
$450$	0	0
$451$	2.29599	0.108114
$452$	0	0
$453$	17.0540	0.801268
$454$	0	0
$455$	2.18167	0.102278
$456$	0	0
$457$	41.4281	1.93793	0.968963	−	0.247204i	$-0.0795119\pi$
$457$	41.4281	1.93793	0.968963	+	0.247204i	$0.0795119\pi$
$458$	0	0
$459$	−2.81956	−0.131606
$460$	0	0
$461$	−14.8920	−0.693589	−0.346795	−	0.937941i	$-0.612730\pi$
$461$	−14.8920	−0.693589	−0.346795	+	0.937941i	$0.612730\pi$
$462$	0	0
$463$	23.6909	1.10101	0.550505	−	0.834832i	$-0.314435\pi$
$463$	23.6909	1.10101	0.550505	+	0.834832i	$0.314435\pi$
$464$	0	0
$465$	45.0736	2.09024
$466$	0	0
$467$	−16.0419	−0.742331	−0.371166	−	0.928567i	$-0.621042\pi$
$467$	−16.0419	−0.742331	−0.371166	+	0.928567i	$0.621042\pi$
$468$	0	0
$469$	−3.89166	−0.179700
$470$	0	0
$471$	36.8784	1.69927
$472$	0	0
$473$	8.19526	0.376818
$474$	0	0
$475$	−0.301307	−0.0138249
$476$	0	0
$477$	10.3679	0.474714
$478$	0	0
$479$	−12.0795	−0.551926	−0.275963	−	0.961168i	$-0.588997\pi$
$479$	−12.0795	−0.551926	−0.275963	+	0.961168i	$0.588997\pi$
$480$	0	0
$481$	4.05681	0.184975
$482$	0	0
$483$	−6.52650	−0.296966
$484$	0	0
$485$	22.3762	1.01605
$486$	0	0
$487$	37.4604	1.69749	0.848745	−	0.528802i	$-0.177359\pi$
$487$	37.4604	1.69749	0.848745	+	0.528802i	$0.177359\pi$
$488$	0	0
$489$	47.1324	2.13140
$490$	0	0
$491$	29.7366	1.34199	0.670996	−	0.741461i	$-0.265867\pi$
$491$	29.7366	1.34199	0.670996	+	0.741461i	$0.265867\pi$
$492$	0	0
$493$	−2.51224	−0.113145
$494$	0	0
$495$	3.06624	0.137817
$496$	0	0
$497$	−8.81605	−0.395454
$498$	0	0
$499$	−11.0838	−0.496178	−0.248089	−	0.968737i	$-0.579803\pi$
$499$	−11.0838	−0.496178	−0.248089	+	0.968737i	$0.579803\pi$
$500$	0	0
$501$	−41.0990	−1.83617
$502$	0	0
$503$	24.4127	1.08851	0.544253	−	0.838921i	$-0.316813\pi$
$503$	24.4127	1.08851	0.544253	+	0.838921i	$0.316813\pi$
$504$	0	0
$505$	9.41331	0.418887
$506$	0	0
$507$	−2.09892	−0.0932162
$508$	0	0
$509$	−3.94696	−0.174946	−0.0874731	−	0.996167i	$-0.527879\pi$
$509$	−3.94696	−0.174946	−0.0874731	+	0.996167i	$0.527879\pi$
$510$	0	0
$511$	−6.01728	−0.266189
$512$	0	0
$513$	4.19654	0.185282
$514$	0	0
$515$	24.1650	1.06484
$516$	0	0
$517$	−7.75485	−0.341058
$518$	0	0
$519$	43.4480	1.90716
$520$	0	0
$521$	11.0421	0.483761	0.241881	−	0.970306i	$-0.422236\pi$
$521$	11.0421	0.483761	0.241881	+	0.970306i	$0.422236\pi$
$522$	0	0
$523$	−25.4603	−1.11330	−0.556651	−	0.830747i	$-0.687914\pi$
$523$	−25.4603	−1.11330	−0.556651	+	0.830747i	$0.687914\pi$
$524$	0	0
$525$	−0.504367	−0.0220124
$526$	0	0
$527$	−8.29250	−0.361227
$528$	0	0
$529$	−13.3312	−0.579619
$530$	0	0
$531$	−0.332945	−0.0144486
$532$	0	0
$533$	−2.29599	−0.0994503
$534$	0	0
$535$	−29.8305	−1.28968
$536$	0	0
$537$	38.1998	1.64844
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−10.1775	−0.437565	−0.218783	−	0.975774i	$-0.570209\pi$
$541$	−10.1775	−0.437565	−0.218783	+	0.975774i	$0.570209\pi$
$542$	0	0
$543$	37.0382	1.58946
$544$	0	0
$545$	33.3677	1.42932
$546$	0	0
$547$	27.1713	1.16176	0.580880	−	0.813989i	$-0.302709\pi$
$547$	27.1713	1.16176	0.580880	+	0.813989i	$0.302709\pi$
$548$	0	0
$549$	−19.3545	−0.826029
$550$	0	0
$551$	3.73912	0.159292
$552$	0	0
$553$	12.1005	0.514566
$554$	0	0
$555$	18.5768	0.788540
$556$	0	0
$557$	−0.350344	−0.0148445	−0.00742227	−	0.999972i	$-0.502363\pi$
$557$	−0.350344	−0.0148445	−0.00742227	+	0.999972i	$0.502363\pi$
$558$	0	0
$559$	−8.19526	−0.346623
$560$	0	0
$561$	−1.76825	−0.0746556
$562$	0	0
$563$	−26.8770	−1.13273	−0.566364	−	0.824155i	$-0.691650\pi$
$563$	−26.8770	−1.13273	−0.566364	+	0.824155i	$0.691650\pi$
$564$	0	0
$565$	−43.6015	−1.83433
$566$	0	0
$567$	11.2411	0.472080
$568$	0	0
$569$	22.9856	0.963607	0.481803	−	0.876279i	$-0.339982\pi$
$569$	22.9856	0.963607	0.481803	+	0.876279i	$0.339982\pi$
$570$	0	0
$571$	−14.9250	−0.624594	−0.312297	−	0.949985i	$-0.601098\pi$
$571$	−14.9250	−0.624594	−0.312297	+	0.949985i	$0.601098\pi$
$572$	0	0
$573$	30.2288	1.26283
$574$	0	0
$575$	0.747200	0.0311604
$576$	0	0
$577$	−37.5568	−1.56351	−0.781756	−	0.623585i	$-0.785676\pi$
$577$	−37.5568	−1.56351	−0.781756	+	0.623585i	$0.785676\pi$
$578$	0	0
$579$	−42.4851	−1.76562
$580$	0	0
$581$	−9.98424	−0.414216
$582$	0	0
$583$	−7.37693	−0.305521
$584$	0	0
$585$	−3.06624	−0.126773
$586$	0	0
$587$	−2.98931	−0.123382	−0.0616909	−	0.998095i	$-0.519649\pi$
$587$	−2.98931	−0.123382	−0.0616909	+	0.998095i	$0.519649\pi$
$588$	0	0
$589$	12.3423	0.508554
$590$	0	0
$591$	25.5532	1.05112
$592$	0	0
$593$	−17.7734	−0.729867	−0.364934	−	0.931033i	$-0.618908\pi$
$593$	−17.7734	−0.729867	−0.364934	+	0.931033i	$0.618908\pi$
$594$	0	0
$595$	−1.83797	−0.0753494
$596$	0	0
$597$	−54.6742	−2.23767
$598$	0	0
$599$	31.7262	1.29630	0.648148	−	0.761514i	$-0.275544\pi$
$599$	31.7262	1.29630	0.648148	+	0.761514i	$0.275544\pi$
$600$	0	0
$601$	32.9615	1.34453	0.672263	−	0.740312i	$-0.265322\pi$
$601$	32.9615	1.34453	0.672263	+	0.740312i	$0.265322\pi$
$602$	0	0
$603$	5.46954	0.222737
$604$	0	0
$605$	−2.18167	−0.0886977
$606$	0	0
$607$	30.8688	1.25292	0.626462	−	0.779452i	$-0.284502\pi$
$607$	30.8688	1.25292	0.626462	+	0.779452i	$0.284502\pi$
$608$	0	0
$609$	6.25903	0.253629
$610$	0	0
$611$	7.75485	0.313728
$612$	0	0
$613$	−20.0803	−0.811037	−0.405518	−	0.914087i	$-0.632909\pi$
$613$	−20.0803	−0.811037	−0.405518	+	0.914087i	$0.632909\pi$
$614$	0	0
$615$	−10.5137	−0.423952
$616$	0	0
$617$	−5.79803	−0.233420	−0.116710	−	0.993166i	$-0.537235\pi$
$617$	−5.79803	−0.233420	−0.116710	+	0.993166i	$0.537235\pi$
$618$	0	0
$619$	−38.8507	−1.56154	−0.780771	−	0.624817i	$-0.785174\pi$
$619$	−38.8507	−1.56154	−0.780771	+	0.624817i	$0.785174\pi$
$620$	0	0
$621$	−10.4068	−0.417612
$622$	0	0
$623$	−10.1233	−0.405583
$624$	0	0
$625$	−23.7408	−0.949630
$626$	0	0
$627$	2.63180	0.105104
$628$	0	0
$629$	−3.41770	−0.136273
$630$	0	0
$631$	14.4194	0.574027	0.287014	−	0.957927i	$-0.407337\pi$
$631$	14.4194	0.574027	0.287014	+	0.957927i	$0.407337\pi$
$632$	0	0
$633$	38.1977	1.51822
$634$	0	0
$635$	1.24946	0.0495834
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	12.3905	0.490161
$640$	0	0
$641$	−35.3170	−1.39494	−0.697468	−	0.716616i	$-0.745690\pi$
$641$	−35.3170	−1.39494	−0.697468	+	0.716616i	$0.745690\pi$
$642$	0	0
$643$	17.4606	0.688580	0.344290	−	0.938863i	$-0.388120\pi$
$643$	17.4606	0.688580	0.344290	+	0.938863i	$0.388120\pi$
$644$	0	0
$645$	−37.5273	−1.47764
$646$	0	0
$647$	24.3378	0.956816	0.478408	−	0.878138i	$-0.341214\pi$
$647$	24.3378	0.956816	0.478408	+	0.878138i	$0.341214\pi$
$648$	0	0
$649$	0.236896	0.00929897
$650$	0	0
$651$	20.6601	0.809733
$652$	0	0
$653$	38.3857	1.50215	0.751074	−	0.660218i	$-0.229536\pi$
$653$	38.3857	1.50215	0.751074	+	0.660218i	$0.229536\pi$
$654$	0	0
$655$	−1.42723	−0.0557665
$656$	0	0
$657$	8.45700	0.329939
$658$	0	0
$659$	18.8427	0.734006	0.367003	−	0.930220i	$-0.380384\pi$
$659$	18.8427	0.734006	0.367003	+	0.930220i	$0.380384\pi$
$660$	0	0
$661$	17.3512	0.674884	0.337442	−	0.941346i	$-0.390438\pi$
$661$	17.3512	0.674884	0.337442	+	0.941346i	$0.390438\pi$
$662$	0	0
$663$	1.76825	0.0686731
$664$	0	0
$665$	2.73557	0.106081
$666$	0	0
$667$	−9.27251	−0.359033
$668$	0	0
$669$	−23.8856	−0.923471
$670$	0	0
$671$	13.7710	0.531624
$672$	0	0
$673$	−48.9403	−1.88651	−0.943255	−	0.332070i	$-0.892253\pi$
$673$	−48.9403	−1.88651	−0.943255	+	0.332070i	$0.892253\pi$
$674$	0	0
$675$	−0.804239	−0.0309552
$676$	0	0
$677$	28.5569	1.09753	0.548765	−	0.835977i	$-0.315098\pi$
$677$	28.5569	1.09753	0.548765	+	0.835977i	$0.315098\pi$
$678$	0	0
$679$	10.2565	0.393607
$680$	0	0
$681$	41.2346	1.58012
$682$	0	0
$683$	−27.6545	−1.05817	−0.529086	−	0.848568i	$-0.677465\pi$
$683$	−27.6545	−1.05817	−0.529086	+	0.848568i	$0.677465\pi$
$684$	0	0
$685$	29.2610	1.11801
$686$	0	0
$687$	5.63640	0.215042
$688$	0	0
$689$	7.37693	0.281039
$690$	0	0
$691$	−3.20892	−0.122073	−0.0610364	−	0.998136i	$-0.519441\pi$
$691$	−3.20892	−0.122073	−0.0610364	+	0.998136i	$0.519441\pi$
$692$	0	0
$693$	1.40545	0.0533887
$694$	0	0
$695$	−34.2874	−1.30059
$696$	0	0
$697$	1.93427	0.0732659
$698$	0	0
$699$	50.0531	1.89318
$700$	0	0
$701$	42.3715	1.60035	0.800174	−	0.599768i	$-0.204740\pi$
$701$	42.3715	1.60035	0.800174	+	0.599768i	$0.204740\pi$
$702$	0	0
$703$	5.08678	0.191851
$704$	0	0
$705$	35.5106	1.33741
$706$	0	0
$707$	4.31472	0.162272
$708$	0	0
$709$	−15.4343	−0.579647	−0.289824	−	0.957080i	$-0.593597\pi$
$709$	−15.4343	−0.579647	−0.289824	+	0.957080i	$0.593597\pi$
$710$	0	0
$711$	−17.0067	−0.637800
$712$	0	0
$713$	−30.6071	−1.14625
$714$	0	0
$715$	2.18167	0.0815900
$716$	0	0
$717$	27.6050	1.03093
$718$	0	0
$719$	10.9417	0.408055	0.204027	−	0.978965i	$-0.434597\pi$
$719$	10.9417	0.408055	0.204027	+	0.978965i	$0.434597\pi$
$720$	0	0
$721$	11.0764	0.412506
$722$	0	0
$723$	47.0265	1.74893
$724$	0	0
$725$	−0.716578	−0.0266131
$726$	0	0
$727$	25.1263	0.931884	0.465942	−	0.884815i	$-0.345716\pi$
$727$	25.1263	0.931884	0.465942	+	0.884815i	$0.345716\pi$
$728$	0	0
$729$	5.27537	0.195384
$730$	0	0
$731$	6.90417	0.255360
$732$	0	0
$733$	8.05168	0.297396	0.148698	−	0.988883i	$-0.452492\pi$
$733$	8.05168	0.297396	0.148698	+	0.988883i	$0.452492\pi$
$734$	0	0
$735$	4.57915	0.168905
$736$	0	0
$737$	−3.89166	−0.143351
$738$	0	0
$739$	−6.19029	−0.227714	−0.113857	−	0.993497i	$-0.536321\pi$
$739$	−6.19029	−0.227714	−0.113857	+	0.993497i	$0.536321\pi$
$740$	0	0
$741$	−2.63180	−0.0966816
$742$	0	0
$743$	15.7238	0.576848	0.288424	−	0.957503i	$-0.406869\pi$
$743$	15.7238	0.576848	0.288424	+	0.957503i	$0.406869\pi$
$744$	0	0
$745$	35.6871	1.30747
$746$	0	0
$747$	14.0324	0.513417
$748$	0	0
$749$	−13.6732	−0.499608
$750$	0	0
$751$	−13.3699	−0.487875	−0.243937	−	0.969791i	$-0.578439\pi$
$751$	−13.3699	−0.487875	−0.243937	+	0.969791i	$0.578439\pi$
$752$	0	0
$753$	−49.4856	−1.80335
$754$	0	0
$755$	17.7264	0.645131
$756$	0	0
$757$	−21.1494	−0.768688	−0.384344	−	0.923190i	$-0.625572\pi$
$757$	−21.1494	−0.768688	−0.384344	+	0.923190i	$0.625572\pi$
$758$	0	0
$759$	−6.52650	−0.236897
$760$	0	0
$761$	−33.1135	−1.20037	−0.600183	−	0.799863i	$-0.704905\pi$
$761$	−33.1135	−1.20037	−0.600183	+	0.799863i	$0.704905\pi$
$762$	0	0
$763$	15.2945	0.553700
$764$	0	0
$765$	2.58318	0.0933949
$766$	0	0
$767$	−0.236896	−0.00855381
$768$	0	0
$769$	−18.3596	−0.662064	−0.331032	−	0.943620i	$-0.607397\pi$
$769$	−18.3596	−0.662064	−0.331032	+	0.943620i	$0.607397\pi$
$770$	0	0
$771$	−62.8261	−2.26263
$772$	0	0
$773$	16.6759	0.599790	0.299895	−	0.953972i	$-0.403048\pi$
$773$	16.6759	0.599790	0.299895	+	0.953972i	$0.403048\pi$
$774$	0	0
$775$	−2.36531	−0.0849646
$776$	0	0
$777$	8.51491	0.305471
$778$	0	0
$779$	−2.87891	−0.103148
$780$	0	0
$781$	−8.81605	−0.315463
$782$	0	0
$783$	9.98033	0.356668
$784$	0	0
$785$	38.3324	1.36814
$786$	0	0
$787$	−23.8476	−0.850075	−0.425038	−	0.905176i	$-0.639739\pi$
$787$	−23.8476	−0.850075	−0.425038	+	0.905176i	$0.639739\pi$
$788$	0	0
$789$	10.5772	0.376558
$790$	0	0
$791$	−19.9854	−0.710598
$792$	0	0
$793$	−13.7710	−0.489023
$794$	0	0
$795$	33.7801	1.19806
$796$	0	0
$797$	−29.9192	−1.05979	−0.529896	−	0.848062i	$-0.677769\pi$
$797$	−29.9192	−1.05979	−0.529896	+	0.848062i	$0.677769\pi$
$798$	0	0
$799$	−6.53314	−0.231126
$800$	0	0
$801$	14.2279	0.502717
$802$	0	0
$803$	−6.01728	−0.212345
$804$	0	0
$805$	−6.78383	−0.239099
$806$	0	0
$807$	−48.7050	−1.71450
$808$	0	0
$809$	3.42618	0.120458	0.0602290	−	0.998185i	$-0.480817\pi$
$809$	3.42618	0.120458	0.0602290	+	0.998185i	$0.480817\pi$
$810$	0	0
$811$	23.3573	0.820184	0.410092	−	0.912044i	$-0.365497\pi$
$811$	23.3573	0.820184	0.410092	+	0.912044i	$0.365497\pi$
$812$	0	0
$813$	12.2355	0.429119
$814$	0	0
$815$	48.9908	1.71607
$816$	0	0
$817$	−10.2759	−0.359509
$818$	0	0
$819$	−1.40545	−0.0491104
$820$	0	0
$821$	40.5031	1.41357	0.706785	−	0.707429i	$-0.250145\pi$
$821$	40.5031	1.41357	0.706785	+	0.707429i	$0.250145\pi$
$822$	0	0
$823$	−43.7647	−1.52554	−0.762770	−	0.646670i	$-0.776161\pi$
$823$	−43.7647	−1.52554	−0.762770	+	0.646670i	$0.776161\pi$
$824$	0	0
$825$	−0.504367	−0.0175598
$826$	0	0
$827$	−26.4185	−0.918660	−0.459330	−	0.888266i	$-0.651910\pi$
$827$	−26.4185	−0.918660	−0.459330	+	0.888266i	$0.651910\pi$
$828$	0	0
$829$	−32.0739	−1.11397	−0.556987	−	0.830521i	$-0.688043\pi$
$829$	−32.0739	−1.11397	−0.556987	+	0.830521i	$0.688043\pi$
$830$	0	0
$831$	−14.5320	−0.504111
$832$	0	0
$833$	−0.842458	−0.0291895
$834$	0	0
$835$	−42.7194	−1.47837
$836$	0	0
$837$	32.9435	1.13870
$838$	0	0
$839$	51.9409	1.79320	0.896600	−	0.442841i	$-0.146029\pi$
$839$	51.9409	1.79320	0.896600	+	0.442841i	$0.146029\pi$
$840$	0	0
$841$	−20.1075	−0.693362
$842$	0	0
$843$	30.5814	1.05328
$844$	0	0
$845$	−2.18167	−0.0750519
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−33.8745	−1.16257
$850$	0	0
$851$	−12.6145	−0.432420
$852$	0	0
$853$	10.7414	0.367777	0.183889	−	0.982947i	$-0.441131\pi$
$853$	10.7414	0.367777	0.183889	+	0.982947i	$0.441131\pi$
$854$	0	0
$855$	−3.84471	−0.131486
$856$	0	0
$857$	50.0792	1.71067	0.855337	−	0.518072i	$-0.173350\pi$
$857$	50.0792	1.71067	0.855337	+	0.518072i	$0.173350\pi$
$858$	0	0
$859$	−56.2222	−1.91828	−0.959138	−	0.282937i	$-0.908691\pi$
$859$	−56.2222	−1.91828	−0.959138	+	0.282937i	$0.908691\pi$
$860$	0	0
$861$	−4.81909	−0.164234
$862$	0	0
$863$	35.6050	1.21201	0.606004	−	0.795462i	$-0.292772\pi$
$863$	35.6050	1.21201	0.606004	+	0.795462i	$0.292772\pi$
$864$	0	0
$865$	45.1611	1.53552
$866$	0	0
$867$	34.1919	1.16122
$868$	0	0
$869$	12.1005	0.410482
$870$	0	0
$871$	3.89166	0.131864
$872$	0	0
$873$	−14.4149	−0.487872
$874$	0	0
$875$	−11.4326	−0.386493
$876$	0	0
$877$	−29.0880	−0.982231	−0.491115	−	0.871095i	$-0.663411\pi$
$877$	−29.0880	−0.982231	−0.491115	+	0.871095i	$0.663411\pi$
$878$	0	0
$879$	34.8040	1.17391
$880$	0	0
$881$	47.1828	1.58963	0.794815	−	0.606852i	$-0.207568\pi$
$881$	47.1828	1.58963	0.794815	+	0.606852i	$0.207568\pi$
$882$	0	0
$883$	11.0815	0.372922	0.186461	−	0.982462i	$-0.440298\pi$
$883$	11.0815	0.372922	0.186461	+	0.982462i	$0.440298\pi$
$884$	0	0
$885$	−1.08478	−0.0364645
$886$	0	0
$887$	−14.9112	−0.500668	−0.250334	−	0.968160i	$-0.580540\pi$
$887$	−14.9112	−0.500668	−0.250334	+	0.968160i	$0.580540\pi$
$888$	0	0
$889$	0.572708	0.0192080
$890$	0	0
$891$	11.2411	0.376590
$892$	0	0
$893$	9.72369	0.325391
$894$	0	0
$895$	39.7060	1.32723
$896$	0	0
$897$	6.52650	0.217914
$898$	0	0
$899$	29.3528	0.978970
$900$	0	0
$901$	−6.21476	−0.207044
$902$	0	0
$903$	−17.2012	−0.572419
$904$	0	0
$905$	38.4985	1.27974
$906$	0	0
$907$	53.5567	1.77832	0.889160	−	0.457596i	$-0.151289\pi$
$907$	53.5567	1.77832	0.889160	+	0.457596i	$0.151289\pi$
$908$	0	0
$909$	−6.06413	−0.201134
$910$	0	0
$911$	−13.1677	−0.436264	−0.218132	−	0.975919i	$-0.569996\pi$
$911$	−13.1677	−0.436264	−0.218132	+	0.975919i	$0.569996\pi$
$912$	0	0
$913$	−9.98424	−0.330430
$914$	0	0
$915$	−63.0595	−2.08468
$916$	0	0
$917$	−0.654190	−0.0216033
$918$	0	0
$919$	46.6671	1.53941	0.769703	−	0.638403i	$-0.220404\pi$
$919$	46.6671	1.53941	0.769703	+	0.638403i	$0.220404\pi$
$920$	0	0
$921$	3.88800	0.128114
$922$	0	0
$923$	8.81605	0.290184
$924$	0	0
$925$	−0.974848	−0.0320528
$926$	0	0
$927$	−15.5673	−0.511297
$928$	0	0
$929$	−26.7137	−0.876449	−0.438225	−	0.898865i	$-0.644393\pi$
$929$	−26.7137	−0.876449	−0.438225	+	0.898865i	$0.644393\pi$
$930$	0	0
$931$	1.25389	0.0410944
$932$	0	0
$933$	−15.7000	−0.513995
$934$	0	0
$935$	−1.83797	−0.0601080
$936$	0	0
$937$	−19.1013	−0.624012	−0.312006	−	0.950080i	$-0.601001\pi$
$937$	−19.1013	−0.624012	−0.312006	+	0.950080i	$0.601001\pi$
$938$	0	0
$939$	6.75228	0.220352
$940$	0	0
$941$	−5.49846	−0.179245	−0.0896223	−	0.995976i	$-0.528566\pi$
$941$	−5.49846	−0.179245	−0.0896223	+	0.995976i	$0.528566\pi$
$942$	0	0
$943$	7.13929	0.232487
$944$	0	0
$945$	7.30168	0.237524
$946$	0	0
$947$	1.77869	0.0577996	0.0288998	−	0.999582i	$-0.490800\pi$
$947$	1.77869	0.0577996	0.0288998	+	0.999582i	$0.490800\pi$
$948$	0	0
$949$	6.01728	0.195329
$950$	0	0
$951$	57.9568	1.87938
$952$	0	0
$953$	−36.5375	−1.18357	−0.591784	−	0.806097i	$-0.701576\pi$
$953$	−36.5375	−1.18357	−0.591784	+	0.806097i	$0.701576\pi$
$954$	0	0
$955$	31.4207	1.01675
$956$	0	0
$957$	6.25903	0.202326
$958$	0	0
$959$	13.4122	0.433102
$960$	0	0
$961$	65.8890	2.12545
$962$	0	0
$963$	19.2170	0.619260
$964$	0	0
$965$	−44.1603	−1.42157
$966$	0	0
$967$	54.2809	1.74556	0.872778	−	0.488118i	$-0.162316\pi$
$967$	54.2809	1.74556	0.872778	+	0.488118i	$0.162316\pi$
$968$	0	0
$969$	2.21718	0.0712262
$970$	0	0
$971$	−50.5366	−1.62180	−0.810899	−	0.585186i	$-0.801021\pi$
$971$	−50.5366	−1.62180	−0.810899	+	0.585186i	$0.801021\pi$
$972$	0	0
$973$	−15.7161	−0.503834
$974$	0	0
$975$	0.504367	0.0161527
$976$	0	0
$977$	−22.2222	−0.710950	−0.355475	−	0.934686i	$-0.615681\pi$
$977$	−22.2222	−0.710950	−0.355475	+	0.934686i	$0.615681\pi$
$978$	0	0
$979$	−10.1233	−0.323543
$980$	0	0
$981$	−21.4957	−0.686306
$982$	0	0
$983$	−2.56705	−0.0818763	−0.0409382	−	0.999162i	$-0.513035\pi$
$983$	−2.56705	−0.0818763	−0.0409382	+	0.999162i	$0.513035\pi$
$984$	0	0
$985$	26.5607	0.846294
$986$	0	0
$987$	16.2768	0.518096
$988$	0	0
$989$	25.4829	0.810308
$990$	0	0
$991$	16.6817	0.529912	0.264956	−	0.964261i	$-0.414643\pi$
$991$	16.6817	0.529912	0.264956	+	0.964261i	$0.414643\pi$
$992$	0	0
$993$	−10.8545	−0.344456
$994$	0	0
$995$	−56.8299	−1.80163
$996$	0	0
$997$	−22.0471	−0.698240	−0.349120	−	0.937078i	$-0.613519\pi$
$997$	−22.0471	−0.698240	−0.349120	+	0.937078i	$0.613519\pi$
$998$	0	0
$999$	13.5774	0.429571

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.f.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.f.1.1	✓	5	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q-2.09892 q^{3} -2.18167 q^{5} -1.00000 q^{7} +1.40545 q^{9} +O(q^{10})\)	\(q-2.09892 q^{3} -2.18167 q^{5} -1.00000 q^{7} +1.40545 q^{9} -1.00000 q^{11} +1.00000 q^{13} +4.57915 q^{15} -0.842458 q^{17} +1.25389 q^{19} +2.09892 q^{21} -3.10946 q^{23} -0.240299 q^{25} +3.34683 q^{27} +2.98203 q^{29} +9.84322 q^{31} +2.09892 q^{33} +2.18167 q^{35} +4.05681 q^{37} -2.09892 q^{39} -2.29599 q^{41} -8.19526 q^{43} -3.06624 q^{45} +7.75485 q^{47} +1.00000 q^{49} +1.76825 q^{51} +7.37693 q^{53} +2.18167 q^{55} -2.63180 q^{57} -0.236896 q^{59} -13.7710 q^{61} -1.40545 q^{63} -2.18167 q^{65} +3.89166 q^{67} +6.52650 q^{69} +8.81605 q^{71} +6.01728 q^{73} +0.504367 q^{75} +1.00000 q^{77} -12.1005 q^{79} -11.2411 q^{81} +9.98424 q^{83} +1.83797 q^{85} -6.25903 q^{87} +10.1233 q^{89} -1.00000 q^{91} -20.6601 q^{93} -2.73557 q^{95} -10.2565 q^{97} -1.40545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - 5 q^{7} + 2 q^{9}+O(q^{10}) \) 5 * q + 3 * q^3 - 5 * q^7 + 2 * q^9	\( 5 q + 3 q^{3} - 5 q^{7} + 2 q^{9} - 5 q^{11} + 5 q^{13} - 3 q^{15} - 9 q^{17} - 4 q^{19} - 3 q^{21} - 4 q^{23} + q^{25} + 3 q^{27} - 8 q^{29} + 7 q^{31} - 3 q^{33} - 10 q^{37} + 3 q^{39} - 18 q^{41} - 22 q^{43} - 26 q^{45} + 12 q^{47} + 5 q^{49} - 7 q^{51} + 7 q^{53} - 6 q^{57} - q^{59} - 26 q^{61} - 2 q^{63} - 8 q^{67} - 12 q^{69} + 18 q^{71} - 25 q^{73} - 16 q^{75} + 5 q^{77} - 6 q^{79} - 7 q^{81} + 11 q^{83} - 7 q^{85} - 5 q^{87} - 14 q^{89} - 5 q^{91} - 41 q^{93} - 22 q^{95} - 33 q^{97} - 2 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - 5 * q^7 + 2 * q^9 - 5 * q^11 + 5 * q^13 - 3 * q^15 - 9 * q^17 - 4 * q^19 - 3 * q^21 - 4 * q^23 + q^25 + 3 * q^27 - 8 * q^29 + 7 * q^31 - 3 * q^33 - 10 * q^37 + 3 * q^39 - 18 * q^41 - 22 * q^43 - 26 * q^45 + 12 * q^47 + 5 * q^49 - 7 * q^51 + 7 * q^53 - 6 * q^57 - q^59 - 26 * q^61 - 2 * q^63 - 8 * q^67 - 12 * q^69 + 18 * q^71 - 25 * q^73 - 16 * q^75 + 5 * q^77 - 6 * q^79 - 7 * q^81 + 11 * q^83 - 7 * q^85 - 5 * q^87 - 14 * q^89 - 5 * q^91 - 41 * q^93 - 22 * q^95 - 33 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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