LMFDB - Embedded newform 4005.2.a.s.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.9800860095$
Analytic rank:	$1$
Dimension:	$10$
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} - x^{9} - 16x^{8} + 15x^{7} + 85x^{6} - 75x^{5} - 163x^{4} + 138x^{3} + 78x^{2} - 67x + 4 $ x^10 - x^9 - 16x^8 + 15x^7 + 85x^6 - 75x^5 - 163x^4 + 138x^3 + 78x^2 - 67x + 4
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			1.7
Root			$-0.875863$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.875863 q^{2} -1.23286 q^{4} -1.00000 q^{5} -3.38461 q^{7} -2.83155 q^{8} +O(q^{10})$	\(q+0.875863 q^{2} -1.23286 q^{4} -1.00000 q^{5} -3.38461 q^{7} -2.83155 q^{8} -0.875863 q^{10} +2.67178 q^{11} +2.37687 q^{13} -2.96445 q^{14} -0.0143162 q^{16} +5.74169 q^{17} -0.615105 q^{19} +1.23286 q^{20} +2.34011 q^{22} +3.22678 q^{23} +1.00000 q^{25} +2.08181 q^{26} +4.17276 q^{28} -6.94067 q^{29} -2.63976 q^{31} +5.65055 q^{32} +5.02893 q^{34} +3.38461 q^{35} +6.16674 q^{37} -0.538748 q^{38} +2.83155 q^{40} +8.24958 q^{41} -10.3534 q^{43} -3.29395 q^{44} +2.82622 q^{46} -3.00688 q^{47} +4.45558 q^{49} +0.875863 q^{50} -2.93036 q^{52} -14.0316 q^{53} -2.67178 q^{55} +9.58368 q^{56} -6.07907 q^{58} -11.9435 q^{59} +8.61089 q^{61} -2.31207 q^{62} +4.97774 q^{64} -2.37687 q^{65} -8.37278 q^{67} -7.07873 q^{68} +2.96445 q^{70} -9.34188 q^{71} -2.05814 q^{73} +5.40122 q^{74} +0.758342 q^{76} -9.04294 q^{77} +8.28582 q^{79} +0.0143162 q^{80} +7.22550 q^{82} -4.56778 q^{83} -5.74169 q^{85} -9.06816 q^{86} -7.56527 q^{88} +1.00000 q^{89} -8.04477 q^{91} -3.97819 q^{92} -2.63362 q^{94} +0.615105 q^{95} -1.11894 q^{97} +3.90248 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.875863	0.619328	0.309664	−	0.950846i	$-0.399783\pi$
$2$	0.875863	0.619328	0.309664	+	0.950846i	$0.399783\pi$
$3$	0	0
$3$	0	0
$4$	−1.23286	−0.616432
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.38461	−1.27926	−0.639631	−	0.768682i	$-0.720913\pi$
$7$	−3.38461	−1.27926	−0.639631	+	0.768682i	$0.720913\pi$
$8$	−2.83155	−1.00110
$9$	0	0
$10$	−0.875863	−0.276972
$11$	2.67178	0.805573	0.402786	−	0.915294i	$-0.368042\pi$
$11$	2.67178	0.805573	0.402786	+	0.915294i	$0.368042\pi$
$12$	0	0
$13$	2.37687	0.659225	0.329612	−	0.944116i	$-0.393082\pi$
$13$	2.37687	0.659225	0.329612	+	0.944116i	$0.393082\pi$
$14$	−2.96445	−0.792283
$15$	0	0
$16$	−0.0143162	−0.00357905
$17$	5.74169	1.39256	0.696282	−	0.717768i	$-0.254836\pi$
$17$	5.74169	1.39256	0.696282	+	0.717768i	$0.254836\pi$
$18$	0	0
$19$	−0.615105	−0.141115	−0.0705574	−	0.997508i	$-0.522478\pi$
$19$	−0.615105	−0.141115	−0.0705574	+	0.997508i	$0.522478\pi$
$20$	1.23286	0.275677
$21$	0	0
$22$	2.34011	0.498914
$23$	3.22678	0.672831	0.336415	−	0.941714i	$-0.390785\pi$
$23$	3.22678	0.672831	0.336415	+	0.941714i	$0.390785\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.08181	0.408277
$27$	0	0
$28$	4.17276	0.788578
$29$	−6.94067	−1.28885	−0.644425	−	0.764668i	$-0.722903\pi$
$29$	−6.94067	−1.28885	−0.644425	+	0.764668i	$0.722903\pi$
$30$	0	0
$31$	−2.63976	−0.474115	−0.237058	−	0.971496i	$-0.576183\pi$
$31$	−2.63976	−0.474115	−0.237058	+	0.971496i	$0.576183\pi$
$32$	5.65055	0.998886
$33$	0	0
$34$	5.02893	0.862455
$35$	3.38461	0.572103
$36$	0	0
$37$	6.16674	1.01381	0.506903	−	0.862003i	$-0.330790\pi$
$37$	6.16674	1.01381	0.506903	+	0.862003i	$0.330790\pi$
$38$	−0.538748	−0.0873964
$39$	0	0
$40$	2.83155	0.447707
$41$	8.24958	1.28837	0.644184	−	0.764870i	$-0.277197\pi$
$41$	8.24958	1.28837	0.644184	+	0.764870i	$0.277197\pi$
$42$	0	0
$43$	−10.3534	−1.57888	−0.789440	−	0.613828i	$-0.789629\pi$
$43$	−10.3534	−1.57888	−0.789440	+	0.613828i	$0.789629\pi$
$44$	−3.29395	−0.496581
$45$	0	0
$46$	2.82622	0.416703
$47$	−3.00688	−0.438599	−0.219299	−	0.975658i	$-0.570377\pi$
$47$	−3.00688	−0.438599	−0.219299	+	0.975658i	$0.570377\pi$
$48$	0	0
$49$	4.45558	0.636512
$50$	0.875863	0.123866
$51$	0	0
$52$	−2.93036	−0.406367
$53$	−14.0316	−1.92740	−0.963698	−	0.266996i	$-0.913969\pi$
$53$	−14.0316	−1.92740	−0.963698	+	0.266996i	$0.913969\pi$
$54$	0	0
$55$	−2.67178	−0.360263
$56$	9.58368	1.28067
$57$	0	0
$58$	−6.07907	−0.798221
$59$	−11.9435	−1.55491	−0.777457	−	0.628936i	$-0.783491\pi$
$59$	−11.9435	−1.55491	−0.777457	+	0.628936i	$0.783491\pi$
$60$	0	0
$61$	8.61089	1.10251	0.551256	−	0.834336i	$-0.314149\pi$
$61$	8.61089	1.10251	0.551256	+	0.834336i	$0.314149\pi$
$62$	−2.31207	−0.293633
$63$	0	0
$64$	4.97774	0.622218
$65$	−2.37687	−0.294814
$66$	0	0
$67$	−8.37278	−1.02290	−0.511449	−	0.859313i	$-0.670891\pi$
$67$	−8.37278	−1.02290	−0.511449	+	0.859313i	$0.670891\pi$
$68$	−7.07873	−0.858422
$69$	0	0
$70$	2.96445	0.354320
$71$	−9.34188	−1.10868	−0.554339	−	0.832291i	$-0.687029\pi$
$71$	−9.34188	−1.10868	−0.554339	+	0.832291i	$0.687029\pi$
$72$	0	0
$73$	−2.05814	−0.240887	−0.120444	−	0.992720i	$-0.538432\pi$
$73$	−2.05814	−0.240887	−0.120444	+	0.992720i	$0.538432\pi$
$74$	5.40122	0.627879
$75$	0	0
$76$	0.758342	0.0869877
$77$	−9.04294	−1.03054
$78$	0	0
$79$	8.28582	0.932227	0.466114	−	0.884725i	$-0.345654\pi$
$79$	8.28582	0.932227	0.466114	+	0.884725i	$0.345654\pi$
$80$	0.0143162	0.00160060
$81$	0	0
$82$	7.22550	0.797923
$83$	−4.56778	−0.501378	−0.250689	−	0.968068i	$-0.580657\pi$
$83$	−4.56778	−0.501378	−0.250689	+	0.968068i	$0.580657\pi$
$84$	0	0
$85$	−5.74169	−0.622774
$86$	−9.06816	−0.977845
$87$	0	0
$88$	−7.56527	−0.806461
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−8.04477	−0.843321
$92$	−3.97819	−0.414755
$93$	0	0
$94$	−2.63362	−0.271637
$95$	0.615105	0.0631085
$96$	0	0
$97$	−1.11894	−0.113611	−0.0568056	−	0.998385i	$-0.518092\pi$
$97$	−1.11894	−0.113611	−0.0568056	+	0.998385i	$0.518092\pi$
$98$	3.90248	0.394210
$99$	0	0
$100$	−1.23286	−0.123286
$101$	−11.1107	−1.10556	−0.552779	−	0.833328i	$-0.686433\pi$
$101$	−11.1107	−1.10556	−0.552779	+	0.833328i	$0.686433\pi$
$102$	0	0
$103$	0.805772	0.0793951	0.0396976	−	0.999212i	$-0.487361\pi$
$103$	0.805772	0.0793951	0.0396976	+	0.999212i	$0.487361\pi$
$104$	−6.73021	−0.659951
$105$	0	0
$106$	−12.2898	−1.19369
$107$	−13.4999	−1.30508	−0.652542	−	0.757752i	$-0.726298\pi$
$107$	−13.4999	−1.30508	−0.652542	+	0.757752i	$0.726298\pi$
$108$	0	0
$109$	11.1153	1.06465	0.532326	−	0.846539i	$-0.321318\pi$
$109$	11.1153	1.06465	0.532326	+	0.846539i	$0.321318\pi$
$110$	−2.34011	−0.223121
$111$	0	0
$112$	0.0484547	0.00457854
$113$	16.7283	1.57366	0.786832	−	0.617167i	$-0.211720\pi$
$113$	16.7283	1.57366	0.786832	+	0.617167i	$0.211720\pi$
$114$	0	0
$115$	−3.22678	−0.300899
$116$	8.55690	0.794489
$117$	0	0
$118$	−10.4609	−0.963003
$119$	−19.4334	−1.78146
$120$	0	0
$121$	−3.86158	−0.351053
$122$	7.54196	0.682817
$123$	0	0
$124$	3.25447	0.292260
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.9842	1.68457	0.842286	−	0.539031i	$-0.181210\pi$
$127$	18.9842	1.68457	0.842286	+	0.539031i	$0.181210\pi$
$128$	−6.94129	−0.613529
$129$	0	0
$130$	−2.08181	−0.182587
$131$	−15.7908	−1.37965	−0.689826	−	0.723975i	$-0.742313\pi$
$131$	−15.7908	−1.37965	−0.689826	+	0.723975i	$0.742313\pi$
$132$	0	0
$133$	2.08189	0.180523
$134$	−7.33341	−0.633510
$135$	0	0
$136$	−16.2579	−1.39410
$137$	−10.2872	−0.878899	−0.439449	−	0.898267i	$-0.644826\pi$
$137$	−10.2872	−0.878899	−0.439449	+	0.898267i	$0.644826\pi$
$138$	0	0
$139$	−6.50136	−0.551439	−0.275719	−	0.961238i	$-0.588916\pi$
$139$	−6.50136	−0.551439	−0.275719	+	0.961238i	$0.588916\pi$
$140$	−4.17276	−0.352663
$141$	0	0
$142$	−8.18221	−0.686636
$143$	6.35047	0.531053
$144$	0	0
$145$	6.94067	0.576391
$146$	−1.80265	−0.149188
$147$	0	0
$148$	−7.60276	−0.624943
$149$	−23.1798	−1.89896	−0.949482	−	0.313821i	$-0.898391\pi$
$149$	−23.1798	−1.89896	−0.949482	+	0.313821i	$0.898391\pi$
$150$	0	0
$151$	10.1261	0.824053	0.412026	−	0.911172i	$-0.364821\pi$
$151$	10.1261	0.824053	0.412026	+	0.911172i	$0.364821\pi$
$152$	1.74170	0.141270
$153$	0	0
$154$	−7.92037	−0.638242
$155$	2.63976	0.212031
$156$	0	0
$157$	1.69594	0.135351	0.0676754	−	0.997707i	$-0.478442\pi$
$157$	1.69594	0.135351	0.0676754	+	0.997707i	$0.478442\pi$
$158$	7.25724	0.577355
$159$	0	0
$160$	−5.65055	−0.446715
$161$	−10.9214	−0.860727
$162$	0	0
$163$	−22.7425	−1.78133	−0.890663	−	0.454664i	$-0.849759\pi$
$163$	−22.7425	−1.78133	−0.890663	+	0.454664i	$0.849759\pi$
$164$	−10.1706	−0.794192
$165$	0	0
$166$	−4.00074	−0.310518
$167$	19.8881	1.53899	0.769494	−	0.638654i	$-0.220508\pi$
$167$	19.8881	1.53899	0.769494	+	0.638654i	$0.220508\pi$
$168$	0	0
$169$	−7.35050	−0.565423
$170$	−5.02893	−0.385702
$171$	0	0
$172$	12.7644	0.973273
$173$	−1.85953	−0.141377	−0.0706887	−	0.997498i	$-0.522520\pi$
$173$	−1.85953	−0.141377	−0.0706887	+	0.997498i	$0.522520\pi$
$174$	0	0
$175$	−3.38461	−0.255852
$176$	−0.0382498	−0.00288318
$177$	0	0
$178$	0.875863	0.0656487
$179$	−18.5991	−1.39016	−0.695081	−	0.718932i	$-0.744631\pi$
$179$	−18.5991	−1.39016	−0.695081	+	0.718932i	$0.744631\pi$
$180$	0	0
$181$	−8.42729	−0.626395	−0.313198	−	0.949688i	$-0.601400\pi$
$181$	−8.42729	−0.626395	−0.313198	+	0.949688i	$0.601400\pi$
$182$	−7.04612	−0.522293
$183$	0	0
$184$	−9.13678	−0.673572
$185$	−6.16674	−0.453388
$186$	0	0
$187$	15.3405	1.12181
$188$	3.70708	0.270366
$189$	0	0
$190$	0.538748	0.0390849
$191$	3.47517	0.251455	0.125727	−	0.992065i	$-0.459874\pi$
$191$	3.47517	0.251455	0.125727	+	0.992065i	$0.459874\pi$
$192$	0	0
$193$	8.50184	0.611976	0.305988	−	0.952035i	$-0.401013\pi$
$193$	8.50184	0.611976	0.305988	+	0.952035i	$0.401013\pi$
$194$	−0.980038	−0.0703626
$195$	0	0
$196$	−5.49313	−0.392366
$197$	−3.01734	−0.214976	−0.107488	−	0.994206i	$-0.534281\pi$
$197$	−3.01734	−0.214976	−0.107488	+	0.994206i	$0.534281\pi$
$198$	0	0
$199$	4.49936	0.318951	0.159475	−	0.987202i	$-0.449020\pi$
$199$	4.49936	0.318951	0.159475	+	0.987202i	$0.449020\pi$
$200$	−2.83155	−0.200221
$201$	0	0
$202$	−9.73147	−0.684704
$203$	23.4915	1.64878
$204$	0	0
$205$	−8.24958	−0.576176
$206$	0.705746	0.0491716
$207$	0	0
$208$	−0.0340277	−0.00235940
$209$	−1.64343	−0.113678
$210$	0	0
$211$	3.98091	0.274057	0.137029	−	0.990567i	$-0.456245\pi$
$211$	3.98091	0.274057	0.137029	+	0.990567i	$0.456245\pi$
$212$	17.2991	1.18811
$213$	0	0
$214$	−11.8241	−0.808276
$215$	10.3534	0.706097
$216$	0	0
$217$	8.93457	0.606518
$218$	9.73547	0.659369
$219$	0	0
$220$	3.29395	0.222078
$221$	13.6472	0.918013
$222$	0	0
$223$	−23.5421	−1.57650	−0.788248	−	0.615357i	$-0.789012\pi$
$223$	−23.5421	−1.57650	−0.788248	+	0.615357i	$0.789012\pi$
$224$	−19.1249	−1.27784
$225$	0	0
$226$	14.6517	0.974615
$227$	6.38612	0.423862	0.211931	−	0.977285i	$-0.432025\pi$
$227$	6.38612	0.423862	0.211931	+	0.977285i	$0.432025\pi$
$228$	0	0
$229$	8.12371	0.536830	0.268415	−	0.963303i	$-0.413500\pi$
$229$	8.12371	0.536830	0.268415	+	0.963303i	$0.413500\pi$
$230$	−2.82622	−0.186355
$231$	0	0
$232$	19.6528	1.29027
$233$	−11.2615	−0.737764	−0.368882	−	0.929476i	$-0.620259\pi$
$233$	−11.2615	−0.737764	−0.368882	+	0.929476i	$0.620259\pi$
$234$	0	0
$235$	3.00688	0.196147
$236$	14.7247	0.958499
$237$	0	0
$238$	−17.0210	−1.10331
$239$	7.94210	0.513732	0.256866	−	0.966447i	$-0.417310\pi$
$239$	7.94210	0.513732	0.256866	+	0.966447i	$0.417310\pi$
$240$	0	0
$241$	−10.4196	−0.671185	−0.335593	−	0.942007i	$-0.608937\pi$
$241$	−10.4196	−0.671185	−0.335593	+	0.942007i	$0.608937\pi$
$242$	−3.38221	−0.217417
$243$	0	0
$244$	−10.6161	−0.679624
$245$	−4.45558	−0.284657
$246$	0	0
$247$	−1.46202	−0.0930264
$248$	7.47461	0.474638
$249$	0	0
$250$	−0.875863	−0.0553944
$251$	−8.44349	−0.532948	−0.266474	−	0.963842i	$-0.585859\pi$
$251$	−8.44349	−0.532948	−0.266474	+	0.963842i	$0.585859\pi$
$252$	0	0
$253$	8.62126	0.542014
$254$	16.6275	1.04330
$255$	0	0
$256$	−16.0351	−1.00219
$257$	24.7981	1.54687	0.773433	−	0.633878i	$-0.218538\pi$
$257$	24.7981	1.54687	0.773433	+	0.633878i	$0.218538\pi$
$258$	0	0
$259$	−20.8720	−1.29692
$260$	2.93036	0.181733
$261$	0	0
$262$	−13.8306	−0.854458
$263$	−9.10151	−0.561223	−0.280612	−	0.959821i	$-0.590537\pi$
$263$	−9.10151	−0.561223	−0.280612	+	0.959821i	$0.590537\pi$
$264$	0	0
$265$	14.0316	0.861957
$266$	1.82345	0.111803
$267$	0	0
$268$	10.3225	0.630548
$269$	0.299276	0.0182472	0.00912359	−	0.999958i	$-0.497096\pi$
$269$	0.299276	0.0182472	0.00912359	+	0.999958i	$0.497096\pi$
$270$	0	0
$271$	13.4021	0.814118	0.407059	−	0.913402i	$-0.366554\pi$
$271$	13.4021	0.814118	0.407059	+	0.913402i	$0.366554\pi$
$272$	−0.0821992	−0.00498406
$273$	0	0
$274$	−9.01021	−0.544327
$275$	2.67178	0.161115
$276$	0	0
$277$	−25.1730	−1.51250	−0.756250	−	0.654282i	$-0.772971\pi$
$277$	−25.1730	−1.51250	−0.756250	+	0.654282i	$0.772971\pi$
$278$	−5.69430	−0.341522
$279$	0	0
$280$	−9.58368	−0.572734
$281$	−2.56597	−0.153073	−0.0765366	−	0.997067i	$-0.524386\pi$
$281$	−2.56597	−0.153073	−0.0765366	+	0.997067i	$0.524386\pi$
$282$	0	0
$283$	24.9990	1.48604	0.743018	−	0.669271i	$-0.233394\pi$
$283$	24.9990	1.48604	0.743018	+	0.669271i	$0.233394\pi$
$284$	11.5173	0.683425
$285$	0	0
$286$	5.56214	0.328896
$287$	−27.9216	−1.64816
$288$	0	0
$289$	15.9670	0.939236
$290$	6.07907	0.356975
$291$	0	0
$292$	2.53741	0.148491
$293$	−1.74445	−0.101912	−0.0509561	−	0.998701i	$-0.516227\pi$
$293$	−1.74445	−0.101912	−0.0509561	+	0.998701i	$0.516227\pi$
$294$	0	0
$295$	11.9435	0.695379
$296$	−17.4614	−1.01492
$297$	0	0
$298$	−20.3023	−1.17608
$299$	7.66964	0.443547
$300$	0	0
$301$	35.0422	2.01980
$302$	8.86910	0.510359
$303$	0	0
$304$	0.00880597	0.000505057 0
$305$	−8.61089	−0.493058
$306$	0	0
$307$	−14.1590	−0.808096	−0.404048	−	0.914738i	$-0.632397\pi$
$307$	−14.1590	−0.808096	−0.404048	+	0.914738i	$0.632397\pi$
$308$	11.1487	0.635257
$309$	0	0
$310$	2.31207	0.131317
$311$	35.0639	1.98829	0.994144	−	0.108059i	$-0.0344636\pi$
$311$	35.0639	1.98829	0.994144	+	0.108059i	$0.0344636\pi$
$312$	0	0
$313$	15.5648	0.879776	0.439888	−	0.898053i	$-0.355018\pi$
$313$	15.5648	0.879776	0.439888	+	0.898053i	$0.355018\pi$
$314$	1.48541	0.0838266
$315$	0	0
$316$	−10.2153	−0.574655
$317$	12.5534	0.705068	0.352534	−	0.935799i	$-0.385320\pi$
$317$	12.5534	0.705068	0.352534	+	0.935799i	$0.385320\pi$
$318$	0	0
$319$	−18.5440	−1.03826
$320$	−4.97774	−0.278264
$321$	0	0
$322$	−9.56565	−0.533073
$323$	−3.53174	−0.196512
$324$	0	0
$325$	2.37687	0.131845
$326$	−19.9193	−1.10323
$327$	0	0
$328$	−23.3591	−1.28979
$329$	10.1771	0.561083
$330$	0	0
$331$	−17.1194	−0.940968	−0.470484	−	0.882408i	$-0.655921\pi$
$331$	−17.1194	−0.940968	−0.470484	+	0.882408i	$0.655921\pi$
$332$	5.63145	0.309066
$333$	0	0
$334$	17.4193	0.953139
$335$	8.37278	0.457454
$336$	0	0
$337$	−17.0678	−0.929743	−0.464871	−	0.885378i	$-0.653899\pi$
$337$	−17.0678	−0.929743	−0.464871	+	0.885378i	$0.653899\pi$
$338$	−6.43803	−0.350182
$339$	0	0
$340$	7.07873	0.383898
$341$	−7.05287	−0.381934
$342$	0	0
$343$	8.61186	0.464997
$344$	29.3161	1.58062
$345$	0	0
$346$	−1.62869	−0.0875590
$347$	5.82150	0.312515	0.156257	−	0.987716i	$-0.450057\pi$
$347$	5.82150	0.312515	0.156257	+	0.987716i	$0.450057\pi$
$348$	0	0
$349$	15.1221	0.809468	0.404734	−	0.914434i	$-0.367364\pi$
$349$	15.1221	0.809468	0.404734	+	0.914434i	$0.367364\pi$
$350$	−2.96445	−0.158457
$351$	0	0
$352$	15.0970	0.804675
$353$	−11.8018	−0.628145	−0.314073	−	0.949399i	$-0.601694\pi$
$353$	−11.8018	−0.628145	−0.314073	+	0.949399i	$0.601694\pi$
$354$	0	0
$355$	9.34188	0.495816
$356$	−1.23286	−0.0653417
$357$	0	0
$358$	−16.2903	−0.860967
$359$	−17.1087	−0.902962	−0.451481	−	0.892281i	$-0.649104\pi$
$359$	−17.1087	−0.902962	−0.451481	+	0.892281i	$0.649104\pi$
$360$	0	0
$361$	−18.6216	−0.980087
$362$	−7.38115	−0.387944
$363$	0	0
$364$	9.91811	0.519850
$365$	2.05814	0.107728
$366$	0	0
$367$	−27.6419	−1.44289	−0.721447	−	0.692470i	$-0.756523\pi$
$367$	−27.6419	−1.44289	−0.721447	+	0.692470i	$0.756523\pi$
$368$	−0.0461953	−0.00240809
$369$	0	0
$370$	−5.40122	−0.280796
$371$	47.4917	2.46564
$372$	0	0
$373$	21.1039	1.09272	0.546358	−	0.837552i	$-0.316014\pi$
$373$	21.1039	1.09272	0.546358	+	0.837552i	$0.316014\pi$
$374$	13.4362	0.694770
$375$	0	0
$376$	8.51412	0.439082
$377$	−16.4971	−0.849642
$378$	0	0
$379$	−17.7915	−0.913889	−0.456945	−	0.889495i	$-0.651056\pi$
$379$	−17.7915	−0.913889	−0.456945	+	0.889495i	$0.651056\pi$
$380$	−0.758342	−0.0389021
$381$	0	0
$382$	3.04378	0.155733
$383$	−17.5838	−0.898489	−0.449244	−	0.893409i	$-0.648307\pi$
$383$	−17.5838	−0.898489	−0.449244	+	0.893409i	$0.648307\pi$
$384$	0	0
$385$	9.04294	0.460871
$386$	7.44644	0.379014
$387$	0	0
$388$	1.37950	0.0700336
$389$	−0.752664	−0.0381616	−0.0190808	−	0.999818i	$-0.506074\pi$
$389$	−0.752664	−0.0381616	−0.0190808	+	0.999818i	$0.506074\pi$
$390$	0	0
$391$	18.5272	0.936960
$392$	−12.6162	−0.637213
$393$	0	0
$394$	−2.64277	−0.133141
$395$	−8.28582	−0.416905
$396$	0	0
$397$	37.6942	1.89182	0.945910	−	0.324430i	$-0.105172\pi$
$397$	37.6942	1.89182	0.945910	+	0.324430i	$0.105172\pi$
$398$	3.94082	0.197535
$399$	0	0
$400$	−0.0143162	−0.000715810 0
$401$	1.33309	0.0665711	0.0332856	−	0.999446i	$-0.489403\pi$
$401$	1.33309	0.0665711	0.0332856	+	0.999446i	$0.489403\pi$
$402$	0	0
$403$	−6.27437	−0.312549
$404$	13.6980	0.681502
$405$	0	0
$406$	20.5753	1.02113
$407$	16.4762	0.816695
$408$	0	0
$409$	−13.0026	−0.642939	−0.321469	−	0.946920i	$-0.604177\pi$
$409$	−13.0026	−0.642939	−0.321469	+	0.946920i	$0.604177\pi$
$410$	−7.22550	−0.356842
$411$	0	0
$412$	−0.993408	−0.0489417
$413$	40.4242	1.98914
$414$	0	0
$415$	4.56778	0.224223
$416$	13.4306	0.658490
$417$	0	0
$418$	−1.43942	−0.0704042
$419$	−18.9659	−0.926544	−0.463272	−	0.886216i	$-0.653325\pi$
$419$	−18.9659	−0.926544	−0.463272	+	0.886216i	$0.653325\pi$
$420$	0	0
$421$	−10.5812	−0.515696	−0.257848	−	0.966185i	$-0.583013\pi$
$421$	−10.5812	−0.515696	−0.257848	+	0.966185i	$0.583013\pi$
$422$	3.48673	0.169732
$423$	0	0
$424$	39.7313	1.92952
$425$	5.74169	0.278513
$426$	0	0
$427$	−29.1445	−1.41040
$428$	16.6435	0.804496
$429$	0	0
$430$	9.06816	0.437306
$431$	22.3563	1.07687	0.538433	−	0.842668i	$-0.319017\pi$
$431$	22.3563	1.07687	0.538433	+	0.842668i	$0.319017\pi$
$432$	0	0
$433$	−21.7473	−1.04511	−0.522555	−	0.852605i	$-0.675021\pi$
$433$	−21.7473	−1.04511	−0.522555	+	0.852605i	$0.675021\pi$
$434$	7.82545	0.375634
$435$	0	0
$436$	−13.7037	−0.656286
$437$	−1.98481	−0.0949464
$438$	0	0
$439$	−10.1758	−0.485665	−0.242832	−	0.970068i	$-0.578076\pi$
$439$	−10.1758	−0.485665	−0.242832	+	0.970068i	$0.578076\pi$
$440$	7.56527	0.360660
$441$	0	0
$442$	11.9531	0.568551
$443$	−17.1712	−0.815829	−0.407915	−	0.913020i	$-0.633744\pi$
$443$	−17.1712	−0.815829	−0.407915	+	0.913020i	$0.633744\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−20.6197	−0.976369
$447$	0	0
$448$	−16.8477	−0.795979
$449$	−31.6369	−1.49304	−0.746518	−	0.665365i	$-0.768276\pi$
$449$	−31.6369	−1.49304	−0.746518	+	0.665365i	$0.768276\pi$
$450$	0	0
$451$	22.0411	1.03787
$452$	−20.6237	−0.970058
$453$	0	0
$454$	5.59337	0.262510
$455$	8.04477	0.377145
$456$	0	0
$457$	12.8795	0.602476	0.301238	−	0.953549i	$-0.402600\pi$
$457$	12.8795	0.602476	0.301238	+	0.953549i	$0.402600\pi$
$458$	7.11526	0.332474
$459$	0	0
$460$	3.97819	0.185484
$461$	1.64108	0.0764328	0.0382164	−	0.999269i	$-0.487832\pi$
$461$	1.64108	0.0764328	0.0382164	+	0.999269i	$0.487832\pi$
$462$	0	0
$463$	32.2433	1.49847	0.749236	−	0.662304i	$-0.230421\pi$
$463$	32.2433	1.49847	0.749236	+	0.662304i	$0.230421\pi$
$464$	0.0993640	0.00461286
$465$	0	0
$466$	−9.86351	−0.456918
$467$	−25.8544	−1.19640	−0.598200	−	0.801347i	$-0.704117\pi$
$467$	−25.8544	−1.19640	−0.598200	+	0.801347i	$0.704117\pi$
$468$	0	0
$469$	28.3386	1.30856
$470$	2.63362	0.121480
$471$	0	0
$472$	33.8186	1.55663
$473$	−27.6621	−1.27190
$474$	0	0
$475$	−0.615105	−0.0282230
$476$	23.9587	1.09815
$477$	0	0
$478$	6.95619	0.318169
$479$	−21.4809	−0.981489	−0.490744	−	0.871304i	$-0.663275\pi$
$479$	−21.4809	−0.981489	−0.490744	+	0.871304i	$0.663275\pi$
$480$	0	0
$481$	14.6575	0.668326
$482$	−9.12613	−0.415684
$483$	0	0
$484$	4.76080	0.216400
$485$	1.11894	0.0508085
$486$	0	0
$487$	25.8281	1.17038	0.585190	−	0.810896i	$-0.301020\pi$
$487$	25.8281	1.17038	0.585190	+	0.810896i	$0.301020\pi$
$488$	−24.3821	−1.10373
$489$	0	0
$490$	−3.90248	−0.176296
$491$	−20.8321	−0.940138	−0.470069	−	0.882630i	$-0.655771\pi$
$491$	−20.8321	−0.940138	−0.470069	+	0.882630i	$0.655771\pi$
$492$	0	0
$493$	−39.8512	−1.79481
$494$	−1.28053	−0.0576139
$495$	0	0
$496$	0.0377914	0.00169688
$497$	31.6186	1.41829
$498$	0	0
$499$	−17.9737	−0.804612	−0.402306	−	0.915505i	$-0.631791\pi$
$499$	−17.9737	−0.804612	−0.402306	+	0.915505i	$0.631791\pi$
$500$	1.23286	0.0551354
$501$	0	0
$502$	−7.39534	−0.330070
$503$	−19.1188	−0.852467	−0.426234	−	0.904613i	$-0.640160\pi$
$503$	−19.1188	−0.852467	−0.426234	+	0.904613i	$0.640160\pi$
$504$	0	0
$505$	11.1107	0.494421
$506$	7.55104	0.335685
$507$	0	0
$508$	−23.4049	−1.03842
$509$	−30.5545	−1.35430	−0.677152	−	0.735843i	$-0.736786\pi$
$509$	−30.5545	−1.35430	−0.677152	+	0.735843i	$0.736786\pi$
$510$	0	0
$511$	6.96601	0.308158
$512$	−0.161968	−0.00715806
$513$	0	0
$514$	21.7198	0.958018
$515$	−0.805772	−0.0355066
$516$	0	0
$517$	−8.03374	−0.353323
$518$	−18.2810	−0.803222
$519$	0	0
$520$	6.73021	0.295139
$521$	−32.8472	−1.43906	−0.719532	−	0.694460i	$-0.755644\pi$
$521$	−32.8472	−1.43906	−0.719532	+	0.694460i	$0.755644\pi$
$522$	0	0
$523$	3.94562	0.172530	0.0862649	−	0.996272i	$-0.472507\pi$
$523$	3.94562	0.172530	0.0862649	+	0.996272i	$0.472507\pi$
$524$	19.4680	0.850462
$525$	0	0
$526$	−7.97168	−0.347582
$527$	−15.1567	−0.660236
$528$	0	0
$529$	−12.5879	−0.547299
$530$	12.2898	0.533835
$531$	0	0
$532$	−2.56669	−0.111280
$533$	19.6082	0.849324
$534$	0	0
$535$	13.4999	0.583652
$536$	23.7079	1.02403
$537$	0	0
$538$	0.262125	0.0113010
$539$	11.9043	0.512756
$540$	0	0
$541$	−38.2187	−1.64315	−0.821575	−	0.570100i	$-0.806904\pi$
$541$	−38.2187	−1.64315	−0.821575	+	0.570100i	$0.806904\pi$
$542$	11.7384	0.504207
$543$	0	0
$544$	32.4437	1.39101
$545$	−11.1153	−0.476127
$546$	0	0
$547$	−5.53747	−0.236765	−0.118383	−	0.992968i	$-0.537771\pi$
$547$	−5.53747	−0.236765	−0.118383	+	0.992968i	$0.537771\pi$
$548$	12.6828	0.541781
$549$	0	0
$550$	2.34011	0.0997828
$551$	4.26924	0.181876
$552$	0	0
$553$	−28.0443	−1.19256
$554$	−22.0481	−0.936735
$555$	0	0
$556$	8.01530	0.339925
$557$	47.1683	1.99858	0.999292	−	0.0376289i	$-0.0119805\pi$
$557$	47.1683	1.99858	0.999292	+	0.0376289i	$0.0119805\pi$
$558$	0	0
$559$	−24.6087	−1.04084
$560$	−0.0484547	−0.00204759
$561$	0	0
$562$	−2.24744	−0.0948026
$563$	−11.0354	−0.465086	−0.232543	−	0.972586i	$-0.574705\pi$
$563$	−11.0354	−0.465086	−0.232543	+	0.972586i	$0.574705\pi$
$564$	0	0
$565$	−16.7283	−0.703764
$566$	21.8957	0.920345
$567$	0	0
$568$	26.4520	1.10990
$569$	−18.1727	−0.761841	−0.380921	−	0.924608i	$-0.624393\pi$
$569$	−18.1727	−0.761841	−0.380921	+	0.924608i	$0.624393\pi$
$570$	0	0
$571$	1.35796	0.0568288	0.0284144	−	0.999596i	$-0.490954\pi$
$571$	1.35796	0.0568288	0.0284144	+	0.999596i	$0.490954\pi$
$572$	−7.82927	−0.327358
$573$	0	0
$574$	−24.4555	−1.02075
$575$	3.22678	0.134566
$576$	0	0
$577$	−43.8436	−1.82523	−0.912617	−	0.408816i	$-0.865942\pi$
$577$	−43.8436	−1.82523	−0.912617	+	0.408816i	$0.865942\pi$
$578$	13.9849	0.581696
$579$	0	0
$580$	−8.55690	−0.355306
$581$	15.4601	0.641395
$582$	0	0
$583$	−37.4895	−1.55266
$584$	5.82773	0.241153
$585$	0	0
$586$	−1.52790	−0.0631171
$587$	−10.6003	−0.437521	−0.218760	−	0.975779i	$-0.570201\pi$
$587$	−10.6003	−0.437521	−0.218760	+	0.975779i	$0.570201\pi$
$588$	0	0
$589$	1.62373	0.0669047
$590$	10.4609	0.430668
$591$	0	0
$592$	−0.0882843	−0.00362846
$593$	−0.417749	−0.0171549	−0.00857746	−	0.999963i	$-0.502730\pi$
$593$	−0.417749	−0.0171549	−0.00857746	+	0.999963i	$0.502730\pi$
$594$	0	0
$595$	19.4334	0.796691
$596$	28.5776	1.17058
$597$	0	0
$598$	6.71755	0.274701
$599$	−25.8964	−1.05810	−0.529048	−	0.848592i	$-0.677451\pi$
$599$	−25.8964	−1.05810	−0.529048	+	0.848592i	$0.677451\pi$
$600$	0	0
$601$	30.8174	1.25707	0.628534	−	0.777782i	$-0.283655\pi$
$601$	30.8174	1.25707	0.628534	+	0.777782i	$0.283655\pi$
$602$	30.6922	1.25092
$603$	0	0
$604$	−12.4841	−0.507973
$605$	3.86158	0.156996
$606$	0	0
$607$	−20.1756	−0.818904	−0.409452	−	0.912332i	$-0.634280\pi$
$607$	−20.1756	−0.818904	−0.409452	+	0.912332i	$0.634280\pi$
$608$	−3.47568	−0.140958
$609$	0	0
$610$	−7.54196	−0.305365
$611$	−7.14696	−0.289135
$612$	0	0
$613$	15.6857	0.633540	0.316770	−	0.948502i	$-0.397402\pi$
$613$	15.6857	0.633540	0.316770	+	0.948502i	$0.397402\pi$
$614$	−12.4013	−0.500477
$615$	0	0
$616$	25.6055	1.03167
$617$	7.52720	0.303034	0.151517	−	0.988455i	$-0.451584\pi$
$617$	7.52720	0.303034	0.151517	+	0.988455i	$0.451584\pi$
$618$	0	0
$619$	−25.1005	−1.00887	−0.504437	−	0.863449i	$-0.668300\pi$
$619$	−25.1005	−1.00887	−0.504437	+	0.863449i	$0.668300\pi$
$620$	−3.25447	−0.130703
$621$	0	0
$622$	30.7111	1.23140
$623$	−3.38461	−0.135602
$624$	0	0
$625$	1.00000	0.0400000
$626$	13.6326	0.544870
$627$	0	0
$628$	−2.09086	−0.0834346
$629$	35.4075	1.41179
$630$	0	0
$631$	1.67962	0.0668647	0.0334323	−	0.999441i	$-0.489356\pi$
$631$	1.67962	0.0668647	0.0334323	+	0.999441i	$0.489356\pi$
$632$	−23.4617	−0.933255
$633$	0	0
$634$	10.9950	0.436668
$635$	−18.9842	−0.753363
$636$	0	0
$637$	10.5903	0.419604
$638$	−16.2420	−0.643025
$639$	0	0
$640$	6.94129	0.274378
$641$	−12.8298	−0.506745	−0.253373	−	0.967369i	$-0.581540\pi$
$641$	−12.8298	−0.506745	−0.253373	+	0.967369i	$0.581540\pi$
$642$	0	0
$643$	43.3386	1.70911	0.854554	−	0.519363i	$-0.173831\pi$
$643$	43.3386	1.70911	0.854554	+	0.519363i	$0.173831\pi$
$644$	13.4646	0.530580
$645$	0	0
$646$	−3.09332	−0.121705
$647$	32.9571	1.29568	0.647839	−	0.761777i	$-0.275673\pi$
$647$	32.9571	1.29568	0.647839	+	0.761777i	$0.275673\pi$
$648$	0	0
$649$	−31.9105	−1.25260
$650$	2.08181	0.0816553
$651$	0	0
$652$	28.0384	1.09807
$653$	33.5599	1.31330	0.656651	−	0.754195i	$-0.271972\pi$
$653$	33.5599	1.31330	0.656651	+	0.754195i	$0.271972\pi$
$654$	0	0
$655$	15.7908	0.616999
$656$	−0.118103	−0.00461113
$657$	0	0
$658$	8.91376	0.347495
$659$	6.17367	0.240492	0.120246	−	0.992744i	$-0.461632\pi$
$659$	6.17367	0.240492	0.120246	+	0.992744i	$0.461632\pi$
$660$	0	0
$661$	12.2661	0.477096	0.238548	−	0.971131i	$-0.423329\pi$
$661$	12.2661	0.477096	0.238548	+	0.971131i	$0.423329\pi$
$662$	−14.9943	−0.582768
$663$	0	0
$664$	12.9339	0.501931
$665$	−2.08189	−0.0807323
$666$	0	0
$667$	−22.3960	−0.867178
$668$	−24.5193	−0.948682
$669$	0	0
$670$	7.33341	0.283314
$671$	23.0064	0.888153
$672$	0	0
$673$	30.2308	1.16531	0.582657	−	0.812718i	$-0.302013\pi$
$673$	30.2308	1.16531	0.582657	+	0.812718i	$0.302013\pi$
$674$	−14.9491	−0.575816
$675$	0	0
$676$	9.06217	0.348545
$677$	−32.0331	−1.23113	−0.615565	−	0.788086i	$-0.711072\pi$
$677$	−32.0331	−1.23113	−0.615565	+	0.788086i	$0.711072\pi$
$678$	0	0
$679$	3.78718	0.145338
$680$	16.2579	0.623460
$681$	0	0
$682$	−6.17735	−0.236543
$683$	−29.9273	−1.14514	−0.572569	−	0.819857i	$-0.694053\pi$
$683$	−29.9273	−1.14514	−0.572569	+	0.819857i	$0.694053\pi$
$684$	0	0
$685$	10.2872	0.393055
$686$	7.54281	0.287986
$687$	0	0
$688$	0.148221	0.00565089
$689$	−33.3514	−1.27059
$690$	0	0
$691$	16.9213	0.643717	0.321859	−	0.946788i	$-0.395692\pi$
$691$	16.9213	0.643717	0.321859	+	0.946788i	$0.395692\pi$
$692$	2.29255	0.0871495
$693$	0	0
$694$	5.09884	0.193549
$695$	6.50136	0.246611
$696$	0	0
$697$	47.3665	1.79414
$698$	13.2449	0.501326
$699$	0	0
$700$	4.17276	0.157716
$701$	−18.7884	−0.709627	−0.354814	−	0.934937i	$-0.615456\pi$
$701$	−18.7884	−0.709627	−0.354814	+	0.934937i	$0.615456\pi$
$702$	0	0
$703$	−3.79320	−0.143063
$704$	13.2994	0.501241
$705$	0	0
$706$	−10.3367	−0.389028
$707$	37.6055	1.41430
$708$	0	0
$709$	−3.14429	−0.118086	−0.0590432	−	0.998255i	$-0.518805\pi$
$709$	−3.14429	−0.118086	−0.0590432	+	0.998255i	$0.518805\pi$
$710$	8.18221	0.307073
$711$	0	0
$712$	−2.83155	−0.106117
$713$	−8.51794	−0.318999
$714$	0	0
$715$	−6.35047	−0.237494
$716$	22.9302	0.856941
$717$	0	0
$718$	−14.9849	−0.559230
$719$	−31.0631	−1.15846	−0.579229	−	0.815165i	$-0.696646\pi$
$719$	−31.0631	−1.15846	−0.579229	+	0.815165i	$0.696646\pi$
$720$	0	0
$721$	−2.72722	−0.101567
$722$	−16.3100	−0.606996
$723$	0	0
$724$	10.3897	0.386130
$725$	−6.94067	−0.257770
$726$	0	0
$727$	−9.07483	−0.336567	−0.168283	−	0.985739i	$-0.553822\pi$
$727$	−9.07483	−0.336567	−0.168283	+	0.985739i	$0.553822\pi$
$728$	22.7791	0.844251
$729$	0	0
$730$	1.80265	0.0667191
$731$	−59.4461	−2.19869
$732$	0	0
$733$	26.2384	0.969139	0.484569	−	0.874753i	$-0.338976\pi$
$733$	26.2384	0.969139	0.484569	+	0.874753i	$0.338976\pi$
$734$	−24.2105	−0.893625
$735$	0	0
$736$	18.2331	0.672081
$737$	−22.3703	−0.824019
$738$	0	0
$739$	10.2337	0.376452	0.188226	−	0.982126i	$-0.439726\pi$
$739$	10.2337	0.376452	0.188226	+	0.982126i	$0.439726\pi$
$740$	7.60276	0.279483
$741$	0	0
$742$	41.5962	1.52704
$743$	18.6803	0.685313	0.342657	−	0.939461i	$-0.388673\pi$
$743$	18.6803	0.685313	0.342657	+	0.939461i	$0.388673\pi$
$744$	0	0
$745$	23.1798	0.849243
$746$	18.4841	0.676750
$747$	0	0
$748$	−18.9128	−0.691521
$749$	45.6919	1.66955
$750$	0	0
$751$	42.4812	1.55016	0.775080	−	0.631863i	$-0.217709\pi$
$751$	42.4812	1.55016	0.775080	+	0.631863i	$0.217709\pi$
$752$	0.0430471	0.00156977
$753$	0	0
$754$	−14.4492	−0.526207
$755$	−10.1261	−0.368528
$756$	0	0
$757$	40.9391	1.48796	0.743978	−	0.668204i	$-0.232937\pi$
$757$	40.9391	1.48796	0.743978	+	0.668204i	$0.232937\pi$
$758$	−15.5829	−0.565998
$759$	0	0
$760$	−1.74170	−0.0631781
$761$	29.9175	1.08451	0.542254	−	0.840215i	$-0.317571\pi$
$761$	29.9175	1.08451	0.542254	+	0.840215i	$0.317571\pi$
$762$	0	0
$763$	−37.6209	−1.36197
$764$	−4.28442	−0.155005
$765$	0	0
$766$	−15.4010	−0.556460
$767$	−28.3882	−1.02504
$768$	0	0
$769$	4.33047	0.156161	0.0780804	−	0.996947i	$-0.475121\pi$
$769$	4.33047	0.156161	0.0780804	+	0.996947i	$0.475121\pi$
$770$	7.92037	0.285430
$771$	0	0
$772$	−10.4816	−0.377241
$773$	37.0126	1.33125	0.665625	−	0.746287i	$-0.268165\pi$
$773$	37.0126	1.33125	0.665625	+	0.746287i	$0.268165\pi$
$774$	0	0
$775$	−2.63976	−0.0948231
$776$	3.16833	0.113736
$777$	0	0
$778$	−0.659230	−0.0236345
$779$	−5.07436	−0.181808
$780$	0	0
$781$	−24.9595	−0.893120
$782$	16.2273	0.580286
$783$	0	0
$784$	−0.0637870	−0.00227811
$785$	−1.69594	−0.0605307
$786$	0	0
$787$	10.9999	0.392104	0.196052	−	0.980594i	$-0.437188\pi$
$787$	10.9999	0.392104	0.196052	+	0.980594i	$0.437188\pi$
$788$	3.71997	0.132518
$789$	0	0
$790$	−7.25724	−0.258201
$791$	−56.6187	−2.01313
$792$	0	0
$793$	20.4670	0.726803
$794$	33.0150	1.17166
$795$	0	0
$796$	−5.54710	−0.196612
$797$	−41.2355	−1.46064	−0.730318	−	0.683108i	$-0.760628\pi$
$797$	−41.2355	−1.46064	−0.730318	+	0.683108i	$0.760628\pi$
$798$	0	0
$799$	−17.2646	−0.610777
$800$	5.65055	0.199777
$801$	0	0
$802$	1.16760	0.0412294
$803$	−5.49891	−0.194052
$804$	0	0
$805$	10.9214	0.384929
$806$	−5.49549	−0.193570
$807$	0	0
$808$	31.4605	1.10678
$809$	55.9394	1.96672	0.983362	−	0.181656i	$-0.0581457\pi$
$809$	55.9394	1.96672	0.983362	+	0.181656i	$0.0581457\pi$
$810$	0	0
$811$	21.2930	0.747697	0.373848	−	0.927490i	$-0.378038\pi$
$811$	21.2930	0.747697	0.373848	+	0.927490i	$0.378038\pi$
$812$	−28.9618	−1.01636
$813$	0	0
$814$	14.4309	0.505802
$815$	22.7425	0.796633
$816$	0	0
$817$	6.36844	0.222803
$818$	−11.3885	−0.398190
$819$	0	0
$820$	10.1706	0.355173
$821$	−10.9696	−0.382841	−0.191420	−	0.981508i	$-0.561309\pi$
$821$	−10.9696	−0.382841	−0.191420	+	0.981508i	$0.561309\pi$
$822$	0	0
$823$	16.4865	0.574683	0.287342	−	0.957828i	$-0.407229\pi$
$823$	16.4865	0.574683	0.287342	+	0.957828i	$0.407229\pi$
$824$	−2.28158	−0.0794826
$825$	0	0
$826$	35.4060	1.23193
$827$	−20.0886	−0.698551	−0.349275	−	0.937020i	$-0.613572\pi$
$827$	−20.0886	−0.698551	−0.349275	+	0.937020i	$0.613572\pi$
$828$	0	0
$829$	−44.2559	−1.53707	−0.768535	−	0.639808i	$-0.779014\pi$
$829$	−44.2559	−1.53707	−0.768535	+	0.639808i	$0.779014\pi$
$830$	4.00074	0.138868
$831$	0	0
$832$	11.8314	0.410181
$833$	25.5826	0.886384
$834$	0	0
$835$	−19.8881	−0.688256
$836$	2.02612	0.0700750
$837$	0	0
$838$	−16.6115	−0.573835
$839$	14.8039	0.511088	0.255544	−	0.966797i	$-0.417745\pi$
$839$	14.8039	0.511088	0.255544	+	0.966797i	$0.417745\pi$
$840$	0	0
$841$	19.1729	0.661134
$842$	−9.26768	−0.319385
$843$	0	0
$844$	−4.90793	−0.168938
$845$	7.35050	0.252865
$846$	0	0
$847$	13.0699	0.449088
$848$	0.200880	0.00689824
$849$	0	0
$850$	5.02893	0.172491
$851$	19.8987	0.682120
$852$	0	0
$853$	5.04936	0.172887	0.0864435	−	0.996257i	$-0.472450\pi$
$853$	5.04936	0.172887	0.0864435	+	0.996257i	$0.472450\pi$
$854$	−25.5266	−0.873502
$855$	0	0
$856$	38.2256	1.30652
$857$	−44.9987	−1.53713	−0.768564	−	0.639773i	$-0.779028\pi$
$857$	−44.9987	−1.53713	−0.768564	+	0.639773i	$0.779028\pi$
$858$	0	0
$859$	−1.64036	−0.0559683	−0.0279841	−	0.999608i	$-0.508909\pi$
$859$	−1.64036	−0.0559683	−0.0279841	+	0.999608i	$0.508909\pi$
$860$	−12.7644	−0.435261
$861$	0	0
$862$	19.5811	0.666934
$863$	−5.80315	−0.197541	−0.0987707	−	0.995110i	$-0.531491\pi$
$863$	−5.80315	−0.197541	−0.0987707	+	0.995110i	$0.531491\pi$
$864$	0	0
$865$	1.85953	0.0632259
$866$	−19.0477	−0.647267
$867$	0	0
$868$	−11.0151	−0.373877
$869$	22.1379	0.750977
$870$	0	0
$871$	−19.9010	−0.674320
$872$	−31.4735	−1.06583
$873$	0	0
$874$	−1.73842	−0.0588030
$875$	3.38461	0.114421
$876$	0	0
$877$	24.8178	0.838036	0.419018	−	0.907978i	$-0.362374\pi$
$877$	24.8178	0.838036	0.419018	+	0.907978i	$0.362374\pi$
$878$	−8.91261	−0.300786
$879$	0	0
$880$	0.0382498	0.00128940
$881$	50.1767	1.69050	0.845248	−	0.534375i	$-0.179453\pi$
$881$	50.1767	1.69050	0.845248	+	0.534375i	$0.179453\pi$
$882$	0	0
$883$	0.572049	0.0192510	0.00962549	−	0.999954i	$-0.496936\pi$
$883$	0.572049	0.0192510	0.00962549	+	0.999954i	$0.496936\pi$
$884$	−16.8252	−0.565893
$885$	0	0
$886$	−15.0396	−0.505266
$887$	−24.0836	−0.808648	−0.404324	−	0.914616i	$-0.632493\pi$
$887$	−24.0836	−0.808648	−0.404324	+	0.914616i	$0.632493\pi$
$888$	0	0
$889$	−64.2540	−2.15501
$890$	−0.875863	−0.0293590
$891$	0	0
$892$	29.0242	0.971803
$893$	1.84955	0.0618928
$894$	0	0
$895$	18.5991	0.621699
$896$	23.4935	0.784864
$897$	0	0
$898$	−27.7096	−0.924680
$899$	18.3217	0.611064
$900$	0	0
$901$	−80.5654	−2.68402
$902$	19.3050	0.642785
$903$	0	0
$904$	−47.3669	−1.57540
$905$	8.42729	0.280132
$906$	0	0
$907$	2.57108	0.0853714	0.0426857	−	0.999089i	$-0.486409\pi$
$907$	2.57108	0.0853714	0.0426857	+	0.999089i	$0.486409\pi$
$908$	−7.87322	−0.261282
$909$	0	0
$910$	7.04612	0.233576
$911$	22.1736	0.734644	0.367322	−	0.930094i	$-0.380275\pi$
$911$	22.1736	0.734644	0.367322	+	0.930094i	$0.380275\pi$
$912$	0	0
$913$	−12.2041	−0.403897
$914$	11.2806	0.373130
$915$	0	0
$916$	−10.0154	−0.330919
$917$	53.4458	1.76494
$918$	0	0
$919$	−38.9128	−1.28362	−0.641808	−	0.766865i	$-0.721815\pi$
$919$	−38.9128	−1.28362	−0.641808	+	0.766865i	$0.721815\pi$
$920$	9.13678	0.301231
$921$	0	0
$922$	1.43736	0.0473370
$923$	−22.2044	−0.730868
$924$	0	0
$925$	6.16674	0.202761
$926$	28.2407	0.928046
$927$	0	0
$928$	−39.2186	−1.28741
$929$	−13.6484	−0.447790	−0.223895	−	0.974613i	$-0.571877\pi$
$929$	−13.6484	−0.447790	−0.223895	+	0.974613i	$0.571877\pi$
$930$	0	0
$931$	−2.74065	−0.0898213
$932$	13.8839	0.454782
$933$	0	0
$934$	−22.6449	−0.740965
$935$	−15.3405	−0.501690
$936$	0	0
$937$	41.9049	1.36897	0.684486	−	0.729026i	$-0.260027\pi$
$937$	41.9049	1.36897	0.684486	+	0.729026i	$0.260027\pi$
$938$	24.8207	0.810426
$939$	0	0
$940$	−3.70708	−0.120912
$941$	11.5174	0.375456	0.187728	−	0.982221i	$-0.439888\pi$
$941$	11.5174	0.375456	0.187728	+	0.982221i	$0.439888\pi$
$942$	0	0
$943$	26.6196	0.866854
$944$	0.170986	0.00556511
$945$	0	0
$946$	−24.2282	−0.787726
$947$	48.4228	1.57353	0.786765	−	0.617253i	$-0.211754\pi$
$947$	48.4228	1.57353	0.786765	+	0.617253i	$0.211754\pi$
$948$	0	0
$949$	−4.89194	−0.158799
$950$	−0.538748	−0.0174793
$951$	0	0
$952$	55.0265	1.78342
$953$	−49.4631	−1.60227	−0.801134	−	0.598485i	$-0.795770\pi$
$953$	−49.4631	−1.60227	−0.801134	+	0.598485i	$0.795770\pi$
$954$	0	0
$955$	−3.47517	−0.112454
$956$	−9.79153	−0.316681
$957$	0	0
$958$	−18.8143	−0.607864
$959$	34.8183	1.12434
$960$	0	0
$961$	−24.0317	−0.775215
$962$	12.8380	0.413913
$963$	0	0
$964$	12.8459	0.413740
$965$	−8.50184	−0.273684
$966$	0	0
$967$	−29.4668	−0.947588	−0.473794	−	0.880636i	$-0.657116\pi$
$967$	−29.4668	−0.947588	−0.473794	+	0.880636i	$0.657116\pi$
$968$	10.9342	0.351440
$969$	0	0
$970$	0.980038	0.0314671
$971$	48.7028	1.56295	0.781474	−	0.623937i	$-0.214468\pi$
$971$	48.7028	1.56295	0.781474	+	0.623937i	$0.214468\pi$
$972$	0	0
$973$	22.0046	0.705434
$974$	22.6218	0.724850
$975$	0	0
$976$	−0.123275	−0.00394594
$977$	31.4998	1.00777	0.503884	−	0.863771i	$-0.331904\pi$
$977$	31.4998	1.00777	0.503884	+	0.863771i	$0.331904\pi$
$978$	0	0
$979$	2.67178	0.0853905
$980$	5.49313	0.175472
$981$	0	0
$982$	−18.2460	−0.582254
$983$	55.5969	1.77326	0.886632	−	0.462475i	$-0.153039\pi$
$983$	55.5969	1.77326	0.886632	+	0.462475i	$0.153039\pi$
$984$	0	0
$985$	3.01734	0.0961403
$986$	−34.9042	−1.11157
$987$	0	0
$988$	1.80248	0.0573445
$989$	−33.4082	−1.06232
$990$	0	0
$991$	−24.4325	−0.776125	−0.388062	−	0.921633i	$-0.626855\pi$
$991$	−24.4325	−0.776125	−0.388062	+	0.921633i	$0.626855\pi$
$992$	−14.9161	−0.473587
$993$	0	0
$994$	27.6936	0.878387
$995$	−4.49936	−0.142639
$996$	0	0
$997$	34.1051	1.08012	0.540060	−	0.841627i	$-0.318402\pi$
$997$	34.1051	1.08012	0.540060	+	0.841627i	$0.318402\pi$
$998$	−15.7425	−0.498319
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.s.1.7		10
3.2	odd	2		1335.2.a.j.1.4	✓	10
15.14	odd	2		6675.2.a.z.1.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.j.1.4	✓	10	3.2	odd	2
4005.2.a.s.1.7		10	1.1	even	1	trivial
6675.2.a.z.1.7		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)

\(f(q)\)	\(=\)	\(q+0.875863 q^{2} -1.23286 q^{4} -1.00000 q^{5} -3.38461 q^{7} -2.83155 q^{8} +O(q^{10})\)	\(q+0.875863 q^{2} -1.23286 q^{4} -1.00000 q^{5} -3.38461 q^{7} -2.83155 q^{8} -0.875863 q^{10} +2.67178 q^{11} +2.37687 q^{13} -2.96445 q^{14} -0.0143162 q^{16} +5.74169 q^{17} -0.615105 q^{19} +1.23286 q^{20} +2.34011 q^{22} +3.22678 q^{23} +1.00000 q^{25} +2.08181 q^{26} +4.17276 q^{28} -6.94067 q^{29} -2.63976 q^{31} +5.65055 q^{32} +5.02893 q^{34} +3.38461 q^{35} +6.16674 q^{37} -0.538748 q^{38} +2.83155 q^{40} +8.24958 q^{41} -10.3534 q^{43} -3.29395 q^{44} +2.82622 q^{46} -3.00688 q^{47} +4.45558 q^{49} +0.875863 q^{50} -2.93036 q^{52} -14.0316 q^{53} -2.67178 q^{55} +9.58368 q^{56} -6.07907 q^{58} -11.9435 q^{59} +8.61089 q^{61} -2.31207 q^{62} +4.97774 q^{64} -2.37687 q^{65} -8.37278 q^{67} -7.07873 q^{68} +2.96445 q^{70} -9.34188 q^{71} -2.05814 q^{73} +5.40122 q^{74} +0.758342 q^{76} -9.04294 q^{77} +8.28582 q^{79} +0.0143162 q^{80} +7.22550 q^{82} -4.56778 q^{83} -5.74169 q^{85} -9.06816 q^{86} -7.56527 q^{88} +1.00000 q^{89} -8.04477 q^{91} -3.97819 q^{92} -2.63362 q^{94} +0.615105 q^{95} -1.11894 q^{97} +3.90248 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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