LMFDB - Embedded newform 4005.2.a.t.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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 19x^{8} - x^{7} + 128x^{6} + 14x^{5} - 358x^{4} - 59x^{3} + 344x^{2} + 71x - 21 $ x^10 - 19x^8 - x^7 + 128x^6 + 14x^5 - 358x^4 - 59x^3 + 344x^2 + 71*x - 21
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			$2.72551$ 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)

$f(q)$	$=$	$q-2.72551 q^{2} +5.42842 q^{4} +1.00000 q^{5} +0.930190 q^{7} -9.34419 q^{8} +O(q^{10})$	\(q-2.72551 q^{2} +5.42842 q^{4} +1.00000 q^{5} +0.930190 q^{7} -9.34419 q^{8} -2.72551 q^{10} -1.45528 q^{11} -0.247367 q^{13} -2.53525 q^{14} +14.6109 q^{16} +7.85189 q^{17} +0.303189 q^{19} +5.42842 q^{20} +3.96637 q^{22} +6.91586 q^{23} +1.00000 q^{25} +0.674203 q^{26} +5.04946 q^{28} +4.32833 q^{29} -5.03008 q^{31} -21.1337 q^{32} -21.4004 q^{34} +0.930190 q^{35} +8.26667 q^{37} -0.826345 q^{38} -9.34419 q^{40} -0.708782 q^{41} +4.86260 q^{43} -7.89985 q^{44} -18.8493 q^{46} +3.15702 q^{47} -6.13475 q^{49} -2.72551 q^{50} -1.34281 q^{52} -10.3222 q^{53} -1.45528 q^{55} -8.69187 q^{56} -11.7969 q^{58} -12.9723 q^{59} -1.39625 q^{61} +13.7095 q^{62} +28.3785 q^{64} -0.247367 q^{65} +13.3812 q^{67} +42.6233 q^{68} -2.53525 q^{70} +10.7654 q^{71} -3.64700 q^{73} -22.5309 q^{74} +1.64584 q^{76} -1.35368 q^{77} -2.82313 q^{79} +14.6109 q^{80} +1.93179 q^{82} -3.03787 q^{83} +7.85189 q^{85} -13.2531 q^{86} +13.5984 q^{88} +1.00000 q^{89} -0.230099 q^{91} +37.5422 q^{92} -8.60450 q^{94} +0.303189 q^{95} -15.3633 q^{97} +16.7203 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) $ 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8	$ 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8} - 4 q^{11} + 15 q^{13} + q^{14} + 22 q^{16} - 11 q^{17} + 14 q^{19} + 18 q^{20} + 10 q^{22} + 8 q^{23} + 10 q^{25} + 14 q^{26} + 36 q^{28} - 5 q^{29} - 8 q^{31} - q^{32} + 2 q^{34} + 9 q^{35} + 23 q^{37} - 8 q^{38} - 3 q^{40} - 13 q^{41} + 25 q^{43} + 8 q^{44} - 14 q^{46} + q^{47} + 21 q^{49} + 51 q^{52} - 9 q^{53} - 4 q^{55} - 15 q^{56} - 8 q^{58} + 15 q^{59} + 8 q^{61} + 8 q^{62} + 9 q^{64} + 15 q^{65} + 52 q^{67} + 28 q^{68} + q^{70} + 22 q^{71} + 34 q^{73} + 18 q^{74} + 14 q^{76} - 4 q^{77} - 3 q^{79} + 22 q^{80} + 17 q^{82} + 10 q^{83} - 11 q^{85} + 6 q^{86} + 4 q^{88} + 10 q^{89} + 18 q^{91} + 14 q^{92} - 43 q^{94} + 14 q^{95} + 34 q^{97} + 38 q^{98}+O(q^{100}) $ 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8 - 4 * q^11 + 15 * q^13 + q^14 + 22 * q^16 - 11 * q^17 + 14 * q^19 + 18 * q^20 + 10 * q^22 + 8 * q^23 + 10 * q^25 + 14 * q^26 + 36 * q^28 - 5 * q^29 - 8 * q^31 - q^32 + 2 * q^34 + 9 * q^35 + 23 * q^37 - 8 * q^38 - 3 * q^40 - 13 * q^41 + 25 * q^43 + 8 * q^44 - 14 * q^46 + q^47 + 21 * q^49 + 51 * q^52 - 9 * q^53 - 4 * q^55 - 15 * q^56 - 8 * q^58 + 15 * q^59 + 8 * q^61 + 8 * q^62 + 9 * q^64 + 15 * q^65 + 52 * q^67 + 28 * q^68 + q^70 + 22 * q^71 + 34 * q^73 + 18 * q^74 + 14 * q^76 - 4 * q^77 - 3 * q^79 + 22 * q^80 + 17 * q^82 + 10 * q^83 - 11 * q^85 + 6 * q^86 + 4 * q^88 + 10 * q^89 + 18 * q^91 + 14 * q^92 - 43 * q^94 + 14 * q^95 + 34 * q^97 + 38 * 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 $

$2$	−2.72551	−1.92723	−0.963614	−	0.267298i	$-0.913869\pi$
$2$	−2.72551	−1.92723	−0.963614	+	0.267298i	$0.913869\pi$
$3$	0	0
$3$	0	0
$4$	5.42842	2.71421
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.930190	0.351579	0.175789	−	0.984428i	$-0.443752\pi$
$7$	0.930190	0.351579	0.175789	+	0.984428i	$0.443752\pi$
$8$	−9.34419	−3.30367
$9$	0	0
$10$	−2.72551	−0.861883
$11$	−1.45528	−0.438782	−0.219391	−	0.975637i	$-0.570407\pi$
$11$	−1.45528	−0.438782	−0.219391	+	0.975637i	$0.570407\pi$
$12$	0	0
$13$	−0.247367	−0.0686074	−0.0343037	−	0.999411i	$-0.510921\pi$
$13$	−0.247367	−0.0686074	−0.0343037	+	0.999411i	$0.510921\pi$
$14$	−2.53525	−0.677573
$15$	0	0
$16$	14.6109	3.65272
$17$	7.85189	1.90436	0.952182	−	0.305532i	$-0.0988344\pi$
$17$	7.85189	1.90436	0.952182	+	0.305532i	$0.0988344\pi$
$18$	0	0
$19$	0.303189	0.0695563	0.0347782	−	0.999395i	$-0.488928\pi$
$19$	0.303189	0.0695563	0.0347782	+	0.999395i	$0.488928\pi$
$20$	5.42842	1.21383
$21$	0	0
$22$	3.96637	0.845634
$23$	6.91586	1.44206	0.721028	−	0.692905i	$-0.243670\pi$
$23$	6.91586	1.44206	0.721028	+	0.692905i	$0.243670\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.674203	0.132222
$27$	0	0
$28$	5.04946	0.954258
$29$	4.32833	0.803750	0.401875	−	0.915695i	$-0.368359\pi$
$29$	4.32833	0.803750	0.401875	+	0.915695i	$0.368359\pi$
$30$	0	0
$31$	−5.03008	−0.903428	−0.451714	−	0.892163i	$-0.649187\pi$
$31$	−5.03008	−0.903428	−0.451714	+	0.892163i	$0.649187\pi$
$32$	−21.1337	−3.73595
$33$	0	0
$34$	−21.4004	−3.67014
$35$	0.930190	0.157231
$36$	0	0
$37$	8.26667	1.35903	0.679516	−	0.733661i	$-0.262190\pi$
$37$	8.26667	1.35903	0.679516	+	0.733661i	$0.262190\pi$
$38$	−0.826345	−0.134051
$39$	0	0
$40$	−9.34419	−1.47745
$41$	−0.708782	−0.110693	−0.0553466	−	0.998467i	$-0.517626\pi$
$41$	−0.708782	−0.110693	−0.0553466	+	0.998467i	$0.517626\pi$
$42$	0	0
$43$	4.86260	0.741539	0.370770	−	0.928725i	$-0.379094\pi$
$43$	4.86260	0.741539	0.370770	+	0.928725i	$0.379094\pi$
$44$	−7.89985	−1.19095
$45$	0	0
$46$	−18.8493	−2.77917
$47$	3.15702	0.460499	0.230249	−	0.973132i	$-0.426046\pi$
$47$	3.15702	0.460499	0.230249	+	0.973132i	$0.426046\pi$
$48$	0	0
$49$	−6.13475	−0.876392
$50$	−2.72551	−0.385446
$51$	0	0
$52$	−1.34281	−0.186215
$53$	−10.3222	−1.41786	−0.708929	−	0.705280i	$-0.750821\pi$
$53$	−10.3222	−1.41786	−0.708929	+	0.705280i	$0.750821\pi$
$54$	0	0
$55$	−1.45528	−0.196229
$56$	−8.69187	−1.16150
$57$	0	0
$58$	−11.7969	−1.54901
$59$	−12.9723	−1.68885	−0.844425	−	0.535673i	$-0.820058\pi$
$59$	−12.9723	−1.68885	−0.844425	+	0.535673i	$0.820058\pi$
$60$	0	0
$61$	−1.39625	−0.178771	−0.0893855	−	0.995997i	$-0.528490\pi$
$61$	−1.39625	−0.178771	−0.0893855	+	0.995997i	$0.528490\pi$
$62$	13.7095	1.74111
$63$	0	0
$64$	28.3785	3.54731
$65$	−0.247367	−0.0306821
$66$	0	0
$67$	13.3812	1.63478	0.817388	−	0.576087i	$-0.195421\pi$
$67$	13.3812	1.63478	0.817388	+	0.576087i	$0.195421\pi$
$68$	42.6233	5.16884
$69$	0	0
$70$	−2.53525	−0.303020
$71$	10.7654	1.27761	0.638807	−	0.769367i	$-0.279428\pi$
$71$	10.7654	1.27761	0.638807	+	0.769367i	$0.279428\pi$
$72$	0	0
$73$	−3.64700	−0.426849	−0.213425	−	0.976960i	$-0.568462\pi$
$73$	−3.64700	−0.426849	−0.213425	+	0.976960i	$0.568462\pi$
$74$	−22.5309	−2.61917
$75$	0	0
$76$	1.64584	0.188790
$77$	−1.35368	−0.154267
$78$	0	0
$79$	−2.82313	−0.317627	−0.158814	−	0.987309i	$-0.550767\pi$
$79$	−2.82313	−0.317627	−0.158814	+	0.987309i	$0.550767\pi$
$80$	14.6109	1.63354
$81$	0	0
$82$	1.93179	0.213331
$83$	−3.03787	−0.333450	−0.166725	−	0.986003i	$-0.553319\pi$
$83$	−3.03787	−0.333450	−0.166725	+	0.986003i	$0.553319\pi$
$84$	0	0
$85$	7.85189	0.851657
$86$	−13.2531	−1.42912
$87$	0	0
$88$	13.5984	1.44959
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−0.230099	−0.0241209
$92$	37.5422	3.91404
$93$	0	0
$94$	−8.60450	−0.887486
$95$	0.303189	0.0311065
$96$	0	0
$97$	−15.3633	−1.55991	−0.779955	−	0.625835i	$-0.784758\pi$
$97$	−15.3633	−1.55991	−0.779955	+	0.625835i	$0.784758\pi$
$98$	16.7203	1.68901
$99$	0	0
$100$	5.42842	0.542842
$101$	9.12188	0.907661	0.453830	−	0.891088i	$-0.350057\pi$
$101$	9.12188	0.907661	0.453830	+	0.891088i	$0.350057\pi$
$102$	0	0
$103$	8.46677	0.834255	0.417128	−	0.908848i	$-0.363037\pi$
$103$	8.46677	0.834255	0.417128	+	0.908848i	$0.363037\pi$
$104$	2.31145	0.226656
$105$	0	0
$106$	28.1332	2.73253
$107$	6.31327	0.610326	0.305163	−	0.952300i	$-0.401289\pi$
$107$	6.31327	0.610326	0.305163	+	0.952300i	$0.401289\pi$
$108$	0	0
$109$	2.29971	0.220272	0.110136	−	0.993917i	$-0.464871\pi$
$109$	2.29971	0.220272	0.110136	+	0.993917i	$0.464871\pi$
$110$	3.96637	0.378179
$111$	0	0
$112$	13.5909	1.28422
$113$	14.4442	1.35880	0.679399	−	0.733769i	$-0.262240\pi$
$113$	14.4442	1.35880	0.679399	+	0.733769i	$0.262240\pi$
$114$	0	0
$115$	6.91586	0.644907
$116$	23.4959	2.18154
$117$	0	0
$118$	35.3562	3.25480
$119$	7.30375	0.669534
$120$	0	0
$121$	−8.88217	−0.807470
$122$	3.80548	0.344532
$123$	0	0
$124$	−27.3053	−2.45209
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.42137	0.392333	0.196167	−	0.980571i	$-0.437151\pi$
$127$	4.42137	0.392333	0.196167	+	0.980571i	$0.437151\pi$
$128$	−35.0784	−3.10052
$129$	0	0
$130$	0.674203	0.0591315
$131$	0.385067	0.0336434	0.0168217	−	0.999859i	$-0.494645\pi$
$131$	0.385067	0.0336434	0.0168217	+	0.999859i	$0.494645\pi$
$132$	0	0
$133$	0.282024	0.0244545
$134$	−36.4707	−3.15059
$135$	0	0
$136$	−73.3695	−6.29139
$137$	−15.5684	−1.33010	−0.665050	−	0.746799i	$-0.731590\pi$
$137$	−15.5684	−1.33010	−0.665050	+	0.746799i	$0.731590\pi$
$138$	0	0
$139$	10.8229	0.917984	0.458992	−	0.888440i	$-0.348211\pi$
$139$	10.8229	0.917984	0.458992	+	0.888440i	$0.348211\pi$
$140$	5.04946	0.426757
$141$	0	0
$142$	−29.3411	−2.46225
$143$	0.359988	0.0301037
$144$	0	0
$145$	4.32833	0.359448
$146$	9.93995	0.822636
$147$	0	0
$148$	44.8749	3.68870
$149$	8.24090	0.675121	0.337560	−	0.941304i	$-0.390398\pi$
$149$	8.24090	0.675121	0.337560	+	0.941304i	$0.390398\pi$
$150$	0	0
$151$	−17.8888	−1.45577	−0.727886	−	0.685698i	$-0.759497\pi$
$151$	−17.8888	−1.45577	−0.727886	+	0.685698i	$0.759497\pi$
$152$	−2.83306	−0.229791
$153$	0	0
$154$	3.68948	0.297307
$155$	−5.03008	−0.404025
$156$	0	0
$157$	−19.5238	−1.55817	−0.779086	−	0.626917i	$-0.784317\pi$
$157$	−19.5238	−1.55817	−0.779086	+	0.626917i	$0.784317\pi$
$158$	7.69448	0.612140
$159$	0	0
$160$	−21.1337	−1.67077
$161$	6.43307	0.506997
$162$	0	0
$163$	7.80978	0.611709	0.305854	−	0.952078i	$-0.401058\pi$
$163$	7.80978	0.611709	0.305854	+	0.952078i	$0.401058\pi$
$164$	−3.84756	−0.300444
$165$	0	0
$166$	8.27976	0.642634
$167$	15.7544	1.21911	0.609554	−	0.792744i	$-0.291348\pi$
$167$	15.7544	1.21911	0.609554	+	0.792744i	$0.291348\pi$
$168$	0	0
$169$	−12.9388	−0.995293
$170$	−21.4004	−1.64134
$171$	0	0
$172$	26.3962	2.01269
$173$	−15.2951	−1.16286	−0.581431	−	0.813596i	$-0.697507\pi$
$173$	−15.2951	−1.16286	−0.581431	+	0.813596i	$0.697507\pi$
$174$	0	0
$175$	0.930190	0.0703158
$176$	−21.2629	−1.60275
$177$	0	0
$178$	−2.72551	−0.204286
$179$	−1.00269	−0.0749446	−0.0374723	−	0.999298i	$-0.511931\pi$
$179$	−1.00269	−0.0749446	−0.0374723	+	0.999298i	$0.511931\pi$
$180$	0	0
$181$	7.29342	0.542116	0.271058	−	0.962563i	$-0.412627\pi$
$181$	7.29342	0.542116	0.271058	+	0.962563i	$0.412627\pi$
$182$	0.627137	0.0464865
$183$	0	0
$184$	−64.6231	−4.76408
$185$	8.26667	0.607778
$186$	0	0
$187$	−11.4267	−0.835601
$188$	17.1376	1.24989
$189$	0	0
$190$	−0.826345	−0.0599494
$191$	6.24685	0.452006	0.226003	−	0.974127i	$-0.427434\pi$
$191$	6.24685	0.452006	0.226003	+	0.974127i	$0.427434\pi$
$192$	0	0
$193$	15.6889	1.12931	0.564655	−	0.825327i	$-0.309009\pi$
$193$	15.6889	1.12931	0.564655	+	0.825327i	$0.309009\pi$
$194$	41.8730	3.00630
$195$	0	0
$196$	−33.3019	−2.37871
$197$	−9.59909	−0.683907	−0.341953	−	0.939717i	$-0.611088\pi$
$197$	−9.59909	−0.683907	−0.341953	+	0.939717i	$0.611088\pi$
$198$	0	0
$199$	18.4790	1.30994	0.654969	−	0.755656i	$-0.272682\pi$
$199$	18.4790	1.30994	0.654969	+	0.755656i	$0.272682\pi$
$200$	−9.34419	−0.660734
$201$	0	0
$202$	−24.8618	−1.74927
$203$	4.02617	0.282581
$204$	0	0
$205$	−0.708782	−0.0495035
$206$	−23.0763	−1.60780
$207$	0	0
$208$	−3.61425	−0.250603
$209$	−0.441224	−0.0305201
$210$	0	0
$211$	−3.52175	−0.242447	−0.121224	−	0.992625i	$-0.538682\pi$
$211$	−3.52175	−0.242447	−0.121224	+	0.992625i	$0.538682\pi$
$212$	−56.0330	−3.84836
$213$	0	0
$214$	−17.2069	−1.17624
$215$	4.86260	0.331626
$216$	0	0
$217$	−4.67893	−0.317626
$218$	−6.26788	−0.424515
$219$	0	0
$220$	−7.89985	−0.532608
$221$	−1.94230	−0.130653
$222$	0	0
$223$	12.4171	0.831507	0.415754	−	0.909477i	$-0.363518\pi$
$223$	12.4171	0.831507	0.415754	+	0.909477i	$0.363518\pi$
$224$	−19.6584	−1.31348
$225$	0	0
$226$	−39.3679	−2.61872
$227$	1.68705	0.111974	0.0559869	−	0.998432i	$-0.482170\pi$
$227$	1.68705	0.111974	0.0559869	+	0.998432i	$0.482170\pi$
$228$	0	0
$229$	12.4269	0.821190	0.410595	−	0.911818i	$-0.365321\pi$
$229$	12.4269	0.821190	0.410595	+	0.911818i	$0.365321\pi$
$230$	−18.8493	−1.24288
$231$	0	0
$232$	−40.4447	−2.65532
$233$	−22.8990	−1.50016	−0.750082	−	0.661345i	$-0.769986\pi$
$233$	−22.8990	−1.50016	−0.750082	+	0.661345i	$0.769986\pi$
$234$	0	0
$235$	3.15702	0.205941
$236$	−70.4191	−4.58389
$237$	0	0
$238$	−19.9065	−1.29034
$239$	−27.1178	−1.75410	−0.877051	−	0.480397i	$-0.840493\pi$
$239$	−27.1178	−1.75410	−0.877051	+	0.480397i	$0.840493\pi$
$240$	0	0
$241$	11.9551	0.770093	0.385047	−	0.922897i	$-0.374185\pi$
$241$	11.9551	0.770093	0.385047	+	0.922897i	$0.374185\pi$
$242$	24.2085	1.55618
$243$	0	0
$244$	−7.57940	−0.485222
$245$	−6.13475	−0.391935
$246$	0	0
$247$	−0.0749991	−0.00477208
$248$	47.0020	2.98463
$249$	0	0
$250$	−2.72551	−0.172377
$251$	−6.99282	−0.441383	−0.220691	−	0.975344i	$-0.570831\pi$
$251$	−6.99282	−0.441383	−0.220691	+	0.975344i	$0.570831\pi$
$252$	0	0
$253$	−10.0645	−0.632749
$254$	−12.0505	−0.756116
$255$	0	0
$256$	38.8497	2.42810
$257$	15.5596	0.970581	0.485291	−	0.874353i	$-0.338714\pi$
$257$	15.5596	0.970581	0.485291	+	0.874353i	$0.338714\pi$
$258$	0	0
$259$	7.68958	0.477807
$260$	−1.34281	−0.0832777
$261$	0	0
$262$	−1.04950	−0.0648386
$263$	−9.81602	−0.605282	−0.302641	−	0.953105i	$-0.597868\pi$
$263$	−9.81602	−0.605282	−0.302641	+	0.953105i	$0.597868\pi$
$264$	0	0
$265$	−10.3222	−0.634085
$266$	−0.768658	−0.0471295
$267$	0	0
$268$	72.6388	4.43712
$269$	9.99631	0.609486	0.304743	−	0.952435i	$-0.401429\pi$
$269$	9.99631	0.609486	0.304743	+	0.952435i	$0.401429\pi$
$270$	0	0
$271$	26.3165	1.59861	0.799307	−	0.600923i	$-0.205200\pi$
$271$	26.3165	1.59861	0.799307	+	0.600923i	$0.205200\pi$
$272$	114.723	6.95610
$273$	0	0
$274$	42.4319	2.56341
$275$	−1.45528	−0.0877565
$276$	0	0
$277$	11.8024	0.709140	0.354570	−	0.935029i	$-0.384627\pi$
$277$	11.8024	0.709140	0.354570	+	0.935029i	$0.384627\pi$
$278$	−29.4979	−1.76916
$279$	0	0
$280$	−8.69187	−0.519439
$281$	−28.9219	−1.72534	−0.862669	−	0.505770i	$-0.831209\pi$
$281$	−28.9219	−1.72534	−0.862669	+	0.505770i	$0.831209\pi$
$282$	0	0
$283$	17.1786	1.02116	0.510581	−	0.859829i	$-0.329430\pi$
$283$	17.1786	1.02116	0.510581	+	0.859829i	$0.329430\pi$
$284$	58.4389	3.46771
$285$	0	0
$286$	−0.981151	−0.0580167
$287$	−0.659302	−0.0389174
$288$	0	0
$289$	44.6522	2.62660
$290$	−11.7969	−0.692738
$291$	0	0
$292$	−19.7974	−1.15856
$293$	−15.9959	−0.934490	−0.467245	−	0.884128i	$-0.654753\pi$
$293$	−15.9959	−0.934490	−0.467245	+	0.884128i	$0.654753\pi$
$294$	0	0
$295$	−12.9723	−0.755277
$296$	−77.2453	−4.48979
$297$	0	0
$298$	−22.4607	−1.30111
$299$	−1.71076	−0.0989357
$300$	0	0
$301$	4.52314	0.260710
$302$	48.7562	2.80560
$303$	0	0
$304$	4.42985	0.254070
$305$	−1.39625	−0.0799488
$306$	0	0
$307$	12.9627	0.739823	0.369911	−	0.929067i	$-0.379388\pi$
$307$	12.9627	0.739823	0.369911	+	0.929067i	$0.379388\pi$
$308$	−7.34836	−0.418712
$309$	0	0
$310$	13.7095	0.778649
$311$	27.1307	1.53844	0.769220	−	0.638984i	$-0.220645\pi$
$311$	27.1307	1.53844	0.769220	+	0.638984i	$0.220645\pi$
$312$	0	0
$313$	−19.2934	−1.09053	−0.545265	−	0.838264i	$-0.683571\pi$
$313$	−19.2934	−1.09053	−0.545265	+	0.838264i	$0.683571\pi$
$314$	53.2125	3.00295
$315$	0	0
$316$	−15.3251	−0.862107
$317$	6.55191	0.367992	0.183996	−	0.982927i	$-0.441097\pi$
$317$	6.55191	0.367992	0.183996	+	0.982927i	$0.441097\pi$
$318$	0	0
$319$	−6.29891	−0.352671
$320$	28.3785	1.58640
$321$	0	0
$322$	−17.5334	−0.977098
$323$	2.38061	0.132461
$324$	0	0
$325$	−0.247367	−0.0137215
$326$	−21.2856	−1.17890
$327$	0	0
$328$	6.62299	0.365694
$329$	2.93663	0.161902
$330$	0	0
$331$	−19.4221	−1.06754	−0.533768	−	0.845631i	$-0.679224\pi$
$331$	−19.4221	−1.06754	−0.533768	+	0.845631i	$0.679224\pi$
$332$	−16.4908	−0.905052
$333$	0	0
$334$	−42.9387	−2.34950
$335$	13.3812	0.731094
$336$	0	0
$337$	12.2846	0.669185	0.334592	−	0.942363i	$-0.391401\pi$
$337$	12.2846	0.669185	0.334592	+	0.942363i	$0.391401\pi$
$338$	35.2649	1.91816
$339$	0	0
$340$	42.6233	2.31157
$341$	7.32015	0.396409
$342$	0	0
$343$	−12.2178	−0.659700
$344$	−45.4370	−2.44980
$345$	0	0
$346$	41.6868	2.24110
$347$	24.0522	1.29119	0.645595	−	0.763680i	$-0.276610\pi$
$347$	24.0522	1.29119	0.645595	+	0.763680i	$0.276610\pi$
$348$	0	0
$349$	−2.27564	−0.121812	−0.0609060	−	0.998144i	$-0.519399\pi$
$349$	−2.27564	−0.121812	−0.0609060	+	0.998144i	$0.519399\pi$
$350$	−2.53525	−0.135515
$351$	0	0
$352$	30.7554	1.63927
$353$	10.6536	0.567034	0.283517	−	0.958967i	$-0.408499\pi$
$353$	10.6536	0.567034	0.283517	+	0.958967i	$0.408499\pi$
$354$	0	0
$355$	10.7654	0.571366
$356$	5.42842	0.287705
$357$	0	0
$358$	2.73285	0.144435
$359$	28.0224	1.47896	0.739482	−	0.673177i	$-0.235071\pi$
$359$	28.0224	1.47896	0.739482	+	0.673177i	$0.235071\pi$
$360$	0	0
$361$	−18.9081	−0.995162
$362$	−19.8783	−1.04478
$363$	0	0
$364$	−1.24907	−0.0654691
$365$	−3.64700	−0.190893
$366$	0	0
$367$	−27.4926	−1.43510	−0.717551	−	0.696506i	$-0.754737\pi$
$367$	−27.4926	−1.43510	−0.717551	+	0.696506i	$0.754737\pi$
$368$	101.047	5.26742
$369$	0	0
$370$	−22.5309	−1.17133
$371$	−9.60157	−0.498489
$372$	0	0
$373$	20.5644	1.06478	0.532392	−	0.846498i	$-0.321293\pi$
$373$	20.5644	1.06478	0.532392	+	0.846498i	$0.321293\pi$
$374$	31.1435	1.61039
$375$	0	0
$376$	−29.4998	−1.52134
$377$	−1.07069	−0.0551431
$378$	0	0
$379$	13.4545	0.691112	0.345556	−	0.938398i	$-0.387690\pi$
$379$	13.4545	0.691112	0.345556	+	0.938398i	$0.387690\pi$
$380$	1.64584	0.0844296
$381$	0	0
$382$	−17.0259	−0.871119
$383$	−29.2679	−1.49552	−0.747761	−	0.663968i	$-0.768871\pi$
$383$	−29.2679	−1.49552	−0.747761	+	0.663968i	$0.768871\pi$
$384$	0	0
$385$	−1.35368	−0.0689901
$386$	−42.7602	−2.17644
$387$	0	0
$388$	−83.3986	−4.23392
$389$	16.7044	0.846948	0.423474	−	0.905908i	$-0.360811\pi$
$389$	16.7044	0.846948	0.423474	+	0.905908i	$0.360811\pi$
$390$	0	0
$391$	54.3026	2.74620
$392$	57.3242	2.89531
$393$	0	0
$394$	26.1624	1.31804
$395$	−2.82313	−0.142047
$396$	0	0
$397$	31.1736	1.56456	0.782279	−	0.622929i	$-0.214057\pi$
$397$	31.1736	1.56456	0.782279	+	0.622929i	$0.214057\pi$
$398$	−50.3646	−2.52455
$399$	0	0
$400$	14.6109	0.730543
$401$	−36.7211	−1.83376	−0.916882	−	0.399158i	$-0.869302\pi$
$401$	−36.7211	−1.83376	−0.916882	+	0.399158i	$0.869302\pi$
$402$	0	0
$403$	1.24428	0.0619818
$404$	49.5173	2.46358
$405$	0	0
$406$	−10.9734	−0.544599
$407$	−12.0303	−0.596319
$408$	0	0
$409$	6.46102	0.319477	0.159738	−	0.987159i	$-0.448935\pi$
$409$	6.46102	0.319477	0.159738	+	0.987159i	$0.448935\pi$
$410$	1.93179	0.0954045
$411$	0	0
$412$	45.9611	2.26434
$413$	−12.0667	−0.593764
$414$	0	0
$415$	−3.03787	−0.149123
$416$	5.22779	0.256313
$417$	0	0
$418$	1.20256	0.0588192
$419$	0.447695	0.0218713	0.0109357	−	0.999940i	$-0.496519\pi$
$419$	0.447695	0.0218713	0.0109357	+	0.999940i	$0.496519\pi$
$420$	0	0
$421$	−30.1587	−1.46984	−0.734922	−	0.678152i	$-0.762781\pi$
$421$	−30.1587	−1.46984	−0.734922	+	0.678152i	$0.762781\pi$
$422$	9.59857	0.467251
$423$	0	0
$424$	96.4522	4.68413
$425$	7.85189	0.380873
$426$	0	0
$427$	−1.29877	−0.0628521
$428$	34.2710	1.65655
$429$	0	0
$430$	−13.2531	−0.639120
$431$	−9.49523	−0.457369	−0.228685	−	0.973501i	$-0.573442\pi$
$431$	−9.49523	−0.457369	−0.228685	+	0.973501i	$0.573442\pi$
$432$	0	0
$433$	−4.16472	−0.200144	−0.100072	−	0.994980i	$-0.531907\pi$
$433$	−4.16472	−0.200144	−0.100072	+	0.994980i	$0.531907\pi$
$434$	12.7525	0.612138
$435$	0	0
$436$	12.4838	0.597864
$437$	2.09681	0.100304
$438$	0	0
$439$	2.00724	0.0958005	0.0479002	−	0.998852i	$-0.484747\pi$
$439$	2.00724	0.0958005	0.0479002	+	0.998852i	$0.484747\pi$
$440$	13.5984	0.648277
$441$	0	0
$442$	5.29377	0.251799
$443$	18.8074	0.893567	0.446783	−	0.894642i	$-0.352569\pi$
$443$	18.8074	0.893567	0.446783	+	0.894642i	$0.352569\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−33.8428	−1.60250
$447$	0	0
$448$	26.3974	1.24716
$449$	−11.9507	−0.563989	−0.281994	−	0.959416i	$-0.590996\pi$
$449$	−11.9507	−0.563989	−0.281994	+	0.959416i	$0.590996\pi$
$450$	0	0
$451$	1.03147	0.0485702
$452$	78.4093	3.68806
$453$	0	0
$454$	−4.59809	−0.215799
$455$	−0.230099	−0.0107872
$456$	0	0
$457$	24.8868	1.16416	0.582078	−	0.813133i	$-0.302240\pi$
$457$	24.8868	1.16416	0.582078	+	0.813133i	$0.302240\pi$
$458$	−33.8696	−1.58262
$459$	0	0
$460$	37.5422	1.75041
$461$	17.1691	0.799643	0.399821	−	0.916593i	$-0.369072\pi$
$461$	17.1691	0.799643	0.399821	+	0.916593i	$0.369072\pi$
$462$	0	0
$463$	29.4222	1.36737	0.683684	−	0.729779i	$-0.260377\pi$
$463$	29.4222	1.36737	0.683684	+	0.729779i	$0.260377\pi$
$464$	63.2406	2.93587
$465$	0	0
$466$	62.4115	2.89116
$467$	−13.0636	−0.604511	−0.302256	−	0.953227i	$-0.597740\pi$
$467$	−13.0636	−0.604511	−0.302256	+	0.953227i	$0.597740\pi$
$468$	0	0
$469$	12.4471	0.574753
$470$	−8.60450	−0.396896
$471$	0	0
$472$	121.216	5.57940
$473$	−7.07643	−0.325374
$474$	0	0
$475$	0.303189	0.0139113
$476$	39.6478	1.81725
$477$	0	0
$478$	73.9098	3.38056
$479$	16.4655	0.752329	0.376165	−	0.926553i	$-0.377243\pi$
$479$	16.4655	0.752329	0.376165	+	0.926553i	$0.377243\pi$
$480$	0	0
$481$	−2.04490	−0.0932396
$482$	−32.5837	−1.48415
$483$	0	0
$484$	−48.2161	−2.19164
$485$	−15.3633	−0.697613
$486$	0	0
$487$	−29.3061	−1.32798	−0.663992	−	0.747740i	$-0.731139\pi$
$487$	−29.3061	−1.32798	−0.663992	+	0.747740i	$0.731139\pi$
$488$	13.0468	0.590600
$489$	0	0
$490$	16.7203	0.755347
$491$	1.04612	0.0472106	0.0236053	−	0.999721i	$-0.492486\pi$
$491$	1.04612	0.0472106	0.0236053	+	0.999721i	$0.492486\pi$
$492$	0	0
$493$	33.9855	1.53063
$494$	0.204411	0.00919688
$495$	0	0
$496$	−73.4938	−3.29997
$497$	10.0138	0.449182
$498$	0	0
$499$	−11.2452	−0.503404	−0.251702	−	0.967805i	$-0.580990\pi$
$499$	−11.2452	−0.503404	−0.251702	+	0.967805i	$0.580990\pi$
$500$	5.42842	0.242766
$501$	0	0
$502$	19.0590	0.850645
$503$	26.0869	1.16316	0.581578	−	0.813490i	$-0.302435\pi$
$503$	26.0869	1.16316	0.581578	+	0.813490i	$0.302435\pi$
$504$	0	0
$505$	9.12188	0.405918
$506$	27.4309	1.21945
$507$	0	0
$508$	24.0011	1.06487
$509$	13.5862	0.602199	0.301099	−	0.953593i	$-0.402646\pi$
$509$	13.5862	0.602199	0.301099	+	0.953593i	$0.402646\pi$
$510$	0	0
$511$	−3.39241	−0.150071
$512$	−35.7285	−1.57899
$513$	0	0
$514$	−42.4079	−1.87053
$515$	8.46677	0.373090
$516$	0	0
$517$	−4.59434	−0.202059
$518$	−20.9580	−0.920843
$519$	0	0
$520$	2.31145	0.101364
$521$	20.9876	0.919482	0.459741	−	0.888053i	$-0.347942\pi$
$521$	20.9876	0.919482	0.459741	+	0.888053i	$0.347942\pi$
$522$	0	0
$523$	−0.683015	−0.0298661	−0.0149331	−	0.999888i	$-0.504754\pi$
$523$	−0.683015	−0.0298661	−0.0149331	+	0.999888i	$0.504754\pi$
$524$	2.09030	0.0913153
$525$	0	0
$526$	26.7537	1.16652
$527$	−39.4956	−1.72046
$528$	0	0
$529$	24.8291	1.07953
$530$	28.1332	1.22203
$531$	0	0
$532$	1.53094	0.0663747
$533$	0.175330	0.00759436
$534$	0	0
$535$	6.31327	0.272946
$536$	−125.037	−5.40076
$537$	0	0
$538$	−27.2451	−1.17462
$539$	8.92775	0.384546
$540$	0	0
$541$	−38.5942	−1.65930	−0.829648	−	0.558287i	$-0.811459\pi$
$541$	−38.5942	−1.65930	−0.829648	+	0.558287i	$0.811459\pi$
$542$	−71.7260	−3.08089
$543$	0	0
$544$	−165.940	−7.11460
$545$	2.29971	0.0985087
$546$	0	0
$547$	−15.6174	−0.667751	−0.333875	−	0.942617i	$-0.608356\pi$
$547$	−15.6174	−0.667751	−0.333875	+	0.942617i	$0.608356\pi$
$548$	−84.5118	−3.61017
$549$	0	0
$550$	3.96637	0.169127
$551$	1.31230	0.0559059
$552$	0	0
$553$	−2.62605	−0.111671
$554$	−32.1677	−1.36667
$555$	0	0
$556$	58.7510	2.49160
$557$	20.3409	0.861871	0.430936	−	0.902383i	$-0.358184\pi$
$557$	20.3409	0.861871	0.430936	+	0.902383i	$0.358184\pi$
$558$	0	0
$559$	−1.20285	−0.0508750
$560$	13.5909	0.574320
$561$	0	0
$562$	78.8270	3.32512
$563$	33.9164	1.42940	0.714702	−	0.699429i	$-0.246562\pi$
$563$	33.9164	1.42940	0.714702	+	0.699429i	$0.246562\pi$
$564$	0	0
$565$	14.4442	0.607673
$566$	−46.8205	−1.96801
$567$	0	0
$568$	−100.594	−4.22081
$569$	40.1345	1.68253	0.841263	−	0.540626i	$-0.181813\pi$
$569$	40.1345	1.68253	0.841263	+	0.540626i	$0.181813\pi$
$570$	0	0
$571$	39.1354	1.63777	0.818883	−	0.573961i	$-0.194594\pi$
$571$	39.1354	1.63777	0.818883	+	0.573961i	$0.194594\pi$
$572$	1.95416	0.0817077
$573$	0	0
$574$	1.79694	0.0750027
$575$	6.91586	0.288411
$576$	0	0
$577$	15.0476	0.626441	0.313221	−	0.949680i	$-0.398592\pi$
$577$	15.0476	0.626441	0.313221	+	0.949680i	$0.398592\pi$
$578$	−121.700	−5.06206
$579$	0	0
$580$	23.4959	0.975616
$581$	−2.82580	−0.117234
$582$	0	0
$583$	15.0216	0.622131
$584$	34.0783	1.41017
$585$	0	0
$586$	43.5970	1.80097
$587$	37.2695	1.53828	0.769139	−	0.639082i	$-0.220686\pi$
$587$	37.2695	1.53828	0.769139	+	0.639082i	$0.220686\pi$
$588$	0	0
$589$	−1.52506	−0.0628392
$590$	35.3562	1.45559
$591$	0	0
$592$	120.783	4.96416
$593$	1.43346	0.0588652	0.0294326	−	0.999567i	$-0.490630\pi$
$593$	1.43346	0.0588652	0.0294326	+	0.999567i	$0.490630\pi$
$594$	0	0
$595$	7.30375	0.299425
$596$	44.7350	1.83242
$597$	0	0
$598$	4.66269	0.190672
$599$	−24.5001	−1.00105	−0.500523	−	0.865723i	$-0.666859\pi$
$599$	−24.5001	−1.00105	−0.500523	+	0.865723i	$0.666859\pi$
$600$	0	0
$601$	12.7040	0.518205	0.259103	−	0.965850i	$-0.416573\pi$
$601$	12.7040	0.518205	0.259103	+	0.965850i	$0.416573\pi$
$602$	−12.3279	−0.502447
$603$	0	0
$604$	−97.1080	−3.95127
$605$	−8.88217	−0.361112
$606$	0	0
$607$	−25.3952	−1.03076	−0.515379	−	0.856962i	$-0.672349\pi$
$607$	−25.3952	−1.03076	−0.515379	+	0.856962i	$0.672349\pi$
$608$	−6.40751	−0.259859
$609$	0	0
$610$	3.80548	0.154080
$611$	−0.780944	−0.0315936
$612$	0	0
$613$	31.3678	1.26693	0.633467	−	0.773769i	$-0.281631\pi$
$613$	31.3678	1.26693	0.633467	+	0.773769i	$0.281631\pi$
$614$	−35.3301	−1.42581
$615$	0	0
$616$	12.6491	0.509646
$617$	−36.7143	−1.47806	−0.739031	−	0.673672i	$-0.764716\pi$
$617$	−36.7143	−1.47806	−0.739031	+	0.673672i	$0.764716\pi$
$618$	0	0
$619$	24.5016	0.984803	0.492402	−	0.870368i	$-0.336119\pi$
$619$	24.5016	0.984803	0.492402	+	0.870368i	$0.336119\pi$
$620$	−27.3053	−1.09661
$621$	0	0
$622$	−73.9450	−2.96492
$623$	0.930190	0.0372673
$624$	0	0
$625$	1.00000	0.0400000
$626$	52.5845	2.10170
$627$	0	0
$628$	−105.984	−4.22920
$629$	64.9090	2.58809
$630$	0	0
$631$	−1.47051	−0.0585400	−0.0292700	−	0.999572i	$-0.509318\pi$
$631$	−1.47051	−0.0585400	−0.0292700	+	0.999572i	$0.509318\pi$
$632$	26.3799	1.04934
$633$	0	0
$634$	−17.8573	−0.709204
$635$	4.42137	0.175457
$636$	0	0
$637$	1.51754	0.0601270
$638$	17.1678	0.679678
$639$	0	0
$640$	−35.0784	−1.38660
$641$	−4.57894	−0.180857	−0.0904286	−	0.995903i	$-0.528824\pi$
$641$	−4.57894	−0.180857	−0.0904286	+	0.995903i	$0.528824\pi$
$642$	0	0
$643$	11.0598	0.436154	0.218077	−	0.975932i	$-0.430022\pi$
$643$	11.0598	0.436154	0.218077	+	0.975932i	$0.430022\pi$
$644$	34.9214	1.37609
$645$	0	0
$646$	−6.48837	−0.255282
$647$	16.8777	0.663531	0.331765	−	0.943362i	$-0.392356\pi$
$647$	16.8777	0.663531	0.331765	+	0.943362i	$0.392356\pi$
$648$	0	0
$649$	18.8783	0.741038
$650$	0.674203	0.0264444
$651$	0	0
$652$	42.3947	1.66031
$653$	2.81421	0.110129	0.0550643	−	0.998483i	$-0.482464\pi$
$653$	2.81421	0.110129	0.0550643	+	0.998483i	$0.482464\pi$
$654$	0	0
$655$	0.385067	0.0150458
$656$	−10.3559	−0.404331
$657$	0	0
$658$	−8.00382	−0.312021
$659$	28.1498	1.09656	0.548281	−	0.836294i	$-0.315283\pi$
$659$	28.1498	1.09656	0.548281	+	0.836294i	$0.315283\pi$
$660$	0	0
$661$	−37.8207	−1.47105	−0.735527	−	0.677496i	$-0.763065\pi$
$661$	−37.8207	−1.47105	−0.735527	+	0.677496i	$0.763065\pi$
$662$	52.9352	2.05739
$663$	0	0
$664$	28.3865	1.10161
$665$	0.282024	0.0109364
$666$	0	0
$667$	29.9341	1.15905
$668$	85.5212	3.30891
$669$	0	0
$670$	−36.4707	−1.40899
$671$	2.03192	0.0784416
$672$	0	0
$673$	−27.3791	−1.05539	−0.527693	−	0.849435i	$-0.676943\pi$
$673$	−27.3791	−1.05539	−0.527693	+	0.849435i	$0.676943\pi$
$674$	−33.4818	−1.28967
$675$	0	0
$676$	−70.2372	−2.70143
$677$	−29.7673	−1.14405	−0.572026	−	0.820236i	$-0.693842\pi$
$677$	−29.7673	−1.14405	−0.572026	+	0.820236i	$0.693842\pi$
$678$	0	0
$679$	−14.2908	−0.548432
$680$	−73.3695	−2.81359
$681$	0	0
$682$	−19.9512	−0.763970
$683$	−36.1034	−1.38146	−0.690728	−	0.723114i	$-0.742710\pi$
$683$	−36.1034	−1.38146	−0.690728	+	0.723114i	$0.742710\pi$
$684$	0	0
$685$	−15.5684	−0.594839
$686$	33.2998	1.27139
$687$	0	0
$688$	71.0468	2.70863
$689$	2.55336	0.0972755
$690$	0	0
$691$	−31.2370	−1.18831	−0.594157	−	0.804349i	$-0.702514\pi$
$691$	−31.2370	−1.18831	−0.594157	+	0.804349i	$0.702514\pi$
$692$	−83.0279	−3.15625
$693$	0	0
$694$	−65.5546	−2.48842
$695$	10.8229	0.410535
$696$	0	0
$697$	−5.56528	−0.210800
$698$	6.20227	0.234760
$699$	0	0
$700$	5.04946	0.190852
$701$	41.2288	1.55719	0.778594	−	0.627528i	$-0.215933\pi$
$701$	41.2288	1.55719	0.778594	+	0.627528i	$0.215933\pi$
$702$	0	0
$703$	2.50636	0.0945293
$704$	−41.2985	−1.55650
$705$	0	0
$706$	−29.0365	−1.09280
$707$	8.48508	0.319114
$708$	0	0
$709$	28.4656	1.06905	0.534523	−	0.845154i	$-0.320491\pi$
$709$	28.4656	1.06905	0.534523	+	0.845154i	$0.320491\pi$
$710$	−29.3411	−1.10115
$711$	0	0
$712$	−9.34419	−0.350188
$713$	−34.7873	−1.30280
$714$	0	0
$715$	0.359988	0.0134628
$716$	−5.44302	−0.203415
$717$	0	0
$718$	−76.3753	−2.85030
$719$	−34.4242	−1.28380	−0.641902	−	0.766786i	$-0.721855\pi$
$719$	−34.4242	−1.28380	−0.641902	+	0.766786i	$0.721855\pi$
$720$	0	0
$721$	7.87570	0.293307
$722$	51.5342	1.91790
$723$	0	0
$724$	39.5917	1.47141
$725$	4.32833	0.160750
$726$	0	0
$727$	−25.1946	−0.934416	−0.467208	−	0.884147i	$-0.654740\pi$
$727$	−25.1946	−0.934416	−0.467208	+	0.884147i	$0.654740\pi$
$728$	2.15009	0.0796875
$729$	0	0
$730$	9.93995	0.367894
$731$	38.1806	1.41216
$732$	0	0
$733$	33.5258	1.23830	0.619151	−	0.785272i	$-0.287477\pi$
$733$	33.5258	1.23830	0.619151	+	0.785272i	$0.287477\pi$
$734$	74.9314	2.76577
$735$	0	0
$736$	−146.158	−5.38745
$737$	−19.4734	−0.717311
$738$	0	0
$739$	−20.9074	−0.769093	−0.384546	−	0.923106i	$-0.625642\pi$
$739$	−20.9074	−0.769093	−0.384546	+	0.923106i	$0.625642\pi$
$740$	44.8749	1.64963
$741$	0	0
$742$	26.1692	0.960702
$743$	25.9136	0.950679	0.475340	−	0.879802i	$-0.342325\pi$
$743$	25.9136	0.950679	0.475340	+	0.879802i	$0.342325\pi$
$744$	0	0
$745$	8.24090	0.301923
$746$	−56.0485	−2.05208
$747$	0	0
$748$	−62.0287	−2.26800
$749$	5.87254	0.214578
$750$	0	0
$751$	−40.7686	−1.48767	−0.743833	−	0.668366i	$-0.766994\pi$
$751$	−40.7686	−1.48767	−0.743833	+	0.668366i	$0.766994\pi$
$752$	46.1268	1.68207
$753$	0	0
$754$	2.91817	0.106273
$755$	−17.8888	−0.651041
$756$	0	0
$757$	−15.3800	−0.558996	−0.279498	−	0.960146i	$-0.590168\pi$
$757$	−15.3800	−0.558996	−0.279498	+	0.960146i	$0.590168\pi$
$758$	−36.6705	−1.33193
$759$	0	0
$760$	−2.83306	−0.102766
$761$	32.0274	1.16099	0.580496	−	0.814263i	$-0.302859\pi$
$761$	32.0274	1.16099	0.580496	+	0.814263i	$0.302859\pi$
$762$	0	0
$763$	2.13917	0.0774430
$764$	33.9105	1.22684
$765$	0	0
$766$	79.7701	2.88221
$767$	3.20893	0.115868
$768$	0	0
$769$	25.2825	0.911710	0.455855	−	0.890054i	$-0.349334\pi$
$769$	25.2825	0.911710	0.455855	+	0.890054i	$0.349334\pi$
$770$	3.68948	0.132960
$771$	0	0
$772$	85.1657	3.06518
$773$	−53.2643	−1.91579	−0.957893	−	0.287127i	$-0.907300\pi$
$773$	−53.2643	−1.91579	−0.957893	+	0.287127i	$0.907300\pi$
$774$	0	0
$775$	−5.03008	−0.180686
$776$	143.558	5.15343
$777$	0	0
$778$	−45.5281	−1.63226
$779$	−0.214895	−0.00769941
$780$	0	0
$781$	−15.6666	−0.560595
$782$	−148.002	−5.29255
$783$	0	0
$784$	−89.6339	−3.20121
$785$	−19.5238	−0.696836
$786$	0	0
$787$	50.3078	1.79328	0.896640	−	0.442761i	$-0.146001\pi$
$787$	50.3078	1.79328	0.896640	+	0.442761i	$0.146001\pi$
$788$	−52.1078	−1.85626
$789$	0	0
$790$	7.69448	0.273757
$791$	13.4359	0.477725
$792$	0	0
$793$	0.345386	0.0122650
$794$	−84.9640	−3.01526
$795$	0	0
$796$	100.311	3.55544
$797$	44.8980	1.59037	0.795185	−	0.606367i	$-0.207374\pi$
$797$	44.8980	1.59037	0.795185	+	0.606367i	$0.207374\pi$
$798$	0	0
$799$	24.7886	0.876957
$800$	−21.1337	−0.747190
$801$	0	0
$802$	100.084	3.53408
$803$	5.30740	0.187294
$804$	0	0
$805$	6.43307	0.226736
$806$	−3.39129	−0.119453
$807$	0	0
$808$	−85.2365	−2.99861
$809$	−1.94749	−0.0684700	−0.0342350	−	0.999414i	$-0.510899\pi$
$809$	−1.94749	−0.0684700	−0.0342350	+	0.999414i	$0.510899\pi$
$810$	0	0
$811$	−53.3106	−1.87199	−0.935994	−	0.352016i	$-0.885496\pi$
$811$	−53.3106	−1.87199	−0.935994	+	0.352016i	$0.885496\pi$
$812$	21.8557	0.766985
$813$	0	0
$814$	32.7887	1.14924
$815$	7.80978	0.273565
$816$	0	0
$817$	1.47429	0.0515787
$818$	−17.6096	−0.615705
$819$	0	0
$820$	−3.84756	−0.134363
$821$	−32.3991	−1.13074	−0.565368	−	0.824839i	$-0.691266\pi$
$821$	−32.3991	−1.13074	−0.565368	+	0.824839i	$0.691266\pi$
$822$	0	0
$823$	−34.0645	−1.18741	−0.593707	−	0.804681i	$-0.702336\pi$
$823$	−34.0645	−1.18741	−0.593707	+	0.804681i	$0.702336\pi$
$824$	−79.1151	−2.75610
$825$	0	0
$826$	32.8880	1.14432
$827$	−44.9625	−1.56350	−0.781749	−	0.623593i	$-0.785672\pi$
$827$	−44.9625	−1.56350	−0.781749	+	0.623593i	$0.785672\pi$
$828$	0	0
$829$	14.3328	0.497797	0.248899	−	0.968530i	$-0.419931\pi$
$829$	14.3328	0.497797	0.248899	+	0.968530i	$0.419931\pi$
$830$	8.27976	0.287395
$831$	0	0
$832$	−7.01990	−0.243371
$833$	−48.1694	−1.66897
$834$	0	0
$835$	15.7544	0.545202
$836$	−2.39515	−0.0828379
$837$	0	0
$838$	−1.22020	−0.0421511
$839$	−7.24937	−0.250276	−0.125138	−	0.992139i	$-0.539937\pi$
$839$	−7.24937	−0.250276	−0.125138	+	0.992139i	$0.539937\pi$
$840$	0	0
$841$	−10.2656	−0.353986
$842$	82.1978	2.83272
$843$	0	0
$844$	−19.1175	−0.658052
$845$	−12.9388	−0.445109
$846$	0	0
$847$	−8.26211	−0.283889
$848$	−150.816	−5.17903
$849$	0	0
$850$	−21.4004	−0.734028
$851$	57.1712	1.95980
$852$	0	0
$853$	39.8314	1.36380	0.681901	−	0.731444i	$-0.261153\pi$
$853$	39.8314	1.36380	0.681901	+	0.731444i	$0.261153\pi$
$854$	3.53983	0.121130
$855$	0	0
$856$	−58.9923	−2.01632
$857$	−2.88902	−0.0986870	−0.0493435	−	0.998782i	$-0.515713\pi$
$857$	−2.88902	−0.0986870	−0.0493435	+	0.998782i	$0.515713\pi$
$858$	0	0
$859$	−6.77722	−0.231236	−0.115618	−	0.993294i	$-0.536885\pi$
$859$	−6.77722	−0.231236	−0.115618	+	0.993294i	$0.536885\pi$
$860$	26.3962	0.900103
$861$	0	0
$862$	25.8794	0.881455
$863$	−27.6490	−0.941183	−0.470592	−	0.882351i	$-0.655959\pi$
$863$	−27.6490	−0.941183	−0.470592	+	0.882351i	$0.655959\pi$
$864$	0	0
$865$	−15.2951	−0.520047
$866$	11.3510	0.385723
$867$	0	0
$868$	−25.3992	−0.862104
$869$	4.10844	0.139369
$870$	0	0
$871$	−3.31008	−0.112158
$872$	−21.4889	−0.727706
$873$	0	0
$874$	−5.71489	−0.193309
$875$	0.930190	0.0314462
$876$	0	0
$877$	−46.1469	−1.55827	−0.779135	−	0.626856i	$-0.784341\pi$
$877$	−46.1469	−1.55827	−0.779135	+	0.626856i	$0.784341\pi$
$878$	−5.47076	−0.184629
$879$	0	0
$880$	−21.2629	−0.716771
$881$	−45.7418	−1.54108	−0.770540	−	0.637392i	$-0.780013\pi$
$881$	−45.7418	−1.54108	−0.770540	+	0.637392i	$0.780013\pi$
$882$	0	0
$883$	20.4170	0.687088	0.343544	−	0.939137i	$-0.388373\pi$
$883$	20.4170	0.687088	0.343544	+	0.939137i	$0.388373\pi$
$884$	−10.5436	−0.354620
$885$	0	0
$886$	−51.2598	−1.72211
$887$	22.1711	0.744432	0.372216	−	0.928146i	$-0.378598\pi$
$887$	22.1711	0.744432	0.372216	+	0.928146i	$0.378598\pi$
$888$	0	0
$889$	4.11272	0.137936
$890$	−2.72551	−0.0913594
$891$	0	0
$892$	67.4049	2.25688
$893$	0.957174	0.0320306
$894$	0	0
$895$	−1.00269	−0.0335163
$896$	−32.6296	−1.09008
$897$	0	0
$898$	32.5718	1.08694
$899$	−21.7718	−0.726130
$900$	0	0
$901$	−81.0485	−2.70012
$902$	−2.81129	−0.0936059
$903$	0	0
$904$	−134.970	−4.48902
$905$	7.29342	0.242442
$906$	0	0
$907$	18.8203	0.624919	0.312459	−	0.949931i	$-0.398847\pi$
$907$	18.8203	0.624919	0.312459	+	0.949931i	$0.398847\pi$
$908$	9.15803	0.303920
$909$	0	0
$910$	0.627137	0.0207894
$911$	24.6463	0.816567	0.408284	−	0.912855i	$-0.366127\pi$
$911$	24.6463	0.816567	0.408284	+	0.912855i	$0.366127\pi$
$912$	0	0
$913$	4.42095	0.146312
$914$	−67.8293	−2.24359
$915$	0	0
$916$	67.4582	2.22888
$917$	0.358185	0.0118283
$918$	0	0
$919$	28.2837	0.932993	0.466497	−	0.884523i	$-0.345516\pi$
$919$	28.2837	0.932993	0.466497	+	0.884523i	$0.345516\pi$
$920$	−64.6231	−2.13056
$921$	0	0
$922$	−46.7945	−1.54109
$923$	−2.66300	−0.0876537
$924$	0	0
$925$	8.26667	0.271806
$926$	−80.1906	−2.63523
$927$	0	0
$928$	−91.4736	−3.00277
$929$	−15.2915	−0.501698	−0.250849	−	0.968026i	$-0.580710\pi$
$929$	−15.2915	−0.501698	−0.250849	+	0.968026i	$0.580710\pi$
$930$	0	0
$931$	−1.85999	−0.0609586
$932$	−124.305	−4.07176
$933$	0	0
$934$	35.6050	1.16503
$935$	−11.4267	−0.373692
$936$	0	0
$937$	36.8342	1.20332	0.601661	−	0.798752i	$-0.294506\pi$
$937$	36.8342	1.20332	0.601661	+	0.798752i	$0.294506\pi$
$938$	−33.9247	−1.10768
$939$	0	0
$940$	17.1376	0.558968
$941$	−21.0371	−0.685791	−0.342896	−	0.939373i	$-0.611408\pi$
$941$	−21.0371	−0.685791	−0.342896	+	0.939373i	$0.611408\pi$
$942$	0	0
$943$	−4.90184	−0.159626
$944$	−189.537	−6.16889
$945$	0	0
$946$	19.2869	0.627071
$947$	21.4116	0.695785	0.347892	−	0.937535i	$-0.386897\pi$
$947$	21.4116	0.695785	0.347892	+	0.937535i	$0.386897\pi$
$948$	0	0
$949$	0.902149	0.0292850
$950$	−0.826345	−0.0268102
$951$	0	0
$952$	−68.2476	−2.21192
$953$	−7.31727	−0.237030	−0.118515	−	0.992952i	$-0.537813\pi$
$953$	−7.31727	−0.237030	−0.118515	+	0.992952i	$0.537813\pi$
$954$	0	0
$955$	6.24685	0.202143
$956$	−147.207	−4.76100
$957$	0	0
$958$	−44.8770	−1.44991
$959$	−14.4816	−0.467635
$960$	0	0
$961$	−5.69833	−0.183817
$962$	5.57341	0.179694
$963$	0	0
$964$	64.8970	2.09019
$965$	15.6889	0.505043
$966$	0	0
$967$	−0.576706	−0.0185456	−0.00927281	−	0.999957i	$-0.502952\pi$
$967$	−0.576706	−0.0185456	−0.00927281	+	0.999957i	$0.502952\pi$
$968$	82.9967	2.66761
$969$	0	0
$970$	41.8730	1.34446
$971$	32.8213	1.05329	0.526643	−	0.850087i	$-0.323451\pi$
$971$	32.8213	1.05329	0.526643	+	0.850087i	$0.323451\pi$
$972$	0	0
$973$	10.0673	0.322744
$974$	79.8740	2.55933
$975$	0	0
$976$	−20.4004	−0.653000
$977$	−41.3952	−1.32435	−0.662174	−	0.749350i	$-0.730366\pi$
$977$	−41.3952	−1.32435	−0.662174	+	0.749350i	$0.730366\pi$
$978$	0	0
$979$	−1.45528	−0.0465108
$980$	−33.3019	−1.06379
$981$	0	0
$982$	−2.85120	−0.0909855
$983$	−24.2842	−0.774545	−0.387273	−	0.921965i	$-0.626583\pi$
$983$	−24.2842	−0.774545	−0.387273	+	0.921965i	$0.626583\pi$
$984$	0	0
$985$	−9.59909	−0.305852
$986$	−92.6280	−2.94988
$987$	0	0
$988$	−0.407126	−0.0129524
$989$	33.6291	1.06934
$990$	0	0
$991$	−54.4286	−1.72898	−0.864491	−	0.502648i	$-0.832359\pi$
$991$	−54.4286	−1.72898	−0.864491	+	0.502648i	$0.832359\pi$
$992$	106.304	3.37516
$993$	0	0
$994$	−27.2928	−0.865676
$995$	18.4790	0.585822
$996$	0	0
$997$	−4.49758	−0.142440	−0.0712200	−	0.997461i	$-0.522689\pi$
$997$	−4.49758	−0.142440	−0.0712200	+	0.997461i	$0.522689\pi$
$998$	30.6489	0.970174
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.t.1.1		10
3.2	odd	2		1335.2.a.i.1.10	✓	10
15.14	odd	2		6675.2.a.ba.1.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.i.1.10	✓	10	3.2	odd	2
4005.2.a.t.1.1		10	1.1	even	1	trivial
6675.2.a.ba.1.1		10	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)

\(f(q)\)	\(=\)	\(q-2.72551 q^{2} +5.42842 q^{4} +1.00000 q^{5} +0.930190 q^{7} -9.34419 q^{8} +O(q^{10})\)	\(q-2.72551 q^{2} +5.42842 q^{4} +1.00000 q^{5} +0.930190 q^{7} -9.34419 q^{8} -2.72551 q^{10} -1.45528 q^{11} -0.247367 q^{13} -2.53525 q^{14} +14.6109 q^{16} +7.85189 q^{17} +0.303189 q^{19} +5.42842 q^{20} +3.96637 q^{22} +6.91586 q^{23} +1.00000 q^{25} +0.674203 q^{26} +5.04946 q^{28} +4.32833 q^{29} -5.03008 q^{31} -21.1337 q^{32} -21.4004 q^{34} +0.930190 q^{35} +8.26667 q^{37} -0.826345 q^{38} -9.34419 q^{40} -0.708782 q^{41} +4.86260 q^{43} -7.89985 q^{44} -18.8493 q^{46} +3.15702 q^{47} -6.13475 q^{49} -2.72551 q^{50} -1.34281 q^{52} -10.3222 q^{53} -1.45528 q^{55} -8.69187 q^{56} -11.7969 q^{58} -12.9723 q^{59} -1.39625 q^{61} +13.7095 q^{62} +28.3785 q^{64} -0.247367 q^{65} +13.3812 q^{67} +42.6233 q^{68} -2.53525 q^{70} +10.7654 q^{71} -3.64700 q^{73} -22.5309 q^{74} +1.64584 q^{76} -1.35368 q^{77} -2.82313 q^{79} +14.6109 q^{80} +1.93179 q^{82} -3.03787 q^{83} +7.85189 q^{85} -13.2531 q^{86} +13.5984 q^{88} +1.00000 q^{89} -0.230099 q^{91} +37.5422 q^{92} -8.60450 q^{94} +0.303189 q^{95} -15.3633 q^{97} +16.7203 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) \) 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8	\( 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8} - 4 q^{11} + 15 q^{13} + q^{14} + 22 q^{16} - 11 q^{17} + 14 q^{19} + 18 q^{20} + 10 q^{22} + 8 q^{23} + 10 q^{25} + 14 q^{26} + 36 q^{28} - 5 q^{29} - 8 q^{31} - q^{32} + 2 q^{34} + 9 q^{35} + 23 q^{37} - 8 q^{38} - 3 q^{40} - 13 q^{41} + 25 q^{43} + 8 q^{44} - 14 q^{46} + q^{47} + 21 q^{49} + 51 q^{52} - 9 q^{53} - 4 q^{55} - 15 q^{56} - 8 q^{58} + 15 q^{59} + 8 q^{61} + 8 q^{62} + 9 q^{64} + 15 q^{65} + 52 q^{67} + 28 q^{68} + q^{70} + 22 q^{71} + 34 q^{73} + 18 q^{74} + 14 q^{76} - 4 q^{77} - 3 q^{79} + 22 q^{80} + 17 q^{82} + 10 q^{83} - 11 q^{85} + 6 q^{86} + 4 q^{88} + 10 q^{89} + 18 q^{91} + 14 q^{92} - 43 q^{94} + 14 q^{95} + 34 q^{97} + 38 q^{98}+O(q^{100}) \) 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8 - 4 * q^11 + 15 * q^13 + q^14 + 22 * q^16 - 11 * q^17 + 14 * q^19 + 18 * q^20 + 10 * q^22 + 8 * q^23 + 10 * q^25 + 14 * q^26 + 36 * q^28 - 5 * q^29 - 8 * q^31 - q^32 + 2 * q^34 + 9 * q^35 + 23 * q^37 - 8 * q^38 - 3 * q^40 - 13 * q^41 + 25 * q^43 + 8 * q^44 - 14 * q^46 + q^47 + 21 * q^49 + 51 * q^52 - 9 * q^53 - 4 * q^55 - 15 * q^56 - 8 * q^58 + 15 * q^59 + 8 * q^61 + 8 * q^62 + 9 * q^64 + 15 * q^65 + 52 * q^67 + 28 * q^68 + q^70 + 22 * q^71 + 34 * q^73 + 18 * q^74 + 14 * q^76 - 4 * q^77 - 3 * q^79 + 22 * q^80 + 17 * q^82 + 10 * q^83 - 11 * q^85 + 6 * q^86 + 4 * q^88 + 10 * q^89 + 18 * q^91 + 14 * q^92 - 43 * q^94 + 14 * q^95 + 34 * q^97 + 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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