LMFDB - Embedded newform 6160.2.a.bm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.90321 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.42864 q^{9} +O(q^{10})$	$q+2.90321 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.42864 q^{9} +1.00000 q^{11} +0.903212 q^{13} -2.90321 q^{15} -0.903212 q^{17} +7.05086 q^{19} -2.90321 q^{21} -1.37778 q^{23} +1.00000 q^{25} +7.05086 q^{27} +3.80642 q^{29} +0.280996 q^{31} +2.90321 q^{33} +1.00000 q^{35} +2.42864 q^{37} +2.62222 q^{39} +2.28100 q^{41} -6.23506 q^{43} -5.42864 q^{45} +1.65878 q^{47} +1.00000 q^{49} -2.62222 q^{51} +5.18421 q^{53} -1.00000 q^{55} +20.4701 q^{57} +2.47457 q^{59} -10.5763 q^{61} -5.42864 q^{63} -0.903212 q^{65} -3.47949 q^{67} -4.00000 q^{69} -5.80642 q^{71} +13.7605 q^{73} +2.90321 q^{75} -1.00000 q^{77} +8.99063 q^{79} +4.18421 q^{81} +0.949145 q^{83} +0.903212 q^{85} +11.0509 q^{87} +4.10171 q^{89} -0.903212 q^{91} +0.815792 q^{93} -7.05086 q^{95} +6.56199 q^{97} +5.42864 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 2 q^{15} + 4 q^{17} + 8 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{25} + 8 q^{27} - 2 q^{29} - 6 q^{31} + 2 q^{33} + 3 q^{35} - 6 q^{37} + 8 q^{39} + 8 q^{43} - 3 q^{45} - 2 q^{47} + 3 q^{49} - 8 q^{51} + 2 q^{53} - 3 q^{55} + 8 q^{57} + 14 q^{59} - 12 q^{61} - 3 q^{63} + 4 q^{65} + 16 q^{67} - 12 q^{69} - 4 q^{71} + 8 q^{73} + 2 q^{75} - 3 q^{77} - q^{81} + 16 q^{83} - 4 q^{85} + 20 q^{87} - 14 q^{89} + 4 q^{91} + 16 q^{93} - 8 q^{95} + 6 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - 4 * q^13 - 2 * q^15 + 4 * q^17 + 8 * q^19 - 2 * q^21 - 4 * q^23 + 3 * q^25 + 8 * q^27 - 2 * q^29 - 6 * q^31 + 2 * q^33 + 3 * q^35 - 6 * q^37 + 8 * q^39 + 8 * q^43 - 3 * q^45 - 2 * q^47 + 3 * q^49 - 8 * q^51 + 2 * q^53 - 3 * q^55 + 8 * q^57 + 14 * q^59 - 12 * q^61 - 3 * q^63 + 4 * q^65 + 16 * q^67 - 12 * q^69 - 4 * q^71 + 8 * q^73 + 2 * q^75 - 3 * q^77 - q^81 + 16 * q^83 - 4 * q^85 + 20 * q^87 - 14 * q^89 + 4 * q^91 + 16 * q^93 - 8 * q^95 + 6 * q^97 + 3 * 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
$3$	2.90321	1.67617	0.838085	−	0.545540i	$-0.183675\pi$
$3$	2.90321	1.67617	0.838085	+	0.545540i	$0.183675\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	5.42864	1.80955
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.903212	0.250506	0.125253	−	0.992125i	$-0.460026\pi$
$13$	0.903212	0.250506	0.125253	+	0.992125i	$0.460026\pi$
$14$	0	0
$15$	−2.90321	−0.749606
$16$	0	0
$17$	−0.903212	−0.219061	−0.109531	−	0.993983i	$-0.534935\pi$
$17$	−0.903212	−0.219061	−0.109531	+	0.993983i	$0.534935\pi$
$18$	0	0
$19$	7.05086	1.61758	0.808789	−	0.588100i	$-0.200124\pi$
$19$	7.05086	1.61758	0.808789	+	0.588100i	$0.200124\pi$
$20$	0	0
$21$	−2.90321	−0.633533
$22$	0	0
$23$	−1.37778	−0.287288	−0.143644	−	0.989629i	$-0.545882\pi$
$23$	−1.37778	−0.287288	−0.143644	+	0.989629i	$0.545882\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	7.05086	1.35694
$28$	0	0
$29$	3.80642	0.706835	0.353418	−	0.935466i	$-0.385019\pi$
$29$	3.80642	0.706835	0.353418	+	0.935466i	$0.385019\pi$
$30$	0	0
$31$	0.280996	0.0504684	0.0252342	−	0.999682i	$-0.491967\pi$
$31$	0.280996	0.0504684	0.0252342	+	0.999682i	$0.491967\pi$
$32$	0	0
$33$	2.90321	0.505384
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	2.42864	0.399266	0.199633	−	0.979871i	$-0.436025\pi$
$37$	2.42864	0.399266	0.199633	+	0.979871i	$0.436025\pi$
$38$	0	0
$39$	2.62222	0.419891
$40$	0	0
$41$	2.28100	0.356232	0.178116	−	0.984010i	$-0.443000\pi$
$41$	2.28100	0.356232	0.178116	+	0.984010i	$0.443000\pi$
$42$	0	0
$43$	−6.23506	−0.950838	−0.475419	−	0.879759i	$-0.657704\pi$
$43$	−6.23506	−0.950838	−0.475419	+	0.879759i	$0.657704\pi$
$44$	0	0
$45$	−5.42864	−0.809254
$46$	0	0
$47$	1.65878	0.241958	0.120979	−	0.992655i	$-0.461397\pi$
$47$	1.65878	0.241958	0.120979	+	0.992655i	$0.461397\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.62222	−0.367184
$52$	0	0
$53$	5.18421	0.712106	0.356053	−	0.934466i	$-0.384122\pi$
$53$	5.18421	0.712106	0.356053	+	0.934466i	$0.384122\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	20.4701	2.71133
$58$	0	0
$59$	2.47457	0.322162	0.161081	−	0.986941i	$-0.448502\pi$
$59$	2.47457	0.322162	0.161081	+	0.986941i	$0.448502\pi$
$60$	0	0
$61$	−10.5763	−1.35415	−0.677077	−	0.735912i	$-0.736754\pi$
$61$	−10.5763	−1.35415	−0.677077	+	0.735912i	$0.736754\pi$
$62$	0	0
$63$	−5.42864	−0.683944
$64$	0	0
$65$	−0.903212	−0.112030
$66$	0	0
$67$	−3.47949	−0.425088	−0.212544	−	0.977152i	$-0.568175\pi$
$67$	−3.47949	−0.425088	−0.212544	+	0.977152i	$0.568175\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−5.80642	−0.689096	−0.344548	−	0.938769i	$-0.611968\pi$
$71$	−5.80642	−0.689096	−0.344548	+	0.938769i	$0.611968\pi$
$72$	0	0
$73$	13.7605	1.61054	0.805272	−	0.592906i	$-0.202019\pi$
$73$	13.7605	1.61054	0.805272	+	0.592906i	$0.202019\pi$
$74$	0	0
$75$	2.90321	0.335234
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	8.99063	1.01153	0.505763	−	0.862673i	$-0.331211\pi$
$79$	8.99063	1.01153	0.505763	+	0.862673i	$0.331211\pi$
$80$	0	0
$81$	4.18421	0.464912
$82$	0	0
$83$	0.949145	0.104182	0.0520911	−	0.998642i	$-0.483411\pi$
$83$	0.949145	0.104182	0.0520911	+	0.998642i	$0.483411\pi$
$84$	0	0
$85$	0.903212	0.0979671
$86$	0	0
$87$	11.0509	1.18478
$88$	0	0
$89$	4.10171	0.434780	0.217390	−	0.976085i	$-0.430246\pi$
$89$	4.10171	0.434780	0.217390	+	0.976085i	$0.430246\pi$
$90$	0	0
$91$	−0.903212	−0.0946823
$92$	0	0
$93$	0.815792	0.0845937
$94$	0	0
$95$	−7.05086	−0.723402
$96$	0	0
$97$	6.56199	0.666269	0.333135	−	0.942879i	$-0.391894\pi$
$97$	6.56199	0.666269	0.333135	+	0.942879i	$0.391894\pi$
$98$	0	0
$99$	5.42864	0.545599
$100$	0	0
$101$	5.33185	0.530539	0.265270	−	0.964174i	$-0.414539\pi$
$101$	5.33185	0.530539	0.265270	+	0.964174i	$0.414539\pi$
$102$	0	0
$103$	−0.709636	−0.0699225	−0.0349612	−	0.999389i	$-0.511131\pi$
$103$	−0.709636	−0.0699225	−0.0349612	+	0.999389i	$0.511131\pi$
$104$	0	0
$105$	2.90321	0.283324
$106$	0	0
$107$	18.9590	1.83283	0.916417	−	0.400224i	$-0.131068\pi$
$107$	18.9590	1.83283	0.916417	+	0.400224i	$0.131068\pi$
$108$	0	0
$109$	−19.4193	−1.86003	−0.930014	−	0.367523i	$-0.880206\pi$
$109$	−19.4193	−1.86003	−0.930014	+	0.367523i	$0.880206\pi$
$110$	0	0
$111$	7.05086	0.669238
$112$	0	0
$113$	7.24443	0.681499	0.340749	−	0.940154i	$-0.389319\pi$
$113$	7.24443	0.681499	0.340749	+	0.940154i	$0.389319\pi$
$114$	0	0
$115$	1.37778	0.128479
$116$	0	0
$117$	4.90321	0.453302
$118$	0	0
$119$	0.903212	0.0827973
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.62222	0.597105
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	11.6128	1.03047	0.515237	−	0.857048i	$-0.327704\pi$
$127$	11.6128	1.03047	0.515237	+	0.857048i	$0.327704\pi$
$128$	0	0
$129$	−18.1017	−1.59377
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−7.05086	−0.611387
$134$	0	0
$135$	−7.05086	−0.606841
$136$	0	0
$137$	0.326929	0.0279315	0.0139657	−	0.999902i	$-0.495554\pi$
$137$	0.326929	0.0279315	0.0139657	+	0.999902i	$0.495554\pi$
$138$	0	0
$139$	8.56199	0.726219	0.363109	−	0.931747i	$-0.381715\pi$
$139$	8.56199	0.726219	0.363109	+	0.931747i	$0.381715\pi$
$140$	0	0
$141$	4.81579	0.405563
$142$	0	0
$143$	0.903212	0.0755304
$144$	0	0
$145$	−3.80642	−0.316106
$146$	0	0
$147$	2.90321	0.239453
$148$	0	0
$149$	−10.8573	−0.889463	−0.444731	−	0.895664i	$-0.646701\pi$
$149$	−10.8573	−0.889463	−0.444731	+	0.895664i	$0.646701\pi$
$150$	0	0
$151$	−3.47949	−0.283157	−0.141579	−	0.989927i	$-0.545218\pi$
$151$	−3.47949	−0.283157	−0.141579	+	0.989927i	$0.545218\pi$
$152$	0	0
$153$	−4.90321	−0.396401
$154$	0	0
$155$	−0.280996	−0.0225702
$156$	0	0
$157$	−16.3684	−1.30634	−0.653171	−	0.757211i	$-0.726562\pi$
$157$	−16.3684	−1.30634	−0.653171	+	0.757211i	$0.726562\pi$
$158$	0	0
$159$	15.0509	1.19361
$160$	0	0
$161$	1.37778	0.108585
$162$	0	0
$163$	7.86665	0.616163	0.308082	−	0.951360i	$-0.400313\pi$
$163$	7.86665	0.616163	0.308082	+	0.951360i	$0.400313\pi$
$164$	0	0
$165$	−2.90321	−0.226015
$166$	0	0
$167$	18.9590	1.46709	0.733545	−	0.679641i	$-0.237864\pi$
$167$	18.9590	1.46709	0.733545	+	0.679641i	$0.237864\pi$
$168$	0	0
$169$	−12.1842	−0.937247
$170$	0	0
$171$	38.2766	2.92708
$172$	0	0
$173$	−9.46520	−0.719626	−0.359813	−	0.933024i	$-0.617160\pi$
$173$	−9.46520	−0.719626	−0.359813	+	0.933024i	$0.617160\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	7.18421	0.539998
$178$	0	0
$179$	20.8573	1.55895	0.779473	−	0.626436i	$-0.215487\pi$
$179$	20.8573	1.55895	0.779473	+	0.626436i	$0.215487\pi$
$180$	0	0
$181$	9.05086	0.672745	0.336372	−	0.941729i	$-0.390800\pi$
$181$	9.05086	0.672745	0.336372	+	0.941729i	$0.390800\pi$
$182$	0	0
$183$	−30.7052	−2.26979
$184$	0	0
$185$	−2.42864	−0.178557
$186$	0	0
$187$	−0.903212	−0.0660494
$188$	0	0
$189$	−7.05086	−0.512874
$190$	0	0
$191$	−17.7146	−1.28178	−0.640890	−	0.767633i	$-0.721435\pi$
$191$	−17.7146	−1.28178	−0.640890	+	0.767633i	$0.721435\pi$
$192$	0	0
$193$	6.42864	0.462744	0.231372	−	0.972865i	$-0.425679\pi$
$193$	6.42864	0.462744	0.231372	+	0.972865i	$0.425679\pi$
$194$	0	0
$195$	−2.62222	−0.187781
$196$	0	0
$197$	10.4286	0.743010	0.371505	−	0.928431i	$-0.378842\pi$
$197$	10.4286	0.743010	0.371505	+	0.928431i	$0.378842\pi$
$198$	0	0
$199$	3.98571	0.282539	0.141270	−	0.989971i	$-0.454882\pi$
$199$	3.98571	0.282539	0.141270	+	0.989971i	$0.454882\pi$
$200$	0	0
$201$	−10.1017	−0.712520
$202$	0	0
$203$	−3.80642	−0.267159
$204$	0	0
$205$	−2.28100	−0.159312
$206$	0	0
$207$	−7.47949	−0.519861
$208$	0	0
$209$	7.05086	0.487718
$210$	0	0
$211$	9.24443	0.636413	0.318206	−	0.948021i	$-0.396919\pi$
$211$	9.24443	0.636413	0.318206	+	0.948021i	$0.396919\pi$
$212$	0	0
$213$	−16.8573	−1.15504
$214$	0	0
$215$	6.23506	0.425228
$216$	0	0
$217$	−0.280996	−0.0190753
$218$	0	0
$219$	39.9496	2.69955
$220$	0	0
$221$	−0.815792	−0.0548761
$222$	0	0
$223$	16.1476	1.08133	0.540663	−	0.841239i	$-0.318173\pi$
$223$	16.1476	1.08133	0.540663	+	0.841239i	$0.318173\pi$
$224$	0	0
$225$	5.42864	0.361909
$226$	0	0
$227$	−6.10171	−0.404985	−0.202492	−	0.979284i	$-0.564904\pi$
$227$	−6.10171	−0.404985	−0.202492	+	0.979284i	$0.564904\pi$
$228$	0	0
$229$	−23.4193	−1.54759	−0.773795	−	0.633437i	$-0.781644\pi$
$229$	−23.4193	−1.54759	−0.773795	+	0.633437i	$0.781644\pi$
$230$	0	0
$231$	−2.90321	−0.191017
$232$	0	0
$233$	2.52051	0.165124	0.0825619	−	0.996586i	$-0.473690\pi$
$233$	2.52051	0.165124	0.0825619	+	0.996586i	$0.473690\pi$
$234$	0	0
$235$	−1.65878	−0.108207
$236$	0	0
$237$	26.1017	1.69549
$238$	0	0
$239$	−19.3461	−1.25140	−0.625699	−	0.780065i	$-0.715186\pi$
$239$	−19.3461	−1.25140	−0.625699	+	0.780065i	$0.715186\pi$
$240$	0	0
$241$	−23.4336	−1.50949	−0.754744	−	0.656019i	$-0.772239\pi$
$241$	−23.4336	−1.50949	−0.754744	+	0.656019i	$0.772239\pi$
$242$	0	0
$243$	−9.00492	−0.577666
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	6.36842	0.405213
$248$	0	0
$249$	2.75557	0.174627
$250$	0	0
$251$	−2.20787	−0.139359	−0.0696796	−	0.997569i	$-0.522198\pi$
$251$	−2.20787	−0.139359	−0.0696796	+	0.997569i	$0.522198\pi$
$252$	0	0
$253$	−1.37778	−0.0866206
$254$	0	0
$255$	2.62222	0.164210
$256$	0	0
$257$	17.6128	1.09866	0.549330	−	0.835606i	$-0.314883\pi$
$257$	17.6128	1.09866	0.549330	+	0.835606i	$0.314883\pi$
$258$	0	0
$259$	−2.42864	−0.150908
$260$	0	0
$261$	20.6637	1.27905
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	−5.18421	−0.318463
$266$	0	0
$267$	11.9081	0.728766
$268$	0	0
$269$	−10.5620	−0.643976	−0.321988	−	0.946744i	$-0.604351\pi$
$269$	−10.5620	−0.643976	−0.321988	+	0.946744i	$0.604351\pi$
$270$	0	0
$271$	19.6414	1.19313	0.596566	−	0.802564i	$-0.296531\pi$
$271$	19.6414	1.19313	0.596566	+	0.802564i	$0.296531\pi$
$272$	0	0
$273$	−2.62222	−0.158704
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	24.5303	1.47389	0.736943	−	0.675955i	$-0.236269\pi$
$277$	24.5303	1.47389	0.736943	+	0.675955i	$0.236269\pi$
$278$	0	0
$279$	1.52543	0.0913250
$280$	0	0
$281$	−9.05086	−0.539929	−0.269964	−	0.962870i	$-0.587012\pi$
$281$	−9.05086	−0.539929	−0.269964	+	0.962870i	$0.587012\pi$
$282$	0	0
$283$	−3.61285	−0.214762	−0.107381	−	0.994218i	$-0.534246\pi$
$283$	−3.61285	−0.214762	−0.107381	+	0.994218i	$0.534246\pi$
$284$	0	0
$285$	−20.4701	−1.21255
$286$	0	0
$287$	−2.28100	−0.134643
$288$	0	0
$289$	−16.1842	−0.952012
$290$	0	0
$291$	19.0509	1.11678
$292$	0	0
$293$	31.1798	1.82154	0.910771	−	0.412913i	$-0.135489\pi$
$293$	31.1798	1.82154	0.910771	+	0.412913i	$0.135489\pi$
$294$	0	0
$295$	−2.47457	−0.144075
$296$	0	0
$297$	7.05086	0.409132
$298$	0	0
$299$	−1.24443	−0.0719673
$300$	0	0
$301$	6.23506	0.359383
$302$	0	0
$303$	15.4795	0.889274
$304$	0	0
$305$	10.5763	0.605596
$306$	0	0
$307$	−2.10171	−0.119951	−0.0599755	−	0.998200i	$-0.519102\pi$
$307$	−2.10171	−0.119951	−0.0599755	+	0.998200i	$0.519102\pi$
$308$	0	0
$309$	−2.06022	−0.117202
$310$	0	0
$311$	12.8430	0.728259	0.364130	−	0.931348i	$-0.381367\pi$
$311$	12.8430	0.728259	0.364130	+	0.931348i	$0.381367\pi$
$312$	0	0
$313$	−31.7146	−1.79261	−0.896306	−	0.443435i	$-0.853760\pi$
$313$	−31.7146	−1.79261	−0.896306	+	0.443435i	$0.853760\pi$
$314$	0	0
$315$	5.42864	0.305869
$316$	0	0
$317$	−7.32741	−0.411548	−0.205774	−	0.978600i	$-0.565971\pi$
$317$	−7.32741	−0.411548	−0.205774	+	0.978600i	$0.565971\pi$
$318$	0	0
$319$	3.80642	0.213119
$320$	0	0
$321$	55.0420	3.07214
$322$	0	0
$323$	−6.36842	−0.354348
$324$	0	0
$325$	0.903212	0.0501012
$326$	0	0
$327$	−56.3783	−3.11772
$328$	0	0
$329$	−1.65878	−0.0914515
$330$	0	0
$331$	−15.3461	−0.843500	−0.421750	−	0.906712i	$-0.638584\pi$
$331$	−15.3461	−0.843500	−0.421750	+	0.906712i	$0.638584\pi$
$332$	0	0
$333$	13.1842	0.722490
$334$	0	0
$335$	3.47949	0.190105
$336$	0	0
$337$	0.0316429	0.00172370	0.000861848	−	1.00000i	$-0.499726\pi$
$337$	0.0316429	0.00172370	0.000861848		1.00000i	$0.499726\pi$
$338$	0	0
$339$	21.0321	1.14231
$340$	0	0
$341$	0.280996	0.0152168
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	16.9906	0.912105	0.456052	−	0.889953i	$-0.349263\pi$
$347$	16.9906	0.912105	0.456052	+	0.889953i	$0.349263\pi$
$348$	0	0
$349$	−5.06515	−0.271131	−0.135566	−	0.990768i	$-0.543285\pi$
$349$	−5.06515	−0.271131	−0.135566	+	0.990768i	$0.543285\pi$
$350$	0	0
$351$	6.36842	0.339921
$352$	0	0
$353$	−12.3684	−0.658304	−0.329152	−	0.944277i	$-0.606763\pi$
$353$	−12.3684	−0.658304	−0.329152	+	0.944277i	$0.606763\pi$
$354$	0	0
$355$	5.80642	0.308173
$356$	0	0
$357$	2.62222	0.138782
$358$	0	0
$359$	8.72393	0.460431	0.230216	−	0.973140i	$-0.426057\pi$
$359$	8.72393	0.460431	0.230216	+	0.973140i	$0.426057\pi$
$360$	0	0
$361$	30.7146	1.61656
$362$	0	0
$363$	2.90321	0.152379
$364$	0	0
$365$	−13.7605	−0.720257
$366$	0	0
$367$	4.14764	0.216505	0.108253	−	0.994123i	$-0.465474\pi$
$367$	4.14764	0.216505	0.108253	+	0.994123i	$0.465474\pi$
$368$	0	0
$369$	12.3827	0.644618
$370$	0	0
$371$	−5.18421	−0.269151
$372$	0	0
$373$	−15.5526	−0.805284	−0.402642	−	0.915357i	$-0.631908\pi$
$373$	−15.5526	−0.805284	−0.402642	+	0.915357i	$0.631908\pi$
$374$	0	0
$375$	−2.90321	−0.149921
$376$	0	0
$377$	3.43801	0.177066
$378$	0	0
$379$	−9.80642	−0.503722	−0.251861	−	0.967763i	$-0.581043\pi$
$379$	−9.80642	−0.503722	−0.251861	+	0.967763i	$0.581043\pi$
$380$	0	0
$381$	33.7146	1.72725
$382$	0	0
$383$	13.1798	0.673454	0.336727	−	0.941602i	$-0.390680\pi$
$383$	13.1798	0.673454	0.336727	+	0.941602i	$0.390680\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	−33.8479	−1.72059
$388$	0	0
$389$	20.1017	1.01920	0.509599	−	0.860412i	$-0.329794\pi$
$389$	20.1017	1.01920	0.509599	+	0.860412i	$0.329794\pi$
$390$	0	0
$391$	1.24443	0.0629336
$392$	0	0
$393$	23.2257	1.17158
$394$	0	0
$395$	−8.99063	−0.452368
$396$	0	0
$397$	−19.9813	−1.00283	−0.501415	−	0.865207i	$-0.667187\pi$
$397$	−19.9813	−1.00283	−0.501415	+	0.865207i	$0.667187\pi$
$398$	0	0
$399$	−20.4701	−1.02479
$400$	0	0
$401$	32.1017	1.60308	0.801541	−	0.597939i	$-0.204014\pi$
$401$	32.1017	1.60308	0.801541	+	0.597939i	$0.204014\pi$
$402$	0	0
$403$	0.253799	0.0126426
$404$	0	0
$405$	−4.18421	−0.207915
$406$	0	0
$407$	2.42864	0.120383
$408$	0	0
$409$	−29.0366	−1.43577	−0.717883	−	0.696164i	$-0.754889\pi$
$409$	−29.0366	−1.43577	−0.717883	+	0.696164i	$0.754889\pi$
$410$	0	0
$411$	0.949145	0.0468179
$412$	0	0
$413$	−2.47457	−0.121766
$414$	0	0
$415$	−0.949145	−0.0465917
$416$	0	0
$417$	24.8573	1.21727
$418$	0	0
$419$	−17.5254	−0.856173	−0.428087	−	0.903738i	$-0.640812\pi$
$419$	−17.5254	−0.856173	−0.428087	+	0.903738i	$0.640812\pi$
$420$	0	0
$421$	−20.4099	−0.994718	−0.497359	−	0.867545i	$-0.665697\pi$
$421$	−20.4099	−0.994718	−0.497359	+	0.867545i	$0.665697\pi$
$422$	0	0
$423$	9.00492	0.437834
$424$	0	0
$425$	−0.903212	−0.0438122
$426$	0	0
$427$	10.5763	0.511822
$428$	0	0
$429$	2.62222	0.126602
$430$	0	0
$431$	−7.35905	−0.354473	−0.177236	−	0.984168i	$-0.556716\pi$
$431$	−7.35905	−0.354473	−0.177236	+	0.984168i	$0.556716\pi$
$432$	0	0
$433$	−25.9081	−1.24507	−0.622533	−	0.782594i	$-0.713896\pi$
$433$	−25.9081	−1.24507	−0.622533	+	0.782594i	$0.713896\pi$
$434$	0	0
$435$	−11.0509	−0.529848
$436$	0	0
$437$	−9.71456	−0.464710
$438$	0	0
$439$	−19.1338	−0.913208	−0.456604	−	0.889670i	$-0.650934\pi$
$439$	−19.1338	−0.913208	−0.456604	+	0.889670i	$0.650934\pi$
$440$	0	0
$441$	5.42864	0.258507
$442$	0	0
$443$	−22.5303	−1.07045	−0.535225	−	0.844710i	$-0.679773\pi$
$443$	−22.5303	−1.07045	−0.535225	+	0.844710i	$0.679773\pi$
$444$	0	0
$445$	−4.10171	−0.194440
$446$	0	0
$447$	−31.5210	−1.49089
$448$	0	0
$449$	−36.6133	−1.72789	−0.863945	−	0.503587i	$-0.832014\pi$
$449$	−36.6133	−1.72789	−0.863945	+	0.503587i	$0.832014\pi$
$450$	0	0
$451$	2.28100	0.107408
$452$	0	0
$453$	−10.1017	−0.474620
$454$	0	0
$455$	0.903212	0.0423432
$456$	0	0
$457$	11.9398	0.558519	0.279260	−	0.960216i	$-0.409911\pi$
$457$	11.9398	0.558519	0.279260	+	0.960216i	$0.409911\pi$
$458$	0	0
$459$	−6.36842	−0.297252
$460$	0	0
$461$	−17.9224	−0.834731	−0.417365	−	0.908739i	$-0.637046\pi$
$461$	−17.9224	−0.834731	−0.417365	+	0.908739i	$0.637046\pi$
$462$	0	0
$463$	3.59994	0.167303	0.0836517	−	0.996495i	$-0.473342\pi$
$463$	3.59994	0.167303	0.0836517	+	0.996495i	$0.473342\pi$
$464$	0	0
$465$	−0.815792	−0.0378314
$466$	0	0
$467$	−33.0879	−1.53113	−0.765563	−	0.643361i	$-0.777539\pi$
$467$	−33.0879	−1.53113	−0.765563	+	0.643361i	$0.777539\pi$
$468$	0	0
$469$	3.47949	0.160668
$470$	0	0
$471$	−47.5210	−2.18965
$472$	0	0
$473$	−6.23506	−0.286689
$474$	0	0
$475$	7.05086	0.323515
$476$	0	0
$477$	28.1432	1.28859
$478$	0	0
$479$	26.3970	1.20611	0.603055	−	0.797700i	$-0.293950\pi$
$479$	26.3970	1.20611	0.603055	+	0.797700i	$0.293950\pi$
$480$	0	0
$481$	2.19358	0.100018
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	−6.56199	−0.297965
$486$	0	0
$487$	18.8256	0.853071	0.426536	−	0.904471i	$-0.359734\pi$
$487$	18.8256	0.853071	0.426536	+	0.904471i	$0.359734\pi$
$488$	0	0
$489$	22.8385	1.03279
$490$	0	0
$491$	3.00937	0.135811	0.0679054	−	0.997692i	$-0.478368\pi$
$491$	3.00937	0.135811	0.0679054	+	0.997692i	$0.478368\pi$
$492$	0	0
$493$	−3.43801	−0.154840
$494$	0	0
$495$	−5.42864	−0.243999
$496$	0	0
$497$	5.80642	0.260454
$498$	0	0
$499$	−5.68598	−0.254539	−0.127270	−	0.991868i	$-0.540621\pi$
$499$	−5.68598	−0.254539	−0.127270	+	0.991868i	$0.540621\pi$
$500$	0	0
$501$	55.0420	2.45909
$502$	0	0
$503$	−33.8064	−1.50735	−0.753677	−	0.657245i	$-0.771722\pi$
$503$	−33.8064	−1.50735	−0.753677	+	0.657245i	$0.771722\pi$
$504$	0	0
$505$	−5.33185	−0.237264
$506$	0	0
$507$	−35.3733	−1.57099
$508$	0	0
$509$	−9.31756	−0.412994	−0.206497	−	0.978447i	$-0.566206\pi$
$509$	−9.31756	−0.412994	−0.206497	+	0.978447i	$0.566206\pi$
$510$	0	0
$511$	−13.7605	−0.608728
$512$	0	0
$513$	49.7146	2.19495
$514$	0	0
$515$	0.709636	0.0312703
$516$	0	0
$517$	1.65878	0.0729531
$518$	0	0
$519$	−27.4795	−1.20622
$520$	0	0
$521$	−26.6450	−1.16734	−0.583669	−	0.811992i	$-0.698383\pi$
$521$	−26.6450	−1.16734	−0.583669	+	0.811992i	$0.698383\pi$
$522$	0	0
$523$	20.2667	0.886201	0.443101	−	0.896472i	$-0.353878\pi$
$523$	20.2667	0.886201	0.443101	+	0.896472i	$0.353878\pi$
$524$	0	0
$525$	−2.90321	−0.126707
$526$	0	0
$527$	−0.253799	−0.0110557
$528$	0	0
$529$	−21.1017	−0.917466
$530$	0	0
$531$	13.4336	0.582967
$532$	0	0
$533$	2.06022	0.0892382
$534$	0	0
$535$	−18.9590	−0.819669
$536$	0	0
$537$	60.5531	2.61306
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	17.3461	0.745769	0.372884	−	0.927878i	$-0.378369\pi$
$541$	17.3461	0.745769	0.372884	+	0.927878i	$0.378369\pi$
$542$	0	0
$543$	26.2766	1.12763
$544$	0	0
$545$	19.4193	0.831830
$546$	0	0
$547$	−22.7556	−0.972958	−0.486479	−	0.873692i	$-0.661719\pi$
$547$	−22.7556	−0.972958	−0.486479	+	0.873692i	$0.661719\pi$
$548$	0	0
$549$	−57.4148	−2.45041
$550$	0	0
$551$	26.8385	1.14336
$552$	0	0
$553$	−8.99063	−0.382321
$554$	0	0
$555$	−7.05086	−0.299292
$556$	0	0
$557$	−32.7338	−1.38697	−0.693487	−	0.720469i	$-0.743927\pi$
$557$	−32.7338	−1.38697	−0.693487	+	0.720469i	$0.743927\pi$
$558$	0	0
$559$	−5.63158	−0.238191
$560$	0	0
$561$	−2.62222	−0.110710
$562$	0	0
$563$	−6.93041	−0.292082	−0.146041	−	0.989279i	$-0.546653\pi$
$563$	−6.93041	−0.292082	−0.146041	+	0.989279i	$0.546653\pi$
$564$	0	0
$565$	−7.24443	−0.304776
$566$	0	0
$567$	−4.18421	−0.175720
$568$	0	0
$569$	34.1116	1.43003	0.715015	−	0.699109i	$-0.246420\pi$
$569$	34.1116	1.43003	0.715015	+	0.699109i	$0.246420\pi$
$570$	0	0
$571$	20.2667	0.848135	0.424068	−	0.905631i	$-0.360602\pi$
$571$	20.2667	0.848135	0.424068	+	0.905631i	$0.360602\pi$
$572$	0	0
$573$	−51.4291	−2.14848
$574$	0	0
$575$	−1.37778	−0.0574576
$576$	0	0
$577$	−7.53972	−0.313883	−0.156941	−	0.987608i	$-0.550163\pi$
$577$	−7.53972	−0.313883	−0.156941	+	0.987608i	$0.550163\pi$
$578$	0	0
$579$	18.6637	0.775637
$580$	0	0
$581$	−0.949145	−0.0393772
$582$	0	0
$583$	5.18421	0.214708
$584$	0	0
$585$	−4.90321	−0.202723
$586$	0	0
$587$	−4.53480	−0.187171	−0.0935855	−	0.995611i	$-0.529833\pi$
$587$	−4.53480	−0.187171	−0.0935855	+	0.995611i	$0.529833\pi$
$588$	0	0
$589$	1.98126	0.0816366
$590$	0	0
$591$	30.2766	1.24541
$592$	0	0
$593$	−19.5669	−0.803517	−0.401758	−	0.915746i	$-0.631601\pi$
$593$	−19.5669	−0.803517	−0.401758	+	0.915746i	$0.631601\pi$
$594$	0	0
$595$	−0.903212	−0.0370281
$596$	0	0
$597$	11.5714	0.473584
$598$	0	0
$599$	−32.3783	−1.32294	−0.661470	−	0.749972i	$-0.730067\pi$
$599$	−32.3783	−1.32294	−0.661470	+	0.749972i	$0.730067\pi$
$600$	0	0
$601$	17.5067	0.714113	0.357056	−	0.934083i	$-0.383780\pi$
$601$	17.5067	0.714113	0.357056	+	0.934083i	$0.383780\pi$
$602$	0	0
$603$	−18.8889	−0.769216
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−26.9590	−1.09423	−0.547116	−	0.837057i	$-0.684274\pi$
$607$	−26.9590	−1.09423	−0.547116	+	0.837057i	$0.684274\pi$
$608$	0	0
$609$	−11.0509	−0.447803
$610$	0	0
$611$	1.49823	0.0606119
$612$	0	0
$613$	1.00937	0.0407680	0.0203840	−	0.999792i	$-0.493511\pi$
$613$	1.00937	0.0407680	0.0203840	+	0.999792i	$0.493511\pi$
$614$	0	0
$615$	−6.62222	−0.267034
$616$	0	0
$617$	−21.1842	−0.852844	−0.426422	−	0.904524i	$-0.640226\pi$
$617$	−21.1842	−0.852844	−0.426422	+	0.904524i	$0.640226\pi$
$618$	0	0
$619$	32.7511	1.31638	0.658189	−	0.752852i	$-0.271323\pi$
$619$	32.7511	1.31638	0.658189	+	0.752852i	$0.271323\pi$
$620$	0	0
$621$	−9.71456	−0.389832
$622$	0	0
$623$	−4.10171	−0.164332
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	20.4701	0.817498
$628$	0	0
$629$	−2.19358	−0.0874636
$630$	0	0
$631$	−17.8983	−0.712520	−0.356260	−	0.934387i	$-0.615948\pi$
$631$	−17.8983	−0.712520	−0.356260	+	0.934387i	$0.615948\pi$
$632$	0	0
$633$	26.8385	1.06674
$634$	0	0
$635$	−11.6128	−0.460842
$636$	0	0
$637$	0.903212	0.0357866
$638$	0	0
$639$	−31.5210	−1.24695
$640$	0	0
$641$	−10.0415	−0.396615	−0.198307	−	0.980140i	$-0.563544\pi$
$641$	−10.0415	−0.396615	−0.198307	+	0.980140i	$0.563544\pi$
$642$	0	0
$643$	−30.7195	−1.21146	−0.605729	−	0.795671i	$-0.707118\pi$
$643$	−30.7195	−1.21146	−0.605729	+	0.795671i	$0.707118\pi$
$644$	0	0
$645$	18.1017	0.712754
$646$	0	0
$647$	26.8943	1.05732	0.528662	−	0.848832i	$-0.322694\pi$
$647$	26.8943	1.05732	0.528662	+	0.848832i	$0.322694\pi$
$648$	0	0
$649$	2.47457	0.0971355
$650$	0	0
$651$	−0.815792	−0.0319734
$652$	0	0
$653$	−39.4104	−1.54225	−0.771124	−	0.636685i	$-0.780305\pi$
$653$	−39.4104	−1.54225	−0.771124	+	0.636685i	$0.780305\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	0	0
$657$	74.7007	2.91435
$658$	0	0
$659$	−10.2351	−0.398701	−0.199351	−	0.979928i	$-0.563883\pi$
$659$	−10.2351	−0.398701	−0.199351	+	0.979928i	$0.563883\pi$
$660$	0	0
$661$	−2.77430	−0.107908	−0.0539540	−	0.998543i	$-0.517182\pi$
$661$	−2.77430	−0.107908	−0.0539540	+	0.998543i	$0.517182\pi$
$662$	0	0
$663$	−2.36842	−0.0919817
$664$	0	0
$665$	7.05086	0.273420
$666$	0	0
$667$	−5.24443	−0.203065
$668$	0	0
$669$	46.8800	1.81249
$670$	0	0
$671$	−10.5763	−0.408293
$672$	0	0
$673$	−27.5812	−1.06318	−0.531589	−	0.847003i	$-0.678405\pi$
$673$	−27.5812	−1.06318	−0.531589	+	0.847003i	$0.678405\pi$
$674$	0	0
$675$	7.05086	0.271388
$676$	0	0
$677$	−38.9131	−1.49555	−0.747775	−	0.663952i	$-0.768878\pi$
$677$	−38.9131	−1.49555	−0.747775	+	0.663952i	$0.768878\pi$
$678$	0	0
$679$	−6.56199	−0.251826
$680$	0	0
$681$	−17.7146	−0.678823
$682$	0	0
$683$	−18.1432	−0.694230	−0.347115	−	0.937823i	$-0.612839\pi$
$683$	−18.1432	−0.694230	−0.347115	+	0.937823i	$0.612839\pi$
$684$	0	0
$685$	−0.326929	−0.0124913
$686$	0	0
$687$	−67.9911	−2.59402
$688$	0	0
$689$	4.68244	0.178387
$690$	0	0
$691$	−42.4657	−1.61547	−0.807735	−	0.589545i	$-0.799307\pi$
$691$	−42.4657	−1.61547	−0.807735	+	0.589545i	$0.799307\pi$
$692$	0	0
$693$	−5.42864	−0.206217
$694$	0	0
$695$	−8.56199	−0.324775
$696$	0	0
$697$	−2.06022	−0.0780365
$698$	0	0
$699$	7.31756	0.276776
$700$	0	0
$701$	−31.8894	−1.20445	−0.602223	−	0.798328i	$-0.705718\pi$
$701$	−31.8894	−1.20445	−0.602223	+	0.798328i	$0.705718\pi$
$702$	0	0
$703$	17.1240	0.645843
$704$	0	0
$705$	−4.81579	−0.181373
$706$	0	0
$707$	−5.33185	−0.200525
$708$	0	0
$709$	−5.07944	−0.190762	−0.0953811	−	0.995441i	$-0.530407\pi$
$709$	−5.07944	−0.190762	−0.0953811	+	0.995441i	$0.530407\pi$
$710$	0	0
$711$	48.8069	1.83040
$712$	0	0
$713$	−0.387152	−0.0144990
$714$	0	0
$715$	−0.903212	−0.0337782
$716$	0	0
$717$	−56.1659	−2.09756
$718$	0	0
$719$	−37.7832	−1.40908	−0.704539	−	0.709666i	$-0.748846\pi$
$719$	−37.7832	−1.40908	−0.704539	+	0.709666i	$0.748846\pi$
$720$	0	0
$721$	0.709636	0.0264282
$722$	0	0
$723$	−68.0326	−2.53016
$724$	0	0
$725$	3.80642	0.141367
$726$	0	0
$727$	20.7926	0.771155	0.385578	−	0.922675i	$-0.374002\pi$
$727$	20.7926	0.771155	0.385578	+	0.922675i	$0.374002\pi$
$728$	0	0
$729$	−38.6958	−1.43318
$730$	0	0
$731$	5.63158	0.208292
$732$	0	0
$733$	37.7891	1.39577	0.697886	−	0.716209i	$-0.254124\pi$
$733$	37.7891	1.39577	0.697886	+	0.716209i	$0.254124\pi$
$734$	0	0
$735$	−2.90321	−0.107087
$736$	0	0
$737$	−3.47949	−0.128169
$738$	0	0
$739$	8.73683	0.321390	0.160695	−	0.987004i	$-0.448627\pi$
$739$	8.73683	0.321390	0.160695	+	0.987004i	$0.448627\pi$
$740$	0	0
$741$	18.4889	0.679205
$742$	0	0
$743$	−33.7975	−1.23991	−0.619956	−	0.784637i	$-0.712849\pi$
$743$	−33.7975	−1.23991	−0.619956	+	0.784637i	$0.712849\pi$
$744$	0	0
$745$	10.8573	0.397780
$746$	0	0
$747$	5.15257	0.188523
$748$	0	0
$749$	−18.9590	−0.692746
$750$	0	0
$751$	−9.60300	−0.350419	−0.175209	−	0.984531i	$-0.556060\pi$
$751$	−9.60300	−0.350419	−0.175209	+	0.984531i	$0.556060\pi$
$752$	0	0
$753$	−6.40990	−0.233590
$754$	0	0
$755$	3.47949	0.126632
$756$	0	0
$757$	23.1811	0.842533	0.421267	−	0.906937i	$-0.361586\pi$
$757$	23.1811	0.842533	0.421267	+	0.906937i	$0.361586\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	46.7422	1.69440	0.847202	−	0.531270i	$-0.178285\pi$
$761$	46.7422	1.69440	0.847202	+	0.531270i	$0.178285\pi$
$762$	0	0
$763$	19.4193	0.703025
$764$	0	0
$765$	4.90321	0.177276
$766$	0	0
$767$	2.23506	0.0807035
$768$	0	0
$769$	17.5353	0.632338	0.316169	−	0.948703i	$-0.397603\pi$
$769$	17.5353	0.632338	0.316169	+	0.948703i	$0.397603\pi$
$770$	0	0
$771$	51.1338	1.84154
$772$	0	0
$773$	31.1526	1.12048	0.560240	−	0.828330i	$-0.310709\pi$
$773$	31.1526	1.12048	0.560240	+	0.828330i	$0.310709\pi$
$774$	0	0
$775$	0.280996	0.0100937
$776$	0	0
$777$	−7.05086	−0.252948
$778$	0	0
$779$	16.0830	0.576232
$780$	0	0
$781$	−5.80642	−0.207770
$782$	0	0
$783$	26.8385	0.959131
$784$	0	0
$785$	16.3684	0.584214
$786$	0	0
$787$	−7.43801	−0.265136	−0.132568	−	0.991174i	$-0.542322\pi$
$787$	−7.43801	−0.265136	−0.132568	+	0.991174i	$0.542322\pi$
$788$	0	0
$789$	23.2257	0.826857
$790$	0	0
$791$	−7.24443	−0.257582
$792$	0	0
$793$	−9.55262	−0.339224
$794$	0	0
$795$	−15.0509	−0.533799
$796$	0	0
$797$	37.1427	1.31566	0.657831	−	0.753165i	$-0.271474\pi$
$797$	37.1427	1.31566	0.657831	+	0.753165i	$0.271474\pi$
$798$	0	0
$799$	−1.49823	−0.0530036
$800$	0	0
$801$	22.2667	0.786755
$802$	0	0
$803$	13.7605	0.485597
$804$	0	0
$805$	−1.37778	−0.0485605
$806$	0	0
$807$	−30.6637	−1.07941
$808$	0	0
$809$	18.8573	0.662987	0.331493	−	0.943458i	$-0.392448\pi$
$809$	18.8573	0.662987	0.331493	+	0.943458i	$0.392448\pi$
$810$	0	0
$811$	17.0607	0.599082	0.299541	−	0.954083i	$-0.403166\pi$
$811$	17.0607	0.599082	0.299541	+	0.954083i	$0.403166\pi$
$812$	0	0
$813$	57.0232	1.99989
$814$	0	0
$815$	−7.86665	−0.275557
$816$	0	0
$817$	−43.9625	−1.53805
$818$	0	0
$819$	−4.90321	−0.171332
$820$	0	0
$821$	5.40943	0.188790	0.0943952	−	0.995535i	$-0.469908\pi$
$821$	5.40943	0.188790	0.0943952	+	0.995535i	$0.469908\pi$
$822$	0	0
$823$	−19.3876	−0.675810	−0.337905	−	0.941180i	$-0.609718\pi$
$823$	−19.3876	−0.675810	−0.337905	+	0.941180i	$0.609718\pi$
$824$	0	0
$825$	2.90321	0.101077
$826$	0	0
$827$	14.7685	0.513550	0.256775	−	0.966471i	$-0.417340\pi$
$827$	14.7685	0.513550	0.256775	+	0.966471i	$0.417340\pi$
$828$	0	0
$829$	22.5531	0.783302	0.391651	−	0.920114i	$-0.371904\pi$
$829$	22.5531	0.783302	0.391651	+	0.920114i	$0.371904\pi$
$830$	0	0
$831$	71.2168	2.47048
$832$	0	0
$833$	−0.903212	−0.0312944
$834$	0	0
$835$	−18.9590	−0.656103
$836$	0	0
$837$	1.98126	0.0684825
$838$	0	0
$839$	−27.9857	−0.966174	−0.483087	−	0.875572i	$-0.660485\pi$
$839$	−27.9857	−0.966174	−0.483087	+	0.875572i	$0.660485\pi$
$840$	0	0
$841$	−14.5111	−0.500384
$842$	0	0
$843$	−26.2766	−0.905012
$844$	0	0
$845$	12.1842	0.419150
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−10.4889	−0.359977
$850$	0	0
$851$	−3.34614	−0.114704
$852$	0	0
$853$	14.7382	0.504627	0.252313	−	0.967646i	$-0.418809\pi$
$853$	14.7382	0.504627	0.252313	+	0.967646i	$0.418809\pi$
$854$	0	0
$855$	−38.2766	−1.30903
$856$	0	0
$857$	10.7382	0.366810	0.183405	−	0.983037i	$-0.441288\pi$
$857$	10.7382	0.366810	0.183405	+	0.983037i	$0.441288\pi$
$858$	0	0
$859$	19.3604	0.660569	0.330285	−	0.943881i	$-0.392855\pi$
$859$	19.3604	0.660569	0.330285	+	0.943881i	$0.392855\pi$
$860$	0	0
$861$	−6.62222	−0.225685
$862$	0	0
$863$	31.0005	1.05527	0.527634	−	0.849472i	$-0.323079\pi$
$863$	31.0005	1.05527	0.527634	+	0.849472i	$0.323079\pi$
$864$	0	0
$865$	9.46520	0.321827
$866$	0	0
$867$	−46.9862	−1.59573
$868$	0	0
$869$	8.99063	0.304986
$870$	0	0
$871$	−3.14272	−0.106487
$872$	0	0
$873$	35.6227	1.20565
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−23.9309	−0.808089	−0.404044	−	0.914739i	$-0.632396\pi$
$877$	−23.9309	−0.808089	−0.404044	+	0.914739i	$0.632396\pi$
$878$	0	0
$879$	90.5215	3.05321
$880$	0	0
$881$	−24.3970	−0.821956	−0.410978	−	0.911645i	$-0.634813\pi$
$881$	−24.3970	−0.821956	−0.410978	+	0.911645i	$0.634813\pi$
$882$	0	0
$883$	15.0638	0.506936	0.253468	−	0.967344i	$-0.418429\pi$
$883$	15.0638	0.506936	0.253468	+	0.967344i	$0.418429\pi$
$884$	0	0
$885$	−7.18421	−0.241495
$886$	0	0
$887$	19.8252	0.665664	0.332832	−	0.942986i	$-0.391996\pi$
$887$	19.8252	0.665664	0.332832	+	0.942986i	$0.391996\pi$
$888$	0	0
$889$	−11.6128	−0.389482
$890$	0	0
$891$	4.18421	0.140176
$892$	0	0
$893$	11.6958	0.391386
$894$	0	0
$895$	−20.8573	−0.697182
$896$	0	0
$897$	−3.61285	−0.120629
$898$	0	0
$899$	1.06959	0.0356729
$900$	0	0
$901$	−4.68244	−0.155995
$902$	0	0
$903$	18.1017	0.602387
$904$	0	0
$905$	−9.05086	−0.300861
$906$	0	0
$907$	1.84791	0.0613589	0.0306794	−	0.999529i	$-0.490233\pi$
$907$	1.84791	0.0613589	0.0306794	+	0.999529i	$0.490233\pi$
$908$	0	0
$909$	28.9447	0.960035
$910$	0	0
$911$	26.0098	0.861745	0.430872	−	0.902413i	$-0.358206\pi$
$911$	26.0098	0.861745	0.430872	+	0.902413i	$0.358206\pi$
$912$	0	0
$913$	0.949145	0.0314121
$914$	0	0
$915$	30.7052	1.01508
$916$	0	0
$917$	−8.00000	−0.264183
$918$	0	0
$919$	−33.4479	−1.10334	−0.551671	−	0.834062i	$-0.686010\pi$
$919$	−33.4479	−1.10334	−0.551671	+	0.834062i	$0.686010\pi$
$920$	0	0
$921$	−6.10171	−0.201058
$922$	0	0
$923$	−5.24443	−0.172623
$924$	0	0
$925$	2.42864	0.0798532
$926$	0	0
$927$	−3.85236	−0.126528
$928$	0	0
$929$	−36.2766	−1.19020	−0.595098	−	0.803654i	$-0.702887\pi$
$929$	−36.2766	−1.19020	−0.595098	+	0.803654i	$0.702887\pi$
$930$	0	0
$931$	7.05086	0.231082
$932$	0	0
$933$	37.2859	1.22069
$934$	0	0
$935$	0.903212	0.0295382
$936$	0	0
$937$	21.6400	0.706949	0.353475	−	0.935444i	$-0.385000\pi$
$937$	21.6400	0.706949	0.353475	+	0.935444i	$0.385000\pi$
$938$	0	0
$939$	−92.0741	−3.00472
$940$	0	0
$941$	46.0973	1.50273	0.751364	−	0.659888i	$-0.229396\pi$
$941$	46.0973	1.50273	0.751364	+	0.659888i	$0.229396\pi$
$942$	0	0
$943$	−3.14272	−0.102341
$944$	0	0
$945$	7.05086	0.229364
$946$	0	0
$947$	7.30465	0.237369	0.118685	−	0.992932i	$-0.462132\pi$
$947$	7.30465	0.237369	0.118685	+	0.992932i	$0.462132\pi$
$948$	0	0
$949$	12.4286	0.403451
$950$	0	0
$951$	−21.2730	−0.689825
$952$	0	0
$953$	−25.9782	−0.841516	−0.420758	−	0.907173i	$-0.638236\pi$
$953$	−25.9782	−0.841516	−0.420758	+	0.907173i	$0.638236\pi$
$954$	0	0
$955$	17.7146	0.573230
$956$	0	0
$957$	11.0509	0.357223
$958$	0	0
$959$	−0.326929	−0.0105571
$960$	0	0
$961$	−30.9210	−0.997453
$962$	0	0
$963$	102.922	3.31660
$964$	0	0
$965$	−6.42864	−0.206945
$966$	0	0
$967$	49.7846	1.60097	0.800483	−	0.599356i	$-0.204576\pi$
$967$	49.7846	1.60097	0.800483	+	0.599356i	$0.204576\pi$
$968$	0	0
$969$	−18.4889	−0.593948
$970$	0	0
$971$	39.8938	1.28025	0.640127	−	0.768269i	$-0.278882\pi$
$971$	39.8938	1.28025	0.640127	+	0.768269i	$0.278882\pi$
$972$	0	0
$973$	−8.56199	−0.274485
$974$	0	0
$975$	2.62222	0.0839781
$976$	0	0
$977$	11.6316	0.372127	0.186064	−	0.982538i	$-0.440427\pi$
$977$	11.6316	0.372127	0.186064	+	0.982538i	$0.440427\pi$
$978$	0	0
$979$	4.10171	0.131091
$980$	0	0
$981$	−105.420	−3.36581
$982$	0	0
$983$	20.4973	0.653763	0.326882	−	0.945065i	$-0.394002\pi$
$983$	20.4973	0.653763	0.326882	+	0.945065i	$0.394002\pi$
$984$	0	0
$985$	−10.4286	−0.332284
$986$	0	0
$987$	−4.81579	−0.153288
$988$	0	0
$989$	8.59057	0.273164
$990$	0	0
$991$	49.5022	1.57249	0.786245	−	0.617914i	$-0.212022\pi$
$991$	49.5022	1.57249	0.786245	+	0.617914i	$0.212022\pi$
$992$	0	0
$993$	−44.5531	−1.41385
$994$	0	0
$995$	−3.98571	−0.126355
$996$	0	0
$997$	−27.9541	−0.885314	−0.442657	−	0.896691i	$-0.645964\pi$
$997$	−27.9541	−0.885314	−0.442657	+	0.896691i	$0.645964\pi$
$998$	0	0
$999$	17.1240	0.541779

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bm.1.3		3
4.3	odd	2		3080.2.a.k.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.k.1.1	✓	3	4.3	odd	2
6160.2.a.bm.1.3		3	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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q+2.90321 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.42864 q^{9} +O(q^{10})\)	\(q+2.90321 q^{3} -1.00000 q^{5} -1.00000 q^{7} +5.42864 q^{9} +1.00000 q^{11} +0.903212 q^{13} -2.90321 q^{15} -0.903212 q^{17} +7.05086 q^{19} -2.90321 q^{21} -1.37778 q^{23} +1.00000 q^{25} +7.05086 q^{27} +3.80642 q^{29} +0.280996 q^{31} +2.90321 q^{33} +1.00000 q^{35} +2.42864 q^{37} +2.62222 q^{39} +2.28100 q^{41} -6.23506 q^{43} -5.42864 q^{45} +1.65878 q^{47} +1.00000 q^{49} -2.62222 q^{51} +5.18421 q^{53} -1.00000 q^{55} +20.4701 q^{57} +2.47457 q^{59} -10.5763 q^{61} -5.42864 q^{63} -0.903212 q^{65} -3.47949 q^{67} -4.00000 q^{69} -5.80642 q^{71} +13.7605 q^{73} +2.90321 q^{75} -1.00000 q^{77} +8.99063 q^{79} +4.18421 q^{81} +0.949145 q^{83} +0.903212 q^{85} +11.0509 q^{87} +4.10171 q^{89} -0.903212 q^{91} +0.815792 q^{93} -7.05086 q^{95} +6.56199 q^{97} +5.42864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 2 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 2 q^{15} + 4 q^{17} + 8 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{25} + 8 q^{27} - 2 q^{29} - 6 q^{31} + 2 q^{33} + 3 q^{35} - 6 q^{37} + 8 q^{39} + 8 q^{43} - 3 q^{45} - 2 q^{47} + 3 q^{49} - 8 q^{51} + 2 q^{53} - 3 q^{55} + 8 q^{57} + 14 q^{59} - 12 q^{61} - 3 q^{63} + 4 q^{65} + 16 q^{67} - 12 q^{69} - 4 q^{71} + 8 q^{73} + 2 q^{75} - 3 q^{77} - q^{81} + 16 q^{83} - 4 q^{85} + 20 q^{87} - 14 q^{89} + 4 q^{91} + 16 q^{93} - 8 q^{95} + 6 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 3 * q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - 4 * q^13 - 2 * q^15 + 4 * q^17 + 8 * q^19 - 2 * q^21 - 4 * q^23 + 3 * q^25 + 8 * q^27 - 2 * q^29 - 6 * q^31 + 2 * q^33 + 3 * q^35 - 6 * q^37 + 8 * q^39 + 8 * q^43 - 3 * q^45 - 2 * q^47 + 3 * q^49 - 8 * q^51 + 2 * q^53 - 3 * q^55 + 8 * q^57 + 14 * q^59 - 12 * q^61 - 3 * q^63 + 4 * q^65 + 16 * q^67 - 12 * q^69 - 4 * q^71 + 8 * q^73 + 2 * q^75 - 3 * q^77 - q^81 + 16 * q^83 - 4 * q^85 + 20 * q^87 - 14 * q^89 + 4 * q^91 + 16 * q^93 - 8 * q^95 + 6 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

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Database

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