LMFDB - Embedded newform 4005.2.a.g.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
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			1.1
Root			$-1.56155$ 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)

$f(q)$	$=$	$q-1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} +2.43845 q^{8} +O(q^{10})$	$q-1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} +2.43845 q^{8} +1.56155 q^{10} -4.00000 q^{11} +2.00000 q^{13} -4.68466 q^{16} -7.12311 q^{17} -5.12311 q^{19} -0.438447 q^{20} +6.24621 q^{22} +1.43845 q^{23} +1.00000 q^{25} -3.12311 q^{26} -9.68466 q^{29} -1.12311 q^{31} +2.43845 q^{32} +11.1231 q^{34} +10.0000 q^{37} +8.00000 q^{38} -2.43845 q^{40} +9.68466 q^{41} -6.24621 q^{43} -1.75379 q^{44} -2.24621 q^{46} +8.00000 q^{47} -7.00000 q^{49} -1.56155 q^{50} +0.876894 q^{52} -8.24621 q^{53} +4.00000 q^{55} +15.1231 q^{58} -6.56155 q^{59} +11.1231 q^{61} +1.75379 q^{62} +5.56155 q^{64} -2.00000 q^{65} +4.80776 q^{67} -3.12311 q^{68} +10.2462 q^{71} -9.68466 q^{73} -15.6155 q^{74} -2.24621 q^{76} -6.56155 q^{79} +4.68466 q^{80} -15.1231 q^{82} +7.68466 q^{83} +7.12311 q^{85} +9.75379 q^{86} -9.75379 q^{88} -1.00000 q^{89} +0.630683 q^{92} -12.4924 q^{94} +5.12311 q^{95} -12.5616 q^{97} +10.9309 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 5 q^{4} - 2 q^{5} + 9 q^{8}+O(q^{10}) $ 2 * q + q^2 + 5 * q^4 - 2 * q^5 + 9 * q^8	$ 2 q + q^{2} + 5 q^{4} - 2 q^{5} + 9 q^{8} - q^{10} - 8 q^{11} + 4 q^{13} + 3 q^{16} - 6 q^{17} - 2 q^{19} - 5 q^{20} - 4 q^{22} + 7 q^{23} + 2 q^{25} + 2 q^{26} - 7 q^{29} + 6 q^{31} + 9 q^{32} + 14 q^{34} + 20 q^{37} + 16 q^{38} - 9 q^{40} + 7 q^{41} + 4 q^{43} - 20 q^{44} + 12 q^{46} + 16 q^{47} - 14 q^{49} + q^{50} + 10 q^{52} + 8 q^{55} + 22 q^{58} - 9 q^{59} + 14 q^{61} + 20 q^{62} + 7 q^{64} - 4 q^{65} - 11 q^{67} + 2 q^{68} + 4 q^{71} - 7 q^{73} + 10 q^{74} + 12 q^{76} - 9 q^{79} - 3 q^{80} - 22 q^{82} + 3 q^{83} + 6 q^{85} + 36 q^{86} - 36 q^{88} - 2 q^{89} + 26 q^{92} + 8 q^{94} + 2 q^{95} - 21 q^{97} - 7 q^{98}+O(q^{100}) $ 2 * q + q^2 + 5 * q^4 - 2 * q^5 + 9 * q^8 - q^10 - 8 * q^11 + 4 * q^13 + 3 * q^16 - 6 * q^17 - 2 * q^19 - 5 * q^20 - 4 * q^22 + 7 * q^23 + 2 * q^25 + 2 * q^26 - 7 * q^29 + 6 * q^31 + 9 * q^32 + 14 * q^34 + 20 * q^37 + 16 * q^38 - 9 * q^40 + 7 * q^41 + 4 * q^43 - 20 * q^44 + 12 * q^46 + 16 * q^47 - 14 * q^49 + q^50 + 10 * q^52 + 8 * q^55 + 22 * q^58 - 9 * q^59 + 14 * q^61 + 20 * q^62 + 7 * q^64 - 4 * q^65 - 11 * q^67 + 2 * q^68 + 4 * q^71 - 7 * q^73 + 10 * q^74 + 12 * q^76 - 9 * q^79 - 3 * q^80 - 22 * q^82 + 3 * q^83 + 6 * q^85 + 36 * q^86 - 36 * q^88 - 2 * q^89 + 26 * q^92 + 8 * q^94 + 2 * q^95 - 21 * q^97 - 7 * 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 $

$2$	−1.56155	−1.10418	−0.552092	−	0.833783i	$-0.686170\pi$
$2$	−1.56155	−1.10418	−0.552092	+	0.833783i	$0.686170\pi$
$3$	0	0
$3$	0	0
$4$	0.438447	0.219224
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	2.43845	0.862121
$9$	0	0
$10$	1.56155	0.493806
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	−4.68466	−1.17116
$17$	−7.12311	−1.72761	−0.863803	−	0.503829i	$-0.831924\pi$
$17$	−7.12311	−1.72761	−0.863803	+	0.503829i	$0.831924\pi$
$18$	0	0
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	−0.438447	−0.0980398
$21$	0	0
$22$	6.24621	1.33170
$23$	1.43845	0.299937	0.149968	−	0.988691i	$-0.452083\pi$
$23$	1.43845	0.299937	0.149968	+	0.988691i	$0.452083\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.12311	−0.612491
$27$	0	0
$28$	0	0
$29$	−9.68466	−1.79840	−0.899198	−	0.437542i	$-0.855849\pi$
$29$	−9.68466	−1.79840	−0.899198	+	0.437542i	$0.855849\pi$
$30$	0	0
$31$	−1.12311	−0.201716	−0.100858	−	0.994901i	$-0.532159\pi$
$31$	−1.12311	−0.201716	−0.100858	+	0.994901i	$0.532159\pi$
$32$	2.43845	0.431061
$33$	0	0
$34$	11.1231	1.90760
$35$	0	0
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	−2.43845	−0.385552
$41$	9.68466	1.51249	0.756245	−	0.654289i	$-0.227032\pi$
$41$	9.68466	1.51249	0.756245	+	0.654289i	$0.227032\pi$
$42$	0	0
$43$	−6.24621	−0.952538	−0.476269	−	0.879300i	$-0.658011\pi$
$43$	−6.24621	−0.952538	−0.476269	+	0.879300i	$0.658011\pi$
$44$	−1.75379	−0.264394
$45$	0	0
$46$	−2.24621	−0.331186
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	−1.56155	−0.220837
$51$	0	0
$52$	0.876894	0.121603
$53$	−8.24621	−1.13270	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	−8.24621	−1.13270	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	0	0
$58$	15.1231	1.98576
$59$	−6.56155	−0.854241	−0.427121	−	0.904195i	$-0.640472\pi$
$59$	−6.56155	−0.854241	−0.427121	+	0.904195i	$0.640472\pi$
$60$	0	0
$61$	11.1231	1.42417	0.712084	−	0.702094i	$-0.247752\pi$
$61$	11.1231	1.42417	0.712084	+	0.702094i	$0.247752\pi$
$62$	1.75379	0.222731
$63$	0	0
$64$	5.56155	0.695194
$65$	−2.00000	−0.248069
$66$	0	0
$67$	4.80776	0.587362	0.293681	−	0.955904i	$-0.405120\pi$
$67$	4.80776	0.587362	0.293681	+	0.955904i	$0.405120\pi$
$68$	−3.12311	−0.378732
$69$	0	0
$70$	0	0
$71$	10.2462	1.21600	0.608001	−	0.793936i	$-0.291972\pi$
$71$	10.2462	1.21600	0.608001	+	0.793936i	$0.291972\pi$
$72$	0	0
$73$	−9.68466	−1.13350	−0.566752	−	0.823889i	$-0.691800\pi$
$73$	−9.68466	−1.13350	−0.566752	+	0.823889i	$0.691800\pi$
$74$	−15.6155	−1.81527
$75$	0	0
$76$	−2.24621	−0.257658
$77$	0	0
$78$	0	0
$79$	−6.56155	−0.738232	−0.369116	−	0.929383i	$-0.620340\pi$
$79$	−6.56155	−0.738232	−0.369116	+	0.929383i	$0.620340\pi$
$80$	4.68466	0.523761
$81$	0	0
$82$	−15.1231	−1.67007
$83$	7.68466	0.843501	0.421750	−	0.906712i	$-0.361416\pi$
$83$	7.68466	0.843501	0.421750	+	0.906712i	$0.361416\pi$
$84$	0	0
$85$	7.12311	0.772609
$86$	9.75379	1.05178
$87$	0	0
$88$	−9.75379	−1.03976
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	0	0
$92$	0.630683	0.0657533
$93$	0	0
$94$	−12.4924	−1.28849
$95$	5.12311	0.525620
$96$	0	0
$97$	−12.5616	−1.27543	−0.637716	−	0.770271i	$-0.720121\pi$
$97$	−12.5616	−1.27543	−0.637716	+	0.770271i	$0.720121\pi$
$98$	10.9309	1.10418
$99$	0	0
$100$	0.438447	0.0438447
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−13.1231	−1.29306	−0.646529	−	0.762889i	$-0.723780\pi$
$103$	−13.1231	−1.29306	−0.646529	+	0.762889i	$0.723780\pi$
$104$	4.87689	0.478219
$105$	0	0
$106$	12.8769	1.25071
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−8.56155	−0.820048	−0.410024	−	0.912075i	$-0.634480\pi$
$109$	−8.56155	−0.820048	−0.410024	+	0.912075i	$0.634480\pi$
$110$	−6.24621	−0.595553
$111$	0	0
$112$	0	0
$113$	2.80776	0.264132	0.132066	−	0.991241i	$-0.457839\pi$
$113$	2.80776	0.264132	0.132066	+	0.991241i	$0.457839\pi$
$114$	0	0
$115$	−1.43845	−0.134136
$116$	−4.24621	−0.394251
$117$	0	0
$118$	10.2462	0.943240
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	−17.3693	−1.57254
$123$	0	0
$124$	−0.492423	−0.0442208
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.2462	−0.909204	−0.454602	−	0.890695i	$-0.650219\pi$
$127$	−10.2462	−0.909204	−0.454602	+	0.890695i	$0.650219\pi$
$128$	−13.5616	−1.19868
$129$	0	0
$130$	3.12311	0.273914
$131$	11.3693	0.993342	0.496671	−	0.867939i	$-0.334556\pi$
$131$	11.3693	0.993342	0.496671	+	0.867939i	$0.334556\pi$
$132$	0	0
$133$	0	0
$134$	−7.50758	−0.648556
$135$	0	0
$136$	−17.3693	−1.48941
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	10.5616	0.895819	0.447910	−	0.894079i	$-0.352169\pi$
$139$	10.5616	0.895819	0.447910	+	0.894079i	$0.352169\pi$
$140$	0	0
$141$	0	0
$142$	−16.0000	−1.34269
$143$	−8.00000	−0.668994
$144$	0	0
$145$	9.68466	0.804267
$146$	15.1231	1.25160
$147$	0	0
$148$	4.38447	0.360401
$149$	21.0540	1.72481	0.862404	−	0.506220i	$-0.168958\pi$
$149$	21.0540	1.72481	0.862404	+	0.506220i	$0.168958\pi$
$150$	0	0
$151$	6.24621	0.508309	0.254155	−	0.967164i	$-0.418203\pi$
$151$	6.24621	0.508309	0.254155	+	0.967164i	$0.418203\pi$
$152$	−12.4924	−1.01327
$153$	0	0
$154$	0	0
$155$	1.12311	0.0902100
$156$	0	0
$157$	17.6847	1.41139	0.705695	−	0.708516i	$-0.250635\pi$
$157$	17.6847	1.41139	0.705695	+	0.708516i	$0.250635\pi$
$158$	10.2462	0.815145
$159$	0	0
$160$	−2.43845	−0.192776
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	4.24621	0.331573
$165$	0	0
$166$	−12.0000	−0.931381
$167$	−7.36932	−0.570255	−0.285127	−	0.958490i	$-0.592036\pi$
$167$	−7.36932	−0.570255	−0.285127	+	0.958490i	$0.592036\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−11.1231	−0.853103
$171$	0	0
$172$	−2.73863	−0.208819
$173$	15.1231	1.14979	0.574894	−	0.818228i	$-0.305043\pi$
$173$	15.1231	1.14979	0.574894	+	0.818228i	$0.305043\pi$
$174$	0	0
$175$	0	0
$176$	18.7386	1.41248
$177$	0	0
$178$	1.56155	0.117043
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−9.36932	−0.696416	−0.348208	−	0.937417i	$-0.613210\pi$
$181$	−9.36932	−0.696416	−0.348208	+	0.937417i	$0.613210\pi$
$182$	0	0
$183$	0	0
$184$	3.50758	0.258582
$185$	−10.0000	−0.735215
$186$	0	0
$187$	28.4924	2.08357
$188$	3.50758	0.255816
$189$	0	0
$190$	−8.00000	−0.580381
$191$	23.6847	1.71376	0.856881	−	0.515514i	$-0.172399\pi$
$191$	23.6847	1.71376	0.856881	+	0.515514i	$0.172399\pi$
$192$	0	0
$193$	0.876894	0.0631202	0.0315601	−	0.999502i	$-0.489952\pi$
$193$	0.876894	0.0631202	0.0315601	+	0.999502i	$0.489952\pi$
$194$	19.6155	1.40831
$195$	0	0
$196$	−3.06913	−0.219224
$197$	0.246211	0.0175418	0.00877091	−	0.999962i	$-0.497208\pi$
$197$	0.246211	0.0175418	0.00877091	+	0.999962i	$0.497208\pi$
$198$	0	0
$199$	24.8078	1.75858	0.879288	−	0.476291i	$-0.158019\pi$
$199$	24.8078	1.75858	0.879288	+	0.476291i	$0.158019\pi$
$200$	2.43845	0.172424
$201$	0	0
$202$	9.36932	0.659223
$203$	0	0
$204$	0	0
$205$	−9.68466	−0.676406
$206$	20.4924	1.42777
$207$	0	0
$208$	−9.36932	−0.649645
$209$	20.4924	1.41749
$210$	0	0
$211$	5.75379	0.396107	0.198054	−	0.980191i	$-0.436538\pi$
$211$	5.75379	0.396107	0.198054	+	0.980191i	$0.436538\pi$
$212$	−3.61553	−0.248315
$213$	0	0
$214$	6.24621	0.426982
$215$	6.24621	0.425988
$216$	0	0
$217$	0	0
$218$	13.3693	0.905484
$219$	0	0
$220$	1.75379	0.118240
$221$	−14.2462	−0.958304
$222$	0	0
$223$	−1.43845	−0.0963255	−0.0481628	−	0.998840i	$-0.515337\pi$
$223$	−1.43845	−0.0963255	−0.0481628	+	0.998840i	$0.515337\pi$
$224$	0	0
$225$	0	0
$226$	−4.38447	−0.291651
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−17.3693	−1.14780	−0.573898	−	0.818927i	$-0.694570\pi$
$229$	−17.3693	−1.14780	−0.573898	+	0.818927i	$0.694570\pi$
$230$	2.24621	0.148111
$231$	0	0
$232$	−23.6155	−1.55044
$233$	−23.1231	−1.51485	−0.757423	−	0.652925i	$-0.773542\pi$
$233$	−23.1231	−1.51485	−0.757423	+	0.652925i	$0.773542\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	−2.87689	−0.187270
$237$	0	0
$238$	0	0
$239$	12.8078	0.828465	0.414233	−	0.910171i	$-0.364050\pi$
$239$	12.8078	0.828465	0.414233	+	0.910171i	$0.364050\pi$
$240$	0	0
$241$	7.12311	0.458840	0.229420	−	0.973328i	$-0.426317\pi$
$241$	7.12311	0.458840	0.229420	+	0.973328i	$0.426317\pi$
$242$	−7.80776	−0.501902
$243$	0	0
$244$	4.87689	0.312211
$245$	7.00000	0.447214
$246$	0	0
$247$	−10.2462	−0.651951
$248$	−2.73863	−0.173903
$249$	0	0
$250$	1.56155	0.0987613
$251$	3.36932	0.212669	0.106335	−	0.994330i	$-0.466089\pi$
$251$	3.36932	0.212669	0.106335	+	0.994330i	$0.466089\pi$
$252$	0	0
$253$	−5.75379	−0.361738
$254$	16.0000	1.00393
$255$	0	0
$256$	10.0540	0.628373
$257$	−15.1231	−0.943353	−0.471677	−	0.881772i	$-0.656351\pi$
$257$	−15.1231	−0.943353	−0.471677	+	0.881772i	$0.656351\pi$
$258$	0	0
$259$	0	0
$260$	−0.876894	−0.0543827
$261$	0	0
$262$	−17.7538	−1.09683
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	8.24621	0.506561
$266$	0	0
$267$	0	0
$268$	2.10795	0.128764
$269$	−2.49242	−0.151966	−0.0759828	−	0.997109i	$-0.524209\pi$
$269$	−2.49242	−0.151966	−0.0759828	+	0.997109i	$0.524209\pi$
$270$	0	0
$271$	22.7386	1.38127	0.690637	−	0.723202i	$-0.257330\pi$
$271$	22.7386	1.38127	0.690637	+	0.723202i	$0.257330\pi$
$272$	33.3693	2.02331
$273$	0	0
$274$	−15.6155	−0.943369
$275$	−4.00000	−0.241209
$276$	0	0
$277$	27.9309	1.67820	0.839102	−	0.543975i	$-0.183081\pi$
$277$	27.9309	1.67820	0.839102	+	0.543975i	$0.183081\pi$
$278$	−16.4924	−0.989150
$279$	0	0
$280$	0	0
$281$	−4.24621	−0.253308	−0.126654	−	0.991947i	$-0.540424\pi$
$281$	−4.24621	−0.253308	−0.126654	+	0.991947i	$0.540424\pi$
$282$	0	0
$283$	14.2462	0.846849	0.423425	−	0.905931i	$-0.360828\pi$
$283$	14.2462	0.846849	0.423425	+	0.905931i	$0.360828\pi$
$284$	4.49242	0.266576
$285$	0	0
$286$	12.4924	0.738692
$287$	0	0
$288$	0	0
$289$	33.7386	1.98463
$290$	−15.1231	−0.888059
$291$	0	0
$292$	−4.24621	−0.248491
$293$	32.4233	1.89419	0.947094	−	0.320955i	$-0.104004\pi$
$293$	32.4233	1.89419	0.947094	+	0.320955i	$0.104004\pi$
$294$	0	0
$295$	6.56155	0.382028
$296$	24.3845	1.41732
$297$	0	0
$298$	−32.8769	−1.90451
$299$	2.87689	0.166375
$300$	0	0
$301$	0	0
$302$	−9.75379	−0.561267
$303$	0	0
$304$	24.0000	1.37649
$305$	−11.1231	−0.636907
$306$	0	0
$307$	−17.9309	−1.02337	−0.511684	−	0.859173i	$-0.670978\pi$
$307$	−17.9309	−1.02337	−0.511684	+	0.859173i	$0.670978\pi$
$308$	0	0
$309$	0	0
$310$	−1.75379	−0.0996085
$311$	−7.36932	−0.417876	−0.208938	−	0.977929i	$-0.567001\pi$
$311$	−7.36932	−0.417876	−0.208938	+	0.977929i	$0.567001\pi$
$312$	0	0
$313$	−27.6155	−1.56092	−0.780461	−	0.625205i	$-0.785016\pi$
$313$	−27.6155	−1.56092	−0.780461	+	0.625205i	$0.785016\pi$
$314$	−27.6155	−1.55843
$315$	0	0
$316$	−2.87689	−0.161838
$317$	30.4924	1.71263	0.856313	−	0.516458i	$-0.172750\pi$
$317$	30.4924	1.71263	0.856313	+	0.516458i	$0.172750\pi$
$318$	0	0
$319$	38.7386	2.16895
$320$	−5.56155	−0.310900
$321$	0	0
$322$	0	0
$323$	36.4924	2.03049
$324$	0	0
$325$	2.00000	0.110940
$326$	−6.24621	−0.345946
$327$	0	0
$328$	23.6155	1.30395
$329$	0	0
$330$	0	0
$331$	0.492423	0.0270660	0.0135330	−	0.999908i	$-0.495692\pi$
$331$	0.492423	0.0270660	0.0135330	+	0.999908i	$0.495692\pi$
$332$	3.36932	0.184915
$333$	0	0
$334$	11.5076	0.629667
$335$	−4.80776	−0.262676
$336$	0	0
$337$	13.3693	0.728273	0.364137	−	0.931346i	$-0.381364\pi$
$337$	13.3693	0.728273	0.364137	+	0.931346i	$0.381364\pi$
$338$	14.0540	0.764435
$339$	0	0
$340$	3.12311	0.169374
$341$	4.49242	0.243278
$342$	0	0
$343$	0	0
$344$	−15.2311	−0.821204
$345$	0	0
$346$	−23.6155	−1.26958
$347$	−21.6155	−1.16038	−0.580191	−	0.814480i	$-0.697022\pi$
$347$	−21.6155	−1.16038	−0.580191	+	0.814480i	$0.697022\pi$
$348$	0	0
$349$	27.1231	1.45187	0.725933	−	0.687765i	$-0.241408\pi$
$349$	27.1231	1.45187	0.725933	+	0.687765i	$0.241408\pi$
$350$	0	0
$351$	0	0
$352$	−9.75379	−0.519879
$353$	−9.68466	−0.515462	−0.257731	−	0.966217i	$-0.582975\pi$
$353$	−9.68466	−0.515462	−0.257731	+	0.966217i	$0.582975\pi$
$354$	0	0
$355$	−10.2462	−0.543812
$356$	−0.438447	−0.0232377
$357$	0	0
$358$	18.7386	0.990368
$359$	17.7538	0.937009	0.468505	−	0.883461i	$-0.344793\pi$
$359$	17.7538	0.937009	0.468505	+	0.883461i	$0.344793\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	14.6307	0.768972
$363$	0	0
$364$	0	0
$365$	9.68466	0.506918
$366$	0	0
$367$	−32.9848	−1.72179	−0.860897	−	0.508779i	$-0.830097\pi$
$367$	−32.9848	−1.72179	−0.860897	+	0.508779i	$0.830097\pi$
$368$	−6.73863	−0.351276
$369$	0	0
$370$	15.6155	0.811813
$371$	0	0
$372$	0	0
$373$	−5.68466	−0.294340	−0.147170	−	0.989111i	$-0.547017\pi$
$373$	−5.68466	−0.294340	−0.147170	+	0.989111i	$0.547017\pi$
$374$	−44.4924	−2.30065
$375$	0	0
$376$	19.5076	1.00603
$377$	−19.3693	−0.997571
$378$	0	0
$379$	15.3693	0.789469	0.394734	−	0.918795i	$-0.370837\pi$
$379$	15.3693	0.789469	0.394734	+	0.918795i	$0.370837\pi$
$380$	2.24621	0.115228
$381$	0	0
$382$	−36.9848	−1.89231
$383$	26.2462	1.34112	0.670559	−	0.741856i	$-0.266054\pi$
$383$	26.2462	1.34112	0.670559	+	0.741856i	$0.266054\pi$
$384$	0	0
$385$	0	0
$386$	−1.36932	−0.0696964
$387$	0	0
$388$	−5.50758	−0.279605
$389$	−10.4924	−0.531987	−0.265993	−	0.963975i	$-0.585700\pi$
$389$	−10.4924	−0.531987	−0.265993	+	0.963975i	$0.585700\pi$
$390$	0	0
$391$	−10.2462	−0.518173
$392$	−17.0691	−0.862121
$393$	0	0
$394$	−0.384472	−0.0193694
$395$	6.56155	0.330148
$396$	0	0
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	−38.7386	−1.94179
$399$	0	0
$400$	−4.68466	−0.234233
$401$	−25.3693	−1.26688	−0.633442	−	0.773790i	$-0.718358\pi$
$401$	−25.3693	−1.26688	−0.633442	+	0.773790i	$0.718358\pi$
$402$	0	0
$403$	−2.24621	−0.111892
$404$	−2.63068	−0.130881
$405$	0	0
$406$	0	0
$407$	−40.0000	−1.98273
$408$	0	0
$409$	7.93087	0.392156	0.196078	−	0.980588i	$-0.437179\pi$
$409$	7.93087	0.392156	0.196078	+	0.980588i	$0.437179\pi$
$410$	15.1231	0.746877
$411$	0	0
$412$	−5.75379	−0.283469
$413$	0	0
$414$	0	0
$415$	−7.68466	−0.377225
$416$	4.87689	0.239109
$417$	0	0
$418$	−32.0000	−1.56517
$419$	−11.0540	−0.540022	−0.270011	−	0.962857i	$-0.587027\pi$
$419$	−11.0540	−0.540022	−0.270011	+	0.962857i	$0.587027\pi$
$420$	0	0
$421$	16.2462	0.791792	0.395896	−	0.918295i	$-0.370434\pi$
$421$	16.2462	0.791792	0.395896	+	0.918295i	$0.370434\pi$
$422$	−8.98485	−0.437375
$423$	0	0
$424$	−20.1080	−0.976528
$425$	−7.12311	−0.345521
$426$	0	0
$427$	0	0
$428$	−1.75379	−0.0847726
$429$	0	0
$430$	−9.75379	−0.470369
$431$	9.75379	0.469823	0.234912	−	0.972017i	$-0.424520\pi$
$431$	9.75379	0.469823	0.234912	+	0.972017i	$0.424520\pi$
$432$	0	0
$433$	−15.1231	−0.726770	−0.363385	−	0.931639i	$-0.618379\pi$
$433$	−15.1231	−0.726770	−0.363385	+	0.931639i	$0.618379\pi$
$434$	0	0
$435$	0	0
$436$	−3.75379	−0.179774
$437$	−7.36932	−0.352522
$438$	0	0
$439$	19.3693	0.924447	0.462224	−	0.886763i	$-0.347052\pi$
$439$	19.3693	0.924447	0.462224	+	0.886763i	$0.347052\pi$
$440$	9.75379	0.464994
$441$	0	0
$442$	22.2462	1.05814
$443$	21.6155	1.02698	0.513492	−	0.858094i	$-0.328351\pi$
$443$	21.6155	1.02698	0.513492	+	0.858094i	$0.328351\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	2.24621	0.106361
$447$	0	0
$448$	0	0
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	−38.7386	−1.82413
$452$	1.23106	0.0579040
$453$	0	0
$454$	6.24621	0.293149
$455$	0	0
$456$	0	0
$457$	31.6155	1.47891	0.739456	−	0.673205i	$-0.235083\pi$
$457$	31.6155	1.47891	0.739456	+	0.673205i	$0.235083\pi$
$458$	27.1231	1.26738
$459$	0	0
$460$	−0.630683	−0.0294058
$461$	32.7386	1.52479	0.762395	−	0.647112i	$-0.224023\pi$
$461$	32.7386	1.52479	0.762395	+	0.647112i	$0.224023\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	45.3693	2.10622
$465$	0	0
$466$	36.1080	1.67267
$467$	−16.4924	−0.763178	−0.381589	−	0.924332i	$-0.624623\pi$
$467$	−16.4924	−0.763178	−0.381589	+	0.924332i	$0.624623\pi$
$468$	0	0
$469$	0	0
$470$	12.4924	0.576232
$471$	0	0
$472$	−16.0000	−0.736460
$473$	24.9848	1.14880
$474$	0	0
$475$	−5.12311	−0.235064
$476$	0	0
$477$	0	0
$478$	−20.0000	−0.914779
$479$	18.8769	0.862507	0.431254	−	0.902231i	$-0.358071\pi$
$479$	18.8769	0.862507	0.431254	+	0.902231i	$0.358071\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	−11.1231	−0.506644
$483$	0	0
$484$	2.19224	0.0996471
$485$	12.5616	0.570391
$486$	0	0
$487$	6.56155	0.297332	0.148666	−	0.988887i	$-0.452502\pi$
$487$	6.56155	0.297332	0.148666	+	0.988887i	$0.452502\pi$
$488$	27.1231	1.22781
$489$	0	0
$490$	−10.9309	−0.493806
$491$	−7.19224	−0.324581	−0.162291	−	0.986743i	$-0.551888\pi$
$491$	−7.19224	−0.324581	−0.162291	+	0.986743i	$0.551888\pi$
$492$	0	0
$493$	68.9848	3.10692
$494$	16.0000	0.719874
$495$	0	0
$496$	5.26137	0.236242
$497$	0	0
$498$	0	0
$499$	23.3693	1.04615	0.523077	−	0.852285i	$-0.324784\pi$
$499$	23.3693	1.04615	0.523077	+	0.852285i	$0.324784\pi$
$500$	−0.438447	−0.0196080
$501$	0	0
$502$	−5.26137	−0.234826
$503$	−44.4924	−1.98382	−0.991910	−	0.126947i	$-0.959482\pi$
$503$	−44.4924	−1.98382	−0.991910	+	0.126947i	$0.959482\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	8.98485	0.399425
$507$	0	0
$508$	−4.49242	−0.199319
$509$	35.6155	1.57863	0.789315	−	0.613988i	$-0.210436\pi$
$509$	35.6155	1.57863	0.789315	+	0.613988i	$0.210436\pi$
$510$	0	0
$511$	0	0
$512$	11.4233	0.504843
$513$	0	0
$514$	23.6155	1.04164
$515$	13.1231	0.578273
$516$	0	0
$517$	−32.0000	−1.40736
$518$	0	0
$519$	0	0
$520$	−4.87689	−0.213866
$521$	−22.4924	−0.985411	−0.492705	−	0.870196i	$-0.663992\pi$
$521$	−22.4924	−0.985411	−0.492705	+	0.870196i	$0.663992\pi$
$522$	0	0
$523$	20.9848	0.917603	0.458802	−	0.888539i	$-0.348279\pi$
$523$	20.9848	0.917603	0.458802	+	0.888539i	$0.348279\pi$
$524$	4.98485	0.217764
$525$	0	0
$526$	−37.4773	−1.63409
$527$	8.00000	0.348485
$528$	0	0
$529$	−20.9309	−0.910038
$530$	−12.8769	−0.559337
$531$	0	0
$532$	0	0
$533$	19.3693	0.838978
$534$	0	0
$535$	4.00000	0.172935
$536$	11.7235	0.506377
$537$	0	0
$538$	3.89205	0.167798
$539$	28.0000	1.20605
$540$	0	0
$541$	20.7386	0.891624	0.445812	−	0.895127i	$-0.352915\pi$
$541$	20.7386	0.891624	0.445812	+	0.895127i	$0.352915\pi$
$542$	−35.5076	−1.52518
$543$	0	0
$544$	−17.3693	−0.744703
$545$	8.56155	0.366737
$546$	0	0
$547$	29.6155	1.26627	0.633134	−	0.774042i	$-0.281768\pi$
$547$	29.6155	1.26627	0.633134	+	0.774042i	$0.281768\pi$
$548$	4.38447	0.187295
$549$	0	0
$550$	6.24621	0.266339
$551$	49.6155	2.11369
$552$	0	0
$553$	0	0
$554$	−43.6155	−1.85305
$555$	0	0
$556$	4.63068	0.196385
$557$	39.4384	1.67106	0.835530	−	0.549444i	$-0.185161\pi$
$557$	39.4384	1.67106	0.835530	+	0.549444i	$0.185161\pi$
$558$	0	0
$559$	−12.4924	−0.528373
$560$	0	0
$561$	0	0
$562$	6.63068	0.279698
$563$	20.1771	0.850363	0.425181	−	0.905108i	$-0.360210\pi$
$563$	20.1771	0.850363	0.425181	+	0.905108i	$0.360210\pi$
$564$	0	0
$565$	−2.80776	−0.118124
$566$	−22.2462	−0.935078
$567$	0	0
$568$	24.9848	1.04834
$569$	−26.1771	−1.09740	−0.548700	−	0.836019i	$-0.684877\pi$
$569$	−26.1771	−1.09740	−0.548700	+	0.836019i	$0.684877\pi$
$570$	0	0
$571$	36.4924	1.52716	0.763580	−	0.645713i	$-0.223440\pi$
$571$	36.4924	1.52716	0.763580	+	0.645713i	$0.223440\pi$
$572$	−3.50758	−0.146659
$573$	0	0
$574$	0	0
$575$	1.43845	0.0599874
$576$	0	0
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	−52.6847	−2.19139
$579$	0	0
$580$	4.24621	0.176314
$581$	0	0
$582$	0	0
$583$	32.9848	1.36609
$584$	−23.6155	−0.977218
$585$	0	0
$586$	−50.6307	−2.09153
$587$	−44.3542	−1.83069	−0.915346	−	0.402668i	$-0.868083\pi$
$587$	−44.3542	−1.83069	−0.915346	+	0.402668i	$0.868083\pi$
$588$	0	0
$589$	5.75379	0.237081
$590$	−10.2462	−0.421830
$591$	0	0
$592$	−46.8466	−1.92538
$593$	8.56155	0.351581	0.175790	−	0.984428i	$-0.443752\pi$
$593$	8.56155	0.351581	0.175790	+	0.984428i	$0.443752\pi$
$594$	0	0
$595$	0	0
$596$	9.23106	0.378119
$597$	0	0
$598$	−4.49242	−0.183709
$599$	30.2462	1.23583	0.617913	−	0.786246i	$-0.287978\pi$
$599$	30.2462	1.23583	0.617913	+	0.786246i	$0.287978\pi$
$600$	0	0
$601$	0.561553	0.0229062	0.0114531	−	0.999934i	$-0.496354\pi$
$601$	0.561553	0.0229062	0.0114531	+	0.999934i	$0.496354\pi$
$602$	0	0
$603$	0	0
$604$	2.73863	0.111433
$605$	−5.00000	−0.203279
$606$	0	0
$607$	−27.0540	−1.09809	−0.549043	−	0.835794i	$-0.685008\pi$
$607$	−27.0540	−1.09809	−0.549043	+	0.835794i	$0.685008\pi$
$608$	−12.4924	−0.506635
$609$	0	0
$610$	17.3693	0.703263
$611$	16.0000	0.647291
$612$	0	0
$613$	15.4384	0.623553	0.311777	−	0.950155i	$-0.399076\pi$
$613$	15.4384	0.623553	0.311777	+	0.950155i	$0.399076\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	0	0
$617$	13.6847	0.550924	0.275462	−	0.961312i	$-0.411169\pi$
$617$	13.6847	0.550924	0.275462	+	0.961312i	$0.411169\pi$
$618$	0	0
$619$	27.1922	1.09295	0.546474	−	0.837476i	$-0.315970\pi$
$619$	27.1922	1.09295	0.546474	+	0.837476i	$0.315970\pi$
$620$	0.492423	0.0197762
$621$	0	0
$622$	11.5076	0.461412
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	43.1231	1.72355
$627$	0	0
$628$	7.75379	0.309410
$629$	−71.2311	−2.84017
$630$	0	0
$631$	18.4233	0.733420	0.366710	−	0.930335i	$-0.380484\pi$
$631$	18.4233	0.733420	0.366710	+	0.930335i	$0.380484\pi$
$632$	−16.0000	−0.636446
$633$	0	0
$634$	−47.6155	−1.89105
$635$	10.2462	0.406608
$636$	0	0
$637$	−14.0000	−0.554700
$638$	−60.4924	−2.39492
$639$	0	0
$640$	13.5616	0.536067
$641$	−10.6307	−0.419887	−0.209943	−	0.977714i	$-0.567328\pi$
$641$	−10.6307	−0.419887	−0.209943	+	0.977714i	$0.567328\pi$
$642$	0	0
$643$	21.4384	0.845450	0.422725	−	0.906258i	$-0.361074\pi$
$643$	21.4384	0.845450	0.422725	+	0.906258i	$0.361074\pi$
$644$	0	0
$645$	0	0
$646$	−56.9848	−2.24204
$647$	19.0540	0.749089	0.374545	−	0.927209i	$-0.377799\pi$
$647$	19.0540	0.749089	0.374545	+	0.927209i	$0.377799\pi$
$648$	0	0
$649$	26.2462	1.03025
$650$	−3.12311	−0.122498
$651$	0	0
$652$	1.75379	0.0686837
$653$	−20.0691	−0.785366	−0.392683	−	0.919674i	$-0.628453\pi$
$653$	−20.0691	−0.785366	−0.392683	+	0.919674i	$0.628453\pi$
$654$	0	0
$655$	−11.3693	−0.444236
$656$	−45.3693	−1.77137
$657$	0	0
$658$	0	0
$659$	−19.3693	−0.754521	−0.377261	−	0.926107i	$-0.623134\pi$
$659$	−19.3693	−0.754521	−0.377261	+	0.926107i	$0.623134\pi$
$660$	0	0
$661$	36.7386	1.42897	0.714484	−	0.699652i	$-0.246662\pi$
$661$	36.7386	1.42897	0.714484	+	0.699652i	$0.246662\pi$
$662$	−0.768944	−0.0298858
$663$	0	0
$664$	18.7386	0.727200
$665$	0	0
$666$	0	0
$667$	−13.9309	−0.539405
$668$	−3.23106	−0.125013
$669$	0	0
$670$	7.50758	0.290043
$671$	−44.4924	−1.71761
$672$	0	0
$673$	−25.6847	−0.990071	−0.495035	−	0.868873i	$-0.664845\pi$
$673$	−25.6847	−0.990071	−0.495035	+	0.868873i	$0.664845\pi$
$674$	−20.8769	−0.804148
$675$	0	0
$676$	−3.94602	−0.151770
$677$	−13.5076	−0.519138	−0.259569	−	0.965725i	$-0.583581\pi$
$677$	−13.5076	−0.519138	−0.259569	+	0.965725i	$0.583581\pi$
$678$	0	0
$679$	0	0
$680$	17.3693	0.666083
$681$	0	0
$682$	−7.01515	−0.268624
$683$	−20.8078	−0.796187	−0.398093	−	0.917345i	$-0.630328\pi$
$683$	−20.8078	−0.796187	−0.398093	+	0.917345i	$0.630328\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	0	0
$687$	0	0
$688$	29.2614	1.11558
$689$	−16.4924	−0.628311
$690$	0	0
$691$	50.9157	1.93693	0.968463	−	0.249159i	$-0.0801542\pi$
$691$	50.9157	1.93693	0.968463	+	0.249159i	$0.0801542\pi$
$692$	6.63068	0.252061
$693$	0	0
$694$	33.7538	1.28128
$695$	−10.5616	−0.400623
$696$	0	0
$697$	−68.9848	−2.61299
$698$	−42.3542	−1.60313
$699$	0	0
$700$	0	0
$701$	−15.6155	−0.589790	−0.294895	−	0.955530i	$-0.595285\pi$
$701$	−15.6155	−0.589790	−0.294895	+	0.955530i	$0.595285\pi$
$702$	0	0
$703$	−51.2311	−1.93222
$704$	−22.2462	−0.838436
$705$	0	0
$706$	15.1231	0.569166
$707$	0	0
$708$	0	0
$709$	−7.12311	−0.267514	−0.133757	−	0.991014i	$-0.542704\pi$
$709$	−7.12311	−0.267514	−0.133757	+	0.991014i	$0.542704\pi$
$710$	16.0000	0.600469
$711$	0	0
$712$	−2.43845	−0.0913847
$713$	−1.61553	−0.0605020
$714$	0	0
$715$	8.00000	0.299183
$716$	−5.26137	−0.196626
$717$	0	0
$718$	−27.7235	−1.03463
$719$	25.7538	0.960454	0.480227	−	0.877144i	$-0.340554\pi$
$719$	25.7538	0.960454	0.480227	+	0.877144i	$0.340554\pi$
$720$	0	0
$721$	0	0
$722$	−11.3153	−0.421113
$723$	0	0
$724$	−4.10795	−0.152671
$725$	−9.68466	−0.359679
$726$	0	0
$727$	11.8617	0.439928	0.219964	−	0.975508i	$-0.429406\pi$
$727$	11.8617	0.439928	0.219964	+	0.975508i	$0.429406\pi$
$728$	0	0
$729$	0	0
$730$	−15.1231	−0.559731
$731$	44.4924	1.64561
$732$	0	0
$733$	−21.6847	−0.800941	−0.400471	−	0.916310i	$-0.631153\pi$
$733$	−21.6847	−0.800941	−0.400471	+	0.916310i	$0.631153\pi$
$734$	51.5076	1.90118
$735$	0	0
$736$	3.50758	0.129291
$737$	−19.2311	−0.708385
$738$	0	0
$739$	−52.4924	−1.93096	−0.965482	−	0.260468i	$-0.916123\pi$
$739$	−52.4924	−1.93096	−0.965482	+	0.260468i	$0.916123\pi$
$740$	−4.38447	−0.161176
$741$	0	0
$742$	0	0
$743$	39.5464	1.45082	0.725408	−	0.688319i	$-0.241651\pi$
$743$	39.5464	1.45082	0.725408	+	0.688319i	$0.241651\pi$
$744$	0	0
$745$	−21.0540	−0.771358
$746$	8.87689	0.325006
$747$	0	0
$748$	12.4924	0.456768
$749$	0	0
$750$	0	0
$751$	28.4924	1.03970	0.519852	−	0.854257i	$-0.325987\pi$
$751$	28.4924	1.03970	0.519852	+	0.854257i	$0.325987\pi$
$752$	−37.4773	−1.36666
$753$	0	0
$754$	30.2462	1.10150
$755$	−6.24621	−0.227323
$756$	0	0
$757$	−13.0540	−0.474455	−0.237227	−	0.971454i	$-0.576239\pi$
$757$	−13.0540	−0.474455	−0.237227	+	0.971454i	$0.576239\pi$
$758$	−24.0000	−0.871719
$759$	0	0
$760$	12.4924	0.453148
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	0	0
$764$	10.3845	0.375697
$765$	0	0
$766$	−40.9848	−1.48084
$767$	−13.1231	−0.473848
$768$	0	0
$769$	30.6695	1.10597	0.552985	−	0.833191i	$-0.313489\pi$
$769$	30.6695	1.10597	0.552985	+	0.833191i	$0.313489\pi$
$770$	0	0
$771$	0	0
$772$	0.384472	0.0138374
$773$	−31.3002	−1.12579	−0.562895	−	0.826529i	$-0.690312\pi$
$773$	−31.3002	−1.12579	−0.562895	+	0.826529i	$0.690312\pi$
$774$	0	0
$775$	−1.12311	−0.0403431
$776$	−30.6307	−1.09958
$777$	0	0
$778$	16.3845	0.587412
$779$	−49.6155	−1.77766
$780$	0	0
$781$	−40.9848	−1.46655
$782$	16.0000	0.572159
$783$	0	0
$784$	32.7926	1.17116
$785$	−17.6847	−0.631193
$786$	0	0
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	0.107951	0.00384558
$789$	0	0
$790$	−10.2462	−0.364544
$791$	0	0
$792$	0	0
$793$	22.2462	0.789986
$794$	9.36932	0.332505
$795$	0	0
$796$	10.8769	0.385521
$797$	15.7538	0.558028	0.279014	−	0.960287i	$-0.409992\pi$
$797$	15.7538	0.558028	0.279014	+	0.960287i	$0.409992\pi$
$798$	0	0
$799$	−56.9848	−2.01598
$800$	2.43845	0.0862121
$801$	0	0
$802$	39.6155	1.39887
$803$	38.7386	1.36706
$804$	0	0
$805$	0	0
$806$	3.50758	0.123549
$807$	0	0
$808$	−14.6307	−0.514706
$809$	−28.8769	−1.01526	−0.507629	−	0.861576i	$-0.669478\pi$
$809$	−28.8769	−1.01526	−0.507629	+	0.861576i	$0.669478\pi$
$810$	0	0
$811$	0.946025	0.0332194	0.0166097	−	0.999862i	$-0.494713\pi$
$811$	0.946025	0.0332194	0.0166097	+	0.999862i	$0.494713\pi$
$812$	0	0
$813$	0	0
$814$	62.4621	2.18930
$815$	−4.00000	−0.140114
$816$	0	0
$817$	32.0000	1.11954
$818$	−12.3845	−0.433013
$819$	0	0
$820$	−4.24621	−0.148284
$821$	−19.1231	−0.667401	−0.333700	−	0.942679i	$-0.608297\pi$
$821$	−19.1231	−0.667401	−0.333700	+	0.942679i	$0.608297\pi$
$822$	0	0
$823$	53.9309	1.87991	0.939956	−	0.341296i	$-0.110866\pi$
$823$	53.9309	1.87991	0.939956	+	0.341296i	$0.110866\pi$
$824$	−32.0000	−1.11477
$825$	0	0
$826$	0	0
$827$	20.1771	0.701626	0.350813	−	0.936446i	$-0.385905\pi$
$827$	20.1771	0.701626	0.350813	+	0.936446i	$0.385905\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	12.0000	0.416526
$831$	0	0
$832$	11.1231	0.385624
$833$	49.8617	1.72761
$834$	0	0
$835$	7.36932	0.255026
$836$	8.98485	0.310747
$837$	0	0
$838$	17.2614	0.596284
$839$	3.19224	0.110208	0.0551041	−	0.998481i	$-0.482451\pi$
$839$	3.19224	0.110208	0.0551041	+	0.998481i	$0.482451\pi$
$840$	0	0
$841$	64.7926	2.23423
$842$	−25.3693	−0.874284
$843$	0	0
$844$	2.52273	0.0868360
$845$	9.00000	0.309609
$846$	0	0
$847$	0	0
$848$	38.6307	1.32658
$849$	0	0
$850$	11.1231	0.381519
$851$	14.3845	0.493093
$852$	0	0
$853$	−40.8769	−1.39960	−0.699799	−	0.714340i	$-0.746727\pi$
$853$	−40.8769	−1.39960	−0.699799	+	0.714340i	$0.746727\pi$
$854$	0	0
$855$	0	0
$856$	−9.75379	−0.333378
$857$	16.7386	0.571781	0.285890	−	0.958262i	$-0.407711\pi$
$857$	16.7386	0.571781	0.285890	+	0.958262i	$0.407711\pi$
$858$	0	0
$859$	−37.1231	−1.26662	−0.633312	−	0.773897i	$-0.718305\pi$
$859$	−37.1231	−1.26662	−0.633312	+	0.773897i	$0.718305\pi$
$860$	2.73863	0.0933866
$861$	0	0
$862$	−15.2311	−0.518772
$863$	−44.4924	−1.51454	−0.757270	−	0.653102i	$-0.773467\pi$
$863$	−44.4924	−1.51454	−0.757270	+	0.653102i	$0.773467\pi$
$864$	0	0
$865$	−15.1231	−0.514201
$866$	23.6155	0.802488
$867$	0	0
$868$	0	0
$869$	26.2462	0.890342
$870$	0	0
$871$	9.61553	0.325810
$872$	−20.8769	−0.706981
$873$	0	0
$874$	11.5076	0.389250
$875$	0	0
$876$	0	0
$877$	23.7538	0.802108	0.401054	−	0.916054i	$-0.368644\pi$
$877$	23.7538	0.802108	0.401054	+	0.916054i	$0.368644\pi$
$878$	−30.2462	−1.02076
$879$	0	0
$880$	−18.7386	−0.631679
$881$	−25.3693	−0.854714	−0.427357	−	0.904083i	$-0.640555\pi$
$881$	−25.3693	−0.854714	−0.427357	+	0.904083i	$0.640555\pi$
$882$	0	0
$883$	10.3845	0.349465	0.174733	−	0.984616i	$-0.444094\pi$
$883$	10.3845	0.349465	0.174733	+	0.984616i	$0.444094\pi$
$884$	−6.24621	−0.210083
$885$	0	0
$886$	−33.7538	−1.13398
$887$	24.8078	0.832963	0.416482	−	0.909144i	$-0.363263\pi$
$887$	24.8078	0.832963	0.416482	+	0.909144i	$0.363263\pi$
$888$	0	0
$889$	0	0
$890$	−1.56155	−0.0523434
$891$	0	0
$892$	−0.630683	−0.0211168
$893$	−40.9848	−1.37151
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	0	0
$898$	53.0928	1.77173
$899$	10.8769	0.362765
$900$	0	0
$901$	58.7386	1.95687
$902$	60.4924	2.01418
$903$	0	0
$904$	6.84658	0.227714
$905$	9.36932	0.311447
$906$	0	0
$907$	−43.5464	−1.44593	−0.722967	−	0.690882i	$-0.757222\pi$
$907$	−43.5464	−1.44593	−0.722967	+	0.690882i	$0.757222\pi$
$908$	−1.75379	−0.0582015
$909$	0	0
$910$	0	0
$911$	−6.73863	−0.223261	−0.111630	−	0.993750i	$-0.535607\pi$
$911$	−6.73863	−0.223261	−0.111630	+	0.993750i	$0.535607\pi$
$912$	0	0
$913$	−30.7386	−1.01730
$914$	−49.3693	−1.63299
$915$	0	0
$916$	−7.61553	−0.251624
$917$	0	0
$918$	0	0
$919$	−38.8769	−1.28243	−0.641215	−	0.767361i	$-0.721569\pi$
$919$	−38.8769	−1.28243	−0.641215	+	0.767361i	$0.721569\pi$
$920$	−3.50758	−0.115641
$921$	0	0
$922$	−51.1231	−1.68365
$923$	20.4924	0.674516
$924$	0	0
$925$	10.0000	0.328798
$926$	49.9697	1.64211
$927$	0	0
$928$	−23.6155	−0.775218
$929$	−29.8617	−0.979732	−0.489866	−	0.871798i	$-0.662954\pi$
$929$	−29.8617	−0.979732	−0.489866	+	0.871798i	$0.662954\pi$
$930$	0	0
$931$	35.8617	1.17532
$932$	−10.1383	−0.332090
$933$	0	0
$934$	25.7538	0.842690
$935$	−28.4924	−0.931802
$936$	0	0
$937$	−16.4233	−0.536526	−0.268263	−	0.963346i	$-0.586450\pi$
$937$	−16.4233	−0.536526	−0.268263	+	0.963346i	$0.586450\pi$
$938$	0	0
$939$	0	0
$940$	−3.50758	−0.114405
$941$	23.3002	0.759564	0.379782	−	0.925076i	$-0.375999\pi$
$941$	23.3002	0.759564	0.379782	+	0.925076i	$0.375999\pi$
$942$	0	0
$943$	13.9309	0.453652
$944$	30.7386	1.00046
$945$	0	0
$946$	−39.0152	−1.26849
$947$	−29.6155	−0.962375	−0.481188	−	0.876618i	$-0.659794\pi$
$947$	−29.6155	−0.962375	−0.481188	+	0.876618i	$0.659794\pi$
$948$	0	0
$949$	−19.3693	−0.628755
$950$	8.00000	0.259554
$951$	0	0
$952$	0	0
$953$	26.8078	0.868389	0.434194	−	0.900819i	$-0.357033\pi$
$953$	26.8078	0.868389	0.434194	+	0.900819i	$0.357033\pi$
$954$	0	0
$955$	−23.6847	−0.766418
$956$	5.61553	0.181619
$957$	0	0
$958$	−29.4773	−0.952367
$959$	0	0
$960$	0	0
$961$	−29.7386	−0.959311
$962$	−31.2311	−1.00693
$963$	0	0
$964$	3.12311	0.100588
$965$	−0.876894	−0.0282282
$966$	0	0
$967$	−3.50758	−0.112796	−0.0563980	−	0.998408i	$-0.517962\pi$
$967$	−3.50758	−0.112796	−0.0563980	+	0.998408i	$0.517962\pi$
$968$	12.1922	0.391873
$969$	0	0
$970$	−19.6155	−0.629817
$971$	26.1080	0.837844	0.418922	−	0.908022i	$-0.362408\pi$
$971$	26.1080	0.837844	0.418922	+	0.908022i	$0.362408\pi$
$972$	0	0
$973$	0	0
$974$	−10.2462	−0.328310
$975$	0	0
$976$	−52.1080	−1.66793
$977$	25.2311	0.807213	0.403607	−	0.914933i	$-0.367756\pi$
$977$	25.2311	0.807213	0.403607	+	0.914933i	$0.367756\pi$
$978$	0	0
$979$	4.00000	0.127841
$980$	3.06913	0.0980398
$981$	0	0
$982$	11.2311	0.358397
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	−0.246211	−0.00784494
$986$	−107.723	−3.43061
$987$	0	0
$988$	−4.49242	−0.142923
$989$	−8.98485	−0.285701
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	−2.73863	−0.0869517
$993$	0	0
$994$	0	0
$995$	−24.8078	−0.786459
$996$	0	0
$997$	4.56155	0.144466	0.0722329	−	0.997388i	$-0.476988\pi$
$997$	4.56155	0.144466	0.0722329	+	0.997388i	$0.476988\pi$
$998$	−36.4924	−1.15515
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.g.1.1		2
3.2	odd	2		1335.2.a.c.1.2	✓	2
15.14	odd	2		6675.2.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.c.1.2	✓	2	3.2	odd	2
4005.2.a.g.1.1		2	1.1	even	1	trivial
6675.2.a.k.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} +2.43845 q^{8} +O(q^{10})\)	\(q-1.56155 q^{2} +0.438447 q^{4} -1.00000 q^{5} +2.43845 q^{8} +1.56155 q^{10} -4.00000 q^{11} +2.00000 q^{13} -4.68466 q^{16} -7.12311 q^{17} -5.12311 q^{19} -0.438447 q^{20} +6.24621 q^{22} +1.43845 q^{23} +1.00000 q^{25} -3.12311 q^{26} -9.68466 q^{29} -1.12311 q^{31} +2.43845 q^{32} +11.1231 q^{34} +10.0000 q^{37} +8.00000 q^{38} -2.43845 q^{40} +9.68466 q^{41} -6.24621 q^{43} -1.75379 q^{44} -2.24621 q^{46} +8.00000 q^{47} -7.00000 q^{49} -1.56155 q^{50} +0.876894 q^{52} -8.24621 q^{53} +4.00000 q^{55} +15.1231 q^{58} -6.56155 q^{59} +11.1231 q^{61} +1.75379 q^{62} +5.56155 q^{64} -2.00000 q^{65} +4.80776 q^{67} -3.12311 q^{68} +10.2462 q^{71} -9.68466 q^{73} -15.6155 q^{74} -2.24621 q^{76} -6.56155 q^{79} +4.68466 q^{80} -15.1231 q^{82} +7.68466 q^{83} +7.12311 q^{85} +9.75379 q^{86} -9.75379 q^{88} -1.00000 q^{89} +0.630683 q^{92} -12.4924 q^{94} +5.12311 q^{95} -12.5616 q^{97} +10.9309 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 5 q^{4} - 2 q^{5} + 9 q^{8}+O(q^{10}) \) 2 * q + q^2 + 5 * q^4 - 2 * q^5 + 9 * q^8	\( 2 q + q^{2} + 5 q^{4} - 2 q^{5} + 9 q^{8} - q^{10} - 8 q^{11} + 4 q^{13} + 3 q^{16} - 6 q^{17} - 2 q^{19} - 5 q^{20} - 4 q^{22} + 7 q^{23} + 2 q^{25} + 2 q^{26} - 7 q^{29} + 6 q^{31} + 9 q^{32} + 14 q^{34} + 20 q^{37} + 16 q^{38} - 9 q^{40} + 7 q^{41} + 4 q^{43} - 20 q^{44} + 12 q^{46} + 16 q^{47} - 14 q^{49} + q^{50} + 10 q^{52} + 8 q^{55} + 22 q^{58} - 9 q^{59} + 14 q^{61} + 20 q^{62} + 7 q^{64} - 4 q^{65} - 11 q^{67} + 2 q^{68} + 4 q^{71} - 7 q^{73} + 10 q^{74} + 12 q^{76} - 9 q^{79} - 3 q^{80} - 22 q^{82} + 3 q^{83} + 6 q^{85} + 36 q^{86} - 36 q^{88} - 2 q^{89} + 26 q^{92} + 8 q^{94} + 2 q^{95} - 21 q^{97} - 7 q^{98}+O(q^{100}) \) 2 * q + q^2 + 5 * q^4 - 2 * q^5 + 9 * q^8 - q^10 - 8 * q^11 + 4 * q^13 + 3 * q^16 - 6 * q^17 - 2 * q^19 - 5 * q^20 - 4 * q^22 + 7 * q^23 + 2 * q^25 + 2 * q^26 - 7 * q^29 + 6 * q^31 + 9 * q^32 + 14 * q^34 + 20 * q^37 + 16 * q^38 - 9 * q^40 + 7 * q^41 + 4 * q^43 - 20 * q^44 + 12 * q^46 + 16 * q^47 - 14 * q^49 + q^50 + 10 * q^52 + 8 * q^55 + 22 * q^58 - 9 * q^59 + 14 * q^61 + 20 * q^62 + 7 * q^64 - 4 * q^65 - 11 * q^67 + 2 * q^68 + 4 * q^71 - 7 * q^73 + 10 * q^74 + 12 * q^76 - 9 * q^79 - 3 * q^80 - 22 * q^82 + 3 * q^83 + 6 * q^85 + 36 * q^86 - 36 * q^88 - 2 * q^89 + 26 * q^92 + 8 * q^94 + 2 * q^95 - 21 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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