LMFDB - Embedded newform 6040.2.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 29 x^{17} + 165 x^{16} + 325 x^{15} - 2208 x^{14} - 1891 x^{13} + 15895 x^{12} + \cdots - 5628 $ x^19 - 5x^18 - 29x^17 + 165x^16 + 325x^15 - 2208x^14 - 1891x^13 + 15895x^12 + 6652x^11 - 67665x^10 - 17345x^9 + 174105x^8 + 41499x^7 - 262172x^6 - 80919x^5 + 206783x^4 + 91643x^3 - 59750x^2 - 40224x - 5628
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.44352$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44352 q^{3} +1.00000 q^{5} -4.90700 q^{7} +2.97081 q^{9} +O(q^{10})$	$q-2.44352 q^{3} +1.00000 q^{5} -4.90700 q^{7} +2.97081 q^{9} -4.55796 q^{11} +3.11931 q^{13} -2.44352 q^{15} -4.54495 q^{17} -0.659498 q^{19} +11.9904 q^{21} +9.23979 q^{23} +1.00000 q^{25} +0.0713283 q^{27} +6.59821 q^{29} -8.71715 q^{31} +11.1375 q^{33} -4.90700 q^{35} +1.00732 q^{37} -7.62211 q^{39} -6.56419 q^{41} +4.24344 q^{43} +2.97081 q^{45} +5.91958 q^{47} +17.0787 q^{49} +11.1057 q^{51} +5.36874 q^{53} -4.55796 q^{55} +1.61150 q^{57} +11.2380 q^{59} +2.04081 q^{61} -14.5778 q^{63} +3.11931 q^{65} -11.9435 q^{67} -22.5776 q^{69} +7.82263 q^{71} +5.78308 q^{73} -2.44352 q^{75} +22.3659 q^{77} -12.3011 q^{79} -9.08672 q^{81} +3.88380 q^{83} -4.54495 q^{85} -16.1229 q^{87} -1.09325 q^{89} -15.3065 q^{91} +21.3006 q^{93} -0.659498 q^{95} +0.995271 q^{97} -13.5408 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.44352	−1.41077	−0.705385	−	0.708825i	$-0.749226\pi$
$3$	−2.44352	−1.41077	−0.705385	+	0.708825i	$0.749226\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.90700	−1.85467	−0.927336	−	0.374229i	$-0.877908\pi$
$7$	−4.90700	−1.85467	−0.927336	+	0.374229i	$0.877908\pi$
$8$	0	0
$9$	2.97081	0.990270
$10$	0	0
$11$	−4.55796	−1.37428	−0.687138	−	0.726527i	$-0.741133\pi$
$11$	−4.55796	−1.37428	−0.687138	+	0.726527i	$0.741133\pi$
$12$	0	0
$13$	3.11931	0.865141	0.432571	−	0.901600i	$-0.357607\pi$
$13$	3.11931	0.865141	0.432571	+	0.901600i	$0.357607\pi$
$14$	0	0
$15$	−2.44352	−0.630915
$16$	0	0
$17$	−4.54495	−1.10231	−0.551156	−	0.834402i	$-0.685813\pi$
$17$	−4.54495	−1.10231	−0.551156	+	0.834402i	$0.685813\pi$
$18$	0	0
$19$	−0.659498	−0.151299	−0.0756496	−	0.997134i	$-0.524103\pi$
$19$	−0.659498	−0.151299	−0.0756496	+	0.997134i	$0.524103\pi$
$20$	0	0
$21$	11.9904	2.61652
$22$	0	0
$23$	9.23979	1.92663	0.963315	−	0.268374i	$-0.0864864\pi$
$23$	9.23979	1.92663	0.963315	+	0.268374i	$0.0864864\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.0713283	0.0137271
$28$	0	0
$29$	6.59821	1.22526	0.612628	−	0.790371i	$-0.290112\pi$
$29$	6.59821	1.22526	0.612628	+	0.790371i	$0.290112\pi$
$30$	0	0
$31$	−8.71715	−1.56565	−0.782823	−	0.622245i	$-0.786221\pi$
$31$	−8.71715	−1.56565	−0.782823	+	0.622245i	$0.786221\pi$
$32$	0	0
$33$	11.1375	1.93879
$34$	0	0
$35$	−4.90700	−0.829435
$36$	0	0
$37$	1.00732	0.165602	0.0828012	−	0.996566i	$-0.473613\pi$
$37$	1.00732	0.165602	0.0828012	+	0.996566i	$0.473613\pi$
$38$	0	0
$39$	−7.62211	−1.22051
$40$	0	0
$41$	−6.56419	−1.02515	−0.512577	−	0.858641i	$-0.671309\pi$
$41$	−6.56419	−1.02515	−0.512577	+	0.858641i	$0.671309\pi$
$42$	0	0
$43$	4.24344	0.647118	0.323559	−	0.946208i	$-0.395121\pi$
$43$	4.24344	0.647118	0.323559	+	0.946208i	$0.395121\pi$
$44$	0	0
$45$	2.97081	0.442862
$46$	0	0
$47$	5.91958	0.863460	0.431730	−	0.902003i	$-0.357903\pi$
$47$	5.91958	0.863460	0.431730	+	0.902003i	$0.357903\pi$
$48$	0	0
$49$	17.0787	2.43981
$50$	0	0
$51$	11.1057	1.55511
$52$	0	0
$53$	5.36874	0.737453	0.368727	−	0.929538i	$-0.379794\pi$
$53$	5.36874	0.737453	0.368727	+	0.929538i	$0.379794\pi$
$54$	0	0
$55$	−4.55796	−0.614595
$56$	0	0
$57$	1.61150	0.213448
$58$	0	0
$59$	11.2380	1.46306	0.731531	−	0.681809i	$-0.238806\pi$
$59$	11.2380	1.46306	0.731531	+	0.681809i	$0.238806\pi$
$60$	0	0
$61$	2.04081	0.261298	0.130649	−	0.991429i	$-0.458294\pi$
$61$	2.04081	0.261298	0.130649	+	0.991429i	$0.458294\pi$
$62$	0	0
$63$	−14.5778	−1.83663
$64$	0	0
$65$	3.11931	0.386903
$66$	0	0
$67$	−11.9435	−1.45913	−0.729564	−	0.683913i	$-0.760277\pi$
$67$	−11.9435	−1.45913	−0.729564	+	0.683913i	$0.760277\pi$
$68$	0	0
$69$	−22.5776	−2.71803
$70$	0	0
$71$	7.82263	0.928375	0.464188	−	0.885737i	$-0.346346\pi$
$71$	7.82263	0.928375	0.464188	+	0.885737i	$0.346346\pi$
$72$	0	0
$73$	5.78308	0.676858	0.338429	−	0.940992i	$-0.390104\pi$
$73$	5.78308	0.676858	0.338429	+	0.940992i	$0.390104\pi$
$74$	0	0
$75$	−2.44352	−0.282154
$76$	0	0
$77$	22.3659	2.54883
$78$	0	0
$79$	−12.3011	−1.38399	−0.691993	−	0.721904i	$-0.743267\pi$
$79$	−12.3011	−1.38399	−0.691993	+	0.721904i	$0.743267\pi$
$80$	0	0
$81$	−9.08672	−1.00964
$82$	0	0
$83$	3.88380	0.426302	0.213151	−	0.977019i	$-0.431627\pi$
$83$	3.88380	0.426302	0.213151	+	0.977019i	$0.431627\pi$
$84$	0	0
$85$	−4.54495	−0.492969
$86$	0	0
$87$	−16.1229	−1.72855
$88$	0	0
$89$	−1.09325	−0.115884	−0.0579421	−	0.998320i	$-0.518454\pi$
$89$	−1.09325	−0.115884	−0.0579421	+	0.998320i	$0.518454\pi$
$90$	0	0
$91$	−15.3065	−1.60455
$92$	0	0
$93$	21.3006	2.20876
$94$	0	0
$95$	−0.659498	−0.0676631
$96$	0	0
$97$	0.995271	0.101054	0.0505272	−	0.998723i	$-0.483910\pi$
$97$	0.995271	0.101054	0.0505272	+	0.998723i	$0.483910\pi$
$98$	0	0
$99$	−13.5408	−1.36090
$100$	0	0
$101$	−12.6516	−1.25888	−0.629441	−	0.777048i	$-0.716716\pi$
$101$	−12.6516	−1.25888	−0.629441	+	0.777048i	$0.716716\pi$
$102$	0	0
$103$	12.9853	1.27948	0.639739	−	0.768592i	$-0.279043\pi$
$103$	12.9853	1.27948	0.639739	+	0.768592i	$0.279043\pi$
$104$	0	0
$105$	11.9904	1.17014
$106$	0	0
$107$	−3.68720	−0.356455	−0.178228	−	0.983989i	$-0.557036\pi$
$107$	−3.68720	−0.356455	−0.178228	+	0.983989i	$0.557036\pi$
$108$	0	0
$109$	−0.860710	−0.0824411	−0.0412206	−	0.999150i	$-0.513125\pi$
$109$	−0.860710	−0.0824411	−0.0412206	+	0.999150i	$0.513125\pi$
$110$	0	0
$111$	−2.46141	−0.233627
$112$	0	0
$113$	−0.987609	−0.0929064	−0.0464532	−	0.998920i	$-0.514792\pi$
$113$	−0.987609	−0.0929064	−0.0464532	+	0.998920i	$0.514792\pi$
$114$	0	0
$115$	9.23979	0.861615
$116$	0	0
$117$	9.26688	0.856723
$118$	0	0
$119$	22.3021	2.04443
$120$	0	0
$121$	9.77498	0.888634
$122$	0	0
$123$	16.0398	1.44626
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.821329	0.0728812	0.0364406	−	0.999336i	$-0.488398\pi$
$127$	0.821329	0.0728812	0.0364406	+	0.999336i	$0.488398\pi$
$128$	0	0
$129$	−10.3689	−0.912934
$130$	0	0
$131$	−9.77119	−0.853713	−0.426856	−	0.904319i	$-0.640379\pi$
$131$	−9.77119	−0.853713	−0.426856	+	0.904319i	$0.640379\pi$
$132$	0	0
$133$	3.23616	0.280611
$134$	0	0
$135$	0.0713283	0.00613896
$136$	0	0
$137$	−20.5205	−1.75318	−0.876591	−	0.481237i	$-0.840188\pi$
$137$	−20.5205	−1.75318	−0.876591	+	0.481237i	$0.840188\pi$
$138$	0	0
$139$	−5.99207	−0.508240	−0.254120	−	0.967173i	$-0.581786\pi$
$139$	−5.99207	−0.508240	−0.254120	+	0.967173i	$0.581786\pi$
$140$	0	0
$141$	−14.4646	−1.21814
$142$	0	0
$143$	−14.2177	−1.18894
$144$	0	0
$145$	6.59821	0.547951
$146$	0	0
$147$	−41.7322	−3.44201
$148$	0	0
$149$	−7.43163	−0.608823	−0.304412	−	0.952541i	$-0.598460\pi$
$149$	−7.43163	−0.608823	−0.304412	+	0.952541i	$0.598460\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−13.5022	−1.09159
$154$	0	0
$155$	−8.71715	−0.700178
$156$	0	0
$157$	19.6466	1.56797	0.783984	−	0.620781i	$-0.213184\pi$
$157$	19.6466	1.56797	0.783984	+	0.620781i	$0.213184\pi$
$158$	0	0
$159$	−13.1186	−1.04038
$160$	0	0
$161$	−45.3397	−3.57327
$162$	0	0
$163$	−20.2618	−1.58703	−0.793513	−	0.608554i	$-0.791750\pi$
$163$	−20.2618	−1.58703	−0.793513	+	0.608554i	$0.791750\pi$
$164$	0	0
$165$	11.1375	0.867052
$166$	0	0
$167$	−18.8685	−1.46009	−0.730043	−	0.683401i	$-0.760500\pi$
$167$	−18.8685	−1.46009	−0.730043	+	0.683401i	$0.760500\pi$
$168$	0	0
$169$	−3.26990	−0.251531
$170$	0	0
$171$	−1.95924	−0.149827
$172$	0	0
$173$	19.6606	1.49477	0.747384	−	0.664392i	$-0.231309\pi$
$173$	19.6606	1.49477	0.747384	+	0.664392i	$0.231309\pi$
$174$	0	0
$175$	−4.90700	−0.370935
$176$	0	0
$177$	−27.4603	−2.06404
$178$	0	0
$179$	−3.10016	−0.231717	−0.115858	−	0.993266i	$-0.536962\pi$
$179$	−3.10016	−0.231717	−0.115858	+	0.993266i	$0.536962\pi$
$180$	0	0
$181$	20.1645	1.49882	0.749408	−	0.662109i	$-0.230338\pi$
$181$	20.1645	1.49882	0.749408	+	0.662109i	$0.230338\pi$
$182$	0	0
$183$	−4.98676	−0.368632
$184$	0	0
$185$	1.00732	0.0740597
$186$	0	0
$187$	20.7157	1.51488
$188$	0	0
$189$	−0.350008	−0.0254594
$190$	0	0
$191$	26.0368	1.88396	0.941980	−	0.335670i	$-0.108963\pi$
$191$	26.0368	1.88396	0.941980	+	0.335670i	$0.108963\pi$
$192$	0	0
$193$	−0.537494	−0.0386896	−0.0193448	−	0.999813i	$-0.506158\pi$
$193$	−0.537494	−0.0386896	−0.0193448	+	0.999813i	$0.506158\pi$
$194$	0	0
$195$	−7.62211	−0.545831
$196$	0	0
$197$	25.7975	1.83800	0.918998	−	0.394262i	$-0.129000\pi$
$197$	25.7975	1.83800	0.918998	+	0.394262i	$0.129000\pi$
$198$	0	0
$199$	20.6942	1.46697	0.733485	−	0.679706i	$-0.237893\pi$
$199$	20.6942	1.46697	0.733485	+	0.679706i	$0.237893\pi$
$200$	0	0
$201$	29.1842	2.05849
$202$	0	0
$203$	−32.3774	−2.27245
$204$	0	0
$205$	−6.56419	−0.458463
$206$	0	0
$207$	27.4497	1.90788
$208$	0	0
$209$	3.00596	0.207927
$210$	0	0
$211$	−18.6906	−1.28672	−0.643359	−	0.765565i	$-0.722460\pi$
$211$	−18.6906	−1.28672	−0.643359	+	0.765565i	$0.722460\pi$
$212$	0	0
$213$	−19.1148	−1.30972
$214$	0	0
$215$	4.24344	0.289400
$216$	0	0
$217$	42.7751	2.90376
$218$	0	0
$219$	−14.1311	−0.954890
$220$	0	0
$221$	−14.1771	−0.953655
$222$	0	0
$223$	12.2346	0.819290	0.409645	−	0.912245i	$-0.365653\pi$
$223$	12.2346	0.819290	0.409645	+	0.912245i	$0.365653\pi$
$224$	0	0
$225$	2.97081	0.198054
$226$	0	0
$227$	−5.43786	−0.360923	−0.180462	−	0.983582i	$-0.557759\pi$
$227$	−5.43786	−0.360923	−0.180462	+	0.983582i	$0.557759\pi$
$228$	0	0
$229$	−23.9167	−1.58046	−0.790230	−	0.612810i	$-0.790039\pi$
$229$	−23.9167	−1.58046	−0.790230	+	0.612810i	$0.790039\pi$
$230$	0	0
$231$	−54.6516	−3.59581
$232$	0	0
$233$	1.60310	0.105022	0.0525112	−	0.998620i	$-0.483277\pi$
$233$	1.60310	0.105022	0.0525112	+	0.998620i	$0.483277\pi$
$234$	0	0
$235$	5.91958	0.386151
$236$	0	0
$237$	30.0581	1.95248
$238$	0	0
$239$	−15.3589	−0.993482	−0.496741	−	0.867899i	$-0.665470\pi$
$239$	−15.3589	−0.993482	−0.496741	+	0.867899i	$0.665470\pi$
$240$	0	0
$241$	−12.6833	−0.817002	−0.408501	−	0.912758i	$-0.633948\pi$
$241$	−12.6833	−0.817002	−0.408501	+	0.912758i	$0.633948\pi$
$242$	0	0
$243$	21.9896	1.41064
$244$	0	0
$245$	17.0787	1.09112
$246$	0	0
$247$	−2.05718	−0.130895
$248$	0	0
$249$	−9.49015	−0.601414
$250$	0	0
$251$	22.5468	1.42314	0.711570	−	0.702615i	$-0.247984\pi$
$251$	22.5468	1.42314	0.711570	+	0.702615i	$0.247984\pi$
$252$	0	0
$253$	−42.1146	−2.64772
$254$	0	0
$255$	11.1057	0.695465
$256$	0	0
$257$	20.0366	1.24985	0.624924	−	0.780686i	$-0.285130\pi$
$257$	20.0366	1.24985	0.624924	+	0.780686i	$0.285130\pi$
$258$	0	0
$259$	−4.94292	−0.307138
$260$	0	0
$261$	19.6020	1.21333
$262$	0	0
$263$	15.3583	0.947035	0.473517	−	0.880784i	$-0.342984\pi$
$263$	15.3583	0.947035	0.473517	+	0.880784i	$0.342984\pi$
$264$	0	0
$265$	5.36874	0.329799
$266$	0	0
$267$	2.67138	0.163486
$268$	0	0
$269$	−8.01918	−0.488938	−0.244469	−	0.969657i	$-0.578614\pi$
$269$	−8.01918	−0.488938	−0.244469	+	0.969657i	$0.578614\pi$
$270$	0	0
$271$	−1.72618	−0.104858	−0.0524290	−	0.998625i	$-0.516696\pi$
$271$	−1.72618	−0.104858	−0.0524290	+	0.998625i	$0.516696\pi$
$272$	0	0
$273$	37.4017	2.26366
$274$	0	0
$275$	−4.55796	−0.274855
$276$	0	0
$277$	−3.90954	−0.234902	−0.117451	−	0.993079i	$-0.537472\pi$
$277$	−3.90954	−0.234902	−0.117451	+	0.993079i	$0.537472\pi$
$278$	0	0
$279$	−25.8970	−1.55041
$280$	0	0
$281$	−25.7769	−1.53772	−0.768862	−	0.639415i	$-0.779177\pi$
$281$	−25.7769	−1.53772	−0.768862	+	0.639415i	$0.779177\pi$
$282$	0	0
$283$	23.4564	1.39434	0.697169	−	0.716907i	$-0.254443\pi$
$283$	23.4564	1.39434	0.697169	+	0.716907i	$0.254443\pi$
$284$	0	0
$285$	1.61150	0.0954570
$286$	0	0
$287$	32.2105	1.90133
$288$	0	0
$289$	3.65655	0.215091
$290$	0	0
$291$	−2.43197	−0.142565
$292$	0	0
$293$	−29.6302	−1.73102	−0.865508	−	0.500895i	$-0.833004\pi$
$293$	−29.6302	−1.73102	−0.865508	+	0.500895i	$0.833004\pi$
$294$	0	0
$295$	11.2380	0.654301
$296$	0	0
$297$	−0.325111	−0.0188649
$298$	0	0
$299$	28.8218	1.66681
$300$	0	0
$301$	−20.8226	−1.20019
$302$	0	0
$303$	30.9145	1.77599
$304$	0	0
$305$	2.04081	0.116856
$306$	0	0
$307$	18.8553	1.07613	0.538064	−	0.842904i	$-0.319156\pi$
$307$	18.8553	1.07613	0.538064	+	0.842904i	$0.319156\pi$
$308$	0	0
$309$	−31.7298	−1.80505
$310$	0	0
$311$	−8.79397	−0.498661	−0.249330	−	0.968419i	$-0.580210\pi$
$311$	−8.79397	−0.498661	−0.249330	+	0.968419i	$0.580210\pi$
$312$	0	0
$313$	−6.25931	−0.353797	−0.176899	−	0.984229i	$-0.556606\pi$
$313$	−6.25931	−0.353797	−0.176899	+	0.984229i	$0.556606\pi$
$314$	0	0
$315$	−14.5778	−0.821364
$316$	0	0
$317$	−25.7840	−1.44817	−0.724087	−	0.689709i	$-0.757739\pi$
$317$	−25.7840	−1.44817	−0.724087	+	0.689709i	$0.757739\pi$
$318$	0	0
$319$	−30.0744	−1.68384
$320$	0	0
$321$	9.00976	0.502876
$322$	0	0
$323$	2.99738	0.166779
$324$	0	0
$325$	3.11931	0.173028
$326$	0	0
$327$	2.10317	0.116305
$328$	0	0
$329$	−29.0474	−1.60143
$330$	0	0
$331$	−15.5427	−0.854302	−0.427151	−	0.904180i	$-0.640483\pi$
$331$	−15.5427	−0.854302	−0.427151	+	0.904180i	$0.640483\pi$
$332$	0	0
$333$	2.99256	0.163991
$334$	0	0
$335$	−11.9435	−0.652542
$336$	0	0
$337$	−7.18169	−0.391211	−0.195606	−	0.980683i	$-0.562667\pi$
$337$	−7.18169	−0.391211	−0.195606	+	0.980683i	$0.562667\pi$
$338$	0	0
$339$	2.41325	0.131070
$340$	0	0
$341$	39.7324	2.15163
$342$	0	0
$343$	−49.4561	−2.67038
$344$	0	0
$345$	−22.5776	−1.21554
$346$	0	0
$347$	−26.2473	−1.40903	−0.704514	−	0.709691i	$-0.748835\pi$
$347$	−26.2473	−1.40903	−0.704514	+	0.709691i	$0.748835\pi$
$348$	0	0
$349$	−31.6220	−1.69269	−0.846344	−	0.532636i	$-0.821202\pi$
$349$	−31.6220	−1.69269	−0.846344	+	0.532636i	$0.821202\pi$
$350$	0	0
$351$	0.222495	0.0118759
$352$	0	0
$353$	−8.96351	−0.477080	−0.238540	−	0.971133i	$-0.576669\pi$
$353$	−8.96351	−0.477080	−0.238540	+	0.971133i	$0.576669\pi$
$354$	0	0
$355$	7.82263	0.415182
$356$	0	0
$357$	−54.4956	−2.88422
$358$	0	0
$359$	30.7553	1.62320	0.811601	−	0.584212i	$-0.198596\pi$
$359$	30.7553	1.62320	0.811601	+	0.584212i	$0.198596\pi$
$360$	0	0
$361$	−18.5651	−0.977109
$362$	0	0
$363$	−23.8854	−1.25366
$364$	0	0
$365$	5.78308	0.302700
$366$	0	0
$367$	3.04193	0.158787	0.0793936	−	0.996843i	$-0.474702\pi$
$367$	3.04193	0.158787	0.0793936	+	0.996843i	$0.474702\pi$
$368$	0	0
$369$	−19.5010	−1.01518
$370$	0	0
$371$	−26.3444	−1.36773
$372$	0	0
$373$	−8.42720	−0.436344	−0.218172	−	0.975910i	$-0.570009\pi$
$373$	−8.42720	−0.436344	−0.218172	+	0.975910i	$0.570009\pi$
$374$	0	0
$375$	−2.44352	−0.126183
$376$	0	0
$377$	20.5819	1.06002
$378$	0	0
$379$	−32.6655	−1.67792	−0.838958	−	0.544196i	$-0.816835\pi$
$379$	−32.6655	−1.67792	−0.838958	+	0.544196i	$0.816835\pi$
$380$	0	0
$381$	−2.00694	−0.102818
$382$	0	0
$383$	−1.55770	−0.0795945	−0.0397973	−	0.999208i	$-0.512671\pi$
$383$	−1.55770	−0.0795945	−0.0397973	+	0.999208i	$0.512671\pi$
$384$	0	0
$385$	22.3659	1.13987
$386$	0	0
$387$	12.6064	0.640821
$388$	0	0
$389$	1.08604	0.0550643	0.0275322	−	0.999621i	$-0.491235\pi$
$389$	1.08604	0.0550643	0.0275322	+	0.999621i	$0.491235\pi$
$390$	0	0
$391$	−41.9944	−2.12375
$392$	0	0
$393$	23.8761	1.20439
$394$	0	0
$395$	−12.3011	−0.618937
$396$	0	0
$397$	10.5760	0.530795	0.265398	−	0.964139i	$-0.414497\pi$
$397$	10.5760	0.530795	0.265398	+	0.964139i	$0.414497\pi$
$398$	0	0
$399$	−7.90763	−0.395877
$400$	0	0
$401$	9.18627	0.458740	0.229370	−	0.973339i	$-0.426333\pi$
$401$	9.18627	0.458740	0.229370	+	0.973339i	$0.426333\pi$
$402$	0	0
$403$	−27.1915	−1.35450
$404$	0	0
$405$	−9.08672	−0.451523
$406$	0	0
$407$	−4.59132	−0.227583
$408$	0	0
$409$	−35.8039	−1.77039	−0.885194	−	0.465222i	$-0.845974\pi$
$409$	−35.8039	−1.77039	−0.885194	+	0.465222i	$0.845974\pi$
$410$	0	0
$411$	50.1422	2.47333
$412$	0	0
$413$	−55.1448	−2.71350
$414$	0	0
$415$	3.88380	0.190648
$416$	0	0
$417$	14.6418	0.717010
$418$	0	0
$419$	−24.7415	−1.20870	−0.604352	−	0.796718i	$-0.706568\pi$
$419$	−24.7415	−1.20870	−0.604352	+	0.796718i	$0.706568\pi$
$420$	0	0
$421$	−2.91112	−0.141879	−0.0709396	−	0.997481i	$-0.522600\pi$
$421$	−2.91112	−0.141879	−0.0709396	+	0.997481i	$0.522600\pi$
$422$	0	0
$423$	17.5859	0.855058
$424$	0	0
$425$	−4.54495	−0.220462
$426$	0	0
$427$	−10.0142	−0.484623
$428$	0	0
$429$	34.7413	1.67732
$430$	0	0
$431$	6.40292	0.308418	0.154209	−	0.988038i	$-0.450717\pi$
$431$	6.40292	0.308418	0.154209	+	0.988038i	$0.450717\pi$
$432$	0	0
$433$	−27.8646	−1.33909	−0.669543	−	0.742774i	$-0.733510\pi$
$433$	−27.8646	−1.33909	−0.669543	+	0.742774i	$0.733510\pi$
$434$	0	0
$435$	−16.1229	−0.773033
$436$	0	0
$437$	−6.09362	−0.291498
$438$	0	0
$439$	−0.0872818	−0.00416573	−0.00208287	−	0.999998i	$-0.500663\pi$
$439$	−0.0872818	−0.00416573	−0.00208287	+	0.999998i	$0.500663\pi$
$440$	0	0
$441$	50.7375	2.41607
$442$	0	0
$443$	16.2609	0.772581	0.386290	−	0.922377i	$-0.373756\pi$
$443$	16.2609	0.772581	0.386290	+	0.922377i	$0.373756\pi$
$444$	0	0
$445$	−1.09325	−0.0518250
$446$	0	0
$447$	18.1594	0.858909
$448$	0	0
$449$	24.7882	1.16983	0.584914	−	0.811095i	$-0.301128\pi$
$449$	24.7882	1.16983	0.584914	+	0.811095i	$0.301128\pi$
$450$	0	0
$451$	29.9193	1.40884
$452$	0	0
$453$	−2.44352	−0.114807
$454$	0	0
$455$	−15.3065	−0.717578
$456$	0	0
$457$	−28.8553	−1.34979	−0.674897	−	0.737912i	$-0.735812\pi$
$457$	−28.8553	−1.34979	−0.674897	+	0.737912i	$0.735812\pi$
$458$	0	0
$459$	−0.324183	−0.0151316
$460$	0	0
$461$	−4.01283	−0.186896	−0.0934481	−	0.995624i	$-0.529789\pi$
$461$	−4.01283	−0.186896	−0.0934481	+	0.995624i	$0.529789\pi$
$462$	0	0
$463$	−0.278363	−0.0129366	−0.00646832	−	0.999979i	$-0.502059\pi$
$463$	−0.278363	−0.0129366	−0.00646832	+	0.999979i	$0.502059\pi$
$464$	0	0
$465$	21.3006	0.987790
$466$	0	0
$467$	23.1235	1.07003	0.535014	−	0.844843i	$-0.320306\pi$
$467$	23.1235	1.07003	0.535014	+	0.844843i	$0.320306\pi$
$468$	0	0
$469$	58.6067	2.70621
$470$	0	0
$471$	−48.0069	−2.21204
$472$	0	0
$473$	−19.3414	−0.889318
$474$	0	0
$475$	−0.659498	−0.0302598
$476$	0	0
$477$	15.9495	0.730278
$478$	0	0
$479$	−0.851379	−0.0389005	−0.0194503	−	0.999811i	$-0.506192\pi$
$479$	−0.851379	−0.0389005	−0.0194503	+	0.999811i	$0.506192\pi$
$480$	0	0
$481$	3.14214	0.143269
$482$	0	0
$483$	110.789	5.04106
$484$	0	0
$485$	0.995271	0.0451929
$486$	0	0
$487$	14.6366	0.663247	0.331623	−	0.943412i	$-0.392404\pi$
$487$	14.6366	0.663247	0.331623	+	0.943412i	$0.392404\pi$
$488$	0	0
$489$	49.5102	2.23893
$490$	0	0
$491$	−38.5616	−1.74026	−0.870131	−	0.492821i	$-0.835966\pi$
$491$	−38.5616	−1.74026	−0.870131	+	0.492821i	$0.835966\pi$
$492$	0	0
$493$	−29.9885	−1.35061
$494$	0	0
$495$	−13.5408	−0.608615
$496$	0	0
$497$	−38.3857	−1.72183
$498$	0	0
$499$	21.6585	0.969568	0.484784	−	0.874634i	$-0.338898\pi$
$499$	21.6585	0.969568	0.484784	+	0.874634i	$0.338898\pi$
$500$	0	0
$501$	46.1056	2.05984
$502$	0	0
$503$	5.07601	0.226328	0.113164	−	0.993576i	$-0.463901\pi$
$503$	5.07601	0.226328	0.113164	+	0.993576i	$0.463901\pi$
$504$	0	0
$505$	−12.6516	−0.562989
$506$	0	0
$507$	7.99008	0.354852
$508$	0	0
$509$	−12.1392	−0.538063	−0.269031	−	0.963131i	$-0.586703\pi$
$509$	−12.1392	−0.538063	−0.269031	+	0.963131i	$0.586703\pi$
$510$	0	0
$511$	−28.3776	−1.25535
$512$	0	0
$513$	−0.0470409	−0.00207691
$514$	0	0
$515$	12.9853	0.572200
$516$	0	0
$517$	−26.9812	−1.18663
$518$	0	0
$519$	−48.0412	−2.10877
$520$	0	0
$521$	−29.1155	−1.27557	−0.637787	−	0.770213i	$-0.720150\pi$
$521$	−29.1155	−1.27557	−0.637787	+	0.770213i	$0.720150\pi$
$522$	0	0
$523$	−14.6340	−0.639901	−0.319950	−	0.947434i	$-0.603666\pi$
$523$	−14.6340	−0.639901	−0.319950	+	0.947434i	$0.603666\pi$
$524$	0	0
$525$	11.9904	0.523303
$526$	0	0
$527$	39.6190	1.72583
$528$	0	0
$529$	62.3737	2.71190
$530$	0	0
$531$	33.3859	1.44883
$532$	0	0
$533$	−20.4757	−0.886903
$534$	0	0
$535$	−3.68720	−0.159412
$536$	0	0
$537$	7.57531	0.326899
$538$	0	0
$539$	−77.8439	−3.35297
$540$	0	0
$541$	−20.4516	−0.879282	−0.439641	−	0.898173i	$-0.644894\pi$
$541$	−20.4516	−0.879282	−0.439641	+	0.898173i	$0.644894\pi$
$542$	0	0
$543$	−49.2724	−2.11448
$544$	0	0
$545$	−0.860710	−0.0368688
$546$	0	0
$547$	−26.5793	−1.13645	−0.568224	−	0.822874i	$-0.692369\pi$
$547$	−26.5793	−1.13645	−0.568224	+	0.822874i	$0.692369\pi$
$548$	0	0
$549$	6.06285	0.258756
$550$	0	0
$551$	−4.35151	−0.185380
$552$	0	0
$553$	60.3617	2.56684
$554$	0	0
$555$	−2.46141	−0.104481
$556$	0	0
$557$	16.4379	0.696495	0.348248	−	0.937403i	$-0.386777\pi$
$557$	16.4379	0.696495	0.348248	+	0.937403i	$0.386777\pi$
$558$	0	0
$559$	13.2366	0.559848
$560$	0	0
$561$	−50.6193	−2.13715
$562$	0	0
$563$	−31.4472	−1.32534	−0.662671	−	0.748910i	$-0.730577\pi$
$563$	−31.4472	−1.32534	−0.662671	+	0.748910i	$0.730577\pi$
$564$	0	0
$565$	−0.987609	−0.0415490
$566$	0	0
$567$	44.5886	1.87254
$568$	0	0
$569$	39.8445	1.67037	0.835184	−	0.549971i	$-0.185361\pi$
$569$	39.8445	1.67037	0.835184	+	0.549971i	$0.185361\pi$
$570$	0	0
$571$	35.8060	1.49843	0.749217	−	0.662325i	$-0.230430\pi$
$571$	35.8060	1.49843	0.749217	+	0.662325i	$0.230430\pi$
$572$	0	0
$573$	−63.6216	−2.65783
$574$	0	0
$575$	9.23979	0.385326
$576$	0	0
$577$	38.8692	1.61815	0.809073	−	0.587708i	$-0.199970\pi$
$577$	38.8692	1.61815	0.809073	+	0.587708i	$0.199970\pi$
$578$	0	0
$579$	1.31338	0.0545822
$580$	0	0
$581$	−19.0578	−0.790651
$582$	0	0
$583$	−24.4705	−1.01346
$584$	0	0
$585$	9.26688	0.383138
$586$	0	0
$587$	−29.9688	−1.23694	−0.618472	−	0.785807i	$-0.712248\pi$
$587$	−29.9688	−1.23694	−0.618472	+	0.785807i	$0.712248\pi$
$588$	0	0
$589$	5.74894	0.236881
$590$	0	0
$591$	−63.0368	−2.59299
$592$	0	0
$593$	17.1766	0.705359	0.352680	−	0.935744i	$-0.385271\pi$
$593$	17.1766	0.705359	0.352680	+	0.935744i	$0.385271\pi$
$594$	0	0
$595$	22.3021	0.914296
$596$	0	0
$597$	−50.5667	−2.06956
$598$	0	0
$599$	8.35518	0.341384	0.170692	−	0.985324i	$-0.445400\pi$
$599$	8.35518	0.341384	0.170692	+	0.985324i	$0.445400\pi$
$600$	0	0
$601$	6.21168	0.253380	0.126690	−	0.991942i	$-0.459565\pi$
$601$	6.21168	0.253380	0.126690	+	0.991942i	$0.459565\pi$
$602$	0	0
$603$	−35.4818	−1.44493
$604$	0	0
$605$	9.77498	0.397409
$606$	0	0
$607$	−22.1313	−0.898281	−0.449141	−	0.893461i	$-0.648270\pi$
$607$	−22.1313	−0.898281	−0.449141	+	0.893461i	$0.648270\pi$
$608$	0	0
$609$	79.1150	3.20590
$610$	0	0
$611$	18.4650	0.747014
$612$	0	0
$613$	−0.776065	−0.0313450	−0.0156725	−	0.999877i	$-0.504989\pi$
$613$	−0.776065	−0.0313450	−0.0156725	+	0.999877i	$0.504989\pi$
$614$	0	0
$615$	16.0398	0.646785
$616$	0	0
$617$	18.0896	0.728262	0.364131	−	0.931348i	$-0.381366\pi$
$617$	18.0896	0.728262	0.364131	+	0.931348i	$0.381366\pi$
$618$	0	0
$619$	−8.50175	−0.341714	−0.170857	−	0.985296i	$-0.554654\pi$
$619$	−8.50175	−0.341714	−0.170857	+	0.985296i	$0.554654\pi$
$620$	0	0
$621$	0.659059	0.0264471
$622$	0	0
$623$	5.36458	0.214927
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.34515	−0.293337
$628$	0	0
$629$	−4.57822	−0.182545
$630$	0	0
$631$	13.3616	0.531916	0.265958	−	0.963985i	$-0.414312\pi$
$631$	13.3616	0.531916	0.265958	+	0.963985i	$0.414312\pi$
$632$	0	0
$633$	45.6710	1.81526
$634$	0	0
$635$	0.821329	0.0325934
$636$	0	0
$637$	53.2737	2.11078
$638$	0	0
$639$	23.2395	0.919342
$640$	0	0
$641$	14.9085	0.588852	0.294426	−	0.955674i	$-0.404871\pi$
$641$	14.9085	0.588852	0.294426	+	0.955674i	$0.404871\pi$
$642$	0	0
$643$	−23.8380	−0.940079	−0.470039	−	0.882645i	$-0.655760\pi$
$643$	−23.8380	−0.940079	−0.470039	+	0.882645i	$0.655760\pi$
$644$	0	0
$645$	−10.3689	−0.408276
$646$	0	0
$647$	0.797347	0.0313470	0.0156735	−	0.999877i	$-0.495011\pi$
$647$	0.797347	0.0313470	0.0156735	+	0.999877i	$0.495011\pi$
$648$	0	0
$649$	−51.2223	−2.01065
$650$	0	0
$651$	−104.522	−4.09654
$652$	0	0
$653$	−24.8601	−0.972850	−0.486425	−	0.873722i	$-0.661699\pi$
$653$	−24.8601	−0.972850	−0.486425	+	0.873722i	$0.661699\pi$
$654$	0	0
$655$	−9.77119	−0.381792
$656$	0	0
$657$	17.1804	0.670272
$658$	0	0
$659$	−4.59726	−0.179084	−0.0895420	−	0.995983i	$-0.528540\pi$
$659$	−4.59726	−0.179084	−0.0895420	+	0.995983i	$0.528540\pi$
$660$	0	0
$661$	−42.3397	−1.64682	−0.823412	−	0.567444i	$-0.807932\pi$
$661$	−42.3397	−1.64682	−0.823412	+	0.567444i	$0.807932\pi$
$662$	0	0
$663$	34.6421	1.34539
$664$	0	0
$665$	3.23616	0.125493
$666$	0	0
$667$	60.9661	2.36062
$668$	0	0
$669$	−29.8956	−1.15583
$670$	0	0
$671$	−9.30191	−0.359096
$672$	0	0
$673$	23.1313	0.891648	0.445824	−	0.895121i	$-0.352911\pi$
$673$	23.1313	0.891648	0.445824	+	0.895121i	$0.352911\pi$
$674$	0	0
$675$	0.0713283	0.00274543
$676$	0	0
$677$	−31.4287	−1.20790	−0.603951	−	0.797022i	$-0.706408\pi$
$677$	−31.4287	−1.20790	−0.603951	+	0.797022i	$0.706408\pi$
$678$	0	0
$679$	−4.88380	−0.187423
$680$	0	0
$681$	13.2875	0.509180
$682$	0	0
$683$	−51.3739	−1.96577	−0.982884	−	0.184226i	$-0.941022\pi$
$683$	−51.3739	−1.96577	−0.982884	+	0.184226i	$0.941022\pi$
$684$	0	0
$685$	−20.5205	−0.784047
$686$	0	0
$687$	58.4411	2.22967
$688$	0	0
$689$	16.7468	0.638001
$690$	0	0
$691$	31.4053	1.19471	0.597356	−	0.801976i	$-0.296218\pi$
$691$	31.4053	1.19471	0.597356	+	0.801976i	$0.296218\pi$
$692$	0	0
$693$	66.4449	2.52403
$694$	0	0
$695$	−5.99207	−0.227292
$696$	0	0
$697$	29.8339	1.13004
$698$	0	0
$699$	−3.91721	−0.148162
$700$	0	0
$701$	30.2348	1.14195	0.570977	−	0.820966i	$-0.306565\pi$
$701$	30.2348	1.14195	0.570977	+	0.820966i	$0.306565\pi$
$702$	0	0
$703$	−0.664326	−0.0250555
$704$	0	0
$705$	−14.4646	−0.544770
$706$	0	0
$707$	62.0815	2.33481
$708$	0	0
$709$	−9.32406	−0.350172	−0.175086	−	0.984553i	$-0.556020\pi$
$709$	−9.32406	−0.350172	−0.175086	+	0.984553i	$0.556020\pi$
$710$	0	0
$711$	−36.5443	−1.37052
$712$	0	0
$713$	−80.5446	−3.01642
$714$	0	0
$715$	−14.2177	−0.531711
$716$	0	0
$717$	37.5298	1.40157
$718$	0	0
$719$	1.57759	0.0588341	0.0294170	−	0.999567i	$-0.490635\pi$
$719$	1.57759	0.0588341	0.0294170	+	0.999567i	$0.490635\pi$
$720$	0	0
$721$	−63.7188	−2.37301
$722$	0	0
$723$	30.9919	1.15260
$724$	0	0
$725$	6.59821	0.245051
$726$	0	0
$727$	16.4423	0.609813	0.304906	−	0.952382i	$-0.401375\pi$
$727$	16.4423	0.609813	0.304906	+	0.952382i	$0.401375\pi$
$728$	0	0
$729$	−26.4720	−0.980446
$730$	0	0
$731$	−19.2862	−0.713326
$732$	0	0
$733$	23.7228	0.876222	0.438111	−	0.898921i	$-0.355648\pi$
$733$	23.7228	0.876222	0.438111	+	0.898921i	$0.355648\pi$
$734$	0	0
$735$	−41.7322	−1.53931
$736$	0	0
$737$	54.4379	2.00524
$738$	0	0
$739$	−27.4891	−1.01120	−0.505601	−	0.862768i	$-0.668729\pi$
$739$	−27.4891	−1.01120	−0.505601	+	0.862768i	$0.668729\pi$
$740$	0	0
$741$	5.02677	0.184663
$742$	0	0
$743$	−22.3533	−0.820064	−0.410032	−	0.912071i	$-0.634483\pi$
$743$	−22.3533	−0.820064	−0.410032	+	0.912071i	$0.634483\pi$
$744$	0	0
$745$	−7.43163	−0.272274
$746$	0	0
$747$	11.5380	0.422154
$748$	0	0
$749$	18.0931	0.661108
$750$	0	0
$751$	27.5418	1.00501	0.502507	−	0.864573i	$-0.332411\pi$
$751$	27.5418	1.00501	0.502507	+	0.864573i	$0.332411\pi$
$752$	0	0
$753$	−55.0936	−2.00772
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−0.0171918	−0.000624848 0	−0.000312424	−	1.00000i	$-0.500099\pi$
$757$	−0.0171918	−0.000624848 0	−0.000312424		1.00000i	$0.500099\pi$
$758$	0	0
$759$	102.908	3.73532
$760$	0	0
$761$	−19.8740	−0.720433	−0.360216	−	0.932869i	$-0.617297\pi$
$761$	−19.8740	−0.720433	−0.360216	+	0.932869i	$0.617297\pi$
$762$	0	0
$763$	4.22351	0.152901
$764$	0	0
$765$	−13.5022	−0.488172
$766$	0	0
$767$	35.0548	1.26575
$768$	0	0
$769$	31.1662	1.12388	0.561940	−	0.827178i	$-0.310055\pi$
$769$	31.1662	1.12388	0.561940	+	0.827178i	$0.310055\pi$
$770$	0	0
$771$	−48.9599	−1.76325
$772$	0	0
$773$	49.5132	1.78087	0.890434	−	0.455113i	$-0.150401\pi$
$773$	49.5132	1.78087	0.890434	+	0.455113i	$0.150401\pi$
$774$	0	0
$775$	−8.71715	−0.313129
$776$	0	0
$777$	12.0782	0.433301
$778$	0	0
$779$	4.32907	0.155105
$780$	0	0
$781$	−35.6552	−1.27584
$782$	0	0
$783$	0.470639	0.0168193
$784$	0	0
$785$	19.6466	0.701217
$786$	0	0
$787$	45.0522	1.60594	0.802968	−	0.596022i	$-0.203253\pi$
$787$	45.0522	1.60594	0.802968	+	0.596022i	$0.203253\pi$
$788$	0	0
$789$	−37.5284	−1.33605
$790$	0	0
$791$	4.84620	0.172311
$792$	0	0
$793$	6.36591	0.226060
$794$	0	0
$795$	−13.1186	−0.465270
$796$	0	0
$797$	0.404496	0.0143280	0.00716399	−	0.999974i	$-0.497720\pi$
$797$	0.404496	0.0143280	0.00716399	+	0.999974i	$0.497720\pi$
$798$	0	0
$799$	−26.9042	−0.951801
$800$	0	0
$801$	−3.24784	−0.114757
$802$	0	0
$803$	−26.3590	−0.930190
$804$	0	0
$805$	−45.3397	−1.59801
$806$	0	0
$807$	19.5950	0.689778
$808$	0	0
$809$	2.67030	0.0938828	0.0469414	−	0.998898i	$-0.485053\pi$
$809$	2.67030	0.0938828	0.0469414	+	0.998898i	$0.485053\pi$
$810$	0	0
$811$	8.37223	0.293989	0.146994	−	0.989137i	$-0.453040\pi$
$811$	8.37223	0.293989	0.146994	+	0.989137i	$0.453040\pi$
$812$	0	0
$813$	4.21796	0.147930
$814$	0	0
$815$	−20.2618	−0.709739
$816$	0	0
$817$	−2.79854	−0.0979084
$818$	0	0
$819$	−45.4726	−1.58894
$820$	0	0
$821$	−38.6690	−1.34956	−0.674778	−	0.738020i	$-0.735761\pi$
$821$	−38.6690	−1.34956	−0.674778	+	0.738020i	$0.735761\pi$
$822$	0	0
$823$	−9.91401	−0.345581	−0.172790	−	0.984959i	$-0.555278\pi$
$823$	−9.91401	−0.345581	−0.172790	+	0.984959i	$0.555278\pi$
$824$	0	0
$825$	11.1375	0.387757
$826$	0	0
$827$	−23.8640	−0.829831	−0.414915	−	0.909860i	$-0.636189\pi$
$827$	−23.8640	−0.829831	−0.414915	+	0.909860i	$0.636189\pi$
$828$	0	0
$829$	30.1061	1.04563	0.522814	−	0.852447i	$-0.324882\pi$
$829$	30.1061	1.04563	0.522814	+	0.852447i	$0.324882\pi$
$830$	0	0
$831$	9.55306	0.331392
$832$	0	0
$833$	−77.6217	−2.68943
$834$	0	0
$835$	−18.8685	−0.652970
$836$	0	0
$837$	−0.621779	−0.0214918
$838$	0	0
$839$	10.5196	0.363177	0.181588	−	0.983375i	$-0.441876\pi$
$839$	10.5196	0.363177	0.181588	+	0.983375i	$0.441876\pi$
$840$	0	0
$841$	14.5364	0.501254
$842$	0	0
$843$	62.9866	2.16937
$844$	0	0
$845$	−3.26990	−0.112488
$846$	0	0
$847$	−47.9658	−1.64813
$848$	0	0
$849$	−57.3163	−1.96709
$850$	0	0
$851$	9.30743	0.319054
$852$	0	0
$853$	26.4542	0.905776	0.452888	−	0.891567i	$-0.350394\pi$
$853$	26.4542	0.905776	0.452888	+	0.891567i	$0.350394\pi$
$854$	0	0
$855$	−1.95924	−0.0670047
$856$	0	0
$857$	38.1461	1.30305	0.651524	−	0.758628i	$-0.274130\pi$
$857$	38.1461	1.30305	0.651524	+	0.758628i	$0.274130\pi$
$858$	0	0
$859$	−26.2063	−0.894148	−0.447074	−	0.894497i	$-0.647534\pi$
$859$	−26.2063	−0.894148	−0.447074	+	0.894497i	$0.647534\pi$
$860$	0	0
$861$	−78.7071	−2.68233
$862$	0	0
$863$	3.93531	0.133959	0.0669797	−	0.997754i	$-0.478664\pi$
$863$	3.93531	0.133959	0.0669797	+	0.997754i	$0.478664\pi$
$864$	0	0
$865$	19.6606	0.668481
$866$	0	0
$867$	−8.93486	−0.303444
$868$	0	0
$869$	56.0681	1.90198
$870$	0	0
$871$	−37.2554	−1.26235
$872$	0	0
$873$	2.95676	0.100071
$874$	0	0
$875$	−4.90700	−0.165887
$876$	0	0
$877$	45.4315	1.53411	0.767057	−	0.641579i	$-0.221720\pi$
$877$	45.4315	1.53411	0.767057	+	0.641579i	$0.221720\pi$
$878$	0	0
$879$	72.4022	2.44206
$880$	0	0
$881$	−13.0379	−0.439257	−0.219628	−	0.975584i	$-0.570484\pi$
$881$	−13.0379	−0.439257	−0.219628	+	0.975584i	$0.570484\pi$
$882$	0	0
$883$	44.1661	1.48631	0.743154	−	0.669121i	$-0.233329\pi$
$883$	44.1661	1.48631	0.743154	+	0.669121i	$0.233329\pi$
$884$	0	0
$885$	−27.4603	−0.923068
$886$	0	0
$887$	−40.1853	−1.34929	−0.674646	−	0.738142i	$-0.735704\pi$
$887$	−40.1853	−1.34929	−0.674646	+	0.738142i	$0.735704\pi$
$888$	0	0
$889$	−4.03026	−0.135171
$890$	0	0
$891$	41.4169	1.38752
$892$	0	0
$893$	−3.90395	−0.130641
$894$	0	0
$895$	−3.10016	−0.103627
$896$	0	0
$897$	−70.4267	−2.35148
$898$	0	0
$899$	−57.5176	−1.91832
$900$	0	0
$901$	−24.4006	−0.812903
$902$	0	0
$903$	50.8804	1.69319
$904$	0	0
$905$	20.1645	0.670291
$906$	0	0
$907$	−44.5559	−1.47946	−0.739728	−	0.672906i	$-0.765046\pi$
$907$	−44.5559	−1.47946	−0.739728	+	0.672906i	$0.765046\pi$
$908$	0	0
$909$	−37.5855	−1.24663
$910$	0	0
$911$	0.356378	0.0118073	0.00590367	−	0.999983i	$-0.498121\pi$
$911$	0.356378	0.0118073	0.00590367	+	0.999983i	$0.498121\pi$
$912$	0	0
$913$	−17.7022	−0.585857
$914$	0	0
$915$	−4.98676	−0.164857
$916$	0	0
$917$	47.9472	1.58336
$918$	0	0
$919$	3.29330	0.108636	0.0543181	−	0.998524i	$-0.482702\pi$
$919$	3.29330	0.108636	0.0543181	+	0.998524i	$0.482702\pi$
$920$	0	0
$921$	−46.0733	−1.51817
$922$	0	0
$923$	24.4012	0.803176
$924$	0	0
$925$	1.00732	0.0331205
$926$	0	0
$927$	38.5768	1.26703
$928$	0	0
$929$	−54.4247	−1.78562	−0.892809	−	0.450436i	$-0.851269\pi$
$929$	−54.4247	−1.78562	−0.892809	+	0.450436i	$0.851269\pi$
$930$	0	0
$931$	−11.2634	−0.369142
$932$	0	0
$933$	21.4883	0.703495
$934$	0	0
$935$	20.7157	0.677475
$936$	0	0
$937$	−18.4535	−0.602851	−0.301425	−	0.953490i	$-0.597462\pi$
$937$	−18.4535	−0.602851	−0.301425	+	0.953490i	$0.597462\pi$
$938$	0	0
$939$	15.2948	0.499126
$940$	0	0
$941$	16.2609	0.530090	0.265045	−	0.964236i	$-0.414613\pi$
$941$	16.2609	0.530090	0.265045	+	0.964236i	$0.414613\pi$
$942$	0	0
$943$	−60.6517	−1.97509
$944$	0	0
$945$	−0.350008	−0.0113858
$946$	0	0
$947$	−27.6991	−0.900099	−0.450050	−	0.893003i	$-0.648594\pi$
$947$	−27.6991	−0.900099	−0.450050	+	0.893003i	$0.648594\pi$
$948$	0	0
$949$	18.0392	0.585578
$950$	0	0
$951$	63.0038	2.04304
$952$	0	0
$953$	−7.04323	−0.228153	−0.114076	−	0.993472i	$-0.536391\pi$
$953$	−7.04323	−0.228153	−0.114076	+	0.993472i	$0.536391\pi$
$954$	0	0
$955$	26.0368	0.842532
$956$	0	0
$957$	73.4874	2.37551
$958$	0	0
$959$	100.694	3.25158
$960$	0	0
$961$	44.9886	1.45125
$962$	0	0
$963$	−10.9540	−0.352987
$964$	0	0
$965$	−0.537494	−0.0173025
$966$	0	0
$967$	−21.9286	−0.705175	−0.352588	−	0.935779i	$-0.614698\pi$
$967$	−21.9286	−0.705175	−0.352588	+	0.935779i	$0.614698\pi$
$968$	0	0
$969$	−7.32418	−0.235287
$970$	0	0
$971$	23.2670	0.746673	0.373336	−	0.927696i	$-0.378214\pi$
$971$	23.2670	0.746673	0.373336	+	0.927696i	$0.378214\pi$
$972$	0	0
$973$	29.4031	0.942620
$974$	0	0
$975$	−7.62211	−0.244103
$976$	0	0
$977$	27.9216	0.893292	0.446646	−	0.894711i	$-0.352618\pi$
$977$	27.9216	0.893292	0.446646	+	0.894711i	$0.352618\pi$
$978$	0	0
$979$	4.98298	0.159257
$980$	0	0
$981$	−2.55701	−0.0816389
$982$	0	0
$983$	−4.15409	−0.132495	−0.0662474	−	0.997803i	$-0.521103\pi$
$983$	−4.15409	−0.132495	−0.0662474	+	0.997803i	$0.521103\pi$
$984$	0	0
$985$	25.7975	0.821977
$986$	0	0
$987$	70.9780	2.25926
$988$	0	0
$989$	39.2085	1.24676
$990$	0	0
$991$	−14.1314	−0.448900	−0.224450	−	0.974486i	$-0.572058\pi$
$991$	−14.1314	−0.448900	−0.224450	+	0.974486i	$0.572058\pi$
$992$	0	0
$993$	37.9788	1.20522
$994$	0	0
$995$	20.6942	0.656049
$996$	0	0
$997$	2.90142	0.0918887	0.0459444	−	0.998944i	$-0.485370\pi$
$997$	2.90142	0.0918887	0.0459444	+	0.998944i	$0.485370\pi$
$998$	0	0
$999$	0.0718505	0.00227325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.p.1.4	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.p.1.4	✓	19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44352 q^{3} +1.00000 q^{5} -4.90700 q^{7} +2.97081 q^{9} +O(q^{10})\)	\(q-2.44352 q^{3} +1.00000 q^{5} -4.90700 q^{7} +2.97081 q^{9} -4.55796 q^{11} +3.11931 q^{13} -2.44352 q^{15} -4.54495 q^{17} -0.659498 q^{19} +11.9904 q^{21} +9.23979 q^{23} +1.00000 q^{25} +0.0713283 q^{27} +6.59821 q^{29} -8.71715 q^{31} +11.1375 q^{33} -4.90700 q^{35} +1.00732 q^{37} -7.62211 q^{39} -6.56419 q^{41} +4.24344 q^{43} +2.97081 q^{45} +5.91958 q^{47} +17.0787 q^{49} +11.1057 q^{51} +5.36874 q^{53} -4.55796 q^{55} +1.61150 q^{57} +11.2380 q^{59} +2.04081 q^{61} -14.5778 q^{63} +3.11931 q^{65} -11.9435 q^{67} -22.5776 q^{69} +7.82263 q^{71} +5.78308 q^{73} -2.44352 q^{75} +22.3659 q^{77} -12.3011 q^{79} -9.08672 q^{81} +3.88380 q^{83} -4.54495 q^{85} -16.1229 q^{87} -1.09325 q^{89} -15.3065 q^{91} +21.3006 q^{93} -0.659498 q^{95} +0.995271 q^{97} -13.5408 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * q^99

Introduction

L-functions

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Database

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